{"text": "```python\nimport holoviews as hv\nhv.extension('bokeh')\nhv.opts.defaults(hv.opts.Curve(width=500), \n hv.opts.Image(width=500, colorbar=True, cmap='Viridis'))\n```\n\n\n```python\nimport numpy as np\nimport scipy.signal\nimport scipy.fft\nfrom IPython.display import Audio\n```\n\n# Dise\u00f1o de sistemas y filtros IIR\n\nUn filtro FIR de buena calidad puede requerir una gran cantidad de coeficientes\n\nEs posible implementar filtros m\u00e1s eficientes usando **recursividad**. Esta es la base de los filtros de respuesta al impulso infinita o IIR que veremos en esta lecci\u00f3n\n\n\n\n## Definici\u00f3n de un sistema IIR \n\nGeneralizando el sistema FIR para incluir versiones pasadas de la salida y asumiendo $a[0] = 1$ llegamos a \n\n$$\n\\begin{align}\ny[n] &= b[0] x[n] + b[1] x[n-1] + b[2] x[n-2] + \\ldots + b[L] x[n-L] \\nonumber \\\\\n& - a[1] y[n-1] - a[2] y[n-2] - \\ldots - a[M] y[n-M] \\nonumber \\\\\n&= \\sum_{l=0}^{L} b[l] x[n-l] - \\sum_{m=1}^{M} a[m] y[n-m] \\nonumber \\\\\n\\sum_{m=0}^{M} a[m] y[n-m] &= \\sum_{l=0}^{L} b[l] x[n-l] \\nonumber \\\\\n(a * y)[n] &= (b * x)[n], \\nonumber\n\\end{align}\n$$\n\nes decir dos convoluciones discretas que definen una **ecuaci\u00f3n de diferencias**\n\nEste tipo de sistema se conoce como \n- sistema *infinite impulse response* (IIR)\n- sistema *auto-regresive moving average* (ARMA)\n - autoregresivo de orden M: incluye valores pasados de la salida\n - media movil de orden L+1: pondera el valor presente y pasados de la entrada\n\nPodemos ver el sistema IIR como una generalizaci\u00f3n del sistema FIR. El caso particular del sistema FIR se recupera si\n\n$a[m] = 0$ para $m=[1, \\ldots, M]$\n\n### Respuesta en frecuencia del sistema IIR\n\nAplicando la transformada de Fourier convertimos las convoluciones en multiplicaciones y encontramos la respuesta en frecuencia como\n\n$$\n\\begin{align}\n\\text{DFT}_N[(a * y)[n]] &= \\text{DFT}_N[(b * x)[n]] \\nonumber \\\\\nA[k] Y[k] &= B[k] X[k] \\nonumber \\\\\nH[k] = \\frac{Y[k]}{X[k]} &= \\frac{B[k]}{A[k]} = \\frac{ \\sum_{l=0}^L b[l]e^{-j \\frac{2\\pi}{N} nl} }{ \\sum_{m=0}^M a[m]e^{-j \\frac{2\\pi}{N} mk}} \\nonumber\n\\end{align}\n$$\n\nque existe siempre que $A[k] \\neq 0$. \n\nLa respuesta en frecuencia tambi\u00e9n suele expresarse como\n\n$$\nH[k] = K \\frac{ \\prod_{l=1}^L (e^{j \\frac{2\\pi}{N} k}- \\beta[l]) }{ \\prod_{m=1}^M (e^{j \\frac{2\\pi}{N} k}- \\alpha[m])} \n$$\n\ndonde \n\n- $K$ se llama **ganancia**\n- las raices del polinomio del numerador $\\alpha$ se llaman conjuntamente **ceros** \n- las raices del polinomio del denominador $\\beta$ se llaman conjuntamente **polos**\n\n### Ejemplo de respuesta al impulso de un sistema IIR\n\nConsideremos el siguiente sistema IIR \n\n$$\n\\begin{align}\ny[n] &= (1-\\gamma) x[n] + \\gamma y[n-1] \\nonumber \\\\\ny[n] - \\gamma y[n-1] &= (1-\\gamma) x[n] \\nonumber\n\\end{align}\n$$\n\nLos coeficientes del sistema son\n\n$a[0] = 1$, $a[1] = -\\gamma$ y $b[0] = (1-\\gamma)$\n\nEs decir que es AR de orden 1 y MA de orden 1\n\n\u00bfC\u00faal es su respuesta al impulso? Asumiendo $y[n]=0, n<0$, tenemos que\n\n$$\n\\begin{matrix}\nn & \\delta[n] & y[n] \\\\\n-2 & 0 & 0 \\\\\n-1 & 0 & 0 \\\\\n0 & 1 & (1-\\gamma) \\\\\n1 & 0 & \\gamma(1-\\gamma) \\\\\n2 & 0 & \\gamma^2(1-\\gamma) \\\\\n3 & 0 & \\gamma^3(1-\\gamma) \\\\\n4 & 0 & \\gamma^4(1-\\gamma) \\\\\n\\end{matrix}\n$$\n\n\u00bfC\u00f3mo cambia la respuesta al impulso con distintos valores de $\\gamma$? \u00bfQu\u00e9 pasa si $\\gamma \\geq 1$?\n\nRespondamos estas preguntas visualizando la respuesta al impulso de este sistema con la funci\u00f3n `scipy.signal.dimpulse`\n\n\n```python\n# Valores de gamma que probaremos:\ngamma = [-1.5, -1, -0.5, 0.5, 1., 1.5]\n\np = []\nfor g in gamma:\n t, y = scipy.signal.dimpulse(([1-g, 0], [1,-g], 1), x0=0, n=30)\n p.append(hv.Curve((t, y[0][:, 0]), label=f\"gamma={g}\"))\n \nhv.Layout(p).cols(3).opts(hv.opts.Curve(width=250, height=200, axiswise=True))\n```\n\nDe las figuras podemos ver que:\n\n- Para $\\gamma < 0$ (primera fila) los coeficientes del sistema son alternantes en signo\n- Para $|\\gamma| < 1$ los coeficientes del sistema tienden a cero\n- Para $|\\gamma| > 1$ los coeficientes del sistema divergen y tienen a infinito\n\n:::{warning}\n\nA diferencia de un sistema FIR, el sistema IIR puede tener configuraciones inestables en que los coeficientes crecen o decrecen infinitamente\n\n:::\n\nPor otro lado consideremos el sistema anterior y asumamos que $|\\gamma|<1$, desenrollando tenemos que \n\n$$\n\\begin{align}\ny[0] &= (1-\\gamma) x[0] \\nonumber \\\\\ny[1] &= (1-\\gamma) (x[1] + \\gamma x[0]) \\nonumber \\\\\ny[2] &= (1-\\gamma) (x[2] + \\gamma x[1] + \\gamma^2 x[0]) \\nonumber \\\\\ny[3] &= (1-\\gamma) (x[3] + \\gamma x[2] + \\gamma^2 x[1] + \\gamma^3 x[0]) \\nonumber \\\\\ny[4] &= (1-\\gamma) (x[4] + \\gamma x[3] + \\gamma^2 x[2] + \\gamma^3 x[1] + \\gamma^4 x[0]) \\nonumber \\\\\ny[5] &= \\ldots \\nonumber \n\\end{align}\n$$\n\n:::{note}\n\nCon un sistema IIR de pocos coeficientes podemos representar un sistema FIR considerablemente m\u00e1s grande\n\n:::\n\nEn el ejemplo anterior, si escogemos $\\gamma$ tal que $\\gamma^{20 }\\approx 0$ entonces aproximamos un sistema FIR de orden 20 con tan s\u00f3lo 3 coeficientes\n\n### Ejemplo de respuesta en frecuencia de un sistema IIR\n\nPara el sistema del ejemplo anterior su respuesta en frecuencia es\n\n$$\n\\begin{align}\nY[k] &= (1-\\gamma) X[k] + \\gamma Y[k] e^{-j \\frac{2\\pi}{N} k} \\nonumber \\\\\nH[k] = \\frac{Y[k]}{X[k]} &= \\frac{1-\\gamma}{1 - \\gamma e^{-j \\frac{2\\pi}{N} k} } \\nonumber \n\\end{align}\n$$\n\nque en notaci\u00f3n de polos y ceros se escribe como\n\n$$\nH[k] = (1-\\gamma)\\frac{e^{j \\frac{2\\pi}{N} k} - 0}{e^{j \\frac{2\\pi}{N} k} - \\gamma }\n$$\n\nes decir que tiene un cero en $0$, un polo en $\\gamma$ y una ganancia de $(1-\\gamma)$\n\nPara entender mejor este sistema estudiemos la magnitud de $|H[k]|$ para $\\gamma < 1$\n\n$$\n\\begin{align}\n| H[k]| &= \\frac{|1-\\gamma|}{|1 - \\gamma e^{-j \\frac{2\\pi}{N} k}|} \\nonumber \\\\\n&= \\frac{1-\\gamma}{\\sqrt{1 - 2\\gamma \\cos(\\frac{2\\pi}{N} k) + \\gamma^2}} \\nonumber\n\\end{align}\n$$\n\n\u00bfC\u00f3mo se ve $|H[k]|$? \u00bfQu\u00e9 funci\u00f3n cumple este sistema?\n\n\n```python\nk = np.arange(-24, 25)/50\nHk = lambda gamma, k : (1-gamma)/np.sqrt(1 - 2*gamma*np.cos(2.0*np.pi*k) + gamma**2)\n```\n\n\n```python\np = []\nfor gamma in [0.25, 0.5, 0.75]:\n p.append(hv.Curve((k, Hk(gamma, k)), 'Frecuencia', 'Respuesta', label=f'gamma={gamma}'))\n \nhv.Overlay(p)\n```\n\n:::{note}\n\nEste sistema atenua las frecuencias altas, es decir que actua como un filtro pasa bajos\n\n:::\n\n## Dise\u00f1o de filtros IIR simples\n\nLos filtros IIR m\u00e1s simples son los de un polo y un cero, es decir filtros de primer orden\n\n$$\nH[k] = \\frac{b[0] + b[1] e^{-j \\frac{2\\pi}{N} k}}{1 + a[1] e^{-j \\frac{2\\pi}{N} k}} = K\\frac{e^{j \\frac{2\\pi}{N} k} - \\beta}{e^{j \\frac{2\\pi}{N} k} - \\alpha } \n$$\n\ndonde podemos reconocer \n\n- $b[0]=K$\n- $\\beta = - b[1] \\cdot K$\n- $\\alpha=-a[1]$\n\nDefinimos la frecuencia de corte $f_c$ como aquella frecuencia en la que el filtro alcanza una atenuaci\u00f3n de 0.7 (-3 dB). Haciendo la equivalencia con el ejemplo anterior tenemos que $\\gamma = e^{-2\\pi f_c}$\n\n### Receta para un filtro pasa bajo IIR con frecuencia de corte $f_c$\n\nAsignamos\n\n- $b[0] = 1 - e^{-2\\pi f_c}$\n- $b[1] = 0$\n- $a[1] = -e^{-2\\pi f_c}$\n\nLo que resulta en la siguiente respuesta en frecuencia\n\n$$\nH[k] = \\frac{1-e^{-2\\pi f_c}}{1 - e^{-2\\pi f_c} e^{-j \\frac{2\\pi}{N} k}} = (1-e^{-2\\pi f_c}) \\frac{(e^{j \\frac{2\\pi}{N} k}- 0)}{(e^{j \\frac{2\\pi}{N} k} - e^{-2\\pi f_c} )}\n$$\n\nEs decir un cero en $0$, un polo en $e^{-2\\pi f_c}$ y ganancia $1-e^{-2\\pi f_c}$\n\n### Receta para un filtro pasa alto IIR con frecuencia de corte $f_c$\n\nAsignamos\n\n- $b[0] = (1 + e^{-2\\pi f_c})/2$\n- $b[1] = -(1 + e^{-2\\pi f_c})/2$\n- $a[1] = -e^{-2\\pi f_c}$\n\nLo que resulta en la siguiente respuesta en frecuencia\n\n$$\nH[k] = \\frac{1+e^{-2\\pi f_c}}{2} \\frac{(e^{j \\frac{2\\pi}{N} k} - 1)}{(e^{j \\frac{2\\pi}{N} k} - e^{-2\\pi f_c})}\n$$\n\nEs decir un cero en $1$, un polo en $e^{-2\\pi f_c}$ y ganancia $\\frac{1+e^{-2\\pi f_c}}{2}$\n\n\n\n### Aplicar un filtro a una se\u00f1al con scipy\n\nPara filtrar una se\u00f1al unidimensional con un filtro IIR (sin variar la fase de la se\u00f1al) podemos utilizar la funci\u00f3n\n\n\n```python\n scipy.signal.filtfilt(b, # Coeficientes del numerador\n a, # Coeficientes del denominador\n x, # Se\u00f1al a filtrar\n ...\n )\n```\n\nLos siguientes ejemplos muestran un se\u00f1al de tipo pulso rectangular filtrada con sistemas IIR de primer orden pasa bajo y pasa-alto dise\u00f1ados con las recetas mostradas anteriormente\n\n\n```python\nn = np.arange(0, 500)\nx = 0.5 + 0.5*scipy.signal.square((n)/(2.*np.pi*5), duty=0.3)\n```\n\n\n```python\ndef iir_low_pass(signal, fc):\n gamma = np.exp(-2*np.pi*(fc))\n b, a = [(1-gamma), 0], [1, -gamma] \n return scipy.signal.filtfilt(b, a, signal)\n\ny = {}\nfor fc in [0.05, 0.02, 0.01]:\n y[fc] = iir_low_pass(x, fc)\n```\n\n\n```python\npx = hv.Curve((n, x))\npy = []\nfor fc, y_ in y.items():\n py.append(hv.Curve((n, y_), label=f'fc={fc}'))\n\nhv.Layout([px, hv.Overlay(py)]).cols(1).opts(hv.opts.Curve(height=200))\n```\n\n\n```python\ndef iir_high_pass(signal, fc):\n gamma = np.exp(-2*np.pi*(fc))\n b, a = [(1+gamma)/2, -(1+gamma)/2], [1, -gamma]\n return scipy.signal.filtfilt(b, a, signal)\n\ny = {}\nfor fc in [0.01, 0.02, 0.05]:\n y[fc] = iir_high_pass(x, fc)\n```\n\n\n```python\npx = hv.Curve((n, x))\npy = []\nfor fc, y_ in y.items():\n py.append(hv.Curve((n, y_), label=f'fc={fc}'))\n\nhv.Layout([px, hv.Overlay(py)]).cols(1).opts(hv.opts.Curve(height=200))\n```\n\n:::{note} \n\nEl filtro pasa-bajos suaviza los cambios de los pulsos rectangulares. El filtro pasa-altos elimina las zonas constantes y resalta los cambios de la se\u00f1al.\n\n:::\n\n## Dise\u00f1o de filtros IIR de segundo orden\n\nLos filtros IIR de segundo orden o **biquad** tienen dos polos y dos ceros.\n\nSu respuesta en frecuencia es\n\n$$\nH[k] = \\frac{b[0] + b[1] W_N^k + b[2] W_N^{2k}}{1 + a[1] W_N^k + a[2] W_N^{2k}} = K \\frac{(W_N^{-k} - \\beta_1) (W_N^{-k} - \\beta_2)}{(W_N^{-k} - \\alpha_1)(W_N^{-k} - \\alpha_2)},\n$$\n\ndonde $W_N = e^{-j \\frac{2 \\pi}{N}}$ y la relaci\u00f3n entreo coeficientes y polos/ceros es: \n\n$$\nb[0] = K, \\quad b[1] = -K (\\beta_1 + \\beta_2), \\quad b[2]= K \\beta_1\\beta_2\n$$\n\n$$\na[1] = - (\\alpha_1 + \\alpha_2), \\quad a[2]=\\alpha_1 \\alpha_2\n$$\n\n\nCon arquitecturas de segundo orden se pueden crear filtros pasabanda y rechaza banda\n\n\n## Dise\u00f1o de filtros IIR de orden mayor\n\nPara crear los coeficientes de filtro IIR de orden mayor podemos usar la funci\u00f3n\n\n```python\nscipy.signal.iirfilter(N, # Orden del filtro\n Wn, # Frecuencias de corte (normalizadas en [0,1])\n fs, # Frecuencia de muestreo\n btype='bandpass', # Tipo de filtro: 'bandpass', 'lowpass', 'highpass', 'bandstop'\n ftype='butter', # Familia del filtro: 'butter', 'ellip', 'cheby1', 'cheby2', 'bessel'\n output='ba', # Retornar coeficientes\n ...\n )\n```\n\nEl filtro Butterworth es \u00f3ptimo en el sentido de tener la banda de paso lo m\u00e1s plana posible. \n\nOtros filtros se dise\u00f1aron con otras consideraciones. \n\nLos filtros IIR digitales est\u00e1n basados en los filtros IIR anal\u00f3gicos.\n\nObserve como al aumentar el orden el filtro pasabajo IIR comienza a cortar de forma m\u00e1s abrupta\n\n\n```python\nHk = {}\nfor order in [1, 2, 5, 20]:\n b, a = scipy.signal.iirfilter(N=order, Wn=0.2, fs=1,\n ftype='butter', btype='lowpass', output='ba')\n freq, response = scipy.signal.freqz(b, a, fs=1)\n Hk[order] = np.abs(response)\n```\n\n\n```python\np = []\nfor order, response in Hk.items():\n p.append(hv.Curve((freq, response), 'Frecuencia', 'Respuesta', label=f'orden={order}'))\nhv.Overlay(p)\n```\n\n## Comparaci\u00f3n de la respuesta en frecuencia de filtros FIR e IIR del orden equivalente\n\nComparemos la respuesta en frecuencia de un filtro IIR y otro FIR ambos pasa-bajo con 20 coeficientes\n\n\n\n```python\nFs = 1\nfc = 0.25\nh = scipy.signal.firwin(numtaps=20, cutoff=fc, pass_zero=True, window='hann', fs=Fs)\nb, a = scipy.signal.iirfilter(N=9, Wn=fc, fs=Fs, ftype='butter', btype='lowpass')\ndisplay(len(h), len(b)+len(a))\n\nfreq_fir, response_fir = scipy.signal.freqz(h, 1, fs=Fs)\nfreq_iir, response_iir = scipy.signal.freqz(b, a, fs=Fs)\n```\n\n\n```python\np1 = hv.Curve((freq_fir, np.abs(response_fir)), 'Frecuencia', 'Respuesta', label='FIR')\np2 = hv.Curve((freq_iir, np.abs(response_iir)), 'Frecuencia', 'Respuesta', label='IIR')\nhv.Overlay([p1, p2])*hv.VLine(fc).opts(color='k', alpha=0.5)\n```\n\nLa linea negra marca la ubicaci\u00f3n de la frecuencia de corte\n\n:::{note}\n \nEl filtro IIR es mucho m\u00e1s abrupto, es decir filtra mejor, que el filtro FIR equivalente\n\n:::\n\nUna desventaja del filtro IIR es que por definici\u00f3n introduce una desfase no constante en la se\u00f1al de salida\n\n\n```python\nfreq_fir, delay_fir = scipy.signal.group_delay(system=(h, 1), fs=Fs)\nfreq_iir, delay_iir = scipy.signal.group_delay(system=(b, a), fs=Fs)\n```\n\n\n```python\np1 = hv.Curve((freq_fir, delay_fir), 'Frecuencia', 'Desfase', label='FIR')\np2 = hv.Curve((freq_iir, delay_iir), 'Frecuencia', 'Desfase', label='IIR')\nhv.Overlay([p1, p2])*hv.VLine(fc).opts(color='k', alpha=0.5)\n```\n\n\u00bfC\u00f3mo se ve una se\u00f1al filtrada donde se preserva la fase versus una donde no se preserva la fase?\n\nConsideremos la se\u00f1al rectangular anterior y apliquemos un filtro pasa-bajo IIR de orden 1\n\nEsta vez compararemos el filtro con la funci\u00f3n `scipy.signal.lfilter` y la funci\u00f3n `scipy.signal.filtfilt`. La primera no preserva la fase mientras que la segunda si lo hace\n\n\n```python\nFs = 1\nfc = 0.01\nn = np.arange(0, 500)\nx = 0.5 + 0.5*scipy.signal.square((n)/(2.*np.pi*5), duty=0.3)\n\nb, a = scipy.signal.iirfilter(N=1, Wn=fc, fs=Fs, ftype='butter', btype='lowpass')\n# No se preserva la fase\ny_lfilter = scipy.signal.lfilter(b, a, x)\n# Se preserva la fase\ny_filtfilt = scipy.signal.filtfilt(b, a, x)\n```\n\n\n```python\npx = hv.Curve((n, x), 'Tiempo', 'Entrada')\npy = []\npy.append(hv.Curve((n, y_filtfilt), 'Tiempo', 'Salida', label=f'Fase constante'))\npy.append(hv.Curve((n, y_lfilter), 'Tiempo', 'Salida', label=f'Fase no constante'))\n\nhv.Layout([px, hv.Overlay(py)]).cols(1).opts(hv.opts.Curve(height=200))\n```\n\n:::{note}\n \nEn el caso donde no se preserva la fase podemos notar que la se\u00f1al de salida est\u00e1 desplazada con respecto a la original. Adem\u00e1s los cambios tienen una transici\u00f3n asim\u00e9trica \n\n:::\n\nLa funci\u00f3n `scipy.signal.filtfilt` \"arregla\" el problema del desfase filtrando la se\u00f1al dos veces. La primera vez se filtra hacia adelante en el tiempo y la segunda vez hacia atr\u00e1s. Por ende no se puede aplicar en un escenario de tipo *streaming* donde los datos van llegando de forma causal.\n\nEn una aplicaci\u00f3n causal donde se necesite preservar la fase debemos usar un filtro FIR.\n\n## Ap\u00e9ndice: Efectos de audio con filtros IIR\n\n\nEl siguiente ejemplo muestra como implementar el conocido filtro Wah-wah usando un sistema IIR\n\nEste es un filtro pasabanda modulado con ancho de pasada fijo $f_b$ [Hz] y una frecuencia central variable $f_c$ [Hz], donde La frecuencia central se modula con una onda lenta\n\n\nSe modela como el siguiente sistema **IIR**\n\n$$\nH[k] = \\frac{(1+c)W_N^{2k} -(1+c) }{W_N^{2k} + d(1-c)W_N^k -c}\n$$\n\ndonde \n\n$$\nd=-\\cos(2\\pi f_c/f_s)\n$$ \n\ny \n\n$$\nc = \\frac{\\tan(\\pi f_b/f_s) -1}{\\tan(2\\pi f_b /f_s)+1}\n$$\n\nVeamos como modifica este filtro una se\u00f1al de audio\n\n\n```python\nimport librosa\ndata, fs = librosa.load(\"../../data/DPSAU.ogg\")\nAudio(data, rate=fs)\n```\n\n\n```python\ndata_wah = []\nzi = np.zeros(shape=(2,))\n# Par\u00e1metros fijos del filtro\nfb, Nw = 200, 5 \nc = (np.tan(np.pi*fb/fs) - 1.)/(np.tan(2*np.pi*fb/fs) +1)\n# Filtramos una ventana de la se\u00f1al moviendo lentamente fc\nfor k in range(len(data)//Nw):\n # C\u00e1lculo de la frecuencia central\n fc = 500 + 2000*(np.cos(2.0*np.pi*k*30./fs) +1)/2\n d = -np.cos(2*np.pi*fc/fs)\n # Coeficientes del filtro\n b, a = [(1+c), 0, -(1+c)], [1, d*(1-c), -c]\n # Filtramos, usando el filtrado anterior como borde (zi)\n data2, zi = scipy.signal.lfilter(b, a, data[k*Nw:(k+1)*Nw], zi=zi)\n # Guardamos\n data_wah.append(data2)\n```\n\n\n```python\nAudio(np.hstack(data_wah), rate=int(fs))\n```\n\nSi quieres profundizar en el tema de los filtros IIR aplicados a efectos de audio recomiendo: https://www.ee.columbia.edu/~ronw/adst-spring2010/lectures/lecture2.pdf\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "4a3f729e7e159ae5250692e0a5b66fb51acf4a2d", "size": 25901, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lectures/unit2/lecture4.ipynb", "max_stars_repo_name": "phuijse/UACH-INFO183", "max_stars_repo_head_hexsha": "0e1b6bef0bd80cda2753bd11e62016268f2de638", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 11, "max_stars_repo_stars_event_min_datetime": "2018-08-27T23:53:15.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-16T23:31:05.000Z", "max_issues_repo_path": "lectures/unit2/lecture4.ipynb", "max_issues_repo_name": "phuijse/UACH-INFO183", "max_issues_repo_head_hexsha": "0e1b6bef0bd80cda2753bd11e62016268f2de638", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "lectures/unit2/lecture4.ipynb", "max_forks_repo_name": "phuijse/UACH-INFO183", "max_forks_repo_head_hexsha": "0e1b6bef0bd80cda2753bd11e62016268f2de638", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 10, "max_forks_repo_forks_event_min_datetime": "2019-01-04T17:43:55.000Z", "max_forks_repo_forks_event_max_datetime": "2021-09-07T16:07:18.000Z", "avg_line_length": 30.2935672515, "max_line_length": 303, "alphanum_fraction": 0.5144589012, "converted": true, "num_tokens": 5737, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.7690802317779601, "lm_q1q2_score": 0.44998973128817304}} {"text": "# Estimating text loss in Middle Dutch chivalric epics\n\nThis English-language, Python notebook accompanies the following publication:\n\n> Mike Kestemont and Folgert Karsdorp, \"Het Atlantis van de Middelnederlandse ridderepiek. Een schatting van het tekstverlies met methodes uit de ecodiversiteit\". *Spiegel der letteren* (2020).\n\nAll figures and numbers were prepared using the code below. Future updates of the code and data will be managed in an open [Github repository](https://github.com/mikekestemont/chivalric_diversity). The code block below loads all (third-party) packages and modules necessary to run the module. These can be installed from the file `requirements.txt`:\n\n pip install -r requirements.txt\n\n\n```python\nfrom functools import partial\nfrom itertools import product\n\nimport numpy as np\nnp.random.seed(12345)\nfrom scipy.special import erfinv\nimport pandas as pd\n\nimport matplotlib.pyplot as plt\nplt.style.use(\"tufte.mplstyle\")\nplt.rcParams[\"text.usetex\"] = False\n%matplotlib inline\n\nimport scipy.stats as stats\nfrom scipy.special import gammaln\n```\n\n## Data\n\nWe load the data from the spreadsheet file `mnl.xlsx`:\n\n\n```python\nmnl = pd.read_excel('mnl.xlsx', header=None, names=('text', 'witness'))\nmnl.head(10)\n```\n\n\n\n\n
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\n\n\n\nWe are only interested in the count data, i.e. the number of witnesses per text (the technical term is \"abundance data\").\n\n\n```python\nmnl.groupby('text').size().sort_values(ascending=False).head()\n```\n\n\n\n\n text\n Historie van Troyen 16\n Roman van Limborch 10\n Madelgijs 10\n Karel ende Elegast 7\n Parthonopeus van Bloys 6\n dtype: int64\n\n\n\nThe counts per text can be plotted as follows:\n\n\n```python\nfig, ax = plt.subplots(figsize=(10,18))\nmnl.groupby('text').size().sort_values(ascending=True).plot.barh(ax=ax);\nax.set(xlabel='aantal handschriften', ylabel='', \n title='Distributie van de (ons bekende) ridderepische teksten over tekstgetuigen')\nax.spines['right'].set_visible(False)\nax.spines['top'].set_visible(False)\nplt.savefig('output/Fig1.jpeg', dpi=300, transparent=True)\n```\n\nYet a different perspective is to list the size of the frequency bins that we can distinguish within the manuscript counts:\n\n\n```python\ntypes = mnl.groupby('text').size().sort_values(ascending=False).value_counts().sort_index()\ntypes = types.to_frame(name='aantal teksten')\ntypes['aantal handschriften'] = types.index\ntypes.to_excel('output/Tab1.xlsx')\ntypes\n```\n\n\n\n\n
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\n\n\n\nFinally, we define the auxiliary function `species_richness` to count the number of unique texts in the data (i.e. the number of non-zero counts):\n\n\n```python\ndef species_richness(counts):\n return np.sum(counts > 0)\n\nprint('# unique texts:', species_richness(mnl.groupby('text').size()))\nprint('# witnesses:', len(mnl))\n```\n\n # unique texts: 74\n # witnesses: 164\n\n\n## Jackknife\n\nThe following function computes the first-order Jackknife estimate, on the basis of the abundance data in our data frame, as well as a confidence interval (.95 be default). This approach is detailed in the following paper:\n\n> K. Burnham & W. Overton, \"Robust Estimation of Population Size When Capture Probabilities Vary Among Animals\". *Ecology* (1979), 927-936.\n\n\n```python\ndef jackknife(data, conf_lvl=0.95):\n jack_stat = species_richness(data)\n \n x = np.array(sum([[i] * c for i, c in enumerate(data, 1)], []))\n index = np.arange(x.shape[0])\n \n vals = []\n for i in range(x.shape[0]):\n t = x[index != i]\n vals.append(species_richness(np.bincount(t)))\n \n mean_jack_stat = np.mean(vals)\n bias = (x.shape[0] - 1) * (mean_jack_stat - jack_stat)\n \n estimate = jack_stat - bias\n \n std_err = np.sqrt(\n (x.shape[0] - 1) * \n np.mean((mean_jack_stat - vals) * \n (mean_jack_stat - vals), axis=0)\n ) \n \n z_score = np.sqrt(2.0) * erfinv(conf_lvl)\n conf_interval = estimate + z_score * np.array((-std_err, std_err))\n \n return estimate, std_err, conf_interval\n\nresults = jackknife(mnl.groupby('text').size())\nprint('jackknife-estimate (order=1):', results[0], results[-1])\n```\n\n jackknife-estimate (order=1): 117.73170731707278 [106.64468284 128.8187318 ]\n\n\nThis implementation is verbose and uses an explicit `for`-loop, which iteratively leaves out observations and tracks the drops in diversity that follow from this operation. In the code blocks below we show that the same estimate can also be obtained in a fully analytical fashion. First we calculate the frequency counts for each unique text:\n\n\n```python\nnum_per_text = mnl.groupby('text').size()\nnum_per_text\n```\n\n\n\n\n text\n Aiol (1) 1\n Aiol (2) 1\n Alexanderroman 1\n Alexanders geesten 5\n Arturs doet 2\n ..\n Van den bere Wisselau 1\n Walewein ende Keye 1\n Willem van Oringen 1\n Willem van Oringen II 1\n Wrake van Ragisel 3\n Length: 74, dtype: int64\n\n\n\nNext, we store the species richness (the number of unique texts) in $t$:\n\n\n```python\nt = species_richness(num_per_text)\nt\n```\n\n\n\n\n 74\n\n\n\nThen we set $s$ to the number of texts that are only attested in a single witness:\n\n\n```python\ns = sum(num_per_text == 1)\ns\n```\n\n\n\n\n 44\n\n\n\nOnly the $s$ texts that occur once will affect the species richness during the iterative Jackknife procedure. We can therefore predict that we will obtain the following deviations when applying the bootstrap:\n\n\n```python\nmu = (((t - s) * t) + (s * (t - 1))) / t\nmu\n```\n\n\n\n\n 73.4054054054054\n\n\n\nThat means that we can calculate the bias as follows:\n\n\n```python\nbias = (t - 1) * (mu - t)\nbias\n```\n\n\n\n\n -43.405405405405546\n\n\n\nTo account for this bias, we can subtract it from the original species richness in the observed data:\n\n\n```python\nt - bias\n```\n\n\n\n\n 117.40540540540555\n\n\n\n## Simple example\n\n\n```python\ncounts = [5, 4, 3, 3, 1, 1, 1, 1, 1]\nnames = 'ABCDEFGHI'\ndata = zip(counts, names)\ndf = pd.DataFrame(zip(names, counts), columns=('naam', 'mss'))\ndf.to_excel('output/Tab2.xlsx')\ndf\n```\n\n\n\n\n
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\n\n\n\n\n```python\nprint('total # of witnesses:', df['mss'].sum())\n```\n\n total # of witnesses: 20\n\n\n\n```python\nspecies_richness(df['mss'])\n```\n\n\n\n\n 9\n\n\n\n\n```python\njackknife(df['mss'])\n```\n\n\n\n\n (13.75, 1.8874586088176875, array([10.0506491, 17.4493509]))\n\n\n\n\n```python\ndata = np.array(df['mss'])\nx = np.array(sum([[i]*c for i, c in enumerate(data, 1)], []))\ntradition = [names[i - 1] for i in x]\nprint(tradition)\n```\n\n ['A', 'A', 'A', 'A', 'A', 'B', 'B', 'B', 'B', 'C', 'C', 'C', 'D', 'D', 'D', 'E', 'F', 'G', 'H', 'I']\n\n\n\n```python\nbootstrap = []\nfor i in range(len(tradition)):\n tradition_ = [tradition[j] for j in range(len(tradition)) if i != j]\n \n bootstrap.append((\n (i + 1), tradition[i], ''.join(tradition_),\n len(set(tradition_)), len(set(tradition_)) - len(set(tradition))))\n```\n\n\n```python\ndf = pd.DataFrame(bootstrap, columns=('iteration', 'leftout', 'imputed tradition', 'richness', 'error'))\ndf.to_excel('output/Tab3.xlsx')\ndf\n```\n\n\n\n\n
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iterationleftoutimputed traditionrichnesserror
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\n\n\n\n\n```python\nmean_estimate = np.mean(df['richness'])\nprint('Average estimate:', mean_estimate)\nprint('Bias:', mean_estimate - 9)\n```\n\n Average estimate: 8.75\n Bias: -0.25\n\n\n\n```python\nbias = 19 * (mean_estimate - 9)\nbias\n```\n\n\n\n\n -4.75\n\n\n\n\n```python\ncorrected = 9 - bias\ncorrected\n```\n\n\n\n\n 13.75\n\n\n\n\n```python\nconf_lvl = .95\n\nstd_err = np.sqrt(\n 19 * np.mean((mean_estimate - df['richness']) * \n (mean_estimate - df['richness']), axis=0))\n \nz_score = np.sqrt(2.0) * erfinv(conf_lvl)\nconf_interval = corrected + z_score * np.array((-std_err, std_err))\nconf_interval\n```\n\n\n\n\n array([10.0506491, 17.4493509])\n\n\n\n## Chao1\n\nIn the paper we eventually opt for the more recent, non-parametric formula \"Chao1\", which is described in this paper:\n\n> A. Chao & L. Jost, \u2018Estimating diversity and entropy profiles via discovery rates of new species\". *Methods in Ecology and Evolution* (2015), 873-882.\n\nBecause we have \"doubletons\" in our data, we use can the following formula, where:\n- $\\hat{f_0}$ is the (theoretical) number of non-observed species/texts;\n- $f_1$ is the number of species/texts attested exactly once (\"singletons\");\n- $f_2$ is the number of species/texts attested exactly twice (\"doubletons\");\n- $n$ is the total number of individuals/manuscripts in the observed data.\n\n\\begin{equation}\n\\hat{f_0} = \\frac{(n - 1)}{n} \\frac{f_1^2}{2f_2}\n\\end{equation}\n\nThe code block below returns the full, theoretical species richness as etimated by Chao1, i.e. it adds the estimated $\\hat{f_0}$ to the species richness that was observed in the sample:\n\n\n```python\ndef chao_richness(x):\n x, n = x[x > 0], x.sum()\n t = x.shape[0]\n f1, f2 = (x == 1).sum(), (x == 2).sum()\n return t + (n - 1) / n * ((f1 ** 2 / 2 / f2) if f2 > 0 else (f1 * (f1 - 1) / 2))\n```\n\nIf we apply this function to our data, we obtain an even higher (but arguably more realistic) estimate of the loss in textual diversity for this literature. Note, however, that this estimate is still a theoretical *minimum estimate*, since the original loss could still be higher.\n\n\n```python\nchao_richness(num_per_text)\n```\n\n\n\n\n 148.00750469043152\n\n\n\nInstead of reporting just this number, we apply a bootstrapped procedure in which we sample from the material using a multinomial distribution (see the Appendix Chao and Jost, 2015) and apply Chao1 to the resulting samples. This procedure allows us to calculate a .95 confidence interval for this value.\n\n\n```python\ndef bt_prob(x):\n x, n = x[x > 0], x.sum()\n f1, f2 = (x == 1).sum(), (x == 2).sum()\n C = 1 - f1 / n * (((n - 1) * f1 / ((n - 1) * f1 + 2 * f2)) if f2 > 0 else\n ((n - 1) * (f1 - 1) / ((n - 1) * (f1 - 1) + 2)) if f1 > 0 else\n 0)\n W = (1 - C) / np.sum(x / n * (1 - x / n) ** n)\n p = x / n * (1 - W * (1 - x / n) ** n)\n f0 = np.ceil(((n - 1) / n * f1 ** 2 / (2 * f2)) if f2 > 0 else\n ((n - 1) / n * f1 * (f1 - 1) / 2))\n p0 = (1 - C) / f0\n p = np.hstack((p, np.array([p0 for i in np.arange(f0)])))\n return p\n\n\ndef bootstrap(x, n_iter=1000, conf=.95):\n # define a multinomial probability distribution\n # for the bootstrap procedure to sample from:\n p, n = bt_prob(x), x.sum()\n data_bt = np.random.multinomial(n, p, n_iter)\n \n pro = np.array([chao_richness(row) for row in data_bt])\n \n pro_mean = pro.mean(0)\n \n lci_pro = -np.quantile(pro, (1 - conf) / 2, axis=0) + pro_mean\n uci_pro = np.quantile(pro, 1 - (1 - conf) / 2, axis=0) - pro_mean\n\n sd_pro = np.std(pro, axis=0)\n\n pro = pro_mean - pro\n return (lci_pro, uci_pro, sd_pro, pro)\n```\n\n\n```python\ndef chao_estimate(x, n_iter=1000, conf=0.95):\n pro = chao_richness(x)\n (lci_pro, uci_pro, sd_pro, bt_pro) = bootstrap(x, n_iter=n_iter, conf=conf)\n lci_pro, uci_pro = pro - lci_pro, pro + uci_pro\n bt_pro = pro - bt_pro\n return (lci_pro, uci_pro, bt_pro, pro)\n```\n\nThe following block applies this bootstrapped procedure to obtain the final estimates:\n\n\n```python\nlci_pro, uci_pro, bt_pro, pro = chao_estimate(num_per_text, n_iter=10000)\nprint('pro:', pro)\nprint('lci_pro:', lci_pro)\nprint('uci_pro:', uci_pro)\n```\n\n pro: 148.00750469043152\n lci_pro: 106.21863495939421\n uci_pro: 219.01578019221017\n\n\nThe array `bt_pro` contains the estimates that were collected during the bootstrap (1,000 iterations by default). Below, we plot the distribution of these numbers using a rainplot:\n\n\n```python\nimport ptitprince as pt\n\nfig, ax = plt.subplots(figsize=(8, 6))\n\nd = list([(x, 'bootstrap') for x in bt_pro])\nbt = pd.DataFrame(d, columns=('bootstrap', 'type'))\n\npt.RainCloud(\n data=bt, x=\"type\", y=\"bootstrap\", ax=ax, \n orient=\"h\", alpha=.8, bw=.2, rain_alpha=.3, palette=\"Greys\"\n)\nax.axvline(pro, c='black', ls='--')\nax.axvline(uci_pro, c='darkgrey', ls='--')\nax.axvline(lci_pro, c='darkgrey', ls='--')\nax.spines['right'].set_visible(False)\nax.spines['top'].set_visible(False)\nax.spines['left'].set_visible(False)\nax.set_yticks([])\nax.set_ylabel('')\nplt.savefig('output/Fig2.png', dpi=300, transparent=True)\n```\n\nThe idea that there were at least 100 texts is not completely unlikely, but it is a very\nconservative estimate, at the very bottom of the probability continuum. The estimate of ~148 manuscripts (or more) is much more plausible, which would mean that *at least half of\nthe chivalric texts have been lost*. Just as 100 is an extremely optimistic\nestimate, ~219 is the most pessimistic estimate: in that\ncase, only a third of the ever available chivalric epics would have been persisted through\ntime, which is quite a dramatic, but not entirely unrealistic figure.\n\n## Species accumulation curve\n\nIn what preceded, we have investigated how many unique texts may have been lost, or, more positively, how many unique texts we may have not yet seen. In this concluding section, we investigate how many texts should be retrieved before we arrive at this diversity estimate. This new estimate provides us with information about the total population size, i.e. the total number of text witnesses. We follow Hsieh, Ma and Chao (2016) to compute this estimate using \"Rarefaction Extrapolation\". For details about this method, see:\n\n> Hsieh, Ma and Chao (2016): iNEXT: an R package for rarefaction and extrapolation ofspecies diversity. *Methods in Ecology and Evolution*, 7, 1451\u20131456.\n\n\n```python\ndef bootstrap_re(x, fn=chao_richness, n_iter=1000, conf=.95):\n # define a multinomial probability distribution\n # for the bootstrap procedure to sample from:\n p, n = bt_prob(x), x.sum()\n data_bt = np.random.multinomial(n, p, n_iter)\n \n Dq = fn(x)\n \n pro = np.array([fn(row) for row in data_bt])\n \n error = stats.norm.ppf(1 - (1 - conf) / 2) * np.std(pro, 0)\n lci_pro = Dq - error\n uci_pro = Dq + error\n\n sd_pro = np.std(pro, axis=0)\n\n return (lci_pro, uci_pro, sd_pro, Dq, )\n\n\ndef rarefaction_extrapolation(x, max_steps):\n x, n = x[x > 0], x.sum()\n def _sub(m):\n if m <= n:\n return np.sum(1 - np.array(\n [np.exp(gammaln(n - i + 1) + gammaln(n - m + 1) - \n gammaln(n - i - m + 1) - gammaln(n + 1)) if i <= (n - m) else\n 0 for i in x]))\n else:\n S = (x > 0).sum()\n f1, f2 = (x == 1).sum(), (x == 2).sum()\n f0 = ((n - 1) / n * f1 * (f1 - 1) / 2) if f2 == 0 else ((n - 1) / n * f1**2 / 2 / f2)\n A = n * f0 / (n * f0 + f1)\n return S if f1 == 0 else (S + f0 * (1 - A**(m - n)))\n return np.array([_sub(mi) for mi in range(1, max_steps)])\n```\n\n\n```python\ncounts = np.bincount(mnl.groupby('text').size())[1:] # ignore zero\nx = np.array(sum([[i] * c for i, c in enumerate(counts, 1)], []))\n```\n\nHere too we use a bootstrap method with 100 samples:\n\n\n```python\nmax_steps = 1000\nlci_pro, uci_pro, sd_pro, Dq = bootstrap_re(\n x, \n fn=partial(rarefaction_extrapolation, max_steps=max_steps), \n n_iter=100\n)\n```\n\n\n```python\nsteps = np.arange(1, max_steps)\ninterpolated = np.arange(1, max_steps) < x.sum()\n\nfig, ax = plt.subplots(figsize=(8, 6))\nax.plot(steps[interpolated], Dq[interpolated], color='C0')\nax.plot(x.sum(), Dq[x.sum() - 1], 'o')\nax.plot(steps[~interpolated], Dq[~interpolated], '--', color='C0')\nax.fill_between(steps, lci_pro, uci_pro, alpha=0.3)\nax.grid()\nax.set(xlabel='# handschriften', ylabel='# teksten', title='Species Accumulation Curve')\nax.spines['right'].set_visible(False)\nax.spines['top'].set_visible(False)\n\nplt.savefig('output/Fig3.png', dpi=300, transparent=True)\n```\n\nThe graph clearly show that a significant part of the text witnesses has been lost (or has not yet been found). It is only at around 600 witnesses (and more), that the curve slowly flattens towards its asymptote.\n", "meta": {"hexsha": "755c8cae889b143d8a0ad8b88c2b4010933f3322", "size": 236651, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "analysis.ipynb", "max_stars_repo_name": "mikekestemont/chivalric_diversity", "max_stars_repo_head_hexsha": "f3f5a62cb4f17ddc447c2e8530e4ca5044447cbb", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "analysis.ipynb", "max_issues_repo_name": "mikekestemont/chivalric_diversity", "max_issues_repo_head_hexsha": "f3f5a62cb4f17ddc447c2e8530e4ca5044447cbb", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "analysis.ipynb", "max_forks_repo_name": "mikekestemont/chivalric_diversity", "max_forks_repo_head_hexsha": "f3f5a62cb4f17ddc447c2e8530e4ca5044447cbb", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-09-17T10:04:33.000Z", "max_forks_repo_forks_event_max_datetime": "2020-09-17T10:04:33.000Z", "avg_line_length": 155.6914473684, "max_line_length": 116708, "alphanum_fraction": 0.8700406928, "converted": true, "num_tokens": 7469, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943925708561, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.4499293228092997}} {"text": "
\n
\n\nThis notebook demonstrates the functionality of the `quantecon.models.solow` module. The code was developed with funding from the Scottish Graduate Programme in Economics (SGPE) and the Scottish Institute for Research in Economics (SIRE). The code has found widespread use in my own research and teaching and it is my fervent hope that others will adapt and contribute to the development of these notebooks for use in their own research and teaching.\n\n\n\n\n```python\n%matplotlib inline\n```\n\n\n```python\nimport matplotlib.pyplot as plt\nimport seaborn as sns\nimport numpy as np\nimport sympy as sym\n\nimport quantecon as qe\nimport solowpy\n```\n\n# 0. Motivation\n\nThe typical economics graduate student places great faith in the analytical mathematical tools that he or she was taught as an undergraduate. In particular this student is likely under the impression that virtually all economic models have closed-form solutions. At best the typical student believes that if he or she were to encounter an economic model without a closed-form solution, then simplifying assumptions could be made that would render the model analytically tractable without sacrificing important economic content. \n\nThe typical economics student is, of course, wrong about general existence of closed-form solutions to economic models. In fact the opposite is true: most economic models, particular dynamic, non-linear models with meaningful constraints (i.e., most any *interesting* model) will fail to have an analytic solution. I the objective of this notebook is to demonstrate this fact and thereby motivate the use of numerical methods in economics more generally using the [Solow model of economic growth](http://faculty.smu.edu/tosang/pdf/Solow_1956.pdf). \n\nEconomics graduate students are very familiar with the Solow growth model. For many students, the Solow model will have been one of the first macroeconomic models taught to them as undergraduates. Indeed, Greg Mankiw's [*Macroeconomics*](http://www.macmillanhighered.com/Catalog/product/macroeconomics-eighthedition-mankiw), the dominant macroeconomics textbook for first and second year undergraduates, devotes two full chapters to motivating and deriving the Solow model. The first few chapters of David Romer's [*Advanced Macroeconomics*](http://highered.mheducation.com/sites/0073511374/index.html), one of the most widely used final year undergraduate and first-year graduate macroeconomics textbook, are also devoted to the Solow growth model and its descendants.\n\n\n## 0.1 The basic Solow growth model\nThe Solow model can be reduced down to a single non-linear differential equation and associated initial condition describing the time evolution of capital stock (per unit effective labor), $k(t)$.\n\n$$ \\dot{k}(t) = sf(k(t)) - (n + g + \\delta)k(t),\\ k(t) = k_0 \\tag {0.1.1} $$\n\nThe parameter $0 < s < 1$ is the fraction of output invested and the parameters $n, g, \\delta$ are the rates of population growth, technological progress, and depreciation of physical capital. The intensive form of the production function $f$ is assumed to be to be strictly concave with \n\n$$ f(0) = 0,\\ lim_{k\\rightarrow 0}\\ f' = \\infty,\\ lim_{k\\rightarrow \\infty}\\ f' = 0. \\tag{0.1.2} $$ \n\nA common choice for the function $f$ which satisfies the above conditions is known as the Cobb-Douglas production function.\n\n$$ f(k(t)) = k(t)^{\\alpha} $$\n\nAssuming a Cobb-Douglas functional form for $f$ also makes the model analytically tractable (and thus contributes to the typical economics student's belief that all such models \"must\" have an analytic solution). [Sato 1963](http://www.jstor.org/stable/2296026) showed that the solution to the model under the assumption of Cobb-Douglas production is\n\n$$ k(t) = \\Bigg[\\bigg(\\frac{s}{n+g+\\delta}\\bigg)\\bigg(1 - e^{-(n+g+\\delta)(1-\\alpha)t}\\bigg)+ k_0^{1-\\alpha}e^{-(n+g+\\delta)(1-\\alpha)t}\\Bigg]^{\\frac{1}{1-\\alpha}}. \\tag{0.1.3} $$\n\nA notable property of the Solow model with Cobb-Douglas production is that the model predicts that the shares of real income going to capital and labor should be constant. Denoting capital's share of income as $\\alpha_K(k)$, the model predicts that \n\n$$ \\alpha_K(k(t)) \\equiv \\frac{\\partial \\ln f(k(t))}{\\partial \\ln k(t)} = \\alpha \\tag{0.1.4} $$\n\nNote that the prediction is that factor shares are constant along both the balanced growth path *and* during the disequilibrium transient (i.e., the period in which $k(t)$ is varying). We can test this implication of the model using data from the newest version of the [Penn World Tables (PWT)](http://www.rug.nl/research/ggdc/data/penn-world-table). \n\n\n```python\nimport pypwt\npwt = pypwt.load_pwt_data()\n```\n\n\n```python\nfig, ax = plt.subplots(1, 1, figsize=(8,6))\n\nfor ctry in pwt.major_axis:\n tmp_data = pwt.major_xs(ctry)\n tmp_data.labsh.plot(color='gray', alpha=0.5)\n \n# plot some specific countries\npwt.major_xs('USA').labsh.plot(color='blue', ax=ax, label='USA')\npwt.major_xs('IND').labsh.plot(color='green', ax=ax, label='IND')\npwt.major_xs('CHN').labsh.plot(color='orange', ax=ax, label='CHN')\n\n# plot global average\navg_labor_share = pwt.labsh.mean(axis=0)\navg_labor_share.plot(color='r', ax=ax)\n\nax.set_title(\"Labor's share has been far from constant!\",\n fontsize=20, family='serif')\nax.set_xlabel('Year', family='serif', fontsize=15)\nax.set_ylabel('Labor share of income', family='serif', fontsize=15)\nax.set_ylim(0, 1)\n\nplt.show()\n```\n\nFrom the above figure it is clear that the prediction of constant factor shares is strongly at odds with the empirical data for most countries. Labor's share of real GDP has been declining, on average, for much of the post-war period. For many countries, such as India, China, and South Korea, the fall in labor's share has been dramatic. Note also that the observed trends in factor shares are inconsistent with an economy being on its long-run balanced growth path. \n\n## 0.2 A more general Solow growth model\n\nWhile the data clearly reject the Solow model with Cobb-Douglas production, they are *not* inconsistent with the Solow model in general. A simple generalization of the Cobb-Douglas production function, known as the constant elasticity of substitution (CES) function:\n\n$$ f(k(t)) = \\bigg[\\alpha k(t)^{\\rho} + (1-\\alpha)\\bigg]^{\\frac{1}{\\rho}} $$\n\nwhere $-\\infty < \\rho < 1$ is the elasticity of substitution between capital and effective labor in production is capable of generating the variable factor shares observed in the data. Note that \n \n$$ \\lim_{\\rho\\rightarrow 0} f(k(t)) = k(t)^{\\alpha} $$\n\nand thus the CES production function nests the Cobb-Douglas functional form as a special case.\n\nTo see that the CES production function also generates variable factor shares note that \n\n$$ \\alpha_K(k(t)) \\equiv \\frac{\\partial \\ln f(k(t))}{\\partial \\ln k(t)} = \\frac{\\alpha k(t)^{\\rho}}{\\alpha k(t)^{\\rho} + (1 - \\alpha)} $$\n\nwhich varies with $k(t)$.\n\nThis seemingly simple generalization of the Cobb-Douglas production function, which is necessary in order for the Solow model generate variable factor share, an economically important feature of the post-war growth experience in most countries, renders the Solow model analytically intractable. To make progress solving a Solow growth model with CES production one needs to resort to computational methods.\n\n# 1. Creating an instance of the `solow.Model` class\n\nWe begin by creating an instance of the `solow.Model` class in the IPython notebook. As always, it is a good idea to read the docstrings...\n\n\n```python\nsolowpy.Model?\n```\n\n From the docsting we see that in order to create an instance of the model we need to specify two primitives: the extensive production function, $F$, and a dictionary of model parameters.\n\n\n```python\n# define model variables\nA, K, L = sym.symbols('A, K, L')\n\n# define production parameters\nalpha, sigma = sym.symbols('alpha, sigma')\n\n# specify some production function\nrho = (sigma - 1) / sigma\nces_output = (alpha * K**rho + (1 - alpha) * (A * L)**rho)**(1 / rho)\n```\n\n\n```python\n# define model parameters\nces_params = {'A0': 1.0, 'L0': 1.0, 'g': 0.02, 'n': 0.03, 's': 0.15,\n 'delta': 0.05, 'alpha': 0.33, 'sigma': 0.95}\n\n# create an instance of the solow.Model class\nces_model = solowpy.CESModel(params=ces_params)\n```\n\nMore details on on how to create instances of the `solow.Model` class can be found in the **Getting started** notebook in the [solowPy](https://github.com/davidrpugh/solowPy) repository.\n\n# 2. Finding the steady state\n\nTraditionally, most analysis of the Solow model focuses almost excusively on the long run steady state of the model. Recall that the steady state of the Solow model is the value of capital stock (per unit effective labor) that solves\n\n$$ 0 = sf(k^*) - (g + n + \\delta)k^*. \\tag{2.0.1} $$\n\nIn words: in the long-run, capital stock (per unit effective labor) converges to the value that balances actual investment, \n$$sf(k),$$ \nwith effective depreciation,\n\n$$(g + n + \\delta)k.$$\n\nGiven the assumption made about the aggregate production technology, $F$, and its intensive form, $f$, there is always a unique value $k^* >0$ satisfying equation 2.0.1.\n\nAlthough it is trivial to derive an analytic expression for the long-run equilibrium of the Solow model for most intensive production functions, the Solow model serves as a good illustrative case for various numerical methods for solving non-linear equations.\n\nThe `solowpy.Model.find_steady_state` method provides a simple interface to the various 1D root finding routines available in `scipy.optimize` and uses them to solve the non-linear equation 2.0.1. To see the list of currently supported methods, check out the docstring for the `Model.find_steady_state` method...\n\n\n```python\nsolowpy.Model.find_steady_state?\n```\n\n\n```python\nk_star, result = ces_model.find_steady_state(1e-6, 1e6, method='bisect', full_output=True)\n```\n\n\n```python\nprint(\"The steady-state value is {}\".format(k_star))\nprint(\"Did the bisection algorithm coverge? {}\".format(result.converged))\n```\n\n The steady-state value is 1.82583173149\n Did the bisection algorithm coverge? True\n\n\nMore details on on how to the various methods of the `solow.Model` class for finding the model's steady state can be found in the accompanying **Finding the steady state** notebook in the [solowPy](https://github.com/davidrpugh/solowPy) repository.\n\n# 3. Graphical analysis using Matplotlib and IPython widgets\n\nGraphical analysis is an important pedagogic and research technique for understanding the behavior of the Solow (or really any!) model and as such the `solow.Model` class has a number of useful, built-in plotting methods.\n\n## Static example: the classic Solow diagram\n\n\n```python\nfig, ax = plt.subplots(1, 1, figsize=(8,6))\nces_model.plot_solow_diagram(ax)\nfig.show()\n```\n\n## Interactive example: the classic Solow diagram\n\n\n```python\nfrom IPython.html.widgets import fixed, interact, FloatSliderWidget\n```\n\n\n```python\n# wrap the static plotting code in a function\ndef interactive_solow_diagram(model, **params):\n \"\"\"Interactive widget for the Solow diagram.\"\"\"\n fig, ax = plt.subplots(1, 1, figsize=(8, 6))\n model.plot_solow_diagram(ax, Nk=1000, **params)\n \n# define some widgets for the various parameters\neps = 1e-2\ntechnology_progress_widget = FloatSliderWidget(min=-0.05, max=0.05, step=eps, value=0.02)\npopulation_growth_widget = FloatSliderWidget(min=-0.05, max=0.05, step=eps, value=0.02)\nsavings_widget = FloatSliderWidget(min=eps, max=1-eps, step=eps, value=0.5)\noutput_elasticity_widget = FloatSliderWidget(min=eps, max=1.0, step=eps, value=0.5)\ndepreciation_widget = FloatSliderWidget(min=eps, max=1-eps, step=eps, value=0.5)\nelasticity_substitution_widget = FloatSliderWidget(min=eps, max=10.0, step=0.01, value=1.0+eps)\n\n# create the widget!\ninteract(interactive_solow_diagram, \n model=fixed(ces_model),\n g=technology_progress_widget,\n n=population_growth_widget,\n s=savings_widget, \n alpha=output_elasticity_widget,\n delta=depreciation_widget,\n sigma=elasticity_substitution_widget,\n )\n```\n\nThere are number of additional plotting methods available (all of which can be turned into interactive plots using IPython widgets). See the **Graphical analysis** notebook in the [solowPy](https://github.com/davidrpugh/solowPy) repository.\n\n# 4. Solving the Solow model\n\nSolving the Solow model requires efficiently and accurately approximating the solution to a non-linear ordinary differential equation (ODE) with a given initial condition (i.e., an non-linear initial value problem). \n\n## 4.1 Solow model as an initial value problem\n\nThe Solow model with can be formulated as an initial value problem (IVP) as follows.\n\n$$ \\dot{k}(t) = sf(k(t)) - (g + n + \\delta)k(t),\\ t\\ge t_0,\\ k(t_0) = k_0 \\tag{4.1.0} $$\n\nThe `quantecon` library has its own module `quantecon.ivp` for solving initial value problems of this form using [finite difference methods](http://en.wikipedia.org/wiki/Finite_difference_method). Upon creation of our instance of the `solow.Model` class, an instance of the `quantecon.ivp.IVP` class was created and stored as an attribute of our model...\n\n\n```python\nces_model.ivp?\n```\n\n...meaning that we can solve this initial value problem by applying the `solve` method of the `ivp` attribute!\n\n\n```python\nces_model.ivp.solve?\n```\n\n\n```python\n# need to specify some initial conditions\nt0, k0 = 0.0, 0.5\nnumeric_soln = ces_model.ivp.solve(t0, k0, T=100, integrator='dopri5')\n```\n\nWe can plot the finite-difference approximation of the solution as follows...\n\n\n```python\nfig, ax = plt.subplots(1, 1, figsize=(8,6))\n\n# plot the finite-difference-approximation\nax.plot(numeric_soln[:,0], numeric_soln[:,1], 'bo', markersize=3.0)\n\n# equilibrium value of capital stock (per unit effective labor)\nk_star = ces_model.steady_state\nax.axhline(k_star, linestyle='dashed', color='k', label='$k^*$')\n\n# axes, labels, title, etc\nax.set_xlabel('Time, $t$', fontsize=15, family='serif')\nax.set_ylabel('$k(t)$', rotation='horizontal', fontsize=20, family='serif')\nax.set_title('Finite-difference approximation',\n fontsize=20, family='serif')\nax.legend(loc=0, frameon=False, bbox_to_anchor=(1.0, 1.0))\nax.grid('on')\n\nplt.show()\n```\n\nFinite-difference methods only return a discrete approximation to the continuous function $k(t)$. To get a continouous approximation of the solution we can combined finite-difference methods with [B-spline interpolation](http://en.wikipedia.org/wiki/B-spline) using the `interpolate` method of the `ivp` attribute.\n\n\n```python\nces_model.ivp.interpolate?\n```\n\n\n```python\n# interpolate!\nti = np.linspace(0, 100, 1000)\ninterpolated_soln = ces_model.ivp.interpolate(numeric_soln, ti, k=3)\n```\n\nWe can graphically compare the discrete, finite-difference approximation with the continuous, B-spline approximation as follows.\n\n\n```python\nfig, ax = plt.subplots(1, 1, figsize=(8,6))\n\n# plot the interpolated and finite difference approximations\nax.plot(ti, interpolated_soln[:,1], 'r-')\nax.plot(numeric_soln[:,0], numeric_soln[:,1], 'bo', markersize=3.0)\n\n# equilibrium value of capital stock (per unit effective labor)\nk_star = ces_model.steady_state\nax.axhline(k_star, linestyle='dashed', color='k', label='$k^*$')\n\n# axes, labels, title, etc\nax.set_xlabel('Time, $t$', fontsize=15, family='serif')\nax.set_ylabel('$k(t)$', rotation='horizontal', fontsize=20, family='serif')\nax.set_title('B-spline approximation of the solution',\n fontsize=20, family='serif')\nax.legend(loc=0, frameon=False, bbox_to_anchor=(1.0, 1.0))\n\nplt.show()\n```\n\n## Accuracy of our numerical methods\n\nWhen doing numerical work it is important to understand the accuracy of the methods that you are using to approximate the solution to your model. Typically one assesses the accuracy of a solution method by computing and evaluatin some residual function:\n\n$$ R(k; \\theta) = sf(\\hat{k}(\\theta)) - (g + n + \\delta)\\hat{k}(\\theta) - \\dot{\\hat{k}}(\\theta) $$\n\nwhere $\\hat{k}(\\theta)$ is out computed solution to the original differential equation. We can assess the accuracy of our finite-difference methods by plotting the residual function for our approximate solution using the `compute_residual` method of the `ivp` attribute.\n\n\n```python\n# compute the residual...\nti = np.linspace(0, 100, 1000)\nresidual = ces_model.ivp.compute_residual(numeric_soln, ti, k=3)\n```\n\nWe can then plot the residual as follows. Our approximation is accurate so long as the residual is everywhere \"close\" to zero.\n\n\n```python\n# extract the raw residuals\ncapital_residual = residual[:, 1]\n\n# typically, normalize residual by the level of the variable\nnorm_capital_residual = np.abs(capital_residual) / interpolated_soln[:,1]\n\n# create the plot\nfig = plt.figure(figsize=(8, 6))\nplt.plot(interpolated_soln[:,1], norm_capital_residual, 'b-', label='$k(t)$')\nplt.axhline(np.finfo('float').eps, linestyle='dashed', color='k', label='Machine eps')\nplt.xlabel('Capital (per unit effective labor), $k$', fontsize=15, family='serif')\nplt.ylim(1e-16, 1)\nplt.ylabel('Residuals (normalized)', fontsize=15, family='serif')\nplt.yscale('log')\nplt.title('Residual', fontsize=20, family='serif')\nplt.legend(loc=0, frameon=False, bbox_to_anchor=(1.0,1.0))\nplt.show()\n```\n\nFor more details behind the numerical methods used in this section see the the **Solving the Solow model** notebook in the [solowPy](https://github.com/davidrpugh/solowPy) repository.\n\n# 5. Impulse response functions\n\nImpulse response functions (IRFs) are a standard tool for analyzing the short run dynamics of dynamic macroeconomic models, such as the Solow growth model, in response to an exogenous shock. The `solow.impulse_response.ImpulseResponse` class has several attributes and methods for generating and analyzing impulse response functions. \n\n### Example: Impact of a change in the savings rate\nOne can analyze the impact of a doubling of the savings rate on model variables as follows.\n\n\n```python\n# 50% increase in the current savings rate...\nces_model.irf.impulse = {'s': 1.5 * ces_model.params['s']}\n\n# in efficiency units...\nces_model.irf.kind = 'efficiency_units'\n```\n\n\n```python\nfig, ax = plt.subplots(1, 1, figsize=(8,6))\nces_model.irf.plot_impulse_response(ax, variable='output')\nplt.show()\n```\n\n### Example: Interactive impulse reponse functions\nUsing IPython widgets makes it extremely easy to analyze the various impulse response functions.\n\n\n```python\ndef interactive_impulse_response(model, shock, param, variable, kind, log_scale):\n \"\"\"Interactive impulse response plotting tool.\"\"\" \n # specify the impulse response\n model.irf.impulse = {param: shock * model.params[param]}\n model.irf.kind = kind\n \n # create the plot\n fig, ax = plt.subplots(1, 1, figsize=(8,6))\n model.irf.plot_impulse_response(ax, variable=variable, log=log_scale)\n \n\nirf_widget = interact(interactive_impulse_response, \n model=fixed(ces_model),\n shock = FloatSliderWidget(min=0.1, max=5.0, step=0.1, value=0.5),\n param = ['g', 'n', 's', 'alpha', 'delta' , 'sigma'],\n variable=['capital', 'output', 'consumption', 'investment'],\n kind=['efficiency_units', 'per_capita', 'levels'],\n log_scale=False,\n )\n```\n\nFor more details and examples see the accompanying **Impulse response function** notebook in the [solowPy](https://github.com/davidrpugh/solowPy) repository.\n\n# 6. The Solow model, declining labor's share, and secular stagnation\n\nRecently there has been much discussion about the reasons for the [Elsby (2013)](http://www.brookings.edu/~/media/Projects/BPEA/Fall%202013/2013b_elsby_labor_share.pdf) has a nice paper that looks at the decline of labor share in the U.S.; as well as much debat about whether or not developed economes are experiencing some sort of [secular stagnation](http://www.economist.com/blogs/buttonwood/2014/11/secular-stagnation) more generally.\n\n\n```python\ndef awesome_interactive_plot(model, iso3_code, **params):\n \"\"\"Interactive widget for the my awesome plot.\"\"\"\n \n # extract the relevant data\n tmp_data = pwt.major_xs(iso3_code)\n actual_labor_share = tmp_data.labsh.values\n actual_capital_share = 1 - tmp_data.labsh\n \n output = tmp_data.rgdpna\n capital = tmp_data.rkna\n labor = tmp_data.emp\n \n # need to update params\n model.params.update(params)\n \n # get new initial condition\n implied_technology = model.evaluate_solow_residual(output, capital, labor)\n k0 = tmp_data.rkna[0] / (implied_technology[0] * labor[0])\n \n # finite difference approximation\n T = actual_labor_share.size\n soln = model.ivp.solve(t0, k0, T=T, integrator='dopri5')\n \n # get predicted labor share\n predicted_capital_share = model.evaluate_output_elasticity(soln[:,1])\n predicted_labor_share = 1 - predicted_capital_share\n \n # get predicted output per unit labor\n predicted_intensive_output = model.evaluate_intensive_output(soln[:,1])\n technology = implied_technology[0] * np.exp(ces_model.params['g'] * soln[:,0])\n predicted_output_per_unit_labor = predicted_intensive_output * technology\n\n # make the plots!\n fig, axes = plt.subplots(1, 2, figsize=(12,6))\n axes[0].plot(soln[:,0], predicted_labor_share, 'b')\n axes[0].plot(soln[:,0], predicted_capital_share, 'g')\n axes[0].plot(actual_labor_share)\n axes[0].plot(actual_capital_share) \n axes[0].set_xlabel('Time, $t$', fontsize=15, family='serif')\n axes[0].set_ylim(0, 1)\n axes[0].set_title('Labor share of income in {}'.format(iso3_code),\n fontsize=20, family='serif')\n axes[0].legend(loc=0, frameon=False, bbox_to_anchor=(1.0, 1.0))\n \n axes[1].set_xlabel('Time, $t$', fontsize=15, family='serif')\n axes[1].set_title('Growth rate of Y/L in {}'.format(iso3_code),\n fontsize=20, family='serif')\n axes[1].legend(loc=0, frameon=False, bbox_to_anchor=(1.0, 1.0))\n axes[1].plot(soln[1:,0], np.diff(np.log(predicted_output_per_unit_labor)),\n 'b', markersize=3.0)\n axes[1].plot(np.log(output / labor).diff().values)\n\n \n# define some widgets for the various parameters\ntechnology_progress_widget = FloatSliderWidget(min=-0.05, max=0.05, step=5e-3, value=0.01)\npopulation_growth_widget = FloatSliderWidget(min=-0.05, max=0.05, step=5e-3, value=0.01)\nsavings_widget = FloatSliderWidget(min=eps, max=1-eps, step=5e-3, value=0.2)\noutput_elasticity_widget = FloatSliderWidget(min=eps, max=1.0, step=5e-3, value=0.15)\ndepreciation_widget = FloatSliderWidget(min=eps, max=1-eps, step=5e-3, value=0.02)\nelasticity_substitution_widget = FloatSliderWidget(min=eps, max=10.0, step=1e-2, value=2.0+eps)\n\n# create the widget!\ninteract(awesome_interactive_plot, \n model=fixed(ces_model),\n iso3_code='USA',\n g=technology_progress_widget,\n n=population_growth_widget,\n s=savings_widget, \n alpha=output_elasticity_widget,\n delta=depreciation_widget,\n sigma=elasticity_substitution_widget,\n )\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "f590dc806dcd290c6ea97aceeffb0556d404fcce", "size": 444269, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/0 Motivation.ipynb", "max_stars_repo_name": "davidrpugh/solowPy", "max_stars_repo_head_hexsha": "91577e04481cec80679ae571ec2bdaa5788151b4", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 31, "max_stars_repo_stars_event_min_datetime": "2016-02-29T00:20:53.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-26T17:40:38.000Z", "max_issues_repo_path": "examples/0 Motivation.ipynb", "max_issues_repo_name": "rfonsek/solowPy", "max_issues_repo_head_hexsha": "91577e04481cec80679ae571ec2bdaa5788151b4", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 11, "max_issues_repo_issues_event_min_datetime": "2015-04-04T20:01:35.000Z", "max_issues_repo_issues_event_max_datetime": "2017-02-20T05:42:49.000Z", "max_forks_repo_path": "examples/0 Motivation.ipynb", "max_forks_repo_name": "rfonsek/solowPy", "max_forks_repo_head_hexsha": "91577e04481cec80679ae571ec2bdaa5788151b4", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 20, "max_forks_repo_forks_event_min_datetime": "2015-08-23T23:42:09.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-23T08:00:53.000Z", "avg_line_length": 377.7797619048, "max_line_length": 141400, "alphanum_fraction": 0.9221170057, "converted": true, "num_tokens": 6043, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.7401743620390163, "lm_q1q2_score": 0.449776699065667}} {"text": "# solve_ivp gives \"wobbly\" results?\n\n## Purpose\n* solve_ivp with RK45 gives some wobbly results can this be improved?\n\n## Methodology\n* Run solve_ivp with various settings.\n* compare the accelerations.\n\n## Results\nDescribe and comment the most important results.\n\n## Setup\n\n\n```python\n# %load imports.py\n## Local packages:\n\n%matplotlib inline\n%load_ext autoreload\n%autoreload 2\n%config Completer.use_jedi = False ## (To fix autocomplete)\n\n## External packages:\nimport pandas as pd\npd.options.display.max_rows = 999\npd.options.display.max_columns = 999\npd.set_option(\"display.max_columns\", None)\n\nimport numpy as np\nimport os\nimport matplotlib.pyplot as plt\n#if os.name == 'nt':\n# plt.style.use('presentation.mplstyle') # Windows\n\nimport plotly.express as px \nimport plotly.graph_objects as go\n\nimport seaborn as sns\nimport sympy as sp\nfrom sympy.physics.mechanics import (dynamicsymbols, ReferenceFrame,\n Particle, Point)\nfrom sympy.physics.vector.printing import vpprint, vlatex\nfrom IPython.display import display, Math, Latex\nfrom src.substitute_dynamic_symbols import run, lambdify\n\nimport pyro\n\nimport sklearn\nimport pykalman\nfrom statsmodels.sandbox.regression.predstd import wls_prediction_std\nimport statsmodels.api as sm\n\nfrom scipy.integrate import solve_ivp\n\n## Local packages:\nfrom src.data import mdl\n\nfrom src.symbols import *\nfrom src.parameters import *\nimport src.symbols as symbols\nfrom src import prime_system\nfrom src.models import regression\nfrom src.visualization.regression import show_pred\nfrom src.visualization.plot import track_plot\n\n## Load models:\n# (Uncomment these for faster loading):\nimport src.models.vmm_simple_nonlinear as vmm\nfrom src.data.case_1 import ship_parameters, df_parameters, ps, ship_parameters_prime\nfrom src.data.transform import transform_to_ship\n```\n\n## Ship parameters\n\n\n```python\nship_parameters\n```\n\n## Brix parameters\n\n\n```python\nmask = df_parameters['prime'].notnull()\nindex = df_parameters.loc[mask,'prime'].index\ncoefficients=vmm.simulator.get_all_coefficients(sympy_symbols=False)\nmissing_coefficients = set(coefficients) - set(index)\nmissing_coefficients\n```\n\n\n```python\nmask = df_parameters['prime'].notnull()\ndf_parameters.loc[mask,'prime']\n```\n\n## Simulate data\n\n\n```python\nparameters=df_parameters['prime'].copy()\n\nt_ = np.linspace(0,70,1000)\ndf_ = pd.DataFrame(index=t_)\n\ndf_['u'] = 2\ndf_['v'] = 0\ndf_['r'] = 0\ndf_['x0'] = 0\ndf_['y0'] = 0\ndf_['psi'] = 0\ndf_['U'] = np.sqrt(df_['u']**2 + df_['v']**2)\ndf_['beta'] = -np.arctan2(df_['v'],df_['u'])\ndf_['thrust'] = 50\n\ndf_['delta'] = 0\ndf_.loc[10:,'delta'] = np.deg2rad(20)\n\nresults = {}\nfor method in ['RK45','Radau','BDF','RK23','DOP853','LSODA']:\n \n result = vmm.simulator.simulate(df_=df_, parameters=parameters, ship_parameters=ship_parameters, \n control_keys=['delta','thrust'], primed_parameters=True,\n prime_system=ps, method=method)\n results[method] = result\n \n```\n\n\n```python\nresults=pd.Series(results)\n```\n\n## Compare\n\n\n```python\nfig,ax=plt.subplots()\nfor method,result in results.items():\n result.result.plot(y='u1d', label=method, ax=ax);\n\nax.set_ylim(results['RK45'].result['u1d'].min(), results['RK45'].result['u1d'].max())\n```\n\n\n```python\nfig,ax=plt.subplots()\nfor method,result in results.loc[['Radau','BDF','LSODA']].items():\n result.result.plot(y='u1d', label=method, ax=ax);\n\nax.set_ylim(results['RK45'].result['u1d'].min(), results['RK45'].result['u1d'].max())\n```\n\n\n```python\nx,y,z = sp.symbols('x y z')\nM = sp.Matrix([sp.sin(x) + y, sp.cos(y) + x, z]) \nM\n```\n\n\n```python\nM.jacobian([x, y, z])\n```\n\n\n```python\neq_acceleration = vmm.simulator.acceleartion_eq.subs([(X_qs,vmm.X_qs_eq.rhs),\n (Y_qs,vmm.Y_qs_eq.rhs),\n (N_qs,vmm.N_qs_eq.rhs),\n ])\nsubs = {value:key for key,value in p.items()} \neq_acceleration = eq_acceleration.subs(subs)\n\njac = eq_acceleration.jacobian([u,v,r])\njac_lambda=lambdify(jac)\n```\n\n\n```python\njac_lambda\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "3d66711c5cf7b998bdc7c2815503e976014e7b0e", "size": 7681, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/13.01_solve_ivp.ipynb", "max_stars_repo_name": "martinlarsalbert/wPCC", "max_stars_repo_head_hexsha": "16e0d4cc850d503247916c9f5bd9f0ddb07f8930", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/13.01_solve_ivp.ipynb", "max_issues_repo_name": "martinlarsalbert/wPCC", "max_issues_repo_head_hexsha": "16e0d4cc850d503247916c9f5bd9f0ddb07f8930", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/13.01_solve_ivp.ipynb", "max_forks_repo_name": "martinlarsalbert/wPCC", "max_forks_repo_head_hexsha": "16e0d4cc850d503247916c9f5bd9f0ddb07f8930", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.4617834395, "max_line_length": 110, "alphanum_fraction": 0.5301393048, "converted": true, "num_tokens": 1117, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.6723316860482762, "lm_q1q2_score": 0.44970939321965014}} {"text": "```python\nclean_up = True\n%run StdPackages.ipynb\nd['gams'] = os.path.join(d['CGE'],'gams')\n```\n\n## Test A: Production version 2\nInvestigate how nested CES and normalized CES sectors work. This goes through identical steps for:\n* CES1: A simple CES sector.\n* CES2: A nested CES sector (2 nests).\n* CES1_norm: Normalized CES sector.\n* CES2_norm: Nested and normalized CES sector.\n* CET1: A simple CET sector.\n* CET2: A nested CET sector. \n* CET1_norm: A normalized CET sector.\n* CET2_norm: A nested, normalized CET sector.\n* CES_CET: A nested CES, CET sector (CES to a intermediate good, that is split into two outputs CET).\n* CES_CET_norm: A nested CES, CET normalized sector (same as CES_CET, but CET sector has normalized technology).\n* CESCET_norm: A nested, normalized CES,CET sector (same as CES_CET, but both sectors have normalized tech).\n* FunkyTree: Mixes normalized input and output trees: Z1 is an input that is split into an output Y1 and an intermediate good X1. Similarly, Z2 is split into output Y2 and intermediate good X2. Finally, X1 and X2 are combined to a single output Y3. \n* ExtremeFunk: Mixes normalized input and output trees to highlight the role of choosing the right $\\mu$s to keep exogenous when calibrating.\n\n## Initialize modules:\n\n*Global settings:*\n\n\n```python\nglob = CGE_globals.SmallOpen(kwargs_vals = {'t': range(1,3)})\n```\n\n*Init nesting structures:*\n\n\n```python\nFunnyName_NS = {n+'_input': n for n in ('Y1','Y2')}\ndata_str = os.path.join(d['data'], 'Nestings.xlsx')\nread_trees = {'CES1': {'CES1': {'f': 'CES'}},\n 'CES2': {'CES2': {'f':'CES'}},\n 'CES1_norm': {'CES1': {'f':'CES_norm'}}, \n 'CES2_norm': {'CES2': {'f':'CES_norm'}},\n 'CET1': {'CET1': {'f': 'CET'}},\n 'CET2': {'CET2': {'f': 'CET'}},\n 'CET1_norm': {'CET1': {'f': 'CET_norm'}},\n 'CET2_norm': {'CET2': {'f': 'CET_norm'}},\n 'CES_CET': {'CES1': {'f': 'CES'}, 'CET1': {'f': 'CET'}},\n 'CES_CET_norm': {'CES1': {'f':'CES'}, 'CET1': {'f':'CET_norm'}},\n 'CESCET_norm': {'CES1': {'f':'CES_norm'}, 'CET1': {'f': 'CET_norm'}},\n 'FunkyTree': {'FunkyTree_CET': {'f':'CET_norm'}, 'FunkyTree_CES': {'f':'CES_norm'}},\n 'ExtremeFunk': {'ExtremeF1': {'f': 'CES_norm'}, 'ExtremeF2': {'f':'CET_norm'}},\n 'FunnyName': {'FunnyNameInp': {'f':'CES_norm'}, 'FunnyNameOut': {'f':'CET_norm'}}}\nTrees = {k: NestingTree.AggTree_from_data(data_str, read_trees = read_trees[k], name = k)(namespace=FunnyName_NS) for k in read_trees} # init trees\n```\n\n*Init:*\n\n\n```python\nws = gams.GamsWorkspace(working_directory=d['work'])\nPs = {k: CGE_Production.Production(tree = Trees[k], glob = glob, ns = {}, s_kwargs = {'ws': ws}) for k in Trees}\n```\n\n#### Calibration subsets\n\nThe default calibration method relies on endogenizing all $\\mu$ parameters and in turn exogenize all $qD$ variables in the nesting tree. This method has to be adjusted in a way that depends on (1) whether or not the model features multiple outputs per sector and (2) if there are nests with scale-preserving technologies.\n\n*NB: We ignore the time index for now.*\n\n##### Without scale-preserving technologies\n\nIn the simple case, all $\\mu$ are endogenized and all $qD$ are exogenized at some level specified by input-output data. Note that even if we know the cost price on ouptuts, $pS[s,n]$, we would leave this variable endogenous. The reason is that for a sector with constant returns to scale (CRS) technology, we have that:\n\n$$\\begin{align}\n \\sum_{n\\in inputs} qD[s,n]pD[s,n] = \\sum_{n\\in outputs} qS[s,n] pS[s,n].\n\\end{align}$$\n\nAll variables on the left-hand-side (LHS) is exogenous when calibration, and so is the supplied quantities $qS[s,n]$. Thus, for the system of equations to be square, we need to leave $pS[s,n]$ endogenous (alternatively, we could remove one of the price index equations from the system when we calibrate). If input-output data is consistent, note that the solution will automatically be the in data.\n\nNote that we still allow for multiple outputs to be produced. Usually, we would distinguish between cost-prices $pS[s,n]$ and equilibrium prices $p[n]$ and assume a price equation in the style of\n\n$$\\begin{align}\n p[n] = (1+m[s])(pS[s,n]+\\tau[s,n])+\\Gamma[s,n]),\n\\end{align}$$\n\nwhere $m[s]$ is a sector-specific mark-up, $\\tau[s,n]$ is a unit tax, and $\\Gamma[s,n]$ is a function capturing e.g. adjustment costs of investments. Note that this equation holds for all (outputs, sector)-combinations. If we observe $p[n],\\tau[s,n]$ and can compute $\\Gamma[s,n]$ from the model, this identifies $m[s]$ and all but one $pS[s,n]$; the last one ($pS$) is endogenous.\n\n##### Scale-preserving technologies\n\nWhen we have scale-preserving technologies, we have to think more carefully about what variables are endogenized/exogenized when calibrating. Consider for instance the simple case with $Y$ being produced by $X_1,X_2$ using some scale-preserving technology. The system of equations read (ignoring taxes+markups etc).:\n\n$$\\begin{align}\n p_Y Y &= p_1X_1+p_2X_2 \\\\ \n X_1 &= \\dfrac{F_1(p_1,p_2,p_Y; \\mu_1)}{F_1(p_1,p_2,p_Y; \\mu_1)+F_2(p_1,p_2,p_Y; \\mu_2)}Y \\\\ \n X_2 &= \\dfrac{F_2(p_1,p_2,p_Y; \\mu_2)}{F_1(p_1,p_2,p_Y; \\mu_1)+F_2(p_1,p_2,p_Y; \\mu_2)}Y,\n\\end{align}$$\n\nwhere $F_1,F_2$ are some price functions with share parameters $\\mu_1,\\mu_2$. In baseline mode, this system is square in $p_Y,X_1,X_2$ taking $Y,p_1,p_2,\\mu_1,\\mu_2$ as given. In calibration mode we would usually exogenize $X_1,X_2$ and endogenize $\\mu_1,\\mu_2$. Given exogenous variables, we can always choose a set of $\\mu_1,\\mu_2$ to induce values $F_1,F_2$, thus, we are essentially solving for $F_1,F_2,p_Y$. Note that solving for $F_1,F_2$ in demand functions yield\n\n$$\\begin{align}\n F_1\\left(1-\\dfrac{X_1}{Y}\\right) &= \\dfrac{X_1}{Y}F_2 \\\\ \n F_2\\left(1-\\dfrac{X_2}{Y}\\right) &= \\dfrac{X_2}{Y}F_1\n\\end{align}$$\n\nUsing that $1-X_1/Y = X_2/Y$ and $1-X_2/Y=X_1/Y$ note that the two conditions are linearly dependent, i.e. that they identify the same restriction:\n\n$$\\begin{align}\n F_1 &= \\dfrac{X_1}{X_2} F_2 \\\\ \n F_2 &= \\dfrac{X_2}{X_1} F_1.\n\\end{align}$$\n\nNaturally, with a scale-preserving function, a nest with $N$ share parameters only identify $N-1$ variables; the final one is residually determined by $X_N = Y-\\sum_{i=1}^{N-1}X_i$. This causality means that we need to keep one of the $\\mu$ fixed in calibration mode in scale-preserving nests, and keep the corresponding quantity endogenous.\n\n##### Identifying the right $\\mu$s to keep exogenous/quantities to keep endogenous\n\nAs it turns out, simply choosing one random element in $\\mu$ for all scale-preserving nests to keep exogenous (and endogenize corresponding quantity), does not always work. In particular, this can be a problem if we have nodes in the nesting tree that are simultaneously a branch in a scale-preserving input tree and another scale-preserving output tree. The nesting tree 'FunkyTree' and 'ExtremeFunk' are examples of this. In this case, we could potentially, randomly, pick the element $qD[s,n]$ to be the endogenous element for two trees, thus exogenizing two $\\mu$ elements, but only endogenizing one quantity. Thus, we go through a couple of steps to make sure that does not happen\n\n##### Identifying calibration subsets, the general case:\n\nThe ```Production``` class includes an algorithm that ensures that the right $\\mu$s are kept exogenous in calibration mode, the right quantities $(qD/qS)$ are kept/made endogenous. Importantly, note that the subset ```exomu``` identifies the same number of elements as ```endo_qS``` and ```endo_qD``` combined.\n\n\n```python\nn = 'ExtremeFunk'\np = Ps[n]\nt = Trees[n]\n```\n\n*Four elements are kept exogenous:*\n\n\n```python\np.get('exomu')\n```\n\n\n\n\n MultiIndex([('s1', 'X1', 'Z2'),\n ('s1', 'Y1', 'Z1'),\n ('s1', 'Y2', 'X3'),\n ('s1', 'Y3', 'Z3')],\n names=['s', 'n', 'nn'])\n\n\n\n*One supply element and three demand elements are endogenized:*\n\n\n```python\np.get('endo_qS')\n```\n\n\n\n\n MultiIndex([('s1', 'Y3')],\n names=['s', 'n'])\n\n\n\n\n```python\np.get('endo_qD')\n```\n\n\n\n\n MultiIndex([('s1', 'X1'),\n ('s1', 'X3'),\n ('s1', 'Z1')],\n names=['s', 'n'])\n\n\n\n#### Test baseline mode\n\n*Init states:*\n\n\n```python\n[P.compile(initDB=True) for P in Ps.values()];\n```\n\n*Write text:*\n\n\n```python\n[P.write() for P in Ps.values()];\n```\n\n*Run models (with same workspace):*\n\n\n```python\nMs = {k: Ps[k].run(exportTo=d['work'], ws=ws,**{'cns': 'CONOPT4'}) for k in Ps}\n```\n\n#### Test calibration mode:\n\n*Change state:*\n\n\n```python\n[setattr(p.s,'state','C') for p in Ps.values()];\n```\n\n*Update database to baseline solution:*\n\n\n```python\n[setattr(Ps[k].s,'db',Ms[k].out_db) for k in Ps];\n```\n\n*Write:*\n\n\n```python\n[P.write() for P in Ps.values()];\n```\n\n*Re-run:*\n\n\n```python\nMs = {k: Ps[k].run(exportTo=d['work'], ws=ws,**{'cns': 'CONOPT4'}) for k in Ps}\n```\n", "meta": {"hexsha": "85e4703b74fd20d014f48f72721fa75d60c5e087", "size": 16416, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "CGE_Generator/Test_Production.ipynb", "max_stars_repo_name": "ChampionApe/GPM_v06", "max_stars_repo_head_hexsha": "643c8cf6a2dc63475582ae2fb90e76f392ef450c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "CGE_Generator/Test_Production.ipynb", "max_issues_repo_name": "ChampionApe/GPM_v06", "max_issues_repo_head_hexsha": "643c8cf6a2dc63475582ae2fb90e76f392ef450c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "CGE_Generator/Test_Production.ipynb", "max_forks_repo_name": "ChampionApe/GPM_v06", "max_forks_repo_head_hexsha": "643c8cf6a2dc63475582ae2fb90e76f392ef450c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.832, "max_line_length": 693, "alphanum_fraction": 0.570297271, "converted": true, "num_tokens": 2702, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370308082623217, "lm_q2_score": 0.7057850278370112, "lm_q1q2_score": 0.4496068067424565}} {"text": "# Planetary Nebula Luminosity Function \u2013 Development \n\n \nThe goal of this project is to measure the planetary nebula luminosity function (PNLF), using integral field unit (IFU) data, that was observed with MUSE for the [PHANGS](https://sites.google.com/view/phangs/home) collaboration. This notebook is used to test the functions for the analysis. The final version of the code is then moved to the `pnlf` packge in the `src` folder. The notebook `PNLF production.ipynb` uses these functions to measure the PNLF for the individual galaxies and `PNLF postproduction.ipynb` is used to create the final figures for the paper.\n\n## Preparation\n \n### Load Basic Packages\n \nFirst we load a bunch of common packages that are used across the project. More specific packages that are only used in one section are loaded later to make it clear where they belong to (this also applies to all custom moduls that were written for this project).\n\n\n```python\n# reload modules after they have been modified\n%load_ext autoreload\n%autoreload 2\n\nfrom pnlf.packages import *\n\nfrom pnlf.constants import tab10, single_column, two_column\n\n%matplotlib inline\n%config InlineBackend.figure_format = 'retina'\n```\n\nwe use the `logging` module to handle informations and warnings (this does not always work as expected in jupyter notebooks).\n\n\n```python\nlogging.basicConfig(stream=sys.stdout,format='%(levelname)s: %(message)s',level=logging.INFO)\nlogger = logging.getLogger(__name__)\nlogging.getLogger('matplotlib').setLevel(logging.WARNING)\n```\n\n### Read in data\n\nthis uses the `ReadLineMaps` class from the `pnlf.io` module. To use it, we first need to specify the path to the data folder\n\n\n```python\nfrom pnlf.io import ReadLineMaps\n\n#with open(basedir / 'data' / 'interim' / 'parameters.json') as json_file:\n# parameters = json.load(json_file)\nwith open(basedir / 'data' / 'interim' / 'parameters.yml') as yml_file:\n parameters = yaml.load(yml_file,Loader=yaml.FullLoader)\n \n# table to save all results\nresults = ascii.read(basedir/'data'/'interim'/ 'results.txt',format='fixed_width_two_line',delimiter_pad=' ',position_char='=')\nresults.add_index('name') \n\nname = 'NGC1385'\n\n# first we need to specify the path to the raw data\nbasedir = Path('..')\ndata_raw = Path('a:')/'Archive'\n#data_raw = Path('d:\\downloads\\MUSEDAP')\n#data_ext = Path('g:\\Archive')\n\nextensions = ['OIII5006', 'HA6562', 'NII6583', 'SII6716']\n\n# read in the data we will be working with and print some information\ngalaxy = ReadLineMaps(data_raw/'MUSE'/'DR2.1'/'MUSEDAP',name,extensions,**parameters[name])\n```\n\n### Star Masks\n\n\n```python\nfrom pnlf.auxiliary import circular_mask\nfrom pnlf.plot import create_RGB\n \nmask = np.zeros(galaxy.shape,dtype=bool)\nmask |= galaxy.star_mask.astype(bool)\n\nif hasattr(galaxy,'mask'):\n mask[circular_mask(*galaxy.shape,galaxy.center,radius=galaxy.mask)] = True\n\nimg = create_RGB(galaxy.HA6562,galaxy.OIII5006_DAP,galaxy.SII6716,weights=[0.6,1,0.6],percentile=[95,99.,95])\nimg[mask,...] = (1,1,1)\n\nfig = plt.figure(figsize=(8,8))\nax = fig.add_subplot()\n\n#norm = simple_norm(galaxy.OIII5006_DAP,clip=False,max_percent=95)\nax.imshow(img,origin='lower')\nplt.show()\n```\n\n## Source Detection\n\nThere are two different approaches to identifying sources in an image. The first utilizes PSF fitting and uses implementations from astropy. The other uses the external `SExtractor` package which detects peaks and classifies them with a neural network.\n\n### Based on IRAFStarFinder or DAOStarFinder\n\nThe sources we are searching for are unresolved. However due to seeing, they will be smeared out. This PSF has the form of a Gaussian (or Moffat). The subsequent algorithms use this and try to fit a theoretical curve to the observed peaks in the image. If the fit aggrees within some threshold, it reports the peak as a source. The advantage is that for crowded fields, the algorithm will try to fit an individual function to each peak and thus enable us correctly identfiy objects that are closeby.\n\nThe following function is based on this tutorial \n\nhttps://photutils.readthedocs.io/en/stable/detection.html\n\nhttps://photutils.readthedocs.io/en/stable/api/photutils.detection.DAOStarFinder.html#photutils.detection.DAOStarFinder\n\n**Requires**\n * A `photutils` starfinder. This can be either `DAOStarFinder` or `IRAFStarFinder`\n * `detect_unresolved_sources`\n \n**Returns**\n * `sources` a table with the position of all identified sources\n\n\n```python\nfrom photutils import DAOStarFinder # DAOFIND routine to detect sources\nfrom photutils import IRAFStarFinder # IRAF starfind routine to detect star\n\nfrom pnlf.detection import detect_unresolved_sources\n```\n\n\n```python\n# we include all sources in this step and reject bad ones later\nsharplo = 0.0 #galaxy.sharplo\nsharphi = 1.0 #galaxy.sharphi\nroundness = 1.0 #galaxy.roundness\n\nsources = detect_unresolved_sources(galaxy,\n 'OIII5006_DAP',\n StarFinder=DAOStarFinder,\n threshold=galaxy.threshold,\n exclude_region=mask,\n oversize=1,\n roundlo=-roundness,\n roundhi=roundness,\n sharplo=sharplo,\n sharphi=sharphi,\n save=False)\n```\n\n#### Compare to Kreckel et al. 2017\n\nAs mentioned in the beginning, we compare the newly detected sources to those from Kreckel et al. (2017). \n\n**Requires**\n * `match_coordinates_sky` from `astropy.coordinates` to compare the two catalogues.\n * `Angle` from `astropy.coordinates` to set a maximum seperation in units of arcseconds.\n\n\n```python\nfrom astropy.coordinates import match_coordinates_sky # match sources against existing catalog\nfrom astropy.coordinates import Angle # work with angles (e.g. 1\u00b02\u20323\u2033)\n\ntolerance = '0.5s'\nID, angle, Quantity = match_coordinates_sky(pn_NGC628_kreckel['SkyCoord'],sources['SkyCoord'])\nwithin_tolerance = len(angle[angle.__lt__(Angle(tolerance))])\n\nprint(f'{within_tolerance} of {len(angle)} match within {tolerance}\": {within_tolerance / len(angle)*100:.1f} %')\nprint(f'mean seperation is {angle[angle.__lt__(Angle(tolerance))].mean().to_string(u.arcsec,decimal=True)}\"')\n```\n\n#### Compare to Herrmann 2008 for NGC628\n\n\n```python\nmatchcoord = search_table(pn_herrmann,'M74')\nmatchcoord['x'],matchcoord['y']= matchcoord['SkyCoord'].to_pixel(wcs=galaxy.wcs)\n\nmatchcoord['in_frame'] = False\nx_dim,y_dim = galaxy.shape\n\nfor row in matchcoord:\n txt,x,y = row['ID'], row['x'], row['y'] \n if 0<=int(x)1$, the first term will be $0$ for $R\\rightarrow \\infty$ and so we end up with \n\n$$\np(r) = \\left[ 1+\\left( \\frac{R}{\\gamma}\\right)^2\\right]^{1-\\alpha} - 1\n$$\n\n\n```python\nfrom pnlf.auxiliary import light_in_gaussian, light_in_moffat, fwhm_moffat\n\nalpha = 4\ngamma = 10\nfwhm = fwhm_moffat(alpha,gamma)\nprint(f'alpha={alpha:.2f}, gamma={gamma:.2f}, fwhm={fwhm:.2f}')\n\nd = np.arange(0,20,0.2)\ng = light_in_gaussian(d,fwhm)\nm = light_in_moffat(d,alpha,gamma)\nplt.plot(d/fwhm,100*g,label='Gaussian')\nplt.plot(d/fwhm,100*m,label='Moffat')\n\nplt.legend()\nplt.xlabel('diameter in fwhm')\nplt.ylabel('light in aperture in %')\nplt.grid()\n```\n\n#### Create an ideal Gaussian/Moffat source and measure growth curve\n\n\n```python\nfrom astropy.modeling import models, fitting \nfrom astropy.nddata import Cutout2D\nfrom astropy.stats import gaussian_sigma_to_fwhm, gaussian_fwhm_to_sigma\n\nfrom pnlf.photometry import growth_curve\nfrom pnlf.auxiliary import fwhm_moffat\n```\n\n\n```python\nsize=64\nfwhm = 10\nbkg = 0.5\nprint(f'fwhm={fwhm}')\n\nstd = fwhm * gaussian_fwhm_to_sigma\ngaussian = models.Gaussian2D(x_mean=size/2,y_mean=size/2,x_stddev=std,y_stddev=std)\nimg = gaussian(*np.indices((size,size))) + np.random.uniform(0,bkg,(size,size))\nplt.imshow(img, origin='lower')\nplt.show()\n\n```\n\n\n```python\nfwhm_fit = growth_curve(img,size/2,size/2,model='gaussian',plot=True)[0]\nprint(f'fwhm={fwhm}, measured={fwhm_fit:.2f}')\n```\n\n\n```python\nsize=64\nalpha = 2.8\ngamma = 4\n#gamma = 6/(2*np.sqrt(2**(1/4.76)-1))\nbkg = 0.01\n\nfwhm = 2*gamma * np.sqrt(2**(1/alpha)-1)\nprint(f'alpha={alpha:.2f}, gamma={gamma:.2f}, fwhm={fwhm:.2f}')\n\nstd = 4 * gaussian_fwhm_to_sigma\nmoffat = models.Moffat2D(x_0=size/2,y_0=size/2,alpha=alpha,gamma=gamma)\nimg = moffat(*np.indices((size,size))) + np.random.uniform(0,bkg,(size,size))\n#plt.imshow(img, origin='lower')\n#plt.show()\n```\n\n\n```python\nfrom pnlf.photometry import measure_single_flux\n\ndef test_growth_curve(alpha=2.8,gamma=4,bkg=0.01):\n '''test the growth curve\n \n measure the flux with the correct FWHM and values that are too large/small\n '''\n \n # create a mock image with a perfect moffat and some background\n size = 64\n moffat = models.Moffat2D(x_0=size/2,y_0=size/2,alpha=alpha,gamma=gamma)\n img = moffat(*np.indices((size,size))) + np.random.uniform(0,bkg,(size,size))\n \n fig, axes = plt.subplots(nrows=1,ncols=3,figsize=(7,3))\n axes_iter = iter(axes.flatten())\n \n for dfwhm in [-0.5,0.0,0.5]:\n fwhm = 2 * gamma * np.sqrt(2**(1/alpha)-1) + dfwhm\n flux = []\n radii = np.arange(1,3.75,0.5)\n for aperture_size in radii:\n flux.append(measure_single_flux(img,[size/2,size/2],aperture_size,alpha,fwhm/(2*np.sqrt(2**(1/alpha)-1))))\n\n flux=np.array(flux) \n \n print(f'{np.max(np.abs((flux[radii>=1.5]-flux[-1])/flux[-1]*100)):.2f}')\n \n ax = next(axes_iter)\n ax.axvline(1.5,color='black',lw=0.8)\n ax.axhline(-1,color='black',lw=0.8)\n ax.axhline(1,color='black',lw=0.8)\n \n ax.scatter(radii,(flux-flux[-1])/flux[-1]*100,s=5)\n\n ax.set_title(f'FWHM={fwhm:.2f}, Ftot={flux[-1]:.2f}')\n ax.set(xlabel='aperture radius / fwhm',ylim=[-15,15])\n \n plt.tight_layout()\n plt.show()\n \ntest_growth_curve(alpha=2.8,gamma=4,bkg=0.01)\n```\n\n\n```python\nfit = growth_curve(img,size/2,size/2,model='moffat',plot=True,length=10)\nfwhm_fit = fit[0]\nfwhm_fit = fwhm_moffat(*fit)\n\n#print(f'fwhm={fwhm:.2f}, fit={fwhm_fit:.2f}')\n```\n\n#### Test our light_in_moffat function\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom astropy.modeling import models, fitting\nfrom photutils import aperture_photometry\nfrom pnlf.auxiliary import light_in_moffat\n\namplitude = 50\nbackground = 0\nx_mean,y_mean = 25,25\ngamma=6\nalpha=2.8\nfwhm = 2*gamma*np.sqrt(2**(1/alpha)-1)\n\n# create some model data\nmodel = models.Moffat2D(amplitude,x_mean,y_mean,alpha=alpha,gamma=gamma) + models.Const2D(background)\ny, x = np.mgrid[0:51, 0:51]\nz = model(x, y) \n\nfor aperture_size in [1,1.5,2,2.5,3,3.5,5,10]:\n r=aperture_size*fwhm/2\n aperture = CircularAperture((25,25),r=r)\n flux = aperture_photometry(z,aperture)['aperture_sum'][0]\n flux_corr = flux / light_in_moffat(r,alpha,gamma)\n print(f'radius={aperture_size:.2f}FWHM: flux={flux_corr:.2f}')\n\n# plot the original image\nfig,ax1=plt.subplots(ncols=1,figsize=(6,6))\nax1.imshow(z)\naperture = CircularAperture((25,25),r=2*fwhm/2)\naperture.plot(axes=ax1,color='red')\nax1.axis('off')\nplt.show()\n\n```\n\n#### Compare PSF and aperture photometry\n\nhttp://web.ipac.caltech.edu/staff/fmasci/home/astro_refs/aperture_phot2.pdf\n\n\n```python\nfrom astropy.modeling import models \nfrom astropy.stats import gaussian_sigma_to_fwhm, gaussian_fwhm_to_sigma\nfrom pnlf.photometry import measure_single_flux\n\n```\n\n\n```python\n# https://github.com/astropy/photutils/issues/558\nfrom photutils.psf import IterativelySubtractedPSFPhotometry, BasicPSFPhotometry, DAOPhotPSFPhotometry\nfrom photutils.psf import IntegratedGaussianPRF, DAOGroup\nfrom photutils.background import MMMBackground, MADStdBackgroundRMS\nfrom astropy.modeling.fitting import LevMarLSQFitter\n```\n\n\n```python\nsize=30\nalpha = 2.8\ngamma = 4\namplitude=5\nbkg = 0.00\n\nfwhm = 2*gamma * np.sqrt(2**(1/alpha)-1)\nmoffat = models.Moffat2D(amplitude=amplitude,x_0=size/2,y_0=size/2,alpha=alpha,gamma=gamma)\nimg = moffat(*np.indices((size,size))) + np.random.uniform(0,bkg,(size,size))\n\nflux_0 = measure_single_flux(img,[size/2,size/2],aperture_size=2,model='Moffat',alpha=alpha,gamma=gamma)\n\npos = Table(names=['x_0', 'y_0'], data=[[size/2],[size/2]])\npos['id'] = np.arange(1,len(pos)+1)\npos['flux_0'] = [flux_0]\n\nprint(f'alpha={alpha:.2f}, gamma={gamma:.2f}, fwhm={fwhm:.2f}, flux0={flux_0:.2f}')\n# the total flux is given by 2*np.pi*amplitude*gamma**2/(2*(alpha-1))\n```\n\n\n```python\nfrom photutils.psf import prepare_psf_model\n\npsffunc = models.Moffat2D(amplitude=1, gamma=gamma, alpha=alpha, x_0=0, y_0=0)\npsffunc.amplitude.fixed=False\npsffunc.gamma.fixed=True\npsffunc.alpha.fixed=True\npsfmodel = prepare_psf_model(psffunc, xname='x_0', yname='y_0', fluxname=None) \n```\n\n\n```python\nphotometry = BasicPSFPhotometry(group_maker = DAOGroup(fwhm*3),\n bkg_estimator = MMMBackground(),\n psf_model = psfmodel,\n fitter = LevMarLSQFitter(),\n fitshape = (15,15))\nresult_tab = photometry(image=img, init_guesses=pos)\n\nprint(f\"aperture={flux_0:.2f}, PSF={result_tab['flux_fit'][0]:.2f}\")\n```\n\n\n```python\nBasicPSFPhotometry?\n```\n\n\n```python\nresult_tab\n```\n\n\n```python\nA = 2*np.pi\nf = \nf\n```\n\n\n```python\nphotometry = DAOPhotPSFPhotometry(crit_separation = fwhm*3,\n threshold = 0.5,\n fwhm = fwhm,\n psf_model = psfmodel,\n fitter = LevMarLSQFitter(),\n fitshape = (11,11),\n aperture_radius = 3*fwhm)\nresult_tab = photometry(image=img)\n```\n\n\n```python\nresult_tab\n```\n\n\n```python\n# do the same for a Gaussian (maybe the fit works here)\n\nfrom photutils.psf import IntegratedGaussianPRF\n\namplitude=10\nsize=64\nfwhm = 4\nbkg = 0.1\n\nstd = fwhm * gaussian_fwhm_to_sigma\ngaussian = models.Gaussian2D(amplitude=amplitude,x_mean=size/2,y_mean=size/2,x_stddev=std,y_stddev=std)\nimg = gaussian(*np.indices((size,size))) + np.random.uniform(0,bkg,(size,size))\n\nflux_0 = measure_single_flux(img,[size/2,size/2],aperture_size=2,model='Gaussian',fwhm=fwhm)\n\npos = Table(names=['x_0', 'y_0'], data=[[size/2],[size/2]])\npos['id'] = np.arange(1,len(pos)+1)\n#pos['flux_0'] = [flux_0]\nfwhm+=0.5\n\nphotometry = BasicPSFPhotometry(group_maker = DAOGroup(fwhm*2),\n bkg_estimator = MMMBackground(),\n psf_model = IntegratedGaussianPRF(sigma=fwhm*gaussian_fwhm_to_sigma),\n fitter = LevMarLSQFitter(),\n finder = DAOStarFinder(threshold=0.5*np.max(img),fwhm=fwhm),\n fitshape = (7,7))\n\n\nresult_tab = photometry(image=img)\nflux_fit = result_tab['flux_fit'][0]\nprint(f\"aperture={flux_0:.2f}, PSF={flux_fit:.2f}, dif={100*(flux_0-flux_fit)/flux_0:.2f}%\")\n```\n\n#### Apply to real data\n\nWe saw that we can predict the aperture size dependence of the flux for mock sources. Now we pick real objects and try to do the same. \n\nStars should have much larger stellar velocities. We use this to identify potential foreground stars in our source catalogue. We cannot use the peaks of the velocity maps as they seem to be displaced from the center of the star.\n\n\n```python\nfrom scipy.signal import convolve\n\nfrom astropy.nddata import Cutout2D\nfrom astropy.stats import gaussian_fwhm_to_sigma\n\nfrom pymuse.plot import single_cutout\nfrom pymuse.photometry import growth_curve, correct_PSF, fwhm_moffat\n```\n\nwe find stars due to their high velocity dispersion\n\n\n```python\n# define kernel for smoothing\nsmoothing_length = 10 \nkernel = np.ones((smoothing_length,smoothing_length))\n\n# find stars by their large velocity dispersion\nstar_mask = np.zeros(NGC628.V_STARS.shape,dtype='f8')\nstar_mask[np.abs(NGC628.V_STARS)>200] = 1\n\n# smooth ouput with convolution\nstar_mask = convolve(star_mask,kernel,mode='same')\nstar_mask[star_mask>0.1] = 1\nstar_mask[star_mask<0.1] = 0\nstar_mask[np.isnan(NGC628.V_STARS)] = np.nan\n```\n\n\n```python\nstars = detect_unresolved_sources(NGC628,\n 'whitelight',\n StarFinder=DAOStarFinder,\n threshold=5,\n oversize_PSF = 1.,\n save=False)\n \nstars = stars[star_mask[stars['y'].astype(int),stars['x'].astype(int)]==1]\n```\n\n\n```python\nstars\n```\n\n\n```python\ni = 0\nsize = 20\nx,y,fwhm = stars[i][['x','y','fwhm']]\nsingle_cutout(NGC628,'whitelight',x,y,size=size)\nprint(fwhm)\n```\n\n\n```python\ndata = NGC628.HA6562\n\naperture = 25\nfit = growth_curve(data,x,y,model='moffat',rmax=15,plot=True,length=10)\n#fwhm_fit=fit[0]\nfwhm_fit = fwhm_moffat(*fit)\n\nradius = np.arange(0,10,0.5)\nplt.plot(radius,light_in_gaussian(radius,fwhm),label='gaussian',ls='--',color='tab:red')\nplt.legend()\n\nprint(f'reported={fwhm:.2f}, measured={fwhm_fit:.2f}, ratio={fwhm_fit/fwhm:.2f}')\n```\n\nwe have 6 objects classified as stars in our field of view. For each of them we do a growth curve analysis\n\n\n```python\nsize = 20\ndata = NGC628.whitelight\n\nfig,ax = plt.subplots(figsize=(8,5))\n\nfor i in range(len(stars)):\n x,y,fwhm = stars[i][['x','y','fwhm']]\n\n fit = growth_curve(data,x,y,model='moffat',rmax=30,plot=True,length=10)\n fwhm_fit = fwhm_moffat(*fit)\n print(f'alpha={fit[0]:.2f}, gamma={fit[1]:.2f}, reported={fwhm:.2f}, measured={fwhm_fit:.2f}, ratio={fwhm_fit/fwhm:.2f}')\n```\n\n#### Fit 2D Function\n\nSo far we didn't fit the PSF shape directly but took a slight detour with the light inside an aperture. Here we try to fit the 2D functions to the observed data\n\n\n```python\nsize = 16\n\n#sub = sources[sources['peak']>500]\n#x,y,fwhm = sub[8][['x','y','fwhm']]\n\nfor line, cmap in zip(['whitelight','OIII5006','HA6562'],[plt.cm.viridis,plt.cm.Blues_r,plt.cm.Reds_r]):\n # defien the size of the cutout region\n star = Cutout2D(getattr(NGC628,line), (x,y), u.Quantity((size, size), u.pixel))\n\n fitter = fitting.LevMarLSQFitter()\n data = star.data\n fig ,(ax1,ax2,ax3,ax4) = plt.subplots(1,4,figsize=(12,3))\n #cmap = plt.cm.Blues\n\n ax1.imshow(data,origin='lower',cmap=cmap)\n ax1.set_title(f'image {line}')\n\n std = fwhm * gaussian_fwhm_to_sigma\n gaussian_theory = models.Gaussian2D(x_mean=size/2,y_mean=size/2,x_stddev=std,y_stddev=std)\n ax2.imshow(gaussian_theory(*np.indices(data.shape)), origin='lower',cmap=cmap)\n ax2.set_title('Gaussian reported')\n \n model = models.Gaussian2D()\n gaussian = fitter(model,*np.indices(data.shape),data,maxiter=1000)\n ax3.imshow(gaussian(*np.indices(data.shape)), origin='lower',cmap=cmap)\n ax3.set_title('Gaussian fit')\n\n model = models.Moffat2D(alpha=4.765,fixed={'alpha':True}) \n moffat = fitter(model,*np.indices(data.shape),data,maxiter=2000)\n fwhm_moffat = 2*moffat.gamma.value * np.sqrt(2**(1/moffat.alpha.value)-1)\n ax4.imshow(moffat(*np.indices(data.shape)), origin='lower',cmap=cmap)\n ax4.set_title('Moffat fit')\n \n plt.tight_layout()\n xstd = gaussian.x_stddev * gaussian_sigma_to_fwhm\n ystd = gaussian.y_stddev * gaussian_sigma_to_fwhm\n\n print(f'{line}: reported: {fwhm:.3f}, gaussian={xstd:.3f} ,{ystd:.3f}, moffat={fwhm_moffat:.3f}')\n\n```\n\n\n```python\ndef fast_photometry(data,x,y,r,r_in,r_out):\n '''\n performe aperture photometry with backround subtraction for one source\n \n Parameters\n ----------\n data : ndarray\n image data\n \n x : float\n x cooridinate of the source\n \n y : float\n y cooridinate of the source\n \n r : float\n radius of the main aperture\n \n r_in : float\n inner radius of the annulus that is used for the background \n \n r_out : float\n outer radius of the annulus that is used for the background\n \n '''\n \n aperture = CircularAperture((x,y), r=r)\n annulus_aperture = CircularAnnulus((x,y), r_in=r_in, r_out=r_out)\n mask = annulus_aperture.to_mask(method='center')\n annulus_data = mask.multiply(data)\n annulus_data_1d = annulus_data[mask.data > 0]\n _, bkg_median, _ = sigma_clipped_stats(annulus_data_1d[~np.isnan(annulus_data_1d)])\n phot = aperture_photometry(data,aperture)\n \n return phot['aperture_sum'][0]-aperture.area*bkg_median\n\ndef aperture_correction(data,positions,r_aperture):\n \n fluxes = [fast_photometry(data,x,y,r,r_in,r_out) for x,y in positions]\n \n apertures = CircularAperture(positions,r_aperture)\n \n```\n\n\n```python\nfrom scipy.spatial import cKDTree\nfrom pymuse.plot import single_cutout\n\npositions = np.transpose([sources['x'],sources['y']])\n\ntree = cKDTree(positions)\ndists = tree.query(positions, 2)\nnn_dist = dists[0][:, 1]\n\nsub = sources[(nn_dist>16)]\n```\n\n### Background subtraction\n\n\n```python\nfrom photutils import Background2D\n\nmask = np.isnan(galaxy.HA6562)\nbkg = Background2D(galaxy.HA6562,(10,10), \n filter_size=(5,5),\n #sigma_clip= SigmaClip(sigma=3.,maxiters=None), \n #bkg_estimator=SExtractorBackground(),\n mask=mask).background\nbkg[mask] = np.nan\n\nfig, (ax1,ax2,ax3) = plt.subplots(nrows=1,ncols=3,figsize=(two_column,two_column/1.618))\n\nnorm = simple_norm(galaxy.HA6562,clip=False,percent=95)\nax1.imshow(galaxy.HA6562,norm=norm,origin='lower',cmap=plt.cm.Reds)\n\n#norm = simple_norm(bkg,clip=False,max_percent=95)\nax2.imshow(bkg,norm=norm,origin='lower',cmap=plt.cm.Reds)\n\n#norm = simple_norm(galaxy.HA6562-bkg,clip=False,max_percent=95)\nax3.imshow(galaxy.HA6562-bkg,norm=norm,origin='lower',cmap=plt.cm.Reds)\n\nplt.savefig(basedir/'reports'/'background.pdf',dpi=600)\n```\n\n### Aperture Photometry\n\nwe use the positions of the previously detected sources to measure the flux of different lines\n\nhttps://photutils.readthedocs.io/en/stable/aperture.html\n\nthe values in the pixels are in units of $10^{-20} \\ \\mathrm{erg} \\ \\mathrm{cm}^{-2} \\ \\mathrm{s}^{-1} / \\mathrm{spaxel}$. For the [OIII] line, this flux is then converted to an apparent magnitude\n$$\nm_{[\\mathrm{O\\ III}]} = -2.5 \\cdot \\log F_{[\\mathrm{O\\ III}]} - 13.74\n$$\n\nwhere $F_{[\\mathrm{O\\ III}]}$ is given in $\\mathrm{erg} \\ \\mathrm{cm}^{-2} \\ \\mathrm{s}^{-1}$. Error propagation gives the error of the magnitude as\n\n$$\n\\Delta m_{[\\mathrm{O\\ III}]} = \\sqrt{\\left(\\frac{-2.5 \\cdot \\Delta F_{[\\mathrm{O\\ III}]}}{\\ln 10 \\cdot F_{[\\mathrm{O\\ III}]}}\\right)^2 }\n$$\n\nWe only correct for extinction in the milky way. therefor we use the extinction function from Cardelli, Clayton & Mathis (1989) with $A_V = 0.2$ and $R_V=3.1$. The extinction is calculated with the following package\n\nhttps://extinction.readthedocs.io/en/latest/\n\n(Note: the DAP products are already extinction corrected).\n\n**Requires**\n * `extinction` a python package to account for the extinction in the Milky Way.\n * `measure_flux` from `pnlf.photometry`\n \n**Returns**\n * `flux` a Table with the measured line fluxes.\n\n\n```python\nfrom astropy.coordinates import match_coordinates_sky # match sources against existing catalog\nfrom astropy.coordinates import Angle # work with angles (e.g. 1\u00b02\u20323\u2033)\nfrom astropy.coordinates import SkyCoord\n\nfrom extinction import ccm89 # calculate extinction Cardelli et al. (1989)\nfrom dust_extinction.parameter_averages import CCM89\n\nfrom pnlf.photometry import measure_flux\n```\n\n\n```python\n# extinction correction with Astropy\n#extinction_model = CCM89(Rv=Rv)\n#extinction = -2.5*np.log10(extinction_model.extinguish(500.7*u.nanometer,Ebv=Ebv))\n#print(f'Av = {extinction:.2f}')\n#flux['mOIII'] = flux['mOIII'] - extinction\n```\n\n\n```python\naperture_size = 2#galaxy.aperturesize\n\nflux = measure_flux(galaxy,\n sources,\n alpha=galaxy.power_index,\n Rv=3.1,\n Ebv=galaxy.Ebv,\n extinction='MW',\n background='local',\n aperture_size=aperture_size)\n\n# calculate astronomical coordinates for comparison\n\n# calculate magnitudes from measured fluxes\nflux['mOIII'] = -2.5*np.log10(flux['OIII5006']*1e-20) - 13.74\nflux['dmOIII'] = np.abs( 2.5/np.log(10) * flux['OIII5006_err'] / flux['OIII5006'] )\n```\n\n\n```python\nsources\n```\n\n### Compare different background\n\n\n```python\n# compare different backgrounds\nloc = -2.5*np.log10(1e-20 * (flux['HA6562_aperture_sum'] - flux['HA6562_bkg_local'])) - 13.74\nglob = -2.5*np.log10(1e-20 * (flux['HA6562_aperture_sum'] - flux['HA6562_bkg_global'])) - 13.74\ncon = -2.5*np.log10(1e-20 * (flux['HA6562_aperture_sum'] - flux['HA6562_bkg_convole'])) - 13.74\n```\n\n\n```python\nfig,ax=plt.subplots(figsize=(4,4))\n\nplt.scatter(loc,glob,marker='o',s=2,color=tab10[0])\nax.set_xlabel(r'$m_{\\mathrm{H}\\alpha}$ (local bkg)')\nax.set_ylabel(r'$m_{\\mathrm{H}\\alpha}$ (global bkg)')\n\nplt.plot([20,30],[19.5,29.5],'gray',ls='--',lw=0.5)\nplt.plot([20,30],[20.5,30.5],'gray',ls='--',lw=0.5)\n\nplt.plot([20,30],[20,30],'black',ls='-',lw=0.5)\nplt.xlim([20,30])\nplt.ylim([20,30])\n\n#plt.savefig('../../notes/img/global_bkg.png',dpi=600)\nplt.show()\n```\n\n\n```python\n# compare DAP vs Kreckel maps\nplt.scatter(-2.5*np.log10(flux['OIII5006']*1e-20) - 13.74,-2.5*np.log10(flux['OIII5006_DAP']*1e-20) - 13.74)\nplt.plot([24,29],[24,29],color='grey',ls='--')\nplt.xlim([24,29])\nplt.ylim([24,29])\nplt.title('mOIII for sources in NGC628')\nplt.xlabel('sum')\nplt.ylabel('fit (DAP)')\nplt.show()\n```\n\n### Compare different FWHM\n\nwe assume an uncertainty of dFWHM = 0.1\"=0.5px\n\n\n```python\nsources['fwhm'] += 0.1\n```\n\n\n```python\naperture_size = 2#galaxy.aperturesize\n\nflux2 = measure_flux(galaxy,sources,alpha=galaxy.power_index,Rv=3.1,Ebv=galaxy.Ebv,\n extinction='MW',background='local',aperture_size=aperture_size)\n\n\nflux2['mOIII'] = -2.5*np.log10(flux2['OIII5006']*1e-20) - 13.74\nflux2['dmOIII'] = np.abs( 2.5/np.log(10) * flux2['OIII5006_err'] / flux2['OIII5006'] )\n```\n\n\n```python\nfig = plt.figure(figsize=(6,6))\n\nax1 = fig.add_subplot(111)\nax1.scatter(flux['mOIII'],flux2['mOIII'])\nxmin,xmax = ax1.get_xlim()\nymin,ymax = ax1.get_ylim()\nlim = max(xmax,ymax)\nax1.plot([20,lim],[20,lim])\nax1.set(xlim=(20,lim),ylim=(20,lim))\n\n\nplt.show()\n```\n\n### Compare aperture sizes\n\n\n```python\nRv = 3.1\nEbv = 0.062\naperture_size2 = 1.5\n\nflux2 = measure_flux(galaxy,\n sources,\n alpha=galaxy.power_index,\n Rv=Rv,\n Ebv=Ebv,\n extinction='MW',\n background='local',\n aperture_size=aperture_size2)\n\n# calculate astronomical coordinates for comparison\n\n# calculate magnitudes from measured fluxes\nflux2['mOIII'] = -2.5*np.log10(flux2['OIII5006']*1e-20) - 13.74\nflux2['dmOIII'] = np.abs( 2.5/np.log(10) * flux2['OIII5006_err'] / flux2['OIII5006'] )\n```\n\n\n```python\nfig,ax=plt.subplots(1,figsize=(4,4))\nax.scatter(flux[tbl['type']=='PN']['mOIII'],flux2[tbl['type']=='PN']['mOIII'],s=2)\nxmin,xmax,ymin,ymax=25.,31,25.,31\ncomp=28\nax.plot([xmin,xmax],[xmin,xmax],color='black',lw=0.4)\nax.plot([xmin,xmax],[xmin-0.2,xmax-0.2],color='gray',lw=0.5,ls='--')\nax.plot([xmin,xmax],[xmin+0.2,xmax+0.2],color='gray',lw=0.5,ls='--')\nax.plot([xmin,comp,comp],[comp,comp,ymin],color='black',lw=0.4)\nax.set(xlabel=f'mOIII aps={aperture_size} FWHM',ylabel=f'mOIII aps={aperture_size2} FWHM',xlim=[xmin,xmax],ylim=[ymin,ymax])\nplt.savefig(basedir/'reports'/galaxy.name/'aperture_size.pdf',dpi=600)\nplt.show()\n```\n\n#### Compare to Kreckel et al. 2017\n\n\n```python\nfrom astropy.coordinates import match_coordinates_sky # match sources against existing catalog\nfrom astropy.coordinates import Angle # work with angles (e.g. 1\u00b02\u20323\u2033)\nfrom astropy.table import hstack\n```\n\n\n```python\nID, angle, Quantity = match_coordinates_sky(pn_NGC628_kreckel['SkyCoord'],tbl['SkyCoord'])\n#tbl[ID][angle.__lt__(Angle('0.5\"'))].show_in_notebook()\n```\n\n\n```python\npn_bright = pn_NGC628_kreckel[pn_NGC628_kreckel['mOIII']<28]\n\nID, angle, Quantity = match_coordinates_sky(pn_bright['SkyCoord'],flux['SkyCoord'])\n\n# for each object from Kreckel et al. 2017, we search for the nearest source\n# and copy our measured quantities to compare the two\npn_bright['mOIII_measured'] = flux[ID]['mOIII']\npn_bright['dmOIII_measured'] = flux[ID]['dmOIII']\npn_bright['sep'] = angle\n\nfig,ax = plt.subplots(figsize=(7,7))\n\n# we only use sources when their position agrees within this tolerance\ntolerance = '0.5\"'\n\n# calculate the difference in magnitude for those objects\ndif = np.mean(np.abs(pn_bright[angleAngle(tolerance)]['mOIII'],pn_bright[angle>Angle(tolerance)]['mOIII'],color='tab:orange')\n\nax.plot([25.5,27.5],[25.5,27.5],color='black',lw=0.4)\nax.plot([25.5,27.5],[25.,27.],color='black',lw=0.2,ls='--')\nax.plot([25.5,27.5],[26.,28.],color='black',lw=0.2,ls='--')\nax.set_xlabel(r'$\\mathrm{m}_{[\\mathrm{OIII}]}$ Kreckel et al. 2017',fontsize=16)\nax.set_ylabel(r'$\\mathrm{m}_{[\\mathrm{OIII}]}$ this work',fontsize=16)\n\nplt.show()\n```\n\n\n```python\nfrom pymuse.plot.plot import single_cutout\nx,y=pn_kreckel[np.argmax(pn_kreckel['mOIII'])]['SkyCoord'].to_pixel(wcs=galaxy.wcs)\n\nsingle_cutout(galaxy,'OIII5006_DAP',x,y)\n```\n\nwhile the OIII magnitudes agree fairly well, we see a huge discrepancy in the H$\\alpha$ fluxes.\n\n\n```python\ntable = hstack([pn_bright,flux[ID]])\n\nfor col in ['OIII5006','HA6562','SII6716','NII6583']:\n table[col][table[col]<0] = table[f'{col}_err'][table[col]<0] \n \ntable['OIII/Ha_measured'] = table['OIII5006'] / table['HA6562']\ntable['Ha/SII_measured'] = table['HA6562'] / table['SII6716']\ntable['Ha/NII_measured'] = table['HA6562'] / table['NII6583']\n\n \n# print all relevant columns\ntable[table['sep']-np.inf)])\n```\n\n\n```python\nlam = 5007\nAlam = ext_model.evaluate(lam*u.angstrom,Rv) * AVmap\n```\n\n## Emission line diagnostics\n\nWe built a catalgoue of possible planetary nebula and measuerd different emission lines. However this catalogue still contains objects that are similar to PN like HII regions or supernova remenants (SNR). In this next step we use emission line diagnostics to eliminate those contanimations. The distance modulus $\\mu$ is defined as the difference between the apparent and the absolute magnitude. By definition of the absolute magnitude, this relates to the distance $d$ in parsec as \n$$\n\\begin{align}\n\\mu = m - M \\\\\nd = 10^{1+\\frac{\\mu}{5}}\n\\end{align}\n$$\n\n 1. filter out HII regions\n $$\n 4 > \\log_{10} \\frac{[\\mathrm{OIII}]}{\\mathrm{H}\\alpha +[\\mathrm{NII}]} > -0.37 M_{[\\mathrm{OIII}]} - 1.16\n $$\n 2. filter out SNR\n $$\n \\mathrm{H}\\alpha / [\\mathrm{SII}] < 2.5\n $$\n \n 3. estimate completness limit and remove fainter sources\n \n\n\n```python\nfrom pnlf.analyse import emission_line_diagnostics\n\nprint(f'emission line diagnostics for {galaxy.name}')\nprint(f'mu={galaxy.mu:.2f}, cl={galaxy.completeness_limit}')\ntbl = emission_line_diagnostics(flux,galaxy.mu,galaxy.completeness_limit) \n\n# create additional columns that are needed for the classification\ntbl['sharp'] = sources['sharpness']\ntbl['round'] = sources['roundness2']\ntbl['SkyCoord'] = SkyCoord.from_pixel(tbl['x'],tbl['y'],galaxy.wcs)\n\ntbl['exclude'] = False\n\ncut=0\nslow = galaxy.sharplo \nshigh = galaxy.sharphi \nr = galaxy.roundness \nif cut>0:\n logger.warning('you are using a cut')\n \n#slope = []\n#for row in tbl:\n# star = Cutout2D(galaxy.OIII5006, (row['x'],row['y']), u.Quantity((size, size), u.pixel),wcs=galaxy.wcs)\n# profile = radial_profile(star.data,star.input_position_cutout)\n# slope.append(np.sum(np.ediff1d(profile)>0) / len(profile))\n#tbl['slope'] = slope \n\n# table contains all detected objects. here we mask all undesired objects.\nc_shape = ((tbl['sharp']>slow) & (tbl['sharp']10*np.abs(tbl['OIII5006_bkg_local']))\nc_PN = (tbl['type']=='PN')\nc_SNR = (tbl['SNRorPN'] & (tbl['type']=='SNR'))\nc_AV = ((tbl['Av']<0.4) | np.isnan(tbl['Av']))\nc_cut = (cuttbl['OIII5006']/tbl['HB4861']) & (tbl['type']=='PN')]\nax.scatter(tmp['MOIII'],tmp['OIII5006']/tmp['HB4861'],label='PN',color='tab:red')\n \nMOIII = np.linspace(-4.5,0)\nax.plot(MOIII,10**(-0.37*MOIII-0.7),color='black')\nax.legend()\n \nax.set(xlim=[-4.5,0],ylim=[1e-1,200],yscale='log')\nax.yaxis.set_major_formatter(mpl.ticker.FuncFormatter(lambda y, _: '{:.16g}'.format(y)))\n\nplt.show()\n```\n\n\n```python\nfrom pnlf.plot.plot import cutout_with_profile\n\ntmp = tbl[(10**(-0.37*tbl['MOIII']-0.7)>tbl['OIII5006']/tbl['HB4861']) & (tbl['type']=='PN')]\ncutout_with_profile(galaxy,table=tmp,size=40,diagnostics=False,filename=None)\n\n```\n\n\n```python\ntmp = tbl[(10**(-0.37*tbl['MOIII']-0.7)27]\n\nfitter = MaximumLikelihood1D(pnlf,\n data,\n #prior=prior,\n #err=err,\n mhigh=completeness)\n\n# a good guess would be mu_guess = min(data)-Mmax\ngalaxy.mu,dp,dm = fitter([28])\nfig = fitter.plot()\n\nplot_pnlf(data,galaxy.mu,completeness,binsize=galaxy.binsize,mhigh=30)\n```\n\n\n```python\n# use data from Kreckel+2017\ncompleteness= 27.5\n\nraw = vstack([pn_kreckel['mOIII'], snr_kreckel['mOIII']])['mOIII']\ndata = raw[raw this assumes Gaussian errors in the magnitudes - which means logarithmic errors in the fluxes - is this the correct assumption?\n\n→\u00a0no. But for small errors (<10%), the distribution of the magnitudes errors is almost Gaussian. Only very few PNe have larger uncertainties, and they do not fall at the bright end. Since this procedure is mainly used to take some weight from the bright end, assuming Gaussian errors should be fine.\n\n\n```python\nfrom scipy.stats import norm\nmu,std = 1e-17,1e-18\nmu_mag = -2.5*np.log10(mu)-13.74\nstd_mag = np.sqrt((2.5/np.log(10)*std/mu)**2)\n\nfluxes = np.random.normal(mu,std,100000)\nmagnitudes = -2.5*np.log10(fluxes)-13.74\nfig,(ax1,ax2)=plt.subplots(ncols=2,figsize=(6,3))\n\nx = np.linspace(0.2e-17,1.8e-17,1000)\nax1.hist(fluxes,bins=np.linspace(0.2e-17,1.8e-17,100),density=True,color='black',alpha=0.2)\nax1.plot(x,norm.pdf(x,mu,std),color='tab:red')\n#ax1.set(xlim=(0.5e-17,1.5e-17))\nax1.set(xlabel='flux / erg s-1 cm-2')\n\nax2.hist(magnitudes,bins=np.linspace(28,29.5,100),density=True,color='black',alpha=0.2)\npdf = lambda x,mu,std: 1/(np.sqrt(2*np.pi)*std)*np.exp(-(x-mu)**2/(2*std**2))\nf = lambda x: pdf(x,mu,std)\ny = np.linspace(28,29.5)\nx_of_y = np.exp(-np.log(10)/2.5*(y+13.74))\nY = f(x_of_y)*np.log(10)/2.5*np.exp(-np.log(10)/2.5*(y+13.74))\nax2.plot(y,Y,color='tab:blue',label='True PDF')\nax2.plot(y,norm.pdf(y,mu_mag,std_mag),color='tab:red',label='Gaussian')\nax2.legend()\nax2.set(xlim=(28,29.5),xlabel='mag')\n\nprint(f'mu = {mu_mag:.2f}+-{std_mag:.2f}')\nprint(f'sample: {np.nanmean(magnitudes):.2f}+-{np.nanstd(magnitudes):.2f}')\nplt.savefig(basedir/'reports'/'benchmarks'/'magnitude_error_propagation_10.png',dpi=200)\nplt.show()\n```\n\n### Plot the fit\n\nto plot the fit we need to bin the data\n\n\n```python\nfrom pnlf.plot.pnlf import plot_pnlf\nfilename = basedir / 'reports' / f'{galaxy.name}_PNLF'\n\nplot_pnlf(tbl[criteria]['mOIII'],galaxy.mu,completeness,binsize=0.5,mhigh=31)\n```\n\n\n```python\nfrom pnlf.plot.plot import plot_sky_with_detected_stars, sample_cutouts\n```\n\n\n```python\nfrom photutils import CircularAperture\n```\n\n\n```python\ntmp = tbl[np.where(criteria & (tbl['mOIII']<28.5))]\npos = np.transpose((tmp['x'],tmp['y']))\n\nsave_file = Path.cwd() / '..' / 'reports' / f'{galaxy.name}_map_PNLF.pdf'\nplot_sky_with_detected_stars(data=galaxy.OIII5006_DAP,\n wcs=galaxy.wcs,\n positions=pos,\n filename=save_file)\n```\n\n### Metallicity dependence of the zeropoint\n\n\n```python\ndef pnlf_Mmax(m,Mmax,mu,mhigh):\n\n m = np.atleast_1d(m)\n mlow = Mmax+mu\n \n normalization = 1/(F(mhigh,mu) - F(mlow,mu)) \n out = normalization * np.exp(0.307*(m-mu)) * (1-np.exp(3*(Mmax-m+mu)))\n out[(m>mhigh) | (mcompletneness] = 0\n out[m 0:\n d_err = 0.2 * np.log(10) * d * mu_err\n print(f'd = ({d/1e6:.2f} + {d_err[0]/1e6:.2f} - {d_err[1]/1e6:.2f}) Mpc')\n \n return d, d_err\n\nd,d_err = distance_modulus_to_parsec(30.033,np.array([0.014,0.015]))\n```\n\n## Playground\n\n\n```python\n \ncompleteness = 29\n\ndata = tbl[(tbl['type']=='PN') & (tbl['mOIII'] 0]\n_, median_sigclip , _ = sigma_clipped_stats(annulus_data_1d[~np.isnan(annulus_data_1d)],sigma=3,maxiters=10,cenfunc='median') \nphot['OIII_bkg_median'] = median_sigclip*aperture.area\n\nannulus_data = annulus_masks.multiply(galaxy.OIII5006_DAP)\nannulus_data_1d = annulus_data[annulus_masks.data > 0]\n_, median_sigclip , _ = sigma_clipped_stats(annulus_data_1d[~np.isnan(annulus_data_1d)],sigma=3,maxiters=10,cenfunc='median') \nphot['OIIIDAP_bkg_median'] = median_sigclip*aperture.area\n\nrc = pyneb.RedCorr(R_V=3.1,E_BV=galaxy.Ebv,law='CCM89')\n\nphot['OIII_flux'] = phot['aperture_sum']-phot['OIII_bkg_median']\nphot['OIIIDAP_flux'] = phot['aperture_sum_DAP']-phot['OIIIDAP_bkg_median']\nphot['OIII_flux'] /= light_in_moffat(r,alpha,gamma)\nphot['OIIIDAP_flux'] /= light_in_moffat(r,alpha,gamma)\nphot['OIII_flux'] *= rc.getCorr(5006)\nphot['mOIII'] = -2.5*np.log10(phot['OIII_flux']*1e-20) - 13.74\nphot['mOIIIDAP'] = -2.5*np.log10(phot['OIIIDAP_flux']*1e-20) - 13.74\n\nphot[['OIII_flux','OIIIDAP_flux','mOIII','mOIIIDAP']]\n\n```\n\n\n```python\nsize = 20\nOIII_cutout = Cutout2D(galaxy.OIII5006,(x,y),size=(size,size))\nHA_cutout = Cutout2D(galaxy.HA6562,(x,y),size=(size,size))\nSII_cutout = Cutout2D(galaxy.SII6716,(x,y),size=(size,size))\n\nfig,(ax1,ax2,ax3)=plt.subplots(ncols=3,figsize=(12,4))\n\nax1.imshow(OIII_cutout.data,origin='lower',cmap=plt.cm.Greens)\nax2.imshow(HA_cutout.data,origin='lower',cmap=plt.cm.Reds)\nax3.imshow(SII_cutout.data,origin='lower',cmap=plt.cm.Blues)\n\naperture = CircularAperture((size/2,size/2),r)\n#aperture.plot(color='black',lw=0.8,axes=ax1)\n#aperture.plot(color='black',lw=0.8,axes=ax2)\n#aperture.plot(color='black',lw=0.8,axes=ax3)\n\nax1.axis('off')\nax2.axis('off')\nax3.axis('off')\n\nax1.set_title('[OIII]5007')\nax2.set_title(r'H$\\alpha$')\nax3.set_title('[SII]6716')\nplt.tight_layout()\nplt.savefig(basedir/'reports'/'NGC1385'/'cutouts_813.pdf',dpi=300)\n\nplt.show()\n```\n\n\n```python\nfilename = data_raw/'MUSE'/'DR2.1'/'datacubes'/'NGC1385_DATACUBE_FINAL_WCS_Pall_mad.fits'\nwith fits.open(filename , memmap=True, mode='denywrite') as hdul:\n cube = hdul[1].data\n cube_wcs = WCS(hdul[1].header)\n```\n\n\n```python\nfrom pnlf.auxiliary import circular_mask, annulus_mask\n\n\nr=aperture_size*(fwhm-PSF_correction)/2\nr_in = 4 * (fwhm-PSF_correction) / 2 \nr_out = np.sqrt(5*r**2+r_in**2)\n\nmask1 = circular_mask(*cube.shape[1:],(x,y),radius=r)\nmask2 = annulus_mask(*cube.shape[1:],(x,y),r_in,r_out)\nspectrum_raw = np.sum(cube[...,mask1],axis=1)\nspectrum_bkg = np.sum(cube[...,mask2],axis=1) * r**2 / (r_out**2-r_in**2)\n\nwavelength = np.linspace(4749.88,9349.88,cube.shape[0]) \n#wavelength = np.arange(cube.shape[0])\nfig,ax=plt.subplots(figsize=(single_column,single_column/1.618))\n\nax.plot(wavelength,spectrum_raw,label='with bkg')\nax.plot(wavelength,spectrum_raw-spectrum_bkg,label='without bkg')\nax.legend()\nax.set(xlim=[4800,7000],ylim=[1,1.1e4],yscale='linear',xlabel=r'Wavelength / $\\AA$',ylabel='flux')\nax.set_title('Spectrum PN 813')\nplt.savefig(basedir/'reports'/'NGC1385'/'spectrum_813.pdf',dpi=300)\nplt.show()\n```\n\n## Copt\n\nuse the convolved-optimised (copt) maps instead. This makes things a lot easier as we do not need to keep track of the different pointings\n\n\n```python\nname = 'NGC1433'\n\ndata_ext = Path('a:')\n\nclass ReadLineMaps:\n \n def __init__(self,filename,extensions=['HB4861','OIII5006','HA6562','NII6583','SII6716','SII6730']):\n \n name, fwhm = filename.stem.split('-')\n fwhm = float(fwhm[:4])*5\n\n logger.info(f'loading {name}')\n\n setattr(self,'name',name) \n setattr(self,'fwhm',fwhm)\n\n with fits.open(filename) as hdul:\n\n # save the white-light image\n header = hdul[f'FLUX'].header\n setattr(self,'header',header)\n setattr(self,'wcs',WCS(header))\n setattr(self,'shape',(header['NAXIS2'],header['NAXIS1']))\n setattr(self,'Ebv_stars',hdul['EBV_STARS'].data)\n setattr(self,'whitelight',hdul['FLUX'].data)\n setattr(self,'whitelight_err',hdul['SNR'].data)\n\n self.lines = []\n \n for line in extensions:\n setattr(self,line,hdul[f'{line}_FLUX'].data)\n setattr(self,f'{line}_err',hdul[f'{line}_FLUX_ERR'].data) \n setattr(self,f'{line}_SIGMA',np.sqrt(hdul[f'{line}_SIGMA'].data**2 - hdul[f'{line}_SIGMA_CORR'].data**2))\n setattr(self,f'{line}_SIGMA_ERR',hdul[f'{line}_SIGMA_ERR'])\n\n # append to list of available lines\n self.lines.append(line)\n \nfilename = [x for x in (data_ext/'MUSE_DR2'/'copt').iterdir() if x.stem.startswith(name)][0]\ngalaxy = ReadLineMaps(filename)\n```\n\n\n```python\nfrom photutils import DAOStarFinder # DAOFIND routine to detect sources\nfrom astropy.stats import sigma_clipped_stats # calcualte statistics of images\n\nthreshold = 3\n\nmean, median, std = sigma_clipped_stats(galaxy.OIII5006, sigma=3.0,maxiters=5)\ncorrect_PSF = lambda lam: 5*3e-5*(lam-6483.58)\n\n# initialize and run StarFinder (DAOPHOT or IRAF)\nfinder = DAOStarFinder(fwhm = (galaxy.fwhm - correct_PSF(5007)), \n threshold = threshold*std)\npeaks = finder(galaxy.OIII5006-median)\nprint(f'{len(peaks)} objects in initial catalogue')\n```\n\n\n```python\nimport re\nfrom photutils import CircularAperture # define circular aperture\nfrom photutils import CircularAnnulus # define annulus\nfrom photutils import aperture_photometry # measure flux in aperture\n\naperture_size = 2\n\ndef light_in_gaussian(x,fwhm):\n\n return 1-np.exp(-4*np.log(2)*x**2 / fwhm**2)\n\nif 'flux' in locals():\n del flux\n\nout = {}\nfor line in galaxy.lines:\n print(line)\n \n wavelength = int(re.findall(r'\\d{4}', line)[0])\n fwhm = galaxy.fwhm - correct_PSF(wavelength)\n r = aperture_size*fwhm/2\n \n data = getattr(galaxy,f'{line}').copy()\n error = getattr(galaxy,f'{line}_err').copy()\n\n positions = np.transpose((peaks['xcentroid'], peaks['ycentroid']))\n aperture = CircularAperture(positions, r=r)\n \n phot = aperture_photometry(data, aperture, error = error)\n\n r_in = 4 * fwhm / 2 \n r_out = np.sqrt(5*r**2+r_in**2)\n annulus_aperture = CircularAnnulus(positions, r_in=r_in, r_out=r_out)\n annulus_masks = annulus_aperture.to_mask(method='center')\n\n bkg_median = []\n bkg_median_no_clip = []\n for mask in annulus_masks:\n annulus_data = mask.multiply(data)\n annulus_data_1d = annulus_data[mask.data > 0]\n median_sigclip,_ , _ = sigma_clipped_stats(annulus_data_1d[~np.isnan(annulus_data_1d)],sigma=3,maxiters=3) \n bkg_median_no_clip.append(np.nanmedian(annulus_data_1d[~np.isnan(annulus_data_1d)])) \n bkg_median.append(median_sigclip)\n phot['bkg_median'] = np.array(bkg_median) \n phot['bkg_local'] = phot['bkg_median'] * aperture.area\n phot['flux'] = phot['aperture_sum'] - phot['bkg_local'] \n phot['flux'] /= light_in_gaussian(r,fwhm)\n \n out[line] = phot\n\nfor k,v in out.items():\n\n # first we create the output table with \n if 'flux' not in locals():\n flux = v[['id','xcenter','ycenter']]\n flux.rename_column('xcenter','x')\n flux.rename_column('ycenter','y')\n flux['x'] = flux['x'].value # we don't want them to be in pixel units\n flux['y'] = flux['y'].value\n flux[k] = v['flux'] \n flux[f'{k}_err'] = v['aperture_sum_err']\nflux['mOIII'] = -2.5*np.log10(flux['OIII5006']*1e-20) - 13.74\nflux['dmOIII'] = np.abs( 2.5/np.log(10) * flux['OIII5006_err'] / flux['OIII5006'] )\n```\n\n\n```python\nfrom pnlf.analyse import emission_line_diagnostics\n\ngalaxy.mu = parameters[name]['mu']\ncompleteness_limit = parameters[name]['completeness_limit']\n\nflux['HA6562_SIGMA'] = 0\nprint(f'emission line diagnostics for {galaxy.name}')\ntbl = emission_line_diagnostics(flux,galaxy.mu,completeness_limit) \n\n# create additional columns that are needed for the classification\ntbl['sharp'] = peaks['sharpness']\ntbl['round'] = peaks['roundness2']\ntbl['SkyCoord'] = SkyCoord.from_pixel(tbl['x'],tbl['y'],galaxy.wcs)\ntbl['exclude'] = False\ntbl['overluminous'] = False\n\nslow = .2 #galaxy.sharplo \nshigh = 1. #galaxy.sharphi \nr = .8 #galaxy.roundness \n\n# table contains all detected objects. here we mask all undesired objects.\nc_shape = ((tbl['sharp']>slow) & (tbl['sharp']=1] = 1\n qi[qi<1] = 0\n \n if Pbad <0: p_bad = 0\n else: p_bad = 2 / (0.02*np.sqrt(2*np.pi)) * np.exp(-(Pbad)**2 / (2*0.02**2))\n\n p_mu = 1 / (std*np.sqrt(2*np.pi)) * np.exp(-(mu-mu0)**2 / (2*std**2))\n p_qi = np.prod((1-Pbad)**qi*Pbad**(1-qi))\n \n if p_mu*p_qi*p_bad <= 0:\n return -np.inf\n else:\n return np.log(p_mu*p_qi*p_bad)\n\ndef log_likelihood(params,data):\n mu, Pbad, Yb, Vb, *qi = params\n qi = np.array(qi)\n qi[qi>=1] = 1\n qi[qi<1] = 0\n \n log_pnlf = qi*np.log(pnlf(data,mu,mhigh=galaxy.completeness_limit))\n log_bad = (1-qi)*np.log(1/np.sqrt(2*np.pi*Vb) * np.exp(-(data-Yb)**2/(2*Vb)))\n \n return np.sum(log_pnlf+log_bad)\n \ndef log_prob_fn(params,data):\n p = log_likelihood(params,data)+log_prior(params)\n if np.isnan(p):\n return -np.inf\n return p\n\n\n#soln = minimize(neg_log_probability,guess,args=(data,),method='Nelder-Mead')\n#if not soln.success:\n# raise RuntimeError('fit was not successful')\n#x = soln.x\n\nndim = len(guess)\npos = np.random.normal(guess,[0.1,0.01,0.5,0.5]+len(data)*[0.001],size=(nwalkers,ndim))\n\nsampler = emcee.EnsembleSampler(nwalkers, ndim,log_prob_fn,args=(data,))\nstate = sampler.run_mcmc(pos, 100)\nsampler.reset()\nsampler.run_mcmc(state,nsteps,progress=True)\n\nflat_samples = sampler.get_chain(discard=100,thin=5, flat=True)\n\n```\n\n\n```python\nfrom IPython.display import display, Math\n\nlabels = ['mu','Pbad','Yb','Vb']\nfor i in range(len(labels)):\n mcmc = np.percentile(flat_samples[:, i], [16, 50, 84])\n q = np.diff(mcmc)\n txt = \"\\mathrm{{{3}}} = {0:.3f}_{{-{1:.3f}}}^{{{2:.3f}}}\"\n txt = txt.format(mcmc[1], q[0], q[1], labels[i])\n display(Math(txt))\n```\n\n\n```python\nqi_p = np.percentile(flat_samples,50,axis=0)\nnp.sum(qi_p>1)\n```\n\n### Mixture model\n\nhttps://dfm.io/posts/mixture-models/\nhttps://emcee.readthedocs.io/en/stable/user/blobs/\n\n\n```python\nimport emcee\n\nmhigh = 28\n\n# Define the probabilistic model...\n# A simple prior:\nbounds = [(27, 32), (0, 0.2), (20, 30), (0.1, 5)]\ndef lnprior(p):\n # We'll just put reasonable uniform priors on all the parameters.\n if not all(b[0] < v < b[1] for v, b in zip(p, bounds)):\n return -np.inf\n return 0\n\n# The \"foreground\" linear likelihood:\ndef lnlike_fg(p,data):\n mu, _, _, _ = p\n \n return np.log(pnlf(data,mu,mhigh=mhigh))\n\n# The \"background\" outlier likelihood:\ndef lnlike_bg(p,data):\n _, Q, M, lnV = p\n \n return -0.5 * ((M - data) ** 2 / np.exp(lnV) + lnV)\n\n# Full probabilistic model.\ndef lnprob(p,data):\n mu, Q, M, lnV = p\n \n # First check the prior.\n lp = lnprior(p)\n if not np.isfinite(lp):\n return -np.inf, (None,None)\n \n # Compute the vector of foreground likelihoods and include the q prior.\n ll_fg = lnlike_fg(p,data)\n arg1 = ll_fg + np.log(Q)\n \n # Compute the vector of background likelihoods and include the q prior.\n ll_bg = lnlike_bg(p,data)\n arg2 = ll_bg + np.log(1.0 - Q)\n \n # Combine these using log-add-exp for numerical stability.\n ll = np.sum(np.logaddexp(arg1, arg2))\n \n # We're using emcee's \"blobs\" feature in order to keep track of the\n # foreground and background likelihoods for reasons that will become\n # clear soon.\n return lp + ll, (arg1, arg2)\n\n# Initialize the walkers at a reasonable location.\nndim, nwalkers = 4, 32\np0 = np.random.normal([29,0.05,25,1],[0.2,0.01,0.5,0.5],size=(nwalkers,ndim))\n\n\n# Set up the sampler.\nsampler = emcee.EnsembleSampler(nwalkers, ndim, lnprob,args=(data,),blobs_dtype=[('args',tuple)])\n\n# Run a burn-in chain and save the final location.\npos, _, _, _ = sampler.run_mcmc(p0, 500)\n\n# Run the production chain.\nsampler.reset()\nsampler.run_mcmc(pos, 1500);\n```\n\n\n```python\nfrom IPython.display import display, Math\n\nflat_samples = sampler.get_chain(discard=100,thin=5, flat=True)\n\nlabels = ['mu','Pbad','Yb','Vb']\nfor i in range(len(labels)):\n mcmc = np.percentile(flat_samples[:, i], [16, 50, 84])\n q = np.diff(mcmc)\n txt = \"\\mathrm{{{3}}} = {0:.3f}_{{-{1:.3f}}}^{{{2:.3f}}}\"\n txt = txt.format(mcmc[1], q[0], q[1], labels[i])\n display(Math(txt))\n```\n\n\n```python\nnorm = 0.0\npost_prob = np.zeros(len(data))\nfor i in range(sampler.chain.shape[1]):\n for j in range(sampler.chain.shape[0]):\n try:\n ll_fg, ll_bg = sampler.blobs[i][j]\n post_prob += np.exp(ll_fg - np.logaddexp(ll_fg, ll_bg))\n norm += 1\n except:\n pass\npost_prob /= norm\n```\n\n\n```python\nprint(\", \".join(map(\"{0:.3f}\".format, post_prob)))\n```\n\n## Compare DR1 and DR2\n\n\n```python\nfrom pnlf.io import read_catalogue\nfrom astropy.coordinates import match_coordinates_sky\n```\n\n\n```python\nfolder = basedir / 'data' / 'DR2'\n```\n\n\n```python\nDR1 = ascii.read(folder/'NGC0628_nebulae_DR1.txt',format='fixed_width',delimiter='\\t')\nDR2 = ascii.read(folder/'NGC0628_nebulae_DR2.txt',format='fixed_width',delimiter='\\t')\n\nDR1['SkyCoord'] = SkyCoord(DR1['RaDec'])\nDR2['SkyCoord'] = SkyCoord(DR2['RaDec'])\n```\n\n\n```python\ntolerance = '0.5\"'\n\nfig = plt.figure(figsize=(9,3))\n\n\nfor i,t in enumerate(['PN','SNR','HII']):\n \n ax = fig.add_subplot(1,3,i+1)\n sub = DR1[DR1['type']==t].copy()\n \n ID, angle, Quantity = match_coordinates_sky(sub['SkyCoord'],DR2['SkyCoord'])\n within_tolerance = len(angle[angle.__lt__(Angle(tolerance))])\n mask = angle.__lt__(Angle(tolerance))\n \n sub['mOIIIDR2'] = DR2[ID]['mOIII']\n sub['typeDR2'] = DR2[ID]['type']\n \n #sub = sub[mask]\n \n for col,ty in zip(['tab:red','tab:blue','tab:orange'],['PN','SNR','HII']):\n subsub = sub[(sub['typeDR2']==ty) & mask]\n ax.scatter(subsub['mOIII'],subsub['mOIIIDR2'],label=f'{ty} in DR2')\n \n ax.scatter(sub[~mask]['mOIII'],sub[~mask]['mOIII'],\n s=15, facecolors='none', edgecolors='gray',label='missing')\n \n xmin,xmax=ax.get_xlim()\n ymin,ymax=ax.get_ylim()\n \n ax.set_title(f'{t} in DR1')\n ax.plot([min(xmin,ymin),max(xmax,ymax)],[min(xmin,ymin),max(xmax,ymax)],'k--')\n\n ax.set_xlabel('mOIII DR1')\n ax.set_ylabel('mOIII DR2')\n \n plt.legend()\n\n print(f'{t}: {within_tolerance} of {len(angle)} match within {tolerance}\": {within_tolerance / len(angle)*100:.1f} %')\n\nplt.show()\n\n```\n\n## Star Mask from GAIA\n\n\n```python\nsample=ascii.read(basedir/'reports'/'sample.txt')\nsample['SkyCoord'] = SkyCoord(sample['R.A.'],sample['Dec.'])\nsample.add_index('Name')\n```\n\n\n```python\nfrom astroquery.gaia import Gaia\n\n\nradius = u.Quantity(4.0, u.arcmin)\nj = Gaia.cone_search_async(sample.loc['NGC0628']['SkyCoord'], radius)\nstars = j.get_results()\nprint(f'{len(stars)} stars found within {radius}')\n```\n\n\n```python\nstars['SkyCoord'] = SkyCoord(stars['ra'],stars['dec'])\nx,y = stars['SkyCoord'].to_pixel(galaxy.wcs)\n```\n\n\n```python\nfig = plt.figure(figsize=(two_column,two_column))\nax = fig.add_subplot(111,projection=galaxy.wcs)\n\nnorm = simple_norm(galaxy.whitelight,clip=False,percent=99)\nax.imshow(galaxy.whitelight,norm=norm)\n\nfor xp,yp in zip(x,y):\n if not np.isnan(galaxy.PSF[int(xp),int(yp)]):\n ax.scatter(xp,yp)\n\n```\n\n## Add additional stuff to parameters.yml\n\n\n```python\nwith open(basedir / 'data' / 'interim' / 'parameters.yml', 'w') as outfile:\n yaml.dump(parameters, outfile, default_flow_style=False)\n```\n", "meta": {"hexsha": "ca08519e30cef18e52805e5c7ccc8b6de8900bc9", "size": 145083, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/PNLF development.ipynb", "max_stars_repo_name": "fschmnn/pymuse", "max_stars_repo_head_hexsha": "91e97d03a3eb1ccc02131f4e731e6bb5df66772c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-05-09T21:24:29.000Z", "max_stars_repo_stars_event_max_datetime": "2020-05-27T15:11:31.000Z", "max_issues_repo_path": "notebooks/PNLF development.ipynb", "max_issues_repo_name": "fschmnn/pymuse", "max_issues_repo_head_hexsha": "91e97d03a3eb1ccc02131f4e731e6bb5df66772c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/PNLF development.ipynb", "max_forks_repo_name": "fschmnn/pymuse", "max_forks_repo_head_hexsha": "91e97d03a3eb1ccc02131f4e731e6bb5df66772c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.3557091882, "max_line_length": 571, "alphanum_fraction": 0.5402218041, "converted": true, "num_tokens": 30559, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307806984444, "lm_q2_score": 0.7057850340255386, "lm_q1q2_score": 0.449606791230567}} {"text": "\n\n# Grid vs. Random Search Hyperparameter Optimization\n\n## Setup\n\n### Installation\n\n\n```python\n!pip install matbench\n!pip install CBFV\n```\n\n Collecting matbench\n Downloading matbench-0.5-py3-none-any.whl (9.9 MB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 9.9 MB 6.8 MB/s \n \u001b[?25hCollecting monty==2021.8.17\n Downloading monty-2021.8.17-py3-none-any.whl (65 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 65 kB 2.1 MB/s \n \u001b[?25hCollecting matminer==0.7.4\n Downloading matminer-0.7.4-py3-none-any.whl (1.4 MB)\n \u001b[K 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pymatgen>=2022.0.11->matminer==0.7.4->matbench) (3.2.2)\n Collecting uncertainties>=3.1.4\n Downloading uncertainties-3.1.6-py2.py3-none-any.whl (98 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 98 kB 10.2 MB/s \n \u001b[?25hRequirement already satisfied: palettable>=3.1.1 in /usr/local/lib/python3.7/dist-packages (from pymatgen>=2022.0.11->matminer==0.7.4->matbench) (3.3.0)\n Requirement already satisfied: networkx>=2.2 in /usr/local/lib/python3.7/dist-packages (from pymatgen>=2022.0.11->matminer==0.7.4->matbench) (2.6.3)\n Requirement already satisfied: kiwisolver>=1.0.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib>=1.5->pymatgen>=2022.0.11->matminer==0.7.4->matbench) (1.3.2)\n Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib>=1.5->pymatgen>=2022.0.11->matminer==0.7.4->matbench) (3.0.7)\n Requirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.7/dist-packages (from matplotlib>=1.5->pymatgen>=2022.0.11->matminer==0.7.4->matbench) (0.11.0)\n Requirement already satisfied: tenacity>=6.2.0 in /usr/local/lib/python3.7/dist-packages (from plotly>=4.5.0->pymatgen>=2022.0.11->matminer==0.7.4->matbench) (8.0.1)\n Requirement already satisfied: certifi>=2017.4.17 in /usr/local/lib/python3.7/dist-packages (from requests>=2.26.0->matminer==0.7.4->matbench) (2021.10.8)\n Requirement already satisfied: urllib3<1.27,>=1.21.1 in /usr/local/lib/python3.7/dist-packages (from requests>=2.26.0->matminer==0.7.4->matbench) (1.24.3)\n Requirement already satisfied: idna<4,>=2.5 in /usr/local/lib/python3.7/dist-packages (from requests>=2.26.0->matminer==0.7.4->matbench) (2.10)\n Requirement already satisfied: charset-normalizer~=2.0.0 in /usr/local/lib/python3.7/dist-packages (from requests>=2.26.0->matminer==0.7.4->matbench) (2.0.11)\n Collecting ruamel.yaml.clib>=0.2.6\n Downloading ruamel.yaml.clib-0.2.6-cp37-cp37m-manylinux1_x86_64.whl (546 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 546 kB 49.1 MB/s \n \u001b[?25hRequirement already satisfied: mpmath>=0.19 in /usr/local/lib/python3.7/dist-packages (from sympy>=1.8->matminer==0.7.4->matbench) (1.2.1)\n Building wheels for collected packages: future, pymatgen\n Building wheel for future (setup.py) ... \u001b[?25l\u001b[?25hdone\n Created wheel for future: filename=future-0.18.2-py3-none-any.whl size=491070 sha256=70bfaeee11de79774a7c0e8aed771e75fb6d7ad171358d66633ff558687e7bd9\n Stored in directory: /root/.cache/pip/wheels/56/b0/fe/4410d17b32f1f0c3cf54cdfb2bc04d7b4b8f4ae377e2229ba0\n Building wheel for pymatgen (PEP 517) ... \u001b[?25l\u001b[?25hdone\n Created wheel for pymatgen: filename=pymatgen-2022.0.17-cp37-cp37m-linux_x86_64.whl size=41841005 sha256=e47f41b2c33b4e17d06ebd93d56ae0085228489168cc4f6c725decd79b788271\n Stored in directory: /root/.cache/pip/wheels/cf/f6/22/58a9be23c5f1b452770e02ff42047175eaf0f9c2f15219fc76\n Successfully built future pymatgen\n Installing collected packages: six, ruamel.yaml.clib, numpy, future, uncertainties, sympy, spglib, scipy, ruamel.yaml, requests, monty, scikit-learn, pymatgen, pint, matminer, matbench\n Attempting uninstall: six\n Found existing installation: six 1.15.0\n Uninstalling six-1.15.0:\n Successfully uninstalled six-1.15.0\n Attempting uninstall: numpy\n Found existing installation: numpy 1.19.5\n Uninstalling numpy-1.19.5:\n Successfully uninstalled numpy-1.19.5\n Attempting uninstall: future\n Found existing installation: future 0.16.0\n Uninstalling future-0.16.0:\n Successfully uninstalled future-0.16.0\n Attempting uninstall: sympy\n Found existing installation: sympy 1.7.1\n Uninstalling sympy-1.7.1:\n Successfully uninstalled sympy-1.7.1\n Attempting uninstall: scipy\n Found existing installation: scipy 1.4.1\n Uninstalling scipy-1.4.1:\n Successfully uninstalled scipy-1.4.1\n Attempting uninstall: requests\n Found existing installation: requests 2.23.0\n Uninstalling requests-2.23.0:\n Successfully uninstalled requests-2.23.0\n Attempting uninstall: scikit-learn\n Found existing installation: scikit-learn 1.0.2\n Uninstalling scikit-learn-1.0.2:\n Successfully uninstalled scikit-learn-1.0.2\n \u001b[31mERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.\n yellowbrick 1.3.post1 requires numpy<1.20,>=1.16.0, but you have numpy 1.21.5 which is incompatible.\n google-colab 1.0.0 requires requests~=2.23.0, but you have requests 2.27.1 which is incompatible.\n google-colab 1.0.0 requires six~=1.15.0, but you have six 1.16.0 which is incompatible.\n datascience 0.10.6 requires folium==0.2.1, but you have folium 0.8.3 which is incompatible.\n albumentations 0.1.12 requires imgaug<0.2.7,>=0.2.5, but you have imgaug 0.2.9 which is incompatible.\u001b[0m\n Successfully installed future-0.18.2 matbench-0.5 matminer-0.7.4 monty-2021.8.17 numpy-1.21.5 pint-0.18 pymatgen-2022.0.17 requests-2.27.1 ruamel.yaml-0.17.20 ruamel.yaml.clib-0.2.6 scikit-learn-1.0 scipy-1.7.3 six-1.16.0 spglib-1.16.3 sympy-1.9 uncertainties-3.1.6\n\n\n\n\n Collecting CBFV\n Downloading CBFV-1.1.0-py3-none-any.whl (539 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 539 kB 8.6 MB/s \n \u001b[?25hRequirement already satisfied: pandas in /usr/local/lib/python3.7/dist-packages (from CBFV) (1.3.5)\n Requirement already satisfied: tqdm in /usr/local/lib/python3.7/dist-packages (from CBFV) (4.62.3)\n Requirement already satisfied: numpy in /usr/local/lib/python3.7/dist-packages (from CBFV) (1.21.5)\n Requirement already satisfied: pytest in /usr/local/lib/python3.7/dist-packages (from CBFV) (3.6.4)\n Requirement already satisfied: pytz>=2017.3 in /usr/local/lib/python3.7/dist-packages (from pandas->CBFV) (2018.9)\n Requirement already satisfied: python-dateutil>=2.7.3 in /usr/local/lib/python3.7/dist-packages (from pandas->CBFV) (2.8.2)\n Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.7/dist-packages (from python-dateutil>=2.7.3->pandas->CBFV) (1.16.0)\n Requirement already satisfied: pluggy<0.8,>=0.5 in /usr/local/lib/python3.7/dist-packages (from pytest->CBFV) (0.7.1)\n Requirement already satisfied: attrs>=17.4.0 in /usr/local/lib/python3.7/dist-packages (from pytest->CBFV) (21.4.0)\n Requirement already satisfied: py>=1.5.0 in /usr/local/lib/python3.7/dist-packages (from pytest->CBFV) (1.11.0)\n Requirement already satisfied: atomicwrites>=1.0 in /usr/local/lib/python3.7/dist-packages (from pytest->CBFV) (1.4.0)\n Requirement already satisfied: more-itertools>=4.0.0 in /usr/local/lib/python3.7/dist-packages (from pytest->CBFV) (8.12.0)\n Requirement already satisfied: setuptools in /usr/local/lib/python3.7/dist-packages (from pytest->CBFV) (57.4.0)\n Installing collected packages: CBFV\n Successfully installed CBFV-1.1.0\n\n\n### Imports\n\n\n```python\nimport numpy as np\nimport pandas as pd\n\nfrom sklearn.linear_model import LogisticRegression\nfrom sklearn.model_selection import GridSearchCV\nfrom sklearn.model_selection import RandomizedSearchCV\nfrom sklearn.tree import DecisionTreeClassifier\n\nfrom scipy.stats import randint\n\nfrom matbench.bench import MatbenchBenchmark\nfrom CBFV.composition import generate_features\n```\n\n### Data\n\n\n```python\nmb = MatbenchBenchmark(subset=[\"matbench_expt_is_metal\"])\ntask = list(mb.tasks)[0]\ntask.load()\nfold0 = task.folds[0]\ntrain_inputs, train_outputs = task.get_train_and_val_data(fold0)\ntest_inputs, test_outputs = task.get_test_data(fold0, include_target=True)\nprint(train_inputs[0:2], train_outputs[0:2])\nprint(train_outputs.shape, test_outputs.shape)\n \n```\n\n 2022-02-09 21:36:57 INFO Initialized benchmark 'matbench_v0.1' with 1 tasks: \n ['matbench_expt_is_metal']\n 2022-02-09 21:36:57 INFO Loading dataset 'matbench_expt_is_metal'...\n Fetching matbench_expt_is_metal.json.gz from https://ml.materialsproject.org/projects/matbench_expt_is_metal.json.gz to /usr/local/lib/python3.7/dist-packages/matminer/datasets/matbench_expt_is_metal.json.gz\n\n\n Fetching https://ml.materialsproject.org/projects/matbench_expt_is_metal.json.gz in MB: 0.034816MB [00:00, 40.50MB/s] \n\n 2022-02-09 21:36:57 INFO Dataset 'matbench_expt_is_metal loaded.\n mbid\n mb-expt-is-metal-0001 Ag(AuS)2\n mb-expt-is-metal-0002 Ag(W3Br7)2\n Name: composition, dtype: object mbid\n mb-expt-is-metal-0001 True\n mb-expt-is-metal-0002 True\n Name: is_metal, dtype: bool\n (3936,) (985,)\n\n\n \n\n\n\n```python\ntrain_inputs.describe()\n```\n\n\n\n\n count 3936\n unique 3936\n top Ag(AuS)2\n freq 1\n Name: composition, dtype: object\n\n\n\n\n```python\ntrain_outputs.describe()\n```\n\n\n\n\n count 3936\n unique 2\n top False\n freq 1976\n Name: is_metal, dtype: object\n\n\n\n\n```python\ntrain_df = pd.DataFrame({\"formula\": train_inputs, \"target\": train_outputs})\ntest_df = pd.DataFrame({\"formula\": test_inputs, \"target\": test_outputs})\ntrain_df\n```\n\n\n\n\n\n
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formulatarget
mbid
mb-expt-is-metal-0001Ag(AuS)2True
mb-expt-is-metal-0002Ag(W3Br7)2True
mb-expt-is-metal-0003Ag0.5Ge1Pb1.75S4False
mb-expt-is-metal-0005Ag2BBrTrue
mb-expt-is-metal-0006Ag2BiO3True
.........
mb-expt-is-metal-4916ZrSiTeTrue
mb-expt-is-metal-4917ZrTaN3False
mb-expt-is-metal-4918ZrTeTrue
mb-expt-is-metal-4920ZrTiF6True
mb-expt-is-metal-4921ZrW2True
\n

3936 rows \u00d7 2 columns

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\n\n\n\n\n\n```python\nX_train, y_train, _, _ = generate_features(train_df)\nX_train\n```\n\n Processing Input Data: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 3936/3936 [00:00<00:00, 15165.79it/s]\n\n\n \tFeaturizing Compositions...\n\n\n Assigning Features...: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 3936/3936 [00:00<00:00, 8136.75it/s]\n\n\n \tCreating Pandas Objects...\n\n\n\n\n\n\n
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avg_Atomic_Numberavg_Atomic_Weightavg_Periodavg_groupavg_familiesavg_Metalavg_Nonmetalavg_Metalliodavg_Mendeleev_Numberavg_l_quantum_numberavg_Atomic_Radiusavg_Miracle_Radius_[pm]avg_Covalent_Radiusavg_Zunger_radii_sumavg_ionic_radiusavg_crystal_radiusavg_Pauling_Electronegativityavg_MB_electonegativityavg_Gordy_electonegativityavg_Mulliken_ENavg_Allred-Rockow_electronegativityavg_metallic_valenceavg_number_of_valence_electronsavg_gilmor_number_of_valence_electronavg_valence_savg_valence_pavg_valence_davg_valence_favg_Number_of_unfilled_s_valence_electronsavg_Number_of_unfilled_p_valence_electronsavg_Number_of_unfilled_d_valence_electronsavg_Number_of_unfilled_f_valence_electronsavg_outer_shell_electronsavg_1st_ionization_potential_(kJ/mol)avg_polarizability(A^3)avg_Melting_point_(K)avg_Boiling_Point_(K)avg_Density_(g/mL)avg_specific_heat_(J/g_K)_avg_heat_of_fusion_(kJ/mol)_...mode_familiesmode_Metalmode_Nonmetalmode_Metalliodmode_Mendeleev_Numbermode_l_quantum_numbermode_Atomic_Radiusmode_Miracle_Radius_[pm]mode_Covalent_Radiusmode_Zunger_radii_summode_ionic_radiusmode_crystal_radiusmode_Pauling_Electronegativitymode_MB_electonegativitymode_Gordy_electonegativitymode_Mulliken_ENmode_Allred-Rockow_electronegativitymode_metallic_valencemode_number_of_valence_electronsmode_gilmor_number_of_valence_electronmode_valence_smode_valence_pmode_valence_dmode_valence_fmode_Number_of_unfilled_s_valence_electronsmode_Number_of_unfilled_p_valence_electronsmode_Number_of_unfilled_d_valence_electronsmode_Number_of_unfilled_f_valence_electronsmode_outer_shell_electronsmode_1st_ionization_potential_(kJ/mol)mode_polarizability(A^3)mode_Melting_point_(K)mode_Boiling_Point_(K)mode_Density_(g/mL)mode_specific_heat_(J/g_K)_mode_heat_of_fusion_(kJ/mol)_mode_heat_of_vaporization_(kJ/mol)_mode_thermal_conductivity_(W/(m_K))_mode_heat_atomization(kJ/mol)mode_Cohesive_energy
047.400000113.1866564.60000013.0000005.2000000.6000000.4000000.00000074.6000001.2000001.378000127.2000001.2900001.9790001.2600001.0340002.4340001.7500006.3004005.6840002.1776004.0640009.0000003.6000001.4000001.6000006.0000005.6000000.6000004.4000004.0000008.4000003.000000902.2000005.180000936.2700002125.43000010.6480000.3822007.967000...4.00.00.00.066.01.00.88103.01.021.1001.000.432.541.195.84585.771.9202.006.02.01.00.00.00.00.02.00.00.01.0890.02.900385.95717.852.070000.1281.717509.80000.26900279.02.85
146.714286110.9316294.61904813.5714296.6666670.3333330.6666670.00000081.0000001.2380951.256667133.2426471.2500001.6945241.2285711.4861902.7395242.4490486.3936866.5285712.2990481.9104766.9047625.9047621.9523813.3333331.6190484.0000000.0476192.6666678.38095210.0000005.2857141014.8095245.6142861288.4452382034.8261908.0942860.36366714.176381...8.00.01.00.095.01.00.94115.01.141.2001.151.822.962.836.40797.592.6850.007.07.02.05.00.00.00.01.010.014.07.01140.03.100265.95331.953.120000.4735.2860015.43800.12200112.01.22
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333.50000076.6128504.00000013.0000005.5000000.5000000.2500000.25000074.2500000.5000001.277500133.2426471.2550001.6862501.3000001.1625002.2150001.7175004.6768505.1900002.1190003.4700008.0000003.5000001.5000001.5000005.0000000.0000000.5000004.5000005.00000014.0000003.000000850.7500005.4750001272.1000002394.3125006.6150000.49075019.521500...4.01.00.00.065.00.01.65144.01.532.3751.601.291.931.073.85224.441.8705.4411.02.01.00.010.00.01.06.00.014.01.0731.07.9001235.152485.1510.500000.23511.30000250.5800429.00000284.02.95
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......................................................................................................................................................................................................................................................
393135.33333382.3031674.33333311.3333335.3333330.3333330.6666670.00000070.6666671.3333331.466667136.0000001.3133331.9716671.3500000.7900001.7766672.0200004.3597004.6333331.7980003.3333334.6666674.0000002.0000002.0000004.0000000.0000000.0000004.0000006.00000014.0000002.666667765.3333339.6000001510.3166672847.0833335.0266670.39333328.313333...4.00.00.00.044.01.01.11110.01.111.4201.100.401.331.703.45213.641.3202.004.02.02.00.00.00.00.02.00.014.02.0640.05.400722.651262.952.330000.20016.9000052.55002.35000197.02.19
393226.80000062.8384243.40000010.8000005.8000000.4000000.6000000.00000067.6000001.4000001.148000103.8000001.0220001.4470000.9900000.5080002.3900002.4380006.1352205.9300002.3716003.6000004.8000004.0000002.0000001.8000001.0000002.8000000.0000004.2000009.00000011.2000003.8000001121.4000006.8600001116.8100002116.0700004.6227500.7060009.916240...7.00.01.00.082.01.00.5672.00.750.5400.650.303.042.856.88347.303.0663.005.05.02.03.00.00.00.03.010.014.05.01402.01.10063.2577.350.001251.0400.360402.79280.02598473.04.92
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\n\n\n\n\n\n```python\nX_test, y_test, _, _ = generate_features(test_df)\nX_test\n```\n\n Processing Input Data: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 985/985 [00:00<00:00, 15770.47it/s]\n\n\n \tFeaturizing Compositions...\n\n\n Assigning Features...: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 985/985 [00:00<00:00, 7830.71it/s]\n\n\n \tCreating Pandas Objects...\n\n\n\n\n\n\n
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avg_Atomic_Numberavg_Atomic_Weightavg_Periodavg_groupavg_familiesavg_Metalavg_Nonmetalavg_Metalliodavg_Mendeleev_Numberavg_l_quantum_numberavg_Atomic_Radiusavg_Miracle_Radius_[pm]avg_Covalent_Radiusavg_Zunger_radii_sumavg_ionic_radiusavg_crystal_radiusavg_Pauling_Electronegativityavg_MB_electonegativityavg_Gordy_electonegativityavg_Mulliken_ENavg_Allred-Rockow_electronegativityavg_metallic_valenceavg_number_of_valence_electronsavg_gilmor_number_of_valence_electronavg_valence_savg_valence_pavg_valence_davg_valence_favg_Number_of_unfilled_s_valence_electronsavg_Number_of_unfilled_p_valence_electronsavg_Number_of_unfilled_d_valence_electronsavg_Number_of_unfilled_f_valence_electronsavg_outer_shell_electronsavg_1st_ionization_potential_(kJ/mol)avg_polarizability(A^3)avg_Melting_point_(K)avg_Boiling_Point_(K)avg_Density_(g/mL)avg_specific_heat_(J/g_K)_avg_heat_of_fusion_(kJ/mol)_...mode_familiesmode_Metalmode_Nonmetalmode_Metalliodmode_Mendeleev_Numbermode_l_quantum_numbermode_Atomic_Radiusmode_Miracle_Radius_[pm]mode_Covalent_Radiusmode_Zunger_radii_summode_ionic_radiusmode_crystal_radiusmode_Pauling_Electronegativitymode_MB_electonegativitymode_Gordy_electonegativitymode_Mulliken_ENmode_Allred-Rockow_electronegativitymode_metallic_valencemode_number_of_valence_electronsmode_gilmor_number_of_valence_electronmode_valence_smode_valence_pmode_valence_dmode_valence_fmode_Number_of_unfilled_s_valence_electronsmode_Number_of_unfilled_p_valence_electronsmode_Number_of_unfilled_d_valence_electronsmode_Number_of_unfilled_f_valence_electronsmode_outer_shell_electronsmode_1st_ionization_potential_(kJ/mol)mode_polarizability(A^3)mode_Melting_point_(K)mode_Boiling_Point_(K)mode_Density_(g/mL)mode_specific_heat_(J/g_K)_mode_heat_of_fusion_(kJ/mol)_mode_heat_of_vaporization_(kJ/mol)_mode_thermal_conductivity_(W/(m_K))_mode_heat_atomization(kJ/mol)mode_Cohesive_energy
046.206897111.0322904.55172414.8965526.1724140.3103450.5517240.13793184.0344830.9310341.226207132.7586211.2686211.5924141.3517240.8306902.2686212.2131035.2284695.1317242.1944142.6193105.5862074.9655171.9310342.9655173.1034483.379310.0689663.0344836.89655210.620694.896552847.5172415.124138668.9465521613.9775866.8524140.26827610.726103...7.00.01.00.089.01.01.03118.01.161.2851.150.562.552.545.76105.892.4342.006.06.02.04.00.00.00.02.010.014.06.0941.03.800490.15958.154.790000.3206.6940037.70000.52000227.02.46
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246.750000107.5061505.00000010.7500004.0000001.0000000.0000000.00000064.2500000.5000001.660000143.5000001.4750002.3937501.5500001.1625001.9975001.3225003.8361504.4425001.8000005.52500010.7500002.2500000.7500000.00000010.0000000.000001.2500006.0000000.00000014.000000.750000749.2500007.1250001383.1500002717.15000010.8750000.23625012.875000...4.01.00.00.065.00.01.65144.01.532.3751.601.291.931.073.85224.441.8705.4411.02.01.00.010.00.01.06.00.014.01.0731.07.9001235.152485.1510.500000.23511.30000250.5800429.00000284.02.95
327.12500061.0840253.50000013.1250005.5000000.5000000.5000000.00000074.8750000.7500001.081250102.7500001.0962501.4487501.0625001.1912502.7187502.3075006.0880755.9975002.6987502.7625008.1250004.3750001.5000002.0000004.6250000.000000.5000004.0000005.37500014.000003.5000001019.8750004.559000813.4500001498.6500005.4882150.5778757.348795...7.00.01.00.087.01.00.4864.00.730.4650.601.213.443.328.37037.543.6100.006.06.02.04.00.00.00.02.010.014.06.01314.00.79354.7590.150.001430.9200.222593.40990.02674249.02.62
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98024.00000051.7853333.66666712.0000006.0000000.3333330.6666670.00000073.3333331.3333331.273333121.3333331.1733331.6750001.1833330.5733332.1633332.3333335.0479005.3600002.1660002.6666675.3333334.6666672.0000002.6666670.6666670.000000.0000003.3333339.33333314.000004.666667880.0000007.900000965.6833332028.6166673.5500000.5633336.778333...7.00.01.00.088.01.00.88103.01.021.1001.000.432.582.655.84586.222.5892.006.06.02.04.00.00.00.02.010.014.06.01000.02.900385.95717.852.070000.7101.717509.80000.26900279.02.85
98145.000000104.6846675.0000009.0000004.6666670.6666670.3333330.00000061.6666671.6666671.723333149.0000001.3733332.3983331.4333330.8600001.8600001.9366674.0380674.3300001.6146674.2600005.6666674.0000001.6666671.0000006.3333330.000000.3333335.0000003.66666714.000001.666667728.00000011.3666671870.8166673682.1500008.5333330.23933320.256667...4.00.00.00.044.01.01.33134.01.261.7651.300.821.331.703.45213.641.3203.004.02.01.00.02.00.00.03.00.014.01.0640.06.600904.152223.156.510000.21016.9000077.140022.70000262.02.75
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\n\n\n\n\n## Train\n\nWe can do hyperparameter tuning in different ways. Two common ways are grid search (less efficient) and random search (more efficient). Below are examples taken/modified from the website https://www.geeksforgeeks.org/hyperparameter-tuning/\n\n\n\n```python\n#Grid search first using logistic regression classifier model\n\n# Creating the hyperparameter grid\nc_space = np.logspace(-5, 8, 15)\nparam_grid = {'C': c_space}\n \n# Instantiating logistic regression classifier\n# https://stats.stackexchange.com/a/184026/293880\nlogreg = LogisticRegression(max_iter=100)\n \n# Instantiating the GridSearchCV object\nlogreg_grid = GridSearchCV(logreg, param_grid, cv = 5)\n \nlogreg_grid.fit(X_train, y_train)\n \n# Print the tuned parameters and score\nprint(\"Grid tuned Logistic Regression Parameters: {}\".format(logreg_grid.best_params_)) \nprint(\"Best score is {}\".format(logreg_grid.best_score_))\n```\n\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n\n\n Grid tuned Logistic Regression Parameters: {'C': 19306.977288832535}\n Best score is 0.8191010003934494\n\n\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n\n\n\n```python\n#Now we can try random search with logistic regression\n \n# Creating the hyperparameter grid \nparam_dist = {\"C\": randint(-5,15)}\n \n# Instantiating Decision Tree classifier\nlogreg = LogisticRegression()\n \n# Instantiating RandomizedSearchCV object\nlogreg_random = RandomizedSearchCV(logreg, param_dist, cv = 5)\n \nlogreg_random.fit(X_train, y_train)\n \n# Print the tuned parameters and score\nprint(\"Random tuned Logistic Regression Parameters: {}\".format(logreg_random.best_params_))\nprint(\"Best score is {}\".format(logreg_random.best_score_))\n\n```\n\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n\n\n Random tuned Logistic Regression Parameters: {'C': 3}\n Best score is 0.820117841317346\n\n\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:818: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG,\n\n\nWe can do the same grid vs random search with another model, like a decision tree classifier\n\n\n```python\n#grid search for decision tree hyperparameters\n \n# Creating the hyperparameter grid \nparam_grid = {\"max_depth\": range(1,10),\n \"max_features\": range(1,10),\n \"min_samples_leaf\": range(1,10),\n \"criterion\": [\"gini\", \"entropy\"]}\n\n# Instantiating Decision Tree classifier\ntree = DecisionTreeClassifier()\n \n# Instantiating GridSearchCV object\ntree_grid = GridSearchCV(tree, param_grid, cv = 5)\n \ntree_grid.fit(X_train, y_train)\n \n# Print the tuned parameters and score\nprint(\"Grid tuned Decision Tree Parameters: {}\".format(tree_grid.best_params_))\nprint(\"Best score is {}\".format(tree_grid.best_score_))\n\n```\n\n Grid tuned Decision Tree Parameters: {'criterion': 'gini', 'max_depth': 6, 'max_features': 7, 'min_samples_leaf': 5}\n Best score is 0.850099652345539\n\n\n\n```python\n#random search for decision tree hyperparameters\n \n# Creating the hyperparameter grid \nparam_dist = {\"max_depth\": randint(1,10),\n \"max_features\": randint(1,10),\n \"min_samples_leaf\": randint(1,10),\n \"criterion\": [\"gini\", \"entropy\"]}\n\n# Instantiating Decision Tree classifier\ntree = DecisionTreeClassifier()\n \n# Instantiating RandomizedSearchCV object\ntree_random = RandomizedSearchCV(tree, param_dist, cv = 5)\n \ntree_random.fit(X_train, y_train)\n \n# Print the tuned parameters and score\nprint(\"Random tuned Decision Tree Parameters: {}\".format(tree_random.best_params_))\nprint(\"Best score is {}\".format(tree_random.best_score_))\n\n```\n\n Random tuned Decision Tree Parameters: {'criterion': 'entropy', 'max_depth': 8, 'max_features': 5, 'min_samples_leaf': 6}\n Best score is 0.8219051335470429\n\n\n## Test\n\n\n```python\n\n```\n\n## Code Graveyard\n\n\n```python\n# from sklearn.preprocessing import StandardScaler\n# scaler = StandardScaler()\n# scaler.fit(X_train)\n# X_train = scaler.transform(X_train)\n# X_test = scaler.transform(X_test)\n# X_train\n```\n", "meta": {"hexsha": "353e5dfcf1dfc08934c1d90135c7be327afa17ec", "size": 201482, "ext": "ipynb", "lang": "Jupyter Notebook", 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YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.6791786926816161, "lm_q1q2_score": 0.4495639134758548}} {"text": "# \u041f\u0440\u0435\u0434\u0435\u043b\u044b\n\n\u0414\u0430\u043d\u043d\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u044f:\n$f(x)=(1+x)^\\frac{1}{x}$\n\n\u042d\u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0430 \u0432 \u0442\u043e\u0447\u043a\u0435 0. \u041d\u043e \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u044e \u0432 \u0442\u043e\u0447\u043a\u0438 \u0441\u043a\u043e\u043b\u044c \u0443\u0433\u043e\u0434\u043d\u043e \u0431\u043b\u0438\u0437\u043a\u043e\u0439 \u043a \u043d\u0443\u043b\u044e\n\n\n```python\n%matplotlib inline\nimport matplotlib.pyplot as plt\nimport numpy as np\n\ndef showPlot(x_start, x_max, step, callback, title='Function'):\n x = np.arange(x_start, x_max, step)\n y = callback(x)\n\n plt.plot(x, y)\n plt.xlabel('X')\n plt.ylabel('Y')\n plt.title(title)\n plt.grid(True)\n plt.show()\n\ndef target_function(x):\n return (1+x)**(1/x)\n\nx = 10.0\nfor i in range(1,10):\n print(x, ': ', target_function(x))\n x /= 10.0\n \n\nshowPlot(0.001, 10, 0.1, target_function)\n```\n\n$\\lim\\limits_{x \\to 0}(1+x)^\\frac{1}{x}=e$\n\n\u0421\u043c\u044b\u0441\u043b \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0442\u044b:\n$e^{ax}$ - \u044d\u0442\u043e \u0440\u0435\u0448\u0435\u043d\u0438\u0435 \u0443\u0440\u043e\u0432\u043d\u0435\u043d\u0438\u044f $f(x+y)=f(x)*f(y)$ , \u043f\u0440\u0438\u043c\u0435\u0440 $f(x)=2^x=e^{x*ln2}; 2^{x+y}=2^x 2^y$\n\n\u041d\u043e \u043d\u0435 \u0432\u0441\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0438\u043c\u0435\u044e\u0442 \u043a\u043e\u043d\u0435\u0447\u043d\u044b\u0439 \u043f\u0440\u0435\u0434\u0435\u043b \u0432 \u0442\u043e\u0447\u043a\u0435 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u044e\u0442 \u0438 \u0431\u0435\u0441\u043a\u043e\u043d\u0435\u0447\u043d\u044b\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u044b\n$\\lim\\limits_{x \\to 0}\\frac{1}{x}=+\\infty$\n\n\n```python\ndef target_function(x):\n return 1/x\n\nx = 10.0\nfor i in range(1,10):\n print(x, ': ', target_function(x))\n x /= 10.0\n \nshowPlot(0.1, 2, 0.01, target_function)\n```\n\n\u0424\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430 \u0432 \u0442\u043e\u0447\u043a\u0435 **a** \u0435\u0441\u043b\u0438\n\n$\\lim\\limits_{x \\to a} f(x) = f(a)$\n\n\n```python\nfrom sympy import *\n\nlimit(sin(x)/x, x, 0)\n```\n\n\n```python\ninit_printing()\nx, y = symbols('x y')\nexpr = Limit((cos(x) - 1)/x, x, 0)\nexpr\n```\n\n\n```python\nexpr.doit()\n```\n\n# \u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435\n\n\u041f\u0440\u0438\u0440\u0430\u0449\u0435\u043d\u0438\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438:\n\n\n$b = \\frac{f(x+h)-f(x)}{h}$, \u0433\u0434\u0435 $b=tg\\theta$\n\n\n\n\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f - \u044d\u0442\u043e\n\n$f'(x)=\\lim\\limits_{\\Delta x \\to 0} \\frac{f(x+\\Delta x)-f(x)}{\\Delta x}$\n\n\u0413\u043b\u0430\u0434\u043a\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f - \u044d\u0442\u043e \u0442\u0430\u043a\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f, \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u043a\u043e\u0442\u043e\u0440\u043e\u0439 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u0438 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430\n\n\n\n$f(x)=x^2; f'(x)=\\lim\\limits_{\\Delta x \\to 0} \\frac{(x+\\Delta x)^2 - x^2}{\\Delta x}=\\lim\\limits_{\\Delta x \\to 0} \\frac{x^2 + 2x\\Delta x + \\Delta x^2 - x^2}{\\Delta x}=\\lim\\limits_{\\Delta x \\to 0} \\frac{\\Delta x(2x+\\Delta x)}{\\Delta x}=2x$\n\n\n```python\nx, y = symbols('x y')\ndiff(cos(x), x)\n```\n\n\n```python\ne = x**3\ndx = diff(e, x)\ndx\n```\n\n\n```python\ndef value_of_expr(x_vector, symbol, expr):\n y = np.empty_like(x_vector)\n for index, element in enumerate(x_vector):\n y[index] = (expr.subs(symbol, element))\n return y\n\nshowPlot(-10, 10, 0.01, lambda i: value_of_expr(i, x, e), title='Target')\nshowPlot(-10, 10, 0.01, lambda i: value_of_expr(i, x, dx), title='Diff')\n```\n\n# \u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u0441\u043b\u043e\u0436\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438\n\n\u0421\u043b\u043e\u0436\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f - $g(x) = h(f(x))$, \u043f\u0440\u0438\u043c\u0435\u0440: $g(x)=sin(x+1), h(x)=sin(x), f(x)=1+x$\n\n$df = f'(x)dx,\\ \u0433\u0434\u0435\\ dx = \\Delta x$\n\n$f'(x)=\\frac{df}{dx}$\n\nChain rule:\n\n\u0414\u0430\u043d\u043e:\n$z=g(y), y=h(x)$\n\n$\\frac{dz}{dx}=\\lim\\limits_{\\sigma x \\to 0}\\frac{\\sigma z}{\\sigma y} \\frac{\\sigma y}{\\sigma x}=\\lim\\limits_{\\sigma y \\to 0}\\frac{\\sigma z}{\\sigma y} \\lim\\limits_{\\sigma x \\to 0}\\frac{\\sigma y}{\\sigma x}=\\frac{dz}{dy} \\frac{dy}{dx}$\n\n\u041d\u0430\u0448 \u043f\u0440\u0438\u043c\u0435\u0440: $g(x)=sin(x+1), g'(x)=1*(cos(x+1))$\n\n\n```python\ne = sin(x+1)\ndiff(e, x)\n```\n\n# \u042d\u043a\u0441\u0442\u0440\u0435\u043c\u0443\u043c\u044b\n\n\u042d\u043a\u0441\u0442\u0440\u0435\u043c\u0443\u043c\u044b - \u044d\u0442\u043e \u0442\u043e\u0447\u043a\u0438, \u0434\u043b\u044f \u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u043e\u043a\u0440\u0435\u0441\u0442\u043d\u043e\u0441\u0442\u044c, \u0433\u0434\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u044f $f(x) >= f(x_0)\\ \u0438\u043b\u0438\\ f(x) <= f(x_0),\\ \u0433\u0434\u0435\\ x_0\\ - \\ \u044d\u043a\u0441\u0442\u0440\u0435\u043c\u0443\u043c,\\ \u0430\\ x \\in U$\n\n\n\n\u041d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u044b\u043c \u0443\u0441\u043b\u043e\u0432\u0438\u0435\u043c \u044d\u043a\u0441\u0442\u0440\u0435\u043c\u0443\u043c\u0430, \u043d\u043e \u043d\u0435 \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u044b\u043c \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0442\u043e, \u0447\u0442\u043e $f'(x_0)=0$\n\n* $f'(x) > 0$ - \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0441\u0442\u0440\u043e\u0433\u043e \u0432\u043e\u0437\u0440\u0430\u0441\u0442\u0430\u0435\u0442\n* $f'(x) < 0$ - \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0441\u0442\u0440\u043e\u0433\u043e \u0443\u0431\u044b\u0432\u0430\u0435\u0442\n* $f''(x) > 0$ - \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0441\u0442\u0440\u043e\u0433\u043e \u0432\u044b\u043f\u0443\u043a\u043b\u0430\n* $f''(x) < 0$ - \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0441\u0442\u0440\u043e\u0433\u043e \u0432\u043e\u0433\u043d\u0443\u0442\u0430\n\n\u0414\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u044b\u043c \u0443\u0441\u043b\u043e\u0432\u0438\u0435\u043c \u043f\u0440\u0438 \u0432\u044b\u043f\u043e\u043b\u043d\u0435\u043d\u0438\u0435 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e\u0433\u043e \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f, \u0442\u043e \u0447\u0442\u043e \u0432\u0442\u043e\u0440\u0430\u044f \u043f\u0440\u043e\u0438\u0437\u043e\u0432\u0434\u043d\u0430\u044f \u0434\u043e\u043b\u0436\u043d\u0430 \u0431\u044b\u0442\u044c \u0441\u0442\u0440\u043e\u0433\u043e \u0431\u043e\u043b\u044c\u0448\u0435 0 \u0438\u043b\u0438 \u043c\u0435\u043d\u044c\u0448\u0435\n\n$f''(x) > 0 - \u043c\u0438\u043d\u0438\u043c\u0443\u043c$\n\n$f''(x) < 0 - \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c$\n\n\n```python\ne = .1*((log(x)/2.)**sin(x))\n\nshowPlot(3, 10, 0.1, lambda i: value_of_expr(i, x, e), title='Target')\n```\n\n\n```python\ndx = diff(e, x)\ndx\n```\n\n\n```python\nshowPlot(3, 10, 0.1, lambda i: value_of_expr(i, x, dx), title='Target')\n```\n\n\n```python\nsecond_dex = diff(e, x, x)\nsecond_dex\n```\n\n\n```python\nshowPlot(3, 10, 0.1, lambda i: value_of_expr(i, x, second_dex), title='Target')\n```\n\n# \u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u043f\u043e \u043d\u0435\u0441\u043a\u043e\u043b\u044c\u043a\u0438\u043c \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u044b\u043c\n\n\u041f\u0443\u0441\u0442\u044c \u0434\u0430\u043d\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043f\u043e \u043d\u0435\u0441\u043a\u043e\u043b\u044c\u043a\u0438\u043c \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u044b\u043c\n$f(x,y,...)$\n\n$f_x'=\\frac{df}{dx}; f_y'=\\frac{df}{dy};...$\n\n\u041f\u0440\u0438\u043c\u0435\u0440:\n$z = x^2 + xy + y^2$\n\n* $z_x' = 2x + y$\n* $z_y' = x + 2y$\n\n\n\n```python\n%matplotlib inline\n\nfrom mpl_toolkits.mplot3d import axes3d, Axes3D\n\n\nfig = plt.figure()\nax = Axes3D(fig)\n\ndef value_of_expr(x_matrix, y_matrix, x_symbol, y_symbol, expr):\n z = np.empty_like(x_matrix)\n for i_1 in range(0, z.shape[0]):\n for i_2 in range(0, z.shape[0]):\n z[i_1, i_2] = expr.subs({x_symbol: x_matrix[i_1, i_2], y_symbol: y_matrix[i_1, i_2]})\n return z\n\ne = x**2 + x*y + y**2\n\n# Make data.\nX = np.arange(-5, 5, 0.25)\nY = np.arange(-5, 5, 0.25)\nX, Y = np.meshgrid(X, Y)\nZ = value_of_expr(X, Y, x, y, e)\n\nsurf = ax.plot_surface(X, Y, Z, linewidth=0)\n\ndx = diff(e,x)\nZ = value_of_expr(X, Y, x, y, dx)\nsurf = ax.plot_surface(X, Y, Z, linewidth=0, color='g')\n\ndy = diff(e,y)\nZ = value_of_expr(X, Y, x, y, dy)\nsurf = ax.plot_surface(X, Y, Z, linewidth=0, color='r')\n\n# Customize the z axis.\nax.set_zlim(0, 20)\n```\n\n\n```python\ndx\n```\n\n\n```python\ndy\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "96738d93ec53788b008369ae838e7b28790a9b85", "size": 311248, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "module_002_math/lesson_002_limits/tutorials_sources/Limits and derivatives.ipynb", "max_stars_repo_name": "CorbenTerminator/ML", "max_stars_repo_head_hexsha": "df083b768be23e53f28d160b23e6dbe8775115f2", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 22, "max_stars_repo_stars_event_min_datetime": "2019-04-06T17:12:33.000Z", "max_stars_repo_stars_event_max_datetime": "2021-07-29T12:35:48.000Z", "max_issues_repo_path": "module_002_math/lesson_002_limits/tutorials_sources/Limits and derivatives.ipynb", "max_issues_repo_name": "CorbenTerminator/ML", "max_issues_repo_head_hexsha": "df083b768be23e53f28d160b23e6dbe8775115f2", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 17, "max_issues_repo_issues_event_min_datetime": "2019-04-01T19:24:38.000Z", "max_issues_repo_issues_event_max_datetime": "2020-01-28T09:10:37.000Z", "max_forks_repo_path": "module_002_math/lesson_002_limits/tutorials_sources/Limits and derivatives.ipynb", "max_forks_repo_name": "CorbenTerminator/ML", "max_forks_repo_head_hexsha": "df083b768be23e53f28d160b23e6dbe8775115f2", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 141, "max_forks_repo_forks_event_min_datetime": "2019-04-01T18:45:05.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-20T12:32:46.000Z", "avg_line_length": 440.8611898017, "max_line_length": 157036, "alphanum_fraction": 0.9387755102, "converted": true, "num_tokens": 2027, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8104789132480439, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.4493866198692647}} {"text": "# IPython\n\n**IPython** (Interactive Python) is an enhanced Python shell which provides a more robust and productive development environment for users. There are several key features that set it apart from the standard Python shell.\n\n* Interactive data analysis and visualization\n* Python kernel for Jupyter notebooks\n* Easy parallel computation\n\n### History\n\nIn IPython, all your inputs and outputs are saved. There are two variables named `In` and `Out` which are assigned as you work with your results. All outputs are saved automatically to variables of the form `_N`, where `N` is the prompt number, and inputs to `_iN`. This allows you to recover quickly the result of a prior computation by referring to its number even if you forgot to store it as a variable. \n\n\n```python\nimport numpy as np\nnp.sin(4)**2\n```\n\n\n```python\n_1\n```\n\n\n```python\n_i1\n```\n\n\n```python\n_1 / 4.\n```\n\n### Introspection\n\nIf you want details regarding the properties and functionality of any Python objects currently loaded into IPython, you can use the `?` to reveal any details that are available:\n\n\n```python\nsome_dict = {}\nsome_dict?\n```\n\nIf available, additional detail is provided with two question marks, including the source code of the object itself.\n\n\n```python\nfrom numpy.linalg import cholesky\ncholesky??\n```\n\nThis syntax can also be used to search namespaces with wildcards (`*`).\n\n\n```python\nimport numpy as np\nnp.random.rand*?\n```\n\n### Tab completion\n\nBecause IPython allows for introspection, it is able to afford the user the ability to tab-complete commands that have been partially typed. This is done by pressing the `` key at any point during the process of typing a command:\n\n\n```python\nnp.ar\n```\n\nThis can even be used to help with specifying arguments to functions, which can sometimes be difficult to remember:\n\n\n```python\nplt.hist\n```\n\n### System commands\n\nIn IPython, you can type `ls` to see your files or `cd` to change directories, just like you would at a regular system prompt:\n\n\n```python\nls /Users/fonnescj/repositories/scientific-python-workshop/\n```\n\nVirtually any system command can be accessed by prepending `!`, which passes any subsequent command directly to the OS.\n\n\n```python\n!locate python | grep pdf\n```\n\nYou can even use Python variables in commands sent to the OS:\n\n\n```python\nfile_type = 'csv'\n!ls ../data/*$file_type\n```\n\nThe output of a system command using the exclamation point syntax can be assigned to a Python variable.\n\n\n```python\ndata_files = !ls ../data/microbiome/\n```\n\n\n```python\ndata_files\n```\n\n## Qt Console\n\nIf you type at the system prompt:\n\n $ ipython qtconsole\n\ninstead of opening in a terminal, IPython will start a graphical console that at first sight appears just like a terminal, but which is in fact much more capable than a text-only terminal. This is a specialized terminal designed for interactive scientific work, and it supports full multi-line editing with color highlighting and graphical calltips for functions, it can keep multiple IPython sessions open simultaneously in tabs, and when scripts run it can display the figures inline directly in the work area.\n\n\n\n# Jupyter Notebook\n\nOver time, the IPython project grew to include several components, including:\n\n* an interactive shell\n* a REPL protocol\n* a notebook document fromat\n* a notebook document conversion tool\n* a web-based notebook authoring tool\n* tools for building interactive UI (widgets)\n* interactive parallel Python\n\nAs each component has evolved, several had grown to the point that they warrented projects of their own. For example, pieces like the notebook and protocol are not even specific to Python. As the result, the IPython team created Project Jupyter, which is the new home of language-agnostic projects that began as part of IPython, such as the notebook in which you are reading this text.\n\nThe HTML notebook that is part of the Jupyter project supports **interactive data visualization** and easy high-performance **parallel computing**.\n\n\n\n```python\n%matplotlib inline\nimport matplotlib.pyplot as plt\nplt.style.use('fivethirtyeight')\n\ndef f(x):\n return (x-3)*(x-5)*(x-7)+85\n\nimport numpy as np\nx = np.linspace(0, 10, 200)\ny = f(x)\nplt.plot(x,y)\n```\n\nThe notebook lets you document your workflow using either HTML or Markdown.\n\nThe Jupyter Notebook consists of two related components:\n\n* A JSON based Notebook document format for recording and distributing Python code and rich text.\n* A web-based user interface for authoring and running notebook documents.\n\nThe Notebook can be used by starting the Notebook server with the command:\n\n $ ipython notebook\n \nThis initiates an **iPython engine**, which is a Python instance that takes Python commands over a network connection.\n\nThe **IPython controller** provides an interface for working with a set of engines, to which one or more **iPython clients** can connect.\n\nThe Notebook gives you everything that a browser gives you. For example, you can embed images, videos, or entire websites.\n\n\n```python\nfrom IPython.display import IFrame\nIFrame('https://jupyter.org', width='100%', height=350)\n```\n\n\n```python\nfrom IPython.display import YouTubeVideo\nYouTubeVideo(\"rl5DaFbLc60\")\n```\n\n## Markdown cells\n\nMarkdown is a simple *markup* language that allows plain text to be converted into HTML.\n\nThe advantages of using Markdown over HTML (and LaTeX):\n\n- its a **human-readable** format \n- allows writers to focus on content rather than formatting and layout\n- easier to learn and use\n\nFor example, instead of writing:\n\n```html\n

In order to create valid \nHTML, you \nneed properly coded syntax that can be cumbersome for \n“non-programmers” to write. Sometimes, you\njust want to easily make certain words bold\n, and certain words italicized without\nhaving to remember the syntax. Additionally, for example,\ncreating lists:

\n\n```\n\nwe can write the following in Markdown:\n\n```markdown\nIn order to create valid [HTML], you need properly\ncoded syntax that can be cumbersome for \n\"non-programmers\" to write. Sometimes, you just want\nto easily make certain words **bold**, and certain \nwords *italicized* without having to remember the \nsyntax. Additionally, for example, creating lists:\n\n* should be easy\n* should not involve programming\n```\n\n### Emphasis\n\nMarkdown uses `*` (asterisk) and `_` (underscore) characters as \nindicators of emphasis. \n\n *italic*, _italic_ \n **bold**, __bold__\n ***bold-italic***, ___bold-italic___\n\n*italic*, _italic_ \n**bold**, __bold__ \n***bold-italic***, ___bold-italic___\n\n### Lists\n\nMarkdown supports both unordered and ordered lists. Unordered lists can use `*`, `-`, or \n`+` to define a list. This is an unordered list: \n\n * Apples\n * Bananas\n * Oranges\n\n* Apples\n* Bananas\n* Oranges\n\nOrdered lists are numbered lists in plain text:\n\n 1. Bryan Ferry\n 2. Brian Eno\n 3. Andy Mackay\n 4. Paul Thompson\n 5. Phil Manzanera\n\n1. Bryan Ferry\n2. Brian Eno\n3. Andy Mackay\n4. Paul Thompson\n5. Phil Manzanera\n\n### Links\n\nMarkdown inline links are equivalent to HTML `` \nlinks, they just have a different syntax. \n\n [Biostatistics home page](http://biostat.mc.vanderbilt.edu \"Visit Biostat!\")\n\n[Biostatistics home page](http://biostat.mc.vanderbilt.edu \"Visit Biostat!\")\n\n### Block quotes\n\nBlock quotes are denoted by a `>` (greater than) character \nbefore each line of the block quote.\n\n > Sometimes a simple model will outperform a more complex model . . . \n > Nevertheless, I believe that deliberately limiting the complexity \n > of the model is not fruitful when the problem is evidently complex. \n\n> Sometimes a simple model will outperform a more complex model . . .\n> Nevertheless, I believe that deliberately limiting the complexity \n> of the model is not fruitful when the problem is evidently complex.\n\n### Images\n\nImages look an awful lot like Markdown links, they just have an extra \n`!` (exclamation mark) in front of them. \n\n \n\n\n\n### Remote Code\n\nUse `%load` to add remote code\n\n\n```python\n# %load http://matplotlib.org/mpl_examples/shapes_and_collections/scatter_demo.py\n\"\"\"\nSimple demo of a scatter plot.\n\"\"\"\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n\nN = 50\nx = np.random.rand(N)\ny = np.random.rand(N)\ncolors = np.random.rand(N)\narea = np.pi * (15 * np.random.rand(N))**2 # 0 to 15 point radiuses\n\nplt.scatter(x, y, s=area, c=colors, alpha=0.5)\nplt.show()\n\n```\n\n### Mathjax Support\n\nMathjax ia a javascript implementation $\\alpha$ of LaTeX that allows equations to be embedded into HTML. For example, this markup:\n\n \"\"\"$$ \\int_{a}^{b} f(x)\\, dx \\approx \\frac{1}{2} \\sum_{k=1}^{N} \\left( x_{k} - x_{k-1} \\right) \\left( f(x_{k}) + f(x_{k-1}) \\right). $$\"\"\"\n \nbecomes this:\n\n$$\n\\int_{a}^{b} f(x)\\, dx \\approx \\frac{1}{2} \\sum_{k=1}^{N} \\left( x_{k} - x_{k-1} \\right) \\left( f(x_{k}) + f(x_{k-1}) \\right).\n$$\n\n## SymPy Support\n\nSymPy is a Python library for symbolic mathematics. It supports:\n\n* polynomials\n* calculus\n* solving equations\n* discrete math\n* matrices\n\n\n```python\nfrom sympy import *\ninit_printing()\nx, y = symbols(\"x y\")\n```\n\n\n```python\neq = ((x+y)**2 * (x+1))\neq\n```\n\n\n```python\nexpand(eq)\n```\n\n\n```python\n(1/cos(x)).series(x, 0, 6)\n```\n\n\n```python\nlimit((sin(x)-x)/x**3, x, 0)\n```\n\n\n```python\ndiff(cos(x**2)**2 / (1+x), x)\n```\n\n### Magic functions\n\nIPython has a set of predefined \u2018magic functions\u2019 that you can call with a command line style syntax. These include:\n\n* `%run`\n* `%edit`\n* `%debug`\n* `%timeit`\n* `%paste`\n* `%load_ext`\n\n\n\n\n```python\n%lsmagic\n```\n\nTiming the execution of code; the `timeit` magic exists both in line and cell form:\n\n\n```python\n%timeit np.linalg.eigvals(np.random.rand(100,100))\n```\n\n\n```python\n%%timeit a = np.random.rand(100, 100)\nnp.linalg.eigvals(a)\n```\n\nIPython also creates aliases for a few common interpreters, such as bash, ruby, perl, etc.\n\nThese are all equivalent to `%%script `\n\n\n```ruby\n%%ruby\nputs \"Hello from Ruby #{RUBY_VERSION}\"\n```\n\n\n```bash\n%%bash\necho \"hello from $BASH\"\n```\n\nIPython has an `rmagic` extension that contains a some magic functions for working with R via rpy2. This extension can be loaded using the `%load_ext` magic as follows:\n\n\n```python\n%load_ext rpy2.ipython\n```\n\nIf the above generates an error, it is likely that you do not have the `rpy2` module installed. You can install this now via:\n\n\n```python\n!pip install rpy2\n```\n\nor, if you are running Anaconda, via `conda`:\n\n\n```python\n!conda install rpy2\n```\n\n\n```python\nx,y = np.arange(10), np.random.normal(size=10)\n%R print(lm(rnorm(10)~rnorm(10)))\n```\n\n\n```r\n%%R -i x,y -o XYcoef\nlm.fit <- lm(y~x)\npar(mfrow=c(2,2))\nprint(summary(lm.fit))\nplot(lm.fit)\nXYcoef <- coef(lm.fit)\n```\n\n\n```python\nXYcoef\n```\n\n## Debugging\n\nThe `%debug` magic can be used to trigger the IPython debugger (`ipd`) for a cell that raises an exception. The debugger allows you to step through code line-by-line and inspect variables and execute code.\n\n\n```python\ndef div(x, y):\n return x/y\n\ndiv(1,0)\n```\n\n\n```python\n%debug\n```\n\n## Exporting and Converting Notebooks\n\nIn Jupyter, one can convert an `.ipynb` notebook document file into various static formats via the `nbconvert` tool. Currently, nbconvert is a command line tool, run as a script using Jupyter.\n\n\n```python\n!jupyter nbconvert --to html \"IPython and Jupyter.ipynb\"\n```\n\nCurrently, `nbconvert` supports HTML (default), LaTeX, Markdown, reStructuredText, Python and HTML5 slides for presentations. Some types can be post-processed, such as LaTeX to PDF (this requires [Pandoc](http://johnmacfarlane.net/pandoc/) to be installed, however).\n\n\n```python\n!jupyter nbconvert --to pdf \"Introduction to pandas.ipynb\"\n```\n\nA very useful online service is the [IPython Notebook Viewer](http://nbviewer.ipython.org) which allows you to display your notebook as a static HTML page, which is useful for sharing with others:\n\n\n```python\nIFrame(\"http://nbviewer.ipython.org/2352771\", width='100%', height=350)\n```\n\nAs of this year, GitHub supports the [rendering of Jupyter Notebooks](https://github.com/fonnesbeck/Bios8366/blob/master/notebooks/Section1_2-Programming-with-Python.ipynb) stored on its repositories.\n\n## Reproducible Research\n\n> reproducing conclusions from a single experiment based on the measurements from that experiment\n\nThe most basic form of reproducibility is a complete description of the data and associated analyses (including code!) so the results can be *exactly* reproduced by others.\n\nReproducing calculations can be onerous, even with one's own work!\n\nScientific data are becoming larger and more complex, making simple descriptions inadequate for reproducibility. As a result, most modern research is irreproducible without tremendous effort.\n\n*** Reproducible research is not yet part of the culture of science in general, or scientific computing in particular. ***\n\n## Scientific Computing Workflow\n\nThere are a number of steps to scientific endeavors that involve computing:\n\n\n\n\nMany of the standard tools impose barriers between one or more of these steps. This can make it difficult to iterate, reproduce work.\n\nThe Jupyter notebook eliminates or reduces these barriers to reproducibility.\n\n## Parallel IPython\n\nThe IPython architecture consists of four components, which reside in the `ipyparallel` package:\n\n1. **Engine** The IPython engine is a Python instance that accepts Python commands over a network connection. When multiple engines are started, parallel and distributed computing becomes possible. An important property of an IPython engine is that it blocks while user code is being executed. \n\n2. **Hub** The hub keeps track of engine connections, schedulers, clients, as well as persist all task requests and results in a database for later use.\n\n3. **Schedulers** All actions that can be performed on the engine go through a Scheduler. While the engines themselves block when user code is run, the schedulers hide that from the user to provide a fully asynchronous interface to a set of engines.\n\n4. **Client** The primary object for connecting to a cluster.\n\n\n(courtesy Min Ragan-Kelley)\n\nThis architecture is implemented using the \u00d8MQ messaging library and the associated Python bindings in `pyzmq`.\n\n### Running parallel IPython\n\nTo enable the IPython Clusters tab in Jupyter Notebook:\n\n ipcluster nbextension enable\n \nWhen you then start a Jupyter session, you should see the following in your **IPython Clusters** tab: \n\n\n\nBefore running the next cell, make sure you have first started your cluster, you can use the [clusters tab in the dashboard](/#tab2) to do so. \n\nSelect the number if IPython engines (nodes) that you want to use, then click **Start**.\n\n\n```python\nfrom ipyparallel import Client\nclient = Client()\ndv = client.direct_view()\n```\n\n\n```python\nlen(dv)\n```\n\n\n```python\ndef where_am_i():\n import os\n import socket\n \n return \"In process with pid {0} on host: '{1}'\".format(\n os.getpid(), socket.gethostname())\n```\n\n\n```python\nwhere_am_i_direct_results = dv.apply(where_am_i)\nwhere_am_i_direct_results.get()\n```\n\nLet's now consider a useful function that we might want to run in parallel. Here is a version of the approximate Bayesian computing (ABC) algorithm.\n\n\n```python\nimport numpy\n\ndef abc(y, N, epsilon=[0.2, 0.8]):\n \n trace = []\n\n while len(trace) < N:\n\n # Simulate from priors\n mu = numpy.random.normal(0, 10)\n sigma = numpy.random.uniform(0, 20)\n\n x = numpy.random.normal(mu, sigma, 50)\n\n #if (np.linalg.norm(y - x) < epsilon):\n if ((abs(x.mean() - y.mean()) < epsilon[0]) &\n (abs(x.std() - y.std()) < epsilon[1])):\n trace.append([mu, sigma])\n\n return trace\n```\n\n\n```python\ny = numpy.random.normal(4, 2, 50)\n```\n\nLet's try running this on one of the cluster engines:\n\n\n```python\ndv0 = client[0]\ndv0.block = True\ndv0.apply(abc, y, 10)\n```\n\nThis fails with a NameError because NumPy has not been imported on the engine to which we sent the task. Each engine has its own namespace, so we need to import whatever modules we will need prior to running our code:\n\n\n```python\ndv0.execute(\"import numpy\")\n```\n\n\n```python\ndv0.apply(abc, y, 10)\n```\n\nAn easier approach is to use the parallel cell magic to import everywhere:\n\n\n```python\n%%px\nimport numpy\n```\n\nThis magic can be used to execute the same code on all nodes.\n\n\n```python\n%%px \nimport os\nprint(os.getpid())\n```\n\n\n```python\n%%px \n%matplotlib inline\nimport matplotlib.pyplot as plt\nimport os\ntsamples = numpy.random.randn(100)\nplt.hist(tsamples)\n_ = plt.title('PID %i' % os.getpid())\n```\n\n## Links and References\n\n* [IPython Notebook Viewer](http://nbviewer.ipython.org) Displays static HTML versions of notebooks, and includes a gallery of notebook examples.\n\n* [A Reference-Free Algorithm for Computational Normalization of Shotgun Sequencing Data](http://ged.msu.edu/papers/2012-diginorm/) A landmark example of reproducible research in genomics: Git repo, iPython notebook, data and scripts.\n\n* Jacques Ravel and K Eric Wommack. 2014. [All Hail Reproducibility in Microbiome Research](http://www.microbiomejournal.com/content/pdf/2049-2618-2-8.pdf). Microbiome, 2:8.\n\n* Benjamin Ragan-Kelley et al.. 2013. [Collaborative cloud-enabled tools allow rapid, reproducible biological insights](http://www.nature.com/ismej/journal/v7/n3/full/ismej2012123a.html). The ISME Journal, 7, 461\u2013464; doi:10.1038/ismej.2012.123;\n", "meta": {"hexsha": "10c809d4f38ebd09d4d3367c7d76bd060d91b60b", "size": 31540, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/IPython and Jupyter.ipynb", "max_stars_repo_name": "fonnesbeck/scientific-python-workshop", "max_stars_repo_head_hexsha": "e8fd00f938bb75c4ff536184a8cbc7289a77cc5d", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 13, "max_stars_repo_stars_event_min_datetime": "2016-04-07T03:26:26.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-05T08:50:27.000Z", "max_issues_repo_path": "notebooks/IPython and Jupyter.ipynb", "max_issues_repo_name": "fonnesbeck/scientific-python-workshop", "max_issues_repo_head_hexsha": "e8fd00f938bb75c4ff536184a8cbc7289a77cc5d", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2016-04-08T03:55:45.000Z", "max_issues_repo_issues_event_max_datetime": "2016-05-03T00:20:06.000Z", "max_forks_repo_path": "notebooks/IPython and Jupyter.ipynb", "max_forks_repo_name": "fonnesbeck/scientific-python-workshop", "max_forks_repo_head_hexsha": "e8fd00f938bb75c4ff536184a8cbc7289a77cc5d", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 17, "max_forks_repo_forks_event_min_datetime": "2016-04-06T22:00:03.000Z", "max_forks_repo_forks_event_max_datetime": "2021-04-23T03:49:33.000Z", "avg_line_length": 26.0661157025, "max_line_length": 522, "alphanum_fraction": 0.5691819911, "converted": true, "num_tokens": 4385, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5312093733737563, "lm_q2_score": 0.84594244507642, "lm_q1q2_score": 0.4493725561593083}} {"text": "   \n\n# Bonus Tutorial: Diving Deeper into Decoding & Encoding\n**Week 2, Day 1: Deep Learning**\n\n**By Neuromatch Academy**\n\n**Content creators**: Jorge A. Menendez, Carsen Stringer\n\n**Content reviewers**: Roozbeh Farhoodi, Madineh Sarvestani, Kshitij Dwivedi, Spiros Chavlis, Ella Batty, Michael Waskom\n\n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n

\n\n---\n# Tutorial Objectives\nIn this tutorial, we'll will dive deeper into our decoding model from Tutorial 1 and we will fit a convolutional neural network directly to neural activities.\n\n\n---\n# Setup\n\n\n\n```python\n# Imports\n\nimport os\nimport numpy as np\n\nimport torch\nfrom torch import nn\nfrom torch import optim\n\nimport matplotlib as mpl\nfrom matplotlib import pyplot as plt\n```\n\n\n```python\n#@title Data retrieval and loading\nimport hashlib\nimport requests\n\nfname = \"W3D4_stringer_oribinned1.npz\"\nurl = \"https://osf.io/683xc/download\"\nexpected_md5 = \"436599dfd8ebe6019f066c38aed20580\"\n\nif not os.path.isfile(fname):\n try:\n r = requests.get(url)\n except requests.ConnectionError:\n print(\"!!! Failed to download data !!!\")\n else:\n if r.status_code != requests.codes.ok:\n print(\"!!! Failed to download data !!!\")\n elif hashlib.md5(r.content).hexdigest() != expected_md5:\n print(\"!!! Data download appears corrupted !!!\")\n else:\n with open(fname, \"wb\") as fid:\n fid.write(r.content)\n```\n\n\n```python\n#@title Figure Settings\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/course-content/master/nma.mplstyle\")\n```\n\n\n```python\n#@title Plotting Functions\n\ndef plot_data_matrix(X, ax):\n \"\"\"Visualize data matrix of neural responses using a heatmap\n\n Args:\n X (torch.Tensor or np.ndarray): matrix of neural responses to visualize\n with a heatmap\n ax (matplotlib axes): where to plot\n\n \"\"\"\n\n cax = ax.imshow(X, cmap=mpl.cm.pink, vmin=np.percentile(X, 1), vmax=np.percentile(X, 99))\n cbar = plt.colorbar(cax, ax=ax, label='normalized neural response')\n\n ax.set_aspect('auto')\n ax.set_xticks([])\n ax.set_yticks([])\n\ndef plot_decoded_results(train_loss, test_loss, test_labels, predicted_test_labels):\n \"\"\" Plot decoding results in the form of network training loss and test predictions\n\n Args:\n train_loss (list): training error over iterations\n test_labels (torch.Tensor): n_test x 1 tensor with orientations of the\n stimuli corresponding to each row of train_data, in radians\n predicted_test_labels (torch.Tensor): n_test x 1 tensor with predicted orientations of the\n stimuli from decoding neural network\n\n \"\"\"\n\n # Plot results\n fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(16, 6))\n\n # Plot the training loss over iterations of GD\n ax1.plot(train_loss)\n # Plot the testing loss over iterations of GD\n ax1.plot(test_loss)\n ax1.legend(['train loss', 'test loss'])\n\n # Plot true stimulus orientation vs. predicted class\n ax2.plot(stimuli_test.squeeze(), predicted_test_labels, '.')\n\n ax1.set_xlim([0, None])\n ax1.set_ylim([0, None])\n ax1.set_xlabel('iterations of gradient descent')\n ax1.set_ylabel('negative log likelihood')\n ax2.set_xlabel('true stimulus orientation ($^o$)')\n ax2.set_ylabel('decoded orientation bin')\n ax2.set_xticks(np.linspace(0, 360, n_classes + 1))\n ax2.set_yticks(np.arange(n_classes))\n class_bins = [f'{i * 360 / n_classes: .0f}$^o$ - {(i + 1) * 360 / n_classes: .0f}$^o$' for i in range(n_classes)]\n ax2.set_yticklabels(class_bins);\n\n # Draw bin edges as vertical lines\n ax2.set_ylim(ax2.get_ylim()) # fix y-axis limits\n for i in range(n_classes):\n lower = i * 360 / n_classes\n upper = (i + 1) * 360 / n_classes\n ax2.plot([lower, lower], ax2.get_ylim(), '-', color=\"0.7\", linewidth=1, zorder=-1)\n ax2.plot([upper, upper], ax2.get_ylim(), '-', color=\"0.7\", linewidth=1, zorder=-1)\n\n plt.tight_layout()\n\ndef visualize_weights(W_in_sorted, W_out):\n plt.figure(figsize=(10,4))\n plt.subplot(1,2,1)\n plt.imshow(W_in_sorted, aspect='auto', cmap='bwr', vmin=-1e-2, vmax=1e-2)\n plt.colorbar()\n plt.xlabel('sorted neurons')\n plt.ylabel('hidden units')\n plt.title('$W_{in}$')\n\n plt.subplot(1,2,2)\n plt.imshow(W_out.T, cmap='bwr', vmin=-3, vmax=3)\n plt.xticks([])\n plt.xlabel('output')\n plt.ylabel('hidden units')\n plt.colorbar()\n plt.title('$W_{out}$')\n\ndef visualize_hidden_units(W_in_sorted, h):\n plt.figure(figsize=(10,8))\n plt.subplot(2,2,1)\n plt.imshow(W_in_sorted, aspect='auto', cmap='bwr', vmin=-1e-2, vmax=1e-2)\n plt.xlabel('sorted neurons')\n plt.ylabel('hidden units')\n plt.colorbar()\n plt.title('$W_{in}$')\n\n plt.subplot(2,2,2)\n plt.imshow(h, aspect='auto')\n plt.xlabel('stimulus orientation ($^\\circ$)')\n plt.ylabel('hidden units')\n plt.colorbar()\n plt.title('$\\mathbf{h}$')\n\n plt.subplot(2,2,4)\n plt.plot(h.T)\n plt.xlabel('stimulus orientation ($^\\circ$)')\n plt.ylabel('hidden unit activity')\n plt.title('$\\mathbf{h}$ tuning curves')\n\n plt.show()\n\ndef plot_weights(weights, channels=[0], colorbar=True):\n \"\"\" plot convolutional channel weights\n Args:\n weights: weights of convolutional filters (conv_channels x K x K)\n channels: which conv channels to plot\n \"\"\"\n wmax = torch.abs(weights).max()\n fig, axs = plt.subplots(1,len(channels), figsize=(12,2.5))\n for i, channel in enumerate(channels):\n im = axs[i].imshow(weights[channel,0], vmin=-wmax, vmax=wmax, cmap='bwr')\n axs[i].set_title('channel %d'%channel)\n\n if colorbar:\n ax = fig.add_axes([1, 0.1, 0.05, 0.8])\n plt.colorbar(im, ax=ax)\n ax.axis('off')\n\ndef plot_tuning(ax, stimuli, respi_train, respi_test, neuron_index, linewidth=2):\n \"\"\"Plot the tuning curve of a neuron\"\"\"\n\n ax.plot(stimuli, respi_train, 'y', linewidth=linewidth) # plot its responses as a function of stimulus orientation\n ax.plot(stimuli, respi_test, 'm', linewidth=linewidth) # plot its responses as a function of stimulus orientation\n ax.set_title('neuron %i' % neuron_index)\n ax.set_xlabel('stimulus orientation ($^o$)')\n ax.set_ylabel('neural response')\n ax.set_xticks(np.linspace(0, 360, 5))\n ax.set_ylim([-0.5, 2.4])\n\n\ndef plot_prediction(ax, y_pred, y_train, y_test):\n \"\"\" plot prediction of neural response + test neural response \"\"\"\n ax.plot(y_train, 'y', linewidth=1)\n ax.plot(y_test,color='m')\n ax.plot(y_pred, 'g', linewidth=3)\n ax.set_xlabel('stimulus bin')\n ax.set_ylabel('response')\n\n\ndef plot_training_curves(train_loss, test_loss):\n \"\"\"\n Args:\n train_loss (list): training error over iterations\n test_loss (list): n_test x 1 tensor with orientations of the\n stimuli corresponding to each row of train_data, in radians\n predicted_test_labels (torch.Tensor): n_test x 1 tensor with predicted orientations of the\n stimuli from decoding neural network\n\n \"\"\"\n\n f, ax = plt.subplots()\n # Plot the training loss over iterations of GD\n ax.plot(train_loss)\n # Plot the testing loss over iterations of GD\n ax.plot(test_loss, '.', markersize=10)\n ax.legend(['train loss', 'test loss'])\n ax.set(xlabel=\"Gradient descent iteration\", ylabel=\"Mean squared error\")\n```\n\n\n```python\n# @title Helper Functions\n\ndef load_data(data_name=fname, bin_width=1):\n \"\"\"Load mouse V1 data from Stringer et al. (2019)\n\n Data from study reported in this preprint:\n https://www.biorxiv.org/content/10.1101/679324v2.abstract\n\n These data comprise time-averaged responses of ~20,000 neurons\n to ~4,000 stimulus gratings of different orientations, recorded\n through Calcium imaging. The responses have been normalized by\n spontaneous levels of activity and then z-scored over stimuli, so\n expect negative numbers. They have also been binned and averaged\n to each degree of orientation.\n\n This function returns the relevant data (neural responses and\n stimulus orientations) in a torch.Tensor of data type torch.float32\n in order to match the default data type for nn.Parameters in\n Google Colab.\n\n This function will actually average responses to stimuli with orientations\n falling within bins specified by the bin_width argument. This helps\n produce individual neural \"responses\" with smoother and more\n interpretable tuning curves.\n\n Args:\n bin_width (float): size of stimulus bins over which to average neural\n responses\n\n Returns:\n resp (torch.Tensor): n_stimuli x n_neurons matrix of neural responses,\n each row contains the responses of each neuron to a given stimulus.\n As mentioned above, neural \"response\" is actually an average over\n responses to stimuli with similar angles falling within specified bins.\n stimuli: (torch.Tensor): n_stimuli x 1 column vector with orientation\n of each stimulus, in degrees. This is actually the mean orientation\n of all stimuli in each bin.\n\n \"\"\"\n with np.load(data_name) as dobj:\n data = dict(**dobj)\n resp = data['resp']\n stimuli = data['stimuli']\n\n if bin_width > 1:\n # Bin neural responses and stimuli\n bins = np.digitize(stimuli, np.arange(0, 360 + bin_width, bin_width))\n stimuli_binned = np.array([stimuli[bins == i].mean() for i in np.unique(bins)])\n resp_binned = np.array([resp[bins == i, :].mean(0) for i in np.unique(bins)])\n else:\n resp_binned = resp\n stimuli_binned = stimuli\n\n # Return as torch.Tensor\n resp_tensor = torch.tensor(resp_binned, dtype=torch.float32)\n stimuli_tensor = torch.tensor(stimuli_binned, dtype=torch.float32).unsqueeze(1) # add singleton dimension to make a column vector\n\n return resp_tensor, stimuli_tensor\n\n\ndef identityLine():\n \"\"\"\n Plot the identity line y=x\n \"\"\"\n ax = plt.gca()\n lims = np.array([ax.get_xlim(), ax.get_ylim()])\n minval = lims[:, 0].min()\n maxval = lims[:, 1].max()\n equal_lims = [minval, maxval]\n ax.set_xlim(equal_lims)\n ax.set_ylim(equal_lims)\n line = ax.plot([minval, maxval], [minval, maxval], color=\"0.7\")\n line[0].set_zorder(-1)\n\ndef get_data(n_stim, train_data, train_labels):\n \"\"\" Return n_stim randomly drawn stimuli/resp pairs\n\n Args:\n n_stim (scalar): number of stimuli to draw\n resp (torch.Tensor):\n train_data (torch.Tensor): n_train x n_neurons tensor with neural\n responses to train on\n train_labels (torch.Tensor): n_train x 1 tensor with orientations of the\n stimuli corresponding to each row of train_data, in radians\n\n Returns:\n (torch.Tensor, torch.Tensor): n_stim x n_neurons tensor of neural responses and n_stim x 1 of orientations respectively\n \"\"\"\n n_stimuli = train_labels.shape[0]\n istim = np.random.choice(n_stimuli, n_stim)\n r = train_data[istim] # neural responses to this stimulus\n ori = train_labels[istim] # true stimulus orientation\n\n return r, ori\n\ndef stimulus_class(ori, n_classes):\n \"\"\"Get stimulus class from stimulus orientation\n\n Args:\n ori (torch.Tensor): orientations of stimuli to return classes for\n n_classes (int): total number of classes\n\n Returns:\n torch.Tensor: 1D tensor with the classes for each stimulus\n\n \"\"\"\n bins = np.linspace(0, 360, n_classes + 1)\n return torch.tensor(np.digitize(ori.squeeze(), bins)) - 1 # minus 1 to accomodate Python indexing\n\ndef grating(angle, sf=1 / 28, res=0.1, patch=False):\n \"\"\"Generate oriented grating stimulus\n\n Args:\n angle (float): orientation of grating (angle from vertical), in degrees\n sf (float): controls spatial frequency of the grating\n res (float): resolution of image. Smaller values will make the image\n smaller in terms of pixels. res=1.0 corresponds to 640 x 480 pixels.\n patch (boolean): set to True to make the grating a localized\n patch on the left side of the image. If False, then the\n grating occupies the full image.\n\n Returns:\n torch.Tensor: (res * 480) x (res * 640) pixel oriented grating image\n\n \"\"\"\n\n angle = np.deg2rad(angle) # transform to radians\n\n wpix, hpix = 640, 480 # width and height of image in pixels for res=1.0\n\n xx, yy = np.meshgrid(sf * np.arange(0, wpix * res) / res, sf * np.arange(0, hpix * res) / res)\n\n if patch:\n gratings = np.cos(xx * np.cos(angle + .1) + yy * np.sin(angle + .1)) # phase shift to make it better fit within patch\n gratings[gratings < 0] = 0\n gratings[gratings > 0] = 1\n xcent = gratings.shape[1] * .75\n ycent = gratings.shape[0] / 2\n xxc, yyc = np.meshgrid(np.arange(0, gratings.shape[1]), np.arange(0, gratings.shape[0]))\n icirc = ((xxc - xcent) ** 2 + (yyc - ycent) ** 2) ** 0.5 < wpix / 3 / 2 * res\n gratings[~icirc] = 0.5\n\n else:\n gratings = np.cos(xx * np.cos(angle) + yy * np.sin(angle))\n gratings[gratings < 0] = 0\n gratings[gratings > 0] = 1\n\n gratings -= 0.5\n\n # Return torch tensor\n return torch.tensor(gratings, dtype=torch.float32)\n\ndef filters(out_channels=6, K=7):\n \"\"\" make example filters, some center-surround and gabors\n Returns:\n filters: out_channels x K x K\n \"\"\"\n grid = np.linspace(-K/2, K/2, K).astype(np.float32)\n xx,yy = np.meshgrid(grid, grid, indexing='ij')\n\n # create center-surround filters\n sigma = 1.1\n gaussian = np.exp(-(xx**2 + yy**2)**0.5/(2*sigma**2))\n wide_gaussian = np.exp(-(xx**2 + yy**2)**0.5/(2*(sigma*2)**2))\n center_surround = gaussian - 0.5 * wide_gaussian\n\n # create gabor filters\n thetas = np.linspace(0, 180, out_channels-2+1)[:-1] * np.pi/180\n gabors = np.zeros((len(thetas), K, K), np.float32)\n lam = 10\n phi = np.pi/2\n gaussian = np.exp(-(xx**2 + yy**2)**0.5/(2*(sigma*0.4)**2))\n for i,theta in enumerate(thetas):\n x = xx*np.cos(theta) + yy*np.sin(theta)\n gabors[i] = gaussian * np.cos(2*np.pi*x/lam + phi)\n\n filters = np.concatenate((center_surround[np.newaxis,:,:],\n -1*center_surround[np.newaxis,:,:],\n gabors),\n axis=0)\n filters /= np.abs(filters).max(axis=(1,2))[:,np.newaxis,np.newaxis]\n filters -= filters.mean(axis=(1,2))[:,np.newaxis,np.newaxis]\n # convert to torch\n filters = torch.from_numpy(filters)\n # add channel axis\n filters = filters.unsqueeze(1)\n\n return filters\n\ndef regularized_MSE_loss(output, target, weights=None, L2_penalty=0, L1_penalty=0):\n \"\"\"loss function for MSE\n\n Args:\n output (torch.Tensor): output of network\n target (torch.Tensor): neural response network is trying to predict\n weights (torch.Tensor): fully-connected layer weights of network (net.out_layer.weight)\n L2_penalty : scaling factor of sum of squared weights\n L1_penalty : scalaing factor for sum of absolute weights\n\n Returns:\n (torch.Tensor) mean-squared error with L1 and L2 penalties added\n\n \"\"\"\n\n loss_fn = nn.MSELoss()\n loss = loss_fn(output, target)\n\n if weights is not None:\n L2 = L2_penalty * torch.square(weights).sum()\n L1 = L1_penalty * torch.abs(weights).sum()\n loss += L1 + L2\n\n return loss\n```\n\n---\n# Section 1: Decoding - Investigating model and evaluating performance\n\nIn this section, we will return to our decoding model from Tutorial 1 and further investigate its performance, and then improve it in the next section. Let's first load the data again and train our model, as we did in Tutorial 1.\n\n\n\n```python\n#@title\n\n#@markdown Execute this cell to load and visualize data\n\n# Load data\nresp_all, stimuli_all = load_data() # argument to this function specifies bin width\nn_stimuli, n_neurons = resp_all.shape\n\nprint(f'{n_neurons} neurons in response to {n_stimuli} stimuli')\n\nfig, (ax1, ax2) = plt.subplots(1, 2, figsize=(2 * 6, 5))\n\n# Visualize data matrix\nplot_data_matrix(resp_all[:100, :].T, ax1) # plot responses of first 100 neurons\nax1.set_xlabel('stimulus')\nax1.set_ylabel('neuron')\n\n# Plot tuning curves of three random neurons\nineurons = np.random.choice(n_neurons, 3, replace=False) # pick three random neurons\nax2.plot(stimuli_all, resp_all[:, ineurons])\nax2.set_xlabel('stimulus orientation ($^o$)')\nax2.set_ylabel('neural response')\nax2.set_xticks(np.linspace(0, 360, 5))\n\nplt.tight_layout()\n```\n\n\n```python\n#@title\n#@markdown Execute this cell to split into training and test sets\n\n# Set random seeds for reproducibility\nnp.random.seed(4)\ntorch.manual_seed(4)\n\n# Split data into training set and testing set\nn_train = int(0.6 * n_stimuli) # use 60% of all data for training set\nishuffle = torch.randperm(n_stimuli)\nitrain = ishuffle[:n_train] # indices of data samples to include in training set\nitest = ishuffle[n_train:] # indices of data samples to include in testing set\nstimuli_test = stimuli_all[itest]\nresp_test = resp_all[itest]\nstimuli_train = stimuli_all[itrain]\nresp_train = resp_all[itrain]\n```\n\n\n```python\n# @markdown Execute this cell to train the network\n\nclass DeepNetReLU(nn.Module):\n \"\"\" network with a single hidden layer h with a RELU \"\"\"\n\n def __init__(self, n_inputs, n_hidden):\n super().__init__() # needed to invoke the properties of the parent class nn.Module\n self.in_layer = nn.Linear(n_inputs, n_hidden) # neural activity --> hidden units\n self.out_layer = nn.Linear(n_hidden, 1) # hidden units --> output\n\n def forward(self, r):\n\n h = torch.relu(self.in_layer(r)) # h is size (n_inputs, n_hidden)\n y = self.out_layer(h) # y is size (n_inputs, 1)\n\n return y\n\n\ndef train(net, loss_fn, train_data, train_labels,\n n_epochs=50, learning_rate=1e-4):\n \"\"\"Run gradient descent to opimize parameters of a given network\n\n Args:\n net (nn.Module): PyTorch network whose parameters to optimize\n loss_fn: built-in PyTorch loss function to minimize\n train_data (torch.Tensor): n_train x n_neurons tensor with neural\n responses to train on\n train_labels (torch.Tensor): n_train x 1 tensor with orientations of the\n stimuli corresponding to each row of train_data\n n_epochs (int, optional): number of epochs of gradient descent to run\n learning_rate (float, optional): learning rate to use for gradient descent\n\n Returns:\n (list): training loss over iterations\n\n \"\"\"\n\n # Initialize PyTorch SGD optimizer\n optimizer = optim.SGD(net.parameters(), lr=learning_rate)\n\n # Placeholder to save the loss at each iteration\n train_loss = []\n\n # Loop over epochs\n for i in range(n_epochs):\n\n # compute network output from inputs in train_data\n out = net(train_data) # compute network output from inputs in train_data\n\n # evaluate loss function\n loss = loss_fn(out, train_labels)\n\n # Clear previous gradients\n optimizer.zero_grad()\n\n # Compute gradients\n loss.backward()\n\n # Update weights\n optimizer.step()\n\n # Store current value of loss\n train_loss.append(loss.item()) # .item() needed to transform the tensor output of loss_fn to a scalar\n\n\n # Track progress\n if (i + 1) % (n_epochs // 5) == 0:\n print(f'iteration {i + 1}/{n_epochs} | loss: {loss.item():.3f}')\n\n return train_loss\n\n\n# Set random seeds for reproducibility\nnp.random.seed(1)\ntorch.manual_seed(1)\n\n# Initialize network with 10 hidden units\nnet = DeepNetReLU(n_neurons, 10)\n\n# Initialize built-in PyTorch MSE loss function\nloss_fn = nn.MSELoss()\n\n# Run gradient descent on data\ntrain_loss = train(net, loss_fn, resp_train, stimuli_train)\n\n# Plot the training loss over iterations of GD\nplt.plot(train_loss)\nplt.xlim([0, None])\nplt.ylim([0, None])\nplt.xlabel('iterations of gradient descent')\nplt.ylabel('mean squared error')\nplt.show()\n```\n\n## Section 1.1: Peering inside the decoding model\n\nWe have built a model to perform decoding that takes as input neural activity and outputs the estimated angle of the stimulus. We can imagine that an animal that needs to determine angles would have a brain area that acts like the hidden layer in our model. It transforms the neural activity from visual cortex and outputs a decision. Decisions about orientations of edges could include figuring out how to jump onto a branch, how to avoid obstacles, or determining the type of an object, e.g. food or predator.\n\nWhat sort of connectivity would this brain area have with visual cortex? Determining this experimentally would be very difficult, perhaps we can look at the model we have and see if its structure constrains the type of connectivity we'd expect.\n\nBelow we will visualize the weights from the neurons in visual cortex to the hidden units $\\mathbf{W}_{in}$, and the weights from the hidden units to the output orientation $\\mathbf{W}_{out}$. \n\n**PyTorch Note**:\n\nAn important thing to note in the code below is the `.detach()` method. The PyTorch `nn.Module` class is special in that, behind the scenes, each of the variables inside it are linked to each other in a computational graph, for the purposes of automatic differentiation (the algorithm used in `.backward()` to compute gradients). As a result, if you want to do anything that is not a `torch` operation to the parameters or outputs of an `nn.Module` class, you'll need to first \"detach\" it from its computational graph. This is what the `.detach()` method does. In this code below, we need to call it on the weights of the network. We also convert the variable from a pytorch tensor to a numpy array using the `.numpy()` method.\n\n\n```python\nW_in = net.in_layer.weight.detach().numpy() # we can run .detach() and .numpy() to get a numpy array\nprint('shape of W_in:')\nprint(W_in.shape)\n\nW_out = net.out_layer.weight.detach().numpy() # we can run .detach() and .numpy() to get a numpy array\nprint('shape of W_out:')\nprint(W_out.shape)\n\nplt.figure(figsize=(10,4))\nplt.subplot(1,2,1)\nplt.imshow(W_in, aspect='auto', cmap='bwr', vmin=-1e-2, vmax=1e-2)\nplt.xlabel('neurons')\nplt.ylabel('hidden units')\nplt.colorbar()\nplt.title('$W_{in}$')\n\nplt.subplot(1,2,2)\nplt.imshow(W_out.T, cmap='bwr', vmin=-3, vmax=3)\nplt.xticks([])\nplt.xlabel('output')\nplt.ylabel('hidden units')\nplt.colorbar()\nplt.title('$W_{out}$')\n\nplt.show()\n```\n\n### Coding Exercise 1.1: Visualizing weights\n\nIt's difficult to see any structure in this weight matrix. How might we visualize it in a better way? \n\nPerhaps we can sort the neurons by their preferred orientation. We will use the `resp_all` matrix which is 360 stimuli (360$^\\circ$ of angles) by number of neurons. How do we find the preferred orientation? \n\nLet's visualize one column of this `resp_all` matrix first as we did at the beginning of the notebook. Can you see how we might want to first process this tuning curve before choosing the preferred orientation?\n\n\n```python\nidx = 235\nplt.plot(resp_all[:,idx])\nplt.ylabel('neural response')\nplt.xlabel('stimulus orientation ($^\\circ$)')\nplt.title(f'neuron {idx}')\nplt.show()\n```\n\nLooking at this tuning curve, there is a bit of noise across orientations, so let's smooth with a gaussian filter and then find the position of the maximum for each neuron. After getting the maximum position aka the \"preferred orientation\" for each neuron, we will re-sort the $\\mathbf{W}_{in}$ matrix. The maximum position in a matrix can be computed using the `.argmax(axis=_)` function in python -- make sure you specify the right axis though! Next, to get the indices of a matrix sorted we will need to use the `.argsort()` function.\n\n\n```python\nfrom scipy.ndimage import gaussian_filter1d\n\n# first let's smooth the tuning curves resp_all to make sure we get\n# an accurate peak that isn't just noise\n# resp_all is size (n_stimuli, n_neurons)\nresp_smoothed = gaussian_filter1d(resp_all, 5, axis=0)\n# resp_smoothed is size (n_stimuli, n_neurons)\n\n############################################################################\n## TO DO for students\n# Fill out function and remove\nraise NotImplementedError(\"Student exercise: find preferred orientation\")\n############################################################################\n\n## find position of max response for each neuron\n## aka preferred orientation for each neuron\npreferred_orientation = ...\n\n## Resort W_in matrix by preferred orientation\nisort = preferred_orientation.argsort()\nW_in_sorted = W_in[:,isort]\n\n# plot resorted W_in matrix\nvisualize_weights(W_in_sorted, W_out)\n```\n\n\n```python\n# to_remove solution\n\nfrom scipy.ndimage import gaussian_filter1d\n\n# first let's smooth the tuning curves resp_all to make sure we get\n# an accurate peak that isn't just noise\n# resp_all is size (n_stimuli, n_neurons)\nresp_smoothed = gaussian_filter1d(resp_all, 5, axis=0)\n# resp_smoothed is size (n_stimuli, n_neurons)\n\n# find position of max response for each neuron\n# aka preferred orientation for each neuron\npreferred_orientation = resp_smoothed.argmax(axis=0)\n\n## Resort W_in matrix by preferred orientation\nisort = preferred_orientation.argsort()\nW_in_sorted = W_in[:,isort]\n\n# plot resorted W_in matrix\nwith plt.xkcd():\n visualize_weights(W_in_sorted, W_out)\n```\n\nWe can plot the activity of hidden units across various stimuli to better understand the hidden units. Recall the formula for the hidden units\n\\begin{equation}\n \\mathbf{h}^{(n)} = \\phi(\\mathbf{W}^{in} \\mathbf{r}^{(n)} + \\mathbf{b}^{in})\n\\end{equation}\nWe can compute the activity $\\mathbf{h}^{(n)}$ directly using $\\mathbf{W}^{in}$ and $\\mathbf{b}^{in}$ or we can modify our network above to return `h` in the `.forward()` method. In this case, we'll compute using the equation, but in practice the second method is recommended.\n\n\n```python\nW_in = net.in_layer.weight.detach() # size (10, n_neurons)\nb_in = net.in_layer.bias.detach().unsqueeze(1) # size (10, 1)\n\n# Compute hidden unit activity h\nh = torch.relu(W_in @ resp_all.T + b_in)\nh = h.detach().numpy() # we can run .detach() and .numpy() to get a numpy array\n\n# Visualize\nvisualize_hidden_units(W_in_sorted, h)\n```\n\n### Think! 1.1: Interpreting weights\n\nWe have just visualized how the model transforms neural activity to hidden layer activity. How should we interpret these matrices? Here are some guiding questions to explore:\n* Why are some of the $\\mathbf{W}_{in}$ weights close to zero for some of the hidden units? Do these correspond to close to zero weights in $\\mathbf{W}_{out}$?\n* Note how each hidden unit seems to have strongest weights to two groups of neurons in $\\mathbf{W}_{in}$, corresponding to two different sets of preferred orientations. Why do you think that is? What does might this tell us about the structure of the tuning curves of the neurons?\n* It appears that there is at least one hidden unit active at each orientation, which is necessary to decode across all orientations. What would happen if some orientations did not activate any hidden units?\n\n\n```python\n# to_remove explanation\n\n\"\"\"\n1. It seems like the model did not learn any weights for those hidden units.\nPerhaps the random initialization of the W_out weights for those hidden units\nwas close to zero and that meant the gradients were small and they remained\nunchanged during training. You could test this hypothesis by trying a\ndifferent random seed when initializing the network.\n\n2. Neurons often have tuning preference to two gratings at 180 degrees apart\nsince these two gratings are the same other than the phase. This is likely why\nthere are two peaks.\n\n3. Any stimulus orientations that don't activate any hidden units would create\nan output vector of zero, and therefore those orientations would not be\ndistinguishable from each other.\n\"\"\";\n```\n\n## Section 1.2: Generalization performance with test data\n\nNote that gradient descent is essentially an algorithm for fitting the network's parameters to a given set of training data. Selecting this training data is thus crucial for ensuring that the optimized parameters **generalize** to unseen data they weren't trained on. In our case, for example, we want to make sure that our trained network is good at decoding stimulus orientations from neural responses to any orientation, not just those in our data set.\n\nTo ensure this, we have split up the full data set into a **training set** and a **testing set**. In Coding Exercise 3.2, we trained a deep network by optimizing the parameters on a training set. We will now evaluate how good the optimized parameters are by using the trained network to decode stimulus orientations from neural responses in the testing set. Good decoding performance on this testing set should then be indicative of good decoding performance on the neurons' responses to any other stimulus orientation. This procedure is commonly used in machine learning (not just in deep learning)and is typically referred to as **cross-validation**.\n\nWe will compute the MSE on the test data and plot the decoded stimulus orientations as a function of the true stimulus.\n\n\n\n```python\n#@title\n#@markdown Execute this cell to evaluate and plot test error\n\nout = net(resp_test) # decode stimulus orientation for neural responses in testing set\nori = stimuli_test # true stimulus orientations\ntest_loss = loss_fn(out, ori) # MSE on testing set (Hint: use loss_fn initialized in previous exercise)\n\nplt.plot(ori, out.detach(), '.') # N.B. need to use .detach() to pass network output into plt.plot()\nidentityLine() # draw the identity line y=x; deviations from this indicate bad decoding!\nplt.title('MSE on testing set: %.2f' % test_loss.item()) # N.B. need to use .item() to turn test_loss into a scalar\nplt.xlabel('true stimulus orientation ($^o$)')\nplt.ylabel('decoded stimulus orientation ($^o$)')\naxticks = np.linspace(0, 360, 5)\nplt.xticks(axticks)\nplt.yticks(axticks)\nplt.show()\n```\n\nIf interested, please see the next section to think more about model criticism, improve the loss function accordingly, and add regularization.\n\n# Section 2: Decoding - Evaluating & improving models\n\n---\n## Section 2.1: Model criticism\n\nLet's now take a step back and think about how our model is succeeding/failing and how to improve it.\n\n\n```python\n#@title\n#@markdown Execute this cell to plot decoding error\n\nout = net(resp_test) # decode stimulus orientation for neural responses in testing set\nori = stimuli_test # true stimulus orientations\nerror = out - ori # decoding error\n\n\nplt.plot(ori, error.detach(), '.') # plot decoding error as a function of true orientation (make sure all arguments to plt.plot() have been detached from PyTorch network!)\n\n# Plotting\nplt.xlabel('true stimulus orientation ($^o$)')\nplt.ylabel('decoding error ($^o$)')\nplt.xticks(np.linspace(0, 360, 5))\nplt.yticks(np.linspace(-360, 360, 9))\nplt.show()\n```\n\n### Think! 2.1: Delving into error problems\n\nIn the cell below, we will plot the *decoding error* for each neural response in the testing set. The decoding error is defined as the decoded stimulus orientation minus true stimulus orientation\n\\begin{equation}\n \\text{decoding error} = y^{(n)} - \\tilde{y}^{(n)}\n\\end{equation}\n\nIn particular, we plot decoding error as a function of the true stimulus orientation.\n\n\n * Are some stimulus orientations harder to decode than others?\n * If so, in what sense? Are the decoded orientations for these stimuli more variable and/or are they biased?\n * Can you explain this variability/bias? What makes these stimulus orientations different from the others?\n * (Will be addressed in next exercise) Can you think of a way to modify the deep network in order to avoid this?\n\n\n```python\n# to_remove explanation\n\n\"\"\"\nIt appears that the errors are larger at 0 and 360 degrees. The errors are biased\nin the positive direction at 0 degrees and in the negative direction at 360 degrees.\nThis is because the 0 degree stimulus and the 360 degree stimulus are in fact the\nsame because orientation is a circular variable. The network therefore has trouble\ndetermining whether the stimulus is 0 or 360 degrees.\n\nWe can modify the deep network to avoid this problem in a few different ways.\nOne approach would be to predict a sine and a cosine of the angle and then taking\nthe predicted angle as the angle of the complex number $sin(\\theta) + i cos(\\theta)$.\n\nAn alternative approach is to bin the stimulus responses and predict the bin of the stimulus.\nThis turns the problem into a classification problem rather than a regression problem,\nand in this case you will need to use a new loss function (see below).\n\"\"\";\n```\n\n## Section 2.2: Improving the loss function \nAs illustrated in the previous exercise, the squared error is not a good loss function for circular quantities like angles, since two angles that are very close (e.g. $1^o$ and $359^o$) might actually have a very large squared error.\n\nHere, we'll avoid this problem by changing our loss function to treat our decoding problem as a **classification problem**. Rather than estimating the *exact* angle of the stimulus, we'll now aim to construct a decoder that classifies the stimulus into one of $C$ classes, corresponding to different bins of angles of width $b = \\frac{360}{C}$. The true class $\\tilde{y}^{(n)}$ of stimulus $i$ is now given by\n\\begin{equation}\n \\tilde{y}^{(n)} =\n \\begin{cases}\n 1 &\\text{if angle of stimulus $n$ is in the range } [0, b] \\\\\n 2 &\\text{if angle of stimulus $n$ is in the range } [b, 2b] \\\\\n 3 &\\text{if angle of stimulus $n$ is in the range } [2b, 3b] \\\\\n \\vdots \\\\\n C &\\text{if angle of stimulus $n$ is in the range } [(C-1)b, 360]\n \\end{cases}\n\\end{equation}\n\nWe have a helper function `stimulus_class` that will extract `n_classes` stimulus classes for us from the stimulus orientations.\n\nTo decode the stimulus class from neural responses, we'll use a deep network that outputs a $C$-dimensional vector of probabilities $\\mathbf{p} = \\begin{bmatrix} p_1, p_2, \\ldots, p_C \\end{bmatrix}^T$, corresponding to the estimated probabilities of the stimulus belonging to each class $1, 2, \\ldots, C$. \n\nTo ensure the network's outputs are indeed probabilities (i.e. they are positive numbers between 0 and 1, and sum to 1), we'll use a [softmax function](https://en.wikipedia.org/wiki/Softmax_function) to transform the real-valued outputs from the hidden layer into probabilities. Letting $\\sigma(\\cdot)$ denote this softmax function, the equations describing our network are\n\\begin{align}\n \\mathbf{h}^{(n)} &= \\phi(\\mathbf{W}^{in} \\mathbf{r}^{(n)} + \\mathbf{b}^{in}), && [\\mathbf{W}^{in}: M \\times N], \\\\\n \\mathbf{p}^{(n)} &= \\sigma(\\mathbf{W}^{out} \\mathbf{h}^{(n)} + \\mathbf{b}^{out}), && [\\mathbf{W}^{out}: C \\times M],\n\\end{align}\nThe decoded stimulus class is then given by that assigned the highest probability by the network:\n\\begin{equation}\n y^{(n)} = \\underset{i}{\\arg\\max} \\,\\, p_i\n\\end{equation}\nThe softmax function can be implemented in PyTorch simply using `torch.softmax()`.\n\nOften *log* probabilities are easier to work with than actual probabilities, because probabilities tend to be very small numbers that computers have trouble representing. We'll therefore actually use the logarithm of the softmax as the output of our network,\n\\begin{equation}\n \\mathbf{l}^{(n)} = \\log \\left( \\mathbf{p}^{(n)} \\right)\n\\end{equation}\nwhich can implemented in PyTorch together with the softmax via an `nn.LogSoftmax` layer. The nice thing about the logarithmic function is that it's *monotonic*, so if one probability is larger/smaller than another, then its logarithm is also larger/smaller than the other's. We therefore have that\n\\begin{equation}\n y^{(n)} = \\underset{i}{\\arg\\max} \\,\\, p_i^{(n)} = \\underset{i}{\\arg\\max} \\, \\log p_i^{(n)} = \\underset{i}{\\arg\\max} \\,\\, l_i^{(n)}\n\\end{equation}\n\nSee the next cell for code for constructing a deep network with one hidden layer that of ReLU's that outputs a vector of log probabilities.\n\n\n```python\n# Deep network for classification\nclass DeepNetSoftmax(nn.Module):\n \"\"\"Deep Network with one hidden layer, for classification\n\n Args:\n n_inputs (int): number of input units\n n_hidden (int): number of units in hidden layer\n n_classes (int): number of outputs, i.e. number of classes to output\n probabilities for\n\n Attributes:\n in_layer (nn.Linear): weights and biases of input layer\n out_layer (nn.Linear): weights and biases of output layer\n\n \"\"\"\n\n def __init__(self, n_inputs, n_hidden, n_classes):\n super().__init__() # needed to invoke the properties of the parent class nn.Module\n self.in_layer = nn.Linear(n_inputs, n_hidden) # neural activity --> hidden units\n self.out_layer = nn.Linear(n_hidden, n_classes) # hidden units --> outputs\n self.logprob = nn.LogSoftmax(dim=1) # probabilities across columns should sum to 1 (each output row corresponds to a different input)\n\n def forward(self, r):\n \"\"\"Predict stimulus orientation bin from neural responses\n\n Args:\n r (torch.Tensor): n_stimuli x n_inputs tensor with neural responses to n_stimuli\n\n Returns:\n torch.Tensor: n_stimuli x n_classes tensor with predicted class probabilities\n\n \"\"\"\n h = torch.relu(self.in_layer(r))\n logp = self.logprob(self.out_layer(h))\n return logp\n```\n\nWhat should our loss function now be? Ideally, we want the probabilities outputted by our network to be such that the probability of the true stimulus class is high. One way to formalize this is to say that we want to maximize the *log* probability of the true stimulus class $\\tilde{y}^{(n)}$ under the class probabilities predicted by the network,\n\\begin{equation}\n \\log \\left( \\text{predicted probability of stimulus } n \\text{ being of class } \\tilde{y}^{(n)} \\right) = \\log p^{(n)}_{\\tilde{y}^{(n)}} = l^{(n)}_{\\tilde{y}^{(n)}}\n\\end{equation}\nTo turn this into a loss function to be *minimized*, we can then simply multiply it by -1: maximizing the log probability is the same as minimizing the *negative* log probability. Summing over a batch of $P$ inputs, our loss function is then given by\n\\begin{equation}\n L = -\\sum_{n=1}^P \\log p^{(n)}_{\\tilde{y}^{(n)}} = -\\sum_{n=1}^P l^{(n)}_{\\tilde{y}^{(n)}}\n\\end{equation}\nIn the deep learning community, this loss function is typically referred to as the **cross-entropy**, or **negative log likelihood**. The corresponding built-in loss function in PyTorch is `nn.NLLLoss()` (documentation [here](https://pytorch.org/docs/master/generated/torch.nn.CrossEntropyLoss.html)).\n\n\n\n### Coding Exercise 2.2: A new loss function\nIn the next cell, we've provided most of the code to train and test a network to decode stimulus orientations via classification, by minimizing the negative log likelihood. Fill in the missing pieces.\n\nOnce you've done this, have a look at the plotted results. Does changing the loss function from mean squared error to a classification loss solve our problems? Note that errors may still occur -- but are these errors as bad as the ones that our network above was making?\n\n\n```python\n#@markdown Run this cell to create train function that uses test_data and L1 and L2 terms for next exercise\ndef train(net, loss_fn, train_data, train_labels,\n n_iter=50, learning_rate=1e-4,\n test_data=None, test_labels=None,\n L2_penalty=0, L1_penalty=0):\n \"\"\"Run gradient descent to opimize parameters of a given network\n\n Args:\n net (nn.Module): PyTorch network whose parameters to optimize\n loss_fn: built-in PyTorch loss function to minimize\n train_data (torch.Tensor): n_train x n_neurons tensor with neural\n responses to train on\n train_labels (torch.Tensor): n_train x 1 tensor with orientations of the\n stimuli corresponding to each row of train_data\n n_iter (int, optional): number of iterations of gradient descent to run\n learning_rate (float, optional): learning rate to use for gradient descent\n test_data (torch.Tensor, optional): n_test x n_neurons tensor with neural\n responses to test on\n test_labels (torch.Tensor, optional): n_test x 1 tensor with orientations of\n the stimuli corresponding to each row of test_data\n L2_penalty (float, optional): l2 penalty regularizer coefficient\n L1_penalty (float, optional): l1 penalty regularizer coefficient\n\n Returns:\n (list): training loss over iterations\n\n \"\"\"\n\n # Initialize PyTorch SGD optimizer\n optimizer = optim.SGD(net.parameters(), lr=learning_rate)\n\n # Placeholder to save the loss at each iteration\n train_loss = []\n test_loss = []\n\n # Loop over epochs\n for i in range(n_iter):\n\n # compute network output from inputs in train_data\n out = net(train_data) # compute network output from inputs in train_data\n\n # evaluate loss function\n if L2_penalty==0 and L1_penalty==0:\n # normal loss function\n loss = loss_fn(out, train_labels)\n else:\n # custom loss function from bonus exercise 3.3\n loss = loss_fn(out, train_labels, net.in_layer.weight,\n L2_penalty, L1_penalty)\n\n # Clear previous gradients\n optimizer.zero_grad()\n # Compute gradients\n loss.backward()\n\n # Update weights\n optimizer.step()\n\n # Store current value of loss\n train_loss.append(loss.item()) # .item() needed to transform the tensor output of loss_fn to a scalar\n\n # Get loss for test_data, if given (we will use this in the bonus exercise 3.2 and 3.3)\n if test_data is not None:\n out_test = net(test_data)\n # evaluate loss function\n if L2_penalty==0 and L1_penalty==0:\n # normal loss function\n loss_test = loss_fn(out_test, test_labels)\n else:\n # (BONUS code) custom loss function from Bonus exercise 3.3\n loss_test = loss_fn(out_test, test_labels, net.in_layer.weight,\n L2_penalty, L1_penalty)\n test_loss.append(loss_test.item()) # .item() needed to transform the tensor output of loss_fn to a scalar\n\n # Track progress\n if (i + 1) % (n_iter // 5) == 0:\n if test_data is None:\n print(f'iteration {i + 1}/{n_iter} | loss: {loss.item():.3f}')\n else:\n print(f'iteration {i + 1}/{n_iter} | loss: {loss.item():.3f} | test_loss: {loss_test.item():.3f}')\n\n if test_data is None:\n return train_loss\n else:\n return train_loss, test_loss\n```\n\n\n```python\ndef decode_orientation(net, n_classes, loss_fn,\n train_data, train_labels, test_data, test_labels,\n n_iter=1000, L2_penalty=0, L1_penalty=0):\n \"\"\" Initialize, train, and test deep network to decode binned orientation from neural responses\n\n Args:\n net (nn.Module): deep network to run\n n_classes (scalar): number of classes in which to bin orientation\n loss_fn (function): loss function to run\n train_data (torch.Tensor): n_train x n_neurons tensor with neural\n responses to train on\n train_labels (torch.Tensor): n_train x 1 tensor with orientations of the\n stimuli corresponding to each row of train_data, in radians\n test_data (torch.Tensor): n_test x n_neurons tensor with neural\n responses to train on\n test_labels (torch.Tensor): n_test x 1 tensor with orientations of the\n stimuli corresponding to each row of train_data, in radians\n n_iter (int, optional): number of iterations to run optimization\n L2_penalty (float, optional): l2 penalty regularizer coefficient\n L1_penalty (float, optional): l1 penalty regularizer coefficient\n\n Returns:\n (list, torch.Tensor): training loss over iterations, n_test x 1 tensor with predicted orientations of the\n stimuli from decoding neural network\n \"\"\"\n\n # Bin stimulus orientations in training set\n train_binned_labels = stimulus_class(train_labels, n_classes)\n test_binned_labels = stimulus_class(test_labels, n_classes)\n\n\n # Run GD on training set data, using learning rate of 0.1\n # (add optional arguments test_data and test_binned_labels!)\n train_loss, test_loss = train(net, loss_fn, train_data, train_binned_labels,\n learning_rate=0.1, test_data=test_data,\n test_labels=test_binned_labels, n_iter=n_iter,\n L2_penalty=L2_penalty, L1_penalty=L1_penalty)\n\n # Decode neural responses in testing set data\n out = net(test_data)\n out_labels = np.argmax(out.detach(), axis=1) # predicted classes\n\n frac_correct = (out_labels==test_binned_labels).sum() / len(test_binned_labels)\n print(f'>>> fraction correct = {frac_correct:.3f}')\n\n return train_loss, test_loss, out_labels\n\n# Set random seeds for reproducibility\nnp.random.seed(1)\ntorch.manual_seed(1)\n\nn_classes = 20\n\n############################################################################\n## TO DO for students\n# Fill out function and remove\nraise NotImplementedError(\"Student exercise: make network and loss\")\n############################################################################\n\n# Initialize network\nnet = ... # use M=20 hidden units\n\n# Initialize built-in PyTorch negative log likelihood loss function\nloss_fn = ...\n\n# Train network and run it on test images\n# this function uses the train function you wrote before\ntrain_loss, test_loss, predicted_test_labels = decode_orientation(net, n_classes, loss_fn,\n resp_train, stimuli_train, resp_test, stimuli_test)\n\n# Plot results\nplot_decoded_results(train_loss, test_loss, stimuli_test, predicted_test_labels)\n```\n\n\n```python\n# to_remove solution\n\ndef decode_orientation(net, n_classes, loss_fn,\n train_data, train_labels, test_data, test_labels,\n n_iter=1000, L2_penalty=0, L1_penalty=0):\n \"\"\" Initialize, train, and test deep network to decode binned orientation from neural responses\n\n Args:\n net (nn.Module): deep network to run\n n_classes (scalar): number of classes in which to bin orientation\n loss_fn (function): loss function to run\n train_data (torch.Tensor): n_train x n_neurons tensor with neural\n responses to train on\n train_labels (torch.Tensor): n_train x 1 tensor with orientations of the\n stimuli corresponding to each row of train_data, in radians\n test_data (torch.Tensor): n_test x n_neurons tensor with neural\n responses to train on\n test_labels (torch.Tensor): n_test x 1 tensor with orientations of the\n stimuli corresponding to each row of train_data, in radians\n n_iter (int, optional): number of iterations to run optimization\n L2_penalty (float, optional): l2 penalty regularizer coefficient\n L1_penalty (float, optional): l1 penalty regularizer coefficient\n\n Returns:\n (list, torch.Tensor): training loss over iterations, n_test x 1 tensor with predicted orientations of the\n stimuli from decoding neural network\n \"\"\"\n\n # Bin stimulus orientations in training set\n train_binned_labels = stimulus_class(train_labels, n_classes)\n test_binned_labels = stimulus_class(test_labels, n_classes)\n\n\n # Run GD on training set data, using learning rate of 0.1\n # (add optional arguments test_data and test_binned_labels!)\n train_loss, test_loss = train(net, loss_fn, train_data, train_binned_labels,\n learning_rate=0.1, test_data=test_data,\n test_labels=test_binned_labels, n_iter=n_iter,\n L2_penalty=L2_penalty, L1_penalty=L1_penalty)\n\n # Decode neural responses in testing set data\n out = net(test_data)\n out_labels = np.argmax(out.detach(), axis=1) # predicted classes\n\n frac_correct = (out_labels==test_binned_labels).sum() / len(test_binned_labels)\n print(f'>>> fraction correct = {frac_correct:.3f}')\n\n return train_loss, test_loss, out_labels\n\n# Set random seeds for reproducibility\nnp.random.seed(1)\ntorch.manual_seed(1)\n\nn_classes = 20\n\n# Initialize network\nnet = DeepNetSoftmax(n_neurons, 20, n_classes) # use M=20 hidden units\n\n# Initialize built-in PyTorch negative log likelihood loss function\nloss_fn = nn.NLLLoss()\n\n# Train network and run it on test images\n# this function uses the train function you wrote before\ntrain_loss, test_loss, predicted_test_labels = decode_orientation(net, n_classes, loss_fn,\n resp_train, stimuli_train, resp_test, stimuli_test)\n\n# Plot results\nwith plt.xkcd():\n plot_decoded_results(train_loss, test_loss, stimuli_test, predicted_test_labels)\n```\n\nHow do the weights $W_{in}$ from the neurons to the hidden layer look now?\n\n\n```python\nW_in = net.in_layer.weight.detach().numpy() # we can run detach and numpy to get a numpy array\nprint('shape of W_in:')\nprint(W_in.shape)\n\nW_out = net.out_layer.weight.detach().numpy()\n\n# plot resorted W_in matrix\nvisualize_weights(W_in[:,isort], W_out)\n```\n\n## Section 2.3: Regularization\n\n\n```python\n# @title Video 1: Regularization\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = 'https://player.bilibili.com/player.html?bvid={0}&page={1}'.format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=\"BV1na4y1a7ug\", width=854, height=480, fs=1)\n print('Video available at https://www.bilibili.com/video/{0}'.format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=\"Qnn5OPHKo5w\", width=854, height=480, fs=1, rel=0)\n print('Video available at https://youtube.com/watch?v=' + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n\n\nAs discussed in the lecture, it is often important to incorporate regularization terms into the loss function to avoid overfitting. In particular, in this case, we will use these terms to enforce sparsity in the linear layer from neurons to hidden units. \n\nHere we'll consider the classic L2 regularization penalty $\\mathcal{R}_{L2}$, which is the sum of squares of each weight in the network $\\sum_{ij} {\\mathbf{W}^{out}_{ij}}^2$ times a constant that we call `L2_penalty`.\n\nWe will also add an L1 regularization penalty $\\mathcal{R}_{L1}$ to enforce sparsity of the weights, which is the sum of the absolute values of the weights $\\sum_{ij} |{\\mathbf{W}^{out}_{ij}}|$ times a constant that we call `L1_penalty`.\n\nWe will add both of these to the loss function:\n\\begin{equation}\n L = (y - \\tilde{y})^2 + \\mathcal{R}_{L2} + \\mathcal{R}_{L1}\n\\end{equation}\n\nThe parameters `L2_penalty` and `L1_penalty` are inputs to the train function.\n\n### Coding Exercise 2.3: Add regularization to training \n\nWe will create a new loss function that adds L1 and L2 regularization. \nIn particular, you will:\n* add L2 loss penalty to the weights \n* add L1 loss penalty to the weights\n\n\nWe will then train the network using this loss function. Full training will take a few minutes: if you want to train for just a few steps to speed up the code while iterating on your code, you can decrease the n_iter input from 500. \n\nHint: since we are using `torch` instead of `np`, we will use `torch.abs` instead of `np.absolute`. You can use `torch.sum` or `.sum()` to sum over a tensor.\n\n\n\n```python\ndef regularized_loss(output, target, weights, L2_penalty=0, L1_penalty=0):\n \"\"\"loss function with L2 and L1 regularization\n\n Args:\n output (torch.Tensor): output of network\n target (torch.Tensor): neural response network is trying to predict\n weights (torch.Tensor): linear layer weights from neurons to hidden units (net.in_layer.weight)\n L2_penalty : scaling factor of sum of squared weights\n L1_penalty : scalaing factor for sum of absolute weights\n\n Returns:\n (torch.Tensor) mean-squared error with L1 and L2 penalties added\n\n \"\"\"\n\n ##############################################################################\n # TO DO: add L1 and L2 regularization to the loss function\n raise NotImplementedError(\"Student exercise: complete regularized_loss\")\n ##############################################################################\n\n loss_fn = nn.NLLLoss()\n loss = loss_fn(output, target)\n\n L2 = L2_penalty * ...\n L1 = L1_penalty * ...\n loss += L1 + L2\n\n return loss\n\n# Set random seeds for reproducibility\nnp.random.seed(1)\ntorch.manual_seed(1)\n\nn_classes = 20\n\n# Initialize network\nnet = DeepNetSoftmax(n_neurons, 20, n_classes) # use M=20 hidden units\n\n# Here you can play with L2_penalty > 0, L1_penalty > 0\ntrain_loss, test_loss, predicted_test_labels = decode_orientation(net, n_classes,\n regularized_loss,\n resp_train, stimuli_train,\n resp_test, stimuli_test,\n n_iter=1000,\n L2_penalty=1e-2,\n L1_penalty=5e-4)\n\n# Plot results\nplot_decoded_results(train_loss, test_loss, stimuli_test, predicted_test_labels)\n```\n\n\n```python\n# to_remove solution\n\ndef regularized_loss(output, target, weights, L2_penalty=0, L1_penalty=0):\n \"\"\"loss function with L2 and L1 regularization\n\n Args:\n output (torch.Tensor): output of network\n target (torch.Tensor): neural response network is trying to predict\n weights (torch.Tensor): linear layer weights from neurons to hidden units (net.in_layer.weight)\n L2_penalty : scaling factor of sum of squared weights\n L1_penalty : scalaing factor for sum of absolute weights\n\n Returns:\n (torch.Tensor) mean-squared error with L1 and L2 penalties added\n\n \"\"\"\n\n loss_fn = nn.NLLLoss()\n loss = loss_fn(output, target)\n\n L2 = L2_penalty * torch.square(weights).sum()\n L1 = L1_penalty * torch.abs(weights).sum()\n loss += L1 + L2\n\n return loss\n\n# Set random seeds for reproducibility\nnp.random.seed(1)\ntorch.manual_seed(1)\n\nn_classes = 20\n\n# Initialize network\nnet = DeepNetSoftmax(n_neurons, 20, n_classes) # use M=20 hidden units\n\n# Here you can play with L2_penalty > 0, L1_penalty > 0\ntrain_loss, test_loss, predicted_test_labels = decode_orientation(net, n_classes,\n regularized_loss,\n resp_train, stimuli_train,\n resp_test, stimuli_test,\n n_iter=1000,\n L2_penalty=1e-2,\n L1_penalty=5e-4)\n\n# Plot results\nwith plt.xkcd():\n plot_decoded_results(train_loss, test_loss, stimuli_test, predicted_test_labels)\n```\n\nIt seems we were overfitting a little because we increased the accuracy a small amount by adding an L1 and L2 regularization penalty. What errors are still being made by the model?\n\nLet's see how the weights look after adding `L1_penalty > 0`.\n\n\n```python\nW_in = net.in_layer.weight.detach().numpy() # we can run detach and numpy to get a numpy array\nprint('shape of W_in:')\nprint(W_in.shape)\n\nW_out = net.out_layer.weight.detach().numpy()\n\nvisualize_weights(W_in[:,isort], W_out)\n```\n\nThe weights appear to be sparser than before.\n\n---\n# Section 3: Encoding - Convolutional Networks for Encoding\n\n\n```python\n# @title Video 2: Convolutional Encoding Model\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = 'https://player.bilibili.com/player.html?bvid={0}&page={1}'.format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=\"BV1Eh41167WP\", width=854, height=480, fs=1)\n print('Video available at https://www.bilibili.com/video/{0}'.format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=\"UNBOPZf0QNQ\", width=854, height=480, fs=1, rel=0)\n print('Video available at https://youtube.com/watch?v=' + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n\n\nIn neuroscience, we often want to understand how the brain represents external stimuli. One approach to discovering these representations is to build an encoding model that takes as input the external stimuli (in this case grating stimuli) and outputs the neural responses. \n\nBecause visual cortex is often thought to be a convolutional network where the same filters are combined across the visual field, we will use a model with a convolutional layer. We learned how to build a convolutional layer in the previous section. We will add to this convolutional layer a fully connected layer from the output of the convolutions to the neurons. We will then visualize the weights of this fully connected layer.\n\n\n```python\n# @markdown Execute this cell to load data\nimport hashlib\nimport requests\n\nfname = \"W3D4_stringer_oribinned6_split.npz\"\nurl = \"https://osf.io/p3aeb/download\"\nexpected_md5 = \"b3f7245c6221234a676b71a1f43c3bb5\"\n\nif not os.path.isfile(fname):\n try:\n r = requests.get(url)\n except requests.ConnectionError:\n print(\"!!! Failed to download data !!!\")\n else:\n if r.status_code != requests.codes.ok:\n print(\"!!! Failed to download data !!!\")\n elif hashlib.md5(r.content).hexdigest() != expected_md5:\n print(\"!!! Data download appears corrupted !!!\")\n else:\n with open(fname, \"wb\") as fid:\n fid.write(r.content)\n\ndef load_data_split(data_name=fname):\n \"\"\"Load mouse V1 data from Stringer et al. (2019)\n\n Data from study reported in this preprint:\n https://www.biorxiv.org/content/10.1101/679324v2.abstract\n\n These data comprise time-averaged responses of ~20,000 neurons\n to ~4,000 stimulus gratings of different orientations, recorded\n through Calcium imaginge. The responses have been normalized by\n spontaneous levels of activity and then z-scored over stimuli, so\n expect negative numbers. The repsonses were split into train and\n test and then each set were averaged in bins of 6 degrees.\n\n This function returns the relevant data (neural responses and\n stimulus orientations) in a torch.Tensor of data type torch.float32\n in order to match the default data type for nn.Parameters in\n Google Colab.\n\n It will hold out some of the trials when averaging to allow us to have test\n tuning curves.\n\n Args:\n data_name (str): filename to load\n\n Returns:\n resp_train (torch.Tensor): n_stimuli x n_neurons matrix of neural responses,\n each row contains the responses of each neuron to a given stimulus.\n As mentioned above, neural \"response\" is actually an average over\n responses to stimuli with similar angles falling within specified bins.\n resp_test (torch.Tensor): n_stimuli x n_neurons matrix of neural responses,\n each row contains the responses of each neuron to a given stimulus.\n As mentioned above, neural \"response\" is actually an average over\n responses to stimuli with similar angles falling within specified bins\n stimuli: (torch.Tensor): n_stimuli x 1 column vector with orientation\n of each stimulus, in degrees. This is actually the mean orientation\n of all stimuli in each bin.\n\n \"\"\"\n with np.load(data_name) as dobj:\n data = dict(**dobj)\n resp_train = data['resp_train']\n resp_test = data['resp_test']\n stimuli = data['stimuli']\n\n # Return as torch.Tensor\n resp_train_tensor = torch.tensor(resp_train, dtype=torch.float32)\n resp_test_tensor = torch.tensor(resp_test, dtype=torch.float32)\n stimuli_tensor = torch.tensor(stimuli, dtype=torch.float32)\n\n return resp_train_tensor, resp_test_tensor, stimuli_tensor\n```\n\n## Section 3.1: Neural tuning curves\n\nIn the next cell, we plot the turning curves of a random subset of neurons. We have binned the stimuli orientations more than in Tutorial 1. We create the gratings images as above for the 60 orientations below, and save them to a variable `grating_stimuli`.\n\nRerun the cell to look at different example neurons and observe the diversity of tuning curves in the population. How can we fit these neural responses with an encoding model?\n\n\n```python\n# @markdown Execute this cell to load data, create stimuli, and plot neural tuning curves\n\n### Load data and bin at 8 degrees\n# responses are split into test and train\nresp_train, resp_test, stimuli = load_data_split()\nn_stimuli, n_neurons = resp_train.shape\nprint('resp_train contains averaged responses of %i neurons to %i binned stimuli' % (n_neurons, n_stimuli))\n#print(resp_train.shape)\n\n# also make stimuli into images\norientations = np.linspace(0, 360, 61)[:-1] - 180\ngrating_stimuli = np.zeros((60, 1, 12, 16), np.float32)\nfor i, ori in enumerate(orientations):\n grating_stimuli[i,0] = grating(ori, res=0.025)#[18:30, 24:40]\n\ngrating_stimuli = torch.from_numpy(grating_stimuli)\nprint('grating_stimuli contains 60 stimuli of size 12 x 16')\n\n# Visualize tuning curves\nfig, axs = plt.subplots(3, 5, figsize=(15,7))\nfor k, ax in enumerate(axs.flatten()):\n neuron_index = np.random.choice(n_neurons) # pick random neuron\n plot_tuning(ax, stimuli, resp_train[:, neuron_index], resp_test[:, neuron_index], neuron_index, linewidth=2)\n if k==0:\n ax.text(1.0, 0.9, 'train', color='y', transform=ax.transAxes)\n ax.text(1.0, 0.65, 'test', color='m', transform=ax.transAxes)\nfig.tight_layout()\nplt.show()\n```\n\n## Section 3.2: Adding a fully-connected layer to create encoding model\n\nWe will build a torch model like above with a convolutional layer. Additionally, we will add a fully connected linear layer from the convolutional units to the neurons. We will use 6 convolutional channels ($C^{out}$) and a kernel size ($K$) of 7 with a stride of 1 and padding of $K/2$ (same as above). The stimulus is size `(12, 16)`. Then the convolutional unit activations will go through a linear layer to be transformed into neural responses.\n\n### Think! 3.2: Number of unis and weights\n\n* How many units will the convolutional layer have?\n* How many weights will the fully connected linear layer from convolutional units to neurons have?\n\n\n```python\n# to_remove explanation\n\n\"\"\"\n1. There are 6 convolutional channels and 12 x 16 positions because the stride\nof the convolution is 1. Therefore, there are 6 * 12 * 16 = 1152 units in the\nconvolutional layer.\n\n2. The fully connected linear layer will have 1152 * n_neurons weights. It will\nalso have n_neurons additive bias terms.\n\"\"\";\n```\n\n### Coding Exercise 3.2: Add linear layer\n\nRemember in Tutorial 1 we used linear layers. Use your knowledge from Tutorial 1 to add a linear layer to the model we created above.\n\n\n```python\n# @markdown Execute to get `train` function for our neural encoding model\n\ndef train(net, custom_loss, train_data, train_labels,\n test_data=None, test_labels=None,\n learning_rate=10, n_iter=500, L2_penalty=0., L1_penalty=0.):\n \"\"\"Run gradient descent for network without batches\n\n Args:\n net (nn.Module): deep network whose parameters to optimize with SGD\n custom_loss: loss function for network\n train_data: training data (n_train x input features)\n train_labels: training labels (n_train x output features)\n test_data: test data (n_train x input features)\n test_labels: test labels (n_train x output features)\n learning_rate (float): learning rate for gradient descent\n n_epochs (int): number of epochs to run gradient descent\n L2_penalty (float): magnitude of L2 penalty\n L1_penalty (float): magnitude of L1 penalty\n\n Returns:\n train_loss: training loss across iterations\n test_loss: testing loss across iterations\n\n \"\"\"\n optimizer = optim.SGD(net.parameters(), lr=learning_rate, momentum=0.9) # Initialize PyTorch SGD optimizer\n train_loss = np.nan * np.zeros(n_iter) # Placeholder for train loss\n test_loss = np.nan * np.zeros(n_iter) # Placeholder for test loss\n\n # Loop over epochs\n for i in range(n_iter):\n if n_iter < 10:\n for param_group in self.optimizer.param_groups:\n param_group['lr'] = np.linspace(0, learning_rate, 10)[n_iter]\n y_pred = net(train_data) # Forward pass: compute predicted y by passing train_data to the model.\n\n if L2_penalty>0 or L1_penalty>0:\n weights = net.out_layer.weight\n loss = custom_loss(y_pred, train_labels, weights, L2_penalty, L1_penalty)\n else:\n loss = custom_loss(y_pred, train_labels)\n\n ### Update parameters\n optimizer.zero_grad() # zero out gradients\n loss.backward() # Backward pass: compute gradient of the loss with respect to model parameters\n optimizer.step() # step parameters in gradient direction\n train_loss[i] = loss.item() # .item() transforms the tensor to a scalar and does .detach() for us\n\n # Track progress\n if (i+1) % (n_iter // 10) == 0 or i==0:\n if test_data is not None and test_labels is not None:\n y_pred = net(test_data)\n if L2_penalty>0 or L1_penalty>0:\n loss = custom_loss(y_pred, test_labels, weights, L2_penalty, L1_penalty)\n else:\n loss = custom_loss(y_pred, test_labels)\n test_loss[i] = loss.item()\n print(f'iteration {i+1}/{n_iter} | train loss: {train_loss[i]:.4f} | test loss: {test_loss[i]:.4f}')\n else:\n print(f'iteration {i+1}/{n_iter} | train loss: {train_loss[i]:.4f}')\n\n return train_loss, test_loss\n```\n\n\n```python\nclass ConvFC(nn.Module):\n \"\"\"Deep network with one convolutional layer + one fully connected layer\n\n Attributes:\n conv (nn.Conv1d): convolutional layer\n dims (tuple): shape of convolutional layer output\n out_layer (nn.Linear): linear layer\n\n \"\"\"\n\n def __init__(self, n_neurons, c_in=1, c_out=6, K=7, b=12*16,\n filters=None):\n \"\"\" initialize layer\n Args:\n c_in: number of input stimulus channels\n c_out: number of convolutional channels\n K: size of each convolutional filter\n h: number of stimulus bins, n_bins\n \"\"\"\n super().__init__()\n self.conv = nn.Conv2d(c_in, c_out, kernel_size=K,\n padding=K//2, stride=1)\n self.dims = (c_out, b) # dimensions of conv layer output\n M = np.prod(self.dims) # number of hidden units\n\n ################################################################################\n ## TO DO for students: add fully connected layer to network (self.out_layer)\n # Fill out function and remove\n raise NotImplementedError(\"Student exercise: add fully connected layer to initialize network\")\n ################################################################################\n self.out_layer = nn.Linear(M, ...)\n\n # initialize weights\n if filters is not None:\n self.conv.weight = nn.Parameter(filters)\n self.conv.bias = nn.Parameter(torch.zeros((c_out,), dtype=torch.float32))\n\n nn.init.normal_(self.out_layer.weight, std=0.01) # initialize weights to be small\n\n def forward(self, s):\n \"\"\" Predict neural responses to stimuli s\n\n Args:\n s (torch.Tensor): n_stimuli x c_in x h x w tensor with stimuli\n\n Returns:\n y (torch.Tensor): n_stimuli x n_neurons tensor with predicted neural responses\n\n \"\"\"\n a = self.conv(s) # output of convolutional layer\n a = a.view(-1, np.prod(self.dims)) # flatten each convolutional layer output into a vector\n\n ################################################################################\n ## TO DO for students: add fully connected layer to forward pass of network (self.out_layer)\n # Fill out function and remove\n raise NotImplementedError(\"Student exercise: add fully connected layer to network\")\n ################################################################################\n y = ...\n\n return y\n\n\ndevice = torch.device('cpu')\n\n# (Optional) To speed up processing, go to \"Runtime\" menu and \"Change runtime\"\n# and select GPU processing, then uncomment line below, otherwise runtime will\n# be ~ 2 minutes\n# device = torch.device('cuda')\n\n# Initialize network\nn_neurons = resp_train.shape[1]\n## Initialize with filters from Tutorial 2\nexample_filters = filters(out_channels=6, K=7)\n\nnet = ConvFC(n_neurons, filters = example_filters)\nnet = net.to(device)\n\n# Run GD on training set data\n# ** this time we are also providing the test data to estimate the test loss\ntrain_loss, test_loss = train(net, regularized_MSE_loss,\n train_data=grating_stimuli.to(device), train_labels=resp_train.to(device),\n test_data=grating_stimuli.to(device), test_labels=resp_test.to(device),\n n_iter=200, learning_rate=2,\n L2_penalty=5e-4, L1_penalty=1e-6)\n\n# Visualize\nplot_training_curves(train_loss, test_loss)\n```\n\n\n```python\n# to_remove solution\nclass ConvFC(nn.Module):\n \"\"\"Deep network with one convolutional layer + one fully connected layer\n\n Attributes:\n conv (nn.Conv1d): convolutional layer\n dims (tuple): shape of convolutional layer output\n out_layer (nn.Linear): linear layer\n\n \"\"\"\n\n def __init__(self, n_neurons, c_in=1, c_out=6, K=7, b=12*16,\n filters=None):\n \"\"\" initialize layer\n Args:\n c_in: number of input stimulus channels\n c_out: number of convolutional channels\n K: size of each convolutional filter\n h: number of stimulus bins, n_bins\n \"\"\"\n super().__init__()\n self.conv = nn.Conv2d(c_in, c_out, kernel_size=K,\n padding=K//2, stride=1)\n self.dims = (c_out, b) # dimensions of conv layer output\n M = np.prod(self.dims) # number of hidden units\n self.out_layer = nn.Linear(M, n_neurons)\n\n # initialize weights\n if filters is not None:\n self.conv.weight = nn.Parameter(filters)\n self.conv.bias = nn.Parameter(torch.zeros((c_out,), dtype=torch.float32))\n\n nn.init.normal_(self.out_layer.weight, std=0.01) # initialize weights to be small\n\n def forward(self, s):\n \"\"\" Predict neural responses to stimuli s\n\n Args:\n s (torch.Tensor): n_stimuli x c_in x h x w tensor with stimuli\n\n Returns:\n y (torch.Tensor): n_stimuli x n_neurons tensor with predicted neural responses\n\n \"\"\"\n a = self.conv(s) # output of convolutional layer\n a = a.view(-1, np.prod(self.dims)) # flatten each convolutional layer output into a vector\n y = self.out_layer(a)\n return y\n\n\ndevice = torch.device('cpu')\n\n# (Optional) To speed up processing, go to \"Runtime\" menu and \"Change runtime\"\n# and select GPU processing, then uncomment line below, otherwise runtime will\n# be ~ 2 minutes\n# device = torch.device('cuda')\n\n# Initialize network\nn_neurons = resp_train.shape[1]\n## Initialize with filters from Tutorial 2\nexample_filters = filters(out_channels=6, K=7)\n\nnet = ConvFC(n_neurons, filters = example_filters)\nnet = net.to(device)\n\n# Run GD on training set data\n# ** this time we are also providing the test data to estimate the test loss\ntrain_loss, test_loss = train(net, regularized_MSE_loss,\n train_data=grating_stimuli.to(device), train_labels=resp_train.to(device),\n test_data=grating_stimuli.to(device), test_labels=resp_test.to(device),\n n_iter=200, learning_rate=2,\n L2_penalty=5e-4, L1_penalty=1e-6)\n\n# Visualize\nwith plt.xkcd():\n plot_training_curves(train_loss, test_loss)\n```\n\nHow well can we fit single neuron tuning curves with this model? What aspects of the tuning curves are we capturing?\n\n\n```python\n# @markdown Execute this cell to examine predictions for random subsets of neurons\n\ny_pred = net(grating_stimuli.to(device))\n# Visualize tuning curves & plot neural predictions\nfig, axs = plt.subplots(2, 5, figsize=(15,6))\nfor k, ax in enumerate(axs.flatten()):\n ineur = np.random.choice(n_neurons)\n plot_prediction(ax, y_pred[:,ineur].detach().cpu(),\n resp_train[:,ineur],\n resp_test[:,ineur])\n if k==0:\n ax.text(.1, 1., 'train', color='y', transform=ax.transAxes)\n ax.text(.1, 0.9, 'test', color='m', transform=ax.transAxes)\n ax.text(.1, 0.8, 'prediction', color='g', transform=ax.transAxes)\n\nfig.tight_layout()\nplt.show()\n```\n\nWe can see if the convolutional channels changed at all from their initialization as center-surround and Gabor filters. If they don't then it means that they were a sufficient basis set to explain the responses of the neurons to orientations to the accuracy seen above.\n\n\n```python\n# get weights of conv layer in convLayer\nout_channels = 6 # how many convolutional channels to have in our layer\nweights = net.conv.weight.detach().cpu()\nprint(weights.shape) # can you identify what each of the dimensions are?\n\nplot_weights(weights, channels = np.arange(0, out_channels))\n```\n", "meta": {"hexsha": "e0f36b86483322a771f741a9c87fe7ee12eb4507", "size": 97321, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W2D1_DeepLearning/W2D1_Tutorial4.ipynb", "max_stars_repo_name": "Beilinson/course-content", "max_stars_repo_head_hexsha": "b74c630bec7002abe2f827ff8e0707f9bbb43f82", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tutorials/W2D1_DeepLearning/W2D1_Tutorial4.ipynb", "max_issues_repo_name": "Beilinson/course-content", "max_issues_repo_head_hexsha": "b74c630bec7002abe2f827ff8e0707f9bbb43f82", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/W2D1_DeepLearning/W2D1_Tutorial4.ipynb", "max_forks_repo_name": "Beilinson/course-content", "max_forks_repo_head_hexsha": "b74c630bec7002abe2f827ff8e0707f9bbb43f82", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 41.4484667802, "max_line_length": 735, "alphanum_fraction": 0.5852899169, "converted": true, "num_tokens": 18102, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.449255793491485}} {"text": "```python\n%run ../Python_files/util.py\n%run ../Python_files/load_dicts.py\n```\n\n No dicts found; please check load_dicts...\n\n\n\n```python\ntmc_capac_dict_AM['129-04138'] * 2.5, \\\ntmc_capac_dict_MD['129-04138'] * 4.75, \\\ntmc_capac_dict_PM['129-04138'] * 2.5, \\\ntmc_capac_dict_NT['129-04138'] * 7\n```\n\n\n\n\n (17500.0, 28500.0, 15000.0, 42000.0)\n\n\n\n## raw data from CTPS capacity dataset\n\nID\tLENGTH\tDIR\tANODE\tBNODE\tCTPS_FUNCT\tSTREETNAME\tROUTENUMBE\tSCEN_00_LA\tSCEN_00_AB\tSCEN_00_BA\tSCEN_00_A1\tSCEN_00_B1\tSCEN_00_A2\tSCEN_00_B2\tSCEN_00_A3\tSCEN_00_B3\tAB_AMCAPAC\tBA_AMCAPAC\tAB_MDCAPAC\tBA_MDCAPAC\tAB_PMCAPAC\tBA_PMCAPAC\tAB_NTCAPAC\tBA_NTCAPAC\tROADINVENT\n\n56573\t0.260468\t1\t73398\t73394\t1\tINTERSTATE 93\t93\t2000\t3.5\t1\t3\t1\t3\t1\t3\t1\t17500\t5000\t28500\t9500\t15000\t5000\t42000\t14000\t1132000\n\n\n```python\nfiltered_tmc_speed_dict = zload('../temp_files/tmc_speed_dict_for_anomaly_detection.pkz')\n```\n\n\n```python\ntmc = '129-04138'\nmonth = 1\nday = 12\n\nfor hour in range(24):\n for minute in range(60):\n key = str(tmc) + '_' + str(month) + '_' + str(day) + '_' + str(hour) + '_' + str(minute)\n \n# print(hour, minute, float(filtered_tmc_speed_dict[key].split('_')[0]))\n```\n\n\n```python\ntmc_ref_speed_dict[tmc]\n```\n\n\n\n\n 73.0\n\n\n\n\n```python\nfrom sympy import Symbol, nsolve\nimport sympy\nx = Symbol('x')\nnsolve(73*x - x**2 - 315, 0)\n```\n\n\n\n\n mpf('4.6056431323658762')\n\n\n\n## Simulating a car accident\nThe road segment has three lanes, and an accident happened in one of the lanes, causing a sudden slow-down of the traffic. The instant flow would not change, while the capacity ($m$) would be reduced to be two thirds of the original value. Thus, using Greenshield's model, we have\n$$4\\left( {\\frac{2}{3}m} \\right)\\left[ {\\frac{{{v_2}}}{{{v_0}}} - {{\\left( {\\frac{{{v_2}}}{{{v_0}}}} \\right)}^2}} \\right] = 4m\\left[ {\\frac{{{v_1}}}{{{v_0}}} - {{\\left( {\\frac{{{v_1}}}{{{v_0}}}} \\right)}^2}} \\right], \\quad (1)$$\nwhere $v_1 = 70$, and $v_0 = 73$.\n\nSolving (1), we obtain $v_2 = 4.6056431323658762$. Note that we only care about the road with heavy congestion in this case.\n\nAssume that the accident happened during 17:15 - 17:45, Jan. 12, 2012.\n\n\n```python\ntmc = '129-04138'\nmonth = 1\nday = 12\n\ntraffic_data_with_anomaly = {}\n\nfor hour in range(24):\n for minute in range(60):\n key = str(tmc) + '_' + str(month) + '_' + str(day) + '_' + str(hour) + '_' + str(minute)\n traffic_data_with_anomaly[key] = float(filtered_tmc_speed_dict[key].split('_')[0])\n if hour == 17 and minute > 15 and minute < 46:\n traffic_data_with_anomaly[key] = 4.6056431323658762\n# print(hour, minute, traffic_data_with_anomaly[key])\n\nzdump(traffic_data_with_anomaly, '../temp_files/traffic_data_with_anomaly.pkz')\n```\n\n\n```python\n# extract reference traffic data, for the purpose of estimating PLs\n\ntmc = '129-04138'\nmonth = 1\nday_list = [2, 3, 4, 5, 6, 9, 10, 11]\n\ntraffic_data_ref = {}\n\nfor hour in range(24):\n for minute in range(60):\n for day in day_list:\n key = str(tmc) + '_' + str(month) + '_' + str(day) + '_' + str(hour) + '_' + str(minute)\n traffic_data_ref[key] = float(filtered_tmc_speed_dict[key].split('_')[0])\n# print(hour, minute, float(filtered_tmc_speed_dict[key].split('_')[0]))\n\nzdump(traffic_data_ref, '../temp_files/traffic_data_ref.pkz')\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "3d203b21d5446bed44ea99d488c576515a1926ab", "size": 6233, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "07_anomaly_detection/extract_data_2_simu.ipynb", "max_stars_repo_name": "jingzbu/InverseVITraffic", "max_stars_repo_head_hexsha": "c0d33d91bdd3c014147d58866c1a2b99fb8a9608", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "07_anomaly_detection/extract_data_2_simu.ipynb", "max_issues_repo_name": "jingzbu/InverseVITraffic", "max_issues_repo_head_hexsha": "c0d33d91bdd3c014147d58866c1a2b99fb8a9608", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "07_anomaly_detection/extract_data_2_simu.ipynb", "max_forks_repo_name": "jingzbu/InverseVITraffic", "max_forks_repo_head_hexsha": "c0d33d91bdd3c014147d58866c1a2b99fb8a9608", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 26.7510729614, "max_line_length": 290, "alphanum_fraction": 0.53730146, "converted": true, "num_tokens": 1151, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.7217432062975979, "lm_q1q2_score": 0.449255793491485}} {"text": "# Cooling Crust Demonstration, INT Workshop 16-2b\nEdward Brown\n20 June 2016\n\n\n```python\n%matplotlib inline\nfrom __future__ import print_function, division\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom matplotlib import rc\nimport scipy.constants as sc\n\n# class to read in dStar output files\nfrom reader import dStarCrustReader, dStarReader\n\n# plot format\ncharsize=14\nmajor_ticklength=0.6*charsize\nmajor_tickwidth=0.9\nminor_ticklength=0.3*charsize\nminor_tickwidth=0.7\nrc('mathtext',**{'fontset':'stixsans'})\nrc('font',**{'size':charsize,'sans-serif':'Bitstream Vera Sans'})\nrc('axes',**{'titlesize':charsize,'labelsize':charsize})\nrc('xtick',**{'major.size':major_ticklength,\n 'major.width':major_tickwidth,'minor.size':minor_ticklength,\n 'minor.width':minor_tickwidth,'labelsize':charsize})\nrc('ytick',**{'major.size':major_ticklength,'major.width':major_tickwidth,\n 'minor.size':minor_ticklength,'minor.width':minor_tickwidth,'labelsize':charsize})\n```\n\n## Auxilliary Functions\nFirst, we define some functions: a routine to compute limits for a logarithmic axis; a routine to compute the thermal time\n\\begin{equation}\n\\newcommand{\\Teff}{T_{\\mathrm{eff}}^\\infty}\n\\tau = \\frac{1}{4}\\left[ \\int \\left(\\frac{\\rho C}{K}\\right)^{1/2}\\mathrm{d}r\\right]^2;\n\\end{equation}\nand a routine to compute the radius from the area.\n\n\n```python\ndef thermal_time(c):\n '''\n Returns an array tau containing the thermal diffusion time to the surface to the location \n of each crust zone.\n \n c: of type dStarCrustReader\n '''\n sc.day\n P = c.crust.pressure[0,:]\n rho = c.crust.density[0,:]\n g = c.crust.gravity[0,:]\n K = c.crust.K[0,:]\n Cp = c.crust.cp[0,:]\n kernel = np.sqrt(rho*Cp/K)/rho/g\n dP = np.zeros_like(kernel)\n dP[1:] = P[1:]-P[0:-1]\n dP[0] = P[0]\n tau = np.zeros_like(kernel)\n tau[0] = 0.25*(kernel[0]*dP[0])**2 / sc.day\n for i in range(1,len(tau)):\n tau[i] = 0.25*(np.dot(kernel[0:i],dP[0:i]))**2 / sc.day\n return tau\n\ndef radius(area):\n return np.sqrt(area/4.0/np.pi)*1.0e-5\n```\n\nNext we define functions to make common plots: the surface effective temperature $\\Teff$ during cooling and theh temperature in the crust during outburst.\n\n\n```python\nMeVfm3 = 10.0*sc.eV*1.0e6/sc.fermi**3\n\ndef loglimits(x,delta=0.05):\n '''\n Returns two limiting values that enclose the array x with a padding \n of delta (in log-space).\n \n x: array-like\n '''\n lx = np.log10(x)\n lmin,lmax = lx.min(),lx.max()\n d = delta*(lmax-lmin)\n return 10.0**(lmin-d),10.0**(lmax+d)\n\ndef grayscale(n,light_to_dark=True,scale=0.9):\n '''\n returns an array of grayscale values (length `n`) ranging from\n `scale` to 0; if `light_to_dark=False`, then the order is reversed.\n '''\n if light_to_dark:\n return [str(scale*(n-i-1)/(n-1)) for i in range(n)]\n else:\n return [str(scale*i/(n-1)) for i in range(n)]\n \ndef cooling_plot(h,ePhi):\n # store cooling times\n t = h.data.time\n quiescent = np.where(t > 0.0)\n t = t[quiescent]\n Teff = 10.0**h.data.lg_Teff[quiescent]*sc.k/sc.eV*ePhi\n\n plt.xlabel(r'$t$ [d]')\n plt.ylabel(r'$T_{\\mathrm{eff}}^\\infty$ [K]')\n plt.xlim(loglimits(t))\n plt.semilogx(t,Teff,color='k')\n plt.tight_layout()\n\ndef outburst_plot(c):\n tout = np.where(c.times <= 0.0)[0]\n clrs = grayscale(len(tout))\n T = c.crust.temperature[tout,:]*1.0e-9\n plt.xlim(loglimits(P))\n plt.xlabel(r'$P\\;[\\mathrm{MeV\\,fm^{-3}}]$')\n plt.ylabel(r'$T$ [GK]')\n for i in range(T.shape[0]):\n plt.semilogx(P,T[i,:],color=clrs[i])\n plt.tight_layout()\n```\n\n## An impure crust\nAs a first case, we set the impurity parameter $Q=100$. Of course, for such a large $Q$ the meaning of impurity breaks down; it is closer to the mark to say that the lattice is very disordered (amporphous?) in this case. As a result, the thermal conductivity is low and the thermal diffusion is slow.\n\n\n```python\nc = dStarCrustReader('LOGS-Q100-H0',dt=10)\ncrust = c.crust\nR = radius(crust.area[0,:])\nP = crust.pressure[0,:]\ndrip = np.where(crust.Xn[0,:] > 0.0)\nPnuc = P/MeVfm3\nePhi = crust.ePhi[0,0]\n```\n\nBefore analyzing the cooling, we plot the radius as a function of pressure and highlight the inner crust.\n\n\n```python\nxmin,xmax = Pnuc.min(),Pnuc.max()\ndx = 0.05*(np.log10(xmax/xmin))\nxmin,xmax = 10.0**(np.log10(xmin)-dx),10.0**(np.log10(xmax)+dx)\nymin,ymax = 11.0,11.85\nplt.xlim(xmin,xmax)\nplt.ylim(ymin,ymax)\nplt.semilogx(Pnuc,R,color='k',label='outer crust')\nplt.semilogx(Pnuc[drip],R[drip],linewidth=4,color='0.4',solid_capstyle='round',\n label='inner crust')\nplt.xlabel(r'$P\\;[\\mathrm{MeV\\,fm^{-3}}]$')\nplt.ylabel(r'$R = \\sqrt{\\mathcal{A}/(4\\pi)}$ [km]')\nplt.legend(loc='lower left',frameon=False,fontsize='small')\nplt.tight_layout()\nplt.savefig('radius.pdf',format='pdf',facecolor='none',edgecolor='none')\n```\n\nNow we read in the history file and plot the observed surface effective temperature $\\Teff$.\n\n\n```python\nh = dStarReader('LOGS-Q100-H0/history.data')\ncooling_plot(h,ePhi)\nplt.savefig('cooling-Q100-H0.pdf',format='pdf',facecolor='none',edgecolor='none')\n```\n\nThe cooling timescale is indeed long, $> 1000\\,\\mathrm{d}$. Let us check the thermal time.\n\n\n```python\ntau = thermal_time(c)\nP = c.crust.pressure[0,:]/MeVfm3\nplt.xlim(loglimits(P))\nplt.xlabel(r'$P\\;[\\mathrm{MeV\\,fm^{-3}}]$')\nplt.ylabel(r'$\\tau$ [d]')\nplt.loglog(P,tau,color='k')\nplt.tight_layout()\nplt.savefig('tau-Q100-H0.pdf',format='pdf',facecolor='none',edgecolor='none')\n```\n\nFinally, let us plot the temperature evolution through the outburst.\n\n\n```python\noutburst_plot(c)\nplt.savefig('profile-Q100-H0.pdf',format='pdf',facecolor='none',edgecolor='none')\ntout = np.where(c.times <= 0)[0][0]\nT0 = c.crust.temperature[tout,:]*1.0e-9\nP0 = P\n```\n\n## A more pure crust\nNow let us set $Q = 4$ and look at the cooling surface temperature.\n\n\n```python\nh4 = dStarReader('LOGS-Q4-H0/history.data')\ncooling_plot(h4,ePhi)\nplt.savefig('cooling-Q4-H0.pdf',format='pdf',facecolor='none',edgecolor='none')\n```\n\nLet's compare the thermal diffusion time with that for the high-impurity crust.\n\n\n```python\nc4 = dStarCrustReader('LOGS-Q4-H0')\nP = c4.crust.pressure[0,:]/MeVfm3\nplt.xlim(loglimits(P))\nplt.xlabel(r'$P\\;[\\mathrm{MeV\\,fm^{-3}}]$')\nplt.ylabel(r'$\\tau$ [d]')\nplt.loglog(P,thermal_time(c4),color='k')\nplt.loglog(P,thermal_time(c),linestyle=':',color='k')\nplt.tight_layout()\nplt.savefig('tau-Q4-H0.pdf',format='pdf',facecolor='none',edgecolor='none')\n```\n\nAs expected from the evolution of the surface temperature, the thermal diffusion time is an order of magnitude less in this case. Because the thermal diffusion time is now less than the duration of the outburst, the crust has time to thermally relax; compare the profile of temperature in crust with that for the high-impurity case.\n\n\n```python\noutburst_plot(c4)\nplt.savefig('profile-Q4-H0.pdf',format='pdf',facecolor='none',edgecolor='none')\n```\n\nThis explains why $\\Teff$ is so constant during the first 100 days: the outer layers of the star are already thermally relaxed, so the flux is unchanging until the inner crust has time to evolve. To see this, let's plot the temperature as a function of $\\tau$.\n\n\n```python\ntout = np.where(c4.times == 0.0)[0][0]\nindcs = [tout,tout+1,tout+3,tout+6,tout+8]\nTs = c4.crust.temperature[indcs,:]*1.0e-9\nclrs = grayscale(len(indcs))\ntimes = c4.times[indcs]\nplt.xlim(loglimits(tau,delta=0.2))\nplt.xlabel(r'$\\tau$ [d]')\nplt.ylabel(r'$T$ [GK]')\nfor i in range(len(indcs)):\n plt.semilogx(tau,Ts[i,:],color=clrs[i])\n if i > len(indcs)-3:\n tstr = r'$t = {0:0.0f}$ d'.format(times[i])\n plt.annotate(s=tstr,xy=(tau[0],Ts[i,0]),verticalalignment='center',\n horizontalalignment='right',color='k',fontsize='small')\nplt.tight_layout()\nplt.savefig('Ttau-Q4-H0.pdf',format='pdf',facecolor='none',edgecolor='none')\n```\n\nTo get $\\Teff$ to cool at early times, we need the temperature to decrease with depth in the outer crust. We achieve this by adding a heat source, of strength $L = 1.7\\,\\mathrm{MeV}\\times \\mathrm{d}\\dot{M}/\\mathrm{d}t$.\n\n\n```python\nh17 = dStarReader('LOGS-Q4-H1.7/history.data')\ncooling_plot(h17,ePhi)\nplt.savefig('cooling-Q4-H1.7.pdf',format='pdf',facecolor='none',edgecolor='none')\n```\n\n\n```python\nc17 = dStarCrustReader('LOGS-Q4-H1.7')\noutburst_plot(c17)\nplt.savefig('profile-Q4-H1.7.pdf',format='pdf',facecolor='none',edgecolor='none')\n```\n\nAt a time $t$ after the end of the outburst, the crust has cooled down to a depth where $\\tau \\approx t$. The evolution of the surface temperature $\\Teff$ is therefore mapping out the crust temperature as a function of depth.\n\n\n```python\ntout = np.where(c17.times == 0.0)[0][0]\nindcs = [tout,tout+1,tout+3,tout+6,tout+8,tout+12,tout+16]\nTs = c17.crust.temperature[indcs,:]*1.0e-9\nclrs = grayscale(len(indcs))\ntimes = c17.times[indcs]\nplt.xlim(loglimits(tau))\nplt.xlabel(r'$\\tau$ [d]')\nplt.ylabel(r'$T$ [GK]')\nfor i in range(len(indcs)):\n plt.semilogx(tau,Ts[i,:],color=clrs[i])\n if i > len(indcs)-5:\n tstr = r'$t = {0:0.0f}$ d'.format(times[i])\n plt.annotate(s=tstr,xy=(1.0,Ts[i,0]),verticalalignment='bottom',\n color='k',fontsize='small')\nplt.tight_layout()\nplt.savefig('Ttau-Q4-H1.7.pdf',format='pdf',facecolor='none',edgecolor='none')\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "8530c840544fbb786d76565ac8ed8d8f7ce2b8f9", "size": 14749, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/INT-16-2b-demo/INT2016-crusts.ipynb", "max_stars_repo_name": "LBJ-Wade/dStar", "max_stars_repo_head_hexsha": "d63b1a611937e8eccfb3c98ebbbdb4cd0c2fd19b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 14, "max_stars_repo_stars_event_min_datetime": "2015-05-19T18:15:17.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-19T08:24:00.000Z", "max_issues_repo_path": "examples/INT-16-2b-demo/INT2016-crusts.ipynb", "max_issues_repo_name": "LBJ-Wade/dStar", "max_issues_repo_head_hexsha": "d63b1a611937e8eccfb3c98ebbbdb4cd0c2fd19b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 8, "max_issues_repo_issues_event_min_datetime": "2015-04-02T20:54:24.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-09T17:12:07.000Z", "max_forks_repo_path": "examples/INT-16-2b-demo/INT2016-crusts.ipynb", "max_forks_repo_name": "LBJ-Wade/dStar", "max_forks_repo_head_hexsha": "d63b1a611937e8eccfb3c98ebbbdb4cd0c2fd19b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 6, "max_forks_repo_forks_event_min_datetime": "2015-04-02T19:01:34.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-06T12:22:52.000Z", "avg_line_length": 30.7912317328, "max_line_length": 339, "alphanum_fraction": 0.5514272154, "converted": true, "num_tokens": 2954, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494678483918, "lm_q2_score": 0.6548947290421275, "lm_q1q2_score": 0.44922469088316413}} {"text": "```python\n%matplotlib inline\n```\n\n# The heat equation in 2D\n\n$$\n\\renewcommand{\\DdQq}[2]{{\\mathrm D}_{#1}{\\mathrm Q}_{#2}}\n\\renewcommand{\\drondt}{\\partial_t}\n\\renewcommand{\\drondx}{\\partial_x}\n\\renewcommand{\\drondtt}{\\partial_{tt}}\n\\renewcommand{\\drondxx}{\\partial_{xx}}\n\\renewcommand{\\drondyy}{\\partial_{yy}}\n\\renewcommand{\\dx}{\\Delta x}\n\\renewcommand{\\dt}{\\Delta t}\n\\renewcommand{\\grandO}{{\\mathcal O}}\n\\renewcommand{\\density}[2]{\\,f_{#1}^{#2}}\n\\renewcommand{\\fk}[1]{\\density{#1}{\\vphantom{\\star}}}\n\\renewcommand{\\fks}[1]{\\density{#1}{\\star}}\n\\renewcommand{\\moment}[2]{\\,m_{#1}^{#2}}\n\\renewcommand{\\mk}[1]{\\moment{#1}{\\vphantom{\\star}}}\n\\renewcommand{\\mke}[1]{\\moment{#1}{e}}\n\\renewcommand{\\mks}[1]{\\moment{#1}{\\star}}\n$$\n\nIn this tutorial, we test a very classical lattice Boltzmann scheme $\\DdQq{2}{5}$ on the heat equation.\n\nThe problem reads\n$$\n\\begin{gathered} \\drondt u = \\mu (\\drondxx+\\drondyy) u, \\quad t>0, \\quad (x, y)\\in(0,1)^2,\\\\ u(0) = u(1) = 0, \\end{gathered}\n$$\n\nwhere $\\mu$ is a constant scalar.\n\n## The scheme $\\DdQq{2}{5}$\n\nThe numerical simulation of this equation by a lattice Boltzmann scheme consists in the approximatation of the solution on discret points of $(0,1)^2$ at discret instants.\n\nTo simulate this system of equations, we use the $\\DdQq{2}{5}$ scheme given by\n\n* five velocities $v_0=(0,0)$, $v_1=(1,0)$, $v_2=(0,1)$, $v_3=(-1,0)$, and $v_4=(0,-1)$ with associated distribution functions $\\fk{i}$, $0\\leq i\\leq 4$,\n* a space step $\\dx$ and a time step $\\dt$, the ration $\\lambda=\\dx/\\dt$ is called the scheme velocity,\n* five moments\n $$ \\mk{0}=\\sum_{i=0}^{4} \\fk{i}, \\quad \\mk{1}= \\sum_{i=0}^{4} v_{ix} \\fk{i}, \\quad \\mk{2}= \\sum_{i=0}^{4} v_{iy} \\fk{i}, \\quad \\mk{3}= \\frac{1}{2} \\sum_{i=0}^{5} (v_{ix}^2+v_{iy}^2) \\fk{i}, \\quad \\mk{4}= \\frac{1}{2} \\sum_{i=0}^{5} (v_{ix}^2-v_{iy}^2) \\fk{i},$$\n \n and their equilibrium values $\\mke{k}$, $0\\leq k\\leq 4$.\n* two relaxation parameters $s_1$ and $s_2$ lying in $[0,2]$ ($s_1$ for the odd moments and $s_2$ for the odd ones).\n\nIn order to use the formalism of the package pylbm, we introduce the five polynomials that define the moments: $P_0 = 1$, $P_1=X$, $P_2=Y$, $P_3=(X^2+Y^2)/2$, and $P_4=(X^2-Y^2)/2$, such that\n$$ \n\\mk{k} = \\sum_{i=0}^4 P_k(v_{ix}, v_{iy}) \\fk{i}.\n$$\n\nThe transformation $(\\fk{0}, \\fk{1}, \\fk{2}, \\fk{3}, \\fk{4})\\mapsto(\\mk{0},\\mk{1}, \\mk{2}, \\mk{3}, \\mk{4})$ is invertible if, and only if, the polynomials $(P_0,P_1,P_2,P_3,P_4)$ is a free set over the stencil of velocities.\n\nThe lattice Boltzmann method consists to compute the distribution functions $\\fk{i}$, $0\\leq i\\leq 4$ in each point of the lattice $x$ and at each time $t^n=n\\dt$.\nA step of the scheme can be read as a splitting between the relaxation phase and the transport phase:\n\n* relaxation: \n$$\n \\begin{aligned}\\mks{1}(t,x,y)&=(1-s_1)\\mk{1}(t,x,y)+s_1\\mke{1}(t,x,y),\\\\ \\mks{2}(t,x,y)&=(1-s_1)\\mk{2}(t,x,y)+s_1\\mke{2}(t,x,y),\\\\ \\mks{3}(t,x,y)&=(1-s_2)\\mk{3}(t,x,y)+s_2\\mke{3}(t,x,y),\\\\ \\mks{4}(t,x,y)&=(1-s_2)\\mk{4}(t,x,y)+s_2\\mke{4}(t,x,y).\\end{aligned}\n$$\n\n* m2f:\n$$\n \\begin{aligned}\\fks{0}(t,x,y)&\\;=\\mk{0}(t,x,y)-2\\mks{3}(t,x,y), \\\\ \\fks{1}(t,x,y)&\\;=\\tfrac{1}{2}(\\phantom{-}\\mks{1}(t,x,y)+\\mks{3}(t,x,y)+\\mks{4}(t,x,y)), \\\\ \\fks{2}(t,x,y)&\\;=\\tfrac{1}{2}(\\phantom{-}\\mks{2}(t,x,y)+\\mks{3}(t,x,y)-\\mks{4}(t,x,y)), \\\\ \\fks{3}(t,x,y)&\\;=\\tfrac{1}{2}(-\\mks{1}(t,x,y)+\\mks{3}(t,x,y)+\\mks{4}(t,x,y)), \\\\ \\fks{4}(t,x,y)&\\;=\\tfrac{1}{2}(-\\mks{2}(t,x,y)+\\mks{3}(t,x,y)-\\mks{4}(t,x,y)).\\end{aligned}\n$$\n\n* transport: \n$$\n \\begin{aligned} \\fk{0}(t+\\dt, x,y)&\\;=\\fks{0}(t,x,y), \\\\ \\fk{1}(t+\\dt, x,y)&\\;=\\fks{1}(t,x-\\dx,y), \\\\ \\fk{2}(t+\\dt, x,y)&\\;=\\fks{2}(t,x,y-\\dx), \\\\ \\fk{3}(t+\\dt, x,y)&\\;=\\fks{3}(t,x+\\dx,y), \\\\ \\fk{4}(t+\\dt, x,y)&\\;=\\fks{4}(t,x,y+\\dx). \\end{aligned}\n$$\n\n* f2m:\n$$\n \\begin{aligned}\\mk{0}(t+\\dt,x,y)&\\;=\\fk{0}(t+\\dt,x,y)+\\fk{1}(t+\\dt,x,y)+\\fk{2}(t+\\dt,x,y)\\\\&\\;\\phantom{=}+\\fk{3}(t+\\dt,x,y)+\\fk{4}(t+\\dt,x,y), \\\\ \\mk{1}(t+\\dt,x,y)&\\;=\\fk{1}(t+\\dt,x,y)-\\fk{3}(t+\\dt,x,y), \\\\ \\mk{2}(t+\\dt,x,y)&\\;=\\fk{2}(t+\\dt,x,y)-\\fk{4}(t+\\dt,x,y), \\\\ \\mk{3}(t+\\dt,x,y)&\\;=\\tfrac{1}{2}(\\fk{1}(t+\\dt,x,y)+\\fk{2}(t+\\dt,x,y)+\\fk{3}(t+\\dt,x,y)+\\fk{4}(t+\\dt,x,y)), \\\\ \\mk{4}(t+\\dt,x,y)&\\;=\\tfrac{1}{2}(\\fk{1}(t+\\dt,x,y)-\\fk{2}(t+\\dt,x,y)+\\fk{3}(t+\\dt,x,y)-\\fk{4}(t+\\dt,x,y)).\\end{aligned}\n$$\n\nThe moment of order $0$, $\\mk{0}$, being conserved during the relaxation phase, \na diffusive scaling $\\dt=\\dx^2$, yields to the following equivalent equation\n$$\n\\drondt\\mk{0} = \\bigl(\\tfrac{1}{s_1}-\\tfrac{1}{2}\\bigr) \\bigl(\\drondxx(\\mke{3}+\\mke{4})+\\drondyy(\\mke{3}-\\mke{4})\\bigr) + \\grandO(\\dx^2),\n$$\n\nif $\\mke{1}=0$.\nIn order to be consistent with the heat equation, the following choice is done:\n$$\n\\mke{3}=\\tfrac{1}{2}u, \\qquad \\mke{4}=0, \\qquad s_1 = \\frac{2}{1+4\\mu}, \\qquad s_2=1.\n$$\n\n\n## Using pylbm\n\npylbm uses Python dictionary to describe the simulation. In the following, we will build this dictionary step by step.\n\n### The geometry\n\nIn pylbm, the geometry is defined by a box and a label for the boundaries. We define here a square $(0, 1)^2$.\n\n\n```python\nimport pylbm\nimport numpy as np\nimport pylab as plt\nxmin, xmax, ymin, ymax = 0., 1., 0., 1.\ndico_geom = {\n 'box': {'x': [xmin, xmax], \n 'y': [ymin, ymax], \n 'label': 0\n },\n}\ngeom = pylbm.Geometry(dico_geom)\nprint(geom)\ngeom.visualize(viewlabel=True);\n```\n\n### The stencil\n\npylbm provides a class stencil that is used to define the discret velocities of the scheme. In this example, the stencil is composed by the velocities $v_0=(0,0)$, $v_1=(1,0)$, $v_2=(-1,0)$, $v_3=(0,1)$, and $v_4=(0,-1)$ numbered by $[0,1,2,3,4]$.\n\n\n```python\ndico_sten = {\n 'dim':2,\n 'schemes': [\n {'velocities': list(range(5))}\n ],\n}\nsten = pylbm.Stencil(dico_sten)\nprint(sten)\nsten.visualize();\n```\n\n###\u00a0The domain\n\nIn order to build the domain of the simulation, the dictionary should contain the space step $\\dx$ and the stencils of the velocities (one for each scheme). \n\nWe construct a domain with $N=10$ points in space. \n\n\n```python\nN = 10\ndx = (xmax-xmin)/N\ndico_dom = {\n 'box': {'x': [xmin, xmax], \n 'y': [ymin, ymax], \n 'label': 0\n },\n 'space_step': dx,\n 'schemes': [\n {'velocities': list(range(5)),}\n ],\n}\ndom = pylbm.Domain(dico_dom)\nprint(dom)\ndom.visualize(view_distance=True);\n```\n\n### The scheme\n\nIn pylbm, a simulation can be performed by using several coupled schemes. In this example, a single scheme is used and defined through a list of one single dictionary. This dictionary should contain:\n\n* 'velocities': a list of the velocities\n* 'conserved_moments': a list of the conserved moments as sympy variables\n* 'polynomials': a list of the polynomials that define the moments\n* 'equilibrium': a list of the equilibrium value of all the moments\n* 'relaxation_parameters': a list of the relaxation parameters ($0$ for the conserved moments)\n* 'init': a dictionary to initialize the conserved moments\n\n(see the documentation for more details)\n\nThe scheme velocity could be taken to $1/\\dx$ and the inital value of $u$ to \n\n$$ u(t=0,x) = \\sin(\\pi x)\\sin(\\pi y).$$\n\n\n```python\nimport sympy as sp\n\ndef solution(x, y, t):\n return np.sin(np.pi*x)*np.sin(np.pi*y)*np.exp(-2*np.pi**2*mu*t)\n\n# parameters\nmu = 1.\nla = 1./dx\ns1 = 2./(1+4*mu)\ns2 = 1.\nu, X, Y, LA = sp.symbols('u, X, Y, LA')\n\ndico_sch = {\n 'dim': 2,\n 'scheme_velocity': la,\n 'schemes': [\n {\n 'velocities': list(range(5)),\n 'conserved_moments': u,\n 'polynomials': [1, X/LA, Y/LA, (X**2+Y**2)/(2*LA**2), (X**2-Y**2)/(2*LA**2)],\n 'equilibrium': [u, 0., 0., .5*u, 0.],\n 'relaxation_parameters': [0., s1, s1, s2, s2],\n }\n ],\n 'parameters': {LA: la},\n}\n\nsch = pylbm.Scheme(dico_sch)\nprint(sch)\n```\n\n +--------------------+\n | Scheme information |\n +--------------------+\n - spatial dimension: 2\n - number of schemes: 1\n - number of velocities: 5\n - conserved moments: [u]\n \n +----------+\n | Scheme 0 |\n +----------+\n - velocities\n (0: 0, 0)\n (1: 1, 0)\n (2: 0, 1)\n (3: -1, 0)\n (4: 0, -1)\n \n - polynomials\n \n \u23a1 1 \u23a4\n \u23a2 \u23a5\n \u23a2 X \u23a5\n \u23a2 \u2500\u2500 \u23a5\n \u23a2 LA \u23a5\n \u23a2 \u23a5\n \u23a2 Y \u23a5\n \u23a2 \u2500\u2500 \u23a5\n \u23a2 LA \u23a5\n \u23a2 \u23a5\n \u23a2 2 2\u23a5\n \u23a2X + Y \u23a5\n \u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n \u23a2 2 \u23a5\n \u23a2 2\u22c5LA \u23a5\n \u23a2 \u23a5\n \u23a2 2 2\u23a5\n \u23a2X - Y \u23a5\n \u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n \u23a2 2 \u23a5\n \u23a3 2\u22c5LA \u23a6\n \n - equilibria\n \n \u23a1 u \u23a4\n \u23a2 \u23a5\n \u23a2 0.0 \u23a5\n \u23a2 \u23a5\n \u23a2 0.0 \u23a5\n \u23a2 \u23a5\n \u23a20.5\u22c5u\u23a5\n \u23a2 \u23a5\n \u23a3 0.0 \u23a6\n \n - relaxation parameters\n \n \u23a10.0\u23a4\n \u23a2 \u23a5\n \u23a20.4\u23a5\n \u23a2 \u23a5\n \u23a20.4\u23a5\n \u23a2 \u23a5\n \u23a21.0\u23a5\n \u23a2 \u23a5\n \u23a31.0\u23a6\n \n - moments matrices\n \n \u23a11 1 1 1 1 \u23a4\n \u23a2 \u23a5\n \u23a2 10 -10 \u23a5\n \u23a20 \u2500\u2500 0 \u2500\u2500\u2500\u2500 0 \u23a5\n \u23a2 LA LA \u23a5\n \u23a2 \u23a5\n \u23a2 10 -10 \u23a5\n \u23a20 0 \u2500\u2500 0 \u2500\u2500\u2500\u2500\u23a5\n \u23a2 LA LA \u23a5\n \u23a2 \u23a5\n \u23a2 50 50 50 50 \u23a5\n \u23a20 \u2500\u2500\u2500 \u2500\u2500\u2500 \u2500\u2500\u2500 \u2500\u2500\u2500 \u23a5\n \u23a2 2 2 2 2 \u23a5\n \u23a2 LA LA LA LA \u23a5\n \u23a2 \u23a5\n \u23a2 50 -50 50 -50 \u23a5\n \u23a20 \u2500\u2500\u2500 \u2500\u2500\u2500\u2500 \u2500\u2500\u2500 \u2500\u2500\u2500\u2500\u23a5\n \u23a2 2 2 2 2 \u23a5\n \u23a3 LA LA LA LA \u23a6\n \n - inverse of moments matrices\n \n \u23a1 2 \u23a4\n \u23a2 -LA \u23a5\n \u23a21 0 0 \u2500\u2500\u2500\u2500\u2500 0 \u23a5\n \u23a2 50 \u23a5\n \u23a2 \u23a5\n \u23a2 2 2 \u23a5\n \u23a2 LA LA LA \u23a5\n \u23a20 \u2500\u2500 0 \u2500\u2500\u2500 \u2500\u2500\u2500 \u23a5\n \u23a2 20 200 200 \u23a5\n \u23a2 \u23a5\n \u23a2 2 2 \u23a5\n \u23a2 LA LA -LA \u23a5\n \u23a20 0 \u2500\u2500 \u2500\u2500\u2500 \u2500\u2500\u2500\u2500\u2500\u23a5\n \u23a2 20 200 200 \u23a5\n \u23a2 \u23a5\n \u23a2 2 2 \u23a5\n \u23a2 -LA LA LA \u23a5\n \u23a20 \u2500\u2500\u2500\u2500 0 \u2500\u2500\u2500 \u2500\u2500\u2500 \u23a5\n \u23a2 20 200 200 \u23a5\n \u23a2 \u23a5\n \u23a2 2 2 \u23a5\n \u23a2 -LA LA -LA \u23a5\n \u23a20 0 \u2500\u2500\u2500\u2500 \u2500\u2500\u2500 \u2500\u2500\u2500\u2500\u2500\u23a5\n \u23a3 20 200 200 \u23a6\n \n \n\n\n### The simulation\n\nA simulation is built by defining a correct dictionary.\n\nWe combine the previous dictionaries to build a simulation. In order to impose the homogeneous Dirichlet conditions in $x=0$, $x=1$, $y=0$, and $y=1$, the dictionary should contain the key 'boundary_conditions' (we use pylbm.bc.Anti_bounce_back function).\n\n\n```python\ndico = {\n 'box': {'x': [xmin, xmax],\n 'y': [ymin, ymax],\n 'label': 0},\n 'space_step': dx,\n 'scheme_velocity': la,\n 'schemes': [\n {\n 'velocities': list(range(5)),\n 'conserved_moments': u,\n 'polynomials': [1, X/LA, Y/LA, (X**2+Y**2)/(2*LA**2), (X**2-Y**2)/(2*LA**2)],\n 'equilibrium': [u, 0., 0., .5*u, 0.],\n 'relaxation_parameters': [0., s1, s1, s2, s2],\n }\n ],\n 'init': {u: (solution, (0.,))},\n 'boundary_conditions': {\n 0: {'method': {0: pylbm.bc.AntiBounceBack,}},\n },\n 'parameters': {LA: la},\n}\n\nsol = pylbm.Simulation(dico)\nprint(sol)\n```\n\n +------------------------+\n | Simulation information |\n +------------------------+\n \n +--------------------+\n | Domain information |\n +--------------------+\n - spatial dimension: 2\n - space step: 0.1\n - with halo:\n bounds of the box: [-0.05 -0.05] x [1.05 1.05]\n number of points: [12, 12]\n - without halo:\n bounds of the box: [0.05 0.05] x [0.95 0.95]\n number of points: [10, 10]\n \n +----------------------+\n | Geometry information |\n +----------------------+\n - spatial dimension: 2\n - bounds of the box: [0. 1.] x [0. 1.]\n - labels: [0, 0, 0, 0]\n \n +--------------------+\n | Scheme information |\n +--------------------+\n - spatial dimension: 2\n - number of schemes: 1\n - number of velocities: 5\n - conserved moments: [u]\n \n +----------+\n | Scheme 0 |\n +----------+\n - velocities\n (0: 0, 0)\n (1: 1, 0)\n (2: 0, 1)\n (3: -1, 0)\n (4: 0, -1)\n \n - polynomials\n \n \u23a1 1 \u23a4\n \u23a2 \u23a5\n \u23a2 X \u23a5\n \u23a2 \u2500\u2500 \u23a5\n \u23a2 LA \u23a5\n \u23a2 \u23a5\n \u23a2 Y \u23a5\n \u23a2 \u2500\u2500 \u23a5\n \u23a2 LA \u23a5\n \u23a2 \u23a5\n \u23a2 2 2\u23a5\n \u23a2X + Y \u23a5\n \u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n \u23a2 2 \u23a5\n \u23a2 2\u22c5LA \u23a5\n \u23a2 \u23a5\n \u23a2 2 2\u23a5\n \u23a2X - Y \u23a5\n \u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n \u23a2 2 \u23a5\n \u23a3 2\u22c5LA \u23a6\n \n - equilibria\n \n \u23a1 u \u23a4\n \u23a2 \u23a5\n \u23a2 0.0 \u23a5\n \u23a2 \u23a5\n \u23a2 0.0 \u23a5\n \u23a2 \u23a5\n \u23a20.5\u22c5u\u23a5\n \u23a2 \u23a5\n \u23a3 0.0 \u23a6\n \n - relaxation parameters\n \n \u23a10.0\u23a4\n \u23a2 \u23a5\n \u23a20.4\u23a5\n \u23a2 \u23a5\n \u23a20.4\u23a5\n \u23a2 \u23a5\n \u23a21.0\u23a5\n \u23a2 \u23a5\n \u23a31.0\u23a6\n \n - moments matrices\n \n \u23a11 1 1 1 1 \u23a4\n \u23a2 \u23a5\n \u23a2 10 -10 \u23a5\n \u23a20 \u2500\u2500 0 \u2500\u2500\u2500\u2500 0 \u23a5\n \u23a2 LA LA \u23a5\n \u23a2 \u23a5\n \u23a2 10 -10 \u23a5\n \u23a20 0 \u2500\u2500 0 \u2500\u2500\u2500\u2500\u23a5\n \u23a2 LA LA \u23a5\n \u23a2 \u23a5\n \u23a2 50 50 50 50 \u23a5\n \u23a20 \u2500\u2500\u2500 \u2500\u2500\u2500 \u2500\u2500\u2500 \u2500\u2500\u2500 \u23a5\n \u23a2 2 2 2 2 \u23a5\n \u23a2 LA LA LA LA \u23a5\n \u23a2 \u23a5\n \u23a2 50 -50 50 -50 \u23a5\n \u23a20 \u2500\u2500\u2500 \u2500\u2500\u2500\u2500 \u2500\u2500\u2500 \u2500\u2500\u2500\u2500\u23a5\n \u23a2 2 2 2 2 \u23a5\n \u23a3 LA LA LA LA \u23a6\n \n - inverse of moments matrices\n \n \u23a1 2 \u23a4\n \u23a2 -LA \u23a5\n \u23a21 0 0 \u2500\u2500\u2500\u2500\u2500 0 \u23a5\n \u23a2 50 \u23a5\n \u23a2 \u23a5\n \u23a2 2 2 \u23a5\n \u23a2 LA LA LA \u23a5\n \u23a20 \u2500\u2500 0 \u2500\u2500\u2500 \u2500\u2500\u2500 \u23a5\n \u23a2 20 200 200 \u23a5\n \u23a2 \u23a5\n \u23a2 2 2 \u23a5\n \u23a2 LA LA -LA \u23a5\n \u23a20 0 \u2500\u2500 \u2500\u2500\u2500 \u2500\u2500\u2500\u2500\u2500\u23a5\n \u23a2 20 200 200 \u23a5\n \u23a2 \u23a5\n \u23a2 2 2 \u23a5\n \u23a2 -LA LA LA \u23a5\n \u23a20 \u2500\u2500\u2500\u2500 0 \u2500\u2500\u2500 \u2500\u2500\u2500 \u23a5\n \u23a2 20 200 200 \u23a5\n \u23a2 \u23a5\n \u23a2 2 2 \u23a5\n \u23a2 -LA LA -LA \u23a5\n \u23a20 0 \u2500\u2500\u2500\u2500 \u2500\u2500\u2500 \u2500\u2500\u2500\u2500\u2500\u23a5\n \u23a3 20 200 200 \u23a6\n \n \n\n\n### Run a simulation\n\nOnce the simulation is initialized, one time step can be performed by using the function one_time_step.\n\nWe compute the solution of the heat equation at $t=0.1$. On the same graphic, we plot the initial condition, the exact solution and the numerical solution.\n\n\n```python\nimport numpy as np\nimport sympy as sp\nimport pylab as plt\n%matplotlib inline\nfrom mpl_toolkits.axes_grid1 import make_axes_locatable\nimport pylbm\n\nu, X, Y = sp.symbols('u, X, Y')\n\ndef solution(x, y, t, k, l):\n return np.sin(k*np.pi*x)*np.sin(l*np.pi*y)*np.exp(-(k**2+l**2)*np.pi**2*mu*t)\n\ndef plot(i, j, z, title):\n im = axarr[i,j].imshow(z)\n divider = make_axes_locatable(axarr[i, j])\n cax = divider.append_axes(\"right\", size=\"20%\", pad=0.05)\n cbar = plt.colorbar(im, cax=cax, format='%6.0e')\n axarr[i, j].xaxis.set_visible(False)\n axarr[i, j].yaxis.set_visible(False)\n axarr[i, j].set_title(title)\n\n# parameters\nxmin, xmax, ymin, ymax = 0., 1., 0., 1.\nN = 128\nmu = 1.\nTf = .1\ndx = (xmax-xmin)/N # spatial step\nla = 1./dx\ns1 = 2./(1+4*mu)\ns2 = 1.\nk, l = 1, 1 # number of the wave\n\ndico = {\n 'box': {'x':[xmin, xmax], \n 'y':[ymin, ymax], \n 'label': 0},\n 'space_step': dx,\n 'scheme_velocity': la,\n 'schemes':[\n {\n 'velocities': list(range(5)),\n 'conserved_moments': u,\n 'polynomials': [1, X/LA, Y/LA, (X**2+Y**2)/(2*LA**2), (X**2-Y**2)/(2*LA**2)],\n 'equilibrium': [u, 0., 0., .5*u, 0.],\n 'relaxation_parameters': [0., s1, s1, s2, s2],\n }\n ],\n 'init': {u: (solution, (0., k, l))},\n 'boundary_conditions': {\n 0: {'method': {0: pylbm.bc.AntiBounceBack,}},\n },\n 'generator': 'cython',\n 'parameters': {LA: la}, \n}\n\nsol = pylbm.Simulation(dico)\nx = sol.domain.x\ny = sol.domain.y\n\nf, axarr = plt.subplots(2, 2)\nf.suptitle('Heat equation', fontsize=20)\n\nplot(0, 0, sol.m[u].copy(), 'initial')\n\nwhile sol.t < Tf:\n sol.one_time_step()\n\nsol.f2m()\nz = sol.m[u]\nze = solution(x[:, np.newaxis], y[np.newaxis, :], sol.t, k, l)\nplot(1, 0, z, 'final')\nplot(0, 1, ze, 'exact')\nplot(1, 1, z-ze, 'error')\n\nplt.show()\n```\n", "meta": {"hexsha": "2de873f88961de84a6fb48e2c3f2551439fe39ab", "size": 101278, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/04_heat_2D.ipynb", "max_stars_repo_name": "bgraille/pylbm", "max_stars_repo_head_hexsha": "fd4419933e05b85be364232fddedfcb4f7275e1f", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 106, "max_stars_repo_stars_event_min_datetime": "2016-09-13T07:19:17.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-19T13:41:55.000Z", "max_issues_repo_path": "notebooks/04_heat_2D.ipynb", "max_issues_repo_name": "gouarin/pylbm", "max_issues_repo_head_hexsha": "fd4419933e05b85be364232fddedfcb4f7275e1f", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 53, "max_issues_repo_issues_event_min_datetime": "2017-09-18T04:51:19.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-19T21:36:23.000Z", "max_forks_repo_path": "notebooks/04_heat_2D.ipynb", "max_forks_repo_name": "gouarin/pylbm", "max_forks_repo_head_hexsha": "fd4419933e05b85be364232fddedfcb4f7275e1f", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 33, "max_forks_repo_forks_event_min_datetime": "2016-06-17T13:21:17.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-11T16:57:46.000Z", "avg_line_length": 126.5975, "max_line_length": 53400, "alphanum_fraction": 0.7949702798, "converted": true, "num_tokens": 6958, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081926, "lm_q2_score": 0.6548947290421275, "lm_q1q2_score": 0.44922468247418534}} {"text": "\u300c Autograd\uff08\u81ea\u52d5\u5fae\u5206\uff09\u300d\n===============================================================\n\u3010\u539f\u984c\u3011Autograd: Automatic Differentiation\n\n\u3010\u539f\u8457\u3011[Soumith Chintala](http://soumith.ch/)\n\n\u3010\u5143URL\u3011https://pytorch.org/tutorials/beginner/blitz/autograd_tutorial.html\n\n\u3010\u7ffb\u8a33\u3011\u96fb\u901a\u56fd\u969b\u60c5\u5831\u30b5\u30fc\u30d3\u30b9ISID AI\u30c8\u30e9\u30f3\u30b9\u30d5\u30a9\u30fc\u30e1\u30fc\u30b7\u30e7\u30f3\u30bb\u30f3\u30bf\u30fc\u3000\u5fb3\u539f\u3000\u5149\n\n\u3010\u65e5\u4ed8\u30112020\u5e7410\u670827\u65e5\n\n\u3010\u30c1\u30e5\u30c8\u30fc\u30ea\u30a2\u30eb\u6982\u8981\u3011\n\nPyTorch\u306b\u3088\u308b\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e\u5b66\u7fd2\u306b\u304a\u3044\u3066\u3001\u91cd\u8981\u306a\u6982\u5ff5\u3067\u3042\u308a\u30d1\u30c3\u30b1\u30fc\u30b8\u3067\u3082\u3042\u308bautograd\u306e\u6a5f\u80fd\u3001\u305d\u3057\u3066\u305d\u306e\u52d5\u4f5c\u5185\u5bb9\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u307e\u3059\u3002\n\n---\n\n\nAutograd: \u81ea\u52d5\u5fae\u5206\n=================\nPyTorch\u306b\u3088\u308b\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u69cb\u7bc9\u306e\u571f\u53f0\u3068\u306a\u3063\u3066\u3044\u308b\u306e\u304c\u3001autograd\uff08\u81ea\u52d5\u5fae\u5206\uff09\u30d1\u30c3\u30b1\u30fc\u30b8\u3067\u3059\u3002\n\n\u3053\u306e\u30d1\u30c3\u30b1\u30fc\u30b8\u306e\u6982\u8981\u3092\u3056\u3063\u304f\u308a\u3068\u78ba\u8a8d\u3057\u3001\u672c\u30c1\u30e5\u30fc\u30c8\u30ea\u30a2\u30eb\u30b7\u30ea\u30fc\u30ba\u3067\u521d\u3081\u3066\u3068\u306a\u308b\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e\u8a13\u7df4\u3092\u4f53\u9a13\u3057\u307e\u3057\u3087\u3046\u3002\n\n\n\n\nautograd\u30d1\u30c3\u30b1\u30fc\u30b8\u306fTensor\u306e\u64cd\u4f5c\u306b\u5bfe\u3059\u308b\u81ea\u52d5\u5fae\u5206\u6a5f\u80fd\u3092\u63d0\u4f9b\u3057\u307e\u3059\u3002\n\nTensor\u64cd\u4f5c\u306b\u57fa\u3065\u304f\u81ea\u52d5\u5fae\u5206\u306fdefine-by-run\u30d5\u30ec\u30fc\u30e0\u30ef\u30fc\u30af\u3067\u3042\u308a\u3001\u30e6\u30fc\u30b6\u30fc\u304c\u5b9f\u884c\u3057\u305f\u30b3\u30fc\u30c9\u306b\u5bfe\u5fdc\u3057\u3066\u8aa4\u5dee\u9006\u4f1d\u642c\u304c\u5b9a\u7fa9\u3055\u308c\u3001\u3059\u3079\u3066\u306e\u30a4\u30c6\u30ec\u30fc\u30b7\u30e7\u30f3\u306e\u8a08\u7b97\u3067\u7570\u306a\u308b\u7d50\u679c\u3092\u751f\u307f\u51fa\u3057\u307e\u3059\u3002\n\n\u3053\u3053\u304b\u3089\u306f\u3001\u3044\u304f\u3064\u304b\u306e\u5b9f\u4f8b\u3092\u901a\u3057\u3066\u3053\u306e\u6a5f\u80fd\u3092\u7c21\u5358\u306b\u898b\u3066\u3044\u304d\u307e\u3057\u3087\u3046\u3002\n\n
\n\n\uff08\u65e5\u672c\u8a9e\u8a33\u6ce8\uff1adefine-by-run\u306f\u30c7\u30fc\u30bf\u3092\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306b\u6d41\u3057\u306a\u304c\u3089\u3001\u30e2\u30c7\u30eb\u306e\u69cb\u7bc9\u3092\u884c\u3063\u3066\u3044\u304f\u624b\u6cd5\u3092\u6307\u3057\u307e\u3059\u3002\u4e00\u65b9\u3067define-and-run\u306f\u5148\u306b\u8aa4\u5dee\u9006\u4f1d\u642c\u306e\u5f62\u3092\u5b9f\u884c\u3059\u308b\u524d\u306b\u69cb\u7bc9\u3057\u307e\u3059\u3002\uff09\n\n\n\n\u30c6\u30f3\u30bd\u30eb\uff08Tensor\uff09\n==================\n``torch.Tensor``\u306f\u30d1\u30c3\u30b1\u30fc\u30b8\u306e\u4e2d\u5fc3\u7684\u306a\u30af\u30e9\u30b9\u3067\u3059\u3002\n\n``.requires_grad``\u5c5e\u6027\u304c``True``\u306b\u8a2d\u5b9a\u3055\u308c\u305f\u5834\u5408\u3001autograd\u30d1\u30c3\u30b1\u30fc\u30b8\u306b\u3088\u3063\u3066\u3059\u3079\u3066\u306e\u64cd\u4f5c\u304c\u8ffd\u8de1\u3055\u308c\u3001\u6f14\u7b97\u304c\u7d42\u4e86\u3057\u305f\u969b\u306f``.backward()``\u3092\u547c\u3073\u51fa\u3059\u3053\u3068\u3067\u3001\u3059\u3079\u3066\u306e\u64cd\u4f5c\u306b\u5bfe\u3059\u308b\u52fe\u914d\u304c\u81ea\u52d5\u7684\u306b\u8a08\u7b97\u3055\u308c\u307e\u3059\u3002\n\n\u3053\u306eTensor\u306b\u5bfe\u3059\u308b\u52fe\u914d\u306f``.grad``\u5c5e\u6027\u306b\u84c4\u7a4d\u3055\u308c\u3066\u3044\u304d\u307e\u3059\u3002\n\n\u8ffd\u8de1\u5c65\u6b74\u304b\u3089Tensor\u3092\u5207\u308a\u96e2\u3057\u3066\u3001\u8ffd\u8de1\u3092\u505c\u6b62\u3059\u308b\u5834\u5408\u306f\u3001 ``.detach()``\u3092\u547c\u3073\u51fa\u3057\u307e\u3059\u3002\n\n\u3053\u308c\u306b\u3088\u308a\u3001\u305d\u306e\u5f8c\u306e\u6f14\u7b97\u3067\u306f\u3053\u306e\u3053\u306eTensor\u306f\u8ffd\u8de1\u3055\u308c\u306a\u3044\u3088\u3046\u306b\u8a2d\u5b9a\u53ef\u80fd\u3067\u3059\u3002\n\n\n\n``with torch.no_grad():``\u3067\u30b3\u30fc\u30c9\u3092\u30d6\u30ed\u30c3\u30af\u306b\u307e\u3068\u3081\u308b\u3053\u3068\u3067\u3001\u8ffd\u8de1\u5c65\u6b74\uff08\u3068\u30e1\u30e2\u30ea\u306e\u5229\u7528\uff09\u3092\u7701\u7565\u3059\u308b\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002\n\n\u3053\u308c\u306f\u30e2\u30c7\u30eb\u3092\u8a55\u4fa1\u3059\u308b\u969b\u3001``requires_grad=True``\u306b\u3088\u308a\u5b66\u7fd2\u53ef\u80fd\u306a\u30d1\u30e9\u30e1\u30fc\u30bf\u3092\u6301\u3063\u3066\u3044\u308b\u304c\u3001\u52fe\u914d\u306e\u8a08\u7b97\u306f\u5fc5\u8981\u306a\u3044\u5834\u5408\u306b\u7279\u306b\u6709\u52b9\u3067\u3059\u3002\n\n\n\n\u305d\u3057\u3066\u3082\u3046\u4e00\u3064\u3001\u81ea\u52d5\u5fae\u5206\u306e\u5b9f\u884c\u306b\u975e\u5e38\u306b\u91cd\u8981\u306a\u30af\u30e9\u30b9\u306b``Function``\u304c\u3042\u308a\u307e\u3059\u3002\n\n``Tensor``\u3068``Function``\u306f \u76f8\u4e92\u306b\u63a5\u7d9a\u3057\u3001\u975e\u5de1\u56de\u30b0\u30e9\u30d5\u3067\u5b8c\u5168\u306a\u8a08\u7b97\u5c65\u6b74\u3092\u8a18\u9332\u3057\u3066\u3044\u307e\u3059\u3002\n\n\u5404Tensor\u306f\u3001\u305d\u306eTensor\u3092\u4f5c\u6210\u3057\u305f``Function`` \u3092\u53c2\u7167\u3059\u308b``.grad_fn``\u5c5e\u6027\u3092\u6301\u3061\u307e\u3059\uff08\u30e6\u30fc\u30b6\u30fc\u304c\u76f4\u63a5\u5b9a\u7fa9\u3057\u305f\u30c6\u30f3\u30bd\u30eb\u306e\u5834\u5408\u306f``grad_fn is None``\u3068\u306a\u308a\u307e\u3059\uff09\u3002\n\n\u5c0e\u95a2\u6570\u3092\u7b97\u51fa\u3059\u308b\u5834\u5408\u306f\u3001``Tensor``\u304c\u6301\u3064\u95a2\u6570``.backward()``\u3092\u547c\u3073\u51fa\u3057\u307e\u3059\u3002\n\n``Tensor``\u304c\u30b9\u30ab\u30e9\u30fc\uff08\u3059\u306a\u308f\u3061\u3001\u8981\u7d20\u6570\u304c1\u3064\u3060\u3051\uff09\u306e\u5834\u5408\u3001``.backward()``\u306b\u5f15\u6570\u3092\u6307\u5b9a\u3059\u308b\u5fc5\u8981\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u3057\u304b\u3057\u3001\u30c6\u30f3\u30bd\u30eb\u304c\u8907\u6570\u8981\u7d20\u3092\u6301\u3064\u5834\u5408\u306f\u3001Tensor\u3068\u540c\u3058\u5927\u304d\u3055\u306eTensor\u3092``gradient``\u306e\u5f15\u6570\u306b\u6307\u5b9a\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\n\n\n\n```python\n %matplotlib inline\n```\n\n\n```python\nimport torch\n```\n\nTensor\u3092\u4f5c\u6210\u3057\u3001``requires_grad=True``\u3068\u6307\u5b9a\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u6f14\u7b97\u3092\u8ffd\u8de1\u3057\u3066\u307f\u307e\u3057\u3087\u3046\u3002\n\n\n\n\n```python\nx = torch.ones(2, 2, requires_grad=True)\nprint(x)\n```\n\n tensor([[1., 1.],\n [1., 1.]], requires_grad=True)\n\n\nTensor\u306e\u8a08\u7b97\u3092\u5b9f\u884c\uff1a\n\n\n\n\n```python\ny = x + 2\nprint(y)\n```\n\n tensor([[3., 3.],\n [3., 3.]], grad_fn=)\n\n\n``y`` \u306f\u8a08\u7b97\u7d50\u679c\u3067\u3042\u308a\u3001 \u8a08\u7b97\u5c65\u6b74\u3068\u3057\u3066``grad_fn``\u3092\u6301\u3063\u3066\u3044\u307e\u3059\u3002\n\n\n\n\n```python\nprint(y.grad_fn)\n```\n\n \n\n\n\u3055\u3089\u306b\u3001``y``\u3092\u7528\u3044\u305f\u8a08\u7b97\u3092\u5b9f\u884c\u3057\u307e\u3059\uff1a\n\n\n\n\n```python\nz = y * y * 3\nout = z.mean()\n\nprint(z, out)\n```\n\n tensor([[27., 27.],\n [27., 27.]], grad_fn=) tensor(27., grad_fn=)\n\n\n``.requires_grad_( ... )`` \u306b\u3088\u3063\u3066\u65e2\u5b58\u306eTensor\u306e ``requires_grad``\u30d5\u30e9\u30b0\u3092\u5909\u66f4\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\n\n\u30c6\u30f3\u30bd\u30eb\u306e\u4f5c\u6210\u6642\u306b\u5f15\u6570\u3067\u4f55\u3082\u6307\u5b9a\u3057\u3066\u3044\u306a\u3044\u5834\u5408\u306f\u3001``requires_grad``\u306f\u30c7\u30d5\u30a9\u30eb\u30c8\u5024\u3068\u3057\u3066 ``False`` \u306b\u8a2d\u5b9a\u3055\u308c\u3066\u3044\u307e\u3059\u3002\n\n\n\n\n```python\na = torch.randn(2, 2)\na = ((a * 3) / (a - 1))\nprint(a.requires_grad)\na.requires_grad_(True)\nprint(a.requires_grad)\nb = (a * a).sum()\nprint(b.grad_fn)\n```\n\n False\n True\n \n\n\n\u52fe\u914d\uff08Gradients\uff09\n==================\n\u3067\u306f\u3001\u8aa4\u5dee\u9006\u4f1d\u642c\u3092\u5b9f\u884c\u3057\u3066\u307f\u307e\u3057\u3087\u3046\u3002\n\n\u5909\u6570``out``\u306f\u30b9\u30ab\u30e9\u30fc\u306e\u5024\u3092\u6301\u3063\u3066\u3044\u308b\u305f\u3081, ``out.backward()`` \u306f``out.backward(torch.tensor(1.))``\u3068\u540c\u3058\u7d50\u679c\u306b\u306a\u308a\u307e\u3059\u3002\n\n\n\n\n```python\nout.backward()\n```\n\n\n```python\nprint(out)\n```\n\n tensor(27., grad_fn=)\n\n\n\u52fe\u914d d(out)/dx\u3092\u8868\u793a\u3002\n\n\n\n\n\n```python\nprint(x.grad)\n```\n\n tensor([[4.5000, 4.5000],\n [4.5000, 4.5000]])\n\n\n\u7d50\u679c\u3068\u3057\u3066\u3059\u3079\u3066\u306e\u8981\u7d20\u304c ``4.5``\u306e\u884c\u5217\u3092\u5f97\u305f\u306e\u3067\u306f\u306a\u3044\u3067\u3057\u3087\u3046\u304b\u3002\n\n\n\n\u3053\u306e\u51fa\u529b\u30c6\u30f3\u30bd\u30eb\u3092\u201c$o$\u201d\u3068\u8a18\u8f09\u3057\u307e\u3059\u3002\n\u30c6\u30f3\u30bd\u30eb\u201c$o$\u201d\u3092\u8a08\u7b97\u3059\u308b\u3068\u3001\n\n$o = \\frac{1}{4}\\sum_i z_i$\n\n$z_i = 3(x_i+2)^2$ \n\n\u305d\u3057\u3066\u3001\n\n $z_i\\bigr\\rvert_{x_i=1} = 27$.\n\n\u306a\u306e\u3067\u3001\n\n$\\frac{\\partial o}{\\partial x_i} = \\frac{3}{2}(x_i+2)$\n\n$\\frac{\\partial o}{\\partial x_i}\\bigr\\rvert_{x_i=1} = \\frac{9}{2} = 4.5$\n\n\u3068\u306a\u308a\u307e\u3059\u3002\n\n\n\u6570\u5b66\u7684\u306b\u306f\u3001\u30d9\u30af\u30c8\u30eb\u95a2\u6570 $\\vec{y}=f(\\vec{x})$\u306b\u304a\u3044\u3066\u3001 $\\vec{x}$\u306b\u95a2\u3059\u308b$\\vec{y}$ \u306e\u52fe\u914d\u306f\u30e4\u30b3\u30d3\u30a2\u30f3\u3068\u547c\u3070\u308c\u3066\u3044\u307e\u3059\u3002\n\n\\begin{align}J=\\left(\\begin{array}{ccc}\n \\frac{\\partial y_{1}}{\\partial x_{1}} & \\cdots & \\frac{\\partial y_{1}}{\\partial x_{n}}\\\\\n \\vdots & \\ddots & \\vdots\\\\\n \\frac{\\partial y_{m}}{\\partial x_{1}} & \\cdots & \\frac{\\partial y_{m}}{\\partial x_{n}}\n \\end{array}\\right)\\end{align}\n\n\n\n\n\n\n\u4e00\u822c\u7684\u306b\u3001 ``torch.autograd`` \u306f\u30d9\u30af\u30c8\u30eb\u306e\u30e4\u30b3\u30d3\u30a2\u30f3\u306e\u7a4d\u3092\u7b97\u51fa\u3059\u308b\u8a08\u7b97\u30a8\u30f3\u30b8\u30f3\u306b\u306a\u308a\u307e\u3059\u3002 \n\n\n\n\u3053\u308c\u306f\u3001$v=\\left(\\begin{array}{cccc} v_{1} & v_{2} & \\cdots & v_{m}\\end{array}\\right)^{T}$\u3068\u3044\u3046\u30d9\u30af\u30c8\u30eb\u306b\u5bfe\u3057\u3066\u3001\u884c\u5217\u7a4d$v^{T}\\cdot J$\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002\n\n \u3082\u3057\u3001 $v$ \u304c\u30b9\u30ab\u30e9\u30fc\u95a2\u6570\u306e\u52fe\u914d $l=g\\left(\\vec{y}\\right)$\u3067\u3042\u3063\u305f\u5834\u5408\u3001\n$v=\\left(\\begin{array}{ccc}\\frac{\\partial l}{\\partial y_{1}} & \\cdots & \\frac{\\partial l}{\\partial y_{m}}\\end{array}\\right)^{T}$\u3068\u8868\u3055\u308c\u307e\u3059\u3002\n\n\u305d\u3057\u3066\u9023\u9396\u5f8b\u306b\u3088\u308a\u3001\u30d9\u30af\u30c8\u30eb\u306e\u30e4\u30b3\u30d3\u30a2\u30f3\u306e\u7a4d\u306f$\\vec{x}$\u306b\u95a2\u3059\u308b$l$\u306e\u52fe\u914d\u3068\u306a\u308b\u306e\u3067\u3059\u3002\n\n\\begin{align}J^{T}\\cdot v=\\left(\\begin{array}{ccc}\n \\frac{\\partial y_{1}}{\\partial x_{1}} & \\cdots & \\frac{\\partial y_{m}}{\\partial x_{1}}\\\\\n \\vdots & \\ddots & \\vdots\\\\\n \\frac{\\partial y_{1}}{\\partial x_{n}} & \\cdots & \\frac{\\partial y_{m}}{\\partial x_{n}}\n \\end{array}\\right)\\left(\\begin{array}{c}\n \\frac{\\partial l}{\\partial y_{1}}\\\\\n \\vdots\\\\\n \\frac{\\partial l}{\\partial y_{m}}\n \\end{array}\\right)=\\left(\\begin{array}{c}\n \\frac{\\partial l}{\\partial x_{1}}\\\\\n \\vdots\\\\\n \\frac{\\partial l}{\\partial x_{n}}\n \\end{array}\\right)\\end{align}\n\n\n\n( $v^{T}\\cdot J$ \u306f\u8ee2\u7f6e\u306e\u516c\u5f0f\u304b\u3089\u3001 $J^{T}\\cdot v$\u3092\u8a08\u7b97\u3057\u3066\u5f97\u3089\u308c\u308b\u5217\u30d9\u30af\u30c8\u30eb\u3001\u3068\u540c\u3058\u6210\u5206\u306e\u884c\u30d9\u30af\u30c8\u30eb\u3092\u4e0e\u3048\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066\u304f\u3060\u3055\u3044)\n\n\u3053\u306e\u30d9\u30af\u30c8\u30eb\u306e\u30e4\u30b3\u30d3\u30a2\u30f3\u306e\u7a4d\u306e\u6027\u8cea\u306f\u3001\u30b9\u30ab\u30e9\u91cf\u3067\u306f\u306a\u3044\u51fa\u529b\u3092\u6301\u3064\u30e2\u30c7\u30eb\u306b\u5bfe\u3057\u3066\u3001\u5916\u90e8\u304b\u3089\u7570\u306a\u308b\u52fe\u914d\u3092\u8ffd\u52a0\u3057\u3066\u8a08\u7b97\u3059\u308b\u969b\u306b\u3001\u975e\u5e38\u306b\u6709\u52b9\u306b\u5229\u7528\u3067\u304d\u307e\u3059\u3002\n\n\u30d9\u30af\u30c8\u30eb\u306e\u30e4\u30b3\u30d3\u30a2\u30f3\u306e\u7a4d\u306e\u4f8b\u3092\u898b\u3066\u307f\u307e\u3057\u3087\u3046\n\n\n```python\nx = torch.randn(3, requires_grad=True)\n\ny = x * 2\nwhile y.data.norm() < 1000:\n y = y * 2\n\nprint(y)\n```\n\n tensor([-100.0453, -496.9997, 898.9301], grad_fn=)\n\n\n\uff08\u65e5\u672c\u8a9e\u8a33\u6ce8\uff1a\u95a2\u6570norm()\u306f\u30ce\u30eb\u30e0\u2252\u8ddd\u96e2\u3092\u4e0e\u3048\u307e\u3059\u3001\u5f15\u6570\u306a\u3057\u306enorm()\u306f2\u4e57\u30ce\u30eb\u30e0\u3067\u3059\u3002\n\u5404\u8981\u7d20\u30922\u4e57\u3057\u3066\u8db3\u3057\u7b97\u3057\u3001\u305d\u306e\u30eb\u30fc\u30c8\u3092\u8a08\u7b97\u3057\u307e\u3059\uff09\n\n\u3088\u3063\u3066\u4e0a\u8a18\u3067\u306f\u3001\u4e71\u6570\u3067\u767a\u751f\u3055\u305b\u305fx\u30922\u500d\u30014\u500d\u30018\u500d\u3001\u30fb\u30fb\u30fb\u3068y\u306e3\u8981\u7d20\u306e2\u4e57\u548c\u306e\u5e73\u5747\u306e\u30eb\u30fc\u30c8\u304c1000\u3092\u8d85\u3048\u308b\u307e\u3067\u5897\u52a0\u3055\u305b\u3066\u3044\u307e\u3059\u3002\n\n\u4ee5\u4e0b\u306e\u8a08\u7b97\u7d50\u679c\u3092\u53c2\u7167\uff09\n\n\n```python\n# \u65e5\u672c\u8a9e\u8a33\u6ce8\ntorch.sqrt(y[0]*y[0] + y[1]*y[1] + y[2]*y[2]) \n```\n\n\n\n\n tensor(1032.0334, grad_fn=)\n\n\n\n\u3053\u306e\u5834\u5408\u3001 ``y`` \u306f\u30b9\u30ab\u30e9\u91cf\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002 ``torch.autograd``\n\u3067\u306f\u76f4\u63a5\u30e4\u30b3\u30d3\u30a2\u30f3\u306e\u5168\u8981\u7d20\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u306f\u3067\u304d\u307e\u305b\u3093\u304c\u3001\u30d9\u30af\u30c8\u30eb\u3068\u30e4\u30b3\u30d3\u30a2\u30f3\u306e\u7a4d\u3092\u8a08\u7b97\u3059\u308b\u3060\u3051\u306e\u5834\u5408\u306b\u306f\u3001\u7c21\u5358\u306b\u5f15\u6570\u3068\u3057\u3066``backward``\u306b\u30d9\u30af\u30c8\u30eb\u3092\u4e0e\u3048\u308b\u3053\u3068\u3067\u52fe\u914d\u304c\u7b97\u51fa\u3067\u304d\u307e\u3059\u3002\n\n\n\n\n```python\nv = torch.tensor([0.1, 1.0, 0.0001], dtype=torch.float)\ny.backward(v)\n\nprint(x.grad)\n```\n\n tensor([5.1200e+01, 5.1200e+02, 5.1200e-02])\n\n\n\n```python\n# \u65e5\u672c\u8a9e\u8a33\u6ce8\nprint(\"x\uff1a\", x)\nprint(\"y\uff1a\", y)\nscale = y/x\nprint(\"\u500d\u6570\uff1a\", scale)\n\n# \u3053\u306e\u30bb\u30eb\u306e\u51fa\u529b\u304b\u3089\u4f55\u500d\u3057\u3066\u3001y\u3092\u6c42\u3081\u305f\u306e\u304b\u5206\u304b\u308a\u307e\u3059\u3002\u500d\u6570:\u3000\u306e\u51fa\u529b\u90e8\u5206\u3067\u3059\u3002\n# \u3053\u306e\u30bb\u30eb\u3067\u306f\u5909\u6570sclae\u3067\u8868\u3057\u3066\u3044\u307e\u3059\u3002 \n# y = scale * x \u306a\u306e\u3067\u3001y\u306ex\u306b\u95a2\u3059\u308b\u52fe\u914d\u306f scale\u3067\u3059\u3002\n# y\u306e\u52fe\u914d\u3092v\u306b\u5bfe\u3057\u3066\u8a08\u7b97\u3057\u305f\u7d50\u679c\u3001x\u306b\u5bfe\u3057\u3066\u6e9c\u307e\u308b\u52fe\u914d\u5024\u304cx.grad\u3067\u3059\u3002\n# \u305d\u306e\u5024x.grad\uff08\u4e0a\u8a18\u306e\u30bb\u30eb\u306e\u51fa\u529b\u7d50\u679c\uff09\u306f\u3001\n# scale\u306bv\uff1d[0.1, 1, 0.0001]\u304c\u304b\u3051\u7b97\u3055\u308c\u305f\u5024\u3068\u306a\u3063\u3066\u3044\u308b\u306f\u305a\u3067\u3059\u3002\n\n```\n\n x\uff1a tensor([-0.1954, -0.9707, 1.7557], requires_grad=True)\n y\uff1a tensor([-100.0453, -496.9997, 898.9301], grad_fn=)\n \u500d\u6570\uff1a tensor([512., 512., 512.], grad_fn=)\n\n\n``with torch.no_grad():``\u3067\u30b3\u30fc\u30c9\u3092\u307e\u3068\u3081\u308b\u3053\u3068\u3067\u3001``.requires_grad=True``\u3068\u306a\u3063\u3066\u3044\u308bTensor\u306e\u8ffd\u8de1\u5c65\u6b74\u304b\u3089autograd\u3092\u505c\u6b62\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\n\n\n\n\n```python\nprint(x.requires_grad)\nprint((x ** 2).requires_grad)\n\nwith torch.no_grad():\n\tprint((x ** 2).requires_grad)\n```\n\n True\n True\n False\n\n\n\u8981\u7d20\u304c\u540c\u3058Tensor\u3092\u4f5c\u6210\u3057\u305f\u969b\u3082\u3001\u52fe\u914d\u304c\u5fc5\u8981\u306a\u3044\u5834\u5408\u306f``.detach()``\u3092\u7528\u3044\u308b\u3053\u3068\u3082\u53ef\u80fd\u3067\u3059\u3002\n\n\uff08\u65e5\u672c\u8a9e\u8a33\u6ce8\uff1a\u4ee5\u4e0b\u306e\u30bb\u30eb\u306e.eq()\u306fTensor\u3068\u3057\u3066\u5024\u304c\u540c\u3058\u3067\u3042\u308c\u3070True\u3092\u8fd4\u3057\u307e\u3059\u3002\u3044\u307e\u306f\u3001x\u3068y\u306e\u8981\u7d20\u306f3\u3064\u3042\u308b\u306e\u3067\u3001\u305d\u306e3\u3064\u306b\u5bfe\u3057\u3066\u6c42\u3081\u305f\u7d50\u679c\u3092.all()\u3067\u307e\u3068\u3081\u3066\u6c42\u3081\u3066\u3044\u307e\u3059\u3000[\u8a73\u7d30](https://pytorch.org/docs/stable/generated/torch.eq.html)\uff09\n\n\n```python\nprint(x.requires_grad)\ny = x.detach()\nprint(y.requires_grad)\nprint(x.eq(y).all())\n```\n\n True\n False\n tensor(True)\n\n\n\n**\u88dc\u8db3:**\n\n``autograd.Function`` \u306e\u30c9\u30ad\u30e5\u30e1\u30f3\u30c8\u306f\nhttps://pytorch.org/docs/stable/autograd.html#function\n\u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\n\n", "meta": {"hexsha": "b8450c0b778189a0cbc36f2a1cc576113241cb2b", "size": 17135, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebook/1_Learning PyTorch/1_2_autograd_tutorial_jp.ipynb", "max_stars_repo_name": "shige-ta/pytorch_tutorials_jp", "max_stars_repo_head_hexsha": "42d8cf317d5a023bd77672861676531bdfef06f9", "max_stars_repo_licenses": ["zlib-acknowledgement"], "max_stars_count": 114, "max_stars_repo_stars_event_min_datetime": "2020-12-18T05:13:19.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T04:36:50.000Z", "max_issues_repo_path": "notebook/1_Learning PyTorch/1_2_autograd_tutorial_jp.ipynb", "max_issues_repo_name": "shige-ta/pytorch_tutorials_jp", "max_issues_repo_head_hexsha": "42d8cf317d5a023bd77672861676531bdfef06f9", "max_issues_repo_licenses": 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YES\n2. YES", "lm_q1_score": 0.7090191337850933, "lm_q2_score": 0.6334102498375401, "lm_q1q2_score": 0.44909998667041223}} {"text": "```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom sympy import *\n```\n\n\n```python\nt = 0\nx = np.linspace(-4,4,100)\n```\n\n\n```python\ny = np.exp(-np.power(x-3*t,2))*np.sin(3*np.pi*(x-t))\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "76ab107931d886d43d6a73eeaa76120d87c0bd28", "size": 2776, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "SimPy/.ipynb_checkpoints/plot_wavepacket-checkpoint.ipynb", "max_stars_repo_name": "nahian-147/my_codes", "max_stars_repo_head_hexsha": "9729c56b227d75354ea49982720de94ed1c21909", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "SimPy/.ipynb_checkpoints/plot_wavepacket-checkpoint.ipynb", "max_issues_repo_name": "nahian-147/my_codes", "max_issues_repo_head_hexsha": "9729c56b227d75354ea49982720de94ed1c21909", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "SimPy/.ipynb_checkpoints/plot_wavepacket-checkpoint.ipynb", "max_forks_repo_name": "nahian-147/my_codes", "max_forks_repo_head_hexsha": "9729c56b227d75354ea49982720de94ed1c21909", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 35.1392405063, "max_line_length": 970, "alphanum_fraction": 0.5731268012, "converted": true, "num_tokens": 76, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7879311956428947, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.4490046809690305}} {"text": "# Viscosity Coefficients\n\n## Prelude\nIn this notebook we will attempt to calculate the viscosity of the Yukawa OCP.\n\nThe YAML input file can be found at [input_file](https://raw.githubusercontent.com/murillo-group/sarkas/master/docs/examples/YOCP/input_files/yocp_viscosity.yaml) and this notebook at [notebook](https://raw.githubusercontent.com/murillo-group/sarkas/master/docs/examples/YOCP/YOCP_viscosity.ipynb).\n\n\n\n```python\n# Import the usual libraries\n%pylab\n%matplotlib inline\n\nimport os\nplt.style.use('MSUstyle')\n\n# Import sarkas\nfrom sarkas.processes import Simulation, PostProcess, PreProcess\n\n\n# Create the file path to the YAML input file\ninput_file_name = os.path.join('input_files', 'yocp_viscosity.yaml')\n```\n\n Using matplotlib backend: Qt5Agg\n Populating the interactive namespace from numpy and matplotlib\n\n\n\n```python\n# pre = PreProcess(input_file_name)\n# pre.setup(read_yaml=True)\n# pre.run()\n```\n\n\n```python\n# sim = Simulation(input_file_name)\n# sim.setup(read_yaml=True)\n# sim.run()\n```\n\n\n```python\npostproc = PostProcess(input_file_name)\npostproc.setup(read_yaml=True)\npostproc.parameters.verbose = True\npostproc.therm.setup(postproc.parameters)\npostproc.therm.temp_energy_plot(postproc)\n```\n\n## Pair Distribution Function\n\nThe first observable to calculate is always the RDF.\n\n\n```python\nrdf = postproc.rdf\nrdf.setup(postproc.parameters, no_slices=1 )\nrdf.parse()\n\n```\n\n\n```python\nrdf.plot(scaling = rdf.a_ws, \n y = ('C-C RDF', 'Mean'),\n xlabel = r'$r /a$')\n```\n\n\n```python\nfrom sarkas.tools.transport import TransportCoefficients\nfrom sarkas.tools.observables import PressureTensor\n```\n\n## Pressure Tensor\n\nThe viscosity is obtained from the autocorrelation function of the Pressure Tensor $\\overleftrightarrow{\\mathcal P}$ whose elements are\n\n\\begin{equation}\n\\mathcal P_{\\alpha\\gamma}(t) = \\frac{1}{V} \\sum_{i}^{N} \\left [ m_i v^{\\alpha}_{i} v^{\\gamma}_{i} - \\sum_{j > i} \\frac{r_{ij}^{\\alpha} r_{ij}^{\\gamma} }{r_{ij}} \\frac{d}{dr}\\phi(r) \\right ],\n\\end{equation}\n\nwhere $r_{ij}^{\\alpha}$ is the $\\alpha$ component of the distance between particles $i$ and $j$. The first term is the kinetic term and the second term is the virial term, but it is often referred to as the potential contribution. The virial is calculated during the simulation phase and saved together with particles corrdinates. \n\nIn order to check that our code are correct, let's verify some laws. \n\nThe pressure of the system is calculated from $\\mathcal P(t)= \\frac1{3} {\\rm Tr} \\overleftrightarrow{\\mathcal P}(t)$ and also from \n\n\\begin{equation}\nP = \\frac{n}{\\beta} - \\frac{2\\pi}{3} n^2 \\int_0^{\\infty} dr \\, r^3 \\frac{d\\phi(r)}{dr} g(r)\n\\end{equation}\n\nwhere $g(r)$ is the pair distribution function that we have already calculated.\n\nLet's calculate the Pressure tensor and the pressure $\\mathcal P$.\n\n\n```python\npt = PressureTensor()\npt.setup(postproc.parameters, no_slices = 2)\n# pt.compute()\npt.parse()\n```\n\nAs usual the data is saved in several dataframes. In this case we have 4 dataframes\n\n* A dataframe for the values of each of the elements of the pressure tensor for each of the slices, `pt.dataframe_slices`\n* A dataframe for the mean and std values of each of the elements of the pressure tensor, `pt.dataframe`\n* A dataframe for the ACF of each pair $\\langle \\mathcal P_{\\alpha\\beta}(t)\\mathcal P_{\\mu\\nu}(0) \\rangle$ for each slice, `pt.dataframe_acf_slices`\n* A dataframe for the mean and std of the ACF of each pair $\\langle \\mathcal P_{\\alpha\\beta}(t)\\mathcal P_{\\mu\\nu}(0) \\rangle$, `pt.dataframe_acf`\n\nLet's look at `pt.dataframe` and at its columns\n\n\n```python\npt.dataframe\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
TimePressureDelta PressurePressure Tensor Kinetic xxPressure Tensor Potential xxPressure Tensor xx...Pressure Tensor Potential zyPressure Tensor zyPressure Tensor Kinetic zzPressure Tensor Potential zzPressure Tensor zz
NaNMeanStdMeanStdMeanStdMeanStdMean...MeanStdMeanStdMeanStdMeanStdMeanStd
00.000000e+001.481050e+134.475378e+099.640046e+084.624121e+097.093304e+111.803769e+101.407423e+133.070777e+101.478356e+13...-1.349294e+109.080726e+081.513634e+096.403260e+097.178245e+112.283417e+091.412591e+131.183573e+101.484373e+139.552317e+09
16.250000e-171.481062e+134.360531e+091.077576e+094.509274e+097.092128e+111.788308e+101.407437e+133.043084e+101.478358e+13...-1.359784e+107.774131e+081.428573e+096.230626e+097.176454e+111.632009e+091.412631e+131.180888e+101.484395e+131.017687e+10
21.250000e-161.481073e+134.238994e+091.189024e+094.387736e+097.091114e+111.771433e+101.407450e+133.012222e+101.478361e+13...-1.368992e+106.259734e+081.336832e+096.048696e+097.174492e+119.777352e+081.412669e+131.176571e+101.484414e+131.078798e+10
31.875000e-161.481084e+134.112669e+091.300868e+094.261411e+097.090262e+111.753208e+101.407463e+132.978323e+101.478365e+13...-1.377276e+104.507040e+081.235023e+095.853759e+097.172359e+113.218677e+081.412707e+131.170957e+101.484430e+131.138770e+10
42.500000e-161.481095e+133.982834e+091.412491e+094.131577e+097.089578e+111.733693e+101.407475e+132.941981e+101.478371e+13...-1.384260e+102.570485e+081.127174e+095.650556e+097.170051e+113.343385e+081.412743e+131.164139e+101.484443e+131.197573e+10
..................................................................
1499959.374688e-121.481016e+134.676448e+096.174698e+084.527706e+097.118162e+112.094178e+101.409873e+133.950421e+091.481055e+13...-9.425949e+096.014705e+09-5.177773e+092.743918e+097.148230e+112.812314e+081.410559e+133.738486e+101.482042e+133.710363e+10
1499969.374750e-121.481018e+134.712473e+096.394380e+084.563731e+097.118652e+112.120674e+101.409889e+134.123326e+091.481076e+13...-9.394952e+096.262046e+09-5.058588e+092.647215e+097.150820e+111.719736e+081.410561e+133.806962e+101.482069e+133.789764e+10
1499979.374812e-121.481020e+134.744467e+096.613843e+084.595724e+097.119236e+112.144853e+101.409906e+134.318283e+091.481098e+13...-9.343740e+096.501897e+09-4.937013e+092.541221e+097.153249e+115.141538e+071.410562e+133.872970e+101.482094e+133.867828e+10
1499989.374875e-121.481023e+134.768360e+096.851347e+084.619617e+097.119910e+112.166620e+101.409922e+134.541414e+091.481121e+13...-9.272190e+096.732523e+09-4.812951e+092.428042e+097.155505e+118.013261e+071.410562e+133.935915e+101.482117e+133.943928e+10
1499999.374937e-121.481025e+134.787700e+097.087402e+084.638958e+097.120667e+112.185886e+101.409938e+134.789867e+091.481144e+13...-9.179610e+096.954353e+09-4.685684e+092.307675e+097.157575e+112.223411e+081.410561e+133.996308e+101.482137e+134.018542e+10
\n

150000 rows \u00d7 59 columns

\n
\n\n\n\nNote that the Pressure $\\mathcal P(t)$ is readily calculated and provided as a column of the dataframe.\n\nNote also that there is a multitude of columns. This is because in dense plasmas it is useful to know the contribution of both the kinetic term and potential term separately, as such the columns of each dataframe contain the kinetic, the potential, and the total value of each $\\mathcal P_{\\alpha\\beta}$ and their ACFs.\n\nLet's plot the Pressure as a function of time\n\n\n```python\n# Let's plot it\np_id = pt.total_num_density / postproc.therm.beta\nax = pt.plot( \n scaling = (1e-12, p_id),\n y = (\"Pressure\", \"Mean\"),\n xlabel = \"Time [ps]\",\n ylabel = r\"$ \\beta P(t)/n$\"\n )\nax.plot(pt.dataframe['Time']*1e12, pt.dataframe[('Pressure','Mean')].expanding().mean()/p_id )\nax.legend(['Pressure', 'Moving avg'])\n```\n\n## Pressure from RDF\n\nLet's now calculate the pressure from the integral of the RDF. This is obtained from the method `compute_from_rdf` of the `Thermodynamics` object. \n\nLooking at the documentation of this [method](:meth:`sarkas.tool.observables.Thermodynamics.compute_from_rdf`) we notice that it returns five values:\nthe Hartree and correlational terms between species :math:`A` and :math:`B` and the ideal pressure $n k_B T$. \n\nThe total pressure is given from the sum of the three terms and should be equal to the \n\n$$ P = n k_BT + P_{\\rm Hartree} + P_{\\rm Corr} = {\\operatorname {Mean} } \\left \\{ \\mathcal P(t) \\right \\} $$\n\n\n```python\nnkT, _, _, p_h, p_c = postproc.therm.compute_from_rdf(rdf, postproc.potential)\n\nP_rdf = nkT + p_h + p_c\nP_trace = pt.dataframe[(\"Pressure\", \"Mean\")].mean()\n\nprint(\"The relative difference between the two methods is = {:.2f} %\".format((P_rdf[0] - P_trace)*100/P_rdf[0] ) )\n```\n\n The relative difference between the two methods is = 0.03 %\n\n\nIt seems that we have done a good job! \n\n### Sum rule\n\nLet's now check that we have calculated the ACF correctly. The equal time ACFs of the elements of $\\overleftrightarrow{\\mathcal P}(t)$ obey the following sum rules\n\n$$\n\\mathcal J_{zzzz}(0) = \\frac 13 \\sum_{\\alpha}\\left \\langle \\mathcal P_{\\alpha\\alpha}(0)\\mathcal P_{\\alpha\\alpha}(0) \\right \\rangle = \\frac{n}{\\beta^2} \\left [ 3 + \\frac{2\\beta}{15} I_1 + \\frac \\beta5 I_2 \\right ] ,\n$$ \n$$\n\\mathcal J_{zzxx}(0) = \\frac 16 \\sum_{\\alpha} \\sum_{\\beta\\neq\\alpha} \\left \\langle \\mathcal P_{\\alpha\\alpha}(0)\\mathcal P_{\\beta\\beta}(0) \\right \\rangle = \\frac{n}{\\beta^2} \\left [ 1 - \\frac{2\\beta}{5} I_1 + \\frac \\beta{15} I_2 \\right ] ,\n$$ \n$$\n\\mathcal J_{xyxy}(0) = \\frac 16 \\sum_{\\alpha}\\sum_{\\beta \\neq \\alpha} \\left \\langle \\mathcal P_{\\alpha\\beta}(0)\\mathcal P_{\\alpha\\beta}(0) \\right \\rangle = \\frac{n}{\\beta^2} \\left [ 1 + \\frac{4\\beta}{15} I_1 + \\frac \\beta{15} I_2 \\right ] ,\n$$ \n\nwhere\n\n$$ \nI_1 = 2\\pi n \\int dr \\, g(r) r^3 \\frac{d\\phi}{dr}, \\quad I_2 = 2\\pi n \\int dr\\, g(r) r^4 \\frac{d^2\\phi}{dr^2}.\n$$\n\nNotice that all three equal time ACF satisfy \n\n$$ \\mathcal J_{zzzz}(0) - \\mathcal J_{zzxx}(0) = 2 \\mathcal J_{xyxy}(0) .$$\n\nLet's look at the dataframe of the ACF first\n\n\n```python\npt.dataframe_acf\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
TimePressure ACFDelta Pressure ACFPressure Tensor Kinetic ACF xxxxPressure Tensor Potential ACF xxxxPressure Tensor Kin-Pot ACF xxxx...Pressure Tensor Kinetic ACF zzzzPressure Tensor Potential ACF zzzzPressure Tensor Kin-Pot ACF zzzzPressure Tensor Pot-Kin ACF zzzzPressure Tensor ACF zzzz
NaNMeanStdMeanStdMeanStdMeanStdMean...MeanStdMeanStdMeanStdMeanStdMeanStd
00.000000e+002.193225e+264.406888e+217.042465e+181.273591e+185.099171e+232.827243e+201.987219e+263.916734e+221.006467e+25...5.096344e+238.893208e+201.986865e+261.015599e+221.006103e+258.459061e+211.006103e+258.459061e+212.193182e+267.651456e+21
16.250000e-172.193225e+264.406887e+217.039566e+181.273605e+185.099170e+232.827093e+201.987219e+263.917093e+221.006467e+25...5.096344e+238.893161e+201.986865e+261.015870e+221.006103e+258.458738e+211.006103e+258.459131e+212.193182e+267.648481e+21
21.250000e-162.193225e+264.406876e+217.030938e+181.273617e+185.099169e+232.826976e+201.987219e+263.917447e+221.006467e+25...5.096342e+238.893180e+201.986865e+261.016138e+221.006103e+258.458483e+211.006103e+258.459218e+212.193182e+267.645642e+21
31.875000e-162.193225e+264.406855e+217.016597e+181.273627e+185.099167e+232.826892e+201.987219e+263.917796e+221.006467e+25...5.096339e+238.893263e+201.986865e+261.016401e+221.006103e+258.458297e+211.006103e+258.459323e+212.193182e+267.642939e+21
42.500000e-162.193225e+264.406825e+216.996570e+181.273635e+185.099164e+232.826843e+201.987219e+263.918140e+221.006467e+25...5.096335e+238.893410e+201.986865e+261.016659e+221.006103e+258.458181e+211.006103e+258.459442e+212.193182e+267.640372e+21
..................................................................
1499959.374688e-122.193499e+267.464009e+21-9.260680e+182.564085e+185.046615e+232.590839e+211.984372e+264.854286e+231.002041e+25...5.131844e+236.688760e+201.992652e+263.809483e+231.010492e+258.977493e+211.011984e+254.153429e+222.200032e+264.141740e+23
1499969.374750e-122.193492e+266.760933e+21-9.466666e+182.228395e+185.047103e+232.612042e+211.984374e+264.891900e+231.002078e+25...5.133477e+239.467156e+201.992626e+263.851990e+231.010650e+258.148359e+211.012135e+254.639596e+222.200038e+264.243933e+23
1499979.374812e-122.193486e+266.052459e+21-9.666614e+181.888858e+185.047653e+232.630856e+211.984376e+264.929383e+231.002119e+25...5.135030e+231.227036e+211.992600e+263.893844e+231.010799e+257.263669e+211.012278e+255.124653e+222.200042e+264.345943e+23
1499989.374875e-122.193479e+265.334349e+21-9.857902e+181.544056e+185.048264e+232.647229e+211.984378e+264.966713e+231.002164e+25...5.136500e+231.509553e+211.992572e+263.934963e+231.010940e+256.324277e+211.012413e+255.608089e+222.200044e+264.447625e+23
1499999.374937e-122.193472e+264.626795e+21-1.004232e+191.194677e+185.048934e+232.661116e+211.984380e+265.003741e+231.002213e+25...5.137886e+231.793975e+211.992544e+263.975644e+231.011072e+255.330746e+211.012540e+256.089548e+222.200043e+264.549231e+23
\n

150000 rows \u00d7 815 columns

\n
\n\n\n\nNotice that in this case we have many more columns since now we have the ACF of the kinetic-kinetic, kinetic-potential, potential-kinetic, potential-potential, and the total ACF of each pair of elements.\n\nLet's verify the sum rules.\n\n\n```python\n# Diagonal terms\ncolumn_zzzz = [\n ('Pressure Tensor ACF xxxx', 'Mean'),\n ('Pressure Tensor ACF yyyy', 'Mean'),\n ('Pressure Tensor ACF zzzz', 'Mean'),\n]\nJ_zzzz_0 = pt.dataframe_acf[column_zzzz].iloc[0].mean()\n\n# Cross-Diagonal Terms\ncolumn_zzxx = [\n ('Pressure Tensor ACF xxyy', 'Mean'),\n ('Pressure Tensor ACF xxzz', 'Mean'),\n ('Pressure Tensor ACF yyxx', 'Mean'),\n ('Pressure Tensor ACF yyzz', 'Mean'),\n ('Pressure Tensor ACF zzxx', 'Mean'),\n ('Pressure Tensor ACF zzyy', 'Mean'),\n]\nJ_zzxx_0 = pt.dataframe_acf[column_zzxx].iloc[0].mean()\n\n# Cross Off Diagonal terms\ncolumn_xyxy = [\n ('Pressure Tensor ACF xyxy', 'Mean'),\n ('Pressure Tensor ACF xzxz', 'Mean'),\n ('Pressure Tensor ACF yxyx', 'Mean'),\n ('Pressure Tensor ACF yzyz', 'Mean'),\n ('Pressure Tensor ACF zxzx', 'Mean'),\n ('Pressure Tensor ACF zyzy', 'Mean'),\n]\nJ_xyxy_0 = pt.dataframe_acf[column_xyxy].iloc[0].mean()\n\n# The units of J's are [Density * Energy]^2\n\nprint('The isotropy condition : (J_zzzz_0 - J_zzxx_0 )/( 2*J_xyxy_0 ) = {:.4f}'.format( (J_zzzz_0 - J_zzxx_0)/(2.0 * J_xyxy_0) ))\n```\n\n The isotropy condition : (J_zzzz_0 - J_zzxx_0 )/( 2*J_xyxy_0 ) = 0.9814\n\n\nNot exactly 1 but pretty close.\n\nLet's now verify the sum rules. These are calculated from the `pt.sum_rule` method\n\n\n```python\n# These sigmas have units of n^2\nsigma_zzzz, sigma_zzxx, sigma_xyxy = pt.sum_rule(postproc.therm.beta, rdf, postproc.potential)\n\n```\n\n\n```python\n\"{:.4e}, {:.4e}\".format(J_zzzz_0, nkT**2)\n```\n\n\n\n\n '2.1932e+26, 5.0955e+23'\n\n\n\n## Viscosity\n\nThe shear viscosity is calculated from the Green-Kubo relation\n\n\\begin{equation}\n\\eta = \\frac{\\beta V}{6} \\sum_{\\alpha} \\sum_{\\gamma \\neq \\alpha} \\int_0^{\\infty} dt \\, \\left \\langle \\mathcal P_{\\alpha\\gamma}(t) \\mathcal P_{\\alpha\\gamma}(0) \\right \\rangle,\n\\end{equation}\n\nwhere $\\beta = 1/k_B T$, $\\alpha,\\gamma = {x, y, z}$ and $\\mathcal J_{\\alpha\\gamma}(t)$ is the autocorrelation function of the $\\alpha,\\gamma$ element of the \n\nThe bulk viscosity is given by a similar relation\n\n\\begin{equation}\n\\eta_V = \\beta V \\int_0^{\\infty}dt \\, \\left \\langle \\delta \\mathcal P(t) \\delta \\mathcal P(0) \\right \\rangle,\n\\end{equation}\n\nwhere\n\n\\begin{equation}\n\\delta \\mathcal P(t) = \\mathcal P(t) - \\left \\langle \\mathcal P \\right \\rangle\n\\end{equation}\n\nis the deviation of the scalar pressure.\n\n\n```python\ntc = TransportCoefficients(params = postproc.parameters, no_slices = 2)\n```\n\n\n```python\ntc.parse(observable = pt, tc_name = \"Viscosities\")\n# tc.viscosities(pt,plot = True)\n```\n\n Data saved in: \n Simulations/yocp_kappa2/PostProcessing/TransportCoefficients/Production/Viscosities_yocp_kappa2.h5\n Simulations/yocp_kappa2/PostProcessing/TransportCoefficients/Production/Viscosities_slices_yocp_kappa2.h5\n \n No. of slices = 2\n No. dumps per slice = 30000\n Time interval of autocorrelation function = 9.3750e-12 [s] ~ 7205 w_p T\n\n\n\n```python\nacf_str = \"Delta Pressure ACF\"\nacf_avg = pt.dataframe_acf[(\"Delta Pressure ACF\", \"Mean\")]\nacf_std = pt.dataframe_acf[(\"Delta Pressure ACF\", \"Std\")]\n\npq = \"Bulk Viscosity\"\ntc_avg = tc.viscosity_df[(pq, \"Mean\")]\ntc_std = tc.viscosity_df[(pq, \"Std\")]\n```\n\n\n```python\nfig, axes = tc.plot_tc(\n time = tc.viscosity_df[\"Time\"].iloc[:,0].to_numpy(),\n acf_data=np.column_stack((acf_avg, acf_std)),\n tc_data=np.column_stack((tc_avg, tc_std)),\n acf_name=acf_str,\n tc_name=\"Bulk Viscosity\",\n figname=\"{}_Plot.png\".format(\"Bulk Viscosity\"),\n show=False\n)\naxes[0].set(ylim = (-1, 1.05))\naxes[1].set(ylim = (-0.5, 1000 ) )\n```\n\n\n```python\npq = \"Shear Viscosity\"\ntc_avg = tc.viscosity_df[(pq, \"Mean\")]\ntc_std = tc.viscosity_df[(pq, \"Std\")]\n\n\nrescale = pt.total_plasma_frequency * pt.a_ws**2 * pt.species_masses[0] * pt.total_num_density * 0.0654\nfig, ax = plt.subplots(1,1)\nax.plot(tc.viscosity_df[\"Time\"].iloc[:,0].to_numpy()*1e12,\n tc_avg / rescale,\n label = r'$\\mu$')\n\nax.fill_between(\n tc.viscosity_df[\"Time\"].iloc[:,0].to_numpy()*1e12,\n (tc_avg - tc_std) / rescale,\n (tc_avg + tc_std) / rescale,\n alpha = 0.2)\n\nax.plot(tc.viscosity_df[\"Time\"].iloc[:,0].to_numpy()*1e12,\n tc_avg.expanding().mean()/rescale,\n label = r'Moving avg')\nax.set(xlabel = r'Time difference $\\tau$ [ps]')\n```\n\nI am missing a factor of 2 somewhere and can't figure out where.\n\nTo be continued.\n", "meta": {"hexsha": "ca8b0d6528158bbb194a1b182d25bccec3129624", "size": 576474, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/examples/YOCP/YOCP_viscosity.ipynb", "max_stars_repo_name": "lucianogsilvestri/sarkas", "max_stars_repo_head_hexsha": "f4ab00014d09976561fbd4349b9d0610e47a61e1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/examples/YOCP/YOCP_viscosity.ipynb", "max_issues_repo_name": "lucianogsilvestri/sarkas", "max_issues_repo_head_hexsha": "f4ab00014d09976561fbd4349b9d0610e47a61e1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/examples/YOCP/YOCP_viscosity.ipynb", "max_forks_repo_name": "lucianogsilvestri/sarkas", "max_forks_repo_head_hexsha": "f4ab00014d09976561fbd4349b9d0610e47a61e1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 371.4394329897, "max_line_length": 214564, "alphanum_fraction": 0.908887478, "converted": true, "num_tokens": 10933, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056040203135, "lm_q2_score": 0.651354857898194, "lm_q1q2_score": 0.4489825537550801}} {"text": "\n\n# AI Feynman 2.0: Learning Regression Equations From\u00a0Data\n\n### Clone repository and install dependencies\n\n\n```\n!git clone https://github.com/SJ001/AI-Feynman.git\n```\n\n Cloning into 'AI-Feynman'...\n remote: Enumerating objects: 28, done.\u001b[K\n remote: Counting objects: 100% (28/28), done.\u001b[K\n remote: Compressing objects: 100% (28/28), done.\u001b[K\n remote: Total 350 (delta 14), reused 0 (delta 0), pack-reused 322\u001b[K\n Receiving objects: 100% (350/350), 31.27 MiB | 9.72 MiB/s, done.\n Resolving deltas: 100% (206/206), done.\n\n\nLook at what we downloaded\n\n\n```\n!ls /content/AI-Feynman\n# %pycat AI-Feynman/requirements.txt if you need to fix the dependencies\n```\n\n Code example_data LICENSE README.md\trequirements.txt\n\n\nFix broken requirements file (may not be needed if later versions fix this).\n\n\n```\n%%writefile AI-Feynman/requirements.txt\ntorch>=1.4.0\nmatplotlib\nsympy==1.4\npandas\nscipy\nsortedcontainers\n```\n\n Overwriting AI-Feynman/requirements.txt\n\n\nInstall dependencies not already installed in Google Collab\n\n\n```\n!pip install -r AI-Feynman/requirements.txt\n```\n\n Requirement already satisfied: torch>=1.4.0 in /usr/local/lib/python3.6/dist-packages (from -r AI-Feynman/requirements.txt (line 1)) (1.5.1+cu101)\n Requirement already satisfied: matplotlib in /usr/local/lib/python3.6/dist-packages (from -r AI-Feynman/requirements.txt (line 2)) (3.2.2)\n Collecting sympy==1.4\n \u001b[?25l Downloading https://files.pythonhosted.org/packages/21/21/f4105795ca7f35c541d82c5b06be684dd2f5cb4f508fb487cd7aea4de776/sympy-1.4-py2.py3-none-any.whl (5.3MB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 5.3MB 2.7MB/s \n \u001b[?25hRequirement already satisfied: pandas in /usr/local/lib/python3.6/dist-packages (from -r AI-Feynman/requirements.txt (line 4)) (1.0.5)\n Requirement already satisfied: scipy in /usr/local/lib/python3.6/dist-packages (from -r AI-Feynman/requirements.txt (line 5)) (1.4.1)\n Requirement already satisfied: sortedcontainers in /usr/local/lib/python3.6/dist-packages (from -r AI-Feynman/requirements.txt (line 6)) (2.2.2)\n Requirement already satisfied: future in /usr/local/lib/python3.6/dist-packages (from torch>=1.4.0->-r AI-Feynman/requirements.txt (line 1)) (0.16.0)\n Requirement already satisfied: numpy in /usr/local/lib/python3.6/dist-packages (from torch>=1.4.0->-r AI-Feynman/requirements.txt (line 1)) (1.18.5)\n Requirement already satisfied: python-dateutil>=2.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib->-r AI-Feynman/requirements.txt (line 2)) (2.8.1)\n Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib->-r AI-Feynman/requirements.txt (line 2)) (2.4.7)\n Requirement already satisfied: kiwisolver>=1.0.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib->-r AI-Feynman/requirements.txt (line 2)) (1.2.0)\n Requirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.6/dist-packages (from matplotlib->-r AI-Feynman/requirements.txt (line 2)) (0.10.0)\n Requirement already satisfied: mpmath>=0.19 in /usr/local/lib/python3.6/dist-packages (from sympy==1.4->-r AI-Feynman/requirements.txt (line 3)) (1.1.0)\n Requirement already satisfied: pytz>=2017.2 in /usr/local/lib/python3.6/dist-packages (from pandas->-r AI-Feynman/requirements.txt (line 4)) (2018.9)\n Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.6/dist-packages (from python-dateutil>=2.1->matplotlib->-r AI-Feynman/requirements.txt (line 2)) (1.12.0)\n Installing collected packages: sympy\n Found existing installation: sympy 1.1.1\n Uninstalling sympy-1.1.1:\n Successfully uninstalled sympy-1.1.1\n Successfully installed sympy-1.4\n\n\nCheck that fortran is installed\n\n\n```\n!gfortran --version\n```\n\n GNU Fortran (Ubuntu 7.5.0-3ubuntu1~18.04) 7.5.0\n Copyright (C) 2017 Free Software Foundation, Inc.\n This is free software; see the source for copying conditions. There is NO\n warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.\n \n\n\nCheck the OS version\n\n\n```\n!lsb_release -a\n```\n\n No LSB modules are available.\n Distributor ID:\tUbuntu\n Description:\tUbuntu 18.04.3 LTS\n Release:\t18.04\n Codename:\tbionic\n\n\nInstall the csh shell\n\n\n```\n!sudo apt-get install csh\n```\n\n Reading package lists... Done\n Building dependency tree \n Reading state information... Done\n The following package was automatically installed and is no longer required:\n libnvidia-common-440\n Use 'sudo apt autoremove' to remove it.\n The following NEW packages will be installed:\n csh\n 0 upgraded, 1 newly installed, 0 to remove and 59 not upgraded.\n Need to get 243 kB of archives.\n After this operation, 358 kB of additional disk space will be used.\n Get:1 http://archive.ubuntu.com/ubuntu bionic-updates/universe amd64 csh amd64 20110502-3ubuntu0.18.04.1 [243 kB]\n Fetched 243 kB in 2s (143 kB/s)\n debconf: unable to initialize frontend: Dialog\n debconf: (No usable dialog-like program is installed, so the dialog based frontend cannot be used. at /usr/share/perl5/Debconf/FrontEnd/Dialog.pm line 76, <> line 1.)\n debconf: falling back to frontend: Readline\n debconf: unable to initialize frontend: Readline\n debconf: (This frontend requires a controlling tty.)\n debconf: falling back to frontend: Teletype\n dpkg-preconfigure: unable to re-open stdin: \n Selecting previously unselected package csh.\n (Reading database ... 144328 files and directories currently installed.)\n Preparing to unpack .../csh_20110502-3ubuntu0.18.04.1_amd64.deb ...\n Unpacking csh (20110502-3ubuntu0.18.04.1) ...\n Setting up csh (20110502-3ubuntu0.18.04.1) ...\n update-alternatives: using /bin/bsd-csh to provide /bin/csh (csh) in auto mode\n Processing triggers for man-db (2.8.3-2ubuntu0.1) ...\n\n\nSet loose permissions to avoid some reported file permissions issues\n\n\n```\n!chmod +777 /content/AI-Feynman/Code/*\n```\n\n chmod: cannot access '/content/AI-Feynman/Code/*': No such file or directory\n\n\n### Compile the fortran code\n\nLook at the code directory\n\n\n```\n!ls -l /content/AI-Feynman/Code\n```\n\n 10ops.txt\t\t\t S_brute_force.py\n 14ops.txt\t\t\t S_change_output.py\n 19ops.txt\t\t\t S_combine_pareto.py\n 7ops.txt\t\t\t S_final_gd.py\n ai_feynman_example.py\t\t S_get_number_DL.py\n ai_feynman_terminal_example.py\t S_get_number_DL_snapped.py\n arity2templates.txt\t\t S_get_symbolic_expr_error.py\n brute_force_oneFile_mdl_v2.scr\t S_NN_eval.py\n brute_force_oneFile_mdl_v3.scr\t S_NN_train.py\n brute_force_oneFile_v1.scr\t S_polyfit.py\n brute_force_oneFile_v2.scr\t S_polyfit_utils.py\n brute_force_oneFile_v3.scr\t S_remove_input_neuron.py\n compile.sh\t\t\t S_run_aifeynman.py\n dimensionalAnalysis.py\t\t S_run_bf_polyfit.py\n get_pareto.py\t\t\t S_separability.py\n getPowers.py\t\t\t S_snap.py\n RPN_to_eq.py\t\t\t S_symmetry.py\n RPN_to_pytorch.py\t\t symbolic_regress1.f\n S_add_bf_on_numbers_on_pareto.py symbolic_regress2.f\n S_add_snap_expr_on_pareto_polyfit.py symbolic_regress3.f\n S_add_snap_expr_on_pareto.py\t symbolic_regress_mdl2.f\n S_add_sym_on_pareto.py\t\t symbolic_regress_mdl3.f\n S_brute_force_number.py\t\t tools.f\n\n\nCompile .f files into .x files\n\n\n```\n!cd /content/AI-Feynman/Code/ && ./compile.sh\n```\n\n### Run the first example from the AI-Feynman repository\n\nChange working directory to the Code directory\n\n\n```\nimport os\nos.chdir(\"/content/AI-Feynman/Code/\")\nprint(os.getcwd())\n```\n\n /content/AI-Feynman/Code\n\n\n\n```\n!pwd\n```\n\n /content/AI-Feynman/Code\n\n\n\n```\n%%writefile ai_feynman_magic.py\nfrom S_run_aifeynman import run_aifeynman\n# Run example 1 as the regression dataset\nrun_aifeynman(\"/content/AI-Feynman/example_data/\",\"example1.txt\",30,\"14ops.txt\", polyfit_deg=3, NN_epochs=400)\n```\n\n Writing ai_feynman_magic.py\n\n\nLook at the first line of the example 1 file\n\n\n```\n!head -n 1 /content/AI-Feynman/example_data/example1.txt\n```\n\n 1.6821347439986711 1.1786188905177983 4.749225735259924 1.3238356535004034 3.462199507094163 \n\n\n\n```\n# Example 1 has data generated from an equation, where the last column is the regression target, and the rest of the columns are the input data\n# The following example shows the relationship between the first line of the file example1.txt and the formula used to make the data\nx=[1.6821347439986711,1.1786188905177983,4.749225735259924,1.3238356535004034,3.462199507094163]\nx0,x1,x2,x3=x[0],x[1],x[2],x[3]\n(x0**2 - 2*x0*x1 + x1**2 + x2**2 - 2*x2*x3 + x3**2)**0.5\n```\n\n\n\n\n 3.4621995070941636\n\n\n\nRun the code. It takes a long time, so go get some coffee.\n\n\n```\n!cd /content/AI-Feynman/Code/ && python3 ai_feynman_magic.py\n```\n\n### Assess the results\n\n\n```\n!cat results.dat \n```\n\n cat: results.dat: No such file or directory\n\n\nWe found a candidate with an excellent fit, let's see what we got\n\n\n```\n!ls -l /content/AI-Feynman/Code/results/\n```\n\n total 68\n drwxr-xr-x 2 root root 4096 Jun 25 19:09 mystery_world_acos\n drwxr-xr-x 2 root root 4096 Jun 25 19:09 mystery_world_asin\n drwxr-xr-x 2 root root 4096 Jun 25 19:09 mystery_world_atan\n drwxr-xr-x 2 root root 4096 Jun 25 19:11 mystery_world_cos\n drwxr-xr-x 2 root root 4096 Jun 25 19:13 mystery_world_exp\n drwxr-xr-x 2 root root 4096 Jun 25 19:15 mystery_world_inverse\n drwxr-xr-x 2 root root 4096 Jun 25 19:18 mystery_world_log\n drwxr-xr-x 2 root root 4096 Jun 25 19:21 mystery_world_sin\n drwxr-xr-x 2 root root 4096 Jun 25 19:23 mystery_world_sqrt\n drwxr-xr-x 2 root root 4096 Jun 25 19:26 mystery_world_squared\n drwxr-xr-x 2 root root 4096 Jun 25 19:30 mystery_world_tan\n drwxr-xr-x 3 root root 4096 Jun 25 17:48 NN_trained_models\n -rw-r--r-- 1 root root 157 Jun 25 19:32 solution_before_snap_example1.txt.txt\n -rw-r--r-- 1 root root 334 Jun 25 19:32 solution_example1.txt\n -rw-r--r-- 1 root root 157 Jun 25 19:32 solution_first_snap_example1.txt.txt\n drwxr-xr-x 2 root root 4096 Jun 25 18:32 translated_data_minus\n drwxr-xr-x 2 root root 4096 Jun 25 19:06 translated_data_multiply\n\n\n\n```\n!ls -l /content/AI-Feynman/Code/results/NN_trained_models/models\n```\n\n total 764\n -rw-r--r-- 1 root root 128288 Jun 25 17:55 example1.txt_train.h5\n -rw-r--r-- 1 root root 127766 Jun 25 18:30 example1.txt_train-translated_minus.h5\n -rw-r--r-- 1 root root 127772 Jun 25 17:58 example1.txt_train-translated_minus_pretrained.h5\n -rw-r--r-- 1 root root 127250 Jun 25 19:04 example1.txt_train-translated_minus-translated_minus.h5\n -rw-r--r-- 1 root root 127254 Jun 25 18:32 example1.txt_train-translated_minus-translated_minus_pretrained.h5\n -rw-r--r-- 1 root root 126731 Jun 25 19:06 example1.txt_train-translated_minus-translated_minus-translated_multiply_pretrained.h5\n\n\n\n```\n!cat /content/AI-Feynman/Code/results/solution_example1.txt\n```\n\n 30.834195568650475 4.946471395876568 593576.5675051882 0.0 30.834454019627426 0\n 29.66356302777754 4.890530251072885 586863.6301287463 1.0 29.661717905473818 1\n 29.66349190678506 4.820679114354003 578481.4937224804 4.0 28.259795376928494 exp(-23.140666108580+exp(pi))\n 0.0 -inf -inf 13.169925001442312 0.0 ((x0-x1)**2 + (x2-x3)**2)**0.5\n\n\nNote in the cell above that the solution with the lowest loss is the formula this data was generated from\n\n### Try our own dataset generation and equation learning\n\nUntil now we were not storing the results in Google Drive. We might want to keep the data in Drive so that the results don't disappear when this Collab instance gets nice and dead.\n\n\n```\nfrom google.colab import drive\ndrive.mount('/content/gdrive', force_remount=True)\n```\n\n Go to this URL in a browser: https://accounts.google.com/o/oauth2/auth?client_id=947318989803-6bn6qk8qdgf4n4g3pfee6491hc0brc4i.apps.googleusercontent.com&redirect_uri=urn%3aietf%3awg%3aoauth%3a2.0%3aoob&response_type=code&scope=email%20https%3a%2f%2fwww.googleapis.com%2fauth%2fdocs.test%20https%3a%2f%2fwww.googleapis.com%2fauth%2fdrive%20https%3a%2f%2fwww.googleapis.com%2fauth%2fdrive.photos.readonly%20https%3a%2f%2fwww.googleapis.com%2fauth%2fpeopleapi.readonly\n \n Enter your authorization code:\n \u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\n Mounted at /content/gdrive\n\n\nMake a directory in the mounted Google Drive where we will do our work\n\n\n```\n!mkdir -p /content/gdrive/My\\ Drive/Lemay.ai_research/AI-Feynman\n```\n\nCopy over the stuff we did so far, and from now on we work out of Google Drive\n\n\n```\n!cp -r /content/AI-Feynman /content/gdrive/My\\ Drive/Lemay.ai_research/\n```\n\nThe code below generates our regression example dataset\n\nWe generate points for 4 columns, where x0 is from the same equation as x1, and x2 is from the same equation as x3\nThe last column is Y\n\n\n```\nimport os\nimport random\n\nos.chdir(\"/content/gdrive/My Drive/Lemay.ai_research/AI-Feynman/example_data\")\n\ndef getY(x01,x23):\n y = -0.5*x01+0.5*x23+3\n return y\n\ndef getRow():\n [x0,x2]=[random.random() for x in range(2)]\n x1=x0\n x3=x2\n y=getY(x1,x3)\n return str(x0)+\" \"+str(x1)+\" \"+str(x2)+\" \"+str(x3)+\" \"+str(y)+\"\\n\"\n\nwith open(\"duplicateVarsExample.txt\", \"w\") as f:\n for _ in range(10000):\n f.write(getRow())\nf.close()\n\n# switch back to the code directory\nos.chdir(\"/content/gdrive/My Drive/Lemay.ai_research/AI-Feynman/Code\")\n```\n\nLet's look at our data\n\n\n```\n!head -n 20 ../example_data/duplicateVarsExample.txt\n```\n\n 0.6377827961278193 0.6377827961278193 0.5283691036027339 0.5283691036027339 2.9452931537374574\n 0.46101048568243563 0.46101048568243563 0.14970328506485298 0.14970328506485298 2.844346399691209\n 0.930127496631396 0.930127496631396 0.1719477165003871 0.1719477165003871 2.6209101099344956\n 0.49439694976966 0.49439694976966 0.04527854391777786 0.04527854391777786 2.775440797074059\n 0.5045856085467867 0.5045856085467867 0.8023019732381558 0.8023019732381558 3.1488581823456845\n 0.6481881660919896 0.6481881660919896 0.03194172397907147 0.03194172397907147 2.691876778943541\n 0.9155914812644674 0.9155914812644674 0.12185793898101016 0.12185793898101016 2.6031332288582716\n 0.2051879503661691 0.2051879503661691 0.33042763022358423 0.33042763022358423 3.0626198399287077\n 0.6719758819552796 0.6719758819552796 0.5321007586743085 0.5321007586743085 2.9300624383595144\n 0.4937648322442755 0.4937648322442755 0.8677873608667648 0.8677873608667648 3.1870112643112445\n 0.2754568587829048 0.2754568587829048 0.5122231062727486 0.5122231062727486 3.118383123744922\n 0.04825794205395173 0.04825794205395173 0.992569726331206 0.992569726331206 3.4721558921386273\n 0.06357405463706156 0.06357405463706156 0.5507102162097997 0.5507102162097997 3.243568080786369\n 0.8125166608435264 0.8125166608435264 0.8967168536653939 0.8967168536653939 3.042100096410934\n 0.908612297146007 0.908612297146007 0.1429574494809186 0.1429574494809186 2.617172576167456\n 0.49574780752769376 0.49574780752769376 0.21176161382235636 0.21176161382235636 2.858006903147331\n 0.2703992647388135 0.2703992647388135 0.3386951046641967 0.3386951046641967 3.0341479199626917\n 0.43877750512732683 0.43877750512732683 0.9017369101062235 0.9017369101062235 3.231479702489448\n 0.6339494882507828 0.6339494882507828 0.2911749626469061 0.2911749626469061 2.828612737198062\n 0.6733412030877575 0.6733412030877575 0.18573221180762323 0.18573221180762323 2.756195504359933\n\n\nLet's also plot the data for x01 and x23 against Y\n\n\n```\n%matplotlib inline\nimport matplotlib.pyplot as plt\nimport pandas as pd\nplt.style.use('seaborn-whitegrid')\nimport numpy as np\n\ndf=pd.read_csv(\"../example_data/duplicateVarsExample.txt\",sep=\" \",header=None)\ndf.plot.scatter(x=0, y=4)\ndf.plot.scatter(x=2, y=4)\n```\n\n\n```\n!pwd\n```\n\n /content/gdrive/My Drive/Lemay.ai_research/AI-Feynman/Code\n\n\nLet's write out the runner file for this experiment\n\n\n```\n%%writefile ai_feynman_duplicate_variables.py\nfrom S_run_aifeynman import run_aifeynman\nrun_aifeynman(\"/content/gdrive/My Drive/Lemay.ai_research/AI-Feynman/example_data/\",\"duplicateVarsExample.txt\",30,\"14ops.txt\", polyfit_deg=3, NN_epochs=400)\n```\n\n Overwriting ai_feynman_duplicate_variables.py\n\n\nDon't forget to lower the file permissions\n\n\n```\n!chmod 777 /content/gdrive/My\\ Drive/Lemay.ai_research/AI-Feynman/Code/*\n!chmod +x /content/gdrive/My\\ Drive/Lemay.ai_research/AI-Feynman/Code/*.scr\n```\n\nNow we run the file, and go get more coffee, because this is not going to be fast...\n\n\n```\n!python3 ai_feynman_duplicate_variables.py\n```\n\nInitial models quickly mapped to x0 and x2 (the system realized x1 and x3 are duplicates and so not needed)\n\nLater on the system found 3.000000000000+log(sqrt(exp((x2-x1)))) which is a bit crazy but looks like a plane\n\nWe can see on Wolfram alpha that an equivalent form of this equation is:\n\n(x2 - x1)/2 + 3.000000000000 \n\nwhich is what we used to generate the dataset!\n\nLink: https://www.wolframalpha.com/input/?i=3.000000000000%2Blog%28sqrt%28exp%28%28x2-x1%29%29%29%29\n\n\n```\n!ls -l /content/gdrive/My\\ Drive/Lemay.ai_research/AI-Feynman/Code/results/\n```\n\n total 64\n drwx------ 2 root root 4096 Jun 26 00:03 mystery_world_acos\n drwx------ 2 root root 4096 Jun 26 00:03 mystery_world_asin\n drwx------ 2 root root 4096 Jun 26 00:03 mystery_world_atan\n drwx------ 2 root root 4096 Jun 26 00:04 mystery_world_cos\n drwx------ 2 root root 4096 Jun 26 00:06 mystery_world_exp\n drwx------ 2 root root 4096 Jun 26 00:07 mystery_world_inverse\n drwx------ 2 root root 4096 Jun 26 00:08 mystery_world_log\n drwx------ 2 root root 4096 Jun 26 00:09 mystery_world_sin\n drwx------ 2 root root 4096 Jun 26 00:11 mystery_world_sqrt\n drwx------ 2 root root 4096 Jun 26 00:12 mystery_world_squared\n drwx------ 2 root root 4096 Jun 26 00:13 mystery_world_tan\n drwx------ 3 root root 4096 Jun 25 21:40 NN_trained_models\n -rw------- 1 root root 266 Jun 26 00:15 solution_before_snap_duplicateVarsExample.txt.txt\n -rw------- 1 root root 157 Jun 25 22:23 solution_before_snap_example1.txt.txt\n -rw------- 1 root root 539 Jun 26 00:15 solution_duplicateVarsExample.txt\n -rw------- 1 root root 334 Jun 25 22:23 solution_example1.txt\n -rw------- 1 root root 266 Jun 26 00:15 solution_first_snap_duplicateVarsExample.txt.txt\n -rw------- 1 root root 157 Jun 25 22:23 solution_first_snap_example1.txt.txt\n drwx------ 2 root root 4096 Jun 26 00:02 translated_data_divide\n drwx------ 2 root root 4096 Jun 25 22:23 translated_data_minus\n drwx------ 2 root root 4096 Jun 25 22:23 translated_data_multiply\n\n\n\n```\n!cat /content/gdrive/My\\ Drive/Lemay.ai_research/AI-Feynman/Code/results/solution_duplicateVarsExample.txt\n```\n\n 31.580135125550523 4.8288606337975075 48288.60633797508 0.0 28.420511876226403 8.91505505199440e-11\n 26.833969862635307 4.745595047207644 47455.95047207644 2.0 26.82665082198086 3.00000000000000\n 26.656936606704242 4.735816831122182 47358.16831122182 15.584962500721156 26.645441468460167 (11 - 3*x1)**0.5\n 21.0914047679013 4.397594700155478 43975.947001554785 24.831703099214298 21.076957194831987 (-3*x1 + 3*x3 + 9)**0.5\n 6.801912219921964e-08 -23.796050345232526 -237960.50345232527 29.0 6.86554578556978e-08 log(sqrt(exp(-x1 + x3))) + 3.0\n\n\nThe solver settled on *log(sqrt(exp(-x1 + x3))) + 3.0* which we know is correct\n\nNow, that was a bit of a softball problem as it has an exact solution. Let's now add noise to the dataset and see how the library holds up\n\n### Let's add small amount of noise to every variabe and see the fit quality\n\nWe do the same thing as before, but now we add or subtract noise to x0,x1,x2,x3 after generating y\n\n\n```\nimport os\nimport random\n\nos.chdir(\"/content/gdrive/My Drive/Lemay.ai_research/AI-Feynman/example_data\")\n\ndef getY(x01,x23):\n y = -0.5*x01+0.5*x23+3\n return y\n\ndef getRow():\n x=[random.random() for x in range(4)]\n x[1]=x[0]\n x[3]=x[2]\n y=getY(x[1],x[3])\n mu=0\n sigma=0.05\n noise=np.random.normal(mu, sigma, 4)\n x=x+noise\n return str(x[0])+\" \"+str(x[1])+\" \"+str(x[2])+\" \"+str(x[3])+\" \"+str(y)+\"\\n\"\n\nwith open(\"duplicateVarsWithNoise100k.txt\", \"w\") as f:\n for _ in range(100000):\n f.write(getRow())\nf.close()\n\n# switch back to the code directory\nos.chdir(\"/content/gdrive/My Drive/Lemay.ai_research/AI-Feynman/Code\")\n```\n\nLet's have a look at the data\n\n\n```\n!head -n 20 ../example_data/duplicateVarsWithNoise100k.txt\n```\n\n 0.6625198249931923 0.7746267346006698 0.9252603835318501 0.9674500116553322 3.115987224559114\n 0.6812134926175376 0.7636599187847101 0.9760315605700961 0.9350752814995479 3.103432362631433\n 0.5083181923447346 0.5327913954314522 0.11759059178639686 0.16996410443663035 2.7988933356602868\n 0.2327635863820598 0.34378888555362863 0.3392341044068419 0.4209338271769121 3.0456857290570944\n 0.33853556069988994 0.4227228753021112 0.574864918065757 0.5257643625852129 3.124623360946375\n 0.7899871443914812 0.7338043487930863 0.4246455367737062 0.35284124988204574 2.832517326918034\n 0.30254885040977764 0.33416064982330146 0.24768029465282182 0.2119893606276008 2.9436014155307717\n 0.337628417696602 0.32131828864656603 0.9501443899830265 0.9565059526859492 3.3022833107216654\n 0.07432257150298417 0.05170573429130186 0.7088134537703481 0.712558554231897 3.3389949654126516\n 0.3577868772203678 0.39209356026845843 0.9396291723006092 0.8402984134521431 3.2572973884818737\n 0.12810214743718631 0.045442753465532705 0.8170629600063886 0.8825925792066633 3.4038366719186985\n 0.7483023142762378 0.8048087165073571 0.12609558649566657 0.14389709368994177 2.676793895563108\n 0.350289420116793 0.41400495446417884 0.13793534619464634 0.2696463626511338 2.9104795373290626\n 0.21898747430173285 0.20001780091821206 0.27834373474121465 0.2675373963192755 3.017834206135602\n 0.5906039149941138 0.4209183281980938 0.2874496045479292 0.2803818704005349 2.8431726977803735\n 0.20593423150883722 0.2800603688869042 0.3678417378115894 0.2609145934578154 3.037039103892987\n 0.4334179428294742 0.4531829384095051 0.020817509082557886 -0.025542567619868606 2.7627438617849696\n 0.47165641921812007 0.5613748177478919 0.7630592802418188 0.6914218500942457 3.097991308929802\n 0.6978200533798706 0.6334439445190119 0.44698678323375945 0.5129192888540692 2.9219085795958932\n 0.29412699838655565 0.33585967631150376 0.5680740078841231 0.5539205896495891 3.113805190448427\n\n\nNow let's plot the data\n\n\n```\n%matplotlib inline\nimport matplotlib.pyplot as plt\nimport pandas as pd\nplt.style.use('seaborn-whitegrid')\nimport numpy as np\n\ndf=pd.read_csv(\"../example_data/duplicateVarsWithNoise100k.txt\",sep=\" \",header=None)\ndf.plot.scatter(x=0, y=4)\ndf.plot.scatter(x=1, y=4)\ndf.plot.scatter(x=2, y=4)\ndf.plot.scatter(x=3, y=4)\n```\n\n\n```\n%%writefile ai_feynman_duplicateVarsWithNoise.py\nfrom S_run_aifeynman import run_aifeynman\nrun_aifeynman(\"/content/gdrive/My Drive/Lemay.ai_research/AI-Feynman/example_data/\",\"duplicateVarsWithNoise100k.txt\",30,\"14ops.txt\", polyfit_deg=3, NN_epochs=600)\n```\n\n Overwriting ai_feynman_duplicateVarsWithNoise.py\n\n\n\n```\n!chmod +777 /content/gdrive/My\\ Drive/Lemay.ai_research/AI-Feynman/Code/*\n!chmod +777 /content/gdrive/My\\ Drive/Lemay.ai_research/AI-Feynman/*\n# switch back to the code directory\nos.chdir(\"/content/gdrive/My Drive/Lemay.ai_research/AI-Feynman/Code/\")\n```\n\n\n```\n!pwd\n```\n\n /content/gdrive/My Drive/Lemay.ai_research/AI-Feynman/Code\n\n\n\n```\n!chmod +x /content/gdrive/My\\ Drive/Lemay.ai_research/AI-Feynman/Code/*.scr\n!ls -l *.scr\n```\n\n -rwx------ 1 root root 653 Jun 25 22:23 brute_force_oneFile_mdl_v2.scr\n -rwx------ 1 root root 654 Jun 25 22:23 brute_force_oneFile_mdl_v3.scr\n -rwx------ 1 root root 541 Jun 25 22:23 brute_force_oneFile_v1.scr\n -rwx------ 1 root root 608 Jun 25 22:23 brute_force_oneFile_v2.scr\n -rwx------ 1 root root 609 Jun 25 22:23 brute_force_oneFile_v3.scr\n\n\n\n```\nprint(os.getcwd())\n!sudo python3 ai_feynman_duplicateVarsWithNoise.py\n```\n\n /content/gdrive/My Drive/Lemay.ai_research/AI-Feynman/Code\n Checking for brute force + \n \n set: Syntax Error.\n Checking for brute force * \n \n set: Syntax Error.\n Checking polyfit \n \n Complexity RMSE Expression\n [51.36220747209003, 26.857860406793748, '3 - 0.268004037730363*x1']\n [102.8220057215417, 25.764121831972535, '-0.268004037730363*x1 + 0.267581809554636*x3 + 3']\n [200.06097637922895, 24.03915719495499, '-0.265982341542862*x0 - 0.268004037730363*x1 + 0.26678281580511*x2 + 0.267581809554636*x3 + 3']\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_atan/duplicateVarsWithNoise100k.txt_train...\n /bin/cp -p results/mystery_world_atan/duplicateVarsWithNoise100k.txt_train mystery.dat\n Number of variables..... 4\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabcd\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.14879262308092922 at j= 1\n Searching for best fit...\n 0.062981887482 -1.912149389642 P 1 0.0276 12.6207 45.8400 12629.0104\n 0.062981887482 -21.911251304432 PE 46 0.0276 18.1443 51.3636 12634.5340\n 0.062981817036 1.229443420950 aPS* 907 0.0276 22.4457 55.6650 12638.8349\n 0.050426421945 1.206233526430 cPE/ 1434 0.0250 22.7858 56.0051 12598.1827\n 0.045130035004 1.211525526305 dPE/ 1435 0.0209 22.6267 55.8460 12526.6887\n 0.044973169701 1.212267751221 dPE>/ 15180 0.0211 26.0247 59.2440 12533.1000\n 0.036207235544 1.263836988520 aPE~/ 15627 0.0159 25.7538 58.9731 12418.0517\n 0.036159256115 1.026911106772 aP>+\\ 19702 0.0156 26.0862 59.3055 12409.3835\n 0.035829127758 -1.877150369952 PEa-L 50136 0.0158 27.4205 60.6398 12416.4613\n 0.035827242227 1.264972421112 abPE-/ 109067 0.0157 28.5417 61.7610 12414.9749\n 0.035755311982 1.265390401260 aPE~>/ 219632 0.0157 29.5487 62.7680 12415.5702\n 0.034633639710 1.218225539043 PaC~/E 308686 0.0153 29.9937 63.2130 12406.8030\n 0.022307859449 1.240627163579 ca-PE/ 418564 0.0085 29.7984 63.0177 12165.8248\n 0.020265359038 1.245919163454 da-PE/ 418565 0.0049 29.6599 62.8792 11939.4705\n 0.019509604913 1.243483373802 db-PE/ 418570 0.0062 29.6051 62.8244 12038.9688\n 0.018458838413 1.244117501261 db-PE<~\\RPSCLE\n Arity 0 : Pabcd\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 1.2924252388523800 at j= 40336\n Searching for best fit...\n 0.101906033496 0.411391719232 P 1 0.0492 13.3150 46.5342 12864.9112\n 0.101906033496 0.602605125527 PL> 186 0.0492 20.8541 54.0734 12872.4504\n 0.054576957867 0.053513442286 cPE+ 759 0.0228 21.9820 55.2013 12560.5050\n 0.051990432075 -0.055826832652 aPE- 1207 0.0221 22.5812 55.8005 12547.6673\n 0.050835705549 -0.058347161991 aPE<- 13377 0.0212 26.0191 59.2384 12535.2026\n 0.050600112914 -0.061515292886 PcPE+- 102321 0.0202 28.9477 62.1669 12517.5074\n 0.026747068219 -0.053491481058 acPE+- 102322 0.0109 28.0279 61.2472 12265.1251\n 0.025708954333 -0.053517155650 bcPE+- 102323 0.0103 27.9708 61.1901 12243.6120\n 0.024744192729 -0.053564182106 bdPE+- 102328 0.0093 27.9157 61.1350 12201.1500\n 0.024056305913 -0.051365540794 acPE>+- 1418567 0.0088 31.6682 64.8875 12181.1835\n 0.022381309029 -0.051389214692 bcPE>+- 1418568 0.0082 31.5641 64.7834 12156.2838\n 0.021926434053 -0.051432574264 bdPE>+- 1418573 0.0073 31.5345 64.7537 12106.9613\n 0.020836990435 -0.053855935070 aPEcS+- 2476052 0.0056 32.2645 65.4838 11998.3225\n 0.019320538482 -0.053881650788 aPEdS+- 2476277 0.0052 32.1557 65.3749 11967.5890\n 0.018029939245 0.053866461531 ca-SPE+ 6911509 0.0041 33.5367 66.7560 11877.6997\n 0.016974884879 0.053891788488 da-SPE+ 6911510 0.0040 33.4498 66.6690 11861.7843\n 0.016224866568 1.082478121228 dPb-+RRR 38105560 0.0036 35.8475 69.0668 11822.3882\n Checking polyfit \n \n Complexity RMSE Expression\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [96.05127279846616, 23.499172324266443, tan(0.0365815024226761*x3 + 0.87662273645401)]\n [96.45193157676024, 23.251340086812633, tan(0.0187665168195963*x2 - 2.25580163260941)]\n [142.88493409902946, 22.82711097251165, tan(-0.00816557370126247*x1 + 0.0305992718786001*x2 + 0.883173022410234)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_cos/duplicateVarsWithNoise100k.txt_train...\n /bin/cp -p results/mystery_world_cos/duplicateVarsWithNoise100k.txt_train mystery.dat\n Number of variables..... 4\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabcd\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.14879262308092922 at j= 1\n Searching for best fit...\n 0.144952083475 -4.088682624876 P 1 0.0435 13.8233 47.0426 12814.2703\n 0.144952083475 -3.088682624876 P< 11 0.0435 17.2827 50.5020 12817.7297\n 0.144952083475 -10.816694834246 PP* 76 0.0435 20.0712 53.2905 12820.5182\n 0.144952083475 -63.850019664652 P>E 511 0.0435 22.8205 56.0397 12823.2675\n 0.144952034236 -0.947089836909 cPS* 909 0.0435 23.6514 56.8707 12824.0984\n 0.137559331853 -0.981483521012 aPE/ 1432 0.0338 24.2316 57.4509 12721.5104\n 0.135023162466 -0.979047731361 bPE/ 1433 0.0345 24.2057 57.4250 12730.6110\n 0.109661197995 -1.218926209600 cP+\\ 1759 0.0414 24.2013 57.4206 12804.7239\n 0.102969465923 -1.898072989182 cd<*C 22674 0.0654 27.7987 61.0180 12995.5292\n 0.101107285815 -1.898226244931 ccE/C 23294 0.0440 27.8113 61.0306 12833.4308\n 0.100094888640 0.011609121341 cC>S~ 68849 0.0436 29.3602 62.5795 12831.8958\n 0.096110549643 -1.003068161793 acP+-E 145322 0.0259 30.3794 63.5987 12620.1504\n 0.089834698759 -1.010041903913 ac>>E/ 235197 0.0209 30.9766 64.1958 12532.7459\n 0.088990594753 -1.005583582791 bc>>E/ 235198 0.0218 30.9630 64.1822 12550.5876\n 0.079909538237 -0.999991890051 acE>E/ 235397 0.0187 30.8089 64.0282 12487.7757\n 0.069415848218 -1.307688402531 Pac<*+\\ 1715881 0.0182 33.4716 66.6908 12480.2998\n 0.068095566477 -1.017107231498 ac>RE-E 4329417 0.0228 34.7791 67.9984 12573.7690\n 0.067569894568 -1.371864129514 acEE/E/ 32984852 0.0266 37.4985 70.7178 12639.5328\n 0.058482178237 -1.716376120144 ac>\\ 95388139 0.0175 38.9967 72.2160 12469.0843\n 0.057491774043 -1.514539755523 ac<*SCRC 97696422 0.0208 39.0309 72.2502 12539.5055\n 0.056249060564 -1.640128615752 ac<*SCSC 97769322 0.0210 39.0005 72.2198 12543.6579\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_cos/duplicateVarsWithNoise100k.txt_train...\n /bin/cp -p results/mystery_world_cos/duplicateVarsWithNoise100k.txt_train mystery.dat\n Number of variables..... 4\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabcd\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.99999999999255185 at j= 74523\n Searching for best fit...\n 0.197862199604 -0.318309877324 P 1 0.0482 14.2722 47.4915 12856.2782\n 0.197862199604 -0.241453001907 P> 6 0.0482 16.8572 50.0765 12858.8632\n 0.197862199604 -0.758546998086 P\\> 166 0.0482 21.6472 54.8665 12863.6532\n 0.172944228023 -0.211684725947 cP>+ 559 0.0542 23.2047 56.4240 12913.6219\n 0.172299119992 -0.211224012199 dP>+ 560 0.0538 23.2019 56.4212 12910.2110\n 0.171914448661 -0.270882757087 PcS+ 696 0.0572 23.5123 56.7316 12935.6264\n 0.167510671221 -0.270254948127 PdS+ 701 0.0570 23.4852 56.7045 12934.1058\n 0.123778857593 -0.244260646241 PaC+ 711 0.0334 23.0692 56.2885 12716.2749\n 0.119672300871 -0.242807883204 PbC+ 716 0.0333 23.0306 56.2499 12714.5608\n 0.111395958715 0.120517069681 bP> 59798 0.0347 29.3101 62.5294 12738.4615\n 0.100724362077 -1.668392040105 aC>SC 83427 0.0401 29.6464 62.8656 12797.9594\n 0.100025570395 -0.326328572668 PbcE>/- 1449011 0.0278 33.7547 66.9740 12652.1257\n 0.095428833406 -0.249238291680 Pac<*>+ 1499881 0.0280 33.7366 66.9559 12654.9146\n 0.091772650865 -0.246813506837 Pbc<*>+ 1499886 0.0290 33.6803 66.8996 12669.5961\n 0.085216610769 -0.246574521645 Pbc<*E+ 1535886 0.0211 33.6076 66.8268 12539.7533\n 0.084007476516 -0.246451111115 Pbd<*E+ 1535911 0.0210 33.5870 66.8062 12538.6470\n 0.077952254330 -0.572444253088 Pbc<*+R 1733886 0.0170 33.6540 66.8732 12451.4108\n 0.077471243735 -0.572236192882 Pbd<*+R 1733911 0.0169 33.6450 66.8643 12449.8148\n 0.073005052816 -1.002709593793 bc 5706217 0.0154 35.2239 68.4432 12413.7697\n 0.067392868939 -1.004181049809 ac<*SCR 5997817 0.0147 35.2344 68.4537 12395.3576\n 0.065135385866 -1.001923563978 bd<*SCR 5997823 0.0135 35.1852 68.4045 12360.7307\n 0.063823667187 -0.553977620491 Padb-*+R 20099361 0.0144 36.9005 70.1198 12387.7602\n 0.062422647738 -0.709034586875 Pca<-S+L 28006171 0.0158 37.3471 70.5664 12425.1195\n 0.062416148251 -0.581643910935 ca<-S>>R 96044239 0.0149 39.1249 72.3442 12403.1874\n 0.060931383804 -0.579351829595 cb<-S>>R 96044244 0.0158 39.0902 72.3094 12427.6392\n 0.059883319008 -0.579154236687 db<-S>>R 96044245 0.0159 39.0651 72.2844 12429.3692\n 0.059530437206 -1.000580938026 ca-SES<~\\RPSCLE\n Arity 0 : Pabcd\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.14879262308092922 at j= 1\n Searching for best fit...\n 16.393614936972 13.548925362924 P 1 5.6739 20.6447 53.8640 14802.0853\n 15.925339372268 16.153426631980 c 4 5.5562 22.6029 55.8222 14795.5289\n 15.818729802708 16.275887185191 d 5 5.4716 22.9151 56.1344 14789.5868\n 15.587806783349 17.486410759303 a~ 17 5.3293 24.6594 57.8787 14780.6012\n 15.548016615215 14.979495044555 cE 49 5.4303 26.1830 59.4023 14789.7881\n 15.257542595539 15.176706185965 dE 50 5.2929 26.1849 59.4042 14779.3582\n 15.248402996698 15.861256266445 dd+ 75 5.2791 26.7690 59.9883 14778.8785\n 15.160446270282 15.003195434378 cP* 79 5.3504 26.8357 60.0549 14784.4284\n 14.658144048727 15.387916619405 dP* 80 5.0728 26.8052 60.0245 14762.7088\n 14.285966765271 12.039475213544 c>E 514 5.0996 29.4518 62.6711 14767.5390\n 13.496376515947 12.575550675609 d>E 515 4.7117 29.3726 62.5919 14735.2663\n 12.324180811175 11.155897275968 cEE 554 4.4129 29.3468 62.5661 14708.6322\n 11.930740755145 12.146498852999 dEE 555 3.8076 29.3026 62.5219 14648.4364\n 11.487999454807 4.047672731599 c>>E 6499 4.8986 32.7977 66.0170 14754.7900\n 10.101496410274 5.504876918813 d>>E 6500 3.8903 32.6124 65.8316 14660.7527\n 8.801028792255 9.204459129615 aCEE 6887 3.0927 32.4970 65.7163 14567.2097\n 8.521876695587 14.640355185081 dEa-E 51040 2.8683 35.3402 68.5594 14539.3650\n 8.364072569628 8.667367657658 caCEE+ 182464 2.8259 37.1511 70.3704 14535.1245\n 8.263632319686 8.789828210869 daCEE+ 182465 2.8082 37.1337 70.3530 14532.5593\n 8.024426119611 13.586566593627 daCEE* 200690 2.2340 37.2286 70.4479 14439.3573\n 7.201955709825 8.406913836447 daC>+E 302165 2.0881 37.6630 70.8823 14412.3923\n 5.940305148226 10.986340515814 ca<<-E 306314 1.8774 37.4048 70.6241 14368.9961\n 5.168403202861 11.643799435957 da<<-E 306315 1.4137 37.2040 70.4233 14253.2470\n 4.475746704080 20.453420827080 PPda-** 1310526 0.8735 39.0935 72.3128 14058.8837\n 4.475746704080 10.583815876697 PPd>a-** 20420231 0.8735 43.0553 76.2745 14062.8455\n 4.325941859764 12.803952775315 dabC>--E 24969315 1.2748 43.2963 76.5156 14217.4013\n 4.173888490739 12.427401795395 dba+C>+E 34320295 1.1055 43.7036 76.9229 14159.7064\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_exp/duplicateVarsWithNoise100k.txt_train...\n /bin/cp -p results/mystery_world_exp/duplicateVarsWithNoise100k.txt_train mystery.dat\n Number of variables..... 4\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabcd\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 33.084133040908405 at j= 40336\n Searching for best fit...\n 20.881362044207 10.531006329689 P 1 13.2654 20.9938 54.2131 15148.6354\n 14.694850987885 16.453888352673 c> 9 5.4843 23.6568 56.8761 14791.3811\n 14.692220017790 8.008535889634 dP+ 55 9.1896 26.2680 59.4873 15004.6262\n 14.391048021624 -10.497872932292 aP- 102 7.8414 27.1292 60.3484 14940.7772\n 14.342022700475 10.988784291062 c>> 154 7.5914 27.7186 60.9379 14928.1475\n 14.272546246509 11.887203503047 cPR+ 659 7.1883 29.8089 63.0282 14907.9813\n 9.096294658049 16.454290628080 caC+ 714 2.7762 29.2747 62.4940 14519.8883\n 8.889942479703 16.629647432885 daC+ 715 2.9551 29.2436 62.4629 14545.3659\n 8.782135999829 16.629248273877 dbC+ 720 2.9572 29.2361 62.4553 14545.6725\n 7.493990570375 7.948662427067 cPa-+ 7179 4.6153 32.3249 65.5442 14730.6280\n 7.477538320656 7.989359807091 dPa-+ 7180 4.7422 32.3220 65.5413 14741.6927\n 7.269638757907 16.372342498009 ca>\\+ 9584 2.3905 32.6979 65.9172 14462.5998\n 4.723875771848 -10.952712631350 ac>>- 12967 1.6965 32.5122 65.7314 14323.1029\n 4.723875771848 10.952712631350 ca<<- 13209 1.6965 32.5388 65.7581 14323.1296\n 3.824452435723 19.860581652235 ca-ER 31559 0.7671 33.4907 66.7099 14000.5171\n 3.658248648165 19.976020617047 cb-ER 31564 0.7462 33.4268 66.6461 13989.2504\n 3.658248648165 19.976020617047 bc-ER\\ 483773 0.7462 37.3648 70.5840 13993.1883\n 3.575780032063 15.170990558946 ca>C+ER 4868509 0.7325 40.6629 73.8822 13988.9394\n 3.343730579052 20.542108722285 ca-PR 80735464 0.9767 44.6065 77.8257 14110.3766\n Checking polyfit \n \n Complexity RMSE Expression\n [43.039100017307746, 1000000, 'acos(-x0**2 + 2*x0*x1 - x1**2 - x2**2 + 2*x2*x3 - x3**2 - 1)']\n [46.46201740344512, 24.34147126490242, -0.969305992126465]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [96.05127279846616, 23.499172324266443, tan(0.0365815024226761*x3 + 0.87662273645401)]\n [96.45193157676024, 23.251340086812633, tan(0.0187665168195963*x2 - 2.25580163260941)]\n [142.88493409902946, 22.82711097251165, tan(-0.00816557370126247*x1 + 0.0305992718786001*x2 + 0.883173022410234)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_inverse/duplicateVarsWithNoise100k.txt_train...\n /bin/cp -p results/mystery_world_inverse/duplicateVarsWithNoise100k.txt_train mystery.dat\n Number of variables..... 4\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabcd\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.14879262308092922 at j= 1\n Searching for best fit...\n 0.069468333651 -2.786332860199 P 1 0.0307 12.7621 45.9814 12672.5981\n 0.069468333651 0.036950003487 P\\ 21 0.0307 17.1545 50.3737 12676.9904\n 0.069468333651 -9.514345069570 PP* 76 0.0307 19.0101 52.2293 12678.8460\n 0.069468333261 666.000000000000 PS\\ 311 0.0307 21.0429 54.2622 12680.8788\n 0.069467394488 -666.000000000000 PEE 551 0.0307 21.8680 55.0873 12681.6957\n 0.042277185671 0.320866243664 aPE/ 1432 0.0185 22.5295 55.7488 12475.6603\n 0.041293892612 0.319312830924 aPE+\\ 33510051 0.0043 35.8868 69.1061 11891.6525\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_inverse/duplicateVarsWithNoise100k.txt_train...\n /bin/cp -p results/mystery_world_inverse/duplicateVarsWithNoise100k.txt_train mystery.dat\n Number of variables..... 4\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabcd\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.39973408779867031 at j= 33304\n Searching for best fit...\n 0.113942540636 0.127239308450 P 1 0.0688 13.4760 46.6953 13001.4238\n 0.083790214358 0.097293018510 aP+ 52 0.0283 18.7330 51.9523 12645.4770\n 0.083367607295 0.097238043016 bP+ 53 0.0282 18.7532 51.9725 12643.2090\n 0.078503132505 -0.126189228678 cP- 104 0.0267 19.6390 52.8583 12622.3779\n 0.075871619652 0.100821772412 PaS+ 686 0.0340 22.3114 55.5307 12722.7738\n 0.075614255978 -0.095911579949 cP>- 1009 0.0251 22.8632 56.0824 12599.9856\n 0.073230543804 0.224593268411 Pc-R 1921 0.0342 23.7459 56.9652 12727.7022\n 0.072485659935 0.100011042270 Pc/+ 95761 0.0113 28.5462 61.7655 12279.2321\n 0.037655540298 0.097764547777 Pad>/+ 95786 0.0126 28.4262 61.6455 12325.8600\n 0.037638071665 0.097769435943 PadE/+ 96786 0.0097 28.4405 61.6598 12220.3206\n 0.037620095720 0.097774461811 Pad<*- 102661 0.0089 28.5248 61.7441 12185.7631\n 0.031128041275 0.196584424524 aPc-+R 136632 0.0137 28.6639 61.8832 12359.8100\n 0.030150925275 0.196529227654 bPc-+R 136633 0.0136 28.6179 61.8372 12356.5592\n 0.027995361901 0.054936493238 aPPc-++ 1298177 0.0112 31.7590 64.9783 12280.9491\n 0.026883031425 0.054918961115 bPPc-++ 1298178 0.0111 31.7005 64.9198 12276.6702\n 0.025618004975 -0.058114517054 caPRE+- 1427059 0.0085 31.7676 64.9868 12169.1730\n 0.024378187904 -0.058094898189 cbPRE+- 1427064 0.0084 31.6960 64.9153 12163.6519\n 0.021805198139 0.585601728153 cPb-+R\\ 2288854 0.0047 32.2167 65.4359 11926.9580\n 0.021080569482 0.168536893973 ac-P/E> 6730517 0.0054 33.7240 66.9433 11986.8956\n 0.020524020576 0.168464824105 bc-P/E> 6730518 0.0053 33.6854 66.9047 11979.1939\n 0.019581979370 0.168102977325 PcPb-+L- 19826771 0.0046 35.1763 68.3955 11921.4878\n 0.018765329475 0.596372215208 cPb-+\\SR 38121559 0.0042 36.0580 69.2772 11886.5955\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545942911475, 8.91505505199440e-11]\n [17.136430453348055, 28.41854594291147, '1/(-666.000000000000+exp(exp(pi)))']\n [32.8450013789013, 28.418212030052608, 7.71490571179178e-5]\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [96.05127279846616, 23.499172324266443, tan(0.0365815024226761*x3 + 0.87662273645401)]\n [96.45193157676024, 23.251340086812633, tan(0.0187665168195963*x2 - 2.25580163260941)]\n [142.88493409902946, 22.82711097251165, tan(-0.00816557370126247*x1 + 0.0305992718786001*x2 + 0.883173022410234)]\n [206.64700957832872, 22.674175327513932, 0.823345422744751*(-0.971410036087036*x1 + x2 + 3.18614816665649)**(-0.76738178730011)]\n [252.9985467754702, 22.3285090516094, 0.778373181819916*sin((-0.984130799770355*x1 + x2 + 3.16531944274902)**(-1.07100188732147))**0.676108956336975]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_log/duplicateVarsWithNoise100k.txt_train...\n /bin/cp -p results/mystery_world_log/duplicateVarsWithNoise100k.txt_train mystery.dat\n Number of variables..... 4\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabcd\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.14879262308092922 at j= 1\n Searching for best fit...\n 0.217586891281 -2.106687042311 P 1 0.0918 14.4093 47.6286 13119.4478\n 0.217586891281 -1.106687042311 P< 11 0.0918 17.8687 51.0880 13122.9072\n 0.217586891281 -22.105788957101 PE 46 0.0918 19.9329 53.1521 13124.9713\n 0.213821123301 0.381862491207 cCC 454 0.0914 23.2107 56.4299 13126.2484\n 0.167733513980 1.009453377489 ccS- 1149 0.0705 24.2000 57.4193 13021.7332\n 0.155171778142 1.023126930609 ddS- 1155 0.0624 24.0953 57.3145 12971.6786\n 0.150554431580 0.005667259552 ddS/ 1380 0.0607 24.3085 57.5277 12960.5542\n 0.137663991643 1.461217658240 aP-\\ 1807 0.0526 24.5682 57.7875 12903.0855\n 0.122044168028 0.677238175923 a>>\\ 4472 0.0512 25.7018 58.9211 12892.7721\n 0.121132708302 0.305267508449 aP+L\\ 30502 0.0508 28.4609 61.6802 12892.7860\n 0.120059302172 0.809254337946 aERE\\ 73897 0.0500 29.7247 62.9440 12887.2810\n 0.119568644462 0.144452353251 aSSCR 76762 0.0506 29.7737 62.9929 12892.3609\n 0.116709165826 1.998558183594 aC<>-/E 25009111 0.0117 36.8697 70.0889 12303.0944\n 0.050033884664 0.741111800435 ad-ERE\\S 121068227 0.0125 39.1399 72.3592 12333.0146\n 0.049917888013 0.694283074286 ad-E>\\>L 122117627 0.0117 39.1490 72.3683 12306.3694\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_log/duplicateVarsWithNoise100k.txt_train...\n /bin/cp -p results/mystery_world_log/duplicateVarsWithNoise100k.txt_train mystery.dat\n Number of variables..... 4\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabcd\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 1.2524925899830208 at j= 40336\n Searching for best fit...\n 0.335536856541 0.398680762669 P 1 0.1706 15.0342 48.2535 13372.1970\n 0.187842818262 0.301637474370 cP+ 54 0.0637 19.9521 53.1714 12976.0561\n 0.179172163154 0.243093361473 cP>+ 559 0.0630 23.2558 56.4750 12974.7626\n 0.178914236019 0.705530741623 Pa-R 1911 0.0710 25.0271 58.2464 13025.2033\n 0.176646304016 0.879766348059 cP+L 2159 0.0661 25.1847 58.4040 12996.2853\n 0.176546622993 0.215160704326 cPER+ 10004 0.0690 27.3961 60.6153 13016.2625\n 0.175970742946 0.314035448781 Pa+- 21393778 0.0121 36.6603 69.8796 12316.0274\n 0.048795256872 1.307594931585 ca-P>/ES 105003314 0.0120 38.8984 72.1177 12315.0274\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545942911475, 8.91505505199440e-11]\n [17.136430453348055, 28.41854594291147, '1/(-666.000000000000+exp(exp(pi)))']\n [32.8450013789013, 28.418212030052608, 7.71490571179178e-5]\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [96.05127279846616, 23.499172324266443, tan(0.0365815024226761*x3 + 0.87662273645401)]\n [96.45193157676024, 23.251340086812633, tan(0.0187665168195963*x2 - 2.25580163260941)]\n [142.88493409902946, 22.82711097251165, tan(-0.00816557370126247*x1 + 0.0305992718786001*x2 + 0.883173022410234)]\n [206.64700957832872, 22.674175327513932, 0.823345422744751*(-0.971410036087036*x1 + x2 + 3.18614816665649)**(-0.76738178730011)]\n [252.9985467754702, 22.3285090516094, 0.778373181819916*sin((-0.984130799770355*x1 + x2 + 3.16531944274902)**(-1.07100188732147))**0.676108956336975]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_sin/duplicateVarsWithNoise100k.txt_train...\n /bin/cp -p results/mystery_world_sin/duplicateVarsWithNoise100k.txt_train mystery.dat\n Number of variables..... 4\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabcd\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 3.8595698840027815E-006 at j= 74523\n Searching for best fit...\n 0.597142829315 -3.141596600582 P 1 0.2424 15.8658 49.0851 13515.4590\n 0.597142829315 -0.318313736896 P\\ 21 0.2424 20.2581 53.4774 13519.8514\n 0.597142829315 -1.772457735137 PR 26 0.2424 20.5662 53.7855 13520.1595\n 0.565360812364 -0.304871824725 aS 32 0.1823 20.7869 54.0062 13404.2102\n 0.534401254104 -0.835141331786 cC 39 0.1974 20.9910 54.2103 13436.9956\n 0.525252251075 -0.829429572146 dC 40 0.1940 21.0027 54.2219 13429.8937\n 0.413950443288 -0.098616193231 aP/ 127 0.1686 22.3259 55.5451 13374.4224\n 0.392316712046 -0.068569730610 bP/ 128 0.1571 22.2597 55.4790 13345.5739\n 0.372089905555 -1.144468709010 a>R 332 0.1573 23.5584 56.7777 13347.4918\n 0.342116586263 -1.102458413164 b>R 333 0.1455 23.4416 56.6609 13315.4413\n 0.318981733404 -0.100586028446 bP>/ 14993 0.1237 28.7911 62.0104 13254.7640\n 0.309595110966 -0.083085045374 bd>>/ 14998 0.1237 28.7906 62.0099 13254.6689\n 0.306213715320 -0.077200517165 bcE>/ 15193 0.1182 28.7934 62.0127 13236.4048\n 0.305742120207 -0.076690905464 bdE>/ 15198 0.1182 28.7916 62.0109 13236.1315\n 0.297665022137 -1.070021272429 acC+R 20767 0.1052 29.2034 62.4227 13189.3557\n 0.264836348350 -1.024964109997 bcC+R 20768 0.0881 29.0349 62.2542 13116.9777\n 0.262162858478 -1.022173980105 bdC+R 20773 0.0876 29.0206 62.2399 13114.4835\n 0.249062533662 0.086771826867 ac-P/ 26667 0.0888 29.3070 62.5263 13120.1918\n 0.233697785570 0.116818289489 bc-P/ 26668 0.0756 29.2152 62.4345 13054.7578\n 0.221618658276 0.120098097916 bd-P/ 26673 0.0748 29.1389 62.3582 13050.3850\n 0.221618658276 0.120098097916 Pbd-/\\ 136166 0.0748 31.4908 64.7101 13052.7369\n 0.191257775429 -0.931855985436 acE\\+R 270497 0.0659 32.2685 65.4878 13001.9141\n 0.189923677343 -0.928778723376 adE\\+R 270502 0.0651 32.2584 65.4777 12996.8343\n 0.146825870718 0.127291055745 ac-P-ESRL 98347327 0.0345 40.3043 73.5236 12745.8983\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_sin/duplicateVarsWithNoise100k.txt_train...\n /bin/cp -p results/mystery_world_sin/duplicateVarsWithNoise100k.txt_train mystery.dat\n Number of variables..... 4\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabcd\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.59713896974525005 at j= 33304\n Searching for best fit...\n 0.947035968153 0.190075232206 P 1 0.4999 16.5311 49.7504 13810.8491\n 0.561459186212 0.617538704192 a 2 0.2258 16.7769 49.9962 13487.5584\n 0.518467004639 -0.581925875337 c< 14 0.2118 19.4693 52.6886 13464.1843\n 0.380360932221 -0.601282570053 ca- 109 0.1516 21.9833 55.2025 13330.8087\n 0.369529327359 -0.599879466785 cb- 114 0.1516 22.0063 55.2256 13330.7909\n 0.349960956205 -0.601805893099 ac<* 822 0.0966 24.7779 57.9972 13149.8976\n 0.349886215050 -0.600363692648 bc<* 823 0.0969 24.7793 57.9986 13150.8294\n 0.349831431283 -0.630734778983 ad<* 827 0.0951 24.7861 58.0054 13143.2300\n 0.309795584533 0.601284372933 acS- 1147 0.1254 25.0827 58.3020 13256.5552\n 0.268295605660 0.304651114163 aac-+ 7232 0.0733 27.5317 60.7510 13040.2440\n 0.237675497985 0.304290503807 bac-+ 7233 0.0700 27.3571 60.5764 13021.5708\n 0.215779122192 0.390593427165 acEC+ 10467 0.0573 27.7508 60.9701 12940.3074\n 0.202989527732 0.390000860086 bcEC+ 10468 0.0572 27.6628 60.8821 12939.5052\n 0.186132110477 -0.226603228432 dcaE-+ 95600 0.0504 30.7288 63.9480 12891.2611\n 0.158288723433 -0.455338185867 caP\\+- 101689 0.0356 30.5841 63.8034 12749.0008\n 0.157163910869 -0.472287728970 daP\\+- 101690 0.0362 30.5738 63.7931 12756.0290\n 0.153564499336 -0.454533091840 cbP\\+- 101694 0.0357 30.5404 63.7597 12750.4182\n 0.153355062153 0.456887923960 bcP\\-S- 1604693 0.0378 34.5185 67.7377 12777.9730\n 0.150919626253 -0.482925581970 caaP+\\+- 19402789 0.0354 38.0913 71.3105 12755.0657\n 0.149648402574 -0.501113584712 dbbP+\\+- 19402820 0.0354 38.0791 71.2983 12754.2592\n 0.147866852375 -0.471292092923 dacP+\\+- 19402840 0.0354 38.0618 71.2811 12754.4316\n 0.143609517706 -0.470388450405 caP\\>L+- 21455514 0.0351 38.1647 71.3840 12750.7736\n 0.143508027982 -0.466773292615 caPRRL+- 21460139 0.0345 38.1640 71.3833 12744.1851\n 0.140304267418 -0.314636919131 cdCaEC-- 40176654 0.0350 39.0361 72.2554 12751.2378\n 0.139807644821 -0.322140901740 dcCbEC-- 40176875 0.0363 39.0310 72.2503 12765.4576\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545942911475, 8.91505505199440e-11]\n [17.136430453348055, 28.41854594291147, '1/(-666.000000000000+exp(exp(pi)))']\n [32.8450013789013, 28.418212030052608, 7.71490571179178e-5]\n [43.65486140660157, 26.808399066180467, 0.138491379080858]\n [43.65486201547717, 26.808398865229687, 0.138491437529828]\n [43.654864729530274, 26.80839764467069, 0.138491698065464]\n [43.654868537895325, 26.8083949094849, 0.138492063650444]\n [44.11035814652141, 26.552667871365905, 'asin(-1.102458413164+sqrt((x1+1)))']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [96.05127279846616, 23.499172324266443, tan(0.0365815024226761*x3 + 0.87662273645401)]\n [96.45193157676024, 23.251340086812633, tan(0.0187665168195963*x2 - 2.25580163260941)]\n [142.88493409902946, 22.82711097251165, tan(-0.00816557370126247*x1 + 0.0305992718786001*x2 + 0.883173022410234)]\n [206.64700957832872, 22.674175327513932, 0.823345422744751*(-0.971410036087036*x1 + x2 + 3.18614816665649)**(-0.76738178730011)]\n [252.9985467754702, 22.3285090516094, 0.778373181819916*sin((-0.984130799770355*x1 + x2 + 3.16531944274902)**(-1.07100188732147))**0.676108956336975]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_sqrt/duplicateVarsWithNoise100k.txt_train...\n /bin/cp -p results/mystery_world_sqrt/duplicateVarsWithNoise100k.txt_train mystery.dat\n Number of variables..... 4\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabcd\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.14879262308092922 at j= 1\n Searching for best fit...\n 0.192827070815 -1.463844017285 P 1 0.0795 14.2350 47.4543 13060.5692\n 0.192827070815 -2.463844017285 P> 6 0.0795 16.8200 50.0393 13063.1542\n 0.192827070815 -21.462945932076 PE 46 0.0795 19.7586 52.9779 13066.0928\n 0.192826160753 -666.000000000000 PEE 551 0.0795 23.3409 56.5602 13069.6720\n 0.146984353293 1.652296402514 ccS- 1149 0.0602 24.0095 57.2288 12957.5723\n 0.141962534249 1.665969955634 ddS- 1155 0.0527 23.9669 57.1862 12902.6264\n 0.139650381662 1.423779527824 aP+\\ 1757 0.0556 24.5484 57.7677 12925.6120\n 0.129058397301 2.104060683265 aP-\\ 1807 0.0437 24.4751 57.6944 12827.5464\n 0.119831790614 1.320081200949 a>>\\ 4472 0.0435 25.6754 58.8947 12827.0667\n 0.105422920922 1.337321499397 aP<+\\ 19727 0.0438 27.6318 60.8511 12831.8615\n 0.104176424835 0.774353481486 aa>/C 23082 0.0443 27.8412 61.0605 12836.2574\n 0.103936637059 1.041474894276 PaP>+/ 105811 0.0436 30.0345 63.2538 12831.7616\n 0.078835664944 1.383665149228 aPc-+\\ 134632 0.0303 29.9833 63.2026 12684.6023\n 0.063788450484 1.393888006133 aPd-+\\ 134657 0.0234 29.6780 62.8973 12577.9313\n 0.063788450484 0.393888006133 aPd-+\\> 2232902 0.0234 33.7296 66.9488 12581.9829\n 0.062944739802 1.505949233867 daP+-ER 2315560 0.0226 33.7628 66.9821 12568.5254\n 0.059349157744 1.382001136615 ad<<<-\\ 3950097 0.0193 34.4485 67.6677 12504.1078\n 0.054639705807 1.729457506315 db-PP+/ 6339015 0.0129 35.0116 68.2308 12340.6265\n 0.043984190861 1.732928701957 da-PL<* 6397510 0.0105 34.7118 67.9311 12255.8731\n 0.043507599518 1.729283386912 da-PLL* 6424510 0.0105 34.7022 67.9215 12256.7552\n 0.042567074440 0.888806195980 da-ERRS 7762310 0.0109 34.9436 68.1629 12272.5995\n 0.041590906514 1.731732618059 da-P<<<* 101169815 0.0102 38.6143 71.8335 12247.9018\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_sqrt/duplicateVarsWithNoise100k.txt_train...\n /bin/cp -p results/mystery_world_sqrt/duplicateVarsWithNoise100k.txt_train mystery.dat\n Number of variables..... 4\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabcd\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 1.8705757945424719 at j= 40336\n Searching for best fit...\n 0.288911147145 0.595422751690 P 1 0.1516 14.8183 48.0376 13323.9516\n 0.288911147145 -666.000000000000 PS 31 0.1516 19.7725 52.9918 13328.9058\n 0.272715324455 0.363055687053 cP>+ 559 0.0904 23.8618 57.0811 13122.2740\n 0.258393267197 0.364555862998 dP>+ 560 0.0874 23.7866 57.0058 13108.4144\n 0.258393267197 0.364555862998 Pd>+ 576 0.0874 23.8272 57.0465 13108.4550\n 0.185249331195 0.451661501833 PaC+ 711 0.0837 23.6509 56.8702 13091.0422\n 0.182009311229 0.451656294014 PbC+ 716 0.0838 23.6355 56.8548 13091.3137\n 0.161712912641 0.917973988953 cP+R 1859 0.0549 24.8415 58.0607 12920.4076\n 0.158936792241 0.171922815699 cPP*+ 7029 0.0673 26.7353 59.9546 13005.5346\n 0.158936792241 0.540111469814 PcP/+ 7291 0.0673 26.7881 60.0073 13005.5874\n 0.158595665354 0.467204073972 PcCC+ 10416 0.0627 27.2996 60.5189 12977.1635\n 0.158140944001 0.824089303578 cP>+R 20604 0.0624 28.2796 61.4988 12975.7945\n 0.157867428680 0.771796335882 cP+L> 27804 0.0592 28.7094 61.9287 12954.8033\n 0.157848770300 1.894192101710 cCR>S 77939 0.0643 30.1963 63.4156 12990.0649\n 0.154679164249 0.074403606967 dcPE++ 93325 0.0775 30.4270 63.6462 13066.4993\n 0.143524861190 -0.077420238071 acPE+- 102322 0.0666 30.4518 63.6711 13005.2511\n 0.143524861190 0.077420238071 caPE-- 104564 0.0666 30.4830 63.7023 13005.2823\n 0.131042932683 0.539952658467 PcaP+/+ 1304091 0.0542 33.9924 67.2116 12924.7816\n 0.102571193018 -0.171766280986 acPP*+- 1317192 0.0396 33.6534 66.8727 12796.0109\n 0.093386578377 -0.171949299217 bcPP*+- 1317193 0.0379 33.5180 66.7373 12778.5186\n 0.092251084000 -0.172285077760 bdPP*+- 1317198 0.0349 33.5004 66.7197 12744.4687\n 0.090357082832 0.824093234920 cPaC++R 1733329 0.0275 33.8665 67.0858 12647.3804\n 0.089672487440 0.825794106489 dPaC++R 1733330 0.0294 33.8556 67.0748 12675.7945\n 0.088161551028 0.824089415869 cPbC++R 1733354 0.0275 33.8311 67.0503 12647.8103\n 0.051935810605 1.309615733263 cPa-+RR 2306829 0.0141 33.4800 66.6993 12374.9609\n 0.051935810605 1.309615733263 Pca-+RR 2306841 0.0141 33.4800 66.6993 12374.9609\n 0.045255077475 -0.133309454797 acPP>*+- 19040897 0.0111 36.3265 69.5457 12280.0708\n 0.043719814723 0.692150127602 cPPa-++R 20096634 0.0105 36.3545 69.5738 12259.1262\n 0.043438409138 0.571682546288 cPa<-+R> 28739659 0.0104 36.8613 70.0806 12255.6934\n 0.042726871312 1.153943106546 bd-ER>\\> 117722928 0.0102 38.8717 72.0910 12248.7062\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545942911475, 8.91505505199440e-11]\n [17.136430453348055, 28.41854594291147, '1/(-666.000000000000+exp(exp(pi)))']\n [32.8450013789013, 28.418212030052608, 7.71490571179178e-5]\n [43.65486140660157, 26.808399066180467, 0.138491379080858]\n [43.65486201547717, 26.808398865229687, 0.138491437529828]\n [43.654864729530274, 26.80839764467069, 0.138491698065464]\n [43.654868537895325, 26.8083949094849, 0.138492063650444]\n [44.11035814652141, 26.552667871365905, 'asin(-1.102458413164+sqrt((x1+1)))']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [96.05127279846616, 23.499172324266443, tan(0.0365815024226761*x3 + 0.87662273645401)]\n [96.45193157676024, 23.251340086812633, tan(0.0187665168195963*x2 - 2.25580163260941)]\n [142.88493409902946, 22.82711097251165, tan(-0.00816557370126247*x1 + 0.0305992718786001*x2 + 0.883173022410234)]\n [194.6710672335131, 22.478676262622, (-0.0591836827349255*x0 + 0.0579931735992432*x3 + 1.34409177303314)**1.85744190216064]\n [252.9985467754702, 22.3285090516094, 0.778373181819916*sin((-0.984130799770355*x1 + x2 + 3.16531944274902)**(-1.07100188732147))**0.676108956336975]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_squared/duplicateVarsWithNoise100k.txt_train...\n /bin/cp -p results/mystery_world_squared/duplicateVarsWithNoise100k.txt_train mystery.dat\n Number of variables..... 4\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabcd\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.14879262308092922 at j= 1\n Searching for best fit...\n 4.320048900614 4.781735875560 P 1 1.6589 18.7207 51.9400 14300.2823\n 3.939356310099 7.386237144616 c 4 1.5572 20.5876 53.8069 14276.4644\n 3.779468551575 7.508697697826 d 5 1.4682 20.8498 54.0690 14252.7668\n 3.595413988874 8.719221271939 a~ 17 1.3275 22.5433 55.7625 14213.4338\n 3.585600176173 6.212305557190 cE 49 1.4623 24.0666 57.2858 14254.4186\n 3.363502100034 6.409516698600 dE 50 1.3179 24.0035 57.2227 14212.0355\n 3.312550098326 7.094066779080 dd+ 75 1.3210 24.5664 57.7857 14213.5885\n 3.255612281535 6.236005947014 cP* 79 1.5202 24.6164 57.8356 14270.9547\n 2.855996631774 6.620727132040 dP* 80 1.2264 24.4456 57.6648 14183.3625\n 2.285809252563 10.423699205298 aP~* 832 0.9005 27.5028 60.7221 14060.6721\n 1.435903907247 8.736376535739 Pca-* 7691 0.3874 30.0406 63.2598 13719.6847\n 1.066344525592 9.121097720766 Pda-* 7696 0.2480 29.6122 62.8315 13537.6280\n 1.066344525592 12.262690461778 Pda>-* 99101 0.2480 33.2989 66.5182 13541.3147\n 1.055554673244 8.865358807920 PdaS-* 99726 0.2415 33.2933 66.5126 13530.5464\n 0.925056348607 9.092289841270 Pda-S* 117601 0.2232 33.3408 66.5601 13498.5936\n 0.925056330803 9.092290096196 PaPd-+S* 19717011 0.2232 40.7302 73.9495 13505.9830\n 0.883314196668 8.372795815369 caP*dE-- 30370389 0.2210 41.2868 74.5061 13502.5812\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_squared/duplicateVarsWithNoise100k.txt_train...\n /bin/cp -p results/mystery_world_squared/duplicateVarsWithNoise100k.txt_train mystery.dat\n Number of variables..... 4\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabcd\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 12.243377517185923 at j= 40336\n Searching for best fit...\n 5.985059467298 3.897187995551 P 1 3.4303 19.1910 52.4103 14596.7372\n 3.862456827044 6.089056844207 c> 9 1.2812 21.7291 54.9484 14198.0328\n 3.712093228344 2.948569517754 cP+ 54 1.9079 24.2568 57.4760 14363.1164\n 3.668989664194 4.066596950388 c>> 154 1.4634 25.7518 58.9711 14256.3841\n 3.634967616528 6.628024719417 cS> 179 1.3512 25.9554 59.1747 14224.0637\n 3.614056066158 15.523117875712 cSCC 5984 1.3380 31.0102 64.2294 14225.1159\n 3.561698645492 2.381133688946 dcP++ 6920 1.3057 31.1988 64.4180 14215.3366\n 1.032830452851 2.941545263735 cPa-+ 7179 0.2919 29.4658 62.6851 13604.1031\n 1.032830452851 2.941545263735 Pca-+ 7191 0.2919 29.4682 62.6875 13604.1055\n 0.950776531870 -3.045136261141 ac>>>- 200772 0.2400 34.1520 67.3713 13529.0613\n 0.944148308023 8.907167566787 daS-P/E 5486935 0.2595 38.9143 72.1336 13565.6988\n 0.940633861258 8.954849692426 da-P/SE 6868510 0.2155 39.2329 72.4522 13490.2666\n 0.911321542383 2.075043906401 cPb>-+RE 30035559 0.2266 41.3158 74.5351 13512.7250\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545942911475, 8.91505505199440e-11]\n [17.136430453348055, 28.41854594291147, '1/(-666.000000000000+exp(exp(pi)))']\n [32.8450013789013, 28.418212030052608, 7.71490571179178e-5]\n [43.65486140660157, 26.808399066180467, 0.138491379080858]\n [43.65486201547717, 26.808398865229687, 0.138491437529828]\n [43.654864729530274, 26.80839764467069, 0.138491698065464]\n [43.654868537895325, 26.8083949094849, 0.138492063650444]\n [44.11035814652141, 26.552667871365905, 'asin(-1.102458413164+sqrt((x1+1)))']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [96.05127279846616, 23.499172324266443, tan(0.0365815024226761*x3 + 0.87662273645401)]\n [96.45193157676024, 23.251340086812633, tan(0.0187665168195963*x2 - 2.25580163260941)]\n [142.88493409902946, 22.82711097251165, tan(-0.00816557370126247*x1 + 0.0305992718786001*x2 + 0.883173022410234)]\n [194.6710672335131, 22.478676262622, (-0.0591836827349255*x0 + 0.0579931735992432*x3 + 1.34409177303314)**1.85744190216064]\n [252.9985467754702, 22.3285090516094, 0.778373181819916*sin((-0.984130799770355*x1 + x2 + 3.16531944274902)**(-1.07100188732147))**0.676108956336975]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_tan/duplicateVarsWithNoise100k.txt_train...\n /bin/cp -p results/mystery_world_tan/duplicateVarsWithNoise100k.txt_train mystery.dat\n Number of variables..... 4\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabcd\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 3.8595698840315281E-006 at j= 74523\n Searching for best fit...\n 0.744438256573 -3.141588881443 P 1 0.2631 16.1839 49.4031 13548.8588\n 0.672750804587 -0.582409798641 c 4 0.2203 18.0378 51.2571 13478.4157\n 0.657514846887 -0.592713620988 d 5 0.2181 18.3267 51.5459 13474.6832\n 0.529319456674 -0.550037413882 cS 34 0.1936 20.7793 53.9986 13428.7567\n 0.515891189445 -0.558613171426 dS 35 0.1918 20.7841 54.0033 13424.9520\n 0.435365266897 -0.952390766529 aC 37 0.1632 20.6194 53.8387 13359.3572\n 0.435210540936 -0.976885825928 bC 38 0.1595 20.6574 53.8766 13349.8412\n 0.435210540936 -4.118478566940 PbC+ 716 0.1595 24.8932 58.1125 13354.0771\n 0.389464590026 0.129372021233 ac<* 822 0.0924 24.9322 58.1515 13131.4223\n 0.388480677799 0.089954480957 bc<* 823 0.0890 24.9303 58.1496 13116.3983\n 0.381967962257 0.126179899217 ad<* 827 0.0916 24.9129 58.1322 13127.9634\n 0.379398687103 0.125644832654 acR 487964 0.0424 32.9911 66.2104 12823.1261\n 0.161968999636 -0.147290021115 ac-P>C* 6416517 0.0396 36.5968 69.8161 12798.9307\n 0.161968999636 -0.687592253419 ac>-P>C* 84914822 0.0396 40.3229 73.5422 12802.6569\n 0.157165267600 -2.129117276408 ac>-EE\\E 98954822 0.0396 40.5003 73.7195 12802.8601\n 0.156894065821 -0.141263794536 ca-cP+R/ 99990114 0.0378 40.5128 73.7321 12784.1120\n 0.152326931398 -0.141068764906 ca-dP+R/ 99990214 0.0377 40.4702 73.6894 12782.9623\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_tan/duplicateVarsWithNoise100k.txt_train...\n /bin/cp -p results/mystery_world_tan/duplicateVarsWithNoise100k.txt_train mystery.dat\n Number of variables..... 4\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabcd\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.74443439700295699 at j= 33304\n Searching for best fit...\n 1.117941457203 -0.236960821587 P 1 0.6332 16.7705 49.9898 13907.2943\n 0.737039637270 -0.769866105167 a 2 0.2916 17.1695 50.3887 13591.8371\n 0.673048368971 0.725468710059 c< 14 0.2718 19.8458 53.0651 13566.0553\n 0.536196164776 0.749600093353 ca- 109 0.1818 22.4787 55.6979 13404.8421\n 0.515464665800 0.747850888581 cb- 114 0.1816 22.4865 55.7058 13404.3647\n 0.373586794377 0.750252503756 ac<* 822 0.0978 24.8722 58.0914 13154.7947\n 0.373493616927 0.748454557755 bc<* 823 0.0978 24.8735 58.0928 13154.9769\n 0.373425319725 0.786317237111 ad<* 827 0.0995 24.8803 58.0996 13161.7091\n 0.371561809882 -0.379798974709 aac-+ 7232 0.0992 28.0015 61.2208 13163.5262\n 0.322709084665 -0.379349413105 bac-+ 7233 0.0952 27.7983 61.0176 13146.8324\n 0.283427868287 -0.486851323627 ac>C+ 10267 0.1339 28.1164 61.3357 13286.7563\n 0.274489031407 -0.486112859382 bc>C+ 10268 0.1335 28.0703 61.2896 13285.2787\n 0.245131462129 -0.486940556817 acEC+ 10467 0.0875 27.9348 61.1541 13113.2044\n 0.227889096216 0.277542201709 ccaE-+ 95599 0.0682 31.0208 64.2400 13014.4851\n 0.222421777513 0.282499127111 dcaE-+ 95600 0.0681 30.9857 64.2050 13014.1895\n 0.209024339134 0.567655813810 caP\\+- 101689 0.0545 30.9852 64.2045 12923.4896\n 0.208284801005 0.569980247348 caP\\S+- 1423559 0.0536 34.7873 68.0066 12920.4392\n 0.206586482853 -0.569587823612 bcP\\-S- 1604693 0.0564 34.9483 68.1676 12940.9442\n 0.204374990157 -0.282330858843 bc-EE-+RC 29711534 0.0465 39.0586 72.2779 12866.5424\n 0.173163514889 -0.146260358935 bc-aP+*> 100049923 0.0409 40.6560 73.8753 12815.4842\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545942911475, 8.91505505199440e-11]\n [17.136430453348055, 28.41854594291147, '1/(-666.000000000000+exp(exp(pi)))']\n [32.8450013789013, 28.418212030052608, 7.71490571179178e-5]\n [43.65486140660157, 26.808399066180467, 0.138491379080858]\n [43.65486201547717, 26.808398865229687, 0.138491437529828]\n [43.654864729530274, 26.80839764467069, 0.138491698065464]\n [43.654868537895325, 26.8083949094849, 0.138492063650444]\n [44.11035814652141, 26.552667871365905, 'asin(-1.102458413164+sqrt((x1+1)))']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [96.05127279846616, 23.499172324266443, tan(0.0365815024226761*x3 + 0.87662273645401)]\n [96.45193157676024, 23.251340086812633, tan(0.0187665168195963*x2 - 2.25580163260941)]\n [142.88493409902946, 22.82711097251165, tan(-0.00816557370126247*x1 + 0.0305992718786001*x2 + 0.883173022410234)]\n [194.6710672335131, 22.478676262622, (-0.0591836827349255*x0 + 0.0579931735992432*x3 + 1.34409177303314)**1.85744190216064]\n [252.9985467754702, 22.3285090516094, 0.778373181819916*sin((-0.984130799770355*x1 + x2 + 3.16531944274902)**(-1.07100188732147))**0.676108956336975]\n Checking for symmetry \n duplicateVarsWithNoise100k.txt_train\n Training a NN on the data... \n \n tensor(0.0260, device='cuda:0', grad_fn=)\n tensor(0.0094, device='cuda:0', grad_fn=)\n tensor(0.0153, device='cuda:0', grad_fn=)\n tensor(0.0118, device='cuda:0', grad_fn=)\n NN loss: tensor(0.0081, device='cuda:0', grad_fn=) \n \n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/translated_data_plus/duplicateVarsWithNoise100k.txt_train-translated_plus...\n /bin/cp -p results/translated_data_plus/duplicateVarsWithNoise100k.txt_train-translated_plus mystery.dat\n Number of variables..... 3\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.29479879300520928 at j= 1\n Searching for best fit...\n 0.684213023160 -0.326751961044 P 1 0.2755 16.0622 49.2814 13567.7330\n 0.684213023160 5.956433520981 P~ 13 0.2755 19.7626 52.9819 13571.4334\n 0.473303044937 2.115193527026 aC 30 0.1566 20.4374 53.6566 13342.1995\n 0.401977879202 2.258014626127 a>\\ 214 0.1562 23.0363 56.2556 13343.9070\n 0.401977879202 1.258014626127 a>\\> 2266 0.1562 26.4408 59.6600 13347.3114\n 0.399572448190 3.794083089707 aC* 8258 0.1491 28.1902 61.4095 13330.2484\n 0.367189163161 3.212787107652 aPCa>E/ 379436 0.0696 33.2417 66.4610 13025.0564\n 0.256480073830 2.472111539795 c>b>E/ 379472 0.0825 33.1802 66.3995 13094.0257\n 0.247861692144 1.876451305042 c>aE/R 406508 0.0560 33.2302 66.4495 12936.0595\n 0.240332491501 1.171355643553 cPa>/+R 1080268 0.0754 34.5957 67.8150 13058.8778\n 0.226446193316 1.486150117093 PacC+-R 1083565 0.0730 34.5142 67.7335 13045.7673\n 0.224385382710 3.000637318088 cba+-P/ 1319700 0.0644 34.7855 68.0048 12994.7155\n 0.216704233933 2.703423791983 caEC+P/ 2640992 0.0557 35.7361 68.9554 12936.7060\n 0.207773535495 3.120205479097 caE-P>/ 3143264 0.0679 35.9266 69.1458 13018.0092\n 0.197069304341 3.206620635244 caE-P/S 3216992 0.0512 35.8837 69.1030 12902.5643\n 0.166327461359 0.565839417586 cPaCC/+R 14435408 0.0463 37.8048 71.0241 12863.7566\n 0.156358608729 2.969385893420 caa+-P>/ 20453012 0.0432 38.2184 71.4377 12835.9359\n 0.149882159900 2.955776186214 cba+-P>/ 20453016 0.0509 38.1574 71.3766 12903.0442\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/translated_data_plus/duplicateVarsWithNoise100k.txt_train-translated_plus...\n /bin/cp -p results/translated_data_plus/duplicateVarsWithNoise100k.txt_train-translated_plus mystery.dat\n Number of variables..... 3\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 3.4990538031282004 at j= 40336\n Searching for best fit...\n 0.997390746302 1.113783386831 P 1 0.5395 16.6059 49.8252 13841.9546\n 0.997390746302 0.844857044605 P> 5 0.5395 18.9278 52.1471 13844.2766\n 0.975905140835 0.680507880964 cP+ 44 0.3438 22.0339 55.2532 13663.5796\n 0.788868000667 -1.110279122716 aP- 74 0.2433 22.4769 55.6962 13523.2235\n 0.760245610016 -1.114377815355 bP- 75 0.2415 22.4430 55.6623 13520.1352\n 0.642251754968 1.749569904374 aC> 130 0.2879 22.9932 56.2125 13592.6875\n 0.641669489117 1.749528129837 bC> 131 0.2881 23.0030 56.2222 13592.9738\n 0.592223052292 -0.842839180718 aP>- 718 0.1927 25.3417 58.5610 13431.3138\n 0.546205363230 1.543092249007 cP+R 1264 0.2114 26.0409 59.2602 13469.9962\n 0.543771406040 2.658654043715 c>RR 3600 0.2211 27.5445 60.7638 13489.7770\n 0.543428540947 1.927723485521 cP>+L 15316 0.2183 29.6326 62.8518 13486.6657\n 0.531637984833 1.758237491062 c>SC> 40352 0.2124 30.9985 64.2178 13476.7591\n 0.386848061198 1.541606548027 cPa-+R 87144 0.1242 31.6506 64.8699 13258.9542\n 0.386848061198 1.541606548027 Pca-+R 87153 0.1242 31.6507 64.8700 13258.9544\n 0.262204805596 -0.294539229654 acPP*+- 867846 0.0880 34.4055 67.6248 13121.5949\n 0.260131026511 -0.294826897267 bcPP*+- 867847 0.0899 34.3940 67.6133 13130.4980\n 0.259879110616 -0.332510569170 acPR>+- 12966830 0.0486 37.9313 71.1506 12883.1760\n 0.189022669720 2.113052571698 c>RaE\\+R 111865732 0.0414 40.9435 74.1627 12821.3420\n Checking polyfit \n \n Complexity RMSE Expression\n [46.893261128102594, 27.327578182802576, '-0.326751961044+pi']\n [48.091811949279766, 26.831252455317497, 2.99970082515141]\n [48.091812250119716, 26.83125070219585, 2.99970145066818]\n [48.09181225032447, 26.831250699084304, 2.99970145109393]\n [52.342670497576094, 26.594939363560307, '2.115193527026+cos(x0)']\n [52.37239106247598, 26.587821185138676, cos(x0) + 2.15921998023987]\n [54.945706215718744, 26.576236070462485, '3.212787107652+(x0/(cos(pi)-1))']\n [58.20515105583124, 26.572480484690878, '3.244863358338+(x0*cos((pi+1)))']\n [60.555514367897636, 26.438279831343824, '2.711104202817+(x2/exp(exp(x0)))']\n [61.974416500029704, 25.235765711280767, '1.812250889593+((x2/pi)+cos(x0))']\n [66.19523291932471, 24.69746030749593, '0.565839417586+sqrt((x2+(pi/cos(cos(x0)))))']\n [69.17189542775445, 24.67167011279005, '2.969385893420+((x2-(x0+x0))/(pi+1))']\n [77.58632198967817, 24.473149377810305, '2.113052571698*sqrt((sqrt((x2+1))+(exp(x0))**(-1)))']\n [149.36966020534734, 24.455621206445624, -0.402340024709702*x0 + 0.24833944439888*x2 + 2.95297111499948]\n [149.37088318455326, 24.454715287659834, -0.402498424053192*x1 + 0.248467355966568*x2 + 2.95279143804132]\n [149.44168397979044, 24.309986678858266, -0.420233070850372*x0 + 0.24925871193409*x2 + 2.96100325966226]\n [149.6337951406249, 24.034098557639915, -0.485634002362974*x0 + 0.243230576273406*x2 + 2.9997386932373]\n [151.26419649600672, 24.022571764940643, '-0.266243472167946*x0 - 0.267882456420088*x1 + 0.257554819236464*x2 + 3']\n [197.01489798607219, 23.74672428832238, -0.243583184367369*x0 - 0.243580780240423*x1 + 0.244451861000069*x2 + 2.99918127059937]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_atan/duplicateVarsWithNoise100k.txt_train-translated_plus...\n /bin/cp -p results/mystery_world_atan/duplicateVarsWithNoise100k.txt_train-translated_plus mystery.dat\n Number of variables..... 3\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.29479879300520928 at j= 1\n Searching for best fit...\n 0.062981887482 -1.912149389642 P 1 0.0276 12.6207 45.8400 12629.0104\n 0.062981887482 -21.911251304432 PE 37 0.0276 17.8302 51.0495 12634.2199\n 0.062981817036 1.229443420950 aPS* 654 0.0276 21.9739 55.1931 12638.3631\n 0.057365918579 0.975474155467 aP+\\ 1198 0.0175 22.7124 55.9317 12452.5825\n 0.053995174954 0.261362785431 aP/C 1438 0.0156 22.8885 56.1077 12405.9428\n 0.036207235544 1.263836988520 aPE~/ 10402 0.0159 25.1666 58.3859 12417.4645\n 0.036159256115 1.026911106772 aP>+\\ 13010 0.0156 25.4875 58.7068 12408.7847\n 0.035829127758 -1.877150369952 PEa-L 32245 0.0158 26.7837 60.0030 12415.8245\n 0.035827242227 1.264972421112 abPE-/ 73054 0.0157 27.9635 61.1828 12414.3968\n 0.035755311982 1.265390401260 aPE~>/ 136742 0.0157 28.8650 62.0843 12414.8866\n 0.034633639710 1.218225539043 PaC~/E 193737 0.0153 29.3217 62.5410 12406.1309\n 0.031107851853 0.947704214038 Pac>/+\\ 1071085 0.0109 31.6337 64.8530 12268.7689\n 0.028289688890 1.004716844182 aPc>\\++\\ 14334830 0.0077 35.2391 68.4584 12132.7457\n 0.025725416168 0.998152517464 aPcE\\++\\ 14335342 0.0069 35.1021 68.3213 12084.4491\n 0.023901319967 1.288913211021 acC+PE~/ 48307946 0.0071 36.7486 69.9679 12101.9013\n 0.021443490439 1.286477421370 bcC+PE~/ 48307947 0.0076 36.5921 69.8114 12126.1136\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_atan/duplicateVarsWithNoise100k.txt_train-translated_plus...\n /bin/cp -p results/mystery_world_atan/duplicateVarsWithNoise100k.txt_train-translated_plus mystery.dat\n Number of variables..... 3\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 1.2924252388523800 at j= 40336\n Searching for best fit...\n 0.101906033496 0.411391719232 P 1 0.0492 13.3150 46.5342 12864.9112\n 0.101906033496 0.602605125527 PL> 133 0.0492 20.3702 53.5895 12871.9665\n 0.068757532089 0.051407218989 cPE+ 560 0.0224 21.8766 55.0959 12553.1129\n 0.051990432075 -0.055826832652 aPE- 846 0.0221 22.0685 55.2878 12547.1546\n 0.050835705549 -0.058347161991 aPE<- 8962 0.0212 25.4412 58.6605 12534.6248\n 0.049602954761 0.054863617978 PEcC- 26269 0.0186 26.9573 60.1766 12482.0368\n 0.048389140668 0.036705495252 cPC>L- 133832 0.0181 29.2705 62.4898 12473.2371\n 0.047917941575 0.321973519348 cPE+R< 211480 0.0196 29.9165 63.1358 12506.5303\n 0.047876216307 1.457472165619 cPE/ES 234952 0.0196 30.0671 63.2864 12506.4275\n 0.047823939003 1.053211876563 cP+RRR 301464 0.0189 30.4251 63.6444 12491.4607\n 0.044922420002 -0.019910089736 acP>E+- 921862 0.0174 31.9474 65.1667 12461.5771\n 0.027686678259 0.257708738507 caPE--R 1084800 0.0090 31.4840 64.7033 12192.7917\n 0.025233641835 -0.053717183774 aPEcS+- 1451114 0.0094 31.7699 64.9891 12211.0998\n 0.020682671475 0.017294591726 cPaPE-*- 11943568 0.0058 34.5239 67.7432 12013.3710\n 0.019379864390 -0.051940655715 aPEc>R+- 22072750 0.0050 35.3161 68.5354 11957.7546\n 0.019379864390 -0.051940655715 ac>R-PE- 42064298 0.0050 36.2464 69.4657 11958.6849\n Checking polyfit \n \n Complexity RMSE Expression\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [99.82550711782861, 23.516791257617538, tan(1.04068541526794*sin(exp(0.0345384214798065*x2)))]\n [144.44234332340804, 23.30665233444474, -1.03783059120178*tan(0.0105930790305138*x0 - 0.882021725177765)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_cos/duplicateVarsWithNoise100k.txt_train-translated_plus...\n /bin/cp -p results/mystery_world_cos/duplicateVarsWithNoise100k.txt_train-translated_plus mystery.dat\n Number of variables..... 3\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.29479879300520928 at j= 1\n Searching for best fit...\n 0.144952083475 -4.088682624876 P 1 0.0435 13.8233 47.0426 12814.2703\n 0.144952083475 -3.088682624876 P< 9 0.0435 16.9932 50.2125 12817.4402\n 0.144952083475 -10.816694834246 PP* 57 0.0435 19.6562 52.8755 12820.1032\n 0.144952083475 -63.850019664652 P>E 393 0.0435 22.4417 55.6610 12822.8887\n 0.144951999816 -0.947089800661 cPS* 656 0.0435 23.1808 56.4001 12823.6276\n 0.137559331853 -0.981483521012 aPE/ 990 0.0338 23.6991 56.9183 12720.9779\n 0.135023162466 -0.979047731361 bPE/ 991 0.0345 23.6737 56.8929 12730.0790\n 0.133664867799 -1.191390644450 cP+\\ 1200 0.0421 23.9352 57.1544 12811.3758\n 0.126595885651 -1.373401843401 Pa-\\ 1233 0.0436 23.8959 57.1152 12825.9351\n 0.121313034895 -1.363398187101 Pb-\\ 1237 0.0399 23.8391 57.0584 12789.2569\n 0.108013776010 -0.905962233857 cPE~/ 10404 0.0379 26.7438 59.9631 12771.5479\n 0.105872694871 -1.143425664026 cP>+\\ 13012 0.0387 27.0376 60.2569 12780.8550\n 0.104689505126 -1.024293303048 cCP-E 32780 0.0387 28.3544 61.5736 12781.3647\n 0.100593558795 -1.898286102201 cSC>E/ 146706 0.0198 30.0263 63.2455 12509.7072\n 0.058866175828 -0.969051522268 acE>E/ 146834 0.0266 29.6870 62.9063 12631.7213\n 0.058621054308 -0.995403127205 aPcEE+/ 943146 0.0172 32.3643 65.5836 12454.5618\n 0.055821983913 -0.963331103834 bacE>E/* 12959859 0.0263 36.0742 69.2934 12632.2002\n 0.054535530764 -0.984966960178 acbCE+E/ 14068522 0.0178 36.1589 69.3782 12474.3343\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_cos/duplicateVarsWithNoise100k.txt_train-translated_plus...\n /bin/cp -p results/mystery_world_cos/duplicateVarsWithNoise100k.txt_train-translated_plus mystery.dat\n Number of variables..... 3\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.99999999999255185 at j= 74523\n Searching for best fit...\n 0.197862199604 -0.318309877324 P 1 0.0482 14.2722 47.4915 12856.2782\n 0.197862199604 -0.241453001907 P> 5 0.0482 16.5941 49.8134 12858.6001\n 0.197862199604 -0.758546998086 P\\> 117 0.0482 21.1426 54.3619 12863.1486\n 0.123778857593 -0.244260646241 PaC+ 529 0.0334 22.6426 55.8619 12715.8483\n 0.119672300871 -0.242807883204 PbC+ 533 0.0333 22.6048 55.8241 12714.1350\n 0.103050129894 -0.245806988807 PcCC+ 7069 0.0368 26.1183 59.3376 12759.0199\n 0.102596582438 -0.720443175723 cCC>R 50132 0.0391 28.9381 62.1574 12786.9541\n 0.098728788954 -1.002696903842 cCC 642220 0.0374 32.5450 65.7643 12772.2160\n 0.096099261523 -0.325335383669 Pbc>>/- 932849 0.0248 33.0616 66.2809 12604.7706\n 0.088638008272 -0.325891910867 PacE>/- 933357 0.0195 32.9458 66.1651 12505.9929\n 0.084309873188 -0.323543734604 PbcE>/- 933361 0.0177 32.8736 66.0928 12467.1373\n 0.079010082432 -0.572980458319 PacE/-R 1085421 0.0175 32.9976 66.2169 12462.7919\n 0.076154705980 -0.570258531519 PbcE/-R 1085425 0.0154 32.9445 66.1638 12410.4239\n 0.074798795152 -1.005475980532 ac>ER/C 2479158 0.0171 34.1102 67.3295 12453.7933\n 0.070113655857 -0.501699478698 acCS*C> 2693958 0.0157 34.1368 67.3561 12419.7802\n 0.067453337758 0.502121017444 bcEE/<< 2717751 0.0168 34.0937 67.3129 12447.7900\n 0.066763354771 -0.501777904440 acC*SC> 3363702 0.0147 34.3865 67.6057 12394.7601\n 0.064932890175 -0.500860595062 bcC*SC> 3363703 0.0142 34.3464 67.5656 12379.8980\n 0.064910306986 -0.500849303466 cCbS*C> 5987452 0.0141 35.1777 68.3970 12376.4764\n 0.064820028523 -2.145037108306 PbcC*C-\\ 16029813 0.0142 36.5965 69.8158 12381.9325\n 0.059755980432 -0.500515749573 acP/-SC> 20845386 0.0132 36.8581 70.0774 12353.3140\n 0.058137206859 -1.000931722451 ac<~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.29479879300520928 at j= 1\n Searching for best fit...\n 16.393614936972 13.548925362924 P 1 5.6739 20.6447 53.8640 14802.0853\n 15.408290755222 15.738795713234 c 4 5.3643 22.5553 55.7746 14781.1866\n 15.408290755222 16.738795713234 c< 12 5.3643 24.1402 57.3595 14782.7716\n 12.664089441422 14.100351004719 cE 40 4.5619 25.5942 58.8135 14718.3956\n 12.484863459844 9.649713945462 c>E 396 4.2682 28.8811 62.1004 14694.5400\n 11.789879965853 13.148628614017 ccE+ 572 4.4139 29.3290 62.5483 14708.7682\n 11.410027758531 15.521890893706 ca-E 1556 4.2289 30.7255 63.9448 14692.7424\n 10.799306361832 11.690709620907 cERE 4928 4.0366 32.3093 65.5286 14675.4210\n 8.801028792255 9.204459129615 aCEE 5062 3.0927 32.0528 65.2721 14566.7656\n 8.056750467395 11.476401683748 caC+E 15992 2.9692 33.5849 66.8042 14551.7987\n 6.768412302270 13.513859994125 ca<-E 16200 2.2372 33.3522 66.5714 14436.3102\n 6.541824079779 21.555740041227 PcaP*-* 865813 1.5458 39.0430 72.2623 14291.1840\n 6.137924513274 15.174172936153 cba<+-E 1120084 2.2647 39.3226 72.5418 14447.4050\n 6.074187452711 14.338021877812 caaC--E 1121552 1.9221 39.3094 72.5287 14380.4682\n 5.969097062538 14.201613108591 cbaC--E 1121556 2.0033 39.2842 72.5035 14397.3567\n 5.907412288121 14.604402490135 caE<<-E 2556032 1.8283 40.4576 73.6769 14361.2556\n 5.876490244789 14.334958183038 cbE<<-E 2556036 2.0076 40.4501 73.6694 14399.4271\n 5.213673315424 6.614292030398 cEaCEE+ 5136096 1.5192 41.2842 74.5035 14286.6831\n 4.616603018619 10.183901458277 c>a<aCS~>/ 79584676 1.1449 44.9177 78.1370 14175.2055\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_exp/duplicateVarsWithNoise100k.txt_train-translated_plus...\n /bin/cp -p results/mystery_world_exp/duplicateVarsWithNoise100k.txt_train-translated_plus mystery.dat\n Number of variables..... 3\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 33.084133040908405 at j= 40336\n Searching for best fit...\n 20.881362044207 10.531006329689 P 1 13.2654 20.9938 54.2131 15148.6354\n 14.374093380510 11.027183236495 c> 8 4.8512 23.4550 56.6743 14741.1632\n 13.906767305192 6.434314685036 cP+ 44 6.8797 25.8668 59.0861 14886.1736\n 13.864453700752 19.100387358970 c>R 252 6.9850 28.3802 61.5995 14894.8868\n 11.047945130257 11.027363917323 caC+ 532 3.8822 29.1306 62.3499 14656.2880\n 11.034502687639 11.027188397256 cbC+ 536 3.8825 29.1397 62.3590 14656.3359\n 7.805440326343 6.421930635187 cPa-+ 5220 3.2523 31.9240 65.1432 14587.3463\n 7.703054988429 6.436412385475 cPb-+ 5236 3.3020 31.9093 65.1286 14593.5409\n 6.415383378209 -8.250099112978 ac>>- 8702 2.0969 32.3783 65.5976 14408.9837\n 6.346064487058 -8.274015030878 bc>>- 8703 2.1117 32.3628 65.5821 14411.8598\n 6.346064487058 8.274015030878 cb<<- 8860 2.1117 32.3886 65.6079 14411.8856\n 4.315313096557 6.395183077849 cPa>/+ 66216 1.1981 34.7340 67.9533 14183.5108\n 3.858820114464 19.005926779609 c>aE/R 406508 0.9003 37.1907 70.4100 14069.5194\n 3.609917174135 12.804829934037 cP+a>>/ 3828028 1.2602 40.3298 73.5491 14210.0003\n 3.581527492698 6.402493911901 cP+aER/ 3839292 0.7645 40.3226 73.5419 14006.0329\n 3.311934685010 19.966132968817 cERaE/R 7778208 0.6473 41.2283 74.4476 13939.1442\n 3.081451543007 20.082185447116 cERbE/R 7778532 0.6414 41.1243 74.3436 13935.3999\n 3.026651849040 -6.411550305126 bPcaE/+- 11936751 0.9775 41.7163 74.9356 14107.9659\n 2.709666510295 20.024075133238 cba+-ERR 21161624 0.5245 42.3827 75.6020 13854.7270\n 2.709666510295 20.024075133238 ca-b-ERR 59890276 0.5245 43.8836 77.1029 13856.2278\n Checking polyfit \n \n Complexity RMSE Expression\n [23.264662506490403, 1000000, 'acos(-x0**2 + 2*x0*x1 - x1**2 - 1)']\n [46.46201740344512, 24.34147126490242, -0.969305992126465]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [99.82550711782861, 23.516791257617538, tan(1.04068541526794*sin(exp(0.0345384214798065*x2)))]\n [144.44234332340804, 23.30665233444474, -1.03783059120178*tan(0.0105930790305138*x0 - 0.882021725177765)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_inverse/duplicateVarsWithNoise100k.txt_train-translated_plus...\n /bin/cp -p results/mystery_world_inverse/duplicateVarsWithNoise100k.txt_train-translated_plus mystery.dat\n Number of variables..... 3\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.28579154716226046 at j= 40336\n Searching for best fit...\n 0.113942540636 -2.855801193850 P 1 0.0543 13.4760 46.6953 12904.7617\n 0.113942540636 -22.854903108641 PE 37 0.0543 18.6855 51.9048 12909.9711\n 0.113942539830 666.000000000000 PS\\ 233 0.0543 21.3402 54.5595 12912.6259\n 0.113942365314 0.285791722028 cPS* 656 0.0543 22.8336 56.0528 12914.1185\n 0.074546279849 0.286220034656 aPE/ 990 0.0319 22.8152 56.0345 12698.3344\n 0.060357795299 0.243004002840 aP-E 1550 0.0248 23.1574 56.3767 12596.4962\n 0.059427843817 0.242505154940 bP-E 1551 0.0252 23.1359 56.3552 12602.6931\n 0.058967699066 -0.623526335573 aC>S 3802 0.0238 24.4183 57.6375 12580.5966\n 0.053355950385 0.372229499831 cPE~/ 10404 0.0196 25.7263 58.9456 12503.3243\n 0.052856745528 0.303784085929 cCPE/ 27740 0.0207 27.1276 60.3468 12525.5731\n 0.049722371079 0.280002239149 acP+-E 91618 0.0245 28.7630 61.9823 12597.4382\n 0.048960604345 0.279934743241 bcP+-E 91619 0.0246 28.7408 61.9600 12598.6476\n 0.041901902818 0.122972242385 cPaC++\\ 1069708 0.0158 32.0616 65.2809 12421.8283\n 0.041826793789 0.122973508340 cPbC++\\ 1069724 0.0158 32.0590 65.2783 12421.9428\n 0.041826793789 0.122973508340 PcbC++\\ 1069733 0.0158 32.0590 65.2783 12421.9428\n 0.031663099667 0.304212573422 acC+PE/ 3161382 0.0127 33.2207 66.4400 12334.9762\n 0.030685697918 0.303711668904 bcC+PE/ 3161383 0.0123 33.1755 66.3948 12321.2215\n 0.023091201086 0.081025690371 Pac>R--\\ 14387249 0.0067 34.9514 68.1707 12077.8996\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_inverse/duplicateVarsWithNoise100k.txt_train-translated_plus...\n /bin/cp -p results/mystery_world_inverse/duplicateVarsWithNoise100k.txt_train-translated_plus mystery.dat\n Number of variables..... 3\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.39973408779867031 at j= 33304\n Searching for best fit...\n 0.113942540636 0.127239308450 P 1 0.0688 13.4760 46.6953 13001.4238\n 0.083790214358 0.097293018510 aP+ 42 0.0283 18.4249 51.6442 12645.1688\n 0.083367607295 0.097238043016 bP+ 43 0.0282 18.4515 51.6708 12642.9073\n 0.079999667836 0.078247917487 aP>+ 430 0.0335 21.7140 54.9333 12716.0533\n 0.075871619652 0.100821772412 PaS+ 513 0.0340 21.8922 55.1114 12722.3545\n 0.075767827737 0.197208863906 aP+R 1262 0.0443 23.1889 56.4081 12832.2482\n 0.073264973224 0.199868405886 cSC> 2396 0.0265 24.0653 57.2846 12622.8397\n 0.073264800454 0.399736811782 cSCR 3692 0.0289 24.6891 57.9084 12658.8214\n 0.072978416206 0.399739593170 cc>/C 15184 0.0275 26.7235 59.9428 12640.5435\n 0.072796826146 -0.417690120838 cP-E< 18612 0.0349 27.0136 60.2329 12738.9244\n 0.072229016048 0.407879449487 cP+RS 20628 0.0332 27.1507 60.3700 12718.2666\n 0.039205086429 0.097172989144 Pac>/+ 66249 0.0098 27.9524 61.1717 12224.1033\n 0.039190700543 0.097173303181 PacE/+ 66761 0.0125 27.9630 61.1823 12320.3907\n 0.035330681877 0.078167611362 aPc>\\++ 877546 0.0116 31.5298 64.7491 12293.3030\n 0.034899450616 0.078132121222 bPc>\\++ 877547 0.0117 31.5121 64.7314 12296.7207\n 0.034899450616 0.078132121222 Pbc>\\++ 877553 0.0117 31.5121 64.7314 12296.7207\n 0.031972401700 0.100749009001 PacE/S+ 972525 0.0093 31.5340 64.7533 12205.8447\n 0.031246953248 0.097160490071 Pac>-E+ 978349 0.0105 31.5095 64.7288 12255.4806\n 0.030693758780 0.097107105296 Pbc>-E+ 978353 0.0106 31.4837 64.7030 12257.7667\n 0.030089440094 0.197083682980 aPcS-+R 1080042 0.0076 31.5977 64.8170 12124.7436\n 0.026868875137 0.083844852955 Pac-ER+ 1152429 0.0085 31.5280 64.7472 12169.6259\n 0.025050815926 0.251980003376 cPb-+\\E 1400220 0.0089 31.7079 64.9271 12188.2429\n 0.020867830306 0.245665632494 a>cP+/> 4872614 0.0054 33.2433 66.4626 11988.8277\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545943297282, '1/(666.000000000000+(sin(pi))**(-1))']\n [24.540090320707122, 28.418544887559722, 2.43951382042255e-7]\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [99.82550711782861, 23.516791257617538, tan(1.04068541526794*sin(exp(0.0345384214798065*x2)))]\n [144.44234332340804, 23.30665233444474, -1.03783059120178*tan(0.0105930790305138*x0 - 0.882021725177765)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_log/duplicateVarsWithNoise100k.txt_train-translated_plus...\n /bin/cp -p results/mystery_world_log/duplicateVarsWithNoise100k.txt_train-translated_plus mystery.dat\n Number of variables..... 3\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.29479879300520928 at j= 1\n Searching for best fit...\n 0.217586891281 -2.106687042311 P 1 0.0918 14.4093 47.6286 13119.4478\n 0.217586891281 -1.106687042311 P< 9 0.0918 17.5792 50.7985 13122.6177\n 0.217586891281 -22.105788957101 PE 37 0.0918 19.6188 52.8380 13124.6572\n 0.217586202654 -666.000000000000 PEE 425 0.0918 23.1406 56.3599 13128.1770\n 0.178846954675 0.993778048696 cPE/ 992 0.0801 24.0806 57.2999 13073.5376\n 0.164215422778 0.780936502798 aP+\\ 1198 0.0675 24.2297 57.4490 13004.0746\n 0.137663991643 1.461217658240 aP-\\ 1230 0.0526 24.0133 57.2326 12902.5306\n 0.122044168028 0.677238175923 a>>\\ 3130 0.0512 25.1871 58.4063 12892.2573\n 0.121132708302 0.305267508449 aP+L\\ 19666 0.0508 27.8277 61.0470 12892.1528\n 0.120059302172 0.809254337946 aERE\\ 49742 0.0500 29.1536 62.3729 12886.7099\n 0.119568644462 0.144452353251 aSSCR 52034 0.0506 29.2127 62.4320 12891.7999\n 0.116709165826 1.998558183594 aC<<~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 1.2524925899830208 at j= 40336\n Searching for best fit...\n 0.335536856541 0.398680762669 P 1 0.1706 15.0342 48.2535 13372.1970\n 0.307551174590 -0.397426405039 aP- 74 0.0963 21.1180 54.3373 13144.8411\n 0.295413815337 -0.398893539427 bP- 75 0.0947 21.0793 54.2985 13138.3010\n 0.219799891750 0.626261687925 aC> 130 0.0863 21.4463 54.6656 13101.0039\n 0.202357936781 0.626246734654 bC> 131 0.0863 21.3380 54.5573 13101.2890\n 0.194980446315 -0.301695797719 aP>- 718 0.0637 23.7389 56.9582 12979.8680\n 0.170454613758 0.552352640538 cP+R 1264 0.0630 24.3609 57.5802 12976.0464\n 0.168859275868 0.105518917889 cPP*+ 5140 0.0715 26.3711 59.5904 13029.4245\n 0.168675615711 0.417489379699 c>>R> 39524 0.0675 29.3124 62.5317 13008.7978\n 0.161544660039 0.551820831205 cPa-+R 87144 0.0615 30.3908 63.6100 12971.9797\n 0.136623005401 0.331320690955 PcaP+/+ 862485 0.0556 33.4561 66.6753 12934.6718\n 0.087201156748 -0.105430845982 acPP*+- 867846 0.0240 32.8172 66.0365 12592.2608\n 0.084597540933 0.383055124440 cPa-+R> 1327500 0.0275 33.3867 66.6060 12648.0684\n 0.084583598791 0.394832352253 cbCE+R> 2682724 0.0289 34.4014 67.6207 12669.6410\n 0.084313146749 0.383317076556 cP+RaC+ 4045628 0.0186 34.9895 68.2088 12489.3700\n 0.076027696937 -0.097245065407 acPP*>+- 12075402 0.0252 36.4179 69.6372 12615.1947\n 0.071849683839 0.363064366003 cPPa-*+R 12363456 0.0202 36.3703 69.5896 12525.4555\n 0.071849683839 0.643514842602 PacP/--R 12372913 0.0202 36.3714 69.5907 12525.4566\n 0.064797085582 0.789773615460 Pac>R--L 14719025 0.0163 36.4729 69.6922 12438.6620\n 0.063868976705 0.830488209185 cPa>/+RR 16989008 0.0174 36.6590 69.8783 12465.6182\n 0.053579627565 0.445560934992 cPa-RE+R 19052816 0.0128 36.5710 69.7903 12340.3039\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545943297282, '1/(666.000000000000+(sin(pi))**(-1))']\n [24.540090320707122, 28.418544887559722, 2.43951382042255e-7]\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [99.82550711782861, 23.516791257617538, tan(1.04068541526794*sin(exp(0.0345384214798065*x2)))]\n [144.44234332340804, 23.30665233444474, -1.03783059120178*tan(0.0105930790305138*x0 - 0.882021725177765)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_sin/duplicateVarsWithNoise100k.txt_train-translated_plus...\n /bin/cp -p results/mystery_world_sin/duplicateVarsWithNoise100k.txt_train-translated_plus mystery.dat\n Number of variables..... 3\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 3.8595698840027815E-006 at j= 74523\n Searching for best fit...\n 0.597142829315 -3.141596600582 P 1 0.2424 15.8658 49.0851 13515.4590\n 0.597142829315 -0.318313736896 P\\ 17 0.2424 19.9533 53.1725 13519.5465\n 0.597142829315 -1.772457735137 PR 21 0.2424 20.2581 53.4774 13519.8514\n 0.565360812364 -0.304871824725 aS 26 0.1823 20.4873 53.7066 13403.9106\n 0.413950443288 -0.098616193231 aP/ 90 0.1686 21.8290 55.0483 13373.9255\n 0.392316712046 -0.068569730610 bP/ 91 0.1571 21.7675 54.9868 13345.0817\n 0.372089905555 -1.144468709010 a>R 250 0.1573 23.1491 56.3684 13347.0826\n 0.342116586263 -1.102458413164 b>R 251 0.1455 23.0337 56.2530 13315.0335\n 0.318981733404 -0.100586028446 bP>/ 9998 0.1331 28.2214 61.4407 13284.0582\n 0.311800899079 -0.067845480380 bc>>/ 9999 0.1221 28.2159 61.4352 13249.0040\n 0.289078530507 -0.073094540088 acE>/ 10126 0.1308 28.1250 61.3442 13276.9351\n 0.248141776933 0.275439655393 ac-P/ 17278 0.0740 28.6755 61.8948 13045.4204\n 0.237276855936 -1.477118494587 bPc-+R 87175 0.0706 30.9459 64.1652 13028.4572\n 0.216301059070 -0.877239396346 ac>\\+R 169170 0.0678 31.7689 64.9881 13013.1153\n 0.203681765307 -0.191253382395 bcC+P/ 201667 0.0645 31.9356 65.1549 12992.5369\n 0.157132123950 -1.824726790468 PcaE--R 1084821 0.0382 33.9887 67.2080 12781.0571\n 0.151529791585 -0.045402102538 aEc-P>/ 6436774 0.0406 36.5052 69.7245 12808.4672\n 0.124602026386 0.134130902175 aac-+P>/ 20452658 0.0301 37.8908 71.1101 12688.6462\n 0.119537842999 0.156922556200 bac-+P>/ 20452659 0.0309 37.8310 71.0502 12699.4373\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_sin/duplicateVarsWithNoise100k.txt_train-translated_plus...\n /bin/cp -p results/mystery_world_sin/duplicateVarsWithNoise100k.txt_train-translated_plus mystery.dat\n Number of variables..... 3\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.59713896974525005 at j= 33304\n Searching for best fit...\n 0.947035968153 0.190075232206 P 1 0.4999 16.5311 49.7504 13810.8491\n 0.561459186212 0.617538704192 a 2 0.2258 16.7769 49.9962 13487.5584\n 0.541996610990 0.725416699544 aS 26 0.2450 20.4264 53.6457 13524.5352\n 0.525891502512 0.308398932395 ba+ 47 0.2244 21.2371 54.4563 13489.5487\n 0.521084695699 -0.297792110783 c<< 148 0.2163 22.8787 56.0980 13476.1909\n 0.346741848300 0.303585871886 acC+ 538 0.1637 24.1530 57.3723 13364.3933\n 0.346146308777 0.303227777457 bcC+ 539 0.1634 24.1532 57.3725 13363.5385\n 0.246572623506 -0.302780099155 ca>- 724 0.0699 24.0896 57.3089 13017.4498\n 0.186132110477 -0.226603228432 caE- 852 0.0504 23.9187 57.1380 12884.4511\n 0.184447424611 0.394989830673 bcERC+ 109827 0.0550 30.9158 64.1351 12926.6256\n 0.162239399843 0.240690306893 bac>C++ 879279 0.0422 33.7318 66.9511 12821.9193\n 0.153979030235 -0.233224392688 caEEC<+- 12973146 0.0426 37.5278 70.7471 12829.4657\n 0.136555829739 0.252982384711 bca>C<+- 12973147 0.0389 37.3662 70.5855 12792.9149\n 0.136544618593 -0.263785466741 cbaE>L+- 13001304 0.0340 37.3692 70.5885 12738.0918\n 0.134678540676 -0.263859992247 cabE>L+- 13001316 0.0340 37.3494 70.5687 12737.7228\n 0.130772747324 0.314318173594 bcaE\\+C+ 13464411 0.0353 37.3574 70.5767 12753.3398\n 0.130026811736 0.314649433067 acbE\\+C+ 13464426 0.0354 37.3492 70.5685 12754.5026\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545943297282, '1/(666.000000000000+(sin(pi))**(-1))']\n [24.540090320707122, 28.418544887559722, 2.43951382042255e-7]\n [43.65486140660157, 26.808399066180467, 0.138491379080858]\n [43.65486201547717, 26.808398865229687, 0.138491437529828]\n [43.654864729530274, 26.80839764467069, 0.138491698065464]\n [43.654868537895325, 26.8083949094849, 0.138492063650444]\n [44.11035814652141, 26.552667871365905, 'asin(-1.102458413164+sqrt((x1+1)))']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [99.82550711782861, 23.516791257617538, tan(1.04068541526794*sin(exp(0.0345384214798065*x2)))]\n [144.44234332340804, 23.30665233444474, -1.03783059120178*tan(0.0105930790305138*x0 - 0.882021725177765)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_sqrt/duplicateVarsWithNoise100k.txt_train-translated_plus...\n /bin/cp -p results/mystery_world_sqrt/duplicateVarsWithNoise100k.txt_train-translated_plus mystery.dat\n Number of variables..... 3\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.29479879300520928 at j= 1\n Searching for best fit...\n 0.192827070815 -1.463844017285 P 1 0.0795 14.2350 47.4543 13060.5692\n 0.192827070815 -2.463844017285 P> 5 0.0795 16.5569 49.7762 13062.8911\n 0.192827070815 -21.462945932076 PE 37 0.0795 19.4445 52.6638 13065.7787\n 0.192826160753 -666.000000000000 PEE 425 0.0795 22.9663 56.1856 13069.2974\n 0.154149598595 1.636621073721 cPE/ 992 0.0683 23.8662 57.0855 13008.3516\n 0.139650381662 1.423779527824 aP+\\ 1198 0.0556 23.9959 57.2152 12925.0595\n 0.129058397301 2.104060683265 aP-\\ 1230 0.0437 23.9202 57.1395 12826.9915\n 0.119831790614 1.320081200949 a>>\\ 3130 0.0435 25.1607 58.3800 12826.5519\n 0.105422920922 1.337321499397 aP<+\\ 13026 0.0438 27.0330 60.2523 12831.2627\n 0.104176424835 0.774353481486 aa>/C 15174 0.0443 27.2361 60.4553 12835.6522\n 0.103936637059 1.041474894276 PaP>+/ 71385 0.0436 29.4667 62.6860 12831.1938\n 0.102334614603 0.961879792762 PaP+/S 88793 0.0432 29.7592 62.9784 12827.9716\n 0.100806873609 2.625862160456 aaSE-S 177738 0.0428 30.7387 63.9580 12825.6179\n 0.100140090262 2.625007630435 aSC<L-+\\ 14347502 0.0240 36.5212 69.7405 12595.7764\n 0.063198340205 2.039357595387 aPcC<--\\ 14385902 0.0218 36.4038 69.6231 12556.4771\n 0.060061542977 1.201357762627 PcaE--R\\ 16910617 0.0200 36.5637 69.7829 12522.2059\n 0.056713375792 0.827694199643 aPcC++RS 17070638 0.0191 36.4945 69.7138 12503.6268\n 0.052599954795 1.163919696951 cP+RaP+/ 60370688 0.0130 38.2082 71.4275 12346.7733\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_sqrt/duplicateVarsWithNoise100k.txt_train-translated_plus...\n /bin/cp -p results/mystery_world_sqrt/duplicateVarsWithNoise100k.txt_train-translated_plus mystery.dat\n Number of variables..... 3\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 1.8705757945424719 at j= 40336\n Searching for best fit...\n 0.288911147145 0.595422751690 P 1 0.1516 14.8183 48.0376 13323.9516\n 0.288911147145 -666.000000000000 PS 25 0.1516 19.4622 52.6815 13328.5954\n 0.185249331195 0.451661501833 PaC+ 529 0.0837 23.2243 56.4436 13090.6156\n 0.182009311229 0.451656294014 PbC+ 533 0.0838 23.2097 56.4290 13090.8879\n 0.154679164249 0.074403606967 cPE+ 560 0.0775 23.0463 56.2656 13059.1186\n 0.153952069447 0.077485659327 cPE<+ 6372 0.0750 26.5477 59.7670 13049.1842\n 0.151587980927 -39.544833484320 cPE-\\ 13428 0.0704 27.6008 60.8201 13024.6652\n 0.151381309157 1.242212642428 cP+RR 20052 0.0603 28.1774 61.3966 12962.2181\n 0.084033902416 -0.074374274002 acPE+- 69602 0.0345 29.1236 62.3429 12736.2498\n 0.082959380711 -0.074408566705 bcPE+- 69603 0.0352 29.1051 62.3243 12744.0007\n 0.078328690864 -0.077453846406 acPE<+- 918342 0.0306 32.7440 65.9633 12690.7518\n 0.071261053505 -39.526285793292 caPE+-\\ 1074432 0.0264 32.8340 66.0533 12631.0718\n 0.046703164354 0.077454003542 cPEaE-+ 1421452 0.0123 32.6282 65.8475 12320.7928\n 0.041633940250 -0.074379229830 bacPE+-+ 11874483 0.0125 35.5249 68.7442 12330.6962\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545943297282, '1/(666.000000000000+(sin(pi))**(-1))']\n [24.540090320707122, 28.418544887559722, 2.43951382042255e-7]\n [43.65486140660157, 26.808399066180467, 0.138491379080858]\n [43.65486201547717, 26.808398865229687, 0.138491437529828]\n [43.654864729530274, 26.80839764467069, 0.138491698065464]\n [43.654868537895325, 26.8083949094849, 0.138492063650444]\n [44.11035814652141, 26.552667871365905, 'asin(-1.102458413164+sqrt((x1+1)))']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [99.82550711782861, 23.516791257617538, tan(1.04068541526794*sin(exp(0.0345384214798065*x2)))]\n [144.44234332340804, 23.30665233444474, -1.03783059120178*tan(0.0105930790305138*x0 - 0.882021725177765)]\n [201.49685622441342, 23.16254036495239, (sin((x0 + cos(x2) + 3.17548894882202)**0.429349541664124) + 0.394926816225052)**1.80531096458435]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_squared/duplicateVarsWithNoise100k.txt_train-translated_plus...\n /bin/cp -p results/mystery_world_squared/duplicateVarsWithNoise100k.txt_train-translated_plus mystery.dat\n Number of variables..... 3\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.29479879300520928 at j= 1\n Searching for best fit...\n 4.320048900614 4.781735875560 P 1 1.6589 18.7207 51.9400 14300.2823\n 3.422307693052 6.971606225870 c 4 1.4101 20.3846 53.6039 14235.9824\n 3.408364344089 6.019883835167 cc+ 56 1.3804 24.1861 57.4054 14231.0877\n 2.838249775550 7.017553107608 cc* 72 1.1890 24.2846 57.5038 14170.5408\n 2.653760841945 7.767498881236 ca- 80 1.0134 24.3396 57.5589 14105.4993\n 2.285809252563 10.423699205298 aP~* 606 0.9005 27.0455 60.2648 14060.2148\n 2.233612213435 9.188024837847 caE- 852 0.7600 27.5037 60.7230 13991.4846\n 2.140874612583 7.775021968316 cca-* 5488 0.7966 30.1299 63.3492 14013.3767\n 1.964665511267 8.563391536603 caa+- 5608 0.6425 30.0372 63.2565 13925.6586\n 1.916315998255 8.507025672033 cba+- 5612 0.6817 30.0023 63.2216 13949.8772\n 1.293270027597 9.471976814595 caP*- 5656 0.3688 29.4462 62.6655 13699.2013\n 1.293270027597 6.330384073582 cPa<*- 69736 0.3688 33.0703 66.2896 13702.8254\n 1.232551933122 9.302918327400 cbaa++- 867668 0.3994 36.6381 69.8574 13738.9799\n 1.037159254112 2.573075150913 cP+aCC/ 3843388 0.2465 38.5362 71.7555 13544.2110\n 1.013515925504 9.027577479242 PcP<~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 12.243377517185923 at j= 40336\n Searching for best fit...\n 5.985059467298 3.897187995551 P 1 3.4303 19.1910 52.4103 14596.7372\n 5.602542206518 4.080807169668 c> 8 1.9593 22.0957 55.3150 14371.1924\n 4.072192900909 12.243979408563 aC 30 1.7887 23.5423 56.7616 14335.9316\n 4.022110083414 12.243394708374 bC 31 1.7899 23.5718 56.7911 14336.2533\n 3.561698645492 2.381133688946 cP+ 44 1.3057 23.9016 57.1209 14208.0395\n 3.525298217646 7.068441324160 c>R 252 1.3683 26.4047 59.6239 14229.6751\n 3.522250899097 1.993442260171 cP>+ 432 1.5182 27.1810 60.4003 14272.8813\n 2.147033946366 2.376550748921 cPa-+ 5220 0.6972 30.0618 63.2811 13958.9350\n 1.993786758187 2.381179224541 PcaC*+ 65457 0.7940 33.6034 66.8227 14015.6228\n 1.630976886765 1.990229191155 cPa<-+ 65704 0.4846 33.3191 66.5384 13814.1065\n 1.604608049949 1.993986313738 cPb<-+ 65720 0.4929 33.2959 66.5152 13821.0554\n 1.228011956846 1.566791658402 cPaCC/+ 895180 0.3633 36.6778 69.8971 13700.3579\n 1.214183149969 1.566704965926 cPbCC/+ 895196 0.3645 36.6615 69.8808 13701.6578\n 1.086566675803 6.098761582740 c>aE/>R 6145936 0.2974 39.2806 72.4999 13621.5047\n 1.022051787613 4.464991170760 c>RaE\\+ 7171648 0.2447 39.4150 72.6343 13542.1609\n 1.000533523514 -2.996786564159 aPcP+ 17966352 0.1987 40.3667 73.5860 13458.4470\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545943297282, '1/(666.000000000000+(sin(pi))**(-1))']\n [24.540090320707122, 28.418544887559722, 2.43951382042255e-7]\n [43.65486140660157, 26.808399066180467, 0.138491379080858]\n [43.65486201547717, 26.808398865229687, 0.138491437529828]\n [43.654864729530274, 26.80839764467069, 0.138491698065464]\n [43.654868537895325, 26.8083949094849, 0.138492063650444]\n [44.11035814652141, 26.552667871365905, 'asin(-1.102458413164+sqrt((x1+1)))']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [99.82550711782861, 23.516791257617538, tan(1.04068541526794*sin(exp(0.0345384214798065*x2)))]\n [144.44234332340804, 23.30665233444474, -1.03783059120178*tan(0.0105930790305138*x0 - 0.882021725177765)]\n [201.49685622441342, 23.16254036495239, (sin((x0 + cos(x2) + 3.17548894882202)**0.429349541664124) + 0.394926816225052)**1.80531096458435]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_tan/duplicateVarsWithNoise100k.txt_train-translated_plus...\n /bin/cp -p results/mystery_world_tan/duplicateVarsWithNoise100k.txt_train-translated_plus mystery.dat\n Number of variables..... 3\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 3.8595698840315281E-006 at j= 74523\n Searching for best fit...\n 0.744438256573 -3.141588881443 P 1 0.2631 16.1839 49.4031 13548.8588\n 0.695571202304 0.309803651174 a~ 14 0.2142 19.8933 53.1125 13468.7527\n 0.435365266897 -0.952390766529 aC 30 0.1632 20.3168 53.5361 13359.0547\n 0.435210540936 -0.976885825928 bC 31 0.1595 20.3636 53.5829 13349.5474\n 0.427319513892 -0.926635978353 cCC 348 0.1661 23.8260 57.0453 13369.7209\n 0.420881255543 0.603641485141 cSC~ 3044 0.1659 26.9329 60.1522 13372.1880\n 0.413497208620 -0.721667700408 bE\\S 3919 0.1561 27.2719 60.4911 13347.8692\n 0.413497208620 -0.721667700408 b~ES 4075 0.1561 27.3282 60.5475 13347.9255\n 0.406366222382 -0.539852822304 bbEC+ 7099 0.1599 28.1039 61.3232 13358.5697\n 0.391440479756 0.116388225314 bP>C* 8259 0.1554 28.2683 61.4875 13347.1153\n 0.391440401237 0.116388257006 bP/ 20453016 0.0330 38.2221 71.4413 12725.9249\n 0.156758592208 -0.156922556200 cb-a-P>/ 59181608 0.0330 39.7549 72.9742 12727.4578\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_tan/duplicateVarsWithNoise100k.txt_train-translated_plus...\n /bin/cp -p results/mystery_world_tan/duplicateVarsWithNoise100k.txt_train-translated_plus mystery.dat\n Number of variables..... 3\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pabc\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.74443439700295699 at j= 33304\n Searching for best fit...\n 1.117941457203 -0.236960821587 P 1 0.6332 16.7705 49.9898 13907.2943\n 0.737039637270 -0.769866105167 a 2 0.2916 17.1695 50.3887 13591.8371\n 0.688868588890 -0.904354213444 aS 26 0.3173 20.7724 53.9917 13630.0129\n 0.686765113039 -0.384471261977 ba+ 47 0.2898 21.6221 54.8414 13593.8861\n 0.683723425876 0.236567676920 cP- 76 0.3899 22.3091 55.5283 13715.6769\n 0.676311762193 0.371248070642 c<< 148 0.2784 23.2549 56.4741 13579.1781\n 0.667304196650 -0.253531931387 ba>+ 435 0.3893 24.7909 58.0102 13717.5450\n 0.400639014465 -0.378470970622 acC+ 538 0.2213 24.3615 57.5808 13487.3419\n 0.334086184418 0.377466439068 ca>- 724 0.0945 24.5278 57.7471 13140.6108\n 0.222421777513 0.282499127111 caE- 852 0.0681 24.1757 57.3950 13007.3795\n 0.213091219657 -0.253081929600 bac<-+ 65739 0.1079 30.3836 63.6029 13201.2960\n 0.213091219657 -0.253081929600 bac-+> 83083 0.1079 30.7214 63.9407 13201.6338\n 0.212119881165 -0.300061045343 bac>C++ 879279 0.0611 34.1186 67.3378 12972.7195\n 0.211738755138 -0.410117947629 bcaC+C+ 972951 0.0737 34.2620 67.4813 13049.3340\n 0.208652556700 -0.410412724001 acbC+C+ 972966 0.0738 34.2408 67.4601 13049.8522\n 0.201705272555 -0.556691661857 acE>LL- 2038790 0.0479 35.2592 68.4785 12875.0455\n 0.199214873588 -0.315385192655 bca>C<+- 12973147 0.0528 37.9111 71.1303 12917.4776\n 0.198688133273 -0.261470361299 acbEC<+- 12973674 0.0517 37.9073 71.1266 12908.7252\n 0.190401827377 0.328853055689 cbaE>L+- 13001304 0.0498 37.8489 71.0682 12893.2978\n 0.182991289789 0.328945964296 cabE>L+- 13001316 0.0499 37.7916 71.0109 12893.9691\n 0.182991289789 -0.328945964296 acbE>L-- 13094634 0.0499 37.8020 71.0212 12893.9794\n 0.180086967702 -0.318660314964 abSc>C++ 21709830 0.0597 38.5083 71.7275 12968.3471\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545943297282, '1/(666.000000000000+(sin(pi))**(-1))']\n [24.540090320707122, 28.418544887559722, 2.43951382042255e-7]\n [43.65486140660157, 26.808399066180467, 0.138491379080858]\n [43.65486201547717, 26.808398865229687, 0.138491437529828]\n [43.654864729530274, 26.80839764467069, 0.138491698065464]\n [43.654868537895325, 26.8083949094849, 0.138492063650444]\n [44.11035814652141, 26.552667871365905, 'asin(-1.102458413164+sqrt((x1+1)))']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [99.82550711782861, 23.516791257617538, tan(1.04068541526794*sin(exp(0.0345384214798065*x2)))]\n [144.44234332340804, 23.30665233444474, -1.03783059120178*tan(0.0105930790305138*x0 - 0.882021725177765)]\n [201.49685622441342, 23.16254036495239, (sin((x0 + cos(x2) + 3.17548894882202)**0.429349541664124) + 0.394926816225052)**1.80531096458435]\n Checking for symmetry \n duplicateVarsWithNoise100k.txt_train-translated_plus\n Found pretrained NN \n \n tensor(0.0153, device='cuda:0', grad_fn=)\n tensor(0.0127, device='cuda:0', grad_fn=)\n tensor(0.0113, device='cuda:0', grad_fn=)\n NN loss after training: tensor(0.0081, device='cuda:0', grad_fn=) \n \n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/translated_data_plus/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/translated_data_plus/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.79103598556975296 at j= 1\n Searching for best fit...\n 0.684213023160 -0.326751961044 P 1 0.2755 16.0622 49.2814 13567.7330\n 0.684213023160 5.956433520981 P~ 10 0.2755 19.3841 52.6034 13571.0549\n 0.557481145073 2.511898142535 bP/ 60 0.2277 21.6735 54.8928 13495.9290\n 0.440978681285 2.274012074379 aSC 245 0.1559 23.3651 56.5843 13343.1689\n 0.384634606413 3.303579955521 aP~/ 572 0.1517 24.3911 57.6104 13333.3912\n 0.379733403635 2.803582318149 aCP/ 1220 0.1562 25.4654 58.6846 13346.3930\n 0.376273106051 2.217294686197 a>LC 2930 0.1501 26.7162 59.9355 13331.3860\n 0.356714005945 3.185572414433 aP~E/ 6741 0.1408 27.7497 60.9690 13306.4046\n 0.224385382710 3.000637318088 ba-P/ 10080 0.0644 27.7529 60.9722 12987.6829\n 0.224385382710 3.318947195414 ba>-P/ 112635 0.0644 31.2350 64.4543 12991.1650\n 0.149882159900 2.955776186214 ba-P>/ 142623 0.0509 30.9934 64.2127 12895.8802\n 0.148369071228 2.649091562426 ab-EE\\S 2333396 0.0424 35.0109 68.2302 12825.3073\n 0.140480938130 2.960765043833 ba-ERRL 2387502 0.0466 34.9652 68.1844 12863.7401\n 0.139866477944 2.564651292297 aPb-+LCS 10468499 0.0414 37.0913 70.3106 12818.0499\n 0.135974321428 2.334898651907 baCE+P>/ 20800413 0.0368 38.0412 71.2604 12771.0268\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/translated_data_plus/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/translated_data_plus/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 3.4990538031282004 at j= 40336\n Searching for best fit...\n 0.997390746302 1.113783386831 P 1 0.5395 16.6059 49.8252 13841.9546\n 0.997390746302 0.844857044605 P> 4 0.5395 18.6059 51.8252 13843.9546\n 0.975905140835 0.680507880964 bP+ 33 0.3438 21.6188 54.8381 13663.1645\n 0.740697675138 0.569708947713 bP>+ 312 0.2482 24.4620 57.6813 13533.3459\n 0.738556840697 0.139177586220 bPE+ 384 0.3931 24.7574 57.9767 13721.3526\n 0.717452925634 -0.151153990358 aPE- 545 0.3803 25.2207 58.4400 13708.3668\n 0.546205363230 1.543092249007 bP+R 780 0.2114 25.3445 58.5638 13469.2997\n 0.543771406040 2.658654043715 b>RR 2364 0.2211 26.9377 60.1570 13489.1703\n 0.543428540947 1.927723485521 bP>+L 9087 0.2183 28.8794 62.0987 13485.9125\n 0.531637984833 1.758237491062 b>SC> 23424 0.2124 30.2139 63.4331 13475.9744\n 0.468296488964 -0.139131986937 abPE+- 43679 0.2380 30.9298 64.1491 13523.4564\n 0.434607890055 0.925685069719 PbaP+/+ 520966 0.1786 34.3983 67.6175 13409.8917\n 0.265348453227 -0.294580783766 abPP*+- 522659 0.1040 33.6911 66.9104 13189.0096\n 0.240127040068 2.322722925146 bPa-+RR 727836 0.0805 34.0248 67.2440 13084.9384\n 0.141069208464 -0.232964733049 abPP>*+- 6756986 0.0412 36.4721 69.6913 12815.3732\n 0.133956073956 1.245400106353 baPRE--R 7759953 0.0314 36.5971 69.8164 12704.9683\n 0.117870449639 2.982206052900 ba-P/ERR 29220057 0.0292 38.3254 71.5447 12676.6473\n 0.103392163605 -0.249581722261 abPP>*<+- 90096509 0.0251 39.7608 72.9801 12615.9376\n Checking polyfit \n \n Complexity RMSE Expression\n [46.893261128102594, 27.327578182802576, '-0.326751961044+pi']\n [48.091811949279766, 26.831252455317497, 2.99970082515141]\n [48.091812250119716, 26.83125070219585, 2.99970145066818]\n [48.09181225032447, 26.831250699084304, 2.99970145109393]\n [55.69223324298765, 26.59193976726203, '2.274012074379+cos(sin(x0))']\n [56.65579385057341, 26.572758653681174, '2.217294686197+cos(log((x0+1)))']\n [63.70190275624589, 25.161781133142636, '3.000637318088+((x1-x0)/pi)']\n [67.24012826274301, 24.417737582213533, '2.334898651907+((x1+exp(cos(x0)))/(pi+1))']\n [74.73485772563441, 24.012198128634623, '2.982206052900*sqrt(sqrt(exp(((x1-x0)/pi))))']\n [102.70931326523065, 23.84771769339375, '-0.257444152219682*x0 + 0.257626825711277*x1 + 3']\n [148.64361064734402, 23.746316947653487, -0.243123829930109*x0 + 0.244686862542045*x1 + 2.99845314025879]\n [163.0766689392833, 23.745070729363103, log(exp(-0.991362750530243*x0 + x1)**0.247284024953842) + 2.99792242050171]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_atan/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/mystery_world_atan/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.79103598556975296 at j= 1\n Searching for best fit...\n 0.062981887482 -1.912149389642 P 1 0.0276 12.6207 45.8400 12629.0104\n 0.062981887482 -21.911251304432 PE 28 0.0276 17.4281 50.6474 12633.8178\n 0.062981752531 1.229443485601 aPS* 437 0.0276 21.3922 54.6115 12637.7811\n 0.062971220621 1.229443408308 PSa/ 1243 0.0276 22.9001 56.1194 12639.2932\n 0.059480794644 -666.000000000000 PPC>/ 6142 0.0208 25.1227 58.3420 12525.7787\n 0.055041845386 1.295794836017 aPE~/ 6323 0.0188 25.0527 58.2720 12485.0313\n 0.048909219336 1.214313334470 bP>E/ 6738 0.0235 24.9740 58.1933 12576.5944\n 0.040858911270 1.220136572069 aP+E\\ 11426 0.0180 25.4765 58.6957 12467.8964\n 0.036621270558 1.220585881124 baP+E/ 49185 0.0132 27.4244 60.6437 12342.8854\n 0.035123086917 0.953986444679 PaP/+\\ 50668 0.0153 27.4070 60.6263 12404.1215\n 0.034623067531 1.079675764916 aP>>+\\ 92057 0.0154 28.2478 61.4670 12406.0266\n 0.034528003934 0.913251824838 aP+R>\\ 160241 0.0153 29.0434 62.2627 12406.1309\n 0.025857775909 1.238722681562 ba-P>E/ 2009430 0.0120 32.2747 65.4940 12310.9637\n 0.015445088119 -0.949873230661 baPE--RR 8792325 0.0055 33.6607 66.8800 11992.7867\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_atan/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/mystery_world_atan/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 1.2924252388523800 at j= 40336\n Searching for best fit...\n 0.101906033496 0.411391719232 P 1 0.0492 13.3150 46.5342 12864.9112\n 0.101906033496 0.602605125527 PL> 88 0.0492 19.7744 52.9937 12871.3707\n 0.068757532089 0.051407218989 bPE+ 384 0.0224 21.3323 54.5515 12552.5685\n 0.064582896057 0.049440677588 bPE>+ 3975 0.0211 24.6137 57.8330 12531.3126\n 0.063841122453 -0.020543653045 aP>E- 6008 0.0283 25.1929 58.4122 12650.9179\n 0.053724250683 0.257759553229 bPE+R 8187 0.0237 25.3905 58.6098 12579.8827\n 0.049602954761 0.054863617978 PEbC- 15103 0.0186 26.1588 59.3780 12481.2383\n 0.048389140668 0.036705495252 bPC>L- 75345 0.0181 28.4417 61.6610 12472.4083\n 0.047917941575 0.321973519348 bPE+R< 117726 0.0196 29.0714 62.2907 12505.6852\n 0.047876216307 1.457472165619 bPE/ES 130929 0.0196 29.2235 62.4428 12505.5839\n 0.047823939003 1.053211876563 bP+RRR 164454 0.0189 29.5508 62.7701 12490.5864\n 0.026265269335 -0.019910603743 abP>E+- 544610 0.0068 30.4138 63.6331 12077.8955\n 0.020754429197 0.257717324398 baPE--R 613350 0.0089 30.2456 63.4648 12184.4270\n 0.017811129077 0.252741458572 baPE>--R 7758117 0.0068 33.6858 66.9051 12081.6069\n 0.017479896630 0.214870801201 baPE--R> 8652357 0.0035 33.8161 67.0354 11813.2858\n 0.015221698443 0.580911007276 baPE--L< 8699013 0.0049 33.6243 66.8436 11947.8297\n 0.013143478186 0.242987041117 bPaPE--+R 90917286 0.0037 36.7982 70.0175 11838.1974\n 0.013096042469 0.570905250531 baPE>--L< 111541716 0.0032 37.0879 70.3072 11774.9671\n Checking polyfit \n \n Complexity RMSE Expression\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [94.16582283646451, 23.248867658173417, tan(0.00980361737310886*x1 + 0.885341703891754)]\n [142.76936460103022, 22.73883853559589, tan(-0.0151103562395357*x0 + 0.0150624554288796*x1 + 0.894917964935303)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_cos/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/mystery_world_cos/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.79103598556975296 at j= 1\n Searching for best fit...\n 0.144952083475 -4.088682624876 P 1 0.0435 13.8233 47.0426 12814.2703\n 0.144952083475 -3.088682624876 P< 7 0.0435 16.6306 49.8499 12817.0777\n 0.144952083475 -10.816694834246 PP* 40 0.0435 19.1452 52.3645 12819.5922\n 0.144952083475 -63.850019664652 P>E 283 0.0435 21.9679 55.1872 12822.4150\n 0.144951999816 -0.947089800661 bPS* 438 0.0435 22.5981 55.8174 12823.0449\n 0.127630410843 -1.013441368510 aPE/ 626 0.0319 22.9297 56.1490 12697.2806\n 0.126848065352 -1.016438179842 aPE+\\ 7791 0.0387 26.2976 59.5169 12780.1150\n 0.104689505126 -1.024293303048 bCP-E 18771 0.0387 27.5500 60.7693 12780.5604\n 0.100593558795 -1.898286102201 bSCEE/ 83234 0.0393 29.5465 62.7657 12789.0870\n 0.091470621513 -0.947092373944 abEEE/ 83306 0.0402 29.5052 62.7245 12799.0376\n 0.081699207672 -1.016505133109 aPb>E*/ 555557 0.0180 32.0797 65.2990 12474.6088\n 0.078644790583 -0.975588615817 abP>+-E 628202 0.0259 32.2020 65.4213 12621.5917\n 0.066264766041 -0.962140439124 aPbE+-E 628430 0.0304 31.9554 65.1747 12687.7731\n 0.065727459097 -0.957437440256 abEREE/ 1121321 0.0336 32.7790 65.9983 12729.7282\n 0.053206644076 -0.986422234212 aabP+E/* 6802604 0.0165 35.0750 68.2943 12440.5943\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_cos/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/mystery_world_cos/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.99999999999255185 at j= 74523\n Searching for best fit...\n 0.197862199604 -0.318309877324 P 1 0.0482 14.2722 47.4915 12856.2782\n 0.197862199604 -0.241453001907 P> 4 0.0482 16.2722 49.4915 12858.2782\n 0.197862199604 -0.758546998086 P\\> 76 0.0482 20.5201 53.7394 12862.5261\n 0.153430706313 -0.041125479695 bPE+ 384 0.0394 22.4903 55.7095 12782.3010\n 0.135469157373 0.044217483331 aPE- 545 0.0334 22.8158 56.0351 12715.7385\n 0.122965454423 -0.216617822335 Pb>R+ 4231 0.0423 25.6327 58.8520 12815.4655\n 0.103050129894 -0.245806988807 PbCC+ 4447 0.0368 25.4497 58.6690 12758.3512\n 0.102596582438 -0.720443175723 bCC>R 30759 0.0391 28.2334 61.4527 12786.2493\n 0.098728788954 -1.002696903842 bCC>/C 1298873 0.0141 33.0456 66.2649 12374.3011\n 0.067801536748 -1.004590383337 abbP++/C 6907907 0.0160 35.4469 68.6662 12430.3591\n 0.067541880693 -1.004330526120 aPbER*/C 7835648 0.0145 35.6232 68.8425 12389.8630\n 0.065685287139 -0.492594432661 PabC*C+R 8418796 0.0145 35.6865 68.9058 12390.1343\n 0.065287821501 -1.002076597029 aPbC//SC 8867372 0.0141 35.7527 68.9720 12377.0229\n 0.057540373352 -0.042585739285 aba-*PE+ 10137692 0.0146 35.7636 68.9829 12391.6966\n Checking polyfit \n \n Complexity RMSE Expression\n [2.0, 27.15815915791503, '3.14159265358979']\n [46.46201740344512, 24.34147126490242, -0.969305992126465]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [94.16582283646451, 23.248867658173417, tan(0.00980361737310886*x1 + 0.885341703891754)]\n [142.76936460103022, 22.73883853559589, tan(-0.0151103562395357*x0 + 0.0150624554288796*x1 + 0.894917964935303)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_exp/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/mystery_world_exp/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.79103598556975296 at j= 1\n Searching for best fit...\n 16.393614936972 13.548925362924 P 1 5.6739 20.6447 53.8640 14802.0853\n 15.408290755222 15.738795713234 b 3 5.3643 22.1402 55.3595 14780.7716\n 15.408290755222 16.738795713234 b< 9 5.3643 23.7252 56.9445 14782.3565\n 14.850223215065 18.225937550100 a~ 11 5.0316 23.9615 57.1808 14756.5214\n 12.664089441422 14.100351004719 bE 30 4.5619 25.1792 58.3985 14717.9806\n 12.484863459844 9.649713945462 b>E 285 4.2682 28.4066 61.6258 14694.0654\n 11.789879965853 13.148628614017 bbE+ 390 4.4139 28.7764 61.9957 14708.2156\n 11.764027202276 21.514180690413 aP~* 410 3.8475 28.8454 62.0647 14652.2542\n 11.718409497757 20.382068437718 baE- 549 3.6299 29.2610 62.4803 14628.9156\n 10.900344365142 16.132685890989 ba-E 945 4.3115 29.9401 63.1594 14699.9142\n 10.799306361832 11.690709620907 bERE 3360 4.0366 31.7568 64.9760 14674.8684\n 9.941140576801 13.872588469910 aCEE 3461 3.4752 31.6800 64.8993 14613.7992\n 8.703377892974 18.524256536323 Pba-* 3646 2.6190 31.5633 64.7826 14498.4523\n 6.453713803265 14.007098661414 baC+E 9468 1.7124 32.5086 65.7279 14326.4389\n 6.137924513274 15.174172936153 ba<-E 9585 2.2647 32.4540 65.6732 14440.5364\n 6.049735549114 15.534332382233 Pbba-+* 521488 2.3729 38.1988 71.4181 14465.3467\n 4.728311962571 23.347919122800 Pbaa+-* 521965 1.5268 37.8446 71.0638 14285.4256\n 4.111171749993 19.691650647245 ba-P>>* 1984158 1.0920 39.5693 72.7886 14150.5831\n 3.340111629711 19.663471148853 bP>+ba-* 23653176 0.7683 42.8451 76.0643 14010.7019\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_exp/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/mystery_world_exp/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 33.084133040908405 at j= 40336\n Searching for best fit...\n 20.881362044207 10.531006329689 P 1 13.2654 20.9938 54.2131 15148.6354\n 14.374093380510 11.027183236495 b> 6 4.8512 23.0400 56.2593 14740.7482\n 13.906767305192 6.434314685036 bP+ 33 6.8797 25.4517 58.6710 14885.7586\n 13.864453700752 19.100387358970 b>R 177 6.9850 27.8706 61.0898 14894.3772\n 13.639414790167 33.415221737414 b>SC 2877 4.8985 31.8697 65.0890 14753.6115\n 6.454271337654 6.424020267247 bPa-+ 3534 1.3948 31.0870 64.3063 14241.3256\n 4.429020648426 5.379475459695 bPa<-+ 42036 1.9306 34.1160 67.3352 14377.5383\n 2.709666510295 20.024075133238 ba-ERR 164619 0.5245 35.3765 68.5958 13847.7208\n 2.709666510295 15.594765394046 ba<-ERR 1853415 0.5245 38.8695 72.0888 13851.2138\n 2.679093106489 20.370784247467 ba-P>/E 2044422 0.5835 38.9946 72.2139 13894.8946\n 2.667482426006 29.963802934764 PaPb-+L-L 93305860 0.7376 44.5006 77.7199 13996.0398\n Checking polyfit \n \n Complexity RMSE Expression\n [2.0, 27.15815915791503, '3.14159265358979']\n [32.85758306839902, 25.12250102174253, 'log(-x0*x1 - 4*x0 + 2*x1**2 + 4*x1 + 20)']\n [46.46201740344512, 24.34147126490242, -0.969305992126465]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [94.16582283646451, 23.248867658173417, tan(0.00980361737310886*x1 + 0.885341703891754)]\n [142.76936460103022, 22.73883853559589, tan(-0.0151103562395357*x0 + 0.0150624554288796*x1 + 0.894917964935303)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_inverse/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/mystery_world_inverse/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.28579154716226046 at j= 40336\n Searching for best fit...\n 0.113942540636 -2.855801193850 P 1 0.0543 13.4760 46.6953 12904.7617\n 0.113942540636 -22.854903108641 PE 28 0.0543 18.2834 51.5027 12909.5690\n 0.113942539830 666.000000000000 PS\\ 163 0.0543 20.8247 54.0440 12912.1104\n 0.113942365314 0.285791722028 bPS* 438 0.0543 22.2508 55.4701 12913.5358\n 0.055941041286 0.286147617631 aPE/ 626 0.0195 21.7397 54.9590 12497.3454\n 0.055782374218 0.286132867829 aPE>/ 6161 0.0197 25.0345 58.2538 12503.7201\n 0.053355950385 0.372229499831 bPE~/ 6324 0.0196 25.0081 58.2273 12502.6061\n 0.052856745528 0.303784085929 bCPE/ 15936 0.0207 26.3279 59.5472 12524.7734\n 0.052669104055 -0.634647579199 bPE/>\\ 122505 0.0206 29.2652 62.4845 12527.3634\n 0.047557564864 0.372260288797 baPE+~/ 588564 0.0179 31.3823 64.6016 12472.1352\n 0.047175277119 0.325437704854 abS-PE/ 1663265 0.0117 32.8694 66.0887 12300.3249\n 0.038455085178 0.324932048864 ab-SPE/ 2119772 0.0122 32.9244 66.1437 12318.1618\n 0.035003640132 0.268322612533 ab-SP-E 2153792 0.0181 32.8117 66.0310 12477.6084\n 0.020784979668 0.165188427402 bPPa-++\\ 6888633 0.0084 33.7371 66.9564 12168.2654\n 0.020784979668 0.165188427402 PbPa-++\\ 6888637 0.0084 33.7371 66.9564 12168.2654\n 0.020762744018 0.163093165256 bPa<<<-+\\ 102603507 0.0069 37.6323 70.8516 12087.5421\n 0.016358305921 -18.627009935232 abPREE--R 103092986 0.0053 37.2952 70.5145 11979.6769\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_inverse/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/mystery_world_inverse/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.79103598556975296 at j= 1\n Searching for best fit...\n 0.069468333651 0.113082729081 P 1 0.0307 12.7621 45.9814 12672.5981\n 0.069468333651 0.657520571271 PCC 247 0.0307 20.7105 53.9298 12680.5464\n 0.048691070313 0.014396913523 aPE+ 383 0.0221 20.8306 54.0499 12547.0870\n 0.041579756297 0.031149418753 aPP*+ 3497 0.0169 23.7936 57.0128 12441.3433\n 0.041579756297 0.097858787840 PaP/+ 3553 0.0169 23.8165 57.0358 12441.3662\n 0.038923581638 0.075046034325 Pa>R+ 4228 0.0175 23.9722 57.1915 12454.7660\n 0.038550919081 0.286029698916 aP+LR 11714 0.0172 25.4285 58.6478 12448.4819\n 0.038478841250 0.259081217442 a>>RR 31550 0.0172 26.8552 60.0745 12450.9095\n 0.033841149225 -0.014974456885 baPE+- 43677 0.0160 27.1391 60.3584 12421.3050\n 0.025894631293 -0.033985422081 baPP*+- 522657 0.0062 30.3339 63.5532 12039.3440\n 0.025894631293 0.033985422081 abPP*-- 523307 0.0062 30.3357 63.5550 12039.3458\n 0.017506885668 -0.026131859601 baPP>*+- 6756984 0.0062 33.4617 66.6809 12045.2222\n 0.017459302471 0.135570861702 aPPb-++R 6893843 0.0057 33.4867 66.7059 12011.3600\n 0.016669622307 0.127630161326 Pab-E>\\- 9320353 0.0044 33.8550 67.0742 11899.4714\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545943297282, '1/(666.000000000000+(sin(pi))**(-1))']\n [2.0, 27.15815915791503, '3.14159265358979']\n [32.85758306839902, 25.12250102174253, 'log(-x0*x1 - 4*x0 + 2*x1**2 + 4*x1 + 20)']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [44.92899598198086, 23.675411015661506, 0.334946516675564]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [94.16582283646451, 23.248867658173417, tan(0.00980361737310886*x1 + 0.885341703891754)]\n [142.76936460103022, 22.73883853559589, tan(-0.0151103562395357*x0 + 0.0150624554288796*x1 + 0.894917964935303)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_log/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/mystery_world_log/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.79103598556975296 at j= 1\n Searching for best fit...\n 0.217586891281 -2.106687042311 P 1 0.0918 14.4093 47.6286 13119.4478\n 0.217586891281 -1.106687042311 P< 7 0.0918 17.2167 50.4359 13122.2551\n 0.217586891281 -22.105788957101 PE 28 0.0918 19.2167 52.4359 13124.2551\n 0.217586202654 -666.000000000000 PEE 307 0.0918 22.6714 55.8907 13127.7078\n 0.178846954675 0.993778048696 bPE/ 627 0.0801 23.4188 56.6380 13072.8757\n 0.133413489652 0.821093983021 aP+\\ 743 0.0582 23.2408 56.4601 12942.7460\n 0.116654482866 0.906451327277 PaE+\\ 7864 0.0527 26.4510 59.6703 12905.9825\n 0.115721047731 1.018462663416 aaE>-E 108170 0.0502 30.2213 63.4406 12889.2890\n 0.113909575874 1.060129533342 ba-PE/ 143487 0.0492 30.6062 63.8254 12881.9524\n 0.111739979651 1.010904013947 bPa>E*/ 555549 0.0494 32.5314 65.7507 12885.6249\n 0.093729847877 0.844693754109 aPbC++\\ 606992 0.0442 32.4056 65.6249 12840.5513\n 0.078456158278 0.823278466707 aPb<-+\\ 607343 0.0316 32.1498 65.3691 12702.7898\n 0.071014023736 -3.714515066017 baPE--R 613350 0.0198 32.0202 65.2395 12511.6459\n 0.068382044217 1.094046571958 ba-PP*/ 1974438 0.0180 33.6524 66.8717 12476.0071\n 0.065389130737 -3.818649451942 baPE>--R 7758117 0.0183 35.5621 68.7814 12483.6523\n 0.062699846050 0.910132331017 Pab<-E+\\ 8385856 0.0217 35.6138 68.8331 12554.0834\n 0.052467404683 -0.684337216913 Pba-P/+R 9064171 0.0143 35.4689 68.6882 12383.5948\n 0.048573567252 0.243427347099 aPb-+RLC 10506083 0.0143 35.5707 68.7900 12383.9057\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_log/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/mystery_world_log/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 1.2524925899830208 at j= 40336\n Searching for best fit...\n 0.335536856541 0.398680762669 P 1 0.1706 15.0342 48.2535 13372.1970\n 0.289092814204 0.203928340519 bP>+ 312 0.0984 23.1046 56.3239 13155.7055\n 0.245669190057 0.309196914642 PbS+ 361 0.1181 23.0803 56.2996 13230.6363\n 0.242481478981 0.049818866825 bPE+ 384 0.1185 23.1505 56.3698 13232.1668\n 0.234646452281 -0.054105842185 aPE- 545 0.1141 23.6083 56.8276 13216.9034\n 0.170454613758 0.552352640538 bP+R 780 0.0630 23.6644 56.8837 12975.3499\n 0.168859275868 0.105518917889 bPP*+ 3498 0.0715 25.8158 59.0351 13028.8693\n 0.168675615711 0.417489379699 b>>R> 22803 0.0675 28.5189 61.7382 13008.0043\n 0.145626551446 -0.049802544480 abPE+- 43679 0.0629 29.2446 62.4639 12980.3592\n 0.133682406480 0.331350632404 PbaP+/+ 520966 0.0466 32.6974 65.9166 12861.4097\n 0.118270512017 -0.105445720351 abPP*+- 522659 0.0594 32.5253 65.7446 12960.8674\n 0.109619921738 0.322514278700 Pba>E/+ 533683 0.0472 32.4458 65.6651 12866.7166\n 0.102785499845 0.622573736638 Pba-S+R 686476 0.0473 32.7162 65.9355 12867.6611\n 0.077667224547 -0.049786232827 aabPE+-+ 6739424 0.0239 35.6073 68.8266 12592.4311\n 0.047203616362 -0.083390144390 abPP>*+- 6756986 0.0143 34.8926 68.1119 12383.5805\n 0.047203616362 0.345368016677 Pba-P>/+ 8957899 0.0143 35.2994 68.5187 12383.9873\n 0.038834172105 1.305296706907 ba-ERRRS 32657769 0.0090 36.8840 70.1033 12194.7985\n 0.036685885343 0.423747127893 baPP<*--R 90939312 0.0084 38.2794 71.4987 12168.3589\n 0.036097833755 0.511961427534 baPP*--R< 93499479 0.0085 38.2962 71.5154 12174.9086\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545943297282, '1/(666.000000000000+(sin(pi))**(-1))']\n [2.0, 27.15815915791503, '3.14159265358979']\n [32.85758306839902, 25.12250102174253, 'log(-x0*x1 - 4*x0 + 2*x1**2 + 4*x1 + 20)']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [44.92899598198086, 23.675411015661506, 0.334946516675564]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [94.16582283646451, 23.248867658173417, tan(0.00980361737310886*x1 + 0.885341703891754)]\n [142.76936460103022, 22.73883853559589, tan(-0.0151103562395357*x0 + 0.0150624554288796*x1 + 0.894917964935303)]\n [143.9438349226117, 22.485357786845263, 1.09423654316956*exp(-0.079430100627574*x0 + 0.0789706392936595*x1)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_sin/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/mystery_world_sin/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 3.8595698840027815E-006 at j= 74523\n Searching for best fit...\n 0.597142829315 -3.141596600582 P 1 0.2424 15.8658 49.0851 13515.4590\n 0.597142829315 -0.318313736896 P\\ 13 0.2424 19.5662 52.7855 13519.1595\n 0.597142829315 -1.772457735137 PR 16 0.2424 19.8658 53.0851 13519.4590\n 0.597142829315 -23.140698515373 PE 28 0.2424 20.6731 53.8924 13520.2664\n 0.379816575635 -0.167182064271 aP/ 59 0.1490 21.0957 54.3149 13322.8776\n 0.331613138224 -0.126816384892 aP>/ 554 0.1435 24.1310 57.3502 13310.5589\n 0.291910800019 -0.059663700476 ab>E/ 6743 0.1034 27.5524 60.7717 13180.4701\n 0.288658905490 -0.082322454215 aPbE+/ 44663 0.1216 30.2639 63.4832 13249.3292\n 0.288284275374 -0.076862767854 aPb>*/ 44690 0.1113 30.2629 63.4821 13213.3139\n 0.282384702161 -0.263750461331 bPa-+\\ 50649 0.1218 30.4136 63.6329 13250.1959\n 0.257998088730 -1.578505502454 aPb-+R 51089 0.0640 30.2958 63.5151 12987.8805\n 0.226543459994 -0.289865716057 abC+P/ 112529 0.0666 31.2474 64.4667 13004.8890\n 0.119537842999 0.156922556200 ab-P>/ 142625 0.0309 30.6670 63.8863 12692.2734\n 0.119537842999 0.398375558108 ab>-P>/ 1655444 0.0309 34.2040 67.4232 12695.8103\n 0.116346400711 -2.039747050958 abPER--R 7759091 0.0274 36.3936 69.6129 12649.0431\n 0.115160917037 -2.013822285711 abPE<~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.79103598556975296 at j= 1\n Searching for best fit...\n 0.670865457737 0.102167430914 P 1 0.2699 16.0337 49.2530 13559.3899\n 0.378990653788 0.209042851535 a 2 0.1590 16.2099 49.4292 13344.4906\n 0.378268278287 -0.332737208672 aC< 113 0.1506 22.0273 55.2466 13327.9750\n 0.374283054766 0.154994413384 aa<+ 323 0.1480 23.5272 56.7465 13322.5143\n 0.314047501545 0.151707879228 abC+ 371 0.1239 23.4740 56.6933 13250.1887\n 0.204119174132 -0.202670364400 ba>- 477 0.0754 23.2150 56.4343 13047.9623\n 0.186637423944 0.275831613249 ab>C+ 4394 0.0558 26.2893 59.5086 12928.3889\n 0.173844113724 0.294336039953 bCaCE\\- 1029555 0.0393 33.6980 66.9173 12792.9458\n 0.135350918173 -0.242616437005 baP>+CE- 8190927 0.0393 36.6900 69.9093 12795.9377\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545943297282, '1/(666.000000000000+(sin(pi))**(-1))']\n [2.0, 27.15815915791503, '3.14159265358979']\n [32.85758306839902, 25.12250102174253, 'log(-x0*x1 - 4*x0 + 2*x1**2 + 4*x1 + 20)']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [44.92899598198086, 23.675411015661506, 0.334946516675564]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [94.16582283646451, 23.248867658173417, tan(0.00980361737310886*x1 + 0.885341703891754)]\n [142.76936460103022, 22.73883853559589, tan(-0.0151103562395357*x0 + 0.0150624554288796*x1 + 0.894917964935303)]\n [143.9438349226117, 22.485357786845263, 1.09423654316956*exp(-0.079430100627574*x0 + 0.0789706392936595*x1)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_sqrt/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/mystery_world_sqrt/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.79103598556975296 at j= 1\n Searching for best fit...\n 0.192827070815 -1.463844017285 P 1 0.0795 14.2350 47.4543 13060.5692\n 0.192827070815 -2.463844017285 P> 4 0.0795 16.2350 49.4543 13062.5692\n 0.192827070815 -21.462945932076 PE 28 0.0795 19.0424 52.2617 13065.3766\n 0.192826160753 -666.000000000000 PEE 307 0.0795 22.4971 55.7164 13068.8282\n 0.154149598595 1.636621073721 bPE/ 627 0.0683 23.2044 56.4236 13007.6897\n 0.111923261482 1.463937008046 aP+\\ 743 0.0475 22.9874 56.2067 12859.6447\n 0.101594572227 1.549294352302 PaE+\\ 7864 0.0437 26.2516 59.4709 12829.2315\n 0.099611033196 1.661305688442 aaE>-E 108170 0.0430 30.0050 63.2243 12826.3899\n 0.089408037560 1.702972558368 ba-PE/ 143487 0.0370 30.2567 63.4760 12765.7391\n 0.072292299258 1.487536779134 aPbC++\\ 606992 0.0322 32.0309 65.2502 12711.4764\n 0.060991585408 2.063777551752 ab-SP-\\ 2134352 0.0193 33.5998 66.8190 12504.5896\n 0.052089357789 -6.216504421779 baP>E--R 7759845 0.0193 35.2344 68.4537 12505.7576\n 0.041492747159 1.552975356042 Pab<-E+\\ 8385856 0.0120 35.0182 68.2375 12313.0722\n 0.040755364973 1.563767576209 Pab-E>*\\ 9565297 0.0115 35.1822 68.4014 12294.1998\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_sqrt/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/mystery_world_sqrt/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 1.8705757945424719 at j= 40336\n Searching for best fit...\n 0.288911147145 0.595422751690 P 1 0.1516 14.8183 48.0376 13323.9516\n 0.288911147145 -666.000000000000 PS 19 0.1516 19.0663 52.2855 13328.1995\n 0.154679164249 0.074403606967 bPE+ 384 0.0775 22.5020 55.7212 13058.5743\n 0.153952069447 0.077485659327 bPE<+ 4056 0.0750 25.8960 59.1153 13048.5325\n 0.151587980927 -39.544833484320 bPE-\\ 8025 0.0704 26.8582 60.0774 13023.9225\n 0.151381309157 1.242212642428 bP+RR 11607 0.0603 27.3886 60.6079 12961.4293\n 0.041633940250 -0.074379229830 abPE+- 43679 0.0125 27.4382 60.6575 12322.6095\n 0.034188665785 -0.071534809388 abPE>+- 542882 0.0084 30.7896 64.0089 12161.5035\n 0.034049094366 -0.068899929096 abPE>>+- 7123889 0.0076 34.4976 67.7169 12124.5979\n 0.032136997112 -0.069482282995 abPEPR++- 94061756 0.0074 38.1371 71.3564 12118.0001\n 0.032136997112 -0.069482282995 abPR+-PE- 124702004 0.0074 38.5439 71.7632 12118.4069\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545943297282, '1/(666.000000000000+(sin(pi))**(-1))']\n [2.0, 27.15815915791503, '3.14159265358979']\n [32.85758306839902, 25.12250102174253, 'log(-x0*x1 - 4*x0 + 2*x1**2 + 4*x1 + 20)']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [44.92899598198086, 23.675411015661506, 0.334946516675564]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [94.16582283646451, 23.248867658173417, tan(0.00980361737310886*x1 + 0.885341703891754)]\n [142.76936460103022, 22.73883853559589, tan(-0.0151103562395357*x0 + 0.0150624554288796*x1 + 0.894917964935303)]\n [143.9438349226117, 22.485357786845263, 1.09423654316956*exp(-0.079430100627574*x0 + 0.0789706392936595*x1)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_squared/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/mystery_world_squared/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.79103598556975296 at j= 1\n Searching for best fit...\n 4.320048900614 4.781735875560 P 1 1.6589 18.7207 51.9400 14300.2823\n 3.422307693052 6.971606225870 b 3 1.4101 19.9696 53.1889 14235.5674\n 2.863300410728 9.458748062736 a~ 11 1.0850 21.5868 54.8060 14130.4922\n 2.838249775550 7.017553107608 bb* 48 1.1890 23.6996 56.9189 14169.9559\n 1.916315998255 8.507025672033 ba- 54 0.6817 23.3029 56.5221 13943.1778\n 1.916315998255 9.507025672033 ba>- 477 0.6817 26.4458 59.6651 13946.3208\n 1.906325869413 7.555303281331 bba-+ 3540 0.6579 29.3300 62.5492 13934.6755\n 1.496650241168 10.042445118197 baa+- 3699 0.4397 29.0443 62.2636 13770.2862\n 1.270194378790 9.506399975452 baaS+- 43605 0.4112 32.3669 65.5862 13746.4914\n 1.261426321222 9.223977787512 baP\\L+- 544374 0.4071 35.9414 69.1607 13746.1260\n 0.932819170798 2.973552896176 bPa-+RE 743388 0.2487 36.0131 69.2323 13545.4671\n 0.846685961919 3.190863723567 ba<<<-RE 20446236 0.2435 40.6549 73.8742 13541.6220\n 0.846685961919 3.190863723567 b>>a->RE 61537929 0.2435 42.2445 75.4638 13543.2117\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_squared/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/mystery_world_squared/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 12.243377517185923 at j= 40336\n Searching for best fit...\n 5.985059467298 3.897187995551 P 1 3.4303 19.1910 52.4103 14596.7372\n 5.602542206518 4.080807169668 b> 6 1.9593 21.6807 54.9000 14370.7773\n 3.561698645492 2.381133688946 bP+ 33 1.3057 23.4866 56.7059 14207.6244\n 3.525298217646 7.068441324160 b>R 177 1.3683 25.8950 59.1143 14229.1654\n 3.522250899097 1.993442260171 bP>+ 312 1.5182 26.7115 59.9308 14272.4118\n 3.519346327415 1.714320089154 bP>>+ 3903 1.7283 30.3553 63.5746 14328.9487\n 3.502330478155 1.797664879640 bPER+ 4299 1.6618 30.4877 63.7070 14313.0660\n 3.477719880032 3.073761648215 PbSSC 36303 1.1852 33.5363 66.7555 14178.2327\n 1.319370711340 1.993453278120 bPaC++ 41685 0.3987 32.3567 65.5760 13733.8782\n 1.149802658285 5.395041786998 bPa-+R 51081 0.4895 32.4515 65.6708 13817.8959\n 0.865839239919 1.476589524974 bPPa-++ 520170 0.2830 35.3904 68.6097 13597.6446\n 0.767634774869 -1.551021424903 abPRE+- 544718 0.1890 35.2833 68.5026 13432.9980\n 0.767634774869 1.551021424903 baPRE-- 549090 0.1890 35.2948 68.5141 13433.0095\n 0.748057967741 -1.565859232414 abPER>+- 7124861 0.1820 38.9553 72.1746 13421.3282\n 0.733901667329 9.085333418461 ba-P/SER 29234313 0.1998 40.9645 74.1837 13461.4639\n 0.660466729603 0.390052583934 baPP*--RE 93826071 0.1573 42.4947 75.7140 13365.5715\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545943297282, '1/(666.000000000000+(sin(pi))**(-1))']\n [2.0, 27.15815915791503, '3.14159265358979']\n [17.93156856932417, 25.29239447970469, '(-x0 + x1 + 9)**0.5']\n [32.85758306839902, 25.12250102174253, 'log(-x0*x1 - 4*x0 + 2*x1**2 + 4*x1 + 20)']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [44.92899598198086, 23.675411015661506, 0.334946516675564]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [94.16582283646451, 23.248867658173417, tan(0.00980361737310886*x1 + 0.885341703891754)]\n [142.76936460103022, 22.73883853559589, tan(-0.0151103562395357*x0 + 0.0150624554288796*x1 + 0.894917964935303)]\n [143.9438349226117, 22.485357786845263, 1.09423654316956*exp(-0.079430100627574*x0 + 0.0789706392936595*x1)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_tan/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/mystery_world_tan/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 3.8595698840315281E-006 at j= 74523\n Searching for best fit...\n 0.744438256573 -3.141588881443 P 1 0.2631 16.1839 49.4031 13548.8588\n 0.744438256573 -0.318306017756 P\\ 13 0.2631 19.8843 53.1036 13552.5592\n 0.744438256573 -23.140690796233 PE 28 0.2631 20.9912 54.2105 13553.6661\n 0.450596789880 -0.374051989054 bP/ 60 0.1832 21.3664 54.5857 13407.0098\n 0.427319513892 -0.926635978353 bCC 249 0.1661 23.3430 56.5623 13369.2379\n 0.382217519681 0.167182064271 aP~/ 572 0.1576 24.3820 57.6013 13348.8424\n 0.363430527803 -0.912214736694 a>LC 2930 0.1556 26.6661 59.8854 13345.9613\n 0.234779987229 -0.206873784353 ba-P/ 10080 0.0799 27.8182 61.0375 13075.9959\n 0.234779987229 -0.525183661679 ba<-P/ 112644 0.0799 31.3004 64.5197 13079.4781\n 0.156758592208 -0.156922556200 ba-P>/ 142623 0.0330 31.0581 64.2774 12718.7610\n 0.129945020038 -2.188949782113 bPa<-+R 611223 0.0325 32.8870 66.1062 12715.1523\n 0.128795925802 -2.156364686024 ba<<<<-R 17067321 0.0333 37.6775 70.8968 12729.6838\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_tan/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus...\n /bin/cp -p results/mystery_world_tan/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.79103598556975296 at j= 1\n Searching for best fit...\n 0.712406740999 -0.107875115821 P 1 0.2878 16.1204 49.3397 13585.4607\n 0.398634925232 -0.220721237865 a 2 0.1713 16.2828 49.5021 13374.9134\n 0.398634910238 -0.220721250432 aPS+ 356 0.1713 23.7585 56.9778 13382.3891\n 0.333695403673 -0.160183190437 abC+ 371 0.1342 23.5615 56.7808 13282.6981\n 0.208511812316 0.213992745412 ba>- 477 0.0804 23.2457 56.4650 13074.2900\n 0.201948317241 -0.291241220025 ab>C+ 4394 0.0556 26.4030 59.6223 12927.0933\n 0.201779717780 -0.226412246311 abERC+ 61841 0.0793 30.2168 63.4361 13075.4787\n 0.200384266780 -0.310779414887 bCaCE\\- 1029555 0.0392 33.7564 66.9756 12791.5063\n 0.140937528547 0.256170445001 baP>+CE- 8190927 0.0392 36.7484 69.9676 12794.4982\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545943297282, '1/(666.000000000000+(sin(pi))**(-1))']\n [2.0, 27.15815915791503, '3.14159265358979']\n [17.93156856932417, 25.29239447970469, '(-x0 + x1 + 9)**0.5']\n [32.85758306839902, 25.12250102174253, 'log(-x0*x1 - 4*x0 + 2*x1**2 + 4*x1 + 20)']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [44.92899598198086, 23.675411015661506, 0.334946516675564]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [94.16582283646451, 23.248867658173417, tan(0.00980361737310886*x1 + 0.885341703891754)]\n [142.76936460103022, 22.73883853559589, tan(-0.0151103562395357*x0 + 0.0150624554288796*x1 + 0.894917964935303)]\n [143.9438349226117, 22.485357786845263, 1.09423654316956*exp(-0.079430100627574*x0 + 0.0789706392936595*x1)]\n Checking for symmetry \n duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus\n Found pretrained NN \n \n tensor(0.0094, device='cuda:0', grad_fn=)\n tensor(0.0097, device='cuda:0', grad_fn=)\n tensor(0.0119, device='cuda:0', grad_fn=)\n NN loss after training: tensor(0.0081, device='cuda:0', grad_fn=) \n \n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/translated_data_minus/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus...\n /bin/cp -p results/translated_data_minus/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.11647158377131471 at j= 1\n Searching for best fit...\n 0.684213023160 -0.326751961044 P 1 0.2755 16.0622 49.2814 13567.7330\n 0.684213023160 5.956433520981 P~ 7 0.2755 18.8695 52.0888 13570.5403\n 0.684213022557 666.000000000000 PS\\ 101 0.2755 22.7204 55.9397 13574.3912\n 0.620190118895 2.143241306888 aCC 158 0.2929 23.2242 56.4435 13599.9939\n 0.224385382710 3.000637318088 aP~/ 316 0.0644 22.7575 55.9768 12982.6875\n 0.180015240181 2.456758434314 aE>\\ 1134 0.0549 24.2830 57.5023 12919.4493\n 0.149882159900 2.955776186214 aP~S 73996 0.0408 29.9552 63.1745 12804.9099\n 0.139866477944 2.564651292297 aP+LCS 74908 0.0414 29.9646 63.1839 12810.9232\n 0.139354862950 2.424345803218 aaP+/EC 332448 0.0398 32.1092 65.3285 12796.3267\n 0.139188345989 2.475935278295 aPL*E>\\ 768862 0.0379 33.3171 66.5364 12778.1096\n 0.138768939649 2.425070994713 aP/>L>C 972890 0.0392 33.6523 66.8716 12791.3947\n 0.138458601955 2.439778272123 aEPL\\+\\ 1189542 0.0359 33.9391 67.1584 12756.1363\n 0.137049530340 2.444915997102 aEPLS+\\ 1190190 0.0397 33.9252 67.1445 12797.7483\n 0.136225271578 0.694861892982 PaE>\\E*R 6670577 0.0359 36.4031 69.6224 12758.7417\n 0.135832538022 0.878102366531 aE>LRSCE 39202284 0.0363 38.9540 72.1733 12765.8739\n 0.135236907914 2.986332336882 aPaSCR~// 41562966 0.0338 39.0320 72.2513 12736.6143\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/translated_data_minus/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus...\n /bin/cp -p results/translated_data_minus/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 3.4990538031282004 at j= 40336\n Searching for best fit...\n 0.997390746302 1.113783386831 P 1 0.5395 16.6059 49.8252 13841.9546\n 0.997390746302 0.844857044605 P> 3 0.5395 18.1908 51.4101 13843.5396\n 0.997390746302 -17.469626636017 PRC 153 0.5395 23.8633 57.0825 13849.2120\n 0.886247143811 0.864533610536 PaS- 293 0.4498 24.6302 57.8495 13775.9435\n 0.468296488964 -0.139131986937 aPE- 304 0.2380 23.7631 56.9823 13516.2896\n 0.265348453227 -0.294580783766 aPP*- 2162 0.1040 25.7738 58.9930 13181.0923\n 0.240127040068 2.322722925146 Pa-RR 5587 0.0805 26.9994 60.2186 13077.9130\n 0.126012279664 -0.128842324451 aaPE-+ 23454 0.0324 28.1388 61.3581 12708.9394\n 0.126012279664 -0.128842324451 aa+PE- 64830 0.0324 29.6057 62.8249 12710.4062\n 0.101089562652 -0.124266574764 aaPE>-+ 271888 0.0249 31.3560 64.5753 12604.4992\n 0.101089562652 -0.124266574764 aa<+PE- 709044 0.0249 32.7389 65.9581 12605.8820\n 0.100978250546 -0.248108899743 aP- 5742136 0.0249 35.7506 68.9698 12608.7065\n 0.100670750089 -0.248342428060 aP>ES\\<<- 78701090 0.0249 39.5272 72.7465 12612.4821\n Checking polyfit \n \n Complexity RMSE Expression\n [9.754887502163468, 23.75790801204378, '3.0 - 0.25*x0']\n [71.99696026886835, 23.745048456789334, 3.00008309382983 - 0.246254071593285*x0]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_atan/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus...\n /bin/cp -p results/mystery_world_atan/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.11647158377131471 at j= 1\n Searching for best fit...\n 0.062981887482 -1.912149389642 P 1 0.0276 12.6207 45.8400 12629.0104\n 0.062981887482 -21.911251304432 PE 19 0.0276 16.8686 50.0879 12633.2584\n 0.062981660867 1.229443402399 aPS* 256 0.0276 20.6207 53.8400 12637.0091\n 0.062978827720 -666.000000000000 PPEE* 2861 0.0276 24.1030 57.3222 12640.4628\n 0.059480794644 -666.000000000000 PPC>/ 3213 0.0208 24.1879 57.4072 12524.8439\n 0.057113863738 1.254667186011 aPE~/ 3294 0.0169 24.1653 57.3845 12439.5626\n 0.053466351693 -1.886601981311 PEa-L 8643 0.0167 25.4617 58.6810 12437.6134\n 0.047701246750 1.097022980840 PaPE+/ 24173 0.0217 26.7809 60.0002 12545.8152\n 0.034908648735 1.083816861542 aPP++\\ 25932 0.0061 26.4318 59.6511 12026.1605\n 0.033735482625 1.404897694841 aPP+-\\ 25996 0.0061 26.3861 59.6053 12025.6376\n 0.018936735698 1.241271430799 aPRS<* 31288 0.0081 25.8203 59.0396 12142.0079\n 0.018606843212 1.241392917339 aPRSL* 33232 0.0079 25.8819 59.1012 12133.0421\n 0.015445088119 -0.949873230661 PEa-RR 110613 0.0055 27.3481 60.5674 11986.4741\n 0.014908249140 1.244293046309 aP\\CR<* 406938 0.0046 29.1763 62.3956 11914.7399\n 0.014907608979 1.244291444494 aP\\CR>E<\\ 33329856 0.0041 35.5180 68.7373 11879.4962\n 0.014690900253 0.316822191281 aER\\>RRS 35757912 0.0044 35.6125 68.8317 11902.2317\n 0.014650209891 0.188927505986 PPa>E+*\\E 45906613 0.0046 35.9689 69.1882 11927.5719\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_atan/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus...\n /bin/cp -p results/mystery_world_atan/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 1.2924252388523800 at j= 40336\n Searching for best fit...\n 0.101906033496 0.411391719232 P 1 0.0492 13.3150 46.5342 12864.9112\n 0.101906033496 0.602605125527 PL> 51 0.0492 18.9874 52.2067 12870.5837\n 0.101905587202 -0.000000112987 aPS\\+ 2346 0.0492 24.5109 57.7302 12876.1052\n 0.026265269335 -0.019910603743 aP>E- 3154 0.0068 22.9819 56.2012 12070.4636\n 0.020754429197 0.257717324398 PEa-R 8211 0.0089 24.0225 57.2418 12178.2040\n 0.017811129077 0.252741458572 PEa<-R 97833 0.0068 27.3766 60.5959 12075.2977\n 0.017479896630 0.214870801201 PEa-R> 105429 0.0035 27.4574 60.6767 11806.9271\n 0.015221698443 0.580911007276 PEa-L< 107157 0.0049 27.2813 60.5006 11941.4866\n 0.015163345765 0.580936604026 PaPE/E- 288663 0.0049 28.7054 61.9247 11939.0610\n 0.013143478186 0.242987041117 PaPE--R 296655 0.0037 28.5386 61.7578 11829.9378\n 0.011722867730 1.372076557796 aPE/>>S 791114 0.0028 29.7886 63.0079 11707.5199\n 0.011670719773 1.372129844058 aPE/S>>S 10511740 0.0027 33.5142 66.7335 11708.8709\n 0.011569482283 -0.869899785160 PaP/-R\\<< 55155243 0.0025 35.8931 69.1124 11681.8957\n 0.011381028038 1.465464252810 PaP/-R>LS 55587243 0.0026 35.8807 69.0999 11685.5129\n Checking polyfit \n \n Complexity RMSE Expression\n [9.754887502163468, 23.75790801204378, '3.0 - 0.25*x0']\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [98.77507424296371, 20.699967368470546, -tan(0.00969398001048327*x0 - 0.895113348960876)]\n [98.77508016991868, 20.699948062984262, -tan(0.00969401983587903*x0 - 0.895113348960876)]\n [102.67524654499746, 20.69988101883838, -tan(sin(0.00969423381824464*x0) - 0.895113348960876)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_cos/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus...\n /bin/cp -p results/mystery_world_cos/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.11647158377131471 at j= 1\n Searching for best fit...\n 0.144952083475 -4.088682624876 P 1 0.0435 13.8233 47.0426 12814.2703\n 0.144952083475 -3.088682624876 P< 5 0.0435 16.1452 49.3645 12816.5922\n 0.144952083475 -10.816694834246 PP* 25 0.0435 18.4671 51.6864 12818.9142\n 0.144952083475 -63.850019664652 P>E 181 0.0435 21.3231 54.5424 12821.7702\n 0.144952080317 666.000000000000 PPS/ 327 0.0435 22.1764 55.3957 12822.6235\n 0.128218880330 -0.972313718504 aPE/ 340 0.0257 22.0557 55.2750 12607.7208\n 0.123292099796 -0.971268850698 aPE>/ 3222 0.0252 25.2435 58.4628 12603.2544\n 0.078644790583 -0.975588615817 aP>-E 4738 0.0259 25.1512 58.3705 12614.5409\n 0.072431560263 -1.266712231375 Pa>C+\\ 43621 0.0230 28.2351 61.4544 12569.7069\n 0.059469921909 -1.229099794186 PaE-E 582978 0.0191 31.5110 64.7303 12497.0607\n 0.046294276905 -1.478135936784 Pa<~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.99999999999255185 at j= 74523\n Searching for best fit...\n 0.197862199604 -0.318309877324 P 1 0.0482 14.2722 47.4915 12856.2782\n 0.197862199604 -0.241453001907 P> 3 0.0482 15.8572 49.0765 12857.8632\n 0.197862199604 -0.758546998086 P\\> 43 0.0482 19.6985 52.9178 12861.7045\n 0.126026531032 0.042033372887 aPE- 304 0.0255 21.8694 55.0886 12604.0307\n 0.123451628137 0.040337837521 aPE>- 2898 0.0253 25.0925 58.3118 12604.7849\n 0.118390963124 -1.012440568592 aP>/C 4486 0.0369 25.6625 58.8818 12759.2164\n 0.103967028603 -1.000029686675 aP+RS 5722 0.0202 25.8261 59.0454 12513.0861\n 0.096097358763 -0.347482093067 PaP-\\+ 24509 0.0158 27.8113 61.0306 12415.2723\n 0.094386509198 -0.450025055215 PaC>+R 44725 0.0368 28.6532 61.8724 12762.1495\n 0.089045436739 -0.785576940712 PaE+\\> 52797 0.0191 28.8085 62.0278 12495.2770\n 0.084577560863 -0.718765460836 PaS-RR 58185 0.0346 28.8744 62.0937 12736.3679\n 0.083253348697 0.505704684300 aP-E<< 69028 0.0192 29.0982 62.3175 12496.7381\n 0.080121102460 -1.022561326715 aP-E>\\ 71476 0.0163 29.0931 62.3124 12429.1878\n 0.060338728115 -1.188417758764 aP+RSS 74716 0.0070 28.7480 61.9673 12085.4789\n 0.059887212488 -0.501555360683 a>P/C> 105720 0.0161 29.2379 62.4572 12424.9609\n 0.050995226465 -0.289108702208 Pa 602591 0.0085 31.3702 64.5895 12166.9012\n 0.042685130776 0.500260110563 a

> 8472817 0.0067 35.0643 68.2836 12071.7894\n Checking polyfit \n \n Complexity RMSE Expression\n [2.0, 27.15815915791503, '3.14159265358979']\n [9.754887502163468, 23.75790801204378, '3.0 - 0.25*x0']\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [98.77507424296371, 20.699967368470546, -tan(0.00969398001048327*x0 - 0.895113348960876)]\n [98.77508016991868, 20.699948062984262, -tan(0.00969401983587903*x0 - 0.895113348960876)]\n [102.67524654499746, 20.69988101883838, -tan(sin(0.00969423381824464*x0) - 0.895113348960876)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_exp/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus...\n /bin/cp -p results/mystery_world_exp/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.11647158377131471 at j= 1\n Searching for best fit...\n 16.393614936972 13.548925362924 P 1 5.6739 20.6447 53.8640 14802.0853\n 13.825141622754 17.274215159398 a~ 8 4.6852 23.3989 56.6181 14726.9481\n 13.319426825379 16.349815851383 aa* 28 5.4412 25.1525 58.3717 14789.8024\n 10.900344365142 16.132685890989 aE\\ 108 4.3115 26.8108 60.0301 14696.7849\n 8.703377892974 18.524256536323 aP~* 244 2.6190 27.6620 60.8812 14494.5509\n 6.137924513274 15.174172936153 a>-* 274416 0.7789 36.2731 69.4924 14009.8324\n 3.026182046892 15.209355653679 aP>-a<* 704646 0.7789 37.6337 70.8530 14011.1929\n 2.949131206214 21.397526500260 Paa/>E 8086925 0.7768 41.1090 74.3282 14013.6092\n 2.898545273273 18.930873445243 aEaPL/SS-E 47945003 0.7415 43.5640 76.7833 13997.2215\n 2.678904359643 -8.401767242348 Pa

>/-E 58792303 0.7951 43.8404 77.0597 14026.0108\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_exp/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus...\n /bin/cp -p results/mystery_world_exp/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 33.084133040908405 at j= 40336\n Searching for best fit...\n 20.881362044207 10.531006329689 P 1 13.2654 20.9938 54.2131 15148.6354\n 6.454271337654 -6.424020267247 aP- 30 1.3948 24.2068 57.4261 14234.4454\n 4.429020648426 -5.379475459695 aP>- 272 1.9306 26.8441 60.0634 14370.2665\n 4.356972541477 35.217414772773 aP+R\\ 5434 1.3603 31.1408 64.3600 14231.7047\n 4.356972541477 35.217414772773 aP+\\R 5562 1.3603 31.1744 64.3936 14231.7383\n 3.464452356811 3.420127932611 Pa-RE 6163 0.8087 30.9917 64.2110 14019.7027\n 2.709666510295 20.024075133238 aERR\\ 15232 0.5245 31.9426 65.1619 13844.2868\n 2.709666510295 20.024075133238 aER\\R 16528 0.5245 32.0604 65.2797 13844.4047\n 2.689449366729 9.965785246892 aaE 40558 0.5409 33.3447 66.5639 13858.3027\n 2.679093106489 20.370784247467 aP~RR 953863 0.5566 37.8766 71.0959 13874.5232\n 2.641062302751 9.088624472137 PPa-E+RR 4126627 0.8817 39.9873 73.2066 14064.3619\n 2.568290024218 -52.802423485742 aPP++LL< 4220560 0.8031 39.9795 73.1987 14026.2892\n 2.540204519218 20.156259058479 aPaCC+~/E 45025238 0.5385 43.3788 76.5981 13866.5971\n Checking polyfit \n \n Complexity RMSE Expression\n [2.0, 27.15815915791503, '3.14159265358979']\n [9.754887502163468, 23.75790801204378, '3.0 - 0.25*x0']\n [43.25789208357, 23.747101876217187, 'log(0.61*x0**2 - 5*x0 + 20.1)']\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [98.77507424296371, 20.699967368470546, -tan(0.00969398001048327*x0 - 0.895113348960876)]\n [98.77508016991868, 20.699948062984262, -tan(0.00969401983587903*x0 - 0.895113348960876)]\n [102.67524654499746, 20.69988101883838, -tan(sin(0.00969423381824464*x0) - 0.895113348960876)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_inverse/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus...\n /bin/cp -p results/mystery_world_inverse/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.11647158377131471 at j= 1\n Searching for best fit...\n 0.069468333651 -2.786332860199 P 1 0.0307 12.7621 45.9814 12672.5981\n 0.069468333651 0.036950003487 P\\ 9 0.0307 15.9321 49.1513 12675.7680\n 0.069468333651 -9.514345069570 PP* 25 0.0307 17.4060 50.6253 12677.2419\n 0.069468333261 666.000000000000 PS\\ 101 0.0307 19.4204 52.6396 12679.2563\n 0.069467394488 -666.000000000000 PEE 197 0.0307 20.3842 53.6035 12680.2119\n 0.050882449139 0.330036046173 aPE/ 340 0.0141 20.7223 53.9416 12362.8637\n 0.046035467019 0.331080913978 aPE>/ 3222 0.0124 23.8223 57.0416 12314.2336\n 0.031744961300 0.345980550622 aP>E/ 3478 0.0151 23.3964 56.6156 12394.3995\n 0.028663648449 0.179805537343 PPa-+\\ 25951 0.0078 26.1485 59.3678 12125.6030\n 0.022115143343 -0.657273493768 aPE/>R 57656 0.0098 26.9260 60.1453 12221.3532\n 0.018827447292 -1.851722019611 aPE+RR 58124 0.0080 26.7055 59.9247 12139.6598\n 0.017254618754 -0.610180724887 aE>>\\C 234526 0.0050 28.5922 61.8114 11950.3369\n 0.016994228409 0.339251587495 aPPL*E/ 291102 0.0055 28.8820 62.1013 11989.1848\n 0.016855856325 -0.589436461975 aER>RRS 2809200 0.0057 32.1408 65.3600 12005.3492\n 0.016667731443 -1.433799540857 PaPP*/+R 3255137 0.0051 32.3371 65.5564 11957.8908\n 0.016615603599 0.511209264553 aPE+\\ 44485325 0.0049 36.0990 69.3183 11947.0679\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_inverse/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus...\n /bin/cp -p results/mystery_world_inverse/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.39973408779867031 at j= 33304\n Searching for best fit...\n 0.113942540636 0.127239308450 P 1 0.0688 13.4760 46.6953 13001.4238\n 0.113942540636 0.077745186741 P>> 37 0.0688 18.6855 51.9048 13006.6332\n 0.095212642000 0.098126181763 PaS+ 221 0.0479 21.0048 54.2241 12861.3508\n 0.054490636705 0.015936981486 aPE+ 232 0.0354 20.2698 53.4891 12738.2392\n 0.026990119997 0.033843988777 aPP*+ 2098 0.0062 22.4330 55.6523 12027.2625\n 0.021281116839 0.014792011449 aaPE++ 23310 0.0083 25.5640 58.7833 12152.4015\n 0.019461451339 0.031202214116 aPP*>+ 24404 0.0062 25.5012 58.7205 12031.5519\n 0.016387667148 0.079998178571 PaP/E+ 24677 0.0035 25.2693 58.4886 11806.1012\n 0.014683568441 0.236566440971 aP/E>R 72776 0.0035 26.6712 59.8904 11802.5369\n 0.014663243132 -7.697976272387 aaPE-+\\ 294928 0.0031 28.6880 61.9073 11757.4693\n 0.014519397878 0.245126403509 aP/ 909546 0.0032 30.2986 63.5178 11765.9952\n 0.014435936578 0.241290322474 aP/S 912490 0.0038 30.2949 63.5142 11837.4117\n 0.014435936520 0.241290329289 PaP/-S 55132331 0.0038 36.2118 69.4311 11843.3285\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545942911475, 8.91505505199440e-11]\n [2.0, 27.15815915791503, '3.14159265358979']\n [9.754887502163468, 23.75790801204378, '3.0 - 0.25*x0']\n [43.25789208357, 23.747101876217187, 'log(0.61*x0**2 - 5*x0 + 20.1)']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [44.928974716409904, 23.675383356059676, 0.334941579543170]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [98.77507424296371, 20.699967368470546, -tan(0.00969398001048327*x0 - 0.895113348960876)]\n [98.77508016991868, 20.699948062984262, -tan(0.00969401983587903*x0 - 0.895113348960876)]\n [102.67524654499746, 20.69988101883838, -tan(sin(0.00969423381824464*x0) - 0.895113348960876)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_log/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus...\n /bin/cp -p results/mystery_world_log/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.11647158377131471 at j= 1\n Searching for best fit...\n 0.217586891281 -2.106687042311 P 1 0.0918 14.4093 47.6286 13119.4478\n 0.217586891281 -1.106687042311 P< 5 0.0918 16.7312 49.9505 13121.7697\n 0.217586891281 -22.105788957101 PE 19 0.0918 18.6572 51.8765 13123.6957\n 0.217586202654 -666.000000000000 PEE 197 0.0918 22.0314 55.2506 13127.0678\n 0.113909575874 1.060129533342 aPE~/ 3294 0.0492 25.1612 58.3805 12876.5074\n 0.078456158278 0.823278466707 aP>+\\ 3946 0.0316 24.8838 58.1031 12695.5238\n 0.071014023736 -3.714515066017 PEa-R 8211 0.0198 25.7972 59.0165 12505.4229\n 0.068382044217 1.094046571958 aPP~*/ 24196 0.0180 27.3019 60.5212 12469.6566\n 0.052467404683 -0.684337216913 PaP/-R 26149 0.0143 27.0317 60.2510 12375.1576\n 0.049622393992 1.087089084752 aPLS<* 31300 0.0140 27.2107 60.4299 12365.7958\n 0.048573567252 0.243427347099 aP+RLC 76300 0.0143 28.4654 61.6846 12376.8003\n 0.048309711892 1.285522133771 PaP/+RC 331863 0.0141 30.5783 63.7976 12373.0729\n 0.048071064345 -0.097985975406 Pa>E+\\E 694063 0.0127 31.6357 64.8549 12331.3201\n 0.047193748072 -0.085825297422 PaE>*\\E 694131 0.0130 31.6092 64.8285 12339.9271\n 0.047161954860 2.058618561972 PaE>\\-L~ 7536629 0.0127 35.0489 68.2682 12334.5288\n 0.046929175979 0.409948393974 a>E>\\<~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 1.2524925899830208 at j= 40336\n Searching for best fit...\n 0.335536856541 0.398680762669 P 1 0.1706 15.0342 48.2535 13372.1970\n 0.335536856541 0.584841630249 PPC+ 223 0.1706 22.8351 56.0544 13379.9979\n 0.145626551446 -0.049802544480 aPE- 304 0.0629 22.0779 55.2972 12973.1925\n 0.118270512017 -0.105445720351 aPP*- 2162 0.0594 24.6080 57.8272 12952.9500\n 0.118270512017 0.331267509627 PaP/- 2179 0.0594 24.6193 57.8385 12952.9613\n 0.102785499845 0.622573736638 PaS-R 4183 0.0473 25.3577 58.5770 12860.3026\n 0.096006509321 0.174225323677 PPaS-+ 23431 0.0455 27.7450 60.9643 12847.7007\n 0.071795785474 -0.046119341322 aaPE-+ 23454 0.0314 27.3272 60.5465 12696.5397\n 0.047203616362 -0.083390144390 aPP>*- 23892 0.0143 26.7489 59.9682 12375.4368\n 0.047203616362 0.345368016677 PaP>/- 24037 0.0143 26.7577 59.9769 12375.4456\n 0.042939201188 -0.078184660404 aPP>*>- 286150 0.0094 30.1945 63.4138 12208.9827\n 0.041629873446 0.347166397184 PaP>/S- 287735 0.0104 30.1578 63.3771 12249.0817\n 0.036685885343 0.620119246549 PaPES> 10054828 0.0087 35.0516 68.2709 12179.2415\n 0.035141930554 0.664383638381 Pa>PEC*+R 58365195 0.0083 37.5776 70.7969 12166.0321\n 0.034735742152 0.127459889907 PPREa-R*> 69090293 0.0084 37.8042 71.0235 12167.4354\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545942911475, 8.91505505199440e-11]\n [2.0, 27.15815915791503, '3.14159265358979']\n [9.754887502163468, 23.75790801204378, '3.0 - 0.25*x0']\n [43.25789208357, 23.747101876217187, 'log(0.61*x0**2 - 5*x0 + 20.1)']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [44.928974716409904, 23.675383356059676, 0.334941579543170]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [97.39915180412565, 22.39003644062657, 1.09437382221222*exp(-0.0745865610942126*x0)]\n [98.77507424296371, 20.699967368470546, -tan(0.00969398001048327*x0 - 0.895113348960876)]\n [98.77508016991868, 20.699948062984262, -tan(0.00969401983587903*x0 - 0.895113348960876)]\n [102.67524654499746, 20.69988101883838, -tan(sin(0.00969423381824464*x0) - 0.895113348960876)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_sin/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus...\n /bin/cp -p results/mystery_world_sin/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 3.8595698840027815E-006 at j= 74523\n Searching for best fit...\n 0.597142829315 -3.141596600582 P 1 0.2424 15.8658 49.0851 13515.4590\n 0.597142829315 -0.318313736896 P\\ 9 0.2424 19.0357 52.2550 13518.6290\n 0.597142829315 -1.772457735137 PR 11 0.2424 19.3252 52.5445 13518.9185\n 0.597142829315 -23.140698515373 PE 19 0.2424 20.1137 53.3330 13519.7070\n 0.328869499861 0.206873784353 aP/ 34 0.0977 20.0927 53.3120 13149.6204\n 0.119537842999 0.156922556200 aP>/ 308 0.0309 21.8120 55.0312 12683.4183\n 0.119537842999 -0.843077443800 aP>/> 3622 0.0309 25.3678 58.5870 12686.9741\n 0.119537842999 -2.984670184813 PaP>/+ 23461 0.0309 28.0632 61.2824 12689.6695\n 0.116346400711 -2.039747050958 aPER+R 44876 0.0274 28.9598 62.1791 12641.6093\n 0.115160917037 -2.013822285711 aPELLP/ 2145414 0.0309 34.5152 67.7345 12696.6069\n 0.114069983308 0.142173491519 aPaCC\\+/ 3341676 0.0270 35.1498 68.3691 12641.8776\n 0.113982605417 -1.988743562707 aPP+R 3693328 0.0277 35.2917 68.5110 12652.4147\n 0.113618223427 -3.723994825762 aPP++LRE 4289680 0.0282 35.5043 68.7236 12659.6400\n 0.113540604818 0.136519835545 aSaCERE/ 14794656 0.0287 37.2895 70.5088 12668.1770\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_sin/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus...\n /bin/cp -p results/mystery_world_sin/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.59713896974525005 at j= 33304\n Searching for best fit...\n 0.947035968153 0.190075232206 P 1 0.4999 16.5311 49.7504 13810.8491\n 0.341531371498 0.307569601718 a 2 0.1509 16.0597 49.2790 13323.1003\n 0.204118732992 0.203006582327 a> 4 0.0756 16.3171 49.5364 13042.0213\n 0.117056562278 0.122290572149 aa>+ 202 0.0294 21.1731 54.3924 12662.8446\n 0.100574440720 0.238315519449 aPR\\+ 2342 0.0248 24.4895 57.7088 12596.3010\n 0.100574440720 0.238315519449 aPR><\\+ 385342 0.0248 31.8518 65.0710 12603.6632\n 0.100501455431 0.238287774329 aP>\\CC+ 395802 0.0248 31.8893 65.1086 12603.5889\n 0.100499810768 0.238287149115 aPP>/>L+ 3576776 0.0248 35.0651 68.2844 12606.7622\n 0.100458682822 -0.273051961508 aaaE+\\P<\\SS- 14480816 0.0248 37.0812 70.3005 12608.6269\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545942911475, 8.91505505199440e-11]\n [2.0, 27.15815915791503, '3.14159265358979']\n [9.754887502163468, 23.75790801204378, '3.0 - 0.25*x0']\n [43.25789208357, 23.747101876217187, 'log(0.61*x0**2 - 5*x0 + 20.1)']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [44.928974716409904, 23.675383356059676, 0.334941579543170]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [97.39915180412565, 22.39003644062657, 1.09437382221222*exp(-0.0745865610942126*x0)]\n [98.77507424296371, 20.699967368470546, -tan(0.00969398001048327*x0 - 0.895113348960876)]\n [98.77508016991868, 20.699948062984262, -tan(0.00969401983587903*x0 - 0.895113348960876)]\n [102.67524654499746, 20.69988101883838, -tan(sin(0.00969423381824464*x0) - 0.895113348960876)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_sqrt/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus...\n /bin/cp -p results/mystery_world_sqrt/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.11647158377131471 at j= 1\n Searching for best fit...\n 0.192827070815 -1.463844017285 P 1 0.0795 14.2350 47.4543 13060.5692\n 0.192827070815 -2.463844017285 P> 3 0.0795 15.8200 49.0393 13062.1542\n 0.192827070815 -21.462945932076 PE 19 0.0795 18.4829 51.7022 13064.8171\n 0.192826160753 -666.000000000000 PEE 197 0.0795 21.8571 55.0763 13068.1881\n 0.183075794542 -666.000000000000 PC>\\ 1129 0.0733 24.3010 57.5203 13037.7611\n 0.089408037560 1.702972558368 aPE~/ 3294 0.0370 24.8118 58.0311 12760.2941\n 0.075349017051 1.406944557504 PaS+\\ 3967 0.0222 24.8332 58.0525 12551.1695\n 0.060991585408 2.063777551752 aSP-\\ 8044 0.0193 25.5481 58.7674 12496.5379\n 0.060083436578 1.729932109778 aPLS<* 31300 0.0167 27.4866 60.7059 12439.2476\n 0.054325127881 1.713662995267 aPS~L/ 38960 0.0205 27.6571 60.8764 12521.7819\n 0.041492747159 1.552975356042 Pa>E+\\ 43693 0.0120 27.4338 60.6530 12305.4878\n 0.040755364973 1.563767576209 PaE>*\\ 43761 0.0115 27.4101 60.6294 12286.4278\n 0.040663281012 1.564014217457 PaE>*\\S 659139 0.0120 31.3197 64.5390 12308.7821\n 0.040479001738 1.565682632654 P\\SaE>/ 1601659 0.0121 32.5941 65.8134 12312.6998\n 0.040190061235 0.157759448579 PaE>\\E+L 6959261 0.0109 34.7031 67.9224 12274.0968\n 0.040125931231 0.936460855052 aE>\\<~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 1.8705757945424719 at j= 40336\n Searching for best fit...\n 0.288911147145 0.595422751690 P 1 0.1516 14.8183 48.0376 13323.9516\n 0.288911147145 -666.000000000000 PS 13 0.1516 18.5188 51.7381 13327.6520\n 0.041633940250 -0.074379229830 aPE- 304 0.0125 20.2715 53.4908 12315.4428\n 0.034188665785 -0.071534809388 aPE>- 2898 0.0084 23.2401 56.4594 12153.9541\n 0.034049094366 -0.068899929096 aPE>>- 33716 0.0076 26.7745 59.9938 12116.8748\n 0.032136997112 -0.069482282995 aPEPR+- 340286 0.0074 30.0264 63.2457 12109.8894\n 0.031585126756 -0.550894083497 aPaPL/+- 3228340 0.0074 33.2474 66.4667 12112.2980\n 0.030932154557 -0.070051839064 aPPRS/E- 3432304 0.0074 33.3056 66.5249 12114.4126\n 0.029403137215 1.732279141832 aaPE>-/E 3547970 0.0072 33.2803 66.4996 12102.1211\n 0.029398084339 -1.732284584796 aPE>/E<< 8590924 0.0072 34.5559 67.7752 12103.4073\n 0.029235174181 1.659673852332 aR 95832874 0.0072 38.0063 71.2256 12107.3140\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545942911475, 8.91505505199440e-11]\n [2.0, 27.15815915791503, '3.14159265358979']\n [9.754887502163468, 23.75790801204378, '3.0 - 0.25*x0']\n [43.25789208357, 23.747101876217187, 'log(0.61*x0**2 - 5*x0 + 20.1)']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [44.928974716409904, 23.675383356059676, 0.334941579543170]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [94.74760051275051, 21.9734402579613, 1.73111939430237 - 0.0711506685888836*x0]\n [98.77507424296371, 20.699967368470546, -tan(0.00969398001048327*x0 - 0.895113348960876)]\n [98.77508016991868, 20.699948062984262, -tan(0.00969401983587903*x0 - 0.895113348960876)]\n [102.67524654499746, 20.69988101883838, -tan(sin(0.00969423381824464*x0) - 0.895113348960876)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_squared/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus...\n /bin/cp -p results/mystery_world_squared/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.11647158377131471 at j= 1\n Searching for best fit...\n 4.320048900614 4.781735875560 P 1 1.6589 18.7207 51.9400 14300.2823\n 1.916315998255 8.507025672033 a~ 8 0.6817 20.5480 53.7673 13940.4229\n 1.916315998255 5.365432931021 Pa- 31 0.6817 22.5022 55.7215 13942.3771\n 1.916315998255 8.507025672033 aPC/ 332 0.6817 25.9230 59.1423 13945.7980\n 1.684824516656 9.090722727494 aa+~ 378 0.4609 25.9245 59.1438 13786.2182\n 1.684824516656 9.090722727494 aPC<* 2602 0.4609 28.7076 61.9269 13789.0014\n 1.316328750325 8.957904724681 aPR~* 2630 0.3017 28.3670 61.5863 13616.0994\n 0.956829856584 8.779578452159 aP\\Ca- 2129794 0.2118 37.2904 70.5097 13481.4995\n 0.788618964758 6.421879302110 a\\ 2522194 0.2090 37.5333 70.7526 13476.2142\n 0.779846283395 9.322089367625 aSS>>Ca- 29642586 0.2081 41.0721 74.2914 13478.0466\n 0.777822766841 8.971295975846 aaaE-/Sa- 49124360 0.2062 41.7971 75.0164 13475.0415\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_squared/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus...\n /bin/cp -p results/mystery_world_squared/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 12.243377517185923 at j= 40336\n Searching for best fit...\n 5.985059467298 3.897187995551 P 1 3.4303 19.1910 52.4103 14596.7372\n 4.138257321480 -2.377324054788 aP- 30 1.7479 23.5656 56.7848 14326.5230\n 2.510250877831 -1.990771492064 aP>- 272 0.9168 26.0249 59.2442 14066.3975\n 2.065035152642 3.025049617879 PaS- 293 0.8848 25.8506 59.0698 14052.0150\n 1.149802658285 5.395041786998 Pa-R 417 0.4895 25.5149 58.7342 13810.9593\n 0.865839239919 1.476589524974 PPa-+ 2109 0.2830 27.4442 60.6634 13589.6983\n 0.767634774869 -1.551021424903 aPRE- 3170 0.1890 27.8584 61.0777 13425.5731\n 0.748057967741 -1.565859232414 aPER>- 33860 0.1820 31.2382 64.4574 13413.6110\n 0.644747121777 2.879363735940 aaP/ESL 820922 0.1559 35.6218 68.8411 13354.9976\n 0.639359719908 -6.587230316572 aP/E>LL< 12151900 0.1518 39.4990 72.7183 13347.9512\n 0.636699711477 -19.540103633149 aPLSL*>C< 104624898 0.1597 42.5990 75.8183 13371.7699\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545942911475, 8.91505505199440e-11]\n [2.0, 27.15815915791503, '3.14159265358979']\n [9.754887502163468, 23.75790801204378, '3.0 - 0.25*x0']\n [40.03306214009066, 23.749032084797616, '(0.06*x0**2 - 1.49*x0 + 9)**0.5']\n [43.25789208357, 23.747101876217187, 'log(0.61*x0**2 - 5*x0 + 20.1)']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [44.928974716409904, 23.675383356059676, 0.334941579543170]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [94.74760051275051, 21.9734402579613, 1.73111939430237 - 0.0711506685888836*x0]\n [98.77507424296371, 20.699967368470546, -tan(0.00969398001048327*x0 - 0.895113348960876)]\n [98.77508016991868, 20.699948062984262, -tan(0.00969401983587903*x0 - 0.895113348960876)]\n [102.67524654499746, 20.69988101883838, -tan(sin(0.00969423381824464*x0) - 0.895113348960876)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_tan/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus...\n /bin/cp -p results/mystery_world_tan/duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 3.8595698840315281E-006 at j= 74523\n Searching for best fit...\n 0.744438256573 -3.141588881443 P 1 0.2631 16.1839 49.4031 13548.8588\n 0.744438256573 -0.318306017756 P\\ 9 0.2631 19.3538 52.5731 13552.0287\n 0.744438256573 -23.140690796233 PE 19 0.2631 20.4318 53.6511 13553.1067\n 0.744438256404 666.000000000000 PS\\ 101 0.2631 22.8421 56.0614 13555.5170\n 0.630455397446 -0.822426032715 aSC 156 0.2990 23.2295 56.4488 13608.3911\n 0.234779987229 -0.206873784353 aP~/ 316 0.0799 22.8228 56.0421 13071.0005\n 0.221921625582 -0.656989770429 aE>\\ 1134 0.0458 24.5850 57.8042 12845.3747\n 0.156758592208 -0.156922556200 aP~<~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.74443439700295699 at j= 33304\n Searching for best fit...\n 1.117941457203 -0.236960821587 P 1 0.6332 16.7705 49.9898 13907.2943\n 0.486323493833 -0.383437361472 a 2 0.1820 16.5696 49.7889 13399.5717\n 0.213091219657 -0.253081929600 a> 4 0.1079 16.3792 49.5985 13187.2915\n 0.172896797284 -0.152455815060 aa>+ 202 0.0460 21.7358 54.9551 12844.5959\n 0.164417407330 -0.297100472429 aPR\\+ 2342 0.0450 25.1986 58.4179 12839.2366\n 0.146912475139 -0.162491735496 aaER+ 2396 0.0407 25.0691 58.2884 12798.1410\n 0.145863851407 -0.383251691683 aaEE\\+ 29146 0.0299 28.6633 61.8826 12676.4368\n 0.137180599308 -0.210134824188 aaSC+ 395396 0.0312 32.3227 65.5420 12697.4126\n 0.135853442140 -0.268806556790 aPaC>S+C- 42163886 0.0312 39.0593 72.2785 12704.1491\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418545942911475, 8.91505505199440e-11]\n [2.0, 27.15815915791503, '3.14159265358979']\n [9.754887502163468, 23.75790801204378, '3.0 - 0.25*x0']\n [40.03306214009066, 23.749032084797616, '(0.06*x0**2 - 1.49*x0 + 9)**0.5']\n [43.25789208357, 23.747101876217187, 'log(0.61*x0**2 - 5*x0 + 20.1)']\n [44.928866696798615, 23.67585519132408, 0.334916502237320]\n [44.928974716409904, 23.675383356059676, 0.334941579543170]\n [46.826322281101795, 23.520733476088242, 1.24775004223988]\n [94.74760051275051, 21.9734402579613, 1.73111939430237 - 0.0711506685888836*x0]\n [98.77507424296371, 20.699967368470546, -tan(0.00969398001048327*x0 - 0.895113348960876)]\n [98.77508016991868, 20.699948062984262, -tan(0.00969401983587903*x0 - 0.895113348960876)]\n [102.67524654499746, 20.69988101883838, -tan(sin(0.00969423381824464*x0) - 0.895113348960876)]\n Checking for symmetry \n duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus\n Just one variable!\n duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus just one variable for ADD \n \n duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus just one variable for ADD \n \n duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus just one variable for ADD \n \n duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus just one variable for ADD \n \n duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus just one variable for ADD\n duplicateVarsWithNoise100k.txt_train-translated_plus-translated_plus-translated_minus just one variable for ADD\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n\n\n\n```\n%%writefile ai_feynman_duplicateVarsWithNoise3.py\nfrom S_run_aifeynman import run_aifeynman\nrun_aifeynman(\"/content/gdrive/My Drive/Lemay.ai_research/AI-Feynman/example_data/\",\"duplicateVarsWithNoise.txt\",30,\"19ops.txt\", polyfit_deg=3, NN_epochs=1000)\n```\n\n Overwriting ai_feynman_duplicateVarsWithNoise3.py\n\n\n\n```\nprint(os.getcwd())\n!sudo python3 ai_feynman_duplicateVarsWithNoise3.py\n```\n\n### No duplicate columns but same noise\n\n\n```\nimport os\nimport random\nimport numpy as np\nos.chdir(\"/content/gdrive/My Drive/Lemay.ai_research/AI-Feynman/example_data\")\n\ndef getY(x01,x23):\n y = -0.5*x01+0.5*x23+3\n return y\n\ndef getRow():\n x=[0 for x in range(4)]\n x[1]=random.random()\n x[3]=random.random()\n y=getY(x[1],x[3])\n mu=0\n sigma=0.05\n noise=np.random.normal(mu, sigma, 4)\n x=x+noise\n return str(x[1])+\" \"+str(x[3])+\" \"+str(y)+\"\\n\"\n\nwith open(\"varsWithNoise.txt\", \"w\") as f:\n for _ in range(100000):\n f.write(getRow())\nf.close()\n\n# switch back to the code directory\nos.chdir(\"/content/gdrive/My Drive/Lemay.ai_research/AI-Feynman/Code\")\n```\n\n\n```\n%matplotlib inline\nimport matplotlib.pyplot as plt\nimport pandas as pd\nplt.style.use('seaborn-whitegrid')\nimport numpy as np\n\ndf=pd.read_csv(\"../example_data/varsWithNoise.txt\",sep=\" \",header=None)\ndf.plot.scatter(x=0, y=2)\ndf.plot.scatter(x=1, y=2)\n```\n\n\n```\n%%writefile ai_feynman_varsWithNoise.py\nfrom S_run_aifeynman import run_aifeynman\nrun_aifeynman(\"/content/gdrive/My Drive/Lemay.ai_research/AI-Feynman/example_data/\",\"varsWithNoise.txt\",30,\"14ops.txt\", polyfit_deg=3, NN_epochs=1000)\n```\n\n Writing ai_feynman_varsWithNoise.py\n\n\n\n```\n!sudo python3 ai_feynman_varsWithNoise.py\n```\n\n Checking for brute force + \n \n set: Syntax Error.\n Checking for brute force * \n \n set: Syntax Error.\n Checking polyfit \n \n Complexity RMSE Expression\n [11.0, 26.922437814458732, '-x0 + x1 + 3']\n [104.69268207809836, 24.299162410776688, '-0.512424382768879*x0 + 0.511797113710875*x1 + 3']\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_atan/varsWithNoise.txt_train...\n /bin/cp -p results/mystery_world_atan/varsWithNoise.txt_train mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.31214542297809916 at j= 1\n Searching for best fit...\n 0.061403642294 -1.910741817479 P 1 0.0267 12.5841 45.8034 12616.0574\n 0.061403642294 4.372443664546 P~ 10 0.0267 15.9060 49.1253 12619.3794\n 0.061403642020 666.000000000000 PS\\ 163 0.0267 19.9328 53.1521 12623.4062\n 0.061403566350 1.230851001359 aPS* 437 0.0267 21.3556 54.5749 12624.8284\n 0.045125544638 1.211160131482 bPE/ 627 0.0212 21.4321 54.6513 12530.9392\n 0.044643642778 0.270731579956 aP/C 878 0.0164 21.9023 55.1216 12425.8009\n 0.038958502866 1.269321052238 aPE~/ 6323 0.0150 24.5541 57.7734 12391.8629\n 0.035625119942 1.222766609733 baP+E/ 49185 0.0175 27.3846 60.6039 12457.8746\n 0.035034023697 0.261316605059 abP+/C 52211 0.0134 27.4466 60.6659 12350.6566\n 0.023778699302 1.249630260186 ba-PE/ 143487 0.0049 28.3460 61.5653 11937.3226\n 0.023392270441 -1.891783909921 baPE--L 625014 0.0048 30.4453 63.6646 11935.2527\n 0.023125579667 1.250478442095 ba-PE>+-\\ 7718537 0.0056 33.9819 67.2012 12003.0010\n 0.020729551543 1.459374782792 abPER+-\\ 7719725 0.0046 33.8976 67.1169 11923.6000\n 0.020366149552 0.467286694116 abP>+-\\E 8927705 0.0050 34.0818 67.3011 11950.9938\n 0.020303507209 1.005632242542 Pa>b>/+\\ 11370256 0.0058 34.4263 67.6456 12015.5368\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_atan/varsWithNoise.txt_train...\n /bin/cp -p results/mystery_world_atan/varsWithNoise.txt_train mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 1.2922545658267337 at j= 46902\n Searching for best fit...\n 0.101649497918 0.411337392322 P 1 0.0490 13.3113 46.5306 12862.9139\n 0.101649497918 1.128872889917 PL 25 0.0490 17.9552 51.1745 12867.5578\n 0.056125554135 0.053578636145 bPE+ 384 0.0240 21.0394 54.2587 12580.3829\n 0.053175125661 -0.055895374255 aPE- 545 0.0232 21.4666 54.6859 12567.3003\n 0.052249058767 -0.058422384921 aPE<- 5513 0.0224 24.7798 57.9991 12555.8311\n 0.023497889594 -0.053626498392 abPE+- 43679 0.0081 26.6130 59.8322 12142.2559\n 0.020670724269 -0.051489755848 abPE>+- 542882 0.0062 30.0637 63.2829 12039.7230\n 0.019559602008 -0.053959432950 aPEbS+- 764429 0.0044 30.4777 63.6970 11895.4042\n 0.018044547281 0.053938461502 ba-SPE+ 2093526 0.0039 31.8149 65.0341 11855.9996\n 0.017767722978 0.047441499111 bPaPE--+ 6740871 0.0040 33.4795 66.6988 11861.2036\n 0.017751970170 1.083362435385 bPa-+RRR 10424427 0.0037 34.1072 67.3265 11834.5867\n 0.017627960689 28.847575908441 ab-SPE+\\ 29850203 0.0042 35.6149 68.8342 11881.9327\n 0.017178522978 0.345767211786 baPP>*--R 90939231 0.0038 37.1848 70.4041 11850.8508\n 0.017125404917 0.523112329716 baPP*>--L 92400471 0.0037 37.2033 70.4226 11833.2000\n 0.016126141473 0.787800699319 bPPa-++RR 93636312 0.0038 37.1358 70.3550 11844.6445\n Checking polyfit \n \n Complexity RMSE Expression\n [11.0, 26.922437814458732, '-x0 + x1 + 3']\n [46.82642955276403, 23.51926784279126, 1.24784282220623]\n [46.826504901765, 23.518700609650345, 1.24790799617767]\n [125.91974286698664, 23.248650485468357, -0.913749694824219*tan(0.019306443631649*x0 + 2.193248927895)]\n [154.37021179561177, 22.372871941531574, tan(0.0405293171426792*x1*exp(-1.0023330450058*x0) + 0.882259964942932)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_cos/varsWithNoise.txt_train...\n /bin/cp -p results/mystery_world_cos/varsWithNoise.txt_train mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.31214542297809916 at j= 1\n Searching for best fit...\n 0.148552080248 -4.092654693322 P 1 0.0417 13.8587 47.0780 12797.3019\n 0.148552080248 -2.723515827877 PR 16 0.0417 17.8587 51.0780 12801.3019\n 0.148551376718 -666.000000000000 PEE 307 0.0417 22.1208 55.3400 12805.5671\n 0.145463038157 -0.989532081014 aPE/ 626 0.0325 23.1184 56.3377 12705.0738\n 0.116227292361 -1.229051979535 bP+\\ 744 0.0356 23.0438 56.2631 12742.1761\n 0.111283341520 -0.050285895697 bbE-S 8679 0.0500 26.5253 59.7445 12884.2488\n 0.102049717303 -1.902468137415 bb>/C 9012 0.0388 26.4546 59.6739 12780.8875\n 0.101616581490 -1.018706245997 abE>E/ 82658 0.0214 29.6457 62.8650 12540.8504\n 0.100500450601 -0.061159087597 bbSE-S 100047 0.0407 29.9052 63.1245 12803.5558\n 0.100055943021 -1.895058535125 ba-C>S 166167 0.0316 30.6308 63.8501 12700.7887\n 0.098306724469 0.010897296405 bCCE>C 463056 0.0382 32.0839 65.3032 12780.3180\n 0.092372541344 -1.327425401716 Pab<*+\\ 607102 0.0181 32.3848 65.6041 12475.6702\n 0.091694615696 -0.976212867105 aPb>*-E 628457 0.0282 32.4241 65.6434 12656.8213\n 0.090446741527 -0.968229547906 aPbE*-E 628673 0.0311 32.4048 65.6241 12697.0669\n 0.089283855971 -1.037220584395 ab>RE-E 1350884 0.0187 33.4897 66.7089 12490.2438\n 0.080898072294 -1.023816104554 abERE-E 1350956 0.0200 33.3474 66.5667 12516.9398\n 0.075502597490 -1.481775805482 baC+S>\\ 1814418 0.0188 33.6734 66.8927 12492.1370\n 0.074292375021 -1.746921208480 ab<*CCR 1858520 0.0188 33.6847 66.9040 12492.2885\n 0.073215982992 -0.994843695785 aabERS*- 7155494 0.0151 35.6086 68.8278 12404.6445\n 0.072226037530 -0.995956954055 abERSL~* 14570804 0.0152 36.6149 69.8342 12410.0274\n 0.072169773576 -0.999040584923 abEE>ER/ 15755024 0.0148 36.7265 69.9458 12399.5413\n 0.068264357162 -0.005840595467 ab<*SCR~ 26213519 0.0159 37.3807 70.6000 12427.8701\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_cos/varsWithNoise.txt_train...\n /bin/cp -p results/mystery_world_cos/varsWithNoise.txt_train mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.99999999999341582 at j= 15150\n Searching for best fit...\n 0.197490127932 -0.318309877324 P 1 0.0481 14.2695 47.4888 12855.8856\n 0.197490127932 -0.241453001907 P> 4 0.0481 16.2695 49.4888 12857.8856\n 0.197490127932 -3.141592740992 P\\ 13 0.0481 17.9699 51.1892 12859.5861\n 0.193341353775 -0.215921577483 bP>+ 312 0.0629 22.5243 55.7435 12973.0072\n 0.176950429545 -0.276857218650 PbS+ 361 0.0643 22.6069 55.8262 12982.2965\n 0.121964805752 -0.242434907370 PaC+ 367 0.0335 22.0938 55.3131 12716.8439\n 0.120352062621 0.103239812942 aPP*- 3713 0.0335 25.4133 58.6326 12719.1761\n 0.120352062621 -0.324337446921 PaP/- 3769 0.0335 25.4349 58.6542 12719.1977\n 0.108165398927 0.120054265921 aP/SC 132803 0.0373 30.3637 63.5830 12768.4474\n 0.103990457060 -0.434631902247 aEE\\>> 343802 0.0402 31.7354 64.9547 12800.9382\n 0.103315776266 -0.504089730811 aSSSC> 357320 0.0376 31.7816 65.0009 12773.7870\n 0.099717132612 -1.748409961955 aC>SCS 455603 0.0397 32.0810 65.3003 12795.7175\n 0.093518025356 -0.325953521812 Pab>>/- 549241 0.0328 32.2581 65.4774 12717.8024\n 0.089975000573 -0.325531347521 PabE>/- 549457 0.0285 32.2030 65.4222 12660.4008\n 0.084359655820 -0.246776859104 Pab<*E+ 568222 0.0212 32.1584 65.3777 12540.3850\n 0.077585310708 -0.572786420625 Pab<*+R 610990 0.0171 32.1424 65.3616 12452.6022\n 0.070504955940 -1.002693047848 ab>R 26563518 0.0151 37.2266 70.4458 12406.8457\n 0.056093596847 -1.000284624375 ba-SES<~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.31214542297809916 at j= 1\n Searching for best fit...\n 16.107160861598 13.760728770157 P 1 5.5401 20.6193 53.8386 14792.3435\n 15.584673876713 16.446662904778 b 3 5.3625 22.1567 55.3759 14780.6325\n 15.244451050753 17.792547012893 a~ 11 5.1386 23.9993 57.2186 14765.1038\n 15.057090859771 15.325109708354 bE 30 5.1985 25.4289 58.6482 14771.2790\n 14.528459189410 15.470827740950 bP* 42 5.0201 25.8628 59.0820 14757.5175\n 13.459790417682 12.615015327944 b>E 285 4.6892 28.5150 61.7343 14732.4590\n 13.124255847326 11.947364360404 PbE* 469 4.5820 29.1972 62.4165 14723.7374\n 12.485802103233 14.414724440228 bb+E 930 4.3513 30.1129 63.3322 14703.6434\n 12.368317848953 23.523183039567 a>E~ 1985 3.8076 31.1931 64.4124 14650.2688\n 10.329504128938 5.248215020268 b>>E 3228 4.0505 31.6348 64.8540 14676.2071\n 7.708016686627 10.373815718198 aCEE 3461 3.1533 31.3130 64.5323 14574.1306\n 7.708016686627 7.232222977185 PaCEE+ 63601 3.1533 35.5128 68.7320 14578.3304\n 7.105494950016 9.918157111806 baCEE+ 63603 2.8970 35.3954 68.6147 14543.7417\n 6.742663662842 8.858571562781 baC>+E 106695 1.8789 36.0661 69.2854 14367.8100\n 5.017940568332 12.117574727798 ba<<-E 108189 1.2091 35.6599 68.8792 14187.9347\n 4.303093734878 21.191325092601 PPba-** 521806 1.1047 37.7082 70.9274 14153.3600\n 4.303093734878 31.060930042984 PPba>-** 6749005 1.1047 41.4013 74.6205 14157.0531\n 4.233456994488 11.297022953188 Pba-S>E* 9285031 0.9878 41.8380 75.0572 14111.8942\n 4.233456994488 11.297022953188 Pab-SS>+E 17890344 1.3100 42.6874 75.9066 14228.0375\n 3.958766280091 13.693854626078 b>aE>S+E 40822368 1.3100 43.8775 77.0968 14229.2276\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_exp/varsWithNoise.txt_train...\n /bin/cp -p results/mystery_world_exp/varsWithNoise.txt_train mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 33.009482372768282 at j= 46902\n Searching for best fit...\n 20.799102387793 10.507244284671 P 1 13.1765 20.9881 54.2074 15145.8939\n 15.591327695152 16.687084367462 b> 6 5.7916 23.1573 56.3766 14813.0437\n 15.289782212288 33.017131866996 aC 23 8.1134 25.0677 58.2870 14952.5441\n 14.747176093366 8.012519172406 bP+ 33 9.1926 25.5364 58.7557 15004.0216\n 14.617704657117 -10.579737116190 aP- 50 8.0324 26.1231 59.3424 14949.5695\n 14.613992526855 11.083904853225 b>> 69 7.8133 26.5874 59.8067 14938.7497\n 14.332221845635 18.043263215137 bS> 84 6.7674 26.8431 60.0624 14880.3890\n 9.134550232435 16.689039005987 baC+ 369 3.0182 28.3285 61.5477 14553.0315\n 8.160075507151 8.054605858698 bPa-+ 3534 4.9304 31.4253 64.6446 14756.5584\n 8.160075507151 -8.054605858698 abP+- 3692 4.9304 31.4884 64.7077 14756.6215\n 7.779057236855 16.866761980153 ba>\\+ 4149 2.3245 31.5878 64.8071 14449.9666\n 5.623620549392 -11.164603724310 ab>>- 5366 2.0824 31.4908 64.7101 14405.4519\n 5.086486426008 -16.866616960328 ba<*< 7236 1.5421 31.7773 64.9966 14283.3087\n 4.245888558127 20.460275712354 ba-ER 11772 0.8507 32.2188 65.4381 14041.3103\n 4.245888558127 20.460275712354 bEaE/R 222276 0.8507 36.4577 69.6770 14045.5492\n 4.223432393308 39.042032380720 ab-S>>\\ 2269460 0.8675 39.8020 73.0213 14056.8796\n 4.210436306156 7.330539097250 ba-SERE 2414934 0.8531 39.8872 73.1064 14050.1006\n 4.182125850281 27.370603948876 ab>C+ER\\ 22345679 0.9138 43.0874 76.3067 14081.3758\n 4.182125850281 5.689789304648 Pab>C+-ER 112518376 0.9138 45.4195 78.6387 14083.7079\n Checking polyfit \n \n Complexity RMSE Expression\n [2.0, 27.146468378239973, '3.14159265358979']\n [11.0, 26.922437814458732, '-x0 + x1 + 3']\n [46.82642955276403, 23.51926784279126, 1.24784282220623]\n [46.826504901765, 23.518700609650345, 1.24790799617767]\n [125.91974286698664, 23.248650485468357, -0.913749694824219*tan(0.019306443631649*x0 + 2.193248927895)]\n [154.37021179561177, 22.372871941531574, tan(0.0405293171426792*x1*exp(-1.0023330450058*x0) + 0.882259964942932)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_inverse/varsWithNoise.txt_train...\n /bin/cp -p results/mystery_world_inverse/varsWithNoise.txt_train mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.28597616922301250 at j= 46902\n Searching for best fit...\n 0.113658339692 -2.855616571790 P 1 0.0540 13.4724 46.6917 12902.7894\n 0.113658339692 -1.855616571790 P< 7 0.0540 16.2798 49.4991 12905.5968\n 0.113658339692 -2.855616571790 P\\\\ 157 0.0540 20.7670 53.9863 12910.0841\n 0.113658339284 666.000000000000 PS\\ 163 0.0540 20.8211 54.0404 12910.1382\n 0.113658099002 -666.000000000000 PEE 307 0.0540 21.7345 54.9538 12911.0499\n 0.074722327000 0.285045932073 aPE/ 626 0.0329 22.1573 55.3766 12709.6908\n 0.056913659456 0.043242306210 bP+\\ 744 0.0234 22.0137 55.2330 12570.6728\n 0.056913659456 1.043242306210 bP+\\< 10599 0.0234 25.8462 59.0655 12574.5053\n 0.056583602399 0.034602764943 b>>>\\ 28392 0.0224 27.2594 60.4786 12557.6959\n 0.056248402326 -0.129571006747 b>R>\\ 28500 0.0225 27.2563 60.4756 12560.7946\n 0.040104920476 0.269373835345 abP+-E 52967 0.0194 27.6624 60.8817 12500.6424\n 0.036753970502 0.327315432009 ab-PE/ 143489 0.0099 28.9743 62.1935 12227.5018\n 0.033402546230 0.089828970890 bPa<-+\\ 607335 0.0048 30.9179 64.1372 11936.1584\n 0.033385552166 0.329182548801 ab-PE-E>RS 26905655 0.0051 35.8063 69.0256 11967.7855\n 0.020788689196 -0.156908843362 bPaS<-+R\\ 112049268 0.0046 37.7611 70.9804 11928.3898\n 0.020488102177 -0.416855546111 bPaS-+RR\\ 126757329 0.0047 37.9181 71.1373 11931.0617\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_inverse/varsWithNoise.txt_train...\n /bin/cp -p results/mystery_world_inverse/varsWithNoise.txt_train mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.39963450891453223 at j= 42311\n Searching for best fit...\n 0.113658339692 0.127207611508 P 1 0.0688 13.4724 46.6917 13001.4090\n 0.080507366102 0.097109640358 aP+ 32 0.0280 17.9749 51.1942 12639.3198\n 0.080024421415 0.199847065727 bC> 87 0.0393 19.4092 52.6284 12779.6232\n 0.079998017978 0.399694135903 bCR 195 0.0386 20.5731 53.7924 12773.4175\n 0.077880387238 0.078125456520 aP>+ 311 0.0331 21.2078 54.4271 12710.8303\n 0.075456230659 0.100700031408 PaS+ 358 0.0337 21.3652 54.5845 12718.6899\n 0.073491599912 -0.095927160443 bP>- 474 0.0251 21.7321 54.9514 12599.4368\n 0.073491599912 0.095927160443 Pb<- 487 0.0251 21.7711 54.9904 12599.4758\n 0.072356887957 0.224598125567 Pb-R 802 0.0342 22.4684 55.6877 12726.4151\n 0.072340773202 0.100008393163 Pb 10293 0.0318 26.1467 59.3659 12700.2707\n 0.072166449364 0.437345826351 Pb-LS 12061 0.0307 26.3752 59.5945 12685.4085\n 0.049301675663 0.078130013322 aPbC++ 41693 0.0184 27.6149 60.8342 12479.0917\n 0.043825693949 0.097116495711 PabC*+ 41938 0.0173 27.4535 60.6728 12454.1368\n 0.039562845009 0.096537709164 Pab>/+ 42262 0.0112 27.3170 60.5363 12276.0383\n 0.039523984642 0.096544768146 PabE/+ 42478 0.0093 27.3229 60.5422 12200.2687\n 0.030334035255 0.196416299906 aPb-+R 51089 0.0134 27.2074 60.4267 12349.3715\n 0.027162533353 0.054884956963 aPPb-++ 520196 0.0109 30.3961 63.6154 12267.8439\n 0.024904146253 -0.058057684837 baPRE+- 544716 0.0082 30.3373 63.5566 12152.3251\n 0.023966234225 0.585088121135 bPa-+R\\ 723948 0.0048 30.6923 63.9116 11935.9414\n 0.020709640901 0.168339267261 ab-P/E> 2054360 0.0051 31.9863 65.2056 11963.3126\n 0.020205028303 0.140693670103 ab>-ES>> 25634279 0.0050 35.5921 68.8114 11957.7380\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418117324017906, 8.91505505199440e-11]\n [2.0, 27.146468378239973, '3.14159265358979']\n [11.0, 26.922437814458732, '-x0 + x1 + 3']\n [44.9284350278788, 23.674076369594914, 0.334816306829453]\n [46.82642955276403, 23.51926784279126, 1.24784282220623]\n [46.826504901765, 23.518700609650345, 1.24790799617767]\n [96.62956053594237, 23.472287322878472, 0.343946248292923*cos(x1)**0.13920846581459]\n [125.91974286698664, 23.248650485468357, -0.913749694824219*tan(0.019306443631649*x0 + 2.193248927895)]\n [154.37021179561177, 22.372871941531574, tan(0.0405293171426792*x1*exp(-1.0023330450058*x0) + 0.882259964942932)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_log/varsWithNoise.txt_train...\n /bin/cp -p results/mystery_world_log/varsWithNoise.txt_train mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.31214542297809916 at j= 1\n Searching for best fit...\n 0.212471204034 -2.102217149018 P 1 0.0891 14.3750 47.5943 13106.9860\n 0.212471204034 2.039375591995 PC 22 0.0891 18.8344 52.0537 13111.4455\n 0.212471204006 666.000000000000 PS\\ 163 0.0891 21.7237 54.9430 13114.3348\n 0.203053081368 0.246131128066 aCR 194 0.0876 21.9095 55.1288 13107.6432\n 0.159913336634 1.023770804510 bbS- 525 0.0622 23.0012 56.2205 12969.4205\n 0.152802506042 0.791348538245 aP+\\ 743 0.0623 23.4366 56.6559 12970.4637\n 0.125770962873 1.483550128494 aP-\\ 761 0.0496 23.1903 56.4095 12877.8150\n 0.122353334282 0.693381828047 a>>\\ 2012 0.0497 24.5532 57.7725 12879.6344\n 0.118675716597 0.709540516163 aP<+\\ 7799 0.0498 26.4638 59.6831 12882.7913\n 0.118396666924 0.813930956289 aE>>\\ 28415 0.0513 28.3257 61.5450 12896.5180\n 0.092408774014 0.759746016783 aPb-+\\ 50657 0.0307 28.8023 62.0216 12688.4611\n 0.090664819225 0.750620950571 PbaS--\\ 609391 0.0350 32.3633 65.5826 12745.3455\n 0.083307059954 -3.725719135340 baPE--R 613350 0.0295 32.2506 65.4699 12675.1714\n 0.068205404468 0.041452469551 bPa-+LR 729132 0.0160 32.2115 65.4308 12426.2259\n 0.063741642515 -0.079182717520 ba<<-RR 1511028 0.0156 33.1651 66.3844 12416.0889\n 0.061753352906 0.711509249007 babE+R-E 8574129 0.0164 35.6239 68.8431 12438.4498\n 0.060026018627 0.021166257555 bPaS-+LR 8801970 0.0170 35.6208 68.8400 12454.6643\n 0.059868278439 0.521000467625 bPaS-+RL 8895282 0.0144 35.6322 68.8515 12387.0140\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_log/varsWithNoise.txt_train...\n /bin/cp -p results/mystery_world_log/varsWithNoise.txt_train mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 1.2518467960290158 at j= 46902\n Searching for best fit...\n 0.334641918738 0.398475200075 P 1 0.1697 15.0303 48.2496 13370.0857\n 0.202231443190 0.303865608701 bP+ 33 0.0651 19.3481 52.5674 12983.8778\n 0.200292501031 0.625995914188 aC> 86 0.0859 20.7161 53.9354 13098.4101\n 0.191879001934 0.244513823711 bP>+ 312 0.0669 22.5133 55.7326 12998.2498\n 0.177583056171 0.315242816784 PbS+ 361 0.0701 22.6121 55.8313 13017.4147\n 0.177579587433 1.017403861533 bSERR 31782 0.0719 29.0721 62.2914 13034.6361\n 0.177178274569 -1.262394207010 bC<<~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 3.6288300847370446E-006 at j= 15150\n Searching for best fit...\n 0.596642468702 -3.141596369843 P 1 0.2417 15.8646 49.0839 13514.3668\n 0.596642468702 -4.141596369843 P> 4 0.2417 17.8646 51.0839 13516.3668\n 0.590561175977 -0.182396155105 aS 20 0.2291 20.1717 53.3910 13496.7357\n 0.368724094936 -0.058387787278 aP/ 59 0.1542 21.0529 54.2722 13336.8168\n 0.325313292903 -1.087854376116 a>R 176 0.1441 22.4490 55.6683 13310.7779\n 0.325313292903 -2.087854376116 a>R> 1391 0.1441 25.4314 58.6507 13313.7604\n 0.324932415006 -0.780895739434 a>>L 2984 0.1446 26.5309 59.7502 13316.2677\n 0.322948218990 0.356907463142 ba>/ 6095 0.1226 27.5047 60.7240 13249.8040\n 0.308023845863 -0.069695590984 abE>/ 6167 0.1167 27.5011 60.7204 13229.9280\n 0.260564682489 -1.032420509912 abC+R 8174 0.0897 27.6662 60.8855 13122.8661\n 0.242486575296 0.097494405657 ab-P/ 10082 0.0829 27.8651 61.0844 13091.1670\n 0.227009866113 -0.892314226891 abE\\+R 95294 0.0667 31.0106 64.2298 13005.3232\n 0.222159744469 -1.301426266790 ab<<-R 96527 0.0673 30.9980 64.2172 13009.2040\n 0.158914041482 0.143020351192 ab-P-P/ 2002412 0.0345 34.8881 68.1073 12740.4773\n 0.158450789889 0.142533236618 ab-PC>* 28013915 0.0346 38.6839 71.9032 12745.4271\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_sin/varsWithNoise.txt_train...\n /bin/cp -p results/mystery_world_sin/varsWithNoise.txt_train mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.59663883987170496 at j= 42311\n Searching for best fit...\n 0.944418800650 0.189916035927 P 1 0.5002 16.5271 49.7464 13811.0655\n 0.550790466425 0.612754854994 a 2 0.2246 16.7492 49.9685 13485.2939\n 0.510708373266 -0.582411871611 b< 9 0.2120 18.8101 52.0294 13463.9065\n 0.361876128870 -0.597758586534 ba- 54 0.1514 20.8981 54.1174 13329.0931\n 0.348743815568 -0.598143597243 ab<* 407 0.0968 23.7588 56.9780 13149.5229\n 0.305767933139 0.597760041402 abS- 524 0.1252 23.9336 57.1529 13254.9586\n 0.267391609106 0.302581915613 aab-+ 3545 0.0730 26.4982 59.7175 13037.7809\n 0.263286186749 0.388843478583 ab>C+ 4394 0.0969 26.7857 60.0049 13153.2559\n 0.207316719278 0.388905593880 abEC+ 4466 0.0568 26.4643 59.6836 12935.4633\n 0.199360407661 -0.221257947941 bbaE-+ 42231 0.0529 29.6491 62.8684 12909.9532\n 0.161696095509 -0.453222604106 baP\\+- 43542 0.0355 29.3911 62.6104 12747.3323\n 0.159643670368 -0.455071354194 baP\\S+- 543960 0.0351 33.0157 66.2350 12745.8116\n 0.159643670368 0.455071354194 abP\\S-- 548336 0.0351 33.0273 66.2465 12745.8232\n 0.159104788510 -0.162361821140 PbaS-*< 600643 0.0382 33.1538 66.3731 12780.2575\n 0.157677054752 -0.456842810175 baP\\SS+- 7135902 0.0348 36.7113 69.9306 12745.7220\n 0.157651939982 0.456865432649 abPCCRL+- 97513517 0.0348 40.4836 73.7028 12749.4535\n 0.156274960436 -0.502532731521 baaCC/<~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.31214542297809916 at j= 1\n Searching for best fit...\n 0.188469292878 -1.460090145112 P 1 0.0772 14.2020 47.4213 13048.4285\n 0.188469292878 2.522973580708 PCS 220 0.0772 21.9834 55.2027 13056.2099\n 0.188468592880 -666.000000000000 PEE 307 0.0772 22.4641 55.6834 13056.6882\n 0.169134895570 1.665897808415 bbS- 525 0.0528 23.0821 56.3013 12902.8703\n 0.167375306799 1.661811803849 bPE/ 627 0.0699 23.3231 56.5424 13017.3366\n 0.129025668049 1.433475542151 aP+\\ 743 0.0511 23.1926 56.4119 12889.3526\n 0.110553104195 2.125677132399 aP-\\ 761 0.0441 23.0042 56.2235 12829.5859\n 0.110527074587 1.351667520069 aP<+\\ 7799 0.0430 26.3612 59.5805 12822.2876\n 0.109130819843 -0.260359788659 PaC+R 8170 0.0439 26.4099 59.6292 12830.8931\n 0.100173394858 0.790370665401 aa>/C 9008 0.0426 26.4272 59.6465 12818.9796\n 0.100119539186 0.456571709190 Pa-RR 11626 0.0428 26.7945 60.0138 12821.2889\n 0.097872371002 2.104446618949 aSP-\\ 17147 0.0429 27.3224 60.5416 12823.0968\n 0.096966158836 1.093290249933 a>>R\\ 29363 0.0428 28.0850 61.3043 12822.4055\n 0.071092232713 1.401873020689 aPb-+\\ 50657 0.0201 28.4240 61.6432 12515.5171\n 0.070889079548 2.050911702797 abP+-\\ 50807 0.0190 28.4241 61.6434 12492.3134\n 0.069348277924 1.392747954476 PbaS--\\ 609391 0.0238 31.9767 65.1959 12588.0454\n 0.059776154042 -3.083592131434 baPE--R 613350 0.0187 31.7717 64.9910 12488.9839\n 0.054659883554 1.740242449474 ba-PLL* 1991718 0.0138 33.3418 66.5611 12366.8457\n 0.054145600423 1.235610745206 ab-ER>\\ 2270864 0.0130 33.5174 66.7367 12342.4402\n 0.052592105180 1.368183448821 PabP+-/E 6918919 0.0141 35.0827 68.3020 12376.4707\n 0.052560178041 1.026387647764 bPa<-+RL 8895174 0.0132 35.4443 68.6636 12351.9814\n 0.052245313841 1.034067004587 Pab-E+L\\ 9945349 0.0144 35.5967 68.8160 12386.3014\n 0.049882051631 1.728041231235 baS-PL<* 23775210 0.0120 36.7873 70.0065 12313.4375\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_sqrt/varsWithNoise.txt_train...\n /bin/cp -p results/mystery_world_sqrt/varsWithNoise.txt_train mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 1.8699718887780803 at j= 46902\n Searching for best fit...\n 0.288110198099 0.595230522520 P 1 0.1508 14.8143 48.0336 13321.7701\n 0.288110198099 1.055018646496 PR 16 0.1508 18.8143 52.0336 13325.7701\n 0.257551154765 0.365247551224 bP>+ 312 0.0862 22.9380 56.1573 13101.8880\n 0.183395410495 0.451535585136 PaC+ 367 0.0832 22.6823 55.9016 13087.5933\n 0.173625190412 0.921298282698 bP+R 780 0.0577 23.6910 56.9103 12939.5106\n 0.170450679356 1.870111852594 bC>S 2517 0.0711 25.3546 58.5738 13026.4554\n 0.164517289342 0.172383378618 bPP*+ 3498 0.0711 25.7783 58.9975 13026.9288\n 0.164517289342 0.541558370938 PbP/+ 3556 0.0711 25.8020 59.0213 13026.9525\n 0.163624460677 -0.189881900987 aPP*- 3713 0.0657 25.8565 59.0758 12994.6254\n 0.161530474547 0.411158286215 Pb>R+ 4231 0.0753 26.0263 59.2456 13050.2879\n 0.158772379846 0.938387224399 PbS+R 8164 0.0678 26.9497 60.1690 13008.7068\n 0.158566786360 1.891778440932 bCC 711860 0.0357 32.4998 65.7191 12752.7707\n 0.085313381192 -1.282848668314 baP+-\\< 715896 0.0357 32.5079 65.7272 12752.7789\n 0.064411136294 1.314275683502 bPa-+RR 727836 0.0188 32.1263 65.3456 12492.4145\n 0.056137680083 -0.133877176056 abPP>*+- 6756986 0.0150 35.1427 68.3620 12403.6613\n 0.055119011095 0.944623829695 bPPa-++L 6909369 0.0114 35.1485 68.3677 12291.6232\n 0.050335854625 0.714902511756 baPRE--R 7759953 0.0107 35.1850 68.4043 12264.9532\n 0.048169129499 0.863946994014 ba-P/ER> 29080089 0.0107 37.0274 70.2467 12268.7370\n 0.047815244319 1.732903037310 ba-P/ERR 29220057 0.0107 37.0237 70.2430 12267.1478\n 0.046868150455 0.148278689777 bPaP>+- 91414865 0.0104 38.6264 71.8457 12255.6957\n 0.045977718363 -0.147224396164 abPE+- 97523831 0.0107 38.7060 71.9253 12267.7495\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418117324017906, 8.91505505199440e-11]\n [2.0, 27.146468378239973, '3.14159265358979']\n [11.0, 26.922437814458732, '-x0 + x1 + 3']\n [43.64577375129211, 26.80747436789829, 0.137621752238064]\n [43.64578331733191, 26.80744749560774, 0.137622664765998]\n [44.9284350278788, 23.674076369594914, 0.334816306829453]\n [46.82642955276403, 23.51926784279126, 1.24784282220623]\n [46.826504901765, 23.518700609650345, 1.24790799617767]\n [96.62956053594237, 23.472287322878472, 0.343946248292923*cos(x1)**0.13920846581459]\n [125.91974286698664, 23.248650485468357, -0.913749694824219*tan(0.019306443631649*x0 + 2.193248927895)]\n [145.85465595328083, 22.80832447440447, 1.09555152691588*exp(-0.154938563704491*x0 + 0.15204918384552*x1)]\n [154.37021179561177, 22.372871941531574, tan(0.0405293171426792*x1*exp(-1.0023330450058*x0) + 0.882259964942932)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_squared/varsWithNoise.txt_train...\n /bin/cp -p results/mystery_world_squared/varsWithNoise.txt_train mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.31214542297809916 at j= 1\n Searching for best fit...\n 4.233095282218 4.852886303402 P 1 1.6142 18.6914 51.9106 14289.1513\n 3.742285230796 7.538820438023 b 3 1.4491 20.0985 53.3178 14246.6855\n 3.407604184135 8.884704546138 a~ 11 1.2344 21.8378 55.0571 14183.1474\n 3.335495717211 6.417267241599 bE 30 1.3143 23.2544 56.4737 14210.1764\n 3.313798603858 7.083161831632 bb+ 39 1.3303 23.6235 56.8428 14215.5017\n 2.905178140849 6.562985274195 bP* 42 1.2682 23.5406 56.7599 14196.0771\n 2.869990306225 10.430157883555 aE~ 146 1.0106 25.3205 58.5398 14105.2486\n 2.243604157981 10.791205018491 aP~* 410 0.9010 26.4549 59.6742 14059.8957\n 2.079911973823 10.385566107002 abP-* 3638 0.7531 29.4951 62.7144 13989.8948\n 1.359007430411 9.359711248271 Pba-* 3646 0.3941 28.8843 62.1036 13725.6487\n 1.359007430411 6.218118507259 Pba<-* 43012 0.3941 32.4446 65.6639 13729.2091\n 1.102096849611 9.004674070639 PbaS-* 43120 0.2523 32.1460 65.3652 13547.2486\n 1.102096849611 5.863081329627 PbaS<-* 538678 0.2523 35.7890 69.0082 13550.8916\n 1.053794231615 8.766171212717 PbaSS-* 539650 0.2862 35.7269 68.9462 13602.3312\n 1.036134804840 9.053697957728 PbSaS-* 759952 0.2462 36.1964 69.4157 13541.3814\n 0.993000472853 2.465889707104 Pa-b>>* 1984228 0.2609 37.5197 70.7389 13566.3452\n 0.993000472853 5.607482448117 Pa-b>*a- 28587463 0.2609 41.3684 74.5877 13570.1939\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_squared/varsWithNoise.txt_train...\n /bin/cp -p results/mystery_world_squared/varsWithNoise.txt_train mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 12.227574326633345 at j= 46902\n Searching for best fit...\n 5.966137056775 3.892157683905 P 1 3.4102 19.1864 52.4057 14594.3409\n 4.233376921373 6.181331839553 b> 6 1.3219 21.2764 54.4957 14210.1956\n 3.945693275809 12.230407899065 aC 23 1.7777 23.1135 56.3328 14333.0305\n 3.689447609929 2.968046351584 bP+ 33 1.9711 23.5374 56.7567 14375.7030\n 3.682900221580 8.693830830291 b>R 177 1.7953 25.9581 59.1774 14339.9981\n 3.649497253948 7.037173981318 bSS> 1434 1.5358 28.9632 62.1824 14279.3193\n 1.350388606762 2.983636359297 bPa-+ 3534 0.3950 28.8301 62.0494 13726.4638\n 1.350388606762 -2.983636359297 abP+- 3692 0.3950 28.8932 62.1125 13726.5269\n 1.338545682217 2.128580083165 Pba-E+ 46333 0.3885 32.5301 65.7493 13723.4307\n 1.263306348441 -3.090409648009 ab>>>- 70193 0.3217 33.0459 66.2652 13647.0091\n 0.988003346522 9.017726340153 ba-P/E 147375 0.2211 33.7614 66.9806 13495.1408\n 0.988003346522 12.397590149066 ba>-P/E 1698210 0.2211 37.2878 70.5071 13498.6672\n 0.988003346522 0.389691254921 Pba-P/+E 9084907 0.2211 39.7073 72.9265 13501.0867\n 0.986057313719 12.728683480783 baS-ER>L 27284778 0.2420 41.2910 74.5103 13539.5397\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418117324017906, 8.91505505199440e-11]\n [2.0, 27.146468378239973, '3.14159265358979']\n [11.0, 26.922437814458732, '-x0 + x1 + 3']\n [36.341478103541235, 24.43618326197655, '(x0**3 - 2*x0**2 - 2*x0 - 2*x1**3 + 3*x1**2 + 2*x1 + 9)**0.5']\n [44.9284350278788, 23.674076369594914, 0.334816306829453]\n [46.82642955276403, 23.51926784279126, 1.24784282220623]\n [46.826504901765, 23.518700609650345, 1.24790799617767]\n [96.62956053594237, 23.472287322878472, 0.343946248292923*cos(x1)**0.13920846581459]\n [125.91974286698664, 23.248650485468357, -0.913749694824219*tan(0.019306443631649*x0 + 2.193248927895)]\n [145.85465595328083, 22.80832447440447, 1.09555152691588*exp(-0.154938563704491*x0 + 0.15204918384552*x1)]\n [154.37021179561177, 22.372871941531574, tan(0.0405293171426792*x1*exp(-1.0023330450058*x0) + 0.882259964942932)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_tan/varsWithNoise.txt_train...\n /bin/cp -p results/mystery_world_tan/varsWithNoise.txt_train mystery.dat\n Number of variables..... 2\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 3.6288300847609377E-006 at j= 15150\n Searching for best fit...\n 0.743469672854 -3.141589112182 P 1 0.2624 16.1820 49.4013 13547.8917\n 0.743469672854 -4.141589112182 P> 4 0.2624 18.1820 51.4013 13549.8917\n 0.578138373028 -0.470373745551 bS 21 0.2306 20.2114 53.4307 13499.5317\n 0.408704637896 -0.983222167393 aC 23 0.1592 19.8423 53.0616 13348.4506\n 0.408704637896 -4.124814908406 PaC+ 367 0.1592 23.8384 57.0577 13352.4467\n 0.403259051090 0.093599102637 ab<* 407 0.0884 23.9683 57.1876 13112.2911\n 0.388858932821 -0.413812832623 ba>/ 558 0.1412 24.3711 57.5904 13304.1937\n 0.363635121939 0.089589812456 ab-E~ 122033 0.0725 31.5368 64.7561 13039.6902\n 0.218780872231 -0.143020351192 ba-PR 163215 0.0443 31.5547 64.7740 12839.4147\n 0.183227441682 -0.153145929873 ba-ERL 173367 0.0386 31.5648 64.7841 12783.5264\n 0.180317288007 -0.173386406160 baS-PR/ 1659375 0.0436 34.8005 68.0197 12835.8234\n 0.177888161591 -0.165490468057 ab-P>C* 1989992 0.0428 35.0430 68.2623 12828.5544\n 0.162424408908 -3.291177319432 PbPa-+*R 6895117 0.0382 36.7046 69.9239 12784.0138\n 0.161897478846 0.265030518107 baPR--L<~\\RPSCLE\n Arity 0 : Pab\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.74346604402369942 at j= 42311\n Searching for best fit...\n 1.114401192648 -0.236652585269 P 1 0.6331 16.7659 49.9852 13907.2234\n 0.713642975598 -0.763548058817 a 2 0.2894 17.1229 50.3422 13588.8384\n 0.661866178382 0.725737952749 b< 9 0.2718 19.1842 52.4035 13565.4148\n 0.509982486440 0.744861349803 ba- 54 0.1811 21.3930 54.6123 13402.1814\n 0.372181480709 0.745341108024 ab<* 407 0.0975 23.8526 57.0719 13152.6591\n 0.365006350631 -0.377044477766 aab-+ 3545 0.0978 26.9472 60.1665 13156.9765\n 0.291039792199 -0.484534199665 ab>C+ 4394 0.1327 26.9302 60.1495 13281.6549\n 0.254022682629 -0.484611600953 abEC+ 4466 0.0860 26.7574 59.9767 13104.7497\n 0.223498393644 0.275707446904 bbaE-+ 42231 0.0670 29.8140 63.0333 13006.0433\n 0.201127702913 0.564756422175 baP\\+- 43542 0.0532 29.7059 62.9252 12911.7281\n 0.201127702913 -0.179767547462 Pab-*> 49471 0.0532 29.8901 63.1094 12911.9123\n 0.199672618104 0.567060132263 baP\\S+- 543960 0.0522 33.3385 66.5578 12908.3340\n 0.197797223493 0.599026937195 baaP+\\+- 6803892 0.0496 36.9697 70.1889 12890.4240\n 0.195042476030 0.563688323104 babP+\\+- 6803901 0.0492 36.9494 70.1687 12887.7286\n 0.193030722533 -1.546593096927 bPa>-+RC 8860155 0.0455 37.3154 70.5347 12856.1324\n 0.193030722533 -1.546593096927 baP<--RC 8862120 0.0455 37.3158 70.5350 12856.1328\n 0.191099965131 -0.587216192697 aPb>>R*C- 100700555 0.0459 40.8075 74.0268 12863.4375\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418117324017906, 8.91505505199440e-11]\n [2.0, 27.146468378239973, '3.14159265358979']\n [11.0, 26.922437814458732, '-x0 + x1 + 3']\n [36.341478103541235, 24.43618326197655, '(x0**3 - 2*x0**2 - 2*x0 - 2*x1**3 + 3*x1**2 + 2*x1 + 9)**0.5']\n [44.9284350278788, 23.674076369594914, 0.334816306829453]\n [46.82642955276403, 23.51926784279126, 1.24784282220623]\n [46.826504901765, 23.518700609650345, 1.24790799617767]\n [96.62956053594237, 23.472287322878472, 0.343946248292923*cos(x1)**0.13920846581459]\n [125.91974286698664, 23.248650485468357, -0.913749694824219*tan(0.019306443631649*x0 + 2.193248927895)]\n [145.85465595328083, 22.80832447440447, 1.09555152691588*exp(-0.154938563704491*x0 + 0.15204918384552*x1)]\n [154.37021179561177, 22.372871941531574, tan(0.0405293171426792*x1*exp(-1.0023330450058*x0) + 0.882259964942932)]\n Checking for symmetry \n varsWithNoise.txt_train\n Training a NN on the data... \n \n tensor(0.0194, device='cuda:0', grad_fn=)\n tensor(0.0154, device='cuda:0', grad_fn=)\n tensor(0.0148, device='cuda:0', grad_fn=)\n tensor(0.0123, device='cuda:0', grad_fn=)\n NN loss: tensor(0.0112, device='cuda:0', grad_fn=) \n \n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/translated_data_minus/varsWithNoise.txt_train-translated_minus...\n /bin/cp -p results/translated_data_minus/varsWithNoise.txt_train-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 5.1292288673140229E-002 at j= 1\n Searching for best fit...\n 0.669343884800 -0.314141760993 P 1 0.2677 16.0305 49.2497 13555.9222\n 0.669343884800 5.969043721032 P~ 7 0.2677 18.8378 52.0571 13558.7296\n 0.459910613898 2.691642525762 a>C 146 0.2337 22.6789 55.8982 13507.6853\n 0.270920748383 2.965777915163 aP~/ 316 0.0849 23.0294 56.2486 13095.4937\n 0.265087597119 4.070147271604 aER~ 1044 0.0659 24.7221 57.9414 12993.7628\n 0.253726663593 2.589232464793 a>E\\ 1262 0.0667 24.9325 58.1518 12999.1543\n 0.238690135465 2.613988473050 aEE\\ 1278 0.0706 24.8625 58.0818 13022.2179\n 0.232631950725 3.062248443653 aP>C* 2758 0.0747 25.9352 59.1544 13046.3768\n 0.177671463957 3.005146130967 aPEC* 2790 0.0473 25.5630 58.7823 12860.2626\n 0.175419138648 3.014445859424 aPP*S* 24852 0.0440 28.6996 61.9189 12833.3455\n 0.174839352929 3.016839763062 aPR\\<* 31216 0.0437 29.0237 62.2430 12830.9690\n 0.174485011551 3.018302819299 aPPL*S* 285054 0.0436 32.2117 65.4310 12833.6496\n 0.172930867417 3.435955926410 aPC 803902 0.0477 33.6946 66.9138 12871.3925\n 0.171148232379 1.245013547196 PaP\\<~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 3.4967948648202607 at j= 46902\n Searching for best fit...\n 0.994508456381 1.113064344455 P 1 0.5366 16.6017 49.8210 13839.7024\n 0.994508456381 0.844311617169 P> 3 0.5366 18.1867 51.4059 13841.2873\n 0.994508456381 0.151110194263 PE 19 0.5366 20.8496 54.0689 13843.9503\n 0.859086674464 -0.853248896386 aP- 30 0.3555 21.2974 54.5167 13676.6046\n 0.508060293243 -0.685886515879 aP>- 272 0.1830 23.7202 56.9395 13408.8048\n 0.223745235292 1.727320572244 Pa-R 417 0.0653 23.1535 56.3728 12989.1086\n 0.197052932104 0.482995742381 PPa-+ 2109 0.0490 25.3086 58.5279 12874.0850\n 0.174143378775 -0.511085601315 aPRE- 3170 0.0371 25.7183 58.9375 12761.0561\n 0.169535948445 -0.516734863269 aPER>- 33860 0.0371 29.0966 62.3159 12764.5738\n 0.165302478848 3.002952936720 aP~/ER 58784 0.0369 29.8560 63.0752 12763.2214\n 0.162545889691 2.487667754459 aEE\\ER 216202 0.0389 31.7106 64.9299 12786.3510\n 0.161146795086 -2.998877991894 aP/>R<< 915434 0.0361 33.7802 66.9995 12758.6117\n 0.160042349137 3.345490215244 aP+RC>R 949470 0.0377 33.8229 67.0422 12775.8535\n 0.159301940682 -0.723120267495 aPaP/E+- 3237572 0.0356 35.5860 68.8052 12754.2572\n 0.159301940682 0.723120267495 Pa-aP/E+ 10942265 0.0356 37.3429 70.5622 12756.0141\n Checking polyfit \n \n Complexity RMSE Expression\n [14.344443782109636, 31.58188268017816, -2.08021320557046e-10]\n [19.042599881712917, 24.230691744700355, '3 - 0.49*x0']\n [98.31044580243638, 24.23055490040669, 3.00018795594202 - 0.485155373811722*x0]\n [98.31184366871774, 24.23050552122283, 3.00019288063049 - 0.485624884663866*x0]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_atan/varsWithNoise.txt_train-translated_minus...\n /bin/cp -p results/mystery_world_atan/varsWithNoise.txt_train-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 5.1292288673140229E-002 at j= 1\n Searching for best fit...\n 0.061403642294 -1.910741817479 P 1 0.0267 12.5841 45.8034 12616.0574\n 0.061403642294 4.372443664546 P~ 7 0.0267 15.3915 48.6107 12618.8648\n 0.061403642020 666.000000000000 PS\\ 101 0.0267 19.2423 52.4616 12622.7156\n 0.061403520672 1.230850961524 aPS* 256 0.0267 20.5841 53.8034 12624.0566\n 0.059394932091 -666.000000000000 PPC>/ 3213 0.0208 24.1858 57.4051 12524.4768\n 0.023778699302 1.249630260186 aPE~/ 3294 0.0049 22.9011 56.1204 11931.8777\n 0.023392270441 -1.891783909921 PEa-L 8643 0.0048 24.2691 57.4884 11929.0764\n 0.021507143688 1.252580244039 aPPE-/ 24316 0.0061 25.6402 58.8595 12029.1919\n 0.021429498108 1.252681079434 aP\\C<* 31320 0.0062 26.0002 59.2195 12035.2588\n 0.021214984991 1.054976863305 PaE>+\\ 43437 0.0058 26.4575 59.6768 12010.5366\n 0.020729551543 1.459374782792 aPER-\\ 44228 0.0046 26.4501 59.6694 11916.1526\n 0.020366149552 0.467286694116 aP>-\\E 63204 0.0050 26.9397 60.1590 11943.8517\n 0.020330233684 1.296687801067 aSP-E~ 108574 0.0055 27.7177 60.9370 11985.5845\n 0.020179998070 0.811477509702 PaE>+R\\ 636783 0.0049 30.2592 63.4784 11946.7323\n 0.020056898558 0.989711026987 Pa<R 12384729 0.0052 34.5236 67.7429 11972.6284\n 0.019855018308 0.751145045213 aSP-E<\\C 19259682 0.0056 35.1544 68.3736 12005.5627\n 0.019843753573 0.607314984877 a>P/C\\SC 19285340 0.0057 35.1555 68.3748 12007.9850\n 0.019642673047 0.194581299095 a<\\<<\\C\\ 33856324 0.0052 35.9527 69.1720 11972.1090\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_atan/varsWithNoise.txt_train-translated_minus...\n /bin/cp -p results/mystery_world_atan/varsWithNoise.txt_train-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 1.2922545658267337 at j= 46902\n Searching for best fit...\n 0.101649497918 0.411337392322 P 1 0.0490 13.3113 46.5306 12862.9139\n 0.101649497918 1.128872889917 PL 17 0.0490 17.3988 50.6181 12867.0014\n 0.023497889594 -0.053626498392 aPE- 304 0.0081 19.4462 52.6655 12135.0891\n 0.020670724269 -0.051489755848 aPE>- 2898 0.0062 22.5142 55.7335 12032.1735\n 0.018044547281 0.053938461502 PEaS- 7095 0.0039 23.6099 56.8292 11847.7947\n 0.017767722978 -0.047441499111 aPPE+- 23884 0.0040 25.3388 58.5581 11853.0628\n 0.017751970170 1.083362435385 Pa-RRR 73285 0.0037 26.9550 60.1743 11827.4345\n 0.017627960689 28.847575908441 PEaS+\\ 96033 0.0042 27.3349 60.5542 11873.6527\n 0.017178522978 0.703667516762 PaP>/-R 296663 0.0038 28.9249 62.1441 11842.5908\n 0.016126141473 0.787800699319 PPa-+RR 329553 0.0038 28.9853 62.2046 11836.4941\n 0.015912637986 0.397091721958 PaERRR<- 5767085 0.0037 33.0954 66.3146 11834.1839\n 0.015793663911 1.483640629140 aP/E>RR\\ 12322972 0.0037 34.1800 67.3993 11832.9802\n 0.015778242437 0.313325170094 PPPa-+*L> 40270713 0.0037 35.8869 69.1062 11830.4525\n 0.015685527205 1.046950488041 PaP<~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 5.1292288673140229E-002 at j= 1\n Searching for best fit...\n 0.148552080248 -4.092654693322 P 1 0.0417 13.8587 47.0780 12797.3019\n 0.148552080248 -2.723515827877 PR 11 0.0417 17.3181 50.5374 12800.7613\n 0.148551376718 -666.000000000000 PEE 197 0.0417 21.4807 54.7000 12804.9270\n 0.127429417720 -0.969841288963 aPE/ 340 0.0267 22.0468 55.2661 12623.1134\n 0.113379195617 -1.017796944327 aP-E 480 0.0222 22.3758 55.5950 12548.4998\n 0.100055943021 -1.895058535125 aC>S 1454 0.0316 23.7943 57.0136 12693.9522\n 0.096547909917 -1.147163170755 aCCP/ 9918 0.0328 26.5129 59.7321 12712.0397\n 0.096216795975 -1.837008549112 aCCRR 16722 0.0322 27.2615 60.4808 12705.3328\n 0.095967200785 -1.166349317490 aR 2642448 0.0167 34.0969 67.3162 12446.3135\n 0.066940940228 -1.736052288211 a>C>SCR 2718824 0.0154 34.0832 67.3025 12413.1530\n 0.064358853930 -1.675241168573 aES*C\\S 9031066 0.0157 35.7320 68.9512 12422.7129\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_cos/varsWithNoise.txt_train-translated_minus...\n /bin/cp -p results/mystery_world_cos/varsWithNoise.txt_train-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.99999999999341582 at j= 15150\n Searching for best fit...\n 0.197490127932 -0.318309877324 P 1 0.0481 14.2695 47.4888 12855.8856\n 0.197490127932 -0.241453001907 P> 3 0.0481 15.8545 49.0737 12857.4706\n 0.197490127932 -3.141592740992 P\\ 9 0.0481 17.4394 50.6587 12859.0555\n 0.131456462628 -0.244197335734 PaC+ 225 0.0362 21.4961 54.7154 12747.1098\n 0.124796549174 -0.338586384651 aC>> 644 0.0356 22.9382 56.1575 12741.5845\n 0.123659743970 -0.505957463999 aCR> 716 0.0356 23.0779 56.2972 12741.9708\n 0.122253352479 -0.511539668039 aSC> 750 0.0375 23.1283 56.3476 12762.9825\n 0.117023386263 -0.323096685524 PaC<+ 2279 0.0355 24.6687 57.8880 12742.4492\n 0.109801051734 -0.243497591291 PaP/C 8502 0.0214 26.1439 59.3632 12537.9186\n 0.081044940664 -0.241893573154 Pa>>S+ 29473 0.0160 27.8316 61.0509 12420.1599\n 0.066930373485 -0.195816697190 Pa>S+ 390855 0.0147 31.0931 64.3124 12389.9962\n 0.056093596847 -1.000284624375 aSE\\S<~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 5.1292288673140229E-002 at j= 1\n Searching for best fit...\n 16.107160861598 13.760728770157 P 1 5.5401 20.6193 53.8386 14792.3435\n 14.751702698722 17.336888406501 a~ 8 4.9501 23.4925 56.7117 14749.3933\n 14.311810398483 16.254776450027 aE\\ 108 4.9149 27.2037 60.4229 14750.2353\n 14.177267658467 17.148040019983 aa<* 242 4.9466 28.3540 61.5733 14754.0235\n 11.950011476847 18.267553715026 aP~* 244 3.6984 28.1193 61.3386 14635.3719\n 10.899117956745 14.867998647604 PaE/ 341 3.5881 28.4694 61.6887 14623.5055\n 9.221661582828 14.991477450271 aE\\E 1998 3.4067 30.7790 63.9983 14604.8799\n 5.799554354533 11.372458637921 Pa-E 4739 0.9383 31.3401 64.5594 14079.9643\n 5.017940568332 12.117574727798 a<CEE 23190 1.2449 33.3306 66.5499 14197.6307\n 4.303093734878 21.191325092601 aPP~** 23620 1.1047 33.2427 66.4620 14148.8946\n 4.233456994488 11.297022953188 PaS\\+E 575547 0.9351 37.7673 70.9866 14085.4883\n 3.895198674933 -7.542341784392 PEaERS/ 1143263 0.9656 38.6961 71.9153 14099.5777\n 3.711828829483 1.640486781771 PREa-ER 1798383 0.8911 39.2800 72.4993 14067.4577\n 3.627202003175 20.404733402503 aPa<<<** 3299156 0.8833 40.1222 73.3414 14064.7553\n 3.582798243113 10.343765664584 PPaSSE/* 3315675 0.9314 40.1116 73.3309 14086.4109\n 3.559297754275 11.559423268934 aE>>SE>E 38533386 0.9522 43.6408 76.8601 14098.9788\n 3.551772063266 20.423231347470 aPaS<<<** 41029950 0.8637 43.7284 76.9476 14059.2428\n 3.551772063266 20.423231347470 PaaS<<<** 41029951 0.8637 43.7284 76.9476 14059.2428\n 3.547110915595 14.097479473099 Pa<~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 33.009482372768282 at j= 46902\n Searching for best fit...\n 20.799102387793 10.507244284671 P 1 13.1765 20.9881 54.2074 15145.8939\n 15.392358609107 -16.870672333195 a< 6 4.7201 23.1387 56.3580 14729.5685\n 8.160075507151 -8.054605858698 aP- 30 4.9304 24.5451 57.7644 14749.6782\n 5.623620549392 -11.164603724310 a<< 58 2.0824 24.9591 58.1784 14398.9202\n 5.376880513904 -12.095492045912 aPR- 288 1.5620 27.2063 60.4256 14283.9043\n 4.245888558127 20.460275712354 aER\\ 1206 0.8507 28.9317 62.1510 14038.0232\n 4.223432393308 39.042032380720 aS>>\\ 14506 0.8675 32.5124 65.7317 14049.5900\n 4.223432393308 12.427464537665 PaS>>/ 36621 0.8675 33.8484 67.0677 14050.9261\n 4.210436306156 7.330539097250 aSER\\E 261556 0.8531 36.6804 69.8997 14046.8938\n 4.201904530459 -9.553778444593 aaCCRE- 453608 0.9249 37.4718 70.6911 14080.6983\n 4.197497756487 10.750858981267 a<< 2272750 0.9805 39.7952 73.0145 14106.8307\n 4.195775288622 -9.982856624081 aSS>L<< 2301828 0.8723 39.8129 73.0322 14059.1459\n 4.184605655775 11.527201904516 PPaS+/E< 3774747 0.8256 40.5227 73.7420 14037.4005\n 4.176561474437 7.200058010244 aaER\\+\\E 8114142 0.8501 41.6240 74.8432 14050.4100\n Checking polyfit \n \n Complexity RMSE Expression\n [2.0, 27.146468378239973, '3.14159265358979']\n [19.042599881712917, 24.230691744700355, '3 - 0.49*x0']\n [46.82642955276403, 23.51926784279126, 1.24784282220623]\n [46.826504901765, 23.518700609650345, 1.24790799617767]\n [99.75565347222093, 21.062376668285015, -tan(0.0191285926138826*x0 - 0.895119249820709)]\n [155.1306094353196, 21.048960435018625, -tan(0.685993492603302*exp(x0)**0.0283585358411074 + 1.56043832990099)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_inverse/varsWithNoise.txt_train-translated_minus...\n /bin/cp -p results/mystery_world_inverse/varsWithNoise.txt_train-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 5.1292288673140229E-002 at j= 1\n Searching for best fit...\n 0.067699281586 -2.787917290203 P 1 0.0298 12.7249 45.9442 12659.5290\n 0.067699281586 -0.646324549191 aa\\* 250 0.0298 20.6907 53.9100 12667.4947\n 0.067699278181 666.000000000000 PPS/ 327 0.0298 21.0781 54.2974 12667.8821\n 0.029173288355 0.334896114156 aPE/ 340 0.0064 19.9198 53.1391 12040.8831\n 0.028520168720 0.334047932247 aPE<~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.39963450891453223 at j= 42311\n Searching for best fit...\n 0.113658339692 0.127207611508 P 1 0.0688 13.4724 46.6917 13001.4090\n 0.093558984794 0.096536615352 aP+ 22 0.0363 17.6511 50.8704 12745.6241\n 0.054109534419 0.077754147676 aP>+ 200 0.0166 20.0455 53.2648 12429.5765\n 0.030334035255 0.196416299906 aP+R 408 0.0134 20.2391 53.4584 12342.4032\n 0.027162533353 0.054884956963 aPP++ 2090 0.0109 22.4367 55.6560 12259.8845\n 0.027162533353 0.054884956963 PaP++ 2091 0.0109 22.4374 55.6567 12259.8852\n 0.024904146253 0.058057684837 aPRE+ 2522 0.0082 22.5825 55.8018 12144.5703\n 0.020709640901 0.168339267261 aP/E> 5078 0.0051 23.3261 56.5454 11954.6524\n 0.020652663952 0.168857630372 aP/SE> 68696 0.0058 27.0801 60.2993 12010.9794\n 0.020205028303 0.140693670103 a> 152360 0.0050 28.1976 61.4169 11950.3436\n 0.020205028303 0.044784184871 Pa>* 5493421 0.0050 33.3698 66.5891 11955.5158\n 0.020205026478 -0.140693667756 Pa<~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 5.1292288673140229E-002 at j= 1\n Searching for best fit...\n 0.212471204034 -2.102217149018 P 1 0.0891 14.3750 47.5943 13106.9860\n 0.212471204034 2.039375591995 PC 15 0.0891 18.2819 51.5012 13110.8929\n 0.212471204006 666.000000000000 PS\\ 101 0.0891 21.0332 54.2525 13113.6443\n 0.212471082413 1.039375629986 aPS* 256 0.0891 22.3750 55.5943 13114.9858\n 0.092408774014 0.759746016783 aP+\\ 392 0.0307 21.7885 55.0078 12681.4473\n 0.084548555711 -0.243318518185 Pa-RR 5587 0.0279 25.4934 58.7127 12645.2760\n 0.068205404468 0.041452469551 Pa-LR 5635 0.0160 25.1959 58.4152 12419.2103\n 0.067904817524 1.102270421243 aaPL*- 23950 0.0152 27.2770 60.4963 12400.7170\n 0.064523137515 1.108539059566 aPP+~/ 25324 0.0171 27.2838 60.5031 12448.6548\n 0.063232502270 -0.103660711817 Pa>-RR 58165 0.0152 28.4543 61.6736 12399.9896\n 0.063071223718 0.192285121159 a>>RRRC 236632 0.0169 30.4524 63.6717 12446.3310\n 0.061625014322 0.488492782934 a<<\\SE 264176 0.0166 30.6004 63.8197 12440.0498\n 0.061625014322 -0.511507217066 a<<\\SE> 2273074 0.0166 33.7055 66.9248 12443.1549\n 0.061599495227 1.888865580017 Pa>-\\>>C 10616361 0.0167 35.9285 69.1478 12446.7741\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_log/varsWithNoise.txt_train-translated_minus...\n /bin/cp -p results/mystery_world_log/varsWithNoise.txt_train-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 1.2518467960290158 at j= 46902\n Searching for best fit...\n 0.334641918738 0.398475200075 P 1 0.1697 15.0303 48.2496 13370.0857\n 0.331284813751 -0.305461698055 aP- 30 0.1494 19.9227 53.1420 13322.7562\n 0.205617948052 -0.245546241783 aP>- 272 0.0873 22.4151 55.6344 13106.7364\n 0.067760026563 0.618377916827 Pa-R 417 0.0131 21.4301 54.6494 12333.9068\n 0.061469117252 0.172911679400 PPa-+ 2109 0.0161 23.6280 56.8473 12419.9004\n 0.061469117252 -0.543218076841 aP/<< 5110 0.0161 24.9048 58.1240 12421.1771\n 0.060211772173 0.726785019005 aE>\\> 10624 0.0145 25.9309 59.1502 12379.8599\n 0.052270945795 -0.162902645552 aPP<*- 23900 0.0119 26.8965 60.1158 12301.7273\n 0.052270945795 0.511773768757 PaP/-< 25765 0.0119 27.0049 60.2242 12301.8357\n 0.052252521988 -0.163245096102 aPEL>\\ 2524890 0.0121 33.6161 66.8354 12314.9184\n 0.052144299627 0.590668300430 PaER>L>/ 5984489 0.0121 34.8611 68.0804 12316.1634\n 0.052132197079 0.959904212216 PaPRE/>>- 43785739 0.0120 37.7319 70.9512 12312.8481\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418117324403838, '1/(666.000000000000+(pi/sin(pi)))']\n [2.0, 27.146468378239973, '3.14159265358979']\n [11.702605602110248, 26.849940214202466, '3.03030303030303']\n [19.042599881712917, 24.230691744700355, '3 - 0.49*x0']\n [44.9284350278788, 23.674076369594914, 0.334816306829453]\n [46.82642955276403, 23.51926784279126, 1.24784282220623]\n [46.826504901765, 23.518700609650345, 1.24790799617767]\n [98.26824164670695, 22.852143124080726, 1.0950038433075*exp(-0.136154734619017*x0)]\n [98.29622998964908, 22.814593299863635, 1.09502410888672*exp(-0.138819361079758*x0)]\n [98.35887925287884, 22.777572958170047, 1.0945246219635*exp(-0.145046580263127*x0)]\n [99.75565347222093, 21.062376668285015, -tan(0.0191285926138826*x0 - 0.895119249820709)]\n [155.1306094353196, 21.048960435018625, -tan(0.685993492603302*exp(x0)**0.0283585358411074 + 1.56043832990099)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_sin/varsWithNoise.txt_train-translated_minus...\n /bin/cp -p results/mystery_world_sin/varsWithNoise.txt_train-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 3.6288300847370446E-006 at j= 15150\n Searching for best fit...\n 0.596642468702 -3.141596369843 P 1 0.2417 15.8646 49.0839 13514.3668\n 0.596642468702 -4.141596369843 P> 3 0.2417 17.4495 50.6688 13515.9517\n 0.242486575296 0.097494405657 aP/ 34 0.0829 19.6531 52.8724 13082.9550\n 0.158914041482 0.143020351192 aP/ 3218 0.0345 25.6067 58.8260 12731.1960\n 0.158450789889 0.142533236618 aPC>* 30984 0.0346 28.8635 62.0828 12735.6067\n 0.157600962025 0.140851734824 aPRSES* 419866 0.0346 32.6236 65.8428 12739.2909\n 0.157576662844 0.140896771626 aP\\>+R/ 3442930 0.0370 35.6587 68.8779 12770.0328\n 0.157542525615 0.139290983661 aPa>>+R/ 3442932 0.0370 35.6587 68.8779 12770.0328\n Checking for brute force * \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_sin/varsWithNoise.txt_train-translated_minus...\n /bin/cp -p results/mystery_world_sin/varsWithNoise.txt_train-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.59663883987170496 at j= 42311\n Searching for best fit...\n 0.944418800650 0.189916035927 P 1 0.5002 16.5271 49.7464 13811.0655\n 0.361876128870 0.597758586534 a 2 0.1514 16.1432 49.3625 13324.3382\n 0.288659591838 0.348260986733 aE< 72 0.1267 20.9870 54.2063 13256.9796\n 0.234639424117 0.199128290313 aa>+ 202 0.0765 22.1764 55.3957 13052.5243\n 0.161696095509 0.453222604106 aP\\+ 212 0.0355 21.7089 54.9282 12739.6501\n 0.159643670368 0.455071354194 aP\\S+ 2410 0.0351 25.1974 58.4167 12737.9933\n 0.157677054752 0.456842810175 aP\\SS+ 29664 0.0348 28.8011 62.0204 12737.8118\n 0.157651939982 0.456865432649 aPCCRL- 450690 0.0348 32.7262 65.9455 12741.6961\n 0.157643772901 0.441219473371 aPaP/-\\+ 3242564 0.0349 35.5731 68.7924 12746.2615\n 0.157592637622 0.456918850063 aPPERC*+ 3279336 0.0347 35.5889 68.8082 12744.4633\n 0.157489231529 0.376565711926 aaP<~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 5.1292288673140229E-002 at j= 1\n Searching for best fit...\n 0.188469292878 -1.460090145112 P 1 0.0772 14.2020 47.4213 13048.4285\n 0.188469292878 2.522973580708 PCS 139 0.0772 21.3210 54.5403 13055.5474\n 0.188468592880 -666.000000000000 PEE 197 0.0772 21.8241 55.0434 13056.0482\n 0.071092232713 1.401873020689 aP+\\ 392 0.0201 21.4102 54.6295 12508.5033\n 0.070889079548 2.050911702797 aP-\\ 400 0.0190 21.4352 54.6545 12485.3245\n 0.060771717718 0.398808485720 Pa-RR 5587 0.0175 25.0170 58.2363 12454.6143\n 0.059776154042 -3.083592131434 PEa-R 8211 0.0187 25.5487 58.7680 12482.7609\n 0.054145600423 1.235610745206 aER>\\ 14584 0.0130 26.2347 59.4540 12335.1575\n 0.052592105180 1.368183448821 PaP-/E 26685 0.0141 27.0644 60.2837 12368.4523\n 0.052245313841 1.034067004587 PaE+L\\ 57261 0.0144 28.1564 61.3756 12378.8610\n 0.051696519412 2.348928873981 PaP>>-/ 279599 0.0137 30.4289 63.6481 12362.5053\n 0.051672744506 3.005788592790 aR~ 2449780 0.0138 33.5594 66.7787 12367.7492\n 0.051599115709 0.261824504390 PPa-*\\E 12831641 0.0137 35.9447 69.1639 12366.2955\n 0.051412785328 -0.402489795025 aP-\\E<~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 1.8699718887780803 at j= 46902\n Searching for best fit...\n 0.288110198099 0.595230522520 P 1 0.1508 14.8143 48.0336 13321.7701\n 0.288110198099 1.055018646496 PR 11 0.1508 18.2738 51.4930 13325.2296\n 0.149563518015 -0.077600843626 aPE- 304 0.0704 22.1164 55.3357 13018.9599\n 0.094264074791 -0.172726137283 aPP*- 2162 0.0312 24.2806 57.4999 12690.2522\n 0.085313381192 1.282848668314 aP+\\> 4986 0.0357 25.3422 58.5615 12745.6132\n 0.064411136294 1.314275683502 Pa-RR 5587 0.0188 25.1010 58.3202 12485.3891\n 0.047710214647 -0.074637857299 aaPE-+ 23454 0.0114 26.7376 59.9569 12284.1772\n 0.046868150455 -0.148278689777 aPP>- 307846 0.0104 30.4124 63.6316 12247.4816\n 0.045977718363 -0.147224396164 aPE- 452218 0.0107 30.9534 64.1727 12259.9969\n 0.045726968288 -0.864709105519 aPRE/<< 610810 0.0106 31.3792 64.5985 12257.3435\n 0.045642628554 -0.146745367449 aPPE- 3626032 0.0105 33.9461 67.1654 12258.1696\n 0.045625421620 2.378178795398 a>>RR>SS 35829428 0.0103 37.2503 70.4695 12251.2822\n 0.045566639380 0.479172104054 aER>\\ERE 39002934 0.0106 37.3708 70.5901 12263.6649\n 0.045510463225 1.730617376042 PaP+/L>RR 55463219 0.0104 37.8770 71.0963 12254.7170\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418117324403838, '1/(666.000000000000+(pi/sin(pi)))']\n [2.0, 27.146468378239973, '3.14159265358979']\n [11.702605602110248, 26.849940214202466, '3.03030303030303']\n [19.042599881712917, 24.230691744700355, '3 - 0.49*x0']\n [44.9284350278788, 23.674076369594914, 0.334816306829453]\n [46.82642955276403, 23.51926784279126, 1.24784282220623]\n [46.826504901765, 23.518700609650345, 1.24790799617767]\n [98.26824164670695, 22.852143124080726, 1.0950038433075*exp(-0.136154734619017*x0)]\n [98.29622998964908, 22.814593299863635, 1.09502410888672*exp(-0.138819361079758*x0)]\n [98.35887925287884, 22.777572958170047, 1.0945246219635*exp(-0.145046580263127*x0)]\n [99.75565347222093, 21.062376668285015, -tan(0.0191285926138826*x0 - 0.895119249820709)]\n [155.1306094353196, 21.048960435018625, -tan(0.685993492603302*exp(x0)**0.0283585358411074 + 1.56043832990099)]\n Checking for brute force + \n \n Trying to solve mysteries with brute force...\n Trying to solve results/mystery_world_squared/varsWithNoise.txt_train-translated_minus...\n /bin/cp -p results/mystery_world_squared/varsWithNoise.txt_train-translated_minus mystery.dat\n Number of variables..... 1\n Functions used.......... +*-/><~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 5.1292288673140229E-002 at j= 1\n Searching for best fit...\n 4.233095282218 4.852886303402 P 1 1.6142 18.6914 51.9106 14289.1513\n 2.935568553091 8.429045939746 a~ 8 1.0258 21.1633 54.3826 14107.1610\n 2.616667331284 7.346933983272 aE\\ 108 1.0008 24.7523 57.9715 14100.8254\n 2.516623178644 8.240197553228 aa<* 242 1.0485 25.8600 59.0793 14120.9955\n 1.359007430411 9.359711248271 aP~* 244 0.3941 24.9829 58.2022 13721.7474\n 1.359007430411 -0.509893702112 PPa-* 2141 0.3941 28.1163 61.3355 13724.8807\n 1.314973538074 9.317144855926 PaS~* 2635 0.3502 28.3683 61.5875 13676.9814\n 1.180382402088 9.057245166683 aPEC/ 3438 0.2890 28.5962 61.8155 13598.9430\n 1.104402546750 4.123478846045 Pa-ER 5651 0.2960 29.2172 62.4365 13609.4666\n 1.017386096557 3.826436690816 Pa<~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 12.227574326633345 at j= 46902\n Searching for best fit...\n 5.966137056775 3.892157683905 P 1 3.4102 19.1864 52.4057 14594.3409\n 1.350388606762 -2.983636359297 aP- 30 0.3950 21.9499 55.1692 13719.5836\n 1.263306348441 -3.090409648009 a<<< 814 0.3217 26.6157 59.8350 13640.5790\n 1.235544262074 -3.278984855101 aPR>- 2882 0.2667 28.4076 61.6269 13565.9606\n 1.200981434330 3.475802078359 PaL< 922714 0.2206 36.4030 69.6223 13496.8069\n 0.976888278109 -4.490156312870 aPaP-/<+ 3242100 0.2185 38.2044 71.4237 13494.7378\n 0.973611035322 -39.050315323280 aP/ESRSL 12755260 0.2498 40.1756 73.3949 13551.3460\n 0.972681392236 -6.867437876960 a<~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 3.6288300847609377E-006 at j= 15150\n Searching for best fit...\n 0.743469672854 -3.141589112182 P 1 0.2624 16.1820 49.4013 13547.8917\n 0.743469672854 -4.141589112182 P> 3 0.2624 17.7669 50.9862 13549.4767\n 0.595531886372 -0.301528451806 aS~ 84 0.2343 22.2542 55.4735 13507.9172\n 0.378633814343 -0.097494405657 aP~/ 316 0.1032 23.5123 56.7316 13175.4931\n 0.347112537116 -1.237759359912 Pa-L 465 0.0992 23.9442 57.1635 13159.8738\n 0.262458050160 0.858005019470 aER~ 1044 0.0440 24.7077 57.9270 12829.2163\n 0.255293145169 0.270824107646 aC* 2758 0.0428 25.5481 58.7674 12819.0594\n 0.162424408908 -3.291177319432 PPa-*R 26111 0.0382 28.6599 61.8791 12775.9690\n 0.162424408908 -3.291177319432 Pa-RPR* 874599 0.0382 33.7257 66.9450 12781.0349\n 0.161897478846 0.265030518107 PRa-LPR~/>S 16378916 0.0401 37.9449 71.1642 12805.5844\n 0.161414546998 -0.148420597647 aaPER/<<~\\RPSCLE\n Arity 0 : Pa\n Arity 1 : ><~\\RSCLE\n Arity 2 : +*-/\n Loading mystery data....\n 80000 rows read from file mystery.dat \n Number of examples...... 80000\n Mystery data has largest magnitude 0.74346604402369942 at j= 42311\n Searching for best fit...\n 1.114401192648 -0.236652585269 P 1 0.6331 16.7659 49.9852 13907.2234\n 0.509982486440 -0.744861349803 a 2 0.1811 16.6382 49.8574 13397.4265\n 0.439416108323 -0.372081521432 a> 4 0.2340 17.4233 50.6426 13503.1343\n 0.302178861492 -0.433964738451 aE< 72 0.1247 21.0530 54.2723 13250.2655\n 0.260899560444 -0.248131888773 aa>+ 202 0.1065 22.3294 55.5487 13187.5691\n 0.201127702913 -0.564756422175 aP\\+ 212 0.0532 22.0237 55.2430 12904.0459\n 0.201127702913 -0.179767547462 aP*> 348 0.0532 22.7388 55.9580 12904.7609\n 0.199672618104 -0.567060132263 aP\\S+ 2410 0.0522 25.5202 58.7394 12900.5157\n 0.194031997062 -0.507602858239 aPa-\\+ 24512 0.0449 28.8252 62.0445 12842.1534\n 0.193299434535 -0.404368578929 aa-RC 60757 0.0455 30.1273 63.3466 12848.9443\n 0.192639941455 -0.260591399315 aaaCC++ 271012 0.0399 32.2816 65.5009 12796.8727\n 0.191149106089 -0.422823412741 aaEP/S+ 346840 0.0415 32.6263 65.8456 12813.8639\n 0.191001102333 0.758214579775 Pa>E>/L 571915 0.0416 33.3467 66.5660 12815.2347\n 0.191001102333 -0.758214579775 a>E>P/L 2040890 0.0416 35.1821 68.4013 12817.0700\n 0.190923145134 -1.028932975961 Paa<+-LC 3825117 0.0453 36.0878 69.3071 12853.0747\n 0.190746926749 -0.317217734409 aaS>\\+ 5334294 0.0469 36.5514 69.7707 12867.8830\n Checking polyfit \n \n Complexity RMSE Expression\n [0.0, 28.418117324403838, '1/(666.000000000000+(pi/sin(pi)))']\n [2.0, 27.146468378239973, '3.14159265358979']\n [11.702605602110248, 26.849940214202466, '3.03030303030303']\n [15.321928094887362, 24.326221263209245, '(9 - 3*x0)**0.5']\n [19.042599881712917, 24.230691744700355, '3 - 0.49*x0']\n [44.9284350278788, 23.674076369594914, 0.334816306829453]\n [46.82642955276403, 23.51926784279126, 1.24784282220623]\n [46.826504901765, 23.518700609650345, 1.24790799617767]\n [98.26824164670695, 22.852143124080726, 1.0950038433075*exp(-0.136154734619017*x0)]\n [98.29622998964908, 22.814593299863635, 1.09502410888672*exp(-0.138819361079758*x0)]\n [98.35887925287884, 22.777572958170047, 1.0945246219635*exp(-0.145046580263127*x0)]\n [99.75565347222093, 21.062376668285015, -tan(0.0191285926138826*x0 - 0.895119249820709)]\n [155.1306094353196, 21.048960435018625, -tan(0.685993492603302*exp(x0)**0.0283585358411074 + 1.56043832990099)]\n Checking for symmetry \n varsWithNoise.txt_train-translated_minus\n Just one variable!\n varsWithNoise.txt_train-translated_minus just one variable for ADD \n \n varsWithNoise.txt_train-translated_minus just one variable for ADD \n \n varsWithNoise.txt_train-translated_minus just one variable for ADD \n \n varsWithNoise.txt_train-translated_minus just one variable for ADD \n \n varsWithNoise.txt_train-translated_minus just one variable for ADD\n varsWithNoise.txt_train-translated_minus just one variable for ADD\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n set: Syntax Error.\n\n\n\n```\n\n```\n", "meta": {"hexsha": "7acd0564d117edefeb1745d92bf399184edbe83d", "size": 782696, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "AI_Feynman.ipynb", "max_stars_repo_name": "alexliberzonlab/AI-Feynman", "max_stars_repo_head_hexsha": "2844118d82cd700215e180ba197f153791406883", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 19, "max_stars_repo_stars_event_min_datetime": "2020-07-04T20:26:55.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-03T07:21:52.000Z", "max_issues_repo_path": "AI_Feynman.ipynb", "max_issues_repo_name": "rodristk/AI-Feynman", "max_issues_repo_head_hexsha": "dbc8b009d4f0e36b83f8d3794dca2e168487dc8b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, 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YES\n2. YES", "lm_q1_score": 0.5774953797290152, "lm_q2_score": 0.7772998611746912, "lm_q1q2_score": 0.44888707849238907}} {"text": "# TakagiTaupin - Calculate the Takagi-Taupin curves\n\nThis file demonstrates the usage of TakagiTaupin that is the main class of the package. TakagiTaupin solves the TT-equations for deformed crystal using the information given to it in TTcrystal and TTscan. For detailed instruction on their use, please refer to the respective documentation and example Jupyter notebooks.\n\n\n```python\nimport sys\nimport os.path\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nsys.path.insert(1, '..')\nfrom pyTTE import TakagiTaupin, TTcrystal, TTscan, Quantity\n```\n\n### First example: rocking curve of a perfect Si crystal in the Bragg case\n\nLet us start with a simple example of calculating the rocking curve of the (111) reflection of a 1 mm thick Si perfect, single crystal. First we define the crystal using TTcrystal class:\n\n\n```python\nxtal = TTcrystal(crystal='Si', hkl=[1,1,1], thickness=Quantity(1,'mm'))\n```\n\nThe given keywords `crystal`, `hkl`, and `thickness` are the required minimum amount of parameters to define a crystal. To avoid conversion errors, all the quantities with physical units are handled in _pyTTE_ using the Quantity class which supports most units commonly used in the context of X-ray diffraction.\n\nFull information of a TTcrystal instance can be seen by passing it to the `print` function:\n\n\n```python\nprint(xtal)\n```\n\n Crystal: Si\n Crystallographic parameters:\n a = 0.543069 nm, b = 0.543069 nm, c = 0.543069 nm\n alpha = 90.0 deg, beta = 90.0 nm, gamma = 90.0 deg\n Direct primitive vectors (before rotations, in nm):\n a1 = [0.5431 0. 0. ]\n a2 = [0. 0.5431 0. ]\n a3 = [0. 0. 0.5431]\n Reciprocal primitive vectors (before rotations, in 1/nm):\n b1 = [11.5698 -0. -0. ]\n b2 = [ 0. 11.5698 -0. ]\n b3 = [ 0. 0. 11.5698]\n \n Reflection: [1, 1, 1]\n Asymmetry angle: 0 deg\n In-plane rotation angle: 0 deg\n Crystal directions parallel to the Cartesian axes (after rotations):\n x || [ 1. -0.2679 -0.7321]\n y || [-0.2679 1. -0.7321]\n z || [1. 1. 1.]\n \n Crystal thickness: 1 mm\n Debye-Waller factor: 1.0\n \n Deformation model: anisotropic toroidal, fixed shape (built-in)\n Meridional bending radius: inf m\n Sagittal bending radius: inf m\n Material elastic isotropy: anisotropic\n Compliance matrix S (with rotations applied):\n [[ 0.0059 -0.0015 -0.001 -0.0012 0.0012 0. ]\n [-0.0015 0.0059 -0.001 0.0012 -0.0012 0. ]\n [-0.001 -0.001 0.0053 0. 0. -0. ]\n [-0.0012 0.0012 0. 0.0173 -0. -0.0024]\n [ 0.0012 -0.0012 0. 0. 0.0173 -0.0024]\n [ 0. 0. -0. -0.0024 -0.0024 0.0149]] GPa^-1\n\n\nIn addition to the crystal parameters, we need to define the scan using the TTscan class. Let's fix the photon energy of the incident beam to 5 keV, use $\\sigma$-polarization and scan the rocking angle from -50 to 150 \u00b5rad in 150 steps. In _pyTTE_, all scans are performed relative to the values fulfilling the kinematical Bragg condition $\\lambda = 2 d \\sin \\theta$. \n\n\n```python\nscan = TTscan(constant = Quantity(5,'keV'), scan = Quantity(np.linspace(-50,150,150),'urad'), polarization = 'sigma')\nprint(scan)\n```\n\n Scan type : angle\n Scan constant : 5 keV\n Polarization : sigma\n Scan points : 150\n Scan range : manual from -50.0 to 150.0 urad\n \n Output type : photon flux\n Integrator : zvode_bdf\n (Minimum) integration step : 1e-10 um\n Alternative starting depth : None\n \n\n\nThe rocking curve is now calculated by initializing the TakagiTaupin instance with the crystal and scan parameter instances, and calling `.run()`.\n\n\n```python\ntt = TakagiTaupin(xtal, scan)\n\nscan_vector, reflectivity, transmission = tt.run()\n```\n\n \n Solving the 1D Takagi-Taupin equation for Si(1,1,1) reflection\n ---------------------------------------------------------------\n \n The direction of diffraction out of the crystal -> Bragg case\n Solving for sigma-polarization\n Asymmetry angle : 0.0 deg\n Wavelength : 0.247968386 nm\n Energy : 5.0 keV\n Bragg angle : 23.292879649001986 deg\n Incidence angle : 23.292879649001986 deg\n Exit angle : 23.292879649001986 deg\n \n Structure factors\n F0 : (115.0489207774186+6.498559984990174j)\n Fh : (46.915823878813626-40.417263893823474j)\n Fb : (40.417263893823446+46.91582387881365j)\n \n Susceptibilities\n chi0 : (-3.961773705462428e-05+2.237815348282492e-06j)\n chih : (-1.615572542160448e-05-1.3917910073321994e-05j)\n chib : (-1.3917910073321984e-05+1.6155725421604487e-05j)\n \n (Mean F and chi values for energy scan)\n \n Transmission in the Bragg case not implemented!\n \n Calculating the TT-curve using 4 cores.\n Solving...0%\n Done.\n\n\n(Note that Jupyter does not update the solving process in the notebook but in the terminal/command prompt.)\n\n`.run()` returns the solution but also stores it in `.solution` with all the relevant information and settings pertaining to it. The stored solution can be quickly visualized by calling `.plot()` and saved as an ASCII file with `.save(filename)` \n\n\n```python\ntt.plot()\n```\n\nFor reference, the same curve was calculated using _XCRYSTAL_ in _XOP_ 2.4 (http://dx.doi.org/10.1117/12.560903) with the EPDL97 anomalous scattering cross-sections, that are also used by _xraylib_. \n\n\n```python\nref = np.loadtxt('reference_spectra/xop_Si_111_bragg.dat')\n\nplt.plot(scan_vector,reflectivity)\nplt.plot(ref[:,0],ref[:,1])\nplt.legend(['pyTTE','XCRYSTAL'])\nplt.ylabel('Reflectivity')\nplt.xlabel('Angle (urad)')\nplt.show()\n```\n\nThe result is practically identical with the one calculated with _pyTTE_. Nearly insignificant deviations are probably explained by slight differences in structure factors: \n\n_PyTTE_ output:\n```\nF0 : (115.0489207774186+6.498559984990174j)\nFh : (46.915823878813626-40.417263893823474j)\nFb : (40.417263893823446+46.91582387881365j)\n```\n\n_Crystal Parameters_ of `XCRYSTAL`:\n```\n Structure factor F(0,0,0) = ( 115.01669312000000 , 6.5506672799999999 )\n Structure factor FH = ( 47.595073721453730 , -41.044406441453731 )\n Structure factor FH_BAR = ( 41.044406441453731 , 47.595073721453730 )\n```\n\n### Second example: The Laue case and automatic scan limits\n\n_PyTTE_ does not require explicit differentiation between the Bragg case (reflection geometry) and the Laue case (transmission geometry) since the diffraction geometry is determined automatically from the propagation direction of the diffracted beam. This can be controlled via the asymmetry angle. For a symmetric Laue case, asymmetry angle is set to 90 degrees:\n\n\n```python\nxtal = TTcrystal(crystal='Ge', hkl=[4,0,0], thickness=Quantity(100,'um'), asymmetry = Quantity(90,'deg'))\n```\n\nIn the previous example we defined the scan points manually but it can be inconvenient to iterate good values for the scan limits, especially with bent crystals. To make the task easier, _pyTTE_ can try to determine the limits automatically by giving `TTscan` the number of scan points instead of a scan vector. \n\n\n```python\nscan = TTscan(constant = Quantity(7,'keV'), scan = 150, polarization = 'sigma')\nprint(scan)\n```\n\n Scan type : angle\n Scan constant : 7 keV\n Polarization : sigma\n Scan points : 150\n Scan range : automatic\n \n Output type : photon flux\n Integrator : zvode_bdf\n (Minimum) integration step : 1e-10 um\n Alternative starting depth : None\n \n\n\n\n```python\ntt = TakagiTaupin(xtal, scan)\n\nscan_vector, diffracted, transmission = tt.run()\ntt.plot()\n```\n\nAlso in this case, the correspondence with _XCRYSTAL_ is practically perfect. \n\n\n```python\nref = np.loadtxt('reference_spectra/xop_Ge_400_laue_transmission.dat')\nref2 = np.loadtxt('reference_spectra/xop_Ge_400_laue_diffraction.dat')\n\nplt.plot(scan_vector,transmission)\nplt.plot(scan_vector,diffracted)\nplt.plot(ref[:,0],ref[:,1])\nplt.plot(ref2[:,0],ref2[:,1])\nplt.legend(['pyTTE transmission', 'pyTTE diffraction', 'XCRYSTAL transmission', 'XCRYSTAL diffraction'])\nplt.ylabel('Reflectivity')\nplt.xlabel('Angle (urad)')\nplt.show()\n```\n\n### Third example: bent crystals and energy scans\n\nDeformation can be introduced by passing the meridional and sagittal bending radii (`Rx` and `Ry`, respectively) to TTcrystal. In the case of spherical bending, a single keyword `R` can be used.\n\n\n```python\nttx = TTcrystal(crystal='Si', hkl=[6,6,0], thickness=Quantity(150,'um'), R = Quantity(100,'cm'), asymmetry = Quantity(5, 'deg'))\ntts = TTscan(scan = 200, constant = Quantity(75, 'deg'), polarization = 'pi')\n\ntt = TakagiTaupin(ttx,tts)\n\ntt.run()\ntt.plot()\n```\n\nBy default _pyTTE_ attempts to calculate the toroidal/spherical/cylindrical bending using elastic constants from a built-in compliance matrix $S$ (Source: CRC Handbook of Chemistry and Physics, 82nd edition). \n\nFor anisotropic crystals, _PyTTE_ has two different deformation models, selected with the TTcrystal keyword `fix_to_axis`. The default value is `'shape'` which assumes that the main axes of curvature are along the coordinate axes $x$ and $y$, and are given by `Rx` and `Ry`. This corresponds to the situation where the crystal is forced to a specific shape _e.g._ by the substrate. The other option is `torques` which assumes that the crystal is bent by two orthogonal torques acting about the aforemention coordinate axes. This applies, for example, when a free standing crystal slab is bent by its ends. In the general case these models are not equal due to non-diagonal elements of $S$. See TTcrystal.ipynb, help(TTcrystal) or the technical docs for more information.\n\nBuilt-in $S$:s are available only for a handful of crystals commonly encountered in X-ray optics. In other cases user needs to define $S$ manually with the keyword `S`. Alternatively, an isotropic deformation model can be used by giving Poisson's ratio $\\nu$ with `nu`. Also Young's modulus $E$ is required but its value is not important here as it is not used in by deformation model. `fix_to_axes` is neglected as the two anisotropic models reduce to the same isotropic model. \n\n\n```python\nttx = TTcrystal(crystal='Si', hkl=[6,6,0], thickness=Quantity(150,'um'), R = Quantity(100,'cm'), asymmetry = Quantity(5, 'deg'), nu = 0.22, E = Quantity(150, 'GPa'))\ntts = TTscan(scan = 200, constant = Quantity(75, 'deg'), polarization = 'pi')\n\ntt = TakagiTaupin(ttx,tts)\n\ntt.run()\ntt.plot()\n```\n\nAlso custom deformation fields can be defined by providing the Jacobian of the displacement vector $\\mathbf{u}$ as a function of $x$ and $z$:\n\n$\\begin{equation}\nJ(x,z) = \\left[\\begin{matrix}\n\\frac{\\partial u_x}{\\partial x}(x,z) & \\frac{\\partial u_x}{\\partial z}(x,z) \\\\\n\\frac{\\partial u_z}{\\partial x}(x,z) & \\frac{\\partial u_z}{\\partial z}(x,z)\n\\end{matrix}\\right]\n\\end{equation}$\n\nFor reference, we do this for the isotropic bending below. Note that the lengths are exceptionally required to be in micrometers.\n\n\n```python\ndef jacobian(x,z):\n nu = 0.22\n R_bend_in_um = 1e6\n thickness_in_um = 150\n\n ux_x = -(z+0.5*thickness_in_um)/R_bend_in_um \n ux_z = -x/R_bend_in_um \n\n uz_x = x/R_bend_in_um \n uz_z = 2*nu/(1-nu)/R_bend_in_um*(z+0.5*thickness_in_um)\n\n return [[ux_x,ux_z],[uz_x,uz_z]]\n\nttx = TTcrystal(crystal='Si', hkl=[6,6,0], thickness=Quantity(150,'um'), asymmetry = Quantity(5, 'deg'))\ntts = TTscan(scan = 200, constant = Quantity(75, 'deg'), polarization = 'pi')\n\n#The custom Jacobian is given to TTcrystal after the initialization\nttx.set_deformation(jacobian = jacobian)\n\ntt = TakagiTaupin(ttx,tts)\n\ntt.run()\ntt.plot()\n```\n\n### Fourth example: Initialization from file\n\nParameters can also be defined in two input files for `TTcrystal` and `TTscan` which can be passed directly to `TakagiTaupin`.\n\n\n```python\n'''\nContents of TTcrystal_init.inp:\n\ncrystal LiF\nhkl 2 0 0\nthickness 200 um\nasymmetry 2.5 deg\nRx 1 m\n\nContents of TTscan_init.inp:\n\nconstant 8 keV\nscan -100 25 150 arcsec\npolarization sigma\nsolver zvode_bdf\nintegration_step 1 nm\n'''\n\ntt = TakagiTaupin('TTcrystal_init.inp','TTscan_init.inp')\n\nprint(tt)\n\ntt.run()\ntt.plot()\nplt.show()\n```\n", "meta": {"hexsha": "1a44899a119d12d0b6f8999971e989c28a78ec1e", "size": 226390, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/TakagiTaupin.ipynb", "max_stars_repo_name": "aripekka/pyTTE", "max_stars_repo_head_hexsha": "198d0232913e38f3d0a8ff034497640a730b4e8b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2019-05-08T03:01:53.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-12T15:00:16.000Z", "max_issues_repo_path": "examples/TakagiTaupin.ipynb", "max_issues_repo_name": "aripekka/pyTTE", "max_issues_repo_head_hexsha": "198d0232913e38f3d0a8ff034497640a730b4e8b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 9, "max_issues_repo_issues_event_min_datetime": "2017-12-21T12:47:04.000Z", "max_issues_repo_issues_event_max_datetime": "2021-06-01T11:01:02.000Z", "max_forks_repo_path": "examples/TakagiTaupin.ipynb", "max_forks_repo_name": "aripekka/pyTTE", "max_forks_repo_head_hexsha": "198d0232913e38f3d0a8ff034497640a730b4e8b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 262.3290845886, "max_line_length": 29692, "alphanum_fraction": 0.9138389505, "converted": true, "num_tokens": 3599, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680199891789, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.44882282507855453}} {"text": "### This notebook computes the joint likelihood in 5 dimensions for the parameters of interest (see below). See inference_demo for more details about basic usage\n\nTo run this notebook, download the pre-computed likelihoods for each lens from dropbox, put the .zip file into lenslikelihood/precomputed_likelihoods/ and unpack it: https://www.dropbox.com/s/gbj4d58kg6cax4e/logprior_likelihoods.zip?dl=0 \n\n\n```python\nfrom lenslikelihood.measurements import flux_measurements, flux_measurement_uncertainties, all_lens_names, all_param_ranges_version2\nimport numpy as np\n\n# Note that the syntax for the uncertainties is\n# {'lens_name': (1-sigma-uncertainty, reference_index, uncertainty_in_ratio)}\n# where reference_index is the reference image with which to compute the flux ratio, and uncertainty_in_ratio specifies\n# whether the measurement uncertainty refers to the flux or the flux ratio\nparam_names = ['LOS_normalization', 'beta', 'log10c0', 'delta_power_law_index', 'sigma_sub']\nparam_ranges = [all_param_ranges_version2[name] for name in param_names]\nfor name in all_lens_names:\n print(name)\n print('fluxes/flux ratios measured: ', flux_measurements[name])\n print('uncertainties: ', flux_measurement_uncertainties[name])\n print('\\n')\n```\n\n WGDJ0405\n fluxes/flux ratios measured: [0.8 0.52 1. 0.94]\n uncertainties: ([0.04, 0.061538461538461535, 0.024, 0.03418803418803419], 0, False)\n \n \n HE0435\n fluxes/flux ratios measured: [0.96 0.976 1. 0.65 ]\n uncertainties: ([0.05, 0.049, 0.048, 0.056], 0, False)\n \n \n WGD2038\n fluxes/flux ratios measured: [0.86 1. 0.79 0.4 ]\n uncertainties: ([0.01, 0.01724137931034483, 0.021739130434782608, 0.021739130434782608], 0, False)\n \n \n B1422\n fluxes/flux ratios measured: [0.88 1. 0.474 0.025]\n uncertainties: ([0.011363636363636364, 0.01, 0.012765957446808512, None], 0, False)\n \n \n WFI2033\n fluxes/flux ratios measured: [1. 0.65 0.5 0.53]\n uncertainties: ([0.03, 0.046875, 0.04, 0.03773584905660377], 0, False)\n \n \n PSJ1606\n fluxes/flux ratios measured: [1. 1. 0.59 0.79]\n uncertainties: ([0.03, 0.03, 0.03333333333333333, 0.02564102564102564], 0, False)\n \n \n WFI2026\n fluxes/flux ratios measured: [1. 0.75 0.31 0.28]\n uncertainties: ([0.02, 0.02666666666666667, 0.06451612903225806, 0.03571428571428571], 0, False)\n \n \n RXJ0911\n fluxes/flux ratios measured: [0.56 1. 0.53 0.24]\n uncertainties: ([0.07142857142857142, 0.05, 0.07547169811320754, 0.16666666666666669], 0, False)\n \n \n RXJ1131\n fluxes/flux ratios measured: [1. 0.61 0.73 0.12]\n uncertainties: ([[0.012345679012345678, 0.024691358024691357], [0.10084033613445378, 0.025210084033613446], None], 1, True)\n \n \n MG0414\n fluxes/flux ratios measured: [1. 0.83 0.36 0.16]\n uncertainties: ([0.06024096385542169, 0.11111111111111112, 0.11764705882352941], 0, True)\n \n \n PG1115\n fluxes/flux ratios measured: [1. 0.93 0.16 0.21]\n uncertainties: ([0.06451612903225806, 0.43750000000000006, 0.1904761904761905], 0, True)\n \n \n\n\n### Models implemented for the halo mass function and concentration-mass relation\n\nThe full set of hyper-parameters we're interested in constraining are defined by the parameterizations of the halo mass function and concentration-mass relation. They are $\\Sigma_{\\rm{sub}}$, $\\delta_{\\rm{LOS}}$, $\\Delta \\alpha$, $q$, $c_8$, and $\\beta$. The first four define to the subhalo and field halo mass functions, and the last two define the concentration-mass relation. \n\nThe field halo mass function is parameterized as\n\\begin{equation}\n\\frac{dN_{\\rm{LOS}}}{dm dV} = \\delta_{\\rm{LOS}} \\left(1+\\xi_{\\rm{2halo}}\\right) \\left(\\frac{m}{10^8}\\right)^{\\Delta \\alpha} \\ \\frac{dN_{\\rm{ShethTormen}}}{dm dV}\n\\end{equation}\nwhere $\\delta_{\\rm{LOS}}$ scales the overall normalization, and $\\Delta \\alpha$ parameterizes deviations from the logarithmic slope predicted by CDM around $10^8 M_{\\odot}$. \n\nThe subhalo mass function is parameterized as\n\\begin{equation}\n\\frac{dN_{\\rm{sub}}}{dm dA} \\sim \\Sigma_{\\rm{sub}} \\ \\left(\\frac{m}{10^8}\\right)^{\\alpha + q \\Delta \\alpha}\n\\end{equation}\nwhere $\\Sigma_{\\rm{sub}}$ is the normalization, $\\alpha$ is the logarithmic slope predicted by CDM, $\\Delta \\alpha$ parameterizes deviations from the value predicted by CDM, and $q$ controls the coupling between the line of sight halo mass function slope and the subhalo mass function slope. When $q=1$ the slopes change in the same way, and when $q=0$ the slopes of the subhalo and field halo mass functions are completely decoupled. \n\nThe concentration-mass relation is parameterized as \n\n\\begin{equation}\nc\\left(M, z\\right) = c_8 \\left(1+z\\right)^{\\zeta} \\left(\\frac{\\nu\\left(M, z\\right)}{\\nu\\left(10^8, z\\right)}\\right)^{-\\beta}\n\\end{equation}\ni.e. it is a power-law in the peak height $\\nu$ with normalization $c_8$ at $10^8$ and a logarithmic slope $\\beta$. The parameter $\\zeta$ modifies the redshift evolution and is marginalized over in the sampling. \n\nThe parameter names used in the python code have the following correspondence: \n\n\n1) sigma_sub = $\\Sigma_{\\rm{sub}}$\n\n2) delta_power_law_index = $\\Delta \\alpha$\n\n3) c0 = $c_8$\n\n4) beta = $\\beta$\n\n5) delta_power_law_index_coupling = $q$\n\n6) LOS_normalization = $\\delta_{\\rm{LOS}}$\n\n### Example inference on three parameters with a subset of lenses\n\nFirst load the model samples, define what parameters we want to look at\n\n\n```python\nfrom trikde.pdfs import DensitySamples, IndepdendentLikelihoods\nimport os\nimport pickle\n\nnbins = 20\nlikelihoods = []\n\nfilename_extension = '_joint_logprior'\nbase_path = './../lenslikelihood/precomputed_likelihoods/'\nprint(all_lens_names)\n\nlens_list = ['B1422',\n 'HE0435',\n 'WGD2038',\n 'WGDJ0405',\n 'WFI2033',\n 'PSJ1606',\n 'WFI2026',\n 'RXJ0911',\n 'RXJ1131',\n 'MG0414',\n 'PG1115']\n\nfor lens in lens_list:\n \n fname = base_path + lens + filename_extension\n print('loading joint likelihoods for lens '+lens+' ...')\n f = open(fname, 'rb')\n single_lens_likelihood = pickle.load(f)\n f.close()\n likelihoods.append(single_lens_likelihood)\n \nlikelihood = IndepdendentLikelihoods(likelihoods)\n```\n\n ['WGDJ0405', 'HE0435', 'WGD2038', 'B1422', 'WFI2033', 'PSJ1606', 'WFI2026', 'RXJ0911', 'RXJ1131', 'MG0414', 'PG1115']\n loading joint likelihoods for lens B1422 ...\n loading joint likelihoods for lens HE0435 ...\n loading joint likelihoods for lens WGD2038 ...\n loading joint likelihoods for lens WGDJ0405 ...\n loading joint likelihoods for lens WFI2033 ...\n loading joint likelihoods for lens PSJ1606 ...\n loading joint likelihoods for lens WFI2026 ...\n loading joint likelihoods for lens RXJ0911 ...\n loading joint likelihoods for lens RXJ1131 ...\n loading joint likelihoods for lens MG0414 ...\n loading joint likelihoods for lens PG1115 ...\n\n\n\n```python\nfrom trikde.triangleplot import TrianglePlot\ntriangle_plot = TrianglePlot([likelihood])\ntriangle_plot.set_cmap('magma')\naxes = triangle_plot.make_triplot(filled_contours=False, show_intervals=False, show_contours=False\n )\nbeta = r'$\\beta$'\nbeta_ticks = [0, 3, 6, 9, 12, 15]\nc0 = r'$\\log_{10}\\left(c_8\\right)$'\nc0_ticks = [0, 1, 2 ,3, 4]\ndelta_power_law_index = r'$\\Delta \\alpha$'\ndpli_ticks = [-0.6, -0.3, 0., 0.3, 0.6, 0.9]\nsigma_sub = r'$\\Sigma_{\\rm{sub}} \\ \\left[\\rm{kpc^{-2}}\\right]$'\nsigma_sub_ticks = [0., 0.025, 0.05, 0.075, 0.1]\ndelta_LOS = r'$\\delta_{\\rm{LOS}}$'\ndlos_ticks = [0., 0.5, 1., 1.5, 2.]\nticksize = 14\nlabelsize = 18\nrotation = 40\n\naxes[5].set_ylabel(beta, fontsize=labelsize)\naxes[5].set_yticks(beta_ticks)\naxes[5].set_yticklabels(beta_ticks, fontsize=ticksize)\n\naxes[10].set_ylabel(c0, fontsize=labelsize)\naxes[10].set_yticks(c0_ticks)\naxes[10].set_yticklabels(c0_ticks, fontsize=ticksize)\n\naxes[15].set_ylabel(delta_power_law_index, fontsize=labelsize)\naxes[15].set_yticks(dpli_ticks)\naxes[15].set_yticklabels(dpli_ticks, fontsize=ticksize)\n\naxes[20].set_ylabel(sigma_sub, fontsize=labelsize)\naxes[20].set_yticks(sigma_sub_ticks)\naxes[20].set_yticklabels(sigma_sub_ticks, fontsize=ticksize)\n\naxes[20].set_xlabel(delta_LOS, fontsize=labelsize)\naxes[20].set_xticks(dlos_ticks)\naxes[20].set_xticklabels(dlos_ticks, fontsize=ticksize, rotation=rotation)\n\naxes[21].set_xlabel(beta, fontsize=labelsize)\naxes[21].set_xticks(beta_ticks)\naxes[21].set_xticklabels(beta_ticks, fontsize=ticksize, rotation=rotation)\n\naxes[22].set_xlabel(c0, fontsize=labelsize)\naxes[22].set_xticks(c0_ticks)\naxes[22].set_xticklabels(c0_ticks, fontsize=ticksize, rotation=rotation)\n\n\naxes[23].set_xlabel(delta_power_law_index, fontsize=labelsize)\naxes[23].set_xticks(dpli_ticks)\naxes[23].set_xticklabels(dpli_ticks, fontsize=ticksize, rotation=rotation)\n\naxes[24].set_xlabel(sigma_sub, fontsize=labelsize)\naxes[24].set_xticks(sigma_sub_ticks)\naxes[24].set_xticklabels(sigma_sub_ticks, fontsize=ticksize, rotation=rotation)\n# can change axis labels\n```\n\n\n```python\nfrom trikde.pdfs import CustomPriorHyperCube\n\ndef couple_mass_functions(samples, sigma_sub_theory=0.05, coupling_strength=0.05):\n \n delta_los_samples = samples[:, 0]\n sigma_sub_samples = samples[:, -1]\n delta_sigma_sub = sigma_sub_samples/sigma_sub_theory\n chi2 = (delta_sigma_sub - delta_los_samples)**2/coupling_strength**2 \n return chi2\n\nkwargs_1 = {'sigma_sub_theory': 0.05}\nkwargs_2 = {'sigma_sub_theory': 0.025}\nprior_on_mass_functions_1 = CustomPriorHyperCube(couple_mass_functions, param_names, param_ranges, nbins, kwargs_1)\nprior_on_mass_functions_2 = CustomPriorHyperCube(couple_mass_functions, param_names, param_ranges, nbins, kwargs_2)\n\nlikelihoods_coupled_with_prior_mass_functions = likelihoods + [prior_on_mass_functions_1]\nlikelihood_coupled_with_prior_mass_functions_1 = IndepdendentLikelihoods(likelihoods_coupled_with_prior_mass_functions)\nlikelihoods_coupled_with_prior_mass_functions = likelihoods + [prior_on_mass_functions_2]\nlikelihood_coupled_with_prior_mass_functions_2 = IndepdendentLikelihoods(likelihoods_coupled_with_prior_mass_functions)\n```\n\n\n```python\ntriangle_plot = TrianglePlot([likelihood_coupled_with_prior_mass_functions_1])\ntriangle_plot.set_cmap('magma')\naxes = triangle_plot.make_triplot(filled_contours=False, show_intervals=False, show_contours=False\n )\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "98aa47a17eb197ed353fbf558e84bf87d360715d", "size": 140015, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/inference_5D_logprior.ipynb", "max_stars_repo_name": "dangilman/lenslikelihood", "max_stars_repo_head_hexsha": "1490ee9756b4d2ed108a2478977609bbe0ba2e17", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/inference_5D_logprior.ipynb", "max_issues_repo_name": "dangilman/lenslikelihood", "max_issues_repo_head_hexsha": "1490ee9756b4d2ed108a2478977609bbe0ba2e17", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/inference_5D_logprior.ipynb", "max_forks_repo_name": "dangilman/lenslikelihood", "max_forks_repo_head_hexsha": "1490ee9756b4d2ed108a2478977609bbe0ba2e17", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 371.3925729443, "max_line_length": 63232, "alphanum_fraction": 0.9292647216, "converted": true, "num_tokens": 3214, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431679972357831, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.44882281133705443}} {"text": "# \u041b\u0430\u0431\u043e\u0440\u0430\u0442\u043e\u0440\u0430\u044f \u0440\u0430\u0431\u043e\u0442\u0430 4.02\n# \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u044f \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f \u0449\u0435\u043b\u044f\u043c\u0438 \u0438\u043d\u0442\u0435\u0440\u0444\u0435\u0440\u0435\u0446\u0438\u043e\u043d\u043d\u044b\u043c \u043c\u0435\u0442\u043e\u0434\u043e\u043c\n\n## \u0412\u044b\u043f\u043e\u043b\u043d\u0438\u043b: \u041a\u043e\u043d\u044f\u0445\u0438\u043d \u0412\u0441\u0435\u0432\u043e\u043b\u043e\u0434 \u0412\u043b\u0430\u0434\u0438\u043c\u0438\u0440\u043e\u0432\u0438\u0447, M32051\n\n## \u041a\u0440\u0430\u0442\u043a\u0438\u0435 \u0442\u0435\u043e\u0440\u0435\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0441\u0432\u0435\u0434\u0435\u043d\u0438\u044f\n\u041f\u0440\u0438 \u043d\u0430\u043b\u043e\u0436\u0435\u043d\u0438\u0438 \u043a\u043e\u0433\u0435\u0440\u0435\u043d\u0442\u043d\u044b\u0445 \u0432\u043e\u043b\u043d \u043f\u0440\u043e\u0438\u0441\u0445\u043e\u0434\u0438\u0442 \u043f\u0435\u0440\u0435\u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0441\u0432\u0435\u0442\u043e\u0432\u043e\u0433\u043e\n\u043f\u043e\u0442\u043e\u043a\u0430 \u0432 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0435, \u0432 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u0435 \u0447\u0435\u0433\u043e \u0432 \u043e\u0434\u043d\u0438\u0445 \u043c\u0435\u0441\u0442\u0430\u0445 \u0432\u043e\u0437\u043d\u0438\u043a\u0430\u044e\u0442 \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c\u044b, \u0430\n\u0432 \u0434\u0440\u0443\u0433\u0438\u0445 \u2014 \u043c\u0438\u043d\u0438\u043c\u0443\u043c\u044b \u0438\u043d\u0442\u0435\u043d\u0441\u0438\u0432\u043d\u043e\u0441\u0442\u0438. \u042d\u0442\u043e \u044f\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u0438\u043d\u0442\u0435\u0440\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0435\u0439\n\u0432\u043e\u043b\u043d. \n\u0421\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u044e\u0442 \u0434\u0432\u0430 \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0445 \u0442\u0438\u043f\u0430 \u0438\u043d\u0442\u0435\u0440\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u043e\u043d\u043d\u044b\u0445 \u0441\u0445\u0435\u043c: \u0441\u0445\u0435\u043c\u0430,\n\u043f\u043e\u0441\u0442\u0440\u043e\u0435\u043d\u043d\u0430\u044f \u043d\u0430 \u043e\u0441\u043d\u043e\u0432\u0435 \u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0432\u043e\u043b\u043d\u043e\u0432\u043e\u0433\u043e \u0444\u0440\u043e\u043d\u0442\u0430, \u0438 \u0441\u0445\u0435\u043c\u0430, \u043f\u043e\u0441\u0442\u0440\u043e\u0435\u043d\u043d\u0430\u044f \u043d\u0430 \u043c\u0435\u0442\u043e\u0434\u0435\n\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0430\u043c\u043f\u043b\u0438\u0442\u0443\u0434\u044b. \n\u0412 \u0434\u0430\u043d\u043d\u043e\u0439 \u0440\u0430\u0431\u043e\u0442\u0435 \u0440\u0430\u0441\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u0435\u0442\u0441\u044f \u043e\u043f\u044b\u0442 \u042e\u043d\u0433\u0430, \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u043d\u044b\u0439 \u043d\u0430 \u0434\u0435\u043b\u0435\u043d\u0438\u0438 \u0432\u043e\u043b\u043d\u043e\u0432\u043e\u0433\u043e \u0444\u0440\u043e\u043d\u0442\u0430. \u0421\u043d\u0438\u0437\u0443 \u043f\u0440\u0438\u0432\u0435\u0434\u0435\u043d\u0430 \u0441\u0445\u0435\u043c\u0430 \u043e\u043f\u044b\u0442\u0430 \u042e\u043d\u0433\u0430. \n\n\n\n## \u0426\u0435\u043b\u044c \u0440\u0430\u0431\u043e\u0442\u044b\n\u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u0442\u044c \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u044f \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f \u0449\u0435\u043b\u044f\u043c\u0438 \u043f\u043e \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u043d\u043e\u0439 \u043e\u0442 \u043d\u0438\u0445 \u0438\u043d\u0442\u0435\u0440\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u043a\u0430\u0440\u0442\u0438\u043d\u0435. \n\n## \u0420\u0430\u0431\u043e\u0447\u0438\u0435 \u0444\u043e\u0440\u043c\u0443\u043b\u044b \u0438 \u0438\u0441\u0445\u043e\u0434\u043d\u044b\u0435 \u0434\u0430\u043d\u043d\u044b\u0435\n\n### \u0424\u043e\u0440\u043c\u0443\u043b\u044b\n$\\Delta x = \\frac{\\lambda}{d}L$ \n$L = x_{\u042d} - x_{\u041e}$\n\n### \u0418\u0441\u0445\u043e\u0434\u043d\u044b\u0435 \u0434\u0430\u043d\u043d\u044b\u0435\n$\\lambda = (632.82 \\pm 0.01) \\space \u043d\u043c$ \n\n### \u0421\u0445\u0435\u043c\u0430 \u0443\u0441\u0442\u0430\u043d\u043e\u0432\u043a\u0438\n\n\n## \u0420\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u044b \u0438\u0437\u043c\u0435\u0440\u0435\u043d\u0438\u0439 \u0438 \u0440\u0430\u0441\u0447\u0435\u0442\u044b\n\n\n```python\nimport sympy\nimport scipy\nimport numpy as np\nimport pandas as pd\nfrom scipy.signal import argrelextrema\nimport matplotlib.pyplot as plt\nplt.rcParams[\"figure.figsize\"] = (10,5)\n%matplotlib inline\n```\n\n### \u041a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u044b \u043f\u043e\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0445 \u043c\u0438\u043d\u0438\u043c\u0443\u043c\u043e\u0432 \u0438\u043d\u0442\u0435\u043d\u0441\u0438\u0432\u043d\u043e\u0441\u0442\u0438 \u043f\u0440\u0438 \u0440\u0430\u0437\u043d\u044b\u0445 \u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u044f\u0445 \u043e\u0431\u044a\u0435\u043a\u0442\u0430 33\n\n\n```python\ndf33 = pd.DataFrame({'X \u043e\u0431\u044a\u0435\u043a\u0442\u0430, \u043c\u043c': [166, 266, 366, 466, 566],\n '$x_1$, \u043c\u043c' : [-25, -25.5, -25.5, -26, -26.5],\n '$x_2$, \u043c\u043c' : [-22.5, -23, -22.5, -24, -24.5],\n '$x_3$, \u043c\u043c' : [-18.5, -19.5, -19.5, -21.5, -22],\n '$x_4$, \u043c\u043c' : [-15, -16, -17, -19, -20],\n '$x_5$, \u043c\u043c' : [-11.5, -13, -14, -17, -18],\n '$x_6$, \u043c\u043c' : [-8.5, -10.5, -11.5, -14, -16],\n '$x_7$, \u043c\u043c' : [-4.5, -6.5, -9, -11.5, -14],\n '$x_8$, \u043c\u043c' : [-0.5, -3, -5.5, -8.5, -12],\n '$x_9$, \u043c\u043c' : [3.5, 0, -3.5, -4, -10],\n '$x_{10}$, \u043c\u043c' : [7, 3.5, 0.5, -2, -8],\n '$x_{11}$, \u043c\u043c' : [11, 6.5, 3, 1, -6],\n '$x_{12}$, \u043c\u043c' : [14.5, 10, 6, 3.5, -3.5],\n '$x_{13}$, \u043c\u043c' : [18, 13.5, 9, 6, -1.5],\n '$x_{14}$, \u043c\u043c' : [21.5, 16.5, 12, 8.5, 1],\n '$x_{15}$, \u043c\u043c' : [25.5, 19.5, 15, 11, 3]})\n\nx_screen = 1090\n\nalpha = 632.82 * 10 ** (-9)\ndelta_alpha = 0.01 * 10 ** (-9)\n\ndf33\n```\n\n\n\n\n

\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
X \u043e\u0431\u044a\u0435\u043a\u0442\u0430, \u043c\u043c$x_1$, \u043c\u043c$x_2$, \u043c\u043c$x_3$, \u043c\u043c$x_4$, \u043c\u043c$x_5$, \u043c\u043c$x_6$, \u043c\u043c$x_7$, \u043c\u043c$x_8$, \u043c\u043c$x_9$, \u043c\u043c$x_{10}$, \u043c\u043c$x_{11}$, \u043c\u043c$x_{12}$, \u043c\u043c$x_{13}$, \u043c\u043c$x_{14}$, \u043c\u043c$x_{15}$, \u043c\u043c
0166-25.0-22.5-18.5-15-11.5-8.5-4.5-0.53.57.011.014.518.021.525.5
1266-25.5-23.0-19.5-16-13.0-10.5-6.5-3.00.03.56.510.013.516.519.5
2366-25.5-22.5-19.5-17-14.0-11.5-9.0-5.5-3.50.53.06.09.012.015.0
3466-26.0-24.0-21.5-19-17.0-14.0-11.5-8.5-4.0-2.01.03.56.08.511.0
4566-26.5-24.5-22.0-20-18.0-16.0-14.0-12.0-10.0-8.0-6.0-3.5-1.51.03.0
\n
\n\n\n\n\n```python\nprint('\u041a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u0430 \u044d\u043a\u0440\u0430\u043d\u0430, {} \u043c\u043c'.format(x_screen))\n```\n\n \u041a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u0430 \u044d\u043a\u0440\u0430\u043d\u0430, 1090 \u043c\u043c\n\n\n\u041f\u043e\u0441\u0447\u0438\u0442\u0430\u0435\u043c \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u0435 \u043e\u0442 \u044d\u043a\u0440\u0430\u043d\u0430 \u0434\u043e \u043e\u0431\u044a\u0435\u043a\u0442\u0430 $L$, \u0448\u0438\u0440\u0438\u043d\u0443 \u0438\u043d\u0442\u0435\u0440\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u043f\u043e\u043b\u043e\u0441\u044b $\\Delta x$: \n\n$L = x_{\u042d} - x_{\u041e}$ \n\n$\\Delta x = \\frac{x_{min\\space max} - x_{min\\space min}}{number\\space of\\space minimums}$\n\n\n```python\ndf33_calc = pd.DataFrame({'X \u043e\u0431\u044a\u0435\u043a\u0442\u0430, \u043c\u043c': [166, 266, 366, 466, 566]})\ndf33_calc['L, \u043c\u043c'] = x_screen - df33_calc['X \u043e\u0431\u044a\u0435\u043a\u0442\u0430, \u043c\u043c']\ndf33_calc['$\\Delta x$, \u043c\u043c'] = (df33['$x_{15}$, \u043c\u043c'] - df33['$x_1$, \u043c\u043c']) / 15\ndf33_calc\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
X \u043e\u0431\u044a\u0435\u043a\u0442\u0430, \u043c\u043cL, \u043c\u043c$\\Delta x$, \u043c\u043c
01669243.366667
12668243.000000
23667242.700000
34666242.466667
45665241.966667
\n
\n\n\n\n### \u041a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u044b \u043f\u043e\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0445 \u043c\u0438\u043d\u0438\u043c\u0443\u043c\u043e\u0432 \u0438\u043d\u0442\u0435\u043d\u0441\u0438\u0432\u043d\u043e\u0441\u0442\u0438 \u043f\u0440\u0438 \u0440\u0430\u0437\u043d\u044b\u0445 \u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u044f\u0445 \u043e\u0431\u044a\u0435\u043a\u0442\u0430 32\n\n\n```python\ndf32 = pd.DataFrame({'X \u043e\u0431\u044a\u0435\u043a\u0442\u0430, \u043c\u043c': [166, 266, 366, 466, 566],\n '$x_1$, \u043c\u043c' : [-25, -27, -25.5, -25, -25.5],\n '$x_2$, \u043c\u043c' : [-22, -23, -22, -23, -23],\n '$x_3$, \u043c\u043c' : [-17, -19, -19, -19.5, -20.5],\n '$x_4$, \u043c\u043c' : [-13, -15, -16, -16.5, -18.5],\n '$x_5$, \u043c\u043c' : [-9, -11.5, -12.5, -14, -16],\n '$x_6$, \u043c\u043c' : [-5, -7.5, -9.5, -11, -13.5],\n '$x_7$, \u043c\u043c' : [-1, -4, -6, -8, -11],\n '$x_8$, \u043c\u043c' : [3, -0.5, -3, -5, -9],\n '$x_9$, \u043c\u043c' : [7, 3.5, 0.5, -2, -6.5],\n '$x_{10}$, \u043c\u043c' : [11, 7, 4, 0.5, -4],\n '$x_{11}$, \u043c\u043c' : [15, 11, 7, 4, -1.5],\n '$x_{12}$, \u043c\u043c' : [19, 14.5, 11.5, 6.5, 0.5]})\n\ndf32_calc = pd.DataFrame({'X \u043e\u0431\u044a\u0435\u043a\u0442\u0430, \u043c\u043c': [166, 266, 366, 466, 566]})\ndf32\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
X \u043e\u0431\u044a\u0435\u043a\u0442\u0430, \u043c\u043c$x_1$, \u043c\u043c$x_2$, \u043c\u043c$x_3$, \u043c\u043c$x_4$, \u043c\u043c$x_5$, \u043c\u043c$x_6$, \u043c\u043c$x_7$, \u043c\u043c$x_8$, \u043c\u043c$x_9$, \u043c\u043c$x_{10}$, \u043c\u043c$x_{11}$, \u043c\u043c$x_{12}$, \u043c\u043c
0166-25.0-22-17.0-13.0-9.0-5.0-13.07.011.015.019.0
1266-27.0-23-19.0-15.0-11.5-7.5-4-0.53.57.011.014.5
2366-25.5-22-19.0-16.0-12.5-9.5-6-3.00.54.07.011.5
3466-25.0-23-19.5-16.5-14.0-11.0-8-5.0-2.00.54.06.5
4566-25.5-23-20.5-18.5-16.0-13.5-11-9.0-6.5-4.0-1.50.5
\n
\n\n\n\n\n```python\ndf32_calc = pd.DataFrame({'X \u043e\u0431\u044a\u0435\u043a\u0442\u0430, \u043c\u043c': [166, 266, 366, 466, 566]})\ndf32_calc['L, \u043c\u043c'] = x_screen - df32_calc['X \u043e\u0431\u044a\u0435\u043a\u0442\u0430, \u043c\u043c']\ndf32_calc['$\\Delta x$, \u043c\u043c'] = (df32['$x_{12}$, \u043c\u043c'] - df32['$x_1$, \u043c\u043c']) / 12\ndf32_calc\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
X \u043e\u0431\u044a\u0435\u043a\u0442\u0430, \u043c\u043cL, \u043c\u043c$\\Delta x$, \u043c\u043c
01669243.666667
12668243.458333
23667243.083333
34666242.625000
45665242.166667
\n
\n\n\n\n### \u041f\u043e\u0441\u0442\u0440\u043e\u0435\u043d\u0438\u044f \u0430\u043f\u043f\u0440\u043e\u043a\u0441\u0438\u043c\u0438\u0440\u0443\u044e\u0449\u0435\u0439 \u043f\u0440\u044f\u043c\u043e\u0439 \n\u0414\u043b\u044f \u044d\u0442\u043e\u0433\u043e \u043e\u0431\u0443\u0447\u0438\u043c \u043b\u0438\u043d\u0435\u0439\u043d\u0443\u044e \u0440\u0435\u0433\u0440\u0435\u0441\u0441\u0438\u044e \u043d\u0430 $\\Delta x$ \u043e\u0442 $L$:\n\n\n```python\nfrom sklearn import linear_model\n\nregr_33 = linear_model.LinearRegression()\nX_33 = np.array(df33_calc['L, \u043c\u043c']).reshape(-1, 1)\ny_33 = np.array(df33_calc['$\\Delta x$, \u043c\u043c'])\nregr_33.fit(X_33, y_33)\n\nregr_32 = linear_model.LinearRegression()\nX_32 = np.array(df32_calc['L, \u043c\u043c']).reshape(-1, 1)\ny_32 = np.array(df32_calc['$\\Delta x$, \u043c\u043c'])\nregr_32.fit(X_32, y_32)\n\nprint('\u0414\u043b\u044f \u043e\u0431\u044a\u0435\u043a\u0442\u0430 33, K = {:.3f}, b = {:.3f}'.format(regr_33.coef_[0], regr_33.intercept_))\nprint('\u0414\u043b\u044f \u043e\u0431\u044a\u0435\u043a\u0442\u0430 32, K = {:.3f}, b = {:.3f}'.format(regr_32.coef_[0], regr_32.intercept_))\n```\n\n \u0414\u043b\u044f \u043e\u0431\u044a\u0435\u043a\u0442\u0430 33, K = 0.003, b = 0.287\n \u0414\u043b\u044f \u043e\u0431\u044a\u0435\u043a\u0442\u0430 32, K = 0.004, b = 0.225\n\n\n\n```python\nfig, ax = plt.subplots(figsize=(20, 5))\nax.set_title('\u0417\u0430\u0432\u0438\u0441\u0438\u043c\u043e\u0441\u0442\u044c \u0448\u0438\u0440\u0438\u043d\u044b \u0438\u043d\u0442\u0435\u0440\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u043f\u043e\u043b\u043e\u0441\u044b \u043e\u0442 \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u044f \u043c\u0435\u0436\u0434\u0443 \u044d\u043a\u0440\u0430\u043d\u043e\u043c \u0438 \u043e\u0431\u044a\u0435\u043a\u0442\u043e\u043c \u0434\u043b\u044f \u0434\u0432\u0443\u0445 \u043e\u0431\u044a\u0435\u043a\u0442\u043e\u0432')\nX_33 = X_33.reshape(5)\nax.scatter(X_33, y_33, c='r')\nk, b = regr_33.coef_[0], regr_33.intercept_\nx = np.linspace(500, 1000, 50)\nax.plot(x, np.polyval([k, b], x), 'r--')\n\nX_32 = X_32.reshape(5)\nax.scatter(X_32, y_32, c='b')\nk, b = regr_32.coef_[0], regr_32.intercept_\nx = np.linspace(500, 1000, 50)\nax.plot(x, np.polyval([k, b], x), 'b--')\n\nplt.show()\n```\n\n\u041d\u0430\u0439\u0434\u0435\u043c $d_{33}$ \u0438 $d_{32}$, \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f $d = \\frac{\\lambda}{K}$\n\n\n```python\nd_32 = alpha / regr_32.coef_[0]\nprint('\u0420\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u0435 \u043c\u0435\u0436\u0434\u0443 \u0449\u0435\u043b\u044f\u043c\u0438 \u0434\u043b\u044f 32-\u0433\u043e \u043e\u0431\u044a\u0435\u043a\u0442\u0430: {:.3f} \u043c\u043a\u043c'.format(d_32 * 10 ** 6))\n\nd_33 = alpha / regr_33.coef_[0]\nprint('\u0420\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u0435 \u043c\u0435\u0436\u0434\u0443 \u0449\u0435\u043b\u044f\u043c\u0438 \u0434\u043b\u044f 33-\u0433\u043e \u043e\u0431\u044a\u0435\u043a\u0442\u0430: {:.3f} \u043c\u043a\u043c'.format(d_33 * 10 ** 6))\n```\n\n \u0420\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u0435 \u043c\u0435\u0436\u0434\u0443 \u0449\u0435\u043b\u044f\u043c\u0438 \u0434\u043b\u044f 32-\u0433\u043e \u043e\u0431\u044a\u0435\u043a\u0442\u0430: 165.083 \u043c\u043a\u043c\n \u0420\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u0435 \u043c\u0435\u0436\u0434\u0443 \u0449\u0435\u043b\u044f\u043c\u0438 \u0434\u043b\u044f 33-\u0433\u043e \u043e\u0431\u044a\u0435\u043a\u0442\u0430: 189.846 \u043c\u043a\u043c\n\n\n\u041d\u0430\u0439\u0434\u0435\u043c $\\overline d$ \u0434\u043b\u044f \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u044f \u0449\u0435\u043b\u0438 \u043f\u043e \u0434\u0432\u0443\u043c \u0442\u043e\u0447\u043a\u0430\u043c\n\n\n```python\nprint('\u0421\u0440\u0435\u0434\u043d\u0435\u0435 \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u0435 \u043c\u0435\u0436\u0434\u0443 \u0449\u0435\u043b\u044f\u043c\u0438: {:.3f} \u043c\u043a\u043c'.format((d_33 + d_32) / 2 * 10 ** 6))\n```\n\n \u0421\u0440\u0435\u0434\u043d\u0435\u0435 \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u0435 \u043c\u0435\u0436\u0434\u0443 \u0449\u0435\u043b\u044f\u043c\u0438: 177.465 \u043c\u043a\u043c\n\n\n\u041d\u0430\u0439\u0434\u0435\u043c \u043f\u043e\u0433\u0440\u0435\u0448\u043d\u043e\u0441\u0442\u044c \u0434\u043b\u044f \u0441\u0440\u0435\u0434\u043d\u0435\u0433\u043e \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u044f \u043c\u0435\u0436\u0434\u0443 \u0449\u0435\u043b\u044f\u043c\u0438: \n$\\Delta K = \\sqrt{\\frac{\\sum(y_i-Kx_i-b)^2}{(n-2)\\sum(x_i-\\overline{x})^2}}$\n\n$\\Delta d = \\sqrt{(\\frac{\\delta d}{\\delta \\lambda}\\cdot \\Delta \\lambda)^2 + (\\frac{\\delta d}{\\delta K}\\cdot \\Delta K)^2 } = \\sqrt{(\\frac{1}{K}\\Delta \\lambda)^2 + (\\frac{\\lambda}{K^2}\\Delta K)^2}$\n\n\n```python\nfrom sklearn.metrics import mean_squared_error\n\nfinal_k = np.array([(regr_33.coef_[0] + regr_32.coef_[0]) / 2]).reshape(1, 1)\nfinal_b = np.array([(regr_33.intercept_ + regr_32.intercept_) / 2])\n\nXs = np.concatenate((X_32, X_33))\nys = np.concatenate((y_32, y_33))\n\nmse = mean_squared_error(ys, (final_k * Xs + final_b).reshape(10))\n\n\ndelta_k = np.sqrt(mse / np.sum((Xs - np.mean(Xs)) ** 2))\ndelta_d = np.sqrt((delta_alpha / final_k[0][0]) ** 2 + (alpha * delta_k / final_k[0][0] ** 2) ** 2)\n\nprint('\u041f\u043e\u0433\u0440\u0435\u0448\u043d\u043e\u0441\u0442\u044c \u0434\u043b\u044f \u0441\u0440\u0435\u0434\u043d\u0435\u0433\u043e \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u044f \u043c\u0435\u0436\u0434\u0443 \u0449\u0435\u043b\u044f\u043c\u0438: {} \u043c\u043a\u043c'.format(delta_d * 10 ** 6))\n```\n\n \u041f\u043e\u0433\u0440\u0435\u0448\u043d\u043e\u0441\u0442\u044c \u0434\u043b\u044f \u0441\u0440\u0435\u0434\u043d\u0435\u0433\u043e \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u044f \u043c\u0435\u0436\u0434\u0443 \u0449\u0435\u043b\u044f\u043c\u0438: 18.444950130275327 \u043c\u043a\u043c\n\n\n\u0422\u043e\u0433\u0434\u0430 $d = \\overline d + \\Delta d = (177 \\pm 18) \\space \u043c\u043a\u043c$\n\n### \u0412\u044b\u0432\u043e\u0434\u044b\n\n\u0412 \u0445\u043e\u0434\u0435 \u0434\u0430\u043d\u043d\u043e\u0439 \u043b\u0430\u0431\u043e\u0440\u0430\u0442\u043e\u0440\u043d\u043e\u0439 \u0440\u0430\u0431\u043e\u0442\u044b \u0431\u044b\u043b \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u043d \u043e\u043f\u044b\u0442 \u042e\u043d\u0433\u0430 \u0434\u043b\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u044f \u043c\u0435\u0436\u0434\u0443 \u0449\u0435\u043b\u044f\u043c\u0438 \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u0434\u043b\u0438\u043d\u044b \u0432\u043e\u043b\u043d\u044b $\\lambda$, \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u044f \u043c\u0435\u0436\u0434\u0443 \u043e\u0431\u044a\u0435\u043a\u0442\u043e\u043c \u0438 \u044d\u043a\u0440\u0430\u043d\u043e\u043c $L$, \u0430 \u0442\u0430\u043a\u0436\u0435 \u0448\u0438\u0440\u0438\u043d\u044b \u0438\u043d\u0442\u0435\u0440\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u043f\u043e\u043b\u043e\u0441\u044b $\\Delta x$. \u0411\u044b\u043b \u043d\u0430\u0439\u0434\u0435\u043d \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0435\u043d\u0442 $K=\\frac{\\lambda}{d}$ \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u043b\u0438\u043d\u0435\u0439\u043d\u043e\u0439 \u0440\u0435\u0433\u0440\u0435\u0441\u0441\u0438\u0438 $\\Delta x(L) = \\frac{\\lambda}{d}\\cdot L$, \u043e\u0442\u043a\u0443\u0434\u0430 \u0431\u044b\u043b\u043e \u0432\u0437\u044f\u0442\u043e \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u0435 $d = \\frac{\\lambda}{K}$. \u042d\u0442\u043e \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u0435 \u043c\u0435\u0436\u0434\u0443 \u0449\u0435\u043b\u044f\u043c\u0438 \u043f\u043e\u043b\u0443\u0447\u0438\u043b\u043e\u0441\u044c \u0440\u0430\u0432\u043d\u044b\u043c $(177 \\pm 18)$ \u043c\u043a\u043c \n", "meta": {"hexsha": "bc15a586eb8f82818b81c2fbceffba121e84a2aa", "size": 63898, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lab4.02/report.ipynb", "max_stars_repo_name": "sevakon/physics-labs", "max_stars_repo_head_hexsha": "2dfe206b13b41d96210f6bcf40def32e6748eceb", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2020-06-05T16:45:14.000Z", "max_stars_repo_stars_event_max_datetime": "2020-06-28T22:34:35.000Z", "max_issues_repo_path": "lab4.02/report.ipynb", "max_issues_repo_name": "sevakon/physics-labs", "max_issues_repo_head_hexsha": "2dfe206b13b41d96210f6bcf40def32e6748eceb", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 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YES\n2. YES", "lm_q1_score": 0.6926419958239131, "lm_q2_score": 0.6477982043529715, "lm_q1q2_score": 0.44869224115418926}} {"text": "Inter-areal balanced amplification enhances signal propagation in a large-scale circuit model of the primate cortex\n======\n\nJoglekar MR, Mejias JF, Yang GR, Wang XJ.. Neuron. 2018 Apr 4;98(1):222-234.e8. doi: 10.1016/j.neuron.2018.02.031. Epub 2018 Mar 22. PMID: 29576389.\n\n\nJournal Club Tuesday 10th April 2018\n------\n\n\n#### Abstract\nThis notebook implements an interative version of the models in the Joglekar et al 2018 paper.\n\n## Introduction\nThe paper discusses signal propagation from \nThis notebook follows the stucture of the Joglekar et al. 2018 paper [1]:\n* Firstly describing the rate model; \n* Secondly, describing the laminar model; \n* Finally, describig the spiking network model.\n\n\n```python\n# LIBRARY\n# vector manipulation\nimport numpy as np\n# math functions\nimport math \nimport sys\n\n# THIS IS FOR PLOTTING\n%matplotlib inline\nimport matplotlib.pyplot as plt # side-stepping mpl backend\nimport warnings\nwarnings.filterwarnings(\"ignore\")\n```\n\n## The Rate Model\nThe rate model is motivated by work in a previous wang paper [2]. The equations describe excitatory (E) and inhibitory (I) firing rates of a neuronal population:\n\\begin{equation} \\tau_E\\frac{dv_E}{dt} =-v_e+\\beta_E[I_E]_{+}, \\end{equation}\n\\begin{equation} \\tau_I\\frac{dv_I}{dt} =-v_I+\\beta_I[I_I]_{+}, \\end{equation}\nwhere $[I]_+=\\max(I,0)$ is the input to the system, $\\tau_E$ and $\\tau_I$ are time constants. $\\beta_E$ and $\\beta_I$ are the slopes of the f-l??? curves.\n\n### Parameters\nThe parameters for the model are:\n\n| Parameter | Excitatory (E) | Inhibitory (I) |\n|------|------|------|\n| $\\tau$ (ms)| 20|10|\n| $\\beta$ | 0.066|0.351|\n\n### Model Input\nThe incoming current is given by \n\\begin{equation}\nI^i_E=(1+\\eta h_i)(w_{EE} v^i_E+I_{lr,E^i})-w_{EI}v_I^i+I_{ext,E^i}\n\\end{equation}\n\n\\begin{equation}\nI^i_I=(1+\\eta h_i)(w_{IE} v^i_E+I_{lr,I^i})-w_{II}v_I^i+I_{ext,I^i}\n\\end{equation}\nwhere $I^i_E$ and $I^i_I$ denote the input currents to the excitatory and inhibitory populations from area i.\n\n$w$ denotes the local-circuit excitatory and inhibitory connections.\n\n$h$ is the hierarchical position, it is normalised between $0$ and $1$.\n\n$\\eta$ scales the exciatory firing rate, it is set as $\\eta=0.68$.\n\nThe local connectivity parameters for the input are:\n\n| Parameter | Value |\n|------|------|\n| $\\mu_{IE}$ | 25.3|\n| $w_{EE}$ | 24.3|\n| $w_{IE}$ | 12.2|\n| $w_{II}$ | 12.5|\n\n\n```python\n#### Model Parameters\n\nmu_IE=25.3\nmu_EE=33.7\n\ndef Input(input_amp,time):\n I=np.zeros((len(time))) # CURRENT (INPUT)\n \n for k in range (0,len(time)):\n if time[k] >0.1 and time[k]<0.35:\n I[k]=input_amp # Input change\n return I\n\n\n# Numerical Solution to the Izhikevich model\ndef DiscreteMillerRateModel(u_E,u_I,I,kI,w):\n tau_E=1000/20\n tau_I=1000/10\n w_EE=w\n w_IE=kI*w\n w_II=w\n w_EI=kI*w\n u_E = u_E + 0.0001*(-(1+w_EE)*u_E+w_EI*u_I+I)/tau_E # Discrete form of membrane potential of the neuron \n u_I = u_I + 0.0001*(-(w_IE)*u_E+(w_II-1)*u_I+I)/tau_I # Discrete form of membrane recovery variable\n # u_E = u_E + 0.0001*(-1*u_E+I)/tau_E # Discrete form of membrane potential of the neuron \n #u_I = u_I + 0.0001*(-1*u_I+I)/tau_I # Discrete form of membrane recovery variable\n return u_E,u_I\n\ntime=np.arange(0,2,0.0001) # time period 1000ms (1s)\ninput_amp=0\nI=Input(input_amp,time)\nfig, ax1 = plt.subplots(figsize=(12,3))\nax1.plot(time, Input(input_amp,time), 'b-') # y label\nax1.set_xlabel('time (s)') # x label\n# Make the y-axis label, ticks and tick labels match the line color.\nax1.set_ylabel('Input mV', color='b')\nax1.set_ylim(0,3*input_amp) # setting the plotting range\nplt.title('Figure 2: Input to the system')\nplt.show() \n```\n\n\n```python\nvE=1*np.ones(len(time))\nvI=1*np.ones(len(time))\n\nfor k in range (0,len(time)-1):\n vE[k+1], vI[k+1]= DiscreteMillerRateModel(vE[k],vI[k],I[k],1.1,4)\n\nfig, ax1 = plt.subplots(figsize=(12,3))\nax1.plot( time,vE, 'b-', label = 'Output')\nax1.set_xlabel('time (ms)')\n# Make the y-axis label, ticks and tick labels match the line color.\nax1.set_ylabel('Output mV', color='b')\nax1.tick_params('y', colors='b')\nax2 = ax1.twinx()\nax2.plot(time, I, 'r', label = 'Input')\nax2.set_ylim(0,input_amp*20)\nax2.set_ylabel('Input (mV)', color='r')\nax2.tick_params('y', colors='r')\nfig.tight_layout()\nax1.legend(loc=1)\nax2.legend(loc=3)\nplt.show()\n```\n\nThe background firing rate are 10Hz for excitatory rate and 35Hz for inhibitory, which has been subtracted for the figures.\n\nThe long-range input currents are given by \n\\begin{equation}\nI_{lr,E}^i=\\mu_{EE}\\sum_{j=1}^{N} FLN_{ij}v_{E}^j\n\\end{equation}\n\\begin{equation}\nI_{lr,I}^i=\\mu_{IE}\\sum_{j=1}^{N} FLN_{ij}v_{E}^j\n\\end{equation}\n\n\nTo model global \n\nThe parameters for the model are:\n\n| Parameter | weak GBA | strong GBA |\n|------|------|------|\n| $w_{EI}$ | 19.7|25.2|\n| $\\mu_{EE}$ | 33.7|51.5|\n\n\n```python\n# Numerical Solution to the Izhikevich model\ndef DiscreteRateModel(u_E,u_I,I_E,I_I):\n tau_E=20\n tau_I=10\n beta_E=0.066\n beta_I=0.351\n u_E = u_E + 0.0001*(-u_E+beta_E*I_E)/tau_E # Discrete form of membrane potential of the neuron \n u_I = u_I + 0.0001*(-u_I+beta_I*I_I)/tau_I # Discrete form of membrane recovery variable\n return u_E,u_I\n```\n\n\n```python\n\n\nFLN=np.random.rand(3,3)\n\ntime=np.arange(0,2,0.0001) # time period 1000ms (1s)\nvE=np.zeros((3,len(time)))\nvI=np.zeros((3,len(time)))\nINTERNAL= np.dot(FLN,vE[:,0])\nINTERNAL= np.dot(FLN,vI[:,0])\nu_E=np.zeros(3)\nu_I=np.zeros(3)\n\nI_E=np.zeros(3)\nI_I=np.zeros(3)\n\n\n\nfor k in range (0,len(time)):\n for i in range(0,3):\n i+k\n u_E, v_E= DiscreteRateModel(uE,u_I,I_E,I_I)\n\n \n \n\n\ndef I_E(input_duration,input_amp,time):\n eta=0.68\n w_EE=24.3\n mu_IE=25.3\n w_IE=12.2\n I_E=(1+eta)*(w_EE+v_e+)\n I=np.zeros((len(time))) # CURRENT (INPUT)\n \n for k in range (0,len(time)):\n if time[k] >0.1 & time[k]<0.35:\n I[k]=input_amp # Input change\n return I\n\n```\n\n#### References\n\n1 Joglekar MR, Mejias JF, Yang GR, Wang XJ. Inter-areal Balanced Amplification Enhances Signal Propagation in a Large-Scale Circuit Model of the Primate Cortex. Neuron. 2018 Apr 4;98(1):222-234.e8. doi: 10.1016/j.neuron.2018.02.031. Epub 2018 Mar 22. PMID: 29576389.\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "694e8234e88f5d0da6c13da2f0931e26010d2860", "size": 38837, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Joglekar et al 2018/Joglekar et al 2018 Neuron - Rate Model.ipynb", "max_stars_repo_name": "beccamb/Decision-Making-Models", "max_stars_repo_head_hexsha": "da14b80ebe1ec658dc48fe1370fb26942eb5e64e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Joglekar et al 2018/Joglekar et al 2018 Neuron - Rate Model.ipynb", "max_issues_repo_name": "beccamb/Decision-Making-Models", "max_issues_repo_head_hexsha": "da14b80ebe1ec658dc48fe1370fb26942eb5e64e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Joglekar et al 2018/Joglekar et al 2018 Neuron - Rate Model.ipynb", "max_forks_repo_name": "beccamb/Decision-Making-Models", "max_forks_repo_head_hexsha": "da14b80ebe1ec658dc48fe1370fb26942eb5e64e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-03-08T11:03:59.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-08T11:03:59.000Z", "avg_line_length": 109.709039548, "max_line_length": 19440, "alphanum_fraction": 0.8395859618, "converted": true, "num_tokens": 2158, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718435083355187, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4486750168891868}} {"text": "```python\n%matplotlib inline\n```\n\n\n```python\nimport os\nos.chdir(\"../\")\n\nfrom markowitz.structs import DBCache\nfrom markowitz.layout_parser import from_file\nfrom markowitz import consumme_window\nfrom matplotlib.pyplot import show\n\ndef main(db, layout_file):\n layout = from_file(layout_file)\n\n for window in layout:\n consumme_window(window, db)\n \n show()\n\n\ndb = DBCache(\"cac40.db\")\n```\n\n# Repr\u00e9sentation graphique de la Th\u00e9orie moderne du portefeuille\n\n## L\u00e9o Duret & Adrien Chaillout-Morlot\n\n## Introduction\n\nLa Th\u00e9orie Moderne du Portefeuille De Markowitz est un mod\u00e8le incontournable des th\u00e9ories de gestion des actifs.\nElle a \u00e9t\u00e9 propos\u00e9 en 1952 par l'\u00e9conomiste am\u00e9ricain Harry Markowitz, le rendant laur\u00e9at d'un prix Nobel en Sciences Economiques. On connait bien les repr\u00e9sentations graphiques classiques des portefeuilles \u00e0 deux titres qui ne font pas parler les nuances des bases de donn\u00e9es. Ainsi dans ce rapport nous essayerons d'apporter de nouveaux \u00e9l\u00e9ments sur ces repr\u00e9sentations graphiques en construisant une application et des outils permettant cette visualisation. \n\n\n\n## 1. Mod\u00e8le\n\nAfin de comprendre l'origine les graphiques qui vont suivre nous proposons au lecteur un rappel sur la th\u00e9orie moderne du portefeuille.\n\n### 1.1 Hypoth\u00e8ses \n\n#### 1.1.1 Rationalit\u00e9 des agents\n\nLa th\u00e9orie moderne du portefeuille ne propose pas de portefeuille optimal, seulement un ensemble paires risque/rendement optimaux (la fronti\u00e8re efficiente). L'hypot\u00e8se de rationnalit\u00e9 implique qu'il existe un seul portefeuille qui maximise l'utilit\u00e9 de l'agent, celui avec le meilleur rendement au risque le plus faible (selon l'aversion au risque de l'agent).\n\n#### 1.1.2 Aversion aux risques\n\nDans le mod\u00e8le de markowitz le rendement se paie au prix du risque. En effet, l'investisseur n'augmentera son risque si et seulement si son \u00e9sperance de gains augmente.\n\n\n\n### 1.2 Rendement-Risque\nLes hypot\u00e8ses en t\u00eates nous pouvons ainsi definir les variables de mesure du rendement et du risque dans le mod\u00e8le de markowitz.\nTout d'abord on proc\u00e8de \u00e0 une analyse individuelle des titres en \u00e9tudiant les variations historiques. Dans notre cas nous avons \u00e9tudi\u00e9 les titres du CAC40 entre le 20/10/2015 et les 19/10/2015.\n\nPour un titre possedant $t$ valeurs on calcule la variation entre les valeurs de fermeture du march\u00e9:\n\n\\begin{align}\n V_t = \\ln(X_t) - \\ln(X_{t-1})\n\\end{align}\n\nainsi que le rendement, soit l'\u00e9sperance de variation:\n\n\\begin{align}\n \\overline{R_{titre}} = E(R_{titre} )=\\frac{1}{N}\\sum\\nolimits_{i=0}^{N}R_i \n\\end{align}\n\nEnfin afin de determiner le risque d'un actif calcule l'\u00e9cart type des variations, que l'on peut interpreter comme \u00e9tant la volatilit\u00e9 du titre durant la p\u00e9riode \u00e9tudi\u00e9e.\n\n\\begin{align}\n \\sigma_{titre}=\\sqrt{\\frac{\\mathrm{\\sum\\nolimits_{i=0}^{N}(R_i-\\overline{R_{titre}})^2} }{\\mathrm{(N-1)} } }\n\\end{align}\n\n### 1.3 Portefeuille\nOn consid\u00e8re un portefeuille comme \u00e9tant une collection d'actifs, ainsi par croisements des donn\u00e9es on peut observer les matrices de covariance et par extension de corr\u00e9lation.\n\nOn rappelle le calcul de la covariance:\n\n\\begin{align}\n Cov(R_A,R_B)= E(R_A \\cdot R_B ) - E(R_A)\\cdot E(R_B)\n\\end{align}\n\nEt du coefficient de corr\u00e9lation $\\varphi$:\n\n\\begin{align} \n \\varphi_{AB}= \\frac{\\mathrm{Cov( R_A \\cdot R_B )} }{\\mathrm{ \\sigma_A \\sigma_B } }\n\\end{align}\n\n\nExemple de matrice de covariance pour 3 titres\n\n\\begin{pmatrix}\n Var(R_A)&Cov(R_A, R_B)&Cov(R_A, R_C)\\\\\n Cov(R_B, R_A)&Var(R_B)&Cov(R_B, R_C)\\\\\n Cov(R_C, R_A)&Cov(R_C, R_B)&Var(R_C)\n\\end{pmatrix} \n\nIdem pour la matrice de corr\u00e9lation\n\n\\begin{pmatrix}\n 1&\\varphi_{AB}&\\varphi_{AC}\\\\\n \\varphi_{BA}&1&\\varphi_{BC}\\\\\n \\varphi_{CA}&\\varphi_{CB}&1\n\\end{pmatrix} \n\n\n```python\n# Nous avons ajout\u00e9 la possibilit\u00e9 d'afficher ces variables dans un r\u00e9capitulatif du portefeuille\n\np = db.load(\"renault\",\"peugeot\",\"lvmh\")\n\np.recap()\n\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
renaultpeugeotlvmh
len515515515
avg0.000355960.0007866890.000911761
stdv0.01973840.02120040.0143329
COVAR
renault0.0003896060.000333820.000152542
peugeot0.000333820.0004494560.000163033
lvmh0.0001525420.0001630330.000205432
CORR
renault10.7977310.539191
peugeot0.79773110.536535
lvmh0.5391910.5365351
\n
\n\n\n\n### 1.4 Base de donn\u00e9es\n\n#### 1.4.1 Aper\u00e7u\nNous avions \u00e0 notre disposition des fichiers Excel d'actifs du CAC 40.\n\nL'interval est 20/10/2015 - 19/10/2017 pour 515 entr\u00e9es par tableau, entr\u00e9es ayant toutes leurs valeurs.\n\n##### Ex: Airliquide\n\n\n| ISIN | JOUR | OUVR | PHAUT | PBAS | CLOT | VOLUME |\n|--------------|------------|----------|----------|----------|----------|---------|\n| FR0000120073 | 20/10/2015 | 100 | 100.1818 | 98.4546 | 99.0909 | 833851 |\n| FR0000120073 | 21/10/2015 | 99.6364 | 100.0455 | 98.3636 | 99.6818 | 714208 |\n| FR0000120073 | 22/10/2015 | 99.5 | 103.4546 | 99.1818 | 103.0455 | 1264656 |\n| FR0000120073 | 23/10/2015 | 104.0454 | 105.9546 | 103.6818 | 105.3636 | 1224405 |\n\n\n\n#### 1.4.2 Transformation\n\nAfin de pouvoir construire un application permettant d'afficher les graphiques il \u00e9tait important de pouvoir recuperer les donn\u00e9es au m\u00eame endroit. Dans cette optique nous avons cr\u00e9\u00e9 un script `docs/script/excel_to_db.py` permettant de transferer un ensemble de fichiers excel sur une base de donn\u00e9e (tel que un fichier soit un tableau). On peut observer ci dessous l'instruction de cr\u00e9ation des tableaux sur une base de donn\u00e9es [SQLite](https://sqlite.org/index.html).\n\n\n\n## 2. Construction de l'application\n\n### 2.1 Avant de commencer\n\n#### 2.1.1 Objectifs\n\nAfin de pouvoir automatiser la production de graphiques de la th\u00e9orie moderne du portefeuille il fallait determiner un certain nombre d'objectifs \u00e0 respecter, nous fournissant ainsi une direction pour la construction de l'application.\n\n- Tout d'abord il est certain qu'il fallait faire en sorte que l'application soit facile d'utilisation. La construction de graphiques simples se doit d'\u00eatre simple, et donc ne doit pas demander \u00e0 l'utilisateur de connaitre les fonctionnalit\u00e9s obscures d'un tableur ou la complexit\u00e9 d'un language de programmation.\n\n- En suite une interface modulable serait de mise car on ne peut pas se contenter de montrer qu'une v\u00e9rit\u00e9. Laisser libre cours \u00e0 l'utilisateur de tordre la donn\u00e9e dans le sens qu'il souhaite \u00e9tait donc un \u00e9l\u00e9ment indispensable.\n\n- Enfin construire un application a part enti\u00e8re est d\u00e9j\u00e0 en soit une belle aventure, cependant le format pousse une utilisation pr\u00e9-mach\u00e9 du logiciel. Ainsi une part du projet est consacr\u00e9 a l'\u00e9laboration d'utilitaires pour faciliter la programme d'outils connexe \u00e0 la th\u00e9orie moderne du portefeuille.\n\n#### 2.1.2 Les outils\n\nPour ce projet nous avons d\u00e9cid\u00e9 d'utiliser python, un language de programmation g\u00e9n\u00e9ral cr\u00e9e par Guido van Rossum et publi\u00e9 en 1991. Cependant, avec maturit\u00e9 du language, nous n'avions pas \u00e0 construire tous les outils permettant l'affichage de graphique. Ainsi, nous avons utilis\u00e9 plusieurs librairie permettant de construire les courbes ainsi que de structurer les informations pendant l'execution du programme.\n\nLes outils utilis\u00e9s sont les suivants:\n- [Numpy](https://www.numpy.org/) La librairie de r\u00e9f\u00e9rence en mati\u00e8re de calculs math\u00e9matiques et de travail avec les listes. les autres librairies d\u00e9pendent de celle-ci.\n- [Pandas](http://pandas.pydata.org/) une librairie permettant l'utilisation de DataFrames (sorte de tableau) semblables \u00e0 celles en R.\n- [Matplotlib](https://matplotlib.org/) la librairie pour l'affichage des graphiques.\n\n\n### 2.2 L'impl\u00e9mentation\n\n#### 2.2.1 Les produits selon markowitz et acc\u00e8s \u00e0 la base de donn\u00e9e\n\n##### 2.2.1.1 Titres\nRappelons nous des m\u00e9thodes de calculs vu pr\u00e9c\u00e9dements pour imaginer un concept de \"titre financier de markowitz\". Nous avons d\u00e9cris un calcul se d\u00e9roulant \u00e0 partir des valeurs pass\u00e9s du titres ainsi la premi\u00e8re chose \u00e0 faire est acceder la base de donn\u00e9e pour r\u00e9cuperer cet historique. Heureusement pour nous la librairie pandas nous facilite la tache en nous permettant de cr\u00e9er une DataFrame directement depuis une requete de la base de donn\u00e9e.\n\n\n```python\n# Nous travaillerons sur les valeurs de clotures ainsi la requete se mat\u00e9rialise de cette mani\u00e8re:\nimport pandas as pd\nimport sqlite3 as sql\n\nconn = sql.connect(\"cac40.db\")\ndf = pd.read_sql(\"SELECT clot FROM AXA\", conn)\n\nprint(df.head(n=5)) # 5 premi\u00e8res valeurs\n```\n\n clot\n 0 22.915\n 1 23.095\n 2 23.820\n 3 24.050\n 4 24.155\n\n\n\n```python\n# On calcule la variation:\n\nprint(df[\"clot\"].pct_change().head(n=5)) # 5 premi\u00e8res valeurs\n```\n\n 0 NaN\n 1 0.007855\n 2 0.031392\n 3 0.009656\n 4 0.004366\n Name: clot, dtype: float64\n\n\nDans notre cas les instances de clotures ayant toutes des valeurs coh\u00e9rentes, il n'est pas necessaire de nettoyer le tableau des valeurs manquantes ou in\u00e9xactes. Pour d'optimiser les op\u00e9rations de lecture de la structure \"Titre\" nous stockerons ind\u00e9pendamment les valeurs constante fr\u00e9quemments utilis\u00e9 calcul\u00e9s \u00e0 partir du tableau des variation, soit l'\u00e9sp\u00e9rance et l'\u00e9quart type. Nous gardons tout de meme le tableau des variations pour le calcul de la corr\u00e9lation des portefeuilles.\n\n##### 2.2.1.2 Portefeuille\n\nComme d\u00e9finit pr\u00e9c\u00e9demment, un portefeuille est un ensemble de titres il est donc important que nous utilisions les \u00e9quations g\u00e9n\u00e9rales et que nous puissions passer autant de titres que nous voulons tout en gardant la possibilit\u00e9 de calculer leur \u00e9sperance. Python nous facilite la tache avec une notation permettant de passer autant d'arguments \u00e0 une fonction/m\u00e9thode `def __init__(self, *args):`. Il ne reste plus qu'\u00e0 implementer les fonctions de calculs de l'\u00e9sp\u00e9rance et de la variance (ci-dessous).\n\n###### L'\u00e9sperance\nSoit $\\{R_1, ..., R_n\\}$ titre dans le portefeuille\n\n\\begin{align}\n E(R_p) = \\sum_{i=0}^{n}\\alpha_i \\cdot E(R_i)\n\\end{align}\n\n\n\n\n```python\n# Esperance\ndef avg(self, dot: m_types.FLOAT_COLLECTION) -> m_types.M_FLOAT: # dot: liste de nombre \u00e0 virgule\n mus = [el.avg for el in self.assets.values()] # Liste des \u00e9sp\u00e9rances des titres\n assert(len(dot) == len(mus)) # Assure le nombre de dotations rapport\u00e9 \u00e0 la taille du portefeuille\n \n # Somme du produit des \u00e9sp\u00e9rances (mus) et des dotations (dot)\n result: float = 0\n for i in range(len(dot)): \n result += dot[i] * mus[i]\n \n return m_types.M_FLOAT(result)\n```\n\n###### L'\u00e9cart type\n\n\\begin{align}\n Var(R_P) = (\\sum_{i=0}^{n}\\alpha_i \\cdot \\sigma_i)^2\n\\end{align}\n\nC'est une forme quadratique d'ordre N donc il existe une matrice sym\u00e9trique $ N \\cdot N $ associ\u00e9e \nPour 3 titres on trouve\n\n\\begin{pmatrix}\n \\alpha^2_{1}& \\frac{\\mathrm{\\alpha_1\\alpha_2} }{\\mathrm{2}} &\\frac{\\mathrm{\\alpha_1\\alpha_3} }{\\mathrm{2}}\\\\\n \\frac{\\mathrm{\\alpha_1\\alpha_2} }{\\mathrm{2}}&\\alpha^2_2&\\frac{\\mathrm{\\alpha_2\\alpha_3} }{\\mathrm{2}}\\\\\n \\frac{\\mathrm{\\alpha_1\\alpha_3} }{\\mathrm{2}}&\\frac{\\mathrm{\\alpha_2\\alpha_3} }{\\mathrm{2}}&\\alpha^2_{3}\n\\end{pmatrix}\n\nAinsi\n\n\\begin{align}\n A = M_{quadratique} \\cdot M_{covar}\\\\\n\\end{align}\n\nLa trace de A est la somme des variances pond\u00e9r\u00e9s par les $\\alpha_i$, soit la variance de $R_p$\n\n\\begin{align}\n trace(A)= Var(R_P)\n\\end{align}\n\n\\begin{align}\n \\sigma_{R_P} = \\sqrt{Var(R_P)} = \\sqrt{trace(A)}\n\\end{align}\n\n\n```python\n# Ecart type\n\ndef stdv(self, dot: m_types.FLOAT_COLLECTION) -> m_types.M_FLOAT:\n assert(len(dot) == len(self.assets)) # Assure le nombre de dotations rapport\u00e9s \u00e0 la taille du portefeuille\n \n product = m_maths.dot(m_maths.symmetric_matrix(dot), self.covar_matrix) # Produit matriciel de la matrice symm\u00e9trique et de covariance\n trace = m_maths.sqrt(m_maths.trace(product)) # Racine carr\u00e9 de la trace du produit matriciel\n \n return m_types.M_FLOAT(trace)\n```\n\n##### 2.2.1.3 L'interface de la base de donn\u00e9e\n\nLes produits ci-dessus sont construits \u00e0 partir d'une base de donn\u00e9e, cependant on remarque que si l'on souhaite afficher plusieurs fois le m\u00eame titres avec l'application, les memes informations seront demand\u00e9 plusieurs fois \u00e0 la base de donn\u00e9e. C'est comme si \u00e0 chaque fois que vous rejoignez votre hotel on vous demandait de repasser par la reception pour prendre une chambre.\n\nPour remedier \u00e0 ce probl\u00e8me et optimiser l'application nous avons mis en place un schema tr\u00e8s simple mais \u00e9fficace. Tout d'abord on va distinguer deux elements dans la base de donn\u00e9e. Ces \u00e9l\u00e9ments vont se comporter comme un gardien et un r\u00e9ceptioniste. En effet si le gardien vous reconnait il vous laisse acceder \u00e0 votre chambre sinon c'est la case reception. Traduisons cela et vous avez une instance d'object servant de gardien dont le travaille est de determiner si le titre \u00e0 d\u00e9j\u00e0 \u00e9t\u00e9 cr\u00e9\u00e9 sinon demander \u00e0 l'autre objet (la r\u00e9c\u00e9ption) de s'en charger et de vous le remettre.\n\nDe cette mani\u00e8re nous sommes assur\u00e9 que chaque titre et portfeuille est charg\u00e9 au maximum une fois. Ceci conclut notre module structure de donn\u00e9e dont l'interface est basique: les services ayant besoin d'un titre n'ont qu'a effectuer une demande \u00e0 l'objet servant de gardien.\n\n#### 2.2.2 L'architecture\n\nConcernant l'architecture du programme elle est plutot simple:\n- D'abord la d\u00e9construction des fichiers de layout en classes connexes contenant les informations des differents graphiques\n- En suite lecture des objects de graphique, chargement de la base de donn\u00e9e et des objets graphiques (prochaine section)\n- Enfin assemblage des titres et des objets de graphiques\n- Affichage des graphiques\n\n\n\n### 2.3 Les graphiques\n\nPour afficher les graphiques nous avons cr\u00e9\u00e9 une syntaxe nous permettant de lire les graphiques directemment depuis un fichier contenant seulement les informations sur les titres et le type de graphique \u00e0 utiliser. Ces type de graphique sont en fait des r\u00e9f\u00e9rence \u00e0 des fichiers contenant les objects du meme noms. Ainsi les graphiques sont g\u00e9n\u00e9r\u00e9 dynamiquement lorsque leur nom est reconnu.\n\nLa syntaxe des fichiers est plut\u00f4t simple\n- Pour les commentaires: `! ...`\n- Nouvelle fenetre: `& nom_fenetre (configuration...) { matrice... }`\n- Pour la configuration: `(nom=valeur,...)`\n- Pour la matrice:\n - ligne: `[...]`\n - separateur de colonne: `|`\n- Les objects: `nom_objet(titre)` ou `nom_objet(titre/titre/...)` pour plusieurs titres\n\nLe chargements de objects est effectu\u00e9 dynamiquement celon le nom de l'objet appel\u00e9 dans le fichier. Enfin la base de donn\u00e9e charge un titre lorsque que l'analyse du fichier ne trouve qu'un `titre`, sinon un portefeuille contenant les titres est g\u00e9n\u00e9r\u00e9 dans le cas o\u00f9 plusieurs titres sont reconnus (`titre/titre`)\n\n#### 2.3.1 Analyse des titres\n\n##### 2.3.1.1 Distribution normale\n\nAfin d'analyser et comparer les titres nous avons fournit un moyen de repr\u00e9senter graphiquement l'\u00e9sperance et l'\u00e9cart type de ces derniers. En effet les titres \u00e9tant compos\u00e9 d'une large quantit\u00e9 de valeurs, on peut estimer que la loi de leur variation peut s'approximer par une loi normale. Ainsi, dans un premier temps nous avons fournit un objet permettant de calculer les points d'une loi normale de param\u00e8tre $N(\\mu, \\sigma)$\n\nEn cr\u00e9ant un ensemble de points entre $-10$ et $10$ leur applicant cette fonction, nous avons cr\u00e9\u00e9 un emble de points $x$ et $y$ que nous affichons sur un graphique.\n\n\n```python\n# NormalGraph.const1 = np.sqrt(2*np.pi)\n# args[0] = mu\n# args[1] = sigma\n\ndef densite(point, *args):\n t1 = 1/(args[1]*NormalGraph.const1)\n x = (point-args[0])/args[1]\n t2 = np.exp(-0.5*x*x)\n return t1*t2\n```\n\n\n```python\nwith open(\"docs/layouts/normale.ly\") as ly:\n print(ly.read())\n```\n\n ! docs/layouts/layout3.ly\n \n & Normale (scale=100) {\n \t[ NormalGraph(axa) | NormalGraph(lvmh) ]\n [ NormalGraph(peugeot) ]\n }\n \n\n\n\n```python\nmain(db, \"docs/layouts/normale.ly\")\n```\n\n##### 2.3.1.1 Estimation de la densit\u00e9 par la m\u00e9thode des noyaux\n\nCependant la distribution normale n'est pas un bon indicateur pour exprimer les series \u00e0 evennements improbables. En effet les \"pattes\" de la loi normale nous montrent que les hautes variations ont tr\u00e8s peut de chance d'arriver. Cependant sur les march\u00e9s financiers ces evennements ne sont pas a sous-estimer. Pour demontrer cette r\u00e9alit\u00e9 il faut trouver un moyen d'afficher la \"vraie\" taille des pattes de la loi normale. Pour ce faire nous avons impl\u00e9ment\u00e9 un moyen d'observer une approximation de la densit\u00e9 des la s\u00e9rie en utilisant la m\u00e9thode de l'\u00e9stimation de la densit\u00e9 par noyau, Kernel Density Estimation en anglais, d'ou le nom KDE pour les graphiques.\n\nPour l'estimation de la densit\u00e9 nous avons choisit un noyau normale car nous cherchions \u00e0 estimer la taille des pattes. Et un parametre $h$ de 1.06 selon les recommendations, le resultat paraissant satisfaisant. Le principe de l'estimation de densit\u00e9 par noyau est de cr\u00e9er un ensemble de loi normale et d'en faire la somme. Ainsi on observe une agr\u00e9gation plus importante autour de l'esperance de la s\u00e9rie mais les valeurs plus extremes sont aussi prisent en compte.\n\n\n```python\n# Kde.const1 = np.sqrt(2*np.pi)\n\ndef _kernel(k):\n \"\"\" Kernel of the KDE \"\"\"\n return (1/Kde.const1)*np.exp(-0.5*k*k)\n\ndef densite(point, df, stdv):\n \"\"\" estimation function \"\"\"\n n = len(df-1) # index 0 is NaN\n h = 1.06 * (stdv)/np.power(n, 0.2)\n pre = 1.0/(n*h)\n results = [\n Kde._kernel((point-xi)/h)\n for xi in df\n if not np.isnan(xi)\n ]\n return np.sum(results) * pre\n```\n\n\n```python\nwith open(\"docs/layouts/kde.ly\") as ly:\n print(ly.read())\n```\n\n ! docs/layouts/layout3.ly\n \n & Kde (scale=100) {\n \t[ NormalGraph(axa) Kde(axa) ]\n }\n \n\n\n\n```python\nmain(db, \"docs/layouts/kde.ly\")\n```\n\n#### 2.3.2 Efficacit\u00e9 des portefeuilles\n\n##### 2.3.2.1 G\u00e9n\u00e9ration du plan \u00e0 deux titre\n\nAfin d'afficher un graphique nous devons sp\u00e9cifier un ensemble de dotations \u00e0 passer pour le calcul de l'\u00e9sp\u00e9rance et de la variance du portefeuille.\nDans le cas ou nous avons seulement 2 dotations il suffit de generer $x$ et donner $y = 1-x$ ainsi \u00e0 partir d'une liste de $x$ nous donnons une liste de $y$, nous pouvons passer ces paires directement dans la fonction de calcul. Le plan \u00e0 deux titres peut s'interpreter de cette mani\u00e8re: _Il n'existe pas de meilleur paires rendement/risque que celles sur la courbe_.\n\n\n```python\nwith open(\"docs/layouts/layout1.ly\") as ly:\n print(ly.read())\n```\n\n ! docs/layouts/layout1.ly\n \n & 2PF (scale=100) {\n \t[ EfficientFrontier(axa/lvmh) ]\n }\n \n\n\n\n```python\nmain(db, \"docs/layouts/layout1.ly\")\n```\n\n##### 2.3.2.2 G\u00e9n\u00e9ration du plan barycentrique (pour 3 titres)\nPour trois titres nous devons trouver une mani\u00e8re efficace de g\u00e9n\u00e9rer un triplet de points dont la somme \u00e9quivaut \u00e0 1. Une mani\u00e8re de faire cela est de calculer des points sur le plan barycentrique. Heureusement, l'industrie du graphisme num\u00e9rique \u00e0 developp\u00e9 ces algorithmes afin de pouvoir remplir des triangles et afficher des mod\u00e8les color\u00e9. L'algo rythme utilis\u00e9 est plutot simple:\n\nOn fixe un point (en l'occurence z) auquel on retire \u00e0 plusieur \u00e9tapes un nombre 's' ( $ 1 / totaletapes$ ) puis on calcule tous les points x, y qui correspondent \u00e0\n$x + y = nbetapes*s$\n\n\n\n\n\n```python\nwith open(\"docs/layouts/layout2.ly\") as ly:\n print(ly.read())\n```\n\n ! docs/layouts/layout2.ly\n \n & 3PF (scale=100) {\n \t[ EfficientFrontier(axa/lvmh/engie) ]\n }\n \n\n\n\n```python\nmain(db, \"docs/layouts/layout2.ly\")\n```\n\n##### 2.3.2.3 G\u00e9n\u00e9ration du plan \u00e0 N titre\n\nPour afficher la fronti\u00e8re efficiente \u00e0 N titres nous avons du utiliser des m\u00e9thodes plus approximatives. En effet l'algorithme au dessus est d\u00e9j\u00e0 couteux pour un grand nombre de points g\u00e9n\u00e9r\u00e9. Ansi pour N titres nous faisons appel \u00e0 la librairie *numpy* qui permet de calculer des points d'une distribution de *Dirichlet*, et cela de mani\u00e8re al\u00e9atoire. Ainsi nous avons un ensemble de points respectant $\\sum_{i=0}^{n}\\alpha_i = 1$\n\n\n```python\nwith open(\"docs/layouts/layout3.ly\") as ly:\n print(ly.read())\n```\n\n ! docs/layouts/layout3.ly\n \n & NPF (scale=100) {\n \t[ EfficientFrontier(axa/lvmh/engie/peugeot/renault) ]\n }\n \n\n\n\n```python\nmain(db, \"docs/layouts/layout3.ly\")\n```\n\n## Conclusion\n\nAinsi \u00e0 travers ces differents graphiques nous avons pu observer les diff\u00e9rents mani\u00e8re d'observe la th\u00e9orie moderne du portefeuille de markowitz, que se soit du point de vu des titres, ou bien encore de la fonti\u00e8re efficiente \u00e0 toutes les dimensions. Par la th\u00e9orie moderne du portefeuille markowitz nous montre que la diversification des actifs, en baissant le rendement, r\u00e9duit consid\u00e9rablement le risque des agents.\n\n### Critiques\n\nOn observe deux critiques majeurs de la th\u00e9orie moderne du portefeuille:\n\n- La premi\u00e8re critique vient de Benoit Mandelbrot qui estime cette th\u00e9orie trop d\u00e9connect\u00e9 des march\u00e9s financiers. En effet la loi de gauss sous \u00e9value la probabilit\u00e9 d'\u00e9venements improbables (notamment les krachs boursiers). Nous avons observ\u00e9 un example en comparant avec l'estimation par noyau.\n\n- La deuxi\u00e8me critique, plus intuitive, sont que les suppositions n\u00e9o-classiques sont irr\u00e9alistes, les individus ne sont pas rationnels et les variations pass\u00e9 ne se r\u00e9p\u00e8tent pas dans le futur.\n\n### Ouverture\n\nAfin de r\u00e9pondre \u00e0 ces critiques et mieux anticiper les \u00e9venements improbables on peut utiliser la loi de pareto ou les lois de puissance qui on moins tendance \u00e0 les sous evaluer.\n\n\n\nLe code est disponibles sur : [github.com/hyyking/markowitz](https://github.com/hyyking/markowitz)\n\nLa bibliographie : [bibliographie](https://github.com/hyyking/markowitz/tree/master/docs/references)\n", "meta": {"hexsha": "3b8ad1016f8c91b87ba2317657e02ddee645f4d7", "size": 899683, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/rapport.ipynb", "max_stars_repo_name": "hyyking/markowitz", "max_stars_repo_head_hexsha": "257fb200723d1ab5e05e8c48ab08a569ed5efb03", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2019-11-26T03:39:50.000Z", "max_stars_repo_stars_event_max_datetime": "2020-12-03T09:23:13.000Z", "max_issues_repo_path": "docs/rapport.ipynb", "max_issues_repo_name": "hyyking/markowitz", "max_issues_repo_head_hexsha": "257fb200723d1ab5e05e8c48ab08a569ed5efb03", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/rapport.ipynb", "max_forks_repo_name": "hyyking/markowitz", "max_forks_repo_head_hexsha": "257fb200723d1ab5e05e8c48ab08a569ed5efb03", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 885.5147637795, "max_line_length": 404424, "alphanum_fraction": 0.9532090748, "converted": true, "num_tokens": 6800, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.7718435083355187, "lm_q1q2_score": 0.4486750168891868}} {"text": "---\ntitle: \"Order statistics \u2014\u00a0Part 3: Proof of expectation formula for uniform RVs\"\ndate: 2021-03-08\ncategories: [statistics, order statistics]\ntags: [statistics, order statistics, uniform distribution, expectation formula, gamma function, beta function]\npreview_image: /enrightward.github.io/assets/img/order-statistics/part3/gamma.png\n---\n\n\n\n## 1. Introduction\n\nIn the [last post](https://enrightward.github.io/enrightward.github.io/posts/order-statistics-part-2/), we proved the following general formulae, after conducting some numerical experiments to gain intuition:\n\n\\begin{align}\nF_{X_{(k)}}(x) &= \\sum_{j=k}^{n} \\binom{n}{j} F_{X}(x)^{j} (1 - F_{X}(x))^{n-j}, \\newline\nf_{X_{(k)}}(x) &= k \\binom{n}{k} f_{X}(x) \\cdot F_{X}(x)^{k-1} \\, (1 - F_{X}(x))^{n-k}, \\newline\n\\mathbb{E} \\left \\lbrack X_{(k)} \\right \\rbrack &=\nk \\binom{n}{k} \\int_{0}^{1} x \\cdot f_{X}(x) \\cdot F_{X}(x)^{k-1} \\, (1 - F_{X}(x))^{n-k} \\, dx,\n\\end{align}\n\nNote they depend only on $n, k, f_{X}$ and $F_{X}$. We will now round out the exposition of $k^{\\textrm{th}}$ order statistics by specialising these general formulae to the case of the uniform distribution $U(0, 1)$. We will show that:\n\n\\begin{align}\nF_{X_{(k)}}(x) &= \\sum_{j=k}^{n} \\binom{n}{j} x^{j} (1 - x)^{n-j}, \\newline\nf_{X_{(k)}}(x) &= k \\binom{n}{k} x^{k-1} \\, (1 - x)^{n-k}, \\newline\n\\mathbb{E} \\left \\lbrack X_{(k)} \\right \\rbrack &= \\frac{k}{n+1} \\cdot\n\\end{align}\n\nThis post is purely theoretical, and has no code. The mathematical tools we use are the beta and gamma functions, introduced below.\n\n## 2. Specialising the formulae to $U(0, 1)$\n\nIn the case where the $X \\sim U(0, 1)$, we have $F_{X}(x) = x$ and $f_{X}(x) = 1$ on the unit interval, so the formulae computed above specialise to:\n\n\\begin{align}\nF_{X_{(k)}}(x) &= \\sum_{j=k}^{n} \\binom{n}{j} x^{j}(1 - x)^{n - j}, \\newline\nf_{X_{(k)}}(x) &= k \\binom{n}{k} x^{k-1} (1 - x)^{n-k}, \\newline\n\\mathbb{E}[X_{(k)}] &= \nk \\binom{n}{k} \\int_{0}^{1} x \\cdot x^{k-1} \\, (1 - x)^{n-k} \\, dx =\nk \\binom{n}{k} \\int_{0}^{1} x^{k} (1 - x)^{n-k} \\, dx.\n\\end{align}\n\n## 3. Proving the expectation formulae using beta and gamma functions\n\nThe integral in the above expectation formula is an example of the [_beta function_](https://en.wikipedia.org/wiki/Beta_function), defined as\n\n\\begin{align}\nB(z, w) := \\int_{0}^{1} x^{z-1} (1 - x)^{w-1} \\, dx.\n\\end{align}\n\nThis formula makes sense for complex numbers $w$ and $z$ with positive real part, but we are only interested in integer-valued arguments, because our exponents for $x$ and $1-x$ are $k$ and $n-k$, respectively. In the new notation of beta functions, our expectations are written:\n\n\\begin{align}\n\\mathbb{E}[X_{(k)}] = k \\binom{n}{k} \\int_{0}^{1} x^{k} (1 - x)^{n-k} \\, dx = k \\binom{n}{k} B(k+1, n-k+1).\n\\end{align}\n\nThis is interesting to us because the beta function can be computed in terms of the [_gamma function_](https://en.wikipedia.org/wiki/Gamma_function)\n\n\\begin{align}\n\\Gamma(z) := \\int_{0}^{\\infty} x^{z-1} e^{-x} \\, dx,\n\\end{align}\n\nand the gamma function is a generalisation to the (positive half) complex plane of the factorial function $n \\mapsto n!$, hence reduces to factorial computations for integer-valued arguments. Concretely, the relations we need are:\n\n\\begin{align}\nB(z, w) &= \\frac{\\Gamma(z) \\Gamma(w)}{\\Gamma(z + w)}, \\quad \\textrm{and} \\newline\n\\Gamma(n) &= (n-1)!\n\\end{align}\n\nWe will show these relations shortly. For now, observe they imply the formulae we conjectured for the expectations of $X_{(k)}$. Indeed, using each of these results in turn, we have\n\n\\begin{align}\nB(k+1, n-k+1) = \\frac{\\Gamma(k+1) \\Gamma(n-k+1)}{\\Gamma(n+2)} = \n\\frac{k!(n-k)!}{(n+1)!} = \\frac{1}{(n+1) \\binom{n}{k}},\n\\end{align}\n\nand hence\n\n\\begin{align}\n\\mathbb{E}[X_{(k)}] &= k \\binom{n}{k} B(k+1, n-k+1) \n= \\frac{k \\binom{n}{k}}{(n+1) \\binom{n}{k}} = \\frac{k}{n+1} \\cdot\n\\end{align}\n\n## 4. Proving the necessary properties of beta and gamma functions\n\nNow we prove the relations for $B(z, w)$ and $\\Gamma(z)$. To see that $\\Gamma(n) = (n-1)!$, it suffices to prove the recursive formula \n\n\\begin{align}\n\\Gamma(z) = (z-1)\\Gamma(z-1),\n\\end{align}\n\nfor any $z$ with positive real part. For then we would have:\n\n\\begin{align}\n\\Gamma(n) &= (n-1)\\Gamma(n-1) \\newline\n&= (n-1)(n-2)\\Gamma(n-2) \\newline\n\\vdots \\newline\n&= (n-1)(n-2)(n-3) \\ldots (2)(1) = (n-1)!\n\\end{align}\n\nWe show this using the definite integral version of the integration by parts formula:\n\n\\begin{align}\n\\int_{a}^{b} u \\, dv = \\left \\lbrack uv \\right \\rbrack_{a}^{b} - \\int_{a}^{b} v \\, du, \n\\end{align}\n\nwith $u = x^{z-1}$ and $dv = e^{-x} dx$, implying that $du = (z-1)x^{z-2} dx$ and $v = -e^{-x}$. Note this formula holds also in the case where $a$ or $b$ is $\\pm \\infty$, provided we interpret the term $\\left \\lbrack uv \\right \\rbrack_{a}^{b}$ as expressing the appropriate limit(s). We have:\n\n\\begin{align}\n\\Gamma(z) &= \\int_{0}^{\\infty} x^{z-1} e^{-x} \\, dx \\newline\n&= \\lim_{b \\rightarrow \\infty} \\left \\lbrack -x^{z-1} e^{-x} \\right \\rbrack_{0}^{b} + (z-1) \\int_{0}^{\\infty} x^{z-2} e^{-x} \\, dx \\newline\n&= 0 + (z-1) \\int_{0}^{\\infty} x^{z-2} e^{-x} \\, dx \\newline\n&= (z-1)\\Gamma(z-1).\n\\end{align}\n\nNow we prove the formula connecting $B(z, w)$ and $\\Gamma(z)$. I'm taking this proof [from Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Relationship_to_the_gamma_function) and filling in details on the variable substitution using [these lecture notes](https://homepage.tudelft.nl/11r49/documents/wi4006/gammabeta.pdf) (Theorem 2, page 3). Note: Curiously, although the Wikipedia article cites Emil Artin's book [Gamma Functions](https://web.archive.org/web/20161112081854/http://www.plouffe.fr/simon/math/Artin%20E.%20The%20Gamma%20Function%20(1931)(23s).pdf), pages 18-19, Artin's proof is not the one written down there.\n\nIn any case, the argument begins like this:\n\n\\begin{align}\n\\Gamma(z) \\, \\Gamma(w) &= \\int_{x=0}^{\\infty} x^{z-1} e^{-x} \\, dx \\int_{y=0}^{\\infty} y^{w-1} e^{-y} \\, dy \\newline\n&= \\int_{x=0}^{\\infty} \\int_{y=0}^{\\infty} e^{-(x+y)} x^{z-1} y^{w-1} \\, dx \\, dy.\n\\end{align}\n\nNow we introduce an implicit change of variables $s, t$, defined by the conditions $x = st$ and $y=s(1 - t)$. This transformation has Jacobian matrix:\n\n\\begin{align}\nJ(x(s, t), y(s, t)) = \\left( \n\\begin{array}{cc}\n\\frac{\\partial x}{\\partial s} & \\frac{\\partial x}{\\partial t} \\newline\n\\frac{\\partial y}{\\partial s} & \\frac{\\partial y}{\\partial t}\n\\end{array} \\right) = \\left( \n\\begin{array}{cc}\nt & s \\newline\n1-t & -s\n\\end{array} \\right),\n\\end{align}\n\nwith determinant $\\det(J) = -st - s(1-t) = -s$. It follows that \n\n\\begin{align}\ndx \\, dy = \\left\\| \\det{J} \\right\\| ds \\, dt = s \\, ds \\, dt. \n\\end{align}\n\nObserve that since $x$ and $y$ range over $[0, \\infty)$ and $x + y = st + s(1-t) = s$, then $s$ must range over \n$[0, \\infty)$, too. On the other hand,\n\n\\begin{align}\nt = \\frac{x}{s} = \\frac{x}{x + y},\n\\end{align}\n\nso $t$ ranges only over the unit interval $[0, 1]$. Making these substitutions gives:\n\n\\begin{align}\n\\Gamma(z) \\cdot \\Gamma(w) &=\n\\int_{x=0}^{\\infty} \\int_{y=0}^{\\infty} e^{-(x+y)} x^{z-1} y^{w-1} \\, dx \\, dy \\newline\n&= \\int_{s=0}^{\\infty} \\int_{t=0}^{1} e^{-s} (st)^{z-1} (s(1 - t))^{w-1} \\, s \\, dt \\, ds \\newline\n&= \\int_{s=0}^{\\infty} \\int_{t=0}^{1} e^{-s} s^{z+w-1} t^{z-1}(1 - t)^{w-1} \\, dt \\, ds \\newline\n&= \\int_{0}^{\\infty} s^{z+w-1} e^{-s} \\left\\{ \\int_{0}^{1} t^{z-1}(1 - t)^{w-1} \\, dt \\, \\right\\} ds \\newline\n&= B(z, w) \\int_{0}^{\\infty} s^{z+w-1} e^{-s} ds \\newline\n&= B(z, w) \\cdot \\Gamma(z+w).\n\\end{align}\n\nDividing both sides of this equation by $\\Gamma(z+w)$ now gives the result.\n\n## 5. Roundup\n\nWe specialised the general formulae:\n\n\\begin{align}\nF_{X_{(k)}}(x) &= \\sum_{j=k}^{n} \\binom{n}{j} F_{X}(x)^{j} (1 - F_{X}(x))^{n-j}, \\newline\nf_{X_{(k)}}(x) &= k \\binom{n}{k} f_{X}(x) \\cdot F_{X}(x)^{k-1} \\, (1 - F_{X}(x))^{n-k}, \\newline\n\\mathbb{E} \\left \\lbrack X_{(k)} \\right \\rbrack &=\nk \\binom{n}{k} \\int_{0}^{1} x \\cdot f_{X}(x) \\cdot F_{X}(x)^{k-1} \\, (1 - F_{X}(x))^{n-k} \\, dx,\n\\end{align}\n\nwhich depend only on $n, k, f_{X}$ and $F_{X}$, to the case of the uniform distribution $U(0, 1)$. We showed that:\n\n\\begin{align}\nF_{X_{(k)}}(x) &= \\sum_{j=k}^{n} \\binom{n}{j} x^{j} (1 - x)^{n-j}, \\newline\nf_{X_{(k)}}(x) &= k \\binom{n}{k} x^{k-1} \\, (1 - x)^{n-k}, \\newline\n\\mathbb{E} \\left \\lbrack X_{(k)} \\right \\rbrack &= \\frac{k}{n+1} \\cdot\n\\end{align}\n\nTo prove the expectation formula, we introduced the beta and gamma function, and proved some identities they satisfy.\n", "meta": {"hexsha": "7762f02011b2d10a24f114d4d5223bbab790e7e2", "size": 11385, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/order-statistics-part-3.ipynb", "max_stars_repo_name": "enrightward/enrightward.github.io", "max_stars_repo_head_hexsha": "80bb007bc90de88e83df287fc80a08c2d9febd64", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/order-statistics-part-3.ipynb", "max_issues_repo_name": "enrightward/enrightward.github.io", "max_issues_repo_head_hexsha": "80bb007bc90de88e83df287fc80a08c2d9febd64", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/order-statistics-part-3.ipynb", "max_forks_repo_name": "enrightward/enrightward.github.io", "max_forks_repo_head_hexsha": "80bb007bc90de88e83df287fc80a08c2d9febd64", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 45.9072580645, "max_line_length": 642, "alphanum_fraction": 0.5232323232, "converted": true, "num_tokens": 3290, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5039061705290805, "lm_q2_score": 0.8902942246666267, "lm_q1q2_score": 0.4486247533959167}} {"text": "# Python Prerequirements for the Deep Learning Workshop\n\nIn this notebook we have some exercises that contain concepts largely unrelated to deep learning but are used frequently through-out the workshop. We would like to confirm understanding of these techniques by completing this notebook.\n\n### Section 1: Tuples and Lists\n\nWe use tuples frequently as a way to store and receive various lists of arguments. Python has the capability to _destructure_ tuples and lists, and this turns out to often by quite convenient.\n\nExercise 1: **Constructing tuples**: Tuples are built typically using the `,` operator. For example, `a = 1,2`.\n\nYou may like to use the `type` inbuilt function to determine if you're completing these exercises correctly.\n\n- (1.1) Construct a tuple containing two elements.\n\n- (1.2) Construct a tuple containing 1 element.\n\n\n```python\n# TODO:\n# Exercise 1.1:\n\n# Exercise 1.2:\n```\n\nExercise 2: **Deconstructing tuples**: (aka _De-tupling_ aka _Unpacking_) Tuples (and lists!) can be _deconstructed_ using standard assignment operators. Destructuring here refers to picking out individual components of the tuple. This can also be performed using standard python indexing, but sometimes destructuring may be more simple.\n\nFor the purposes of these exercises, let:\n\n```\na = (1, 2, \"Hello\")\n```\n\n- (2.1) Using only the variable `a` as so-defined, define three new variables that take on each of the values in the tuple.\n\n- (2.2) Again using only the variable `a`, define a single new variable that takes on the value `\"Hello\"`. (Hint: Oftentimes we will use the special variable name `_` to indicate that we do not care about a value.)\n\n\n```python\na = (1, 2, \"Hello\")\n\n# TODO:\n# Exercise 2.1:\n\n# Exercise 2.2:\n```\n\nExercise 3: **Lists and Destructuring**: We typically build a list with square brackets, like `a = [1,2]`. Can we do the same de-structuring operations with a list?\n\n- (3.1) Repeat exercises 1.1, 1.2, 2.1 and 2.2 using lists instead of tuples. Does everything just work?\n\n\n```python\n# TODO:\n\n# Exercise 3.1\n```\n\n### Section 2: Slicing\n\nSlicing is a great tool. It allows us to access ranges of elements in lists/tuples/arrays in a compact fashion. It also is supported on numpy arrays, which we typically use quite often. We won't attempt to cover _all_ the details of Python slicing here, only the parts that we use in the workshop.\n\nIn order to understand slicing, note that in Python (and most programming languages) when we index into an array, we start at `0`. So given:\n\n```\na = [2, 3, 5, 7, 11, 13] # Prime numbers ...\n```\n\nThen `a[0] == 2` and `a[2] == 5`.\n\nExercise 4: **Slicing**.\n\n- (4.1) Confirm the above statements.\n\n- (4.2) We can also use negative numbers to pick out elements starting from the end of the list. What item does `-1` return?\n\n\n```python\n# TODO:\n\n# Exercise 4.1\n\n# Exercise 4.2\n```\n\nExercise 5: **Selecting segments with slicing**: While we can pick out specific values with indexing, slicing lets us pick out specific resgions of our list.\n\nThe syntax for a slice is `m:n` for some integers `m` and `n`. We use it like `a[0:2]` to pick out all the elements from `a` starting with `0` up to but not including `2`.\n\n- (5.1) Before running the above slice, can you guess how many elements will be returned? Test your guess.\n\n- (5.2) Try other numbers in the slice operation. What happens? Can you generate an Exception? What happens when you try negative numbers?\n\n\n```python\n# TODO:\n\n# Exercise 5.1\n\n# Exercise 5.2\n```\n\n### Section 3: `numpy` Arrays\n\nNumpy is used to manage data coming in and going out of TensorFlow. There are a few concepts in numpy and TensorFlow that will be used frequently. In particular, let us look at the \"Shape\".\n\nConsider:\n\n\n```python\nimport numpy as np\n\na = np.array([2, 3, 5, 7, 11, 13])\n```\n\nThen we can get the _shape_ with `.shape`:\n\n\n```python\na.shape\n```\n\nHere the shape is hinting that the array can actually be multi-dimensional. And indeed, they can, and it turns out to be very useful!\n\nExercise 6: **Shapes**.\n\n- (6.1) Use the function [`np.ones`](https://docs.scipy.org/doc/numpy/reference/generated/numpy.ones.html) to construct a multi-dimensional array of size `1280x1024x3` (i.e. a standard RGB image of height 1280, width 1024 and 3 colour channels). Check the shape with `.shape`.\n\n- (6.2) *Multi-dimensional slicing* We can perform a slice on this variable, say `b`, like `b[:, :, 1]` to pick out the second RGB dimension (Green) from this faux-image. Confirm this by doing it, and checking the shape. Try out other multi-dimensional slices!\n\n### Section 4: TensorFlow, The Computation Graph & The Session\n\nHere we will not use any deep-learning functionality in TensorFlow, but we will introduce two key ideas.\n\nThe first is the _Computation Graph_. This is a global object that we construct indirectly by defining various operations, variables, and placeholders via the TensorFlow API.\n\nFor example:\n\n\n```python\nimport tensorflow as tf\n\na = tf.constant(2)\nb = tf.constant(9)\nc = tf.add(tf.square(a), b)\n\nprint(c)\n```\n\nThe above establishes the computation graph to consider the expression:\n\n$$\n\\begin{align}\na &= 2 \\\\\nb &= 9 \\\\\nc &= a^2 + b\n\\end{align}\n$$\n\nIn diagram form, the flow of data for this expression looks like:\n\n\n\n\nClearly, TensorFlow is very verbose! And `c` doesn't even take on a concrete value, evidently (we'll get into this when we discuss the _Session_).\n\nWe can simplify it a bit (if we are only interested in the value of `c`), by utilising operator overloading:\n\n\n```python\na = 2\nb = 9\nc = tf.square(a) + b\n\nprint(c)\n```\n\nExercise 7: **Building graphs by using operations**: We refer to the process of definine such expressions as \"building the graph\". TensorFlow supports many mathematical operations (after all, we're going to do deep learning later!) but for now, stick to the ones you know to build up some interesting TensorFlow graphs. Note that the result of all of the expressions is a _Tensor_.\n\n- (7.1) Define your own mathematical expression using the TensorFlow operations:\n\n - [tf.add](https://www.tensorflow.org/api_docs/python/tf/add)\n - [tf.square](https://www.tensorflow.org/api_docs/python/tf/square)\n - [tf.subtract](https://www.tensorflow.org/api_docs/python/tf/subtract)\n\n\n```python\n# TODO:\n\n# Exercise 7.1\n```\n\nAll of this is well-and-good, but we're interested in _evaluating_ such expression. To perform evaluation, we need to use a _Session_. This sets up a manager of all the resources we may need. This tends to be useful if we are performing computations across GPUs, where we don't want to worry about memory-management: TensorFlow takes care of it for us.\n\nHere's an example:\n\n\n```python\nsess = tf.InteractiveSession()\n\n# Note: We could also use the `resource` form here, like:\n#\n# with tf.Session() as sess:\n# a = 2\n# b = 9\n# c = tf.square(a) + b\n# c_value = sess.run(c)\n#\n# but it turns out to be more convenient to use the \n# `InteractiveSession` one here, as it let's us re-use\n# the session between cells.\n\n\n\na = 2\nb = 9\nc = tf.square(a) + b\n\nc_value = sess.run(c)\n\nprint(c_value)\n```\n\nIndeed, we've computed the value finally! We often call this \"evaluating\" the graph. Note that TensorFlow only computes those values (which it calls nodes) that are required for the particular value you are seeking. So for example, `c` depends on `a` and `b`, so those values are required.\n\nExercise 8: **Sessions and running them**.\n\n- (8.1) Run your own expressions with `sess.run`.\n\n- (8.2) Try defining two different equations for `c` and `d`, then running them with `sess.run([c, d])`. What is the return value here? How does it relate to what we've done with tuples/lists above?\n\n### Section 5: Tensorflow - Feeding in values with `feed_dict`\n\nSo now we have the ability to compute arbitrary math functions.\n\nConsider now that we'd like to set up a graph to compute the following expression:\n\n$$\nc(x, y) = x^2 + y\n$$\n\nNote how this differs notationally from our earlier example. Here we are calling out that `c` depends on `x` and `y`, and that these variables need to be provided. In TensorFlow we accomplish this with the `feed_dict` argument to `sess.run( c, feed_dict=... )`, and [_placeholders_](https://www.tensorflow.org/api_docs/python/tf/placeholder).\n\nLet's see an example of a placeholder:\n\n\n```python\na = tf.placeholder(tf.int32, shape=(1,))\nc = a + 2\n\na_value, = sess.run(c, feed_dict={ a: (9,)})\n\nprint(a_value)\n```\n\nExercise 9: **Placeholders and feed dicts**. We can set up expressions with values that we later fill-in by using placeholders. The `feed_dict` argument to [`Session.run( ... )`](https://www.tensorflow.org/api_docs/python/tf/Session) lets us provide values for the placeholders by constructing a dictionary like `{ placholderVariable: placeholderValue }`. Note that matching up shapes is crucial, and TensorFlow doesn't let us provide a value that isn't in an array of some kind!\n\n- (9.1) Using placeholders, define a graph where the value $c(x,y)$ can be computed by providing the value for two placeholders named `x` and `y`, and compute them for a few different values of `x` and `y` using the feed_dict.\n\n\n```python\n# TODO:\n\n# Exercise 9.1\n```\n\n### That's it!\n\nThanks for completing the pre-requistites! See you at the workshop!\n\n\n```python\n\n```\n", "meta": {"hexsha": "6b9eb8de429822f9bb2ea059036e5bd0cb003f1f", "size": 14218, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Pre-Requesites.ipynb", "max_stars_repo_name": "silverpond/dl-workshop-pre-req", "max_stars_repo_head_hexsha": "2073c5ccf4d4e91a4daef70ba8ae0b63e2758b1a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Pre-Requesites.ipynb", "max_issues_repo_name": "silverpond/dl-workshop-pre-req", "max_issues_repo_head_hexsha": "2073c5ccf4d4e91a4daef70ba8ae0b63e2758b1a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Pre-Requesites.ipynb", "max_forks_repo_name": "silverpond/dl-workshop-pre-req", "max_forks_repo_head_hexsha": "2073c5ccf4d4e91a4daef70ba8ae0b63e2758b1a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.0225225225, "max_line_length": 488, "alphanum_fraction": 0.5879870587, "converted": true, "num_tokens": 2386, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.863391617003942, "lm_q1q2_score": 0.4485503542282431}} {"text": "```python\n%pylab inline\nnumpy.random.seed(0)\n\nimport seaborn; seaborn.set_style('whitegrid')\nfrom tqdm import tqdm_notebook as tqdm\n```\n\n Populating the interactive namespace from numpy and matplotlib\n\n\n## 1. Submodular Selection\nAuthor: Jacob Schreiber \n\nAs an increasing amount of data in the world has enabled increasingly complex machine learning models that perform tasks like translate text between languages, classify objects in videos, and enable precision medicine. However, training these models comes at an increased computational cost. While some of this cost can be mitigated through parallelized implementations and the utilization of GPUs, this hardware costs money that individuals or small organizations don't always have.\n\nSubmodular selection is one approach to decreasing the computational cost of training complex models through the selection of a representative subset of the data to instead train models on. This subset is selected in a manner that reduces redundancy in the data set and increased the diversity of the chosen examples. Correspondingly, the models that are trained using this subset almost always outperform random subsets and frequently can achieve optimal performance in only a fraction of the samples.\n\nBriely, the way that submodular search works is that one defines a function that returns approximately how useful a set of samples are, and then greedily optimizes that function. Optimally, this function would be the performance by some machine learning model. One would then be able to identify a minimal set that yielded top performance, where adding additional examples didn't improve the performance of the model. However, because this would require retraining a model over and over again, frequently an approximation of model performance is instead used.\n\napricot implements two forms of submodular selection: feature based functions and facility location functions. These functions will be described in more detail below, but feature based functions force a diversity of feature values by modeling the saturation of each feature in the growing subset, whereas facility location functions force diversity in the original space by measuring the distance from all points to their nearest representative.\n\n### Feature Based Submodularity\n\nA problem with some submodular functions is that they do not scale well across massive data sets. For example, the facility location function first involves calculating some pairwise information across the data set, and second involves recalculating this pairwise measurement each time a sample is potentially added to the growing set. While optimizations exist that can greatly speed this up when compared to a naive implementation, it still can be prohibitively slow in some cases.\n\nA different type of submodular function, called a \"feature based\" function, does not involve calculating pairwise information and is much more scalable to large data sets. These functions involve calculating how \"saturated\" a particular feature is and preferentially selecting samples that have large feature values that have not yet been seen. These function takes the following form:\n\n\\begin{equation}\nf(X) = \\sum\\limits_{u \\in U} w_{u} \\phi_{u} \\left( \\sum\\limits_{x \\in X} m_{u}(x) \\right)\n\\end{equation}\n\nIn this equation, $U$ refers to all features, or dimensions, of a sample, and $u$ refers to a specific feature. $X$ refers to the original data set that we are selecting from and $x$ refers to a single sample from that data set. $w$ is a vector of weights that indicate how important each feature is, with $w_{u}$ being a scalar referring to how important feature $u$ is. Frequently these weights are uniform. $\\phi$ refers to a set of saturating functions, such as $sqrt(X)$ or $log(X + 1)$ that take some \n\n\nFeature based functions perform best when the value of each feature corresponds roughly to some notion of how important it is rather than being an arbitrary value. For example, when considering audio data, a spike in noise indicates that something is happening in an otherwise silent background, and when considering financial portfolios the more money a person has in a stock or fund, the more important that fund is for them. Examples of features where feature based functions are not likely to work well on are those without this notion, including categorical variables such as car color or bimodally distributed variables.\n\nThe reason that feature based funtions don't perform well when thre is no notion of importance can easily be visualized by looking at two Gaussian blobs. One might expect that a good summarization function would select a similar number of points from each blob, perhaps near the boundary between the two. Let's see what a feature based function does.\n\n\n```python\nfrom apricot import FeatureBasedSelection\n\nX = numpy.concatenate([numpy.random.normal(5, 1, size=(50, 2)), numpy.random.normal(9, 1, size=(35, 2))])\nXi = FeatureBasedSelection(10).fit_transform(X)\n\nplt.figure(figsize=(8, 6))\nplt.title(\"Submodular selection on Gaussian blobs\", fontsize=16)\nplt.scatter(X[:,0], X[:,1])\nplt.scatter(Xi[:,0], Xi[:,1], label=\"Selected Samples\")\nplt.legend(fontsize=14)\nplt.show()\n```\n\nThe feature based selection chooses those samples with the highest absolute magnitude. This is not surprising because the function is trying to maximize the sum of the saturated values over each feature. You can maximize this value by always selecting the biggest values.\n\n#### 20 newsgroups Data Set\n\nThe setting in which feature based functions perform the best is when the value of the feature corresponds to a quantity or a significance, where a higher value of that feature corresponds to having a large amount of that feature. A natural example of this is using word counts or tf-idf to represent sentences where each feature is a word. Let's see that in action here with an example from the 20newsgroups data set.\n\n\n```python\nfrom sklearn.datasets import fetch_20newsgroups\nfrom sklearn.feature_extraction.text import TfidfVectorizer\n\ncats = ('sci.med', 'sci.space')\nstop = ('headers', 'footers', 'quotes')\n\ntrain_data = fetch_20newsgroups(subset='train', categories=cats, remove=stop)\ntest_data = fetch_20newsgroups(subset='test', categories=cats, remove=stop)\n\nX_train, y_train = train_data.data, train_data.target\nX_test, y_test = test_data.data, test_data.target\n\nvectorizer = TfidfVectorizer(stop_words='english', max_features=1000)\n\nX_train = vectorizer.fit_transform(X_train).toarray()\nX_test = vectorizer.transform(X_test).toarray()\n\nX_train.shape, X_test.shape\n```\n\n\n\n\n ((1187, 1000), (790, 1000))\n\n\n\nWe've downloading 1,187 samples for our training set and 790 for our test set. This is just a toy example and so training on the entire training set is feasible, but let's see how good models trained using small subsets can perform when using random selection versus using a feature based function. We'll also perform our random selection 20 times, showing the best, worst, and average performance across the 20 times.\n\n\n```python\nfrom sklearn.linear_model import LogisticRegressionCV\n\nsubmodular_accuracy, random_accuracy = [], []\n\nselection = FeatureBasedSelection(1000)\nselection.fit(X_train)\n\nnumpy.random.seed(0)\nrandom_idxs = []\nfor i in range(20):\n idxs = numpy.arange(X_train.shape[0])\n numpy.random.shuffle(idxs)\n random_idxs.append(idxs)\n\nx = [20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000]\n \nfor n in tqdm(x):\n Xi, yi = selection.transform(X_train, y_train)\n Xi, yi = Xi[:n], yi[:n]\n\n model = LogisticRegressionCV().fit(Xi, yi)\n submodular_accuracy.append(model.score(X_test, y_test))\n\n accs = []\n for i in range(20):\n idx = random_idxs[i][:n]\n Xi, yi = X_train[idx], y_train[idx]\n\n model = LogisticRegressionCV().fit(Xi, yi)\n accs.append(model.score(X_test, y_test))\n \n random_accuracy.append(accs)\n\nrandom_accuracy = numpy.array(random_accuracy)\n```\n\n\n A Jupyter Widget\n\n\n \n\n\n\n```python\nplt.title(\"20 newsgroups space vs medicine articles\", fontsize=14)\nplt.scatter(x, submodular_accuracy, label=\"Feature Based Selection\")\nplt.scatter(x, random_accuracy.mean(axis=1), color='m', label=\"Random Selection\")\n\nplt.plot(x, submodular_accuracy)\nplt.plot(x, random_accuracy.mean(axis=1), color='m')\nplt.plot(x, random_accuracy.min(axis=1), alpha=0.3, color='m')\nplt.plot(x, random_accuracy.max(axis=1), alpha=0.3, color='m')\n\nplt.fill_between(x, random_accuracy.min(axis=1), random_accuracy.max(axis=1), color='m', alpha=0.2)\nplt.xlabel(\"Samples Used\", fontsize=14)\nplt.ylabel(\"Accuracy\", fontsize=14)\nplt.legend(fontsize=12)\nplt.xlim(0, 1050)\nplt.savefig(\"img/20newsgroups.png\")\nplt.show()\n```\n\nIt looks like submodular selection through a feature based method distinctly outperforms using random selection for a small number of samples. Using only 100 samples selected using submodular selection one can achieve high accuracy. \n\n### Facility Location\n\nAnother approach to subset selection comes in the form of facility location functions. These functions operate primarily on pairwise similarity scores between two samples rather than their feature values directly. Because of this, the feature values of the samples can be negative as long as the similarity score is strictly non-negative. \n\nFacility location functions are named because an early use was in placing facilities for companies, such as distribution warehouses or stores. The question was, \"if we have facilities at some locations, what is the next best location to put a facility?\" The answer is typically an underserved region. This results in a greedy selection algorithm where the first facility is at a central location and then subsequent facilities are spaced out based on where locations exist.\n\n\n```python\nfrom apricot import FacilityLocationSelection\n\nnumpy.random.seed(0)\nX = numpy.concatenate([numpy.random.normal((1, 1), 0.5, size=(15, 2)),\n numpy.random.normal((6, 3), 0.5, size=(25, 2)),\n numpy.random.normal((5, 7), 0.5, size=(40, 2)),\n numpy.random.normal((1, 7), 0.5, size=(30, 2)),\n numpy.random.normal((10, 4), 0.5, size=(15, 2)),\n numpy.random.normal((3, 4), 0.5, size=(15, 2))])\n\nXi = FacilityLocationSelection(6, 'euclidean').fit_transform(X)\nXr = numpy.random.choice(numpy.arange(X.shape[0]), size=6)\nXr = X[Xr]\n\nplt.figure(figsize=(8, 6))\nplt.title(\"Facility location for exemplar identification\", fontsize=16)\nplt.scatter(X[:,0], X[:,1], s=10)\nplt.scatter(Xi[:,0], Xi[:,1], color=\"#FF6600\", label=\"Submodular Selection\")\nplt.scatter(Xr[:,0], Xr[:,1], color=\"#8A2BE2\", label=\"Random Selection\", alpha=0.6)\nplt.legend(fontsize=14, loc=1)\nplt.xlim(-1, 14)\nplt.axis('off')\nplt.show()\n```\n\nIn this example an example of selecting representative points from six clusters using either a facility location function or random selection. You can see immediately that the points are more representative of the total space than a feature based function would be. A random selection of points is unlikely to sample each of the clusters only once, and this skew worsens with the skew of samples per cluster, whereas a facility location function is more stable towards that. An animation of the selection process can give some insight into how it works.\n\n\n\nThe first selected sample is at a central location because it is representative of the full data set. In the context where facility location is actually selection store locations, this would be the optimal location to put a single store to service everyone. The second location is in the largest cluster of points but skewed towards the second largest cluster of points. At this point something interesting happens: the next selected samples seem to be central to the cluster they're selected in. The third location is at a central location in a cluster of samples that is furthest from the selected items, and thus least served. The next three points are also at the center of their respective clusters. Essentially, once the space is sparsely covered then the next selected sample can focus on serving a local neighborhood.\n\nFacility location can be thought of as a greedy version of k-means clustering. K-means is an iterative procedure that would've resulted in the centroids lying at the center of each of the clusters above. However, it would be significantly more time intensive to calculate these. Correspondingly, facility location functions can be a good approximation of k-means clustering, and sometimes are even used as an initialization to the k-means algorithm.\n\n#### Digits Data Set\n\nWe can use facility location for sample selection for machine learning algorithms as well. In fact, it works better than feature based methods in cases where the features do not have the nice properties seen before. Let's load up the digits data set. A feature based function would try to select samples that had a diversity of high pixel values. That's not exactly what we want because the digit can exist in several places in the image with different sizes and thicknesses.\n\n\n```python\nfrom sklearn.datasets import load_digits\n\ndata = load_digits()\nX, y = data.data, data.target \n\nidx = numpy.arange(X.shape[0])\nnumpy.random.shuffle(idx)\n\nX, y = X[idx], y[idx]\nX_train, y_train = X[:1250], y[:1250]\nX_test, y_test = X[1250:], y[1250:]\nX_train.shape, X_test.shape\n```\n\n\n\n\n ((1250, 64), (547, 64))\n\n\n\nLet's take 1,250 samples for the training set and 547 samples for the test set. We can try using a facility location function on the data that calculates the euclidean distance between samples and compare that both to random selection and to using a feature based selection algorithm. \n\n\n```python\nselector = FacilityLocationSelection(X_train.shape[0], 'euclidean', verbose=True)\nselector.fit(X_train)\nr1 = selector.ranking\n\nselector2 = FeatureBasedSelection(X_train.shape[0], verbose=True)\nselector2.fit(X_train)\nr2 = selector2.ranking\n```\n\n 0%| | 0/1250 [00:00,) in ignored\n \n 1%| | 12/1250 [00:00<00:10, 113.90it/s]\u001b[A\n 46%|\u2588\u2588\u2588\u2588\u258c | 570/1250 [00:00<00:04, 161.30it/s]\u001b[A\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 1250/1250 [00:00<00:00, 3771.99it/s]\u001b[A\n\n\nNow let's try training logistic regression models like before in these settings.\n\n\n```python\nfrom sklearn.linear_model import LogisticRegression\n\nnumpy.random.seed(0)\nrandom_idxs = []\nfor i in range(20):\n idxs = numpy.arange(X_train.shape[0])\n numpy.random.shuffle(idxs)\n random_idxs.append(idxs)\n\nfl_acc, fb_acc, random_acc = [], [], []\nn_samples = range(25, 251, 25) + range(300, 1251, 100)\n\nfor n in tqdm(n_samples):\n Xi, yi = selector.transform(X_train, y_train)\n model = LogisticRegression().fit(Xi[:n], yi[:n])\n fl_acc.append(model.score(X_test, y_test))\n \n Xi, yi = selector2.transform(X_train, y_train)\n model = LogisticRegression().fit(Xi[:n], yi[:n])\n fb_acc.append(model.score(X_test, y_test))\n\n accs = []\n for i in range(20):\n idx = random_idxs[i][:n]\n Xi, yi = X_train[idx], y_train[idx]\n\n model = LogisticRegression().fit(Xi, yi)\n accs.append(model.score(X_test, y_test))\n \n random_acc.append(accs)\n\nrandom_acc = numpy.array(random_acc)\n```\n\n\n A Jupyter Widget\n\n\n \n\n\n\n```python\nplt.title(\"MNIST Performance using Facility Location\", fontsize=14)\nplt.scatter(n_samples, fl_acc, c='b', label=\"Facility Location\")\nplt.scatter(n_samples, fb_acc, c='g', label=\"Feature Based\")\nplt.scatter(n_samples, random_acc.mean(axis=1), color='m', label=\"Random Selection\")\n\nplt.plot(n_samples, fl_acc, c='b')\nplt.plot(n_samples, fb_acc, c='g')\nplt.plot(n_samples, random_acc.mean(axis=1), color='m')\nplt.plot(n_samples, random_acc.min(axis=1), alpha=0.3, color='m')\nplt.plot(n_samples, random_acc.max(axis=1), alpha=0.3, color='m')\n\nplt.fill_between(n_samples, random_acc.min(axis=1), random_acc.max(axis=1), color='m', alpha=0.2)\nplt.xlabel(\"Number of Selected Points\", fontsize=14)\nplt.ylabel(\"Accuracy\", fontsize=14)\nplt.legend(fontsize=12)\nplt.show()\n```\n\nIt looks like the facility location function outperforms both the feature based function and the random selection. Similar to the case of the feature based function before the model performance appears to saturate with just a small percentage of the samples. In fact, the feature based function performs worse than the random selection. This is not surprising in the case where optimizing a diversity of features irrespective of the structure of the features may not be the best idea.\n", "meta": {"hexsha": "0ac73530795e9c581b9bdb8e5d3fbf6d42144334", "size": 160584, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/1. Introduction to Submodular Selection.ipynb", "max_stars_repo_name": "domoritz/apricot", "max_stars_repo_head_hexsha": "6dff8d08dee9145ec6c7e3e79d77efa0bbf19474", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tutorials/1. Introduction to Submodular Selection.ipynb", "max_issues_repo_name": "domoritz/apricot", "max_issues_repo_head_hexsha": "6dff8d08dee9145ec6c7e3e79d77efa0bbf19474", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/1. Introduction to Submodular Selection.ipynb", "max_forks_repo_name": "domoritz/apricot", "max_forks_repo_head_hexsha": "6dff8d08dee9145ec6c7e3e79d77efa0bbf19474", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 297.9294990724, "max_line_length": 44716, "alphanum_fraction": 0.9088950331, "converted": true, "num_tokens": 3945, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.629774621301746, "lm_q2_score": 0.712232184238947, "lm_q1q2_score": 0.4485457541079982}} {"text": "# Handwritten Music Symbol Recognition with Deep Ensemble\n\nIn ancient times, there was no system to record or document the music. Later, the musical pieces from the classical and post-classical period of European music were documented as scores using western staff notations. These notations are used by most of the modern genres of music due to its versatility. Hence, it is very important to develop a method that can store such sheets of handwritten music scores digitally. Optical Music Recognition (OMR) is a system which automatically interprets the scanned handwritten music scores. In this work, we have proposed a classifier ensemble of deep transfer learning models with Support Vector Machine (SVM) as the aggregation function for handwritten music symbol recognition. We have applied three pre-trained deep learning models, namely ResNet50, GoogleNet and DenseNet161 (each trained on ImageNet) and fine-tuned on our target datasets i.e., music symbol image datasets. The proposed ensemble is able to capture a more complex association of the base learners, thus improving the overall performance. We have evaluated the proposed model on three publicly available standard datasets, namely Handwritten Online Music Symbols (HOMUS), Capitan_Score_Non-uniform and Rebelo_real,and achieved state-of-the-art results for all three datasets.\n

\nHyperparameter Initialization\n\n\n```python\n#hyper params\nlr = 1e-4\nbs = 32\nval_split = 0.85\nnum_epoch = 20\nnum_classes = 32\n```\n\nWe use pytorch to implement the project. Here we include relevant modules and check for GPU.\n\n\n```python\n#imports\nimport torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport torch.optim as optim\nfrom torch.optim import lr_scheduler\nfrom torch.utils import data\nimport numpy as np\nimport torchvision\nfrom numpy import exp,absolute\nfrom torchvision import datasets, transforms\nimport matplotlib.pyplot as plt\nimport time\nimport os\nimport copy\nimport math\nfrom sklearn import svm\nimport sklearn.model_selection as model_selection\nfrom sklearn.metrics import accuracy_score,f1_score,precision_score ,recall_score \n\ndevice = torch.device(\"cuda:0\" if torch.cuda.is_available() else \"cpu\")\n```\n\nThis function gives us training, validation and test set and takes the path to folder as input. This folder must be arranged as per `torchvision.datasets.Imagefolder` specification.\n\n\n```python\ndef get_TVT(path):\n data_transforms = transforms.Compose([\n transforms.ToTensor(),\n transforms.Normalize([0.485, 0.456, 0.406], [0.229, 0.224, 0.225])\n ])\n\n dataset = datasets.ImageFolder(path+'train/',transform=data_transforms)\n\n train_size = math.floor(len(dataset)*val_split)\n val_size = len(dataset) - train_size\n trainset, valset = data.random_split(dataset,lengths=[train_size,val_size])\n testset = datasets.ImageFolder(path+'test/',transform=data_transforms)\n return trainset,valset,testset\n```\n\nThis is the function to train the model\n\n\n```python\ndef train_model(trainset, valset, model, criterion, optimizer, scheduler, num_epochs):\n dataloaders = {\n 'train': data.DataLoader(trainset,batch_size=bs,shuffle=True),\n 'val' : data.DataLoader(valset,batch_size=bs,shuffle=True)\n }\n dataset_sizes = {'train':len(trainset),'val':len(valset)}\n since = time.time()\n\n best_model_wts = copy.deepcopy(model.state_dict())\n best_acc = 0.0\n\n for epoch in range(num_epochs):\n print('Epoch {}/{}'.format(epoch, num_epochs - 1))\n print('-' * 10)\n\n # Each epoch has a training and validation phase\n for phase in ['train', 'val']:\n if phase == 'train':\n model.train() # Set model to training mode\n else:\n model.eval() # Set model to evaluate mode\n\n running_loss = 0.0\n running_corrects = 0\n\n # Iterate over data.\n for inputs, labels in dataloaders[phase]:\n inputs = inputs.to(device)\n labels = labels.to(device)\n # print('bruh')\n\n # zero the parameter gradients\n optimizer.zero_grad()\n\n # forward\n # track history if only in train\n with torch.set_grad_enabled(phase == 'train'):\n outputs = model(inputs)\n _, preds = torch.max(outputs, 1) \n loss = criterion(outputs, labels)\n\n # backward + optimize only if in training phase\n if phase == 'train':\n loss.backward()\n optimizer.step()\n\n # statistics\n running_loss += loss.item() * inputs.size(0)\n running_corrects += torch.sum(preds == labels.data)\n if phase == 'train':\n scheduler.step()\n\n epoch_loss = running_loss / dataset_sizes[phase]\n epoch_acc = running_corrects.double() / dataset_sizes[phase]\n \n print('{} Loss: {:.4f} Acc: {:.4f}'.format(\n phase, epoch_loss, epoch_acc))\n\n # deep copy the model\n if phase == 'val' and epoch_acc > best_acc:\n best_acc = epoch_acc\n best_model_wts = copy.deepcopy(model.state_dict())\n\n print()\n\n time_elapsed = time.time() - since\n print('Training complete in {:.0f}m {:.0f}s'.format(\n time_elapsed // 60, time_elapsed % 60))\n print('Best val Acc: {:4f}'.format(best_acc))\n\n # load best model weights\n model.load_state_dict(best_model_wts)\n return model\n```\n\nThis function calculates the model accuracy on test set.\n\n\n```python\ndef test_acc(model, testset):\n running_corrects = 0\n testloader = data.DataLoader(testset,batch_size=bs,shuffle=True)\n for inputs, labels in testloader:\n inputs = inputs.to(device)\n labels = labels.to(device)\n\n with torch.set_grad_enabled(False):\n outputs = model(inputs)\n _, preds = torch.max(outputs, 1)\n\n running_corrects += torch.sum(preds == labels.data)\n return (running_corrects/len(testset))\n```\n\nThis function returns a pair of set of data X and label Y. The elements in X represent the concatenated score from the models. If size of dataset is N, number of classes is c and number of trained model is k then the shape of X is (N,ck). The samples are also given weight based on total number of unique classification made on them (Explained later).\n\n\n```python\ndef get_weighted_score_ft(models,dataset):\n num_models = len(models)\n X = np.empty((0,num_models*num_classes))\n Y = np.empty((0),dtype=int)\n dataloader = data.DataLoader(dataset,batch_size=1,shuffle=True)\n for inputs,labels in dataloader:\n inputs,labels = inputs.to(device),labels.to(device)\n predictions = set()\n with torch.set_grad_enabled(False):\n x = models[0](inputs)\n _, preds = torch.max(x, 1)\n predictions.add(preds)\n for i in range(1,num_models):\n x1 = models[i](inputs)\n _, preds = torch.max(x1, 1)\n predictions.add(preds)\n x = torch.cat((x,x1),dim=1)\n if len(predictions) > 1:\n X = np.append(X,x.cpu().numpy()*3,axis=0)\n else:\n X = np.append(X,x.cpu().numpy(),axis=0)\n Y = np.append(Y,labels.cpu().numpy(),axis=0) \n return X,Y\n\n```\n\nWe load the models with pretrained weights\n\n\n```python\ndef get_models():\n googlenet = torchvision.models.googlenet(pretrained=True)\n resnet = torchvision.models.resnet50(pretrained=True)\n densenet = torchvision.models.densenet161(pretrained=True)\n\n densenet.classifier = nn.Linear(2208,num_classes)\n resnet.fc = nn.Linear(2048,num_classes)\n googlenet.fc = nn.Linear(1024,num_classes)\n densenet = densenet.to(device)\n resnet = resnet.to(device)\n googlenet = googlenet.to(device)\n\n return [densenet,googlenet,resnet]\n```\n\nThis is the main code cell where all the functions are utilised together. Now let us consider there are $K$ number of base classifiers $\\{CF_1, CF_2, \\dots, CF_K\\}$ to deal with an $n$-class classification problem. Hence, the output of any classifier (say, $CF_i$) is an $n$-dimensional vector $O_i = {s_1^i, s_2^i, \\dots, s_n^i}$. Here, $s_j^i$ is confidence score produced by $i_{th}$ classifier for the $j_{th}$ class. We concatenate all the output vectors produced by the classifiers $\\{CF_1, CF_2, \\dots, CF_K\\}$ to get a vector $S$ of length $nK$. $S$ is represented by\n\n\\begin{equation}\n \\label{equ:final_vector}\n S = \\{s_1^1, s_1^2, \\dots, s_2^1, s_2^2, \\dots, s_n^K\\}\n\\end{equation}\n\nOne such vector $S$ is produced for every sample of the dataset. Let us consider that we have $N$ such samples with corresponding labels $y_i$ in the dataset to be used for training. Thus obtained the set $\\{(S_1, y_1), (S_2, y_2),\\\\ \\dots, (S_N, y_N)\\}$ on which we train the SVM model. To introduce weights on samples, we consider the total number of unique predictions made on a sample by the base classifiers. For example, if there are three base classifiers and for some sample two of the classifiers are predicting the label 'class-x' and the remaining one is predicting the label 'class-y', then the total number of unique predictions of that sample is $2$. If the total number of prediction is greater than $1$, it suggests that there is a conflict among the classifiers on the correct class. So we propose that the SVM must put more emphasis on these samples in order to approximate a better decision boundary or support vectors.\n\nA sample is assigned with $\\mathcal{W}$ times more weight if the number of unique predictions regarding the corresponding image is greater than $\\lambda$, which is an integer and whose value lies between $[1, K]$. In this work, we choose the values of both $K$ and $\\lambda$ to be 3. The value of $\\mathcal{W}$ is taken as 3 which is decided experimentally.\n\nWhile testing and image it is first passed through all of the three DL models and the three output vectors are obtained. Then these output vectors are concatenated, during this concatenation the order of the models are maintained (as same as during training). We pass this vector through the trained SVM classifier, which predicts the final class of our test image.\n\n\n\n\n\n```python\ncriterion = nn.CrossEntropyLoss()\nensemble_accuracy=[]\nfor fold in ['Fold_1','Fold_2','Fold_3','Fold_4','Fold_5']:\n for folder in ['HOMUS']: #['Capitan_Score_Non-uniform','Capitan_Score_Uniform','Fornes_Music_Symbols_labelled']['Rebelo_Syn_labelled']:\n trainset,valset,testset = get_TVT('/content/homus/'+fold+'/',folder)\n models = get_models()\n for model in models:\n optimizer = optim.Adam(model.parameters(),lr=lr)\n exp_lr_scheduler = lr_scheduler.StepLR(optimizer, step_size=9, gamma=0.1)\n model = train_model(trainset, valset, model, criterion, optimizer, exp_lr_scheduler,num_epoch)\n\n print(test_acc(model,testset))\n train_X, train_Y = get_weighted_score_ft(models,trainset)\n test_X, test_Y = get_weighted_score_ft(models,testset)\n clf = svm.SVC(kernel='rbf',break_ties=True).fit(train_X, train_Y)\n pred = clf.predict(test_X)\n acc = accuracy_score(test_Y, pred)\n ensemble_accuracy.append(acc)\n print('Ensemble on '+fold+': '+str(acc))\nprint(\"Average Ensemble Accuracy:\",sum(ensemble_accuracy)/len(ensemble_accuracy))\n```\n", "meta": {"hexsha": "0227e801ee4631bdabc93e5bcb86a9c8372966be", "size": 17505, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "homus.ipynb", "max_stars_repo_name": "ashis0013/Music-Symbol-Recognition", "max_stars_repo_head_hexsha": "2d48371612c7703ed7e9022d313cd89f0f29eb14", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "homus.ipynb", "max_issues_repo_name": "ashis0013/Music-Symbol-Recognition", "max_issues_repo_head_hexsha": "2d48371612c7703ed7e9022d313cd89f0f29eb14", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "homus.ipynb", "max_forks_repo_name": "ashis0013/Music-Symbol-Recognition", "max_forks_repo_head_hexsha": "2d48371612c7703ed7e9022d313cd89f0f29eb14", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2021-07-02T18:14:09.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-12T19:46:06.000Z", "avg_line_length": 44.6556122449, "max_line_length": 1299, "alphanum_fraction": 0.5405884033, "converted": true, "num_tokens": 2693, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.712232184238947, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.44854574420987064}} {"text": "# Dynamic Simulation: The Basic Procedure\nOnce a set of dynamic mass balance equations has been formulated, they can be numerically solved, and thus the behavior of a network can be simulated in response to environmental and genetic changes. Simulation results can be obtained using a number of different software packages. Dynamic simulation generates the time dependent behavior of the concentrations, i.e., $\\textbf{x}$(t). This solution can be obtained in response to several different types of perturbations and the results graphically displayed. The basic principles and procedures associated with dynamic simulation are covered in this chapter. The following three chapters then apply the simulation process to a set of simple but progressively more complex and relevant examples. \n\n## Numerical Solutions\nNetwork dynamics are described by a set of ordinary differential equations (ODEs): the dynamic mass balance equations; see Eq. (1.1). To obtain the dynamic solutions, we need three things: first, the equations themselves; second, the numerical values for the kinetic constants that are in the equations; and third, the initial conditions and parameters that are being perturbed. We describe each briefly. \n\n**1.** To formulate the mass balances we have to specify the system boundary, the fluxes in and out of the system, and the reactions that take place in the network. From the set of reactions that are taking place, a stoichiometric matrix is formed. This matrix is then put into Eq. (1.1) . One can multiply out the individual dynamic mass balances, as was done in Eq. (2.13) for the adenosine phosphate network, to prevent a large number of numerical operations that involve multiplication of reaction rates by the zero elements in $\\textbf{S}$. The reaction rate laws for the reactions are then identified and substituted into the equations. Typically, one would use elementary kinetics as shown in Eq. (2.6), or apply more complex rate laws if they are appropriate and available. This process leads to the definition of the dynamic mass balances. \n\n**2.** The numerical values for the kinetic parameters in the rate laws have to be specified, as do any imposed fluxes across the system boundary. Obtaining numerical values for the kinetic constants is typically difficult. They are put into a parameter vector designated by $\\textbf{k}$. In select cases, detailed kinetic characterization has been carried out. More often, though, one only knows these values approximately. It is important to make sure that all units are consistent throughout the equations and that the numerical values used are in the appropriate units. \n\n**3.** With the equations and numerical values for the kinetic constants specified $(\\textbf{k})$, we can simulate the responses of the network that they represent. To do so, we have to set initial conditions $(x_0)$. This leads to the numerical solution of\n\n$$\\begin{equation} \\frac{d\\textbf{x}}{dt} = \\textbf{Sv(x;k)},\\ \\textbf{x}(t = 0) = \\textbf{x}_0 \\tag{3.1} \\end{equation}$$\n\nThere are three conditions that are typically considered. \n\n1. First, the initial conditions for the concentrations are set, and the motion of the network into a steady state (open system) or equilibrium (closed system) is simulated. This scenario is typically physiologically unrealistic since individual concentrations cannot simply change individually in a living cell.\n2. Second, a change in an input flux is imposed on a network that is in a steady state. This scenario can be used to simulate the response of a cell to a change in its environment. \n3. Third, a change in a kinetic parameter is implemented at the initial time. The initial concentrations are typically set at the steady state values with the nominal value of the parameter. The equations are then simulated to a long time to obtain the steady state values that correspond to the altered kinetic parameters. These are set as the initial conditions when examining the responses of the system with the altered kinetic properties. \n\n**4.** Once the solution has been obtained it can be graphically displayed and the results analyzed. There are several ways to accomplish this step, as detailed in the next two sections. The analysis of the results can lead to post-processing of the output to form an alternative set of dynamic variables. \n\nThe simulation is implemented using a numerical solver. Currently, such implementation is carried out using standard and readily available software, such as Mathematica or MATLAB. Specialized simulation packages are also available (Table\u00a03.1). After the simulation is set up and the conditions specified, the software computes the concentrations as a function of time. The output is a file that contains the numerical values of the concentrations at a series of time points (Figure 3.1). This set of numbers is typically graphically displayed, and/or used for subsequent computations. \n\n**Table 3.1:** Available software for dynamic simulation. Assembled by Neema Jamshidi.\n\n\n\nWe will be using iPython notebooks using the python software package **MASSpy** as our simulation software.\n\n## Graphically Displaying the Solution\nThe simulation procedure described in the previous section results in a file that contains the concentrations as a function of time (Figure 3.1). These results are graphically displayed, typically in two ways: by plotting the concentrations as a function of time, or by plotting two concentrations against one another with time as a parameter along the trajectory. \n\n\n\n**Figure 3.1:** The fundamental structure of the file $\\textbf{x}$(t) that results from a numerical simulation. The two vertical bars show the list of values that would be used to compute $\\sigma_{12}$(2) (see Eq. 3.8); that is, the correlation between $x_1$ and $x_2$ with a time lag of 2.\n\nBefore describing these methods, we observe certain fundamental aspects of the equations that we are solving. The dynamic mass balances can be expanded as: \n\n$$\\begin{equation} \\frac{d\\textbf{x}}{dt} = \\textbf{Sv(x)} = \\sum\\textbf{s}_i v_i(\\textbf{x}) \\tag{3.2} \\end{equation}$$\n\nIn other words, the time derivatives are linear combinations of the reaction vectors $(\\textbf{s}_i)$ weighted by the reaction rates, that in turn change with the concentrations that are time varying. Thus, the motions are linear combinations of the directions specified by $\\textbf{s}_i$. This characteristic is important because if the $v_i$. have different time constants, the motion can be decomposed in time along these reaction vectors. \n\n### Time Profiles\nThe simulation results in a file that contains the vector $\\textbf{x}$(t) and the time points at which the numerical values for the concentrations are given. These time points can be specified by the user or are automatically generated by the solver used. Typically, the user specifies the initial time, the final time, and sometimes the time increment between the time points where the simulator stores the computed concentration values in the file. The results can then be graphically displayed depending on a few features of the solution. Some of these are shown in Figure 3.2 and are now briefly described: \n\n* Panel A: The most common way to display a dynamic solution is to plot the concentration as a function of time. \n\n* Panel B: If there are many concentration variables they are often displayed on the same graph. \n\n* Panel C: In many cases there are different response times and one plots multiple time profiles where the x-axis on each plot is scaled to a particular response time. Alternatively, one can use a logarithmic scale for time. \n\n* Panel D: If a variable moves on many time scales changing over many orders of magnitude, the y-axis is often displayed on a logarithmic scale. \n\n\n\n**Figure 3.2:** Graphing concentrations over time. (a) A single concentration shown as a function of time. (b) Many concentrations shown as a function of time. (c) A single concentration shown as a function of time separately on different time scales. (d) The logarithm of a single concentration shown as a function of time to distinguish the decay on different time scales.\n\nThe solution can thus be displayed in different ways depending on the characteristics of the time profiles. One normally plays with these representations to get an understanding of the responses of the network that they have formulated and to represent the features in which one is interested. \n\n### Dynamic phase portraits\nDynamic phase portraits represent trajectories formed when two concentrations plotted against each other, parameterized with respect to time (Figure 3.3). The dynamic trajectories in the diagram move from an initial state to a final state. Analysis of these trajectories can point to key dynamic relationships between compounds in a biochemical reaction network. For example, if a system is dynamically stable, the dynamic trajectories will converge to a single point in the plane, known as an attracting fixed pointattracting fixed point. A stable steady-state point would represent a homeostatic state. Conversely, if the system is unstable, the trajectories will not approach a fixed point but diverge away from it. The former is essentially always the case for biochemical reaction networks representing real cells. The way the trajectories converge on the steady state is highly informative as different dynamic characteristics are evident from the trajectory. \n\n\n\n**Figure 3.3:** A dynamic phase portrait.\n\n### Characteristic features of phase portraits\nA trajectory in the phase portraitphase portrait may indicate the presence of one or more general dynamic features. Namely, the shapes of the trajectories contain significant information about the dynamic characteristics of a network. Some important features of trajectories in a phase portrait are shown in Figure 3.4\n\n\n\n**Figure 3.4:** General features of dynamic phase portraits. Dynamic phase portraits are formed by graphing the time dependent concentrations of two concentrations $(x_1$ and $x_2)$ against one another. Phase portraits have certain characteristic features. (a) Conservation relationship. (b) A pair of concentrations that could be in quasi-equilibrium with one another. (c) Motion of the two concentrations dynamically independent of one another. (d) Closed loop traces representing either a periodic motion or a return to the original steady state. Modified from Kauffman 2002 [64].\n\n1. When the trajectory has a negative slope, it indicates that one concentration is increasing while the other is decreasing. The concentrations are moving on the same time scales but in opposite directions; that is, one is consumed while the other is produced. This feature might represent the substrate concentration versus the product concentration of a given reaction. Such behavior helps define aggregate concentration variablesaggregate concentration variables. \n\n2. When a trajectory in the phase portrait between two concentrations is a straight line with a positive slope, it means that the two concentrations are moving in tandem; i.e., as one increases so does the other. This feature is observed when two or more concentrations move on the same time scales and are in quasi-equilibrium with one another. Such behavior helps define aggregate concentration variables. \n\n3. When a trajectory is vertical or horizontal, it indicates that one of the concentrations is changing while the other remains constant. This feature implies either that the motions of the concentrations during the trajectory are independent of one another or that the dynamic motions of the concentrations progress on different characteristic time scales. Such behavior helps define time scale decomposition.\n\n4. When a trajectory forms a closed loop, it implies one of two possibilities. The system never converges to a steady state over time but oscillates forming a closed loop trajectory. On the other hand, if the orbit begins at one point, moves away from it, then returns to the same point after a sufficiently long time interval, then it implies that a change in another variable in the system forced it away from its steady state temporarily, but it returned to the original steady state. Such behavior helps define disturbance rejection characteristics. \n\n\n\n**Figure 3.5:** A schematic of a tiled phase portrait.The matrix is symmetric, making it possible to display statistical information about a phase portrait in the mirror position.The diagonal elements are meaningless.Originally developed in Kauffman 2002 [64].\n\nThe qualitative characteristics of dynamic phase portraitsphase portrait can provide insight into the dynamic features of a network. A trajectory may have more than one of these basic features. For instance, there can be a fast independent motion (i.e., a horizontal phase portrait trajectory) followed by a line with a positive slope after an equilibrium state has been reached.\n\n### Tiling dynamic phase portraits\nPhase portraits show the dynamic relationships between two variables on multiple time scales, see Figure 3.5. If a system has $n$ variables, then there are $n^2$ dynamic phase portraits. All pair-wise phase portraits can be tiled in a matrix form where the $\\textit{i}$, $\\textit{j}$ entry represents the dynamic phase portrait between variables $x_i$ and $x_j$. Note that such an array is symmetric and that the diagonal elements are un-informative. Thus, there are $(n^2-n)/2$ phase portraits of interest. This feature of this graphical representation opens the possibility of putting the phase portrait in the $\\textit{i}$, $\\textit{j}$ position in the array and showing other information (such as a regression coefficient or a slope) in the corresponding $\\textit{j}$, $\\textit{i}$ position. \n\nSince the time scales in biochemical reaction networks typically vary over many orders of magnitude, it often makes sense to make a series of tiled phase portraits, each of which represents a key time scale. For instance, rapid equilibration leads to straight lines with positive slopes in the phase portrait (Figure 3.4b) where the slope is the equilibrium constant of the reaction. This may be one of many dynamic events taking place. If a phase portrait is graphed separately on this time scale alone, the positive line will show up with a high regression coefficient and a slope that corresponds to the equilibrium constant.\n\n## Post-Processing the Solution \n\nThe initial suggestions obtained from graphing and visualizing the concentration vector $\\textbf{x}$(t) can lead to a more formal analysis of the results. We describe three post-processing procedures of $\\textbf{x}$(t). \n\n### Computing the fluxes from the concentration variables:\nThe solution for the concentrations $\\textbf{x}$(t)can be used to compute the fluxes from\n\n$$\\begin{equation} \\textbf{v}(t)= \\textbf{v}(\\textbf{x}(t)) \\tag{3.3} \\end{equation}$$\n\nand subsequently we can plot the fluxes in the same way as the concentrations. Graphical information about both the $\\textbf{x}$(t) and $\\textbf{v}$(t) is useful. \n\n### Combining concentrations to form aggregate variables:\nThe graphical and statistical multi-time scale analysis discussed above may lead to the identification of aggregate variables. Pooled variables, p, are computed from\n\n$$\\begin{equation} \\textbf{p}(t)= \\textbf{Px}(t)) \\tag{3.4} \\end{equation}$$\n\nwhere the pool transformation matrix, $\\textbf{P}$, defines the linear combination of the concentration variables that forms the aggregate variables. For instance, if we find that a logical way to pool two variables, $x_1$ and $x_2$, into new aggregate variables is $p_1 = x_1 + x_2$ and $p_2 = x_1 - x_2$, then we form the following matrix equation describing these relationships as: \n\n$$\\begin{equation} \\textbf{p}(t) = \\textbf{Px}(t) = \\begin{pmatrix} {p_1(t)} \\\\ {p_2(t)} \\end{pmatrix} = \\begin{pmatrix} {1} & {1} \\\\ {1} & {-1} \\end{pmatrix} \\begin{pmatrix} {x_1(t)} \\\\ {x_2(t)} \\end{pmatrix} = \\begin{pmatrix} {x_1(t) + x_2(t)} \\\\ {x_1(t) - x_2(t)} \\end{pmatrix} \\end{equation}$$\n\nThe dynamic variables, $\\textbf{p}$(t), can be graphically studied as described in the previous section. \n\n#### Example: The Phosphorylated Adenosines\nThe pool formation discussed in Chapter\u00a02 can be described by the pool transformation matrix: \n\n$$\\begin{equation}\\textbf{P} = \\begin{pmatrix} {1} & {1} & {0} \\\\ {2} & {1} & {0} \\\\ {1} & {1} & {1} \\end{pmatrix} \\end{equation}$$\n$$\\tag{3.5}$$\n\nand thus \n\n$$\\begin{equation}\\\\textbf{p} = \\textbf{Px} = \\textbf{P}\\begin{pmatrix} {\\text{ATP}} \\\\ {\\text{ADP}} \\\\ {\\text{AMP}} \\end{pmatrix} = \\begin{pmatrix} {\\text{ATP} + \\text{ADP}} \\\\ {2 \\text{ATP} + \\text{ADP}} \\\\ {\\text{ATP} + \\text{ADP} + \\text{AMP}} \\end{pmatrix}\\end{equation}$$\n$$\\tag{3.6}$$\n\nThe pool sizes $p_i$(t) can then be graphed as a function of time. \n\n### Correlating concentrations over time:\nOne can construct the time-separated correlation matrix, $\\textbf{R}$, based on a time scale structure of a system. In this matrix, we compute the correlation between two concentrations on a time scale as: \n\n$$\\begin{equation} \\textbf{R}(\\tau) = (r_{ij}) = \\frac{\\sigma_{ij}(\\tau)}{\\sqrt{\\sigma_{ii}\\sigma_{jj}}} \\tag{3.7} \\end{equation}$$\n\nin which $\\sigma_{ii}$ is the variance of the dataset $x_i(k)$ and $\\sigma_{ij}(\\tau)$ is the time-lagged covariance between the discrete, uniformly sampled datasets $x_i(k)$ and $x_j(k + \\tau)$, determined as, \n\n$$\\begin{equation} \\sigma_{ij}(\\tau) = \\frac{1}{n}\\sum\\limits_{k=1}^{n-\\tau} (x_i(k) - \\bar{x_i})(x_j(k + \\tau) - \\bar{x_j}) \\tag{3.8} \\end{equation}$$\n\nin which $n$ is the number of data points in the series, and $\\bar{x_i}$ indicates the average value of the series $x_i$. The values in $\\textbf{R}$ range from -1 to 1, indicating perfect anti-correlation or correlation, respectively, between two datasets with a delay of time steps. Elements in $\\textbf{R}$ equal to zero indicate that the two corresponding datasets are completely uncorrelated. If such correlation computations were done for the cases shown in Figure 3.4, one would expect to find a strong negative correlation for the data shown in Figure 3.4a, a strong positive correlation for Figure 3.4b, and no correlation for Figure 3.4c, \n\nThe correlation computations can be performed with an increment, $\\tau$, offset in time between two concentrations. An example of a time offset is shown in Figure 3.2 showing the values used from the output file to compute the correlation between $x_1$ and $x_2$ with a time lag of 2. \n\nThe matrix of phase portraits is symmetric with uninformative diagonal elements. One can therefore enter a correlation coefficient corresponding to a particular phase portrait in the transpose position to the phase portrait in the matrix. A correlation coefficient provides a quantitative description of the phase portrait's linearity between the two variables over the time scale displayed. In addition to the correlation coefficient, the slope can be computed and displayed, giving the equilibrium constant between the two compounds displayed. \n\n## Demonstration of the Simulation Procedure in MASSpy\n### Setting up the model\nThe following builds the model of three reactions in series that is described on pages 51-56 in the book. We show how the model is built, simulated, solutions graphically displayed, solutions post processed and analyzed mathematically.\n\nTo construct a model in **MASSpy**, the `MassModel`, `MassReaction`, and `MassMetabolite` objects need to be imported into the environment. \n\n\n```python\nfrom mass import MassModel, MassMetabolite, MassReaction\n```\n\n#### Defining metabolites and reactions\nOne method for creating the model is to objects that represent the metabolites and reactions. \nMetabolite are represented by `MassMetabolite` objects, and can be created by providing a unique identifier for that object. Therefore we can define the four metabolites, $x_1, x_2, x_3$, and $x_4$ by the following;\n\n\n```python\nx1 = MassMetabolite('x1')\nx2 = MassMetabolite('x2')\nx3 = MassMetabolite('x3')\nx4 = MassMetabolite('x4')\n```\n\nReactions are represented by `MassReaction` objects, and like metabolites, they can be also created by providing a unique identifier for that object.\n\n\n```python\nv1 = MassReaction('v1')\nv2 = MassReaction('v2')\n```\n\nBy default, a reaction is considered reversible. However, if we wish to make an irreversible reaction, we set the `reversible` argument to `False`. \n\n\n```python\nv3 = MassReaction('v3', reversible=False)\n```\n\nOnce the `MassReaction` objects have been created, metabolites can be added to the reaction using the `MassReaction.add_metabolites` method. \nTo quickly see how this method is used, we can use the `help()` function. Alternatively, we can go to the [API documentation](https://masspy.readthedocs.io/en/latest/autoapi/index.html) and read about how the [MassReaction.add_metabolites](https://masspy.readthedocs.io/en/latest/autoapi/mass/core/mass_reaction/index.html#mass.core.mass_reaction.MassReaction.add_metabolites) method works.\n\n\n```python\nhelp(MassReaction.add_metabolites)\n```\n\n Help on function add_metabolites in module mass.core.mass_reaction:\n \n add_metabolites(self, metabolites_to_add, combine=True, reversibly=True)\n Add metabolites and their coefficients to the reaction.\n \n If the final coefficient for a metabolite is 0 then it is removed from\n the reaction.\n \n The change is reverted upon exit when using the :class:`~.MassModel`\n as a context.\n \n Notes\n -----\n * A final coefficient of < 0 implies a reactant and a final\n coefficient of > 0 implies a product.\n \n * Extends :meth:`~cobra.core.reaction.Reaction.add_metabolites` of the\n :class:`cobra.Reaction ` by first\n ensuring that the metabolites to be added are\n :class:`.MassMetabolite`\\ s and not\n :class:`cobra.Metabolites `.\n and error message raised reflects the :mod:`mass` object.\n \n * If a :class:`cobra.Metabolite ` is\n provided. a warning is raised and a :class:`.MassMetabolite`\n will be instantiated using the\n :class:`cobra.Metabolite `.\n \n Parameters\n ----------\n metabolites_to_add : dict\n A ``dict`` with :class:`.MassMetabolite`\\ s or metabolite\n identifiers as keys and stoichiometric coefficients as values. If\n keys are strings (id of a metabolite), the reaction must already\n be part of a :class:`~.MassModel` and a metabolite with the given\n id must already exist in the :class:`~.MassModel`.\n combine : bool\n Describes the behavior of existing metabolites.\n If ``True``, the metabolite coefficients are combined together.\n If ``False`` the coefficients are replaced.\n reversibly : bool\n Whether to add the change to the context to make the change\n reversible (primarily intended for internal use).\n \n See Also\n --------\n :meth:`subtract_metabolites`\n \n\n\nTo use `MassReaction.add_metabolites`, a dictionary input is required, where the `MassMetabolite` objects are keys and the value is their stoichiometric coefficient. Reactants are defined with negative coefficients, while products are defined with positive coefficients. \n\n\n```python\nv1.add_metabolites({x1 : -1, x2 : 1})\nv2.add_metabolites({x2 : -1, x3 : 1})\nv3.add_metabolites({x3 : -1, x4 : 1})\n```\n\nReactions, e.g., $v_1$ can be used to define any kind of chemical transformation, association, activation etc. A series of methods are provided for inspection of the reaction.\n\n\n```python\nv1.id\n```\n\n\n\n\n 'v1'\n\n\n\n\n```python\nv1.reactants\n```\n\n\n\n\n []\n\n\n\n\n```python\nv1.products\n```\n\n\n\n\n []\n\n\n\n\n```python\nv1.stoichiometry\n```\n\n\n\n\n [-1, 1]\n\n\n\n\n```python\nv1.reversible\n```\n\n\n\n\n True\n\n\n\nCheck the documentation for the `MassReaction` class for further details.\n\n#### Model Setup\nTo construct a model capable of dynamic simulation, a `MassModel` object must be created. The minimal input for creating a `MassModel` object is a unique identifier. \n\n\n```python\nmodel = MassModel('Model')\n```\n\n Set parameter Username\n\n\nTo add reactions and their corresponding metabolites to the model, the `MassModel.add_reactions` method can be used by providing a list of reactions to add to the model. \n\n\n```python\nmodel.add_reactions([v1, v2, v3])\n```\n\n#### Model Inspection\nSimilar to the `MassReaction` object, the `MassModel` object also has various methods that can be used to inspect the model. For example, to obtain the list of reactions and species in the system:\n\n\n```python\nmodel.reactions\n```\n\n\n\n\n [,\n ,\n ]\n\n\n\n\n```python\nmodel.metabolites\n```\n\n\n\n\n [,\n ,\n ,\n ]\n\n\n\nIn some circumstances, it is helpful to iterate through a reaction and its associated metabolites using a loop:\n\n\n```python\nprint(\"Model ID: %s\" % model.id)\nfor rxn in model.reactions:\n print(\"\\nReaction: %s\\n------------\" % rxn.id)\n for metab, stoichiometry in rxn.metabolites.items():\n print(\"%s: %s \" % (metab.id, stoichiometry))\n```\n\n Model ID: Model\n \n Reaction: v1\n ------------\n x1: -1 \n x2: 1 \n \n Reaction: v2\n ------------\n x3: 1 \n x2: -1 \n \n Reaction: v3\n ------------\n x4: 1 \n x3: -1 \n\n\nTo examine the stoichiometric matrix:\n\n\n```python\nmodel.S\n```\n\n\n\n\n array([[-1., 0., 0.],\n [ 1., -1., 0.],\n [ 0., 1., -1.],\n [ 0., 0., 1.]])\n\n\n\nThe stoichiometric matrix can also be viewed as a `pandas.DataFrame` with annotated information about the metabolites and reactions.\n\nNote: The `update_model` argument can be used to store matrix as the specified `array_type` for the next time the stoichiometric matrix is viewed. \n\n\n```python\nmodel.update_S(array_type=\"DataFrame\", update_model=True)\n```\n\n\n\n\n
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\n\n\n\nThe rate equations can be examined,\n\n\n```python\nfor rxn, rate in model.rates.items():\n print(\"%s: %s\" % (rxn.id, rate))\n```\n\n v1: kf_v1*(x1(t) - x2(t)/Keq_v1)\n v2: kf_v2*(x2(t) - x3(t)/Keq_v2)\n v3: kf_v3*x3(t)\n\n\nor just one rate equation can be called out:\n\n\n```python\nprint(model.rates[v2])\n```\n\n kf_v2*(x2(t) - x3(t)/Keq_v2)\n\n\nThe ordinary differential equations can be also be listed in full,\n\n\n```python\nfor metab, ode in model.odes.items():\n print(\"%s: %s\" % (metab.id, ode))\n```\n\n x1: -kf_v1*(x1(t) - x2(t)/Keq_v1)\n x2: kf_v1*(x1(t) - x2(t)/Keq_v1) - kf_v2*(x2(t) - x3(t)/Keq_v2)\n x3: kf_v2*(x2(t) - x3(t)/Keq_v2) - kf_v3*x3(t)\n x4: kf_v3*x3(t)\n\n\nor just one ordiniary differential equation can be called out:\n\n\n```python\nprint(model.odes[x3])\n```\n\n kf_v2*(x2(t) - x3(t)/Keq_v2) - kf_v3*x3(t)\n\n\nNote that none of these expressions have been provided during the model construction process. Instead the expresions have been generated automatically from the provided list of reactions and their metabolites. \n\n#### Set parameters and initial condtions\nWhen using Jupyter notebooks, an overview of the model is rendered as a table when only the model object is called. Note that this also applies to metabolites and reactions.\n\n\n```python\nmodel\n```\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
NameModel
Memory address0x07fedd07132b0
Stoichiometric Matrix4x3
Matrix Rank3
Number of metabolites4
Initial conditions defined0/4
Number of reactions3
Number of genes0
Number of enzyme modules0
Number of groups0
Objective expression0
Compartments
\n\n\n\n\nFrom the model overview it can be seen that no parameters or initial conditions have been defined. Parameters can be defined directly for a specific reaction:\n\n\n```python\nv1.forward_rate_constant = 1\nv2.kf = 0.01 # Shorthand method\nv3.kf = 0.0001\n\nv1.equilibrium_constant = 1\nv2.Keq = 1 # Shorthand method\n\nfor param_type, param_dict in model.parameters.items():\n print(\"%s: %s\" %(param_type, param_dict))\n```\n\n kf: {'kf_v1': 1, 'kf_v2': 0.01, 'kf_v3': 0.0001}\n Keq: {'Keq_v1': 1, 'Keq_v2': 1, 'Keq_v3': inf}\n kr: {}\n v: {}\n Custom: {}\n Boundary: {}\n\n\nInitial conditions for metabolites can be defined directly for a specific metabolite,\n\n\n```python\nx1.initial_condition = 1\nx2.ic = 0 # Shorthand method\nmodel.initial_conditions\n```\n\n\n\n\n {: 1,\n : 0}\n\n\n\nor a dictionary can be used to define them in a model directly. The `update_metabolites` argument will subsequently update the initial condition in the metabolite object as well. \n\n\n```python\nmodel.update_initial_conditions({x3: 0, x4:0})\nmodel.initial_conditions\n```\n\n\n\n\n {: 1,\n : 0,\n : 0,\n : 0}\n\n\n\nCheck the documentation for further details on the `MassModel` class. \n\n### Simulating Dynamic Responses\n#### Simulate\nSimulating the model once it is set up properly is very simple. To set up the simulation, we use a `Simulation` object. The simulation object requires a `MassModel` for initialization.\n\n\n```python\nfrom mass import Simulation\n```\n\n\n```python\nsim = Simulation(model, verbose=True)\n```\n\n \u001b[93mWARNING:\u001b[0m \u001b[93mNo compartments found in model. Therefore creating compartment 'compartment' for entire model.\u001b[0m\n\n\n Successfully loaded MassModel 'Model' into RoadRunner.\n\n\nThe `Simulation.simulate` method from the will integrate the ordinary differential equations of the system in the provided time interval and return the dynamic responses of concentrations and fluxes.\n\n\n```python\nt0 = 0\ntf = 1e6\n\nconc_sol, flux_sol = sim.simulate(\n model, time=(t0, tf), interpolate=True, verbose=True)\n```\n\n Getting time points\n Setting output selections\n Setting simulation values for 'Model'\n Simulating 'Model'\n Simulation for 'Model' successful\n Adding 'Model' simulation solutions to output\n Updating stored solutions\n\n\nNote: If a model is unable to be simulated, a warning will be raised. By setting the `verbose` argument to `True`, a QC/QA report outlining inconsistencies, missing values, and other issues will also be generated and displayed to assist in diagnosing the reason why a model could not be simulated. \n\n#### Inspect the solution\nAs the default setting, the Simulation object utilizes scipy interpolating functions to capture the concentration and flux responses (see documentation for scipy.interpolate for additional information). The `Simulation.simulate_model` method returns two `cobra.DictLists` containing specialized dictionaries known as `MassSolution` objects.\n\nThe first `MassSolution` object contains the `MassMetabolite` identifiers as keys, and their corresponding concentration solutions as values.\n\n\n```python\nfor metabolite, solution in conc_sol.items():\n print(metabolite, solution)\n```\n\n x1 \n x2 \n x3 \n x4 \n\n\nSimilarly, the second `MassSolution` object contains the `MassReaction` identifiers as keys, and their corresponding flux solutions as values.\n\n\n```python\nfor reaction, solution in flux_sol.items():\n print(reaction, solution)\n```\n\n v1 \n v2 \n v3 \n\n\n#### Query time responses\nThe interpolating functions are functions of time. Therefore, we can evaluate the interpolating function at a specific time point using the following: \n\n\n```python\ntime_points = 100;\nfor metabolite, interpolating_function in conc_sol.items():\n print(\"%s: %s\" % (metabolite, interpolating_function(time_points)))\nprint()\nfor reaction, interpolating_function in flux_sol.items():\n print(\"%s: %s\" % (reaction, interpolating_function(time_points)))\n```\n\n x1: 0.3710242389082219\n x2: 0.3704524448547136\n x3: 0.2569363810507253\n x4: 0.0015869284328310024\n \n v1: 0.0005717940535082393\n v2: 0.0011351606380398832\n v3: 2.5693638105072534e-05\n\n\nIt is also possible to get values for multiple time points at once:\n\n\n```python\ntime_points = [0.01, 0.1, 1, 10, 100, 1000];\nfor metabolite, interpolating_function in conc_sol.items():\n print(\"%s: %s\" % (metabolite, interpolating_function(time_points)))\nprint()\nfor reaction, interpolating_function in flux_sol.items():\n print(\"%s: %s\" % (reaction, interpolating_function(time_points)))\n```\n\n x1: [0.99009934 0.90936384 0.56699581 0.4790072 0.37102424 0.32389592]\n x2: [0.00990017 0.09058937 0.43018534 0.47682748 0.37045244 0.32388517]\n x3: [4.96651050e-07 4.67952432e-05 2.81873928e-03 4.41437817e-02\n 2.56936381e-01 3.21735095e-01]\n x4: [1.65778860e-13 1.58583691e-10 1.07541682e-07 2.15291648e-05\n 1.58692843e-03 3.04838051e-02]\n \n v1: [9.80199167e-01 8.18774471e-01 1.36810476e-01 2.17971609e-03\n 5.71794054e-04 1.07505761e-05]\n v2: [9.89967148e-05 9.05425714e-04 4.27366598e-03 4.32683701e-03\n 1.13516064e-03 2.15007648e-05]\n v3: [4.96651050e-11 4.67952432e-09 2.81873928e-07 4.41437817e-06\n 2.56936381e-05 3.21735095e-05]\n\n\nFor example, a `pandas.Dataframe` of concentration values at different time points could be generated using this method: \n\n\n```python\nimport pandas as pd\n```\n\n\n```python\ndata = [interpolating_function(time_points) \n for interpolating_function in conc_sol.values()]\nindex_col = [metabolite for metabolite in conc_sol.keys()]\npd.DataFrame(data, index=index_col, columns=time_points)\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
0.010.101.0010.00100.001000.00
x19.900993e-019.093638e-015.669958e-010.4790070.3710240.323896
x29.900168e-039.058937e-024.301853e-010.4768270.3704520.323885
x34.966510e-074.679524e-052.818739e-030.0441440.2569360.321735
x41.657789e-131.585837e-101.075417e-070.0000220.0015870.030484
\n
\n\n\n\nThe same can be done for the fluxes: \n\n\n```python\ndata = [interpolating_function(time_points) \n for interpolating_function in flux_sol.values()]\nindex_col = [reaction for reaction in flux_sol.keys()]\npd.DataFrame(data, index=index_col, columns=time_points)\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
0.010.101.0010.00100.001000.00
v19.801992e-018.187745e-011.368105e-010.0021800.0005720.000011
v29.899671e-059.054257e-044.273666e-030.0043270.0011350.000022
v34.966510e-114.679524e-092.818739e-070.0000040.0000260.000032
\n
\n\n\n\n#### Filtering for specific species and fluxes\nBecause concentration and flux `MassSolution` objects are specialized dictionaries, they can be handled like any other dictionary. Therefore, obtaining the solution for individual species and fluxes can be done easily by using the `MassMetabolite` or `MassReaction` identifiers as keys.\n\n\n```python\nprint(x1.id, conc_sol[x1.id])\n```\n\n x1 \n\n\n\n```python\nfor flux in [v1, v2]:\n print(flux.id, flux_sol[flux.id])\n```\n\n v1 \n v2 \n\n\n#### Switching between numerical arrays and interpolating functions\nSuppose that instead of working with interpolating functions, we would rather work with the original time points and the corresponding solutions utilized by the ODE solver. One way this can be done would be to access the original time point values stored in the Solution object, and use those in the interpolating function:\n\n\n```python\ntime_points = conc_sol.t\n# Get a slice of the first 50 points\nprint(conc_sol[\"x1\"](time_points)[:50])\n```\n\n [1. 1. 0.99999943 0.99999734 0.99999525 0.99999316\n 0.99998821 0.99997787 0.99995537 0.99990413 0.99978077 0.99942553\n 0.99907055 0.99871581 0.99806461 0.99741426 0.99676475 0.9961161\n 0.9948862 0.99290131 0.98987466 0.9868666 0.983877 0.98090577\n 0.97795277 0.97334374 0.96877916 0.96425858 0.95727034 0.94370569\n 0.92186543 0.90109967 0.88135539 0.86258222 0.84473223 0.82775987\n 0.81162185 0.78756746 0.76536568 0.7448733 0.72595816 0.70849829\n 0.69238106 0.67750262 0.66376717 0.64105858 0.62146897 0.6045664\n 0.58997873 0.57738534]\n\n\nTo quickly convert an entire `MassSolution` object from interpolating functions to numerical arrays or vice-versa, we use the `MassSolution.interpolate` setter method:\n\n\n```python\nconc_sol.interpolate = False\n# Get a slice of the first 50 points\nconc_sol[\"x1\"][:50]\n```\n\n\n\n\n array([1. , 1. , 0.99999943, 0.99999734, 0.99999525,\n 0.99999316, 0.99998821, 0.99997787, 0.99995537, 0.99990413,\n 0.99978077, 0.99942553, 0.99907055, 0.99871581, 0.99806461,\n 0.99741426, 0.99676475, 0.9961161 , 0.9948862 , 0.99290131,\n 0.98987466, 0.9868666 , 0.983877 , 0.98090577, 0.97795277,\n 0.97334374, 0.96877916, 0.96425858, 0.95727034, 0.94370569,\n 0.92186543, 0.90109967, 0.88135539, 0.86258222, 0.84473223,\n 0.82775987, 0.81162185, 0.78756746, 0.76536568, 0.7448733 ,\n 0.72595816, 0.70849829, 0.69238106, 0.67750262, 0.66376717,\n 0.64105858, 0.62146897, 0.6045664 , 0.58997873, 0.57738534])\n\n\n\n\n```python\nconc_sol.interpolate = True\nconc_sol[\"x1\"]\n```\n\n\n\n\n \n\n\n\n\n```python\nfor key, value in conc_sol.items():\n print(key, value)\n```\n\n x1 \n x2 \n x3 \n x4 \n\n\n\n```python\nconc_sol[\"x1\"]\n```\n\n\n\n\n \n\n\n\n\n```python\nconc_sol.x1\n```\n\n\n\n\n \n\n\n\n### Visualizing the Solution Graphically\nOnce the model has been simulated, the solutions can be visualized using the visualization tools in **MASSpy**. \n\n\n```python\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfrom mass.visualization import (\n plot_phase_portrait, plot_time_profile, plot_tiled_phase_portraits)\n```\n\nAll visualization tools utilize the matplotlib python package. See documentation for the visualization class for more details on the available plotting kwargs. \n\n#### Draw time course\nPlotting the dynamic responses is straightforward using the `plot_time_profile` function:\n\n\n```python\nplot_time_profile(conc_sol);\n```\n\nFor this model and simulation, plotting on a linear scale does not provide us information about the dyanmics at various time scales. Therefore, we can use the `plot_function` kwarg to change the scale. Let us keep a linear scale on the y-axis, but change the x-axis to a logarithmic scale. \n\n\n```python\nplot_time_profile(conc_sol, plot_function=\"semilogx\");\n```\n\nThe `observable` argument allows one to specify particular solutions from the solution profile to observe while filtering out all other solutions. For example, only the solutions for $x_1$ and $x_2$ can be observed by setting observable to an list of these two keys in the solution profile. \n\n\n```python\nplot_time_profile(conc_sol, observable=[\"x1\", \"x2\"], \n plot_function=\"semilogx\");\n```\n\nThough the dynamic behavior is clear, the above plots do not provide any other information. Let us add axes labels, a title, and a legend to the plot.\n\n\n```python\nplot_time_profile(\n conc_sol, legend=\"right outside\", plot_function=\"semilogx\",\n xlabel=\"Time\", ylabel=\"Concentration\", \n title=(\"Concentration Solutions\", {\"size\": \"large\"}));\n```\n\n#### Draw phase portraits\nPlotting the dynamic responses against one another is also straightforward by using the `plot_phase_portrait` function:\n\n\n```python\nplot_phase_portrait(conc_sol, x=\"x1\", y=\"x2\",\n xlabel=\"x1\", ylabel=\"x2\");\n```\n\n$x_1$ vs $x_2$: note that you can use the `annotate_time_points` argument to highlight particular time points of interest. This argument can be utilized either by providing iterable of time points of interest. The `annotate_time_points_color` can be used to set the color of the time points. To use color to distinguish time points, the number of colors should equal the number of time points specified.\n\n\n```python\nplot_phase_portrait(\n conc_sol, x=\"x1\", y=\"x2\", xlabel=\"x1\", ylabel=\"x2\",\n annotate_time_points=[t0, 1e-1, 1e0, 1e1, 1e3, tf],\n annotate_time_points_color= [\n \"red\", \"green\", \"purple\", \"yellow\", \"cyan\", \"blue\"],\n annotate_time_points_legend=\"lower outside\");\n```\n\nAll pairwise phase portraits can be generated and viewed at once in a tiled format using the `plot_tiled_phase_portrait` function:\n\n\n```python\nplot_tiled_phase_portraits(conc_sol,\n annotate_time_points_legend=\"right outside\");\n```\n\nThis method is particularly useful for looking at correlations at various time scales. For example, looking at the overall behavior, a fast time timescale of (0, 1), an intermediate timescale of (3, 100), and a slow timescale of (300, 10000), we can generate the following:\n\n\n```python\ncorrelations = [\n np.empty((3, 3)).astype(str),\n np.empty((3, 3)).astype(str),\n np.empty((3, 3)).astype(str),\n np.empty((3, 3)).astype(str)]\nfor entry in correlations:\n entry.fill(\"\")\n\nfmt_str = \"{0:.2f}\\n{1:.2f}\"\ncorrelations[1][0, 1] = fmt_str.format(*[1, 1])\ncorrelations[2][0, 1] = fmt_str.format(*[1, 1.02])\ncorrelations[2][0, 2] = fmt_str.format(*[1, 0.51])\ncorrelations[2][1, 2] = fmt_str.format(*[1, 0.5])\ncorrelations[3][0, 1] = fmt_str.format(*[1, 1.02])\ncorrelations[3][0, 2] = fmt_str.format(*[1, 1.02])\ncorrelations[3][1, 2] = fmt_str.format(*[1, 1.02])\n\nfig, axes = plt.subplots(2, 2, figsize=(10, 10))\naxes = axes.flatten()\n\ntimes = [(0, 400000), (0, 1), (3, 100), (300, 10000)]\ntitles = [\"{0}\\nt0={1}; tf={2}\".format(label, *time)\n for label, time in zip([\"(a)\", \"(b)\", \"(c)\", \"(d)\"], times)]\n\nfor i, ax in enumerate(axes.flatten()):\n plot_tiled_phase_portraits(\n conc_sol, observable=[\"x1\", \"x2\", \"x3\"], ax=ax,\n plot_tile_placement=\"lower\", additional_data=correlations[i],\n time_vector=np.linspace(*times[i], int(1e6)),\n tile_xlabel_fontdict={\"size\": \"large\"},\n tile_ylabel_fontdict={\"size\": \"large\"},\n title=titles[i])\n```\n\n### Post process the solution\n#### Analyze pool behavior\nIn order to analyze the behavior of pools, pools can be created using the `MassSolution.make_aggregate_solution` method using the string representation of the pooling formulas. Additional parameters can also be incorporated into the pool formulation using a dictionary input for the `parameters` argument. \n\n\n```python\npools = [\"x1 - x2\", \"x1 + x2 - 2*x3\", \"x1 + x2 + x3\"]\n\nfor i, equation_str in enumerate(pools):\n pool_id = \"p\" + str(i + 1)\n conc_sol.make_aggregate_solution(\n pool_id, equation=equation_str, update=True)\n print(pool_id, conc_sol[pool_id])\n```\n\n p1 \n p2 \n p3 \n\n\nThis method utilizes the solutions for the individual metabolites over the time range input, and then creates new solutions to represent the behavior of those pools. \n\n\n```python\nplot_time_profile(\n conc_sol, observable=[\"p1\", \"p2\", \"p3\"], legend=\"best\",\n plot_function=\"semilogx\",\n xlabel=\"time\", ylabel=\"Concentrations\",\n title=(\"Pool profile\", {\"size\": \"large\"}));\n```\n\n#### Compute and plot the fluxes\nA similar process as above can be utilized to obtain behavior of the net flux through a group of reactions. Note that the `MassSolution.make_aggregate_solution` method relies on the `sympy.sympify` function and can therefore utilize specific methods, such as the absolute value function, in the string as well. \n\n\n```python\nflux_sol.make_aggregate_solution(\n \"v_net\", equation='Abs(v1) + Abs(v2) + Abs(v3)', update=True)\n```\n\n\n\n\n {'v_net': }\n\n\n\nAgain, this method obtains the solutions for the individual fluxes over the time range input, and then creates new solutions to represent the behavior of various flux combinations. \n\n\n```python\nplot_time_profile(\n flux_sol, observable=[\"v_net\"], legend=\"best\",\n plot_function=\"semilogx\", xlabel=\"time\", ylabel=\"Fluxes\", \n title=(\"Net Flux\", {\"size\": \"large\"}));\n```\n\n#### Plot phase portraits of pools\n\n\n```python\nfig, ax = plt.subplots(1, 1, figsize=(5, 5))\nplot_tiled_phase_portraits(\n conc_sol, observable=[\"p1\", \"p2\", \"p3\"], ax=ax,\n plot_tile_placement=\"lower\",\n annotate_time_points_legend=\"right outside\");\n```\n\nHere, we can see that all of the defined pools are dynamically independent of one another.\n\n## Summary\n\n* Network dynamics are described by dynamic mass balances $(d\\textbf{x}/dt = \\textbf{Sv}(\\textbf{x}; \\textbf{k}))$ that are formulated after applying a series of simplifying assumptions \n\n* To simulate the dynamic mass balances we have to specify the numerical values of the kinetic constants $(\\textbf{k})$, the initial conditions $(\\textbf{x}_0)$, and any fixed boundary fluxes.\n\n* The equations with the initial conditions can be integrated numerically. \n\n* The solution contains numerical values for the concentration variables at discrete time points. The solution is graphically displayed as concentrations over time, or in a phase portrait.\n\n* The solution can be post-processed following its initial analysis to bring out special dynamic features of the network. Such features will be described in more detail in the subsequent notebooks.\n\n$\\tiny{\\text{\u00a9 B. \u00d8. Palsson 2011;}\\ \n\\text{This publication is in copyright.}\\\\ \n\\text{Subject to statutory exception and to the provisions of relevant collective licensing agreements,}\\\\ \\text{no reproduction of any part may take place without the written permission of Cambridge University Press.}}$\n", "meta": {"hexsha": "074fa78acdfb1035ac124d8733cbffba608e7603", "size": 300888, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/education/sb2/chapters/sb2_chapter3.ipynb", "max_stars_repo_name": "z-haiman/MASSpy", "max_stars_repo_head_hexsha": "aeeed1e3f9d1058e9485247a86f85cb94eeecbc9", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/education/sb2/chapters/sb2_chapter3.ipynb", "max_issues_repo_name": "z-haiman/MASSpy", "max_issues_repo_head_hexsha": "aeeed1e3f9d1058e9485247a86f85cb94eeecbc9", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/education/sb2/chapters/sb2_chapter3.ipynb", "max_forks_repo_name": "z-haiman/MASSpy", "max_forks_repo_head_hexsha": "aeeed1e3f9d1058e9485247a86f85cb94eeecbc9", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 119.4, "max_line_length": 44292, "alphanum_fraction": 0.852167584, "converted": true, "num_tokens": 13465, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321720225279, "lm_q2_score": 0.6297745935070808, "lm_q1q2_score": 0.44854572661815273}} {"text": "# Week 04: Bottom-Up Parsing I & II\n\n## Predictive Parser: LL(1)\n\n### Predictive Parsing\n\n- **Concept**\n - Look at the next? tokens.\n - No backtracking.\n - Accept LL(K) grammars. (Left-to-Right, Leftmost derivation, k tokens)\n - At each step, only one choice.\n - To avoid ambiguousness, need to **left-factor** the grammar.\n- **Parsing Table**\n - **Leftmode Non-terminal** x **Next Input Token**\n - Use stack to record frontier of parse tree\n- **Differnce** from Recursive Descent\n - For the leftmost non-terminal **S**\n - Look at the next input token **a**\n - Choose the production shown at **[S,a]**\n- **Reject** on reaching error state\n- **Accept** on (end of input && empty stack)\n\n### First Set\n\n**Def**. (First Set)\n\n$$First(X)=\\{t\\ |\\ X\\rightarrow^*t\\alpha\\}\\cup\\{\\epsilon\\ |\\ X\\rightarrow^*\\epsilon\\}$$\n\n**Algo**\n\n1. $First(X)=\\{t\\}, if\\ t\\ is\\ a\\ terminal.$\n2. $\\epsilon \\in First(X), if\n\\begin{cases}\nX\\rightarrow \\epsilon \\\\\nX\\rightarrow A_1A_2\\dots A_n, and\\ \\epsilon \\in First(A_i), \\forall 1\\leq i\\leq n\n\\end{cases}$\n3. $First(\\alpha) \\subseteq First(X), if\\ X\\rightarrow A_1A_2\\dots A_n\\alpha, and\\ \\epsilon \\in First(A_i), \\forall 1\\leq i\\leq n$\n\n### Follow Set\n\n**Def**. (Follow Set)\n\n$$Follow(X)=\\{t\\ |\\ S\\rightarrow^*\\beta Xt\\delta\\}$$\n\n**Algo**\n\n1. $\\$ \\in Follow(S)$\n2. $First(\\beta)-\\{\\epsilon\\}\\subseteq Follow(X), for\\ each\\ A\\rightarrow\\alpha X\\beta$\n3. $Follow(A) \\subseteq Follow(X), for\\ each A\\rightarrow\\alpha X\\beta\\ where\\ \\epsilon\\in First(\\beta)$\n\n### How to Construct LL(1) Parsing Tables\n\n- Construct a parsing table **T** for CFG **G**\n- For each production $A \\rightarrow \\alpha$ in G do:\n - For each terminal $t \\in First(\\alpha)$, $T[A, t] = \\alpha$\n - If $\\epsilon \\in First(\\alpha),$, for each $t \\in Follow(A)$, $T[A,t] = \\alpha$\n - If $\\epsilon \\in First(\\alpha)$ and $ \\$ \\in Follow(A)$, $T[A,\\$] = \\alpha$\n- If any entry is multiply defined, then G is not LL(1).\n\n## Bottom-Up Parsing \n\n### Shift-Reduce Parsing\n\n- Concept\n - Don't need **left-factored** grammars.\n - Reduce the string to the start symbol. (Inverting production)\n - A bottom-up parser traces a rightmost derivation in reverse. $$\n\\begin{align}\n& int*int + int \\\\\n& \\textbf{int*T} + int \\\\\n& \\textbf{T} + int \\\\\n& T + \\textbf{T} \\\\\n& T + \\textbf{E} \\\\\n& \\textbf{E}\n\\end{align}\n$$\n - Thm. \n - In some step, let string as $\\alpha \\beta \\gamma$.\n - Assume the next reduction is by $X \\rightarrow \\beta$. ($\\alpha \\beta \\gamma \\rightarrow \\alpha X \\gamma$)\n - Then $\\gamma$ is a string which contains only terminals.\n - Split the string as $L_{Str} | R_{Str}$.\n - $L_{Str}$ contains non-terminals and terminals. $R_{Str}$ contains only terminals.\n- Actions\n - **Shift**: Move | one place to the right. Shift a terminal to the left string.\n - **Reduce**: Apply an inverse production at the right of $L_{str}$. $$\nfor\\ A \\rightarrow xy, Cbxy|ijk \\Rightarrow CbA|ijk $$\n- Implementation\n - Use a stack to maintain $L_{Str}$.\n- If is's legal to shift or reduce, there is a **shift-reduce** conflict.\n- If is's legal to reduce by two productions, there is a **reduce-reduce** conflict.\n\n### Term. (Handle)\n\n- **Scenario**\n - Grammar\n - $E \\rightarrow T+E|E$\n - $T \\rightarrow int*T|int|(E)$\n - At the steop $int|*int+int$\n - If we reduce by ($T \\rightarrow int$) given ($T | *int+int$), it will fail.\n- **Target**\n - Want to reduce only if the result can still be reduced to the start symbol.\n- A **handle** is the rhs of the production that you can trace back in the parse tree.\n - $\\Rightarrow$ Handles are the **children of some internal node** of some AST.\n- Assume a rightmost derivation $$S \\rightarrow^* \\alpha X \\omega \\rightarrow \\alpha\\beta\\omega$$\n - Then $\\beta$ is a **handle** of $\\alpha \\beta \\omega$.\n- Thm.\n - In shift-reduce parsing, handles appear only **at the top of the stack**, never inside.\n- Bottom-up parsing algorithms are based on recognizing handles\n\n### Recognizing Handles\n\n- There are no known efficient algo to recognize handles. But there are good heuristics.\n- **Viable Prefix**: $\\alpha_1\\dots\\alpha_n$ is a **viable prefix** if there is an $\\omega$ s.t. $\\alpha_1\\dots\\alpha_n\\ |\\ \\omega$ is a state of a shift-reduce parser.\n - A viable prefix is always the prefix of some handle.\n - If we can maintain the stack's contents are viable prefixes all the time, no parsing error will occur.\n- Thm. For any grammar, the set of viable prefixes is a **regular language**.\n- **Item**: An **item** is a production with '.' somewhere on the rhs.\n - Example. The items for the production rule $T \\rightarrow (E)$ are\n - $T \\rightarrow .(E)$\n - $T \\rightarrow (.E)$\n - $T \\rightarrow (E.)$\n - $T \\rightarrow (E).$\n - Example. The items for ($X \\rightarrow \\epsilon$) are $X \\rightarrow .$\n - Items are often called **LR(0) items**.\n- Target: To describe the states of the production rule.\n - If we need to use some production rule $R$ to reduce, the top of the stack will be the left string split by '.' on some item of $R$.\n - Example.\n - Input: $(int)$\n - Grammar\n - $E \\rightarrow T+E\\ |\\ T$\n - $T \\rightarrow int*T\\ |\\ int\\ |\\ (E)$\n - $(E$ is the left string split by '.' on the item $T \\rightarrow (E.)$\n\n### Recognizing Viable Prefixes\n\n1. Add a dummy production $S' \\rightarrow S$ to $G$\n2. The NFA states are the items of $G$\n3. For item $E \\rightarrow \\alpha .X\\beta$ add transition: $(E \\rightarrow \\alpha . X \\beta) \\rightarrow^X (E \\rightarrow \\alpha X.\\beta)$\n4. For item $E \\rightarrow \\alpha .X\\beta$ and production $X \\rightarrow \\gamma$, add transition: $(E \\rightarrow \\alpha . X \\beta) \\rightarrow^\\epsilon (X \\rightarrow .\\gamma)$\n5. Every state is an accepting state\n6. Start state is $S' \\rightarrow .S$\n\n### Valid Items\n\n- Concept\n - Item $X \\rightarrow \\beta .\\gamma$ is **valid** for a viable prefix $\\alpha\\beta$ if $$S' \\rightarrow^* \\alpha X\\omega \\rightarrow \\alpha\\beta\\gamma\\omega$$ by a right-most derivation.\n - An item $I$ is valid for a viable prefix $\\alpha(\\alpha_1\\dots\\alpha_n)$ if the DFA accepts $\\alpha_1\\dots\\alpha_n$ and terminates on some state $s$ containing $I$.\n - The items in $s$ describe what the top of the item stack might be after reading input $\\alpha_1\\dots\\alpha_n$.\n \n\n### LR(0) Parsing\n\n- **Assume**\n - Stack contains $\\alpha_1\\dots\\alpha_k$\n - Next input is $t$\n - DFA on input $\\alpha_1\\dots\\alpha_k$ teminates in state $s$\n- **Action**\n - Reduce by $X \\rightarrow \\beta$ if $s$ contains item $X \\rightarrow \\beta.$\n - Shift if $s$ contains item $X \\rightarrow \\beta .t\\omega$\n- **Conflict**\n - **Reduce/Reduce** if any state contains two reduce rule.\n - **Shift/Reduce** if any state has a reduced item and a shift item\n \n### Simple LR Paring (SLR(1) Parsing)\n\n- **Assume**\n - Stack contains $\\alpha_1\\dots\\alpha_k$\n - Next input is $t$\n - DFA on input $\\alpha_1\\dots\\alpha_k$ teminates in state $s$\n- **Action**\n - Reduce by $X \\rightarrow \\beta$ if $s$ contains item $X \\rightarrow \\beta.$ and $t \\in Follow(X)$\n - Shift if $s$ contains item $X \\rightarrow \\beta .t\\omega$\n- If there are conflicts under these rules, the grammar is not SLR.\n- We can use **precedence declarations** to resolve conflicts. \\* is grater than \\+.\n\n### LR(1)\n\n- More powerful than SLR(1)\n- Build by (item x lookahead).\n- SLR(1) only checks whether lookahead is in follow set.\n\n\n```python\n\n```\n", "meta": {"hexsha": "b8b347e7a90db55f591a64079ca50d5ebf3d5522", "size": 10119, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "OCW/[Stanford]CS143_Compilers/Notes/Week04.ipynb", "max_stars_repo_name": "Quexint/MOOC-Assignment-Tutorial", "max_stars_repo_head_hexsha": "0b1cb34797d3df56913f0f860dae21ef35a24dfb", "max_stars_repo_licenses": ["Unlicense", "MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "OCW/[Stanford]CS143_Compilers/Notes/Week04.ipynb", "max_issues_repo_name": "Quexint/MOOC-Assignment-Tutorial", "max_issues_repo_head_hexsha": "0b1cb34797d3df56913f0f860dae21ef35a24dfb", "max_issues_repo_licenses": ["Unlicense", "MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "OCW/[Stanford]CS143_Compilers/Notes/Week04.ipynb", "max_forks_repo_name": "Quexint/MOOC-Assignment-Tutorial", "max_forks_repo_head_hexsha": "0b1cb34797d3df56913f0f860dae21ef35a24dfb", "max_forks_repo_licenses": ["Unlicense", "MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 43.2435897436, "max_line_length": 210, "alphanum_fraction": 0.5419507857, "converted": true, "num_tokens": 2228, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331319177487, "lm_q2_score": 0.7520125793176222, "lm_q1q2_score": 0.44852521792395383}} {"text": "- To navigate in this slideshow, the spacebar moves to the next slide and shift+spacebar moves to the previous slide.\n- If the slides do not fit in your screen, reduce the zoom in your browser until you can see them correctly.\n\n

\n The \u201cBasic Reproduction Number\u201d and Modeling the Spread of an Epidemic\n

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~~~or~~~

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\n How to stop the spread of Covid19 using linear ordinary differential equations\n

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(at least, in a mathematical model\u2026)

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\nBasic Reproduction Number \u2014 Modeling the Spread of an Epidemic. Peter Thomas. CWRU. March 5, 2020.

\n\n

\u201cBasic Reproduction Number\u201d and Modeling the Spread of an Epidemic

\n\nDefinition: The basic reproduction number (R0) of a disease is the average number of secondary infections produced through directly direct contact with a single infected individual, in a fully susceptible population.\n\nR0 depends on the transmission mechanism, the duration of the contagious period of an infected person, and properties of the susceptible population (hygiene, detection capabilities, quarantine policies, etc.)\n\n- If R0>1, the epidemic spreads. \n- If R0<1, the epidemic dies out.\n\n\n

\nSource: https://en.wikipedia.org/wiki/Basic_reproduction_number

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\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Values of R0 of well-known infectious diseases\n
DiseaseTransmissionR0\n
MeaslesAirborne12\u201318\n
DiphtheriaSaliva6\u20137\n
SmallpoxAirborne droplet5\u20137\n
PolioFecal\u2013oral route5\u20137\n
RubellaAirborne droplet5\u20137\n
MumpsAirborne droplet4\u20137\n
PertussisAirborne droplet5.5\n
HIVSexual contact2\u20135\n
\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Values of R0 of well-known infectious diseases\n
DiseaseTransmissionR0\n
SARSAirborne droplet2\u20135\n
COVID-19Airborne droplet1.4\u20133.8\n
Influenza
(1918 pandemic strain)		Airborne droplet2\u20133\n
Ebola
(2014 Ebola outbreak)		Body fluids1.5\u20132.5\n
MERSAirborne droplet0.3-0.8\n
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\nSee also: Lipsitch, Marc, et al. \"Transmission dynamics and control of severe acute respiratory syndrome.\" Science 300.5627 (2003): 1966-1970.

\n
\n\n### BRN for a basic Susceptible-Infected-Recovered model\n\n$S$ = Susceptible, $I$ = Infected, $R=1-(S+I)$ = Recovered\n\n\\begin{align*}\n\\frac{dS}{dt} &= -\\beta SI \\\\\n\\frac{dI}{dt} &= \\beta SI -\\gamma I\n\\end{align*}\n\n$\\beta$: Rate of new infection per infected idividual, per day.\n\n$\\gamma$: Rate of recovery of infected individual, per day.\n\n__Question:__ Given the BRN for a basic Susceptible-Infected-Recovered model, \n\n\\begin{align*}\n\\frac{dS}{dt} &= -\\beta SI \\\\\n\\frac{dI}{dt} &= \\beta SI -\\gamma I\n\\end{align*}\n\nWhat is the basic reproduction number $R_0$ for this model?\n\n__Answer:__ $R_0 = \\beta/\\gamma$.\n\nRecall $S+I+R\\equiv 1$. For a total population of size $N\\gg 1$, starting with $S(0)=(N-1)/N$ and $I(0)=1/N$, the initial growth rate is $\\beta-\\gamma$, and $I(t)\\approx \\frac{1}{N}e^{(\\beta-\\gamma)t}$ for small $t$. If $\\beta>\\gamma$, then $R_0>1$ and $I(t)$ increases.\n\n

Other Epidemic Models

\n\n

\nI0 = infected but undiagnosed / asymptomatic (latent)
\nI1 = infected and diagnosed / with symptoms
\nD = deceased\n

\n\nTo find the initial growth rate $\\lambda$, consider the system of two linear equations\n\\begin{align}\n\\frac{dI_0}{dt}&=\\beta_0I_0+\\beta_1I_1-\\gamma_0I_0-\\delta I_0\\\\\n\\frac{dI_1}{dt}&=\\delta I_0 -\\gamma_1I_1-\\mu I_1.\n\\end{align}\nEquivalently: $\\frac{d\\mathbf{Y}}{dt}=A\\mathbf{Y}$, with\n$A=\\begin{pmatrix}\\beta_0-\\gamma_0-\\delta & \\beta_1 \\\\ \\delta & -\\gamma_1-\\mu \\end{pmatrix}$ and $\\mathbf{Y}=\\begin{pmatrix}I_0 \\\\ I_1\\end{pmatrix}$.\n\n### Your assignment\n$\\frac{d\\mathbf{Y}}{dt}=A\\mathbf{Y}$, with\n$A=\\begin{pmatrix}\\beta_0-\\gamma_0-\\delta & \\beta_1 \\\\ \\delta & -\\gamma_1-\\mu \\end{pmatrix}$ and $\\mathbf{Y}=\\begin{pmatrix}I_0 \\\\ I_1\\end{pmatrix}$.\nSuppose the matrix $A$ has two real eigenvalues. The larger is $\\lambda$. If $\\lambda>0$, the epidemic spreads. If $\\lambda<0$, it dies out. Your assignment:\n\n1. Find $\\lambda$ for the parameters below. For these parameters, does the epidemic spread or not? (See next slide.) \n2. The parameter $\\beta_1$ could be reduced through public policy such as hygiene and quarantine practices. If $\\lambda>0$ with the original parameters, how much must you reduce $\\beta_1$\nto control the epidemic?\n\nEquations: $\\frac{d\\mathbf{Y}}{dt}=A\\mathbf{Y}$, with\n$A=\\left(\\begin{matrix}\\beta_0-\\gamma_0-\\delta & \\beta_1 \\\\ \\delta & -\\gamma_1-\\mu \\end{matrix}\\right)$ and $\\mathbf{Y}=\\left(\\begin{matrix}I_0 \\\\ I_1\\end{matrix}\\right)$.\n\n\n| Parameter | Value | Interpretation |\n| :---------:| :----:| :-----------------------------------------------------|\n| $\\beta_0$ | 0.2 | Coefficient of infectivity from state $I_0$ (per day) |\n| $\\beta_1$ | 0.4 | Coefficient of infectivity from state $I_1$ (per day) |\n| $\\gamma_0$ | 0.2 | Recovery rate from state $I_0$ (per day) |\n| $\\gamma_0$ | 0.1 | Recovery rate from state $I_1$ (per day) |\n| $\\delta$ | 0.2 | Transition rate from $I_0$ to $I_1$ (per day) |\n| $\\mu$ | 0.03 | Mortality rate (per day) |\n\n*Nota bene:* Prof.~Thomas made up these parameter values for purposes of illustration only, so take them with a grain of salt.\n\n# Your mission, should you accept it...\n\n

Question 1

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Find the eigenvalues of the matrix $A=\\begin{pmatrix} -0.2 & 0.4 \\\\ 0.2 & -0.13 \\end{pmatrix}$
\nClick on the cell below and run it using SHIFT+Enter/Return
\nThe cell will create the matrix $A$ and it will store its eigenvectors and eigenvalues.\n

\n\n\n```octave\n# Definition of the parameters:\n\nb0=.2; % coefficient of infectivity from state I0 (per day)\nb1=.4; % coefficient of infectivity from state I1 (per day)\ng0=.2; % recovery rate from state I0 (per day)\ng1=.1; % recovery rate from state I1 (per day)\ndelta=.2; % transition rate from I0 to I1 (per day)\nmu=.03; % mortality rate (per day)\n\n# Construction of the matrix\nA=[b0-g0-delta,b1;\n delta,-g1-mu];\n\n# Eigenvectors and Eigenvalues\n[V,dd]=eig(A);\n```\n\n

Question 1 (continued)

\n

Run the cell below to see the eigenvalues.\n

\n\n\n```octave\ndiag(dd)\n```\n\n

Run the cell below to see the eigenvectors (in matrix form).\n

\n\n\n```octave\nV\n```\n\n

Question 1 (continued)

\n\nFollow up questions:\n1. What does the eigenvector that goes with the largest eigenvalue tell you?\n\n2. As of March 5, 2020, there have been 95,000 cases of Covid19 since the first cases appeared in early December 2019, about 90 days ago. If the infecton where still in the linear growth phase, how large would you expect $\\lambda$ to be (in units of days$^{-1})$?\n\n

Question 2

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(Numerically) find the eigenvalues of $A=\\begin{pmatrix} -0.2 & \\beta_1 \\\\ 0.2 & -0.13 \\end{pmatrix}$
\nHow small must $\\beta_1$ be to stop the epidemic?\n

\nIn the next cell you will find a snippet of code that will plot the real part of the eigenvalues depending on the parameter $\\beta_1$.\n \nIf you cannot see the plot after running the cell, zoom out in your browser.\n

\n\n\n```octave\nb1min=0.1; % adjust to taste. Try +0.1 first. \nb1max=.4; % adjust to taste. Try +0.4 first.\nfor b1=linspace(b1min,b1max,201)\nA=[b0-g0-delta,b1;\n delta,-g1-mu];\ndd=eigs(A);\nplot([b1,b1],real(dd),'o','MarkerSize',10)\nhold on\nend\nset(line([b1min b1max],[0 0]),'Color','k','LineWidth',10)\nset(gca,'FontSize',20)\nxlabel('Parameter \\beta_1')\ntitle('Growth Rate of Epidemic')\nylabel('Real(\\lambda_\\pm)')\ngrid on\nxlim([b1min,b1max])\n```\n\n\nIn this SLIR model, reducing $\\beta_1$ by a factor of four, stops the epidemic.\n\n\nThe characteristic polynomial $P(\\lambda) = \\text{det}(A-\\lambda I)$ always has two roots, even for non-physical parameter values ($\\beta_1<0$). However, sometimes the roots form a complex conjugate pair, so here we plot the real part of $\\lambda_{\\pm}$.\n\n## Question 3\nFor the SIR model, $R_0>1$ if and only if the growth rate $\\beta-\\gamma>0$.\nCalculate $R_0$ for the SLIR model, and investigate whether $R_0>1$ iff $\\lambda_+>0$ for this model as well.\n\n# THE END\n\n- Original Presentation: Professor Peter J. Thomas (CWRU)\n- Adaptation to Octave and Binder Creation: Dr. Daniel Balague Guardia (CWRU)\n\n## What now?\n\n- You can go back to the previous slides with Octave code and modify them with different paramters.\n- Play with the models and the code\n- Please share/copy/modify!\n", "meta": {"hexsha": "3d23b114c9929590fe45c8d3ecdf966dc62eefaa", "size": 14655, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "index.ipynb", "max_stars_repo_name": "dbalague/materials_epidemics", "max_stars_repo_head_hexsha": "2bf0ec93317fba7f537fa13c63f91916a51f1dda", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "index.ipynb", "max_issues_repo_name": "dbalague/materials_epidemics", "max_issues_repo_head_hexsha": "2bf0ec93317fba7f537fa13c63f91916a51f1dda", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "index.ipynb", "max_forks_repo_name": "dbalague/materials_epidemics", "max_forks_repo_head_hexsha": "2bf0ec93317fba7f537fa13c63f91916a51f1dda", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 14655.0, "max_line_length": 14655, "alphanum_fraction": 0.6730126237, "converted": true, "num_tokens": 3481, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583270090337583, "lm_q2_score": 0.8031737963569016, "lm_q1q2_score": 0.44843362345423776}} {"text": "# CNTK 203: Reinforcement Learning Basics\n\n\nReinforcement learning (RL) is an area of machine learning inspired by behaviorist psychology, concerned with how [software agents](https://en.wikipedia.org/wiki/Software_agent) ought to take [actions](https://en.wikipedia.org/wiki/Action_selection) in an environment so as to maximize some notion of cumulative reward. In machine learning, the environment is typically formulated as a [Markov decision process](https://en.wikipedia.org/wiki/Markov_decision_process) (MDP) as many reinforcement learning algorithms for this context utilize [dynamic programming](https://en.wikipedia.org/wiki/Dynamic_programming) techniques. \n\nIn some machine learning settings, we do not have immediate access to labels, so we cannot rely on supervised learning techniques. If, however, there is something we can interact with and thereby get some feedback that tells us occasionally, whether our previous behavior was good or not, we can use RL to learn how to improve our behavior.\n\nUnlike in supervised learning, in RL, labeled correct input/output pairs are never presented and sub-optimal actions are never explicitly corrected. This mimics many of the online learning paradigms which involves finding a balance between exploration (of conditions or actions never learnt before) and exploitation (of already learnt conditions or actions from previous encounters). Multi-arm bandit problems is one of the category of RL algorithms where exploration vs. exploitation trade-off have been thoroughly studied. See figure below for [reference](http://www.simongrant.org/pubs/thesis/3/2.html).\n\n\n\n**Problem**\n\nWe will use the [CartPole](https://gym.openai.com/envs/CartPole-v0) environment from OpenAI's [gym](https://github.com/openai/gym) simulator to teach a cart to balance a pole. As described in the link above, in the CartPole example, a pole is attached by an un-actuated joint to a cart, which moves along a frictionless track. The system is controlled by applying a force of +1 or -1 to the cart. A reward of +1 is provided for every timestep that the pole remains upright. The episode ends when the pole is more than 15 degrees from vertical, or the cart moves more than 2.4 units from the center. See figure below for reference.\n\n**Goal**\nOur goal is to prevent the pole from falling over as the cart moves with the pole in upright position (perpendicular to the cart) as the starting state. More specifically if the pole is less than 15 degrees from vertical while the cart is within 2.4 units of the center we will collect reward. In this tutorial, we will train till we learn a set of actions (policies) that lead to an average reward of 200 or more over last 50 batches. \n\nIn, RL terminology, the goal is to find _policies_ $a$, that maximize the _reward_ $r$ (feedback) through interaction with some environment (in this case the pole being balanced on the cart). So given a series of experiences $$s \\xrightarrow{a} r, s'$$ we then can learn how to choose action $a$ in a given state $s$ to maximize the accumulated reward $r$ over time: \n\\begin{align}\nQ(s,a) &= r_0 + \\gamma r_1 + \\gamma^2 r_2 + \\ldots \\newline\n&= r_0 + \\gamma \\max_a Q^*(s',a)\n\\end{align}\nwhere $\\gamma \\in [0,1)$ is the discount factor that controls how much we should value reward that is further away. This is called the [*Bellmann*-equation](https://en.wikipedia.org/wiki/Bellman_equation). \n\nIn this tutorial we will show how to model the state space, how to use the received reward to figure out which action yields the highest future reward. \n\nWe present two different popular approaches here:\n\n**Deep Q-Networks**: DQNs have become famous in 2015 when they were successfully used to train how to play Atari just form raw pixels. We train neural network to learn the $Q(s,a)$ values (thus _Q-Network _). From these $Q$ functions values we choose the best action.\n\n**Policy gradient**: This method directly estimates the policy (set of actions) in the network. The outcome is a learning of an ordered set of actions which leads to maximize reward by probabilistically choosing a subset of actions. In this tutorial, we learn the actions using a gradient descent approach to learn the policies.\n\nIn this tutorial, we focus how to implement RL in CNTK. We choose a straight forward shallow network. One can extend the approaches by replacing our shallow model with deeper networks that are introduced in other CNTK tutorials. \n\nAdditionally, this tutorial is in its early stages and will be evolving in future updates.\n\n## Before we start...\nPlease run the following cell from the menu above or select the cell below and hit `Shift + Enter` to ensure the environment is ready. Verify that the following imports work in your notebook.\n\n\n```python\nfrom __future__ import print_function\nimport matplotlib.pyplot as plt\nfrom matplotlib import style\nimport numpy as np\nimport pandas as pd\nimport seaborn as sns\n\nstyle.use('ggplot')\n%matplotlib inline\n```\n\nWe use the following construct to install the OpenAI gym package if it is not installed. For users new to Jupyter environment, this construct can be used to install any python package. \n\n\n```python\ntry:\n import gym\nexcept:\n !pip install gym\n import gym\n```\n\n# CartPole: Data and Environment\n\nWe will use the [CartPole](https://gym.openai.com/envs/CartPole-v0) environment from OpenAI's [gym](https://github.com/openai/gym) simulator to teach a cart to balance a pole. Please follow the links to get more details.\n\nIn every time step, the agent\n * gets an observation $(x, \\dot{x}, \\theta, \\dot{\\theta})$, corresponding to *cart position*, *cart velocity*, *pole angle with the vertical*, *pole angular velocity*,\n * performs an action `LEFT` or `RIGHT`, and\n * receives \n * a reward of +1 for having survived another time step, and\n * a new state $(x', \\dot{x}', \\theta', \\dot{\\theta}')$\n \nThe episode ends, if \n * the pole is more than 15 degrees from vertical and/or\n * the cart is moving more than 2.4 units from center.\n \nThe task is considered done, if\n * the agent achieved and averaged reward of 200 over the last 50 episodes (if you manage to get a reward of 200 averaged over the last 100 episode you can consider submitting it to OpenAI)\n\n# Part 1: DQN\n\nAfter a transition $(s,a,r,s')$, we are trying to move our value function $Q(s,a)$ closer to our target $r+\\gamma \\max_{a'}Q(s',a')$, where $\\gamma$ is a discount factor for future rewards and ranges in value between 0 and 1.\n\nDQNs\n * learn the _Q-function_ that maps observation (state, action) to a `score`\n * use memory replay (previously recorded $Q$ values corresponding to different $(s,a)$ to decorrelate experiences (sequence state transitions)\n * use a second network to stabilize learning (*not* part of this tutorial)\n\n### Setting up the model\n\\begin{equation}\nl_1 = relu( x W_1 + b_1) \\\\\nQ(s,a) = l_1 W_2 + b_2 \\\\\n\\end{equation}\n\nWe will start with a slightly modified version for Keras, https://github.com/jaara/AI-blog/blob/master/CartPole-basic.py, published by Jarom\u00edr Janisch in his [AI blog](https://jaromiru.com/2016/09/27/lets-make-a-dqn-theory/), and will then incrementally convert it to use CNTK. \n\nWe use a simple two-layer densely connected network, for simpler illustrations. More advance networks can be substituted.\n\n**CNTK** concepts: The commented out code is meant to be an illustration of the similarity of concepts between CNTK API/abstractions against Keras. \n\n\n```python\nimport random, numpy, math\n\n#from keras.models import Sequential\n#from keras.layers import *\n#from keras.optimizers import *\nfrom cntk import *\nfrom cntk.models import Sequential\nfrom cntk.layers import *\n```\n\nSTATE_COUNT = 4 (corresponding to $(x, \\dot{x}, \\theta, \\dot{\\theta})$),\n\nACTION_COUNT = 2 (corresponding to `LEFT` or `RIGHT`)\n\n\n```python\nenv = gym.make('CartPole-v0')\n\nSTATE_COUNT = env.observation_space.shape[0]\nACTION_COUNT = env.action_space.n\n\nSTATE_COUNT, ACTION_COUNT\n```\n\nNote: in the cell below we highlight how one would do it in Keras. And a marked similarity with CNTK. While CNTK allows for more compact representation, we present a slightly verbose illustration for ease of learning.\n\nAdditionally, you will note that, CNTK model doesn't need to be compiled explicitly and is implicitly done when data is processed during training.\n\nCNTK effectively uses available memory on the system between minibatch execution. Thus the learning rates are stated as **rates per sample** instead of **rates per minibatch** (as with other toolkits).\n\n\n```python\nBATCH_SIZE_BASELINE = 50 # calculate average reward over these many episodes\nH = 64 # hidden layer size\n\nclass Brain:\n def __init__(self):\n self.params = {}\n self.model, self.trainer, self.loss = self._create()\n # self.model.load_weights(\"cartpole-basic.h5\")\n \n def _create(self):\n observation = input_variable(STATE_COUNT, np.float32, name=\"s\")\n q_target = input_variable(ACTION_COUNT, np.float32, name=\"q\")\n\n # model = Sequential()\n # model.add(Dense(output_dim=64, activation='relu', input_dim=STATE_COUNT))\n # model.add(Dense(output_dim=ACTION_COUNT, activation='linear'))\n\n # Following a style similar to Keras \n l1 = Dense(H, activation=relu)\n l2 = Dense(ACTION_COUNT)\n unbound_model = Sequential([l1, l2])\n model = unbound_model(observation)\n\n self.params = dict(W1=l1.W, b1=l1.b, W2=l2.W, b2=l2.b) \n \n lr = 0.00025\n # opt = RMSprop(lr=0.00025)\n # model.compile(loss='mse', optimizer=opt)\n\n # loss='mse'\n loss = reduce_mean(square(model - q_target), axis=0)\n meas = reduce_mean(square(model - q_target), axis=0)\n\n # optimizer=opt\n lr_schedule = learning_rate_schedule(lr, UnitType.minibatch)\n learner = sgd(model.parameters, lr_schedule, gradient_clipping_threshold_per_sample=10)\n trainer = Trainer(model, loss, meas, learner)\n\n # CNTK: return trainer and loss as well\n return model, trainer, loss\n\n def train(self, x, y, epoch=1, verbose=0):\n #self.model.fit(x, y, batch_size=64, nb_epoch=epoch, verbose=verbose)\n arguments = dict(zip(self.loss.arguments, [y,x]))\n updated, results =self.trainer.train_minibatch(arguments, outputs=[self.loss.output])\n\n def predict(self, s):\n return self.model.eval([s])\n```\n\nThe `Memory` class stores the different states, actions and rewards.\n\n\n```python\nclass Memory: # stored as ( s, a, r, s_ )\n samples = []\n\n def __init__(self, capacity):\n self.capacity = capacity\n\n def add(self, sample):\n self.samples.append(sample) \n\n if len(self.samples) > self.capacity:\n self.samples.pop(0)\n\n def sample(self, n):\n n = min(n, len(self.samples))\n return random.sample(self.samples, n)\n```\n\nThe `Agent` uses the `Brain` and `Memory` to replay the past actions to choose optimal set of actions that maximize the rewards.\n\n\n```python\nMEMORY_CAPACITY = 100000\nBATCH_SIZE = 64\n\nGAMMA = 0.99 # discount factor\n\nMAX_EPSILON = 1\nMIN_EPSILON = 0.01 # stay a bit curious even when getting old\nLAMBDA = 0.0001 # speed of decay\n\nclass Agent:\n steps = 0\n epsilon = MAX_EPSILON\n\n def __init__(self):\n self.brain = Brain()\n self.memory = Memory(MEMORY_CAPACITY)\n \n def act(self, s):\n if random.random() < self.epsilon:\n return random.randint(0, ACTION_COUNT-1)\n else:\n return numpy.argmax(self.brain.predict(s))\n\n def observe(self, sample): # in (s, a, r, s_) format\n self.memory.add(sample) \n\n # slowly decrease Epsilon based on our eperience\n self.steps += 1\n self.epsilon = MIN_EPSILON + (MAX_EPSILON - MIN_EPSILON) * math.exp(-LAMBDA * self.steps)\n\n def replay(self): \n batch = self.memory.sample(BATCH_SIZE)\n batchLen = len(batch)\n\n no_state = numpy.zeros(STATE_COUNT)\n\n \n # CNTK: explicitly setting to float32\n states = numpy.array([ o[0] for o in batch ], dtype=np.float32)\n states_ = numpy.array([(no_state if o[3] is None else o[3]) for o in batch ], dtype=np.float32)\n\n p = agent.brain.predict(states)\n p_ = agent.brain.predict(states_)\n\n # CNTK: explicitly setting to float32\n x = numpy.zeros((batchLen, STATE_COUNT)).astype(np.float32)\n y = numpy.zeros((batchLen, ACTION_COUNT)).astype(np.float32)\n \n for i in range(batchLen):\n s, a, r, s_ = batch[i]\n \n # CNTK: [0] because of sequence dimension\n t = p[0][i]\n if s_ is None:\n t[a] = r\n else:\n t[a] = r + GAMMA * numpy.amax(p_[0][i])\n\n x[i] = s\n y[i] = t\n\n self.brain.train(x, y)\n```\n\n### Brain surgery\n\nAs any learning experiences, we expect to see the initial state of actions to be wild exploratory and over the iterations the system learns the range of actions that yield longer runs and collect more rewards. The tutorial below implements the [$\\epsilon$-greedy](https://en.wikipedia.org/wiki/Reinforcement_learning) approach. \n\n\n```python\ndef plot_weights(weights, figsize=(7,5)):\n '''Heat map of weights to see which neurons play which role'''\n sns.set(style=\"white\")\n f, ax = plt.subplots(len(weights), figsize=figsize)\n cmap = sns.diverging_palette(220, 10, as_cmap=True)\n\n for i, data in enumerate(weights):\n axi = ax if len(weights)==1 else ax[i]\n if isinstance(data, tuple): \n w, title = data\n axi.set_title(title)\n else:\n w = np.asarray(data)\n sns.heatmap(w, cmap=cmap, square=True, center=True, #annot=True,\n linewidths=.5, cbar_kws={\"shrink\": .25}, ax=axi)\n```\n\n### Exploration - exploitation trade-off\n\nNote that the initial $\\epsilon$ is set to 1 which implies we are entirely exploring but as steps increase we reduce exploration and start leveraging the learnt space to collect rewards (a.k.a. exploitation) as well.\n\n\n```python\ndef epsilon(steps):\n return MIN_EPSILON + (MAX_EPSILON - MIN_EPSILON) * np.exp(-LAMBDA * steps)\nplt.plot(range(10000), [epsilon(x) for x in range(10000)], 'r')\nplt.xlabel('step');plt.ylabel('$\\epsilon$')\n```\n\nWe are now ready to train our agent using **DQN**. Note this would take anywhere between 2-10 min and we stop whenever the learner hits the average reward of 200 over past 50 batches. One would get better results if they could train the learner until say one hits a reward of 200 or higher for say larger number of runs. This is left as an exercise.\n\n\n```python\ndef run(agent):\n s = env.reset()\n R = 0 \n\n while True: \n #env.render()\n\n # CNTK: explicitly setting to float32\n a = agent.act(s.astype(np.float32))\n\n s_, r, done, info = env.step(a)\n\n if done: # terminal state\n s_ = None\n\n agent.observe((s, a, r, s_))\n agent.replay() \n\n s = s_\n R += r\n\n if done:\n return R\n\nagent = Agent()\n\nepisode_number = 0\nreward_sum = 0\nwhile episode_number<3000:\n reward_sum += run(agent)\n episode_number += 1\n if episode_number % BATCH_SIZE_BASELINE == 0:\n print('Episode: %d, Average reward for episode %f.' % (episode_number, \n reward_sum / BATCH_SIZE_BASELINE))\n if episode_number%200==0:\n plot_weights([(agent.brain.params['W1'], 'Episode %i $W_1$'%episode_number)], figsize=(14,5))\n if reward_sum / BATCH_SIZE_BASELINE > 200:\n print('Task solved in %d episodes' % episode_number)\n plot_weights([(agent.brain.params['W1'], 'Episode %i $W_1$'%episode_number)], figsize=(14,5)) \n break\n reward_sum = 0\nagent.brain.model.save_model('dqn.mod')\n```\n\nIf you run it, you should see something like\n\n```[2016-10-26 22:06:25,436] Making new env: CartPole-v0\nEpisode: 50, Average reward for episode 23.700000.\nEpisode: 100, Average reward for episode 18.720000.\nEpisode: 150, Average reward for episode 17.960000.\n...\nEpisode: 1750, Average reward for episode 100.180000.\nEpisode: 1800, Average reward for episode 111.380000.\nEpisode: 1850, Average reward for episode 207.240000.\nTask solved in 1850 episodes```\n\n#### Task 1.1\nRewrite the model without using the layer lib.\n#### Task 1.2\nPlay with different [learners](https://cntk.ai/pythondocs/cntk.learner.html?highlight=learner#module-cntk.learner). Which one works better? Worse? Think about which parameters you would need to adapt when switching from one learner to the other.\n\n### Running the DQN model\n\n\n```python\nimport cntk as C\nenv = gym.make('CartPole-v0')\n\nnum_episodes = 10 # number of episodes to run\n\nmodelPath = 'dqn.mod'\nroot = C.load_model(modelPath)\nobservation = env.reset() # reset environment for new episode\ndone = False\nfor i_episode in range(num_episodes):\n print(i_episode)\n while not done:\n env.render()\n action = np.argmax(root.eval([observation.astype(np.float32)]))\n observation, reward, done, info = env.step(action)\n if done:\n observation = env.reset() # reset environment for new episode \n```\n\n# Part 2: Policy gradient\n**Goal:** \n\\begin{equation}\\text{maximize } E [R | \\pi_\\theta]\n\\end{equation}\n\n**Approach:**\n1. Collect experience (sample a bunch of trajectories through $(s,a)$ space)\n2. Update the policy so that _good_ experiences become more probable\n\n**Difference to DQN: **\n * we don't consider single $(s,a,r,s')$ transitions, but rather use whole episodes for the gradient updates\n * our parameters directly model the policy (output is an action probability), whereas in DQN they model the value function (output is raw score)\n\n#### Rewards:\nRemember, we get +1 reward for every time step, in which we still were in the game. \n\nThe problem: we normally do not know, which action led to a continuation of the game, and which was actually a bad one. Our simple heuristic: actions in the beginning of the episode are good, and those towards the end are likely bad (they led to losing the game after all).\n\n\n```python\ndef discount_rewards(r, gamma=0.999):\n \"\"\"Take 1D float array of rewards and compute discounted reward \"\"\"\n discounted_r = np.zeros_like(r)\n running_add = 0\n for t in reversed(range(0, r.size)):\n running_add = running_add * gamma + r[t]\n discounted_r[t] = running_add\n return discounted_r\n```\n\n\n```python\ndiscounted_epr = discount_rewards(np.ones(10))\nf, ax = plt.subplots(1, figsize=(5,2))\nsns.barplot(list(range(10)), discounted_epr, color=\"steelblue\")\n```\n\nWe normalize the rewards so that they tank below zero towards the end. gamma controls how late the rewards tank.\n\n\n```python\ndiscounted_epr_cent = discounted_epr - np.mean(discounted_epr)\ndiscounted_epr_norm = discounted_epr_cent/np.std(discounted_epr_cent)\nf, ax = plt.subplots(1, figsize=(5,2))\nsns.barplot(list(range(10)), discounted_epr_norm, color=\"steelblue\")\n```\n\n\n```python\ndiscounted_epr = discount_rewards(np.ones(10), gamma=0.5)\ndiscounted_epr_cent = discounted_epr - np.mean(discounted_epr)\ndiscounted_epr_norm = discounted_epr_cent/np.std(discounted_epr_cent)\nf, ax = plt.subplots(2, figsize=(5,3))\nsns.barplot(list(range(10)), discounted_epr, color=\"steelblue\", ax=ax[0])\nsns.barplot(list(range(10)), discounted_epr_norm, color=\"steelblue\", ax=ax[1])\n```\n\n### Setting up the model\n\\begin{equation}\nl_1 = relu( x W_1 + b_1) \\\\\nl_2 = l_1 W_2 + b_2 \\\\\n\\pi(a|s) = sigmoid(l_2)\n\\end{equation}\n\nNote: in policy gradient approach, the output of the dense layer is mapped into to a 0-1 range via the sigmoid function.\n\n\n```python\nimport cntk as C \n\nTOTAL_EPISODES = 10000\nD = 4 # input dimensionality\nH = 10 # number of hidden layer neurons\n\nobservations = C.input_variable(STATE_COUNT, np.float32, name=\"obs\")\n\nW1 = C.parameter(shape=(STATE_COUNT, H), init=C.glorot_uniform(), name=\"W1\")\nb1 = C.parameter(shape=H, name=\"b1\")\nlayer1 = C.relu(C.times(observations, W1) + b1)\n\nW2 = C.parameter(shape=(H, ACTION_COUNT), init=C.glorot_uniform(), name=\"W2\")\nb2 = C.parameter(shape=ACTION_COUNT, name=\"b2\")\nscore = C.times(layer1, W2) + b2\n# Until here it was similar to DQN\n\nprobability = C.sigmoid(score, name=\"prob\")\n```\n\n**Policy Search**: The optimal policy search can be carried out with either gradient free approaches or by computing gradients over the policy space ($\\pi_\\theta$) which is parameterized by $\\theta$. In this tutorial, we use the classic forward (`loss.forward`) and back (`loss.backward`) propagation of errors over the parameterized space $\\theta$. In this case, $\\theta = \\{W_1, b_1, W_2, b_2\\}$, our model parameters. \n\n\n```python\ninput_y = C.input_variable(1, np.float32, name=\"input_y\")\nadvantages = C.input_variable(1, np.float32, name=\"advt\")\n\nloss = -C.reduce_mean(C.log(C.square(input_y - probability) + 1e-4) * advantages, axis=0, name='loss')\n\nlr = 0.001 \nlr_schedule = learning_rate_schedule(lr, UnitType.sample) \nsgd = C.sgd([W1, W2], lr_schedule)\n\ngradBuffer = dict((var.name, np.zeros(shape=var.shape)) for var in loss.parameters if var.name in ['W1', 'W2', 'b1', 'b2'])\n\nxs, hs, label, drs = [], [], [], []\nrunning_reward = None\nreward_sum = 0\nepisode_number = 1\n\nobservation = env.reset()\n\nwhile episode_number <= TOTAL_EPISODES:\n x = np.reshape(observation, [1, STATE_COUNT]).astype(np.float32)\n\n # Run the policy network and get an action to take.\n prob = probability.eval(arguments={observations: x})[0][0][0]\n action = 1 if np.random.uniform() < prob else 0\n\n xs.append(x) # observation\n # grad that encourages the action that was taken to be taken\n\n y = 1 if action == 0 else 0 # a \"fake label\"\n label.append(y)\n\n # step the environment and get new measurements\n observation, reward, done, info = env.step(action)\n reward_sum += float(reward)\n\n # Record reward (has to be done after we call step() to get reward for previous action)\n drs.append(float(reward))\n\n if done:\n # Stack together all inputs, hidden states, action gradients, and rewards for this episode\n epx = np.vstack(xs)\n epl = np.vstack(label).astype(np.float32)\n epr = np.vstack(drs).astype(np.float32)\n xs, label, drs = [], [], [] # reset array memory\n\n # Compute the discounted reward backwards through time.\n discounted_epr = discount_rewards(epr)\n # Size the rewards to be unit normal (helps control the gradient estimator variance)\n discounted_epr -= np.mean(discounted_epr)\n discounted_epr /= np.std(discounted_epr)\n\n # Forward pass\n arguments = {observations: epx, input_y: epl, advantages: discounted_epr}\n state, outputs_map = loss.forward(arguments, outputs=loss.outputs, \n keep_for_backward=loss.outputs)\n \n # Backward psas\n root_gradients = {v: np.ones_like(o) for v, o in outputs_map.items()}\n vargrads_map = loss.backward(state, root_gradients, variables=set([W1, W2]))\n\n for var, grad in vargrads_map.items():\n gradBuffer[var.name] += grad\n\n # Wait for some batches to finish to reduce noise\n if episode_number % BATCH_SIZE_BASELINE == 0:\n grads = {W1: gradBuffer['W1'].astype(np.float32), \n W2: gradBuffer['W2'].astype(np.float32)}\n updated = sgd.update(grads, BATCH_SIZE_BASELINE)\n\n # reset the gradBuffer\n gradBuffer = dict((var.name, np.zeros(shape=var.shape)) \n for var in loss.parameters if var.name in ['W1', 'W2', 'b1', 'b2'])\n\n print('Episode: %d. Average reward for episode %f.' % (episode_number, reward_sum / BATCH_SIZE_BASELINE))\n\n if reward_sum / BATCH_SIZE_BASELINE > 200:\n print('Task solved in: %d ' % episode_number)\n break\n\n reward_sum = 0\n\n observation = env.reset() # reset env\n episode_number += 1\nprobability.save_model('pg.mod')\n```\n\n# Solutions\n#### Solution 1.1\n\n\n```python\nobservation = input_variable(STATE_COUNT, np.float32, name=\"s\")\n\nW1 = parameter(shape=(STATE_COUNT, H), init=glorot_uniform(), name=\"W1\")\nb1 = parameter(shape=H, name=\"b1\")\nlayer1 = relu(times(observation, W1) + b1)\nW2 = parameter(shape=(H, ACTION_COUNT), init=glorot_uniform(), name=\"W2\")\nb2 = parameter(shape=ACTION_COUNT, name=\"b2\")\nmodel = times(layer1, W2) + b2\nW1.shape, b1.shape, W2.shape, b2.shape, model.shape\n```\n", "meta": {"hexsha": "09b37650223a91439dc7de3b9dc625efdced942e", "size": 33739, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Tutorials/CNTK_203_Reinforcement_Learning_Basics.ipynb", "max_stars_repo_name": "trevhartzell/CNTK", "max_stars_repo_head_hexsha": "dcdbd0d787580206d99d00d49c7db2fe4236431a", "max_stars_repo_licenses": ["RSA-MD"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-07-16T05:38:50.000Z", "max_stars_repo_stars_event_max_datetime": "2021-07-16T05:38:50.000Z", "max_issues_repo_path": "Tutorials/CNTK_203_Reinforcement_Learning_Basics.ipynb", "max_issues_repo_name": "trevhartzell/CNTK", "max_issues_repo_head_hexsha": "dcdbd0d787580206d99d00d49c7db2fe4236431a", "max_issues_repo_licenses": ["RSA-MD"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Tutorials/CNTK_203_Reinforcement_Learning_Basics.ipynb", "max_forks_repo_name": "trevhartzell/CNTK", "max_forks_repo_head_hexsha": "dcdbd0d787580206d99d00d49c7db2fe4236431a", "max_forks_repo_licenses": ["RSA-MD"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 39.6462984724, "max_line_length": 639, "alphanum_fraction": 0.5829455526, "converted": true, "num_tokens": 6188, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936430859597, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.4483592950281939}} {"text": "```python\nimport torch\nimport torch.nn.functional as F\n\nimport torchsde\nimport math\nimport matplotlib.pyplot as plt\n\nimport numpy as np\n\nfrom tqdm.notebook import tqdm\n\nfrom torch import _vmap_internals\n```\n\n\n```python\ncd ..\n```\n\n C:\\Users\\vargf\\OneDrive\\Documents\\Projects\\ControlledFollMerDrift\n\n\n\n```python\nfrom cfollmer.objectives import log_g, relative_entropy_control_cost, stl_relative_entropy_control_cost\nfrom cfollmer.sampler_utils import FollmerSDE\nfrom cfollmer.trainers import basic_batched_trainer\n```\n\n# The Model\n\n\\begin{align}\n\\theta &\\sim \\mathcal{N}(\\theta | 0, \\sigma_w^2 \\mathbb{I}) \\\\\ny_i | x_i, \\theta &\\sim \\mathrm{Bernouli}\\left[\\mathrm{NN}_{\\theta}\\left(x_i \\right)\\right]\n\\end{align}\n\nWe want samples from $p(\\theta | \\{(y_i, x_i)\\})$. Note $f(x; \\theta)$ is a neural net with params $\\theta$\n\n## Loading the iris dataset\n\n\n```python\nfrom sklearn.datasets import load_iris\nfrom sklearn.model_selection import train_test_split\nfrom sklearn.preprocessing import StandardScaler\n\ndevice = \"cuda\" if torch.cuda.is_available() else \"cpu\"\n\niris = load_iris()\nX = iris['data']\ny = iris['target']\n\n# Binary classification\nX = X[~(y==2)][:,[0,1]]\ny = y[~(y==2)]\n\n# dummy dims \nX = np.concatenate((torch.ones(X.shape[0],1), torch.tensor(X) ), axis=1)\n\nnames = iris['target_names']\nfeature_names = iris['feature_names']\n\n# Scale data to have mean 0 and variance 1 \n# which is importance for convergence of the neural network\nscaler = StandardScaler()\nX_scaled = scaler.fit_transform(X)\n\n# Split the data set into training and testing\nX_train, X_test, y_train, y_test = train_test_split(\n X_scaled, y, test_size=0.2, random_state=2)\n\n\nX_train, X_test, y_train, y_test = \\\n torch.tensor(X_train, dtype=torch.float32, device=device), \\\n torch.tensor(X_test, dtype=torch.float32, device=device), \\\n torch.tensor(y_train, dtype=torch.float32, device=device), \\\n torch.tensor(y_test, dtype=torch.float32, device=device) \n```\n\n\n```python\nfig, ax1 = plt.subplots(1, 1, figsize=(16, 6))\nfor target, target_name in enumerate(names[0:2]):\n X_plot = X[y == target]\n ax1.plot(X_plot[:, 1], X_plot[:, 2], \n linestyle='none', \n marker='o', \n label=target_name)\nax1.set_xlabel(feature_names[0])\nax1.set_ylabel(feature_names[1])\nax1.axis('equal')\nax1.legend();\n```\n\n$$\\DeclareMathOperator*{\\argmin}{arg\\,min}$$\n$$\\def\\E{{\\mathbb{E}}}$$\n$$\\def\\rvu{{\\mathbf{u}}}$$\n$$\\def\\rvTheta{{\\bm{\\Theta}}}$$\n$$\\def\\gU{{\\mathcal{U}}}$$\n$$\\def\\mX{{\\mathbf{X}}}$$\n\n## Controlled Schrodinger Follmer Sampler\n\nThe objevtive we are trying to implement is:\n\n\\begin{align}\n \\mathbf{u}_t^{*}= \\argmin_{\\rvu_t \\in \\mathcal{U}}\\mathbb{E}\\left[\\frac{1}{2\\gamma}\\int_0^1||\\rvu(t, \\Theta_t)||^2 dt - \\ln\\left(\\frac{ p(\\mX | \\Theta_1)p(\\Theta_1)}{\\mathcal{N}(\\Theta_1|\\mathbf{0}, \\gamma \\mathbb{I} )}\\right)\\right] \\\n\\end{align}\n\nWhere:\n\\begin{align}\nd\\Theta_t = \\rvu(t, \\Theta_t)dt + \\sqrt{\\gamma} dB_t\n\\end{align}\n\nTo do so we use the EM discretisation.\n\n\n```python\nimport torch.nn.functional as F\n\n\nclass OnedRegressionForwardNet(object):\n \n def __init__(\n self, input_dim=1, output_dim=1, depth=None,\n width=20, width_seq=None, device=\"cpu\", activation=F.relu\n ):\n \n self.device = device\n self.output_dim = output_dim\n self.input_dim = input_dim \n self.activation = activation\n \n self.depth = depth\n if not self.depth:\n self.depth = 1\n if not width_seq:\n self.width = width\n self.width_seq = [self.width] * (self.depth + 1)\n self.shapes = [(self.width_seq[i-1], self.width_seq[i]) for i in range(1,self.depth)]\n self.shapes += [(self.width_seq[-1], self.output_dim)]\n self.shapes = [(self.input_dim, self.width_seq[0])] + self.shapes\n \n self.dim = sum([wx * wy + wy for wx, wy in self.shapes])\n \n def forward(self, x, \u0398):\n index = 0\n n, d = x.shape\n \n# dim_bl = sum([wx * wy + wy for wx, wy in self.shapes[:-1]])\n# \u0398[:dim_bl] = (\u0398[:dim_bl] - \u0398[:dim_bl].mean()) / \u0398[:dim_bl].std()\n# \u03c3_\u0398, \u03bc_\u0398 = \u0398.std(), \u0398.mean()\n# \u0398 = (\u0398 - \u03bc_\u0398) / \u03c3_\u0398\n\n for wx, wy in self.shapes[:-1]:\n x = F.linear(\n x,\n \u0398[index: index + wx * wy].reshape(wy, wx),\n \u0398[index + wx * wy: index + wx * wy + wy].reshape(1,wy)\n )\n x = self.activation(x)\n index += wx * wy + wy\n wx, wy = self.shapes[-1]\n x = F.linear(\n x,\n \u0398[index: index + wx * wy].reshape(wy, wx), #* \u03c3_\u0398 + \u03bc_\u0398,\n \u0398[index + wx * wy: index + wx * wy + wy].reshape(1,wy) # * \u03c3_\u0398 + \u03bc_\u0398\n )\n return x.to(self.device)\n \n def map_forward(self, x, \u0398):\n preds_func = lambda \u03b8: self.forward(x, \u03b8)\n batched_preds = torch._vmap_internals.vmap(preds_func)\n preds = torch.hstack(list(map(preds_func, \u0398)))\n return preds\n \n```\n\n\n```python\nnet = OnedRegressionForwardNet(\n 3,1, device=device, depth=1, width=30, activation=F.tanh\n)\n\n\ndef gaussian_prior(\u0398, \u03c3_w=1.8):\n \"\"\"\n Logistic regresion bayesian prior\n \"\"\"\n return -0.5 * (\u0398**2).sum(axis=1) / \u03c3_w\n\n\n# def log_likelihood_vmap(\u0398, X, y):\n# \"\"\"\n# Hoping this implementation is less buggy / faster\n \n# still feels a bit slow.\n# \"\"\"\n# logits = X.mm(\u0398.T)\n \n# pos_weights = torch.ones(logits.shape[0], device=device)\n \n# loss = torch.nn.BCEWithLogitsLoss(pos_weight=pos_weights, reduction=\"sum\")\n \n# # TODO: Double check this is right, changed to a minus sign here\n# loss_ = lambda x: -1.0 * loss(x, y)\n \n# batched_loss = torch._vmap_internals.vmap(loss_)\n\n# return batched_loss(logits.T)\n\n\ndef log_likelihood_vmap_nn(\u0398, X, y, net=net):\n \"\"\"\n Hoping this implementation is less buggy / faster\n \n still feels a bit slow.\n \"\"\"\n pos_weights = torch.ones(X.shape[0], device=device)\n \n def loss(\u03b8):\n preds = net.forward(X, \u03b8)\n bce = torch.nn.BCEWithLogitsLoss(pos_weight=pos_weights, reduction=\"sum\")\n ll_bcs = -1.0 * bce(preds.reshape(-1), y.reshape(-1))\n return ll_bcs\n \n batched_loss = torch._vmap_internals.vmap(loss)\n\n return batched_loss(\u0398)\n```\n\n\n```python\n\u03b3 = 0.2\n\u0394t=0.05\n\ndim= net.dim\n\nsde, losses = basic_batched_trainer(\n \u03b3, \u0394t, gaussian_prior, log_likelihood_vmap_nn, dim, X_train, y_train,\n method=\"euler\", stl=True, adjoint=False, optimizer=None,\n num_steps=400, batch_size_data=X_train.shape[0], batch_size_\u0398=20,\n batchnorm=True, device=device, lr=0.001\n)\n```\n\n LOS STL?: \n\n\n\n 0%| | 0/400 [00:00 0.5).float().flatten()== y_train).float().mean()\n```\n\n\n\n\n tensor(0.9875, device='cuda:0')\n\n\n\n\n```python\npred_test = predc(X_test.float(), \u0398_1)\n```\n\n\n```python\n((pred_test > 0.5).float().flatten() == y_test).float().mean()\n```\n\n\n\n\n tensor(1., device='cuda:0')\n\n\n\n\n```python\ny_test, (pred_test < 0.5).long().flatten()\n```\n\n\n\n\n (tensor([1., 0., 1., 0., 0., 0., 0., 0., 0., 0., 1., 1., 1., 0., 0., 0., 0., 0.,\n 1., 1.], device='cuda:0'),\n tensor([0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0],\n device='cuda:0'))\n\n\n\n\n```python\nplt.clf()\n\nscaler = StandardScaler()\nX_scaled2 = scaler.fit_transform(X)\n\nplt.plot(X_scaled2[y==1, 1], X_scaled2[y==1, 2], 'bx')\nplt.plot(X_scaled2[y==0, 1], X_scaled2[y==0, 2], 'ro')\nplt.legend(('y=1', 'y=0'))\n\n\n# # Overlay contour plot of approximate predictive distribution:\nx_grid = np.arange(-4, 4, 0.009)\nX1, X2 = np.meshgrid(x_grid, x_grid)\nNG = X1.size\nX_test_2 = np.hstack((np.ones((NG,1)), X1.reshape(NG,1), X2.reshape(NG,1)))\nX_test_2.shape\n\nX_test_2_tt = torch.tensor(X_test_2).to(device).float()\np_test = (predc(X_test_2_tt.cpu(),\u0398_1.cpu())).mean(axis=1).detach().cpu().numpy()\n\n# kappa = 1.0 / np.sqrt(1 + (np.pi/8)*np.sum(np.dot(X_test,V)*X_test, 1))\n# p_test = 1.0 / (1+np.exp(-np.dot(X_test,mm)*kappa))\nP = np.reshape(p_test, X1.shape)\nCS = plt.contour(X1, X2, P, [0.1,0.175,0.25,0.375, 0.5,0.625,0.75,0.825,0.9])\nplt.clabel(CS)\nplt.xlabel('x_1')\nplt.ylabel('x_2')\nplt.title('Contours of p(y=1|x,D)')\nplt.show()\n```\n\n\n```python\ntorch.nn.Softplus()\n```\n\n\n\n\n Softplus(beta=1, threshold=20)\n\n\n\n## MAP Baseline\n\nWe run the point estimate approximation (Maximum a posteriori) to double check what the learned weights look like. We get the exact same training accuracy as with the controlled model and similarly large weights for the non bias weights. \n\n\n```python\nX.shape\n```\n\n\n\n\n (100, 3)\n\n\n\n\n```python\n\u0398_map = torch.zeros((1, dim), requires_grad=True, device=device)\noptimizer_map = torch.optim.Adam([\u0398_map], lr=0.05)\n# optimizer = torch.optim.LBFGS(gpr.parameters(), lr=0.01)\n\nlosses_map = []\nnum_steps = 1000\nfor i in tqdm(range(num_steps)):\n optimizer_map.zero_grad()\n\n if isinstance(optimizer_map, torch.optim.LBFGS):\n def closure_map():\n loss_map = log_likelihood_vmap()\n optimizer_map.zero_grad()\n loss_map.backward()\n return loss\n\n optimizer_map.step(closure_map)\n losses_map.append(closure_map().item())\n else:\n loss_map = -(log_likelihood_vmap(\u0398_map, X_train, y_train) + gaussian_prior(\u0398_map))\n optimizer_map.zero_grad()\n loss_map.backward()\n print(loss_map.item())\n optimizer_map.step()\n losses_map.append(loss_map.item())\n\n\u0398_map\npred_map = torch.sigmoid(X_train.mm(\u0398_map.T)).mean(axis=1)\n((pred_map < 0.5).float() == y_train).float().mean(), \u0398_map\n```\n\n\n 0%| | 0/1000 [00:00 1e-10 else f\"{0:8.4g}\"})\n# make cells nice and wide\nfrom IPython.display import display, HTML\ndisplay(HTML(data=\"\"\"\n\n\"\"\"))\ndisplay(HTML(\"\"))\n%matplotlib notebook\n```\n\n\n\n\n\n\n\n\n\n\n\nimporting libraries\n\n\n```python\nfrom posixpath import join\nfrom cvxopt import solvers\nimport random\nimport matplotlib.pyplot as plt\nfrom numpy.lib.function_base import average\nimport torch\nfrom autograd import grad\nfrom autograd import jacobian\nimport autograd.numpy as np\nimport autograd.numpy as jnp\nimport scipy.optimize as optim\nfrom scipy.optimize import minimize, Bounds,LinearConstraint\nfrom scipy.optimize import LinearConstraint,NonlinearConstraint\nfrom scipy.optimize import BFGS\nfrom autograd.numpy import cos,sin\nimport pdb\n```\n\nLoading DeLan Model and defining analytic form of Value function\n\n\n```python\nt = 0.02\nq_prev = None\n\ndevice = 'cpu'\nmodel = torch.load('models/model_750_model_epoch_20000.pth', map_location=torch.device('cpu')) # loaded trained model #TODO\nq_dim = 6 # q_dim is the dimension of joint space\nq_dim_changed = int(0.5 * q_dim)\n\n#Weight for cost function \nw_des_vel = 0.003\nweight_orient = 0.2\n#Desired final Orientation of end effector\nroll_des= -3.141105126296926\npitch_des= 0.00046035505135551175\nyaw_des = -2.355906195444897\norient_desired = np.asarray([ roll_des , pitch_des , yaw_des ])\n\n#value function defnation\nweight = []\nfor key in (model.keys()):\n # print(key)\n weight.append(model[key].cpu().numpy()) # load weight and bias\n\n\ndef leaky_relu(z):\n return np.maximum(0.01 * z, z)\n\n\ndef softplus(z, beta=1):\n return (1 / beta) * np.log(1 + np.exp(z * beta))\n\n\ndef assemble_lower_triangular_matrix(Lo, Ld):\n Lo = Lo.squeeze(0)\n Ld = Ld.squeeze(0)\n\n assert (2 * Lo.shape[0] == (Ld.shape[0] ** 2 - Ld.shape[0]))\n diagonal_matrix = np.identity(len(Ld)) * np.outer(np.ones(len(Ld)), Ld)\n\n L = np.tril(np.ones(diagonal_matrix.shape)) - np.eye(q_dim_changed)\n\n # Set off diagonals\n\n L = np.array([[0, 0, 0], [1, 0, 0], [0, 0, 0]]) * Lo.reshape(3)[0] + np.array([[0, 0, 0], [0, 0, 0], [1, 0, 0]]) * \\\n Lo.reshape(3)[1] + np.array([[0, 0, 0], [0, 0, 0], [0, 1, 0]]) * Lo.reshape(3)[2]\n # Add diagonals\n L = L + diagonal_matrix\n return L\n\n\ndef value(x1):\n global weight, goal\n fc1_w = weight[0]\n fc1_b = weight[1]\n fc2_w = weight[2]\n fc2_b = weight[3]\n fc_Ld_w = weight[4]\n fc_Ld_b = weight[5]\n fc_Lo_w = weight[6]\n fc_Lo_b = weight[7]\n #pdb.set_trace()\n net_input = np.concatenate([np.squeeze(x1), np.squeeze(goal)], axis=0)\n net_input = np.array([net_input])\n\n z1 = np.dot(net_input, fc1_w.transpose()) + fc1_b\n hidden1 = leaky_relu(z1)\n z2 = np.dot(hidden1, fc2_w.transpose()) + fc2_b\n hidden2 = leaky_relu(z2)\n hidden3 = np.dot(hidden2, fc_Ld_w.transpose()) + fc_Ld_b\n Ld = softplus(hidden3)\n Lo = np.dot(hidden2, fc_Lo_w.transpose()) + fc_Lo_b\n L = assemble_lower_triangular_matrix(Lo, Ld)\n\n H = L @ L.transpose() + 1e-9 * np.eye(3)\n return H\n```\n\n\n```python\n#Analytical funtion for forward kinematics of Franka Panda\ndef fk_franka_orient(q):\n \n q_1 = q[0]\n q_2 = q[1]\n q_3 = q[2]\n q_4 = q[3]\n q_5 = q[4]\n q_6 = q[5]\n q_7 = q[6]\n\n x = -0.107*(((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.cos(q_4) - np.sin(q_2)*np.sin(q_4)*np.cos(q_1))*np.cos(q_5) + (np.sin(q_1)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1)*np.cos(q_2))*np.sin(q_5))*np.sin(q_6) - 0.088*(((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.cos(q_4) - np.sin(q_2)*np.sin(q_4)*np.cos(q_1))*np.cos(q_5) + (np.sin(q_1)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1)*np.cos(q_2))*np.sin(q_5))*np.cos(q_6) + 0.088*((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.sin(q_4) + np.sin(q_2)*np.cos(q_1)*np.cos(q_4))*np.sin(q_6) - 0.107*((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.sin(q_4) + np.sin(q_2)*np.cos(q_1)*np.cos(q_4))*np.cos(q_6) + 0.384*(np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.sin(q_4) + 0.0825*(np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.cos(q_4) - 0.0825*np.sin(q_1)*np.sin(q_3) - 0.0825*np.sin(q_2)*np.sin(q_4)*np.cos(q_1) + 0.384*np.sin(q_2)*np.cos(q_1)*np.cos(q_4) + 0.316*np.sin(q_2)*np.cos(q_1) + 0.0825*np.cos(q_1)*np.cos(q_2)*np.cos(q_3)\n\n y = 0.107*(((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.cos(q_4) + np.sin(q_1)*np.sin(q_2)*np.sin(q_4))*np.cos(q_5) - (np.sin(q_1)*np.sin(q_3)*np.cos(q_2) - np.cos(q_1)*np.cos(q_3))*np.sin(q_5))*np.sin(q_6) + 0.088*(((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.cos(q_4) + np.sin(q_1)*np.sin(q_2)*np.sin(q_4))*np.cos(q_5) - (np.sin(q_1)*np.sin(q_3)*np.cos(q_2) - np.cos(q_1)*np.cos(q_3))*np.sin(q_5))*np.cos(q_6) - 0.088*((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.sin(q_4) - np.sin(q_1)*np.sin(q_2)*np.cos(q_4))*np.sin(q_6) + 0.107*((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.sin(q_4) - np.sin(q_1)*np.sin(q_2)*np.cos(q_4))*np.cos(q_6) - 0.384*(np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.sin(q_4) - 0.0825*(np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.cos(q_4) - 0.0825*np.sin(q_1)*np.sin(q_2)*np.sin(q_4) + 0.384*np.sin(q_1)*np.sin(q_2)*np.cos(q_4) + 0.316*np.sin(q_1)*np.sin(q_2) + 0.0825*np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + 0.0825*np.sin(q_3)*np.cos(q_1)\n\n z = -0.107*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.sin(q_6) - 0.088*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.cos(q_6) + 0.088*(np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.sin(q_6) - 0.107*(np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.cos(q_6) + 0.384*np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + 0.0825*np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - 0.0825*np.sin(q_2)*np.cos(q_3) - 0.0825*np.sin(q_4)*np.cos(q_2) + 0.384*np.cos(q_2)*np.cos(q_4) + 0.316*np.cos(q_2) + 0.33\n\n cq1 = np.cos(q_1)\n cq2 = np.cos(q_2)\n cq3 = np.cos(q_3)\n cq4 = np.cos(q_4)\n cq5 = np.cos(q_5)\n cq6 = np.cos(q_6)\n cq7 = np.cos(q_7)\n\n sq1 = np.sin(q_1)\n sq2 = np.sin(q_2)\n sq3 = np.sin(q_3)\n sq4 = np.sin(q_4)\n sq5 = np.sin(q_5)\n sq6 = np.sin(q_6)\n sq7 = np.sin(q_7)\n ax1 = -0.5*(((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.cos(q_4) + np.sin(q_1)*np.sin(q_2)*np.sin(q_4))*np.cos(q_5) - (np.sin(q_1)*np.sin(q_3)*np.cos(q_2) - np.cos(q_1)*np.cos(q_3))*np.sin(q_5))*np.sin(q_6) + 0.5*(((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.cos(q_6) - (np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.sin(q_6))*np.sin(q_7) - 0.5*((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.sin(q_4) - np.sin(q_1)*np.sin(q_2)*np.cos(q_4))*np.cos(q_6) - 0.5*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.sin(q_5) + np.sin(q_2)*np.sin(q_3)*np.cos(q_5))*np.cos(q_7)\n\n ax2 = -0.5*(((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.cos(q_4) - np.sin(q_2)*np.sin(q_4)*np.cos(q_1))*np.cos(q_5) + (np.sin(q_1)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1)*np.cos(q_2))*np.sin(q_5))*np.sin(q_6) + 0.5*(((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.cos(q_6) - (np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.sin(q_6))*np.cos(q_7) - 0.5*((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.sin(q_4) + np.sin(q_2)*np.cos(q_1)*np.cos(q_4))*np.cos(q_6) + 0.5*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.sin(q_5) + np.sin(q_2)*np.sin(q_3)*np.cos(q_5))*np.sin(q_7)\n\n ax3 = -0.5*np.sin(q_2)*np.sin(q_4)*np.sin(q_5)*np.cos(q_1 + q_7) + 0.5*np.sin(q_2)*np.sin(q_4)*np.sin(q_1 + q_7)*np.cos(q_5)*np.cos(q_6) + 0.5*np.sin(q_2)*np.sin(q_6)*np.sin(q_1 + q_7)*np.cos(q_4) - 0.5*np.sin(q_3)*np.sin(q_4)*np.sin(q_6)*np.cos(q_1 + q_7) - 0.5*np.sin(q_3)*np.sin(q_5)*np.sin(q_1 + q_7)*np.cos(q_2)*np.cos(q_6) + 0.5*np.sin(q_3)*np.sin(q_5)*np.sin(q_1 + q_7)*np.cos(q_4) - 0.5*np.sin(q_3)*np.cos(q_2)*np.cos(q_5)*np.cos(q_1 + q_7) + 0.5*np.sin(q_3)*np.cos(q_4)*np.cos(q_5)*np.cos(q_6)*np.cos(q_1 + q_7) - 0.5*np.sin(q_4)*np.sin(q_6)*np.sin(q_1 + q_7)*np.cos(q_2)*np.cos(q_3) - 0.5*np.sin(q_5)*np.cos(q_2)*np.cos(q_3)*np.cos(q_4)*np.cos(q_1 + q_7) + 0.5*np.sin(q_5)*np.cos(q_3)*np.cos(q_6)*np.cos(q_1 + q_7) + 0.5*np.sin(q_1 + q_7)*np.cos(q_2)*np.cos(q_3)*np.cos(q_4)*np.cos(q_5)*np.cos(q_6) - 0.5*np.sin(q_1 + q_7)*np.cos(q_3)*np.cos(q_5)\n\n r_32 = -cq7*(cq5*sq2*sq3 - sq5*(cq2*sq4 - cq3*cq4*sq2)) - sq7*(cq6*(cq5*(cq2*sq4 - cq3*cq4*sq2) + sq2*sq3*sq5) + sq6*(cq2*cq4 + cq3*sq2*sq4))\n r_33 = -cq6*(cq2*cq4 + cq3*sq2*sq4) + sq6*(cq5*(cq2*sq4 - cq3*cq4*sq2) + sq2*sq3*sq5)\n r_31 = cq7*(cq6*(cq5*(cq2*sq4 - cq3*cq4*sq2) + sq2*sq3*sq5) + sq6*(cq2*cq4 + cq3*sq2*sq4)) - sq7*(cq5*sq2*sq3 - sq5*(cq2*sq4 - cq3*cq4*sq2))\n r_21 = cq7*(cq6*(cq5*(cq4*(cq1*sq3 + cq2*cq3*sq1) + sq1*sq2*sq4) + sq5*(cq1*cq3 - cq2*sq1*sq3)) + sq6*(cq4*sq1*sq2 - sq4*(cq1*sq3 + cq2*cq3*sq1))) - sq7*(cq5*(cq1*cq3 - cq2*sq1*sq3) - sq5*(cq4*(cq1*sq3 + cq2*cq3*sq1) + sq1*sq2*sq4))\n r_11 = cq7*(cq6*(cq5*(cq1*sq2*sq4 + cq4*(cq1*cq2*cq3 - sq1*sq3)) - sq5*(cq1*cq2*sq3 + cq3*sq1)) + sq6*(cq1*cq4*sq2 - sq4*(cq1*cq2*cq3 - sq1*sq3))) + sq7*(cq5*(cq1*cq2*sq3 + cq3*sq1) + sq5*(cq1*sq2*sq4 + cq4*(cq1*cq2*cq3 - sq1*sq3)))\n roll_ = np.arctan2(r_32, r_33)\n pitch_=-np.arcsin(r_31)\n yaw_ = np.arctan2(r_21,r_11)\n \n cartpos = np.array([x,y,z])\n pose = [x,y,z,roll_,pitch_,yaw_]\n return pose\n\n#function to compute joint and Cart Space Trajectory Cost \ndef traj_cost(trajectory):\n #pdb.set_trace()\n cost = 0\n cart_cost = 0\n for i in range(len(trajectory) - 1):\n cost += np.linalg.norm(np.asarray(trajectory[i]) - np.asarray(trajectory[i + 1]), ord=2)\n # pdb.set_trace()\n current = np.asarray(fk_franka_orient(trajectory[i]))[0:3] \n next = np.asarray(fk_franka_orient(trajectory[i+1]))[0:3] \n cart_cost += np.linalg.norm(current - next, ord=2)\n return cost, cart_cost\n```\n\n\n```python\n#plotting the traj/ trajc_dot\n#orientResidual_tracker, cart_cost_tracker , trajectory,init_joint, end_cart,qdottrajc,plotno\ndef plot(orientResidual_tracker, cost_tracker , TrajectoryValue,init_joint, end_cartesian,qdottraj , plotno):\n x_fin = end_cartesian.item(0)\n y_fin = end_cartesian.item(1)\n z_fin = end_cartesian.item(2)\n pos = fk_franka_orient(init_joint)[0:3]\n # pos = init_joint\n x_init = pos[0]\n y_init = pos[1]\n z_init = pos[2]\n countValue = len(TrajectoryValue)\n xopt_solValue = np.zeros(countValue - 1)\n yopt_solValue = np.zeros(countValue - 1)\n zopt_solValue = np.zeros(countValue - 1)\n for i in range(0, countValue - 1):\n pos = fk_franka_orient(TrajectoryValue[i])[0:3]\n xopt_solValue[i] = pos[0]\n yopt_solValue[i] = pos[1]\n zopt_solValue[i] = pos[2]\n JointCostValue, CartCostValue = traj_cost(TrajectoryValue)\n\n fig = plt.figure(1, figsize=(12, 12), dpi=100)\n plt.title(\"3d-Trajectory Plot in cartesian coordinate\")\n ax = fig.add_subplot(111, projection='3d')\n ax.plot(xopt_solValue, yopt_solValue, zopt_solValue, '--or', linewidth=2.0, markersize=6.0,\n label=('Traj cost in : joint-Space=', \"{:.2f}\".format(JointCostValue), 'cart-Space=',\n \"{:.2f}\".format(CartCostValue), 'Orient_cost_residual', \"{:.2f}\".format(orientResidual_tracker[-1]), 'euclid_cost_residual', \"{:.2f}\".format(cost_tracker[-1])))\n ax.plot(x_init * np.ones(1), y_init * np.ones(1), z_init * np.ones(1), 'om', markersize=15)\n ax.plot(x_fin * np.ones(1), y_fin * np.ones(1), z_fin * np.ones(1), 'og', markersize=10)\n ax.set_xlim3d(-1.0, 1.0)\n ax.set_ylim3d(-1.0, 1.0)\n ax.set_zlim3d(-0.3, 1.2)\n ax.legend(loc='upper left', frameon=False)\n ax.set_xlabel('X Axis')\n ax.set_ylabel('Y Axis')\n ax.set_zlabel('Z Axis')\n plt.show()\n #plt.close()\n fig = plt.figure(2, figsize=(12, 8), dpi=100)\n plt.title(\"cart-Cost / Orientation-residual cost VS iteration\")\n plt.plot(orientResidual_tracker, label=(\n 'orient residual'))\n plt.plot(cost_tracker, label=(\n 'euclid-cost residual'))\n plt.legend(loc='upper left', frameon=False)\n plt.show()\n fig =plt.figure(3, figsize=(12, 8), dpi=100)\n trajct = np.asarray(TrajectoryValue)\n plt.plot(trajct[:-1, 0], '-o', linewidth=1.0, markersize=3.0 , label= ('joint1'))\n plt.plot(trajct[:-1, 1], '-o', linewidth=1.0, markersize=3.0,label= ('joint2'))\n plt.plot(trajct[:, 2], '-o', linewidth=1.0, markersize=3.0,label= ('joint3'))\n plt.plot(trajct[:, 3], '-o', linewidth=1.0, markersize=3.0,label= ('joint4'))\n plt.plot(trajct[:, 4], '-o', linewidth=1.0, markersize=3.0,label= ('joint5'))\n plt.plot(trajct[:, 5], '-o', linewidth=1.0, markersize=3.0,label= ('joint6'))\n plt.plot(trajct[:, 6], '-o', linewidth=1.0, markersize=3.0,label= ('joint7'))\n plt.title(\"joint angle trajc VS iteration\")\n plt.legend(loc='upper left', frameon=False)\n plt.show()\n fig =plt.figure(4, figsize=(12, 8), dpi=100)\n trajct = np.asarray(qdottrajc)\n plt.plot(trajct[:-1, 0], '-o', linewidth=1.0, markersize=3.0 , label= ('joint1'))\n plt.plot(trajct[:-1, 1], '-o', linewidth=1.0, markersize=3.0,label= ('joint2'))\n plt.plot(trajct[:, 2], '-o', linewidth=1.0, markersize=3.0,label= ('joint3'))\n plt.plot(trajct[:, 3], '-o', linewidth=1.0, markersize=3.0,label= ('joint4'))\n plt.plot(trajct[:, 4], '-o', linewidth=1.0, markersize=3.0,label= ('joint5'))\n plt.plot(trajct[:, 5], '-o', linewidth=1.0, markersize=3.0,label= ('joint6'))\n plt.plot(trajct[:, 6], '-o', linewidth=1.0, markersize=3.0,label= ('joint7'))\n plt.title(\"q-dot trajc VS iteration \")\n plt.legend(loc='upper left', frameon=False)\n plt.show()\n return\n```\n\nDeLan Metric IK formulation:\n\nLets say we the current joint angle is $\\textbf{q}_{init}$ and the corresponding end-effector position is $x_{init}$. We want to solve for the $\\dot{\\textbf{q}}$ that will take us to $\\textbf{x}_{next}$ at the next instant. We can formulate the following optimization problem for it at iteration $k$.\n\n\\begin{align}\n\\min_{\\dot{\\textbf{q}}}\\frac{1}{2}(\\textbf{x}_{next}-\\textbf{x}_{fin})^T\\textbf{H}(\\textbf{x}_{next}, \\textbf{x}_{fin})(\\textbf{x}_{next}-\\textbf{x}_{fin})+\\Vert \\dot{\\textbf{q}}\\Vert_2^2 + orientation_{cost} --eqn(1)\\\\\nf_{fk}(\\dot{\\textbf{q}}t+\\textbf{q}_{init}) = \\textbf{x}_{next}---eqn(2)\\\\\n{\\textbf{q}}_{min}\\leq {\\textbf{q}} \\leq {\\textbf{q}}_{max}\\\\\n{\\text{or we can write}}\\\\\n-{\\textbf{q}}_{max}\\leq \\dot{\\textbf{q}}*t +\\textbf{q}_{init} \\leq {\\textbf{q}}_{max}\\\\\n-\\dot{\\textbf{q}}_{max}\\leq \\dot{\\textbf{q}} \\leq \\dot{\\textbf{q}}_{max}\\\\\n-\\ddot{\\textbf{q}}_{max}t \\leq \\dot{\\textbf{q}}-\\dot{\\textbf{q}}^{k-1} \\leq \\ddot{\\textbf{q}}_{max}t\n\\end{align} \n\nSimplification: The $\\textbf{H}$ matrix is a function of $\\textbf{x}_{next}$ itself which is not known. However, we can make one simplification, we can replace $\\textbf{x}_{next}$ with $\\textbf{x}_{init}$ , i.e the current position in $\\textbf{H}$. This makes the first term convex and gets rid of the non-smoothness aspect. \n\nScipy can solve the optimization problem. The only problem is the non-convex equality constraints from forward kinematics but we can code it as non-linear equality constraints in scipy SLSQP.\n\nWe solve the optimization in MPC setting. That is, we solve $\\dot{\\textbf{q}}$ and updated $\\textbf{q}_{init}$ and so on.\n\n\n```python\n# Cost function defination eqn-1\n\ndef costfxn(solverVariable,x_position,val):\n global goal,diffinCart,roll_des,pitch_des,yaw_des ,w_des_vel,weight_orient\n diff = solverVariable[7:] - goal\n cost = np.matmul(diff.transpose(),np.matmul(val, diff ))\n diffprev = x_position - goal\n cost_prev =np.matmul(diffprev.transpose(),np.matmul(val, diffprev ))\n\n smoothness_cost = np.sum(solverVariable[0:7]**2,axis = 0)\n global q_prev, t ,maxCartdist,iteration\n\n roll_des= -3.141105126296926\n pitch_des= 0.00046035505135551175\n yaw_des = -2.355906195444897\n orient_desired = np.asarray([ roll_des , pitch_des , yaw_des ])\n q = solverVariable[:7]*t + q_prev \n q_1 = q[0]\n q_2 = q[1]\n q_3 = q[2]\n q_4 = q[3]\n q_5 = q[4]\n q_6 = q[5]\n q_7 = q[6]\n\n cq1 = np.cos(q_1)\n cq2 = np.cos(q_2)\n cq3 = np.cos(q_3)\n cq4 = np.cos(q_4)\n cq5 = np.cos(q_5)\n cq6 = np.cos(q_6)\n cq7 = np.cos(q_7)\n\n sq1 = np.sin(q_1)\n sq2 = np.sin(q_2)\n sq3 = np.sin(q_3)\n sq4 = np.sin(q_4)\n sq5 = np.sin(q_5)\n sq6 = np.sin(q_6)\n sq7 = np.sin(q_7)\n ax1 = -0.5*(((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.cos(q_4) + np.sin(q_1)*np.sin(q_2)*np.sin(q_4))*np.cos(q_5) - (np.sin(q_1)*np.sin(q_3)*np.cos(q_2) - np.cos(q_1)*np.cos(q_3))*np.sin(q_5))*np.sin(q_6) + 0.5*(((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.cos(q_6) - (np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.sin(q_6))*np.sin(q_7) - 0.5*((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.sin(q_4) - np.sin(q_1)*np.sin(q_2)*np.cos(q_4))*np.cos(q_6) - 0.5*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.sin(q_5) + np.sin(q_2)*np.sin(q_3)*np.cos(q_5))*np.cos(q_7)\n\n ax2 = -0.5*(((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.cos(q_4) - np.sin(q_2)*np.sin(q_4)*np.cos(q_1))*np.cos(q_5) + (np.sin(q_1)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1)*np.cos(q_2))*np.sin(q_5))*np.sin(q_6) + 0.5*(((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.cos(q_6) - (np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.sin(q_6))*np.cos(q_7) - 0.5*((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.sin(q_4) + np.sin(q_2)*np.cos(q_1)*np.cos(q_4))*np.cos(q_6) + 0.5*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.sin(q_5) + np.sin(q_2)*np.sin(q_3)*np.cos(q_5))*np.sin(q_7)\n\n ax3 = -0.5*np.sin(q_2)*np.sin(q_4)*np.sin(q_5)*np.cos(q_1 + q_7) + 0.5*np.sin(q_2)*np.sin(q_4)*np.sin(q_1 + q_7)*np.cos(q_5)*np.cos(q_6) + 0.5*np.sin(q_2)*np.sin(q_6)*np.sin(q_1 + q_7)*np.cos(q_4) - 0.5*np.sin(q_3)*np.sin(q_4)*np.sin(q_6)*np.cos(q_1 + q_7) - 0.5*np.sin(q_3)*np.sin(q_5)*np.sin(q_1 + q_7)*np.cos(q_2)*np.cos(q_6) + 0.5*np.sin(q_3)*np.sin(q_5)*np.sin(q_1 + q_7)*np.cos(q_4) - 0.5*np.sin(q_3)*np.cos(q_2)*np.cos(q_5)*np.cos(q_1 + q_7) + 0.5*np.sin(q_3)*np.cos(q_4)*np.cos(q_5)*np.cos(q_6)*np.cos(q_1 + q_7) - 0.5*np.sin(q_4)*np.sin(q_6)*np.sin(q_1 + q_7)*np.cos(q_2)*np.cos(q_3) - 0.5*np.sin(q_5)*np.cos(q_2)*np.cos(q_3)*np.cos(q_4)*np.cos(q_1 + q_7) + 0.5*np.sin(q_5)*np.cos(q_3)*np.cos(q_6)*np.cos(q_1 + q_7) + 0.5*np.sin(q_1 + q_7)*np.cos(q_2)*np.cos(q_3)*np.cos(q_4)*np.cos(q_5)*np.cos(q_6) - 0.5*np.sin(q_1 + q_7)*np.cos(q_3)*np.cos(q_5)\n\n r_32 = -cq7*(cq5*sq2*sq3 - sq5*(cq2*sq4 - cq3*cq4*sq2)) - sq7*(cq6*(cq5*(cq2*sq4 - cq3*cq4*sq2) + sq2*sq3*sq5) + sq6*(cq2*cq4 + cq3*sq2*sq4))\n r_33 = -cq6*(cq2*cq4 + cq3*sq2*sq4) + sq6*(cq5*(cq2*sq4 - cq3*cq4*sq2) + sq2*sq3*sq5)\n r_31 = cq7*(cq6*(cq5*(cq2*sq4 - cq3*cq4*sq2) + sq2*sq3*sq5) + sq6*(cq2*cq4 + cq3*sq2*sq4)) - sq7*(cq5*sq2*sq3 - sq5*(cq2*sq4 - cq3*cq4*sq2))\n r_21 = cq7*(cq6*(cq5*(cq4*(cq1*sq3 + cq2*cq3*sq1) + sq1*sq2*sq4) + sq5*(cq1*cq3 - cq2*sq1*sq3)) + sq6*(cq4*sq1*sq2 - sq4*(cq1*sq3 + cq2*cq3*sq1))) - sq7*(cq5*(cq1*cq3 - cq2*sq1*sq3) - sq5*(cq4*(cq1*sq3 + cq2*cq3*sq1) + sq1*sq2*sq4))\n r_11 = cq7*(cq6*(cq5*(cq1*sq2*sq4 + cq4*(cq1*cq2*cq3 - sq1*sq3)) - sq5*(cq1*cq2*sq3 + cq3*sq1)) + sq6*(cq1*cq4*sq2 - sq4*(cq1*cq2*cq3 - sq1*sq3))) + sq7*(cq5*(cq1*cq2*sq3 + cq3*sq1) + sq5*(cq1*sq2*sq4 + cq4*(cq1*cq2*cq3 - sq1*sq3)))\n f_orient_cost = np.linalg.norm( ((r_32/r_33)-np.tan(roll_des))**2+(-np.sin(r_31)-np.sin(pitch_des))**2 + ((r_21 / r_11) -np.tan(yaw_des ) )**2 ) \n totalcost = np.add(cost , w_des_vel*smoothness_cost ) \n weight = weight_orient/cost_prev\n if f_orient_cost< 0.08999999999999999999999 :\n \tweight = 0\n totalcost = totalcost + weight*f_orient_cost\n return totalcost\njac_cost = jacobian(costfxn) #jaccost has a shape of (3,)\n\n\n#Defination of Non linear constraint function eqn-2\ndef constraintfxn(qdot_x_next):\n global t, q_prev\n q = qdot_x_next[:7]*t + q_prev\n x_next = qdot_x_next[7]\n y_next = qdot_x_next[8]\n z_next = qdot_x_next[9]\n\n # x,y,z= robot.fkine(q) #TODO\n q_1 = q[0]\n q_2 = q[1]\n q_3 = q[2]\n q_4 = q[3]\n q_5 = q[4]\n q_6 = q[5]\n q_7 = q[6]\n\n\n x = -0.107*(((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.cos(q_4) - np.sin(q_2)*np.sin(q_4)*np.cos(q_1))*np.cos(q_5) + (np.sin(q_1)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1)*np.cos(q_2))*np.sin(q_5))*np.sin(q_6) - 0.088*(((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.cos(q_4) - np.sin(q_2)*np.sin(q_4)*np.cos(q_1))*np.cos(q_5) + (np.sin(q_1)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1)*np.cos(q_2))*np.sin(q_5))*np.cos(q_6) + 0.088*((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.sin(q_4) + np.sin(q_2)*np.cos(q_1)*np.cos(q_4))*np.sin(q_6) - 0.107*((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.sin(q_4) + np.sin(q_2)*np.cos(q_1)*np.cos(q_4))*np.cos(q_6) + 0.384*(np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.sin(q_4) + 0.0825*(np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.cos(q_4) - 0.0825*np.sin(q_1)*np.sin(q_3) - 0.0825*np.sin(q_2)*np.sin(q_4)*np.cos(q_1) + 0.384*np.sin(q_2)*np.cos(q_1)*np.cos(q_4) + 0.316*np.sin(q_2)*np.cos(q_1) + 0.0825*np.cos(q_1)*np.cos(q_2)*np.cos(q_3)\n\n y = 0.107*(((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.cos(q_4) + np.sin(q_1)*np.sin(q_2)*np.sin(q_4))*np.cos(q_5) - (np.sin(q_1)*np.sin(q_3)*np.cos(q_2) - np.cos(q_1)*np.cos(q_3))*np.sin(q_5))*np.sin(q_6) + 0.088*(((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.cos(q_4) + np.sin(q_1)*np.sin(q_2)*np.sin(q_4))*np.cos(q_5) - (np.sin(q_1)*np.sin(q_3)*np.cos(q_2) - np.cos(q_1)*np.cos(q_3))*np.sin(q_5))*np.cos(q_6) - 0.088*((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.sin(q_4) - np.sin(q_1)*np.sin(q_2)*np.cos(q_4))*np.sin(q_6) + 0.107*((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.sin(q_4) - np.sin(q_1)*np.sin(q_2)*np.cos(q_4))*np.cos(q_6) - 0.384*(np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.sin(q_4) - 0.0825*(np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.cos(q_4) - 0.0825*np.sin(q_1)*np.sin(q_2)*np.sin(q_4) + 0.384*np.sin(q_1)*np.sin(q_2)*np.cos(q_4) + 0.316*np.sin(q_1)*np.sin(q_2) + 0.0825*np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + 0.0825*np.sin(q_3)*np.cos(q_1)\n\n z = -0.107*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.sin(q_6) - 0.088*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.cos(q_6) + 0.088*(np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.sin(q_6) - 0.107*(np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.cos(q_6) + 0.384*np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + 0.0825*np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - 0.0825*np.sin(q_2)*np.cos(q_3) - 0.0825*np.sin(q_4)*np.cos(q_2) + 0.384*np.cos(q_2)*np.cos(q_4) + 0.316*np.cos(q_2) + 0.33\n pos_residual = np.array([x - x_next,y-y_next,z - z_next])\n return pos_residual\n\n#Analytic function for jacobian of Nonlinear constraint function defined above \ndef compute_franka_jac(qdot_x_next):\n\n global q_prev,t\n qdot = qdot_x_next[:7]\n qinit = q_prev \n qdot_1 = qdot[0]\n qdot_2 = qdot[1]\n qdot_3 = qdot[2]\n qdot_4 = qdot[3]\n qdot_5 = qdot[4]\n qdot_6 = qdot[5]\n qdot_7 = qdot[6]\n \n \n qinit_1 = qinit[0]\n qinit_2 = qinit[1]\n qinit_3 = qinit[2]\n qinit_4 = qinit[3]\n qinit_5 = qinit[4]\n qinit_6 = qinit[5]\n qinit_7 = qinit[6]\n \n const_1_q_1 = 0.0825*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4) - 0.384*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) - 0.316*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2) - 0.0825*t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - 0.0825*t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1) + (0.088*(-t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4) + (-t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5) + 0.088*(t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) - t*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*sin(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) + (0.107*(-t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4) + (-t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5) + 0.107*(t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) - t*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*sin(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + (-0.088*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) + 0.088*(t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6) + (0.107*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) - 0.107*(t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_4*t + qinit_4))*cos(qdot_6*t + qinit_6) + (0.0825*t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + 0.0825*t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4) + (0.384*t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + 0.384*t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_4*t + qinit_4)\n const_1_q_2 = 0.384*t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) + 0.0825*t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) - 0.0825*t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) - 0.0825*t*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2) + 0.384*t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) + 0.316*t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2) + (0.088*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*sin(qdot_5*t + qinit_5)*cos(qdot_1*t + qinit_1) + 0.088*(-t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + t*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*cos(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) + (0.107*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*sin(qdot_5*t + qinit_5)*cos(qdot_1*t + qinit_1) + 0.107*(-t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + t*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*cos(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + (-0.107*t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) - 0.107*t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*cos(qdot_6*t + qinit_6) + (0.088*t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) + 0.088*t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6)\n const_1_q_3 = -0.0825*t*sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) - 0.0825*t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2) + (0.088*(t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_5*t + qinit_5) + 0.088*(-t*sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) - t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*cos(qdot_4*t + qinit_4)*cos(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) + (0.107*(t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_5*t + qinit_5) + 0.107*(-t*sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) - t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*cos(qdot_4*t + qinit_4)*cos(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + (0.0825*t*sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) + 0.0825*t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*cos(qdot_4*t + qinit_4) + (0.384*t*sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) + 0.384*t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*sin(qdot_4*t + qinit_4) + 0.088*(t*sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) + t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*sin(qdot_4*t + qinit_4)*sin(qdot_6*t + qinit_6) - 0.107*(t*sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) + t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*sin(qdot_4*t + qinit_4)*cos(qdot_6*t + qinit_6)\n const_1_q_4 = -t*(0.0825*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - 0.0825*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4) + t*(0.384*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - 0.384*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4) - 0.384*t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1) - 0.0825*t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_4*t + qinit_4) + 0.107*(-t*(-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4) + t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6)*cos(qdot_5*t + qinit_5) + 0.088*(-t*(-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4) + t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5)*cos(qdot_6*t + qinit_6) + (-0.107*t*(sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4) + 0.107*t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1))*cos(qdot_6*t + qinit_6) + (0.088*t*(sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4) - 0.088*t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1))*sin(qdot_6*t + qinit_6)\n const_1_q_5 = (-0.107*t*((-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4) + sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1))*sin(qdot_5*t + qinit_5) + 0.107*t*(-sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) - sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*cos(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + (-0.088*t*((-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4) + sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1))*sin(qdot_5*t + qinit_5) + 0.088*t*(-sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) - sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*cos(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6)\n const_1_q_6 = -t*(0.088*((-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4) + sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1))*cos(qdot_5*t + qinit_5) - 0.088*(sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*sin(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + t*(0.107*((-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4) + sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1))*cos(qdot_5*t + qinit_5) - 0.107*(sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*sin(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) + t*(-0.088*(-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4) + 0.088*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_4*t + qinit_4))*cos(qdot_6*t + qinit_6) - t*(0.107*(-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4) - 0.107*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6)\n const_1_q_7 = 0\n const_2_q_1 = -0.0825*t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - 0.0825*t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1) + 0.384*t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_4*t + qinit_4) + 0.316*t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1) + 0.0825*t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + (0.088*(-t*sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) - t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*sin(qdot_5*t + qinit_5) + 0.088*(t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1) + (-t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) + (0.107*(-t*sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) - t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*sin(qdot_5*t + qinit_5) + 0.107*(t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1) + (-t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + (0.0825*t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - 0.0825*t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4) + (0.384*t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - 0.384*t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4) + (-0.107*t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_4*t + qinit_4) - 0.107*(t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4))*cos(qdot_6*t + qinit_6) + (0.088*t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_4*t + qinit_4) + 0.088*(t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6)\n const_2_q_2 = 0.384*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_3*t + qinit_3) + 0.0825*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) - 0.0825*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - 0.0825*t*sin(qdot_1*t + qinit_1)*sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2) + 0.384*t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) + 0.316*t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2) + (0.088*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*sin(qdot_5*t + qinit_5) + 0.088*(-t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + t*sin(qdot_1*t + qinit_1)*sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2))*cos(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) + (0.107*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*sin(qdot_5*t + qinit_5) + 0.107*(-t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + t*sin(qdot_1*t + qinit_1)*sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2))*cos(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + (-0.107*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_3*t + qinit_3) - 0.107*t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*cos(qdot_6*t + qinit_6) + (0.088*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_3*t + qinit_3) + 0.088*t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6)\n const_2_q_3 = -0.0825*t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) + 0.0825*t*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) + (0.088*(-t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) + t*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4)*cos(qdot_5*t + qinit_5) + 0.088*(-t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) + (0.107*(-t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) + t*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4)*cos(qdot_5*t + qinit_5) + 0.107*(-t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + (0.0825*t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) - 0.0825*t*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4) + (0.384*t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) - 0.384*t*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4) + 0.088*(t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) - t*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4)*sin(qdot_6*t + qinit_6) - 0.107*(t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) - t*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4)*cos(qdot_6*t + qinit_6)\n const_2_q_4 = t*(-0.384*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - 0.384*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4) - t*(-0.0825*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - 0.0825*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_4*t + qinit_4) - 0.384*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4) - 0.0825*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) + (-0.107*t*(-sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4) + 0.107*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4))*cos(qdot_6*t + qinit_6) + (0.088*t*(-sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4) - 0.088*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6) + 0.107*(-t*(sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_4*t + qinit_4) + t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6)*cos(qdot_5*t + qinit_5) + 0.088*(-t*(sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_4*t + qinit_4) + t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5)*cos(qdot_6*t + qinit_6)\n const_2_q_5 = (-0.107*t*((sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4) + sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4))*sin(qdot_5*t + qinit_5) + 0.107*t*(-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) + cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*cos(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + (-0.088*t*((sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4) + sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4))*sin(qdot_5*t + qinit_5) + 0.088*t*(-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) + cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*cos(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6)\n const_2_q_6 = -t*(0.088*((sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4) + sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5) - 0.088*(sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) - cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*sin(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + t*(0.107*((sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4) + sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5) - 0.107*(sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) - cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*sin(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) + t*(-0.088*(sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_4*t + qinit_4) + 0.088*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*cos(qdot_6*t + qinit_6) - t*(0.107*(sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_4*t + qinit_4) - 0.107*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6)\n const_2_q_7 = 0\n const_3_q_1 = 0\n const_3_q_2 = 0.0825*t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4) - 0.384*t*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) - 0.316*t*sin(qdot_2*t + qinit_2) + 0.384*t*sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + 0.0825*t*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) - 0.0825*t*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + (-0.088*t*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) + 0.088*t*sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_6*t + qinit_6) + (0.107*t*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) - 0.107*t*sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_6*t + qinit_6) + (0.088*t*sin(qdot_3*t + qinit_3)*sin(qdot_5*t + qinit_5)*cos(qdot_2*t + qinit_2) + 0.088*(-t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4) - t*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) + (0.107*t*sin(qdot_3*t + qinit_3)*sin(qdot_5*t + qinit_5)*cos(qdot_2*t + qinit_2) + 0.107*(-t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4) - t*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6)\n const_3_q_3 = -0.088*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*sin(qdot_4*t + qinit_4)*sin(qdot_6*t + qinit_6) + 0.107*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*sin(qdot_4*t + qinit_4)*cos(qdot_6*t + qinit_6) - 0.384*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*sin(qdot_4*t + qinit_4) - 0.0825*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + 0.0825*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3) + (0.088*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4)*cos(qdot_5*t + qinit_5) + 0.088*t*sin(qdot_2*t + qinit_2)*sin(qdot_5*t + qinit_5)*cos(qdot_3*t + qinit_3))*cos(qdot_6*t + qinit_6) + (0.107*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4)*cos(qdot_5*t + qinit_5) + 0.107*t*sin(qdot_2*t + qinit_2)*sin(qdot_5*t + qinit_5)*cos(qdot_3*t + qinit_3))*sin(qdot_6*t + qinit_6)\n const_3_q_4 = -0.0825*t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_3*t + qinit_3) + 0.384*t*sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) - 0.384*t*sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2) - 0.0825*t*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) + 0.107*(t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_3*t + qinit_3) + t*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6)*cos(qdot_5*t + qinit_5) + 0.088*(t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_3*t + qinit_3) + t*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5)*cos(qdot_6*t + qinit_6) + (-0.107*t*sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + 0.107*t*sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2))*cos(qdot_6*t + qinit_6) + (0.088*t*sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) - 0.088*t*sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2))*sin(qdot_6*t + qinit_6)\n const_3_q_5 = (-0.107*t*(-sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2))*sin(qdot_5*t + qinit_5) + 0.107*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*cos(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + (-0.088*t*(-sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2))*sin(qdot_5*t + qinit_5) + 0.088*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*cos(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6)\n const_3_q_6 = -t*(0.088*(-sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2))*cos(qdot_5*t + qinit_5) + 0.088*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*sin(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + t*(0.107*(-sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2))*cos(qdot_5*t + qinit_5) + 0.107*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*sin(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) - t*(-0.107*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_3*t + qinit_3) - 0.107*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6) + t*(0.088*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_3*t + qinit_3) + 0.088*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*cos(qdot_6*t + qinit_6)\n const_3_q_7 = 0\n const_1_x = -1\n const_2_x = 0\n const_3_x = 0\n const_1_y = 0\n const_2_y = -1\n const_3_y = 0\n const_1_z = 0\n const_2_z = 0\n const_3_z = -1\n const_1_jac = np.hstack(( const_1_q_1, const_1_q_2, const_1_q_3, const_1_q_4, const_1_q_5, const_1_q_6, const_1_q_7, const_1_x, const_1_y, const_1_z ))\n const_2_jac = np.hstack(( const_2_q_1, const_2_q_2, const_2_q_3, const_2_q_4, const_2_q_5, const_2_q_6, const_2_q_7, const_2_x, const_2_y, const_2_z ))\n const_3_jac = np.hstack(( const_3_q_1, const_3_q_2, const_3_q_3, const_3_q_4, const_3_q_5, const_3_q_6, const_3_q_7, const_3_x, const_3_y, const_3_z ))\n # const_3_jac = np.hstack(( const_3_q_1, const_3_q_2, const_3_q_3, const_3_q_4, const_3_q_5, const_3_q_6, const_3_q_7, const_2_x, const_2_y, const_2_z )) orignal MISTAKE?\n const_jac = np.vstack(( const_1_jac, const_2_jac, const_3_jac )) \n return const_jac\n\njac_constraintSympy = compute_franka_jac\n\n\n\n```\n\n\n```python\ndef orient_Residual(q):\n global roll_des,pitch_des, yaw_des\n q_1 = q[0]\n q_2 = q[1]\n q_3 = q[2]\n q_4 = q[3]\n q_5 = q[4]\n q_6 = q[5]\n q_7 = q[6]\n\n cq1 = np.cos(q_1)\n cq2 = np.cos(q_2)\n cq3 = np.cos(q_3)\n cq4 = np.cos(q_4)\n cq5 = np.cos(q_5)\n cq6 = np.cos(q_6)\n cq7 = np.cos(q_7)\n\n sq1 = np.sin(q_1)\n sq2 = np.sin(q_2)\n sq3 = np.sin(q_3)\n sq4 = np.sin(q_4)\n sq5 = np.sin(q_5)\n sq6 = np.sin(q_6)\n sq7 = np.sin(q_7)\n ax1 = -0.5*(((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.cos(q_4) + np.sin(q_1)*np.sin(q_2)*np.sin(q_4))*np.cos(q_5) - (np.sin(q_1)*np.sin(q_3)*np.cos(q_2) - np.cos(q_1)*np.cos(q_3))*np.sin(q_5))*np.sin(q_6) + 0.5*(((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.cos(q_6) - (np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.sin(q_6))*np.sin(q_7) - 0.5*((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.sin(q_4) - np.sin(q_1)*np.sin(q_2)*np.cos(q_4))*np.cos(q_6) - 0.5*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.sin(q_5) + np.sin(q_2)*np.sin(q_3)*np.cos(q_5))*np.cos(q_7)\n\n ax2 = -0.5*(((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.cos(q_4) - np.sin(q_2)*np.sin(q_4)*np.cos(q_1))*np.cos(q_5) + (np.sin(q_1)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1)*np.cos(q_2))*np.sin(q_5))*np.sin(q_6) + 0.5*(((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.cos(q_6) - (np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.sin(q_6))*np.cos(q_7) - 0.5*((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.sin(q_4) + np.sin(q_2)*np.cos(q_1)*np.cos(q_4))*np.cos(q_6) + 0.5*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.sin(q_5) + np.sin(q_2)*np.sin(q_3)*np.cos(q_5))*np.sin(q_7)\n\n ax3 = -0.5*np.sin(q_2)*np.sin(q_4)*np.sin(q_5)*np.cos(q_1 + q_7) + 0.5*np.sin(q_2)*np.sin(q_4)*np.sin(q_1 + q_7)*np.cos(q_5)*np.cos(q_6) + 0.5*np.sin(q_2)*np.sin(q_6)*np.sin(q_1 + q_7)*np.cos(q_4) - 0.5*np.sin(q_3)*np.sin(q_4)*np.sin(q_6)*np.cos(q_1 + q_7) - 0.5*np.sin(q_3)*np.sin(q_5)*np.sin(q_1 + q_7)*np.cos(q_2)*np.cos(q_6) + 0.5*np.sin(q_3)*np.sin(q_5)*np.sin(q_1 + q_7)*np.cos(q_4) - 0.5*np.sin(q_3)*np.cos(q_2)*np.cos(q_5)*np.cos(q_1 + q_7) + 0.5*np.sin(q_3)*np.cos(q_4)*np.cos(q_5)*np.cos(q_6)*np.cos(q_1 + q_7) - 0.5*np.sin(q_4)*np.sin(q_6)*np.sin(q_1 + q_7)*np.cos(q_2)*np.cos(q_3) - 0.5*np.sin(q_5)*np.cos(q_2)*np.cos(q_3)*np.cos(q_4)*np.cos(q_1 + q_7) + 0.5*np.sin(q_5)*np.cos(q_3)*np.cos(q_6)*np.cos(q_1 + q_7) + 0.5*np.sin(q_1 + q_7)*np.cos(q_2)*np.cos(q_3)*np.cos(q_4)*np.cos(q_5)*np.cos(q_6) - 0.5*np.sin(q_1 + q_7)*np.cos(q_3)*np.cos(q_5)\n\n r_32 = -cq7*(cq5*sq2*sq3 - sq5*(cq2*sq4 - cq3*cq4*sq2)) - sq7*(cq6*(cq5*(cq2*sq4 - cq3*cq4*sq2) + sq2*sq3*sq5) + sq6*(cq2*cq4 + cq3*sq2*sq4))\n r_33 = -cq6*(cq2*cq4 + cq3*sq2*sq4) + sq6*(cq5*(cq2*sq4 - cq3*cq4*sq2) + sq2*sq3*sq5)\n r_31 = cq7*(cq6*(cq5*(cq2*sq4 - cq3*cq4*sq2) + sq2*sq3*sq5) + sq6*(cq2*cq4 + cq3*sq2*sq4)) - sq7*(cq5*sq2*sq3 - sq5*(cq2*sq4 - cq3*cq4*sq2))\n r_21 = cq7*(cq6*(cq5*(cq4*(cq1*sq3 + cq2*cq3*sq1) + sq1*sq2*sq4) + sq5*(cq1*cq3 - cq2*sq1*sq3)) + sq6*(cq4*sq1*sq2 - sq4*(cq1*sq3 + cq2*cq3*sq1))) - sq7*(cq5*(cq1*cq3 - cq2*sq1*sq3) - sq5*(cq4*(cq1*sq3 + cq2*cq3*sq1) + sq1*sq2*sq4))\n r_11 = cq7*(cq6*(cq5*(cq1*sq2*sq4 + cq4*(cq1*cq2*cq3 - sq1*sq3)) - sq5*(cq1*cq2*sq3 + cq3*sq1)) + sq6*(cq1*cq4*sq2 - sq4*(cq1*cq2*cq3 - sq1*sq3))) + sq7*(cq5*(cq1*cq2*sq3 + cq3*sq1) + sq5*(cq1*sq2*sq4 + cq4*(cq1*cq2*cq3 - sq1*sq3)))\n #pdb.set_trace()\n f_orient_cost = ((r_32/r_33)-np.tan(roll_des))**2+(-np.sin(r_31)-np.sin(pitch_des))**2 + ((r_21 / r_11) -np.tan(yaw_des ) )**2\n return f_orient_cost\n```\n\n\n```python\n#if you want to go from A to B pass the joint position of A fk(q_a) , x_fin\n#main function to compute Delan Metric-Based IK\nimport time\ndef trajMetricBased(init_joint,end_cartesian,seq):\n start_cart = np.asarray(fk_franka_orient(init_joint))[0:3]\n global t,q_prev,goal\n print(\"Start : \",start_cart)\n print(\"Goal : \", end_cartesian)\n goal = np.squeeze(end_cartesian)\n x_des = end_cartesian\n num_dof = 7 #franka panda\n qdot = np.zeros(num_dof) ########## initial velocity\n qdotprev = np.zeros(num_dof)\n q_min = np.array([-165, -100, -165, -165, -165, -1.0, -165]) * np.pi / 180 # TODO\n #q_min = q_min.reshape(7,1)\n q_max = np.array([165, 101, 165, 1.0, 165, 214, 165]) * np.pi / 180 #TODO\n #q_max = q_max.reshape(7,1)\n\n\n qdot_max = np.array([2.1750\t,2.1750\t,2.1750\t,2.1750,\t2.6100,\t2.6100,\t2.6100]) #TODO\n qdot_min = -1*qdot_max\n qacc_max = np.array([15,\t7.5,\t10\t,12.5\t,15\t,20,\t20]) #TODO\n qacc_min = -1*qacc_max\n\n x_pos = np.asarray(fk_franka_orient(init_joint))[0:3] ########### initial positions\n\n q = init_joint\n q_prev = init_joint\n ######### delta t for which the computed velocity is commanded to the robot 3msec\n x_next = fk_franka_orient(qdot*t + q)[0:3]\n\n mpc_iter =300\n q_1 = []\n q_2 = []\n q_3 = []\n q_4 = []\n q_5 = []\n q_6 = []\n q_7 = []\n cost_tracker = []\n trajectory = []\n qdottrajc =[]\n orientResidual =[]\n #pdb.set_trace()\n solverVariable = np.hstack((qdot,x_next))\n x_min = -np.ones(3,)*np.Inf \n x_max = np.ones(3,)*np.Inf \n solver_minbounds= np.hstack((qdot_min , x_min))\n solver_maxbounds = np.hstack((qdot_max , x_max))\n Amat = np.hstack((np.identity(7),np.zeros((7,3)))) \n Bmat = np.hstack((np.identity(7),np.zeros((7,3))))\n Total_time = 0\n pose = fk_franka_orient(q)\n for i in range(0, mpc_iter):\n print(f\"mpc-itr={i}\")\n orientResidual.append( orient_Residual(q))\n cost_tracker.append(np.linalg.norm(x_pos - x_des))\n print(\"itr=\",i, \"euclid-norm\",np.linalg.norm(x_pos - x_des) , \"orient-norm\" , orient_Residual(q) )\n if np.linalg.norm(x_pos - x_des) < 0.05 and orient_Residual(q) < 0.1 :\n break\n bnds = Bounds(solver_minbounds,solver_maxbounds)\n nonlinear_constraint = NonlinearConstraint(constraintfxn , np.zeros((3,)), np.zeros((3,)) , jac= jac_constraintSympy, hess=BFGS() )\n linear_constraint_A = LinearConstraint(Amat*t, q_min-q_prev, q_max-q_prev)\n linear_constraint_B = LinearConstraint(Bmat ,qacc_min*t + qdotprev, qacc_max*t + qdotprev )\n defaultopts={ 'maxiter': 80 , 'ftol': 1e-08, 'iprint': 1, 'disp': False, 'eps': 1.4901161193847656e-08, 'finite_diff_rel_step': None}\n val = value(x_pos)\n currentTime = time.time()\n res = minimize(costfxn , solverVariable, args =(x_pos,val) , method='SLSQP', jac=jac_cost,\n constraints=[nonlinear_constraint,linear_constraint_A, linear_constraint_B], options=defaultopts ,bounds=bnds) \n timeTaken = (time.time() - currentTime)\n Total_time += timeTaken\n solverVariable = np.asarray(res['x']).squeeze()\n trajectory.append([q[0], q[1], q[2], q[3], q[4], q[5], q[6]])\n qdottrajc.append(qdotprev)\n q = q + solverVariable[0:7] * t\n x_pos = fk_franka_orient(q)[0:3] #solverVariable[7:] #or x_next\n solverVariable[7:] = x_pos\n qdotprev = solverVariable[0:7]\n q_prev = q\n pose = fk_franka_orient(q)\n if len(trajectory) != 0 :\n JointCost, CartCost = traj_cost(trajectory)\n \n averageTime = Total_time/(i+1)\n executionTime = (i+1)*0.02\n #OrienResidual = orienResidual(q) \n cart_residual = cost_tracker[-1]\n pose = fk_franka_orient(q)\n orient_residual = orient_Residual(q)\n return JointCost, CartCost, cost_tracker, trajectory,qdottrajc,orientResidual\n else:\n return 0, 0, 0 , [], 0 ,0 \n\n```\n\n\n```python\n#example 1\ninit_joint =np.asarray([-0.6581236205695661, -0.685344738206894, -0.07143761874163594, -2.7322892472673206, -0.050369729249887224, 2.0478311535710176, 1.6687292468254735])\nend_cart = np.asarray([ -0.27332807, 0.27789939, -0.15650605 ])\n\n#example 2 \n#init_joint = np.asarray([-1.6202005289676729,-1.2697694275822582,-2.600571094528167,-2.176462031576798,-2.09104048608945,2.28241996926819,-0.05311600216515573])\n#end_cart = np.asarray([ 0.6367232 , -0.40827509, 0.39513954])\nJointCost, CartCost, cart_cost_tracker, trajectory,qdottrajc ,orientResidual_tracker = trajMetricBased(init_joint , end_cart ,\"example2\")\n```\n\n Start : [ 0.24605026 -0.22180353 0.4166907 ]\n Goal : [-0.27332807 0.27789939 -0.15650605]\n mpc-itr=0\n itr= 0 euclid-norm 0.9208753334845314 orient-norm 3.9766230535877584e-05\n mpc-itr=1\n itr= 1 euclid-norm 0.9179723472154594 orient-norm 0.0004519908770243149\n mpc-itr=2\n itr= 2 euclid-norm 0.9120976667148027 orient-norm 0.0030620813372865966\n mpc-itr=3\n itr= 3 euclid-norm 0.9031492458091034 orient-norm 0.011439786915084309\n mpc-itr=4\n itr= 4 euclid-norm 0.8909917527970994 orient-norm 0.03033629403579955\n mpc-itr=5\n itr= 5 euclid-norm 0.8754707552953916 orient-norm 0.06515339926209383\n mpc-itr=6\n itr= 6 euclid-norm 0.8564408125051055 orient-norm 0.1166745795015591\n mpc-itr=7\n itr= 7 euclid-norm 0.8342988091976554 orient-norm 0.19100033159558477\n mpc-itr=8\n itr= 8 euclid-norm 0.8142300634616978 orient-norm 0.2779530383111696\n mpc-itr=9\n itr= 9 euclid-norm 0.7955952423145648 orient-norm 0.3780345558241193\n mpc-itr=10\n itr= 10 euclid-norm 0.7781454879724871 orient-norm 0.4925455123708523\n mpc-itr=11\n itr= 11 euclid-norm 0.7620799481550798 orient-norm 0.6179568713066785\n mpc-itr=12\n itr= 12 euclid-norm 0.747784588435236 orient-norm 0.7471461044877954\n mpc-itr=13\n itr= 13 euclid-norm 0.7349717532152321 orient-norm 0.8786999425207335\n mpc-itr=14\n itr= 14 euclid-norm 0.7236372403382607 orient-norm 1.007364707416616\n mpc-itr=15\n itr= 15 euclid-norm 0.7135904412826327 orient-norm 1.1310251291303102\n mpc-itr=16\n itr= 16 euclid-norm 0.7046070380556929 orient-norm 1.2489406059291137\n mpc-itr=17\n itr= 17 euclid-norm 0.6964554602860829 orient-norm 1.3615921778415165\n mpc-itr=18\n itr= 18 euclid-norm 0.6888838257699009 orient-norm 1.4709585367266005\n mpc-itr=19\n itr= 19 euclid-norm 0.6817449700739754 orient-norm 1.578041967806966\n mpc-itr=20\n itr= 20 euclid-norm 0.6727599519390013 orient-norm 1.6964282763560394\n mpc-itr=21\n itr= 21 euclid-norm 0.6611864515151117 orient-norm 1.8281557904009247\n mpc-itr=22\n itr= 22 euclid-norm 0.647276449211088 orient-norm 1.9709231609285582\n mpc-itr=23\n itr= 23 euclid-norm 0.6312606162840506 orient-norm 2.1219394729393315\n mpc-itr=24\n itr= 24 euclid-norm 0.6139797186035865 orient-norm 2.269642324041856\n mpc-itr=25\n itr= 25 euclid-norm 0.5960466379112828 orient-norm 2.403818568919186\n mpc-itr=26\n itr= 26 euclid-norm 0.5783086575537171 orient-norm 2.502669081841128\n mpc-itr=27\n itr= 27 euclid-norm 0.5610102669883865 orient-norm 2.5623084300321994\n mpc-itr=28\n itr= 28 euclid-norm 0.5440819160116853 orient-norm 2.5867086561390904\n mpc-itr=29\n itr= 29 euclid-norm 0.5291106245802388 orient-norm 2.5791355850356164\n mpc-itr=30\n itr= 30 euclid-norm 0.5159027711132875 orient-norm 2.545451844341975\n mpc-itr=31\n itr= 31 euclid-norm 0.5045605740424741 orient-norm 2.4895923325978067\n mpc-itr=32\n itr= 32 euclid-norm 0.4946239398332391 orient-norm 2.4221529072508643\n mpc-itr=33\n itr= 33 euclid-norm 0.48546392878498607 orient-norm 2.356131301729767\n mpc-itr=34\n itr= 34 euclid-norm 0.47695239038927423 orient-norm 2.2947562298304645\n mpc-itr=35\n itr= 35 euclid-norm 0.4680186365745303 orient-norm 2.2314553499096137\n mpc-itr=36\n itr= 36 euclid-norm 0.4584777324782933 orient-norm 2.167621586356325\n mpc-itr=37\n itr= 37 euclid-norm 0.44930311130135314 orient-norm 2.0899562876827673\n mpc-itr=38\n itr= 38 euclid-norm 0.44035528394021456 orient-norm 2.004697353005917\n mpc-itr=39\n itr= 39 euclid-norm 0.43159364652770155 orient-norm 1.9257706899525497\n mpc-itr=40\n itr= 40 euclid-norm 0.42311537792933973 orient-norm 1.8487016268231566\n mpc-itr=41\n itr= 41 euclid-norm 0.414772351516169 orient-norm 1.7690732218969927\n mpc-itr=42\n itr= 42 euclid-norm 0.4068075506793391 orient-norm 1.685924228845298\n mpc-itr=43\n itr= 43 euclid-norm 0.39935543414452085 orient-norm 1.5982917238081242\n mpc-itr=44\n itr= 44 euclid-norm 0.3923395196718938 orient-norm 1.5084458989350096\n mpc-itr=45\n itr= 45 euclid-norm 0.38570923783878386 orient-norm 1.4179036761349004\n mpc-itr=46\n itr= 46 euclid-norm 0.37922254403963773 orient-norm 1.3302096716708365\n mpc-itr=47\n itr= 47 euclid-norm 0.3725946634480924 orient-norm 1.2476052214237896\n mpc-itr=48\n itr= 48 euclid-norm 0.3658476479256193 orient-norm 1.1685638500753806\n mpc-itr=49\n itr= 49 euclid-norm 0.35899993179248185 orient-norm 1.0921637779227682\n mpc-itr=50\n itr= 50 euclid-norm 0.3520664208309003 orient-norm 1.0178020572379325\n mpc-itr=51\n itr= 51 euclid-norm 0.34505704284341604 orient-norm 0.9451096253474756\n mpc-itr=52\n itr= 52 euclid-norm 0.337983879956387 orient-norm 0.8738779131474643\n mpc-itr=53\n itr= 53 euclid-norm 0.3308659018658105 orient-norm 0.8038540235747833\n mpc-itr=54\n itr= 54 euclid-norm 0.32371030694896785 orient-norm 0.7350372366544702\n mpc-itr=55\n itr= 55 euclid-norm 0.3165222664482285 orient-norm 0.6675547890504009\n mpc-itr=56\n itr= 56 euclid-norm 0.30928809985699907 orient-norm 0.6017903948380554\n mpc-itr=57\n itr= 57 euclid-norm 0.3020195312051132 orient-norm 0.5380015354812621\n mpc-itr=58\n itr= 58 euclid-norm 0.29474138451964627 orient-norm 0.476596468363466\n mpc-itr=59\n itr= 59 euclid-norm 0.2873664838803182 orient-norm 0.4187081354637925\n mpc-itr=60\n itr= 60 euclid-norm 0.27973755133798794 orient-norm 0.36590478700996687\n mpc-itr=61\n itr= 61 euclid-norm 0.2718597908784527 orient-norm 0.3184604360898108\n mpc-itr=62\n itr= 62 euclid-norm 0.2637315097536895 orient-norm 0.2766971577700479\n mpc-itr=63\n itr= 63 euclid-norm 0.25536000375691265 orient-norm 0.24076666597288668\n mpc-itr=64\n itr= 64 euclid-norm 0.24677382296859798 orient-norm 0.2105159811814169\n mpc-itr=65\n itr= 65 euclid-norm 0.23803114806937145 orient-norm 0.18543174226034315\n mpc-itr=66\n itr= 66 euclid-norm 0.22922061002504532 orient-norm 0.16468917954712126\n mpc-itr=67\n itr= 67 euclid-norm 0.22051318553311408 orient-norm 0.14711993729971684\n mpc-itr=68\n itr= 68 euclid-norm 0.2118961896016381 orient-norm 0.13213957070849155\n mpc-itr=69\n itr= 69 euclid-norm 0.2034162976759297 orient-norm 0.11896337788925818\n mpc-itr=70\n itr= 70 euclid-norm 0.19522807188048905 orient-norm 0.1067313718572154\n mpc-itr=71\n itr= 71 euclid-norm 0.18741777775485705 orient-norm 0.09505409300217262\n mpc-itr=72\n itr= 72 euclid-norm 0.1778661653899078 orient-norm 0.09000127397492907\n mpc-itr=73\n itr= 73 euclid-norm 0.16719765419853955 orient-norm 0.0900065115778486\n mpc-itr=74\n itr= 74 euclid-norm 0.15728823291821506 orient-norm 0.09000158565180366\n mpc-itr=75\n itr= 75 euclid-norm 0.1480533855059605 orient-norm 0.09000044160684158\n mpc-itr=76\n itr= 76 euclid-norm 0.13944977907274064 orient-norm 0.09000000837303676\n mpc-itr=77\n itr= 77 euclid-norm 0.13143345251003236 orient-norm 0.09000001491421611\n mpc-itr=78\n itr= 78 euclid-norm 0.12401235105514666 orient-norm 0.09000006649950953\n mpc-itr=79\n itr= 79 euclid-norm 0.11713471136034516 orient-norm 0.09000009333261272\n mpc-itr=80\n itr= 80 euclid-norm 0.11074781218281415 orient-norm 0.09000009635271981\n mpc-itr=81\n itr= 81 euclid-norm 0.1048021675002188 orient-norm 0.09000008719230083\n mpc-itr=82\n itr= 82 euclid-norm 0.0992609458165895 orient-norm 0.09000007817876803\n mpc-itr=83\n itr= 83 euclid-norm 0.09407864633643875 orient-norm 0.09000006650326561\n mpc-itr=84\n itr= 84 euclid-norm 0.08919591528095895 orient-norm 0.0900003089754385\n mpc-itr=85\n itr= 85 euclid-norm 0.08463782361466685 orient-norm 0.09000005534119065\n mpc-itr=86\n itr= 86 euclid-norm 0.08037342568159457 orient-norm 0.09000003651708405\n mpc-itr=87\n itr= 87 euclid-norm 0.07638364004034548 orient-norm 0.09000003004821502\n mpc-itr=88\n itr= 88 euclid-norm 0.07264463777427155 orient-norm 0.09000002456430273\n mpc-itr=89\n itr= 89 euclid-norm 0.06913351460561037 orient-norm 0.09000002197942364\n mpc-itr=90\n itr= 90 euclid-norm 0.06583061836312387 orient-norm 0.09000002116259825\n mpc-itr=91\n itr= 91 euclid-norm 0.06271943875515563 orient-norm 0.09000001856434771\n mpc-itr=92\n itr= 92 euclid-norm 0.059784276002300324 orient-norm 0.09000001827711174\n mpc-itr=93\n itr= 93 euclid-norm 0.05701176344037878 orient-norm 0.09000001802489015\n mpc-itr=94\n itr= 94 euclid-norm 0.054396622752763434 orient-norm 0.09000000709038067\n mpc-itr=95\n itr= 95 euclid-norm 0.05192116313775348 orient-norm 0.09000000976190008\n mpc-itr=96\n itr= 96 euclid-norm 0.04957272768324672 orient-norm 0.09000001013087859\n\n\n\n```python\nplotno =1\n%matplotlib inline\nplot(orientResidual_tracker, cart_cost_tracker , trajectory,init_joint, end_cart,qdottrajc,plotno)\n```\n\n\n```python\n%matplotlib notebook\nimport roboticstoolbox as rtb\ntrajc = np.reshape(trajectory , ( len(trajectory) ,7))\nrobot = rtb.models.DH.Panda()\nrobot.plot(trajc, dt = 0.02 , movie = 'example.gif') #for smooth visualization click an play on the saved gif file in current dir (might need to refresh the directory)\n```\n\n\n \n\n\n\n\n\n\n\n", "meta": {"hexsha": "27189809312e9928deee72b090fb5a67b38f4d6f", "size": 668341, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/DeLan_IK.ipynb", "max_stars_repo_name": "prajwalresearch/rearrangement_latest", "max_stars_repo_head_hexsha": "1c4cd2b777ba74419e79599de08b750614462bfd", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/DeLan_IK.ipynb", "max_issues_repo_name": "prajwalresearch/rearrangement_latest", "max_issues_repo_head_hexsha": "1c4cd2b777ba74419e79599de08b750614462bfd", 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YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.6442251201477015, "lm_q1q2_score": 0.44835383556246905}} {"text": "# 10. Gyakorlat - 2 DoF nemline\u00e1ris rendszer\n2021.04.16.\n\n## Feladat:\n\n
\n\nA mell\u00e9kelt \u00e1br\u00e1n egy k\u00e9t szabads\u00e1gfok\u00fa mechanikai leng\u0151rendszer l\u00e1that\u00f3, mely egy $m_1$ t\u00f6meg\u0171 \u00e9s $r$ sugar\u00fa korongb\u00f3l \u00e1ll, ami egy $R$ bels\u0151 sugar\u00fa mozg\u00f3 fel\u00fcleten g\u00f6rd\u00fcl. Ez az $m_2$ t\u00f6meg a $k_1$, $k_2$ \u00e9s $k_3$ rug\u00f3merevs\u00e9g\u0171 rug\u00f3k \u00e1ltal van al\u00e1t\u00e1masztva. Ennek v\u00edzszintes ir\u00e1ny\u00fa elmozdul\u00e1s\u00e1t az $y$ \u00e1ltal\u00e1nos\u00edtott koordin\u00e1ta \u00edrja le, valamint a g\u00f6rd\u00fcl\u0151 korong poz\u00edci\u00f3j\u00e1nak le\u00edr\u00e1s\u00e1ra a $\\varphi$ \u00e1ltal\u00e1nos koordin\u00e1ta haszn\u00e1latos. A gravit\u00e1ci\u00f3s mez\u0151ben elhelyezett rendszer egyens\u00falyi \u00e1llapota az $y = 0$ \u00e9s a $\\varphi = 0$ \u00e1llapothoz tartozik. Itt egyed\u00fcl a r\u00fag\u00f3kat terheli statikus deform\u00e1ci\u00f3. \n### Adatok:\nAz $m_1$ \u00e9s $m_2$ t\u00f6megek, az $r$ \u00e9s $R$ sugarak, valamint a $k_1$, $k_2$, $k_3$ rug\u00f3merevs\u00e9gek ismertnek tekinthet\u0151ek.\n### R\u00e9szfeladatok:\n\n1. Hat\u00e1rozza meg a rendszer nemline\u00e1ris mozg\u00e1segyenleteit a m\u00e1sodfaj\u00fa Lagrange-egyenlettel, tov\u00e1bb\u00e1 lineariz\u00e1lja az $y = 0$ \u00e9s $\\varphi = 0$ egyens\u00falyi helyzet k\u00f6r\u00fcli kis kit\u00e9r\u00e9sek mellett.\n2. Hat\u00e1rozza meg a lineariz\u00e1lt mozg\u00e1segyenleteket m\u00e1trix egy\u00fctthat\u00f3s alakban is.\n\n## Megold\u00e1s:\n\n###\u00a01. Feladat\n\n\n```python\nimport sympy as sp\nfrom IPython.display import Math\nsp.init_printing()\n# a k\u00e9t \u00e1ltal\u00e1nos koordin\u00e1ta felv\u00e9tele\n\nt = sp.Symbol('t')\ny = sp.Function('y')(t)\n\u03c6 = sp.Function('\u03c6')(t)\n\nm1, m2, r, R, k1, k2, k3, g = sp.symbols('m_1, m_2, r, R, k_1, k_2, k_3, g')\n# T\u00e1roljuk ezeket a `q` \u00e1ltal\u00e1nos\u00edtott koord. vektorban\nq = sp.Matrix([[y],[\u03c6]]) # a m\u00e1sodik r\u00e9teg sz\u00f6gletes z\u00e1r\u00f3jel nem k\u00f6telez\u0151 itt\ndisplay(Math('\\mathbf{{q}} = {}'.format(sp.latex(q))))\n\n# Figyelj\u00fcnk r\u00e1, hogy mostant\u00f3l `y` `q[0]` lesz, `\u03c6` pedig `q[1]`.\n```\n\n\n$\\displaystyle \\mathbf{q} = \\left[\\begin{matrix}y{\\left(t \\right)}\\\\\u03c6{\\left(t \\right)}\\end{matrix}\\right]$\n\n\nA kinetikus energia meghat\u00e1roz\u00e1sa:\n\n$$T = \\frac{1}{2}m_2\\dot{y}^2 + \\frac{1}{2}m_1v_{\\mathrm{S}}^2 + \\frac{1}{2}\\Theta_{\\mathrm{1Sz}}\\omega_{\\mathrm{1z}}^2$$\n\n\n```python\n\"\"\" A kinetikus energi\u00e1t az \u00e1ltal\u00e1nos\u00edtott koordin\u00e1kkal kell fel\u00edrni, \n\u00edgy `v_S` \u00e9s `\u03c9_1z`-t azok seg\u00edts\u00e9g\u00e9vel kell kifejezni. \n\"\"\"\n\n# egy kis LaTeX-es form\u00e1z\u00e1s: a vektort f\u00e9lk\u00f6v\u00e9rrel \u00edratjuk ki (bold font)\n# valamint az index \u00e1ll\u00f3 bet\u0171 legyen (roman).\nv_O = sp.MatrixSymbol('\\mathbf{v}_\\mathrm{O}',3,1) # Hasonl\u00f3, mint az `sp.Symbol`, csak jelen esetben \n # 3x1-es m\u00e1trixk\u00e9nt kezeli. \u00cdgy \u00f6sszeadhat\u00f3\n # szimbolikusan egy megfelel\u0151 dimenzi\u00f3j\u00fa m\u00e1trixszal.\nv_S = sp.MatrixSymbol('\\mathbf{v}_\\mathrm{S}',3,1)\nr_OSx,r_OSy = sp.symbols('\\mathbf{r}_{\\mathrm{OS}z},\\mathbf{r}_{\\mathrm{OS}y}')\nr_OS = sp.Matrix([[r_OSx],[r_OSy],[0]])\nv_OS_red = v_O + sp.Matrix([[0],[0],[q[1].diff(t)]]).cross(r_OS)\neq_vos = sp.Eq(v_S,v_OS_red)\neq_vos\n```\n\n\n\n\n$\\displaystyle \\mathbf{v}_\\mathrm{S} = \\left[\\begin{matrix}- \\mathbf{r}_{\\mathrm{OS}y} \\frac{d}{d t} \u03c6{\\left(t \\right)}\\\\\\mathbf{r}_{\\mathrm{OS}z} \\frac{d}{d t} \u03c6{\\left(t \\right)}\\\\0\\end{matrix}\\right] + \\mathbf{v}_\\mathrm{O}$\n\n\n\n\n```python\n\"\"\" Sajnos a m\u00e1trix szimb\u00f3lumos \u00fat nem j\u00e1rhat\u00f3 \na jelenlegi `sympy` verzi\u00f3 mellett (1.7.1). \nRem\u00e9lhet\u0151leg a j\u00f6v\u0151ben jobban ki lesz dolgozva \nez a megk\u00f6zel\u00edt\u00e9s is. Az egyik legf\u0151bb hi\u00e1nyoss\u00e1got\naz al\u00e1bbi k\u00f3dsor szeml\u00e9lteti:\n\"\"\"\nsp.solve(eq_vos,v_O) # Azaz egyszer\u0171en fejezz\u00fck ki a `v_O` vektort.\n\n\"\"\" L\u00e1that\u00f3, hogy hib\u00e1t kapunk. A `sympy solve` \nmet\u00f3dusa nincs felk\u00e9sz\u00edtve a m\u00e1trix szimb\u00f3lumokkal\nval\u00f3 sz\u00e1mol\u00e1sra. Ez is j\u00f3l mutatja, hogy a `sympy` \nk\u00f6zel sem t\u00f6k\u00e9letes modul, b\u00e1r folyamatosan fejlesztik.\nRem\u00e9lhet\u0151leg egy j\u00f6v\u0151beli verzi\u00f3ba beleker\u00fcl ez a funkci\u00f3 is.\n\"\"\"\n```\n\n\n```python\n# Most teh\u00e1t minden szimbolikus vektort/m\u00e1trixot a benne l\u00e9v\u0151 elemek\n# szimbolikuss\u00e1 t\u00e9tel\u00e9vel fogunk kezelni\n\nv_Sx, v_Sy = sp.symbols('v_{\\mathrm{S}x}, v_{\\mathrm{S}y}')\nv_Ox, v_Oy = sp.symbols('v_{\\mathrm{O}x}, v_{\\mathrm{O}y}')\nv_Bx, v_By = sp.symbols('v_{\\mathrm{B}x}, v_{\\mathrm{B}y}')\nr_OSx, r_OSy = sp.symbols('r_{\\mathrm{OS}x}, r_{\\mathrm{OS}y}')\nr_BSx, r_BSy = sp.symbols('r_{\\mathrm{BS}x}, r_{\\mathrm{BS}y}')\n\u03c9_1z = sp.Symbol('\u03c9_1z')\n\nv_S = sp.Matrix([[v_Sx],[v_Sy],[0]])\nv_O = sp.Matrix([[v_Ox],[v_Oy],[0]])\nv_B = sp.Matrix([[v_Bx],[v_By],[0]])\nr_OS = sp.Matrix([[r_OSx],[r_OSy],[0]])\nr_BS = sp.Matrix([[r_BSx],[r_BSy],[0]])\n```\n\n2 sebess\u00e9gredukci\u00f3s k\u00e9plet seg\u00edts\u00e9g\u00e9vel sz\u00e1m\u00edthatjuk ki a megfelel\u0151 sebess\u00e9geket a kinetikus energia kifejez\u00e9s\u00e9ben, valamint ezek felhaszn\u00e1l\u00e1s\u00e1val hat\u00e1rozhatjuk meg a kapcsolatot az \u00e1ltal\u00e1nos koordin\u00e1t\u00e1k \u00e9s ezen sebess\u00e9gek k\u00f6z\u00f6tt. Az \u00e1bra alapj\u00e1n:\n\n\n$$\\mathbf{v}_\\mathrm{S} = \\mathbf{v}_\\mathrm{O} + \\begin{bmatrix}0 \\\\ 0 \\\\ \\dot\\varphi\\end{bmatrix} \\times \\mathbf{r}_\\mathrm{OS},$$ valamint \n$$\\mathbf{v}_\\mathrm{S} = \\mathbf{v}_\\mathrm{B} + \\begin{bmatrix}0 \\\\ 0 \\\\ \\omega_{1z}\\end{bmatrix} \\times \\mathbf{r}_\\mathrm{BS}.$$\n\nMeg\u00e1llap\u00edthat\u00f3 tov\u00e1bb\u00e1, hogy \n\n$$\\mathbf{v}_\\mathrm{O} = \\mathbf{v}_\\mathrm{B} = \\begin{bmatrix}0 \\\\ \\dot{y} \\\\ 0\\end{bmatrix}.$$\n\n\n```python\nv_OS_red = sp.Matrix([[0],[q[0].diff(t)],[0]]) + sp.Matrix([[0],[0],[q[1].diff(t)]]).cross(r_OS)\n\nv_BS_red = sp.Matrix([[0],[q[0].diff(t)],[0]]) + sp.Matrix([[0],[0],[\u03c9_1z]]).cross(r_BS)\n\n# A helyvektorok koordin\u00e1t\u00e1i az \u00e1br\u00e1r\u00f3l leolvashat\u00f3ak:\nvekt_koord = [(r_OSx, (R-r) * sp.sin(q[1])), (r_OSy, -(R-r) * sp.cos(q[1])),\n (r_BSx, -r*sp.sin(q[1])), (r_BSy, r*sp.cos(q[1]))]\n\n# A fentiek alapj\u00e1n az al\u00e1bbi egyenl\u0151s\u00e9g \u00e1ll fenn:\nsp.Eq(v_OS_red.subs(vekt_koord),v_BS_red.subs(vekt_koord))\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}- \\left(- R + r\\right) \\cos{\\left(\u03c6{\\left(t \\right)} \\right)} \\frac{d}{d t} \u03c6{\\left(t \\right)}\\\\\\left(R - r\\right) \\sin{\\left(\u03c6{\\left(t \\right)} \\right)} \\frac{d}{d t} \u03c6{\\left(t \\right)} + \\frac{d}{d t} y{\\left(t \\right)}\\\\0\\end{matrix}\\right] = \\left[\\begin{matrix}- r \u03c9_{1z} \\cos{\\left(\u03c6{\\left(t \\right)} \\right)}\\\\- r \u03c9_{1z} \\sin{\\left(\u03c6{\\left(t \\right)} \\right)} + \\frac{d}{d t} y{\\left(t \\right)}\\\\0\\end{matrix}\\right]$\n\n\n\n\n```python\n#\u00a0Fejezz\u00fck ki `\u03c9_1z` \u00e9s `\u03c6.diff(t)` k\u00f6z\u00f6tti kapcsolatot, felhaszn\u00e1lva\n#\u00a0(pl.) az x komponensek egyenl\u0151s\u00e9g\u00e9t:\n\ntmp = sp.Eq(v_OS_red.subs(vekt_koord)[0],v_BS_red.subs(vekt_koord)[0])\ndisplay(tmp)\n\n\u03c9_1z_expr = sp.solve(tmp, \u03c9_1z)[0]\n\ndisplay(Math('\u03c9_{{1z}} = {}'.format(sp.latex(\u03c9_1z_expr))))\n```\n\n\n```python\n# Ezt felhaszn\u00e1lva a B \u00e9s S pontok k\u00f6z\u00f6tti sebess\u00e9gred. k\u00e9pletb\u0151l:\nv_S_expr = v_BS_red.subs(vekt_koord).subs(\u03c9_1z,\u03c9_1z_expr)\ndisplay(Math('\\mathbf{{v}}_\\mathrm{{S}} = {}'.format(sp.latex(v_S_expr))))\n\n# Mostm\u00e1r fel\u00edrhat\u00f3 a kinetikus energia az \u00e1ltal\u00e1nos\u00edtott\n# koordin\u00e1t\u00e1k seg\u00edts\u00e9g\u00e9vel\n\n\u0398_1Sz = sp.Rational(1,2)*m1*r**2\nT = (sp.Rational(1,2)*m2 * q[0].diff(t)**2 \n + sp.Rational(1,2)*m1 * v_S_expr.dot(v_S_expr)\n + sp.Rational(1,2)*\u0398_1Sz * \u03c9_1z_expr**2)\n```\n\n\n$\\displaystyle \\mathbf{v}_\\mathrm{S} = \\left[\\begin{matrix}- \\left(- R + r\\right) \\cos{\\left(\u03c6{\\left(t \\right)} \\right)} \\frac{d}{d t} \u03c6{\\left(t \\right)}\\\\- \\left(- R + r\\right) \\sin{\\left(\u03c6{\\left(t \\right)} \\right)} \\frac{d}{d t} \u03c6{\\left(t \\right)} + \\frac{d}{d t} y{\\left(t \\right)}\\\\0\\end{matrix}\\right]$\n\n\n\n```python\n## A potenci\u00e1lis energia\n\n\"\"\"A potenci\u00e1lis energia a rug\u00f3kban felhalmoz\u00f3d\u00f3 potenci\u00e1lis energia\n\u00e9s a gravit\u00e1ci\u00f3s er\u0151 potenci\u00e1lis energi\u00e1j\u00e1b\u00f3l tev\u0151dik \u00f6ssze.\nMivel a rug\u00f3 el\u0151terhelt \u00e1llapotban van az egyens\u00falyi poz\u00edci\u00f3ban, ez\u00e9rt\nc\u00e9lszer\u0171 a potenci\u00e1lis energi\u00e1nak bevezetni egy \u00faj koordin\u00e1ta rendszert,\naminek a f\u00fcgg\u0151leges nullpontja ott van, ahol a rug\u00f3 hossza megegyegyzik\na terheletlen hossz\u00e1val. Ezzel a transzform\u00e1ci\u00f3val \naz \u00faj koordin\u00e1ta-rendszerben a nullszintt\u0151l val\u00f3 elt\u00e9r\u00e9st a z koordin\u00e1ta m\u00e9ri.\"\"\"\n\nz_st = sp.symbols(\"z_st\")\nz = q[0] - z_st\n\nke = k1 + k2 + k3\nU = sp.Rational(1,2)*ke*z**2 + m1*g*(z+r_OSy) + m2*g*z\n\n# Fejezz\u00fck ki a statikus deform\u00e1ci\u00f3t az egyens\u00falyi egyenletb\u0151l:\n\ne_egy = sp.Eq((m1 + m2)*g, ke*z_st) # azaz egyens\u00falyban `(m1 + m2)*g = ke*z_st`\nz_st_expr = sp.solve(e_egy,z_st)[0]\n\n# \u00cdrjuk vissza U-ba:\nU = U.subs(z_st,z_st_expr).subs(vekt_koord)\n```\n\n\n```python\n# Minden adott a Lagrange-egyenletben; \u00edrjuk fel a mozg\u00e1segyenleteket\n\neom_y = T.diff(q[0].diff(t)).diff(t) - T.diff(q[0]) + U.diff(q[0]) # `y`-hoz tartoz\u00f3\neom_\u03c6 = T.diff(q[1].diff(t)).diff(t) - T.diff(q[1]) + U.diff(q[1]) # `\u03c6`-hez tartoz\u00f3\neom_\u03c6 = eom_\u03c6.simplify()\neom_y = eom_y.simplify()\n```\n\n\n```python\ndisplay(sp.Eq(eom_y,0),sp.Eq(eom_\u03c6,0))\n```\n\n\n```python\n\"\"\"A lineariz\u00e1l\u00e1shoz \u00edrjuk fel az egyens\u00falyi helyzet k\u00f6r\u00fcli Taylor sorfejt\u00e9shez\ntart\u00f3z\u00f3 Taylor polinomot az els\u0151fok\u00fa tagig. Ehhez a `sympy` `sp.series` met\u00f3dusa\nhaszn\u00e1lhat\u00f3. Demonstr\u00e1ci\u00f3: lineariz\u00e1ljuk a `sin(a(t))` f\u00fcggv\u00e9nyt `a(t) = 0` k\u00f6r\u00fcl.\"\"\" \n\na = sp.Function('a')(t)\na_sym = sp.Symbol('a_sym')\nexpr = sp.sin(a)\n\n\"\"\" Jelen helyzetben `a` `t`-nek a f\u00fcggv\u00e9nye, \u00edgy a `sympy` nem szimb\u00f3lumk\u00e9nt kezeli.\nL\u00e9tre kell hozni egy \u00e1tmeneti szimb\u00f3lumot, amire kicser\u00e9lj\u00fck a f\u00fcggv\u00e9ny\u00fcnket. Ez csak\nformais\u00e1g. A `sp.series` az argumentum\u00e1ba el\u0151sz\u00f6r egy f\u00fcggv\u00e9nyt v\u00e1r (itt t\u00f6rt\u00e9nik az eml\u00edtett csere),\nmajd v\u00e1rja a f\u00fcggv\u00e9ny argumentum\u00e1t. Ezt k\u00f6vet\u0151en meg kell adni, hogy mely pont k\u00f6r\u00fcl fejt\u00fcnk \nsorba (itt a_sym = 0), valamint hogy hanyadrendig szeretn\u00e9nk megtenni a sorfejt\u00e9st.\"\"\"\n\na_lin_sym = sp.series(expr.subs(a,a_sym),a_sym,0,2)\n\n# Alternat\u00edv f\u00fcggv\u00e9ny h\u00edv\u00e1s:\nexpr.subs(a,a_sym).series(a_sym,0,2) # azaz k\u00f6zvetlen\u00fcl a `sympy` objektumra\n```\n\n\n```python\n\"\"\"A magasabb rend\u0171 tagokat a `.removeO()` met\u00f3dussal hagyhatjuk el (figyelem, nem .remove0()!),\nmajd cser\u00e9lj\u00fck vissza a szimb\u00f3lumot a f\u00fcggv\u00e9ny\u00fcnkre:\"\"\"\n\na_lin = a_lin_sym.removeO().subs(a_sym,a)\na_lin\n```\n\n\n```python\n\"\"\"Lineariz\u00e1ljuk a mozg\u00e1segyenletetek bal oldal\u00e1t (mint f\u00fcggv\u00e9nyeket), a \nq(t) = 0, \u00e9s q.diff(t) = 0 egyens\u00falyi helyzet k\u00f6r\u00fcl. Arra kell itt figyelni, hogy az \u00e1lt. koordin\u00e1t\u00e1k\nderiv\u00e1ltjait k\u00fcl\u00f6n v\u00e1ltoz\u00f3k\u00e9nt kell kezelni.\"\"\" \n\n# Cser\u00e9lj\u00fck ki a deriv\u00e1ltakat szimb\u00f3lumokra. Fontos, hogy a legmagasabb rend\u0171 deriv\u00e1lttal\n# kezdj\u00fcnk, mert ha pl. az els\u0151 deriv\u00e1lttal kezd\u00fcnk, a m\u00e1sodik deriv\u00e1ltat is kicser\u00e9li annak a deriv\u00e1ltj\u00e1ra.\n\u03c6, d\u03c6, dd\u03c6 = sp.symbols('\u03c6, d\u03c6, dd\u03c6')\n\u03c6_csere = [(q[1].diff(t,2),dd\u03c6),(q[1].diff(t),d\u03c6),(q[1],\u03c6)]\n\u03c6_csere_vissza = [(elem[1], elem[0]) for elem in \u03c6_csere] # hasznos lesz, ha a vissza kell cser\u00e9lni\n\neom1_csere = eom_y.subs(\u03c6_csere)\n\neom1_taylor = eom1_csere.series(d\u03c6,0,2).removeO().series(\u03c6,0,2).removeO() # egym\u00e1s ut\u00e1n t\u00f6bbsz\u00f6r is meg lehet h\u00edvni\neom1_taylor = eom1_taylor.subs(\u03c6_csere_vissza).simplify() # visszacsere\n\ndisplay(eom1_taylor)\n# Van m\u00e9g egy m\u00e1sodrendben kicsi tagunk, amit k\u00e9zzel kell null\u00e1zni. Ejts\u00fck ki azt a tagot \u00fagy, hogy\n# pl. \u03c6 hely\u00e9re 0-t helyettes\u00edt\u00fcnk. (Ha lenne m\u00e1shol is \u03c6 ez nem lenne j\u00f3 d\u00f6nt\u00e9s.)\n\neom1_lin = sp.Eq(eom1_taylor.subs(q[1],0).simplify().collect(q[0]),0) \neom1_lin # megk\u00fczd\u00f6tt\u00fcnk vele (:\n```\n\n\n```python\n# V\u00e9gezz\u00fcnk hasonl\u00f3t a m\u00e1sik mozg\u00e1segyenlet\u00fcnk eset\u00e9ben:\n\neom2_taylor = eom_\u03c6.subs(\u03c6_csere).series(\u03c6,0,2).removeO().subs(\u03c6_csere_vissza)\ndisplay(eom2_taylor)\n\n# M\u00e1sodrendben kicsi tagunk itt akkor lesz, ha `\u03c6`-t szorozzunk y''-tal.\n# K\u00e9zzel ezeket a tagokat jelen esetben pl. \u00fagy tudjuk elt\u00fcntetni, hogy\n# y'' hely\u00e9re 0-t helyettes\u00edt\u00fcnk:\neom2_lin = sp.Eq(eom2_taylor.subs(q[0].diff(t,2),0).collect(q[1]),0)\neom2_lin\n```\n\n### 2. Feladat\n\nAdjuk meg a mozg\u00e1segyenleteket m\u00e1trix egy\u00fctthat\u00f3s alakban, melynek alakja jelen esetben (disszipat\u00edv potenci\u00e1l, \u00e9s nem potenci\u00e1los akt\u00edv er\u0151k n\u00e9lk\u00fcl):\n\n$$\\mathbf{M}\\mathbf{\\ddot{q}} + \\mathbf{K}\\mathbf{q} = \\mathbf{0}.$$\n\n\n```python\n# Tegy\u00fck meg ezt k\u00e9tf\u00e9lek\u00e9ppen:\n\n# Els\u0151nek parci\u00e1lisan deriv\u00e1ljuk a kinetikus energi\u00e1t a 2 \u00e1lt. sebess\u00e9g szerint,\n# majd helyettes\u00edts\u00fcnk 0-t a hely\u00fckre.\n# Szintax: egym\u00e1sba \u00e1gyazott 2 db `list comprehension`. Olyan mint 2 egym\u00e1sba\n# \u00e1gyazott `for` ciklus. \nM_array = [[T.expand().diff(q1.diff()).diff(q2.diff()).simplify().subs([(q1,0),(q2,0)]) for q1 in q] for q2 in q]\n\nK_array = [[U.expand().diff(q1).diff(q2).simplify().subs([(q1,0),(q2,0)]) for q1 in q] for q2 in q]\n\nM = sp.Matrix(M_array)\nK = sp.Matrix(K_array)\n\ndisplay(Math('\\mathbf{{M}} = {}'.format(sp.latex(M))))\ndisplay(Math('\\mathbf{{K}} = {}'.format(sp.latex(K))))\n\n# R\u00e1n\u00e9zve erre a cell\u00e1ra, gondoljuk meg, mennyire egyszer\u0171 fel\u00edrni a m\u00e1trix egy\u00fctthat\u00f3s\n# mozg\u00e1segyenletet (2 \u00e9rdemi sor), ha ismert a kinetikus \u00e9s a potenci\u00e1lis energia f\u00fcggv\u00e9nye \n# (+ a disszipat\u00edv potenci\u00e1l, ha relev\u00e1ns), valamint ha nincs nem potenci\u00e1los akt\u00edv er\u0151 (homog\u00e9n eset). \n```\n\n\n$\\displaystyle \\mathbf{M} = \\left[\\begin{matrix}m_{1} + m_{2} & 0\\\\0 & \\frac{3 m_{1} \\left(- R + r\\right)^{2}}{2}\\end{matrix}\\right]$\n\n\n\n$\\displaystyle \\mathbf{K} = \\left[\\begin{matrix}k_{1} + k_{2} + k_{3} & 0\\\\0 & g m_{1} \\left(R - r\\right)\\end{matrix}\\right]$\n\n\n\n```python\n# M\u00e1sik m\u00f3dszer, a m\u00e1r lineariz\u00e1lt mozg\u00e1segyenletekb\u0151l (amit a Lagrange-egyenletekb\u0151l kaptunk)\n\n# f\u0171zz\u00fck \u0151ket \u00f6ssze egy vektorba:\neom_12 = sp.Matrix([[eom1_lin.lhs],[eom2_lin.lhs]]) # kell a `left hand side` mert egyenlet alakban vannak\nM2 = eom_12.jacobian(q.diff(t,2)) # Jacobi-m\u00e1trix az \u00e1ltal\u00e1nos gyorsul\u00e1sok szerint\nK2 = eom_12.jacobian(q)\n\ndisplay(Math('\\mathbf{{M}} = {}'.format(sp.latex(M2))))\ndisplay(Math('\\mathbf{{K}} = {}'.format(sp.latex(K2))))\n```\n\n\n$\\displaystyle \\mathbf{M} = \\left[\\begin{matrix}m_{1} + m_{2} & 0\\\\0 & \\frac{3 R^{2} m_{1}}{2} - 3 R m_{1} r + \\frac{3 m_{1} r^{2}}{2}\\end{matrix}\\right]$\n\n\n\n$\\displaystyle \\mathbf{K} = \\left[\\begin{matrix}k_{1} + k_{2} + k_{3} & 0\\\\0 & R g m_{1} - g m_{1} r\\end{matrix}\\right]$\n\n\n\n```python\ndisplay(sp.simplify(M-M2))\ndisplay(sp.simplify(K-K2)) # Ha nem b\u00edzn\u00e1nk magunkban (:\n```\n\n\n$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & 0\\end{matrix}\\right]$\n\n\n\n$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & 0\\end{matrix}\\right]$\n\n\nK\u00e9sz\u00edtette: \n\n Csuzdi Domonkos (Alkalmazott Mechanika Szakoszt\u00e1ly) \n Balogh Tam\u00e1s (BME MM) kidolgoz\u00e1sa alapj\u00e1n.\n\n Hib\u00e1k, javaslatok:\n amsz.bme@gmail.com\n csuzdi02@gmail.com\n\n 2021.04.04\n \n", "meta": {"hexsha": "a26b8c153b0ffddce625b88355fc312b72e1641c", "size": 129735, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "10_tizedik_het/gyak_10.ipynb", "max_stars_repo_name": "DomonkosCs/RezgestanPython", "max_stars_repo_head_hexsha": "eed1c6653fe9a4914155b2fdce773c44378d4190", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "10_tizedik_het/gyak_10.ipynb", "max_issues_repo_name": "DomonkosCs/RezgestanPython", "max_issues_repo_head_hexsha": "eed1c6653fe9a4914155b2fdce773c44378d4190", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 12, "max_issues_repo_issues_event_min_datetime": "2021-03-29T19:12:39.000Z", "max_issues_repo_issues_event_max_datetime": "2021-04-26T18:06:02.000Z", "max_forks_repo_path": "10_tizedik_het/gyak_10.ipynb", "max_forks_repo_name": "DomonkosCs/RezgestanPython", "max_forks_repo_head_hexsha": "eed1c6653fe9a4914155b2fdce773c44378d4190", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2021-03-29T19:29:08.000Z", "max_forks_repo_forks_event_max_datetime": "2021-04-10T20:58:06.000Z", "avg_line_length": 151.2062937063, "max_line_length": 9468, "alphanum_fraction": 0.7428296142, "converted": true, "num_tokens": 5780, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863698, "lm_q2_score": 0.6959583313396339, "lm_q1q2_score": 0.44835383011735186}} {"text": "\n# PHY321: Two-body scattering and Variational calculus\n\n \n**[Morten Hjorth-Jensen](http://mhjgit.github.io/info/doc/web/)**, Department of Physics and Astronomy and Facility for Rare Ion Beams (FRIB), Michigan State University, USA and Department of Physics, University of Oslo, Norway\n\nDate: **Apr 9, 2021**\n\nCopyright 1999-2021, [Morten Hjorth-Jensen](http://mhjgit.github.io/info/doc/web/). Released under CC Attribution-NonCommercial 4.0 license\n\n\n\n\n## Aims and Overarching Motivation\n\n\n### Monday\n\n* Two-body scattering and [Rutherford scattering](https://www.youtube.com/watch?v=dNp-vP17asI&ab_channel=TylerDeWitt).\n\n**Reading suggestion**: Taylor sections 14.1-14.6 and Lecture notes\n\n\n### Wednesday\n\n* Lagrangian formalism, top down approach first and derivation of the Euler-Lagrange equations\n\n* Principle of Least Action, watch [Feynman Lecture](https://www.feynmanlectures.caltech.edu/II_19.html).\n\n**Reading suggestion**: Taylor sections 6.1-6.3\n\n### Friday\n\n* [Variational Calculus](https://en.wikipedia.org/wiki/Calculus_of_variations)\n\n**Reading suggestion**: Taylor sections 6.3-6.4\n\n\n\n\n## Scattering and Cross Sections\n\nScattering experiments don't measure entire trajectories. For elastic\ncollisions, they measure the distribution of final scattering angles\nat best. Most experiments use targets thin enough so that the number\nof scatterings is typically zero or one. The cross section, $\\sigma$,\ndescribes the cross-sectional area for particles to scatter with an\nindividual target atom or nucleus. Cross section measurements form the\nbasis for MANY fields of physics. BThe cross section, and the\ndifferential cross section, encapsulates everything measurable for a\ncollision where all that is measured is the final state, e.g. the\noutgoing particle had momentum $\\boldsymbol{p}_f$. y studying cross sections,\none can infer information about the potential interaction between the\ntwo particles. Inferring, or constraining, the potential from the\ncross section is a classic {\\it inverse} problem. Collisions are\neither elastic or inelastic. Elastic collisions are those for which\nthe two bodies are in the same internal state before and after the\ncollision. If the collision excites one of the participants into a\nhigher state, or transforms the particles into different species, or\ncreates additional particles, the collision is inelastic. Here, we\nconsider only elastic collisions.\n\n## Scattering: Coulomb forces\n\nFor Coulomb forces, the cross section is infinite because the range of\nthe Coulomb force is infinite, but for interactions such as the strong\ninteraction in nuclear or particle physics, there is no long-range\nforce and cross-sections are finite. Even for Coulomb forces, the part\nof the cross section that corresponds to a specific scattering angle,\n$d\\sigma/d\\Omega$, which is a function of the scattering angle\n$\\theta_s$ is still finite.\n\nIf a particle travels through a thin target, the chance the particle\nscatters is $P_{\\rm scatt}=\\sigma dN/dA$, where $dN/dA$ is the number\nof scattering centers per area the particle encounters. If the density\nof the target is $\\rho$ particles per volume, and if the thickness of\nthe target is $t$, the areal density (number of target scatterers per\narea) is $dN/dA=\\rho t$. Because one wishes to quantify the collisions\nindependently of the target, experimentalists measure scattering\nprobabilities, then divide by the areal density to obtain\ncross-sections,\n\n$$\n\\begin{eqnarray}\n\\sigma=\\frac{P_{\\rm scatt}}{dN/dA}.\n\\end{eqnarray}\n$$\n\n## Scattering, more details\n\nInstead of merely stating that a particle collided, one can measure\nthe probability the particle scattered by a given angle. The\nscattering angle $\\theta_s$ is defined so that at zero the particle is\nunscattered and at $\\theta_s=\\pi$ the particle is scattered directly\nbackward. Scattering angles are often described in the center-of-mass\nframe, but that is a detail we will neglect for this first discussion,\nwhere we will consider the scattering of particles moving classically\nunder the influence of fixed potentials $U(\\boldsymbol{r})$. Because the\ndistribution of scattering angles can be measured, one expresses the\ndifferential cross section,\n\n\n
\n\n$$\n\\begin{equation}\n\\frac{d^2\\sigma}{d\\cos\\theta_s~d\\phi}.\n\\label{_auto1} \\tag{1}\n\\end{equation}\n$$\n\nUsually, the literature expresses differential cross sections as\n\n\n
\n\n$$\n\\begin{equation}\nd\\sigma/d\\Omega=\\frac{d\\sigma}{d\\cos\\theta d\\phi}=\\frac{1}{2\\pi}\\frac{d\\sigma}{d\\cos\\theta},\n\\label{_auto2} \\tag{2}\n\\end{equation}\n$$\n\nwhere the last equivalency is true when the scattering does not depend\non the azimuthal angle $\\phi$, as is the case for spherically\nsymmetric potentials.\n\nThe differential solid angle $d\\Omega$ can be thought of as the area\nsubtended by a measurement, $dA_d$, divided by $r^2$, where $r$ is the\ndistance to the detector,\n\n$$\n\\begin{eqnarray}\ndA_d=r^2 d\\Omega.\n\\end{eqnarray}\n$$\n\nWith this definition $d\\sigma/d\\Omega$ is independent of the distance\nfrom which one places the detector, or the size of the detector (as\nlong as it is small).\n\n\n## Differential scattering cross sections\n\nDifferential scattering cross sections are calculated by assuming a\nrandom distribution of impact parameters $b$. These represent the\ndistance in the $xy$ plane for particles moving in the $z$ direction\nrelative to the scattering center. An impact parameter $b=0$ refers to\nbeing aimed directly at the target's center. The impact parameter\ndescribes the transverse distance from the $z=0$ axis for the\ntrajectory when it is still far away from the scattering center and\nhas not yet passed it. The differential cross section can be expressed\nin terms of the impact parameter,\n\n\n
\n\n$$\n\\begin{equation}\nd\\sigma=2\\pi bdb,\n\\label{_auto3} \\tag{3}\n\\end{equation}\n$$\n\nwhich is the area of a thin ring of radius $b$ and thickness $db$. In\nclassical physics, one can calculate the trajectory given the incoming\nkinetic energy $E$ and the impact parameter if one knows the mass and\npotential.\n\n\n## More on Differential Cross Sections\n\nFrom the trajectory, one then finds the scattering angle\n$\\theta_s(b)$. The differential cross section is then\n\n\n
\n\n$$\n\\begin{equation}\n\\frac{d\\sigma}{d\\Omega}=\\frac{1}{2\\pi}\\frac{d\\sigma}{d\\cos\\theta_s}=b\\frac{db}{d\\cos\\theta_s}=\\frac{b}{(d/db)\\cos\\theta_s(b)}.\n\\label{_auto4} \\tag{4}\n\\end{equation}\n$$\n\nTypically, one would calculate $\\cos\\theta_s$ and $(d/db)\\cos\\theta_s$\nas functions of $b$. This is sufficient to plot the differential cross\nsection as a function of $\\theta_s$.\n\nThe total cross section is\n\n\n
\n\n$$\n\\begin{equation}\n\\sigma_{\\rm tot}=\\int d\\Omega\\frac{d\\sigma}{d\\Omega}=2\\pi\\int d\\cos\\theta_s~\\frac{d\\sigma}{d\\Omega}. \n\\label{_auto5} \\tag{5}\n\\end{equation}\n$$\n\nEven if the total cross section is infinite, e.g. Coulomb forces, one\ncan still have a finite differential cross section as we will see\nlater on.\n\n\n## Rutherford Scattering\n\nThis refers to the calculation of $d\\sigma/d\\Omega$ due to an inverse\nsquare force, $F_{12}=\\pm\\alpha/r^2$ for repulsive/attractive\ninteraction. Rutherford compared the scattering of $\\alpha$ particles\n($^4$He nuclei) off of a nucleus and found the scattering angle at\nwhich the formula began to fail. This corresponded to the impact\nparameter for which the trajectories would strike the nucleus. This\nprovided the first measure of the size of the atomic nucleus. At the\ntime, the distribution of the positive charge (the protons) was\nconsidered to be just as spread out amongst the atomic volume as the\nelectrons. After Rutherford's experiment, it was clear that the radius\nof the nucleus tended to be roughly 4 orders of magnitude smaller than\nthat of the atom, which is less than the size of a football relative\nto Spartan Stadium.\n\n\n## Rutherford Scattering, more details\n\n\n\nIn order to calculate differential cross section, we must find how the\nimpact parameter is related to the scattering angle. This requires\nanalysis of the trajectory. We consider our previous expression for\nthe trajectory where we derived the elliptic form for the trajectory,\nFor that case we considered an attractive\nforce with the particle's energy being negative, i.e. it was\nbound. However, the same form will work for positive energy, and\nrepulsive forces can be considered by simple flipping the sign of\n$\\alpha$. For positive energies, the trajectories will be hyperbolas,\nrather than ellipses, with the asymptotes of the trajectories\nrepresenting the directions of the incoming and outgoing\ntracks.\n\n## Rutherford Scattering, final trajectories\n\n\nWe have\n\n\n
\n\n$$\n\\begin{equation}\\label{eq:ruthtraj} \\tag{6}\nr=\\frac{1}{\\frac{m\\alpha}{L^2}+A\\cos\\theta}.\n\\end{equation}\n$$\n\nOnce $A$ is large enough, which will happen when the energy is\npositive, the denominator will become negative for a range of\n$\\theta$. This is because the scattered particle will never reach\ncertain angles. The asymptotic angles $\\theta'$ are those for which\nthe denominator goes to zero,\n\n\n
\n\n$$\n\\begin{equation}\n\\cos\\theta'=-\\frac{m\\alpha}{AL^2}.\n\\label{_auto6} \\tag{7}\n\\end{equation}\n$$\n\n## Rutherford Scattering, Closest Approach\n\n\n\nThe trajectory's point of closest approach is at $\\theta=0$ and the\ntwo angles $\\theta'$, which have this value of $\\cos\\theta'$, are the\nangles of the incoming and outgoing particles. From\nFig (**to come**), one can see that the scattering angle\n$\\theta_s$ is given by,\n\n\n
\n\n$$\n\\begin{eqnarray}\n\\label{eq:sthetover2} \\tag{8}\n2\\theta'-\\pi&=&\\theta_s,~~~\\theta'=\\frac{\\pi}{2}+\\frac{\\theta_s}{2},\\\\\n\\nonumber\n\\sin(\\theta_s/2)&=&-\\cos\\theta'\\\\\n\\nonumber\n&=&\\frac{m\\alpha}{AL^2}.\n\\end{eqnarray}\n$$\n\nNow that we have $\\theta_s$ in terms of $m,\\alpha,L$ and $A$, we wish\nto re-express $L$ and $A$ in terms of the impact parameter $b$ and the\nenergy $E$. This will set us up to calculate the differential cross\nsection, which requires knowing $db/d\\theta_s$. It is easy to write\nthe angular momentum as\n\n\n
\n\n$$\n\\begin{equation}\nL^2=p_0^2b^2=2mEb^2.\n\\label{_auto7} \\tag{9}\n\\end{equation}\n$$\n\n## Rutherford Scattering, getting there\n\n\nFinding $A$ is more complicated. To accomplish this we realize that\nthe point of closest approach occurs at $\\theta=0$, so from\nEq. ([6](#eq:ruthtraj))\n\n\n
\n\n$$\n\\begin{eqnarray}\n\\label{eq:rminofA} \\tag{10}\n\\frac{1}{r_{\\rm min}}&=&\\frac{m\\alpha}{L^2}+A,\\\\\n\\nonumber\nA&=&\\frac{1}{r_{\\rm min}}-\\frac{m\\alpha}{L^2}.\n\\end{eqnarray}\n$$\n\nNext, $r_{\\rm min}$ can be found in terms of the energy because at the\npoint of closest approach the kinetic energy is due purely to the\nmotion perpendicular to $\\hat{r}$ and\n\n\n
\n\n$$\n\\begin{equation}\nE=-\\frac{\\alpha}{r_{\\rm min}}+\\frac{L^2}{2mr_{\\rm min}^2}.\n\\label{_auto8} \\tag{11}\n\\end{equation}\n$$\n\n## Rutherford Scattering, More Manipulations\n\nOne can solve the quadratic equation for $1/r_{\\rm min}$,\n\n\n
\n\n$$\n\\begin{equation}\n\\frac{1}{r_{\\rm min}}=\\frac{m\\alpha}{L^2}+\\sqrt{(m\\alpha/L^2)^2+2mE/L^2}.\n\\label{_auto9} \\tag{12}\n\\end{equation}\n$$\n\nWe can plug the expression for $r_{\\rm min}$ into the expression for $A$, Eq. ([10](#eq:rminofA)),\n\n\n
\n\n$$\n\\begin{equation}\nA=\\sqrt{(m\\alpha/L^2)^2+2mE/L^2}=\\sqrt{(\\alpha^2/(4E^2b^4)+1/b^2}\n\\label{_auto10} \\tag{13}\n\\end{equation}\n$$\n\n## Rutherford Scattering, final expression\n\nFinally, we insert the expression for $A$ into that for the scattering angle, Eq. ([8](#eq:sthetover2)),\n\n\n
\n\n$$\n\\begin{eqnarray}\n\\label{eq:scattangle} \\tag{14}\n\\sin(\\theta_s/2)&=&\\frac{m\\alpha}{AL^2}\\\\\n\\nonumber\n&=&\\frac{a}{\\sqrt{a^2+b^2}}, ~~a\\equiv \\frac{\\alpha}{2E}\n\\end{eqnarray}\n$$\n\n## Rutherford Scattering, the Differential Cross Section\n\n\n\nThe differential cross section can now be found by differentiating the\nexpression for $\\theta_s$ with $b$,\n\n\n
\n\n$$\n\\begin{eqnarray}\n\\label{eq:rutherford} \\tag{15}\n\\frac{1}{2}\\cos(\\theta_s/2)d\\theta_s&=&\\frac{ab~db}{(a^2+b^2)^{3/2}}=\\frac{bdb}{a^2}\\sin^3(\\theta_s/2),\\\\\n\\nonumber\nd\\sigma&=&2\\pi bdb=\\frac{\\pi a^2}{\\sin^3(\\theta_s/2)}\\cos(\\theta_s/2)d\\theta_s\\\\\n\\nonumber\n&=&\\frac{\\pi a^2}{2\\sin^4(\\theta_s/2)}\\sin\\theta_s d\\theta_s\\\\\n\\nonumber\n\\frac{d\\sigma}{d\\cos\\theta_s}&=&\\frac{\\pi a^2}{2\\sin^4(\\theta_s/2)},\\\\\n\\nonumber\n\\frac{d\\sigma}{d\\Omega}&=&\\frac{a^2}{4\\sin^4(\\theta_s/2)}.\n\\end{eqnarray}\n$$\n\nwhere $a= \\alpha/2E$. This the Rutherford formula for the differential\ncross section. It diverges as $\\theta_s\\rightarrow 0$ because\nscatterings with arbitrarily large impact parameters still scatter to\narbitrarily small scattering angles. The expression for\n$d\\sigma/d\\Omega$ is the same whether the interaction is positive or\nnegative.\n\n\n## Rutherford Scattering, Example\n\n\nConsider a particle of mass $m$ and charge $z$ with kinetic energy $E$\n(Let it be the center-of-mass energy) incident on a heavy nucleus of\nmass $M$ and charge $Z$ and radius $R$. We want to find the angle at which the\nRutherford scattering formula breaks down.\n\nLet $\\alpha=Zze^2/(4\\pi\\epsilon_0)$. The scattering angle in Eq. ([14](#eq:scattangle)) is\n\n$$\n\\sin(\\theta_s/2)=\\frac{a}{\\sqrt{a^2+b^2}}, ~~a\\equiv \\frac{\\alpha}{2E}.\n$$\n\nThe impact parameter $b$ for which the point of closest approach\nequals $R$ can be found by using angular momentum conservation,\n\n$$\n\\begin{eqnarray*}\np_0b&=&b\\sqrt{2mE}=Rp_f=R\\sqrt{2m(E-\\alpha/R)},\\\\\nb&=&R\\frac{\\sqrt{2m(E-\\alpha/R)}}{\\sqrt{2mE}}\\\\\n&=&R\\sqrt{1-\\frac{\\alpha}{ER}}.\n\\end{eqnarray*}\n$$\n\n## Rutherford Scattering, Example, wrapping up\n\n\n\nPutting these together\n\n$$\n\\theta_s=2\\sin^{-1}\\left\\{\n\\frac{a}{\\sqrt{a^2+R^2(1-\\alpha/(RE))}}\n\\right\\},~~~a=\\frac{\\alpha}{2E}.\n$$\n\nIt was from this departure of the experimentally measured\n$d\\sigma/d\\Omega$ from the Rutherford formula that allowed Rutherford\nto infer the radius of the gold nucleus, $R$.\n\n\n\n## Variational Calculus\n\nThe calculus of variations involves \nproblems where the quantity to be minimized or maximized is an integral. \n\n\nThe usual minimization problem one faces involves taking a function\n${\\cal L}(x)$, then finding the single value $x$ for which ${\\cal L}$\nis either a maximum or minimum. In multivariate calculus one also\nlearns to solve problems where you minimize for multiple variables,\n${\\cal L}(x_1,x_2,\\cdots x_n)$, and finding the points $(x_1\\cdots\ny_n)$ in an $n$-dimensional space that maximize or minimize the\nfunction. Here, we consider what seems to be a much more ambitious\nproblem. Imagine you have a function ${\\cal L}(x(t),\\dot{x}(t),t)$,\nand you wish to find the extrema for an infinite number of values of\n$x$, i.e. $x$ at each point $t$. The function ${\\cal L}$ will not only\ndepend on $x$ at each point $t$, but also on the slope at each point,\nplus an additional dependence on $t$. Note we are NOT finding an\noptimum value of $t$, we are finding the set of optimum values of $x$\nat each point $t$, or equivalently, finding the function $x(t)$.\n\n\n## Variational Calculus, introducing the action\n\nOne treats the function $x(t)$ as being unknown while minimizing the action\n\n$$\nS=\\int_{t_1}^{t_2}dt~{\\cal L}(x(t),\\dot{x}(t),t).\n$$\n\nThus, we are minimizing $S$ with respect to an infinite number of\nvalues of $x(t_i)$ at points $t_i$. As an additional criteria, we will\nassume that $x(t_1)$ and $x(t_2)$ are fixed, and that that we will\nonly consider variations of $x$ between the boundaries. The dependence\non the derivative, $\\dot{x}=dx/dt$, is crucial because otherwise the\nsolution would involve simply finding the one value of $x$ that\nminimized ${\\cal L}$, and $x(t)$ would equal a constant if there were no\nexplicit $t$ dependence. Furthermore, $x$ wouldn't need to be\ncontinuous at the boundary.\n\n\n## Variational Calculus, general Action\n\n\nIn the general case we have an integral of the type\n\n$$\nS[q]= \\int_{t_1}^{t_2} {\\cal L}(q(t),\\dot{q}(t),t)dt,\n$$\n\nwhere $S$ is the quantity which is sought minimized or maximized. The\nproblem is that although ${\\cal L}$ is a function of the general variables\n$q(t),\\dot{q}(t),t$ (note our change of variables), the exact dependence of $q$ on $t$ is not known.\nThis means again that even though the integral has fixed limits $t_1$\nand $t_2$, the path of integration is not known. In our case the unknown\nquantities are the positions and general velocities of a given number\nof objects and we wish to choose an integration path which makes the\nfunctional $S[q]$ stationary. This means that we want to find minima,\nor maxima or saddle points. In physics we search normally for minima.\nOur task is therefore to find the minimum of $S[q]$ so that its\nvariation $\\delta S$ is zero subject to specific constraints. The\nconstraints can be treated via the technique of Lagrangian multipliers\nas we will see below.\n\n\n## Variational Calculus, Optimal Path\n\n\nWe assume the existence of an optimum path, that is a path for which\n$S[q]$ is stationary. There are infinitely many such paths. The\ndifference between two paths $\\delta q$ is called the variation of\n$q$.\n\nWe call the variation $\\eta(t)$ and it is scaled by a factor $\\alpha$.\nThe function $\\eta(t)$ is arbitrary except for\n\n$$\n\\eta(t_1)=\\eta(t_2)=0,\n$$\n\nand we assume that we can model the change in $q$ as\n\n$$\nq(t,\\alpha) = q(t)+\\alpha\\eta(t),\n$$\n\nand\n\n$$\n\\delta q = q(t,\\alpha) -q(t,0)=\\alpha\\eta(t).\n$$\n\n## Variational Calculus, Condition for an Extreme Value\n\n\nWe choose $q(t,\\alpha=0)$ as the unkonwn path that will minimize $S$. The value\n$q(t,\\alpha\\ne 0)$ describes a neighbouring path.\n\nWe have\n\n$$\nS[q(\\alpha)]= \\int_{t_1}^{t_2} {\\cal L}(q(t,\\alpha),\\dot{q}(t,\\alpha),t)dt.\n$$\n\nThe condition for an extreme of\n\n$$\nS[q(\\alpha)]= \\int_{t_1}^{t_2} {\\cal L}(q(t,\\alpha),\\dot{q}(t,\\alpha),t)dt,\n$$\n\nis\n\n$$\n\\left[\\frac{\\partial S[q(\\alpha)]}{\\partial t}\\right]_{\\alpha=0} =0.\n$$\n\n## Variational Calculus. $\\alpha$ Dependence\n\n\nThe $\\alpha$ dependence is contained in $q(t,\\alpha)$ and $\\dot{q}(t,\\alpha)$ meaning that\n\n$$\n\\left[\\frac{\\partial E[q(\\alpha)]}{\\partial \\alpha}\\right]=\\int_{t_1}^{t_2} \\left( \\frac{\\partial {\\cal l}}{\\partial q}\\frac{\\partial q}{\\partial \\alpha}+\\frac{\\partial {\\cal L}}{\\partial \\dot{q}}\\frac{\\partial \\dot{q}}{\\partial \\alpha}\\right)dt.\n$$\n\nWe have defined\n\n$$\n\\frac{\\partial q(x,\\alpha)}{\\partial \\alpha}=\\eta(x)\n$$\n\nand thereby\n\n$$\n\\frac{\\partial \\dot{q}(t,\\alpha)}{\\partial \\alpha}=\\frac{d(\\eta(t))}{dt}.\n$$\n\n## INtegrating by Parts\n\nUsing\n\n$$\n\\frac{\\partial q(t,\\alpha)}{\\partial \\alpha}=\\eta(t),\n$$\n\nand\n\n$$\n\\frac{\\partial \\dot{q}(t,\\alpha)}{\\partial \\alpha}=\\frac{d(\\eta(t))}{dt},\n$$\n\nin the integral gives\n\n$$\n\\left[\\frac{\\partial S[q(\\alpha)]}{\\partial \\alpha}\\right]=\\int_{t_1}^{t_2} \\left( \\frac{\\partial {\\cal L}}{\\partial q}\\eta(t)+\\frac{\\partial {\\cal L}}{\\partial \\dot{q}}\\frac{d(\\eta(t))}{dt}\\right)dt.\n$$\n\nIntegrating the second term by parts\n\n$$\n\\int_{t_1}^{t_2} \\frac{\\partial {\\cal L}}{\\partial \\dot{q}}\\frac{d(\\eta(t))}{dt}dt =\\eta(t)\\frac{\\partial {\\cal L}}{\\partial \\dot{q}}|_{t_1}^{t_2}-\n\\int_a^b \\eta(t)\\frac{d}{dx}\\frac{\\partial {\\cal L}}{\\partial \\dot{q}}dt,\n$$\n\nand since the first term dissappears due to $\\eta(a)=\\eta(b)=0$, we obtain\n\n$$\n\\left[\\frac{\\partial S[q(\\alpha)]}{\\partial \\alpha}\\right]=\\int_{t_1}^{t_2} \\left( \\frac{\\partial {\\cal L}}{\\partial q}-\\frac{d}{dx}\\frac{\\partial {\\cal L}}{\\partial \\dot{q}}\n\\right)\\eta(t)dt=0.\n$$\n\n## Euler-Lagrange Equations\n\n\nThe latter can be written as\n\n$$\n\\left[\\frac{\\partial S[q(\\alpha)]}{\\partial \\alpha}\\right]_{\\alpha=0}=\\int_{t_1}^{t_2} \\left( \\frac{\\partial {\\cal L}}{\\partial q}-\\frac{d}{\\\ndx}\\frac{\\partial {\\cal L}}{\\partial \\dot{q}}\\right)\\delta q(t)dt=\\delta S = 0.\n$$\n\nThe condition for a stationary value is thus a partial differential equation\n\n$$\n\\frac{\\partial {\\cal L}}{\\partial q}-\\frac{d}{dx}\\frac{\\partial {\\cal L}}{\\partial \\dot{q}}=0,\n$$\n\nknown as the **Euler-Lagrange** equation.\n", "meta": {"hexsha": "dd23034006f36579ffceecfc12cf54694be19400", "size": 32215, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "doc/pub/week14/ipynb/week14.ipynb", "max_stars_repo_name": "Shield94/Physics321", "max_stars_repo_head_hexsha": "9875a3bf840b0fa164b865a3cb13073aff9094ca", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 20, "max_stars_repo_stars_event_min_datetime": "2020-01-09T17:41:16.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-09T00:48:58.000Z", "max_issues_repo_path": "doc/pub/week14/ipynb/week14.ipynb", "max_issues_repo_name": "Shield94/Physics321", "max_issues_repo_head_hexsha": "9875a3bf840b0fa164b865a3cb13073aff9094ca", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2020-01-08T03:47:53.000Z", "max_issues_repo_issues_event_max_datetime": "2020-12-15T15:02:57.000Z", "max_forks_repo_path": "doc/pub/week14/ipynb/week14.ipynb", "max_forks_repo_name": "Shield94/Physics321", "max_forks_repo_head_hexsha": "9875a3bf840b0fa164b865a3cb13073aff9094ca", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 33, "max_forks_repo_forks_event_min_datetime": "2020-01-10T20:40:55.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-11T20:28:41.000Z", "avg_line_length": 30.886864813, "max_line_length": 284, "alphanum_fraction": 0.5590563402, "converted": true, "num_tokens": 6086, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.8376199572530448, "lm_q1q2_score": 0.4482091229123938}} {"text": "# Deep Learning Bootcamp November 2017, GPU Computing for Data Scientists\n\n\n\n## 18 PyTorch NUMER.AI Deep Learning Binary Classification using BCELoss \n\nWeb: https://www.meetup.com/Tel-Aviv-Deep-Learning-Bootcamp/events/241762893/\n\nNotebooks: On GitHub\n\n*Shlomo Kashani*\n\n\n\n# Introduction\n\n- This tutorial was written in order to demonstrate a **fully working** example of a PyTorch NN on a real world use case, namely a Binary Classification problem. If you are interested in the sk-learn version of this problem please refer to: https://github.com/QuantScientist/deep-ml-meetups/tree/master/hacking-kaggle/python/numer-ai \n\n- For the scientific foundation behind Binary Classification and Logistic Regression, refer to: https://github.com/QuantScientist/deep-ml-meetups/blob/master/data-science-interviews/imp-ans.pdf\n\n- Every step, from reading the CSV into numpy arrays, converting to GPU based tensors, training and validation, are meant to aid newcomers in their first steps in PyTorch. \n\n- Additionally, commonly used Kaggle metrics such as ROC_AUC and LOG_LOSS are logged and plotted both for the training set as well as for the validation set. \n\n- Thus, the NN architecture is naive and by any means **unoptimized**. Hopefully, I will improve it over time and I am working on a second CNN based version of the same problem. \n\n\n## Data\n- Download from https://numer.ai/leaderboard\n\n\n\n\n# PyTorch Imports\n\n\n\n```python\n# !pip install pycuda\n%reset -f\n# %%timeit\n\nimport torch\nfrom torch.autograd import Variable\nimport numpy as np\nimport pandas\nimport numpy as np\nimport pandas as pd\nfrom sklearn import cross_validation\nfrom sklearn import metrics\nfrom sklearn.metrics import roc_auc_score, log_loss, roc_auc_score, roc_curve, auc\nimport matplotlib.pyplot as plt\nfrom sklearn import cross_validation\nfrom sklearn import metrics\nfrom sklearn.metrics import roc_auc_score, log_loss, roc_auc_score, roc_curve, auc\nfrom sklearn.cross_validation import StratifiedKFold, ShuffleSplit, cross_val_score, train_test_split\nimport logging\nimport numpy\nimport numpy as np\nfrom __future__ import print_function\nfrom __future__ import division\nimport math\nimport numpy as np\nimport matplotlib.pyplot as plt\n%matplotlib inline\nimport pandas as pd\nimport os\nimport torch\nfrom torch.utils.data.dataset import Dataset\nfrom torch.utils.data import DataLoader\nfrom torchvision import transforms\nfrom torch import nn\nimport torch.nn.functional as F\nimport torch.optim as optim\nfrom torch.autograd import Variable\nfrom sklearn.preprocessing import MultiLabelBinarizer\nimport time\nfrom sklearn.preprocessing import PolynomialFeatures\nimport pandas as pd\nimport numpy as np\nimport scipy\n%matplotlib inline\nfrom pylab import rcParams\nrcParams['figure.figsize'] = (6, 6) # setting default size of plots\nimport tensorflow as tf \nprint(\"tensorflow:\" + tf.__version__)\n!set \"KERAS_BACKEND=tensorflow\"\nimport torch\nimport sys\nprint('__Python VERSION:', sys.version)\nprint('__pyTorch VERSION:', torch.__version__)\nprint('__CUDA VERSION')\nfrom subprocess import call\nprint('__CUDNN VERSION:', torch.backends.cudnn.version())\nprint('__Number CUDA Devices:', torch.cuda.device_count())\nprint('__Devices')\n\n# !pip install http://download.pytorch.org/whl/cu75/torch-0.2.0.post1-cp27-cp27mu-manylinux1_x86_64.whl\n# !pip install torchvision \n# ! pip install cv2\n# import cv2\n\nprint(\"OS: \", sys.platform)\nprint(\"Python: \", sys.version)\nprint(\"PyTorch: \", torch.__version__)\nprint(\"Numpy: \", np.__version__)\n\nhandler=logging.basicConfig(level=logging.INFO)\nlgr = logging.getLogger(__name__)\n%matplotlib inline\n\n# !pip install psutil\nimport psutil\ndef cpuStats():\n print(sys.version)\n print(psutil.cpu_percent())\n print(psutil.virtual_memory()) # physical memory usage\n pid = os.getpid()\n py = psutil.Process(pid)\n memoryUse = py.memory_info()[0] / 2. ** 30 # memory use in GB...I think\n print('memory GB:', memoryUse)\n\ncpuStats()\n```\n\n /usr/local/lib/python2.7/dist-packages/sklearn/cross_validation.py:44: DeprecationWarning: This module was deprecated in version 0.18 in favor of the model_selection module into which all the refactored classes and functions are moved. Also note that the interface of the new CV iterators are different from that of this module. This module will be removed in 0.20.\n \"This module will be removed in 0.20.\", DeprecationWarning)\n\n\n tensorflow:1.2.1\n __Python VERSION: 2.7.12 (default, Nov 19 2016, 06:48:10) \n [GCC 5.4.0 20160609]\n __pyTorch VERSION: 0.2.0+42448cf\n __CUDA VERSION\n __CUDNN VERSION: None\n __Number CUDA Devices: 1\n __Devices\n OS: linux2\n Python: 2.7.12 (default, Nov 19 2016, 06:48:10) \n [GCC 5.4.0 20160609]\n PyTorch: 0.2.0+42448cf\n Numpy: 1.13.1\n 2.7.12 (default, Nov 19 2016, 06:48:10) \n [GCC 5.4.0 20160609]\n 0.0\n svmem(total=67469099008, available=63134949376, percent=6.4, used=3730501632, free=61150445568, active=4437622784, inactive=1223413760, buffers=284803072, cached=2303348736, shared=55218176)\n memory GB: 0.221778869629\n\n\n# CUDA\n\n\n```python\n# %%timeit\nuse_cuda = torch.cuda.is_available()\n# use_cuda = False\n\nFloatTensor = torch.cuda.FloatTensor if use_cuda else torch.FloatTensor\nLongTensor = torch.cuda.LongTensor if use_cuda else torch.LongTensor\nTensor = FloatTensor\n\nlgr.info(\"USE CUDA=\" + str (use_cuda))\n\n# ! watch -n 0.1 'ps f -o user,pgrp,pid,pcpu,pmem,start,time,command -p `lsof -n -w -t /dev/nvidia*`'\n# sudo apt-get install dstat #install dstat\n# sudo pip install nvidia-ml-py #install Python NVIDIA Management Library\n# wget https://raw.githubusercontent.com/datumbox/dstat/master/plugins/dstat_nvidia_gpu.py\n# sudo mv dstat_nvidia_gpu.py /usr/share/dstat/ #move file to the plugins directory of dstat\n```\n\n INFO:__main__:USE CUDA=True\n\n\n# Global params\n\n\n```python\n# NN params\nDROPOUT_PROB = 0.75\nN_EPOCHS = 50\nBATCH_SIZE = 4\nLR = 0.005\nMOMENTUM= 0.9\nPIN_MEMORY=use_cuda # True IF CUDA\n\n# Data params\nTARGET_VAR= 'target'\nTOURNAMENT_DATA_CSV = 'numerai_tournament_data.csv'\nTRAINING_DATA_CSV = 'numerai_training_data.csv'\nBASE_FOLDER = 'numerai/'\n\n# fix seed\nseed=17*19\nnp.random.seed(seed)\ntorch.manual_seed(seed)\nif use_cuda:\n torch.cuda.manual_seed(seed)\n```\n\n# Load a CSV file for Binary classification (numpy)\n\n\n```python\n# %%timeit\ndf_train = pd.read_csv(BASE_FOLDER + TRAINING_DATA_CSV)\ndf_train.head(5)\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
ideradata_typefeature1feature2feature3feature4feature5feature6feature7...feature13feature14feature15feature16feature17feature18feature19feature20feature21target
0135682era1train0.533520.643360.465770.530010.557340.457730.41169...0.512240.504840.419290.509540.473830.487970.383730.462330.333410
1110546era1train0.541960.815760.466320.623200.524270.643780.55662...0.526430.638090.671210.494210.452910.469320.544450.309970.190230
276047era1train0.491580.691310.578160.540100.430640.499860.61902...0.433100.722860.762570.366000.553300.565660.675280.349600.257211
366098era1train0.545190.424730.634720.390030.374850.438100.59557...0.416580.634170.501890.408830.587050.637850.562250.559890.586420
488227era1train0.443070.740760.522100.565430.511250.664570.42263...0.458510.588050.498600.480230.526060.532530.383610.438290.250140
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\n
\n\n\n\n# Feature enrichement\n- This would be usually not required when using NN's; it is here for demonstration purposes. \n\n\n```python\ndef genBasicFeatures(inDF):\n print('Generating basic features ...')\n df_copy=inDF.copy(deep=True)\n magicNumber=21\n feature_cols = list(inDF.columns)\n\n inDF['x_mean'] = np.mean(df_copy.ix[:, 0:magicNumber], axis=1)\n inDF['x_median'] = np.median(df_copy.ix[:, 0:magicNumber], axis=1)\n inDF['x_std'] = np.std(df_copy.ix[:, 0:magicNumber], axis=1)\n inDF['x_skew'] = scipy.stats.skew(df_copy.ix[:, 0:magicNumber], axis=1)\n inDF['x_kurt'] = scipy.stats.kurtosis(df_copy.ix[:, 0:magicNumber], axis=1)\n inDF['x_var'] = np.var(df_copy.ix[:, 0:magicNumber], axis=1)\n inDF['x_max'] = np.max(df_copy.ix[:, 0:magicNumber], axis=1)\n inDF['x_min'] = np.min(df_copy.ix[:, 0:magicNumber], axis=1) \n\n return inDF\n\ndef addPolyFeatures(inDF, deg=2):\n print('Generating poly features ...')\n df_copy=inDF.copy(deep=True)\n poly=PolynomialFeatures(degree=deg)\n p_testX = poly.fit(df_copy)\n # AttributeError: 'PolynomialFeatures' object has no attribute 'get_feature_names'\n target_feature_names = ['x'.join(['{}^{}'.format(pair[0],pair[1]) for pair in tuple if pair[1]!=0]) for tuple in [zip(df_copy.columns,p) for p in poly.powers_]]\n df_copy = pd.DataFrame(p_testX.transform(df_copy),columns=target_feature_names)\n \n return df_copy\n```\n\n# Train / Validation / Test Split\n- Numerai provides a data set that is allready split into train, validation and test sets. \n\n\n```python\n# Train, Validation, Test Split\ndef loadDataSplit():\n df_train = pd.read_csv(BASE_FOLDER + TRAINING_DATA_CSV)\n # TOURNAMENT_DATA_CSV has both validation and test data provided by NumerAI\n df_test_valid = pd.read_csv(BASE_FOLDER + TOURNAMENT_DATA_CSV)\n\n answers_1_SINGLE = df_train[TARGET_VAR]\n df_train.drop(TARGET_VAR, axis=1,inplace=True)\n df_train.drop('id', axis=1,inplace=True)\n df_train.drop('era', axis=1,inplace=True)\n df_train.drop('data_type', axis=1,inplace=True) \n \n # Add polynomial features \n df_train=genBasicFeatures(df_train)\n# df_train = addPolyFeatures(df_train)\n\n df_train.to_csv(BASE_FOLDER + TRAINING_DATA_CSV + 'clean.csv', header=False, index = False) \n df_train= pd.read_csv(BASE_FOLDER + TRAINING_DATA_CSV + 'clean.csv', header=None, dtype=np.float32) \n df_train = pd.concat([df_train, answers_1_SINGLE], axis=1)\n feature_cols = list(df_train.columns[:-1])\n# print (feature_cols)\n target_col = df_train.columns[-1]\n trainX, trainY = df_train[feature_cols], df_train[target_col]\n \n \n # TOURNAMENT_DATA_CSV has both validation and test data provided by NumerAI\n # Validation set\n df_validation_set=df_test_valid.loc[df_test_valid['data_type'] == 'validation'] \n df_validation_set=df_validation_set.copy(deep=True)\n answers_1_SINGLE_validation = df_validation_set[TARGET_VAR]\n df_validation_set.drop(TARGET_VAR, axis=1,inplace=True) \n df_validation_set.drop('id', axis=1,inplace=True)\n df_validation_set.drop('era', axis=1,inplace=True)\n df_validation_set.drop('data_type', axis=1,inplace=True)\n \n # Add polynomial features \n df_validation_set=genBasicFeatures(df_validation_set)\n# df_validation_set = addPolyFeatures(df_validation_set)\n \n df_validation_set.to_csv(BASE_FOLDER + TRAINING_DATA_CSV + '-validation-clean.csv', header=False, index = False) \n df_validation_set= pd.read_csv(BASE_FOLDER + TRAINING_DATA_CSV + '-validation-clean.csv', header=None, dtype=np.float32) \n df_validation_set = pd.concat([df_validation_set, answers_1_SINGLE_validation], axis=1)\n feature_cols = list(df_validation_set.columns[:-1])\n\n target_col = df_validation_set.columns[-1]\n valX, valY = df_validation_set[feature_cols], df_validation_set[target_col]\n \n # Test set for submission (not labeled) \n df_test_set = pd.read_csv(BASE_FOLDER + TOURNAMENT_DATA_CSV)\n# df_test_set=df_test_set.loc[df_test_valid['data_type'] == 'live'] \n df_test_set=df_test_set.copy(deep=True)\n df_test_set.drop(TARGET_VAR, axis=1,inplace=True)\n tid_1_SINGLE = df_test_set['id']\n df_test_set.drop('id', axis=1,inplace=True)\n df_test_set.drop('era', axis=1,inplace=True)\n df_test_set.drop('data_type', axis=1,inplace=True) \n \n # Add polynomial features \n df_test_set=genBasicFeatures(df_test_set)\n# df_test_set = addPolyFeatures(df_test_set)\n \n \n feature_cols = list(df_test_set.columns) # must be run here, we dont want the ID \n# print (feature_cols)\n df_test_set = pd.concat([tid_1_SINGLE, df_test_set], axis=1) \n testX = df_test_set[feature_cols].values\n \n return trainX, trainY, valX, valY, testX, df_test_set\n```\n\n\n```python\n# %%timeit\ntrainX, trainY, valX, valY, testX, df_test_set = loadDataSplit()\n# X, y = loadDataSplit(999)\n\n\n# # Number of features for the input layer\nN_FEATURES=trainX.shape[1]\n# print (trainX.head(3))\n# print (df_test_set.head(3))\nprint (trainX.shape)\nprint (trainY.shape)\nprint (valX.shape)\nprint (valY.shape)\nprint (testX.shape)\nprint (df_test_set.shape)\n```\n\n Generating basic features ...\n\n\n /usr/local/lib/python2.7/dist-packages/ipykernel_launcher.py:7: DeprecationWarning: \n .ix is deprecated. Please use\n .loc for label based indexing or\n .iloc for positional indexing\n \n See the documentation here:\n http://pandas.pydata.org/pandas-docs/stable/indexing.html#ix-indexer-is-deprecated\n import sys\n\n\n Generating basic features ...\n Generating basic features ...\n (108405, 29)\n (108405,)\n (16686, 29)\n (16686,)\n (45647, 29)\n (45647, 30)\n\n\n# Create PyTorch GPU tensors from numpy arrays\n\n- Note how we transfrom the np arrays\n\n\n```python\n# Convert the np arrays into the correct dimention and type\n# Note that BCEloss requires Float in X as well as in y\ndef XnumpyToTensor(x_data_np):\n x_data_np = np.array(x_data_np.values, dtype=np.float32) \n print(x_data_np.shape)\n print(type(x_data_np))\n\n if use_cuda:\n lgr.info (\"Using the GPU\") \n X_tensor = Variable(torch.from_numpy(x_data_np).cuda()) # Note the conversion for pytorch \n else:\n lgr.info (\"Using the CPU\")\n X_tensor = Variable(torch.from_numpy(x_data_np)) # Note the conversion for pytorch\n \n print(type(X_tensor.data)) # should be 'torch.cuda.FloatTensor'\n print(x_data_np.shape)\n print(type(x_data_np)) \n return X_tensor\n\n\n# Convert the np arrays into the correct dimention and type\n# Note that BCEloss requires Float in X as well as in y\ndef YnumpyToTensor(y_data_np): \n y_data_np=y_data_np.reshape((y_data_np.shape[0],1)) # Must be reshaped for PyTorch!\n print(y_data_np.shape)\n print(type(y_data_np))\n\n if use_cuda:\n lgr.info (\"Using the GPU\") \n # Y = Variable(torch.from_numpy(y_data_np).type(torch.LongTensor).cuda())\n Y_tensor = Variable(torch.from_numpy(y_data_np)).type(torch.FloatTensor).cuda() # BCEloss requires Float \n else:\n lgr.info (\"Using the CPU\") \n # Y = Variable(torch.squeeze (torch.from_numpy(y_data_np).type(torch.LongTensor))) # \n Y_tensor = Variable(torch.from_numpy(y_data_np)).type(torch.FloatTensor) # BCEloss requires Float \n\n print(type(Y_tensor.data)) # should be 'torch.cuda.FloatTensor'\n print(y_data_np.shape)\n print(type(y_data_np)) \n return Y_tensor\n```\n\n# The NN model\n\n### MLP model\n- A multilayer perceptron is a logistic regressor where instead of feeding the input to the logistic regression you insert a intermediate layer, called the hidden layer, that has a nonlinear activation function (usually tanh or sigmoid) . One can use many such hidden layers making the architecture deep.\n\n- Here we define a simple MLP structure. We map the input feature vector to a higher space, then later gradually decrease the dimension, and in the end into a 1-dimension space. Because we are calculating the probability of each genre independently, after the final layer we need to use a sigmoid layer. \n\n### Initial weights selection\n\n- There are many ways to select the initial weights to a neural network architecture. A common initialization scheme is random initialization, which sets the biases and weights of all the nodes in each hidden layer randomly.\n\n- Before starting the training process, an initial value is assigned to each variable. This is done by pure randomness, using for example a uniform or Gaussian distribution. But if we start with weights that are too small, the signal could decrease so much that it is too small to be useful. On the other side, when the parameters are initialized with high values, the signal can end up to explode while propagating through the network.\n\n- In consequence, a good initialization can have a radical effect on how fast the network will learn useful patterns.For this purpose, some best practices have been developed. One famous example used is **Xavier initialization**. Its formulation is based on the number of input and output neurons and uses sampling from a uniform distribution with zero mean and all biases set to zero.\n\n- In effect (according to theory) initializing the weights of the network to values that would be closer to the optimal, and therefore require less epochs to train.\n\n### References: \n* **`nninit.xavier_uniform(tensor, gain=1)`** - Fills `tensor` with values according to the method described in [\"Understanding the difficulty of training deep feedforward neural networks\" - Glorot, X. and Bengio, Y.](http://jmlr.org/proceedings/papers/v9/glorot10a/glorot10a.pdf), using a uniform distribution.\n* **`nninit.xavier_normal(tensor, gain=1)`** - Fills `tensor` with values according to the method described in [\"Understanding the difficulty of training deep feedforward neural networks\" - Glorot, X. and Bengio, Y.](http://jmlr.org/proceedings/papers/v9/glorot10a/glorot10a.pdf), using a normal distribution.\n* **`nninit.kaiming_uniform(tensor, gain=1)`** - Fills `tensor` with values according to the method described in [\"Delving deep into rectifiers: Surpassing human-level performance on ImageNet classification\" - He, K. et al.](https://arxiv.org/abs/1502.01852) using a uniform distribution.\n* **`nninit.kaiming_normal(tensor, gain=1)`** - Fills `tensor` with values according to the method described in [\"Delving deep into rectifiers: Surpassing human-level performance on ImageNet classification\" - He, K. et al.]\n\n\n\n```python\n# p is the probability of being dropped in PyTorch\n# At each layer, DECREASE dropout\ndropout = torch.nn.Dropout(p=1 - (DROPOUT_PROB +0.20))\n\n# class Net(torch.nn.Module):\n# def __init__(self, n_feature, n_hidden, n_output,initKernel='uniform'):\n# super(Net, self).__init__()\n# self.hidden = torch.nn.Linear(n_feature, n_hidden) # hidden layer\n# self.out = torch.nn.Linear(n_hidden, n_output) # output layer \n \n# # xavier initializer\n# if initKernel == 'uniform':\n# nn.init.xavier_uniform(self.hidden.weight, gain=np.sqrt(2.0))\n# else:\n# nn.init.kaiming_normal(self.hidden.weight) \n\n# def forward(self, x):\n# x = F.relu(self.hidden(x)) # activation function for hidden layer\n# x = self.out(x)\n# return F.sigmoid(x)\n\nclass Net2(nn.Module):\n def __init__(self, n_feature, n_hidden, n_output,initKernel='uniform'):\n super(Net2, self).__init__()\n self.dis = nn.Sequential(\n nn.Linear(n_feature, n_hidden),\n dropout,\n nn.LeakyReLU(0.1),\n nn.Linear(n_hidden, n_hidden),\n dropout,\n nn.LeakyReLU(0.1),\n nn.Linear(n_hidden, 1),\n dropout,\n nn.Sigmoid()\n ) \n def forward(self, x):\n x = self.dis(x)\n return x\n\n\nhiddenLayer1Size=1024\nhiddenLayer2Size=int(hiddenLayer1Size/8)\nhiddenLayer3Size=int(hiddenLayer1Size/16)\nhiddenLayer4Size=int(hiddenLayer1Size/32)\nhiddenLayer5Size=int(hiddenLayer1Size/64)\n\n# # Hypothesis using sigmoid\nlinear1=torch.nn.Linear(N_FEATURES, hiddenLayer1Size, bias=True) \ntorch.nn.init.xavier_uniform(linear1.weight)\n\nlinear2=torch.nn.Linear(hiddenLayer1Size, hiddenLayer2Size)\ntorch.nn.init.xavier_uniform(linear2.weight)\n\nlinear3=torch.nn.Linear(hiddenLayer2Size, hiddenLayer3Size)\ntorch.nn.init.xavier_uniform(linear3.weight)\n\nlinear4=torch.nn.Linear(hiddenLayer3Size, hiddenLayer4Size)\ntorch.nn.init.xavier_uniform(linear4.weight)\n\nlinear5=torch.nn.Linear(hiddenLayer4Size, hiddenLayer5Size)\ntorch.nn.init.xavier_uniform(linear5.weight)\n\nlinear6=torch.nn.Linear(hiddenLayer5Size, 1)\ntorch.nn.init.xavier_uniform(linear6.weight)\n\nsigmoid = torch.nn.Sigmoid()\ntanh=torch.nn.Tanh()\nrelu=torch.nn.LeakyReLU()\n\nnet = torch.nn.Sequential(linear1,dropout,tanh,nn.BatchNorm1d(hiddenLayer1Size),\n linear2,dropout,tanh,\n linear3,dropout,relu,\n linear4,dropout,tanh,\n linear5,dropout,relu,\n linear6,sigmoid\n )\n\n# net = Net(n_feature=N_FEATURES, n_hidden=1024, n_output=1) # define the network\n# net = Net2(n_feature=N_FEATURES, n_hidden=2048, n_output=1) # define the network\n\nlgr.info(net) # net architecture\n```\n\n INFO:__main__:Sequential (\n (0): Linear (29 -> 1024)\n (1): Dropout (p = 0.05)\n (2): Tanh ()\n (3): BatchNorm1d(1024, eps=1e-05, momentum=0.1, affine=True)\n (4): Linear (1024 -> 128)\n (5): Dropout (p = 0.05)\n (6): Tanh ()\n (7): Linear (128 -> 64)\n (8): Dropout (p = 0.05)\n (9): LeakyReLU (0.01)\n (10): Linear (64 -> 32)\n (11): Dropout (p = 0.05)\n (12): Tanh ()\n (13): Linear (32 -> 16)\n (14): Dropout (p = 0.05)\n (15): LeakyReLU (0.01)\n (16): Linear (16 -> 1)\n (17): Sigmoid ()\n )\n\n\n## Print the full net architecture\n\n\n```python\n# Taken from https://stackoverflow.com/questions/42480111/model-summary-in-pytorch/42616812\nfrom torch.nn.modules.module import _addindent\nimport torch\nimport numpy as np\ndef torch_summarize(model, show_weights=True, show_parameters=True):\n \"\"\"Summarizes torch model by showing trainable parameters and weights.\"\"\"\n tmpstr = model.__class__.__name__ + ' (\\n'\n for key, module in model._modules.items():\n # if it contains layers let call it recursively to get params and weights\n if type(module) in [\n torch.nn.modules.container.Container,\n torch.nn.modules.container.Sequential\n ]:\n modstr = torch_summarize(module)\n else:\n modstr = module.__repr__()\n modstr = _addindent(modstr, 2)\n\n params = sum([np.prod(p.size()) for p in module.parameters()])\n weights = tuple([tuple(p.size()) for p in module.parameters()])\n\n tmpstr += ' (' + key + '): ' + modstr \n if show_weights:\n tmpstr += ', weights={}'.format(weights)\n if show_parameters:\n tmpstr += ', parameters={}'.format(params)\n tmpstr += '\\n' \n\n tmpstr = tmpstr + ')'\n return tmpstr\n\nlgr.info(torch_summarize(net))\n```\n\n INFO:__main__:Sequential (\n (0): Linear (29 -> 1024), weights=((1024L, 29L), (1024L,)), parameters=30720\n (1): Dropout (p = 0.05), weights=(), parameters=0\n (2): Tanh (), weights=(), parameters=0\n (3): BatchNorm1d(1024, eps=1e-05, momentum=0.1, affine=True), weights=((1024L,), (1024L,)), parameters=2048\n (4): Linear (1024 -> 128), weights=((128L, 1024L), (128L,)), parameters=131200\n (5): Dropout (p = 0.05), weights=(), parameters=0\n (6): Tanh (), weights=(), parameters=0\n (7): Linear (128 -> 64), weights=((64L, 128L), (64L,)), parameters=8256\n (8): Dropout (p = 0.05), weights=(), parameters=0\n (9): LeakyReLU (0.01), weights=(), parameters=0\n (10): Linear (64 -> 32), weights=((32L, 64L), (32L,)), parameters=2080\n (11): Dropout (p = 0.05), weights=(), parameters=0\n (12): Tanh (), weights=(), parameters=0\n (13): Linear (32 -> 16), weights=((16L, 32L), (16L,)), parameters=528\n (14): Dropout (p = 0.05), weights=(), parameters=0\n (15): LeakyReLU (0.01), weights=(), parameters=0\n (16): Linear (16 -> 1), weights=((1L, 16L), (1L,)), parameters=17\n (17): Sigmoid (), weights=(), parameters=0\n )\n\n\n# Loss and Optimizer\n\n### BCELoss\n- In addition, we will calculate the binary cross entropy loss (BCELoss). Luckily we have one loss function already present. For details please checkout http://pytorch.org/docs/master/nn.html. \n\n- ** NOTE this BCELoss may not be numerical stable, although it's fine during my training process.**\n\n### Optimization\n\n- if return F.log_softmax(x) then loss = F.nll_loss(output, target) (MNIST)\n- print(nn.BCEWithLogitsLoss()(o, t)) is equivalent to print(nn.BCELoss()(sigmoid(o), t))\n\n\n```python\n# ! pip install sympy\nimport sympy as sp\nsp.interactive.printing.init_printing(use_latex=True)\nfrom IPython.display import display, Math, Latex\nmaths = lambda s: display(Math(s))\nlatex = lambda s: display(Latex(s))\n\n#the loss function is as follows:\nmaths(\"\\mathbf{Loss Function:} J(x, z) = -\\sum_k^d[x_k \\log z_k + (1-x_k)log(1-z_k)]\")\n```\n\n\n$$\\mathbf{Loss Function:} J(x, z) = -\\sum_k^d[x_k \\log z_k + (1-x_k)log(1-z_k)]$$\n\n\n\n```python\n# optimizer = torch.optim.SGD(net.parameters(), lr=0.02)\n# optimizer = optim.SGD(model.parameters(), lr=0.001, momentum=0.9)\n# optimizer = optim.SGD(net.parameters(), lr=LR, momentum=MOMENTUM, weight_decay=5e-4)\n\n#L2 regularization can easily be added to the entire model via the optimizer\noptimizer = torch.optim.Adam(net.parameters(), lr=LR,weight_decay=5e-4) # L2 regularization\n# optimizer = torch.optim.Adagrad(net.parameters(), lr=1e-6, weight_decay=5e-4)\n# loss_func = torch.nn.CrossEntropyLoss() # the target label is NOT an one-hotted\n# loss_func = torch.nn.NLLLoss()\nloss_func=torch.nn.BCELoss() # Binary cross entropy: http://pytorch.org/docs/nn.html#bceloss\n# http://andersonjo.github.io/artificial-intelligence/2017/01/07/Cost-Functions/\n# use_cuda=True\nif use_cuda:\n lgr.info (\"Using the GPU\") \n net.cuda()\n loss_func.cuda()\n# cudnn.benchmark = True\n #net = torch.nn.DataParallel(net, device_ids=range(torch.cuda.device_count()))\n\nlgr.info (optimizer)\nlgr.info (loss_func)\n```\n\n INFO:__main__:Using the GPU\n INFO:__main__:\n INFO:__main__:BCELoss (\n )\n\n\n# Training in batches + Measuring the performance of the deep learning model\n\n\n```python\nimport time\nstart_time = time.time() \nepochs=250 # change to 1500 for better results\nall_losses = []\n\nX_tensor_train= XnumpyToTensor(trainX)\nY_tensor_train= YnumpyToTensor(trainY)\n\nprint(type(X_tensor_train.data), type(Y_tensor_train.data)) # should be 'torch.cuda.FloatTensor'\n\n# From here onwards, we must only use PyTorch Tensors\nfor step in range(epochs): \n out = net(X_tensor_train) # input x and predict based on x\n cost = loss_func(out, Y_tensor_train) # must be (1. nn output, 2. target), the target label is NOT one-hotted\n\n optimizer.zero_grad() # clear gradients for next train\n cost.backward() # backpropagation, compute gradients\n optimizer.step() # apply gradients\n \n \n if step % 50 == 0: \n loss = cost.data[0]\n all_losses.append(loss)\n print(step, cost.data.cpu().numpy())\n # RuntimeError: can't convert CUDA tensor to numpy (it doesn't support GPU arrays). \n # Use .cpu() to move the tensor to host memory first. \n prediction = (net(X_tensor_train).data).float() # probabilities \n# prediction = (net(X_tensor).data > 0.5).float() # zero or one\n# print (\"Pred:\" + str (prediction)) # Pred:Variable containing: 0 or 1\n# pred_y = prediction.data.numpy().squeeze() \n pred_y = prediction.cpu().numpy().squeeze()\n target_y = Y_tensor_train.cpu().data.numpy()\n \n tu = ((pred_y == target_y).mean(),log_loss(target_y, pred_y),roc_auc_score(target_y,pred_y ))\n print ('ACC={}, LOG_LOSS={}, ROC_AUC={} '.format(*tu)) \n \nend_time = time.time()\nprint ('{} {:6.3f} seconds'.format('GPU:', end_time-start_time))\n\n%matplotlib inline\nimport matplotlib.pyplot as plt\nplt.plot(all_losses)\nplt.show()\n\nfalse_positive_rate, true_positive_rate, thresholds = roc_curve(target_y,pred_y)\nroc_auc = auc(false_positive_rate, true_positive_rate)\n\nplt.title('LOG_LOSS=' + str(log_loss(target_y, pred_y)))\nplt.plot(false_positive_rate, true_positive_rate, 'b', label='AUC = %0.6f' % roc_auc)\nplt.legend(loc='lower right')\nplt.plot([0, 1], [0, 1], 'r--')\nplt.xlim([-0.1, 1.2])\nplt.ylim([-0.1, 1.2])\nplt.ylabel('True Positive Rate')\nplt.xlabel('False Positive Rate')\nplt.show()\n```\n\n# Performance of the deep learning model on the Validation set\n\n\n```python\nnet.eval()\n# Validation data\nprint (valX.shape)\nprint (valY.shape)\n\nX_tensor_val= XnumpyToTensor(valX)\nY_tensor_val= YnumpyToTensor(valY)\n\n\nprint(type(X_tensor_val.data), type(Y_tensor_val.data)) # should be 'torch.cuda.FloatTensor'\n\npredicted_val = (net(X_tensor_val).data).float() # probabilities \n# predicted_val = (net(X_tensor_val).data > 0.5).float() # zero or one\npred_y = predicted_val.cpu().numpy()\ntarget_y = Y_tensor_val.cpu().data.numpy() \n\nprint (type(pred_y))\nprint (type(target_y))\n\ntu = (str ((pred_y == target_y).mean()),log_loss(target_y, pred_y),roc_auc_score(target_y,pred_y ))\nprint ('\\n')\nprint ('acc={} log_loss={} roc_auc={} '.format(*tu))\n\nfalse_positive_rate, true_positive_rate, thresholds = roc_curve(target_y,pred_y)\nroc_auc = auc(false_positive_rate, true_positive_rate)\n\nplt.title('LOG_LOSS=' + str(log_loss(target_y, pred_y)))\nplt.plot(false_positive_rate, true_positive_rate, 'b', label='AUC = %0.6f' % roc_auc)\nplt.legend(loc='lower right')\nplt.plot([0, 1], [0, 1], 'r--')\nplt.xlim([-0.1, 1.2])\nplt.ylim([-0.1, 1.2])\nplt.ylabel('True Positive Rate')\nplt.xlabel('False Positive Rate')\nplt.show()\n\n# print (pred_y)\n```\n\n# Submission on Test set\n\n\n```python\n# testX, df_test_set\n# df[df.columns.difference(['b'])]\n# trainX, trainY, valX, valY, testX, df_test_set = loadDataSplit()\n\nprint (df_test_set.shape)\ncolumns = ['id', 'probability']\ndf_pred=pd.DataFrame(data=np.zeros((0,len(columns))), columns=columns)\ndf_pred.id.astype(int)\n\nfor index, row in df_test_set.iterrows():\n rwo_no_id=row.drop('id') \n# print (rwo_no_id.values) \n x_data_np = np.array(rwo_no_id.values, dtype=np.float32) \n if use_cuda:\n X_tensor_test = Variable(torch.from_numpy(x_data_np).cuda()) # Note the conversion for pytorch \n else:\n X_tensor_test = Variable(torch.from_numpy(x_data_np)) # Note the conversion for pytorch\n \n X_tensor_test=X_tensor_test.view(1, trainX.shape[1]) # does not work with 1d tensors \n predicted_val = (net(X_tensor_test).data).float() # probabilities \n p_test = predicted_val.cpu().numpy().item() # otherwise we get an array, we need a single float\n \n df_pred = df_pred.append({'id':row['id'].astype(int), 'probability':p_test},ignore_index=True)\n\ndf_pred.head(5)\n```\n\n (45647, 30)\n\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
idprobability
097040.00.519708
165399.00.517178
2147258.00.518780
3129573.00.519850
4134978.00.522069
\n
\n\n\n\n# Create a CSV with the ID's and the coresponding probabilities. \n\n\n```python\ndf_pred.id=df_pred.id.astype(int)\n\ndef savePred(df_pred, loss):\n# csv_path = 'pred/p_{}_{}_{}.csv'.format(loss, name, (str(time.time())))\n csv_path = 'pred/pred_{}_{}.csv'.format(loss, (str(time.time())))\n df_pred.to_csv(csv_path, columns=('id', 'probability'), index=None)\n print (csv_path)\n \nsavePred (df_pred, log_loss(target_y, pred_y))\n```\n\n pred/pred_0.693924313567_1504720448.33.csv\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "aa0716d0b3ab8c77244cee773ff53e41f8b6e20d", "size": 108639, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "day 02 PyTORCH and PyCUDA/PyTorch/18-PyTorch-NUMER.AI-Binary-Classification-BCELoss.ipynb", "max_stars_repo_name": "koriavinash1/Deep-Learning-Boot-Camp", "max_stars_repo_head_hexsha": "472e58c6ef9205cee6f1ade539241d7b1375d930", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2017-09-07T14:32:56.000Z", "max_stars_repo_stars_event_max_datetime": "2017-09-07T14:32:56.000Z", "max_issues_repo_path": "day 02 PyTORCH and PyCUDA/PyTorch/18-PyTorch-NUMER.AI-Binary-Classification-BCELoss.ipynb", "max_issues_repo_name": "kookbal/Deep-Learning-Boot-Camp", "max_issues_repo_head_hexsha": "472e58c6ef9205cee6f1ade539241d7b1375d930", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "day 02 PyTORCH and PyCUDA/PyTorch/18-PyTorch-NUMER.AI-Binary-Classification-BCELoss.ipynb", "max_forks_repo_name": "kookbal/Deep-Learning-Boot-Camp", "max_forks_repo_head_hexsha": "472e58c6ef9205cee6f1ade539241d7b1375d930", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-05-17T15:09:57.000Z", "max_forks_repo_forks_event_max_datetime": "2020-05-17T15:09:57.000Z", "avg_line_length": 70.4076474401, "max_line_length": 20932, "alphanum_fraction": 0.740894154, "converted": true, "num_tokens": 9888, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358411176238, "lm_q2_score": 0.6992544147913993, "lm_q1q2_score": 0.4479674401750999}} {"text": "

Table of Contents

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\n\n\n```python\n# code for loading the format for the notebook\nimport os\n\n# path : store the current path to convert back to it later\npath = os.getcwd()\nos.chdir(os.path.join('..', '..', 'notebook_format'))\n\nfrom formats import load_style\nload_style(css_style = 'custom2.css', plot_style = False)\n```\n\n\n\n\n\n\n\n\n\n\n\n\n```python\nos.chdir(path)\n\n# 1. magic for inline plot\n# 2. magic to print version\n# 3. magic so that the notebook will reload external python modules\n# 4. magic to enable retina (high resolution) plots\n# https://gist.github.com/minrk/3301035\n%matplotlib inline\n%load_ext watermark\n%load_ext autoreload\n%autoreload 2\n%config InlineBackend.figure_format = 'retina'\n\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nfrom pathlib import Path\nfrom sklearn.ensemble import RandomForestClassifier\nfrom sklearn.model_selection import train_test_split\n\n%watermark -a 'Ethen' -d -t -v -p numpy,pandas,matplotlib,sklearn\n```\n\n Ethen 2018-01-24 16:15:24 \n \n CPython 3.5.2\n IPython 6.2.1\n \n numpy 1.13.3\n pandas 0.21.1\n matplotlib 2.1.0\n sklearn 0.19.1\n\n\n# Partial Dependence Plot\n\nDuring the talk, [Youtube: PyData - Random Forests Best Practices for the Business World](https://www.youtube.com/watch?v=E7VLE-U07x0&index=51&list=PLGVZCDnMOq0oqs6RTJk4zZde86DZrgnzm), one of the best practices that the speaker mentioned when using tree-based models is to check for directional relationships. When using non-linear machine learning algorithms, such as popular tree-based models random forest and gradient boosted trees, it can be hard to understand the relations between predictors and model outcome as they do not give us handy coefficients like linear-based models. For example, in terms of random forest, all we get is the feature importance. Although based on that information, we can tell which feature is significantly influencing the outcome based on the importance calculation, it does not inform us in which direction is the predictor influencing outcome. In this notebook, we'll be exploring **Partial dependence plot (PDP)**, a model agnostic technique that gives us an approximate directional influence for a given feature that was used in the model. Note much of the explanation is \"borrowed\" from the blog post at the following link, [Blog: Introducing PDPbox](https://towardsdatascience.com/introducing-pdpbox-2aa820afd312), this documentation aims to improve upon it by giving a cleaner implementation.\n\n**Partial dependence plot (PDP)** aims to visualize the marginal effect of a given predictor towards the model outcome by plotting out the average model outcome in terms of different values of the predictor. Let's first gain some intuition of how it works with a made up example. Assume we have a data set that only contains three data points and three features (A, B, C) as shown below.\n\n\n\nIf we wish to see how feature A is influencing the prediction Y, what PDP does is to generate a new data set as follow. (here we assume that feature A only has three unique values: A1, A2, A3)\n\n\n\nWe then perform the prediction as usual with this new set of data. As we can imagine, PDP would generate **num_rows * num_grid_points** (here, the number of grid point equals the number of unique values of the target feature, more on this later) number of predictions and average them for each unique value of Feature A.\n\n\n\nIn the end, PDP would only plot out the average predictions for each unique value of our target feature.\n\n\n\nLet's now formalize this idea with some notation. The partial dependence function is defined as:\n\n$$\n\\begin{align}\n\\hat{f}_{x_S}(x_S) = E_{x_C} \\left[ f(x_S, x_C) \\right]\n\\end{align}\n$$\n\nThe term $x_S$ denotes the set of features for which the partial dependence function should be plotting and $x_C$ are all other features that were used in the machine learning model $f$. In other words, if there were $p$ predictors, $S$ is a subset of our $p$ predictors, $S \\subset \\left\\{ x_1, x_2, \\ldots, x_p \\right\\}$, $C$ would be complementing $S$ such that $S \\cup C = \\left\\{x_1, x_2, \\ldots, x_p\\right\\}$. The function above is then estimated by calculating averages in the training data, which is also known as Monte Carlo method:\n\n$$\n\\begin{align}\n\\hat{f}_{x_S}(x_S) = \\frac{1}{n} \\sum_{i=1}^n f(x_S, x_{Ci})\n\\end{align}\n$$\n\nWhere $\\left\\{x_{C1}, x_{C2}, \\ldots, x_{CN}\\right\\}$ are the values of $X_C$ occurring over all observations in the training data. In other words, in order to calculate the partial dependence of a given variable (or variables), the entire training set must be utilized for every set of joint values. For classification, where the machine learning model outputs probabilities, the partial dependence function displays the probability for a certain class given different values for features $x_s$, a straightforward way to handle multi-class problems is to plot one line per class.\n\n## Individual Conditional Expectation (ICE) Plot\n\nAs an extension of a PDP, ICE plot visualizes the relationship between a feature and the predicted responses for each observation. While a PDP visualizes the averaged relationship between features and predicted responses, a set of ICE plots disaggregates the averaged information and visualizes an individual dependence for each observation. Hence, instead of only plotting out the average predictions, ICEbox displays all individual lines. (three lines in total in this case)\n\n\n\nThe authors of the [Paper: A. Goldstein, A. Kapelner, J. Bleich, E. Pitkin Peeking Inside the Black Box: Visualizing Statistical Learning with Plots of Individual Conditional Expectation](https://arxiv.org/abs/1309.6392) claims with everything displayed in its raw state, any interesting discovers wouldn\u2019t be shielded because of the averaging inherented with PDP. A vivid example from the paper is shown below:\n\n\n\nIn this example, if we only look at the PDP in Figure b, we would think that on average, the feature X2 is not meaningfully associated with the our target response variable Y. However, if judging from the scatter plot showed in Figure a, this conclusion is plainly wrong. Now if we were to plot out the individual estimated conditional expectation curves, everything becomes more obvious.\n\n\n\nAfter having an understand of the procedure for PDP and ICE plot, we can observe that:\n\n- PDP is a global method, it takes into account all instances and makes a statement about the global relationship of a feature with the predicted outcome.\n- One of the main advantage of PDP is that it can be used to interpret the result of any \"black box\" learning methods.\n- PDP can be quite computationally expensive when the data set becomes large.\n- Owing to the limitations of computer graphics, and human perception, the size of the subsets $x_S$ must be small (l \u2248 1,2,3). There are of course a large number of such subsets, but only those chosen from among the usually much smaller set of highly relevant predictors are likely to be informative.\n- PDP can obfuscate relationship that comes from interactions. PDPs show us how the average relationship between feature $x_S$ and $\\hat{y}$ looks like. This works well only in cases where the interactions between $x_S$ and the remaining features $x_C$ are weak. In cases where interactions do exist, the ICE plot may give a lot more insight of the underlying relationship.\n\n\n## Implementation\n\nWe'll be using the [titanic dataset](https://www.kaggle.com/c/titanic/data) (details of the dataset is listed in the link) to test our implementation.\n\n\n```python\n# we download the training data and store it\n# under the `data` directory\ndata_dir = Path('data')\ndata_path = data_dir / 'train.csv'\ndata = pd.read_csv(data_path)\nprint('dimension: ', data.shape)\nprint('features: ', data.columns)\ndata.head()\n```\n\n dimension: (891, 12)\n features: Index(['PassengerId', 'Survived', 'Pclass', 'Name', 'Sex', 'Age', 'SibSp',\n 'Parch', 'Ticket', 'Fare', 'Cabin', 'Embarked'],\n dtype='object')\n\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
PassengerIdSurvivedPclassNameSexAgeSibSpParchTicketFareCabinEmbarked
0103Braund, Mr. Owen Harrismale22.010A/5 211717.2500NaNS
1211Cumings, Mrs. John Bradley (Florence Briggs Th...female38.010PC 1759971.2833C85C
2313Heikkinen, Miss. Lainafemale26.000STON/O2. 31012827.9250NaNS
3411Futrelle, Mrs. Jacques Heath (Lily May Peel)female35.01011380353.1000C123S
4503Allen, Mr. William Henrymale35.0003734508.0500NaNS
\n
\n\n\n\n\n```python\n# some naive feature engineering\ndata['Age'] = data['Age'].fillna(data['Age'].median())\ndata['Embarked'] = data['Embarked'].fillna('S')\ndata['Sex'] = data['Sex'].apply(lambda x: 1 if x == 'male' else 0)\ndata = pd.get_dummies(data, columns = ['Embarked'])\n\n# features/columns that are used\nlabel = data['Survived']\nfeatures = [\n 'Pclass', 'Sex',\n 'Age', 'SibSp',\n 'Parch', 'Fare',\n 'Embarked_C', 'Embarked_Q', 'Embarked_S']\ndata = data[features]\n\nX_train, X_test, y_train, y_test = train_test_split(\n data, label, test_size = 0.2, random_state = 1234, stratify = label)\n```\n\n\n```python\n# fit a baseline random forest model and show its top 2 most important features\nrf = RandomForestClassifier(n_estimators = 50, random_state = 1234)\nrf.fit(X_train, y_train)\n\nprint('top 2 important features:')\nimp_index = np.argsort(rf.feature_importances_)\nprint(features[imp_index[-1]])\nprint(features[imp_index[-2]])\n```\n\n top 2 important features:\n Fare\n Sex\n\n\nAforementioned, tree-based models lists out the top important features, but it is not clear whether they have a positive or negative impact on the result. This is where tools such as partial dependence plots can aid us communicate the results better to others.\n\n\n```python\nfrom partial_dependence import PartialDependenceExplainer\nplt.rcParams['figure.figsize'] = 16, 9\n\n\n# we specify the feature name and its type to fit the partial dependence\n# result, after fitting the result, we can call .plot to visualize it\n# since this is a binary classification model, when we call the plot\n# method, we tell it which class are we targeting, in this case 1 means\n# the passenger did indeed survive (more on centered argument later)\npd_explainer = PartialDependenceExplainer(estimator = rf, verbose = 0)\npd_explainer.fit(data, feature_name = 'Sex', feature_type = 'cat')\npd_explainer.plot(centered = False, target_class = 1)\nplt.show()\n```\n\nHopefully, we can agree that the partial dependence plot makes intuitive sense, as for the categorical feature `Sex`, 1 indicates that the passenger was a male. And we know that during the titanic accident, the majority of the survivors were female passenger, thus the plot is telling us male passengers will on average have around 40% chance lower of surviving when compared with female passengers. Also instead of only plotting the \"partial dependence\" plot, the plot also fills between the standard deviation range. This is essentially borrowing the idea from ICE plot that only plotting the average may obfuscate the relationship.\n\nCentered plot can be useful when we are not interested in seeing the absolute change of a predicted value, but rather the difference in prediction compared to a fixed point of the feature range.\n\n\n```python\n# centered = True is actually the default\npd_explainer.plot(centered = True, target_class = 1)\nplt.show()\n```\n\nWe can perform the same process for numerical features such as `Fare`. We know that more people from the upper class survived, and people from the upper class generally have to pay more Fare to get onboard the titanic. The partial dependence plot below also depicts this trend.\n\n\n```python\npd_explainer.fit(data, feature_name = 'Fare', feature_type = 'num')\npd_explainer.plot(target_class = 1)\nplt.show()\n```\n\nIf you prefer to create your own visualization, you can call the `results_` attribute to access the partial dependence result. And for those that are interested in the implementation details, the code can be obtained at the following [link](https://github.com/ethen8181/machine-learning/tree/master/model_selection/partial_dependence/partial_dependence.py).\n\nWe'll conclude our discussion on parital dependence plot by providing a link to another blog that showcases this method's usefulness in ensuring the behavior of the new machine learning model does intuitively and logically match our intuition and does not differ significantly from a baseline model. [Blog: Using Partial Dependence to Compare Sort Algorithms](http://techblog.hotwire.com/2016/06/13/partial-dependence-compare-sort/)\n\n# Reference\n\n- [Blog: Introducing PDPbox](https://towardsdatascience.com/introducing-pdpbox-2aa820afd312)\n- [Online Book: Partial Dependence Plot (PDP)](https://christophm.github.io/interpretable-ml-book/pdp.html)\n- [Mathworks Documentation: plotPartialDependence](https://www.mathworks.com/help/stats/regressiontree.plotpartialdependence.html?requestedDomain=true#mw_79dadf51-f451-45a9-a801-2e9ccec37aae)\n- [Github: PDPbox - python partial dependence plot toolbox](https://github.com/SauceCat/PDPbox)\n", "meta": {"hexsha": "977dac6683ed870d0c4816070b257c01d0fe84a9", "size": 341832, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "model_selection/partial_dependence/partial_dependence.ipynb", "max_stars_repo_name": "certara-ShengnanHuang/machine-learning", "max_stars_repo_head_hexsha": "d21dfbeabf2876ffe49fcef444ca4516c4d36df0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2019-10-18T14:48:04.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-21T14:53:17.000Z", "max_issues_repo_path": "model_selection/partial_dependence/partial_dependence.ipynb", "max_issues_repo_name": "certara-ShengnanHuang/machine-learning", "max_issues_repo_head_hexsha": "d21dfbeabf2876ffe49fcef444ca4516c4d36df0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "model_selection/partial_dependence/partial_dependence.ipynb", "max_forks_repo_name": "certara-ShengnanHuang/machine-learning", "max_forks_repo_head_hexsha": "d21dfbeabf2876ffe49fcef444ca4516c4d36df0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-03-03T21:07:43.000Z", "max_forks_repo_forks_event_max_datetime": "2020-03-03T21:07:43.000Z", "avg_line_length": 366.7725321888, "max_line_length": 108408, "alphanum_fraction": 0.9199811603, "converted": true, "num_tokens": 5740, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241632752914, "lm_q2_score": 0.7154239957834733, "lm_q1q2_score": 0.4479442507469928}} {"text": "\n\n# ECE-6524 / CS-6524 Deep Learning\n# Assignment 3\n\nIn this assignment, **you need to complete the Yolo loss function, and train an object detector. Yay!**\n\nThis assignment is inspired and adapted from [UIUC CS498](http://slazebni.cs.illinois.edu/fall18/assignment3_part2.html)\n## Submission guideline\n\n1. Click the Save button at the top of the Jupyter Notebook.\n2. Please make sure to have entered your Virginia Tech PID below.\n3. Select Cell -> All Output -> Clear. This will clear all the outputs from all cells (but will keep the content of cells).\n4. Select Cell -> Run All. This will run all the cells in order.\n5. Once you've rerun everything, select File -> Download as -> PDF via LaTeX\n6. Look at the PDF file and make sure all your solutions are displayed correctly there. \n7. Zip the all the files along with this notebook (Please don't include the data)\n8. Name your PDF file as Assignment2_[YOUR ID NUMBER].\n9. Submit your zipped file and the PDF **INDEPENDENTLY**.\n10. **PLEASE DO NOT ZIP YOUR DATASET. ONLY NOTEBOOK/CODE/PDF.**\n\n**While you are encouraged to discuss with your peers, all work submitted is expected to be your own. If you use any information from other resources (e.g. online materials), you are required to cite it below you VT PID. Any violation will result in a 0 mark for the assignment.**\n\n813808996### Please Write Your VT PID Here: 813808996\n\n---\n\n\n### Reference (if any): https://github.com/xiongzihua/pytorch-YOLO-v1\n\nIn this homework, you would need to use **Python 3.6+** along with the following packages:\n```\n1. pytorch 1.2\n2. torchvision\n3. numpy\n4. matplotlib\n5. tqdm (for better, cuter progress bar. Yay!)\n```\nTo install pytorch, please follow the instructions on the [Official website](https://pytorch.org/). In addition, the [official document](https://pytorch.org/docs/stable/) could be very helpful when you want to find certain functionalities. \n\n\nNote that, on a high-end GPU, it sill takes 3-4 hours to train. **SO START EARLY. IT'S IMPOSSIBLE TO FINISH IT AT THE LAST MINUTE!**\n\n\n\n```python\nfrom google.colab import drive\ndrive.mount('/content/drive')\n%cd /content/drive/My Drive/Colab Notebooks/Assignment_3\n```\n\n Drive already mounted at /content/drive; to attempt to forcibly remount, call drive.mount(\"/content/drive\", force_remount=True).\n /content/drive/My Drive/Colab Notebooks/Assignment_3\n\n\n\n```python\nimport os\nimport random\n\nimport cv2\nimport numpy as np\n\nimport torch\nfrom torch.utils.data import DataLoader\nfrom torchvision import models\n\nfrom resnet_yolo import resnet50\n#from yolo_loss import YoloLoss\nfrom dataset import VocDetectorDataset\nfrom eval_voc import evaluate\nfrom predict import predict_image\nfrom config import VOC_CLASSES, COLORS\nfrom kaggle_submission import output_submission_csv\nimport matplotlib.pyplot as plt\nfrom tqdm import tqdm\n\n%matplotlib inline\n%load_ext autoreload\n%autoreload 2\n```\n\n## Initialization\n\n\n```python\ndevice = torch.device(\"cuda:0\" if torch.cuda.is_available() else \"cpu\")\n```\n\n# You Only Look Once: Unified, Real-Time Object Detection [100 pts]\nIn this assignment, you need to implement the loss function and train the **YOLO object detector** (specfically, YOLO-v1). Here we provide a list of recommend readings for you:\n- [YOLO original paper](http://slazebni.cs.illinois.edu/fall18/lec09_detection.pdf) (recommended)\n- [Great post about YOLO](https://medium.com/adventures-with-deep-learning/yolo-v1-part-1-cfb47135f81f) on Medium\n- [Differences between YOLO, YOLOv2 and YOLOv3\n](https://medium.com/@jonathan_hui/real-time-object-detection-with-yolo-yolov2-28b1b93e2088)\n- [Great explanation of the Yolo Loss function](https://stats.stackexchange.com/questions/287486/yolo-loss-function-explanation)\n- [YOLO on SNL, suggested by UIUC CS498](https://www.youtube.com/watch?v=xZGahvrep3o)\n\nWe adopt a variant of YOLO, which:\n1. Use pretrained ResNet50 classifier as detector backbone. The pretrained model is offered in `torchvision.models`.\n2. Instead of using a $7\\times7$ detection grid, we use $14\\times14$ to get a more finegrained detection.\n\nIn general, the backbone models are usually pretrained on ImageNet dataset (> 1 million images) with numerous classes. As a result, having these pretrained backbone can greatly shorten the required training time, as well as improve the performance. **But still, it takes at least 3-4 hours to train, not to mention that you might need to debug after one training run. So START EARLY, DON'T GO #YOLO!**\n\n\nYou are supposed to get a reasonable detector (like the ... above?) after training the model correctly.\n\n\n```python\n# YOLO network hyperparameters\nB = 2 # number of bounding box predictions per cell\nS = 14 # width/height of network output grid (larger than 7x7 from paper since we use a different network)\n```\n\n## Load the pretrained ResNet classifier\nLoad the pretrained classifier. By default, it would use the pretrained model provided by `Pytorch`.\n\n\n```python\nload_network_path = None\npretrained = True\n\n# use to load a previously trained network\nif load_network_path is not None:\n print('Loading saved network from {}'.format(load_network_path))\n net = resnet50().to(device)\n net.load_state_dict(torch.load(load_network_path))\nelse:\n print('Load pre-trained model')\n net = resnet50(pretrained=pretrained).to(device)\n```\n\n Load pre-trained model\n\n\nSome basic hyperparameter settings that you probably don't have to tune.\n\n\n```python\nlearning_rate = 0.001\nnum_epochs = 30\nbatch_size = 12\n\n# Yolo loss component coefficients (as given in Yolo v1 paper)\nlambda_coord = 5\nlambda_noobj = 0.5\n```\n\n## Implement the YOLO-v1 loss [50 pts]\nNow, you have to implement the `YoloLoss` for training your object detector. Please read closely to the [YOLO original paper](http://slazebni.cs.illinois.edu/fall18/lec09_detection.pdf) so that you can implement it.\n\nIn general, there are 4 components in the YOLO loss. Consider that we have our prediction grid of size$(N, S, S, 5B+c)$ ( (x, y, w, h, C) for each bounding box, and c is the number of classes), where $N$ is the batch size, $S$ is the grid size, $B$ is the number of bounding boxes. We have :\n1. Bounding box regression loss on the bounding box$(x, y, w, h)$\n - $l_{coord}=\\sum_{i=0}^{S^2}\\sum_{j=0}^B\\mathbb{1}^{obj}_{ij}\\left[(x_i-\\hat{x}_i)^2+(y_i-\\hat{y}_i)^2\\right]$ + $\\sum_{i=0}^{S^2}\\sum_{j=0}^B\\mathbb{1}^{obj}_{ij}\\left[(\\sqrt{w_i}-\\sqrt{\\hat{w}_i})^2+(\\sqrt{h_i}-\\sqrt{\\hat{h}_i})^2\\right]$\n - $\\mathbb{1}^{obj}_{ij}$: equals to 1 when object appears in cell $i$, and the bounding box $j$ is responsible for the prediction. 0 otherwise.\n2. Contain object loss on the confidence prediction $c$ (only calculate for those boxes that actually have objects)\n - $l_{contain}=\\sum_{i=0}^{S^2}\\sum_{j=0}^B\\mathbb{1}^{obj}_{ij}(C_i-\\hat{C}_i)^2$\n - $C_i$ the predicted confidence score for cell $i$ from predicted box $j$\n - For each grid cell, you only calculate the contain object loss for the predicted bounding box that has maximum overlap (iou) with the gruond truth box.\n - We say that this predicted box with maximum iou is **responsible** for the prediction.\n3. No object loss on the confidence prediction $c$ (only calculate for those boxes that don't have objects)\n - $l_{noobj}=\\sum_{i=0}^{S^2}\\sum_{j=0}^B\\mathbb{1}^{noobj}_{ij}(C_i-\\hat{C}_i)^2$\n - $\\mathbb{1}^{obj}_{ij}$: equals to 1 when **no object appears** in cell $i$.\n4. Classification error loss.\n - $l_{class}=\\sum_{i=0}^{S^2}\\mathbb{1}_i^{obj}\\sum_{c\\in classes}\\left(p_i(c)-\\hat{p_i}(c)\\right)^2$\n - $p_i(c)$ is the predicted score for class $c$\n \nPutting them together, we get the yolo loss:\n\\begin{equation}\nyolo=\\lambda_{coord}l_{coord}+l_{contain}+\\lambda_{noobj}l_{noobj}+l_{class}\n\\end{equation}\nwhere $\\lambda$ are hyperparameters. We have provided detailed comments to gudie you through implementing the loss. So now, please complete the YoloLoss in the code block below. **If you have any problem with regard to implementation, post and discuss it on Piazza.**\n\n\n```python\nimport torch.nn as nn\nimport torch.nn.functional as F\nfrom torch.autograd import Variable\n \nclass YoloLoss(nn.Module):\n def __init__(self,S,B,l_coord,l_noobj):\n super(YoloLoss,self).__init__()\n self.S = S\n self.B = B\n self.l_coord = l_coord\n self.l_noobj = l_noobj\n \n def compute_iou(self, box1, box2): \n '''Compute the intersection over union of two set of boxes, each box is [x1,y1,x2,y2].\n Args:\n box1: (tensor) bounding boxes, sized [N,4].\n box2: (tensor) bounding boxes, sized [M,4].\n Return:\n (tensor) iou, sized [N,M].\n '''\n N = box1.size(0)\n M = box2.size(0)\n \n lt = torch.max(\n box1[:,:2].unsqueeze(1).expand(N,M,2), # [N,2] -> [N,1,2] -> [N,M,2]\n box2[:,:2].unsqueeze(0).expand(N,M,2), # [M,2] -> [1,M,2] -> [N,M,2]\n ) \n \n rb = torch.min(\n box1[:,2:].unsqueeze(1).expand(N,M,2), # [N,2] -> [N,1,2] -> [N,M,2]\n box2[:,2:].unsqueeze(0).expand(N,M,2), # [M,2] -> [1,M,2] -> [N,M,2]\n ) \n \n wh = rb - lt # [N,M,2]\n wh[wh<0] = 0 # clip at 0\n inter = wh[:,:,0] * wh[:,:,1] # [N,M]\n \n area1 = (box1[:,2]-box1[:,0]) * (box1[:,3]-box1[:,1]) # [N,]\n area2 = (box2[:,2]-box2[:,0]) * (box2[:,3]-box2[:,1]) # [M,]\n area1 = area1.unsqueeze(1).expand_as(inter) # [N,] -> [N,1] -> [N,M]\n area2 = area2.unsqueeze(0).expand_as(inter) # [M,] -> [1,M] -> [N,M]\n \n iou = inter / (area1 + area2 - inter)\n return iou \n \n def get_class_prediction_loss(self, classes_pred, classes_target):\n \"\"\" \n Parameters:\n classes_pred : (tensor) size (batch_size, S, S, 20) \n classes_target : (tensor) size (batch_size, S, S, 20)\n \n Returns:\n class_loss : scalar\n \"\"\"\n \n ##### CODE #####\n class_loss = F.mse_loss(classes_pred,classes_target,size_average=False)\n\n return class_loss\n \n \n def get_regression_loss(self, box_pred_response, box_target_response):\n \"\"\"\n Parameters:\n box_pred_response : (tensor) size (-1, 5)\n box_target_response : (tensor) size (-1, 5)\n Note : -1 corresponds to ravels the tensor into the dimension specified \n See : https://pytorch.org/docs/stable/tensors.html#torch.Tensor.view_as\n \n Returns:\n reg_loss : scalar\n \n \"\"\"\n \n ##### CODE #####\n reg_loss = F.mse_loss(box_pred_response[:,:2],box_target_response[:,:2],size_average=False) + F.mse_loss(torch.sqrt(box_pred_response[:,2:4]),torch.sqrt(box_target_response[:,2:4]),size_average=False)\n\n return reg_loss\n \n def get_contain_object_loss(self, box_pred_response, box_target_response_iou):\n \"\"\"\n Parameters:\n box_pred_response : (tensor) size ( -1 , 5)\n box_target_response_iou : (tensor) size ( -1 , 5)\n Note : -1 corresponds to ravels the tensor into the dimension specified \n See : https://pytorch.org/docs/stable/tensors.html#torch.Tensor.view_as\n \n Returns:\n contain_loss : scalar\n \n \"\"\"\n \n ##### CODE #####\n contain_loss = F.mse_loss(box_pred_response[:,4],box_target_response_iou[:,4],size_average=False)\n\n return contain_loss\n \n def get_no_object_loss(self, target_tensor, pred_tensor, no_object_mask):\n \"\"\" \n Parameters:\n target_tensor : (tensor) size (batch_size, S , S, 30)\n pred_tensor : (tensor) size (batch_size, S , S, 30)\n no_object_mask : (tensor) size (batch_size, S , S)\n \n Returns:\n no_object_loss : scalar\n \n Hints:\n 1) Create 2 tensors no_object_prediction and no_object_target which only have the \n values which have no object. \n 2) Have another tensor no_object_prediction_mask of the same size such that \n mask with respect to both confidences of bounding boxes set to 1. \n 3) Create 2 tensors which are extracted from no_object_prediction and no_object_target using\n the mask created above to find the loss. \n \"\"\"\n \n ##### CODE #####\n no_object_prediction = pred_tensor[no_object_mask].view(-1, 30)\n no_object_target = target_tensor[no_object_mask].view(-1, 30)\n no_object_prediction_mask = torch.cuda.ByteTensor(no_object_prediction.size())\n no_object_prediction_mask.zero_()\n no_object_prediction_mask[:, 4] = 1\n no_object_prediction_mask[:, 9] = 1\n noo_pred_c = no_object_prediction[no_object_prediction_mask]\n noo_target_c = no_object_target[no_object_prediction_mask]\n no_object_loss = F.mse_loss(noo_pred_c, noo_target_c, reduction='sum')\n \n return no_object_loss\n \n \n \n def find_best_iou_boxes(self, box_target, box_pred):\n \"\"\"\n Parameters: \n box_target : (tensor) size (-1, 5)\n box_pred : (tensor) size (-1, 5)\n Note : -1 corresponds to ravels the tensor into the dimension specified \n See : https://pytorch.org/docs/stable/tensors.html#torch.Tensor.view_as\n \n Returns: \n box_target_iou: (tensor)\n contains_object_response_mask : (tensor)\n \n Hints:\n 1) Find the iou's of each of the 2 bounding boxes of each grid cell of each image.\n 2) Set the corresponding contains_object_response_mask of the bounding box with the max iou\n of the 2 bounding boxes of each grid cell to 1.\n 3) For finding iou's use the compute_iou function\n 4) Before using compute preprocess the bounding box coordinates in such a way that \n if for a Box b the coordinates are represented by [x, y, w, h] then \n x, y = x/S - 0.5*w, y/S - 0.5*h ; w, h = x/S + 0.5*w, y/S + 0.5*h\n Note: Over here initially x, y are the center of the box and w,h are width and height. \n We perform this transformation to convert the correct coordinates into bounding box coordinates.\n 5) Set the confidence of the box_target_iou of the bounding box to the maximum iou\n \n \"\"\"\n \n ##### CODE #####\n coo_response_mask = torch.cuda.ByteTensor(box_target.size())\n coo_response_mask.zero_()\n coo_not_response_mask = torch.cuda.ByteTensor(box_target.size())\n coo_not_response_mask.zero_()\n \n box_target_iou = torch.zeros(box_target.size()).cuda()\n \n for i in range(0,box_target.size()[0],2): #choose the best iou box\n box1 = box_pred[i:i+2]\n box1_xyxy = Variable(torch.FloatTensor(box1.size()))\n box1_xyxy[:,:2] = box1[:,:2]/14. -0.5*box1[:,2:4]\n box1_xyxy[:,2:4] = box1[:,:2]/14. +0.5*box1[:,2:4]\n box2 = box_target[i].view(-1,5)\n box2_xyxy = Variable(torch.FloatTensor(box2.size()))\n box2_xyxy[:,:2] = box2[:,:2]/14. -0.5*box2[:,2:4]\n box2_xyxy[:,2:4] = box2[:,:2]/14. +0.5*box2[:,2:4]\n iou = self.compute_iou(box1_xyxy[:,:4],box2_xyxy[:,:4]) #[2,1]\n max_iou,max_index = iou.max(0)\n max_index = max_index.data.cuda()\n \n coo_response_mask[i+max_index]=1\n coo_not_response_mask[i+1-max_index]=1\n\n #####\n # we want the confidence score to equal the\n # intersection over union (IOU) between the predicted box\n # and the ground truth\n #####\n box_target_iou[i+max_index,torch.LongTensor([4]).cuda()] = (max_iou).data.cuda()\n box_target_iou = Variable(box_target_iou).cuda()\n \n \n\n return box_target_iou, coo_response_mask\n \n\n \n def forward(self, pred_tensor,target_tensor):\n '''\n pred_tensor: (tensor) size(batchsize,S,S,Bx5+20=30)\n where B - number of bounding boxes this grid cell is a part of = 2\n 5 - number of bounding box values corresponding to [x, y, w, h, c]\n where x - x_coord, y - y_coord, w - width, h - height, c - confidence of having an object\n 20 - number of classes\n \n target_tensor: (tensor) size(batchsize,S,S,30)\n \n Returns:\n Total Loss\n '''\n \n N = pred_tensor.size(0)\n total_loss = None\n \n # Create 2 tensors contains_object_mask and no_object_mask \n # of size (Batch_size, S, S) such that each value corresponds to if the confidence of having \n # an object > 0 in the target tensor.\n\n ##### CODE #####\n batch_size = pred_tensor.size()[0]\n \n contains_object_mask = target_tensor[:, :, :, 4] > 0\n no_object_mask = target_tensor[:, :, :, 4] == 0\n contains_object_mask = contains_object_mask.unsqueeze(-1).expand_as(target_tensor)\n no_object_mask = no_object_mask.unsqueeze(-1).expand_as(target_tensor)\n \n # Create a tensor contains_object_pred that corresponds to \n # to all the predictions which seem to confidence > 0 for having an object\n # Then, split this tensor into 2 tensors : \n # 1) bounding_box_pred : Contains all the Bounding box predictions (x, y, w, h, c) of all grid \n # cells of all images\n # 2) classes_pred : Contains all the class predictions for each grid cell of each image\n # Hint : Use contains_object_mask\n \n ##### CODE #####\n contains_object_pred = pred_tensor[contains_object_mask].view(-1, 30)\n bounding_box_pred = contains_object_pred[:, :10].contiguous().view(-1, 5) \n classes_pred = contains_object_pred[:, 10:]\n\n ##### CODE ##### \n \n # Similarly, create 2 tensors bounding_box_target and classes_target\n # using the contains_object_mask.\n \n ##### CODE #####\n contains_object_target = target_tensor[contains_object_mask].view(-1, 30)\n bounding_box_target = contains_object_target[:, :10].contiguous().view(-1, 5)\n classes_target = contains_object_target[:, 10:]\n \n \n #Compute the No object loss here\n # Instruction: finish your get_no_object_loss\n \n ##### CODE #####\n no_object_loss = self.get_no_object_loss(pred_tensor,target_tensor,no_object_mask)\n \n # Compute the iou's of all bounding boxes and the mask for which bounding box \n # of 2 has the maximum iou the bounding boxes for each grid cell of each image.\n # Instruction: finish your find_best_iou_boxes and use it.\n ##### CODE #####\n \n \n box_target_iou, coo_response_mask = self.find_best_iou_boxes(bounding_box_target, bounding_box_pred)\n \n \n # Create 3 tensors :\n # 1) box_prediction_response - bounding box predictions for each grid cell which has the maximum iou\n # 2) box_target_response_iou - bounding box target ious for each grid cell which has the maximum iou\n # 3) box_target_response - bounding box targets for each grid cell which has the maximum iou\n # Hint : Use coo_response_mask\n \n ##### CODE #####\n box_pred_response = bounding_box_pred[coo_response_mask].view(-1,5)\n box_target_response_iou = box_target_iou[coo_response_mask].view(-1,5)\n box_target_response = bounding_box_target[coo_response_mask].view(-1,5)\n \n \n # Find the class_loss, containing object loss and regression loss\n \n ##### CODE #####\n \n class_loss = self.get_class_prediction_loss(classes_pred, classes_target)\n \n contain_loss = self.get_contain_object_loss(box_pred_response, box_target_response_iou)\n \n reg_loss = self.get_regression_loss(box_pred_response, box_target_response)\n \n total_loss = self.l_coord*reg_loss + contain_loss + self.l_noobj*no_object_loss + class_loss\n \n return total_loss / N\n```\n\n\n```python\ncriterion = YoloLoss(S, B, lambda_coord, lambda_noobj)\noptimizer = torch.optim.SGD(net.parameters(), lr=learning_rate, momentum=0.9, weight_decay=5e-4)\n```\n\n## Reading Pascal Data\n\nSince Pascal is a small dataset (5000 in train+val) we have combined the train and val splits to train our detector. This is not typically a good practice, but we will make an exception in this case to be able to get reasonable detection results with a comparatively small object detection dataset. Use `download_data.sh` to download the dataset.\n\nThe train dataset loader also using a variety of data augmentation techniques including random shift, scaling, crop, and flips. Data augmentation is slightly more complicated for detection dataset since the bounding box annotations must be kept consistent through the transformations.\n\nSince the output of the dector network we train is a $(S, S, 5B+c)$ tensor, we use an encoder to convert the original bounding box coordinates into relative grid bounding box coordinates corresponding to the the expected output. We also use a decoder which allows us to convert the opposite direction into image coordinate bounding boxes.\n\n\n```python\n#!sh download_data.sh\n```\n\n\n```python\nfile_root_train = 'VOCdevkit_2007/VOC2007/JPEGImages/'\nannotation_file_train = 'voc2007.txt'\n\ntrain_dataset = VocDetectorDataset(root_img_dir=file_root_train,dataset_file=annotation_file_train,train=True, S=S)\ntrain_loader = DataLoader(train_dataset,batch_size=batch_size,shuffle=True,num_workers=4)\nprint('Loaded %d train images' % len(train_dataset))\n```\n\n Initializing dataset\n Loaded 5011 train images\n\n\n\n```python\nfile_root_test = 'VOCdevkit_2007/VOC2007test/JPEGImages/'\nannotation_file_test = 'voc2007test.txt'\n\ntest_dataset = VocDetectorDataset(root_img_dir=file_root_test,dataset_file=annotation_file_test,train=False, S=S)\ntest_loader = DataLoader(test_dataset,batch_size=batch_size,shuffle=False,num_workers=4)\nprint('Loaded %d test images' % len(test_dataset))\n```\n\n Initializing dataset\n Loaded 4950 test images\n\n\n\n```python\nimport warnings\nwarnings.filterwarnings('ignore')\nwarnings.simplefilter('ignore')\n```\n\n## Train detector\nNow, train your detector.\n\n\n```python\nbest_test_loss = np.inf\n\nfor epoch in range(num_epochs):\n net.train()\n \n # Update learning rate late in training\n if epoch == 30 or epoch == 40:\n learning_rate /= 10.0\n\n for param_group in optimizer.param_groups:\n param_group['lr'] = learning_rate\n \n print('\\n\\nStarting epoch %d / %d' % (epoch + 1, num_epochs))\n print('Learning Rate for this epoch: {}'.format(learning_rate))\n \n total_loss = 0.\n \n for i, (images, target) in enumerate(tqdm(train_loader, total=len(train_loader))):\n images, target = images.to(device), target.to(device)\n\n pred = net(images)\n loss = criterion(pred,target)\n total_loss += loss.item()\n \n optimizer.zero_grad()\n loss.backward()\n optimizer.step()\n print('Epoch [%d/%d], average_loss: %.4f'\n % (epoch+1, num_epochs, total_loss / (i+1)))\n \n # evaluate the network on the test data\n with torch.no_grad():\n test_loss = 0.0\n net.eval()\n for i, (images, target) in enumerate(tqdm(test_loader, total=len(test_loader))):\n images, target = images.to(device), target.to(device)\n\n pred = net(images)\n loss = criterion(pred,target)\n test_loss += loss.item()\n test_loss /= len(test_loader)\n \n if best_test_loss > test_loss:\n best_test_loss = test_loss\n print('Updating best test loss: %.5f' % best_test_loss)\n torch.save(net.state_dict(),'best_detector.pth')\n\n torch.save(net.state_dict(),'detector.pth')\n \n\n```\n\n 0%| | 0/418 [00:00

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Kontakt: prof. dr. Andrej Ko\u0161ir, andrej.kosir@lucami.fe.uni-lj.si, skype=akosir_sid

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\n\n\n## \u25a0 Cilji\n\n- Cilj: \n - Modeliranje z markovskimi verigami\n - Osnove analize stanj markovske verige\n\n\n- Slu\u010dajni proces na kon\u010dno stanjih, ki je najbolje poznan in uporaben. Omejimo se na:\n - Diskreten \u010das\n - Diskreten prostor stanj (kon\u010dno ali \u0161tevno neskon\u010dno stanj)\n\n\n- Primer: uporabniki in njihovo obna\u0161anje\n- Primeri uporabe\n - Analiza kon\u010dnega avtomata: analizaobna\u0161anja TK sistema, ...\n - Ekonomski modeli\n\n\n- Kaj nas zanima:\n - Pogostnost obiskov\n - Stanje po dolgem \u010dasu\n - Pojavi kot je absorbcija\n\n\n
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\n\n\n## \u25a0 Poglavja \n\n\n10.1. Definicija, osnovne lastnosti\n\n\u25a0 Definicija markovske verige $\\large{*}$\n\n\u25a0 O prehodni matriki, porazdelitev stanj $\\large{*}$\n\n\n10.2. Klasifikacija stanj\n\n\u25a0 Povezanost stanj in perioda $\\large{*}$\n\n\u25a0 Povrnljivo in minljivo stanje $\\large{*}$\n\n\u25a0 Ergodijska mar. veriga in \u010dasi vra\u010danja v stanje\n\n\u25a0 Stacionarna (limitna) porazdelitev $\\large{*}$\n\n\u25a0 Kon\u010dna markovska veriga $\\large{*}$\n\n\n10.3. Ocenjevanje prehodne matrike\n\n\u25a0 Ocene prehodnih verjetnosti\n\n\u25a0 Interval zaupanja in grafikov kvantilov\n\n\u25a0 Dolo\u010dane velikosti vzorca\n\n\u25a0 Numeri\u010dni vidik\n\n\u25a0 Primer: razredi uporabnikov TK storitev\n\n\n

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\n\n\n## 10.1. Definicija, osnovne lastnosti\n\n\u25a0 Definicija markovske verige\n\n\u25a0 O prehodni matriki, porazdelitev stanj\n\n\n

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\n\n## \u25a0 Definicija markovske verige\n\n\n- Prostor stanj $S$, $m=|S|$\n - Kon\u010den, neskon\u010den (\u0161teven)\n- \u010cas $T$\n - Diskreten, zvezen\n\n- Slu\u010dajni proces: \n$$ X_n : T \\to S $$\n\n- Diskretna Markovska veriga je slu\u010dajni proces na diskretnem \u010dasu in diskretnim prostorom stanj, za katerega velja pravilo prehodne verjetnosti enega koraka\n$$ P[X_{n+1}|_{X_n, X_{n-1}, \\ldots, X_0}] = P[X_{n+1}|_{X_n}] = P[X_1|_{X_0}] $$\n\n\n- Prehodna verjetnost enega koraka\n$$ p_{ij} = P[X_{n+1}=j|_{X_n=i}] $$\n\n\n- Kon\u010den prostor stanj \u2013 matrika prehodnih verjetnosti = prehodna matrija\n$$ P = [p_{ij}], \\; i,j\\in S. $$\n\nVrsti\u010dne vsote ima enake 1.\n\n\n

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\n\n\n\n```python\nimport numpy as np\nfrom sympy import Matrix\nfrom sympy.functions import re\n\n# Case 1 \n# Transition matrix\nP = np.array([[0.7, 0.2, 0.1, 0.0], \n [0.2, 0.5, 0.2, 0.1], \n [0.05, 0.15, 0.6, 0.2], \n [0.0, 0.2, 0.0, 0.8]])\n\nP\n```\n\n\n\n\n array([[0.7 , 0.2 , 0.1 , 0. ],\n [0.2 , 0.5 , 0.2 , 0.1 ],\n [0.05, 0.15, 0.6 , 0.2 ],\n [0. , 0.2 , 0. , 0.8 ]])\n\n\n\n
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\n\n\n## \u25a0 O prehodni matriki, porazdelitev stanj\n\n\n- Prehodna matrika za $n$ korakov\n - $p_{ij}^{(n)} = P[X_{k+n}=j|_{X_k=i}]$, $P^{(n)}=[p_{ij}^{(n)}]$\n - Formula o popolni verjetnosti\n $$ P^{(n)} = P^n $$\n \n \n \n- Porazdelitev stanj: \n $$ \\pi_i^n = P[X_n=i], \\pi^n = [\\pi_1^n, \\ldots, \\pi_i^n] $$\n - To je dele\u017e \u010dasa sistema v danem stanju na dolgi rok? Ne vedno.\n - Za\u010detna porazdelitev: $\\pi^0$. \u010ce je stanje $X_0=i_0$ znano, je \n $$ \\pi^0=\\delta_{i_0 i} $$ \n - Formula o popolni verjetnosti:\n $$ \\pi^n = P^{(n)} \\pi^0 $$\n - O limitni porazdelitvi kasneje\n\n

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\n\n\n## 10.2. Klasifikacija stanj\n\n\n\u25a0 Povezanost stanj in perioda\n\n\u25a0 Povrnljivo in minljivo stanje\n\n\u25a0 Ergodijska mar. veriga in \u010dasi vra\u010danja v stanje\n\n\u25a0 Stacionarna (limitna) porazdelitev\n\n\u25a0 Kon\u010dna markovska veriga\n\n\n

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10. Markovske verige
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10.2. Klasifikacija stanj
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\n\n\n## \u25a0 Povezanost stanj in perioda\n\n\n- Dosegljivo stanje\n - Stanje $j$ je dosegljivo iz stanja $i$, \u010de $\\exists n: p_{ij}^{(n)}>0$\n - Stanji sta povezani, \u010de sta vzajemno dosegljivi, ozna\u010dimo $i\\leftrightarrow j$\n - Povezanost je ekvivalen\u010dna relacija\n\n\n- Markovska veriga je nerazcepna, \u010de ima en sam razred stanj;\n - Prehodna matrika je blo\u010dno diagonalna\n\n\n- Perioda stanja $d(i)$ je najve\u010dji skupni delitelj vseh $n$, za katere velja $P_{ii}^{(n)} > 0$ \n - \u010ce je $i\\leftrightarrow j$, potem $d(i) = d(j)$;\n - Stanje je neperiodi\u010dno, \u010de je $d(i)=1$;\n \n \n- Stanje je ni\u010delno, \u010de je \n$$ \\lim_{n\\to\\infty} P_{ii}^{(n)} = 0. $$\n\n\n

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\n\n\n\n```python\n# ---------------------------------------------------------------------------------------------------\n# Classification of states\nprint ('Transition matrix P=\\n', P)\nprint ('\\nTransition matrix P^2=\\n', np.linalg.matrix_power(P, 2))\n\n# All entries of P^2 are non-zero\n# => All atates are connected\n```\n\n Transition matrix P=\n [[0.7 0.2 0.1 0. ]\n [0.2 0.5 0.2 0.1 ]\n [0.05 0.15 0.6 0.2 ]\n [0. 0.2 0. 0.8 ]]\n \n Transition matrix P^2=\n [[0.535 0.255 0.17 0.04 ]\n [0.25 0.34 0.24 0.17 ]\n [0.095 0.215 0.395 0.295]\n [0.04 0.26 0.04 0.66 ]]\n\n\n\n```python\n# ---------------------------------------------------------------------------------------------------\n# Decompose transition matrix\n# Use Jordan decomposition\n#mP = Matrix(P)\n#mB, mJ = mP.jordan_form()\n#B, J = np.array(mB), np.array(mJ)\nB = np.array([[-0.5,-0.776313,0.659526,-0.413722],\n [-0.5,-0.212847,-0.11514,0.830469],\n [-0.5,0.283275,-0.73692,-0.123153],\n [-0.5,0.521335,0.0933625,-0.352121]])\nJ = np.array([[1.,0,0,0],\n [0,0.718345,0,0],\n [0,0,0.553349,0],\n [0,0,0,0.328305]])\n\n# Test decomposition\nerr_mat = P - np.dot(B, np.dot(J, np.linalg.inv(B)))\nprint ('\\nDecomposition error norm = ', np.linalg.norm(err_mat))\n```\n\n \n Decomposition error norm = 7.249920892047682e-07\n\n\n
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10. Markovske verige
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10.2. Klasifikacija stanj
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\n\n\n## \u25a0 Povrnljivo in minljivo stanje\n\n\n- Verjetnost prve vrnitve po $n$ korakih\n$$ f_{ij}^n = P[X_n=j|_{X_{n-1}\\ne j, \\ldots, X_1\\ne j, X_0=i}] $$\n\n\n- Slu\u010dajna spremenljivka \u0161tevila korakov prehoda iz $i$ v $j$ ozna\u010dimo s $T_{ij}$ in ima porazdelitev \n$$ T_{ij} \\sim \\left(\\matrix{0 & 1 & 2 & \\ldots \\cr f_{ij}^0 & f_{ij}^1 & f_{ij}^2 & \\ldots}\\right) $$\n - Posebej ozna\u010dimo $T_i = T_{ii}$\n - Ozna\u010dimo: $\\sum_{n=0}^\\infty f_{ii}^n = f_{ii}^*$. Vrednost \n $$ 1- f_{ii}^* $$\n je verjetnost, da se nikoli ve\u010d ne vrnemo v stanje $i$ potem ko smo stanje $i$ zapustili.\n\n- Stanje $i$ je **povrnljivo**, \u010de je $f_{ii}^* = 1$. Stanje je **minljivo**, \u010de ni povrnljivo.\n - Obe lastnosti se ohranjata pri povezanosti, torej definirata ekvivalen\u010dne razrede.\n\n- Velja: stanje $i$ povrnljivo je natanko tedaj, ko je\n$$ \\sum_{n=1}^\\infty P_{ii}^{(n)} = \\infty $$\n\n\n- Velja : \u010de $i\\leftrightarrow j$, sta obe stanji ali minljivi ali povarnljivi\n\n\n

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\n\n\n\n```python\n# Recurent, transient? \n\n# Eigenvalues\neigVals, eigVecs = np.linalg.eig(P.T)\n\n# Get it sorted\nidx = eigVals.argsort()[::-1] \neigVa = eigVals[idx]\neigVc = eigVecs[:,idx]\n\nlas_bf = (1./(1-eigVa[1:4]))-1\nprint ('las_bf', las_bf)\n \nsumJn = np.diag(np.insert(las_bf, 0, np.inf))\nprint ('\\nSum J^n\\n', sumJn)\n\nsumPn = np.dot(B, np.dot(sumJn, np.linalg.inv(B)))\nprint ('\\nSum P^n\\n', sumPn)\n\n# Since all diagonal elements of sum P^n are infinity\n# => all states are recurrent\n\n```\n\n las_bf [2.55044912 1.23888596 0.48877142]\n \n Sum J^n\n [[ inf 0. 0. 0. ]\n [0. 2.55044912 0. 0. ]\n [0. 0. 1.23888596 0. ]\n [0. 0. 0. 0.48877142]]\n \n Sum P^n\n [[inf inf inf inf]\n [inf inf inf inf]\n [inf inf inf inf]\n [inf inf inf inf]]\n\n\n\n```python\ndef inf_sum(a_lst):\n return [(la/(1.0-la) if la < 1 else np.infty) for la in a_lst]\n\n# ---------------------------------------------------------------------------------------------------\n# Are states recurrent?\n# Compute sum of P^n\nsumJ = np.diag(inf_sum(np.diag(J)))\nsumP = np.dot(B, np.dot(sumJ, np.linalg.inv(B)))\n\nprint ('\\nSum of P^n = \\n', sumP)\n# => All states are reccurent \n```\n\n \n Sum of P^n = \n [[inf inf inf inf]\n [inf inf inf inf]\n [inf inf inf inf]\n [inf inf inf inf]]\n\n\n
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10. Markovske verige
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10.2. Klasifikacija stanj
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\n\n\n## \u25a0 Ergodijska mar. veriga in \u010dasi vra\u010danja v stanje\n\n\n- Markovska veriga je Ergodijska, \u010de je\n - Nerazcepna\n - Povrnljiva (vsa stanja so povrnljiva)\n - Neperiodi\u010dna (vsa stanja so nepriodi\u010dna)\n - Neni\u010delna (vsa stanja so neni\u010delna)\n \n \n- Pri\u010dakovan \u010das vra\u010danja: \n$$ E(T_i) = \\sum_{n=0}^\\infty n f_{ii}^{(n)} $$\n\n\n- Za vsa stanja Ergodijske markovske verige velja\n$$ \\lim_{n\\to\\infty} P_{ii}^{(n)} = \\frac{1}{E(T_i)} $$\n$$ \\lim_{n\\to\\infty} P_{ij}^{(n)} = \\lim_{n\\to\\infty} P_{ii}^{(n)} $$ \n\n\n\n- Povrnljivo **ni\u010delno** stanje $i$\n$$ \\lim_{n\\to\\infty} P_{ii}^{(n)} = 0 \\; \\Leftrightarrow \\; E(T_i) = \\infty $$\n\n\n- **Povrnljivo pozitivno** = **ergodijsko stanje** $i$: \n$$ \\lim_{n\\to\\infty} P_{ii}^{(n)} > 0 \\; \\Leftrightarrow \\; E(T_i) < \\infty $$\n\n\n

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\n\n\n\n```python\ndef inf_lim(a_lst):\n return [(0 if la < 1 else 1) for la in a_lst]\n\n# ---------------------------------------------------------------------------------------------------\n# Zero or positive states?\n# Compute lim of P^n\nlimJ = np.diag(inf_lim(np.diag(J)))\nlimP = np.dot(B, np.dot(limJ, np.linalg.inv(B)))\n\nprint ('\\nlim of P^n = \\n', limP)\n\n# Since all lim P^n are positive\n# => all states are positive\n\n# Expected times of visits\nETi = 1.0/np.diag(limP)\n\nprint ('\\nExpected number of visits py states:\\n', ETi)\n\n```\n\n \n lim of P^n = \n [[0.21301781 0.27218919 0.18934915 0.32544385]\n [0.21301781 0.27218919 0.18934915 0.32544385]\n [0.21301781 0.27218919 0.18934915 0.32544385]\n [0.21301781 0.27218919 0.18934915 0.32544385]]\n \n Expected number of visits py states:\n [4.69444311 3.67391519 5.28124903 3.07272668]\n\n\n
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10. Markovske verige
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10.2. Klasifikacija stanj
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\n\n\n## \u25a0 Stacionarna (limitna) porazdelitev\n\n\n- Spomnimo se \n$$ \\pi_i^n = P[X_n=i], \\qquad \\pi^n = [\\pi_1^n, \\ldots, \\pi_i^n] $$\n\n\n- Stacionarna (limitna) porazdelitev: \n $$ \\pi = \\lim_{n\\to\\infty} \\pi^n $$\n - Enoli\u010dno dolo\u010dena: vsa stanja so povrnljiva neni\u010delna\n\n\n- Za ergodijsko markovsko verigo obstaja limitna porazdelitev\n- Velja: limitna porazdelitev Ergodijske Markovske verige je levi lastni vektor prehodne matrike:\n $$ \\pi^\\top = \\pi^\\top P. $$\n\n\n\n- Opazimo: V ergodijsk verigi je limitna porazdelitev neodvisna od za\u010detne porazdelitve $\\pi^0$.\n\n\n

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\n\n\n\n```python\n# ---------------------------------------------------------------------------------------------------\n# Limit distribuion\neigVals, eigVecs = np.linalg.eig(P.T)\npi = eigVecs[:, 0]/sum(eigVecs[:, 0])\n\nprint ('\\neigVals=\\n', eigVals)\n#print '\\neigVecs=\\n', eigVecs\nprint ('\\nLimit distribution =\\n', pi)\n```\n\n \n eigVals=\n [1. 0.32830522 0.71834549 0.55334929]\n \n Limit distribution =\n [0.21301775 0.27218935 0.18934911 0.32544379]\n\n\n
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10. Markovske verige
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10.2. Klasifikacija stanj
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\n\n\n\n## \u25a0 Verjetnost absorbcije\n\n\n- Absorbcija: veriga dane podmno\u017eice stanj $S_0 \\subset S$ ne zapusti ve\u010d \n- Verjetnost absorpcije: verjetnost, da se to zgodi\n- \u010ce je veriga na za\u010detku v minljivem stanje, se vedno zgodi:\n 1. Slu\u010dajni proces z verjetnostjo 1 zapusti mno\u017eico minljivih stanj\n 2. Slu\u010dajni proces z verjetnostjo 1 ostane v mno\u017eici povrnljivih stanj, v katera je vstopil;\n\n\n- Absorbcijsko stanje $i$: edino stanje povrnljivega razreda. \n Velja: $i$ absorbirajo\u010de \u010de in samo \u010de $P_{ii}=1$.\n \n \n- Naj bo $i\\in S$ minljivo stanje, $S_e$ razred povrnljivih stanj:
\n**Verjetnost absorbcije:** $\\pi_i(S_e)$ je verjetnost, da je veriga z $X_0=i$ absorbirana v $S_e$ \n\n\n- Velja \u2013 formula za izra\u010dun - za vsak $i$\n$$ \\lim_{n\\to\\infty} P_{ij}^{(n)} = \\pi_i(S_e)\\pi_j. $$\n\n\n

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10.2. Klasifikacija stanj
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\n\n\n## \u25a0 Kon\u010dna markovska veriga\n\n- Ima kon\u010dno mno\u017eico stanj $|S|=m < \\infty$.\n- Vsaka kon\u010dna Markovska veriga stanja vsebuje tudi povrnljiva stanja.\n- Vsa povrnljiva stanja so pozitivna, torej **ergodijska** in imajo kon\u010den pri\u010dakovan \u010das vrnitve\n$$ \\lim_{n\\to\\infty} P_{ii}^{(n)} > 0. $$\n\n\n- Klasifikacija stanj: vsa stanja so $S=S_m\\cup S_e$, kjer je \n - Minljiva stanja: $S_m$\n - Ergodijska (povrnljiva neni\u010delna): $S_e$\n \n \n- Blo\u010dna razdelitev prehodne matrike: uredimo indekse stanj\n - Ergodisjka stanja: $1, \\ldots, m-K$\n - Minljiva stanja: $m-K+1, \\ldots, m$\n $$ P = \\left[\\matrix{S & 0 \\cr R & Q}\\right] $$\n \n \n- Fundamentalna matrika: \n $$ N = (I - Q)^{-1}. $$\n\n\n

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\n\n\n\n```python\nimport numpy as np\n# Case 2\nc2P = np.array(\n [[0.7, 0.3, 0.0, 0.0], \n [0.2, 0.8, 0.0, 0.0], \n [0.05,0.15, 0.6, 0.2], \n [0.0, 0.2, 0.0, 0.8]])\nprint ('Transition matrix P=\\n', c2P)\n\n```\n\n Transition matrix P=\n [[0.7 0.3 0. 0. ]\n [0.2 0.8 0. 0. ]\n [0.05 0.15 0.6 0.2 ]\n [0. 0.2 0. 0.8 ]]\n\n\n\n```python\n# ---------------------------------------------------------------------------------------------------\n# Classification of states\nprint ('Transition matrix P=\\n', c2P)\nprint ('\\nTransition matrix P^2=\\n', np.linalg.matrix_power(c2P, 2))\nprint ('\\nTransition matrix P^3=\\n', np.linalg.matrix_power(c2P, 3))\n\n# Matrix multiplications by blocks\n# => 2 x 2 block of zeros stayes the same\n# => There are two connected classes of states\n\n```\n\n Transition matrix P=\n [[0.7 0.3 0. 0. ]\n [0.2 0.8 0. 0. ]\n [0.05 0.15 0.6 0.2 ]\n [0. 0.2 0. 0.8 ]]\n \n Transition matrix P^2=\n [[0.55 0.45 0. 0. ]\n [0.3 0.7 0. 0. ]\n [0.095 0.265 0.36 0.28 ]\n [0.04 0.32 0. 0.64 ]]\n \n Transition matrix P^3=\n [[0.475 0.525 0. 0. ]\n [0.35 0.65 0. 0. ]\n [0.1375 0.3505 0.216 0.296 ]\n [0.092 0.396 0. 0.512 ]]\n\n\n\n```python\n# ---------------------------------------------------------------------------------------------------\n# Decompose transition matrix\n# Use Jordan decomposition\n#mP = Matrix(c2P)\n#mB, mJ = mP.jordan_form()\n#B, J = np.array(mB), np.array(mJ)\nc2B = np.array([[1.0, 0, 0, 10.0/4],\n [1.0, 0, 0, -3.0/2],\n [1.0, 1.0, 1.0, -7.0/8],\n [1.0, 1.0, 0, 1.0]])\nc2J = np.array([[1.0,0,0,0],\n [0,4.0/5,0,0],\n [0,0,3.0/5,0],\n [0,0,0,1.0/2]])\n\n# Test decomposition\nerr_mat = c2P - np.dot(c2B, np.dot(c2J, np.linalg.inv(c2B)))\nprint ('\\nDecomposition error norm = ', np.linalg.norm(err_mat))\n```\n\n \n Decomposition error norm = 1.9675159943996247e-16\n\n\n\n```python\ndef prd(a,b):\n if np.isinf(a) and b == 0:\n return 0\n if np.isinf(b) and a == 0:\n return 0\n return a*b\n\ndef myDot(A,B):\n nA1, nA2, nB2 = A.shape[0], A.shape[1], B.shape[1]\n AB = np.zeros((nA1,nB2))\n for i in range(nA1):\n for j in range(nB2):\n curr = 0\n for k in range(nA2):\n curr += prd(A[i,k], B[k,j])\n AB[i, j] = curr\n return AB\n\n# Classification of states\n# Recurent, transient? \n\n# Eigenvalues\neigVals, eigVecs = np.linalg.eig(c2P.T)\n\n# Get it sorted\nidx = eigVals.argsort()[::-1] \neigVa = eigVals[idx]\neigVc = eigVecs[:,idx]\n\nlas_bf = (1./(1-eigVa[1:4]))-1\nsumJn = np.diag(np.insert(las_bf, 0, np.inf))\n\nsumPn = myDot(c2B, myDot(sumJn, np.linalg.inv(c2B)))\nprint ('\\nSum P^n\\n', sumPn)\n\n# Since diagonal elements of sum P^n i=1 and i=2 are infinity\n# => states i=1 and i=2 are recurrent\n# Since diagonal elements of sum P^n i=3 and i=4 are finite\n# => states i=3 and i=4 are transient\n\n```\n\n \n Sum P^n\n [[inf inf 0. 0. ]\n [inf inf 0. 0. ]\n [inf inf 1.5 2.5]\n [inf inf 0. 4. ]]\n\n\n\n```python\ndef inf_lim(a_lst):\n return [(0 if la < 1 else 1) for la in a_lst]\n\n# ---------------------------------------------------------------------------------------------------\n# Zero or positive states?\n# Compute lim of P^n\nlimJ = np.diag(inf_lim(np.diag(c2J)))\nlimP = np.dot(c2B, np.dot(limJ, np.linalg.inv(c2B)))\n\nprint ('\\nlim of P^n = \\n', limP)\n\n# Since all lim P^n are positive\n# => all states are positive\n\n# Expected times of visits\nETi = 1.0/np.diag(limP)\n\nprint ('\\nExpected number of visits py states:\\n', ETi)\n```\n\n \n lim of P^n = \n [[0.4 0.6 0. 0. ]\n [0.4 0.6 0. 0. ]\n [0.4 0.6 0. 0. ]\n [0.4 0.6 0. 0. ]]\n \n Expected number of visits py states:\n [2.5 1.66666667 inf inf]\n\n\n /home/nbuser/anaconda2_501/lib/python2.7/site-packages/ipykernel/__main__.py:16: RuntimeWarning: divide by zero encountered in divide\n\n\n\n```python\n# ---------------------------------------------------------------------------------------------------\n# Limit distribuion\neigVals, eigVecs = np.linalg.eig(c2P.T)\n\n\n# Get it sorted\nidx = eigVals.argsort()[::-1] \neigenValues = eigVals[idx]\neigenVectors = eigVecs[:,idx]\n\n# Normalize eigenvector\npi = eigenVectors[:, 0]/sum(eigenVectors[:, 0])\n\nprint ('\\neigVals=\\n', eigenValues)\n#print '\\neigVecs=\\n', eigenVectors\nprint ('\\nLimit distribution =', pi)\n```\n\n \n eigVals=\n [1. 0.8 0.6 0.5]\n \n Limit distribution = [ 0.4 0.6 -0. -0. ]\n\n\n
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10. Markovske verige
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10.2. Klasifikacija stanj
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\n\n\n\n## \u25a0 \u0160tevilo obiskov minljivega stanja\n\n- Slu\u010dajna spremenljivka $n_i$: \u0161tevilo obiskov stanja $i$ v poljubno dolgem \u010dasu\n- Matemati\u010dno upanje \u0161tevila obiskov $j$ \u010de je za\u010detno stanje $i$: \n $$ E_i[n_j] < \\infty $$\n\n\n- Velja: \u010de je stanje $i$ minljivo, je \n $$ E_i[n_j] = \\sum_{k=1}^\\infty P_{ij}^{(k)} $$\n\n\n- Velja: fundamentalna matrika vsebuje pri\u010dakovana \u0161tevila vra\u010danj:\n$$ \\left[E_i[n_j]\\right] = N $$ \n\n\n\n

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\n\n\n\n```python\n# Expected number of visits of recurent states\nm = 4 # |S|\nK = 2 # number of recurent states\n\nQ = c2P[K:m, K:m]\nprint ('Part of transition matrix Q = \\n', Q)\n\nN = np.linalg.inv(np.eye(m-K)-Q)\nprint ('\\nFundamental matrix N =\\n', N)\n```\n\n Part of transition matrix Q = \n [[0.6 0.2]\n [0. 0.8]]\n \n Fundamental matrix N =\n [[2.5 2.5]\n [0. 5. ]]\n\n\n
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10. Markovske verige
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10.3 Ocenjevanje prehodne matrike
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\n\n\n## 10.3 Ocenjevanje prehodne matrike\n\n\u25a0 Ocene prehodnih verjetnosti\n\n\u25a0 Interval zaupanja in grafikov kvantilov\n\n\u25a0 Dolo\u010dane velikosti vzorca\n\n\u25a0 Numeri\u010dni vidik\n\n\u25a0 Primer: razredi uporabnikov TK storitev\n\n\n

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10.3 Ocenjevanje prehodne matrike
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\n\n\n\n## \u25a0 Ocene prehodnih verjetnosti\n\n- Preodna verjetnost\n$$ p_{ij} = \\frac{n_k}{n} $$\nje statisti\u010dna ocena;\n\n\n- Ocena parametra (npr. povpre\u010dje ali dele\u017e): \n - ocenimo in dolo\u010dimo\n - povpre\u010dje - pri\u010dakovano vrednost: $\\overline{x}$\n - standardni odklon: $\\sigma$\n - za stopnjo tveganja $\\alpha$ dolo\u010dimo $z_\\alpha$ (za normalno porazdelitev je $z_{0.05}=1.96$)\n - Interval zaupanja: \n $$ CI = [\\overline{x} - z_\\alpha\\frac{\\sigma}{\\sqrt{n}}, \\overline{x} + z_\\alpha\\frac{\\sigma}{\\sqrt{n}}] $$\n - Velikost vzorca: dolo\u010dimo ga glede na predpisano velikost intervala zaupanja\n $$ |CI| = 2 z_\\alpha\\frac{\\sigma}{\\sqrt{n}} $$\n\n\n\n\n\n\n

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10. Markovske verige
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10.3 Ocenjevanje prehodne matrike
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\n\n\n\n## \u25a0 Interval zaupanja in grafikov kvantilov\n\n\n- Vizualizacija intervala zaupanja\n\n\n\n\n\n

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10.3 Ocenjevanje prehodne matrike
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\n\n\n\n## \u25a0 Dolo\u010dane velikosti vzorca\n\n- Iz zahteve po \u0161irini intervala zaupanja\n- Graf velikosti vzorca za oceno dele\u017ea $\\alpha=0.05$ ($z_\\alpha=1.96$)\n$$ CI = [p_0 - z_\\alpha\\sqrt{p_0(1-p_0)/n}, p_0 + z_\\alpha\\sqrt{p_0(1-p_0)/n}] $$\n\n\n- Za primer $p=0.65$ in $|CI|=\\Delta p$ pri $\\alpha=0.05$ velja neena\u010dba\n$$ n\\geq \\frac{4 z_\\alpha^2}{\\Delta p^2} p_0 (1-p_0) $$\n\n- Od tod dobimo:\n - Za $\\Delta p=0.01:$ $n\\geq 34959$,\n - Za $\\Delta p=0.02:$ $n\\geq 175$,\n - Za $\\Delta p=0.04:$ $n\\geq 88$.\n\n\n

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10.3 Ocenjevanje prehodne matrike
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\n\n\n\n## \u25a0 Numeri\u010dni vidik - matri\u010dni ra\u010dun\n\n\n- Potrebujemo izra\u010dun spektra in potenc \n$$ P^n = P^{(n)} $$\n\n\n- Jordanova dekompozicija\n$$ P = B J B^{-1}, $$\nkjer je $J$ Jordanova kanoni\u010dna forma.\n\n\n- Za vsak razred stanj posebej velja\n$$ J = \\lambda I + R $$\n\n\n- Velja\n$$ P^n = B J^n B^{-1} $$\n\n\n

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10.3 Ocenjevanje prehodne matrike
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\n\n\n\n## \u25a0 Primer: razredi uporabnikov TK storitev\n\n\n- Mno\u017eica uporabnikov TK storitve $U$\n- Razdelimo jih po zadovoljstvu: $U=U_1 \\cup U_2 \\cup U_3 \\cup U_4$\n - Zelo zadovoljni: $U_1$\n - Zadovoljni: $U_2$\n - Nezadovoljni: $U_3$\n - Zapu\u0161\u010dajo ponudnika: $U_4$\n\n\n- Za\u010detna porazdelitev: ocena trenutnega stanja uporabnikov\n- Rezultati:\n - Povrnljivost - minljivost stanj in razredi teh stanj\n - Limitna porazdelitev\n\n\n\n

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10. Markovske verige
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10.3 Ocenjevanje prehodne matrike
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\n\n\n\n## \u25a0 Zaklju\u010dek\n\n- Preprosta in uporabna analiza kon\u010dnih stanj\n- Preverjanje pogojev pri oceni za\u010detne porazdelitve in prehodne matrike\n\n

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\n\n", "meta": {"hexsha": "54681ba14e74bdbfe2ffef00659bf91add8bb158", "size": 44191, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "AKosir-OPvTK-Lec10_MarkovChains_SLO.ipynb", "max_stars_repo_name": "andrejkk/OPvTK", "max_stars_repo_head_hexsha": "769b4dc144fa07e4604d945df5672f14741154b5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "AKosir-OPvTK-Lec10_MarkovChains_SLO.ipynb", "max_issues_repo_name": "andrejkk/OPvTK", "max_issues_repo_head_hexsha": "769b4dc144fa07e4604d945df5672f14741154b5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "AKosir-OPvTK-Lec10_MarkovChains_SLO.ipynb", "max_forks_repo_name": "andrejkk/OPvTK", "max_forks_repo_head_hexsha": "769b4dc144fa07e4604d945df5672f14741154b5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 31.7692307692, "max_line_length": 194, "alphanum_fraction": 0.4985630558, "converted": true, "num_tokens": 9915, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878696277513, "lm_q2_score": 0.7279754548076477, "lm_q1q2_score": 0.44776887163892937}} {"text": "```python\nfrom decodes.core import *\nfrom decodes.io.jupyter_out import JupyterOut\nimport math\n\nout = JupyterOut.unit_square( )\n```\n\n# Lines and Planes in Cartesian Space\n\nThe equation for a line in $\\mathbb{R}^2$ is $y = mx + b$, which describes a line with slope $m$ and y-intercept $b$. It may seem odd at first that this same equation in $\\mathbb{R}^3$ does not describe a line, but rather a plane. Odd, that is, until we consider a line as something like a lower-dimensional section through a higher-dimensional plane.\n\nHere we present a representation of lines and planes from the point of view of vectors. Such an approach will lead us away from the familiar ***general equations***, and to the so-called ***parametric equation*** of a line and the ***normal form*** equation of a plane.\n\n## Line Representation\n\nWe know that lines are usually represented as an equation such as\n\n\\begin{align}\ny = mx + b\n\\end{align}\n\nPoints along the line can then be easily plotted by picking values for $x$ and solving for the corresponding coordinate value $y$.\n\nFor example, we can easily see that the line \n\n\\begin{align}\ny = 3x - 1 \n\\end{align}\n\ncontains the points $(0, -1)$, $(1,2)$, $(2,5)$, as each of these combinations of coordinates satisfy the equation. Conversely, given any two points, this equation can be found by solving for the slope $m$, the ratio with the easy mnemonic \u201crise over run\u201d. \n\nFor two points $(3,1)$ and $(2,5)$, $m = (5 - 1) (2 - 3) = -4$ and the equation for the line can be solved for by using one of the given points to get $y - 1 =- 4(x - 3)$, which can then be put in the standard form $y =- 4x + 13$.\n\nAn important restriction to this representation is that ***these equations only describe lines in two dimensions***. In $\\mathbb{R}^3$, the equation describes a plane.\n\nA different formulation for a line may be found by returning to our fundamental description of a line: ***an object determined by two points***.\n\nWe begin with a line $L$ and two points $P_{0}$ and $P_{1}$ that lie upon it. \n\nFollowing the convention used in a previous section, remember that $\\vec{p_{i}}$ refers to the vector from the origin to the point $P_{i}$, with both vector and point having the same coordinates. \n\nIt then follows that the vector $\\vec{v} = \\vec{p_{1}}-\\vec{p_{0}}$ is in the direction of the line, which, together with a point on the line (we will use $P_{0}$ for this), is enough to determine $L$.\n\nWith this in hand, ***to arrive at a functional formulation, we need only to be able to express any point $P$ on the line in terms of quantities $\\vec{v}$ and $P_{0}$***.\n\n\n\n\nTo this end, we begin by forming a triangle with vertices at the origin, $P_{0}$ and $P_{1}$. \n\nThe nearby diagram demonstrates that we can get to $P$ by starting at the origin, moving along $\\vec{p_{0}}$ to $P_{0}$, and then moving again along a suitable multiple of $\\vec{v}$. \n\nIn vector notation this can be written as seen below, a relationship that works for any point on the line given a suitable choice of $t$.\n\n\\begin{align}\n\\vec{p} = \\vec{p_{0}} + t\\vec{v}\n\\end{align}\n\nAll vectors found in this manner will fall on Line $L$.\n\n\n\nTo further differentiate between those vectors that fall on Line $L$, by vector addition, the diagram below demonstrates that values of $t$ between $0$ and $1$ will completely fill out the line segment that spans between the two given points. \n\nWhen $t < 0$ points fall on the side of $P_{0}$ away from $P_{1}$, and which we may read as falling \u201cbefore\u201d the start-point of the segment. \n\nFor values $t > 1$, the points fall on the side of $P_{1}$ away from $P_{0}$, or \u201cafter\u201d the end-point of the segment.\n\n\n\nThe variable $t$ is the only variable in this equation, since $\\vec{v}$ and $P_{0}$ are both fixed quantities for a given line. Applying such an equation, evaluating points along the line is a matter of picking values of $t$, which we call the ***parameter*** for evaluating a line, and this vector equation as a ***parametric equation*** for a line.\n\nNote the ***inherent directionality*** to this representation for plotting points on lines corresponding to increasing value of $t$. Since only ***one parameter is required to determine a position on a line***, we may consider it a single dimensional object.\n\nStarting with just two points on a line, we have now arrived at a representation that allows us to express this line in terms of a ***point on the line*** and a ***vector***, and to plot points by varying a parameter ***t***. \n\nNote that our account thus far captures an infinite line. Building on the diagrams above, we can further define two related entities by simply constraining the value of the parameter.\n\nA ***segment*** can be described by limiting the values of t to be in the range $0 <= t <= 1$.\n\nSimilarly, a ***ray*** can be achieved by letting the parameter vary in the interval $0 <= t < \\infty$ or $-\\infty < t <= 1$.\n\n## Plane Representation\n\nOur formulation of planes takes a similar route as our presentation of lines, again starting with the plotting of points using elementary geometry.\n\nIn three dimensions, a plane has the general scalar equation $ax + by + cz = d$. Points on a plane can be plotted by picking two coordinate values (say $x$ and $y$) and then solving for the third ($z$).\n\nFor instance, the plane $-x + 3y - z = 0$ contains the points $(1,0, -1)$, $(0,1,3)$ and $(1,1,2)$, as each of these combinations of coordinates satisfy the equation.\n\nTo represent a plane from a vector point of view, we start by defining a new entity: the ***normal vector***. \n\nGiven a plane, ***a vector $\\vec{n}$ is said to be normal to the plane if it is perpendicular to any vector which lies in the plane***. \n\nThis definition can be formulated more precisely by drawing $\\vec{n}$ with its tail at any fixed point $P_{0}$ in the plane, and then taking another vector that connects $P_{0}$ to any point $P$ in the plane. \n\nThis perpendicularity condition yields the normal form equation for a plane:\n\n\\begin{align}\n\\vec{n} \\bullet (\\vec{p} - \\vec{p_{0}}) = 0\n\\end{align}\n\nThis equation states that ***a plane in three dimensions is determined by a point and vector*** - the same two entities that form a line. The text demonstrates the association between the normal (vector) form and the general (scalar) form of this equation.\n\n\n\nHere we describe how to find the equation for a plane that encompasses three points. \n\nGiven three arbitrary (but non-collinear) points, we can find the normal vector, and, using this vector direction, derive the equation of the plane. Let $P_{0}$, $P_{1}$ and $P_{2}$ be points not all lying along the same line. Forming the vectors $\\vec{v_{1}} = \\vec{p_{1}} - \\vec{p_{0}}$ and $\\vec{v_{2}} = \\vec{p_{2}} - \\vec{p_{0}}$ both lying in the plane, the cross product gives a vector that is perpendicular to the plane determined by these two vectors, which is precisely the normal vector of the desired plane.\n\n\n\nLike a line, there also exists ***a parametric equation for a plane*** that allows for the evaluation of points that fall upon it. However, it arrives with a caveat that will prevent us from incorporating an evaluation method of planes in code. As a two-dimensional object, ***two parameters are required to determine a position on a plane***.\n\nWhich is to say, evaluating points on the plane is a calculation that requires ***pairs of parameter values***, a process which brings to mind the discussion of coordinates. In fact, when the two vectors in the equation above are orthonormal, the action demonstrated here is exactly the process for evaluating a coordinate system. Given that we will define a Plane as the pairing of one Point and a single Vec, the evaluation of a Plane would require a supplementary vector, which is equivlient to a CS. \n\nFor this reason, we do not evaluate coordinates on a Plane.\n", "meta": {"hexsha": "74c1ef958b2bff0af0c6453a4a8f19c7055ab980", "size": 11088, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "106 - Lines and Planes/209 - Lines and Planes in Cartesian Space.ipynb", "max_stars_repo_name": "ksteinfe/decodes_ipynb", "max_stars_repo_head_hexsha": "2e4bb6b398472fc61ef8b88dad7babbdeb2a5754", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-05-15T14:31:23.000Z", "max_stars_repo_stars_event_max_datetime": "2018-05-15T14:31:23.000Z", "max_issues_repo_path": "106 - Lines and Planes/209 - Lines and Planes in Cartesian Space.ipynb", "max_issues_repo_name": "ksteinfe/decodes_ipynb", "max_issues_repo_head_hexsha": "2e4bb6b398472fc61ef8b88dad7babbdeb2a5754", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "106 - Lines and Planes/209 - Lines and Planes in Cartesian Space.ipynb", "max_forks_repo_name": "ksteinfe/decodes_ipynb", "max_forks_repo_head_hexsha": "2e4bb6b398472fc61ef8b88dad7babbdeb2a5754", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2020-05-19T05:40:18.000Z", "max_forks_repo_forks_event_max_datetime": "2020-06-28T02:18:08.000Z", "avg_line_length": 49.5, "max_line_length": 531, "alphanum_fraction": 0.6387987013, "converted": true, "num_tokens": 1952, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632683808532, "lm_q2_score": 0.8244619306896955, "lm_q1q2_score": 0.44757009834979655}} {"text": "## Word Embeddings\n\n**Word Embeddings transform text into a sequence of vectors.**\n\n### Objectives:\n\n* Introduction:\n\n* Understand the concepts in word embeddings\n\n* Use a pretrained word2vec model (SpaCy)\n\n* Train a word2vec model on your own corpus (gensim)\n\n* Visualize the results using PCA\n\n*prerequisites: spacy, gensim*\n\n---\n\n## Introduction\n\n* **Word vectors** are dense representations of words (in contrast to a huge sparse binary matrix), that allows to feed text into neural networks.\n\n - A **Word embedding** is the transformation of all input words into word vectors. \n \n**This approach is also referred to as Word2Vec.**\n\n\n\n* What: Word embeddings are dense representations of words in a low-dimensional vector space. Embeddings translate discrete variables (words/phrases) to continuous vectors. Word embedding models (Genism / Glove) are to NLP as VGGNET is to image recognition; a pre-trained model, ready for use out the box.\n\n\n* Why: three main uses for word embeddings:\n\n 1. Finding nearest neighbors in the embedding space. These can be used to find similarities in terms, make recommendations based on user interests, cluster categories etc.\n\n 2. For visualization of concepts and relations between categories.\n\n 3. As input to a machine learning model for a supervised task.\n\n* How: Optimize a supervised learning problem (neural word embedding), then use the weights of the activation layer vector as the word embedding representation of a particular word.\n\nsee ` Efficient Estimation of Word Representations in Vector Space by Mikolov et al.,2013` for more information\n\n---\n\n### Understand the concepts\n\nCheck out the `Tensorflow embedding projector` to see a dimensionality reduced representation of the similarity between words\n\n### The task:\n* Predict context\n\n\n\n#### Input to the model:\n* Tokenize text\n * Split your corpus\n * Enumerate through the set of words\n * Create a word dictionary and an id dictionary\n * Also works with CountVectorizer's vocabulary_ attribute\n\n\n```python\nimport pandas as pd\nfrom sklearn.feature_extraction.text import CountVectorizer\n```\n\n\n```python\nc = CountVectorizer()\n```\n\n\n```python\ncorpus = ['The cat sat on the mat',\n 'She sells sea shells on the sea shore',\n 'Backstreets back alright',\n 'The sun is shining']\n```\n\n\n```python\nvec_corpus = c.fit_transform(corpus)\n```\n\n\n```python\nvec_corpus.todense()\n```\n\n\n\n\n matrix([[0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2],\n [0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 1, 0, 1, 0, 1],\n [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1]], dtype=int64)\n\n\n\n\n```python\nc.vocabulary_\n```\n\n\n\n\n {'the': 15,\n 'cat': 3,\n 'sat': 7,\n 'on': 6,\n 'mat': 5,\n 'she': 10,\n 'sells': 9,\n 'sea': 8,\n 'shells': 11,\n 'shore': 13,\n 'backstreets': 2,\n 'back': 1,\n 'alright': 0,\n 'sun': 14,\n 'is': 4,\n 'shining': 12}\n\n\n\n\n```python\ndf = pd.DataFrame(vec_corpus.todense(), columns = list(sorted(c.vocabulary_)))\ndf\n```\n\n\n\n\n
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\n\n\n\n### The model\n\n\n\n## CBOW - Continuous Bag of Words\n\n### Output from the model:\n\n* A vectorized representation of the predictor word\n\n* Lets see a pretrained model's implementation\n \n\n---\n\n### Use a pretrained word2vec model (spacy)\n\n\n```python\ncorpus.append('US political parties can succeed')\n```\n\n\n```python\nimport spacy\n\nnlp = spacy.load('en_core_web_sm')\n```\n\n\n```python\ndoc = nlp(corpus[-1])\n```\n\n\n```python\ndoc\n```\n\n\n\n\n US political parties can succeed\n\n\n\n\n```python\npolitical = doc[1]\nparties = doc[2]\nsucceed = doc[4]\n```\n\n\n```python\npolitical.vector.shape\n```\n\n\n\n\n (96,)\n\n\n\n\n```python\nprint(political.similarity(parties))\nprint(political.similarity(succeed))\n```\n\n 0.09397331\n -0.11618046\n\n\n /Users/maximcondon/anaconda3/lib/python3.7/runpy.py:193: ModelsWarning: [W007] The model you're using has no word vectors loaded, so the result of the Token.similarity method will be based on the tagger, parser and NER, which may not give useful similarity judgements. This may happen if you're using one of the small models, e.g. `en_core_web_sm`, which don't ship with word vectors and only use context-sensitive tensors. You can always add your own word vectors, or use one of the larger models instead if available.\n \"__main__\", mod_spec)\n /Users/maximcondon/anaconda3/lib/python3.7/runpy.py:193: ModelsWarning: [W007] The model you're using has no word vectors loaded, so the result of the Token.similarity method will be based on the tagger, parser and NER, which may not give useful similarity judgements. This may happen if you're using one of the small models, e.g. `en_core_web_sm`, which don't ship with word vectors and only use context-sensitive tensors. You can always add your own word vectors, or use one of the larger models instead if available.\n \"__main__\", mod_spec)\n\n\n---\n\n### Train a word2vec model on your own corpus (gensim)\n\n\n```python\nfrom gensim.models import Word2Vec\n```\n\n\n```python\nnew_corpus = [each.lower().split() for each in corpus]\nnew_corpus\n```\n\n\n\n\n [['the', 'cat', 'sat', 'on', 'the', 'mat'],\n ['she', 'sells', 'sea', 'shells', 'on', 'the', 'sea', 'shore'],\n ['backstreets', 'back', 'alright'],\n ['the', 'sun', 'is', 'shining'],\n ['us', 'political', 'parties', 'can', 'succeed']]\n\n\n\n\n```python\nmodel = Word2Vec(new_corpus, min_count=1, size=len(corpus), window=3)\n```\n\n- size : int, optional - Dimensionality of the word vectors.\n- window : int, optional - Maximum distance between the current and predicted word within a sentence.\n- min_count : int, optional - Ignores all words with total frequency lower than this.\n\n\n```python\nwords = list(model.wv.vocab)\n```\n\n\n```python\nwords\n```\n\n\n\n\n ['the',\n 'cat',\n 'sat',\n 'on',\n 'mat',\n 'she',\n 'sells',\n 'sea',\n 'shells',\n 'shore',\n 'backstreets',\n 'back',\n 'alright',\n 'sun',\n 'is',\n 'shining',\n 'us',\n 'political',\n 'parties',\n 'can',\n 'succeed']\n\n\n\n\n```python\ntype(model.wv.vocab)\n```\n\n\n\n\n dict\n\n\n\n---\n\n### And visualise the results\n\nGoing to perform dimensionality reduction with PCA\n\n\n```python\nfrom sklearn.decomposition import PCA\n```\n\n\n```python\npca = PCA(n_components=2) # 2 dimensions\n\nX = model[model.wv.vocab]\nprint(X.shape)\n```\n\n (21, 5)\n\n\n /Users/maximcondon/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:3: DeprecationWarning: Call to deprecated `__getitem__` (Method will be removed in 4.0.0, use self.wv.__getitem__() instead).\n This is separate from the ipykernel package so we can avoid doing imports until\n\n\n\n```python\nnew_x = pca.fit_transform(X)\n```\n\n\n```python\nfrom matplotlib import pyplot as plt\n%matplotlib inline\n\nplt.figure(figsize=(8,8))\nplt.scatter(new_x[:,0], new_x[:, 1])\n\nfor i, word in enumerate(words):\n plt.annotate(word, fontsize=12, xy=(new_x[i,0], new_x[i, 1]))\nplt.show()\n```\n\n## Skip-Grams\n\n**Skip-grams** are simply pairs of words that are 1 or more words apart. Skip grams are frequently used to calculate word embeddings:\n\n\n\n1. Tokenizing the data follows the same principles as in a Bag-of-words. To represent spatial relationships between words.\n2. Consider words and their context by generating pairs of words up to N words apart.\n3. Count the co-occurence of words X (a huge sparse matrix)\n4. Optimize a loss function for the word vectors w over words i and j\n\n\\begin{align}\nJ \\approx \\sum_{ij}(w_i^T w_j - log X_{ij})^2\n\\end{align}\n\n### Pretrained embeddings\n\n- Because calculating a word embedding from scratch is costly (millions of tokens, 1 million parameters and more), it is often useful to start with pretrained word vectors. The two systems most frequently used are **GenSim** and **GloVe**.\n\n### Embedding Layers\nAn embedding layer transforms one-hot encoded input (usually with a huge number of dimensions) to much fewer dimensions, e.g. representing a word as a 50 or 200-dimensional vector.\n\nThere are plenty of uses and architectures for Word Embeddings. The majority of these have an Embedding Layer either as the first layer (to understand text) or as the last layer (to produce text).\n\nHere is a typical architecture\n\n\nTo condense the information in a longer text sequence, you can use one or more **1D convolutional layers**.\n\n\n## Cosine Similarity\n- You can compare word vectors using the cosine similarity:\n\n\n```python\nimport numpy as np\n\ndef cosine_similarity(a, b):\n \"\"\"Returns cosine similarity of two word vectors\"\"\"\n return a.dot(b) / (np.linalg.norm(a) * np.linalg.norm(b))\n```\n\n\n```python\nmodel = Word2Vec(new_corpus, min_count=1, size=len(corpus), window=3)\n```\n\n\n```python\nsea = model.wv['sea']\nsea\n```\n\n\n\n\n array([ 0.08907603, -0.05653713, -0.01306491, -0.09036548, 0.00158615],\n dtype=float32)\n\n\n\n\n```python\nshells = model.wv['shells']\nshells\n```\n\n\n\n\n array([-0.07308082, -0.0746229 , 0.08016061, 0.03841601, -0.0146901 ],\n dtype=float32)\n\n\n\n\n```python\ncosine_similarity(sea, shells)\n```\n\n\n\n\n -0.35500664\n\n\n\n### Embeddings for other types of data\n- Embeddings can be used for other types of sparse data as well, e.g. to build recommender systems (see the parallel to NMF).\n\n## Examples\n\n### Create GenSim Vectors\n\n\n\n```python\nfrom gensim.models import word2vec\n\ncorpus = [\n \"She loves you yeah yeah yeah\",\n \"see you later alligator\",\n \"see you later crocodile\",\n \"i just call to say i love you\",\n \"and it seems to me you lived your life like a candle in the wind\",\n \"baby you can drive my car\",\n \"we all live in the yellow submarine\"\n]\n```\n\n\n```python\ntokenised = [s.lower().split() for s in corpus]\n\nwv = word2vec.Word2Vec(tokenised, size=7, window=5, min_count=1)\nwords = list(wv.wv.vocab)\nlen(words)\n```\n\n\n\n\n 37\n\n\n\n\n```python\nwords\n```\n\n\n\n\n ['she',\n 'loves',\n 'you',\n 'yeah',\n 'see',\n 'later',\n 'alligator',\n 'crocodile',\n 'i',\n 'just',\n 'call',\n 'to',\n 'say',\n 'love',\n 'and',\n 'it',\n 'seems',\n 'me',\n 'lived',\n 'your',\n 'life',\n 'like',\n 'a',\n 'candle',\n 'in',\n 'the',\n 'wind',\n 'baby',\n 'can',\n 'drive',\n 'my',\n 'car',\n 'we',\n 'all',\n 'live',\n 'yellow',\n 'submarine']\n\n\n\n\n```python\nprint(wv['crocodile'])\n```\n\n [-0.05908774 -0.05743772 0.06097939 -0.01397748 -0.01745179 -0.04459087\n -0.00569117]\n\n\n /Users/maximcondon/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:1: DeprecationWarning: Call to deprecated `__getitem__` (Method will be removed in 4.0.0, use self.wv.__getitem__() instead).\n \"\"\"Entry point for launching an IPython kernel.\n\n\n\n```python\nwv.wv.most_similar('crocodile')\n```\n\n\n\n\n [('and', 0.8792561292648315),\n ('yellow', 0.7760361433029175),\n ('in', 0.7684190273284912),\n ('loves', 0.7585940361022949),\n ('wind', 0.7297528982162476),\n ('just', 0.5329340100288391),\n ('we', 0.43828681111335754),\n ('baby', 0.4327443540096283),\n ('your', 0.3974326550960541),\n ('all', 0.3879947364330292)]\n\n\n\n#### Pretty terrible results!\n\n### Download Pretrained GenSim Vectors\n\n\n\n```python\nimport os\nfrom keras.utils import get_file\nimport gensim\nimport subprocess\n\n\nMODEL = 'GoogleNews-vectors-negative300.bin'\npath = get_file(MODEL + '.gz', 'https://s3.amazonaws.com/dl4j-distribution/%s.gz' % MODEL)\nif not os.path.isdir('generated'):\n os.mkdir('generated')\n\nunzipped = os.path.join('generated', MODEL)\nif not os.path.isfile(unzipped):\n with open(unzipped, 'wb') as fout:\n zcat = subprocess.Popen(['zcat'],\n stdin=open(path),\n stdout=fout\n )\n zcat.wait()\n\n\nmodel = gensim.models.KeyedVectors.load_word2vec_format(unzipped, binary=True)\n```\n\n Downloading data from https://s3.amazonaws.com/dl4j-distribution/GoogleNews-vectors-negative300.bin.gz\n 1647050752/1647046227 [==============================] - 336s 0us/step\n\n\n### Similarity of Words\n\n\n```python\nmodel.most_similar(positive=['espresso'])\n```\n\n\n\n\n [('cappuccino', 0.6888186931610107),\n ('mocha', 0.6686208248138428),\n ('coffee', 0.6616826057434082),\n ('latte', 0.6536753177642822),\n ('caramel_macchiato', 0.6491267681121826),\n ('ristretto', 0.6485546827316284),\n ('espressos', 0.6438628435134888),\n ('macchiato', 0.6428250074386597),\n ('chai_latte', 0.6308027505874634),\n ('espresso_cappuccino', 0.6280542612075806)]\n\n\n\n### Vector Addition\n\n\n```python\ndef A_is_to_B_as_C_is_to(a, b, c, topn=1):\n a, b, c = map(lambda x:x if type(x) == list else [x], (a, b, c))\n res = model.most_similar(positive=b + c, negative=a, topn=topn)\n if len(res):\n if topn == 1:\n return res[0][0]\n return [x[0] for x in res]\n return None\n\n\nfor country in 'Italy', 'France', 'India', 'China':\n print('%s is the capital of %s' % \n (A_is_to_B_as_C_is_to('Germany', 'Berlin', country), country))\n```\n\n Rome is the capital of Italy\n Paris is the capital of France\n Delhi is the capital of India\n Beijing is the capital of China\n\n\n### Exercises\n\n1. Load a set of pretrained word vectors into a Python program.\n\n2. Calculate cosine similarities between pairs of words.\n\n\n```python\nmodel\n```\n\n\n\n\n \n\n\n\n\n```python\nlen(model.wv.vocab)\n```\n\n /Users/maximcondon/anaconda3/envs/deep_learning/lib/python3.6/site-packages/ipykernel_launcher.py:1: DeprecationWarning: Call to deprecated `wv` (Attribute will be removed in 4.0.0, use self instead).\n \"\"\"Entry point for launching an IPython kernel.\n\n\n\n\n\n 3000000\n\n\n\n\n```python\nmodel.wv['crocodile'][:20]\n```\n\n /Users/maximcondon/anaconda3/envs/deep_learning/lib/python3.6/site-packages/ipykernel_launcher.py:1: DeprecationWarning: Call to deprecated `wv` (Attribute will be removed in 4.0.0, use self instead).\n \"\"\"Entry point for launching an IPython kernel.\n\n\n\n\n\n array([-0.0703125 , 0.06542969, -0.33203125, 0.05078125, -0.09814453,\n -0.11962891, -0.27148438, -0.00601196, 0.13671875, 0.12597656,\n 0.15722656, -0.44140625, -0.27734375, -0.13964844, -0.15429688,\n 0.08984375, -0.05712891, 0.13476562, -0.19433594, -0.18457031],\n dtype=float32)\n\n\n\n\n```python\nmodel.most_similar(positive=['crocodile'])\n```\n\n\n\n\n [('croc', 0.8339575529098511),\n ('crocodiles', 0.8061199188232422),\n ('alligator', 0.7101820707321167),\n ('crocs', 0.7064967751502991),\n ('reptile', 0.6930339932441711),\n ('python', 0.6591362953186035),\n ('saltwater_crocodile', 0.6548329591751099),\n ('Nile_crocodile', 0.653219997882843),\n ('shark', 0.648021936416626),\n ('stingray', 0.6332049369812012)]\n\n\n\n\n```python\n# the least similar - negative\nmodel.most_similar(negative=['crocodile'])\n```\n\n\n\n\n [('C####_SAN_DIEGO', 0.25290921330451965),\n ('TRENDING_DOWN', 0.2514353394508362),\n ('Tadahito_Iguchi_Juan_Uribe', 0.24043317139148712),\n ('TRENDING_UP', 0.23945780098438263),\n ('Our_highligh', 0.22059018909931183),\n ('Mike_Lysaght_Correspondent', 0.218318909406662),\n ('Negotiated_Settlement_Agreement', 0.21281158924102783),\n ('MOHNTON_Penn', 0.20657244324684143),\n ('DENVILLE_TWP', 0.20519186556339264),\n ('theSan_Francisco', 0.2050630897283554)]\n\n\n\n\n```python\nsea = model.wv['sea']\n```\n\n /Users/maximcondon/anaconda3/envs/deep_learning/lib/python3.6/site-packages/ipykernel_launcher.py:1: DeprecationWarning: Call to deprecated `wv` (Attribute will be removed in 4.0.0, use self instead).\n \"\"\"Entry point for launching an IPython kernel.\n\n\n\n```python\nshells = model.wv['shells']\n```\n\n /Users/maximcondon/anaconda3/envs/deep_learning/lib/python3.6/site-packages/ipykernel_launcher.py:1: DeprecationWarning: Call to deprecated `wv` (Attribute will be removed in 4.0.0, use self instead).\n \"\"\"Entry point for launching an IPython kernel.\n\n\n\n```python\ncosine_similarity(sea, shells)\n```\n\n\n\n\n 0.2764841\n\n\n\n\n```python\ndoctor = model.wv['doctor']\nnurse = model.wv['nurse']\n\ncosine_similarity(doctor, nurse)\n```\n\n /Users/maximcondon/anaconda3/envs/deep_learning/lib/python3.6/site-packages/ipykernel_launcher.py:1: DeprecationWarning: Call to deprecated `wv` (Attribute will be removed in 4.0.0, use self instead).\n \"\"\"Entry point for launching an IPython kernel.\n /Users/maximcondon/anaconda3/envs/deep_learning/lib/python3.6/site-packages/ipykernel_launcher.py:2: DeprecationWarning: Call to deprecated `wv` (Attribute will be removed in 4.0.0, use self instead).\n \n\n\n\n\n\n 0.6319524\n\n\n\n\n```python\nbutt = model.wv['butt']\nass = model.wv['ass']\n\ncosine_similarity(butt, ass)\n```\n\n /Users/maximcondon/anaconda3/envs/deep_learning/lib/python3.6/site-packages/ipykernel_launcher.py:1: DeprecationWarning: Call to deprecated `wv` (Attribute will be removed in 4.0.0, use self instead).\n \"\"\"Entry point for launching an IPython kernel.\n /Users/maximcondon/anaconda3/envs/deep_learning/lib/python3.6/site-packages/ipykernel_launcher.py:2: DeprecationWarning: Call to deprecated `wv` (Attribute will be removed in 4.0.0, use self instead).\n \n\n\n\n\n\n 0.7703797\n\n\n\n3. Calculate the word vector resulting from\n\n`glove - hand + foot`\n\n - Calculate the cosine similarity between the resulting vector and what you think might be the real answer.\n\n\n```python\nglove = model.wv['glove']\nhand = model.wv['hand']\nfoot = model.wv['foot']\n\nwordvec_res = glove - hand + foot\nsock = model.wv['sock']\n\n\ncosine_similarity(wordvec_res, sock)\n```\n\n /Users/maximcondon/anaconda3/envs/deep_learning/lib/python3.6/site-packages/ipykernel_launcher.py:1: DeprecationWarning: Call to deprecated `wv` (Attribute will be removed in 4.0.0, use self instead).\n \"\"\"Entry point for launching an IPython kernel.\n /Users/maximcondon/anaconda3/envs/deep_learning/lib/python3.6/site-packages/ipykernel_launcher.py:2: DeprecationWarning: Call to deprecated `wv` (Attribute will be removed in 4.0.0, use self instead).\n \n /Users/maximcondon/anaconda3/envs/deep_learning/lib/python3.6/site-packages/ipykernel_launcher.py:3: DeprecationWarning: Call to deprecated `wv` (Attribute will be removed in 4.0.0, use self instead).\n This is separate from the ipykernel package so we can avoid doing imports until\n /Users/maximcondon/anaconda3/envs/deep_learning/lib/python3.6/site-packages/ipykernel_launcher.py:6: DeprecationWarning: Call to deprecated `wv` (Attribute will be removed in 4.0.0, use self instead).\n \n\n\n\n\n\n 0.2992683\n\n\n\n\n```python\n\n```\n\n4. Calculate cosine similarities between word pairs like (eagle, good) to find out whether there are good or evil animals.\n - How do you interpret the results?\n\n\n```python\ngood = model.wv['good']\nevil = model.wv['evil']\neagle = model.wv['eagle']\nstalin = model.wv['stalin']\n\ncosine_similarity(good, eagle)\n```\n\n /Users/maximcondon/anaconda3/envs/deep_learning/lib/python3.6/site-packages/ipykernel_launcher.py:1: DeprecationWarning: Call to deprecated `wv` (Attribute will be removed in 4.0.0, use self instead).\n \"\"\"Entry point for launching an IPython kernel.\n /Users/maximcondon/anaconda3/envs/deep_learning/lib/python3.6/site-packages/ipykernel_launcher.py:2: DeprecationWarning: Call to deprecated `wv` (Attribute will be removed in 4.0.0, use self instead).\n \n /Users/maximcondon/anaconda3/envs/deep_learning/lib/python3.6/site-packages/ipykernel_launcher.py:3: DeprecationWarning: Call to deprecated `wv` (Attribute will be removed in 4.0.0, use self instead).\n This is separate from the ipykernel package so we can avoid doing imports until\n /Users/maximcondon/anaconda3/envs/deep_learning/lib/python3.6/site-packages/ipykernel_launcher.py:4: DeprecationWarning: Call to deprecated `wv` (Attribute will be removed in 4.0.0, use self instead).\n after removing the cwd from sys.path.\n\n\n\n\n\n 0.039407644\n\n\n\n\n```python\ncosine_similarity(eagle, evil)\n```\n\n\n\n\n 0.14356834\n\n\n\n\n```python\ncosine_similarity(stalin, evil)\n```\n\n\n\n\n 0.35903656\n\n\n\n\n```python\ncosine_similarity(stalin, good)\n```\n\n\n\n\n 0.075627156\n\n\n\n\n```python\ncommunism = model.wv['communism']\n\ncosine_similarity(stalin, communism)\n```\n\n /Users/maximcondon/anaconda3/envs/deep_learning/lib/python3.6/site-packages/ipykernel_launcher.py:1: DeprecationWarning: Call to deprecated `wv` (Attribute will be removed in 4.0.0, use self instead).\n \"\"\"Entry point for launching an IPython kernel.\n\n\n\n\n\n 0.31829628\n\n\n\n\n```python\ncosine_similarity(good, communism)\n```\n\n\n\n\n 0.08336561\n\n\n\n\n```python\ncosine_similarity(evil, communism)\n```\n\n\n\n\n 0.2972171\n\n\n", "meta": {"hexsha": "10205ec8384c657bfe37d78836b40c54ccea767a", "size": 367117, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "02_WordEmbeddings.ipynb", "max_stars_repo_name": "maximcondon/Project_SentimentAnalysis", "max_stars_repo_head_hexsha": "c1b0f84e6d2c2c9e84600ec9fba05abc609fa618", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "02_WordEmbeddings.ipynb", "max_issues_repo_name": "maximcondon/Project_SentimentAnalysis", "max_issues_repo_head_hexsha": "c1b0f84e6d2c2c9e84600ec9fba05abc609fa618", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "02_WordEmbeddings.ipynb", "max_forks_repo_name": "maximcondon/Project_SentimentAnalysis", "max_forks_repo_head_hexsha": "c1b0f84e6d2c2c9e84600ec9fba05abc609fa618", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 211.4729262673, "max_line_length": 131188, "alphanum_fraction": 0.9112517263, "converted": true, "num_tokens": 6839, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300698514778, "lm_q2_score": 0.6370308082623217, "lm_q1q2_score": 0.44753329822607224}} {"text": "# Assignment 4: Word Embeddings \n\nWelcome to the fourth programming assignment of Course 2. In this assignment we will show you how to compute the word embeddings. In Natural Language Processing (NLP) we can not only rely on counting the number of positive words and negative words, as we did in the last course using logistic regression. Instead we will try to find a way to represent each word by a vector. The vector could then represent syntactic (i.e. parts of speech) and semantic (i.e. meaning) structures. In this assignment you will explore a classic way of learning embeddings, or representations, of words by using a famous model called the continuous bag of words (CBOW) model. By completing this assignment you will:\n\n- Train word vectors from scratch.\n- Learn how to create batches of data.\n- Understand how backpropagation works.\n- Plot and visualize your learned word vectors.\n\nBecause it will take a while to train your CBOW model, you will code the model and make sure to get the expected outputs. We will give you some slightly pre-trained vectors and then you will fine tune them on the Shakespeare dataset. \n\nKnowing how to train these models will give you a better understanding of word vectors, which are building blocks to many applications in natural language processing.\n\n\n# 1.0 The Continuous bag of words model\n\nLet's take a look at the following sentence: **'I am happy because I am learning'**. In continuous bag of words modeling we try to predict the center word given a few context words. For example, if you were to choose a context (say $C = 2$), then you would try to predict the word **happy** given the context: {I, am, because, I}. In other words, you have\n\n$$context = [I,am, because, I]$$\n$$target = happy$$\n\nThe structure of your model will look like this:\n\n\n
Figure 1
\n\nWhere $\\bar x$ is the average one hot vector for all the context word encodings. \n\n
Figure 2
\n\nOnce you have encoded all the context words, you can use $\\bar x$ as the input to your model. The architecture you will be implementing is as follows:\n\n\\begin{align}\n h &= W_1 \\ X + b_1 \\tag{1} \\\\\n a &= ReLU(h) \\tag{2} \\\\\n z &= W_2 \\ a + b_2 \\tag{3} \\\\\n \\hat y &= softmax(z) \\tag{4} \\\\\n\\end{align}\n\n\n```python\n# Import Python libraries and helper functions (in utils2) \nimport nltk\nfrom nltk.tokenize import word_tokenize\nimport numpy as np\nfrom collections import Counter\nfrom utils2 import sigmoid, get_batches, compute_pca, get_dict\n```\n\n\n```python\n# Download sentence tokenizer\nnltk.data.path.append('.')\n```\n\n\n```python\n# Load, tokenize and process the data\nimport re # Load the Regex-modul\ndata = open('shakespeare.txt').read() # Read in the data\ndata = re.sub(r'[,!?;-]', '.',data) # Punktuations are replaced by .\ndata = nltk.word_tokenize(data) # Tokenize string to words\ndata = [ ch.lower() for ch in data if ch.isalpha() or ch == '.'] # Lower case and drop non-alphabetical tokens\nprint(\"Number of tokens:\", len(data),'\\n', data[:15]) # print data sample\n```\n\n Number of tokens: 60996 \n ['o', 'for', 'a', 'muse', 'of', 'fire', '.', 'that', 'would', 'ascend', 'the', 'brightest', 'heaven', 'of', 'invention']\n\n\n\n```python\n# Compute the frequency distribution of the words in the dataset (vocabulary)\nfdist = nltk.FreqDist(word for word in data)\nprint(\"Size of vocabulary: \",len(fdist) )\nprint(\"Most frequent tokens: \",fdist.most_common(20) ) # print the 20 most frequent words and their freq.\n```\n\n Size of vocabulary: 5778\n Most frequent tokens: [('.', 9630), ('the', 1521), ('and', 1394), ('i', 1257), ('to', 1159), ('of', 1093), ('my', 857), ('that', 781), ('in', 770), ('a', 752), ('you', 748), ('is', 630), ('not', 559), ('for', 467), ('it', 460), ('with', 441), ('his', 434), ('but', 417), ('me', 417), ('your', 397)]\n\n\n#### Mapping words to indices and indices to words\nWe provide a helper function to create a dictionary that maps words to indices and indices to words.\n\n\n```python\n# get_dict creates two dictionaries, converting words to indices and viceversa.\nword2Ind, Ind2word = get_dict(data)\nV = len(word2Ind)\nprint(\"Size of vocabulary: \", V)\n```\n\n Size of vocabulary: 5778\n\n\n\n```python\n# example of word to index mapping\nprint(\"Index of the word 'king' : \",word2Ind['king'] )\nprint(\"Word which has index 2743: \",Ind2word[2743] )\n```\n\n Index of the word 'king' : 2745\n Word which has index 2743: kindness\n\n\n# 2.0 Training the Model\n\n### Initializing the model\n\nYou will now initialize two matrices and two vectors.
The first matrix ($W_1$) is of dimension $V \\times N$, where $V$ is the number of words in your vocabulary and $N$ is the dimension of your word vector.
\nThe second matrix ($W_2$) is of dimension $N \\times V$. \nThe two vectors, $b_1$ and $b_2$ are of dimension $N\\times 1$ and $V\\times 1$ respectively (column vectors). $b_1$ and $b_2$ are the bias vectors of the linear layers from matrices $W_1$ and $W_2$.\nThe overall structure of the model will look as in Figure 1, but at this stage we are just initializing the parameters. \n\n\n\n```python\nnp.random.seed(1)\n# UNQ_C1 (UNIQUE CELL IDENTIFIER, DO NOT EDIT)\n# UNIT TEST COMMENT: Candidate for Table Driven Tests\n# GRADED FUNCTION: initialize_model\ndef initialize_model(N,V):\n ### START CODE HERE ###\n W1 = np.random.random((N, V))\n W2 = np.random.random((V, N))\n b1 = np.random.random((N, 1))\n b2 = np.random.random((V, 1))\n ### END CODE HERE ###\n return W1, W2, b1, b2\n```\n\n\n```python\n# Test your function example.\nN = 4\nV = 10\nW1, W2, b1, b2 = initialize_model(N,V)\nassert W1.shape == ((N,V))\nassert W2.shape == ((V,N))\nprint(W1.shape, W2.shape, b1.T)\n```\n\n (4, 10) (10, 4) [[0.88330609 0.62367221 0.75094243 0.34889834]]\n\n\n* **Expected Output:** (4, 10) (10, 4) [[0.88330609 0.62367221 0.75094243 0.34889834]]\n\n### Softmax\nBefore we can start training the model, we need to implement the softmax function as defined in equation 5: \n\n
\n$$ \\text{softmax}(z_i) = \\frac{e^{z_i} }{\\sum_{i} e^{z_i} } \\tag{5} $$\n\n\n**Instructions**: Implement the softmax function below. \n\n\n```python\n# UNQ_C2 (UNIQUE CELL IDENTIFIER, DO NOT EDIT)\n# UNIT TEST COMMENT: Candidate for Table Driven Tests\n# GRADED FUNCTION: softmax\ndef softmax(x):\n \"\"\"Compute softmax values for each sets of scores in x.\"\"\"\n\n ### START CODE HERE (REPLACE INSTANCES OF 'None' WITH YOUR OWN CODE) ###\n e_x = np.exp(x)\n sum_e_x = np.sum(e_x)\n s = e_x / sum_e_x\n ### END CODE HERE ###\n return s\n```\n\n\n```python\n# testing the softmax function \nprint(softmax([0,3,-2])) \n```\n\n [0.04712342 0.94649912 0.00637746]\n\n\n**Expected Ouput:** array([0.04712342, 0.94649912, 0.00637746])\n\n### Forward propagation\n\nImplement the forward propagation $z$ according to equations (1) to (3).
\nFor that, you will use as activation the Rectified Linear Unit (ReLU) given by:\n\n$$f(h)=\\max (0,h) \\tag{6}$$\n\n\n```python\n# UNQ_C3 (UNIQUE CELL IDENTIFIER, DO NOT EDIT)\n# UNIT TEST COMMENT: Candidate for Table Driven Tests\n# GRADED FUNCTION: forward_prop\ndef forward_prop(x, W1, W2, b1, b2):\n '''\n Inputs: \n x: average one hot vector for the context \n W1, W2, b1, b2: matrices and biases to be learned\n Outputs: \n z: output score vector\n h: output of first hidden layer\n '''\n ### START CODE HERE (REPLACE INSTANCES OF 'None' WITH YOUR OWN CODE) ###\n ### (YOU WILL NEED TO ADD CODE IN ADDITION TO JUST REPLACING THE 'None' VALUES) ###\n \n # Calculate h and then used it to calculate z\n h = np.dot(W1, x) + b1\n a = h.copy()\n a[a < 0] = 0\n z = np.dot(W2, a) + b2\n ### END CODE HERE ###\n\n return z, h\n```\n\n## Cost function\n\nWe have implemented the *cross-entropy* cost function for you.
\nIf you want to understand it better, we refer to a [good explanation](https://cs224d.stanford.edu/lecture_notes/notes1.pdf).\n\n\n```python\ntry:\n from scipy.misc import logsumexp\nexcept ImportError:\n from scipy.special import logsumexp\n\n# compute_cost: cross-entropy cost function,\ndef compute_cost(z, C, y, yhat, batch_size):\n z_hat = logsumexp(z, axis=-1, keepdims=True) \n cost = (-np.sum(y*np.log(yhat)) + np.sum(2.0*C*z_hat)) / batch_size\n return cost\n```\n\n## Training the Model\n\nNow that you have understood how the CBOW model works, you will train it.
\nYou created a function for the forward propagation. Now you will implement a function that computes the gradients to backpropagate the errors.\n\n\n```python\n# UNQ_C4 (UNIQUE CELL IDENTIFIER, DO NOT EDIT)\n# UNIT TEST COMMENT: Candidate for Table Driven Tests\n# GRADED FUNCTION: back_prop\ndef back_prop(x, z, y, h, W1, W2, b1, b2, batch_size, m):\n '''\n Inputs: \n x: average one hot vector for the context \n z: score vector\n y: target vector\n h: hidden vector (see eq. 1)\n W1, W2, b1, b2: matrices and biases \n batch_size: batch size \n m: number of context words\n Outputs: \n grad_W1, grad_W2, grad_b1, grad_b2: gradients of matrices and biases \n '''\n ### START CODE HERE (REPLACE INSTANCES OF 'None' WITH YOUR OWN CODE) ###\n ### (YOU WILL NEED TO ADD CODE IN ADDITION TO JUST REPLACING THE 'None' VALUES) ###\n def relu(k):\n result = k.copy()\n result[result < 0] = 0\n return result\n \n yhat = softmax(z)\n grad_W1 = np.dot(relu(np.dot(W2.T, yhat - y)), x.T)\n grad_W2 = np.dot(yhat - y, h.T)\n grad_b1 = relu(np.dot(W2.T, yhat - y))\n grad_b2 = yhat - y\n ### END CODE HERE ###\n\n return grad_W1, grad_W2, grad_b1, grad_b2\n```\n\nNow that you have implemented a function to compute the gradients, you will implement batch gradient descent over your training set. \n\n**Hint:** For that, you will use initialize_model and the back_prop function that you just created (and the compute_cost function). You can also use the provided get_batches helper function:\n\n```for x, y in get_batches(data, word2Ind, V, C, batch_size):```\n\n```...```\n\nAlso: print the cost after each batch is processed (use batch size = 128)\n\n\n```python\n# UNQ_C5 (UNIQUE CELL IDENTIFIER, DO NOT EDIT)\n# UNIT TEST COMMENT: Candidate for Table Driven Tests\n# GRADED FUNCTION: gradient_descent\ndef gradient_descent(data, word2Ind, C, N, V, num_iters, alpha=0.03):\n \n '''\n This is the gradient_descent function\n \n Inputs: \n data: text\n word2Ind: words to Indices\n C: context window\n N: dimension of hidden vector \n V: dimension of vocabulary \n num_iters: number of iterations \n Outputs: \n W1, W2, b1, b2: updated matrices and biases \n\n '''\n W1, W2, b1, b2 = initialize_model(N,V)\n m = (2*C)\n batch_size = 128\n iters = 0\n for x, y in get_batches(data, word2Ind, V, C, batch_size):\n ### START CODE HERE (REPLACE INSTANCES OF 'None' WITH YOUR OWN CODE) ###\n z, h = forward_prop(x, W1, W2, b1, b2)\n yhat = softmax(z)\n cost = compute_cost(z, C, y, yhat, batch_size)\n print('iters', iters + 1 , ' cost',cost)\n grad_W1, grad_W2, grad_b1, grad_b2 = back_prop(x, z, y, h, W1, W2, b1, b2, batch_size, m)\n W1 = W1 - alpha * grad_W1\n W2 = W2 - alpha * grad_W2\n b1 = b1 - alpha * grad_b1\n b2 = b2 - alpha * grad_b2\n iters += 1 \n if iters == num_iters: \n break\n if iters % 100 == 0:\n alpha *= 0.66\n \n ### END CODE HERE ###\n\n return W1, W2, b1, b2\n```\n\n\n```python\n# test your function\nC = 2\nN = 50\nword2Ind, Ind2word = get_dict(data)\nV = len(word2Ind)\nnum_iters = 15\nW1, W2, b1, b2 = gradient_descent(data, word2Ind, C, N, V, num_iters)\n```\n\n iters 1 cost 5670.685215911188\n iters 2 cost 5664.065992365718\n iters 3 cost 5670.990718728146\n iters 4 cost 5677.9154450905735\n iters 5 cost nan\n iters 6 cost nan\n iters 7 cost nan\n iters 8 cost nan\n iters 9 cost nan\n iters 10 cost nan\n iters 11 cost nan\n iters 12 cost nan\n iters 13 cost nan\n iters 14 cost nan\n iters 15 cost nan\n\n\n**Expected Output:** iters 15 cost 41.66082959286652\n\n## 3.0 Visualizing the word vectors\n\nIn this part you will visualize the word vectors trained using the function you just coded above. \n\n\n```python\n# visualizing the word vectors here\nfrom matplotlib import pyplot\n%config InlineBackend.figure_format = 'svg'\nwords = ['king', 'queen','lord','man', 'woman','dog','horse',\n 'rich','happy','sad']\n\nembs = (W1.T + W2)/2.0\n \n# given a list of words and the embeddings, it returns a matrix with all the embeddings\nidx = [word2Ind[word] for word in words]\nX = embs[idx, :]\nprint(X.shape, idx) # X.shape: Number of words of dimension N each \n```\n\n (10, 50) [2745, 3951, 2961, 3023, 5675, 1452, 2472, 4191, 2316, 4278]\n\n\n\n```python\nresult= compute_pca(X, 2)\npyplot.scatter(result[:, 0], result[:, 1])\nfor i, word in enumerate(words):\n pyplot.annotate(word, xy=(result[i, 0], result[i, 1]))\npyplot.show()\n```\n\nYou can see that woman and queen are next to each other. However, we have to be carefull with the interpretation of this projected word vectors, since the PCA depends on the projection -- as shown in the following illustration.\n\n\n```python\nresult= compute_pca(X, 4)\npyplot.scatter(result[:, 3], result[:, 1])\nfor i, word in enumerate(words):\n pyplot.annotate(word, xy=(result[i, 3], result[i, 1]))\npyplot.show()\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "9b3bc0840e226fdc081ab7999e78200d9e5684fa", "size": 28589, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Natural Language Processing with Probabilistic Models/Week 4/C2_W4_Assignment.ipynb", "max_stars_repo_name": "PastaSardini/Natural-Language-Processing-Specialization-deeplearning.ai", "max_stars_repo_head_hexsha": "7f5a578d89b135d1967a2708db14b701919a8169", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 12, "max_stars_repo_stars_event_min_datetime": "2020-06-18T02:26:12.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-09T01:08:35.000Z", "max_issues_repo_path": "Natural Language Processing with Probabilistic Models/Week 4/C2_W4_Assignment.ipynb", "max_issues_repo_name": "PastaSardini/Natural-Language-Processing-Specialization-deeplearning.ai", "max_issues_repo_head_hexsha": "7f5a578d89b135d1967a2708db14b701919a8169", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Natural Language Processing with Probabilistic Models/Week 4/C2_W4_Assignment.ipynb", "max_forks_repo_name": "PastaSardini/Natural-Language-Processing-Specialization-deeplearning.ai", "max_forks_repo_head_hexsha": "7f5a578d89b135d1967a2708db14b701919a8169", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2020-07-18T12:30:59.000Z", "max_forks_repo_forks_event_max_datetime": "2021-12-26T23:34:58.000Z", "avg_line_length": 44.8103448276, "max_line_length": 2002, "alphanum_fraction": 0.5891076988, "converted": true, "num_tokens": 3987, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370308082623217, "lm_q2_score": 0.7025300698514777, "lm_q1q2_score": 0.4475332982260722}} {"text": "# Example 4: Quantum-to-classical transfer learning.\n\nThis is an example of a hybrid network for quantum state classification, developed according to the *quantum-to-classical transfer learning* scheme presented in [1]. \n\n## The initial pre-trained quantum network\n\nOur starting point is the pre-trained continuous variable (CV) quantum network presented in [_Killoran et al._](https://arxiv.org/abs/1806.06871) [2], _Section IV.D, Experiment C_. The original aim of this network was to encode 7 different 4X4 images, representing the (L,O,T,I,S,J,Z) [_tetrominos_](https://en.wikipedia.org/wiki/Tetromino) (popularized by the video game _Tetris_), in the Fock basis of 7 two-mode quantum states. The input of the quanutm network is one of the following combinations of two-mode coherent states:\n\n\\begin{align}\n|\\varphi_1\\rangle &= |\\alpha\\rangle|\\alpha\\rangle \\\\\n|\\varphi_2\\rangle &= |-\\alpha\\rangle|\\alpha\\rangle \\\\\n|\\varphi_3\\rangle &= |\\alpha\\rangle|-\\alpha\\rangle \\\\\n|\\varphi_4\\rangle &= |-\\alpha\\rangle|-\\alpha\\rangle \\\\\n|\\varphi_5\\rangle &= |i \\alpha\\rangle| i\\alpha\\rangle \\\\\n|\\varphi_6\\rangle &= |-i \\alpha\\rangle|i \\alpha\\rangle \\\\\n|\\varphi_7\\rangle &= |i \\alpha\\rangle|-i \\alpha\\rangle \\\\\n\\end{align}\n\nwhere the parameter $\\alpha=1.4$ is a fixed constant.\n\nThe task of the network is to generate an optimal unitary transformation $|\\tilde{\\varphi}_j\\rangle=U|\\varphi_j\\rangle$, such that the probability of finding $i$ photons in the first mode and $j$ photons in the second mode is proportional to the amplitude of pixel $(i,j)$. More precisely, the network is trained to reproduce the tetromino images after projecting the quantum state on the subspace of up to 3 photons. A simulation of the photon number probability distribution and its renormalized subspace projection, are respectively reported in the first and second row of the following figure (taken from Figure 10 of [_Killoran et al._](https://arxiv.org/abs/1806.06871)) [2].\n\n\n\n## Quantum-to-classical transfer learning\n\nWith respect to the problem above, we are going to change the dataset and the task in order to give proof-of-principle demonstration of the _quantum-to-classical transfer learning_ method.\n\nWe assume that the 7 combinations coherent states discussed above are subject to a Gaussian additive noise (random displacements) and that our goal is to correctly guess the original quantum state:\n\n\\begin{align}\n|\\varphi_j & \\rangle \\xrightarrow{\\text{random displacement}} \\hat D (\\delta\\alpha_1,\\delta\\alpha_2)|\\varphi_j \\rangle \\\\\n&\\xrightarrow{\\text{quantum-classical network}} \\text{Outcome: } \\textit{\"the state belongs to class } j\\text{\"}\n\\end{align}\n\n\nIn a machine learning language this can be seen as a classification problem with 7 classes and a quantum dataset consisting of randomly displaced coherent states.\n\nThe starting point of our hybrid model is the CV quantum network presented in the previous section despite it was pre-trained for a quite different task. **The motivation behind this approach is that the image pixels which are produced by the pre-trained network can be considered as classical features possessing a strong correlation with the input state. It is then reasonable to assume that such features can be efficiently post-processed and classified.**\n\nThe code presented in the next sections is a practical implementation of this idea, which can be summarized in 4 operational steps:\n\n1. Remove some quantum layers (0, 1, 2, or more) from the previously described pre-trained quantum network.\n2. Measure the system in the Fock basis. In this way the quantum circuit acts as a feature extractor.\n3. Add some final classical layers to post-process the estimated photon-number probability distribution.\n4. Train only such classical layers to classify the input quantum states.\n\n## General setup\n\nThe main imported modules are: the `tensorflow` machine learning framework, the quantum CV \nsoftware `strawberryfields` [3] and the python plotting library `matplotlib`. All modules should be correctly installed in the system before running this notebook.\n\n\n```python\n# Plotting\n%matplotlib inline\nimport matplotlib.pyplot as plt\n\n# TensorFlow\nimport tensorflow as tf\n\n# Strawberryfields (simulation of CV quantum circuits)\nimport strawberryfields as sf\nfrom strawberryfields.ops import Dgate, BSgate, Kgate, Sgate, Rgate\n\n# Other modules\nimport numpy as np\nimport time\n\n# System variables\nimport os\nos.environ['TF_CPP_MIN_LOG_LEVEL'] = '2' # avoid warning messages\nos.environ['OMP_NUM_THREADS'] = '1' # set number of threads.\nos.environ['CUDA_VISIBLE_DEVICES'] = '1' # select the GPU unit.\n\n```\n\nSetting of the main parameters of the network model and of the training process.
\n\n\n```python\ncutoff = 11 # must be larger than or equal to 3. Optimal value 11.\nim_dim = 4 # subspace dimension\ndump = 100 # loggin period\nnum_images = 7 # number of images\ndisp_clip = 1 # max displacement\nsq_clip = 1 # max squeezing\nkerr_clip = 1 # max kerr non-linearity \nsdev = 0.1 # initial variance of random weights\nnum_test_batches=1000 # number of test batches used to estimate accuracy\n\nnoise_scale = 0.6 # noise strength (mean deviation of random displacements)\nsub_space = True # If True, the state is projected in the 0-3 photons subspace.\nfine_tune = False # If True, also the quantum paramenters alre trained. Suggested value is False.\nnum_epochs = 1000 # Number of training iterations (number of batches of 7 quantum states).\nq_depth = 15 # Number of quantum layers (Max=25). \nc_depth = 1 # Number of classical layers. \nstep = 0.01 # Learning rate\nalpha = 1.4 # Amplitude of coherent states. Note that the quantum network is pre-trained with alpha=1.4.\n\ntf.reset_default_graph() # reset tensorflow graph. Useful to re-run the code.\ntf.set_random_seed(1) # tensorflow random seed\nrng_data = np.random.RandomState(100) # numpy random seed\n```\n\n## Quantum dataset\n\nWe define the quantum dataset consisting in the 7 two-mode coherent input states definied at the beginning of this notebook (same as in [_Killoran et al._](https://arxiv.org/abs/1806.06871)), which here we assume to be subject to random displacements sampled from a Gaussian distribution with zero mean and variance `noise_scale`.\n\n\n```python\n# function which generates a batch of random noise (real values).\ndef noise_sample():\n return rng_data.normal(scale=noise_scale, size=num_images)\n\nnoise_alpha=tf.placeholder(dtype=tf.complex64, shape=[num_images])\nnoise_beta=tf.placeholder(dtype=tf.complex64, shape=[num_images])\n\ndisps_alpha = tf.constant([alpha, -alpha, alpha, -alpha, 1.0j*alpha, -1.0j*alpha, 1.0j*alpha], dtype = tf.complex64) + noise_alpha\ndisps_beta = tf.constant([alpha, alpha, -alpha, -alpha, 1.0j*alpha, 1.0j*alpha, -1.0j*alpha], dtype = tf.complex64) + noise_beta\n \nlabels = [0, 1, 2, 3, 4, 5, 6]\n\n# convert labels into TensorFlow format\nlabels_holder = tf.constant(labels,dtype = tf.int64)\n```\n\n## Loading of pre-trained quantum weights\n\nAs discussed in the introduction we make use of a pre-trained CV quantum neural network. Such network, originally presented [_Killoran et al._](https://arxiv.org/abs/1806.06871), was trained to reproduce tetronomos images. The corresponding optimal weights can be loaded from the numpy file `pre_trained\trained_params.npy` that must be present in the notebook directory.\n\n\n```python\n# loading of pre-trained weights\ntrained_params_npy = np.load('pre_trained/trained_params.npy')\n# conversion into TensorFlow format\ntrained_params = tf.constant(trained_params_npy)\n\n# initialization of the variational parameters of the quantum circuit defined later.\nwith tf.name_scope('variables'):\n r1 = tf.get_variable(\"r1\", trainable=fine_tune, initializer=trained_params[0])\n r2 = tf.get_variable(\"r2\", trainable=fine_tune, initializer=trained_params[1])\n \n theta1 = tf.get_variable(\"theta1\", trainable=fine_tune, initializer=trained_params[2])\n phi1 = tf.get_variable(\"phi1\", trainable=fine_tune, initializer=trained_params[3])\n \n theta2 = tf.get_variable(\"theta2\", trainable=fine_tune,initializer=trained_params[4])\n phi2 = tf.get_variable(\"phi2\", trainable=fine_tune, initializer=trained_params[5])\n\n sqr1 = tf.get_variable(\"sqr1\", trainable=fine_tune, initializer=trained_params[6])\n sqphi1 = tf.get_variable(\"sqphi1\", trainable=fine_tune, initializer=trained_params[7])\n\n sqr2 = tf.get_variable(\"sqr2\", trainable=fine_tune, initializer=trained_params[8])\n sqphi2 = tf.get_variable(\"sqphi2\", trainable=fine_tune, initializer=trained_params[9])\n\n dr1 = tf.get_variable(\"dr1\", trainable=fine_tune, initializer=trained_params[10])\n dphi1 = tf.get_variable(\"dphi1\", trainable=fine_tune, initializer=trained_params[11])\n\n dr2 = tf.get_variable(\"dr2\", trainable=fine_tune, initializer=trained_params[12])\n dphi2 = tf.get_variable(\"dphi2\", trainable=fine_tune, initializer=trained_params[13])\n\n kappa1 = tf.get_variable(\"kappa1\", trainable=fine_tune, initializer=trained_params[14])\n kappa2 = tf.get_variable(\"kappa2\", trainable=fine_tune, initializer=trained_params[15])\n\nParameters = [r1, r2, theta1, phi1, theta2, phi2, sqr1, sqphi1, sqr2, sqphi2, dr1, dphi1, dr2, dphi2, kappa1, kappa2]\n\n```\n\n## Hybrid transfer learning model (quantum-to-classical).\n\nWe first instantiate a _StrawberryFields_ quantum simulator, taylored for simulating a two-mode quantum optical system.\n\n\n```python\nprog = sf.Program(2)\neng = sf.Engine('tf', backend_options={'cutoff_dim': cutoff, 'batch_size': num_images})\n```\n\n### First block: pre-trained quantum neural network.\n\nNow we can define, via _StrawberryFields_, the CV quantum neural network model proposed in [_Killoran et al._](https://arxiv.org/abs/1806.06871). \n\nWith respect to the original version of the network which was designed to have 25 variaitonal layers, here we allow for the possibility of applying only a number `q_depth` of such layers (form 0 up to 25). This is motivated by the idea that the final features of a pre-trained network can be too task-specific, while intermediate features can be more suitable for a transfer learning operation.\n\n\n```python\n# Definition of a single variational quantum layer composed of: \n# beam splitters, squeezing, displacements, Kerr non-linearities, etc.\n\ndef layer(l):\n with tf.name_scope('layer_{}'.format(l)):\n BSgate(theta1[l], phi1[l]) | (qMode[0], qMode[1])\n Rgate(r1[l]) | qMode[0]\n\n Sgate(tf.clip_by_value(sqr1[l], -sq_clip, sq_clip), sqphi1[l]) | qMode[0]\n Sgate(tf.clip_by_value(sqr2[l], -sq_clip, sq_clip), sqphi2[l]) | qMode[1]\n\n BSgate(theta2[l], phi2[l]) | (qMode[0], qMode[1])\n Rgate(r2[l]) | qMode[0]\n\n Dgate(tf.clip_by_value(dr1[l], -disp_clip, disp_clip), dphi1[l]) | qMode[0]\n Dgate(tf.clip_by_value(dr2[l], -disp_clip, disp_clip), dphi2[l]) | qMode[1]\n\n Kgate(tf.clip_by_value(kappa1[l], -kerr_clip, kerr_clip)) | qMode[0]\n Kgate(tf.clip_by_value(kappa2[l], -kerr_clip, kerr_clip)) | qMode[1]\n\n# Definition of the complete quantum circuit: state preparation + quantum layers\nwith prog.context as qMode:\n Dgate(disps_alpha) | qMode[0]\n Dgate(disps_beta) | qMode[1]\n for i in range(q_depth):\n layer(i)\n \n# Simulation of the quantum state evolution \nresults = eng.run(prog, run_options={\"eval\": False}) \nket = results.state.ket()\n\n# Projection on the subspace of up to im_dim-1 photons for each mode.\nket_reduced = ket[:, :im_dim, :im_dim]\nnorm = tf.sqrt(tf.abs(tf.reduce_sum(tf.conj(ket_reduced) * ket_reduced, axis=[1, 2])))\n\n# Since norm has shape [num_images] while ket_reduced has shape [num_images,im_dim,im_dim]\n# we need to add 2 extra dimensions to the norm tensor.\nnorm_extended = tf.reshape(norm, [num_images, 1, 1])\nket_processed = ket_reduced / tf.cast(norm_extended, dtype=tf.complex64)\n\n# Convert the state coefficients into images for features visualization.\nimages_out = tf.abs(ket_processed) ** 2\nimages_out_big = tf.abs(ket) ** 2\n\n# Definition of the classical output of the quantum circuit, i.e. the probabilities of photon number detections.\nif sub_space == True:\n ket_abs = tf.abs(ket_processed)\n num_features = (cutoff + 1) * (cutoff + 1)\nelse:\n ket_abs = tf.abs(ket)\n num_features = im_dim * im_dim\n \n# Flatten to get a classical vector of features\nket_abs_flatten = tf.contrib.layers.flatten(ket_abs)\n```\n\n### Second block: trainable classical network.\nBy following the transfer learning method, we connect the pre-trained quantum block to a final trainable classical network. Depending on the parameter `c_depth`, the classical block is a simple linear classfier (if `c_depth` is 1) or a more complex neural network with `c_depth` dense layers and non-linear activations (_ReLU_). \n\n\n```python\n# sequence of fully connected classical layers \nc_in = ket_abs_flatten\nfor _ in range(c_depth - 1):\n c_in = tf.contrib.layers.fully_connected(c_in,num_features, activation_fn=tf.nn.relu)\nc_out = tf.contrib.layers.fully_connected(c_in, num_images, activation_fn=None) \n```\n\n### Loss function, accuracy, and optimizer.\n\n\n```python\n# Definition of the loss function to minimize\nloss = tf.reduce_mean(tf.nn.sparse_softmax_cross_entropy_with_logits(labels = labels_holder, logits = c_out))\n# Convert logits to labels\npredictions = tf.argmax(c_out, 1)\n# Batch accuracy\naccuracy = tf.reduce_mean(tf.cast(tf.equal(predictions, labels_holder), tf.float32))\n# Optimization algorithm\noptim = tf.train.AdamOptimizer(learning_rate=step)\ntraining = optim.minimize(loss)\n```\n\n## Training and testing\n\nUp to now we just defined the analytic graph of the hybrid network without evaluating it. Now, after initializing a _TensorFlow session_, we can finally run the actual training and testing phases. \n\n\n```python\nsaver = tf.train.Saver()\n\ntest_accuracy = 0.0\ntrain_loss = 0.0\ntest_loss = 0.0\ntrain_loss_sum = 0.0\ntest_loss_sum = 0.0\ntrain_accuracy_sum = 0.0\ntest_accuracy_sum = 0.0\n\nwith tf.Session() as sess:\n sess.run(tf.global_variables_initializer())\n \n #### training phase ####\n for i in range(num_epochs):\n # generate random displacements\n noise_dict = {noise_alpha: noise_sample() + 1.0j * noise_sample(), noise_beta: noise_sample() + 1.0j * noise_sample()}\n rep_time = time.time()\n [_training,_loss,_accuracy] = sess.run([training, loss, accuracy], feed_dict=noise_dict)\n train_loss_sum += _loss\n train_loss = train_loss_sum / (i + 1)\n train_accuracy_sum += _accuracy\n train_accuracy = train_accuracy_sum / (i + 1)\n if (i % dump == 0) and (i != 0):\n print('Train batch: {:d}, Running loss: {:.3f}, Running accuracy {:.3f}, Single batch time {:.3f}'.format(i,train_loss,train_accuracy,time.time()-rep_time))\n\n #### test phase ####\n for i in range(num_test_batches):\n # generate random displacements\n noise_dict = {noise_alpha: noise_sample() + 1.0j * noise_sample(), noise_beta: noise_sample() + 1.0j * noise_sample()}\n rep_time = time.time()\n [_loss,_accuracy] = sess.run([loss,accuracy], feed_dict = noise_dict) ## same as before without training\n test_loss_sum += _loss\n test_loss = test_loss_sum / (i + 1)\n test_accuracy_sum += _accuracy\n test_accuracy = test_accuracy_sum / (i + 1)\n if (i % dump == 0) and (i != 0):\n print('Test batch: {:d}, Running loss: {:.3f}, Running accuracy {:.3f}, Single batch time {:.3f}'.format(i,test_loss,test_accuracy,time.time()-rep_time))\n \n #### Save model to file ####\n save_path = saver.save(sess, \"./model_q2c.ckpt\")\n print(\"Model saved in path: %s\" % save_path)\n\nprint('Training and testing phases completed.')\nprint('RESULTS:')\nprint('{:>11s}{:>11s}{:>11s}{:>11s}'.format('train_loss', 'train_acc', 'test_loss', 'test_acc'))\nprint('{:11f}{:11f}{:11f}{:11f}'.format(train_loss, train_accuracy, test_loss, test_accuracy))\n \n \n```\n\n Train batch: 100, Running loss: 1.670, Running accuracy 0.485, Single batch time 1.883\n Train batch: 200, Running loss: 1.493, Running accuracy 0.597, Single batch time 2.052\n Train batch: 300, Running loss: 1.370, Running accuracy 0.645, Single batch time 2.054\n Train batch: 400, Running loss: 1.276, Running accuracy 0.675, Single batch time 2.171\n Train batch: 500, Running loss: 1.202, Running accuracy 0.698, Single batch time 2.318\n Train batch: 600, Running loss: 1.143, Running accuracy 0.714, Single batch time 2.341\n Train batch: 700, Running loss: 1.095, Running accuracy 0.724, Single batch time 2.317\n Train batch: 800, Running loss: 1.065, Running accuracy 0.729, Single batch time 2.049\n Train batch: 900, Running loss: 1.029, Running accuracy 0.737, Single batch time 2.312\n Test batch: 100, Running loss: 0.719, Running accuracy 0.805, Single batch time 2.121\n Test batch: 200, Running loss: 0.732, Running accuracy 0.803, Single batch time 2.309\n Test batch: 300, Running loss: 0.735, Running accuracy 0.804, Single batch time 2.012\n Test batch: 400, Running loss: 0.734, Running accuracy 0.802, Single batch time 2.250\n Test batch: 500, Running loss: 0.730, Running accuracy 0.805, Single batch time 2.312\n Test batch: 600, Running loss: 0.726, Running accuracy 0.807, Single batch time 3.231\n Test batch: 700, Running loss: 0.730, Running accuracy 0.807, Single batch time 2.797\n Test batch: 800, Running loss: 0.726, Running accuracy 0.808, Single batch time 2.347\n Test batch: 900, Running loss: 0.722, Running accuracy 0.810, Single batch time 2.334\n Model saved in path: ./model_q2c.ckpt\n Training and testing phases completed.\n RESULTS:\n train_loss train_acc test_loss test_acc\n 1.001114 0.741857 0.725259 0.806714\n\n\n## Load model and visualize predictions\n\n\n```python\nsaver = tf.train.Saver()\nwith tf.Session() as sess:\n # Restore variables from disk.\n saver.restore(sess, \"./model_q2c.ckpt\")\n print(\"Model restored.\")\n \n noise_dict={noise_alpha: noise_sample() + 1.0j * noise_sample(), noise_beta: noise_sample() + 1.0j * noise_sample()}\n [_predictions, _images_out, _images_out_big] = sess.run([predictions, images_out, images_out_big], feed_dict=noise_dict) \n```\n\n INFO:tensorflow:Restoring parameters from ./model_q2c.ckpt\n Model restored.\n\n\nNow we represent the Fock state probabilities as 4X4 images. These are the _features_ extracted by the quantum network and successively processed and classified by by the final classical network. \n\n\n```python\n# plot features as images\nfig, axs = plt.subplots(nrows=1, ncols=7, figsize = (15, 3))\nfor idx, ax in enumerate(axs.flat):\n ax.axis('off')\n if idx < 7:\n ax.imshow(_images_out[idx], cmap='gray')\n ax.set_title('class ' + str(idx + 1) + '\\n' + 'pred. ' + str(_predictions[idx] + 1) , fontsize=22)\n #else:\n # ax.imshow(_KetImageOutBig[idx-7], cmap='gray')\nplt.tight_layout()\nplt.show() \n```\n\n## References\n\n[1] Andrea Mari, Thomas R. Bromley, Josh Izaac, Maria Schuld, and Nathan Killoran. _Transfer learning in hybrid classical-quantum neural networks_. [arXiv:1912.08278](https://arxiv.org/abs/1912.08278), (2019).\n\n[2] Nathan Killoran, Thomas R. Bromley, Juan Miguel Arrazola, Maria Schuld, Nicol\u00e1s Quesada, and Seth Lloyd. _Continuous-variable quantum neural networks_. [arXiv:1806.06871](https://arxiv.org/abs/1806.06871), (2018).\n\n[3] Nathan Killoran, Josh Izaac, Nicol\u00e1s Quesada, Ville Bergholm, Matthew Amy, and Christian Weedbrook. _Strawberry Fields: A Software Platform for Photonic Quantum Computing_. [Quantum, 3, 129 (2019)](https://doi.org/10.22331/q-2019-03-11-129).\n", "meta": {"hexsha": "c9209a52bd8eeff1ccc4c219c3940e91d1a25c74", "size": 37353, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "q2c_transfer_learning.ipynb", "max_stars_repo_name": "Jerry2001Qu/quantum-transfer-learning", "max_stars_repo_head_hexsha": "231def697d689e761af997634318bb31d6785c01", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 43, "max_stars_repo_stars_event_min_datetime": "2019-12-17T04:58:49.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-27T17:02:07.000Z", "max_issues_repo_path": "q2c_transfer_learning.ipynb", "max_issues_repo_name": "Jerry2001Qu/quantum-transfer-learning", "max_issues_repo_head_hexsha": "231def697d689e761af997634318bb31d6785c01", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 9, "max_issues_repo_issues_event_min_datetime": "2020-01-20T19:43:31.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-12T07:13:38.000Z", "max_forks_repo_path": "q2c_transfer_learning.ipynb", "max_forks_repo_name": "pkino/quantum-transfer-learning", "max_forks_repo_head_hexsha": "e92df0c0794c283dd51dde7be5f2d8496ebb85b5", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 26, "max_forks_repo_forks_event_min_datetime": "2020-01-20T04:23:21.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-23T06:05:31.000Z", "avg_line_length": 63.7423208191, "max_line_length": 10100, "alphanum_fraction": 0.695713865, "converted": true, "num_tokens": 5248, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300698514777, "lm_q2_score": 0.6370307875894138, "lm_q1q2_score": 0.4475332837027327}} {"text": "# Building Set 2 - More Frequency Domain, the SigSys Abstraction\n\n\n```python\nimport sys\nsys.path.append(\"code\") # for thinkdsp code\n\nfrom __future__ import print_function, division\n\nimport thinkdsp\nimport thinkplot\n\nimport numpy as np\n\nimport sympy\n\nimport pandas\n\nimport warnings\nwarnings.filterwarnings('ignore')\n\nfrom ipywidgets import interact, interactive, fixed\nimport ipywidgets as widgets\nfrom IPython.display import display\n\n%matplotlib inline\n```\n\n\n```python\n# https://gist.github.com/Pretz/1773870\n\nimport wave\nimport numpy\nimport struct\nimport sys\nimport csv\n\ndef write_wav(data, filename, framerate, amplitude):\n wavfile = wave.open(filename, \"w\")\n nchannels = 1\n sampwidth = 2\n framerate = framerate\n nframes = len(data)\n comptype = \"NONE\"\n compname = \"not compressed\"\n wavfile.setparams((nchannels,\n sampwidth,\n framerate,\n nframes,\n comptype,\n compname))\n print(\"Please be patient whilst the file is written\")\n frames = []\n for s in data:\n mul = int(s * amplitude)\n # print \"s: %f mul: %d\" % (s, mul)\n frames.append(struct.pack('h', mul))\n # frames = (struct.pack('h', int(s*self.amp)) for s in sine_list)\n\n frames = b''.join(frames)\n wavfile.writeframes(frames)\n wavfile.close()\n print(\"%s written\" %(filename))\n\nfname = 'BSet2Files/handel.csv'\n\ndata = []\n\nfor value in csv.reader(open(fname, 'U'), delimiter=','):\n try:\n data.append(float(value[0]))\n except ValueError:\n pass # Just skip it\narr = numpy.array(data)\n# Normalize data\narr /= numpy.max(numpy.abs(data))\nfilename_head, extension = fname.rsplit(\".\", 1)\nwrite_wav(arr, \"BSet2Files/handel.wav\", 8192, 32700)\n\n\n```\n\n Please be patient whilst the file is written\n BSet2Files/handel.wav written\n\n\n## Frequency-Domain Analysis\n### Discrete-time Fourier Transform (DTFT) (appx 3.5 hours including reading and exercises)\n\nIn this section, we are going to develop the Discrete-Time Fourier Transform (DTFT), which will provide better frequency resolution than the DFT, and serve as a bridge to the CT Fourier transform (CTFT), enabling us to do frequency analysis on continuous signals.\n\nThe Discrete-Time Fourier Transform (DTFT) of a DT signal is defined for any angular frequency $\\Omega$ radians per sample as follows\n\n\\begin{align}\nX(\\Omega) = \\sum_{n=-\\infty}^\\infty x[n]e^{-j\\Omega n}$\n\\end{align}\n\n\nThe DTFT of $x[n]$, denoted by $X(\\Omega)$ tells us \"how much\" in magnitude and phase, of a complex exponential $e^{j\\Omega n}$ is present in the signal $x[n]$. Compared to the DFT, the DTFT evaluates the frequency content in $x[n]$ at a continuum of frequencies $\\Omega$. $\\Omega$ here is in units of radians/sample. \n\nSince $\\Omega$ can take a continuum of values, the inverse transform cannot take the form of a summation. The Inverse DTFT takes the form of an integral over a $2\\pi$ interval. The IDFT is defined as follows\n\n\\begin{align}\nx[n] = \\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi} X(\\Omega) e^{j\\Omega n} d\\Omega\n\\end{align}\n\n$x[n]$ is therefore made up of a dense sum (i.e. an integral) of complex exponentials (frequency components), where the frequency component corresponding to $e^{j\\Omega n}$ is weighted by $X(\\Omega)$. In general $X(\\Omega)$ can be complex.\n\nThe DTFT has a number of useful properties which are carried over from the DFT. If $W(\\Omega)$ is the DTFT of $w[n]$, $X(\\Omega)$ is the DTFT of $x[n]$, $Y(\\Omega)$ is the DTFT of $y[n]$, we have the following properties\n\n\n- If $y[n] = x[n]+w[n]$, then $Y(\\Omega) = X(\\Omega) + W(\\Omega)$. This property enables us to decompose a signal into the sum of simpler signals, perform the DTFT on the simpler signals, and re-assemble them. \n- If $y[n] = x[n-n_0]$, then $Y(\\Omega) = e^{-j\\Omega n_0}X(\\Omega)$. This property is useful in analyzing the effects of difference equations in the frequency domain. \n- If $y[n] = x[n]e^{j\\Omega_0 n}$, then $Y(\\Omega) = X(\\Omega-\\Omega_0)$. This property is useful for moving things around in frequency, like what we did in class on Monday. \n\n\nThe following example of a moving-average filter helps illustrate how the DTFT can be used to do understand the effects of a difference equation in the frequency domain.\n\n### 1.\nSuppose that we have a signal $x[n]$. A 3-point moving average of this signal is given by $y[n]$, which is defined as follows\n\n\\begin{align}\ny[n] = \\frac13(x[n+1]+x[n]+x[n-1]).\n\\end{align}\n\nThe result of the moving average at time $n$ is the average of the original signal at the current time, one sample ahead, and one sample before. The moving average filter has the effect of smoothing out a signal.\n\n(a) Using the properties of the DTFT in the bullet points above, find the DTFT of $y[n]$, denoted by $Y(\\Omega)$, in terms of $X(\\Omega)$. Simplify your expression using $\\cos\\theta =\\frac12 e^{j\\theta}+\\frac12 e^{-j\\theta}$.\n\n\n$y[n] = \\frac13(x[n+1]+x[n]+x[n-1]$\n\n\n\n(b) How is the frequency content of $y[n]$ related to the frequency content of $x[n]$?\n\nHint: think about how $Y(\\Omega)$ is related to $X(\\Omega)$ at $\\Omega = 0$ (low frequencies) vs $\\Omega = \\pi$ (high frequencies).\n\n\n```python\n\n```\n\n(c) How might you actually apply the moving average filter to a given vector? In other words, in MATLAB, how would you operate on a vector $x[n]$ to produce $y[n]$? Think about three different approaches:\n\ni. Use a for loop.\n\nii. Use vector indices and vector addition.\n\niii. Use a matrix multiply.\n\nChoose one that makes sense to you, and implement it. Note that you'll have to think about how to deal with the terms at the edges: there is not a $n-1$ term if $n=0$!\n\n\n```python\nfor n in range(2):\n print(n)\n```\n\n 0\n 1\n\n\n(d) At the MATLAB prompt, type `load handel` to load an audio segment provided with MATLAB, which you used in the last BSet. Write code to implement the moving average filter, and apply it to the audio sample in `y`. Using a for loop may be the simplest approach here.\n\n(Unfortunately, MATLAB calls this signal `y`, but this will be the input to our moving average filter). Plot the original signal `y`, and the result of applying the moving average to it on the same axes. Zoom in on a segment of the audio signal to see the effect of the moving average in the time domain. Do you see what you expected to see? Play the original signal and the result of the moving average. Does the filtered signal sound as expected? Please turn in your plots with your answers to the questions.\n\n\n```python\nhandel = thinkdsp.read_wave('BSet2Files/handel.wav')\n\ndir(handel)\n\nhandel.plot(color='r')\n\ndef moving_average(sound):\n x = sound.ys\n y = [0]\n for n in range(1, len(x) - 1):\n y.append(1/3 * (x[n + 1] + x[n] + x[n - 1]))\n y.append(0)\n \n sound.ys = y\n return sound\n\nmoving_average(handel)\n# handel.make_audio()\n\nhandel.plot()\n\n\n```\n\n(e) On the same axes, plot the magnitude of the DTFT of $x[n]$ and $y[n]$. You can use the following code segment to do this.\n\n```\nX = fft(x); % Take the FFT of x\nY = fft(y); % Take the FFT of y\nNx = length(x); % Find the length of x\nNy = length(y); % Find the length of y\n% make vectors for the frequency axis points in radians \n% per sample\nwx = linspace(-pi, pi*(Nx-1)/Nx, Nx); \nwy = linspace(-pi, pi*(Ny-1)/Ny, Ny); % shouldn't this be -pi to pi??\n% plot the magnitudes on the same axes \n% use fftshift to reorder the FFT\nplot(wx, fftshift(abs(X)));\nhold on;\nplot(wy, fftshift(abs(Y)), 'm'); % shouldn't this be wy and abs(Y)?\n```\n\nPlease submit all plots you made, and write a sentence or two explaining qualitatively how the frequency content of the signal before and after the moving average filter is applied.\n\n\n```python\n\n```\n\n### 2.\n\n\n\n\n## The Sig-Sys Abstraction and Block Diagrams (3.5 hours)\n\nYou've dealt with a variety of different types of abstraction in the past -- for example, you've represented physical situations using stock and flow diagrams, using free body diagrams, using differential equations, etc. \n\nA key type of abstraction that gets widely used in engineering is the so-called \"sig-sys\" abstraction. This abstraction says that you can think of things as being broken down into _signals_ (which we discussed a bit last time), and _systems_ which operate on signals (ie., they take signals as input, and produce signals as output).\n\nWe tend to use \"block diagrams\" to represent the sig-sys abstraction, as we discussed in class. You're actually quite used to doing this already: you did so for the bad modem. Block diagrams are frequently used in designing and representing systems, from communication systems to control systems. \n\nOne of the very handy aspects of block diagrams is the extent to which they lend themselves to different _levels of abstraction_ - i.e., you can always \"drill down\" to ask \"what is the block diagram of that block?\", and you can also always draw a dotted line around a set of blocks and call it a new single block. The ability to express different levels of abstraction conveniently is powerful since different situations in a design process require different levels of granularity. \n\nOften, block diagrams can also be translated to hardware and software more directly than mathematical expressions. As such, block diagrams using the SigSys abstraction are powerful tools for the design of real-world systems, and can offer insight that may not be as apparent by examining mathematical relationships alone. Those of you in PoE will learn to appreciate this fact in a month or so. \n\nLets think first about block diagrams for discrete signals. In the following discussions $x[n]$ represents the input signal, $y[n]$ represents the output signal, $n$ is an integer and represents the time index of the signal. Consider the following relationship between the input and output.\n\n\\begin{align}\ny[n] + ay[n-1] = bx[n]\n\\end{align}\n\n$a$ and $b$ are constant coefficients here. With appropriate choices for $a$ and $b$, this expression can implement a digital filter (you've seen this kind of idea above). The above equation involves three basic operations.\n\n- Addition\n- Multiplication by a coefficient\n- A delay which takes $y[n]$ as its input and outputs the previous sample $y[n-1]$.\n\n### 4.\n\n\n(c) (updated) the first one, because the second one should sort-of be zero because the noise is hopefully centered around zero, and ehhhhhhhh.\n\n### 7.\nConsider a simple model for the balance in the bank account. A certain amount is deposited every month into the account and assume that there are no withdrawals from the account. The balance amount in the account earns interest at a rate of 1% each month, which is calculated every month. Write the input - output relationship for such a bank account and implement it ung a block diagram. Please indicate clearly which signals are the inputs and outputs in the diagram. \n\n x[n] ---> (+) ------> y[n]\n ^ |\n | v\n /a\\ [ D ]\n --- |\n ^ |\n |_____|\n\n$a = 0.01$\n\n### 8.\n\nConsider the circuit described in the Figure below. Apply Kirchoff's voltage law (KVL) to note that \n \\begin{align}\n v_{in} (t)-v_R (t)-v_{out} (t)=0 \\\\\n v_{out}(t) = \\frac1C \\int_{-\\infty}^t i(\\tau) d\\tau\\,.\n \\end{align}\n\n\n\nNote that $v_{R}(t)$ is the voltage across the resistor, and $i(\\tau)$ is the current flowing clockwise in the circuit. \n\n(a) Find an expression relating $v_{out}$ to $v_{in}$, and draw a block diagram representation of this system.\n\n\n$v_{in}(t) = v_{out}(t) + RC\\frac{d}{dt}V_{out}(t)$\n\n\nDepending on how you approached the previous part, you may have come up with the following differential equation relating $v_{in}(t)$ and $v_{out}(t)$.\n\n\\begin{align}\nv_{in}(t) = v_{out}(t) + RC \\frac{d}{dt}v_{out}(t)\n\\end{align}\n\nBy taking the CTFT of each term in the expression above, find the equation relating $V_{in}(\\omega)$ and $V_{out}(\\omega)$. In other words, find the relationship between the input voltage and the output voltage in frequency space. You will need to use the CTFT properties derived in the previous section for this. \n\n(b) \nThe ratio of $V_{out}(\\omega)$ to $V_{in}(\\omega)$ is the frequency response of this system (again, you will see more of this later). Plot the magnitude of the frequency response in the range $\\omega = 1$ to $\\omega = 10^4$ radians/second. You should use logarithmic axes here. You can use $C = 1\\mu F$ and $R = 1k\\Omega$, as nominal values here. Does this picture look familiar? You will find MATLAB's `loglog` function useful to make a log-log scale plot. \n\n\n\n$V_{out}(\\omega) = \\frac{1}{1 + RCj\\omega}V_{in}(\\omega)$\n\n\n\n\n```python\nfrom sympy import symbols\nfrom sympy.plotting import plot\nimport math\n\nR = 1000\nC = 1e-6\n\nv_in = symbols('v_in')\n\nv_out = lambda w: 1/(1 + R * C * math.I * w) * v_in\n\nv_in = v_out + R*C*(dt)*v_out\n\n\n\np2 = plot((v_out, (v_out, 1, 10e4)), xscale='log', yscale='log')\n\n```\n\n## Mark's Lecture\n\nParticular tools:\n\nyou can think in discrete or continuous time\nand you can think in discrete or continous frequency\n\n\n\n\n\n\n\n
Discrete FreqContinuous Freq
Discrete TimeDFTDTFT
Continuous TimeFSCTFT
\n\n### Notch filter\n\n\n\n```python\nd_275 = thinkdsp.read_wave('BSet2Files/Disturbance275.wav');\nd_261 = thinkdsp.read_wave('BSet2Files/Disturbance261.wav');\nd_orig = thinkdsp.read_wave('BSet2Files/DisturbanceOrig.wav');\n\nd_261.plot()\n\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "f66c6517233cd734ab2cb8d9e8e809ccadcb78a4", "size": 77548, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "BSet2.ipynb", "max_stars_repo_name": "wolfd/QEA", "max_stars_repo_head_hexsha": "e46e33030d21c0af2b7da344f3f7b2087f204f5d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "BSet2.ipynb", "max_issues_repo_name": "wolfd/QEA", "max_issues_repo_head_hexsha": "e46e33030d21c0af2b7da344f3f7b2087f204f5d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "BSet2.ipynb", "max_forks_repo_name": "wolfd/QEA", "max_forks_repo_head_hexsha": "e46e33030d21c0af2b7da344f3f7b2087f204f5d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 128.8172757475, "max_line_length": 24708, "alphanum_fraction": 0.8475395884, "converted": true, "num_tokens": 3609, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.7025300573952054, "lm_q1q2_score": 0.4475332806088168}} {"text": "# invariant\n> ### coordinate independent or coordinate invariant\n>> ### \uad00\ucc30\uc790\uac00 \uce21\uc815\uac12\uc5d0 \uc601\ud5a5\uc744 \uc8fc\uc9c0 \uc54a\uc740 \uac12\n>> ### \uc88c\ud45c \ub3c5\ub9bd \ud2b9\uc131\n>>> #### \uc124\uc815\ud55c \uc88c\ud45c\uacc4\uc640\ub294 \uc0c1\uad00\uc5c6\uc774 \ub3c5\ub9bd\uc801\uc73c\ub85c \uc874\uc7ac\ud574\uc57c \ud55c\ub2e4.\n\n> ## position vector\n>> ### origin \uc774 \ub2e4\ub974\uba74 \ub2ec\ub77c\uc9c0\ub294 \uac12\uc774\ub2e4.\n> ## different among position vectors\n>> ### invariant\n> ## differential \n>> ### $ du^{i} = \\sum_{j=1}^{3} \\frac{\\partial u^i}{\\partial x^j}dx^j = \\frac{\\partial u^i}{\\partial x^j}dx^j$\n>>> ### $i$ is free index, $j$ is dummy index\n\n# Vector\n> ## $(\\,V\\,,\\,S\\,,\\,+\\,,\\,\\cdot\\,)$\n>> ## V = basis vector: $ \\quad \\sum_{i=1}^{n}\\vec{e_i} $\n>> ## S = scalar field: $ \\quad S \\in \\mathbb{F}$\n>> ## + = addition (binary operator)\n>> ## $\\cdot$ = scalar multiply (binary operator)\n\n### $ \\vec{u} = (u^1,\\, u^2,\\, u^3, ...) = u^1\\,e_1 + u^2\\,e_2 + u^3\\,e_3 + ... \\iff (B.x,\\, B.y,\\, B.z) = B.x\\,B.i + B.y\\,B.j + B.z\\,B.k$\n### $ \\vec{v} = (v^1,\\, v^2,\\, v^3, ...) = v^1\\,e_1 + v^2\\,e_2 + v^3\\,e_3 + ... \\iff (B.x,\\, B.y,\\, B.z) = B.x\\,B.i + B.y\\,B.j + B.z\\,B.k$\n### $ \\vec{w} = (w^1,\\, w^2,\\, w^3, ...) = w^1\\,e_1 + w^2\\,e_2 + w^3\\,e_3 + ... \\iff (B.x,\\, B.y,\\, B.z) = B.x\\,B.i + B.y\\,B.j + B.z\\,B.k$\n> ### $ \\sum_{i=1}^{n} v^ie_i \\iff \\sum_{s=x}^{z}B.s\\,\\sum_{b=i}^{k}B.b $\n\n# Cordinate basis\n> ### basis\ub97c \ud589\uc5f4 F\ub85c \ubcc0\ud658 \uc2dc\ud0a4\uba74 \ubcf8\ub798 \ubca1\ud130\ub294 \uadf8\ub300\ub85c \uc778\ub370 \ubca1\ud130\uc758 \uc131\ubd84\ub9cc \ub2ec\ub77c\uc9c4\ub2e4.\n> ### Old basis: {$\\vec{e_1}, \\vec{e_2}$}, New basis: {$\\tilde{\\vec{e_1}},\\tilde{\\vec{e_2}}$}\n>> #### $\\tilde{\\vec{e_1}} = 2\\vec{e_1} + 1\\vec{e_2} \\\\\n\\tilde{\\vec{e_2}} = \\frac{-1}{2}\\vec{e_1} + \\frac{1}{4}\\vec{e_2} \\\\\nF = \\begin{bmatrix} 2 & \\frac{-1}{2} \\\\ 1 & \\frac{1}{4}\n\\end{bmatrix}, \\because \\text{ firstly inner product}\n$\n>> #### \ud589\uc5f4 F\ub294 \uae30\ubcf8\uc88c\ud45c\uacc4\uc758 elements\ub97c \ubcc0\ud658 \uc2dc\ud0a8\ud6c4 \ubcc0\ud658\ub41c \uae30\ubcf8 \uc88c\ud45c\uac12\uc744 \ub118\uaca8\uc900\ub2e4.\n>> #### \uae30\ubcf8\uc88c\ud45c\uacc4\uc5d0\uc11c vector v\uc758 elements\uac00 \ubcc0\ud658\ub41c \uc88c\ud45c\ub97c \uae30\ubcf8\ucd95\uc73c\ub85c \ud588\uc744\ub54c \uc5b4\ub5bb\uac8c \uc77d\ud600\uc9c0\ub294\uac00 \ud558\uace0\ub294 \ub2e4\ub978 \uac83\uc774\ub2e4.\n\n\n\n\n```python\n# matrix product \n## firstly inner product (same element * same eleemnt)\n\n```\n\n# Tensors\n> ## in an m-dimentional space, \n>> ### a tensor of rank n is\n>>> ### a mathmatical object\n>>>> ### that has n indices, $m^n$ components, and\n>>>>> ### obeys certain transformation rules.\n> ### transformation rules:\n>> ### A tensor is an object that transformation like a tensor.\n>> ### A tensor is an object that is invariant under a change of coordinate systems, with components that change according to a special set of mathmatical formulae.\n> ### cordinate system change Intuition:\n>> ### To specify component of rank-n tensor,\n>> ### need:\n>>> ### magnitude of componet\n>>> ### n basis vecotors\n>> ### Suppose N is a coordinate transformation operator applied to a tensor\n>>> ### magnitude of tensor components will transform according to L,\n>>> ### basis vectors will transform according to $L^{-1}$.\n>>>> ### Effects of L and $L^{-1}$ cancle $\\to$ same overall tensor as under old coordinates(but writtern differently).\n\n> ## a collection of vectors and covectors combined together using the tensor product\n>> ### Tensors as partial derivatives and gradients that transform with the Jacobian Matrix\n\n\n```python\n#M = N.create_new('M',transformation=lambda x,y,z:(2*x+y,-1/2*x + 1/4*y,z))\n```\n\n## Matices\n> ### representations of rank-2 tensors. Just arrays of numbers.\n## Tensors\n> ### required detailed specification and are invariant under change of coordinate system.\n\n\n\n## [forward backward](https://www.youtube.com/watch?v=uPbBDToXjBw&list=PLJHszsWbB6hrkmmq57lX8BV-o-YIOFsiG&index=5)\n> ### Old basis: {$\\vec{e_1}, \\vec{e_2}$}, New basis: {$\\tilde{\\vec{e_1}},\\tilde{\\vec{e_2}}$}\n>> #### $\\tilde{\\vec{e_1}} = 2\\vec{e_1} + 1\\vec{e_2} \\\\\n\\tilde{\\vec{e_2}} = \\frac{-1}{2}\\vec{e_1} + \\frac{1}{4}\\vec{e_2} \\\\\nF = \\begin{bmatrix} 2 & \\frac{-1}{2} \\\\ 1 & \\frac{1}{4}\n\\end{bmatrix}, \\because \\text{ firstly inner product}\n$\n>>> ### $F' = \\begin{bmatrix} 2 & 1 \\\\ -\\frac{1}{2} & \\frac{1}{4}\n\\end{bmatrix}, \\because \\text{ what different?}\n$\n\n\n```python\n\n```\n\n\n```python\n#####################\nimport matplotlib.pyplot as plt\n%matplotlib widget\nimport numpy as np\nimport sympy as sm\nF = sm.Matrix([[2,-1/2],[1,1/4]])\nF1 = sm.Matrix([[2,1],[-1/2,1/4]])\na = sm.Matrix([[1],[1]])\n\nfig = plt.figure(figsize=(4,4))\n#ax = fig.add_subplot(projection='3d')\n#\n#ax.set_xlim(-2,3)\n#ax.set_ylim(-2,3)\n#ax.set_zlim(-2,3)\n#ax.set_yticks(np.arange(-3,3,1/4))\n#ax.set_xticks(np.arange(-3,3,1/4))\n#ax.set_xticklabels(ax.get_xticks(),rotation=90)\n#ax.set_yticklabels(ax.get_yticks(),rotation=90)\n#ax.grid()\n##ax.quiver(0,0,2,1,units='xy',color='red',scale=1)\n#ax.quiver3D(0,0,0,2,1,0,arrow_length_ratio=0.1,color='red')\n#ax.quiver3D(0,0,0,-0.5,1/4,0,color='red')\n#ax.quiver3D(0,0,0,1,0,0)\n#ax.quiver3D(0,0,0,0,1,0)\n#ax.quiver3D(0,0,0,1,1.5,0,color='green')\nax = fig.add_subplot()\nax.set_xlim(-0.75,3.25)\nax.set_ylim(-0.75,3.25)\nax.set_xticks(np.arange(-0.75,3.25,1/4))\nax.set_yticks(np.arange(-0.75,3.25,1/4))\nax.set_xticklabels(ax.get_xticks(),rotation=90)\nax.grid()\nax.quiver(0,0,1,0,units='xy',scale=1,color='r')\nax.quiver(0,0,0,1,units='xy',scale=1,color='r')\n# \\tilde \\vec{e_1} = 2 \\vec{e_1} + 1 \\vec{e_2}\nax.quiver(0,0,2,1,units='xy',scale=1,color='r')\nax.text(2,1,r'$\\tilde\\vec{e_1}$',fontsize=16)\nax.quiver(0,0,-1/2,1/4,units='xy',scale=1,color='r')\nax.text(-0.75,1/4,r'$\\tilde\\vec{e_2}$',fontsize=16)\nax.quiver(0,0,1,3/2,units='xy',scale=1,color='g')\n# useless\nF1*a\nax.quiver(0,0,3,-0.25,units='xy',scale=1,color='b')\nax.text(1.5,0.,r'$F_1(1.1)\\to(3,-0.25)$',fontsize=12)\n\n## F(1,1) -> B(1.5,1.25)\nF*a\nax.quiver(0,0,1.5,1.25,units='xy',scale=1,color='g')\nax.text(1,1.30,r'$F(1.1)\\to(1.5,1.25)$',fontsize=12)\n\n# F^{-1}*(1,1) => if B(1,1) then F(0.75,1)\nb = sm.Matrix([[0.75],[1]])\nax.quiver(0,0,0.75,1,units='xy',scale=1,color='magenta')\nF*b\n```\n\n\n```python\n#### \\vec{v} base_vectors(basis) -> B_1 + B_2 + B_3.. = \\sum_{i=1}^{n}B_i\n#### \\vec{v} base_scalars(components) -> B^1 + B^2 + B^3.. = \\sum_{i=1}^{n}B^i\n\n#### New.base_vector \\vec{e_n}(N.i, N.j, N.k)\n#--- wrt = with respect to --- \n## taking as independent variable\n##### represent New basis wrt Basis #####\n### (1,0) wrt New -> (?,?) wrt Basis\n# N.i = x*B.i + y*B.j + z*B.k\n# N.j = x*B.i + y*B.j + z*B.k\n# N.k = x*B.i + y*B.j + z*B.k\n## N.i = ( 2, 1, 0)\n## N.j = (-1/2, 1/4, 0)\n## N.k = ( 0, 0, 1)\n# v = B(1, 1.5, 0) = B.i + 1.5*B.j \n# v = N(1, 2, 0) = N.i + 2*N.j\n##############################\nimport sympy as sm\nimport sympy.vector\nB = sm.vector.CoordSys3D('B')\nT = sm.vector.CoordSys3D('T')\n# B frame = Coordinate system\nBs = B.x*B.i + B.y*B.j + B.z*B.k\nTs = T.x*T.i + T.y*T.j + T.z*T.k\n\nT.i = Bs.subs({B.x:2,B.y:1,B.z:0}) # components expression :(2,1,0) wrt Ti\nT.i = 2*B.i + 1*B.j + 0*B.k # vector expression\n# v_1\nT.j = Bs.subs({B.x:-1/2,B.y:1/4,B.z:0})\nT.j = (-1/2)*B.i + 1/4*B.j + 0*B.k\n# v_2\nT.k = Bs.subs({B.x:0,B.y:0,B.z:1})\nT.k = 1*B.k\n```\n\n# Matrix\n## $\\text{Matrix only need all coefficients of basis vectors}$\n> ### $ \\left \\{ \\begin{aligned}\nB.i = \\vec{e_1}, && B.x = v^1 \\\\\nB.j = \\vec{e_2}, && B.y = v^2 \\\\\nB.k = \\vec{e_3}, && B.z = v^3 \\end{aligned} \n\\right .\n\\quad \\left \\{ \\begin{aligned}\nT.i = \\tilde{\\vec{e_1}}, && T.x = \\tilde{v}^1 \\\\\nT.j = \\tilde{\\vec{e_2}}, && T.y = \\tilde{v}^2 \\\\\nT.k = \\tilde{\\vec{e_3}}, && T.z = \\tilde{v}^3 \\end{aligned} \n\\right .\n\\\\\n\\left \\{ \\begin{aligned}\n\\vec{v} & = v^1\\; \\vec{e_1} + v^2 \\; \\vec{e_2} + v^3\\; \\vec{e_3} &= B.x\\; B.i + B.y\\; B.j + B.z\\; B.k \\\\\n\\tilde{\\vec{v}} & = \\tilde{v}^1 \\; \\tilde{\\vec{e_1}} + \\tilde{v}^2 \\; \\tilde{\\vec{e_2}} + \\tilde{v}^3 \\; \\tilde{\\vec{e_3}} &= T.x \\; T.i + T.y \\; T.j + T.z \\; T.k \n\\end{aligned} \\right .\n\\\\\nF = \\begin{bmatrix} \nF_{11} & F_{12} & \\cdots & F_{1k} & \\cdots & F_{1n}\\\\\nF_{21} & F_{22} & \\cdots & F_{2k} & \\cdots & F_{2n}\\\\\n\\vdots & \\ddots & \\ddots & \\ddots & \\ddots & \\vdots \\\\\nF_{k1} & F_{k2} & \\cdots & F_{kk} & \\cdots & F_{kn}\\\\\n\\vdots & \\ddots & \\ddots & \\ddots & \\ddots & \\vdots \\\\\nF_{n1} & F_{n2} & \\cdots & F_{nk} & \\cdots & F_{nn}\\\\\n\\end{bmatrix}_T \nF^{-1} = \\begin{bmatrix} \nB_{11} & B_{12} & \\cdots & B_{1k} & \\cdots & B_{1n}\\\\\nB_{21} & B_{22} & \\cdots & B_{2k} & \\cdots & B_{2n}\\\\\n\\vdots & \\ddots & \\ddots & \\ddots & \\ddots & \\vdots \\\\\nB_{k1} & B_{k2} & \\cdots & B_{kk} & \\cdots & B_{kn}\\\\\n\\vdots & \\ddots & \\ddots & \\ddots & \\ddots & \\vdots \\\\\nB_{n1} & B_{n2} & \\cdots & B_{nk} & \\cdots & B_{nn}\\\\\n\\end{bmatrix}_{T^{-1}}\nF\\;F^{-1} = \\begin{bmatrix} \nF_{11}B_{11} + F_{12}B_{21} + \\cdots + F_{1k}B_{k1} + \\cdots + F_{1n}B_{n1} & \\cdots\\\\\nF_{21}B_{12} + F_{22}B_{22} + \\cdots + F_{2k}B_{k2} + \\cdots + F_{2n}B_{n2} & \\cdots\\\\\n\\ddots & \\cdots\\\\\nF_{k1}B_{1k} + F_{k2}B_{2k} + \\cdots + F{kk}B_{kk} + \\cdots + F_{kn}B_{nk} & \\cdots\\\\\n\\vdots & \\cdots\\\\\nF_{n1}B_{1n} + F_{n2}B_{2n} + \\cdots + F_{nk}B_{kn} + \\cdots + F_{nn}B_{nn} & \\cdots\\\\\n\\end{bmatrix}\n$\n\n> ### $\n(FB)_{ij} = (FF^{-1})_{ij} = \\sum_{k=1}^{n}F_{ik}B_{kj} = \\delta_{ij} \\begin{cases}1 & \\text{if}& i = j \\\\ 0 & \\text{if} &i \\neq j \\end{cases}\n$\n\n> ### $\n\\left \\{ \\begin{aligned}\n\\tilde{\\vec{e_i}} & = F_{1i}\\vec{e_1} + F_{2i}\\vec{e_2} + F_{3i}\\vec{e_3} + ... + F_{ni}\\vec{e_n} & = \\sum_{j=1}^{n} F_{ji}\\,\\vec{e_j} \\\\\n\\vec{e_i} & = B_{1i}\\tilde{\\vec{e_1}} + B_{2i}\\tilde{\\vec{e_2}} + B_{3i}\\tilde{\\vec{e_3}} + ... + B_{ni}\\tilde{\\vec{e_n}} & = \\sum_{j=1}^{n} B_{ji}\\,\\tilde{\\vec{e_j}} \\\\\n\\end{aligned} \\right .\n\\\\\n\\left \\{ \\begin{aligned}\n\\tilde{v}^i & = F_{i1}\\vec{e_i} + F_{i2}\\vec{e_i} + F_{i3}\\vec{e_i} + ... + F_{in}\\vec{e_i} & = \\sum_{j=1}^{n} F_{ij}\\,\\vec{e_i} \\\\\nv^i & = B_{i1}\\tilde{\\vec{e_i}} + B_{i2}\\tilde{\\vec{e_i}} + B_{i3}\\tilde{\\vec{e_i}} + ... + B_{in}\\tilde{\\vec{e_i}} & = \\sum_{j=1}^{n} B_{ij}\\,\\tilde{\\vec{e_i}} \\\\\n\\end{aligned} \\right .\n\\\\\n$\n\n> ## $\\text{coefficient of } \\tilde{\\vec{e_i}} $\n>> ### $\n\\begin{bmatrix}\\tilde{\\vec{e_1}} \\\\ \\tilde{\\vec{e_2}} \\\\ \\tilde{\\vec{e_3}} \\end{bmatrix} \n= \n\\begin{bmatrix} u^1\\vec{e_1} + u^2\\vec{e_2} + u^3\\vec{e_3} \\\\ v^1\\vec{e_1} + v^2\\vec{e_2} + v^3\\vec{e_3} \\\\ w^1\\vec{e_1} + w^2\\vec{e_2} + w^3\\vec{e_3} \\end{bmatrix} \\\\\n\\iff\n\\begin{bmatrix}\\tilde{\\vec{e_1}} & \\tilde{\\vec{e_2}} & \\tilde{\\vec{e_3}} \\end{bmatrix} \n= \n\\begin{bmatrix} u^1\\vec{e_1} + u^2\\vec{e_2} + u^3\\vec{e_3} & v^1\\vec{e_1} + v^2\\vec{e_2} + v^3\\vec{e_3} & w^1\\vec{e_1} + w^2\\vec{e_2} + w^3\\vec{e_3} \\end{bmatrix} \n$\n\n> ## $ \\therefore \\vec{T}\\; =\\; T.x\\; T.i \\;+\\; T.x\\; T.j \\;+\\; T.z\\; T.k \\: = \\: (\\;T.x,\\; T.y,\\; T.z\\;)$ \n>> ### $\n\\begin{bmatrix} T.i & T.j & T.k \\end{bmatrix} \\begin{bmatrix}T.x \\\\ T.y \\\\ T.z \\end{bmatrix}\n= \n\\begin{bmatrix} \nu^1\\vec{e_1} + u^2\\vec{e_2} + u^3\\vec{e_3} & \nv^1\\vec{e_1} + v^2\\vec{e_2} + v^3\\vec{e_3} & \nw^1\\vec{e_1} + w^2\\vec{e_2} + w^3\\vec{e_3} \n\\end{bmatrix}\n\\begin{bmatrix}T.x \\\\ T.y \\\\ T.z \\end{bmatrix}\\\\\n$\n> ## $\\therefore \\text{ coefficient of } \\vec{B}(\\text{ wrt }\\vec{T}) = (B.x, B.y, B.z) = B.x\\: B.i + B.y\\: B.j + B.z\\: B.k$\n>> ## $ \\begin{bmatrix} \nu^1 & v^1 & w^1 \\\\ \nu^2 & v^2 & w^2 \\\\\nu^3 & v^3 & w^3 \\\\\n\\end{bmatrix}\n\\begin{bmatrix}T.x \\\\ T.y \\\\ T.z \\end{bmatrix}\n=\n\\begin{bmatrix} \\vec{e_1} \\\\ \\vec{e_2} \\\\ \\vec{e_3} \\end{bmatrix}\n$\n\n\n```python\n\n```\n\n> ### $ \\tilde{\\vec{e_i}} \\quad \\left \\{ \\begin{aligned}\n\\: & T.i = 2(B.i) + 1(B.j) + 0(B.k) \\\\ \n& T.j = \\frac{-1}{2}(B.i) + \\frac{1}{4}(B.j) + 0(B.k) \\\\\n& T.k = 0(B.i) + 0(B.j) + 1(B.k) \\end{aligned} \\right.$\n\n> ### $ \\vec{T} \\quad \\left \\{ \\begin{aligned}\n\\: & T.x\\big(T.i\\big) = T.x\\big(2(B.i) + 1(B.j) + 0(B.k)\\big) \\\\ \n& T.y\\big(T.j\\big) = T.y\\big(\\frac{-1}{2}(B.i) + \\frac{1}{4}(B.j) + 0(B.k)\\big) \\\\\n& T.z\\big(T.k\\big) = T.z\\big(0(B.i) + 0(B.j) + 1(B.k)\\big) \\end{aligned}\\right.$\n> ### $ \\vec{T} \\text{ wrt } \\vec{B} \\quad \\left\\{\n\\begin{aligned}\n\\;& \\Big( 2 + \\frac{-1}{2} + 0 \\Big)B.i\\\\\n& \\Big( 1 + \\frac{1}{4} + 0 \\Big)B.j \\\\\n& \\Big( 0 + 0 + 1 \\Big)B.k \n\\end{aligned}\\right.\n$\n\n>> ### $\n\\begin{bmatrix} 2 & \\frac{-1}{2} & 0 \\\\ 1 & \\frac{1}{4} & 0 \\\\ 0 & 0 & 1 \\\\ \\end{bmatrix} \n\\begin{pmatrix} T.x \\\\ T.y \\\\ T.z \\\\ \\end{pmatrix}\n= \n\\begin{pmatrix} B.i \\\\ B.j \\\\ B.k \\\\ \\end{pmatrix} $\n\n> ### $(T.x\\; u^1 + T.y\\;v^1 + t.z\\; w^1)\\; e_1 + (T.x\\; u^2 + T.y\\; v^2 + t.z\\;w^2)\\;e_2 + (T.x\\; u^3 + T.y\\; v^3 + T.z\\;w^3)\\; e_3 \\\\\n\\begin{bmatrix}\nT.x\\; u^1 & T.x\\; u^2 & T.x\\; u^3 \\\\ \nT.y\\; v^1 & T,y\\; v^2 & T.y\\; v^3 \\\\\nT.z\\; w^1 & T,z\\; v^2 & T.z\\; w^3 \\\\ \n\\end{bmatrix} \\iff \n\\begin{bmatrix}\ne_1 \\\\\ne_2 \\\\\ne_3 \\\\\n\\end{bmatrix}^{\\text{ cofficient of } e_i} \\\\\n\\begin{bmatrix}\nu^1 & v^1 & w^1 \\\\ \nu^2 & v^2 & w^2 \\\\\nu^3 & v^3 & w^3 \\\\ \n\\end{bmatrix} \n\\begin{bmatrix}\nT.x \\\\\nT.y \\\\\nT.z \\\\\n\\end{bmatrix}\n\\iff\n\\begin{bmatrix}\ne_1 \\\\\ne_2 \\\\\ne_3 \\\\\n\\end{bmatrix}^{\\text{ coefficient of } e_i} \\\\\nu^1 v^1 w^1] [T.x] [e_1]\nu^2 v^2 w^2] [T.y] [e_2]\nu^3 v^3 w^3] [T.z] [e_3]\n$\n\n\n```python\nT.i = Bs.subs({B.x:2,B.y:1,B.z:0}) # components expression :(2,1,0) wrt Ti\nT.i = 2*B.i + 1*B.j + 0*B.k # vector expression\n# v_1\nT.j = Bs.subs({B.x:-1/2,B.y:1/4,B.z:0}) #(2,1,0)(-1/2,1/4,0)(0,0,1)\nT.j = (-1/2)*B.i + 1/4*B.j + 0*B.k\n# v_2\nT.k = Bs.subs({B.x:0,B.y:0,B.z:1}) #(0,0,1)\nT.k = 1*B.k\n\n# T.x\nF11 = T.i.coeff(B.i) \nF21 = T.i.coeff(B.j) \nF31 = T.i.coeff(B.k) \n# T.y\nF12 = T.j.coeff(B.i) \nF22 = T.j.coeff(B.j) \nF32 = T.j.coeff(B.k) \n#T.z\nF13 = T.k.coeff(B.i) \nF23 = T.k.coeff(B.j) \nF33 = T.k.coeff(B.k) \n# \\tield{\\vec{e}_{i}} = \\sum_{j=1}^{n}F_{ji}\\vec{e}_{i}\n# v^i = \\sum(j=1}^{n} F_{ij}\\tilde{\\vec{e}_{i}}\nF = sm.Matrix([[F11, F12, F13],\n [F21, F22, F23],\n [F31, F32, F33]])\nvi = T.x*T.i # v^0\nvj = T.y*T.j # v^1\nvk = T.z*T.k # v^2\nv = vi + vj + vk # sum_{i=1}^{3} v^i\n\nm1x = v.to_matrix(B)[0].coeff(T.x)\nm1y = v.to_matrix(B)[0].coeff(T.y)\nm1z = v.to_matrix(B)[0].coeff(T.z)\nm1x\nm2x = v.to_matrix(B)[1].coeff(T.x)\nm2y = v.to_matrix(B)[1].coeff(T.y)\nm2z = v.to_matrix(B)[1].coeff(T.z)\nm2x\nm3x = v.to_matrix(B)[2].coeff(T.x)\nm3y = v.to_matrix(B)[2].coeff(T.y)\nm3z = v.to_matrix(B)[2].coeff(T.z)\nm3x\n\nTm = sm.Matrix([[m1x, m1y, m1z],[m2x, m2y, m2z],[m3x, m3y, m3z]])\n# Tm * b_i = t_i\nFv = T.x*T.i + T.y*T.j + T.z*T.k\nFv.subs({T.x:1,T.y:2,T.z:0})\nF*sm.Matrix([[1],[2],[0]])\n#(2,1,0)(-1/2,1/4,0)(0,0,1)\nTm\n\n```\n\n\n```python\n\n```\n\n\n\n\n$\\displaystyle \\mathbf{\\hat{0}}$\n\n\n\n\n```python\nF*(sm.Matrix([1,2,0]))\n\n# N(1,2,0) wrt Matrix and vector\nv = 1*B.i + 2*B.j\nF*v.to_matrix(B) \n\n# Backward Matrix\nBm = F.inv()\nBm*sm.Matrix([1,1.5,0])\n```\n\n\n```python\nM = sm.vector.CoordSys3D('M')\n# represent Basis wrt M\nB.i = 1/4*M.i + (-1)*M.j\nB.j = 1/2*M.i + 2*M.j\nB.k = M.k\n\n# B(1, 1.5, 0) wrt vector\n1*B.i + 1.5*B.j\n\n# B(1, 1.5, 0) wrt Matrix\nBm = F.inv()\nBm * sm.Matrix( [1, 1.5, 0] )\n\n# wrt matrix and vector\nv = 1*M.i + 1.5*M.j\nBm * v.to_matrix(M)\n```\n\n---\n> ### $\n\\tilde{\\vec{e_1}} = F_{11}\\; \\vec{e_1} + F_{21}\\; \\vec{e_2} + \\dots + F_{n1}\\;\\vec{e_n}\\\\\n\\tilde{\\vec{e_2}} = F_{12}\\; \\vec{e_1} + F_{22}\\; \\vec{e_2} + \\dots + F_{n2}\\;\\vec{e_n}\\\\\n\\tilde{\\vec{e_n}} = F_{1n}\\; \\vec{e_1} + F_{2n}\\; \\vec{e_2} + \\dots + F_{nn}\\;\\vec{e_n}\\\\\nF = \\begin{bmatrix} \nF_{11} & F_{12} & F_{13} & \\dots & F_{1n} \\\\ \nF_{21} & F_{22} & F_{23} & \\dots & F_{2n} \\\\ \n\\vdots & \\vdots & \\ddots & \\vdots \\\\ \nF_{n1} & F_{n2} & F_{n3} & \\dots & F_{nn} \\\\ \n\\end{bmatrix}$ \n>> ### $ \\tilde{\\vec{e_i}} = \\sum_{j=1}^{n}F_{ji}\\;\\vec{e_{j}}\n$\n\n\n```python\ni,j,k,n = sm.symbols('i j k n', postive=True)\n\nF = sm.MatrixSymbol('F',n,n)\nB = sm.MatrixSymbol('B',n,n)\nsm.Sum(F[i,j]*B[j,k],(j,1,n))\n#sm.Sum(f[k,j]*b[j,i],(j,1,n)).subs({i:3,k:3})\n```\n\n> ### $\n\\vec{e_1} = B_{11}\\;\\tilde{ \\vec{e_1}} + B_{21}\\;\\tilde{ \\vec{e_2}} + \\dots + B_{n1}\\;\\tilde{\\vec{e_n}}\\\\\n\\vec{e_2} = B_{12}\\;\\tilde{ \\vec{e_1}} + B_{22}\\;\\tilde{ \\vec{e_2}} + \\dots + B_{n2}\\;\\tilde{\\vec{e_n}}\\\\\n\\vec{e_n} = B_{1n}\\;\\tilde{ \\vec{e_1}} + B_{2n}\\;\\tilde{ \\vec{e_2}} + \\dots + B_{nn}\\;\\tilde{\\vec{e_n}}\\\\\nB = \\begin{bmatrix} \nB_{11} & B_{12} & \\dots & B_{1n} \\\\ \nB_{21} & B_{22} & \\dots & B_{2n} \\\\ \n\\vdots & \\vdots & \\ddots & \\vdots \\\\ \nB_{n1} & B_{n2} & \\dots & B_{nn} \\\\ \n\\end{bmatrix}$ \n>> ### $$ \\vec{e_i} = \\sum_{j=1}^{n}B_{ji}\\;\\tilde{\\vec{e_{j}}}\n$$\n\n---\n### $$ \n\\vec{e_i} = \\sum_{j=1}^{n} B_{ji}\\; \\tilde{\\vec{e_j}} \\quad\n\\because \\tilde{\\vec{e_j}} = \\sum_{k=1}^{n}F_{kj}\\; \\vec{e_k}, \\\\\n\\vec{e_i} = \\sum_{j} B_{ji}\\; \\sum_{k} F_{kj}\\; \\vec{e_{k}} \\\\\n\\vec{e_i} = \\sum_{k} \\Big(\\sum_{j} B_{ji}\\; F_{kj}\\Big)\\; \\vec{e_{k}} \\\\\n\\quad \\because \\sum_{j}B_{ji}F_{jk} = \n\\begin{cases} \n1 & \\text{if } i = k \\\\\n0 & \\text{if } i \\neq k \\\\\n\\therefore \\delta_{ik} =: \\text{Kronecker Delta}\n\\end{cases}\n$$ \n$\nF = \\begin{bmatrix} \nF_{11} & F_{12} & \\dots & F_{1n} \\\\ \nF_{21} & F_{22} & \\dots & F_{2n} \\\\ \n\\vdots & \\vdots & \\ddots & \\vdots \\\\ \nF_{n1} & F_{n2} & \\dots & F_{nn} \\\\ \n\\end{bmatrix} \\;\nB = \\begin{bmatrix} \nB_{11} & B_{12} & \\dots & B_{1n} \\\\ \nB_{21} & B_{22} & \\dots & B_{2n} \\\\ \n\\vdots & \\vdots & \\ddots & \\vdots \\\\ \nB_{n1} & B_{n2} & \\dots & B_{nn} \\\\ \n\\end{bmatrix}\\;\nI(FB) = \\begin{bmatrix} \n1 & 0 & \\dots & 0 \\\\ \n0 & 1 & \\dots & 0 \\\\ \n\\vdots & \\vdots & 1 & \\vdots \\\\ \n0 & 0 & \\dots & 1\\\\ \n\\end{bmatrix}$ \n\n\n```python\n\n```\n\n# Contra Variant\n> ### $\n\\vec{v} \n= v_1\\vec{e_1} + v_2\\vec{e_2} + ... + v_n\\vec{e_n} \n= \\sum_{j=1}^n v_j\\vec{e_j}\\\\\n\\vec{v} \n= \\tilde{v_1}\\tilde{\\vec{e_1}} + \\tilde{v_2}\\tilde{\\vec{e_2}} + ... + \\tilde{v_n}\\tilde{\\vec{e_n}} \n= \\sum_{j=1}^n \\tilde{v_j}\\tilde{\\vec{e_j}}\\\\\n\\vec{v} \n\\begin{cases} \n\\sum_{j=1}^{n}v_j\\vec{e_j} \\\\\n\\sum_{i=1}^{n} \\tilde{v_i}\\tilde{\\vec{e_i}}\n\\end{cases}\n$\n\n> ### $ \\because\n\\tilde{\\vec{e_j}} = \\sum_{i=1}^n F_{ij}\\vec{e_i}, \\quad\n\\vec{e_j} = \\sum_{i=1}^n B_{ij}\\tilde{\\vec{e_i}}\n$\n>> ### $\\sum_{j=1}^{n}v_j\\vec{e_j}\n=\\sum_{j=1}^{n}v_j \\Big(\\sum_{i=1}^n B_{ij}\\tilde{\\vec{e_i}}\\Big)\n=\\sum_{i=1}^{n}\\Big(\\sum_{j=1}^n B_{ij}\\; v_j\\Big)\\tilde{\\vec{e_i}}\\\\\n\\quad \\because \\vec{v} = \\sum_{i=1}^{n}\\tilde{v_i}\\tilde{\\vec{e_i}} \\\\\n\\quad =\\sum_{i=1}^{n}\\Big(\\tilde{v_i}\\Big)\\tilde{\\vec{e_i}} \\\\\n\\therefore \\tilde{v_j} = \\sum_{j=1}^{n}B_{ij}\\; \\vec{v_j} \\\\\n$\n## Components vectors Transformation \n- ### Contra Varient\n> ### behave contray to the basis vecotrs\n>> ### therefore we gona wirte the indices above the V\n>> ### $\n\\quad \\therefore \\tilde{v^j} = \\sum_{j=1}^{n}B_{ij}\\; v^j \\\\\n\\quad \\therefore v^j = \\sum_{j=1}^{n}F_{ij}\\; \\tilde{v^j} \\\\\n\\vec{v} = \\sum_{i=1}^{n} v^{i}\\; \\vec{e_i} = \\sum_{i=1}^{n}\\tilde{v^i}\\; \\tilde{\\vec{e_i}}\n$\n\n## Basis vectors trasformation\n> ### $ \n\\quad \\therefore \\tilde{\\vec{e_j}} = \\sum_{i=1}^n F_{ij}\\vec{e_i} \\\\\n\\quad \\therefore \\vec{e_j} = \\sum_{i=1}^n B_{ij}\\tilde{\\vec{e_i}}\n$\n\n\n```python\n\n```\n\n# CoVectors\n> - ### covectors are invariant\n> - ### covector components are not invariant\n> - ### covectors are functions $\\alpha : V \\to \\mathbb{R}$\n> - ### covectors don't live in the vector space V\n> - ### so we can't use basis vectors in V like {$\\vec{e_1}, \\vec{e_2}$} to measure covectors\n\n> ### take the basis{$\\vec{e_1}, \\vec{e_2}$} for V.\n> ### when we write $ \\begin{bmatrix}2 \\\\ 1 \\end{bmatrix}_{\\vec{e_i}} = 2\\; \\vec{e_1} + 1\\; \\vec{e}_2$\n> ### introduce two special covectors $\\epsilon^1, \\epsilon^2: V \\to \\mathbb{R}$\n>> ### $\\epsilon^1 (\\vec{e_1}) = 1, \\quad \\epsilon^1 (\\vec{e_2}) = 0$\n>> ### $\\epsilon^2 (\\vec{e_1}) = 1, \\quad \\epsilon^2 (\\vec{e_2}) = 0$\n>> ### $\\epsilon^i (\\vec{e_j}) = \\delta_{ij} = \n\\begin{cases} \n1 & if \\;i = j \\\\\n0 & if \\;i \\neq j \\\\\n\\end{cases}\n$\n>>> ### $\\epsilon^1 (\\vec{v}) \n= \\epsilon^1 ( {v}^1 \\vec{e_1} + {v}^2 \\vec{e_2}) \n= v^1 \\epsilon^1 (\\vec{e_1}) + v^2 \\epsilon^1(\\vec{e_2}) = v^1 $\n>>> ### $\\epsilon^2 (\\vec{v}) \n= \\epsilon^2 ( {v}^1 \\vec{e_1} + {v}^2 \\vec{e_2}) \n= v^1 \\epsilon^2 (\\vec{e_1}) + v^2 \\epsilon^2(\\vec{e_2}) = v^2 $\n>> ### $\\therefore \\epsilon^i (\\vec{v}) = v^i$\n>> ### $\\therefore v^i = \\epsilon^i (\\vec{v}) $\n\n\n```python\nF = Bm.inv()\n\nsm.Matrix([[2,1,0]])*F\nsm.Matrix([[5,-3/4,0]])*Bm\nBm\n```\n\n> ### $\\alpha (\\vec{v}) \\begin{cases}\n\\alpha ( {v}^1 \\vec{e_1} + {v}^2 \\vec{e_2}) \\\\\n\\quad \\because \\text{ addition ruls}\\\\\nv^1 \\alpha(\\vec{e_1}) + {v}^2 \\alpha (\\vec{e_2}) \\\\\n\\quad \\because v^1 = \\epsilon^1(\\vec{v}), \\quad \\epsilon^2(\\vec{v}) = v^2 \\\\\n\\epsilon^1(\\vec{v})\\alpha(\\vec{e_1}) + \\epsilon^2(\\vec{v})\\alpha(\\vec{e_1}) \\\\\n\\quad \\text{define: } \\alpha(\\vec{e_1}) = \\alpha_1, \\quad \\alpha(\\vec{e_2}) = \\alpha_2 \\\\\n\\epsilon^1 \\alpha_1(\\vec{v}) + \\epsilon^2\\alpha_1 (\\vec{v})\\\\\n(\\epsilon^1 \\alpha_1 + \\epsilon^2\\alpha_1) (\\vec{v})\\\\\n\\quad \\because \\text{cancel } \\vec{v}\n\\end{cases}\n$\n> ## $\\therefore \\alpha = \\alpha_1 \\epsilon^1 + \\alpha_2 \\epsilon^2 \n\\begin{cases} \n\\alpha (\\vec{e_1}) = \\alpha_1 \\\\\n\\alpha (\\vec{e_2}) = \\alpha_2 \\\\\n\\epsilon^i (\\vec{e_j}) = \\delta_{ij} \\\\\n\\epsilon^i (\\vec{v_i}) = v^i\n\\end{cases}$\n>> ### $\\epsilon$ is called Dual Basis(\uc591\ucabd\uc5d0\uc11c Basis)\n>>> ### $\\because \\epsilon$ Covector form a basis for the set of all Covectors \n>> ### $\\alpha$ is General Covector\n\n\n```python\n#\n```\n\n> ### A covector(row vector) is \n>> ### Linearity\n>>> ### A fucntion that takes a vector and produces a scalar/number \n$ \\alpha: V \\to \\mathbb{R}$\n>>> ### We can add inputs or add output and get the same answer\n$ \\alpha(\\vec{v} + \\vec{w}) = \\alpha(\\vec{v}) + \\alpha \\vec{w}$\n>>> ### We can scale inputs or scale outputs and get the same answer\n$ \\alpha(n \\vec{v} + m \\vec{w}) = n \\alpha(\\vec{v}) + m \\alpha(\\vec{w})$\n\n\n\n```python\n#\n```\n\n---\n# Vector Space\n> ### $(V,S, +,\\cdot)$\n# Dual Vector Space\n> ### $(V^*, S, + , \\cdot)$\n>> ### Elements of $V^*$ ard covectors, $\\alpha: V \\to \\mathbb{R}$\n>>> $(n \\cdot \\alpha)(\\vec{v}) = n\\alpha(\\vec v) \\\\\n(\\beta + \\gamma)(\\vec v) = \\beta(\\vec v) + \\gamma(\\vec v)\\\\\n\\alpha(\\vec v + \\vec w) = \\alpha(\\vec v) + \\alpha(\\vec w)\\\\\n\\alpha(n\\vec v) = n \\alpha(\\vec v)\n$\n\n\n\n```python\nimport matplotlib.pyplot as plt\nimport numpy as np\nx = np.linspace(-10,10,100)\nfig = plt.figure(figsize=(10,10))\nax = fig.add_subplot()\nax.set_aspect('equal')\nax.set_xlim(xmin=-10,xmax=10)\nax.set_ylim(ymin=-10,ymax=10)\nax.set_xticks(np.arange(-10,10,1))\nax.set_yticks(np.arange(-10,10,1))\nax.spines['left'].set_position('center')\nax.spines['bottom'].set_position('center')\nax.grid()\n\n#ax.axhline(0)\n#ax.axvline(0)\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "487d9a5b3edf059217df054432e27bbab8d004df", "size": 128769, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "python/Vectors/Tensors.ipynb", "max_stars_repo_name": "karng87/nasm_game", "max_stars_repo_head_hexsha": "a97fdb09459efffc561d2122058c348c93f1dc87", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "python/Vectors/Tensors.ipynb", "max_issues_repo_name": "karng87/nasm_game", "max_issues_repo_head_hexsha": "a97fdb09459efffc561d2122058c348c93f1dc87", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "python/Vectors/Tensors.ipynb", "max_forks_repo_name": "karng87/nasm_game", "max_forks_repo_head_hexsha": "a97fdb09459efffc561d2122058c348c93f1dc87", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 127.3679525223, "max_line_length": 37655, "alphanum_fraction": 0.8199644324, "converted": true, "num_tokens": 10252, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952052, "lm_q2_score": 0.6370307875894139, "lm_q1q2_score": 0.44753327576770374}} {"text": "# Multi-Class Classification - Language Classification\n\nThis notebook implements the method presented in Goldberg's [2017] book \"Neural Network Methods for Natural Language Processing\". It shows the steps you need to go through in order to successfully train a classifier, and it should also, so I hope, illustrate the notational differences between Goldberg and standard machine learning literature.\n\n## Getting and cleaning the data\n\nThe data consists of downloaded Wikipedia articles about the Second World War in German, English, French, Spanish, Italian and Finnish (instead of \"O\" in Goldberg). The data is in HTML, so we need to some preprocessing to get the text out of it. We also restrict ourselfes to the characters from a to z in the alphabet (as described in Goldberg). In this fashion, we get rid of all the Umlauts (\u00e4, \u00f6, \u00fc) and all other characters with diacritics (as, e.g., the \u00e9 or \u00e7 in French). Note however, that if these characters ocurring in bigrams would probably be good features. In some way, we still keep the information \"special character\" by not fully deleting the character, but by replacing it by the dollar sign \"\\$\". Furthermore, we replace all punctuation marks and digits by dollar signs as well. As such, all special characters, digits, and punctuation marks are mapped to $. The space will be replaced by an underscore \"\\_\". We then represent each langauge by 28 characters, as is suggested by Goldberg.\n\n### Cleaning HTML\nWe first strip the HTML to get only the text of the Wikipedia page.\n\n#### Get the html files\n\n\n```python\nimport re\nfrom bs4 import BeautifulSoup\nfrom urllib.request import urlopen\nfrom collections import defaultdict\n\narticle_dict = defaultdict(lambda: defaultdict(str))\n\nregex = r'[\\n ]{2,}'\npattern = re.compile(regex)\n\nurls = open('urls.txt', 'r').readlines()\n\nfor index, url in enumerate(urls):\n language = url[8:10]\n doc_id = 'doc_%d' % index\n html = urlopen(url.strip()).read() \n soup = BeautifulSoup(html, 'html.parser')\n raw = soup.body.get_text() # only get text from the text body (this excludes headers and should exclude navigation bars)\n raw = re.sub(pattern, ' ', raw) # replace multiple breaks and spaces by only one space\n raw = re.sub(r'\\n', ' ', raw) # replace every line break with a space\n article_dict[language][doc_id] = raw.lower() # assign each text to its language and lower all uppercase characters\n```\n\n### Preprocessing --> prepare the text\nreplace special characters and digits\n\n\n```python\npreprocessed_dict = defaultdict(lambda: defaultdict(str))\n\nabc = r'[a-z]'\nabc_pattern = re.compile(abc)\n\nfor lang, doc in article_dict.items():\n for doc, text in doc.items():\n for char in text:\n if re.match(abc_pattern, char):\n preprocessed_dict[lang][doc] += char\n elif re.match(' ', char):\n preprocessed_dict[lang][doc] += '_'\n else:\n preprocessed_dict[lang][doc] += '$'\n```\n\n### Count bigrams --> Feature extraction\n\nThe distribution of bigrams will be our only feature. We could extend this by taking into account other n-grams.\n\n\n```python\ncharset = 'abcdefghijklmnopqrstuvwxyz$_' # define the character set we want to use\n```\n\n\n```python\nfrom itertools import combinations_with_replacement, permutations\n\ndef bigrams(text):\n \"\"\"\n Function to extract bigrams from text and calculate their distribution\n :param text: text string\n :return: dictionary containing bigrams as keys, and the normalised count as values\n \"\"\"\n combs = combinations_with_replacement(charset, 2)\n perms = permutations(charset, 2)\n bigram_dict = dict()\n \n for comb in set(list(combs) + list(perms)):\n bigram_dict[''.join(comb)] = 0\n \n doc_length = len(text)\n \n for index in range(0, len(text)-1):\n bigram = text[index] + text[index+1]\n bigram_dict[bigram] += 1\n \n for bigram, count in bigram_dict.items():\n bigram_dict[bigram] = count/doc_length\n\n return bigram_dict \n```\n\n### Put data into pandas dataframe\nThe pandas dataframe allows us to conveniently represent all the data we need in one table. So let's do this. But first we need to extract the features.\n\n\n```python\nbigram_dict_full = defaultdict(lambda: defaultdict(dict))\n\nfor lang, doc in preprocessed_dict.items():\n for doc, text in sorted(doc.items()):\n bigram_dict = bigrams(text)\n bigram_dict_full[lang][doc] = bigram_dict\n```\n\n\n```python\nimport pandas as pd\n\ncol_names = ['y'] + sorted(bigram_dict_full['en']['doc_0'].keys())\nmy_df = dict()\n\nfor col in col_names:\n my_df[col] = list()\n \ndf = pd.DataFrame(my_df)\n\nfor lang, doc in bigram_dict_full.items():\n for key, value in doc.items():\n df_obj = value\n df_obj['y'] = lang\n df = df.append(df_obj, ignore_index=True)\n \ndf.head()\n \n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
$$$_$a$b$c$d$e$f$g$h...zqzrzsztzuzvzwzxzyzz
00.0686040.0252860.0009440.0009440.0004470.0002480.0013410.0002480.0000990.001490...0.00.00.00.00.0000500.00.0000000.00.00.0
10.0704150.0345380.0020640.0003170.0002490.0002950.0010210.0001360.0001130.001066...0.00.00.00.00.0000230.00.0000000.00.00.0
20.0535900.0313920.0006110.0002620.0001750.0003200.0006400.0001750.0000290.001571...0.00.00.00.00.0000000.00.0000000.00.00.0
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\n\n\n\n\n```python\ndf.shape\n```\n\n\n\n\n (60, 785)\n\n\n\nNow we split the data into the label vector \\begin{equation}\\mathbf{y}\\end{equation} and a training data matrix \\begin{equation}\\mathbf{X}\\end{equation}. But first, we shuffle the df and split it into a training and a test set.\n\n\n```python\ntrain = df.sample(frac=0.9,random_state=200)\ntest = df.drop(train.index)\n\ny = train.y\nX = train.drop('y', axis=1)\n```\n\n\n```python\nX.shape\n```\n\n\n\n\n (54, 784)\n\n\n\n\n```python\ny.shape\n```\n\n\n\n\n (54,)\n\n\n\n\n```python\nsum(X.iloc[2])\n```\n\n\n\n\n 0.9999709065518464\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "f076d34c7136e6e575f040e06be3e78c52c13f6e", "size": 16164, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lib/.ipynb_checkpoints/language_classification-checkpoint.ipynb", "max_stars_repo_name": "pstroe/goldberg_nnm4nlp", "max_stars_repo_head_hexsha": "2b66842e4c85b882c3c22959b3522daff3465493", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2019-05-19T03:22:27.000Z", "max_stars_repo_stars_event_max_datetime": "2021-01-08T06:03:36.000Z", "max_issues_repo_path": "lib/.ipynb_checkpoints/language_classification-checkpoint.ipynb", "max_issues_repo_name": "pstroe/goldberg_nnm4nlp", "max_issues_repo_head_hexsha": "2b66842e4c85b882c3c22959b3522daff3465493", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "lib/.ipynb_checkpoints/language_classification-checkpoint.ipynb", "max_forks_repo_name": "pstroe/goldberg_nnm4nlp", "max_forks_repo_head_hexsha": "2b66842e4c85b882c3c22959b3522daff3465493", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2018-10-01T13:51:49.000Z", "max_forks_repo_forks_event_max_datetime": "2019-06-18T09:50:15.000Z", "avg_line_length": 30.5557655955, "max_line_length": 1022, "alphanum_fraction": 0.4503835684, "converted": true, "num_tokens": 2778, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307806984444, "lm_q2_score": 0.7025300573952052, "lm_q1q2_score": 0.44753327092659057}} {"text": "```python\nfrom IPython.core.display import display_html\nfrom urllib.request import urlopen\n\ncssurl = 'http://j.mp/1DnuN9M'\ndisplay_html(urlopen(cssurl).read(), raw=True)\n```\n\n\n\n\n\n\n\n\n\n\n# An\u00e1lisis de estabilidad para sistema bajo realimentaci\u00f3n integral\n\n## Ejemplo 1\n\nDado el sistema:\n\n$$\n\\dot{x}(t) = A x(t) + B u(t - h)\n$$\n\ncon la ley de control:\n\n$$\nu(t) = k \\left[ x(t) + \\int_{-h}^0 e^{-A(\\theta + h)} B u(t + \\theta) d\\theta \\right]\n$$\n\npor lo que el sistema en lazo cerrado es:\n\n$$\n\\dot{x}(t) = \\left( A + e^{-Ah} B K \\right) x(t)\n$$\n\nel sistema para el que haremos este desarrollo es:\n\n$$\n\\dot{x}(t) =\n\\begin{pmatrix}\n0 & 0 \\\\\n1 & 1\n\\end{pmatrix} x(t) +\n\\begin{pmatrix}\n1 \\\\\n0\n\\end{pmatrix} u(t - h)\n$$\n\ncon $h = 1$.\n\n\n```python\nfrom sympy import var, sin, cos, Matrix, Integer, eye, Function, Rational, exp, Symbol, I, solve\nfrom sympy.physics.mechanics import mechanics_printing\nmechanics_printing()\n```\n\n\n```python\nvar(\"t h \u03b8 s \u03c9\")\n```\n\nPara analizar la estabilidad del sistema bajo realimentaci\u00f3n, utilizamos:\n\n$$\n\\dot{x}(t) = \\left( A + e^{-Ah} B K \\right) x(t)\n$$\n\npor lo que la funci\u00f3n de transferencia del sistema realimentado ser\u00e1:\n\n$$\n\\det{\\left( sI - A - e^{-Ah} B K \\right)}\n$$\n\n\n```python\nA1 = Matrix([[0, 0], [1, 1]])\nB1 = Matrix([[1], [0]])\nK1 = Matrix([[1 - 4*exp(h), -4*exp(h)]])\n```\n\n\n```python\n(s*eye(2) - A1 - exp(-A1*h)*B1*K1).det()\n```\n\nPor lo que observamos que esta realimentaci\u00f3n coloca dos polos en $-1$, sin embargo queremos analizar la estabilidad bajo los parametros que establecimos, por lo que notamos que este polinomio puede ser escrito como:\n\n$$\ns^2 - \\left( 1 + \\alpha_1 + \\alpha_2 \\right)s + \\alpha_1\n$$\n\n\n```python\n\u03b11, \u03b12 = K1[0] - K1[1], exp(-h)*K1[1]\n\u03b11, \u03b12\n```\n\n\n```python\ns**2 - (1 + \u03b11 + \u03b12)*s + \u03b11\n```\n\nEste polinomio caracteristico esta libre de retardos, por lo que podemos analizarlo con Routh-Hurwitz y obtener las siguientes condiciones:\n\n$$\n\\begin{align}\n\\alpha_1 &> 0 \\\\\n\\alpha_1 &< -1 - \\alpha_2\n\\end{align}\n$$\n\nPor otro lado, si hacemos un analisis de D-particiones, al sustituir $s = 0$ y $s = j \\omega$ obtenemos que:\n\n$$\n\\begin{align}\n\\alpha_1 &= 0 \\\\\n\\alpha_1 &= -1 - \\alpha_2\n\\end{align}\n$$\n\n\n```python\nvar(\"\u03b1_1 \u03b1_2 \u03c9\")\n```\n\n\n```python\n(s**2 - (1 + \u03b1_1 + \u03b1_2)*s + \u03b1_1).subs(s, 0)\n```\n\n\n```python\n(s**2 - (1 + \u03b1_1 + \u03b1_2)*s + \u03b1_1).subs(s, 1j*\u03c9).coeff(-1j*\u03c9)\n```\n\nLo cual es consistente con los resultados de Routh-Hurwitz. Al graficar estas curvas limite de las D-particiones, obtenemos:\n\n\n```python\nfrom numpy import linspace, zeros, concatenate, column_stack\n```\n\n\n```python\n%matplotlib inline\nfrom matplotlib.pyplot import plot, style, figure, legend, fill, Polygon\nstyle.use(\"ggplot\")\n```\n\n\n```python\nx = linspace(-4, -1, 100)\nalpha1 = linspace(-4, 4, 100)\nalpha2 = -alpha1 - 1\n```\n\n\n```python\nf = figure(figsize=(8, 8))\nplot(zeros(len(alpha1)), alpha1)\nplot(alpha1, alpha2)\n\nax = f.gca()\nax.set_xlim(-4, 4)\nax.set_ylim(-4, 4)\n\nax.fill_betweenx(alpha2, 0, alpha1, where=alpha1>0, alpha=0.3, facecolor='purple')\n\nax.set_xlabel(r\"$\u03b1_1$\", fontsize=20)\nax.set_ylabel(r\"$\u03b1_2$\", fontsize=20);\n```\n\nPor otro lado, para nalizar la estabilidad del controlador empezamos calculando $e^{-A (\\theta + h)}$:\n\n\n```python\nexp(-A1*(\u03b8+ h))\n```\n\nSustituyendo $A$, $B$ y $e^{-A(\\theta + h)}$ en $u(t)$, tenemos:\n\n$$\n\\begin{align}\nu(t) &=\n\\begin{pmatrix}\nk_1 & k_2\n\\end{pmatrix} x(t) +\n\\begin{pmatrix}\nk_1 & k_2\n\\end{pmatrix}\n\\int_{-h}^0 e^{-A(\\theta + h)} B u(t + \\theta) d\\theta \\\\\n&=\n\\begin{pmatrix}\nk_1 & k_2\n\\end{pmatrix} x(t) +\n\\begin{pmatrix}\nk_1 & k_2\n\\end{pmatrix}\n\\int_{-h}^0\n\\begin{pmatrix}\n1 & 0 \\\\\ne^{-(\\theta + h)} - 1 & e^{-(\\theta + h)}\n\\end{pmatrix}\n\\begin{pmatrix}\n1 \\\\\n0\n\\end{pmatrix}\nu(t + \\theta) d\\theta \\\\\n&=\n\\begin{pmatrix}\nk_1 & k_2\n\\end{pmatrix} x(t) +\n\\begin{pmatrix}\nk_1 & k_2\n\\end{pmatrix}\n\\int_{-h}^0\n\\begin{pmatrix}\n1 \\\\\ne^{-(\\theta + h)} - 1\n\\end{pmatrix}\nu(t + \\theta) d\\theta\n\\end{align}\n$$\n\nSustituyendo $k_1 = 1 - 4 e^h$ y $k_2 = -4e^h$ tenemos:\n\n$$\nu(t)=\n\\begin{pmatrix}\n1 - 4 e^h & -4e^h\n\\end{pmatrix} x(t) +\n\\begin{pmatrix}\n1 - 4 e^h & -4e^h\n\\end{pmatrix}\n\\int_{-h}^0\n\\begin{pmatrix}\n1 \\\\\ne^{-(\\theta + h)} - 1\n\\end{pmatrix}\nu(t + \\theta) d\\theta\n$$\n\ny podemos meter estas ganancias a la integral, para obtener:\n\n$$\n\\begin{align}\nu(t) &=\n\\begin{pmatrix}\n1 - 4 e^h & -4e^h\n\\end{pmatrix} x(t) +\n\\int_{-h}^0\n\\left( 1 - 4e^{-\\theta} \\right)\nu(t + \\theta) d\\theta \\\\\n&=\n\\begin{pmatrix}\n1 - 4 e^h & -4e^h\n\\end{pmatrix} x(t) +\n\\int_{-h}^0 u(t + \\theta) d\\theta -\n\\int_{-h}^0 4e^{-\\theta} u(t + \\theta) d\\theta\n\\end{align}\n$$\n\n\n```python\n(K1*exp(-A1*(\u03b8 + h))*B1)[0].simplify()\n```\n\n\n```python\nx1 = Function(\"x1\")(t)\nx2 = Function(\"x2\")(t)\n\nX = Matrix([[x1], [x2]])\n\nu = Function(\"u\")(t + \u03b8)\n```\n\n\n```python\nA1*X\n```\n\n\n```python\n((K1*exp(-A1*(\u03b8 + h))*B1)[0].simplify()*u).integrate((\u03b8, -h, 0))\n```\n\n\n```python\n(K1*X)[0] + ((K1*exp(-A1*(\u03b8 + h))*B1)[0].simplify()*u).integrate((\u03b8, -h, 0))\n```\n\nSi aplicamos la transformada de Laplace a esto, obtendremos:\n\n$$\n\\begin{align}\nu(t) - \\int_{-h}^0 u(t + \\theta) d\\theta + \\int_{-h}^0 4e^{-\\theta} u(t + \\theta) d\\theta &=\n(1 - 4 e^h) x_1(t) - 4e^h x_2(t) \\\\\n\\left[ 1 - \\frac{1 - e^{-hs}}{s} + 4 \\frac{1 - e^{-h(s-1)}}{s-1} \\right] u(s) &=\n(1 - 4 e^h) x_1(s) - 4e^h x_2(s)\n\\end{align}\n$$\n\nPor lo que el polinomio caracteristico del controlador del sistema es:\n\n$$\n1 - \\frac{1 - e^{-hs}}{s} + 4\\frac{1 - e^{-h(s-1)}}{s-1} = 0\n$$\n\nSi ahora introducimos los parametros $\\alpha_1 = k_1 - k_2 = 1$ y $\\alpha_2 = e^{-h} k_2 = -4$, este polinomio caracteristico queda de la forma:\n\n$$\n1 - \\alpha_1 \\frac{1 - e^{-hs}}{s} - \\alpha_2 \\frac{1 - e^{-h(s-1)}}{s-1} = 0\n$$\n\nSustituyendo $s = j \\omega$, tendremos:\n\n$$\n- \\alpha_{1} \\left(- \u03c9 \\operatorname{sin}\\left(h \u03c9\\right) + \\operatorname{cos}\\left(h \u03c9\\right) - 1\\right) + \\alpha_{2} \u03c9 e^{h} \\operatorname{sin}\\left(h \u03c9\\right) - \u03c9^{2} + j \\left( - \\alpha_{1} \\left(- \u03c9 \\operatorname{cos}\\left(h \u03c9\\right) + \u03c9 - \\operatorname{sin}\\left(h \u03c9\\right)\\right) - \\alpha_{2} \u03c9 \\left(- e^{h} \\operatorname{cos}\\left(h \u03c9\\right) + 1\\right) - \u03c9 \\right) = 0\n$$\n\n\n```python\nh = Symbol(\"h\", real=True, imag=False)\n\u03c9 = Symbol(\"\u03c9\", real=True, imag=False)\n\u03b1_1 = Symbol(\"\u03b1_1\", real=True, imag=False)\n\u03b1_2 = Symbol(\"\u03b1_2\", real=True, imag=False)\n```\n\n\n```python\nr, i = (s**2 - s - \u03b1_1*(s - 1)*(1 - exp(-h*s)) - \u03b1_2*s*(1 - exp(-h*(s - 1)))).subs(s, I*\u03c9).as_real_imag()\nr + i*I\n```\n\n\n```python\nr\n```\n\n\n```python\ni\n```\n\nPor lo que al separar en parte real e imaginaria, obtenemos dos expresiones de donde se puede obtener $\\alpha_1$ y $\\alpha_2$, en terminos de $\\omega$:\n\n\n```python\nal2 = solve(r, \u03b1_2)[0]\nal2\n```\n\n\n```python\nal1 = solve(i.subs(\u03b1_2, al2), \u03b1_1)[0]\nal1\n```\n\nCreando funciones parametricas para estos valores, podemos graficar las D-particiones del controlador:\n\n\n```python\ndef par1(\u03c9, h):\n from numpy import sin, cos, exp\n num = \u03c9*(\u03c9*cos(\u03c9*h)*exp(h) - \u03c9 - sin(\u03c9*h)*exp(h))\n den = \u03c9*sin(\u03c9*h)*(exp(h) - 1) + (cos(\u03c9*h) - 1)*(exp(h) + 1)\n return num/den\n \ndef par2(\u03b11, \u03c9, h):\n from numpy import sin, cos, exp\n num = \u03c9**2 - \u03b11*(\u03c9*sin(\u03c9*h) - cos(\u03c9*h) + 1)\n den = \u03c9*sin(\u03c9*h)*exp(h)\n return num/den\n```\n\n\n```python\nfrom numpy import pi\n\u03c4 = 2*pi # o mas bien \u03c0 = 1/2 \u03c4\n\u025b = 0.0001\n```\n\n\n```python\noms = linspace(\u025b, \u03c4 - \u025b, 1.0/\u025b)\nalpha_1_1 = [par1(om, 1.0) for om in oms]\nalpha_2_1 = [par2(alpha1, om, 1.0) for om, alpha1 in zip(oms, alpha_1_1)]\n\noms = linspace(\u03c4 + \u025b, 1.3*\u03c4 - \u025b, 0.3/\u025b)\nalpha_1_2 = [par1(om, 1.0) for om in oms]\nalpha_2_2 = [par2(alpha1, om, 1.0) for om, alpha1 in zip(oms, alpha_1_2)]\n\nal1 = concatenate((alpha_1_1[::-1], alpha_1_2))\nal2 = concatenate((alpha_2_1[::-1], alpha_2_2))\n```\n\n\n```python\nf = figure(figsize=(8, 8))\nplot(alpha_1_1, alpha_2_1)\nplot(alpha_1_2, alpha_2_2)\n\nax = f.gca()\nax.set_xlim(-4, 4)\nax.set_ylim(-4, 4)\n\np = Polygon(column_stack((al1, al2)), facecolor='cyan', alpha=0.2, edgecolor='none')\n\nax.add_artist(p)\n\nax.set_xlabel(r\"$k_1$\", fontsize=20)\nax.set_ylabel(r\"$k_2$\", fontsize=20);\n```\n\nEn donde el punto $(\\alpha_1, \\alpha_2) = (0, 0)$ es trivialmente estable por Routh - Hurwitz, por lo que podemos considerar la region central de esta gr\u00e1fica, como estable. Si ahora juntamos las dos gr\u00e1ficas de D-particiones, obtenemos:\n\n\n```python\nf = figure(figsize=(8, 8))\n\nplot(zeros(len(alpha1)), alpha1)\nplot(alpha1, alpha2)\n\nplot(alpha_1_1, alpha_2_1)\nplot(alpha_1_2, alpha_2_2)\n\nax = f.gca()\nax.set_xlim(-4, 4)\nax.set_ylim(-4, 4)\n\nax.fill_betweenx(alpha2, 0, alpha1, where=alpha1>0, alpha=0.3, facecolor='purple')\np = Polygon(column_stack((al1, al2)), facecolor='cyan', alpha=0.2, edgecolor='none')\nax.add_artist(p)\n\nax.set_xlabel(r\"$k_1$\", fontsize=20)\nax.set_ylabel(r\"$k_2$\", fontsize=20);\n```\n\nY el sistema bajo realimentaci\u00f3n ser\u00e1 estable, siempre y cuando escojamos una realimentaci\u00f3n dentro de las dos regiones de estabilidad.\n\n## Ejemplo 2\n\nPara el sistema:\n\n$$\n\\dot{x}(t) =\n\\begin{pmatrix}\n0 & 1 \\\\\n0 & 0\n\\end{pmatrix}\nx(t) +\n\\begin{pmatrix}\n0 \\\\\n1\n\\end{pmatrix}\nu(t - h)\n$$\n\ncon $h = 1$.\n\n\n```python\nvar(\"k1 k2\")\n```\n\n\n```python\nA2 = Matrix([[0, 1], [0, 0]])\nB2 = Matrix([[0], [1]])\nK2 = Matrix([[k1, k2]])\n```\n\nTiene un polinomio caracteristico:\n\n\n```python\n((s*eye(2) - A2 - exp(-A2*h)*B2*K2).det()).collect(s)\n```\n\no bien:\n\n$$\ns^2 + \\left( h k_1 - k_2 \\right) s - k_1\n$$\n\nAl cual podemos aplicar el criterio de estabilidad de Routh-Hurwitz y obtener:\n\n$$\n\\begin{align}\nk_1 &< 0 \\\\\nk_2 &< h k_1\n\\end{align}\n$$\n\nPor lo que la gr\u00e1fica de D-particiones se ver\u00e1:\n\n\n```python\nh = 1\nx = linspace(-4, -1, 100)\nK_1 = linspace(-4, 4, 100)\nK_2 = h*K_1\n```\n\n\n```python\nf = figure(figsize=(8, 8))\nplot(zeros(len(K_1)), K_1)\nplot(K_1, K_2)\n\nax = f.gca()\nax.set_xlim(-4, 4)\nax.set_ylim(-4, 4)\n\nax.fill_betweenx(K_2, 0, K_1, where=K_1<0, alpha=0.3, facecolor='green')\n\nax.set_xlabel(r\"$k_1$\", fontsize=20)\nax.set_ylabel(r\"$k_2$\", fontsize=20);\n```\n\nPor otro lado, para analizar el comportamiento del controlador, sustituimos los datos en la ecuaci\u00f3n del controlador:\n\n$$\n\\begin{align}\nu(t) &=\n\\begin{pmatrix}\nk_1 & k_2\n\\end{pmatrix} x(t) +\n\\begin{pmatrix}\nk_1 & k_2\n\\end{pmatrix}\n\\int_{-h}^0 e^{-A(\\theta + h)} B u(t + \\theta) d\\theta \\\\\n&=\n\\begin{pmatrix}\nk_1 & k_2\n\\end{pmatrix}\n\\begin{pmatrix}\nx_1(t) \\\\\nx_2(t)\n\\end{pmatrix} +\n\\begin{pmatrix}\nk_1 & k_2\n\\end{pmatrix}\n\\int_{-h}^0 e^{-A(\\theta + h)} B u(t + \\theta) d\\theta \\\\\n\\end{align}\n$$\n\n\n```python\nexp(-A2*(\u03b8 + h))\n```\n\n$$\n\\begin{align}\nu(t) &=\n\\begin{pmatrix}\nk_1 & k_2\n\\end{pmatrix}\n\\begin{pmatrix}\nx_1(t) \\\\\nx_2(t)\n\\end{pmatrix} +\n\\int_{-h}^0\n\\begin{pmatrix}\nk_1 & k_2\n\\end{pmatrix}\n\\begin{pmatrix}\n1 & - (\\theta +h) \\\\\n0 & 1\n\\end{pmatrix}\n\\begin{pmatrix}\n0 \\\\\n1\n\\end{pmatrix} u(t + \\theta) d\\theta \\\\\n&= k_1 x_1(t) + k_2 x_2(t) - \\int_{-h}^0 k_1 \\theta u(t + \\theta) d\\theta - \\int_{-h}^0 k_1 h u(t + \\theta) d\\theta + \\int_{-h}^0 k_2 u(t + \\theta) d\\theta \\\\\n\\end{align}\n$$\n\ny al aplicar la transformada de Laplace, tenemos:\n\n$$\nu(s) = k_1 x_1(s) + k_2 x_2(s) - h k_1 \\frac{e^{-hs}}{s} u(s) + k_1 \\frac{1 - e^{-hs}}{s^2} u(s) - h k_1 \\frac{1 - e^{-hs}}{s} u(s) + k_2 \\frac{1 - e^{-hs}}{s} u(s)\n$$\n\npor lo que al pasar a un solo lado todos los terminos de $u(s)$:\n\n$$\n\\begin{align}\n\\left[ 1 + h k_1 \\frac{e^{-hs}}{s} - k_1 \\frac{1 - e^{-hs}}{s^2} + h k_1 \\frac{1 - e^{-hs}}{s} - k_2 \\frac{1 - e^{-hs}}{s} \\right] u(s) &= k_1 x_1(s) + k_2 x_2(s) \\\\\n\\left[ 1 + \\frac{h k_1 e^{-hs}}{s} - \\frac{k_1}{s^2} + \\frac{k_1 e^{-hs}}{s^2} + \\frac{h k_1}{s} - \\frac{h k_1 e^{-hs}}{s} - \\frac{k_2}{s} + \\frac{k_2 e^{-hs}}{s} \\right] u(s) &= k_1 x_1(s) + k_2 x_2(s) \\\\\n\\left[ 1 - \\frac{k_1}{s^2} + \\frac{k_1 e^{-hs}}{s^2} + \\frac{h k_1}{s} - \\frac{k_2}{s} + \\frac{k_2 e^{-hs}}{s} \\right] u(s) &= k_1 x_1(s) + k_2 x_2(s) \\\\\n\\left[ 1 + \\frac{k_1 e^{-hs} - k_1}{s^2} + \\frac{h k_1 + k_2 e^{-hs} - k_2}{s} \\right] u(s) &= k_1 x_1(s) + k_2 x_2(s)\n\\end{align}\n$$\n\nobtenemos el polinomio caracteristico de la ecuaci\u00f3n de control:\n\n$$\n1 + \\frac{k_1 e^{-hs} - k_1}{s^2} + \\frac{h k_1 + k_2 e^{-hs} - k_2}{s}\n$$\n\ny al sustituir $s = j \\omega$, obtendremos dos ecuaciones, correspondientes a la parte real e imaginaria:\n\n$$\n\\begin{align}\nk_1 \\left[ \\omega h - \\sin{(\\omega h)} \\right] - k_2 \\left[ \\omega - \\cos{(\\omega h)} \\right] &= 0 \\\\\n- k_1 \\left[ 1 - \\cos{(\\omega h)} \\right] + k_2 \\left[ \\omega \\sin{(\\omega h)} \\right] - \\omega^2 &= 0 \\\\\n\\end{align}\n$$\n\npor lo que podemos despejar $k_2$ de ambas ecuaciones y obtener:\n\n$$\nk_2 = \\frac{k_1 \\left[ \\omega h - \\sin{(\\omega h)} \\right]}{\\omega - \\cos{(\\omega h)}} = \\frac{k_1 \\left[ 1 - \\cos{(\\omega h)} \\right] + \\omega^2}{\\omega \\sin{(\\omega h)}}\n$$\n\ny haciendo un poco de algebra, podemos obtener:\n\n$$\n\\frac{k_1 \\left[ \\omega h - \\sin{(\\omega h)} \\right]}{\\omega - \\cos{(\\omega h)}} = \\frac{k_1 \\left[ 1 - \\cos{(\\omega h)} \\right] + \\omega^2}{\\omega \\sin{(\\omega h)}}\n$$\n\n$$\n\\frac{k_1 \\left[ \\omega h - \\sin{(\\omega h)} \\right] \\left[ \\omega \\sin{(\\omega h)} \\right]}{\\omega - \\cos{(\\omega h)}} - k_1 \\left[ 1 - \\cos{(\\omega h)} \\right] = \\omega^2\n$$\n\n$$\nk_1 \\frac{\\left[ \\omega h - \\sin{(\\omega h)} \\right] \\left[ \\omega \\sin{(\\omega h)} \\right] - \\left[ 1 - \\cos{(\\omega h)} \\right] \\left[ \\omega - \\cos{(\\omega h)} \\right]}{\\omega - \\cos{(\\omega h)}} = \\omega^2\n$$\n\n$$\nk_1 = \\frac{\\omega^2 \\left[ \\omega - \\cos{(\\omega h)} \\right]}{\\left[ \\omega h - \\sin{(\\omega h)} \\right] \\left[ \\omega \\sin{(\\omega h)} \\right] - \\left[ 1 - \\cos{(\\omega h)} \\right] \\left[ \\omega - \\cos{(\\omega h)} \\right]}\n$$\n\nSi sustituimos un punto por debajo de esta curva, $(k_1, k_2) = (0, 0)$, podemos ver que el polinomio caracteristico es trivialmente estable por el criterio de Routh-Hurwitz:\n\n$$\nP(s) = 1\n$$\n\npor lo que la gr\u00e1fica de D-particiones para el controlador queda:\n\n\n```python\ndef par1(\u03c9, h):\n from numpy import sin, cos\n num = \u03c9**2*(\u03c9 - cos(\u03c9*h))\n den = (\u03c9*h - sin(\u03c9*h))*(\u03c9*sin(\u03c9*h)) - (1 - cos(\u03c9*h))*(\u03c9 - cos(\u03c9*h))\n return num/den\n \ndef par2(k1, \u03c9, h):\n from numpy import sin, cos\n num = k1*(\u03c9*h - sin(\u03c9*h))\n den = \u03c9 - cos(\u03c9*h)\n return num/den\n```\n\n\n```python\noms = linspace(-0.22*\u03c4, 0.13*\u03c4, 10000)\nk_1 = [par1(om, 1.0) for om in oms]\nk_2 = [par2(k1, om, 1.0) for om, k1 in zip(oms, k_1)]\n```\n\n\n```python\nf = figure(figsize=(8, 8))\nplot(k_1, k_2)\n\nax = f.gca()\nax.set_xlim(-4, 4)\nax.set_ylim(-4, 4)\n\nax.fill_between(k_1, -4, k_2, alpha=0.3, facecolor='orange')\n\nax.set_xlabel(r\"$k_1$\", fontsize=20)\nax.set_ylabel(r\"$k_2$\", fontsize=20);\n```\n\nY el sistema con este controlador ser\u00e1 estable para los valores de $k_1$ y $k_2$ escogidos tal que se encuentren en la intersecci\u00f3n de estas dos regiones:\n\n\n```python\nf = figure(figsize=(8, 8))\nplot(zeros(len(K_1)), K_1)\nplot(K_1, K_2)\nplot(k_1, k_2)\n\nax = f.gca()\nax.set_xlim(-4, 4)\nax.set_ylim(-4, 4)\n\nax.fill_betweenx(K_2, 0, K_1, where=K_1<0, alpha=0.3, facecolor='green')\nax.fill_between(k_1, -4, k_2, alpha=0.3, facecolor='orange')\n\nax.set_xlabel(r\"$k_1$\", fontsize=20)\nax.set_ylabel(r\"$k_2$\", fontsize=20);\n```\n\nPuedes acceder a este notebook a traves de la p\u00e1gina\n\nhttp://bit.ly/1wMAK3L\n\no escaneando el siguiente c\u00f3digo:\n\n\n\n\n```python\n# Codigo para generar codigo :)\nfrom qrcode import make\nimg = make(\"http://bit.ly/1wMAK3L\")\nimg.save(\"codigos/codigo11.jpg\")\n```\n", "meta": {"hexsha": "9344cca2f4cbdb88c45b3282feff417066cc2bf5", "size": 172260, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "IPythonNotebooks/Sistemas con retardo en la entrada/Tarea 11.ipynb", "max_stars_repo_name": "robblack007/DCA", "max_stars_repo_head_hexsha": "0ea5f8b613e2dabe1127b857c7bfe9be64c52d20", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "IPythonNotebooks/Sistemas con retardo en la entrada/Tarea 11.ipynb", "max_issues_repo_name": "robblack007/DCA", "max_issues_repo_head_hexsha": "0ea5f8b613e2dabe1127b857c7bfe9be64c52d20", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "IPythonNotebooks/Sistemas con retardo en la entrada/Tarea 11.ipynb", "max_forks_repo_name": "robblack007/DCA", "max_forks_repo_head_hexsha": "0ea5f8b613e2dabe1127b857c7bfe9be64c52d20", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-03-20T12:44:13.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-20T12:44:13.000Z", "avg_line_length": 106.5306122449, "max_line_length": 22902, "alphanum_fraction": 0.8265180541, "converted": true, "num_tokens": 6926, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526660244838, "lm_q2_score": 0.7853085859124002, "lm_q1q2_score": 0.44751019133409864}} {"text": "# Troubleshooting of logD estimation\n\nIn our current SAMPL7 analysis, we use something like (following Bannan et al. 10.1007/s10822-016-9954-8), for a basic solute:\n\n$ \\log D = \\log P - log( 1+ 10^{pK_a - pH})$ (eq. 3)\n\nor for an acidic solute:\n$ \\log D = \\log P - log( 1+ 10^{pH - pK_a})$ (eq. 4)\n\nThese are for a compound with a single change in protonation; accounting for other states could include Equation 5 there.\n\n\nPion, whose instruments yielded the logD analysis (in Francisco et al., https://doi.org/10.1016/j.ejmech.2021.113399) says that their logD analysis uses a general equation which originates from Avdeef, A., \"Assessment of distribution-pH profiles\", *Lipophilicity in drug action and toxicology*, Pliska, V.; Testa, B.; Van de Waterbeemd, H. eds., VCH, Weinheim; p109-139 (1996). The equation given is \n\\begin{equation}\nD = \\frac{P^0 + [H+]\\beta_1 P^1 + [H+]^2 \\beta_2 P^2 + ...}{1 + [H+]\\beta_1 + [H+]^2 \\beta_2 + ...}\n\\end{equation}\n\nwhere I assume [H+] is found from the pH and $\\beta_1 = K_{a1}$, $\\beta_2 = K_{a1}K_{a2}$, etc. $\\beta$ is a \"cumulative\" protonation constant. \n\n\n\nTransition networks: https://docs.google.com/presentation/d/16SlgjA3mxwmi6bdbAkhiZrNEc6wJ94hLW8pvTBkI3Uo/edit#slide=id.g98334362cf_0_199\n\nNote there are several cases where our current approach disagrees profoundly with that used by Pion. These are SM42, SM25, SM26, SM41 and SM43, in increasing order of disagreement. From the transition networks, \n- SM42 has two neutral states and a +1 and -1\n- SM25 is complex, with three neutral states, a +1 and two -1 states\n- SM26 is complex, with three neutral states, a +1 state and two -1 states\n- SM41 has a single neutral state, a +1 and -1\n- SM43 is complex, with two neutral states, two +1 states, a +2 state, and a -1 state \n\n**The best starting point for simple troubleshooting, then, is SM41**\n\n## Analysis of SM41\n\nExperimental values for the challenge are found here: https://github.com/samplchallenges/SAMPL7/blob/master/physical_property/experimental_data/Experimental_Properties_of_SAMPL7_Compounds.csv\n\nSM41 has a pKa of 5.22 and a logP of 0.58. The reported logD7.4 is -0.42.\n\n### Using the Bannan expression for a base\n\n\n\n```python\nimport numpy as np\npKa = 5.22\nlogP = 0.58\npH = 7.4\n\nlogD = logP - np.log10(1.+10**(pKa-pH))\nprint(logD)\n```\n\n 0.57714008208903\n\n\n### Using the Bannan expression for an acid\n\n\n```python\nimport numpy as np\npKa = 5.22\nlogP = 0.58\npH = 7.4\n\nlogD = logP - np.log10(1.+10**(pH-pKa))\nprint(logD)\n```\n\n -1.6028599179109704\n\n\n### Analyze using Pion's expression\n\nThis involves solving for the proton concentration from the pH.\n\n\n\n\n```python\npKa = 5.22\nlogP = 0.58\npH = 7.4\nconcH = 10**(-pH)\nP = 10**(logP)\nKa = 10**(-pKa)\n\nD = ( P + concH*Ka*P)/(1 + concH*Ka)\nlogD = np.log10(D)\nprint(logD)\nprint(Ka, P, concH)\nprint(concH*Ka, concH*Ka*P)\nprint(D)\n```\n\n 0.58\n 6.025595860743581e-06 3.8018939632056115 3.981071705534969e-08\n 2.39883291901949e-13 9.120108393559093e-13\n 3.801893963205612\n\n\n# Throw that out and analyze a different way\n\nThe Comer et al. paper, and Sirius's slide deck, provide more context to indicate that P^0 is the partition of the neutral species and P^1 the partition of hte +1 species, etc. This means the above analysis, where I was exponentiating instead, is completely a mistake. \n\nMy key question to proceed is how to get P^0 and P^1 etc for an example case, like SM41, but Figure 4 in the Comer paper suggests a path forward as the graphs look a lot like those in the Sirius report for this particular compound; the caption says \"Partial lipophilicity profiles derived using equations 8 or 9, after calculating logP^N from the data in graphs (a) and (b) using Eqns 10 or 11.\" This basically describes data I have from Sirius, and I have equations 10 and 11 (though one concern is these are for the monoprotic case only, but still I can try).\n\nThe other concern is that Eq. 10-11 have r, a volume fraction, which does not occur in the Sirius report, but that's a minor/surmountable issue I think. I am also not sure I know $p_o K_a$, the pKa in octanol, but perhaps this can be figured out. \n\nSo let's try that. \n\n- Begin with $p_o K_a$. Where do I get it? Sirius slide deck (e.g. ch 7 slide 2) indicates it's the shifted pKa in the presence of octanol. (There is also a special version of this, the limiting one, called the Scherrer $p_o K_a$, which occurs in octanol saturation, but that is not what we need here.) \n - **But where exactly does the $p_o K_a$ come from?** \n - It's PROBABLY the rightmost shifted pKa in the \"mean molecular charge\" graph. Checking by viewing underlying data\n - XLS spreadsheet has theoretical mean molecular charge as based on pKa in columns\n - Next to it there's the actual one \"shifted due to solvent\" run at three solvent concentrations (not themselves stated)\n - So probably use the rightmost of these \n - pH that concentration hits 0.5 is about 7.3 (between 7.18 and 7.4)\n- Now what about $r$?\n - Could use just a placeholder value of some kind\n - There is no data in the paper on what $r$ is, nor in the Sirius report\n - Emailed Karol/Carlo asking\n - In the interim probably try some limits, e.g. volume fraction of 1, 0.1, 0.5\n\n\n```python\npKa = 5.22\nlogP = 0.58\npH = 7.4\nconcH = 10**(-pH)\npoKa = 7.3\nr = 2/1.09995\n\nP0 = 10**(logP)\nP1 = (10**(poKa-pKa))/r - 1\n\nKa = 10**(-pKa)\n\nD = (P0 + P1*concH*Ka)/(1+concH*Ka)\nprint(D)\n```\n\n 3.8018939632203215\n\n\n\n```python\nprint(P0, P1)\n```\n\n 3.8018939632056115 65.12153824287118\n\n\n\n```python\nprint(concH*Ka)\n\n```\n\n 2.39883291901949e-13\n\n\n\n```python\nprint(1+concH*Ka)\n```\n\n 1.0000000000002398\n\n\n\n```python\n10**5.22\n```\n\n\n\n\n 165958.69074375596\n\n\n\n\n```python\nprint(np.log10(D))\n```\n\n 0.5800000000016803\n\n\nThe Sirius analysis gives a negative value for logD here; the only way this could happen is if P^1 is negative, which will happen only if the $P^1 = 10^{p_o K_a -pKa}/r$ term is less than 1. Note that for a poKa-pKa gap of 2.1, as here, we're dealing with a factor of about 120 so $r$ would have to mean that the partition solvent is far in excess of the aqueous phase. That seems... odd. \n\nSirius has slides on this in Ch 7 though, e.g. slide 16-17, basically the point is that you want an r such that you get a shift in pKa as you modulate the amount of octanol. They have an example for a logP between 0 and 2.5 where you might be using, as a third octanol concentration, an octanol volume of 1.2 and a water volume of 1.\n\nLooks like I'm looking at data point 114, which has a water volume of 1.09995 mL and an octanol volume of 2 mL.\n\n\n```python\nprint(10**(poKa -pKa)/r)\n```\n\n 66.12153824287118\n\n\n\n```python\npoKa-pKa\n```\n\n\n\n\n 2.08\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "58d2e7d4fdaefad1bc96fa650d3227cb3019a6c0", "size": 11235, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "LogD troubleshooting.ipynb", "max_stars_repo_name": "samplchallenges/SAMPL7", "max_stars_repo_head_hexsha": "1feed16ed8502a3519559fbdcc23812f21c64be1", "max_stars_repo_licenses": ["CC-BY-4.0", "MIT"], "max_stars_count": 29, "max_stars_repo_stars_event_min_datetime": "2019-10-23T17:59:16.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-18T18:42:14.000Z", "max_issues_repo_path": "LogD troubleshooting.ipynb", "max_issues_repo_name": "samplchallenges/SAMPL7", "max_issues_repo_head_hexsha": "1feed16ed8502a3519559fbdcc23812f21c64be1", "max_issues_repo_licenses": ["CC-BY-4.0", "MIT"], "max_issues_count": 39, "max_issues_repo_issues_event_min_datetime": "2019-10-16T18:42:05.000Z", "max_issues_repo_issues_event_max_datetime": "2021-10-05T23:28:04.000Z", "max_forks_repo_path": "LogD troubleshooting.ipynb", "max_forks_repo_name": "samplchallenges/SAMPL7", "max_forks_repo_head_hexsha": "1feed16ed8502a3519559fbdcc23812f21c64be1", "max_forks_repo_licenses": ["CC-BY-4.0", "MIT"], "max_forks_count": 22, "max_forks_repo_forks_event_min_datetime": "2019-10-07T08:47:41.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-10T14:15:07.000Z", "avg_line_length": 29.801061008, "max_line_length": 572, "alphanum_fraction": 0.5690253672, "converted": true, "num_tokens": 2174, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802471698041, "lm_q2_score": 0.6688802669716107, "lm_q1q2_score": 0.44740079829897555}} {"text": "## \u6d4b\u8bd5appmode\n\n\n```python\nfrom ipywidgets import interact\n```\n\n\n```python\n@interact(x=5)\ndef add3(x):\n return x + 3\n```\n\n\n

Failed to display Jupyter Widget of type interactive.

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\n If you're reading this message in the Jupyter Notebook or JupyterLab Notebook, it may mean\n that the widgets JavaScript is still loading. If this message persists, it\n likely means that the widgets JavaScript library is either not installed or\n not enabled. See the Jupyter\n Widgets Documentation for setup instructions.\n

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\n If you're reading this message in another frontend (for example, a static\n rendering on GitHub or NBViewer),\n it may mean that your frontend doesn't currently support widgets.\n

\n\n\n\n\n```python\ninteract(add3, x=5)\n```\n\n\n

Failed to display Jupyter Widget of type interactive.

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\n If you're reading this message in the Jupyter Notebook or JupyterLab Notebook, it may mean\n that the widgets JavaScript is still loading. If this message persists, it\n likely means that the widgets JavaScript library is either not installed or\n not enabled. See the Jupyter\n Widgets Documentation for setup instructions.\n

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\n\n\n\n\n\n\n \n\n\n\n\n```python\nlen?\n```\n\n\n```python\nthis?\n\n```\n\n Object `this` not found.\n\n\n\n```python\nadd3?\n```\n\n Object `add3` not found.\n\n\n\n```python\ndef aaa():\n \"\"\"aaa\"\"\"\n pass\n\n```\n\n\n```python\naaa?\n```\n\n\n```python\nadd3?\n```\n\n Object `add3` not found.\n\n\n\n```python\naaa??\n```\n\n\n```python\n%magic\n```\n\n\n```python\nOut\n```\n\n\n\n\n {}\n\n\n\n\n```python\nIn\n```\n\n\n\n\n ['',\n \"get_ipython().run_line_magic('pinfo', 'len')\",\n \"get_ipython().run_line_magic('pinfo', 'this')\",\n \"get_ipython().run_line_magic('pinfo', 'add3')\",\n 'def aaa():\\n \"\"\"aaa\"\"\"\\n pass',\n \"get_ipython().run_line_magic('pinfo', 'aaa')\",\n \"get_ipython().run_line_magic('pinfo', 'add3')\",\n \"get_ipython().run_line_magic('pinfo2', 'aaa')\",\n \"get_ipython().run_line_magic('magic', '')\",\n 'Out',\n 'In']\n\n\n\n\n```python\n%debug\n```\n\n > \u001b[0;32m\u001b[0m(1)\u001b[0;36m\u001b[0;34m()\u001b[0m\n \u001b[0;32m----> 1 \u001b[0;31m\u001b[0ma\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n \u001b[0m\n ipdb> b=2\n *** The specified object '=2' is not a function or was not found along sys.path.\n ipdb> print(b)\n 0\n ipdb> bye\n *** NameError: name 'bye' is not defined\n ipdb> quit\n\n\n\n```python\na=1\nb=0\n```\n\n\n```python\na/b\n```\n\n\n```python\nimport numpy\n```\n\n\n```python\nnumpy.__version__\n```\n\n\n\n\n '1.13.0'\n\n\n\n\n```python\nimport numpy as np\n```\n\n\n```python\nnp?\n```\n\n\n```python\nnp.zeros(10, dtype = int)\n```\n\n\n\n\n array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])\n\n\n\n\n```python\nnp.full((3, 5), 10)\n```\n\n\n\n\n array([[10, 10, 10, 10, 10],\n [10, 10, 10, 10, 10],\n [10, 10, 10, 10, 10]])\n\n\n\n\n```python\nnp.arange(0, 100, 1)\n```\n\n\n\n\n array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,\n 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,\n 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50,\n 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67,\n 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84,\n 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99])\n\n\n\n\n```python\nnp.linspace(0, 1, 10)\n```\n\n\n\n\n array([ 0. , 0.11111111, 0.22222222, 0.33333333, 0.44444444,\n 0.55555556, 0.66666667, 0.77777778, 0.88888889, 1. ])\n\n\n\n\n```python\nfrom sympy import *\n```\n\n\n```python\n%matplotlib inline\n```\n\n\n```python\nfrom sympy import *\n```\n\n\n```python\nI**2\n```\n\n\n\n\n -1\n\n\n\n\n```python\npi\n```\n\n\n\n\n pi\n\n\n\n\n```python\na = Symbol('a')\n```\n\n\n```python\n(pi + a)**2\n```\n\n\n\n\n (a + pi)**2\n\n\n\n\n```python\ninit_printing()\n```\n\n\n```python\nexpand((pi + a)**2)\n```\n\n\n```python\n4/5\n```\n\n\n```python\nr1 = Rational(4,5)\n```\n\n\n```python\nr1\n```\n\n\n```python\npi.evalf(n=100)\n```\n\n\n```python\nx = Symbol('x')\n```\n\n\n```python\nexpand((x+1)*(x**2+2)*(x**3+5))\n```\n\n\n```python\nfactor(6*x**2+5*x+1)\n```\n\n\n```python\nfactor(x**2+2*x+1)\n```\n\n\n```python\nfactor(121*x**2+33*x+11)\n```\n\n\n```python\nfactor((x+2)**2-9)\n```\n\n\n```python\nI**2\n```\n\n\n```python\nI\n```\n\n\n```python\nexpand((2+I)**2)\n```\n\n\n```python\nsolve(x**2-1,x)\n```\n\n\n```python\ny = Symbol('y')\n```\n\n\n```python\nsolve([3*x+5*y-2,2*x+3*y],[x,y])\n```\n\n\n```python\nimport matplotlib.pyplot as plt\n```\n\n\n```python\nfrom scipy import *\n```\n\n\n```python\n3//2\n```\n\n\n```python\n9**(1/2)\n```\n\n\n```python\nfrom fractions import Fraction\n```\n\n\n```python\nFraction(3,2)\n```\n\n\n\n\n Fraction(3, 2)\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "223eae0e9b777c3030573f4b1ce684a40cc0eec1", "size": 49294, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "appmode.ipynb", "max_stars_repo_name": "roadlabs/mynotebooks", "max_stars_repo_head_hexsha": "6e64b93a1c3996408200f839ba22afc261cf7778", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "appmode.ipynb", "max_issues_repo_name": "roadlabs/mynotebooks", "max_issues_repo_head_hexsha": "6e64b93a1c3996408200f839ba22afc261cf7778", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "appmode.ipynb", "max_forks_repo_name": "roadlabs/mynotebooks", "max_forks_repo_head_hexsha": "6e64b93a1c3996408200f839ba22afc261cf7778", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 38.4808743169, "max_line_length": 6288, "alphanum_fraction": 0.6919706252, "converted": true, "num_tokens": 1844, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.6723317057447908, "lm_q1q2_score": 0.4473766159828398}} {"text": "```python\n# import numpy as np\n\n# # !/usr/bin/env python3\n# # -*- coding: utf-8 -*-\n# \"\"\"\n# Created on 20181219\n\n# @author: zhangji\n\n# Trajection of a ellipse, Jeffery equation. \n# \"\"\"\n\n# %pylab inline\n# pylab.rcParams['figure.figsize'] = (25, 11)\n# fontsize = 40\n\n# import numpy as np\n# import scipy as sp\n# from scipy.optimize import leastsq, curve_fit\n# from scipy import interpolate\n# from scipy.interpolate import interp1d\n# from scipy.io import loadmat, savemat\n# # import scipy.misc\n\n# import matplotlib\n# from matplotlib import pyplot as plt\n# from matplotlib import animation, rc\n# import matplotlib.ticker as mtick\n# from mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\n# from mpl_toolkits.mplot3d import Axes3D, axes3d\n\n# from sympy import symbols, simplify, series, exp\n# from sympy.matrices import Matrix\n# from sympy.solvers import solve\n\n# from IPython.display import display, HTML\n# from tqdm import tqdm_notebook as tqdm\n# import pandas as pd\n# import re\n# from scanf import scanf\n# import os\n# import glob\n\n# from codeStore import support_fun as spf\n# from src.support_class import *\n# from src import stokes_flow as sf\n\n# rc('animation', html='html5')\n# PWD = os.getcwd()\n# font = {'size': 20}\n# matplotlib.rc('font', **font)\n# np.set_printoptions(linewidth=90, precision=5)\n\n%load_ext autoreload\n%autoreload 2\n\nfrom tqdm import tqdm_notebook\nimport os\nimport glob\nimport natsort \nimport numpy as np\nimport scipy as sp\nfrom scipy.optimize import leastsq, curve_fit\nfrom scipy import interpolate, integrate\nfrom scipy import spatial, signal\n# from scipy.interpolate import interp1d\nfrom scipy.io import loadmat, savemat\n# import scipy.misc\nfrom IPython.display import display, HTML\nimport pandas as pd\nimport pickle\nimport re\nfrom scanf import scanf\n\nimport matplotlib\n# matplotlib.use('agg')\nfrom matplotlib import pyplot as plt\nimport matplotlib.colors as colors\nfrom matplotlib import animation, rc\nimport matplotlib.ticker as mtick\nfrom mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\nfrom mpl_toolkits.mplot3d import Axes3D, axes3d\nfrom mpl_toolkits.axes_grid1.axes_divider import make_axes_locatable\nfrom mpl_toolkits.mplot3d.art3d import Line3DCollection\nfrom matplotlib import cm\n\nfrom tqdm.notebook import tqdm as tqdm_notebook\nfrom tqdm import tqdm\nfrom time import time\nfrom src.support_class import *\nfrom src import jeffery_model as jm\nfrom codeStore import support_fun as spf\nfrom codeStore import support_fun_table as spf_tb\n\n# %matplotlib notebook\n%matplotlib inline\nrc('animation', html='html5')\nfontsize = 40\nPWD = os.getcwd()\n```\n\n\n```python\nfig = plt.figure(figsize=(2, 2))\nfig.patch.set_facecolor('white')\nax0 = fig.add_subplot(1, 1, 1)\n```\n\n\n```python\njob_dir = 'ecoC01B05_th1.285_ph2.999_ps0.000'\n\nt_headle = '(.*?).pickle'\n```\n\n\n```python\ndef get_major_fre(*arg, **kwargs):\n return spf_tb.get_major_fre(*arg, **kwargs)\n \ndef load_rand_data_pickle_dir_v3(t_dir, t_headle='(.*?).pickle', n_load=None, rand_mode=False): \n t_path = os.listdir(t_dir)\n filename_list = [filename for filename in t_path if re.match(t_headle, filename) is not None]\n mean_eta_list = []\n dx_list = []\n dy_list = []\n dz_list = []\n wt_list = []\n pickle_path_list = []\n \n n_load = len(filename_list) if n_load is None else n_load\n assert n_load <= len(filename_list)\n if rand_mode:\n tidx = np.random.choice(len(filename_list), n_load, replace=False)\n else:\n tidx = np.arange(n_load)\n use_filename_list = np.array(filename_list)[tidx]\n\n for tname in tqdm_notebook(use_filename_list):\n tpath = os.path.join(t_dir, tname)\n with open(tpath, 'rb') as handle:\n tpick = pickle.load(handle)\n pickle_path_list.append(tpath)\n wt_list.append(tpick['omega_tail'])\n\n # fft rule\n tx = tpick['Table_t']\n tmin = np.max((0, tx.max() - 1000))\n idx = tx > tmin\n freq_pk = get_major_fre(tx[idx], tpick['Table_theta'][idx])\n idx = tx > (tx.max() - 1 / freq_pk * 10)\n mean_eta_list.append(np.mean(tpick['Table_eta'][idx]))\n for i0, tlist in enumerate((dx_list, dy_list, dz_list)):\n tpoly = np.polyfit(tx[idx], tpick['Table_X'][idx, i0], 1, w=np.blackman(idx.sum()))\n tlist.append(tpoly[0])\n\n dx_list = np.hstack(dx_list)\n dy_list = np.hstack(dy_list)\n dz_list = np.hstack(dz_list)\n wt_list = np.hstack(wt_list)\n mean_eta_list = np.hstack(mean_eta_list)\n pickle_path_list = np.array(pickle_path_list)\n return dx_list, dy_list, dz_list, wt_list, mean_eta_list, pickle_path_list\n```\n\n\n```python\nn_load = None\nrand_mode=False\n\nt_dir = os.path.join(PWD, job_dir)\n_ = load_rand_data_pickle_dir_v3(t_dir, t_headle, n_load=n_load, rand_mode=rand_mode)\ndx_list, dy_list, dz_list, wt_list, mean_eta_list, pickle_path_list = _\n```\n\n\n HBox(children=(FloatProgress(value=0.0, max=111.0), HTML(value='')))\n\n\n \n\n\n\n```python\nwt_list.max()\n```\n\n\n\n\n 10.0\n\n\n\n\n```python\n# %matplotlib notebook\n%matplotlib inline\n\nfigsize = np.array((16, 9)) * 0.3\ndpi = 800 if 'inline' in matplotlib.get_backend() else 100\nlinthreshx = 1e-3\nlinscalex = 0.5\n\ntidx = np.argsort(wt_list)\nfig, axi = plt.subplots(1, 1, figsize=figsize, dpi=dpi)\nfig.patch.set_facecolor('white')\naxi_pos = axi.get_position().bounds\ntfct = axi_pos[2] / axi_pos[3] * figsize[0] / figsize[1]\ndpos = np.array(((axi_pos[2] * 0.1, axi_pos[3] * -0.01, axi_pos[2] * -0.98, axi_pos[3] * 0.1)))\n\naxi3 = fig.add_axes(axi_pos + dpos)\naxi3.set_xticklabels([])\naxi3.set_yticklabels([])\nfor t1 in axi3.spines:\n axi3.spines[t1].set_visible(False)\naxi3.tick_params(axis=u'both', which=u'both',length=0)\ndpos = np.array(((axi_pos[2] * 0.1, 0, axi_pos[2] * -0.98, 0)))\naxi4 = fig.add_axes(axi_pos + dpos)\naxi4.set_xticklabels([])\naxi4.set_yticklabels([])\nfor t1 in axi4.spines:\n axi4.spines[t1].set_visible(False)\naxi4.tick_params(axis=u'both', which=u'both',length=0)\ntd = 0.02 # how big to make the diagonal lines in axes coordinates\nkwargs = dict(transform=axi4.transAxes, color='k', clip_on=False)\naxi4.plot((-td * tfct,+td * tfct), (1-td,1+td), **kwargs)\naxi4.plot((-td * tfct,+td * tfct), (-td,+td), **kwargs)\naxi4.plot((1-td * tfct,1+td * tfct), (-td,+td), **kwargs)\naxi4.plot((1-td * tfct,1+td * tfct),(1-td,1+td), **kwargs)\n\naxi1 = fig.add_axes(axi_pos)\naxi1.plot(wt_list[tidx], mean_eta_list[tidx] / np.pi, '--C0', label='th1.285, ph2.999, ps0.000')\naxi1.set_xscale('symlog', linthreshx=linthreshx, linscalex=linscalex)\naxi1.patch.set_alpha(0)\naxi1.set_xlabel('$\\\\omega_t / \\\\tau_s$')\naxi1.set_ylabel('$\\\\langle \\\\eta / \\\\pi \\\\rangle$')\naxi2 = axi1.twinx()\naxi2.plot(wt_list[tidx], dy_list[tidx], '.-C1')\naxi2.set_ylabel('$u_2$')\naxi2.set_xscale('symlog', linthreshx=linthreshx, linscalex=linscalex)\n# axi.legend()\nfor t1 in axi.spines:\n axi1.spines[t1].set_visible(False)\nfor t1 in axi2.spines:\n axi2.spines[t1].set_visible(False)\n\naxi.set_xlim(axi1.get_xlim())\naxi.set_xscale('symlog', linthreshx=linthreshx, linscalex=linscalex)\naxi.tick_params(axis=u'both', which=u'both',length=0)\naxi.set_xticklabels([])\naxi.set_yticklabels([])\naxi.axvspan(0, 0.015, alpha=0.05, color='gray')\naxi.axvspan(0.015, 0.6, alpha=0.1, color='gray')\naxi.axvspan(0.6, wt_list.max(), alpha=0.2, color='gray')\n```\n\n\n```python\ndpi\n```\n\n\n\n\n 150\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "c7e631809c273eee1c31b4a6344948256b4f1998", "size": 317426, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "head_Force/do_calculate_table/phase_map_v5_ecoC01B05_th1.285_ph2.999_ps0.000.ipynb", "max_stars_repo_name": "pcmagic/stokes_flow", "max_stars_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-11-11T05:00:53.000Z", "max_stars_repo_stars_event_max_datetime": "2018-11-11T05:00:53.000Z", "max_issues_repo_path": "head_Force/do_calculate_table_loc/phase_map_v5_ecoC01B05_th1.285_ph2.999_ps0.000.ipynb", "max_issues_repo_name": "pcmagic/stokes_flow", "max_issues_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "head_Force/do_calculate_table_loc/phase_map_v5_ecoC01B05_th1.285_ph2.999_ps0.000.ipynb", "max_forks_repo_name": "pcmagic/stokes_flow", "max_forks_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 807.6997455471, "max_line_length": 303448, "alphanum_fraction": 0.9493771777, "converted": true, "num_tokens": 2278, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646140788307, "lm_q2_score": 0.6619228625116081, "lm_q1q2_score": 0.4473040477351118}} {"text": "

Modelos Matem\u00e1ticos de Ingenier\u00eda

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Sesi\u00f3n 06 - Diferencias finitas

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2021/02

\n

MEDELL\u00cdN - COLOMBIA

\n\n\n \n
\n Text provided under a Creative Commons Attribution license, CC-BY. All code is made available under the FSF-approved MIT license.(c) Carlos Alberto Alvarez Henao
\n\n*** \n\n***Docente:*** Carlos Alberto \u00c1lvarez Henao, I.C. D.Sc.\n\n***e-mail:*** carlosalvarezh@gmail.com\n\n***skype:*** carlos.alberto.alvarez.henao\n\n***Linkedin:*** https://www.linkedin.com/in/carlosalvarez5/\n\n***github:*** https://github.com/carlosalvarezh/Dinamica\n\n***Herramienta:*** [Jupyter](http://jupyter.org/)\n\n***Kernel:*** Python 3.9\n\n\n***\n\n

Tabla de Contenidos

\n
\n\n## Diferencias Finitas\n\nEn [an\u00e1lisis num\u00e9rico](https://en.wikipedia.org/wiki/Numerical_analysis), los m\u00e9todos de diferencias finitas (FDM) son una clase de t\u00e9cnicas num\u00e9ricas para resolver ecuaciones diferenciales mediante la aproximaci\u00f3n de derivadas con diferencias finitas. Tanto el dominio espacial como el intervalo de tiempo (si corresponde) se discretizan o se dividen en un n\u00famero finito de pasos, y el valor de la soluci\u00f3n en estos puntos discretos se aproxima resolviendo ecuaciones algebraicas que contienen diferencias finitas y valores de puntos cercanos.\n\nLos m\u00e9todos de diferencias finitas convierten las ecuaciones diferenciales ordinarias (EDO) o las ecuaciones diferenciales parciales (PDE), que pueden ser no lineales, en un sistema de ecuaciones lineales que pueden resolverse mediante t\u00e9cnicas de \u00e1lgebra matricial. Las computadoras modernas pueden realizar estos c\u00e1lculos de \u00e1lgebra lineal de manera eficiente, lo que, junto con su relativa facilidad de implementaci\u00f3n, ha llevado al uso generalizado de FDM en el an\u00e1lisis num\u00e9rico moderno. Hoy en d\u00eda, FDM es uno de los enfoques m\u00e1s comunes para la soluci\u00f3n num\u00e9rica de PDE, junto con los m\u00e9todos de elementos finitos.\n\n### Series de Taylor\n\nLa [serie de Taylor](https://en.wikipedia.org/wiki/Taylor_series) de una funci\u00f3n es una suma infinita de t\u00e9rminos que se expresan en t\u00e9rminos de las derivadas de la funci\u00f3n en un solo punto. Para las funciones m\u00e1s comunes, la funci\u00f3n y la suma de su serie de Taylor son iguales cerca de este punto. Las series de Taylor llevan el nombre de [Brook Taylor](https://en.wikipedia.org/wiki/Brook_Taylor), quien las present\u00f3 en 1715.\n\nLa suma parcial formada por los primeros $n + 1$ t\u00e9rminos de una serie de Taylor es un polinomio de grado $n$ que se llama el *$n$-\u00e9simo polinomio de Taylor de la funci\u00f3n*. Los polinomios de Taylor son aproximaciones de una funci\u00f3n, que generalmente se vuelven mejores a medida que $n$ aumenta. El [teorema de Taylor](https://en.wikipedia.org/wiki/Taylor%27s_theorem) da estimaciones cuantitativas sobre el error introducido por el uso de tales aproximaciones. Si la serie de Taylor de una funci\u00f3n es convergente, su suma es el l\u00edmite de la secuencia infinita de los polinomios de Taylor.\n\n### Esquemas para la primera derivada en 1D\n\nDe la [serie de Taylor](https://en.wikipedia.org/wiki/Taylor_series) \n\n\n\\begin{equation*}\nf(x_{i \\pm 1}) = f(x_i) \\pm f'(x_i)h + \\frac{f''(x_i)h^2}{2!} \\pm \\frac{f'''(x_i)h^3}{3!} + \\ldots\n\\label{eq:Ec6_1} \\tag{6.1}\n\\end{equation*}\n\ncon $h=\\Delta x = x_{i+1}-x_i$ siendo el tama\u00f1o de paso.\n\nDada que la serie contiene infinitos t\u00e9rminos, partir de la ecuaci\u00f3n [(6.1)](#Ec6_1) se pueden obtener infinitos esquemas num\u00e9ricos para determinar cada una de las infinitas derivadas de dicho polinomio.\n\n#### Primer orden hacia adelante (forward)\n\nDe la ecuaci\u00f3n [(6.1)](#Ec6_1) tomando los valores positivos, que involucran \u00fanicamente t\u00e9rminos hacia adelante, se trunca la serie hasta la primera derivada y se realiza un despeje algebraico para llegar a:\n\n\n\\begin{equation*}\nf'(x_i) = \\frac{f(x_{i+1})-f(x_i)}{h} + \\mathcal{O}(h)\n\\label{eq:Ec6_2} \\tag{6.2}\n\\end{equation*}\n\nse puede observar que el t\u00e9rmino $\\mathcal{O}(h)$ indica que el error es de orden lineal, es decir, si se reduce el tama\u00f1o de paso, $h$, a la mitad, el error se reducir\u00e1 a la mitad. Si se reduc el tama\u00f1o de paso a una cuarta parte, el error se reducir\u00e1, linealmente, una cuarta parte.\n\n#### Primer orden hacia atr\u00e1s (backward)\n\nDe la ecuaci\u00f3n [(6.1)](#Ec6_1) tomando los valores negativos, que involucran \u00fanicamente t\u00e9rminos hacia atr\u00e1s (backward). Se trunca la serie hasta la primera derivada y se realiza un despeje algebraico para llegar a:\n\n\n\\begin{equation*}\nf'(x_i) = \\frac{f(x_{i})-f(x_{i-1})}{h} + \\mathcal{O}(h)\n\\label{eq:Ec6_3} \\tag{6.3}\n\\end{equation*}\n\nse observa que se llega a una expresi\u00f3n similar a la de la ecuaci\u00f3n [(6.2)](#Ec6_2), pero de esta vez, se tiene en cuenta es el valor anterior al punto $x_i$. Tambi\u00e9n se observa que el error es de orden lineal, por lo que se mantiene un esquema de primer orden.\n\n\n\n\n#### Segundo orden (central)\n\nUna forma de aumentar el orden de estos esquemas, es realizar el truncamiento de la *serie de Taylor* hasta la segunda derivada, hacia adelante y hacia atras, y realizar su resta aritm\u00e9tica.\n\n\n\\begin{equation*}\n\\begin{split}\nf(x_{i+1}) & = f(x_i) + f'(x_i)h + \\frac{f''(x_i)h^2}{2!} \\\\\n& - \\\\\nf(x_{i-1}) & = f(x_i) - f'(x_i)h + \\frac{f''(x_i)h^2}{2!} \\\\\n\\hline \\\\\nf(x_{i+1}) - f(x_{i-1}) & = 2 f'(x_i)h\n\\end{split}\n\\label{eq:Ec6_4} \\tag{6.4}\n\\end{equation*}\n \nde la anterior ecuaci\u00f3n, despejando el t\u00e9rmino que corresponde a la primera derivada queda:\n\n\n\\begin{equation*}\n\\begin{split}\nf'(x_i) = \\frac{f(x_{i+1}) - f(x_{i-1})}{2h} + \\mathcal{O}(h^2)\n\\end{split}\n\\label{eq:Ec6_5} \\tag{6.5}\n\\end{equation*}\n\nse llega al esquema de diferencias finitas central para la primera derivada, que es de orden dos, es decir, si se disminuye el tama\u00f1o de paso, $h$, a la mitad, el error se disminuye una cuarta partes. En principio, esta es una mejor aproximaci\u00f3n que los dos esquemas anteriores. La selecci\u00f3n del esquema depender\u00e1 de la disponibilidad de puntos y del fen\u00f3meno f\u00edsico a tratar.\n\n#### Resumen esquemas para la primera derivada\n\nComo la serie de Taylor es infinita, se podr\u00edan determinar infinitos esquemas de diferentes ordenes para la primera derivada. En la siguiente tabla se presentan algunos esquemas de diferencias finitas para la primera derivada de diferentes \u00f3rdenes. Se deja al estudiante la consulta de otros esquemas.\n\n|***Esquema***|***Funci\u00f3n***|***Error***|\n|:-----:|:-----:|:---:|\n|***Forward***|$$f\u00b4(x_0)=\\frac{f(x_0+h)-f(x_0)}{h}$$|$$\\mathcal{O}(h)$$|\n| |$$f\u00b4(x_0)=\\frac{-3f(x_0)+4f(x_0+h)-f(x_0+2h)}{2h}$$|$$\\mathcal{O}(h^2)$$|\n|***Central***|$$f\u00b4(x_0)=\\frac{f(x_0+h)-f(x_0-h)}{2h}$$|$$\\mathcal{O}(h^2)$$|\n| |$$f\u00b4(x_0)=\\frac{f(x_0-2h)-8f(x_0-h)+8f(x_0+h)-f(x_0+2h)}{12h}$$|$$\\mathcal{O}(h^4)$$|\n|***Backward***|$$f\u00b4(x_0)=\\frac{f(x_0)-f(x_0-h)}{h}$$|$$\\mathcal{O}(h)$$|\n| |$$f\u00b4(x_0)=\\frac{f(x_0-2h)-4f(x_0-h)+3f(x_0)}{2h}$$|$$\\mathcal{O}(h^2)$$|\n\n\n\n### Esquemas para la segunda derivada en 1D\n\nSiguiendo con la misma forma de abordar el problema para la primera derivada, si se amplian los t\u00e9rminos en la serie de Taylor hasta la tercera derivada tanto hacia adelante como hacia atr\u00e1s, y se suman, se llega a:\n\n\n\\begin{equation*}\n\\begin{split}\nf(x_{i+1}) & = f(x_i) + f'(x_i)h + \\frac{f''(x_i)h^2}{2!} + \\frac{f'''(x_i)h^3}{3!}\\\\\n& + \\\\\nf(x_{i-1}) & = f(x_i) - f'(x_i)h + \\frac{f''(x_i)h^2}{2!} - \\frac{f'''(x_i)h^3}{3!}\\\\\n\\hline \\\\\nf(x_{i+1}) + f(x_{i-1}) & = 2 f(x_i) + 2f''(x_i)\\frac{h^2}{2!} + \\mathcal{O}(h^3)\n\\end{split}\n\\label{eq:Ec6_6} \\tag{6.6}\n\\end{equation*}\n \nDespejando para el t\u00e9rmino de la segunda derivada, se llega a:\n\n\n\\begin{equation*}\n\\begin{split}\nf''(x_i) = \\frac{f(x_{i+1}) - 2f(x_i) + f(x_{i-1})}{h^2} + \\mathcal{O}(h^3)\n\\end{split}\n\\label{eq:Ec6_7} \\tag{6.7}\n\\end{equation*}\n\nQue corresponde a un esquema de diferencias finitas de segundo orden para la segunda derivada. A este esquema tambi\u00e9n se le llama \"*mol\u00e9cula de tres puntos*\"\n\nIgual que para la primera derivada, se pueden determinar infinitos esquemas de diferentes \u00f3rdenes para la segunda derivada, y derivadas superiores. A continuaci\u00f3n se muestra un cuadro resumen de algunos esquemas de diferencias finitas para la segunda derivada. Se deja al estudiante la revisi\u00f3n de esquemas de mayor orden para la segunda derivada y derivadas superiores.\n\n|***Esquema***|***Funci\u00f3n***|***Error***|\n|:-----:|:-----:|:---:|\n|***Forward***|$$f''(x_0)=\\frac{f(x_0)-2f(x_0+h)+f(x_0+2h)}{h^2}$$|$$\\mathcal{O}(h)$$|\n| |$$f''(x_0)=\\frac{2f(x_0)-5f(x_0+h)+4f(x_0+2h)-f(x_0+3h)}{h^2}$$|$$\\mathcal{O}(h^2)$$|\n|***Central***|$$f''(x_0)=\\frac{f(x_0-h)-2f(x_0)+f(x_0+h)}{h^2}$$|$$\\mathcal{O}(h^2)$$|\n| |$$f''(x_0)=\\frac{-f(x_0-2h)+16f(x_0-h)-30f(x_0)+16f(x_0+h)-f(x_0+2h)}{12h^2}$$|$$\\mathcal{O}(h^4)$$|\n|***Backward***|$$f''(x_0)=\\frac{f(x_0-2h)-2f(x_0-h)+f(x_0)}{h}$$|$$\\mathcal{O}(h^2)$$|\n| |$$f''(x_0)=\\frac{-f(x_0-3h)+4f(x_0-2h)-5f(x_0-h)+2f(x_0)}{h^2}$$|$$\\mathcal{O}(h^2)$$|\n\n\n### Ejemplos de implmentaci\u00f3n computacional\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport sympy as sym\n\nsym.init_printing(use_latex='mathjax')\n```\n\n\n```python\ndef f(x):\n return x**2 - x + 1\n```\n\n\n```python\ndef d1forward(x0, h):\n return (f(x0 + h) - f(x0)) / h\n```\n\n\n```python\ndef d1backward(x0, h):\n return (f(x0) - f(x0 - h)) / h\n```\n\n\n```python\ndef d1central(x0, h):\n return (f(x0 + h) - f(x0 - h) ) / (2 * h)\n```\n\n\n```python\nx = 3.5\nh = 0.1\n```\n\n\n```python\nd1f = d1forward(x, h)\nprint(\"El valor de la primera derivada en {0:4.2f} usando forward con un tama\u00f1o de paso {2:6.5f} es: {1:6.5f}\".format(x,d1f,h))\nd1b = d1backward(x, h)\nprint(\"El valor de la primera derivada en {0:4.2f} usando backward con un tama\u00f1o de paso {2:6.5f} es: {1:6.5f}\".format(x,d1b,h))\nd1c = d1central(x, h)\nprint(\"El valor de la primera derivada en {0:4.2f} usando central con un tama\u00f1o de paso {2:6.5f} es: {1:6.5f}\".format(x,d1c,h))\n```\n\n### Esquemas en 2D\n\nDe la serie truncada de Taylor en 2D, sea $\\phi=\\phi(x,y)$\n\n\n\\begin{equation*}\n\\begin{split}\n\\phi(x_{i+1},y_{i+1})=\\phi(x_i,y_i)+ \\left.\\Delta x \\frac{\\partial \\phi}{\\partial x}\\right|_{i,j}+ \\left.\\Delta y \\frac{\\partial \\phi}{\\partial y}\\right|_{i,j} + \\left.\\frac{(\\Delta x)^2}{2!}\\frac{\\partial^2\\phi}{\\partial x^2}\\right|_{i,j} + \\left. 2\\frac{(\\Delta x)(\\Delta y)}{2!}\\frac{\\partial^2 \\phi}{\\partial x \\partial y} \\right|_{i,j}+ \\left.\\frac{(\\Delta y)^2}{2!}\\frac{\\partial^2\\phi}{\\partial y^2}\\right|_{i,j} + R_N\n\\end{split}\n\\label{eq:Ec6_8} \\tag{6.8}\n\\end{equation*}\n\nDe la anterior ecuaci\u00f3n, el \u00fanico t\u00e9rmino que faltar\u00eda por determinar es el correspondiente a la derivada mixta en $x$ y $y$, que est\u00e1 dada por:\n\n\n\\begin{equation*}\n\\begin{split}\n\\frac{\\partial^2 \\phi}{\\partial x \\partial y} = \\frac{\\phi_{i+1,j+1}-\\phi_{i+1,j-1}-\\phi_{i-1,j+1}+\\phi_{i-1,j-1}}{4(\\Delta x)(\\Delta y)} + O \\left[ (\\Delta x)^2,(\\Delta y)^2 \\right ]\n\\end{split}\n\\label{eq:Ec6_9} \\tag{6.9}\n\\end{equation*}\n\n\n### T\u00e9rmino difusivo no uniforme en 1D\n\n\n\\begin{equation*}\n\\begin{split}\n\\frac{\\partial}{\\partial x}\\left( \\Gamma \\frac{\\partial \\phi}{\\partial x} \\right)\n\\end{split}\n\\label{eq:Ec6_10} \\tag{6.10}\n\\end{equation*}\n\ndonde $\\Gamma$ es el coeficiente de difusi\u00f3n o viscosidad\n\n\n\\begin{equation*}\n\\begin{split}\n\\left. \\frac{\\partial}{\\partial x}\\left( \\Gamma \\frac{\\partial \\phi}{\\partial x} \\right) \\right |_{i}=\\frac{\\left(\\Gamma \\frac{\\partial \\phi}{\\partial x}\\right)_{i+1/2}-\\left(\\Gamma \\frac{\\partial \\phi}{\\partial x}\\right)_{i-1/2}}{\\Delta x}+O(\\Delta x)^2\n\\end{split}\n\\label{eq:Ec6_11} \\tag{6.11}\n\\end{equation*}\n\nSe realiza entonces una aproximaci\u00f3n en los puntos $x_{i \\pm 1/2}$\n\n\n\\begin{equation*}\n\\begin{split}\n\\left. \\frac{\\partial}{\\partial x}\\left( \\Gamma \\frac{\\partial \\phi}{\\partial x} \\right) \\right |_{i} = \\frac{\\Gamma_{i+1/2}\\frac{\\phi_{i+1}-\\phi_i}{\\Delta x}-\\Gamma_{i-1/2}\\frac{\\phi_{i}-\\phi_{i-1}}{\\Delta x}}{\\Delta x}+O(\\Delta x)^2\n\\end{split}\n\\label{eq:Ec6_12} \\tag{6.12}\n\\end{equation*}\n\nsimplificando,\n\n\n\\begin{equation*}\n\\begin{split}\n\\left. \\frac{\\partial}{\\partial x}\\left( \\Gamma \\frac{\\partial \\phi}{\\partial x} \\right) \\right |_{i} = \\Gamma_{i+1/2} \\frac{\\phi_{i+1}-\\phi_i}{(\\Delta x)^2}-\\Gamma_{i-1/2}\\frac{\\phi_{i}-\\phi_{i-1}}{(\\Delta x)^2}+O(\\Delta x)^2\n\\end{split}\n\\label{eq:Ec6_13} \\tag{6.13}\n\\end{equation*}\n\nAhora, promediando los valores de $\\Gamma_{i \\pm1/2}$,\n\n\n\\begin{equation*}\n\\begin{split}\n\\Gamma_{i+1/2}=\\frac{\\Gamma_i+\\Gamma_{i+1}}{2} \\quad\\quad \\Gamma_{i-1/2}=\\frac{\\Gamma_i+\\Gamma_{i-1}}{2} \n\\end{split}\n\\label{eq:Ec6_14} \\tag{6.14}\n\\end{equation*}\n\n## Mol\u00e9cula computacional de una aproximaci\u00f3n en Diferencias Finitas\n\n### Ejemplo Ecuaci\u00f3n de Laplace\n\nEn c\u00e1lculo vectorial, la [ecuaci\u00f3n de Laplace](https://en.wikipedia.org/wiki/Laplace%27s_equation) es una ecuaci\u00f3n en derivadas parciales de segundo orden de tipo el\u00edptico, que recibe ese nombre en honor al f\u00edsico y matem\u00e1tico Pierre-Simon Laplace. La ecuaci\u00f3 de Laplace en dos dimensiones se expresa como:\n\n\n\\begin{equation*}\n\\begin{split}\n\\nabla^2 \\phi & = 0 \\\\\n& = \\left. \\frac{\\partial^2 \\phi}{\\partial x^2} \\right|_{i,j} + \\left. \\frac{\\partial^2 \\phi}{\\partial y^2} \\right|_{i,j}\n\\end{split}\n\\label{eq:Ec6_15} \\tag{6.15}\n\\end{equation*}\n\nEmpleando un esquema de diferencias finitas central para la segunda derivada, se tiene la discretizaci\u00f3n espacial como:\n\n\n\\begin{equation*}\n\\begin{split}\n\\nabla^2 \\phi & = \\frac{\\phi_{i+1,j}-2\\phi_{i,j}+\\phi_{i-1,j}}{(\\Delta x)^2} + \\frac{\\phi_{i,j+1}-2\\phi_{i,j}+\\phi_{i,j-1}}{(\\Delta y)^2} + O \\left[ (\\Delta x)^2,(\\Delta y)^2 \\right ]\n\\end{split}\n\\label{eq:Ec6_16} \\tag{6.16}\n\\end{equation*}\n\nEl anterior esquema genera lo que se denomina como *mol\u00e9cula de cinco puntos* o *f\u00f3rmula de cinco puntos*\n\n

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Fuente: Propio
\n\n## Discretizaci\u00f3n de ecuaciones por el m\u00e9todo de Diferencias Finitas\n\n### Estado estacionario\n\n[Ecuaciones en estado estacionario](https://en.wikipedia.org/wiki/Steady_state), son ecuaciones que no dependel del tiempo. A manera de ejemplo, consid\u00e9rese la ecuaci\u00f3n de Laplace en $2D$ discretizada por la f\u00f3rmula de cinco puntos en una regi\u00f3n $\\Omega$ cuadrada con condiciones de contorno Dirichlet $\\phi(x,y)=\\Theta(x,y)$ en la frontera $\\delta \\Omega$\n\n\n\\begin{equation*}\n\\begin{split}\n\\nabla^2 \\phi & = \\frac{\\partial^2 \\phi}{\\partial x^2} + \\frac{\\partial^2 \\phi}{\\partial y^2}=0\n\\end{split}\n\\label{eq:Ec6_17} \\tag{6.17}\n\\end{equation*}\n\nDiscretizando esta ecuaci\u00f3n con la ecuaci\u00f3n [(6.16)](#6_16), y reordenando los t\u00e9rminos, se llega a:\n\n\n\\begin{equation*}\n\\begin{split}\n\\phi_{i+1,j}+\\phi_{i-1,j}-2(1+\\beta^2)\\phi_{i,j}+\\beta^2(\\phi_{i,j+1}+\\phi_{i,j-1})=0\n\\end{split}\n\\label{eq:Ec6_18} \\tag{6.18}\n\\end{equation*}\n\ndonde $\\beta=\\frac{\\Delta x}{\\Delta y}$.\n\nTeniendo una discretizaci\u00f3n del dominio como se muestra en la figura:\n\n

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Fuente: Propio
\n\nLa ecuaci\u00f3n [(6.18)](#Ec6_18) ser\u00e1 v\u00e1lida para todos los puntos discretos $(i,j)$ al interior de la regi\u00f3n (nodos en color naranja). Dando valores a los sub\u00edndices $(i,j)$ desde $0$ hasta $4$ y reemplazando en la ecuaci\u00f3n [(6.18)](#Ec6_18), se llega al siguente sistema de ecuaciones:\n\n\n\\begin{equation*}\n\\begin{split}\n(1,1) : \\phi_{2,1} + \\phi_{0,1} - 2(1+\\beta^2)\\phi_{1,1} + \\beta^2(\\phi_{1,2} + \\phi_{1,0})=0 \\\\\n(2,1) : \\phi_{3,1} + \\phi_{1,1} - 2(1+\\beta^2)\\phi_{2,1} + \\beta^2(\\phi_{2,2} + \\phi_{2,0})=0 \\\\\n(3,1) : \\phi_{4,1} + \\phi_{2,1} - 2(1+\\beta^2)\\phi_{3,1} + \\beta^2(\\phi_{3,2} + \\phi_{3,0})=0 \\\\\n(1,2) : \\phi_{2,2} + \\phi_{0,2} - 2(1+\\beta^2)\\phi_{1,2} + \\beta^2(\\phi_{1,3} + \\phi_{1,1})=0 \\\\\n(2,2) : \\phi_{3,2} + \\phi_{1,2} - 2(1+\\beta^2)\\phi_{2,2} + \\beta^2(\\phi_{2,3} + \\phi_{2,1})=0 \\\\\n(3,2) : \\phi_{4,2} + \\phi_{2,2} - 2(1+\\beta^2)\\phi_{3,2} + \\beta^2(\\phi_{3,3} + \\phi_{3,1})=0 \\\\\n(1,3) : \\phi_{2,3} + \\phi_{0,3} - 2(1+\\beta^2)\\phi_{1,3} + \\beta^2(\\phi_{1,4} + \\phi_{1,2})=0 \\\\\n(2,3) : \\phi_{3,3} + \\phi_{1,3} - 2(1+\\beta^2)\\phi_{2,3} + \\beta^2(\\phi_{2,4} + \\phi_{2,2})=0 \\\\\n(3,3) : \\phi_{4,3} + \\phi_{2,3} - 2(1+\\beta^2)\\phi_{3,3} + \\beta^2(\\phi_{3,4} + \\phi_{3,2})=0 \n\\end{split}\n\\label{eq:Ec6_19} \\tag{6.19}\n\\end{equation*}\n\nExpresando el anterior sistema de ecuaciones lineales en forma matricial se llega a un sistema $Ax=b$\n\n\\begin{align*}\n\\left[\\begin{array}{ccccccccc}\n \\gamma & 1 & 0 &\\beta^2 & 0 & 0 & 0 & 0 & 0 \\\\\n 1 & \\gamma & 1 & 0 & \\beta^2 & 0 & 0 & 0 & 0 \\\\\n 0 & 1 & \\gamma & 0 & 0 & \\beta^2 & 0 & 0 & 0 \\\\\n \\beta^2& 0 & 0 & \\gamma & 1 & 0 & \\beta^2 & 0 & 0 \\\\\n 0 & \\beta^2& 0 & 1 & \\gamma & 1 & 0 & \\beta^2 & 0 \\\\\n 0 & 0 & \\beta^2& 0 & 1 & \\gamma & 0 & 0 & \\beta^2 \\\\\n 0 & 0 & 0 & \\beta^2& 0 & 0 & \\gamma & 1 & 0 \\\\\n 0 & 0 & 0 & 0 & \\beta^2 & 0 & 1 & \\gamma & 1 \\\\\n 0 & 0 & 0 & 0 & 0 & \\beta^2 & 0 & 1 & \\gamma\n\\end{array}\\right]\n\\begin{Bmatrix}\n\\phi_{1,1}\\\\\n\\phi_{2,1}\\\\\n\\phi_{3,1}\\\\\n\\phi_{1,2}\\\\\n\\phi_{2,2}\\\\\n\\phi_{3,2}\\\\\n\\phi_{1,3}\\\\\n\\phi_{2,3}\\\\\n\\phi_{3,3}\n\\end{Bmatrix}\n= \\begin{Bmatrix}\n-\\phi_{0,1}-\\beta^2\\phi_{1,0}\\\\\n-\\beta^2\\phi_{2,0}\\\\\n-\\phi_{4,1}-\\beta^2\\phi_{3,0}\\\\\n-\\phi_{0,2}\\\\\n0\\\\\n-\\phi_{4,2}\\\\\n-\\phi_{0,3}-\\beta^2\\phi_{1,4}\\\\\n-\\beta^2\\phi_{2,4}\\\\\n-\\phi_{4,3}-\\beta^2\\phi_{3,4}\n\\end{Bmatrix}\n\\end{align*}\n\ndonde $\\gamma=-2(1+\\beta^2)$\n\n\n### Ejemplo: ecuaci\u00f3n de transporte en estado estacionario en 1D\n\nResuelva la ecuaci\u00f3n de transporte de convecci\u00f3n - difusi\u00f3n homog\u00e9nea en estado estacionario en 1D dada por\n\n$$\\frac{d^2\\phi}{dx^2}-\\beta \\frac{d\\phi}{dx}=0$$\n\nsometida a las condiciones de contorno\n\n$$\\phi(0)=0 \\quad \\quad \\phi(1) = 1$$\n\nSe va a resolver num\u00e9ricamente esta ecuaci\u00f3n utilizando un esquema de diferencias finitas centrales (Leapfrog). Discretizando la ecuaci\u00f3n dada por los esquemas indicados, se llega a:\n\n$$-\\frac{1}{h^2} \\left[ \\phi(x_{i-1})-2\\phi(x_i)+\\phi(x_{i+1})\\right]+\\frac{\\beta}{2h}\\left[ \\phi(x_{i+1})-\\phi(x_{i-1})\\right]=0$$\n\norganizando t\u00e9rminos,\n\n$$(2+\\beta h)\\phi(x_{i-1})-4\\phi(x_i)+(2-\\beta h)\\phi(x_{i+1})=0$$\n\ntomando como tama\u00f1o de paso inicialmente el valor de $h=0.1$ y por las condiciones de contorno dadas, se tiene la siguiente distribuci\u00f3n de nodos y elementos (\"malla\"):\n\n

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Fuente: Propio
\n\nDando valores a $i$ desde $1$ hasta $9$, se tiene el siguiente conjunto de ecuaciones:\n\n\\begin{align*}\n\\begin{array}{cccc}\ni=1:& -4\\phi(x_1)+(2-\\beta h)\\phi(x_2)&=-(2+\\beta h)\\phi(x_0) \\\\\ni=2:& (2+\\beta h)\\phi(x_1)-4\\phi(x_2)+(2-\\beta h)\\phi(x_3)&=0 \\\\\ni=3:& (2+\\beta h)\\phi(x_2)-4\\phi(x_3)+(2-\\beta h)\\phi(x_4)&=0 \\\\\ni=4:& (2+\\beta h)\\phi(x_3)-4\\phi(x_4)+(2-\\beta h)\\phi(x_5)&=0 \\\\\ni=5:& (2+\\beta h)\\phi(x_4)-4\\phi(x_5)+(2-\\beta h)\\phi(x_6)&=0 \\\\\ni=6:& (2+\\beta h)\\phi(x_5)-4\\phi(x_6)+(2-\\beta h)\\phi(x_7)&=0 \\\\\ni=7:& (2+\\beta h)\\phi(x_6)-4\\phi(x_7)+(2-\\beta h)\\phi(x_8)&=0 \\\\\ni=8:& (2+\\beta h)\\phi(x_7)-4\\phi(x_8)+(2-\\beta h)\\phi(x_9)&=0 \\\\\ni=9:& (2+\\beta h)\\phi(x_8)-4\\phi(x_9)&=-(2-\\beta h)\\phi(x_{10})\n\\end{array}\n\\end{align*}\n\nlos nodos $i=0$ e $i=10$ corresponder\u00e1n al primer y \u00faltimo nodo de la malla, cuyos valores $\\phi(x_0)$ y $\\phi(x_{10})$ son dados por las condiciones de contorno\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\n```\n\n\n```python\n# Establecer las CCs - Izquierda\n\nx0 = 0\nu0 = 0\n\n# Establecer las CCs - Derecha\n\nx1 = 1\nu1 = 1\n\n```\n\n\n```python\n# Ingreso de datos\nn = int(input(\"Ingrese el n\u00famero de nodos: \"))\n```\n\n\n```python\nbeta = float(input(\"Ingrese el valor de la velocidad: \"))\n```\n\n\n```python\n# Inicializaci\u00f3n de arreglos\nA = np.zeros((n-1, n-1)) # Matriz de coeficientes\nb = np.zeros((n-1, 1)) # Vector de t\u00e9rinos independientes\nu_lf = np.zeros((n-1, 1)) # Vector de inc\u00f3gnitas\nuex = np.zeros((n-1, 1)) # Vector con la soluci\u00f3n exacta\n```\n\n\n```python\n# c\u00e1lculos iniciales\nh = (x1 - x0) / n # Tama\u00f1o de paso Dx\nx = np.linspace(x0, x1, n + 1)\n```\n\n\n```python\n# Ensamblaje de la matriz de coeficientes\n\nfor i in range(1, n-1):\n A[i-1, i] = 2 - beta * h\n A[i-1, i-1] = -4\n A[i, i - 1] = 2 + beta * h\n b[i] = 0\n\nA[n-2,n-2] = -4\nb[0] = -(2 + beta * h) * u0\nb[n-2] = -(2 - beta * h ) * u1\n```\n\n\n```python\n# Soluci\u00f3n del SEL Au=b\nu_lf = np.linalg.solve(A,b)\n```\n\n\n```python\n# Adicionando las CC en x0 y x1\n\nu_lf = np.insert(u_lf, 0, u0)\nu_lf = np.insert(u_lf, n, u1)\n```\n\n\n```python\n# definici\u00f3n de la funci\u00f3n exacta\ndef exacta(x):\n return (np.exp(beta * x) - 1) / (np.exp(beta) - 1)\n```\n\n\n```python\n# Graficaci\u00f3n\nplt.xlabel(\"x\")\nplt.ylabel(\"u\")\nplt.title(\"Exacta vs Leapfrog\")\nplt.plot(x, u_lf, 'r', label=\"Leapfrog\")\nplt.plot(x, exacta(x), 'b', label=('Exacta'))\nplt.legend(loc='center left')\nplt.grid(True)\nplt.show()\n```\n\n### Ecuaciones No estacionarias\n\n#### Introducci\u00f3n\n\nLas ecuaciones [no estacionarias](https://en.wikipedia.org/wiki/Transient_state) (transient, en ingl\u00e9s) o dependientes del tiempo, exigen el c\u00e1lculo de la soluci\u00f3n en intervalos sucesivos de tiempo\n\n\n\\begin{equation*}\n\\begin{split}\nt_0+\\Delta t, t_0 + 2\\Delta t, t_0 + 3\\Delta t, \\ldots, t_f - \\Delta t, t_f \n\\end{split}\n\\label{eq:Ec6_20} \\tag{6.20}\n\\end{equation*}\n\nUna de las ecuaciones m\u00e1s representativas de este tipo es la [ecuaci\u00f3n de calor](https://en.wikipedia.org/wiki/Heat_equation), dada por\n\n\n\\begin{equation*}\n\\begin{split}\n\\frac{\\partial T}{\\partial t}=\\alpha\\frac{\\partial^2T}{\\partial x^2}\n\\end{split}\n\\label{eq:Ec6_21} \\tag{6.21}\n\\end{equation*}\n\nsometida a las condiciones auxiliares:\n\n- ***Iniciales***\n\n\n\\begin{equation*}\n\\begin{split}\nT_i^0=T(i\\Delta x,t_0)=T_{inic}\n\\end{split}\n\\label{eq:Ec6_22} \\tag{6.22}\n\\end{equation*}\n\n\n- ***Contorno:*** tipo Dirichlet\n\n\n\\begin{equation*}\n\\begin{split}\nT(0,t) &= T_0 \\\\\nT(IM,t) &= T_{IM}\n\\end{split}\n\\label{eq:Ec6_23} \\tag{6.23}\n\\end{equation*}\n\n\n#### Discretizaci\u00f3n de ecuaciones no estacionarias\n\nLa ecuaci\u00f3n [(6.21)](#Ec6_21) tiene dos variables independientes, $x$ y $t$. Se busca determinar un vector $T^{(n)}$ que contenga las temperaturas, para cada nivel de tiempo $n$, y para todos los nodos de la malla.\n\n

\n \n

\n\n
Fuente: Propio
\n\nFilas del mismo color representan la discretizaci\u00f3n espacial en $1D$. La direcci\u00f3n vertical representa el tiempo.\n\nAhora, discretizando la ecuaci\u00f3n [(6.21)](#Ec6_21) en la coordenada espacial en un punto $x_i$, empleando un [esquema de diferencias finitas central para la segunda derivada](#DFC_d2y) dado por la ecuaci\u00f3n [(6.7)](#Ec6_7):\n\n\n\\begin{equation*}\n\\begin{split}\n\\left. \\frac{\\partial T}{\\partial t} \\right|_{i}=\\alpha\\frac{T_{i-1}-2T_i+T_{i+1}}{(\\Delta x)^2} + \\mathcal{O}(\\Delta x)^2\n\\end{split}\n\\label{eq:Ec6_24} \\tag{6.24}\n\\end{equation*}\n\nPara discretizar la ecuaci\u00f3n [(6.21)](#Ec6_21) en la coordenada temporal se emplear\u00e1n dos enfoques diferentes: [expl\u00edcita e impl\u00edcita](https://en.wikipedia.org/wiki/Explicit_and_implicit_methods):\n\n#### Discretizaci\u00f3n expl\u00edcita\n\nLa ecuaci\u00f3n [(6.24)](#Ec6_24) discretizada en un nivel de tiempo $n$ toma la forma:\n\n\n\\begin{equation*}\n\\begin{split}\n\\left. \\frac{\\partial T}{\\partial t} \\right|_{i}^{(n)}=\\alpha\\frac{T_{i-1}^{(n)}-2T_i^{(n)}+T_{i+1}^{(n)}}{(\\Delta x)^2} + \\mathcal{O}(\\Delta x)^2\n\\end{split}\n\\label{eq:Ec6_25} \\tag{6.25}\n\\end{equation*}\n\nRelacionando los valores de $T_i^{(n+1)}$ con los de $T_i^{(n)}$, utilizando un esquema en Diferencias Finitas hacia adelante (Forward) para la primera derivada:\n\n\n\\begin{equation*}\n\\begin{split}\n\\left. \\frac{\\partial T}{\\partial t} \\right|_{i}^{(n)}=\\frac{T_{i}^{(n+1)}-T_{i}^{(n)}}{\\Delta t} + \\mathcal{O}(\\Delta t)\n\\end{split}\n\\label{eq:Ec6_26} \\tag{6.26}\n\\end{equation*}\n\nIgualando las ecuaciones [(6.25)](#Ec6_25) y [(6.26)](#Ec6_26), se llega al siguiente esquema:\n\n\n\\begin{equation*}\n\\begin{split}\n\\frac{T_{i}^{(n+1)}-T_{i}^{(n)}}{\\Delta t} = \\alpha\\frac{T_{i-1}^{(n)}-2T_i^{(n)}+T_{i+1}^{(n)}}{(\\Delta x)^2} + \\mathcal{O} \\left[ \\Delta t, (\\Delta x)^2 \\right]\n\\end{split}\n\\label{eq:Ec6_27} \\tag{6.27}\n\\end{equation*}\n\nPor \u00faltimo, de la anterior ecuaci\u00f3n se despeja la variable temporal en el tiempo siguiente, $T_i^{(n+1)}$ y se llega al esquema llamado *M\u00e9todo de Euler expl\u00edcito*:\n\n\n\\begin{equation*}\n\\begin{split}\nT_{i}^{(n+1)} = T_{i}^{(n)} + \\alpha (\\Delta t) \\frac{T_{i-1}^{(n)}-2T_i^{(n)}+T_{i+1}^{(n)}}{(\\Delta x)^2}\n\\end{split}\n\\label{eq:Ec6_28} \\tag{6.28}\n\\end{equation*}\n\nEl lado derecho de la ecuaci\u00f3n [(6.28)](#Ec6_28) solamente depende de los valores de $T$ en el nivel de tiempo $n$.\n\nLa mol\u00e9cula de diferencias finitas resultante se observa en la siguiente figura\n\n

\n \n

\n\n
Fuente: Propio
\n\n***Observaciones:***\n\n- Por las C.Aux. cuando $n=0$ y $x=0$, los valores de $T_i^{(0)}$ son conocidos\n\n\n- Avanzando en el tiempo se obtiene un sistema de ecuaciones lineales, cuyas inc\u00f3gnitas son los valores de $T^{(n+1)}$.\n\n\n- Solamente se puede calcular $T^{(n+1)}$ despu\u00e9s de conocer los valores en $T^{(n)}$.\n\n\n***Dificultades:***\n\n- Excluyen valores de C.C. en el nivel de tiempo $n$, que tiene influencia sobre la soluci\u00f3n del problema en el nivel de tiempo $n+1$.\n\n

\n \n

\n\n
Fuente: Propio
\n\n- La soluci\u00f3n en los nodos en amarillo no tiene efecto sobre la soluci\u00f3n en el punto interno $(x,t)$ en el nivel de tiempo $n+2$.\n\n\n- La soluci\u00f3n en el punto $(x,t)$ es influenciada solamente por las C.C. del nivel de tiempo $n$, y no por el valor de la C.C. en el nivel de tiempo $n+2$.\n\n\n- Efecto exclusivamente num\u00e9rico, equivalente a un atraso en la propagaci\u00f3n de las C.C. adentro del dominio.\n\n\n- F\u00edsicamente incorrecto.\n\n\n- Este fen\u00f3meno causa grandes errores cuando las C.C. presentan variaciones considerables entre instantes sucesivos de tiempo.\n\n#### Discretizaci\u00f3n impl\u00edcita\n\nProceso de Marcha en el tiempo (igual que en el Expl\u00edcito). $T^{(1)}$ es calculado a partir de las condiciones iniciales. $T^{(2)}, T^{(3)}, \\ldots$, son calculados secuencialmente, precisando de todos los valores del paso anterior. La diferencia est\u00e1 en la forma como es calculado $T^{(n+1)}$. La discretizaci\u00f3n espacial se realiza en el nivel de tiempo $n+1$.\n\n\n\\begin{equation*}\n\\begin{split}\n\\frac{T_{i}^{(n+1)}-T_{i}^{(n)}}{\\Delta t} = \\alpha\\frac{T_{i-1}^{(n+1)}-2T_i^{(n+1)}+T_{i+1}^{(n+1)}}{(\\Delta x)^2} + \\mathcal{O} \\left[ \\Delta t, (\\Delta x)^2 \\right]\n\\end{split}\n\\label{eq:Ec6_29} \\tag{6.29}\n\\end{equation*}\n\nreorganizando la ecuaci\u00f3n para dejar a un lado los t\u00e9rminos evaluados en $T^{(n+1)}$ y al otro lado los t\u00e9rminos evaluados en $T^{(n)}$, obteni\u00e9ndose la expresi\u00f3n del *m\u00e9todo de Euler impl\u00edcito*.\n\n\n\\begin{equation*}\n\\begin{split}\n-sT_{i-1}^{(n+1)}+(1+2s)T_{i}^{(n+1)}-sT_{i+1}^{(n+1)}=T_{i}^{(n)}\n\\end{split}\n\\label{eq:Ec6_30} \\tag{6.30}\n\\end{equation*}\n\ndonde $s=\\frac{\\alpha (\\Delta t)}{(\\Delta x)^2}$\n\nLa mol\u00e9cula de diferencias finitas resultante para el m\u00e9todo implicito es:\n\n

\n \n

\n\n
Fuente: Propio
\n\nLa ecuaci\u00f3n [(6.30)](#Ec6_30) por s\u00ed sola no permite determinar el valor de $T_{i}^{(n+1)}$. A cada paso de tiempo aparecen tres inc\u00f3gnitas nuevas: $T_{i-1}^{(n+1)}, T_{i}^{(n+1)}$ y $T_{i+1}^{(n+1)}$. El valor $T_{i}^{(n+1)}$ est\u00e1 definido \u201cimpl\u00edcitamente\u201d. \n\nEscribiendo las ecuaciones asociadas a cada nodo de la malla discretizada\n\n\n\\begin{equation*}\n\\begin{split}\n-sT_{0}^{(n+1)}+(1+2s)T_{1}^{(n+1)}-sT_{2}^{(n+1)} &= T_{1}^{(n)}\\\\\n-sT_{1}^{(n+1)}+(1+2s)T_{2}^{(n+1)}-sT_{3}^{(n+1)} &= T_{2}^{(n)}\\\\\n-sT_{2}^{(n+1)}+(1+2s)T_{3}^{(n+1)}-sT_{4}^{(n+1)} &= T_{3}^{(n)}\\\\\n& \\vdots \\\\\n-sT_{IM-3}^{(n+1)}+(1+2s)T_{IM-2}^{(n+1)}-sT_{IM-1}^{(n+1)} &= T_{IM-2}^{(n)}\\\\\n-sT_{IM-2}^{(n+1)}+(1+2s)T_{IM-1}^{(n+1)}-sT_{IM}^{(n+1)} &= T_{IM-1}^{(n)}\n\\end{split}\n\\label{eq:Ec6_31} \\tag{6.31}\n\\end{equation*}\n\naplicando las condiciones de contorno en la ecuaci\u00f3n [(6.31)](#Ec6_31)\n\n\n\\begin{equation*}\n\\begin{split}\n(1+2s)T_{1}^{(n+1)}-sT_{2}^{(n+1)} &= T_{1}^{(n)} + sT_{0}^{(n+1)}\\\\\n-sT_{1}^{(n+1)}+(1+2s)T_{2}^{(n+1)}-sT_{3}^{(n+1)} &= T_{2}^{(n)}\\\\\n-sT_{2}^{(n+1)}+(1+2s)T_{3}^{(n+1)}-sT_{4}^{(n+1)} &= T_{3}^{(n)}\\\\\n& \\vdots \\\\\n-sT_{IM-3}^{(n+1)}+(1+2s)T_{IM-2}^{(n+1)}-sT_{IM-1}^{(n+1)} &= T_{IM-2}^{(n)}\\\\\n-sT_{IM-2}^{(n+1)}+(1+2s)T_{IM-1}^{(n+1)} &= T_{IM-1}^{(n)} + sT_{IM}^{(n+1)}\n\\end{split}\n\\label{eq:Ec6_32} \\tag{6.32}\n\\end{equation*}\n\nrepresentando la anterior ecuaci\u00f3n en forma matricial, se tiene \n\n\n\\begin{align*}\n\\left[\\begin{array}{ccccccccc}\n 1 + 2s & -s & 0 & 0 & \\ldots & 0 \\\\\n -s & 1 + 2s & -s & 0 & \\ldots & 0 \\\\\n 0 & -s & 1 + 2s & -s & \\ldots & 0 \\\\\n & & & \\ddots & & \\\\\n 0 & \\ldots & 0 & -s & 1+2s & -s \\\\\n 0 & \\ldots & 0 & 0 & -s & 1+2s \n \\end{array}\\right]\n\\begin{Bmatrix}\nT_{1}^{(n+1)}\\\\\nT_{2}^{(n+1)}\\\\\nT_{3}^{(n+1)}\\\\\n\\ldots\\\\\nT_{IM-2}^{(n+1)}\\\\\nT_{IM-1}^{(n+1)}\n\\end{Bmatrix}\n= \\begin{Bmatrix}\nT_{1}^{(n)}+ sT_{0}^{(n+1)}\\\\\nT_{2}^{(n)}\\\\\nT_{3}^{(n)}\\\\\n\\vdots\\\\\nT_{IM-2}^{(n)}\\\\\nT_{IM-1}^{(n)}+sT_{IM}^{(n+1)}\n\\end{Bmatrix}\n\\label{eq:Ec6_33} \\tag{6.33}\n\\end{align*}\n\n\n\n#### Resumen\n\n***M\u00e9todos expl\u00edcitos:***\n\n- Las ecuaciones resultantes son independientes, por lo tanto, la soluci\u00f3n es directa.\n\n\n***M\u00e9todos impl\u00edcitos:***\n\n- Las ecuaciones resultantes son acopladas, por lo que se debe resolver un sistema de ecuaciones lineales en cada instante de tiempo!\n\n### Discretizaci\u00f3n multidimensional\n\nComo ejemplo, tomemos la ecuaci\u00f3n de difusi\u00f3n no permanente en 2D\n\n\n\\begin{equation*}\n\\begin{split}\n\\frac{\\partial \\phi}{\\partial t} = \\alpha \\nabla^2 \\phi = \\alpha \\left( \\frac{\\partial^2 \\phi}{\\partial x^2} + \\frac{\\partial^2 \\phi}{\\partial y^2} \\right)\n\\end{split}\n\\label{eq:Ec6_34} \\tag{6.34}\n\\end{equation*}\n\nun dominio computacional en 2D, discretizado por una serie de nodos y elementos, se observa en la siguiente figura.\n\n

\n \n

\n\n
Fuente: Propio
\n\n- Empleando el m\u00e9todo de Euler expl\u00edcito y diferencias finitas centrales de segundo orden, se llega a\n\n\n\\begin{equation*}\n\\begin{split}\n\\frac{\\phi_{i,j}^{(n+1)}-\\phi_{i,j}^{(n)}}{\\Delta t} = \\alpha \\frac{\\phi_{i+1,j}^{(n)}-2\\phi_{i,j}^{(n)}+\\phi_{i-1,j}^{(n)}}{(\\Delta x)^2} + \\alpha \\frac{\\phi_{i,j+1}^{(n)}-2\\phi_{i,j}^{(n)}+\\phi_{i,j-1}^{(n)}}{(\\Delta y)^2} + \\mathcal{O} \\left[(\\Delta t),(\\Delta x)^2,(\\Delta y)^2\\right]\n\\end{split}\n\\label{eq:Ec6_35} \\tag{6.35}\n\\end{equation*}\n\norganizando t\u00e9rminos,\n\n\n\\begin{equation*}\n\\begin{split}\n\\phi_{i,j}^{(n+1)}=\\phi_{i,j}^{(n)} + s_x \\left( \\phi_{i+1,j}^{(n)} - 2\\phi_{i,j}^{(n)} + \\phi_{i-1,j}^{(n)} \\right) + s_y \\left( \\phi_{i,j+1}^{(n)} - 2\\phi_{i,j}^{(n)} + \\phi_{i,j-1}^{(n)} \\right)\n\\end{split}\n\\label{eq:Ec6_36} \\tag{6.36}\n\\end{equation*}\n\n\n- Ahora empleando el m\u00e9todo de Euler impl\u00edcito y diferencias finitas centrales de segundo orden,\n\n\n\\begin{equation*}\n\\begin{split}\n-s_x \\left( \\phi_{i+1,j}^{(n+1)} + \\phi_{i-1,j}^{(n+1)} \\right) + \\left[1+2\\left( s_x+s_y \\right) \\right] \\phi_{i,j}^{(n+1)} - s_y \\left( \\phi_{i,j+1}^{(n+1)} + \\phi_{i,j-1}^{(n+1)} \\right) = \\phi_{i,j}^{(n)} \n\\end{split}\n\\label{eq:Ec6_37} \\tag{6.37}\n\\end{equation*}\n\ndonde $s_x=\\frac{\\alpha (\\Delta t)}{(\\Delta x)^2}$ y $s_y=\\frac{\\alpha (\\Delta t)}{(\\Delta y)^2}$\n\n## Consistencia, Convergencia y Estabilidad\n\n### Introducci\u00f3n\n\nLa soluci\u00f3n calculada se aproxima, de alguna forma, a la soluci\u00f3n \u201creal\u201d de la EDP?\n\nEl *Error Local de Truncamiento (ELT)* influencia en la calidad de la aproximaci\u00f3n num\u00e9rica. Cu\u00e1ndo y sobre qu\u00e9 condiciones la soluci\u00f3n de las ecuaciones lineales es representativa de la soluci\u00f3n real de la EDP?\n\nDepende de principalmente de tres elementos: [Consistencia, Estabilidad y Convergencia](https://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations#Analysis)\n\n- Consistencia de las ecuaciones de diferencias finitas.\n\n\n- Estabilidad y convergencia del m\u00e9todo num\u00e9rico empleado.\n\n\n### Consistencia\n\nEn la discretizaci\u00f3n, el $ELT \\rightarrow 0$ cuando $\\Delta x, \\Delta t \\rightarrow 0$, es decir, si\n\n\n\\begin{equation*}\n\\begin{split}\n\\lim \\limits_{\\Delta x \\to 0}\\frac{\\phi_{i+1}-\\phi_i}{\\Delta x}=\\frac{\\partial \\phi}{\\partial x}\n\\end{split}\n\\label{eq:Ec6_38} \\tag{6.38}\n\\end{equation*}\n\nVerificando:\n\n- Se substituye la expansi\u00f3n en series de Taylor en la E.D.F.\n\n\n- Se hace que $\\Delta x, \\Delta t \\rightarrow 0$\n\n\n- $ELT \\rightarrow 0$?, la discretizaci\u00f3n es consistente con la EDP.\n\n\n- $ELT\\nrightarrow 0$?, la discretizaci\u00f3n no es consistente con la EDP.\n\n

\n \n

\n\n
Fuente: Propio
\n\nLa *consistencia* es una condici\u00f3n necesaria para la *convergencia*.\n\n### Convergencia\n\n*C\u00f3mo afecta la discretizaci\u00f3n hecha a la EDP en un determinado n\u00famero arbitrario de pasos de tiempo?*\n\nSi la soluci\u00f3n num\u00e9rica se aproxima a la soluci\u00f3n exacta de la EDP, conforme $\\Delta x, \\Delta t \\rightarrow 0$, se dice que el m\u00e9todo es convergente. \n\n

\n \n

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Fuente: Propio
\n\n### Estabilidad\n\nUn m\u00e9todo num\u00e9rico es estable si cualquier error o perturbaci\u00f3n en la soluci\u00f3n no se amplifican sin l\u00edmites. Ese crecimiento es relativo al m\u00e9todo num\u00e9rico exclusivamente y no a la f\u00edsica del problema. La estabilidad trata sobre el crecimiento, o disminuci\u00f3n, de los errores introducidos en los c\u00e1lculos.\n\n- En los m\u00e9todos no\u2013permanentes, la estabilidad garantiza que la soluci\u00f3n num\u00e9rica sea limitada.\n\n\n- Para los m\u00e9todos iterativos de la soluci\u00f3n de sistemas lineales un m\u00e9todo estable es aquel que no diverge conforme las iteraciones avanzan.\n\n\n***Cu\u00e1les son estos errores?***\n\n- Condiciones de contorno e iniciales aproximadas de forma incorrecta\n\n\n- Acumulaci\u00f3n de errores de redondeo durante los c\u00e1lculos.\n\n\n***C\u00f3mo se evitan?***\n\n- Correcta discretizaci\u00f3n de las condiciones auxiliares.\n\n\n- No puede ser evitado, pero s\u00ed controlado.\n\n\n***Con relaci\u00f3n a la estabilidad los m\u00e9todos num\u00e9ricos pueden clasificarse en:***\n\n\n- ***Condicionalmente estables:*** Deben satisfacer alguna condici\u00f3n de estabilidad. M\u00e9todos expl\u00edcitos.\n\n\n- ***Incondicionalmente estables:*** No necesitan satisfacer condiciones de estabilidad. M\u00e9todos Impl\u00edcitos.\n\n\n- ***Incondicionalmente inestables:**** No existen valores de $\\Delta t$ que permitan soluciones estables. Deben evitarse.\n\n\n#### Teorema de Lax\n\nEl [Teorema de Lax](https://en.wikipedia.org/wiki/Lax_equivalence_theorem) (en honor al matem\u00e1tico [Peter Lax](https://en.wikipedia.org/wiki/Peter_Lax)) establece que:\n\n\n> *Dado un problema de valores iniciales bien planteado matem\u00e1ticamente y una aproximaci\u00f3n en diferencias finitas que satisface la condici\u00f3n de Consistencia, entonces la Estabilidad es condici\u00f3n necesaria y suficiente para la Convergencia.*\n\n#### Condici\u00f3n CFL\n\nla condici\u00f3n de [Courant-Friedrichs-Lewy](https://en.wikipedia.org/wiki/Courant%E2%80%93Friedrichs%E2%80%93Lewy_condition) (condici\u00f3n CFL) es una condici\u00f3n de convergencia de ecuaciones diferenciales en derivadas parciales solucionadas mediante ciertos algortimos (no confundir con estabilidad num\u00e9rica). Como consecuencia de esta condici\u00f3n, el paso de tiempo debe ser inferior a un cierto valor, si no, la simulaci\u00f3n producir\u00e1 resultados incorrectos. Est\u00e1 condici\u00f3n se llama as\u00ed en honor a [Richard Courant](https://en.wikipedia.org/wiki/Richard_Courant), [Kurt Friedrichs](https://en.wikipedia.org/wiki/Kurt_Otto_Friedrichs) y [Hans Lewy](https://en.wikipedia.org/wiki/Hans_Lewy) que la describieron en un [art\u00edculo en 1928](https://web.stanford.edu/class/cme324/classics/courant-friedrichs-lewy.pdf).\n\nLa condici\u00f3n *CFL* se representa com\u00fanmente para esquemas de advecci\u00f3n puros (es decir ignorando los t\u00e9rminos de difusi\u00f3n y reacci\u00f3n) como:\n\n\n\\begin{equation*}\n\\begin{split}\n\\frac{u\\cdot \\Delta t}{\\Delta x} < C\n\\end{split}\n\\label{eq:Ec6_39} \\tag{6.39}\n\\end{equation*}\n\nEn un caso bidimensional la ecuaci\u00f3n anterior se transforma en:\n\n\n\\begin{equation*}\n\\begin{split}\n\\frac{u_{x}\\cdot \\Delta t}{\\Delta x} + \\frac{u_{y}\\cdot \\Delta t}{\\Delta y} < C\n\\end{split}\n\\label{eq:Ec6_40} \\tag{6.40}\n\\end{equation*}\n\nLa condici\u00f3n CFL puede ser muy limitante para el intervalo de tiempo $\\Delta t$, hasta el punto que para ciertas ecuaciones diferenciales en derivadas parciales de cuarto orden es:\n\n\n\\begin{equation*}\n\\begin{split}\n\\frac{\\Delta t}{\\Delta x^4} < C\n\\end{split}\n\\label{eq:Ec6_41} \\tag{6.41}\n\\end{equation*}\n\npor lo que se utilizan mejor m\u00e9todos impl\u00edcitos para evitarla.\n\n#### Relaci\u00f3n Consistencia - Convergencia - Estabilidad\n\n

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Fuente: Propio
\n\n### Esquema Upwind\n\nCon la intenci\u00f3n de suavizar las oscilaciones presentes debido al t\u00e9rmino convectivo, se plantear\u00e1 un equema num\u00e9rico tipo [upwind](https://en.wikipedia.org/wiki/Upwind_scheme), que consiste principalmente en adicionar una cierta cantidad de difusi\u00f3n extra al t\u00e9rmino difusivo teniendo en cuenta la direcci\u00f3n del flujo. El esquema de diferenciaci\u00f3n [upwind](https://en.wikipedia.org/wiki/Upwind_differencing_scheme_for_convection) supera la incapacidad del esquema de diferenciaci\u00f3n [central](#Ej01_TransportEq). Este esquema est\u00e1 desarrollado para flujos convectivos fuertes con efectos de difusi\u00f3n suprimidos. \n\nVolviendo al enunciado del ejemplo visto en el numeral [3.2](#Ej01_TransportEq), considere la ecuaci\u00f3n de transporte de convecci\u00f3n - difusi\u00f3n homog\u00e9nea en estado estacionario en 1D dada por\n\n$$\\frac{d^2\\phi}{dx^2}-\\beta \\frac{d\\phi}{dx}=0$$\n\nsometida a las condiciones de contorno\n\n$$\\phi(0)=0 \\quad \\quad \\phi(1) = 1$$\n\nEmpleando alguno de los esquemas de discretizacion para el t\u00e9rmino de la primera derivada (advectivo) vistos,\n\n\n\n\\begin{equation*}\n\\begin{split}\n\\text{Forward:} \\quad\\quad \\phi'(x_i) = \\frac{\\phi(x_{i+1})-\\phi(x_{i})}{h}\n\\end{split}\n\\label{eq:Ec6_42} \\tag{6.42}\n\\end{equation*}\n\no \n\n\n\\begin{equation*}\n\\begin{split}\n\\text{Backward:} \\quad\\quad \\phi'(x_i) = \\frac{\\phi(x_{i})-\\phi(x_{i-1})}{h}\n\\end{split}\n\\label{eq:Ec6_43} \\tag{6.43}\n\\end{equation*}\n\n\nSi se aplica un [an\u00e1lisis de estabilidad de VonNeumann](https://en.wikipedia.org/wiki/Von_Neumann_stability_analysis) se encontrar\u00e1 que el esquema de discretizaci\u00f3n es inestable.\n\nUna posible soluci\u00f3n para intentar evitar las dificultades encontradas en tales formulaciones es el [esquema Upwind](https://en.wikipedia.org/wiki/Upwind_scheme). La Ec. [(6.42)](#Ec6_42) puede estabilizarse sustituyendo la discretizaci\u00f3n espacial hacia adelante adelante por un esquema de discretizaci\u00f3n hacia atr\u00e1s, siempre que la velocidad $\\beta>0$. Si $\\beta<0$ es negativo, se debe utilizar un esquema de diferencias hacia adelante para asegurar la estabilidad. En esta formulaci\u00f3n el t\u00e9rmino de difusi\u00f3n permanece sin cambios y solo el t\u00e9rmino convectivo (en forma conservadora) se calcula de la siguiente manera:\n\n\n\\begin{equation*}\n\\begin{split}\n-\\beta\\frac{\\phi(x_{i})-\\phi(x_{i-1})}{h} + \\text{T\u00e9rmino viscoso, si } \\beta > 0\n\\end{split}\n\\label{eq:Ec6_44} \\tag{6.44}\n\\end{equation*}\n\no \n\n\n\\begin{equation*}\n\\begin{split}\n-\\beta\\frac{\\phi(x_{i+1})-\\phi(x_{i})}{h} + \\text{T\u00e9rmino viscoso, si } \\beta < 0\n\\end{split}\n\\label{eq:Ec6_45} \\tag{6.45}\n\\end{equation*}\n\nUna formulaci\u00f3n en diferencias finitas de una ecuaci\u00f3n de flujo posee la propiedad de transporte si el efecto de una perturbaci\u00f3n se convecta (advecta) solo en la direcci\u00f3n de la velocidad.\n\n\n### Soluci\u00f3n de la ecuaci\u00f3n de convecci\u00f3n-difusi\u00f3n en estado estacionario en 1D empleando el esquema Upwind\n\nEn la secci\u00f3n [3.2](#Ej01_TransportEq) se present\u00f3 la ecuaci\u00f3n de transporte de convecci\u00f3n - difusi\u00f3n homoeg\u00e9nea en estado estacionario en 1D dada por\n\n$$-\\frac{d^2\\phi}{dx^2}+\\beta \\frac{d\\phi}{dx}=0$$\n\nsometida a las condiciones de contorno\n\n$$\\phi(0)=0 \\quad \\quad \\phi(1) = 1$$\n\ny se observ\u00f3 que para valores de $\\beta$ altos ($>50$ en el ejemplo) se presentaron oscilaciones debidas a la incapacidad del esquema de discretizaci\u00f3n empleado (de segundo orden tanto para el t\u00e9rmino convectivo como para el t\u00e9rmino difusivo) de capturarlas. Ya con el conocimiento de los elementos de *estabilidad, consistencia* y *convergencia* dados en el numeral anterior, se pueden plantear esquemas de discretizaci\u00f3n que cumplan con dichas restricciones.\n\nAhora, se va a resolver num\u00e9ricamente esta ecuaci\u00f3n utilizando un esquema de diferencias finitas centrales para el t\u00e9rmino difusivo y hacia atr\u00e1s, forward, para el t\u00e9rmino advectivo (seg\u00fan lo indicado para el esquema *Upwind*). Discretizando la ecuaci\u00f3n dada por los esquemas indicados, se llega a:\n\n$$\\frac{1}{h^2} \\left[ \\phi(x_{i-1})-2\\phi(x_i)+\\phi(x_{i+1})\\right]-\\frac{\\beta}{h}\\left[ \\phi(x_{i})-\\phi(x_{i-1})\\right]=0$$\n\norganizando t\u00e9rminos,\n\n$$(1+\\beta h)\\phi(x_{i-1})-(2+\\beta h)\\phi(x_i)+\\phi(x_{i+1})=0$$\n\nSe realiza el mismo procedimiento anterior para determinar el sistema de ecuaciones resultantes. A continuaci\u00f3n se presenta una codificaci\u00f3n del esquema *Upwind* propuesta por el autor.\n\n\n```python\n# Esquema Upwind\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\ndef exacta(x):\n return (np.exp(beta * x) - 1) / (np.exp(beta) - 1)\n\n# establecimiento de las condiciones de contorno a la izquierda\n\na0 = 0\nu0 = 0\n\n# establecimiento de las condiciones de contorno a la derecha\n\na1 = 1\nu1 = 1\n\n# Ingreso de datos\nn = int(input(\"Ingrese la cantidad de nodos: \"))\nbeta = float(input(\"ingrese el valor de la velocidad, beta: \"))\n\n# Inicializaci\u00f3n de los diferentes arreglos a emplear\nA = np.zeros((n-1, n-1)) # matriz de coeficientes\nb = np.zeros((n-1, 1)) # vector de t\u00e9rminos independietes\nu_upw = np.zeros((n-1, 1)) # vector de inc\u00f3gnitas\nuex = np.zeros((n-1,1)) # vector de la soluci\u00f3n exacta\n\nh = (a1 - a0) / n # establece el tama\u00f1o de paso Delta x\nx = np.linspace(a0, a1, n+1) # Genera n+1 puntos entre a0 y a1 igualmente espaciados\n\n# ensamblaje de la matriz de coeficientes y establecimiento de las c.c.\nfor i in range(1,n-1):\n A[i-1, i] = 1 # phi{i+1}\n A[i-1, i-1] = -2 - beta * h # phi{i}\n A[i, i-1] = 1 + beta * h # phi{i-1}\n b[i] = 0\n\nA[n-2,n-2] = -2 - beta * h\nb[0] = -(1 + beta * h) * u0\nb[n-2] = -1 * u1\n\n# Soluci\u00f3n del SEL Au = b\nu_upw = np.linalg.solve(A, b)\n\n# Adicionando las CC\nu_upw = np.insert(u_upw,0,u0) # c.c. a laizquierda\nu_upw = np.insert(u_upw,n,u1) # c.c. a la derecha\n\nplt.xlabel (r\"$x$\")\nplt.ylabel (r\"$\\phi(x)$\")\nplt.title (r\"Exacta vs Upwind\")\nplt.plot(x, u_upw,'r', label='Upwind')\nplt.plot(x, exacta(x),'b', label='Exacta')\nplt.legend(loc='center left')\nplt.grid(True)\nplt.show()\n```\n\n### Esquema de Viscosidad artificial\n\nEl esquema de [viscosidad artificial](https://nptel.ac.in/courses/112/104/112104030/) consiste de la inclusi\u00f3n, de forma consistente, de un t\u00e9rmino que adiciona difusi\u00f3n artificial a la ecuaci\u00f3n, y est\u00e1 dado como:\n\n\n\\begin{equation*}\n\\begin{split}\n-\\beta\\frac{d\\phi}{dx}+\\frac{d^2\\phi}{dx^2}+\\nu_{ex}\\frac{d^2\\phi}{dx^2}=0\n\\end{split}\n\\label{eq:Ec6_46} \\tag{6.46}\n\\end{equation*}\n\njuntando t\u00e9rminos, queda\n\n\n\\begin{equation*}\n\\begin{split}\n-\\beta\\frac{d\\phi}{dx}+(1+\\nu_{ex}) \\frac{d^2\\phi}{dx^2}=0\n\\end{split}\n\\label{eq:Ec6_47} \\tag{6.47}\n\\end{equation*}\n\ndonde $\\nu_{ex}=\\frac{1}{2}\\beta h$\n\nEmpleando un esquema de discretizaci\u00f3n de diferencias finitas central para la segunda derivada y backward para el t\u00e9rmino de la primera derivada, como en upwind, se llega al siguiente esquema iterativo\n\n\n\\begin{equation*}\n\\begin{split}\n-(2+\\beta h) \\phi_{i+1}+4(1+\\beta h) \\phi_{i}-(2+3\\beta h) \\phi_{i-1}\n\\end{split}\n\\label{eq:Ec6_48} \\tag{6.48}\n\\end{equation*}\n\n\n```python\n# Esquema Viscosidad artificial\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\ndef exacta(x):\n return (np.exp(beta * x) - 1) / (np.exp(beta) - 1)\n\n# establecimiento de las condiciones de contorno a la izquierda\n\na0 = 0\nu0 = 0\n\n# establecimiento de las condiciones de contorno a la derecha\n\na1 = 1\nu1 = 1\n\n# Ingreso de datos\nn = int(input(\"Ingrese la cantidad de nodos: \"))\nbeta = float(input(\"ingrese el valor de la velocidad, beta: \"))\n\n# Inicializaci\u00f3n de los diferentes arreglos a emplear\nA = np.zeros((n-1, n-1)) # matriz de coeficientes\nb = np.zeros((n-1, 1)) # vector de t\u00e9rminos independietes\nu_visc = np.zeros((n-1, 1)) # vector de inc\u00f3gnitas\nuex = np.zeros((n-1,1)) # vector de la soluci\u00f3n exacta\n\nh = (a1 - a0) / n # establece el tama\u00f1o de paso Delta x\nx = np.linspace(a0, a1, n+1) # Genera n+1 puntos entre a0 y a1 igualmente espaciados\nbh = beta * h\n\n# ensamblaje de la matriz de coeficientes y establecimiento de las c.c.\nfor i in range(1,n-1):\n A[i-1, i] = -(2 + bh) # phi{i+1}\n A[i-1, i-1] = 4 * (1 + bh) # phi{i}\n A[i, i-1] = -(2 + 3 * bh) # phi{i-1}\n b[i] = 0\n\nA[n-2,n-2] = 4 * (1 + bh)\nb[0] = (2 + 3 * bh) * u0\nb[n-2] = (2 + bh) * u1\n\n# Soluci\u00f3n del SEL Au = b\nu_visc = np.linalg.solve(A, b)\n\n# Adicionando las CC\nu_visc = np.insert(u_visc,0,u0) # c.c. a laizquierda\nu_visc = np.insert(u_visc,n,u1) # c.c. a la derecha\n\nplt.xlabel (r\"$x$\")\nplt.ylabel (r\"$\\phi(x)$\")\nplt.title (r\"Exacta vs Viscosidad artificial\")\nplt.plot(x, u_visc,'r', label='Viscosidad artificial')\nplt.plot(x, exacta(x),'b', label='Exacta')\nplt.legend(loc='center left')\nplt.grid(True)\nplt.show()\n```\n\nAhoroa, visualizando las tres soluciones en una \u00fanica gr\u00e1fica\n\n\n```python\nplt.xlabel (r\"$x$\")\nplt.ylabel (r\"$\\phi(x)$\")\nplt.title (r\"Exacta vs Esquemas de DFF\")\nplt.plot(x, exacta(x),'b', label='Exacta')\nplt.plot(x, u_lf,'g', label='Leapfrog')\nplt.plot(x, u_upw,'k', label='Upwind')\nplt.plot(x, u_visc,'r', label='Viscosidad artificial')\nplt.legend(loc='center left')\nplt.grid(True)\nplt.show()\n```\n\n### Ecuaci\u00f3n de convecci\u00f3n difusi\u00f3n dependiente del tiempo\n\nSea la ecuaci\u00f3n de [Convecci\u00f3n lineal](https://en.wikipedia.org/wiki/Advection) en 1D:\n\n$$\\frac{d\\phi}{dt}+c \\frac{d\\phi}{dx}=0$$\n\nDetermine el esquema de diferencias finitas empleando:\n\n- Euler Expl\u00edcito\n\n\n- Euler Impl\u00edcito\n\ny realice un c\u00f3digo en python que resuelva dicha ecuaci\u00f3n, sometida a las siguientes condiciones de contorno e iniciales, para $x\\in [0,2]$:\n\n$$\\phi(x_i^{(0)})=2; \\quad 0.5 \\le x_i \\le 1$$\n\ny \n\n$$\\phi(x_i^{(0)})=1; \\quad \\text{ en el resto de puntos } x_i $$\n\nEsta es la llamada funci\u00f3n \"sombrero\", que gr\u00e1ficamente quedar\u00eda:\n\n

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Fuente: Propio
\n\n\n```python\n\n```\n", "meta": {"hexsha": "3eb6b23474acaf2d6bf17d056e0ec249b21a2d1e", "size": 78705, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Sesion06_DiffFin.ipynb", "max_stars_repo_name": "carlosalvarezh/Modelos_Matematicos", "max_stars_repo_head_hexsha": "be46d3961ccd6a1e47fdd3d9a5409cafd2bc97c1", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Sesion06_DiffFin.ipynb", "max_issues_repo_name": "carlosalvarezh/Modelos_Matematicos", "max_issues_repo_head_hexsha": "be46d3961ccd6a1e47fdd3d9a5409cafd2bc97c1", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Sesion06_DiffFin.ipynb", "max_forks_repo_name": "carlosalvarezh/Modelos_Matematicos", "max_forks_repo_head_hexsha": "be46d3961ccd6a1e47fdd3d9a5409cafd2bc97c1", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-09-18T14:56:46.000Z", "max_forks_repo_forks_event_max_datetime": "2021-09-18T14:56:46.000Z", "avg_line_length": 41.7976633032, "max_line_length": 7849, "alphanum_fraction": 0.5595070199, "converted": true, "num_tokens": 20634, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621764862150636, "lm_q2_score": 0.795658104908603, "lm_q1q2_score": 0.44730027764605484}} {"text": "# Overriding root-finding in the generated C++ code -- Chemical kinetics\nThis is an advanced notebook exploring how the user can override the root-finding routine in the native C++ code which is exported.\n\nIf you only want to do some simple modelling of chemical kinetics you may want look into [ChemPy](https://github.io/bjodah/chempy) or [Cantera](http://www.cantera.org/docs/sphinx/html/index.html) (Cantera is more advanced and is the preferred choice when extensive thermodynamic data is available for the reacting species).\n\n\n```python\nfrom operator import mul\nfrom functools import reduce\nimport subprocess\nimport numpy as np\nimport sympy as sp\nimport matplotlib.pyplot as plt\nfrom pyodesys.results import Result\nfrom pyodesys.symbolic import SymbolicSys\nfrom pyodesys.native import native_sys\nfrom pyodesys.native.util import parse_standalone_output\nfrom _chem_kinet import get_odesys\nsp.init_printing()\n%matplotlib inline\n```\n\n\n```python\n# Fe+3 + SCN- <-> FeSCN+2\nstoich_reac, stoich_prod = [(1, 1, 0), (0, 0, 1)], [(0, 0, 1), (1, 1, 0)]\nkineticsys = get_odesys(stoich_reac, stoich_prod, 'Fe+3 SCN- FeSCN+2'.split(), SymbolicSys,\n steady_state_root=True, latex_names=['%s' % s for s in 'Fe^{3+} SCN^- FeSCN^{2+}'.split()])\n```\n\n\n```python\ndef integrate_and_plot(odesys, plot=True, **kwargs):\n tend = 2\n result = odesys.integrate(tend, [1e-2, 2e-3, 0], [800, 8], integrator='cvode', **kwargs)\n if plot:\n fig, axes = plt.subplots(1, 2, figsize=(14, 4))\n if result.xout[-1] != tend:\n axes[0].axvline(result.xout[-1], linestyle='--', label='t = %.4f' % result.xout[-1])\n result.plot(ax=axes[0])\n result.plot(ax=axes[1], deriv=True)\n axes[1].set_yscale('symlog', linthreshy=1e-9)\n axes[1].axhline(1e-9, linestyle='--')\n axes[1].axhline(-1e-9, linestyle='--')\n for ax in axes:\n ax.set_xlim([0, tend])\n return result\n```\n\n\n```python\nintegrate_and_plot(kineticsys)\n```\n\n\n```python\nintegrate_and_plot(kineticsys, atol=1e-14, rtol=1e-14)\n```\n\n\n```python\nintegrate_and_plot(kineticsys, atol=1e-14, rtol=1e-14, return_on_root=True)\n```\n\n\n```python\nkineticsys.roots\n```\n\n\n```python\nnative_override = {\n 'p_nroots': \"\"\" return 1; \"\"\",\n 'p_roots': \"\"\"\n const indextype ny = get_ny();\n std::vector f(ny);\n realtype tot=0.0;\n rhs(x, y, &f[0]);\n for (indextype i=0; inrev++;\n return AnyODE::Status::success;\n\"\"\"\n}\nnative_extend={\n 'p_constructor': [\n 'if (special_settings.size() != 1) std::cerr << \"len(special_settings) != 1\" << std::endl;'\n ]\n}\nnativesys = native_sys['cvode'].from_other(\n kineticsys, namespace_override=native_override, namespace_extend=native_extend)\nfor path in nativesys._native._written_files:\n if path.endswith('.cpp'):\n print(path)\n print('...\\n' + ''.join(open(path).readlines()[-20:]))\n print(\"\")\n```\n\nThere are some linking issues with boost's program options in the below (commented) cells.\n\n\n```python\n#standalone_prog = nativesys.as_standalone('chem_kinet', compile_kwargs=dict(options=['warn', 'pic', 'openmp', 'debug']))\n#standalone_prog\n```\n\n\n```python\n#p = subprocess.Popen([standalone_prog, '--return-on-root', '1', '--special-settings', '1000'],\n# stdin=subprocess.PIPE, stdout=subprocess.PIPE, stderr=subprocess.PIPE)\n#out, err = p.communicate(input='2 1e-2 2e-3 0 800 8 0 0 0 0 1'.encode('utf-8'))\n#retc = p.wait()\n#assert retc == 0\n#print(err.decode('utf-8'))\n```\n\n\n```python\n#res_sa, = [Result(*args, kineticsys) for args in parse_standalone_output(out.decode('utf-8').split('\\n'))]\n#res_sa.plot()\n```\n\n## Time to reach steady state\nIf we define steady state to occur when the change in chemical concentrations is below a certain threshold, then the obtained time will depend on that threshold. Here we investigate how that choice affects the answer (with respect to numerical precision etc.)\n\n\n```python\nnative = native_sys['cvode'].from_other(kineticsys, namespace_override=native_override)\n```\n\n\n```python\ndef plot_tss_conv(factor, tols, ax):\n tol_kw = dict(plot=False, return_on_root=True, nsteps=2000, special_settings=[factor])\n tss = [integrate_and_plot(native, atol=tol, rtol=tol, **tol_kw).xout[-1] for tol in tols]\n ax.semilogx(tols, tss, label=factor)\n```\n\n\n```python\nfig, ax = plt.subplots(figsize=(14, 6))\ntols = np.logspace(-15, -10, 200)\nfor factor in [1e2, 1e3, 1e4, 1.1e4, 1e5, 1e6, 1e7]:\n plot_tss_conv(factor, tols, ax)\nax.legend()\n```\n", "meta": {"hexsha": "34f86ec603d41744c603fee19255bd687de5a47f", "size": 7531, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/_native_override_chemical_kinetics.ipynb", "max_stars_repo_name": "slayoo/pyodesys", "max_stars_repo_head_hexsha": "8e1afb195dadf6c6f8e765873bc9dd0fae067c39", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": 82, "max_stars_repo_stars_event_min_datetime": "2015-09-29T16:51:03.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-02T13:26:50.000Z", "max_issues_repo_path": "examples/_native_override_chemical_kinetics.ipynb", "max_issues_repo_name": "slayoo/pyodesys", "max_issues_repo_head_hexsha": "8e1afb195dadf6c6f8e765873bc9dd0fae067c39", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": 28, "max_issues_repo_issues_event_min_datetime": "2015-09-29T14:40:45.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-18T19:29:50.000Z", "max_forks_repo_path": "examples/_native_override_chemical_kinetics.ipynb", "max_forks_repo_name": "slayoo/pyodesys", "max_forks_repo_head_hexsha": "8e1afb195dadf6c6f8e765873bc9dd0fae067c39", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": 13, "max_forks_repo_forks_event_min_datetime": "2016-03-18T14:00:39.000Z", "max_forks_repo_forks_event_max_datetime": "2021-09-17T13:54:29.000Z", "avg_line_length": 30.6138211382, "max_line_length": 329, "alphanum_fraction": 0.5557030939, "converted": true, "num_tokens": 1409, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593171945417, "lm_q2_score": 0.7185943925708562, "lm_q1q2_score": 0.44729577493948164}} {"text": "# Molecular graph generation with PyTorch and PyGeometric\n> We use [GraphVAE](https://arxiv.org/abs/1802.03480) for molecular generation with one shot generation of a probabilistic graph with predefined maximum size. \n\n- toc: true \n- badges: true\n- comments: false\n- author: Anirudh Jain\n- categories: [graph generation, pytorch, pygeometric, tutorial]\n\n# Requirements\n\nThe following packages need to be installed:\n- rdkit\n- pytorch\n- torch_geometric\n- networkx\n\n\n```python\n#collapse-hide\n\n#Initial imports\n\nimport numpy as np\nimport torch\nimport matplotlib.pyplot as plt\nfrom glob import glob\nimport tqdm\nfrom rdkit import Chem \nfrom rdkit.Chem import AllChem\nfrom rdkit.Chem import Draw\nfrom rdkit.Chem.Draw import IPythonConsole\nfrom torch.utils.data import Dataset, DataLoader\nimport torch\nfrom torch_geometric.data import Batch\nfrom torch.utils.data import random_split\n```\n\n# Introduction\n\nWe represent a molecule as graph $G = (\\mathcal{X, A})$ using PyGeometric framework. Each molecule is represented by a feature matrix $\\mathcal{X}$ and adjacency matrix $\\mathcal{A}$. We use QM9 dataset from [MoleculeNet:A Benchmark for Molecular Machine Learning](https://arxiv.org/abs/1703.00564) implemented in `torch_geometric.datasets.QM9`. PyGeometric relies on rdkit to process the SMILES string and convert them into graphs.\n\nWe modify the data processing script in two ways:\n- We strip hydrogen atoms from the molecules to keep only the heavy atoms\n- We kekulize the molecules to convert aromatic rings to Kekule form\n\nThe modified script can be found [here](https://gist.github.com/sponde25/7dfa5492c21c007cf1e60a02dced1334)\n\nAfter processing the dataset, we have a set of molecules with 4 heavy atoms (C, N, O, F) and 3 bond types (SINGLE, DOUBLE and TRIPLE) with maximum graph size of 9. \n\nThe decoder outputs the graph as one-hot encoded vectors for atoms `[9 x 5]` and bonds `[9 x 4]`. The label 0 represents empty atom or edge. \n\n\n```python\n#Imports for data pre-processing\n\nimport torch_geometric\nfrom qm9_modified import QM9\nfrom torch_geometric.utils.convert import to_networkx\nimport networkx\n```\n\n\n```python\n# Setting up variables for the dataset\n\nMAX_ATOM = 5 \nMAX_EDGE = 4 \npath = '/scratch/project_2002655/datasets/qm9_noH' # Change the path for your local directory structure\ndataset = QM9(path)\n\n# Store the max. graph size\nMAX_N = -1\nfor data in dataset:\n if MAX_N < data.x.shape[0]: MAX_N = data.x.shape[0]\nMAX_E = int(MAX_N * (MAX_N - 1))\nprint('MAX ATOMS: {}'.format(MAX_N)) # Maximum number of atoms in a graph in the dataset\nprint('MAX EDGE: {}'.format(MAX_E)) # Corresponding size of upper triangle adjacency matrix \n```\n\n MAX ATOMS: 9\n MAX EDGE: 72\n\n\n`torch_geometric` stores the graph as `torch_geometric.data.Data` and we generate the one-hot representation of the graph $G$ as described above. For each graph $G$, we create a vector $\\mathcal{X}$ as one-hot encoded for atom of dimension `[MAX_N x MAX_ATOM]` and vector bond of dimension `[MAX_E x MAX_EDGE]`.\n\n\n\n\n```python\n# We create a matrix to map the index of the edge vector $\\mathcal{A}$ to the upper triangular adjacency matrix.\n\nindex_array = torch.zeros([MAX_N, MAX_N], dtype=int)\nidx = 0\nfor i in range(MAX_N):\n for j in range(MAX_N):\n if i < j:\n index_array[i, j] = idx\n idx+=1\n\nprint(index_array)\n```\n\n tensor([[ 0, 0, 1, 2, 3, 4, 5, 6, 7],\n [ 0, 0, 8, 9, 10, 11, 12, 13, 14],\n [ 0, 0, 0, 15, 16, 17, 18, 19, 20],\n [ 0, 0, 0, 0, 21, 22, 23, 24, 25],\n [ 0, 0, 0, 0, 0, 26, 27, 28, 29],\n [ 0, 0, 0, 0, 0, 0, 30, 31, 32],\n [ 0, 0, 0, 0, 0, 0, 0, 33, 34],\n [ 0, 0, 0, 0, 0, 0, 0, 0, 35],\n [ 0, 0, 0, 0, 0, 0, 0, 0, 0]])\n\n\nWe process the `torch_geometric.dataset` to generate matrices $\\mathcal{X}$ and $\\mathcal{A}$ which act as the ground truth for our decoder. We will also setup utility functions to convert between the vector representation $(\\mathcal{X}, \\mathcal{A})$ and `torch_geometric.data` representation $\\mathcal{G}$. We use the following key for atoms and bonds:\n```\n C: 1 SINGLE: 1\n N: 2 DOUBLE: 2\n O: 3 TRIPLE: 3\n F: 4\n```\n`0` is the placeholder label for empty entry.\n\n\n```python\n# Initialize the labels with -1\nedge_labels = torch.ones(len(dataset), MAX_E) * -1\natom_labels = torch.ones(len(dataset), MAX_N) * -1\nidx = 0\nfor data in dataset:\n edge_attr = data.edge_attr # One hot encoded bond labels\n edge_index = data.edge_index # Bond indices as one hot adjacency list\n upper_index = edge_index[0] < edge_index[1] # Bond indices as upper triangular adjacency matrix\n _, edge_label = torch.max(edge_attr, dim=-1)# Bond labels from one hot vectors\n x = data.x[:, 1:5] # One hot encoded atom labels\n _, atom_label = torch.max(x, dim=-1) # Atom labels from one hot vectors\n # Expand the label vectors to size [MAX_N x MAX_ATOM] and [MAX_E x MAX_EDGE]\n atom_labels[idx][:len(atom_label)] = atom_label\n a0 = edge_index[0,upper_index]\n a1 = edge_index[1,upper_index]\n up_idx = index_array[a0, a1]\n edge_labels[idx][up_idx] = edge_label[upper_index].float()\n idx += 1\n\natom_labels += 1\nedge_labels += 1\n```\n\nNow that we have the dataset represented as $(\\mathcal{X}, \\mathcal{A})$ let's plot some graphs to visually check if the molecules are as we expected. We use `rdkit` to plot the molecules which does a lot of having lifting for us. The function `graphToMol` takes in the vectors $(\\mathcal{X}, \\mathcal{A})$ and returns an object of type `rdkit.Mol`. We can also obtain visualizations for the graphs $\\mathcal{G}$ by using `torch_geometric.utils.convert.to_networkx` and then ploting the netowrkx graph. But `rdkit` plots the molecules in a canonical orientation and is built to minimize intramolecular clashes, i.e. to maximize the clarity of the drawing.\n\n\n```python\n#collapse-hide\n\ndef get_index(index, index_array):\n for i in range(9):\n for j in range(9):\n if i < j:\n if(index_array[i, j] == index):\n return [i, j]\n\ndef graphToMol(atom, edge):\n\n possible_atoms = {\n 0: 'H',\n 1: 'C',\n 2: 'N',\n 3: 'O',\n 4: 'F'\n }\n possible_edges = {\n 1: Chem.rdchem.BondType.SINGLE, \n 2: Chem.rdchem.BondType.DOUBLE, \n 3: Chem.rdchem.BondType.TRIPLE\n }\n max_n = 9\n \n mol = Chem.RWMol()\n rem_idxs = []\n for a in atom: \n atom_symbol = possible_atoms[a.item()]\n mol.AddAtom(Chem.Atom(atom_symbol))\n for a in mol.GetAtoms():\n if a.GetAtomicNum() == 1:\n rem_idxs.append(a.GetIdx())\n for i, e in enumerate(edge):\n e = e.item()\n if e != 0:\n a0, a1 = get_index(i, index_array)\n if a0 in rem_idxs or a1 in rem_idxs:\n return None\n bond_type = possible_edges[e]\n mol.AddBond(a0, a1, order=bond_type)\n rem_idxs.sort(reverse=True)\n for i in rem_idxs:\n mol.RemoveAtom(i)\n return mol\n```\n\n\n```python\n# We pick 9 random molecules from QM9 dataset to plot\nmols = []\nfor i in np.random.randint(0, 100, size=9):\n mols.append(graphToMol(atom_labels[i], edge_labels[i]))\nChem.Draw.IPythonConsole.ShowMols(mols)\n```\n\nThe final step is to create a custom dataset for PyTorch which combines the two representations $\\mathcal{G}$ and $(\\mathcal{X}, \\mathcal{A})$. The dataset returns a list containing the graph, atom label and edge label. We also define a `collate_fn` which creates mini batches for the data loader that we will need for training and testing our model.\n\n\n```python\n#collapse-hide\n\n# Utility function to combine the torch_geometric.data and torch.tensor into one mini batch\ndef collate_fn(datas):\n graphs, atoms, edges = [], [], []\n for data in datas:\n graph, atom, edge = data\n graphs.append(graph)\n atoms.append(atom)\n edges.append(edge)\n graphs = Batch.from_data_list(graphs)\n atoms = torch.stack(atoms)\n edges = torch.stack(edges)\n return graphs, atoms, edges\n\n# Returns the dataloader with a given batch size\ndef get_dataloader(dataset, batch_size=32, shuffle=True):\n return DataLoader(dataset, batch_size=batch_size, shuffle=shuffle, collate_fn=collate_fn)\n\n# Create the dataset from the QM9 dataset from PyGeometric and the label matrices\nclass GraphTensorDataset(Dataset):\n\n def __init__(self):\n\n self.graph_dataset = QM9('/scratch/project_2002655/datasets/qm9_noH')\n self.atom_labels = torch.load('/scratch/project_2002655/drug_discovery/saved/atom_labels_noH.tensor')\n self.edge_labels = torch.load('/scratch/project_2002655/drug_discovery/saved/edge_labels_noH.tensor')\n\n def __len__(self):\n return len(self.graph_dataset)\n\n def __getitem__(self, idx):\n if torch.is_tensor(idx):\n idx = idx.tolist()\n\n graphs = self.graph_dataset[idx]\n atoms = self.atom_labels[idx]\n edges = self.edge_labels[idx]\n\n return graphs, atoms, edges\n \n # return the subset of data with the specified size\n def get_splits(self, train_size, test_size, cv_size=None):\n total_data = len(self.graph_dataset)\n if cv_size is not None:\n trainset, testset = random_split(self, [train_size+cv_size, total_data - train_size - cv_size])\n trainset, cvset = random_split(trainset, [train_size, cv_size])\n testset, _ = random_split(testset, [test_size, len(testset) - test_size])\n self.trainset = trainset\n self.cvset = cvset\n self.testset = testset\n return self.trainset, self.testset, self.cvset\n else:\n trainset, testset = random_split(self, [train_size, total_data - train_size])\n testset, _ = random_split(testset, [test_size, len(testset) - test_size])\n self.trainset = trainset\n self.testset = testset\n return self.trainset, self.testset\n\n```\n\nNow we wrap up this section by creating the train and test loaders with a mini batch of size `32`. We train the model on a subset of 500 molecules and test the reconstruction on 10000 molecules. To monitor training we look at three metrics averaged over the minibatch:\n- cross-entropy loss over the atom and edge labels\n- Atom label accuracy\n- Edge label accuracy \n\n\n```python\ndataset = GraphTensorDataset()\ntrain_size = int(500)\ntest_size = int(10000)\ntorch.manual_seed(0) # We set the random seed before the random split so we can get the same subset for reproducibility \ntrainset, testset = dataset.get_splits(train_size, test_size)\ntrainloader = get_dataloader(trainset)\ntestloader = get_dataloader(testset)\n```\n\n# Model\n\nOur generative model is a VAE with an encoder $q_\\phi (\\textbf{z} | \\mathcal{G})$ and a decoder $p_\\theta (\\mathcal{X}, \\mathcal{A} | \\textbf{z})$ parameterized by $\\phi$ and $\\theta$ respectively. We place a prior over the latent vector $\\textbf{z} \\sim \\mathcal{N}(0, I)$ which act as a regularizer to our model. Our goal is to map the graph $\\mathcal{G}$ into the latent vector $\\textbf{z}$ and generate new graphs by using the decoder to map new latent vectors sampled from the prior.\n\n***Encoder***: For a graph $\\mathcal{G}(\\mathcal{V}, \\mathcal{E})$ with upto N nodes, we aim to map the graph to a latent vector $\\textbf{z} \\in \\mathbb{R}^D$. We use graph convolution network(GCN) to learn graph level feature vector $h_{\\mathcal{G}}$. For each node $\\mathcal{v} \\in \\mathcal{V}$, GCN update the node level features as,\n\\begin{equation}\n h_{\\mathcal{v}} = \\sum_{\\mathcal{u} \\in Nbd(\\mathcal{v}, \\mathcal{G})}\\Phi (W_l^T X_{\\mathcal{u}})\n\\end{equation}\nwhere $Nbd(\\mathcal{v}, \\mathcal{G})$ is the neighborhood of node $\\mathcal{v}$ in graph $\\mathcal{G}$, $W_l$ are the shared weights analogous to the kernel in convolution layer and $X_\\mathcal{u}$ is the feature vector of node $\\mathcal{u}$ with $\\Phi$ as the non-linearity.\n\nAfter 2 graph convolution layers, we create a graph level vector representation by taking the mean of all node feature vectors $h_{\\mathcal{v}}$.\n\\begin{equation}\n h_{\\mathcal{G}} = 1 / |\\mathcal{V}| \\sum_{\\mathcal{v} \\in \\mathcal{V}} h_{\\mathcal{v}}\n\\end{equation}\nNow we use fully connected layers to learn the latent vector $\\textbf{z}$ with the inference model as,\n\\begin{equation}\n q(\\textbf{z}|\\mathcal{G}) = \\mathcal{N}(\\textbf{z}|\\mu(h_{\\mathcal{G}}), \\sigma^2(h_{\\mathcal{G}}))\n\\end{equation}\nwhere $\\mu(h_{\\mathcal{G}})$ and $\\sigma^2(h_{\\mathcal{G}})$ are parameterized by neural networks to learn the latent space.\n\n\n\n***Decoder***: The primary problem with graph generation is that order of nodes can be arbritary which makes comparing two graphs a combatorial problem. We sidestep this problem by framing the graph generation problem as multi class classification problem. Keeping in the spirit of a toy example, we consider two independent classification problems. \n\nWe train a decoder to predict atom and edge label vectors $(\\mathcal{X}, \\mathcal{A})$ respectively. We can train the model by using the standard VAE loss,\n\\begin{equation}\n \\mathcal{L} = \\alpha_{\\mathcal{a}} \\mathcal{L}_{\\mathcal{a}} + \\alpha_\\mathcal{e} \\mathcal{L}_{\\mathcal{e}} + \\alpha_{KL} KL[q(\\textbf{z}|\\mathcal{G}) | p(\\textbf{z})]\n\\end{equation}\n\n$\\mathcal{L}_{\\mathcal{a}}$ and $\\mathcal{L}_{\\mathcal{e}}$ are the cross entropy loss for multiclass vectors for atoms and edges respectively. This restricts the decoder to memorize the ordering for the atom labels and independently learn the edges based only on the latent vector $\\textbf{z}$. We set $\\alpha_\\mathcal{a} = 1/|\\mathcal{X}|$, $\\alpha_\\mathcal{e} = 1/|{\\mathcal{A}}|$ and $\\alpha_KL = 1/ 128$ so all three loses are on the same scale. $\\mathcal{X}$ is the number of atoms in a mini batch, $|\\mathcal{A}|$ is the number of edges in a mini batch and we set the dimension of the latent vector as $128$.\n\nWe use a single sample from the latent distribution to train the VAE and 100 samples during testing.\n\n\n\n\n```python\n#collapse-hide\n\nimport torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport torch_geometric\nfrom torch_geometric.nn import GCNConv, BatchNorm, GlobalAttention\nfrom torch_geometric.utils import add_self_loops \n\nclass VAE(nn.Module):\n def __init__(self, max_n, max_atoms, max_edges, in_features):\n super(VAE, self).__init__()\n n_output = int(max_n * (max_n - 1) / 2)\n self.n_output = n_output\n self.max_n = max_n\n self.max_atoms = max_atoms\n self.max_edges = max_edges\n\n self.gc1 = GCNConv(in_features, 32)\n self.gc2 = GCNConv(32, 64)\n self.gc1_bn = nn.BatchNorm1d(32)\n self.gc2_bn = nn.BatchNorm1d(64)\n\n attn1 = nn.Linear(64, 1)\n attn2 = nn.Linear(64, 64)\n self.graph_pool = GlobalAttention(attn1, attn2)\n\n self.fc1 = nn.Linear(64, 128)\n self.fc1_bn = nn.BatchNorm1d(128)\n\n self.mu = nn.Linear(128, 128)\n self.logv = nn.Linear(128, 128)\n\n self.dec1 = nn.Linear(128, 128)\n self.dec1_bn = nn.BatchNorm1d(128)\n self.dec2 = nn.Linear(128, 256)\n self.dec2_bn = nn.BatchNorm1d(256)\n self.dec3 = nn.Linear(256, 512)\n self.dec3_bn = nn.BatchNorm1d(512)\n\n self.atoms = nn.Linear(512, max_atoms*max_n)\n self.edges = nn.Linear(512, max_edges*n_output)\n\n def encode(self, data):\n x, edge_index, batch = data.x, data.edge_index, data.batch\n edge_index, _ = add_self_loops(edge_index)\n\n x = self.gc1(x, edge_index)\n x = F.relu(self.gc1_bn(x))\n x = self.gc2(x, edge_index)\n x = F.relu(self.gc2_bn(x))\n #x = self.graph_pool(x, batch)\n #x = F.tanh(x)\n x = torch_geometric.nn.global_add_pool(x, data.batch)\n x = self.fc1(x)\n x = F.relu(self.fc1_bn(x))\n return self.mu(x), self.logv(x)\n\n def sample(self, mu, logv):\n eps = torch.randn_like(logv).to(mu.device)\n return mu + eps * torch.exp(logv)\n\n def decode(self, sample):\n x = self.dec1(sample)\n x = F.relu(self.dec1_bn(x))\n x = self.dec2(x)\n x = F.relu(self.dec2_bn(x))\n x = self.dec3(x)\n x = F.relu(self.dec3_bn(x))\n \n atoms = self.atoms(x)\n atoms = atoms.view(-1, self.max_n, self.max_atoms)\n atoms = F.softmax(atoms, dim=-1)\n\n edges = self.edges(x)\n edges = edges.view(-1, self.n_output, self.max_edges)\n edges = F.softmax(edges, dim=-1)\n return atoms, edges\n\n def forward(self, data, num_samples=1):\n mu, logv = self.encode(data)\n _atoms, _edges = [], []\n for i in range(num_samples):\n sample = self.sample(mu, logv)\n atoms, edges = self.decode(sample)\n _atoms.append(atoms)\n _edges.append(edges) \n return mu, logv, _atoms, _edges\n```\n\nNow that we have the dataloader and the model, we can setup utility functions `train` and `test` along with the metrics that we need to evaluate the model performance. We use multi class accuracy to monitor as an indicator for model performance. \n\n\n```python\n#collapse-hide\n\nfrom pathlib import Path\n\n# KL loss for the latent vector\ndef get_vae_loss(mu, logv):\n kl_loss = 0.5 * torch.sum(torch.exp(logv) + mu**2 - 1.0 - logv)\n return kl_loss/128\n\n# Reconstruction loss for atom and edge vectors\ndef get_recon_loss(atom_true, atom_prob, edge_true, edge_prob):\n ashape = atom_prob.shape\n num_atoms = (atom_true!=0).int().sum()\n atom_prob = torch.log(atom_prob)\n atom_loss = F.nll_loss(atom_prob.view(-1, ashape[-1]), \n atom_true.view(-1).long(), reduction='sum') / num_atoms\n eshape = edge_prob.shape\n edge_prob = torch.log(edge_prob)\n num_edges = (edge_true!=0).int().sum()\n edge_loss = F.nll_loss(edge_prob.view(-1, eshape[-1]), \n edge_true.view(-1).long(), reduction='sum') / num_edges\n return atom_loss, edge_loss\n\n# Return the correct predictions for multi class label classification\ndef get_accuracy(y_true, y_prob, threshold = 0.5):\n if len(y_prob.shape) <=2:\n y_pred = (y_prob >= threshold).int()\n else:\n _, y_pred = torch.max(y_prob, dim=-1)\n index = y_true != 0\n correct = y_true[index] == y_pred[index]\n return correct.int().sum(), index.int().sum()\n\n# Class to run and store training statistics we can monitor for model performance\nclass Experiment():\n def __init__(self, model, optimizer, path = './saved/tmp/'):\n self.model = model\n self.optimizer = optimizer\n self.path = path\n\n Path(path+'models/').mkdir(parents=True, exist_ok=True)\n Path(path+'metrics/').mkdir(parents=True, exist_ok=True)\n Path(path+'figs/').mkdir(parents=True, exist_ok=True)\n\n self.trainLoss = []\n self.trainAcc = []\n self.trainAUC = []\n self.trainLossBatch = []\n self.trainAccBatch = []\n self.betaBatch = []\n self.testLoss = []\n self.testAcc = []\n self.testAUC = []\n self.testLossBatch = []\n self.testAccBatch = []\n\n def train(self, dataloader, epoch, cyclic = False, device = 'cuda'):\n self.model.train()\n \n total_data = len(dataloader.dataset)\n total_batches = len(dataloader)\n batch_idx = epoch * total_batches\n if cyclic:\n batches_cycle = int(total_batches / 3)\n batches_half_cycle = int(batches_cycle / 2)\n betas = np.ones(batches_cycle)\n betas[:batches_half_cycle] = np.linspace(0, 1, num=batches_half_cycle)\n atom_correct = atom_total = edge_correct = edge_total = 0.\n atom_auc = edge_auc = 0.\n kl_loss = atom_loss = edge_loss = 0.\n beta = 1\n\n for data in dataloader:\n graph = data[0].to(device)\n atom_true = data[1].to(device)\n edge_true = data[2].to(device)\n batch_size = atom_true.shape[0]\n\n # Core Training\n self.optimizer.zero_grad()\n mu, logv, atom_probs, edge_probs = self.model(graph)\n _kl_loss = get_vae_loss(mu, logv)\n aloss = eloss = 0.\n for atom_prob, edge_prob in zip(atom_probs, edge_probs):\n _atom_loss, _edge_loss = get_recon_loss(atom_true, atom_prob, edge_true, edge_prob)\n aloss += _atom_loss\n eloss += _edge_loss\n C = len(atom_probs)\n _atom_loss = aloss / C\n _edge_loss = eloss / C\n _recon_loss = _atom_loss + _edge_loss\n if cyclic:\n beta_idx = batch_idx % batches_cycle\n beta = betas[beta_idx]\n loss = _kl_loss * beta + _recon_loss \n loss.backward()\n self.optimizer.step()\n batch_idx += 1\n\n with torch.no_grad():\n self.trainLossBatch.append([_kl_loss, _atom_loss, _edge_loss])\n self.betaBatch.append(beta)\n\n kl_loss += _kl_loss.item() * batch_size\n atom_loss += _atom_loss.item() * batch_size\n edge_loss += _edge_loss.item() * batch_size\n _atom_correct = _edge_correct = _atom_auc = _edge_auc = 0.\n C = len(atom_probs)\n for atom_prob, edge_prob in zip(atom_probs, edge_probs):\n acorrect, _atom_total = get_accuracy(atom_true, atom_prob)\n ecorrect, _edge_total = get_accuracy(edge_true, edge_prob)\n aauc = get_multi_auc(atom_true.cpu(), atom_prob.cpu(), atom_prob.shape[-1]) * batch_size\n eauc = get_multi_auc(edge_true.cpu(), edge_prob.cpu(), edge_prob.shape[-1]) * batch_size\n _atom_correct += acorrect\n _edge_correct += ecorrect\n _atom_auc += aauc \n _edge_auc += eauc \n atom_auc += _atom_auc.item() / C\n edge_auc += _edge_auc.item() / C\n atom_correct += _atom_correct.item() / C\n atom_total += _atom_total.item() \n edge_correct += _edge_correct.item() / C\n edge_total += _edge_total.item()\n\n self.trainAccBatch.append([_atom_correct/C/_atom_total, _edge_correct/C/_edge_total])\n \n self.trainLoss.append([kl_loss/total_data, atom_loss/total_data, edge_loss/total_data])\n self.trainAcc.append([atom_correct/atom_total, edge_correct/edge_total])\n self.trainAUC.append([atom_auc/total_data, edge_auc/total_data])\n\n def test(self, dataloader, epoch, num_samples = 1, device = 'cuda'):\n self.model.eval()\n\n with torch.no_grad():\n kl_loss = 0.\n atom_loss = 0.\n edge_loss = 0.\n total_data = len(dataloader.dataset)\n total_batches = len(dataloader)\n batch_idx = epoch * total_batches\n atom_correct = atom_total = edge_correct = edge_total = 0.\n atom_auc = edge_auc = 0.\n\n for data in dataloader:\n graph = data[0].to(device)\n atom_true = data[1].to(device)\n edge_true = data[2].to(device)\n batch_size = atom_true.shape[0]\n\n mu, logv, atom_probs, edge_probs = self.model(graph, num_samples)\n _kl_loss = get_vae_loss(mu, logv)\n aloss = eloss = 0.\n for atom_prob, edge_prob in zip(atom_probs, edge_probs):\n _atom_loss, _edge_loss = get_recon_loss(atom_true, atom_prob, edge_true, edge_prob)\n aloss += _atom_loss\n eloss += _edge_loss\n C = len(atom_probs)\n _atom_loss = aloss / C\n _edge_loss = eloss / C\n\n kl_loss += _kl_loss.item() * batch_size\n atom_loss += _atom_loss.item() * batch_size\n edge_loss += _edge_loss.item() * batch_size\n _atom_correct = _edge_correct = _atom_auc = _edge_auc = 0.\n C = len(atom_probs)\n for atom_prob, edge_prob in zip(atom_probs, edge_probs):\n acorrect, _atom_total = get_accuracy(atom_true, atom_prob)\n ecorrect, _edge_total = get_accuracy(edge_true, edge_prob)\n aauc = get_multi_auc(atom_true.cpu(), atom_prob.cpu(), atom_prob.shape[-1]) * batch_size\n eauc = get_multi_auc(edge_true.cpu(), edge_prob.cpu(), edge_prob.shape[-1]) * batch_size\n _atom_correct += acorrect\n _edge_correct += ecorrect\n _atom_auc += aauc \n _edge_auc += eauc\n atom_auc += _atom_auc.item() / C\n edge_auc += _edge_auc.item() / C\n atom_correct += _atom_correct.item() / C\n atom_total += _atom_total.item() \n edge_correct += _edge_correct.item() / C\n edge_total += _edge_total.item()\n\n self.testLossBatch.append([_kl_loss, _atom_loss, _edge_loss])\n self.testAccBatch.append([_atom_correct/_atom_total, _edge_correct/_edge_total])\n\n self.testLoss.append([kl_loss/total_data, atom_loss/total_data, edge_loss/total_data])\n self.testAcc.append([atom_correct/atom_total, edge_correct/edge_total])\n self.testAUC.append([atom_auc/total_data, edge_auc/total_data])\n\n def save_state(self, epoch):\n metrics = {}\n trainLoss = np.array(self.trainLoss)\n trainAcc = np.array(self.trainAcc)\n trainAUC = np.array(self.trainAUC)\n testLoss = np.array(self.testLoss)\n testAcc = np.array(self.testAcc)\n testAUC = np.array(self.testAUC)\n\n metrics['Train KL Loss'] = trainLoss[:, 0]\n metrics['Train Atom Loss'] = trainLoss[:, 1]\n metrics['Train Edge Loss'] = trainLoss[:, 2]\n\n metrics['Train Atom Acc'] = trainAcc[:, 0]\n metrics['Train Edge Acc'] = trainAcc[:, 1]\n\n metrics['Train Atom AUC'] = trainAUC[:, 0]\n metrics['Train Edge AUC'] = trainAUC[:, 1]\n\n metrics['Test KL Loss'] = testLoss[:, 0]\n metrics['Test Atom Loss'] = testLoss[:, 1]\n metrics['Test Edge Loss'] = testLoss[:, 2]\n\n metrics['Test Atom Acc'] = testAcc[:, 0]\n metrics['Test Edge Acc'] = testAcc[:, 1]\n\n metrics['Test Atom AUC'] = testAUC[:, 0]\n metrics['Test Edge AUC'] = testAUC[:, 1]\n\n torch.save(metrics, self.path + 'metrics/epoch_{}.metric'.format(epoch))\n\n batch_metrics = {}\n trainLossBatch = np.array(self.trainLossBatch)\n trainAccBatch = np.array(self.trainAccBatch)\n testLossBatch = np.array(self.testLossBatch)\n testAccBatch = np.array(self.testAccBatch)\n\n batch_metrics['Train KL Loss'] = trainLossBatch[:, 0]\n batch_metrics['Train Atom Loss'] = trainLossBatch[:, 1]\n batch_metrics['Train Edge Loss'] = trainLossBatch[:, 2]\n\n batch_metrics['Train Atom Acc'] = trainAccBatch[:, 0]\n batch_metrics['Train Edge Acc'] = trainAccBatch[:, 1]\n\n batch_metrics['Test KL Loss'] = testLossBatch[:, 0]\n batch_metrics['Test Atom Loss'] = testLossBatch[:, 1]\n batch_metrics['Test Edge Loss'] = testLossBatch[:, 2]\n\n batch_metrics['Test Atom Acc'] = testAccBatch[:, 0]\n batch_metrics['Test Edge Acc'] = testAccBatch[:, 1]\n \n torch.save(batch_metrics, self.path + 'metrics/batches.metric')\n \n torch.save(self.model.state_dict(), self.path + 'models/gvae_{}.model'.format(epoch))\n```\n\nWe can use many tricks from the VAE literature to improve performance and obtain better results. [Cyclic Annealing Schedule](https://arxiv.org/abs/1903.10145) for VAE training has been show to significantly improve performance on autoregressive NLP tasks such as language modeling, dialog response generation etc. We can train the model using cyclic annealing schedule by setting the flag `cyclic=True`.\n\nWe use the mean of 40 samples during testing to calculate the loss and accuracy on the test set.\n\n\n```python\nmodel = VAE(MAX_N, MAX_ATOM, MAX_EDGE, 13)\nlr = 1e-3\noptimizer = torch.optim.Adam(model.parameters(), lr=lr, betas=(0.9, 0.999))\nepochs = 100\ndevice = 'cuda'\nmodel = model.to(device)\npath = './saved/test/'\nexperiment = Experiment(model, optimizer, path)\n\nfor epoch in range(epochs):\n \n experiment.train(trainloader, epoch, cyclic=False)\n experiment.test(testloader, epoch, 40)\n experiment.save_state(epoch)\n```\n\n# Results\n\nWe train the model on the dataset of 500 molecules and test on a dataset of 10,000 molecules for 100 epochs. The small size of the trainset leads to a large generalization gap and the model overfits on trainset with the atom and edge cross entropy loss decreasing but a flat test loss. The test atom classification accuracy converges at around 80% while the edge classification accuracy converges at around 75%. We only consider the labels with non-zero entry in the ground truth to calculate accuracy.\n\n\n\nNow we consider metrics to check the generative performance for molecules. We sample $n_s = 10,000$ latent vectors $\\textbf{z}$ from the prior $\\mathcal{N}(0, I)$ and use the trained decoder to generate new molecules. We use the following three metrics to test the quality of the learned latent space:\n- Validity: Let list $|V|$ represent chemically valid molecules, validity is the ratio $|V|/n_s$\n- Uniqueness: The set of chemically valid molecules $U = set(V)$, uniqueness is the ratio $|U|/|V|$\n- Novelty: Novelty is the set of elements in $U$ not in the trainset $\\mathcal{D}_{train}$ defined as $1 - (U \\cap \\mathcal{D}_{train})/|U|$\n\nWe obtain the following results from our trained model:\n\n\n| Validity | Uniqueness | Novelty| \n|----------|-----------|---------|\n| 62.33%| 76.40% | 66.71% |\n\nEven the simple toy example can generate a substantial number of unique and novel valid molecules. This is without any graph matching in the loss function so the model can learn to predict nodes in arbritary order and is restricted by the order of nodes in the trainset. Also these results are obtained from a very small subset of data and is comparable to [GraphAF](https://arxiv.org/abs/2001.09382) SOTA method without validity checks during training (67% in Table 3).\n\n\n\n\n\nThe model predicts a lot of disconnected graphs and molecules with invalid chemical valency rules\n\n# Conclusion\n\nThe GraphVAE presents a simple model to learn a generative model over graphs and gives a reasonable performance in modelling the structure behind molecules. This gives us a simple but suprisingly effective baseline for structured generation. Several work has improved upon this baseline using [reinforcement learning](https://arxiv.org/abs/1806.02473) and [flows](https://arxiv.org/abs/2001.09382) for goal directed generation of molecules. Both of these methods use an oracle during autoregressive generation to prune chemically invalid additions to the sub-graph. Going forward, we will look at implementing the SOTA methods for improved baseline. We will also explore ways to develop algorithms for structured data without depending on an oracle but learn the underlying rules implicitly from the data.\n", "meta": {"hexsha": "2814cc8fa9adb907d95dd4a04f9b51cfb2e0d3cc", "size": 66293, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "_notebooks/2020-06-24-graphvae.ipynb", "max_stars_repo_name": "sponde25/approxbayes", "max_stars_repo_head_hexsha": "6e262b1df6d6ed57d944bc02cf5f8bdc3a6bfd55", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "_notebooks/2020-06-24-graphvae.ipynb", "max_issues_repo_name": "sponde25/approxbayes", "max_issues_repo_head_hexsha": "6e262b1df6d6ed57d944bc02cf5f8bdc3a6bfd55", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-09-28T03:24:56.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-28T03:24:56.000Z", "max_forks_repo_path": "_notebooks/2020-06-24-graphvae.ipynb", "max_forks_repo_name": "sponde25/approxbayes", "max_forks_repo_head_hexsha": "6e262b1df6d6ed57d944bc02cf5f8bdc3a6bfd55", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 76.6393063584, "max_line_length": 25499, "alphanum_fraction": 0.7183262185, "converted": true, "num_tokens": 8211, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791787121629466, "lm_q2_score": 0.6584175072643413, "lm_q1q2_score": 0.44718315464933284}} {"text": "```python\n# import numpy as np\n\n# # !/usr/bin/env python3\n# # -*- coding: utf-8 -*-\n# \"\"\"\n# Created on 20181219\n\n# @author: zhangji\n\n# Trajection of a ellipse, Jeffery equation. \n# \"\"\"\n\n# %pylab inline\n# pylab.rcParams['figure.figsize'] = (25, 11)\n# fontsize = 40\n\n# import numpy as np\n# import scipy as sp\n# from scipy.optimize import leastsq, curve_fit\n# from scipy import interpolate\n# from scipy.interpolate import interp1d\n# from scipy.io import loadmat, savemat\n# # import scipy.misc\n\n# import matplotlib\n# from matplotlib import pyplot as plt\n# from matplotlib import animation, rc\n# import matplotlib.ticker as mtick\n# from mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\n# from mpl_toolkits.mplot3d import Axes3D, axes3d\n\n# from sympy import symbols, simplify, series, exp\n# from sympy.matrices import Matrix\n# from sympy.solvers import solve\n\n# from IPython.display import display, HTML\nfrom tqdm import tqdm\nfrom tqdm.notebook import tqdm as tqdm_notebook\n# import pandas as pd\n# import re\n# from scanf import scanf\n# import os\n# import glob\n\n# from codeStore import support_fun as spf\n# from src.support_class import *\n# from src import stokes_flow as sf\n\n# rc('animation', html='html5')\n# PWD = os.getcwd()\n# font = {'size': 20}\n# matplotlib.rc('font', **font)\n# np.set_printoptions(linewidth=90, precision=5)\n\nimport os\nimport glob\nimport natsort \nimport numpy as np\nimport scipy as sp\nfrom scipy.optimize import leastsq, curve_fit\nfrom scipy import interpolate, integrate\nfrom scipy import spatial, signal\n# from scipy.interpolate import interp1d\nfrom scipy.io import loadmat, savemat\n# import scipy.misc\nimport importlib\nfrom IPython.display import display, HTML\nimport pandas as pd\nimport pickle\nimport re\nfrom scanf import scanf\n\nimport matplotlib\n# matplotlib.use('agg')\nfrom matplotlib import pyplot as plt\nimport matplotlib.colors as colors\nfrom matplotlib import animation, rc\nimport matplotlib.ticker as mtick\nfrom mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\nfrom mpl_toolkits.mplot3d import Axes3D, axes3d\nfrom mpl_toolkits.axes_grid1.axes_divider import make_axes_locatable\nfrom mpl_toolkits.mplot3d.art3d import Line3DCollection\nfrom matplotlib import cm\n\nfrom tqdm import tqdm\nfrom tqdm.notebook import tqdm as tqdm_notebook\nfrom time import time\nfrom src.support_class import *\nfrom src import jeffery_model as jm\nfrom codeStore import support_fun as spf\nfrom codeStore import support_fun_table as spf_tb\n\n# %matplotlib notebook\n%matplotlib inline\nrc('animation', html='html5')\nfontsize = 40\nPWD = os.getcwd()\n```\n\n\n```python\nfig = plt.figure(figsize=(2, 2))\nfig.patch.set_facecolor('white')\nax0 = fig.add_subplot(1, 1, 1)\n```\n\n\n```python\njob_dir = 'ecoliB01_a'\ntable_name = 'planeShearRatex_1d'\n```\n\n\n```python\n# show phase map of theta-phi, load date\nimportlib.reload(spf_tb)\n\nt_headle = '(.*?).pickle'\nt_path = os.listdir(os.path.join(PWD, job_dir))\nfilename_list = [filename for filename in os.listdir(os.path.join(PWD, job_dir)) \n if re.match(t_headle, filename) is not None]\n\nfor tname in tqdm_notebook(filename_list[:]):\n tpath = os.path.join(PWD, job_dir, tname)\n with open(tpath, 'rb') as handle:\n tpick = pickle.load(handle)\n Table_t = tpick['Table_t']\n if 'Table_dt' not in tpick.keys():\n Table_dt = np.hstack((np.diff(tpick['Table_t']), 0))\n else:\n Table_dt = tpick['Table_dt']\n Table_X = tpick['Table_X']\n Table_P = tpick['Table_P']\n Table_P2 = tpick['Table_P2']\n Table_theta = tpick['Table_theta']\n Table_phi = tpick['Table_phi']\n Table_psi = tpick['Table_psi']\n Table_eta = tpick['Table_eta']\n \n save_name = '%s.jpg' % (os.path.splitext(os.path.basename(tname))[0])\n idx = Table_t > 0\n fig = spf_tb.save_table_result(os.path.join(PWD, job_dir, save_name), \n Table_t[idx], Table_dt[idx], Table_X[idx], Table_P[idx], Table_P2[idx], \n Table_theta[idx], Table_phi[idx], Table_psi[idx], Table_eta[idx])\n plt.close(fig)\n```\n\n\n HBox(children=(FloatProgress(value=0.0, max=1128.0), HTML(value='')))\n\n\n \n\n\n\n```python\nfilename_list\n```\n\n\n\n\n ['th1.503_ph0.000_ps0.000_D20190714_T225427.pickle',\n 'th0.546_ph5.214_ps0.000_D20190715_T040610.pickle',\n 'th0.683_ph1.337_ps0.000_D20190715_T011428.pickle',\n 'th2.595_ph5.481_ps0.000_D20190715_T043702.pickle',\n 'th2.459_ph3.877_ps0.000_D20190715_T032737.pickle',\n 'th0.546_ph1.337_ps0.000_D20190715_T011429.pickle',\n 'th2.732_ph5.214_ps0.000_D20190715_T040610.pickle',\n 'th2.185_ph4.412_ps0.000_D20190715_T035349.pickle',\n 'th1.093_ph2.807_ps0.000_D20190715_T023653.pickle',\n 'th2.732_ph1.203_ps0.000_D20190715_T011130.pickle',\n 'th0.137_ph3.342_ps0.000_D20190715_T031151.pickle',\n 'th1.366_ph6.016_ps0.000_D20190715_T045021.pickle',\n 'th2.868_ph0.936_ps0.000_D20190715_T005118.pickle',\n 'th1.366_ph2.005_ps0.000_D20190715_T020546.pickle',\n 'th1.366_ph2.406_ps0.000_D20190715_T021007.pickle',\n 'th1.229_ph0.668_ps0.000_D20190715_T005119.pickle',\n 'th1.229_ph5.882_ps0.000_D20190715_T044438.pickle',\n 'th0.410_ph5.347_ps0.000_D20190715_T042035.pickle',\n 'th2.595_ph3.075_ps0.000_D20190715_T024550.pickle',\n 'th1.776_ph1.604_ps0.000_D20190715_T020546.pickle',\n 'th0.546_ph6.016_ps0.000_D20190715_T045020.pickle',\n 'th2.732_ph4.412_ps0.000_D20190715_T035348.pickle',\n 'th2.595_ph2.005_ps0.000_D20190715_T020546.pickle',\n 'th1.229_ph1.069_ps0.000_D20190715_T005116.pickle',\n 'th1.503_ph2.540_ps0.000_D20190715_T021007.pickle',\n 'th3.142_ph4.011_ps0.000_D20190715_T032724.pickle',\n 'th2.322_ph1.203_ps0.000_D20190715_T011130.pickle',\n 'th1.503_ph4.144_ps0.000_D20190715_T142926.pickle',\n 'th1.503_ph4.679_ps0.000_D20190715_T040314.pickle',\n 'th1.503_ph5.214_ps0.000_D20190715_T040610.pickle',\n 'th1.503_ph2.139_ps0.000_D20190715_T021010.pickle',\n 'th2.595_ph0.936_ps0.000_D20190715_T005118.pickle',\n 'th0.546_ph5.347_ps0.000_D20190715_T042036.pickle',\n 'th1.366_ph0.668_ps0.000_D20190715_T005119.pickle',\n 'th2.049_ph5.748_ps0.000_D20190715_T043918.pickle',\n 'th0.820_ph3.476_ps0.000_D20190715_T031732.pickle',\n 'th1.912_ph2.540_ps0.000_D20190715_T021007.pickle',\n 'th1.639_ph3.342_ps0.000_D20190715_T031150.pickle',\n 'th2.732_ph4.278_ps0.000_D20190715_T033910.pickle',\n 'th0.820_ph0.000_ps0.000_D20190714_T225427.pickle',\n 'th2.732_ph0.936_ps0.000_D20190715_T005118.pickle',\n 'th2.459_ph3.342_ps0.000_D20190715_T031151.pickle',\n 'th0.410_ph2.005_ps0.000_D20190715_T020546.pickle',\n 'th1.366_ph0.936_ps0.000_D20190715_T005118.pickle',\n 'th1.503_ph4.813_ps0.000_D20190715_T040331.pickle',\n 'th1.639_ph2.406_ps0.000_D20190715_T021007.pickle',\n 'th1.912_ph0.802_ps0.000_D20190715_T005124.pickle',\n 'th2.185_ph3.342_ps0.000_D20190715_T031150.pickle',\n 'th1.366_ph1.203_ps0.000_D20190715_T011130.pickle',\n 'th0.273_ph1.872_ps0.000_D20190715_T020546.pickle',\n 'th1.639_ph5.615_ps0.000_D20190715_T043817.pickle',\n 'th2.185_ph1.337_ps0.000_D20190715_T011428.pickle',\n 'th1.229_ph3.743_ps0.000_D20190715_T032737.pickle',\n 'th2.322_ph1.069_ps0.000_D20190715_T005116.pickle',\n 'th0.000_ph4.946_ps0.000_D20190715_T040521.pickle',\n 'th2.049_ph3.877_ps0.000_D20190715_T032737.pickle',\n 'th3.142_ph2.273_ps0.000_D20190715_T021007.pickle',\n 'th2.595_ph2.674_ps0.000_D20190715_T023606.pickle',\n 'th3.142_ph0.267_ps0.000_D20190714_T225427.pickle',\n 'th2.595_ph4.011_ps0.000_D20190715_T032724.pickle',\n 'th2.595_ph5.214_ps0.000_D20190715_T040610.pickle',\n 'th2.185_ph3.609_ps0.000_D20190715_T032653.pickle',\n 'th2.049_ph4.144_ps0.000_D20190715_T032725.pickle',\n 'th3.005_ph5.347_ps0.000_D20190715_T042035.pickle',\n 'th0.000_ph6.150_ps0.000_D20190715_T045619.pickle',\n 'th1.093_ph2.941_ps0.000_D20190715_T023808.pickle',\n 'th3.005_ph3.743_ps0.000_D20190715_T032737.pickle',\n 'th0.000_ph0.267_ps0.000_D20190714_T225427.pickle',\n 'th0.683_ph0.668_ps0.000_D20190715_T005119.pickle',\n 'th0.956_ph0.535_ps0.000_D20190715_T005110.pickle',\n 'th1.229_ph2.807_ps0.000_D20190715_T023652.pickle',\n 'th0.410_ph1.069_ps0.000_D20190715_T005116.pickle',\n 'th2.868_ph2.941_ps0.000_D20190715_T023808.pickle',\n 'th1.776_ph4.946_ps0.000_D20190715_T040521.pickle',\n 'th2.322_ph3.342_ps0.000_D20190715_T031151.pickle',\n 'th2.185_ph2.139_ps0.000_D20190715_T021010.pickle',\n 'th2.185_ph4.679_ps0.000_D20190715_T040313.pickle',\n 'th1.093_ph4.412_ps0.000_D20190715_T035348.pickle',\n 'th2.595_ph1.069_ps0.000_D20190715_T005116.pickle',\n 'th2.185_ph3.075_ps0.000_D20190715_T024550.pickle',\n 'th0.410_ph4.813_ps0.000_D20190715_T040330.pickle',\n 'th1.229_ph4.545_ps0.000_D20190715_T040118.pickle',\n 'th0.273_ph4.813_ps0.000_D20190715_T040330.pickle',\n 'th0.820_ph4.011_ps0.000_D20190715_T032724.pickle',\n 'th0.137_ph4.144_ps0.000_D20190715_T032725.pickle',\n 'th2.868_ph4.412_ps0.000_D20190715_T035349.pickle',\n 'th0.956_ph2.273_ps0.000_D20190715_T021007.pickle',\n 'th2.459_ph0.000_ps0.000_D20190714_T225427.pickle',\n 'th2.868_ph3.609_ps0.000_D20190715_T032653.pickle',\n 'th0.000_ph2.941_ps0.000_D20190715_T023808.pickle',\n 'th2.459_ph4.011_ps0.000_D20190715_T032724.pickle',\n 'th1.366_ph1.069_ps0.000_D20190715_T005116.pickle',\n 'th3.142_ph2.807_ps0.000_D20190715_T023652.pickle',\n 'th0.410_ph5.214_ps0.000_D20190715_T040610.pickle',\n 'th1.366_ph1.471_ps0.000_D20190715_T013943.pickle',\n 'th2.185_ph0.401_ps0.000_D20190714_T225427.pickle',\n 'th0.820_ph6.283_ps0.000_D20190715_T045619.pickle',\n 'th0.000_ph3.877_ps0.000_D20190715_T032737.pickle',\n 'th1.093_ph4.144_ps0.000_D20190715_T032725.pickle',\n 'th1.229_ph3.208_ps0.000_D20190715_T025014.pickle',\n 'th2.049_ph3.075_ps0.000_D20190715_T024550.pickle',\n 'th1.229_ph1.471_ps0.000_D20190715_T013943.pickle',\n 'th1.366_ph3.743_ps0.000_D20190715_T032737.pickle',\n 'th2.595_ph0.535_ps0.000_D20190715_T005110.pickle',\n 'th0.683_ph5.481_ps0.000_D20190715_T043702.pickle',\n 'th2.595_ph1.471_ps0.000_D20190715_T013943.pickle',\n 'th0.820_ph0.802_ps0.000_D20190715_T005124.pickle',\n 'th2.868_ph1.738_ps0.000_D20190715_T020546.pickle',\n 'th1.093_ph3.609_ps0.000_D20190715_T032653.pickle',\n 'th0.546_ph6.150_ps0.000_D20190715_T045619.pickle',\n 'th1.229_ph5.347_ps0.000_D20190715_T042035.pickle',\n 'th2.185_ph5.748_ps0.000_D20190715_T043918.pickle',\n 'th3.142_ph3.476_ps0.000_D20190715_T031731.pickle',\n 'th2.595_ph1.203_ps0.000_D20190715_T011130.pickle',\n 'th2.322_ph5.080_ps0.000_D20190715_T040605.pickle',\n 'th1.776_ph2.139_ps0.000_D20190715_T021010.pickle',\n 'th0.683_ph4.813_ps0.000_D20190715_T040330.pickle',\n 'th0.683_ph2.674_ps0.000_D20190715_T023606.pickle',\n 'th2.459_ph2.139_ps0.000_D20190715_T021010.pickle',\n 'th0.820_ph5.748_ps0.000_D20190715_T043918.pickle',\n 'th0.683_ph3.208_ps0.000_D20190715_T025014.pickle',\n 'th1.776_ph4.813_ps0.000_D20190715_T040330.pickle',\n 'th1.366_ph5.214_ps0.000_D20190715_T040610.pickle',\n 'th3.142_ph4.144_ps0.000_D20190715_T032725.pickle',\n 'th0.546_ph0.668_ps0.000_D20190715_T005119.pickle',\n 'th2.459_ph5.347_ps0.000_D20190715_T042036.pickle',\n 'th1.776_ph6.150_ps0.000_D20190715_T045619.pickle',\n 'th1.912_ph5.882_ps0.000_D20190715_T044438.pickle',\n 'th1.639_ph2.005_ps0.000_D20190715_T020546.pickle',\n 'th0.683_ph0.936_ps0.000_D20190715_T005118.pickle',\n 'th2.322_ph1.604_ps0.000_D20190715_T020546.pickle',\n 'th2.185_ph5.080_ps0.000_D20190715_T040604.pickle',\n 'th1.912_ph3.877_ps0.000_D20190715_T032737.pickle',\n 'th0.820_ph2.674_ps0.000_D20190715_T023605.pickle',\n 'th2.459_ph1.738_ps0.000_D20190715_T020546.pickle',\n 'th1.229_ph4.412_ps0.000_D20190715_T035349.pickle',\n 'th1.229_ph0.267_ps0.000_D20190714_T225427.pickle',\n 'th1.776_ph0.936_ps0.000_D20190715_T005118.pickle',\n 'th1.229_ph3.075_ps0.000_D20190715_T024550.pickle',\n 'th1.776_ph5.214_ps0.000_D20190715_T040610.pickle',\n 'th0.273_ph1.203_ps0.000_D20190715_T011130.pickle',\n 'th1.366_ph4.011_ps0.000_D20190715_T032724.pickle',\n 'th2.868_ph2.674_ps0.000_D20190715_T023605.pickle',\n 'th2.459_ph5.748_ps0.000_D20190715_T043918.pickle',\n 'th2.322_ph4.144_ps0.000_D20190715_T032725.pickle',\n 'th3.142_ph2.139_ps0.000_D20190715_T021010.pickle',\n 'th0.546_ph2.273_ps0.000_D20190715_T021007.pickle',\n 'th2.185_ph1.203_ps0.000_D20190715_T011131.pickle',\n 'th2.459_ph5.481_ps0.000_D20190715_T043702.pickle',\n 'th0.273_ph6.016_ps0.000_D20190715_T045020.pickle',\n 'th1.503_ph0.267_ps0.000_D20190715_T142926.pickle',\n 'th0.000_ph3.075_ps0.000_D20190715_T024550.pickle',\n 'th2.459_ph3.476_ps0.000_D20190715_T031731.pickle',\n 'th1.912_ph2.807_ps0.000_D20190715_T023652.pickle',\n 'th1.776_ph5.347_ps0.000_D20190715_T042035.pickle',\n 'th2.732_ph3.877_ps0.000_D20190715_T032737.pickle',\n 'th0.683_ph5.080_ps0.000_D20190715_T040605.pickle',\n 'th0.000_ph1.872_ps0.000_D20190715_T020546.pickle',\n 'th2.049_ph1.738_ps0.000_D20190715_T020546.pickle',\n 'th1.912_ph2.005_ps0.000_D20190715_T020546.pickle',\n 'th0.137_ph2.406_ps0.000_D20190715_T021007.pickle',\n 'th1.912_ph1.069_ps0.000_D20190715_T005116.pickle',\n 'th3.005_ph5.615_ps0.000_D20190715_T043817.pickle',\n 'th1.503_ph2.674_ps0.000_D20190715_T023605.pickle',\n 'th1.639_ph1.471_ps0.000_D20190715_T013943.pickle',\n 'th1.776_ph0.535_ps0.000_D20190715_T005110.pickle',\n 'th0.683_ph0.802_ps0.000_D20190715_T005124.pickle',\n 'th2.322_ph1.337_ps0.000_D20190715_T011428.pickle',\n 'th0.410_ph1.203_ps0.000_D20190715_T011131.pickle',\n 'th2.459_ph6.016_ps0.000_D20190715_T045020.pickle',\n 'th0.956_ph5.481_ps0.000_D20190715_T043702.pickle',\n 'th2.732_ph3.075_ps0.000_D20190715_T024550.pickle',\n 'th1.776_ph1.337_ps0.000_D20190715_T011428.pickle',\n 'th1.366_ph6.150_ps0.000_D20190715_T045619.pickle',\n 'th0.000_ph5.882_ps0.000_D20190715_T044438.pickle',\n 'th0.683_ph1.738_ps0.000_D20190715_T020546.pickle',\n 'th0.683_ph5.347_ps0.000_D20190715_T042035.pickle',\n 'th0.137_ph3.208_ps0.000_D20190715_T025014.pickle',\n 'th3.142_ph2.005_ps0.000_D20190715_T020547.pickle',\n 'th1.776_ph0.134_ps0.000_D20190714_T225427.pickle',\n 'th3.142_ph3.342_ps0.000_D20190715_T031150.pickle',\n 'th0.273_ph0.535_ps0.000_D20190715_T005110.pickle',\n 'th0.683_ph3.877_ps0.000_D20190715_T032737.pickle',\n 'th3.142_ph5.481_ps0.000_D20190715_T043702.pickle',\n 'th1.912_ph5.481_ps0.000_D20190715_T043702.pickle',\n 'th0.546_ph1.203_ps0.000_D20190715_T011130.pickle',\n 'th1.366_ph4.946_ps0.000_D20190715_T040522.pickle',\n 'th0.000_ph3.743_ps0.000_D20190715_T032737.pickle',\n 'th0.000_ph3.476_ps0.000_D20190715_T031732.pickle',\n 'th2.049_ph1.203_ps0.000_D20190715_T011131.pickle',\n 'th0.273_ph3.609_ps0.000_D20190715_T032653.pickle',\n 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"size": 74096, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "head_Force/do_calculate_table/pickle2jpg.ipynb", "max_stars_repo_name": "pcmagic/stokes_flow", "max_stars_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-11-11T05:00:53.000Z", "max_stars_repo_stars_event_max_datetime": "2018-11-11T05:00:53.000Z", "max_issues_repo_path": "head_Force/do_calculate_table_loc/pickle2jpg.ipynb", "max_issues_repo_name": "pcmagic/stokes_flow", "max_issues_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "head_Force/do_calculate_table_loc/pickle2jpg.ipynb", "max_forks_repo_name": "pcmagic/stokes_flow", "max_forks_repo_head_hexsha": 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YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.6584175139669997, "lm_q1q2_score": 0.447183150650403}} {"text": "```python\n%matplotlib inline\n\nimport matplotlib\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nplt.rcParams[\"figure.figsize\"] = (12, 9)\nplt.rcParams[\"font.size\"] = 18\n```\n\n# In Core Fuel Management\n\nIn core fuel management focuses on the study of requirements and operational considerations impacting fuel performance in the reactor core, power history, core loading patterns, and refuelling activities.\n\n\n## Learning Objectives\n\nAt the end of this lesson, you will be equipped to:\n\n- List safety constraints driving in core fuel management decisions.\n- Calculate capacity and availability factors.\n- Calculate the mass required for each reactor year of operation.\n- Calculate core and assembly discharge burnup based on power output.\n- Analyze the reactivity evolution of a core based on burnup.\n- Apply burnup calculations to multiple batch cores.\n- Recognize the relationship between the number of batches and the final burnup. \n- Understand the goals driving choices in various fuel loading patterns. \n- Apply these lessons to pebble-fuelled and liquid-fueled advanced reactor designs.\n- Recognize the impact of extended burnup on fuel utilization, SWU utilization, and fuel cycle cost.\n- Understand how isotopic activities can be used to determine fuel burnup.\n- Calculate burnup based on key activity ratios.\n\n## Safety Constraints\n\n\n- $\\frac{P_{peak}}{P_{avg}}$, peak to average power ratio.\n- $T_{max}$, maximimum core temperature.\n- Departure from Nucleate Boiling Ratio (DNBR)\n- $\\rho$, reactivity in the core.\n- $\\alpha_T$, temperature coefficient of reactivity \n\nPrimarily, there is a loss of coolant accident (LOCA) peak clad temp (PCT) limit of 1205 $^\\circ C$, which limits the maximum pellet linear power density to approx 48 kW/m at Hot Full Power(HFP).\n\n- Critical Heat Flux (CHF), which denotes departure from nuclear boiling (DNB) for a PWR and Dryout for a BWR, not being exceeded during anticipated transients, which limits the maximum average fuel pin linear power density to approximately 29 kW/m at HFP.\n- Fuel cladding strain limit not exceeded during anticipated transients\n\n### Safety Variables\n\n- Fuel enrichment\n- Re-load batch size & number of assemblies\n- Fuel loading pattern of fresh and partially spent fuel assemblies\n- Control mechanisms\n\n\n## Mass Required\n\nThe simplest possible representation of the mass of fuel that must be added into a reactor is:\n\n\\begin{align}\nM(t) &= \\frac{Q}{BU}\n\\end{align}\n\nwhere\n\\begin{align}\nM &= \\mbox{mass of heavy metal (e.g., uranium) in the core }[MTHM/yr]\\\\\nQ &= \\mbox{annual thermal energy output }[GWd/yr]\\\\\nBU &= \\mbox{burnup }[GWd/MTIHM]\n\\end{align}\n\n\nBut, Q itself typically needs to be back-calculated from energy produced.\n\n\\begin{align}\nQ &= \\frac{P_0\\cdot CF\\cdot T}{\\eta_{th}}\n\\end{align}\n\nwhere\n\\begin{align}\nP_0 &= \\mbox{installed electric capacity }[GWe]\\\\\nCF &= \\mbox{capacity factor }[-]\\\\\nT &= \\mbox{time in core } [days]\\\\\n\\eta_{th} &= \\mbox{thermal efficiency }[GWe/GWth]\\\\\n\\end{align}\n\n\n\n\n```python\ndef m(q, bu):\n return q/bu\n\ndef q(p0, cf, t, eta_th):\n return p0*cf*t/eta_th\n\np0 = 1500 # installed electric capacity GWe\ncf = 0.9 # capacity factor\nt = 365 # days per year\neta_th = 0.33 # thermal efficiency GWe/GWth\nbu = 50 # burnup GWd/MTIHM\n\nprint(m(q(p0, cf, t, eta_th), bu))\n\n```\n\n 29863.636363636364\n\n\n## Capacity and Availability Factors\n\nThe capacity factor is representative of the plant's tendency to acheive its rated power capacity.\n\n\\begin{align}\nCF &= \\frac{\\mbox{actual power generated over time T}}{\\mbox{rated power potential over time T}}\\\\\n &=\\frac{\\int_0^T P(t)dt}{P_0T}\\\\\nP(t) &= \\mbox{ thermal power at time t during period T}\n\\end{align}\n\nThe capacity factor, integrated over time, gives Effective Full Power Days (EFPD), the equivalent number of days at full power.\n\n\\begin{align}\nEFPD &= \\int_0^TCF(t)dt\\\\\n &= \\int_0^T \\frac{\\int_0^T P(t)dt}{P_0T}\\\\\n\\end{align}\n\n\n\nThe availability factor is always greater than the capacity factor. \n\n\\begin{align}\nAF &= \\frac{\\mbox{time during which the reactor was operational during time period T}}{T}\n\\end{align}\n\n\n\n\n```python\n# The reactor shuts down:\n# for a few days during the 10th month\n# for one month during month 18\nshutdowns = {10:10.1,\n 18.5:19.5}\n\nimport numpy as np\ndef A(t, shutdowns):\n to_ret = 1.0*(t > 0)\n for start,stop in shutdowns.items():\n if start < t and t < stop:\n to_ret = 0\n return to_ret\n\ntimes = np.arange(0.0, 20.0, 0.01)\nhist = np.arange(0.0, 20.0, 0.01)\ncf = np.arange(0.0, 20.0, 0.01)\n\nfor i in range(0, times.size):\n hist[i] = A(times[i], shutdowns)\n cf[i] = A(times[i], shutdowns)*(1.-0.1*np.random.random())\n \nplt.plot(times, hist, label='Availability')\nplt.plot(times, cf, label='Capacity')\n\nplt.ylim([-0.5, 1.5])\nplt.title('Capacity and Availabilty')\nplt.xlabel('Time (months)')\nplt.ylabel('Factor [-]')\nplt.legend()\n\n```\n\nWe can do a quick numeric integral to get each factor as an integral over the 20 month cycle.\n\n\\begin{align}\nAF &= \\frac{\\int_0^{20}A(t)dt}{T}\\\\\nCF &= \\frac{\\int_0^{20}P(t)dt}{P_0T}\\\\\n\\end{align}\n\n\n\n```python\nprint(\"Availability Factor = \", hist.sum()/hist.shape[0])\nprint(\"Capacity Factor = \", cf.sum()/cf.shape[0])\n```\n\n Availability Factor = 0.9455\n Capacity Factor = 0.898687813941\n\n\n## Simple Reactivity Model\n- On each cycle (1/n)th of the fuel is replaced\n- Each fuel batch experiences a discharge burnup of Bd\n- Each fuel batch on each cycle experiences a burnup of Bd/n\n- $k_{reactor}$ is the uncontrolled multiplication factor (excess reactivity)\n- $k_i$ is the infinite multiplication factor of a fuel batch (excess reactivity)\n \nEach batch of fuel will have a different burn-up and $k_i(B)$ since each batch has been in the reactor a different length of time. The reactivity of the reactor is found by summing over the reactivities of all the batches of fuel, for n batches:\n\n\\begin{align}\nk_{reactor} = \\frac{1}{n}\\sum_{i=1}^{n}k_i(B)\n\\end{align}\n\n\n\\begin{align}\nk_i(B) = k_0 - \\alpha B_n\n\\end{align}\n\n- $k_0$ is the uncontrolled infinite multiplication factor of the fuel batch when it is fresh.\n- $B_n$ is the burnup of the batch in a single cycle. The n refers to the number of batches that the reload scheme includes.\n- $\\alpha$ is a constant of proportionality with units of 1/Bn. Uniform linear depletion.\n- $k_F$ is the uncontrolled infinite multiplication factor necessary to sustain a chain reaction at the end of an operating cycle\n\n\n\n\n```python\n\ndef ki(k0, alpha, b):\n return k0 - alpha*b\n\ndef k(ki, n):\n return (1/n)*np.sum(ki)\n\nn=3\nk0 = 4.5\nalpha = (k0 - 1)/20000\nbu = np.arange(0, 30000., 1000.)\nplt.plot(bu, ki(k0, alpha, bu))\nplt.plot(bu, np.zeros(bu.shape), color='r')\nplt.ylabel(r'$k_i(B)$')\nplt.xlabel(r'$B$')\nplt.title('Excess Reactivity Using Linear Depletion Model')\n```\n\nThis approximation is somewhat accurate and gives an intuition for the impact of reloading on excess reactivity in the core. \n\n\n\n## Single Cycle Refuelling\n\n\n\n\n\\begin{align}\nk_{reactor} = k_1(B_1)\n\\end{align}\n\n\n\\begin{align}\nk_1(B_1) = k_0 - \\alpha B_1\n\\end{align}\n\nTherefore the fuel burnup capability is:\n\n\\begin{align}\nB_1 &= \\frac{k_0-k_F}{\\alpha}\n\\end{align}\n\n\n## Two Cycle Refuelling\n\n\n\nAt the end of each cycle one batch of fuel has been burned for one cycle and the other batch has been burned for two cycles. Thus:\n\n\\begin{align}\nk_F &= \\frac{k_0 - \\alpha B_2}{2} + \\frac{k_0 - 2\\alpha B_2}{2}\\\\\n &= k_0 - \\frac{3\\alpha B_2}{2}\\\\\nB_2 &= \\frac{2(k_0 - k_F)}{3\\alpha}\\\\\n &= \\frac{2}{3}B_1\n\\end{align}\n\n- Each batch in the two cycle reload scheme is burned for $2B_2$. \n\nSo, in terms of the single cycle reload burnup:\n\n\\begin{align}\n2B_2 &= 2\\left(\\frac{2}{3}B_1\\right)\\\\\n &= \\frac{4}{3}B_1\\\\\n\\end{align}\n\n**This means there is 1/3 more burnup in the two cycle reload, for the same initial and final multiplication factors $k_0$ and $k_F$ (exactly the same fuel.)**\n\n\n## N Cycle Reload Scheme\n\nThe relation between end-of-cycle core multiplication factor kF and the fresh fuel batch infinite multiplication factor k0 and the batch burnup in general is\n\n\n\\begin{align}\nk_F &= k_0 - \\frac{1}{n}\\sum_{i=1}^{n}i\\alpha B_n\\\\\n\\end{align}\n\nRecall from your geometric series intution:\n\\begin{align}\n\\sum_{i=1}^{n}i &= \\frac{n(n + 1)}{2}\\\\\n\\end{align}\n\nTherefore:\n\n\\begin{align}\nk_F &= k_0 - \\left(\\frac{n + 1}{2}\\right)\\alpha B_n\\\\\n\\end{align}\n\nThe batch burnup in a single cycle is then the result of solving for $B_n$:\n\n\\begin{align}\nB_n &= \\frac{2(k_0 - k_F)}{\\alpha(n + 1)}\n\\end{align}\n\n\nThe discharge burnup of batch n, is the batch burnup in a cycle times the number of cycles:\n\n\\begin{align}\nB_n^d &= nB_n\\\\\n&= \\frac{2n(k_0 - k_F)}{\\alpha(n + 1)}\\\\\n &= \\left(\\frac{2n}{n + 1}\\right)\\frac{k_0 - k_F}{\\alpha} \\\\\n &= \\left(\\frac{2n}{n + 1}\\right)B_1 \\\\\n\\end{align}\n\n\n\n```python\ndef bd(n, b1):\n num = 2*n*b1\n denom = n+1\n return num/denom\n\nb1 = 12000\nn = np.arange(1,50)\nplt.plot(n, bd(n, b1))\n```\n\n### Discussion: What is the primary drawback of many batches per core?\n \n \n\n## Fuel Loading Patterns \n\nVarious fuel loading patterns are used to acheive improved fuel utilization (higher burnup), better core control, and lower leakage to the pressure vessel. \n\n\n\n\n\n\n\n## Many and $\\infty$ Batch Reactor Designs\n\nInfinite batch refuelling (a.k.a. online refuelling) is possible in liquid fuelled cores with online reprocessing.\n\n\n\nWhat exactly is a pebble core, then, in terms of batches?\n\n\n
Aufiero, 2016
\n\n\n\n\n## Determining Burnup\n\n- Direct methods occur while the fuel is still in the core (using ion chambers and in-core flux probes)\n- Indirect methods use measurements of activity after the fuel has been removed.\n\n\\begin{align}\nBU &= a + bA(^{137}Cs)\\\\\nBU &= c(e, r) + d(e, r) \\left[A(^{134}Cs)/A(^{137}Cs)\\right]\\\\\nBU &= a\\cdot exp{\\left[b\\cdot ln\\left(\\frac{A(^{106}Ru)A(^{137}Cs)}{[A(^{134}Cs)^2}\\right)\\right]}\\\\\na, b, c, d &= \\mbox{calibration constants}\\\\\ne &= \\mbox{enrichment}\\\\\nr &= \\mbox{power rating}\n\\end{align}\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "1edc70bb52c8ef91c1872fb1094b01940df8b544", "size": 128011, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "in-core/in-core.ipynb", "max_stars_repo_name": "atomicaristides/NPRE412", "max_stars_repo_head_hexsha": "b2ae552303f3e4894628c8401d3bedd2db85a551", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "in-core/in-core.ipynb", "max_issues_repo_name": "atomicaristides/NPRE412", "max_issues_repo_head_hexsha": "b2ae552303f3e4894628c8401d3bedd2db85a551", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "in-core/in-core.ipynb", "max_forks_repo_name": "atomicaristides/NPRE412", "max_forks_repo_head_hexsha": "b2ae552303f3e4894628c8401d3bedd2db85a551", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 221.4723183391, "max_line_length": 56216, "alphanum_fraction": 0.8958683238, "converted": true, "num_tokens": 2989, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175139669997, "lm_q2_score": 0.6791786926816161, "lm_q1q2_score": 0.44718314637478657}} {"text": "

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\n\n\n```python\n# code for loading the format for the notebook\nimport os\n\n# path : store the current path to convert back to it later\npath = os.getcwd()\nos.chdir(os.path.join('..', 'notebook_format'))\nfrom formats import load_style\nload_style()\n```\n\n\n\n\n\n\n\n\n\n\n\n```python\nos.chdir(path)\n\n# 1. magic for inline plot\n# 2. magic to print version\n# 3. magic so that the notebook will reload external python modules\n# 4. magic to enable retina (high resolution) plots\n# https://gist.github.com/minrk/3301035\n%matplotlib inline\n%load_ext watermark\n%load_ext autoreload\n%autoreload 2\n%config InlineBackend.figure_format = 'retina'\n\nimport warnings\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nfrom scipy.stats import chi2_contingency\nfrom sklearn.linear_model import LinearRegression\nfrom sklearn.metrics import r2_score, mean_squared_error\n\n%watermark -a 'Ethen' -d -t -v -p scipy,numpy,pandas,sklearn,matplotlib\n```\n\n Ethen 2017-12-23 09:28:58 \n \n CPython 3.5.2\n IPython 6.2.1\n \n scipy 1.0.0\n numpy 1.13.3\n pandas 0.20.3\n sklearn 0.19.1\n matplotlib 2.1.0\n\n\n# Multicollearity\n\nMany machine learning models have either some inherent internal ranking of features or it is not extremely complicated to generate/access the ranking from the structure of the model. This document discusses the issue of multicollinearity, i.e. how multicollinearity can affect the feature ranking and potential methods that can be used to address them. As we'll be looking at the coefficients of linear regression model for selecting and interpreting features, the next section contains a introduction to this method/algorithm.\n\n\n## Linear Regression\n\nUnlike a lot of other tutorials, we'll introduce linear regression from a **maximum likelihood** perspective. The principle of maximum likelihood is at the heart of machine learning. It guides us to find the best model in a search space of all models. In simple terms, Maximum Likelihood Estimation (MLE) lets us choose a model (parameters) that explains the data (training set) better than all other models.\n\n### Maximum Likelihood - Primer\n\nThe process of sampling from a normal distribution is expressed as, $x \\sim \\mathcal{N}(\\mu, \\sigma^{2})$. $x$ is a random variable sampled or generated or simulated from the gaussian distribution. As we sample from this distribution, most samples will fall around the center, near the mean, because of higher probability density in the center.\n\n\n\nLet's consider 3 data points, $y1=1,y2=0.5,y3=1.5$, which are independent and drawn from a gaussian with unknown mean $\\theta$ and constant variance of 1. Suppose we now have two choices for $\\theta$: {1, 2.5}. Which one should we choose? Which model $\\theta$ would explain the data better? In general, any data point drawn from a gaussian with mean $\\theta$ and and variance 1, can be written as,\n\n$$\n\\begin{align}\ny_i \\sim \\mathcal{N}(\\theta, 1) = \\theta + \\mathcal{N}(0,1)\n\\end{align}\n$$\n\nThis can be read as $\\theta$ the mean, shifts the center of the standard normal distribution ($\\mu = 0$ and $\\sigma^2=1$) The likelihood of data (y1,y2,y3) having been drawn from $\\mathcal{N}(\\theta,1)$, can be expressed as:\n\n$$\n\\begin{align}\nP(y1,y2,y3 \\vert \\theta) = P(y1 \\vert \\theta) P(y2 \\vert \\theta) P(y3 \\vert \\theta)\n\\end{align}\n$$\n\nas the data points are assumed to be independent of one another.\n\nNow, we have two normal distributions defined by $\\theta = 1$ and $\\theta = 2.5$. Let us draw both and plot the data points. In the figure below, notice the dotted lines that connect the bell curve to the data points. Consider the point $y2=0.5$ in the first distribution $\\mathcal{N}(\\mu=1, \\sigma^2=1)$. The length of the dotted line gives the probability of the $y2=0.5$ being drawn from $\\mathcal{N}(\\mu=1, \\sigma^2=1)$. And the same goes for the second distribution $\\mathcal{N}(\\mu=2.5, \\sigma^2=1)$.\n\n\n\nKnowing that the likelihood of data (y1,y2,y3) having been drawn from $\\mathcal{N}(\\mu=1,\\sigma^2=1)$ is given by:\n\n$$\n\\begin{align}\nP(y1,y2,y3 \\vert \\theta=1) = P(y1 \\vert \\theta=1) P(y2 \\vert \\theta=1) P(y3 \\vert \\theta=1)\n\\end{align}\n$$\n\nThe individual probabilities in the equation above are equal to the heights of corresponding dotted lines in the figure. We see that the likelihood, computed by the product of individual probabilities of data points given model, is essentially the product of lengths of dotted lines. In this toy example, the likelihood of model $\\theta = 1$ seems to higher than $\\theta = 2.5$, so that's the model we'll be going with.\n\n### Maximum Likelihood - Linear Regression\n\nFor linear regression we assume the relationship between our input variable $X$ and our output label $Y$ can be modeled by a linear function.\n\n$$\n\\begin{align}\nY = \\theta_0 + \\theta_1X_1 + \\theta_2X_2 + \\ldots + \\theta_pX_p + \\epsilon\n\\end{align}\n$$\n\nThe model assumes the label for each observation, $y_i$, is gaussian distributed with mean, $x_i^T\\theta$ and variance, $\\sigma^2$, which can be written as:\n\n$$\n\\begin{align}\ny_i &= \\mathcal{N}(x_i^{T}\\theta, \\sigma^{2}) = x_i^{T}\\theta + \\mathcal{N}(0, \\sigma^{2})\\\\\nprediction, \\hat{y_i} &= x_i^T\\theta\n\\end{align}\n$$\n\nThe mean $x_i^{T}\\theta$ represents the best fitted line with all data points varying around that line, and the term $\\epsilon$, captures this varying variance $\\mathcal{N}(0, \\sigma^{2})$.\n\n\n\nNow, recall that the formula for the gaussian/normal distribution is:\n\n$$\n\\begin{align}\np(x) = \\frac{1}{\\sigma\\sqrt{2\\pi}}e^{-(x-\\mu)^2/2\\sigma^2}\n\\end{align}\n$$\n\nGiven that linear regression assumes each point $y_i$ to be gaussian distributed, the process of learning becomes the process of maximizing the product of the individual probabilities:\n\n$$\n\\begin{align}\np(y \\vert X, \\theta, \\sigma)\n&= \\prod_{i=1}^{n} p(y_i \\vert x_i, \\theta, \\sigma) \\\\\n&= \\prod_{i=1}^{n} (2\\pi\\sigma^2)^{-1/2} e^{-\\frac{1}{2\\sigma^2}(y_i - x_i^T\\theta)^2}\n\\end{align}\n$$\n\nNext we rewrite the equation in vector form and due to the fact that the original maximization problem is equivalent to maximizing its log likelihood (log is a monotonic transformation thus does not affect that learned parameters). Thus we take the log to make the derivation later easier.\n\n$$\n\\begin{align}\np(y \\vert X, \\theta, \\sigma)\n&= (2\\pi\\sigma^2)^{-n/2} e^{-\\frac{1}{2\\sigma^2}(y - X\\theta)^2} \\\\\n&= -\\frac{n}{2}log(2\\pi\\sigma^2) -\\frac{1}{2\\sigma^2}(y-X\\theta)^2\n\\end{align}\n$$\n\nOur current interest right now is to solve for the unknown parameter $\\theta$ (we can use similar idea to solve for the unknown $\\sigma$), thus we can further simplify the equation above, to remove all the terms that are not relevant to $\\theta$, and in machine learning problems, we're often interested in minimizing the objective function, thus we negate the negative sign and turn the maximization problem into a minimization one.\n\n$$\n\\begin{align}\nL = \\frac{1}{2}(y-X\\theta)^2\n\\end{align}\n$$\n\nWhen introducing linear regression, an alternative way of viewing it is from a least squares perspective. The objective of least squares is to minimize the squared distance between the prediciton and the ground truth. So, we want to minimize the mean squared error: $\\frac{1}{2} (y - X\\theta)^2$. We now see that we can come to the same objective from two different perspective. The following section lists out the derivation for solving $\\theta$.\n\nWe'll first expand this equation:\n\n$$\n\\begin{align}\nL &= \\frac{1}{2} (y - X\\theta)^2 \\\\\n &= \\frac{1}{2} (y - X\\theta)^T(y - X\\theta) \\\\\n &= \\frac{1}{2} (y^Ty - 2\\theta^TX^Ty + \\theta^TX^TX\\theta)\n\\end{align}\n$$\n\nUsing the standard rule of minimization in calculus, if we wish to solve for the weight $\\theta$, we would take the derivative w.r.t. $\\theta$ and set it to zero.\n\n$$\n\\begin{align}\n\\frac{\\partial}{\\partial{\\theta}} \\frac{1}{2} (y^Ty - 2\\theta^TX^Ty + \\theta^TX^TX\\theta) &= 0 \\\\\n\\frac{1}{2} (2X^Ty - 2X^TX\\theta) &= 0 \\\\\nX^Ty - X^TX\\theta &= 0\n\\end {align}\n$$\n\nIn the steps above, $y^Ty$ vanished as there\u2019s no $\\theta$ dependence, and $\\theta^TX^TX\\theta$ becomes $2X^TX\\theta$ as $\\theta^T\\theta$ is analogous to $\\theta^2$. As the final step, we will perform some rearrangement of the formula:\n\n$$\n\\begin{align}\nX^Ty - X^TX\\theta &= 0 \\\\\nX^TX\\theta &= X^Ty \\\\\n\\theta &= (X^TX)^{-1}X^Ty\n\\end {align}\n$$\n\nMatrix calculus can feel a bit handwavy sometimes, if you're not convinced by the derivation above, the following link walks through each individual steps in much more detail. [Blog: The Normal Equation and matrix calculus](https://eli.thegreenplace.net/2015/the-normal-equation-and-matrix-calculus/)\n\n---\n\nAfter solving for the coefficients of the regression model, we can use it for selecting and interpreting features, if all features are on the same scale, the most important features should have the highest coefficients in the model, while features uncorrelated with the output variables should have coefficient values close to zero. This approach can work well when the data is not very noisy (or there is a lot of data compared to the number of features) and the features are (relatively) independent:\n\n\n```python\n# sklearn's LinearRegression may give harmless errors\n# https://github.com/scipy/scipy/issues/5998\nwarnings.filterwarnings(\n action = 'ignore', module = 'scipy', message = '^internal gelsd')\n \n\ndef pretty_print_linear(estimator, names = None, sort = False):\n \"\"\"A helper method for pretty-printing linear models' coefficients\"\"\"\n coef = estimator.coef_\n if names is None:\n names = ['X%s' % x for x in range(1, len(coef) + 1)]\n\n info = zip(coef, names)\n if sort:\n info = sorted(info, key = lambda x: -np.abs(x[0]))\n \n output = ['{} * {}'.format(round(coef, 3), name) for coef, name in info]\n output = ' + '.join(output)\n return output\n \n\n# A dataset with 3 features\nsize = 5000\nnp.random.seed(0)\nX = np.random.normal(0, 1, (size, 3))\n\n# y = X0 + 2 * X1 + noise\ny = X[:, 0] + 2 * X[:, 1] + np.random.normal(0, 2, size)\nlinear = LinearRegression()\nlinear.fit(X, y)\nprint('Linear model:', pretty_print_linear(linear))\n```\n\n Linear model: 0.984 * X1 + 1.995 * X2 + -0.041 * X3\n\n\nAs we can see in this example, the model indeed recovers the underlying structure of the data very well, despite quite significant noise in the data. Given that the the predictors are on the same scale, we can compare the coefficients directly to determine variable importance, we can see here that when using linear regression, X2 is the most important predictor for this given dataset. To be explicit, standardized coefficients represent the mean change in the response given one standard deviation change in the predictor.\n\n\n## R-squared\n\nAfter fitting our predictive model, we would most likely wish to evaluate its performance and **R-squared** is a statistic that is often used to evaluate a regression model's performance. It takes a value ranging from 0 to 1 and is usually interpreted as summarizing the percent of variation in the response that the regression model is capable of explaining. So a R-squared of 0.65 means the model explains about 65% of the variation in our dependent variable. Given this logic, we prefer our regression models have a high R-squared, since we want the model we've trained to capture the output's variance as much as possible. One way to compute R-squared is the sum of squared fitted-value deviations divided by the sum of squared original-value deviations:\n\n$$\n\\begin{align}\nR^{2} = \\frac{\\sum (\\hat{y} - \\bar{\\hat{y}})^{2}}{\\sum (y - \\bar{y})^{2}}\n\\end{align}\n$$\n\n- $y$: original reponse variable.\n- $\\hat{y}$: predicted value for the reponse variable.\n- $\\bar{y}$: The average of reponse variable (pronounced as y bar).\n\nAn alternative form is:\n\n$$\n\\begin{align}\nR^2 = 1 - \\frac{RSS}{TSS} = 1- \\frac{\\sum (y - \\hat{y})^2}{\\sum (y - \\bar{y})^2}\n\\end{align}\n$$\n\n- RSS: Stands for Residual Sum of Squares or referred to as sum of squared error (the measurement that linear model tries of minimize). This value captures the variance that is left between our prediction and the actual values of the output.\n- TSS: Stands for Total Sum of Squares, which measures the total variance in the output variable.\n\n\n```python\ndef rsquared_score(y_true, y_pred):\n \"\"\"rsquared evaluation metric\"\"\"\n rss = np.sum((y_true - y_pred) ** 2)\n tss = np.sum((y_true - np.mean(y_true)) ** 2)\n rsquared = 1 - rss / tss\n return rsquared\n\n\ny_pred = linear.predict(X)\nprint('rsquared:', rsquared_score(y, y_pred))\n\n# we can use scikit-learn's r2_score function\n# by passing in the predicted value and the true label\nprint('rsquared:', r2_score(y, y_pred))\n\n# or for regression models, the default evaluation\n# metric is set to be rsquared and can be accessed\n# through the .score method\nprint('rsquared:', linear.score(X, y))\n```\n\n rsquared: 0.551603126683\n rsquared: 0.551603126683\n rsquared: 0.551603126683\n\n\nThough widely used, this is actually a measurement that requires some context for it to be a valid evaluation metric. We'll give some examples of why:\n\n> R-squared can be arbitrarily close to 1 when the model is totally wrong.\n\n\n```python\n# change default style figure and font size\nplt.rcParams['figure.figsize'] = 8, 6\nplt.rcParams['font.size'] = 12\n\n# generate some exponential data and fit a linear regression to it\nrstate = np.random.RandomState(1)\nx = rstate.exponential(scale = 1 / 0.005, size = 50)\ny = (x - 1) ** 2 * rstate.uniform(low = 0.8, high = 1.2, size = 50)\n\n# scikit-learn model expects a 2d ndarray\n# even if the data only contains 1 feature\nX = x.reshape(-1, 1)\nlinear = LinearRegression()\nlinear.fit(X, y)\nprint('rsquared:', linear.score(X, y))\n\ny_pred = linear.predict(X)\nplt.plot(x, y_pred, 'r')\nplt.scatter(x, y)\nplt.show()\n```\n\nWhen checking the R-squared value for this model, it\u2019s very high at about 0.90, but the model is completely wrong as this data follows a nonlinear distribution. Using R-squared to justify the \"goodness\" of our model in this instance would be a mistake. Hopefully one would plot the data first and recognize that a simple linear regression in this case would be inappropriate.\n\n> We\u2019re better off using Mean Square Error (MSE) or other error-based metric as a measure of prediction error. As R-squared can be anywhere between 0 and 1 just by changing the range of X.\n\nLet\u2019s demonstrate this statement by first generating data that meets all simple linear regression assumptions and then regressing y on x to assess both R-squared and MSE.\n\n\n```python\nx = np.linspace(1, 10, 100)\ny = 2 + 1.2 * x + rstate.normal(loc = 0, scale = 0.9, size = 100)\nX = x.reshape(-1, 1)\nlinear = LinearRegression()\nlinear.fit(X, y)\n\ny_pred = linear.predict(X)\nprint('rsquared:', r2_score(y, y_pred))\nprint('mse:', mean_squared_error(y, y_pred))\n\nplt.plot(x, y_pred, 'r')\nplt.scatter(x, y)\nplt.show()\n```\n\nWe repeat the above code, but this time with a different range of x. Leaving everything else the same:\n\n\n```python\n# smaller range for x\nx = np.linspace(1, 2, 100)\ny = 2 + 1.2 * x + rstate.normal(loc = 0, scale = 0.9, size = 100)\nX = x.reshape(-1, 1)\nlinear = LinearRegression()\nlinear.fit(X, y)\n\ny_pred = linear.predict(X)\nprint('rsquared:', r2_score(y, y_pred))\nprint('mse:', mean_squared_error(y, y_pred))\n\nplt.plot(x, y_pred, 'r')\nplt.scatter(x, y)\nplt.show()\n```\n\nR-squared falls from around 0.9 to around 0.2, but the MSE remains fairly the same. In other words the predictive ability is the same for both data sets, but the R-squared would lead you to believe the first example somehow had a model with more predictive power.\n\n---\n\nThe problem we just tackled was particularly well suited for a linear model: purely linear relationship between features and the response variable, and no correlations between features. The issue arises when there are multiple (linearly) correlated features (as is the case with very many real life datasets), the model then becomes unstable, meaning small changes in the data can cause large changes in the model (i.e. coefficient values), making model interpretation very difficult.\n\nFor example, assume we have a dataset where the \"true\" model for the data is $Y = X1 + X2$, while we observe $\\hat{Y} = X1 + X2 + \\epsilon$, with $\\epsilon$ being the error term. On top of that let's say $X1$ and $X2$ are linearly correlated such that $X1 \\approx X2$. Ideally the learned model will be $Y = X1 + X2$. But depending on the amount of noise $\\epsilon$, the amount of data at hand and the correlation between $X1$ and $X2$, it could also be $Y = 2X1$ (i.e. using only $X1$ as the predictor) or $Y = \u2212X1 + 3X2$ (shifting of the coefficients might happen to give a better fit in the noisy training set) etc.\n\n\n```python\ndef generate_random_data(size, seed):\n \"\"\"Example of collinear features existing within the data\"\"\"\n rstate = np.random.RandomState(seed)\n X_seed = rstate.normal(0, 1, size)\n X1 = X_seed + rstate.normal(0, .1, size)\n X2 = X_seed + rstate.normal(0, .1, size)\n X3 = X_seed + rstate.normal(0, .1, size)\n y = X1 + X2 + X3 + rstate.normal(0, 1, size)\n X = np.array([X1, X2, X3]).T\n return X, y\n \n\nseed = 5\nsize = 100\nX, y = generate_random_data(size, seed)\n\nlinear = LinearRegression()\nlinear.fit(X, y)\nprint('Linear model:', pretty_print_linear(linear))\nprint('rsquared:', linear.score(X, y))\n```\n\n Linear model: -1.291 * X1 + 1.591 * X2 + 2.747 * X3\n rsquared: 0.907885896631\n\n\nThe coefficients of our fitted linear model sums up to ~3, so we can expect it to perform well. On the other hand, if we were to interpret the coefficients at face value, then according to the model $X3$ has a strong positive impact on the output variable, while $X1$ has a negative one, when in fact all the features are correlated and should have equal effects to the output variable. This multicollearity issue also applies to other methods/algorithms and should be addressed before feeding our data to a machine learning method/algorithm.\n\n## Variance Inflation Factor\n\nOne of the most widely used statistical measure of detecting multicollinearity amongst numerical variable is the **Variance Inflation Factor (VIF)**. The VIF may be calculated for each predictor by performing a linear regression of that predictor on all the other predictors, i.e. if we wish to calculate the VIF for predictor $x_k$, then we would use that column as the response variable and use all other columns excluding $x_k$ as the input. After fitted the linear regression, we would then obtain the rsquared value, $R^2$, which tells us how much variance in our predictor $x_k$ can be explained by all the other predictors. Lastly the VIF can be computed using:\n\n$$\n\\begin{align}\nVIF = \\frac{1}{1 - R^2}\n\\end{align}\n$$\n\nIt\u2019s called the variance inflation factor because it estimates how much the variance of a coefficient is \"inflated\" because of linear dependence with other predictors. Thus, a VIF of 1.8 tells us that the variance (the square of the standard error) of a particular coefficient is 80% larger than it would be if that predictor was completely uncorrelated with all the other predictors.\n\n\n```python\ndef remove_collinearity(X, colnames = None, threshold = 5.0):\n \"\"\"\n Identify multi-collinearity between the numeric variables\n using variance inflation factor (vif)\n \"\"\"\n if colnames is None:\n colnames = ['feature' + str(j) for j in range(1, X.shape[1] + 1)]\n\n while True:\n n_features = X.shape[1]\n if n_features == 1:\n break\n\n vif = [compute_vif(X, index) for index in range(n_features)]\n max_index = np.argmax(vif)\n if vif[max_index] >= threshold:\n removed = colnames[max_index]\n colnames.remove(removed)\n X = np.delete(X, max_index, axis = 1)\n else:\n break\n\n return X, colnames\n\n\ndef compute_vif(X, target_index):\n \"\"\"\n Similar implementation as statsmodel's variance_inflation_factor\n with some enhancemants:\n 1. includes the intercept by default\n 2. prevents float division errors (dividing by 0)\n\n References\n ----------\n http://www.statsmodels.org/dev/generated/statsmodels.stats.outliers_influence.variance_inflation_factor.html\n \"\"\"\n n_features = X.shape[1]\n X_target = X[:, target_index]\n mask = np.arange(n_features) != target_index\n X_not_target = X[:, mask]\n\n linear = LinearRegression()\n linear.fit(X_not_target, X_target)\n rsquared = linear.score(X_not_target, X_target)\n vif = 1. / (1. - rsquared + 1e-5)\n return vif\n```\n\n\n```python\n# removing collinearity, thus redundant features\n# while still retaining predictive power\nX, colnames = remove_collinearity(X)\nprint('remaining feature:', colnames)\n\nlinear = LinearRegression()\nlinear.fit(X, y)\nprint('Linear model:', pretty_print_linear(linear))\nprint('rsquared:', linear.score(X, y))\n```\n\n remaining feature: ['feature3']\n Linear model: 3.024 * X1\n rsquared: 0.903735567258\n\n\n## Cramer's V\n\nNow that we've discussed the method for detecting collinearity amongst numerical variables, we will shift our gears towards categorical variables. **Cramer\u2019s V** is a statistic measuring the strength of association or dependency between two (nominal) categorical variables.\n\nSuppose $X$ and $Y$ are two categorical variables that are to be analyzed in a some experimental or observational data with the following information: \n\n- $X$ has $M$ distinct categories or classes, labeled $X_1,\\ldots,X_M$.\n- $Y$ has $N$ distinct categories, labeled $Y_1,\\ldots,Y_N$.\n- Form a $M\\times N$ contingency table such that cell $(i,j)$ contains the count $n_{ij}$ of occurrences of category $X_i$ in $X$ and category $Y_j$ in $Y$. This would give us $n$ total pairs of observations.\n\nWe start of with the null hypothesis that $X$ and $Y$ are independent random variables, then based on the table and the null hypothesis, the chi-squared statistic $\\chi^2$ can be computed. After that, Cramer's V is defined to be:\n\n$$\n\\begin{align}\nV=V(X,Y)=\\sqrt{\\frac{\\chi^2}{n\\operatorname{min}(M-1,N-1)}}\n\\end{align}\n$$\n\nRemarks:\n\n- $0\\leq V\\leq 1$. The closer $V$ is to 0, the smaller the association between the categorical variables $X$ and $Y$. On the other hand, $V$ being close to 1 is an indication of a strong association between $X$ and $Y$. If $X=Y$, then $V(X,Y)=1$.\n- In order for $V$ to make sense, each categorical variable must have at least 2 categories.\n- If one of the categorical variables is dichotomous, i.e. either $M$ or $N=2$, Cramer's V is equal to the **phi statistic** ($\\Phi$), which is defined to be $\\Phi=\\sqrt{\\frac{\\chi^2}{n}}$.\n- Cramer's V is a chi-square based measure of association. The chi-square value depends on the strength of the relationship and sample size, while eliminates the sample size by dividing chi-square by $n$, the sample size, and taking the square root.\n\n\n```python\n# generate a correlated categorical variable\n# and see if cramer's v method will detect it\ndf = pd.DataFrame(index = range(1, 8))\ndf['a'] = ['chicken', 'chicken', 'chicken', 'chicken', 'chat', 'chat', 'chat']\ndf['b'] = ['dog', 'dog', 'dog', 'dog', 'cat', 'cat', 'cat']\nobserved = pd.crosstab(df['a'], df['b'])\nobserved\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
bcatdog
a
chat30
chicken04
\n
\n\n\n\n\n```python\ndef compute_cramersv(observed, correction = False):\n \"\"\"\n Parameters\n ----------\n observed : 2d ndarray\n The contingency table. The table contains the observed frequencies\n (i.e. number of occurrences) for each category.\n\n correction : bool, default False\n If True, and the degrees of freedom is 1, apply Yates\u2019 correction for continuity.\n The effect of the correction is to adjust each observed value by 0.5 towards the\n corresponding expected value. This is set to False by defualt as the effect of\n Yates' correction is to prevent overestimation of statistical significance for small\n data. i.e. It is chiefly used when at least one cell of the table has an expected\n count smaller than 5. And most people probably aren't working with a data size that's\n that small.\n\n Returns\n -------\n cramersv : float\n \"\"\"\n n_obs = observed.sum()\n n_row, n_col = observed.shape\n chi2 = chi2_contingency(observed, correction = correction)[0]\n cramersv = np.sqrt(chi2 / (n_obs * min(n_row - 1, n_col - 1)))\n return cramersv\n\n\ncorrection = False\nobserved = observed.values\ncompute_cramersv(observed, correction)\n```\n\n\n\n\n 1.0\n\n\n\n# Reference\n\n\n- [Blog: Cramer\u2019s V](http://planetmath.org/cramersv)\n- [Blog: Is R-squared Useless?](http://data.library.virginia.edu/is-r-squared-useless/)\n- [Blog: Deriving normal equation](https://wiseodd.github.io/techblog/2017/04/14/normal-equation/)\n- [Blog: The Principle of Maximum Likelihood](http://suriyadeepan.github.io/2017-01-22-mle-linear-regression/)\n- [Blog: The Normal Equation and matrix calculus](https://eli.thegreenplace.net/2015/the-normal-equation-and-matrix-calculus/)\n- [Blog: When Can You Safely Ignore Multicollinearity?](https://statisticalhorizons.com/multicollinearity)\n- [Blog: Selecting good features \u2013 Part II: linear models and regularization\n](http://blog.datadive.net/selecting-good-features-part-ii-linear-models-and-regularization/)\n- [Blog: How to Identify the Most Important Predictor Variables in Regression Models](http://blog.minitab.com/blog/adventures-in-statistics-2/how-to-identify-the-most-important-predictor-variables-in-regression-models)\n- [Github: Duplicate (and highly correlated) categoricals](https://github.com/JosPolfliet/pandas-profiling/issues/40)\n", "meta": {"hexsha": "5f8f5fd7c5b88f92c636f9084db596ca40da0638", "size": 189571, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "model_selection/collinearity.ipynb", "max_stars_repo_name": "certara-ShengnanHuang/machine-learning", "max_stars_repo_head_hexsha": "d21dfbeabf2876ffe49fcef444ca4516c4d36df0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2104, "max_stars_repo_stars_event_min_datetime": "2016-04-15T13:35:55.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-28T10:39:51.000Z", "max_issues_repo_path": "model_selection/collinearity.ipynb", "max_issues_repo_name": "certara-ShengnanHuang/machine-learning", "max_issues_repo_head_hexsha": "d21dfbeabf2876ffe49fcef444ca4516c4d36df0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 10, "max_issues_repo_issues_event_min_datetime": "2017-04-07T14:25:23.000Z", "max_issues_repo_issues_event_max_datetime": "2021-05-18T03:16:15.000Z", "max_forks_repo_path": "model_selection/collinearity.ipynb", "max_forks_repo_name": "certara-ShengnanHuang/machine-learning", "max_forks_repo_head_hexsha": "d21dfbeabf2876ffe49fcef444ca4516c4d36df0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 539, "max_forks_repo_forks_event_min_datetime": "2015-12-10T04:23:44.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-31T07:15:28.000Z", "avg_line_length": 165.5641921397, "max_line_length": 51742, "alphanum_fraction": 0.8554050989, "converted": true, "num_tokens": 8929, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443134, "lm_q2_score": 0.7690802423634963, "lm_q1q2_score": 0.44706872183937796}} {"text": "```python\nimport arviz as az\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport pandas as pd\n\nimport warnings\nwarnings.simplefilter(action='ignore', category=FutureWarning)\nwarnings.simplefilter(action='ignore', category=UserWarning)\n\n\nimport pymc3 as pm\nimport theano\n\nprint('Running on PyMC3 v{}'.format(pm.__version__))\n```\n\n Running on PyMC3 v3.9.3\n\n\n\n```python\n%config InlineBackend.figure_format = 'retina'\naz.style.use('default')\n```\n\n# A Tour of Model Checking techniques using PyMC3\n\n
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\n\nRob Zinkov\n\n## Outline\n\n1. Introduction\n2. Setting up example problem\n3. Motivate why model checking is necessary\n4. Cross validation and predictive accuracy / LOO\n5. Prior & Posterior Predictive Checks / Population Predictive Checks\n6. BIC / DIC / WAIC\n7. Likelihood on heldout data\n8. Marginal Likelihood\n9. Likelihood ratios, Bayes factors\n10. MMD / relative MMD / Kernel Stein Discrepancy / Witness function\n11. Posterior dispersion indices\n12. Flowchart of when to use different techniques\n\n\n# Introduction\n\nOne of the ongoing challenges when fitting models is figuring out\nif what we have is any good.\n\nThe issue is there a good guidelines on the model fitting side,\nbut less so on model evaluation side\n\nI'll go over several methods that exist to give an overview of\nhow to evaluate and compare different models.\n\nI will cover both Bayesian *and* Frequentist methods. They each\nhave their places and evaluating models is a place where both can shine\n\n## Model Evaluation vs Model Selection\n\nWhile I consider these to be two aspects of the same process sometimes it'll help to distinguish\n\n*Model Evaluation* asks is my model of the data any good?\n\n(Sometimes is called Model checking or Model criticism)\n\n*Model Selection* asks which of these models is the best fit for my data\n\n(Sometimes is called Model comparison)\n\n\n \n\n# Assumptions and Cavaets\n\n* Assuming you have run and fit a few models in pymc3\n* I won't nor can cover all goodness of fit tests\n* Not all tests are or should be natively supported by pymc3\n\n# Running example\n\n\n\nRadon is an odorless colorless gas that in high concentrations is known to cause\nlung cancer. Levels vary from household to household but also geographically.\n\nThe EPA conducted a study and found out that the presence of a basement as well\nas general levels of uranium in the level are predictive of radon levels.\n\n\n\n
\n\n## Running example\n\nGelman et al. (2013) use a hierarchical Bayesian model to fit and predict\nradon levels. These and other models have been ported to pymc3 and the radon\ndata comes included with the library.\n\nSo we'll be using these models as a running example of picking between different models\n\nhttps://docs.pymc.io/notebooks/multilevel_modeling.html\nby Chris Fonnesbeck covers the models themselves in more detail\n\n\n```python\n\n```\n\n\n```python\n# Read data\nsrrs2 = pd.read_csv(pm.get_data(\"srrs2.dat\"))\nsrrs2.columns = srrs2.columns.map(str.strip)\nsrrs_mn = srrs2[srrs2.state == \"MN\"].copy()\n\n# Add county information\nsrrs_mn.county = srrs_mn.county.map(str.strip)\nmn_counties = srrs_mn.county.unique()\ncounties = len(mn_counties)\ncounty_lookup = dict(zip(mn_counties, range(counties)))\n\ncounty = srrs_mn[\"county_code\"] = srrs_mn.county.replace(county_lookup).values\nradon = srrs_mn[\"radon\"] = srrs_mn.activity\nsrrs_mn[\"log_radon\"] = log_radon = np.log(radon + 0.1).values\nfloor = srrs_mn.floor.values\n```\n\n## Radon prediction\n\nThe way this works is we will try to predict radon levels based on:\n\n1. What county the house is located in\n2. Whether the measurement was taken in the Basement (floor 0) or Ground Level (floor 1)\n\n\n```python\nsrrs_mn[['floor', 'county', 'radon']].head(n=10).style.hide_index()\n```\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
floor county radon
1AITKIN2.200000
0AITKIN2.200000
0AITKIN2.900000
0AITKIN1.000000
0ANOKA3.100000
0ANOKA2.500000
0ANOKA1.500000
0ANOKA1.000000
0ANOKA0.700000
0ANOKA1.200000
\n\n\n\n\n```python\nsrrs_mn.groupby(['county']).size().sort_values().reset_index(name='counts')\n```\n\n\n\n\n
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countycounts
0WILKIN1
1MAHNOMEN1
2MURRAY1
3YELLOW MEDICINE2
4LAC QUI PARLE2
.........
80WASHINGTON46
81ANOKA52
82DAKOTA63
83HENNEPIN105
84ST LOUIS116
\n

85 rows \u00d7 2 columns

\n
\n\n\n\n## Radon prediction\n\n\n```python\nsrrs_mn.log_radon.hist(bins=25, rwidth=0.9);\n```\n\n## Radon prediction models\n\n**Complete pooling:**\n\nTreat all counties the same, and estimate a single radon level.\n\n$$ y_i = \\alpha +\\beta x_i + \\epsilon_i $$\n\n**No pooling:**\n\nModel radon in each county independently.\n\n$$ y_i = \\alpha_{j[i]} +\\beta x_i + \\epsilon_i $$\n\nwhere $j=1,\\ldots,85$\n\n## Radon prediction models (continued)\n\n\n**Partial pooling**\n\n$$ y_i = \\alpha_{j[i]} +\\beta x_i + \\epsilon_i $$\n\n$$ \\alpha_{j[i]} \\sim \\mathcal{N}(\\mu_\\alpha, \\sigma^2_\\alpha) $$\n\nwhere $j=1,\\ldots,85$\n\n# Predictive accuracy\n\nThe best way to evaluate models is on some downstream task\n\n\n\n## Predictive accuracy\n\nOne way to evaluate this is to set aside some data and see\nif our model can predict the true value from this unseen data.\n\n\n\n\n```python\n\n```\n\n\n```python\ndef split_data(df, fraction=0.2):\n out_sample = df.sample(frac = fraction) \n in_sample = df.drop(out_sample.index)\n return in_sample, out_sample\n```\n\n\n```python\n# Set aside some of the data\nsrrs_in_sample, srrs_out_sample = split_data(srrs_mn, fraction=0.1)\n\ncounty_in = srrs_in_sample.county_code.values\nradon_in = srrs_in_sample.log_radon.values\nfloor_in = srrs_in_sample.floor.values\n\ncounty_out = srrs_out_sample.county_code.values\nradon_out = srrs_out_sample.log_radon.values\nfloor_out = srrs_out_sample.floor.values\n```\n\n\n```python\n\n```\n\n\n```python\ncoords = {\n \"Level\": [\"Basement\", \"Floor\"],\n \"obs_id\": np.arange(floor_in.size)\n}\nwith pm.Model(coords=coords) as pooled_model:\n floor_idx = pm.Data(\"floor_idx\", floor_in, dims=\"obs_id\")\n b = pm.Normal(\"b\", 0.0, sigma=10.0, dims=\"Level\")\n\n theta = b[floor_idx]\n sigma = pm.Exponential(\"sigma\", 1.0)\n\n radon_shared = pm.Data(\"radon\", radon_in, dims=\"obs_id\")\n y = pm.Normal(\"y\", theta, sigma=sigma, observed=radon_shared, dims=\"obs_id\")\n pooled_trace = pm.sample(random_seed=42)\n```\n\n Auto-assigning NUTS sampler...\n Initializing NUTS using jitter+adapt_diag...\n Multiprocess sampling (2 chains in 2 jobs)\n NUTS: [sigma, b]\n\n\n\n\n
\n \n \n 100.00% [4000/4000 00:02<00:00 Sampling 2 chains, 0 divergences]\n
\n\n\n\n Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 3 seconds.\n\n\n\n```python\naz.summary(pooled_trace)\n```\n\n\n\n\n
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meansdhdi_3%hdi_97%mcse_meanmcse_sdess_meaness_sdess_bulkess_tailr_hat
b[0]1.3510.0301.2951.4060.0010.0002366.02366.02358.01301.01.0
b[1]0.7760.0650.6470.8900.0010.0012753.02753.02757.01248.01.0
sigma0.7680.0200.7310.8050.0000.0002511.02511.02502.01430.01.0
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\n\n\n\n\n```python\ncoords = {\n \"Level\": [\"Basement\", \"Floor\"],\n \"obs_id\": np.arange(floor_in.size),\n \"County\": mn_counties\n}\nwith pm.Model(coords=coords) as unpooled_model:\n floor_idx = pm.Data(\"floor_idx\", floor_in, dims=\"obs_id\")\n county_idx = pm.Data(\"county_idx\", county_in, dims=\"obs_id\")\n a = pm.Normal(\"a\", 0.0, sigma=10.0, dims=(\"County\", \"Level\"))\n\n theta = a[county_idx, floor_idx]\n sigma = pm.Exponential(\"sigma\", 1.0)\n\n radon_shared = pm.Data(\"radon\", radon_in, dims=\"obs_id\")\n y = pm.Normal(\"y\", theta, sigma=sigma, observed=radon_shared, dims=\"obs_id\")\n unpooled_trace = pm.sample(random_seed=42)\n```\n\n Auto-assigning NUTS sampler...\n Initializing NUTS using jitter+adapt_diag...\n Multiprocess sampling (2 chains in 2 jobs)\n NUTS: [sigma, a]\n\n\n\n\n
\n \n \n 100.00% [4000/4000 00:09<00:00 Sampling 2 chains, 0 divergences]\n
\n\n\n\n Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 9 seconds.\n\n\n\n```python\naz.summary(unpooled_trace)\n```\n\n\n\n\n
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meansdhdi_3%hdi_97%mcse_meanmcse_sdess_meaness_sdess_bulkess_tailr_hat
a[0,0]0.6670.397-0.0451.4560.0060.0064175.02421.04194.01313.01.00
a[0,1]0.8140.687-0.3902.1770.0120.0133122.01316.03132.01267.01.00
a[1,0]0.9640.1020.7691.1480.0020.0012488.02469.02487.01529.01.00
a[1,1]-0.1300.392-0.8390.6050.0070.0092953.01044.02988.01490.01.00
a[2,0]1.5010.6960.2142.8600.0120.0103446.02648.03452.01316.01.00
....................................
a[83,0]1.6730.1981.3072.0260.0040.0033134.03120.03126.01368.01.01
a[83,1]-0.2639.747-18.24617.8400.1790.2492970.0765.02959.01447.01.00
a[84,0]1.2020.4800.3282.1380.0080.0073192.02722.03214.01468.01.00
a[84,1]0.08510.109-19.44419.2650.1680.2753613.0677.03606.01146.01.00
sigma0.6930.0190.6570.7270.0000.0002293.02293.02251.01394.01.00
\n

171 rows \u00d7 11 columns

\n
\n\n\n\n\n```python\ncoords = {\n \"Level\": [\"Basement\", \"Floor\"],\n \"obs_id\": np.arange(floor_in.size),\n \"County\": mn_counties\n}\nwith pm.Model(coords=coords) as partial_pooling:\n floor_idx = pm.Data(\"floor_idx\", floor_in, dims=\"obs_id\")\n county_idx = pm.Data(\"county_idx\", county_in, dims=\"obs_id\")\n # Hyperpriors:\n a = pm.Normal(\"a\", mu=0.0, sigma=10.0)\n sigma_a = pm.Exponential(\"sigma_a\", 1.0)\n\n # Varying intercepts:\n a_county = pm.Normal(\"a_county\", mu=a, sigma=sigma_a, dims=\"County\")\n # Common slope:\n b = pm.Normal(\"b\", mu=0.0, sigma=10.0)\n\n # Expected value per county:\n theta = a_county[county_idx] + b * floor_idx\n # Model error:\n sigma = pm.Exponential(\"sigma\", 1.0)\n\n radon_shared = pm.Data(\"radon\", radon_in, dims=\"obs_id\")\n y = pm.Normal(\"y\", theta, sigma=sigma, observed=radon_shared, dims=\"obs_id\")\n partial_pooling_trace = pm.sample(random_seed=42)\n```\n\n Auto-assigning NUTS sampler...\n Initializing NUTS using jitter+adapt_diag...\n Multiprocess sampling (2 chains in 2 jobs)\n NUTS: [sigma, b, a_county, sigma_a, a]\n\n\n\n\n
\n \n \n 100.00% [4000/4000 00:06<00:00 Sampling 2 chains, 0 divergences]\n
\n\n\n\n Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 7 seconds.\n\n\n\n```python\n# Hack since pm.find_MAP diverges\n\nbest_ll = -np.infty\nbest = None\n\npp_logp = partial_pooling.logp\nfor p in partial_pooling_trace:\n ll = pp_logp(p)\n if ll > best_ll:\n best = p\n best_ll = ll\n \npartial_pooled_point_estimate = best\n```\n\n## Predictive accuracy\n\nFor continuous data root mean squared error (rmse) is a popular choice\n\n$$ \\text{RMSE}(Y) = \\sqrt{\\frac{1}{n}\\sum_{i=1}^n(Y_i-\\mathbb{E}[Y_i \\mid \\theta])^2}$$\n\n\n```python\ndef rmse(test_data, predicted):\n \"\"\"Calculate root mean squared error.\n Ignoring missing values in the test data.\n \"\"\"\n I = ~np.isnan(test_data) # indicator for missing values\n N = I.sum() # number of non-missing values\n sqerror = abs(test_data - predicted) ** 2 # squared error array\n mse = sqerror[I].sum() / N # mean squared error\n return np.sqrt(mse)\n```\n\n\n```python\nwith pooled_model:\n pooled_point_estimate = pm.find_MAP(model=pooled_model)\n # change the value and shape of the data\n pm.set_data({\n \"floor_idx\": floor_out,\n # use dummy values with the same shape:\n \"radon\": radon_out,\n })\n\n post_pred = pm.sample_posterior_predictive(pooled_trace)\n```\n\n\n\n
\n \n \n 100.00% [14/14 00:00<00:00 logp = -960.74, ||grad|| = 8.0476]\n
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\n \n \n 100.00% [2000/2000 00:20<00:00]\n
\n\n\n\n\n```python\npreds = post_pred[\"y\"].mean(axis=0)\n# lower is better\nrmse(preds, radon_out)\n```\n\n\n\n\n 0.9721695872119326\n\n\n\n\n```python\nwith unpooled_model:\n unpooled_point_estimate = pm.find_MAP(model=unpooled_model)\n # change the value and shape of the data\n pm.set_data({\n \"floor_idx\": floor_out,\n \"county_idx\": county_out,\n # use dummy values with the same shape:\n \"radon\": radon_out,\n })\n\n post_pred = pm.sample_posterior_predictive(unpooled_trace)\n```\n\n\n\n
\n \n \n 100.00% [75/75 00:00<00:00 logp = -1,343.4, ||grad|| = 0.041008]\n
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\n\n\n\n\n```python\npreds = post_pred[\"y\"].mean(axis=0)\n# lower is better\nrmse(preds, radon_out)\n```\n\n\n\n\n 1.0243686247273727\n\n\n\n\n```python\nwith partial_pooling:\n # change the value and shape of the data\n pm.set_data({\n \"floor_idx\": floor_out,\n \"county_idx\": county_out,\n # use dummy values with the same shape:\n \"radon\": radon_out,\n })\n\n post_pred = pm.sample_posterior_predictive(partial_pooling_trace)\n```\n\n\n\n
\n \n \n 100.00% [2000/2000 00:19<00:00]\n
\n\n\n\n\n```python\npreds = post_pred[\"y\"].mean(axis=0)\n# lower is better\nrmse(preds, radon_out)\n```\n\n\n\n\n 0.9299338549155656\n\n\n\n\n```python\n\n```\n\n# Cross validation\n\n## Cross validation\n\nPredictive accuracy is based on a single split which introduces bias. We can do multiple\nsplits and average the accuracy across these splits. This is called *cross-validation*\n\nExercise 1:\n\nImplement cross-validation and report results on radon models\n\n# Likelihood on held-out data\n\n## Held-out likelihood\n\nIn the same spirit of predictive accuracy, we can simply ask on unseen data, which of the models is least \u201csurprised\u201d\n\n\n```python\npooled_point_estimate[\"y\"] = radon_out\nunpooled_point_estimate[\"y\"] = radon_out\npartial_pooled_point_estimate[\"y\"] = radon_out\n\nprint(\"Pooled model LL:\", pooled_model.logp(pooled_point_estimate))\nprint(\"Unpooled model LL:\", unpooled_model.logp(unpooled_point_estimate))\nprint(\"Partially-pooled model LL:\", partial_pooling.logp(partial_pooled_point_estimate))\n```\n\n Pooled model LL: -141.3818492829713\n Unpooled model LL: -710.3533170768483\n Partially-pooled model LL: -128.2446075503398\n\n\n# Marginal Likelihood\n\nMarginal likelihood measures the probability of observing our data under the model family. This is also more Bayesian as we don't use a point estimate of our parameters\n\n\n```python\ncoords = {\n \"Level\": [\"Basement\", \"Floor\"],\n \"obs_id\": np.arange(floor.size)\n}\nwith pm.Model(coords=coords) as pooled_model:\n floor_idx = pm.Data(\"floor_idx\", floor, dims=\"obs_id\")\n b = pm.Normal(\"b\", 0.0, sigma=10.0, dims=\"Level\")\n\n theta = b[floor_idx]\n sigma = pm.Exponential(\"sigma\", 1.0)\n\n radon_shared = pm.Data(\"radon\", log_radon, dims=\"obs_id\")\n y = pm.Normal(\"y\", theta, sigma=sigma, observed=radon_shared, dims=\"obs_id\")\n pooled_trace = pm.sample(random_seed=42)\n \ncoords = {\n \"Level\": [\"Basement\", \"Floor\"],\n \"obs_id\": np.arange(floor.size),\n \"County\": mn_counties\n}\nwith pm.Model(coords=coords) as unpooled_model:\n floor_idx = pm.Data(\"floor_idx\", floor, dims=\"obs_id\")\n county_idx = pm.Data(\"county_idx\", county, dims=\"obs_id\")\n a = pm.Normal(\"a\", 0.0, sigma=10.0, dims=(\"County\", \"Level\"))\n\n theta = a[county_idx, floor_idx]\n sigma = pm.Exponential(\"sigma\", 1.0)\n\n radon_shared = pm.Data(\"radon\", log_radon, dims=\"obs_id\")\n y = pm.Normal(\"y\", theta, sigma=sigma, observed=radon_shared, dims=\"obs_id\")\n unpooled_trace = pm.sample(random_seed=42)\n \ncoords = {\n \"Level\": [\"Basement\", \"Floor\"],\n \"obs_id\": np.arange(floor.size),\n \"County\": mn_counties\n}\nwith pm.Model(coords=coords) as partial_pooling:\n floor_idx = pm.Data(\"floor_idx\", floor, dims=\"obs_id\")\n county_idx = pm.Data(\"county_idx\", county, dims=\"obs_id\")\n # Hyperpriors:\n a = pm.Normal(\"a\", mu=0.0, sigma=10.0)\n sigma_a = pm.Exponential(\"sigma_a\", 1.0)\n\n # Varying intercepts:\n a_county = pm.Normal(\"a_county\", mu=a, sigma=sigma_a, dims=\"County\")\n # Common slope:\n b = pm.Normal(\"b\", mu=0.0, sigma=10.0)\n\n # Expected value per county:\n theta = a_county[county_idx] + b * floor_idx\n # Model error:\n sigma = pm.Exponential(\"sigma\", 1.0)\n\n radon_shared = pm.Data(\"radon\", log_radon, dims=\"obs_id\")\n y = pm.Normal(\"y\", theta, sigma=sigma, observed=radon_shared, dims=\"obs_id\")\n partial_pooling_trace = pm.sample(random_seed=42)\n```\n\n Auto-assigning NUTS sampler...\n Initializing NUTS using jitter+adapt_diag...\n Multiprocess sampling (2 chains in 2 jobs)\n NUTS: [sigma, b]\n\n\n\n\n
\n \n \n 100.00% [4000/4000 00:02<00:00 Sampling 2 chains, 0 divergences]\n
\n\n\n\n Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 2 seconds.\n Auto-assigning NUTS sampler...\n Initializing NUTS using jitter+adapt_diag...\n Multiprocess sampling (2 chains in 2 jobs)\n NUTS: [sigma, a]\n\n\n\n\n
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\n\n\n\n Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 12 seconds.\n Auto-assigning NUTS sampler...\n Initializing NUTS using jitter+adapt_diag...\n Multiprocess sampling (2 chains in 2 jobs)\n NUTS: [sigma, b, a_county, sigma_a, a]\n\n\n\n\n
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\n\n\n\n Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 8 seconds.\n\n\n\n```python\nwith pooled_model:\n pooled_smc_trace = pm.sample_smc(4000)\n```\n\n Initializing SMC sampler...\n Sampling 2 chains in 2 jobs\n Stage: 0 Beta: 0.000\n Stage: 1 Beta: 0.000\n Stage: 2 Beta: 0.001\n Stage: 3 Beta: 0.003\n Stage: 4 Beta: 0.007\n Stage: 5 Beta: 0.015\n Stage: 6 Beta: 0.037\n Stage: 7 Beta: 0.092\n Stage: 8 Beta: 0.238\n Stage: 9 Beta: 0.608\n Stage: 10 Beta: 1.000\n Stage: 0 Beta: 0.000\n Stage: 1 Beta: 0.000\n Stage: 2 Beta: 0.001\n Stage: 3 Beta: 0.003\n Stage: 4 Beta: 0.007\n Stage: 5 Beta: 0.015\n Stage: 6 Beta: 0.036\n Stage: 7 Beta: 0.090\n Stage: 8 Beta: 0.230\n Stage: 9 Beta: 0.588\n Stage: 10 Beta: 1.000\n\n\n\n```python\nwith unpooled_model:\n unpooled_smc_trace = pm.sample_smc(4000)\n```\n\n Initializing SMC sampler...\n Sampling 2 chains in 2 jobs\n Stage: 0 Beta: 0.000\n Stage: 1 Beta: 0.000\n Stage: 2 Beta: 0.000\n Stage: 3 Beta: 0.001\n Stage: 4 Beta: 0.003\n Stage: 5 Beta: 0.006\n Stage: 6 Beta: 0.011\n Stage: 7 Beta: 0.019\n Stage: 8 Beta: 0.028\n Stage: 9 Beta: 0.039\n Stage: 10 Beta: 0.050\n Stage: 11 Beta: 0.063\n Stage: 12 Beta: 0.076\n Stage: 13 Beta: 0.089\n Stage: 14 Beta: 0.104\n Stage: 15 Beta: 0.120\n Stage: 16 Beta: 0.137\n Stage: 17 Beta: 0.154\n Stage: 18 Beta: 0.173\n Stage: 19 Beta: 0.195\n Stage: 20 Beta: 0.216\n Stage: 21 Beta: 0.239\n Stage: 22 Beta: 0.264\n Stage: 23 Beta: 0.292\n Stage: 24 Beta: 0.323\n Stage: 25 Beta: 0.354\n Stage: 26 Beta: 0.390\n Stage: 27 Beta: 0.427\n Stage: 28 Beta: 0.468\n Stage: 29 Beta: 0.513\n Stage: 30 Beta: 0.561\n Stage: 31 Beta: 0.614\n Stage: 32 Beta: 0.673\n Stage: 33 Beta: 0.737\n Stage: 34 Beta: 0.807\n Stage: 35 Beta: 0.882\n Stage: 36 Beta: 0.964\n Stage: 37 Beta: 1.000\n Stage: 0 Beta: 0.000\n Stage: 1 Beta: 0.000\n Stage: 2 Beta: 0.000\n Stage: 3 Beta: 0.001\n Stage: 4 Beta: 0.003\n Stage: 5 Beta: 0.006\n Stage: 6 Beta: 0.011\n Stage: 7 Beta: 0.018\n Stage: 8 Beta: 0.027\n Stage: 9 Beta: 0.038\n Stage: 10 Beta: 0.050\n Stage: 11 Beta: 0.062\n Stage: 12 Beta: 0.075\n Stage: 13 Beta: 0.090\n Stage: 14 Beta: 0.104\n Stage: 15 Beta: 0.119\n Stage: 16 Beta: 0.136\n Stage: 17 Beta: 0.154\n Stage: 18 Beta: 0.173\n Stage: 19 Beta: 0.193\n Stage: 20 Beta: 0.216\n Stage: 21 Beta: 0.239\n Stage: 22 Beta: 0.264\n Stage: 23 Beta: 0.292\n Stage: 24 Beta: 0.322\n Stage: 25 Beta: 0.353\n Stage: 26 Beta: 0.387\n Stage: 27 Beta: 0.424\n Stage: 28 Beta: 0.464\n Stage: 29 Beta: 0.510\n Stage: 30 Beta: 0.557\n Stage: 31 Beta: 0.611\n Stage: 32 Beta: 0.668\n Stage: 33 Beta: 0.731\n Stage: 34 Beta: 0.803\n Stage: 35 Beta: 0.881\n Stage: 36 Beta: 0.965\n Stage: 37 Beta: 1.000\n\n\n\n```python\nwith partial_pooling:\n partial_smc_trace = pm.sample_smc(4000)\n```\n\n Initializing SMC sampler...\n Sampling 2 chains in 2 jobs\n Stage: 0 Beta: 0.000\n Stage: 1 Beta: 0.000\n Stage: 2 Beta: 0.001\n Stage: 3 Beta: 0.003\n Stage: 4 Beta: 0.006\n Stage: 5 Beta: 0.011\n Stage: 6 Beta: 0.029\n Stage: 7 Beta: 0.054\n Stage: 8 Beta: 0.089\n Stage: 9 Beta: 0.139\n Stage: 10 Beta: 0.201\n Stage: 11 Beta: 0.276\n Stage: 12 Beta: 0.363\n Stage: 13 Beta: 0.461\n Stage: 14 Beta: 0.572\n Stage: 15 Beta: 0.696\n Stage: 16 Beta: 0.830\n Stage: 17 Beta: 0.969\n Stage: 18 Beta: 1.000\n Stage: 0 Beta: 0.000\n Stage: 1 Beta: 0.000\n Stage: 2 Beta: 0.001\n Stage: 3 Beta: 0.003\n Stage: 4 Beta: 0.005\n Stage: 5 Beta: 0.011\n Stage: 6 Beta: 0.019\n Stage: 7 Beta: 0.034\n Stage: 8 Beta: 0.063\n Stage: 9 Beta: 0.111\n Stage: 10 Beta: 0.171\n Stage: 11 Beta: 0.243\n Stage: 12 Beta: 0.326\n Stage: 13 Beta: 0.419\n Stage: 14 Beta: 0.522\n Stage: 15 Beta: 0.633\n Stage: 16 Beta: 0.751\n Stage: 17 Beta: 0.875\n Stage: 18 Beta: 1.000\n\n\n\n```python\nprint(\"pooled marginal_likelihood:\", pooled_smc_trace.report.log_marginal_likelihood)\nprint(\"unpooled marginal likelihood:\", unpooled_smc_trace.report.log_marginal_likelihood)\nprint(\"partial_pooling marginal_likelihood:\", partial_smc_trace.report.log_marginal_likelihood)\n```\n\n pooled marginal_likelihood: [-1101.22611358 -1101.18021879]\n unpooled marginal likelihood: [-2283.75780614 -2288.24476468]\n partial_pooling marginal_likelihood: [-1075.34181204 -1070.76515456]\n\n\n## Bayes factors\n\nMarginal likelihoods are nice for evaluating models but they can be used to compare models \n\n## Bayes factors\n\nThe intuition is suppose we have two models $p(M_0 \\mid y)$ and $p(M_1 \\mid y)$ and we want to choose between them\n\n\n## Bayes factors\n\nThanks to some algebra we don't need to compute the normalising constants and just need\nthe likelihoods and the priors for the models.\n\n$$\n\\begin{align}\nBF &= \\frac{p(y|M_0)}{p(y|M_1)} \\\\\n&= \\frac{\\int p(\\theta_0|M_0)p(D|\\theta_0,M_0)\\,d\\theta_0}\n{\\int p(\\theta_1|M_1)p(y|\\theta_1,M_1)\\,d\\theta_1} \\\\\n&= \\frac{\\frac{p(M_0|y)p(y)}{p(M_0)}}{\\frac{p(M_1|y)p(y)}{p(M_1)}} \\\\\n&= \\frac{p(M_0|y)}{p(M_1|y)}\\frac{p(M_1)}{p(M_0)}\n\\end{align}\n$$\n\nWe assume each model is equally likely\n\n## Bayes factors\n\nThis ratio has some informal intuition over how much you said favour $M_0$ over $M_1$\n\n* 1-3: anecdotal\n* 3-10: moderate\n* 10-30: strong\n* 30-100: very strong\n* 100+: extreme\n\n## Bayes factors drawbacks\n\n1. Must be a pair of models\n2. Results are sensitive to latent variables\n3. Made for discrete set of models\n\n\n```python\nBF_smc = np.exp(partial_smc_trace.report.log_marginal_likelihood - \n pooled_smc_trace.report.log_marginal_likelihood)\nprint(round(BF_smc[0]))\n```\n\n 174344931409\n\n\n# Prior & Posterior Predictive Checks\n\n## Prior Predictive Checks\n\nPrior Predictive Checks are there to check if your priors are reasonable by comparing them to your data\n\n\n```python\n\n```\n\n\n```python\nwith pooled_model:\n prior_check = pm.sample_prior_predictive(samples=50, random_seed=42)\n```\n\n\n```python\nidata = az.from_pymc3(pooled_trace, prior=prior_check)\naz.plot_ppc(idata, group='prior');\n```\n\nExercise 2:\n\nAdjust radon models to use a more realistic prior. Does this lead to improved models? Why or why not?\n\n## Posterior Predictive Checks\n\nPosterior Predictive Checks allow us to see if after our model is fitted does it generate replicated data\nthat resembles our observed data\n\n\n```python\nwith pooled_model:\n ppc = pm.sample_posterior_predictive(pooled_trace, random_seed=42)\nidata = az.from_pymc3(pooled_trace, posterior_predictive=ppc)\naz.plot_ppc(idata);\n```\n\n\n```python\nwith unpooled_model:\n ppc = pm.sample_posterior_predictive(unpooled_trace, random_seed=42)\nidata = az.from_pymc3(unpooled_trace, posterior_predictive=ppc)\naz.plot_ppc(idata);\n```\n\n\n```python\nwith partial_pooling:\n ppc = pm.sample_posterior_predictive(partial_pooling_trace, random_seed=42)\n \nidata = az.from_pymc3(partial_pooling_trace, posterior_predictive=ppc)\naz.plot_ppc(idata);\n```\n\n## Posterior Predictive Checks\n\nWe usually in PyMC3 and Stan we usually plot our observed data and compare to posterior predictive distribution visually.\n\nBut any discrepancy measure can be used here.\n\n# Information Criterions\n\n## Information Criterions\n\nSometimes though, we want a single number to just be able to score our models. These are called *information criterions*\n\n## Akaike information criterion (AIC)\n\n$$ -2\\log p(y \\mid \\hat{\\theta}) + 2\\text{dim}(\\hat{\\theta}) $$\n\nwhere $\\hat{\\theta} = \\text{argmax}_\\theta \\, p(y \\mid \\theta)$\n\nNot very Bayesian\n\n## Bayesian information criterion (BIC)\n\n$$ -2\\log p(y \\mid \\hat{\\theta}) + \\text{dim}(\\hat{\\theta}) \\log n $$\n\nwhere $\\hat{\\theta} = \\text{argmax}_\\theta \\, p(y \\mid \\theta)$\n\nStill not very Bayesian\n\n\n```python\n\n```\n\n## Deviance information criterion (DIC)\n\n$$ 2\\left(\\log p(y \\mid \\mathbb{E}[\\theta \\mid y]) - \\mathbb{E}[\\log p(y \\mid \\theta)]\\right) $$\n\nwhere both expectations are over $p(\\theta \\mid y)$\n\nKinda unstable\n\nExercise 3:\n\nImplement AIC, BIC, and DIC for the radon models?\n\nHint: `model.ndim` gives the number of parameters\n\n## Watanabe-Akaike or widely available information criterion (WAIC)\n\nThis measure is actually Bayesian and more stable than DIC\n\n$$-2\\frac{1}{n} \\sum_{i=1}^n \\log \\mu(y_i) + \\sigma^2_{log}(y_i)$$\n\nwhere $\\mu(y_i) = \\mathbb{E}[p(y_i \\mid \\theta)]$ and $\\sigma^2_{log}(y_i) = \\mathbb{V}[\\log p(y_i \\mid \\theta)]$\n\nboth $\\mu$ and $\\sigma^2_{log}$ and wrt to the $p(\\theta \\mid y)$ distribution\n\n\n```python\npm.compare({\n 'pooled': pooled_trace,\n 'unpooled': unpooled_trace,\n 'partially_pooled': partial_pooling_trace,\n },\n scale=\"log\",\n ic=\"waic\")\n```\n\n\n\n\n
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rankwaicp_waicd_waicweightsedsewarningwaic_scale
partially_pooled0-1036.3948.540400.99999925.80180Truelog
unpooled1-1080.25131.96743.85582.10673e-0730.798812.12Truelog
pooled2-1089.913.7628453.51344.19402e-0728.986910.6924Falselog
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\n\n\n\n## Leave one out Cross-Validations (LOO-cv)\n\nThe idea behind LOO-cv is that we can repeatedly\n\n1. Set aside one of our data points $y_i$\n2. Fit a model on the rest of data $p(\\theta \\mid y_{-i})$\n3. Evaluate $\\log \\mathbb{E}[y_i \\mid \\theta]$\n4. Repeat for each data point and average the results\n\n\n```python\npm.compare({\n 'pooled': pooled_trace,\n 'unpooled': unpooled_trace,\n 'partially_pooled': partial_pooling_trace,\n },\n scale=\"log\",\n ic=\"loo\")\n```\n\n\n\n\n
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rankloop_lood_looweightsedsewarningloo_scale
partially_pooled0-1036.7748.9170130.87890Falselog
pooled1-1089.913.7624753.13631.60083e-0927.418710.7068Falselog
unpooled2-1095.06146.77558.28682.9066e-1324.19212.5854Truelog
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\n\n\n\nWhile LOO-CV is expensive to run, WAIC and other IC can be seen as fast approximations of it\n\n# Posterior Dispersion Indices\n\n
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\n\nFurther reference:\n\nEvaluating Bayesian Models with Posterior Dispersion Indices\n\nAlp Kucukelbir, Yixin Wang, David M. Blei\n\n\n```python\nfrom arviz.stats.stats_utils import get_log_likelihood as get_log_likelihood\nfrom arviz.stats.stats_utils import logsumexp as logsumexp\nfrom arviz.stats.stats_utils import make_ufunc as make_ufunc\nfrom arviz.stats.stats_utils import stats_variance_2d as svar\nfrom arviz.stats.stats_utils import wrap_xarray_ufunc as wrap_xarray_ufunc\n\ndef WAPDI(idata, model=None, varnames=None):\n log_likelihood = get_log_likelihood(idata, 'y')\n log_likelihood = log_likelihood.stack(sample=(\"chain\", \"draw\"))\n shape = log_likelihood.shape\n n_samples = shape[-1]\n n_data_points = np.product(shape[:-1])\n\n ufunc_kwargs = {\"n_dims\": 1, \"ravel\": False}\n kwargs = {\"input_core_dims\": [[\"sample\"]]}\n lppd_i = wrap_xarray_ufunc(\n logsumexp,\n log_likelihood,\n func_kwargs={\"b_inv\": n_samples},\n ufunc_kwargs=ufunc_kwargs,\n **kwargs,\n )\n\n vars_lpd = log_likelihood.var(dim=\"sample\")\n return vars_lpd / lppd_i\n\ny_pdi = WAPDI(az.from_pymc3(pooled_trace))\n```\n\n\n```python\ni = y_pdi.argsort()\n#srrs_mn.iloc[int(i)]\n\ndata = srrs_mn[['county', 'radon', 'log_radon']]\n\ndata.iloc[i[:10]]\n```\n\n\n\n\n
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countyradonlog_radon
5557MCLEOD0.0-2.302585
5224COTTONWOOD0.0-2.302585
5186CARVER0.0-2.302585
5556MCLEOD25.43.238678
5685REDWOOD20.73.034953
5716SCOTT19.32.965273
5124ANOKA0.2-1.203973
5318FARIBAULT0.1-1.609438
5183CARVER14.72.694627
5162BLUE EARTH14.32.667228
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\n\n\n\n\n```python\ndata[data.county == \"CARVER\"]\n#data[data.county == \"MCLEOD\"]\n#data[data.county == \"COTTONWOOD\"]\n```\n\n\n\n\n
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countyradonlog_radon
5181CARVER1.60.530628
5182CARVER12.92.564949
5183CARVER14.72.694627
5184CARVER4.71.568616
5185CARVER9.62.272126
5186CARVER0.0-2.302585
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\n\n\n\n# Two-Sample tests\n\n# MMD / Relative MMD / Kernel Stein Discrepancy / Witness function\n\n
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\n\n
\n\n\n```python\n# Adapted from https://github.com/eugenium/MMD\nfrom sklearn.metrics.pairwise import rbf_kernel\nfrom scipy import stats\n\ndef grbf(x1, x2, sigma):\n '''Calculates the Gaussian radial base function kernel'''\n n, nfeatures = x1.shape\n m, mfeatures = x2.shape\n\n k1 = np.sum((x1*x1), 1)\n q = np.tile(k1, (m, 1)).transpose()\n del k1\n\n k2 = np.sum((x2*x2), 1)\n r = np.tile(k2.T, (n, 1))\n del k2\n\n h = q + r\n del q, r\n\n # The norm\n h = h - 2*np.dot(x1, x2.transpose())\n h = np.array(h, dtype=float)\n\n return np.exp(-1.*h/(2.*pow(sigma, 2)))\n\n\ndef kernelwidthPair(x1, x2):\n '''Implementation of the median heuristic. See Gretton 2012\n Pick sigma such that the exponent of exp(- ||x-y|| / (2*sigma2)),\n in other words ||x-y|| / (2*sigma2), equals 1 for the median distance x\n and y of all distances between points from both data sets X and Y.\n '''\n n, nfeatures = x1.shape\n m, mfeatures = x2.shape\n\n k1 = np.sum((x1*x1), 1)\n q = np.tile(k1, (m, 1)).transpose()\n del k1\n\n k2 = np.sum((x2*x2), 1)\n r = np.tile(k2, (n, 1))\n del k2\n\n h = q + r\n del q, r\n\n # The norm\n h = h - 2*np.dot(x1, x2.transpose())\n h = np.array(h, dtype=float)\n\n mdist = np.median([i for i in h.flat if i])\n\n sigma = np.sqrt(mdist/2.0)\n if not sigma:\n sigma = 1\n\n return sigma\n\n\ndef kernelwidth(Zmed):\n '''Alternative median heuristic when we cant partition the points\n '''\n m = Zmed.shape[0]\n k1 = np.expand_dims(np.sum((Zmed*Zmed), axis=1), 1)\n q = np.kron(np.ones((1, m)), k1)\n r = np.kron(np.ones((m, 1)), k1.T)\n del k1\n\n h = q + r\n del q, r\n\n # The norm\n h = h - 2.*Zmed.dot(Zmed.T)\n h = np.array(h, dtype=float)\n\n mdist = np.median([i for i in h.flat if i])\n\n sigma = np.sqrt(mdist/2.0)\n if not sigma:\n sigma = 1\n\n return sigma\n\n\ndef MMD_Test(X, Y, sigma=-1, SelectSigma=2):\n if(sigma < 0):\n # Similar heuristics\n if(SelectSigma > 1):\n siz = np.min((1000, X.shape[0]))\n sigma = kernelwidthPair(X[0:siz], Y[0:siz])\n else:\n siz = np.min((1000, X.shape[0]*3))\n Zem = np.r_[X[0:siz/2], Y[0:siz/2]]\n sigma = kernelwidth(Zem)\n\n Kyy = grbf(Y, Y, sigma)\n Kxy = grbf(X, Y, sigma)\n Kyynd = Kyy-np.diag(np.diagonal(Kyy))\n m = Kxy.shape[0]\n n = Kyy.shape[0]\n\n u_yy = np.sum(Kyynd)*(1./(n*(n-1)))\n u_xy = np.sum(Kxy)/(m*n)\n\n # Compute the test statistic\n\n Kxx = grbf(X, X, sigma)\n Kxxnd = Kxx-np.diag(np.diagonal(Kxx))\n u_xx = np.sum(Kxxnd)*(1./(m*(m-1)))\n MMDXY = u_xx+u_yy-2.*u_xy\n return MMDXY\n```\n\n\n```python\nnorm1 = np.random.normal(1., 1., (1000, 2))\nnorm2 = np.random.normal(0.2, 1., (1000, 2))\n\nn = norm1.shape[0]\ntt = MMD_Test(norm1, norm2)\npvalue = stats.norm.sf(tt, scale=(2.0/n)**0.5)\npvalue\n```\n\n\n\n\n 3.248337337392221e-05\n\n\n\n# Relative MMD\n\n\n```python\n# Adapted from https://github.com/eugenium/MMD\nimport scipy as sp\n\ndef MMD_Diff_Var(Kyy, Kzz, Kxy, Kxz):\n '''\n Compute the variance of the difference statistic MMDXY-MMDXZ\n See http://arxiv.org/pdf/1511.04581.pdf Appendix for derivations\n '''\n m = Kxy.shape[0]\n n = Kyy.shape[0]\n r = Kzz.shape[0]\n\n Kyynd = Kyy-np.diag(np.diagonal(Kyy))\n Kzznd = Kzz-np.diag(np.diagonal(Kzz))\n\n u_yy = np.sum(Kyynd)*(1./(n*(n-1)))\n u_zz = np.sum(Kzznd)*(1./(r*(r-1)))\n u_xy = np.sum(Kxy)/(m*n)\n u_xz = np.sum(Kxz)/(m*r)\n\n # compute zeta1\n t1 = (1./n**3)*np.sum(Kyynd.T.dot(Kyynd))-u_yy**2\n t2 = (1./(n**2*m))*np.sum(Kxy.T.dot(Kxy))-u_xy**2\n t3 = (1./(n*m**2))*np.sum(Kxy.dot(Kxy.T))-u_xy**2\n t4 = (1./r**3)*np.sum(Kzznd.T.dot(Kzznd))-u_zz**2\n t5 = (1./(r*m**2))*np.sum(Kxz.dot(Kxz.T))-u_xz**2\n t6 = (1./(r**2*m))*np.sum(Kxz.T.dot(Kxz))-u_xz**2\n t7 = (1./(n**2*m))*np.sum(Kyynd.dot(Kxy.T))-u_yy*u_xy\n t8 = (1./(n*m*r))*np.sum(Kxy.T.dot(Kxz))-u_xz*u_xy\n t9 = (1./(r**2*m))*np.sum(Kzznd.dot(Kxz.T))-u_zz*u_xz\n\n zeta1 = (t1+t2+t3+t4+t5+t6-2.*(t7+t8+t9))\n\n zeta2 = (1/m/(m-1))*np.sum((Kyynd-Kzznd-Kxy.T-Kxy+Kxz+Kxz.T)**2) - \\\n (u_yy - 2.*u_xy - (u_zz-2.*u_xz))**2\n\n data = dict({'t1': t1,\n 't2': t2,\n 't3': t3,\n 't4': t4,\n 't5': t5,\n 't6': t6,\n 't7': t7,\n 't8': t8,\n 't9': t9,\n 'zeta1': zeta1,\n 'zeta2': zeta2,\n })\n # TODO more precise version for zeta2\n # xx=(1/m^2)*sum(sum(Kxxnd.*Kxxnd))-u_xx^2;\n # yy=(1/n^2)*sum(sum(Kyynd.*Kyynd))-u_yy^2;\n # xy=(1/(n*m))*sum(sum(Kxy.*Kxy))-u_xy^2;\n # xxy=(1/(n*m^2))*sum(sum(Kxxnd*Kxy))-u_xx*u_xy;\n # yyx=(1/(n^2*m))*sum(sum(Kyynd*Kxy'))-u_yy*u_xy;\n #zeta2=(xx+yy+xy+xy-2*(xxy+xxy +yyx+yyx))\n\n Var = (4.*(m-2)/(m*(m-1)))*zeta1\n Var_z2 = Var+(2./(m*(m-1)))*zeta2\n\n return Var, Var_z2, data\n\n\ndef relative_MMD_Test(X, Y, Z, sigma=-1, SelectSigma=2, computeMMDs=False):\n '''Performs the relative MMD test which returns a test statistic for whether Y is closer to X or than Z.\n See http://arxiv.org/pdf/1511.04581.pdf\n The bandwith heuristic is based on the median heuristic (see Smola,Gretton).\n '''\n if(sigma < 0):\n # Similar heuristics\n if(SelectSigma > 1):\n siz = np.min((1000, X.shape[0]))\n sigma1 = kernelwidthPair(X[0:siz], Y[0:siz])\n sigma2 = kernelwidthPair(X[0:siz], Z[0:siz])\n sigma = (sigma1+sigma2)/2.\n else:\n siz = np.min((1000, X.shape[0]*3))\n Zem = np.r_[X[0:siz/3], Y[0:siz/3], Z[0:siz/3]]\n sigma = kernelwidth(Zem)\n\n Kyy = grbf(Y, Y, sigma)\n Kzz = grbf(Z, Z, sigma)\n Kxy = grbf(X, Y, sigma)\n Kxz = grbf(X, Z, sigma)\n Kyynd = Kyy-np.diag(np.diagonal(Kyy))\n Kzznd = Kzz-np.diag(np.diagonal(Kzz))\n m = Kxy.shape[0]\n n = Kyy.shape[0]\n r = Kzz.shape[0]\n\n u_yy = np.sum(Kyynd)*(1./(n*(n-1)))\n u_zz = np.sum(Kzznd)*(1./(r*(r-1)))\n u_xy = np.sum(Kxy)/(m*n)\n u_xz = np.sum(Kxz)/(m*r)\n # Compute the test statistic\n t = u_yy - 2.*u_xy - (u_zz-2.*u_xz)\n Diff_Var, Diff_Var_z2, data = MMD_Diff_Var(Kyy, Kzz, Kxy, Kxz)\n\n pvalue = sp.stats.norm.cdf(-t/np.sqrt((Diff_Var)))\n # pvalue_z2=sp.stats.norm.cdf(-t/np.sqrt((Diff_Var_z2)))\n tstat = t/np.sqrt(Diff_Var)\n\n if(computeMMDs):\n Kxx = grbf(X, X, sigma)\n Kxxnd = Kxx-np.diag(np.diagonal(Kxx))\n u_xx = np.sum(Kxxnd)*(1./(m*(m-1)))\n MMDXY = u_xx+u_yy-2.*u_xy\n MMDXZ = u_xx+u_zz-2.*u_xz\n else:\n MMDXY = None\n MMDXZ = None\n return pvalue, tstat #, sigma, MMDXY, MMDXZ\n```\n\n\n```python\ntrue_model = np.random.normal(1., 1., (1000, 2))\nmodel1 = np.random.normal(1.3, 1., (1000, 2))\nmodel2 = np.random.normal(1.1, 1., (1000, 2))\n\npvalue, tstat = relative_MMD_Test(true_model, model1, model2)\npvalue\n```\n\n\n\n\n 0.0018383511949117198\n\n\n\n## Kernel Stein Discrepancy\n\n
\n\n
\n\n\n```python\nclass PyMCgof(object):\n def __init__(self, model):\n self.model = model\n self.dlogp = model.dlogp()\n \n def grad_log(self, x):\n res = self.dlogp({\"x\": x})\n return res.reshape(x.shape)\n```\n\n\n```python\n# Install kgof with pip install git+https://github.com/wittawatj/kernel-gof.git\n\nimport kgof.goftest as gof\nfrom kgof import data, kernel, util\n\n# sample true data\nmu_r = 0.1\nvar_r = 1\nn = 500\n \nX = np.random.normal(loc=mu_r, scale=var_r, size=(n, 2))\n\nwith pm.Model() as model:\n x = pm.Normal(\"x\", [0., 0.1], 1., shape=X.shape)\n \np = PyMCgof(model)\n\nmed = util.meddistance(X, subsample=1000)\nk = kernel.KGauss(sigma2=med**2/2.0)\nV = util.fit_gaussian_draw(X, 5)\nfssd = gof.FSSD(p=p, k=k, V=V) # Finite Set Stein Discrepancy\nfssd.perform_test(data.Data(X))\n```\n\n\n\n\n {'alpha': 0.01,\n 'pvalue': 0.014666666666666666,\n 'test_stat': 0.8537068071404676,\n 'h0_rejected': False,\n 'n_simulate': 3000,\n 'time_secs': 0.008444786071777344}\n\n\n\n
\n\n# Conclusions\n\n
\n\n
\n", "meta": {"hexsha": "927e127feba85a28be81779f6140dacef9586ad1", "size": 892061, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Tour.ipynb", "max_stars_repo_name": "zaxtax/pymc2020_model_checking", "max_stars_repo_head_hexsha": "792ea72c36e6fdccb2c031fc4fec0d817703a34b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2020-10-25T21:16:42.000Z", "max_stars_repo_stars_event_max_datetime": "2020-12-24T16:14:54.000Z", "max_issues_repo_path": "Tour.ipynb", "max_issues_repo_name": "zaxtax/pymc2020_model_checking", "max_issues_repo_head_hexsha": "792ea72c36e6fdccb2c031fc4fec0d817703a34b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Tour.ipynb", "max_forks_repo_name": "zaxtax/pymc2020_model_checking", "max_forks_repo_head_hexsha": "792ea72c36e6fdccb2c031fc4fec0d817703a34b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 234.7528947368, "max_line_length": 245436, "alphanum_fraction": 0.9021490683, "converted": true, "num_tokens": 19260, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.682573734412324, "lm_q2_score": 0.6548947357776795, "lm_q1q2_score": 0.44701394544674294}} {"text": "```python\n# initial setup\ntry:\n # settings colab:\n import google.colab\nexcept ModuleNotFoundError: \n # settings local:\n %run \"../../../common/0_notebooks_base_setup.py\" \n```\n\n---\n\n\n\n\n#### Distribuci\u00f3n Normal\n\n\u00bfPor qu\u00e9 es tan popular esta distribuci\u00f3n?\n\nLa importancia de la distribuci\u00f3n normal proviene principalmente del hecho de que las distribuciones de muchos fen\u00f3menos naturales est\u00e1n al menos aproximadamente distribuidos normalmente.\n\nUna de las primeras aplicaciones de la distribuci\u00f3n normal fue el an\u00e1lisis de errores de medici\u00f3n realizados en observaciones astron\u00f3micas, errores debidos a instrumentos imperfectos y observadores imperfectos.\n\nGalileo en el siglo XVII observ\u00f3 que estos errores eran sim\u00e9tricos y que los errores peque\u00f1os ocurr\u00edan con mayor frecuencia que los errores grandes.\n\nEsto condujo a varias distribuciones hipot\u00e9ticas de errores, pero no fue hasta principios del siglo XIX que se descubri\u00f3 que estos errores seguian una distribuci\u00f3n normal.\n\nIndependientemente, los matem\u00e1ticos Adrain en 1808 y Gauss en 1809 desarrollaron la f\u00f3rmula para la distribuci\u00f3n normal y mostraron que los errores se ajustaban bien a esta distribuci\u00f3n.\n\n\nLa distribuci\u00f3n normal tiene dos par\u00e1metros: su media $\\mu$ y su varianza $\\sigma^2$\n\nSu funci\u00f3n de densidad de probabilidad es\n\n\\begin{equation}\n f(x) = \\frac{1}{\\sigma \\sqrt{2\\pi}}exp^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}\n\\end{equation}\n\nLos valores posibles de X son los reales entre menos y m\u00e1s infinito. \n\nEl gr\u00e1fico de la densidad para una variable normal tiene forma de campana. \n\nLa probabilidad de que una variable aleatoria X con distribuci\u00f3n normal y par\u00e1metros $\\mu$, $\\sigma^2$ sea menor o igual que un valor $a$ fijo es \n\n\\begin{equation}\n P(X \\le a) = \\int^a_{-\\infty}\\frac{1}{\\sigma \\sqrt{2\\pi}}exp^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}dx \n\\end{equation}\n\n\nLa distribuci\u00f3n normal es sim\u00e9trica respecto a su media. \n\nEn particular cuando la media es cero y el desv\u00edo est\u00e1ndar es igual a 1 decimos se se trata de una **normal est\u00e1ndar**. \n\n#### Algunas propiedades de la distribuci\u00f3n normal\n\n* Cualquier funci\u00f3n lineal de una variable aleatoria normal tambi\u00e9n es normal. \nSi X es una variable aleatoria normal con par\u00e1metros $\\mu$, $\\sigma^2$ podemos definir Y=a+bX con b distinto de cero (\u00bfPor qu\u00e9?). \nEntonces Y tambi\u00e9n tendr\u00e1 distribuci\u00f3n normal (\u00bfCon qu\u00e9 par\u00e1metros?)\n\n* La variable aleatoria generada por una suma finita de variables aleatorias normales independientes es normal. \n\n* En el caso de que se sumen infinitas variables aleatorias, independientes y tales que tanto la suma de las medias como de las varianzas sean finitas tambi\u00e9n vale la propiedad.\n\n
\n
Funci\u00f3n de densidad de probabilidad:
\n
\nFunci\u00f3n de distribuci\u00f3n de probabilidad:
\n
\nDatos con distribuci\u00f3n normal:
\n
\n
\n\n\n**Ejemplos**:\n\n* X: longitud de los clavos producidos en una f\u00e1brica\n\n* X: altura de los alumnos de DH\n\n* X: tiempo vida de un modelo de l\u00e1mpara\n\n* X: tiempo que utiliza un empleado en realizar determinada tarea en una empresa.\n\n\n\n
\n
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\n
\n\n#### Referencias\n\nhttps://luminousmen.com/post/data-science-probability-distributions\n\nGr\u00e1ficos: https://es.wikipedia.org/wiki/Distribuci%C3%B3n_normal\n", "meta": {"hexsha": "453ec3263147be3eaa828b8d56c48e8e6716a4f1", "size": 5670, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Code/3-numpy/probabilidad/4.6_normal.ipynb", "max_stars_repo_name": "Flor91/Data-Science", "max_stars_repo_head_hexsha": "f67ec537341e8b2d8213a56ef8ee63028e46e1b2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-10-06T12:59:39.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-08T10:58:14.000Z", "max_issues_repo_path": "Code/3-numpy/probabilidad/4.6_normal.ipynb", "max_issues_repo_name": "Flor91/Data-Science", "max_issues_repo_head_hexsha": "f67ec537341e8b2d8213a56ef8ee63028e46e1b2", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Code/3-numpy/probabilidad/4.6_normal.ipynb", "max_forks_repo_name": "Flor91/Data-Science", "max_forks_repo_head_hexsha": "f67ec537341e8b2d8213a56ef8ee63028e46e1b2", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 39.1034482759, "max_line_length": 388, "alphanum_fraction": 0.6202821869, "converted": true, "num_tokens": 993, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.817574478416099, "lm_q1q2_score": 0.44699915943900254}} {"text": "#### [**NICOLAS CACHANOSKY**](http://www.ncachanosky.com) | Department of Economics | Metropolitan State University of Denver | ncachano@msudenver.edu\n\n# A SIMPLE RAMSEY MODEL\n---\n\nThis note illustrates how to code a simple Ramsey model in Python. The purpose of the note is to walk through Python applications, not to offer a detailed discussion of Ramsey model or to show best coding practices. The note also assumes familiarity with Ramsey model and a beginner experience with Python.\n\nFor a more complete and detailed discussion of Python applications see the material in [Quant Econ](https://quantecon.org/).\n\n---\n\n## TABLE OF CONTENTS\n1. [Description of the model](#1.-DESCRIPTION-OF-THE-MODEL)\n2. [The respresentative household](#2.-THE-REPRESENTATIVE-HOUSEHOLD)\n3. [The Representative Firm](#3.-THE-REPRESENTATIVE-FIRM)\n4. [Equilibrium](#4.-EQUILIBRIUM)\n5. [The code](#5.-THE-CODE)\n\n## 1. DESCRIPTION OF THE MODEL\n\nA known issue with the Solow model is that the savings rate is exogenous (and constant). This means the ratio of consumption to income $Y/C$ is also constant. While the Solow model can be useful to highlight some features of long-run growth, this assumption also limits the applicability of this model. Yet, the Solow model can be interpreted as a specIal case of Ramsey model (or Ramsey-Kass-Koopmans model). Ramsey model can be considered as the building block of macro models with a household that optimizes its utility and, therefore, endogenously sets the savings rate.\n\nThis note presents a \"simple\" Ramsey model (it offers a simplified discussion, does not discuss shocks, and does not add other features). For a more complete and detailed discussion see an advanced macroeconomics textbook. \n\nThe model is populated by a representative household that maximizes its utility and by a representative firm that maximizes its profits. The model assumes a competitive market (no monopoly power, agents are price takers, etc.) The household provides labor services to the firm in exchange of a wage. The wage can be used to consume goods or save and accumulate assets. The model assumes a closed economy with no government. Finally, the model is presented in continuous (rather than discrete) time. A \"dot\" on top of a variable denotes its instantaneous time-change $(\\dot x (t) = \\frac{\\partial x}{\\partial t})$\n\n# 2. THE REPRESENTATIVE HOUSEHOLD\n\nAssume a large number of identicial infinitely-lived households. Because these households are identical, we can use a **representative** household to populate the model. The assumptions that this households are infinitely lived captures a continuum of generations. The representative household has $L(t)$ individuals with a population growth rate of $n$. Therefore, $L(t) = L(0) \\cdot e^{nt}$\n\n## 2.1. THE HOUSEHOLD UTILITY\n\nThe household utility $(U)$ is the present value of the utility (felicity function) of all household individuals $(u(c))$.\n\n\\begin{equation}\n U = \\int_{0}^{\\infty} u[c(t)] \\cdot e^{nt} \\cdot \\underbrace{e^{-\\rho t}}_{L(t)} \\cdot dt = \\int_{0}^{\\infty} u[c(t)] \\cdot e^{(n-\\rho)t} \\cdot dt\n\\end{equation}\n\nwhere $\\rho > 0$ is the time preference of the household individual and where $n -\\rho >0$. Also, $u(c)$ is concave $(u^{'}(c) > 0, u^{''}(c) < 0)$ and satisfies the Inada conditions $(u^{'}(c) \\to \\infty \\text{ as } c \\to 0 \\text{, and } u^{'}(c) \\to 0 \\text{ as } c \\to \\infty$.\n\nThe parameter $\\rho$ can be interpreted as the time preference of the household or as capturing a discount of the future generation by the present generations. If $\\rho = 1$, then present generation assigns equal \"value\" to the next generatio (i.e. parents with respect to their kids). If $\\rho < 0$, then the present generation is altruistic and values the utility of the next generation more than their own.\n\n## 2.2. THE HOUSEHOLD BUDGET CONSTRAINT\n\nThe household budget constrained is composed of assets $(\\Lambda)$ and wages $(w)$. The non-consumed wage becames an increase in assets. At some period in time, assets can be negative, which means that in that period the household has a net debt. Assets have a rate of return of $r$. Therefore, the stock of assets increase by the return of the assets plus wages, and decrease by the amount of consumption. Since population growths at rate $n$, the change in household's assets can be represented as follows.\n\n\\begin{equation}\n \\frac{d\\Lambda}{dt} = r(t) \\cdot \\Lambda(t) + w(t)L(t) - C(t)\n\\end{equation}\n\n\nDividing by $L(t)$ we get the change in assets per capita $(a)$ (and dropping the $t$ variable for notation simplicity):\n\n\\begin{align}\n \\dot{\\left(\\frac{A}{L} \\right)} &= \\frac{\\dot{A}L - A \\dot{L}}{L^2} \\\\\n \\dot{a} &= \\frac{\\dot{A}}{L} - \\frac{A}{L}n \\\\\n \\dot{a} &= ra + w - c - na \\\\\n \\dot{a} &= (r - n)a + w - c\n\\end{align}\n\nTo avoid the situation where the household borrows money indifenetely, the model also imposses a \"no Ponzi game condition\". Intuitively, at the end of the household life $(T)$, assets have to be at least zero $a(T) \\geq 0$. A household may have debt at any point in time, but at the end of its lifetime all debts are paid. The present value of the assets have to be asymptotically nonnegative.\n\n\\begin{equation}\n \\lim\\limits_{t \\to \\infty} a(t) e^{-\\tilde{r}(t)} \\geq 0\n\\end{equation}\n\nwhere $\\tilde{r}(t) = \\int_{0}^{t} (r(\\tau) - n) d\\tau$. This expression means that any moment $\\tau \\in (0, \\infty)$, the present value of $a$ cannot be negative. Note that $\\tilde{r}$ is taking into consideration the growht rate of the household as well $n$. In a more simple notation, this condition can be written as $\\lim\\limits_{t \\to \\infty} a(t) \\geq 0$.\n\n## 2.3. THE HOUSEHOLD OPTIMIZATION PROBLEM\n\nThe household is facing a dynamic optimization problem. It needs to set the path of consumption that maximizes the present value of its life utility. The optimization problem can be described with the following equations (where $n - \\rho > 0$).\n\n\\begin{align}\n \\max_{c(t)} U &= \\int_{t=0}^{\\infty} u[c(t)] \\cdot e^{-(n-\\rho)t} \\\\\n \\text{subject to:} \\\\\n \\dot{a} &= (r(t) - n)a(t) + w(t) - c(t) \\\\\n a(0) &= a_{0} \\\\\n \\lim\\limits_{t \\to \\infty}a(t) &\\geq 0 \\\\\n \\text{transversality condition:} \\\\\n \\lim\\limits_{t \\to \\infty} \\mu(t) \\cdot a(t) = 0\n\\end{align}\n\nThe optimization problem is given an initial asset value of $a_0$ and a terminal condition for which assets cannot be negative. Let $\\mathscr{H}$ be the present-value Hamiltonian:\n\n\\begin{equation}\n \\mathscr{H} = e^{-(\\rho - n)t} \\cdot u[c(t)] + \\mu(t) \\cdot \\left[r(t) - n)a(t) + w(t) - c(t) \\right] \n\\end{equation}\n\nwhere $\\mu(t)$ is the present-value shadow price. Since the household faces an optimization problem at each moment $t$, the shadow price may vary over time as well. The \"transversality condition\" states that in the limit, either the shadow price or the assets have to be zero. \n\nThe first-order-conditions (FOC) of the hamiltonian are:\n\n\\begin{align}\n \\frac{\\partial \\mathscr{H}}{\\partial c} &= e^{-(\\rho - n)t} \\cdot u'_{c} - \\mu(t) = 0 \\\\\n \\frac{\\partial\\mathscr{H}}{\\partial\\mu} &= (r(t) - n)a(t) + w(t) - c(t) = \\dot{a} \\\\\n \\frac{\\partial \\mathscr{H}}{\\partial a} &= - (r(t) - n)\\cdot \\mu (t) = \\dot{\\mu}(t)\n\\end{align}\n\nNow we proceed to use the first equation to get $\\dot{\\mu}(t)$.\n\n\\begin{align}\n \\mu(t) &= e^{-(\\rho - n)t} \\cdot u'_{c} \\\\\n \\dot{\\mu}(t) &= -(\\rho - n) \\cdot e^{-(\\rho - n)t} \\cdot u'_{c} + e^{-(\\rho - n)t} \\cdot u''_c \\cdot \\dot{c}\n\\end{align} \n\nMake now two substitutions: (1) The first FOC in the third FOC, and (2) this last derivative in the third FOC as well.\n\n\\begin{align}\n -[r(t) -n] \\cdot \\mu(t) &= -(\\rho - n) \\cdot e^{-(\\rho - n)t} \\cdot u'(c) + e^{-(\\rho - n)t} \\cdot u''(c) \\cdot \\dot{c} \\\\[10pt]\n -[r(t) -n] \\cdot e^{-(\\rho - n)t} \\cdot u'(c) &= e^{-(\\rho - n)t} \\left[-(\\rho - n) \\cdot u'(c) + u''(c) \\cdot \\dot{c} \\right] \\\\[10pt]\n -[r(t) -n] \\cdot u'(c) &= -(\\rho - n) \\cdot u'(c) + u''(c) \\cdot \\dot{c} \\\\[10pt]\n r(t) &= \\rho - \\frac{u''(c)\\cdot c}{u'(c)} \\cdot \\frac{\\dot{c}}{c}\n\\end{align}\n \nThe last equation is the **Euler equation**, which defines the path for $c$ given the rate of return $r(t)$. Recall that $u''(c)<0$ and therefore $\\frac{u''(c)c}{u'(c)}>0$. The right-hand side of the equation can be interpreted as the \"rate of return of consumption.\" Consumers would like to smooth consumption in time. Therefore, the rate of return $(r)$ has to be equal to the time preference $(\\rho)$ minus the change of consumption over time. Or, put it differently, if $r = \\rho$, then consumption will be flat. There is no opportunity to increase utility by moving consumption over time (extend a loan and postpone consumption or take a debt and advance consumption). Consumers will be willing to postpone consumption if $r$ is sufficiently above $\\rho$. The required compensation to postpone consumption is the second term in the righ-hand side of the equation.\n\n\\begin{align}\n r &= \\rho \\to \\frac{\\dot{c}}{c} = 0 \\\\\n r &> \\rho \\to \\frac{\\dot{c}}{c} > 0 \\text{ (saving today: postpone present consumption)} \\\\\n r &< \\rho \\to \\frac{\\dot{c}}{c} < 0 \\text{ (consuming today: advance future consumption)}\n\\end{align}\n\nThe Euler equation also includes the elasticity of marginal utility: $\\frac{u''(c)\\cdot c}{u'(c)}$. For $r$ and $\\frac{\\dot{c}}{c}$ to be constant, this elasticity of marginal utility has to be constant. A common practice is to assume a CRRA (constant relative risk aversion) utility function that depicts constant intertemporal elasticity of substitution (CIES).\n\n\\begin{equation}\n u(c) = \\frac{c^{1-\\theta}-1}{1-\\theta}\n\\end{equation}\n\nwhere $\\theta > 0$ is the coefficient of relative risk aversion. The $1 - \\theta$ denominator assures the function will keep its desired properties of the function $(u'>0, u''<0)$ even if $\\theta > 1$ (see below). The inclution of the term $-1/(1-\\theta)$ is for convenience; when $\\theta \\to 1$ then the CRRA function becomes $(u(c) = log(c))$. Consider now the first and second derivative of the CRRA function and the elasticity of substitution $(\\sigma)$.\n\n\\begin{align}\n u'(c) &= c^{-\\theta} > 0 \\\\\n u''(c) &= -\\theta \\cdot c^{-(\\theta + 1)} < 0 \\\\\n \\sigma &= \\left[ \\frac{u''(c) \\cdot c}{u'(c)} \\right]^{-1} = \\left[\\frac{\\theta \\cdot c^{-(\\theta + 1)} \\cdot c}{c^{-\\theta}} \\right]^{-1} = \\frac{1}{\\theta}\n\\end{align}\n\nThen, the Euler equation from the household maximization problem becomes\n\n\\begin{align}\n r(t) &= \\rho - \\frac{u''(c)\\cdot c}{u'(c)} \\cdot \\frac{\\dot{c}}{c} \\\\\n r(t) &= \\rho - \\theta \\cdot \\frac{\\dot{c}}{c} \\\\\n\\end{align}\n\nWe can rewrite the Euler equation to more conveniently depict the consumption path that maximizes the household utility.\n\n\\begin{equation}\n \\frac{\\dot{c}}{c} = \\frac{1}{\\theta} \\cdot \\left(r(t) - \\rho \\right)\n\\end{equation}\n\nThis is one of the two equations that describes Ramsey model. The second equation is derived from the firms profit maximization problem (next section).\n\n---\n\nThe following code plots the utility function for different levels of $\\theta$. The function also allows for $\\theta = 1$, in such case the utility function becomes the log of consumption.\n\n\n```python\n\"1|IMPORT PACKAGES\"\nimport numpy as np # Package for scientific computing with Python\nimport matplotlib.pyplot as plt # Matplotlib is a 2D plotting library\n\n\n\"2|DEFINE PARAMETERS AND ARRAYS\"\n# PARAMETERS\nsize = 5 # Model domain\nsteps = size*100 # Number of \"dots\" in the domain\n\n# ARRAY\nc = np.linspace(0, size, steps) # Create array of consumption\n\n\n\"3|DEFINE UTILITY FUNCTION\"\n# Production function\ndef u(x, CRRA):\n if CRRA == 1:\n u = np.log(x)\n else:\n u = ((x)**(1-CRRA)-1)/(1-CRRA)\n return u\n\n\n\"4|CALCULATE UTILITY FUNCTIONS FOR DIFFERENT VALUES OF THETA\"\ntheta = [0.25, 0.75, 1, 1.5, 2]\n\n# Ignore c[0] = 0 to avoid dividing by zero\nu1 = u(c[1:], theta[0])\nu2 = u(c[1:], theta[1])\nu3 = u(c[1:], theta[2])\nu4 = u(c[1:], theta[3])\nu5 = u(c[1:], theta[4])\n\n\n\"5|PLOT UTILITY FUNCTIONS\"\n### AXIS RANGE\nv = [0, size, -10, 5] \n### BUILD PLOT AND POPULIATE WITH LOCI LINES\nfig, ax = plt.subplots(figsize=(10, 7))\nax.plot(c[1:], u1, alpha = 0.8, label = r'$\\theta=%.2f$' %theta[0])\nax.plot(c[1:], u2, alpha = 0.8, label = r'$\\theta=%.2f$' %theta[1])\nax.plot(c[1:], u3, alpha = 0.8, label = r'$\\theta=%.2f$' %theta[2])\nax.plot(c[1:], u4, alpha = 0.8, label = r'$\\theta=%.2f$' %theta[3])\nax.plot(c[1:], u5, alpha = 0.8, label = r'$\\theta=%.2f$' %theta[4])\n### AXIS\nax.axhline(0, color = 'k') # Add horizontal axis\nax.axvline(0, color = 'k') # Add vertical axis\nax.yaxis.set_major_locator(plt.NullLocator()) # Hide ticks\nax.xaxis.set_major_locator(plt.NullLocator()) # Hide ticks\n### TEXT \nplt.text(size-0.10, -0.5, r'$c$', color = 'k') # x-axis label\nplt.text( -0.25, 4.5, r'$u(c)$') # y-axis label\n# SETTINGS\nplt.box(False) # Hide axis\nplt.legend(loc=0, frameon=False)\nplt.axis(v)\nplt.show()\n\n```\n\n# 3. THE REPRESENTATIVE FIRM\n\nThe representative firm uses a labor-technology augmenting production function $(Y = F(K; AL)$ with constant returns to scale (CRS). The production function depicts the typical properties of diminishing marginal returns and the Inada conditions. Technology $(A)$ grows at rage $g$: $A(t) = A(0)\\cdot e^{gt}$.\n\nIn terms of effective labor $(AL)$, and normalizing the price of the good to one $(p = 1)$, the profit of the firm becomes (where the $\"\\hat{}\"$ symbol denotes the variable is in per effective labor units):\n\n\\begin{equation}\n \\hat{\\pi} = f(\\hat{k}) - R \\hat{k} - \\hat{w}\n\\end{equation}\n\nThe rate of return on capital equals the gross rate of return $(R)$ minus the depreciation rate $(\\delta)$: $r = R - \\delta$. In this model there is no intertermporal optimization. The firm rents capital from the household one period at the time.\n\nThe FOC for the firm with respect to capital per effective unit of labor utilization is: \n\n\\begin{equation}\n \\frac{\\partial \\pi}{\\partial \\hat{k}} = f'(\\hat{k}) - R = 0\n\\end{equation}\n\nIn equilibrium the firm's profit is zero, therefore:\n\n\\begin{align}\n \\hat{w} &= f(\\hat{w}) - R\\hat{k} \\\\\n \\hat{w} &= f(\\hat{w}) - (r + \\delta)k\n\\end{align}\n\n# 4. EQUILIBRIUM\n\nIn equilibrium, the assets owned by the household coincide with the capital per unit of labor. Recall that in this model households own the capital that is rented to the firms. Also, note that assets $(a)$ are expressed in per labor terms, while $(k)$ is expressed in per effective labor. Therefore: $a = Ak$. Then:\n\n\\begin{align}\n \\dot{a} &= ra + w - c - na \\\\[5pt]\n \\dot{(Ak)} &= r \\cdot Ak + Aw - Ac - n \\cdot Ak \\\\[5pt]\n \\dot{A}k + A \\dot{k} &= r \\cdot Ak + Aw - Ac - n \\cdot Ak \\\\[5pt]\n \\frac{\\dot{A}}{A}k + \\dot{k} &= rk + w - c - nk \\\\[5pt]\n gk + \\dot{k} &= rk + w - c - n k \\\\[5pt]\n \\dot{k} &= rk + f(k) - (r + \\delta)k - c - (n + g)k \\\\[5pt]\n \\dot{k} &= f(k) - c - (n + g + \\delta)k\n\\end{align}\n\nKnowing that $R = r + \\delta$ and that $f'(k) = R + \\delta$, then the two motion equations that model Ramsey model are:\n\n\\begin{cases}\n \\frac{\\dot{c}}{c} &= \\frac{1}{\\theta} \\cdot (f'(k) - \\delta - \\rho) \\\\[10pt]\n \\dot{k} &= f(k) - c - (n + g + \\delta)\n\\end{cases}\n\nIn equilibrium, $\\hat{c} = 0$ and $\\hat{k} = 0$. Assuming the equilibrium conditions, we can rewrite the first of Ramsey model equations as an equilibrium **value** of $\\hat{k}$, and the second equation can be represented as $\\hat{c}$ in terms of $\\hat{k}$. First, assume the production following, $Y = A \\cdot \\left[K^{\\alpha} (AL)^{1 - \\alpha} \\right] (\\alpha \\in (0, 1))$. In terms of effective labor: $\\hat{y} = \\hat{k}^{\\alpha}$.\n\n\\begin{align}\n \\hat{k} &= (\\delta + \\rho)^{1/\\alpha} \\\\\n \\hat{c} &= \\hat{k}^{\\alpha} - (n + g + \\delta) \\cdot \\hat{k}\n\\end{align}\n\nFrom the second equation we can also derive the value of $\\hat{k}$ that maximizes $\\hat{c}$; the *golden-rule*.\n\n\\begin{align}\n \\hat{c} &= \\hat{k}^{\\alpha} - (n + g + \\delta) \\cdot \\hat{k} \\\\[10pt]\n \\frac{\\partial \\hat{c}}{\\partial \\hat{k}} &= \\alpha \\cdot \\hat{k}^{-(1 - \\alpha)} - (n + g + \\delta) \\hat{k} = 0 \\\\[8pt]\n \\hat{k}^{*}_{g} &= \\left(\\frac{\\alpha}{n + g + \\delta} \\right)^{\\frac{1}{1-\\alpha}}\n\\end{align}\n\nThe model has one stable saddle path that converges to equilibrium. This saddle-path is unstable, if the model is outside this path by the smallest distance, then the model diverges away from equilibirum.\n\n# 5. THE CODE\n\nThe Python code has the following structure. Following the above discussion, after importing the required packages and defining parameters and arrays, the equilibrium, gold values, and _loci_ lines are calculated. The code plots two graphs. \n\nThe first one shows the _loci_ lines with five sample paths (in red). Each one of these sample paths depicts the unstable dynamics of the model. The second graph plots the stable-path (in blue). \n\n## 5.1 ESTIMATING THE STABLE-PATH\n\nA common way to estimate the stable-path is with a \"shooting\" methodology. This approach shoots a number of paths until it finds one that is close enough the actual stable path. Let $\\tilde{k}$ and $\\tilde{c}$ denote estimated values of the path produced by the \"shooting\" algorithm.\n\nThis note uses a **simple** code and methodology to illustrate this type of numerical method. It is not intended to be the more efficient one. The stable path is estimated in two halves. One estimation for the stable path section to the left of the equilirium point, and another estimation for the stable path to the right of the equilibrium point. These two estimations are coded as two different functions: (1) backward shoot and (2) forward shoot respectively. Alternatively, one simple \"shoot\" function could include both initial locations. \n\nThese type of iterations can be time consuming depending on how many shoots the code is going to perform (which depends on how big a correction is when a shoot misses the steady-state), what the tolerance level is, etc. A more sofisticated code could reduce the number of attempts and save computational time.\n\nSince we know that is very unlikely that our \"shoots\" will exactly hit the steady state $(k^*, c(k^*)$, we need to define a success criteria. The success criteria is when the shoots hits close enough to the target. We need to define circle around the equilibrium that plays the role of the tolerance level. Once we find a \"shoot\" that hits inside this circle, the shoots is considered a success and an approximation good enough for the estimation purposes.\n\nThe distance of a point to the steady-state is: $distance = \\sqrt[2]{(\\tilde{k} - k^*)^2 + (\\tilde{c} - c^*)^2}$, where $\\tilde{k}, \\tilde{c}$ are the shoot estimations.\n\nThe shooting functions have two inputs. The initial level of $k_0$ and a \"step\" size. The \"step\" size is how big the correction of the initial shoot will be after it misses the tolerance level (the target). Given $k_0$, the function estimates a level of $c(k_0)$ (the first shoot). If the shoot misses the target, then the algorithm performs a correction to the initial level of $c_0$ and shoots again. This sequence repeats until the shoot hits the target. \n\n### 5.1.1 \"FORWARD\" SHOOT\nThe first shoot of the forward shoot function is located one step below the _loci_ function. We know is very likely that this initial shoot is too high, and therefore the path will go above $c^*$. The loop has two break points to define a succesfull shoot. The first one is if the path produces a point $(k, c)$ inside the tolerance distance from the steady-state (the above discussion). The second one is the path produces a value of $\\tilde{k} > k_*$. This break point is in place because the step size (how far the new shoot starting point moves) is a fixed amount without a correction or feedback mechanism. It is possible, then, that after missing the target the next iteration starts at a value of $c$ that is too low. Such path may still be an estimation good enough even if it does not satisfy the tolerance level (for the purposes of this notebook, this is enough). \n\nThere is a third break, which is a cap on the number of iterations the loop can perform. When building a code like this, this type of break can help avoid a mistake that sends the code into an infinite loop.\n\nIf the path produces a value of $\\tilde{c} > c^*$, then the loop resets and starts again from a new initial $c_0$ that is located one \"step size\" below the previous shoot.\n\n### 5.1.2 \"BACKWARD\" SHOOT\n\nThe backward shoot function follow a similar structure than the forward shoot. Yet, because the starting point is different, the break points have to be modified. However, there is a difference in the starting point. Rather than starting at the _loci_ level given the initial on step above $c_0 = c(k_0)$, the code starts with $c_0 = c(k_0)\\cdot 1.5$. This is an example of an easy way to save some computational power. Looking at the first figure (the unstable paths) we know that the level of consumption of the stable path will be several steps above $c(k_0)$. Therefore, we can save computation time by avoiding shoot attempts between $c(k_0)$ and $c(k_0)\\cdot 1.5$. \n\nThe forward shoot loop has two stop points of success. The first one is if the shoot produces a point inside the tolerance circle. The second one is if the path produces a value of $\\tilde{k} < k^*$ (for similar reasons than the forward shoot break). There is also the third break built on the number of iterations the loop will perform. Since the starting point is farther away from the equilibrium point, this break has a larger limit of iterations than the forward shoot loop.\n\nIf the shoot produces a value of $\\tilde{c} < c^*$, then the code resets and calculates a new shoot with a new $c_0$ one step above the previous shoot.\n\n\n```python\n\"1|IMPORT PACKAGES\"\nimport numpy as np # Package for scientific computing with Python\nimport matplotlib.pyplot as plt # Matplotlib is a 2D plotting library\n\n\"2|DEFINE PARAMETERS AND ARRAYS\"\n# PARAMETERS\nk_size = 150 # Model domain\nsteps = k_size*100 # Number of \"dots\" in the domain\nalpha = 0.30 # Output elasticity of capital\ndelta = 0.35 # Depreciation rate\nrho = 0.35 # Time preference\nn = 0.05 # Population growth rate\ng = 0.05 # TFP growth rate\ntheta = 0.8 # Coefficient or Relative Risk Aversion\n# ARRAY\nk_array = np.linspace(0, k_size, steps) # Create array of k\n\n\n\"3|DEFINE FUNCTIONS\"\n# PRODUCTION FUNCTION\ndef y(k):\n y = k**alpha\n return y\n\n# MARGINAL PRODUCT OF CAPITAL\ndef y_prime(k):\n y_prime = alpha*k**(alpha-1)\n return y_prime\n\n# CONSUMPTION\ndef consumption(k):\n c = y(k) - (n + g + delta)*k\n return c\n\n# MOTION FUNCTION OF CONSUMPTION\ndef cdot(k, c):\n cdot = 1/theta * (y_prime(k) - delta - rho) * c\n return cdot\n\n# MOTION FUNCTION OF CAPITAL\ndef kdot(k, c):\n kdot = y(k) - c - (n + g + delta)*k \n return kdot\n \n\n\"4|CALCULATE STEADY STATE AND GOLD VALUES\"\n# STEADY STATE VALUES\nk_star = (delta + rho)**(1/alpha)\nc_star = consumption(k_star)\ny_star = y(k_star)\ns_star = (y_star - c_star)/y_star\n\n# GOLD VALUES\nk_gold = (alpha/(n+g+delta))**(1/(1-alpha))\nc_gold = consumption(k_gold)\ny_gold = y(k_gold)\ns_gold = (y_gold - c_gold)/y_gold\n\n\n\"5|CALCULATE LOCI FUNCTIONS\"\nk_loci = k_star\nc_loci = consumption(k_array)\n\n\n\"6|CALCULATE SAMPLE PATHS\"\nk00, k01, k02 = k_star*0.25, k_star*0.50, k_star*2.2\nc0 = [consumption(k00)*0.25, consumption(k00), c_star*1.25,\n consumption(k01)*0.40, consumption(k01)*1.50]\n\ndef sample_path(k0, c0, n):\n path = np.zeros(shape=(n, 2))\n path[0, 0] = k0\n path[0, 1] = c0\n \n for j in range(n-1):\n if path[j,0] < 0: # Stop if motion goes off-chart\n break\n else:\n path[j+1, 0] = path[j, 0] + kdot(path[j, 0], path[j, 1])*0.25\n path[j+1, 1] = path[j, 1] + cdot(path[j, 0], path[j, 1])*0.25\n return path\n\npath1 = sample_path(k00, c0[0], 30)\npath2 = sample_path(k00, c0[1], 30)\npath3 = sample_path(k01, c0[2], 30)\npath4 = sample_path(k02, c0[3], 30)\npath5 = sample_path(k02, c0[4], 30)\n\n\n\"7|PLOT RAMSEY MODEL WITH SAMPLE PATHS\"\n# Value of k such that c = 0\nk_zero = (1/(n + g + delta))**(1/(1-alpha))\n# Axis range\ny_max = np.max(c_loci)*1.7\nx_max = k_zero*1.2\nv = [0, x_max, -0.1, y_max] \n\n### BUILD PLOT AND POPULIATE WITH LOCI LINES\nfig, ax = plt.subplots(figsize=(10, 7))\nax.plot(k_array, c_loci, color=\"K\", alpha = 0.8)\nax.axvline(k_loci, color=\"K\", alpha = 0.8)\n### SAMPLE PATHS\nax.plot(path1[:,0], path1[:,1], \"r--\", alpha = 0.7)\nax.plot(path2[:,0], path2[:,1], \"r--\", alpha = 0.7)\nax.plot(path3[:,0], path3[:,1], \"r--\", alpha = 0.7)\nax.plot(path4[:,0], path4[:,1], \"r--\", alpha = 0.7)\nax.plot(path5[:,0], path5[:,1], \"r--\", alpha = 0.7)\n### ADD DOTS AT BEGINNING OF SAMPLE PATHS\nplt.plot(k00, c0[0], 'ro', alpha = 0.7)\nplt.plot(k00, c0[1], 'ro', alpha = 0.7)\nplt.plot(k01, c0[2], 'ro', alpha = 0.7)\nplt.plot(k02, c0[3], 'ro', alpha = 0.7)\nplt.plot(k02, c0[4], 'ro', alpha = 0.7)\n### ADD MOTION ARROWS\nplt.arrow(k00, c0[0], 0, 0.10, head_width=0.03,head_length=0.02,fc='r',ec='r')\nplt.arrow(k00, c0[0], 0.18, 0, head_width=0.02,head_length=0.03,fc='r',ec='r')\nplt.arrow(k01, c0[2], 0, 0.10, head_width=0.03,head_length=0.02,fc='r',ec='r')\nplt.arrow(k01, c0[2],-0.08, 0, head_width=0.02,head_length=0.03,fc='r',ec='r')\nplt.arrow(k02, c0[3], 0,-0.08, head_width=0.03,head_length=0.02,fc='r',ec='r')\nplt.arrow(k02, c0[3], 0.22, 0, head_width=0.02,head_length=0.03,fc='r',ec='r')\nplt.arrow(k02, c0[4], 0,-0.12, head_width=0.03,head_length=0.02,fc='r',ec='r')\nplt.arrow(k02, c0[4],-0.12, 0, head_width=0.02,head_length=0.03,fc='r',ec='r')\n### AXIS\nax.axvline(0, color=\"k\") # y-axis\nax.axhline(0, color=\"k\") # x-axis\nax.yaxis.set_major_locator(plt.NullLocator()) # Hide ticks\nax.xaxis.set_major_locator(plt.NullLocator()) # Hide ticks\n### TEXT \nplt.text(x_max*0.98, -0.05 , r'$\\hat{k}$', color = 'k') # x-axis label\nplt.text(-0.15 , y_max*0.95 , r'$\\hat{c}$', color = 'k') # y-axis label\nplt.text(k_star*1.05, y_max*0.95, r'$\\hat{c}=0$', color = \"k\")\nplt.text(k_zero*0.95, 0.07 , r'$\\hat{k}=0$', color = \"k\")\n# SETTINGS\nplt.axis(v)\nplt.box(False) # Hide axis\nplt.show()\n\n\n\"8|STABLE-PATH: FORWARD AND BACKWARD SHOOTING\"\ndef forward_shoot(k0, step):\n # Set conditions for initial shoot\n tol = 1.0e-10\n c0 = consumption(k0) - step\n error = np.abs(((k0 - k_star)**2 + (c0 - c_star)**2)**0.5)\n path = np.zeros(shape=(1, 2))\n path[0,0] = k0\n path[0,1] = c0\n\n # Start loop (forward shoot)\n count = 0\n while 1:\n k1 = path[count, 0]\n c1 = path[count, 1]\n k2 = k1 + kdot(k1, c1)*0.1\n c2 = c1 + cdot(k1, c1)*0.1\n error = np.abs(((k2 - k_star)**2 + (c2 - c_star)**2)**0.5)\n path = np.append(path, [[k2, c2]], axis = 0)\n count = count + 1\n\n # Check if this is the stable-path\n if error < tol:\n break\n \n # Set up code break and reset the shoot\n if c2 > c_star:\n path = np.zeros(shape=(1, 2))\n c0 = c0 - step\n path[0,0] = k0\n path[0,1] = c0\n count = 0\n \n # Break code when k < k*\n if k2 > k_star:\n break\n \n # Put a limit of iterations to the code\n if count > 100:\n break\n \n return path\n\n\ndef backward_shoot(k0, step):\n # Set conditions for initial shoot\n tol = 1.0e-10\n c0 = consumption(k_gold) * 1.5\n error = np.abs(((k0 - k_star)**2 + (c0 - c_star)**2)**0.5)\n path = np.zeros(shape=(1, 2))\n path[0,0] = k0\n path[0,1] = c0\n\n # Start loop (forward shoot)\n count = 0\n while 1: # Infinite loop with break conditions inside\n k1 = path[count, 0]\n c1 = path[count, 1]\n k2 = k1 + kdot(k1, c1)*0.01\n c2 = c1 + cdot(k1, c1)*0.01\n error = np.abs(((k2 - k_star)**2 + (c2 - c_star)**2)**0.5)\n path = np.append(path, [[k2, c2]], axis = 0)\n count = count + 1\n\n # Check if this is the stable-path\n if error < tol:\n break\n \n # Set up code break and reset the shoot\n if c2 < c_star:\n path = np.zeros(shape=(1, 2))\n c0 = c0 + step\n path[0,0] = k0\n path[0,1] = c0\n count = 0\n \n # Break code when k < k*\n if k2 < k_star:\n break\n \n # Put a limit of iterations to the code\n if count > 400:\n break\n \n return path\n\nstable_L = forward_shoot(k00, 0.000001)\nstable_R = backward_shoot(k_star*2.00, 0.000001)\n\n\n\n\"9|PLOT RAMSEY MODEL WITH STABLE PATH\"\n# Value of k such that c = 0\nk_zero = (1/(n + g + delta))**(1/(1-alpha))\n# Axis range\ny_max = np.max(c_loci)*1.7\nx_max = k_zero*1.2\nv = [0, x_max, -0.1, y_max] \n\n### BUILD PLOT AND POPULIATE WITH LOCI LINES\nfig, ax = plt.subplots(figsize=(10, 7))\nax.plot(k_array, c_loci, color=\"K\", alpha = 0.8)\nax.axvline(k_loci, color=\"K\", alpha = 0.8)\n### STABLE PATH\nax.plot(stable_L[:,0], stable_L[:,1], \"b--\", alpha = 0.7)\nax.plot(stable_R[:,0], stable_R[:,1], \"b--\", alpha = 0.7)\n### AXIS\nax.axvline(0, color=\"k\") # y-axis\nax.axhline(0, color=\"k\") # x-axis\nax.yaxis.set_major_locator(plt.NullLocator()) # Hide ticks\nax.xaxis.set_major_locator(plt.NullLocator()) # Hide ticks\n### TEXT \nplt.text(x_max*0.98, -0.05 , r'$\\hat{k}$', color = 'k') # x-axis label\nplt.text(-0.15 , y_max*0.95 , r'$\\hat{c}$', color = 'k') # y-axis label\nplt.text(k_star*1.05, y_max*0.95, r'$\\hat{c}=0$', color = \"k\")\nplt.text(k_zero*0.95, 0.07 , r'$\\hat{k}=0$', color = \"k\")\nplt.text(k_star*1.05, -0.05 , r'$k^*$' , color = 'k')\nplt.text(-0.15, c_star , r'$c^*$' , color = 'k')\nax.axhline(c_star, 0, k_star/v[1], linestyle=\":\", color = \"grey\")\n# SETTINGS\nplt.axis(v)\nplt.box(False) # Hide axis\nplt.show()\n```\n", "meta": {"hexsha": "9638e0188744a392049fa88af2bc77ad116beabb", "size": 106509, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ramsey.ipynb", "max_stars_repo_name": "devmacro/nb-open", "max_stars_repo_head_hexsha": "98304713c9e6d4196edf4f28001f550c42b5bf58", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "ramsey.ipynb", "max_issues_repo_name": "devmacro/nb-open", "max_issues_repo_head_hexsha": "98304713c9e6d4196edf4f28001f550c42b5bf58", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ramsey.ipynb", "max_forks_repo_name": "devmacro/nb-open", "max_forks_repo_head_hexsha": "98304713c9e6d4196edf4f28001f550c42b5bf58", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-08-14T10:21:50.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-14T10:21:50.000Z", "avg_line_length": 150.8626062323, "max_line_length": 27016, "alphanum_fraction": 0.8177994348, "converted": true, "num_tokens": 9349, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585903489891, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.44695604356120544}} {"text": "Loading of libraries and initialization\n\n\n```python\nimport random\n\nimport cirq\nfrom cirq.contrib.svg import SVGCircuit\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport sympy\nimport tensorflow as tf\nimport tensorflow_quantum as tfq\n\nif not tf.config.list_physical_devices('GPU'):\n print(\"Warning: GPU was not found, so simulations can be very slow\")\n```\n\n\n```python\nqubit = cirq.GridQubit(0, 0)\n\n# Define some circuits.\ncircuit1 = cirq.Circuit(cirq.X(qubit))\ncircuit2 = cirq.Circuit(cirq.H(qubit))\n\n# Convert to a tensor.\ninput_circuit_tensor = tfq.convert_to_tensor([circuit1, circuit2])\n\n# Define a circuit that we want to append\ny_circuit = cirq.Circuit(cirq.Y(qubit))\n\n# Instantiate our layer\ny_appender = tfq.layers.AddCircuit()\n\n# Run our circuit tensor through the layer and save the output.\noutput_circuit_tensor = y_appender(input_circuit_tensor, append=y_circuit)\n\nprint(tfq.from_tensor(input_circuit_tensor))\nprint(tfq.from_tensor(output_circuit_tensor))\n```\n\n [cirq.Circuit([\n cirq.Moment(\n cirq.X(cirq.GridQubit(0, 0)),\n ),\n ])\n cirq.Circuit([\n cirq.Moment(\n cirq.H(cirq.GridQubit(0, 0)),\n ),\n ])]\n [cirq.Circuit([\n cirq.Moment(\n cirq.X(cirq.GridQubit(0, 0)),\n ),\n cirq.Moment(\n cirq.Y(cirq.GridQubit(0, 0)),\n ),\n ])\n cirq.Circuit([\n cirq.Moment(\n cirq.H(cirq.GridQubit(0, 0)),\n ),\n cirq.Moment(\n cirq.Y(cirq.GridQubit(0, 0)),\n ),\n ])]\n\n", "meta": {"hexsha": "283fd217430b6470a56cfa18bcfdea519b2cab16", "size": 2936, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/qcnn.ipynb", "max_stars_repo_name": "AgenttiX/fys2029-project", "max_stars_repo_head_hexsha": "26dc885064721f40db10fad4405f3366f2cfdf4a", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-05-21T14:39:07.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-21T14:39:07.000Z", "max_issues_repo_path": "examples/qcnn.ipynb", "max_issues_repo_name": "AgenttiX/fys2029-project", "max_issues_repo_head_hexsha": "26dc885064721f40db10fad4405f3366f2cfdf4a", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-05-25T12:52:49.000Z", "max_issues_repo_issues_event_max_datetime": "2021-05-25T12:52:49.000Z", "max_forks_repo_path": "examples/qcnn.ipynb", "max_forks_repo_name": "AgenttiX/fys2029-project", "max_forks_repo_head_hexsha": "26dc885064721f40db10fad4405f3366f2cfdf4a", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.0655737705, "max_line_length": 83, "alphanum_fraction": 0.5071525886, "converted": true, "num_tokens": 424, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.731058578630005, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.4469560363964298}} {"text": "\n\n# Lambda School Data Science Module 142\n## Sampling, Confidence Intervals, and Hypothesis Testing\n\n## Prepare - examine other available hypothesis tests\n\nIf you had to pick a single hypothesis test in your toolbox, t-test would probably be the best choice - but the good news is you don't have to pick just one! Here's some of the others to be aware of:\n\n\n```\nimport numpy as np\nfrom scipy.stats import chisquare # One-way chi square test\n\n# Chi square can take any crosstab/table and test the independence of rows/cols\n# The null hypothesis is that the rows/cols are independent -> low chi square\n# The alternative is that there is a dependence -> high chi square\n# Be aware! Chi square does *not* tell you direction/causation\n\nind_obs = np.array([[1, 1], [2, 2]]).T\nprint(ind_obs)\nprint(chisquare(ind_obs, axis=None))\n\ndep_obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T\nprint(dep_obs)\nprint(chisquare(dep_obs, axis=None))\n```\n\n [[1 2]\n [1 2]]\n Power_divergenceResult(statistic=0.6666666666666666, pvalue=0.8810148425137847)\n [[16 32]\n [18 24]\n [16 16]\n [14 28]\n [12 20]\n [12 24]]\n Power_divergenceResult(statistic=23.31034482758621, pvalue=0.015975692534127565)\n\n\n\n```\n# Distribution tests:\n# We often assume that something is normal, but it can be important to *check*\n\n# For example, later on with predictive modeling, a typical assumption is that\n# residuals (prediction errors) are normal - checking is a good diagnostic\n\nfrom scipy.stats import normaltest\n# Poisson models arrival times and is related to the binomial (coinflip)\nsample = np.random.poisson(5, 1000)\nprint(normaltest(sample)) # Pretty clearly not normal\n```\n\n NormaltestResult(statistic=14.975113494112255, pvalue=0.0005600095353013921)\n\n\n\n```\n# Kruskal-Wallis H-test - compare the median rank between 2+ groups\n# Can be applied to ranking decisions/outcomes/recommendations\n# The underlying math comes from chi-square distribution, and is best for n>5\nfrom scipy.stats import kruskal\n\nx1 = [1, 3, 5, 7, 9]\ny1 = [2, 4, 6, 8, 10]\nprint(kruskal(x1, y1)) # x1 is a little better, but not \"significantly\" so\n\nx2 = [1, 1, 1]\ny2 = [2, 2, 2]\nz = [2, 2] # Hey, a third group, and of different size!\nprint(kruskal(x2, y2, z)) # x clearly dominates\n```\n\n KruskalResult(statistic=0.2727272727272734, pvalue=0.6015081344405895)\n KruskalResult(statistic=7.0, pvalue=0.0301973834223185)\n\n\nAnd there's many more! `scipy.stats` is fairly comprehensive, though there are even more available if you delve into the extended world of statistics packages. As tests get increasingly obscure and specialized, the importance of knowing them by heart becomes small - but being able to look them up and figure them out when they *are* relevant is still important.\n\n## Live Lecture - let's explore some more of scipy.stats\n\nCandidate topics to explore:\n\n- `scipy.stats.chi2` - the Chi-squared distribution, which we can use to reproduce the Chi-squared test\n- Calculate the Chi-Squared test statistic \"by hand\" (with code), and feed it into `chi2`\n- Build a confidence interval with `stats.t.ppf`, the t-distribution percentile point function (the inverse of the CDF) - we can write a function to return a tuple of `(mean, lower bound, upper bound)` that you can then use for the assignment (visualizing confidence intervals)\n\n\n```\nimport pandas as pd\nimport numpy as np\nfrom scipy import stats\n\ndf = pd.read_csv('https://raw.githubusercontent.com/ryanleeallred/datasets/master/adult.csv', na_values=\" ?\")\nprint(df.shape)\ndf.head(20)\n```\n\n (32561, 15)\n\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
ageworkclassfnlwgteducationeducation-nummarital-statusoccupationrelationshipracesexcapital-gaincapital-losshours-per-weekcountrysalary
039State-gov77516Bachelors13Never-marriedAdm-clericalNot-in-familyWhiteMale2174040United-States<=50K
150Self-emp-not-inc83311Bachelors13Married-civ-spouseExec-managerialHusbandWhiteMale0013United-States<=50K
238Private215646HS-grad9DivorcedHandlers-cleanersNot-in-familyWhiteMale0040United-States<=50K
353Private23472111th7Married-civ-spouseHandlers-cleanersHusbandBlackMale0040United-States<=50K
428Private338409Bachelors13Married-civ-spouseProf-specialtyWifeBlackFemale0040Cuba<=50K
537Private284582Masters14Married-civ-spouseExec-managerialWifeWhiteFemale0040United-States<=50K
649Private1601879th5Married-spouse-absentOther-serviceNot-in-familyBlackFemale0016Jamaica<=50K
752Self-emp-not-inc209642HS-grad9Married-civ-spouseExec-managerialHusbandWhiteMale0045United-States>50K
831Private45781Masters14Never-marriedProf-specialtyNot-in-familyWhiteFemale14084050United-States>50K
942Private159449Bachelors13Married-civ-spouseExec-managerialHusbandWhiteMale5178040United-States>50K
1037Private280464Some-college10Married-civ-spouseExec-managerialHusbandBlackMale0080United-States>50K
1130State-gov141297Bachelors13Married-civ-spouseProf-specialtyHusbandAsian-Pac-IslanderMale0040India>50K
1223Private122272Bachelors13Never-marriedAdm-clericalOwn-childWhiteFemale0030United-States<=50K
1332Private205019Assoc-acdm12Never-marriedSalesNot-in-familyBlackMale0050United-States<=50K
1440Private121772Assoc-voc11Married-civ-spouseCraft-repairHusbandAsian-Pac-IslanderMale0040NaN>50K
1534Private2454877th-8th4Married-civ-spouseTransport-movingHusbandAmer-Indian-EskimoMale0045Mexico<=50K
1625Self-emp-not-inc176756HS-grad9Never-marriedFarming-fishingOwn-childWhiteMale0035United-States<=50K
1732Private186824HS-grad9Never-marriedMachine-op-inspctUnmarriedWhiteMale0040United-States<=50K
1838Private2888711th7Married-civ-spouseSalesHusbandWhiteMale0050United-States<=50K
1943Self-emp-not-inc292175Masters14DivorcedExec-managerialUnmarriedWhiteFemale0045United-States>50K
\n
\n\n\n\n\n```\ndf.isnull().sum()\n```\n\n\n\n\n age 0\n workclass 1836\n fnlwgt 0\n education 0\n education-num 0\n marital-status 0\n occupation 1843\n relationship 0\n race 0\n sex 0\n capital-gain 0\n capital-loss 0\n hours-per-week 0\n country 583\n salary 0\n dtype: int64\n\n\n\n\n```\ndf.describe()\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
agefnlwgteducation-numcapital-gaincapital-losshours-per-week
count32561.0000003.256100e+0432561.00000032561.00000032561.00000032561.000000
mean38.5816471.897784e+0510.0806791077.64884487.30383040.437456
std13.6404331.055500e+052.5727207385.292085402.96021912.347429
min17.0000001.228500e+041.0000000.0000000.0000001.000000
25%28.0000001.178270e+059.0000000.0000000.00000040.000000
50%37.0000001.783560e+0510.0000000.0000000.00000040.000000
75%48.0000002.370510e+0512.0000000.0000000.00000045.000000
max90.0000001.484705e+0616.00000099999.0000004356.00000099.000000
\n
\n\n\n\n\n```\ndf.describe(exclude='number')\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
workclasseducationmarital-statusoccupationrelationshipracesexcountrysalary
count307253256132561307183256132561325613197832561
unique816714652412
topPrivateHS-gradMarried-civ-spouseProf-specialtyHusbandWhiteMaleUnited-States<=50K
freq22696105011497641401319327816217902917024720
\n
\n\n\n\n\n```\ndf['hours-per-week']\n```\n\n\n```\ncut_points = [0, 9, 19, 29, 39, 49, 1000]\nlabel_names = ['0-9', '10-19', '20-29', '30-39', '40-49', '50+']\ndf['hours_per_week_categories'] = pd.cut(df['hours-per-week'], cut_points, labels=label_names)\n\ndf.head()\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
ageworkclassfnlwgteducationeducation-nummarital-statusoccupationrelationshipracesexcapital-gaincapital-losshours-per-weekcountrysalaryhours_per_week_categories
039State-gov77516Bachelors13Never-marriedAdm-clericalNot-in-familyWhiteMale2174040United-States<=50K40-49
150Self-emp-not-inc83311Bachelors13Married-civ-spouseExec-managerialHusbandWhiteMale0013United-States<=50K10-19
238Private215646HS-grad9DivorcedHandlers-cleanersNot-in-familyWhiteMale0040United-States<=50K40-49
353Private23472111th7Married-civ-spouseHandlers-cleanersHusbandBlackMale0040United-States<=50K40-49
428Private338409Bachelors13Married-civ-spouseProf-specialtyWifeBlackFemale0040Cuba<=50K40-49
\n
\n\n\n\n\n```\ndf['sex'].iloc[0]\n```\n\n\n\n\n ' Male'\n\n\n\n\n```\ndf['hours_per_week_categories'].iloc[0]\n```\n\n\n\n\n '40-49'\n\n\n\n\n```\ndf['sex'].value_counts()\n```\n\n\n\n\n Male 21790\n Female 10771\n Name: sex, dtype: int64\n\n\n\n\n```\ndf['hours_per_week_categories'].value_counts()\n\n```\n\n\n\n\n 40-49 18336\n 50+ 6462\n 30-39 3667\n 20-29 2392\n 10-19 1246\n 0-9 458\n Name: hours_per_week_categories, dtype: int64\n\n\n\n\n```\ndf = df.sort_values(by='hours_per_week_categories', ascending=True)\n\ndf.head()\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
ageworkclassfnlwgteducationeducation-nummarital-statusoccupationrelationshipracesexcapital-gaincapital-losshours-per-weekcountrysalaryhours_per_week_categories
3129055Self-emp-not-inc41938Bachelors13Married-civ-spouseProf-specialtyWifeWhiteFemale008United-States<=50K0-9
517232NaN134886HS-grad9Married-civ-spouseNaNWifeWhiteFemale002United-States>50K0-9
2292817NaN33266610th6Never-marriedNaNOwn-childWhiteFemale004United-States<=50K0-9
790235Private359131Bachelors13Married-civ-spouseProf-specialtyWifeWhiteFemale729808NaN>50K0-9
660441Private406603HS-grad9Never-marriedOther-serviceNot-in-familyWhiteMale006Iran<=50K0-9
\n
\n\n\n\n\n```\ncontingency_table = pd.crosstab(df['sex'], df['hours_per_week_categories'], margins=True)\n\ncontingency_table\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
hours_per_week_categories0-910-1920-2930-3940-4950+All
sex
Female235671128719145636102810771
Male2235751105175312700543421790
All45812462392366718336646232561
\n
\n\n\n\n\n```\nfemalecount = contingency_table.iloc[0][0:6].values\nfemalecount\n```\n\n\n\n\n array([ 235, 671, 1287, 1914, 5636, 1028])\n\n\n\n\n```\nmalecount = contingency_table.iloc[1][0:6].values\nmalecount\n```\n\n\n\n\n array([ 223, 575, 1105, 1753, 12700, 5434])\n\n\n\n\n```\nimport matplotlib.pyplot as plt\nimport seaborn as sns\n\n#Plots the bar chart\nfig = plt.figure(figsize=(10, 5))\nsns.set(font_scale=1.8)\ncategories = [\"0-9\",\"10-19\",\"20-29\",\"30-39\",\"40-49\",\"50+\"]\np1 = plt.bar(categories, malecount, 0.55, color='#d62728')\np2 = plt.bar(categories, femalecount, 0.55, bottom=malecount)\nplt.legend((p2[0], p1[0]), ('Male', 'Female'))\nplt.xlabel('Hours per Week Worked')\nplt.ylabel('Count')\nplt.show()\n```\n\n### Expected Value Calculation\n\\begin{align}\nexpected_{i,j} =\\frac{(row_{i} \\text{total})(column_{j} \\text{total}) }{(\\text{total observations})} \n\\end{align}\n\n\n```\n# Get Row Sums\nrow_sums = contingency_table.iloc[0:2, 6].values\ncol_sums = contingency_table.iloc[2, 0:6].values\n\nprint(row_sums)\nprint(col_sums)\n```\n\n [10771 21790]\n [ 458 1246 2392 3667 18336 6462]\n\n\n\n```\ntotal = contingency_table.loc['All','All']\ntotal\n```\n\n\n\n\n 32561\n\n\n\n\n```\nexpected = []\nfor i in range(len(row_sums)):\n expected_row = []\n for column in col_sums:\n expected_val = column*row_sums[i]/total\n expected_row.append(expected_val)\n expected.append(expected_row)\n \nprint(np.array(expected))\n```\n\n [[ 151.50388502 412.16995793 791.26046497 1213.02346365\n 6065.44811277 2137.59411566]\n [ 306.49611498 833.83004207 1600.73953503 2453.97653635\n 12270.55188723 4324.40588434]]\n\n\n\n```\nobserved = pd.crosstab(df['sex'], df['hours_per_week_categories']).values\nobserved\n```\n\n\n\n\n array([[ 235, 671, 1287, 1914, 5636, 1028],\n [ 223, 575, 1105, 1753, 12700, 5434]])\n\n\n\n## Chi-Squared Statistic with Numpy\n\n\\begin{align}\n\\chi^2 = \\sum \\frac{(observed_{i}-expected_{i})^2}{(expected_{i})}\n\\end{align}\n\nFor the $observed$ values we will just use a version of our contingency table without the margins as a numpy array. In this way, if our observed values array and our expected values array are the same shape, then we can subtract them and divide them directly which makes the calculations a lot cleaner. No for loops!\n\n\n```\nchi_squared = ((observed - expected)**2/(expected)).sum()\nprint(f\"Chi-Squared: {chi_squared}\")\n```\n\n Chi-Squared: 2287.190943926107\n\n\n\n```\n# Calculate Degrees of Freedom\ndof = (len(row_sums)-1)*(len(col_sums)-1)\nprint(f\"Degrees of Freedom: {dof}\") \n```\n\n Degrees of Freedom: 5\n\n\n\n```\nchi_squared, p_value, dof, expected = stats.chi2_contingency(observed)\n\nprint(f\"Chi-Squared: {chi_squared}\")\nprint(f\"P-value: {p_value}\")\nprint(f\"Degrees of Freedom: {dof}\") \nprint(\"Expected: \\n\", np.array(expected))\n```\n\n Chi-Squared: 2287.190943926107\n P-value: 0.0\n Degrees of Freedom: 5\n Expected: \n [[ 151.50388502 412.16995793 791.26046497 1213.02346365\n 6065.44811277 2137.59411566]\n [ 306.49611498 833.83004207 1600.73953503 2453.97653635\n 12270.55188723 4324.40588434]]\n\n\n# Confidence Interval Example\n\n\n```\n#confidence_interval = [lower_bound, upper_bound]\n\ncoinflips = np.random.binomial(n=1, p=.5, size=100)\nprint(coinflips)\n```\n\n [1 1 0 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0\n 0 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 0 0 0 1 0 0 0 0 0 1 1\n 0 1 1 1 1 0 0 1 0 1 1 1 1 0 1 1 1 1 0 0 1 1 0 0 1 1]\n\n\n\n```\ncoinflips.mean()\n```\n\n\n\n\n 0.52\n\n\n\n\n```\nstats.ttest_1samp(coinflips, .5)\n```\n\n\n\n\n Ttest_1sampResult(statistic=0.39831375340784614, pvalue=0.6912566363051549)\n\n\n\n\n```\ncoinflips_100 = np.random.binomial(n=1, p=.5, size=100)\ncoinflips_1000 = np.random.binomial(n=1, p=.5, size=1000)\n\nprint(\"100 Coinflips Mean:\", coinflips_100.mean())\nprint(\"1000 Coinflips Mean:\", coinflips_1000.mean())\n\nprint(\"100 Coinflips Standard Deviation:\", np.std(coinflips_100))\nprint(\"1000 Coinflips Standard Deviation:\", np.std(coinflips_1000))\n```\n\n 100 Coinflips Mean: 0.57\n 1000 Coinflips Mean: 0.488\n 100 Coinflips Standard Deviation: 0.4950757517794625\n 1000 Coinflips Standard Deviation: 0.49985597925802594\n\n\n\n```\nprint(\"100 Coinflips Standard Error:\", stats.sem(coinflips_100))\nprint(\"1000 Coinflips Standard :\", stats.sem(coinflips_1000))\n```\n\n 100 Coinflips Standard Error: 0.0497569851956243\n 1000 Coinflips Standard : 0.01581474331458169\n\n\n\n```\n0.4950757517794625/np.sqrt(100-1)\n```\n\n\n\n\n 0.0497569851956243\n\n\n\n\n```\n0.49985597925802594/np.sqrt(1000-1)\n```\n\n\n\n\n 0.015814743314581686\n\n\n\n## Confidence Interval Equation\n\n\n\n```\n# Confidence intervals!\n# Similar to hypothesis testing, but centered at sample mean\n# Generally better than reporting the \"point estimate\" (sample mean)\n# Why? Because point estimates aren't always perfect\n\nimport numpy as np\nfrom scipy import stats\n\ndef confidence_interval(data, confidence=0.95):\n \"\"\"\n Calculate a confidence interval around a sample mean for given data.\n Using t-distribution and two-tailed test, default 95% confidence. \n \n Arguments:\n data - iterable (list or numpy array) of sample observations\n confidence - level of confidence for the interval\n \n Returns:\n tuple of (mean, lower bound, upper bound)\n \"\"\"\n data = np.array(data)\n mean = np.mean(data)\n n = len(data)\n stderr = stats.sem(data)\n interval = stderr * stats.t.ppf((1 + confidence) / 2.0, n - 1)\n return (mean, mean - interval, mean + interval)\n\ndef report_confidence_interval(confidence_interval):\n \"\"\"\n Return a string with a pretty report of a confidence interval.\n \n Arguments:\n confidence_interval - tuple of (mean, lower bound, upper bound)\n \n Returns:\n None, but prints to screen the report\n \"\"\"\n #print('Mean: {}'.format(confidence_interval[0]))\n #print('Lower bound: {}'.format(confidence_interval[1]))\n #print('Upper bound: {}'.format(confidence_interval[2]))\n s = \"our mean lies in the interval [{:.5}, {:.5}]\".format(\n confidence_interval[1], confidence_interval[2])\n return s\n```\n\n\n```\nreport_confidence_interval(confidence_interval(coinflips_100))\n```\n\n\n\n\n 'our mean lies in the interval [0.47, 0.67]'\n\n\n\n\n```\nreport_confidence_interval(confidence_interval(coinflips_1000))\n```\n\n\n\n\n 'our mean lies in the interval [0.46, 0.52]'\n\n\n\n\n```\nstats.t.ppf(0.05, 5) # p-value, dof\n```\n\n\n\n\n -2.0150483726691575\n\n\n\n\n```\ncoinflips_1M = np.random.binomial(n=1, p=.5, size=1000000)\nreport_confidence_interval(confidence_interval(coinflips_1M))\n```\n\n\n\n\n 'our mean lies in the interval [0.49912, 0.50108]'\n\n\n\n## Assignment - Build a confidence interval\n\nA confidence interval refers to a neighborhood around some point estimate, the size of which is determined by the desired p-value. For instance, we might say that 52% of Americans prefer tacos to burritos, with a 95% confidence interval of +/- 5%.\n\n52% (0.52) is the point estimate, and +/- 5% (the interval $[0.47, 0.57]$) is the confidence interval. \"95% confidence\" means a p-value $\\leq 1 - 0.95 = 0.05$.\n\nIn this case, the confidence interval includes $0.5$ - which is the natural null hypothesis (that half of Americans prefer tacos and half burritos, thus there is no clear favorite). So in this case, we could use the confidence interval to report that we've failed to reject the null hypothesis.\n\nBut providing the full analysis with a confidence interval, including a graphical representation of it, can be a helpful and powerful way to tell your story. Done well, it is also more intuitive to a layperson than simply saying \"fail to reject the null hypothesis\" - it shows that in fact the data does *not* give a single clear result (the point estimate) but a whole range of possibilities.\n\nHow is a confidence interval built, and how should it be interpreted? It does *not* mean that 95% of the data lies in that interval - instead, the frequentist interpretation is \"if we were to repeat this experiment 100 times, we would expect the average result to lie in this interval ~95 times.\"\n\nFor a 95% confidence interval and a normal(-ish) distribution, you can simply remember that +/-2 standard deviations contains 95% of the probability mass, and so the 95% confidence interval based on a given sample is centered at the mean (point estimate) and has a range of +/- 2 (or technically 1.96) standard deviations.\n\nDifferent distributions/assumptions (90% confidence, 99% confidence) will require different math, but the overall process and interpretation (with a frequentist approach) will be the same.\n\nYour assignment - using the data from the prior module ([congressional voting records](https://archive.ics.uci.edu/ml/datasets/Congressional+Voting+Records)):\n\n1. Generate and numerically represent a confidence interval\n2. Graphically (with a plot) represent the confidence interval\n3. Interpret the confidence interval - what does it tell you about the data and its distribution?\n\nStretch goals:\n\n1. Write a summary of your findings, mixing prose and math/code/results. *Note* - yes, this is by definition a political topic. It is challenging but important to keep your writing voice *neutral* and stick to the facts of the data. Data science often involves considering controversial issues, so it's important to be sensitive about them (especially if you want to publish).\n2. Apply the techniques you learned today to your project data or other data of your choice, and write/discuss your findings here.\n3. Refactor your code so it is elegant, readable, and can be easily run for all issues.\n\n\n```\nimport matplotlib.pyplot as plt\nimport seaborn as sns\n```\n\n\n```\ncongress_data = 'https://archive.ics.uci.edu/ml/machine-learning-databases/voting-records/house-votes-84.data'\ncongress = pd.read_csv(congress_data, names=columns)\n\n```\n\n\n```\ncolumns = ['party','handicapped-infants' \n ,'water-project-cost-sharing' \n ,'adoption-of-the-budget-resolution'\n ,'physician-fee-freeze'\n ,'el-salvador-aid'\n ,'religious-groups-in-schools'\n ,'anti-satellite-test-ban'\n ,'aid-to-nicaraguan-contras' \n ,'mx-missile'\n ,'immigration'\n ,'synfuels-corporation-cutback'\n ,'education-spending'\n ,'superfund-right-to-sue'\n ,'crime'\n ,'duty-free-exports'\n ,'export-administration-act-south-africa']\n```\n\n\n```\ncongress.head()\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
partyhandicapped-infantswater-project-cost-sharingadoption-of-the-budget-resolutionphysician-fee-freezeel-salvador-aidreligious-groups-in-schoolsanti-satellite-test-banaid-to-nicaraguan-contrasmx-missileimmigrationsynfuels-corporation-cutbackeducation-spendingsuperfund-right-to-suecrimeduty-free-exportsexport-administration-act-south-africa
0republicannynyyynnnyyyyny
1republicannynyyynnnnnyyyn
2democratyyyynnnnynyynn
3democratnyynynnnnynynny
4democratyyynyynnnnyyyyy
\n
\n\n\n\n\n```\ncongress.head(2)\n```\n\n\n```\ncongress = congress.sort_values(by='', ascending=True)\n\ncongress.head()\n```\n\n\n```\ncongress[columns] = congress[columns].replace({\"?\": \" \"})\n```\n\n\n```\ncongress[columns] = congress[columns].replace({\"y\": 1, \"n\": 0})\n```\n\n\n```\ncongress.head()\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
partyhandicapped-infantswater-project-cost-sharingadoption-of-the-budget-resolutionphysician-fee-freezeel-salvador-aidreligious-groups-in-schoolsanti-satellite-test-banaid-to-nicaraguan-contrasmx-missileimmigrationsynfuels-corporation-cutbackeducation-spendingsuperfund-right-to-suecrimeduty-free-exportsexport-administration-act-south-africa
0republican010111000111101
1republican010111000001110
2democrat11110000101100
3democrat011010000101001
4democrat111011000011111
\n
\n\n\n\nlet's see if party is important for who supports the education spending bill\n\n\n\n\n\n\n```\ncontinge_table = pd.crosstab(congress['party'], congress['education-spending'], margins=True)\n```\n\n\n```\ncontingency = (continge_table)\ncontingency\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
education-spendingnyAll
party
democrat1821336267
republican1320135168
All31233171435
\n
\n\n\n\n\n```\nrepcount = contingency.iloc[1][0:4].values\nrepcount\n```\n\n\n\n\n array([ 13, 20, 135, 168])\n\n\n\n\n```\ndemcount = contingency.iloc[0][0:4].values\ndemcount\n```\n\n\n\n\n array([ 18, 213, 36, 267])\n\n\n\n\n```\ncontingencyedit = contingency.\n```\n\n\n```\nno = contingency['n']\nno\n```\n\n\n\n\n party\n democrat 213\n republican 20\n All 233\n Name: n, dtype: int64\n\n\n\n\n```\nimport matplotlib.pyplot as plt\nimport seaborn as sns\n\n\nfig = plt.figure(figsize=(10, 4))\nsns.set(font_scale=1.8)\np1 = plt.bar(yes, demcount, 0.55, color='#d62728')\np2 = plt.bar(yes, repcount, 0.55, bottom=demcount)\nplt.legend((p2[0], p1[0]), ('Democrat', 'Republican'))\nplt.xlabel('Support for the Education Bill')\nplt.ylabel('Count')\nplt.show()\n```\n\n\n```\nfig = plt.figure(figsize=(10, 4))\nsns.set(font_scale=1.2)\ncontingency.plot.bar(stacked=True)\nplt.legend(title='vote')\n\nplt.show()\n```\n\n## Resources\n\n- [Interactive visualize the Chi-Squared test](https://homepage.divms.uiowa.edu/~mbognar/applets/chisq.html)\n- [Calculation of Chi-Squared test statistic](https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test)\n- [Visualization of a confidence interval generated by R code](https://commons.wikimedia.org/wiki/File:Confidence-interval.svg)\n- [Expected value of a squared standard normal](https://math.stackexchange.com/questions/264061/expected-value-calculation-for-squared-normal-distribution) (it's 1 - which is why the expected value of a Chi-Squared with $n$ degrees of freedom is $n$, as it's the sum of $n$ squared standard normals)\n", "meta": {"hexsha": "6d3ad24a5441755d33f9032e5c7ae9635976479a", "size": 151977, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Copy_of_LS_DS5_142_Sampling_Confidence_Intervals_and_Hypothesis_Testing.ipynb", "max_stars_repo_name": "eliza0shrug/DS-Unit-1-Sprint-3-Statistical-Tests-and-Experiments", "max_stars_repo_head_hexsha": "e14153d3c3476407629e9fdeb7060685bac8d0e2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Copy_of_LS_DS5_142_Sampling_Confidence_Intervals_and_Hypothesis_Testing.ipynb", "max_issues_repo_name": "eliza0shrug/DS-Unit-1-Sprint-3-Statistical-Tests-and-Experiments", "max_issues_repo_head_hexsha": "e14153d3c3476407629e9fdeb7060685bac8d0e2", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Copy_of_LS_DS5_142_Sampling_Confidence_Intervals_and_Hypothesis_Testing.ipynb", "max_forks_repo_name": "eliza0shrug/DS-Unit-1-Sprint-3-Statistical-Tests-and-Experiments", "max_forks_repo_head_hexsha": "e14153d3c3476407629e9fdeb7060685bac8d0e2", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 47.9271523179, "max_line_length": 29902, "alphanum_fraction": 0.5301262691, "converted": true, "num_tokens": 14941, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819591324418, "lm_q2_score": 0.7310585903489891, "lm_q1q2_score": 0.4469560332081661}} {"text": "---\nlayout: page\ntitle: Verossimilhan\u00e7a\nnav_order: 16\n---\n\n[](https://colab.research.google.com/github/icd-ufmg/icd-ufmg.github.io/blob/master/_lessons/16-vero.ipynb)\n\n# Verossimilhan\u00e7a\n\n{: .no_toc .mb-2 }\n\nEntendimento de rela\u00e7\u00e3o entre dados.\n{: .fs-6 .fw-300 }\n\n{: .no_toc .text-delta }\nResultados Esperados\n\n1. Revisitar os m\u00ednimos quadrados (ALC)\n1. Entender a regress\u00e3o linear do ponto de vista probabil\u00edstico\n1. Entender o conceito de verossimilhan\u00e7a\n\n---\n**Sum\u00e1rio**\n1. TOC\n{:toc}\n---\n\n\n```python\n# -*- coding: utf8\n\nfrom scipy import stats as ss\n\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport pandas as pd\nimport seaborn as sns\n```\n\n\n```python\nplt.style.use('seaborn-colorblind')\nplt.rcParams['figure.figsize'] = (16, 10)\nplt.rcParams['axes.labelsize'] = 20\nplt.rcParams['axes.titlesize'] = 20\nplt.rcParams['legend.fontsize'] = 20\nplt.rcParams['xtick.labelsize'] = 20\nplt.rcParams['ytick.labelsize'] = 20\nplt.rcParams['lines.linewidth'] = 4\n```\n\n\n```python\nplt.ion()\n```\n\n\n```python\ndef despine(ax=None):\n if ax is None:\n ax = plt.gca()\n # Hide the right and top spines\n ax.spines['right'].set_visible(False)\n ax.spines['top'].set_visible(False)\n\n # Only show ticks on the left and bottom spines\n ax.yaxis.set_ticks_position('left')\n ax.xaxis.set_ticks_position('bottom')\n```\n\n## Introdu\u00e7\u00e3o\n\nContinuando da aula passada. Vamos ver mais uma forma de entender um modelo de regress\u00e3o linear. Lembre-se at\u00e9 agora falamos de correla\u00e7\u00e3o e covari\u00e2ncia cobrindo os seguintes t\u00f3picos:\n\n1. Covari\u00e2ncia\n1. Coeficiente de Pearson (Covari\u00e2ncia Normalizada)\n1. Coeficiente de Pearson como sendo a fra\u00e7\u00e3o do desvio de y capturado por x\n1. M\u00ednimos Quadrados\n\nTodos os passos acima chegam no mesmo local de tra\u00e7ar a \"melhor\" reta no gr\u00e1fico de dispers\u00e3o. Melhor aqui significa a reta que que minimiza o erro abaixo:\n\n$$\\Theta = [\\alpha, \\beta]$$\n$$L(\\Theta) = \\sum_i (y_i - \\hat{y}_i)^2$$\n$$L(\\Theta) = \\sum_i (y_i - \\beta x_i + \\alpha)^2$$\n\nChegamos em:\n\n\\begin{align}\n \\alpha & = \\bar{y} - \\beta\\,\\bar{x}, \\\\[5pt]\n \\beta &= \\frac{ \\sum_{i=1}^n (x_i - \\bar{x})(y_i - \\bar{y}) }{ \\sum_{i=1}^n (x_i - \\bar{x})^2 } \\\\[6pt]\n &= \\frac{ \\operatorname{Cov}(x, y) }{ \\operatorname{Var}(x) } \\\\[5pt]\n &= r_{xy} \\frac{s_y}{s_x}. \\\\[6pt]\n\\end{align}\n\n## Vis\u00e3o probab\u00edlistica\n\nVamos aprender uma \u00faltima forma de pensar na regress\u00e3o. Em particular, vamos fazer uso de uma vis\u00e3o probab\u00edlistica. Para tal, exploraremos o caso dos apartamentos de BH abaixo.\n\nInicialmente, vamos observar os dados al\u00e9m do resultado da melhor regress\u00e3o.\n\n\n```python\ndf = pd.read_csv('https://raw.githubusercontent.com/icd-ufmg/material/master/aulas/17-Verossimilhanca/aptosBH.txt', index_col=0)\ndf['preco'] = df['preco'] / 1000\nplt.scatter(df['area'], df['preco'], edgecolors='k', s=80, alpha=0.6)\nplt.title('Preco de Apartamentos em BH')\nplt.ylabel(r'Pre\u00e7o * $10^3$ (R\\$)')\nplt.xlabel(r'\u00c1rea ($M^2$)')\ndespine()\n```\n\nO seaborn tem uma fun\u00e7\u00e3o regplot que plota a melhor reta al\u00e9m de um IC (estmado via bootstrap -- aula passada).\n\n\n```python\nsns.regplot(x='area', y='preco', data=df, n_boot=10000,\n line_kws={'color':'magenta', 'lw':4},\n scatter_kws={'edgecolor':'k', 's':80, 'alpha':0.8})\nplt.title('Preco de Apartamentos em BH')\nplt.ylabel(r'Pre\u00e7o * $10^3$ (R\\$)')\nplt.xlabel(r'\u00c1rea ($M^2$)')\ndespine()\n```\n\nA reta pode ser recuperada usando scipy.\n\n\n```python\nmodel = ss.linregress(df['area'], df['preco'])\n# beta = slope\n# alpha = intercept\nmodel \n```\n\n\n\n\n LinregressResult(slope=3.5357191563336516, intercept=200.5236136898945, rvalue=0.694605637960645, pvalue=1.917920339304322e-32, stderr=0.2503210673009473)\n\n\n\nUsando esta reta podemos prever o pre\u00e7o de um apartamento usando apenas a \u00e1rea do mesmo.\n\n\n```python\nbeta = model.slope\nalpha = model.intercept\nnovo_apt_area = 225\npreco = beta * novo_apt_area + alpha\npreco\n```\n\n\n\n\n 996.0604238649662\n\n\n\nOu seja, quando um apartamento de 225m2 entra no mercado o mesmo custa em torno de 1M de reais.\n\n## Erros Normais\n\nAgora, ser\u00e1 que conseguimos chegar no mesmo pensando na regress\u00e3o como um modelo probabil\u00edstico?\n\n[Discuss\u00e3o nos Slides](https://docs.google.com/presentation/d/1nSmN9ch1x6ABczaAzB292XgTqbXcIo1UIX1xjCrF1d8/edit#slide=id.g5a114266c3_0_26)\n\n\n```python\nx = np.linspace(-5, 5, 100)\nplt.plot(x, ss.distributions.norm.pdf(x, scale=1))\nplt.xlabel(r'$\\epsilon_i$')\nplt.ylabel(r'$p(\\epsilon_i \\mid \\mu=0, \\sigma=1)$')\n\ndespine()\n```\n\n\n```python\nbeta = 1\nalpha = 1\n\nfig = plt.figure(figsize=(36, 10))\n\nx = np.array([2, 8, 5])\ny = np.array([0, 1, 3])\n\nplt.subplot(121)\nplt.scatter(x, y, edgecolors='k', s=80, alpha=0.6)\nplt.title('3 Pontinhos')\nplt.ylabel(r'Y')\nplt.xlabel(r'X')\n\ny_bar = x * beta + alpha\nplt.plot(x, y_bar, color='magenta')\n\ny_min = [min(y_i, y_bar_i) for y_i, y_bar_i in zip(y, y_bar)]\ny_max = [max(y_i, y_bar_i) for y_i, y_bar_i in zip(y, y_bar)]\nplt.vlines(x, ymin=y_min, ymax=y_max, color='magenta', lw=1)\n\ndespine()\n\nplt.subplot(122)\nplt.title('PDF da Normal')\nei_x = np.linspace(-10, 10, 100)\nsigma = (y - y_bar).std(ddof=1)\nplt.plot(ei_x, ss.distributions.norm.pdf(ei_x, scale=sigma))\nplt.xlabel(r'$\\epsilon_i$')\nplt.ylabel(r'$p(\\epsilon_i \\mid \\mu=0, \\sigma={})$'.format(np.round(sigma, 2)))\ndespine()\n```\n\n\n```python\nbeta = 3.535719156333653\nalpha = 200.52361368989432\n\nfig = plt.figure(figsize=(36, 10))\n\nx = df['area']\ny = df['preco']\n\nplt.subplot(121)\nplt.scatter(x, y, edgecolors='k', s=80, alpha=0.6)\nplt.title('Preco de Apartamentos em BH')\nplt.ylabel(r'Pre\u00e7o * $10^3$ (R\\$)')\nplt.xlabel(r'\u00c1rea ($M^2$)')\n\ny_bar = x * beta + alpha\nplt.plot(x, y_bar, color='magenta')\n\ny_min = [min(y_i, y_bar_i) for y_i, y_bar_i in zip(y, y_bar)]\ny_max = [max(y_i, y_bar_i) for y_i, y_bar_i in zip(y, y_bar)]\nplt.vlines(x, ymin=y_min, ymax=y_max, color='magenta', lw=1)\n\ndespine()\n\nplt.subplot(122)\nplt.title('PDF da Normal')\nei_x = np.linspace(-1000, 1000, 100)\nsigma = (y - y_bar).std(ddof=1)\nplt.plot(ei_x, ss.distributions.norm.pdf(ei_x, scale=sigma))\nplt.xlabel(r'$\\epsilon_i$')\nplt.ylabel(r'$p(\\epsilon_i \\mid \\mu=0, \\sigma={})$'.format(np.round(sigma, 2)))\ndespine()\n```\n\n\n```python\nsns.residplot(x='area', y='preco', data=df,\n line_kws={'color':'magenta', 'lw':4},\n scatter_kws={'edgecolor':'k', 's':80, 'alpha':0.8})\nplt.ylabel(r'$\\epsilon_i$')\nplt.xlabel(r'\u00c1rea ($M^2$)')\ndespine()\n```\n\n\n```python\nss.probplot(y - y_bar, plot=plt.gca());\n```\n\n## Close Nova Dataset\n\nAbaixo temos a dispers\u00e3o dos dados\n\n\n```python\ndf = pd.read_csv('https://media.githubusercontent.com/media/icd-ufmg/material/master/aulas/17-Verossimilhanca/close_novas.csv')\nx = df.values[:, 0]\ny = df.values[:, 1]\n\nplt.scatter(x, y, alpha=0.8, edgecolors='k', s=80)\nplt.xlabel(df.columns[0])\nplt.ylabel(df.columns[1])\nplt.xlim((0, 300))\nplt.ylim((0, 0.03))\nplt.title('Close Nova Dataset')\ndespine()\n```\n\n\n```python\n1e6 / (ss.pearsonr(x, y)[0] * y.std(ddof=1) / x.std(ddof=1))\n```\n\n\n\n\n 14612822334.220728\n\n\n\n\n```python\nx = df['Distance (million parsecs)']\ny = df['Speed (parsecs/year)']\n\nmodel = ss.linregress(x, y)\nbeta = model.slope\nalpha = model.intercept\n\nplt.scatter(x, y, edgecolors='k', s=80, alpha=0.6)\nplt.title('Closed Novas')\nplt.ylabel(r'Speed (parsecs/year)')\nplt.xlabel(r'Distance (million parsecs)')\n\ny_bar = x * beta + alpha\nplt.plot(x, y_bar, color='magenta')\n\ny_min = [min(y_i, y_bar_i) for y_i, y_bar_i in zip(y, y_bar)]\ny_max = [max(y_i, y_bar_i) for y_i, y_bar_i in zip(y, y_bar)]\nplt.vlines(x, ymin=y_min, ymax=y_max, color='magenta', lw=1)\n\ndespine()\n```\n\n\n```python\nsns.residplot(x='Distance (million parsecs)', y='Speed (parsecs/year)', data=df,\n line_kws={'color':'magenta', 'lw':4},\n scatter_kws={'edgecolor':'k', 's':80, 'alpha':0.8})\nplt.ylabel(r'$\\epsilon_i$')\ndespine()\n```\n\n\n```python\nss.probplot(y - y_bar, plot=plt);\n```\n\n\n```python\nimport statsmodels.api as sm\n\nstocks = {'Year': [2017,2017,2017,2017,2017,2017,2017,2017,2017,2017,2017,2017,2016,2016,2016,2016,2016,2016,2016,2016,2016,2016,2016,2016],\n 'Month': [12, 11,10,9,8,7,6,5,4,3,2,1,12,11,10,9,8,7,6,5,4,3,2,1],\n 'Interest_Rate': [2.75,2.5,2.5,2.5,2.5,2.5,2.5,2.25,2.25,2.25,2,2,2,1.75,1.75,1.75,1.75,1.75,1.75,1.75,1.75,1.75,1.75,1.75],\n 'Unemployment_Rate': [5.3,5.3,5.3,5.3,5.4,5.6,5.5,5.5,5.5,5.6,5.7,5.9,6,5.9,5.8,6.1,6.2,6.1,6.1,6.1,5.9,6.2,6.2,6.1],\n 'Stock_Index_Price': [1464,1394,1357,1293,1256,1254,1234,1195,1159,1167,1130,1075,1047,965,943,958,971,949,884,866,876,822,704,719] \n }\n\ndf = pd.DataFrame(stocks, columns=['Year','Month', 'Interest_Rate', 'Unemployment_Rate', 'Stock_Index_Price'])\n```\n\n\n```python\nx = df['Interest_Rate']\ny = df['Stock_Index_Price']\nmodel = ss.linregress(x, y)\nbeta = model.slope\nalpha = model.intercept\n\nplt.scatter(x, y, edgecolors='k', s=80, alpha=0.6)\nplt.title('Stocks')\nplt.ylabel(r'Stock_Index_Price')\nplt.xlabel(r'Interest_Rate')\n\ny_bar = x * beta + alpha\nplt.plot(x, y_bar, color='magenta')\n\ny_min = [min(y_i, y_bar_i) for y_i, y_bar_i in zip(y, y_bar)]\ny_max = [max(y_i, y_bar_i) for y_i, y_bar_i in zip(y, y_bar)]\nplt.vlines(x, ymin=y_min, ymax=y_max, color='magenta', lw=1)\n\ndespine()\n```\n\n\n```python\nsns.residplot(x='Interest_Rate', y='Stock_Index_Price', data=df,\n line_kws={'color':'magenta', 'lw':4},\n scatter_kws={'edgecolor':'k', 's':80, 'alpha':0.8})\nplt.ylabel(r'$\\epsilon_i$')\ndespine()\n```\n\n\n```python\nss.probplot(y - y_bar, plot=plt);\n```\n\n\n```python\nx = df['Unemployment_Rate']\ny = df['Stock_Index_Price']\nmodel = ss.linregress(x, y)\nbeta = model.slope\nalpha = model.intercept\n\nplt.scatter(x, y, edgecolors='k', s=80, alpha=0.6)\nplt.title('Stocks')\nplt.ylabel(r'Unemployment_Rate')\nplt.xlabel(r'Stock_Index_Price')\n\ny_bar = x * beta + alpha\nplt.plot(x, y_bar, color='magenta')\n\ny_min = [min(y_i, y_bar_i) for y_i, y_bar_i in zip(y, y_bar)]\ny_max = [max(y_i, y_bar_i) for y_i, y_bar_i in zip(y, y_bar)]\nplt.vlines(x, ymin=y_min, ymax=y_max, color='magenta', lw=1)\n\ndespine()\n```\n\n\n```python\nsns.residplot(x='Unemployment_Rate', y='Stock_Index_Price', data=df,\n line_kws={'color':'magenta', 'lw':4},\n scatter_kws={'edgecolor':'k', 's':80, 'alpha':0.8})\nplt.ylabel(r'$\\epsilon_i$')\ndespine()\n```\n\n\n```python\nss.probplot(y - y_bar, plot=plt);\n```\n\n\n```python\ndf = pd.read_csv('http://www.statsci.org/data/oz/dugongs.txt', sep='\\t')\ndf\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
AgeLength
01.01.80
11.51.85
21.51.87
31.51.77
42.52.02
54.02.27
65.02.15
75.02.26
87.02.35
98.02.47
108.52.19
119.02.26
129.52.40
139.52.39
1410.02.41
1512.02.50
1612.02.32
1713.02.43
1813.02.47
1914.52.56
2015.52.65
2115.52.47
2216.52.64
2317.02.56
2422.52.70
2529.02.72
2631.52.57
\n
\n\n\n\n\n```python\nx = df['Age']\ny = df['Length']\nmodel = ss.linregress(x, y)\nbeta = model.slope\nalpha = model.intercept\n\nplt.scatter(x, y, edgecolors='k', s=80, alpha=0.6)\nplt.title('Dugongos')\nplt.ylabel(r'Length')\nplt.xlabel(r'Age')\n\ny_bar = x * beta + alpha\nplt.plot(x, y_bar, color='magenta')\n\ny_min = [min(y_i, y_bar_i) for y_i, y_bar_i in zip(y, y_bar)]\ny_max = [max(y_i, y_bar_i) for y_i, y_bar_i in zip(y, y_bar)]\nplt.vlines(x, ymin=y_min, ymax=y_max, color='magenta', lw=1)\n\ndespine()\n```\n\n\n```python\nsns.residplot(x='Age', y='Length', data=df,\n line_kws={'color':'magenta', 'lw':4},\n scatter_kws={'edgecolor':'k', 's':80, 'alpha':0.8})\nplt.ylabel(r'$\\epsilon_i$')\ndespine()\n```\n\n\n```python\ndf = pd.read_csv('http://www.statsci.org/data/oz/dugongs.txt', sep='\\t')\ny, x = df.values.T\n```\n", "meta": {"hexsha": "76f6d1e15094592c9fa26896cf70361dc279f518", "size": 1047858, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "_lessons/.ipynb_checkpoints/16-vero-checkpoint.ipynb", "max_stars_repo_name": "icd-ufmg/icd-ufmg.github.io", "max_stars_repo_head_hexsha": "5bc96e818938f8dec09dc93d786e4b291d298a02", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2019-02-25T18:25:49.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-20T19:22:24.000Z", "max_issues_repo_path": "_lessons/.ipynb_checkpoints/16-vero-checkpoint.ipynb", "max_issues_repo_name": "thiagomrs/icd-ufmg.github.io", "max_issues_repo_head_hexsha": "f72c0eca5a0f97d83be214aff52715c986b078a7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "_lessons/.ipynb_checkpoints/16-vero-checkpoint.ipynb", "max_forks_repo_name": "thiagomrs/icd-ufmg.github.io", "max_forks_repo_head_hexsha": "f72c0eca5a0f97d83be214aff52715c986b078a7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2021-06-05T20:49:02.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-11T20:21:44.000Z", "avg_line_length": 679.9857235561, "max_line_length": 129092, "alphanum_fraction": 0.9412773486, "converted": true, "num_tokens": 5052, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.611381973294151, "lm_q2_score": 0.7310585727705126, "lm_q1q2_score": 0.44695603281404167}} {"text": "

Table of Contents

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\n\n# Imports\n\n\n```python\n# %matplotlib notebook\nfrom NearFieldOptics.Materials import *\nfrom NearFieldOptics.Materials.material_types import *\nfrom NearFieldOptics.Materials.TransferMatrixMedia import *\nfrom common.baseclasses import AWA\nimport sympy\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport matplotlib.ticker as ticker\nimport matplotlib as mpl\nimport scipy.special as ss\nfrom scipy.optimize import curve_fit\nimport mpmath as mp\nfrom mpmath.ctx_fp import FPContext\nfrom sys import stdout\n\nsympy.init_printing(use_unicode=True)\n```\n\n :\n \tLoading tabulated material data from file \"Bi2Se3_epsilon.pickle\"...\n :\n \tLoading tabulated material data from file \"PMMA_epsilon.pickle\"...\n :\n \tLoading tabulated material data from file \"sio2_300nm_extracted_epsilon_cone_A=2a.pickle\"...\n :\n \tLoading tabulated material data from file \"TaS2_eps_230K.csv\"...\n :\n \tLoading tabulated material data from file \"TaS2_eps_30K.csv\"...\n :\n \tLoading tabulated material data from file \"Erik_BSTS_epsilon.pickle\"...\n :\n \tLoading tabulated material data from file \"VO2_Insulating.pickle\"...\n :\n \tLoading tabulated material data from file \"VO2_Metallic.pickle\"...\n\n\n# Material\n\n\n```python\nLM = LayeredMediaTM((BN_Caldwell,30e-7),(SiO2_300nm,60e-7))\n```\n\n# rp numerics\n\n## freq and q arrays\n\n\n```python\nfreq = np.linspace(200,2000,500)\nq = np.linspace(1,1e6,400)\n\nrp = LM.reflection_p(freq,q)\nabs(rp).plot(vmax=1)\nplt.ticklabel_format(style='sci', axis='x', scilimits=(0,0))\n```\n\n\n```python\nabs(rp).cslice[1e5].plot()\n```\n\n## q array and single freq\n\n\n```python\nq = np.linspace(1,1e6,400)\nfreq = 1400\nrp = LM.reflection_p(freq,q)\nabs(rp).plot()\nplt.ticklabel_format(style='sci', axis='x', scilimits=(0,0))\n```\n\n## freq array and single q\n\n\n```python\nfreq = np.linspace(200,2000,500)\nq = 1/30e-7\nrp = LM.reflection_p(freq,q)\nabs(rp).plot()\n```\n\n# rs numerics\n\n## freq and q arrays\n\n\n```python\nfreq = np.linspace(200,2000,500)\n# q = 1/30e-7\nq = np.linspace(1,1e6,400)\n\nrs = LM.reflection_s(freq,q)\nabs(rs).plot(vmax=1)\nplt.ticklabel_format(style='sci', axis='x', scilimits=(0,0))\n```\n\n\n```python\nabs(rs).cslice[1e5].plot()\n```\n\n## q array and single freq\n\n\n```python\nq = np.linspace(1,1e6,400)\nfreq = 1400\nrs = LM.reflection_s(freq,q)\nabs(rs).plot()\nplt.ticklabel_format(style='sci', axis='x', scilimits=(0,0))\n```\n\n## freq array and single q\n\n\n```python\nfreq = np.linspace(200,2000,500)\nq = 1/30e-7\nrs = LM.reflection_s(freq,q)\nabs(rs).plot()\n```\n\n# tp numerics\n\n## freq and q arrays\n\n\n```python\nfreq = np.linspace(200,2000,500)\n# q = 1/30e-7\nq = np.linspace(1,1e6,400)\n\ntp = LM.transmission_p(freq,q)\nabs(tp).plot(vmax=1)\nplt.ticklabel_format(style='sci', axis='x', scilimits=(0,0))\n```\n\n\n```python\nabs(tp).cslice[1e5].plot()\n```\n\n## q array and single freq\n\n\n```python\nq = np.linspace(1,1e6,400)\nfreq = 1400\ntp = LM.transmission_p(freq,q)\nabs(tp).plot()\nplt.ticklabel_format(style='sci', axis='x', scilimits=(0,0))\n```\n\n## freq array and single q\n\n\n```python\nfreq = np.linspace(200,2000,500)\nq = 1/30e-7\ntp = LM.transmission_p(freq,q)\nabs(tp).plot()\n```\n\n# ts numerics\n\n## freq and q arrays\n\n\n```python\nfreq = np.linspace(200,2000,500)\n# q = 1/30e-7\nq = np.linspace(1,1e6,400)\n\nts = LM.transmission_s(freq,q)\nabs(ts).plot(vmax=1)\nplt.ticklabel_format(style='sci', axis='x', scilimits=(0,0))\n```\n\n\n```python\nabs(ts).cslice[1e5].plot()\n```\n\n## q array and single freq\n\n\n```python\nq = np.linspace(1,1e6,400)\nfreq = 1400\nts = LM.transmission_s(freq,q)\nabs(ts).plot()\nplt.ticklabel_format(style='sci', axis='x', scilimits=(0,0))\n```\n\n## freq array and single q\n\n\n```python\nfreq = np.linspace(200,2000,500)\nq = 1/30e-7\nts = LM.transmission_s(freq,q)\nabs(ts).plot()\n```\n\n# h numerics\n\n## freq and q arrays\n\n\n```python\nfreq = np.linspace(200,2000,500)\n# q = 1/30e-7\nq = np.linspace(1,1e6,400)\n\nh = LM.h_field(freq,q,index=1)\nabs(h).plot(vmax=1e-11)\nplt.ticklabel_format(style='sci', axis='x', scilimits=(0,0))\n```\n\n\n```python\nabs(h).cslice[1e5].plot()\n```\n\n## q array and single freq\n\n\n```python\nq = np.linspace(1,1e6,400)\nfreq = 1400\nh = LM.h_field(freq,q)\nabs(h).plot()\nplt.ticklabel_format(style='sci', axis='x', scilimits=(0,0))\n```\n\n## freq array and single q\n\n\n```python\nfreq = np.linspace(200,2000,500)\nq = 1/30e-7\nh = LM.h_field(freq,q)\nabs(h).plot()\n```\n\n# k numerics\n\n## freq and q arrays\n\n\n```python\nfreq = np.linspace(200,2000,500)\n# q = 1/30e-7\nq = np.linspace(1,1e6,400)\n\nk = LM.Coulomb_kernel(freq,q,index=1)\nabs(k).plot(vmax=5e-5)\nplt.ticklabel_format(style='sci', axis='x', scilimits=(0,0))\n```\n\n\n```python\nabs(k).cslice[1e5].plot()\n```\n\n## q array and single freq\n\n\n```python\nq = np.linspace(1,1e6,400)\nfreq = 1400\nk = LM.Coulomb_kernel(freq,q)\nabs(k).plot()\nplt.ticklabel_format(style='sci', axis='x', scilimits=(0,0))\n```\n\n## freq array and single q\n\n\n```python\nfreq = np.linspace(200,2000,500)\nq = 1/30e-7\nk = LM.Coulomb_kernel(freq,q)\nabs(k).plot()\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "72dbcb6844e6799e49b5b416c3900ccedd142cfd", "size": 505457, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "LayeredMediaTM example use case.ipynb", "max_stars_repo_name": "Leo-Lo/TransferMatrixMedia", "max_stars_repo_head_hexsha": "dbce6f69ba794cf469884a0357d903293cf288ed", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "LayeredMediaTM example use case.ipynb", "max_issues_repo_name": "Leo-Lo/TransferMatrixMedia", "max_issues_repo_head_hexsha": "dbce6f69ba794cf469884a0357d903293cf288ed", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "LayeredMediaTM example use case.ipynb", "max_forks_repo_name": "Leo-Lo/TransferMatrixMedia", "max_forks_repo_head_hexsha": "dbce6f69ba794cf469884a0357d903293cf288ed", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 498.9703849951, "max_line_length": 49332, "alphanum_fraction": 0.9458074574, "converted": true, "num_tokens": 2867, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081926, "lm_q2_score": 0.6513548782017745, "lm_q1q2_score": 0.4467965237194349}} {"text": "## Configurations for Colab\n\n\n```python\nimport sys\nIN_COLAB = \"google.colab\" in sys.modules\n\nif IN_COLAB:\n !apt install python-opengl\n !apt install ffmpeg\n !apt install xvfb\n !pip install PyVirtualDisplay==3.0\n !pip install gym==0.21.0\n from pyvirtualdisplay import Display\n \n # Start virtual display\n dis = Display(visible=0, size=(400, 400))\n dis.start()\n```\n\n# 05. Noisy Networks for Exploration\n\n[M. Fortunato et al., \"Noisy Networks for Exploration.\" arXiv preprint arXiv:1706.10295, 2017.](https://arxiv.org/pdf/1706.10295.pdf)\n\n\nNoisyNet is an exploration method that learns perturbations of the network weights to drive exploration. The key insight is that a single change to the weight vector can induce a consistent, and potentially very complex, state-dependent change in policy over multiple time steps.\n\nFirstly, let's take a look into a linear layer of a neural network with $p$ inputs and $q$ outputs, represented by\n\n$$\ny = wx + b,\n$$\n\nwhere $x \\in \\mathbb{R}^p$ is the layer input, $w \\in \\mathbb{R}^{q \\times p}$, and $b \\in \\mathbb{R}$ the bias.\n\nThe corresponding noisy linear layer is defined as:\n\n$$\ny = (\\mu^w + \\sigma^w \\odot \\epsilon^w) x + \\mu^b + \\sigma^b \\odot \\epsilon^b,\n$$\n\nwhere $\\mu^w + \\sigma^w \\odot \\epsilon^w$ and $\\mu^b + \\sigma^b \\odot \\epsilon^b$ replace $w$ and $b$ in the first linear layer equation. The parameters $\\mu^w \\in \\mathbb{R}^{q \\times p}, \\mu^b \\in \\mathbb{R}^q, \\sigma^w \\in \\mathbb{R}^{q \\times p}$ and $\\sigma^b \\in \\mathbb{R}^q$ are learnable, whereas $\\epsilon^w \\in \\mathbb{R}^{q \\times p}$ and $\\epsilon^b \\in \\mathbb{R}^q$ are noise random variables which can be generated by one of the following two ways:\n\n1. **Independent Gaussian noise**: the noise applied to each weight and bias is independent, where each random noise entry is drawn from a unit Gaussian distribution. This means that for each noisy linear layer, there are $pq + q$ noise variables (for $p$ inputs to the layer and $q$ outputs).\n2. **Factorised Gaussian noise:** This is a more computationally efficient way. It produces 2 random Gaussian noise vectors ($p, q$) and makes $pq + q$ noise entries by outer product as follows:\n\n$$\n\\begin{align}\n\\epsilon_{i,j}^w &= f(\\epsilon_i) f(\\epsilon_j),\\\\\n\\epsilon_{j}^b &= f(\\epsilon_i),\\\\\n\\text{where } f(x) &= sgn(x) \\sqrt{|x|}.\n\\end{align}\n$$\n\nIn all experiements of the paper, the authors used Factorised Gaussian noise, so we will go for it as well.\n\n\n```python\nimport math\nimport os\nfrom typing import Dict, List, Tuple\n\nimport gym\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport torch.optim as optim\nfrom IPython.display import clear_output\n```\n\n## Replay buffer\n\nPlease see *01.dqn.ipynb* for detailed description.\n\n\n```python\nclass ReplayBuffer:\n \"\"\"A simple numpy replay buffer.\"\"\"\n\n def __init__(self, obs_dim: int, size: int, batch_size: int = 32):\n self.obs_buf = np.zeros([size, obs_dim], dtype=np.float32)\n self.next_obs_buf = np.zeros([size, obs_dim], dtype=np.float32)\n self.acts_buf = np.zeros([size], dtype=np.float32)\n self.rews_buf = np.zeros([size], dtype=np.float32)\n self.done_buf = np.zeros(size, dtype=np.float32)\n self.max_size, self.batch_size = size, batch_size\n self.ptr, self.size, = 0, 0\n\n def store(\n self,\n obs: np.ndarray,\n act: np.ndarray, \n rew: float, \n next_obs: np.ndarray, \n done: bool,\n ):\n self.obs_buf[self.ptr] = obs\n self.next_obs_buf[self.ptr] = next_obs\n self.acts_buf[self.ptr] = act\n self.rews_buf[self.ptr] = rew\n self.done_buf[self.ptr] = done\n self.ptr = (self.ptr + 1) % self.max_size\n self.size = min(self.size + 1, self.max_size)\n\n def sample_batch(self) -> Dict[str, np.ndarray]:\n idxs = np.random.choice(self.size, size=self.batch_size, replace=False)\n return dict(obs=self.obs_buf[idxs],\n next_obs=self.next_obs_buf[idxs],\n acts=self.acts_buf[idxs],\n rews=self.rews_buf[idxs],\n done=self.done_buf[idxs])\n\n def __len__(self) -> int:\n return self.size\n```\n\n## Noisy Layer\n\n**References:**\n- https://github.com/higgsfield/RL-Adventure/blob/master/5.noisy%20dqn.ipynb\n- https://github.com/Kaixhin/Rainbow/blob/master/model.py\n\n\n```python\nclass NoisyLinear(nn.Module):\n \"\"\"Noisy linear module for NoisyNet.\n \n Attributes:\n in_features (int): input size of linear module\n out_features (int): output size of linear module\n std_init (float): initial std value\n weight_mu (nn.Parameter): mean value weight parameter\n weight_sigma (nn.Parameter): std value weight parameter\n bias_mu (nn.Parameter): mean value bias parameter\n bias_sigma (nn.Parameter): std value bias parameter\n \n \"\"\"\n\n def __init__(self, in_features: int, out_features: int, std_init: float = 0.5):\n \"\"\"Initialization.\"\"\"\n super(NoisyLinear, self).__init__()\n \n self.in_features = in_features\n self.out_features = out_features\n self.std_init = std_init\n\n self.weight_mu = nn.Parameter(torch.Tensor(out_features, in_features))\n self.weight_sigma = nn.Parameter(\n torch.Tensor(out_features, in_features)\n )\n self.register_buffer(\n \"weight_epsilon\", torch.Tensor(out_features, in_features)\n )\n\n self.bias_mu = nn.Parameter(torch.Tensor(out_features))\n self.bias_sigma = nn.Parameter(torch.Tensor(out_features))\n self.register_buffer(\"bias_epsilon\", torch.Tensor(out_features))\n\n self.reset_parameters()\n self.reset_noise()\n\n def reset_parameters(self):\n \"\"\"Reset trainable network parameters (factorized gaussian noise).\"\"\"\n mu_range = 1 / math.sqrt(self.in_features)\n self.weight_mu.data.uniform_(-mu_range, mu_range)\n self.weight_sigma.data.fill_(\n self.std_init / math.sqrt(self.in_features)\n )\n self.bias_mu.data.uniform_(-mu_range, mu_range)\n self.bias_sigma.data.fill_(\n self.std_init / math.sqrt(self.out_features)\n )\n\n def reset_noise(self):\n \"\"\"Make new noise.\"\"\"\n epsilon_in = self.scale_noise(self.in_features)\n epsilon_out = self.scale_noise(self.out_features)\n\n # outer product\n self.weight_epsilon.copy_(epsilon_out.ger(epsilon_in))\n self.bias_epsilon.copy_(epsilon_out)\n\n def forward(self, x: torch.Tensor) -> torch.Tensor:\n \"\"\"Forward method implementation.\n \n We don't use separate statements on train / eval mode.\n It doesn't show remarkable difference of performance.\n \"\"\"\n return F.linear(\n x,\n self.weight_mu + self.weight_sigma * self.weight_epsilon,\n self.bias_mu + self.bias_sigma * self.bias_epsilon,\n )\n \n @staticmethod\n def scale_noise(size: int) -> torch.Tensor:\n \"\"\"Set scale to make noise (factorized gaussian noise).\"\"\"\n x = torch.randn(size)\n\n return x.sign().mul(x.abs().sqrt())\n```\n\n## Noisy Network\n\nWe use NoisyLinear for the last two FC layers, and there is a method to reset noise at every step.\nThese are the only differences from the example of *01.dqn.ipynb*.\n\n\n```python\nclass Network(nn.Module):\n def __init__(self, in_dim: int, out_dim: int):\n \"\"\"Initialization.\"\"\"\n super(Network, self).__init__()\n\n self.feature = nn.Linear(in_dim, 128)\n self.noisy_layer1 = NoisyLinear(128, 128)\n self.noisy_layer2 = NoisyLinear(128, out_dim)\n\n def forward(self, x: torch.Tensor) -> torch.Tensor:\n \"\"\"Forward method implementation.\"\"\"\n feature = F.relu(self.feature(x))\n hidden = F.relu(self.noisy_layer1(feature))\n out = self.noisy_layer2(hidden)\n \n return out\n \n def reset_noise(self):\n \"\"\"Reset all noisy layers.\"\"\"\n self.noisy_layer1.reset_noise()\n self.noisy_layer2.reset_noise()\n```\n\n## DQN + NoisyNet Agent (w/o DuelingNet)\n\nHere is a summary of DQNAgent class.\n\n| Method | Note |\n| --- | --- |\n|select_action | select an action from the input state. |\n|step | take an action and return the response of the env. |\n|compute_dqn_loss | return dqn loss. |\n|update_model | update the model by gradient descent. |\n|target_hard_update| hard update from the local model to the target model.|\n|train | train the agent during num_frames. |\n|test | test the agent (1 episode). |\n|plot | plot the training progresses. |\n\nIn the paper, NoisyNet is used as a component of the Dueling Network Architecture, which includes Double-DQN and Prioritized Experience Replay. However, we don't implement them to simplify the tutorial. One thing to note is that NoisyNet is an alternertive to $\\epsilon$-greedy method, so all $\\epsilon$ related lines are removed. Please check all comments with *NoisyNet*.\n\n\n```python\nclass DQNAgent:\n \"\"\"DQN Agent interacting with environment.\n \n Attribute:\n env (gym.Env): openAI Gym environment\n memory (ReplayBuffer): replay memory to store transitions\n batch_size (int): batch size for sampling\n target_update (int): period for target model's hard update\n gamma (float): discount factor\n dqn (Network): model to train and select actions\n dqn_target (Network): target model to update\n optimizer (torch.optim): optimizer for training dqn\n transition (list): transition information including\n state, action, reward, next_state, done\n \"\"\"\n\n def __init__(\n self, \n env: gym.Env,\n memory_size: int,\n batch_size: int,\n target_update: int,\n gamma: float = 0.99,\n ):\n \"\"\"Initialization.\n \n Args:\n env (gym.Env): openAI Gym environment\n memory_size (int): length of memory\n batch_size (int): batch size for sampling\n target_update (int): period for target model's hard update\n gamma (float): discount factor\n \"\"\"\n # NoisyNet: All attributes related to epsilon are removed\n obs_dim = env.observation_space.shape[0]\n action_dim = env.action_space.n\n \n self.env = env\n self.memory = ReplayBuffer(obs_dim, memory_size, batch_size)\n self.batch_size = batch_size\n self.target_update = target_update\n self.gamma = gamma\n \n # device: cpu / gpu\n self.device = torch.device(\n \"cuda\" if torch.cuda.is_available() else \"cpu\"\n )\n print(self.device)\n\n # networks: dqn, dqn_target\n self.dqn = Network(obs_dim, action_dim).to(self.device)\n self.dqn_target = Network(obs_dim, action_dim).to(self.device)\n self.dqn_target.load_state_dict(self.dqn.state_dict())\n self.dqn_target.eval()\n \n # optimizer\n self.optimizer = optim.Adam(self.dqn.parameters())\n\n # transition to store in memory\n self.transition = list()\n \n # mode: train / test\n self.is_test = False\n\n def select_action(self, state: np.ndarray) -> np.ndarray:\n \"\"\"Select an action from the input state.\"\"\"\n # NoisyNet: no epsilon greedy action selection\n selected_action = self.dqn(\n torch.FloatTensor(state).to(self.device)\n ).argmax()\n selected_action = selected_action.detach().cpu().numpy()\n \n if not self.is_test:\n self.transition = [state, selected_action]\n \n return selected_action\n\n def step(self, action: np.ndarray) -> Tuple[np.ndarray, np.float64, bool]:\n \"\"\"Take an action and return the response of the env.\"\"\"\n next_state, reward, done, _ = self.env.step(action)\n\n if not self.is_test:\n self.transition += [reward, next_state, done]\n self.memory.store(*self.transition)\n \n return next_state, reward, done\n\n def update_model(self) -> torch.Tensor:\n \"\"\"Update the model by gradient descent.\"\"\"\n samples = self.memory.sample_batch()\n\n loss = self._compute_dqn_loss(samples)\n\n self.optimizer.zero_grad()\n loss.backward()\n self.optimizer.step()\n \n # NoisyNet: reset noise\n self.dqn.reset_noise()\n self.dqn_target.reset_noise()\n\n return loss.item()\n \n def train(self, num_frames: int, plotting_interval: int = 200):\n \"\"\"Train the agent.\"\"\"\n self.is_test = False\n \n state = self.env.reset()\n update_cnt = 0\n losses = []\n scores = []\n score = 0\n\n for frame_idx in range(1, num_frames + 1):\n action = self.select_action(state)\n next_state, reward, done = self.step(action)\n\n state = next_state\n score += reward\n \n # NoisyNet: removed decrease of epsilon\n\n # if episode ends\n if done:\n state = self.env.reset()\n scores.append(score)\n score = 0\n\n # if training is ready\n if len(self.memory) >= self.batch_size:\n loss = self.update_model()\n losses.append(loss)\n update_cnt += 1\n \n # if hard update is needed\n if update_cnt % self.target_update == 0:\n self._target_hard_update()\n\n # plotting\n if frame_idx % plotting_interval == 0:\n self._plot(frame_idx, scores, losses)\n \n self.env.close()\n \n def test(self, video_folder: str) -> None:\n \"\"\"Test the agent.\"\"\"\n self.is_test = True\n \n # for recording a video\n naive_env = self.env\n self.env = gym.wrappers.RecordVideo(self.env, video_folder=video_folder)\n \n state = self.env.reset()\n done = False\n score = 0\n \n while not done:\n action = self.select_action(state)\n next_state, reward, done = self.step(action)\n\n state = next_state\n score += reward\n \n print(\"score: \", score)\n self.env.close()\n \n # reset\n self.env = naive_env\n\n def _compute_dqn_loss(self, samples: Dict[str, np.ndarray]) -> torch.Tensor:\n \"\"\"Return dqn loss.\"\"\"\n device = self.device # for shortening the following lines\n state = torch.FloatTensor(samples[\"obs\"]).to(device)\n next_state = torch.FloatTensor(samples[\"next_obs\"]).to(device)\n action = torch.LongTensor(samples[\"acts\"].reshape(-1, 1)).to(device)\n reward = torch.FloatTensor(samples[\"rews\"].reshape(-1, 1)).to(device)\n done = torch.FloatTensor(samples[\"done\"].reshape(-1, 1)).to(device)\n \n # G_t = r + gamma * v(s_{t+1}) if state != Terminal\n # = r otherwise\n curr_q_value = self.dqn(state).gather(1, action)\n next_q_value = self.dqn_target(next_state).max(\n dim=1, keepdim=True\n )[0].detach()\n mask = 1 - done\n target = (reward + self.gamma * next_q_value * mask).to(self.device)\n\n # calculate dqn loss\n loss = F.smooth_l1_loss(curr_q_value, target)\n\n return loss\n\n def _target_hard_update(self):\n \"\"\"Hard update: target <- local.\"\"\"\n self.dqn_target.load_state_dict(self.dqn.state_dict())\n \n def _plot(\n self, \n frame_idx: int, \n scores: List[float], \n losses: List[float], \n ):\n \"\"\"Plot the training progresses.\"\"\"\n clear_output(True)\n plt.figure(figsize=(20, 5))\n plt.subplot(131)\n plt.title('frame %s. score: %s' % (frame_idx, np.mean(scores[-10:])))\n plt.plot(scores)\n plt.subplot(132)\n plt.title('loss')\n plt.plot(losses)\n plt.show()\n```\n\n## Environment\n\nYou can see the [code](https://github.com/openai/gym/blob/master/gym/envs/classic_control/cartpole.py) and [configurations](https://github.com/openai/gym/blob/master/gym/envs/__init__.py#L53) of CartPole-v0 from OpenAI's repository.\n\n\n```python\n# environment\nenv_id = \"CartPole-v0\"\nenv = gym.make(env_id)\nif IN_COLAB:\n env = gym.wrappers.Monitor(env, \"videos\", force=True)\n```\n\n## Set random seed\n\n\n```python\nseed = 777\n\ndef seed_torch(seed):\n torch.manual_seed(seed)\n if torch.backends.cudnn.enabled:\n torch.backends.cudnn.benchmark = False\n torch.backends.cudnn.deterministic = True\n\nnp.random.seed(seed)\nseed_torch(seed)\nenv.seed(seed)\n```\n\n\n\n\n [777]\n\n\n\n## Initialize\n\n\n```python\n# parameters\nnum_frames = 20000\nmemory_size = 10000\nbatch_size = 128\ntarget_update = 150\n\n# train\nagent = DQNAgent(env, memory_size, batch_size, target_update)\n```\n\n cpu\n\n\n## Train\n\n\n```python\nagent.train(num_frames)\n```\n\n## Test\n\nRun the trained agent (1 episode).\n\n\n```python\nvideo_folder=\"videos/noisy_net\"\nagent.test(video_folder=video_folder)\n```\n\n score: 200.0\n\n\n## Render\n\n\n```python\nimport base64\nimport glob\nimport io\nimport os\n\nfrom IPython.display import HTML, display\n\n\ndef ipython_show_video(path: str) -> None:\n \"\"\"Show a video at `path` within IPython Notebook.\"\"\"\n if not os.path.isfile(path):\n raise NameError(\"Cannot access: {}\".format(path))\n\n video = io.open(path, \"r+b\").read()\n encoded = base64.b64encode(video)\n\n display(HTML(\n data=\"\"\"\n \n \"\"\".format(encoded.decode(\"ascii\"))\n ))\n\n\ndef show_latest_video(video_folder: str) -> str:\n \"\"\"Show the most recently recorded video from video folder.\"\"\"\n list_of_files = glob.glob(os.path.join(video_folder, \"*.mp4\"))\n latest_file = max(list_of_files, key=os.path.getctime)\n ipython_show_video(latest_file)\n return latest_file\n\n\nlatest_file = show_latest_video(video_folder=video_folder)\nprint(\"Played:\", latest_file)\n```\n\n\n\n\n\n\n\n Played: videos/noisy_net/rl-video-episode-0.mp4\n\n", "meta": {"hexsha": "3eac66744ff4cd0966406abf172e2951837098a5", "size": 128113, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "05.noisy_net.ipynb", "max_stars_repo_name": "Curt-Park/2nd_dlcat_rainbow", "max_stars_repo_head_hexsha": "6ee8d96f90a2aee2f9b82a96713c9d8d8731e31d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-06-23T12:00:08.000Z", "max_stars_repo_stars_event_max_datetime": "2019-06-23T12:00:08.000Z", "max_issues_repo_path": "05.noisy_net.ipynb", "max_issues_repo_name": "Curt-Park/2nd_dlcat_rainbow", "max_issues_repo_head_hexsha": "6ee8d96f90a2aee2f9b82a96713c9d8d8731e31d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "05.noisy_net.ipynb", "max_forks_repo_name": "Curt-Park/2nd_dlcat_rainbow", "max_forks_repo_head_hexsha": "6ee8d96f90a2aee2f9b82a96713c9d8d8731e31d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 167.0312907432, "max_line_length": 51528, "alphanum_fraction": 0.8676715087, "converted": true, "num_tokens": 4333, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.6513548646660543, "lm_q1q2_score": 0.44679651443461493}} {"text": "# Kinematics of particle\n\n> Marcos Duarte, Renato Naville Watanabe \n> [Laboratory of Biomechanics and Motor Control](http://pesquisa.ufabc.edu.br/bmclab) \n> Federal University of ABC, Brazil\n\n## Biomechanics & Mechanics\n\n**A good knowledge of Mechanics is a necessary condition, although not sufficient!, to master Biomechanics**\n\nFor this reason, we will review principles of Classical Mechanics in the context of Biomechanics. \n\nThe book [*Introduction to Statics and Dynamics*](http://ruina.tam.cornell.edu/Book/index.html) , written by Andy Ruina and Rudra Pratap, is an excellent reference (a rigorous and yet didactic presentation of Mechanics for undergraduate students) on Classical Mechanics and we will use this book as the main reference on Mechanics and Mathematics for this brief review. The preface and first chapter of the book are a good read on how someone should study Mechanics. You should read them!\n\nAs we argued in the notebook [Biomechanics](http://nbviewer.jupyter.org/github/demotu/BMC/blob/master/notebooks/Biomechanics.ipynb), we will start with a branch of Classical Mechanics that is simpler to measure its related quantities on biological systems, the Kinematics. \n\nThere are some relevant cases in the study of human movement where modeling the human body or one of its segments as a particle might be all we need to explore the phenomenon. The concept of kinematics of a particle, for instance, can be applied to study the performance in the 100-m race; to describe spatial and temporal characteristics of a movement pattern, and to conjecture about how voluntary movements are planned (the minimum jerk hypothesis). \n\nNow, let's review the concept of kinematics of a particle and later apply to the study of human movement.\n\n## Kinematics\n\n**Kinematics** is the branch of Classical Mechanics that describes the motion of objects without consideration of the causes of motion ([Wikipedia](http://en.wikipedia.org/wiki/Kinematics)). \n\nKinematics of a particle is the description of the motion when the object is considered a particle. \n\nA particle as a physical object does not exist in nature; it is a simplification to understand the motion of a body or it is a conceptual definition such as the center of mass of a system of objects.\n\n### Vectors in Kinematics\n\nSome mechanical quantities in Kinematics (position and its derivatives) are represented as vectors and others, such as time and distance, are scalars. \nA vector in Mechanics is a physical quantity with magnitude, direction, and satisfies some elementary vector arithmetic, whereas a scalar is a physical quantity that is fully expressed by a magnitude (a number) only. \n\nFor a review about scalars and vectors, see chapter 1 of [Ruina and Rudra's book](http://ruina.tam.cornell.edu/Book/index.html). \n\n For how to use Python to work with scalars and vectors, see the notebook [Scalar and Vector](http://nbviewer.jupyter.org/github/BMCLab/BMC/blob/master/notebooks/ScalarVector.ipynb).\n\n## Position\n\nConsider a point in the three-dimensional Euclidean space described in a Cartesian coordinate system (see the notebook [Frame of reference](http://nbviewer.jupyter.org/github/demotu/BMC/blob/master/notebooks/ReferenceFrame.ipynb) for an introduction on coordinate systems in Mechanics and Biomechanics): \n
\n
Figure. Representation of a point $\\mathbf{P}$ and its position vector $\\overrightarrow{\\mathbf{r}}$ in a Cartesian coordinate system. The versors $\\hat{\\mathbf{i}},\\, \\hat{\\mathbf{j}},\\, \\hat{\\mathbf{k}}\\,$ form a basis for this coordinate system and are usually represented in the color sequence RGB (red, green, blue) for easier visualization.
\n\nThe position of this point in space can be represented as a triple of values each representing the coordinate at each axis of the Cartesian coordinate system following the $ \\mathbf{X, Y, Z} $ convention order (which is omitted):\n\n\n\\begin{equation}\n(x,\\, y,\\, z)\n\\label{eq_xyz}\n\\end{equation}\n\n\nThe position of a particle in space can also be represented by a vector in the Cartesian coordinate system, with the origin of the vector at the origin of the coordinate system and the tip of the vector at the point position:\n\n\n\\begin{equation}\n\\overrightarrow{\\mathbf{r}}(t) = x\\,\\hat{\\mathbf{i}} + y\\,\\hat{\\mathbf{j}} + z\\,\\hat{\\mathbf{k}}\n\\label{eq_rxyz}\n\\end{equation} \n\n\nWhere $\\hat{\\mathbf{i}},\\, \\hat{\\mathbf{j}},\\, \\hat{\\mathbf{k}}\\,$ are unit vectors in the directions of the axes $ \\mathbf{X, Y, Z} $. \n\nFor a review on vectors, see the notebook [Scalar and vector](http://nbviewer.ipython.org/github/demotu/BMC/blob/master/notebooks/ScalarVector.ipynb).\n\nWith this new notation, the coordinates of a point representing the position of a particle that vary with time would be expressed by the following position vector $\\overrightarrow{\\mathbf{r}}(t)$: \n\n\n\\begin{equation}\n\\overrightarrow{\\mathbf{r}}(t) = x(t)\\,\\hat{\\mathbf{i}} + y(t)\\,\\hat{\\mathbf{j}} + z(t)\\,\\hat{\\mathbf{k}}\n\\label{eq_rxyzijk}\n\\end{equation}\n\n\nA vector can also be represented in matrix form:\n\n\n\\begin{equation}\n\\overrightarrow{\\mathbf{r}}(t) = \\begin{bmatrix} x(t) \\\\y(t) \\\\z(t) \\end{bmatrix}\n\\label{eq_rxyzmatrix}\n\\end{equation}\n\n\nAnd the unit vectors in each Cartesian coordinate in matrix form are given by:\n\n\n\\begin{equation}\n\\hat{\\mathbf{i}} = \\begin{bmatrix}1\\\\0\\\\0 \\end{bmatrix},\\; \\hat{\\mathbf{j}} = \\begin{bmatrix}0\\\\1\\\\0 \\end{bmatrix},\\; \\hat{\\mathbf{k}} = \\begin{bmatrix} 0 \\\\ 0 \\\\ 1 \\end{bmatrix}\n\\label{eq_ijk}\n\\end{equation}\n\n\n## Basis\n\nIn [linear algebra](http://en.wikipedia.org/wiki/Linear_algebra), a set of unit linearly independent vectors as the three vectors above (orthogonal in the Euclidean space) that can represent any vector via [linear combination](http://en.wikipedia.org/wiki/Linear_combination) is called a basis. A basis is the foundation of creating a reference frame and we will study how to do that other time. \n\n## Displacement\n\nThe shortest distance between two positions of a particle. \n\nAs the difference between two vectors, displacement is also a vector quantity.\n\n\n\\begin{equation}\n\\mathbf{\\overrightarrow{d}} = \\mathbf{\\overrightarrow{r}_2} - \\mathbf{\\overrightarrow{r}_1}\n\\label{eq_distance}\n\\end{equation}\n\n\n
Figure. Representation of the displacement vector $\\mathbf{\\overrightarrow{d}}$ between two positions $\\mathbf{\\overrightarrow{r}_1}$ and $\\mathbf{\\overrightarrow{r}_2}$.
\n\n## Velocity\n\nVelocity is the rate (with respect to time) of change of the position of a particle. \n\nThe average velocity between two instants is:\n\n\n\\begin{equation}\n\\overrightarrow{\\mathbf{v}}(t) = \\frac{\\overrightarrow{\\mathbf{r}}(t_2)-\\overrightarrow{\\mathbf{r}}(t_1)}{t_2-t_1} = \\frac{\\Delta \\overrightarrow{\\mathbf{r}}}{\\Delta t}\n\\label{eq_velocity}\n\\end{equation}\n \n\nThe instantaneous velocity of the particle is obtained when $\\Delta t$ approaches to zero, which from calculus is the first-order [derivative](http://en.wikipedia.org/wiki/Derivative) of the position vector:\n\n\n\\begin{equation}\n\\overrightarrow{\\mathbf{v}}(t) = \\lim_{\\Delta t \\to 0} \\frac{\\Delta \\overrightarrow{\\mathbf{r}}}{\\Delta t} = \\lim_{\\Delta t \\to 0} \\frac{\\overrightarrow{\\mathbf{r}}(t+\\Delta t)-\\overrightarrow{\\mathbf{r}}(t)}{\\Delta t} = \\frac{\\mathrm{d}\\overrightarrow{\\mathbf{r}}}{dt}\n\\label{eq_velocityderiv}\n\\end{equation}\n \n\nFor the movement of a particle described with respect to an [inertial Frame of reference](http://nbviewer.jupyter.org/github/demotu/BMC/blob/master/notebooks/ReferenceFrame.ipynb), the derivative of a vector is obtained by differentiating each vector component of the Cartesian coordinates (since the base versors $\\hat{\\mathbf{i}}, \\hat{\\mathbf{j}}, \\hat{\\mathbf{k}}$ are constant): \n\n\n\\begin{equation}\n\\overrightarrow{\\mathbf{v}}(t) = \\frac{\\mathrm{d}\\overrightarrow{\\mathbf{r}}(t)}{dt} = \\frac{\\mathrm{d}x(t)}{\\mathrm{d}t}\\hat{\\mathbf{i}} + \\frac{\\mathrm{d}y(t)}{\\mathrm{d}t}\\hat{\\mathbf{j}} + \\frac{\\mathrm{d}z(t)}{\\mathrm{d}t}\\hat{\\mathbf{k}}\n\\label{eq_velocityderiv2}\n\\end{equation}\n\n\nOr in matrix form (and using the Newton's notation for differentiation):\n\n\n\\begin{equation} \n\\overrightarrow{\\mathbf{v}}(t) = \\begin{bmatrix}\n\\dot x(t) \\\\\n\\dot y(t) \\\\\n\\dot z(t)\n\\end{bmatrix}\n\\label{eq_velocityderiv3}\n\\end{equation}\n\n\n## Acceleration \n\nAcceleration is the rate (with respect to time) of change of the velocity of a particle, which can also be given by the second-order rate of change of the position.\n\nThe average acceleration between two instants is:\n\n\n\\begin{equation}\n\\overrightarrow{\\mathbf{a}}(t) = \\frac{\\overrightarrow{\\mathbf{v}}(t_2)-\\overrightarrow{\\mathbf{v}}(t_1)}{t_2-t_1} = \\frac{\\Delta \\overrightarrow{\\mathbf{v}}}{\\Delta t}\n\\label{eq_acc}\n\\end{equation}\n\n\nLikewise, instantaneous acceleration is the first-order derivative of the velocity or the second-order derivative of the position vector: \n\n\n\\begin{equation} \n\\overrightarrow{\\mathbf{a}}(t) = \\frac{\\mathrm{d}\\overrightarrow{\\mathbf{v}}(t)}{\\mathrm{d}t} = \\frac{\\mathrm{d}^2\\overrightarrow{\\mathbf{r}}(t)}{\\mathrm{d}t^2} = \\frac{\\mathrm{d}^2x(t)}{\\mathrm{d}t^2}\\hat{\\mathbf{i}} + \\frac{\\mathrm{d}^2y(t)}{\\mathrm{d}t^2}\\hat{\\mathbf{j}} + \\frac{\\mathrm{d}^2z(t)}{\\mathrm{d}t^2}\\hat{\\mathbf{k}}\n\\label{eq_accderiv}\n\\end{equation}\n\n\nAnd in matrix form:\n\n\n\\begin{equation} \n\\mathbf{a}(t) = \\begin{bmatrix}\n\\ddot x(t) \\\\\n\\ddot y(t) \\\\\n\\ddot z(t)\n\\end{bmatrix}\n\\label{eq_accderiv2}\n\\end{equation}\n\n\nFor curiosity, see [Notation for differentiation](https://en.wikipedia.org/wiki/Notation_for_differentiation) on the origin of the different notations for differentiation.\n\nWhen the base versors change in time, for instance when the basis is attached to a rotating frame or reference, the components of the vector\u2019s derivative is not the derivatives of its components; we will also have to consider the derivative of the basis with respect to time.\n\n## The antiderivative\n\nAs the acceleration is the derivative of the velocity which is the derivative of position, the inverse mathematical operation is the [antiderivative](http://en.wikipedia.org/wiki/Antiderivative) (or integral):\n\n\n\\begin{equation} \n\\begin{array}{l l}\n\\mathbf{r}(t) = \\mathbf{r}_0 + \\int \\mathbf{v}(t) \\:\\mathrm{d}t \\\\\n\\mathbf{v}(t) = \\mathbf{v}_0 + \\int \\mathbf{a}(t) \\:\\mathrm{d}t \n\\end{array}\n\\label{eq_antiderivative}\n\\end{equation}\n\n\n**This part of the kinematics is presented in chapter 12, pages 605-620, of the [Ruina and Rudra's book](http://ruina.tam.cornell.edu/Book/index.html).** \n\n## Some cases of motion of a particle\n\nTo deduce some trivial cases of motion of a particle (at rest, at constant speed, and at constant acceleration), we can start from the equation for its position and differentiate it to obtain expressions for the velocity and acceleration or the inverse approach, start with the equation for acceleration, and then integrate it to obtain the velocity and position of the particle. Both approachs are valid in Mechanics. For the present case, it probaly makes more sense to start with the expression for acceleration.\n\n#### Particle at rest\n\n\n\\begin{equation} \n\\begin{array}{l l}\n\\overrightarrow{\\mathbf{a}}(t) = 0 \\\\\n\\overrightarrow{\\mathbf{v}}(t) = 0 \\\\\n\\overrightarrow{\\mathbf{r}}(t) = \\overrightarrow{\\mathbf{r}}_0\n\\end{array}\n\\label{eq_rest}\n\\end{equation}\n\n\n#### Particle at constant speed\n\n\n\\begin{equation} \n\\begin{array}{l l}\n\\overrightarrow{\\mathbf{a}}(t) = 0 \\\\\n\\overrightarrow{\\mathbf{v}}(t) = \\overrightarrow{\\mathbf{v}}_0 \\\\\n\\overrightarrow{\\mathbf{r}}(t) = \\overrightarrow{\\mathbf{r}}_0 + \\overrightarrow{\\mathbf{v}}_0t\n\\end{array}\n\\label{eq_constantspeed}\n\\end{equation}\n\n\n#### Particle at constant acceleration\n\n\n\\begin{equation} \n\\begin{array}{l l}\n\\overrightarrow{\\mathbf{a}}(t) = \\overrightarrow{\\mathbf{a}}_0 \\\\\n\\overrightarrow{\\mathbf{v}}(t) = \\overrightarrow{\\mathbf{v}}_0 + \\overrightarrow{\\mathbf{a}}_0t \\\\\n\\overrightarrow{\\mathbf{r}}(t) = \\overrightarrow{\\mathbf{r}}_0 + \\overrightarrow{\\mathbf{v}}_0t + \n\\frac{1}{2}\\overrightarrow{\\mathbf{a}}_0 t^2 \n\\end{array}\n\\label{eq_constantacceleration}\n\\end{equation}\n\n\n### Visual representation of these cases\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\n%matplotlib inline\nimport seaborn as sns\nsns.set_context(\"notebook\", font_scale=1.2, rc={\"lines.linewidth\": 2, \"lines.markersize\": 10})\n```\n\n\n```python\nt = np.arange(0, 2.0, 0.02)\nr0 = 1; v0 = 2; a0 = 4\n\nplt.rc('axes', labelsize=14, titlesize=14) \nplt.rc('xtick', labelsize=10)\nplt.rc('ytick', labelsize=10) \nf, axarr = plt.subplots(3, 3, sharex = True, sharey = True, figsize=(14,7))\nplt.suptitle('Scalar kinematics of a particle', fontsize=20);\n\ntones = np.ones(np.size(t))\n\naxarr[0, 0].set_title('at rest', fontsize=14);\naxarr[0, 0].plot(t, r0*tones, 'g', linewidth=4, label='$r(t)=1$')\naxarr[1, 0].plot(t, 0*tones, 'b', linewidth=4, label='$v(t)=0$')\naxarr[2, 0].plot(t, 0*tones, 'r', linewidth=4, label='$a(t)=0$')\naxarr[0, 0].set_ylabel('r(t) [m]')\naxarr[1, 0].set_ylabel('v(t) [m/s]')\naxarr[2, 0].set_ylabel('a(t) [m/s$^2$]')\n\naxarr[0, 1].set_title('at constant speed');\naxarr[0, 1].plot(t, r0*tones+v0*t, 'g', linewidth=4, label='$r(t)=1+2t$')\naxarr[1, 1].plot(t, v0*tones, 'b', linewidth=4, label='$v(t)=2$')\naxarr[2, 1].plot(t, 0*tones, 'r', linewidth=4, label='$a(t)=0$')\n\naxarr[0, 2].set_title('at constant acceleration');\naxarr[0, 2].plot(t, r0*tones+v0*t+1/2.*a0*t**2,'g', linewidth=4,\n label='$r(t)=1+2t+\\\\frac{1}{2}4t^2$')\naxarr[1, 2].plot(t, v0*tones+a0*t, 'b', linewidth=4,\n label='$v(t)=2+4t$')\naxarr[2, 2].plot(t, a0*tones, 'r', linewidth=4,\n label='$a(t)=4$')\n\nfor i in range(3): \n axarr[2, i].set_xlabel('Time [s]');\n for j in range(3):\n axarr[i,j].set_ylim((-.2, 10))\n axarr[i,j].legend(loc = 'upper left', frameon=True, framealpha = 0.9, fontsize=16)\n \nplt.subplots_adjust(hspace=0.09, wspace=0.07)\n```\n\n## Symbolic programming\n\nWe can use [Sympy](http://www.sympy.org/en/index.html), a Python library for symbolic mathematics, to deduce the expressions for the cases of motion of a particle we just visualized. \nLet's show how to integrate with Sympy for the case of particle with constant acceleration:\n\n\n```python\nfrom sympy import Symbol, integrate, init_printing\ninit_printing(use_latex='mathjax')\n\nt = Symbol('t', real=True, positive=True)\ng = Symbol('g', real=True, positive=True)\nv0 = Symbol('v0', real=True, positive=True, constant = True)\nr0 = Symbol('r0', real=True, positive=True, constant = True)\n```\n\n\n```python\nv = integrate(g, t) + v0 # a constant has to be added\nv\n```\n\n\n\n\n$\\displaystyle g t + v_{0}$\n\n\n\n\n```python\nr = integrate(v, t) + r0 # a constant has to be added\nr\n```\n\n\n\n\n$\\displaystyle \\frac{g t^{2}}{2} + r_{0} + t v_{0}$\n\n\n\n## Kinematics of human movement\n\n### Kinematics of the 100-m race\n\nAn example where the analysis of some aspects of the human body movement can be reduced to the analysis of a particle is the study of the Biomechanics of the 100-m race. \n\nA technical report with the kinematic data for the 100-m world record by Usain Bolt can be downloaded from the [website for Research Projects](http://www.iaaf.org/development/research) from the International Association of Athletics Federations. \n[Here is a direct link for that report](http://www.iaaf.org/download/download?filename=76ade5f9-75a0-4fda-b9bf-1b30be6f60d2.pdf&urlSlug=1-biomechanics-report-wc-berlin-2009-sprint). In particular, the following table shows the data for the three medalists in that race: \n
\n
Figure. Data from the three medalists of the 100-m dash in Berlin, 2009 (IAAF report).
\n\nThe column **RT** in the table above refers to the reaction time of each athlete. The IAAF has a very strict rule about reaction time: any athlete with a reaction time less than 100 ms is disqualified from the competition! See the website [Reaction Times and Sprint False Starts](http://condellpark.com/kd/reactiontime.htm) for a discussion about this rule.\n\nYou can measure your own reaction time in a simple way visiting this website: [http://www.humanbenchmark.com/tests/reactiontime](http://www.humanbenchmark.com/tests/reactiontime).\n\nThe article [A Kinematics Analysis Of Three Best 100 M Performances Ever](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3661886/) by Krzysztof and Mero presents a detailed kinematic analysis of 100-m races. \n\n### Spatial and temporal characteristics of a movement pattern\n\nSee the notebook [Spatial and temporal characteristics](http://nbviewer.jupyter.org/github/demotu/BMC/blob/master/notebooks/SpatialTemporalCharacteristcs.ipynb) about how the simple measurement of spatial and temporal kinematic variables can be very useful to describe the human gait.\n\n### The minimum jerk hypothesis\n\nSee the notebook [The minimum jerk hypothesis](http://nbviewer.jupyter.org/github/demotu/BMC/blob/master/notebooks/MinimumJerkHypothesis.ipynb) about the conjecture that movements are performed (organized) with the smoothest trajectory possible.\n\n## Problems\n\n1. Read the preface and first chapter of the [Ruina and Rudra's book](http://ruina.tam.cornell.edu/Book/index.html) about how someone should study Mechanics. \n\n2. Consider the data for the three medalists of the 100-m dash in Berlin, 2009, shown previously. \n a. Calculate the average velocity and acceleration. \n b. Plot the graphs for the displacement, velocity, and acceleration versus time. \n c. Plot the graphs velocity and acceleration versus partial distance (every 20m). \n d. Calculate the average velocity and average acceleration and the instants and values of the peak velocity and peak acceleration. \n\n3. The article \"Biomechanical Analysis of the Sprint and Hurdles Events at the 2009 IAAF World Championships in Athletics\" by Graubner and Nixdorf lists the 10-m split times for the three medalists of the 100-m dash in Berlin, 2009: \n
\n
\n\n a. Repeat the same calculations performed in problem 2 and compare the results. \n\n4. A body attached to a spring has its position (in cm) described by the equation $x(t) = 2\\sin(4\\pi t + \\pi/4)$. \n a. Calculate the equation for the body velocity and acceleration. \n b. Plot the position, velocity, and acceleration in the interval [0, 1] s. \n \n5. There are some nice free software that can be used for the kinematic analysis of human motion. Some of them are: [Kinovea](http://www.kinovea.org/), [Tracker](http://www.cabrillo.edu/~dbrown/tracker/), and [SkillSpector](http://video4coach.com/index.php?option=com_content&task=view&id=13&Itemid=45). Visit their websites and explore these software to understand in which biomechanical applications they could be used. \n\n6. (Sample 12.1 of Ruina and Rudra's book) The position vector of a particle is given as a functions time: $\\overrightarrow{\\mathbf{r}}(t) = (C_1+C_2t+C_3t^2)\\hat{\\mathbf{i}}+C_4t\\,\\hat{\\mathbf{j}}$ . Where $C_1=1m, C_2=3m/s,C_3=1m/s^2, C_4=2m/s$. \n a. Find the position, velocity, and acceleration of the particle at $t=2s$. \n b. Find the change in the position of the particle between $t=2s$ and $t=3s$. \n\n7. From Ruina and Rudra's book, solve the problems **12.1.1** to **12.1.14**. \n\n## References\n\n- Graubner R, Nixdorf E (2011) [Biomechanical Analysis of the Sprint and Hurdles Events at the 2009 IAAF World Championships in Athletics ](http://www.meathathletics.ie/devathletes/pdf/Biomechanics%20of%20Sprints.pdf). [New Studies in Athletics](http://www.iaaf.org/development/new-studies-in-athletics), 1/2, 19-53.\n- Krzysztof M, Antti Mero A (2013) [A Kinematics Analysis Of Three Best 100 M Performances Ever](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3661886/). Journal of Human Kinetics, 36, 149\u2013160.\n- [Research Projects](http://www.iaaf.org/development/research) from the International Association of Athletics Federations. \n- Ruina A, Rudra P (2019) [Introduction to Statics and Dynamics](http://ruina.tam.cornell.edu/Book/index.html). Oxford University Press. \n", "meta": {"hexsha": "e72422bf0c23d7d2f9bfd838c5b63a9f53f5ac4a", "size": 78528, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/.ipynb_checkpoints/KinematicsParticle-checkpoint.ipynb", "max_stars_repo_name": "ofgod2/Biomec-nica-y-Comtrol-de-Motor", "max_stars_repo_head_hexsha": "4fb332309e3aa648041f1db6c325d57014b2a1df", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-03-18T01:39:12.000Z", "max_stars_repo_stars_event_max_datetime": "2020-04-12T19:16:48.000Z", "max_issues_repo_path": "notebooks/.ipynb_checkpoints/KinematicsParticle-checkpoint.ipynb", "max_issues_repo_name": "erichuang2013/BMC", "max_issues_repo_head_hexsha": "18c08d9b581672fcf8e1132e37da2ee978f315dc", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/.ipynb_checkpoints/KinematicsParticle-checkpoint.ipynb", "max_forks_repo_name": "erichuang2013/BMC", "max_forks_repo_head_hexsha": "18c08d9b581672fcf8e1132e37da2ee978f315dc", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 93.5971394517, "max_line_length": 46416, "alphanum_fraction": 0.8159637327, "converted": true, "num_tokens": 6018, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660542, "lm_q2_score": 0.6859494421679929, "lm_q1q2_score": 0.4467965060710884}} {"text": "# Asynchronous communication\n\nIn general speaking, asynchronous communication is when you send a message without expecting an immediate response. For example, you send an email. I open and respond to the email several hours later. In contrast, synchronous communication is when you send a message and the recipient processes the information immediately. All previous communications are synchronous communications because all agents have to actively involved with the communication operation in the same time. It is common in cluster and HPC since the computers are dedicated to do one thing at one time. Then, why do we care about the asynchronous communication?\n\nThe main advantages of asynchronous communication over the synchronous communication is it can avoid the idel times.\nIn the synchronous communication pattern, all agents has to start and end the communication at the same time. If some agents is delayed, all others have to wait for that one. On the other hand, the asynchronous communication eventible bought more complexity into the code and you may also face some data racing problem.\nBluefog provided the convenient functionality for asynchronous communication. We will use the examples to show the traits of asynchronous communication.\n\nThere are two main goals of this notebook. The first is to introduce the asynchronous communication in BlueFog. In order to achieve that, we have to introduce the concept the `window`, whihc is an area of memory, which follows some protocols that allow others to modify it. The modification operation, i.e. data transfer, is called one-sided communication. Second, we will revisit the average consensus and push-sum algorithm and exam them under the asynchronous situation. You see the behavior of the same algorithm under synchronous and asynchronous scenario may differ.\n\n\n```\nimport torch\nimport networkx as nx\nimport ipyparallel as ipp\nimport matplotlib.pyplot as plt\n\n%matplotlib inline\n```\n\n\n```\nrc = ipp.Client(profile=\"bluefog\")\nrc.block = True\nrc.ids\n```\n\n\n\n\n [0, 1, 2, 3]\n\n\n\n\n```\n%%px\nimport torch\nimport bluefog.torch as bf\n\nbf.init()\n```\n\n# Asynchronous Operation in BlueFog\n\nAsynchronous operation may have different meaning in different context. \n> In BlueFog, *asynchronous operation* means the operation executing in agent $k$ is independent from the operation running in other agent $k'$. \n\n*Note the difference between the asynchronous operation and nonblocking operation introduced before.*\n\nFor example, `neighbor_allreduce()` is NOT asynchronous operation because it will be finished in agent $k$ only when the other agents $k'$ finished the call as well. In other words, all collective communication operation in BlueFog is classified as *synchronous* operation. Similarly, the basic send-recieve communication pair can be the asynchronous operation since the send operation in agent $k$ is dependent in the recieve operation of other agent.\n\n\nBlueFog built the asynchronous operation and algorithm is based upon the *One-sided communication ops*, which is introduced in MPI-2. The most notable feature of one-sided communication is indicated by the name that allows the communication ops of one process to be decoupled from the behavior of another process. [Here](https://pages.tacc.utexas.edu/~eijkhout/pcse/html/mpi-onesided.html) is a nice introduction for the MPI one-sided communication ops. \n\nThis decoupled behavior can bring us several of benefits. For instance, under heterogenous environment, some agent may have better hardware and some get worse one. For example, you may have several GPUs but some were bought with latest model but some were bought years ago. Hence, the computation time can vary among agents. Under the one-sided communications, every one can immediately continue the process without awaiting.\n\n\nNow, we formally introduce the asynchronous operation in BlueFog. The first concept is called **window**. A window in each processor (agent) is an area of memory, available to one-sided transfers.\n\n\n\nTherefore, creating a window is always the first step to use the one-sided communication.\nIn Bluefog, each window must be associated with a unique name, which will be the identity to operate the window in the future. The following example shows how you can create a window named `x`. The memory size of that window is exactly the same size as the the shape of tensor you provided.\n\n\n```\n%%px\ntopo = bf.load_topology()\nx = torch.FloatTensor([bf.rank()])\nbf.win_create(x, name=\"x\")\nprint(x)\n```\n\n [stdout:0] tensor([0.])\n [stdout:1] tensor([2.])\n [stdout:2] tensor([1.])\n [stdout:3] tensor([3.])\n\n\nThe shape and the name of one window is immutable, i.e., as long as you create it, you cannot modify it.\nIf you want to change it, you have to manually free it through `bf.win_free` then create a new one through `bf.win_create` again:\n\n\n```\n%%px\nx = torch.FloatTensor([bf.rank(), bf.rank()])\n# If you uncomment the following code, you will encouter\n# the assertion failure. It is because we require each\n# window has a unique name.\n# assert bf.win_create(x, name=\"x\")\n\n# However, you can free previous window named \"x\".\n# and create a new one\nassert bf.win_free(name=\"x\")\nassert bf.win_create(x, name=\"x\")\nprint(x)\n```\n\n [stdout:0] tensor([0., 0.])\n [stdout:1] tensor([2., 2.])\n [stdout:2] tensor([1., 1.])\n [stdout:3] tensor([3., 3.])\n\n\nA unique property of `win_create` in BlueFog is that it is related with the underlying topology actually. Let's plot the default topology first.\n\n\n```\ntopo = rc[0].pull(\"topo\", targets=0)\nnx.draw_circular(topo)\n```\n\nWe assume there are four agents and the topology used is Exponential 2 graph. What happened after `win_create` called is illustrated:\n \n\nAfter this call, each process will allocate the number of incoming neighbor\u2019s windows as buffer, which is illustrated in the figure as red square. Each buffer is dedicated to one neighbor. You don\u2019t need to know which one is dedicated to which neighbor because these buffers are invisible to the python frontend. The only way to interact with them is through the win_update.\n\n## Interacte with window object in BlueFog\n\nTwo basic operation to interact the data in the window is `win_get` and `win_put` operation. As their name indicate, they allow to put the data in local memory to window of the remote windows or get the data in remote memory into local window.\n\n\n\nwin_get is one of three main methods to exchange the information between the processes in window. By default, it will get (fetch) the incoming neighbor\u2019s local value into the its own buffer. Note it doesn\u2019t need the sender to do anything.\n\n\n```\n%%px\n\nif bf.rank() == 0:\n bf.win_get(name=\"x\")\n print(x) # the value of x is not changed.\n```\n\n [stdout:0] tensor([0., 0.])\n\n\nAs you see before, the value of `x` is still 0 afte the `win_get` operations from the neighbors. Why it don't update the value? It is because when you get the value from your neighbors, BlueFog do not update that value into the main window memory. Instead, it just put that value into the buffer of that window for future usage as shown in the following figure.\n\nThe main reason to design like this style is like non-blocking function:\n\n> ```python\nhandle = bf.win_get_nonblocking(name=\"x\")\nSomeOperation(x)\nbf.win_wait(handle)\n> ```\n\nNow, you should not worry about data racing problem -- whether the value `x` is applied some operation first or replaced by win_get value or in the mixing situation.\n\n\n \n\nIn order to interact with the value in the buffer, you have to explicitly call the `win_update` function, which is the bridge to connect the value of buffers (corresponding to the neighbor value) with the local value. It has two functionalities. One is to update the buffer to make sure that the neighbor value, which may be changed through win_put or win_get, is synchronized and visible to local memory. Another is it updates the local value to the average of self and neighbor\u2019s value.\n\n \n\n\n```\n%%px\nbf.win_update(name=\"x\")\nprint(f\"{bf.rank()}: {x}\")\n```\n\n [stdout:0] 0: tensor([1.6667, 1.6667])\n [stdout:1] 2: tensor([2., 2.])\n [stdout:2] 1: tensor([1., 1.])\n [stdout:3] 3: tensor([3., 3.])\n\n\nNow you see that the value of `x` in rank 0 is updated. (In four neighbor and exp2 graph case, it's value is 1.66 because $(0+2+3)/3$. Meanwhile, the value of `x` in other ranks are unchanged.\n\nSimilar as the `neighbor_allreduce` function, `win_get()` has `src_weights` argument to control the weight of receiving information and which neighbor to recieve the information.\n\n\n```\n%%px\ny = torch.FloatTensor([bf.rank()])\nassert bf.win_create(y, name=\"y\")\nif bf.rank() == 0:\n bf.win_get(src_weights={bf.size() - 1: 0.5}, name=\"y\")\n bf.win_update(name=\"y\", self_weight=1.0, neighbor_weights={bf.size() - 1: 0.4})\nprint(f\"Rank {bf.rank()}: {y}\")\nassert bf.win_free(name=\"y\")\n```\n\n [stdout:0] Rank 0: tensor([0.6000])\n [stdout:1] Rank 2: tensor([2.])\n [stdout:2] Rank 1: tensor([1.])\n [stdout:3] Rank 3: tensor([3.])\n\n\nLet's see the status of window object in above code step-by-step (assuming 4 nodes under exponential-2 graph):\n1. All ranks create create a window with initial value: \\[rank\\]\n2. For rank 0: \n The window intialized \\[0\\] for self and \\[0,0\\] for neighbors.\n It called `win_get` the value from agent 3 only with weight 0.5. Hence, after this step the window object became:\n \\[0\\] for self and \\[0, 1.5\\].\n3. For rank 0:\n It updates local value with self value 1.0 and neighbor weights 0.4 for rank 3, this produces $(0 \\times 1.0 + 1.5 \\times 0.4)=0.6$.\n4. Last, all agents printed their local value. You saw the value of `x` is 0.4 as we calculated and freed the window.\n\n\nThere are another operations `win_put` shown in the next figure, which is very similar as `win_get`. So we leave the user to explore this function.\n\n \n\n# Asynchronous \"Average Consensus\" algorithm\n\nNow you can use these window operation to build simple asynchonous Average Consensus. \n\n\n```\n%%px\nbf.win_free() # free all windows in case \"x\" is used.\n# Set up the average consensus problem.\nx = torch.randn(1, dtype=torch.double)\nx_bar = bf.allreduce(x, average=True)\n\nassert bf.win_create(x, name=\"x\", zero_init=True)\n```\n\n\n```\n%%px\ndef async_consensus_step(x):\n bf.win_put(x, name=\"x\")\n bf.win_update(name=\"x\") # notice it is inplace update\n```\n\n\n```\nx_bar = rc[0].pull(\"x_bar\", block=True)\nx_list = []\nfor i in range(15):\n %px async_consensus_step(x)\n x_list.append(rc[:].pull(\"x\", block=True))\n```\n\n\n```\nplt.plot(x_list)\nplt.axhline(x_bar, linestyle=\"--\", color=\"k\")\nplt.legend([f\"agent {i}\" for i in range(len(x_list[0]))])\nplt.xlabel(\"relative time\")\nplt.ylabel(\"value\")\n```\n\nYou see from the above the figure that all agents reached some consensus value but that value is not the same as the mean of initial value. That is mainly because the asynchronous essense breaks the unbias property of average consensus algorithm. Fortunately, it is easy to correct its bias and the algorithm is the same push-sum algorithm we introduced in previous section. But before we present the push-sum algorithm in asynchronous style, let's exam the key feature of asynchronous communication -- time.\n\n# Asynchronous versus synchronous communication in time\n\nAs mentioned before, in the case that some agents work fast and some agents work slow, the synchronous algorithm has to wait until the slowerest one finished meanwhile asynchronous one do not. The speed of agents usually depends on lots of factors including hardward, system scheduling, network congestion, etc. For this tutorial, we use `sleep` function to illustrate the slow worker behavior.\n\n\n```\n%%px\nimport time\n\n\ndef async_consensus_step_with_sleep(x):\n if bf.rank() == 0:\n time.sleep(0.5)\n bf.win_put(x, name=\"x\")\n bf.win_update(name=\"x\") # notice it is inplace update\n\n\ndef sync_consensus_step_with_sleep(x):\n if bf.rank() == 0:\n time.sleep(0.5)\n y = bf.neighbor_allreduce(x)\n return y\n```\n\n\n```\n%%px\n\nx = torch.randn(1, dtype=torch.double)\nt_start = time.time()\nfor _ in range(20):\n x = sync_consensus_step_with_sleep(x)\nduration = time.time() - t_start\nprint(f\"Rank {bf.rank()} took {duration} seconds\")\n```\n\n [stdout:0] Rank 3 took 10.731921195983887 seconds\n [stdout:1] Rank 0 took 10.72654104232788 seconds\n [stdout:2] Rank 1 took 10.72762393951416 seconds\n [stdout:3] Rank 2 took 10.725753784179688 seconds\n\n\n\n```\n%%px\n\nx = torch.randn(1, dtype=torch.double)\nx_bar = bf.allreduce(x, average=True)\nbf.win_free()\nassert bf.win_create(x, name=\"x\", zero_init=True)\nt_start = time.time()\nfor _ in range(20):\n async_consensus_step_with_sleep(x)\nduration = time.time() - t_start\nprint(f\"Rank {bf.rank()} took {duration} seconds\")\n```\n\n [stdout:0] Rank 3 took 0.42594003677368164 seconds\n [stdout:1] Rank 0 took 10.355370044708252 seconds\n [stdout:2] Rank 1 took 0.5432538986206055 seconds\n [stdout:3] Rank 2 took 0.451002836227417 seconds\n\n\n# Asynchronous Push-Sum Consensus Algorithm\n\nRecall the Push-Sum consensus algorithm in synchronized behavior is\n\nFor each agent $i$, run it in parallel:\n$$\n x_{i, k+1} = \\sum_{j\\in \\mathcal{N}_i} w_{ij}^{(k)} x_{j,k}\\\\\n p_{i, k+1} = \\sum_{j\\in \\mathcal{N}_i} w_{ij}^{(k)} p_{j,k}\\\\\n y_{i, k+1} = x_{i, k+1} / p_{i, k+1}\n$$\n\nIn asynchronous style, the mathametical equation becomes tricky to represent it. we are no longer able to use the global iteration/time $k$. Instead, we will use the logical event counter $e$, which acts similar as iteration $k$ but with different definition.\n\n[TODO] Add a figure of global counter\n\nThere are two ways to descibe it -- with *push* view or *pull* view, one is based on the events that the data is pushed to the neighbors and another is based on the events that the data is recieved from neighbors. It is easy to describe the `push` view in words but not easy to write in words. So we will first use words to describte the push mode in words then give the equation description in equations: \n\n- Push view:
\n When at agent $i$ is ready to push a information: Split the data $x_i$ into $|\\mathcal{N}_i|$ pieces. Each piece only holds the partial of $x_i$ according to the weights, i.e. $a_{ij} x_i$ and do the same thing for $p_i$ as well. Then push this information to the corresponding neighbors for adding. \n \n- Pull view:
\nLet's use $e$ to denote the global event counter and assume that $j$ is the agent rank that triggers the event $e$.\n\\begin{align}\n x_{i, e+1} =&\\; x_{i, e} + w_{ij}^{(k)} x_{j,k}\\;\\; {\\rm if}\\; i\\neq j\\;\\; {\\rm else}\\; w_{jj} x_{j, e}\\\\\n p_{i, e+1} =&\\; x_{i, e} + w_{ij}^{(k)} p_{j,k}\\;\\; {\\rm if}\\; i\\neq j\\;\\; {\\rm else}\\; w_{jj} p_{j, e}\\\\\n y_{i, e+1} =&\\; x_{i, e+1} / p_{i, e_i+1}\n\\end{align}\n\nIf you do not get the intuition of equations, it will be much clear if we re-write above equations over all agents by introducing the stacking matrix:\n\n$$\n X_{e} = \\left[ \\begin{array}{c}\n -x^T_{1,e}- \\\\ \n -x^T_{2,e}- \\\\ \n \\cdots \\\\\n -x^T_{N, e}-\n \\end{array} \\right]\n$$\n\nUnder this notation, the first equation of **synchronous** mode is equivalent to \n$$\n X_{k+1} = W^{(k)} X_k\n$$\nwhere matrix $W^{(k)} = [w_{ij}]$ (it is doubly stochastic matrix).\n\nThe first equation of **asynchronous** mode is\n$$\n X_{k+1} = W^{(e)} X_k\n$$\nwhere matrix $W^{(e)}$ is (again assuming $i'$ is the agent that trigger the event $e$.)\n$$\n W^{(e)} = \\left[ \\begin{array}{ccccc}\n 1 & & w_{1i'} & &\\\\\n &\\ddots& \\vdots & &\\\\\n & & w_{i'i'} & &\\\\\n & & \\vdots & \\ddots&\\\\\n & & w_{Ni'} & & 1\n \\end{array} \\right]\n$$\nNote now that $W^{(e)}$ is no longer a doubly stochastic matrix but a column stochastic matrix, i.e. the summation of each column is 1 but not the sum of rows. One key property that you should verify, we call it mass conservation property, is \n$$\n \\sum_{i} x_{i, e} = \\sum_{i} x_{i, e'},\\;\\; \\forall e, e'\n$$\n\nIt should be clear now that we need a new function called `win_accumulate`, which is similar to `win_put` that sends data to remote window object. But it accumulate (add) the value onto the remove window object instead of overwriting the data.\n \n\nNow let's use a simple example to illustrate the behavior of `win_accunulate`.\n\n\n```\n%%px\ny = torch.DoubleTensor([bf.rank() + 1])\n# zero_init will create the buffer with 0.\nbf.win_create(y, name=\"y\", zero_init=True)\nif bf.rank() == 0:\n bf.win_accumulate(y, name=\"y\")\nif bf.rank() == 1:\n # neighbor_weights: Dict[int: float] -- rank to weight.\n bf.win_update(name=\"y\", self_weight=1.0, neighbor_weights={0: 1.0})\n print(y)\nbf.barrier()\nif bf.rank() == 1:\n bf.win_update(name=\"y\", self_weight=1.0, neighbor_weights={0: 1.0})\n print(y)\n```\n\n [stdout:2] \n tensor([2.], dtype=torch.float64)\n tensor([3.], dtype=torch.float64)\n\n\nFirst, the argument `self_weight` and `neighbor_weights` in `win_update` is the same as previous `neighbot_allreduce` argument. It controls the weights to do the update. In this case, we meant update the main memory of window `y` equals to its previous value times $1.0$ plus the buffer memory value for agent `0` with weights 1.0.\n\nHence, the first print you saw is $2.0$ because self value in main memory is initialized as $2$ and buffer value for neighbor is $0$ due to zero initialization. You may curious that isn't `bf.accumulate` of agent $0$ will accumulate value $1$ into the buffer? Actually it is not likely because note in agent `0` and `1` these two function is executed almost simultaneously. Hence, the buffer hasn't been updated yet. However, after we added `barrier` function, it is guaranteed that `win_update` in agent $0$ is finished. Hence, the second print funciton, you will see the value is 3 because the buffer value is updated into 1 at that time.\n\nNow, let see how to build asynchronous push-sum algorithm:\n\n\n```\n%%px\n# Set up the average consensus problem.\nx = torch.randn(10, dtype=torch.double)\nx_bar = bf.allreduce(x, average=True)\nmse = [torch.norm(x - x_bar, p=2) / torch.norm(x_bar, p=2)]\np = torch.DoubleTensor([1.0])\nx_ext = torch.cat([x, p], 0)\n# Instead of initalize the buffer with same value as x_ext,\n# we initialize it as zero for accumulation.\nbf.win_create(x_ext, name=\"x_ext\", zero_init=True)\noutdegree = len(bf.out_neighbor_ranks())\nindegree = len(bf.in_neighbor_ranks())\n\ndst_weights = {rank: 1.0 / (outdegree + 1) for rank in bf.out_neighbor_ranks()}\nself_weight = 1 / (1 + outdegree)\n```\n\nBut it is not enough with accumulate only. \nWe designate the local buffer of one agent for all incoming neighbors. This means the `win_accumulate` from different agents sends to common destination agent is independent. This property is very crucial to algorithm implementation because you no longer need to consider the data racing problem between the two agents sends to common destinations.\nOn the other hand, it is possible that the `win_update` and the corresponding `win_accumulate` from other agents operate on the same window object *simulatenously*.\nSince agent $k$ cannot know when the incoming neighbor agent $k'$ will execute the `win_accumulate`, it is impossible to write the code completely avoid that. Consider the following example, the `win_update` at Node 1 may read the old value or the updated value from Node 2, but you cannot control.\n\n \n\nThis data racing can be problematic for the push-sum algorithm. It can easily break the key mass conservation property of push-sum algorithm, i.e., the sum of $p$ and $x$ over all agent is always constant.\n\n\nFortunately, it is very easy to avoid this situation through the **distributed mutex**. It is same usage as normal mutext that when agent $k$ want to update its local window object it has to acquire the mutex first. If it cannot acquire the mutex. it has to wait until the mutex is available. The difference from the normal mutex is that this mutex can also be acquired by neighbor agents. Hence, when an agent $k'$ wants to accumulate some value to the remote window object at agent $k$, it has to acquire the mutex of agent $k'$ first as well.\nTo use this distributed mutex, you just need to set the `require_mutex` argument in `win_accumulate` to be True. (`win_put` and `win_get` have the same argument as well.)\n\nBesides the `win_accumulate` which is the writing side of function, we also need to change the behavior of `win_update` for asynchronous push-sum as well, which is the reading side. One obvious thing is that we need to set `require_mutex` as true. A less obvious thing is that we need clean the buffer in the window. Note by default `win_update` only compute the average value between self and neighbors and leave the value in the buffer untouched. In this case, `win_accumulate` will continue adding the values. Instead, what we want is move the value from buffer to local value, i.e. let self value to be the sum of self and neighbor, and let the neighbor value to be zero. It is doable though `win_update` with the proper `self_weight`, `neighbor_weigth` and `reset` argument. But it is boilerplate, so we introduce a new function `win_update_then_collect`.\n\nNow we are ready to present the asynchronous push-sum algorithm:\n\n\n```\n%%px\ndef push_sum_alg(x, self_weight, dst_weights, name):\n global mse\n bf.win_accumulate(\n x,\n self_weight=self_weight,\n dst_weights=dst_weights,\n name=name,\n require_mutex=True,\n )\n bf.win_update_then_collect(name=\"x_ext\")\n x, associated_p = x_ext[:-1], x_ext[-1]\n mse.append(torch.norm(x / associated_p - x_bar, p=2) / torch.norm(x_bar, p=2))\n return x_ext\n```\n\n\n```\nfor i in range(100):\n if i % 5 == 0:\n print(f\"iteration {i}\", end=\"\\r\")\n %px x_ext = push_sum_alg(x_ext, self_weight, dst_weights, name=\"x_ext\")\n```\n\n iteration 95\r\n\n\n```\n%%px\n# Do not forget to sync at last!\nbf.barrier()\nbf.win_update_then_collect(name=\"x_ext\")\nx, associated_p = x_ext[:-1], x_ext[-1]\nprint(f\"associated p at {bf.rank()} is {associated_p}\")\nmse.append(torch.norm(x / associated_p - x_bar, p=2) / torch.norm(x_bar, p=2))\nassert bf.win_free(name=\"x_ext\")\n```\n\n [stdout:0] associated p at 0 is 1.00427521234524\n [stdout:1] associated p at 2 is 0.991455734193527\n [stdout:2] associated p at 1 is 1.0000002021018388\n [stdout:3] associated p at 3 is 1.004276265183896\n\n\n\n```\nmse_list = rc[:].pull(\"mse\")\nfor i, mse in enumerate(mse_list):\n plt.semilogy(mse, label=\"agent \" + str(i))\nplt.legend()\nplt.xlabel(\"Local update counts\")\n```\n\n# Asynchronous push-sum algorithm over the dynamic topology\n\nLast, let's present the push-sum combining the asynchronous operation over dynamic topology.\nThe nice convergence property will gurantee that under very mild condition, push-sum algorithm will converge to the mean of all agents no matter what the order of asynchronous operations between agents is and how different the dynamic topologies change with time.\n\n\n```\n%%px\nbf.win_free()\n\n# Set up the average consensus problem.\nx = torch.randn(10, dtype=torch.double)\nx_bar = bf.allreduce(x, average=True)\nmse = [torch.norm(x - x_bar, p=2) / torch.norm(x_bar, p=2)]\np = torch.DoubleTensor([1.0])\nx_ext = torch.cat([x, p], 0)\n# Instead of initalize the buffer with same value as x_ext,\n# we initialize it as zero for accumulation.\nbf.win_create(x_ext, name=\"x_ext\", zero_init=True)\noutdegree = len(bf.out_neighbor_ranks())\nindegree = len(bf.in_neighbor_ranks())\n```\n\n\n```\n%%px\ndef dynamic_push_sum_alg(i, x, self_weight, dst_weights, name):\n global mse\n num_out_neighbors = len(bf.out_neighbor_ranks())\n sent_neighbor = bf.out_neighbor_ranks()[i % num_out_neighbors]\n dst_weights = {sent_neighbor: 0.5}\n self_weight = 0.5\n bf.win_accumulate(\n x,\n self_weight=self_weight,\n dst_weights=dst_weights,\n name=name,\n require_mutex=True,\n )\n bf.win_update_then_collect(name=\"x_ext\")\n x, associated_p = x_ext[:-1], x_ext[-1]\n mse.append(torch.norm(x / associated_p - x_bar, p=2) / torch.norm(x_bar, p=2))\n return x_ext\n```\n\n\n```\nfor i in range(100):\n if i % 5 == 0:\n print(f\"iteration {i}\", end=\"\\r\")\n rc[:].push({\"i\": i})\n %px x_ext = dynamic_push_sum_alg(i, x_ext, self_weight, dst_weights, name=\"x_ext\")\n```\n\n iteration 95\r\n\n\n```\n%%px\n# Do not forget to sync at last!\nbf.barrier()\nbf.win_update_then_collect(name=\"x_ext\")\nx, associated_p = x_ext[:-1], x_ext[-1]\nprint(f\"associated p at {bf.rank()} is {associated_p}\")\nmse.append(torch.norm(x / associated_p - x_bar, p=2) / torch.norm(x_bar, p=2))\nassert bf.win_free(name=\"x_ext\")\n```\n\n [stdout:0] associated p at 0 is 0.832190677523613\n [stdout:1] associated p at 2 is 1.5\n [stdout:2] associated p at 1 is 0.667809322476387\n [stdout:3] associated p at 3 is 1.0\n\n\n\n```\nmse_list = rc[:].pull(\"mse\")\nfor i, mse in enumerate(mse_list):\n plt.semilogy(mse, label=\"agent \" + str(i))\nplt.legend()\nplt.xlabel(\"Local update counts\")\n```\n\n# Other features\n\nSimilar as the `neighbor_allreduce`, we also provided the nonblocking version of all window functions, such as `win_put_nonblocking`, etc. We didn't illustrated in this notebook. The reader can try to use the nonblocking version of window operation to build an asynchronous Adapt-With-Combination algorithm, which combines nonblocking, asynchrnous, gradient, consensus communication features.\n", "meta": {"hexsha": "8f3e6dee4e33c7deea09f45bb724c887f711f4c9", "size": 121887, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Section 6/Asynchronous Communication.ipynb", "max_stars_repo_name": "Bluefog-Lib/bluefog-tutorial", "max_stars_repo_head_hexsha": "a9b371376dafd40d4dee8c61e9060f4e2cec773d", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-10-01T07:51:59.000Z", "max_stars_repo_stars_event_max_datetime": "2021-10-01T07:51:59.000Z", "max_issues_repo_path": "Section 6/Asynchronous Communication.ipynb", "max_issues_repo_name": "Bluefog-Lib/bluefog-tutorial", "max_issues_repo_head_hexsha": "a9b371376dafd40d4dee8c61e9060f4e2cec773d", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Section 6/Asynchronous Communication.ipynb", "max_forks_repo_name": "Bluefog-Lib/bluefog-tutorial", "max_forks_repo_head_hexsha": "a9b371376dafd40d4dee8c61e9060f4e2cec773d", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 110.907188353, "max_line_length": 35812, "alphanum_fraction": 0.8474898882, "converted": true, "num_tokens": 6686, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011686727231, "lm_q2_score": 0.7634837581726991, "lm_q1q2_score": 0.44671523916948896}} {"text": "\n\n\n```python\nfrom math import *\nimport numpy as np \nimport scipy as sp\nfrom pylab import *\nimport matplotlib.pyplot as plt\n```\n\nAuteurs : AL AMMIRI Anass, WANG Ruiqi\n\n# Exercice 1 (Tirage d'importance auto-normalis\u00e9)\n\nEn cours vous avez vu la notion de tirage d'importance afin de tenter d'am\u00e9liorer les performances de l'estimateur de Monte Carlo na\u00eff pour l'estimation d'une esp\u00e9rance. Nous allons revenir sur cette probl\u00e9matique en g\u00e9n\u00e9ralisant son cadre d'application : la densit\u00e9 \"naturelle\" $f$ est connue \u00e0 une constante multiplicative pr\u00e8s (et m\u00eame parfois la densit\u00e9 d'importance $p$ \u00e9galement). Nous sommes donc dans la situtation o\u00f9\n\n$$f(x) = c_f f_0(x), \\qquad p(x) = c_p p_0(x), \\qquad x \\in \\mathbb{R},$$\n\no\u00f9 les constantes de normalisation $c_f$ et $c_p$ sont \u00e9videmment donn\u00e9es par\n$$c_f^{-1} = \\int_\\mathbb{R} f_0(x) \\mbox{d$x$}, \\qquad c_p^{-1} = \\int_\\mathbb{R} p_0(x) \\mbox{d$x$}.$$\n\nDans la suite, notre objectif sera d'estimer la quantit\u00e9 (suppos\u00e9e finie)\n$$I = \\int_\\mathbb{R} h(x) f(x) \\mbox{d$x$},$$\npour une fonction donn\u00e9e $h$.\n\n1. Montrez que l'on peut \u00e9crire\n\n$$I = \\frac{\\mathbb{E}[h(X) w(X)]}{\\mathbb{E}[w(X)]}, \\qquad X \\sim p,$$\no\u00f9 $w: x \\mapsto w(x)$ est une fonction connue que vous d\u00e9terminerez et qui fera intervenir les fonctions $p_0$ et $f_0$.\n\n$$I = \\int_\\mathbb{R} h(x) f(x) \\mbox{d$x$} = \\frac{\\int_\\mathbb{R} h(x) \\frac{f_0(x)}{p_0(x)} p(x) \\mbox{d$x$}}{\\int_\\mathbb{R} \\frac{f_0(x)}{p_0(x)} p(x) \\mbox{d$x$}} = \\frac{\\int_\\mathbb{R} h(x) w(x) p(x) \\mbox{d$x$}}{\\int_\\mathbb{R} w(x) p(x) \\mbox{d$x$}} = \\frac{\\mathbb{E}[h(X) w(X)]}{\\mathbb{E}[w(X)]},\\qquad w(x) = \\frac{f_0(x)}{p_0(x)}$$\n\n2. Donnez deux estimateurs de Monte Carlo \"na\u00effs\", not\u00e9s $\\hat{N}_n $ et $\\hat{D}_n$, pour le num\u00e9rateur et le d\u00e9nominateur de l'expression pr\u00e9c\u00e9dente, utilisant $n$ variables al\u00e9atoires ind\u00e9pendantes $X_1,\\dots,X_n$ de m\u00eame loi que $X$. En d\u00e9duire un estimateur $\\hat{I}_n$ pour $I$, o\u00f9 $\\hat{N}_n$ et $\\hat D_n$ sont fonction des m\u00eames $n$ variables al\u00e9atoires. \n\n$$\\hat{N}_n = \\frac{1}{n} {\\sum_{i=1}^n h(X_i) w(X_i)}, \\qquad \\hat{D}_n = \\frac{1}{n} {\\sum_{i=1}^n w(X_i)}, \\qquad \\hat{I}_n = \\frac{\\frac{1}{n} {\\sum_{i=1}^n h(X_i) w(X_i)}}{\\frac{1}{n} {\\sum_{i=1}^n w(X_i)}}$$\n\n3. En quoi ce nouvel estimateur se distingue de l'estimateur par tirage d'importance que vous avez vu en cours ? En quoi r\u00e9pond-il au cadre de travail pos\u00e9 en introduction ?\n\nL'estimateur par tirage d'importance est sous la forme : \n$ \\hat{I}_n = \\frac{1}{n} \\sum_{i=1}^n h(Z_i) \\frac{f_0(Z_{i})}{p_0(Z_{i})}$. Avec Z de densit\u00e9 $p_0$. \\\\\nSauf que dans notre cas $f_0$ est connue \u00e0 une constante multiplicative pr\u00e8s. D'o\u00f9 l'int\u00e9r\u00eat du terme ${\\mathbb{E}[w(X)]}$.\n\n\n4. Montrez que, sous des hypoth\u00e8ses que vous pr\u00e9ciserez, $\\hat I_n$ est un estimateur fortement convergent, i.e., $\\hat I_n \\to I$ p.s. lorsque $n \\to \\infty$.\n\nPour $n$ variables al\u00e9atoires ind\u00e9pendantes $X_1,\\dots,X_n$ de m\u00eame loi que $X$, et $h$ et $w$ deux fonctions r\u00e9elles, on a selon la loi forte des grands nombres : $\\hat{N}_n$ converge ps vers $\\mathbb{E}[w(X)]$, de m\u00eame $\\hat{D}_n$ vers $\\mathbb{E}[w(X)]$ (qui est strictement postive, sinon $f_0$ est nulle). D'o\u00f9 la convergence presque s\u00fbre de $\\hat{I}_n$.\n \n$$\\hat{N}_n = \\frac{1}{n} {\\sum_{i=1}^n h(X_i) w(X_i)} \\to \\mathbb{E}[h(X) w(X)], \\qquad \\hat{D}_n = \\frac{1}{n} {\\sum_{i=1}^n w(X_i)}\\to \\mathbb{E}[w(X)]$$\n\n$$\\hat{I}_n = \\frac{\\frac{1}{n} {\\sum_{i=1}^n h(X_i) w(X_i)}}{\\frac{1}{n} {\\sum_{i=1}^n w(X_i)}}\\to I $$ \n\n5. Selon vous $\\hat I_n$ est-il sans bians pour $I$ ?\n\nOui\n\ud83d\udfe5\n\n6. Calculez $\\mbox{Var}(\\hat N_n)$, $\\mbox{Var}(\\hat D_n)$ et $\\mbox{Cov}(\\hat N_n, \\hat D_n)$. En d\u00e9duire que l'expression suivante\n$$\\sum_{i=1}^n \\omega_i^2 \\{h(X_i) - \\hat I_n\\}^2, \\qquad X_1, \\ldots, X_n \\stackrel{iid}{\\sim} p,$$\nest un estimateur de la variance (approch\u00e9e) de $\\hat I_n$ avec\n$$\\omega_i = \\frac{w(X_i)}{\\sum_{j=1}^n w(X_j)}.$$\n\nAstuce : La m\u00e9thode delta est votre meilleur ami pour obtenir la derni\u00e8re expression !\n\n* Pour $\\mbox{Var}(\\hat N_n) $ \\begin{align}\n\\mbox{Var}(\\hat N_n) & = & Var(\\frac{1}{n} {\\sum_{i=1}^n h(X_i) w(X_i)}) \\\\\n & = & \\frac{1}{n^2} Var( {\\sum_{i=1}^n h(X_i) w(X_i)}) \\\\\n & = & \\frac{1}{n^2} \\sum_{i=1}^n Var( { h(X_i) w(X_i)}) \\text{ (par ind\u00e9pendance)}\\\\ \n & = & \\frac{1}{n^2} \\sum_{i=1}^n Var( { h(X_i) w(X_i)}) \\\\\n & = & \\frac{1}{n} Var( { h(X) w(X)}) \\\\\n & = & \\frac{1}{n} Var( { h(X) \\frac{f_0(X)}{p_0(X)}}) \\\\\n\\end{align}\n\n* Pour $\\mbox{Var}(\\hat D_n) $ \\begin{align*}\n\\mbox{Var}(\\hat D_n) & = & Var(\\frac{1}{n} {\\sum_{i=1}^n w(X_i)}) \\\\\n & = & \\frac{1}{n^2} Var( {\\sum_{i=1}^n w(X_i)}) \\\\\n & = & \\frac{1}{n^2} \\sum_{i=1}^n Var( { w(X_i)}) \\text{ (par ind\u00e9pendance)}\\\\ \n & = & \\frac{1}{n^2} \\sum_{i=1}^n Var( {w(X_i)}) \\\\\n & = & \\frac{1}{n} Var({w(X)}) \\\\\n & = & \\frac{1}{n} Var( { \\frac{f_0(X)}{p_0(X)}}) \\\\\n\\end{align*}\n\n* Pour $\\mbox{Cov}(\\hat N_n, \\hat D_n)$ \\begin{align*}\n\\mbox{Cov}(\\hat N_n, \\hat D_n) & = & Cov(\\frac{1}{n} {\\sum_{i=1}^n h(X_i) w(X_i)}, \\frac{1}{n} {\\sum_{i=1}^n w(X_i)}) \\\\\n & = & \\frac{1}{n^2}Cov( {\\sum_{i=1}^n h(X_i) w(X_i)}, {\\sum_{i=1}^n w(X_i)}) \\\\\n & = & \\frac{1}{n^2} \\sum_{i=1}^n \\sum_{j=1}^n Cov( h(X_i) w(X_i), w(X_j)) \\\\ \n & = & \\frac{1}{n^2} \\sum_{i=1}^n \\sum_{j=1}^n Cov( h(X_i) w(X_i), w(X_i)) \\\\ \\text{ ( pour $ i \\neq j$ la covariance est nulle par ind\u00e9pendance)}\\\\\n & = & \\frac{1}{n} Cov( h(X) w(X), w(X)) \\\\\n\\end{align*}\n\n\n\nPour trouver l'expression de l'estimateur on utilise la d\u00e9lta m\u00e9thode. \\\\\nOn note :\n\n* $\\Psi = ( w(X),h(X)w(X))$\n* $\\hat{\\Psi}_n = \\mbox{Var} (\\hat D_n, \\hat N_n )$\n* $\\mathbb{E}_n = \\mathbb{E} (h(X)w(X))$\n* $\\mathbb{E}_d = \\mathbb{E} (w(X))$\n* $V_n = \\mbox{Var} (h(X)w(X))$\n* $V_d = \\mbox{Var} (w(X))$\n* $\\theta = (\\mathbb{E}_d,\\mathbb{E}_n)$\n* $Cov= Cov( h(X) w(X), w(X)) $\n\nLa suite des variables al\u00e9atoires $\\hat{\\Psi}_1, \\ldots, \\hat{\\Psi}_n$. Par le th\u00e9or\u00e8me de central limite: $\\sqrt{n}(\\hat{\\Psi}_n - \\mathbb{E}(\\Psi)) \\xrightarrow{L} \\mathcal{N}(0,\\Sigma)$. Avec $\\Sigma $ la matrice de covariance de $\u03a8$. On pose la fonction $g$, telle que $ g(x,y)= \\frac{x}{y}$. Par la delta m\u00e9thode :\n${\\sqrt{n}[g(\\hat{\\Psi}_n )-g(\\Psi)]\\,\\xrightarrow{L}\\,\\mathcal{N}\\left(0,\\nabla g(\\theta)\\Sigma \\nabla g(\\theta)^T\\right)}$ avec $\\nabla g(\\theta)$ la matrice jacobienne de $g$ en $\\theta$.\n\nDonc : ${\\sqrt{n}[\\hat{I}_n -I]\\,\\xrightarrow{L}\\,\\mathcal{N}\\left(0,\\nabla g(\\theta)\\Sigma \\nabla g(\\theta)^T\\right)}$\nAlors : $\\hat{I}_n \\,\\xrightarrow{L}\\,\\mathcal{N}\\left(I,\\frac{1}{n}\\nabla g(\\theta)\\Sigma \\nabla g(\\theta)^T\\right)$\n \nDonc l'expression de la variance (approch\u00e9e ) de $\\hat{I}_n$ est : $\\frac{1}{n}\\nabla g(\\theta)\\Sigma \\nabla g(\\theta)^T = \\frac{1}{n \\mathbb{E}_d^2} \\left( V_n - 2 \\frac{\\mathbb{E}_n}{\\mathbb{E}_d}Cov+ \\frac{\\mathbb{E}_n^2}{\\mathbb{E}_d^2}V_d \\right) = \\frac{1}{n \\mathbb{E}(w(X))^2} \\left( V(h(X)w(X)) - 2 \\frac{\\mathbb{E}(h(X)w(X))}{\\mathbb{E}(w(X))}Cov(h(X)w(X),w(X))+ \\left( \\frac{\\mathbb{E}(h(X)w(X))}{\\mathbb{E}(w(X))} \\right)^2 V(w(X)) \\right) = \\frac{1}{n \\mathbb{E}(w(X))^2} \\left( V(h(X)w(X)) - 2 \\frac{\\mathbb{E}(h(X)w(X))}{\\mathbb{E}(w(X))}Cov(h(X)w(X),w(X))+ \\left( \\frac{\\mathbb{E}(h(X)w(X))}{\\mathbb{E}(w(X))} \\right)^2 V(w(X)) \\right) = \\frac{1}{n \\mathbb{E}(w(X))^2} \\left( V(h(X)w(X)) - 2 I.Cov(h(X)w(X),w(X))+ I^2 V(w(X)) \\right) = \\frac{1}{n \\mathbb{E}(w(X))^2} \\left( V(h(X)w(X)) - 2 Cov(h(X)w(X),w(X)I)+ V(w(X) I) \\right) = \\frac{1}{n \\mathbb{E}(w(X))^2} \\left( V(h(X)w(X)) - 2 Cov(h(X)w(X),w(X)I)+ V(w(X) I) \\right)= \\frac{ Var(h(X)w(X)- w(X)I) }{n \\mathbb{E}(w(X))^2} = \\frac{ Var(h(X)w(X)- w(X)I) + \\stackrel{\\mathbb{E}(h(X)w(X)- w(X)I)=0}{\\mathbb{E}(h(X)w(X)- w(X)I)^2} }{n \\mathbb{E}(w(X))^2} = \\frac{ \\mathbb{E}\\left( \\left( h(X)w(X)- w(X)I \\right)^2 \\right) }{n \\mathbb{E}(w(X))^2} \\stackrel{empiriquement}{=} \\frac{1}{n} \\frac{\\frac{1}{n}\\sum_{i=1}^n \\{h(X_i).w(X_i) - \\hat I_n.w(X_i)\\}^2}{\\left( \\frac{1}{n} \\sum_{i=1}^n w(X_i) \\right) ^2} = \\frac{\\sum_{i=1}^n w(X_i) ^2 \\{h(X_i) - \\hat I_n\\}^2}{\\left(\\sum_{i=1}^n w(X_i) \\right) ^2}= \\sum_{i=1}^n \\omega_i^2 \\{h(X_i) - \\hat I_n\\}^2$ \n\nPour $\\omega_i = \\frac{w(X_i)}{\\sum_{j=1}^n w(X_j)}$ . Cqfd \n\n7. En d\u00e9duire un intervalle de confiance (sym\u00e9trique) pour $I$ de niveau asymptotique $1 - \\alpha$.\n\nPar le th\u00e9or\u00e8m centrale limite (d\u00e9j\u00e0 appliqu\u00e9e) et par le th\u00e9or\u00e8me de Slutsky on a comme IC asymptotique $\u039e_n= \\left[\\hat I_n - z_{1-\u03b1/2} \\sqrt{\\frac{\\sum_{i=1}^n \\omega_i^2 \\{h(X_i) - \\hat I_n\\}^2}{n}} , \\hat I_n + z_{1-\u03b1/2} \\sqrt{\\frac{\\sum_{i=1}^n \\omega_i^2 \\{h(X_i) - \\hat I_n\\}^2}{n}} \\right]$ \n\nAvec $\\omega_i = \\frac{w(X_i)}{\\sum_{j=1}^n w(X_j)}$, et $z_{1-\u03b1/2}$ le quantile $1-\u03b1/2$ de la loi normale.\n\n8. Application num\u00e9rique. On consid\u00e8re le cas o\u00f9 $h(x) = 1_{\\{x \\in A\\}}$, $A \\subset \\mathbb{R}$, de sorte que $I$ est la probabilit\u00e9 d'appartenir \u00e0 $A$ sous la densit\u00e9 $f$. Pour cette application num\u00e9rique nous allons consid\u00e9rer le cas o\u00f9 $A = (4, \\infty)$ et $f$ est la densit\u00e9 d'une $N(0,1)$.\n\n a) D\u00e9finissez un estimateur de Monte Carlo na\u00eff pour $I$.\n \n b) D\u00e9finissez un estimateur par tirage d'importance pour $I$ avec une loi d'importance que vous choisirez.\n \n c) D\u00e9finissez un estimateur par tirage d'importance auto-normalis\u00e9 pour $I$ bas\u00e9 sur celui de b).\n \n d) Commentez les r\u00e9sultats obtenus\n\n\n```python\nfrom scipy.stats import norm\n\ndef f(x):\n return (1/sqrt(2*np.pi)*np.exp(-x**2/2))\n\ndef h(x):\n return(x>4).astype(int) \n\ndef q(x):\n return np.exp(-(x-4))\n\ndef w(x):\n return np.divide(f(x),q(x))\n\ndef hw(x):\n return np.multiply(h(x),w(x))\n```\n\n a) D\u00e9finissez un estimateur de Monte Carlo na\u00eff pour $I$.\n\n $\\hat I_n = \\frac{1}{n} \\sum_{i=1}^n \ud835\udfd9_{\\{X_n \\geq 4 \\}} f(X_n)$\n\n\n```python\nn=50\ndef MC1(n):\n X=np.random.normal(size=n)\n I=mean(np.multiply((X>4).astype(int) ,X))\n return I\nMC1(n)\n```\n\n\n\n\n 0.0\n\n\n\n\n```python\n\nn_values=list(range(0,1000,100))\nI_values = [MC1(n) for n in n_values]\nprint(I_values)\n#plt.plot(n_values,I_values)\n#plt.title(\"L'estimation de I pour diff\u00e9rentes valeurs de n \")\n#plt.show()\n```\n\n [nan, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]\n\n\n /usr/local/lib/python3.7/dist-packages/numpy/core/fromnumeric.py:3373: RuntimeWarning: Mean of empty slice.\n out=out, **kwargs)\n /usr/local/lib/python3.7/dist-packages/numpy/core/_methods.py:170: RuntimeWarning: invalid value encountered in double_scalars\n ret = ret.dtype.type(ret / rcount)\n\n\nNous pouvons voir qu'il y a un probl\u00e8me ici, car presque tous les \u00e9chantillons sont plus petits que 4. Et pour la variance :\n\n\n```python\nn=10000\nX=np.random.normal(size=n)\nnp.var(np.multiply((X>4).astype(int) ,X))\n```\n\n\n\n\n 0.0021943521469248617\n\n\n\n b) D\u00e9finissez un estimateur par tirage d'importance pour $I$ avec une loi d'importance que vous choisirez.\n\nOn prend la loi d'importance $ \\begin{equation}\n q(x)=\n \\begin{cases}\n exp(-(x-4)) = exp(-x).exp(4) , & \\text{if}\\ x \\geq 4 \\\\\n 0 & \\text{otherwise}\n \\end{cases}\n \\end{equation} $ , pour ce choix on a utilis\u00e9 cette [r\u00e9f\u00e9rence](https://www.math.arizona.edu/~tgk/mc/book_chap6.pdf) o\u00f9 la variance est aussi calcul\u00e9.\n\n\n\n $\\hat I_n = \\frac{1}{n} \\sum_{i=1}^n \ud835\udfd9_{\\{Z_n \\geq 4 \\}} \\frac{ f(Z_n)}{q(Z_n)}$\n\n\n```python\ndef MC2(n):\n Z= np.random.exponential(scale=1.0, size=n)\n I=mean(np.multiply((Z>4).astype(int) ,np.divide(f(Z),q(Z))))\n return I\n\nn=10000\nMC2(n)\n```\n\n\n\n\n 5.483400921877393e-07\n\n\n\nLa variance:\n\n\n```python\nn=10000\nZ=np.random.exponential(scale=1.0, size=n)\nnp.var(np.multiply((Z>4).astype(int) ,np.divide(f(Z),q(Z))))\n```\n\n\n\n\n 5.070188557023406e-11\n\n\n\nLa variance est plus petite par rapport \u00e0 la premi\u00e8re m\u00e9thode \n\nc) D\u00e9finissez un estimateur par tirage d'importance auto-normalis\u00e9 pour I bas\u00e9 sur celui de b)\n\nOn utilise les r\u00e9sultats des questions pr\u00e9c\u00e9dentes.\n\n\n```python\ndef MC3(n):\n X=np.random.normal(scale=1.0, size=n)\n return np.mean(hw(X))/np.mean(w(X))\nn=10000\nMC3(n)\n```\n\n\n\n\n 0.0\n\n\n\nPour la variance :\n\n\n```python\nn=10000\nZ=np.random.normal(scale=1.0, size=n)\nf_z=f(Z)\nq_z=q(Z)\nnp.var([np.mean(f_z[0:n])/np.mean(q_z[0:n]) for n in range(2,len(f_z)) ])\n```\n\n\n\n\n 7.45230411377179e-09\n\n\n\nd) Commentez les r\u00e9sultats obtenus\n\n\n```python\nn=1000000\nI_exact=1-norm.cdf(4)\nprint(f\"Valeur exacte: I={Iexact}\")\nI=MC1(n)\nprint(f\"Estimation na\u00efve: I={I}\")\nerreur = np.linalg.norm(I_exact-I)\nprint(f\"Erreur: {I}\")\nI=MC2(n)\nprint(f\"Estimation par tirage d'importance: I={I}\")\nerreur = np.linalg.norm(I_exact-I)\nprint(f\"Erreur: {I}\")\nI=MC3(n)\nprint(f\"Estimation par tirage d'importance auto-normalis\u00e9: I={I}\")\nerreur = np.linalg.norm(I_exact-I)\nprint(f\"Erreur: {I}\")\n```\n\n Valeur exacte: I=3.167124183311998e-05\n Estimation na\u00efve: I=0.00014548439116039618\n Erreur: 0.00014548439116039618\n Estimation par tirage d'importance: I=5.88678965280222e-07\n Erreur: 5.88678965280222e-07\n Estimation par tirage d'importance auto-normalis\u00e9: I=2.1298953023364217e-07\n Erreur: 2.1298953023364217e-07\n\n\nOn remarque que l'erreur la plus petite est celle du tirage auto-normalis\u00e9 (ceci pour plusieurs essais)\n\nPour la variance (empirique):\n\n\n```python\nn=1000000\n\nn_values=list(range(0,1000000,100000))\nI_values_1 =[]\nI_values_2 =[]\nI_values_3 = []\nfor n in n_values :\n X=np.random.normal(size=n)\n I_values_1.append(np.var(np.multiply((X>4).astype(int) ,X)))\n Z=np.random.exponential(scale=1.0, size=n)\n I_values_2.append(np.var(np.multiply((Z>4).astype(int) ,np.divide(f(Z),q(Z)))))\n Z=np.random.normal(scale=1.0, size=n)\n f_z=f(X)\n q_z=q(X)\n I_values_3.append(np.var([np.mean(f_z[0:n])/np.mean(q_z[0:n]) for n in range(2,len(f_z)) ]))\n\nplt.plot(n_values,I_values_1, label = \"Monte Carlo naif\")\nplt.plot(n_values,I_values_2, label = \"Echantillonnage d\u2019importance\")\nplt.plot(n_values,I_values_3, label = \"Tirage d'importance auto-normalis\u00e9 \")\nplt.title(\"Variance empirique pour diff\u00e9rentes valeurs de n pour les trois m\u00e9thodes\")\nplt.show()\n\n\n```\n\n\n```python\nplt.plot(n_values,I_values_1, label = \"Monte Carlo naif\")\nplt.plot(n_values,I_values_2, label = \"Echantillonnage d\u2019importance\")\nplt.plot(n_values,I_values_3, label = \"Tirage d'importance auto-normalis\u00e9 \")\nplt.title(\"Variance empirique pour diff\u00e9rentes valeurs de n pour les trois m\u00e9thodes\")\nplt.legend()\n```\n\n\n```python\n\nplt.plot(n_values,I_values_2, label = \"Echantillonnage d\u2019importance\")\nplt.plot(n_values,I_values_3, label = \"Tirage d'importance auto-normalis\u00e9 \")\nplt.title(\"Variance empirique pour diff\u00e9rentes valeurs de n pour les trois m\u00e9thodes\")\nplt.legend()\n```\n\nLa troisi\u00e8me m\u00e9thode est meilleure en termes de pr\u00e9cision.\nDans ce cas, la variance empirique est plus petite pour la secondes m\u00e9thode, alors qu'elle d\u00e9cro\u00eet avec n pour la troisi\u00e8me m\u00e9thode (qui est celle qui est prouv\u00e9e). Il existe d'autres m\u00e9thodes pour diminuer la variance asymptotiquement \u00e0 chaque fois (mise \u00e0 jour des poids, estimations cumul\u00e9es..).\nBien que l'exemple que nous avons soit particulier, si nous n'ayons pas choisi une bonne loi d'importance ou si nous ne connaissions pas la constante multiplicative on aurait pas le m\u00eame r\u00e9sultat. Il est pr\u00e9f\u00e9rable d'utiliser la troisi\u00e8me m\u00e9thode (mem.\n\nNous avons \u00e9galement confirm\u00e9 que la m\u00e9thode du tirage est meilleure que la m\u00e9thode naive.\n\n# Exercice 2 (D\u00e9composition en valeurs singuli\u00e8res randomis\u00e9e)\n\nLa **d\u00e9composition en valeurs singuli\u00e8res (SVD)** d'une matrice $A \\in \\mathbb{R}^{m\\times n}$, avec $m \\ge n$, s'\u00e9crit sous la forme \n$$\n\\tag{1}\nA = U S V^T \n$$\no\u00f9 $U = [u_1, \\dots, u_n]\\in \\mathbb{R}^{m\\times m}$ et $V= [v_1, \\dots, v_n]\\in \\mathbb{R}^{n\\times n}$ sont des matrices orthonorm\u00e9es. Les $u_1, \\dots, u_m$ sont appel\u00e9s vecteurs singuliers \u00e0 gauche, ils forment une base pour l'espace engendr\u00e9 par les colonnes de $A$. De m\u00eame les $v_1, \\dots, v_n$ sont les vecteurs singuliers \u00e0 droite et forment une base pour l'espace engendr\u00e9 par les lignes de $A$. \nLa matrice $S = \\mathrm{diag}(\\sigma_1, \\dots, \\sigma_n)$ de $\\mathbb{R}^{m \\times n}$ est une matrice rectangulaire form\u00e9e des valeurs singuli\u00e8res de la matrice $A$ not\u00e9es $\\sigma_1\\ge \\dots \\ge \\sigma_n \\ge 0$. \n\nDans certaines applications (e.g. compression d'image, r\u00e9duction de mod\u00e8les), il peut \u00eatre int\u00e9ressant de consid\u00e9rer la SVD tronqu\u00e9e \u00e0 $r$ termes, avec $r\\le n$, de la matrice $A$. On la note \n\n$$\n\\tag{2}\nA_r = \\sum_{i=1}^r [u_1, \\dots, u_r]\\mathrm{diag}(\\sigma_1, \\dots, \\sigma_{r}) [v_1, \\dots, v_r]^T = \\sum_{i=1}^r\\sigma_i u_iv_i^T,\n$$\nqui est une matrice de rang au plus $r$. \n\nLe theor\u00e8me d'Eckart-Young assure que $A_r$ est solution du probl\u00e8me de meilleure approximation de $A$ par une matrice de rang au plus $r$, \n$$\n\\tag{3}\n \\| A-A_r \\| := \\min_{Z \\in R^{m \\times n}, \\mathrm{rang}(Z)\\le r}\\| A-Z \\|,\n$$\navec comme norme matricielle la norme spectrale ou la norme de Frobenius. \n- Pour la norme spectrale $\\|\\cdot\\|:=\\|\\cdot\\|_2$ l'erreur d'approximation est donn\u00e9e par \n$$\n\\| A-A_r \\|_2 = \\sigma_{r+1},\n$$\n- Pour la norme de Frobenius $\\|\\cdot\\|:=\\|\\cdot\\|_F$ l'erreur d'approximation est donn\u00e9e par \n$$\n\\| A-A_r \\|_F = \\sqrt{\\sum_{i=r+1}^n\\sigma^2_{i}}.\n$$\n\n\n\n**Question 1.** \n\n Consid\u00e9rer une matrice dont les entr\u00e9es sont des variables al\u00e9atoires ind\u00e9pendantes qui suivent une loi uniforme discrete dans $\\{0,\\dots,9\\}$.\n- En utilisant la commande `numpy.linalg.svd()` de Python, tester la d\u00e9composition en valeurs singuli\u00e8res de cette matrice. Afficher les valeurs singuli\u00e8res de $A$. \n- Calculer $A_r$ une approximation de rang $r$ de $A$ et donner les erreurs d'approximation en norme spectrale et de Frobenius.\n\nOn prendra par la suite $m=20, n=15$ et $r=5$ pour les tests num\u00e9riques.\n\n\n\n\n```python\n#definition de la matrice A\nn=15\nm=20\nentries = np.random.randint(0, 9+1, size= n*m) \nA= np.reshape(entries, (m, n))\nA\n```\n\n\n\n\n array([[8, 5, 7, 7, 0, 2, 0, 6, 0, 8, 6, 3, 5, 2, 5],\n [0, 2, 8, 3, 6, 3, 5, 2, 3, 4, 2, 6, 3, 9, 3],\n [8, 0, 2, 0, 1, 4, 3, 2, 0, 9, 6, 1, 1, 0, 8],\n [8, 0, 8, 8, 7, 0, 1, 5, 0, 7, 3, 5, 5, 4, 8],\n [3, 2, 7, 3, 8, 7, 8, 6, 8, 8, 8, 9, 8, 2, 9],\n [4, 7, 0, 6, 8, 5, 2, 2, 7, 2, 9, 5, 7, 9, 5],\n [9, 8, 7, 7, 0, 2, 9, 7, 5, 5, 9, 5, 4, 2, 2],\n [5, 2, 2, 2, 6, 5, 7, 1, 0, 9, 9, 8, 0, 5, 7],\n [2, 3, 2, 1, 3, 9, 0, 5, 4, 0, 8, 1, 2, 6, 7],\n [9, 7, 8, 2, 8, 5, 6, 4, 6, 5, 1, 4, 7, 9, 3],\n [6, 0, 7, 5, 5, 4, 3, 7, 1, 2, 2, 0, 2, 4, 3],\n [7, 9, 5, 6, 0, 0, 8, 7, 2, 3, 6, 0, 4, 2, 7],\n [2, 5, 0, 4, 4, 2, 1, 7, 2, 4, 9, 7, 1, 1, 5],\n [6, 0, 2, 4, 0, 2, 7, 1, 5, 9, 0, 6, 3, 7, 3],\n [3, 8, 4, 2, 0, 4, 6, 7, 7, 6, 3, 0, 4, 2, 7],\n [2, 3, 9, 1, 9, 9, 6, 6, 8, 0, 6, 2, 2, 9, 7],\n [9, 7, 2, 8, 3, 3, 9, 8, 9, 2, 9, 2, 3, 3, 8],\n [7, 4, 3, 1, 8, 2, 1, 1, 1, 7, 3, 9, 9, 9, 5],\n [7, 7, 9, 6, 9, 1, 2, 9, 1, 0, 5, 8, 2, 6, 4],\n [8, 6, 4, 8, 3, 4, 6, 1, 3, 2, 9, 6, 4, 9, 1]])\n\n\n\n\n```python\n# d\u00e9composition en valeurs singuli\u00e8res \nU,S,Vh = np.linalg.svd(A)\nprint(U, \"\\n\", S, \"\\n\", Vh)\n```\n\n [[-2.08774734e-01 2.34620877e-01 -2.88850371e-01 1.22543532e-01\n 1.39100581e-02 1.96469208e-01 2.51839557e-01 -3.13431996e-02\n -1.23290778e-01 5.25220081e-01 3.48876746e-03 -1.86897703e-02\n -1.53495321e-01 -1.64456482e-01 6.04106200e-02 1.85410790e-01\n 4.21311501e-02 -9.33343593e-02 5.37674578e-01 1.41716736e-01]\n [-1.85281262e-01 -3.20948591e-01 5.08064583e-03 9.78451304e-02\n 1.37814296e-01 -1.09698642e-01 -2.96477995e-01 2.88796061e-02\n -1.04774215e-01 2.58784707e-01 -3.79439936e-01 2.59434647e-01\n -5.44962730e-02 -5.00791726e-02 -2.40801107e-01 -5.62360601e-01\n 1.94764940e-01 -5.15255854e-03 1.00580076e-01 1.27619504e-01]\n [-1.51647230e-01 1.55934833e-01 -2.80537532e-01 -3.93367623e-01\n 1.52181971e-01 1.49431353e-01 1.96130270e-01 -3.29336193e-01\n -1.37232307e-01 -2.02247777e-01 1.72400583e-01 -1.45791413e-02\n -2.42884733e-02 -2.53683680e-01 -2.09944047e-01 -4.23462180e-01\n -3.58553519e-01 -4.87044675e-02 -1.08861310e-01 7.76696942e-02]\n [-2.25825414e-01 -5.70258713e-02 -3.76039027e-01 1.79168995e-01\n 1.47445902e-01 3.23833570e-01 7.04739542e-02 -1.72894901e-02\n 4.55763782e-01 -1.59646844e-02 -1.39319097e-01 2.30635009e-01\n -5.21671853e-02 -2.22182836e-01 6.01547604e-02 1.57761674e-01\n 2.32198669e-01 7.86705958e-02 -4.62698588e-01 -7.07320501e-02]\n [-3.00823578e-01 -1.14811681e-01 9.29402798e-03 -3.57556218e-01\n 2.02983748e-01 6.53211077e-02 -2.42618638e-01 4.60828889e-01\n 2.07326562e-01 1.71930064e-01 4.42244065e-01 1.62574530e-01\n 1.69937377e-01 1.18861040e-01 2.76431304e-01 -5.68723183e-02\n -1.34637229e-01 -8.95594479e-02 6.50825300e-02 6.40356737e-02]\n [-2.46023329e-01 -1.79230079e-01 2.21858535e-01 -5.10828878e-02\n -4.48206774e-01 -9.38793566e-02 3.25458411e-01 9.52830629e-02\n 2.75312133e-01 4.57579841e-02 -2.81961918e-02 1.58007012e-01\n -4.10119244e-01 -2.11953910e-03 -1.04218699e-01 -9.08160699e-02\n -2.39101852e-01 -2.35127320e-01 3.35089406e-02 -3.55349343e-01]\n [-2.61669373e-01 3.64656393e-01 -1.89118583e-02 1.36862564e-01\n -6.62392096e-02 -1.97580806e-01 -2.81503273e-01 2.52737514e-02\n -1.80778943e-01 1.98362589e-01 3.64192164e-01 -8.44211207e-02\n 3.71950112e-02 -8.69176025e-02 -3.94703790e-01 2.54771111e-02\n 2.16303951e-01 -1.91153685e-03 -2.20057520e-01 -4.28685084e-01]\n [-2.23168062e-01 -1.02302589e-01 -2.17298036e-01 -4.04623068e-01\n -1.67533753e-01 5.11956390e-02 -3.35707160e-01 -2.50558893e-01\n -2.65398844e-01 -2.67252148e-01 -1.29451617e-01 1.22528330e-01\n -2.71021005e-01 1.34100048e-01 2.39760511e-01 2.22843794e-01\n 2.56664752e-01 4.84260831e-02 1.75036380e-01 -2.27054779e-01]\n [-1.69587929e-01 -7.17453753e-02 3.47288621e-01 -2.25326526e-01\n -6.18381541e-02 2.47204903e-01 3.00155530e-01 -2.72758261e-01\n -1.03002592e-01 2.66758965e-01 -9.04246207e-02 -1.65456108e-01\n 4.25943469e-01 8.95510895e-02 2.04934227e-01 -1.30632943e-01\n 3.44132451e-01 -1.63983211e-01 -1.75293215e-01 -1.52998920e-01]\n [-2.64545236e-01 -2.06316440e-01 1.18501417e-02 2.33980287e-01\n 2.21216067e-01 -2.96631002e-01 2.47861272e-01 2.96757304e-02\n -2.29562973e-01 -2.43070484e-01 2.76727018e-01 -1.82579311e-01\n -3.05134352e-01 -1.12659016e-02 1.92041221e-01 -2.26355536e-02\n 2.91085772e-01 -3.09557368e-01 -1.49253726e-01 2.79579795e-01]\n [-1.64778167e-01 -6.33818807e-03 1.01497051e-02 2.22131564e-01\n 2.84125697e-01 2.02666094e-01 -1.70279068e-02 -3.02650559e-01\n 2.05613929e-01 2.56064148e-02 1.07790728e-01 -1.73773990e-01\n -1.84052454e-01 7.15103427e-01 -1.24724114e-01 -9.74593971e-02\n -1.18387897e-01 1.57512538e-01 9.11600251e-02 -8.65248518e-02]\n [-2.16519339e-01 4.09560634e-01 4.19753315e-02 1.41047945e-01\n 5.66471563e-02 -5.70319786e-02 7.25844628e-02 4.43130523e-02\n -1.69499731e-01 -2.15488213e-01 -2.54822887e-01 5.18726491e-01\n 2.52852577e-01 2.97503618e-01 -2.13647079e-02 1.18264174e-01\n -1.46021844e-01 -3.77583049e-01 -4.59379444e-02 8.92456244e-02]\n [-1.75196771e-01 1.05972121e-01 2.03749254e-02 -1.63793315e-01\n -3.40014342e-01 3.18425915e-01 -1.56507937e-01 2.88384573e-01\n -7.10158493e-02 2.91734049e-02 -1.92864194e-01 -3.60306247e-01\n -1.66770357e-01 1.66380387e-01 -2.69584227e-01 8.56371268e-02\n -3.19483749e-04 -8.37201346e-02 -2.72580947e-01 4.65233469e-01]\n [-1.73074698e-01 -5.90561287e-02 -2.86275563e-01 -1.21123344e-01\n 1.24357531e-01 -4.78369517e-01 -1.07695575e-01 -9.79472123e-02\n 2.61250727e-01 1.02464589e-01 -3.57043727e-01 -4.46462410e-01\n 2.13337216e-01 5.58436989e-03 2.93847145e-02 1.49558418e-01\n -2.14802709e-01 -2.76021666e-01 -4.83355369e-03 -8.80469935e-02]\n [-1.96998641e-01 2.43637862e-01 1.61237092e-01 -1.32502065e-01\n 2.91146808e-01 -1.57440389e-01 2.29449890e-01 2.99559224e-01\n -2.21910617e-01 8.33639099e-02 -3.21408480e-01 -7.53939162e-02\n -2.17796750e-01 -5.03646536e-03 2.23219181e-01 -2.71092998e-02\n -1.27516139e-01 5.21270809e-01 -1.77200186e-01 -1.54121566e-01]\n [-2.49747124e-01 -2.69832259e-01 4.63476131e-01 -4.44368597e-02\n 3.06813388e-01 9.46965709e-02 -1.01252642e-01 -2.07265602e-01\n -5.06488378e-02 -1.82165357e-02 -8.68339584e-03 6.50363425e-02\n 9.41718579e-03 -2.84873717e-01 -3.21565990e-01 4.95238781e-01\n -1.97690123e-01 4.89981054e-02 6.04359599e-02 9.73360744e-02]\n [-2.74478087e-01 3.58149231e-01 2.41821194e-01 -4.29968233e-02\n -7.34786894e-02 -1.04843127e-01 -7.53480126e-03 -3.02867780e-02\n 4.54985712e-01 -3.63276204e-01 -2.73257878e-02 -1.04840193e-01\n 8.87154936e-02 -1.62735998e-01 -3.04329513e-02 -1.62125436e-01\n 3.19019651e-01 1.89498539e-01 3.70662743e-01 1.77034737e-01]\n [-2.23940276e-01 -3.58423947e-01 -3.06916546e-01 2.05353984e-02\n -1.80752675e-01 -7.82550970e-02 3.58673432e-01 2.22421731e-01\n -1.56569556e-01 -1.83032254e-01 6.01136161e-02 1.22824837e-02\n 4.05648181e-01 1.59975367e-01 -3.18641860e-01 1.03439599e-01\n 5.76130935e-02 3.46588561e-01 1.47479087e-01 -2.24241449e-02]\n [-2.47919618e-01 -8.01553924e-02 2.89721655e-02 4.58330983e-01\n -1.04496576e-01 3.50211126e-01 -2.24406901e-01 1.20422937e-01\n -1.85531259e-01 -2.54620667e-01 -7.40961127e-02 -2.67255374e-01\n 1.59354378e-01 -2.19213534e-01 2.85981803e-01 -1.72120078e-01\n -2.98062657e-01 -4.79378515e-02 1.17559292e-01 -2.38303828e-01]\n [-2.39235543e-01 -9.38940215e-03 1.56061483e-02 1.65947763e-01\n -4.07068437e-01 -2.65049510e-01 -1.15638726e-01 -3.81285285e-01\n 1.06742045e-03 2.17536484e-01 1.47982051e-01 1.54227045e-01\n 8.50715175e-02 2.49790946e-03 2.90194554e-01 2.08710875e-03\n -2.36519152e-01 3.33614444e-01 -2.11341386e-01 3.51874392e-01]] \n [81.14251504 22.33329813 19.92933606 17.94414652 15.83212029 15.65264764\n 11.61541834 11.2170188 8.72086849 7.99833555 6.96043009 6.09387527\n 3.63729034 3.12147252 1.7207727 ] \n [[-0.31350191 -0.24372383 -0.27013956 -0.23613004 -0.25288055 -0.20053127\n -0.25793457 -0.26065103 -0.21085107 -0.24536732 -0.31606018 -0.24679151\n -0.21833357 -0.27657486 -0.29210497]\n [ 0.23878531 0.26469916 -0.08981383 0.24396718 -0.51849299 -0.18692987\n 0.1924538 0.28564061 0.00874485 0.04254616 0.21281435 -0.29247458\n -0.1155718 -0.48050606 0.07721237]\n [-0.33415327 0.23071191 -0.04277932 -0.09537178 0.08434827 0.30499434\n 0.0932124 0.19088353 0.42618107 -0.59888631 0.17371963 -0.29320517\n -0.11316316 0.09822258 -0.00548057]\n [ 0.21877162 0.23548388 0.40161433 0.32650863 0.09254078 -0.33312413\n -0.1339512 0.17850909 -0.15937526 -0.38718168 -0.31087109 -0.05903465\n 0.07936016 0.1931191 -0.37471508]\n [-0.031724 -0.26782042 0.51419471 -0.23200777 -0.02386439 0.13425985\n 0.22165485 0.22480197 0.14606787 0.19433309 -0.49900185 -0.35226083\n -0.00376818 -0.1347955 0.20100809]\n [-0.08810087 -0.16987254 0.17451895 0.03308471 0.30415517 0.05009905\n -0.49881647 0.38365681 -0.36544855 -0.12524468 0.26070935 0.00934138\n -0.18166355 -0.28353645 0.33182725]\n [ 0.27722177 0.23168339 -0.24074605 -0.12964543 -0.06024995 0.12437601\n -0.51425241 -0.05240606 0.04480352 0.07207315 -0.14568477 -0.43502768\n 0.45206464 0.16218758 0.24507271]\n [-0.40289921 0.30565658 -0.07012055 -0.1155291 0.13291834 -0.32517406\n -0.07364287 0.21482896 0.25780893 0.08981442 -0.20661549 0.34666549\n 0.41079543 -0.37871259 0.03777692]\n [ 0.05428482 -0.56400838 -0.24899899 0.57797391 0.14696497 -0.15083099\n -0.02016742 -0.01607608 0.39060744 -0.14065713 -0.09696082 -0.07690468\n 0.14982558 -0.07486426 0.14883826]\n [-0.38895357 -0.12183006 0.33010045 0.2980194 -0.48508519 0.29892305\n -0.32157564 0.00986814 0.07765515 0.1893101 0.18673209 0.05597537\n 0.23607912 0.08231073 -0.25600903]\n [ 0.37930455 -0.16816826 0.1568249 -0.22667472 0.18161261 0.24245224\n 0.02300469 -0.16286992 0.08818231 -0.17132089 0.23834561 0.01759871\n 0.35063808 -0.4957636 -0.41240277]\n [-0.29935043 0.0788777 0.257303 0.12903502 0.10102726 -0.21541665\n 0.2696949 -0.48783359 -0.29981289 -0.12513405 0.23001787 -0.29996943\n 0.3311563 -0.05792381 0.30418602]\n [ 0.14479589 -0.25095912 0.03500787 -0.26480817 -0.47538546 -0.06982907\n 0.09038322 0.12444344 -0.10768094 -0.47991833 -0.01011793 0.37787997\n 0.26873577 0.15269858 0.33141187]\n [-0.1644352 -0.10928829 -0.36647903 0.09867367 0.09962025 0.26163756\n 0.34122578 0.42729173 -0.48630419 0.03711768 -0.09096173 -0.13217603\n 0.34858568 0.07323263 -0.22156958]\n [ 0.03888073 0.26793681 -0.02841155 0.33642402 -0.03390119 0.53710828\n 0.01690946 -0.26946926 -0.14393739 -0.17425854 -0.42982195 0.2662602\n -0.09599482 -0.29528041 0.21689784]]\n\n\n\n```python\nprint(\"les valeurs singuli\u00e8res: \", S)\n```\n\n les valeurs singuli\u00e8res: [81.14251504 22.33329813 19.92933606 17.94414652 15.83212029 15.65264764\n 11.61541834 11.2170188 8.72086849 7.99833555 6.96043009 6.09387527\n 3.63729034 3.12147252 1.7207727 ]\n\n\nV\u00e9rification :\n\n\n```python\nimport scipy.linalg as la\nU@la.diagsvd(S,*(U.shape[1],Vh.shape[0]))@Vh\n```\n\n\n\n\n array([[ 8.00000000e+00, 5.00000000e+00, 7.00000000e+00,\n 7.00000000e+00, -1.47846521e-14, 2.00000000e+00,\n 5.26029070e-15, 6.00000000e+00, -2.73483985e-15,\n 8.00000000e+00, 6.00000000e+00, 3.00000000e+00,\n 5.00000000e+00, 2.00000000e+00, 5.00000000e+00],\n [ 2.97930644e-15, 2.00000000e+00, 8.00000000e+00,\n 3.00000000e+00, 6.00000000e+00, 3.00000000e+00,\n 5.00000000e+00, 2.00000000e+00, 3.00000000e+00,\n 4.00000000e+00, 2.00000000e+00, 6.00000000e+00,\n 3.00000000e+00, 9.00000000e+00, 3.00000000e+00],\n [ 8.00000000e+00, 1.13878268e-14, 2.00000000e+00,\n 5.87809901e-15, 1.00000000e+00, 4.00000000e+00,\n 3.00000000e+00, 2.00000000e+00, -2.80585842e-15,\n 9.00000000e+00, 6.00000000e+00, 1.00000000e+00,\n 1.00000000e+00, -1.08158142e-14, 8.00000000e+00],\n [ 8.00000000e+00, 1.13674421e-14, 8.00000000e+00,\n 8.00000000e+00, 7.00000000e+00, -3.36414170e-15,\n 1.00000000e+00, 5.00000000e+00, 8.41067758e-16,\n 7.00000000e+00, 3.00000000e+00, 5.00000000e+00,\n 5.00000000e+00, 4.00000000e+00, 8.00000000e+00],\n [ 3.00000000e+00, 2.00000000e+00, 7.00000000e+00,\n 3.00000000e+00, 8.00000000e+00, 7.00000000e+00,\n 8.00000000e+00, 6.00000000e+00, 8.00000000e+00,\n 8.00000000e+00, 8.00000000e+00, 9.00000000e+00,\n 8.00000000e+00, 2.00000000e+00, 9.00000000e+00],\n [ 4.00000000e+00, 7.00000000e+00, 2.43271144e-15,\n 6.00000000e+00, 8.00000000e+00, 5.00000000e+00,\n 2.00000000e+00, 2.00000000e+00, 7.00000000e+00,\n 2.00000000e+00, 9.00000000e+00, 5.00000000e+00,\n 7.00000000e+00, 9.00000000e+00, 5.00000000e+00],\n [ 9.00000000e+00, 8.00000000e+00, 7.00000000e+00,\n 7.00000000e+00, -1.47039110e-14, 2.00000000e+00,\n 9.00000000e+00, 7.00000000e+00, 5.00000000e+00,\n 5.00000000e+00, 9.00000000e+00, 5.00000000e+00,\n 4.00000000e+00, 2.00000000e+00, 2.00000000e+00],\n [ 5.00000000e+00, 2.00000000e+00, 2.00000000e+00,\n 2.00000000e+00, 6.00000000e+00, 5.00000000e+00,\n 7.00000000e+00, 1.00000000e+00, 9.03503125e-16,\n 9.00000000e+00, 9.00000000e+00, 8.00000000e+00,\n 5.85860544e-16, 5.00000000e+00, 7.00000000e+00],\n [ 2.00000000e+00, 3.00000000e+00, 2.00000000e+00,\n 1.00000000e+00, 3.00000000e+00, 9.00000000e+00,\n 5.78036559e-15, 5.00000000e+00, 4.00000000e+00,\n 4.12327313e-15, 8.00000000e+00, 1.00000000e+00,\n 2.00000000e+00, 6.00000000e+00, 7.00000000e+00],\n [ 9.00000000e+00, 7.00000000e+00, 8.00000000e+00,\n 2.00000000e+00, 8.00000000e+00, 5.00000000e+00,\n 6.00000000e+00, 4.00000000e+00, 6.00000000e+00,\n 5.00000000e+00, 1.00000000e+00, 4.00000000e+00,\n 7.00000000e+00, 9.00000000e+00, 3.00000000e+00],\n [ 6.00000000e+00, 1.11727574e-14, 7.00000000e+00,\n 5.00000000e+00, 5.00000000e+00, 4.00000000e+00,\n 3.00000000e+00, 7.00000000e+00, 1.00000000e+00,\n 2.00000000e+00, 2.00000000e+00, -5.03092605e-15,\n 2.00000000e+00, 4.00000000e+00, 3.00000000e+00],\n [ 7.00000000e+00, 9.00000000e+00, 5.00000000e+00,\n 6.00000000e+00, -1.34637851e-14, -4.20529909e-15,\n 8.00000000e+00, 7.00000000e+00, 2.00000000e+00,\n 3.00000000e+00, 6.00000000e+00, -7.24328196e-15,\n 4.00000000e+00, 2.00000000e+00, 7.00000000e+00],\n [ 2.00000000e+00, 5.00000000e+00, -2.70453764e-17,\n 4.00000000e+00, 4.00000000e+00, 2.00000000e+00,\n 1.00000000e+00, 7.00000000e+00, 2.00000000e+00,\n 4.00000000e+00, 9.00000000e+00, 7.00000000e+00,\n 1.00000000e+00, 1.00000000e+00, 5.00000000e+00],\n [ 6.00000000e+00, 7.72801445e-15, 2.00000000e+00,\n 4.00000000e+00, -8.71558572e-15, 2.00000000e+00,\n 7.00000000e+00, 1.00000000e+00, 5.00000000e+00,\n 9.00000000e+00, 2.19603744e-15, 6.00000000e+00,\n 3.00000000e+00, 7.00000000e+00, 3.00000000e+00],\n [ 3.00000000e+00, 8.00000000e+00, 4.00000000e+00,\n 2.00000000e+00, -8.98472074e-15, 4.00000000e+00,\n 6.00000000e+00, 7.00000000e+00, 7.00000000e+00,\n 6.00000000e+00, 3.00000000e+00, -3.34535897e-15,\n 4.00000000e+00, 2.00000000e+00, 7.00000000e+00],\n [ 2.00000000e+00, 3.00000000e+00, 9.00000000e+00,\n 1.00000000e+00, 9.00000000e+00, 9.00000000e+00,\n 6.00000000e+00, 6.00000000e+00, 8.00000000e+00,\n 3.69514014e-15, 6.00000000e+00, 2.00000000e+00,\n 2.00000000e+00, 9.00000000e+00, 7.00000000e+00],\n [ 9.00000000e+00, 7.00000000e+00, 2.00000000e+00,\n 8.00000000e+00, 3.00000000e+00, 3.00000000e+00,\n 9.00000000e+00, 8.00000000e+00, 9.00000000e+00,\n 2.00000000e+00, 9.00000000e+00, 2.00000000e+00,\n 3.00000000e+00, 3.00000000e+00, 8.00000000e+00],\n [ 7.00000000e+00, 4.00000000e+00, 3.00000000e+00,\n 1.00000000e+00, 8.00000000e+00, 2.00000000e+00,\n 1.00000000e+00, 1.00000000e+00, 1.00000000e+00,\n 7.00000000e+00, 3.00000000e+00, 9.00000000e+00,\n 9.00000000e+00, 9.00000000e+00, 5.00000000e+00],\n [ 7.00000000e+00, 7.00000000e+00, 9.00000000e+00,\n 6.00000000e+00, 9.00000000e+00, 1.00000000e+00,\n 2.00000000e+00, 9.00000000e+00, 1.00000000e+00,\n 1.98316377e-15, 5.00000000e+00, 8.00000000e+00,\n 2.00000000e+00, 6.00000000e+00, 4.00000000e+00],\n [ 8.00000000e+00, 6.00000000e+00, 4.00000000e+00,\n 8.00000000e+00, 3.00000000e+00, 4.00000000e+00,\n 6.00000000e+00, 1.00000000e+00, 3.00000000e+00,\n 2.00000000e+00, 9.00000000e+00, 6.00000000e+00,\n 4.00000000e+00, 9.00000000e+00, 1.00000000e+00]])\n\n\n\n\n```python\nnp.allclose(A,U@la.diagsvd(S,*(U.shape[1],Vh.shape[0]))@Vh)# np.dot(U[:, :len(S)] * S, Vh))\n```\n\n\n\n\n True\n\n\n\n\n```python\nr=5\n\n```\n\nCalculons la matrice $A_r$ :\n\nOn a $A_r = \\sum_{i=1}^r [u_1, \\dots, u_r]\\mathrm{diag}(\\sigma_1, \\dots, \\sigma_{r}) [v_1, \\dots, v_r]^T = \\sum_{i=1}^r\\sigma_i u_iv_i^T$ :\n\n\n```python\n#Ur\nUr=U[np.ix_(list(range(len(U))),list(range(r)))]\n#Sr\nSr=S[:r]\n#Vr\nVhr=Vh[np.ix_(list(range(r)),list(range(Vh.shape[1])))]\nSr\n```\n\n\n\n\n array([81.14251504, 22.33329813, 19.92933606, 17.94414652, 15.83212029])\n\n\n\n\n```python\n#Vr exemple\nVhr=Vh[np.ix_(list(range(r)),list(range(Vh.shape[1])))]\nVhr\n```\n\n\n\n\n array([[-0.31350191, -0.24372383, -0.27013956, -0.23613004, -0.25288055,\n -0.20053127, -0.25793457, -0.26065103, -0.21085107, -0.24536732,\n -0.31606018, -0.24679151, -0.21833357, -0.27657486, -0.29210497],\n [ 0.23878531, 0.26469916, -0.08981383, 0.24396718, -0.51849299,\n -0.18692987, 0.1924538 , 0.28564061, 0.00874485, 0.04254616,\n 0.21281435, -0.29247458, -0.1155718 , -0.48050606, 0.07721237],\n [-0.33415327, 0.23071191, -0.04277932, -0.09537178, 0.08434827,\n 0.30499434, 0.0932124 , 0.19088353, 0.42618107, -0.59888631,\n 0.17371963, -0.29320517, -0.11316316, 0.09822258, -0.00548057],\n [ 0.21877162, 0.23548388, 0.40161433, 0.32650863, 0.09254078,\n -0.33312413, -0.1339512 , 0.17850909, -0.15937526, -0.38718168,\n -0.31087109, -0.05903465, 0.07936016, 0.1931191 , -0.37471508],\n [-0.031724 , -0.26782042, 0.51419471, -0.23200777, -0.02386439,\n 0.13425985, 0.22165485, 0.22480197, 0.14606787, 0.19433309,\n -0.49900185, -0.35226083, -0.00376818, -0.1347955 , 0.20100809]])\n\n\n\nOn d\u00e9finit la d\u00e9composition en une fonction :\n\n\n```python\ndef rSVD_1(A):\n U,S,Vh = np.linalg.svd(A) \n #Ur\n Ur=U[np.ix_(list(range(len(U))),list(range(r)))]\n #Sr\n Sr=S[:r]\n #Vr\n Vhr=Vh[np.ix_(list(range(r)),list(range(Vh.shape[1])))]\n return Ur,Sr,Vhr\n \nUr,Sr,Vhr=rSVD_1(A)\nAr=Ur@la.diagsvd(Sr,*(Ur.shape[1],Vhr.shape[0]))@Vhr\nAr\n```\n\n\n\n\n array([[ 8.95974685, 4.64650981, 5.34831695, 6.49441131, 1.27977213,\n -0.04106605, 4.59565044, 5.2554752 , 0.84610491, 7.01853742,\n 4.67582335, 4.1287213 , 3.91821448, 1.99707888, 4.60482914],\n [ 3.28272648, 1.61932382, 6.52781938, 1.85869933, 7.63727795,\n 4.09364879, 2.7562393 , 2.69448837, 3.18932714, 3.06751539,\n 1.60929908, 4.90478764, 4.23052459, 7.65717373, 3.61823309],\n [ 4.93679228, 0.32348869, 1.65450161, 1.42473104, 0.12373696,\n 2.78623937, 4.80253997, 2.41645655, 1.71914168, 9.71694979,\n 4.65106531, 3.22549369, 2.34755221, -0.50720302, 7.02318192],\n [ 8.57402756, 2.53178219, 7.87656098, 5.23902404, 4.90382365,\n 0.86932971, 3.86951549, 4.08056095, 0.48719814, 8.13895897,\n 2.05425192, 6.07993759, 5.2423606 , 5.25005042, 4.55978806],\n [ 5.4726843 , 2.94165002, 5.89203813, 2.28012079, 6.84737421,\n 7.9994919 , 7.39162526, 5.24243701, 6.69527372, 8.87798676,\n 7.59232947, 5.966431 , 5.08352333, 6.32909938, 9.98132263],\n [ 3.84973845, 6.51061785, 1.54624756, 4.66266668, 7.58112736,\n 5.45260711, 3.34082755, 3.14515471, 5.16814532, 1.0558863 ,\n 10.05164673, 7.35477785, 4.2748377 , 8.65840717, 4.41509175],\n [ 9.29758464, 8.10280391, 5.46750037, 8.08161804, 1.3671976 ,\n 1.66156387, 6.4473874 , 7.99122889, 3.84289836, 4.62730913,\n 8.13827622, 3.19304197, 3.93605772, 2.53776567, 5.70194912],\n [ 5.07427237, 1.8101711 , 1.00243806, 2.37628107, 4.79001184,\n 4.80015606, 4.21205884, 1.34836389, 2.72230735, 9.23527234,\n 8.06548542, 7.76996187, 4.14158487, 4.6361782 , 7.32438713],\n [ 0.76517494, 3.83658684, 1.43790247, 1.10530725, 4.54361177,\n 6.38539899, 4.21075486, 3.50837627, 6.33855961, 0.53847634,\n 6.95607722, 2.41890865, 2.08920903, 4.60676198, 5.17623476],\n [ 6.35783885, 4.11727243, 9.68959675, 4.48038432, 8.14225278,\n 4.30950183, 4.88592969, 5.86083861, 4.42888334, 3.98454927,\n 2.79206184, 5.09439998, 5.51251958, 8.5128847 , 5.04394291],\n [ 4.81960052, 3.00180175, 7.52979266, 3.36116189, 3.73311336,\n 2.04548285, 3.90347882, 5.20597637, 2.9259514 , 2.48440371,\n 0.74711618, 1.46193248, 3.21207759, 3.94924643, 3.30415302],\n [ 7.93774557, 7.25193893, 5.36639451, 6.91861053, -0.0163558 ,\n 1.34571917, 6.22971434, 8.00516497, 3.86855562, 3.39334551,\n 6.41040236, 0.95002613, 2.88158346, 0.91406883, 5.06550926],\n [ 4.41393936, 4.9345021 , -0.33803866, 4.18475529, 2.25853171,\n 2.78852604, 3.36060456, 2.72412425, 2.87331003, 3.43748421,\n 8.66718338, 4.76688766, 2.57136891, 2.99245125, 4.35233113],\n [ 5.45628231, 0.71828533, 4.29576665, 2.37205467, 3.50588336,\n 2.31103768, 3.56426671, 2.24934654, 1.15209549, 8.03070276,\n 2.86006321, 4.95920087, 3.68436753, 3.27237269, 5.24185057],\n [ 4.57046353, 4.28315997, 5.10728113, 2.94979904, 1.16205333,\n 4.57932366, 6.80998473, 6.94589962, 5.83973015, 4.04561164,\n 5.20740976, -0.0720133 , 2.29150944, 1.04161256, 6.88928493],\n [ 1.49913057, 3.98627176, 7.79796944, 1.0467368 , 8.83861141,\n 8.92523135, 6.11178053, 6.27356628, 8.99336734, 0.43693187,\n 4.55110977, 2.39147502, 3.99417123, 8.59896681, 6.67880492],\n [ 7.1499432 , 8.78717005, 4.18391121, 6.76881652, 1.84773554,\n 4.54171506, 7.57876389, 8.61060615, 6.77293899, 2.99151441,\n 10.39903455, 2.19938558, 3.33605478, 2.79760924, 7.15216895],\n [ 6.00053735, 1.75187608, 4.56585429, 3.70543793, 8.33200367,\n 2.76769761, 1.89258104, 0.70471419, 0.67786263, 7.08239401,\n 4.29047881, 9.6054066 , 5.62468957, 8.72812687, 4.01001263],\n [ 7.53800585, 6.94210046, 8.02276271, 7.32754264, 6.86459706,\n 1.58293172, 3.4297571 , 5.93856029, 2.91965987, 1.00822665,\n 4.34629967, 5.41619303, 5.19264706, 8.29197751, 2.32049927],\n [ 6.78766233, 7.17471815, 3.13158827, 6.97048374, 5.47328849,\n 2.16955958, 3.16831462, 4.14204261, 2.80783669, 2.16255001,\n 8.43505542, 6.85534017, 4.48797113, 6.94402132, 3.24122537]])\n\n\n\nV\u00e9rification des normes :\n\n\n```python\n#norme spectrale\nnp.linalg.norm(A-Ar,ord=2)\n```\n\n\n\n\n 15.652647638313617\n\n\n\n\n```python\nS[r]\n```\n\n\n\n\n 15.652647638313615\n\n\n\n\n```python\nnp.isclose(S[r],np.linalg.norm(A-Ar,ord=2))\n```\n\n\n\n\n True\n\n\n\n\n```python\n#norme de Frobenus\nsqrt(sum(S[r:]**2))\n```\n\n\n\n\n 27.51888777204114\n\n\n\n\n```python\nlinalg.norm(A-Ar, ord='fro')\n```\n\n\n\n\n 27.51888777204115\n\n\n\n\n```python\nnp.isclose(sqrt(sum(S[r:]**2)),linalg.norm(A-Ar, ord='fro'))\n```\n\n\n\n\n True\n\n\n\nOn peut aussi v\u00e9rifer avec la ```TruncatedSVD``` de Sklearn :\n\n\n```python\nfrom sklearn.decomposition import TruncatedSVD\n\nsvd = TruncatedSVD(n_components=r, n_iter=10, random_state=42)\nsvd.fit(A)\nAr_sk = svd.transform(A)\n\nprint(svd.singular_values_)\n\n```\n\n [81.14251504 22.33329813 19.92933606 17.94414652 15.83212029]\n\n\n\n```python\nS[:r]\n```\n\n\n\n\n array([81.14251504, 22.33329813, 19.92933606, 17.94414652, 15.83212029])\n\n\n\n**Question 2.** On consid\u00e8re d\u00e9sormais une version randomis\u00e9e de la SVD. \n1. On g\u00e9n\u00e8re une matrice gaussienne $\\Omega$ de taille $n \\times k$ dont les entr\u00e9es sont i.i.d. et suivent la loi $\\cal{N}(0,1/k)$.\n2. On en d\u00e9duit $Y = A \\Omega$. \n3. On calcule la d\u00e9composition $QR$ de $Y$.\n4. On forme la matrice $B =Q^T A$ de taille $k \\times n$.\n5. On calcule la SVD de la matrice $B$ not\u00e9e $\\tilde USV^T$\n6. Alors la SVD randomis\u00e9e de la matrice $A$ est donn\u00e9e par \n$$\\hat A_k = U S V^T$$ avec $U =Q \\tilde U \\in \\mathbb{R}^{m\\times k}$, $S \\in \\mathbb{R}^{k \\times k}$ et $V \\in \\mathbb{R}^{n\\times k}$.\n\n*Remarque : Ici SVD randomis\u00e9e retourne une d\u00e9composition tronqu\u00e9e \u00e0 $k$ termes.*\n\n*Question 2.a* \n\nD\u00e9finir une fonction `rSVD(A,Omega)` qui prend en argument une matrice $A \\in \\mathbb{R}^{m \\times n}$ et une matrice al\u00e9atoire gaussienne $\\Omega \\in \\mathbb{R}^{n \\times k}$ et retourne $U,S,V^T$. Pour la factorisation $QR$ de la matrice $Y$ utiliser `np.linalg.qr()`.\n\n\n\n\n\n```python\n\n#definition de la matrice A\nn=15\nm=20\nentries = np.random.randint(0, 9+1, size= n*m) \nA= np.reshape(entries, (m, n))\n```\n\n\n```python\nk=10\nentries = np.random.normal(0, 1/k, size= n*k)\nOmega= np.reshape(entries, (n, k))\n#Omega\n\n\ndef rSVD(A,Omega):\n Y=A@Omega \n Q,R = np.linalg.qr(Y)\n B = np.transpose(Q)@A\n U_ , S , Vh = np.linalg.svd(B)\n U = Q @ U_ \n return U,S,Vh\n```\n\n\n```python\nrSVD(A,Omega)\n```\n\n\n\n\n (array([[-0.24295666, 0.20911214, -0.24293062, 0.05276083, -0.31596681,\n 0.30214748, 0.16522719, 0.08847456, -0.1399176 , -0.07546655],\n [-0.16848859, 0.18517951, -0.047516 , -0.15466959, -0.35063234,\n -0.05480719, 0.16554326, -0.27024564, 0.04138071, -0.04251701],\n [-0.17548436, 0.0928677 , 0.26968528, -0.34463618, 0.48983575,\n 0.26704732, 0.06048475, -0.22841078, -0.07647665, -0.33760688],\n [-0.2968976 , 0.08205571, -0.27648518, -0.36144564, 0.19105236,\n 0.01605749, -0.00316294, 0.19875134, -0.08575054, 0.56527614],\n [-0.20902202, -0.37712235, 0.09696073, -0.23110745, -0.08425176,\n -0.05900037, -0.41447744, 0.19981562, -0.08020377, 0.08899209],\n [-0.22436721, -0.36921733, -0.11799808, -0.11087539, 0.22197215,\n 0.29083374, 0.02287996, -0.02094831, 0.3066016 , -0.02220439],\n [-0.18689386, 0.40479101, 0.01683913, -0.04302421, -0.09224616,\n 0.08114749, -0.09034164, 0.11882648, -0.14240549, -0.37934202],\n [-0.14230039, 0.18873909, -0.13036354, 0.17969913, 0.0569992 ,\n -0.32739414, -0.16790684, -0.3217911 , -0.17486032, 0.17275734],\n [-0.26179171, -0.41174869, -0.0999043 , 0.36893015, 0.04010438,\n -0.21916585, 0.03279012, -0.18012159, 0.09510267, -0.16676408],\n [-0.19856871, 0.10808812, -0.3159143 , -0.26095169, 0.15532888,\n -0.3942888 , 0.3479804 , -0.0676543 , 0.38489553, -0.09952426],\n [-0.23791391, -0.1419586 , 0.21077888, -0.03649549, -0.39590134,\n 0.24418668, 0.07248369, 0.25287732, 0.21735436, -0.0467456 ],\n [-0.20260179, 0.0623463 , 0.46208712, -0.21030492, -0.18286413,\n -0.25465885, -0.07148665, -0.01896655, 0.26976586, 0.04581718],\n [-0.27791119, 0.23899929, -0.00132358, 0.08518556, -0.1267616 ,\n 0.15156236, -0.48517319, -0.34975753, 0.19666836, 0.23101747],\n [-0.24919191, -0.1731812 , -0.22774647, -0.03864886, -0.13778046,\n -0.19280983, 0.08252423, 0.27239413, -0.42686884, -0.1133825 ],\n [-0.22184961, -0.23142576, -0.02511646, -0.11693985, -0.12572896,\n 0.14963216, 0.19555557, -0.44102734, -0.3291617 , -0.06942797],\n [-0.14573509, -0.03348083, 0.22664695, 0.30406075, 0.1611378 ,\n 0.2730162 , 0.2752955 , -0.12671524, -0.1953835 , 0.39118034],\n [-0.24041097, 0.11870321, 0.45774606, 0.02628102, 0.18982602,\n -0.31267192, 0.09608713, 0.20470138, -0.29104032, 0.03551051],\n [-0.26887731, -0.09125861, 0.11879997, 0.28772346, -0.01943043,\n -0.16910973, 0.00130477, -0.04768803, 0.07929625, -0.12891901],\n [-0.20256003, 0.20545566, 0.05199864, 0.34608409, 0.09772623,\n 0.09496305, 0.30579857, 0.26993282, 0.28011614, 0.14067005],\n [-0.24063663, 0.13008079, -0.22582298, 0.23693143, 0.30651923,\n 0.10892196, -0.37609545, 0.20969682, 0.01164535, -0.27034729]]),\n array([79.86161996, 21.2792864 , 20.15592485, 16.08372274, 15.21831086,\n 13.77627942, 10.99120537, 9.23170503, 8.49802838, 6.75765844]),\n array([[-2.65586335e-01, -2.60742118e-01, -2.43437639e-01,\n -2.70152489e-01, -2.68031708e-01, -3.12649838e-01,\n -2.16589485e-01, -2.37874220e-01, -2.58504949e-01,\n -3.03146493e-01, -2.78807116e-01, -2.49214041e-01,\n -2.35036355e-01, -2.61347673e-01, -1.84496020e-01],\n [ 4.51585180e-01, 1.53111506e-01, -2.86639331e-01,\n 4.08242765e-01, -1.91582538e-01, 2.65808809e-02,\n -2.09351449e-01, 1.43661834e-01, 2.43800476e-01,\n 1.41816454e-01, -3.37589760e-01, -2.85631795e-01,\n -3.82027890e-01, 2.37237028e-02, -1.77538051e-02],\n [-4.24801713e-03, 3.36226889e-01, 2.83071004e-01,\n -1.31092351e-02, -2.65128536e-01, 1.58219200e-01,\n -2.34589533e-01, 8.74959305e-02, 2.76464787e-02,\n -3.68543557e-01, -3.44548064e-01, -9.24442895e-02,\n 4.32375863e-01, -1.97988465e-01, 3.98273561e-01],\n [ 2.04932466e-01, -5.74864681e-02, -1.78802392e-01,\n 2.46604718e-02, -8.31185952e-02, -1.79099895e-01,\n -4.66195758e-01, 4.03900853e-01, -3.64347481e-01,\n 4.92857314e-02, 7.04795637e-03, 5.50349572e-01,\n 1.93270834e-01, 2.09440823e-02, -1.63569351e-01],\n [-3.76303982e-02, 7.43564971e-03, -1.56917762e-01,\n 4.79772036e-01, 1.72259041e-01, -3.13401574e-02,\n -9.66617804e-02, -5.27860888e-01, -4.51221475e-01,\n 2.66271753e-01, -7.55386426e-02, -1.23483762e-01,\n 3.19551335e-01, 3.66289630e-02, 1.61795382e-01],\n [-4.98475759e-01, 1.62442330e-01, -3.98960238e-01,\n 3.74587432e-01, -9.70254520e-02, 5.96436922e-02,\n 2.99769224e-01, 1.33547963e-01, 1.04984954e-01,\n -9.42607158e-02, -5.67142626e-02, 3.91083208e-01,\n -1.21587566e-01, -3.08189649e-01, 1.38374618e-01],\n [ 3.85925629e-01, -8.29449880e-02, 2.82174568e-01,\n 1.73280043e-01, 1.21656563e-01, -5.61618134e-01,\n 4.42359949e-01, -2.18216209e-02, -1.29007294e-01,\n -2.49825909e-01, -1.96423866e-01, 1.90561401e-01,\n -1.13353551e-01, -1.19855670e-01, 1.64295659e-01],\n [ 6.40072418e-02, -5.78707192e-02, -1.83660840e-01,\n -1.90951954e-01, 6.05172578e-01, 3.27366512e-01,\n -7.08640048e-02, -1.40827820e-01, 5.80814322e-02,\n -2.50058452e-01, -5.10923301e-01, 2.52215334e-01,\n -9.13773042e-02, 1.58651743e-01, -5.01717871e-03],\n [-7.82972234e-02, -3.89554050e-02, 2.15704099e-01,\n 1.48342081e-01, 3.56093603e-01, -2.08886857e-01,\n -2.80994808e-01, -4.87461567e-02, 4.22489473e-01,\n 1.66572079e-01, -3.76738393e-02, -3.63256088e-03,\n 1.78972911e-01, -5.65636394e-01, -3.35254810e-01],\n [ 2.42212938e-01, 7.12328600e-01, -1.81603123e-01,\n -2.37169834e-01, -5.24658171e-02, -9.37278090e-02,\n 1.81481794e-02, -4.20387487e-01, 8.33410039e-03,\n -9.83748779e-02, 2.13951716e-01, 1.82601021e-01,\n -4.55597584e-02, -9.81680089e-02, -2.35628693e-01],\n [-1.56646829e-02, 2.87819942e-01, -5.63354632e-03,\n -2.80623128e-01, 2.97066056e-01, 5.67260108e-02,\n 1.27780038e-01, 3.65983235e-01, -4.38685104e-01,\n 3.64754343e-01, -9.78888149e-02, -2.50685490e-01,\n -1.77448817e-01, -3.79880844e-01, 1.50606606e-01],\n [-2.50780191e-02, -3.97796524e-02, 5.01214867e-01,\n 5.11869697e-02, -3.25690371e-01, 3.58916032e-01,\n 2.23560408e-02, -2.40680008e-01, -1.76333441e-01,\n 2.84209165e-01, -2.62532457e-01, 3.39157102e-01,\n -3.07572275e-01, -1.20509381e-01, -2.13748103e-01],\n [ 2.09090247e-01, -1.53990774e-01, -2.81365933e-01,\n -2.00874088e-01, -2.39388084e-01, 8.64320180e-02,\n 4.65863633e-01, 2.69016572e-02, 8.91568137e-02,\n 2.30117352e-01, -3.20401939e-01, -3.25596549e-02,\n 5.19885937e-01, -1.01071749e-01, -2.92239280e-01],\n [-4.05895278e-01, 3.31339442e-01, 1.36863231e-01,\n -6.04977547e-02, 1.60315386e-04, -4.40929695e-01,\n -5.70151407e-03, 8.68031713e-02, 1.09922037e-01,\n 2.75404604e-01, -3.90280884e-01, 8.05682559e-03,\n 3.80132435e-02, 4.96429792e-01, -1.13504097e-01],\n [ 8.03004082e-02, -1.69393703e-01, -1.43791203e-01,\n -3.44898807e-01, -1.13787312e-01, -1.75964322e-01,\n -1.54287756e-01, -2.36588148e-01, 2.88872640e-01,\n 4.06156093e-01, -1.76753425e-02, 2.53778599e-01,\n -6.70260677e-02, -1.02794474e-01, 6.11024955e-01]]))\n\n\n\n*Question 2.b* \n\nReprendre les diff\u00e9rents points de la Question 1. pour la fonction la SVD randomis\u00e9e `rSVD()`.\nOn consid\u00e8rera une matrice al\u00e9atoire de taille $k=l+r$ avec $l=4$. \n\n\n\n```python\nl = 4\nn=15\nk=r + l\nentries = np.random.randint(0, 9+1, size= n*m) \nA= np.reshape(entries, (m, n))\n\nentries = np.random.normal(0, 1/k, size= n*k)\nOmega= np.reshape(entries, (n, k))\n\n```\n\n\n```python\n# Valeurs sing. S\n#par la m\u00e9thode de la question 1 \nU_1,S_1,Vh_1 = rSVD_1(A)\nprint(S_1)\nU_2,S_2,Vh_2 = rSVD(A,Omega)\nprint(S_2)\nU_3,S_3,Vh_3 = np.linalg.svd(A) \nprint(S_3)\n```\n\n [78.2028406 21.80831211 19.88334755 17.5464853 16.91267418]\n [77.68442211 21.57704051 19.31971216 16.79758938 14.67452894 12.65156491\n 11.51871842 10.66549451 6.51861085]\n [78.2028406 21.80831211 19.88334755 17.5464853 16.91267418 14.07579029\n 12.90014707 11.56092787 10.34131357 8.70682192 7.56925519 6.75349791\n 5.45240487 4.01244576 3.12538132]\n\n\nNous avons extrait avec cette m\u00e9thode les valeurs singuli\u00e8res sans utiliser la m\u00e9thode svd impl\u00e9ment\u00e9e, avec une pr\u00e9cision acceptable. Il s'agit donc d'un moyen rapide de r\u00e9duire la dimensionnalit\u00e9.\n\n**Question 3** \n\nOn va essayer d'am\u00e9liorer la SVD randomis\u00e9e en effectuant des puissances it\u00e9r\u00e9es de la matrice $A$. Ainsi l'\u00e9tape 2. de l'algorithme devient :\n\n2. Calculer $Y = A \\Omega$\\\n Pour $i=1,\\dots,p$ faire \\\n     $Y = A (A^T Y)$\\\n Fin pour\n\n*Question 3.a*\n\nEcrire une fonction `power_iteration(A,Omega,p=3)` renvoyant une matrice $Y$ obtenue par des puissances it\u00e9r\u00e9es de $A$. Cette fonction prendra en argument $A, \\Omega$ et $p=3$ un param\u00e8tre correspondant au nombre de puissances it\u00e9r\u00e9s. Ecrire une nouvelle fonction `rSVD2(A,Omega)` o\u00f9 la matrice $Y$ est obtenue par puissances it\u00e9r\u00e9es.\n\n\n\n\n```python\ndef power_iteration(A,Omega,p=3):\n Y= A@Omega\n for i in range(p):\n Y = A @ (np.transpose(A)@Y)\n return Y\n```\n\n*Question 3.b* Reprendre les diff\u00e9rents points de la Question 2.c. Comparer les r\u00e9sultats obtenus avec `np.linalg.svd()`et `rSVD()`.\n\n\n```python\ndef rSVD_power(A,Omega,p):\n Y=power_iteration(A,Omega,p=3)\n Q,R = np.linalg.qr(Y)\n B = np.transpose(Q)@A\n U_ , S , Vh = np.linalg.svd(B)\n U = Q @ U_ \n return U,S,Vh\n```\n\n\n```python\nU_1,S_1,Vh_1 = rSVD_power(A,Omega,3)\nprint(\"rSVD power :\" , S_1)\nU_2,S_2,Vh_2 = rSVD(A,Omega)\nprint(\"rSVD :\" , S_2)\nU_3,S_3,Vh_3 = np.linalg.svd(A) \nprint(\"SVD :\" , S_3)\n```\n\n rSVD power : [78.2028406 21.80830967 19.8832032 17.54415591 16.9085791 14.05367681\n 12.89877685 11.55815385 9.7562184 ]\n rSVD : [77.68442211 21.57704051 19.31971216 16.79758938 14.67452894 12.65156491\n 11.51871842 10.66549451 6.51861085]\n SVD : [78.2028406 21.80831211 19.88334755 17.5464853 16.91267418 14.07579029\n 12.90014707 11.56092787 10.34131357 8.70682192 7.56925519 6.75349791\n 5.45240487 4.01244576 3.12538132]\n\n\nNous pouvons voir clairement que rSVD_power est bien meilleur que rSVD .\nOn peut comparer les distances : \n\n\n```python\nlinalg.norm(A-U_1@la.diagsvd(S_1,*(U_1.shape[1],Vh_1.shape[0]))@Vh_1, ord='fro')\n\n```\n\n\n\n\n 15.716715113498228\n\n\n\n\n```python\nlinalg.norm(A-U_2@la.diagsvd(S_2,*(U_2.shape[1],Vh_2.shape[0]))@Vh_2, ord='fro')\n\n```\n\n\n\n\n 24.491259024169523\n\n\n\n**Question 4.**\n\nOn s'int\u00e9resse dans cette derni\u00e8re question \u00e0 la compression d'une image de taille $256 \\times 256$ : `lena256x256.png`\n\n\n\n\n- Charger l'image en utilisant `imageio.imread()`. Celle-ci est alors stock\u00e9e sous forme d'une matrice $A$ de taille $256\\times 256$.\n- Afin de compresser cette image, il suffit d'appliquer une SVD tronqu\u00e9e de rang $r$ \u00e0 la matrice $A$. Calculer les approximations de rang $r=50$ pour les trois m\u00e9thodes \u00e9tudi\u00e9es pr\u00e9c\u00e9demment. Afficher les images compress\u00e9es obtenues.\n- Comparer les erreurs associ\u00e9es (en norme $\\|\\cdot\\|_2$ et $\\|\\cdot\\|_F$) et les temps d'ex\u00e9cution de chacune d'elle.\n\nCommmenter.\n\n\n\n\n```python\nimport os\nfrom google.colab import drive\n\ndrive.mount(\"/content/gdrive\")\n```\n\n Mounted at /content/gdrive\n\n\n\n```python\nimage_path = \"/content/gdrive/My Drive/Google Collab/lena256x256.png\"\n```\n\n\n```python\nimport imageio as iio\n\nim = iio.imread(image_path)\n```\n\n\n```python\nr = 50\n```\n\n\n```python\nnp.shape(im)\n```\n\n\n\n\n (256, 256)\n\n\n\nM\u00e9thode 1 (rSVD): \n\n\n```python\nimport time\nn=256\nstart = time.time()\n\nk=r\nentries = np.random.normal(0, 1/k, size= n*k)\nOmega= np.reshape(entries, (n, k))\nU,S,Vh = rSVD(im,Omega)\nA_k = U@la.diagsvd(S,*(U.shape[1],Vh.shape[0]))@Vh\n\nend = time.time()\nprint(end - start)\n```\n\n 0.018517017364501953\n\n\nM\u00e9thode 2 avec k=r+l\n\n\n\n```python\n\nstart = time.time()\nk= r + l\nentries = np.random.normal(0, 1/k, size= n*k)\nOmega= np.reshape(entries, (n, k))\nU_2,S_2,Vh_2 = rSVD(im,Omega)\nA_k_2 = U_2@la.diagsvd(S_2,*(U_2.shape[1],Vh_2.shape[0]))@Vh_2\nend = time.time()\nprint(end - start)\n```\n\n 0.02601337432861328\n\n\nM\u00e9thode 3 (power):\n\n\n```python\nimport scipy.linalg as la\n\nstart = time.time()\np =3\nU_3,S_3,Vh_3 = rSVD_power(im,Omega,p)\n\nA_k_3 = U_3@la.diagsvd(S_3,*(U_3.shape[1],Vh_3.shape[0]))@Vh_3\nend = time.time()\nprint(end - start)\n```\n\n 0.021928071975708008\n\n\nSVD originale :\n\n\n```python\nstart = time.time()\nU_4,S_4,Vh_4 = np.linalg.svd(im) \nA_k_4 = U_4@la.diagsvd(S_4,*(U_4.shape[1],Vh_4.shape[0]))@Vh_4\nend = time.time()\nprint(end - start)\n```\n\n 0.051738739013671875\n\n\nComme pr\u00e9vu, moins nous utilisons d'informations et plus la dimension est petite, plus l'algorithme est rapide, m\u00eame si nous avons gagn\u00e9 beaucoup de temps par rapport aux informations perdues.\n\nComparaison :\n\n\n```python\nimport matplotlib.pyplot as plt\nimgplot = plt.imshow(im)\n```\n\n\n```python\nimgplot = plt.imshow(A_k_4)\n```\n\nLa SVD compl\u00e8te donne les m\u00eames r\u00e9sultats que l'original, comme pr\u00e9vu. \n\n\n```python\nimgplot = plt.imshow(A_k)\n```\n\nrSVD, est rapide et donne un bon r\u00e9sultat\n\n\n```python\nimgplot = plt.imshow(A_k_2)\n```\n\n\u00e9galement bon r\u00e9sultat pour k>r\n\n\n\n```python\nimgplot = plt.imshow(A_k_3)\n```\n\nVoici \u00e9galement une bonne image presque identique \u00e0 l'original.\nNous concluons que les m\u00e9thodes rSVD pourraient nous donner un gain \u00e9norme en termes de temps de calcul (et aussi un peu de gain de m\u00e9moire).\n", "meta": {"hexsha": "c79971d1ae349d5ef01420a032d7eb5ae2b1f32b", "size": 746198, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "DM_MNP_2021_2022.ipynb", "max_stars_repo_name": "RandomAnass/Jupyter-books", "max_stars_repo_head_hexsha": "90898dac796f69c204d0282c27fb8fb343063275", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "DM_MNP_2021_2022.ipynb", "max_issues_repo_name": "RandomAnass/Jupyter-books", "max_issues_repo_head_hexsha": "90898dac796f69c204d0282c27fb8fb343063275", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "DM_MNP_2021_2022.ipynb", "max_forks_repo_name": "RandomAnass/Jupyter-books", "max_forks_repo_head_hexsha": "90898dac796f69c204d0282c27fb8fb343063275", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 278.9525233645, "max_line_length": 124310, "alphanum_fraction": 0.8995159462, "converted": true, "num_tokens": 27585, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804196836383, "lm_q2_score": 0.7217432003123989, "lm_q1q2_score": 0.4466005603931184}} {"text": "_Lambda School Data Science \u2014\u00a0Linear Models_\n\n# Understanding Linear Regression\n\n#### Objectives\n- understand how ordinary least squares regression minimizes the sum of squared errors\n- understand how linear algebra can solve ordinary least squares regression\n- get and interpret coefficients of a linear model\n- visualize a line of best fit in 2D, and hyperplane in 3D\n- use regression metrics: MSE, RMSE, MAE, R^2\n\n#### Extra Links\n- [Statistics 101: Simple Linear Regression](https://www.youtube.com/watch?v=ZkjP5RJLQF4) (20 minute video)\n- [_An Introduction to Statistical Learning_](http://www-bcf.usc.edu/~gareth/ISL/ISLR%20Seventh%20Printing.pdf), Chapter 3.1, Simple Linear Regression, & 3.2, Multiple Linear Regression\n- Priceonomics, [The Discovery of Statistical Regression](https://priceonomics.com/the-discovery-of-statistical-regression/)\n- Priceonomics, [Why the Father of Modern Statistics Didn\u2019t Believe Smoking Caused Cancer](https://priceonomics.com/why-the-father-of-modern-statistics-didnt-believe/)\n- Harvard Business Review, [When to Act on a Correlation, and When Not To](https://hbr.org/2014/03/when-to-act-on-a-correlation-and-when-not-to)\n- [xkcd 552: Correlation](https://www.explainxkcd.com/wiki/index.php/552:_Correlation)\n- [xkcd 1725: Linear Regression](https://www.explainxkcd.com/wiki/index.php/1725:_Linear_Regression)\n\n## What is Linear Regression?\n\n[Linear Regression](https://en.wikipedia.org/wiki/Linear_regression) is a statistical model that seeks to describe the relationship between some y variable and one or more x variables. \n\nIn the simplest case, linear regression seeks to fit a straight line through a cloud of points. This line is referred to as the \"regression line\" or \"line of best fit.\" This line tries to summarize the relationship between our X and Y in a way that enables us to use the equation for that line to make predictions.\n\n\n```python\n%matplotlib inline\nimport seaborn as sns\nx = [10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5]\ny = [ 8, 6, 7, 8, 8, 9, 7, 4, 10, 4, 5]\nsns.regplot(x, y);\n```\n\n\n\n### Synonyms for \"y variable\" \n- Dependent Variable\n- Response Variable\n- Outcome Variable \n- Predicted Variable\n- Measured Variable\n- Explained Variable\n- Label\n- Target\n\n### Synonyms for \"x variable\"\n- Independent Variable\n- Explanatory Variable\n- Regressor\n- Covariate\n- Feature\n\n## Simple Linear Regresion\n\n#### Making Predictions\n\nSay that we were trying to create a model that captured the relationship between temperature outside and ice cream sales. In Machine Learning our goal is often different that of other flavors of Linear Regression Analysis, because we're trying to fit a model to this data with the intention of making **predictions** on new data (in the future) that we don't have yet.\n\n#### What are we trying to predict?\n\nSo if we had measured ice cream sales and the temprature outside on 11 different days, at the end of our modeling **what would be the thing that we would want to predict? - Ice Cream Sales or Temperature?**\n\nWe would probably want to be measuring temperature with the intention of using that to **forecast** ice cream sales. If we were able to successfully forecast ice cream sales from temperature, this might help us know beforehand how much ice cream to make or how many cones to buy or on which days to open our store, etc. Being able to make predictions accurately has a lot of business implications. This is why making accurate predictions is so valuable (And in large part is why data scientists are paid so well).\n\n#### y variable intuition\n\nWe want the thing that we're trying to predict to serve as our **y** variable. This is why it's sometimes called the \"predicted variable.\" We call it the \"dependent\" variable because our prediction for how much ice cream we're going to sell \"depends\" on the temperature outside. \n\n#### x variable intuition\n\nAll other variables that we use to predict our y variable (we're going to start off just using one) we call our **x** variables. These are called our \"independent\" variables because they don't *depend* on y, they \"explain\" y. Hence they are also referred to as our \"explanatory\" variables.\n\n## Example: Predict presidential election voting\n\n#### Douglas Hibbs, [Background Information on the \u2018Bread and Peace\u2019 Model of Voting in Postwar US Presidential Elections](https://douglas-hibbs.com/background-information-on-bread-and-peace-voting-in-us-presidential-elections/)\n\n> Aggregate two-party vote shares going to candidates of the party holding the presidency during the postwar era are well explained by just two fundamental determinants:\n\n> (1) Positively by weighted-average growth of per capita real disposable personal income over the term. \n> (2) Negatively by cumulative US military fatalities (scaled to population) owing to unprovoked, hostile deployments of American armed forces in foreign wars.\n\n#### Data sources\n- 1952-2012: Douglas Hibbs, [2014 lecture at Deakin University Melbourne](http://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf), Slide 40\n- 2016, Vote Share: [The American Presidency Project](https://www.presidency.ucsb.edu/statistics/elections)\n- 2016, Recent Growth in Personal Incomes: [The 2016 election economy: the \"Bread and Peace\" model final forecast](https://angrybearblog.com/2016/11/the-2016-election-economy-the-bread-and-peace-model-final-forecast.html)\n- 2016, US Military Fatalities: Assumption that Afghanistan War fatalities in 2012-16 occured at the same rate as 2008-12\n\n\n```python\nimport pandas as pd\n\ncolumns = ['Year','Incumbent Party Candidate','Other Candidate','Incumbent Party Vote Share']\n\ndata = [[1952,\"Stevenson\",\"Eisenhower\",44.6],\n [1956,\"Eisenhower\",\"Stevenson\",57.76],\n [1960,\"Nixon\",\"Kennedy\",49.91],\n [1964,\"Johnson\",\"Goldwater\",61.34],\n [1968,\"Humphrey\",\"Nixon\",49.60],\n [1972,\"Nixon\",\"McGovern\",61.79],\n [1976,\"Ford\",\"Carter\",48.95],\n [1980,\"Carter\",\"Reagan\",44.70],\n [1984,\"Reagan\",\"Mondale\",59.17],\n [1988,\"Bush, Sr.\",\"Dukakis\",53.94],\n [1992,\"Bush, Sr.\",\"Clinton\",46.55],\n [1996,\"Clinton\",\"Dole\",54.74],\n [2000,\"Gore\",\"Bush, Jr.\",50.27],\n [2004,\"Bush, Jr.\",\"Kerry\",51.24],\n [2008,\"McCain\",\"Obama\",46.32],\n [2012,\"Obama\",\"Romney\",52.00], \n [2016,\"Clinton\",\"Trump\",48.2]]\n \nvotes = pd.DataFrame(data=data, columns=columns)\n```\n\n\n```python\ncolumns = ['Year','Average Recent Growth in Personal Incomes']\n\ndata = [[1952,2.40],\n [1956,2.89],\n [1960, .85],\n [1964,4.21],\n [1968,3.02],\n [1972,3.62],\n [1976,1.08],\n [1980,-.39],\n [1984,3.86],\n [1988,2.27],\n [1992, .38],\n [1996,1.04],\n [2000,2.36],\n [2004,1.72],\n [2008, .10],\n [2012, .95], \n [2016, .10]]\n \ngrowth = pd.DataFrame(data=data, columns=columns)\n```\n\n\n```python\n\"\"\"\nFatalities denotes the cumulative number of American military\nfatalities per millions of US population the in Korea, Vietnam,\nIraq and Afghanistan wars during the presidential terms\npreceding the 1952, 1964, 1968, 1976 and 2004, 2008 and\n2012 elections.\n\nhttp://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf\n\"\"\"\n\ncolumns = ['Year','US Military Fatalities per Million']\n\ndata = [[1952,190],\n [1956, 0],\n [1960, 0],\n [1964, 1],\n [1968,146],\n [1972, 0],\n [1976, 2],\n [1980, 0],\n [1984, 0],\n [1988, 0],\n [1992, 0],\n [1996, 0],\n [2000, 0],\n [2004, 4],\n [2008, 14],\n [2012, 5], \n [2016, 5]]\n \ndeaths = pd.DataFrame(data=data, columns=columns)\n```\n\n\n```python\ndf = votes.merge(growth).merge(deaths)\n```\n\n### Plot univariate correlations\n[Seaborn tutorial: Visualizing linear relationships](https://seaborn.pydata.org/tutorial/regression.html)\n\n\n```python\n%matplotlib inline\nimport seaborn as sns\n\ntarget = 'Incumbent Party Vote Share'\nfeatures = ['Average Recent Growth in Personal Incomes', \n 'US Military Fatalities per Million']\n\nfor feature in features:\n sns.lmplot(x=feature, y=target, data=df)\n```\n\nWe can see from the scatterplot that these data points seem to follow a somewhat linear relationship for the \"Average Recent Growth in Personal Incomes\" feature. This means that we could probably summarize their relationship well by fitting a line of best fit to these points. Lets do it.\n\n\n## The Equation for a Line\n\nA common equation for a line is:\n\n\\begin{align}\ny = mx + b\n\\end{align}\n\nWhere $m$ is the slope of our line and $b$ is the y-intercept. \n\nIf we want to plot a line through our cloud of points we figure out what these two values should be. Linear Regression seeks to **estimate** the slope and intercept values that describe a line that best fits the data points.\n\n## The Anatomy of Linear Regression\n\n- Intercept: The $b$ value in our line equation $y=mx+b$\n- Slope: The $m$ value in our line equation $y=mx+b$. These two values together define our regression line.\n- $\\hat{y}$ : A prediction\n- Line of Best Fit (Regression Line)\n- Predicted (fitted) Values: Points on our regression line\n- Observed Values: Points from our dataset\n- Error: The distance between predicted and observed values.\n\n\nOrdinary Least Squares Regression is a way to solve for $m$ and $b$.\n\nLet's start by seeing what would happen if we just guessed and checked some values for $m$ and $b$. \n\nWhat's the line of \"best\" fit look like? What's the error?\n\n\n\n```python\n# TODO\n```\n\n## R Squared: $R^2$\n\nOne final attribute of linear regressions that we're going to talk about today is a measure of goodness of fit known as $R^2$ or R-squared. $R^2$ is a statistical measure of how close the data are fitted to our regression line. A helpful interpretation for the $R^2$ is the percentage of the dependent variable that is explained by the model.\n\nIn other words, the $R^2$ is the percentage of y that is explained by the x variables included in the model. For this reason the $R^2$ is also known as the \"coefficient of determination,\" because it explains how much of y is explained (or determined) by our x varaibles. We won't go into the calculation of $R^2$ today, just know that a higher $R^2$ percentage is nearly always better and indicates a model that fits the data more closely. \n\n## Residual Error \n\nThe residual error is the distance between points in our dataset and our regression line.\n\n\n```python\ndef regression_residuals(df, feature, target, m, b):\n x = df[feature]\n y = df[target]\n y_pred = m*x + b\n \n plt.scatter(x, y, label='y_true')\n plt.plot(x, y_pred, label='y_pred')\n plt.legend()\n \n # Plot residual errors\n for x, y1, y2 in zip(x, y, y_pred):\n plt.plot((x, x), (y1, y2), color='grey')\n \n mae = mean_absolute_error(y, y_pred) \n r2 = r2_score(y, y_pred)\n print('Mean Absolute Error:', mae)\n print('R^2:', r2)\n \nregression_residuals(df, feature='Average Recent Growth in Personal Incomes', \n target='Incumbent Party Vote Share', m=3, b=46)\n```\n\n\n```python\nfrom ipywidgets import interact, fixed\n\ninteract(regression_residuals, \n df=fixed(df), \n feature=fixed('Average Recent Growth in Personal Incomes'), \n target=fixed('Incumbent Party Vote Share'), \n m=(-10,10,0.5), \n b=(40,60,0.5));\n```\n\n## More Formal Notation\n\n\n\nWe have talked about a line of regression being represented like a regular line $y=mx+b$ but as we get to more complicated versions we're going to need to extend this equation. So lets establish the proper terminology.\n\n**X** - Independent Variable, predictor variable, explanatory variable, regressor, covariate\n\n**Y** - Response variable, predicted variable, measured vairable, explained variable, outcome variable\n\n$\\beta_0$ - \"Beta Naught\" or \"Beta Zero\", the intercept value. This is how much of y would exist if X were zero. This is sometimes represented by the letter \"a\" but I hate that. So it's \"Beta 0\" during my lecture.\n\n$\\beta_1$ - \"Beta One\" The primary coefficient of interest. This values is the slope of the line that is estimated by \"minimizing the sum of the squared errors/residuals\" - We'll get to that. \n\n$\\epsilon$ - \"Epsilon\" The \"error term\", random noise, things outside of our model that affect y.\n\n## Minimizing the Sum of the Squared Error\n\nThe most common method of estimating our $\\beta$ parameters is what's known as \"Ordinary Least Squares\" (OLS). (There are different methods of arriving at a line of best fit). OLS estimates the parameters that minimize the squared distance between each point in our dataset and our line of best fit. \n\n\\begin{align}\nSSE = \\sum(y_i - \\hat{y})^2\n\\end{align}\n\n\n\n```python\nfrom matplotlib.patches import Rectangle\nimport numpy as np\nfrom sklearn.metrics import mean_squared_error\n\ndef regression_squared_errors(df, feature, target, m, b):\n x = df[feature]\n y = df[target]\n y_pred = m*x + b\n \n fig = plt.figure(figsize=(7,7))\n ax = plt.axes()\n ax.scatter(x, y, label='y_true')\n ax.plot(x, y_pred, label='y_pred')\n ax.legend()\n \n # Plot square errors\n xmin, xmax = ax.get_xlim()\n ymin, ymax = ax.get_ylim()\n scale = (xmax-xmin)/(ymax-ymin)\n for x, y1, y2 in zip(x, y, y_pred):\n bottom_left = (x, min(y1, y2))\n height = abs(y1 - y2)\n width = height * scale\n ax.add_patch(Rectangle(xy=bottom_left, width=width, height=height, alpha=0.1))\n \n mse = mean_squared_error(y, y_pred)\n rmse = np.sqrt(mse)\n mae = mean_absolute_error(y, y_pred)\n r2 = r2_score(y, y_pred)\n print('Mean Squared Error:', mse)\n print('Root Mean Squared Error:', rmse)\n print('Mean Absolute Error:', mae)\n print('R^2:', r2)\n \nregression_squared_errors(df, feature='Average Recent Growth in Personal Incomes', \n target='Incumbent Party Vote Share', m=3, b=46)\n```\n\n\n```python\ninteract(regression_squared_errors, \n df=fixed(df), \n feature=fixed('Average Recent Growth in Personal Incomes'), \n target=fixed('Incumbent Party Vote Share'), \n m=(-10,10,0.5), \n b=(40,60,0.5));\n```\n\n## Hypotheses\n\n\n```python\nb = 46\nms = np.arange(-10,10,0.5)\nsses = []\nfeature = 'Average Recent Growth in Personal Incomes'\n\nfor m in ms:\n predictions = m * df[feature] + b\n errors = predictions - df[target]\n square_errors = errors ** 2\n sse = square_errors.sum()\n sses.append(sse)\n \nhypotheses = pd.DataFrame({'Slope': ms})\nhypotheses['Intercept'] = b\nhypotheses['Sum of Square Errors'] = sses\n\nhypotheses\n```\n\n\n```python\nhypotheses.plot(x='Slope', y='Sum of Square Errors', \n title=f'Intercept={b}');\n```\n\n## Scikit-learn\n\n#### Jake VanderPlas, [Python Data Science Handbook, Chapter 5.2, Introducing Scikit-Learn](https://jakevdp.github.io/PythonDataScienceHandbook/05.02-introducing-scikit-learn.html), Scikit-Learn's Estimator API\n\n> Most commonly, the steps in using the Scikit-Learn estimator API are as follows (we will step through a handful of detailed examples in the sections that follow).\n\n> 1. Choose a class of model by importing the appropriate estimator class from Scikit-Learn. \n> 2. Choose model hyperparameters by instantiating this class with desired values. \n> 3. Arrange data into a features matrix and target vector following the discussion above.\n> 4. Fit the model to your data by calling the `fit()` method of the model instance.\n> 5. Apply the Model to new data: For supervised learning, often we predict labels for unknown data using the `predict()` method.\n\n\n```python\n# TODO\n```\n\n## Linear Algebra!\n\nThe same result that is found by minimizing the sum of the squared errors can be also found through a linear algebra process known as the \"Least Squares Solution:\"\n\n\\begin{align}\n\\hat{\\beta} = (X^{T}X)^{-1}X^{T}y\n\\end{align}\n\nBefore we can work with this equation in its linear algebra form we have to understand how to set up the matrices that are involved in this equation. \n\n### The $\\beta$ vector\n\nThe $\\beta$ vector represents all the parameters that we are trying to estimate, our $y$ vector and $X$ matrix values are full of data from our dataset. The $\\beta$ vector holds the variables that we are solving for: $\\beta_0$ and $\\beta_1$\n\nNow that we have all of the necessary parts we can set them up in the following equation:\n\n\\begin{align}\ny = X \\beta + \\epsilon\n\\end{align}\n\nSince our $\\epsilon$ value represents **random** error we can assume that it will equal zero on average.\n\n\\begin{align}\ny = X \\beta\n\\end{align}\n\nThe objective now is to isolate the $\\beta$ matrix. We can do this by pre-multiplying both sides by \"X transpose\" $X^{T}$.\n\n\\begin{align}\nX^{T}y = X^{T}X \\beta\n\\end{align}\n\nSince anything times its transpose will result in a square matrix, if that matrix is then an invertible matrix, then we should be able to multiply both sides by its inverse to remove it from the right hand side. (We'll talk tomorrow about situations that could lead to $X^{T}X$ not being invertible.)\n\n\\begin{align}\n(X^{T}X)^{-1}X^{T}y = (X^{T}X)^{-1}X^{T}X \\beta\n\\end{align}\n\nSince any matrix multiplied by its inverse results in the identity matrix, and anything multiplied by the identity matrix is itself, we are left with only $\\beta$ on the right hand side:\n\n\\begin{align}\n(X^{T}X)^{-1}X^{T}y = \\hat{\\beta}\n\\end{align}\n\nWe will now call it \"beta hat\" $\\hat{\\beta}$ because it now represents our estimated values for $\\beta_0$ and $\\beta_1$\n\n### Lets calculate our $\\beta$ coefficients with numpy!\n\n\n```python\nfrom statsmodels.api import add_constant\n\nX = add_constant(df[feature].values)\nprint('X')\nprint(X)\n\ny = df[target].values[:, np.newaxis]\nprint('y')\nprint(y)\n\nX_transpose = X.T\nprint('X Transpose')\nprint(X_transpose)\n\nX_transpose_X = X_transpose @ X\nprint('X Transpose X')\nprint(X_transpose_X)\n\nX_transpose_X_inverse = np.linalg.inv(X_transpose_X)\nprint('X Transpose X Inverse')\nprint(X_transpose_X_inverse)\n\nX_transpose_y = X_transpose @ y\nprint('X Transpose y')\nprint(X_transpose_y)\n\nbeta_hat = X_transpose_X_inverse @ X_transpose_y\nprint('Beta Hat')\nprint(beta_hat)\n```\n\n# Multiple Regression\n\nSimple or bivariate linear regression involves a single $x$ variable and a single $y$ variable. However, we can have many $x$ variables. A linear regression model that involves multiple x variables is known as **Multiple** Regression (NOT MULTIVARIATE).\n\n\\begin{align}\ny = \\beta_0 + \\beta_1 X_1 + \\beta_2 X_2 + ... + \\beta_n X_n + \\epsilon\n\\end{align}\n\n\n```python\n# TODO\n```\n\n## Visualize hyperplane of best fit in 3D\n\n\n```python\n# https://stackoverflow.com/a/47230966\n# Plotly notebook mode with google colaboratory\n# You need to define this function\n# And call it in each offline plotting cell\n\ndef configure_plotly_browser_state():\n import IPython\n display(IPython.core.display.HTML('''\n \n \n '''))\n```\n\n\n```python\nimport itertools\nimport plotly.graph_objs as go\nfrom plotly.offline import init_notebook_mode, iplot\ninit_notebook_mode(connected=True)\n\ndef viz3D(fitted_model, X, features, target='', num=100):\n \"\"\"\n Visualize model predictions in 3D, for regression model fit on 2 features\n \n Parameters\n ----------\n fitted_model : scikit-learn model, already fitted\n X : pandas dataframe, which was used to fit model\n features : list of strings, column names of the 2 features\n target : string, name of target\n num : int, number of grid points for each feature\n \n References\n ----------\n https://plot.ly/python/3d-charts/\n \"\"\"\n feature1, feature2 = features\n min1, max1 = X[feature1].min(), X[feature1].max()\n min2, max2 = X[feature2].min(), X[feature2].max()\n x1 = np.linspace(min1, max1, num)\n x2 = np.linspace(min2, max2, num)\n combos = list(itertools.product(x1, x2))\n Z = fitted_model.predict(combos).reshape(num, num)\n \n configure_plotly_browser_state()\n data = [go.Surface(x=x1, y=x2, z=Z)]\n layout = go.Layout(\n scene={'xaxis': {'title': feature1, 'range': [min1,max1], 'showticklabels': True}, \n 'yaxis': {'title': feature2, 'range': [min2,max2], 'showticklabels': True}, \n 'zaxis': {'title': target, 'showticklabels': True}}, \n )\n fig = go.Figure(data=data, layout=layout)\n iplot(fig)\n```\n\n\n```python\n# TODO\n```\n\n## Dimensionality in Linear Regression\n\nMuliple Regression is simply an extension of the bivariate case. The reason why we see the bivariate case demonstrated so often is simply because it's easier to graph and all of the intuition from the bivariate case is the same as we keep on adding explanatory variables.\n\nAs we increase the number of $x$ values in our model we are simply fitting a n-1-dimensional plane to an n-dimensional cloud of points within an n-dimensional hypercube. \n\n## Interpreting Coefficients\n\nOne of Linear Regression's strengths is that the parameters of the model (coefficients) are readily interpretable and useful. Not only do they describe the relationship between x and y but they put a number on just how much x is associated with y. We should be careful to not speak about this relationshiop in terms of causality because these coefficients are in fact correlative measures. We would need a host of additional techniques in order to estimate a causal effect using linear regression (econometrics).\n\n\\begin{align}\n\\hat{\\beta} = \\frac{Cov(x,y)}{Var(y)}\n\\end{align}\n\n## Why is Linear Regression so Important?\n\n### Popularity \n\nLinear Regression is an extremely popular technique that every data scientist **needs** to understand. It's not the most advanced technique and there are supervised learning techniques that will obtain a higher accuracy, but where it lacks in accuracy it makes up for it in interpretability and simplicity.\n\n### Interpretability\n\nFew other models possess coefficients that are so directly linked to their variables with a such a clear interpretation. Tomorrow we're going to learn about ways to make them even easier to interpret.\n\n### Simplicity\n\nA linear regression model can be communicated just by writing out its equation. It's kind of incredible that such high dimensional relationships can be described from just a linear combination of variables and coefficients. \n\n# Assignment\n- Continue to predict New York City apartment rents. This is your last assignment with this dataset.\n- You may select any number of features. You are encouraged to engineer new features.\n- Get and plot your model's coefficients.\n- Report your Root Mean Squared Error, Mean Absolute Error, and R^2 Score, for your Train and Test sets. Share your scores with your cohort on Slack!\n- Fit a model with 2 features, and visualize the plane of best fit in 3D.\n- Commit your notebook to your fork of the repo.\n\n## Stretch Goals\n\nStudy more about Linear Regression. Here are two helpful links. If you find more links, share your favorites with your cohort on Slack.\n\n1. Watch this 20 minute video that just hit 1 million views: Brandon Foltz, Statistics 101: Simple Linear Regression (https://www.youtube.com/watch?v=ZkjP5RJLQF4)\n2. Skim _An Introduction to Statistical Learning_, Chapter 3.1, Simple Linear Regression, & Chapter 3.2, Multiple Linear Regression (http://www-bcf.usc.edu/~gareth/ISL/ISLR%20Seventh%20Printing.pdf)\n\nIn your 3D visualization, can you include the actual datapoints, like in [this notebook](https://nbviewer.jupyter.org/urls/s3.amazonaws.com/datarobotblog/notebooks/multiple_regression_in_python.ipynb)? Can you also include the residual lines from the datapoints to the plane of the best fit, like in _An Introduction to Statistical Learning?_ This would be hard to do, but awesome!\n\n\nCan you get creative with feature engineering? Share with your cohort on Slack. We mentioned some feature ideas at the end of last lesson, but didn't demonstrate how to engineer them. So here are some example solutions:\n\n```python\n# Does apartment have a non-empty description?\ndf['description'] = df['description'].str.strip().fillna('')\ndf['has_description'] = df['description'] != ''\n\n# How long is the description?\ndf['description_length'] = df['description'].str.len()\n\n# How many total perks does each apartment have?\nperk_cols = ['elevator', 'cats_allowed', 'hardwood_floors', 'dogs_allowed',\n 'doorman', 'dishwasher', 'no_fee', 'laundry_in_building',\n 'fitness_center', 'pre-war', 'laundry_in_unit', 'roof_deck',\n 'outdoor_space', 'dining_room', 'high_speed_internet', 'balcony',\n 'swimming_pool', 'new_construction', 'exclusive', 'terrace', \n 'loft', 'garden_patio', 'common_outdoor_space', \n 'wheelchair_access']\ndf['perk_count'] = df[perk_cols].sum(axis=1)\n\n# Are pets allowed?\ndf['pets_allowed'] = (df['cats_allowed']==1) | (df['dogs_allowed']==1)\n```\n\n", "meta": {"hexsha": "9f6ef6cde0f93478c165232330df12a26ee9d3ac", "size": 55815, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "module3-understanding-linear-regression/understanding_linear_regression.ipynb", "max_stars_repo_name": "unburied/DS-Unit-2-Linear-Models", "max_stars_repo_head_hexsha": "2c5c6d41d2b79b894859a44d7c2abdf41883efaa", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "module3-understanding-linear-regression/understanding_linear_regression.ipynb", "max_issues_repo_name": "unburied/DS-Unit-2-Linear-Models", "max_issues_repo_head_hexsha": "2c5c6d41d2b79b894859a44d7c2abdf41883efaa", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-02-11T04:57:37.000Z", "max_issues_repo_issues_event_max_datetime": "2020-02-11T04:57:37.000Z", "max_forks_repo_path": "module3-understanding-linear-regression/understanding_linear_regression.ipynb", "max_forks_repo_name": "unburied/DS-Unit-2-Linear-Models", "max_forks_repo_head_hexsha": "2c5c6d41d2b79b894859a44d7c2abdf41883efaa", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2019-12-10T03:23:43.000Z", "max_forks_repo_forks_event_max_datetime": "2020-12-02T13:55:28.000Z", "avg_line_length": 54.7742885182, "max_line_length": 14902, "alphanum_fraction": 0.6439129266, "converted": true, "num_tokens": 6490, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.7090191276365463, "lm_q1q2_score": 0.4465222427495256}} {"text": "### Functions\n\n\n```\n# Detect Intersection #\nimport math\nfrom sklearn.preprocessing import MinMaxScaler\nfrom sklearn.cluster import KMeans\nfrom sympy import Line\n\n\ndef line(p1, p2):\n A = (p1[1] - p2[1])\n B = (p2[0] - p1[0])\n C = (p1[0]*p2[1] - p2[0]*p1[1])\n return A, B, -C\n\ndef intersection2(L1, L2):\n D = L1[0] * L2[1] - L1[1] * L2[0]\n Dx = L1[2] * L2[1] - L1[1] * L2[2]\n Dy = L1[0] * L2[2] - L1[2] * L2[0]\n if D != 0:\n x = Dx / D\n y = Dy / D\n return x,y\n else:\n return False\n\n\ndef regression(img, x, y, color=(255, 0, 0)):\n\n y_at_border = np.array([0, img.shape[0]])\n p = np.polyfit(y, x, deg=1)\n x_at_border = np.poly1d(p)(y_at_border)\n\n cv2.line(img, (int(x_at_border[0]), int(y_at_border[0])), (int(x_at_border[1]), int(y_at_border[1])), color, 2)\n\n return x_at_border, y_at_border\n\n\ndef drawLines(img, lines, color=(255,0,0)):\n \"\"\"\n Draw lines on an image\n \"\"\"\n\n centroids = list()\n r_xs = list()\n r_ys = list()\n for line_ in lines:\n for rho,theta in line_:\n a = np.cos(theta)\n b = np.sin(theta)\n x0 = a*rho\n y0 = b*rho\n x1 = int(x0 + 1000*(-b))\n y1 = int(y0 + 1000*(a))\n x2 = int(x0 - 1000*(-b))\n y2 = int(y0 - 1000*(a))\n\n slope = (y1 - y0) / float(x1 - x0)\n angle = math.degrees(math.atan(slope))\n if abs(angle) > 80:\n # print(img.shape[1])\n h_layout = line((0, 0), (img.shape[1], 0))\n h_layout_lower = line((0, img.shape[0]), (img.shape[1], img.shape[0]))\n r = intersection2(h_layout, line((x1, y1), (x2, y2)))\n r_lower = intersection2(h_layout_lower, line((x1, y1), (x2, y2)))\n # cv2.line(img, (int(x1), int(y1)), (int(x2), int(y2)), color, 2)\n cv2.line(img, (int(r[0]), int(r[1])), (int(r_lower[0]), int(r_lower[1])), color, 2)\n center_p = (int((r[0] + r_lower[0]) / 2), int((r[1] + r_lower[1])/ 2))\n centroids.append(center_p)\n\n r_xs.append((r[0], r_lower[0]))\n r_ys.append((r[1], r_lower[1]))\n cv2.circle(img, center_p, 10, (255, 0, 255), -1)\n\n cv2.line(img, (int(0), int(0)), (int(0), int(img.shape[0])), color, 2)\n cv2.line(img, (int(img.shape[1]), int(0)), (int(img.shape[1]), int(img.shape[0])), color, 2)\n cv2.circle(img, (0, int(img.shape[0] / 2)), 10, (255, 0, 255), -1)\n cv2.circle(img, (img.shape[1], int(img.shape[0] / 2)), 10, (255, 0, 255), -1)\n centroids.append((0, int(img.shape[0] / 2)))\n centroids.append((img.shape[1], int(img.shape[0] / 2)))\n\n return r_xs, r_ys, centroids \n\nfrom scipy.spatial import distance as sci_dist\n\ndef order_points(pts):\n\n xSorted = pts[np.argsort(pts[:, 0]), :]\n leftMost = xSorted[:2, :]\n rightMost = xSorted[2:, :]\n leftMost = leftMost[np.argsort(leftMost[:, 1]), :]\n (tl, bl) = leftMost\n D = sci_dist.cdist(tl[np.newaxis], rightMost, \"euclidean\")[0]\n (br, tr) = rightMost[np.argsort(D)[::-1], :]\n\n return np.array([tl, tr, br, bl], dtype=\"float32\")\n\n\n```\n\n\n```\nfrom PIL import Image\nimport os\nimport matplotlib.pyplot as plt\nimport numpy as np\nfrom XiaohuLuVPDetection.lu_vp_detect.vp_detection import VPDetection\nimport time\n# import os\nimport cv2\n# import pylab as pl\nfrom skimage import morphology as mp\nimport sys\n\n\nlength_thresh = 50\nprincipal_point = None\nfocal_length = 1300 # 1102.79\nseed = 1300\nvpd = VPDetection(length_thresh, principal_point, focal_length, seed)\n\n# org_path = 'datasets/lsun/images/' # -> original image\n# layout_path = 'drn_d_105_024_val_ms/images/' # -> layout image (result image frome above model file)\n# refer_path = '../refer_data/wall/myxkehu1kfzggursepnk0tfiyps8zbs5umvzv8d92r6hhxejgawebwsufssgov5q-.jpg'\n# val_image_list = os.listdir(layout_path)\n\n# Refering for one image data #\ndef refering(org_path, layout_path, refer_path):\n\n # print(image)\n img = Image.open(layout_path + image)\n # print(type(img))\n img_np = np.invert(np.asarray(img))\n # print(img_np.max(), img_np.min())\n ret, thr = cv2.threshold(img_np, 254, 255, cv2.THRESH_BINARY_INV)\n\n org = Image.open(org_path + image)\n org_np = np.asarray(org)\n\n vps = vpd.find_vps(org_path + image)\n vl_img = vpd.create_debug_VP_image()\n\n # print(image)\n # plt.subplot(131)\n # plt.imshow(org)\n # plt.show()\n img_size = (org_np.shape[1], org_np.shape[0])\n \n scale_factor = 6\n print('scale_factor :', scale_factor)\n\n refer = Image.open(refer_path)\n refer = np.asarray(refer)\n\n # refer_size\uc640 img_size\uac00 \ub3d9\uc77c\ud558\uac70\ub098 refer_size\uac00 \uc791\uc740 \uacbd\uc6b0\ub97c \uace0\ub824\ud574\uc57c\ud55c\ub2e4. #\n refer = np.tile(refer, (scale_factor, scale_factor, 1))\n size_ratio = math.floor(min((refer.shape[0] / (org_np.shape[0] * 1.5)), (refer.shape[1] / (org_np.shape[1] * 1.5))))\n refer = Image.fromarray(refer).resize((int(refer.shape[1] / size_ratio), int(refer.shape[0] / size_ratio)))\n\n tun = thr\n skl = mp.medial_axis(tun).astype(np.uint8) * 255\n rho = 1\n theta = np.pi/180\n thresh = 50\n lines = cv2.HoughLines(skl, rho, theta, thresh)\n\n # print(lines)\n # plt.imshow(skl)\n # plt.show()\n\n # Draw all Hough lines in red\n img_with_all_lines = np.copy(skl)\n img_with_all_lines = cv2.cvtColor(img_with_all_lines, cv2.COLOR_GRAY2RGB)\n # plt.imshow(img_with_all_lines)\n # plt.show()\n org_np = org_np.astype(np.uint8)\n org_np2 = org_np.copy()\n\n start = time.time()\n\n r_x_list, r_y_list, centroids_list = drawLines(org_np2, lines)\n drawLines(img_with_all_lines, lines)\n\n vps = vpd.find_vps(skl)\n vl_img, vl_list = vpd.create_debug_VP_image()\n\n centroids_data = np.array(centroids_list)[:, [0]]\n # print('len(r_x_list), len(r_y_list) :', len(r_x_list), len(r_y_list))\n print('centroids_data.shape :', centroids_data.shape)\n mms = MinMaxScaler()\n cen_data =mms.fit_transform(centroids_data)\n # print('cen_data :', cen_data)\n K = range(2, 6)\n s_dist = list()\n for k in K:\n if cen_data.shape[0] < k:\n break\n km = KMeans(n_clusters=k)\n km = km.fit(cen_data)\n inertia = km.inertia_\n s_dist.append(inertia)\n \n # if inertia <= 0:\n # print('vertical line number =', k)\n # break\n mms2 = MinMaxScaler()\n mms_s_dist = mms2.fit_transform(np.array(s_dist).reshape(-1, 1))\n k_thresh = 0.1\n # plt.plot(range(len(mms_s_dist)), mms_s_dist)\n # plt.show()\n\n if cen_data.shape[0] > 2:\n for i in range(len(mms_s_dist)):\n if mms_s_dist[i] < k_thresh:\n k = i + 2\n break\n else:\n k = 2\n\n # k Confirmation : Comparing k.cluster_centers_ dist #\n while True: \n\n km = KMeans(n_clusters=k)\n km = km.fit(cen_data)\n if k <= 2:\n break\n\n cluster_centroids = km.cluster_centers_\n # print('cluster_centroids :', cluster_centroids)\n error_exist = False\n for i in range(len(cluster_centroids) - 1):\n for j in range(i + 1, len(cluster_centroids)):\n if abs(cluster_centroids[i] - cluster_centroids[j]) < 0.05:\n error_exist = True\n if error_exist:\n k -= 1\n else:\n break\n\n print('k is', k)\n\n predict_cen = km.predict(cen_data)\n print(predict_cen)\n\n keys = list(range(k))\n for rm in predict_cen[-2:]:\n keys.remove(rm)\n print('keys :', keys)\n\n skl_rgb = cv2.cvtColor(skl, cv2.COLOR_GRAY2RGB)\n skl_copy = np.copy(skl_rgb)\n\n # black_plane = np.zeros(skl_rgb.shape).astype(np.uint8)\n black_plane2 = np.zeros(skl_rgb.shape).astype(np.uint8)\n\n \n reg_xs = list()\n reg_ys = list()\n for key in keys:\n temp_rx = tuple()\n temp_ry = tuple()\n for pred_key, r_x, r_y in zip(predict_cen[:-2], r_x_list, r_y_list):\n if key == pred_key:\n temp_rx += r_x\n temp_ry += r_y\n\n # Regression Multiple Lines #\n border_x, border_y = regression(org_np2, temp_rx, temp_ry)\n # regression(black_plane, temp_rx, temp_ry)\n # print(border_x, border_y)\n\n reg_xs.append(border_x)\n reg_ys.append(border_y)\n\n\n # VL Segmentation #\n # Angle between line1 and x-axis\n # print(len(vl_list))\n v_lines = list()\n h_lines = list()\n\n for vl in vl_list:\n x0, y0, x1, y1 = vl\n slope = (y1 - y0) / float(x1 - x0)\n angle = math.degrees(math.atan(slope))\n # print(angle)\n \n if abs(angle) > 80:\n # cv2.line(skl_copy, (int(x1), int(y1)), (int(x0), int(y0)), (255, 0, 0), 3,\n # cv2.LINE_AA)\n v_lines.append(vl)\n \n elif abs(angle) < 70:\n # cv2.line(skl_copy, (int(x1), int(y1)), (int(x0), int(y0)), (0, 0, 255), 2,\n # cv2.LINE_AA)\n h_lines.append(vl)\n\n # print('len(v_lines) :', len(v_lines))\n # print('len(h_lines) :', len(h_lines))\n \n # Extend v_lines and draw h_lines #\n all_closest_inters = list()\n all_centroid_inters = list()\n h_border = line((0, 0), (skl_copy.shape[1], 0))\n h_border_lower = line((0, skl_copy.shape[0]), (skl_copy.shape[1], skl_copy.shape[0]))\n for reg_x, reg_y in zip(reg_xs, reg_ys):\n\n left_vh_intersections = list()\n right_vh_intersections = list()\n\n left_angle = list()\n right_angle = list()\n\n vline = line((reg_x[0], reg_y[0]), (reg_x[1], reg_y[1]))\n intersections = list()\n for h_line in h_lines:\n hline = line(h_line[:2], h_line[2:])\n center_h_x = (h_line[0] + h_line[2]) / 2\n center_v_x = (reg_x[0] + reg_x[1]) / 2\n\n slope = (h_line[3] - h_line[1]) / float(h_line[2] - h_line[0])\n angle = math.degrees(math.atan(slope))\n \n if center_h_x < center_v_x:\n left_vh_intersections.append(intersection2(vline, hline))\n left_angle.append(angle)\n else:\n right_vh_intersections.append(intersection2(vline, hline))\n right_angle.append(angle)\n \n print('len(left_vh_intersections) :', len(left_vh_intersections))\n print('len(right_vh_intersections) :', len(right_vh_intersections))\n\n # Find Close Intersection Points in vline #\n close_points = list()\n close_thr = 30\n angle_gap = 10\n for (ix, iy), l_angle in zip(left_vh_intersections, left_angle):\n for (jx, jy), r_angle in zip(right_vh_intersections, right_angle):\n dist = math.hypot(jx - ix, jy - iy)\n # print(dist)\n if dist < close_thr:\n # print(l_angle, r_angle)\n if abs(l_angle - r_angle) > angle_gap:\n intersections.append((ix, iy, jx, jy))\n # cv2.circle(black_plane, (int(ix), int(iy)), 5, (255, 0, 255), -1)\n # cv2.circle(black_plane, (int(jx), int(jy)), 5, (255, 255, 0), -1)\n\n # Find intersections between horizontal border and vertical lines #\n r = intersection2(h_border, vline)\n r_lower = intersection2(h_border_lower, vline)\n # print(r)\n # print(r + r)\n\n intersections.append(r + r)\n intersections.append(r_lower + r_lower)\n intersections_ = intersections.copy()\n\n imms = MinMaxScaler()\n intersections = np.array(intersections)[:, [1, 3]]\n inter_data =imms.fit_transform(intersections)\n K = range(2, 5)\n s_dist = list()\n for k in K:\n if inter_data.shape[0] < k:\n break\n km = KMeans(n_clusters=k)\n km = km.fit(inter_data)\n inertia = km.inertia_\n s_dist.append(inertia)\n # plt.plot(s_dist)\n # plt.show()\n\n print(s_dist)\n mms2 = MinMaxScaler()\n mms_s_dist = mms2.fit_transform(np.array(s_dist).reshape(-1, 1))\n k_thresh = 0.025\n # print(mms_s_dist)\n # plt.plot(range(len(mms_s_dist)), mms_s_dist)\n # plt.show()\n\n if inter_data.shape[0] > 2:\n for i in range(len(mms_s_dist)):\n if mms_s_dist[i] < k_thresh:\n k = i + 2\n break\n elif i == len(mms_s_dist) - 1:\n k = 4\n else:\n k = 2\n \n # k Confirmation : Comparing k.cluster_centers_ dist #\n while True: \n km = KMeans(n_clusters=k)\n km = km.fit(inter_data) \n if k <= 2:\n break\n cluster_centroids = km.cluster_centers_[:, [1]]\n # print('cluster_centroids :', cluster_centroids)\n error_exist = False\n for i in range(len(cluster_centroids) - 1):\n for j in range(i + 1, len(cluster_centroids)):\n if abs(cluster_centroids[i] - cluster_centroids[j]) < 0.05:\n error_exist = True\n if error_exist:\n k -= 1\n else:\n break\n\n print('k is ', k)\n predict_inter = km.predict(inter_data)\n print(predict_inter)\n\n keys = list(range(k))\n for rm in predict_inter[-2:]:\n keys.remove(rm)\n print('keys :', keys)\n\n # \ud574\ub2f9 \ud0a4 \uc548\uc5d0\uc11c\uc758 closest intersection \ub450\uc30d\uc758 centroid\ub97c \uad6c\ud558\uba74 \ub41c\ub2e4. #\n centroid_inters = list()\n closest_inters = list()\n temp_black = black_plane2.copy()\n for key in keys:\n temp_inter_left = list()\n temp_inter_right = list()\n \n for pred_key, inter_point in zip(predict_inter[:-2], intersections_[:-2]):\n if key == pred_key:\n temp_inter_left.append(inter_point[:2])\n temp_inter_right.append(inter_point[2:])\n # else: \n # cv2.circle(black_plane2, (int(inter_point[0]), int(inter_point[1])), 5, (255, 0, 255), -1)\n # plt.imshow(temp_black)\n # plt.show()\n\n print('len(temp_inter_left) :', len(temp_inter_left))\n print('len(temp_inter_right) :', len(temp_inter_right))\n min_dist = close_thr\n closest_p = None\n closest_inter = None\n for ix, iy in temp_inter_left:\n for jx, jy in temp_inter_right:\n dist = math.hypot(jx - ix, jy - iy)\n # print(dist)\n if dist < min_dist:\n min_dist = dist\n closest_p = ((ix + jx) / 2, (iy + jy) / 2)\n closest_inter = [ix, iy, jx, jy]\n # closest_inter = (ix, iy, jx, jy)\n \n # print('min_dist :', min_dist)\n \n if closest_p:\n centroid_inters.append(closest_p)\n closest_inters.append(closest_inter)\n # cv2.circle(black_plane, (int(closest_p[0]), int(closest_p[1])), 5, (0, 255, 0), -1)\n\n # \uc5ec\uae30\uc11c append\ud558\ub294 closeest_inters\ub294 \ud56d\uc0c1 2\uac1c\ub97c \uc720\uc9c0\ud574\uc57c\ud55c\ub2e4. #\n # \ub2e8, intersection\uc774 1\uac1c\uc774\uba74 \ub098\uc911\uc5d0 middle section refering \ud63c\ub780\uc744 \ubc29\uc9c0\ud558\uae30 \uc704\ud574 \uc880 \ub2e4\ub978 \ud615\uc2dd\uc73c\ub85c #\n # border intersection\uc744 \uc228\uaca8\uc11c \ubcf4\ub0b4\uc900\ub2e4. #\n # intersection\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294 vline\uc740 \uc5c6\uc564\ub2e4. #\n if len(closest_inters) != 0:\n if len(closest_inters) == 1:\n # check ceil / floor type inter #\n # ceil condition #\n if (closest_inters[0][1] + closest_inters[0][3]) / 2 < (1 / 3) * org_np.shape[0]:\n opposite_inters = r_lower\n else:\n opposite_inters = r\n closest_inters = [closest_inters[0] + list(opposite_inters)]\n # print('closest_inters :', closest_inters)\n\n all_centroid_inters.append(centroid_inters)\n all_closest_inters.append(closest_inters)\n\n h_intersections = list()\n v_border = line((0, 0), (0, skl_copy.shape[0]))\n v_border_lower = line((skl_copy.shape[1], 0), (skl_copy.shape[1], skl_copy.shape[0]))\n for h_line in h_lines:\n\n # Find Intersection between v_line, h_line #\n vh_intersections_x = list()\n vh_intersections_y = list()\n\n hline = line(h_line[:2], h_line[2:])\n for reg_x, reg_y in zip(reg_xs, reg_ys):\n vline = line((reg_x[0], reg_y[0]), (reg_x[1], reg_y[1]))\n\n # Extract only x - coordination #\n vh_intersections_x.append(intersection2(vline, hline)[0])\n vh_intersections_y.append(intersection2(vline, hline)[1])\n\n # h_x = np.array([h_line[0], h_line[2]])\n # h_y = np.array([h_line[1], h_line[3]])\n # ex_h_x, ex_h_y = extended(h_x, h_y, 500)\n # ex_h_line = line((int(h_x[0]), int(h_y[0])), (int(h_x[1]), int(h_y[1])))\n r = intersection2(v_border, hline)\n r_lower = intersection2(v_border_lower, hline)\n\n vh_intersections_x.append(r[0])\n vh_intersections_y.append(r[1])\n vh_intersections_x.append(r_lower[0])\n vh_intersections_y.append(r_lower[1])\n\n sorted_vh_inter_x = sorted(vh_intersections_x)\n # print('vh_intersections_x :', vh_intersections_x)\n # print('sorted_vh_inter_x :', sorted_vh_inter_x)\n\n center_h_x = (h_line[0] + h_line[2]) / 2\n # print('center_h_x :', center_h_x)\n \n # hline \uc0c1\uc758 \uad50\ucc28\uc810\uc744 \ucc3e\uc544\ub0b4\uc11c \ubc94\uc704 \ub0b4\uc5d0\uc11c \uc5f0\uacb0\ud55c\ub2e4. #\n for i in range(1, len(sorted_vh_inter_x)):\n\n if sorted_vh_inter_x[i - 1] <= center_h_x <= sorted_vh_inter_x[i]:\n lx, ly = sorted_vh_inter_x[i - 1], vh_intersections_y[vh_intersections_x.index(sorted_vh_inter_x[i - 1])]\n rx, ry = sorted_vh_inter_x[i], vh_intersections_y[vh_intersections_x.index(sorted_vh_inter_x[i])]\n # print('lx, ly, rx, ry :', lx, ly, rx, ry)\n # \uc774\uacf3\uc758 lx, ly, rx, ry \ub294 close_p \ub2e4. \uc88c\uc6b0\ub85c \ub098\ub258\uc5b4\uc9c4 lx, ly\uac00 \uc544\ub2c8\ub2e4. #\n \n # \uc815\uc81c\ub41c \uad50\ucc28\uc810\uc744 \ub9cc\ub4dc\ub294 hline\ub9cc \uc0ac\uc6a9\ud574\uc57c\ud55c\ub2e4. #\n for inters in all_closest_inters:\n for inter in inters:\n if lx in inter or rx in inter:\n h_intersections.append((lx, ly, rx, ry))\n # cv2.line(black_plane, (int(lx), int(ly)), (int(rx), int(ry)), (0, 0, 255), 1, cv2.LINE_AA)\n \n\n # vline \uae30\uc900\uc73c\ub85c \uad6c\ud68d\ud574\uc57c\ud55c\ub2e4. -> \uac00\uc7a5 \uc67c\ucabd / \uc624\ub978\ucabd vline + \uad6c\uc5ed \uba3c\uc800 \ucc3e\uc544 \uc791\uc5c5\ud558\uae30 #\n # vline \ubcc4\ub85c \uad6c\ud68d \ub098\ub204\uae30 #\n h_intersections = list(set(h_intersections))\n print('np.array(all_closest_inters).shape :', np.array(all_closest_inters).shape)\n all_closest_inters = np.array(all_closest_inters)\n print('len(all_closest_inters) :', len(all_closest_inters))\n\n center_xs = list()\n for inters in all_closest_inters:\n # all_closest_inter\uac00 \uc815\ub82c\ub418\uc9c0 \uc54a\uc558\ub2e4\uba74 inters type = list() #\n sum_x = 0\n for inter in inters:\n sum_x += inter[0] + inter[2]\n center_xs.append(sum_x / len(inters) * 2)\n sorted_center_xs = sorted(center_xs)\n\n sorted_index = list()\n for center_x in sorted_center_xs:\n # print('center_x :', center_x)\n sorted_index.append(center_xs.index(center_x))\n # print('sorted_index :', sorted_index)\n # sorted_center_xs \uc21c\uc11c\ub85c all_closest_inter\ub97c \uc815\ub82c\ud55c\ub2e4. #\n # index list\ub97c \ucd94\ucd9c\ud574\uc11c for \ubb38\uc73c\ub85c all_closest_inter\uc758 inters\ub97c \ucd94\ucd9c\ud574\n # sorted_all ... \uc5d0 append \uc2dc\ud0a8\ub2e4. #\n sorted_all_closest_inters = list()\n sorted_all_centroid_inters = list()\n for s_index in sorted_index:\n sorted_all_closest_inters.append(all_closest_inters[s_index])\n sorted_all_centroid_inters.append(all_centroid_inters[s_index])\n all_closest_inters = sorted_all_closest_inters\n all_centroid_inters = sorted_all_centroid_inters\n\n\n for inters_i, inters in enumerate(all_closest_inters):\n\n # print('inters :', inters)\n inter_x = np.array(inters)[:, [0, 2]]\n inter_y = np.array(inters)[:, [1, 3]]\n\n # vline \ubcc4\ub85c \uc591\uc606\uc73c\ub85c \uc791\uc5c5\uc744 \ud558\uba74 len(vline) = 1\uc758 \uc791\uc5c5\uc744 \ubc18\ubcf5\ud560 \ud544\uc694\uac00 \uc5c6\uc5b4\uc9c4\ub2e4. #\n iter = False\n while True:\n\n four_inters = list()\n find_pair = True\n centroid_inters = all_centroid_inters[inters_i]\n\n if not iter:\n # vline \uc6b0\uce21 session #\n # vline \uc6b0\ud3b8 \uc88c\ud45c #\n final_xs = inter_x[:, [1]].reshape(-1, )\n final_ys = inter_y[:, [1]].reshape(-1, )\n # print(final_xs)\n # four_inters.append([final_xs[0], final_ys[0]])\n four_inters.append(centroid_inters[0])\n # print(four_inters)\n \n # intersection\uc774 1\uac1c\uc774\uace0 \uc624\ub978\ucabd \ub05d vline\uc774\uba74, border intersection\uc744 \ucd94\uac00\ud574\uc900\ub2e4. #\n # \uc624\ub978\ucabd \ub05d\uc774 \uc544\ub2c8\uace0 \ud604\uc7ac vline\uc758 \uad50\ucc28\uc810\uacfc \ub2e4\uc74c vline\uc758 \uad50\ucc28\uc810\uc774\uac19\uc740 \uc704\uce58\uc5d0 \uc5c6\uc73c\uba74 \ud3c9\ud589 copy, #\n # \ud604\uc7ac \uad50\ucc28\uc810\uc774 \uc5c6\uace0 \ub2e4\uc74c \uad50\ucc28\uc810\ub3c4 \uc5c6\uc73c\uba74 border intersection #\n # \ub458\ub2e4 \uc788\uc73c\uba74 \ucd94\uac00 #\n # border inter parallel copy #\n\n if len(inters[0]) == 6:\n print('len(inter[0]) == 6')\n # \uc624\ub978\ucabd \ub05d vline \uc774\uba74 #\n if inters_i == len(all_closest_inters) - 1:\n print('inters_i == len(all_closest_inters) - 1')\n four_inters.append(inters[0][-2:])\n four_inters.append([org_np.shape[1], inters[0][-1]])\n else:\n find_pair = False\n next_inters = np.array(all_closest_inters[inters_i + 1])\n next_centroid_inters = np.array(all_centroid_inters[inters_i + 1])\n print(next_centroid_inters)\n four_inters.append(next_centroid_inters[0])\n if len(next_inters) == 2:\n print('len(next_inters) == 2')\n # \uc5c6\ub294 \ubd80\ubd84 \ud3c9\ud589 copy #\n four_inters.append(next_centroid_inters[1])\n\n # 1. \uc5c6\ub294 \ubd80\ubd84\uc774 \uc5b4\ub514\uc778\uc9c0 \ud655\uc778\ud574\uc57c\ud55c\ub2e4. #\n # 2. copy\ud560 \ubd80\ubd84\uc758 \uc778\ub371\uc2a4 \ubc88\ud638\ub97c \ud655\uc778\ud574\uc57c\ud55c\ub2e4. #\n if inters[0][-1] == 0: # -> \ucc9c\uc7a5 \ubd80\ubd84 \uad50\ucc28\uc810\uc774 \uc5c6\ub2e4.\n print(type(next_inters))\n if np.mean(next_inters[[0], [1, 3]]) < np.mean(next_inters[[1], [1, 3]]):\n y_in = np.mean(next_inters[[0], [1, 3]])\n else:\n y_in = np.mean(next_inters[[1], [1, 3]])\n else: # -> \ubc14\ub2e5 \uad50\ucc28\uc810\uc774 \uc5c6\ub2e4. \n if np.mean(next_inters[[0], [1, 3]]) < np.mean(next_inters[[1], [1, 3]]):\n y_in = np.mean(next_inters[[1], [1, 3]])\n else:\n y_in = np.mean(next_inters[[0], [1, 3]])\n\n x = (inters[0][0], inters[0][2])\n y = (inters[0][1], inters[0][3])\n p = np.polyfit(y, x , deg=1)\n x_out = np.poly1d(p)(y_in)\n four_inters.append([x_out, y_in])\n\n else:\n if inters[0][-1] == next_inters[0][-1]:\n print('inters[0][-1] == next_inters[0][-1]')\n # \uc5c6\ub294 \ubd80\ubd84 border intersection #\n four_inters.append(inters[0][-2:])\n four_inters.append(next_inters[0][-2:])\n\n else: # \ub2e4\ub978 \uc704\uce58\n # \uc5c6\ub294 \ubd80\ubd84 \ud3c9\ud589 copy #\n x = (inters[0][0], inters[0][2])\n y = (inters[0][1], inters[0][3])\n p = np.polyfit(y, x , deg=1)\n y_in = np.mean(next_inters[0, [1, 3]])\n x_out = np.poly1d(p)(y_in)\n four_inters.append([x_out, y_in])\n\n x = (next_inters[0][0], next_inters[0][2])\n y = (next_inters[0][1], next_inters[0][3])\n p = np.polyfit(y, x , deg=1)\n y_in = np.mean(inters[0, [1, 3]])\n x_out = np.poly1d(p)(y_in)\n four_inters.append([x_out, y_in])\n\n # len vline inters = 2 #\n else:\n # four_inters.append([final_xs[1], final_ys[1]])\n four_inters.append(centroid_inters[1])\n\n # \uc624\ub978\ucabd \ub05d vline \uc774\uba74 #\n if inters_i == len(all_closest_inters) - 1:\n print('inters_i == len(all_closest_inters) - 1')\n else:\n find_pair = False\n inters = np.array(inters)\n next_inters = np.array(all_closest_inters[inters_i + 1])\n next_centroid_inters = np.array(all_centroid_inters[inters_i + 1])\n four_inters.append(next_centroid_inters[0])\n if len(next_inters) == 2:\n print('len(next_inters) == 2')\n four_inters.append(next_centroid_inters[1])\n else:\n # 1. \uc5c6\ub294 \ubd80\ubd84\uc774 \uc5b4\ub514\uc778\uc9c0 \ud655\uc778\ud574\uc57c\ud55c\ub2e4. #\n # 2. copy\ud560 \ubd80\ubd84\uc758 \uc778\ub371\uc2a4 \ubc88\ud638\ub97c \ud655\uc778\ud574\uc57c\ud55c\ub2e4. #\n if next_inters[0][-1] == 0: # -> \ucc9c\uc7a5 \ubd80\ubd84 \uad50\ucc28\uc810\uc774 \uc5c6\ub2e4.\n print('next_inters[0] :', next_inters[0])\n print('type(next_inters) :', type(next_inters))\n inters = np.array(inters) \n if np.mean(inters[[0], [1, 3]]) < np.mean(inters[[1], [1, 3]]):\n y_in = np.mean(inters[[0], [1, 3]])\n else:\n y_in = np.mean(inters[[1], [1, 3]]) \n else: # -> \ubc14\ub2e5 \uad50\ucc28\uc810\uc774 \uc5c6\ub2e4. \n if np.mean(inters[0, [1, 3]]) < np.mean(inters[[[1]], [1, 3]]):\n y_in = np.mean(inters[[1], [1, 3]])\n else:\n y_in = np.mean(inters[[0], [1, 3]])\n\n x = (next_inters[0][0], next_inters[0][2])\n y = (next_inters[0][1], next_inters[0][3])\n p = np.polyfit(y, x , deg=1)\n x_out = np.poly1d(p)(y_in)\n four_inters.append([x_out, y_in])\n\n # i = 0 \uc5d0 \ud55c\ud574\uc11c\ub9cc \uc67c\ucabd\uc73c\ub85c\ub3c4 refering \uc9c4\ud589, \ub098\uba38\uc9c0\ub294 \uc624\ub978\ucabd\uc73c\ub85c\ub9cc #\n else:\n # \ud55c vline\uc5d0 \ub300\ud574 \ubd84\ud3ec\ud558\ub294 \ubaa8\ub4e0 intersection\uc5d0 \ub300\ud55c pair\ub294 \ucc3e\uc544\uc8fc\uc5b4\uc57c \ud55c\ub2e4. #\n # vline \uc88c\ud3b8 \uc88c\ud45c #\n final_xs = inter_x[:, [0]].reshape(-1, )\n final_ys = inter_y[:, [0]].reshape(-1, )\n # print(final_xs)\n four_inters.append(centroid_inters[0])\n # print(four_inters) \n\n # intersection\uc774 1\uac1c\uc774\uba74, border intersection\uc744 \ucd94\uac00\ud574\uc900\ub2e4. #\n # border inter parallel copy #\n if len(inters[0]) == 6:\n print('inters[0][-2:] :', inters[0][-2:])\n four_inters.append(inters[0][-2:])\n four_inters.append([0, inters[0][-1]])\n else:\n four_inters.append(centroid_inters[1])\n # print(four_inters)\n\n print('four_inters :', four_inters) \n print('h_intersections :', h_intersections)\n # Find intersection pairs by h_intersections #\n if find_pair:\n # \ub9e8 \uc88c\uc6b0 vline\uc77c \uacbd\uc6b0\uc5d0\ub9cc \ud574\ub2f9\ud558\ub294\ub370 => pair\ub97c \ucc3e\uc544\uc8fc\ub294\uac8c \uc544\ub2c8\ub77c v_border\uc640\uc758 #\n # \uad50\ucc28\uc810\uc744 \ucc3e\uc544\uc8fc\ub294\uac8c \ub9de\ub294 \ubc29\ud5a5\uc774\ub2e4. #\n for h_inter in h_intersections:\n for final_x, final_y in zip(final_xs, final_ys):\n if final_x in h_inter and final_y in h_inter:\n hline = line(h_inter[:2], h_inter[2:])\n if iter:\n print('intersection2(v_border, hline) :', intersection2(v_border, hline))\n four_inters.append(list(intersection2(v_border, hline)))\n else:\n four_inters.append(list(intersection2(v_border_lower, hline)))\n\n # Four Intersection \uc644\uc131 #\n print('four_inters :', four_inters) \n if len(four_inters) != 4:\n print('Error in four_inter == 4')\n raise Exception\n break\n\n # Refering #\n # Find Top left / right & Bottom left / right #\n # -> tl, tr, bl, br #\n four_inters = np.array(four_inters)\n [tl, tr, br, bl] = order_points(four_inters)\n # print(four_inters.shape)\n top_length = math.hypot(tl[0] - tr[0], tl[1] - tr[1])\n bottom_length = math.hypot(bl[0] - br[0], bl[1] - br[1])\n max_hlength = max(top_length, bottom_length)\n\n # 1. Compare left / right height #\n l_height = bl[1] - tl[1]\n r_height = br[1] - tr[1]\n\n # 2. Extend Shorter Height #\n if l_height <= r_height:\n shorter_points = [bl, tl]\n longer_points = [br, tr]\n else:\n shorter_points = [br, tr]\n longer_points = [bl, tl]\n\n ex_shorter_points = np.zeros_like(shorter_points)\n ex_longer_points = np.zeros_like(longer_points)\n\n print('shorter_points :', shorter_points)\n x = (shorter_points[0][0], shorter_points[1][0])\n y = (shorter_points[0][1], shorter_points[1][1])\n print('shorter x, y :', x, y)\n\n y_ext = np.array([org_np.shape[0], 0])\n p = np.polyfit(y, x , deg=1)\n x_ext = np.poly1d(p)(y_ext)\n\n ex_shorter_points[0] = [x_ext[0], y_ext[0]]\n ex_shorter_points[1] = [x_ext[1], y_ext[1]]\n\n # Find intersection between (longer + shorter points)'s parallel line and longer line #\n long_short_lower = Line(longer_points[0], shorter_points[0])\n long_short = Line(longer_points[1], shorter_points[1])\n p_line_lower = long_short_lower.parallel_line(ex_shorter_points[0])\n p_line = long_short.parallel_line(ex_shorter_points[1])\n longer_line = Line(longer_points[0], longer_points[1])\n\n # print('ex_shorter_points[1] in p_line ? :', ex_shorter_points[1] in p_line)\n # print('p_line_lower.intersection(longer_line) :', p_line_lower.intersection(longer_line)[0][0])\n ex_longer_points[0] = [p_line_lower.intersection(longer_line)[0][0], p_line_lower.intersection(longer_line)[0][1]]\n ex_longer_points[1] = [p_line.intersection(longer_line)[0][0], p_line.intersection(longer_line)[0][1]]\n\n if l_height <= r_height:\n ex_bl, ex_tl = ex_shorter_points\n ex_br, ex_tr = ex_longer_points\n else:\n ex_br, ex_tr = ex_shorter_points\n ex_bl, ex_tl = ex_longer_points\n\n for ex_p in [ex_bl, ex_br, ex_tl, ex_tr]:\n print(ex_p)\n # cv2.circle(black_plane2, (int(ex_p[0]), int(ex_p[1])), 10, (255, 0, 255), -1)\n\n min_x, max_x = min(ex_bl[0], ex_tl[0]), max(ex_br[0], ex_tr[0])\n min_y, max_y = min(ex_tl[1], ex_tr[1]), max(ex_bl[1], ex_br[1])\n src_x = max((max_x - min_x), max_hlength) * math.sqrt(2)\n print('src_x :', src_x)\n\n refered = np.asarray(refer)[:int(max_y - min_y), :int(src_x)]\n\n # plt.imshow(refered)\n # plt.title('refered')\n # plt.show()\n\n # tl, tr, br, bl #\n # refer\ub97c \uc704\ud574 src_x => 0 \uc73c\ub85c \ub9de\ucdb0\uc900\ub2e4. #\n src = np.array([\n [0, 0],\n [src_x, 0],\n [src_x, max_y - min_y],\n [0, max_y - min_y]], dtype = \"float32\")\n dst = np.array([\n [ex_tl[0] - min_x, ex_tl[1] - min_y],\n [ex_tr[0] - min_x, ex_tr[1] - min_y],\n [ex_br[0] - min_x, ex_br[1] - min_y],\n [ex_bl[0] - min_x, ex_bl[1] - min_y]], dtype = \"float32\")\n\n print('src :', src)\n print('dst :', dst)\n\n # compute the perspective transform matrix and then apply it\n matrix = cv2.getPerspectiveTransform(src, dst)\n refered = cv2.warpPerspective(refered, matrix, (refered.shape[1], refered.shape[0]))\n\n # print('refered.min() :', refered.min())\n # print(refered[[0], [0]])\n # print('refered.shape :', refered.shape)\n # plt.imshow(refered)\n # plt.title('warfed')\n # plt.xlim(min_x, max_x)\n # plt.xlim(0, max_x - min_x)\n # plt.ylim(max_y, min_y)\n # plt.xlim(-1000, 1000)\n # plt.ylim(1000, -1000)\n # plt.show()\n \n refer_crop = refered[int(abs(min_y)): int(abs(min_y)) + org_np.shape[0], :]\n # print('refer_crop.shape :', refer_crop.shape)\n\n # plt.imshow(refer_crop)\n # plt.title('cropped refer')\n # plt.show()\n\n # Crop Section From Cropped Refer #\n # src_x => 0\uc73c\ub85c \ub9de\ucdb0\uc92c\ub358 \uac83\uc744 \ub418\ub3cc\ub9b0\ub2e4. #\n print('int(min_x) :', int(min_x))\n print('int(min_y) :', int(min_y))\n\n for i in range(refer_crop.shape[1]):\n for j in range(refer_crop.shape[0]):\n if sum(refer_crop[j][i]) != 0:\n if i + int(min_x) < black_plane2.shape[1]:\n black_plane2[j][i + int(min_x)] = refer_crop[j][i]\n\n # i != 0 \uc778 \uacbd\uc6b0 break #\n if inters_i == 0 and not iter:\n iter = True\n print('iter :', iter)\n else:\n break\n\n if len(all_closest_inters) == 0:\n refered = np.asarray(refer)[:black_plane2.shape[1], :black_plane2.shape[0]]\n black_plane2 = refered\n \n else:\n # Refer\uc758 \ube48 \ubd80\ubd84\uc740 original image\ub85c \ucc44\uc6b4\ub2e4. #\n for i in range(black_plane2.shape[1]):\n for j in range(black_plane2.shape[0]):\n if sum(black_plane2[j][i]) == 0:\n black_plane2[j][i] = org_np[j][i]\n \n print('elapsed time :', time.time() - start)\n\n \n # plt.figure(figsize=(10, 8))\n\n # plt.subplot(121)\n # plt.imshow(org_np2)\n # plt.xlim(-10, org_np.shape[0] + 10)\n \n # # plt.subplot(132)\n # # plt.imshow(black_plane)\n\n # plt.subplot(122)\n # plt.imshow(black_plane2)\n # plt.show()\n \n return black_plane2\n```\n\n\n Output hidden; open in https://colab.research.google.com to view.\n\n\n\n```\n\n```\n", "meta": {"hexsha": "44062acab9d0f4fad81a55c00efe26b49503b277", "size": 36303, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Layout Estimation Func Final.ipynb", "max_stars_repo_name": "jaytoone/Indoor_Segmentation", "max_stars_repo_head_hexsha": "d99e502f99cb74000483974d931c2da058071bec", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Layout Estimation Func Final.ipynb", "max_issues_repo_name": "jaytoone/Indoor_Segmentation", "max_issues_repo_head_hexsha": "d99e502f99cb74000483974d931c2da058071bec", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Layout Estimation Func Final.ipynb", "max_forks_repo_name": "jaytoone/Indoor_Segmentation", "max_forks_repo_head_hexsha": "d99e502f99cb74000483974d931c2da058071bec", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-04-29T15:27:56.000Z", "max_forks_repo_forks_event_max_datetime": "2021-04-29T15:27:56.000Z", "avg_line_length": 36303.0, "max_line_length": 36303, "alphanum_fraction": 0.5394870947, "converted": true, "num_tokens": 9824, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8221891305219504, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.4463362907838934}} {"text": "## Demo: Expected Free Energy minimization for controlling a Duffing oscillator\n\nThis project considers a [Duffing oscillator](hkps://en.wikipedia.org/wiki/Duffing_equation), a driven damped harmonic oscillator with a cubic nonlinearity in its spring stiffness component. The continuous-time dynamics of the system are:\n\n$$\\begin{align}\nm \\frac{d^2 x(t)}{dt^2} + c \\frac{d x(t)}{dt} + a x(t) + b x^3(t) =&\\ u(t) + w(t) \\\\\ny(t) =&\\ x(t) + e(t)\n\\end{align}$$\n\nwhere\n$$\\begin{align}\nm =&\\ \\text{mass} \\\\\nc =&\\ \\text{damping} \\\\\na =&\\ \\text{linear stiffness} \\\\\nb =&\\ \\text{nonlinear stiffness} \\\\\ny(t) =&\\ \\text{observation (displacement)} \\\\\nx(t) =&\\ \\text{state (displacement)} \\\\\nu(t) =&\\ \\text{force} \\\\\nv(t) =&\\ \\text{measurement noise} \\\\\nw(t) =&\\ \\text{process noise}\n\\end{align}$$\n\nThe process noise is a Wiener process, where the increment is Gaussian distributed $w(t) \\sim \\mathcal{N}(0, \\zeta^{-1}dt)$ with $\\zeta$ representing the precision of the process. The measurement noise is also a Wiener process, with $v(t) \\sim \\mathcal{N}(0, \\xi^{-1}dt)$ and $\\xi$ as precision parameter.\n\nWe cast this to the following discrete-time system:\n\n$$\\begin{align}\nx_k =&\\ \\theta_1 x_{k-1} + \\theta_2 x_{k-1}^3 + \\theta_3 x_{k-2} + \\eta u_k + w_k \\\\\ny_k =&\\ x_k + e_k\n\\end{align}$$\n\nwhere the coefficients $\\theta = (\\theta_1, \\dots, \\theta_3)$ and $\\eta$ are nonlinear combinations of the physical parameters (i.e. mass, damping and stiffness).\n\n\n```julia\nusing Pkg\nPkg.activate(\".\")\nPkg.instantiate()\n```\n\n \u001b[32m\u001b[1m Activating\u001b[22m\u001b[39m environment at `C:\\Syndr\\Wouter\\Onderzoek\\Demonstraties\\tue\\actinf-oscillator\\Project.toml`\n \u001b[32m\u001b[1m Updating\u001b[22m\u001b[39m registry at `C:\\Users\\Wouter Kouw\\.julia\\registries\\General`\n \u001b[32m\u001b[1m Installed\u001b[22m\u001b[39m Functors \u2500\u2500\u2500\u2500\u2500\u2500\u2500 v0.2.2\n \u001b[32m\u001b[1m 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[cf7118a7] \u001b[39m\u001b[92m+ UUIDs\u001b[39m\n \u001b[90m [4ec0a83e] \u001b[39m\u001b[92m+ Unicode\u001b[39m\n \u001b[90m [e66e0078] \u001b[39m\u001b[92m+ CompilerSupportLibraries_jll\u001b[39m\n \u001b[90m [deac9b47] \u001b[39m\u001b[92m+ LibCURL_jll\u001b[39m\n \u001b[90m [29816b5a] \u001b[39m\u001b[92m+ LibSSH2_jll\u001b[39m\n \u001b[90m [c8ffd9c3] \u001b[39m\u001b[92m+ MbedTLS_jll\u001b[39m\n \u001b[90m [14a3606d] \u001b[39m\u001b[92m+ MozillaCACerts_jll\u001b[39m\n \u001b[90m [83775a58] \u001b[39m\u001b[92m+ Zlib_jll\u001b[39m\n \u001b[90m [8e850ede] \u001b[39m\u001b[92m+ nghttp2_jll\u001b[39m\n \u001b[90m [3f19e933] \u001b[39m\u001b[92m+ p7zip_jll\u001b[39m\n\n\n\n```julia\nusing Plots\npyplot()\nviz = true;\n```\n\n \u250c Info: Precompiling Plots [91a5bcdd-55d7-5caf-9e0b-520d859cae80]\n \u2514 @ Base loading.jl:1317\n\n\n### Simulation environment\n\n\n```julia\n# Dynamical parameters\nmass = 1.2\ndamping = 0.4\nstiffness_lin = 0.9\nstiffness_nnl = 0.8\n\n# Process noise precision\n\u03b6_true = 1e4\n\n# Measurement noise precision\n\u03be_true = 1e3\n\n# Pack parameters\nsys_params = (mass, damping, stiffness_lin, stiffness_nnl, \u03b6_true, \u03be_true);\n\n# State transition coefficients\n\u03b81 = (2*mass + damping - stiffness_lin)/(mass + damping)\n\u03b82 = -stiffness_nnl/(mass + damping)\n\u03b83 = -mass/(mass + damping)\n\u03b8_true = Array{Float64}([\u03b81, \u03b82, \u03b83])\n\n# Control coefficient\n\u03b7_true = 1/(mass + damping)\n\n# Pack substituted variables\nsubs_params = (\u03b8_true, \u03b7_true, \u03b6_true, \u03be_true);\n```\n\n\n```julia\nfunction sim_sys(input, state, sys_params)\n \"Simulate dynamic system\"\n \n # Unpack state\n x_kmin1, x_kmin2 = state\n \n # Unpack system parameters\n m,c,a,b,\u03b6,\u03be = sys_params\n \n # Draw noises\n w_k = sqrt(inv(\u03b6))*randn(1,)[1]\n v_k = sqrt(inv(\u03be))*randn(1,)[1]\n \n # State transition\n x_k = (2*m + c - a)/(m+c)*x_kmin1 + (-b)/(m+c)*x_kmin1^3 + -m/(m+c)*x_kmin2 + 1/(m+c)*input + w_k\n \n # Generate observation\n y_k = x_k + v_k\n \n return y_k, x_k \nend\n```\n\n\n\n\n sim_sys (generic function with 1 method)\n\n\n\n\n```julia\n# Test simulation environment\n\n# Time\nT = 200\n\n# Input signal\ninput = sin.((1:T)./ (6*\u03c0))\n\n# Preallocate arrays\nstates = zeros(T,)\noutput = zeros(T,)\n\nfor k = 4:T\n output[k], states[k] = sim_sys(input[k], [states[k-1], states[k-2]], sys_params)\nend\n\np100 = plot(1:T, states, linewidth=2, color=\"blue\", label=\"states\", size=(700,400))\nplot!(1:T, input, linewidth=2, color=\"red\", label=\"input\", size=(700,400))\nscatter!(1:T, output, color=\"black\", label=\"output\")\n```\n\n\n```julia\nsavefig(p100, \"figures/example-input-output_seq1.png\")\n# savefig(p100, \"figures/example-input-output_seq1.pdf\")\n```\n\n## Model\n\nWe cast the above system into matrix form:\n\n$$ \\underbrace{\\begin{bmatrix} x_{k} \\\\ x_{k-1} \\end{bmatrix}}_{z_k} = \\underbrace{\\begin{bmatrix} 0 & 0 \\\\ 1 & 0 \\end{bmatrix}}_{S} \\underbrace{\\begin{bmatrix} x_{k-1} \\\\ x_{k-2} \\end{bmatrix}}_{z_{k-1}} + \\underbrace{\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}}_{s} g(\\theta, z_{k-1}) + \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\eta u_t + \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\tilde{w}_t \\, ,$$\n\nwith $g(\\theta, z_{k-1}) = \\theta_1 x_{k-1} + \\theta_2 x_{k-2}$ and $\\tilde{w}_k \\sim \\mathcal{N}(0, \\zeta^{-1})$. The system is now a nonlinear autoregressive process:\n\n$$z_k = f(\\theta, z_{k-1}, \\eta, u_k) + \\tilde{w}_k$$\n\nwhere $f(\\theta, z_{k-1}, \\eta, u_k) = S z_{k-1} + s g(\\theta, z_{k-1}) + s \\eta u_k$.\n\n### Likelihood\n\nIntegrating out $\\tilde{w}_t$ and $v_t$ produces a Gaussian state transition node:\n\n$$\\begin{align}\nz_k \\sim&\\ \\mathcal{N}(f(\\theta, z_{k-1}, \\eta, u_k), V(\\zeta)) \\\\\ny_k \\sim&\\ \\mathcal{N}(s^{\\top} z_k, \\xi^{-1}) \\, ,\n\\end{align}$$\n\nwhere $V(\\zeta) = \\begin{bmatrix} \\zeta^{-1} & 0 \\\\ 0 & \\epsilon \\end{bmatrix}$ and $W(\\zeta) = V(\\zeta)^{-1} = \\begin{bmatrix} \\zeta & 0 \\\\ 0 & \\epsilon^{-1} \\end{bmatrix}$.\n\n### Approximating the nonlinearity\n\nThe nonlinearity is approximated using a first-order Taylor expansion;\n\n$$ g(\\theta, x) = g(m_{\\theta}, m_x) + J_{x}(m_{\\theta}, m_x)^{\\top}(x - m_x) + J_{\\theta}(m_{\\theta}, m_x)^{\\top}(\\theta - m_{\\theta}) \\, ,$$\n\nwhere $J_x$ denotes the partial derivative of $g$ with respect to $x$ and $J_{\\theta}$ w.r.t. $\\theta$. Note that our current $g$ is linear in $\\theta$ and one could argue that the approximation is unnecessary. However, this form is more general and the first-order Taylor is exact anyway.\n\n### Priors\n\nWe choose the following priors:\n\n$$\\begin{align}\np(\\theta) = \\text{Normal}(m^{0}_{\\theta}, V^{0}_{\\theta}) \\, , \\quad \np(\\eta) = \\text{Normal}(m^{0}_{\\eta}, v^{0}_{\\eta}) \\, , \\quad \np(\\zeta)= \\text{Gamma}(a^{0}_\\zeta, b^{0}_\\zeta) \\, , \\quad\np(\\xi)= \\text{Gamma}(a^{0}_\\xi, b^{0}_\\xi) \\, .\n\\end{align}$$\n\n### Recognition model\n\nThe recognition model will follow the generative model:\n\n$$\\begin{align}\nq(\\theta) = \\text{Normal}(m_{\\theta}, V_{\\theta}) \\ , \\quad \nq(\\eta) = \\text{Normal}(m_{\\eta}, v_{\\eta}) \\ , \\quad \nq(\\zeta)= \\text{Gamma}(a_\\zeta, b_\\zeta) \\, , \\quad\nq(\\xi)= \\text{Gamma}(a_\\xi, b_\\xi) \\, .\n\\end{align}$$\n\n### Variational free energy\n\n#TODO\n\n## Implementation\n\nWe implemented the model in [ForneyLab.jl](hkps://github.com/biaslab/ForneyLab.jl) with a custom node called Nonlinear Latent Autoregressive eXogenous input (NLARX) model.\n\n\n```julia\nusing LinearAlgebra\nusing ProgressMeter\nusing Zygote\nusing Flux: update!, Descent, ADAM\n\nusing ForneyLab\nusing NLARX\n\ninclude(\"util.jl\");\n```\n\n \u250c Info: Precompiling Zygote [e88e6eb3-aa80-5325-afca-941959d7151f]\n \u2514 @ Base loading.jl:1317\n \u250c Info: Precompiling Flux [587475ba-b771-5e3f-ad9e-33799f191a9c]\n \u2514 @ Base loading.jl:1317\n WARNING: both ExprTools and LLVM export \"parameters\"; uses of it in module GPUCompiler must be qualified\n \u250c Warning: Error requiring `CUDA` from `Zygote`\n \u2502 exception = (ArgumentError(\"Package Zygote does not have CUDA in its dependencies:\\n- If you have Zygote checked out for development and have\\n added CUDA as a dependency but haven't updated your primary\\n environment's manifest file, try `Pkg.resolve()`.\\n- Otherwise you may need to report an issue with Zygote\"), Union{Ptr{Nothing}, Base.InterpreterIP}[Ptr{Nothing} @0x00000000610a3edb, Ptr{Nothing} @0x000000000277da06, Ptr{Nothing} @0x000000000277f72c, Ptr{Nothing} @0x00000000027617b2, Ptr{Nothing} @0x00000000027621c5, Base.InterpreterIP in top-level CodeInfo for Zygote at statement 8, Ptr{Nothing} @0x000000000277e7c7, Ptr{Nothing} @0x000000000278024f, Ptr{Nothing} @0x00000000641d808f, Ptr{Nothing} @0x00000000641d80b3, Ptr{Nothing} @0x000000000276103c, Ptr{Nothing} @0x0000000002760c34, Ptr{Nothing} @0x0000000002761588, Ptr{Nothing} @0x0000000002761b75, Ptr{Nothing} @0x0000000002762060, Base.InterpreterIP in MethodInstance for err(::Any, ::Module, ::String) at statement 2, Ptr{Nothing} @0x00000000641d7fe2, Ptr{Nothing} @0x00000000641d8003, Ptr{Nothing} @0x000000000276103c, Ptr{Nothing} @0x0000000002760c34, Ptr{Nothing} @0x00000000027619da, Ptr{Nothing} @0x0000000002761b75, Ptr{Nothing} @0x0000000002762060, Base.InterpreterIP in MethodInstance for withpath(::Any, ::String) at statement 10, Ptr{Nothing} @0x00000000641d7f46, Ptr{Nothing} @0x00000000641d7f73, Ptr{Nothing} @0x00000000027554e6, Ptr{Nothing} @0x0000000064647d11, Ptr{Nothing} @0x000000000276103c, Ptr{Nothing} @0x0000000002760c34, Ptr{Nothing} @0x0000000002761588, Ptr{Nothing} @0x0000000002762060, Base.InterpreterIP in MethodInstance for loadpkg(::Base.PkgId) at statement 6, Ptr{Nothing} @0x00000000027554e6, Ptr{Nothing} @0x000000006109fe88, Ptr{Nothing} @0x00000000610a064e, Ptr{Nothing} @0x00000000610a1ac3, Ptr{Nothing} @0x00000000610a3073, Ptr{Nothing} @0x00000000610a3d53, Ptr{Nothing} @0x000000000277da06, Ptr{Nothing} @0x000000000277dbe4, Ptr{Nothing} @0x000000000277f185, Ptr{Nothing} @0x000000000277f301, Ptr{Nothing} @0x000000000278024f, Ptr{Nothing} @0x00000000610e4ec5, Ptr{Nothing} @0x00000000610e529f, Ptr{Nothing} @0x000000006465127d, Ptr{Nothing} @0x00000000027554e6, Ptr{Nothing} @0x00000000610d5ca2, Ptr{Nothing} @0x00000000610d5ed6, Ptr{Nothing} @0x00000000610d5ef3, Ptr{Nothing} @0x0000000002766284])\n \u2514 @ Requires C:\\Users\\Wouter Kouw\\.julia\\packages\\Requires\\7Ncym\\src\\require.jl:49\n \u250c Info: Precompiling ForneyLab [9fc3f58a-c2cc-5bff-9419-6a294fefdca9]\n \u2514 @ Base loading.jl:1317\n \u250c Info: Precompiling NLARX [63ebac55-cc67-47e6-977a-6a1f35aa9955]\n \u2514 @ Base loading.jl:1317\n\n\n\n```julia\ngraph1 = FactorGraph()\n\n# Static parameters\n@RV \u03b8 ~ GaussianMeanPrecision(placeholder(:m_\u03b8, dims=(3,)), placeholder(:w_\u03b8, dims=(3,3)))\n@RV \u03b7 ~ GaussianMeanPrecision(placeholder(:m_\u03b7), placeholder(:w_\u03b7))\n@RV \u03b6 ~ Gamma(placeholder(:a_\u03b6), placeholder(:b_\u03b6))\n@RV \u03be ~ Gamma(placeholder(:a_\u03be), placeholder(:b_\u03be))\n\n# Nonlinearity\ng(\u03b8, x) = \u03b8[1]*x[1] + \u03b8[2]*x[1]^3 + \u03b8[3]*x[2]\n\n# State prior\n@RV z_kmin1 ~ GaussianMeanPrecision(placeholder(:m_z, dims=(2,)), placeholder(:w_z, dims=(2, 2)), id=:z_kmin1)\n\n# Autoregressive node\n@RV z_k ~ NLatentAutoregressiveX(\u03b8, z_kmin1, \u03b7, placeholder(:u_k), \u03b6, g=g, id=:z_k)\n\n# Specify likelihood\n@RV y_k ~ GaussianMeanPrecision(dot([1. , 0.], z_k), \u03be, id=:y_k)\n\n# Placeholder for observation\nplaceholder(y_k, :y_k);\n\n# Draw time-slice subgraph\n# ForneyLab.draw(graph1)\n```\n\n\n```julia\n# Specify recognition model\nq1 = PosteriorFactorization(z_k, z_kmin1, \u03b8, \u03b7, \u03b6, \u03be, ids=[:z_k, :z_kmin1, :\u03b8, :\u03b7, :\u03b6, :\u03be])\nalgo1 = messagePassingAlgorithm([z_k, z_kmin1, \u03b8, \u03b7, \u03b6, \u03be], q1, free_energy=true)\n\n# Compile inference algorithm\nsource_code1 = algorithmSourceCode(algo1, free_energy=true)\neval(Meta.parse(source_code1));\n# println(source_code1)\n```\n\n### Infer parameters on training data\n\n\n```julia\n# Goal state (mean and std dev)\ngoal_state = (-.35, 0.001)\n\n# Time horizon\nT_trn = 300\n\n# Planning horizon\nplan_horizon = 2\n\n# Number of gradient steps in EFE min\nnum_steps = 100\n\n# Number of variational updates in VFE min\nnum_iterations = 10\n\n# Initialize input-output arrays\ninputs = zeros(T_trn,)\nstates = zeros(T_trn,)\noutputs = zeros(T_trn,)\n\n# Initialize marginal and data dicts\ndata = Dict()\nmarginals = Dict()\n\n# Initialize free energy tracking array\nfree_energy_trn = zeros(T_trn, num_iterations)\n\n# Initialize arrays of parameterizations\nparams_z = (zeros(2,T_trn+1), repeat(eye(2), outer=(1,1,T_trn+1)))\nparams_\u03b8 = (zeros(3,T_trn+1), repeat(eye(3), outer=(1,1,T_trn+1)))\nparams_\u03b7 = (ones(1,T_trn+1), ones(1,T_trn+1))\nparams_\u03b6 = (1e2*ones(1,T_trn+1), 1e0*ones(1,T_trn+1))\nparams_\u03be = (1e4*ones(1,T_trn+1), 1e0*ones(1,T_trn+1))\n\n# Perform inference at each time-step\n@showprogress for k = 1:T_trn\n \n \"Control\"\n \n # Pack current model params\n model_params = (params_\u03b8[1][:,k], params_\u03b7[1][1,k], params_\u03b6[1][k]/params_\u03b6[2][k], params_\u03be[1][k]/params_\u03be[2][k])\n \n # Previous state\n state_kmin1 = [params_z[1][:,k], params_z[2][:,:,k]]\n \n # Minimize EFE to select input signal\n policy, G = minEFE(state_kmin1, goal_state, model_params, g, num_iters=num_steps, plan_horizon=plan_horizon)\n \n # Only execute first action in policy\n inputs[k] = policy[1]\n \n # Execute input and observe output\n outputs[k], states[k] = sim_sys(inputs[k], params_z[1][:,k], sys_params)\n \n# \"Test identification in case of designed control\"\n# inputs[k] = sin(k ./ (6*\u03c0))\n# outputs[k], states[k] = sim_sys(inputs[k], params_z[1][:,k], sys_params)\n \n \"Identification\"\n\n # Initialize marginals\n marginals[:z_kmin1] = ProbabilityDistribution(Multivariate, GaussianMeanPrecision, m=params_z[1][:,k], w=params_z[2][:,:,k])\n marginals[:z_k] = ProbabilityDistribution(Multivariate, GaussianMeanPrecision, m=params_z[1][:,k], w=params_z[2][:,:,k])\n marginals[:\u03b8] = ProbabilityDistribution(Multivariate, GaussianMeanPrecision, m=params_\u03b8[1][:,k], w=params_\u03b8[2][:,:,k])\n marginals[:\u03b7] = ProbabilityDistribution(Univariate, GaussianMeanPrecision, m=params_\u03b7[1][1,k], w=params_\u03b7[2][1,k])\n marginals[:\u03b6] = ProbabilityDistribution(Univariate, Gamma, a=params_\u03b6[1][1,k], b=params_\u03b6[2][1,k])\n marginals[:\u03be] = ProbabilityDistribution(Univariate, Gamma, a=params_\u03be[1][1,k], b=params_\u03be[2][1,k])\n \n data = Dict(:y_k => outputs[k],\n :u_k => inputs[k],\n :m_z => params_z[1][:,k],\n :w_z => params_z[2][:,:,k],\n :m_\u03b8 => params_\u03b8[1][:,k],\n :w_\u03b8 => params_\u03b8[2][:,:,k],\n :m_\u03b7 => params_\u03b7[1][1,k],\n :w_\u03b7 => params_\u03b7[2][1,k],\n :a_\u03b6 => params_\u03b6[1][1,k],\n :b_\u03b6 => params_\u03b6[2][1,k],\n :a_\u03be => params_\u03be[1][1,k],\n :b_\u03be => params_\u03be[2][1,k])\n\n # Iterate variational parameter updates\n for i = 1:num_iterations\n\n # Update parameters\n step\u03b7!(data, marginals)\n step\u03b8!(data, marginals)\n \n # Update states\n stepz_k!(data, marginals)\n stepz_kmin1!(data, marginals)\n \n # Update noise\n step\u03b6!(data, marginals)\n step\u03be!(data, marginals)\n \n # Compute free energy\n free_energy_trn[k, i] = freeEnergy(data, marginals)\n \n end \n\n # Store current parameterizations of marginals\n params_z[1][:,k+1] = ForneyLab.unsafeMean(marginals[:z_k])\n params_z[2][:,:,k+1] = marginals[:z_k].params[:w]\n params_\u03b8[1][:,k+1] = ForneyLab.unsafeMean(marginals[:\u03b8])\n params_\u03b8[2][:,:,k+1] = marginals[:\u03b8].params[:w]\n params_\u03b7[1][1,k+1] = ForneyLab.unsafeMean(marginals[:\u03b7])\n params_\u03b7[2][1,k+1] = marginals[:\u03b7].params[:w]\n params_\u03b6[1][1,k+1] = marginals[:\u03b6].params[:a]\n params_\u03b6[2][1,k+1] = marginals[:\u03b6].params[:b]\n params_\u03be[1][1,k+1] = marginals[:\u03be].params[:a]\n params_\u03be[2][1,k+1] = marginals[:\u03be].params[:b]\n\nend\n```\n\n \u001b[32mProgress: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| Time: 0:00:14\u001b[39m\n\n\n\n```julia\np1 = plot(1:T_trn, inputs, color=\"red\", label=\"inputs\", ylims=[-1, 1.])\np2 = plot(1:T_trn, states, color=\"blue\", label=\"states\", ylims=[-1, 1.])\nplot!(1:T_trn, params_z[1][1,2:end], color=\"purple\", label=\"inferred\", ylims=[-1, 1.])\nplot!(1:T_trn, repeat([goal_state[1]], outer=(1,T_trn))', ribbon=[goal_state[2], goal_state[2]], label=\"goal state\")\nplot(p1, p2, layout=(2,1), size=(800,600))\n```\n\n\n```julia\n# Scatter FE during training\np24 = plot(1:T_trn, free_energy_trn[:,end], color=\"black\", xlabel=\"time (t)\", ylabel=\"F[q]\", label=\"\", title=\"Free energy at training time\")\n```\n\n\n```julia\n# Scatter FE during training\np34 = plot(mean(free_energy_trn, dims=1)', color=\"black\", xlabel=\"time (t)\", ylabel=\"F[q]\", label=\"\", title=\"Average free energy over iterations\")\n```\n\n### Validate identified system\n\nWe validate the identified system by computing simulation error on the validation set.\n\n\n```julia\n# Time horizon\nT_val = 300\n\n# Input signal\n# input_val = [mean(sin.(t./ (0.1:.1:6 .*\u03c0))) for t in 1:T_val] \ninput_val = sin.((1:T_val) ./ (6*\u03c0))\n\n# Preallocate arrays\nstates_val = zeros(T_val,)\noutput_val = zeros(T_val,)\n\nfor k = 4:T_val\n output_val[k], states_val[k] = sim_sys(input_val[k], [states_val[k-1], states_val[k-2]], sys_params)\nend\n\n# Write to file\n```\n\n\n```julia\n# Prediction graph\ngraph2 = FactorGraph()\n\n# Autoregressive node\n@RV z_pred ~ NLatentAutoregressiveX(placeholder(:\u03b8, dims=(3,)), placeholder(:z_tmin1, dims=(2,)), placeholder(:\u03b7), placeholder(:u_t), placeholder(:\u03b6), g=g, id=:z_pred_t)\n\n# Draw time-slice subgraph\n# ForneyLab.draw(graph2)\n\n# Inference algorithm\nq2 = PosteriorFactorization(z_pred, ids=[:z_pred])\nalgo2 = messagePassingAlgorithm([z_pred], q2, free_energy=true)\nsource_code2 = algorithmSourceCode(algo2, free_energy=true)\neval(Meta.parse(source_code2));\n# println(source_code2)\n```\n\n\n```julia\n# Initialize free energy tracking array\nfree_energy_pred = zeros(T_val, num_iterations)\n\n# Initialize future state arrays\nparams_preds = (zeros(2, T_val), repeat(.1 .*float(eye(2)), outer=(1,1,T_val)))\n\n# Start simulation with known output\nparams_preds[1][1,2] = output_val[2]\nparams_preds[1][2,2] = output_val[1]\n\n@showprogress for k = 3:T_val\n\n # Initialize marginals\n marginals[:z_pred] = ProbabilityDistribution(Multivariate, GaussianMeanPrecision, m=params_preds[1][:,k], w=params_preds[2][:,:,k])\n\n # Clamp data\n data = Dict(:u_t => input_val[k],\n :z_tmin1 => params_preds[1][:,k-1],\n :\u03b8 => params_\u03b8[1][:,end],\n :\u03b7 => params_\u03b7[1][end],\n :\u03b6 => params_\u03b6[1][end]/params_\u03b6[2][end])\n\n # Iterate variational parameter updates\n for i = 1:num_iterations\n\n # Make prediction\n stepz_pred!(data, marginals)\n\n # Compute free energy\n free_energy_pred[k, i] = freeEnergy(data, marginals)\n end\n\n # Store current parameterizations of marginals\n params_preds[1][:,k] = ForneyLab.unsafeMean(marginals[:z_pred])\n params_preds[2][:,:,k] = marginals[:z_pred].params[:w]\n\nend\n```\n\n \u001b[32mProgress: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| Time: 0:00:00\u001b[39m\n\n\n### Visualize results\n\n\n```julia\n# Mean and std dev of predictions\npredictions_mean = params_preds[1][1,:]\npredictions_std = sqrt.(inv.(params_preds[2][1,1,:]))\n\n# Plot predictions\np23 = scatter(1:T_val, output_val, label=\"observations\", xlabel=\"time (t)\", color=\"black\")\nplot!(1:T_val, predictions_mean, ribbon=[predictions_std, predictions_std], label=\"predictions\", color=\"red\")\n```\n\n\n```julia\nPlots.savefig(p23, \"figures/simulation_error.png\")\n```\n\n\n```julia\n# Compute prediction error\nsq_pred_error = (predictions_mean[2:end] .- output_val[2:end]).^2\n\n# Simulation error\nMSE_sim = mean(sq_pred_error)\n\n# Scatter error over time\np24 = scatter(1:T_val, sq_pred_error, color=\"black\", xlabel=\"time (t)\", ylabel=\"Prediction error\", label=\"\")\ntitle!(\"MSE = \"*string(MSE_sim))\n```\n\n\n```julia\nPlots.savefig(p24, \"figures/pred-errors.png\")\n```\n\n\n```julia\n# Scatter FE during validation\np24 = plot(1:T_val, free_energy_pred[:,end], color=\"black\", xlabel=\"time (t)\", ylabel=\"F[q]\", label=\"\", title=\"Free energy of predictions\")\n```\n\n\n```julia\n\n```\n", "meta": {"hexsha": "1e01a7543132cc17ae408549f86428a30e906fbb", "size": 387182, "ext": "ipynb", "lang": "Jupyter Notebook", 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"max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 370.154875717, "max_line_length": 106901, "alphanum_fraction": 0.9190380751, "converted": true, "num_tokens": 14586, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8221891305219504, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.4463362907838934}} {"text": "# Exercises\n\n\n\n\n\n## Exercise 1: Stabilizing the Crank-Nicolson method by Rannacher time stepping\n
\n\nmathcal{I}_t is well known that the Crank-Nicolson method may give rise to\nnon-physical oscillations in the solution of diffusion equations\nif the initial data exhibit jumps (see the section [diffu:pde1:analysis:CN](#diffu:pde1:analysis:CN)).\nRannacher [[Rannacher_1984]](#Rannacher_1984) suggested a stabilizing technique\nconsisting of using the Backward Euler scheme for the first two\ntime steps with step length $\\frac{1}{2}\\Delta t$. One can generalize\nthis idea to taking $2m$ time steps of size $\\frac{1}{2}\\Delta t$ with\nthe Backward Euler method and then continuing with the\nCrank-Nicolson method, which is of second-order in time.\nThe idea is that the high frequencies of the initial solution are\nquickly damped out, and the Backward Euler scheme treats these\nhigh frequencies correctly. Thereafter, the high frequency content of\nthe solution is gone and the Crank-Nicolson method will do well.\n\nTest this idea for $m=1,2,3$ on a diffusion problem with a\ndiscontinuous initial condition. Measure the convergence rate using\nthe solution ([diffu:analysis:pde1:step:erf:sol](#diffu:analysis:pde1:step:erf:sol)) with the boundary\nconditions\n([diffu:analysis:pde1:p1:erf:uL](#diffu:analysis:pde1:p1:erf:uL))-([diffu:analysis:pde1:p1:erf:uR](#diffu:analysis:pde1:p1:erf:uR))\nfor $t$ values such that the conditions are in the vicinity of $\\pm 1$.\nFor example, $t< 5a 1.6\\cdot 10^{-2}$ makes the solution diffusion from\na step to almost a straight line. The\nprogram `diffu_erf_sol.py` shows how to compute the analytical\nsolution.\n\n\n\n\n\n\n\n\n## Project 2: Energy estimates for diffusion problems\n
\n\n\nThis project concerns so-called *energy estimates* for diffusion problems\nthat can be used for qualitative analytical insight and for\nverification of implementations.\n\n\n**a)**\nWe start with a 1D homogeneous diffusion equation with zero Dirichlet\nconditions:\n\n\n
\n\n$$\n\\begin{equation}\nu_t = \\alpha u_xx, x\\in \\Omega =(0,L),\\ t\\in (0,T],\n\\label{diffu:exer:estimates:p1:pdf} \\tag{1} \n\\end{equation}\n$$\n\n\n
\n\n$$\n\\begin{equation} \nu(0,t) = u(L,t) = 0, t\\in (0,T],\n\\label{diffu:exer:estimates:p1:bc} \\tag{2}\n\\end{equation}\n$$\n\n\n
\n\n$$\n\\begin{equation} \nu(x,0) = I(x), x\\in [0,L]\n\\label{diffu:exer:estimates:p1:ic} \\tag{3}\n\\thinspace .\n\\end{equation}\n$$\n\nThe energy estimate for this problem reads\n\n\n
\n\n$$\n\\begin{equation}\n||u||_{L^2} \\leq ||I||_{L^2},\n\\label{diffu:exer:estimates:p1:result} \\tag{4}\n\\end{equation}\n$$\n\nwhere the $||\\cdot ||_{L^2}$ norm is defined by\n\n\n
\n\n$$\n\\begin{equation}\n||g||_{L^2} = \\sqrt{\\int_0^L g^2dx}\\thinspace .\n\\label{diffu:exer:estimates:L2} \\tag{5}\n\\end{equation}\n$$\n\nThe quantify $||u||_{L^2}$ or $\\frac{1}{2} ||u||_{L^2}$ is known\nas the *energy* of the solution, although it is not the physical\nenergy of the system. A mathematical tradition has introduced the\nnotion *energy* in this context.\n\nThe estimate ([4](#diffu:exer:estimates:p1:result)) says that the\n\"size of $u$\" never exceeds that of the initial condition,\nor more precisely, it says that the area under the $u$ curve decreases\nwith time.\n\nTo show ([4](#diffu:exer:estimates:p1:result)), multiply the PDE\nby $u$ and integrate from $0$ to $L$. Use that $uu_t$ can be\nexpressed as the time derivative of $u^2$ and that $u_xxu$ can\nintegrated by parts to form an integrand $u_x^2$. Show that\nthe time derivative of $||u||_{L^2}^2$ must be less than or equal\nto zero. Integrate this expression and derive\n([4](#diffu:exer:estimates:p1:result)).\n\n\n\n**b)**\nNow we address a slightly different problem,\n\n\n
\n\n$$\n\\begin{equation}\nu_t = \\alpha u_xx + f(x,t), x\\in \\Omega =(0,L),\\ t\\in (0,T],\n\\label{diffu:exer:estimates:p2:pdf} \\tag{6} \n\\end{equation}\n$$\n\n\n
\n\n$$\n\\begin{equation} \nu(0,t) = u(L,t) = 0, t\\in (0,T],\n\\label{diffu:exer:estimates:p2:bc} \\tag{7}\n\\end{equation}\n$$\n\n\n
\n\n$$\n\\begin{equation} \nu(x,0) = 0, x\\in [0,L]\n\\label{diffu:exer:estimates:p2:ic} \\tag{8}\n\\thinspace .\n\\end{equation}\n$$\n\nThe associated energy estimate is\n\n\n
\n\n$$\n\\begin{equation}\n||u||_{L^2} \\leq ||f||_{L^2}\\thinspace .\n\\label{diffu:exer:estimates:p2:result} \\tag{9}\n\\end{equation}\n$$\n\n(This result is more difficult to derive.)\n\nNow consider the compound problem with an initial condition $I(x)$ and\na right-hand side $f(x,t)$:\n\n\n
\n\n$$\n\\begin{equation}\nu_t = \\alpha u_xx + f(x,t), x\\in \\Omega =(0,L),\\ t\\in (0,T],\n\\label{diffu:exer:estimates:p3:pdf} \\tag{10} \n\\end{equation}\n$$\n\n\n
\n\n$$\n\\begin{equation} \nu(0,t) = u(L,t) = 0, t\\in (0,T],\n\\label{diffu:exer:estimates:p3:bc} \\tag{11}\n\\end{equation}\n$$\n\n\n
\n\n$$\n\\begin{equation} \nu(x,0) = I(x), x\\in [0,L]\n\\label{diffu:exer:estimates:p3:ic} \\tag{12}\n\\thinspace .\n\\end{equation}\n$$\n\nShow that if $w_1$ fulfills\n([1](#diffu:exer:estimates:p1:pdf))-([3](#diffu:exer:estimates:p1:ic))\nand $w_2$ fulfills\n([6](#diffu:exer:estimates:p2:pdf))-([8](#diffu:exer:estimates:p2:ic)),\nthen $u=w_1 + w_2$ is the solution of\n([10](#diffu:exer:estimates:p3:pdf))-([12](#diffu:exer:estimates:p3:ic)).\nUsing the triangle inequality for norms,\n\n$$\n||a + b|| \\leq ||a|| + ||b||,\n$$\n\nshow that the energy estimate for\n([10](#diffu:exer:estimates:p3:pdf))-([12](#diffu:exer:estimates:p3:ic))\nbecomes\n\n\n
\n\n$$\n\\begin{equation}\n||u||_{L^2} \\leq ||I||_{L^2} + ||f||_{L^2}\\thinspace .\n\\label{diffu:exer:estimates:p3:result} \\tag{13}\n\\end{equation}\n$$\n\n**c)**\nOne application of ([13](#diffu:exer:estimates:p3:result)) is to prove uniqueness\nof the solution.\nSuppose $u_1$ and $u_2$ both fulfill\n([10](#diffu:exer:estimates:p3:pdf))-([12](#diffu:exer:estimates:p3:ic)).\nShow that $u=u_1 - u_2$ then fulfills\n([10](#diffu:exer:estimates:p3:pdf))-([12](#diffu:exer:estimates:p3:ic))\nwith $f=0$ and $I=0$. Use ([13](#diffu:exer:estimates:p3:result))\nto deduce that the energy must be zero for all times and therefore\nthat $u_1=u_2$, which proves that the solution is unique.\n\n**d)**\nGeneralize ([13](#diffu:exer:estimates:p3:result)) to a 2D/3D\ndiffusion equation $u_t = \\nabla\\cdot (\\alpha \\nabla u)$ for $x\\in\\Omega$.\n\n\n\n**Hint.**\nUse integration by parts in multi dimensions:\n\n$$\n\\int_\\Omega u \\nabla\\cdot (\\alpha\\nabla u)\\dx =\n- \\int_\\Omega \\alpha \\nabla u\\cdot\\nabla u\\dx\n+ \\int_{\\partial\\Omega} u \\alpha\\frac{\\partial u}{\\partial n},\n$$\n\nwhere $\\frac{\\partial u}{\\partial n} = \\boldsymbol{n}\\cdot\\nabla u$,\n$\\boldsymbol{n}$ being the outward unit normal to the boundary $\\partial\\Omega$\nof the domain $\\Omega$.\n\n\n\n**e)**\nNow we also consider the multi-dimensional PDE $u_t =\n\\nabla\\cdot (\\alpha \\nabla u)$. Integrate both sides over $\\Omega$\nand use Gauss' divergence theorem, $\\int_\\Omega \\nabla\\cdot\\boldsymbol{q}\\dx\n= \\int_{\\partial\\Omega}\\boldsymbol{q}\\cdot\\boldsymbol{n}\\ds$ for a vector field\n$\\boldsymbol{q}$. Show that if we have homogeneous Neumann conditions\non the boundary, $\\partial u/\\partial n=0$, area under the\n$u$ surface remains constant in time and\n\n\n
\n\n$$\n\\begin{equation}\n\\int_{\\Omega} u\\dx = \\int_{\\Omega} I\\dx\n\\thinspace .\n\\label{diffu:exer:estimates:p4:result} \\tag{14}\n\\end{equation}\n$$\n\n**f)**\nEstablish a code in 1D, 2D, or 3D that can solve a diffusion equation with a\nsource term $f$, initial condition $I$, and zero Dirichlet or\nNeumann conditions on the whole boundary.\n\nWe can use ([13](#diffu:exer:estimates:p3:result))\nand ([14](#diffu:exer:estimates:p4:result)) as a partial verification\nof the code. Choose some functions $f$ and $I$ and\ncheck that ([13](#diffu:exer:estimates:p3:result)) is obeyed at any\ntime when zero Dirichlet conditions are used.\nmathcal{I}_terate over the same $I$ functions and check that\n([14](#diffu:exer:estimates:p4:result)) is fulfilled\nwhen using zero Neumann conditions.\n\n**g)**\nMake a list of some possible bugs in the code, such as indexing errors\nin arrays, failure to set the correct boundary conditions,\nevaluation of a term at a wrong time level, and similar.\nFor each of the bugs, see if the verification tests from the previous\nsubexercise pass or fail. This investigation shows how strong\nthe energy estimates and the estimate ([14](#diffu:exer:estimates:p4:result))\nare for pointing out errors in the implementation.\n\nFilename: `diffu_energy`.\n\n\n\n\n\n\n\n\n## Exercise 3: Splitting methods and preconditioning\n
\n\n\nIn the section [diffu:2D:direct_vs_iter](#diffu:2D:direct_vs_iter), we outlined a class of\niterative methods for $Au=b$ based on splitting $A$ into $A=M-N$\nand introducing the iteration\n\n$$\nMu^{k} = Nu^k + b\\thinspace .\n$$\n\nThe very simplest splitting is $M=I$, where $I$ is the identity\nmatrix. Show that this choice corresponds to the iteration\n\n\n
\n\n$$\n\\begin{equation}\nu^k = u^{k-1} + r^{k-1},\\quad r^{k-1} = b - Au^{k-1},\n\\label{diffu:exer:splitting_prec:simplest} \\tag{15}\n\\end{equation}\n$$\n\nwhere $r^{k-1}$ is the residual in the linear system in iteration\n$k-1$. The formula ([15](#diffu:exer:splitting_prec:simplest)) is known\nas Richardson's iteration.\nShow that if we apply the simple iteration method\n([15](#diffu:exer:splitting_prec:simplest)) to the *preconditioned*\nsystem $M^{-1}Au=M^{-1}b$, we arrive at the Jacobi method by choosing\n$M=D$ (the diagonal of $A$) as preconditioner and the SOR method by\nchoosing $M=\\omega^{-1}D + L$ ($L$ being the lower triangular part of\n$A$). This equivalence shows that we can apply one iteration of the\nJacobi or SOR method as preconditioner.\n\n\n\n**Solution.**\nInserting $M=I$ and $N=I-A$ in the iterative method leads to\n\n$$\nu^{k} = (I-A)u^{k-1} + b = u^{k-1} + (b - Au^{k-1}),\n$$\n\nwhich is ([15](#diffu:exer:splitting_prec:simplest)).\nReplacing $A$ by $M^{-1}A$ and $b$ by $M^{-1}b$ in this equation\ngives\n\n$$\nu^k = u^{k-1} + M^{-1}r^{k-1},\\quad r^{k-1}=b-Au^{k-1},\n$$\n\nwhich we after multiplication by $M$ and reordering can write\nas\n\n$$\nMu^k = (M-A)u^{k-1} + b = Nu^{k-1} + b,\n$$\n\nwhich is the standard form for the Jacobi and SOR methods. Choosing $M=D$\ngives Jacobi and $M=\\omega^{-1}D+L$ gives SOR. We have shown that we may\nview $M$ as a preconditioner of a simplest possible iteration method.\n\n\n\n\n\n\n\n\n\n\n## Problem 4: Oscillating surface temperature of the earth\n
\n\nConsider a day-and-night or seasonal variation in temperature at\nthe surface of the earth. How deep down in the ground will the\nsurface oscillations reach? For simplicity, we model only the\nvertical variation along a coordinate $x$, where $x=0$ at the\nsurface, and $x$ increases as we go down in the ground.\nThe temperature is governed by the heat equation\n\n$$\n\\varrho c_v\\frac{\\partial T}{\\partial t} = \\nabla\\cdot(k\\nabla T),\n$$\n\nin some spatial domain $x\\in [0,L]$, where $L$ is chosen large enough such\nthat we can assume that $T$ is approximately constant, independent of the surface\noscillations, for $x>L$. The parameters $\\varrho$, $c_v$, and $k$ are the\ndensity, the specific heat capacity at constant volume, and the\nheat conduction coefficient, respectively.\n\n\n**a)**\nDerive the mathematical model for computing $T(x,t)$.\nAssume the surface oscillations to be sinusoidal around some mean\ntemperature $T_m$. Let $T=T_m$ initially. At $x=L$, assume $T\\approx T_m$.\n\n\n\n**Solution.**\nThe surface temperature is set as\n\n$$\nT(0,t) = T_m + A\\sin(\\omega t)\\thinspace .\n$$\n\nWith only one \"active\" spatial coordinate we get the initial-boundary\nvalue problem\n\n$$\n\\begin{alignat*}{2}\n\\varrho c_v \\frac{\\partial T}{\\partial t} &= \\frac{\\partial}{\\partial x}\n\\left(k(x)\\frac{\\partial T}{\\partial x}\\right), & x\\in (0,L),\\ t\\in (0,T],\\\\\nT(x,0)&= T_m, & x\\in [0,L],\\\\\nT(0,t)&= T_m + A\\sin(\\omega t), & t\\in (0,T],\\\\\nT(L,t) &= T_m, & t\\in (0,T].\n\\end{alignat*}\n$$\n\n\n\n**b)**\nScale the model in a) assuming $k$ is constant. Use a time scale\n$t_c = \\omega^{-1}$ and a length scale $x_c = \\sqrt{2\\dfc/\\omega}$,\nwhere $\\dfc = k/(\\varrho c_v)$. The primary unknown can be scaled\nas $\\frac{T-T_m}{2A}$.\n\nShow that the scaled PDE is\n\n$$\n\\frac{\\partial u}{\\partial \\bar t} =\n\\frac{1}{2}\\frac{\\partial^2 u}{\\partial x^2},\n$$\n\nwith initial condition $u(\\bar x,0) = 0$,\nleft boundary condition\n$u(0,\\bar t) = \\sin(\\bar t)$,\nand right boundary condition\n$u(\\bar L,\\bar t) = 0$. The bar indicates a dimensionless quantity.\n\nShow that $u(\\bar x, \\bar t)=e^{-\\bar x}\\sin (\\bar x - \\bar t)$ is a\nsolution that fulfills the PDE and the boundary condition at $\\bar x\n=0$ (this is the solution we will experience as $\\bar\nt\\rightarrow\\infty$ and $L\\rightarrow\\infty$). Conclude that an\nappropriate domain for $x$ is $[0,4]$ if a damping $e^{-4}\\approx\n0.18$ is appropriate for implementing $\\bar u\\approx\\hbox{const}$;\nincreasing to $[0,6]$ damps $\\bar u$ to 0.0025.\n\n\n\n**Solution.**\nChapter 3.2.4 in the book [[Langtangen_scaling]](#Langtangen_scaling) describes the\nscaling of this problem in detail.\nInserting dimensionless variables $\\bar t = \\omega t$, $\\bar x =\n\\sqrt{\\omega/(2\\dfc)} x$, and\n\n$$\nu = \\frac{T-T_m}{2A},\n$$\n\nleads to\n\n$$\n\\begin{alignat*}{2}\n\\frac{\\partial u}{\\partial \\bar t} &=\n\\frac{1}{2}\\frac{\\partial^2 u}{\\partial x^2},\n\\quad & \\bar x\\in (0,\\bar L),\\ \\bar t\\in (0,\\bar T],\n\\\\\nu(\\bar x,0) &= 0,\n\\quad &\\bar x\\in [0,1],\n\\\\\nu(0,\\bar t) & = \\sin(\\bar t),\n\\quad &\\bar t\\in (0,\\bar T],\n\\\\\nu(\\bar L,\\bar t) & = 0,\n\\quad &\\bar t\\in (0,\\bar T].\n\\end{alignat*}\n$$\n\nThe domain lengths $\\bar L$ and $\\bar T$ follows from straightforward\nscaling of $L$ and $T$.\n\nInserting $u(\\bar x, \\bar t)=e^{-\\bar x}\\sin (\\bar t - \\bar x)$ in the\nPDE shows that this is a solution. mathcal{I}_t also obeys\nthe boundary condition $\\bar u(0,\\bar t)=sin(\\bar t)$. As\n$\\bar t\\rightarrow\\infty$, the initial condition has no longer impact\non the solution and is \"forgotten\" and of no interest.\nThe boundary condition at $\\bar x=\\bar L$ is never compatible with the\ngiven solution unless $\\bar u$ is damped to zero, which happens\nmathematically as $\\bar L\\rightarrow\\infty$. For a numerical solution,\nhowever, we may use a small finite value such as $\\bar L=4$.\n\n\n\n**c)**\nCompute the scaled temperature and make animations comparing two solutions\nwith $\\bar L=4$ and $\\bar L=8$, respectively (keep $\\Delta x$ the same).\n\n\n\n**Solution.**\nWe can use the `viz` function in `diff1D_vc.py` to do the number\ncrunching. Appropriate calls and visualization go here:\n\n\n```python\n%matplotlib inline\n\nimport sys, os\nsys.path.insert(0, os.path.join(os.pardir, 'src-diffu'))\nfrom diffu1D_vc import viz\n\nsol = [] # store solutions\nfor Nx, L in [[20, 4], [40, 8]]:\n dt = 0.1\n dx = float(L)/Nx\n D = dt/dx**2\n from math import pi, sin\n T = 2*pi*6\n from numpy import zeros\n a = zeros(Nx+1) + 0.5\n cpu, u_ = viz(\n I=lambda x: 0, a=a, L=L, Nx=Nx, D=D, T=T,\n umin=-1.1, umax=1.1, theta=0.5,\n u_L=lambda t: sin(t),\n u_R=0,\n animate=False, store_u=True)\n sol.append(u_)\n print('computed solution for Nx=%d in [0,%g]' % (Nx, L))\n\nprint sol[0].shape\nprint sol[1].shape\nimport scitools.std as plt\ncounter = 0\nfor u0, u1 in zip(sol[0][2:], sol[1][2:]):\n x0 = sol[0][0]\n x1 = sol[1][0]\n plt.plot(x0, u0, 'r-', x1, u1, 'b-',\n legend=['short', 'long'],\n savefig='tmp_%04d.png' % counter,\n axis=[x1[0], x1[-1], -1.1, 1.1])\n counter += 1\n```\n\n\n\n\n\n```python\nfrom IPython.display import HTML\n_s = \"\"\"\n
\n\n
\n

\n\n\n\n\n\"\"\"\nHTML(_s)\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n## Problem 5: Oscillating and pulsating flow in tubes\n
\n\nWe consider flow in a straight tube with radius $R$ and straight walls.\nThe flow is driven by a pressure gradient $\\beta(t)$. The effect of\ngravity can be neglected. The mathematical problem reads\n\n\n
\n\n$$\n\\begin{equation}\n\\varrho\\frac{\\partial u}{\\partial t} =\n\\mu\\frac{1}{r}\\frac{\\partial}{\\partial r}\\left(\nr\\frac{\\partial u}{\\partial r}\\right) + \\beta(t),\\quad\n r\\in [0,R],\\ t\\in (0,T],\n\\label{_auto1} \\tag{16}\n\\end{equation}\n$$\n\n\n
\n\n$$\n\\begin{equation} \nu(r,0) = I(r),\\quad r\\in [0,R],\n\\label{_auto2} \\tag{17}\n\\end{equation}\n$$\n\n\n
\n\n$$\n\\begin{equation} \nu(R,t) = 0,\\quad t\\in (0,T],\n\\label{_auto3} \\tag{18}\n\\end{equation}\n$$\n\n\n
\n\n$$\n\\begin{equation} \n\\frac{\\partial u}{\\partial r}(0,t) = 0,\\quad t\\in (0,T].\n\\label{_auto4} \\tag{19}\n\\end{equation}\n$$\n\nWe consider two models for $\\beta(t)$. One plain, sinusoidal oscillation:\n\n\n
\n\n$$\n\\begin{equation}\n\\beta = A\\sin(\\omega t),\n\\label{_auto5} \\tag{20}\n\\end{equation}\n$$\n\nand one with periodic pulses,\n\n\n
\n\n$$\n\\begin{equation}\n\\beta = A\\sin^{16}(\\omega t),\n\\label{_auto6} \\tag{21}\n\\end{equation}\n$$\n\nNote that both models can be written as $\\beta = A\\sin^m(\\omega t)$, with\n$m=1$ and $m=16$, respectively.\n\n\n**a)**\nScale the mathematical model, using the viscous time scale $\\varrho R^2/\\mu$.\n\n\n\n**Solution.**\nWe can introduce\n\n$$\n\\bar r = \\frac{r}{R}, \\quad \\bar t = \\frac{t}{\\varrho R^2/\\mu},\\quad u = \\frac{u}{u_c}\\thinspace .\n$$\n\nInserted in the PDE, we get\n\n$$\n\\frac{\\partial\\bar u}{\\partial\\bar t} =\n\\frac{1}{\\bar r}\\frac{\\partial}{\\partial\\bar r}\\left(\n\\bar r\\frac{\\partial\\bar u}{\\partial\\bar r}\\right) +\n\\frac{R^2 A}{u_c \\mu}\\sin^m (\\alpha\\bar t)\n$$\n\nwhere $\\alpha$ is a dimensionless number\n\n$$\n\\alpha = \\frac{\\omega\\varrho R^2}{\\mu} = \\frac{\\varrho R^2/\\mu}{1/\\omega},\n$$\n\nreflecting the ratio of the viscous diffusion time scale and the\ntime scale of the oscillating pressure gradient.\nWe may choose $u_c$ such that the coefficient in the pressure gradient\nterm equals unity:\n\n$$\nu_c = \\frac{R^2 A}{\\mu}\\thinspace .\n$$\n\nThe governing PDE, dropping the bars, then reads\n\n$$\n\\frac{\\partial u}{\\partial t} =\n\\frac{1}{r}\\frac{\\partial}{\\partial r}\\left(\nr\\frac{\\partial u}{\\partial r}\\right) +\n\\sin^m (\\alpha\\bar t),\\quad r\\in (0,1),\\ t\\in (0,T]\\thinspace .\n$$\n\n\n\n**b)**\nImplement the scaled model from a), using the unifying $\\theta$ scheme\nin time and centered differences in space.\n\n\n\n**Solution.**\nWe need to take into account extensions below: a coefficient in front of\nthe viscous term, and an extra source term.\n\nA preliminary and unfinished code:\n\n\n```python\n\"\"\"\nSolve the diffusion equation for axi-symmetric case:\n\n u_t = 1/r * (r*a(r)*u_r)_r + f(r,t)\n\non (0,R) with boundary conditions u(0,t)_r = 0 and u(R,t) = 0,\nfor t in (0,T]. Initial condition: u(r,0) = I(r). \nPressure gradient f.\n\nThe following naming convention of variables are used.\n\n===== ==========================================================\nName Description\n===== ==========================================================\nNx The total number of mesh cells; mesh points are numbered\n from 0 to Nx.\nT The stop time for the simulation.\nI Initial condition (Python function of x).\na Variable coefficient (constant).\nR Length of the domain ([0,R]).\nr Mesh points in space.\nt Mesh points in time.\nn Index counter in time.\nu Unknown at current/new time level.\nu_1 u at the previous time level.\ndr Constant mesh spacing in r.\ndt Constant mesh spacing in t.\n===== ==========================================================\n\n``user_action`` is a function of ``(u, r, t, n)``, ``u[i]`` is the\nsolution at spatial mesh point ``r[i]`` at time ``t[n]``, where the\ncalling code can add visualization, error computations, data analysis,\nstore solutions, etc.\n\"\"\"\n\nimport scipy.sparse\nimport scipy.sparse.linalg\nfrom numpy import linspace, zeros, random, array, ones, sum, log, sqrt\nimport time, sys\nimport sympy as sym \n\n\ndef solver_theta(I, a, R, Nr, D, T, theta=0.5, u_L=None, u_R=0,\n user_action=None, f=0):\n \"\"\"\n The array a has length Nr+1 and holds the values of\n a(x) at the mesh points.\n\n Method: (implicit) theta-rule in time.\n\n Nr is the total number of mesh cells; mesh points are numbered\n from 0 to Nr.\n D = dt/dr**2 and implicitly specifies the time step.\n T is the stop time for the simulation.\n I is a function of r.\n u_L = None implies du/dr = 0, i.e. a symmetry condition \n f(r,t) is pressure gradient with radius.\n\n user_action is a function of (u, x, t, n) where the calling code\n can add visualization, error computations, data analysis,\n store solutions, etc.\n \n r*alpha is needed midway between spatial mesh points, - use\n arithmetic mean of successive mesh values (i.e. of r_i*alpha_i)\n \"\"\"\n import time\n t0 = time.perf_counter()\n\n r = linspace(0, R, Nr+1) # mesh points in space\n dr = r[1] - r[0]\n dt = D*dr**2 \n Nt = int(round(T/float(dt)))\n t = linspace(0, T, Nt+1) # mesh points in time\n\n if isinstance(u_L, (float,int)):\n u_L_ = float(u_L) # must take copy of u_L number\n u_L = lambda t: u_L_\n if isinstance(u_R, (float,int)):\n u_R_ = float(u_R) # must take copy of u_R number\n u_R = lambda t: u_R_\n if isinstance(f, (float,int)):\n f_ = float(f) # must take copy of f number\n f = lambda r, t: f_\n\n ra = r*a # help array in scheme\n\n inv_r = zeros(len(r)-2) # needed for inner mesh points\n inv_r = 1.0/r[1:-1]\n\n u = zeros(Nr+1) # solution array at t[n+1]\n u_1 = zeros(Nr+1) # solution at t[n]\n\n Dl = 0.5*D*theta\n Dr = 0.5*D*(1-theta)\n\n # Representation of sparse matrix and right-hand side\n diagonal = zeros(Nr+1)\n lower = zeros(Nr)\n upper = zeros(Nr)\n b = zeros(Nr+1)\n\n # Precompute sparse matrix (scipy format)\n diagonal[1:-1] = 1 + Dl*(ra[2:] + 2*ra[1:-1] + ra[:-2])*inv_r\n lower[:-1] = -Dl*(ra[1:-1] + ra[:-2])*inv_r\n upper[1:] = -Dl*(ra[2:] + ra[1:-1])*inv_r\n # Insert boundary conditions\n if u_L == None: # symmetry axis, du/dr = 0\n diagonal[0] = 1 + 8*a[0]*Dl\n upper[0] = -8*a[0]*Dl\n else:\n diagonal[0] = 1\n upper[0] = 0\n diagonal[Nr] = 1\n lower[-1] = 0\n\n A = scipy.sparse.diags(\n diagonals=[diagonal, lower, upper],\n offsets=[0, -1, 1],\n shape=(Nr+1, Nr+1),\n format='csr')\n #print A.todense()\n\n # Set initial condition\n for i in range(0,Nr+1):\n u_1[i] = I(r[i])\n\n if user_action is not None:\n user_action(u_1, r, t, 0)\n\n # Time loop\n for n in range(0, Nt):\n b[1:-1] = u_1[1:-1] + Dr*(\n (ra[2:] + ra[1:-1])*(u_1[2:] - u_1[1:-1]) -\n (ra[1:-1] + ra[0:-2])*(u_1[1:-1] - u_1[:-2]))*inv_r + \\\n dt*theta*f(r[1:-1], t[n+1]) + \\\n dt*(1-theta)*f(r[1:-1], t[n])\n \n # Boundary conditions\n if u_L == None: # symmetry axis, du/dr = 0\n b[0] = u_1[0] + 8*a[0]*Dr*(u_1[1] - u_1[0]) + \\\n dt*theta*f(0, (n+1)*dt) + \\\n dt*(1 - theta)*f(0, n*dt)\n else: \n b[0] = u_L(t[n+1]) \n b[-1] = u_R(t[n+1])\n #print b \n \n # Solve\n u[:] = scipy.sparse.linalg.spsolve(A, b)\n \n if user_action is not None:\n user_action(u, r, t, n+1)\n\n # Switch variables before next step\n u_1, u = u, u_1\n\n t1 = time.perf_counter()\n # return u_1, since u and u_1 are switched\n return u_1, t, t1-t0\n\ndef compute_rates(h_values, E_values):\n m = len(h_values)\n q = [log(E_values[i+1]/E_values[i])/\n log(h_values[i+1]/h_values[i])\n for i in range(0, m-1, 1)]\n q = [round(q_, 2) for q_ in q]\n return q\n\ndef make_a(alpha, r):\n \"\"\"\n alpha is a func, generally of r, - but may be constant.\n Note: when solution is to be axi-symmetric, alpha\n must be so too.\n \"\"\"\n a = alpha(r)*ones(len(r))\n return a\n\ndef tests_with_alpha_and_u_exact():\n '''\n Test solver performance when alpha is either const or \n a fu of r, combined with a manufactured sol u_exact \n that is either a fu of r only, or a fu of both r and t.\n Note: alpha and u_e are defined as symb expr here, since \n test_solver_symmetric needs to automatically generate \n the source term f. After that, test_solver_symmetric\n redefines alpha, u_e and f as num functions.\n '''\n R, r, t = sym.symbols('R r t')\n\n # alpha const ...\n \n # ue = const\n print('Testing with alpha = 1.5 and u_e = R**2 - r**2...')\n test_solver_symmetric(alpha=1.5, u_exact=R**2 - r**2)\n \n # ue = ue(t)\n print('Testing with alpha = 1.5 and u_e = 5*t*(R**2 - r**2)...')\n test_solver_symmetric(alpha=1.5, u_exact=5*t*(R**2 - r**2))\n \n # alpha function of r ...\n \n # ue = const \n print('Testing with alpha = 1 + r**2 and u_e = R**2 - r**2...')\n test_solver_symmetric(alpha=1+r**2, u_exact=R**2 - r**2)\n \n # ue = ue(t)\n print('Testing with alpha = 1+r**2 and u_e = 5*t*(R**2 - r**2)...')\n test_solver_symmetric(alpha=1+r**2, u_exact=5*t*(R**2 - r**2))\n\n\n\ndef test_solver_symmetric(alpha, u_exact):\n '''\n Test solver performance for manufactured solution\n given in the function u_exact. Parameter alpha is \n either a const or a function of r. In the latter \n case, an \"exact\" sol can not be achieved, so then\n testing switches to conv. rates.\n R is tube radius and T is duration of simulation.\n alpha constant:\n Compares the manufactured solution with the \n solution from the solver at each time step. \n alpha function of r:\n convergence rates are tested (using the sol\n at the final point in time only).\n ''' \n \n def compare(u, r, t, n): # user_action function\n \"\"\"Compare exact and computed solution.\"\"\"\n u_e = u_exact(r, t[n])\n diff = abs(u_e - u).max()\n #print diff\n tol = 1E-12\n assert diff < tol, 'max diff: %g' % diff\n\n def pde_source_term(a, u):\n '''Return the terms in the PDE that the source term\n must balance, here du/dt - (1/r) * d/dr(r*a*du/dr).\n a, i.e. alpha, is either const or a fu of r.\n u is a symbolic Python function of r and t.'''\n \n return sym.diff(u, t) - \\\n (1.0/r)*sym.diff(r*a*sym.diff(u, r), r)\n \n R, r, t = sym.symbols('R r t')\n\n # fit source term\n f = sym.simplify(pde_source_term(alpha, u_exact)) \n\n R = 1.0 # radius of tube\n T = 2.0 # duration of simulation \n \n if sym.diff(alpha, r) == 0: \n alpha_is_const = True\n else:\n alpha_is_const = False \n\n # make alpha, f and u_exact numerical functions\n alpha = sym.lambdify([r], alpha, modules='numpy') \n f = sym.lambdify([r, t], f.subs('R', R), modules='numpy') \n u_exact = sym.lambdify(\n [r, t], u_exact.subs('R', R), modules='numpy') \n\n I = lambda r: u_exact(r, 0)\n\n # some help variables\n FE = 0 # Forward Euler method\n BE = 1 # Backward Euler method\n CN = 0.5 # Crank-Nicolson method\n\n # test all three schemes \n for theta in (FE, BE, CN):\n print('theta: ', theta)\n E_values = []\n dt_values = []\n for Nr in (2, 4, 8, 16, 32, 64):\n print('Nr:', Nr)\n r = linspace(0, R, Nr+1) # mesh points in space\n dr = r[1] - r[0]\n a_values = make_a(alpha, r) \n if theta == CN:\n dt = dr\n else: # either FE or BE\n # use most conservative dt as decided by FE\n K = 1.0/(4*a_values.max()) \n dt = K*dr**2 \n D = dt/dr**2\n\n if alpha_is_const: \n u, t, cpu = solver_theta(\n I, a_values, R, Nr, D, T, \n theta, u_L=None, u_R=0,\n user_action=compare, f=f) \n else: # alpha depends on r\n u, t, cpu = solver_theta(\n I, a_values, R, Nr, D, T, \n theta, u_L=None, u_R=0,\n user_action=None, f=f) \n \n # compute L2 error at t = T\n u_e = u_exact(r, t[-1])\n e = u_e - u\n E = sqrt(dr*sum(e**2))\n E_values.append(E)\n dt_values.append(dt)\n \n if alpha_is_const is False: \n q = compute_rates(dt_values, E_values) \n print('theta=%g, q: %s' % (theta, q))\n expected_rate = 2 if theta == CN else 1\n tol = 0.1\n diff = abs(expected_rate - q[-1])\n print('diff:', diff)\n assert diff < tol\n \n \nif __name__ == '__main__':\n tests_with_alpha_and_u_exact() \n print('This is just a start. More remaining for this Exerc.')\n```\n\n\n\n**c)**\nVerify the implementation in b) using a manufactured solution that is\nquadratic in $r$ and linear in $t$. Make a corresponding test function.\n\n\n\n**Hint.**\nYou need to include an extra source term\nin the equation to allow for such tests. Let the spatial variation be\n$1-r^2$ such that the boundary condition is fulfilled.\n\n\n\n**d)**\nMake animations for $m=1,16$ and $\\alpha=1,0.1$. Choose $T$ such that\nthe motion has reached a steady state (non-visible changes from period to\nperiod in $u$).\n\n**e)**\nFor $\\alpha\\gg 1$, the scaling in a) is not good, because the\ncharacteristic time for changes (due to the pressure) is much smaller\nthan the viscous diffusion time scale ($\\alpha$ becomes large).\nWe should in this case base\nthe short time scale on $1/\\omega$. Scale the model again, and\nmake an animation for $m=1,16$ and $\\alpha = 10$.\n\n\n\n**Solution.**\nNow the governing PDE becomes\n\n$$\n\\frac{\\partial u}{\\partial t} =\n\\alpha^{-1}\\frac{1}{r}\\frac{\\partial}{\\partial r}\\left(\nr\\frac{\\partial u}{\\partial r}\\right) +\n\\sin^m t,\\quad r\\in (0,1),\\ t\\in (0,T]\\thinspace .\n$$\n\nIn this case,\n\n$$\nu_c = \\frac{A}{\\varrho\\omega}\\thinspace .\n$$\n\nWe see that for $\\alpha\\gg 1$, we can neglect the viscous term, and we\nbasically have a balance between the acceleration and the driving pressure\ngradient:\n\n$$\n\\frac{\\partial u}{\\partial t} = \\sin^m t\\thinspace .\n$$\n\n[hpl 1: This may be a great challenge numerically, since we have a plug\nindependent of r that oscillates back and forth. CN is probably very\nunstable. Can make a point out of this. Try $\\alpha=1$ and increase\ngently.]\n\n\n\nFilename: `axisymm_flow`.\n\n\n\n\n\n\n\n\n## Problem 6: Scaling a welding problem\n
\n\nWelding equipment makes a very localized heat source that moves in\ntime. We shall investigate the heating due to welding and choose, for\nmaximum simplicity, a one-dimensional heat equation with a fixed\ntemperature at the ends, and we neglect melting. We shall scale the\nproblem, and besides solving such a problem numerically, the aim is to\ninvestigate the appropriateness of alternative scalings.\n\nThe governing PDE problem reads\n\n$$\n\\begin{alignat*}{2}\n\\varrho c\\frac{\\partial u}{\\partial t} &= k\\frac{\\partial^2 u}{\\partial x^2}\n+ f, & x\\in (0,L),\\ t\\in (0,T),\\\\\nu(x,0) &= U_s, & x\\in [0,L],\\\\\nu(0,t) = u(L,t) &= 0, & t\\in (0,T].\n\\end{alignat*}\n$$\n\nHere, $u$ is the temperature, $\\varrho$ the density of the material,\n$c$ a heat capacity, $k$ the heat conduction coefficient, $f$ is\nthe heat source from the welding equipment, and $U_s$ is the\ninitial constant (room) temperature in the material.\n\nA possible model for the heat source is a moving Gaussian function:\n\n$$\nf = A\\exp{\\left(-\\frac{1}{2}\\left(\\frac{x-vt}{\\sigma}\\right)^2\\right)},\n$$\n\nwhere $A$ is the strength, $\\sigma$ is a parameter governing how\npeak-shaped (or localized in space) the heat source is, and\n$v$ is the velocity (in positive $x$ direction) of the source.\n\n\n**a)**\nLet $x_c$, $t_c$, $u_c$, and $f_c$ be scales, i.e., characteristic\nsizes, of $x$, $t$, $u$, and $f$, respectively. The natural choice of\n$x_c$ and $f_c$ is $L$ and $A$, since these make the scaled $x$ and\n$f$ in the interval $[0,1]$. If each of the three terms in the PDE\nare equally important, we can find $t_c$ and $u_c$ by demanding that\nthe coefficients in the scaled PDE are all equal to unity. Perform\nthis scaling. Use scaled quantities in the arguments for the\nexponential function in $f$ too and show that\n\n$$\n\\bar f= e^{-\\frac{1}{2}\\beta^2(\\bar x -\\gamma \\bar t)^2},\n$$\n\nwhere $\\beta$ and $\\gamma$ are dimensionless numbers. Give an\ninterpretation of $\\beta$ and $\\gamma$.\n\n\n\n**Solution.**\nWe introduce\n\n$$\n\\bar x=\\frac{x}{L},\\quad \\bar t = \\frac{t}{t_c},\\quad \\bar u = \\frac{u-U_s}{u_c},\n\\quad \\bar f=\\frac{f}{A}\\thinspace .\n$$\n\nInserted in the PDE and dividing by $\\varrho c u_c/t_c$ such that the\ncoefficient in front of $\\partial\\bar u/\\partial\\bar t$ becomes unity,\nand thereby all terms become dimensionless, we get\n\n$$\n\\frac{\\partial\\bar u}{\\partial\\bar t} =\n\\frac{k t_c}{\\varrho c L^2}\\frac{\\partial^2\\bar u}{\\partial\\bar x^2}\n+ \\frac{A t_c}{\\varrho c u_c}\\bar f\\thinspace .\n$$\n\nDemanding that all three terms are equally important, it follows that\n\n$$\n\\frac{k t_c}{\\varrho c L^2} = 1,\\quad \\frac{A t_c}{\\varrho c u_c}=1\\thinspace .\n$$\n\nThese constraints imply the *diffusion time scale*\n\n$$\nt_c = \\frac{\\varrho cL^2}{k},\n$$\n\nand a scale for $u_c$,\n\n$$\nu_c = \\frac{AL^2}{k}\\thinspace .\n$$\n\nThe scaled PDE reads\n\n$$\n\\frac{\\partial\\bar u}{\\partial\\bar t} =\n\\frac{\\partial^2\\bar u}{\\partial\\bar x^2}\n+ \\bar f\\thinspace .\n$$\n\nScaling $f$ results in\n\n$$\n\\begin{align*}\n\\bar f &= \\exp{\\left(-\\frac{1}{2}\\left(\\frac{x-vt}{\\sigma}\\right)^2\\right)}\\\\\n&= \\exp{\\left(-\\frac{1}{2}\\frac{L^2}{\\sigma^2}\n\\left(\\bar x- \\frac{vt_c}{L}t\\right)^2\\right)}\\\\\n&= \\exp{\\left(-\\frac{1}{2}\\beta^2\\left(\\bar x-\\gamma \\bar t\\right)^2\\right)},\n\\end{align*}\n$$\n\nwhere $\\beta$ and $\\gamma$ are dimensionless numbers:\n\n$$\n\\beta = \\frac{L}{\\sigma},\\quad\n\\gamma = \\frac{vt_c}{L} = \\frac{v\\varrho cL}{k}\\thinspace .\n$$\n\nThe $\\sigma$ parameter measures the width of the Gaussian peak, so\n$\\beta$ is the ratio of the domain and the width of the heat source (large\n$\\beta$ implies a very peak-formed heat source). The $\\gamma$\nparameter arises from $t_c/(L/v)$, which is the ratio of the diffusion\ntime scale and the time it takes for the heat source to travel through\nthe domain. Equivalently, we can multiply by $t_c/t_c$ to get $\\gamma\n= v/(t_cL)$ as the ratio between the velocity of the heat source and\nthe diffusion velocity.\n\n\n\n**b)**\nArgue that for large $\\gamma$ we should base the time scale on the\nmovement of the heat source. Show that this gives rise to the scaled\nPDE\n\n$$\n\\frac{\\partial\\bar u}{\\partial\\bar t} =\n\\gamma^{-1}\\frac{\\partial^2\\bar u}{\\partial\\bar x^2}\n+ \\bar f,\n$$\n\nand\n\n$$\n\\bar f = \\exp{(-\\frac{1}{2}\\beta^2(\\bar x - \\bar t)^2)}\\thinspace .\n$$\n\nDiscuss when the scalings in a) and b) are appropriate.\n\n\n\n**Solution.**\nWe perform the scaling as in a), but this time we determine $t_c$ such\nthat the heat source moves with unit velocity. This means that\n\n$$\n\\frac{vt_c}{L} = 1\\quad\\Rightarrow\\quad t_c = \\frac{L}{v}\\thinspace .\n$$\n\nScaling of the PDE gives, as before,\n\n$$\n\\frac{\\partial\\bar u}{\\partial\\bar t} =\n\\frac{k t_c}{\\varrho c L^2}\\frac{\\partial^2\\bar u}{\\partial\\bar x^2}\n+ \\frac{A t_c}{\\varrho c u_c}\\bar f\\thinspace .\n$$\n\nInserting the expression for $t_c$, we have\n\n$$\n\\frac{\\partial\\bar u}{\\partial\\bar t} =\n\\frac{k L}{\\varrho c L^2v}\\frac{\\partial^2\\bar u}{\\partial\\bar x^2}\n+ \\frac{A L}{v\\varrho c u_c}\\bar f\\thinspace .\n$$\n\nWe recognize the first coefficient as $\\gamma^{-1}$, while $u_c$ can\nbe determined from demanding the second coefficient to be unity:\n\n$$\nu_c = \\frac{AL}{v\\varrho c}\\thinspace .\n$$\n\nThe scaled PDE is therefore\n\n$$\n\\frac{\\partial\\bar u}{\\partial\\bar t} =\n\\gamma^{-1}\\frac{\\partial^2\\bar u}{\\partial\\bar x^2}\n+ \\bar f\\thinspace .\n$$\n\nIf the heat source moves very fast, there is little time for the\ndiffusion to transport the heat away from the source, and the heat\nconduction term becomes insignificant. This is reflected in the\ncoefficient $\\gamma^{-1}$, which is small when $\\gamma$, the ratio of\nthe heat source velocity and the diffusion velocity, is large.\n\nThe scaling in a) is therefore appropriate if diffusion is a\nsignificant process, i.e., the welding equipment moves at a slow speed\nso heat can efficiently spread out by diffusion. For large $\\gamma$,\nthe scaling in b) is appropriate, and $t=1$ corresponds to having the\nheat source traveled through the domain (with the scaling in a), the\nheat source will leave the domain in short time).\n\n\n\n**c)**\nOne aim with scaling is to get a solution that lies in the interval\n$[-1,1]$. This is not always the case when $u_c$ is based on a scale\ninvolving a source term, as we do in a) and b). However, from the\nscaled PDE we realize that if we replace $\\bar f$ with $\\delta\\bar f$,\nwhere $\\delta$ is a dimensionless factor, this corresponds to\nreplacing $u_c$ by $u_c/\\delta$. So, if we observe that $\\bar\nu\\sim1/\\delta$ in simulations, we can just replace $\\bar f$ by $\\delta\n\\bar f$ in the scaled PDE.\n\nUse this trick and implement the two scaled models. Reuse software for\nthe diffusion equation (e.g., the `solver` function in\n`diffu1D_vc.py`). Make a function `run(gamma, beta=10, delta=40,\nscaling=1, animate=False)` that runs the model with the given\n$\\gamma$, $\\beta$, and $\\delta$ parameters as well as an indicator\n`scaling` that is 1 for the scaling in a) and 2 for the scaling in\nb). The last argument can be used to turn screen animations on or off.\n\nExperiments show that with $\\gamma=1$ and $\\beta=10$, $\\delta =20$\nis appropriate. Then $\\max |\\bar u|$ will be larger than 4 for $\\gamma\n=40$, but that is acceptable.\n\nEquip the `run` function with visualization, both animation of $\\bar u$\nand $\\bar f$, and plots with $\\bar u$ and $\\bar f$ for $t=0.2$ and $t=0.5$.\n\n\n\n**Hint.**\nSince the amplitudes of $\\bar u$ and $\\bar f$ differs by a factor $\\delta$,\nit is attractive to plot $\\bar f/\\delta$ together with $\\bar u$.\n\n\n\n\n\n**Solution.**\nHere is a possible `run` function:\n\n\n```python\n# from .diffu1D_vc import solver\nimport numpy as np\n\ndef run(gamma, beta=10, delta=40, scaling=1, animate=False):\n \"\"\"Run the scaled model for welding.\"\"\"\n if scaling == 1:\n v = gamma\n a = 1\n elif scaling == 2:\n v = 1\n a = 1.0/gamma\n\n b = 0.5*beta**2\n L = 1.0\n ymin = 0\n # Need gloal to be able change ymax in closure process_u\n global ymax\n ymax = 1.2\n\n I = lambda x: 0\n f = lambda x, t: delta*np.exp(-b*(x - v*t)**2)\n\n import time\n import scitools.std as plt\n plot_arrays = []\n\n def process_u(u, x, t, n):\n global ymax\n if animate:\n plt.plot(x, u, 'r-',\n x, f(x, t[n])/delta, 'b-',\n axis=[0, L, ymin, ymax], title='t=%f' % t[n],\n xlabel='x', ylabel='u and f/%g' % delta)\n if t[n] == 0:\n time.sleep(1)\n plot_arrays.append(x)\n dt = t[1] - t[0]\n tol = dt/10.0\n if abs(t[n] - 0.2) < tol or abs(t[n] - 0.5) < tol:\n plot_arrays.append((u.copy(), f(x, t[n])/delta))\n if u.max() > ymax:\n ymax = u.max()\n\n Nx = 100\n D = 10\n T = 0.5\n u_L = u_R = 0\n theta = 1.0\n cpu = solver(\n I, a, f, L, Nx, D, T, theta, u_L, u_R, user_action=process_u)\n x = plot_arrays[0]\n plt.figure()\n for u, f in plot_arrays[1:]:\n plt.plot(x, u, 'r-', x, f, 'b--', axis=[x[0], x[-1], 0, ymax],\n xlabel='$x$', ylabel=r'$u, \\ f/%g$' % delta)\n plt.hold('on')\n plt.legend(['$u,\\\\ t=0.2$', '$f/%g,\\\\ t=0.2$' % delta,\n '$u,\\\\ t=0.5$', '$f/%g,\\\\ t=0.5$' % delta])\n filename = 'tmp1_gamma%g_s%d' % (gamma, scaling)\n s = 'diffusion' if scaling == 1 else 'source'\n plt.title(r'$\\beta = %g,\\ \\gamma = %g,\\ $' % (beta, gamma)\n + 'scaling=%s' % s)\n plt.savefig(filename + '.pdf'); plt.savefig(filename + '.png')\n return cpu\n```\n\nNote that we have dropped the bar notation in the plots. mathcal{I}_t is common\nto drop the bars as soon as the scaled problem is established.\n\n\n\n**d)**\nUse the software in c) to investigate $\\gamma=0.2,1,5,40$ for the\ntwo scalings. Discuss the results.\n\n\n\n**Solution.**\nFor these investigations, we compare the two scalings for each of\nthe different $\\gamma$ values. An appropriate function for automating\nthe tasks is\n\n\n```python\ndef investigate():\n \"\"\"Do scienfic experiments with the run function above.\"\"\"\n # Clean up old files\n import glob\n for filename in glob.glob('tmp1_gamma*') + \\\n glob.glob('welding_gamma*'):\n os.remove(filename)\n\n gamma_values = 1, 40, 5, 0.2, 0.025\n for gamma in gamma_values:\n for scaling in 1, 2:\n run(gamma=gamma, beta=10, delta=20, scaling=scaling)\n\n # Combine images\n for gamma in gamma_values:\n for ext in 'pdf', 'png':\n cmd = 'doconce combine_images -2 '\\\n 'tmp1_gamma%(gamma)g_s1.%(ext)s '\\\n 'tmp1_gamma%(gamma)g_s2.%(ext)s '\\\n 'welding_gamma%(gamma)g.%(ext)s' % vars()\n os.system(cmd)\n # pdflatex doesn't like 0.2 in filenames...\n if '.' in str(gamma):\n os.rename(\n 'welding_gamma%(gamma)g.%(ext)s' % vars(),\n ('welding_gamma%(gamma)g' % vars()).replace('.', '_')\n + '.' + ext)\n```\n\nWe run here a Backward Euler scheme with $N_x=100$ and quite long\ntime steps.\n\nRunning the `investigate` function, we get the following plots:\n\n\n\n\n

\n\n\n\n\n\n\n\n\n

\n\n\n\n\n\n\n\n\n

\n\n\n\n\n\n\n\n\n

\n\n\n\n\n\n\n\n\n

\n\n\n\n\n\nFor $\\gamma\\ll 1$ as in $\\gamma = 0.025$, the heat source moves very\nslowly on the diffusion time scale and has hardly entered the medium,\nwhile the scaling in b) is not inappropriate, but a larger $\\delta$ is\nneeded to bring $\\bar u$ around unity. We see that for $\\gamma=0.2$,\neach of the scalings work, but with the diffusion time scale, the heat\nsource has not moved much into the domain. For $\\gamma=1$, the\nmathematical problems are identical and hence the plots too. For\n$\\gamma=5$, the time scale based on the source is clearly the best\nchoice, and for $\\gamma=40$, only this scale is appropriate.\n\nA conclusion is that the scaling in b) works well for a range of $\\gamma$\nvalues, also in the case $\\gamma\\ll 1$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFilename: `welding`.\n\n\n\n\n\n\n\n\n## Exercise 7: Implement a Forward Euler scheme for axi-symmetric diffusion\n
\n\nBased on the discussion in the section [diffu:fd2:radial](#diffu:fd2:radial), derive in detail\nthe discrete equations for a Forward Euler in time, centered in space,\nfinite difference method for axi-symmetric diffusion. The\ndiffusion coefficient may be a function of the radial coordinate.\nAt the outer boundary $r=R$, we may have either a Dirichlet or Robin\ncondition.\nImplement this scheme. Construct appropriate test problems.\n\n\n\n**Solution.**\nWe start with the equation at $r=0$. According to the section [diffu:fd2:radial](#diffu:fd2:radial),\nwe get\n\n$$\n\\frac{u^{n+1}_0-u^n_0}{\\Delta t} = 4\\dfc(0)\\frac{u_1^n - u^n_0}{\\Delta r^2}\n+ f_0^n\\thinspace .\n$$\n\nFor $i>0$, we have\n\n$$\n\\begin{align*}\n\\frac{u^{n+1}_i-u^n_i}{\\Delta t} &= \\frac{1}{r_i\\Delta r^2}(\n\\frac{1}{2}(r_i + r_{i+1})\\frac{1}{2}(\\dfc_i + \\dfc_{i+1})(u^n_{i+1} - u^n_i) -\\\\\n&\\qquad\\frac{1}{2}(r_{i-1} + r_{i})\\frac{1}{2}(\\dfc_{i-1} + \\dfc_{i})(u^n_{i} - u^n_{i-1}))\n+ f_i^n\n\\end{align*}\n$$\n\nSolving with respect to $u^{n+1}_i$ and introducing $D=\\Delta t/\\Delta r^2$\nresults in\n\n$$\n\\begin{align*}\nu^{n+1}_0 &= u^n_0 + 4D\\dfc(0)(u_1^n - u^n_0)\n+ f_0^n,\\\\\nu^{n+1}_i &= u^n_i + D\\frac{1}{r_i}(\n\\frac{1}{2}(r_i + r_{i+1})\\frac{1}{2}(\\dfc_i + \\dfc_{i+1})(u^n_{i+1} - u^n_i) -\\\\\n&\\qquad\\frac{1}{2}(r_{i-1} + r_{i})\\frac{1}{2}(\\dfc_{i-1} + \\dfc_{i})(u^n_{i} - u^n_{i-1}))\n+ \\Delta t f_i^n,\\\\\n&\\qquad i = 1,\\ldots,N_r-1,\n\\end{align*}\n$$\n\nand $u^{n+1}_i$ at the end point $i=N_r$ is assumed known in case of\na Dirichlet condition. A Robin condition\n\n$$\n-\\dfc\\frac{\\partial u}{\\partial n} = h_T(u-U_s),\n$$\n\ncan be discretized at $i=N_r$ by\n\n$$\n-\\alpha_i\\frac{u_{i+1}^n-u_{i-1}^n}{2\\Delta r} = h_T(u_i^n - U_s)\\thinspace .\n$$\n\nSolving with respect to the value at the fictitious point $i+1$ gives\n\n$$\nu_{i+1}^n = u_{i-1}^n - 2\\Delta r \\frac{h_T}{\\alpha_i}(u_i^n - U_s)\\thinspace .\n$$\n\nThis value is then inserted for $u_{i+1}^n$ in the discrete PDE at $i=N_r$.\n\n\nFilename: `FE_axisym`.\n\n\n", "meta": {"hexsha": "77f5d638b02b060b661838ddd3996e2e732f95a1", "size": 74434, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "fdm-devito-notebooks/03_diffu/diffu_exer.ipynb", "max_stars_repo_name": "devitocodes/devito_book", "max_stars_repo_head_hexsha": "30405c3d440a1f89df69594fd0704f69650c1ded", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2020-07-17T13:19:15.000Z", "max_stars_repo_stars_event_max_datetime": "2021-03-27T05:21:09.000Z", "max_issues_repo_path": "fdm-devito-notebooks/03_diffu/diffu_exer.ipynb", "max_issues_repo_name": "devitocodes/devito_book", "max_issues_repo_head_hexsha": "30405c3d440a1f89df69594fd0704f69650c1ded", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 73, "max_issues_repo_issues_event_min_datetime": "2020-07-14T15:38:52.000Z", "max_issues_repo_issues_event_max_datetime": "2020-09-25T11:54:59.000Z", "max_forks_repo_path": "fdm-devito-notebooks/03_diffu/diffu_exer.ipynb", "max_forks_repo_name": "devitocodes/devito_book", "max_forks_repo_head_hexsha": "30405c3d440a1f89df69594fd0704f69650c1ded", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-03-27T05:21:14.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-27T05:21:14.000Z", "avg_line_length": 31.9048435491, "max_line_length": 239, "alphanum_fraction": 0.5002015208, "converted": true, "num_tokens": 15332, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.531209388216861, "lm_q2_score": 0.8397339616560072, "lm_q1q2_score": 0.44607456403620865}} {"text": "```python\n%matplotlib inline\n```\n\n\n\n# Tutorial 04: Block-sparse reduction \n\nIn many cases, the interaction radius $R$ is much smaller than the size of the domain. Consequently, the sums in the local averages (see `tuto_averages`) contain only a small fraction of non zero terms. To gain in efficiency, we can follow the classical strategy:\n\n* Subdivide the domain into a fixed number of cells of size at least $R$.\n* For a particle in a given cell, only look at the contiguous cells to compute the local averages. In dimension $d$, there are $3^d$ contiguous cells (including the cell itself). \n\nA practical implementation is called the *Verlet list method*. However, the implementation below is different than the classical one. It is adapted from the `block-sparse reduction method `_ implemented in the `KeOps `_ library. \n\nWe illustrate the gain in efficency for the Vicsek model. \n\n

Note

The method is sub-optimal for moderate numbers of particles. As a rule of thumb, the block-sparse reduction method becomes useful for systems with at least $10^4$ particles.

\n\n\n## Set up and benchmarks\n\nFirst, some standard imports...\n\n\n\n\n\n```python\nimport copy\nimport time \nimport torch\nfrom matplotlib import pyplot as plt\n\nuse_cuda = torch.cuda.is_available()\ndtype = torch.cuda.FloatTensor if use_cuda else torch.FloatTensor\n```\n\nLet the $N$ particles be uniformly scattered in a box of size $L$ with interaction radius $R$ and uniformly sampled velocities. \n\n\n\n\n\n```python\nfrom sisyphe.models import Vicsek\n\nN = 100000\nL = 100. \nR = 1.\n\npos = L*torch.rand((N,2)).type(dtype)\nvel = torch.randn(N,2).type(dtype)\nvel = vel/torch.norm(vel,dim=1).reshape((N,1))\n\nsimu=Vicsek(pos=pos,vel=vel,\n v=1.,\n sigma=1.,nu=3.,\n interaction_radius=R,\n box_size=L)\n\nsimu.__next__() #GPU warmup...\n```\n\nWithout block-sparse reduction, let us compute the simulation time of 100 iterations.\n\n\n\n\n```python\nsimu_copy = copy.deepcopy(simu) # Make a new deepcopy\ns = time.time()\nfor k in range(100):\n simu_copy.__next__()\ne = time.time()\n\nsimulation_time = e-s\n\nprint(\"Average simulation time without block-sparse reduction: \" + str(simulation_time) + \" seconds.\")\n```\n\nThen with block-sparse reduction... First, turn on the attribute :attr:`blocksparse `. \n\n\n\n\n```python\nsimu.blocksparse = True\n```\n\nThen, we need to define the maximum number of cells. This can be set by the keyword argument ``number_of_cells`` when an instance of the class :class:`sisyphe.particles.Particles` is created. The number of cells has a strong influence on the efficiency of the method and should be chosen wisely. When the optimal value is not known a priori, it is recommanded to use the method :meth:`best_blocksparse_parameters() ` which will time 100 iterations of the simulation for various numbers of cells and automatically choose the best one. Below, we test all the numbers of cells which are powers of the dimension (here $d=2$) between $10^2$ and $70^2$. \n\n\n\n\n\n```python\nncell_min = 10\nncell_max = 70\nfastest, nb_cells, average_simu_time, simulation_time = simu.best_blocksparse_parameters(ncell_min, ncell_max, step=1, nb_calls=100)\n```\n\nWe plot the average simulation time as a function of the square root of the number of cells and print the best. \n\n\n\n\n```python\nplt.plot(nb_cells,average_simu_time) \nplt.xlabel(\"Square root of the number of cells\") \nplt.ylabel(\"Simulation time\") \n\nprint(\"Average simulation time with block-sparse reduction: \" + str(average_simu_time.min()) + \" seconds.\")\n```\n\nSame experiment with one million particles. \n\n\n\n\n```python\nN = 1000000\nL = 100. \nR = 1.\n\npos = L*torch.rand((N,2)).type(dtype)\nvel = torch.randn(N,2).type(dtype)\nvel = vel/torch.norm(vel,dim=1).reshape((N,1))\n\n\nsimu=Vicsek(pos=pos,vel=vel,\n v=1.,\n sigma=1.,nu=3.,\n interaction_radius=R,\n box_size=L,\n block_sparse_reduction=False)\n\nsimu_copy = copy.deepcopy(simu) # Make a new deepcopy\ns = time.time()\nfor k in range(100):\n simu_copy.__next__()\ne = time.time()\n\nsimulation_time = e-s\n\nprint(\"Average simulation time without block-sparse reduction: \" + str(simulation_time) + \" seconds.\")\n```\n\nWith block-sparse reduction...\n\n\n\n\n```python\nsimu.blocksparse = True\n\nfastest, nb_cells, average_simu_time, simulation_time = simu.best_blocksparse_parameters(30, 100, nb_calls=100)\n```\n\nWe plot the average simulation time as a function of the square root of the number of cells and print the best. \n\n\n\n\n```python\nplt.plot(nb_cells,average_simu_time) \nplt.xlabel(\"Square root of the number of cells\") \nplt.ylabel(\"Simulation time\") \n\nprint(\"Average simulation time with block-sparse reduction: \" + str(average_simu_time.min()) + \" seconds.\")\n```\n\n

Note

The optimal parameters chosen initially may not stay optimal in the course of the simulation. This may be the case in particular if there is a strong concentration of particles.

\n\n\n\n## How does it work \n\n### Cell size and number of cells\n\nThe cells have a rectangular shape. The length of the cells along each dimension cannot be smaller than the interaction radius $R$. The maximum number of cells is thus equal to: \n\n\\begin{align}n_\\mathrm{max} = \\prod_{k=1}^d \\left\\lfloor \\frac{L_k}{R} \\right\\rfloor,\\end{align}\n\nwhere $L_k$ is the length of the (rectangular) domain along dimension $k$. This corresponds to rectangular cells with a length along dimension $k$ equal to: \n\n\\begin{align}\\varepsilon_k = \\frac{L_k}{\\left\\lfloor \\frac{L_k}{R} \\right\\rfloor}.\\end{align}\n\nIf the number of cells demanded $n_0$ exceeds $n_\\mathrm{max}$, this will be the chosen value. Otherwise, we first compute the typical length: \n\n\\begin{align}\\varepsilon_0 = \\left(\\frac{\\prod_{k=1}^d L_k}{n_0}\\right)^{1/d}\\end{align}\n\nThen the length of the cells along dimension $k$ is set to\n\n\\begin{align}\\varepsilon_k = \\frac{L_k}{\\left\\lfloor\\frac{L_k}{\\varepsilon_0}\\right\\rfloor}.\\end{align}\n\nIn particular, in a square domain $L_k=L$ for all $k$ and when $n_0$ is a power of $d$, then there are exactly $n_0$ square cells with length $L/n_0^{1/d}$. \n\n\n\n\n\n### The block-sparse parameters\n\nThe initialisation or the method :meth:`best_blocksparse_parameters() ` define three attributes which are used to speed up the computations. Given a number of cells, they are computed by the method :meth:`compute_blocksparse_parameters() `.\n\n* :attr:`centroids ` : the coordinates of the centers of the cells. \n* :attr:`keep ` : a square BoolTensor which indicates whether two cells are contiguous. \n* :attr:`eps ` : the length of the cells along each dimension. \n\nThe particles are clustered into the cells using the method :meth:`uniform_grid_separation() `. \n\n

Note

A drawback of the method is the high memory cost needed to store the boolean mask :attr:`keep `. As a consequence, unlike the classical Verlet list method, the optimal number of cells is often **not** the maximum one. In the examples presented in this documentation, the optimal number of cells is always smaller than $10^4$.

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YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.7879312006227324, "lm_q1q2_score": 0.4459836640117845}} {"text": "[](https://colab.research.google.com/github/MridulaMaddukuri/Pytorch_Tutorial/blob/master/Pytorch_Basics.ipynb)\n\n## Pytorch Basics\n\n\n#### 1. What are tensors?\n#### 2. Creating Tensors\n#### 3. Tensor Data types\n#### 4. Indexing Tensors\n#### 5. Tensor Manipulation \n#### 6. Operations on tensors\n#### 7. Matrix/Vector Operations \n#### 8. Conversion from and to Numpy\n#### 9. Performance : Numpy arrays vs tensors on CPU vs tensors on GPU\n#### 10. What are Variables?\n#### 11. Activation Funtions\n\n\n\n\n\n\n\n\n\n\n\n\n---\n\n\n\n\n\n\n\n\n\n\n\n## Pytorch \n\nA Deep Learning framework that provides \n\n- Tensors (a replacement for numpy arrays) with strong GPU acceleration \n- Dynamic Neural Networks (with GPU acceleration)\n\nBoth features useful in Deep Learning projects \n\n\n#### Comparing with the other popular DL framework: TensorFlow\n\n> - Relatively new when compared to ***_TensorFlow_*** but quickly gaining momentum. \nEasier to adapt to Pytorch due to similarities with numpy. \n\n> - More pythonic \n\n> - Offers flexibility to change the NN as you go : change and execute nodes as you go\n\n\n\n\nMore Pros and Cons can be found in this blog:\n\nhttps://towardsdatascience.com/pytorch-vs-tensorflow-spotting-the-difference-25c75777377b\n\n\n\n\n\n```\n# install pytorch\n!pip3 install http://download.pytorch.org/whl/cu80/torch-0.3.0.post4-cp36-cp36m-linux_x86_64.whl\n```\n\n Collecting torch==0.3.0.post4 from http://download.pytorch.org/whl/cu80/torch-0.3.0.post4-cp36-cp36m-linux_x86_64.whl\n \u001b[?25l Downloading http://download.pytorch.org/whl/cu80/torch-0.3.0.post4-cp36-cp36m-linux_x86_64.whl (592.3MB)\n \u001b[K 100% |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 592.3MB 45.9MB/s \n \u001b[?25hRequirement already satisfied: pyyaml in /usr/local/lib/python3.6/dist-packages (from torch==0.3.0.post4) (3.13)\n Requirement already satisfied: numpy in /usr/local/lib/python3.6/dist-packages (from torch==0.3.0.post4) (1.14.6)\n Installing collected packages: torch\n Successfully installed torch-0.3.0.post4\n\n\n\n```\n# import libraries\nimport torch\nimport numpy as np\n```\n\n### 1. What are tensors? \n\nA replacement for NumPy to use the power of GPUs \n\n\n### 2. Creating Tensors\n\n\n```\nx = torch.Tensor(2,2)\nprint(\"Shape of tensor x:\" + str(x.shape))\nprint(\"Datatype of x:\" + str(type(x)))\nprint(x) # uninitialized tensor holding garbage values\n```\n\n Shape of tensor x:torch.Size([2, 2])\n Datatype of x:\n \n 1.00000e-19 *\n 0.0000 0.0000\n 1.3563 1.3994\n [torch.FloatTensor of size 2x2]\n \n\n\n\n```\ntorch.manual_seed(2) # to make results reproducible\n\n\nx = torch.rand(2,3) # returns tensor of shape (2,3) filled with random numbers from a uniform distribution on interval [0,1)\n#torch.rand? # uncomment to know more\n\ny = torch.randn(2,3) # returns a tensor of shape (2,3) filled with random numbers from standard normal distribution : mean =0 , variance = 1\n\n#torch.randn? # uncomment to know more\n\nprint(x,y)\n# if you want these tensors on GPU\nx,y = x.cuda(),y.cuda()\nprint(x,y)\n```\n\n \n 0.6147 0.3810 0.6371\n 0.4745 0.7136 0.6190\n [torch.FloatTensor of size 2x3]\n \n -2.1409 -0.5534 -0.5000\n -0.0815 -0.1633 1.5277\n [torch.FloatTensor of size 2x3]\n \n \n 0.6147 0.3810 0.6371\n 0.4745 0.7136 0.6190\n [torch.cuda.FloatTensor of size 2x3 (GPU 0)]\n \n -2.1409 -0.5534 -0.5000\n -0.0815 -0.1633 1.5277\n [torch.cuda.FloatTensor of size 2x3 (GPU 0)]\n \n\n\n\n```\n# Initialize with a range of values\n\nx = torch.arange(5)\nprint(\"arange starting with 0: \")\nprint(x)\n\nx = torch.arange(2,20, step = 4)\nprint(\"arange starting with a specific value: \")\nprint(x)\n\n\n\n# Initialize on a scale\nx = torch.linspace(0,20, steps = 6) # includes 0 and 20\nprint(\"linspace between a and b with n steps: \")\nprint(x)\n\n\n```\n\n arange starting with 0: \n \n 0\n 1\n 2\n 3\n 4\n [torch.FloatTensor of size 5]\n \n arange starting with a specific value: \n \n 2\n 6\n 10\n 14\n 18\n [torch.FloatTensor of size 5]\n \n linspace between a and b with n steps: \n \n 0\n 4\n 8\n 12\n 16\n 20\n [torch.FloatTensor of size 6]\n \n\n\n### 3. Tensor Data Types \n\n(https://pytorch.org/docs/stable/tensor_attributes.html) \n\n\nCommon ones we work with: \n\n**FloatTensor** : Equivalent to numpy.float32\n\n\n**LongTensor** : Equivalent to numpy.int64\n\n\n\n```\nx = torch.randn([2,3])\nprint(\"Original x: \")\nprint(x)\n# convert floattensor to long tensor\n\nx = x.type(torch.LongTensor)\nprint(\"Datatype modified x: \")\nprint(x)\n\n\n# converting data type of a tensor based on another tensor\ny = torch.rand([2,2])\nprint(\"Original y: \")\nprint(y)\n\ny = y.type_as(x) # tensor x used as reference to change dtype\nprint(\"Datatype of y modified based on x: \")\nprint(y)\n\n\n\n\n\n```\n\n Original x: \n \n -0.4023 0.0972 -0.5682\n -1.2692 0.5789 -1.5181\n [torch.FloatTensor of size 2x3]\n \n Datatype modified x: \n \n 0 0 0\n -1 0 -1\n [torch.LongTensor of size 2x3]\n \n Original y: \n \n 0.0458 0.1755\n 0.6177 0.8291\n [torch.FloatTensor of size 2x2]\n \n Datatype of y modified based on x: \n \n 0 0\n 0 0\n [torch.LongTensor of size 2x2]\n \n\n\n**_Why is this important?_**\n\n-- For operations between tensors, they should strictly have the same data type.\n\n-- Certain functions are also very specific about the datatypes\n\n### 4. Indexing Tensors\n\nVery similar to numpy indexing\n\n\n```\n## 2D tensor\n\nx = torch.randn(5,3).type(torch.FloatTensor)\nprint(x)\n\nprint(\"Indexing the second column\")\nprint(x[:,1])\n\nprint(\"Indexing the 2nd element of the 4th row\")\nprint(x[3,1])\n\n```\n\n \n 1.1125 -0.7474 1.4220\n -0.5941 0.7546 0.5748\n -0.8063 0.6410 0.7828\n 1.6251 0.1756 -0.8472\n -0.8448 -0.8347 -0.7278\n [torch.FloatTensor of size 5x3]\n \n Indexing the second column\n \n -0.7474\n 0.7546\n 0.6410\n 0.1756\n -0.8347\n [torch.FloatTensor of size 5]\n \n Indexing the 2nd element of the 4th row\n 0.17558613419532776\n\n\n\n```\n## 3D tensor\n\ny = torch.randn(1,5,3).type(torch.FloatTensor)\nprint(y)\n\nprint(\"Indexing the 5X3 tensor from y: \")\nprint(y[0,:,:])\n\nprint(\"Indexing the 1X3 tensor from y:\")\nprint(y[:,0,:])\n\nprint(\"Indexing the size 3 tensor from y:\")\nprint(y[0,0,:])\n```\n\n \n (0 ,.,.) = \n -0.0602 -0.8961 0.8293\n -0.5403 0.5956 -0.7712\n 0.0709 -0.8890 -1.6091\n 0.5055 0.9108 0.8569\n -1.0442 0.1758 0.6451\n [torch.FloatTensor of size 1x5x3]\n \n Indexing the 5X3 tensor from y: \n \n -0.0602 -0.8961 0.8293\n -0.5403 0.5956 -0.7712\n 0.0709 -0.8890 -1.6091\n 0.5055 0.9108 0.8569\n -1.0442 0.1758 0.6451\n [torch.FloatTensor of size 5x3]\n \n Indexing the 1X3 tensor from y:\n \n -0.0602 -0.8961 0.8293\n [torch.FloatTensor of size 1x3]\n \n Indexing the 3 tensor from y:\n \n -0.0602\n -0.8961\n 0.8293\n [torch.FloatTensor of size 3]\n \n\n\n### 5. Tensor Manipulation \n\n\n```\n\nx = torch.arange(9)\nprint(x)\nprint(x.shape)\n\n\n# Squeeze and unsqueeze : to add or remove a dimension\nprint(\"Unsqueeze on the first axis to get a row vector: \")\nx = torch.unsqueeze(x,0) \nprint(x) \n\nprint(\"Squeeze to get rid of the additional dimension: \")\nx= torch.squeeze(x,0) # specify squeeze dimension: the dimension we want to get rid of \nprint(x)\n\nprint(\"Unsqueeze on the second axis to get a column vector: \")\nx = torch.unsqueeze(x,1) \nprint(x)\n\n# View : similar to numpy.reshape\nprint(\"Reshape the tensor: \")\nx = x.view(3,3) \nprint(x)\n\n\n```\n\n \n 0\n 1\n 2\n 3\n 4\n 5\n 6\n 7\n 8\n [torch.FloatTensor of size 9]\n \n torch.Size([9])\n Unsqueeze on the first axis to get a row vector: \n \n 0 1 2 3 4 5 6 7 8\n [torch.FloatTensor of size 1x9]\n \n Squeeze to get rid of the additional dimension: \n \n 0\n 1\n 2\n 3\n 4\n 5\n 6\n 7\n 8\n [torch.FloatTensor of size 9]\n \n Unsqueeze on the second axis to get a column vector: \n \n 0\n 1\n 2\n 3\n 4\n 5\n 6\n 7\n 8\n [torch.FloatTensor of size 9x1]\n \n Reshape the tensor: \n \n 0 1 2\n 3 4 5\n 6 7 8\n [torch.FloatTensor of size 3x3]\n \n\n\n\n```\n# Concatenation\nprint(\"Concatenate in the 0th dimension:\")\nprint(torch.cat((x,x),0))\n\nprint(\"Concatenate in the 1st dimension:\")\nprint(torch.cat((x,x),1))\n```\n\n Concatenate in the 0th dimension:\n \n 0 1 2\n 3 4 5\n 6 7 8\n 0 1 2\n 3 4 5\n 6 7 8\n [torch.FloatTensor of size 6x3]\n \n Concatenate in the 1st dimension:\n \n 0 1 2 0 1 2\n 3 4 5 3 4 5\n 6 7 8 6 7 8\n [torch.FloatTensor of size 3x6]\n \n\n\n\n```\n# Stacking\nprint(\"Stacking on 0th dimension: \")\nprint(torch.stack((x,x),0))\n\nprint(\"Stacking on first dimension: \")\nprint(torch.stack((x,x),1))\n```\n\n Stacking on 0th dimension: \n \n (0 ,.,.) = \n 0 1 2\n 3 4 5\n 6 7 8\n \n (1 ,.,.) = \n 0 1 2\n 3 4 5\n 6 7 8\n [torch.FloatTensor of size 2x3x3]\n \n Stacking on first dimension: \n \n (0 ,.,.) = \n 0 1 2\n 0 1 2\n \n (1 ,.,.) = \n 3 4 5\n 3 4 5\n \n (2 ,.,.) = \n 6 7 8\n 6 7 8\n [torch.FloatTensor of size 3x2x3]\n \n\n\n### 6. Operations on Tensors\n\n\n```\nx= torch.randn(2,3)\nprint(x)\n\nprint(\"Get absolute values\")\nprint(torch.abs(x))\n\n\nprint(\"Get the over all mean\")\nprint(torch.mean(x))\n\n\nprint(\"Get the means in the first dimension: \")\nprint(torch.mean(x,0))\n\nprint(\"Get element wise square: \")\nprint(x.pow(2))\n```\n\n \n -1.7490 1.3883 -0.1098\n 1.2384 0.5407 0.5787\n [torch.FloatTensor of size 2x3]\n \n Get absolute values\n \n 1.7490 1.3883 0.1098\n 1.2384 0.5407 0.5787\n [torch.FloatTensor of size 2x3]\n \n Get the over all mean\n 0.3145650302370389\n Get the means in the first dimension: \n \n -0.2553\n 0.9645\n 0.2345\n [torch.FloatTensor of size 3]\n \n Get element wise square: \n \n 3.0590 1.9274 0.0121\n 1.5336 0.2924 0.3349\n [torch.FloatTensor of size 2x3]\n \n\n\n### 7. Matrix/Vector Operations\n\n\n```\n## MATRICES \nx = torch.ones(2,2)\n\ny = torch.rand(2,2)\n\nprint(x,y)\n\n\n# addition\nprint(\"Sum of x and y: \")\nSum = x+y\nprint(Sum)\n\n\n# element wise multiplication\nprint(\"Element wise multiplication of x and y:\")\nmul = torch.mul(x,y)\nprint(mul)\n\n# Matrix multiplication \nprint(\"Matrix multiplication of x and y: \")\nMatMul = x.mm(y)\nprint(MatMul)\n\n\n\n```\n\n \n 1 1\n 1 1\n [torch.FloatTensor of size 2x2]\n \n 0.8177 0.8756\n 0.0064 0.5755\n [torch.FloatTensor of size 2x2]\n \n Sum of x and y: \n \n 1.8177 1.8756\n 1.0064 1.5755\n [torch.FloatTensor of size 2x2]\n \n Element wise multiplication of x and y:\n \n 0.8177 0.8756\n 0.0064 0.5755\n [torch.FloatTensor of size 2x2]\n \n Matrix multiplication of x and y: \n \n 0.8241 1.4511\n 0.8241 1.4511\n [torch.FloatTensor of size 2x2]\n \n\n\n\n```\n## VECTORS:\n\nx = torch.arange(5)\n#x= torch.unsqueeze(x,1)\nprint(x)\n\ny = torch.arange(1,6)\nprint(y)\n\n# inner product\nprint(\"Inner Product: \")\ninner = torch.dot(x,y) # will produce a scalar. Make sure x and y are of the same size\nprint(inner)\n\n\n# outer product\nprint(\"\\nOuter Product: \")\nouter = torch.ger(x,y) # will produce a matrix of size size(x) X size(y)\nprint(outer)\n\n\n```\n\n \n 0\n 1\n 2\n 3\n 4\n [torch.FloatTensor of size 5]\n \n \n 1\n 2\n 3\n 4\n 5\n [torch.FloatTensor of size 5]\n \n Inner Product: \n 40.0\n \n Outer Product: \n \n 0 0 0 0 0\n 1 2 3 4 5\n 2 4 6 8 10\n 3 6 9 12 15\n 4 8 12 16 20\n [torch.FloatTensor of size 5x5]\n \n\n\n### 8. Conversion from and to Numpy\n\n\n```\nx = np.array([[1,2],[3,4]])\nprint(x)\nprint(type(x))\n\n\n\n# From numpy ndarray to Tensor\nprint(\"\\nConverted to : \")\nx_tensor = torch.from_numpy(x)\nprint(x_tensor)\n\n\n# From Tensor to numpy ndarray\nprint(\"Converted back to : \")\nprint(x_tensor.numpy())\nprint(type(x_tensor.numpy()))\n```\n\n [[1 2]\n [3 4]]\n \n \n Converted to : \n \n 1 2\n 3 4\n [torch.LongTensor of size 2x2]\n \n Converted back to : \n [[1 2]\n [3 4]]\n \n\n\n### 9. Performance: Numpy arrays vs Tensors on CPU vs Tensors on GPU \n\n\n```\n\n# numpy ndarray on CPU\nprint(\"Numpy array on CPU\")\nx = np.random.random((1,64))\ny = np.random.random((1000, 64))\n%timeit z = (x*y).sum(axis=1)\n\n# torch Tensor on CPU\nprint(\"Tensor on CPU\")\nx = torch.from_numpy(x)\ny = torch.from_numpy(y)\n%timeit z=(x*y).sum(dim=1)\n\n# torch Tensor on GPU\nprint(\"Tensor on GPU\")\nx, y = x.cuda(), y.cuda()\n%timeit z = (x*y).sum(dim=1)\n\n```\n\n Numpy array on CPU\n 1000 loops, best of 3: 185 \u00b5s per loop\n Tensor on CPU\n 1000 loops, best of 3: 235 \u00b5s per loop\n Tensor on GPU\n 10000 loops, best of 3: 75.8 \u00b5s per loop\n\n\n### 10. What are Variables?\n\n\nAutograd package: \n\n\n```\nfrom torch.autograd import Variable\n\n\nx = torch.Tensor([4.0]) #[1,2,3],[3,4,5]\nprint(\"When the shape of x is: \"+str(x.shape))\nvar = Variable(x,requires_grad = True) # can differentiate wrt this # FOCUS MORE\nprint(var)\n\n\n# when the output is scalar\ny = torch.mean(2*var)\n\nprint(y)\n\ny.backward() # This will throw an error\n\n\nprint(var.grad)\n\n```\n\n When the shape of x is: torch.Size([1])\n Variable containing:\n 4\n [torch.FloatTensor of size 1]\n \n Variable containing:\n 8\n [torch.FloatTensor of size 1]\n \n Variable containing:\n 2\n [torch.FloatTensor of size 1]\n \n\n\n#### When x is an m X n Tensor\n\n\\begin{equation}\nx = \n\\begin{bmatrix}\n x_{11} & x_{12} \\\\\n x_{21} & x_{22} \n\\end{bmatrix} \n\\end{equation}\n\n\n\n\\begin{equation}\ny = mean(2*x)\n\\end{equation}\n\n\n\n\\begin{equation}\n y = 2 \\times 1/4 \\times (x_{11} +x_{12} +x_{21}+x_{22} )\n\\end{equation}\n\n\\begin{equation}\n\\frac{dy}{dx} = \n\\begin{bmatrix}\n \\frac{\\partial y}{\\partial x_{11}} & \\frac{\\partial y}{\\partial x_{12}} \\\\\n \\frac{\\partial y}{\\partial x_{21}} & \\frac{\\partial y}{\\partial x_{22}} \n\\end{bmatrix} \n\\end{equation}\n\n\n\n\n```\nx = torch.Tensor([[1,2],[3,4]]) #[1,2,3],[3,4,5]\nprint(\"When the shape of x is: \"+str(x.shape))\nprint(\"Wrapping Tensor x into Variable x:\")\nvar = Variable(x,requires_grad = True) # can differentiate wrt this # FOCUS MORE\nprint(var)\nprint(\"Variables essentially tensors but with an ability to \\\n utilize autograd package to compute gradients \")\n\n\n# when the output is scalar\ny = torch.mean(2*var.pow(2))\n\nprint(y)\n\ny.backward() # This will throw an error\n\n\nprint(var.grad)\n```\n\n When the shape of x is: torch.Size([2, 2])\n Wrapping Tensor x into Variable x:\n Variable containing:\n 1 2\n 3 4\n [torch.FloatTensor of size 2x2]\n \n Variables essentially tensors but with an ability to utilize autograd package to compute gradients \n Variable containing:\n 15\n [torch.FloatTensor of size 1]\n \n Variable containing:\n 1 2\n 3 4\n [torch.FloatTensor of size 2x2]\n \n\n\n#### 11. Activation Functions\n\nActivation functions can be found here :** Torch.nn.functional**\n\n\n```\nimport torch.nn.functional as F\nimport matplotlib.pyplot as plt\nx = torch.linspace(-5,5,1000)\nx = Variable(x)\n\n# using activation function \ny_relu = F.relu(x)\ny_sigmoid = F.sigmoid(x)\ny_tanh = F.tanh(x)\n\n\n\n# plt to visualize these activation function\nplt.plot(x.data.numpy(), y_relu.data.numpy(), c='red', label='relu')\nplt.plot(x.data.numpy(), y_sigmoid.data.numpy(), c='blue', label='sigmoid')\nplt.plot(x.data.numpy(), y_tanh.data.numpy(), c='green', label='tanh')\nplt.plot()\nplt.ylim((-1.5, 1.5))\nplt.title(\"Activation Functions\")\nplt.legend(loc='best');\n\n```\n\n\n```\n\"\"\"# When the output is non scalar\n\nx = Variable(torch.randn(10), requires_grad=True)\nprint(x)\ny = x ** 2\nprint(y)\n\n#grad = torch.randn(10)\n#print(grad)\n\ntorch.autograd.backward([y], [x])\n\n\"\"\"\n```\n", "meta": {"hexsha": "d32c4ed7bff28189a3c159db3d176c5cf6ca8dea", "size": 62937, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Pytorch_Basics.ipynb", "max_stars_repo_name": "MridulaMaddukuri/Pytorch_Tutorial", "max_stars_repo_head_hexsha": "7c7422d7fe8ddef4dd107e69bd9bd79d57205238", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Pytorch_Basics.ipynb", "max_issues_repo_name": "MridulaMaddukuri/Pytorch_Tutorial", "max_issues_repo_head_hexsha": "7c7422d7fe8ddef4dd107e69bd9bd79d57205238", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Pytorch_Basics.ipynb", "max_forks_repo_name": "MridulaMaddukuri/Pytorch_Tutorial", "max_forks_repo_head_hexsha": "7c7422d7fe8ddef4dd107e69bd9bd79d57205238", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 45.5075921909, "max_line_length": 24772, "alphanum_fraction": 0.6177447289, "converted": true, "num_tokens": 5432, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.7122321842389469, "lm_q1q2_score": 0.445945790352128}} {"text": "\n\n\n```\n#@title Licensed under the Apache License, Version 2.0 (the \"License\");\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n# https://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\n```\n\n# Introduction to Control Systems v1a\n\nHello everyone, in this occasion I would like to share my notebook that I used to create the poster for this year EuroPython.\n\nDon't forget to follow me on github then :)\n\nInstall the require library first, if you already installed it then skip to the next section.\n\n\n```\n!pip install control\n!pip install slycot\n!pip install scipy\n!pip install numpy\n!pip install matplotlib\n```\n\n Collecting control\n \u001b[?25l Downloading https://files.pythonhosted.org/packages/e8/b0/32a903138505dd4ea523f8a3fc156c4272aa58b10100ef24ff74ced2fae8/control-0.8.3.tar.gz (249kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 256kB 3.3MB/s \n \u001b[?25hRequirement already satisfied: numpy in /usr/local/lib/python3.6/dist-packages (from control) (1.18.5)\n Requirement already satisfied: scipy in /usr/local/lib/python3.6/dist-packages (from control) (1.4.1)\n Requirement already satisfied: matplotlib in /usr/local/lib/python3.6/dist-packages (from control) (3.2.2)\n Requirement already satisfied: kiwisolver>=1.0.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib->control) (1.2.0)\n Requirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.6/dist-packages (from matplotlib->control) (0.10.0)\n Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib->control) (2.4.7)\n Requirement already satisfied: python-dateutil>=2.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib->control) (2.8.1)\n Requirement already satisfied: six in /usr/local/lib/python3.6/dist-packages (from cycler>=0.10->matplotlib->control) (1.12.0)\n Building wheels for collected packages: control\n Building wheel for control (setup.py) ... \u001b[?25l\u001b[?25hdone\n Created wheel for control: filename=control-0.8.3-py2.py3-none-any.whl size=260982 sha256=0a6c52a9e379f850f1dbd04c38b1d1cea598de1914909142665e138bafb36f0f\n Stored in directory: /root/.cache/pip/wheels/c2/d9/cc/90b28cb139a6320a3af2285428b6da87eee8d8920c78bb0223\n Successfully built control\n Installing collected packages: control\n Successfully installed control-0.8.3\n Collecting slycot\n \u001b[?25l Downloading https://files.pythonhosted.org/packages/85/21/4e7110462f3529b2fbcff8a519b61bf64e0604b8fcbe9a07649c9bed9d7a/slycot-0.4.0.0.tar.gz (1.5MB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 1.6MB 3.4MB/s \n \u001b[?25h Installing build dependencies ... \u001b[?25l\u001b[?25hdone\n Getting requirements to build wheel ... \u001b[?25l\u001b[?25hdone\n Preparing wheel metadata ... \u001b[?25l\u001b[?25hdone\n Requirement already satisfied: numpy in /usr/local/lib/python3.6/dist-packages (from slycot) (1.18.5)\n Building wheels for collected packages: slycot\n Building wheel for slycot (PEP 517) ... \u001b[?25l\u001b[?25hdone\n Created wheel for slycot: filename=slycot-0.4.0-cp36-cp36m-linux_x86_64.whl size=1413148 sha256=056b26702cf834f59b6978482606dd82235755a9573fa98f837e577056f6b59f\n Stored in directory: /root/.cache/pip/wheels/a2/46/56/f82cbb2fd06556f4f3952a2eb2396e8fd29264fffecbaad3cf\n Successfully built slycot\n Installing collected packages: slycot\n Successfully installed slycot-0.4.0\n Requirement already satisfied: scipy in /usr/local/lib/python3.6/dist-packages (1.4.1)\n Requirement already satisfied: numpy>=1.13.3 in /usr/local/lib/python3.6/dist-packages (from scipy) (1.18.5)\n Requirement already satisfied: numpy in /usr/local/lib/python3.6/dist-packages (1.18.5)\n Requirement already satisfied: matplotlib in /usr/local/lib/python3.6/dist-packages (3.2.2)\n Requirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.6/dist-packages (from matplotlib) (0.10.0)\n Requirement already satisfied: python-dateutil>=2.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib) (2.8.1)\n Requirement already satisfied: numpy>=1.11 in /usr/local/lib/python3.6/dist-packages (from matplotlib) (1.18.5)\n Requirement already satisfied: kiwisolver>=1.0.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib) (1.2.0)\n Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib) (2.4.7)\n Requirement already satisfied: six in /usr/local/lib/python3.6/dist-packages (from cycler>=0.10->matplotlib) (1.12.0)\n\n\nCaveat: \n* Sorry for the array mess because the python control systems library always print that kind of array and I still try to figure out the way to get rid of it.\n* In case the equation got broken or not displayed properly try to visit the button below to view the notebook immediately from the Google Colab.\n\n\n\n\n\n\n# Tutorial on Control Systems Design with Python\n\nThe LTI (Linear Time-Invariant) were assumed to be used here because nonlinear or other complex systems are difficult to design and need a more advanced understanding of the control systems field.\n\n## Library Importing \n\nFirst of all, we need to import several essential libraries for designing the control systems, as listed below\n\n\n```\nimport control # This is python control library (https://python-control.readthedocs.io/en/latest/intro.html)\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport scipy\nfrom control.matlab import * # To import matlab like function in designing control systems\n```\n\n## Defining Transfer Function\n\nLet's assume we have an arbitrary transfer function equal to\n\nContinuous Transfer Function\n\n\\begin{align}\n\\frac{s}{s^2 + 2s + 6}\n\\end{align}\n\nDiscrete Transfer Function\n\n\\begin{align}\n\\frac{z}{z^2 + 2z + 6}\n\\end{align}\n\nImport the Function from Python Control\n\n\n```\nfrom control import TransferFunction, pole, zero # Transfer Function function import\n```\n\n### Python Control Function\n\n\n```\n# Continuous Time Systems\ns = TransferFunction.s\nsysc = s / (s**2 + 2*s + 6)\n# Discrete Time Systems\nz = TransferFunction.z\nsysd = z / (z**2 + 2*z + 6)\n```\n\n### MATLAB like Function\n\n\n```\n# Continuous Time Systems\ns = tf('s')\nsysc = s/(s**2 + 2*s + 6)\n# Discrete Time Systems\nz = tf('z')\nsysd = z / (z**2 + 2*z + 6)\n```\n\n### Stability Check\n\nIn order to get the specified output, the various parameters of the system must be controlled. Along with this, the system must be stable enough so that the output must not get affected by the undesirable variations in the parameter of the system or disturbances.\n\nThus we can say that a stable system is designed so as to get the desired response of the system without any intolerable variation with the changes in the system parameters.\n\nSource: https://electronicscoach.com/stability-of-control-system.html\n\n\n\nSource: https://flylib.com/books/en/2.729.1/the_z_transform.html\n\nIn the continuous systems, it called unstable if\n\n* The system poles were located on right half plane of s-plane \n\nIn the discrete systems, it caled unstable if\n\n\n* The systems poles were located outside the unitary circle\n\n\n\n```\n# The function of python control libraries are the same as MATLAB\n# From this we can analyze the systems stability by find out the poles\n\n# Continuous Systems\n\nprint('Continuous Systems')\n# The poles\npc = pole(sysc)\nzc = zero(sysc)\n\nprint('Pole of The Systems'),\nprint(pc),\n\nprint()\n\n# Discrete Systems\n\nprint('Discrete systems')\n\n# The poles\npd = pole(sysd)\nzd = zero(sysd)\n\nprint('Pole of The Systems'),\nprint(pd)\n\n```\n\n Continuous Systems\n Pole of The Systems\n [-1.+2.23606798j -1.-2.23606798j]\n \n Discrete systems\n Pole of The Systems\n [-1.+2.23606798j -1.-2.23606798j]\n\n\n## Defining State Space Matrix of System\n\nImporting The Main Function \n\n\n```\nfrom control import StateSpace\n```\n\n### Convert Transfer Function to State Space Form and Vice Versa\n\n\n```\n# In this case Python Control Function as same as MATLAB Like Function\nfrom control import tf2ss\nsysc = tf2ss(sysc)\nsysd = tf2ss(sysd)\nsysc\n```\n\n\n\n\n A = [[-2. -6.]\n [ 1. 0.]]\n \n B = [[-1.]\n [ 0.]]\n \n C = [[-1. 0.]]\n \n D = [[0.]]\n\n\n\n\n```\n# Assume we have systems as below\nA = np.array([[-2, -6], [1, 0]])\nB = np.array([[-1], [0]])\nC = np.array([[-1, 0]])\nD = np.array([[0]])\n```\n\n### Python Control Function\n\n\n```\n# Continuous Systems\nsysc = StateSpace(A,B,C,D)\nsysc\n# Discrete Systems\nts = 0.01 # The Sampling Time\nsysd = StateSpace(A,B,C,D,ts)\nsysd\n```\n\n\n\n\n A = [[-2. -6.]\n [ 1. 0.]]\n \n B = [[-1.]\n [ 0.]]\n \n C = [[-1. 0.]]\n \n D = [[0.]]\n \n dt = 0.01\n\n\n\n### MATLAB like Function\n\n\n```\n# Continuous Systems\nsysc = ss(A,B,C,D)\nsysc\n# Discrete Systems\nts = 0.01 # The Sampling Time\nsysd = ss(A,B,C,D,ts)\nsysd\n```\n\n\n\n\n A = [[-2. -6.]\n [ 1. 0.]]\n \n B = [[-1.]\n [ 0.]]\n \n C = [[-1. 0.]]\n \n D = [[0.]]\n \n dt = 0.01\n\n\n\n### Stability Check\n\nThe aim of stability check in state space form is same as in transfer function, it's to make the system is can reach the desired point or reference without any intolerable variation or change.\n\nThe way to check it also the same but in this case instead of using pole, the stability is checked using eigenvalue of the state space matrix.\n\n\n\n\n\n```\n# Check the systems stability by viewing the eigenvalue\n# Continuous Systems\neigs, eigvs = np.linalg.eig(sysc.A)\nprint('Continuous Systems Eigenvalues'),\nprint(eigs)\n# Discrete Systems\neigd, eigvd = np.linalg.eig(sysd.A)\nprint('Discrete Systems Eigenvalues'),\nprint(eigd)\n```\n\n Continuous Systems Eigenvalues\n [-1.+2.23606798j -1.-2.23606798j]\n Discrete Systems Eigenvalues\n [-1.+2.23606798j -1.-2.23606798j]\n\n\n### Controllability and Observability Check\n\nThe intuition according to the controllability and observability checking\n* Controllability:In order to be able to do whatever we want with the given dynamic system under control input,the system must be controllable.\n* Observability:In order to see what is going on inside the system under observation,the system must be observable.\n\nSource: https://www.ece.rutgers.edu/~gajic/psfiles/chap5.pdf\n\n\n\n\n```\nfrom control import obsv, ctrb\n# In this case the function for control libraries and MATLAB are same\n# Continuous Systems\n# Controllability Check\ncc = ctrb(sysc.A, sysc.B)\nrankcc = np.linalg.matrix_rank(cc)\nprint('Continuous Systems', '\\n')\nprint('The Controllability Matrix'),\nprint(cc),\nprint('Rank of Controllability Matrix'),\nprint(rankcc),\n# Observability Check\noc = obsv(sysc.A, sysc.C)\nrankoc = np.linalg.matrix_rank(oc)\nprint('The Observability Matrix'),\nprint(oc),\nprint('Rank of Observability Matrix'),\nprint(rankoc),\nprint()\n# Discrete Systems\n# Controllability Check\ncd = ctrb(sysd.A, sysc.B)\nrankcd = np.linalg.matrix_rank(cd)\nprint('Discrete Systems', '\\n')\nprint('The Controllability Matrix'),\nprint(cd),\nprint('Rank of Controllability Matrix'),\nprint(rankcd),\n# Observability Check\nod = obsv(sysd.A, sysc.C)\nrankod = np.linalg.matrix_rank(od)\nprint('The Observability Matrix'),\nprint(od),\nprint('Rank of Observability Matrix'),\nprint(rankod)\n```\n\n Continuous Systems \n \n The Controllability Matrix\n [[-1. 2.]\n [ 0. -1.]]\n Rank of Controllability Matrix\n 2\n The Observability Matrix\n [[-1. 0.]\n [ 2. 6.]]\n Rank of Observability Matrix\n 2\n \n Discrete Systems \n \n The Controllability Matrix\n [[-1. 2.]\n [ 0. -1.]]\n Rank of Controllability Matrix\n 2\n The Observability Matrix\n [[-1. 0.]\n [ 2. 6.]]\n Rank of Observability Matrix\n 2\n\n\n## Analyze the Control Systems\n\n\n```\nfrom control import bode_plot, nyquist_plot, root_locus, pzmap\n```\n\n##### Pole Zero Map\n\n\n```\n# Continuous\npzmap(sysc)\n```\n\n\n```\n# Discrete\npzmap(sysd)\n```\n\n##### Root Locus\n\n\n```\n# Continuous\nroot_locus(sysc)\n```\n\n\n```\n# Discrete\nroot_locus(sysd)\n```\n\n##### Bode Plot\n\n\n```\n# continuous\nbode_plot(sysc)\n```\n\n\n```\n# Discrete\nbode_plot(sysd)\n```\n\n##### Nyquist Plot\n\n\n```\n# Continuous\nnyquist_plot(sysc)\n```\n\n\n```\n# Discrete\nnyquist_plot(sysd)\n```\n\n## Test the Systems Response\n\nActually, we can analyze the response of systems from two approach, first time domain approach, then frequency domain approach\n\n#### Time Domain Approach\n\n##### Step Response\n\n\n```\nfrom control import step_response # Step Response function import\n```\n\n###### Python Control Function\n\n\n```\n# Continuous Time Systems\ntc, yc = step_response(sysc)\nplt.subplot(2,1,1)\nplt.plot(tc,yc)\nplt.title('Continuous Step Response')\nplt.xlabel('Time (s)')\nplt.ylabel('Amplitude')\n# Discrete Time Systems\ntd, yd = step_response(sysd)\nplt.subplot(2,1,2)\nplt.plot(td,yd)\nplt.title('Discrete Step Response')\nplt.xlabel('Time (s)')\nplt.ylabel('Amplitude')\nplt.tight_layout(1)\n```\n\n###### MATLAB like function\n\n\n```\n# Continuous Time Systems\nyc, tc = step(sysc)\nplt.subplot(2,1,1)\nplt.plot(tc,yc)\nplt.title('Continuous Step Response')\nplt.xlabel('Time (s)')\nplt.ylabel('Amplitude')\n# Discrete Time Systems\nyd, td = step(sysd)\nplt.subplot(2,1,2)\nplt.plot(td,yd)\nplt.title('Discrete Step Response')\nplt.xlabel('Time (s)')\nplt.ylabel('Amplitude')\nplt.tight_layout(1)\n```\n\n##### Impulse Response\n\n\n```\nfrom control import impulse_response # Impulse Response function import\n```\n\n###### Python Control Function\n\n\n```\n# Continuous Time Systems\ntc, yc = impulse_response(sysc)\nplt.subplot(2,1,1)\nplt.plot(tc,yc)\nplt.title('Continuous Step Response')\nplt.xlabel('Time (s)')\nplt.ylabel('Amplitude')\n# Discrete Time Systems\ntd, yd = impulse_response(sysd)\nplt.subplot(2,1,2)\nplt.plot(td,yd)\nplt.title('Discrete Step Response')\nplt.xlabel('Time (s)')\nplt.ylabel('Amplitude')\nplt.tight_layout(1)\n```\n\n###### MATLAB like Function\n\n\n```\n# Continuous Time Systems\nyc, tc = impulse(sysc)\nplt.subplot(2,1,1)\nplt.plot(tc,yc)\nplt.title('Continuous Step Response')\nplt.xlabel('Time (s)')\nplt.ylabel('Amplitude')\n# Discrete Time Systems\nyd, td = impulse(sysd)\nplt.subplot(2,1,2)\nplt.plot(td,yd)\nplt.title('Discrete Step Response')\nplt.xlabel('Time (s)')\nplt.ylabel('Amplitude')\nplt.tight_layout(1)\n```\n\n#### Frequency Domain Approach\n\n##### Frequency Response\n\n\n```\nfrom control import freqresp\n```\n\n\n```\n# Continuous\nfreqresp(sysc, [10])\n```\n\n\n\n\n (array([[[0.10405382]]]), array([[[-1.36115648]]]), array([10]))\n\n\n\n\n```\n# Discrete\nfreqresp(sysd, [12])\n```\n\n\n\n\n (array([[[0.11148702]]]), array([[[0.06678141]]]), array([12]))\n\n\n\n## Okay Then, Let's Simulate The Real World Systems!\n\nIn this occasion, the design of the control algorithm was using MATLAB like function, so the code also compatible with MATLAB. In case you want to try the simulation then it will be also worked on MATLAB but you must revise it with slight several changes on the syntax.\n\n### DC Motor Speed Control\n\n\n\n\nSource : http://ctms.engin.umich.edu/CTMS/index.php?example=MotorSpeed§ion=SystemModeling (CTMS, University of Michigan)\n\n**The controlling aim for this system is to drive the DC motor speed into the desired speed by regulating a voltage to the motor.**\n\nThe system characteristics are continuous and linear in nature. This is because in this tutorial I only limit the system to Linear System.\n\nAs well, in this case the name for the controlling goal is tracking. Since, the goal of our control problem is to drive the DC motor speed to the desired speed.\n\n\n```\nJ = 0.08; # The motor moment of inertia\nb = 0.05;\nK = 0.01;\nR = 0.5;\nL = 0.5;\ns = tf('s');\nsys_tf = K/((J*s+b)*(L*s+R)+K**2)\n# Also change the form of transfer function into state space model for controlling through the modern control algorithm\nsys_ss = tf2ss(sys_tf)\n```\n\n\n```\n## stability Checking\npzmap(sys_ss)\npole(sys_tf)\n```\n\n\n```\n# Stability Checking\neig = np.linalg.eig(sys_ss.A)\nprint('System Eigenvalue'),\nprint(eig[0], \"\\n\"),\n\n# Controllability Check\nctrM = ctrb(sys_ss.A, sys_ss.B)\nrankCtr = np.linalg.matrix_rank(ctrM)\nprint('Controllability Rank'),\nprint(rankCtr, \"\\n\"),\n\n# Observabiity Check\nobsM = obsv(sys_ss.A, sys_ss.C)\nrankObs = np.linalg.matrix_rank(obsM)\nprint('Observability Rank'),\nprint(rankObs)\n```\n\n System Eigenvalue\n [-0.9932104 -0.6317896] \n \n Controllability Rank\n 2 \n \n Observability Rank\n 2\n\n\nIt's stable due to pole or the eigenvalue located in left half plane. The system were controllable and observable because both of controllability and observability matrix have a full rank.\n\n#### PID Controller\n\nThe equation of PID controller is denoted below,\n\n\\begin{align}\nu(t) = K_p*e(t) + K_d*\\frac{de(t)}{dt} + K_i * \\int{e(t)} \n\\end{align}\n\nWhere Kp, Ki, Kd, e(t) are proportional gain, integral gain, derivative gain, and system error, respectively.\n\nIf we want to write it as transfer function form then, \n\\begin{align}\nU(s)=K_p*E(s) + K_d * sE(s) + K_i * \\frac{1}{s} * E(s) \n\\end{align}\n\nIf we assume\n\\begin{align}\n\\tau_i = \\frac{K_p}{K_i} ;\n\\tau_d = \\frac{K_d}{K_p}\n\\end{align}\n\nThus, the final equation for the controller is,\n\\begin{align}\nU(s) = K_p * E(s)(1 + s * \\tau_d + \\frac{1}{s*\\tau_i})\n\\end{align}\n\nIn this test I select the gain arbitrary as follows, Kp = 160, Ki = 4, and Kd = 80. As well try to experiment with the other gain combination to see the effect of gain changing.\n\n\n\n\n```\nKp = 160; # The proportional gain\nKi = 4; # The integral gain\nKd = 80; # The derivative gain\nTi = Kp / Ki\nTd = Kd / Kp\npid = Kp * (1 + 1 / (Ti * s) + Td * s)\nisys = feedback(sys_tf * pid, 1)\nt = np.linspace(0, 10)\ny, t = step(isys, t)\nplt.plot(t, y)\nplt.title('PID Controller Response')\nplt.xlabel('Time (s)')\nplt.ylabel('Motor Speed (RPM)')\n```\n\n#### State Feedback Controller (Pole Placement Method)\n\nIn this design, I would like to place all of the two pole to the left half plane coordinate and place it to specific pole location as this (-3, -4). You can also try the other combination to test how the system response change\n\n\n```\ndesiredPole = np.array([-3,-4])\nppGain = place(sys_ss.A, sys_ss.B, desiredPole)\nfeedbackMech = sys_ss.A - sys_ss.B * ppGain\nnewSys = ss(feedbackMech, sys_ss.B, sys_ss.C, 0)\nt = np.linspace(0,10)\nscaleSF = 1 / 0.02083333 # Because there are large steady state error we should a precompensator for scaling the reference signal\nyref = 20*(np.sin(2*np.pi*t*0.1))\ny, t = step(newSys * scaleSF, t)\nplt.plot(t,y)\nplt.title('State-Feedback Controller Response')\nplt.xlabel('Time (s)')\nplt.ylabel('Motor Speed (RPM)')\n```\n\n#### LQR Controller\n\nLQR (Linear Quadratic Regulator) is the one of Optimal Control Algorithm variant, it uses calculus variation such as Algebraic Riccati Equation (ARE) to determine the optimal gain for the controller. \n\n\n```\n# Defining the LQR Parameter\nQ = np.array([[0,0],\n [0,100]])\nR = 10\ngainLQR, X, E = lqr(sys_ss.A, sys_ss.B, Q, R)\nfeedbackMech = sys_ss.A - sys_ss.B * gainLQR\nnewSysqr = ss(feedbackMech, sys_ss.B, sys_ss.C, 0)\nt = np.linspace(0,10)\nscaleLQR = 1 / 0.07754505 # Because there are large steady state error we should a precompensator for scaling the reference signal\ny,t = step(newSysqr * scaleLQR, t)\nplt.plot(t,y)\nplt.title('LQR Controller Response')\nplt.xlabel('Time (s)')\nplt.ylabel('Motor Speed (RPM)')\n```\n\n#### Compare it Together\n\nBefore we can use the variations of reference signal such as square wave, sinusoidal signal, etc. We ought to import the forced_response function first from the Python Control Systems Library and the signal function from SciPy library\n\n\n```\nfrom control import forced_response\nfrom scipy import signal\n```\n\n##### Step Speed Reference of 1200 RPM\n\n\n```\n# The Step Signal with 1500 RPM amplitude\nmaxSim = 10 # The simulation time is 10 second\nt = np.linspace(0, maxSim)\namp = 1200 # Because the reference signal is on 1500 RPM \nref = amp * np.ones(np.shape(t))\nisys # The transfer function \ny1 = forced_response(isys, T = t, U = ref)\nres1 = y1[1]\ny2 = forced_response(newSys * scaleSF, T = t, U = ref)\nres2 = y2[1]\ny3 = forced_response(newSysqr * scaleLQR, T = t, U = ref)\nres3 = y3[1]\nplt.plot(t,ref)\nplt.plot(t,res1)\nplt.plot(t,res2)\nplt.plot(t,res3)\nplt.title('1200 RPM Speed Reference')\nplt.xlabel('Time (s)')\nplt.ylabel('Motor Speed (RPM)')\nplt.legend(['Reference','PID', 'State Feedback','LQR'])\nfrom google.colab import files\nplt.savefig('step.png',type = 'png',DPI = 1200)\nfiles.download('step.png')\n```\n\n##### Sinusoidal Speed Reference \n\n\n```\n# The Sinusoidal Signal with 1000 RPM amplitude\nmaxSim = 10 # The simulation time is 10 second\nt = np.linspace(0, maxSim)\namp = 1000 # Because the reference signal is on 1500 RPM\nf = 0.1 # The sinusoidal signal frequency \nref = amp * np.sin(2 * np.pi * f * t)\nisys # The transfer function \ny1 = forced_response(isys, T = t, U = ref)\nres1 = y1[1]\ny2 = forced_response(newSys * scaleSF, T = t, U = ref)\nres2 = y2[1]\ny3 = forced_response(newSysqr * scaleLQR, T = t, U = ref)\nres3 = y3[1]\nplt.plot(t,ref)\nplt.plot(t,res1)\nplt.plot(t,res2)\nplt.plot(t,res3)\nplt.title('Sinusoidal Speed Reference')\nplt.xlabel('Time (s)')\nplt.ylabel('Motor Speed (RPM)')\nplt.legend(['Reference','PID', 'State Feedback','LQR'])\nplt.savefig('')\n```\n\n##### Square Speed Reference\n\n\n```\n# The Square Signal with 500 RPM amplitude\nmaxSim = 10 # The simulation time is 10 second\nt = np.linspace(0, maxSim, endpoint = True)\namp = 500 # Because the reference signal is on 500 RPM\nf = 0.1 # The square signal frequency in Hz\nref = amp * signal.square(2 * np.pi * f * t)\nisys # The transfer function \ny1 = forced_response(isys, T = t, U = ref)\nres1 = y1[1]\ny2 = forced_response(newSys * scaleSF, T = t, U = ref)\nres2 = y2[1]\ny3 = forced_response(newSysqr * scaleLQR, T = t, U = ref)\nres3 = y3[1]\nplt.plot(t,ref)\nplt.plot(t,res1)\nplt.plot(t,res2)\nplt.plot(t,res3)\nplt.title('Square Wave Speed Reference')\nplt.xlabel('Time (s)')\nplt.ylabel('Motor Speed (RPM)')\nplt.legend(['Reference','PID', 'State Feedback','LQR'])\nplt.savefig('square.png',type = 'png',DPI = 1200)\nfiles.download('square.png')\n```\n\n##### Sawtooth Speed Reference\n\n\n```\n# The Sawtooth Signal with 800 RPM amplitude\nmaxSim = 20 # The simulation time is 20 second\nt = np.linspace(0, maxSim)\namp = 800 # Because the reference signal is on 800 RPM\nf = 0.2 # The sawtooth signal frequency \nref = amp * signal.sawtooth(2 * np.pi * f * t)\ny1 = forced_response(isys, T = t, U = ref)\nres1 = y1[1]\ny2 = forced_response(newSys * scaleSF, T = t, U = ref)\nres2 = y2[1]\ny3 = forced_response(newSysqr * scaleLQR, T = t, U = ref)\nres3 = y3[1]\nplt.plot(t,ref)\nplt.plot(t,res1)\nplt.plot(t,res2)\nplt.plot(t,res3)\nplt.title('Sawtooth Signal Speed Reference')\nplt.xlabel('Time (s)')\nplt.ylabel('Motor Speed (RPM)')\nplt.legend(['Reference','PID', 'State Feedback','LQR'])\n```\n\n# More Advanced Control Systems Design (Optional)\n\nComing Soon on V2\n\n\n## Robust Control Design\n\n\n```\n\n```\n\n

Copyright © 2020 - brilianputraa

\n", "meta": {"hexsha": "d59d54cbe680e883e777ec1307a3e58e72f2d614", "size": 508635, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "EuroPython/cspythonv1.ipynb", "max_stars_repo_name": "brilianputraa/dyncontrol", "max_stars_repo_head_hexsha": "429b4e8c8287ab6552275202c0237af3f357f867", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2020-07-26T19:12:17.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-18T09:25:44.000Z", "max_issues_repo_path": "EuroPython/cspythonv1.ipynb", "max_issues_repo_name": "brilianputraa/dyncontrol", "max_issues_repo_head_hexsha": "429b4e8c8287ab6552275202c0237af3f357f867", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "EuroPython/cspythonv1.ipynb", "max_forks_repo_name": "brilianputraa/dyncontrol", "max_forks_repo_head_hexsha": "429b4e8c8287ab6552275202c0237af3f357f867", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2020-07-26T19:24:23.000Z", "max_forks_repo_forks_event_max_datetime": "2020-09-04T13:43:35.000Z", "avg_line_length": 172.0686738836, "max_line_length": 43670, "alphanum_fraction": 0.8540328526, "converted": true, "num_tokens": 6813, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241632752916, "lm_q2_score": 0.7122321781307374, "lm_q1q2_score": 0.4459457765898464}} {"text": "```python\nfrom IPython.core.display import HTML, Image\ncss_file = 'style.css'\nHTML(open(css_file, 'r').read())\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n```python\nfrom sympy import init_printing, Matrix, symbols, eye, Rational\ninit_printing()\n```\n\n# Columnspace and nullspace of a matrix\n\n## Columnspaces of matrices\n\nWe saw in the previous lecture that columns of a matrix can be regarded as vectors.\n\nConsider the following example matrix.\n\n\n```python\nA = Matrix([[1, 1, 2], [2, 1, 3], [3, 1, 4], [4, 1, 5]])\nA\n```\n\n\n```python\nb = Matrix([1, 2, 3, 4])\nb\n```\n\nEach of the column spaces are vectors (column space) in $\\mathbb{R}^4$. The linear combinations of all the column vectors form a subspace. Is it the whole $V= \\mathbb{R}^4$, though? The reason why we ask, is because we want to bring it back to a system of linear equations and ask the question: _Is there (always) a solution to_ (1)_?_\n\n$${A}\\underline{x}=\\underline{b}\\tag{1}$$\n\nIn other words, which right-hand sides $\\underline{b}$ are allowed? In our example above we are in $\\mathbb{R}^4$ and we ask if linear combination of all of them fill $\\mathbb{R}^4$.\n\nFrom our example above some right-hand sides will be allowed (they form a subspace). Let's look at an example for $\\underline{b}$.\n\n\n```python\nx1, x2, x3 = symbols('x1, x2, x3')\nvec_x = Matrix([x1, x2, x3])\nb = Matrix([1, 2, 3, 4])\nA, vec_x, b\n```\n\n\n```python\nA * vec_x\n```\n\nWhile we can view this as a system of linear equation, we will prefer the column view. By viewing the matrix of coefficients, with the columns as vectors, we are asking: _How many, $x_1$, of column $1$ plus how many, $x_2$, of columns $2$, plus how many, $x_3$ of column $3$ equals $\\underline{b}$_? The question is easier to visualize when written as in (2).\n\n$$x_1\\begin{bmatrix}1\\\\2\\\\3\\\\4\\end{bmatrix}+x_2\\begin{bmatrix}1\\\\1\\\\1\\\\1\\end{bmatrix}+x_3\\begin{bmatrix}2\\\\3\\\\4\\\\5\\end{bmatrix}=\\underline{b}\\tag{2}$$\n\nWell, since $\\underline{b}$ is the same as the first column, $\\underline{x}$ would be as shown in (3).\n\n$$\\begin{bmatrix}1\\\\0\\\\0\\end{bmatrix}\\tag{3}$$\n\n\n```python\nA * Matrix([1, 0, 0]) == b\n```\n\n\n\n\n True\n\n\n\n### Linear independence\n\nWe really need to know if the columns above are linearly independent (not constant multiples or additions of each other). We note that column three above is a linear combination of the first two, so adds nothing _new_. Actually, we could also throw away the first one because it is column $3$ plus $-1$ times column $2$. Same for column$2$. We thus have two columns left and we say that the column space is of dimension $2$ (a 2-dimensional subspace of $\\mathbb{R}^4$).\n\n## The nullspace\n\nThe null space contains all solutions of $\\underline{x}$ for $A\\underline{x}=\\underline{0}$. This solution(s) is (are) in $\\mathbb{R}^3$.\n\n\n```python\nzero_b = Matrix([0, 0, 0, 0])\nA, vec_x, zero_b\n```\n\n\n\n\n$$\\begin{pmatrix}\\left[\\begin{matrix}1 & 1 & 2\\\\2 & 1 & 3\\\\3 & 1 & 4\\\\4 & 1 & 5\\end{matrix}\\right], & \\left[\\begin{matrix}x_{1}\\\\x_{2}\\\\x_{3}\\end{matrix}\\right], & \\left[\\begin{matrix}0\\\\0\\\\0\\\\0\\end{matrix}\\right]\\end{pmatrix}$$\n\n\n\nSome possible solutions are shown in (4).\n\n$$\\begin{bmatrix}0\\\\0\\\\0\\end{bmatrix},\\quad\\begin{bmatrix}1\\\\1\\\\-2\\end{bmatrix},\\quad\\begin{bmatrix}2\\\\2\\\\-2\\end{bmatrix}\\tag{4}$$\n\nIn fact, fro any constant, $c$, these can all be written as in (5).\n\n$$ {c}\\begin{bmatrix}1\\\\1\\\\-1\\end{bmatrix}\\tag{5}$$\n\n__Always__ remember, for any space the rules of addition and scalar multiplication must hold for vectors to remain in that space.\n\n\n```python\n\n```\n", "meta": {"hexsha": "23e1021c41178cc1dd42567ea3a329de61ff72ac", "size": 20952, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Mathematics/Linear Algebra/0.0 MIT-18.06 - Jupyter/Lecture_6_Column_and_null_spaces.ipynb", "max_stars_repo_name": "okara83/Becoming-a-Data-Scientist", "max_stars_repo_head_hexsha": "f09a15f7f239b96b77a2f080c403b2f3e95c9650", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Mathematics/Linear Algebra/0.0 MIT-18.06 - Jupyter/Lecture_6_Column_and_null_spaces.ipynb", "max_issues_repo_name": "okara83/Becoming-a-Data-Scientist", "max_issues_repo_head_hexsha": "f09a15f7f239b96b77a2f080c403b2f3e95c9650", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Mathematics/Linear Algebra/0.0 MIT-18.06 - Jupyter/Lecture_6_Column_and_null_spaces.ipynb", "max_forks_repo_name": "okara83/Becoming-a-Data-Scientist", "max_forks_repo_head_hexsha": "f09a15f7f239b96b77a2f080c403b2f3e95c9650", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2022-02-09T15:41:33.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-11T07:47:40.000Z", "avg_line_length": 45.0580645161, "max_line_length": 4176, "alphanum_fraction": 0.6717735777, "converted": true, "num_tokens": 1740, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.7931059487389968, "lm_q1q2_score": 0.44586552709380034}} {"text": "\n\n# Week 4: Linear Algebra IV\n\n# Tutorial 1\n\n# [insert your name]\n\n**Important reminders**: Before starting, click \"File -> Save a copy in Drive\". Produce a pdf for submission by \"File -> Print\" and then choose \"Save to PDF\".\n\nTo complete this tutorial, you should have watched Videos 4.1, 4.2, 4.3, and 4.4.\n\n\n**Credits**: Video 4.3 is the Week 1 Day 5 Intro Video from NMA (Neuromatch Academy) (https://github.com/NeuromatchAcademy/course-content). Exercise 2/3 in this tutorial are modified from content in NMA W1D5 tutorials.\n\nWe are again using code for visualizing linear transformations from https://openedx.seas.gwu.edu/courses/course-v1:GW+EngComp4+2019/about. In particular, we are using their `plot_linear_transformation` and `plot_linear_transformations` functions.\n\nThe interactive demo in Exercise 2A is based on matlab code from: https://stats.stackexchange.com/questions/2691/making-sense-of-principal-component-analysis-eigenvectors-eigenvalues\n\n\n```\n# @markdown Imports\n\n# Imports\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport ipywidgets as widgets # interactive display\n\n```\n\n\n```\n# @markdown Plotting functions\nimport numpy\nfrom numpy.linalg import inv, eig\nfrom math import ceil\nfrom matplotlib import pyplot, ticker, get_backend, rc\nfrom mpl_toolkits.mplot3d import Axes3D\nfrom itertools import cycle\n\n\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/course-content/master/nma.mplstyle\")\n\nclassic = 'k'\n\n_int_backends = ['GTK3Agg', 'GTK3Cairo', 'MacOSX', 'nbAgg',\n 'Qt4Agg', 'Qt4Cairo', 'Qt5Agg', 'Qt5Cairo',\n 'TkAgg', 'TkCairo', 'WebAgg', 'WX', 'WXAgg', 'WXCairo']\n_backend = get_backend() # get current backend name\n\n# shrink figsize and fontsize when using %matplotlib notebook\nif _backend in _int_backends:\n fontsize = 4\n fig_scale = 0.75\nelse:\n fontsize = 5\n fig_scale = 1\n\ngrey = '#808080'\ngold = '#cab18c' # x-axis grid\nlightblue = '#0096d6' # y-axis grid\ngreen = '#008367' # x-axis basis vector\nred = '#E31937' # y-axis basis vector\ndarkblue = '#004065'\n\npink, yellow, orange, purple, brown = '#ef7b9d', '#fbd349', '#ffa500', '#a35cff', '#731d1d'\n\nquiver_params = {'angles': 'xy',\n 'scale_units': 'xy',\n 'scale': 1,\n 'width': 0.012}\n\ngrid_params = {'linewidth': 0.5,\n 'alpha': 0.8}\ndef plot_sample_images(X):\n \"\"\"\n Plots 9 images from the data.\n\n Args:\n X (numpy array of floats) : Data matrix each column corresponds to a\n different random variable\n\n Returns:\n Nothing.\n\n \"\"\"\n im_size = int(np.sqrt(X.shape[1]))\n fig, ax = plt.subplots()\n k = 0\n for k1 in range(3):\n for k2 in range(3):\n k = k + 1\n plt.imshow(np.reshape(X[k, :], (im_size, im_size)),\n extent=[(k1 + 1) * im_size, k1 * im_size, (k2+1) * im_size, k2 * im_size],\n vmin=0, vmax=255, cmap='gray')\n plt.xlim((3 * im_size, 0))\n plt.ylim((3 * im_size, 0))\n plt.tick_params(axis='both', which='both', bottom=False, top=False,\n labelbottom=False)\n plt.clim([0, 250])\n ax.set_xticks([])\n ax.set_yticks([])\n plt.show()\n\n\ndef plot_variance_explained(variance_explained):\n \"\"\"\n Plots eigenvalues.\n\n Args:\n variance_explained (numpy array of floats) : Vector of variance explained\n for each PC\n\n Returns:\n Nothing.\n\n \"\"\"\n\n plt.figure()\n plt.plot(np.arange(1, len(variance_explained) + 1), variance_explained,\n '--k')\n plt.xlabel('Number of components')\n plt.ylabel('Variance explained')\n plt.show()\n\ndef plot_reconstructions(X, X_reconstructed):\n \"\"\"\n Plots 9 images in the dataset side-by-side with the reconstructed\n images.\n\n Args:\n X (numpy array of floats) : Data matrix each column\n corresponds to a different\n random variable\n X_reconstructed (numpy array of floats) : Data matrix each column\n corresponds to a different\n random variable\n\n Returns:\n Nothing.\n \"\"\"\n\n im_size = int(np.sqrt(X.shape[1]))\n\n plt.figure()\n ax = plt.subplot(121)\n k = 0\n for k1 in range(3):\n for k2 in range(3):\n k = k + 1\n plt.imshow(np.reshape(X[k, :], (im_size, im_size)),\n extent=[(k1 + 1) * im_size, k1 * im_size, (k2 + 1) * im_size, k2 * im_size],\n vmin=0, vmax=255, cmap='gray')\n plt.xlim((3 * im_size, 0))\n plt.ylim((3 * im_size, 0))\n plt.tick_params(axis='both', which='both', bottom=False, top=False,\n labelbottom=False)\n ax.set_xticks([])\n ax.set_yticks([])\n plt.title('Data')\n plt.clim([0, 250])\n ax = plt.subplot(122)\n k = 0\n for k1 in range(3):\n for k2 in range(3):\n k = k + 1\n plt.imshow(np.reshape(np.real(X_reconstructed[k, :]), (im_size, im_size)),\n extent=[(k1 + 1) * im_size, k1 * im_size, (k2 + 1) * im_size, k2 * im_size],\n vmin=0, vmax=255, cmap='gray')\n plt.xlim((3 * im_size, 0))\n plt.ylim((3 * im_size, 0))\n plt.tick_params(axis='both', which='both', bottom=False, top=False,\n labelbottom=False)\n ax.set_xticks([])\n ax.set_yticks([])\n plt.clim([0, 250])\n plt.title('Reconstructed')\n plt.tight_layout()\n\ndef plot_principal_components(weights):\n \"\"\"\n Visualize PCA basis vector weights. Red = positive weights,\n blue = negative weights, white = zero weight.\n\n Args:\n weights (numpy array of floats) : PCA basis vector\n\n Returns:\n Nothing.\n \"\"\"\n im_size = int(np.sqrt(X.shape[1]))\n\n fig, ax = plt.subplots()\n cmap = plt.cm.get_cmap('seismic')\n plt.imshow(np.real(np.reshape(weights, (im_size, im_size))), cmap=cmap)\n plt.tick_params(axis='both', which='both', bottom=False, top=False,\n labelbottom=False)\n plt.clim(-.15, .15)\n plt.colorbar(ticks=[-.15, -.1, -.05, 0, .05, .1, .15])\n ax.set_xticks([])\n ax.set_yticks([])\n plt.show()\n\n\ndef plot_pca_transformation(data, transformed_data):\n\n\n fig, axes = plt.subplots(1, 2)\n axes[0].scatter(data[:,0], data[:, 1], s=1, c='#63BA79');\n for j in range(2):\n axes[j].spines['right'].set_visible(False)\n axes[j].spines['top'].set_visible(False)\n\n orig_correlation = round(np.corrcoef(data[:, 0], data[:, 1])[0, 1], 2)\n axes[0].set(title='Data in original coordinates \\n Correlation = ' + str(orig_correlation), xlabel='Neuron 1 activity', ylabel='Neuron 2 activity', xlim=[-5, 15], ylim=[-5, 15]);\n\n axes[1].scatter(transformed_data[:,0], transformed_data[:, 1], s=1, c='#63BA79');\n pca_correlation = round(np.corrcoef(transformed_data[:, 0], transformed_data[:, 1])[0, 1], 2)\n axes[1].set(title='Data in PC coordinates \\n Correlation = ' + str(pca_correlation), xlabel='PC 1', ylabel='PC 2');\n\n plt.tight_layout()\n\ndef plot_data_and_PCs(X, W):\n \"\"\"\n Plots bivariate data as well as new basis vectors.\n\n Args:\n X (numpy array of floats) : Data matrix each column corresponds to a\n different random variable\n W (numpy array of floats) : Square matrix representing new orthonormal\n basis each column represents a basis vector\n\n Returns:\n Nothing.\n \"\"\"\n\n plt.figure()\n plt.scatter(X[:, 0], X[:, 1], s=1, color='#63BA79')\n plt.axis('equal')\n plt.xlabel('Neuron 1 activity')\n plt.ylabel('Neuron 2 activity')\n colors = plt.rcParams['axes.prop_cycle'].by_key()['color']\n plt.plot([0, W[0, 0]], [0, W[1, 0]], color=colors[4], linewidth=1,\n label='Component 1')\n plt.plot([0, W[0, 1]], [0, W[1, 1]], color=colors[3], linewidth=1,\n label='Component 2')\n plt.legend()\n plt.show()\n\n\ndef plot_vector(vectors, tails=None):\n ''' Draw 2d vectors based on the values of the vectors and the position of their tails.\n \n Parameters\n ----------\n vectors : list.\n List of 2-element array-like structures, each represents a 2d vector.\n \n tails : list, optional.\n List of 2-element array-like structures, each represents the coordinates of the tail\n of the corresponding vector in vectors. If None (default), all tails are set at the\n origin (0,0). If len(tails) is 1, all tails are set at the same position. Otherwise,\n vectors and tails must have the same length.\n \n Examples\n --------\n >>> v = [(1, 3), (3, 3), (4, 6)]\n >>> plot_vector(v) # draw 3 vectors with their tails at origin\n >>> t = [numpy.array((2, 2))]\n >>> plot_vector(v, t) # draw 3 vectors with their tails at (2,2)\n >>> t = [[3, 2], [-1, -2], [3, 5]]\n >>> plot_vector(v, t) # draw 3 vectors with 3 different tails\n\n ''' \n vectors = numpy.array(vectors)\n assert vectors.shape[1] == 2, \"Each vector should have 2 elements.\" \n if tails is not None:\n tails = numpy.array(tails)\n assert tails.shape[1] == 2, \"Each tail should have 2 elements.\"\n else:\n tails = numpy.zeros_like(vectors)\n \n # tile vectors or tails array if needed\n nvectors = vectors.shape[0]\n ntails = tails.shape[0]\n if nvectors == 1 and ntails > 1:\n vectors = numpy.tile(vectors, (ntails, 1))\n elif ntails == 1 and nvectors > 1:\n tails = numpy.tile(tails, (nvectors, 1))\n else:\n assert tails.shape == vectors.shape, \"vectors and tail must have a same shape\"\n\n # calculate xlimit & ylimit\n heads = tails + vectors\n limit = numpy.max(numpy.abs(numpy.hstack((tails, heads))))\n limit = numpy.ceil(limit * 1.2) # add some margins\n \n figsize = numpy.array([2,2]) * fig_scale\n figure, axis = pyplot.subplots(figsize=figsize)\n axis.quiver(tails[:,0], tails[:,1], vectors[:,0], vectors[:,1], color=darkblue, \n angles='xy', scale_units='xy', scale=1)\n axis.set_xlim([-limit, limit])\n axis.set_ylim([-limit, limit])\n axis.set_aspect('equal')\n\n # if xticks and yticks of grid do not match, choose the finer one\n xticks = axis.get_xticks()\n yticks = axis.get_yticks()\n dx = xticks[1] - xticks[0]\n dy = yticks[1] - yticks[0]\n base = max(int(min(dx, dy)), 1) # grid interval is always an integer\n loc = ticker.MultipleLocator(base=base)\n axis.xaxis.set_major_locator(loc)\n axis.yaxis.set_major_locator(loc)\n axis.grid(True, **grid_params)\n \n # show x-y axis in the center, hide frames\n axis.spines['left'].set_position('center')\n axis.spines['bottom'].set_position('center')\n axis.spines['right'].set_color('none')\n axis.spines['top'].set_color('none')\n\ndef plot_transformation_helper(axis, matrix, *vectors, unit_vector=True, unit_circle=False, title=None):\n \"\"\" A helper function to plot the linear transformation defined by a 2x2 matrix.\n \n Parameters\n ----------\n axis : class matplotlib.axes.Axes.\n The axes to plot on.\n\n matrix : class numpy.ndarray.\n The 2x2 matrix to visualize.\n\n *vectors : class numpy.ndarray.\n The vector(s) to plot along with the linear transformation. Each array denotes a vector's\n coordinates before the transformation and must have a shape of (2,). Accept any number of vectors. \n \n unit_vector : bool, optional.\n Whether to plot unit vectors of the standard basis, default to True.\n \n unit_circle: bool, optional.\n Whether to plot unit circle, default to False.\n \n title: str, optional.\n Title of the plot.\n\n \"\"\"\n assert matrix.shape == (2,2), \"the input matrix must have a shape of (2,2)\"\n grid_range = 20\n x = numpy.arange(-grid_range, grid_range+1)\n X_, Y_ = numpy.meshgrid(x,x)\n I = matrix[:,0]\n J = matrix[:,1]\n X = I[0]*X_ + J[0]*Y_\n Y = I[1]*X_ + J[1]*Y_\n origin = numpy.zeros(1)\n \n # draw grid lines\n for i in range(x.size):\n axis.plot(X[i,:], Y[i,:], c=gold, **grid_params)\n axis.plot(X[:,i], Y[:,i], c=lightblue, **grid_params)\n \n # draw (transformed) unit vectors\n if unit_vector:\n axis.quiver(origin, origin, [I[0]], [I[1]], color=green, **quiver_params)\n axis.quiver(origin, origin, [J[0]], [J[1]], color=red, **quiver_params)\n\n # draw optional vectors\n color_cycle = cycle([pink, darkblue, orange, purple, brown])\n if vectors:\n for vector in vectors:\n color = next(color_cycle)\n vector_ = matrix @ vector.reshape(-1,1)\n axis.quiver(origin, origin, [vector_[0]], [vector_[1]], color=color, **quiver_params)\n\n # draw optional unit circle\n if unit_circle:\n alpha = numpy.linspace(0, 2*numpy.pi, 41)\n circle = numpy.vstack((numpy.cos(alpha), numpy.sin(alpha)))\n circle_trans = matrix @ circle\n axis.plot(circle_trans[0], circle_trans[1], color=red, lw=0.8)\n\n # hide frames, set xlimit & ylimit, set title\n limit = 4\n axis.spines['left'].set_position('center')\n axis.spines['bottom'].set_position('center')\n axis.spines['left'].set_linewidth(0.3)\n axis.spines['bottom'].set_linewidth(0.3)\n axis.spines['right'].set_color('none')\n axis.spines['top'].set_color('none')\n axis.set_xlim([-limit, limit])\n axis.set_ylim([-limit, limit])\n if title is not None:\n axis.set_title(title)\n\ndef plot_linear_transformation(matrix, *vectors, unit_vector=True, unit_circle=False):\n \"\"\" Plot the linear transformation defined by a 2x2 matrix using the helper\n function plot_transformation_helper(). It will create 2 subplots to visualize some\n vectors before and after the transformation.\n \n Parameters\n ----------\n matrix : class numpy.ndarray.\n The 2x2 matrix to visualize.\n\n *vectors : class numpy.ndarray.\n The vector(s) to plot along with the linear transformation. Each array denotes a vector's\n coordinates before the transformation and must have a shape of (2,). Accept any number of vectors.\n \n unit_vector : bool, optional.\n Whether to plot unit vectors of the standard basis, default to True.\n \n unit_circle: bool, optional.\n Whether to plot unit circle, default to False.\n \n \"\"\"\n with plt.rc_context({\"figure.dpi\": 200, 'font.family':'serif', 'axes.axisbelow':True, 'font.size':fontsize, \"axes.titlesize\":5, \"lines.linewidth\":1}):\n figsize = numpy.array([4,2]) * fig_scale\n figure, (axis1, axis2) = pyplot.subplots(1, 2, figsize=figsize)\n plot_transformation_helper(axis1, numpy.identity(2), *vectors, unit_vector=unit_vector, unit_circle=unit_circle, title='Before transformation')\n plot_transformation_helper(axis2, matrix, *vectors, unit_vector=unit_vector, unit_circle=unit_circle, title='After transformation')\n\ndef plot_linear_transformations(*matrices, unit_vector=True, unit_circle=False):\n \"\"\" Plot the linear transformation defined by a sequence of n 2x2 matrices using the helper\n function plot_transformation_helper(). It will create n+1 subplots to visualize some\n vectors before and after each transformation.\n\n Parameters\n ----------\n *matrices : class numpy.ndarray.\n The 2x2 matrices to visualize. Accept any number of matrices.\n \n unit_vector : bool, optional.\n Whether to plot unit vectors of the standard basis, default to True.\n \n unit_circle: bool, optional.\n Whether to plot unit circle, default to False.\n \n \"\"\"\n nplots = len(matrices) + 1\n nx = 2\n ny = ceil(nplots/nx)\n with plt.rc_context({\"figure.dpi\": 200, 'font.family':'serif', 'axes.axisbelow':True, 'font.size':fontsize, \"axes.titlesize\":5, \"lines.linewidth\":1}):\n figsize = numpy.array([2*nx, 2*ny]) * fig_scale\n figure, axes = pyplot.subplots(nx, ny, figsize=figsize)\n\n for i in range(nplots): # fig_idx \n if i == 0:\n matrix_trans = numpy.identity(2)\n title = 'Before transformation'\n else:\n matrix_trans = matrices[i-1] @ matrix_trans\n if i == 1:\n title = 'After {} transformation'.format(i)\n else:\n title = 'After {} transformations'.format(i)\n plot_transformation_helper(axes[i//nx, i%nx], matrix_trans, unit_vector=unit_vector, unit_circle=unit_circle, title=title)\n # hide axes of the extra subplot (only when nplots is an odd number)\n if nx*ny > nplots:\n axes[-1,-1].axis('off')\n```\n\n\n```\n# @markdown Helper functions\ndef sort_evals_descending(evals, evectors):\n \"\"\"\n Sorts eigenvalues and eigenvectors in decreasing order. Also aligns first two\n eigenvectors to be in first two quadrants (if 2D).\n\n Args:\n evals (numpy array of floats) : Vector of eigenvalues\n evectors (numpy array of floats) : Corresponding matrix of eigenvectors\n each column corresponds to a different\n eigenvalue\n\n Returns:\n (numpy array of floats) : Vector of eigenvalues after sorting\n (numpy array of floats) : Matrix of eigenvectors after sorting\n \"\"\"\n\n index = np.flip(np.argsort(evals))\n evals = evals[index]\n evectors = evectors[:, index]\n if evals.shape[0] == 2:\n if np.arccos(np.matmul(evectors[:, 0],\n 1 / np.sqrt(2) * np.array([1, 1]))) > np.pi / 2:\n evectors[:, 0] = -evectors[:, 0]\n if np.arccos(np.matmul(evectors[:, 1],\n 1 / np.sqrt(2) * np.array([-1, 1]))) > np.pi / 2:\n evectors[:, 1] = -evectors[:, 1]\n return evals, evectors\n\ndef calculate_cov_matrix(var_1, var_2, corr_coef):\n \"\"\"\n Calculates the covariance matrix based on the variances and\n correlation coefficient.\n\n Args:\n var_1 (scalar) : variance of the first random variable\n var_2 (scalar) : variance of the second random variable\n corr_coef (scalar) : correlation coefficient\n\n Returns:\n (numpy array of floats) : covariance matrix\n \"\"\"\n cov = corr_coef * np.sqrt(var_1 * var_2)\n cov_matrix = np.array([[var_1, cov], [cov, var_2]])\n return cov_matrix\n\ndef get_variance_explained(evals):\n \"\"\"\n Plots eigenvalues.\n\n Args:\n (numpy array of floats) : Vector of eigenvalues\n\n Returns:\n Nothing.\n\n \"\"\"\n\n # cumulatively sum the eigenvalues\n csum = np.cumsum(evals)\n # normalize by the sum of eigenvalues\n variance_explained = csum / np.sum(evals)\n\n return variance_explained\n\ndef add_noise(X, frac_noisy_pixels):\n \"\"\"\n Randomly corrupts a fraction of the pixels by setting them to random values.\n\n Args:\n X (numpy array of floats) : Data matrix\n frac_noisy_pixels (scalar) : Fraction of noisy pixels\n\n Returns:\n (numpy array of floats) : Data matrix + noise\n\n \"\"\"\n\n X_noisy = np.reshape(X, (X.shape[0] * X.shape[1]))\n N_noise_ixs = int(X_noisy.shape[0] * frac_noisy_pixels)\n noise_ixs = np.random.choice(X_noisy.shape[0], size=N_noise_ixs,\n replace=False)\n X_noisy[noise_ixs] = np.random.uniform(0, 255, noise_ixs.shape)\n X_noisy = np.reshape(X_noisy, (X.shape[0], X.shape[1]))\n\n return X_noisy\n\n```\n\n# Key concept review & coding tips\n\n## Special matrices\n\n### Diagonal matrices\n* Have only nonzero entries on the diagonal \n* Can be rectangular\n* Scales space\n* Inverse is diagonal matrix with inverse entries\n\n### Orthogonal matrices\n* Square matrix where every column is a unit vector and every pair of columns is orthogonal\n* Rotates space\n* Its inverse is its transpose\n\n### Symmetric matrices\n* Square matrix where $a_{ij} = a_{ji}$\n* Equals its own transpose\n* Eigenvalues are always real (not complex)\n* Eigenvectors associated with different eigenvalues are orthogonal\n\n## Matrix decomposition\n\n* Factorization of a matrix into a product of matrices \n* Product matrices might be more compact/ordered, could make computations easier, could shed light on matrix structure\n\n### Eigendecomposition\n* $A = VDV^{-1}$ where V has eigenvectors as columns and D is diagonal matrix with eigenvalues on the diagonal\n* Can only do this if A is square and if eigenvectors of A form a basis for space\n* $A^n = VD^nV^{-1}$\n* $A^{-1} = VD^{-1}V^{-1}$\n\n### Singular value decomposition\n* `np.linalg.svd`\n* $A = USV^T$ where U/V are orthogonal matrices and S is diagonal\n* Can decompose any matrix this way\n* Diagonal entries of S are singular values, columns of U are left singular values, columns of V are right singular values\n* Decomposes transformation that matrix enacts into a rotation, then a scaling, then a rotation\n* Columns of V associated with zero or non-existant singular values form an orthogonal basis for the nullspace of A\n* Columns of U associated with non-zero singular values form an orthogonal basis for the column space of A\n* rank(A) = number of non-zero singular values\n* SVD factorizes A into sum of outer products with decreasing influence, can use first K sums to form rank K approximation of A\n\n## Dimensionality Reduction\n* Transform data from high D to low D while keeping as much information as possible about the data (finding new representation for the data)\n* Can help with data visualization, noise reduction, data preprocessing for further analyses, and scientific findings, among other things\n\n### Principal components analysis\n* `sklearn.decomposition.pca` for PCA, see info at the end of this tutorial\n* Find axes in space that maximize the variance of the data (and minimize the residuals) while being orthongal to each other. Project data onto these axes and keep first K components\n* Can think of PCA as a change of basis where the new basis vectors are the principal components\n* $U = XV$ where U is the transformed data (# data points x reduced dim), X is the data matrix (# data points x orig dim), and V is the components matrix (orig dim x reduced dim)\n* Always center your data first!\n* Can find principal components as the eigenvectors of the covariance matrix ($\\frac{1}{n}X^TX$), eigenvalues tell you the variance explained by that component (plot these to make a scree plot) \n* Could also use SVD to find PCA - the columns of V are the eigenvectors of the covariance matrix and the squared singular values over N are the eigenvalues\n\n# Exercise 1: Delving into SVD\n\nWe'll explore SVD by hand in this problem to help solidify our understanding of how the matrices interact with each other. Let $$A = \\begin{bmatrix}\n2 & -4 \\\\\n3 & 1 \\\\\n\\end{bmatrix}, \\bar{v} = \\begin{bmatrix}\n2 \\\\\n1 \\\\\n\\end{bmatrix}$$\n\nIn the following plot, we'll see the transformation that this matrix enacts.\n\n\n\n```\nA = np.array([[2, -4], [3, 1]])\nplot_linear_transformation(A)\n```\n\n## A) Computing the SVD\n\nUse `np.linalg.svd` to get the SVD of the matrix A. Note that the outputs are not quite the U, S, V we've been discussing. This function outputs $V^T$ directly. Get S/V from the outputs.\n\n\n\n\n```\nU, s, VT = ...\nS = ...\nV = ...\n```\n\n## B) SVD step by step transformations\n\nMultiply out the operations of $V^T$, S, and U with vector $\\bar{v}$ one at a time. In other words, get $V^T\\bar{v}$, then $SV^T\\bar{v}$, then $USV^t\\bar{v}$. You do not need to do this by hand - use code - but make sure you understand the matrix vector multiplication! \n\nMake sure $USV^t\\bar{v}$ = $A\\bar{v}$.\n\nExecute the following cell to visualize the vectors.\n\n\n```\nv = ...\nVTv = ...\nSVTv = ...\nUSVTv = ...\nAv = ...\nprint(USVTv)\nprint(Av)\n```\n\n\n```\n# @title\n# @markdown Execute to visualize vector transforms\nvec_names = [r'$\\bar{v}$', r'$SV^T\\bar{v}$', r'$V^T\\bar{v}$', r'A$\\bar{v}$']\nvecs = np.array([v, \n SVTv,\n VTv,\n USVTv])\n\nfig, axes = plt.subplots(1, 1)\ncolors = plt.rcParams['axes.prop_cycle'].by_key()['color']\n\naxes.set(xlim=[-8, 8], ylim=[-8, 8])\naxes.axis('Off')\n\nfor i_vec, vec in enumerate(vecs): \n axes.arrow(0, 0, vec[0], vec[1], head_width=.2, facecolor=colors[i_vec], edgecolor=colors[i_vec], length_includes_head=True);\n axes.annotate(vec_names[i_vec], xy=(vec[0]+np.sign(vec[0])*.15, vec[1]+np.sign(vec[1])*.15), color=colors[i_vec]);\n\naxes.plot([0, 0], [-8, 8], classic, alpha=.4);\naxes.plot([-8, 8], [0, 0], classic, alpha=.4);\n\n\n```\n\nWhat transformation is happening to $\\bar{v}$ at each step?\n\n\n**Your text answer**\n\n## C) Low rank approximation \n\nWe'll explore successful low rank approximations of receptive fields in Tutorial 2 and this will make the concept much more intuitive - the goal of this problem is just to understand the computation involved and what a rank 1 approximation means.\n\nCalculate a rank 1 approximation of A by hand. Specifically, compute:\n\n$$\\text{Rank 1 approx } = B = s_1\\bar{u}_1\\bar{v}_1^T $$\n\nwhere $s_1$ is the first (highest) singular value and $\\bar{u}_1$ and $\\bar{v}_1$ are the corresponding columns of U and V. \n\nShow your work for the computation! You should round to 2 places after the decimal.\n\n**Your math answer** show your work!\n\nCompare B to the original matrix A. What does a rank 1 approximation mean? What is the computation \"trying to do\"? What is happening with the columns/rows of B? \n\n**Your text answer here**\n\nNote that the rank 1 approximation here is not great because our matrix is not anywhere close to rank 1! We would fully recover our matrix with a rank 2 approximation - $ A = s_1\\bar{u}_1\\bar{v}_1^T + s_2\\bar{u}_2\\bar{v}_2^T$ - since A is 2 x 2 and has maximum rank of 2.\n\n\n\n## (Optional) Extra info: Orthogonal matrices can also reflect\n\nExecute the next cell to visualize the transformation at each step of SVD (by $V^T$, then $S$, then $U$). You will notice that it isn't simply rotation, then scaling, then a rotation. Both $V^T$ and $U$ enact a reflection in addition to a rotation. Orthogonal matrices can reflect in addition to rotating space. \n\nWe could get an SVD without reflection if we hadn't ordered our columns by the size of the singular values. If we switched the columns in U, S, and V, we would see just a rotation, then scaling, then another rotation (show below).\n\n\n```\n# @markdown Execute this cell to visualize transformations\n\nplot_linear_transformations(VT, S, U, unit_vector=True, unit_circle=False)\n\n```\n\n\n```\n# @markdown Execute this cell to visualize transformations with permuted columns\n\nplot_linear_transformations(V[:, [1, 0]].T, np.diag(s[::-1]), U[:, [1, 0]], unit_vector=True, unit_circle=False)\n\n```\n\n# Exercise 2: PCA implementation and correlation exploration\n\n### Modified from NMA W1D5 T2\n\nIn this exercise, you will implement PCA, apply it to 2 dimensional data, and examine the effects of correlations between dimensions.\n\n\n```\n# @markdown Execute this cell to generate some data (X)\nnp.random.seed(123)\nvariance_1 = 1\nvariance_2 = 1\ncorr_coef = 0.8\ncov_matrix = calculate_cov_matrix(variance_1, variance_2, corr_coef)\nX = np.random.multivariate_normal([5, 10], cov_matrix, size=1000)\n\nfig, ax = plt.subplots()\nax.scatter(X[:,0], X[:, 1], s=1, color='#63BA79');\nax.spines['right'].set_visible(False)\nax.spines['top'].set_visible(False)\n\nax.set(xlabel='Neuron 1 activity', ylabel='Neuron 2 activity', xlim=[-5, 15], ylim=[-5, 15]);\n```\n\n## A) Interactive Demo: Identifying first principal component\n\nLet's take a subset of our data as shown below and mean subtract it. Play with the interactive demo. About which value of theta represents the first principal component? Why?\n\n\n\n```\n# @markdown Make sure you execute this cell to enable the widget!\n\ndef plot_potential_component(theta=180):\n n_points = 30\n\n mean_subtracted_X = X - np.mean(X, 0)\n\n fig, ax = plt.subplots()\n ax.spines['right'].set_visible(False)\n ax.spines['top'].set_visible(False)\n\n ax.set(xlabel='Neuron 1 activity', ylabel='Neuron 2 activity', xlim=[-5, 5], ylim=[-5, 5]);\n\n w = np.asarray([np.cos(theta*np.pi/180), np.sin(theta*np.pi/180)])[None, :];\n z = mean_subtracted_X[:n_points, :] @ w.T @ w;\n for i in range(n_points):\n ax.plot([mean_subtracted_X[i,0], z[i,0]], [mean_subtracted_X[i,1], z[i,1]], 'r')\n\n ax.plot(w[0, 0]*5*np.array([-1, 1]), w[0, 1]*5*np.array([-1, 1]), 'k')\n ax.scatter(z[:,0], z[:, 1], color='r')\n ax.scatter(mean_subtracted_X[:n_points,0], mean_subtracted_X[:n_points, 1], color='#63BA79');\n\n\n_ = widgets.interact(plot_potential_component, theta = (0, 180, 1), fontsize=60)\n```\n\n**Your text answer**\n\n## B) Implement PCA \n\nLet's first implement PCA! We will build a function that takes in data and returns the transformed data, the principal components, and the variance explained by each component.\n\nWe will use an implementation involving the eigenvectors/eigenvalues of the covariance matrix (as opposed to using SVD).\n\n\n\n```\ndef pca(X):\n \"\"\"\n Sorts eigenvalues and eigenvectors in decreasing order.\n\n Args:\n X (numpy array of floats): Data matrix each column corresponds to a\n different random variable \n\n Returns:\n (numpy array of floats) : Data projected onto the new basis\n (numpy array of floats) : Vector of eigenvalues\n (numpy array of floats) : Corresponding matrix of eigenvectors\n\n \"\"\"\n\n # Subtract the mean of X\n X = ...\n\n # Calculate the sample covariance matrix\n cov_matrix = ... # hint: covariance matrix = (1/n)X^TX\n\n # Calculate the eigenvalues and eigenvectors\n evals, evectors = ... # hint: use np.linalg.eig\n\n # Sort the eigenvalues in descending order using a helper function\n evals, evectors = sort_evals_descending(evals, evectors)\n\n # Project the data onto the new eigenvector basis\n transformed_data = ... # hint: remember U = XV\n \n return transformed_data, evectors, evals\n\n# Uncomment below once you have filled in the above function\n\n# Perform PCA on the data matrix X\n#X_pca, evectors, evals = pca(X)\n\n# Plot the data projected into the new basis\n#plot_pca_transformation(X, X_pca)\n```\n\nNote that the correlation between dimensions goes to 0 after the transformation to the principal components basis! This is a property of PCA: it decorrelates the data.\n\n## C) Visualize variance explained\n\nWe want to create a plot telling us the percent of variance explained by each principal component (here we have just two). Determine what information you need for this (the inputs) and complete the function below.\n\n\n```\ndef plot_variances(...):\n\n percent_explained_variance = ...\n\n fig, ax = plt.subplots() \n colors = plt.rcParams['axes.prop_cycle'].by_key()['color']\n ax.plot(percent_explained_variance, '-o', color=colors[4])\n ax.set(ylim=[0, 1], ylabel='% Explained Variance', xlabel='Component number', xticks=np.arange(len(percent_explained_variance)))\n \nplot_variances(...)\n```\n\n## D) Interactive Demo: Exploration of the correlation coefficient\n\nRun the following cell and use the slider to change the correlation coefficient of the data. This will update a plot of the data with the principal components overlaid and a plot of percentage of explained variance.\n\n**Questions:**\n* Can you find a correlation coefficient value for which the components have equal explained variance?\n* Can you find a value for which only only one component explains any variance?\n\n\n```\n# @markdown Make sure you execute this cell to enable the widget!\n\ndef refresh(corr_coef=.8):\n cov_matrix = calculate_cov_matrix(variance_1, variance_2, corr_coef)\n X = X = np.random.multivariate_normal([0, 0], cov_matrix, size=1000)\n score, evectors, evals = pca(X)\n plot_data_and_PCs(X, evectors)\n plot_variances(evals)\n\n_ = widgets.interact(refresh, corr_coef=(-1, 1, .1), fontsize=60)\n```\n\n**Your text answer**\n\n## Optional advanced challenge: PCA implementation with SVD\n\nTake the PCA function from part A and implement with SVD instead of with the eigenvectors of the covariance matrix.\n\n\n\n\n```\ndef pca_with_SVD(X):\n\n ...\n```\n\n# Exercise 3: PCA of images \n## Modified from NMA W1D5 T3\n\nIn this exercise, we will look at the PCA of images. We will use images from the MNIST dataset, which is a dataset of handdrawn numbers (0-9). We're using this data instead of more neuroscience related data because it's a small dataset that's easy to interpret. Everything we will learn here could be applied to, for example, the frames of a video of a mouse performing a task. \n\n\n\nThe MNIST dataset consists of a 70,000 images of individual handwritten digits. Each image is a 28x28 pixel grayscale image. For convenience, each 28x28 pixel image is often unravelled into a single 784 (=28*28) element vector (a process called flattening the images), so that the whole dataset is represented as a 70,000 x 784 matrix. Each row represents a different image, and each column represents a different pixel.\n \nExecute the following cell to load the MNIST dataset and plot the first nine images.\n\n\n```\nfrom sklearn.datasets import fetch_openml\nmnist = fetch_openml(name='mnist_784')\nX = mnist.data\nplot_sample_images(X)\n```\n\n## A) Explained variances\n\nWe will first perform PCA and plot the cumulative percentage explained variance over components. Note that this is related to our earlier plots but now we are plotting the percentage of explained variance **cumulatively**. Execute the next cell to do this. \n\n- How many principal components are required to explain 90% of the variance?\n- How does the intrinsic dimensionality of this dataset compare to its extrinsic dimensionality? \n\n\n\n```\ntransformed_data, evectors, evals = pca(X)\nvariance_explained = get_variance_explained(evals)\nplot_variance_explained(variance_explained)\n```\n\n**Your text answer**\n\n## B) PCA Reconstruction\n\nLet's try projecting down onto our reduced dimensionality PCA space and then **reconstructing** our images from the low-D space.\n\nTo see this, recall that to perform PCA we projected the data $\\bf X$ onto the eigenvectors of the covariance matrix:\n\\begin{equation}\n\\bf U = X V\n\\end{equation}\nwhere $U$ is the transformed data, $X$ is the data matrix, and $V$ is the components matrix.\n\nSince $\\bf V$ is an orthogonal matrix, ${\\bf V}^{-1} = {\\bf V}^T$. We can reconstruct by:\n \n\\begin{equation}\n\\bf X = U V^T\n\\end{equation}\n\nTo reconstruct the data from a low-dimensional approximation, we just have to truncate these matrices. Let's call ${\\bf U}_{1:K}$ and ${\\bf V}_{1:K}$ as keeping only the first $K$ columns of this matrix. Then our reconstruction is:\n\\begin{equation}\n{\\bf \\hat X = U}_{1:K} ({\\bf V}_{1:K})^T.\n\\end{equation}\n\nComplete the following function to reconstruct the images from the top K components.\n\n\n```\ndef reconstruct_data(transformed_data, evectors, X_mean, K):\n \"\"\"\n Reconstruct the data based on the top K components.\n\n Args:\n transformed_data (numpy array of floats) : data projected onto PCA basis\n evectors (numpy array of floats) : Matrix of eigenvectors\n X_mean (numpy array of floats) : Vector corresponding to data mean\n K (scalar) : Number of components to include\n\n Returns:\n (numpy array of floats) : Matrix of reconstructed data\n\n \"\"\"\n\n #################################################\n ## TO DO for students: Reconstruct the original data in X_reconstructed\n # Comment once you've filled in the function\n raise NotImplementedError(\"Student exercise: reconstructing data function!\")\n #################################################\n\n # Reconstruct the data from the score and eigenvectors\n # Don't forget to add the mean!!\n X_reconstructed = ...\n\n return X_reconstructed\n\n\nK = 100\n\n# Fill in below then uncomment the last line\n\n# Reconstruct the data based on all components\nX_mean = ...\nX_reconstructed = ...\n\n# Plot the data and reconstruction\n# plot_reconstructions(X, X_reconstructed)\n```\n\n## C) Interactive Demo: Reconstruct the data matrix using different numbers of PCs\n\nNow run the code below and experiment with the slider to reconstruct the data matrix using different numbers of principal components.\n\n\n- How many principal components are necessary to reconstruct the numbers (by eye)? How does this relate to the intrinsic dimensionality of the data?\n- Do you see any information in the data with only a single principal component?\n\n\n```\n# @markdown Make sure you execute this cell to enable the widget!\n\n\ndef refresh(K=100):\n X_reconstructed = reconstruct_data(transformed_data, evectors, X_mean, K)\n plot_reconstructions(X, X_reconstructed)\n plt.title('Reconstructed, K={}'.format(K))\n\n\n_ = widgets.interact(refresh, K=(1, 784, 10))\n```\n\n**Your text answer**\n\n## D) Visualization of the principal components\n\nWe can visualize the principal components as images by reversing the flattening. Here we plot using a differenet colormap than black & white as it highlights more structure.\n\n* What structure do you see in the first principal component? What kinds of images would this basis vector differentiate?\n* Try visualizing the second and third basis vectors. Do you see any structure? What about the 100th basis vector? 500th? 700th?\n\n\n```\n plot_principal_components(evectors[:, 0])\n```\n\n**Your text answer**\n\n## (Read only) E) Denoising with PCA\n\nWe will add some noise to our images to see how PCA reconstructions can be helpful for reducing noise. In this case, we will set 20% of the pixels to random values. We will then visualize some of the noisy images and the resulting cumulative variance explained plot.\n\nIn the next cell, we will project the images onto the original PCA space (from the clean, not noisy, data) and then reconstruct from the top 50 components. Observe that this removes the noise quite effectively!\n\n\n```\n# @markdown Execute this cell to visualize noisy data\nnp.random.seed(2020) # set random seed\nX_noisy = add_noise(X, .2)\nscore_noisy, evectors_noisy, evals_noisy = pca(X_noisy)\nvariance_explained_noisy = get_variance_explained(evals_noisy)\n\n\nplot_sample_images(X_noisy)\nplot_variance_explained(variance_explained_noisy)\n```\n\n\n```\n# @markdown Execute to visualize denoised reconstructions\nX_noisy_mean = np.mean(X_noisy, 0)\nprojX_noisy = np.matmul(X_noisy - X_noisy_mean, evectors)\nX_reconstructed = reconstruct_data(projX_noisy, evectors, X_noisy_mean, 50)\n\nplot_reconstructions(X_noisy, X_reconstructed)\n```\n\n# Extra info: PCA & Sklearn\n\nIn this tutorial, we created our own functions to compute PCA and reconstruct images so we could better understand the algorithms. Usually though, you would use `sklearn.decomposition.pca` to perform PCA. Sklearn is a class based package - I have a video explaining the basics of class based programming (object oriented programming) and a video on sklearn as part of my Pandemic Python for Neuroscientists course so check that out if interested. \nI'll demonstrate the basics here using some data with 3 features (X). \n\nSee docs here: https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.PCA.html\n\n\n\n```\n# @markdown Execute to generate some data X\nnp.random.seed(123)\nvariance_1 = 1\nvariance_2 = 1\ncorr_coef = 0.8\ncov_matrix = np.array([[1, .2, .3], [.2, 1, .3], [.3, .3, 1]])\nX = np.random.multivariate_normal([5, 10, 2], cov_matrix, size=1000)\n\n```\n\n\n```\n# Import PCA class\nfrom sklearn.decomposition import PCA\n\n# Set up model, tell it the number of components you'll want to keep (if not set, all components will be kept)\npca = PCA(n_components=2)\n\n# Fit the model to your data, aka find the principal components\npca.fit(X)\n\n# Now you can access the principal components\nprint(pca.components_)\n\n# And the % of explained variance for each component\nprint(pca.explained_variance_ratio_)\n\n# You can transform your data now\ntransformed_data = pca.transform(X)\n\n# You could have fit and transformed at the same time if you had wanted\ntransformed_data = pca.fit_transform(X)\n\n# You can also reconstruct into the original space\nreconstruction = pca.inverse_transform(transformed_data)\n```\n", "meta": {"hexsha": "f9d0aa4f60e1075b7f2b067910a7b1e9aa9318fe", "size": 61668, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Week4/Week4Tutorial1.ipynb", "max_stars_repo_name": "hugoladret/MathToolsforNeuroscience", "max_stars_repo_head_hexsha": "fad301909da9274bb6c40cac96e2c62ed85b3956", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 395, "max_stars_repo_stars_event_min_datetime": "2020-11-11T23:58:31.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-13T07:51:59.000Z", "max_issues_repo_path": "Week4/Week4Tutorial1.ipynb", "max_issues_repo_name": "likeajumprope/MathToolsforNeuroscience", "max_issues_repo_head_hexsha": "bfde76bbfbacc581a9fada98b41c9df4a32973d0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2021-03-05T14:28:10.000Z", "max_issues_repo_issues_event_max_datetime": "2021-07-12T21:56:18.000Z", "max_forks_repo_path": "Week4/Week4Tutorial1.ipynb", "max_forks_repo_name": "likeajumprope/MathToolsforNeuroscience", "max_forks_repo_head_hexsha": "bfde76bbfbacc581a9fada98b41c9df4a32973d0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 51, "max_forks_repo_forks_event_min_datetime": "2020-11-12T04:22:41.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-08T16:14:35.000Z", "avg_line_length": 40.491135916, "max_line_length": 461, "alphanum_fraction": 0.5132970098, "converted": true, "num_tokens": 10413, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.6959583313396339, "lm_q1q2_score": 0.44585587008084626}} {"text": "```python\nimport snap \nimport pandas as pd \nimport numpy as np\nimport matplotlib.pyplot as plt \nimport networkx as nx\n\nfrom matplotlib.pyplot import loglog \n\n%matplotlib inline \n\n```\n\n\n```python\ndf = pd.read_csv('/home/merchantsameer2014/project/dnc-temporalGraph/out.dnc-temporalGraph', sep='\\t', header=None)\n```\n\n\n```python\ndf.head(2)\n```\n\n\n\n\n
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\n\n\n\n\n```python\ndf.columns = ['src', 'dst', 'weight', 'timestamp']\n```\n\n\n```python\ndf.head(2)\n```\n\n\n\n\n
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041946511463507482
186945311462337903
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\n\n\n\n\n```python\ndef ProcessEdgeRow(row, Graph):\n src = row['src']\n dst = row['dst']\n timestamp = row['timestamp']\n \n if not Graph.IsNode(src):\n Graph.AddNode(src)\n \n if not Graph.IsNode(dst):\n Graph.AddNode(dst)\n \n if not Graph.IsEdge(src, dst):\n EId = Graph.AddEdge(src, dst) \n else:\n EId = Graph.GetEI(src, dst) \n \n Graph.AddIntAttrDatE(EId, timestamp, 'timestamp')\n \n return\n \n```\n\n\n```python\ndef GenerateDirectedGraph(df):\n '''\n Returns a TNEANet Graph object \n '''\n Graph = snap.TNEANet.New()\n df.apply(ProcessEdgeRow, axis=1, Graph=Graph)\n\n return Graph\n```\n\n\n```python\nG = GenerateDirectedGraph(df)\n```\n\n\n```python\nprint \"Nodes: \", G.GetNodes()\nprint \"Edges: \", G.GetEdges()\n```\n\n Nodes: 1891\n Edges: 5598\n\n\n## Get Raw Statistics \n\n 1. Degree Centrality \n 2. Clustering Coefficient \n 3. Mean shortest path from each node \n 4. Betweeness Centralitily \n 5. Hub and Authority Score \n\n\n```python\nNodeAttributes = dict()\n```\n\n#### Step 1: Degree Centrality\n\n\n```python\n#\n# 1. Degree Centrality \n# Get In Degree and Out Degree for each node \n#\nInDegV = snap.TIntPrV()\nOutDegV = snap.TIntPrV()\n\nsnap.GetNodeOutDegV(G, OutDegV)\nsnap.GetNodeInDegV(G, InDegV)\n\nInDegreeList = [ (item.GetVal1(), item.GetVal2()) for item in InDegV ]\nOutDegreeList = [ (item.GetVal1(), item.GetVal2()) for item in OutDegV ]\n```\n\n\n```python\nInDegreeList.sort(key=lambda x: x[1], reverse=True)\nOutDegreeList.sort(key=lambda x: x[1], reverse=True)\n\nminOutDegree = min(OutDegreeList, key = lambda x: x[1])[1]\nmaxOutDegree = max(OutDegreeList, key = lambda x: x[1])[1]\nminInDegree = min(InDegreeList, key = lambda x: x[1])[1]\nmaxInDegree = max(InDegreeList, key = lambda x: x[1])[1]\n```\n\n\n```python\n#\n# Sanity Check \nprint maxOutDegree, minOutDegree, maxInDegree, minInDegree\nprint InDegreeList[0], InDegreeList[-1]\n```\n\n 331 0 249 0\n (1874, 249) (893, 0)\n\n\n\n```python\nfor (nodeId, Degree) in InDegreeList:\n if not NodeAttributes.get(nodeId, None):\n NodeAttributes[nodeId] = dict()\n NodeAttributes[nodeId]['InDegree'] = Degree\n normalizedDegree = (float(Degree) - float(minInDegree))/(float(maxInDegree - float(minInDegree)))\n NodeAttributes[nodeId]['NormInDegree'] = normalizedDegree\n\nfor (nodeId, Degree) in OutDegreeList: \n NodeAttributes[nodeId]['OutDegree'] = Degree\n normalizedDegree = (float(Degree) - float(minOutDegree))/(float(maxOutDegree - float(minOutDegree)))\n NodeAttributes[nodeId]['NormOutDegree'] = normalizedDegree\n```\n\n\n```python\n#\n# Sanity Check \n#\nprint NodeAttributes[1874]\nprint NodeAttributes[893]\n```\n\n {'InDegree': 249, 'NormInDegree': 1.0, 'NormOutDegree': 1.0, 'OutDegree': 331}\n {'InDegree': 0, 'NormInDegree': 0.0, 'NormOutDegree': 0.0030211480362537764, 'OutDegree': 1}\n\n\n\n```python\n\n#\n# Get Degree Distribution \n# \nOutDegToCntV = snap.TIntPrV()\nsnap.GetOutDegCnt(G, OutDegToCntV)\ncount = 0\nnodeList = []\ndegreeList = []\nfor item in OutDegToCntV:\n (n, d) = (item.GetVal2(), item.GetVal1())\n nodeList.append(n)\n degreeList.append(d)\nx = np.array( [ np.log10(item.GetVal1()) for itemm in OutDegToCntV if item.GetVal1() > 0 ] )\ny = np.array( [ np.log10(item.GetVal2()) for item in OutDegToCntV if item.GetVal2() > 0 ] )\n#\n# Plot Degree Distribution\n#\nplt.figure(figsize=(15,15))\nloglog(degreeList, nodeList, 'bo')\n#plt.plot(x_plot, 10**b*x_plot**a, 'r-')\nplt.title(\"LogLog plot of out-degree distribution\")\nplt.show()\n```\n\n#### Step 2: Clustering Coefficient\n\n\n```python\nNIdCCfH = snap.TIntFltH()\nsnap.GetNodeClustCf(G, NIdCCfH)\nClusterCoeffList = list()\nfor nodeId in NIdCCfH:\n NodeAttributes[nodeId]['ClusterCoeff'] = NIdCCfH[nodeId]\n ClusterCoeffList.append((nodeId, NIdCCfH[nodeId]))\n```\n\n\n```python\nClusterCoeffList.sort(key=lambda x: x[1], reverse=True)\nminClusterCoeff = min(ClusterCoeffList, key=lambda x: x[1])[1]\nmaxClusterCoeff = max(ClusterCoeffList, key=lambda x: x[1])[1]\n```\n\n\n```python\n#\n# Sanity Check \n#\nprint ClusterCoeffList[1], maxClusterCoeff, ClusterCoeffList[-1], minClusterCoeff\n```\n\n (1753, 1.0) 1.0 (1381, 0.0) 0.0\n\n\n\n```python\nNIdCCfH = snap.TIntFltH()\nsnap.GetNodeClustCf(G, NIdCCfH)\nClusterCoeffList = list()\nfor nodeId in NIdCCfH:\n clusterCoeff = NIdCCfH[nodeId]\n normClusterCoeff = (clusterCoeff - minClusterCoeff)/(maxClusterCoeff - minClusterCoeff)\n NodeAttributes[nodeId]['NormClusterCoeff'] = normClusterCoeff\n```\n\n\n```python\nprint NodeAttributes[2012]\n```\n\n {'NormOutDegree': 0.0, 'ClusterCoeff': 0.6666666666666666, 'InDegree': 3, 'OutDegree': 0, 'NormInDegree': 0.012048192771084338, 'NormClusterCoeff': 0.6666666666666666}\n\n\n#### Step 3: Avergate shortest path length\n\n\n```python\nnodeCount = float(G.GetNodes() - 1)\navgShortPathLenList = list()\nfor src in G.Nodes():\n srcId = src.GetId()\n totalShortPathLength = 0 \n \n for dst in G.Nodes():\n dstId = dst.GetId()\n #\n # Skip Self Edges\n #\n if srcId == dstId:\n continue\n \n #\n # Compute Shortest Path \n #\n l = snap.GetShortPath(G, srcId, dstId, True)\n \n #\n # Skip nodes that cannot be reached \n #\n if l < 0:\n continue\n \n totalShortPathLength += float(l) \n NodeAttributes[srcId]['AvgShortPathLen'] = totalShortPathLength/nodeCount\n avgShortPathLenList.append((srcId, totalShortPathLength/nodeCount))\n```\n\n\n```python\nminAvgShortPathLength = min(avgShortPathLenList, key=lambda x: x[1])[1]\nmaxAvgShortPathLength = max(avgShortPathLenList, key=lambda x: x[1])[1]\n```\n\n\n```python\nfor (node, spLen) in avgShortPathLenList:\n normAvgShortPath = (spLen - minAvgShortPathLength)/(maxAvgShortPathLength - minAvgShortPathLength)\n NodeAttributes[node]['normAvgShortPathLen'] = normAvgShortPath\n```\n\n\n```python\nprint NodeAttributes[480]\n```\n\n {'NormOutDegree': 0.0030211480362537764, 'ClusterCoeff': 0.3333333333333333, 'normAvgShortPathLen': 1.0, 'InDegree': 2, 'OutDegree': 1, 'NormInDegree': 0.008032128514056224, 'NormClusterCoeff': 0.3333333333333333, 'AvgShortPathLen': 4.321164021164021}\n\n\n#### Step 4: Betweeness Centrality\n\n\n```python\nNodes = snap.TIntFltH()\nEdges = snap.TIntPrFltH()\nBetweenessNodeList = list()\nBetweenessEdgeList = list()\n\nsnap.GetBetweennessCentr(G, Nodes, Edges, 1.0)\nfor node in Nodes:\n NodeAttributes[node]['Betweeness'] = Nodes[node]\n BetweenessNodeList.append((node, Nodes[node]))\n\nfor edge in Edges:\n #print \"edge: (%d, %d) centrality: %f\" % (edge.GetVal1(), edge.GetVal2(), Edges[edge])\n BetweenessEdgeList.append((edge.GetVal1(), edge.GetVal2(), Edges[edge]))\n\nBetweenessNodeList.sort(key=lambda x: x[1], reverse=True) \nBetweenessEdgeList.sort(key=lambda x: x[2], reverse=True)\n```\n\n\n```python\nprint BetweenessNodeList[0], BetweenessNodeList[-1]\n```\n\n (1669, 556845.3564806876) (884, 0.0)\n\n\n\n```python\nminBetweeness = BetweenessNodeList[-1][1]\nmaxBetweeness = BetweenessNodeList[0][1]\nfor (node, betweeness) in BetweenessNodeList:\n normBetweeness = (betweeness - minBetweeness)/(maxBetweeness - minBetweeness)\n NodeAttributes[node]['normBetweeness'] = normBetweeness\n \n```\n\n\n```python\nprint NodeAttributes[1669]\nprint NodeAttributes[884]\n```\n\n {'NormOutDegree': 0.7311178247734139, 'ClusterCoeff': 0.0007700181235972993, 'normAvgShortPathLen': 0.33304763070895066, 'InDegree': 222, 'normBetweeness': 1.0, 'OutDegree': 242, 'NormInDegree': 0.891566265060241, 'NormClusterCoeff': 0.0007700181235972993, 'Betweeness': 556845.3564806876, 'AvgShortPathLen': 1.439153439153439}\n {'NormOutDegree': 0.0, 'ClusterCoeff': 0.0, 'normAvgShortPathLen': 0.0, 'InDegree': 1, 'normBetweeness': 0.0, 'OutDegree': 0, 'NormInDegree': 0.004016064257028112, 'NormClusterCoeff': 0.0, 'Betweeness': 0.0, 'AvgShortPathLen': 0.0}\n\n\n#### Step 5: Get Auth and Hub Score\n\n\n```python\nNIdHubH = snap.TIntFltH()\nNIdAuthH = snap.TIntFltH()\nsnap.GetHits(G, NIdHubH, NIdAuthH)\nHubNodes = []\nfor nodeId in NIdHubH:\n HubNodes.append((nodeId, NIdHubH[nodeId]))\n NodeAttributes[nodeId]['HubScore'] = NIdHubH[nodeId]\n \nHubNodes.sort(key = lambda x: x[1], reverse=True)\n\nAuthNodes = []\nfor nodeId in NIdAuthH:\n AuthNodes.append((nodeId, NIdAuthH[nodeId])) \n NodeAttributes[nodeId]['AuthScore'] = NIdAuthH[nodeId]\nAuthNodes.sort(key = lambda x: x[1], reverse=True)\n\n```\n\n\n```python\nprint AuthNodes[0], AuthNodes[-1]\nprint HubNodes[0], HubNodes[-1]\n```\n\n (1874, 0.30510831720503034) (893, 0.0)\n (1874, 0.38410704002084095) (884, 0.0)\n\n\n\n```python\nminAuthNodes = AuthNodes[-1][1]\nmaxAuthNodes = AuthNodes[0][1]\nminHubNodes = HubNodes[-1][1]\nmaxHubNodes = HubNodes[0][1]\n```\n\n\n```python\nfor (node, hubScore) in HubNodes:\n normHubScore = (hubScore - minHubNodes)/(maxHubNodes - minHubNodes)\n NodeAttributes[node]['normHubScore'] = normHubScore\n \nfor (node, authScore) in AuthNodes:\n normAuthScore = (authScore - minAuthNodes)/(maxAuthNodes - minAuthNodes)\n NodeAttributes[node]['normAuthScore'] = normAuthScore\n```\n\n\n```python\nprint NodeAttributes[1874]\nprint NodeAttributes[893]\n```\n\n {'NormOutDegree': 1.0, 'ClusterCoeff': 0.0014452513597956258, 'normAvgShortPathLen': 0.33696583812905595, 'InDegree': 249, 'normBetweeness': 0.8890896997683746, 'normAuthScore': 1.0, 'OutDegree': 331, 'AuthScore': 0.30510831720503034, 'NormInDegree': 1.0, 'normHubScore': 1.0, 'NormClusterCoeff': 0.0014452513597956258, 'HubScore': 0.38410704002084095, 'Betweeness': 495085.4708108281, 'AvgShortPathLen': 1.456084656084656}\n {'NormOutDegree': 0.0030211480362537764, 'ClusterCoeff': 0.0, 'normAvgShortPathLen': 0.0001224439818782907, 'InDegree': 0, 'normBetweeness': 0.0, 'normAuthScore': 0.0, 'OutDegree': 1, 'AuthScore': 0.0, 'NormInDegree': 0.0, 'normHubScore': 0.00048173348087729835, 'NormClusterCoeff': 0.0, 'HubScore': 0.00018503722141871546, 'Betweeness': 0.0, 'AvgShortPathLen': 0.0005291005291005291}\n\n\n## Automatic Social Hierarchy Detection From Email Network \n\n### Steps \n\n 1. Compute node's importance on response time for email \n 2. Get all Cliques (Algorithm 457)\n 3. Number of clique each node is part of \n 4. Raw Clique Score computed using \n \n $$R*2^{n-1}$$\n \n 5. Weighted Clique Score (Based on importance of a person) \n \n $$W = t*2^{n-1}$$\n \n\n### Use networkx library for Clique analysis \n\n 1. Build a multigraph with edges for each (src, dst, timestamp) entry in the email data set \n 2. Build undirected graph with edges between nodes only when email count exceeds N=4 between two those nodes \n \n\n\n```python\ndef ProcessNxEdgeRow(row, Graph):\n src = row['src']\n dst = row['dst']\n timestamp = row['timestamp']\n \n Graph.add_node(src)\n Graph.add_node(dst)\n Graph.add_edge(src, dst, timestamp=timestamp)\n \n return\n \n```\n\n\n```python\ndef GenerateDirectedNxGraph(df):\n '''\n Returns a TNEANet Graph object \n '''\n Graph = nx.MultiDiGraph(name=\"DNC Email Network\")\n df.apply(ProcessNxEdgeRow, axis=1, Graph=Graph)\n\n return Graph\n```\n\n\n```python\nGNx = GenerateDirectedNxGraph(df)\n```\n\n\n```python\nprint \"Networkx Nodes: \", GNx.number_of_nodes()\nprint \"Networkx Edges: \", GNx.number_of_edges()\n```\n\n Networkx Nodes: 1891\n Networkx Edges: 39264\n\n\n#### Prune all edges with N <=4 emails exchanged\nAs per the paper - consider edges between nodes only if the nodes have exchanged > N messages\nN is a tunable parameter \n\nDNC email has a median of message exchanged = 2 \n\n\n```python\nedgeCount = dict()\nfor edge in GNx.edges():\n if not edgeCount.get(edge, None):\n edgeCount[edge] = 0\n edgeCount[edge] += 1 \n```\n\n\n```python\nemailDist= [ v for k, v in edgeCount.iteritems() ]\nemailDist.sort(reverse=True)\n```\n\n\n```python\nprint emailDist[0], emailDist[-1], np.median(emailDist), len(emailDist)\n```\n\n 731 1 2.0 5598\n\n\n\n```python\nN = 4\npruneList = [ k for k, v in edgeCount.iteritems() if v <= N ]\nprunedEdgeList = [ k for k, v in edgeCount.iteritems() if v > N ]\n```\n\n### Generate undirected graph with edges between nodes with email count > N\n\n\n```python\nuGNx = nx.Graph()\n\nfor edge in prunedEdgeList:\n (src, dst) = edge\n uGNx.add_edge(src, dst)\n \nprint \"Networkx Nodes: \", uGNx.number_of_nodes()\nprint \"Networkx Edges: \", uGNx.number_of_edges() \n```\n\n Networkx Nodes: 452\n Networkx Edges: 1139\n\n\n### Step 1: Compute Nodes importance on email response time \n\n### Approach 1: Based on Rowe et. al. paper \n\nUse response time to measure importance of a node \n\n 1. For each node get all the out bound email timestamps \n 2. For each email sent - check the response time from the receiver \n 3. Consider responses within a day for computing time score. \n 4. Consider response time for nodes that have exchanged at least 100 emails \n (fewer emails with high response time can falsely promote node based on time score)\n \n\n\n```python\ntemporalMap = dict()\nfor n, nbrs in GNx.adjacency():\n temporalMap[n] = dict()\n for nbr, edict in nbrs.items():\n t1 = [d['timestamp'] for d in edict.values()]\n t1.sort()\n temporalMap[n][nbr] = t1 \n```\n\n\n```python\navgNodeResponse = list()\n\nfor src, destinations in temporalMap.iteritems():\n totalRequestResp = 0\n totalResponseTime = 0.0\n \n for dst, reqTimestamps in destinations.iteritems():\n responseTime = None\n #\n # Walk through ALl requests sent to a destination\n #\n for req in reqTimestamps:\n reqTime = datetime.fromtimestamp(req)\n \n #\n # Look for response time from the destinaton to ths source\n #\n for resp in temporalMap[dst].get(src, list()):\n respTime = datetime.fromtimestamp(resp)\n \n if resp < req:\n #\n # Look for first response after the request time \n #\n continue\n \n deltaTime = respTime - reqTime\n \n if deltaTime.total_seconds() > 86400:\n #\n # If response time exceeds a day don't \n # consider it as a response\n break\n \n #print \"Found Response time: src: %d, dst: %d req: %s, resp: %s\" % (src, dst, reqTime, respTime)\n \n totalRequestResp += 1\n totalResponseTime += deltaTime.total_seconds()\n break\n #\n # Compute average across all dst response times \n #\n if totalRequestResp > 0 and totalResponseTime > 0:\n avgResponse = totalResponseTime/totalRequestResp\n else:\n #\n # Set default response to 7 day \n #\n avgResponse = float(7*86400)\n #\n # Lower response Higher the timeScore \n # \n timeScore = 1/avgResponse\n avgNodeResponse.append((src, timeScore, totalResponseTime, totalRequestResp ))\n \n```\n\n\n```python\navgNodeTimeScore = list()\n\n#\n# Ignore response time for email exchanges fewer than 10\n#\nfor x in avgNodeResponse:\n (src, timeScore, totalResponseTime, totalRequestResp) = x\n \n if totalRequestResp <= 200:\n timeScore = 1.0/float(7*86400)\n \n avgNodeTimeScore.append((src, timeScore, totalResponseTime, totalRequestResp))\n \navgNodeTimeScore.sort(key=lambda x: x[1], reverse=True)\n\n```\n\n\n```python\nprint avgNodeTimeScore[-1]\nprint avgNodeTimeScore[:2]\n```\n\n (2029, 1.6534391534391535e-06, 0.0, 0)\n [(1625, 0.0020221266801294163, 368424.0, 745), (895, 0.00011885108532413246, 1918367.0, 228)]\n\n\n\n```python\nprint NodeAttributes[1625]\n```\n\n {'NormOutDegree': 0.030211480362537766, 'ClusterCoeff': 0.05714285714285714, 'normAvgShortPathLen': 0.4491245255295702, 'InDegree': 7, 'normBetweeness': 0.007319923714214854, 'normAuthScore': 0.12096933772423019, 'OutDegree': 10, 'AuthScore': 0.036908751066446865, 'NormInDegree': 0.028112449799196786, 'normHubScore': 0.09633108380044303, 'NormClusterCoeff': 0.05714285714285714, 'HubScore': 0.037001447460587755, 'Betweeness': 4076.0655300534095, 'AvgShortPathLen': 1.9407407407407407}\n\n\n\n```python\nnormTimeScore = dict()\nminAvgTimeScore = avgNodeTimeScore[-1][1]\nmaxAvgTimeScore = avgNodeTimeScore[0][1]\n\nfor (node, timeScore, _, _) in avgNodeTimeScore:\n normTimeScore[node] = (timeScore - minAvgTimeScore)/(maxAvgTimeScore - minAvgTimeScore)\n```\n\n\n```python\nnormTimeScore[1625]\n```\n\n\n\n\n 1.0\n\n\n\n\n```python\nnodeCliqueCount = dict()\nrawCliqueScore = dict()\nweightedCliqueScore = dict()\n\n#\n# Find all maximal cliques \n# \nfor clique in nx.find_cliques(uGNx):\n for node in clique:\n \n if not nodeCliqueCount.get(node, None):\n nodeCliqueCount[node] = 0\n \n if not rawCliqueScore.get(node, None):\n rawCliqueScore[node] = 0\n \n if not weightedCliqueScore.get(node, None):\n weightedCliqueScore[node] = 0\n \n nodeCliqueCount[node] += 1 \n \n n = len(clique)\n \n rawCliqueScore[node] += 2**n\n \n weightedCliqueScore[node] += (2**n)*normTimeScore[node]\n```\n\n\n```python\n#\n# Get a sorted list of nodes based on their clique count \n#\nnodeCliqueList = [ (k, v) for k, v in nodeCliqueCount.iteritems() ]\nnodeCliqueList.sort(key=lambda x: x[1], reverse=True )\n```\n\n\n```python\nrawCliqueList = [ (k, v) for k, v in rawCliqueScore.iteritems() ]\nrawCliqueList.sort(key=lambda x: x[1], reverse=True )\n```\n\n\n```python\nweightedCliqueList = [ (k, v) for k, v in weightedCliqueScore.iteritems() ]\nweightedCliqueList.sort(key=lambda x: x[1], reverse=True )\n```\n\n\n```python\nprint nodeCliqueList[:10]\nprint rawCliqueList[:10]\nprint weightedCliqueList[:10]\n```\n\n [(1874, 157), (1839, 123), (1669, 104), (999, 79), (1258, 79), (1369, 76), (453, 75), (585, 66), (1144, 57), (1159, 46)]\n [(1874, 29604), (999, 27636), (1258, 27616), (511, 22656), (1369, 21044), (1974, 18560), (1998, 18416), (585, 17972), (1144, 12248), (1899, 10304)]\n [(1258, 1133.2113553681547), (1874, 1079.971912453243), (999, 887.6525051646298), (511, 671.973655462151), (1998, 455.752744276308), (585, 335.72489139325376), (1669, 246.35443082589936), (1440, 223.6014465859277), (1287, 134.0397246718862), (453, 98.89859633344226)]\n\n\n\n```python\nminNodeCliqueCount = nodeCliqueList[-1][1]\nmaxNodeCliqueCount = nodeCliqueList[0][1]\n\nminRawCliqueScore = rawCliqueList[-1][1]\nmaxRawCliqueScore = rawCliqueList[0][1]\n\nminWeightedCliqueScore = weightedCliqueList[-1][1]\nmaxWeightedCliqueScore = weightedCliqueList[0][1]\n```\n\n\n```python\nfor node, score in nodeCliqueList:\n NodeAttributes[node]['nodeClique'] = score\n NodeAttributes[node]['normNodeClique'] = float(score - minNodeCliqueCount)/float(maxNodeCliqueCount - minNodeCliqueCount)\n```\n\n\n```python\nfor node, score in rawCliqueList:\n NodeAttributes[node]['rawClique'] = score\n NodeAttributes[node]['normRawClique'] = float(score - minRawCliqueScore)/float(maxRawCliqueScore - minRawCliqueScore)\n```\n\n\n```python\nfor node, score in weightedCliqueList:\n NodeAttributes[node]['weigthedClique'] = score\n NodeAttributes[node]['normWeightedClique'] = float(score - minWeightedCliqueScore)/float(maxWeightedCliqueScore - minWeightedCliqueScore)\n```\n\n\n```python\nNodeAttributes[1874]\n```\n\n\n\n\n {'AuthScore': 0.30510831720503034,\n 'AvgShortPathLen': 1.456084656084656,\n 'Betweeness': 495085.4708108281,\n 'ClusterCoeff': 0.0014452513597956258,\n 'HubScore': 0.38410704002084095,\n 'InDegree': 249,\n 'NormClusterCoeff': 0.0014452513597956258,\n 'NormInDegree': 1.0,\n 'NormOutDegree': 1.0,\n 'OutDegree': 331,\n 'nodeClique': 157,\n 'normAuthScore': 1.0,\n 'normAvgShortPathLen': 0.33696583812905595,\n 'normBetweeness': 0.8890896997683746,\n 'normHubScore': 1.0,\n 'normNodeClique': 1.0,\n 'normRawClique': 1.0,\n 'normWeightedClique': 0.9530189645005671,\n 'rawClique': 29604,\n 'weigthedClique': 1079.971912453243}\n\n\n\n## Compute Weighted Social Score \n\n\n```python\n#\n# Weights:\n#\n# W = [ Weighted Clique, RawClique, NumClique, OutDegree, InDegree, Cluster Coeff, Betweeness, Avg ShortPath, Auth, Hub]\n#\nw_weightedClique = 0.9\nw_rawClique = 0.8\nw_numClique = 0.7\nw_outDegree = 0.6\nw_inDegree = 0.6\nw_clusterCoeff = 0.3 \nw_betweeness = 0.3 \nw_shortpath = 0.5 \nw_auth = 0.2 \nw_hub = 0.1 \n\nw = np.array([w_weightedClique,\\\n w_rawClique, \\\n w_numClique, \\\n w_outDegree, \\\n w_inDegree, \\\n w_clusterCoeff, \\\n w_betweeness, \\\n w_shortpath, \\\n w_auth, \\\n w_hub])\n\nw_sum = np.sum(w)\n```\n\n### Social Score is computed as follows \n\n\\begin{align}\nw_x * C_x & = w_x * 100 * \\left[ \\frac{x_i - infx}{sup\\ x - inf\\ x} \\right] \\\\\n\\\\\nscore & = \\frac{\\Sigma_{all\\ x} w * C_x}{\\Sigma_{all\\ x} w}\n\\end{align}\n\n\n```python\nsocialScore = list()\n\nfor node, attributes in NodeAttributes.iteritems():\n weightedClique = attributes.get('normWeightedClique', 0.0)\n rawClique = attributes.get('normRawClique', 0.0)\n numClique = attributes.get('normNodeClique', 0.0)\n outDeg = attributes.get('NormOutDegree', 0.0)\n inDeg = attributes.get('NormInDegree', 0.0)\n clusterCoeff = attributes.get('NormClusterCoeff', 0.0)\n betweeness = attributes.get('normBetweeness', 0.0)\n shortestPath = attributes.get('normAvgShortPathLen', 0.0)\n authScore = attributes.get('normAuthScore', 0.0)\n hubSccore = attributes.get('normHubScore', 0.0)\n \n C_x = np.array([weightedClique, \\\n rawClique, \\\n numClique, \\\n outDeg, \\\n inDeg, \\\n clusterCoeff, \\\n betweeness, \\\n shortestPath, \\\n authScore, \\\n hubSccore])\n \n score = 100.0 * np.dot(w, C_x)/w_sum\n socialScore.append((node, score))\n \n```\n\n\n```python\nsocialScore.sort(key=lambda x: x[1], reverse=True)\n```\n\n\n```python\nprint socialScore[:10]\n```\n\n [(1874, 85.86720944906979), (1258, 55.9778907924103), (999, 50.61051396942534), (1669, 47.27542700847994), (511, 40.31705218340864), (585, 33.064089627423826), (1998, 32.41884981873541), (1839, 28.873591736745578), (1369, 28.7430390082418), (453, 28.201871267917863)]\n\n\n\n```python\nwith open('socialScore.txt', 'w') as fd:\n for (node, score) in socialScore:\n fd.write(\"%r %r\\n\" % (node, score))\n```\n", "meta": {"hexsha": "b2fb75b4438d9216fe67136254b7e23d206add0c", "size": 57673, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "socialScore/DncSocialScore.ipynb", "max_stars_repo_name": "ppsekhar/CS224W", "max_stars_repo_head_hexsha": "8992a92a5654f66c7c045b7b3b53560b53fd247b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-04-22T04:26:23.000Z", "max_stars_repo_stars_event_max_datetime": "2019-04-22T04:26:23.000Z", "max_issues_repo_path": "socialScore/DncSocialScore.ipynb", "max_issues_repo_name": "ppsekhar/CS224W", "max_issues_repo_head_hexsha": "8992a92a5654f66c7c045b7b3b53560b53fd247b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "socialScore/DncSocialScore.ipynb", "max_forks_repo_name": "ppsekhar/CS224W", "max_forks_repo_head_hexsha": "8992a92a5654f66c7c045b7b3b53560b53fd247b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2019-07-25T01:31:23.000Z", "max_forks_repo_forks_event_max_datetime": "2019-07-25T01:31:23.000Z", "avg_line_length": 39.3672354949, "max_line_length": 16912, "alphanum_fraction": 0.6464723527, "converted": true, "num_tokens": 7534, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.6959583376458152, "lm_q1q2_score": 0.4458558645706812}} {"text": "\n\n# Neuromatch Academy: Week 2, Day 5, Tutorial 1\n# Learning to Predict\n\n__Content creators:__ Marcelo Mattar and Eric DeWitt with help from Matt Krause\n\n__Content reviewers:__ Byron Galbraith and Michael Waskom\n\n\n---\n\n# Tutorial objectives\n \nIn this tutorial, we will learn how to estimate state-value functions in a classical conditioning paradigm using Temporal Difference (TD) learning and examine TD-errors at the presentation of the conditioned and unconditioned stimulus (CS and US) under different CS-US contingencies. These exercises will provide you with an understanding of both how reward prediction errors (RPEs) behave in classical conditioning and what we should expect to see if Dopamine represents a \"canonical\" model-free RPE. \n\nAt the end of this tutorial: \n* You will learn to use the standard tapped delay line conditioning model\n* You will understand how RPEs move to CS\n* You will understand how variability in reward size effects RPEs\n* You will understand how differences in US-CS timing effect RPEs\n\n\n```python\n# Imports\nimport numpy as np \nimport matplotlib.pyplot as plt\n```\n\n\n```python\n#@title Figure settings\nimport ipywidgets as widgets # interactive display\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/course-content/master/nma.mplstyle\")\n```\n\n\n```python\n# @title Helper functions\nfrom matplotlib import ticker\n\ndef plot_value_function(V, ax=None, show=True):\n \"\"\"Plot V(s), the value function\"\"\"\n if not ax:\n fig, ax = plt.subplots()\n\n ax.stem(V, use_line_collection=True)\n ax.set_ylabel('Value')\n ax.set_xlabel('State')\n ax.set_title(\"Value function: $V(s)$\")\n \n if show:\n plt.show()\n\ndef plot_tde_trace(TDE, ax=None, show=True, skip=400):\n \"\"\"Plot the TD Error across trials\"\"\"\n if not ax:\n fig, ax = plt.subplots()\n\n indx = np.arange(0, TDE.shape[1], skip)\n im = ax.imshow(TDE[:,indx])\n positions = ax.get_xticks()\n # Avoid warning when setting string tick labels\n ax.xaxis.set_major_locator(ticker.FixedLocator(positions))\n ax.set_xticklabels([f\"{int(skip * x)}\" for x in positions])\n ax.set_title('TD-error over learning')\n ax.set_ylabel('State')\n ax.set_xlabel('Iterations')\n ax.figure.colorbar(im)\n if show:\n plt.show()\n\ndef learning_summary_plot(V, TDE):\n \"\"\"Summary plot for Ex1\"\"\"\n fig, (ax1, ax2) = plt.subplots(nrows = 2, gridspec_kw={'height_ratios': [1, 2]})\n \n plot_value_function(V, ax=ax1, show=False)\n plot_tde_trace(TDE, ax=ax2, show=False)\n plt.tight_layout()\n\ndef reward_guesser_title_hint(r1, r2):\n \"\"\"\"Provide a mildly obfuscated hint for a demo.\"\"\"\n if (r1==14 and r2==6) or (r1==6 and r2==14):\n return \"Technically correct...(the best kind of correct)\"\n \n if ~(~(r1+r2) ^ 11) - 1 == (6 | 24): # Don't spoil the fun :-)\n return \"Congratulations! You solved it!\"\n\n return \"Keep trying....\"\n\n#@title Default title text\nclass ClassicalConditioning:\n \n def __init__(self, n_steps, reward_magnitude, reward_time):\n \n # Task variables\n self.n_steps = n_steps \n self.n_actions = 0\n self.cs_time = int(n_steps/4) - 1\n\n # Reward variables\n self.reward_state = [0,0]\n self.reward_magnitude = None\n self.reward_probability = None\n self.reward_time = None\n \n self.set_reward(reward_magnitude, reward_time)\n \n # Time step at which the conditioned stimulus is presented\n\n # Create a state dictionary\n self._create_state_dictionary()\n \n def set_reward(self, reward_magnitude, reward_time):\n \n \"\"\"\n Determine reward state and magnitude of reward\n \"\"\"\n if reward_time >= self.n_steps - self.cs_time:\n self.reward_magnitude = 0\n \n else:\n self.reward_magnitude = reward_magnitude\n self.reward_state = [1, reward_time]\n \n def get_outcome(self, current_state):\n \n \"\"\"\n Determine next state and reward\n \"\"\"\n # Update state\n if current_state < self.n_steps - 1: \n next_state = current_state + 1\n else:\n next_state = 0\n \n # Check for reward\n if self.reward_state == self.state_dict[current_state]:\n reward = self.reward_magnitude\n else:\n reward = 0\n \n return next_state, reward\n \n def _create_state_dictionary(self):\n \n \"\"\"\n This dictionary maps number of time steps/ state identities\n in each episode to some useful state attributes:\n \n state - 0 1 2 3 4 5 (cs) 6 7 8 9 10 11 12 ...\n is_delay - 0 0 0 0 0 0 (cs) 1 1 1 1 1 1 1 ...\n t_in_delay - 0 0 0 0 0 0 (cs) 1 2 3 4 5 6 7 ...\n \"\"\"\n d = 0\n\n self.state_dict = {}\n for s in range(self.n_steps):\n if s <= self.cs_time:\n self.state_dict[s] = [0,0]\n else: \n d += 1 # Time in delay \n self.state_dict[s] = [1,d]\n \nclass MultiRewardCC(ClassicalConditioning):\n \"\"\"Classical conditioning paradigm, except that one randomly selected reward, \n magnitude, from a list, is delivered of a single fixed reward.\"\"\"\n def __init__(self, n_steps, reward_magnitudes, reward_time=None):\n \"\"\"\"Build a multi-reward classical conditioning environment\n Args:\n - nsteps: Maximum number of steps\n - reward_magnitudes: LIST of possible reward magnitudes.\n - reward_time: Single fixed reward time\n Uses numpy global random state.\n \"\"\"\n super().__init__(n_steps, 1, reward_time)\n self.reward_magnitudes = reward_magnitudes\n \n def get_outcome(self, current_state):\n next_state, reward = super().get_outcome(current_state)\n if reward:\n reward=np.random.choice(self.reward_magnitudes)\n return next_state, reward\n \n\nclass ProbabilisticCC(ClassicalConditioning):\n \"\"\"Classical conditioning paradigm, except that rewards are stochastically omitted.\"\"\"\n def __init__(self, n_steps, reward_magnitude, reward_time=None, p_reward=0.75):\n \"\"\"\"Build a multi-reward classical conditioning environment\n Args:\n - nsteps: Maximum number of steps\n - reward_magnitudes: Reward magnitudes.\n - reward_time: Single fixed reward time.\n - p_reward: probability that reward is actually delivered in rewarding state\n Uses numpy global random state.\n \"\"\"\n super().__init__(n_steps, reward_magnitude, reward_time)\n self.p_reward = p_reward\n \n def get_outcome(self, current_state):\n next_state, reward = super().get_outcome(current_state)\n if reward:\n reward*= int(np.random.uniform(size=1)[0] < self.p_reward)\n return next_state, reward\n\n```\n\n---\n# Section 1: TD-learning\n\n\n```python\n#@title Video 1: Introduction\nfrom IPython.display import IFrame\nclass BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\nvideo = BiliVideo(id='BV13f4y1d7om', width=854, height=480, fs=1)\nprint(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\nvideo\n```\n\n Video available at https://www.bilibili.com/video/BV13f4y1d7om\n\n\n\n\n\n\n\n\n\n\n\n__Environment:__\n\n- The agent experiences the environment in episodes or trials. \n- Episodes terminate by transitioning to the inter-trial-interval (ITI) state and they are initiated from the ITI state as well. We clamp the value of the terminal/ITI states to zero. \n- The classical conditioning environment is composed of a sequence of states that the agent deterministically transitions through. Starting at State 0, the agent moves to State 1 in the first step, from State 1 to State 2 in the second, and so on. These states represent time in the tapped delay line representation\n- Within each episode, the agent is presented a CS and US (reward). \n- The CS is always presented at 1/4 of the total duration of the trial. The US (reward) is then delivered after the CS. The interval between the CS and US is specified by `reward_time`.\n- The agent's goal is to learn to predict expected rewards from each state in the trial. \n\n\n**General concepts**\n\n* Return $G_{t}$: future cumulative reward, which can be written in arecursive form\n\\begin{align}\nG_{t} &= \\sum \\limits_{k = 0}^{\\infty} \\gamma^{k} r_{t+k+1} \\\\\n&= r_{t+1} + \\gamma G_{t+1}\n\\end{align}\nwhere $\\gamma$ is discount factor that controls the importance of future rewards, and $\\gamma \\in [0, 1]$. $\\gamma$ may also be interpreted as probability of continuing the trajectory.\n* Value funtion $V_{\\pi}(s_t=s)$: expecation of the return\n\\begin{align}\nV_{\\pi}(s_t=s) &= \\mathbb{E} [ G_{t}\\; | \\; s_t=s, a_{t:\\infty}\\sim\\pi] \\\\\n& = \\mathbb{E} [ r_{t+1} + \\gamma G_{t+1}\\; | \\; s_t=s, a_{t:\\infty}\\sim\\pi]\n\\end{align}\nWith an assumption of **Markov process**, we thus have:\n\\begin{align}\nV_{\\pi}(s_t=s) &= \\mathbb{E} [ r_{t+1} + \\gamma V_{\\pi}(s_{t+1})\\; | \\; s_t=s, a_{t:\\infty}\\sim\\pi] \\\\\n&= \\sum_a \\pi(a|s) \\sum_{r, s'}p(s', r)(r + V_{\\pi}(s_{t+1}=s'))\n\\end{align}\n\n**Temporal difference (TD) learning**\n\n* With a Markovian assumption, we can use $V(s_{t+1})$ as an imperfect proxy for the true value $G_{t+1}$ (Monte Carlo bootstrapping), and thus obtain the generalised equation to calculate TD-error:\n\\begin{align}\n\\delta_{t} = r_{t+1} + \\gamma V(s_{t+1}) - V(s_{t})\n\\end{align}\n\n* Value updated by using the learning rate constant $\\alpha$:\n\\begin{align}\nV(s_{t}) \\leftarrow V(s_{t}) + \\alpha \\delta_{t}\n\\end{align}\n\n (Reference: https://web.stanford.edu/group/pdplab/pdphandbook/handbookch10.html)\n\n\n\n__Definitions:__\n\n* TD-error:\n\\begin{align}\n\\delta_{t} = r_{t+1} + \\gamma V(s_{t+1}) - V(s_{t})\n\\end{align}\n\n* Value updates:\n\\begin{align}\nV(s_{t}) \\leftarrow V(s_{t}) + \\alpha \\delta_{t}\n\\end{align}\n\n\n## Exercise 1: TD-learning with guaranteed rewards\n \nImplement TD-learning to estimate the state-value function in the classical-conditioning world with guaranteed rewards, with a fixed magnitude, at a fixed delay after the conditioned stimulus, CS. Save TD-errors over learning (i.e., over trials) so we can visualize them afterwards. \n\nIn order to simulate the effect of the CS, you should only update $V(s_{t})$ during the delay period after CS. This period is indicated by the boolean variable `is_delay`. This can be implemented by multiplying the expression for updating the value function by `is_delay`.\n\nUse the provided code to estimate the value function.\n\n\n```python\ndef td_learner(env, n_trials, gamma=0.98, alpha=0.001):\n \"\"\" Temporal Difference learning\n\n Args:\n env (object): the environment to be learned\n n_trials (int): the number of trials to run\n gamma (float): temporal discount factor\n alpha (float): learning rate\n \n Returns:\n ndarray, ndarray: the value function and temporal difference error arrays\n \"\"\"\n V = np.zeros(env.n_steps) # Array to store values over states (time)\n TDE = np.zeros((env.n_steps, n_trials)) # Array to store TD errors\n\n for n in range(n_trials):\n state = 0 # Initial state\n for t in range(env.n_steps):\n # Get next state and next reward\n next_state, reward = env.get_outcome(state)\n # Is the current state in the delay period (after CS)?\n is_delay = env.state_dict[state][0]\n \n ########################################################################\n ## TODO for students: implement TD error and value function update \n # Fill out function and remove\n raise NotImplementedError(\"Student excercise: implement TD error and value function update\")\n #################################################################################\n # Write an expression to compute the TD-error\n TDE[state, n] = ...\n\n # Write an expression to update the value function\n V[state] += ...\n\n # Update state\n state = next_state\n\n return V, TDE\n\n\n# Uncomment once the td_learner function is complete\n# env = ClassicalConditioning(n_steps=40, reward_magnitude=10, reward_time=10)\n# V, TDE = td_learner(env, n_trials=20000)\n# learning_summary_plot(V, TDE)\n```\n\n\n```python\n#to_remove solution\ndef td_learner(env, n_trials, gamma=0.98, alpha=0.001):\n \"\"\" Temporal Difference learning\n\n Args:\n env (object): the environment to be learned\n n_trials (int): the number of trials to run\n gamma (float): temporal discount factor\n alpha (float): learning rate\n \n Returns:\n ndarray, ndarray: the value function and temporal difference error arrays\n \"\"\"\n V = np.zeros(env.n_steps) # Array to store values over states (time)\n TDE = np.zeros((env.n_steps, n_trials)) # Array to store TD errors\n\n for n in range(n_trials):\n state = 0 # Initial state\n for t in range(env.n_steps):\n # Get next state and next reward\n next_state, reward = env.get_outcome(state)\n # Is the current state in the delay period (after CS)?\n is_delay = env.state_dict[state][0]\n \n # Write an expression to compute the TD-error\n TDE[state, n] = (reward + gamma * V[next_state] - V[state]) \n \n # Write an expression to update the value function\n V[state] += alpha * TDE[state, n] * is_delay\n \n # Update state\n state = next_state\n\n return V, TDE\n\n\nenv = ClassicalConditioning(n_steps=40, reward_magnitude=10, reward_time=10)\nV, TDE = td_learner(env, n_trials=20000)\nwith plt.xkcd():\n learning_summary_plot(V, TDE)\n```\n\n## Interactive Demo 1: US to CS Transfer \n\nDuring classical conditioning, the subject's behavioral response (e.g., salivating) transfers from the unconditioned stimulus (US; like the smell of tasty food) to the conditioned stimulus (CS; like Pavlov ringing his bell) that predicts it. Reward prediction errors play an important role in this process by adjusting the value of states according to their expected, discounted return.\n\nUse the widget below to examine how reward prediction errors change over time. Before training (orange line), only the reward state has high reward prediction error. As training progresses (blue line, slider), the reward prediction errors shift to the conditioned stimulus, where they end up when the trial is complete (green line). \n\nDopamine neurons, which are thought to carry reward prediction errors _in vivo_, show exactly the same behavior!\n\n\n\n```python\n#@title \n\n#@markdown Make sure you execute this cell to enable the widget!\n\nn_trials = 20000\n\n@widgets.interact\ndef plot_tde_by_trial(trial = widgets.IntSlider(value=5000, min=0, max=n_trials-1 , step=1, description=\"Trial #\")):\n if 'TDE' not in globals():\n print(\"Complete Exercise 1 to enable this interactive demo!\")\n else:\n\n fig, ax = plt.subplots()\n ax.axhline(0, color='k') # Use this + basefmt=' ' to keep the legend clean.\n ax.stem(TDE[:, 0], linefmt='C1-', markerfmt='C1d', basefmt=' ', \n label=\"Before Learning (Trial 0)\",\n use_line_collection=True)\n ax.stem(TDE[:, -1], linefmt='C2-', markerfmt='C2s', basefmt=' ', \n label=\"After Learning (Trial $\\infty$)\",\n use_line_collection=True)\n ax.stem(TDE[:, trial], linefmt='C0-', markerfmt='C0o', basefmt=' ', \n label=f\"Trial {trial}\",\n use_line_collection=True)\n \n ax.set_xlabel(\"State in trial\")\n ax.set_ylabel(\"TD Error\")\n ax.set_title(\"Temporal Difference Error by Trial\")\n ax.legend()\n```\n\n## Interactive Demo 2: Learning Rates and Discount Factors\n\nOur TD-learning agent has two parameters that control how it learns: $\\alpha$, the learning rate, and $\\gamma$, the discount factor. In Exercise 1, we set these parameters to $\\alpha=0.001$ and $\\gamma=0.98$ for you. Here, you'll investigate how changing these parameters alters the model that TD-learning learns.\n\nBefore enabling the interactive demo below, take a moment to think about the functions of these two parameters. $\\alpha$ controls the size of the Value function updates produced by each TD-error. In our simple, deterministic world, will this affect the final model we learn? Is a larger $\\alpha$ necessarily better in more complex, realistic environments?\n\nThe discount rate $\\gamma$ applies an exponentially-decaying weight to returns occuring in the future, rather than the present timestep. How does this affect the model we learn? What happens when $\\gamma=0$ or $\\gamma \\geq 1$?\n\nUse the widget to test your hypotheses.\n\n\n\n\n\n```python\n#@title \n\n#@markdown Make sure you execute this cell to enable the widget!\n\n@widgets.interact\ndef plot_summary_alpha_gamma(alpha = widgets.FloatSlider(value=0.0001, min=0.001, max=0.1, step=0.0001, description=\"alpha\"),\n gamma = widgets.FloatSlider(value=0.980, min=0, max=1.1, step=0.010, description=\"gamma\")):\n env = ClassicalConditioning(n_steps=40, reward_magnitude=10, reward_time=10) \n try:\n V_params, TDE_params = td_learner(env, n_trials=20000, gamma=gamma, alpha=alpha)\n except NotImplementedError:\n print(\"Finish Exercise 1 to enable this interactive demo\")\n \n learning_summary_plot(V_params,TDE_params)\n\n\n```\n\n\n```python\n#to_remove explanation\n\"\"\"\nAlpha determines how fast the model learns. In the simple, deterministic world\nwe're using here, this allows the moodel to quickly converge onto the \"true\" \nmodel that heavily values the conditioned stimulus. In more complex environments,\nhowever, excessively large values of alpha can slow, or even prevent, learning, \nas we'll see later. \n\nGamma effectively controls how much the model cares about the future: larger values of\ngamma cause the model to weight future rewards nearly as much as present ones. At gamma=1,\nthe model weights all rewards, regardless of when they occur, equally and when greater than one, it\nstarts to *prefer* rewards in the future, rather than the present (this is rarely good). \nWhen gamma=0, however, the model becomes greedy and only considers rewards that \ncan be obtained immediately.\n \"\"\";\n```\n\n---\n# Section 2: TD-learning with varying reward magnitudes\n\nIn the previous exercise, the environment was as simple as possible. On every trial, the CS predicted the same reward, at the same time, with 100% certainty. In the next few exercises, we will make the environment more progressively more complicated and examine the TD-learner's behavior. \n\n\n## Interactive Demo 3: Match the Value Functions\n\nFirst, will replace the environment with one that dispenses one of several rewards, chosen at random. Shown below is the final value function $V$ for a TD learner that was trained in an enviroment where the CS predicted a reward of 6 or 14 units; both rewards were equally likely). \n\nCan you find another pair of rewards that cause the agent to learn the same value function? Assume each reward will be dispensed 50% of the time. \n\nHints:\n* Carefully consider the definition of the value function $V$. This can be solved analytically.\n* There is no need to change $\\alpha$ or $\\gamma$. \n* Due to the randomness, there may be a small amount of variation.\n\n\n```python\n#@title \n\n#@markdown Make sure you execute this cell to enable the widget!\n\nn_trials = 20000\nnp.random.seed(2020)\nrng_state = np.random.get_state()\nenv = MultiRewardCC(40, [6, 14], reward_time=10)\nV_multi, TDE_multi = td_learner(env, n_trials, gamma=0.98, alpha=0.001)\n\n@widgets.interact\ndef reward_guesser_interaction(r1 = widgets.IntText(value=0, min=0, max=50, description=\"Reward 1\"),\n r2 = widgets.IntText(value=0, min=0, max=50, description=\"Reward 2\")): \n try:\n env2 = MultiRewardCC(40, [r1, r2], reward_time=10)\n V_guess, _ = td_learner(env2, n_trials, gamma=0.98, alpha=0.001)\n fig, ax = plt.subplots()\n m, l, _ = ax.stem(V_multi, linefmt='y-', markerfmt='yo', basefmt=' ', label=\"Target\", \n use_line_collection=True)\n m.set_markersize(15)\n m.set_markerfacecolor('none')\n l.set_linewidth(4)\n m, _, _ = ax.stem(V_guess, linefmt='r', markerfmt='rx', basefmt=' ', label=\"Guess\",\n use_line_collection=True)\n m.set_markersize(15)\n\n ax.set_xlabel(\"State\")\n ax.set_ylabel(\"Value\")\n ax.set_title(\"Guess V(s)\\n\" + reward_guesser_title_hint(r1, r2))\n ax.legend()\n except NotImplementedError:\n print(\"Please finish Exercise 1 first!\")\n```\n\n## Section 2.1 Examining the TD Error\n\nRun the cell below to plot the TD errors from our multi-reward environment. A new feature appears in this plot? What is it? Why does it happen?\n\n\n```python\nplot_tde_trace(TDE_multi)\n```\n\n\n```python\n#to_remove explanation\n\"\"\"\nThe TD trace now takes on negative values because the reward delivered is\nsometimes larger than the expected reward and sometimes smaller.\n\"\"\";\n```\n\n---\n# Section 3: TD-learning with probabilistic rewards\n\nIn this environment, we'll return to delivering a single reward of ten units. However, it will be delivered intermittently: on 20 percent of trials, the CS will be shown but the agent will not receive the usual reward; the remaining 80% will proceed as usual.\n\n Run the cell below to simulate. How does this compare with the previous experiment?\n\nEarlier in the notebook, we saw that changing $\\alpha$ had little effect on learning in a deterministic environment. What happens if you set it to an large value, like 1, in this noisier scenario? Does it seem like it will _ever_ converge?\n\n\n```python\nnp.random.set_state(rng_state) # Resynchronize everyone's notebooks\nn_trials = 20000\ntry:\n env = ProbabilisticCC(n_steps=40, reward_magnitude=10, reward_time=10, \n p_reward=0.8)\n V_stochastic, TDE_stochastic = td_learner(env, n_trials*2, alpha=1)\n learning_summary_plot(V_stochastic, TDE_stochastic)\nexcept NotImplementedError: \n print(\"Please finish Exercise 1 first\")\n```\n\n\n```python\n#to_remove explanation\n\"\"\"\nThe multi-reward and probabilistic reward enviroments are the same. You \ncould simulate a probabilistic reward of 10 units, delivered 50% of the time, \nby having a mixture of 10 and 0 unit rewards, or vice versa. The takehome message \nfrom these last three exercises is that the *average* or expected reward is \nwhat matters during TD learning.\n\nLarge values of alpha prevent the TD Learner from converging. As a result, the \nvalue function seems implausible: one state may have an extremely low value while\nthe neighboring ones remain high. This pattern persists even if training continues \nfor hundreds of thousands of trials.\n\"\"\";\n```\n\n---\n# Summary\n\nIn this notebook, we have developed a simple TD Learner and examined how its state representations and reward prediction errors evolve during training. By manipualting its environment and parameters ($\\alpha$, $\\gamma$), you developed an intuition for how it behaves. \n\nThis simple model closely resembles the behavior of subjects undergoing classical conditioning tasks and the dopamine neurons that may underlie that behavior. You may have implemented TD-reset or used the model to recreate a common experimental error. The update rule used here has been extensively studied for [more than 70 years](https://www.pnas.org/content/108/Supplement_3/15647) as a possible explanation for artificial and biological learning. \n\nHowever, you may have noticed that something is missing from this notebook. We carefully calculated the value of each state, but did not use it to actually do anything. Using values to plan _**Actions**_ is coming up next!\n\n# Bonus\n\n## Exercise 2: Removing the CS\n\nIn Exercise 1, you (should have) included a term that depends on the conditioned stimulus. Remove it and see what happens. Do you understand why?\nThis phenomena often fools people attempting to train animals--beware!\n\n\n```python\n#to_remove explanation\n\n\"\"\"\nYou should only be updating the V[state] once the conditioned stimulus appears.\n\nIf you remove this term the Value Function becomes periodic, dropping towards zero \nright after the reward and gradually rising towards the end of the trial. This\nbehavior is actually correct, because the model is learning the time until the \n*next* reward, and State 37 is closer to a reward than State 21 or 22. \n\nIn an actual experiment, the animal often just wants rewards; it doesn't care about \n/your/ experiment or trial structure! \n\"\"\";\n```\n", "meta": {"hexsha": "6d8f10be9c0e52edb94ca607322cc7492ba4b412", "size": 498553, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W2D5_ReinforcementLearning/W2D5_Tutorial1.ipynb", "max_stars_repo_name": "erlichlab/course-content", "max_stars_repo_head_hexsha": "af199353593127ebeb97111e31754394a7fe733f", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 26, "max_stars_repo_stars_event_min_datetime": "2020-07-01T20:38:44.000Z", "max_stars_repo_stars_event_max_datetime": "2021-06-20T06:37:27.000Z", "max_issues_repo_path": "tutorials/W2D5_ReinforcementLearning/W2D5_Tutorial1.ipynb", "max_issues_repo_name": "erlichlab/course-content", "max_issues_repo_head_hexsha": "af199353593127ebeb97111e31754394a7fe733f", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2020-06-23T03:46:36.000Z", "max_issues_repo_issues_event_max_datetime": "2020-07-07T05:26:01.000Z", "max_forks_repo_path": "tutorials/W2D5_ReinforcementLearning/W2D5_Tutorial1.ipynb", "max_forks_repo_name": "erlichlab/course-content", "max_forks_repo_head_hexsha": "af199353593127ebeb97111e31754394a7fe733f", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 16, "max_forks_repo_forks_event_min_datetime": "2020-07-06T06:48:02.000Z", "max_forks_repo_forks_event_max_datetime": "2021-07-30T08:18:52.000Z", "avg_line_length": 249.526026026, "max_line_length": 129952, "alphanum_fraction": 0.912627143, "converted": true, "num_tokens": 6099, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.6959583313396339, "lm_q1q2_score": 0.44585586053071535}} {"text": "```python\n%matplotlib inline\n```\n\n\n\u5e8f\u5217\u6a21\u578b\u548c LSTM \u7f51\u7edc\uff08\u957f\u77ed\u8bb0\u5fc6\u7f51\u7edc\uff09\n===================================================\n\n\u4e4b\u524d\u6211\u4eec\u5df2\u7ecf\u5b66\u8fc7\u4e86\u8bb8\u591a\u7684\u524d\u9988\u7f51\u7edc. \u6240\u8c13\u524d\u9988\u7f51\u7edc, \u5c31\u662f\u7f51\u7edc\u4e2d\u4e0d\u4f1a\u4fdd\u5b58\u72b6\u6001. \u7136\u800c\u6709\u65f6\n\u8fd9\u5e76\u4e0d\u662f\u6211\u4eec\u60f3\u8981\u7684\u6548\u679c. \u5728\u81ea\u7136\u8bed\u8a00\u5904\u7406 (NLP, Natural Language Processing)\n\u4e2d, \u5e8f\u5217\u6a21\u578b\u662f\u4e00\u4e2a\u6838\u5fc3\u7684\u6982\u5ff5. \u6240\u8c13\u5e8f\u5217\u6a21\u578b, \u5373\u8f93\u5165\u4f9d\u8d56\u4e8e\u65f6\u95f4\u4fe1\u606f\u7684\u6a21\u578b. \u4e00\u4e2a\u5178\u578b\n\u7684\u5e8f\u5217\u6a21\u578b\u662f\u9690\u9a6c\u5c14\u79d1\u592b\u6a21\u578b (HMM, Hidden Markov Model). \u53e6\u4e00\u4e2a\u5e8f\u5217\u6a21\u578b\u7684\u4f8b\u5b50\n\u662f\u6761\u4ef6\u968f\u673a\u573a (CRF, Conditional Random Field).\n\n\u9012\u5f52\u795e\u7ecf\u7f51\u7edc\u662f\u6307\u53ef\u4ee5\u4fdd\u5b58\u67d0\u79cd\u72b6\u6001\u7684\u795e\u7ecf\u7f51\u7edc. \u6bd4\u5982\u8bf4, \u7f51\u7edc\u4e0a\u4e2a\u65f6\u523b\u7684\u8f93\u51fa\u53ef\u4ee5\u4f5c\u4e3a\u4e0b\u4e2a\n\u65f6\u523b\u7684\u8f93\u5165, \u8fd9\u6837\u4fe1\u606f\u5c31\u53ef\u4ee5\u901a\u8fc7\u5e8f\u5217\u5728\u7f51\u7edc\u4e2d\u4e00\u76f4\u5f80\u540e\u4f20\u9012. \u5bf9\u4e8eLSTM (Long-Short \nTerm Memory) \u6765\u8bf4, \u5e8f\u5217\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u6709\u4e00\u4e2a\u76f8\u5e94\u7684\u9690\u72b6\u6001 $h_t$, \u8be5\u9690\u72b6\u6001\n\u539f\u5219\u4e0a\u53ef\u4ee5\u5305\u542b\u5e8f\u5217\u5f53\u524d\u7ed3\u70b9\u4e4b\u524d\u7684\u4efb\u4e00\u8282\u70b9\u7684\u4fe1\u606f. \u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u9690\u85cf\u72b6\u6001\u6765\u9884\u6d4b\u8bed\u8a00\u6a21\u578b\n\u4e2d\u7684\u5355\u8bcd, \u8bcd\u6027\u6807\u7b7e\u4ee5\u53ca\u5176\u4ed6\u5404\u79cd\u5404\u6837\u7684\u4e1c\u897f.\n\n\nPytorch \u4e2d\u7684 LSTM\n~~~~~~~~~~~~~~~~~~\n\n\u5f00\u59cb\u4f8b\u5b50\u4e4b\u524d,\u6709\u51e0\u4e2a\u70b9\u8bf4\u660e\u4e00\u4e0b. Pytorch \u4e2d, LSTM \u7684\u6240\u6709\u7684\u5f62\u5f0f\u56fa\u5b9a\u4e3a3D \u7684 tensor.\n\u6bcf\u4e2a\u7ef4\u5ea6\u6709\u56fa\u5b9a\u7684\u8bed\u4e49\u542b\u4e49, \u4e0d\u80fd\u4e71\u6389. \u5176\u4e2d\u7b2c\u4e00\u7ef4\u662f\u5e8f\u5217\u672c\u8eab, \u7b2c\u4e8c\u7ef4\u4ee5 mini-batch \u5f62\u5f0f\n\u6765\u7d22\u5f15\u5b9e\u4f8b, \u800c\u7b2c\u4e09\u7ef4\u5219\u7d22\u5f15\u8f93\u5165\u7684\u5143\u7d20. \u56e0\u4e3a\u6211\u4eec\u6ca1\u6709\u8ba8\u8bba\u8fc7 mini-batch, \u6240\u4ee5\u5728\u8fd9\u91cc\u6211\u4eec\n\u5047\u8bbe\u7b2c\u4e8c\u7ef4\u7684\u7ef4\u5ea6\u603b\u662f1. \u5982\u679c\u6211\u4eec\u60f3\u5728\u53e5\u5b50 \"The cow jumped\" \u4e0a\u8fd0\u884c\u4e00\u4e2a\u5e8f\u5217\u6a21\u578b, \u6a21\u578b\n\u7684\u8f93\u5165\u7c7b\u4f3c\u8fd9\u6837:\n\n\\begin{align}\\begin{bmatrix}\n \\overbrace{q_\\text{The}}^\\text{row vector} \\\\\n q_\\text{cow} \\\\\n q_\\text{jumped}\n \\end{bmatrix}\\end{align}\n\n\u9664\u4e86\u6709\u4e00\u4e2a\u989d\u5916\u7684\u5927\u5c0f\u4e3a1\u7684\u7b2c\u4e8c\u7ef4\u5ea6.\n\n\u6b64\u5916, \u4f60\u8fd8\u53ef\u4ee5\u5411\u7f51\u7edc\u9010\u4e2a\u8f93\u5165\u5e8f\u5217, \u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b, \u7b2c\u4e00\u4e2a\u8f74\u7684\u5927\u5c0f\u4e5f\u662f1.\n\n\u6765\u770b\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50.\n\n\n\n\n```python\n# \u4f5c\u8005: Robert Guthrie\n\nimport torch\nimport torch.autograd as autograd\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport torch.optim as optim\n\ntorch.manual_seed(1)\n```\n\n\n\n\n \n\n\n\n\n```python\nlstm = nn.LSTM(3, 3) # \u8f93\u5165\u7ef4\u5ea6\u662f3, \u8f93\u51fa\u7ef4\u5ea6\u4e5f\u662f3\ninputs = [autograd.Variable(torch.randn((1, 3)))\n for _ in range(5)] # \u6784\u9020\u4e00\u4e2a\u957f\u5ea6\u4e3a5\u7684\u5e8f\u5217\n\n# \u521d\u59cb\u5316\u9690\u85cf\u72b6\u6001\nhidden = (autograd.Variable(torch.randn(1, 1, 3)),\n autograd.Variable(torch.randn((1, 1, 3))))\nfor i in inputs:\n # \u5c06\u5e8f\u5217\u7684\u5143\u7d20\u9010\u4e2a\u8f93\u5165\u5230LSTM\n # \u7ecf\u8fc7\u6bcf\u6b65\u64cd\u4f5c,hidden \u7684\u503c\u5305\u542b\u4e86\u9690\u85cf\u72b6\u6001\u7684\u4fe1\u606f\n out, hidden = lstm(i.view(1, 1, -1), hidden)\n\n# \u53e6\u5916, \u6211\u4eec\u8fd8\u53ef\u4ee5\u4e00\u6b21\u5bf9\u6574\u4e2a\u5e8f\u5217\u8fdb\u884c\u8bad\u7ec3. LSTM \u8fd4\u56de\u7684\u7b2c\u4e00\u4e2a\u503c\u8868\u793a\u6240\u6709\u65f6\u523b\u7684\u9690\u72b6\u6001\u503c,\n# \u7b2c\u4e8c\u4e2a\u503c\u8868\u793a\u6700\u8fd1\u7684\u9690\u72b6\u6001\u503c (\u56e0\u6b64\u4e0b\u9762\u7684 \"out\"\u7684\u6700\u540e\u4e00\u4e2a\u503c\u548c \"hidden\" \u7684\u503c\u662f\u4e00\u6837\u7684).\n# \u4e4b\u6240\u4ee5\u8fd9\u6837\u8bbe\u8ba1, \u662f\u4e3a\u4e86\u901a\u8fc7 \"out\" \u7684\u503c\u6765\u83b7\u53d6\u6240\u6709\u7684\u9690\u72b6\u6001\u503c, \u800c\u7528 \"hidden\" \u7684\u503c\u6765\n# \u8fdb\u884c\u5e8f\u5217\u7684\u53cd\u5411\u4f20\u64ad\u8fd0\u7b97, \u5177\u4f53\u65b9\u5f0f\u5c31\u662f\u5c06\u5b83\u4f5c\u4e3a\u53c2\u6570\u4f20\u5165\u540e\u9762\u7684 LSTM \u7f51\u7edc.\n\n# \u589e\u52a0\u989d\u5916\u7684\u7b2c\u4e8c\u4e2a\u7ef4\u5ea6\ninputs = torch.cat(inputs).view(len(inputs), 1, -1)\nhidden = (autograd.Variable(torch.randn(1, 1, 3)), autograd.Variable(\n torch.randn((1, 1, 3)))) # \u6e05\u7a7a\u8f93\u51fa\u9690\u72b6\u6001\nout, hidden = lstm(inputs, hidden)\nprint(out)\nprint(hidden)\n```\n\n tensor([[[-0.0187, 0.1713, -0.2944]],\n \n [[-0.3521, 0.1026, -0.2971]],\n \n [[-0.3191, 0.0781, -0.1957]],\n \n [[-0.1634, 0.0941, -0.1637]],\n \n [[-0.3368, 0.0959, -0.0538]]], grad_fn=)\n (tensor([[[-0.3368, 0.0959, -0.0538]]], grad_fn=), tensor([[[-0.9825, 0.4715, -0.0633]]], grad_fn=))\n\n\n\u4f8b\u5b50: \u7528 LSTM \u6765\u8fdb\u884c\u8bcd\u6027\u6807\u6ce8\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n\u5728\u8fd9\u90e8\u5206, \u6211\u4eec\u5c06\u4f1a\u4f7f\u7528\u4e00\u4e2a LSTM \u7f51\u7edc\u6765\u8fdb\u884c\u8bcd\u6027\u6807\u6ce8. \u5728\u8fd9\u91cc\u6211\u4eec\u4e0d\u4f1a\u7528\u5230\u7ef4\u7279\u6bd4\u7b97\u6cd5,\n\u524d\u5411\u540e\u5411\u7b97\u6cd5\u6216\u8005\u4efb\u4f55\u7c7b\u4f3c\u7684\u7b97\u6cd5, \u800c\u662f\u5c06\u8fd9\u90e8\u5206\u5185\u5bb9\u4f5c\u4e3a\u4e00\u4e2a (\u6709\u6311\u6218) \u7684\u7ec3\u4e60\u7559\u7ed9\u8bfb\u8005,\n\u5e0c\u671b\u8bfb\u8005\u5728\u4e86\u89e3\u4e86\u8fd9\u90e8\u5206\u7684\u5185\u5bb9\u540e\u80fd\u591f\u5b9e\u73b0\u5982\u4f55\u5c06\u7ef4\u7279\u6bd4\u7b97\u6cd5\u5e94\u7528\u5230 LSTM \u7f51\u7edc\u4e2d\u6765.\n\n\u6574\u4e2a\u6a21\u578b\u7684\u53c2\u6570\u5b9a\u4e49\u5982\u4e0b: \u8f93\u5165\u7684\u53e5\u5b50\u5b9a\u4e49\u4e3a $w_1, \\dots, w_M$, \u5176\u4e2d\u52a8\u8bcd\u5b9a\u4e49\n\u4e3a $w_1, \\dots, w_M$, \u6807\u7b7e\u96c6\u5408\u5b9a\u4e49\u4e3a $T$, \u5355\u8bcd $w_i$ \u7684\u5b9e\u9645\n\u6807\u7b7e\u4e3a $y_i$. \u5b9a\u4e49\u5355\u8bcd $w_i$ \u7684\u9884\u6d4b\u6807\u7b7e\u4e3a $\\hat{y}_i$.\n\n\u8fd9\u662f\u4e00\u4e2a\u7ed3\u6784\u9884\u6d4b\u6a21\u578b, \u6211\u4eec\u7684\u8f93\u51fa\u662f\u4e00\u4e2a\u5e8f\u5217 $\\hat{y}_1, \\dots, \\hat{y}_M$,\n\u5176\u4e2d $\\hat{y}_i \\in T$.\n\n\u5728\u8fdb\u884c\u9884\u6d4b\u65f6, \u9700\u5c06\u53e5\u5b50\u6bcf\u4e2a\u8bcd\u8f93\u5165\u5230\u4e00\u4e2a LSTM \u7f51\u7edc\u4e2d. \u5c06\u65f6\u523b $i$ \u7684\u9690\u72b6\u6001\u6807\u8bb0\n\u4e3a $h_i$. \u540c\u6837\u5730, \u5bf9\u6bcf\u4e2a\u6807\u7b7e\u8d4b\u4e00\u4e2a\u72ec\u4e00\u65e0\u4e8c\u7684\u7d22\u5f15 (\u7c7b\u4f3c word embeddings \u90e8\u5206\nword\\_to\\_ix \u7684\u8bbe\u7f6e). \u7136\u540e\u5c31\u5f97\u5230\u4e86 $\\hat{y}_i$ \u7684\u9884\u6d4b\u89c4\u5219:\n\n\\begin{align}\\hat{y}_i = \\text{argmax}_j \\ (\\log \\text{Softmax}(Ah_i + b))_j\\end{align}\n\n\u5373\u5148\u5bf9\u9690\u72b6\u6001\u8fdb\u884c\u4e00\u4e2a\u4eff\u5c04\u53d8\u6362, \u7136\u540e\u8ba1\u7b97\u4e00\u4e2a\u5bf9\u6570 softmax, \u6700\u540e\u5f97\u5230\u7684\u9884\u6d4b\u6807\u7b7e\u5373\u4e3a\u5bf9\u6570\nsoftmax \u4e2d\u6700\u5927\u7684\u503c\u5bf9\u5e94\u7684\u6807\u7b7e. \u6ce8\u610f, \u8fd9\u4e5f\u610f\u5473\u7740 $A$ \u7a7a\u95f4\u7684\u7ef4\u5ea6\u662f $|T|$.\n\n\n\u51c6\u5907\u6570\u636e:\n\n\n\n\n```python\ndef prepare_sequence(seq, to_ix):\n idxs = [to_ix[w] for w in seq]\n tensor = torch.LongTensor(idxs)\n return autograd.Variable(tensor)\n\n\ntraining_data = [\n (\"The dog ate the apple\".split(), [\"DET\", \"NN\", \"V\", \"DET\", \"NN\"]),\n (\"Everybody read that book\".split(), [\"NN\", \"V\", \"DET\", \"NN\"])\n]\nword_to_ix = {}\nfor sent, tags in training_data:\n for word in sent:\n if word not in word_to_ix:\n word_to_ix[word] = len(word_to_ix)\nprint(word_to_ix)\ntag_to_ix = {\"DET\": 0, \"NN\": 1, \"V\": 2}\n\n# \u5b9e\u9645\u4e2d\u901a\u5e38\u4f7f\u7528\u66f4\u5927\u7684\u7ef4\u5ea6\u598232\u7ef4, 64\u7ef4.\n# \u8fd9\u91cc\u6211\u4eec\u4f7f\u7528\u5c0f\u7684\u7ef4\u5ea6, \u4e3a\u4e86\u65b9\u4fbf\u67e5\u770b\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u6743\u91cd\u7684\u53d8\u5316.\nEMBEDDING_DIM = 6\nHIDDEN_DIM = 6\n```\n\n {'The': 0, 'dog': 1, 'ate': 2, 'the': 3, 'apple': 4, 'Everybody': 5, 'read': 6, 'that': 7, 'book': 8}\n\n\n\u6784\u9020\u6a21\u578b:\n\n\n\n\n```python\nclass LSTMTagger(nn.Module):\n\n def __init__(self, embedding_dim, hidden_dim, vocab_size, tagset_size):\n super(LSTMTagger, self).__init__()\n self.hidden_dim = hidden_dim\n\n self.word_embeddings = nn.Embedding(vocab_size, embedding_dim)\n\n # LSTM \u4ee5 word_embeddings \u4f5c\u4e3a\u8f93\u5165, \u8f93\u51fa\u7ef4\u5ea6\u4e3a hidden_dim \u7684\u9690\u72b6\u6001\u503c\n self.lstm = nn.LSTM(embedding_dim, hidden_dim)\n\n # \u7ebf\u6027\u5c42\u5c06\u9690\u72b6\u6001\u7a7a\u95f4\u6620\u5c04\u5230\u6807\u6ce8\u7a7a\u95f4\n self.hidden2tag = nn.Linear(hidden_dim, tagset_size)\n self.hidden = self.init_hidden()\n\n def init_hidden(self):\n # \u5f00\u59cb\u65f6\u523b, \u6ca1\u6709\u9690\u72b6\u6001\n # \u5173\u4e8e\u7ef4\u5ea6\u8bbe\u7f6e\u7684\u8be6\u60c5,\u8bf7\u53c2\u8003 Pytorch \u6587\u6863\n # \u5404\u4e2a\u7ef4\u5ea6\u7684\u542b\u4e49\u662f (num_layers, minibatch_size, hidden_dim)\n return (autograd.Variable(torch.zeros(1, 1, self.hidden_dim)),\n autograd.Variable(torch.zeros(1, 1, self.hidden_dim)))\n\n def forward(self, sentence):\n embeds = self.word_embeddings(sentence)\n lstm_out, self.hidden = self.lstm(\n embeds.view(len(sentence), 1, -1), self.hidden)\n tag_space = self.hidden2tag(lstm_out.view(len(sentence), -1))\n tag_scores = F.log_softmax(tag_space, dim=1)\n return tag_scores\n```\n\n\u8bad\u7ec3\u6a21\u578b:\n\n\n\n\n```python\nmodel = LSTMTagger(EMBEDDING_DIM, HIDDEN_DIM, len(word_to_ix), len(tag_to_ix))\nloss_function = nn.NLLLoss()\noptimizer = optim.SGD(model.parameters(), lr=0.1)\n\n# \u67e5\u770b\u4e0b\u8bad\u7ec3\u524d\u5f97\u5206\u7684\u503c\n# \u6ce8\u610f: \u8f93\u51fa\u7684 i,j \u5143\u7d20\u7684\u503c\u8868\u793a\u5355\u8bcd i \u7684 j \u6807\u7b7e\u7684\u5f97\u5206\ninputs = prepare_sequence(training_data[0][0], word_to_ix)\ntag_scores = model(inputs)\nprint(tag_scores)\n\nfor epoch in range(300): # \u518d\u6b21\u8bf4\u660e\u4e0b, \u5b9e\u9645\u60c5\u51b5\u4e0b\u4f60\u4e0d\u4f1a\u8bad\u7ec3300\u4e2a\u5468\u671f, \u6b64\u4f8b\u4e2d\u6211\u4eec\u53ea\u662f\u6784\u9020\u4e86\u4e00\u4e9b\u5047\u6570\u636e\n for sentence, tags in training_data:\n # Step 1. \u8bf7\u8bb0\u4f4f Pytorch \u4f1a\u7d2f\u52a0\u68af\u5ea6\n # \u6bcf\u6b21\u8bad\u7ec3\u524d\u9700\u8981\u6e05\u7a7a\u68af\u5ea6\u503c\n model.zero_grad()\n\n # \u6b64\u5916\u8fd8\u9700\u8981\u6e05\u7a7a LSTM \u7684\u9690\u72b6\u6001\n # \u5c06\u5176\u4ece\u4e0a\u4e2a\u5b9e\u4f8b\u7684\u5386\u53f2\u4e2d\u5206\u79bb\u51fa\u6765\n model.hidden = model.init_hidden()\n\n # Step 2. \u51c6\u5907\u7f51\u7edc\u8f93\u5165, \u5c06\u5176\u53d8\u4e3a\u8bcd\u7d22\u5f15\u7684 Variables \u7c7b\u578b\u6570\u636e\n sentence_in = prepare_sequence(sentence, word_to_ix)\n targets = prepare_sequence(tags, tag_to_ix)\n\n # Step 3. \u524d\u5411\u4f20\u64ad\n tag_scores = model(sentence_in)\n\n # Step 4. \u8ba1\u7b97\u635f\u5931\u548c\u68af\u5ea6\u503c, \u901a\u8fc7\u8c03\u7528 optimizer.step() \u6765\u66f4\u65b0\u68af\u5ea6\n loss = loss_function(tag_scores, targets)\n loss.backward()\n optimizer.step()\n\n# \u67e5\u770b\u8bad\u7ec3\u540e\u5f97\u5206\u7684\u503c\ninputs = prepare_sequence(training_data[0][0], word_to_ix)\ntag_scores = model(inputs)\n# \u53e5\u5b50\u662f \"the dog ate the apple\", i,j \u8868\u793a\u5bf9\u4e8e\u5355\u8bcd i, \u6807\u7b7e j \u7684\u5f97\u5206.\n# \u6211\u4eec\u91c7\u7528\u5f97\u5206\u6700\u9ad8\u7684\u6807\u7b7e\u4f5c\u4e3a\u9884\u6d4b\u7684\u6807\u7b7e. \u4ece\u4e0b\u9762\u7684\u8f93\u51fa\u6211\u4eec\u53ef\u4ee5\u770b\u5230, \u9884\u6d4b\u5f97\n# \u5230\u7684\u7ed3\u679c\u662f0 1 2 0 1. \u56e0\u4e3a \u7d22\u5f15\u662f\u4ece0\u5f00\u59cb\u7684, \u56e0\u6b64\u7b2c\u4e00\u4e2a\u503c0\u8868\u793a\u7b2c\u4e00\u884c\u7684\n# \u6700\u5927\u503c, \u7b2c\u4e8c\u4e2a\u503c1\u8868\u793a\u7b2c\u4e8c\u884c\u7684\u6700\u5927\u503c, \u4ee5\u6b64\u7c7b\u63a8. \u6240\u4ee5\u6700\u540e\u7684\u7ed3\u679c\u662f DET\n# NOUN VERB DET NOUN, \u6574\u4e2a\u5e8f\u5217\u90fd\u662f\u6b63\u786e\u7684!\nprint(tag_scores)\n```\n\n tensor([[-1.1389, -1.2024, -0.9693],\n [-1.1065, -1.2200, -0.9834],\n [-1.1286, -1.2093, -0.9726],\n [-1.1190, -1.1960, -0.9916],\n [-1.0137, -1.2642, -1.0366]], grad_fn=)\n tensor([[-0.0858, -2.9355, -3.5374],\n [-5.2313, -0.0234, -4.0314],\n [-3.9098, -4.1279, -0.0368],\n [-0.0187, -4.7809, -4.5960],\n [-5.8170, -0.0183, -4.1879]], grad_fn=)\n\n\n\u7ec3\u4e60: \u4f7f\u7528\u5b57\u7b26\u7ea7\u7279\u5f81\u6765\u589e\u5f3a LSTM \u8bcd\u6027\u6807\u6ce8\u5668\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n\u5728\u4e0a\u9762\u7684\u4f8b\u5b50\u4e2d, \u6bcf\u4e2a\u8bcd\u90fd\u6709\u4e00\u4e2a\u8bcd\u5d4c\u5165, \u4f5c\u4e3a\u5e8f\u5217\u6a21\u578b\u7684\u8f93\u5165. \u63a5\u4e0b\u6765\u8ba9\u6211\u4eec\u4f7f\u7528\u6bcf\u4e2a\u7684\u5355\u8bcd\u7684\n\u5b57\u7b26\u7ea7\u522b\u7684\u8868\u8fbe\u6765\u589e\u5f3a\u8bcd\u5d4c\u5165. \u6211\u4eec\u671f\u671b\u8fd9\u4e2a\u64cd\u4f5c\u5bf9\u7ed3\u679c\u80fd\u6709\u663e\u8457\u63d0\u5347, \u56e0\u4e3a\u50cf\u8bcd\u7f00\u8fd9\u6837\u7684\u5b57\u7b26\u7ea7\n\u4fe1\u606f\u5bf9\u4e8e\u8bcd\u6027\u6709\u5f88\u5927\u7684\u5f71\u54cd. \u6bd4\u5982\u8bf4, \u50cf\u5305\u542b\u8bcd\u7f00 *-ly* \u7684\u5355\u8bcd\u57fa\u672c\u4e0a\u90fd\u662f\u88ab\u6807\u6ce8\u4e3a\u526f\u8bcd.\n\n\u5177\u4f53\u64cd\u4f5c\u5982\u4e0b. \u7528 $c_w$ \u6765\u8868\u793a\u5355\u8bcd $w$ \u7684\u5b57\u7b26\u7ea7\u8868\u8fbe, \u540c\u4e4b\u524d\u4e00\u6837, \u6211\u4eec\u4f7f\n\u7528 $x_w$ \u6765\u8868\u793a\u8bcd\u5d4c\u5165. \u5e8f\u5217\u6a21\u578b\u7684\u8f93\u5165\u5c31\u53d8\u6210\u4e86 $x_w$ \u548c $c_w$ \n\u7684\u62fc\u63a5. \u56e0\u6b64, \u5982\u679c $x_w$ \u7684\u7ef4\u5ea6\u662f5, $c_w$ \u7684\u7ef4\u5ea6\u662f3, \u90a3\u4e48\u6211\u4eec\u7684 LSTM\n\u7f51\u7edc\u7684\u8f93\u5165\u7ef4\u5ea6\u5927\u5c0f\u5c31\u662f8.\n\n\u4e3a\u4e86\u5f97\u5230\u5b57\u7b26\u7ea7\u522b\u7684\u8868\u8fbe, \u5c06\u5355\u8bcd\u7684\u6bcf\u4e2a\u5b57\u7b26\u8f93\u5165\u4e00\u4e2a LSTM \u7f51\u7edc, \u800c $c_w$ \u5219\u4e3a\u8fd9\u4e2a\nLSTM \u7f51\u7edc\u6700\u540e\u7684\u9690\u72b6\u6001. \u4e00\u4e9b\u63d0\u793a:\n\n* \u65b0\u6a21\u578b\u4e2d\u9700\u8981\u4e24\u4e2a LSTM, \u4e00\u4e2a\u8ddf\u4e4b\u524d\u4e00\u6837, \u7528\u6765\u8f93\u51fa\u8bcd\u6027\u6807\u6ce8\u7684\u5f97\u5206, \u53e6\u5916\u4e00\u4e2a\u65b0\u589e\u52a0\u7684\u7528\u6765\n \u83b7\u53d6\u6bcf\u4e2a\u5355\u8bcd\u7684\u5b57\u7b26\u7ea7\u522b\u8868\u8fbe.\n* \u4e3a\u4e86\u5728\u5b57\u7b26\u7ea7\u522b\u4e0a\u8fd0\u884c\u5e8f\u5217\u6a21\u578b, \u4f60\u9700\u8981\u7528\u5d4c\u5165\u7684\u5b57\u7b26\u6765\u4f5c\u4e3a\u5b57\u7b26 LSTM \u7684\u8f93\u5165.\n\n\n\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "8cbf60ff48eb2e85ffb215d16e54c9f3b433fe09", "size": 11748, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "20pytorch/sequence_models_tutorial.ipynb", "max_stars_repo_name": "KEVINYZY/python-tutorial", "max_stars_repo_head_hexsha": "ae43536908eb8af56c34865f52a6e8644edc4fa3", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-01-04T10:44:44.000Z", 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"lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.6406358411176238, "lm_q1q2_score": 0.4458558469406185}} {"text": " **Chapter 4: [Spectroscopy](CH4-Spectroscopy.ipynb)** \n\n
\n\n\n\n# Chemical Composition in Core-Loss Spectra\n\n[Download](https://raw.githubusercontent.com/gduscher/MSE672-Introduction-to-TEM/main/Spectroscopy/CH4_08-Chemical_Compostion.ipynb)\n \n[](\n https://colab.research.google.com/github/gduscher/MSE672-Introduction-to-TEM/blob/main/Spectroscopy/CH4_08-Chemical_Compostion.ipynb)\n\n\npart of \n\n **[MSE672: Introduction to Transmission Electron Microscopy](../_MSE672_Intro_TEM.ipynb)**\n\nby Gerd Duscher, Spring 2021\n\nMicroscopy Facilities
\nJoint Institute of Advanced Materials
\nMaterials Science & Engineering
\nThe University of Tennessee, Knoxville\n\nBackground and methods to analysis and quantification of data acquired with transmission electron microscopes.\n\n\n## Content\n\nQuantitative determination of composition in a core-loss EELS spectrum\n\nPlease cite:\n\n[M. Tian et al. *Measuring the areal density of nanomaterials by electron energy-loss spectroscopy*\nUltramicroscopy Volume 196, 2019, pages 154-160](https://doi.org/10.1016/j.ultramic.2018.10.009)\n\nas a reference of the here introduced model-based quantification method.\n\n\n## Load important packages\n\n### Check Installed Packages\n\n\n\n```python\nimport sys\nfrom pkg_resources import get_distribution, DistributionNotFound\n\ndef test_package(package_name):\n \"\"\"Test if package exists and returns version or -1\"\"\"\n try:\n version = get_distribution(package_name).version\n except (DistributionNotFound, ImportError) as err:\n version = '-1'\n return version\n\n# Colab setup ------------------\nif 'google.colab' in sys.modules:\n !pip install pyTEMlib -q\n# pyTEMlib setup ------------------\nelse:\n if test_package('sidpy') < '0.0.5':\n print('installing sidpy')\n !{sys.executable} -m pip install --upgrade pyTEMlib -q\n if test_package('pyTEMlib') < '0.2021.4.20':\n print('installing pyTEMlib')\n !{sys.executable} -m pip install --upgrade pyTEMlib -q\n# ------------------------------\nprint('done')\n```\n\n done\n\n\n### Import all relevant libraries\n\nPlease note that the EELS_tools package from pyTEMlib is essential.\n\n\n```python\nimport sys\nif 'google.colab' in sys.modules:\n %pylab --no-import-all inline\nelse: \n %pylab --no-import-all notebook\n %gui qt\n \nimport warnings\nwarnings.filterwarnings('ignore')\n\n## We need to import a few important additional function from matplotlib, \n## because we want to demonstrate a few more hidden functionalities of the EELS_tools of pytTEMlib.\nfrom matplotlib.widgets import Cursor\nfrom matplotlib.patches import Rectangle\nfrom matplotlib.widgets import SpanSelector\nfrom scipy.ndimage.filters import gaussian_filter\n\n## import the configuration files of pyTEMlib (we need access to the data folder)\nimport pyTEMlib\nimport pyTEMlib.file_tools as ft\nimport pyTEMlib.eels_tools as eels\nimport pyTEMlib.interactive_eels as ieels\n\n# For archiving reasons it is a good idea to print the version numbers out at this point\nprint('pyTEM version: ',pyTEMlib.__version__)\n```\n\n Populating the interactive namespace from numpy and matplotlib\n pyTEM version: 0.2021.04.02\n\n\n## Chemical Composition\n\n>\n> We discuss first the conventional method of EELS chemical compoisition determination\n>\n\nIn this chapter we use the area under the ionization edge to determine the chemical composition of a (small) sample volume. \nThe equation used to determine the number of atoms per unit volume $N$ (also called areal density) is:\n\\begin{equation}\nI_{edge}(\\beta, \\Delta E) = N * I_{0}(\\beta) * \\sigma_{edge}(\\beta, \\Delta E)\n\\end{equation}\n\n$I_0$ is the number of electrons hitting the sample, and so directly comparable to the beam current.\n\nThe equation can be approximated assuming that the spectrum has not been corrected for single scattering:\n\\begin{equation} \nI_{edge}(\\beta, \\Delta E) = N * I_{low-loss}(\\beta,\\Delta E) * \\sigma_{edge}(\\beta, \\Delta E)\n\\end{equation}\nwhere $\\beta$ is the collection angle and $\\sigma_{edge}$ is the **partial** cross--section (for energy window $\\Delta E$) for the core--loss excitation.\n\nThe integration interval $\\Delta E$ which defines $I_{edge}(\\beta, \\Delta E)$ and $I_{low-loss}$ is shown in figure below. \n\n\n\n*Valence--loss, core--loss edges and background of SrTiO$_3$]{\\label{fig:edge2} Ti-L$_{2,3}$ and O-K (right) core--loss edges and background of SrTiO$_3$. The valence-loss spectrum with the zero--loss $I_{zero-loss}$ and low--loss intensities $I_{low-loss} $ to be used in the quantification are displayed in the background.*\n\nIf we cannot determine the intensity of the zero-loss peak $I_{zero-loss}$ or of the low-loss area $I_{low-loss}$, we still can determine relative chemical compositions of two elements $a$ and $b$ considering that:\n\n\\begin{equation}\n\\frac{N_a}{N_b} = \\frac{I_{e_a}(\\beta, \\Delta E)}{I_0 \\sigma_{e_a}(\\beta, \\Delta E)} \\frac{I_0 \\sigma_{e_b}(\\beta, \\Delta E) } {I_{e_b}(\\beta, \\Delta E)} \\nonumber \n\\end{equation}\n\n\\begin{equation}\n\\frac{N_a}{N_b}= \\frac{I_{e_a}(\\beta, \\Delta E)\\sigma_{e_b}(\\beta, \\Delta E) } \n{I_{e_b}(\\beta, \\Delta E)\\sigma_{e_a}(\\beta, \\Delta E) } \n\\end{equation}\n\nand the value $I_0$ cancels out.\n\nThe integration windows for the two edges $\\Delta E$ should be the same, but can be chosen differently as long as we use a different cross-section $\\sigma$ as well. For that case we get:\n\n\\begin{equation} \\\n\\frac{N_a}{N_b} = \\frac{I_{e_a}(\\beta, \\Delta E_a)\\sigma_{e_b}(\\beta, \\Delta E_b) } \n{I_{e_b}(\\beta, \\Delta E_a)\\sigma_{e_a}(\\beta, \\Delta E_a) } \n\\end{equation}\n\nNote, that the use of different integration windows usually results in a larger error of the quantification.\n\nIn order to use the above equation we first have to determine the background under the edge.\nThis background is then subtracted from the spectrum.\nThen we integrate over the different parts of the spectrum to determine the integrals (sums) of $I_{edge}$, and $I_{zero-loss}$, or $I_{low-loss} $ (depending whether we did a SSD first or not).\nAfter that we have to determine the cross-section ([notebook](CH4_07-Working_with_X-Sections.ipynb)) for each edge for the parameter $\\beta$ and $\\Delta E$.\n\n### Background Fitting\n\nThe core-loss edges occur usually at energy-losses higher than 100 eV, superimposed to a monotonic decreasing background. For quantification of the chemical composition we need the area under the edge of a spectrum, and therefore, need to subtract the intensity of the background under the edge. Here we discuss several methods of how to determine the intensity of the background under the edge.\n\nThe high energy ``tail`` of the plasmon peak follows the power law $A E^{-r}$. The parameter is varies widely and is associated with the intensity of the background. $E$ is the energy loss. The exponent $r$ gives the slope and should be between 2-6. The value $r$ usually decreases with increasing specimen thickness, because of plural-scattering contributions. $r$ also decreases with increasing collection angles $\\beta$, but increases with increasing energy--loss.\n\n>\n>Consequently, we have to determine the parameters $A$ and $r$ for each ionization edge.\n>\n\nThe fitting of the power law $A E^{-r}$ (to determine $A$ and $r$) is usually done in an area just before the edge, assuming that the background follows the same power law under the edge.\nThis fit can only work if the detector dark current and gain variation is corrected prior to the analysis.\n\n\nA standard technique is to match the pre--edge background $J(E)$ to a function $F(E)$ (here the power law) whose parameter (here $A$ and $r$) minimize the quantity:\n\\begin{equation} \n\\chi^2 = \\sum \\limits_{i} \\left[ \\frac{J_i - F_i}{\\sigma_i} \\right]^2\n\\end{equation}\nwhere $i$ is the index of the channel within the fitting region and $\\sigma_i$ represents the statistical error (standard deviation) of the intensity in that channel.\n\nThe value $\\sigma_i$ is often considered constant (for example $1/3$ or $e^{-1}$). For our problem, the quantum mechanical shot noise is adequate \n\\begin{equation} \n\\sigma_i = \\ln(J_i - \\sqrt{J_i}) - \\ln(J_i) \\approx \\sqrt{J_i} \n\\end{equation}\nwhere we assume that $J_i$ is in number of electrons and not in counts (number of electrons times a conversion factor).\n\nIn the figure below, we see the result and the output of the background fit (plus subtraction).\n\n\n\n* Background fit on a spectrum of SrTiO$_3$]{\\label{fig:background} Background fit on a spectrum of SrTiO$_3$. The $A$ and $r$ parameter together with the normalized (reduced) $\\chi^2_n$ parameter is displayed in the {\\bf Results} window.*\n\n### Background Fitting: Spatial Difference\n\n\n\nWe can use also an experimentally determined background, if impurity or dopant atoms are present in confined areas only. Then, we can take two spectra: one next to this area and one at this area. The near by area will result in a spectrum without this edge and can be used as a background for quantification of the other spectrum. This method is highly accurate if the sample thickness does not change between these areas. The measurement of ppm levels of dopants and impurity atoms can achieved with this method. This Method will be more closely examined in Section Spatial-Difference\n\n### Background Subtraction Errors\n\nIn addition to possible systematic errors, any fitted background to noisy data will be subject to to a random or statistical error. The signal noise ration (SNR) can be defined as:\n\n\\begin{equation}\nSNR = I_{edge} [\\mbox{var}(I_{edge}]^{-1/2} = I_{edge}/(I_{edge}+ h \\quad I_{background})^{1/2}\n\\end{equation}\n\nwhere the dimensionless parameter $h$ represents the uncertainty of the background fit due to noise.\nIf the width of the integration window is sufficiently small (for example the same width as the fitting window) then this factor $h$ is relatively small. For equal windows we can approximate $h = 10$.\n\n\n### Cross-Section\n\n\nThe cross--section gives us the weight of the edge intensity to compare to different elements or to the total number of electrons (to compute areal density).\nSuch a cross--sections is compared to an edge intensity in the figure below.\n\n\n\n*The shape of a calculated cross-section (black) is compared to the intensity of a Si-L$_{2,3}$ and Si-$L_{1}$ edge after background subtraction (gray). The SSD--corrected spectrum (red) and the extrapolated background (light blue) are also shown. In the results window, we see the integrated edge intensity, the integrated cross-sections and the determined areal density.*\n\nThere are different methods of how to use cross-sections in the quantification of the chemical composition from spectra:\n\nEgerton gives in his book a tabulated list of generalized oscillator strength (GOS) for different elements. The values for different integration windows $\\Delta E$ can be linearly extrapolated for other integration window widths. The GOS have to be extrapolated for the chosen integration window and converted to cross--sections. The GOS are calculated with the Bethe theory in Hydrogenic approximation (see below in chapter \\ref{sec:cross-section-calculate}\n### Calculation of the Cross--Section\n\nThere are two methods in the literature to calculate the cross--section. One is the one where we assume s states in free atoms and is called Hydrogenic approximation and one which approximates the free atoms a little more detailed: the Hatree-Slater method. \n\nBoth methods are based on free atom calculations, because of the strong bonding of the core--electrons to the nucleus, band-structure (collective) effects can be neglected.\n\n\n\nThe figure below compares the cross--sections of these two approximations (with a background added) to an experimental spectrum.\n\n\n\n*The shape of a Hydrogenic (green) and Hatree--Slater (blue) cross-section (with a background added) is compared to an experimental (SSD-corrected) spectrum of Si.*\n\n### Summary\nThe power law is not really correct for any larger energy interval, which results in a change of the $r$ exponent throughout the spectrum's energy range.\n\t\nA polynomial of 2$^{nd}$ order can be used to fit the spectrum, but often leads to strong oscillations in the extrapolated high energy part. \n\nThe exact slope of the extrapolated background depends on pre-edge features and noise\n\t\n>\n>Generally the above described classic method of quantification is often non-reproducible, and results in errors often in excess of 100%.\n>\n\nIn the following we will work with a model based method that reduces the artefacts, increases reproducibility and improves the error to about 3 atom % absolute. \n\n## Load and plot a spectrum\n\nAs an example we load the spectrum **1EELS Acquire (high-loss).dm3** from the *example data* folder.\n\nPlease see [Loading an EELS Spectrum](LoadEELS.ipynb) for details on storage and plotting.\n\n\n```python\ntry:\n current_channel.file.close()\nexcept:\n pass\n# Load file\nfile_name = '../example_data/1EELS Acquire (high-loss).dm3'\nmain_dataset = ft.open_file(file_name)#os.path.join(current_directory,filename))\ncurrent_channel = main_dataset.h5_dataset.parent.parent\n\n\nif main_dataset.data_type.name != 'SPECTRUM':\n print('NOT what we want here')\nelse:\n main_dataset.metadata = eels.read_dm3_eels_info(main_dataset.original_metadata)\n main_dataset.metadata['exposure_time'] = main_dataset.metadata['single_exposure_time'] * main_dataset.metadata['number_of_frames']\n main_dataset.view_metadata()\n\nmain_dataset.plot() \n```\n\n single_exposure_time : 3.0\n exposure_time : 63.0\n number_of_frames : 21\n collection_angle : 33.0\n convergence_angle : 30.0\n acceleration_voltage : 200000.0\n microscope : Unknown\n\n\n\n \n\n\n\n\n\n\n### Which elements are present\n\nTo determine which elements are present we add a cursor to the above plot (see [Working with Cross-Sections](CH4_07-Working_with_X-Sections.ipynb) for details) and with a left (right) mouse-click, we will get the major (all) edges in the vincinity of the cursor.\n\nIn the example we note that the N-K edge of this boron nitride sample is not at 400keV. We have to adjust the energy-scale.
(THIS SHOULD NOT HAPPEN IN NORMAL SPECTRA AND IS FOR DEMONSTRATION ONLY)\n\n\n```python\nmaximal_chemical_shift = 5\nenergy_scale = main_dataset.energy_loss\ncursor = ieels.EdgesAtCursor(main_dataset.view.axis, energy_scale,main_dataset,maximal_chemical_shift)\n```\n\nLet's correct the energy scale of the example spectrum.\n\nAgain a shift of the enrrgy scale is normal but not a discripancy of the dispersion.\n\n### Probability scale of y-axis\n\nWe need to know the total amount of electrons involved in the EELS spectrum \n\nThere are three possibilities:\n- the intensity of the low loss will give us the counts per acquisition time\n- the intensity of the beam in an image\n- a direct measurement of the incident beam current\n\nHere we got the low-loss spectrum. For the example please load **1EELS Acquire (low-loss).dm3** from the *example data* folder.\n\n\n```python\nll_dataset = ft.open_file('../example_data/1EELS Acquire (low-loss).dm3')\nll_dataset.h5_dataset.file.close()\nll_dataset.metadata = eels.read_dm3_eels_info(ll_dataset.original_metadata)\nll_dataset.metadata['exposure_time'] = ll_dataset.metadata['single_exposure_time'] * ll_dataset.metadata['number_of_frames']\nll_dataset.view_metadata()\nll_dataset.plot()\n```\n\n single_exposure_time : 0.01\n exposure_time : 0.21\n number_of_frames : 21\n collection_angle : 33.0\n convergence_angle : 30.0\n acceleration_voltage : 200000.0\n microscope : Unknown\n\n\n\n \n\n\n\n\n\n\n### Intensity to Probability Calibration\n\n We need to calibrate the number of counts with the integration time of the spectrum.\n\n\n```python\nprint(f\"{ll_dataset.sum():.0f} counts in {ll_dataset.metadata['exposure_time']:.2f}sec\")\nI0 = ll_dataset.sum()/ll_dataset.metadata['exposure_time']\n\nI0 = main_dataset.sum()/ll_dataset.metadata['exposure_time']*main_dataset.metadata['exposure_time']\nprint(f\"incident beam current of core--loss spectrum is {I0:.0f} counts\")\n\nmain_dataset.metadata['intentsity_scale_ppm'] = 1e6/I0\nmain_dataset.metadata['incident_beam_current_counts'] = I0\n\nspectrum = main_dataset*main_dataset.metadata['intentsity_scale_ppm']\nspectrum.quantity = 'probability'\nspectrum.units = 'ppm'\nspectrum.plot()\n```\n\n 115700152 counts in 0.21sec\n incident beam current of core--loss spectrum is 11721591600 counts\n\n\n\n \n\n\n\n\n\n\n## Components of a core loss spectrum\n\n-background\n\n-absorption edges\n\n### Plotting of cross sections and spectrum\nplease note that spectrum and cross sections are not on the same scale\n\n\n```python\n\nB_Xsection = eels.xsec_xrpa(energy_scale, 200, 5, 10. )/1e10 \nN_Xsection = eels.xsec_xrpa(energy_scale, 200, 7, 10. ,shift=-6)/1e10 # xsec is in barns = 10^28 m2 = 10^10 nm2\n\nfig, ax1 = plt.subplots()\n\nax1.plot(energy_scale, Ti_Xsection, label='Ti X-section' )\nax1.plot(energy_scale, O_Xsection, label='O X-section' )\nax1.set_xlabel('energy_loss [eV]')\nax1.set_ylabel('probability [atoms/nm$^{2}$]')\n\nax2 = ax1.twinx()\nax2.plot(energy_scale, spectrum, c='r', label='spectrum')\nax2.tick_params('y', colors='r')\nax2.set_ylabel('probability [ppm]')\n#plt.xlim(100,500)\nplt.legend();\nfig.tight_layout();\n\n\n\n```\n\n\n \n\n\n\n\n\n\n### Background\nThe other ingredient in a core loss spectrum is the background\n\nThe backgrund consists of\n- ionization edges to the left of the beginning of the spectrum (offset)\n- tail of the plasmon peak (generally a power_law with $\\approx A* E^{-3}$)\n\nHere we approximate the background in an energy window before the first ionization edge in the spectrum as a power law with exponent $r\\approx 3$\n\n\n```python\nfrom scipy.optimize import leastsq ## leastsqure fitting routine fo scipy\n\n\n# Determine energy window in pixels\nbgdStart = 120\nbgdWidth = 40\noffset = energy_scale[0]\ndispersion = energy_scale[1]-energy_scale[0]\nstartx = int((bgdStart-offset)/dispersion)\nendx = startx + int(bgdWidth/dispersion) \n\nx = np.array(energy_scale[startx:endx])\ny = np.array(spectrum[startx:endx])\n\n# Initial values of parameters\np0 = np.array([1.0E+20,3])\n\n## background fitting \ndef bgdfit(p, y, x):\n err = y - (p[0]* np.power(x,(-p[1])))\n return err\np, lsq = leastsq(bgdfit, p0, args=(y, x), maxfev=2000)\nprint(f'Power-law background with amplitude A: {p[0]:.1f} and exponent -r: {p[1]:.2f}')\n\n#Calculate background over the whole energy scale\nbackground = p[0]* np.power(energy_scale,(-p[1]))\n\nplt.figure()\n\nplt.xlabel('energy_loss [eV]')\nplt.ylabel('probability [ppm]')\n\nplt.plot(energy_scale, spectrum, label='spectrum')\nplt.plot(energy_scale, background, label='background')\nplt.plot(energy_scale, spectrum-background, label='spectrum-background')\nplt.legend();\nplt.axhline(0, color='gray')\n```\n\n Power-law background with amplitude A: 46400388.8 and exponent -r: 3.27\n\n\n\n \n\n\n\n\n\n\n\n\n\n \n\n\n\n## Fitting a Spectrum\n\nWe are revisiting the above the fundamental equation of the chemical composition:\n\nWe already calibrated the cross section in per $nm^2$ and so if we start again from the fundamental equation:\n\n\\begin{equation}\nI_{edge}(\\beta, \\Delta E) = N I_{0}(\\beta) \\sigma_{edge}(\\beta, \\Delta E)\n\\end{equation}\n\nand as above we calibrate the intensity of the spectrum by $I_{spectrum}/I_0$ then we get::\n\n\\begin{equation}\n\\frac{I_{edge}(\\beta, \\Delta E)}{I_0} = I^{norm}_{edge} = N \\sigma_{edge}(\\beta, \\Delta E)\n\\end{equation}\n\nand if we fit the calibrated intensity with the cross section then we can replace$I^{norm}_{edge}$ by a fitting value $q_{edge}$ multiplied by cross section $\\sigma$:\n\n\n$$\nN = \\frac{I_{edge}(\\beta, \\Delta E)/I_0}{\\sigma_{edge}(\\beta, \\Delta E)} = \\frac{I^{norm}_{edge}}{\\sigma_{edge}(\\beta, \\Delta E)} = \\frac{q_{edge} * \\sigma_{edge}(\\beta, \\Delta E)}{\\sigma_{edge}(\\beta, \\Delta E)} = q_{edge}\n$$ \n\nand N is in atoms per nm$^2$.\n\nSo a fit to a callibrated spectrum as above, will get us a ``fitting parameter`` which is an ``areal density`` (which is a legitimate thermodynamic quantity).\n\nAnd for the relative composition we get:\n$$\n\\frac{N_a}{N_b}= \\frac{q_a}{q_b}\n$$\n\nIn the following we will do this kind of a fit by:\n- calibrate the intensity in the spectrum (in ppm)\n- using cross section in units of nm$^2$\n\n\n>\n> Please note that for the relative composition , the $I_0$ will fall out and so a fit to a spectrum without calibrated intensity will still give the relative intensity accurately.\n>\n\n### Preparing the fitting mask\n\nOur theoretical cross sections do not include any solid state effects (band structure) and so the fine structure at the onset of the spectra must be omitted in a quantification.\n\nThese parts of the spectrum will be simply set to zero. We plot the masked spectrum that will be evaluated.\n\n\n```python\nenergy_scale = main_dataset.energy_loss*1.03-3\n\ndispersion = (energy_scale[1] - energy_scale[0])\noffset = energy_scale[0]\nstartx = int((bgdStart-offset)/dispersion)\n\nmask = np.ones(len(energy_scale))\nmask[0 : int(startx)] = 0.0;\n\nedges = {}\nedges['1'] = {}\nedges['1']['Z']=5\nedges['1']['symmetry']= 'K1'\nedges['2'] = {}\nedges['2']['Z']=7\nedges['2']['symmetry']= 'K1'\n\nfor key in edges:\n print((eels.get_x_sections(edges[key]['Z']))[edges[key]['symmetry']])\n edges[key]['onset'] = (eels.get_x_sections(edges[key]['Z']))[edges[key]['symmetry']]['onset']\n if key == '2':\n edges[key]['onset'] = 390\n print('onset')\n edges[key]['onset_pixel'] = int((edges[key]['onset'] -offset)/dispersion)\n edges[key]['start_exclude'] = int((edges[key]['onset']-5 - offset)/dispersion)\n edges[key]['end_exclude'] = int((edges[key]['onset']+50 - offset)/dispersion)\n print(key)\n if key == '2':\n edges[key]['onset'] = 400\n print('onset')\n mask[edges[key]['start_exclude']:edges[key]['end_exclude']] = 0.0\n\n\n\nplt.figure()\nplt.plot(energy_scale, spectrum, label='spectrum')\nplt.plot(energy_scale, spectrum*mask, label='spectrum')\nplt.xlabel('energy-loss [eV]')\nplt.ylabel('probability [ppm]'); \n```\n\n### The Fit\n\nThe function **model** just sums the weighted cross-sections and the background.\n\nThe background consists of the power-lawbackground before plus a polynomial component allowing for *a variation of the exponent $r$ of the power-law*.\n\nThe least square fit is weighted by the noise according to Poison statistic $\\sqrt{I(\\Delta E)}$.\n\n>\n>Please note that the cross sections are for single atoms only and do not cover any solid state effects vsible as strong peaks in the first 50 eV or so past the onset.\n>\n> We exclude those parts from the fits.\n\n\n```python\nplt.figure()\nplt.plot(energy_scale, spectrum, label='spectrum')\nplt.plot(energy_scale, spectrum*mask, label='spectrum')\nplt.xlabel('energy-loss [eV]')\nplt.ylabel('probability [ppm]'); \n\nregions = ieels.RegionSelector(plt.gca())\n\nfor key in edges:\n print(key)\n regions.set_regions(str(key),edges[key]['onset']-5, 50.)\n \nregions.set_regions('fit region',bgdStart, energy_scale[-1]-bgdStart)\n```\n\n\n \n\n\n\n\n\n\n 1\n 2\n\n\n\n```python\nregion_tags = regions.get_regions()\nstartx = int((region_tags['fit_area']['start_x']-offset)/dispersion)\n\nprint(region_tags)\n```\n\n {'1': {'start_x': 183.0, 'width_x': 50.0}, '2': {'start_x': 395, 'width_x': 50.0}, 'fit_area': {'start_x': 120, 'width_x': 507.10249999999996}}\n\n\n\n```python\nedges\n```\n\n\n\n\n {'1': {'Z': 5,\n 'symmetry': 'K1',\n 'onset': 188.0,\n 'onset_pixel': 341,\n 'start_exclude': 322,\n 'end_exclude': 535},\n '2': {'Z': 7,\n 'symmetry': 'K1',\n 'onset': 400,\n 'onset_pixel': 1126,\n 'start_exclude': 1106,\n 'end_exclude': 1320}}\n\n\n\n\n```python\nregion_tags = regions.get_regions()\n\nmask = np.ones(main_dataset.shape)\n\n#startx = np.searchsorted(tags['energy_scale'],region_tags['fit_area']['start_x'])\n \nmask[0:startx] = 0.0\n\nfor key in region_tags:\n end = region_tags[key]['start_x']+region_tags[key]['width_x']\n startx = np.searchsorted(energy_scale,region_tags[key]['start_x'])\n endx = np.searchsorted(energy_scale,end)\n if key == 'fit_area':\n mask[0:startx] = 0.0\n mask[endx:-1] = 0.0\n else:\n mask[startx:endx] = 0.0\n\n\n\npin = np.array([1.0,1.0,.0,0.0,0.0,0.0, 1.0,1.0,0.001,5,3])\nx = energy_scale\n\nblurred = gaussian_filter(spectrum, sigma=5)\n\ny = blurred*1e-6 ## now in probability\ny[np.where(y<1e-8)]=1e-8\n\n\nB_Xsection = eels.xsec_xrpa(energy_scale, 200, 5, 10. )/1e10 \nN_Xsection = eels.xsec_xrpa(energy_scale, 200, 7, 10. )/1e10 # xsec is in barns = 10^-28 m2 = 10^-10 nm2\n\nxsec = np.array([B_Xsection, N_Xsection])\nnumberOfEdges = 2\n\ndef residuals(p, x, y ):\n err = (y-model(x,p))*mask/np.sqrt(np.abs(y))\n return err \n\ndef model(x, p): \n y = (p[9]* np.power(x,(-p[10]))) +p[7]*x+p[8]*x*x\n for i in range(numberOfEdges):\n y = y + p[i] * xsec[i,:]\n return y\n\np, cov = leastsq(residuals, pin, args = (x,y) )\n \nprint(f\"B/N ratio is {p[0]/p[1]:.3f}\")\n\n#the B atom areal density of a single layer of h-BN (18.2 nm\u22122) \nprint(f\" B areal density is {p[0]:.0f} atoms per square nm, which equates {abs(p[0])/18.2:.1f} atomic layers\")\nprint(f\" N areal density is {p[1]:.0f} atoms per square nm, which equates {abs(p[1])/18.2:.1f} atomic layers\")\n\n\n```\n\n B/N ratio is 1.071\n B areal density is 420 atoms per square nm, which equates 23.1 atomic layers\n N areal density is 393 atoms per square nm, which equates 21.6 atomic layers\n\n\nThis result let's us pause a little. How big of and error do we have?\n\n>\n>Why is the number of B atoms not equal to the number of N atoms? \n>\n\nNaively, we would assume an error of 7 atom%\n\nHowever, the number of single atomic layers gives us a hint.\n\nThere are 23 layers of B and 24 layers of N. \n\nThis result suggests that BN is terminated on the B and N face with a layer of B and the error is 2.5 atom%\n\n\n\n```python\nprint(f'The expected ratio between 23 layers of B and 22 layers of N is {23/22:.3f}')\n```\n\n The expected ratio between 23 layers of B and 22 layers of N is 1.045\n\n\n### Plotting of the fit\n\n\n\n```python\nmodel_spectrum = model(x, p)*1e6 # in ppm\nmodel_background = ((p[9]* np.power(x,-p[10])) +p[7]*x+p[8]*x*x)*1e6 # in ppm\n\nplt.figure()\n#plt.plot(energy_scale, spectrum, label='spectrum')\nplt.plot(energy_scale, blurred, label='blurred spectrum')\nplt.plot(x,model_spectrum, label='model')\nplt.plot(x,spectrum-model_spectrum, label='difference')\nplt.plot(x,(spectrum-model_spectrum), label='difference')\nplt.plot(x,model_background, label='background')\nplt.plot([x[0],x[-1]],[0,0],c='black')\n\nplt.xlabel('energy-loss [eV]')\nplt.ylabel('probability [ppm]')\nplt.legend();\n```\n\n\n \n\n\n\n\n\n\n## Summary\n\nWe use a cross section in unsits of nm$^2$ and a calibrated spectrum to fit a cross section to each edge.\n\nThe fitting parameter is then the areal density of the element.\n\nWe only fit the part of the spectrum we know which is the single atom part of the edge, and avoid to fit any solid state effects at the onset of the edge.\n\nThe interpreation of solid state effects at the onset are discussed in the [energy-loss near-edge structure (ELNES)](CH4_10-ELNES.ipynb) notebook.\n\n\n## Navigation\n- **Up Chapter 4: [Imaging](CH4_00-Spectroscopy.ipynb)** \n- **Back: [Introduction to Core-Loss](CH4_07-Introduction_Core_Loss.ipynb)** \n- **Next: [Analysis of Core-Loss](CH4_09-Analysis_Core_Loss.ipynb)** \n- **List of Content: [Front](../_MSE672_Intro_TEM.ipynb)** \n\n\n```python\n\n```\n", "meta": {"hexsha": "85388e7884b602fe63af33e777ad2ea6908095b2", "size": 777315, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Spectroscopy/CH4_08-Chemical_Composition.ipynb", "max_stars_repo_name": "ahoust17/MSE672-Introduction-to-TEM", "max_stars_repo_head_hexsha": "6b412a3ad07ee273428a95a7158aa09058d7e2ee", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2021-01-22T18:09:53.000Z", "max_stars_repo_stars_event_max_datetime": "2021-07-26T20:17:34.000Z", "max_issues_repo_path": "Spectroscopy/CH4_08-Chemical_Composition.ipynb", "max_issues_repo_name": "ahoust17/MSE672-Introduction-to-TEM", "max_issues_repo_head_hexsha": "6b412a3ad07ee273428a95a7158aa09058d7e2ee", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Spectroscopy/CH4_08-Chemical_Composition.ipynb", "max_forks_repo_name": "ahoust17/MSE672-Introduction-to-TEM", "max_forks_repo_head_hexsha": "6b412a3ad07ee273428a95a7158aa09058d7e2ee", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2021-01-26T16:10:11.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-03T14:53:16.000Z", "avg_line_length": 96.789316399, "max_line_length": 106831, "alphanum_fraction": 0.740275178, "converted": true, "num_tokens": 7523, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358411176238, "lm_q2_score": 0.6959583250334526, "lm_q1q2_score": 0.4458558469406185}} {"text": "\n# PHY321: Two-body problems and Gravitational Forces\n\n \n**[Morten Hjorth-Jensen](http://mhjgit.github.io/info/doc/web/)**, Department of Physics and Astronomy and Facility for Rare Ion Beams (FRIB), Michigan State University, USA and Department of Physics, University of Oslo, Norway\n\nDate: **Mar 19, 2021**\n\nCopyright 1999-2021, [Morten Hjorth-Jensen](http://mhjgit.github.io/info/doc/web/). Released under CC Attribution-NonCommercial 4.0 license\n\n\n\n\n## Aims and Overarching Motivation\n\n\n### Monday\n\nDefinition of the two-body problem, rewriting the equations in relative and center-of-mass coordinates\n\n**Reading suggestion**: Taylor sections 8.2-8.3\n\n\n### Wednesday\n\nPreparing the ground for the gravitional force and its solution in two dimensions\n\n**Reading suggestion**: Taylor chapter 8.4\n\n### Friday\n\nHarmonic Oscillator example. Begin Kepler's laws and computing with classes.\n\n**Reading suggestion**: Taylor section 8.5-8.6 \n\n\n\n# Two-body Problems\n\n**Note: more text to be added**.\n\nThe gravitational potential energy and forces involving two masses $a$ and $b$ are\n\n$$\n\\begin{eqnarray}\nV_{ab}&=&-\\frac{Gm_am_b}{|\\boldsymbol{r}_a-\\boldsymbol{r}_b|},\\\\\n\\nonumber\nF_{ba}&=&-\\frac{Gm_am_b}{|\\boldsymbol{r}_a-\\boldsymbol{r}_b|^2}\\hat{r}_{ab},\\\\\n\\nonumber\n\\hat{r}_{ab}&=&\\frac{\\boldsymbol{r}_b-\\boldsymbol{r}_a}{|\\boldsymbol{r}_a-\\boldsymbol{r}_b|}.\n\\end{eqnarray}\n$$\n\nHere $G=6.67\\times 10^{-11}$ Nm$^2$/kg$^2$, and $F_{ba}$ is the force\non $b$ due to $a$. By inspection, one can see that the force on $b$\ndue to $a$ and the force on $a$ due to $b$ are equal and opposite. The\nnet potential energy for a large number of masses would be\n\n\n
\n\n$$\n\\begin{equation}\nV=\\sum_{a\n
\n\n$$\n\\begin{eqnarray}\n\\label{eq:radialeqofmotion} \\tag{2}\n\\frac{d}{dt}r^2&=&\\frac{d}{dt}(x^2+y^2)=2x\\dot{x}+2y\\dot{y}=2r\\dot{r},\\\\\n\\nonumber\n\\dot{r}&=&\\frac{x}{r}\\dot{x}+\\frac{y}{r}\\dot{y},\\\\\n\\nonumber\n\\ddot{r}&=&\\frac{x}{r}\\ddot{x}+\\frac{y}{r}\\ddot{y}\n+\\frac{\\dot{x}^2+\\dot{y}^2}{r}\n-\\frac{\\dot{r}^2}{r}.\n\\end{eqnarray}\n$$\n\nRecognizing that the numerator of the third term is the velocity squared, and that it can be written in polar coordinates,\n\n\n
\n\n$$\n\\begin{equation}\nv^2=\\dot{x}^2+\\dot{y}^2=\\dot{r}^2+r^2\\dot{\\theta}^2,\n\\label{_auto2} \\tag{3}\n\\end{equation}\n$$\n\none can write $\\ddot{r}$ as\n\n\n
\n\n$$\n\\begin{eqnarray}\n\\label{eq:radialeqofmotion2} \\tag{4}\n\\ddot{r}&=&\\frac{F_x\\cos\\theta+F_y\\sin\\theta}{m}+\\frac{\\dot{r}^2+r^2\\dot{\\theta}^2}{r}-\\frac{\\dot{r}^2}{r}\\\\\n\\nonumber\n&=&\\frac{F}{m}+\\frac{r^2\\dot{\\theta}^2}{r}\\\\\n\\nonumber\nm\\ddot{r}&=&F+\\frac{L^2}{mr^3}.\n\\end{eqnarray}\n$$\n\nThis derivation used the fact that the force was radial,\n$F=F_r=F_x\\cos\\theta+F_y\\sin\\theta$, and that angular momentum is\n$L=mrv_{\\theta}=mr^2\\dot{\\theta}$. The term $L^2/mr^3=mv^2/r$ behaves\nlike an additional force. Sometimes this is referred to as a\ncentrifugal force, but it is not a force. Instead, it is the\nconsequence of considering the motion in a rotating (and therefore\naccelerating) frame.\n\nNow, we switch to the particular case of an attractive inverse square\nforce, $F=-\\alpha/r^2$, and show that the trajectory, $r(\\theta)$, is\nan ellipse. To do this we transform derivatives w.r.t. time to\nderivatives w.r.t. $\\theta$ using the chain rule combined with angular\nmomentum conservation, $\\dot{\\theta}=L/mr^2$.\n\n\n
\n\n$$\n\\begin{eqnarray}\n\\label{eq:rtotheta} \\tag{5}\n\\dot{r}&=&\\frac{dr}{d\\theta}\\dot{\\theta}=\\frac{dr}{d\\theta}\\frac{L}{mr^2},\\\\\n\\nonumber\n\\ddot{r}&=&\\frac{d^2r}{d\\theta^2}\\dot{\\theta}^2\n+\\frac{dr}{d\\theta}\\left(\\frac{d}{dr}\\frac{L}{mr^2}\\right)\\dot{r}\\\\\n\\nonumber\n&=&\\frac{d^2r}{d\\theta^2}\\left(\\frac{L}{mr^2}\\right)^2\n-2\\frac{dr}{d\\theta}\\frac{L}{mr^3}\\dot{r}\\\\\n\\nonumber\n&=&\\frac{d^2r}{d\\theta^2}\\left(\\frac{L}{mr^2}\\right)^2\n-\\frac{2}{r}\\left(\\frac{dr}{d\\theta}\\right)^2\\left(\\frac{L}{mr^2}\\right)^2\n\\end{eqnarray}\n$$\n\nEquating the two expressions for $\\ddot{r}$ in Eq.s ([4](#eq:radialeqofmotion2)) and ([5](#eq:rtotheta)) eliminates all the derivatives w.r.t. time, and provides a differential equation with only derivatives w.r.t. $\\theta$,\n\n\n
\n\n$$\n\\begin{equation}\n\\label{eq:rdotdot} \\tag{6}\n\\frac{d^2r}{d\\theta^2}\\left(\\frac{L}{mr^2}\\right)^2\n-\\frac{2}{r}\\left(\\frac{dr}{d\\theta}\\right)^2\\left(\\frac{L}{mr^2}\\right)^2\n=\\frac{F}{m}+\\frac{L^2}{m^2r^3},\n\\end{equation}\n$$\n\nthat when solved yields the trajectory, i.e. $r(\\theta)$. Up to this\npoint the expressions work for any radial force, not just forces that\nfall as $1/r^2$.\n\nThe trick to simplifying this differential equation for the inverse\nsquare problems is to make a substitution, $u\\equiv 1/r$, and rewrite\nthe differential equation for $u(\\theta)$.\n\n$$\n\\begin{eqnarray}\nr&=&1/u,\\\\\n\\nonumber\n\\frac{dr}{d\\theta}&=&-\\frac{1}{u^2}\\frac{du}{d\\theta},\\\\\n\\nonumber\n\\frac{d^2r}{d\\theta^2}&=&\\frac{2}{u^3}\\left(\\frac{du}{d\\theta}\\right)^2-\\frac{1}{u^2}\\frac{d^2u}{d\\theta^2}.\n\\end{eqnarray}\n$$\n\nPlugging these expressions into Eq. ([6](#eq:rdotdot)) gives an\nexpression in terms of $u$, $du/d\\theta$, and $d^2u/d\\theta^2$. After\nsome tedious algebra,\n\n\n
\n\n$$\n\\begin{equation}\n\\frac{d^2u}{d\\theta^2}=-u-\\frac{F m}{L^2u^2}.\n\\label{_auto3} \\tag{7}\n\\end{equation}\n$$\n\nFor the attractive inverse square law force, $F=-\\alpha u^2$,\n\n\n
\n\n$$\n\\begin{equation}\n\\frac{d^2u}{d\\theta^2}=-u+\\frac{m\\alpha}{L^2}.\n\\label{_auto4} \\tag{8}\n\\end{equation}\n$$\n\nThe solution has two arbitrary constants, $A$ and $\\theta_0$,\n\n\n
\n\n$$\n\\begin{eqnarray}\n\\label{eq:Ctrajectory} \\tag{9}\nu&=&\\frac{m\\alpha}{L^2}+A\\cos(\\theta-\\theta_0),\\\\\n\\nonumber\nr&=&\\frac{1}{(m\\alpha/L^2)+A\\cos(\\theta-\\theta_0)}.\n\\end{eqnarray}\n$$\n\nThe radius will be at a minimum when $\\theta=\\theta_0$ and at a\nmaximum when $\\theta=\\theta_0+\\pi$. The constant $A$ is related to the\neccentricity of the orbit. When $A=0$ the radius is a constant\n$r=L^2/(m\\alpha)$, and the motion is circular. If one solved the\nexpression $mv^2/r=-\\alpha/r^2$ for a circular orbit, using the\nsubstitution $v=L/(mr)$, one would reproduce the expression\n$r=L^2/(m\\alpha)$.\n\nThe form describing the elliptical trajectory in\nEq. ([9](#eq:Ctrajectory)) can be identified as an ellipse with one\nfocus being the center of the ellipse by considering the definition of\nan ellipse as being the points such that the sum of the two distances\nbetween the two foci are a constant. Making that distance $2D$, the\ndistance between the two foci as $2a$, and putting one focus at the\norigin,\n\n$$\n\\begin{eqnarray}\n2D&=&r+\\sqrt{(r\\cos\\theta-2a)^2+r^2\\sin^2\\theta},\\\\\n\\nonumber\n4D^2+r^2-4Dr&=&r^2+4a^2-4ar\\cos\\theta,\\\\\n\\nonumber\nr&=&\\frac{D^2-a^2}{D+a\\cos\\theta}=\\frac{1}{D/(D^2-a^2)-a\\cos\\theta/(D^2-a^2)}.\n\\end{eqnarray}\n$$\n\nBy inspection, this is the same form as Eq. ([9](#eq:Ctrajectory)) with $D/(D^2-a^2)=m\\alpha/L^2$ and $a/(D^2-a^2)=A$.\n\n\nLet us remind ourselves about what an ellipse is before we proceed.\n\n\n```python\n%matplotlib inline\n\nimport numpy as np\nfrom matplotlib import pyplot as plt\nfrom math import pi\n\nu=1. #x-position of the center\nv=0.5 #y-position of the center\na=2. #radius on the x-axis\nb=1.5 #radius on the y-axis\n\nt = np.linspace(0, 2*pi, 100)\nplt.plot( u+a*np.cos(t) , v+b*np.sin(t) )\nplt.grid(color='lightgray',linestyle='--')\nplt.show()\n```\n\n## Effective or Centrifugal Potential\n\nThe total energy of a particle is\n\n$$\n\\begin{eqnarray}\nE&=&V(r)+\\frac{1}{2}mv_\\theta^2+\\frac{1}{2}m\\dot{r}^2\\\\\n\\nonumber\n&=&V(r)+\\frac{1}{2}mr^2\\dot{\\theta}^2+\\frac{1}{2}m\\dot{r}^2\\\\\n\\nonumber\n&=&V(r)+\\frac{L^2}{2mr^2}+\\frac{1}{2}m\\dot{r}^2.\n\\end{eqnarray}\n$$\n\nThe second term then contributes to the energy like an additional\nrepulsive potential. The term is sometimes referred to as the\n\"centrifugal\" potential, even though it is actually the kinetic energy\nof the angular motion. Combined with $V(r)$, it is sometimes referred\nto as the \"effective\" potential,\n\n$$\n\\begin{eqnarray}\nV_{\\rm eff}(r)&=&V(r)+\\frac{L^2}{2mr^2}.\n\\end{eqnarray}\n$$\n\nNote that if one treats the effective potential like a real potential, one would expect to be able to generate an effective force,\n\n$$\n\\begin{eqnarray}\nF_{\\rm eff}&=&-\\frac{d}{dr}V(r) -\\frac{d}{dr}\\frac{L^2}{2mr^2}\\\\\n\\nonumber\n&=&F(r)+\\frac{L^2}{mr^3}=F(r)+m\\frac{v_\\perp^2}{r},\n\\end{eqnarray}\n$$\n\nwhich is indeed matches the form for $m\\ddot{r}$ in Eq. ([4](#eq:radialeqofmotion2)), which included the **centrifugal** force.\n\nThe following code plots this effective potential for a simple choice of parameters, with a standard gravitational potential $-\\alpha/r$. Here we have chosen $L=m=\\alpha=1$.\n\n\n```python\n# Common imports\nimport numpy as np\nfrom math import *\nimport matplotlib.pyplot as plt\n\nDeltax = 0.01\n#set up arrays\nxinitial = 0.3\nxfinal = 5.0\nalpha = 1.0 # spring constant\nm = 1.0 # mass, you can change these\nAngMom = 1.0 # The angular momentum\nn = ceil((xfinal-xinitial)/Deltax)\nx = np.zeros(n)\nfor i in range(n):\n x[i] = xinitial+i*Deltax\nV = np.zeros(n)\nV = -alpha/x+0.5*AngMom*AngMom/(m*x*x)\n# Plot potential\nfig, ax = plt.subplots()\nax.set_xlabel('r[m]')\nax.set_ylabel('V[J]')\nax.plot(x, V)\nfig.tight_layout()\nplt.show()\n```\n\n### Gravitational force example\n\nUsing the above parameters, we can now study the evolution of the system using for example the velocity Verlet method.\nThis is done in the code here for an initial radius equal to the minimum of the potential well. We seen then that the radius is always the same and corresponds to a circle (the radius is always constant).\n\n\n```python\n# Common imports\nimport numpy as np\nimport pandas as pd\nfrom math import *\nimport matplotlib.pyplot as plt\nimport os\n\n# Where to save the figures and data files\nPROJECT_ROOT_DIR = \"Results\"\nFIGURE_ID = \"Results/FigureFiles\"\nDATA_ID = \"DataFiles/\"\n\nif not os.path.exists(PROJECT_ROOT_DIR):\n os.mkdir(PROJECT_ROOT_DIR)\n\nif not os.path.exists(FIGURE_ID):\n os.makedirs(FIGURE_ID)\n\nif not os.path.exists(DATA_ID):\n os.makedirs(DATA_ID)\n\ndef image_path(fig_id):\n return os.path.join(FIGURE_ID, fig_id)\n\ndef data_path(dat_id):\n return os.path.join(DATA_ID, dat_id)\n\ndef save_fig(fig_id):\n plt.savefig(image_path(fig_id) + \".png\", format='png')\n\n\n# Simple Gravitational Force -alpha/r\n \nDeltaT = 0.01\n#set up arrays \ntfinal = 100.0\nn = ceil(tfinal/DeltaT)\n# set up arrays for t, v and r\nt = np.zeros(n)\nv = np.zeros(n)\nr = np.zeros(n)\n# Constants of the model, setting all variables to one for simplicity\nalpha = 1.0\nAngMom = 1.0 # The angular momentum\nm = 1.0 # scale mass to one\nc1 = AngMom*AngMom/(m*m)\nc2 = AngMom*AngMom/m\nrmin = (AngMom*AngMom/m/alpha)\n# Initial conditions\nr0 = rmin\nv0 = 0.0\nr[0] = r0\nv[0] = v0\n# Start integrating using the Velocity-Verlet method\nfor i in range(n-1):\n # Set up acceleration\n a = -alpha/(r[i]**2)+c1/(r[i]**3)\n # update velocity, time and position using the Velocity-Verlet method\n r[i+1] = r[i] + DeltaT*v[i]+0.5*(DeltaT**2)*a\n anew = -alpha/(r[i+1]**2)+c1/(r[i+1]**3)\n v[i+1] = v[i] + 0.5*DeltaT*(a+anew)\n t[i+1] = t[i] + DeltaT\n # Plot position as function of time\nfig, ax = plt.subplots(2,1)\nax[0].set_xlabel('time')\nax[0].set_ylabel('radius')\nax[0].plot(t,r)\nax[1].set_xlabel('time')\nax[1].set_ylabel('Velocity')\nax[1].plot(t,v)\nsave_fig(\"RadialGVV\")\nplt.show()\n```\n\nChanging the value of the initial position to a value where the energy is positive, leads to an increasing radius with time, a so-called unbound orbit. Choosing on the other hand an initial radius that corresponds to a negative energy and different from the minimum value leads to a radius that oscillates back and forth between two values. \n\n### Harmonic Oscillator in two dimensions\n\nConsider a particle of mass $m$ in a 2-dimensional harmonic oscillator with potential\n\n$$\nV=\\frac{1}{2}kr^2=\\frac{1}{2}k(x^2+y^2).\n$$\n\nIf the orbit has angular momentum $L$, we can find the radius and angular velocity of the circular orbit as well as the b) the angular frequency of small radial perturbations.\n\nWe consider the effective potential. The radius of a circular orbit is at the minimum of the potential (where the effective force is zero).\nThe potential is plotted here with the parameters $k=m=0.1$ and $L=1.0$.\n\n\n```python\n# Common imports\nimport numpy as np\nfrom math import *\nimport matplotlib.pyplot as plt\n\nDeltax = 0.01\n#set up arrays\nxinitial = 0.5\nxfinal = 3.0\nk = 1.0 # spring constant\nm = 1.0 # mass, you can change these\nAngMom = 1.0 # The angular momentum\nn = ceil((xfinal-xinitial)/Deltax)\nx = np.zeros(n)\nfor i in range(n):\n x[i] = xinitial+i*Deltax\nV = np.zeros(n)\nV = 0.5*k*x*x+0.5*AngMom*AngMom/(m*x*x)\n# Plot potential\nfig, ax = plt.subplots()\nax.set_xlabel('r[m]')\nax.set_ylabel('V[J]')\nax.plot(x, V)\nfig.tight_layout()\nplt.show()\n```\n\n$$\n\\begin{eqnarray*}\nV_{\\rm eff}&=&\\frac{1}{2}kr^2+\\frac{L^2}{2mr^2}\n\\end{eqnarray*}\n$$\n\nThe effective potential looks like that of a harmonic oscillator for\nlarge $r$, but for small $r$, the centrifugal potential repels the\nparticle from the origin. The combination of the two potentials has a\nminimum for at some radius $r_{\\rm min}$.\n\n$$\n\\begin{eqnarray*}\n0&=&kr_{\\rm min}-\\frac{L^2}{mr_{\\rm min}^3},\\\\\nr_{\\rm min}&=&\\left(\\frac{L^2}{mk}\\right)^{1/4},\\\\\n\\dot{\\theta}&=&\\frac{L}{mr_{\\rm min}^2}=\\sqrt{k/m}.\n\\end{eqnarray*}\n$$\n\nFor particles at $r_{\\rm min}$ with $\\dot{r}=0$, the particle does not\naccelerate and $r$ stays constant, i.e. a circular orbit. The radius\nof the circular orbit can be adjusted by changing the angular momentum\n$L$.\n\nFor the above parameters this minimum is at $r_{\\rm min}=1$.\n\n Now consider small vibrations about $r_{\\rm min}$. The effective spring constant is the curvature of the effective potential.\n\n$$\n\\begin{eqnarray*}\nk_{\\rm eff}&=&\\left.\\frac{d^2}{dr^2}V_{\\rm eff}(r)\\right|_{r=r_{\\rm min}}=k+\\frac{3L^2}{mr_{\\rm min}^4}\\\\\n&=&4k,\\\\\n\\omega&=&\\sqrt{k_{\\rm eff}/m}=2\\sqrt{k/m}=2\\dot{\\theta}.\n\\end{eqnarray*}\n$$\n\nBecause the radius oscillates with twice the angular frequency,\nthe orbit has two places where $r$ reaches a minimum in one\ncycle. This differs from the inverse-square force where there is one\nminimum in an orbit. One can show that the orbit for the harmonic\noscillator is also elliptical, but in this case the center of the\npotential is at the center of the ellipse, not at one of the foci.\n\nThe solution is also simple to write down exactly in Cartesian coordinates. The $x$ and $y$ equations of motion separate,\n\n$$\n\\begin{eqnarray*}\n\\ddot{x}&=&-kx,\\\\\n\\ddot{y}&=&-ky.\n\\end{eqnarray*}\n$$\n\nThe general solution can be expressed as\n\n$$\n\\begin{eqnarray*}\nx&=&A\\cos\\omega_0 t+B\\sin\\omega_0 t,\\\\\ny&=&C\\cos\\omega_0 t+D\\sin\\omega_0 t.\n\\end{eqnarray*}\n$$\n\nThe code here finds the solution for $x$ and $y$ using the code we\ndeveloped in homework 5 and 6 and the midterm. Note that this code is\ntailored to run in Cartesian coordinates. There is thus no angular\nmomentum dependent term.\n\nHere we have chose initial conditions that\ncorrespond to the minimum of the effective potential\n$r_{\\mathrm{min}}$. We have chosen $x_0=r_{\\mathrm{min}}$ and\n$y_0=0$. Similarly, we use the centripetal acceleration to determine\nthe initial velocity so that we have a circular motion (see back to the\nlast question of the midterm). This means that we set the centripetal\nacceleration $v^2/r$ equal to the force from the harmonic oscillator $-k\\boldsymbol{r}$. Taking the\nmagnitude of $\\boldsymbol{r}$ we have then\n$v^2/r=k/mr$, which gives $v=\\pm\\omega_0r$. \n\nSince the code here solves the equations of motion in cartesian\ncoordinates and the harmonic oscillator potential leads to forces in\nthe $x$- and $y$-directions that are decoupled, we have to select the initial velocities and positions so that we don't get that for example $y(t)=0$.\n\nWe set $x_0$ to be different from zero and $v_{y0}$ to be different from zero.\n\n\n```python\n\nDeltaT = 0.001\n#set up arrays \ntfinal = 5.0\nn = ceil(tfinal/DeltaT)\n# set up arrays\nt = np.zeros(n)\nv = np.zeros((n,2))\nr = np.zeros((n,2))\nradius = np.zeros(n)\n# Constants of the model\nk = 1.0 # spring constant\nm = 1.0 # mass, you can change these\nomega02 = k/m # Frequency\nAngMom = 1.0 # The angular momentum\n# Potential minimum\nrmin = (AngMom*AngMom/k/m)**0.25\n# Initial conditions as compact 2-dimensional arrays, x0=rmin and y0 = 0\nx0 = rmin; y0= 0.0\nr0 = np.array([x0,y0])\nvy0 = sqrt(omega02)*rmin; vx0 = 0.0\nv0 = np.array([vx0,vy0])\nr[0] = r0\nv[0] = v0\n# Start integrating using the Velocity-Verlet method\nfor i in range(n-1):\n # Set up the acceleration\n a = -r[i]*omega02 \n # update velocity, time and position using the Velocity-Verlet method\n r[i+1] = r[i] + DeltaT*v[i]+0.5*(DeltaT**2)*a\n anew = -r[i+1]*omega02 \n v[i+1] = v[i] + 0.5*DeltaT*(a+anew)\n t[i+1] = t[i] + DeltaT\n# Plot position as function of time\nradius = np.sqrt(r[:,0]**2+r[:,1]**2)\nfig, ax = plt.subplots(3,1)\nax[0].set_xlabel('time')\nax[0].set_ylabel('radius squared')\nax[0].plot(t,r[:,0]**2+r[:,1]**2)\nax[1].set_xlabel('time')\nax[1].set_ylabel('x position')\nax[1].plot(t,r[:,0])\nax[2].set_xlabel('time')\nax[2].set_ylabel('y position')\nax[2].plot(t,r[:,1])\n\nfig.tight_layout()\nsave_fig(\"2DimHOVV\")\nplt.show()\n```\n\nWe see that the radius (to within a given error), we obtain a constant radius.\n\n\nThe following code shows first how we can solve this problem using the radial degrees of freedom only.\nHere we need to add the explicit centrifugal barrier. Note that the variable $r$ depends only on time. There is no $x$ and $y$ directions\nsince we have transformed the equations to polar coordinates.\n\n\n```python\nDeltaT = 0.01\n#set up arrays \ntfinal = 10.0\nn = ceil(tfinal/DeltaT)\n# set up arrays for t, v and r\nt = np.zeros(n)\nv = np.zeros(n)\nr = np.zeros(n)\nE = np.zeros(n)\n# Constants of the model\nAngMom = 1.0 # The angular momentum\nm = 1.0\nk = 1.0\nomega02 = k/m\nc1 = AngMom*AngMom/(m*m)\nc2 = AngMom*AngMom/m\nrmin = (AngMom*AngMom/k/m)**0.25\n# Initial conditions\nr0 = rmin\nv0 = 0.0\nr[0] = r0\nv[0] = v0\nE[0] = 0.5*m*v0*v0+0.5*k*r0*r0+0.5*c2/(r0*r0)\n# Start integrating using the Velocity-Verlet method\nfor i in range(n-1):\n # Set up acceleration\n a = -r[i]*omega02+c1/(r[i]**3) \n # update velocity, time and position using the Velocity-Verlet method\n r[i+1] = r[i] + DeltaT*v[i]+0.5*(DeltaT**2)*a\n anew = -r[i+1]*omega02+c1/(r[i+1]**3)\n v[i+1] = v[i] + 0.5*DeltaT*(a+anew)\n t[i+1] = t[i] + DeltaT\n E[i+1] = 0.5*m*v[i+1]*v[i+1]+0.5*k*r[i+1]*r[i+1]+0.5*c2/(r[i+1]*r[i+1])\n # Plot position as function of time\nfig, ax = plt.subplots(2,1)\nax[0].set_xlabel('time')\nax[0].set_ylabel('radius')\nax[0].plot(t,r)\nax[1].set_xlabel('time')\nax[1].set_ylabel('Energy')\nax[1].plot(t,E)\nsave_fig(\"RadialHOVV\")\nplt.show()\n```\n\nWith some work using double angle formulas, one can calculate\n\n$$\n\\begin{eqnarray*}\nr^2&=&x^2+y^2\\\\\n\\nonumber\n&=&(A^2+C^2)\\cos^2(\\omega_0t)+(B^2+D^2)\\sin^2\\omega_0t+(AB+CD)\\cos(\\omega_0t)\\sin(\\omega_0t)\\\\\n\\nonumber\n&=&\\alpha+\\beta\\cos 2\\omega_0 t+\\gamma\\sin 2\\omega_0 t,\\\\\n\\alpha&=&\\frac{A^2+B^2+C^2+D^2}{2},~~\\beta=\\frac{A^2-B^2+C^2-D^2}{2},~~\\gamma=AB+CD,\\\\\nr^2&=&\\alpha+(\\beta^2+\\gamma^2)^{1/2}\\cos(2\\omega_0 t-\\delta),~~~\\delta=\\arctan(\\gamma/\\beta),\n\\end{eqnarray*}\n$$\n\nand see that radius oscillates with frequency $2\\omega_0$. The\nfactor of two comes because the oscillation $x=A\\cos\\omega_0t$ has two\nmaxima for $x^2$, one at $t=0$ and one a half period later.\n\n\n\n\n## Stability of Orbits\n\nThe effective force can be extracted from the effective potential, $V_{\\rm eff}$. Beginning from the equations of motion, Eq. ([2](#eq:radialeqofmotion)), for $r$,\n\n$$\n\\begin{eqnarray}\nm\\ddot{r}&=&F+\\frac{L^2}{mr^3}\\\\\n\\nonumber\n&=&F_{\\rm eff}\\\\\n\\nonumber\n&=&-\\partial_rV_{\\rm eff},\\\\\n\\nonumber\nF_{\\rm eff}&=&-\\partial_r\\left[V(r)+(L^2/2mr^2)\\right].\n\\end{eqnarray}\n$$\n\nFor a circular orbit, the radius must be fixed as a function of time,\nso one must be at a maximum or a minimum of the effective\npotential. However, if one is at a maximum of the effective potential\nthe radius will be unstable. For the attractive Coulomb force the\neffective potential will be dominated by the $-\\alpha/r$ term for\nlarge $r$ because the centrifugal part falls off more quickly, $\\sim\n1/r^2$. At low $r$ the centrifugal piece wins and the effective\npotential is repulsive. Thus, the potential must have a minimum\nsomewhere with negative potential. The circular orbits are then stable\nto perturbation.\n\n\nThe effective potential is sketched for two cases, a $1/r$ attractive\npotential and a $1/r^3$ attractive potential. The $1/r$ case has a\nstable minimum, whereas the circular orbit in the $1/r^3$ case is\nunstable.\n\n\nIf one considers a potential that falls as $1/r^3$, the situation is\nreversed and the point where $\\partial_rV$ disappears will be a local\nmaximum rather than a local minimum. **Fig to come here with code**\n\nThe repulsive centrifugal piece dominates at large $r$ and the attractive\nCoulomb piece wins out at small $r$. The circular orbit is then at a\nmaximum of the effective potential and the orbits are unstable. It is\nthe clear that for potentials that fall as $r^n$, that one must have\n$n>-2$ for the orbits to be stable.\n\n\nConsider a potential $V(r)=\\beta r$. For a particle of mass $m$ with\nangular momentum $L$, find the angular frequency of a circular\norbit. Then find the angular frequency for small radial perturbations.\n\n\nFor the circular orbit you search for the position $r_{\\rm min}$ where the effective potential is minimized,\n\n$$\n\\begin{eqnarray*}\n\\partial_r\\left\\{\\beta r+\\frac{L^2}{2mr^2}\\right\\}&=&0,\\\\\n\\beta&=&\\frac{L^2}{mr_{\\rm min}^3},\\\\\nr_{\\rm min}&=&\\left(\\frac{L^2}{\\beta m}\\right)^{1/3},\\\\\n\\dot{\\theta}&=&\\frac{L}{mr_{\\rm min}^2}=\\frac{\\beta^{2/3}}{(mL)^{1/3}}\n\\end{eqnarray*}\n$$\n\nNow, we can find the angular frequency of small perturbations about the circular orbit. To do this we find the effective spring constant for the effective potential,\n\n$$\n\\begin{eqnarray*}\nk_{\\rm eff}&=&\\partial_r^2 \\left.V_{\\rm eff}\\right|_{r_{\\rm min}}\\\\\n&=&\\frac{3L^2}{mr_{\\rm min}^4},\\\\\n\\omega&=&\\sqrt{\\frac{k_{\\rm eff}}{m}}\\\\\n&=&\\frac{\\beta^{2/3}}{(mL)^{1/3}}\\sqrt{3}.\n\\end{eqnarray*}\n$$\n\nIf the two frequencies, $\\dot{\\theta}$ and $\\omega$, differ by an\ninteger factor, the orbit's trajectory will repeat itself each time\naround. This is the case for the inverse-square force,\n$\\omega=\\dot{\\theta}$, and for the harmonic oscillator,\n$\\omega=2\\dot{\\theta}$. In this case, $\\omega=\\sqrt{3}\\dot{\\theta}$,\nand the angles at which the maxima and minima occur change with each\norbit.\n\n\n### Code example with gravitional force\n\nThe code example here is meant to illustrate how we can make a plot of the final orbit. We solve the equations in polar coordinates (the example here uses the minimum of the potential as initial value) and then we transform back to cartesian coordinates and plot $x$ versus $y$. We see that we get a perfect circle when we place ourselves at the minimum of the potential energy, as expected.\n\n\n```python\n\n# Simple Gravitational Force -alpha/r\n \nDeltaT = 0.01\n#set up arrays \ntfinal = 8.0\nn = ceil(tfinal/DeltaT)\n# set up arrays for t, v and r\nt = np.zeros(n)\nv = np.zeros(n)\nr = np.zeros(n)\nphi = np.zeros(n)\nx = np.zeros(n)\ny = np.zeros(n)\n# Constants of the model, setting all variables to one for simplicity\nalpha = 1.0\nAngMom = 1.0 # The angular momentum\nm = 1.0 # scale mass to one\nc1 = AngMom*AngMom/(m*m)\nc2 = AngMom*AngMom/m\nrmin = (AngMom*AngMom/m/alpha)\n# Initial conditions, place yourself at the potential min\nr0 = rmin\nv0 = 0.0 # starts at rest\nr[0] = r0\nv[0] = v0\nphi[0] = 0.0\n# Start integrating using the Velocity-Verlet method\nfor i in range(n-1):\n # Set up acceleration\n a = -alpha/(r[i]**2)+c1/(r[i]**3)\n # update velocity, time and position using the Velocity-Verlet method\n r[i+1] = r[i] + DeltaT*v[i]+0.5*(DeltaT**2)*a\n anew = -alpha/(r[i+1]**2)+c1/(r[i+1]**3)\n v[i+1] = v[i] + 0.5*DeltaT*(a+anew)\n t[i+1] = t[i] + DeltaT\n phi[i+1] = t[i+1]*c2/(r0**2)\n# Find cartesian coordinates for easy plot \nx = r*np.cos(phi)\ny = r*np.sin(phi)\nfig, ax = plt.subplots(3,1)\nax[0].set_xlabel('time')\nax[0].set_ylabel('radius')\nax[0].plot(t,r)\nax[1].set_xlabel('time')\nax[1].set_ylabel('Angle $\\cos{\\phi}$')\nax[1].plot(t,np.cos(phi))\nax[2].set_ylabel('y')\nax[2].set_xlabel('x')\nax[2].plot(x,y)\n\nsave_fig(\"Phasespace\")\nplt.show()\n```\n\nTry to change the initial value for $r$ and see what kind of orbits you get.\nIn order to test different energies, it can be useful to look at the plot of the effective potential discussed above.\n\nHowever, for orbits different from a circle the above code would need modifications in order to allow us to display say an ellipse. For the latter, it is much easier to run our code in cartesian coordinates, as done here. In this code we test also energy conservation and see that it is conserved to numerical precision. The code here is a simple extension of the code we developed for homework 4.\n\n\n```python\n# Common imports\nimport numpy as np\nimport pandas as pd\nfrom math import *\nimport matplotlib.pyplot as plt\n\nDeltaT = 0.01\n#set up arrays \ntfinal = 10.0\nn = ceil(tfinal/DeltaT)\n# set up arrays\nt = np.zeros(n)\nv = np.zeros((n,2))\nr = np.zeros((n,2))\nE = np.zeros(n)\n# Constants of the model\nm = 1.0 # mass, you can change these\nalpha = 1.0\n# Initial conditions as compact 2-dimensional arrays\nx0 = 0.5; y0= 0.\nr0 = np.array([x0,y0]) \nv0 = np.array([0.0,1.0])\nr[0] = r0\nv[0] = v0\nrabs = sqrt(sum(r[0]*r[0]))\nE[0] = 0.5*m*(v[0,0]**2+v[0,1]**2)-alpha/rabs\n# Start integrating using the Velocity-Verlet method\nfor i in range(n-1):\n # Set up the acceleration\n rabs = sqrt(sum(r[i]*r[i]))\n a = -alpha*r[i]/(rabs**3)\n # update velocity, time and position using the Velocity-Verlet method\n r[i+1] = r[i] + DeltaT*v[i]+0.5*(DeltaT**2)*a\n rabs = sqrt(sum(r[i+1]*r[i+1]))\n anew = -alpha*r[i+1]/(rabs**3)\n v[i+1] = v[i] + 0.5*DeltaT*(a+anew)\n E[i+1] = 0.5*m*(v[i+1,0]**2+v[i+1,1]**2)-alpha/rabs\n t[i+1] = t[i] + DeltaT\n# Plot position as function of time\nfig, ax = plt.subplots(3,1)\nax[0].set_ylabel('y')\nax[0].set_xlabel('x')\nax[0].plot(r[:,0],r[:,1])\nax[1].set_xlabel('time')\nax[1].set_ylabel('y position')\nax[1].plot(t,r[:,0])\nax[2].set_xlabel('time')\nax[2].set_ylabel('y position')\nax[2].plot(t,r[:,1])\n\nfig.tight_layout()\nsave_fig(\"2DimGravity\")\nplt.show()\nprint(E)\n```\n", "meta": {"hexsha": "9cfcff1df9dfa914230a3f0107933365fb8e321a", "size": 135797, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "doc/pub/week11/ipynb/week11.ipynb", "max_stars_repo_name": "mhjensen/Physics321", "max_stars_repo_head_hexsha": "f858db36328c9fc127ccb44f62934d8f8749dd9f", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 20, "max_stars_repo_stars_event_min_datetime": "2020-01-09T17:41:16.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-09T00:48:58.000Z", "max_issues_repo_path": "doc/pub/week11/ipynb/week11.ipynb", "max_issues_repo_name": "mhjensen/Physics321", "max_issues_repo_head_hexsha": "f858db36328c9fc127ccb44f62934d8f8749dd9f", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2020-01-08T03:47:53.000Z", "max_issues_repo_issues_event_max_datetime": "2020-12-15T15:02:57.000Z", "max_forks_repo_path": "doc/pub/week11/ipynb/week11.ipynb", "max_forks_repo_name": "mhjensen/Physics321", "max_forks_repo_head_hexsha": "f858db36328c9fc127ccb44f62934d8f8749dd9f", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 33, "max_forks_repo_forks_event_min_datetime": "2020-01-10T20:40:55.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-11T20:28:41.000Z", "avg_line_length": 92.2533967391, "max_line_length": 25020, "alphanum_fraction": 0.819723558, "converted": true, "num_tokens": 10649, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.8152324871074607, "lm_q1q2_score": 0.4457187034389536}} {"text": "```python\nfrom IPython.core.display import HTML, Image\ncss_file = 'style.css'\nHTML(open(css_file, 'r').read())\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n```python\nfrom sympy import init_printing, Matrix, symbols\ninit_printing()\n```\n\n# The four fundamental subspaces\n# Introducing the matrix space\n\n## The four fundamental subspaces\n\nIn this section, we bring together the four fundamentals spaces in linear algebra: \n\n* The column space, $C\\left( A \\right)$\n* The nullspace, $N \\left( A \\right)$\n* The rowspace:\n * All linear combinations of rows\n * All linear combinations of $A^T$ and $C \\left( A^T \\right)$\n* The nullspace of $A^T$, $N \\left( A^T \\right)$ (also termed the left nullspace of A)\n\nFor a matrix (with dimensions noted) $A_{m \\times n}$ we that $C \\left( A \\right)$ is in $\\mathbb{R}^m$, $N \\left( A \\right)$ is in $\\mathbb{R}^n$, $C \\left( A^T \\right)$ is in $\\mathbb{R}^n$, and $N \\left( A^T \\right)$ is in $\\mathbb{R}^m$.\n\n## Calculating basis and dimension\n\nWe also need to know the bases and dimensions of these fundamental spaces:\n\n* $C \\left( A \\right)$\n * The bases are the picot columns\n * The dimension is the rank, $r$\n* $N \\left( A \\right)$\n * The bases are the special solutions (one for every free variable, $n-r$)\n * The dimensions are $n-r$\n* $ C \\left( A ^T \\right)$\n * If $A$ is reduced to row-echelon form, $R$, then $ C\\left( R \\right) \\ne C \\left( A \\right)$\n * We do have, though, that the rowspace of $R$ is equal to the rowspace of $R$, in other words $ C \\left( R^T \\right) = C \\left( A^T \\right)$\n* $ N \\left( A^T \\right) $\n * The bases are such that:\n * $A^T \\underline{y} = \\underline{0}$\n * $ \\underline{y}^T {\\left( A^T \\right)}^T = \\underline{0}^T $\n * $ \\underline{y}^T A = \\underline{0}^T$\n * These are the pivot columns of $R^T$ \n * The dimensions are $m-r$\n\n## Example problems\n\n### Example 1\n\nConsider this example matrix and calculate the bases and dimension for all four fundamental spaces\n\n\n```python\nA = Matrix([[1, 2, 3, 1], [1, 1, 2, 1], [1, 2, 3, 1]]) # We note that rows 1 and three are identical and that\n# columns 3 is the addtion of columns 1 and 2 and column 1 equals column 4\nA\n```\n\n#### Columnspace\n\n\n```python\nA.rref() # Remember that the columnspace contains the pivot columns as a basis\n```\n\n* The bases are the pivot columns: \n$ \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\\\ 0 & 0 \\end{bmatrix} $ \n* This is indeed $\\mathbb{R}^3$ (each column vector in the matrix has three components) as $A_{m \\times n} = A_{3 \\times 4}$\n\n* The rank (no of columns with pivots) is $2$, thus $\\text{dim} \\left( A \\right) = 2$\n\n#### Nullspace\n\n\n```python\nA.nullspace() # Calculating the nullspace vectors\n```\n\n* The basis is in $\\mathbb{R}^4$ ($A$ has $n = 4$ columns)\n\n\n```python\nA.rref() # No pivots for columns 3 and 4\n```\n\n\n\n\n$$\\begin{pmatrix}\\left[\\begin{matrix}1 & 0 & 1 & 1\\\\0 & 1 & 1 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right], & \\begin{bmatrix}0, & 1\\end{bmatrix}\\end{pmatrix}$$\n\n\n\n* The dimension is $2$ (there are $2$ column vectors, which is indeed $n-r=4-2=2$\n\n#### Rowspace C(AT)\n\n* Here, we are looking for the pivot columns of $A^T$\n\n\n```python\nA_t = A.transpose()\nA_t.rref()\n```\n\n* The pivot rows of $A$ (columns in $A^T$) are rows $1$ and $2$ (columns $1$ and $2$ of $A^T$)\n\n* As stated above, it is in $\\mathbb{R}^4$\n\n* The dimension are $n-r=4-2=2$\n\n#### Nullspace of AT\n\n\n```python\nA_t.nullspace()\n```\n\n* Which is in $\\mathbb{R}^3$\n\n* The dimension is $1$, since $m-r=3-2=1$ (remember that the rank is the number of pivot columns)\n\n### Example 2\n\nConsider this example matrix (in LU form) and calculate the bases and dimension for all four fundamental spaces\n\n\n```python\nL = Matrix([[1, 0, 0], [2, 1, 0], [-1, 0, 1]])\nU = Matrix([[5, 0, 3], [0, 1, 1], [0, 0, 0]])\nA = L * U\nL, U, A\n```\n\n#### Columnspace of A\n\n\n```python\nA.rref()\n```\n\n* The basis is thus: \n$ \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\\\ 0 & 0 \\end{bmatrix} $\n* Another basis would be the pivot columns of L: \n$ \\begin{bmatrix} 1 & 0 \\\\ 2 & 1 \\\\ -1 & 0 \\end{bmatrix} $\n* It is in $\\mathbb{R}^3$, since $m=3$\n* It has a rank of $2$ (two pivot columns)\n* Since the dimension of the columnspace is equal to the rank, $\\text{dim} \\left( A \\right) = 2$\n * Note that it is also equal to the number of pivot columns in U\n\n#### Nullspace of A\n\n\n```python\nA.nullspace()\n```\n\n\n\n\n$$\\begin{bmatrix}\\left[\\begin{matrix}- \\frac{3}{5}\\\\-1\\\\1\\end{matrix}\\right]\\end{bmatrix}$$\n\n\n\n* The nullspace is in $\\mathbb{R}^3$, since $n=3$\n* The basis is the special solution(s), which is one column vector for every free variable\n * Since we only have a single free variable, we have a single nullspace column vector\n * This fits in with the fact that it needs to be $n-r$\n * It can also be calculated by taking $U$, setting the free variable to $1$ and solving for the other rows by setting each equal to $0$\n* The dimension of the nullspace is also $1$ ($n-r$, i.e. a single column)\n * It is also the number of free variables\n\n#### The rowspace\n\n* This is the columnspace of $A^T$\n* Don't take the transpose first!\n* Row reduce, identify the rows with pivots and transpose them\n\n\n```python\nA.rref()\n```\n\n* The basis can also be written down by identifying the rows with pivots in $U$ and writing them down as columns (getting their transpose)\n$$ \\begin{bmatrix} 5 & 0 \\\\ 0 & 1 \\\\ 3 & 1 \\end{bmatrix} $$\n* It is in $\\mathbb{R}^3$, since $n=3$\n* The rank is $2$, which is equal to the dimension, i.e. $\\text{dim} \\left( A^T \\right) = 2$\n\n#### The nullspace of AT\n\n\n```python\nA.transpose().nullspace()\n```\n\n\n\n\n$$\\begin{bmatrix}\\left[\\begin{matrix}1\\\\0\\\\1\\end{matrix}\\right]\\end{bmatrix}$$\n\n\n\n* It is indeed in $\\mathbb{R}^3$, since $m=3$\n* A good way to do it is to take the inverse of $L$, such that $L^{-1}A=U$\n * Now the free variable row in $U$ is row three\n * Take the corresponding row in $L^{-1}$ and transpose it\n* The dimension in $m-2=3-2=1$\n\n## The matrix space\n\nNote square matrix is also a _vector_ space, because they obey the vector space rules of addition and scalar multiplication. Subspaces (of same) would include:\n* Upper triangular matrices\n* Symmetric matrices\n\n\n```python\n\n```\n", "meta": {"hexsha": "be5c54f79ffab18c8ffe8178c3fb0aee037c1a3c", "size": 38121, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Mathematics/Linear Algebra/0.0 MIT-18.06 - Jupyter/Lecture_10_Four_Fundamental_Subspaces.ipynb", "max_stars_repo_name": "okara83/Becoming-a-Data-Scientist", "max_stars_repo_head_hexsha": "f09a15f7f239b96b77a2f080c403b2f3e95c9650", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Mathematics/Linear Algebra/0.0 MIT-18.06 - Jupyter/Lecture_10_Four_Fundamental_Subspaces.ipynb", "max_issues_repo_name": "okara83/Becoming-a-Data-Scientist", "max_issues_repo_head_hexsha": "f09a15f7f239b96b77a2f080c403b2f3e95c9650", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Mathematics/Linear Algebra/0.0 MIT-18.06 - Jupyter/Lecture_10_Four_Fundamental_Subspaces.ipynb", "max_forks_repo_name": "okara83/Becoming-a-Data-Scientist", "max_forks_repo_head_hexsha": "f09a15f7f239b96b77a2f080c403b2f3e95c9650", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2022-02-09T15:41:33.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-11T07:47:40.000Z", "avg_line_length": 50.0932982917, "max_line_length": 4572, "alphanum_fraction": 0.6993782954, "converted": true, "num_tokens": 2646, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269796369905, "lm_q2_score": 0.7981867729389246, "lm_q1q2_score": 0.44564921012118613}} {"text": "\n\n\n# PHY321: Two-body problems and Gravitational Forces\n**[Morten Hjorth-Jensen](http://mhjgit.github.io/info/doc/web/)**, Department of Physics and Astronomy and Facility for Rare Ion Beams (FRIB), Michigan State University, USA and Department of Physics, University of Oslo, Norway\n\nDate: **Mar 21, 2022**\n\nCopyright 1999-2022, [Morten Hjorth-Jensen](http://mhjgit.github.io/info/doc/web/). Released under CC Attribution-NonCommercial 4.0 license\n\n## Aims and Overarching Motivation\n\n### Monday March 21\n\nDefinition of the two-body problem, rewriting the equations in relative and center-of-mass coordinates\n\n**Reading suggestion**: Taylor sections 8.2-8.3\n\n### Wednesday March 23\n\nPreparing the ground for the gravitational force and its solution in two dimensions. \nHarmonic Oscillator example in two dimensions (if we get time).\n\n**Reading suggestion**: Taylor chapter 8.4-8.5\n\n### Friday March 25\n\nDiscussion and work on homework 7\n\n## Videos of possible interest\n\n* [Video on solving differential equations numerically](https://youtu.be/7nYIfV0z1VM)\n\n* [Video on Fourier aanalysis](https://youtu.be/neXZ4fb-4Rs)\n\n* [Handwritten notes for Fourier analysis](https://github.com/mhjensen/Physics321/blob/master/doc/HandWrittenNotes/Spring2022/NotesFourierAnalysisMarch21.pdf)\n\n# Two-body Problems\n\nTwo-body problems play a central role in physics. Some of these problems can, with appropriate transformations, be solved analytically. The gravitional force problem (as well as the Coulomb potential problem for two particles) is an example of this.\n\nThere are several small steps which we need to do in order to reach this solution. These are\n1. Rewriting the equations in terms of the degrees of freedom of the relative motion and the center of mass motion\n\n2. Making a transformation either to polar (two dimensions) or to spherical coordinates (three dimensions)\n\n3. This gives us uncoupled differential equations for each coordinate that we can solve analytically\n\nThe advantage in doing so is that we can extract a lot of interesting insights about the motion and the physics of these important physics problems. These insights can be transferred to other physics problems where the potentials are given by expressions proportional with the inverse relative distance.\n\n## The gravitational force\n\nThe gravitational potential energy and forces involving two masses $a$ and $b$ are\n\n$$\n\\begin{eqnarray}\nV_{ab}&=&-\\frac{Gm_am_b}{|\\boldsymbol{r}_a-\\boldsymbol{r}_b|},\\\\\n\\nonumber\nF_{ba}&=&-\\frac{Gm_am_b}{|\\boldsymbol{r}_a-\\boldsymbol{r}_b|^2}\\hat{r}_{ab},\\\\\n\\nonumber\n\\hat{r}_{ab}&=&\\frac{\\boldsymbol{r}_b-\\boldsymbol{r}_a}{|\\boldsymbol{r}_a-\\boldsymbol{r}_b|}.\n\\end{eqnarray}\n$$\n\nHere $G=6.67\\times 10^{-11}$ Nm$^2$/kg$^2$, and $F_{ba}$ is the force\non $b$ due to $a$. By inspection, one can see that the force on $b$\ndue to $a$ and the force on $a$ due to $b$ are equal and opposite. The\nnet potential energy for a large number of masses would be\n\n\n
\n\n$$\n\\begin{equation}\nV=\\sum_{a\n
\n\n$$\n\\begin{eqnarray}\n\\label{eq:radialeqofmotion} \\tag{2}\n\\frac{d}{dt}r^2&=&\\frac{d}{dt}(x^2+y^2)=2x\\dot{x}+2y\\dot{y}=2r\\dot{r},\\\\\n\\nonumber\n\\dot{r}&=&\\frac{x}{r}\\dot{x}+\\frac{y}{r}\\dot{y},\\\\\n\\nonumber\n\\ddot{r}&=&\\frac{x}{r}\\ddot{x}+\\frac{y}{r}\\ddot{y}\n+\\frac{\\dot{x}^2+\\dot{y}^2}{r}\n-\\frac{\\dot{r}^2}{r}.\n\\end{eqnarray}\n$$\n\n## Reordering the equations\nRecognizing that the numerator of the third term is the velocity squared, and that it can be written in polar coordinates,\n\n\n
\n\n$$\n\\begin{equation}\nv^2=\\dot{x}^2+\\dot{y}^2=\\dot{r}^2+r^2\\dot{\\theta}^2,\n\\label{_auto2} \\tag{3}\n\\end{equation}\n$$\n\none can write $\\ddot{r}$ as\n\n\n
\n\n$$\n\\begin{eqnarray}\n\\label{eq:radialeqofmotion2} \\tag{4}\n\\ddot{r}&=&\\frac{F_x\\cos\\theta+F_y\\sin\\theta}{m}+\\frac{\\dot{r}^2+r^2\\dot{\\theta}^2}{r}-\\frac{\\dot{r}^2}{r}\\\\\n\\nonumber\n&=&\\frac{F}{m}+\\frac{r^2\\dot{\\theta}^2}{r}\\\\\n\\nonumber\nm\\ddot{r}&=&F+\\frac{L^2}{mr^3}.\n\\end{eqnarray}\n$$\n\n## Force depends on $r$ only\n\nThis derivation used the fact that the force was radial,\n$F=F_r=F_x\\cos\\theta+F_y\\sin\\theta$, and that angular momentum is\n$L=mrv_{\\theta}=mr^2\\dot{\\theta}$. The term $L^2/mr^3=mv^2/r$ behaves\nlike an additional force. Sometimes this is referred to as a\ncentrifugal force, but it is not a force. Instead, it is the\nconsequence of considering the motion in a rotating (and therefore\naccelerating) frame.\n\nNow, we switch to the particular case of an attractive inverse square\nforce, $F=-\\alpha/r^2$, and show that the trajectory, $r(\\theta)$, is\nan ellipse. To do this we transform derivatives w.r.t. time to\nderivatives w.r.t. $\\theta$ using the chain rule combined with angular\nmomentum conservation, $\\dot{\\theta}=L/mr^2$.\n\n\n
\n\n$$\n\\begin{eqnarray}\n\\label{eq:rtotheta} \\tag{5}\n\\dot{r}&=&\\frac{dr}{d\\theta}\\dot{\\theta}=\\frac{dr}{d\\theta}\\frac{L}{mr^2},\\\\\n\\nonumber\n\\ddot{r}&=&\\frac{d^2r}{d\\theta^2}\\dot{\\theta}^2\n+\\frac{dr}{d\\theta}\\left(\\frac{d}{dr}\\frac{L}{mr^2}\\right)\\dot{r}\\\\\n\\nonumber\n&=&\\frac{d^2r}{d\\theta^2}\\left(\\frac{L}{mr^2}\\right)^2\n-2\\frac{dr}{d\\theta}\\frac{L}{mr^3}\\dot{r}\\\\\n\\nonumber\n&=&\\frac{d^2r}{d\\theta^2}\\left(\\frac{L}{mr^2}\\right)^2\n-\\frac{2}{r}\\left(\\frac{dr}{d\\theta}\\right)^2\\left(\\frac{L}{mr^2}\\right)^2\n\\end{eqnarray}\n$$\n\n## Further manipulations\n\nEquating the two expressions for $\\ddot{r}$ in Eq.s ([4](#eq:radialeqofmotion2)) and ([5](#eq:rtotheta)) eliminates all the derivatives w.r.t. time, and provides a differential equation with only derivatives w.r.t. $\\theta$,\n\n\n
\n\n$$\n\\begin{equation}\n\\label{eq:rdotdot} \\tag{6}\n\\frac{d^2r}{d\\theta^2}\\left(\\frac{L}{mr^2}\\right)^2\n-\\frac{2}{r}\\left(\\frac{dr}{d\\theta}\\right)^2\\left(\\frac{L}{mr^2}\\right)^2\n=\\frac{F}{m}+\\frac{L^2}{m^2r^3},\n\\end{equation}\n$$\n\nthat when solved yields the trajectory, i.e. $r(\\theta)$. Up to this\npoint the expressions work for any radial force, not just forces that\nfall as $1/r^2$.\n\n## Final manipulations, part 1\n\nThe trick to simplifying this differential equation for the inverse\nsquare problems is to make a substitution, $u\\equiv 1/r$, and rewrite\nthe differential equation for $u(\\theta)$.\n\n$$\n\\begin{eqnarray}\nr&=&1/u,\\\\\n\\nonumber\n\\frac{dr}{d\\theta}&=&-\\frac{1}{u^2}\\frac{du}{d\\theta},\\\\\n\\nonumber\n\\frac{d^2r}{d\\theta^2}&=&\\frac{2}{u^3}\\left(\\frac{du}{d\\theta}\\right)^2-\\frac{1}{u^2}\\frac{d^2u}{d\\theta^2}.\n\\end{eqnarray}\n$$\n\nPlugging these expressions into Eq. ([6](#eq:rdotdot)) gives an\nexpression in terms of $u$, $du/d\\theta$, and $d^2u/d\\theta^2$. After\nsome tedious algebra,\n\n\n
\n\n$$\n\\begin{equation}\n\\frac{d^2u}{d\\theta^2}=-u-\\frac{F m}{L^2u^2}.\n\\label{_auto3} \\tag{7}\n\\end{equation}\n$$\n\n## Final manipulations, part 2\n\nFor the attractive inverse square law force, $F=-\\alpha u^2$,\n\n\n
\n\n$$\n\\begin{equation}\n\\frac{d^2u}{d\\theta^2}=-u+\\frac{m\\alpha}{L^2}.\n\\label{_auto4} \\tag{8}\n\\end{equation}\n$$\n\nThe solution has two arbitrary constants, $A$ and $\\theta_0$,\n\n\n
\n\n$$\n\\begin{eqnarray}\n\\label{eq:Ctrajectory} \\tag{9}\nu&=&\\frac{m\\alpha}{L^2}+A\\cos(\\theta-\\theta_0),\\\\\n\\nonumber\nr&=&\\frac{1}{(m\\alpha/L^2)+A\\cos(\\theta-\\theta_0)}.\n\\end{eqnarray}\n$$\n\nThe radius will be at a minimum when $\\theta=\\theta_0$ and at a\nmaximum when $\\theta=\\theta_0+\\pi$. The constant $A$ is related to the\neccentricity of the orbit. When $A=0$ the radius is a constant\n$r=L^2/(m\\alpha)$, and the motion is circular. If one solved the\nexpression $mv^2/r=-\\alpha/r^2$ for a circular orbit, using the\nsubstitution $v=L/(mr)$, one would reproduce the expression\n$r=L^2/(m\\alpha)$.\n\n## Final manipulations, part 3\n\nThe form describing the elliptical trajectory in\nEq. ([9](#eq:Ctrajectory)) can be identified as an ellipse with one\nfocus being the center of the ellipse by considering the definition of\nan ellipse as being the points such that the sum of the two distances\nbetween the two foci are a constant. Making that distance $2D$, the\ndistance between the two foci as $2a$, and putting one focus at the\norigin,\n\n$$\n\\begin{eqnarray}\n2D&=&r+\\sqrt{(r\\cos\\theta-2a)^2+r^2\\sin^2\\theta},\\\\\n\\nonumber\n4D^2+r^2-4Dr&=&r^2+4a^2-4ar\\cos\\theta,\\\\\n\\nonumber\nr&=&\\frac{D^2-a^2}{D+a\\cos\\theta}=\\frac{1}{D/(D^2-a^2)-a\\cos\\theta/(D^2-a^2)}.\n\\end{eqnarray}\n$$\n\nBy inspection, this is the same form as Eq. ([9](#eq:Ctrajectory)) with $D/(D^2-a^2)=m\\alpha/L^2$ and $a/(D^2-a^2)=A$.\n\n## Ellipse reminder\n\nLet us remind ourselves about what an ellipse is before we proceed.\n\n\n```\n%matplotlib inline\n\nimport numpy as np\nfrom matplotlib import pyplot as plt\nfrom math import pi\n\nu=1. #x-position of the center\nv=0.5 #y-position of the center\na=2. #radius on the x-axis\nb=1.5 #radius on the y-axis\n\nt = np.linspace(0, 2*pi, 100)\nplt.plot( u+a*np.cos(t) , v+b*np.sin(t) )\nplt.grid(color='lightgray',linestyle='--')\nplt.show()\n```\n\n## Effective or Centrifugal Potential\n\nThe total energy of a particle is\n\n$$\n\\begin{eqnarray}\nE&=&V(r)+\\frac{1}{2}mv_\\theta^2+\\frac{1}{2}m\\dot{r}^2\\\\\n\\nonumber\n&=&V(r)+\\frac{1}{2}mr^2\\dot{\\theta}^2+\\frac{1}{2}m\\dot{r}^2\\\\\n\\nonumber\n&=&V(r)+\\frac{L^2}{2mr^2}+\\frac{1}{2}m\\dot{r}^2.\n\\end{eqnarray}\n$$\n\nThe second term then contributes to the energy like an additional\nrepulsive potential. The term is sometimes referred to as the\n\"centrifugal\" potential, even though it is actually the kinetic energy\nof the angular motion. Combined with $V(r)$, it is sometimes referred\nto as the \"effective\" potential,\n\n$$\n\\begin{eqnarray}\nV_{\\rm eff}(r)&=&V(r)+\\frac{L^2}{2mr^2}.\n\\end{eqnarray}\n$$\n\nNote that if one treats the effective potential like a real potential, one would expect to be able to generate an effective force,\n\n$$\n\\begin{eqnarray}\nF_{\\rm eff}&=&-\\frac{d}{dr}V(r) -\\frac{d}{dr}\\frac{L^2}{2mr^2}\\\\\n\\nonumber\n&=&F(r)+\\frac{L^2}{mr^3}=F(r)+m\\frac{v_\\perp^2}{r},\n\\end{eqnarray}\n$$\n\nwhich is indeed matches the form for $m\\ddot{r}$ in Eq. ([4](#eq:radialeqofmotion2)), which included the **centrifugal** force.\n\n## Code example\n\nThe following code plots this effective potential for a simple choice of parameters, with a standard gravitational potential $-\\alpha/r$. Here we have chosen $L=m=\\alpha=1$.\n\n\n```\n# Common imports\nimport numpy as np\nfrom math import *\nimport matplotlib.pyplot as plt\n\nDeltax = 0.01\n#set up arrays\nxinitial = 0.3\nxfinal = 5.0\nalpha = 1.0 # spring constant\nm = 1.0 # mass, you can change these\nAngMom = 1.0 # The angular momentum\nn = ceil((xfinal-xinitial)/Deltax)\nx = np.zeros(n)\nfor i in range(n):\n x[i] = xinitial+i*Deltax\nV = np.zeros(n)\nV = -alpha/x+0.5*AngMom*AngMom/(m*x*x)\n# Plot potential\nfig, ax = plt.subplots()\nax.set_xlabel('r[m]')\nax.set_ylabel('V[J]')\nax.plot(x, V)\nfig.tight_layout()\nplt.show()\n```\n\n## Gravitational force example\n\nUsing the above parameters, we can now study the evolution of the system using for example the velocity Verlet method.\nThis is done in the code here for an initial radius equal to the minimum of the potential well. We seen then that the radius is always the same and corresponds to a circle (the radius is always constant).\n\n\n```\n# Common imports\nimport numpy as np\nimport pandas as pd\nfrom math import *\nimport matplotlib.pyplot as plt\nimport os\n\n# Where to save the figures and data files\nPROJECT_ROOT_DIR = \"Results\"\nFIGURE_ID = \"Results/FigureFiles\"\nDATA_ID = \"DataFiles/\"\n\nif not os.path.exists(PROJECT_ROOT_DIR):\n os.mkdir(PROJECT_ROOT_DIR)\n\nif not os.path.exists(FIGURE_ID):\n os.makedirs(FIGURE_ID)\n\nif not os.path.exists(DATA_ID):\n os.makedirs(DATA_ID)\n\ndef image_path(fig_id):\n return os.path.join(FIGURE_ID, fig_id)\n\ndef data_path(dat_id):\n return os.path.join(DATA_ID, dat_id)\n\ndef save_fig(fig_id):\n plt.savefig(image_path(fig_id) + \".png\", format='png')\n\n\n# Simple Gravitational Force -alpha/r\n \nDeltaT = 0.01\n#set up arrays \ntfinal = 100.0\nn = ceil(tfinal/DeltaT)\n# set up arrays for t, v and r\nt = np.zeros(n)\nv = np.zeros(n)\nr = np.zeros(n)\n# Constants of the model, setting all variables to one for simplicity\nalpha = 1.0\nAngMom = 1.0 # The angular momentum\nm = 1.0 # scale mass to one\nc1 = AngMom*AngMom/(m*m)\nc2 = AngMom*AngMom/m\nrmin = (AngMom*AngMom/m/alpha)\n# Initial conditions\nr0 = rmin\nv0 = 0.0\nr[0] = r0\nv[0] = v0\n# Start integrating using the Velocity-Verlet method\nfor i in range(n-1):\n # Set up acceleration\n a = -alpha/(r[i]**2)+c1/(r[i]**3)\n # update velocity, time and position using the Velocity-Verlet method\n r[i+1] = r[i] + DeltaT*v[i]+0.5*(DeltaT**2)*a\n anew = -alpha/(r[i+1]**2)+c1/(r[i+1]**3)\n v[i+1] = v[i] + 0.5*DeltaT*(a+anew)\n t[i+1] = t[i] + DeltaT\n # Plot position as function of time\nfig, ax = plt.subplots(2,1)\nax[0].set_xlabel('time')\nax[0].set_ylabel('radius')\nax[0].plot(t,r)\nax[1].set_xlabel('time')\nax[1].set_ylabel('Velocity')\nax[1].plot(t,v)\nsave_fig(\"RadialGVV\")\nplt.show()\n```\n\nChanging the value of the initial position to a value where the energy is positive, leads to an increasing radius with time, a so-called unbound orbit. Choosing on the other hand an initial radius that corresponds to a negative energy and different from the minimum value leads to a radius that oscillates back and forth between two values.\n\n## Harmonic Oscillator in two dimensions\n\nConsider a particle of mass $m$ in a 2-dimensional harmonic oscillator with potential\n\n$$\nV=\\frac{1}{2}kr^2=\\frac{1}{2}k(x^2+y^2).\n$$\n\nIf the orbit has angular momentum $L$, we can find the radius and angular velocity of the circular orbit as well as the b) the angular frequency of small radial perturbations.\n\nWe consider the effective potential. The radius of a circular orbit is at the minimum of the potential (where the effective force is zero).\nThe potential is plotted here with the parameters $k=m=0.1$ and $L=1.0$.\n\n\n```\n# Common imports\nimport numpy as np\nfrom math import *\nimport matplotlib.pyplot as plt\n\nDeltax = 0.01\n#set up arrays\nxinitial = 0.5\nxfinal = 3.0\nk = 1.0 # spring constant\nm = 1.0 # mass, you can change these\nAngMom = 1.0 # The angular momentum\nn = ceil((xfinal-xinitial)/Deltax)\nx = np.zeros(n)\nfor i in range(n):\n x[i] = xinitial+i*Deltax\nV = np.zeros(n)\nV = 0.5*k*x*x+0.5*AngMom*AngMom/(m*x*x)\n# Plot potential\nfig, ax = plt.subplots()\nax.set_xlabel('r[m]')\nax.set_ylabel('V[J]')\nax.plot(x, V)\nfig.tight_layout()\nplt.show()\n```\n\n$$\n\\begin{eqnarray*}\nV_{\\rm eff}&=&\\frac{1}{2}kr^2+\\frac{L^2}{2mr^2}\n\\end{eqnarray*}\n$$\n\n## Harmonic oscillator in two dimensions and effective potential\nThe effective potential looks like that of a harmonic oscillator for\nlarge $r$, but for small $r$, the centrifugal potential repels the\nparticle from the origin. The combination of the two potentials has a\nminimum for at some radius $r_{\\rm min}$.\n\n$$\n\\begin{eqnarray*}\n0&=&kr_{\\rm min}-\\frac{L^2}{mr_{\\rm min}^3},\\\\\nr_{\\rm min}&=&\\left(\\frac{L^2}{mk}\\right)^{1/4},\\\\\n\\dot{\\theta}&=&\\frac{L}{mr_{\\rm min}^2}=\\sqrt{k/m}.\n\\end{eqnarray*}\n$$\n\nFor particles at $r_{\\rm min}$ with $\\dot{r}=0$, the particle does not\naccelerate and $r$ stays constant, i.e. a circular orbit. The radius\nof the circular orbit can be adjusted by changing the angular momentum\n$L$.\n\nFor the above parameters this minimum is at $r_{\\rm min}=1$.\n\n Now consider small vibrations about $r_{\\rm min}$. The effective spring constant is the curvature of the effective potential.\n\n$$\n\\begin{eqnarray*}\nk_{\\rm eff}&=&\\left.\\frac{d^2}{dr^2}V_{\\rm eff}(r)\\right|_{r=r_{\\rm min}}=k+\\frac{3L^2}{mr_{\\rm min}^4}\\\\\n&=&4k,\\\\\n\\omega&=&\\sqrt{k_{\\rm eff}/m}=2\\sqrt{k/m}=2\\dot{\\theta}.\n\\end{eqnarray*}\n$$\n\nBecause the radius oscillates with twice the angular frequency,\nthe orbit has two places where $r$ reaches a minimum in one\ncycle. This differs from the inverse-square force where there is one\nminimum in an orbit. One can show that the orbit for the harmonic\noscillator is also elliptical, but in this case the center of the\npotential is at the center of the ellipse, not at one of the foci.\n\nThe solution is also simple to write down exactly in Cartesian coordinates. The $x$ and $y$ equations of motion separate,\n\n$$\n\\begin{eqnarray*}\n\\ddot{x}&=&-kx,\\\\\n\\ddot{y}&=&-ky.\n\\end{eqnarray*}\n$$\n\nThe general solution can be expressed as\n\n$$\n\\begin{eqnarray*}\nx&=&A\\cos\\omega_0 t+B\\sin\\omega_0 t,\\\\\ny&=&C\\cos\\omega_0 t+D\\sin\\omega_0 t.\n\\end{eqnarray*}\n$$\n\nThe code here finds the solution for $x$ and $y$ using the code we\ndeveloped in homework 5 and 6 and the midterm. Note that this code is\ntailored to run in Cartesian coordinates. There is thus no angular\nmomentum dependent term.\n\nHere we have chose initial conditions that\ncorrespond to the minimum of the effective potential\n$r_{\\mathrm{min}}$. We have chosen $x_0=r_{\\mathrm{min}}$ and\n$y_0=0$. Similarly, we use the centripetal acceleration to determine\nthe initial velocity so that we have a circular motion (see back to the\nlast question of the midterm). This means that we set the centripetal\nacceleration $v^2/r$ equal to the force from the harmonic oscillator $-k\\boldsymbol{r}$. Taking the\nmagnitude of $\\boldsymbol{r}$ we have then\n$v^2/r=k/mr$, which gives $v=\\pm\\omega_0r$. \n\nSince the code here solves the equations of motion in cartesian\ncoordinates and the harmonic oscillator potential leads to forces in\nthe $x$- and $y$-directions that are decoupled, we have to select the initial velocities and positions so that we don't get that for example $y(t)=0$.\n\nWe set $x_0$ to be different from zero and $v_{y0}$ to be different from zero.\n\n\n```\n\nDeltaT = 0.001\n#set up arrays \ntfinal = 10.0\nn = ceil(tfinal/DeltaT)\n# set up arrays\nt = np.zeros(n)\nv = np.zeros((n,2))\nr = np.zeros((n,2))\nradius = np.zeros(n)\n# Constants of the model\nk = 1.0 # spring constant\nm = 1.0 # mass, you can change these\nomega02 = k/m # Frequency\nAngMom = 1.0 # The angular momentum\n# Potential minimum\nrmin = (AngMom*AngMom/k/m)**0.25\n# Initial conditions as compact 2-dimensional arrays, x0=rmin and y0 = 0\nx0 = rmin; y0= 0.0\nr0 = np.array([x0,y0])\nvy0 = sqrt(omega02)*rmin; vx0 = 0.0\nv0 = np.array([vx0,vy0])\nr[0] = r0\nv[0] = v0\n# Start integrating using the Velocity-Verlet method\nfor i in range(n-1):\n # Set up the acceleration\n a = -r[i]*omega02 \n # update velocity, time and position using the Velocity-Verlet method\n r[i+1] = r[i] + DeltaT*v[i]+0.5*(DeltaT**2)*a\n anew = -r[i+1]*omega02 \n v[i+1] = v[i] + 0.5*DeltaT*(a+anew)\n t[i+1] = t[i] + DeltaT\n# Plot position as function of time\nradius = np.sqrt(r[:,0]**2+r[:,1]**2)\nfig, ax = plt.subplots(3,1)\nax[0].set_xlabel('time')\nax[0].set_ylabel('radius squared')\nax[0].plot(t,r[:,0]**2+r[:,1]**2)\nax[1].set_xlabel('time')\nax[1].set_ylabel('x position')\nax[1].plot(t,r[:,0])\nax[2].set_xlabel('time')\nax[2].set_ylabel('y position')\nax[2].plot(t,r[:,1])\n\nfig.tight_layout()\nsave_fig(\"2DimHOVV\")\nplt.show()\n```\n\nWe see that the radius (to within a given error), we obtain a constant radius.\n\nThe following code shows first how we can solve this problem using the radial degrees of freedom only.\nHere we need to add the explicit centrifugal barrier. Note that the variable $r$ depends only on time. There is no $x$ and $y$ directions\nsince we have transformed the equations to polar coordinates.\n\n\n```\nDeltaT = 0.01\n#set up arrays \ntfinal = 10.0\nn = ceil(tfinal/DeltaT)\n# set up arrays for t, v and r\nt = np.zeros(n)\nv = np.zeros(n)\nr = np.zeros(n)\nE = np.zeros(n)\n# Constants of the model\nAngMom = 1.0 # The angular momentum\nm = 1.0\nk = 1.0\nomega02 = k/m\nc1 = AngMom*AngMom/(m*m)\nc2 = AngMom*AngMom/m\nrmin = (AngMom*AngMom/k/m)**0.25\n# Initial conditions\nr0 = rmin\nv0 = 0.0\nr[0] = r0\nv[0] = v0\nE[0] = 0.5*m*v0*v0+0.5*k*r0*r0+0.5*c2/(r0*r0)\n# Start integrating using the Velocity-Verlet method\nfor i in range(n-1):\n # Set up acceleration\n a = -r[i]*omega02+c1/(r[i]**3) \n # update velocity, time and position using the Velocity-Verlet method\n r[i+1] = r[i] + DeltaT*v[i]+0.5*(DeltaT**2)*a\n anew = -r[i+1]*omega02+c1/(r[i+1]**3)\n v[i+1] = v[i] + 0.5*DeltaT*(a+anew)\n t[i+1] = t[i] + DeltaT\n E[i+1] = 0.5*m*v[i+1]*v[i+1]+0.5*k*r[i+1]*r[i+1]+0.5*c2/(r[i+1]*r[i+1])\n # Plot position as function of time\nfig, ax = plt.subplots(2,1)\nax[0].set_xlabel('time')\nax[0].set_ylabel('radius')\nax[0].plot(t,r)\nax[1].set_xlabel('time')\nax[1].set_ylabel('Energy')\nax[1].plot(t,E)\nsave_fig(\"RadialHOVV\")\nplt.show()\n```\n\nWith some work using double angle formulas, one can calculate\n\n$$\n\\begin{eqnarray*}\nr^2&=&x^2+y^2\\\\\n\\nonumber\n&=&(A^2+C^2)\\cos^2(\\omega_0t)+(B^2+D^2)\\sin^2\\omega_0t+(AB+CD)\\cos(\\omega_0t)\\sin(\\omega_0t)\\\\\n\\nonumber\n&=&\\alpha+\\beta\\cos 2\\omega_0 t+\\gamma\\sin 2\\omega_0 t,\\\\\n\\alpha&=&\\frac{A^2+B^2+C^2+D^2}{2},~~\\beta=\\frac{A^2-B^2+C^2-D^2}{2},~~\\gamma=AB+CD,\\\\\nr^2&=&\\alpha+(\\beta^2+\\gamma^2)^{1/2}\\cos(2\\omega_0 t-\\delta),~~~\\delta=\\arctan(\\gamma/\\beta),\n\\end{eqnarray*}\n$$\n\nand see that radius oscillates with frequency $2\\omega_0$. The\nfactor of two comes because the oscillation $x=A\\cos\\omega_0t$ has two\nmaxima for $x^2$, one at $t=0$ and one a half period later.\n\n## Stability of Orbits\n\nThe effective force can be extracted from the effective potential, $V_{\\rm eff}$. Beginning from the equations of motion, Eq. ([2](#eq:radialeqofmotion)), for $r$,\n\n$$\n\\begin{eqnarray}\nm\\ddot{r}&=&F+\\frac{L^2}{mr^3}\\\\\n\\nonumber\n&=&F_{\\rm eff}\\\\\n\\nonumber\n&=&-\\partial_rV_{\\rm eff},\\\\\n\\nonumber\nF_{\\rm eff}&=&-\\partial_r\\left[V(r)+(L^2/2mr^2)\\right].\n\\end{eqnarray}\n$$\n\nFor a circular orbit, the radius must be fixed as a function of time,\nso one must be at a maximum or a minimum of the effective\npotential. However, if one is at a maximum of the effective potential\nthe radius will be unstable. For the attractive Coulomb force the\neffective potential will be dominated by the $-\\alpha/r$ term for\nlarge $r$ because the centrifugal part falls off more quickly, $\\sim\n1/r^2$. At low $r$ the centrifugal piece wins and the effective\npotential is repulsive. Thus, the potential must have a minimum\nsomewhere with negative potential. The circular orbits are then stable\nto perturbation.\n\nThe effective potential is sketched for two cases, a $1/r$ attractive\npotential and a $1/r^3$ attractive potential. The $1/r$ case has a\nstable minimum, whereas the circular orbit in the $1/r^3$ case is\nunstable.\n\nIf one considers a potential that falls as $1/r^3$, the situation is\nreversed and the point where $\\partial_rV$ disappears will be a local\nmaximum rather than a local minimum. **Fig to come here with code**\n\nThe repulsive centrifugal piece dominates at large $r$ and the attractive\nCoulomb piece wins out at small $r$. The circular orbit is then at a\nmaximum of the effective potential and the orbits are unstable. It is\nthe clear that for potentials that fall as $r^n$, that one must have\n$n>-2$ for the orbits to be stable.\n\nConsider a potential $V(r)=\\beta r$. For a particle of mass $m$ with\nangular momentum $L$, find the angular frequency of a circular\norbit. Then find the angular frequency for small radial perturbations.\n\nFor the circular orbit you search for the position $r_{\\rm min}$ where the effective potential is minimized,\n\n$$\n\\begin{eqnarray*}\n\\partial_r\\left\\{\\beta r+\\frac{L^2}{2mr^2}\\right\\}&=&0,\\\\\n\\beta&=&\\frac{L^2}{mr_{\\rm min}^3},\\\\\nr_{\\rm min}&=&\\left(\\frac{L^2}{\\beta m}\\right)^{1/3},\\\\\n\\dot{\\theta}&=&\\frac{L}{mr_{\\rm min}^2}=\\frac{\\beta^{2/3}}{(mL)^{1/3}}\n\\end{eqnarray*}\n$$\n\nNow, we can find the angular frequency of small perturbations about the circular orbit. To do this we find the effective spring constant for the effective potential,\n\n$$\n\\begin{eqnarray*}\nk_{\\rm eff}&=&\\partial_r^2 \\left.V_{\\rm eff}\\right|_{r_{\\rm min}}\\\\\n&=&\\frac{3L^2}{mr_{\\rm min}^4},\\\\\n\\omega&=&\\sqrt{\\frac{k_{\\rm eff}}{m}}\\\\\n&=&\\frac{\\beta^{2/3}}{(mL)^{1/3}}\\sqrt{3}.\n\\end{eqnarray*}\n$$\n\nIf the two frequencies, $\\dot{\\theta}$ and $\\omega$, differ by an\ninteger factor, the orbit's trajectory will repeat itself each time\naround. This is the case for the inverse-square force,\n$\\omega=\\dot{\\theta}$, and for the harmonic oscillator,\n$\\omega=2\\dot{\\theta}$. In this case, $\\omega=\\sqrt{3}\\dot{\\theta}$,\nand the angles at which the maxima and minima occur change with each\norbit.\n\n### Code example with gravitional force\n\nThe code example here is meant to illustrate how we can make a plot of the final orbit. We solve the equations in polar coordinates (the example here uses the minimum of the potential as initial value) and then we transform back to cartesian coordinates and plot $x$ versus $y$. We see that we get a perfect circle when we place ourselves at the minimum of the potential energy, as expected.\n\n\n```\n\n# Simple Gravitational Force -alpha/r\n \nDeltaT = 0.01\n#set up arrays \ntfinal = 8.0\nn = ceil(tfinal/DeltaT)\n# set up arrays for t, v and r\nt = np.zeros(n)\nv = np.zeros(n)\nr = np.zeros(n)\nphi = np.zeros(n)\nx = np.zeros(n)\ny = np.zeros(n)\n# Constants of the model, setting all variables to one for simplicity\nalpha = 1.0\nAngMom = 1.0 # The angular momentum\nm = 1.0 # scale mass to one\nc1 = AngMom*AngMom/(m*m)\nc2 = AngMom*AngMom/m\nrmin = (AngMom*AngMom/m/alpha)\n# Initial conditions, place yourself at the potential min\nr0 = rmin\nv0 = 0.0 # starts at rest\nr[0] = r0\nv[0] = v0\nphi[0] = 0.0\n# Start integrating using the Velocity-Verlet method\nfor i in range(n-1):\n # Set up acceleration\n a = -alpha/(r[i]**2)+c1/(r[i]**3)\n # update velocity, time and position using the Velocity-Verlet method\n r[i+1] = r[i] + DeltaT*v[i]+0.5*(DeltaT**2)*a\n anew = -alpha/(r[i+1]**2)+c1/(r[i+1]**3)\n v[i+1] = v[i] + 0.5*DeltaT*(a+anew)\n t[i+1] = t[i] + DeltaT\n phi[i+1] = t[i+1]*c2/(r0**2)\n# Find cartesian coordinates for easy plot \nx = r*np.cos(phi)\ny = r*np.sin(phi)\nfig, ax = plt.subplots(3,1)\nax[0].set_xlabel('time')\nax[0].set_ylabel('radius')\nax[0].plot(t,r)\nax[1].set_xlabel('time')\nax[1].set_ylabel('Angle $\\cos{\\phi}$')\nax[1].plot(t,np.cos(phi))\nax[2].set_ylabel('y')\nax[2].set_xlabel('x')\nax[2].plot(x,y)\n\nsave_fig(\"Phasespace\")\nplt.show()\n```\n\nTry to change the initial value for $r$ and see what kind of orbits you get.\nIn order to test different energies, it can be useful to look at the plot of the effective potential discussed above.\n\nHowever, for orbits different from a circle the above code would need modifications in order to allow us to display say an ellipse. For the latter, it is much easier to run our code in cartesian coordinates, as done here. In this code we test also energy conservation and see that it is conserved to numerical precision. The code here is a simple extension of the code we developed for homework 4.\n\n\n```\n# Common imports\nimport numpy as np\nimport pandas as pd\nfrom math import *\nimport matplotlib.pyplot as plt\n\nDeltaT = 0.01\n#set up arrays \ntfinal = 10.0\nn = ceil(tfinal/DeltaT)\n# set up arrays\nt = np.zeros(n)\nv = np.zeros((n,2))\nr = np.zeros((n,2))\nE = np.zeros(n)\n# Constants of the model\nm = 1.0 # mass, you can change these\nalpha = 1.0\n# Initial conditions as compact 2-dimensional arrays\nx0 = 0.5; y0= 0.\nr0 = np.array([x0,y0]) \nv0 = np.array([0.0,1.0])\nr[0] = r0\nv[0] = v0\nrabs = sqrt(sum(r[0]*r[0]))\nE[0] = 0.5*m*(v[0,0]**2+v[0,1]**2)-alpha/rabs\n# Start integrating using the Velocity-Verlet method\nfor i in range(n-1):\n # Set up the acceleration\n rabs = sqrt(sum(r[i]*r[i]))\n a = -alpha*r[i]/(rabs**3)\n # update velocity, time and position using the Velocity-Verlet method\n r[i+1] = r[i] + DeltaT*v[i]+0.5*(DeltaT**2)*a\n rabs = sqrt(sum(r[i+1]*r[i+1]))\n anew = -alpha*r[i+1]/(rabs**3)\n v[i+1] = v[i] + 0.5*DeltaT*(a+anew)\n E[i+1] = 0.5*m*(v[i+1,0]**2+v[i+1,1]**2)-alpha/rabs\n t[i+1] = t[i] + DeltaT\n# Plot position as function of time\nfig, ax = plt.subplots(3,1)\nax[0].set_ylabel('y')\nax[0].set_xlabel('x')\nax[0].plot(r[:,0],r[:,1])\nax[1].set_xlabel('time')\nax[1].set_ylabel('y position')\nax[1].plot(t,r[:,0])\nax[2].set_xlabel('time')\nax[2].set_ylabel('y position')\nax[2].plot(t,r[:,1])\n\nfig.tight_layout()\nsave_fig(\"2DimGravity\")\nplt.show()\nprint(E)\n```\n\nCentral forces are forces which are directed towards or away from a\nreference point. A familiar force is the gravitional\nforce with the motion of our Earth around the Sun as a classic. The Sun, being\napproximately sixth order of magnitude heavier than the Earth serves\nas our origin. A force like the gravitational force is a function of the\nrelative distance $\\boldsymbol{r}=\\boldsymbol{r}_1-\\boldsymbol{r}_2$ only, where \n$\\boldsymbol{r}_1$ and $\\boldsymbol{r}_2$ are the positions relative to a defined\norigin for object one and object two, respectively.\n\nThese forces depend on the spatial degrees of freedom only (the\npositions of the interacting objects/particles). As discussed earlier, from such forces we can infer\nthat the total internal energy, the total linear momentum and total angular momentum are so-called constants of the motion, that is they stay constant over time. We say that energy, linear and anuglar momentum are conserved.\n\nWith a scalar potential $V(\\boldsymbol{r})$ we define the force as the gradient of the potential\n\n$$\n\\boldsymbol{F}(\\boldsymbol{r})=-\\boldsymbol{\\nabla}V(\\boldsymbol{r}).\n$$\n\nIn general these potentials depend only on the magnitude of the\nrelative position and we will write the potential as $V(r)$ where $r$\nis defined as,\n\n$$\nr = |\\boldsymbol{r}_1-\\boldsymbol{r}_2|.\n$$\n\nIn three dimensions our vectors are defined as (for a given object/particle $i$)\n\n$$\n\\boldsymbol{r}_i = x_i\\boldsymbol{e}_1+y_i\\boldsymbol{e}_2+z_i\\boldsymbol{e}_3,\n$$\n\nwhile in two dimensions we have\n\n$$\n\\boldsymbol{r}_i = x_i\\boldsymbol{e}_1+y_i\\boldsymbol{e}_2.\n$$\n\nIn two dimensions the radius $r$ is defined as\n\n$$\nr = |\\boldsymbol{r}_1-\\boldsymbol{r}_2|=\\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}.\n$$\n\nIf we consider the gravitational potential involving two masses $1$ and $2$, we have\n\n$$\nV_{12}(r)=V(r)=-\\frac{Gm_1m_2}{|\\boldsymbol{r}_1-\\boldsymbol{r}_2|}=-\\frac{Gm_1m_2}{r}.\n$$\n\nCalculating the gradient of this potential we obtain the force\n\n$$\n\\boldsymbol{F}(\\boldsymbol{r})=-\\frac{Gm_1m_2}{|\\boldsymbol{r}_1-\\boldsymbol{r}_1|^2}\\hat{\\boldsymbol{r}}_{12}=-\\frac{Gm_am_b}{r^2}\\hat{\\boldsymbol{r}},\n$$\n\nwhere we have the unit vector\n\n$$\n\\hat{\\boldsymbol{r}}=\\hat{\\boldsymbol{r}}_{12}=\\frac{\\boldsymbol{r}_2-\\boldsymbol{r}_1}{|\\boldsymbol{r}_1-\\boldsymbol{r}_2|}.\n$$\n\nHere $G=6.67\\times 10^{-11}$ Nm$^2$/kg$^2$, and $\\boldsymbol{F}$ is the force\non $2$ due to $1$. By inspection, one can see that the force on $2$\ndue to $1$ and the force on $1$ due to $2$ are equal and opposite. The\nnet potential energy for a large number of masses would be\n\n$$\nV=\\sum_{i\n
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Functions

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Named sequences of statements that performs some useful operations.

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\n\n\n- [Overview](#Overview)\n- [Function calls](#Function-calls)\n- [Composition](#Composition)\n- [Creating new functions](#Creating-new-functions)\n- [Flow of execution](#Flow-of-execution)\n- [Function arguments](#Function-arguments)\n- [Function namespace](#Function-namespace)\n- [Return objects](#Return-objects)\n- [Incremental development](#Incremental-development)\n- [Notebook namespace](#Notebook-namespace)\n- [Why functions?](#Why-functions?)\n- [Debugging](#Debugging)\n- [Glossary](#Glossary)\n- [Exercises](#Exercises)\n\n## Overview\n\nIn the context of programming, a **function** is a named sequence of programming statements that performs a specific task or computation when called. When you define a function, you specify the name of the function and the sequence of statements that get executed when the function is called. Functions can be called, by name, from other programs or from other functions.\n\n## Function calls\n\nWe have already seen **function calls** (the use of functions) in previous chapters, for example the ``radians`` function:\n\n\n```python\nfrom numpy import radians\n\nradians(90)\n```\n\nThe name of this function is ``radians`` and it was defined in the ``numpy`` module. This function is given a single argument (``90`` degrees) and returns the equivalent angle in radians. It is common to say that a function \"takes\" an argument and \"returns\" a result. The result is also called the return object.\n\n
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Take Note

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It is the parenthese - () - after the function name that instructs Python to call the function. The expressions or objects inside the parenthese are the arguments to the function. Omitting the parenthese is interpreted by Python as \"lookup the object bound to the variable name radians\". Consider the output from running the following Code Cell:

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\n\n\n```python\nfrom numpy import radians\n\nradians\n```\n\nIn the example above we do not know how the ``radians`` function works or what statements are inside this function, we simply take for granted that it works and will correctly return the equivalent angle in radians for an input angle in degrees.\n\nFor the sake of interest we can verify that this ``radians`` function works correctly, by manually converting ``90`` degrees to radians and comparing the answer to that of the ``radians`` function:\n\n\n```python\nfrom numpy import pi\n\n90 * pi / 180\n```\n\n## Composition\n\nSo far, we have looked at the elements of a program - variables, objects, expressions, and statements - in isolation, without talking much about how to combine them. One of the most useful features of programming languages is their ability to take small building blocks and **compose** them. For example, the arguments of a function can be any kind of expression, including arithmetic operators or even other function calls:\n\n\n```python\nimport numpy as np\n\ndegrees = 90\nx = np.sin(degrees / 360 * 2 * np.pi)\nprint(x)\n\nx = np.exp(np.log(x + 1))\nprint(x)\n```\n\n
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Take Note

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The expression inside the function parentheses, for example (degrees / 360 \\* 2 \\* np.pi), is evaluated first and the result thereof is sent as an argument to the function np.sin. Almost anywhere you can put a value, you can put an arbitrary expression.

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Keep in mind that everthing on the right hand side of the assignment (numbers, variables, objects, etc.) must be \"known\" (can be computed). Therefore, make sure that a name has been bound to an object before you use that name on the right of the assignment (equal sign).

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Only a variable name is allowed on the left hand side of the equal sign. Any other expression on the left hand side, including function calls, gives a syntax error, for example:

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\n\n\n```python\nimport numpy as np\n\ndegrees = 90\nnp.sin(degrees / 360 * 2 * np.pi) = x\nprint(x)\n```\n\n## Creating new functions\n\nProgramming would be a very difficult task if we could only make use of existing functions.\nNot only would our programs become very long and difficult to keep track of, but it would also be difficult for a group of programmers to develop different components of a new program simultaneously.\nSo far, we have only been using the functions that come with *Python* or can be found in the ``numpy`` module.\nFortunately, almost all programming languages allow for the creating of new functions.\n\n### Example - Tongue twister\n\nAs an example, lets create a function that prints a tongue twister to the screen:\n\n\n```python\ndef print_tongue_twister():\n print('Fuzzy Wuzzy was a bear.')\n print('Fuzzy Wuzzy had no hair.')\n print(\"Fuzzy Wuzzy wasn't fuzzy, was he?\")\n```\n\nThe **function definition** specifies the name of the new function (``print_tongue_twister``) and the sequence of programming statements that run when the function is called.\n\nThe ``def`` keyword is used to define the new function with a specific function name (in this case ``print_tongue_twister``). The empty parentheses, after the function name, indicates that this function does not take any arguments. The first line of the function definition is called the **function header** and the rest is called the **function body**. The function body can contain any number of programming statements.\n\n
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Take Note

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The rules for function names are the same as for variable names: Letters, numbers and underscores are legal, but the first character can't be a number. You should not use a keyword as either a function name or variable name, and you should avoid having a variable and a function with the same name.

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The function header must end with a double colon (:). This double colon instructs Python where the statements inside the function body start. If the double colon is left out you will see an error message. Try it out - delete the double colon in the Code Cell above and run it to see the error message.

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The function body must be indented. It is this indentation that instructs Python as to which statements belong to the function body. Indentation means the amount of white space placed in front of your code. *Python* uses 4 spaces as 1 indentation.

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Executing the Code Cell above only defines the function and does not give any output. This is because the statements in the function body are not executed until the function is called. The function definition must be executed first, to define the function, before the function can be called, otherwise you get a NameError.

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More Information

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Most editors (like *Jupyter Notebook*) will automatically add 4 spaces when you push the `tab` key. And `Shift + tab` will automatically remove 4 spaces. This can also be used if you have highlighted several lines of code. Try it !!

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The strings in the print statements are enclosed in single and double quotes. Single quotes and double quotes do the same thing; most people use single quotes except in cases like this where a single quote (which is also an apostrophe) appears in the string.

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All quotation marks (single and double) must be \u201cstraight quotes\u201d, usually located next to Enter on the keyboard. \u201cCurly quotes\u201d, like the ones in this sentence, are not legal in Python.

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For more information on valid names, see the PEP8 standard: http://www.python.org/dev/peps/pep-0008/

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To see a list of keywords, that should not be used as function names, execute help('keywords') in a Code Cell.

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\n\nExecuting the *Code Cell* above (with the ``print_tongue_twister`` function) only defines the function (no output is given). Defining a function creates a **function object**, with a specified name, which has type ``function`` (note the output of the following two *Code Cell*):\n\n\n```python\nprint(print_tongue_twister)\n```\n\n\n```python\ntype(print_tongue_twister)\n```\n\nThe syntax for calling this new function is the same as for built-in functions (keeping in mind this function does not take any arguments):\n\n\n```python\nprint_tongue_twister()\n```\n\nOnce a function has been defined, it can be called (as shown in the *Code Cell* above) or it can be used inside another function. For example, to repeat the tongue twister, we could write a function called ``repeat_twister``:\n\n\n```python\ndef repeat_twister():\n print_tongue_twister()\n print_tongue_twister()\n```\n\nAnd then call this ``repeat_twister`` function:\n\n\n```python\nrepeat_twister()\n```\n\nPulling together the code fragments above, the whole program looks like this:\n\n\n```python\n%reset -f\n\ndef print_tongue_twister():\n print(\"Fuzzy Wuzzy was a bear.\")\n print(\"Fuzzy Wuzzy had no hair.\")\n print(\"Fuzzy Wuzzy wasn't fuzzy, was he?\")\n\n\ndef repeat_twister():\n print_tongue_twister()\n print_tongue_twister()\n\n\nrepeat_twister()\n```\n\n
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Take Note

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This program contains two function definitions: print_tongue_twister and repeat_twister. Function definitions get executed just like other statements, but the effect is to create function objects. The statements inside the function do not run until the function is called, and the function definition generates no output.

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As you might expect, you have to create a function before you can run it. In other words, the function definition has to run before the function gets called.

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As an exercise, move the last line (line 14) of this program to the top (just after the %reset -f line), so the function call appears before the definitions. Run the Code Cell and see what error message you get.

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Now move the function call back to the bottom and move the definition of print_tongue_twister after the definition of repeat_twister. What happens when you run this program and why?

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\n\n## Flow of execution\n\nTo ensure that a function is defined before its first use, you have to know the order statements run in, which is called the flow of execution. Execution always begins at the first statement of the program. Statements are run one at a time, in order from top to bottom.\n\nFunction definitions do not alter the flow of execution of the program, but remember that statements inside the function don\u2019t run until the function is called.\n\nA function call is like a detour in the flow of execution. Instead of going to the next statement, the flow jumps to the body of the function, runs the statements there, and then comes back to pick up where it left off.\n\nThat sounds simple enough, until you remember that one function can call another. While in the middle of one function, the program might have to run the statements in another function. Then, while running that new function, the program might have to run yet another function!\n\nFortunately, *Python* is good at keeping track of where it is, so each time a function completes, the program picks up where it left off in the function that called it. When it gets to the end of the program, it terminates.\n\nIn summary, when you read a program, you don\u2019t always want to read from top to bottom. Sometimes it makes more sense if you follow the flow of execution as if you were *Python*.\n\nThere is also the added complexity of *Code Cells* and the order in which you execute these *Code Cells* that will be explained in the following sections. For now, execute the following two *Code Cells* and step through the line-by-line execution to ensure you understand the explaination given above:\n\n\n```python\n%load_ext nbtutor\n```\n\n\n```python\n%%nbtutor -r -f --depth 3\n\ndef print_tongue_twister():\n print(\"Fuzzy Wuzzy was a bear.\")\n print(\"Fuzzy Wuzzy had no hair.\")\n print(\"Fuzzy Wuzzy wasn't fuzzy, was he?\")\n\n\ndef repeat_twister():\n print_tongue_twister()\n print_tongue_twister()\n\n\nrepeat_twister()\n```\n\n## Function arguments\n\nIn the tongue twister example above, none on the functions we created required any arguments, but many of the functions we have seen before do require arguments. For example, when we call the ``numpy.radians`` function we passed a single number as an argument.\n\nInside the function, the arguments are assigned to variable names called parameters. Here is a definition for a function that takes a single argument:\n\n\n```python\ndef print_twice(thing):\n print(thing)\n print(thing)\n```\n\nWhen this function is called, the single argument is assigned to the parameter named ``thing`` and the value of the object (whatever is is) is printed twice:\n\n\n```python\nprint_twice(\"Hello\")\n```\n\n\n```python\nprint_twice(42)\n```\n\nThe same rules of composition that apply to built-in functions also apply to programmer-defined functions, so we can use any kind of expression as an argument for ``print_twice``:\n\n\n```python\nimport numpy as np\n\ndegrees = 90\nprint_twice(np.sin(np.radians(degrees)))\n```\n\n
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Take Note

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The argument is evaluated before the function is called, so in the example above the expression np.sin(np.radians(degrees)) is only evaluated once and the result thereof is sent as input to the print_twice function and assigned to the parameter named thing.

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The variable name we pass as an argument has nothing to do with the parameter name (thing). It does not matter what the object was called \"back home\" (in the caller); in print_twice, everybody is called thing. For example consider the following Code Cell:

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\n\n\n```python\n%%nbtutor -r -f \ndef print_twice(thing):\n print(thing)\n print(thing)\n\n\nbird = \"Robin\"\nprint_twice(bird)\n```\n\nA function can take as many arguments as needed for the specific task or computation it is designed for. Here is a definition for a function that takes two arguments:\n\n\n```python\ndef print_cat(part1, part2):\n print(part1 + part2)\n```\n\nThis function prints the result of adding two parameters together. The order of the arguments passed in to the function is mapped to the order of the parameters, for example:\n\n\n```python\none = \"Here's\"\ntwo = \" Johnny\"\n\nprint_cat(one, two)\nprint_cat(two, one)\n```\n\n## Function namespace\n\nEach function gets its own namespace where variables and parameters may exist / get created. For example, when you create a variable inside a function, it is local, which means that it only exists inside that function namespace. Function parameters are also local, which means they only exist inside the function namespace where they were created.\n\n\n```python\n%%nbtutor -r -f --depth 3\ndef print_twice(thing):\n print(thing)\n print(thing)\n\n\ndef cat_twice(part1, part2):\n cat = part1 + part2\n print_twice(cat)\n\n\nthis = \"Bing tiddle \"\nthat = \"tiddle bang.\"\ncat_twice(this, that)\n```\n\nRunning the example above (using ``nbtutor`` to visualize the execution) will show how each function gets its own namespace. The ``Global frame`` is the namespace that is global to this notebook, I.e. global variables and function definitions get added to the notebook namespace. Function parameters and local function variables get created only in the function namespace where they were specifed.\n\nStepping through the example above you should notice that the function parameters ``part1`` and ``part2`` and the variable ``cat`` only exist inside the ``cat_twice`` function namespace. The function parameter ``thing`` only exists inside the ``print_twice`` function namespace.\n\nOnce the ``print_twice`` function finishes running (terminates), the ``thing`` parameter no longer exists (I.e. the ``print_twice`` namespace is deleted along with its variables and parameters). Once the ``cat_twice`` function terminates, the ``part`` and ``part2`` parameters and the ``cat`` variable no longer exist.\n\nFor example, if we tried to print the variable ``cat``, we would get an error:\n\n\n```python\nprint(cat)\n```\n\n
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Take Note

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When a function is called, a new namespace is created where local variable names and parameters are created. Once the function terminates (finishes) this namespace is deleted along with the local variable names and parameters. Only the variable and paramater names are deleted, not the objects. The objects only get deleted if there are no names pointing to them anymore.

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Objects send as arguments to functions are not copied! Consider the example above and take note of how parameters and/or variables in different namespaces point to the same objects.

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\n\n## Return objects\n\nWhen calling a function that generates a return object, we usually assign that object to a variable or use it as part of an expression:\n\n\n```python\nimport numpy as np\n\nradius = 1\ne = np.exp(1.0)\nradians = np.radians(90)\nheight = radius * np.sin(radians)\n```\n\nMany of the functions we have used, such as the ``numpy`` functions, return objects. Other functions, like ``print_twice``, perform an action but do not return any objects. The truth is, in *Python*, every function returns something to the caller. If we do not specify what the function should return, by default it will return ``None`` (a special object for \"nothing\"). For example, in the ``print_twice`` function we did not specify what this function should return:\n\n\n```python\ndef print_twice(thing):\n print(thing)\n print(thing)\n```\n\nAnd if we use this function, expecting it to return something (similar to how the ``numpy.radians`` function returned an answer) we see that ``None`` is returned:\n\n\n```python\nsomething = print_twice(\"Hello\")\nprint(something)\n```\n\nThe object ``None`` is not the same as the string ``'None'`` . It is a special object that has its own type:\n\n\n```python\ntype(None)\n```\n\n### Example - Circle area\n\nAs an example, lets create a function that computes and returns the area of a circle from a given radius:\n\n\n```python\nimport numpy as np\n\ndef area(radius):\n ans = np.pi * radius**2\n return ans\n```\n\nThe ``return`` statement means: \u201cReturn immediately from this function and use the following expression as a return object.\u201d\n\n
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Take Note

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The return statement tells Python what object/s must be returned from the function as output. The return statement exits the function immediately and no code (in the function body) after the return statement will be executed.

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\n\n The expression (following the ``return`` keyword) can be arbitrarily complicated, so we could have written this function more concisely:\n\n\n```python\nimport numpy as np\n\ndef area(radius):\n return np.pi * radius**2\n```\n\nOn the other hand, temporary variables like ``ans`` can make debugging easier.\n\n## Incremental development\n\nAs you write larger functions, you might find yourself spending more time debugging. To deal with increasingly complex programs, you might want to try a process called incremental development. The goal of incremental development is to avoid long debugging sessions by adding and testing only a small amount of code at a time.\n\n### Example - Distance\n\nAs an example, suppose you want to find the distance between two points, given by the coordinates $(x_1, y_1)$ and $(x_2, y_2)$. By the Pythagorean theorem, the distance is:\n$$\n\\text{distance} = \\sqrt{(x_2 \u2212 x_1)^2 + (y_2 \u2212 y_1)^2}\n$$\n\nThe first step is to consider what a distance function should look like in *Python*. In other words, what are the arguments (parameters) and what is the output (return object)?\n\nIn this case, the arguments are the two points, which you can represent using four numbers. The return object is the distance represented by a floating-point number.\n\nImmediately you can write an outline of the function:\n\n\n```python\ndef distance(x1, y1, x2, y2):\n return 0.0\n```\n\nObviously, this version doesn\u2019t compute distances; it always returns zero. But it is syntactically correct, and it runs, which means that you can test it before you make it more complicated. To test the new function, call it with sample arguments:\n\n\n```python\ndistance(1, 2, 4, 6)\n```\n\nI chose these values so that the horizontal distance is 3 and the vertical distance is 4; that way, the result is 5, the hypotenuse of a 3-4-5 triangle. When testing a function, it is useful to know the right answer.\n\nAt this point we have confirmed that the function is syntactically correct, and we can start adding code to the body. A reasonable next step is to find the differences $x_2 \u2212 x_1$ and $y_2 \u2212 y_1$.\n\nThe next version stores those values in temporary variables and prints them:\n\n\n```python\ndef distance(x1, y1, x2, y2):\n dx = x2 - x1\n dy = y2 - y1\n print('dx is', dx)\n print('dy is', dy)\n return 0.0\n```\n\nIf the function is working, it should display ``dx`` is 3 and ``dy`` is 4 . If so, we know that the function is getting the right arguments and performing the first computation correctly. If not, there are only a few lines to check.\n\n\n```python\ndistance(1, 2, 4, 6)\n```\n\nNext we compute the sum of squares of ``dx`` and ``dy``:\n\n\n```python\ndef distance(x1, y1, x2, y2):\n dx = x2 - x1\n dy = y2 - y1\n dsquared = dx**2 + dy**2\n print('dsquared is: ', dsquared)\n return 0.0\n```\n\nAgain, you would run the program at this stage and check the output (which should be\n25).\n\n\n```python\ndistance(1, 2, 4, 6)\n```\n\n Finally, you can use ``np.sqrt`` to compute and return the result:\n\n\n```python\nimport numpy as np\n\ndef distance(x1, y1, x2, y2):\n dx = x2 - x1\n dy = y2 - y1\n dsquared = dx**2 + dy**2\n result = np.sqrt(dsquared)\n return result\n```\n\nIf that works correctly, you are done. Otherwise, you might want to print the value of result before the return statement.\n\n\n```python\ndistance(1, 2, 4, 6)\n```\n\n
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The final version of the function doesn\u2019t display anything when it runs; it only returns an object. The print statements we wrote are useful for debugging, but once you get the function working, you should remove them. Code like that is called scaffolding because it is helpful for building the program but is not part of the final product.

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\n\nWhen you start out, you should add only a line or two of code at a time. As you gain more experience, you might find yourself writing and debugging bigger chunks. Either way, incremental development can save you a lot of debugging time.\n\nThe key aspects of the process are:\n1. Start with a working program and make small incremental changes. At any point, if there is an error, you should have a good idea where it is.\n2. Use variables to hold intermediate objects so you can display and check them.\n3. Once the program is working, you might want to remove some of the scaffolding or consolidate multiple statements into compound expressions, but only if it does not make the program difficult to read.\n\nAs an exercise, use incremental development to write a function called ``hypotenuse`` that returns the length of the hypotenuse of a right triangle given the lengths of the other two legs as arguments. Record each stage of the development process as you go.\n\n### Example - Circle area\n\nIncremental development not only helps from a single function point of view, but also from a function structure point of view. As an example, lets write a function that takes two points, the center of a circle and a point on the perimeter, and computes the area of the circle.\n\nAssume that the center point is stored in the variables ``xc`` and ``yc``, and the perimeter point is in ``xp`` and ``yp``. The first step is to find the radius of the circle, which is the distance between the two points. We just wrote a function, distance, that does that:\n```python\nradius = distance(xc, yc, xp, yp)\n```\n\nThe next step is to find the area of a circle with that radius; we just wrote that, too:\n```python\nresult = area(radius)\n```\n\nEncapsulating these steps in a function, we get:\n\n\n```python\ndef circle_area(xc, yc, xp, yp):\n radius = distance(xc, yc, xp, yp)\n result = area(radius)\n return result\n```\n\n
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Take Note

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It is often useful, when you come to a function call, instead of following the flow of execution, you assume that the function works correctly and returns the right result.

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In fact, you are already practicing this leap of faith when you use built-in functions. When you call ``numpy.cos`` or ``numpy.exp``, you don\u2019t examine the bodies of those functions. You just assume that they work because the people who wrote the built-in functions were good programmers.

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The same is true when you call one of your own functions. Once we have convinced ourselves that this function is correct \u2014 by examining the code and testing \u2014 we can use the function without looking at the body again.

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\n\nThe temporary variables ``radius`` and ``result`` are useful for development and debugging, but once the program is working, we can make it more concise by composing the function calls:\n\n\n```python\ndef circle_area(xc, yc, xp, yp):\n radius = distance(xc, yc, xp, yp)\n return area(radius)\n```\n\nPutting the pieces together in one *Code Cell*, we have:\n\n\n```python\nimport numpy as np\n\ndef area(radius):\n return np.pi * radius**2\n\n\ndef distance(x1, y1, x2, y2):\n return np.sqrt((x2 - x1)**2 + (y2 - y1)**2)\n\n\ndef circle_area(xc, yc, xp, yp):\n radius = distance(xc, yc, xp, yp)\n return area(radius)\n\n\nresult = circle_area(1, 2, 4, 6)\nprint(result)\n```\n\n## Notebook namespace\n\nAs mentioned previously, each function gets its own namespace where variables and parameters may exist / get created. For example, when you create a variable inside a function, it is local, which means that it only exists inside that function namespace. There is also a global (or Notebook) namespace. Global variables and function definitions get added to this Notebook namespace and are accessible from inside local function namespaces.\n\nAll variables and function definitions that exist in the Notebook namespace can be displayed by executing ``%whos`` in a *Code Cell*, for example:\n\n\n```python\n%whos\n```\n\nThe Notebook namespace can be cleared (I.e. all variables and function definitions be deleted) by executing ``%reset`` in a *Code Cell*, for example:\n\n\n```python\n%reset\n```\n\n
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Take Note

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After executing the *Code Cell* above with the ``%reset`` statement, re-execute the *Code Cell* above with the ``%whos`` statement to see how the Notebook namespace was cleared.

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\n\n### Example - Falling object\n\nAs an example, lets consider the example of an object falling from a height above the ground.\nThe equations describing the motion of this object, assuming constant acceleration, are:\n\n$$ \n\\begin{align}\n a(t) &= g \\: \\text{(constant)} \\\\\n v(t) &= \\int a(t) \\: \\mathrm{d}t = v_0 + gt \\\\\n s(t) &= \\int v(t) \\: \\mathrm{d}t = s_0 + v_0t + 0.5 gt^2\n\\end{align}\n$$\n\nwhere $g$ is the gravitationaly acceleration $[m/s^2]$, $v_0$ the initial object velocity $[m/s]$, $s_0$ the initial object height $[m]$ above the ground, and $t$ is the time $[s]$.\n\nThe following function (named ``height``) computes the objects height $s(t)$ above the ground for any initial height $s_0$, velocity $v_0$, and time $t$ values:\n\n\n```python\ndef height(s0, v0, t):\n g = -9.81\n st = s0 + v0 * t + 0.5 * g * t**2\n return st\n```\n\nWhen creating new functions, you as the programmer will decide what inputs are needed for the function and what value/s the function returns. In this example, the equation we are using requires four variables in order to solve for the height $s(t)$, these variable being the initial height $s_0$, initial velocity $v_0$, gravitational acceleration $g$, and time $t$.\n\nI decided that the ``height`` function should take the initial height $s_0$, initial velocity $v_0$, and the time $t$ values as input as these are likely to change. I also decided the gravitational acceleration $g$ would likely remain constant and as such defined $g$ as a local variable inside the ``height``function namespace.\n\nI could have decided that the gravitational acceleration $g$ should be a global variable, for example:\n\n\n```python\ndef height(s0, v0, t):\n st = s0 + v0 * t + 0.5 * g * t**2\n return st\n\n\ng = -9.81\ns = height(100, 0, 2)\nprint(s)\n```\n\nBut wait... how is this code not crashing? The statements inside the function ``height`` rely on a variable ``g`` which is not only created outside the function but also created after the function, how is this code not crashing? Try and answer this question yourself before continuing to read further.\n\n\n```python\n%%nbtutor -rf \n\ndef height(s0, v0, t):\n st = s0 + v0 * t + 0.5 * g * t**2\n return st\n\n\ng = -9.81\ns = height(100, 0, 2)\nprint(s)\n```\n\nWhen a function is called, only then will the statements inside the function be executed and only when these statements are executed will *Python* start looking for what variables are. When line 4 gets executed *Python* will first look in the local function (``height``) namespace for the variables ``s0``, ``v0``, ``t``, and ``g``. The variables ``s0``, ``v0``, and ``t`` exist in the local namespace as parameters passed in as arguments, but ``g`` does not exist in the local namespace at all. If *Python* can't find a variable in the local namespace it will then look at the next namespace above, in this case the Notebook namespace.\n\nStepping through this example line-by-line should make it clearer. The function object and name gets created in the Notebook namespace (line 3), but the statements inside the function only get executed when the function is called (line 9) and this function call is after the definition of ``g`` (line 8), so this means that when the statements inside the function get executed *Python* knows what ``g`` is because it was defined before the function call in the Notebook namespace.\n\nThis is bad programming. All variables used in a function should either be passed as arguments or should be defined inside the function.\n\nYou may be thinking to yourself, but why? In this example it makes sense to have gravity as a \"global\" constant, it is not going to change. As a hypothetical lets say you have this exact example above and now you continue to work on another example, lets say computing the numerical gradient of a function:\n\n\n```python\ndef func(x):\n return x**2 + 5\n\n\ndx = 0.0001\ny1 = func(2)\ny2 = func(2 + dx)\ng = (y2 - y1) / dx\nprint(g)\n```\n\nOh no... I've used ``g`` as a variable for the gradient, which now means that the ``height`` function will now not give me the correct result because ``g`` is no longer ``-9.81``. Thus all variables used in a function should either be passed as arguments or should be defined inside the function to avoid possible bugs like this.\n\n\n```python\ns = height(100, 0, 2)\nprint(g)\nprint(s)\n```\n\n## Why functions?\n\nIt may not be clear why it is worth the trouble to divide a program into functions. There are several reasons:\n- Creating a new function gives you an opportunity to name a group of statements, which makes your program easier to read and debug.\n- Functions can make a program smaller by eliminating repetitive code. Later, if you make a change, you only have to make it in one place.\n- Dividing a long program into functions allows you to debug the parts one at a time and then assemble them into a working whole.\n- Well-designed functions are often useful for many programs. Once you write and debug one, you can reuse it.\n\n## Debugging\n\nOne of the most important skills you will acquire is debugging. Although it can be frustrating, debugging is one of the most intellectually rich, challenging, and interesting parts of programming.\n\nIn some ways debugging is like detective work. You are confronted with clues and you have to infer the processes and events that led to the results you see.\n\nDebugging is also like an experimental science. Once you have an idea about what is going wrong, you modify your program and try again. If your hypothesis was correct, you can predict the result of the modification, and you take a step closer to a working program. If your hypothesis was wrong, you have to come up with a new one. As Sherlock Holmes pointed out, \u201cWhen you have eliminated the impossible, whatever remains, however improbable, must be the truth.\u201d (A. Conan Doyle, The Sign of Four)\n\nFor some people, programming and debugging are the same thing. That is, programming is the process of gradually debugging a program until it does what you want. The idea is that you should start with a working program and make small modifications, debugging them as you go.\n\nBreaking a large program into smaller functions creates natural checkpoints for debugging. If a function is not working, there are three possibilities to consider:\n- There is something wrong with the arguments the function is getting; a precondition is violated.\n- There is something wrong with the function; a postcondition is violated.\n- There is something wrong with the return object or the way it is being used.\n\nTo rule out the first possibility, you can add a print statement at the beginning of the function and display the values of the parameters (and maybe their types). Or you can write code that checks the preconditions explicitly.\n\nIf the parameters look good, add a print statement before each return statement and display the return object. If possible, check the result by hand. Consider calling the function with values that make it easy to check the result.\n\nIf the function seems to be working, look at the function call to make sure the return object is being used correctly (or used at all!).\n\nAdding print statements at the beginning and end of a function can help make the flow of execution more visible. It takes some time to develop effective scaffolding, but a little bit of scaffolding can save a lot of debugging.\n\n## Glossary\n\n**argument**: An object provided to a function when the function is called. This object is assigned to the corresponding parameter in the function.\n\n**composition**: Using an expression as part of a larger expression, or a statement as part of a larger statement.\n\n**flow of execution**: The order statements run in.\n\n**function**: A named sequence of statements that performs some useful operation. Functions may or may not take arguments and may or may not produce a result.\n\n**function body**: The sequence of statements inside a function definition.\n\n**function call**: A statement that runs a function. It consists of the function name followed by an argument list in parentheses.\n\n**function definition**: A statement that creates a new function, specifying its name, parameters, and the statements it contains.\n\n**function header**: The first line of a function definition.\n\n**function object**: An object created by a function definition. The name of the function is a variable that refers to a function object.\n\n**incremental development**: A program development plan intended to avoid debugging by adding and testing only a small amount of code at a time.\n\n**local variable**: A variable defined inside a function. A local variable can only be used inside its function.\n\n**namespace**: A mapping from names to objects.\n\n**None**: A special object returned by functions when no ``return`` is specified.\n\n**parameter**: A name used inside a function to refer to the object passed as an argument.\n\n**return object**: The result of a function. If a function call is used as an expression, the return object is the object of the expression.\n\n**scaffolding**: Code that is used during program development but is not part of the final version.\n\n**temporary variable**: A variable used to store an intermediate object in a complex calculation.\n\n## Exercises\n\n1) Write a function (named ``area``) that takes a single argument, namely the radius $r$ of a sphere. This function must compute and return the surface area of a sphere given by:\n\n$$\n S = 4 \\pi r^2\n$$\n\n2) Write a function (named ``volume``) that takes a single argument, namely the radius $r$ of a sphere. This function must compute and return the volume of a sphere given by:\n\n$$\n V = \\frac{4}{3} \\pi r^3\n$$\n\n3) Consider a rectangular section with a width $b$ (along the $x$-axis) and a height $h$ (along the $y$-axis). Write three functions that compute the moment of inertia around the centroidal $x$-axis, $y$-axis, and polar resectively:\n\n$$\n\\begin{aligned}\n I_x &= \\frac{1}{12}bh^3 \\\\\n I_y &= \\frac{1}{12}hb^3 \\\\\n I_z &= \\frac{bh}{12}\\left(b^2 + h^2\\right) \\\\\n\\end{aligned}\n$$\n\nEach function takes two arguments, namely the width $b$ and height $h$ of the rectangular section.\n\n4) Redo number 3, however this time the function that computes $I_y$ must make use of the function that computes $I_x$ and must only contain a single statement in the function body.\n\n5a) Write a function that computes the terminal velocity of a falling body in a non-dense medium given by:\n\n$$\n v_t = \\sqrt{\\frac{2mg}{\\rho A C_d}}\n$$\n\nwhere:\n\n- $g=9.81 \\; m/s^2$ is the gravitational acceleration\n- $m$ is the body mass - $kg$\n- $A$ is the body cross sectional area - $m^2$\n- $\\rho$ is the medium density - $kg / m^3$\n- $C_d$ is the coefficient of drag\n\n5b) Write a function that computes the velocity (of the falling object) at a given time $t$:\n\n$$\n v(t) = \\sqrt{\\frac{2mg}{\\rho A C_d}} \\tanh \\left(t\\sqrt{\\frac{g\\rho C_d A}{2m}}\\right)\n$$\n\n*Hint: Make use of the function you created in 5a*\n\n6) A function object can be assigned to a variable or passed as an argument. For example, ``do_twice`` is a function that takes another function object as an argument and calls it twice:\n\n\n```python\ndef do_twice(func):\n func()\n func()\n```\n\nHere\u2019s an example that uses ``do_twice`` to call a function named ``print_spam`` twice.\n\n\n```python\ndef print_spam():\n print('spam')\n\n\ndo_twice(print_spam)\n```\n\n6a) Modify ``do_twice`` so that it takes two arguments, namely a function object and a value, and calls the function twice, passing the value as an argument.\n\nRecall the ``print_twice`` function from earlier in this chapter:\n\n\n```python\ndef print_twice(thing):\n print(thing)\n print(thing)\n```\n\n6b) Use the modified version of ``do_twice`` to call ``print_twice`` twice, passing `spam` as an argument.\n\n6c) Define a new function (called ``do_four``) that takes a function object and a value and calls the function four times, passing the value as an argument. There should be only two statements in the body of this function, not four.\n\n\n```python\n\n```\n", "meta": {"hexsha": "9576e67f9eae86bd919e0b4fda4fc49e0c253b3d", "size": 57779, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "book/Chapter04_Functions.ipynb", "max_stars_repo_name": "lgpage/solve-it-with-python", "max_stars_repo_head_hexsha": "05e1f31ed3d114d55d3c10196555a2b8c2888ebe", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "book/Chapter04_Functions.ipynb", "max_issues_repo_name": "lgpage/solve-it-with-python", "max_issues_repo_head_hexsha": "05e1f31ed3d114d55d3c10196555a2b8c2888ebe", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "book/Chapter04_Functions.ipynb", "max_forks_repo_name": "lgpage/solve-it-with-python", "max_forks_repo_head_hexsha": "05e1f31ed3d114d55d3c10196555a2b8c2888ebe", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 37.5920624593, "max_line_length": 645, "alphanum_fraction": 0.6135447135, "converted": true, "num_tokens": 9722, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.7662936484231889, "lm_q1q2_score": 0.4454488661695065}} {"text": "# Online Drift Detection on the Wine Quality Dataset\n\nIn the context of deployed models, data (model queries) usually arrive sequentially and we wish to detect it as soon as possible after its occurence. One approach is to perform a test for drift every $W$ time-steps, using the $W$ samples that have arrived since the last test. Such a strategy could be implemented using any of the offline detectors implemented in `alibi-detect`, but being both sensitive to slight drift and responsive to severe drift is difficult. If the window size $W$ is too small then slight drift will be undetectable. If it is too large then the delay between test-points hampers responsiveness to severe drift.\n\nAn alternative strategy is to perform a test each time data arrives. However the usual offline methods are not applicable because the process for computing p-values is too expensive and doesn't account for correlated test outcomes when using overlapping windows of test data. \n\nOnline detectors instead work by computing the test-statistic once using the first $W$ data points and then updating the test-statistic sequentially at low cost. When no drift has occured the test-statistic fluctuates around its expected value and once drift occurs the test-statistic starts to drift upwards. When it exceeds some preconfigured threshold value, drift is detected.\n\nUnlike offline detectors which require the specification of a threshold p-value (a false positive rate), the online detectors in `alibi-detect` require the specification of an expected run-time (ERT) (an inverted FPR). This is the number of time-steps that we insist our detectors, on average, should run for in the absense of drift before making a false detection. Usually we would like the ERT to be large, however this results in insensitive detectors which are slow to respond when drift does occur. There is a tradeoff between the expected run time and the expected detection delay. \n\nTo target the desired ERT, thresholds are configured during an initial configuration phase via simulation. This configuration process is only suitable when the amount reference data (most likely the training data of the model of interest) is relatively large (ideally around an order of magnitude larger than the desired ERT). Configuration can be expensive (less so with a GPU) but allows the detector to operate at low-cost during deployment. \n\nThis notebook demonstrates online drift detection using two different two-sample distance metrics for the test-statistic, the maximum mean discrepency (MMD) and least-squared density difference (LSDD), both of which can be updated sequentially at low cost. \n\n### Backend\n\nThe online detectors are implemented in both the *PyTorch* and *TensorFlow* frameworks with support for CPU and GPU. Various preprocessing steps are also supported out-of-the box in Alibi Detect for both frameworks and an example will be given in this notebook. Alibi Detect does however not install PyTorch for you. Check the [PyTorch docs](https://pytorch.org/) how to do this. \n\n### Dataset\n\nThe [Wine Quality Data Set](https://archive.ics.uci.edu/ml/datasets/wine+quality) consists of 4898 and 1599 samples of white and red wine respectively. Each sample has an associated quality (as determined by experts) and 11 numeric features indicating its acidity, density, pH etc. We consider the regression problem of tring to predict the quality of white wine samples given these features. We will then consider whether the model remains suitable for predicting the quality of red wine samples or whether the associated change in the underlying distribution should be considered as drift.\n\n## Online detection with MMD and Pytorch\n\nThe Maximum Mean Discepency (MMD) is a distance-based measure between 2 distributions *p* and *q* based on the mean embeddings $\\mu_{p}$ and $\\mu_{q}$ in a reproducing kernel Hilbert space $F$:\n\n\\begin{align}\nMMD(F, p, q) & = || \\mu_{p} - \\mu_{q} ||^2_{F} \\\\\n\\end{align}\n\nGiven reference samples $\\{X_i\\}_{i=1}^{N}$ and test samples $\\{Y_i\\}_{i=t}^{t+W}$ we may compute an unbiased estimate $\\widehat{MMD}^2(F, \\{X_i\\}_{i=1}^N, \\{Y_i\\}_{i=t}^{t+W})$ of the squared MMD between the two underlying distributions. Depending on the size of the reference and test windows, $N$ and $W$ respectively, this can be relatively expensive. However, once computed it is possible to update the statistic to estimate to the squared MMD between the distributions underlying $\\{X_i\\}_{i=1}^{N}$ and $\\{Y_i\\}_{i=t+1}^{t+1+W}$ at a very low cost, making it suitable for online drift detection.\n\nBy default we use a [radial basis function kernel](https://en.wikipedia.org/wiki/Radial_basis_function_kernel), but users are free to pass their own kernel of preference to the detector.\n\n\n```python\nfrom functools import partial\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport torch\nimport tensorflow as tf\nfrom torch import nn\nimport pandas as pd\nimport scipy\n\nfrom alibi_detect.models.pytorch.trainer import trainer\nfrom alibi_detect.cd.utils import encompass_batching\n\nnp.random.seed(0)\ntorch.manual_seed(0)\ntf.random.set_seed(0)\n```\n\n### Load data\n\nFirst we load in the data:\n\n\n```python\nred = pd.read_csv(\n \"https://storage.googleapis.com/seldon-datasets/wine_quality/winequality-red.csv\", sep=';'\n)\nwhite = pd.read_csv(\n \"https://storage.googleapis.com/seldon-datasets/wine_quality/winequality-white.csv\", sep=';'\n)\nwhite.describe()\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
fixed acidityvolatile aciditycitric acidresidual sugarchloridesfree sulfur dioxidetotal sulfur dioxidedensitypHsulphatesalcoholquality
count4898.0000004898.0000004898.0000004898.0000004898.0000004898.0000004898.0000004898.0000004898.0000004898.0000004898.0000004898.000000
mean6.8547880.2782410.3341926.3914150.04577235.308085138.3606570.9940273.1882670.48984710.5142675.877909
std0.8438680.1007950.1210205.0720580.02184817.00713742.4980650.0029910.1510010.1141261.2306210.885639
min3.8000000.0800000.0000000.6000000.0090002.0000009.0000000.9871102.7200000.2200008.0000003.000000
25%6.3000000.2100000.2700001.7000000.03600023.000000108.0000000.9917233.0900000.4100009.5000005.000000
50%6.8000000.2600000.3200005.2000000.04300034.000000134.0000000.9937403.1800000.47000010.4000006.000000
75%7.3000000.3200000.3900009.9000000.05000046.000000167.0000000.9961003.2800000.55000011.4000006.000000
max14.2000001.1000001.66000065.8000000.346000289.000000440.0000001.0389803.8200001.08000014.2000009.000000
\n
\n\n\n\nWe can see that the data for both red and white wine samples take the same format.\n\n\n```python\nred.describe()\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
fixed acidityvolatile aciditycitric acidresidual sugarchloridesfree sulfur dioxidetotal sulfur dioxidedensitypHsulphatesalcoholquality
count1599.0000001599.0000001599.0000001599.0000001599.0000001599.0000001599.0000001599.0000001599.0000001599.0000001599.0000001599.000000
mean8.3196370.5278210.2709762.5388060.08746715.87492246.4677920.9967473.3111130.65814910.4229835.636023
std1.7410960.1790600.1948011.4099280.04706510.46015732.8953240.0018870.1543860.1695071.0656680.807569
min4.6000000.1200000.0000000.9000000.0120001.0000006.0000000.9900702.7400000.3300008.4000003.000000
25%7.1000000.3900000.0900001.9000000.0700007.00000022.0000000.9956003.2100000.5500009.5000005.000000
50%7.9000000.5200000.2600002.2000000.07900014.00000038.0000000.9967503.3100000.62000010.2000006.000000
75%9.2000000.6400000.4200002.6000000.09000021.00000062.0000000.9978353.4000000.73000011.1000006.000000
max15.9000001.5800001.00000015.5000000.61100072.000000289.0000001.0036904.0100002.00000014.9000008.000000
\n
\n\n\n\nWe shuffle and normalise the data such that each feature takes a value in \\[0,1\\], as does the quality we seek to predict. We assue that our model was trained on white wine samples, which therefore forms the reference distribution, and that red wine samples can be considered to be drawn from a drifted distribution.\n\n\n```python\nwhite, red = np.asarray(white, np.float32), np.asarray(red, np.float32)\nn_white, n_red = white.shape[0], red.shape[0]\n\ncol_maxes = white.max(axis=0)\nwhite, red = white / col_maxes, red / col_maxes\nwhite, red = white[np.random.permutation(n_white)], red[np.random.permutation(n_red)]\nX = white[:, :-1]\nX_corr = red[:, :-1]\n```\n\nAlthough it may not be necessary on this relatively low-dimensional data for which individual features are semantically meaningful, we demonstrate how a preprocessing stage can be defined to project raw data onto a lower dimensional representation which more concisely captures the factors of variation in the data. As not to bias the detector it is necessary to learn the projection using a split of the data which isn't then passed as reference data. We additionally split off some white wine samples to act as undrifted data during deployment.\n\n\n```python\nX_train = X[:(n_white//2)]\nX_ref = X[(n_white//2):(3*n_white//4)]\nX_h0 = X[(3*n_white//4):]\n\nX_train_ds = torch.utils.data.TensorDataset(torch.tensor(X_train), torch.tensor(X_train))\nX_train_dl = torch.utils.data.DataLoader(X_train_ds, batch_size=32, shuffle=True, drop_last=True)\n```\n\nNow we define and fit an autoencder so that we can use the encoder as a preprocessing function which projects the 11-D data onto a 2-D representation.\n\n\n```python\nencoder = nn.Sequential(\n nn.Linear(11, 16),\n nn.ReLU(),\n nn.Linear(16, 2)\n)\ndecoder = nn.Sequential(\n nn.Linear(2, 16),\n nn.ReLU(),\n nn.Linear(16, 11)\n)\n\ndevice = torch.device('cuda' if torch.cuda.is_available() else 'cpu')\nae = nn.Sequential(encoder, decoder).to(device)\n\ntrainer(ae, nn.MSELoss(), X_train_dl, device, torch.optim.Adam, learning_rate=0.001, epochs=10)\n```\n\n /home/oliver/Projects/alibi-detect/.venv/lib/python3.8/site-packages/torch/cuda/__init__.py:52: UserWarning: CUDA initialization: Found no NVIDIA driver on your system. Please check that you have an NVIDIA GPU and installed a driver from http://www.nvidia.com/Download/index.aspx (Triggered internally at /pytorch/c10/cuda/CUDAFunctions.cpp:100.)\n return torch._C._cuda_getDeviceCount() > 0\n Epoch 1/10: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 76/76 [00:00<00:00, 161.73it/s, loss=0.104]\n Epoch 2/10: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 76/76 [00:00<00:00, 191.40it/s, loss=0.0121]\n Epoch 3/10: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 76/76 [00:00<00:00, 179.78it/s, loss=0.00488]\n Epoch 4/10: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 76/76 [00:00<00:00, 172.70it/s, loss=0.00453]\n Epoch 5/10: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 76/76 [00:00<00:00, 151.17it/s, loss=0.00548]\n Epoch 6/10: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 76/76 [00:00<00:00, 175.39it/s, loss=0.00517]\n Epoch 7/10: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 76/76 [00:00<00:00, 185.24it/s, loss=0.00426]\n Epoch 8/10: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 76/76 [00:00<00:00, 136.51it/s, loss=0.00441]\n Epoch 9/10: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 76/76 [00:00<00:00, 155.20it/s, loss=0.00467]\n Epoch 10/10: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 76/76 [00:00<00:00, 211.75it/s, loss=0.00448]\n\n\nThe autoencoder was fit to data from the reference distribution (white wine samples) and should therefore more accurately reconstruct white wine samples than red.\n\n\n```python\nae = ae.eval()\nae_fn = encompass_batching(ae, backend='pytorch', batch_size=32)\nrecon_ref = ae_fn(X_ref)\nrecon_corr = ae_fn(X_corr)\n\nref_mse = np.square(recon_ref - X_ref).mean()\ncorr_mse = np.square(recon_corr - X_corr).mean()\n\nprint(f'MSE when reconstructing unseen white wine samples: {ref_mse}')\nprint(f'MSE when reconstructing unseen red wine samples: {corr_mse}')\n```\n\n MSE when reconstructing unseen white wine samples: 0.005474736448377371\n MSE when reconstructing unseen red wine samples: 0.024097805842757225\n\n\nHopefully the learned preprocessing step has learned a projection such that in the lower dimensional space the two samples are distinguishable.\n\n\n```python\nencoder_fn = encompass_batching(encoder, backend='pytorch', batch_size=32)\nenc_h0 = encoder_fn(X_h0)\nenc_h1 = encoder_fn(X_corr)\n\nplt.scatter(enc_h0[:,0], enc_h0[:,1], alpha=0.2, color='green', label='white wine')\nplt.scatter(enc_h1[:,0], enc_h1[:,1], alpha=0.2, color='red', label='red wine')\nplt.legend(loc='upper right')\nplt.show()\n```\n\nNow we can define our online drift detector. We specify an expected run-time (in the absence of drift) of 50 time-steps, and a window size of 10 time-steps. Upon initialising the detector thresholds will be computed using 2500 boostrap samples. These values of `ert`, `window_size` and `n_bootstraps` are lower than a typical use-case in order to demonstrate the average behaviour of the detector over a large number of runs in a reasonable time. \n\n\n```python\nfrom alibi_detect.cd import MMDDriftOnline\n\nert = 50\nwindow_size = 10\n\ncd = MMDDriftOnline(\n X_ref, ert, window_size, backend='pytorch', preprocess_fn=encoder_fn, n_bootstraps=2500\n)\n```\n\n No GPU detected, fall back on CPU.\n 1%|\u258f | 32/2500 [00:00<00:07, 312.22it/s]Generating permutations of kernel matrix..\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 2500/2500 [00:07<00:00, 342.63it/s]\n Computing thresholds: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 10/10 [00:04<00:00, 2.18it/s]\n\n\nWe now define a function which will simulate a single run and return the run-time. Note how the detector acts on single instances at a time, the run-time is considered as the time elapsed after the test-window has been filled, and that the detector is stateful and must be reset between detections.\n\n\n```python\ndef time_run(cd, X, window_size):\n n = X.shape[0]\n perm = np.random.permutation(n)\n t = 0\n cd.reset()\n while True:\n pred = cd.predict(X[perm[t%n]])\n if pred['data']['is_drift'] == 1:\n return t\n else:\n t += 1\n```\n\nNow we look at the distribution of run-times when operating on the held-out data from the reference distribution of white wine samples. We report the average run-time, however note that the targeted run-time distribution, a Geometric distribution with mean `ert`, is very high variance so the empirical average may not be that close to `ert` over a relatively small number of runs. We can see that the detector accurately targets the desired Geometric distribution however by inspecting the linearity of a [Q-Q plot](https://en.wikipedia.org/wiki/Q%E2%80%93Q_plot).\n\n\n```python\nn_runs = 250\ntimes_h0 = [time_run(cd, X_h0, window_size) for _ in range(n_runs)]\nprint(f\"Average run-time under no-drift: {np.mean(times_h0)}\")\n_ = scipy.stats.probplot(np.array(times_h0), dist=scipy.stats.geom, sparams=1/ert, plot=plt)\n```\n\nIf we run the detector in an identical manner but on data from the drifted distribution of red wine samples the average run-time is much lower.\n\n\n```python\nn_runs = 250\ntimes_h1 = [time_run(cd, X_corr, window_size) for _ in range(n_runs)]\nprint(f\"Average run-time under drift: {np.mean(times_h1)}\")\n```\n\n Average run-time under drift: 5.456\n\n\n## Online detection with LSDD and TensorFlow\n\nHere we address the same problem but using the least squares density difference (LSDD) as the two-sample distance in a manner similar to [Bu et al. (2017)](https://ieeexplore.ieee.org/abstract/document/7890493). The LSDD between two distributions $p$ and $q$ on $\\mathcal{X}$ is defined as $$LSDD(p,q) = \\int_{\\mathcal{X}} (p(x)-q(x))^2 \\,dx$$ and also has an empirical estimate $\\widehat{LSDD}(\\{X_i\\}_{i=1}^N, \\{Y_i\\}_{i=t}^{t+W})$ that can be updated at low cost as the test window is updated to $\\{Y_i\\}_{i=t+1}^{t+1+W}$.\n\nWe additionally show that TensorFlow can also be used as the backend and that sometimes it is not necessary to perform preprocessing, making definition of the drift detector simpler. Moreover, in the absence of a learned preprocessing stage we may use all of the reference data available.\n\n\n```python\nX_ref = np.concatenate([X_train, X_ref], axis=0)\n```\n\nAnd now we define the LSDD-based online drift detector, again with an `ert` of 50 and `window_size` of 10.\n\n\n```python\nfrom alibi_detect.cd import LSDDDriftOnline\n\ncd = LSDDDriftOnline(\n X_ref, ert, window_size, backend='tensorflow', n_bootstraps=2500,\n)\n```\n\n Computing thresholds: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 9/9 [00:10<00:00, 1.12s/it]\n\n\nWe run this new detector on the held out reference data and again see that in the absence of drift the distribution of run-times follows a Geometric distribution with mean `ert`.\n\n\n```python\nn_runs = 250\ntimes_h0 = [time_run(cd, X_h0, window_size) for _ in range(n_runs)]\nprint(f\"Average run-time under no-drift: {np.mean(times_h0)}\")\n_ = scipy.stats.probplot(np.array(times_h0), dist=scipy.stats.geom, sparams=1/ert, plot=plt)\n```\n\nAnd when drift has occured the detector is very fast to respond.\n\n\n```python\nn_runs = 250\ntimes_h1 = [time_run(cd, X_corr, window_size) for _ in range(n_runs)]\nprint(f\"Average run-time under drift: {np.mean(times_h1)}\")\n```\n\n Average run-time under drift: 4.176\n\n", "meta": {"hexsha": "18fe5c61428c8671d3aa7287152ba1563e2d9d38", "size": 661844, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/cd_online_wine.ipynb", "max_stars_repo_name": "tim5go/alibi-detect", "max_stars_repo_head_hexsha": "a2fc6125eaf8d15068a3069e64d8fbc9cd1b41cb", "max_stars_repo_licenses": ["ECL-2.0", "Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "examples/cd_online_wine.ipynb", "max_issues_repo_name": "tim5go/alibi-detect", "max_issues_repo_head_hexsha": "a2fc6125eaf8d15068a3069e64d8fbc9cd1b41cb", "max_issues_repo_licenses": ["ECL-2.0", "Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "examples/cd_online_wine.ipynb", "max_forks_repo_name": "tim5go/alibi-detect", "max_forks_repo_head_hexsha": "a2fc6125eaf8d15068a3069e64d8fbc9cd1b41cb", "max_forks_repo_licenses": ["ECL-2.0", "Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 1079.6802610114, "max_line_length": 442720, "alphanum_fraction": 0.6810033784, "converted": true, "num_tokens": 6698, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154240079185319, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.44532234949471966}} {"text": "# Model Project\n\nThis project will create and simulate an Small Open Economy using an AS-AD-model. We examine the effects on aggregate demand and supply from changes in interest rates both domestic and abroad, and furhter examine effects from changes in the effective exchange rate.\n\nThe Model is structured the following way\n\n 1. Defining the AS-AD Model\n 2. Defining the Aggregate Demand (AD)\n 3. Defining Money Supply \n 4. Defining the Long Run Aggregate Supply (LRAS)\n 5. Defining the Short Run Aggregate Supply (LRAS)\n 6. Combining all elements\n 7. Simulating a shock to domestic and foreing interest rate.\n\n# Defining the Aggregate Demand\n\n//DEFINER AD teoretisk, aka ligninger og s\u00e5 videre\n\n\n\n\n```python\n#Importing Packeges used\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy import optimize\nimport sympy as sm\n```\n\n\n```python\n#Creating the AD\n\nclass AD\n #Using an ___init___ method, we define the parameters of the model\n def ___init___ ( )\n \n \n #We assign values to the parameters\n self.a = \n \n \n \n #HER SKAL INDG\u00c5 2 FUNCTIONER, EN FOR MONEY SUPPLY OG EN FOR MONEY MULTIPLIER\n \n \n def AD_curve(self)\n #Kald p\u00e5 self.parametrene\n \n #Return AD-kurven\n \n \n```\n", "meta": {"hexsha": "2374c3741bc4e0680b990f8dee8434b4847a6a34", "size": 3196, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "modelproject/ModelProject.ipynb", "max_stars_repo_name": "NumEconCopenhagen/projects-2022-git-good", "max_stars_repo_head_hexsha": "df457732b3da0d52c481b0adcb18e1cef63a5089", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "modelproject/ModelProject.ipynb", "max_issues_repo_name": "NumEconCopenhagen/projects-2022-git-good", "max_issues_repo_head_hexsha": "df457732b3da0d52c481b0adcb18e1cef63a5089", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "modelproject/ModelProject.ipynb", "max_forks_repo_name": "NumEconCopenhagen/projects-2022-git-good", "max_forks_repo_head_hexsha": "df457732b3da0d52c481b0adcb18e1cef63a5089", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.96875, "max_line_length": 297, "alphanum_fraction": 0.5541301627, "converted": true, "num_tokens": 305, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.6654105454764747, "lm_q1q2_score": 0.44507997891191925}} {"text": "   \n\n# Example Model Project: the Train Illusion\n\n**By Neuromatch Academy**\n\n__Content creators:__ Marius t'Hart, Megan Peters, Paul Schrater, Gunnar Blohm\n\n__Production editors:__ Spiros Chavlis\n\n
\n\n**Disclaimer**: this is a \"toy\" model used to demonstrate the [10 step procedure of how-to-model](https://doi.org/10.1523/ENEURO.0352-19.2019). It is not meant to be state of the art research.\n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n

\n\n---\n# Phenomenon\n*Part of Steps 1-2*\n\nThe train illusion occurs when sitting on a train and viewing another train outside the window. Suddenly, the other train *seems* to move, i.e. you experience visual motion of the other train relative to your train. But which train is actually moving?\n\nOften people have the wrong percept. In particular, they think their own train might be moving when it's the other train that moves; or vice versa. The illusion is usually resolved once you gain vision of the surroundings that lets you disambiguate the relative motion; or if you experience strong vibrations indicating that it is indeed your own train that is in motion.\n\n---\n# Question\n\n*Part of Step 1*\n\nWe asked the following (arbitrary) question for our demo project: \"How do noisy vestibular estimates of motion lead to illusory percepts of self motion?\"\n\n---\n# Background\n*Part of Step 2*\n\nYou have learned all about the vestibular system in the Intro video. This is also where you would do a literature search to learn more about what's known about self-motion perception and vestibular signals. You would also want to examine any attempts to model self-motion, perceptual decision making and vestibular processing. \n\n---\n# Ingredients\n\n*Part of step 3*\n\nWe determined that we probably needed the following ingredients for our model:\n* Vestibular input: $v(t)$\n* Binary decision output: $d$ - time dependent?\n* Decision threshold: $\\theta$\n* A filter (maybe running average?): $f$\n* An integration mechanism to get from vestibular acceleration to sensed velocity: $\\int$\n\n---\n# Hypotheses\n\n*Part of step 4*\n\nOur main hypothesis is that the strength of the illusion has a linear relationship to the amplitude of vestibular noise. \n\nMathematically, this would write as $S = k \\cdot N$, where $S$ is the illusion strength and $N$ is the noise level, and $k$ is a free parameter.\n\n>we could simply use the frequency of occurance across repetitions as the \"strength of the illusion\"\n\nWe would get the noise as the standard deviation of $v(t)$, i.e., $N = \\mathbb{E}[v(t)^2]$, where $\\mathbb{E}[\\cdot]$ stands for the expected value.\n\nDo we need to take the average across time points?\n\n> doesn't really matter because we have the generative process, so we can just use the $\\sigma$ that we define\n\n---\n# Selected toolkit\n\n*Part of step 5*\n\nWe chose to use a [Drift-Diffusion Model (DDM)](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2474742/) because it is a well-established framework that allows us to model decision making in the case of 2 alternative choices (here: self-motion vs. other train motion).\n\nFor our purposes simplest equation looks something like this:\n\n\\begin{align}\n\\dot e = \\frac{de}{dt}= -c \\cdot e + v \\, ,\n\\end{align}\n\nwhere $e$ is the accumulated evidence and $v$ is our vestibular input already containing the noise (so we don't need to add more noise?). $c$ is the leakage constant, i.e., $c=0$ means perfect integration; $c=1$ means no integration (perfect leakage).\n\n---\n# Model draft\n\n*Part of step 6*\n\nBrainstorming on the whiteboard, we came up with this...\n\n\n\n---\n# Model implementation\n\n*Part of step 7*\n\nWe need at least 3 functions:\n1. vestibular signal generator\n2. integrator (or drift-diffusion mechanism)\n3. decision mechanism (threshold comparison)\n\n**Note:** we did not add a filter (yet). We're not sure if we need one...\n\nSo let's go step by step... first we need to get set up...\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.stats import norm\n!pip install pandas --quiet\nplt.style.use('dark_background')\n```\n\n## 1. Vestibular signal generator\n\n\n```python\ndef vestibular_signal(sig, mov):\n \"\"\"\n Computes a vestibular signal that corresponds to the acceleration of the\n train with different amplitudes of noise\n Args:\n sig: scalar SD of noise\n mov: 0 means no self-motion; 1 means self-motion (or scaling or motion signal)\n Returns: vector array of vestibular signal v\n \"\"\"\n # create white noise series for 10s with 1ms resolution\n x = np.linspace(-7, 14, 1001)\n z = 1/(1 + np.exp(-x))\n noise = norm.rvs(size=1000)\n v = sig*noise + mov*np.diff(z)/0.001\n\n return v\n```\n\nLet's see if that works... (*unit test*)\n\n\n```python\nv = vestibular_signal(1,1)\n\n# plot signal\nt = np.linspace(0, 10, 1000)\nplt.plot(t,v)\nplt.xlabel(\"time (s)\")\nplt.ylabel(\"vestibular signal (a.u.)\")\n```\n\n## 2. Integrator (DDM mechanism)\n\n\n```python\ndef ddm(v, c):\n \"\"\"\n Leaky integration of vestibular signal\n Args:\n v: array of vestibular signal\n c: leakage constant\n Outputs: time series of integrated vestibular signal (with leakage)\n = evidence e\n \"\"\"\n e = np.random.normal(0, 0.1)\n E = []\n for i in range(len(v)):\n e += -c*e + v[i]*0.001\n E.append(e)\n return E\n```\n\nLet's test the DDM function... (*unit test*)\n\n\n```python\ne = ddm(v, c=0.001)\n\n# plot result\nplt.plot(t, e)\nplt.xlabel(\"time (s)\")\nplt.ylabel(\"integrated evidence\")\nplt.show()\n```\n\n## 3. Thresholding (decision mechanism)\n\n\n```python\ndef threshold(e, thr):\n \"\"\"\n Thresholding of motion evidence\n Args:\n motion evidence: e (array)\n threshold: thr\n Output: decision d if threshold was reached\n \"\"\"\n d = any(np.array(e) > thr)*1\n return d\n```\n\nNow let's test this function... (*unit test*)\n\n\n```python\nd = threshold(e, .6)\nprint(d)\n```\n\n 1\n\n\n## Assembling the model\n\n\n```python\ndef run_model(sig, c, thr, mov):\n \"\"\"\n runs the full model to simulate self-motion decision, e.g., for train illusion\n Args:\n sig: SD of vestibular noise\n c: leakage constant\n thr: decision threshold\n mov: self-motion? ) no selfmotion; 1 self-motion\n Output: decision d (0: no self-motion; 1: self-motion)\n \"\"\"\n v = vestibular_signal(sig, mov)\n e = ddm(v, c)\n d = threshold(e, thr)\n return d\n```\n\nLet's run the model and see if it works...\n\n\n```python\nd = run_model(200, 0.001, 0.8, 1)\nprint(d)\n```\n\n 1\n\n\n---\n# Model completion\n\n*Part of step 8*\n\nSo the model seems to work. Running different parameters gives us different results. Are we done?\n* **can we answer our question**: yes, in our model the illusion arises because integrating very noisy vestibular signals representing motion evidence sometimes accumulate to a decision threshold and sometimes do not reach that threshold.\n* **can we speak to our hypothesis**: yes, we can now simulate different trials with different noise levels (and leakage and thrshold parameters) and evaluate the hypothesized linear relationship between vestibular noise and how often our perceptual system is fooled...\n* **does the model reach our goals**: yes, we wanted to generate a mechanistic model to be able to make some specific predictions that can then be tested experimentally later...\n\n\n\n---\n# Model evaluation & testing\n\n*Part of step 9*\n\nOk, so we still need to actually evaluate and test our model performance. Since this is a conceptual model and we don't have actual data (yet), we will evaluate how our model behaves as a function of the 3 parameters. If we had data with different conditions, we could try to fit the model to the data and evaluate the goodness of fit, etc. If other alterative models existed, we could evaluate our model against those alternatives too. \n\nSo let's run out model in different parameter regimes and analyze the result to get some insight into the model performance\n\n\n```python\nimport itertools # to automatically generate possible combinations of parameters\n\n# define parameter list\nparams = {\n 'sig': np.linspace(1, 21, 5)**2,\n 'c': np.exp(np.linspace(-10, -1, 5)),\n 'thr': np.linspace(0, 2, 5),\n 'mov': np.linspace(0, 1, 2),\n }\n\n# run combination of parameters\nkeys = list(params)\nD = []\nfor i in range(0,100):\n for values in itertools.product(*map(params.get, keys)):\n d = run_model(**dict(zip(keys, values)))\n temp = list(values)\n temp.append(d)\n D.append(temp)\n```\n\nNow let's explicitly test our hypothsis for different parameter combinations...\n\n\n```python\n# want data frames:\nimport pandas as pd\ndf = pd.DataFrame(D, columns=['Sig', 'c', 'Thr', 'Mov', 'Decisions'])\n# multi panel layout:\naxs = plt.figure(figsize=(12,12), constrained_layout=True).subplots(5, 5)\n# plot for movement absent/present\nMov_s = np.unique(df['Mov'])\n# plot for leakage parameter & threshold values:\nc_s = np.unique(df['c'])\nThr_s = np.unique(df['Thr'])\n# plot for data for both movement condition for each leakage/threshold combination\nSig_s = np.unique(df['Sig'])\nfor Thr_n in range(len(Thr_s)):\n for c_n in range(len(c_s)):\n subdf0 = df[(df.Mov == 0) & (df.c == c_s[c_n]) & (df.Thr == Thr_s[Thr_n])].groupby(['Sig'])['Decisions'].mean()\n subdf1 = df[(df.Mov == 1) & (df.c == c_s[c_n]) & (df.Thr == Thr_s[Thr_n])].groupby(['Sig'])['Decisions'].mean()\n im0 = axs[Thr_n, c_n].plot(Sig_s, subdf0, label=\"no motion\")\n im1 = axs[Thr_n, c_n].plot(Sig_s, subdf1, label=\"motion\")\n axs[Thr_n, c_n].set_title(f\"Thr = {Thr_s[Thr_n]}; c = {c_s[c_n]:.4f}\")\n axs[Thr_n, c_n].set_ylim(0, 1.1)\n axs[Thr_n, c_n].set_xlim(0, 450)\naxs[4, 2].set_xlabel(\"Noise level $\\sigma$\")\naxs[2, 0].set_ylabel(\"Proportion motion judgment\")\naxs[3, 1].set_facecolor('grey')\naxs[4, 4].legend()\nplt.show()\n```\n\nThere seems to be some parameter redundancy, i.e., we could chose different parameter combinations to make the model do something sensible...\n\nBut it looks like $c=0.0004$ works well for $\\theta = 1.5$ and $\\sigma=50$ (highlighted plot). Lets run a few trials on that to analyze those results more specifically...\n\n\n```python\n# run \"best\" parameter combination\nsig = 50\nc = 0.0004\nthr = 1.5\nd0 = []\nd1 = []\nfor i in range(0, 1000):\n d0.append(run_model(sig, c, thr, 0))\n d1.append(run_model(sig, c, thr, 1))\nprint(f\"\\n Motion detected for no-motion: {sum(d0)/10}% and motion: {sum(d1)/10}%\")\n```\n\n \n Motion detected for no-motion: 27.4% and motion: 59.2%\n\n\nThis does indeed result in roughly 50% likelihood of experiencing the illusion both ways.\n\nFinally, let's explicitly evaluate our hypothesis...\n\n\n```python\nsig = np.linspace(1, 201, 20)\nc = 0.0004\nthr = 1.5\nD0 = []\nD1 = []\nfor s in range(len(sig)):\n d0 = []\n d1 = []\n for i in range(0, 100):\n d0.append(run_model(sig[s], c, thr, 0))\n d1.append(run_model(sig[s], c, thr, 1))\n D0.append(sum(d0) / 100)\n D1.append(sum(d1) / 100)\n```\n\n\n```python\nplt.plot(sig, D0, label=\"no motion\")\nplt.plot(sig, D1, label=\"motion\")\nplt.xlabel(\"Noise level\")\nplt.ylabel(\"% motion decisions\")\nplt.legend()\nplt.show()\n```\n\nOur **hypothesis** of linear increase of illusion strength with noise only holds true in a limited range of noise... It's monotonic but saturating of course...\n\n**And regarding our original question**: it is really the noise that pushes the integrated signal over the threshold. The less leaky the integration and the lower the threshold, the more motion decisions we get...\n\n---\n# Summary\n*Part of Step 10*\n\nLet's write a simple abstract following the guidelines...\n\n**A. What is the phenomena**? Here summarize the part of the phenomena which your modeling addresses.\n\n_The \"train illusion\" occurs when sitting in a stationary train and experiencing relative visual motion of an adjacent train outside the window; sometimes we feel like we're moving even if we're not (and vice versa). Previous literature has suggested that vestibular signals are used to disambiguate self-motion from motion of an adjacent object._\n\n**B. What is the key scientific question?**: Clearly articulate the question which your modeling tries to answer.\n\n_How noisy vestibular estimates of motion lead to illusory percepts of self motion is currently unknown._\n\n**C. What was our hypothesis?**: Explain the key relationships which we relied on to simulate the phenomena.\n\n_We hypothesized that noisy vestibular signals are integrated leading the brain to decide whether self-motion is occurring or not, and that larger noise is linearly associated with more frequent errors in self-motion judgment._\n\n**D. How did your modeling work?** Give an overview of the model, it's main components, and how the modeling works. ''Here we ... ''\n\n_To investigate this hypothesis, we constructed a drift diffusion model and simulated self-motion decisions under varying noise conditions, when true self motion was occurring or not._\n\n**E. What did you find? Did the modeling work?** Explain the key outcomes of your modeling evaluation. \n\n_We observed that higher noise did indeed lead to more frequent errors in self-motion perception but this relationship was not linear._\n\n**F. What can you conclude?** Conclude as much as you can _with reference to the hypothesis_, within the limits of the modeling. \n\n_We conclude that accumulated noisy vestibular information can explain the occurrence of the train illusion, and the higher the noise (or the lower the signal-to-noise ratio), the more frequently such illusions will occur._\n\n**G. What are the limitations and future directions?** What is left to be learned? Briefly argue the plausibility of the approach and/or what you think is essential that may have been left out.\n\n_Future research should investigate whether trial-by-trial variations of noisy vestibular signals actually correlate with self-motion judgments._\n\n>If we put this all in one paragraph, we have our final complete abstract. But, first, do not include the letters in _your_ abstract, and second, we did paraphrase the answers a little so they fit together.\n\n
\n\n## Abstract\n(A) The \"train illusion\" occurs when sitting in a stationary train and experiencing relative visual motion of an adjacent train outside the window; sometimes we feel like we're moving even if we're not (and vice versa). Previous literature has suggested that vestibular signals are used to disambiguate self-motion from motion of an adjacent object. (B) How noisy vestibular estimates of motion lead to illusory percepts of self motion is currently unknown. (C) We hypothesized that noisy vestibular signals are integrated leading the brain to decide whether self-motion is occurring or not, and that larger noise is linearly associated with more frequent errors in self-motion judgment. (D) To investigate this hypothesis, we constructed a drift diffusion model and simulated self-motion decisions under varying noise conditions, when true self motion was occurring or not. (E) We observed that higher noise did indeed lead to more frequent errors in self-motion perception but this relationship was not linear. (F) We conclude that accumulated noisy vestibular information can explain the occurrence of the train illusion, and the higher the noise (or the lower the signal-to-noise ratio), the more frequently such illusions will occur. (G) Future research should investigate whether trial-by-trial variations of noisy vestibular signals actually correlate with self-motion judgments.\n\n---\n# Final thoughts\n\nNote that the model we built here was extremely simple and used artificial data on purpose. It allowed us to go through all the steps of building a model, and hopefully you noticed that it is not always a linear process, you will go back to different steps if you hit a roadblock somewhere.\n\nThere are many issues that we did not address in this model. However, if you're interested in how to actually approach modeling a similar phenomenon in a probabilistic way, we encourage you to read the paper by [Dokka et al., 2019](https://doi.org/10.1073/pnas.1820373116), where the authors model how judgments of heading direction are influenced by objects that are also moving.\n", "meta": {"hexsha": "52e1d2fed82c484a4ae7a9b9857826396b0081e1", "size": 214433, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "projects/modelingsteps/TrainIllusionModelingProjectDL.ipynb", "max_stars_repo_name": "justynaekert/course-content-dl", "max_stars_repo_head_hexsha": "aa64d9feb1ae92ad4b7afaf13b13616b3a020c20", "max_stars_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_stars_count": 473, "max_stars_repo_stars_event_min_datetime": "2021-04-13T18:27:42.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-28T14:14:35.000Z", "max_issues_repo_path": "projects/modelingsteps/TrainIllusionModelingProjectDL.ipynb", "max_issues_repo_name": "justynaekert/course-content-dl", "max_issues_repo_head_hexsha": "aa64d9feb1ae92ad4b7afaf13b13616b3a020c20", "max_issues_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_issues_count": 399, "max_issues_repo_issues_event_min_datetime": "2021-06-07T20:56:59.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-26T23:05:06.000Z", "max_forks_repo_path": "projects/modelingsteps/TrainIllusionModelingProjectDL.ipynb", "max_forks_repo_name": "justynaekert/course-content-dl", "max_forks_repo_head_hexsha": "aa64d9feb1ae92ad4b7afaf13b13616b3a020c20", "max_forks_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_forks_count": 170, "max_forks_repo_forks_event_min_datetime": "2021-04-16T11:09:32.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-31T12:13:52.000Z", "avg_line_length": 260.8673965937, "max_line_length": 125468, "alphanum_fraction": 0.9196112539, "converted": true, "num_tokens": 4096, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.7279754548076478, "lm_q1q2_score": 0.44507107007000685}} {"text": "# NOTE\nThis notebook doesn't work yet. I have previously written my own version of `lambdify` [here](https://github.com/bjodah/sym/blob/222d1c6a3d1db03c67e8467ff562d3ac0f124616/sym/_sympy_Lambdify.py#L214).\nDon't know if that's the path to go, or wait for [next release of numba](https://github.com/numba/numba/issues/2328#issuecomment-303382911).\n\nIn this notebook we will use `numba` the increase the performance of our callbacks produced by lambdify in SymPy.\n\n\n```python\nimport json\nimport numpy as np\nimport sympy as sym\nfrom scipy2017codegen.odesys import ODEsys\nfrom scipy2017codegen.chem import mk_rsys\n```\n\nThe `ODEsys` class and convenience functions from previous notebook (35) has been put in two modules for easy importing. Recapping what we did last:\n\n\n```python\nwatrad_data = json.load(open('../scipy2017codegen/data/radiolysis_300_Gy_s.json'))\nwatrad = mk_rsys(ODEsys, **watrad_data)\ntout = np.logspace(-6, 3, 200) # close to one hour of operation\nc0 = {'H2O': 55.4e3, 'H+': 1e-4, 'OH-': 1e-4}\ny0 = [c0.get(symb.name, 0) for symb in watrad.y]\n```\n\n\n```python\n%timeit yout, info = watrad.integrate_odeint(tout, y0)\n```\n\nso that is the benchmark to beat.\n\n\n```python\nfrom numba import njit\nwatrad_numba = mk_rsys(ODEsys, **watrad_data, lambdify=lambda *args: njit(sym.lambdify(*args, modules=\"numpy\")))\nwatrad_numba.integrate_odeint(tout, y0)\n```\n\n\n```python\n%timeit watrad_numba.integrate_odeint(tout, y0)\n```\n\n\n```python\nimport matplotlib.pyplot as plt\n%matplotlib inline\n```\n\nJust to see that everything looks alright:\n\n\n```python\nfig, ax = plt.subplots(1, 1, figsize=(14, 6))\nwatrad_numba.plot_result(tout, *watrad_numba.integrate_odeint(tout, y0), ax=ax)\nax.set_xscale('log')\nax.set_yscale('log')\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "78103ec010feca4e0681d01f838c7393442fc603", "size": 3824, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/_37-chemical-kinetics-numba.ipynb", "max_stars_repo_name": "gvvynplaine/scipy-2017-codegen-tutorial", "max_stars_repo_head_hexsha": "4bd0cdb1bdbdc796bb90c08114a00e390b3d3026", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 56, "max_stars_repo_stars_event_min_datetime": "2017-05-31T21:01:22.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-14T04:26:01.000Z", "max_issues_repo_path": "notebooks/_37-chemical-kinetics-numba.ipynb", "max_issues_repo_name": "gvvynplaine/scipy-2017-codegen-tutorial", "max_issues_repo_head_hexsha": "4bd0cdb1bdbdc796bb90c08114a00e390b3d3026", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 26, "max_issues_repo_issues_event_min_datetime": "2017-06-06T19:05:04.000Z", "max_issues_repo_issues_event_max_datetime": "2018-08-29T23:15:19.000Z", "max_forks_repo_path": "notebooks/_37-chemical-kinetics-numba.ipynb", "max_forks_repo_name": "gvvynplaine/scipy-2017-codegen-tutorial", "max_forks_repo_head_hexsha": "4bd0cdb1bdbdc796bb90c08114a00e390b3d3026", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 29, "max_forks_repo_forks_event_min_datetime": "2017-06-06T14:45:12.000Z", "max_forks_repo_forks_event_max_datetime": "2021-09-25T14:11:06.000Z", "avg_line_length": 23.3170731707, "max_line_length": 208, "alphanum_fraction": 0.574790795, "converted": true, "num_tokens": 564, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723317123102955, "lm_q2_score": 0.6619228825191872, "lm_q1q2_score": 0.44503174502149173}} {"text": "\n\nCopyright: Thomas Klinder\n\n\n#
Increase Your Property Value and Stretch Your House Buying Budget Further
\n\n**Phase 2 Final Project**
\n* Student name: Elena Kazakova\n* Student pace: full time\n* Cohort: DS02222021\n* Scheduled project review date/time: 05/07/2021\n* Instructor name: James Irving\n* Blog post URL: The Power of Color\n\n## Table of Contents \n\n*Click to jump to matching Markdown Header.*
\n\n- **[Introduction](#INTRODUCTION)
**\n- **[Obtain](#Obtain)**
\n- **[Scrub](#Scrub)**
\n- **[Explore](#Explore)**
\n- **[Model](#Model)**
\n- **[iNterpret](#iNterpret)**
\n- **[Conclusions/Recommendations](#Conclusions-and-Recommendation)
**\n___\n\n# Introduction\n\n## Business Problem\nThis project is the Inference Analysis project of King County, WA house prices, and various factors that might affect the sales price.
\nThis study aims to build a model(s) of house sale prices depending on the features of the property in the dataset provided. This information can be helpful for house owners, house buyers, and real estate agents in the county.\n\n# Obtain\n\n## Data Understanding\n\nThe dataset used in this project has been downloaded from [KAGGLE site]( https://www.kaggle.com/harlfoxem/housesalesprediction?select=kc_house_data.csv). The dataset includes the information about properties sold in King County of Washington State between May 2014 and May 2015. The area consists of Seattle city area but does not include the inner city. The dataset consists of 21 dependent and independent variables and 21597 records.\n\n### Importing Python tools and utilities\n\n\n```python\nimport pandas as pd\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport seaborn as sns\nimport folium\nimport json\n\nimport plotly.express as px\n\nimport statsmodels.api as sm\nimport statsmodels.formula.api as smf\nimport statsmodels.stats.api as sms\nimport sklearn.metrics as metrics\n#import plotly.graph_objects as go\n\nimport scipy.stats as stats\n\nimport math\n#import pickle\nimport scipy.stats\n\nfrom matplotlib import style\n\nfrom statsmodels.formula.api import ols\nfrom statsmodels.stats.outliers_influence import variance_inflation_factor\n\n#from pandasql import sqldf\n\nfrom sklearn.model_selection import train_test_split\nfrom sklearn.linear_model import LinearRegression\nfrom sklearn.metrics import mean_squared_error\n#from sklearn.preprocessing import OneHotEncoder, PolynomialFeatures, StandardScaler, RobustScaler, MinMaxScaler\nfrom sklearn.preprocessing import OneHotEncoder, PolynomialFeatures, RobustScaler, MinMaxScaler\n\nfrom sklearn.feature_selection import RFE\nfrom sklearn.metrics import make_scorer\nfrom sklearn.model_selection import cross_val_score\nfrom sklearn.experimental import enable_iterative_imputer\nfrom sklearn.impute import IterativeImputer\n\nfrom scipy.stats import pearsonr\nfrom IPython.display import display, Math, Latex\nfrom folium import plugins\nfrom plotly.subplots import make_subplots\n\n#from mlxtend.evaluate import bias_variance_decomp\n\nfrom warnings import filterwarnings\nfilterwarnings('ignore')\n\n%matplotlib inline\n```\n\n### Functions used\n\n\n```python\ndef count_unique_records(df):\n unique_records=[]\n for column in df.columns:\n n = df[column].nunique()\n unique_records.append((column,n))\n print(unique_records)\n return None\n\ndef count_dups_field(field1):\n dups = df.pivot_table(index = [field1], aggfunc ='size')\n return dups\n\ndef count_dups_fields(field1, field2):\n dups = df.pivot_table(index = [field1, field2], aggfunc ='size')\n return dups\n\ndef remove_columns(df, y_columns=['price'], x_columns=[], exclude_columns=[], add_constant=True):\n \n if x_columns==[]:\n x_columns=list(df.drop(columns=y_columns, axis=1))\n \n [x_columns.remove(columns) for columns in exclude_columns]\n \n df_x=df[x_columns]\n df_y=df[y_columns]\n \n return df_x, df_y\n \n# Function to convert geo coordinates to distance from center. I am using coordinates\ndef distance_from_center(lat_coord,lon_coord):\n R = 3959.999\n\n# I am using geo coordinates of Seattle 47.6062\u00b0 N, 122.3321\u00b0 W, from Wikipedia\n\n lat1 = math.radians(47.6062)\n lon1 = math.radians(122.3321)\n lat2 = math.radians(lat_coord)\n lon2 = math.radians(lon_coord)\n\n dlon = lon2 - lon1\n dlat = lat2 - lat1\n\n a = math.sin(dlat / 2)**2 + math.cos(lat1) * math.cos(lat2) * math.sin(dlon / 2)**2\n c = 2 * math.atan2(math.sqrt(a), math.sqrt(1 - a))\n d=R*c\n return round(d,1)\n\ndef jointplot(df):\n sns.set_style('white')\n for col in df.columns:\n g=sns.jointplot(x=col, y='price', data=df, size=5, kind='reg', marginal_ticks=True,\n joint_kws={'line_kws':{'color':'green'}}, height=15, space=0.7)\n name=col\n R2,p= scipy.stats.pearsonr(x=df[col], y=df.price)\n g.fig.suptitle('For {}: R2 coefficient is {}, p-value is {}'.format(name, round(R2,4),p))\n g.fig.tight_layout()\n g.fig.subplots_adjust(top=0.85)\n \n return None\n\ndef r2_p(df):\n for col in df.columns:\n name=col\n R2,p= scipy.stats.pearsonr(x=df[col], y=df.price)\n print('For {}: R2 coefficient is {}, p-value is {}'.format(name, round(R2,4),p))\n return None\n\n# This is a snippet from https://www.analyticsvidhya.com/blog/2020/03/what-is-multicollinearity/\ndef calc_vif(X):\n\n # Calculating VIF\n vif = pd.DataFrame()\n vif[\"variables\"] = X.columns\n vif[\"VIF\"] = [variance_inflation_factor(X.values, i) for i in range(X.shape[1])]\n\n return(vif)\n\n# This is a snippet from https://atmamani.github.io/cheatsheets/seaborn/seaborn_cheat_sheet_1/\ndef distribution(column):\n col_mean = column.mean()\n col_sd = column.std()\n skew_val = stats.skew(column, bias=False)\n kurt_val = stats.kurtosis(column,bias=False)\n \n ax = sns.distplot(column, kde_kws={\"color\": \"r\", \"lw\": 2, \"label\": \"KDE\", \"bw_adjust\": 3})\n \n ax.axvline(x=col_mean, color='black', linestyle='dashed')\n\n ax.axvline(x=col_mean + col_sd, color='red', linestyle='dotted')\n ax.axvline(x=col_mean - col_sd, color='red', linestyle='dotted')\n\n ax.set_title('$\\mu = {}$ | $\\sigma = {}$ | Skew = {} | Kurtosis = {}'.\n format(round(col_mean, 2), round(col_sd, 2), round(skew_val,2), round(kurt_val,2)))\n \n plt.subplots_adjust(top=0.5)\n plt.tight_layout()\n \n return None\n\ndef boxen_plot(df,colname):\n ax = sns.catplot(x=df[colname], y=df.price/1000000, kind=\"boxen\",\n data=df.sort_values(colname), height=7, aspect=8/5)\n ax.set_xticklabels(fontsize=12)\n ax.set_yticklabels(fontsize=12)\n plt.ylabel('Price in millions', fontsize=15)\n plt.xlabel(colname,fontsize=15)\n plt.grid()\n plt.show()\n return None\n\n#Zipcode choropleth maps with average values per a zipcode (King County)\ndef map_choropleth_zip(df, column, title, column_name):\n fig=px.choropleth_mapbox(data_frame=df, locations='zipcode', geojson=KC_zip_json, color=column, \n mapbox_style='open-street-map', zoom=8.5, height=900, featureidkey='properties.ZCTA5CE10', \n center={'lat': 47.403768, 'lon': -122.005863}, opacity=0.4,\n color_continuous_scale=px.colors.sequential.YlOrRd,\n title=title,\n template = \"plotly_dark\", \n labels={\n column: column_name})\n fig.update_layout(\n font_family=\"Arial\",\n font_size=16,\n font_color=\"white\",\n title_font_family=\"Arial\",\n title_font_color=\"white\",\n title_font_size=20)\n \n fig.update_layout(\n title={\n 'y':0.98,\n 'x':0.5,\n 'xanchor': 'center',\n 'yanchor': 'top',\n })\n \n fig.show()\n return None\n\n```\n\n### Importing data\n\n\n```python\n# Importing raw data\ndf=pd.read_csv('data/kc_house_data.csv')\npd.set_option('display.width', 1000)\ndf.head()\n```\n\n\n```python\ndf.info()\n```\n\n\n```python\n# Displaying tuples of fields and unique records in them\ncount_unique_records(df)\ndf.nunique()\n```\n\n\n```python\n# Finding all duplicate IDs\n\ndf_a=pd.DataFrame(count_dups_field('id'))\ndf_a.columns=['count']\ndf_a = df_a[df_a['count'] > 1].sort_values('count', ascending=False)\ndf_a.reset_index(level=0, inplace=True)\ndf_a\n```\n\n\n```python\n# Listing duplicate IDs and associated dataset records to find out the reason they are in the file twice\n\ndf_dup=pd.DataFrame()\nfor i in range(len(df_a)):\n df_dup=df_dup.append(df[df.id==df_a.id[i]], ignore_index=True)\ndf_dup\n```\n\n*The inspection of the records shows that the records with duplicate IDs have different sale dates and sale prices. However, all other features remain the same. Records with the same set of predictors but different prices would introduce additional \"noise\" to the data. Because there are only 353 records with that problem, I decided to drop them from the dataset.*\n\n\n```python\n#listing datatype, number of null values, min and max values in the fields of the dataset\n\nfields1=['bedrooms','bathrooms','floors','waterfront', 'view','condition','grade']\nfields2=['sqft_lot15', 'sqft_living15','yr_renovated','yr_built','sqft_above','sqft_living','sqft_lot']\nfor column in df.columns:\n type_=df[column].dtypes\n num_nulls=df[column].isna().sum()\n min_=0\n max_=0\n unique_=[0]\n if column in fields1:\n unique_=df[column].unique()\n unique_.sort()\n else:\n if column in fields2:\n min_=df[column].min()\n max_=df[column].max()\n else:\n continue\n print('Column name:', column)\n print('Type:', type_)\n print('Number of null values', num_nulls)\n print('Unique values:',unique_)\n print('Min value: ', min_, 'Max value:', max_)\n print('***********************************')\n```\n\n### Description of the fields\n\nThe file has 21597 records with 21 columns, out of which 11 columns have integer values, 8 are real numbers, and 2 are strings.
\n The annotation to the fields and associated data
(link to the definitions [here](https://github.com/emilyageller/king_county_house_prices))\n* id: Unique ID for each home sold\n\n 1. no NULL values
\n 2.integer numbers
\n 3.176 duplicate records\n>353 rows to be dropped\n\n* **date:** Date of the sale\n>no NULL values
\nstring
\n\n>Convert to DateTime type\n\n* **price** - Price of each home sold\n>no NULL values
\nReal numbers
\nMinimum price: 78000
\nMaximum price: 7700000\n\n* **bedrooms** - Number of bedrooms\n>no NULL values
\n>Integer numbers, between 1 and 33\n\n* **bathrooms** - Number of bathrooms, where .5 accounts for a room with a toilet but no shower\n>no NULL values
\n>Real numbers, between 0.5 and 8.0\n\n* **sqft_living** - Square footage of the house interior living space\n>No NULL values
\nInteger numbers
\nMinimum value: 370
\nMaximum value: 13540\n\n* **sqft_lot** - Square footage of the land lot\n>No NULL values
\nInteger numbers
\nMinimum value: 520
\nMaximum value: 1651359\n\n* **floors** - Number of floors\n>no NULL values
\n>Real numbers, between 1.0 and 3.5\n\n* **waterfront** - A categorical variable for whether the house was overlooking the waterfront or not\n>**2376** NULL values
\n>Real numbers, only two values 1.0 and 0.0
\n\n>Convert to a categorical variable
\nWaterfront, not Waterfront

\n>Replace NULL values with \"Missing\" category\n\n* **view** - A categorical variable describing how good the view of the property was\n>**63** NULL values
\n>Real numbers: 1.0, 2.0, 3.0, 4.0
\n\n>Convert to a categorical variable
\nPoor, Fair, Good Excellent\n\n>Replace NULL values with \"Missing\" category\n\n* **condition** - A categorical variable describing the condition of the house\n>no NULL values
\n>Integer numbers, between 1 and 5
\n\n>Convert to a categorical variable
\nPoor, Fair, Good, Very Good, Excellent\n\n* **grade** - A categorical variable describing the quality of construction, from 1 to 13; 1-3 falls short of building construction and design, 7 has an average level of construction and design, and 11-13 have a high quality level of construction and design.\n>no NULL values
\n>Integer numbers, between 3 and 13
\n\n* **sqft_above** - The square footage of the interior housing space that is above ground level\n>No NULL values
\nInteger numbers
\nMinimum value: 370
\nMaximum value: 9410\n\n* **sqft_basement** - The square footage of the interior housing space that is below ground level\n>No NULL values
\nString
\n\n>Convert to integer
\n\n* **yr_built** - The year the house was initially built\n>No NULL values
\nInteger numbers
\nMinimum value: 1900
\nMaximum value: 2015\n\n* **yr_renovated** - The year of the last house renovation\n>**3842** NULL values
\n>Real numbers, between 0.0 and 2015.0
\n\n>Convert to integer
\n\n* **zipcode** - What zipcode area the house is in\n>no NULL values
\nInteger numbers, 70 unique values
\n\n>Convert to categorical variable or drop\n\n* **lat** - Lattitude\n>no NULL values
\n>Real numbers\n\n* **long** - Longitude\n>no NULL values
\n>Real numbers\n\n* **sqft_living15** - The square footage of interior housing living space for the nearest 15 neighbors\n>No NULL values
\nInteger numbers
\nMinimum value: 399
\nMaximum value: 6210\n\n* **sqft_lot15** - The square footage of the land lots of the nearest 15 neighbors\n>No NULL values
\nInteger numbers
\nMinimum value: 651
\nMaximum value: 871200\n\n### Initial cleaning of the data\n\n\n```python\ndf.info()\n```\n\n#### 1. Dropping duplicate rows for houses sold twice in the timeframe of the dataset\n\n\n```python\n#Dropping duplicate rows for houses sold twice in the timeframe of the dataset\ndf = df.drop_duplicates(subset='id', keep=\"first\")\ndf\n```\n\n#### 2. Converting the 'date' field to DateTime formate and making sure it worked\n\n\n```python\n# date string to datetime type\ndf['date'] = pd.to_datetime(df['date']) \n```\n\n\n```python\n#Checking if the conversion went OK\nmask = (df['date'] > '9/24/2014') & (df['date'] <= '4/22/2015')\ndf.loc[mask]\n```\n\n#### 3. Locations of the houses with missing 'waterfront' values\n\n\n```python\n# Possible strategies:\n# 1. Check if there are waterfront properties among neighbors within a certain distance range\n# 2. Make a map and place properties with missing values on it visually\n# 3. What is the longitude of the bay shore? Any house with a missing value too far away from it \n# should have their waterfront value set to 0. Hopefully, it will eliminate most of the missing values in this field\n```\n\n\n```python\n# Leaving just coordinates of the waterfront NaN properties\nmask = (df['waterfront'].isna())\ndf_wf_na_coord=df.loc[mask]\ndf_wf_na_coord.drop(df_wf_na_coord.columns[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,19,20]], axis=1, inplace=True)\n\ndf_wf_na_coord\n```\n\n##### 3.1 Visual assesment of the houses with Null value in the 'waterfront column\n\n\n```python\n# Initializing the map based on the Wikipedia location of Seattle\nKCMap = folium.Map(location=[47.6171,-122.3249], tiles='Stamen Terrain', zoom_start=9)\n\n# for each row in the KC_house dataset with missing waterfront value, \n# plot the corresponding latitude and longitude on the map\nfor index, row in df_wf_na_coord.iterrows():\n folium.CircleMarker((row['lat'], row['long']), radius=1, weight=2, \n color='red', fill_color='red', fill_opacity=.5).add_to(KCMap)\n \ndisplay(KCMap)\n# adding the heatmap\nKCMap.add_child(plugins.HeatMap(data=df_wf_na_coord[['lat', 'long']], radius=20, blur=10))\n\n```\n\n\n```python\n# Displaying\n# the percentage of waterfront properties in the dataset with missing waterfront values,\n# the percentage of waterfront properties in the full dataset\n# The systemic error introduced to the full dataset if missing waterfront values would be replaced with 0\n\na=round((20/len(df_wf_na_coord))*100,3)\n\nb=round((len(df[df['waterfront'] == 1])/len(df)*100),2)\n\nc=round(round(((len(df[df['waterfront'] == 1])+20)/len(df)*100),3)-round((len(df[df['waterfront'] == 1])/len(df)*100),3),3)\n\nprint('{}%, {}%, {}%'.format(a, b, c))\n```\n\n*It is self-evident from the visuals above that the vast majority of the houses are located inland. Simple zooming in the maps allows a rough counting of alleged waterfront properties. The estimate is approximately 20 waterfront houses. It is 0.85% of all properties with no value in 'waterfront column (2353). In the primary dataset, the percentage of waterfront properties out of the total number of properties is 0.68%. The numbers above indicate that replacing the NaN values with 0 would introduce a systemic error of 0.01% to the whole system.*
\nConclusion: The NULL values in the 'waterfront' column will be replaced with 0.\n\n\n##### 3.2 Replacing NULL values in 'waterfront' column and converting the column to the integer datatype to make it categorical\n\n\n```python\n#Replacing NaN values in 'waterfront' column\n\ndf.loc[df.waterfront.isna(),'waterfront']=0\ndf.waterfront=df.waterfront.astype('int64')\n\n\nsubset_df = df[df['waterfront'] == 1]\ncount = len(subset_df)\nprint(count)\n#df.info()\n```\n\n\n```python\ndf.info()\n```\n\n##### 3.3 Replacing NULL values in waterfront and view field using IterativeImputer\n\n\n```python\n# Dropping columns with str datatypes to use IterativeImputer on the rest\n\ndf_to_II=df.drop(['date','id','zipcode','sqft_basement'],axis=1)\ndf_to_II.info()\n```\n\n\n```python\n# Using IterativeImputer to fill in Nan cells\n\nimp = IterativeImputer(max_iter=10,random_state=0)\nimp.fit(df_to_II)\nimputed_df_to_II = imp.transform(df_to_II)\nimputed_df = pd.DataFrame(imputed_df_to_II, columns=df_to_II.columns)\nimputed_df.info()\n```\n\n\n```python\ndf.head(20)\n```\n\n\n```python\nimputed_df.head(20)\n```\n\n\n```python\n# Creating a subset of records with imputed NaN values\n\nsubset_df = imputed_df[(imputed_df['yr_renovated'] < 1934) & (imputed_df['yr_renovated'] != 0)]\ncount = len(subset_df)\nprint(count)\n```\n\n\n```python\n# Listing Imputed NaN values in yr_renovated column\nlist_unique=list(df.yr_renovated.unique())\n\ninverse_boolean_series = ~imputed_df.yr_renovated.isin(list_unique)\ninverse_filtered_df = imputed_df[inverse_boolean_series]\ninverse_filtered_df.yr_renovated.sort_values()\n```\n\n\n```python\n# Listing Imputed NaN values in waterfront column\ninverse_boolean_series = ~imputed_df.waterfront.isin([0,1])\ninverse_filtered_df = imputed_df[inverse_boolean_series]\ninverse_filtered_df.waterfront.sort_values() \n\n```\n\n\n```python\n# Listing Imputed NaN values in view column\ninverse_boolean_series = ~imputed_df.view.isin([0,1,2,3,4])\ninverse_filtered_df = imputed_df[inverse_boolean_series]\ninverse_filtered_df.view.sort_values() \n```\n\n**Conclusion:** Based on the results of IterativeImputer the original approach of replacing missing values in 'waterfront', 'yr_renovated', and 'view' with 0 will be taken
\n\n#### 4. Replacing NULL values in yr_renovated column with 0. value and changing the type to integer\n\n\n```python\ndf.loc[df.yr_renovated.isna(),'yr_renovated']=0.0\ndf.yr_renovated=df.yr_renovated.astype('int64')\n```\n\n#### 5. Replacing '?' values in 'sqft_basement' column with 0. value and changing the type to integer\n\n\n```python\ndf.loc[(df.sqft_basement == '?')]\n```\n\n\n```python\n# Converting '?' sqft_basement values to 0 and listing unique values\ndf.loc[(df.sqft_basement == '?'),'sqft_basement']='0.0'\n\ndf.sqft_basement=df.sqft_basement.astype('float64')\n\ndf.sqft_basement=df.sqft_basement.astype('int64')\n\n#Checking the result\ndf_temp=df.sqft_basement.unique()\nlist1 = df_temp.tolist()\nlist1.sort()\nprint(list1)\n```\n\n#### 6. Replacing NULL values in 'view' column with 0.0 and making it integer\n\n\n```python\ndf.loc[df.view.isna(),'view']=0\ndf.view=df.view.astype('int64')\n\ndf.info()\n```\n\n# Scrub and Explore\n\n## Additional data cleaning\n\n### Dropping non-needed fields\n\n\n```python\n# Dropping 'zipcode' variable because I think there are better indicators of a location.\n# Zipcodes boundaries are usually drawn out of convenience for postal services or\n# other more formal reasons than geographic location\n\n# Dropping 'id' field\n# Resettingh index because of the removal of the duplicates\n\ndf_1=df\ndf_1=df_1.drop(['id','zipcode'], axis=1)\n#df_1.tail()\n\n\ndf_1.reset_index()\n```\n\n **21420** records out of the original **21597** left\n\n\n\n## Exploring distributions and correlation of original variables\n\n### Numerical variables: Investigating distributions and correlations between the original, minimally processed predictors and the target (price) \n\n#### Histograms and pair correlations of the original predictor variables\n\n\n```python\ndf_1.info()\n```\n\n\n```python\n# Investigating histograms\n\ndf_1.hist(figsize=(20, 30), bins='auto')\n```\n\n**Based on the histograms above**\n> 1. The following variables should be considered categorical:\n>>Waterfront
\nCondition
\nView
\n> 2. sqft_basement, sqft_lot, sqft_lot15, and yr_renovated have a large number of zeros and are strong candidates for removal of outliers and/or engineered variables
\n> 3. Latitudes and Longitudes can be used as descriptors of a geographic location of a property. However, I think there is a better variable to describe the location of a property, a distance from the center of the city, which can be calculated from geocoordinates.
\n >4. The target variable, the price of the property, has a strong positive skew attributed to outliers in the higher price bracket. The strategy is to **remove the outliers and to transform the variable** to make it more normally distributed\n\n\n```python\n# Checking for correlations between the variables with Pearson coefficient between 1 and 0.3\n# I am using the same approach and reusing the code from Lesson 19\n\ndf_coeff=df_1.corr().abs().stack().reset_index().sort_values(0, ascending=False)\ndf_coeff['pairs'] = list(zip(df_coeff.level_0, df_coeff.level_1))\ndf_coeff.set_index(['pairs'], inplace = True)\ndf_coeff.drop(columns=['level_1', 'level_0'], inplace = True)\ndf_coeff.columns = ['cc']\ndf_coeff.drop_duplicates(inplace=True)\ndf_coeff[((df_coeff.cc>.3) & (df_coeff.cc <1))]\n```\n\n **The variables which have the strongest correlations with the price are**
\n* sqft_living
\n* grade
\n* sqft_above
\n* sqft_living15
\n* bathrooms
\n* view
\n* bedrooms
\n*lat\n\n#### Establishing an intermediate dataframe and removing some outliers from it\n\n\n```python\n# Removing extreme values from sqft_lot, sqft_lot15, bathrooms, and price\n# On one hand, it can be left to the step of outlier removal, on the other doing it now\n# will help a visual investigation of the distributions of the variables mentioned above\n\ndf_2=df_1[(df_1.sqft_lot >100) & (df_1.sqft_lot<40000)]\n\ndf_2=df_2[(df_2.sqft_lot15 >100) & (df_2.sqft_lot15<40000)]\n\ndf_2=df_2[df_2.price < 2500000]\n\ndf_2=df_2[df_2.bedrooms < 30]\n\ndf_2.info()\n\n```\n\n\n```python\n# Creating a new categorical variable: month\ndf_2['month'] = pd.to_datetime(df_2['date']).dt.month\n\n# Creating a new numerical variable: distance (distance from center)\ndf_2['distance'] = df_2.apply(lambda row: distance_from_center(row.lat,abs(row.long)), axis=1) \n\n# Creating a new categorical variable: basement_exists (1/0), integer datatype\ndf_2.loc[(df_2['sqft_basement'] > 50), 'basement_exists'] = 1\ndf_2.loc[(df_2['sqft_basement'] <= 50), 'basement_exists'] = 0\ndf_2.basement_exists=df_2.basement_exists.astype('int64')\n```\n\n\n```python\n# Creating a new categorical variable (integer datatype) renovation_done with values [0,1,2,3,4]\n\n# 0 representing renovation never done on houses more than 9 years old (yr_built between 2015 and 2006)\n# 1 representing renovation done more than or equal 50 years ago\n# 2 representing renovation done between 30 and 49 years ago\n# 3 representing renovation done between 29 and 10 years ago\n# 4 representing renovation done between 9 and 1 year ago OR houses built less or equal 9 years ago\n# (yr_built between 2015 and 2006)\n\ndf_2.loc[((df_2['yr_renovated'] == 0) & (df_2['yr_built'] < 2006)), 'renovation_done'] = 0\ndf_2.loc[((2015-df_2['yr_renovated'] >= 50) & (df_2['yr_renovated'] != 0)), 'renovation_done'] = 1\ndf_2.loc[((2015-df_2['yr_renovated'] < 50) & (2015-df_2['yr_renovated'] >= 30)), 'renovation_done'] = 2\ndf_2.loc[((2015-df_2['yr_renovated'] < 30) & (2015-df_2['yr_renovated'] >= 10)), 'renovation_done'] = 3\ndf_2.loc[((2015-df_2['yr_renovated'] < 10) | (df_2['yr_built'] >= 2006)), 'renovation_done'] = 4\n\ndf_2.renovation_done=df_2.renovation_done.astype('int64')\n```\n\n\n```python\n# Dropping sqft_basement variable\n# Dropping latitude & longtitude variable\n# Dropping date variable# Resetting index due to removal of records with extreme values in \n# sqft_lot, sqft_lot15, bathrooms and price\n\ndf_2=df_2.drop(['sqft_basement','date', 'lat','long','yr_renovated'], axis=1)\n\ndf_2=df_2.reset_index(drop=True)\n\ndf_2\n```\n\n**20015** records out of the original **21597** left
\n*Index reset*\n\n\n#### Plotting numerical variables against the target variable\n\n\n```python\n# Dropping categorical variables to simplify the analysis of the numerical ones\n\ndf_num1=df_2.drop(['waterfront','view','condition','basement_exists','renovation_done','month'], axis=1)\n\n```\n\n\n```python\n# Displaying the Coefficients of Determination and p-values for the remaining variables\nr2_p(df_num1)\n```\n\n\n```python\n# Joint plot of the original (not altered) numerical variables\njointplot(df_num1)\n```\n\n\n```python\ndf_num1.info()\n```\n\n\n```python\n# Removing extreme values from bedrooms, bathrooms, floors, and distance\n# The observations with the extreme values of these variables are visually identifiable \n\ndf_num2=df_num1[(df_num1.bedrooms < 9) & (df_num1.bathrooms < 5.5) &(df_num1.distance < 30) \n & (df_num1.floors<3.5)]\ndf_num2.drop(['yr_built'], axis=1, inplace=True)\ndf_num2.reset_index(drop=True, inplace=True)\n```\n\n\n```python\ndf_num2.info()\n```\n\n*df_num2 DataFrame*
**19811** records out of original **21597** left
\n*Index reset*\n\n\n```python\n# Joint plot of numerical variables after adjustments\njointplot(df_num2)\n```\n\n\n```python\nr2_p(df_num2)\n```\n\n\n```python\ndf_num2.info()\n```\n\n#### Visualizing numerical predictors correlation with the price and with each other using heat map\n\n\n```python\nfig, ax = plt.subplots(figsize=(15,10))\nsns.heatmap(df_num2.corr(), cmap=\"cubehelix\", annot=True)\n```\n\n\n```python\n# Bedrooms is relatively highly correlated with bathrooms and sqft_living; correlations of floors, sqft_lot and sqft_lot15 \n# with prices are low. Dropping these variables\n\ndf_num3=df_num2.drop(['bedrooms','sqft_lot','sqft_lot15','floors'], axis=1)\n```\n\n\n```python\nfig, ax = plt.subplots(figsize=(15,10))\nsns.heatmap(df_num3.corr(), cmap=\"cubehelix\", annot=True)\n```\n\n\n```python\nr2_p(df_num3)\n```\n\n#### Exploring the distribution of the remaining numerical predictors\n\n\n```python\n# Plotting KDE and distribution plots of the remaining variables\n\ncontinuous=['price','sqft_living','sqft_living15','sqft_above','distance']\nfor col in continuous:\n fig, ax =plt.subplots(figsize=(5, 5))\n distribution(df_num3[col])\n\ndiscrete=['bathrooms','grade']\nfor col in discrete:\n fig, ax =plt.subplots(figsize=(5, 5))\n ax = sns.distplot(df_num3[col], bins=10, kde_kws={\"color\": \"r\", \"lw\": 2, \"label\": \"KDE\", \"bw_adjust\": 4})\n skew_val = stats.skew(df_num3[col], bias=False)\n kurt_val = stats.kurtosis(df_num3[col],bias=False)\n print('Column: {} | Skewness = {} | Kurtosis = {}'.\n format(col, round(skew_val,2), round(kurt_val,2)))\n```\n\n*All but one variable (distance) are left shifted, which is indicated by the skewness values >1, with price has the most skewed distribution. Because the skewness it to the left (positive values), the log transformation might be needed to normalize the variables. Kurtosis values for all variables are different from 0 (Pearson's definition of kurtosis of a normal distribution).
\n Price distribution is highly Leptokurtic, other variables are slightly Leptokurtic (sqft_living, sqft_above, sqft_living15, grade), slightly Platykurtic (distance) or almost Mesokurtic (bathrooms)*\n\n\n\n```python\n# Diamond box plots for descrete numerical variables\n\ndiscrete=['bathrooms','grade']\nfor col in discrete:\n\n ax = sns.catplot(x=col, y='price', kind=\"boxen\",\n data=df_num3.sort_values(col), height=9, aspect=12/9)\n ax.set_xticklabels(rotation=45, ha='right', fontsize=12)\n ax.set_yticklabels(fontsize=12)\n plt.ylabel('Price in millions', fontsize=15)\n plt.xlabel(col, fontsize=15);\n plt.grid()\n plt.show()\n```\n\n*Based on the distribution plots, variable 'bathroom' has a symmetrical distribution (Skewness is 0.28), with a very low kurtosis (0.09) indicative of a Mesokurtic curve (Gaussian distribution has a kurtosis of 0 by Pearson's definition used by scipy.stats.kurtosis method)
\nVariable 'grade' is slightly skewed to the right (0.73) and relatively low kurtosis, slightly above 1.
\nThe pronounced correlation of these variables with the price is identifiable in the box plots above. The plots show that the numbers of outliers in the distribution of the variables are reasonable. Both variables have a wider range of values in the higher price brackets.*\n\n### Numerical variables: Exploring Mutual Correlation Coefficients and Variance Inflation Factor\n\n#### Using VIF as an indicator of collinearity between independent variables\n\n\n```python\ncalc_vif(df_num3)\n```\n\n\n```python\n# Dropping fields that have high VIF. I decided to leave in sqft_living versus sqft_living15 due to the fact that it is\n# easier to interpret in a model and it is a feature under control of a property owner (versus sqft_living15 which is\n# a charecteristic of a neighborhood)\n\ndf_num4=df_num3.drop(['price','sqft_above','sqft_living15'],axis=1)\n\ncalc_vif(df_num4)\n```\n\n#### Pearson coefficients analysis of the remaining independent variables\n\n\n```python\ndf_num4['price']=df_num3['price']\n```\n\n\n```python\n# Displaying R2s for the remaining variables\nr2_p(df_num4)\n```\n\n\n```python\n# Displaying correlations between predictors and the target\njointplot(df_num4)\n```\n\n\n```python\n# Using heatmap to visualize mutual relationships between predictors and their correlations with the target\nfig, ax = plt.subplots(figsize=(15,10))\nsns.heatmap(df_num4.corr(), cmap=\"Reds\", annot=False)\n```\n\n\n```python\n# Checking for correlations between the variables with pearson coefficient between 1 and 0.3\n\n\ndf_coeff=df_num4.corr().abs().stack().reset_index().sort_values(0, ascending=False)\ndf_coeff['pairs'] = list(zip(df_coeff.level_0, df_coeff.level_1))\ndf_coeff.set_index(['pairs'], inplace = True)\ndf_coeff.drop(columns=['level_1', 'level_0'], inplace = True)\ndf_coeff.columns = ['cc']\ndf_coeff.drop_duplicates(inplace=True)\ndf_coeff[((df_coeff.cc>.3) & (df_coeff.cc <1))]\n```\n\n*Mutual correlation coefficients between the remaining independent variables are \nslightly higher or below 0.7. I am leaving sqft_living, grade, and bathroom variables in because of their logical connection with a property price despite their multicollinearity (0.74 & 0.73 are above the 0.7 threshold).*
\n\nTherefore the remaining numerical variables for modeling are

\n1. grade
\n2. bathrooms
\n3. sqft_living
\n4. distance
\n\n\n\n```python\ndf_num4.describe()\n```\n\n### Categorical variables: Investigating distributions and the raw correlations between the original, minimally processed predictor and the target (price)\n\n#### Original categorical variables: waterfront, view, condition, basement_exists, renovation_done, month\n\n\n```python\n#Creating a DataFrame with categorical variables\n\ndf_cat1 = df_2.filter(['price','waterfront','view','condition','basement_exists','renovation_done','month'], axis=1)\n\n```\n\n#### Visual investigation of the box plots\n\n\n```python\ndf_cat1\n```\n\n\n```python\nnames=['waterfront','view','condition','basement_exists','renovation_done','month']\nfor col in names:\n boxen_plot(df_cat1, col)\n```\n\n*Based on the plots, it is quite clear that 'month', 'condition' and 'basement_exists' variables do not affect the price of the properties and can be dropped from the categorical variables*\n\n#### Dumming out variables in the categorical DataFrame\n\n\n```python\n# Using OheHotEncoder to dummy categorical out. Based on the boxplots above, I am leaving \n# 'month','basement_exists','condition' variables out, they seem not to have too much effect on the target.\n# The reason for removing 'waterfront' variable is a tiny porting of all properties in the dataset.\n\nohe=OneHotEncoder(drop='first')\ndf_cat_transform=ohe.fit_transform(df_cat1.drop(['price','month','basement_exists','condition','waterfront'], axis=1))\ndf_cat1_trsfm=pd.DataFrame(df_cat_transform.todense(), \n columns=ohe.get_feature_names(['view','renovation_done']))\ndf_cat1_trsfm.info()\n```\n\n\n```python\n# Adding a price column and resetting the index \n# Due to the removal of the extreme values in numerical values, some rows have Nan price values; these rows are dropped\n\ndf_cat1_trsfm['price']=df_num4['price']\n\ndf_cat1_trsfm=df_cat1_trsfm[df_cat1_trsfm.price.notna()]\n\ndf_cat1_trsfm=df_cat1_trsfm.reset_index()\n\ndf_cat1_trsfm=df_cat1_trsfm.drop('index', axis=1)\n\ndf_cat1_trsfm\n```\n\n*df_cat1_trsfm DataFrame*
**19811** records out of original **21597** left
\n**Index reset**\n\n\n```python\ndf_cat1_trsfm.info()\n```\n\n# Model\n\n## Data Modeling\n\n\n### Baseline model\n\n>\u201cEverything should be made as simple as possible, but no simpler.\u201d
Albert Einstein
\n\nThe chosen baseline model is a model with only one numerical variable, grade.\n\n#### Creating a model\n\n\n```python\n## Create the formula and the model\n\nf = 'price~grade'\n\nmodel_baseline = smf.ols(f, df_num4).fit()\nmodel_baseline.summary()\n```\n\n\n```python\nfig = sm.graphics.qqplot(model_baseline.resid,dist=stats.norm,fit=True,line='45')\n```\n\n*The baseline model has a coefficient of determination of 0.452, indicating that roughly 45% of the observations fit the model. F-statistics is very high that indicates that the baseline model is a significant improvement of the \"intercept only model\"
\n The Skewness and the Kurtosis values indicate non-normal distribution of the target variable
\n QQ plot is also indicative of the abnormal distribution of the residuals, especially in the upper Quantile*\n\n### Model 1 (all numerical variables considered significant, see Explore section)\n\n\n```python\n## Create a formula including the remaining numerical variables\n\nvariables_to_include = ' + '.join(df_num4.drop('price',axis=1).columns)\n\n## Create the formula and the model\nf = \"price~\" + variables_to_include\n\nmodel_1 = smf.ols(f, df_num4).fit()\nmodel_1.summary()\n```\n\n#### QQ plot: to access normality of the target variable\n\n\n```python\nfig = sm.graphics.qqplot(model_1.resid,dist=stats.norm,fit=True,line='45')\n```\n\n*The summary of the model above indicates that
\n 1. All the independent variables coefficients and the intercept value are significant (p-values< 0.05)
\n 2. The coefficient of determination (R2) is not very high, but it is significantly higher than R2 of the baseline model. It indicates that about 66.8 percent of the observations fall within the regression line
\n 3. The skew and the Kurtosis values indicate the highly non-normal distribution of the target variable
\n 4. The high value of JB coefficient also indicates that the data is highly non-normal
*\n\n*From the model's QQ plot, it is also quite obvious that the 'price' target variable is not normally distributed. A steep swing up indicates that the higher-priced houses are less likely to fit the baseline model and are more spread out. One possible reason might be an unusually large number of outliers in the dataset*\n\n*There are two potential approaches that can be taken
\n 1. Normalization by either log or square root transformation
\n 2. Removal of outliers*\n\n#### Plot regression results against each regressor: accessing linearity of there relationship with the target and their homoscedasticity \n\n\n```python\nfor col in (df_num4.drop('price',axis=1).columns):\n fig = sm.graphics.plot_regress_exog(model_1, col, fig=plt.figure(figsize=(12,8)))\n```\n\n*The results indicate that all of the predictors display linear relationship with the target. The distance, sqft_living and bathrooms variables display less heteroscedasticity than the grade variable does.

\nThere might be several appropriate ways to address this issue
\n 1. Log transformation of the target and/or independent variables
\n 2. Using either Generalized Least Squares or Weighted Least squares
\n 3. Bootstrapping
*\n\n### Model 2 (adding categorical variables)\n\n#### Creating a model with all numerical variables and all categorical variables\n\n\n```python\ndf_num4\n```\n\n\n```python\ndf_cat1_trsfm\n```\n\n\n```python\n# DatyaFrames to concatinate: df_cat1_trsfm AND df_num4\ndf_num_cat_1 = pd.concat([df_num4, df_cat1_trsfm.drop('price',axis=1)],axis=1)\n```\n\n\n```python\n# Visualizing multicollinearity in the new dataset\nfig, ax = plt.subplots(figsize=(15,10))\nsns.heatmap(df_num_cat_1.corr(), cmap=\"cubehelix\", annot=True)\n```\n\n*The matrix indicates no strong correlation between the price and any of the categorical variables. However, renovation_done_4 correlation is slightly higher than the rest of the categorical variables. Conceptually this variables indicative of a recent renovation or a newer property.
\nThere is no high expectations that adding the categorical variables to the mix will significantly improve the model.*\n\n\n```python\ndf_num_cat_1\n```\n\n\n```python\n# Create a formula for the numerical variables from the basemodel\n# AND the categorical variables from the previous section\n\nvariables_to_include = ' + '.join(df_num_cat_1.drop('price',axis=1).columns)\n\n## Create the formula and the model\nf = \"price~\" + variables_to_include\n\n\nmodel_2 = smf.ols(f, df_num_cat_1).fit()\nmodel_2.summary()\n```\n\n\n```python\nfig = sm.graphics.qqplot(model_2.resid,dist=stats.norm,fit=True,line='45')\n```\n\n*The summary indicates a very slight improvement over the previous model, 66.9% versus 66.8% of all of the observations fall within the results of the line formed by the regression equation.

\n It is also evident that p-values of most of the categorical values are very high, indicating their insignificance in the model. However, because they describe the same feature, I am leaving them in for now

\nThe residuals normality did not improve*\n\n### Model 3 (preprocessing and removal of outliers)\n\n#### Scaling with Robust Scaler\n\n*I ruled out the scaling of the data in this step of the process because it would not change the statistics of the mode, though the correlation coefficients would become more compatible with each other.\n

I am leaving the snippets in the notebook in case I reconsider.\n
The next step is a removal of ourliers*\n\n\n```python\n# Using RobustScaler to scale the data\n\"\"\"# Using RobustScaler, which transforms the feature vector by subtracting the median and then dividing by the\n# interquartile range (%25-75%). It is the most robust to the ourliers\n\ndf_for_scalers=df_num_cat_2.copy()\n\ncols=['bathrooms','grade','sqft_living15','distance','waterfront_1','renovation_done_4']\nscaler = RobustScaler()\nrobust_df = scaler.fit_transform(df_for_scalers.drop('price',axis=1))\nrobust_df = pd.DataFrame(robust_df, columns=cols)\n\ndf_num_cat_2.describe()\"\"\"\n\n\"\"\"robust_df.describe()\"\"\"\n\n\"\"\"fig, (ax1, ax2) = plt.subplots(ncols = 2, figsize =(20, 5))\nax1.set_title('Before Scaling')\n \nsns.kdeplot(df_for_scalers['distance'], ax = ax1, color ='blue')\n\nax2.set_title('After Robust Scaling')\n \nsns.kdeplot(robust_df['distance'], ax = ax2, color ='red')\n\n\nplt.show()\"\"\"\n```\n\n\n```python\n# Building and summary of the model with scaled data\n\"\"\"robust_df['price']=df_for_scalers['price']\nrobust_df.info()\n\n\nnum_cat_var2_robust = ' + '.join(robust_df.drop('price',axis=1).columns)\n\n## Create the formula and the model\nf = \"price~\" + num_cat_var2_robust\n\n\nmodel_num_cat_2_robust = smf.ols(f, robust_df).fit()\nmodel_num_cat_2_robust.summary()\n\ncoeffs=model_num_cat_2.params\ncoeffs.sort_values().round(2)\n\ncoeffs=model_num_cat_2_robust.params\ncoeffs.sort_values().round(2)\"\"\"\n```\n\n#### Removal of outliers\n\n##### IQR method\n\n###### Using IQR\n\n\n```python\ndf_num_cat_1\n```\n\n\n```python\ndf_num_cat_2=df_num_cat_1.copy()\n\n# Due to the fact that discrete variables are not suitable for IQR method outlier removal, they are being dropped \n# from the DataFrame. They will be added back to the Dataframe for modeling\n\ndf_num_cat_2=df_num_cat_2.drop(['view_1','view_2','view_3','view_4',\n 'renovation_done_1','renovation_done_2','renovation_done_3','renovation_done_4',\n 'grade','bathrooms'], axis=1)\n\nQ1 = df_num_cat_2.quantile(q=.25)\nQ3 = df_num_cat_2.quantile(q=.75)\nIQR = df_num_cat_2.apply(stats.iqr)\n\ndf_num_cat_3 = df_num_cat_2[~((df_num_cat_2 < (Q1-1.5*IQR)) | (df_num_cat_2 > (Q3+1.5*IQR))).any(axis=1)]\n\ndf_num_cat_3\n```\n\n\n```python\ndf_num_cat_3['grade']=df_num_cat_1['grade']\ndf_num_cat_3['bathrooms']=df_num_cat_1['bathrooms']\ndf_num_cat_3['view_1']=df_num_cat_1['view_1']\ndf_num_cat_3['view_2']=df_num_cat_1['view_2']\ndf_num_cat_3['view_3']=df_num_cat_1['view_3']\ndf_num_cat_3['view_4']=df_num_cat_1['view_4']\ndf_num_cat_3['renovation_done_1']=df_num_cat_1['renovation_done_1']\ndf_num_cat_3['renovation_done_2']=df_num_cat_1['renovation_done_2']\ndf_num_cat_3['renovation_done_3']=df_num_cat_1['renovation_done_3']\ndf_num_cat_3['renovation_done_4']=df_num_cat_1['renovation_done_4']\n```\n\n\n```python\ndf_num_cat_3.info()\n```\n\n\n```python\ndf_num_cat_3=df_num_cat_3.reset_index()\n\ndf_num_cat_3.info()\n```\n\n\n```python\ndf_num_cat_3=df_num_cat_3.drop('index', axis=1)\ndf_num_cat_3.info()\n```\n\n*df_num_cat_3 DataFrame

18619 records out of the original 21597 left
\nIndex reset*\n\n###### Building the model\n\n\n```python\n## Formula is the same, model is for the cleaned DF\n\nvariables_to_include_3_1 = ' + '.join(df_num_cat_3.drop('price',axis=1).columns)\nf = \"price~\" + variables_to_include_3_1\n\n\nmodel_3_1 = smf.ols(f, df_num_cat_3).fit()\nmodel_3_1.summary()\n```\n\n\n```python\nfig = sm.graphics.qqplot(model_3_1.resid,dist=stats.norm,fit=True,line='45')\n```\n\n*While the IQR removal of outliers decreased the R squared of the model, it made the distribution more normal (Skew and Kurtosis values are almost within the normality ranges). This fact is also reflected by the QQ plot of the model residuals. Unfortunately, the Coefficient of determination dropped*\n\n##### Z-score method\n\n###### Using Z scores\n\n\n```python\ndf_num_cat_1\n```\n\n\n```python\ndf_num_cat_4=df_num_cat_1.copy()\n\ndf_num_cat_4=df_num_cat_4.drop(['view_1','view_2','view_3','view_4',\n 'renovation_done_1','renovation_done_2','renovation_done_3','renovation_done_4',\n 'grade','bathrooms'], axis=1)\n\ndf_num_cat_4['z_sqft_living']=stats.zscore(df_num_cat_4['sqft_living'])\ndf_num_cat_4['z_distance']=stats.zscore(df_num_cat_4['distance'])\ndf_num_cat_4['z_price']=stats.zscore(df_num_cat_4['price'])\n\ndf_num_cat_4\n```\n\n\n```python\ndf_num_cat_5=df_num_cat_4[(abs(df_num_cat_4.z_price) < 3)]\n\ndf_num_cat_5=df_num_cat_5[(abs(df_num_cat_5.z_distance) < 3)]\n\ndf_num_cat_5=df_num_cat_5[(abs(df_num_cat_5.z_sqft_living) < 3)]\ndf_num_cat_5\n```\n\n\n```python\ndf_num_cat_5['grade']=df_num_cat_1['grade']\ndf_num_cat_5['bathrooms']=df_num_cat_1['bathrooms']\ndf_num_cat_5['view_1']=df_num_cat_1['view_1']\ndf_num_cat_5['view_2']=df_num_cat_1['view_2']\ndf_num_cat_5['view_3']=df_num_cat_1['view_3']\ndf_num_cat_5['view_4']=df_num_cat_1['view_4']\ndf_num_cat_5['renovation_done_1']=df_num_cat_1['renovation_done_1']\ndf_num_cat_5['renovation_done_2']=df_num_cat_1['renovation_done_2']\ndf_num_cat_5['renovation_done_3']=df_num_cat_1['renovation_done_3']\ndf_num_cat_5['renovation_done_4']=df_num_cat_1['renovation_done_4']\n```\n\n\n```python\ndf_num_cat_5=df_num_cat_5.drop(['z_sqft_living','z_distance','z_price'], axis=1)\n```\n\n\n```python\ndf_num_cat_5=df_num_cat_5.reset_index()\ndf_num_cat_5=df_num_cat_5.drop('index', axis=1)\ndf_num_cat_5.info()\n```\n\n*df_num_cat_5 DataFrame

19261 records out of the original 21597 left\n
\nIndex reset*\n\n##### Building the model\n\n\n```python\n## Formula is the same, model is for the cleaned DF\n\nvariables_to_include_3_2 = ' + '.join(df_num_cat_5.drop('price',axis=1).columns)\nf = \"price~\" + variables_to_include_3_2\n\n\nmodel_3_2 = smf.ols(f, df_num_cat_5).fit()\nmodel_3_2.summary()\n```\n\n\n```python\nfig = sm.graphics.qqplot(model_3_2.resid,dist=stats.norm,fit=True,line='45')\n```\n\n*The R squared of the model is 0.654 and F-statistics is higher than for the previous model\n
The IQR method of outliers removal made the residual distribution more normal than Z-score method due to the former having more strict criteria. The decision is to use the dataset compiled after Z-score outlier removal. \n
\nThe next step is Log transformation of the target variable*\n\n\n### Model 4 (Using log and square root transformations on the target variable)\n\n#### Log Transformation\n\n\n```python\ndf_num_cat_5.info()\n```\n\n\n```python\n# Log transform\ndf_num_cat_5_log=df_num_cat_5.copy()\n\ndf_num_cat_5_log['log_price'] = df_num_cat_5['price'].map(lambda x: np.log(x))\ndf_num_cat_5_log\n```\n\n\n```python\ndf_num_cat_5_log.info()\n```\n\n\n```python\n# Histogram of log_price and price\ncontinuous=['price','log_price']\nfor col in continuous:\n fig, ax =plt.subplots(figsize=(5, 5))\n distribution(df_num_cat_5_log[col])\n\n```\n\n*The transformation worked well, improving the normality of the 'price' variable. Log_price distribution looks more symmetrical. Skewness improved dramatically (from 1.18 to -0.01, 0 being perfectly symmetrical)\n
\n Kurtosis value decreased, making the curve more Mesokurtic\u00a0(close to a Gaussian curve). It is an expected effect of log transform.\n
\nThe next step is test a square root transformation.*\n\n\n\n```python\n# Square transform\ndf_num_cat_5_sqrt=df_num_cat_5.copy()\n\ndf_num_cat_5_sqrt['sqrt_price'] = df_num_cat_5['price'].map(lambda x: np.sqrt(x))\ndf_num_cat_5_sqrt\n```\n\n\n```python\n# Histogram of log_price and price\ncontinuous=['price','sqrt_price']\nfor col in continuous:\n fig, ax =plt.subplots(figsize=(5, 5))\n distribution(df_num_cat_5_sqrt[col])\n\n```\n\n*The square root transformation also worked well in improving the normality of the 'price' variable. \nsqrt_price distribution looks more symmetrical. Skewness improved dramatically (from 1.18 to -0.59, 0 being perfectly symmetrical)\n
\n However, both of the parameters are worse that the parameters of log_price distribution.\n
\nThe next step is create two separate models and to see if the transformations made a difference*\n\n\n#### Model using log transformed target variable\n\n\n```python\n## Formula is the same, model is for the cleaned DF\n\nvariables_to_include_4_1 = ' + '.join(df_num_cat_5_log.drop(['price','log_price'],axis=1).columns)\nf = \"log_price~\" + variables_to_include_4_1\n\n\nmodel_4_1 = smf.ols(f, df_num_cat_5_log).fit()\nmodel_4_1.summary()\n```\n\n#### Model using square root transformed target variable\n\n\n```python\n## Formula is the same, model is for the cleaned DF\n\nvariables_to_include_4_2 = ' + '.join(df_num_cat_5_sqrt.drop(['price','sqrt_price'],axis=1).columns)\nf = \"sqrt_price~\" + variables_to_include_4_2\n\n\nmodel_4_2= smf.ols(f, df_num_cat_5_sqrt).fit()\nmodel_4_2.summary()\n```\n\n\n```python\nfig = sm.graphics.qqplot(model_4_1.resid,dist=stats.norm,fit=True,line='45')\n```\n\n\n```python\nfig = sm.graphics.qqplot(model_4_2.resid,dist=stats.norm,fit=True,line='45')\n```\n\n*Both models have improved the R squared and the F-statistics of the previous models. The residuals of both models display a close-to-normal distribution. Log transformation helped improve the upper part of the distribution, while square root transformation worked better in the lower part of the distribution. \n
\nR squared of the square root transformation-based model is slightly higher, while its kurtosis value is slightly worse than the kurtosis value of the log-transformed price model. The decision is to use the log-transformed target variable.\n
\nThe next step is to remove unnecessary categorical variables, scale the remaining variables, and built the last model with coefficients in the regression model, which are easy to compare and interpret*\n\n\n\n```python\n# Removing all categorical values with high p-values and renaming the renovation_done_4\n# to a more prominent name, reflective of the definition of the field (see Categorical variables subsection \n# in Exploring distributions and correlations section)\n\ndf_num_cat_5_log_cropped=df_num_cat_5_log.copy()\ndf_num_cat_5_log_cropped=df_num_cat_5_log_cropped.drop(['view_1','view_2','view_3','view_4','renovation_done_1',\n 'renovation_done_2','renovation_done_3'], axis=1)\ndf_num_cat_5_log_cropped.rename(columns={'renovation_done_4': 'recent_renovation_new'}, inplace=True)\n```\n\n\n```python\ndf_num_cat_5_log_cropped\n```\n\n\n```python\n# Standardizing the independent variables. I decided to do it manually due to a better control of the utput\n\n\ndf_num_cat_5_log_cropped_st=df_num_cat_5_log_cropped.copy()\n\n\nlv_min=df_num_cat_5_log_cropped_st.sqft_living.min()\nlv_range=df_num_cat_5_log_cropped_st.sqft_living.max()-df_num_cat_5_log_cropped_st.sqft_living.min()\ndistance_range=df_num_cat_5_log_cropped_st.distance.max()-df_num_cat_5_log_cropped_st.distance.min()\ndistance_min=df_num_cat_5_log_cropped_st.distance.min()\nbathroom_range=df_num_cat_5_log_cropped_st.bathrooms.max()-df_num_cat_5_log_cropped_st.bathrooms.min()\nbathroom_min=df_num_cat_5_log_cropped_st.bathrooms.min()\ngrade_range=df_num_cat_5_log_cropped_st.grade.max()-df_num_cat_5_log_cropped_st.grade.min()\ngrade_min=df_num_cat_5_log_cropped_st.grade.min()\n\n\ndf_num_cat_5_log_cropped_st['sqft_living_st']=df_num_cat_5_log_cropped_st.apply(lambda row: round((row.sqft_living-lv_min)/lv_range,3), axis=1)\ndf_num_cat_5_log_cropped_st['distance_st']=df_num_cat_5_log_cropped_st.apply(lambda row: round((row.distance-distance_min)/distance_range,3), axis=1)\ndf_num_cat_5_log_cropped_st['bathrooms_st']=df_num_cat_5_log_cropped_st.apply(lambda row: round((row.bathrooms-bathroom_min)/bathroom_range,3), axis=1)\ndf_num_cat_5_log_cropped_st['grade_st']=df_num_cat_5_log_cropped_st.apply(lambda row: (row.grade-grade_min)/grade_range, axis=1)\n#df_num_cat_5_sqrt_cropped_st['recent_renovation_new_str']=df_num_cat_5_sqrt_cropped_st['recent_renovation_new'].astype('str')\n\ndf_num_cat_5_log_cropped_st\n```\n\n\n```python\n## Formula is the same, model is for the cleaned DF\n\nvariables_to_include_4_3 = ' + '.join(df_num_cat_5_log_cropped_st.drop(\n ['price','log_price','sqft_living','distance','grade','bathrooms'],axis=1).columns)\nf = \"log_price~\" + variables_to_include_4_3\nprint(f)\nmodel_4_3= smf.ols(f, df_num_cat_5_log_cropped_st).fit()\nmodel_4_3.summary()\n```\n\n\n```python\nfig = sm.graphics.qqplot(model_4_3.resid,dist=stats.norm,fit=True,line='45')\n\n# Removing variable to simplify the display (categorical variable with only 2 values does not have\n# much use when plotted by regress_exog)\n# print(df_num_cat_5_log_cropped_st.drop(['price','log_price','recent_renovation_new',\n# 'sqft_living', 'distance', 'grade', 'bathrooms'],axis=1).columns)\n\n\nfor col in (df_num_cat_5_log_cropped_st.drop(['price','log_price','recent_renovation_new',\n 'sqft_living', 'distance', 'grade', 'bathrooms'],axis=1).columns):\n fig = sm.graphics.plot_regress_exog(model_4_3, col, fig=plt.figure(figsize=(12,8)))\n```\n\n*The final linear regression model of a log of the price variable versus grade, bathrooms, distance from the center of the city, sqft_living space, and the indicator if a house has been renovated recently or a newer house has a Coefficient of Determination of 0.663. It is indicative of the fact that 66.3% of the sold properties fall within the results of the line formed by the regression equation. F-statistics displays the high value and the overall p-value much lower than the confidence interval \n
\nThe independent variables used in the equation display a clear linear relationship with the target and homoscedasticity.\n
\nThe next step is to validate the model by using training and test datasets*\n\n### Train the model\n\n\n```python\n# Define X and y\ny=df_num_cat_5_log_cropped_st[['log_price']]\nX=df_num_cat_5_log_cropped_st.drop(['price','log_price','recent_renovation_new',\n 'sqft_living', 'distance', 'grade', 'bathrooms'], axis=1)\n```\n\n\n```python\ny\n```\n\n\n```python\nX\n```\n\n\n```python\n# Split the data into training and test sets. Use the default split size\n# X_train, X_test, y_train, y_test = train_test_split(X, y)\n\nX_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state=100)\n\nprint(len(X_train), len(X_test), len(y_train), len(y_test))\n```\n\n\n```python\n# Fit the model to train data\nlinreg = LinearRegression()\nmodel=linreg.fit(X_train, y_train)\n\n# Calculate predictions on training and test sets for y_hat\ny_hat_train=linreg.predict(X_train)\ny_hat_test=linreg.predict(X_test)\n\n# Calculate residuals\ntrain_residuals=y_hat_train-y_train\ntest_residuals=y_hat_test-y_test\n\n# Calculate training and test RMSE\ntrain_mse = mean_squared_error(y_train, y_hat_train, squared=False)\ntest_mse = mean_squared_error(y_test, y_hat_test, squared=False)\nprint('Train MSE:', round(train_mse,3))\nprint('Test MSE:', round(test_mse,3))\n```\n\n\n```python\nprint(model.coef_, model.intercept_, model.score(X_test, y_test))\n```\n\n### Validation\n\n\n```python\nprice_target=df_num_cat_5_log_cropped_st[['log_price']]\nprice_predictors= df_num_cat_5_log_cropped_st.drop(['price','log_price','recent_renovation_new',\n 'sqft_living', 'distance', 'grade', 'bathrooms'], axis=1)\n```\n\n\n```python\nmetrics.r2_score(price_target, linreg.predict(price_predictors))\n```\n\n\n```python\nmetrics.mean_absolute_error(price_target, linreg.predict(price_predictors))\n```\n\n\n```python\nmean_squared_error(price_target, linreg.predict(price_predictors))\n```\n\n>Train MSE: 0.272
\nTest MSE: 0.272
\nAre equal down to the third decimal digit indicating a good agreement between the training and the test sets

\nR squared for for the prediction on the full dataset is the same as in model_4_3: 0.663
\nMean Absolute Error for the prediction on the full dataset is 0.213 which is not great but acceptable. The best possible theoretical value is 0.
\nMean Squared Error for the prediction on the full dataset is 0.074. The best possible theoretical value is 0.\n\n\n\n```python\n# Not sure if the results of this function are indicative of anything, but leaving it here for a possible future use\n\"\"\"mse, bias, var = bias_variance_decomp(model, X_train.values, y_train.values.flatten(), \n X_test.values, y_test.values, num_rounds=200, random_seed=1, loss='mse')\n# summarize results\nprint('MSE: %.3f' % mse)\nprint('Bias: %.3f' % bias)\nprint('Variance: %.3f' % var)\"\"\"\n```\n\n# iNterpret\n\n\n```latex\n%%latex\n\n\\begin{align}\\ln\\left( {Price} \\right) = 12.366 + 1.265 \\cdot \\left( grade \\right) + \n1.080 \\cdot \\left( sqft\\_living \\right) - 0.989 \\cdot \\left( distance \\right) \n+ 0.060 \\cdot \\left( bathrooms \\right) - 0.035 \\cdot \\left( recent\\_renovation\\_new \\right) \\end{align}\n```\n\nThe final model has a reasonable predictive ability tested in the final step of the model validation. MSE, MAE, and R2 score along with the model p-values for all predictors indicate a good fit.

\nThe most influential predictor is a **building grade**, following by a **living space footage**. Both factors are **positively correlated** with the price of the property. Both factors are within property owners' control when they are renovating their houses.
A **distance** from the center of the city is **negatively correlated** with the price of property, meaning the further away a property is, the lower is the price. It is not a controllable variable but is helpful for home buyers if the living space and the number of bedrooms/bathrooms are important.
A **number of bathrooms** has a **positive effect** on the price of a property, but it is not as strong as the first two factors. This fact indicates that the convenience of having multiple bathrooms is essential for potential buyers and should be taken into account when owners are planning a renovation.
\nThe last predictor in the model is an indicator of whether a property **has been renovated recently or a new construction**. It is very **weakly negatively correlated** with the price variable. The negative correlation (reduction of the price) might be related to the following factors: newer properties, on the average, are of less building quality.
\n**The intercept** of the model is a **bias** of the model and can be interpreted as an offset of the model due to other factors not taken into account for various reasons.\n\n# Conclusions and Recommendation\n\n**Recommendations to property owners planning a renovation to their properties:**\n* Increase the living space of your property\n* Do the renovation with higher building quality\n* Consider adding a bathroom\n\n**Recommendations to potential buyers:**\n* Look for properties further away from the city center to make the best out of your property buying budget\n* Properties in some zipcodes of the city are more affordable than others at the same distance from the city center\n* Properties in some zipcodes of the city are more affordable than others with a better view, more considerable property lots, and with older houses of better quality construction if these factors are essential to a buyer\n\n**Limitations of the model:**\n* The original dataset does not include other important factors, and therefore the model is biased\n* Multiple linear regression models, while easily interpretable, are limited in their predictive ability\n* Some variables in the dataset are strongly correlated with each other, and that affect the predictive power of the model\n\n**Suggestion for future improvements**:\n* Add variables to the original dataset like kitchen renovation, average commute time, crime index, average nearby public school quality, etc.\n* Update the dataset with more current data\n\n# Appendix\n\n>Exporting DataFrames to CSV files\n\n\n```python\ndf_test=X_test.copy()\ndf_test['log_price']=y_test['log_price']\ndf_test['recent_renovation_new']=df_num_cat_5_log_cropped_st['recent_renovation_new']\ndf_test['sqft_living']=df_num_cat_5_log_cropped_st['sqft_living']\ndf_test['distance']=df_num_cat_5_log_cropped_st['distance']\ndf_test['grade']=df_num_cat_5_log_cropped_st['grade']\ndf_test['bathrooms']=df_num_cat_5_log_cropped_st['bathrooms']\ndf_test['price']=df_test.apply(lambda row: math.exp(row.log_price), axis=1)\ndf_test['recent_renovation_new_str']=df_test['recent_renovation_new'].astype('str')\n#df_test=df_test.reset_index(drop='index')\ndf_test\n```\n\n\n```python\ndf_test.to_csv('data/df_test.csv', encoding='utf-8')\n```\n\n\n```python\ndf_zipcode_viz=df.groupby('zipcode').mean()\n\ndf_zipcode_viz=df_zipcode_viz.reset_index()\n\ndf_zipcode_viz=df_zipcode_viz.drop(['id','sqft_above','sqft_basement','yr_renovated','lat','long'], axis=1)\ndf_zipcode_viz\n```\n\n\n```python\ndf_zipcode_viz.to_csv('data/df_zipcode_vs.csv', encoding='utf-8')\n```\n\n> Link to Visualization Notebook\n\n[Visualization notebook](./visualization_appendix.ipynb)\n", "meta": {"hexsha": "2878d39e77b5756b445622a49de7ab437bbdb098", "size": 110586, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "project2_OSEMN_plus_mvp_05112021.ipynb", "max_stars_repo_name": "sealaurel/dsc-phase-2-project", "max_stars_repo_head_hexsha": "a8660d503e68805a380a2501541b807827c1df3c", "max_stars_repo_licenses": ["Fair"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "project2_OSEMN_plus_mvp_05112021.ipynb", "max_issues_repo_name": "sealaurel/dsc-phase-2-project", "max_issues_repo_head_hexsha": "a8660d503e68805a380a2501541b807827c1df3c", "max_issues_repo_licenses": ["Fair"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "project2_OSEMN_plus_mvp_05112021.ipynb", "max_forks_repo_name": "sealaurel/dsc-phase-2-project", "max_forks_repo_head_hexsha": "a8660d503e68805a380a2501541b807827c1df3c", "max_forks_repo_licenses": ["Fair"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.5655058043, "max_line_length": 886, "alphanum_fraction": 0.5929231548, "converted": true, "num_tokens": 16002, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.66192288918838, "lm_q2_score": 0.6723316991792861, "lm_q1q2_score": 0.4450317408136859}} {"text": "\n\n# Tutorial 1: Optimization techniques\n**Week 1, Day 4: Optimization**\n\n**By Neuromatch Academy**\n\n__Content creators:__ Jose Gallego-Posada, Ioannis Mitliagkas\n\n__Content reviewers:__ Piyush Chauhan, Vladimir Haltakov, Siwei Bai, Kelson Shilling-Scrivo\n\n__Content editors:__ Charles J Edelson, Gagana B, Spiros Chavlis\n\n__Production editors:__ Arush Tagade, Spiros Chavlis\n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n

\n\n---\n# Tutorial Objectives\n\nObjectives:\n* Necessity and importance of optimization\n* Introduction to commonly used optimization techniques\n* Optimization in non-convex loss landscapes \n* 'Adaptive' hyperparameter tuning \n* Ethical concerns\n\n\n\n\n```python\n# @title Tutorial slides\n\n# @markdown These are the slides for the videos in this tutorial\nfrom IPython.display import IFrame\nIFrame(src=f\"https://mfr.ca-1.osf.io/render?url=https://osf.io/ft2sz/?direct%26mode=render%26action=download%26mode=render\", width=854, height=480)\n```\n\n---\n# Setup\n\n\n```python\n# @title Install dependencies\n!pip install git+https://github.com/NeuromatchAcademy/evaltools --quiet\nfrom evaltools.airtable import AirtableForm\n\n# generate airtable form\natform = AirtableForm('appn7VdPRseSoMXEG','W1D4_T1','https://portal.neuromatchacademy.org/api/redirect/to/9548a279-c9f9-4586-b89c-f0ceceba5c14')\n```\n\n\n```python\n# Imports\nimport time\nimport copy\nimport torch\nimport torchvision\n\nimport numpy as np\nimport ipywidgets as widgets\nimport matplotlib.pyplot as plt\n\nimport torch.nn as nn\nimport torch.optim as optim\nimport torch.nn.functional as F\nimport torchvision.datasets as datasets\n\nfrom tqdm.auto import tqdm\n```\n\n\n```python\n# @title Figure settings\nimport ipywidgets as widgets # interactive display\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/content-creation/main/nma.mplstyle\")\nplt.rc('axes', unicode_minus=False)\n```\n\n\n```python\n# @title Helper functions\ndef print_params(model):\n for name, param in model.named_parameters():\n if param.requires_grad:\n print(name, param.data)\n```\n\n\n```python\n# @title Set random seed\n\n# @markdown Executing `set_seed(seed=seed)` you are setting the seed\n\n# for DL its critical to set the random seed so that students can have a\n# baseline to compare their results to expected results.\n# Read more here: https://pytorch.org/docs/stable/notes/randomness.html\n\n# Call `set_seed` function in the exercises to ensure reproducibility.\nimport random\nimport torch\n\ndef set_seed(seed=None, seed_torch=True):\n if seed is None:\n seed = np.random.choice(2 ** 32)\n random.seed(seed)\n np.random.seed(seed)\n if seed_torch:\n torch.manual_seed(seed)\n torch.cuda.manual_seed_all(seed)\n torch.cuda.manual_seed(seed)\n torch.backends.cudnn.benchmark = False\n torch.backends.cudnn.deterministic = True\n\n print(f'Random seed {seed} has been set.')\n\n\n# In case that `DataLoader` is used\ndef seed_worker(worker_id):\n worker_seed = torch.initial_seed() % 2**32\n np.random.seed(worker_seed)\n random.seed(worker_seed)\n```\n\n\n```python\n# @title Set device (GPU or CPU). Execute `set_device()`\n# especially if torch modules used.\n\n# inform the user if the notebook uses GPU or CPU.\n\ndef set_device():\n device = \"cuda\" if torch.cuda.is_available() else \"cpu\"\n if device != \"cuda\":\n print(\"WARNING: For this notebook to perform best, \"\n \"if possible, in the menu under `Runtime` -> \"\n \"`Change runtime type.` select `GPU` \")\n else:\n print(\"GPU is enabled in this notebook.\")\n\n return device\n```\n\n\n```python\nSEED = 2021\nset_seed(seed=SEED)\nDEVICE = set_device()\n```\n\n Random seed 2021 has been set.\n GPU is enabled in this notebook.\n\n\n---\n# Section 1. Introduction\n\n*Time estimate: ~15 mins*\n\n\n```python\n# @title Video 1: Introduction\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1VB4y1K7Vr\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"zm9oekdkJbQ\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 1: Introduction')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n## Discuss: Unexpected consequences\n\nCan you think of examples from your own experience/life where poorly chosen incentives or objectives have lead to unexpected consequences?\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content-dl/tree/main//tutorials/W1D4_Optimization/solutions/W1D4_Tutorial1_Solution_b8bbba6f.py)\n\n\n\n---\n# Section 2: Case study: successfully training an MLP for image classification\n\n*Time estimate: ~40 mins*\n\nMany of the core ideas (and tricks) in modern optimization for deep learning can be illustrated in the simple setting of training an MLP to solve an image classification task. In this tutorial we will guide you through the key challenges that arise when optimizing high-dimensional, non-convex$^\\dagger$ problems. We will use these challenges to motivate and explain some commonly used solutions.\n\n**Disclaimer:** Some of the functions you will code in this tutorial are already implemented in Pytorch and many other libraries. For pedagogical reasons, we decided to bring these simple coding tasks into the spotlight and place a relatively higher emphasis in your understanding of the algorithms, rather than the use of a specific library. \n\nIn 'day-to-day' research projects you will likely to rely on the community-vetted, optimized libraries rather than the 'manual implementations' you will write today. In Section 8 you will have a chance to 'put it all together' and use the full power of Pytorch to tune the parameters of an MLP to classify handwritten digits.\n\n$^\\dagger$: A **convex** function has one, global minimum - a nice property, as an optimization algorithm won't get stuck in a local minimum that isn't a global one (e.g., $f(x)=x^2 + 2x + 1$). A **non-convex** function is wavy - has some 'valleys' (local minima) that aren't as deep as the overall deepest 'valley' (global minimum). Thus, the optimization algorithms can get stuck in the local minimum, and it can be hard to tell when this happens (e.g., $f(x) = x^4 + x^3 - 2x^2 - 2x$). See also **Section 5** for more details.\n\n\n```python\n# @title Video 2: Case Study - MLP Classification\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1GB4y1K7Ha\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"pJc2ENhYbqA\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 2: Case Study - MLP Classification')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n## Section 2.1: Data\n\nWe will use the MNIST dataset of handwritten digits. We load the data via the Pytorch `datasets` module, as you learned in W1D1.\n\n**Note:** Although we can download the MNIST dataset directly from `datasets` using the optional argument `download=True`, we are going to download them from NMA directory on OSF to ensure network reliability.\n\n\n\n```python\n# @title Download MNIST dataset\nimport tarfile, requests, os\n\nfname = 'MNIST.tar.gz'\nname = 'MNIST'\nurl = 'https://osf.io/y2fj6/download'\n\nif not os.path.exists(name):\n print('\\nDownloading MNIST dataset...')\n r = requests.get(url, allow_redirects=True)\n with open(fname, 'wb') as fh:\n fh.write(r.content)\n print('\\nDownloading MNIST completed.')\n\nif not os.path.exists(name):\n with tarfile.open(fname) as tar:\n tar.extractall()\n os.remove(fname)\nelse:\n print('MNIST dataset has been dowloaded.')\n```\n\n \n Downloading MNIST dataset...\n \n Downloading MNIST completed.\n\n\n\n```python\ndef load_mnist_data(change_tensors=False, download=False):\n \"\"\"Load training and test examples for the MNIST digits dataset\n\n Returns:\n train_data (tensor): training input tensor of size (train_size x 784)\n train_target (tensor): training 0-9 integer label tensor of size (train_size)\n test_data (tensor): test input tensor of size (70k-train_size x 784)\n test_target (tensor): training 0-9 integer label tensor of size (70k-train_size)\n\n \"\"\"\n # Load train and test sets\n train_set = datasets.MNIST(root='.', train=True, download=download,\n transform=torchvision.transforms.ToTensor())\n test_set = datasets.MNIST(root='.', train=False, download=download,\n transform=torchvision.transforms.ToTensor())\n\n # Original data is in range [0, 255]. We normalize the data wrt its mean and std_dev.\n ## Note that we only used *training set* information to compute mean and std\n ## This is done to avoid data leakage\n mean = train_set.data.float().mean()\n std = train_set.data.float().std()\n\n if change_tensors:\n # Apply normalization directly to the tensors containing the dataset\n # We use the same mean, std, to transform the test_set as well, we assume i.i.d.\n # We cannot compute for the test dataset because most of the times the test dataset is an online stream\n train_set.data = (train_set.data.float() - mean) / std\n test_set.data = (test_set.data.float() - mean) / std\n else:\n tform = torchvision.transforms.Compose([torchvision.transforms.ToTensor(),\n torchvision.transforms.Normalize(mean=[mean / 255.], std=[std / 255.])\n ])\n train_set = datasets.MNIST(root='.', train=True, download=download,\n transform=tform)\n test_set = datasets.MNIST(root='.', train=False, download=download,\n transform=tform)\n\n return train_set, test_set\n\n\ntrain_set, test_set = load_mnist_data(change_tensors=True)\n```\n\nAs we are just getting started, we will concentrate on a small subset of only 500 examples out of the 60.000 data points contained in the whole training set.\n\n\n\n\n```python\n# Sample a random subset of 500 indices\nsubset_index = np.random.choice(len(train_set.data), 500)\n\n# We will use these symbols to represent the training data and labels, to stay\n# as close to the mathematical expressions as possible.\nX, y = train_set.data[subset_index, :], train_set.targets[subset_index]\n```\n\nRun the following cell to visualize the content of three examples in our training set. Note how the pre-processing we applied to the data changes the range of pixel values after normalization.\n\n\n\n```python\n# @title Run me!\nnum_figures = 3\nfig, axs = plt.subplots(1, num_figures, figsize=(5 * num_figures, 5))\n\nfor sample_id, ax in enumerate(axs):\n # Plot the pixel values for each image\n ax.matshow(X[sample_id, :], cmap='gray_r')\n # 'Write' the pixel value in the corresponding location\n for (i, j), z in np.ndenumerate(X[sample_id, :]):\n text = '{:.1f}'.format(z)\n ax.text(j, i, text, ha='center',\n va='center', fontsize=6, c='steelblue')\n\n ax.set_title('Label: ' + str(y[sample_id].item()))\n ax.axis('off')\n\nplt.show()\n```\n\n## Section 2.2: Model\n\nAs you will see next week, there are specific model architectures that are better suited to image-like data, such as Convolutional Neural Networks (CNNs). For simplicity, in this tutorial we will focus exclusively on Multi-Layer Perceptron (MLP) models as they allow us to highlight many important optimization challenges shared with more advanced neural network designs.\n\n\n```python\nclass MLP(nn.Module):\n \"\"\" This class implements MLPs in Pytorch of an arbitrary number of hidden\n layers of potentially different sizes. Since we concentrate on classification\n tasks in this tutorial, we have a log_softmax layer at prediction time.\n \"\"\"\n\n def __init__(self, in_dim=784, out_dim=10, hidden_dims=[], use_bias=True):\n \"\"\"Constructs a MultiLayerPerceptron\n\n Args:\n in_dim (int): dimensionality of input data\n out_dim (int): number of classes\n hidden_dims (list): contains the dimensions of the hidden layers, an empty\n list corresponds to a linear model (in_dim, out_dim)\n \"\"\"\n\n super(MLP, self).__init__()\n\n self.in_dim = in_dim\n self.out_dim = out_dim\n\n # If we have no hidden layer, just initialize a linear model (e.g. in logistic regression)\n if len(hidden_dims) == 0:\n layers = [nn.Linear(in_dim, out_dim, bias=use_bias)]\n else:\n # 'Actual' MLP with dimensions in_dim - num_hidden_layers*[hidden_dim] - out_dim\n layers = [nn.Linear(in_dim, hidden_dims[0], bias=use_bias), nn.ReLU()]\n\n # Loop until before the last layer\n for i, hidden_dim in enumerate(hidden_dims[:-1]):\n layers += [nn.Linear(hidden_dim, hidden_dims[i + 1], bias=use_bias),\n nn.ReLU()]\n\n # Add final layer to the number of classes\n layers += [nn.Linear(hidden_dims[-1], out_dim, bias=use_bias)]\n\n self.main = nn.Sequential(*layers)\n\n def forward(self, x):\n # Flatten the images into 'vectors'\n transformed_x = x.view(-1, self.in_dim)\n hidden_output = self.main(transformed_x)\n output = F.log_softmax(hidden_output, dim=1)\n return output\n```\n\nLinear models constitute a very special kind of MLPs: they are equivalent to an MLP with *zero* hidden layers. This is simply an affine transformation, in other words a 'linear' map $W x$ with an 'offset' $b$; followed by a softmax function.\n\n$$f(x) = \\text{softmax}(W x + b)$$\n\nHere $x \\in \\mathbb{R}^{784}$, $W \\in \\mathbb{R}^{10 \\times 784}$ and $b \\in \\mathbb{R}^{10}$. Notice that the dimensions of the weight matrix are $10 \\times 784$ as the input tensors are flattened images, i.e., $28 \\times 28 = 784$-dimensional tensors and the output layer consists of $10$ nodes.\n\n\n```python\n# Empty hidden_dims means we take a model with zero hidden layers.\nmodel = MLP(in_dim=784, out_dim=10, hidden_dims=[])\n\n# We print the model structure with 784 inputs and 10 outputs\nprint(model)\n```\n\n MLP(\n (main): Sequential(\n (0): Linear(in_features=784, out_features=10, bias=True)\n )\n )\n\n\n## Section 2.3: Loss\n\nWhile we care about the accuracy of the model, the 'discrete' nature of the 0-1 loss makes it challenging to optimize. In order to learn good parameters for this model, we will use the cross entropy loss (negative log-likelihood), which you saw in last lecture, as a surrogate objective to be minimized. \n\nThis particular choice of model and optimization objective leads to a *convex* optimization problem with respect to the parameters $W$ and $b$. \n\n\n```python\nloss_fn = F.nll_loss\n```\n\n## Section 2.4: Interpretability\n\nIn last lecture, you saw that inspecting the weights of a model can provide insights on what 'concepts' the model has learned. Here we show the weights of a partially trained model. The weights corresponding to each class 'learn' to _fire_ when an input of the class is detected.\n\n\n\n```python\n#@markdown Run _this cell_ to train the model. If you are curious about how the training\n#@markdown takes place, double-click this cell to find out. At the end of this tutorial\n#@markdown you will have the opportunity to train a more complex model on your own.\n\ncell_verbose = False\npartial_trained_model = MLP(in_dim=784, out_dim=10, hidden_dims=[])\n\nif cell_verbose:\n print('Init loss', loss_fn(partial_trained_model(X), y).item()) # This matches around np.log(10 = # of classes)\n\noptimizer = optim.Adam(partial_trained_model.parameters(), lr=7e-4)\nfor _ in range(200):\n loss = loss_fn(partial_trained_model(X), y)\n optimizer.zero_grad()\n loss.backward()\n optimizer.step()\n\nif cell_verbose:\n print('End loss', loss_fn(partial_trained_model(X), y).item()) # This should be less than 1e-2\n```\n\n\n```python\n# Show class filters of a trained model\nW = partial_trained_model.main[0].weight.data.numpy()\n\nfig, axs = plt.subplots(1, 10, figsize=(15, 4))\nfor class_id in range(10):\n axs[class_id].imshow(W[class_id, :].reshape(28, 28), cmap='gray_r')\n axs[class_id].axis('off')\n axs[class_id].set_title('Class ' + str(class_id) )\n\nplt.show()\n```\n\n---\n# Section 3: High dimensional search\n\n*Time estimate: ~25 mins*\n\nWe now have a model with its corresponding trainable parameters as well as an objective to optimize. Where do we go to next? How do we find a 'good' configuration of parameters?\n\nOne idea is to choose a random direction and move only if the objective is reduced. However, this is inefficient in high dimensions and you will see how gradient descent (with a suitable step-size) can guarantee consistent improvement in terms of the objective function.\n\n\n```python\n# @title Video 3: Optimization of an Objective Function\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1aL411H7Ce\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"aSJTRdjRvvw\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 3: Optimization of an Objective Function')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n## Coding Exercise 3: Implement gradient descent\n\nIn this exercise you will use PyTorch automatic differentiation capabilities to compute the gradient of the loss with respect to the parameters of the model. You will then use these gradients to implement the update performed by the gradient descent method. \n\n\n```python\ndef zero_grad(params):\n \"\"\"Clear up gradients as Pytorch automatically accumulates gradients from\n successive backward calls\n \"\"\"\n for par in params:\n if not(par.grad is None):\n par.grad.data.zero_()\n\n\ndef random_update(model, noise_scale=0.1, normalized=False):\n \"\"\" Performs a random update on the parameters of the model\n \"\"\"\n for par in model.parameters():\n noise = torch.randn_like(par)\n if normalized:\n noise /= torch.norm(noise)\n par.data += noise_scale * noise\n```\n\n\n```python\ndef gradient_update(loss, params, lr=1e-3):\n \"\"\"Perform a gradient descent update on a given loss over a collection of parameters\n\n Args:\n loss (tensor): A scalar tensor containing the loss whose gradient will be computed\n params (iterable): Collection of parameters with respect to which we compute gradients\n lr (float): Scalar specifying the learning rate or step-size for the update\n \"\"\"\n # Clear up gradients as Pytorch automatically accumulates gradients from\n # successive backward calls\n zero_grad(params)\n\n # Compute gradients on given objective\n loss.backward()\n\n with torch.no_grad():\n for par in params:\n #################################################\n ## TODO for students: update the value of the parameter ##\n #raise NotImplementedError(\"Student exercise: implement gradient update\")\n #################################################\n # Here we work with the 'data' attribute of the parameter rather than the\n # parameter itself.\n par.data -= lr * par.grad.data\n\n\n# add event to airtable\natform.add_event('Coding Exercise 3: Implement gradient descent')\n\n\nset_seed(seed=SEED)\nmodel1 = MLP(in_dim=784, out_dim=10, hidden_dims=[])\nprint('\\n The model1 parameters before the update are: \\n')\nprint_params(model1)\nloss = loss_fn(model1(X), y)\n\n## Uncomment below to test your function\ngradient_update(loss, list(model1.parameters()), lr=1e-1)\nprint('\\n The model1 parameters after the update are: \\n')\nprint_params(model1)\n```\n\n Random seed 2021 has been set.\n \n The model1 parameters before the update are: \n \n main.0.weight tensor([[-0.0264, 0.0010, 0.0173, ..., 0.0297, 0.0278, -0.0221],\n [-0.0040, -0.0295, -0.0086, ..., -0.0070, 0.0254, -0.0233],\n [ 0.0240, -0.0231, 0.0342, ..., 0.0124, 0.0270, -0.0180],\n ...,\n [-0.0005, 0.0157, 0.0111, ..., 0.0144, -0.0301, -0.0144],\n [ 0.0181, 0.0303, 0.0255, ..., -0.0110, -0.0175, 0.0205],\n [ 0.0208, -0.0353, -0.0183, ..., -0.0271, 0.0099, 0.0003]])\n main.0.bias tensor([-0.0290, -0.0033, 0.0100, -0.0320, 0.0022, 0.0221, 0.0307, 0.0243,\n 0.0159, -0.0064])\n \n The model1 parameters after the update are: \n \n main.0.weight tensor([[-0.0263, 0.0010, 0.0174, ..., 0.0298, 0.0278, -0.0220],\n [-0.0047, -0.0302, -0.0093, ..., -0.0077, 0.0248, -0.0240],\n [ 0.0234, -0.0237, 0.0335, ..., 0.0117, 0.0263, -0.0187],\n ...,\n [-0.0006, 0.0156, 0.0110, ..., 0.0143, -0.0302, -0.0145],\n [ 0.0164, 0.0286, 0.0238, ..., -0.0127, -0.0191, 0.0188],\n [ 0.0206, -0.0354, -0.0184, ..., -0.0272, 0.0098, 0.0002]])\n main.0.bias tensor([-0.0292, -0.0018, 0.0115, -0.0370, 0.0054, 0.0155, 0.0317, 0.0246,\n 0.0198, -0.0061])\n\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content-dl/tree/main//tutorials/W1D4_Optimization/solutions/W1D4_Tutorial1_Solution_1c2b3d1a.py)\n\n\n\n```\n The model1 parameters after the update are: \n\nmain.0.weight tensor([[-0.0263, 0.0010, 0.0174, ..., 0.0298, 0.0278, -0.0220],\n [-0.0047, -0.0302, -0.0093, ..., -0.0077, 0.0248, -0.0240],\n [ 0.0234, -0.0237, 0.0335, ..., 0.0117, 0.0263, -0.0187],\n ...,\n [-0.0006, 0.0156, 0.0110, ..., 0.0143, -0.0302, -0.0145],\n [ 0.0164, 0.0286, 0.0238, ..., -0.0127, -0.0191, 0.0188],\n [ 0.0206, -0.0354, -0.0184, ..., -0.0272, 0.0098, 0.0002]])\nmain.0.bias tensor([-0.0292, -0.0018, 0.0115, -0.0370, 0.0054, 0.0155, 0.0317, 0.0246,\n 0.0198, -0.0061])\n```\n\n## Comparing updates\n\nThese plots compare the effectiveness of updating random directions for the problem of optimizing the parameters of a high-dimensional linear model. We contrast the behavior at initialization and during an intermediate stage of training by showing the histograms of change in loss over 100 different random directions vs the changed in loss induced by the gradient descent update\n\n**Remember:** since we are trying to minimize, here negative is better!\n\n\n\n```python\n# @markdown _Run this cell_ to visualize the results\nfig, axs = plt.subplots(1, 2, figsize=(10, 4))\n\nfor id, (model_name, my_model) in enumerate([('Initialization', model),\n ('Partially trained', partial_trained_model)]):\n # Compue the loss we will be comparing to\n base_loss = loss_fn(my_model(X), y)\n\n # Compute the improvement via gradient descent\n dummy_model = copy.deepcopy(my_model)\n loss1 = loss_fn(dummy_model(X), y)\n gradient_update(loss1, list(dummy_model.parameters()), lr=1e-2)\n gd_delta = loss_fn(dummy_model(X), y) - base_loss\n\n deltas = []\n for trial_id in range(100):\n # Compute the improvement obtained with a random direction\n dummy_model = copy.deepcopy(my_model)\n random_update(dummy_model, noise_scale=1e-2)\n deltas.append((loss_fn(dummy_model(X), y) - base_loss).item())\n\n # Plot histogram for random direction and vertical line for gradient descent\n axs[id].hist(deltas, label='Random Directions', bins=20)\n axs[id].set_title(model_name)\n axs[id].set_xlabel('Change in loss')\n axs[id].set_ylabel('% samples')\n axs[id].axvline(0, c='green', alpha=0.5)\n axs[id].axvline(gd_delta.item(), linestyle='--', c='red', alpha=1,\n label='Gradient Descent')\n\n\nhandles, labels = axs[id].get_legend_handles_labels()\nfig.legend(handles, labels, loc='upper center',\n bbox_to_anchor=(0.5, 1.05),\n fancybox=False, shadow=False, ncol=2)\n\nplt.show()\n```\n\n## Think! 3: Gradient descent vs. random search\n\nCompare the behavior of gradient descent and random search based on the histograms above. Is any of the two methods more reliable? How can you explain the changes between behavior of the methods at initialization vs during training?\n\n\n```python\n# @title Student Response\nfrom ipywidgets import widgets\n\n\ntext=widgets.Textarea(\n value='Type your answer here and click on `Submit!`',\n placeholder='Type something',\n description='',\n disabled=False\n)\n\nbutton = widgets.Button(description=\"Submit!\")\n\ndisplay(text,button)\n\ndef on_button_clicked(b):\n atform.add_answer('q1' , text.value)\n print(\"Submission successful!\")\n\n\nbutton.on_click(on_button_clicked)\n```\n\n\n Textarea(value='Type your answer here and click on `Submit!`', placeholder='Type something')\n\n\n\n Button(description='Submit!', style=ButtonStyle())\n\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content-dl/tree/main//tutorials/W1D4_Optimization/solutions/W1D4_Tutorial1_Solution_2de57667.py)\n\n\n\n---\n# Section 4: Poor conditioning\n\n*Time estimate: ~30 mins*\n\nAlready in this 'simple' logistic regression problem, the issue of bad conditioning is haunting us. Not all parameters are created equal and the sensitivity of the network to changes on the parameters will have a big impact in the dynamics of the optimization.\n\n\n\n```python\n# @title Video 4: Momentum\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1NL411H71t\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"3ES5O58Y_2M\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 4: Momentum')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nWe illustrate this issue in a 2-dimensional setting. We freeze all but two parameters of the network: one of them is an element of the weight matrix (filter) for class 0, while the other is the bias for class 7. This results in an optimization with two decision variables.\n\nHow much difference is there in the behavior of these two parameters under gradient descent? What is the effect of momentum in bridging that gap?\n\n\n\n```python\n# @markdown _Run this cell_ to setup some helper functions.\n\ndef loss_2d(model, u, v, mask_idx=(0, 378), bias_id=7):\n \"\"\"Defines a 2-dim function by freezing all but two parameters of a linear\n model.\n\n Args:\n model (torch module): a pytorch 0-hidden layer (linear) model\n u (scalar): first free parameter\n v (scalar): second free parameter\n mask_idx (tuple): selects parameter in weight matrix replaced by u\n bias_idx (int): selects parameter in bias vector replaced by v\n\n Returns:\n scalar: loss of the 'new' model over inputs X, y (defined externally)\n \"\"\"\n\n # We zero out the element of the weight tensor that will be\n # replaced by u\n mask = torch.ones_like(model.main[0].weight)\n mask[mask_idx[0], mask_idx[1]] = 0.\n masked_weights = model.main[0].weight * mask\n\n # u is replacing an element of the weight matrix\n masked_weights[mask_idx[0], mask_idx[1]] = u\n\n res = X.reshape(-1, 784) @ masked_weights.T + model.main[0].bias\n\n # v is replacing a bias for class 7\n res[:, 7] += v - model.main[0].bias[7]\n res = F.log_softmax(res, dim=1)\n\n return loss_fn(res, y)\n\n\ndef plot_surface(U, V, Z, fig):\n \"\"\" Plot a 3D loss surface given meshed inputs U, V and values Z\n \"\"\"\n ax = fig.add_subplot(1, 2, 2, projection='3d')\n ax.view_init(45, -130)\n\n surf = ax.plot_surface(U, V, Z, cmap=plt.cm.coolwarm,\n linewidth=0, antialiased=True, alpha=0.5)\n\n # Select certain level contours to plot\n # levels = Z.min() * np.array([1.005, 1.1, 1.3, 1.5, 2.])\n # plt.contour(U, V, Z)# levels=levels, alpha=0.5)\n\n ax.set_xlabel('Weight')\n ax.set_ylabel('Bias')\n ax.set_zlabel('Loss', rotation=90)\n\n return ax\n\n\ndef plot_param_distance(best_u, best_v, trajs, fig, styles, labels,\n use_log=False, y_min_v=-12.0, y_max_v=1.5):\n \"\"\" Plot the distance to each of the two parameters for a collection of 'trajectories'\n \"\"\"\n ax = fig.add_subplot(1, 1, 1)\n\n for traj, style, label in zip(trajs, styles, labels):\n d0 = np.array([np.abs(_[0] - best_u) for _ in traj])\n d1 = np.array([np.abs(_[1] - best_v) for _ in traj])\n if use_log:\n d0 = np.log(1e-16 + d0)\n d1 = np.log(1e-16 + d1)\n ax.plot(range(len(traj)), d0, style, label='weight - ' + label)\n ax.plot(range(len(traj)), d1, style, label='bias - ' + label)\n ax.set_xlabel('Iteration')\n if use_log:\n ax.set_ylabel('Log distance to optimum (per dimension)')\n ax.set_ylim(y_min_v, y_max_v)\n else:\n ax.set_ylabel('Abs distance to optimum (per dimension)')\n ax.legend(loc='right', bbox_to_anchor=(1.5, 0.5),\n fancybox=False, shadow=False, ncol=1)\n\n return ax\n\n\ndef run_optimizer(inits, eval_fn, update_fn, max_steps=500,\n optim_kwargs={'lr':1e-2}, log_traj=True):\n \"\"\"Runs an optimizer on a given objective and logs parameter trajectory\n\n Args:\n inits list(scalar): initialization of parameters\n eval_fn (callable): function computing the objective to be minimized\n update_fn (callable): function executing parameter update\n max_steps (int): number of iterations to run\n optim_kwargs (dict): customize optimizer hyperparameters\n\n Returns:\n list[list]: trajectory information [*params, loss] for each optimization step\n \"\"\"\n\n # Initialize parameters and optimizer\n params = [nn.Parameter(torch.tensor(_)) for _ in inits]\n # Methods like momentum and rmsprop keep and auxiliary vector of parameters\n aux_tensors = [torch.zeros_like(_) for _ in params]\n if log_traj:\n traj = np.zeros((max_steps, len(params)+1))\n for _ in range(max_steps):\n # Evaluate loss\n loss = eval_fn(*params)\n # Store 'trajectory' information\n if log_traj:\n traj[_, :] = [_.item() for _ in params] + [loss.item()]\n # Perform update\n if update_fn == gradient_update:\n gradient_update(loss, params, **optim_kwargs)\n else:\n update_fn(loss, params, aux_tensors, **optim_kwargs)\n if log_traj:\n return traj\n\n\nL = 4.\nxs = np.linspace(-L, L, 30)\nys = np.linspace(-L, L, 30)\nU, V = np.meshgrid(xs, ys)\n```\n\n## Coding Exercise 4: Implement momentum\n\nIn this exercise you will implement the momentum update given by:\n\n\\begin{equation}\nw_{t+1} = w_t - \\eta \\nabla J(w_t) + \\beta (w_t - w_{t-1})\n\\end{equation}\n\nIt is convenient to re-express this update rule in terms of a recursion. For that, we define 'velocity' as the quantity:\n\\begin{equation}\nv_{t-1} := w_{t} - w_{t-1}\n\\end{equation}\n\nwhich leads to the two-step update rule:\n\n\\begin{equation}\nv_t = - \\eta \\nabla J(w_t) + \\beta (\\underbrace{w_t - w_{t-1}}_{v_{t-1}})\n\\end{equation}\n\n\\begin{equation}\nw_{t+1} \\leftarrow w_t + v_{t}\n\\end{equation}\n\nPay attention to the positive sign of the update in the last equation, given the definition of $v_t$, above. \n\n\n```python\ndef momentum_update(loss, params, grad_vel, lr=1e-3, beta=0.8):\n \"\"\"Perform a momentum update over a collection of parameters given a loss and 'velocities'\n\n Args:\n loss (tensor): A scalar tensor containing the loss whose gradient will be computed\n params (iterable): Collection of parameters with respect to which we compute gradients\n grad_vel (iterable): Collection containing the 'velocity' v_t for each parameter\n lr (float): Scalar specifying the learning rate or step-size for the update\n beta (float): Scalar 'momentum' parameter\n \"\"\"\n # Clear up gradients as Pytorch automatically accumulates gradients from\n # successive backward calls\n zero_grad(params)\n # Compute gradients on given objective\n loss.backward()\n\n with torch.no_grad():\n for (par, vel) in zip(params, grad_vel):\n #################################################\n ## TODO for students: update the value of the parameter ##\n #raise NotImplementedError(\"Student exercise: implement momentum update\")\n #################################################\n # Update 'velocity'\n vel.data = - lr * par.grad + beta * vel.data\n # Update parameters\n par.data += vel.data\n\n\n# add event to airtable\natform.add_event('Coding Exercise 4: Implement momentum')\n\n\nset_seed(seed=SEED)\nmodel2 = MLP(in_dim=784, out_dim=10, hidden_dims=[])\nprint('\\n The model2 parameters before the update are: \\n')\nprint_params(model2)\nloss = loss_fn(model2(X), y)\ninitial_vel = [torch.randn_like(p) for p in model2.parameters()]\n\n## Uncomment below to test your function\nmomentum_update(loss, list(model2.parameters()), grad_vel=initial_vel, lr=1e-1, beta=0.9)\nprint('\\n The model2 parameters after the update are: \\n')\nprint_params(model2)\n```\n\n Random seed 2021 has been set.\n \n The model2 parameters before the update are: \n \n main.0.weight tensor([[-0.0264, 0.0010, 0.0173, ..., 0.0297, 0.0278, -0.0221],\n [-0.0040, -0.0295, -0.0086, ..., -0.0070, 0.0254, -0.0233],\n [ 0.0240, -0.0231, 0.0342, ..., 0.0124, 0.0270, -0.0180],\n ...,\n [-0.0005, 0.0157, 0.0111, ..., 0.0144, -0.0301, -0.0144],\n [ 0.0181, 0.0303, 0.0255, ..., -0.0110, -0.0175, 0.0205],\n [ 0.0208, -0.0353, -0.0183, ..., -0.0271, 0.0099, 0.0003]])\n main.0.bias tensor([-0.0290, -0.0033, 0.0100, -0.0320, 0.0022, 0.0221, 0.0307, 0.0243,\n 0.0159, -0.0064])\n \n The model2 parameters after the update are: \n \n main.0.weight tensor([[ 1.5898, 0.0116, -2.0239, ..., -1.0871, 0.4030, -0.9577],\n [ 0.4653, 0.6022, -0.7363, ..., 0.5485, -0.2747, -0.6539],\n [-1.4117, -1.1045, 0.6492, ..., -1.0201, 0.6503, 0.1310],\n ...,\n [-0.5098, 0.5075, -0.0718, ..., 1.1192, 0.2900, -0.9657],\n [-0.4405, -0.1174, 0.7542, ..., 0.0792, -0.1857, 0.3537],\n [-1.0824, 1.0080, -0.4254, ..., -0.3760, -1.7491, 0.6025]])\n main.0.bias tensor([ 0.4147, -1.0440, 0.8720, -1.6201, -0.9632, 0.9430, -0.5180, 1.3417,\n 0.6574, 0.3677])\n\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content-dl/tree/main//tutorials/W1D4_Optimization/solutions/W1D4_Tutorial1_Solution_ba72f88a.py)\n\n\n\n```\n The model2 parameters after the update are: \n\nmain.0.weight tensor([[ 1.5898, 0.0116, -2.0239, ..., -1.0871, 0.4030, -0.9577],\n [ 0.4653, 0.6022, -0.7363, ..., 0.5485, -0.2747, -0.6539],\n [-1.4117, -1.1045, 0.6492, ..., -1.0201, 0.6503, 0.1310],\n ...,\n [-0.5098, 0.5075, -0.0718, ..., 1.1192, 0.2900, -0.9657],\n [-0.4405, -0.1174, 0.7542, ..., 0.0792, -0.1857, 0.3537],\n [-1.0824, 1.0080, -0.4254, ..., -0.3760, -1.7491, 0.6025]])\nmain.0.bias tensor([ 0.4147, -1.0440, 0.8720, -1.6201, -0.9632, 0.9430, -0.5180, 1.3417,\n 0.6574, 0.3677])\n```\n\n## Interactive Demo 4: Momentum vs. GD\n\nThe plots below show the distance to the optimum for both variables across the two methods, as well as the parameter trajectory over the loss surface.\n\n\n```python\n# @markdown Run this cell to enable the widget!\nfrom matplotlib.lines import Line2D\n\n# Find the optimum of this 2D problem using Newton's method\ndef run_newton(func, init_list=[0., 0.], max_iter=200):\n\n par_tensor = torch.tensor(init_list, requires_grad=True)\n t_g = lambda par_tensor: func(par_tensor[0], par_tensor[1])\n\n for _ in tqdm(range(max_iter)):\n eval_loss = t_g(par_tensor)\n eval_grad = torch.autograd.grad(eval_loss, [par_tensor])[0]\n eval_hess = torch.autograd.functional.hessian(t_g, par_tensor)\n # Newton's update is: - inverse(Hessian) x gradient\n par_tensor.data -= torch.inverse(eval_hess) @ eval_grad\n\n return par_tensor.data.numpy()\n\n\nset_seed(2021)\nmodel = MLP(in_dim=784, out_dim=10, hidden_dims=[])\n# Define 2d loss objectives and surface values\ng = lambda u, v: loss_2d(copy.deepcopy(model), u, v)\nZ = np.fromiter(map(g, U.ravel(), V.ravel()), U.dtype).reshape(V.shape)\n\nbest_u, best_v = run_newton(func=g)\n\n# Initialization of the variables\nINITS = [2.5, 3.7]\n\n# Used for plotting\nLABELS = ['GD', 'Momentum']\nCOLORS = ['black', 'red']\nLSTYLES = ['-', '--']\n\n\n@widgets.interact_manual\ndef momentum_experiment(max_steps=widgets.IntSlider(300, 50, 500, 5),\n lr=widgets.FloatLogSlider(value=1e-1, min=-3, max=0.7, step=0.1),\n beta=widgets.FloatSlider(value=9e-1, min=0, max=1., step=0.01)\n ):\n\n # Execute both optimizers\n sgd_traj = run_optimizer(INITS, eval_fn=g, update_fn=gradient_update,\n max_steps=max_steps, optim_kwargs={'lr': lr})\n mom_traj = run_optimizer(INITS, eval_fn=g, update_fn=momentum_update,\n max_steps=max_steps, optim_kwargs={'lr': lr, 'beta':beta})\n\n TRAJS = [sgd_traj, mom_traj]\n\n # Plot distances\n fig = plt.figure(figsize=(9,4))\n plot_param_distance(best_u, best_v, TRAJS, fig,\n LSTYLES, LABELS, use_log=True, y_min_v=-12.0, y_max_v=1.5)\n\n # # Plot trajectories\n fig = plt.figure(figsize=(12, 5))\n ax = plot_surface(U, V, Z, fig)\n for traj, c, label in zip(TRAJS, COLORS, LABELS):\n ax.plot3D(*traj.T, c, linewidth=0.3, label=label)\n ax.scatter3D(*traj.T, '.-', s=1, c=c)\n\n # Plot optimum point\n ax.scatter(best_u, best_v, Z.min(), marker='*', s=80, c='lime', label='Opt.');\n lines = [Line2D([0], [0],\n color=c,\n linewidth=3,\n linestyle='--') for c in COLORS]\n lines.append(Line2D([0], [0], color='lime', linewidth=0, marker='*'))\n ax.legend(lines, LABELS + ['Optimum'], loc='right',\n bbox_to_anchor=(.8, -0.1), ncol=len(LABELS) + 1)\n```\n\n Random seed 2021 has been set.\n\n\n\n 0%| | 0/200 [00:00 num_batches:\n break\n # Extract minibatch data\n data, labels = batch[0].to(device), batch[1].to(device)\n # Evaluate model and loss on minibatch\n preds = model(data)\n loss_log.append(loss_fn(preds, labels).item())\n acc_log.append(torch.mean(1. * (preds.argmax(dim=1) == labels)).item())\n\n return np.mean(loss_log), np.mean(acc_log)\n```\n\nWe define an optimizer in the following steps:\n\n1. Load the corresponding class that implements the parameter updates and other internal management activities, including:\n - create auxiliary variables,\n - update moving averages,\n - adjust learning rate.\n2. Pass the parameters of the Pytorch model that the optimizer has control over. Note that different parameter groups can potentially be controlled by different optimizers.\n3. Specify hyperparameters, including learning rate, momentum, moving average factors, etc.\n\n\n\n## Exercise 8: Train your own model\n\nNow, train the model with your preferred optimizer and find a good combination of hyperparameter settings.\n\n\n```python\n#################################################\n## TODO for students: adjust training settings ##\n\n# The three parameters below are in your full control\nMAX_EPOCHS = 2 # select number of epochs to train\nLR = 1e-5 # choose the step size\nBATCH_SIZE = 64 # number of examples per minibatch\n\n# Define the model and associated optimizer -- you may change its architecture!\nmodel = MLP(in_dim=784, out_dim=10, hidden_dims=[200, 100, 50]).to(DEVICE)\n\n# You can take your pick from many different optimizers\n# Check the optimizer documentation and hyperparameter meaning before using!\n# More details on Pytorch optimizers: https://pytorch.org/docs/stable/optim.html\n# optimizer = torch.optim.SGD(model.parameters(), lr=LR, momentum=0.9)\n# optimizer = torch.optim.RMSprop(model.parameters(), lr=LR, alpha=0.99)\n# optimizer = torch.optim.Adagrad(model.parameters(), lr=LR)\noptimizer = torch.optim.Adam(model.parameters(), lr=LR)\n#################################################\n```\n\n\n```python\nset_seed(seed=SEED)\n# Print trainig stats every LOG_FREQ minibatches\nLOG_FREQ = 200\n# Frequency for evaluating the validation metrics\nVAL_FREQ = 200\n# Load data using a Pytorch Dataset\ntrain_set_orig, test_set_orig = load_mnist_data(change_tensors=False)\n\n# We separate 10,000 training samples to create a validation set\ntrain_set_orig, val_set_orig = torch.utils.data.random_split(train_set, [50000, 10000])\n\n# Create the corresponding DataLoaders for training and test\ng_seed = torch.Generator()\ng_seed.manual_seed(SEED)\n\ntrain_loader = torch.utils.data.DataLoader(train_set_orig,\n shuffle=True,\n batch_size=BATCH_SIZE,\n worker_init_fn=seed_worker,\n generator=g_seed)\nval_loader = torch.utils.data.DataLoader(val_set_orig,\n shuffle=True,\n batch_size=256,\n worker_init_fn=seed_worker,\n generator=g_seed)\ntest_loader = torch.utils.data.DataLoader(test_set_orig,\n batch_size=256,\n worker_init_fn=seed_worker,\n generator=g_seed)\n\n# Run training\nmetrics = {'train_loss':[],\n 'train_acc':[],\n 'val_loss':[],\n 'val_acc':[],\n 'val_idx':[]}\n\nstep_idx = 0\nfor epoch in tqdm(range(MAX_EPOCHS)):\n\n running_loss, running_acc = 0., 0.\n\n for batch_id, batch in enumerate(train_loader):\n step_idx += 1\n # Extract minibatch data and labels\n data, labels = batch[0].to(DEVICE), batch[1].to(DEVICE)\n # Just like before, refresh gradient accumulators.\n # Note that this is now a method of the optimizer.\n optimizer.zero_grad()\n # Evaluate model and loss on minibatch\n preds = model(data)\n loss = loss_fn(preds, labels)\n acc = torch.mean(1.0 * (preds.argmax(dim=1) == labels))\n # Compute gradients\n loss.backward()\n # Update parameters\n # Note how all the magic in the update of the parameters is encapsulated by\n # the optimizer class.\n optimizer.step()\n # Log metrics for plotting\n metrics['train_loss'].append(loss.cpu().item())\n metrics['train_acc'].append(acc.cpu().item())\n\n if batch_id % VAL_FREQ == (VAL_FREQ - 1):\n # Get an estimate of the validation accuracy with 100 batches\n val_loss, val_acc = eval_model(model, val_loader,\n num_batches=100,\n device=DEVICE)\n metrics['val_idx'].append(step_idx)\n metrics['val_loss'].append(val_loss)\n metrics['val_acc'].append(val_acc)\n\n print(f\"[VALID] Epoch {epoch + 1} - Batch {batch_id + 1} - \"\n f\"Loss: {val_loss:.3f} - Acc: {100*val_acc:.3f}%\")\n\n # print statistics\n running_loss += loss.cpu().item()\n running_acc += acc.cpu().item()\n # Print every LOG_FREQ minibatches\n if batch_id % LOG_FREQ == (LOG_FREQ-1):\n print(f\"[TRAIN] Epoch {epoch + 1} - Batch {batch_id + 1} - \"\n f\"Loss: {running_loss / LOG_FREQ:.3f} - \"\n f\"Acc: {100 * running_acc / LOG_FREQ:.3f}%\")\n\n running_loss, running_acc = 0., 0.\n```\n\n Random seed 2021 has been set.\n\n\n\n 0%| | 0/2 [00:00\n \n \n \"\"\" )\n```\n\n\n\n\n\n
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Table of Contents

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\n\n\n```python\n# code for loading the format for the notebook\nimport os\n\n# path : store the current path to convert back to it later\npath = os.getcwd()\nos.chdir(os.path.join('..', '..', 'notebook_format'))\n\nfrom formats import load_style\nload_style(plot_style=False)\n```\n\n\n\n\n\n\n\n\n\n\n\n```python\nos.chdir(path)\n\n# 1. magic for inline plot\n# 2. magic to print version\n# 3. magic so that the notebook will reload external python modules\n# 4. magic to enable retina (high resolution) plots\n# https://gist.github.com/minrk/3301035\n%matplotlib inline\n%load_ext watermark\n%load_ext autoreload\n%autoreload 2\n%config InlineBackend.figure_format='retina'\n\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nfrom sklearn.ensemble import RandomForestClassifier\nfrom sklearn.model_selection import train_test_split\nfrom sklearn.metrics import roc_auc_score, roc_curve, auc\nfrom sklearn.metrics import precision_recall_curve, average_precision_score\n\n%watermark -a 'Ethen' -d -t -v -p numpy,pandas,matplotlib,sklearn\n```\n\n Ethen 2017-09-22 16:18:29 \n \n CPython 3.5.2\n IPython 6.1.0\n \n numpy 1.13.1\n pandas 0.20.3\n matplotlib 2.0.0\n sklearn 0.18.1\n\n\n# ROC/AUC for Binary Classification\n\nFor this documentation, we'll be working with a human resource dataset. Our goal is to find out the employees that are likely to leave in the future and act upon our findings, i.e. retain them before they choose to leave. This dataset contains 12000 observations and 7 variables, each representing :\n\n- `S` The satisfaction level on a scale of 0 to 1\n- `LPE` Last project evaluation by a client on a scale of 0 to 1\n- `NP` Represents the number of projects worked on by employee in the last 12 month\n- `ANH` Average number of hours worked in the last 12 month for that employee\n- `TIC` Amount of time the employee spent in the company, measured in years\n- `Newborn` This variable will take the value 1 if the employee had a newborn within the last 12 month and 0 otherwise\n- `left` 1 if the employee left the company, 0 if they're still working here. This is our response variable\n\n\n```python\nfilename = 'HR.csv'\ndata = pd.read_csv(filename)\nprint('dimensions: ', data.shape)\ndata.head()\n```\n\n dimensions: (12000, 7)\n\n\n\n\n\n
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\n\n\n\nTo train and evaluate the model, we\u2019ll perform a simple train/test split. 80 percent of the dataset will be used to actually train the model, while the rest will be used to evaluate the accuracy of this model, i.e. out of sample error. Note that the best practice is to split it in three ways train/validation/test split.\n\n\n```python\nlabel_col = 'left'\nlabel = data[label_col].values\ndata = data.drop(label_col, axis = 1)\nprint('labels distribution:', np.bincount(label) / label.size)\n\ntest_size = 0.2\nrandom_state = 1234\ndata_train, data_test, y_train, y_test = train_test_split(\n data, label, test_size = test_size, random_state = random_state, stratify = label)\n```\n\n labels distribution: [ 0.83333333 0.16666667]\n\n\nThis probability table tells you that around 16 percent of the employees who became a staff member of yours have left! If those employees are all the ones that are performing well in the company, then this is probably not a good sign. We'll leave the exploratory analysis part to you ...\n\n## Sklearn Transformer\n\nWe then convert perform some generic data preprocessing including standardizing the numeric columns and one-hot-encode the categorical columns (the \"Newborn\" variable is treated as a categorical variable) and convert everything into a numpy array that sklearn expects. This generic preprocessing step is written as a custom sklearn Transformer. You don't have to follow this structure if you prefer your way of doing it.\n\nTo roll out our own Transformer a adheres to the sklearn API, we need to \n\n- Ensure that all arguments to the `__init__` method should be explicit: i.e. `*args` or `**kwargs` should be avoided, as they will not be correctly handled within cross-validation routines\n- Subclass/Inherit [`BaseEstimator`](http://scikit-learn.org/stable/modules/generated/sklearn.base.BaseEstimator.html) to get some free stuff. It will give us class representations that are more informative when printing the class object. And provides us a `get_params` and `set_params` functions. These functionalities are used in sklearn's methods such as GridSearch and RandomSearch.\n- Subclass/Inherit an appropriate class for your task (one of ClassifierMixin, RegressorMixin, ClusterMixin, TransformerMixin). In our case, we will be implementing a Transformer, thus we'll be subclassing [`TransformerMixin`](http://scikit-learn.org/stable/modules/generated/sklearn.base.TransformerMixin.html). For transformer, we need to implement a `.fit` method which fits some stuff on the training data and a `.transform` method that can perform transformation on both the training and test data. Note that we don't need to subclass [`TransformerMixin`](http://scikit-learn.org/stable/modules/generated/sklearn.base.TransformerMixin.html) this to work, but it does give the end-user the idea that this is a Transformer and we get the `.fit_transform` method that does the fitting and transformer on the training data in one shot for free\n- In the fit implementation, you'll notice results that were learned during the `.fit` method is stored with a trailing underscore (e.g., self.colnames_). This is a convention used in sklearn so that we can quickly scan the members of an estimator and distinguish which members are fitting during training time\n\nIf you would like to read more on this topic. The following two link might be of interest to you.\n\n- [Blog: Creating your own estimator in scikit-learn](http://danielhnyk.cz/creating-your-own-estimator-scikit-learn/)\n- [scikit-learn Documentation: Rolling your own estimator](http://scikit-learn.org/dev/developers/contributing.html#rolling-your-own-estimator)\n\n\n```python\nfrom collections import defaultdict\nfrom sklearn.base import BaseEstimator, TransformerMixin\nfrom sklearn.preprocessing import LabelEncoder, OneHotEncoder, StandardScaler\n\n\nclass Preprocess(BaseEstimator, TransformerMixin):\n \"\"\"\n Generic data preprocessing including:\n - standardize numeric columns\n - one-hot encode categorical columns\n\n Parameters\n ----------\n num_cols : list\n Numeric column's name\n\n cat_cols : list\n Categorical column's name\n\n Attributes\n ----------\n colnames_ : list\n Column name of the transformed numpy array\n\n label_encode_dict_ : dict of sklearn's LabelEncoder\n LabelEncoder that was used to encode the value\n of the categorical columns into with value between\n 0 and n_classes-1. Categorical columns will go through\n this encoding process before being one-hot encoded\n\n cat_encode_ : sklearn's OneHotEncoder\n OneHotEncoder that was used to one-hot encode the\n categorical columns\n\n scaler_ : sklearn's StandardScaler\n Scaler that was used to standardize the numeric columns\n \"\"\"\n\n def __init__(self, num_cols = None, cat_cols = None):\n self.num_cols = num_cols\n self.cat_cols = cat_cols\n\n def fit(self, data):\n \"\"\"\n Fit the Preprocess Transformer\n\n Parameters\n ----------\n data : DataFrame\n \"\"\"\n data = data.copy()\n\n # Label encoding across multiple columns in scikit-learn\n # https://stackoverflow.com/questions/24458645/label-encoding-across-multiple-columns-in-scikit-learn\n if self.cat_cols is not None:\n self.label_encode_dict_ = defaultdict(LabelEncoder)\n label_encoded = (data[self.cat_cols].\n apply(lambda x: self.label_encode_dict_[x.name].fit_transform(x)))\n\n self.cat_encode_ = OneHotEncoder(sparse = False)\n self.cat_encode_.fit(label_encoded)\n\n if self.num_cols is not None:\n self.scaler_ = StandardScaler().fit(data[self.num_cols])\n\n # store the column names (numeric columns comes before the\n # categorical columns) so we can refer to them later\n if self.num_cols is not None:\n colnames = self.num_cols.copy()\n else:\n colnames = []\n\n if self.cat_cols is not None:\n for col in self.cat_cols:\n cat_colnames = [col + '_' + str(classes)\n for classes in self.label_encode_dict_[col].classes_]\n colnames += cat_colnames\n\n self.colnames_ = colnames\n return self\n\n def transform(self, data):\n \"\"\"\n Trasform the data using the fitted Preprocess Transformer\n\n Parameters\n ----------\n data : DataFrame\n \"\"\"\n if self.cat_cols is not None:\n label_encoded = (data[self.cat_cols].\n apply(lambda x: self.label_encode_dict_[x.name].transform(x)))\n cat_encoded = self.cat_encode_.transform(label_encoded)\n\n if self.num_cols is not None:\n scaled = self.scaler_.transform(data[self.num_cols])\n\n # combine encoded categorical columns and scaled numerical\n # columns, it's the same as concatenate it along axis 1\n if self.cat_cols is not None and self.num_cols is not None:\n X = np.hstack((scaled, cat_encoded))\n elif self.num_cols is None:\n X = cat_encoded\n else:\n X = scaled\n\n return X\n```\n\n\n```python\nnum_cols = ['S', 'LPE', 'NP', 'ANH', 'TIC']\ncat_cols = ['Newborn']\n\npreprocess = Preprocess(num_cols, cat_cols)\nX_train = preprocess.fit_transform(data_train)\nX_test = preprocess.transform(data_test)\n\nprint('colnames', preprocess.colnames_)\nX_train\n```\n\n colnames ['S', 'LPE', 'NP', 'ANH', 'TIC', 'Newborn_0', 'Newborn_1']\n\n\n\n\n\n array([[ 0.24997745, 0.61402599, 0.16885833, ..., 0.72182646,\n 1. , 0. ],\n [ 0.12545568, -0.27568766, 1.02655144, ..., -0.22007024,\n 1. , 0. ],\n [ 0.58203549, -1.34334405, 0.16885833, ..., -0.22007024,\n 1. , 0. ],\n ..., \n [-1.32729826, -0.75020161, -0.68883478, ..., -0.22007024,\n 1. , 0. ],\n [ 0.58203549, -0.21637342, -0.68883478, ..., -1.16196694,\n 0. , 1. ],\n [ 0.45751372, 0.25814053, -0.68883478, ..., -0.22007024,\n 1. , 0. ]])\n\n\n\n\n```python\n# pick your favorite classfication model\ntree = RandomForestClassifier(max_depth = 4)\ntree.fit(X_train, y_train)\n```\n\n\n\n\n RandomForestClassifier(bootstrap=True, class_weight=None, criterion='gini',\n max_depth=4, max_features='auto', max_leaf_nodes=None,\n min_impurity_split=1e-07, min_samples_leaf=1,\n min_samples_split=2, min_weight_fraction_leaf=0.0,\n n_estimators=10, n_jobs=1, oob_score=False, random_state=None,\n verbose=0, warm_start=False)\n\n\n\nAfter training our model, we need to evaluate whether its any good or not and the most straightforward and intuitive metric for a supervised classifier's performance is accuracy. Unfortunately, there are circumstances where simple accuracy does not work well. For example, with a disease that only affects 1 in a million people, a completely bogus screening test that always reports \"negative\" will be 99.9999% accurate. Unlike accuracy, ROC curves are less sensitive to class imbalance; the bogus screening test would have an AUC of 0.5, which is like not having a test at all.\n\n## ROC curves\n\n**ROC curve (Receiver Operating Characteristic)** is a commonly used way to visualize the performance of a binary classifier and AUC (Area Under the ROC Curve) is used to summarize its performance in a single number. Most machine learning algorithms have the ability to produce probability scores that tells us the strength in which it thinks a given observation is positive. Turning these probability scores into yes or no predictions requires setting a threshold; cases with scores above the threshold are classified as positive, and vice versa. Different threshold values can lead to different result:\n\n- A higher threshold is more conservative about labeling a case as positive; this makes it less likely to produce false positive (an observation that has a negative label but gets classified as positive by the model) results but more likely to miss cases that are in fact positive (lower true positive rate)\n- A lower threshold produces positive labels more liberally, so it creates more false positives but also generate more true positives\n\nA quick refresher on terminology:\n\n\\begin{align}\n[\\text{true positive rate}]\n&= \\frac{[\\text{# positive data points with positive predictions}]}{\\text{[# all positive data points]}} \\\\\n&= \\frac{[\\text{# true positives}]}{[\\text{# true positives}] + [\\text{# false negatives}]}\n\\end{align}\n\ntrue positive rate is also known as **recall** or **sensitivity**\n\n\\begin{align}\n[\\text{false positive rate}]\n&= \\frac{[\\text{# positive data points with positive predictions}]}{\\text{[# all negative data points]}} \\\\\n&= \\frac{[\\text{# false positives}]}{[\\text{# false positives}] + [\\text{# true negatives}]}\n\\end{align}\n\nThe ROC curve is created by plotting the true positive rate (when it's actually a yes, how often does it predict yes?) on the y axis against the false positive rate (when it's actually a no, how often does it predict yes?) on the x axis at various cutoff settings, giving us a picture of the whole spectrum of the trade-off we're making between the two measures.\n\nIf all these true/false positive terminology is confusing to you, consider reading the material at the following link. [Blog: Simple guide to confusion matrix terminology](http://www.dataschool.io/simple-guide-to-confusion-matrix-terminology/)\n\n### Implementation\n\nThere are packages to plot ROC curves and to compute metrics from them, but it can still be worthwhile to work through how these curves are calculated from scratch to try and understand better what exactly are they showing us.\n\n\n```python\ndef _binary_clf_curve(y_true, y_score):\n \"\"\"\n Calculate true and false positives per binary classification\n threshold (can be used for roc curve or precision/recall curve);\n the calcuation makes the assumption that the positive case\n will always be labeled as 1\n\n Parameters\n ----------\n y_true : 1d ndarray, shape = [n_samples]\n True targets/labels of binary classification\n\n y_score : 1d ndarray, shape = [n_samples]\n Estimated probabilities or scores\n\n Returns\n -------\n tps : 1d ndarray\n True positives counts, index i records the number\n of positive samples that got assigned a\n score >= thresholds[i].\n The total number of positive samples is equal to\n tps[-1] (thus false negatives are given by tps[-1] - tps)\n\n fps : 1d ndarray\n False positives counts, index i records the number\n of negative samples that got assigned a\n score >= thresholds[i].\n The total number of negative samples is equal to\n fps[-1] (thus true negatives are given by fps[-1] - fps)\n\n thresholds : 1d ndarray\n Predicted score sorted in decreasing order\n\n References\n ----------\n Github: scikit-learn _binary_clf_curve\n - https://github.com/scikit-learn/scikit-learn/blob/ab93d65/sklearn/metrics/ranking.py#L263\n \"\"\"\n\n # sort predicted scores in descending order\n # and also reorder corresponding truth values\n desc_score_indices = np.argsort(y_score)[::-1]\n y_score = y_score[desc_score_indices]\n y_true = y_true[desc_score_indices]\n\n # y_score typically consists of tied values. Here we extract\n # the indices associated with the distinct values. We also\n # concatenate a value for the end of the curve\n distinct_indices = np.where(np.diff(y_score))[0]\n end = np.array([y_true.size - 1])\n threshold_indices = np.hstack((distinct_indices, end))\n\n thresholds = y_score[threshold_indices]\n tps = np.cumsum(y_true)[threshold_indices]\n\n # (1 + threshold_indices) = the number of positives\n # at each index, thus number of data points minus true\n # positives = false positives\n fps = (1 + threshold_indices) - tps\n return tps, fps, thresholds\n```\n\n\n```python\n# we'll work with some toy data so it's easier to\n# show and confirm the calculated result\ny_true = np.array([1, 0, 1, 0, 1])\ny_score = np.array([0.45, 0.4, 0.35, 0.35, 0.8])\n\ntps, fps, thresholds = _binary_clf_curve(y_true, y_score)\nprint('thresholds:', thresholds)\nprint('true positive count:', tps)\nprint('false positive count:', fps)\n```\n\n thresholds: [ 0.8 0.45 0.4 0.35]\n true positive count: [1 2 2 3]\n false positive count: [0 0 1 2]\n\n\nFrom the result above, we can see that the function will compute the true/false positive count for all unique threshold in the predicted score `y_score`. We can validate the result by hand to confirm that the calculation this in fact correct.\n\nRecall that ROC curve plots that true positive rate on the y-axis and false positive rate on the x-axis. Thus all we need to do is to convert the count into rate and we have our ROC curve.\n\n\n```python\n# convert count to rate, append 0 to\n# both true positive and false positive\n# so the visualization will start from origin (0, 0)\ntpr = np.hstack((0, tps / tps[-1]))\nfpr = np.hstack((0, fps / fps[-1]))\nprint('true positive rate:', tpr)\nprint('false positive rate:', fpr)\n\nplt.rcParams['figure.figsize'] = 8, 6\nplt.rcParams['font.size'] = 12\n\nfig = plt.figure()\nplt.plot(fpr, tpr, marker = 'o', lw = 1)\nplt.xlabel('false positive rate')\nplt.ylabel('true positive rate')\nplt.title('Receiver Operator Characteristic')\nplt.show()\n```\n\nNow to calculate the AUC (Area Under the Curve) for the ROC curve, we need sum up the rectangular area and the triangular area under the curve. Depicted by the visualization below:\n\n- For the rectangular area (the plot on the left illustrates one of them), the height are the TPR (true positive rate) and widths are in the difference in the FPR (false positive rate), so the total area of all the rectangles is the dot product of TPR and FPR's differences\n- For the triangular area (the plot on the right illustrates one of them), the height are the difference in TPR (true positive rate) and widths are in the difference in the FPR (false positive rate), so the total area of all the rectangles is the dot product of both TPR's and FPR's difference. But only half the area of each rectangle is below its segment of the ROC curve, thus we divide the rectangle by 2 to obtain the triangular area\n\n\n```python\nimport matplotlib.patches as patches\n\n\nfig, ax = plt.subplots(1, 2, figsize = (10, 6))\nfig.suptitle('Receiver Operator Characteristic', y = 1.02)\n\n# this part is hard-coded for illustration purpose\nfpr_diff = fpr[3] - fpr[0]\ntpr_diff = tpr[3] - tpr[0]\nrect1 = patches.Rectangle(xy = (0, 0), width = fpr_diff,\n height = tpr_diff, alpha = 0.3)\nax[0].add_patch(rect1)\n\nfpr_diff = fpr[-1] - fpr[-2]\ntpr_diff = tpr[-1] - tpr[-2]\nrect2 = patches.Rectangle(xy = (fpr[-2], tpr[-2]), width = fpr_diff,\n height = tpr_diff, alpha = 0.3)\nax[1].add_patch(rect2)\n\nfor i in range(len(ax)):\n ax[i].plot(fpr, tpr, marker = 'o', lw = 1)\n ax[i].set_xlabel('false positive rate')\n ax[i].set_ylabel('true positive rate')\n\nplt.tight_layout()\nplt.show()\n```\n\n\n```python\ndef _roc_auc_score(y_true, y_score):\n \"\"\"\n Compute Area Under the Curve (AUC) from prediction scores\n\n Parameters\n ----------\n y_true : 1d ndarray, shape = [n_samples]\n True targets/labels of binary classification\n\n y_score : 1d ndarray, shape = [n_samples]\n Estimated probabilities or scores\n\n Returns\n -------\n auc : float\n \"\"\"\n\n # ensure the target is binary\n if np.unique(y_true).size != 2:\n raise ValueError('Only two class should be present in y_true. ROC AUC score '\n 'is not defined in that case.')\n \n tps, fps, _ = _binary_clf_curve(y_true, y_score)\n\n # convert count to rate\n tpr = tps / tps[-1]\n fpr = fps / fps[-1]\n\n # compute AUC using the trapezoidal rule;\n # appending an extra 0 is just to ensure the length matches\n zero = np.array([0])\n tpr_diff = np.hstack((np.diff(tpr), zero))\n fpr_diff = np.hstack((np.diff(fpr), zero))\n auc = np.dot(tpr, fpr_diff) + np.dot(tpr_diff, fpr_diff) / 2\n return auc\n```\n\n\n```python\nauc_score = _roc_auc_score(y_true, y_score)\nprint('auc score:', auc_score)\n\n# confirm with scikit-learn's result\nauc_score = roc_auc_score(y_true, y_score)\nprint('package auc socre:', auc_score)\n```\n\n auc score: 0.75\n package auc socre: 0.75\n\n\nAfter working through the implementation of ROC curve and AUC score from sratch, we now pull back and visualize:\n\n- The ROC curve of our original model\n- Dotted line represents the ROC curve of a purely random classifier and a perfect classifier\n\n\n```python\n# calling the roc_curve, extract the probability of \n# the positive class from the predicted probability\ntree_test_pred = tree.predict_proba(X_test)[:, 1]\nfpr, tpr, thresholds = roc_curve(y_test, tree_test_pred, pos_label = 1)\n\n# AUC score that summarizes the ROC curve\nroc_auc = auc(fpr, tpr)\n\nplt.plot(fpr, tpr, lw = 2, label = 'ROC AUC: {:.2f}'.format(roc_auc))\nplt.plot([0, 1], [0, 1],\n linestyle = '--',\n color = (0.6, 0.6, 0.6),\n label = 'random guessing')\nplt.plot([0, 0, 1], [0, 1, 1],\n linestyle = ':',\n color = 'black', \n label = 'perfect performance')\n\nplt.xlim([-0.05, 1.05])\nplt.ylim([-0.05, 1.05])\nplt.xlabel('false positive rate')\nplt.ylabel('true positive rate')\nplt.title('Receiver Operator Characteristic')\nplt.legend(loc = \"lower right\")\nplt.tight_layout()\nplt.show()\n```\n\nThe goal of visualizing the ROC curve is to let us know how well can our classifier be expected to perform in general, at a variety of different baseline probabilities (percentage of the majority class)?\n\nThe diagonal line depicts a completely random classifier and ideally our model's ROC curve should be toward the top-left corner and stay as far away from the diagonal line as possible.\n\nSide note: Apart from comparing the model's ROC curve against the ROC curve of a classifier that does random guessing, it's also useful to plot the ROC curve of different classifiers to compare performance against each other.\n\n### AUC probabilistic interpretation\n\nThe probabilistic interpretation of the AUC metric is that if we randomly choose a positive case and a negative case, the probability that the positive case outranks the negative case according to the classifier's prediction. Hopefully, this is evident from the ROC curve figure, where plot is enumerating all possible combinations of positive and negative cases, and the fraction under the curve comprises of the area where the positive case outranks the negative one. I personally find this interpretation extremely useful when conveying what AUC is measuring to a non-technical audience.\n\n\n```python\ndef auc_probability(y_true, y_score, size = 100000):\n \"\"\"probabilistic interpretation of AUC\"\"\"\n labels = y_true.astype(np.bool)\n pos = np.random.choice(y_score[labels], size = size, replace = True)\n neg = np.random.choice(y_score[~labels], size = size, replace = True)\n auc = np.sum(pos > neg) + np.sum(pos == neg) / 2\n auc /= size\n return auc\n\n\n# we want this be close the score returned\n# by roc_auc_score\nauc_probability(y_true = y_test, y_score = tree_test_pred)\n```\n\n\n\n\n 0.97633000000000003\n\n\n\n## Precision Recall Curve\n\nApart from ROC curve, there is also the **precision recall curve**. Instead of plotting true positive rate (a.k.a recall) versus false positive rate. We now plot precision versus recall.\n\n\\begin{align}\n[\\text{precision}]\n&= \\frac{[\\text{# positive data points with positive predicitions}]}{\\text{[# all data points with positive predictions]}} \\\\\n&= \\frac{[\\text{# true positives}]}{[\\text{# true positives}] + [\\text{# false positives}]}\n\\end{align}\n\n\n```python\ntree_test_pred = tree.predict_proba(X_test)[:, 1]\nprecision, recall, thresholds = precision_recall_curve(\n y_test, tree_test_pred, pos_label = 1)\n\n# AUC score that summarizes the precision recall curve\navg_precision = average_precision_score(y_test, tree_test_pred)\n\nlabel = 'Precision Recall AUC: {:.2f}'.format(avg_precision)\nplt.plot(recall, precision, lw = 2, label = label)\nplt.xlabel('Recall') \nplt.ylabel('Precision') \nplt.title('Precision Recall Curve')\nplt.legend()\nplt.tight_layout()\nplt.show()\n```\n\nA classifier with high recall but low precision flags many positive results, but most of its predicted labels are incorrect when compared to its corresponding labels. On the other hand, a classifier with high precision but low recall is just the opposite, returning very few results, but most of its predicted labels are correct when compared to the training labels. An ideal system with high precision and high recall will return many results, with all results labeled correctly.\n\nPrecision recall curve answers a fundamentally different question compared to ROC curve. By definition, precision directly answers the question, \"What is the probability that this is a real hit given my classifier says it is? Thus it is useful in practice for needle-in-haystack type problems or problems where the \"positive\" class is more interesting than the negative class.\n\nYou can also think about it in the following way. ROC AUC looks at TPR and FPR, the entire confusion matrix for all thresholds. On the other hand, Precision-Recall AUC looks at Precision and Recall (TPR), it doesn't look at True Negative Rate (TNR). Because of that PR AUC can be a better choice when you care only about the \"positive\" while ROC AUC cares about both \"positive\" and \"negative\". Since PR AUC doesn't use TNR directly it can also be better for highly imbalanced problems. You may want to take a look at this [Blog: F1 Score vs ROC AUC vs Accuracy vs PR AUC: Which Evaluation Metric Should You Choose?](https://neptune.ml/blog/f1-score-accuracy-roc-auc-pr-auc). \n\nAlthough ROC curve is presumably the more popular choice when evaluating binary classifiers, it is highly recommended to use precision recall curve as a supplement to ROC curves to get a full picture when evaluating and comparing classifiers. For more discussion on this topic, consider taking a look at another documentation. [Notebook: Evaluating Imbalanced Datasets](http://nbviewer.jupyter.org/github/ethen8181/machine-learning/blob/master/model_selection/imbalanced/imbalanced_metrics.ipynb)\n\n# Thresholding via Cost\n\nLots of real world binary classification needs a threshold to convert the model into a business decision. i.e. All cases with a model score above the threshold get some sort of special treatment. For example:\n\n- **Fraud Prevention**: We're a social network company and we'd like to delete fake accounts. We build a classifier that assigns a \"fraud score\" between 0 and 1 to each account and, after some research, decide that all accounts whose score is above 0.9 should be sent to our fraud team, which will review each case and delete the accounts that are actually fake\n- **Response/Propensity Modeling**: We're at an enterprise software company and we'd like to improve our outbound sales program. We buy a large database of potential customers and build a classifier to predict which ones are likely buy our product if contacted by our sales team. We decide that all customers with a \"response score\" of above threshold 0.7 should get a call\n- **Shopping Cart Abandonment**: We're at an e-commerce company, and we'd like to email a 10% off coupon to users who abandoned their shopping carts and won't return organically. (We don't want to give a 10% off coupon to users who are going to return anyway, since then you'd be losing 10% of the sale price.) We build a classifier to predict which users will never return to their carts, and decide that all users with a \"abandonment score\" above 0.85 should get the 10% off coupon\n- Etc.\n\nThresholding is popular because of its simplicity and ease of implementation: We translate a continuous score to a binary yes/no decision, and act on it in a predetermined way. The biggest question for the thresholding pattern is: Where should I set the threshold point?\n\nUp until this point, we've been using AUC to give us a single-number summary of classifier performance. This might be suitable in some circumstances, but for binary classifiers, evaluation metrics that take into account the actual costs of false positive and false negative errors may be much more appropriate than AUC. If we know these costs, we can use them not only to tie the evaluation metric more directly to the business value but also choose an appropriate final cutoff threshold for the classifier.\n\nIn real world application, the cost that comes along with making these two mistakes (false positive and false negative) are usually a whole lot different. Take our case for example, a false negative (FN) means an employee left our company but our model failed to detect that, while a false positive (FP) means an employee is still currently working at our company and our model told us that they will be leaving. The former mistake would be a tragedy, since, well, the employee left and we didn't do anything about it! As for conducting the latter mistake, we might be wasting like 20 minutes of a HR manager's time when we arrange a face to face interview with a employee, questioning about how the company can do better to retain him/her, while he/she is perfectly fine with the current situation.\n\nIn the code cell below, we assign a cost for a false negative (FN) and false positive (FP) to be 100 and 1000 respectively, given the cost associated with the two mistakes we will multiply them with the false negative and false positive rate at each threshold to figure out where's the best cutoff value. Note that the cost associated with the mistake can just be a back of the envelope number as long as we're sure about which one is more \"expensive\" than the other.\n\n\n```python\nfpr, tpr, thresholds = roc_curve(y_test, tree_test_pred, pos_label = 1)\nfnr = tpr[-1] - tpr\n\n# fn = false negative, meaning the employee left and we\n# didn't do anything about it, in our case this will be\n# substantially more expensive than a false positive\nfp_cost = 100\nfn_cost = 1000\n\ncosts = (fpr * fp_cost + fnr * fn_cost) * y_test.size\n```\n\n\n```python\nbest = np.argmin(costs)\nbest_cost = np.round(costs[best])\nbest_threshold = np.round(thresholds[best], 2)\n\nplt.plot(thresholds, costs, label = 'cost ($)')\nlabel = 'best cost: {} at threshold {}'.format(best_cost, best_threshold)\nplt.axvline(best_threshold, linestyle = '--', lw = 2, color = 'black', label = label)\nplt.xlabel('threshold') \nplt.ylabel('$') \nplt.title('Cost as a Function of Threshold')\nplt.legend()\nplt.tight_layout()\nplt.show()\n```\n\nJust to hit the notion home, when executing on a project, if we are able to compute expected cost for each mistake, consider maximizing that directly instead of AUC or another general-purpose metrics.\n\n# Reference\n\n- [Scikit-learn Documentation: Precision Recall](http://scikit-learn.org/stable/auto_examples/model_selection/plot_precision_recall.html)\n- [StackExchange: ROC vs precision-and-recall curves](https://stats.stackexchange.com/questions/7207/roc-vs-precision-and-recall-curves)\n- [Blog: Calculating AUC: the area under a ROC Curve](http://blog.revolutionanalytics.com/2016/11/calculating-auc.html)\n- [Blog: F1 Score vs ROC AUC vs Accuracy vs PR AUC: Which Evaluation Metric Should You Choose?](https://neptune.ml/blog/f1-score-accuracy-roc-auc-pr-auc)\n- [Blog: Machine Learning Meets Economics](http://blog.mldb.ai/blog/posts/2016/01/ml-meets-economics/)\n- [Blog: Visualizing Machine Learning Thresholds to Make Better Business Decisions](http://blog.insightdatalabs.com/visualizing-classifier-thresholds/)\n", "meta": {"hexsha": "25b7c979cf566c3485d0239bef3f0f4566bb1426", "size": 363666, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "model_selection/auc/auc.ipynb", "max_stars_repo_name": "anhnongdan/machine_learning", "max_stars_repo_head_hexsha": "ad247554026b53f285ea96491c4834c8f3057435", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2104, "max_stars_repo_stars_event_min_datetime": "2016-04-15T13:35:55.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-28T10:39:51.000Z", "max_issues_repo_path": "model_selection/auc/auc.ipynb", "max_issues_repo_name": "aniruddhachoudhury/machine-learning", "max_issues_repo_head_hexsha": "02744a34709fe908c169aefdcd48dc3f528991f0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 10, "max_issues_repo_issues_event_min_datetime": "2017-04-07T14:25:23.000Z", "max_issues_repo_issues_event_max_datetime": "2021-05-18T03:16:15.000Z", "max_forks_repo_path": "model_selection/auc/auc.ipynb", "max_forks_repo_name": "aniruddhachoudhury/machine-learning", "max_forks_repo_head_hexsha": "02744a34709fe908c169aefdcd48dc3f528991f0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 539, "max_forks_repo_forks_event_min_datetime": "2015-12-10T04:23:44.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-31T07:15:28.000Z", "avg_line_length": 255.9225897255, "max_line_length": 84714, "alphanum_fraction": 0.8927614899, "converted": true, "num_tokens": 10000, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.596433160611502, "lm_q2_score": 0.7461389873857265, "lm_q1q2_score": 0.44502203450193445}} {"text": "```python\n%matplotlib inline\n\nimport matplotlib.pyplot as plt\nimport seaborn as sns\nsns.set(color_codes=True)\nsns.set(font='SimHei')\n#plt.rcParams['axes.grid'] = False\n\nimport numpy as np\n\nimport pandas as pd\npd.options.display.max_rows = 10\n\nfrom IPython.display import Image\n```\n\n\u51b3\u7b56\u6811\u539f\u7406\u4e0e\u5b9e\u73b0\u7b80\u4ecb\n=================\n\n### \u524d\u8a00\n\n\u4e3a\u4ec0\u4e48\u8bb2\u51b3\u7b56\u6811?\n\n1. \u539f\u7406\u7b80\u5355\uff0c\u76f4\u767d\u6613\u61c2\u3002\n2. \u53ef\u89e3\u91ca\u6027\u597d\u3002\n3. \u53d8\u79cd\u5728\u5de5\u4e1a\u4e0a\u5e94\u7528\u591a\uff1a\u968f\u673a\u68ee\u6797\u3001GBDT\u3002\n\n\u6df1\u5316\u62d3\u5c55\n\n1. \u7406\u8bba\uff0c\u8003\u53e4\uff1aID3, C4.5, CART\n2. \u5de5\u7a0b\uff0c\u5b9e\u73b0\u7ec6\u8282\uff1a\n + demo\n + scikit-learn\n + spark\n + xgboost\n3. \u5e94\u7528\uff0c\u8c03\u53c2\u5206\u6790\n4. \u6f14\u793a\n\n### \u7406\u8bba\n\n\u7b97\u6cd5\uff1a\n\n1. ID3\n2. C4.5\n3. C5.0\n4. **CART**\n5. CHAID\n6. MARS\n\n#### \u884c\u4e1a\u9ed1\u8bdd\n\n+ \u5206\u7c7b\u95ee\u9898 vs \u56de\u5f52\u95ee\u9898\n\n+ \u6837\u672c = \uff08\u7279\u5f81$x$\uff0c\u771f\u5b9e\u503c$y$\uff09\n\n+ \u76ee\u7684\uff1a\u627e\u5230\u6a21\u578b$h(\\cdot)$\uff0c\u4f7f\u5f97\u9884\u6d4b\u503c$\\hat{y} = h(x)$ $\\to$ \u771f\u5b9e\u503c$y$\n\n\n```python\nfrom sklearn.datasets import load_iris\ndata = load_iris()\n\n# \u51c6\u5907\u7279\u5f81\u6570\u636e\nX = pd.DataFrame(data.data, \n columns=[\"sepal_length\", \"sepal_width\", \"petal_length\", \"petal_width\"])\n\n# \u51c6\u5907\u6807\u7b7e\u6570\u636e\ny = pd.DataFrame(data.target, columns=['target'])\ny.replace(to_replace=range(3), value=data.target_names, inplace=True)\n\n# \u7ec4\u5efa\u6837\u672c [\u7279\u5f81\uff0c\u6807\u7b7e]\nsamples = pd.concat([X, y], axis=1, keys=[\"x\", \"y\"])\nsamples.head(5)\n```\n\n\n\n\n
\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
xy
sepal_lengthsepal_widthpetal_lengthpetal_widthtarget
05.13.51.40.2setosa
14.93.01.40.2setosa
24.73.21.30.2setosa
34.63.11.50.2setosa
45.03.61.40.2setosa
\n
\n\n\n\n\n```python\nsamples[\"y\", \"target\"].value_counts()\n```\n\n\n\n\n versicolor 50\n virginica 50\n setosa 50\n Name: (y, target), dtype: int64\n\n\n\n\n```python\nsamples[\"x\"].describe()\n```\n\n\n\n\n
\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
sepal_lengthsepal_widthpetal_lengthpetal_width
count150.000000150.000000150.000000150.000000
mean5.8433333.0540003.7586671.198667
std0.8280660.4335941.7644200.763161
min4.3000002.0000001.0000000.100000
25%5.1000002.8000001.6000000.300000
50%5.8000003.0000004.3500001.300000
75%6.4000003.3000005.1000001.800000
max7.9000004.4000006.9000002.500000
\n
\n\n\n\n#### \u4e09\u5206\u949f\u660e\u767d\u51b3\u7b56\u6811\n\n\n```python\nImage(url=\"https://upload.wikimedia.org/wikipedia/commons/f/f3/CART_tree_titanic_survivors.png\")\n```\n\n\n\n\n\n\n\n\n\n```python\nImage(url=\"http://scikit-learn.org/stable/_images/iris.svg\")\n```\n\n\n\n\n\n\n\n\n\n```python\nImage(url=\"http://scikit-learn.org/stable/_images/sphx_glr_plot_iris_0011.png\")\n```\n\n\n\n\n\n\n\n\n\n```python\nsamples = pd.concat([X, y], axis=1)\nsamples.head(3)\n```\n\n\n\n\n
\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
sepal_lengthsepal_widthpetal_lengthpetal_widthtarget
05.13.51.40.2setosa
14.93.01.40.2setosa
24.73.21.30.2setosa
\n
\n\n\n\n### \u5de5\u7a0b\n\n#### Demo\u5b9e\u73b0\n\n\u5176\u4e3b\u8981\u95ee\u9898\u662f\u5728\u6bcf\u6b21\u51b3\u7b56\u65f6\u627e\u5230\u4e00\u4e2a\u5206\u5272\u70b9\uff0c\u8ba9\u751f\u6210\u7684\u5b50\u96c6\u5c3d\u53ef\u80fd\u5730\u7eaf\u51c0\u3002\u8fd9\u91cc\u6d89\u53ca\u5230\u56db\u4e2a\u95ee\u9898:\n\n1. \u5982\u4f55\u5206\u5272\u6837\u672c\uff1f\n2. \u5982\u4f55\u8bc4\u4ef7\u5b50\u96c6\u7684\u7eaf\u51c0\u5ea6\uff1f\n3. \u5982\u4f55\u627e\u5230\u5355\u4e2a\u6700\u4f73\u7684\u5206\u5272\u70b9\uff0c\u5176\u5b50\u96c6\u6700\u4e3a\u7eaf\u51c0\uff1f\n4. \u5982\u4f55\u627e\u5230\u6700\u4f73\u7684\u5206\u5272\u70b9\u5e8f\u5217\uff0c\u5176\u6700\u7ec8\u5206\u5272\u5b50\u96c6\u603b\u4f53\u6700\u4e3a\u7eaf\u51c0\uff1f\n\n\n```python\nImage(url=\"https://upload.wikimedia.org/wikipedia/commons/f/f3/CART_tree_titanic_survivors.png\")\n```\n\n\n\n\n\n\n\n\n#### 1.0 \u5982\u4f55\u5206\u5272\u6837\u672c\n\u51b3\u7b56\u6811\u7684\u5206\u5272\u65b9\u6cd5\u662f\u53d6\u4e00\u4e2a\u7279\u5f81 $f$ \u548c\u9608\u503c $t$\uff0c\u4ee5\u6b64\u4e3a\u754c\u5c06\u6837\u672c $X$ \u62c6\u5206\u4e3a\u4e24\u4e2a\u5b50\u96c6 $X_l, X_r$\u3002\u5176\u6570\u5b66\u8868\u8fbe\u5f62\u540c\uff1a\n\n\\begin{align}\n X = \\begin{cases}\n X_l, \\ \\text{if } X[f] < t \\\\\n X_r, \\ \\text{if } X[f] \\geq t\n \\end{cases}\n\\end{align}\n\n\n```python\ndef splitter(samples, feature, threshold):\n # \u6309\u7279\u5f81 f \u548c\u9608\u503c t \u5206\u5272\u6837\u672c\n \n left_nodes = samples.query(\"{f} < {t}\".format(f=feature, t=threshold))\n right_nodes = samples.query(\"{f} >= {t}\".format(f=feature, t=threshold))\n \n return {\"left_nodes\": left_nodes, \"right_nodes\": right_nodes}\n```\n\n\n```python\nsplit = splitter(samples, \"sepal_length\", 5)\n\n# \u5de6\u5b50\u96c6\nx_l = split[\"left_nodes\"].loc[:, \"target\"].value_counts()\nx_l\n```\n\n\n\n\n setosa 20\n versicolor 1\n virginica 1\n Name: target, dtype: int64\n\n\n\n\n```python\n# \u53f3\u5b50\u96c6\nx_r = split[\"right_nodes\"].loc[:, \"target\"].value_counts()\nx_r\n```\n\n\n\n\n versicolor 49\n virginica 49\n setosa 30\n Name: target, dtype: int64\n\n\n\n#### 2. \u5982\u4f55\u8bc4\u4ef7\u5b50\u96c6\u7684\u7eaf\u51c0\u5ea6\uff1f\n\n\u5e38\u7528\u7684\u8bc4\u4ef7\u51fd\u6570\u6b63\u662f\u8ba1\u7b97\u5404\u6807\u7b7e $c_k$ \u5728\u5b50\u96c6\u4e2d\u7684\u5360\u6bd4 $p_k = c_k / \\sum (c_k)$\uff0c\u5e76\u901a\u8fc7\u7ec4\u5408 $p_k$ \u6765\u63cf\u8ff0\u5360\u6bd4\u96c6\u4e2d\u6216\u5206\u6563\u3002\n\n\n```python\ndef calc_class_proportion(node):\n # \u8ba1\u7b97\u5404\u6807\u7b7e\u5728\u96c6\u5408\u4e2d\u7684\u5360\u6bd4\n \n y = node[\"target\"]\n return y.value_counts() / y.count()\n```\n\n\n```python\ncalc_class_proportion(split[\"left_nodes\"])\n```\n\n\n\n\n setosa 0.909091\n versicolor 0.045455\n virginica 0.045455\n Name: target, dtype: float64\n\n\n\n\n```python\ncalc_class_proportion(split[\"right_nodes\"])\n```\n\n\n\n\n versicolor 0.382812\n virginica 0.382812\n setosa 0.234375\n Name: target, dtype: float64\n\n\n\n\u4e3b\u8981\u7684\u8bc4\u4ef7\u51fd\u6570\u6709\u4e09\u79cd\uff0c\u5b83\u4eec\u8bc4\u4ef7\u7684\u662f\u96c6\u5408\u7684\u4e0d\u7eaf\u5ea6\uff08\u503c\u8d8a\u5927\uff0c\u96c6\u5408\u8d8a\u6df7\u6742\uff09\u3002\n\n\u5148\u505a\u4e9b\u6570\u5b66\u5b9a\u4e49\u4ee5\u4fbf\u4e8e\u63cf\u8ff0\uff1a \n\u5047\u8bbe\u5bf9\u4e8e\u96c6\u5408 $m$ \u6709 $N_m$ \u4e2a\u6837\u672c\uff0c\u53ef\u5206\u5272\u6210 $R_m$ \u5b50\u96c6\u3002 \n\u82e5\u603b\u7684\u6807\u7b7e\u7c7b\u522b\u6709 $K$ \u79cd\uff0c\u5219\u6807\u7b7e $k$ \u5728\u6b64\u96c6\u5408\u4e2d\u7684\u5360\u6bd4\u4e3a\uff1a\n\n\\begin{equation}\n \\hat{p}_{m k} = \\frac{1}{N_m} \\displaystyle \\sum_{x_i \\in R_m} I(y_i = k)\n\\end{equation}\n\n\u4e14\u4ee4\u6807\u7b7e $k$ \u662f\u5360\u6bd4\u6700\u5927\u7684\u6807\u7b7e\uff0c\u5373 $k(m) = \\operatorname{arg max}_k \\hat{p}_{m k}$.\n\n##### 1. Misclassification error\n\u6211\u4eec\u4e00\u822c\u628a\u96c6\u5408\u7684\u5206\u7c7b\u7ed3\u679c\u5b9a\u4e49\u4e3a\u5360\u6bd4\u6700\u5927\u7684\u6807\u7b7e\uff0c\u90a3\u4e48\u843d\u5728\u6b64\u96c6\u5408\u4e2d\u7684\u5176\u5b83\u6807\u7b7e\u5c31\u662f\u8bef\u5206\u7c7b\u3002\u5176\u6bd4\u7387\u662f $1 - \\hat{p}_{m k}(m)$.\n\n#### scikit-learn\u5b9e\u73b0\u7b80\u4ecb\n\n### \u5e94\u7528\n\n\u8bc4\u4ef7\uff1a\u8fc7\u62df\u5408\n\n\u5982\u4f55\u6a21\u578b\u89e3\u91ca\n\n\u53c2\u6570\u7684\u542b\u4e49\n\n### \u6f14\u793a\n\n\u4f7f\u7528sklearn\u7684\u51b3\u7b56\u6811\uff0c\u8fdb\u884c\u4e00\u4e2a\u5c0f\u6837\u672c\u7684\u5206\u6790\n\n\n```python\n\n```\n", "meta": {"hexsha": "e6d0a032292ecf034a3e7ef6e17af7bad05fa93e", "size": 19260, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "machine_learning/tree/decision_tree/presentation.ipynb", "max_stars_repo_name": "ningchi/book_notes", "max_stars_repo_head_hexsha": "c6f8001f7d5f873896c4b3a8b1409b21ef33c328", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2017-12-31T12:10:42.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-10T15:49:34.000Z", "max_issues_repo_path": "machine_learning/tree/decision_tree/presentation.ipynb", "max_issues_repo_name": "ningchi/book_notes", "max_issues_repo_head_hexsha": "c6f8001f7d5f873896c4b3a8b1409b21ef33c328", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2017-12-05T13:04:14.000Z", "max_issues_repo_issues_event_max_datetime": "2017-12-07T16:24:50.000Z", "max_forks_repo_path": "machine_learning/tree/decision_tree/presentation.ipynb", "max_forks_repo_name": "ningchi/book_notes", "max_forks_repo_head_hexsha": "c6f8001f7d5f873896c4b3a8b1409b21ef33c328", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2017-06-27T07:19:28.000Z", "max_forks_repo_forks_event_max_datetime": "2017-11-19T08:57:35.000Z", "avg_line_length": 22.9558998808, "max_line_length": 107, "alphanum_fraction": 0.4125129803, "converted": true, "num_tokens": 3119, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646255, "lm_q2_score": 0.7461389873857264, "lm_q1q2_score": 0.4450220237971705}} {"text": "```python\n# %load ../../../preconfig.py\n%matplotlib inline\n\nimport matplotlib.pyplot as plt\nimport seaborn as sns\nsns.set(color_codes=True)\n#sns.set(font='SimHei')\n#plt.rcParams['axes.grid'] = False\n\nimport numpy as np\n\nimport pandas as pd\npd.options.display.max_rows = 20\n\n#import sklearn\n\n#import itertools\n\nimport logging\nlogger = logging.getLogger()\n```\n\n\u51b3\u7b56\u6811\u7b80\u4ecb\u548c Python \u5b9e\u73b0\n======================\n\u53c2\u8003\uff1a\n\n+ [Building a decision tree from scratch - a beginner tutorial](http://www.patricklamle.com/Tutorials/Decision%20tree%20python/tuto_decision%20tree.html)\n\n+ 9.2 Tree-Based Methods - The Elements of Statistical Learning\n\n+ [Classification and Regression Trees (CART) Theory and Applications](http://edoc.hu-berlin.de/master/timofeev-roman-2004-12-20/PDF/timofeev.pdf)\n\n#### 0. \u57fa\u672c\u4ecb\u7ecd\n\u672c\u6587\u4e3b\u8981\u662f\u53c2\u7167 Tree-Based Methods - The Elements of Statistical Learning \u6765\u5b9e\u73b0\u4e00\u4e2a\u7b80\u5316\u7248\u8303\u4f8b\uff0c\u5176\u7b97\u6cd5\u662f CART\u3002\n\n\u51b3\u7b56\u6811\u7684\u601d\u60f3\u672c\u8eab\u975e\u5e38\u6734\u7d20\uff0c\u5173\u4e8e\u5b83\u7684\u57fa\u672c\u4ecb\u7ecd\u5728\u7f51\u4e0a\u5df2\u7ecf\u975e\u5e38\u4e30\u5bcc\uff0c\u6bd4\u5982\uff1a\n\n+ [\u7b97\u6cd5\u6742\u8d27\u94fa\u2014\u2014\u5206\u7c7b\u7b97\u6cd5\u4e4b\u51b3\u7b56\u6811(Decision tree)](http://www.cnblogs.com/leoo2sk/archive/2010/09/19/decision-tree.html)\n\n\u5176\u4e3b\u8981\u95ee\u9898\u662f\u5728\u6bcf\u6b21\u51b3\u7b56\u65f6\u627e\u5230\u4e00\u4e2a\u5206\u5272\u70b9\uff0c\u8ba9\u751f\u6210\u7684\u5b50\u96c6\u5c3d\u53ef\u80fd\u5730\u7eaf\u51c0\u3002\u8fd9\u91cc\u6d89\u53ca\u5230\u56db\u4e2a\u95ee\u9898:\n\n1. \u5982\u4f55\u5206\u5272\u6837\u672c\uff1f\n\n2. \u5982\u4f55\u8bc4\u4ef7\u5b50\u96c6\u7684\u7eaf\u51c0\u5ea6\uff1f\n\n3. \u5982\u4f55\u627e\u5230\u5355\u4e2a\u6700\u4f73\u7684\u5206\u5272\u70b9\uff0c\u5176\u5b50\u96c6\u6700\u4e3a\u7eaf\u51c0\uff1f\n\n4. \u5982\u4f55\u627e\u5230\u6700\u4f73\u7684\u5206\u5272\u70b9\u5e8f\u5217\uff0c\u5176\u6700\u7ec8\u5206\u5272\u5b50\u96c6\u603b\u4f53\u6700\u4e3a\u7eaf\u51c0\uff1f\n\n\u63a5\u4e0b\u6765\uff0c\u56f4\u7ed5\u4e0a\u8ff0\u95ee\u9898\uff0c\u4e00\u4e00\u6982\u8981\u8bf4\u660e\uff0c\u5e76\u52a0\u4ee5\u6f14\u793a\u3002\n\n#### \u52a0\u8f7d\u6570\u636e\n\n\u53e4\u8bdd\u8bf4\uff0c\u300c\u4e09\u519b\u672a\u52a8\uff0c\u7cae\u8349\u5148\u884c\u300d\u3002\n\n\u6211\u4eec\u5148\u52a0\u8f7d\u6f14\u793a\u6570\u636e\uff0c\u4f7f\u7528\u7684\u662f sklearn \u81ea\u5e26\u7684\u6d4b\u8bd5\u7528\u4f8b\u3002\n\n\n```python\nfrom sklearn.datasets import load_iris\ndata = load_iris()\n```\n\n\n```python\n# \u51c6\u5907\u7279\u5f81\u6570\u636e\nX = pd.DataFrame(data.data, \n columns=[\"sepal_length\", \"sepal_width\", \"petal_length\", \"petal_width\"])\nX.head(2)\n```\n\n\n\n\n
\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
sepal_lengthsepal_widthpetal_lengthpetal_width
05.13.51.40.2
14.93.01.40.2
\n
\n\n\n\n\n```python\n# \u51c6\u5907\u6807\u7b7e\u6570\u636e\ny = pd.DataFrame(data.target, columns=['target'])\ny.replace(to_replace=range(3), value=data.target_names, inplace=True)\ny.head(3)\n```\n\n\n\n\n
\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
target
0setosa
1setosa
2setosa
\n
\n\n\n\n\n```python\n# \u7ec4\u5efa\u6837\u672c [\u7279\u5f81\uff0c\u6807\u7b7e]\nsamples = pd.concat([X, y], axis=1) #, keys=[\"x\", \"y\"])\nsamples.head(3)\n```\n\n\n\n\n
\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
sepal_lengthsepal_widthpetal_lengthpetal_widthtarget
05.13.51.40.2setosa
14.93.01.40.2setosa
24.73.21.30.2setosa
\n
\n\n\n\n#### 1.0 \u5982\u4f55\u5206\u5272\u6837\u672c\n\u51b3\u7b56\u6811\u7684\u5206\u5272\u65b9\u6cd5\u662f\u53d6\u4e00\u4e2a\u7279\u5f81 $f$ \u548c\u9608\u503c $t$\uff0c\u4ee5\u6b64\u4e3a\u754c\u5c06\u6837\u672c $X$ \u62c6\u5206\u4e3a\u4e24\u4e2a\u5b50\u96c6 $X_l, X_r$\u3002\u5176\u6570\u5b66\u8868\u8fbe\u5f62\u540c\uff1a\n\n\\begin{align}\n X = \\begin{cases}\n X_l, \\ \\text{if } X[f] < t \\\\\n X_r, \\ \\text{if } X[f] \\geq t\n \\end{cases}\n\\end{align}\n\n\n```python\ndef splitter(samples, feature, threshold):\n # \u6309\u7279\u5f81 f \u548c\u9608\u503c t \u5206\u5272\u6837\u672c\n \n left_nodes = samples.query(\"{f} < {t}\".format(f=feature, t=threshold))\n right_nodes = samples.query(\"{f} >= {t}\".format(f=feature, t=threshold))\n \n return {\"left_nodes\": left_nodes, \"right_nodes\": right_nodes}\n```\n\n\n```python\nsplit = splitter(samples, \"sepal_length\", 5)\n\n# \u5de6\u5b50\u96c6\nx_l = split[\"left_nodes\"].loc[:, \"target\"].value_counts()\nx_l\n```\n\n\n\n\n setosa 20\n versicolor 1\n virginica 1\n Name: target, dtype: int64\n\n\n\n\n```python\n# \u53f3\u5b50\u96c6\nx_r = split[\"right_nodes\"].loc[:, \"target\"].value_counts()\nx_r\n```\n\n\n\n\n virginica 49\n versicolor 49\n setosa 30\n Name: target, dtype: int64\n\n\n\n#### 2. \u5982\u4f55\u8bc4\u4ef7\u5b50\u96c6\u7684\u7eaf\u51c0\u5ea6\uff1f\n\n\u4ece\u5e38\u7406\u6765\u8bf4\uff0c\u6211\u4eec\u5e0c\u671b\u5206\u5272\u5b50\u96c6\u5c3d\u53ef\u80fd\u5730\u7eaf\u51c0\uff0c\u6700\u597d\u662f\u5355\u4e2a\u5b50\u96c6\u5c31\u53ea\u542b\u6709\u4e00\u7c7b\u6807\u7b7e\uff0c\u4ece\u800c\u4fdd\u8bc1\u51b3\u7b56\u7ed3\u679c\u7cbe\u51c6\u3002\n\n\u90a3\u4e48\u4ec0\u4e48\u6837\u7684\u8bc4\u4ef7\u51fd\u6570\uff0c\u53ef\u4ee5\u7528\u6765\u5ea6\u91cf\u5404\u5b50\u96c6\u7684\u7eaf\u51c0\u5ea6\u5462\uff1f\n\n\u4ee5\u521a\u624d\u8ba1\u7b97\u7ed3\u679c\u4e3a\u4f8b\uff0c $x_l$ \u4e3b\u8981\u6807\u7b7e\u662f setosa\uff0c\u975e\u5e38\u7eaf\u51c0\uff0c\u800c $x_r$ \u5219\u4e09\u79cd\u6807\u7b7e\u52bf\u5747\u529b\u654c\uff0c\u975e\u5e38\u6df7\u6742\u3002\u6240\u4ee5\u601d\u8def\u662f\uff0c\u82e5\u4e00\u79cd\u6807\u7b7e\u5728\u5b50\u96c6\u4e2d\u5360\u6bd4\u975e\u5e38\u5927\uff0c\u5219\u6b64\u5b50\u96c6\u5c31\u8f83\u7eaf\u51c0\uff1b\u82e5\u5404\u6807\u7b7e\u5360\u6bd4\u5dee\u522b\u4e0d\u5927\uff0c\u5c31\u8f83\u4e3a\u6df7\u6742\u3002\n\n\u5e38\u7528\u7684\u8bc4\u4ef7\u51fd\u6570\u6b63\u662f\u8ba1\u7b97\u5404\u6807\u7b7e $c_k$ \u5728\u5b50\u96c6\u4e2d\u7684\u5360\u6bd4 $p_k = c_k / \\sum (c_k)$\uff0c\u5e76\u901a\u8fc7\u7ec4\u5408 $p_k$ \u6765\u63cf\u8ff0\u5360\u6bd4\u96c6\u4e2d\u6216\u5206\u6563\u3002\n\n\n```python\ndef calc_class_proportion(node):\n # \u8ba1\u7b97\u5404\u6807\u7b7e\u5728\u96c6\u5408\u4e2d\u7684\u5360\u6bd4\n \n y = node[\"target\"]\n return y.value_counts() / y.count()\n```\n\n\n```python\ncalc_class_proportion(split[\"left_nodes\"])\n```\n\n\n\n\n setosa 0.909091\n versicolor 0.045455\n virginica 0.045455\n Name: target, dtype: float64\n\n\n\n\n```python\ncalc_class_proportion(split[\"right_nodes\"])\n```\n\n\n\n\n virginica 0.382812\n versicolor 0.382812\n setosa 0.234375\n Name: target, dtype: float64\n\n\n\n\u4e3b\u8981\u7684\u8bc4\u4ef7\u51fd\u6570\u6709\u4e09\u79cd\uff0c\u5b83\u4eec\u8bc4\u4ef7\u7684\u662f\u96c6\u5408\u7684\u4e0d\u7eaf\u5ea6\uff08\u503c\u8d8a\u5927\uff0c\u96c6\u5408\u8d8a\u6df7\u6742\uff09\u3002\n\n\u5148\u505a\u4e9b\u6570\u5b66\u5b9a\u4e49\u4ee5\u4fbf\u4e8e\u63cf\u8ff0\uff1a \n\u5047\u8bbe\u5bf9\u4e8e\u96c6\u5408 $m$ \u6709 $N_m$ \u4e2a\u6837\u672c\uff0c\u53ef\u5206\u5272\u6210 $R_m$ \u5b50\u96c6\u3002 \n\u82e5\u603b\u7684\u6807\u7b7e\u7c7b\u522b\u6709 $K$ \u79cd\uff0c\u5219\u6807\u7b7e $k$ \u5728\u6b64\u96c6\u5408\u4e2d\u7684\u5360\u6bd4\u4e3a\uff1a\n\n\\begin{equation}\n \\hat{p}_{m k} = \\frac{1}{N_m} \\displaystyle \\sum_{x_i \\in R_m} I(y_i = k)\n\\end{equation}\n\n\u4e14\u4ee4\u6807\u7b7e $k$ \u662f\u5360\u6bd4\u6700\u5927\u7684\u6807\u7b7e\uff0c\u5373 $k(m) = \\operatorname{arg max}_k \\hat{p}_{m k}$.\n\n##### 1. Misclassification error\n\u6211\u4eec\u4e00\u822c\u628a\u96c6\u5408\u7684\u5206\u7c7b\u7ed3\u679c\u5b9a\u4e49\u4e3a\u5360\u6bd4\u6700\u5927\u7684\u6807\u7b7e\uff0c\u90a3\u4e48\u843d\u5728\u6b64\u96c6\u5408\u4e2d\u7684\u5176\u5b83\u6807\u7b7e\u5c31\u662f\u8bef\u5206\u7c7b\u3002\u5176\u6bd4\u7387\u662f $1 - \\hat{p}_{m k}(m)$.\n\n\n```python\ndef misclassification_error(node):\n p_mk = calc_class_proportion(node)\n \n return 1 - p_mk.max()\n```\n\n\n```python\nmisclassification_error(split[\"left_nodes\"])\n```\n\n\n\n\n 0.090909090909090939\n\n\n\n\n```python\nmisclassification_error(split[\"right_nodes\"])\n```\n\n\n\n\n 0.6171875\n\n\n\n\u5bf9\u4e8e\u4e8c\u5206\u7c7b\u95ee\u9898\uff0c\n\n\n```python\nbinary_class = pd.Series(np.arange(0, 1.01, 0.01)).to_frame(name=\"p\")\nbinary_class[\"1-p\"] = 1 - binary_class[\"p\"]\nbinary_class.head(3)\n```\n\n\n\n\n
\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
p1-p
00.001.00
10.010.99
20.020.98
\n
\n\n\n\n\u8bef\u5206\u7c7b\u7387\u548c\u5360\u6bd4 $p$ \u7684\u5173\u7cfb\u53ef\u5212\u56fe\u4e3a\uff1a\n\n\n```python\nbinary_class[\"misclass\"] = binary_class.apply(lambda x: 1 - x.max(), axis=1)\nbinary_class.plot(x=\"p\", y=\"misclass\")\n```\n\n\u5f53 $p=0.5$\uff0c\u4e24\u79cd\u6807\u7b7e\u5404\u5360\u4e00\u534a\uff0c\u4e0d\u7eaf\u5ea6\u6700\u9ad8\uff1b\u5f53 $p=0$ \u6216 $p=1$, \u53ea\u542b\u6709\u5176\u4e2d\u4e00\u79cd\u6807\u7b7e\u65f6\uff0c\u4e0d\u7eaf\u5ea6\u6700\u4f4e\u3002\n\n##### 2. Gini index\n\u8fd9\u91cc\u7684\u57fa\u5c3c\u7cfb\u6570\u5e76\u975e\u662f\u7ecf\u6d4e\u4e0a\u6d4b\u91cf\u5206\u914d\u516c\u5e73\u7684\u6307\u6807\u3002\n\n\u5b83\u7684\u601d\u8def\u662f\u4ece\u96c6\u5408\u4e2d\u968f\u673a\u62bd\u53d6\u5143\u7d20 $a \\in K_p$\uff0c\u518d\u4ee5 $K_p$ \u5728\u96c6\u5408\u4e2d\u7684\u5206\u5e03\u4e3a\u53c2\u8003\u968f\u673a\u7ed9 $a$ \u5206\u914d\u6807\u7b7e\uff0c\u90a3\u4e48\u8bef\u5206\u914d\u7387\u5c31\u662f\u57fa\u5c3c\u7cfb\u6570\u3002\n\n\u5177\u4f53\u5230\u51b3\u7b56\u6811\u7684\u8282\u70b9 $m$ \u4e0a\uff0c\u6807\u7b7e $k_i$ \u7684\u5360\u6bd4\u4e3a $p_{k_i m}$\u3002\u5219\u62bd\u4e2d\u5c5e\u4e8e\u6807\u7b7e $k_i$ \u7684\u5143\u7d20\u6982\u7387\u662f $p_{k_i m}$\uff0c\u8bef\u5206\u914d\u5230\u5176\u5b83\u6807\u7b7e\u7684\u6982\u7387\u662f $\\sum_{k' \\neq k_i} p_{k_i m} p_{k' m}$\u3002\u5bf9\u4e8e\u6574\u4e2a\u96c6\u5408\u7684\u6807\u7b7e\u5219\u662f\uff1a\n\n\\begin{equation}\n G(m) = \\displaystyle \\sum_{k \\neq k'} p_{k m} p_{k' m} \\, \\overset{\u4e58\u6cd5\u5206\u914d\u5f8b}{=} \\sum_{k = 1}^{K} p_{k m} (1 - p_{k m})\n\\end{equation}\n\n\n```python\ndef gini_index(node):\n p_mk = calc_class_proportion(node)\n \n return (p_mk * (1 - p_mk)).sum()\n```\n\n\n```python\ngini_index(split[\"left_nodes\"])\n```\n\n\n\n\n 0.1694214876033058\n\n\n\n\n```python\ngini_index(split[\"right_nodes\"])\n```\n\n\n\n\n 0.6519775390625\n\n\n\n\u5728\u4e8c\u5206\u7c7b\u4e2d\uff0c\u57fa\u5c3c\u7cfb\u6570\u548c\u5360\u6bd4 $p$ \u7684\u5173\u7cfb\u53ef\u5212\u56fe\u4e3a\uff1a\n\n\n```python\nbinary_class[\"gini\"] = (binary_class[\"p\"] * binary_class[\"1-p\"] * 2)\nbinary_class.plot(x=\"p\", y=\"gini\")\n```\n\n##### 3. Cross-entropy\nref: \n[Qualitively what is Cross Entropy](http://stats.stackexchange.com/questions/80967/qualitively-what-is-cross-entropy)\n\n\u8fd9\u4e2a\u635f\u5931\u51fd\u6570\u7684\u601d\u8def\u6765\u6e90\u4e8e\u4fe1\u606f\u8bba\uff1a\u82e5\u67d0\u4e8b\u4ef6\u7684\u53d1\u751f\u6982\u7387\u662f $p$\uff0c\u5219\u9700\u81f3\u5c11 $\\log_2 (1/p)$ \u4f4d\u7f16\u7801\u3002\u90a3\u4e48\u5bf9\u4e8e\u6240\u6709\u4e8b\u4ef6\uff0c\u5176\u6700\u4f18\u7f16\u7801\u7684\u5e73\u5747\u5b57\u957f\u4e3a $\\sum_i p_i \\log_2 (1 / p_i)$\u3002\n\n\u501f\u7528\u5176\u601d\u8def\uff0c\u5bf9\u4e8e\u8282\u70b9\u6765\u8bf4\uff0c\u5176\u5185\u5bb9\u8d8a\u6df7\u6742\uff0c\u5c31\u9700\u8981\u8d8a\u591a\u5b57\u957f\u6765\u533a\u5206\u3002\u6240\u4ee5\u8fd9\u91cc cross-entropy \u5b9a\u4e49\u4e3a\uff1a\n\n\\begin{equation}\n C(m) = \\displaystyle \\sum_{k=1}^K p_{m k} \\log (1 / p_{m k}) \\, = - \\sum_{k=1}^K p_{m k} \\log p_{m k}\n\\end{equation}\n\n\n```python\ndef cross_entropy(node):\n p_mk = calc_class_proportion(node)\n \n return - (p_mk * p_mk.apply(np.log)).sum()\n```\n\n\n```python\ncross_entropy(split[\"left_nodes\"])\n```\n\n\n\n\n 0.36764947740014225\n\n\n\n\n```python\ncross_entropy(split[\"right_nodes\"])\n```\n\n\n\n\n 1.075199711851601\n\n\n\n\u5728\u4e8c\u5206\u7c7b\u4e2d\uff0ccross-entropy \u548c\u5360\u6bd4 $p$ \u7684\u5173\u7cfb\u53ef\u5212\u56fe\u4e3a\uff1a\n\n\n```python\nx = binary_class[[\"p\", \"1-p\"]]\nbinary_class[\"cross_entropy\"] = -(x * np.log(x)).sum(axis=1)\nbinary_class.plot(x=\"p\", y=\"cross_entropy\")\n```\n\n\u5728\u4e8c\u5206\u7c7b\u95ee\u9898\u4e2d\uff0c\u4e09\u79cd\u8bc4\u4ef7\u51fd\u6570\u7684\u6bd4\u8f83\u5982\u56fe:\n\n\n```python\nbinary_class.plot(x=\"p\", y=[\"misclass\", \"gini\", \"cross_entropy\"])\n```\n\n\u4e3a\u4e86\u4fbf\u4e8e\u6bd4\u8f83\uff0c\u6211\u4eec\u5c06 cross_entropy \u4e5f\u653e\u7f29\u5230 $(0.5, 0.5)$\u3002\n\n\n```python\nbinary_class[\"cross_entropy_scaled\"] = binary_class[\"cross_entropy\"] / binary_class[\"cross_entropy\"].max() * 0.5\nbinary_class.plot(x=\"p\", y=[\"misclass\", \"gini\", \"cross_entropy_scaled\"], ylim=[0,0.55])\n```\n\n\u53ef\u4ee5\u770b\u5230\uff0c\u8bc6\u5206\u7c7b\u7387\u5728\u6574\u4e2a\u533a\u95f4\u662f\u5747\u4e00\u7684\uff0c\u800c cross_entropy \u8d8a\u9760\u8fd1\u7eaf\u51c0\uff0c\u5176\u503c\u53d8\u5316\u8d8a\u5267\u70c8\u3002\u6240\u4ee5 cross_entropy \u5bf9\u7eaf\u51c0\u66f4\u654f\u611f\u7684\u7279\u6027\uff0c\u6709\u5229\u4e8e\u8ba9\u7ed3\u679c\u5b50\u96c6\u66f4\u7eaf\u51c0\uff0c\u5176\u4f7f\u7528\u76f8\u5bf9\u8f83\u591a\u3002\n\n#### 3. \u5982\u4f55\u627e\u5230\u5355\u4e2a\u6700\u4f73\u7684\u5206\u5272\u70b9\uff0c\u5176\u5b50\u96c6\u6700\u4e3a\u7eaf\u51c0\uff1f\n\u5355\u4e2a\u6700\u4f73\u5206\u5272\u70b9\uff0c\u6d89\u53ca\u4e09\u4e2a\u95ee\u9898\uff1a\n\n1. \u5bf9\u4e8e\u5355\u6b21\u5206\u5272\uff0c\u5206\u5272\u524d\u548c\u5206\u5272\u540e\uff0c\u96c6\u5408\u7684\u7eaf\u51c0\u5ea6\u63d0\u5347\u4e86\u591a\u5c11\uff1f\n\n2. \u7ed9\u5b9a\u4e00\u4e2a\u7279\u5f81\uff0c\u7eaf\u51c0\u5ea6\u63d0\u5347\u6700\u5927\u7684\u9608\u503c\u662f\u591a\u5c11\uff1f\n\n3. \u5bf9\u4e8e\u591a\u4e2a\u7279\u5f81\uff0c\u54ea\u4e00\u4e2a\u7279\u5f81\u7684\u6700\u4f73\u9608\u503c\u5bf9\u7eaf\u51c0\u5ea6\u63d0\u5347\u6700\u5927\uff1f\n\n\n##### 3.1 \u5bf9\u4e8e\u5355\u6b21\u5206\u5272\uff0c\u5206\u5272\u524d\u548c\u5206\u5272\u540e\uff0c\u96c6\u5408\u7684\u7eaf\u51c0\u5ea6\u63d0\u5347\u4e86\u591a\u5c11\uff1f\n\u4ee4\u6d4b\u91cf\u4e0d\u7eaf\u5ea6\u7684\u51fd\u6570\u4e3a $G$, \n\u5bf9\u4e00\u4e2a\u8282\u70b9 $m$ \u6765\u8bf4\uff0c\u82e5\u5176\u6309\u5206\u5272\u65b9\u6cd5 $f$ \u5f97\u5230\u5b50\u96c6 $m_l$ \u548c $m_r$\uff0c\u5219\u603b\u7684\u4e0d\u7eaf\u5ea6\u51cf\u5c11\u91cf\u4e3a\uff1a\n\n\\begin{equation}\n G(m) - G(m_l) - G(m_r)\n\\end{equation}\n\n\n```python\ndef calc_impurity_measure(node, feathure, threshold, measure, min_nodes=5):\n child = splitter(node, feathure, threshold)\n left = child[\"left_nodes\"]\n right = child[\"right_nodes\"]\n \n if left.shape[0] <= min_nodes or right.shape[0] <= min_nodes:\n return 0\n \n \n impurity = pd.DataFrame([], \n columns=[\"score\", \"rate\"],\n index=[])\n \n impurity.loc[\"all\"] = [measure(node), node.shape[0]]\n impurity.loc[\"left\"] = [-measure(left), left.shape[0]]\n impurity.loc[\"right\"] = [-measure(right), right.shape[0]]\n \n impurity[\"rate\"] /= impurity.at[\"all\", \"rate\"]\n \n logger.info(impurity)\n \n return (impurity[\"score\"] * impurity[\"rate\"]).sum()\n```\n\n\n```python\ncalc_impurity_measure(samples, \"sepal_length\", 5, gini_index)\n```\n\n\n\n\n 0.08546401515151514\n\n\n\n\n```python\ncalc_impurity_measure(samples, \"sepal_length\", 1, gini_index)\n```\n\n\n\n\n 0\n\n\n\n##### 3.2. \u7ed9\u5b9a\u4e00\u4e2a\u7279\u5f81\uff0c\u7eaf\u51c0\u5ea6\u63d0\u5347\u6700\u5927\u7684\u9608\u503c\u662f\u591a\u5c11\uff1f\n\n\u5bf9\u4e8e\u4e00\u4e2a\u7ed9\u5b9a\u7684\u7279\u5f81\uff0c\u7406\u8bba\u4e0a\u901a\u8fc7\u679a\u53d6\u6240\u6709\u53ef\u80fd\u7684\u9608\u503c\uff0c\u4ece\u4e2d\u627e\u5230\u6700\u5927\u51cf\u5c11\u91cf\u7684\u9608\u503c\u70b9\uff0c\u5c31\u662f\u6b64\u7279\u5f81\u7684\u6700\u4f73\u5206\u9694\u70b9\u3002\n\n\u4f46\u73b0\u5b9e\u4e2d\uff0c\u5f88\u591a\u7279\u5f81\u662f\u8fde\u7eed\u7684\uff0c\u6216\u8005\u9608\u503c\u70b9\u592a\u591a\uff0c\u5168\u90e8\u7a77\u5c3d\u5e76\u4e0d\u73b0\u5b9e\uff0c\u5f80\u5f80\u9700\u8981\u7528\u5230\u6700\u4f18\u5316\u7684\u5bfb\u4f18\u65b9\u6cd5\u3002\u8fd9\u91cc\u4e3a\u4e86\u7b80\u6613\u8d77\u89c1\uff0c\u6211\u4eec\u5bf9\u7279\u5f81\u7684\u503c\u7531\u5c0f\u5230\u5927\u8bbe\u4e8610\u4e2a\u5206\u4f4d\u70b9\uff0c\u8fdb\u884c\u8ba1\u7b97\u3002\n\n\n```python\ndef find_best_threshold(node, feature, measure):\n threshold_candidates = node[feature].quantile(np.arange(0, 1, 0.2))\n \n res = pd.Series([], name=feature)\n for t in threshold_candidates:\n res[t] = calc_impurity_measure(node, feature, t, measure)\n \n logger.info(res)\n \n if res.max() == 0:\n return None\n else:\n return res.argmax()\n```\n\n\n```python\nfind_best_threshold(samples, \"sepal_width\", gini_index)\n```\n\n\n\n\n 3.3999999999999999\n\n\n\n\n```python\nfind_best_threshold(samples, \"sepal_length\", gini_index)\n```\n\n\n\n\n 5.5999999999999996\n\n\n\n##### 3.3. \u5bf9\u4e8e\u591a\u4e2a\u7279\u5f81\uff0c\u54ea\u4e00\u4e2a\u7279\u5f81\u7684\u6700\u4f73\u9608\u503c\u5bf9\u7eaf\u51c0\u5ea6\u63d0\u5347\u6700\u5927\uff1f\n\u663e\u7136\uff0c\u6700\u66b4\u529b\u7684\u65b9\u6cd5\u662f\uff1a\u6bcf\u6b21\u5206\u5272\uff0c\u6211\u4eec\u7a77\u5c3d\u6240\u6709\u7279\u5f81\uff0c\u5373\u53ef\u627e\u5230\u5bf9\u6b64\u8282\u70b9\u6700\u4f73\u5206\u5272\u70b9\n\n\n```python\ndef find_best_split(node, measure):\n if node[\"target\"].unique().shape[0] <= 1:\n return None\n \n purity_gain = pd.Series([], name=\"feature\")\n \n for f in node.drop(\"target\", axis=1).columns:\n purity_gain[f] = find_best_threshold(node, f, measure)\n \n if pd.isnull(purity_gain.max()):\n return None\n else:\n best_split = {\"feature\": purity_gain.argmax(), \"threshold\": purity_gain.max()}\n best_split[\"child\"] = splitter(node, **best_split)\n\n return best_split\n```\n\n\n```python\nbest_split = find_best_split(samples, gini_index)\n[best_split[x] for x in [\"feature\", \"threshold\"]]\n```\n\n\n\n\n ['sepal_length', 5.5999999999999996]\n\n\n\n#### 4. \u5982\u4f55\u627e\u5230\u6700\u4f73\u7684\u5206\u5272\u70b9\u5e8f\u5217\uff0c\u5176\u6700\u7ec8\u5206\u5272\u5b50\u96c6\u603b\u4f53\u6700\u4e3a\u7eaf\u51c0\uff1f\n\u641c\u7d22\u5168\u5c40\u6700\u4f18\u89e3\u5728\u76ee\u524d\u8fd8\u6ca1\u6709\u6709\u6548\u7684\u65b9\u6cd5\uff0c\u6240\u4ee5\u9000\u4e00\u6b65\uff0c\u6211\u4eec\u7528\u8d2a\u5a6a\u7684\u601d\u60f3\uff0c\u5728\u6bcf\u6b21\u5206\u5272\u65f6\u53d6\u6700\u4f18\uff0c\u5e0c\u671b\u7531\u5c40\u90e8\u6700\u4f18\u7684\u5206\u5272\u5e8f\u5217\u80fd\u591f\u8fbe\u5230\u5168\u5c40\u6700\u4f18\u7684\u6548\u679c\u3002\n\n\u6211\u4eec\u4f7f\u7528\u9012\u5f52\u7684\u65b9\u6cd5\u7531\u4e0a\u800c\u4e0b\u4f9d\u6b21\u8ba1\u7b97\uff0c\u5728\u5904\u7406\u8282\u70b9\u987a\u5e8f\u65f6\u4f7f\u7528\u6df1\u5ea6\u4f18\u5148\u65b9\u6cd5\u7ec4\u5efa\u51fa\u51b3\u7b56\u6811\u3002\n\n\n```python\nclass BinaryNode:\n def __init__(self, samples, max_depth, measure=gini_index):\n self.samples = samples\n self.max_depth = max_depth\n self.measure = measure\n \n self.is_leaf = False\n self.class_ = None\n \n self.left = None\n self.right = None\n \n self.best_split = None\n \n def split(self, depth):\n if depth > self.max_depth:\n self.is_leaf = True\n self.class_ = self.samples[\"target\"].value_counts().argmax()\n return\n \n best_split = find_best_split(self.samples, self.measure)\n if pd.isnull(best_split):\n self.is_leaf = True\n self.class_ = self.samples[\"target\"].value_counts().argmax()\n return\n\n self.best_split = best_split\n left = self.best_split[\"child\"][\"left_nodes\"]\n self.left = BinaryNode(left.drop(best_split[\"feature\"], axis=1), self.max_depth)\n \n right = self.best_split[\"child\"][\"right_nodes\"]\n self.right = BinaryNode(right.drop(best_split[\"feature\"], axis=1), self.max_depth)\n\n # \u5148\u5e8f\u6df1\u5ea6\u4f18\u5148\n self.left.split(depth+1)\n self.right.split(depth+1)\n```\n\n\n```python\nbinaryNode = BinaryNode(samples, 3)\nbinaryNode.split(0)\n```\n\n\n```python\ndef show(node, depth):\n if node.left:\n show(node.left, depth+1)\n \n if node.is_leaf:\n print(\"{}{}\".format(\"\\t\"*(depth+2), node.class_))\n return \n else:\n print(\"{}{}: {}\".format(\"\\t\"*depth,\n node.best_split[\"feature\"],\n node.best_split[\"threshold\"]))\n if node.right:\n show(node.right, depth+1)\n```\n\n\n```python\nshow(binaryNode, 0)\n```\n\n \t\t\t\tversicolor\n \tsepal_width: 2.8200000000000003\n \t\t\t\t\tsetosa\n \t\tpetal_length: 1.6\n \t\t\t\t\t\tsetosa\n \t\t\tpetal_width: 0.4\n \t\t\t\t\t\tsetosa\n sepal_length: 5.6\n \t\t\t\t\tversicolor\n \t\tsepal_width: 3.1\n \t\t\t\t\tversicolor\n \tpetal_length: 4.8\n \t\t\t\t\t\tversicolor\n \t\t\tpetal_width: 1.8\n \t\t\t\t\t\tvirginica\n \t\tsepal_width: 2.9\n \t\t\t\t\t\tvirginica\n \t\t\tpetal_width: 2.0\n \t\t\t\t\t\tvirginica\n\n\n\u89c2\u5bdf\u53ef\u77e5\uff0c\u8fd9\u9897\u6811\u662f\u6709\u95ee\u9898\u7684\uff0c\u5982 petal_width: 0.4 \u4e0b\u4e24\u4e2a\u53f6\u5b50\u7684\u7c7b\u522b\u5747\u662f setosa\u3002\u5728\u6811\u751f\u6210\u540e\uff0c\u53ef\u4ee5\u901a\u8fc7\u540e\u7eed\u7684\u526a\u679d\u64cd\u4f5c\uff0c\u8ba9\u6574\u9897\u6811\u66f4\u7cbe\u7b80\u3002\u8fd9\u91cc\u4e0d\u518d\u8be6\u8ff0\u3002\n", "meta": {"hexsha": "79603f8fd54f2979bed726eaf1f6e0000f60e31c", "size": 184436, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "machine_learning/tree/decision_tree/demo.ipynb", "max_stars_repo_name": "ningchi/book_notes", "max_stars_repo_head_hexsha": "c6f8001f7d5f873896c4b3a8b1409b21ef33c328", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2017-12-31T12:10:42.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-10T15:49:34.000Z", "max_issues_repo_path": "machine_learning/tree/decision_tree/demo.ipynb", "max_issues_repo_name": "ningchi/book_notes", "max_issues_repo_head_hexsha": "c6f8001f7d5f873896c4b3a8b1409b21ef33c328", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": 1, 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YES\n2. YES", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.6334102498375401, "lm_q1q2_score": 0.44498973917307844}} {"text": "# Priors: Theano Graph\n\n\n```python\nwith gp.G3Model() as model:\n g3 = gp.G3GP(gp.Constant(), gp.SE(x,gp.ARD_L2), gp.Identity(y))\n g3.observed(x_obs,y_obs)\n mu, var, median, I_up, I_down, m1, m2 = g3.compilate_functions(model.vars,x)\n \nmodel.use_prior()\ngp.show_graph(model.logpt)\n```\n\n\n```python\ngp.show_graph(model.fastlogp.f)\n```\n\n\n```python\nmodel.use_prior(False)\ngp.show_graph(model.logpt)\n```\n\n\n```python\ngp.show_graph(model.fastlogp.f)\n```\n\n# Mapping\n\n\n```python\nwith g3.Model() as model:\n mean = g3.Bias()\n ker = g3.SE(x)\n trans = g3.Identity(y_obs) @ g3.LogShifted(y_obs)\n tgp = g3.TGP(x, mean, ker, trans, noise=True, hidden=y)\n tgp.describe(str(k),'SUNACTIVITY','YEAR')\n tgp.observed(x_obs,y_obs)\n tgp.testing(x_test,y_test)\n tgp.compile()\n#gp.plot_gp(gp.find_default(), samples=10)\n```\n\n# Transformation\n\n\n```python\nimport numpy as np\nfrom scipy.stats import gamma, norm\nimport matplotlib.pyplot as plt\n\nfig, ax = plt.subplots(1, 1)\na = 2.0\nb = 2.0\nN = 10000\nmean, var, skew, kurt = gamma.stats(a, moments='mvsk')\nx = np.linspace(gamma.ppf(0.01, a,scale=b), gamma.ppf(0.99, a,scale=b), N)\nrv = gamma(a,scale=b)\nax.plot(x, rv.pdf(x), 'k-', lw=2, label='pdf')\nvals = gamma.ppf([0.001, 0.5, 0.999], a)\nnp.allclose([0.001, 0.5, 0.999], gamma.cdf(vals, a))\nr = gamma.rvs(a, scale = b,size=N)\nax.hist(r,50, normed=True, histtype='stepfilled', alpha=0.2)\nax.legend(loc='best', frameon=False)\n\n\n## transformar a gaussiana\n\ng = r**(1.0/3.0)\ns = np.std(g)\nm = np.mean(g)\nx = np.linspace(min(g), max(g), N)\nfig, ax = plt.subplots(1, 1)\nax.plot(x, norm.pdf(x,m,s), 'k-', lw=2, label='pdf')\nax.hist(g,50, normed=True, histtype='stepfilled', alpha=0.2)\nax.legend(loc='best', frameon=False)\n\nplt.show()\n\n```\n\nhttp://localhost:8888/notebooks/GaussianProcesses/BoxCoxGP/sunspots-GP-vs-TGP.ipynb\n\n\n```python\nhidden_var = np.linspace(-10,5,501)\nparams = {'ARD_L2_log_': np.array(2.5, dtype=np.float32),\n 'COV_SE_log_': np.array([ 2, -3 ], dtype=np.float32),\n 'Mean_Constant': np.array(3.0, dtype=np.float32),\n 'T_Shift': np.array(0.0, dtype=np.float32),\n 'T_Power_log_': np.array(-0.4, dtype=np.float32),\n }\ntrans, dtrans, dist_gp, dist_tgp = g3.compile_distributions(model.point_to_In(params))\nparams\n```\n\n\n```python\ngrilla = np.linspace(-5,15,501)\ngp.style_normal()\nplt.figure(1,(10,10))\n\ndesfase = 5\n\n\nyyy = np.zeros_like(grilla)\ni = 0\nfor xxx in grilla:\n yyy[i] = dist_gp(np.array([xxx]))\n i+=1\nplt.plot(grilla,yyy*50-desfase,label='GaussianDist')\n\n\nplt.plot(trans(grilla),grilla,label='BoxCoxTrans')\n\n\nyyy = np.zeros_like(grilla)\ni = 0\nfor xxx in trans(grilla):\n yyy[i] = dist_gp(np.array([xxx]))\n i+=1\nplt.plot(yyy*30-desfase,grilla,label='BoxCox(Gaussian)')\n\n\ni = 0\nfor xxx in grilla:\n yyy[i] = dtrans(np.array([xxx]))\n i+=1\nplt.plot((yyy-np.min(yyy))-desfase,grilla,label='Jacobian(BoxCox)')\n\n\ni = 0\nfor xxx in grilla:\n yyy[i] = dist_tgp(np.array([xxx]))\n i+=1\nplt.plot(yyy*40-desfase,grilla,label='BoxCox-GaussianDist')\n\n\n\n#location_gp = grilla[np.argmax(yyy*100)]\n#min_gp = np.min(yyy*100)\n#max_gp = trans(np.array([location_gp]))\n#plt.plot(np.array([location_gp,location_gp]),np.array([min_gp,max_gp]))\n\n\n\ngp.plot('Box-Cox Change of Variable','GP','Box-Cox GP',loc=1,ncol=1)\nplt.axis([-5,12,-5,12])\ngp.save('change_of_variable.pdf')\n```\n\n\n```python\nimport theano.sandbox.linalg as sT\ntrans = tt.log(sT.det(tt.jacobian(g3.mapping.inv(value),value)))\nftrans = th.function([value]+model.point_to_In(init),trans,allow_input_downcast=True)\nyyy = np.zeros_like(grilla)\ni = 0\nfor xxx in grilla:\n yyy[i] = ftrans(np.array([xxx]))\n i+=1\nplt.plot(grilla,yyy)\n```\n\n\n```python\nvalue,dist = gp.marginal_tgp(np.array([0.0]))\nfdist = th.function([value]+gp_model.point_to_In(start_bfgs),dist,allow_input_downcast=True)\ngrilla = np.linspace(-200,200,301)\nyyy = np.zeros_like(grilla)\ni = 0\nfor xxx in grilla:\n yyy[i] = fdist(np.array([xxx]))\n i+=1\nplt.plot(grilla,yyy)\n```\n\n# Plot Cov\n\n\n```python\nker=LaplaceKernel(noise,h,np.array([l]))\nplt.matshow(ker.cov(x_obs[:,np.newaxis]))\nplt.matshow(sp.linalg.inv(ker.cov(x_obs[:,np.newaxis])))\nplt.matshow(CovCholesky(ker.cov(x_obs[:,np.newaxis])))\nplt.matshow(CovCholesky(ker.cov(x_obs[:,np.newaxis])).T)\n```\n\n\n```python\nplt.imshow(Jxx,cmap=cm.seismic,vmax=v,vmin=-v)\n```\n\n\n```python\n#Graficar Covarianza\nplt.figure(1, figsize=(20,6))\nKxx=InfoMatrix(GaussKer,np.expand_dims(np.arange(-2,2,0.1),1))\n\nplt.subplot(121)\nv=np.max(np.abs(Kxx))\nplt.imshow(Kxx,cmap=cm.seismic,vmax=v,vmin=-v)\nplt.colorbar()\n\n#Graficar Precision\nplt.subplot(122)\nJxx=np.linalg.solve(Kxx,np.eye(len(Kxx)))\nv=np.max(np.abs(Jxx))\nplt.imshow(Jxx,cmap=cm.seismic,vmax=v,vmin=-v)\nplt.colorbar()\n\n\nplt.figure(2, figsize=(20,6))\nplt.plot(Kxx[len(Kxx)/2,:],'b')\nplt.plot(Jxx[len(Jxx)/2,:],'r')\n\n#plt.axis([550,650,-0.1,14])\n#show(np.linalg.norm(Jxx.dot(Kxx)- np.eye(len(Kxx))))\n```\n\n\n```python\n_ = plt.plot(gp.compiles['covariance'](**gp.get_params())[0])\n_ = plt.plot(gp.compiles['covariance'](**gp.get_params())[len(x)//2])\n_ = plt.plot(gp.compiles['covariance'](**gp.get_params())[-1])\n```\n\n# GP 2D\nhttp://localhost:8888/notebooks/GaussianProcesses/BoxCoxGP/Presentation-TGP.ipynb\n\n\n```python\n# Parametros grilla\nx = np.linspace(0, 50, 51)\ny = np.linspace(0,200, 201)\n\n# Parametros GP\nl = 30\nsigma = 10*np.log(1.5)\ngamma_x = 0.0*np.log(2.0/l**2)\ngamma_y = 0.8*np.log(2.0/l**2)\n\n\nx2d, y2d = np.meshgrid(x, y)\nxy = np.zeros((len(x)*len(y),2))\nfor i in range(len(x)):\n for j in range(len(y)):\n xy[i*len(y)+j,:] = x[i],y[j]\nx2d = x2d.T\ny2d = y2d.T\n\n\nKerxy = GaussianKernel(-np.inf,sigma,np.array([gamma_x,gamma_y]))\nfxy2d_hidden = Kerxy.sample(xy).reshape((len(x),len(y)))\n\nfig = plt.figure(1,[20,10])\nax = fig.gca(projection='3d')\nax.plot_surface(x2d, y2d, fxy2d_hidden, alpha=0.4,cmap=cm.RdBu_r)\ncset = ax.contourf(x2d, y2d, fxy2d_hidden, zdir='z', offset=np.min(fxy2d_hidden), cmap=cm.RdBu_r)\n```\n\n# Sympy\n\n\n```python\n%matplotlib inline\nimport sympy as sy\nimport seaborn as sb\nimport matplotlib.pyplot as plt\nfrom IPython.display import display\n\nfrom sympy import init_printing\nfrom sympy import symbols,diff,log,sign,Abs\nfrom sympy.plotting import plot\n\ninit_printing() \nplt.rcParams['figure.figsize'] = (20, 10)\n#init_session(quiet=True)\n\nx,y = symbols('x,y',real=True)\n\n\n```\n\n\n```python\nphi = sign(x)*log(Abs(x))\ndphi = diff(phi,x)\nphi_1 = sy.solve(phi-y,x,domain=sy.S.Reals)\n```\n\n\n```python\ndisplay(phi)\ndisplay(dphi)\ndisplay(phi_1)\n```\n\n\n```python\nplot(phi,(x,-2,2))\n```\n\n\n```python\nplot(dphi,(x,-2,2))\n```\n\n# Widgets\n\n\n```python\ntgp.widget_params()\n```\n\n# Traces\n\n\n```python\ntraces = tgp.sample_hypers(start=tgp.get_params(), samples=5000,advi=False)\ng3.save_trace(traces)\ng3.style_seaborn()\ntraces.varnames.clear()\nfor v in gp.model.vars:\n traces.varnames.append(v.name)\ntraces.varnames\ng3.traceplot(traces)\ndatatraces = g3.datatrace(model, traces)\ndatatraces.describe().T\n\nitems_ll = ['niter','ll']\nitems_mt = ['TGP_Bias_Constant','TGP_BoxCoxShifted_shift','TGP_SE_ARD_L2_Scales']\nitems_k = ['TGP_BoxCoxShifted_power', 'TGP_Noise_Var','TGP_SE_Var']\n\ng3.plot_datatrace(datatraces,items_mt+items_k)\ng3.plot_datatrace(datatraces,items_ll+items_mt)\ng3.plot_datatrace(datatraces,items_ll+items_k)\n\ng3.style_seaborn()\ntgp.widget_trace(traces)\ntgp.plot_tgp(tgp.get_params(), samples=15)\n```\n\n\n```python\nitems_ll = ['niter','ll']\nitems_k1 = ['GP_SM1_M','GP_SM1_S_log_','GP_SM1_Var_log_']\nitems_k2 = ['GP_SM3_M','GP_SM2_S_log_','GP_SM2_Var_log_']\nitems_m = ['GP_Bias_Constant','GP_Noise_Var_log_']\n\ng3.plot_datatrace(datatraces,items_ll+items_k1)\ng3.plot_datatrace(datatraces,items_ll+items_k2)\ng3.plot_datatrace(datatraces,items_ll+items_m)\n\n\ng3.plot_datatrace(datatraces,items_k1+items_k2)\ng3.plot_datatrace(datatraces,items_k1+items_m)\ng3.plot_datatrace(datatraces,items_k2+items_m)\n\n\n```\n\n\n```python\ntgp.widget_trace(traces)\ntgp.plot_tgp(tgp.get_params(), samples=10)\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "c16581a835776156f913d98f9b2c875158d4e520", "size": 14650, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "sandbox/Misc.ipynb", "max_stars_repo_name": "griosd/g3py", "max_stars_repo_head_hexsha": "10402f045d10f1df6d3adf5320e9fb9103b5a6b5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2016-12-20T19:04:56.000Z", "max_stars_repo_stars_event_max_datetime": "2020-07-07T23:09:04.000Z", "max_issues_repo_path": "sandbox/Misc.ipynb", "max_issues_repo_name": "griosd/g3py", "max_issues_repo_head_hexsha": "10402f045d10f1df6d3adf5320e9fb9103b5a6b5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 49, "max_issues_repo_issues_event_min_datetime": "2016-12-20T05:44:12.000Z", "max_issues_repo_issues_event_max_datetime": "2017-09-16T04:13:38.000Z", "max_forks_repo_path": "sandbox/Misc.ipynb", "max_forks_repo_name": "griosd/g3py", "max_forks_repo_head_hexsha": "10402f045d10f1df6d3adf5320e9fb9103b5a6b5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 5, "max_forks_repo_forks_event_min_datetime": "2017-02-15T17:06:12.000Z", "max_forks_repo_forks_event_max_datetime": "2020-05-23T03:06:40.000Z", "avg_line_length": 24.3760399334, "max_line_length": 103, "alphanum_fraction": 0.529556314, "converted": true, "num_tokens": 2721, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757645879592642, "lm_q2_score": 0.6584175139669997, "lm_q1q2_score": 0.4449352400310726}} {"text": "# Understanding moving time vs ellapsed time, distance and speed/pace calculations \n> How the tracking apps analyse the GPS activity data\n\n- toc: false \n- badges: true\n- comments: true\n- author: Marcel Caraciolo\n- categories: [calculation, gps, metrics, jupyter, features, tracking]\n- image: images/post-lat-lon.png\n\nSport tracking applications and the social networks are quite popular among the runners nowadays. Everyone wants to perform the best run looking for a better pace, distance or time. But have you wondered how these metrics are calculated ?\n\nIn this post we\u2019ll use gpx-file downloaded from Strava social application, extract, and analyse the gpx data of one single route. We\u2019ll start with extracting the data from the gpx-file into a convenient ``runpandas.Activity`` dataframe. From there we\u2019ll explore the data and try to replicate the stats and graphs that the interface of one most popular running application provides us with.\n\n\n## Getting the data\n\nMost of the popular tracking apps allow you to download your activity as a gpx or tcx file. In this post, we will analyse a 12km run from [Strava](http://strava.com/). The gpx-file, short for GPS Exchange Format, can usually be obtained by clicking on export. The screenshot below shows you where you can download your gpx-file in Strava. You can download the file used in this article [here](https://github.com/corriporai/pandasrunner/blob/master/_notebooks/data/post-metrics.gpx).\n\n \n\nNow, we want to load our gpx data into a ``runpandas.Activity``. Take a look at your start point and end point and make sure everything makes sense (i.e. start time < end time). If not, there might be something wrong with your data, or you might have forgotten about some tracks or segments.\n\n\n\n```python\nimport runpandas\nactivity = runpandas.read_file('./data/post-metrics.gpx')\nprint('Start', activity.index[0],'End:', activity.index[-1])\nprint(activity.iloc[0]['lat'], activity.iloc[-1]['lat'])\n```\n\n Start 0 days 00:00:00 End: 0 days 01:25:27\n -8.045075 -8.036119\n\n\nThe head of the activity (frame) should look like this:\n\n\n\n```python\nactivity.head(8)\n```\n\n\n\n\n
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\n\n\n\nIt is important to notice that the time interval between two data points isn't constant. This happens because the device which records the data can't always provide the GPS-data due to connectivity problems or even to hardware limitation. In case of such a failure/limitation the data point is skipped (without an error) and the app will collect the data at the next time interval. So keep in mind that for further analysis we can't assume that the interval between all points is the same.\n\n## Plotting the data\n\nNow with the data loaded, we can explore it with some basic plots. Strava provides some interesting graphs to the runner such as the 2D map (longitude vs latitude) and elevation gain during the activity (altitude vs time). Using native plotting tools we can come to quite similar results.\n\n\n```python\n#hide\nfrom IPython.display import set_matplotlib_formats\nset_matplotlib_formats('png')\n```\n\n\n```python\nimport matplotlib.pyplot as plt\nplt.plot(activity['lon'], activity['lat'])\n```\n\n \n\n\n```python\naxis = activity.plot.area(y='alt')\naxis.set_ylim(0,100)\n```\n\n \n\n## Calculating the total distance\n\nNow let's going through the key metrics of the activity such as distance and speed. A quick look can misguide you into incorrect calculations. The first one is the distance between two LL-points (latitude, longitude) which it isn't a straight line, but spherical.\n\n \n\nThere are two main approaches to calculate the distance between two points on a spherical surface: the Haversine distance and the Vincenty distance. The two formulas take a different approach on calculating the distance, but this is outside the scoop of this post. The maths behind both approaches can be found [here](https://www.neovasolutions.com/2019/10/04/haversine-vs-vincenty-which-is-the-best/).\n\nAnother issue about the distance calculation is if the data have been calculated solely based on latitude and longitude, without taking into consideration the elevation gain or loss, then the distance covered can be underestimated. The easiest way to do this, is to compute the spherical 2d distance and then use the Euclidean formula to add the altitude measurements. The formula below shows this last step.\n\nIf the uncorrected distance covered at time point $t_{i}$ is $d_{2,i}$, to correct the distance covered between time point $t_{i-1}$ and time point $t_{i}$ to\n\\begin{equation}\nd_{i} - d_{i-1} = \\sqrt{ \\left(d_{2,i} - d_{2,i-1} \\right)^{\\!\\!2} + \\left(a_{i} - a_{i-1} \\right)^{\\!\\!2}}\n\\end{equation}\n,\nwhere $d_{i}$ and $a_{i}$ are the corrected cumulative distance and the altitude at time $t_{i}$, respectively.\n\nNow with the theoretical background presented, we can start implementing these formulas in our code. We will apply in all data points all possible implementations of our distance formula (Haversine or Vicenty and 2d or 3d) and compute the total distance for every data point by using the dataframe function `cumsum`.\n\nThere are geodesic python libraries already available with Vicenty and Haversine formulas, so we will import them here for our distance calculations.\n\n\n\n```python\n#pip install haversine & pip install vincenty\nfrom vincenty import vincenty as vn \nfrom haversine import haversine as hv \nfrom haversine import Unit\nimport numpy as np\n```\n\n\n```python\nactivity['shift_lon'] = activity.shift(1)['lon']\nactivity['shift_lat'] = activity.shift(1)['lat']\n#compute the vincenty 2d distance\nactivity['distance_vin_2d'] = activity[['lat', 'lon', 'shift_lat', 'shift_lon']].apply(lambda x: vn( (x[0], x[1]),(x[2], x[3])), axis=1)\n#transform the value in meters\nactivity['distance_vin_2d'] = activity['distance_vin_2d'] * 1000\n#compute the haversine 2d distance\nactivity['distance_hav_2d'] = activity[['lat', 'lon', 'shift_lat', 'shift_lon']].apply(lambda x: hv((x[0], x[1]),(x[2], x[3]), unit=Unit.METERS), axis=1)\n#compute the total distance\nactivity['distance_vin_no_alt'] = activity['distance_vin_2d'].cumsum()\nactivity['distance_hav_no_alt'] = activity['distance_hav_2d'].cumsum()\n#compute the vincenty and haversine 3d\nactivity['shift_alt'] = activity.shift(1)['alt']\nactivity['alt_dif'] = activity[['alt', 'shift_alt']].apply(lambda x: x[1] - x[0], axis=1)\nactivity['distance_vin_3d'] = activity[['distance_vin_2d', 'alt_dif']].apply(lambda x: np.sqrt(x[0]**2 + (x[1])**2), axis=1)\nactivity['distance_hav_3d'] = activity[['distance_hav_2d', 'alt_dif']].apply(lambda x: np.sqrt(x[0]**2 + (x[1])**2), axis=1)\n\n#compute the total distances for vincenty and haversined 3D\nactivity['distance_vin'] = activity['distance_vin_3d'].cumsum()\nactivity['distance_hav'] = activity['distance_hav_3d'].cumsum()\n\nactivity.drop(['shift_lon', 'shift_lat', 'shift_alt'], axis=1)\n#present the results\nactivity[['distance_vin', 'distance_hav', 'distance_hav_3d', 'distance_vin_3d']]\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
distance_vindistance_havdistance_hav_3ddistance_vin_3d
time
00:00:00NaNNaNNaNNaN
00:00:040.0000000.0000000.0000000.000000
00:00:065.0990005.1181625.1181625.099000
00:00:0812.56800012.6091537.4909917.469000
00:00:1019.33000019.3908836.7817306.762000
...............
01:25:1712544.27215012564.1723936.4375926.421000
01:25:2112559.22248512579.20383415.03144114.950334
01:25:2312564.86848512584.8749825.6711485.646000
01:25:2512570.23248512590.2677055.3927235.364000
01:25:2712575.31348512595.3611435.0934375.081000
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1697 rows \u00d7 4 columns

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\n\n\n\nFor futher convenice, we can extract the data in our previously created activity. Let's check the results with the following print comand.\n\n\n```python\nprint('Vicenty 2D: ', activity['distance_vin_no_alt'].max())\nprint('Haversine 2D: ', activity['distance_hav_no_alt'].max())\nprint('Vincenty 3D: ', activity['distance_vin'].max())\nprint('Haversine 3D: ', activity['distance_hav'].max())\n```\n\n Vicenty 2D: 12574.600999999993\n Haversine 2D: 12594.649752172338\n Vincenty 3D: 12575.313484827602\n Haversine 3D: 12595.361142855156\n\n\n\nIf we compare the distance calculations between the app and ours, we will see they are almost similar.\n\n \n\nBut why do we have 50m of difference between our internal calculations and the distance showed in the app ? It is explained in an [article](https://support.strava.com/hc/en-us/articles/216919487-How-Distance-is-Calculated) from Strava's support team.\n\n
\nA flat surface is assumed, and vertical speed from topography is not accounted for. \u2014 Strava\n
\n\nIn this scenario our 2d calculations are right and the we might conclude the app doesn\u2019t take elevation into account. The difference between the distance proposed by the app and our estimate is about 50m (0.03%). Note that this difference will increase if you undertake more altitude-intense activities (mountain biking or hiking).\n\nNow let's going forward to our next metric, the total activity time ellapsed x moving time.\n\n## Calculating the time ellapsed x moving time\n\n\n```python\ndef strfdelta(tdelta, fmt):\n d = {\"days\": tdelta.days}\n d[\"hours\"], rem = divmod(tdelta.seconds, 3600)\n d[\"minutes\"], d[\"seconds\"] = divmod(rem, 60)\n return fmt.format(**d)\n\nprint('Total time: ' , strfdelta(activity.index[-1], \"{hours} hour {minutes} min {seconds} sec\"))\n```\n\n Total time: 1 hour 25 min 27 sec\n\n\nThe total time elapsed totally agrees with our calculations, but the moving time seems to be diferent. Let us explain the basic concepts: **Elapsed time** is the duration from the moment you hit start on your device or phone to the moment you finish the activity. It includes stoplights, coffee breaks, bathroom stops and stopping for photos. **Moving time**, on the other hand, is a measure of how long you were active, this can be realistic when we have to slow down and stop for a traffic light, for example.\n\nLet\u2019s see if we can figure out which threshold Strava uses to stop the timer (and therefore boost our average speed). To do so, we need to create a new variable that calculates our movement in meters per second (and not just movement per data point, hence why we created the time difference variable). Let\u2019s do this for our haversine 2d distance, since that\u2019s the closest approximation of the distance proposed by the app.\n\n\n\n```python\nimport pandas as pd\nimport numpy as np\nactivity['time'] = activity.index\nactivity['time_dif'] = (activity.time - activity.time.shift(1).fillna(activity.time.iloc[0]))/np.timedelta64(1,'s')\nactivity['distance_dif_per_sec'] = activity['distance_hav_2d'] / activity['time_dif']\nactivity['distance_dif_per_sec']\n```\n\n\n\n\n time\n 00:00:00 NaN\n 00:00:04 0.000000\n 00:00:06 2.559081\n 00:00:08 3.745496\n 00:00:10 3.390865\n ... \n 01:25:17 3.218796\n 01:25:21 3.757777\n 01:25:23 2.835574\n 01:25:25 2.696362\n 01:25:27 2.546719\n Name: distance_dif_per_sec, Length: 1697, dtype: float64\n\n\n\nWith this new variable we can iterate through a list of thresholds. Let's assume values between 50 cm and 1 meter, and try to evaluate which one adds up to a timer time-out closest to 640 seconds (~=10 minutes).\n\n\n\n```python\nfor treshold in [0.5, 0.6, 0.7, 0.8, 0.9, 1]:\n print(treshold, 'm', ' : Time:', \n sum(activity[activity['distance_dif_per_sec'] < treshold]['time_dif']),\n ' seconds')\n```\n\n 0.5 m : Time: 574.0 seconds\n 0.6 m : Time: 577.0 seconds\n 0.7 m : Time: 640.0 seconds\n 0.8 m : Time: 640.0 seconds\n 0.9 m : Time: 640.0 seconds\n 1 m : Time: 640.0 seconds\n\n\nAs we can see at the table above, the movement per second was less than 80-70 centimeters, the application didn't consider it as movement and discard those intervals (it's about 2.9 km/h , a speed far below that most people do in their walking pace). Since we don't have the algorithm used for the real calculation in the app, we can get to an approximate moving time.\n\n\n```python\ntotal_time = activity['time_dif'].sum()\nstopped_time = sum(activity[activity['distance_dif_per_sec'] < 0.8]['time_dif'])\npd.Timedelta(seconds=total_time - stopped_time)\n\n```\n\n\n\n\n Timedelta('0 days 01:14:47')\n\n\n\n### Calculating the speed and pace\n\nWith the moving time and distance calculated, we can now calculate the pace and speed. **Speed** is calculated by dividing the distance traveled in meters by the time it took in seconds, and then converted to km/h.\n\n\n```python\nactivity['speed'] = (activity['distance_hav_2d'] / activity\n['time_dif']) * 3.6\n```\n\nNext we will filter out all the data where the movement per second is larger than 80 centimeters, based on the threshold we evaluated above.\n\n\n```python\nactivity_with_timeout = activity[activity['distance_dif_per_sec'] > 0.8]\n```\n\nFinally, we compute the weighted average speed and convert it to minutes and seconds per kilometers to get the **Pace metric**.\n\n\n```python\ndef pace(speed, fmt):\n d = {\"hours\": 0}\n d[\"minutes\"] = np.floor(60 / speed)\n d[\"seconds\"] = round(((60 / speed - np.floor(60 / speed))*60), 0)\n return fmt.format(**d)\n\navg_km_h = (sum((activity_with_timeout['speed'] * \n activity_with_timeout['time_dif'])) / \n sum(activity_with_timeout['time_dif']))\n\nprint('Speed:', avg_km_h , 'km/h')\nprint('Cadence:' , pace(avg_km_h, \"{hours} hour {minutes} min {seconds} sec\"))\n```\n\n Speed: 10.065838469772745 km/h\n Cadence: 0 hour 5.0 min 58.0 sec\n\n\nThe results compared to the proposed by our app show similar values, with average speed of 5 minutes and 58 seconds per kilometer, a difference of only just 2 secs.\n\nLet's plot our average speed for every 10 seconds to see our speed floats during the run. For this plot we will need the cumulative sum of our time differente to 10 seconds and plot the aggregated speed against it.\n\n\n```python\nactivity['time_10s'] = list(map(lambda x: round(x, -1), np.cumsum(activity['time_dif'])))\nplt.plot(activity.groupby(['time_10s']).mean()['speed'])\n```\n\nThe result shown is a smooth line plot where we check the speed (km/h) vs the time in seconds.\n\n### Calculating the elevation gain\n\nThe last metric we will explore is the elevation gain. According to the apps [documentation](https://support.strava.com/hc/en-us/articles/216919447-Elevation-for-Your-Activity), the cumulative elevation gain refers to the sum of every gain elevation throughout an entire Activity. \n\nBased on that, we can compute the elevation gain by mapping over our altitude difference column of our dataframe.\n\n\n\n```python\nactivity_with_timeout.loc[activity_with_timeout['alt_dif'] > 0]['alt_dif'].sum()\n```\n\n\n\n\n 38.0\n\n\n\nThe elevation calculated is far away from what the app showed (about 25m). Checking the altitude difference column values we can see that there are measures down to 10 centimeters. After reading the docummentation, we found out that the elevation data go through a noise correction algorithm. It is based on the a threshold where climbing needs to occur consistently for more than 10 meters for activities without strong barometric or two meters for an activity with the barometric data before it's added to the total elevation gain.\n\n\n```python\nelevations = zip(list(activity_with_timeout['alt']), list(activity_with_timeout['distance_hav_2d']))\n\nTHRESHOLD_METERS = 10\ndef calculate_elevation_gain(elevations):\n elevations = list(elevations)\n thresholdStartingPoint = elevations[0][0]\n diff_distance = 0\n count_diff = 0\n gain = 0\n valueAgainstThreshold = 0\n for elevation, distance in elevations:\n diff = elevation - thresholdStartingPoint\n diff_distance+=distance\n if diff > 0:\n valueAgainstThreshold += diff\n if abs(diff_distance) >= THRESHOLD_METERS:\n gain += valueAgainstThreshold\n diff_distance = 0\n valueAgainstThreshold = 0\n else:\n diff_distance = 0\n valueAgainstThreshold = 0\n thresholdStartingPoint = elevation\n\n return gain\n\n#plt.plot(activity['alt_dif'])\nprint(calculate_elevation_gain(elevations))\n```\n\n 25.30000000000002\n\n\nThe new result is 25.2m, very close to the elevation propose by the app. Based on that, we could recalculate the 3rd distances to get these new elevation values into account to get the distance closest to the 2d distance.\n\n\n## What's next ?\n\nIn this article we presented some popular metrics in runner's world such as moving time, cadence, pace and elevation gain. The calculations behind the metrics were also detailed and we gave a possible approximation of some of the algorithms used by a famous tracking running application, since we don't have access, of course,how they implemented it.\n\nThe next steps will be implement these metrics as properties, methods of the ``runpandas.Activity`` dataframe. We will release it soon with examples detailed at one of our future posts. Keep following here!\n\n\n\u2014 Please feel free to bring any inconsistencies or mistakes by leaving a comment below.-\n\n", "meta": {"hexsha": "bcc316f514426f71f632bb08b6d1e09131f32987", "size": 78578, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "_notebooks/2021-01-29-metrics.ipynb", "max_stars_repo_name": "corriporai/pandasrunner", "max_stars_repo_head_hexsha": "63e0642402ddf625f7e3c36c30117c3cf2ed01d5", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2020-08-03T13:49:54.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-22T05:25:01.000Z", "max_issues_repo_path": "_notebooks/2021-01-29-metrics.ipynb", "max_issues_repo_name": "corriporai/pandasrunner", "max_issues_repo_head_hexsha": "63e0642402ddf625f7e3c36c30117c3cf2ed01d5", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 7, "max_issues_repo_issues_event_min_datetime": "2020-12-26T10:41:17.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-26T08:56:43.000Z", "max_forks_repo_path": "_notebooks/2021-01-29-metrics.ipynb", "max_forks_repo_name": "corriporai/pandasrunner", "max_forks_repo_head_hexsha": "63e0642402ddf625f7e3c36c30117c3cf2ed01d5", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 98.2225, "max_line_length": 21587, "alphanum_fraction": 0.80747792, "converted": true, "num_tokens": 5879, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.6757646010190476, "lm_q1q2_score": 0.44493523957102427}} {"text": "# 2. logistic_regression\uff08\u903b\u8f91\u56de\u5f52\uff09\n\n# \u5907\u6ce8\n\u8fd0\u884c\u73af\u5883\uff1apython 3.6 \n\u73b0\u5728\u6211\u77e5\u9053\u6211\u5e94\u8be5\u8003\u8651\u5217\u5411\u91cf\uff0c\u800cTensorflow\u5bf9\u6570\u636e\u7684\u5f62\u72b6\u975e\u5e38\u6311\u5254\u3002 \u4f46\u662f\u5728numpy\u4e2d\uff0c\u6b63\u5e38\u7684\u4e00\u7ef4ndarray\u5df2\u7ecf\u88ab\u8868\u793a\u4e3a\u5217\u5411\u91cf\u3002 \u5982\u679c\u6211\u91cd\u65b0\u5851\u9020$\\mathbb{R}^n$ \u4e3a $\\mathbb{R}^{n\\times1}$\uff0c\u5b83\u4e0d\u518d\u662f\u5217\u5411\u91cf\u4e86\uff0c\u800c\u662f\u662f1\u5217\u7684\u77e9\u9635,\u90a3\u4f7f\u7528scipy\u4f1a\u6709\u9ebb\u70e6\u3002\n*\u6240\u4ee5\u6211\u4eec\u5e94\u8be5\u628aTensorFlow\u7684\u6570\u636e\u89c6\u4e3a\u7279\u6b8a\u60c5\u51b5\u3002 \u6211\u4eec\u7ee7\u7eed\u4f7f\u7528numpy\u7684\u60ef\u4f8b\u3002\n\n\n```python\nimport pandas as pd\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport seaborn as sns\nplt.style.use('fivethirtyeight')\nimport matplotlib.pyplot as plt\n# import tensorflow as tf\nfrom sklearn.metrics import classification_report#\u8fd9\u4e2a\u5305\u662f\u8bc4\u4ef7\u62a5\u544a\n```\n\n# \u51c6\u5907\u6570\u636e\n\n\n```python\ndata = pd.read_csv('ex2data1.txt', names=['exam1', 'exam2', 'admitted'])\ndata.head()#\u770b\u524d\u4e94\u884c\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
exam1exam2admitted
034.62366078.0246930
130.28671143.8949980
235.84740972.9021980
360.18259986.3085521
479.03273675.3443761
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\n\n\n\n\n```python\ndata.describe()\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
exam1exam2admitted
count100.000000100.000000100.000000
mean65.64427466.2219980.600000
std19.45822218.5827830.492366
min30.05882230.6032630.000000
25%50.91951148.1792050.000000
50%67.03298867.6823811.000000
75%80.21252979.3606051.000000
max99.82785898.8694361.000000
\n
\n\n\n\n\n```python\n#sns.set(context=\"notebook\", style=\"darkgrid\", palette=sns.color_palette(\"RdBu\", 2))\nsns.set(context=\"notebook\", style=\"darkgrid\", palette=sns.color_palette(\"RdBu\", 2), color_codes=False)\n\nsns.lmplot('exam1', 'exam2', hue='admitted', data=data, \n size=6, \n fit_reg=False, \n scatter_kws={\"s\": 50}\n )\nplt.show()#\u770b\u4e0b\u6570\u636e\u7684\u6837\u5b50\n```\n\n\n```python\ndef get_X(df):#\u8bfb\u53d6\u7279\u5f81\n# \"\"\"\n# use concat to add intersect feature to avoid side effect\n# not efficient for big dataset though\n# \"\"\"\n ones = pd.DataFrame({'ones': np.ones(len(df))})#ones\u662fm\u884c1\u5217\u7684dataframe\n data = pd.concat([ones, df], axis=1) # \u5408\u5e76\u6570\u636e\uff0c\u6839\u636e\u5217\u5408\u5e76\n #return data.iloc[:, :-1].as_matrix() # \u8fd9\u4e2a\u64cd\u4f5c\u8fd4\u56de ndarray,\u4e0d\u662f\u77e9\u9635\n #return data.iloc[:, :-1].iloc[:, :].values # \u8fd9\u4e2a\u64cd\u4f5c\u8fd4\u56de ndarray,\u4e0d\u662f\u77e9\u9635\n return data.iloc[:, :-1].values # \u8fd9\u4e2a\u64cd\u4f5c\u8fd4\u56de ndarray,\u4e0d\u662f\u77e9\u9635\n\n\ndef get_y(df):#\u8bfb\u53d6\u6807\u7b7e\n# '''assume the last column is the target'''\n return np.array(df.iloc[:, -1])#df.iloc[:, -1]\u662f\u6307df\u7684\u6700\u540e\u4e00\u5217\n\n\ndef normalize_feature(df):\n# \"\"\"Applies function along input axis(default 0) of DataFrame.\"\"\"\n return df.apply(lambda column: (column - column.mean()) / column.std())#\u7279\u5f81\u7f29\u653e\n```\n\n\n```python\nX = get_X(data)\nprint(X.shape)\n\ny = get_y(data)\nprint(y.shape)\n```\n\n (100, 3)\n (100,)\n\n\n# sigmoid \u51fd\u6570\ng \u4ee3\u8868\u4e00\u4e2a\u5e38\u7528\u7684\u903b\u8f91\u51fd\u6570\uff08logistic function\uff09\u4e3aS\u5f62\u51fd\u6570\uff08Sigmoid function\uff09\uff0c\u516c\u5f0f\u4e3a\uff1a \\\\[g\\left( z \\right)=\\frac{1}{1+{{e}^{-z}}}\\\\] \n\u5408\u8d77\u6765\uff0c\u6211\u4eec\u5f97\u5230\u903b\u8f91\u56de\u5f52\u6a21\u578b\u7684\u5047\u8bbe\u51fd\u6570\uff1a \n\t\\\\[{{h}_{\\theta }}\\left( x \\right)=\\frac{1}{1+{{e}^{-{{\\theta }^{T}}X}}}\\\\] \n\n\n\n```python\ndef sigmoid(z):\n return 1 / (1 + np.exp(-z))\n```\n\n\n```python\nfig, ax = plt.subplots(figsize=(8, 6))\nax.plot(np.arange(-10, 10, step=0.01),\n sigmoid(np.arange(-10, 10, step=0.01)))\nax.set_ylim((-0.1,1.1))\nax.set_xlabel('z', fontsize=18)\nax.set_ylabel('g(z)', fontsize=18)\nax.set_title('sigmoid function', fontsize=18)\nplt.show()\n```\n\n# cost function(\u4ee3\u4ef7\u51fd\u6570)\n> * $max(\\ell(\\theta)) = min(-\\ell(\\theta))$ \n> * choose $-\\ell(\\theta)$ as the cost function\n\n$$\\begin{align}\n & J\\left( \\theta \\right)=-\\frac{1}{m}\\sum\\limits_{i=1}^{m}{[{{y}^{(i)}}\\log \\left( {{h}_{\\theta }}\\left( {{x}^{(i)}} \\right) \\right)+\\left( 1-{{y}^{(i)}} \\right)\\log \\left( 1-{{h}_{\\theta }}\\left( {{x}^{(i)}} \\right) \\right)]} \\\\ \n & =\\frac{1}{m}\\sum\\limits_{i=1}^{m}{[-{{y}^{(i)}}\\log \\left( {{h}_{\\theta }}\\left( {{x}^{(i)}} \\right) \\right)-\\left( 1-{{y}^{(i)}} \\right)\\log \\left( 1-{{h}_{\\theta }}\\left( {{x}^{(i)}} \\right) \\right)]} \\\\ \n\\end{align}$$\n\n\n\n```python\ntheta = theta=np.zeros(3) # X(m*n) so theta is n*1\ntheta\n```\n\n\n\n\n array([0., 0., 0.])\n\n\n\n\n```python\n\n```\n\n\n```python\ndef cost(theta, X, y):\n ''' cost fn is -l(theta) for you to minimize'''\n return np.mean(-y * np.log(sigmoid(X @ theta)) - (1 - y) * np.log(1 - sigmoid(X @ theta)))\n\n# X @ theta\u4e0eX.dot(theta)\u7b49\u4ef7\n```\n\n\n```python\ncost(theta, X, y)\n```\n\n\n\n\n 0.6931471805599453\n\n\n\n# gradient descent(\u68af\u5ea6\u4e0b\u964d)\n* \u8fd9\u662f\u6279\u91cf\u68af\u5ea6\u4e0b\u964d\uff08batch gradient descent\uff09 \n* \u8f6c\u5316\u4e3a\u5411\u91cf\u5316\u8ba1\u7b97\uff1a $\\frac{1}{m} X^T( Sigmoid(X\\theta) - y )$\n$$\\frac{\\partial J\\left( \\theta \\right)}{\\partial {{\\theta }_{j}}}=\\frac{1}{m}\\sum\\limits_{i=1}^{m}{({{h}_{\\theta }}\\left( {{x}^{(i)}} \\right)-{{y}^{(i)}})x_{_{j}}^{(i)}}$$\n\n\n```python\ndef gradient(theta, X, y):\n# '''just 1 batch gradient'''\n return (1 / len(X)) * X.T @ (sigmoid(X @ theta) - y)\n```\n\n\n```python\ngradient(theta, X, y)\n```\n\n\n\n\n array([ -0.1 , -12.00921659, -11.26284221])\n\n\n\n# \u62df\u5408\u53c2\u6570\n> * \u8fd9\u91cc\u6211\u4f7f\u7528 [`scipy.optimize.minimize`](http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html#scipy.optimize.minimize) \u53bb\u5bfb\u627e\u53c2\u6570 \n\n\n\n```python\nimport scipy.optimize as opt\n```\n\n\n```python\nres = opt.minimize(fun=cost, x0=theta, args=(X, y), method='Newton-CG', jac=gradient)\n```\n\n\n```python\nprint(res)\n```\n\n fun: 0.20349771447597362\n jac: array([9.71219458e-06, 6.96602898e-04, 8.53722702e-04])\n message: 'Optimization terminated successfully.'\n nfev: 73\n nhev: 0\n nit: 29\n njev: 253\n status: 0\n success: True\n x: array([-25.17064404, 0.20630619, 0.20154693])\n\n\n# \u7528\u8bad\u7ec3\u96c6\u9884\u6d4b\u548c\u9a8c\u8bc1\n\n\n```python\ndef predict(x, theta):\n prob = sigmoid(x @ theta)\n return (prob >= 0.5).astype(int)\n```\n\n\n```python\nfinal_theta = res.x\ny_pred = predict(X, final_theta)\n\nprint(classification_report(y, y_pred))\n```\n\n precision recall f1-score support\n \n 0 0.87 0.85 0.86 40\n 1 0.90 0.92 0.91 60\n \n accuracy 0.89 100\n macro avg 0.89 0.88 0.88 100\n weighted avg 0.89 0.89 0.89 100\n \n\n\n# \u5bfb\u627e\u51b3\u7b56\u8fb9\u754c\nhttp://stats.stackexchange.com/questions/93569/why-is-logistic-regression-a-linear-classifier\n> $X \\times \\theta = 0$ (this is the line)\n\n\n```python\nprint(res.x) # this is final theta\n```\n\n [-25.17064404 0.20630619 0.20154693]\n\n\n\n```python\ncoef = -(res.x / res.x[2]) # find the equation\nprint(coef)\n\nx = np.arange(130, step=0.1)\ny = coef[0] + coef[1]*x\n```\n\n [124.88725955 -1.02361365 -1. ]\n\n\n\n```python\ndata.describe() # find the range of x and y\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
exam1exam2admitted
count100.000000100.000000100.000000
mean65.64427466.2219980.600000
std19.45822218.5827830.492366
min30.05882230.6032630.000000
25%50.91951148.1792050.000000
50%67.03298867.6823811.000000
75%80.21252979.3606051.000000
max99.82785898.8694361.000000
\n
\n\n\n\n> you know the intercept would be around 125 for both x and y\n\n\n```python\nsns.set(context=\"notebook\", style=\"ticks\", font_scale=1.5)\n\nsns.lmplot('exam1', 'exam2', hue='admitted', data=data, \n size=6, \n fit_reg=False, \n scatter_kws={\"s\": 25}\n )\n\nplt.plot(x, y, 'grey')\nplt.xlim(0, 130)\nplt.ylim(0, 130)\nplt.title('Decision Boundary')\nplt.show()\n```\n\n# 3- \u6b63\u5219\u5316\u903b\u8f91\u56de\u5f52\n\n\n```python\ndf = pd.read_csv('ex2data2.txt', names=['test1', 'test2', 'accepted'])\ndf.head()\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
test1test2accepted
00.0512670.699561
1-0.0927420.684941
2-0.2137100.692251
3-0.3750000.502191
4-0.5132500.465641
\n
\n\n\n\n\n```python\nsns.set(context=\"notebook\", style=\"ticks\", font_scale=1.5)\n\nsns.lmplot('test1', 'test2', hue='accepted', data=df, \n size=6, \n fit_reg=False, \n scatter_kws={\"s\": 50}\n )\n\nplt.title('Regularized Logistic Regression')\nplt.show()\n```\n\n# feature mapping\uff08\u7279\u5f81\u6620\u5c04\uff09\n\npolynomial expansion\n\n```\nfor i in 0..i\n for p in 0..i:\n output x^(i-p) * y^p\n```\n\n\n\n```python\ndef feature_mapping(x, y, power, as_ndarray=False):\n# \"\"\"return mapped features as ndarray or dataframe\"\"\"\n # data = {}\n # # inclusive\n # for i in np.arange(power + 1):\n # for p in np.arange(i + 1):\n # data[\"f{}{}\".format(i - p, p)] = np.power(x, i - p) * np.power(y, p)\n\n data = {\"f{}{}\".format(i - p, p): np.power(x, i - p) * np.power(y, p)\n for i in np.arange(power + 1)\n for p in np.arange(i + 1)\n }\n\n if as_ndarray:\n #return pd.DataFrame(data).as_matrix()\n return pd.DataFrame(data).values\n else:\n return pd.DataFrame(data)\n\n```\n\n\n```python\nx1 = np.array(df.test1)\nx2 = np.array(df.test2)\n```\n\n\n```python\ndata = feature_mapping(x1, x2, power=6)\nprint(data.shape)\ndata.head()\n```\n\n (118, 28)\n\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
f00f10f01f20f11f02f30f21f12f03...f23f14f05f60f51f42f33f24f15f06
01.00.0512670.699560.0026280.0358640.4893840.0001350.0018390.0250890.342354...0.0009000.0122780.1675421.815630e-082.477505e-070.0000030.0000460.0006290.0085890.117206
11.0-0.0927420.684940.008601-0.0635230.469143-0.0007980.005891-0.0435090.321335...0.002764-0.0204120.1507526.362953e-07-4.699318e-060.000035-0.0002560.001893-0.0139810.103256
21.0-0.2137100.692250.045672-0.1479410.479210-0.0097610.031616-0.1024120.331733...0.015151-0.0490770.1589709.526844e-05-3.085938e-040.001000-0.0032380.010488-0.0339730.110047
31.0-0.3750000.502190.140625-0.1883210.252195-0.0527340.070620-0.0945730.126650...0.017810-0.0238510.0319402.780914e-03-3.724126e-030.004987-0.0066790.008944-0.0119780.016040
41.0-0.5132500.465640.263426-0.2389900.216821-0.1352030.122661-0.1112830.100960...0.026596-0.0241280.0218901.827990e-02-1.658422e-020.015046-0.0136500.012384-0.0112350.010193
\n

5 rows \u00d7 28 columns

\n
\n\n\n\n\n```python\ndata.describe()\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
f00f10f01f20f11f02f30f21f12f03...f23f14f05f60f51f42f33f24f15f06
count118.0118.000000118.000000118.000000118.000000118.0000001.180000e+02118.000000118.000000118.000000...118.0000001.180000e+02118.0000001.180000e+02118.0000001.180000e+02118.0000001.180000e+02118.0000001.180000e+02
mean1.00.0547790.1831020.247575-0.0254720.3013705.983333e-020.0306820.0154830.142350...0.0182784.089084e-030.1157107.837118e-02-0.0007031.893340e-02-0.0017052.259170e-02-0.0063021.257256e-01
std0.00.4966540.5197430.2485320.2240750.2845362.746459e-010.1347060.1501430.326134...0.0585139.993907e-020.2990921.938621e-010.0582713.430092e-020.0374434.346935e-020.0906212.964416e-01
min1.0-0.830070-0.7697400.000040-0.4840960.000026-5.719317e-01-0.358121-0.483743-0.456071...-0.142660-4.830370e-01-0.2702226.472253e-14-0.2039712.577297e-10-0.1134482.418097e-10-0.4826841.795116e-14
25%1.0-0.372120-0.2543850.043243-0.1782090.061086-5.155632e-02-0.023672-0.042980-0.016492...-0.001400-7.449462e-03-0.0010728.086369e-05-0.0063811.258285e-04-0.0057493.528590e-04-0.0166622.298277e-04
50%1.0-0.0063360.2134550.165397-0.0165210.252195-2.544062e-070.006603-0.0000390.009734...0.001026-8.972096e-090.0004444.527344e-03-0.0000043.387050e-03-0.0000053.921378e-03-0.0000201.604015e-02
75%1.00.4789700.6465620.3899250.1007950.4641891.099616e-010.0863920.0795100.270310...0.0211482.751341e-020.1130205.932959e-020.0021042.090875e-020.0010242.103622e-020.0012891.001215e-01
max1.01.0709001.1089001.1468270.5683071.2296591.228137e+000.4492510.5055771.363569...0.2873234.012965e-011.6767251.508320e+000.2505772.018260e-010.1835482.556084e-010.4362091.859321e+00
\n

8 rows \u00d7 28 columns

\n
\n\n\n\n# regularized cost\uff08\u6b63\u5219\u5316\u4ee3\u4ef7\u51fd\u6570\uff09\n$$J\\left( \\theta \\right)=\\frac{1}{m}\\sum\\limits_{i=1}^{m}{[-{{y}^{(i)}}\\log \\left( {{h}_{\\theta }}\\left( {{x}^{(i)}} \\right) \\right)-\\left( 1-{{y}^{(i)}} \\right)\\log \\left( 1-{{h}_{\\theta }}\\left( {{x}^{(i)}} \\right) \\right)]}+\\frac{\\lambda }{2m}\\sum\\limits_{j=1}^{n}{\\theta _{j}^{2}}$$\n\n\n```python\ntheta = np.zeros(data.shape[1])\nX = feature_mapping(x1, x2, power=6, as_ndarray=True)\nprint(X.shape)\n\ny = get_y(df)\nprint(y.shape)\n```\n\n (118, 28)\n (118,)\n\n\n\n```python\ndef regularized_cost(theta, X, y, l=1):\n# '''you don't penalize theta_0'''\n theta_j1_to_n = theta[1:]\n regularized_term = (l / (2 * len(X))) * np.power(theta_j1_to_n, 2).sum()\n\n return cost(theta, X, y) + regularized_term\n#\u6b63\u5219\u5316\u4ee3\u4ef7\u51fd\u6570\n```\n\n\n```python\nregularized_cost(theta, X, y, l=1)\n```\n\n\n\n\n 0.6931471805599454\n\n\n\nthis is the same as the not regularized cost because we init theta as zeros...\n\u56e0\u4e3a\u6211\u4eec\u8bbe\u7f6etheta\u4e3a0\uff0c\u6240\u4ee5\u8fd9\u4e2a\u6b63\u5219\u5316\u4ee3\u4ef7\u51fd\u6570\u4e0e\u4ee3\u4ef7\u51fd\u6570\u7684\u503c\u76f8\u540c\n\n# regularized gradient(\u6b63\u5219\u5316\u68af\u5ea6)\n$$\\frac{\\partial J\\left( \\theta \\right)}{\\partial {{\\theta }_{j}}}=\\left( \\frac{1}{m}\\sum\\limits_{i=1}^{m}{\\left( {{h}_{\\theta }}\\left( {{x}^{\\left( i \\right)}} \\right)-{{y}^{\\left( i \\right)}} \\right)} \\right)+\\frac{\\lambda }{m}{{\\theta }_{j}}\\text{ }\\text{ for j}\\ge \\text{1}$$\n\n\n```python\ndef regularized_gradient(theta, X, y, l=1):\n# '''still, leave theta_0 alone'''\n theta_j1_to_n = theta[1:]\n regularized_theta = (l / len(X)) * theta_j1_to_n\n\n # by doing this, no offset is on theta_0\n regularized_term = np.concatenate([np.array([0]), regularized_theta])\n\n return gradient(theta, X, y) + regularized_term\n```\n\n\n```python\nregularized_gradient(theta, X, y)\n```\n\n\n\n\n array([8.47457627e-03, 1.87880932e-02, 7.77711864e-05, 5.03446395e-02,\n 1.15013308e-02, 3.76648474e-02, 1.83559872e-02, 7.32393391e-03,\n 8.19244468e-03, 2.34764889e-02, 3.93486234e-02, 2.23923907e-03,\n 1.28600503e-02, 3.09593720e-03, 3.93028171e-02, 1.99707467e-02,\n 4.32983232e-03, 3.38643902e-03, 5.83822078e-03, 4.47629067e-03,\n 3.10079849e-02, 3.10312442e-02, 1.09740238e-03, 6.31570797e-03,\n 4.08503006e-04, 7.26504316e-03, 1.37646175e-03, 3.87936363e-02])\n\n\n\n# \u62df\u5408\u53c2\u6570\n\n\n```python\nimport scipy.optimize as opt\n```\n\n\n```python\nprint('init cost = {}'.format(regularized_cost(theta, X, y)))\n\nres = opt.minimize(fun=regularized_cost, x0=theta, args=(X, y), method='Newton-CG', jac=regularized_gradient)\nres\n```\n\n init cost = 0.6931471805599454\n\n\n\n\n\n fun: 0.5290027297127542\n jac: array([-4.71952568e-08, -1.14457746e-08, 2.45828726e-08, -3.37674508e-08,\n 1.50839836e-08, -6.26344651e-08, 1.47936000e-08, 1.44056576e-08,\n 9.25470477e-09, -2.89079664e-08, -4.15440254e-08, -5.39192035e-09,\n -4.31597056e-08, 7.74514234e-09, -7.33884638e-08, 2.48858272e-09,\n 1.14058975e-08, -1.27449854e-08, -8.06033750e-09, 1.32958062e-08,\n -4.66425956e-08, -3.00496891e-08, 6.99938544e-10, -1.90907362e-08,\n -5.59017882e-09, -1.83826946e-08, 2.74986858e-09, -7.07341836e-08])\n message: 'Optimization terminated successfully.'\n nfev: 7\n nhev: 0\n nit: 6\n njev: 66\n status: 0\n success: True\n x: array([ 1.27273914, 0.62527176, 1.18108862, -2.01996089, -0.91742465,\n -1.43166351, 0.12400766, -0.36553408, -0.35723919, -0.17512987,\n -1.4581576 , -0.05099003, -0.61555683, -0.27470695, -1.19281634,\n -0.24218765, -0.20600567, -0.04473145, -0.27778483, -0.29537771,\n -0.45635687, -1.04320334, 0.02777145, -0.29243213, 0.01556617,\n -0.32738012, -0.1438876 , -0.92465245])\n\n\n\n# \u9884\u6d4b\n\n\n```python\nfinal_theta = res.x\ny_pred = predict(X, final_theta)\n\nprint(classification_report(y, y_pred))\n```\n\n precision recall f1-score support\n \n 0 0.90 0.75 0.82 60\n 1 0.78 0.91 0.84 58\n \n accuracy 0.83 118\n macro avg 0.84 0.83 0.83 118\n weighted avg 0.84 0.83 0.83 118\n \n\n\n# \u4f7f\u7528\u4e0d\u540c\u7684 $\\lambda$ \uff08\u8fd9\u4e2a\u662f\u5e38\u6570\uff09\n# \u753b\u51fa\u51b3\u7b56\u8fb9\u754c\n* \u6211\u4eec\u627e\u5230\u6240\u6709\u6ee1\u8db3 $X\\times \\theta = 0$ \u7684x\n* instead of solving polynomial equation, just create a coridate x,y grid that is dense enough, and find all those $X\\times \\theta$ that is close enough to 0, then plot them\n\n\n```python\ndef draw_boundary(power, l):\n# \"\"\"\n# power: polynomial power for mapped feature\n# l: lambda constant\n# \"\"\"\n density = 1000\n threshhold = 2 * 10**-3\n\n final_theta = feature_mapped_logistic_regression(power, l)\n x, y = find_decision_boundary(density, power, final_theta, threshhold)\n\n df = pd.read_csv('ex2data2.txt', names=['test1', 'test2', 'accepted'])\n sns.lmplot('test1', 'test2', hue='accepted', data=df, size=6, fit_reg=False, scatter_kws={\"s\": 100})\n\n plt.scatter(x, y, c='R', s=10)\n plt.title('Decision boundary')\n plt.show()\n```\n\n\n```python\ndef feature_mapped_logistic_regression(power, l):\n# \"\"\"for drawing purpose only.. not a well generealize logistic regression\n# power: int\n# raise x1, x2 to polynomial power\n# l: int\n# lambda constant for regularization term\n# \"\"\"\n df = pd.read_csv('ex2data2.txt', names=['test1', 'test2', 'accepted'])\n x1 = np.array(df.test1)\n x2 = np.array(df.test2)\n y = get_y(df)\n\n X = feature_mapping(x1, x2, power, as_ndarray=True)\n theta = np.zeros(X.shape[1])\n\n res = opt.minimize(fun=regularized_cost,\n x0=theta,\n args=(X, y, l),\n method='TNC',\n jac=regularized_gradient)\n final_theta = res.x\n\n return final_theta\n```\n\n\n```python\ndef find_decision_boundary(density, power, theta, threshhold):\n t1 = np.linspace(-1, 1.5, density)\n t2 = np.linspace(-1, 1.5, density)\n\n cordinates = [(x, y) for x in t1 for y in t2]\n x_cord, y_cord = zip(*cordinates)\n mapped_cord = feature_mapping(x_cord, y_cord, power) # this is a dataframe\n \n #inner_product = mapped_cord.as_matrix() @ theta\n inner_product = mapped_cord.values @ theta\n\n decision = mapped_cord[np.abs(inner_product) < threshhold]\n\n return decision.f10, decision.f01\n#\u5bfb\u627e\u51b3\u7b56\u8fb9\u754c\u51fd\u6570\n```\n\n\n```python\ndraw_boundary(power=6, l=1)#lambda=1\n```\n\n\n```python\ndraw_boundary(power=6, l=0) # no regularization, over fitting\uff0c#lambda=0,\u6ca1\u6709\u6b63\u5219\u5316\uff0c\u8fc7\u62df\u5408\u4e86\n```\n\n\n```python\ndraw_boundary(power=6, l=100) # underfitting\uff0c#lambda=100,\u6b20\u62df\u5408\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "ac2c33653db0cd80592eb1a8d75af4e4d66f513f", "size": 180148, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "MachineLearning/ex2-logistic-regression/1. logistic_regression_v1.ipynb", "max_stars_repo_name": "DLonng/Go", "max_stars_repo_head_hexsha": "a67ac6d6501f9fadadec6a6cf766d4b4a356d572", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 23, "max_stars_repo_stars_event_min_datetime": "2020-04-10T01:53:46.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-31T03:43:10.000Z", "max_issues_repo_path": "MachineLearning/ex2-logistic-regression/1. logistic_regression_v1.ipynb", "max_issues_repo_name": "DLonng/Go", "max_issues_repo_head_hexsha": "a67ac6d6501f9fadadec6a6cf766d4b4a356d572", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-12-10T07:08:37.000Z", "max_issues_repo_issues_event_max_datetime": "2021-04-14T07:47:01.000Z", "max_forks_repo_path": "MachineLearning/ex2-logistic-regression/1. logistic_regression_v1.ipynb", "max_forks_repo_name": "DLonng/Go", "max_forks_repo_head_hexsha": "a67ac6d6501f9fadadec6a6cf766d4b4a356d572", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2020-04-05T11:49:22.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-04T10:23:37.000Z", "avg_line_length": 96.6977992485, "max_line_length": 38936, "alphanum_fraction": 0.7961453916, "converted": true, "num_tokens": 10911, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791787121629466, "lm_q2_score": 0.6548947290421276, "lm_q1q2_score": 0.4447905586731341}} {"text": "# Exercise Session 1: Getting Started with Computer Vision\n\nThe goals of this exercise are:\n* getting started with Python for image manipulation\n* getting familiar with the basic image manipulation functions\n* implementing some simple real-world Computer Vision algorithms\n\n\n```python\n%matplotlib inline\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport cv2\n\nfrom skimage import measure, color\nfrom os import listdir \n\nplt.rcParams['figure.figsize'] = (10, 10)\nplt.rcParams['image.cmap'] = 'gray'\n```\n\nIf you have some missing packages, you can use the corresponding ```conda install``` command from below list to install it,\n\n\n```python\nconda install -c conda-forge scikit-image\nconda install -c conda-forge matplotlib\nconda install -c conda-forge opencv\nconda install -c anaconda numpy\n```\n\n## Exercise 1: Image Segmentation\n\nIn many vision applications, it is useful to separate out the regions of the image corresponding to objects in which we are interested in the regions of the image that correspond to the background. Thresholding often provides an easy and convenient way to perform this segmentation on the basis of the different intensities or colours in the foreground and background regions of an image.\n\nThe input to a thresholding operation is typically a grayscale or colour image. In the simplest implementation, the output is a binary image representing the segmentation. Black pixels correspond to background and white pixels correspond to foreground (or vice versa). Multiple thresholds can be specified, so that a band of intensity values can be set to white while everything else is set to black.\n\nIf it is possible to separate out the foreground of an image on the basis of pixel intensity, then the intensity of pixels within foreground objects must be distinctly different from the intensity of pixels within the background. In this case, we expect to see a distinct peak in the histogram corresponding to foreground objects such that thresholds can be chosen to isolate this peak accordingly. If such a peak does not exist, then it is unlikely that simple thresholding will produce a good segmentation.\n \n\n\n* Read and display the image \"wdg.png\" using function ```cv2.imread()```. Convert it from color to greyscale if necessary using function ```cv2.cvtColor()```. Visualize the results using ```plt.imshow()``` function.\n\n\n```python\n#add your code here\n```\n\n* Write a function to threshold a gray scale image by using two threshold values as shown above. The values must satisfy the following conditions:\n\\begin{align}\nTh1 < Th2 \n\\newline \nTh1 > 0 \n\\newline \nTh2 < I_{max}\n\\end{align}\nwhere $I_{max}$ is the maximum intensity of the image.\n\n\n```python\n#add your code here\n```\n\n* Take a look at the pixels intensity histogram using function ```matplotlib.pyplot.hist()``` and choose the best threshold values and segment the image.\n\n\n```python\n#add your code here\n```\n\n* Repeat the same steps for images \"brain.png\" and \"shading.png\". What do you notice? What are the drawbacks of this segmentation method? \n\n\n```python\n#add your code here\n```\n\n## Exercise 2: Background Substraction\n\nBackground subtraction is an important preprocessing step of many algorithms, e.g. object detection. In the following exercises we will try to subtract the scene background using multiple images.\n\n### 2.1 Extracting a moving object\n\n* Load the \"street1.tiff\" and the \"street2.tiff\" images. Visualize them.\n\n\n```python\n#add your code here. \n```\n\n* Transform the 8-bit images into float images. You can use image attribute ```dtype``` to check the type of image. To perform the type casting you can use ```np.float32()``` function. \n\n\n```python\n#add your code here. \n```\n\n* Subtract the second image from the first one using basic matrix arithmetic operations. Visualize the results. Why was it important to do the casting before subtracting the images? \n\n\n```python\n# add your code here. Assign the difference to 'image_diff'\nplt.imshow(image_diff) \n```\n\n### 2.2 Building a background model\nFor this exercise, you are given a sequence of images that contains pedestrians and we wish to segment with a background subtraction algorithm.\n\n* Load and create a stack of images from the images inside ```images/sequence1```. Build a \"background model\" by averaging out the set of given images. Detect pedestrians\n subtracting the background model from the original images and applying the right threshold.\n\n\n```python\nsq_of_images = listdir('images/sequence1/') \nsq_of_images = [img for img in sq_of_images if img.endswith(\".jpg\")]\n\n# 'sq_of_images' holds the list of image names. Create an image stack using them.\n \n# Compute the mean image using the stack and assign it to 'mean_image'.\nplt.imshow(mean_image) \nplt.title('Background model')\n```\n\n\n```python\nT = 0.1 \nplt.figure(2)\nplt.suptitle('Pedestrians')\n\n# First convert the mean image to grayscale.\n# When subtracting, each image must also be converted to grayscale.\n \nfor i in range(len(stack)):\n # Compute the foreground image here. Assign it to 'foreground' variable.\n plt.subplot(6,5,i+1)\n plt.imshow(foreground)\n plt.axis('off')\n plt.title('Image: ' + str(i+1)) \n```\n\n* Create a more sophisticated background model, where each pixel can be modeled with a Gaussian distribution. We can classify a pixel as background if its current intensity ($I_t$) lies within some confidence interval of its distribution\u2019s mean ($\\mu(t)$):\n\n\n\\begin{align}\n\\frac{\\mid{(I_t - \\mu_t)}\\mid}{\\sigma_t} > T \\rightarrow Foreground \n\\newline\n\\frac{\\mid{(I_t - \\mu_t)}\\mid}{\\sigma_t} < T \\rightarrow Background \n\\end{align}\n\n$\\sigma_t$ is the standard deviation of the pixel $t$ in the background model. $T$ is the threshold.\n\n\n```python\n# Add your code here. Before computing the model convert the images into gray-scale images. \n```\n\nWhat difference do you notice between the two approaches? How does changing\nthe threshold affect them?\n\n## Exercise 3: Connected Components\n\nSegmentation can be also done for colour images. It is also often a first step for the further analysis e.g. measuring properties of the object. Here our goal is to count the number of apples in the image below.\n\n\n\n* Read and display\"apples.jpg\" image.\n\n\n```python\n#add your code here\n```\n\n* Check the size of the image. Compared to the previous images it should have an additional dimension corresponding to three colour channels: red, green and blue. Visualize those 3 channels separately.\n\n\n```python\nfig, axes = plt.subplots(1, 3)\nfig.set_size_inches(18.5, 10.5)\n# add your code for visualizing three channels\n```\n\n* Try to obtain a binary image such that binary image == 1 for pixels representing apples and 0 otherwise. Which channel(s) would you use for that?\n\n\n```python\nbin_img = np.zeros(img_apples.shape[0:2])\n#add your code fot thresholding the image\nplt.imshow(bin_img)\n```\n\n* Count the number of connected components in your binary image (here corresponding to apples). For this, you can use function ```measure.label()```. Its output is an array of the same size as input binary image, with each pixel assigned to a different connected component (ID). Visualize the image with detected connected components.\n\n\n```python\n#add your code to find connected components\n# labels = \n```\n\n\n```python\nplt.imshow(labels,cmap=\"jet\")\n```\n\n* Simple thresholding sometimes leads to detecting also noise in the background that is detected as seperate connected components. Try to suppress the noise by removing all connected components smaller than a user-defined threshold.\n\n\n```python\ndef remove_noise(label_img,threshold):\n\n#add your code here\n \n return label_img_new\n```\n\n\n```python\nlabels_new = remove_noise(labels,50)\nplt.imshow(labels_new,cmap=\"jet\")\n```\n", "meta": {"hexsha": "c2bed73f22a04c169503715b117e59414f990b7a", "size": 13563, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ExerciseSession1/.ipynb_checkpoints/exercise_session_1-checkpoint.ipynb", "max_stars_repo_name": "PhilipLGQ/CS442-Computer-Vision", "max_stars_repo_head_hexsha": "668d5f40d7ed9e23ef09b1466fcc6d7cb326252e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "ExerciseSession1/.ipynb_checkpoints/exercise_session_1-checkpoint.ipynb", "max_issues_repo_name": "PhilipLGQ/CS442-Computer-Vision", "max_issues_repo_head_hexsha": "668d5f40d7ed9e23ef09b1466fcc6d7cb326252e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ExerciseSession1/.ipynb_checkpoints/exercise_session_1-checkpoint.ipynb", "max_forks_repo_name": "PhilipLGQ/CS442-Computer-Vision", "max_forks_repo_head_hexsha": "668d5f40d7ed9e23ef09b1466fcc6d7cb326252e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 27.5111561866, "max_line_length": 517, "alphanum_fraction": 0.599941016, "converted": true, "num_tokens": 1734, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.6791787121629466, "lm_q1q2_score": 0.444790558673134}} {"text": "# Graph Attention Network\n\n\\[[paper](https://arxiv.org/abs/1710.10903)\\] , \\[[original code](https://github.com/PetarV-/GAT)\\] , \\[[all other implementations](https://paperswithcode.com/paper/graph-attention-networks)\\]\n\nFrom [Graph Convolutional Network (GCN)](https://arxiv.org/abs/1609.02907), we learned that combining local graph structure and node-level features yields good performance on node classification task. Hwever, the way GCN aggregates is structure-dependent, which may hurt its generalizability.\n\nOne workaround is to simply average over all neighbor node features as in GraphSAGE. Graph Attention Network proposes an alternative way by weighting neighbor features with feature dependent and structure free normalization, in the style of attention.\n\nThe goal of this tutorial:\n\n- Explain what is Graph Attention Network.\n- Understand the attentions learnt.\n- Introduce to inductive learning.\n\nIntroducing Attention to GCN\n----------------------------\n\nThe key difference between GAT and GCN is how the information from the one-hop neighborhood is aggregated.\n\nFor GCN, a graph convolution operation produces the normalized sum of the node features of neighbors:\n\n\n$$h_i^{(l+1)}=\\sigma\\left(\\sum_{j\\in \\mathcal{N}(i)} {\\frac{1}{c_{ij}} W^{(l)}h^{(l)}_j}\\right)$$\n\nwhere $\\mathcal{N}(i)$ is the set of its one-hop neighbors (to include $v_i$ in the set, simply add a self-loop to each node),\n$c_{ij}=\\sqrt{|\\mathcal{N}(i)|}\\sqrt{|\\mathcal{N}(j)|}$ is a normalization constant based on graph structure, $\\sigma$ is an activation function (GCN uses ReLU), and $W^{(l)}$ is a shared weight matrix for node-wise feature transformation. Another model proposed in\n[GraphSAGE](https://www-cs-faculty.stanford.edu/people/jure/pubs/graphsage-nips17.pdf)\nemploys the same update rule except that they set\n$c_{ij}=|\\mathcal{N}(i)|$.\n\nGAT introduces the attention mechanism as a substitute for the statically\nnormalized convolution operation. Below are the equations to compute the node\nembedding $h_i^{(l+1)}$ of layer $l+1$ from the embeddings of\nlayer $l$:\n\n\n\n\n```python\n!pip install -q --upgrade git+https://github.com/mlss-skoltech/tutorials_week2.git#subdirectory=graph_neural_networks\n```\n\n\n```python\nimport pkg_resources\n\nZIP_PATH = pkg_resources.resource_filename('gnnutils', 'data/data.zip')\nDATA_PATH = './data'\n\n!unzip -u {ZIP_PATH} -d ./\n```\n\n Archive: /anaconda3/lib/python3.7/site-packages/gnnutils/data/data.zip\n inflating: ./data/ind.cora.ally \n inflating: ./data/ind.cora.test.index \n inflating: ./data/ind.pubmed.graph \n inflating: ./data/ind.cora.allx \n inflating: ./data/ind.citeseer.test.index \n inflating: ./data/ind.citeseer.ty \n inflating: ./data/ind.citeseer.tx \n inflating: ./data/ind.citeseer.graph \n inflating: ./data/preprocessed_MNIST.dump \n inflating: ./data/ind.cora.tx \n inflating: ./data/ind.pubmed.test.index \n inflating: ./data/ind.cora.ty \n inflating: ./data/ind.cora.graph \n inflating: ./data/ind.citeseer.y \n inflating: ./data/ind.citeseer.ally \n inflating: ./data/ind.citeseer.allx \n inflating: ./data/ind.citeseer.x \n inflating: ./data/ind.pubmed.ally \n inflating: ./data/ind.pubmed.allx \n inflating: ./data/ind.pubmed.x \n inflating: ./data/ind.pubmed.tx \n inflating: ./data/ind.cora.x \n inflating: ./data/ind.pubmed.ty \n inflating: ./data/ind.pubmed.y \n inflating: ./data/ind.cora.y \n\n\n\n```python\nimport os,sys,inspect\nimport os\nimport joblib\nimport tensorflow as tf\nimport numpy as np\nimport h5py\nimport scipy.sparse.linalg as la\nimport scipy.sparse as sp\nimport scipy\nimport time\nimport pickle\n\nimport matplotlib.pyplot as plt\nfrom matplotlib.backends.backend_pdf import PdfPages\n%matplotlib inline\n\nimport scipy.io as sio\n\nfrom gnnutils import process_data\n```\n\n\n```python\ndef count_no_weights():\n total_parameters = 0\n for variable in tf.trainable_variables():\n # shape is an array of tf.Dimension\n shape = variable.get_shape()\n variable_parameters = 1\n for dim in shape:\n variable_parameters *= dim.value\n total_parameters += variable_parameters\n print('#weights in the model: %d' % (total_parameters,))\n\ndef frobenius_norm(tensor):\n square_tensor = tf.square(tensor)\n tensor_sum = tf.reduce_sum(square_tensor)\n frobenius_norm = tf.sqrt(tensor_sum)\n return frobenius_norm\n\n```\n\n\n```python\nclass GAT:\n \n \"\"\"\n The neural network model.\n \"\"\"\n def __init__(self, idx_rows, idx_cols, A_shape, X, Y, num_hidden_feat, n_heads, learning_rate=5e-2, gamma=1e-3, idx_gpu = '/gpu:3'):\n \n self.num_hidden_feat = num_hidden_feat\n self.learning_rate = learning_rate\n self.gamma=gamma\n with tf.Graph().as_default() as g:\n self.graph = g\n \n with tf.device(idx_gpu):\n \n # list of weights' tensors l2-loss \n self.regularizers = []\n \n #definition of constant matrices\n self.X = tf.constant(X, dtype=tf.float32) \n self.Y = tf.constant(Y, dtype=tf.float32)\n \n #placeholder definition\n self.idx_nodes = tf.placeholder(tf.int32)\n self.keep_prob = tf.placeholder(tf.float32)\n \n #model definition\n \n self.X0 = []\n for k in range(n_heads):\n with tf.variable_scope('GCL_1_{}'.format(k+1)):\n self.X0.append(self.GAT_layer(self.X, num_hidden_feat, idx_rows, idx_cols, A_shape, tf.nn.elu))\n self.X0 = tf.concat(self.X0, 1)\n \n with tf.variable_scope('GCL_2'):\n self.logits = self.GAT_layer(self.X0, Y.shape[1], idx_rows, idx_cols, A_shape, tf.identity)\n \n self.l_out = tf.gather(self.logits, self.idx_nodes)\n self.c_Y = tf.gather(self.Y, self.idx_nodes)\n \n #loss function definition\n self.l2_reg = tf.reduce_sum(self.regularizers)\n self.data_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(logits=self.l_out, labels=self.c_Y)) \n \n self.loss = self.data_loss + self.gamma*self.l2_reg\n \n #solver definition\n self.optimizer = tf.train.AdamOptimizer(learning_rate=self.learning_rate)\n self.opt_step = self.optimizer.minimize(self.loss)\n \n #predictions and accuracy extraction\n self.c_predictions = tf.argmax(tf.nn.softmax(self.l_out), 1)\n self.accuracy = tf.contrib.metrics.accuracy(self.c_predictions, tf.argmax(self.c_Y, 1))\n \n #gradients computation\n self.trainable_variables = tf.trainable_variables()\n self.var_grad = tf.gradients(self.loss, tf.trainable_variables())\n self.norm_grad = frobenius_norm(tf.concat([tf.reshape(g, [-1]) for g in self.var_grad], 0))\n \n #session creation\n config = tf.ConfigProto(allow_soft_placement = True)\n config.gpu_options.allow_growth = True\n self.session = tf.Session(config=config)\n\n #session initialization\n init = tf.global_variables_initializer()\n self.session.run(init)\n \n count_no_weights()\n\n def GAT_layer(self, X, Fout, idx_rows, idx_cols, A_shape, activation):\n X = tf.nn.dropout(X, self.keep_prob)\n \n W = tf.get_variable(\"W\", shape=[X.shape[1], Fout], initializer=tf.glorot_uniform_initializer())\n self.regularizers.append(tf.nn.l2_loss(W))\n X_w = tf.matmul(X, W)\n\n # simplest possible attention mechanism\n W_att1 = tf.get_variable(\"W_att1\", shape=[X_w.shape[1], 1], initializer=tf.glorot_uniform_initializer())\n b_att1 = tf.get_variable(\"b_att1\", shape=[1,], initializer=tf.zeros_initializer())\n self.regularizers.append(tf.nn.l2_loss(W_att1))\n W_att2 = tf.get_variable(\"W_att2\", shape=[X_w.shape[1], 1], initializer=tf.glorot_uniform_initializer())\n b_att2 = tf.get_variable(\"b_att2\", shape=[1,], initializer=tf.zeros_initializer())\n self.regularizers.append(tf.nn.l2_loss(W_att2))\n \n X_att_1 = tf.squeeze(tf.matmul(X_w, W_att1)) + b_att1\n X_att_2 = tf.squeeze(tf.matmul(X_w, W_att2)) + b_att2\n \n logits = tf.gather(X_att_1, idx_rows) + tf.gather(X_att_2, idx_cols)\n \n A_att = tf.SparseTensor(indices=np.vstack([idx_rows, idx_cols]).T, \n values=tf.nn.leaky_relu(logits), \n dense_shape=A_shape)\n A_att = tf.sparse_reorder(A_att)\n A_att = tf.sparse_softmax(A_att)\n \n # apply dropout\n A_att = tf.SparseTensor(indices=A_att.indices,\n values=tf.nn.dropout(A_att.values, self.keep_prob),\n dense_shape=A_shape)\n A_att = tf.sparse_reorder(A_att)\n\n X_w = tf.nn.dropout(X_w, self.keep_prob)\n res = tf.sparse_tensor_dense_matmul(A_att, X_w)\n res = tf.contrib.layers.bias_add(res)\n\n return activation(res)\n \n```\n\nMulti-head Attention\n^^^^^^^^^^^^^^^^^^^^\n\nAnalogous to multiple channels in ConvNet, GAT introduces **multi-head\nattention** to enrich the model capacity and to stabilize the learning\nprocess. Each attention head has its own parameters and their outputs can be\nmerged in two ways:\n\n\\begin{align}\\text{concatenation}: h^{(l+1)}_{i} =||_{k=1}^{K}\\sigma\\left(\\sum_{j\\in \\mathcal{N}(i)}\\alpha_{ij}^{k}W^{k}h^{(l)}_{j}\\right)\\end{align}\n\nor\n\n\\begin{align}\\text{average}: h_{i}^{(l+1)}=\\sigma\\left(\\frac{1}{K}\\sum_{k=1}^{K}\\sum_{j\\in\\mathcal{N}(i)}\\alpha_{ij}^{k}W^{k}h^{(l)}_{j}\\right)\\end{align}\n\nwhere $K$ is the number of heads. The authors suggest using\nconcatenation for intermediary layers and average for the final layer.\n\n\n\n```python\n#learning parameters and path dataset\n\nlearning_rate = 5e-3\nval_test_interval = 1\nnum_hidden_feat = 8\nn_heads = 8\ngamma = 5e-4\npatience = 100\npath_dataset = './CORA/dataset.pickle'\n \n#dataset loading\n#ds = Dataset(path_dataset, normalize_feat=1)\n\nA, X, Y, train_idx, val_idx, test_idx = process_data.load_data(\"cora\", path_to_data=DATA_PATH)\nX = process_data.preprocess_features(X)\n```\n\n (2708, 2708)\n (2708, 1433)\n\n\n\n```python\n# extracts rows and cols of adjacency matrix\nA = sp.csr_matrix(A)\nA.setdiag(1)\n\nidx_rows, idx_cols = A.nonzero()\n```\n\n\n```python\nfrom tqdm import tqdm\n```\n\n\n```python\n# num_exp = 10 #number of times training GCN over the given dataset\nnum_exp = 1 #number of times training GCN over the given dataset\n\nlist_all_acc = []\nlist_all_cost_val_avg = []\nlist_all_data_cost_val_avg = []\nlist_all_acc_val_avg = []\nlist_all_cost_test_avg = []\nlist_all_acc_test_avg = []\n\nnum_done = 0\n```\n\n\n```python\nnum_total_iter_training = int(10e4)\n\nGCNN = GAT(idx_rows, idx_cols, A.shape, X, Y, num_hidden_feat, n_heads, learning_rate=learning_rate, gamma=gamma)\n\ncost_train_avg = []\ngrad_norm_train_avg = []\nacc_train_avg = []\ncost_test_avg = []\ngrad_norm_test_avg = []\nacc_test_avg = []\ncost_val_avg = []\ndata_cost_val_avg = []\nacc_val_avg = []\niter_test = []\nlist_training_time = list()\n\nmax_val_acc = 0\nmin_val_loss = np.inf\n\n#Training code\nfor i in tqdm(range(num_total_iter_training)):\n if (len(cost_train_avg) % val_test_interval) == 0:\n #Print last training performance\n if (len(cost_train_avg)>0):\n tqdm.write(\"[TRN] epoch = %03i, cost = %3.2e, |grad| = %.2e, acc = %3.2e (%03.2fs)\" % \\\n (len(cost_train_avg), cost_train_avg[-1], grad_norm_train_avg[-1], acc_train_avg[-1], time.time() - tic))\n\n #Validate the model\n tic = time.time()\n\n feed_dict = {GCNN.idx_nodes: val_idx, GCNN.keep_prob:1.0}\n acc_val, cost_val, data_cost_val = GCNN.session.run([GCNN.accuracy, GCNN.loss, GCNN.data_loss], feed_dict)\n\n data_cost_val_avg.append(data_cost_val)\n cost_val_avg.append(cost_val)\n acc_val_avg.append(acc_val)\n tqdm.write(\"[VAL] epoch = %03i, data_cost = %3.2e, cost = %3.2e, acc = %3.2e (%03.2fs)\" % \\\n (len(cost_train_avg), data_cost_val_avg[-1], cost_val_avg[-1], acc_val_avg[-1], time.time() - tic))\n\n #Test the model\n tic = time.time()\n\n feed_dict = {GCNN.idx_nodes: test_idx, GCNN.keep_prob:1.0}\n acc_test, cost_test = GCNN.session.run([GCNN.accuracy, GCNN.loss], feed_dict)\n\n cost_test_avg.append(cost_test)\n acc_test_avg.append(acc_test)\n tqdm.write(\"[TST] epoch = %03i, cost = %3.2e, acc = %3.2e (%03.2fs)\" % \\\n (len(cost_train_avg), cost_test_avg[-1], acc_test_avg[-1], time.time() - tic))\n iter_test.append(len(cost_train_avg))\n\n\n if acc_val_avg[-1] >= max_val_acc or data_cost_val_avg[-1] <= min_val_loss:\n max_val_acc = np.maximum(acc_val_avg[-1], max_val_acc)\n min_val_loss = np.minimum(data_cost_val_avg[-1], min_val_loss)\n if acc_val_avg[-1] >= max_val_acc and data_cost_val_avg[-1] <= min_val_loss:\n best_model_test_acc = acc_test_avg[-1]\n curr_step = 0\n else:\n curr_step += 1\n if curr_step == patience:\n tqdm.write('Early stop! Min loss: ', min_val_loss, ', Max accuracy: ', max_val_acc)\n break\n\n tic = time.time()\n feed_dict = {GCNN.idx_nodes: train_idx, GCNN.keep_prob: 0.4}\n\n _, current_training_loss, norm_grad, current_acc_training = GCNN.session.run([GCNN.opt_step, GCNN.loss, GCNN.norm_grad, GCNN.accuracy], feed_dict) \n\n training_time = time.time() - tic \n\n cost_train_avg.append(current_training_loss)\n grad_norm_train_avg.append(norm_grad)\n acc_train_avg.append(current_acc_training)\n\n\n#Compute and print statistics of the last realized experiment\nlist_all_acc.append(100*best_model_test_acc)\nlist_all_cost_val_avg.append(cost_val_avg)\nlist_all_data_cost_val_avg.append(data_cost_val_avg)\nlist_all_acc_val_avg.append(acc_val_avg)\nlist_all_cost_test_avg.append(cost_test_avg)\nlist_all_acc_test_avg.append(acc_test_avg)\n\nprint('Num done: %d' % num_done)\nprint('Max accuracy on test set achieved: %f%%' % np.max(np.asarray(acc_test_avg)*100))\nprint('Max suggested accuracy: %f%%' % (100*best_model_test_acc))#(np.asarray(acc_test_avg)[np.asarray(data_cost_val_avg)==np.min(data_cost_val_avg)]),))\nprint('Current mean: %f%%' % np.mean(list_all_acc))\nprint('Current std: %f' % np.std(list_all_acc))\n\nnum_done += 1\n```\n\n\n```python\n#Print average performance\nprint(np.mean(list_all_acc))\nprint(np.std(list_all_acc))\n```\n\n 83.25999975204468\n 0.5765412469708043\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "f42e06e73cae6dbefa5be0ca0a92723050c4609b", "size": 21156, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Machine Learning Summer School 2019 (Moscow, Russia)/tutorials/graph_neural_networks/citation_networks/CORA_GAT.ipynb", "max_stars_repo_name": "xuedong/rlss2019", "max_stars_repo_head_hexsha": "d7468c2fcf269d8afd6fb0f44993aa9797867944", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Machine Learning Summer School 2019 (Moscow, Russia)/tutorials/graph_neural_networks/citation_networks/CORA_GAT.ipynb", "max_issues_repo_name": "xuedong/rlss2019", "max_issues_repo_head_hexsha": "d7468c2fcf269d8afd6fb0f44993aa9797867944", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Machine Learning Summer School 2019 (Moscow, Russia)/tutorials/graph_neural_networks/citation_networks/CORA_GAT.ipynb", "max_forks_repo_name": "xuedong/rlss2019", "max_forks_repo_head_hexsha": "d7468c2fcf269d8afd6fb0f44993aa9797867944", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 39.6923076923, "max_line_length": 301, "alphanum_fraction": 0.5302514653, "converted": true, "num_tokens": 3843, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791787121629466, "lm_q2_score": 0.6548947155710233, "lm_q1q2_score": 0.4447905495238468}} {"text": "# Optimizaci\u00f3n media-varianza\n\n\n\n\nLa **teor\u00eda de portafolios** es uno de los avances m\u00e1s importantes en las finanzas modernas e inversiones.\n- Apareci\u00f3 por primera vez en un [art\u00edculo corto](https://www.math.ust.hk/~maykwok/courses/ma362/07F/markowitz_JF.pdf) llamado \"Portfolio Selection\" en la edici\u00f3n de Marzo de 1952 de \"the Journal of Finance\".\n- Escrito por un desconocido estudiante de la Universidad de Chicago, llamado Harry Markowitz.\n- Escrito corto (s\u00f3lo 14 p\u00e1ginas), poco texto, f\u00e1cil de entender, muchas gr\u00e1ficas y unas cuantas referencias.\n- No se le prest\u00f3 mucha atenci\u00f3n hasta los 60s.\n\nFinalmente, este trabajo se convirti\u00f3 en una de las m\u00e1s grandes ideas en finanzas, y le di\u00f3 a Markowitz el Premio Nobel casi 40 a\u00f1os despu\u00e9s.\n- Markowitz estaba incidentalmente interesado en los mercados de acciones e inversiones.\n- Estaba m\u00e1s bien interesado en entender c\u00f3mo las personas tomaban sus mejores decisiones cuando se enfrentaban con \"trade-offs\".\n- Principio de conservaci\u00f3n de la miseria. O, dir\u00edan los instructores de gimnasio: \"no pain, no gain\".\n- Si queremos m\u00e1s de algo, tenemos que perder en alg\u00fan otro lado.\n- El estudio de este fen\u00f3meno era el que le atra\u00eda a Markowitz.\n\nDe manera que nadie se hace rico poniendo todo su dinero en la cuenta de ahorros. La \u00fanica manera de esperar altos rendimientos es si se toma bastante riesgo. Sin embargo, riesgo significa tambi\u00e9n la posibilidad de perder, tanto como ganar.\n\nPero, \u00bfqu\u00e9 tanto riesgo es necesario?, y \u00bfhay alguna manera de minimizar el riesgo mientras se maximizan las ganancias?\n- Markowitz b\u00e1sicamente cambi\u00f3 la manera en que los inversionistas pensamos acerca de esas preguntas.\n- Alter\u00f3 completamente la pr\u00e1ctica de la administraci\u00f3n de inversiones.\n- Incluso el t\u00edtulo de su art\u00edculo era innovador. Portafolio: una colecci\u00f3n de activos en lugar de tener activos individuales.\n- En ese tiempo, un portafolio se refer\u00eda a una carpeta de piel.\n- En el resto de este m\u00f3dulo, nos ocuparemos de la parte anal\u00edtica de la teor\u00eda de portafolios, la cual puede ser resumida en dos frases:\n - No pain, no gain.\n - No ponga todo el blanquillo en una sola bolsa.\n \n\n**Objetivos:**\n- \u00bfQu\u00e9 es la l\u00ednea de asignaci\u00f3n de capital?\n- \u00bfQu\u00e9 es el radio de Sharpe?\n- \u00bfC\u00f3mo deber\u00edamos asignar nuestro capital entre un activo riesgoso y un activo libre de riesgo?\n\n*Referencia:*\n- Notas del curso \"Portfolio Selection and Risk Management\", Rice University, disponible en Coursera.\n___ \n\n## 1. L\u00ednea de asignaci\u00f3n de capital\n\n### 1.1. Motivaci\u00f3n\n\nEl proceso de construcci\u00f3n de un portafolio tiene entonces los siguientes dos pasos:\n1. Escoger un portafolio de activos riesgosos.\n2. Decidir qu\u00e9 tanto de tu riqueza invertir\u00e1s en el portafolio y qu\u00e9 tanto invertir\u00e1s en activos libres de riesgo.\n\nAl paso 2 lo llamamos **decisi\u00f3n de asignaci\u00f3n de activos**.\n\nPreguntas importantes:\n1. \u00bfQu\u00e9 es el portafolio \u00f3ptimo de activos riesgosos?\n - \u00bfCu\u00e1l es el mejor portafolio de activos riesgosos?\n - Es un portafolio eficiente en media-varianza.\n2. \u00bfQu\u00e9 es la distribuci\u00f3n \u00f3ptima de activos?\n - \u00bfC\u00f3mo deber\u00edamos distribuir nuestra riqueza entre el portafolo riesgoso \u00f3ptimo y el activo libre de riesgo?\n - Concepto de **l\u00ednea de asignaci\u00f3n de capital**.\n - Concepto de **radio de Sharpe**.\n\nDos suposiciones importantes:\n- Funciones de utilidad media-varianza.\n- Inversionista averso al riesgo.\n\nLa idea sorprendente que saldr\u00e1 de este an\u00e1lisis, es que cualquiera que sea la actitud del inversionista de cara al riesgo, el mejor portafolio de activos riesgosos es id\u00e9ntico para todos los inversionistas.\n\nLo que nos importar\u00e1 a cada uno de nosotros en particular, es simplemente la desici\u00f3n \u00f3ptima de asignaci\u00f3n de activos.\n___\n\n### 1.2. L\u00ednea de asignaci\u00f3n de capital\n\nSean:\n- $r_s$ el rendimiento del activo riesgoso,\n- $r_f$ el rendimiento libre de riesgo, y\n- $w$ la fracci\u00f3n invertida en el activo riesgoso.\n\n Realizar deducci\u00f3n de la l\u00ednea de asignaci\u00f3n de capital en el tablero.\n\n**Tres doritos despu\u00e9s...**\n\n#### L\u00ednea de asignaci\u00f3n de capital (LAC):\n$E[r_p]$ se relaciona con $\\sigma_p$ de manera af\u00edn. Es decir, mediante la ecuaci\u00f3n de una recta:\n\n$$E[r_p]=r_f+\\frac{E[r_s-r_f]}{\\sigma_s}\\sigma_p.$$\n\n- La pendiente de la LAC es el radio de Sharpe $\\frac{E[r_s-r_f]}{\\sigma_s}=\\frac{E[r_s]-r_f}{\\sigma_s}$,\n- el cual nos dice qu\u00e9 tanto rendimiento obtenemos por unidad de riesgo asumido en la tenencia del activo (portafolio) riesgoso.\n\nAhora, la pregunta es, \u00bfd\u00f3nde sobre esta l\u00ednea queremos estar?\n___\n\n### 1.3. Resolviendo para la asignaci\u00f3n \u00f3ptima de capital\n\nRecapitulando de la clase pasada, tenemos las curvas de indiferencia: **queremos estar en la curva de indiferencia m\u00e1s alta posible, que sea tangente a la LAC**.\n\n Ver en el tablero.\n\nAnal\u00edticamente, el problema es\n\n$$\\max_{w} \\quad E[U(r_p)]\\equiv\\max_{w} \\quad E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2,$$\n\ndonde los puntos $(\\sigma_p,E[r_p])$ se restringen a estar en la LAC, esto es $E[r_p]=r_f+\\frac{E[r_s-r_f]}{\\sigma_s}\\sigma_p$ y $\\sigma_p=w\\sigma_s$. Entonces el problema anterior se puede escribir de la siguiente manera:\n\n$$\\max_{w} \\quad r_f+wE[r_s-r_f]-\\frac{1}{2}\\gamma w^2\\sigma_s^2.$$\n\n Encontrar la $w$ que maximiza la anterior expresi\u00f3n en el tablero.\n\n**Tres doritos despu\u00e9s...**\n\nLa soluci\u00f3n es entonces:\n\n$$w^\\ast=\\frac{E[r_s-r_f]}{\\gamma\\sigma_s^2}.$$\n\nDe manera intuitiva:\n- $w^\\ast\\propto E[r_s-r_f]$: a m\u00e1s exceso de rendimiento que se obtenga del activo riesgoso, m\u00e1s querremos invertir en \u00e9l.\n- $w^\\ast\\propto \\frac{1}{\\gamma}$: mientras m\u00e1s averso al riesgo seas, menos querr\u00e1s invertir en el activo riesgoso.\n- $w^\\ast\\propto \\frac{1}{\\sigma_s^2}$: mientras m\u00e1s riesgoso sea el activo, menos querr\u00e1s invertir en \u00e9l.\n___\n\n## 2. Ejemplo de asignaci\u00f3n \u00f3ptima de capital: acciones y billetes de EU\n\nPongamos algunos n\u00fameros con algunos datos, para ilustrar la derivaci\u00f3n que acabamos de hacer.\n\nEn este caso, consideraremos:\n- **Portafolio riesgoso**: mercado de acciones de EU (representados en alg\u00fan \u00edndice de mercado como el S&P500).\n- **Activo libre de riesgo**: billetes del departamento de tesorer\u00eda de EU (T-bills).\n\nTenemos los siguientes datos:\n\n$$E[r_{US}]=11.9\\%,\\quad \\sigma_{US}=19.15\\%, \\quad r_f=1\\%.$$\n\nRecordamos que podemos escribir la expresi\u00f3n de la LAC como:\n\n\\begin{align}\nE[r_p]&=r_f+\\left[\\frac{E[r_{US}-r_f]}{\\sigma_{US}}\\right]\\sigma_p\\\\\n &=0.01+\\text{S.R.}\\sigma_p,\n\\end{align}\n\ndonde $\\text{S.R}=\\frac{0.119-0.01}{0.1915}\\approx0.569$ es el radio de Sharpe (\u00bfqu\u00e9 es lo que es esto?).\n\nGrafiquemos la LAC con estos datos reales:\n\n\n```python\n# Importamos librer\u00edas que vamos a utilizar\nimport numpy as np\nimport matplotlib.pyplot as plt\n%matplotlib inline\n```\n\n\n```python\n# Datos\nErs = 0.119\nss = 0.1915\nrf = 0.01\n# Radio de Sharpe para este activo\nSR = (Ers - rf) / ss\n# Vector de volatilidades del portafolio (sugerido: 0% a 50%)\nsp = np.linspace(0, 0.5, 100)\n# LAC\nErp = rf + SR * sp\n```\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(6, 4))\nplt.plot(sp, Erp, lw=3, label='LAC')\nplt.plot(0, rf, 'ob', ms=5, label='Libre de riesgo')\nplt.plot(ss, Ers, 'or', ms=5, label='Portafolio/activo riesgoso')\nplt.legend(loc='best')\nplt.xlabel('Volatilidad $\\sigma$')\nplt.ylabel('Rendimiento esperado $E[r]$')\nplt.grid()\n```\n\nBueno, y \u00bfen qu\u00e9 punto de esta l\u00ednea querr\u00edamos estar?\n- Pues ya vimos que depende de tus preferencias.\n- En particular, de tu actitud de cara al riesgo, medido por tu coeficiente de aversi\u00f3n al riesgo.\n\nSoluci\u00f3n al problema de asignaci\u00f3n \u00f3ptima de capital:\n\n$$\\max_{w} \\quad E[U(r_p)]$$\n\n$$w^\\ast=\\frac{E[r_s-r_f]}{\\gamma\\sigma_s^2}$$\n\nDado que ya tenemos datos, podemos intentar para varios coeficientes de aversi\u00f3n al riesgo:\n\n\n```python\n# importar pandas\nimport pandas as pd\n```\n\n\n```python\n# Crear un DataFrame con los pesos, rendimiento\n# esperado y volatilidad del portafolio \u00f3ptimo \n# entre los activos riesgoso y libre de riesgo\n# cuyo \u00edndice sean los coeficientes de aversi\u00f3n\n# al riesgo del 1 al 10 (enteros)\ngamma = np.arange(1, 11)\nw = (Ers - rf) / (gamma * ss**2)\npd.DataFrame({'$\\gamma$': gamma, '$w$': w})\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
$\\gamma$$w$
012.972275
121.486137
230.990758
340.743069
450.594455
560.495379
670.424611
780.371534
890.330253
9100.297227
\n
\n\n\n\n\u00bfC\u00f3mo se interpreta $w^\\ast>1$?\n- Cuando $01$, tenemos $1-w^\\ast<0$. Lo anterior implica una posici\u00f3n corta en el activo libre de riesgo (suponiendo que se puede) y una posici\u00f3n larga (de m\u00e1s del 100%) en el mercado de activos: apalancamiento.\n\n# Anuncios parroquiales.\n\n## 1. Quiz la siguiente clase.\n\n## 2. Pueden consultar sus calificaciones en el siguiente [enlace](https://docs.google.com/spreadsheets/d/1qDJGx731VGLk_LlxrYibolp4UwIMMNCsiQDTnm4QWew/edit?usp=sharing)\n\n\n\n\n
\nCreated with Jupyter by Esteban Jim\u00e9nez Rodr\u00edguez.\n
\n", "meta": {"hexsha": "6aa62c8f60b3259d5c2c7e74f3de76fa88118f32", "size": 38399, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Modulo3/Clase12_OptimizacionMediaVarianza.ipynb", "max_stars_repo_name": "Noesns/porinvp2020", "max_stars_repo_head_hexsha": "a4ddfecc3b1aa75ff9ce0c7f2708fcfc75e9ff97", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Modulo3/Clase12_OptimizacionMediaVarianza.ipynb", "max_issues_repo_name": "Noesns/porinvp2020", "max_issues_repo_head_hexsha": "a4ddfecc3b1aa75ff9ce0c7f2708fcfc75e9ff97", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Modulo3/Clase12_OptimizacionMediaVarianza.ipynb", "max_forks_repo_name": "Noesns/porinvp2020", "max_forks_repo_head_hexsha": "a4ddfecc3b1aa75ff9ce0c7f2708fcfc75e9ff97", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-03-03T18:17:18.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-03T18:17:18.000Z", "avg_line_length": 77.2615694165, "max_line_length": 21972, "alphanum_fraction": 0.7864788146, "converted": true, "num_tokens": 3206, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.831143054132195, "lm_q1q2_score": 0.44474334223006023}} {"text": "# Background\n\nThis post will introduce data scientists who are interested in cryptocurrency exchange models that can be used to predict buy and sell opportunities based on several strategies with in depth descriptions and access to code that is being used to simulate a variety of models with varying performance.\n\n## What is Forex\n\nForex is short for foreign currency exchange and is the trading of currencies with the goal making profits by timing the buy and sell of specific currency pairs while using candlestick charts. Strategies for trading are created by looking for patterns that can be used to predict future currency exchange price fluctuations. \n\n## Candlestick Charts\n\nA candlestick chart is the standard plot used in visualizing trading activity where a candle is represented by a box plot that visualizes 4 prices within a given period: the high, low, open and close price. The box, or body of the candle, is colored based on if the open price is greater than the close and differently if vice versa. In the below chart, a white candlestick means the close price is higher than the open price meaning the price is going up. The lines coming out of the candlestick body are called \"shadows\" or \"wicks\" and represent the price spread for the given period by extending out to the high and low price. An individual candlestick can represent a period as short as a second to days or weeks or more. The chart below is a 15 minute candlestick chart so each candlestick represents a 15 minute period.\n\n

\n
Figure X - Here is an example 15-minute candlestick chart for the Ethereum/Bitcoin cryptocurrency exchange rate.
This visualization was rendered using the Python library mplfinance.
\n\n# Feature Engineering\n\n## Lookbacks\n\nEach record in the dataset represents a single candlestick's data. Since context likely matters, a lookback feature is the candlestick data for a previous candlestick included with the current record. A lookback can be for any feature already in the dataset. For example, if a candlestick data is represented just by `open`, `close`, `high`, `low` price data. A lookback of 1 could be something like this for a 2-record dataset:\n\n|index|open|close|high|low|open_1|close_1|high_1|low_1|\n|:-|-:|-:|-:|-:|-:|-:|-:|-:|\n|2021-07-29 01:00:00|0.057756|0.057657|0.057776|0.057576|--|--|--|--|\n|2021-07-29 01:00:00|0.057657|0.057697|0.057788|0.057612|0.057756|0.057657|0.057776|0.057576|\n\n\n## Indicators\n\nIndicators are non-Boolean numerical features that are calculated based on a window of data.\n\n### Moving Averages (MA)\n\nMoving averages are common indicators used in forex and can give insight into price trend or momentum by smoothing out the variations. A moving average operates on a window which represents the average over a given number of records. A typical value of `14` is used in forex and for this work three are used: `14`, `30` and `90`.\n\nThe equation for moving average is the following where *k* is the window size on a dataset containing *N* items:\n\n

\n\n### Average True Range (ATR)\n\nThe ATR is an indicator of volatility in the market. A higher ATR and the price is more likely to jump around which can result in higher gains but at higher risk. It is calculated by looking at the true range (TR) of the current record and then averaging out over a window typically of size `14`. This is also useful determining target and stop loss prices which will be discussed in the different strategies.\n\n

\n\n### Relative Strength Index (RSI)\n\nThe RSI is a momentum indicator that measures the magnitude of recent price changes. The value is limited between `0` and `100`. Traditionally, if the RSI value is above `70`, it is considered \"overvalued\" and the price may soon reverse. A value below `30` is thus considered \"undervalued\". The calculation for this is more complicated. First the relative strength is found by comparing the exponential moving averages of the positive differences in closing price divided by the exponential moving averages of the negative differences in the closing price. Then the value is normalized into an index between `0` and `100`.\n\n

\n\n### Support and Resistance Lines\n\nSupport and resistance are levels that act as barriers that prevent a price from either dropping below (support) or raising above (resistance) an expected price. The levels can be horizontal or they can represent a trendline. While there is no agreed formula, this feature dataset will use the `min` and `max` in a given window. Again, three windows will be used resulting in three features `14`, `30` and `90`.\n\n

\n\n## Signals\n\nSignals are Boolean features represented by either a `1` (True) or `0` (False).\n\n### Upward Trend\n\nIs the current record within the context of an upward trend? To calculate this, the difference in moving average is used for a given window.\n\n

\n\n### Candlestick Patterns\n\nThere are numerous candlestick patterns and each pattern tells a story of how the market is behaving and what that might mean for the future. Their names often derive from their shape and the table below provides links to more information about the ones supported in this dataset. The candlestick pattern signals are implemented according to the below algorithms.\n\n|Pattern Name|Indication|Image|Algorithm|\n|:-|:-|:-:|:-|\n|Shooting Star|Bearish Reversal|| |\n|Hammer|Bullish Reversal|| |\n|Bearish Harami|Bearish Reversal|| |\n|Bullish Harami|Bullish Reversal|| |\n|Engulfing Bullish|Bullish Reversal|| |\n|Engulfing Bearish|Bearish Reversal|| |\n\n## Other Features\n\n### Date and Time\n\nIt's possible that certain days-of-the-week (DOW) or time-of-the-day (TOD) impact trading trends. For this reason, DOW and TOD are added as features in a numerical format based on the candlestick's open time. The DOW is represented by a number between `0` and `6` and the TOD is normalized between `0` and `1`.\n\n### Trading Volume\n\nWithin the candlestick data that is coming from Binance there are five features that relate to trading volume. This data is included, unaltered, as available features. They include `number_of_trades`, `volume`, `quote_asset_volume`, `taker_buy_base_asset_volume`, and `taker_buy_quote_asset_volume`.\n\n\\begin{align}\n\\text{MA} &= \\frac{1}{k}\\sum_{i=n-k+1}^{N} p_{i}\n\\end{align}\n\n\\begin{align}\n\\text{TR} &= \\text{max}\\left[\\left(\\text{high}-\\text{low}\\right), \\text{abs}\\left(\\text{high}-\\text{close}_{\\text{prev}}\\right), \\text{abs}\\left(\\text{low}-\\text{close}_{\\text{prev}}\\right)\\right]\n\\\\\n&= \\text{max}\\left[\\left(\\text{high}, \\text{close}_{\\text{prev}}\\right)\\right] - \\text{min}\\left[\\left(\\text{low}, \\text{close}_{\\text{prev}}\\right)\\right]\n\\\\\n\\text{ATR} &= \\frac{1}{k}\\sum_{i=n-k+1}^{N} \\text{TR}_{i}\n\\end{align}\n\n\\begin{align}\nU &= \\begin{cases}\n\\text{close}-\\text{close}_{\\text{prev}} & \\text{if close}_{\\text{prev}} < \\text{close} \\\\\n0 & \\text{if close}_{\\text{prev}} \\geq \\text{close}\n\\end{cases}\n\\\\\n\\\\\nD &= \\begin{cases}\n\\text{close}_{\\text{prev}}-\\text{close} & \\text{if close}_{\\text{prev}} > \\text{close} \\\\\n0 & \\text{if close}_{\\text{prev}} \\leq \\text{close}\n\\end{cases}\n\\\\\n\\\\\n\\text{EMA}_{i} &= \\left[p_{i} \\times \\left(\\frac{s}{1+d}\\right)\\right] + \\text{EMA}_{i-1} \\times \\left[1-\\left(\\frac{s}{1+d}\\right)\\right]\n\\\\\n\\text{RS} &= \\frac{\\text{EMA}_{U}}{\\text{EMA}_{D}}\n\\\\\n\\text{RSI} &= 100-\\frac{100}{1+RS}\n\\end{align}\n\n\\begin{align}\n\\text{support} &= \\text{min}\\left[\\text{low}_{n-k+1}, \\text{low}_{n-k+2} \\ldots, \\text{low}_{n}\\right]\n\\\\\n\\text{resistence} &= \\text{max}\\left[\\text{high}_{n-k+1}, \\text{high}_{n-k+2} \\ldots, \\text{high}_{n}\\right]\n\\end{align}\n\n\\begin{align}\nt_{j} &= \\text{MA}_{j} - \\text{MA}_{j-1}\n\\\\\nr_{j} &= \\begin{cases}\n1 & \\text{if } t_{j} > 0 \\\\\n0 & \\text{if } t_{j} \\leq 0 \\\\\n\\end{cases}\n\\end{align}\n\n\\begin{align}\nr_{\\text{ss}} &= \\begin{cases}\n1 & \\text{if} \\text{ low} + 0.382\\left(\\text{high}-\\text{low}\\right) \\geq \\text{close } \\tiny{\\text{and}} \\\\\n & \\hspace{0.45cm} \\text{open } \\leq \\text{close} \\\\\n0 & \\text{else} \\\\\n\\end{cases}\n\\end{align}\n\n\\begin{align}\nr_{\\text{ssr}} &= \\begin{cases}\n1 & \\text{if } \\text{low} + 0.382\\left(\\text{high}-\\text{low}\\right) \\geq \\text{open } \\tiny{\\text{and}} \\\\\n & \\hspace{0.45cm} \\text{open } > \\text{close} \\\\\n0 & \\text{else} \\\\\n\\end{cases}\n\\end{align}\n\n\\begin{align}\nr_{\\text{hm}} &= \\begin{cases}\n1 & \\text{if } \\text{high} - 0.382\\left(\\text{high}-\\text{low}\\right) \\leq \\text{close } \\tiny{\\text{and}} \\\\\n & \\hspace{0.45cm} \\text{open } \\geq \\text{close} \\\\\n0 & \\text{else} \\\\\n\\end{cases}\n\\end{align}\n\n\\begin{align}\nr_{\\text{hmr}} &= \\begin{cases}\n1 & \\text{if } \\text{high} - 0.382\\left(\\text{high}-\\text{low}\\right) \\leq \\text{open } \\tiny{\\text{and}} \\\\\n & \\hspace{0.45cm} \\text{open} < \\text{close} \\\\\n0 & \\text{else} \\\\\n\\end{cases}\n\\end{align}\n\n\\begin{align}\nr_{\\text{brh}} &= \\begin{cases}\n1 & \\text{if } \\text{open} > \\text{close} \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{open}_{\\text{prev}} < \\text{close}_{\\text{prev}} \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{close}_{\\text{prev}} \\geq \\text{open } \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{open}_{\\text{prev}} \\leq \\text{close} \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{high}_{\\text{prev}} \\geq \\text{high} \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{low}_{\\text{prev}} \\leq \\text{low} \\\\\n0 & \\text{else} \\\\\n\\end{cases}\n\\end{align}\n\n\\begin{align}\nr_{\\text{buh}} &= \\begin{cases}\n1 & \\text{if } \\text{open} < \\text{close} \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{open}_{\\text{prev}} > \\text{close}_{\\text{prev}} \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{close}_{\\text{prev}} \\leq \\text{open } \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{open}_{\\text{prev}} \\geq \\text{close} \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{high}_{\\text{prev}} \\geq \\text{high} \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{low}_{\\text{prev}} \\leq \\text{low} \\\\\n0 & \\text{else} \\\\\n\\end{cases}\n\\end{align}\n\n\\begin{align}\nr_{\\text{ebu}} &= \\begin{cases}\n1 & \\text{if } \\text{open} < \\text{close} \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{open}_{\\text{prev}} > \\text{close}_{\\text{prev}} \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{close}_{\\text{prev}} \\geq \\text{open } \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{open}_{\\text{prev}} \\leq \\text{close} \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{high}_{\\text{prev}} \\leq \\text{high} \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{low}_{\\text{prev}} \\geq \\text{low} \\\\\n0 & \\text{else} \\\\\n\\end{cases}\n\\end{align}\n\n\\begin{align}\nr_{\\text{ebr}} &= \\begin{cases}\n1 & \\text{if } \\text{open} > \\text{close} \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{open}_{\\text{prev}} < \\text{close}_{\\text{prev}} \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{close}_{\\text{prev}} \\leq \\text{open } \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{open}_{\\text{prev}} \\geq \\text{close} \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{high}_{\\text{prev}} \\leq \\text{high} \\hspace{0.2cm}\\tiny{\\text{and}} \\\\\n &\\hspace{0.4cm}\\text{low}_{\\text{prev}} \\geq \\text{low} \\\\\n0 & \\text{else} \\\\\n\\end{cases}\n\\end{align}\n\n# Strategies\n\n## Target and Stop Loss\n\nIn traditional forex trading, stop and limit orders are methods to protect an investor that can be used to buy and sell currencies when a price reaches a certain level. Using this, a predictive model can focus only on buy opportunities and then rely on a simple strategy to determine when to sell. A sell strategy defines two sell prices for a given buy opportunity. The first sell price is called the **target**, which is the high price that results in a profit and the next is the **stop loss**, which is the low resulting in a loss. When a buy opportunity is identified, and a target and stop loss is calculated, the purchase can be made and the sell will be automatic either by the exchange or by another system that monitors the market price.\n\nIn the example below, a buy opportunity is identified at the close of the 4:00am candlestick at a price of `0.060497` Bitcoin (`BTC`) per 1.0 Etherum (`ETH`). Buying ETH at this price, a target and stop loss is calculated with a `1.0% : 0.5%` ratio, thus `0.061102` for a target and `0.060195` for a stop loss. The price reaches the target price eight candlesticks later or 2 hours later at 6:00am, thus securing `1.0%` profit (assuming no fees).\n\n

\n
Figure X - Example ETH buy opportunity.
\n\n### Identifying Buying Opportunities\n\nUsing a target and stop loss approach simplifies the model to a binary classification problem but a new problem is created, there is no labeled data to train on. The goal here is to create a label for each record. A record being the data for one candlestick. This includes the low, high, open and close prices as well as some additional features such as volume and number of trades. Using the close price for each record, a target and stop loss price is calculated using the same threshold ratio that will be used on the deployed model. Using the example above, a ratio of `1.0% : 0.5%` returns a target price `1.0%` higher than the close price and a stop loss of `0.5%` below the close price. The next step is to peek into the future and see what happens first. Does the price reach the target price first or the stop loss? If it reaches the target, the record's label will be a `1` meaning \"buy\". Another consideration is how far in the future it should look. This is called the \"window\". Typically, 15 candles in the future is used. If the price reaches stop loss first or if price hovers between the target and stop loss within the window, the record will be a `0` meaning \"not buy\".\n\nA common question is why not make the stop loss as small as possible? Setting the stop loss too small can result in being \"wicked out\" of a trade. Looking at figure above, if the stoploss is made too small, the wick of the next candle after the buy could poke through resulting in the stop loss being breached before the target price, thus resulting in a \"not buy\". Therefore, setting a higher stop loss gives some buffer for the price to fluctuate before a gain is achieved while minimizing losses.\n\nFor the remainder of the target stop loss strategy discussion, the strategy will focus on `BTC` buy opportunities with the starting coin being `ETH`. In other words, `ETH` will be used to buy `BTC` and will be sold back to `ETH` when the price reaches a target or stop loss price. This can cause a bit of confusion because the price is the number of BTC within 1 ETH which means, a profit is made when the price actually drops thus the target will be lower than stop loss (opposite the figure above). It is for this reason the `reverse` flag is set to `True` (see below).\n\n### Determining Ideal Ratios\n\nIn the example above, a `1.0% : 0.5%` ratio is used but is this a good ratio to use? Setting a ratio of `10% : 5%` might be too high because it would be unlikely to gain `10%` resulting in a very sparsely labeled dataset. Likewise, using a ratio of `0.1% : 0.005%` could be too low, especially when considering transaction fees (to be discussed later). It's also worth mentioning that using a percentage might result in inconsistencies since some currency pairs are more volatile than others and volatility for a given pair can change over time. For this reason, forex traders sometimes use a ratio of the ATR. For example, using an ATR `2:1` ratio is a good place to start.\n\nModels generally perform better on balanced data so getting half of the labels to be `1` is ideal. But achieving this with a ratio that is consistent and profitable may not be practical. To find a good ratio, different multiples are generated and the percent of `1`'s is plotted. On the below ATR ratio figure, the multiple of `2x` means the numerator is `2` times the denominator, where the denominator is the `x-axis` value. Therefore, when `x-axis = 3` the ratio is `6:3`. When the multiple is `4x` and `x-axis = 2`, the ratio is `8:2`, etc. For the percentage ratio, `x-axis` represents the numerator, and the denominator is then the numerator divided by the legend's label. For example, when `x-axis = 0.01` for the `/2` line, the ratio is `1.0% : 0.5%`.\n\n

\n
Figure X - Finding the best ratios to maximize label data using a window of 30 on ETHBTC 15-minute candles.
\n\nUnsurprisingly, as the ratio grows or ratio multiple grows, fewer buy opportunities can be found in the data because there are fewer windows where high profits can be achieved and fewer windows where smaller stop losses don't get wicked out by the volatility of the market. None of the plots reaches the goal of `50%` but the results provide plenty of options to avoid sparsely labeled data. From this analysis, the ATR Ratio is maximized at `2:1` with approximately `40%` of the labels being `1`. The percentage ratio is maximized at `1.0% : 0.5%` with approximately `27%` of the labels being `1`.\n\n### Building Xy Datasets\n\nThe imbalance of a dataset is not the only criteria for determining if one ratio is better than another, but it does give a sense. To explore this further, several labeled datasets are generated with different labelling strategies by changing ratios, the window, and the whether the ratio represents a percentage or ATR ratio. The following table shows 12 different labeled datasets generated that are used to compare different model performances.\n\n|dataset|use_atr|ratio|reverse|window|size|true_labels|imbalance|train_imbal|test_imbal|\n|:-|:-:|:-:|:-:|:-:|-:|-:|-:|-:|-:|\n|20210806a|False|(0.01, 0.005)|True|30|141174|37979|0.269023|0.262033|0.319201|\n|20210806b|False|(0.01, 0.0025)|True|30|141174|23782|0.168459|0.169634|0.196954|\n|20210806c|False|(0.0075, 0.0025)|True|30|141174|29824|0.211257|0.220094|0.233808|\n|20210806d|True|(2, 1)|True|15|141174|44769|0.317119|0.317522|0.341973|\n|20210806e|True|(4, 2)|True|15|141174|17024|0.120589|0.123430|0.124245|\n|20210806f|True|(4, 1)|True|15|141174|15304|0.108405|0.110710|0.111111|\n|20210806g|True|(4, 2)|True|30|141174|31640|0.224121|0.227442|0.238302|\n|20210806h|True|(4, 1)|True|30|141174|26488|0.187627|0.189435|0.201498|\n|20210806i|True|(2, 1)|True|30|141174|55315|0.391821|0.391990|0.409738|\n|20210806j|False|(0.01, 0.005)|True|15|141174|29035|0.205668|0.187259|0.265218|\n|20210806k|False|(0.01, 0.0025)|True|15|141174|19184|0.135889|0.129528|0.174831|\n|20210806l|False|(0.0075, 0.0025)|True|15|141174|25995|0.184134|0.183938|0.224619|\n\n\nIt is during this time that feature engineering was initially performed but later not necessary since the AWS pipeline has already completed this step (discussed later). For this reason, each dataset already includes all the features discussed in the feature engineering section plus `14` lookbacks resulting in `542` features (a lookback is the previous records features).\n\nEach dataset is also split into train and validation sets according to the table below.\n\n|Purpose|Start Date|End Date|Number of Records|\n|:-|:-:|:-:|-:|\n|*not used*|--|2017/12/31|16353|\n|Train|2018/01/01|2020/12/31|104796|\n|Validation|2021/01/01|2021/07/29|20025|\n|Test|2021/07/30|--|--|\n\nIn 2017, Binance started reporting figures and it took some time for these to develop robustness. For this reason, 2017 is excluded from the training and testing sets. The final test set analysis is performed on AWS in simulation using near-live data.\n\n### Simulating Trades on Labeled Data\n\nTo get a sense of what an ideal profit would look like for each of the labelling strategies, it is necessary to run them through a simulator. Why is this necessary? Why can't this be calculated from the above figures? To answer this, imagine having two candles, one after another, where the labeling has marked both as `1`. In a deployed model, when a buy signal is received, all available currency will be spent. When evaluating the next candlestick data, the model will be evaluating for selling, not buying, so that candlestick will not be evaluated for a buy opportunity.\n\nThe simulator logic is the same logic as in the deployed model pipeline but instead of looking at the predictions, it looks at the validation labeled data which is already guaranteed to contain profitable trades assuming it uses the same ratio and other hyperparameters as the dataset's labelling strategy. The simulator works, in short, by progressing through the labeled data, looking for the next buy opportunity, calculates the target and stop loss prices, finds the next record that surpasses one of these, calculating the profit/loss along with any fees, and then repeats the process until it reaches the end. The last sell indicates the maximized profit achievable for the dataset. The below table shows how each dataset's number of trades and maximized profit based on a starting value of `1 ETH` and a fee of `0.1%` for each buy or sell transaction using the validation data.\n\n|dataset|sim_num_trades|sim_max_profit|sim_bad_trades|\n|:-|-:|-:|-:|\n|20210806a|1179.0|13228.993600|0.0|\n|20210806b|1093.0|6620.481356|0.0|\n|20210806c|1456.0|3126.592203|0.0|\n|20210806d|1128.0|17342.974458|1.0|\n|20210806e|335.0|416.560613|0.0|\n|20210806f|340.0|467.521090|1.0|\n|20210806g|368.0|1084.998178|0.0|\n|20210806h|376.0|1295.216356|2.0|\n|20210806i|1033.0|9843.078247|2.0|\n|20210806j|1199.0|15539.693313|0.0|\n|20210806k|1097.0|6837.112001|0.0|\n|20210806l|1497.0|3921.842442|0.0|\n\nWhile the label data guarantees a trade is profitable, it doesn't guarantee the profit surpasses the fee amount. For this reason, some labels result in bad trades. In the above table, only the ATR ratio datasets result in bad trades which makes sense since all the percentage-based datasets are larger than the fee. Keeping an eye on this number will be important when using an ATR ratio dataset.\n\n### Comparing Datasets with Base Classifiers\n\nFor each of the datasets, a set of base classifiers are trained. Below is a table of the base classifiers used.\n\n|Name|Parameters|\n|:-|:-|\n|GaussianNB|*none*|\n|LogisticRegression|`random_state=42, max_iter=10000`|\n|RandomForestClassifier|`random_state=42, n_jobs=-1`|\n|AdaBoostClassifier|`random_state=42`|\n|GradientBoostingClassifier|`random_state=42`|\n|XGBClassifier|`n_jobs=-1, random_state=42, use_label_encoder=False`|\n|MLPClassifier|`random_state=42`|\n\nOne note about the `MLPClassifier`: since this classifier is sensitive to scaling, `make_pipeline()` with `StandardScaler()` is used.\n\nThis exercise determines the F1-score, precision, recall and the simulator's profit on the validation set for each dataset/classifier combination resulting in `84` results. This was performed two more times on the same datasets and classifiers but first reducing the number of lookbacks from `14` to `3` and then to `0` thus reducing the number of features each time and ending up with a total of 252 trained model results.\n\n### Identifying Best Performing Dataset/Classifier Combinations\n\nWhen it comes to ranking best performance, precision is a good starting point. A high precision means the model has reduced the number of false positives (FP). In other words, it reduced the chance of predicting a buy that is unprofitable--the absolute worst case that should be avoided. A low recall, on the other hand, just means the model is predicting fewer buy opportunities than expected. This is generally fine so long as it does predict buys often enough (one true positive every couple of days on average).\n\nLooking at precision alone can be misleading. On the extreme side, a precision of `1.0` is perfect precision but if recall was very low, such as having only one `1` true positive (TP), the model would be ineffective since it so rarely makes predictions. For this reason, the F1-score is not a good show of performance, and several factors must be considered. Maximizing precision and the number of TPs is the overall goal. Ranking based on the number TPs does little in the way of explaining performance if the number of FPs is still high. Therefore, ranking is performed first on precision, second on the difference between TPs and FPs and then on the ratio between. The top 10 models based on this ranking is shown in the table below.\n\n|Rank|Classifier|Dataset|Lookbacks|TP|FP|Diff|Ratio|Precision|Recall|Sim. Profit|\n|-:|:-|:-|-:|-:|-:|-:|-:|-:|-:|-:|\n|1|LogisticRegression|20210806i|0|453|277|176|1.64|0.6204|0.0082|1.3210|\n|2|AdaBoostClassifier|20210806g|14|98|13|85|7.54|0.8824|0.0031|1.0243|\n|3|LogisticRegression|20210806i|3|840|631|209|1.33|0.5708|0.0152|1.6651|\n|4|RandomForestClassifier|20210806g|14|2075|1856|219|1.12|0.5278|0.0656|1.2348|\n|5|LogisticRegression|20210806i|14|1731|1535|196|1.13|0.5299|0.0313|1.0503|\n|6|LogisticRegression|20210806d|0|40|26|14|1.54|0.6000|0.0009|1.0045|\n|7|GradientBoostingClassifier|20210806d|0|58|38|20|1.53|0.6000|0.0013|1.0126|\n|8|GradientBoostingClassifier|20210806i|3|2710|2460|250|1.10|0.5241|0.0490|0.6239|\n|9|GradientBoostingClassifier|20210806i|14|1919|1770|149|1.08|0.5201|0.0347|0.8400|\n|10|AdaBoostClassifier|20210806a|14|60|54|6|1.11|0.5263|0.0016|0.9951|\n\n\nReviewing the simulated profit for each of these, the top 7 all produce profits suggesting the ranking is robust. The highest performing models have then been deployed to AWS for live simulations which is discussed below.\n\n### Building a Logistic Regression Ensemble\n\nMany logistic regression models outperformed other models and are easy to train and tune so it seems logical to ask if performance could be improved further with an ensemble. To build an ensemble, the prediction from each model in the ensemble is weighted and summed and if that sum is greater than or equal to some threshold, the prediction would be considered a `1` or else `0`. The weights will be the precision of each model on the validation set. The equation for this can be explained as follows:\n\n

\n\nWhere *r_j* is the prediction result for the *j*th record, *p_m[i]* is the prediction of *i*th model in *m*, *c_m[i]* is the precision of said model, and *t* is the hyperparameter threshold. Any model that has a validation set precision of `0` would always be zeroed out so these models will not be included in the ensemble.\n\nFinding a good value for *t* can be achieved by trying out by measuring the precision, recall and simulated profit. Another issue that needs to be considered is that the datasets use different ratios so which ratio should be used on the ensemble? It stands to reason that the model/dataset with the highest precision should be used since that carries the most weight. However, simulations show this is not always the case as will be shown later with the scaled version shown later. For the logistic regression, it so happens that these are aligned. In the below figure, profit is maximized at threshold of `0.12` with a value of `1.86` which surpasses any individual model simulation performance.\n\n

\n
Figure X - Simulated precision, recall and profit for varying thresholds on a Logistic Regression ensemble.
\n\n### Scaling Data in Isolation\n\nThere is a forex theory that a good strategy is generalizable, in that it can be applied to any currency pair, even opposite pairs, and be profitable. All models previously train (except for the MLP) have not been scaled so it is impractical to expect one of these models perform well for both ETH to BTC and the opposite BTC to ETH. Likewise, using standard scaling, like done for the MLP, is also not practical since the dataset for BTC to ETH trades is scaled very differently. So, can the data be scaled in isolation? The answer is yes, but at the cost of zeroing-out one of the features. By defining the open price as the mean and using the close, high, and low in the calculation of a standard deviation, all price data can be scaled such that one standard deviation difference is -1 or 1. The formula can be described as the following:\n\n

\n\nUsing this scaling algorithm, each record is individually scaled independent of the other data in the dataset. Repeating the same model/dataset comparison using this scaler produces another 252 trained models. Many of the logistic regression models returned profits in simulation of `ETHBTC` suggesting again an ensemble might outperform a single model.\n\n### Ensemble with Custom Scaling\n\nUsing the scaler discussed previously, a new ensemble of logistic regression models can be produced with the goal of having a model that be able to perform on both `ETHBTC` and `BTCETH` trading. Again, a threshold and ratio are brute forced. In the below figure, profit is maximized at threshold of `0.10` with a value of `1.35` again surpassing any individual model simulation performance. This time, the profit is maximized with a ratio of `4:2` in contrast with the highest precision dataset being `2:1`. \n\n

\n
Figure X - Simulated precision, recall and profit for varying thresholds on a scaled Logistic Regression ensemble.
\n\n### Deep Neural Network\n\nTo see how a deep neural network can perform on the same datasets used to train base classifiers, the dataset `20210806i` is chosen for its consistent performance. A variety of ResNet28 models are built using PyTorch and trained with varying hyperparameters or model changes over 20 epochs on a GPU to allow for quicker iteration. A subset of these run configurations is shown in the table below.\n\n|Model Name|Scaler|Width|Optimizer|Learning Rate|\n|:-|:-|-:|:-|:-|\n|nm_torch1_alpha33|CustomScaler1|128|AdamW|0.006|\n|nm_torch1_alpha34|*none*|128|AdamW|0.006|\n|nm_torch1_alpha35|*none*|32|AdamW|0.001|\n|nm_torch1_alpha36|*none*|32|AdamW|0.006|\n|nm_torch1_alpha37|CustomScaler1|32|AdamW|0.006|\n|nm_torch1_alpha38|CustomScaler1|128|AdamW|0.03|\n\nBelow is a figure of the training and validations results of this subset of models. While 20 epochs is still quite young for a comprehensive analysis, some trends do start to appear which only become more pronounced with further epochs and as many as 100 when the training set begins to converge.\n\n

\n
Figure X - Simulated precision, recall and profit for varying thresholds on a scaled Logistic Regression ensemble.
\n\nThe general trend that is consistent across all models is that the training loss drops predictably and consistently but the validation loss steeply rises after just a few epochs. The recall on the validation set generally does slowly improve which also improves the F1-score but the precision remains erratic, averaging around `0.5`. This unexpected behavior is likely due to a shift in the how the `ETHBTC` market behaves over time, so the model is learning a strategy that is no longer profitable in 2021. To validate this, each model was simulated with the results in the below table.\n\n|Model Name|Buys|Starting Value|Ending Value|\n|:-|-:|:-:|:-|\n|nm_torch1_alpha33|3989|1.0|0.3472|\n|nm_torch1_alpha34|0|1.0|0.0|\n|nm_torch1_alpha35|51|1.0|1.0391|\n|nm_torch1_alpha36|0|1.0|0.0|\n|nm_torch1_alpha37|3083|1.0|0.4240|\n|nm_torch1_alpha38|236|1.0|0.9454|\n\n\n### Ensemble\n\\begin{align}\nx_{j} &= \\sum_{i=0}^M \\left( p_{m[i]} \\cdot c_{m[i]}\\right)\n\\\\\nr_{j} &= \\begin{cases}\n1 & \\text{if } x_{j} \\geq t \\\\\n0 & \\text{if } x_{j} < t \\\\\n\\end{cases}\n\\end{align}\n\n\n### Scaler\n\n\\begin{align}\n\\mu_{j} &= p_{j}^\\text{open}\n\\\\\n\\sigma_{j} &= \\sqrt{\\left(p_{j}^\\text{close} - p_{j}^\\text{open}\\right)^2 + \\left(p_{j}^\\text{high} - p_{j}^\\text{open}\\right)^2 + \\left(p_{j}^\\text{low} - p_{j}^\\text{open}\\right)^2}\n\\\\\np_{j} &= \\begin{cases}\n\\frac{\\left(p_{j} - \\mu_{j}\\right)}{\\sigma_{j}} & \\text{if } p_{j} \\text{ is a price} \\\\\n\\frac{p_{j}}{p_{j}^\\text{ATR}} & \\text{if } p_{j} \\text{ is an ATR difference} \\\\\n\\frac{p_{j} - p_{j}^\\text{ATR}}{p_{j}^\\text{ATR}} & \\text{if } p_{j} \\text{ is an ATR lookback} \\\\\n\\frac{p_{j}}{p_{j}^\\text{RSI}} & \\text{if } p_{j} \\text{ is an RSI difference} \\\\\n\\frac{p_{j} - 50}{20} & \\text{if } p_{j} \\text{ is an RSI or RSI lookback} \\\\\n\\end{cases}\n\\end{align}\n\nConverted to images with [LaTeX to Image converter](https://latex2image.joeraut.com/)\n", "meta": {"hexsha": "aff85d3de5c1be092bee5dea259257de0214f9fe", "size": 39163, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "playground/nicks/report/report_sections.ipynb", "max_stars_repo_name": "mads-swaps/swap-for-profit", "max_stars_repo_head_hexsha": "543fe8f5b0a990423f3373f29653d57775ea4c25", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-12-16T15:15:58.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-30T06:10:25.000Z", "max_issues_repo_path": "playground/nicks/report/report_sections.ipynb", "max_issues_repo_name": "mads-swaps/swap-for-profit", "max_issues_repo_head_hexsha": "543fe8f5b0a990423f3373f29653d57775ea4c25", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "playground/nicks/report/report_sections.ipynb", "max_forks_repo_name": "mads-swaps/swap-for-profit", "max_forks_repo_head_hexsha": "543fe8f5b0a990423f3373f29653d57775ea4c25", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 81.41995842, "max_line_length": 1213, "alphanum_fraction": 0.6567934019, "converted": true, "num_tokens": 9601, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548511303336, "lm_q2_score": 0.6825737279551494, "lm_q1q2_score": 0.4445977089577032}} {"text": "```python\nfrom sympy import symbols,Matrix,init_printing\ninit_printing()\n```\n\n\n```python\na,b,c,d,e,f,g,h,i = symbols('a,b,c,d,e,f,g,h,i')\n```\n\n\n```python\nR = Matrix([[a,b,c],[d,e,f],[g,h,i]])\nR\n```\n\n\n\n\n$$\\left[\\begin{matrix}a & b & c\\\\d & e & f\\\\g & h & i\\end{matrix}\\right]$$\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "1aa758c0bdb4dca1ce28c9391a96002126caeb49", "size": 1620, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "python/Cap1/jupyterNotebook/Sympy.ipynb", "max_stars_repo_name": "silvajhonatan/robotics", "max_stars_repo_head_hexsha": "d1097809e88c744658dab6d661092b6ea8f0e13a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2017-11-16T18:34:27.000Z", "max_stars_repo_stars_event_max_datetime": "2021-01-28T15:33:46.000Z", "max_issues_repo_path": "python/Cap1/jupyterNotebook/Sympy.ipynb", "max_issues_repo_name": "sjhonatan/robotics", "max_issues_repo_head_hexsha": "d1097809e88c744658dab6d661092b6ea8f0e13a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "python/Cap1/jupyterNotebook/Sympy.ipynb", "max_forks_repo_name": "sjhonatan/robotics", "max_forks_repo_head_hexsha": "d1097809e88c744658dab6d661092b6ea8f0e13a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 18.202247191, "max_line_length": 91, "alphanum_fraction": 0.4580246914, "converted": true, "num_tokens": 110, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149868676283, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.44456123042008694}} {"text": "

Re:Invent 2019 | IoT-309

\n

Combining IoT and Machine Learning for Predictive Maintenance

\n

\nA multivariate LSTM neural network for prediction future pollution levels to efficiently operate air filters

\n\n# About\n\nPredicting failure, remaining useful life or operating conditions is a classic request of IoT systems. By predicting which devices will fail, proactive maintenance can be scheduled to increase device uptime, optimize asset utilization, avoid costly catastrophic device failure and optimize field service efficiency. In this Notebook template, we will show how to implement a multivariate LSTM algorithm to predict pollution levels using the [Beijing PM2.5 data set](https://archive.ics.uci.edu/ml/datasets/Beijing+PM2.5+Data).\n\nThis Jupyter notebook is part of a larger solution intended to be deployed to your AWS account. Run through this notebook after air pollution data has been published into your account and stored in IoT Analytics. After finishing all steps in this notebook, you will move on to the next stage of the solution which is deploying your LSTM model to an edge solution with IoT Greengrass.\n\n

\nArchitecture diagram

\n\n\n# Set-up: import required notebook libraries\n\n

This notebook requires a few basic Python libraries including `pandas`, `numpy`, `keras`, `tensorflow`, `scikit-learn` and `matplotlib`.

\n\n\n```python\nfrom pandas import read_csv\nfrom datetime import datetime\nfrom math import sqrt\nfrom numpy import concatenate\nfrom matplotlib import pyplot\nfrom pandas import to_datetime\nfrom pandas import DataFrame\nfrom pandas import concat\nfrom sklearn.preprocessing import MinMaxScaler\nfrom sklearn.preprocessing import LabelEncoder\nfrom sklearn.metrics import mean_squared_error\nfrom keras.models import Sequential\nfrom keras.layers import Dense\nfrom keras.layers import LSTM\nfrom joblib import dump, load\nimport numpy\nimport sagemaker\nimport pickle\nimport warnings\nimport tensorflow\nfrom sagemaker.tensorflow import TensorFlow\nimport json\nfrom sagemaker import get_execution_role\nfrom urllib.parse import urlparse\nimport os\nimport boto3\nimport tarfile\nfrom keras.models import model_from_json\nfrom IPython.display import display, Markdown\nimport traceback\n \n###Turn off warnings\nwarnings.simplefilter(action='ignore', category=FutureWarning)\nif type(tensorflow.contrib) != type(tensorflow): tensorflow.contrib._warning = None\n```\n\n Using TensorFlow backend.\n\n\n# Background: problem description and approach\n\nIn order to support business decision making, we must be able to take action on predictive analytics. For predictive operational analysis, we want to know what would be the operating condition of equipment to optimize the parameters for increasing the life of that equipment. To solve this for real-world equipment that has multiple operation modes and reports time-series measurements from multiple sensors, our primary approach is an Long Short-Term Memory neural network in TensorFlow. LSTM provides multivariate time series forecasting to predict the future pollution levels which could be used to automatically alter the parameters of the air filter to improve efficiency and increase remaining useful life (RUL) of air filters.\n\nIn the [CRISP-DM lifecycle](https://en.wikipedia.org/wiki/Cross-industry_standard_process_for_data_mining), we have already completed the Business Understanding phase in knowing our objective is to increase RUL of our air filtration equipment and that we are going to try using pollution forecasting to drive improvements. The next phase, which starts in this notebook, is Data Understanding. We need to evaluate the available data, look for trends, and assess whether this data will be useful to our objective.\n\nBEST PRACTICES NOTE Different modeling approaches provide different business trade-offs, and different teams may opt for different approaches. For example, although the field service team is interested in the most precise prediction for any given filter in order to eliminate false positives and unnecessary truck rolls, a supervisor is interested in all possible devices that aren't operating at full efficiency to increase operational equipment effectiveness (OEE).\n\n

Step 1 | Loading data

\n\n# Data set description\n\nThe Beijing PM2.5 is an hourly data set that contains the PM2.5 data of US Embassy in Beijing along with meteorological data from Beijing Capital International Airport. The data is collected for time period starting from Jan 1st, 2010 to Dec 31st, 2014. \n\nThe data contain features like PM2.5 concentration (pollution level), dew point, temperature, pressure, wind direction, wind speed, cumulated hours of snow and cumulated hours of rain.\n\n

Data Set Attribution:

Song Xi Chen, Guanghua School of Management, Center for Statistical Science, Peking University\n\nThe data can be downloaded from this page: https://archive.ics.uci.edu/ml/datasets/Beijing+PM2.5+Data\n\nThe data has been split into training, validation and test data sets. The first 2 years of data is used for training data, followed by the next 2 year of data as validation to check the training accuracy. The last 1 year of data will later be used for device simulation testing.\n\nTo load data already ingested from your device to the cloud, we need to set up an IoT Analytics SDK client to access your data set. This code initializes the client, fetches content from your data set, sorts by time, and previews the first five rows.\n\n\n```python\ns3Bucket = \"XYZ-S3Bucket-XYZ\"\ndataset = \"XYZ-IotAnalyticsDataset-XYZ\"\nuse_local_data = True\n\nif use_local_data:\n dataset_url = \"source_pollution.csv\"\nelse:\n iotanalytics_client = boto3.client('iotanalytics')\n dataset_url = iotanalytics_client.get_dataset_content(datasetName = dataset)['entries'][0]['dataURI']\n\ndataset = read_csv(dataset_url, header=0, index_col='date')\nif dataset.empty:\n raise Exception('No data found')\ndataset.sort_index(inplace=True)\ndataset = dataset[['pollution', 'dew', 'temp', 'press', 'wnd_dir', 'wnd_spd', 'snow', 'rain']]\ndataset.drop_duplicates(inplace=True)\ndataset.to_csv('pollution.csv')\n\n# displays the first few entries of the DataFrame\ndataset.head() \n```\n\n\n\n\n
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pollutiondewtemppresswnd_dirwnd_spdsnowrain
date
2010-01-02 00:00:00129.0-16-4.01020.021.7900
2010-01-02 01:00:00148.0-15-4.01020.022.6800
2010-01-02 02:00:00159.0-11-5.01021.023.5700
2010-01-02 03:00:00181.0-7-5.01022.025.3610
2010-01-02 04:00:00138.0-7-5.01022.026.2520
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\n\n\n\nStart a new session in SageMaker and make the training data available in S3.\n\n\n```python\n###Upload this queried data to s3\nsagemaker_session = sagemaker.Session()\nuploaded_data_path = sagemaker_session.upload_data(path='pollution.csv', bucket=s3Bucket, key_prefix='data')\n```\n\nPlot the columns to understand statistical properties of the data set.\n\n\n```python\nvalues = dataset.values\n# specify columns to plot\ncolumns_to_plot = [0, 1, 2, 3, 5, 6, 7]\nfor column_index in columns_to_plot:\n\tpyplot.figure()\n\tpyplot.subplot(1, 1, 1)\n\tpyplot.plot(values[:, column_index])\n\tpyplot.title(dataset.columns[column_index])\n \n```\n\nWe can visually observe some seasonal values in dew, temp, pressure and snow attributes in the above plots. However, we don't observe any particular statistical features in the pollution readings. We need to further analyze the features of our pollution readings. \n\nWe are using a library `statsmodels` to see if there is any seasonality in the pollution readings. Seasonality or periodicity in data is often useful in picking the suitable ML algorithm.\n\nLet's first observe the first month of pollution data. \n\n\n```python\n###Exploring short term seasonality in data\nimport statsmodels.api as sm\ntrend_dataset = dataset.reset_index()\ntrend_dataset = trend_dataset[['date', 'pollution']]\ntrend_dataset['date'] = to_datetime(trend_dataset['date'])\ntrend_dataset = trend_dataset.set_index('date')\n\npyplot.rcParams[\"figure.figsize\"] = (20,10)\nnumber_of_monthly_datapoints = 24*30\ndecomposition = sm.tsa.seasonal_decompose(trend_dataset[:number_of_monthly_datapoints], model='additive')\nfig = decomposition.plot()\n```\n\nIf we observe the seasonal plot above, we can find that our pollution dataset has a daily seasonality in it. \n\nLet's try to look for long term seasonality by forcing the model to ignore short term seasonality we found above. By setting the `freq` to a larger number. We are setting it here to look for monthly seasonality in the first year of pollution data. \n\n\n```python\n###Exploring long seasonality in data\nnumber_of_yearly_datapoints = 24*365\ndecomposition = sm.tsa.seasonal_decompose(trend_dataset[:number_of_yearly_datapoints], model='additive', freq=number_of_monthly_datapoints)\nfig = decomposition.plot()\n```\n\nIf we observe the seasonal plot above, we can find that our pollution dataset also has a monthly seasonality in it (observe the 12 similar segments in the 1 year of data).\n\nWe can observe that pollution data has both long (monthly) and short (daily) term periodicity. \n\n

Step 2 | Processing data

\n\n# Sensor measurement normalization\n\nThe next step of the CRISP-DM lifecycle is Data Preparation. These next steps are about manipulating the desired data into a format best fit for machine learning.\n\nMultivariate problems require normalization of sensor measurements and operational settings to remove the effects of different scales being used across the units of measure. For example, wind speed versus cumulated hours of snow/rain operate in different orders of magnitude. The 'MinMaxScaler' transforms each value into a given range, in our case between 0 and 1.\n\nBelow is the transformation we are performing on each sensor measurement. [Learn more](http://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.MinMaxScaler.html) about preprocessing with **sklearn**.\n\n\\begin{align}\nX_{std} & = (X - X_{min})/(X_{max} - X_{min}) \\\\\nX_{scaled} & = X_{std} * (feature\\_range_{max} - feature\\_range_{min}) + feature\\_range_{min}\n\\end{align}\n\n\n```python\n#Transform all columns as float\nvalues = values.astype('float32')\n\n#Normalize features\nscaler = MinMaxScaler(feature_range=(0, 1))\nnormalized_values = scaler.fit_transform(values)\n```\n\n# Target parameter determination\n\nIn order to model a LSTM, we need a target parameter that puts all the data on the same basis for supervised training and the subsequent classification test. For time-series forecasting of pollution levels, we will assume attribute data from the previous timestamp being associated with pollution levels of the current timestamp. \n\nTo mark these observations into supervised learning, we first need to transform the data set where the all previous timestamp's attributes (including the pollution level) are the variables and current pollution level is the target value.\n\nBEST PRACTICES NOTE In this example, only the last timestamp's attributes are used for training and prediction. A deeper lookback into the history may increase model accuracy while increasing the complexity and training time. \n\n\n```python\n#Convert series to supervised learning\ndef transform_to_supervised_series(data, columns, n_in=1, n_out=1, dropnan=True):\n\tn_vars = 1 if type(data) is list else data.shape[1]\n\tdf = DataFrame(data)\n\tcols, names = list(), list()\n\t# input sequence (t-n, ... t-1)\n\tfor i in range(n_in, 0, -1):\n\t\tcols.append(df.shift(i))\n\t\tnames += [('%s(t-%d)' % (columns[j], i)) for j in range(n_vars)]\n\t# forecast sequence (t, t+1, ... t+n)\n\tfor i in range(0, n_out):\n\t\tcols.append(df.shift(-i))\n\t\tif i == 0:\n\t\t\tnames += [('%s(t)' % (columns[j])) for j in range(n_vars)]\n\t\telse:\n\t\t\tnames += [('%s(t+%d)' % (columns[j], i)) for j in range(n_vars)]\n\t# put it all together\n\tagg = concat(cols, axis=1)\n\tagg.columns = names\n\t# drop rows with NaN values\n\tif dropnan:\n\t\tagg.dropna(inplace=True)\n\treturn agg\n\n#Frame as supervised learning\nsupervised_series_dataset = transform_to_supervised_series(normalized_values, dataset.columns, 1, 1)\n#Drop the unnecessary columns\nsupervised_series_dataset.drop(['dew(t)', 'temp(t)', 'press(t)', 'wnd_dir(t)', 'wnd_spd(t)', 'snow(t)', 'rain(t)'], axis=1, inplace=True)\n```\n\nThe data has been prepared for supervised learning as described in the previous cell. Let's take a look at the first five rows.\n\n\n```python\nsupervised_series_dataset.head()\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
pollution(t-1)dew(t-1)temp(t-1)press(t-1)wnd_dir(t-1)wnd_spd(t-1)snow(t-1)rain(t-1)pollution(t)
10.1297790.3529410.2459020.5272730.6666670.0022900.0000000.00.148893
20.1488930.3676470.2459020.5272730.6666670.0038110.0000000.00.159960
30.1599600.4264710.2295080.5454540.6666670.0053320.0000000.00.182093
40.1820930.4852940.2295080.5636370.6666670.0083910.0370370.00.138833
50.1388330.4852940.2295080.5636370.6666670.0099120.0740740.00.109658
\n
\n\n\n\n# Split the data into training, validation and test data sets\n\nWe will now split the data into training, validation and test data sets. The first 2 years of data is used for training data, followed by the next 2 years of data as validation to check the training accuracy. The last 1 year of data has been used for device simulation testing.\n\n\n```python\n######Splitting data into training and test data\nvalues = supervised_series_dataset.values\nn_train_hours = 24*365*2\nn_validation_hours = 24*365*4 \ntrain = values[:n_train_hours, :]\nvalidation = values[n_train_hours:n_validation_hours, :]\n```\n\n# Split the training and validation data sets into input and output data sets\n\nLSTM model takes input in a 3D format where the 1st axis are the samples, 2nd axis is the timesteps that you want to take and the last axis is the features. We would need to transform out input data into this format. \n\n\n```python\n#Split into input and outputs\ntrain_X, train_y = train[:, :-1], train[:, -1]\nvalidation_X, validation_y = validation[:, :-1], validation[:, -1]\n#Reshape input to be 3D [samples, timesteps, features]\ntrain_X = train_X.reshape((train_X.shape[0], 1, train_X.shape[1]))\nvalidation_X = validation_X.reshape((validation_X.shape[0], 1, validation_X.shape[1]))\n```\n\n

Step 3 | Train and test model

\n\n# Time-series forecasting via LSTM\n\nLSTM or 'Long Short-Term Memory' is a recurrent neural network that is particularly forgiving of unknown time gaps between significant events in time-series data, such as the unknown periodicity in the pollution levels. LSTM is capable of handling large data sets. This makes it very useful for prediction with time-series IoT data. \n\nKeras LSTM layers expect an input in the following shape: (batch_size, timesteps, input_dim)\n
batch_size - number of sequences
\n
timesteps - the sequence length to look back, look back cycles in our example
\n
input_dim - number of features of each sequence at each time step
\n\nHUMAN LEARNING NOTE Keras is a neural network (NN) framework that makes running NN fast and easy on TensorFlow. Alternatives to Keras include Theano and CNTK.\n\n# LSTM model setup\nNow that we have the data structured for training we can set the parameters in our LSTM model. Here we are using a sequential LSTM model for time-series forecasting. This model configuration choice makes our model capable of learning higher-level temporal representations. We could see in the earlier data visualization that our data has long-term temporal periodicity.\n\n`Epoch` is a hyperparameter in ML model to control the number of complete passes through the training dataset. Larger epoch number usually allows the learning algorithm to run until the error from the model has been sufficiently minimized. \n\n`Batch Size` is another hyperparameter in ML model that defines the number of samples to work through before updating the internal model parameters.\n\n`Mean Absolute Error (MAE)` is one of the most commonly used regression loss function. MAE is the sum of absolute differences between our target and predicted variables. So, it measures the average magnitude of errors in a set of predictions, without considering their directions. The range is also 0 to \u221e\n\n\n`Adam optimizer` is an adaptive learning rate optimization algorithm that\u2019s been designed specifically for training deep neural networks. The algorithms leverage the power of adaptive learning rates methods to find individual learning rates for each parameter. It also has advantages of Adagrad, which works really well in settings with sparse gradients, but struggles in non-convex optimization of neural networks, and RMSprop, which tackles to resolve some of the problems of Adagrad and works really well in on-line settings. Its name is derived from adaptive moment estimation and uses estimations of first and second moments of gradient to adapt the learning rate for each weight of the neural network. \n\nHUMAN LEARNING NOTE You can read about different supported optimizers and losses on Keras `optimizers` and `losses`.\n\n\n```python\n#LSTM for time-series predictions\nmodel = Sequential()\nmodel.add(LSTM(50, input_shape=(train_X.shape[1], train_X.shape[2])))\nmodel.add(Dense(1))\nmodel.compile(loss='mae', optimizer='adam')\n\nprint(model.summary())\n```\n\n _________________________________________________________________\n Layer (type) Output Shape Param # \n =================================================================\n lstm_2 (LSTM) (None, 50) 11800 \n _________________________________________________________________\n dense_2 (Dense) (None, 1) 51 \n =================================================================\n Total params: 11,851\n Trainable params: 11,851\n Non-trainable params: 0\n _________________________________________________________________\n None\n\n\nWe are using an LSTM with 50 neurons in the first hidden layer and 1 neuron in the output layer for predicting pollution. The input shape will be 1 time step with 8 features. We reached to this model after tuning these parameters based on some experimentation, you can try to play around with these to see if you can tune them better. \n\n# Model training\n\nThe next step of the CRISP-DM lifecycle is Modeling, where we start the process of training and testing models.\n\nSince LSTM is a recurrent neural network we can observe the incremental training at each iteration or epoch on the training data set. \n\n`loss` is the value of cost function on our training data.\n`val_loss` is the value of cost function for your cross-validation data.\n\nThese values are useful in determining if our model is overfitting to training data. If we have `loss` noticeably lower than `val_loss` then it is the sign of overfitting.\n\nPRODUCTION NOTE we have chosen to train over 50 epochs. This allows us to test the model within a reasonable timeframe (~15 mins). \n\n\n```python\n### fit network\n#history = model.fit(train_X, train_y, epochs=50, batch_size=72, validation_data=(validation_X, validation_y), verbose=2, shuffle=False)\n#history = history.history\n\n###Instead of training here, we are calling it through a training job, since sagemaker will pack the model for us and \n###upload it on to s3 to make it easier for us use\n\n###We are performing a m5.xlarge instance for training , you can do local training if you want to. \n###You can change the train_instance_type to a different p type sagemaker instance for faster training.\n###Note: Local training doesn't result into a training job that can be used for green grass directly\nuse_local_training = False \n\nif use_local_training:\n train_instance_type = \"local\"\nelse: \n train_instance_type = \"ml.m5.xlarge\"\n \ntf_estimator = TensorFlow(entry_point='trainLSTM.py', role=get_execution_role(),\n train_instance_count=1, train_instance_type=train_instance_type,\n framework_version='1.12', py_version='py3', script_mode=True,\n output_path = 's3://' + s3Bucket, base_job_name = \"pollution-forecasting-lstm\",\n hyperparameters={'batch_size': 72,\n 'epochs': 50,\n 'n_train_hours': n_train_hours,\n 'n_validation_hours': n_validation_hours})\n\ntf_estimator.fit(uploaded_data_path)\n```\n\n 2019-11-26 19:49:34 Starting - Starting the training job...\n 2019-11-26 19:49:35 Starting - Launching requested ML instances......\n 2019-11-26 19:50:57 Starting - Preparing the instances for training......\n 2019-11-26 19:51:58 Downloading - Downloading input data\n 2019-11-26 19:51:58 Training - Downloading the training image..\u001b[31m2019-11-26 19:52:12,958 sagemaker-containers INFO Imported framework sagemaker_tensorflow_container.training\u001b[0m\n \u001b[31m2019-11-26 19:52:12,963 sagemaker-containers INFO No GPUs detected (normal if no gpus installed)\u001b[0m\n \u001b[31m2019-11-26 19:52:13,229 sagemaker-containers INFO No GPUs detected (normal if no gpus installed)\u001b[0m\n \u001b[31m2019-11-26 19:52:13,243 sagemaker-containers INFO No GPUs detected (normal if no gpus installed)\u001b[0m\n \u001b[31m2019-11-26 19:52:13,253 sagemaker-containers INFO Invoking user script\n \u001b[0m\n \u001b[31mTraining Env:\n \u001b[0m\n \u001b[31m{\n \"additional_framework_parameters\": {},\n \"channel_input_dirs\": {\n \"training\": \"/opt/ml/input/data/training\"\n },\n \"current_host\": \"algo-1\",\n \"framework_module\": \"sagemaker_tensorflow_container.training:main\",\n \"hosts\": [\n \"algo-1\"\n ],\n \"hyperparameters\": {\n \"batch_size\": 72,\n \"n_train_hours\": 17520,\n \"model_dir\": \"s3://iot309demo-sagemaker-232skq6dqo4r/pollution-forecasting-lstm-2019-11-26-19-49-33-760/model\",\n \"epochs\": 50,\n \"n_validation_hours\": 35040\n },\n \"input_config_dir\": \"/opt/ml/input/config\",\n \"input_data_config\": {\n \"training\": {\n \"TrainingInputMode\": \"File\",\n \"S3DistributionType\": \"FullyReplicated\",\n \"RecordWrapperType\": \"None\"\n }\n },\n \"input_dir\": \"/opt/ml/input\",\n \"is_master\": true,\n \"job_name\": \"pollution-forecasting-lstm-2019-11-26-19-49-33-760\",\n \"log_level\": 20,\n \"master_hostname\": \"algo-1\",\n \"model_dir\": \"/opt/ml/model\",\n \"module_dir\": \"s3://iot309demo-sagemaker-232skq6dqo4r/pollution-forecasting-lstm-2019-11-26-19-49-33-760/source/sourcedir.tar.gz\",\n \"module_name\": \"trainLSTM\",\n \"network_interface_name\": \"eth0\",\n \"num_cpus\": 4,\n \"num_gpus\": 0,\n \"output_data_dir\": \"/opt/ml/output/data\",\n \"output_dir\": \"/opt/ml/output\",\n \"output_intermediate_dir\": \"/opt/ml/output/intermediate\",\n \"resource_config\": {\n \"current_host\": \"algo-1\",\n \"hosts\": [\n \"algo-1\"\n ],\n \"network_interface_name\": \"eth0\"\n },\n \"user_entry_point\": \"trainLSTM.py\"\u001b[0m\n \u001b[31m}\n \u001b[0m\n \u001b[31mEnvironment variables:\n \u001b[0m\n \u001b[31mSM_HOSTS=[\"algo-1\"]\u001b[0m\n \u001b[31mSM_NETWORK_INTERFACE_NAME=eth0\u001b[0m\n \u001b[31mSM_HPS={\"batch_size\":72,\"epochs\":50,\"model_dir\":\"s3://iot309demo-sagemaker-232skq6dqo4r/pollution-forecasting-lstm-2019-11-26-19-49-33-760/model\",\"n_train_hours\":17520,\"n_validation_hours\":35040}\u001b[0m\n \u001b[31mSM_USER_ENTRY_POINT=trainLSTM.py\u001b[0m\n \u001b[31mSM_FRAMEWORK_PARAMS={}\u001b[0m\n \u001b[31mSM_RESOURCE_CONFIG={\"current_host\":\"algo-1\",\"hosts\":[\"algo-1\"],\"network_interface_name\":\"eth0\"}\u001b[0m\n \u001b[31mSM_INPUT_DATA_CONFIG={\"training\":{\"RecordWrapperType\":\"None\",\"S3DistributionType\":\"FullyReplicated\",\"TrainingInputMode\":\"File\"}}\u001b[0m\n \u001b[31mSM_OUTPUT_DATA_DIR=/opt/ml/output/data\u001b[0m\n \u001b[31mSM_CHANNELS=[\"training\"]\u001b[0m\n \u001b[31mSM_CURRENT_HOST=algo-1\u001b[0m\n \u001b[31mSM_MODULE_NAME=trainLSTM\u001b[0m\n \u001b[31mSM_LOG_LEVEL=20\u001b[0m\n \u001b[31mSM_FRAMEWORK_MODULE=sagemaker_tensorflow_container.training:main\u001b[0m\n \u001b[31mSM_INPUT_DIR=/opt/ml/input\u001b[0m\n \u001b[31mSM_INPUT_CONFIG_DIR=/opt/ml/input/config\u001b[0m\n \u001b[31mSM_OUTPUT_DIR=/opt/ml/output\u001b[0m\n \u001b[31mSM_NUM_CPUS=4\u001b[0m\n \u001b[31mSM_NUM_GPUS=0\u001b[0m\n \u001b[31mSM_MODEL_DIR=/opt/ml/model\u001b[0m\n \u001b[31mSM_MODULE_DIR=s3://iot309demo-sagemaker-232skq6dqo4r/pollution-forecasting-lstm-2019-11-26-19-49-33-760/source/sourcedir.tar.gz\u001b[0m\n \u001b[31mSM_TRAINING_ENV={\"additional_framework_parameters\":{},\"channel_input_dirs\":{\"training\":\"/opt/ml/input/data/training\"},\"current_host\":\"algo-1\",\"framework_module\":\"sagemaker_tensorflow_container.training:main\",\"hosts\":[\"algo-1\"],\"hyperparameters\":{\"batch_size\":72,\"epochs\":50,\"model_dir\":\"s3://iot309demo-sagemaker-232skq6dqo4r/pollution-forecasting-lstm-2019-11-26-19-49-33-760/model\",\"n_train_hours\":17520,\"n_validation_hours\":35040},\"input_config_dir\":\"/opt/ml/input/config\",\"input_data_config\":{\"training\":{\"RecordWrapperType\":\"None\",\"S3DistributionType\":\"FullyReplicated\",\"TrainingInputMode\":\"File\"}},\"input_dir\":\"/opt/ml/input\",\"is_master\":true,\"job_name\":\"pollution-forecasting-lstm-2019-11-26-19-49-33-760\",\"log_level\":20,\"master_hostname\":\"algo-1\",\"model_dir\":\"/opt/ml/model\",\"module_dir\":\"s3://iot309demo-sagemaker-232skq6dqo4r/pollution-forecasting-lstm-2019-11-26-19-49-33-760/source/sourcedir.tar.gz\",\"module_name\":\"trainLSTM\",\"network_interface_name\":\"eth0\",\"num_cpus\":4,\"num_gpus\":0,\"output_data_dir\":\"/opt/ml/output/data\",\"output_dir\":\"/opt/ml/output\",\"output_intermediate_dir\":\"/opt/ml/output/intermediate\",\"resource_config\":{\"current_host\":\"algo-1\",\"hosts\":[\"algo-1\"],\"network_interface_name\":\"eth0\"},\"user_entry_point\":\"trainLSTM.py\"}\u001b[0m\n \u001b[31mSM_USER_ARGS=[\"--batch_size\",\"72\",\"--epochs\",\"50\",\"--model_dir\",\"s3://iot309demo-sagemaker-232skq6dqo4r/pollution-forecasting-lstm-2019-11-26-19-49-33-760/model\",\"--n_train_hours\",\"17520\",\"--n_validation_hours\",\"35040\"]\u001b[0m\n \u001b[31mSM_OUTPUT_INTERMEDIATE_DIR=/opt/ml/output/intermediate\u001b[0m\n \u001b[31mSM_CHANNEL_TRAINING=/opt/ml/input/data/training\u001b[0m\n \u001b[31mSM_HP_BATCH_SIZE=72\u001b[0m\n \u001b[31mSM_HP_N_TRAIN_HOURS=17520\u001b[0m\n \u001b[31mSM_HP_MODEL_DIR=s3://iot309demo-sagemaker-232skq6dqo4r/pollution-forecasting-lstm-2019-11-26-19-49-33-760/model\u001b[0m\n \u001b[31mSM_HP_EPOCHS=50\u001b[0m\n \u001b[31mSM_HP_N_VALIDATION_HOURS=35040\u001b[0m\n \u001b[31mPYTHONPATH=/opt/ml/code:/usr/local/bin:/usr/lib/python36.zip:/usr/lib/python3.6:/usr/lib/python3.6/lib-dynload:/usr/local/lib/python3.6/dist-packages:/usr/lib/python3/dist-packages\n \u001b[0m\n \u001b[31mInvoking script with the following command:\n \u001b[0m\n \u001b[31m/usr/bin/python trainLSTM.py --batch_size 72 --epochs 50 --model_dir s3://iot309demo-sagemaker-232skq6dqo4r/pollution-forecasting-lstm-2019-11-26-19-49-33-760/model --n_train_hours 17520 --n_validation_hours 35040\n \n \u001b[0m\n \u001b[31mUsing TensorFlow backend.\u001b[0m\n \u001b[31mTrain on 17520 samples, validate on 17520 samples\u001b[0m\n \u001b[31mEpoch 1/50\u001b[0m\n \u001b[31m - 3s - loss: 0.0480 - val_loss: 0.0474\u001b[0m\n \u001b[31mEpoch 2/50\u001b[0m\n \n 2019-11-26 19:52:10 Training - Training image download completed. Training in progress.\u001b[31m - 1s - loss: 0.0188 - val_loss: 0.0272\u001b[0m\n \u001b[31mEpoch 3/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0150 - val_loss: 0.0196\u001b[0m\n \u001b[31mEpoch 4/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0146 - val_loss: 0.0173\u001b[0m\n \u001b[31mEpoch 5/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0144 - val_loss: 0.0166\u001b[0m\n \u001b[31mEpoch 6/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0144 - val_loss: 0.0157\u001b[0m\n \u001b[31mEpoch 7/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0144 - val_loss: 0.0158\u001b[0m\n \u001b[31mEpoch 8/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0143 - val_loss: 0.0162\u001b[0m\n \u001b[31mEpoch 9/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0143 - val_loss: 0.0161\u001b[0m\n \u001b[31mEpoch 10/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0143 - val_loss: 0.0163\u001b[0m\n \u001b[31mEpoch 11/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0143 - val_loss: 0.0158\u001b[0m\n \u001b[31mEpoch 12/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0142 - val_loss: 0.0156\u001b[0m\n \u001b[31mEpoch 13/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0143 - val_loss: 0.0160\u001b[0m\n \u001b[31mEpoch 14/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0142 - val_loss: 0.0160\u001b[0m\n \u001b[31mEpoch 15/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0143 - val_loss: 0.0153\u001b[0m\n \u001b[31mEpoch 16/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0143 - val_loss: 0.0151\u001b[0m\n \u001b[31mEpoch 17/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0141 - val_loss: 0.0157\u001b[0m\n \u001b[31mEpoch 18/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0142 - val_loss: 0.0157\u001b[0m\n \u001b[31mEpoch 19/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0142 - val_loss: 0.0154\u001b[0m\n \u001b[31mEpoch 20/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0142 - val_loss: 0.0154\u001b[0m\n \u001b[31mEpoch 21/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0142 - val_loss: 0.0158\u001b[0m\n \u001b[31mEpoch 22/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0142 - val_loss: 0.0154\u001b[0m\n \u001b[31mEpoch 23/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0142 - val_loss: 0.0146\u001b[0m\n \u001b[31mEpoch 24/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0142 - val_loss: 0.0153\u001b[0m\n \u001b[31mEpoch 25/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0142 - val_loss: 0.0149\u001b[0m\n \u001b[31mEpoch 26/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0142 - val_loss: 0.0148\u001b[0m\n \u001b[31mEpoch 27/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0142 - val_loss: 0.0149\u001b[0m\n \u001b[31mEpoch 28/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0142 - val_loss: 0.0151\u001b[0m\n \u001b[31mEpoch 29/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0141 - val_loss: 0.0147\u001b[0m\n \u001b[31mEpoch 30/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0141 - val_loss: 0.0148\u001b[0m\n \u001b[31mEpoch 31/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0141 - val_loss: 0.0152\u001b[0m\n \u001b[31mEpoch 32/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0141 - val_loss: 0.0150\u001b[0m\n \u001b[31mEpoch 33/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0141 - val_loss: 0.0150\u001b[0m\n \u001b[31mEpoch 34/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0141 - val_loss: 0.0152\u001b[0m\n \u001b[31mEpoch 35/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0141 - val_loss: 0.0151\u001b[0m\n \u001b[31mEpoch 36/50\u001b[0m\n \u001b[31m - 1s - loss: 0.0142 - val_loss: 0.0147\u001b[0m\n \u001b[31mEpoch 37/50\u001b[0m\n\n\n#### Download the training model and history created using training job\nThis next code downloads the training model that we have created the through the SageMaker training job. \nIt copies the trained model in tar.gz-format from S3 to this notebook instance and unpacks it. The model files will be stored in the local file directory **[job\\_name]/model**. [job\\_name] is a placeholder that is different every time you run this notebook. When copying and unpacking are finished the names of your model files will be printed below the cell.\n\n\n```python\nprint(\"job_name: {}\".format(tf_estimator.latest_training_job.name))\n\ns3_model_path = '{}'.format(tf_estimator.model_data)\nprint(\"s3_model_path: {}\".format(s3_model_path))\ns3_bucket = urlparse(s3_model_path).netloc\nprint(s3_bucket)\ns3_key = urlparse(s3_model_path).path.lstrip('/')\nprint(\"s3_key: {}\".format(s3_key))\n\nlocal_model_dir = 'model'\nif not os.path.exists(local_model_dir):\n try:\n os.makedirs(local_model_dir)\n except Exception as e:\n print(\"ERROR: failed to created directory {}: {}\".format(local_model_dir, e))\n\nif not os.path.exists(local_model_dir + '/model.tar.gz'):\n s3_client = boto3.client('s3')\n s3_client.download_file(s3_bucket, s3_key, local_model_dir + '/model.tar.gz')\n \ntar = tarfile.open(local_model_dir + '/model.tar.gz')\ntar.extractall(path=local_model_dir)\ntar.close()\n\nmodel_files = os.listdir(local_model_dir)\nprint(\"model_files: {}\".format(model_files))\n```\n\n### Read the model file and history\nThe model files that we have downloaded above need to be loaded here in the notebook.\n\n\n```python\nscaler = load('%s/scaler.model' % local_model_dir)\n\nwith open('%s/model.json' % local_model_dir, 'r') as json_file:\n loaded_model_json = json_file.read()\nloaded_model = model_from_json(loaded_model_json)\nloaded_model.load_weights('%s/model.h5' % local_model_dir)\n\nwith open('%s/history.json' % local_model_dir) as f:\n history = json.load(f)\nloaded_model.compile(loss='mae', optimizer='adam')\n```\n\n /home/ec2-user/anaconda3/envs/tensorflow_p36/lib/python3.6/site-packages/sklearn/base.py:306: UserWarning: Trying to unpickle estimator MinMaxScaler from version 0.21.1 when using version 0.21.3. This might lead to breaking code or invalid results. Use at your own risk.\n UserWarning)\n\n\n WARNING:tensorflow:From /home/ec2-user/anaconda3/envs/tensorflow_p36/lib/python3.6/site-packages/keras/backend/tensorflow_backend.py:174: The name tf.get_default_session is deprecated. Please use tf.compat.v1.get_default_session instead.\n \n WARNING:tensorflow:From /home/ec2-user/anaconda3/envs/tensorflow_p36/lib/python3.6/site-packages/keras/backend/tensorflow_backend.py:184: The name tf.ConfigProto is deprecated. Please use tf.compat.v1.ConfigProto instead.\n \n\n\n#### Model accuracy and overfit check\nThe next step in CRISP-DM is Evaluation. Now that we've trained a model, we must inspect its performance in forecasting pollution of time N against the actual time N. Let's compare the `loss` and `val_loss` of the model to see if we ended overfitting or not.\n\n\n\n```python\n# plot accuracy as we learn\npyplot.plot(history['loss'], label='loss (Train)')\npyplot.plot(history['val_loss'], label='val_loss (Validation)')\npyplot.legend()\npyplot.show()\n```\n\nAs you may have noticed, the loss of the train dataset is similar to the val_loss of the validation data. This corresponds that we have not overfitted out model onto training data.\n\n

Step 4 | Results summary

\n\n### Apply model to test data\n\nLet's make inferences from the trained model onto our validation dataset\n\n\n```python\nyhat = loaded_model.predict(validation_X)\n#If trained within the notebook use yhat = model.predict(validation_X)\n```\n\n### Calculate the root mean square error \nWe can calculate the root mean square error (RMSE) on the validation data set to evaluate the performance of our model. Unlike classification problems, in regressions problems like this one it is hard to calculate the accuracy of a model, so we use RMSE to benchmark the model's performance.\n\n\n```python\nvalidation_X = validation_X.reshape((validation_X.shape[0], validation_X.shape[2]))\n# invert scaling for forecast\ninv_yhat = concatenate((yhat, validation_X[:, 1:]), axis=1)\ninv_yhat = scaler.inverse_transform(inv_yhat)\ninv_yhat = inv_yhat[:,0]\n# invert scaling for actual\nvalidation_y = validation_y.reshape((len(validation_y), 1))\ninv_y = concatenate((validation_y, validation_X[:, 1:]), axis=1)\ninv_y = scaler.inverse_transform(inv_y)\ninv_y = inv_y[:,0]\n\nprint(\"\\n\\nLSTM Metrics\")\n# calculate RMSE\nrmse = sqrt(mean_squared_error(inv_y, inv_yhat))\nprint('Val RMSE: %.3f' % rmse)\n```\n\n \n \n LSTM Metrics\n Val RMSE: 27.931\n\n\nIn this case, our RMSE is approximately 30 (your exact number may vary by a few percent).\n\n### Benchmarking with a Persistence Model\nThe persistence forecast is where the observation from the prior time step (t-1) is used to predict the observation at the current time step (t).\n\n\n```python\n#Let's create a naive persistence model to benchmark the performance of our model\n\n##Creating a dateframe with the previous timestep values and using that as prediction for next step\npersistence_model_df = DataFrame()\npersistence_model_df['pollution(t-1)'] = dataset['pollution']\npersistence_model_df['predicted_pollution(t)'] = persistence_model_df['pollution(t-1)']\npersistence_model_df.head(5)\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
pollution(t-1)predicted_pollution(t)
date
2010-01-02 00:00:00129.0129.0
2010-01-02 01:00:00148.0148.0
2010-01-02 02:00:00159.0159.0
2010-01-02 03:00:00181.0181.0
2010-01-02 04:00:00138.0138.0
\n
\n\n\n\n### Calculate the root mean square error for persistence model for validation data\n\n\n```python\n#Let's extract the actual values\npersistence_model_y = persistence_model_df[['pollution(t-1)']].values[1:].astype('float32')\n#Extract the predicted values\npersistence_model_y_hat = persistence_model_df[['predicted_pollution(t)']].values[:-1].astype('float32')\n\n#Filter out for validation dataset only\npersistence_model_y = persistence_model_y[n_train_hours:n_validation_hours, :]\npersistence_model_y_hat = persistence_model_y_hat[n_train_hours:n_validation_hours, :]\n\npersistence_model_y = persistence_model_y.reshape((len(persistence_model_y)))\npersistence_model_y_hat = persistence_model_y_hat.reshape((len(persistence_model_y_hat)))\npersistence_model_rmse = sqrt(mean_squared_error(persistence_model_y, persistence_model_y_hat))\nprint('Val RMSE for Persistence Model: %.3f' % persistence_model_rmse)\n```\n\n Val RMSE for Persistence Model: 27.436\n\n\nIn this case, our vanilla LSTM model is performing similar to a persistence model. We can experiment with hyperparameter tuning to further enhance the model and reduce its error.\n\n

Summary

\n\nAt this stage you have successfully trained and tested a LSTM model to forecast air pollution! The next stage of the CRISP-DM lifecycle is Deployment. That happens outside the scope of this notebook. Resume the next step of this solution from the associated README file.\n\nThe code in this notebook is not tightly correlated to the air pollution data set. You could continue experimenting beyond the scope of this solution by bringing an alternate data set, changing the window of the Beijing 2.5PM used to train, or experimenting with the LSTM hyperparameters to see the effect they have on model accuracy (RMSE).\n", "meta": {"hexsha": "9e41eea61bb5baccf9c440d6a16f4fc6a4731df4", "size": 703148, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "sagemaker/pollutionPredictionLSTM-Training.ipynb", "max_stars_repo_name": "ashishbajaj91/amazon-sagemaker-aws-greengrass-custom-timeseries-forecasting", "max_stars_repo_head_hexsha": "3c8cfcf54e9354583938da3239e2e26d7cd41566", "max_stars_repo_licenses": ["MIT-0"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2020-05-26T14:55:46.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-31T17:21:16.000Z", "max_issues_repo_path": "sagemaker/pollutionPredictionLSTM-Training.ipynb", "max_issues_repo_name": "ashishbajaj91/amazon-sagemaker-aws-greengrass-custom-timeseries-forecasting", "max_issues_repo_head_hexsha": "3c8cfcf54e9354583938da3239e2e26d7cd41566", "max_issues_repo_licenses": ["MIT-0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-02-24T14:59:22.000Z", "max_issues_repo_issues_event_max_datetime": "2021-02-24T14:59:22.000Z", "max_forks_repo_path": "sagemaker/pollutionPredictionLSTM-Training.ipynb", "max_forks_repo_name": "ashishbajaj91/amazon-sagemaker-aws-greengrass-custom-timeseries-forecasting", "max_forks_repo_head_hexsha": "3c8cfcf54e9354583938da3239e2e26d7cd41566", "max_forks_repo_licenses": ["MIT-0"], "max_forks_count": 13, "max_forks_repo_forks_event_min_datetime": "2019-12-01T19:59:09.000Z", "max_forks_repo_forks_event_max_datetime": "2021-09-14T11:18:04.000Z", "avg_line_length": 465.044973545, "max_line_length": 275728, "alphanum_fraction": 0.9317967199, "converted": true, "num_tokens": 12258, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7853085859124002, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.44449921547505855}} {"text": "# \u30ca\u30fc\u30b9\u30b9\u30b1\u30b8\u30e5\u30fc\u30ea\u30f3\u30b0\u554f\u984c\u3092D-Wave 2000Q\u3067\u89e3\u304f(\u691c\u8a3c\u7de8)\n\n## \u6982\u8981\n\u8a18\u4e8b\u300c[\u30ca\u30fc\u30b9\u30b9\u30b1\u30b8\u30e5\u30fc\u30ea\u30f3\u30b0\u554f\u984c\u3092D-Wave 2000Q\u3067\u89e3\u304f](https://qard.is.tohoku.ac.jp/T-Wave/?p=1756)\u300d\u3067\u306f\u3001\u30ca\u30fc\u30b9\u30b9\u30b1\u30b8\u30e5\u30fc\u30ea\u30f3\u30b0\u554f\u984c(NSP)\u3092D-Wave 2000Q\u3067\u89e3\u304f\u3053\u3068\u306b\u3088\u308a\u3001\u91cf\u5b50\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0(QA)\u306e\u8a08\u7b97\u6027\u80fd\u3092\u8a55\u4fa1\u3059\u308b\u8ad6\u6587\u3092\u7d39\u4ecb\u3057\u307e\u3057\u305f\u3002\u7d50\u679c\u3068\u3057\u3066\u3001NSP\u3092\u89e3\u304f\u3053\u3068\u306b\u95a2\u3057\u3066\u306f\u3001QA\u3088\u308a\u3082\u30b7\u30df\u30e5\u30ec\u30fc\u30c6\u30c3\u30c9\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0(SA)\u306e\u65b9\u304c\u6709\u7528\u3067\u3057\u305f\u3002\u3057\u304b\u3057\u3001\u3053\u306e\u8ad6\u6587\u3067\u306f\u5236\u7d04\u306e\u4fc2\u6570\u3092\u4eba\u6570\u3084\u65e5\u6570\u3068\u3044\u3063\u305f\u8a2d\u5b9a\u6bce\u306b\u5909\u3048\u3066\u3044\u306a\u3044\u305f\u3081\u3001\u4fc2\u6570\u3092\u8abf\u6574\u3059\u308b\u3053\u3068\u3067\u3001\u3088\u308a\u826f\u3044\u7d50\u679c\u3092\u5f97\u3089\u308c\u308b\u53ef\u80fd\u6027\u304c\u3042\u308a\u307e\u3059\u3002\u305d\u3053\u3067\u672c\u8a18\u4e8b\u3067\u306f\u307e\u305a\u3001\u6700\u9069\u3060\u3068\u8003\u3048\u3089\u308c\u308b\u4fc2\u6570\u3092\u63a2\u7d22\u3057\u307e\u3059\u3002\u305d\u3057\u3066\u3001\u305d\u306e\u4fc2\u6570\u3092\u7528\u3044\u308b\u3053\u3068\u3067\u3001\u8ad6\u6587\u306e\u7d50\u679c\u3088\u308a\u3082\u57fa\u5e95\u72b6\u614b\u306e\u89e3\u3092\u898b\u3064\u3051\u308b\u78ba\u7387\u304c\u5411\u4e0a\u3059\u308b\u306e\u304b\u691c\u8a3c\u3057\u307e\u3059\u3002\n\n## \u554f\u984c\n\nNSP\u306e\u8a73\u7d30\u306b\u3064\u3044\u3066\u306f\u3001\u5148\u884c\u7814\u7a76\u306e\u89e3\u8aac\u8a18\u4e8b\u300c[\u30ca\u30fc\u30b9\u30b9\u30b1\u30b8\u30e5\u30fc\u30ea\u30f3\u30b0\u554f\u984c\u3092D-Wave 2000Q\u3067\u89e3\u304f](https://qard.is.tohoku.ac.jp/T-Wave/?p=1756)\u300d\u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\n## \u5b9a\u5f0f\u5316\n\n| \u5b9a\u6570 | \u8aac\u660e |\n| ---- | ---- |\n| $n = 1 , \\cdots, N$ | \u770b\u8b77\u5e2b\u306e\u4eba\u6570 |\n| $d = 1 , \\cdots, D$ | \u65e5\u6570 |\n| \u884c\u5217$J_{i(n, d), j\\left(n, d^{\\prime}\\right)}=\\left\\{\\begin{array}{cc} a>0 & d^{\\prime}=d \\pm 1 \\\\ 0 & \\text { otherwise } \\end{array}\\right.$ | \u770b\u8b77\u5e2b$n$\u304c2\u65e5\u4ee5\u4e0a\u9023\u7d9a\u3067\u52e4\u52d9\u3059\u308b\u3068\u30da\u30ca\u30eb\u30c6\u30a3\u3092\u8ab2\u3059 |\n| $W(d)$ | $d$\u65e5\u306b\u5fc5\u8981\u306a\u52b4\u50cd\u529b |\n| $E(n)$ | \u770b\u8b77\u5e2b$n$\u306e\u6301\u3064\u52b4\u50cd\u529b |\n| $F(n)$ | \u770b\u8b77\u5e2b$n$\u304c\u5e0c\u671b\u3059\u308b\u52e4\u52d9\u65e5\u6570 |\n| $G(n, d) = h_{1}(n)h_{2}(d)$ | \u770b\u8b77\u5e2b$n$\u304c$d$\u65e5\u306b\u52e4\u52d9\u5e0c\u671b\u3059\u308b |\n\n\u5909\u6570\u304a\u3088\u3073\u76ee\u7684\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002\u5b9a\u5f0f\u5316\u306e\u8a73\u7d30\u306f\u3001\u8a18\u4e8b\u300c[\u30ca\u30fc\u30b9\u30b9\u30b1\u30b8\u30e5\u30fc\u30ea\u30f3\u30b0\u554f\u984c\u3092D-Wave 2000Q\u3067\u89e3\u304f](nsp.md)\u300d\u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\n**\u5909\u6570:**\n\n$q_{n,d}$: \u770b\u8b77\u5e2b$n$\u304c$d$\u65e5\u306b\u50cd\u304f\u5834\u5408\u306b1\u3001\u50cd\u304b\u306a\u3044\u5834\u5408\u306b0\n\n**\u76ee\u7684\u95a2\u6570:**\n\n$$\n\\begin{align}\nH(\\mathbf{q}) &= \\alpha \\sum_{n, n^{\\prime}}^{N} \\sum_{d, d^{\\prime}}^{D} J_{i(n, d), j\\left(n^{\\prime}, d^{\\prime}\\right)} q_{i(n, d)} q_{j\\left(n^{\\prime}, d^{\\prime}\\right)}\\\\\n&+ \\lambda \\sum_{d}^{D}\\left(\\sum_{n}^{N} E(n) q_{i(n, d)}-W(d)\\right)^{2}\\\\\n&+ \\gamma \\sum_{n}^{N}\\left(\\sum_{d}^{D} h_{1}(n) h_{2}(d) q_{i(n, d)}-F(n)\\right)^{2}\n\\end{align}\n$$\n\n\u305f\u3060\u3057\u3001\u5f8c\u8ff0\u3059\u308b\u5b9f\u9a13\u306e\u305f\u3081\u306b\u3001\u7b2c\u4e00\u9805\u306e\u4fc2\u6570\u3092$\\alpha$\u3068\u3057\u3066\u3044\u307e\u3059\u3002\n\n\n\u8ad6\u6587\u306e\u554f\u984c\u8a2d\u5b9a\u306f\u6b21\u306e\u901a\u308a\u3067\u3057\u305f\u3002\n\n| \u5b9a\u6570 | \u8aac\u660e |\n| ---- | ---- |\n| $N = 3, 4$ | \u770b\u8b77\u5e2b\u306e\u4eba\u6570 |\n| $D = 5, \\cdots, 14$ | \u65e5\u6570 |\n| $h_{1}(n) = 1$ | \u5168\u54e1\u6687 |\n| $h_{2}(d) = 1$ | \u5e38\u306b\u5e73\u65e5 |\n| $E(n) = 1$ | \u5168\u54e1\u304c1\u4eba\u5206\u306e\u52b4\u50cd\u529b\u3092\u6301\u3064 |\n| $W(d) = 1$ | \u5404\u65e5\u306b1\u4eba\u5206\u306e\u52b4\u50cd\u529b\u304c\u5fc5\u8981 |\n\n| \u76ee\u7684\u95a2\u6570\u30fbD-Wave\u30de\u30b7\u30f3\u306e\u8a2d\u5b9a | \u8aac\u660e |\n| ---- | ---- |\n| $a = 7/2$ | 2\u65e5\u4ee5\u4e0a\u52e4\u52d9\u3059\u308b\u3068\u8ab2\u3055\u308c\u308b\u30da\u30ca\u30eb\u30c6\u30a3 |\n| $\\lambda = 0.3, \\gamma = 1.3$ | \u5236\u7d04\u9805\u306e\u4fc2\u6570 |\n| 200 $\\mu s$ | \u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u6642\u9593 |\n| 1000 | \u30b5\u30f3\u30d7\u30eb\u6570 |\n\n\u672c\u8a18\u4e8b\u3067\u306f\u3001\u4e0a\u8a18\u306e\u8a2d\u5b9a\u3067\u5b9f\u9a13\u3092\u884c\u3044\u307e\u3057\u305f\u3002\n\u4ee5\u4e0b\u306b\u3001[PyQUBO](https://pyqubo.readthedocs.io/en/latest/)\u3092\u7528\u3044\u305fQUBO\u751f\u6210\u304b\u3089\u6700\u9069\u5316\u307e\u3067\u306ePython\u30b3\u30fc\u30c9\u3092\u793a\u3057\u307e\u3059\u3002\n\n\u307e\u305a\u3001\u5fc5\u8981\u306a\u30e9\u30a4\u30d6\u30e9\u30ea\u3092\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u3057\u307e\u3059\u3002\n\n\n```python\n!pip install numpy\n!pip install matplotlib\n!pip install dwave-ocean-sdk # D-Wave Ocean SDK\u306bPyQUBO\u304c\u542b\u307e\u308c\u3066\u3044\u307e\u3059\n```\n\n\u6b21\u306b\u521d\u671f\u8a2d\u5b9a\u3092\u3057\u307e\u3059\u3002\n\n\n```python\nN = 3 # \u770b\u8b77\u5e2b\u306e\u6570\nD = 14 # \u65e5\u6570\na = 7 / 2 # 2\u65e5\u4ee5\u4e0a\u9023\u7d9a\u3067\u52e4\u52d9\u3057\u305f\u6642\u306e\u30da\u30ca\u30eb\u30c6\u30a3\nF = [4, 5, 5] # \u5404\u770b\u8b77\u5e2b\u304c\u5e0c\u671b\u3059\u308b\u52e4\u52d9\u65e5\u6570\n```\n\n$J_{i(n, d), j\\left(n, d^{\\prime}\\right)}=\\left\\{\\begin{array}{cc} a>0 & d^{\\prime}=d \\pm 1 \\\\ 0 & \\text { otherwise } \\end{array}\\right.$ \u306f\u3001\u6b21\u306e\u3088\u3046\u306b\u751f\u6210\u3057\u307e\u3059\u3002\n\n\n```python\nfrom itertools import product\nimport numpy as np\n\nJ = np.zeros((N, D, N, D))\nfor n1, d1, n2, d2 in product(range(N), range(D), range(N), range(D)):\n if n1 == n2 and d1 + 1 == d2:\n J[n1, d1, n2, d2] = a\n```\n\n\u5b9a\u6570\u306e\u8a2d\u5b9a\u304c\u7d42\u308f\u3063\u305f\u306e\u3067\u3001\u5b9a\u5f0f\u5316\u3057\u3066\u3044\u304d\u307e\u3059\u3002\n\n\n```python\nfrom itertools import product\nfrom pyqubo import Array, Constraint, Placeholder\n\n# \u30d0\u30a4\u30ca\u30ea\u5909\u6570\nq = Array.create('q', shape=(N, D), vartype='BINARY')\n\n# 2\u65e5\u4ee5\u4e0a\u9023\u7d9a\u3067\u52e4\u52d9\u3059\u308b\u306e\u3092\u907f\u3051\u308b\u305f\u3081\u306e\u9805\nH1 = np.sum([J[n1, d1, n2, d2] * q[n1, d1] * q[n2, d2]\n for n1, n2, d1, d2 in product(range(N), range(N), range(D), range(D))])\n\n# \u5404d\u306b1\u4eba\u306e\u770b\u8b77\u5e2b\u3092\u78ba\u4fdd\u3059\u308b\u305f\u3081\u306e\u9805\nH2 = np.sum([(np.sum([q[n,d] for n in range(N)]) - 1)**2 for d in range(D)])\n\n# \u5168\u54e1\u306e\u51fa\u52e4\u56de\u6570\u3092\u5747\u7b49\u306b\u3059\u308b\u305f\u3081\u306e\u9805\nH3 = np.sum([(np.sum([q[n,d] for d in range(D)]) - F[n])**2 for n in range(N)])\n\n# \u6700\u5c0f\u5316\u3057\u305f\u3044QUBO\nH = Placeholder('alpha') * Constraint(H1, 'H1') + Placeholder('lam') * Constraint(H2, 'H2') + Placeholder('gamma') * H3\nmodel = H.compile()\n```\n\n\n```python\nfeed_dict = {'alpha': 1.0, 'lam': 1.3, 'gamma': 0.3} # \u5236\u7d04\u9805\u306e\u4fc2\u6570\nbqm = model.to_bqm(feed_dict=feed_dict)\n```\n\n\u3053\u3053\u3067\u3001\u30b5\u30f3\u30d7\u30e9\u30fc\u306e\u8a2d\u5b9a\u3092\u3057\u307e\u3059\u3002\n\n\n```python\n# SA\u306e\u5834\u5408\n# from neal import SimulatedAnnealingSampler\n# sampler = SimulatedAnnealingSampler()\n\n# D-Wave\u30de\u30b7\u30f3\u306e\u5834\u5408\nfrom dwave.system import DWaveSampler, EmbeddingComposite\nsampler_config = {'solver': 'DW_2000Q_6', 'token': 'YOUR_TOKEN'}\nsampler = EmbeddingComposite(DWaveSampler(**sampler_config))\n```\n\n\u5168\u3066\u306e\u8a2d\u5b9a\u304c\u7d42\u308f\u308a\u307e\u3057\u305f\u3002\n\u3044\u3088\u3044\u3088\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u3092\u5b9f\u884c\u3057\u307e\u3059\u3002\n\n\n\n\n```python\nnum_reads = 1000\nsampleset = sampler.sample(bqm, num_reads=num_reads)\n```\n\n\u5f97\u3089\u308c\u305f\u89e3\u306f\u3001\u6b21\u306e\u3088\u3046\u306b\u78ba\u8a8d\u3067\u304d\u307e\u3059\u3002\n\n\n```python\nsampleset.record[:10]\n```\n\n\n\n\n rec.array([([0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0], 1.76214598e-12, 1),\n ([1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0], 1.76214598e-12, 1),\n ([1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1], 1.75504056e-12, 1),\n ([0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0], 1.75504056e-12, 1),\n ([0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0], 6.00000000e-01, 1),\n ([0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0], 1.75504056e-12, 1),\n ([1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0], 1.75504056e-12, 1),\n ([0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0], 1.75504056e-12, 1),\n ([0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0], 1.75504056e-12, 1),\n ([0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1], 1.75504056e-12, 1)],\n dtype=[('sample', 'i1', (42,)), ('energy', 'Email me with your thoughts, comments, suggestions and corrections \n
Linear Algebra OCW MIT18.06 IPython notebook [2] study notes by Dr Juan H Klopper is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.\n\n+ [1] OCW MIT 18.06\n+ [2] Fernando P\u00e9rez, Brian E. Granger, IPython: A System for Interactive Scientific Computing, Computing in Science and Engineering, vol. 9, no. 3, pp. 21-29, May/June 2007, doi:10.1109/MCSE.2007.53. URL: http://ipython.org\n\n\n```python\nfrom IPython.core.display import HTML, Image\ncss_file = 'style.css'\nHTML(open(css_file, 'r').read())\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n```python\n#import numpy as np\nfrom sympy import init_printing, Matrix, symbols\n#import matplotlib.pyplot as plt\n#import seaborn as sns\n#from IPython.display import Image\nfrom warnings import filterwarnings\n\ninit_printing(use_latex = 'mathjax') # Pretty Latex printing to the screen\n#%matplotlib inline\nfilterwarnings('ignore')\n```\n\n# Matrix spaces\n\n## New *vector* / matrix spaces\n\n### Square matrices\n\n* Consider *M* to be all 3×3 matrices (with real elements)\n\n* Subspaces would be:\n * Upper or lower triangular matrices\n * Symmetric matrices\n\n* Basis would be:\n$$ \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix},\\begin{bmatrix} 0 & 1 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix},\\begin{bmatrix} 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix},\\begin{bmatrix} 0 & 0 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix},\\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix},\\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 0 & 0 \\end{bmatrix},\\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 1 & 0 & 0 \\end{bmatrix},\\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 1 & 0 \\end{bmatrix},\\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix} $$\n\n* The dimension would be 9\n\n* For upper and lower triangular matrices the dimensions would be 6 and the basis:\n$$ \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix},\\begin{bmatrix} 0 & 1 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix},\\begin{bmatrix} 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix},\\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix},\\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 0 & 0 \\end{bmatrix},\\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix} $$\n\n* For symmetric matrices the dimension would also be six ( Knowing the diagonal and entries on one of the two sides)\n\n* These are unique cases where the bases for the subspaces are contained in the basis of the 3×3 matrix *M*\n\n### Other square matrices that are subspaces of *M*\n\n* The intersection of symmetric and upper triangular matrices (that is symmetric AND upper triangular, *S*∩*U*)\n * This is a diagonal matrix\n * The dimension is 3\n * The basis is\n$$ \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix},\\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix},\\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix} $$\n\n* The union of symmetric and upper triangular matrices (that is symmetric OR upper triangular, *S*∪*U*)\n * It is **NOT** a subspace\n\n* The addition (sum) of symmetric and upper triangular matrices\n * It **IS** a subspace\n * It is actually all 3×3 matrices\n * The dimension is 9\n\n* This gives the equation: dim(*S*) + dim(*U*) = dim(*S*∩*U*) + dim(*S*+*U*) = 12\n\n## Example problems\n\n### Example problem 1\n\n* Show that the set of 2×3 matrices whose nullspace contains the column vector below is a vector subspace and find a basis for it\n$$ \\begin{bmatrix} 2 \\\\ 1 \\\\ 1 \\end{bmatrix} $$\n\n#### Solution\n\n* In essence we have to show the following\n$$ A\\begin{bmatrix} 2 \\\\ 1 \\\\ 1 \\end{bmatrix}=\\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}\\\\ \\begin{bmatrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\\\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\end{bmatrix}\\begin{bmatrix} 2 \\\\ 1 \\\\ 1 \\end{bmatrix}=\\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix} $$\n* ... and ...\n$$ B\\begin{bmatrix} 2 \\\\ 1 \\\\ 1 \\end{bmatrix}=\\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix} $$\n\n* This can be shown by addition:\n$$ \\left( A+B \\right) \\begin{bmatrix} 2 \\\\ 1 \\\\ 1 \\end{bmatrix}=\\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}=A\\begin{bmatrix} 2 \\\\ 1 \\\\ 1 \\end{bmatrix}+B\\begin{bmatrix} 2 \\\\ 1 \\\\ 1 \\end{bmatrix}=\\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix} $$\n* Therefor (by virtue of the fact that addition remains in the nullspace) the set is vector subspace\n\n* We also need to look at scalar multiplication (if we multiply a matrix in the set by a scalar, does it remain in the set)\n$$ \\left( cA \\right) \\begin{bmatrix} 2 \\\\ 1 \\\\ 1 \\end{bmatrix}=c\\left( A\\begin{bmatrix} 2 \\\\ 1 \\\\ 1 \\end{bmatrix}= \\right) c\\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}=\\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix} $$\n\n#### Example problem 2\n\n* Find a basis for the nullspace above\n\n#### Solution\n\n* Let's look at the first row:\n$$ \\begin{bmatrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\end{bmatrix}\\begin{bmatrix} 2 \\\\ 1 \\\\ 1 \\end{bmatrix}=\\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}\\\\ 2{ a }_{ 11 }+{ a }_{ 12 }+{ a }_{ 13 }=0\\\\ { a }_{ 13 }=-2{ a }_{ 11 }-{ a }_{ 12 } $$\n\n* From this we can make the following row vectors\n$$ \\begin{bmatrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\end{bmatrix}=\\begin{bmatrix} { a }_{ 11 } & { a }_{ 12 } & \\left( -2{ a }_{ 11 }-{ a }_{ 12 } \\right) \\end{bmatrix}\\\\ =\\quad \\begin{bmatrix} { a }_{ 11 } & 0 & { -2a }_{ 11 } \\end{bmatrix}+\\begin{bmatrix} 0 & { a }_{ 12 } & { -a }_{ 12 } \\end{bmatrix}\\\\ ={ a }_{ 11 }\\begin{bmatrix} 1 & 0 & -2 \\end{bmatrix}+{ a }_{ 12 }\\begin{bmatrix} 0 & 1 & -1 \\end{bmatrix} $$\n\n* From this we can construct 4 basis:\n$$ \\begin{bmatrix} 1 & 0 & -2 \\\\ 0 & 0 & 0 \\end{bmatrix},\\begin{bmatrix} 0 & 1 & -1 \\\\ 0 & 0 & 0 \\end{bmatrix},\\begin{bmatrix} 0 & 0 & 0 \\\\ 1 & 0 & -2 \\end{bmatrix},\\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 1 & -1 \\end{bmatrix} $$\n\n#### Example problem 3\n\n* What about the set of those whose column space contains the following column vector?\n$$ \\begin{bmatrix} 2 \\\\ 1 \\end{bmatrix} $$\n\n#### Solution\n\n* Well, any subspace must contain the zero matrix\n$$ \\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix} $$\n* It does not contain the above column vector, which is therefor not a subspace\n\n\n```python\n\n```\n", "meta": {"hexsha": "bdafb8f9fc5862a2d5e3a25ed047c691e6fd4e68", "size": 14576, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "forks/MIT_OCW_Linear_Algebra_18_06-master/I_12_Matrix_spaces.ipynb", "max_stars_repo_name": "solomonxie/jupyter-notebooks", "max_stars_repo_head_hexsha": "65999f179e037242138de72f512dda4bf00c7379", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-02-13T05:52:05.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-08T09:52:35.000Z", "max_issues_repo_path": "forks/MIT_OCW_Linear_Algebra_18_06-master/I_12_Matrix_spaces.ipynb", "max_issues_repo_name": "solomonxie/jupyter-notebooks", "max_issues_repo_head_hexsha": "65999f179e037242138de72f512dda4bf00c7379", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, 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YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.8128673110375457, "lm_q1q2_score": 0.44442557144537004}} {"text": "\n\n\n```python\n%matplotlib inline\n```\n\n\n# Sequence-to-Sequence Modeling with nn.Transformer and TorchText\n\nThis is a tutorial on how to train a sequence-to-sequence model\nthat uses the\n[`nn.Transformer`](https://pytorch.org/docs/master/nn.html?highlight=nn%20transformer#torch.nn.Transformer) module.\n\nPyTorch 1.2 release includes a standard transformer module based on the paper [Attention is All You Need](https://arxiv.org/pdf/1706.03762.pdf). \nThe transformer model has been proved to be superior in quality for many sequence-to-sequence problems while being more parallelizable. \nThe `nn.Transformer` module relies entirely on an attention mechanism (another module recently implemented as [`nn.MultiheadAttention`](https://pytorch.org/docs/master/nn.html?highlight=multiheadattention#torch.nn.MultiheadAttention) to draw global dependencies between input and output. \nThe `nn.Transformer` module is now highly modularized such that a single component (like [`nn.TransformerEncoder`](https://pytorch.org/docs/master/nn.html?highlight=nn%20transformerencoder#torch.nn.TransformerEncoder) in this tutorial) can be easily adapted/composed.\n\n\n\n\n\n## Define the model\n\n\n\n\nIn this tutorial, we train ``nn.TransformerEncoder`` model on a language modeling task. \nThe language modeling task is to assign a probability for the likelihood of a given word (or a sequence of words) to follow a sequence of words. \nA sequence of tokens are passed to the embedding layer first, followed by a positional encoding layer to account for the order of the word (see the next paragraph for more details). \nThe ``nn.TransformerEncoder`` consists of multiple layers of [`nn.TransformerEncoderLayer`](https://pytorch.org/docs/master/nn.html?highlight=transformerencoderlayer#torch.nn.TransformerEncoderLayer). \nAlong with the input sequence, a square attention mask is required because the self-attention layers in ``nn.TransformerEncoder`` are only allowed to attend the earlier positions in the sequence. \nFor the language modeling task, any tokens on the future positions should be masked. To have the actual words, the output of ``nn.TransformerEncoder`` model is sent to the final Linear layer, which is followed by a log-Softmax function.\n\n\n\n\n\n```python\nimport math\nimport torch\nimport torch.nn as nn\nimport torch.nn.functional as F\n\nclass TransformerModel(nn.Module):\n\n def __init__(self, ntoken, ninp, nhead, nhid, nlayers, dropout=0.5):\n super(TransformerModel, self).__init__()\n from torch.nn import TransformerEncoder, TransformerEncoderLayer\n self.model_type = 'Transformer'\n self.src_mask = None\n self.pos_encoder = PositionalEncoding(ninp, dropout)\n encoder_layers = TransformerEncoderLayer(ninp, nhead, nhid, dropout)\n self.transformer_encoder = TransformerEncoder(encoder_layers, nlayers)\n self.encoder = nn.Embedding(ntoken, ninp)\n self.ninp = ninp\n self.decoder = nn.Linear(ninp, ntoken)\n\n self.init_weights()\n\n def _generate_square_subsequent_mask(self, sz):\n mask = (torch.triu(torch.ones(sz, sz)) == 1).transpose(0, 1)\n mask = mask.float().masked_fill(mask == 0, float('-inf')).masked_fill(mask == 1, float(0.0))\n return mask\n\n def init_weights(self):\n initrange = 0.1\n self.encoder.weight.data.uniform_(-initrange, initrange)\n self.decoder.bias.data.zero_()\n self.decoder.weight.data.uniform_(-initrange, initrange)\n\n def forward(self, src):\n if self.src_mask is None or self.src_mask.size(0) != len(src):\n device = src.device\n mask = self._generate_square_subsequent_mask(len(src)).to(device)\n self.src_mask = mask\n\n src = self.encoder(src) * math.sqrt(self.ninp)\n src = self.pos_encoder(src)\n output = self.transformer_encoder(src, self.src_mask)\n output = self.decoder(output)\n return output\n```\n\n``PositionalEncoding`` module injects some information about the relative or absolute position of the tokens in the sequence. \nThe positional encodings have the same dimension as the embeddings so that the two can be summed. Here, we use ``sine`` and ``cosine`` functions of different frequencies.\n\n\n\n\n\n```python\nclass PositionalEncoding(nn.Module):\n\n def __init__(self, d_model, dropout=0.1, max_len=5000):\n super(PositionalEncoding, self).__init__()\n self.dropout = nn.Dropout(p=dropout)\n\n pe = torch.zeros(max_len, d_model)\n position = torch.arange(0, max_len, dtype=torch.float).unsqueeze(1)\n div_term = torch.exp(torch.arange(0, d_model, 2).float() * (-math.log(10000.0) / d_model))\n pe[:, 0::2] = torch.sin(position * div_term)\n pe[:, 1::2] = torch.cos(position * div_term)\n pe = pe.unsqueeze(0).transpose(0, 1)\n self.register_buffer('pe', pe)\n\n def forward(self, x):\n x = x + self.pe[:x.size(0), :]\n return self.dropout(x)\n```\n\nLoad and batch data\n-------------------\n\n\n\n\nThe training process uses Wikitext-2 dataset from ``torchtext``. \nThe vocab object is built based on the train dataset and is used to numericalize tokens into tensors. \nStarting from sequential data, the ``batchify()`` function arranges the dataset into columns, trimming off any tokens remaining after the data has been divided into batches of size ``batch_size``.\nFor instance, with the alphabet as the sequence (total length of 26) and a batch size of 4, we would divide the alphabet into 4 sequences of length 6:\n\n\\begin{align}\\begin{bmatrix}\n \\text{A} & \\text{B} & \\text{C} & \\ldots & \\text{X} & \\text{Y} & \\text{Z}\n \\end{bmatrix}\n \\Rightarrow\n \\begin{bmatrix}\n \\begin{bmatrix}\\text{A} \\\\ \\text{B} \\\\ \\text{C} \\\\ \\text{D} \\\\ \\text{E} \\\\ \\text{F}\\end{bmatrix} &\n \\begin{bmatrix}\\text{G} \\\\ \\text{H} \\\\ \\text{I} \\\\ \\text{J} \\\\ \\text{K} \\\\ \\text{L}\\end{bmatrix} &\n \\begin{bmatrix}\\text{M} \\\\ \\text{N} \\\\ \\text{O} \\\\ \\text{P} \\\\ \\text{Q} \\\\ \\text{R}\\end{bmatrix} &\n \\begin{bmatrix}\\text{S} \\\\ \\text{T} \\\\ \\text{U} \\\\ \\text{V} \\\\ \\text{W} \\\\ \\text{X}\\end{bmatrix}\n \\end{bmatrix}\\end{align}\n\nThese columns are treated as independent by the model, which means that the dependence of ``G`` and ``F`` can not be learned, but allows more efficient batch processing.\n\n\n\n\n\n```python\nimport torchtext\nfrom torchtext.data.utils import get_tokenizer\n#TEXT = torchtext.data.Field(tokenize=get_tokenizer(\"basic_english\"),\nTEXT = torchtext.data.Field(tokenize=get_tokenizer(\"spacy\"),\n#TEXT = torchtext.data.Field(tokenize=get_tokenizer(\"moses\"),\n init_token='',\n eos_token='',\n lower=True)\ntrain_txt, val_txt, test_txt = torchtext.datasets.WikiText2.splits(TEXT)\nTEXT.build_vocab(train_txt)\ndevice = torch.device(\"cuda\" if torch.cuda.is_available() else \"cpu\")\n\ndef batchify(data, bsz):\n data = TEXT.numericalize([data.examples[0].text])\n # Divide the dataset into bsz parts.\n nbatch = data.size(0) // bsz\n # Trim off any extra elements that wouldn't cleanly fit (remainders).\n data = data.narrow(0, 0, nbatch * bsz)\n # Evenly divide the data across the bsz batches.\n data = data.view(bsz, -1).t().contiguous()\n return data.to(device)\n\nbatch_size = 20\neval_batch_size = 10\ntrain_data = batchify(train_txt, batch_size)\nval_data = batchify(val_txt, eval_batch_size)\ntest_data = batchify(test_txt, eval_batch_size)\n```\n\n downloading wikitext-2-v1.zip\n\n\n wikitext-2-v1.zip: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 4.48M/4.48M [00:00<00:00, 8.66MB/s]\n\n\n extracting\n\n\n### Functions to generate input and target sequence\n\n\n\n\n``get_batch()`` function generates the input and target sequence for the transformer model. \nIt subdivides the source data into chunks of length ``bptt``. \nFor the language modeling task, the model needs the following words as ``Target``. \nFor example, with a ``bptt`` value of 2, we\u2019d get the following two Variables for ``i`` = 0:\n\n\n\n\nIt should be noted that the chunks are along dimension 0, consistent with the ``S`` dimension in the Transformer model. The batch dimension ``N`` is along dimension 1.\n\n\n\n\n\n```python\nbptt = 35\ndef get_batch(source, i):\n seq_len = min(bptt, len(source) - 1 - i)\n data = source[i:i+seq_len]\n target = source[i+1:i+1+seq_len].view(-1)\n return data, target\n```\n\nInitiate an instance\n--------------------\n\n\n\n\nThe model is set up with the hyperparameter below. The vocab size is\nequal to the length of the vocab object.\n\n\n\n\n\n```python\nntokens = len(TEXT.vocab.stoi) # the size of vocabulary\nemsize = 200 # embedding dimension\nnhid = 200 # the dimension of the feedforward network model in nn.TransformerEncoder\nnlayers = 2 # the number of nn.TransformerEncoderLayer in nn.TransformerEncoder\nnhead = 2 # the number of heads in the multiheadattention models\ndropout = 0.2 # the dropout value\nmodel = TransformerModel(ntokens, emsize, nhead, nhid, nlayers, dropout).to(device)\n```\n\nRun the model\n-------------\n\n\n\n\n[`CrossEntropyLoss`](https://pytorch.org/docs/master/nn.html?highlight=crossentropyloss#torch.nn.CrossEntropyLoss) is applied to track the loss and [`SGD`](https://pytorch.org/docs/master/optim.html?highlight=sgd#torch.optim.SGD) implements stochastic gradient descent method as the optimizer. \nThe initial learning rate is set to 5.0. \n[`StepLR`](https://pytorch.org/docs/master/optim.html?highlight=steplr#torch.optim.lr_scheduler.StepLR) is applied to adjust the learn rate through epochs. \nDuring the training, we use [`nn.utils.clip_grad_norm_`](https://pytorch.org/docs/master/nn.html?highlight=nn%20utils%20clip_grad_norm#torch.nn.utils.clip_grad_norm_) function to scale all the gradient together to prevent exploding.\n\n\n\n\n\n```python\ncriterion = nn.CrossEntropyLoss()\nlr = 5.0 # learning rate\noptimizer = torch.optim.SGD(model.parameters(), lr=lr)\nscheduler = torch.optim.lr_scheduler.StepLR(optimizer, 1.0, gamma=0.95)\n\nimport time\ndef train():\n model.train() # Turn on the train mode\n total_loss = 0.\n start_time = time.time()\n ntokens = len(TEXT.vocab.stoi)\n for batch, i in enumerate(range(0, train_data.size(0) - 1, bptt)):\n data, targets = get_batch(train_data, i)\n optimizer.zero_grad()\n output = model(data)\n loss = criterion(output.view(-1, ntokens), targets)\n loss.backward()\n torch.nn.utils.clip_grad_norm_(model.parameters(), 0.5)\n optimizer.step()\n\n total_loss += loss.item()\n log_interval = 200\n if batch % log_interval == 0 and batch > 0:\n cur_loss = total_loss / log_interval\n elapsed = time.time() - start_time\n print('| epoch {:3d} | {:5d}/{:5d} batches | '\n 'lr {:02.2f} | ms/batch {:5.2f} | '\n 'loss {:5.2f} | ppl {:8.2f}'.format(\n epoch, batch, len(train_data) // bptt, scheduler.get_lr()[0],\n elapsed * 1000 / log_interval,\n cur_loss, math.exp(cur_loss)))\n total_loss = 0\n start_time = time.time()\n\ndef evaluate(eval_model, data_source):\n eval_model.eval() # Turn on the evaluation mode\n total_loss = 0.\n ntokens = len(TEXT.vocab.stoi)\n with torch.no_grad():\n for i in range(0, data_source.size(0) - 1, bptt):\n data, targets = get_batch(data_source, i)\n output = eval_model(data)\n output_flat = output.view(-1, ntokens)\n total_loss += len(data) * criterion(output_flat, targets).item()\n return total_loss / (len(data_source) - 1)\n```\n\nLoop over epochs. Save the model if the validation loss is the best\nwe've seen so far. Adjust the learning rate after each epoch.\n\n\n\n\n```python\nbest_val_loss = float(\"inf\")\nepochs = 10 # The number of epochs\nbest_model = None\n\nfor epoch in range(1, epochs + 1):\n epoch_start_time = time.time()\n train()\n val_loss = evaluate(model, val_data)\n print('-' * 89)\n print('| end of epoch {:3d} | time: {:5.2f}s | valid loss {:5.2f} | '\n 'valid ppl {:8.2f}'.format(epoch, (time.time() - epoch_start_time),\n val_loss, math.exp(val_loss)))\n print('-' * 89)\n\n if val_loss < best_val_loss:\n best_val_loss = val_loss\n best_model = model\n\n scheduler.step()\n```\n\n /usr/local/lib/python3.6/dist-packages/torch/optim/lr_scheduler.py:351: UserWarning: To get the last learning rate computed by the scheduler, please use `get_last_lr()`.\n \"please use `get_last_lr()`.\", UserWarning)\n\n\n | epoch 1 | 200/ 3195 batches | lr 2.44 | ms/batch 11.64 | loss 4.30 | ppl 73.64\n | epoch 1 | 400/ 3195 batches | lr 2.44 | ms/batch 11.55 | loss 4.33 | ppl 75.62\n | epoch 1 | 600/ 3195 batches | lr 2.44 | ms/batch 11.55 | loss 4.22 | ppl 67.98\n | epoch 1 | 800/ 3195 batches | lr 2.44 | ms/batch 11.60 | loss 4.19 | ppl 66.25\n | epoch 1 | 1000/ 3195 batches | lr 2.44 | ms/batch 11.51 | loss 4.33 | ppl 76.24\n | epoch 1 | 1200/ 3195 batches | lr 2.44 | ms/batch 11.47 | loss 4.29 | ppl 72.90\n | epoch 1 | 1400/ 3195 batches | lr 2.44 | ms/batch 11.47 | loss 4.32 | ppl 75.56\n | epoch 1 | 1600/ 3195 batches | lr 2.44 | ms/batch 11.57 | loss 4.27 | ppl 71.31\n | epoch 1 | 1800/ 3195 batches | lr 2.44 | ms/batch 11.50 | loss 4.33 | ppl 75.59\n | epoch 1 | 2000/ 3195 batches | lr 2.44 | ms/batch 11.48 | loss 4.36 | ppl 78.14\n | epoch 1 | 2200/ 3195 batches | lr 2.44 | ms/batch 11.49 | loss 4.31 | ppl 74.23\n | epoch 1 | 2400/ 3195 batches | lr 2.44 | ms/batch 11.55 | loss 4.17 | ppl 64.78\n | epoch 1 | 2600/ 3195 batches | lr 2.44 | ms/batch 11.59 | loss 4.28 | ppl 72.48\n | epoch 1 | 2800/ 3195 batches | lr 2.44 | ms/batch 11.51 | loss 4.27 | ppl 71.81\n | epoch 1 | 3000/ 3195 batches | lr 2.44 | ms/batch 11.47 | loss 4.23 | ppl 68.58\n -----------------------------------------------------------------------------------------\n | end of epoch 1 | time: 38.80s | valid loss 4.92 | valid ppl 137.19\n -----------------------------------------------------------------------------------------\n | epoch 2 | 200/ 3195 batches | lr 2.32 | ms/batch 11.55 | loss 4.26 | ppl 70.52\n | epoch 2 | 400/ 3195 batches | lr 2.32 | ms/batch 11.51 | loss 4.28 | ppl 72.45\n | epoch 2 | 600/ 3195 batches | lr 2.32 | ms/batch 11.54 | loss 4.18 | ppl 65.23\n | epoch 2 | 800/ 3195 batches | lr 2.32 | ms/batch 11.48 | loss 4.15 | ppl 63.51\n | epoch 2 | 1000/ 3195 batches | lr 2.32 | ms/batch 11.47 | loss 4.29 | ppl 73.00\n | epoch 2 | 1200/ 3195 batches | lr 2.32 | ms/batch 11.46 | loss 4.25 | ppl 69.88\n | epoch 2 | 1400/ 3195 batches | lr 2.32 | ms/batch 11.51 | loss 4.28 | ppl 72.57\n | epoch 2 | 1600/ 3195 batches | lr 2.32 | ms/batch 11.49 | loss 4.22 | ppl 68.28\n | epoch 2 | 1800/ 3195 batches | lr 2.32 | ms/batch 11.50 | loss 4.28 | ppl 72.16\n | epoch 2 | 2000/ 3195 batches | lr 2.32 | ms/batch 11.47 | loss 4.31 | ppl 74.81\n | epoch 2 | 2200/ 3195 batches | lr 2.32 | ms/batch 11.56 | loss 4.26 | ppl 70.64\n | epoch 2 | 2400/ 3195 batches | lr 2.32 | ms/batch 11.54 | loss 4.14 | ppl 62.55\n | epoch 2 | 2600/ 3195 batches | lr 2.32 | ms/batch 11.53 | loss 4.24 | ppl 69.16\n | epoch 2 | 2800/ 3195 batches | lr 2.32 | ms/batch 11.53 | loss 4.23 | ppl 68.72\n | epoch 2 | 3000/ 3195 batches | lr 2.32 | ms/batch 11.51 | loss 4.19 | ppl 65.97\n -----------------------------------------------------------------------------------------\n | end of epoch 2 | time: 38.74s | valid loss 4.94 | valid ppl 139.39\n -----------------------------------------------------------------------------------------\n | epoch 3 | 200/ 3195 batches | lr 2.20 | ms/batch 11.69 | loss 4.21 | ppl 67.57\n | epoch 3 | 400/ 3195 batches | lr 2.20 | ms/batch 11.55 | loss 4.24 | ppl 69.67\n | epoch 3 | 600/ 3195 batches | lr 2.20 | ms/batch 11.53 | loss 4.14 | ppl 62.55\n | epoch 3 | 800/ 3195 batches | lr 2.20 | ms/batch 11.49 | loss 4.11 | ppl 61.23\n | epoch 3 | 1000/ 3195 batches | lr 2.20 | ms/batch 11.50 | loss 4.25 | ppl 70.45\n | epoch 3 | 1200/ 3195 batches | lr 2.20 | ms/batch 11.47 | loss 4.21 | ppl 67.59\n | epoch 3 | 1400/ 3195 batches | lr 2.20 | ms/batch 11.49 | loss 4.24 | ppl 69.69\n | epoch 3 | 1600/ 3195 batches | lr 2.20 | ms/batch 11.43 | loss 4.18 | ppl 65.42\n | epoch 3 | 1800/ 3195 batches | lr 2.20 | ms/batch 11.49 | loss 4.25 | ppl 69.82\n | epoch 3 | 2000/ 3195 batches | lr 2.20 | ms/batch 11.52 | loss 4.28 | ppl 72.38\n | epoch 3 | 2200/ 3195 batches | lr 2.20 | ms/batch 11.45 | loss 4.22 | ppl 68.36\n | epoch 3 | 2400/ 3195 batches | lr 2.20 | ms/batch 11.48 | loss 4.10 | ppl 60.41\n | epoch 3 | 2600/ 3195 batches | lr 2.20 | ms/batch 11.47 | loss 4.20 | ppl 66.84\n | epoch 3 | 2800/ 3195 batches | lr 2.20 | ms/batch 11.49 | loss 4.20 | ppl 66.59\n | epoch 3 | 3000/ 3195 batches | lr 2.20 | ms/batch 11.55 | loss 4.15 | ppl 63.73\n -----------------------------------------------------------------------------------------\n | end of epoch 3 | time: 38.69s | valid loss 4.95 | valid ppl 141.09\n -----------------------------------------------------------------------------------------\n | epoch 4 | 200/ 3195 batches | lr 2.09 | ms/batch 11.50 | loss 4.19 | ppl 65.77\n | epoch 4 | 400/ 3195 batches | lr 2.09 | ms/batch 11.45 | loss 4.21 | ppl 67.21\n | epoch 4 | 600/ 3195 batches | lr 2.09 | ms/batch 11.42 | loss 4.10 | ppl 60.55\n | epoch 4 | 800/ 3195 batches | lr 2.09 | ms/batch 11.46 | loss 4.09 | ppl 59.59\n | epoch 4 | 1000/ 3195 batches | lr 2.09 | ms/batch 11.39 | loss 4.22 | ppl 68.14\n | epoch 4 | 1200/ 3195 batches | lr 2.09 | ms/batch 11.46 | loss 4.18 | ppl 65.41\n | epoch 4 | 1400/ 3195 batches | lr 2.09 | ms/batch 11.50 | loss 4.21 | ppl 67.52\n | epoch 4 | 1600/ 3195 batches | lr 2.09 | ms/batch 11.45 | loss 4.15 | ppl 63.51\n | epoch 4 | 1800/ 3195 batches | lr 2.09 | ms/batch 11.46 | loss 4.21 | ppl 67.24\n | epoch 4 | 2000/ 3195 batches | lr 2.09 | ms/batch 11.47 | loss 4.25 | ppl 69.80\n | epoch 4 | 2200/ 3195 batches | lr 2.09 | ms/batch 11.47 | loss 4.19 | ppl 66.05\n | epoch 4 | 2400/ 3195 batches | lr 2.09 | ms/batch 11.50 | loss 4.07 | ppl 58.29\n | epoch 4 | 2600/ 3195 batches | lr 2.09 | ms/batch 11.53 | loss 4.17 | ppl 64.53\n | epoch 4 | 2800/ 3195 batches | lr 2.09 | ms/batch 11.55 | loss 4.16 | ppl 64.11\n | epoch 4 | 3000/ 3195 batches | lr 2.09 | ms/batch 11.43 | loss 4.12 | ppl 61.39\n -----------------------------------------------------------------------------------------\n | end of epoch 4 | time: 38.57s | valid loss 4.94 | valid ppl 140.24\n -----------------------------------------------------------------------------------------\n | epoch 5 | 200/ 3195 batches | lr 1.99 | ms/batch 11.55 | loss 4.15 | ppl 63.13\n | epoch 5 | 400/ 3195 batches | lr 1.99 | ms/batch 11.44 | loss 4.18 | ppl 65.10\n | epoch 5 | 600/ 3195 batches | lr 1.99 | ms/batch 11.51 | loss 4.07 | ppl 58.53\n | epoch 5 | 800/ 3195 batches | lr 1.99 | ms/batch 11.47 | loss 4.05 | ppl 57.53\n | epoch 5 | 1000/ 3195 batches | lr 1.99 | ms/batch 11.48 | loss 4.19 | ppl 66.02\n | epoch 5 | 1200/ 3195 batches | lr 1.99 | ms/batch 11.49 | loss 4.14 | ppl 62.98\n | epoch 5 | 1400/ 3195 batches | lr 1.99 | ms/batch 11.46 | loss 4.18 | ppl 65.26\n | epoch 5 | 1600/ 3195 batches | lr 1.99 | ms/batch 11.49 | loss 4.11 | ppl 61.23\n | epoch 5 | 1800/ 3195 batches | lr 1.99 | ms/batch 11.46 | loss 4.18 | ppl 65.20\n | epoch 5 | 2000/ 3195 batches | lr 1.99 | ms/batch 11.52 | loss 4.22 | ppl 67.92\n | epoch 5 | 2200/ 3195 batches | lr 1.99 | ms/batch 11.52 | loss 4.16 | ppl 64.12\n | epoch 5 | 2400/ 3195 batches | lr 1.99 | ms/batch 11.59 | loss 4.03 | ppl 56.27\n | epoch 5 | 2600/ 3195 batches | lr 1.99 | ms/batch 11.56 | loss 4.13 | ppl 62.20\n | epoch 5 | 2800/ 3195 batches | lr 1.99 | ms/batch 11.50 | loss 4.13 | ppl 62.10\n | epoch 5 | 3000/ 3195 batches | lr 1.99 | ms/batch 11.49 | loss 4.08 | ppl 59.44\n -----------------------------------------------------------------------------------------\n | end of epoch 5 | time: 38.68s | valid loss 4.95 | valid ppl 141.03\n -----------------------------------------------------------------------------------------\n | epoch 6 | 200/ 3195 batches | lr 1.89 | ms/batch 11.49 | loss 4.11 | ppl 60.95\n | epoch 6 | 400/ 3195 batches | lr 1.89 | ms/batch 11.47 | loss 4.14 | ppl 62.83\n | epoch 6 | 600/ 3195 batches | lr 1.89 | ms/batch 11.46 | loss 4.04 | ppl 56.84\n | epoch 6 | 800/ 3195 batches | lr 1.89 | ms/batch 11.47 | loss 4.03 | ppl 56.07\n | epoch 6 | 1000/ 3195 batches | lr 1.89 | ms/batch 11.55 | loss 4.16 | ppl 64.04\n | epoch 6 | 1200/ 3195 batches | lr 1.89 | ms/batch 11.54 | loss 4.12 | ppl 61.46\n | epoch 6 | 1400/ 3195 batches | lr 1.89 | ms/batch 11.44 | loss 4.15 | ppl 63.38\n | epoch 6 | 1600/ 3195 batches | lr 1.89 | ms/batch 11.41 | loss 4.08 | ppl 59.31\n | epoch 6 | 1800/ 3195 batches | lr 1.89 | ms/batch 11.48 | loss 4.15 | ppl 63.35\n | epoch 6 | 2000/ 3195 batches | lr 1.89 | ms/batch 11.41 | loss 4.18 | ppl 65.42\n | epoch 6 | 2200/ 3195 batches | lr 1.89 | ms/batch 11.51 | loss 4.13 | ppl 62.19\n | epoch 6 | 2400/ 3195 batches | lr 1.89 | ms/batch 11.46 | loss 4.00 | ppl 54.56\n | epoch 6 | 2600/ 3195 batches | lr 1.89 | ms/batch 11.47 | loss 4.10 | ppl 60.26\n | epoch 6 | 2800/ 3195 batches | lr 1.89 | ms/batch 11.46 | loss 4.09 | ppl 60.02\n | epoch 6 | 3000/ 3195 batches | lr 1.89 | ms/batch 11.45 | loss 4.06 | ppl 57.82\n -----------------------------------------------------------------------------------------\n | end of epoch 6 | time: 38.57s | valid loss 4.95 | valid ppl 140.76\n -----------------------------------------------------------------------------------------\n | epoch 7 | 200/ 3195 batches | lr 1.79 | ms/batch 11.53 | loss 4.09 | ppl 59.48\n | epoch 7 | 400/ 3195 batches | lr 1.79 | ms/batch 11.43 | loss 4.12 | ppl 61.42\n | epoch 7 | 600/ 3195 batches | lr 1.79 | ms/batch 11.45 | loss 4.01 | ppl 55.27\n | epoch 7 | 800/ 3195 batches | lr 1.79 | ms/batch 11.48 | loss 3.99 | ppl 54.23\n | epoch 7 | 1000/ 3195 batches | lr 1.79 | ms/batch 11.43 | loss 4.13 | ppl 62.08\n | epoch 7 | 1200/ 3195 batches | lr 1.79 | ms/batch 11.44 | loss 4.09 | ppl 59.74\n | epoch 7 | 1400/ 3195 batches | lr 1.79 | ms/batch 11.38 | loss 4.12 | ppl 61.53\n | epoch 7 | 1600/ 3195 batches | lr 1.79 | ms/batch 11.39 | loss 4.05 | ppl 57.58\n | epoch 7 | 1800/ 3195 batches | lr 1.79 | ms/batch 11.40 | loss 4.11 | ppl 61.25\n | epoch 7 | 2000/ 3195 batches | lr 1.79 | ms/batch 11.47 | loss 4.16 | ppl 64.01\n | epoch 7 | 2200/ 3195 batches | lr 1.79 | ms/batch 11.43 | loss 4.10 | ppl 60.42\n | epoch 7 | 2400/ 3195 batches | lr 1.79 | ms/batch 11.42 | loss 3.97 | ppl 53.14\n | epoch 7 | 2600/ 3195 batches | lr 1.79 | ms/batch 11.41 | loss 4.07 | ppl 58.66\n | epoch 7 | 2800/ 3195 batches | lr 1.79 | ms/batch 11.53 | loss 4.06 | ppl 58.17\n | epoch 7 | 3000/ 3195 batches | lr 1.79 | ms/batch 11.49 | loss 4.03 | ppl 56.04\n -----------------------------------------------------------------------------------------\n | end of epoch 7 | time: 38.48s | valid loss 4.96 | valid ppl 142.14\n -----------------------------------------------------------------------------------------\n | epoch 8 | 200/ 3195 batches | lr 1.70 | ms/batch 11.46 | loss 4.06 | ppl 57.71\n | epoch 8 | 400/ 3195 batches | lr 1.70 | ms/batch 11.48 | loss 4.09 | ppl 59.51\n | epoch 8 | 600/ 3195 batches | lr 1.70 | ms/batch 11.42 | loss 3.98 | ppl 53.61\n | epoch 8 | 800/ 3195 batches | lr 1.70 | ms/batch 11.39 | loss 3.97 | ppl 52.86\n | epoch 8 | 1000/ 3195 batches | lr 1.70 | ms/batch 11.44 | loss 4.10 | ppl 60.52\n | epoch 8 | 1200/ 3195 batches | lr 1.70 | ms/batch 11.46 | loss 4.06 | ppl 57.78\n | epoch 8 | 1400/ 3195 batches | lr 1.70 | ms/batch 11.45 | loss 4.09 | ppl 59.78\n | epoch 8 | 1600/ 3195 batches | lr 1.70 | ms/batch 11.42 | loss 4.03 | ppl 55.99\n | epoch 8 | 1800/ 3195 batches | lr 1.70 | ms/batch 11.39 | loss 4.09 | ppl 59.86\n | epoch 8 | 2000/ 3195 batches | lr 1.70 | ms/batch 11.45 | loss 4.13 | ppl 62.31\n | epoch 8 | 2200/ 3195 batches | lr 1.70 | ms/batch 11.43 | loss 4.07 | ppl 58.80\n | epoch 8 | 2400/ 3195 batches | lr 1.70 | ms/batch 11.47 | loss 3.94 | ppl 51.55\n | epoch 8 | 2600/ 3195 batches | lr 1.70 | ms/batch 11.54 | loss 4.04 | ppl 56.69\n | epoch 8 | 2800/ 3195 batches | lr 1.70 | ms/batch 11.42 | loss 4.04 | ppl 56.65\n | epoch 8 | 3000/ 3195 batches | lr 1.70 | ms/batch 11.43 | loss 4.00 | ppl 54.75\n -----------------------------------------------------------------------------------------\n | end of epoch 8 | time: 38.48s | valid loss 4.99 | valid ppl 146.84\n -----------------------------------------------------------------------------------------\n | epoch 9 | 200/ 3195 batches | lr 1.62 | ms/batch 11.42 | loss 4.03 | ppl 56.29\n | epoch 9 | 400/ 3195 batches | lr 1.62 | ms/batch 11.43 | loss 4.06 | ppl 58.20\n | epoch 9 | 600/ 3195 batches | lr 1.62 | ms/batch 11.42 | loss 3.96 | ppl 52.43\n | epoch 9 | 800/ 3195 batches | lr 1.62 | ms/batch 11.42 | loss 3.95 | ppl 51.71\n | epoch 9 | 1000/ 3195 batches | lr 1.62 | ms/batch 11.41 | loss 4.08 | ppl 59.05\n | epoch 9 | 1200/ 3195 batches | lr 1.62 | ms/batch 11.46 | loss 4.04 | ppl 56.62\n | epoch 9 | 1400/ 3195 batches | lr 1.62 | ms/batch 11.45 | loss 4.07 | ppl 58.45\n | epoch 9 | 1600/ 3195 batches | lr 1.62 | ms/batch 11.47 | loss 4.00 | ppl 54.56\n | epoch 9 | 1800/ 3195 batches | lr 1.62 | ms/batch 11.42 | loss 4.07 | ppl 58.37\n | epoch 9 | 2000/ 3195 batches | lr 1.62 | ms/batch 11.38 | loss 4.10 | ppl 60.50\n | epoch 9 | 2200/ 3195 batches | lr 1.62 | ms/batch 11.41 | loss 4.05 | ppl 57.59\n | epoch 9 | 2400/ 3195 batches | lr 1.62 | ms/batch 11.42 | loss 3.92 | ppl 50.42\n | epoch 9 | 2600/ 3195 batches | lr 1.62 | ms/batch 11.35 | loss 4.02 | ppl 55.44\n | epoch 9 | 2800/ 3195 batches | lr 1.62 | ms/batch 11.38 | loss 4.01 | ppl 55.34\n | epoch 9 | 3000/ 3195 batches | lr 1.62 | ms/batch 11.40 | loss 3.97 | ppl 53.18\n -----------------------------------------------------------------------------------------\n | end of epoch 9 | time: 38.39s | valid loss 4.99 | valid ppl 146.35\n -----------------------------------------------------------------------------------------\n | epoch 10 | 200/ 3195 batches | lr 1.54 | ms/batch 11.44 | loss 4.01 | ppl 55.16\n | epoch 10 | 400/ 3195 batches | lr 1.54 | ms/batch 11.47 | loss 4.03 | ppl 56.49\n | epoch 10 | 600/ 3195 batches | lr 1.54 | ms/batch 11.41 | loss 3.93 | ppl 51.11\n | epoch 10 | 800/ 3195 batches | lr 1.54 | ms/batch 11.45 | loss 3.92 | ppl 50.48\n | epoch 10 | 1000/ 3195 batches | lr 1.54 | ms/batch 11.40 | loss 4.06 | ppl 57.76\n | epoch 10 | 1200/ 3195 batches | lr 1.54 | ms/batch 11.38 | loss 4.01 | ppl 55.28\n | epoch 10 | 1400/ 3195 batches | lr 1.54 | ms/batch 11.36 | loss 4.04 | ppl 56.75\n | epoch 10 | 1600/ 3195 batches | lr 1.54 | ms/batch 11.37 | loss 3.97 | ppl 53.20\n | epoch 10 | 1800/ 3195 batches | lr 1.54 | ms/batch 11.45 | loss 4.04 | ppl 56.95\n | epoch 10 | 2000/ 3195 batches | lr 1.54 | ms/batch 11.39 | loss 4.08 | ppl 59.32\n | epoch 10 | 2200/ 3195 batches | lr 1.54 | ms/batch 11.38 | loss 4.02 | ppl 55.94\n | epoch 10 | 2400/ 3195 batches | lr 1.54 | ms/batch 11.43 | loss 3.89 | ppl 49.08\n | epoch 10 | 2600/ 3195 batches | lr 1.54 | ms/batch 11.41 | loss 3.99 | ppl 54.14\n | epoch 10 | 2800/ 3195 batches | lr 1.54 | ms/batch 11.41 | loss 3.99 | ppl 53.91\n | epoch 10 | 3000/ 3195 batches | lr 1.54 | ms/batch 11.41 | loss 3.95 | ppl 52.15\n -----------------------------------------------------------------------------------------\n | end of epoch 10 | time: 38.42s | valid loss 4.98 | valid ppl 145.14\n -----------------------------------------------------------------------------------------\n\n\nEvaluate the model with the test dataset\n-------------------------------------\n\nApply the best model to check the result with the test dataset.\n\n\n\n\n```python\ntest_loss = evaluate(best_model, test_data)\nprint('=' * 89)\nprint('| End of training | test loss {:5.2f} | test ppl {:8.2f}'.format(\n test_loss, math.exp(test_loss)))\nprint('=' * 89)\n```\n\n =========================================================================================\n | End of training | test loss 4.82 | test ppl 124.40\n =========================================================================================\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "127da282151e7e6449caf680cab67affe08f5636", "size": 42376, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/2020_0722transformer_tutorial.ipynb", "max_stars_repo_name": "project-ccap/project-ccap.github.io", "max_stars_repo_head_hexsha": "867e32af5459ae55d864d9d022d69eac17fbb450", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, 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"alphanum_fraction": 0.474891448, "converted": true, "num_tokens": 10442, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833841649233, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.4443617622669087}} {"text": "```python\n%matplotlib nbagg\nimport warnings\nimport inspect\nimport matplotlib.pyplot as plt\nimport IPython.display\nfrom cued_sf2_lab.familiarisation import load_mat_img, plot_image\nimport numpy as np\nfrom typing import Tuple\n```\n\n
\n\n\n\n
Figure 5: An $L$ level binary discrete wavelet transform.
\n\n# 9 The Discrete Wavelet Transform (DWT)\n\n
\n \nThis notebook is incomplete!
\n\nThe final method of energy compaction that we shall investigate, is the\ndiscrete wavelet transform. In some ways this attempts to combine the best features of\nthe Laplacian pyramid and the DCT:\n\n* Like the pyramid, the DWT analyses the image at a range of different\n scales (levels) and employs symmetrical filters;\n\n* Like the DCT, the DWT avoids any expansion in the number of coefficients.\n\nWavelet theory was evolved by mathematicians during the 1980's. As with the LBT, we shall not attempt to teach this theory here, just illustrate a relatively simple form of it.\n\nWavelets are short waveforms which are usually the impulse responses of\nfilters. Wavelet transforms employ banks of bandpass filters, whose impulse\nresponses are scaled versions of each other, in\norder to get pass-bands in different parts of the frequency spectrum. If the\nimpulse response of a filter is scaled in time by a factor $a$, then the\nfilter frequency response is scaled by the factor $1/a$. Typically $a = 2$\nfrom one filter to the next, and each bandpass filter is designed to pass a\n2:1 range of frequencies (one octave). We can split an image up using wavelets by a process known as a _binary wavelet tree_.\n\n## 9.1 The binary wavelet tree\n\n\nWe start in 1-D with the\nsimplest possible pair of filters, operating on just two input samples, $x_n$\nand $x_{n-1}$. The two filter outputs, $u_n$ and $v_n$ at time $n$ are\ngiven by:\n$$\n u_n = \\tfrac{1}{2} (x_n + x_{n-1}) \\quad \\mbox{and} \\quad\n v_n = \\tfrac{1}{2} (x_n - x_{n-1})\n$$\n\nThe first filter averages adjacent samples, and so rejects the higher\nfrequency components of $x$, while the second filter differences these\nsamples, and so rejects the lower frequency components. These filters are\nknown as the _analysis_ filter pair, $H_1(z) = \\tfrac{1}{2} (1 + z^{-1})$\nand $H_2(z) = \\tfrac{1}{2} (1 - z^{-1})$. It is clear that we can recover the\ntwo input samples from the filter outputs using:\n$$\n x_n = u_n + v_n \\quad \\mbox{and} \\quad x_{n-1} = u_n - v_n\n$$\n\nNext it is important to note that we need only retain the samples of $u_n$\nand $v_n$ at even values of $n$ in order to be able to recover all the\noriginal samples of $x$. Hence $u$ and $v$ may be decimated 2:1 and still\nallow perfect reconstruction of $x$. If $x$ is a finite length vector (e.g. a\nrow of image pixels), then $u$ and $v$ are each half as long as $x$, so the\ntotal number of samples is preserved by the transformation.\n\nA wavelet binary tree may be constructed using these filters, by using an\nidentical pair, $H_1$ and $H_2$, to filter the decimated lowpass signal\n$u_{2n}$, to give a pair of outputs, $uu_{2n}$ and $uv_{2n}$, representing\nthe lower and upper halves of the first low band. These may again be\ndecimated 2:1 and still permit perfect reconstruction of $u$. This process\nmay be continued as often as desired: each time splitting the lowest band in\ntwo, and decimating the sample rate of the filter outputs by 2:1. At each\nstage the bandwidth of the two lowest filters is halved, and their impulse\nresponses are doubled in length. The total number of output samples remains\nconstant, however many stages are used.\n\nFor example, if $f_s$ is the input sample rate, a 3-stage binary tree will\nsplit the input signal bandwidth of 0 to $f_s/2$ into the following four\nbands:\n$$\n0 \\rightarrow f_s/16; \\ \\ f_s/16 \\rightarrow f_s/8; \\ \\ f_s/8 \\rightarrow\nf_s/4; \\ \\ f_s/4 \\rightarrow f_s/2.\n$$\n\nThe very simple filters, given above, do not generate a filter tree with\ngood characteristics, since the wavelets turn out to be just a pair of\nsquare pulses. These generate _blocking_ artefacts when used for image\ncompression (in fact they are equivalent to the 2 point ($N=2$)\nDCT). A better set of filters are the LeGall 5 and 3 tap pair,\ngiven by:\n$$\n u_n = \\tfrac{1}{8} (-x_{n+2} + 2 x_{n+1} + 6 x_n + 2 x_{n-1} - x_{n-2})\n \\quad \\mbox{ and } \\quad\n v_{n+1} = \\tfrac{1}{4} (-x_{n+2} + 2 x_{n+1} - x_n)\n$$\n\nIf $u$ and $v$ are decimated by 2 by choosing even $n$ only, the lowband outputs\n$u_n$ are centred on the even samples, and the highband outputs $v_{n+1}$ are\ncentred on the odd samples. This is very important to allow perfect\nreconstruction of $x$ from $u$ and $v$.\n\nThe equations for reconstruction may be obtained by solving the above to get:\n\n\\begin{align}\n x_n &= \\tfrac{1}{2} (-v_{n+1} + 2 u_n - v_{n-1}) \\quad \\mbox{and} \\\\\n x_{n+1} &= \\tfrac{1}{2} (x_{n+2} + 4 v_{n+1} + x_n) =\n \\tfrac{1}{4} (-v_{n+3} + 2 u_{n+2} + 6 v_{n+1} + 2 u_n - v_{n-1})\n\\end{align}\n\nIn general, most analysis filters will not yield such simple reconstruction\nsolutions, and the design of suitable filters is a non-trivial topic that we\nshall not cover here.\n\n\n# 9.2 Applying the DWT to images\n\n\nAs with the DCT, the 2-D DWT may be obtained by applying a 1-D transform to\nfirst the rows and then the columns of an image.\n\nStart by loading the Lighthouse image and defining the two LeGall\nfilters given above:\n\n\n```python\nX, _ = load_mat_img(img='lighthouse.mat', img_info='X', cmap_info={'map', 'map2'})\nX = X - 128.0\nh1 = np.array([-1, 2, 6, 2, -1])/8\nh2 = np.array([-1, 2, -1])/4\n```\n\nWe can use the function `rowdec` from the pyramid work, to\nproduce a decimated and lowpass filtered version of the rows of\n`X` (remembering to subtract 128 as before) using:\n\n\n```python\nfrom cued_sf2_lab.laplacian_pyramid import rowdec\nU = rowdec(X, h1)\n```\n\nTo get the high-pass image `V`, it is important to align the decimated\nsamples with the odd columns of `X` (assuming the first column is $n = 0$)\nwhereas `U` is aligned with the even columns. To do this we use a\nslightly modified version of `rowdec`, called `rowdec2`.\n\n\n```python\nfrom cued_sf2_lab.laplacian_pyramid import rowdec2\nV = rowdec2(X, h2)\n```\n\n
\n\nDisplay `U` and `V` to see the outputs of the first filter pair\nand comment on their relative energies (or standard deviations). Note that `U` and `V` are half the width of `X`, but that `U` is otherwise similar to `X`.
\n\n\n```python\n# your code here\n```\n\nNow filter the columns of `U` and `V` using `rowdec\n/ rowdec2` with the transpose operator:\n\n\n```python\nUU = rowdec(U.T, h1).T\nUV = rowdec2(U.T, h2).T\nVU = rowdec(V.T, h1).T\nVV = rowdec2(V.T, h2).T\n```\n\n
\n \nDisplay `np.block([[UU, VU], [UV, VV]])`, and comment\non what sort of edges or features are selected by each filter. You may need to multiply the high-pass images by a factor $k > 1$ to display them clearly. Why is this?
\n\n\n```python\n# your code here\n```\n\nWe must now check that it is possible to recover the image from\nthese sub-images, using reconstruction filters, `g1` and `g2`, and the functions, `rowint` and `rowint` (which\nis modified in a similar way to `rowdec2` to allow correct\nalignment of the high-pass samples). To reconstruct `Ur` and\n`Vr` from `UU`, `UV`, `VU` and `VV` use:\n\n\n```python\nfrom cued_sf2_lab.laplacian_pyramid import rowint, rowint2\n\ng1 = np.array([1, 2, 1])/2\ng2 = np.array([-1, -2, 6, -2, -1])/4\nUr = rowint(UU.T, g1).T + rowint2(UV.T, g2).T\nVr = rowint(VU.T, g1).T + rowint2(VV.T, g2).T\n```\n\nNote the gain of 2 in the reconstruction filters, `g1` and\n`g2` (to compensate for losing half the samples in the\ndecimation / interpolation processes). These filters are also\nnot quite the same as those that might be inferred from the\nequations for $x_n$ and $x_{n+1}$ on the previous page. This is\nbecause `g1` defines how {\\it only} the $u$ samples contribute\nboth to the even and odd samples of $x$, while `g2` defines\nhow the $v$ samples contribute.\n\nCheck that `Ur` and `Vr` are the same as `U` and\n`V`, and then reconstruct `Xr` from these:\n\n\n```python\n# your code here to check Ur and Vr\n```\n\n\n```python\n# demonstrator answer here\nnp.testing.assert_equal(Ur, U)\nnp.testing.assert_equal(Vr, V)\n```\n\n\n```python\nXr = rowint(Ur,g1) + rowint2(Vr,g2)\n```\n\nCheck that `Xr` is the same as `X`.\n\n\n```python\n# your code here\n```\n\nThe above operations are a bit tedious to repeat if we want to\napply the DWT recursively to obtain several levels of filtering,\nso we have written a pair of functions, `dwt` and `idwt`, to perform the 2-D analysis and reconstruction\noperations. Examine these to see that they perform the same\noperations as above, except that the transformed sub-images are\nstored as parts of a single matrix, the same size as `X`,\nrather than as separate matrices.\n\n\n```python\nfrom cued_sf2_lab.dwt import dwt\nIPython.display.Code(inspect.getsource(dwt), language=\"python\")\n```\n\n\n\n\n
def dwt(X: np.ndarray, h1: np.ndarray = h1, h2: np.ndarray = h2) -> np.ndarray:\n    """\n    Return a 1-level 2-D discrete wavelet transform of X.\n\n    Default h1 and h2 are the LeGall filter pair.\n\n    Parameters:\n        X: Image matrix (Usually 256x256)\n        h1, h2: Filter coefficients\n    Returns:\n        Y: 1-level 2D DWT of X\n    """\n    m, n = X.shape\n    if m % 2 or n % 2:\n        raise ValueError("Image dimensions must be even")\n    Y = np.concatenate([rowdec(X, h1), rowdec2(X, h2)], axis=1)\n    Y = np.concatenate([rowdec(Y.T, h1).T, rowdec2(Y.T, h2).T], axis=0)\n    return Y\n
\n\n\n\n\n\n```python\nfrom cued_sf2_lab.dwt import idwt\nIPython.display.Code(inspect.getsource(idwt), language=\"python\")\n```\n\n\n\n\n
def idwt(X: np.ndarray, g1: np.ndarray = g1, g2: np.ndarray = g2)-> np.ndarray:\n    """\n    Return a 1-level 2-D inverse discrete wavelet transform on X.\n\n    If filters G1 and G2 are given, then they are used, otherwise the LeGall\n    filter pair are used.\n    """\n    m, n = X.shape\n    if m % 2 or n % 2:\n        raise ValueError("Image dimensions must be even")\n    m2 = m//2\n    n2 = n//2\n    Y = rowint(X[:m2, :].T, g1).T + rowint2(X[m2:, :].T,g2).T;\n    Y = rowint(Y[:, :n2], g1) + rowint2(Y[:, n2:], g2)\n    return Y\n
\n\n\n\n\nYou can check their operation as below::\n\n\n```python\nY = dwt(X)\nXr = idwt(Y)\n\nfig, axs = plt.subplots(1, 2)\nplot_image(Y, ax=axs[0])\naxs[0].set(title=\"Y\")\nplot_image(Xr, ax=axs[1])\naxs[1].set(title=\"Xr\");\n```\n\n\n \n\n\n\n\n\n\n`Y` should be the same as the composite `[UU VU; UV VV]` image that\nyou displayed earlier, and `Xr` should be the same as `X`.\n\nNow implement a multilevel DWT by first applying `dwt` to\n`X` using:\n\n```python\nm=256\nY=dwt(X)\nplot_image(Y, ax=some_axis)\n```\n\nand then iteratively apply `dwt` to the top left sub-image\nof `Y` by repeating:\n```python\nm = m//2\nY[:m,:m] = dwt(Y[:m,:m])\nplot_image(Y, ax=some_axis)\n```\n\n\n```python\n# your code here\n```\n\nWe now have the image split using a binary wavelet tree (stricly a\nquaternary tree in 2-D). Write\nsimilar iterative code to that given above, which can reconstruct\nthe image from the final set of `Y` sub-images after a 4-level\nwavelet transform. Check that your reconstructed image is the\nsame as `X`.\n\n\n```python\n# your code here\n```\n\n## 9.3 Quantisation and coding efficiency\n\nFirst rewrite the sequences of operations required to perform\n$n$ levels of DWT and inverse DWT as two separate M-files, `nlevdwt` and `nlevidwt`. `nlevdwt` should transform\n`X` into `Y`, and `nlevidwt` should inverse\ntransform a quantised set of sub-images `Yq` into the\nreconstructed image `Z`. Check your functions by ensuring\nthat `Z` is the same as `X` if `Yq = Y`.\n\n\n```python\ndef nlevdwt(X, n):\n # your code here\n pass\n\ndef nlevidwt(Y, n):\n # your code here\n pass\n```\n\n\n```python\n# your code here to test `nlevdwt` and `nlevidwt`\n```\n\nNow design a function, `quantdwt`, which will quantise the\nsub-images of `Y` to give `Yq` and calculate their\nentropy. The sub-images at each level `i` of the DWT should\nbe quantised according to a $3 \\times (n+1)$ matrix `dwtstep[k,i]` of\nstep-sizes, where $\\mathtt{k}=\\left\\{0,1,2\\right\\}$ corresponds to each of the three high-pass images at level `i` (top right, bottom left, and bottom right, respectively), and the final low-pass image is quantised with `dwtstep[0,n]`. This matrix will be populated either with the same number in all elements (for equal-step-size quantisation) or a range of different numbers (for equal-MSE quantisation). The entropies for each sub-image should be stored in a similar $3 \\times (n+1)$ matrix `dwtent[k,i]`.\n\n\n```python\ndef quantdwt(Y: np.ndarray, dwtstep: np.ndarray) -> Tuple[np.ndarray, np.ndarray]:\n \"\"\"\n Parameters:\n Y: the output of `dwt(X, n)`\n dwtstep: an array of shape `(3, n+1)`\n Returns:\n Yq: the quantized version of `Y`\n dwtenc: an array of shape `(3, n+1)` containing the entropies\n \"\"\"\n # your code here\n Yq = None\n dwtent = None\n return Yq, dwtent\n```\n\nUsing these functions, for a given number of levels $n$ (typically\nbetween 3 and 5), you should generate `Y`, quantise it to\ngive `Yq` and reconstruct `Z` from `Yq`.\n\n\n```python\n# your code here\n```\n\nAll of our experiments thus far have been performed on only one image. At this stage it is worth starting to experiment with the additional `Bridge` image, as well as Lighthouse. Bridge contains a lot more fine detail and may not lead to the same conclusions regarding performance.\n\n\n```python\nXb, _ = load_mat_img(img='bridge.mat', img_info='X', cmap_info={'map'})\nXb = Xb - 128.0\n```\n\n\n```python\nfig, ax = plt.subplots()\nplot_image(Xb, ax=ax)\nax.set(title=\"bridge.mat\");\n```\n\n\n \n\n\n\n\n\n\n\n
\n\nInvestigate the performance of\nboth an equal-step-size and an equal-MSE scheme (follow a similar procedure as you used for the Laplacian Pyramid to find the appropriate step-size ratios). Hence determine how many levels of DWT are reasonably optimal for the Lighthouse and Bridge images. Also evaluate the subjective quality of your reconstructed images, and comment on how this depends on $n$ and on the way that step-sizes are assigned\nto the different levels. Once again, for each image choose quantisation steps such that you match the rms error to that for direct quantisation with a step-size of 17.
\n\n\n```python\n# your code here\n```\n\n## 9.4 Second Interim Report\n\nThis report should include the new results from the DCT, LBT and DWT energy\ncompaction methods in a format that will allow them to be compared with each other and contrasted to the\nLaplacian pyramid work in your first report. Again try to answer questions\nraised in the text, and also include discussion of any topics that have led to\nunexpected results or have proved particularly interesting.\n", "meta": {"hexsha": "6d3bd56c5a87c5135ea980dec20aa93c376da59d", "size": 525557, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "9-dwt.ipynb", "max_stars_repo_name": "sigproc/cued_sf2_lab", "max_stars_repo_head_hexsha": "d31f5e6725e9c1be64145006d20ddb08ae68e70e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-05-13T10:00:02.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-17T11:03:13.000Z", "max_issues_repo_path": "9-dwt.ipynb", "max_issues_repo_name": "sigproc/cued_sf2_lab", "max_issues_repo_head_hexsha": "d31f5e6725e9c1be64145006d20ddb08ae68e70e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-05-17T11:57:29.000Z", "max_issues_repo_issues_event_max_datetime": "2021-05-17T11:57:29.000Z", "max_forks_repo_path": "9-dwt.ipynb", "max_forks_repo_name": "sigproc/cued_sf2_lab", "max_forks_repo_head_hexsha": "d31f5e6725e9c1be64145006d20ddb08ae68e70e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2021-05-17T11:06:55.000Z", "max_forks_repo_forks_event_max_datetime": "2021-05-20T20:22:51.000Z", "avg_line_length": 180.7279917469, "max_line_length": 231795, "alphanum_fraction": 0.8426602633, "converted": true, "num_tokens": 9184, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982315512488, "lm_q2_score": 0.6859494550081925, "lm_q1q2_score": 0.44435684388785}} {"text": "```python\nfrom sympy import init_printing, S, symbols, Eq, solveset\n```\n\n\n```python\ninit_printing()\n```\n\n\n```python\n# length dimensions\nm, cm, mm, Km, nm, um = symbols('m, cm, mm, Km, nm, um')\n\n# Time\ns, ms, us = symbols('s, ms, us')\n\n# Energy\ncm_1, eV, J, kJ = symbols('cm_1, eV, J, kJ')\n\n# Frequency\nhz, khz, mhz, ghz, thz = symbols('hz, khz, mhz, ghz, thz')\n\n\n```\n\n\n```python\ndef make_double_symbols(x):\n lower = x.split(',')\n upper = []\n for i in lower:\n temp = i.capitalize()\n print(temp)\n \n upper.append(temp)\n \n return tuple(lower), tuple(upper)\n\ntest = 'hz, khz, mhz, ghz, thz'\ntemp = [i.strip().capitalize() for i in test.split(',')]\ntemp\n```\n\n\n\n\n ['Hz', 'Khz', 'Mhz', 'Ghz', 'Thz']\n\n\n\n\n```python\ntest.split(',')\n```\n\n\n\n\n ['hz', ' khz', ' mhz', ' ghz', ' thz']\n\n\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "8b2b84275639f71298af8d65eac1e2d683b021f1", "size": 2667, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "testing/archives/unit_conversion.ipynb", "max_stars_repo_name": "aravindhnivas/FELion-Spectrum-Analyser", "max_stars_repo_head_hexsha": "430f16884482089b2f717ea7dd50625078971e48", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "testing/archives/unit_conversion.ipynb", "max_issues_repo_name": "aravindhnivas/FELion-Spectrum-Analyser", "max_issues_repo_head_hexsha": "430f16884482089b2f717ea7dd50625078971e48", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "testing/archives/unit_conversion.ipynb", "max_forks_repo_name": "aravindhnivas/FELion-Spectrum-Analyser", "max_forks_repo_head_hexsha": "430f16884482089b2f717ea7dd50625078971e48", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2019-01-25T20:37:57.000Z", "max_forks_repo_forks_event_max_datetime": "2019-01-25T20:37:57.000Z", "avg_line_length": 19.3260869565, "max_line_length": 67, "alphanum_fraction": 0.455568054, "converted": true, "num_tokens": 297, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081926, "lm_q2_score": 0.647798211152541, "lm_q1q2_score": 0.44435682989536757}} {"text": "# Tutorial 1b: Data frames\n\n(c) 2017 Justin Bois. This work is licensed under a [Creative Commons Attribution License CC-BY 4.0](https://creativecommons.org/licenses/by/4.0/). All code contained herein is licensed under an [MIT license](https://opensource.org/licenses/MIT).\n\n\n```python\nimport numpy as np\nimport pandas as pd\n\nimport bokeh.io\nimport bokeh.plotting\n\nfrom bokeh.models import Legend\nfrom bokeh.plotting import figure, show, output_file\n\nbokeh.io.output_notebook()\n```\n\n\n\n
\n \n Loading BokehJS ...\n
\n\n\n\n\nIn this tutorial, we will learn how to load data stored on disk into a Python data structure. We will use Pandas to read in CSV (comma separated value) files and store the results in the very handy Pandas DataFrame. This incredibly flexible and powerful data structure will be a centerpiece in the rest of this course and beyond. In this tutorial, we will learn about what a data frame is and how to use it.\n
\n
\nThe data set we will use comes from a fun paper about the adhesive properties of frog tongues. The reference is Kleinteich and Gorb, Tongue adhesion in the horned frog Ceratophrys sp., *Sci. Rep.*, 4, 5225, 2014. \n\n## The data file\nThe data are contained in the file `frog_tongue_adhesion.csv`. Let's look at its contents:\n\n\n```python\nwith open('data/frog_tongue_adhesion.csv', 'r') as f:\n for _ in range(20):\n print(next(f), end='')\n```\n\n # These data are from the paper,\n # Kleinteich and Gorb, Sci. Rep., 4, 5225, 2014.\n # It was featured in the New York Times.\n # http://www.nytimes.com/2014/08/25/science/a-frog-thats-a-living-breathing-pac-man.html\n #\n # The authors included the data in their supplemental information.\n #\n # Importantly, the ID refers to the identifites of the frogs they tested.\n # I: adult, 63 mm snout-vent-length (SVL) and 63.1 g body weight,\n # Ceratophrys cranwelli crossed with Ceratophrys cornuta\n # II: adult, 70 mm SVL and 72.7 g body weight,\n # Ceratophrys cranwelli crossed with Ceratophrys cornuta\n # III: juvenile, 28 mm SVL and 12.7 g body weight, Ceratophrys cranwelli\n # IV: juvenile, 31 mm SVL and 12.7 g body weight, Ceratophrys cranwelli\n date,ID,trial number,impact force (mN),impact time (ms),impact force / body weight,adhesive force (mN),time frog pulls on target (ms),adhesive force / body weight,adhesive impulse (N-s),total contact area (mm2),contact area without mucus (mm2),contact area with mucus / contact area without mucus,contact pressure (Pa),adhesive strength (Pa)\n 2013_02_26,I,3,1205,46,1.95,-785,884,1.27,-0.290,387,70,0.82,3117,-2030\n 2013_02_26,I,4,2527,44,4.08,-983,248,1.59,-0.181,101,94,0.07,24923,-9695\n 2013_03_01,I,1,1745,34,2.82,-850,211,1.37,-0.157,83,79,0.05,21020,-10239\n 2013_03_01,I,2,1556,41,2.51,-455,1025,0.74,-0.170,330,158,0.52,4718,-1381\n 2013_03_01,I,3,493,36,0.80,-974,499,1.57,-0.423,245,216,0.12,2012,-3975\n\n\n## Loading a data set\nWe use `pd.read_csv()` to load the data set. The data are stored in a **DataFrame**. Let's load a `DataFrame`\n\n\n```python\ndf = pd.read_csv('data/frog_tongue_adhesion.csv', comment='#')\n\n# Look at the DataFrame\ndf.head()\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
dateIDtrial numberimpact force (mN)impact time (ms)impact force / body weightadhesive force (mN)time frog pulls on target (ms)adhesive force / body weightadhesive impulse (N-s)total contact area (mm2)contact area without mucus (mm2)contact area with mucus / contact area without mucuscontact pressure (Pa)adhesive strength (Pa)
02013_02_26I31205461.95-7858841.27-0.290387700.823117-2030
12013_02_26I42527444.08-9832481.59-0.181101940.0724923-9695
22013_03_01I11745342.82-8502111.37-0.15783790.0521020-10239
32013_03_01I21556412.51-45510250.74-0.1703301580.524718-1381
42013_03_01I3493360.80-9744991.57-0.4232452160.122012-3975
\n
\n\n\n\n\n```python\n# We can access a column of data (slice the data) using the column name\ndf.head()['impact force (mN)']\n```\n\n\n\n\n 0 1205\n 1 2527\n 2 1745\n 3 1556\n 4 493\n Name: impact force (mN), dtype: int64\n\n\n\nClearly, the indexing of the rows is preserved. The data were interpreted as integer type (`dtype = int64`), so we want to convert them into floats using `.astype()` method.\n\n\n```python\n# Use df.astype() method to convert it to a NumPy float 64 data type.\ndf['impact force (mN)'] = df['impact force (mN)'].astype(float)\n\n# Let's check if it worked well\ndf['impact force (mN)'].dtype\n```\n\n\n\n\n dtype('float64')\n\n\n\nNow let's select only the specific impact forces, say, with the impact force above two Newtons. Pandas `DataFrame` can be conveniently sliced with the Booleans.\n\n\n```python\n# Generate True/False array of rows for indexing \ninds = df['impact force (mN)'] >= 2000.0\n\n# Take a look\ninds.head()\n```\n\n\n\n\n 0 False\n 1 True\n 2 False\n 3 False\n 4 False\n Name: impact force (mN), dtype: bool\n\n\n\nNow we have an array of Booleans that is of the same length as the `DataFrame` itself. Now, we can use the `.loc` featore of a `DataFrame` to slice what we want out of the `DataFrame`.\n\n\n```python\n# Slice out rows we want (with big force)\ndf_big_force = df.loc[inds,:]\n\n# Let's look at the sliced rows; \n# there will be just a couple of high-force valuea\ndf_big_force\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
dateIDtrial numberimpact force (mN)impact time (ms)impact force / body weightadhesive force (mN)time frog pulls on target (ms)adhesive force / body weightadhesive impulse (N-s)total contact area (mm2)contact area without mucus (mm2)contact area with mucus / contact area without mucuscontact pressure (Pa)adhesive strength (Pa)
12013_02_26I42527.0444.08-9832481.59-0.181101940.0724923-9695
52013_03_01I42276.0313.68-5929690.96-0.1763411060.696676-1737
82013_03_05I32641.0504.27-6904911.12-0.2392692240.179824-2568
172013_03_15I32032.0603.28-6524861.05-0.2571471340.0913784-4425
\n
\n\n\n\nUsing `.loc` allows us to index by row and column. We chose all columns (using `:`) and put an array of Booleans for the rows. We get back a `DataFrame` with the rows associated with `True` in the Booleans' array.\n
\n
\n\nNow we only have the strikes of high force. It can be noted that the original indexing of rowas was retained. This is a good idea, as the indices do not have to be integers. \n
\n To overcome this, we can use `iloc` attribute of a `DataFrame`, which give the indexing with sequential integers.\n\n\n```python\n# Indexing using iloc, which enables indexing by the corresponding\n# integer sequence\ndf_big_force['impact force (mN)'].iloc[3]\n```\n\n\n\n\n 2032.0\n\n\n\n## Tidy data\nThe data in our `DataFrame` are tidy. This concept comes from the development of databases, but has been generalized to data processing in the recent years. Tidy data refers to data in a tabular form with the following format:
\n
\n1. Each variable forms a column,\n2. Each observation forms a row.\n3. Each type of observational unit forms a separate table.\n\n## More data extraction\nTo extract a single observation (i. e. a single experiment), we can extract a row and see all of the measured quantities from a given strike using `.loc`.\n\n\n```python\n# Slice out experiment with index 42\ndf.loc[42,:]\n```\n\n\n\n\n date 2013_05_27\n ID III\n trial number 3\n impact force (mN) 324\n impact time (ms) 105\n impact force / body weight 2.61\n adhesive force (mN) -172\n time frog pulls on target (ms) 619\n adhesive force / body weight 1.38\n adhesive impulse (N-s) -0.079\n total contact area (mm2) 55\n contact area without mucus (mm2) 23\n contact area with mucus / contact area without mucus 0.37\n contact pressure (Pa) 5946\n adhesive strength (Pa) -3149\n Name: 42, dtype: object\n\n\n\n`DataFrame` conveniently arranges the indices describing the elements of the row as column headings. This is a Pandas `Series` object.\n
\n
\nSlicing out a single index is not very meaningful, because the indices are arbitrary. We can instead look at our lab notebook and look at a trial number 3 on May 27, 2013, of frog III. This is a more common and indeed meaningful, use case.\n\n\n```python\n# Set up Boolean slicing\ndate = df['date'] == '2013_05_27'\ntrial = df['trial number'] == 3\nID = df['ID'] == 'III'\n\n# Slice out the row\ndf.loc[date & trial & ID]\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
dateIDtrial numberimpact force (mN)impact time (ms)impact force / body weightadhesive force (mN)time frog pulls on target (ms)adhesive force / body weightadhesive impulse (N-s)total contact area (mm2)contact area without mucus (mm2)contact area with mucus / contact area without mucuscontact pressure (Pa)adhesive strength (Pa)
422013_05_27III3324.01052.61-1726191.38-0.07955230.375946-3149
\n
\n\n\n\nThe difference is that the returned object (slice) is a `DataFrame` instead of `Series`. We can easily make it a `Series` object using `iloc`. Note that in most use cases, this is jsut a matter of convenience for viewing and nothing more.\n\n\n\n```python\ndf.loc[date & trial & ID, :].iloc[0]\n```\n\n\n\n\n date 2013_05_27\n ID III\n trial number 3\n impact force (mN) 324\n impact time (ms) 105\n impact force / body weight 2.61\n adhesive force (mN) -172\n time frog pulls on target (ms) 619\n adhesive force / body weight 1.38\n adhesive impulse (N-s) -0.079\n total contact area (mm2) 55\n contact area without mucus (mm2) 23\n contact area with mucus / contact area without mucus 0.37\n contact pressure (Pa) 5946\n adhesive strength (Pa) -3149\n Name: 42, dtype: object\n\n\n\n## Renaming columns\nThe lengthy syntax of access column names can be annoying, so it can be useful to change the names of the columns. For instance, now, to access the ratio of contact area with mucus to contact area without mucus, we would have to do following.\n\n\n```python\n# Set up criteria for our search in the DataFrame\ndate = df['date'] == '2013_05_27'\ntrial = df['trial number'] == 3\nID = df['ID'] == 'III'\n\n# When indexind DataFrames, use & for Boolean and (and / for or; - for not)\ndf.loc[date & trial & ID, 'contact area with mucus / \\\ncontact area without mucus']\n\n```\n\n\n\n\n 42 0.37\n Name: contact area with mucus / contact area without mucus, dtype: float64\n\n\n\nThe reason to use the verbose nature of the column headings is to avoid ambiguity. Also, many plotting packages, including *HoloViews* can automatically label axes based on headers in `DataFrame`s. To use shorter names, we do this.\n\n\n```python\n# Make a dictionary to rename columns\nrename_dict = {'trial number' : 'trial',\n 'contact area with mucus / contact area without mucus' :\n 'ca_ratio'}\n\n# Rename the columns\ndf = df.rename(columns=rename_dict)\n\n# Try out the new column name\ndf.loc[date & trial & ID, 'ca_ratio']\n\n# Indexing of dictionaries looks syntactically similar to cols in DataFrames\nrename_dict['trial number']\n```\n\n\n\n\n 'trial'\n\n\n\n## Computing with DataFrames and adding columns\nThe `DataFrame`s are convenient for the organization and selection of data, but how about computations and storing the results of computations in the new columns?\n
\nAs a simple example, let's say we want to have a column with the impact force i nunits of Newtons instead of milliNewtons, so we can divide `impact force (mN)` columns elementwise by 1000, just as with Numpy arrays.\n\n\n```python\n# Add a new columns with impact force in units of Newtons\ndf['impact force (N)'] = df['impact force (mN)'] / 1000\n\n# Take a look\ndf.head()\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
dateIDtrialimpact force (mN)impact time (ms)impact force / body weightadhesive force (mN)time frog pulls on target (ms)adhesive force / body weightadhesive impulse (N-s)total contact area (mm2)contact area without mucus (mm2)ca_ratiocontact pressure (Pa)adhesive strength (Pa)impact force (N)
02013_02_26I31205.0461.95-7858841.27-0.290387700.823117-20301.205
12013_02_26I42527.0444.08-9832481.59-0.181101940.0724923-96952.527
22013_03_01I11745.0342.82-8502111.37-0.15783790.0521020-102391.745
32013_03_01I21556.0412.51-45510250.74-0.1703301580.524718-13811.556
42013_03_01I3493.0360.80-9744991.57-0.4232452160.122012-39750.493
\n
\n\n\n\nThe new column was created by assigning it to Pandas `Series` we calculated.\n
\nWe can do other calculations on `DataFrame`s besides elementwise calculations. For example, if we wanted the mean impact force in units of milliNewtons, we can do this...\n\n\n```python\ndf['impact force (mN)'].mean()\n```\n\n\n\n\n 801.6875\n\n\n\nTo compute all sorts of useful summary statistics all at once about the `DataFrame`, we can use `describe()` method, which gives the count, mean, standard deviation, minimum, 25th percentile, median, 75th percentiel, and maximum for each column.\n\n\n```python\ndf.describe()\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
trialimpact force (mN)impact time (ms)impact force / body weightadhesive force (mN)time frog pulls on target (ms)adhesive force / body weightadhesive impulse (N-s)total contact area (mm2)contact area without mucus (mm2)ca_ratiocontact pressure (Pa)adhesive strength (Pa)impact force (N)
count80.00000080.00000080.00000080.00000080.00000080.00000080.00000080.00000080.00000080.00000080.00000080.00000080.00000080.000000
mean2.400000801.68750039.0625002.920375-397.7625001132.4500001.444875-0.187462166.47500061.4000000.5690006073.162500-3005.8750000.801687
std1.164887587.83614319.5586391.581092228.675562747.1726950.6598580.13405898.69605258.5328210.3418625515.2657062525.4684210.587836
min1.00000022.0000006.0000000.170000-983.000000189.0000000.220000-0.76800019.0000000.0000000.010000397.000000-17652.0000000.022000
25%1.000000456.00000029.7500001.470000-567.750000682.2500000.990000-0.277250104.75000016.7500000.2800002579.250000-3443.2500000.456000
50%2.000000601.00000034.0000003.030000-335.000000927.0000001.320000-0.165000134.50000043.0000000.6650004678.000000-2185.5000000.601000
75%3.0000001005.00000042.0000004.277500-224.5000001381.2500001.772500-0.081250238.25000092.5000000.8850007249.750000-1736.0000001.005000
max5.0000002641.000000143.0000006.490000-92.0000004251.0000003.400000-0.001000455.000000260.0000001.00000028641.000000-678.0000002.641000
\n
\n\n\n\nNow let's say we want to compute the mean impact force of each frog. We can combine Boolean indexing with looping to do that.\n\n\n```python\nfor frog in df['ID'].unique():\n inds = df['ID'] == frog\n mean_impact_force = df.loc[inds, 'impact force (mN)'].mean()\n print('{0:s}: {1:.1f} mN'.format(frog, mean_impact_force))\n```\n\n I: 1530.2 mN\n II: 707.4 mN\n III: 550.1 mN\n IV: 419.1 mN\n\n\nWe used the `.unique()` method to get the unique entries in the `ID` column. We can also use Panda's `.groupby()` function to do this much more cleanly and efficiently, but for now, it is good to appreciate the ability to pull out the data needed and do computations with it.\n
\n## Creating a DataFrame from scratch\nLet's construct our own `DataFrame` from scratch, which would contain information about each frog.\n
\nOne way to do this is to first create a dictionary with the respective fields, and then convert it into a `DataFrame` by instantiating a `pd.DataFrame` class with it.\n\n\n\n```python\n# Create a dictionary with the appropriate fields\ndata_dict = {'ID': ['I', 'II', 'III', 'IV'],\n 'age': ['adult', 'adult', 'juvenile', 'juvenile'],\n ' SVL (mm)': [63, 70, 28, 31],\n 'weight (g)': [63.1, 72.7, 12.7, 12.7],\n 'species': ['cross', 'cross', 'cranwelli', 'cranwelli']}\n\n# Make it into a DataFrame\ndf_frog_info = pd.DataFrame(data=data_dict)\n\n# Take a look\ndf_frog_info\n\n\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
IDageSVL (mm)weight (g)species
0Iadult6363.1cross
1IIadult7072.7cross
2IIIjuvenile2812.7cranwelli
3IVjuvenile3112.7cranwelli
\n
\n\n\n\nIn some instances, the data sets are not small enough to construct a dictionary by hand. Oftentimes, we have a two-dimensional array of data that we want to make into a `DataFrame`. For instance, we can have a Numpy array with two columns, one for snout-vent length and one for the weight.\n\n\n```python\ndata = np.array([[63, 70, 28, 31], \\\n [63.1, 72.7, 12.7, 12.7]]).transpose()\n\n# Let's verify this\ndata\n```\n\n\n\n\n array([[63. , 63.1],\n [70. , 72.7],\n [28. , 12.7],\n [31. , 12.7]])\n\n\n\nTo turn the data above into `DataFrame`, we specify the `column` keyword argument too.\n\n\n```python\ndf_demo = pd.DataFrame(data=data, columns=['SVL (mm)', 'weight (g)'])\n\n# Take a look\ndf_demo\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
SVL (mm)weight (g)
063.063.1
170.072.7
228.012.7
331.012.7
\n
\n\n\n\nIn general, any two-dimensional Numpy array can be converted into a `DataFrame` in this way. We only need to supply column names.\n
\n## Merging DataFrames\nFor each row in the `DataFrame` we can add the relevant value in each column. We will do these operations on a copy of `df` using the `copy()` method.\n\n\n```python\n# Make a copy of df\ndf_copy = df.copy()\n\n# Build each column\nfor col in df_frog_info.columns[df_frog_info.columns != 'ID']:\n # Make a new column with empty values\n df_copy[col] = np.empty(len(df_copy))\n \n # Add in each entry, row by row\n for i, r in df_copy.iterrows():\n ind = df_frog_info['ID'] == r['ID']\n df_copy.loc[i, col] = df_frog_info.loc[ind, col].iloc[0]\n \n# Take a look at the updated DataFrame\ndf_copy.head()\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
dateIDtrialimpact force (mN)impact time (ms)impact force / body weightadhesive force (mN)time frog pulls on target (ms)adhesive force / body weightadhesive impulse (N-s)total contact area (mm2)contact area without mucus (mm2)ca_ratiocontact pressure (Pa)adhesive strength (Pa)impact force (N)ageSVL (mm)weight (g)species
02013_02_26I31205.0461.95-7858841.27-0.290387700.823117-20301.205adult63.063.1cross
12013_02_26I42527.0444.08-9832481.59-0.181101940.0724923-96952.527adult63.063.1cross
22013_03_01I11745.0342.82-8502111.37-0.15783790.0521020-102391.745adult63.063.1cross
32013_03_01I21556.0412.51-45510250.74-0.1703301580.524718-13811.556adult63.063.1cross
42013_03_01I3493.0360.80-9744991.57-0.4232452160.122012-39750.493adult63.063.1cross
\n
\n\n\n\nWe used the `iterrows()` method of the `df_copy` data frame. The iterator gives an index (which we called `i`) and a row of a `DataFrame` (which we called `r`). This methods, and the analogous one for iterating over columns, `iteritems()`, can be useful.\n
\nHowever, much better method is to use the Pandas's built-in `merge()` method. Called with all the default kwargs, this function finds a common columns between two `DataFramee`S (in this case, the `ID` column), and then uses those columns to merge them ,filling in values that amtch in the common column. This is exactly what we want.\n\n\n```python\ndf = df.merge(df_frog_info)\n\n# Check it \ndf.head()\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
dateIDtrialimpact force (mN)impact time (ms)impact force / body weightadhesive force (mN)time frog pulls on target (ms)adhesive force / body weightadhesive impulse (N-s)total contact area (mm2)contact area without mucus (mm2)ca_ratiocontact pressure (Pa)adhesive strength (Pa)impact force (N)ageSVL (mm)weight (g)species
02013_02_26I31205.0461.95-7858841.27-0.290387700.823117-20301.205adult6363.1cross
12013_02_26I42527.0444.08-9832481.59-0.181101940.0724923-96952.527adult6363.1cross
22013_03_01I11745.0342.82-8502111.37-0.15783790.0521020-102391.745adult6363.1cross
32013_03_01I21556.0412.51-45510250.74-0.1703301580.524718-13811.556adult6363.1cross
42013_03_01I3493.0360.80-9744991.57-0.4232452160.122012-39750.493adult6363.1cross
\n
\n\n\n\n## Plotting how impact forces correlate with other metrics\nLet's say that we want to know how the impact forces correlate with other measures about the impact. Let's make a scatter plot of adhesion forces vs. impact forces.\n\n\n```python\n# Set up the plot\np = bokeh.plotting.figure(plot_height=300,\n plot_width=500,\n x_axis_label='impact force (mN)',\n y_axis_label='adhesive force (mN)')\n\n# Add a scatter plot\np.circle(df['impact force (mN)'], df['adhesive force (mN)'])\n\nbokeh.io.show(p)\n```\n\n\n\n
\n
\n
\n\n\n\n\nWe can see some correlation between the adhesive forces and the impact forces. The stronger the impact force, the stronger the adhesive force, with some exceptions.\n
\n## Making subplots\nNow let's learn how to make subplots in Bokeh. Let's say we want to plot the adhesive force, total contact area, impact time, and contact pressure against impact force. To lay these out as subplots, we make the respective `bokeh.plotting.figure.Figure` objects one by one and put them in a list. We then use the utilities from `bokeh.layouts` to make the subplots. Let's start by making the list of figure objects. It will help to wirte a quick function to generate a plot.\n\n\n```python\ndef scatter(df, x, y, p=None, color='#30a2da',\n plot_height=300, plot_width=500):\n \"\"\"Populate a figure with a scatter plot.\"\"\"\n p = bokeh.plotting.figure(plot_height=plot_height,\n plot_width=plot_width,\n x_axis_label=x,\n y_axis_label=y)\n \n p.circle(df[x], df[y], color=color)\n \n return p\n```\n\nNow we can build a list of plots\n\n\n```python\n# To be plotted\ncols = ['impact time (ms)',\n 'adhesive force (mN)',\n 'total contact area (mm2)',\n 'contact pressure (Pa)']\n\nplots = []\nfor col in cols:\n plots.append(scatter(df,\n 'impact force (mN)',\n col,\n plot_height=200,\n plot_width=400))\n \n# Line up the plots in a column\nbokeh.io.show(bokeh.layouts.column(plots))\n```\n\n\n\n
\n
\n
\n\n\n\n\nThere are some things that are to be cleared:
\n* Plots share the same x-axis, so we want to link the axes. This is accomplisehed by setting the `x_range` attribute of the respective `bokeh.plotting.figure.Figure` objects.\n* All the x-axes are the same, so we do not need to label them all.\n\n\n```python\nfor p in plots[:-1]:\n # Set each plot's x_range to be that of the last plot\n p.x_range = plots[-1].x_range\n \n # Only have x_axis label on bottom plot\n p.xaxis.axis_label = None\n \n# To show only a single toolbar for a set of subplots, we should do this\nbokeh.io.show(bokeh.layouts.gridplot(plots, ncols=1))\n```\n\n\n\n
\n
\n
\n\n\n\n\nTo arrange the plots in a 2x2 gird, we use ncols=2 as the kwarg. However, we have to supply an axis label for the plot in the lower left corner.\n\n\n```python\nplots[2].xaxis.axis_label = 'impact force (mN)'\nbokeh.io.show(bokeh.layouts.gridplot(plots, ncols=2))\n```\n\n\n\n
\n
\n
\n\n\n\n\nTo have the blank spaces on the two-dimensional lsit of plots, we can use `bokeh.layouts.gridplot()` again. Let's leave out the contact pressure plot\n\n\n```python\n# Make 2D list of plots (put None for no plot)\nplots = [[plots[0], plots[1]],\n [plots[2], None]]\n\n# Show using gridplot without an ncols kwatg\nbokeh.io.show(bokeh.layouts.gridplot(plots))\n```\n\n\n\n
\n
\n
\n\n\n\n\n## Plotting the distribution of impact forces\nTo simply plot the distribution of impact forces, we can do a couple of things. Probably the most familiar thing is to plot them as a histogram. To do that with Bokeh, we can first compute the edges and heights of the bars of the histogram using Numpy, and then add them to the plot using the `quad()` method of Bokeh figures, which plots the filled rectangles. \n\n\n```python\n# Compute the histogram\nheights, edges = np.histogram(df['impact force (mN)'])\n\n# Set up the plot\np = bokeh.plotting.figure(plot_height=300,\n plot_width=500,\n x_axis_label='impact force (mN)',\n y_axis_label='count')\n\n# Generate the histogram\np.quad(top=heights, bottom=0, left=edges[:-1], right=edges[1:])\n\nbokeh.io.show(p)\n```\n\n\n\n
\n
\n
\n\n\n\n\nFrom looking at the histogram, there might be some bimodalidy. However, there is a better way to visualize the data - use of ECDF...\n## Empirical cumulative distribution functions (ECDFs)\nThe primary reason why histogram might not be the best way to display the distributions of measurements is the **binning biais**. To create a histogram, we necessarily have to consider not the exact values of the measurements, but rather place them in bins. We do not plot all of the data, and the choice of bins can change what we can infer from the plot.\n
\n
\nInstead, we should look at the **empirical cumulative distribution function** (or ECDF). The ECDF evaluated at $x$ for a set of measurements is defined as \n\\begin{align}\nECDF(x) = fraction \\ of \\ measurements \\leq x\n\\end{align}\nLet's write a function that will generate $x$ values of ECDFs and $y$ values of ECDFs.\n\n\n```python\ndef ecdf(data):\n \"\"\"Return the ECDF of the data.\"\"\"\n x = np.sort(data)\n y = np.arange(1, len(data)+1) / len(data)\n \n return x, y\n```\n\n\n```python\n# Get the values of the impact force for ECDF calculation\n# For adult frogs\ninds_ad = df['age'] == 'adult'\nvals_ad = df.loc[inds_ad,'impact force (mN)']\n\n# For juvenile frogs\ninds_juv = df['age'] == 'juvenile'\nvals_juv = df.loc[inds_juv,'impact force (mN)']\n\n# Calculate the ECDFs for both adult and juvenile frogs\nxad, yad = ecdf(vals_ad)\nxjuv, yjuv = ecdf(vals_juv)\n\n# Plot the ECDFs for the adult and juvenile frogs\n# Set up the plot\np = bokeh.plotting.figure(plot_height=300,\n plot_width=600,\n x_axis_label='impact force (mN)',\n y_axis_label='ECDF')\n\n# Add a scatter plot\np = figure(toolbar_location=\"above\")\n\np1 = p.circle(xad, yad, color='blue')\np2 = p.circle(xjuv, yjuv, color='red')\n\nlegend = Legend(items=[\n ('adult', [p1]),\n ('juvenile', [p2])\n ])\n# Add legend\np.add_layout(legend)\n\np.xaxis.axis_label = 'impact force (mN)'\np.yaxis.axis_label = 'ECDF'\n\nbokeh.io.show(p)\n```\n\n\n\n
\n
\n
\n\n\n\n\nFrom the ECDF graph above, it is apparent that there is a difference in the impact forces between adult frogs and the juvenile frogs, explaining the slight bimodality of the histogram.\n## Conclusions: what we learned?\n* How to lead data from CSVfiles into Pandas `DataFrame`s.\n* `DataFrame`s are useful objects for looking at data.\n\n", "meta": {"hexsha": "cd3b0743a0b0817d46290aee5332fcf61c9214e5", "size": 279059, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": ".ipynb_checkpoints/t1b_dataframe-checkpoint.ipynb", "max_stars_repo_name": "MiroGasparek/DataAnalysis_intro", "max_stars_repo_head_hexsha": "585757815d28661a5e3ccea8f03f9b7c12bc0888", "max_stars_repo_licenses": ["CC-BY-4.0", "MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": ".ipynb_checkpoints/t1b_dataframe-checkpoint.ipynb", "max_issues_repo_name": "MiroGasparek/DataAnalysis_intro", "max_issues_repo_head_hexsha": "585757815d28661a5e3ccea8f03f9b7c12bc0888", "max_issues_repo_licenses": ["CC-BY-4.0", "MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": ".ipynb_checkpoints/t1b_dataframe-checkpoint.ipynb", "max_forks_repo_name": "MiroGasparek/DataAnalysis_intro", "max_forks_repo_head_hexsha": "585757815d28661a5e3ccea8f03f9b7c12bc0888", "max_forks_repo_licenses": ["CC-BY-4.0", "MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 90.8986970684, "max_line_length": 36350, "alphanum_fraction": 0.5582726234, "converted": true, "num_tokens": 14477, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982043529716, "lm_q2_score": 0.6859494614282923, "lm_q1q2_score": 0.4443568293901357}} {"text": "# Conclusion\n> Concluding the research\n\n- toc: true \n- badges: false\n- comments: true\n- hide: false\n- categories: [jupyter]\n- permalink: /conclusion\n- image: images/conclusion.jpg\n- sectionnr: 7\n\nIn the previous chapters I have first explained photon bunching and photon statistics qualitatively and then explored its impact on the DESHIMA spectrometer quantitatively. I have shown a model for the probabilistics behind photon bunching, showing that it is not the detection itself that triggers bunching, but rather a change in the underlying photon probability. I have also discussed the coherence time, defined as the timescale below which the non-stochastic effects of photon bunching take hold. The coherence time is related to the bandwidth by\n\n$$\\begin{equation}\n\\tau_\\mathrm{coh}\\approx\\frac{1}{\\Delta\\nu}\n\\end{equation}$$\n\nAfter exploring photon statistics, I discussed photon noise induced by both Poisson statistics and photon bunching. The total noise equivalent power induced by photon noise is given by{%cite zmuidzinas_2003%}:\n\n$$\\begin{equation}\n\\mathrm{NEP}_{\\tau,\\mathrm{ph}}^2=\\frac{1}{\\tau}\\int_0^\\infty h\\nu\\eta\\left(\\nu\\right)\\mathrm{PSD} + \\eta^2\\left(\\nu\\right)\\mathrm{PSD}^2d\\nu\n\\end{equation}$$\n\nWhereas the approximation given by {%cite zmuidzinas_2003%} and used previously in calculating the sensitivity of the DESHIMA system{% cite endo_2019%} is found by approximating a very narrow bandwidth ($\\nu\\gg\\Delta\\nu$) and then approximating the integral by\n\n$$\\begin{equation}\n\\mathrm{NEP}_{\\tau,\\mathrm{ph}}^2=\\frac{1}{\\tau}\\left(h\\nu\\eta_0\\mathrm{PSD}\\Delta\\nu+\\eta_0^2\\mathrm{PSD}^2\\Delta\\nu\\right)\\label{NEP_int}\n\\end{equation}$$\n\nthis approximation overestimates the photon bunching effects for a filter $\\eta\\left(\\nu\\right)$ with a Lorentzian shape. I have shown that, in the case of a Lorentzian filter and a flat $\\mathrm{PSD}$, eq. \\eqref{NEP_int} collapses to \n\n$$\\begin{equation}\n\\mathrm{NEP}_{\\tau,\\mathrm{ph}}^2=\\frac{1}{\\tau}\\left(h\\nu\\eta_0\\mathrm{PSD}\\Delta\\nu+\\frac{2}{\\pi}\\eta_0^2\\mathrm{PSD}^2\\Delta\\nu\\right)\n\\end{equation}$$\n\nwith $\\Delta\\nu$ the $\\mathrm{FWHM}$ of the filter. This factor of $\\pi/2$ is explained by the width of the Lorentzian filter. Previously the bandwidth of the filters was assumed to be negligible, resulting in an overestimation of the bunching. Because the photons that are impinging on the detector span a bigger bandwidth they bunch less.\n\nThe definition for the noise equivalent power $\\mathrm{NEP}_{\\tau=0.5s}$ means that it is defined at an integration time of $\\tau=0.5\\:s$. For different integration times the $\\mathrm{NEP}_\\tau$ is defined as:\n\n$$\\begin{equation}\n\\mathrm{NEP}_{\\tau} = \\frac{1}{\\sqrt{2\\tau}}\\mathrm{NEP}_{\\tau=0.5\\mathrm{s}}\n\\end{equation}$$\n\nHowever, this assumes that the integration time is much bigger than the coherence time of the detected photons $\\tau\\gg t_\\mathrm{coh}$. Due to correlation of photons within the coherence time the $\\mathrm{NEP}_{\\tau}$ drops when the integration time approaches and subceeds the coherence time.\n\nBesides this algebraic result for a Lorentzian filter I have also modified the existing `deshima-sensitivity`{% cite deshima-sensitivity %} model to calculate the integral in eq. \\eqref{NEP_int} not just for mathematical filters, but for arbitrary filter shapes loaded in via a file. This allows researchers to compare the sensitivity of various designs in software.\n\nTo verify the changes to the model I have compared it with the old model and confirmed that in the case of perfect Lorentzian filter the latter overestimated the bunching noise by a factor of $\\pi/2$ , even on average for a non-flat $\\mathrm{PSD}$. Other than this, the changes affect the power and noise in local extrema of the $\\mathrm{PSD}$, where the old model didn't integrate over the full range of the filter and therefore took the $\\mathrm{PSD}$ as a flat spectrum locally.\n\n## The future of this research\nBecause the model now more closely resembles the physics that is occurring inside the filter section of a DESHIMA spectrometer, it can be used to compare different filter designs in the `deshima-sensitivity` package itself. This is an important tool to compare filter topologies differing from Lorentzian shapes. Paired with methods to accurately describe the coupling of a source to the detector it can be used to very accurately calculate the signal to noise ratio of a specific filter profile, aiding in the experimental design of different filter profiles. Such research is in progress and as such the model will immediately be put to use. \n\nThe improved accuracy of the approximation in the bunching term for a Lorentzian filter can prove useful when designing the DESHIMA spectrometer, as it gives a more physically rigorous target of the maximum sensitivity the DESHIMA spectrometer can strive towards. \n\nFinally, this thesis gives a thorough overview of photon statistics in astronomical measurement and of photon bunching in particular and as such can be a great teaching tool. My thesis supervisor, dr. Akira Endo, has been very interested in using this as teaching material for courses he gives on the subject and I am looking forward to helping other students understand photon statistics and photon bunching better.\n\n\n\n## Bibliography\n{% bibliography --cited_in_order %}\n", "meta": {"hexsha": "c96e114a955a8f91134291123f5282ec5b77a920", "size": 7593, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "_notebooks/2022-01-02-Conclusion.ipynb", "max_stars_repo_name": "Joristiebosch/thesis", "max_stars_repo_head_hexsha": "e1fc8eb041bdb7cde7a6f70921c4ed5b2fad8bad", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "_notebooks/2022-01-02-Conclusion.ipynb", "max_issues_repo_name": "Joristiebosch/thesis", "max_issues_repo_head_hexsha": "e1fc8eb041bdb7cde7a6f70921c4ed5b2fad8bad", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-12-20T20:32:56.000Z", "max_issues_repo_issues_event_max_datetime": "2021-12-20T20:32:56.000Z", "max_forks_repo_path": "_notebooks/2022-01-02-Conclusion.ipynb", "max_forks_repo_name": "Joristiebosch/thesis", "max_forks_repo_head_hexsha": "e1fc8eb041bdb7cde7a6f70921c4ed5b2fad8bad", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 54.6258992806, "max_line_length": 653, "alphanum_fraction": 0.6637692612, "converted": true, "num_tokens": 1477, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982043529715, "lm_q2_score": 0.6859494485880928, "lm_q1q2_score": 0.4443568210722775}} {"text": "# The Variational Quantum Eigensolver algorithm\n\n\n```python\n!pip install qiskit\n```\n\n Requirement already satisfied: qiskit in /usr/local/lib/python3.7/dist-packages (0.32.0)\n Requirement already satisfied: qiskit-aer==0.9.1 in /usr/local/lib/python3.7/dist-packages (from qiskit) (0.9.1)\n Requirement already satisfied: qiskit-ignis==0.6.0 in /usr/local/lib/python3.7/dist-packages (from qiskit) (0.6.0)\n Requirement already satisfied: qiskit-aqua==0.9.5 in /usr/local/lib/python3.7/dist-packages (from qiskit) (0.9.5)\n Requirement already satisfied: qiskit-ibmq-provider==0.18.0 in /usr/local/lib/python3.7/dist-packages (from qiskit) (0.18.0)\n Requirement already satisfied: qiskit-terra==0.18.3 in /usr/local/lib/python3.7/dist-packages (from qiskit) (0.18.3)\n Requirement already satisfied: scipy>=1.0 in /usr/local/lib/python3.7/dist-packages (from qiskit-aer==0.9.1->qiskit) (1.4.1)\n Requirement already satisfied: numpy>=1.16.3 in /usr/local/lib/python3.7/dist-packages (from qiskit-aer==0.9.1->qiskit) (1.19.5)\n Requirement already satisfied: yfinance>=0.1.62 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit) (0.1.66)\n Requirement already satisfied: psutil>=5 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit) (5.4.8)\n Requirement already satisfied: scikit-learn>=0.20.0 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit) (1.0.1)\n Requirement already satisfied: fastdtw<=0.3.4 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit) (0.3.4)\n Requirement already satisfied: setuptools>=40.1.0 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit) (57.4.0)\n Requirement already satisfied: quandl in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit) (3.7.0)\n Requirement already satisfied: pandas in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit) (1.1.5)\n Requirement already satisfied: dlx<=1.0.4 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit) (1.0.4)\n Requirement already satisfied: docplex>=2.21.207 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit) (2.22.213)\n Requirement already satisfied: retworkx>=0.8.0 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit) (0.10.2)\n Requirement already satisfied: sympy>=1.3 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit) (1.7.1)\n Requirement already satisfied: h5py<3.3.0 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit) (3.1.0)\n Requirement already satisfied: websocket-client>=1.0.1 in /usr/local/lib/python3.7/dist-packages (from qiskit-ibmq-provider==0.18.0->qiskit) (1.2.1)\n Requirement already satisfied: python-dateutil>=2.8.0 in /usr/local/lib/python3.7/dist-packages (from qiskit-ibmq-provider==0.18.0->qiskit) (2.8.2)\n Requirement already satisfied: urllib3>=1.21.1 in /usr/local/lib/python3.7/dist-packages (from qiskit-ibmq-provider==0.18.0->qiskit) (1.24.3)\n Requirement already satisfied: requests>=2.19 in /usr/local/lib/python3.7/dist-packages (from qiskit-ibmq-provider==0.18.0->qiskit) (2.23.0)\n Requirement already satisfied: requests-ntlm>=1.1.0 in /usr/local/lib/python3.7/dist-packages (from qiskit-ibmq-provider==0.18.0->qiskit) (1.1.0)\n Requirement already satisfied: python-constraint>=1.4 in /usr/local/lib/python3.7/dist-packages (from qiskit-terra==0.18.3->qiskit) (1.4.0)\n Requirement already satisfied: fastjsonschema>=2.10 in /usr/local/lib/python3.7/dist-packages (from qiskit-terra==0.18.3->qiskit) (2.15.1)\n Requirement already satisfied: tweedledum<2.0,>=1.1 in /usr/local/lib/python3.7/dist-packages (from qiskit-terra==0.18.3->qiskit) (1.1.1)\n Requirement already satisfied: ply>=3.10 in /usr/local/lib/python3.7/dist-packages (from qiskit-terra==0.18.3->qiskit) (3.11)\n Requirement already satisfied: jsonschema>=2.6 in /usr/local/lib/python3.7/dist-packages (from qiskit-terra==0.18.3->qiskit) (2.6.0)\n Requirement already satisfied: dill>=0.3 in /usr/local/lib/python3.7/dist-packages (from qiskit-terra==0.18.3->qiskit) (0.3.4)\n Requirement already satisfied: symengine>0.7 in /usr/local/lib/python3.7/dist-packages (from qiskit-terra==0.18.3->qiskit) (0.8.1)\n Requirement already satisfied: six in /usr/local/lib/python3.7/dist-packages (from docplex>=2.21.207->qiskit-aqua==0.9.5->qiskit) (1.15.0)\n Requirement already satisfied: cached-property in /usr/local/lib/python3.7/dist-packages (from h5py<3.3.0->qiskit-aqua==0.9.5->qiskit) (1.5.2)\n Requirement already satisfied: certifi>=2017.4.17 in /usr/local/lib/python3.7/dist-packages (from requests>=2.19->qiskit-ibmq-provider==0.18.0->qiskit) (2021.10.8)\n Requirement already satisfied: chardet<4,>=3.0.2 in /usr/local/lib/python3.7/dist-packages (from requests>=2.19->qiskit-ibmq-provider==0.18.0->qiskit) (3.0.4)\n Requirement already satisfied: idna<3,>=2.5 in /usr/local/lib/python3.7/dist-packages (from requests>=2.19->qiskit-ibmq-provider==0.18.0->qiskit) (2.10)\n Requirement already satisfied: cryptography>=1.3 in /usr/local/lib/python3.7/dist-packages (from requests-ntlm>=1.1.0->qiskit-ibmq-provider==0.18.0->qiskit) (35.0.0)\n Requirement already satisfied: ntlm-auth>=1.0.2 in /usr/local/lib/python3.7/dist-packages (from requests-ntlm>=1.1.0->qiskit-ibmq-provider==0.18.0->qiskit) (1.5.0)\n Requirement already satisfied: cffi>=1.12 in /usr/local/lib/python3.7/dist-packages (from cryptography>=1.3->requests-ntlm>=1.1.0->qiskit-ibmq-provider==0.18.0->qiskit) (1.15.0)\n Requirement already satisfied: pycparser in /usr/local/lib/python3.7/dist-packages (from cffi>=1.12->cryptography>=1.3->requests-ntlm>=1.1.0->qiskit-ibmq-provider==0.18.0->qiskit) (2.21)\n Requirement already satisfied: threadpoolctl>=2.0.0 in /usr/local/lib/python3.7/dist-packages (from scikit-learn>=0.20.0->qiskit-aqua==0.9.5->qiskit) (3.0.0)\n Requirement already satisfied: joblib>=0.11 in /usr/local/lib/python3.7/dist-packages (from scikit-learn>=0.20.0->qiskit-aqua==0.9.5->qiskit) (1.1.0)\n Requirement already satisfied: mpmath>=0.19 in /usr/local/lib/python3.7/dist-packages (from sympy>=1.3->qiskit-aqua==0.9.5->qiskit) (1.2.1)\n Requirement already satisfied: lxml>=4.5.1 in /usr/local/lib/python3.7/dist-packages (from yfinance>=0.1.62->qiskit-aqua==0.9.5->qiskit) (4.6.4)\n Requirement already satisfied: multitasking>=0.0.7 in /usr/local/lib/python3.7/dist-packages (from yfinance>=0.1.62->qiskit-aqua==0.9.5->qiskit) (0.0.10)\n Requirement already satisfied: pytz>=2017.2 in /usr/local/lib/python3.7/dist-packages (from pandas->qiskit-aqua==0.9.5->qiskit) (2018.9)\n Requirement already satisfied: more-itertools in /usr/local/lib/python3.7/dist-packages (from quandl->qiskit-aqua==0.9.5->qiskit) (8.11.0)\n Requirement already satisfied: inflection>=0.3.1 in /usr/local/lib/python3.7/dist-packages (from quandl->qiskit-aqua==0.9.5->qiskit) (0.5.1)\n\n\nThe goal of this notebook is to guide through a VQE implemenation for a Ising type Hamiltonian. This notebook is a squeleton code and some parts must be improved. The \"examples\" has to be taken as inspiration and improved. The VQE algorithms permit to compute the ground state energy of physical systems. First some import.\n\n\n```python\n# Comment !!!\nfrom qiskit.opflow import Z, I, X, Y\nfrom qiskit.opflow import CircuitStateFn, StateFn, CircuitSampler, PauliExpectation, ListOp\nfrom qiskit.providers.aer import AerSimulator\nfrom qiskit.utils import QuantumInstance\nfrom qiskit.circuit import QuantumCircuit, QuantumRegister, Parameter, ParameterVector\nfrom qiskit.algorithms.optimizers import ADAM, SPSA\nfrom qiskit.circuit.library import TwoLocal\nfrom qiskit.circuit.library.n_local import EfficientSU2, RealAmplitudes\nfrom qiskit import Aer, execute\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n```\n\nThe first step is to implement the Transverse Ising Hamiltonian with periodic boundary conditions wand an external field using the qiskit.opflow module. We wish to implement the following Hamiltonian \n\\begin{equation}\nH=J_z\\sum_{i=0}^{N}\\sigma_i^z\\sigma_{i+1}^z + h\\sum_{i=0}^{N}\\sigma_i^x,\n\\end{equation}\nIn the opflow module, the Pauli operators are given by I, X, Y and Z and you may use @ or * for the matrix multiplication and ^ for the tensor product.\n\n\n```python\n# implement the Transverse Ising Hamiltonian H with periodic bounday conditions and parameters N, J and h.\nn_qubits = 3\n\nN = 3\n\nh = 1\nJ = 1\n\nH = 0\nfor i in range(N-1):\n H += (J * (I^i)^(Z^Z)^(I^(N-2-i))) + (h * (I^i)^X^(I^(N-1-i)))\n# Boundary Condition\nH += (J * Z^(I^(N-2))^Z) + (h * (I^(N-1))^X)\n\n# to contract similar terms together\nH = H.reduce()\nprint(H)\n```\n\n 1.0 * ZZI\n + 1.0 * ZIZ\n + 1.0 * IZZ\n + 1.0 * XII\n + 1.0 * IXI\n + 1.0 * IIX\n\n\n### The Ansatz\nNow that we have the Hamiltonian operator, we will need an ansatz for our wavefunction. In quantum computing, an ansatz is a circuit that at the end will produce a qubit state $\\left |\\psi \\right >$. In this case, we need a variational ansatz, so some gates will contains parameters that are going to be iteratively optimized (as an example, rotation on the $x,y,z$ axis).\n\nThere is no analytical way to choose an ansatz for the system: there are empirical rules based on similarity with what we are studying.\nSome ansätze come from classical computational chemistry, such as the highly accurate [q-UCCSD](https://arxiv.org/pdf/1506.00443.pdf), but mostly we have to consider some circuits that can be run on current devices, so they have to contain few two qubits gates and be relatively shallow: these ansätze are called hardware-efficient.\n\n\nWhat we are going to consider is one of the so-called hardware-efficient ansätze. \nThe system does not contain many qubits, so our trial ansatz will be very simple : a layer of rotations around the $y$-axis followed by CNOTs and again a layer of rotations. This simple structure can be easily extended both in depth (adding more CNOTs and rotation) and in width, to study bigger system, therefore is widely used.\n\nAs an introduction, try to implement the circuit defined above, with 3 qubits. \nOne you get an understanding, you can try to reliate to the template class [TwoLocal](https://qiskit.org/documentation/stubs/qiskit.circuit.library.TwoLocal.html), [EfficientSU2](https://qiskit.org/documentation/stubs/qiskit.circuit.library.EfficientSU2.html) or [RealAmplitude](https://qiskit.org/documentation/stubs/qiskit.circuit.library.RealAmplitudes.html) to design quickly any quantum circuit you like.\n\n\n\nHint: Try to work in a general way in order to to remain flexible. \nHint: You can use the ParameterVector class to create parametrized circuits and bind numerical values at the end.\n\n\n```python\nn_qubits = 3\nn_layers = 2\nparams = ParameterVector('\u03b8',(n_qubits*6))\n\nqr = QuantumRegister(n_qubits)\nansatz = QuantumCircuit(qr)\n\n# Example for the first layer\n\nfor k in range(n_layers-1):\n for i in range(n_qubits):\n ansatz.ry(params[i + n_qubits*k], i)\n ansatz.barrier()\n for i in range(n_qubits):\n ansatz.cx(i, (i+1) % n_qubits)\n ansatz.barrier()\nfor i in range(n_qubits):\n ansatz.ry(params[i + n_qubits*(n_layers-1)], i)\nansatz.barrier()\n \n\nprint(ansatz.draw())\n\n \n```\n\n \u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510 \u2591 \u250c\u2500\u2500\u2500\u2510 \u2591 \u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510 \u2591 \n q84222_0: \u2524 Ry(\u03b8[0]) \u251c\u2500\u2591\u2500\u2500\u2500\u25a0\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2524 X \u251c\u2500\u2591\u2500\u2524 Ry(\u03b8[3]) \u251c\u2500\u2591\u2500\n \u251c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2524 \u2591 \u250c\u2500\u2534\u2500\u2510 \u2514\u2500\u252c\u2500\u2518 \u2591 \u251c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2524 \u2591 \n q84222_1: \u2524 Ry(\u03b8[1]) \u251c\u2500\u2591\u2500\u2524 X \u251c\u2500\u2500\u25a0\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2591\u2500\u2524 Ry(\u03b8[4]) \u251c\u2500\u2591\u2500\n \u251c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2524 \u2591 \u2514\u2500\u2500\u2500\u2518\u250c\u2500\u2534\u2500\u2510 \u2502 \u2591 \u251c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2524 \u2591 \n q84222_2: \u2524 Ry(\u03b8[2]) \u251c\u2500\u2591\u2500\u2500\u2500\u2500\u2500\u2500\u2524 X \u251c\u2500\u2500\u25a0\u2500\u2500\u2500\u2591\u2500\u2524 Ry(\u03b8[5]) \u251c\u2500\u2591\u2500\n \u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2518 \u2591 \u2514\u2500\u2500\u2500\u2518 \u2591 \u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2518 \u2591 \n\n\n### The VQE algorithm\n\nThe core of the Variational Quantum Eigensolver is to optimize a parametrized trial wavefunction to minimize the energy expectation of the Hamiltonian of interest. By the variational theorem, this would yield a good estimate of the true groundstate and its energy. \nIn the following, we will implement the VQE algorithm. For this purpose, we need the expectation value of the Hamiltonian, its gradient with respect to the ansatz's parameters and an optimizer to update them.\nWe remember, that for an involutive quantum gate (i.e. any single qubit rotation), the gradient of the expectation value with respect to its parameter $\\theta$ is given by\n\n$$\\frac{d}{d\\theta}\\left<\\psi(\\theta)|H|\\psi(\\theta)\\right> =$$\n$$\\frac{1}{2} \\left[\\left<\\psi(\\theta+\\pi/2)|H|\\psi(\\theta+\\pi/2)\\right> - \\left<\\psi(\\theta-\\pi/2)|H|\\psi(\\theta-\\pi/2)\\right>\\right].$$\n\n\nWe will compute the expectation value by sampling the wavefunction. For example, $$ \\left< ZZ\\right>= [\\text{counts}(00) + \\text{counts}(11) - \\text{counts}(01) - \\text{counts}(10) ]/ \\text{shots}.$$ Note that we can only measure in the Z basis. To measure another Pauli string P, we will have to append a unitary U after the ansatz to perform a basis rotation. U has to be chosen such that \n$$U^\\dagger Z^N U = P.$$\n\nIf the identity is present in the Pauli string, we will act on the remaining subspace. That means, you can perform the same technique by forgetting the dimension where $I$ is present. \nNote that in qiskit, we read the qubits from right to left. That means that XZ = $X_1Z_0$\n\n\n\n```python\nnoise_model = None # the noise model used to simulate the hardware, for now, leave it to None (no noise)\n\n\nclass VQE:\n def __init__(self,ansatz, qr, H):\n self.H = H\n self.ansatz = ansatz\n self.qr = qr\n self.backend = Aer.get_backend('qasm_simulator',noise = noise_model)\n self.shots = 2**12 # number of sample used to approximate the expectation value\n #choose some classical optimizer. You may take a gradient based or gradient free optimizer \n # and play with the hyperparameters as well.\n self.optimizer = ADAM(maxiter=1, tol=1e-06, lr=0.1, beta_1=0.9, beta_2=0.99, amsgrad=False)\n \n\n \n \n def expectation_value(self,parameters):\n \n '''\n return the expectation value of the quantum circuit wrt to the observable H\n \n '''\n \n result = 0\n \n for i in range(len(self.H)): #loop over the pauli terms\n ansatz = self.ansatz.bind_parameters(parameters) #insert the parameters\n coeff = self.H[i].primitive.coeffs # get the coefficient\n pauli = self.H[i].primitive.table.to_labels()[0] #get the pauli string\n \n observable, measure_which = self.basis_change(pauli) #get which qubits to measure and the basis change\n \n ansatz.append(observable,self.qr) # append the basis change\n ansatz.measure_all() # measure all the qubits\n \n job = execute(ansatz, self.backend, shots=self.shots) \n counts = job.result().get_counts()\n result = result + coeff * self.expectation_from_counts(counts,measure_which)\n \n \n \n return result.real\n \n def basis_change(self,pauli):\n '''transform the measurement basis as function of the observable\n return: -quantum circuit to append to the ansatz\n -qubit to measure\n '''\n observable = QuantumCircuit(len(pauli)) #the basis transformation circuit U to append to the ansatz\n \n measure_which = [] #which qubit we will have to measure\n for i in range(len(pauli)):\n if pauli[i] == 'X':\n observable.h(i)\n measure_which += [i]\n elif pauli[i] == 'Y':\n observable.s(i)\n observable.h(i)\n observable.s(i)\n measure_which += [i]\n elif pauli[i] == 'Z':\n measure_which += [i]\n \n return observable, measure_which\n \n def expectation_from_counts(self,counts,measure_which):\n '''compute the expectation value from counts\n Tipp: look at the parity of the state\n remember: in qiskit we read the qubits from right to left\n\n '''\n \n total_counts = np.sum([counts[key] for key in counts])\n expectation = 0\n \n if len(measure_which)==0:\n # return the expectation value when the pauli is the identity\n # just sum the probabilities\n probs = np.array(counts.values()) / total_counts\n return np.sum(probs)\n \n for state in counts:\n parity = (-1) ** np.bitwise_xor.reduce([int(b) for b in state]) # your code goes here, compute the parity of the state\n expectation += parity * counts[state] # your code goes here\n \n return expectation / total_counts\n \n def gradient(self,parameters):\n '''\n return the gradient of the quantum circuit \n \n '''\n \n gradients = np.zeros_like(parameters)\n for i,p in enumerate(parameters):\n \n shift = np.zeros_like(parameters)\n shift[i] = np.pi/2\n gradients[i] = 0.5 * (self.expectation_value(p + shift) - self.expectation_value(p - shift))\n \n return gradients\n \n def update(self,parameters):\n '''\n update the parameters with the classical optimizer\n \n '''\n parameters, loss, it = self.optimizer.optimize(parameters.size,\n lambda param: self.expectation_value(param),\n gradient_function= lambda param: self.gradient(param),initial_point=parameters)\n \n return loss, parameters\n \n\n \n```\n\n\n```python\n%matplotlib inline\nplt.ion()\nfig = plt.figure()\nax = fig.add_subplot(111)\n\n# Here we choose some initial weights and optimize them\nnp.random.seed(1)\nweights = np.pi * np.random.normal(0,1,size = len(ansatz.parameters))\n\nvqe = VQE(ansatz, qr, H)\nexact_energy = min(np.real(np.linalg.eig(H.to_matrix())[0])) # exact smallest eigenvalue\nloss = []\nepoch = 10\nfor i in range(epoch):\n # update the parameters and save the loss in loss\n loss_tmp, weights = vqe.update(weights)\n loss.append(loss_tmp)\n print(i, loss_tmp)\n \n#plot the learning curve\nax.clear()\nax.plot(loss,'b.',label='VQE')\nax.plot(exact_energy*np.ones_like(range(epoch)),'k--',label='exact')\nax.plot(range(epoch), loss, label='loss')\nax.legend()\nax.set_xlim([-0.2, epoch])\nfig.show()\nfig.canvas.draw()\n```\n\nOnce you have a working script, you can try to imrpove it. For example you could try to use an ansatz with less parameters as possible, or with less CNOT gates as possible. CNOT gates are expensive and with a relativ high error rates so it is good to design ansatz with few of them. You could also try to explore more complexe Hamiltonian, for example with more qubits, more interaction or let vary the constant J. Another thing which is worth exploring is the optimizer itself. The SPSA optimizer is a solid choice as it uses a fix number of points to estimate the gradients and is therefore quicker than gradient descent and also robust to shot noise. A last thing you could play with, is to add noise to the hardware, as shown below.\n\n\n```python\nimport qiskit.providers.aer.noise as noise\nerror_1 = noise.depolarizing_error(0.001, 1) #error rate of single qubit gates\nerror_2 = noise.depolarizing_error(0.01, 2) #error rate of double qubit gates\n\n\nnoise_model = noise.NoiseModel()\nnoise_model.add_all_qubit_quantum_error(error_1, ['u1,u2,u3']) # add the single-qubit gates where you want to have noise\nnoise_model.add_all_qubit_quantum_error(error_2, ['cx']) # add the double-qubits gates where you want to have noise\n\n#Then you can just use this noise model in the quantum instance of the VQE \n#to automatically incoporate it in the computations\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "d14a4bc204d226712a21ec4c77c5b2e634e90cea", "size": 43616, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Guided_exercises/VQE_Notebook.ipynb", "max_stars_repo_name": "Sager611/Quantum_Hackathon_2021", "max_stars_repo_head_hexsha": "eb37ef16090231461e7651ecc8dbf7ead25a383e", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Guided_exercises/VQE_Notebook.ipynb", "max_issues_repo_name": "Sager611/Quantum_Hackathon_2021", "max_issues_repo_head_hexsha": "eb37ef16090231461e7651ecc8dbf7ead25a383e", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Guided_exercises/VQE_Notebook.ipynb", "max_forks_repo_name": "Sager611/Quantum_Hackathon_2021", "max_forks_repo_head_hexsha": "eb37ef16090231461e7651ecc8dbf7ead25a383e", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 77.3333333333, "max_line_length": 14906, "alphanum_fraction": 0.6926815847, "converted": true, "num_tokens": 5831, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585786300049, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.444237373223837}} {"text": "
\n \n

\n Histograms of Oriented Gradients (HOG)\n

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Introduction

\n\nAs we saw with the ORB algorithm, we can use keypoints in images to do keypoint-based matching to detect objects in images. These type of algorithms work great when you want to detect objects that have a lot of consistent internal features that are not affected by the background. For example, these algorithms work well for facial detection because faces have a lot of consistent internal features that don\u2019t get affected by the image background, such as the eyes, nose, and mouth. However, these type of algorithms don\u2019t work so well when attempting to do more general object recognition, say for example, pedestrian detection in images. The reason is that people don\u2019t have consistent internal features, like faces do, because the body shape and style of every person is different (see Fig. 1). This means that every person is going to have a different set of internal features, and so we need something that can more generally describe a person. \n\n
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Fig. 1. - Pedestrians.
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\n\nOne option is to try to detect pedestrians by their contours instead. Detecting objects in images by their contours (boundaries) is very challenging because we have to deal with the difficulties brought about by the contrast between the background and the foreground. For example, suppose you wanted to detect a pedestrian in an image that is walking in front of a white building and she is wearing a white coat and black pants (see Fig. 2). We can see in Fig. 2, that since the background of the image is mostly white, the black pants are going to have a very high contrast, but the coat, since it is white as well, is going to have very low contrast. In this case, detecting the edges of pants is going to be easy but detecting the edges of the coat is going to be very difficult. This is where **HOG** comes in. HOG stands for **Histograms of Oriented Gradients** and it was first introduced by Navneet Dalal and Bill Triggs in 2005.\n\n
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Fig. 2. - High and Low Contrast.
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\n\nThe HOG algorithm works by creating histograms of the distribution of gradient orientations in an image and then normalizing them in a very special way. This special normalization is what makes HOG so effective at detecting the edges of objects even in cases where the contrast is very low. These normalized histograms are put together into a feature vector, known as the HOG descriptor, that can be used to train a machine learning algorithm, such as a Support Vector Machine (SVM), to detect objects in images based on their boundaries (edges). Due to its great success and reliability, HOG has become one of the most widely used algorithms in computer vison for object detection.\n\n\nIn this notebook, you will learn:\n\n* How the HOG algorithm works\n* How to use OpenCV to create a HOG descriptor\n* How to visualize the HOG descriptor. \n\n# The HOG Algorithm\n\nAs its name suggests, the HOG algorithm, is based on creating histograms from the orientation of image gradients. The HOG algorithm is implemented in a series of steps:\n\n1. Given the image of particular object, set a detection window (region of interest) that covers the entire object in the image (see Fig. 3).\n\n2. Calculate the magnitude and direction of the gradient for each individual pixel in the detection window.\n\n3. Divide the detection window into connected *cells* of pixels, with all cells being of the same size (see Fig. 3). The size of the cells is a free parameter and it is usually chosen so as to match the scale of the features that want to be detected. For example, in a 64 x 128 pixel detection window, square cells 6 to 8 pixels wide are suitable for detecting human limbs.\n\n4. Create a Histogram for each cell, by first grouping the gradient directions of all pixels in each cell into a particular number of orientation (angular) bins; and then adding up the gradient magnitudes of the gradients in each angular bin (see Fig. 3). The number of bins in the histogram is a free parameter and it is usually set to 9 angular bins.\n\n5. Group adjacent cells into *blocks* (see Fig. 3). The number of cells in each block is a free parameter and all blocks must be of the same size. The distance between each block (known as the stride) is a free parameter but it is usually set to half the block size, in which case you will get overlapping blocks (*see video below*). The HOG algorithm has been shown empirically to work better with overlapping blocks.\n\n6. Use the cells contained within each block to normalize the cell histograms in that block (see Fig. 3). If you have overlapping blocks this means that most cells will be normalized with respect to different blocks (*see video below*). Therefore, the same cell may have several different normalizations.\n\n7. Collect all the normalized histograms from all the blocks into a single feature vector called the HOG descriptor.\n\n8. Use the resulting HOG descriptors from many images of the same type of object to train a machine learning algorithm, such as an SVM, to detect those type of objects in images. For example, you could use the HOG descriptors from many images of pedestrians to train an SVM to detect pedestrians in images. The training is done with both positive a negative examples of the object you want detect in the image.\n\n9. Once the SVM has been trained, a sliding window approach is used to try to detect and locate objects in images. Detecting an object in the image entails finding the part of the image that looks similar to the HOG pattern learned by the SVM.\n\n
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Fig. 3. - HOG Diagram.
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Vid. 1. - HOG Animation.
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\n\n# Why The HOG Algorithm Works\n\nAs we learned above, HOG creates histograms by adding the magnitude of the gradients in particular orientations in localized portions of the image called *cells*. By doing this we guarantee that stronger gradients will contribute more to the magnitude of their respective angular bin, while the effects of weak and randomly oriented gradients resulting from noise are minimized. In this manner the histograms tell us the dominant gradient orientation of each cell. \n\n\n### Dealing with contrast \n\nNow, the magnitude of the dominant orientation can vary widely due to variations in local illumination and the contrast between the background and the foreground.\n\nTo account for the background-foreground contrast differences, the HOG algorithm tries to detect edges locally. In order to do this, it defines groups of cells, called **blocks**, and normalizes the histograms using this local group of cells. By normalizing locally, the HOG algorithm can detect the edges in each block very reliably; this is called **block normalization**.\n\nIn addition to using block normalization, the HOG algorithm also uses overlapping blocks to increase its performance. By using overlapping blocks, each cell contributes several independent components to the final HOG descriptor, where each component corresponds to a cell being normalized with respect to a different block. This may seem redundant but, it has been shown empirically that by normalizing each cell several times with respect to different local blocks, the performance of the HOG algorithm increases dramatically. \n\n### Loading Images and Importing Resources\n\nThe first step in building our HOG descriptor is to load the required packages into Python and to load our image. \n\nWe start by using OpenCV to load an image of a triangle tile. Since, the `cv2.imread()` function loads images as BGR we will convert our image to RGB so we can display it with the correct colors. As usual we will convert our BGR image to Gray Scale for analysis.\n\n\n```python\nimport cv2\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n\n# Set the default figure size\nplt.rcParams['figure.figsize'] = [17.0, 7.0]\n\n# Load the image \nimage = cv2.imread('./images/triangle_tile.jpeg')\n\n# Convert the original image to RGB\noriginal_image = cv2.cvtColor(image, cv2.COLOR_BGR2RGB)\n\n# Convert the original image to gray scale\ngray_image = cv2.cvtColor(image, cv2.COLOR_BGR2GRAY)\n\n# Print the shape of the original and gray scale images\nprint('The original image has shape: ', original_image.shape)\nprint('The gray scale image has shape: ', gray_image.shape)\n\n# Display the images\nplt.subplot(121)\nplt.imshow(original_image)\nplt.title('Original Image')\nplt.subplot(122)\nplt.imshow(gray_image, cmap='gray')\nplt.title('Gray Scale Image')\nplt.show()\n```\n\n The original image has shape: (250, 250, 3)\n The gray scale image has shape: (250, 250)\n\n\n\n \n\n\n# Creating The HOG Descriptor\n\nWe will be using OpenCV\u2019s `HOGDescriptor` class to create the HOG descriptor. The parameters of the HOG descriptor are setup using the `HOGDescriptor()` function. The parameters of the `HOGDescriptor()` function and their default values are given below:\n\n`cv2.HOGDescriptor(win_size = (64, 128), \n block_size = (16, 16), \n block_stride = (8, 8), \n cell_size = (8, 8), \n nbins = 9, \n win_sigma = DEFAULT_WIN_SIGMA, \n threshold_L2hys = 0.2, \n gamma_correction = true, \n nlevels = DEFAULT_NLEVELS)`\n\nParameters:\n\n* **win_size** \u2013 *Size* \nSize of detection window in pixels (*width, height*). Defines the region of interest. Must be an integer multiple of cell size.\n\n\n* **block_size** \u2013 *Size* \nBlock size in pixels (*width, height*). Defines how many cells are in each block. Must be an integer multiple of cell size and it must be smaller than the detection window. The smaller the block the finer detail you will get.\n\n\n* **block_stride** \u2013 *Size* \nBlock stride in pixels (*horizontal, vertical*). It must be an integer multiple of cell size. The `block_stride` defines the distance between adjecent blocks, for example, 8 pixels horizontally and 8 pixels vertically. Longer `block_strides` makes the algorithm run faster (because less blocks are evaluated) but the algorithm may not perform as well.\n\n\n* **cell_size** \u2013 *Size* \nCell size in pixels (*width, height*). Determines the size fo your cell. The smaller the cell the finer detail you will get.\n\n\n* **nbins** \u2013 *int* \nNumber of bins for the histograms. Determines the number of angular bins used to make the histograms. With more bins you capture more gradient directions. HOG uses unsigned gradients, so the angular bins will have values between 0 and 180 degrees.\n\n\n* **win_sigma** \u2013 *double* \nGaussian smoothing window parameter. The performance of the HOG algorithm can be improved by smoothing the pixels near the edges of the blocks by applying a Gaussian spatial window to each pixel before computing the histograms.\n\n\n* **threshold_L2hys** \u2013 *double* \nL2-Hys (Lowe-style clipped L2 norm) normalization method shrinkage. The L2-Hys method is used to normalize the blocks and it consists of an L2-norm followed by clipping and a renormalization. The clipping limits the maximum value of the descriptor vector for each block to have the value of the given threshold (0.2 by default). After the clipping the descriptor vector is renormalized as described in *IJCV*, 60(2):91-110, 2004.\n\n\n* **gamma_correction** \u2013 *bool* \nFlag to specify whether the gamma correction preprocessing is required or not. Performing gamma correction slightly increases the performance of the HOG algorithm.\n\n\n* **nlevels** \u2013 *int* \nMaximum number of detection window increases.\n\nAs we can see, the `cv2.HOGDescriptor()`function supports a wide range of parameters. The first few arguments (`block_size, block_stride, cell_size`, and `nbins`) are probably the ones you are most likely to change. The other parameters can be safely left at their default values and you will get good results. \n\nIn the code below, we will use the `cv2.HOGDescriptor()`function to set the cell size, block size, block stride, and the number of bins for the histograms of the HOG descriptor. We will then use `.compute(image)`method to compute the HOG descriptor (feature vector) for the given `image`. \n\n\n```python\n# Specify the parameters for our HOG descriptor\n\n# Cell Size in pixels (width, height). Must be smaller than the size of the detection window\n# and must be chosen so that the resulting Block Size is smaller than the detection window.\ncell_size = (6, 6)\n\n# Number of cells per block in each direction (x, y). Must be chosen so that the resulting\n# Block Size is smaller than the detection window\nnum_cells_per_block = (2, 2)\n\n# Block Size in pixels (width, height). Must be an integer multiple of Cell Size.\n# The Block Size must be smaller than the detection window\nblock_size = (num_cells_per_block[0] * cell_size[0],\n num_cells_per_block[1] * cell_size[1])\n\n# Calculate the number of cells that fit in our image in the x and y directions\nx_cells = gray_image.shape[1] // cell_size[0]\ny_cells = gray_image.shape[0] // cell_size[1]\n\n# Horizontal distance between blocks in units of Cell Size. Must be an integer and it must\n# be set such that (x_cells - num_cells_per_block[0]) / h_stride = integer.\nh_stride = 1\n\n# Vertical distance between blocks in units of Cell Size. Must be an integer and it must\n# be set such that (y_cells - num_cells_per_block[1]) / v_stride = integer.\nv_stride = 1\n\n# Block Stride in pixels (horizantal, vertical). Must be an integer multiple of Cell Size\nblock_stride = (cell_size[0] * h_stride, cell_size[1] * v_stride)\n\n# Number of gradient orientation bins\nnum_bins = 9 \n\n\n# Specify the size of the detection window (Region of Interest) in pixels (width, height).\n# It must be an integer multiple of Cell Size and it must cover the entire image. Because\n# the detection window must be an integer multiple of cell size, depending on the size of\n# your cells, the resulting detection window might be slightly smaller than the image.\n# This is perfectly ok.\nwin_size = (x_cells * cell_size[0] , y_cells * cell_size[1])\n\n# Print the shape of the gray scale image for reference\nprint('\\nThe gray scale image has shape: ', gray_image.shape)\nprint()\n\n# Print the parameters of our HOG descriptor\nprint('HOG Descriptor Parameters:\\n')\nprint('Window Size:', win_size)\nprint('Cell Size:', cell_size)\nprint('Block Size:', block_size)\nprint('Block Stride:', block_stride)\nprint('Number of Bins:', num_bins)\nprint()\n\n# Set the parameters of the HOG descriptor using the variables defined above\nhog = cv2.HOGDescriptor(win_size, block_size, block_stride, cell_size, num_bins)\n\n# Compute the HOG Descriptor for the gray scale image\nhog_descriptor = hog.compute(gray_image)\n```\n\n \n The gray scale image has shape: (250, 250)\n \n HOG Descriptor Parameters:\n \n Window Size: (246, 246)\n Cell Size: (6, 6)\n Block Size: (12, 12)\n Block Stride: (6, 6)\n Number of Bins: 9\n \n\n\n# Number of Elements In The HOG Descriptor\n\nThe resulting HOG Descriptor (feature vector), contains the normalized histograms from all cells from all blocks in the detection window concatenated in one long vector. Therefore, the size of the HOG feature vector will be given by the total number of blocks in the detection window, multiplied by the number of cells per block, times the number of orientation bins:\n\n\n\\begin{equation}\n\\mbox{total_elements} = (\\mbox{total_number_of_blocks})\\mbox{ } \\times \\mbox{ } (\\mbox{number_cells_per_block})\\mbox{ } \\times \\mbox{ } (\\mbox{number_of_bins})\n\\end{equation}\n\n\nIf we don\u2019t have overlapping blocks (*i.e.* the `block_stride`equals the `block_size`), the total number of blocks can be easily calculated by dividing the size of the detection window by the block size. However, in the general case we have to take into account the fact that we have overlapping blocks. To find the total number of blocks in the general case (*i.e.* for any `block_stride` and `block_size`), we can use the formula given below:\n\n\n\\begin{equation}\n\\mbox{Total}_i = \\left( \\frac{\\mbox{block_size}_i}{\\mbox{block_stride}_i} \\right)\\left( \\frac{\\mbox{window_size}_i}{\\mbox{block_size}_i} \\right) - \\left [\\left( \\frac{\\mbox{block_size}_i}{\\mbox{block_stride}_i} \\right) - 1 \\right]; \\mbox{ for } i = x,y\n\\end{equation}\n\n\nWhere Total$_x$, is the total number of blocks along the width of the detection window, and Total$_y$, is the total number of blocks along the height of the detection window. This formula for Total$_x$ and Total$_y$, takes into account the extra blocks that result from overlapping. After calculating Total$_x$ and Total$_y$, we can get the total number of blocks in the detection window by multiplying Total$_x$ $\\times$ Total$_y$. The above formula can be simplified considerably because the `block_size`, `block_stride`, and `window_size`are all defined in terms of the `cell_size`. By making all the appropriate substitutions and cancelations the above formula reduces to:\n\n\n\\begin{equation}\n\\mbox{Total}_i = \\left(\\frac{\\mbox{cells}_i - \\mbox{num_cells_per_block}_i}{N_i}\\right) + 1\\mbox{ }; \\mbox{ for } i = x,y\n\\end{equation}\n\n\nWhere cells$_x$ is the total number of cells along the width of the detection window, and cells$_y$, is the total number of cells along the height of the detection window. And $N_x$ is the horizontal block stride in units of `cell_size` and $N_y$ is the vertical block stride in units of `cell_size`. \n\nLet's calculate what the number of elements for the HOG feature vector should be and check that it matches the shape of the HOG Descriptor calculated above.\n\n\n```python\n# Calculate the total number of blocks along the width of the detection window\ntot_bx = np.uint32(((x_cells - num_cells_per_block[0]) / h_stride) + 1)\n\n# Calculate the total number of blocks along the height of the detection window\ntot_by = np.uint32(((y_cells - num_cells_per_block[1]) / v_stride) + 1)\n\n# Calculate the total number of elements in the feature vector\ntot_els = (tot_bx) * (tot_by) * num_cells_per_block[0] * num_cells_per_block[1] * num_bins\n\n# Print the total number of elements the HOG feature vector should have\nprint('\\nThe total number of elements in the HOG Feature Vector should be: ',\n tot_bx, 'x',\n tot_by, 'x',\n num_cells_per_block[0], 'x',\n num_cells_per_block[1], 'x',\n num_bins, '=',\n tot_els)\n\n# Print the shape of the HOG Descriptor to see that it matches the above\nprint('\\nThe HOG Descriptor has shape:', hog_descriptor.shape)\nprint()\n```\n\n \n The total number of elements in the HOG Feature Vector should be: 40 x 40 x 2 x 2 x 9 = 57600\n \n The HOG Descriptor has shape: (57600, 1)\n \n\n\n# Visualizing The HOG Descriptor\n\nWe can visualize the HOG Descriptor by plotting the histogram associated with each cell as a collection of vectors. To do this, we will plot each bin in the histogram as a single vector whose magnitude is given by the height of the bin and its orientation is given by the angular bin that its associated with. Since any given cell might have multiple histograms associated with it, due to the overlapping blocks, we will choose to average all the histograms for each cell to produce a single histogram for each cell.\n\nOpenCV has no easy way to visualize the HOG Descriptor, so we have to do some manipulation first in order to visualize it. We will start by reshaping the HOG Descriptor in order to make our calculations easier. We will then compute the average histogram of each cell and finally we will convert the histogram bins into vectors. Once we have the vectors, we plot the corresponding vectors for each cell in an image. \n\nThe code below produces an interactive plot so that you can interact with the figure. The figure contains:\n* the grayscale image, \n* the HOG Descriptor (feature vector), \n* a zoomed-in portion of the HOG Descriptor, and \n* the histogram of the selected cell. \n\n**You can click anywhere on the gray scale image or the HOG Descriptor image to select a particular cell**. Once you click on either image a *magenta* rectangle will appear showing the cell you selected. The Zoom Window will show you a zoomed in version of the HOG descriptor around the selected cell; and the histogram plot will show you the corresponding histogram for the selected cell. The interactive window also has buttons at the bottom that allow for other functionality, such as panning, and giving you the option to save the figure if desired. The home button returns the figure to its default value.\n\n**NOTE**: If you are running this notebook in the Udacity workspace, there is around a 2 second lag in the interactive plot. This means that if you click in the image to zoom in, it will take about 2 seconds for the plot to refresh. \n\n\n```python\n%matplotlib notebook\n\nimport copy\nimport matplotlib.patches as patches\n\n# Set the default figure size\nplt.rcParams['figure.figsize'] = [9.8, 9]\n\n# Reshape the feature vector to [blocks_y, blocks_x, num_cells_per_block_x, num_cells_per_block_y, num_bins].\n# The blocks_x and blocks_y will be transposed so that the first index (blocks_y) referes to the row number\n# and the second index to the column number. This will be useful later when we plot the feature vector, so\n# that the feature vector indexing matches the image indexing.\nhog_descriptor_reshaped = hog_descriptor.reshape(tot_bx,\n tot_by,\n num_cells_per_block[0],\n num_cells_per_block[1],\n num_bins).transpose((1, 0, 2, 3, 4))\n\n# Print the shape of the feature vector for reference\nprint('The feature vector has shape:', hog_descriptor.shape)\n\n# Print the reshaped feature vector\nprint('The reshaped feature vector has shape:', hog_descriptor_reshaped.shape)\n\n# Create an array that will hold the average gradients for each cell\nave_grad = np.zeros((y_cells, x_cells, num_bins))\n\n# Print the shape of the ave_grad array for reference\nprint('The average gradient array has shape: ', ave_grad.shape) \n\n# Create an array that will count the number of histograms per cell\nhist_counter = np.zeros((y_cells, x_cells, 1))\n\n# Add up all the histograms for each cell and count the number of histograms per cell\nfor i in range (num_cells_per_block[0]):\n for j in range(num_cells_per_block[1]):\n ave_grad[i:tot_by + i,\n j:tot_bx + j] += hog_descriptor_reshaped[:, :, i, j, :]\n \n hist_counter[i:tot_by + i,\n j:tot_bx + j] += 1\n\n# Calculate the average gradient for each cell\nave_grad /= hist_counter\n \n# Calculate the total number of vectors we have in all the cells.\nlen_vecs = ave_grad.shape[0] * ave_grad.shape[1] * ave_grad.shape[2]\n\n# Create an array that has num_bins equally spaced between 0 and 180 degress in radians.\ndeg = np.linspace(0, np.pi, num_bins, endpoint = False)\n\n# Each cell will have a histogram with num_bins. For each cell, plot each bin as a vector (with its magnitude\n# equal to the height of the bin in the histogram, and its angle corresponding to the bin in the histogram). \n# To do this, create rank 1 arrays that will hold the (x,y)-coordinate of all the vectors in all the cells in the\n# image. Also, create the rank 1 arrays that will hold all the (U,V)-components of all the vectors in all the\n# cells in the image. Create the arrays that will hold all the vector positons and components.\nU = np.zeros((len_vecs))\nV = np.zeros((len_vecs))\nX = np.zeros((len_vecs))\nY = np.zeros((len_vecs))\n\n# Set the counter to zero\ncounter = 0\n\n# Use the cosine and sine functions to calculate the vector components (U,V) from their maginitudes. Remember the \n# cosine and sine functions take angles in radians. Calculate the vector positions and magnitudes from the\n# average gradient array\nfor i in range(ave_grad.shape[0]):\n for j in range(ave_grad.shape[1]):\n for k in range(ave_grad.shape[2]):\n U[counter] = ave_grad[i,j,k] * np.cos(deg[k])\n V[counter] = ave_grad[i,j,k] * np.sin(deg[k])\n \n X[counter] = (cell_size[0] / 2) + (cell_size[0] * i)\n Y[counter] = (cell_size[1] / 2) + (cell_size[1] * j)\n \n counter = counter + 1\n\n# Create the bins in degress to plot our histogram. \nangle_axis = np.linspace(0, 180, num_bins, endpoint = False)\nangle_axis += ((angle_axis[1] - angle_axis[0]) / 2)\n\n# Create a figure with 4 subplots arranged in 2 x 2\nfig, ((a,b),(c,d)) = plt.subplots(2,2)\n\n# Set the title of each subplot\na.set(title = 'Gray Scale Image\\n(Click to Zoom)')\nb.set(title = 'HOG Descriptor\\n(Click to Zoom)')\nc.set(title = 'Zoom Window', xlim = (0, 18), ylim = (0, 18), autoscale_on = False)\nd.set(title = 'Histogram of Gradients')\n\n# Plot the gray scale image\na.imshow(gray_image, cmap = 'gray')\na.set_aspect(aspect = 1)\n\n# Plot the feature vector (HOG Descriptor)\nb.quiver(Y, X, U, V, color = 'white', headwidth = 0, headlength = 0, scale_units = 'inches', scale = 5)\nb.invert_yaxis()\nb.set_aspect(aspect = 1)\nb.set_facecolor('black')\n\n# Define function for interactive zoom\ndef onpress(event):\n \n #Unless the left mouse button is pressed do nothing\n if event.button != 1:\n return\n \n # Only accept clicks for subplots a and b\n if event.inaxes in [a, b]:\n \n # Get mouse click coordinates\n x, y = event.xdata, event.ydata\n \n # Select the cell closest to the mouse click coordinates\n cell_num_x = np.uint32(x / cell_size[0])\n cell_num_y = np.uint32(y / cell_size[1])\n \n # Set the edge coordinates of the rectangle patch\n edgex = x - (x % cell_size[0])\n edgey = y - (y % cell_size[1])\n \n # Create a rectangle patch that matches the the cell selected above \n rect = patches.Rectangle((edgex, edgey),\n cell_size[0], cell_size[1],\n linewidth = 1,\n edgecolor = 'magenta',\n facecolor='none')\n \n # A single patch can only be used in a single plot. Create copies\n # of the patch to use in the other subplots\n rect2 = copy.copy(rect)\n rect3 = copy.copy(rect)\n \n # Update all subplots\n a.clear()\n a.set(title = 'Gray Scale Image\\n(Click to Zoom)')\n a.imshow(gray_image, cmap = 'gray')\n a.set_aspect(aspect = 1)\n a.add_patch(rect)\n\n b.clear()\n b.set(title = 'HOG Descriptor\\n(Click to Zoom)')\n b.quiver(Y, X, U, V, color = 'white', headwidth = 0, headlength = 0, scale_units = 'inches', scale = 5)\n b.invert_yaxis()\n b.set_aspect(aspect = 1)\n b.set_facecolor('black')\n b.add_patch(rect2)\n\n c.clear()\n c.set(title = 'Zoom Window')\n c.quiver(Y, X, U, V, color = 'white', headwidth = 0, headlength = 0, scale_units = 'inches', scale = 1)\n c.set_xlim(edgex - cell_size[0], edgex + (2 * cell_size[0]))\n c.set_ylim(edgey - cell_size[1], edgey + (2 * cell_size[1]))\n c.invert_yaxis()\n c.set_aspect(aspect = 1)\n c.set_facecolor('black')\n c.add_patch(rect3)\n\n d.clear()\n d.set(title = 'Histogram of Gradients')\n d.grid()\n d.set_xlim(0, 180)\n d.set_xticks(angle_axis)\n d.set_xlabel('Angle')\n d.bar(angle_axis,\n ave_grad[cell_num_y, cell_num_x, :],\n 180 // num_bins,\n align = 'center',\n alpha = 0.5,\n linewidth = 1.2,\n edgecolor = 'k')\n\n fig.canvas.draw()\n\n# Create a connection between the figure and the mouse click\nfig.canvas.mpl_connect('button_press_event', onpress)\nplt.show()\n```\n\n The feature vector has shape: (57600, 1)\n The reshaped feature vector has shape: (40, 40, 2, 2, 9)\n The average gradient array has shape: (41, 41, 9)\n\n\n\n \n\n\n\n\n\n\n# Understanding The Histograms\n\nLet's take a look at a couple of snapshots of the above figure to see if the histograms for the selected cell make sense. Let's start looking at a cell that is inside a triangle and not near an edge:\n\n
\n
\n \n
Fig. 4. - Histograms Inside a Triangle.
\n
\n
\n\nIn this case, since the triangle is nearly all of the same color there shouldn't be any dominant gradient in the selected cell. As we can clearly see in the Zoom Window and the histogram, this is indeed the case. We have many gradients but none of them clearly dominates over the other.\n\nNow let\u2019s take a look at a cell that is near a horizontal edge:\n\n
\n
\n \n
Fig. 5. - Histograms Near a Horizontal Edge.
\n
\n
\n\nRemember that edges are areas of an image where the intensity changes abruptly. In these cases, we will have a high intensity gradient in some particular direction. This is exactly what we see in the corresponding histogram and Zoom Window for the selected cell. In the Zoom Window, we can see that the dominant gradient is pointing up, almost at 90 degrees, since that\u2019s the direction in which there is a sharp change in intensity. Therefore, we should expect to see the 90-degree bin in the histogram to dominate strongly over the others. This is in fact what we see. \n\nNow let\u2019s take a look at a cell that is near a vertical edge:\n\n
\n
\n \n
Fig. 6. - Histograms Near a Vertical Edge.
\n
\n
\n\nIn this case we expect the dominant gradient in the cell to be horizontal, close to 180 degrees, since that\u2019s the direction in which there is a sharp change in intensity. Therefore, we should expect to see the 170-degree bin in the histogram to dominate strongly over the others. This is what we see in the histogram but we also see that there is another dominant gradient in the cell, namely the one in the 10-degree bin. The reason for this, is because the HOG algorithm is using unsigned gradients, which means 0 degrees and 180 degrees are considered the same. Therefore, when the histograms are being created, angles between 160 and 180 degrees, contribute proportionally to both the 10-degree bin and the 170-degree bin. This results in there being two dominant gradients in the cell near the vertical edge instead of just one. \n\nTo conclude let\u2019s take a look at a cell that is near a diagonal edge.\n\n
\n
\n \n
Fig. 7. - Histograms Near a Diagonal Edge.
\n
\n
\n\nTo understand what we are seeing, let\u2019s first remember that gradients have an *x*-component, and a *y*-component, just like vectors. Therefore, the resulting orientation of a gradient is going to be given by the vector sum of its components. For this reason, on vertical edges the gradients are horizontal, because they only have an x-component, as we saw in Figure 4. While on horizontal edges the gradients are vertical, because they only have a y-component, as we saw in Figure 3. Consequently, on diagonal edges, the gradients are also going to be diagonal because both the *x* and *y* components are non-zero. Since the diagonal edges in the image are close to 45 degrees, we should expect to see a dominant gradient orientation in the 50-degree bin. This is in fact what we see in the histogram but, just like in Figure 4., we see there are two dominant gradients instead of just one. The reason for this is that when the histograms are being created, angles that are near the boundaries of bins, contribute proportionally to the adjacent bins. For example, a gradient with an angle of 40 degrees, is right in the middle of the 30-degree and 50-degree bin. Therefore, the magnitude of the gradient is split evenly into the 30-degree and 50-degree bin. This results in there being two dominant gradients in the cell near the diagonal edge instead of just one.\n\nNow that you know how HOG is implemented, in the workspace you will find a notebook named *Examples*. In there, you will be able set your own paramters for the HOG descriptor for various images. Have fun!\n\n\n```python\n\n```\n", "meta": {"hexsha": "c6542fe0eeaa240a7f9954de644baa5400a9e1aa", "size": 453914, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "1. 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YES", "lm_q1_score": 0.6076631556226292, "lm_q2_score": 0.731058584489497, "lm_q1q2_score": 0.4442373663959002}} {"text": "# OpenQEMIST and Rigetti example\n\nThis notebook shows how OpenQEMIST can be combined with the Rigetti stack to use the Variational Quantum Eigensolver (VQE) as an electronic structure solver, and combine it with a problem-decomposition technique such as Density Matrix Embedding Theory (DMET).\n\n## VQE Example\n\nThis tutorial assumes that the user has correctly set up and configured the OpenQEMIST package. The Variational Quantum Eigensolver (VQE)$^{1,2}$ is a hybrid quantum-classical algorithm for simulating quantum systems. We here focus on VQE within the context of solving the molecular electronic structure problem for the ground-state energy of a molecular system. In VQE, we first prepare the trial wavefunction (quantum state) $\\vert \\Psi(\\vec{\\theta}) \\rangle = U(\\vec{\\theta}) \\vert 0 \\rangle$ based on an ansatz that depends on $m$ parameters defining $\\vec{\\theta}=(\\theta_1, \\theta_2, \\ldots, \\theta_m)$. The expectation value of the Hamiltonian ($\\hat{H}$), $\\langle \\Psi(\\vec{\\theta}) \\vert \\hat{H} \\vert \\Psi(\\vec{\\theta}) \\rangle$, will then be simulated. \n\nThe expectation value can be minimized based on the variational principle, \n\n\\begin{equation}\nE = \\min_{\\vec{\\theta}} \\frac{\\langle \\Psi(\\vec{\\theta}) \\vert \\hat{H} \\vert \\Psi(\\vec{\\theta}) \\rangle}{\\langle \\Psi(\\vec{\\theta}) \\vert \\Psi(\\vec{\\theta}) \\rangle} \\geq E_{\\text{gs}}\\nonumber\n\\end{equation}\n\nwhich ensures that the energy computed will be an upper bound to the true ground-state energy $E_{\\text{gs}}$. This allows us using classical minimizers to find optimal parameters $\\vec{\\theta}$ for the ground-state energy $E_{\\text{gs}}$. \n\nVQE can be performed using OpenQEMIST in conjuction with the Rigetti stack for calculating the ground state energy of a molecular system. The unitary coupled-cluster ansatz can be used to prepare the trial wavefunction $\\vert \\Psi(\\vec{\\theta}) \\rangle$. In this notebook, we will show you an example using a small molecule, the hydrogen molecule (H$_\\text{2}$), for a simulation using VQE. \n\n\n```python\nfrom openqemist.electronic_structure_solvers import VQESolver, FCISolver\nfrom openqemist.quantum_solvers import RigettiParametricSolver\n\nfrom pyscf import gto\n\n# Build the molecule\nH2 = [['H', [0.0, 0.0, 0.0]], ['H', [0.0, 0.0, 0.74137727]]]\nmol = gto.Mole()\nmol.atom = H2\nmol.basis = \"sto-3g\"\nmol.charge = 0\nmol.spin = 0\nmol.build()\n\n# Configure the solver object\nvqe_solver = VQESolver()\nvqe_solver.hardware_backend_type = RigettiParametricSolver\nvqe_solver.ansatz_type = RigettiParametricSolver.Ansatze.UCCSD\n```\n\nWe can now simulate the molecule and get its energy.\n\n\n```python\nenergy_fci = FCISolver().simulate(mol)\nenergy_vqe = vqe_solver.simulate(mol)\n\nprint(\"\\nFCI energy = \", energy_fci)\nprint(\"VQE energy = \", energy_vqe)\n```\n\nIt is possible to use different initial parameters for the optimization:\n\n\n```python\n# Using custom initial parameters\n# Getting the dimension of the initial parameters vector\nnum_var_params = vqe_solver.hardware_backend.amplitude_dimension\n# Set the intial parameters for the solver\nvqe_solver.initial_var_params = [0.01 for i in range(num_var_params)]\n\nvqe_solver.simulate(mol)\n```\n\n## Using the QVM shot-based simulator\nTo use the QVM, we can use the `backend_parameters` attribute of the `VQESolver` object. The VQE object then configures the hardware backend automatically. Because the QVM is slower than the default wavefunction simulator backend, we specify an optimizer function that returns after a few iterations, in the interest of showing the usage of the solver in a reasonable time. See the documentation for more details about using custom optimizers. This interface is what would also be used to target a QPU backend in the future.\n\n\n```python\ndef quick_optimizer(backend, amplitudes):\n from scipy.optimize import minimize\n\n print(\"Running using custom optimizer.\")\n \n # We use a force the optimizer to return after 2 iterations.\n result = minimize(backend, amplitudes, method='COBYLA',\n options={'disp':True, 'maxiter':2})\n\n return result.fun\n\nvqe = VQESolver()\nvqe.optimizer = quick_optimizer\n```\n\nTo use the QVM, we can use the `backend_parameters` attribute of the `VQESolver` object. The VQE object then configures the hardware backend automatically. We can then run the simulation with the object. The number of shots can also be set with this parameter.\n\nNote that because we restricted the optimizer to 2 iterations and reduced the number of shots, the resulting energy will not be accurate. \n\n\n```python\nvqe.hardware_backend_type = RigettiParametricSolver\nvqe.ansatz_type = RigettiParametricSolver.Ansatze.UCCSD\nvqe.backend_parameters = {'backend': '4q-qvm', 'n_shots': 10}\n\nenergy = vqe.simulate(mol)\nprint(\"Unconverged QMV energy: \", energy)\n```\n\n## DMET Example\n\nAt the current early stage of quantum hardware, the available computational resource is yet very limited. Thus, it is still challenging to perform accurate electronic structure calculations on actual quantum hardware. Simulation on classical computer requires large computational cost as well. Therefore, we need to reduce the problem size while maintaining the accuracy of electronic structure calculation to solve a problem for small sized molecules to perform quantum simulations. \n\nDensity Matrix Embedding Theory (DMET)$^{3,4}$ is a powerful problem decomposition technique to reduce the problem size, while maintaining the accuracy of the electronic structure calculation. The DMET method decomposes a molecule into fragments, and each fragment is treated as an open quantum system that is entangled with each of the other fragments, all taken together to be that fragment's surrounding environment (or \"bath\"). VQE algorithm can be used with DMET using OpenQEMIST in conjuction with the Rigetti stack. \n\nIn this notebook, we will show you an example of H$_\\text{4}$ molecule for DMET simulation using VQE as an electronic structure solver.\n\n\n```python\nfrom openqemist.problem_decomposition import DMETProblemDecomposition\nfrom openqemist.problem_decomposition.electron_localization import meta_lowdin_localization\n\nH4 = [['H', [0.7071067811865476, 0.0, 0.0]],\n ['H', [0.0, 0.7071067811865476, 0.0]],\n ['H', [-1.0071067811865476, 0.0, 0.0]],\n ['H', [0.0, -1.0071067811865476, 0.0]]]\n\nmol = gto.Mole()\nmol.atom = H4\nmol.basis = \"minao\"\nmol.charge = 0\nmol.spin = 0\nmol.build()\n\ndmet = DMETProblemDecomposition()\ndmet.verbose = True\ndmet.electron_localization_method = meta_lowdin_localization\n# Set the DMET object to use the solver that we configured above\ndmet.electronic_structure_solver = vqe_solver\nenergy_vqe = dmet.simulate(mol, [1,1,1,1])\n\nprint(\"The DMET energy is: \", energy_vqe)\n```\n\n## References\n1. Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Al\u00e1n Aspuru-Guzik, and Jeremy L. O'Brien, \"A variational eigenvalue solver on a photonic quantum processor\", Nat. Commun., 5, 4213 (2014).\n2. Jarrod R. McClean, Jonathan Romero, Ryan Babbush, and Al\u00e1n Aspuru-Guzik, \"The theory of variational hybrid quantum-classical algorithms\", New J. Phys., 18, 023023 (2016).\n3. Gerald Knizia and Garnet K.-L. Chan, \"Density Matrix Embedding: A Simple Alternative to Dynamical Mean-Field Theory\", Phys. Rev. Lett., 109, 186404 (2012).\n4. Sebastian Wouters, Carlos A. Jim\u00e9nez-Hoyos, Qiming Sun, and Garnet K.-L. Chan, \"A Practical Guide to Density Matrix Embedding Theory in Quantum Chemistry\", J. Chem. Theory Comput., 12, pp. 2706–2719 (2016).\n", "meta": {"hexsha": "8fd82e1390c01239df57793300c570b1e31df6d6", "size": 10420, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/rigetti_example.ipynb", "max_stars_repo_name": "1QB-Information-Technologies/openqemist", "max_stars_repo_head_hexsha": "e2ab887af31d78d03dcb92cfa3a0705b2436823d", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 35, "max_stars_repo_stars_event_min_datetime": "2019-05-31T22:37:23.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-06T12:01:18.000Z", "max_issues_repo_path": "examples/rigetti_example.ipynb", "max_issues_repo_name": "1QB-Information-Technologies/openqemist", "max_issues_repo_head_hexsha": "e2ab887af31d78d03dcb92cfa3a0705b2436823d", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2021-03-23T22:34:23.000Z", "max_issues_repo_issues_event_max_datetime": "2021-06-23T13:09:46.000Z", "max_forks_repo_path": "examples/rigetti_example.ipynb", "max_forks_repo_name": "1QB-Information-Technologies/openqemist", "max_forks_repo_head_hexsha": "e2ab887af31d78d03dcb92cfa3a0705b2436823d", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 10, "max_forks_repo_forks_event_min_datetime": "2019-06-06T23:14:18.000Z", "max_forks_repo_forks_event_max_datetime": "2021-12-02T21:56:13.000Z", "avg_line_length": 43.5983263598, "max_line_length": 801, "alphanum_fraction": 0.6401151631, "converted": true, "num_tokens": 1987, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802476562641, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.44414027847744886}} {"text": "## Note: the example below requires Cantera 2.5.0a2 or newer\n\nIf the stable release of 2.5.0 is not yet available, you might consider [building from source](https://cantera.org/install/compiling-install.html). Note also that 2.5.0, once released, is more recent than and supercedes 2.5.0a2, which is an alpha release.\n\n\n```python\nimport cantera as ct\nprint('Runnning Cantera version: ' + ct.__version__)\n```\n\n Runnning Cantera version: 2.5.0a2\n\n\n# Lithium Ion Battery Example\n## Open circuit calculations as a function of anode and cathode lithium content.\n \nIn this example we will illustrate how to calculate the open circuit voltage (voltage when the external applied current is 0) for a lithium ion battery. The thermodynamics are based on a graphite anode and a LiCoO2 cathode (the typical/standard active materials for commercial batteries, as of 2019), and are modeled using the 'BinarySolutionTabulatedThermo' class.\n\nFor the sake of simplicity, we're going to assume that the anode and cathode capacities are perfectly balanced (i.e. if the cathode lithium content is X percent of it's max possible (i.e. its capacity), then we will assume that the anode is at 1-X percent. Without loss of generality, we will define the anode composition:\n\nThe routine below returns the cell voltage (in Volt) of a lithium-ion cell for a given cell current and active material lithium stoichiometries.\n\nNote that the function 'E_cell' below has even greater capabilities than what we use, here. It calculates the steady state cell voltage, at a given composition and cell current, for a given electrolyte ionic resistance. This functionality is presented in greater detail in the reference (which also describes the derivation of the BinarySolutionTabulatedThermo class): M. Mayur, S. DeCaluwe, B. L. Kee, W. G. Bessler, \"Modeling thermodynamics and kinetics of intercalation phases for lithium-ion batteries in open source software\", currently under review at _Electrochimica Acta_. In the future, this example may be developed further to demonstrate simulating charge-discharge of the lithium ion battery.\n\n\nOther than the typical Cantera dependencies, plotting functions require that you have matplotlib installed, and finding the electrode potentials uses Scipy's fsolve. See https://matplotlib.org/ and https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fsolve.html for additional info.\n\nThe open circuit (i.e. equilibrium) voltage, here, is calculated via two means: kinetically and thermodynamically.\n\nThe system here, can be thought of as consisting of two particles--anode and cathode--connected by a liquid electrolyte:\n\n\n\n### Kinetic equilibrium calculations\nIn the kinetic equilibrium problem, steady state is achieved when the faradaic current at each electrode (anode and cathode) surface and the ionic current through the electrolyte are all zero. There are, essentially, four relevant electric potentials which must be determined:\n\n- $\\phi_{\\rm anode}$: electric potential of graphite anode.\n- $\\phi_{\\rm elyte,\\, anode}$: electric potential of electrolyte at anode interface.\n- $\\phi_{\\rm elyte,\\, cathode}$: electric potential of electrolyte at cathode interface.\n- $\\phi_{\\rm cathode}$: electric potential of LCO cathode.\n\nSetting one of these four to the reference potential of zero (because it is the difference in electric potential which drives currents, the actual potential values are irrelevant. Let's assume the anode electric potential is zero), there is only one distribution of electric potentials across the cell such that the current is invariant across the cell. I.e. we want the potentials such that:\n\n\\begin{equation}\ni_{\\rm Far,\\, anode} = i_{\\rm ionic} = i_{\\rm Far,\\,cathode}= i_{\\rm app}\n\\end{equation}\n\nwhere $i_{\\rm app}$ is the user input for the applied current. For this example, we assume an applied current of 0, to calculate the equilibrium voltage.\n\n#### Faradaic current\nThe Faradaic current for this model is calculated using Butler-Volmer kinetics. For a Li-ion battery, this is:\n\n\\begin{equation}\ni = S_{\\rm elde}i_\\circ\\left[\\exp\\left(\\frac{F\\beta\\eta}{RT}\\right) - \\exp\\left(-\\frac{F\\beta\\eta}{RT}\\right) \\right]\n\\end{equation}\n\nwhere $S_{\\rm elde}$ is the specific surface area of the electrode in question, $F$ is Faraday's constant, $\\beta$ is the charge-transfer symmetry parameter, $R$ the universal gas constant, $T$ the temperature, and $\\eta$ the overpotential, which is the electric potential difference between the electrode and electrolyte, $\\Delta \\phi = \\phi_{\\rm elde} - \\phi_{\\rm elyte}$, relative to that at equilibrium, $\\Delta \\phi_{\\rm eq}$:\n\n\\begin{equation}\n\\eta = \\Delta \\phi - \\Delta \\phi_{\\rm eq}\n\\end{equation}\n\n$i_\\circ$ is known as the \"exchange current density,\" which is equal to the rate of the forward and reverse current at equilibrium (which are equal). $i_\\circ$ and $\\beta$ are provided as user inputs in the cti file. At any particular state, (user-specified electric potentials, pressure, temperature, and chemical compositions), Cantera calculates $\\eta$ as part of the routine to evaluate reaction rates of progress $\\left(\\dot{q} = \\frac{i_{\\rm Far}}{F}\\right)$. The user simply sets the state values mentioned above.\n\n#### Ionic current\nThe electrolyte is modeled as a resistor with user-defined ionic resistance $R_{\\rm io}$, and hence the ionic current is calculated as:\n\n\\begin{equation}\ni_{\\rm ionic} = \\frac{\\phi_{\\rm elyte,\\,ca} - \\phi_{\\rm elyte,\\,an}}{R_{\\rm io}}\n\\end{equation}\n\nwhere positive current is defined as delivering Li$^+$ to the anode interface. Given $i_{\\rm app}$, this equation can be inverted, to calculate the electric potential of the electrolyte at the cathode interface, relative to that at the anode interface:\n\n\\begin{equation}\n\\phi_{\\rm elyte,\\,ca} = \\phi_{\\rm elyte,\\,an} + R_{\\rm io}i_{\\rm app} \n\\end{equation}\n\nAgain: in this example, $i_{\\rm app} = 0$ and hence the two electric potential values in the electrolyte are equal.\n\n#### Numerical routine\nFor the kinetic routine, there are three processes to determine the cell voltage $\\phi_{\\rm cathode} - \\phi_{\\rm anode}$ which corresponds to the user-provided $i_{\\rm app}$:\n1. Determine the $\\phi_{\\rm elyte,\\,anode}$ value which corresponds to $i_{\\rm app}$, given $X_{\\rm Li, anode}$, the percentage of Li in the anode active material.\n2. Determine $\\phi_{\\rm elyte,\\,cathode}$, given $\\phi_{\\rm elyte,\\,anode}$ and $i_{\\rm app}$.\n3. Determine the $\\phi_{\\rm cathode}$ which corresponds to $i_{\\rm app}$, given $phi_{\\rm elyte,\\,cathode}$ and $X_{\\rm Li, anode}$, the percentage of Li in the anode active material.\n\nThe routines below are written generally such that an interested user may set $i_{\\rm app}$ to any value of interest.\n\n#### Import necessary packages:\n\n\n```python\nfrom __future__ import division\nfrom __future__ import print_function\n\nimport numpy as np\n\nfrom scipy.optimize import fsolve\n\n# Used for timing our calculations:\nimport time\n\n# Plotting:\nimport matplotlib.pyplot as plt\nimport matplotlib as mpl\n\nplt.rcParams['axes.labelsize'] = 16\nplt.rcParams['xtick.labelsize'] = 12\nplt.rcParams['ytick.labelsize'] = 12\nplt.rcParams['figure.autolayout'] = True\n```\n\n### Define the phases\n\nThe phase thermodynamics are defined according to experimentally-measured open circuit voltage values, as desscribed in the reference provided above:\n\n\n```python\ninputCTI = 'lithium_ion_battery.cti'\nanode = ct.Solution(inputCTI, 'anode');\ncathode = ct.Solution(inputCTI,'cathode');\n# The 'elde' electrode phase is needed as a source/sink for electrons:\nelde = ct.Solution(inputCTI,'electron');\nelyte = ct.Solution(inputCTI,'electrolyte');\nanode_interface = ct.Interface(inputCTI, 'edge_anode_electrolyte', [anode, elde, elyte]);\ncathode_interface = ct.Interface(inputCTI, 'edge_cathode_electrolyte', [cathode, elde, elyte]);\n```\n\n### Define battery conditions : temperature, pressure, stoichiometry, electrolyte resistance:\n\nInputs are:\n- Stoichiometries X_Li_ca and X_Li_an [-] (can be vectors)\n- Temperature T [K]\n- Pressure P [Pa]\n- Externally-applied current i_app [A]\n- Electrolyte resistance R_elyte [Ohm]\n- Anode total surface area S_an [m^2]\n- Cathode total surface area S_ca [m^2]\n\n\n\n```python\n# Array of lithium mole fractions in the anode:\nX_Li_an = np.arange(0.005, 0.995, 0.02)\n# Assume that the cathode and anode capacities are balanced:\nX_Li_ca = 1. - X_Li_an;\n\n# I_app = 0: Open circuit\nI_app = 0.;\n\n# At zero current, electrolyte resistance is irrelevant:\nR_elyte = 0.;\n\n# Temperature and pressure\nT = 300 # K\nP = ct.one_atm\n\nF = ct.faraday\n\nS_ca = 1.1167; # [m^2] Cathode total active material surface area\nS_an = 0.7824; # [m^2] Anode total active material surface area\n```\n\n### Set phase temperatures and pressures:\n\n\n```python\nphases = [anode, elde, elyte, cathode, anode_interface, cathode_interface];\nfor ph in phases:\n ph.TP = T,P\n\n```\n\n### Helper Functions:\n\n\n```python\ndef anode_curr(phi_l,I_app,phi_s,X_Li_an):\n\n # Set the active material mole fraction\n anode.X = 'Li[anode]:'+str(X_Li_an)+', V[anode]:'+str(1-X_Li_an)\n\n # Set the electrode and electrolyte potential\n elde.electric_potential = phi_s\n elyte.electric_potential = phi_l\n\n # Get the net product rate of electrons in the anode (per m2^ interface)\n r_elec = anode_interface.get_net_production_rates(elde)\n\n anCurr = r_elec*ct.faraday*S_an;\n diff = I_app + anCurr\n \n return diff\n\ndef cathode_curr(phi_s,I_app,phi_l,X_Li_ca):\n \n # Set the active material mole fractions\n cathode.X = 'Li[cathode]:'+str(X_Li_ca)+', V[cathode]:'+str(1-X_Li_ca)\n\n # Set the electrode and electrolyte potential\n elde.electric_potential = phi_s\n elyte.electric_potential = phi_l\n \n # Get the net product rate of electrons in the cathode (per m2^ interface)\n r_elec = cathode_interface.get_net_production_rates(elde)\n \n caCurr = r_elec*ct.faraday*S_an;\n diff = I_app - caCurr\n \n return diff\n```\n\n### Run the calculations for all stoichiometries:\n\n\n```python\n#Tic\nt0 = time.time()\n\n# Initialize array of OCVs:\nE_cell_kin = np.zeros_like(X_Li_ca)\n\nfor i,X_an in enumerate(X_Li_an):\n #Set anode electrode potential to 0:\n phi_s_an = 0\n E_init = 3.0\n \n phi_l_an = fsolve(anode_curr,E_init,args=(I_app,phi_s_an,X_an))\n \n # Calculate electrolyte potential at cathode interface:\n phi_l_ca = phi_l_an + I_app*R_elyte;\n \n # Calculate cathode electrode potential\n phi_s_ca = fsolve(cathode_curr,E_init,args=(I_app,phi_l_ca,X_Li_ca[i]))\n \n # Calculate cell voltage\n E_cell_kin[i] = phi_s_ca - phi_s_an\n \n \n#Toc\nt1 = time.time()\nprint('{:d} cell voltages calculated in {:3.2f} seconds.'.format(i,t1-t0))\n```\n\n 49 cell voltages calculated in 0.36 seconds.\n\n\n### Plot cell voltage, as a function of the cathode stoichiometry:\n\n\n```python\nplt.figure()\nplt.plot(100*X_Li_ca, E_cell_kin,color='b',linewidth=2.5)\nplt.ylim([2.5,4.3])\nplt.xlabel('Li Fraction in Cathode (%)',fontname='Times New Roman',fontsize=18)\nplt.ylabel('Open Circuit Potential (V)',fontname='Times New Roman',fontsize=18)\n\nax = plt.gca()\n\nfor tick in ax.xaxis.get_major_ticks():\n tick.label1.set_fontsize(14)\n tick.label1.set_fontname('Times New Roman')\nfor tick in ax.yaxis.get_major_ticks():\n tick.label1.set_fontsize(14)\n tick.label1.set_fontname('Times New Roman')\n \nplt.show()\n\n```\n\n### Thermodynamic Equilibrium Calculation\nFor the I_app = 0 case, we can also calcualte the voltage using thermodynamics. At equilibrium, the net electrochemical potential change of the reaction must be zero:\n\n\\begin{equation}\n\\sum_k\\nu_k\\tilde{\\mu}_k = 0\n\\end{equation}\n\nwhere $\\tilde{\\mu}_k = \\mu_k + z_kF\\Phi_k$, where, in turn $\\mu_k = \\frac{\\partial g_k}{\\partial n_k}$ is the chemical potential, $\\nu_k$ the net stoichiometric coefficient, $z_k$ the net elementary charge, and $\\Phi_k$ the phase electric potential for species $k$.\n\nFrom this, we can calculate the equilibrium electric potential difference $\\Delta \\Phi_{\\rm eq} = \\left(\\Phi_{\\rm elde} - \\Phi_{\\rm elyte}\\right)_{\\rm eq}$ as:\n\n\\begin{equation}\n\\Delta \\Phi_{\\rm eq} = -\\frac{\\Delta g_{\\rm rxn}}{n_{\\rm charge}F}\n\\end{equation}\n\nwhere $\\Delta g_{\\rm rxn} = \\sum_k \\nu_k\\mu_k$ is the chemical potential of the reaction and and $n_{\\rm charge} = \\sum_{k,\\,{\\rm elde}} \\nu_k z_k$ is the net elementary charge transferred from the electrolyte to the electrode.\n\n\n```python\n#Tic\nt0 = time.time()\n\n# Initialize array of OCVs:\nE_cell_therm = np.zeros_like(X_Li_ca)\n\nfor i,X_an in enumerate(X_Li_an):\n #Set anode electrode potential to 0:\n anode.X = 'Li[anode]:'+str(X_an)+', V[anode]:'+str(1-X_an)\n dG_an = anode_interface.delta_gibbs[0]\n n_charge = -1.\n E_eq_an = -dG_an/n_charge/ct.faraday\n \n cathode.X = 'Li[cathode]:'+str(1.-X_an)+', V[cathode]:'+str(X_an)\n dG_ca = cathode_interface.delta_gibbs[0]\n n_charge = 1.\n E_eq_ca = -dG_ca/n_charge/ct.faraday\n \n E_cell_therm[i] = E_eq_ca - E_eq_an\n \n#Toc\nt1 = time.time()\nprint('{:d} cell voltages calculated in {:3.2f} seconds.'.format(i,t1-t0))\n```\n\n 49 cell voltages calculated in 0.01 seconds.\n\n\n### Plot thermodynamic OCV, and compare to results from kinetic method\n\n\n```python\nplt.figure()\nplt.plot(100*X_Li_ca, E_cell_therm,color='b',linewidth=2.5)\nplt.plot(100*X_Li_ca, E_cell_kin,linewidth=0.,marker='o',markerfacecolor='none',markeredgecolor='r')\nplt.ylim([2.5,4.3])\nplt.xlabel('Li Fraction in Cathode (%)',fontname='Times New Roman',fontsize=18)\nplt.ylabel('Open Circuit Potential (V)',fontname='Times New Roman',fontsize=18)\nplt.legend(['Thermodynamic','Kinetic'])\n\nax = plt.gca()\n\nfor tick in ax.xaxis.get_major_ticks():\n tick.label1.set_fontsize(14)\n tick.label1.set_fontname('Times New Roman')\nfor tick in ax.yaxis.get_major_ticks():\n tick.label1.set_fontsize(14)\n tick.label1.set_fontname('Times New Roman')\n \nplt.show()\n```\n\nAs one would expect, the two approaches give identical results. While both methods are incredibly fast, the thermodynamic method is roughly 30 times faster.\n\nA large part of this is that the thermodynamic approach is an analytical approach (i.e. the answer is known from theory), while the kinetic approach relies on the root-finding fzero method to fit the correct voltage. Note also that the kinetic method, because of the use of Butler-Volmer kinetics, calculates the thermodynamic voltage, in order to calculate the overpotential $\\eta = \\Delta \\Phi - \\Delta \\Phi_{\\rm eq}$.\n\nHowever, it is at last important to note that, while slower, the kinetic method is of course more robust, and can be used to find results away from equilibrium. The thermodynamic method is only applicable at equilibrium (zero current).\n\n\n```python\n\n```\n", "meta": {"hexsha": "92ba943d0f3baefe28131461dc2575331ec05893", "size": 63512, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "electrochemistry/lithium_ion_battery.ipynb", "max_stars_repo_name": "funnelferry/cantera-jupyter", "max_stars_repo_head_hexsha": "66b4d26ff542a45eddd7c9d0ad87fecc93203ce3", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "electrochemistry/lithium_ion_battery.ipynb", "max_issues_repo_name": "funnelferry/cantera-jupyter", "max_issues_repo_head_hexsha": "66b4d26ff542a45eddd7c9d0ad87fecc93203ce3", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "electrochemistry/lithium_ion_battery.ipynb", "max_forks_repo_name": "funnelferry/cantera-jupyter", "max_forks_repo_head_hexsha": "66b4d26ff542a45eddd7c9d0ad87fecc93203ce3", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 123.0852713178, "max_line_length": 24052, "alphanum_fraction": 0.8530041567, "converted": true, "num_tokens": 4056, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8006920020959544, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.44396006669223925}} {"text": "\n\n\n# 1D Alfven Wave `GiRaFFEfood` Initial Data for `GiRaFFE`\n\n## This module provides another initial data option for `GiRaFFE`, drawn from [this paper](https://arxiv.org/abs/1310.3274) .\n\n**Notebook Status:** Self-Validated \n\n**Validation Notes:** This tutorial notebook has been confirmed to be self-consistent with its corresponding NRPy+ module, as documented [below](#code_validation). **Additional validation tests may have been performed, but are as yet, undocumented. (TODO)**\n\n### NRPy+ Source Code for this module: [GiRaFFEfood_NRPy/GiRaFFEfood_NRPy_1D_tests.py](../edit/GiRaFFEfood_NRPy/GiRaFFEfood_NRPy_1D_tests.py)\n\n## Introduction:\n\n### Alfvén Wave:\n\n This is a flat-spacetime test with initial data \n\\begin{align}\nA_x &= 0 \\\\\nA_y &= \\left \\{ \\begin{array}{lll}\\gamma_\\mu x - 0.015 & \\mbox{if} & x \\leq -0.1/\\gamma_\\mu \\\\\n1.15 \\gamma_\\mu x - 0.03g(x) & \\mbox{if} & -0.1/\\gamma_\\mu \\leq x \\leq 0.1/\\gamma_\\mu \\\\ \n1.3 \\gamma_\\mu x - 0.015 & \\mbox{if} & x \\geq 0.1/\\gamma_\\mu \\end{array} \\right. , \\\\\n A_z = &\\ y - \\gamma_\\mu (1-\\mu)x ,\n\\end{align}\nwhich generates the magnetic field in the wave frame,\n\\begin{align}\nB'^{x'}(x') = &\\ 1.0,\\ B'^y(x') = 1.0, \\\\\nB'^z(x') = &\\ \\left \\{ \\begin{array}{lll} 1.0 & \\mbox{if} & x' \\leq -0.1 \\\\\n\t\t\t\t1.0+0.15 f(x') & \\mbox{if} & -0.1 \\leq x' \\leq 0.1 \\\\\n\t\t\t\t1.3 & \\mbox{if} & x' \\geq 0.1 \\end{array} \\right. .\n\\end{align}\nThe electric field in the wave frame is then given by\n$$E'^{x'}(x') = -B'^z(0,x') \\ \\ , \\ \\ E'^y(x') = 0.0 \\ \\ , \\ \\ E'^z(x') = 1.0 .$$\n\nThese are converted to the grid frame by \n\\begin{align}\n B^x(0,x) = &\\ B'^{x'}(\\gamma_\\mu x) , \\\\\n B^y(0,x) = &\\ \\gamma_\\mu [ B'^y(\\gamma_\\mu x) - \\mu E'^z(\\gamma_\\mu x) ] , \\\\ \n B^z(0,x) = &\\ \\gamma_\\mu [ B'^z(\\gamma_\\mu x) + \\mu E'^y(\\gamma_\\mu x) ] , \n\\end{align}\nand\n\\begin{align}\n E^x(0,x) = &\\ E'^{x'}(\\gamma_\\mu x) , \\\\ \n E^y(0,x) = &\\ \\gamma_\\mu [ E'^y(\\gamma_\\mu x) + \\mu B'^z(\\gamma_\\mu x) ] ,\\\\ \n E^z(0,x) = &\\ \\gamma_\\mu [ E'^z(\\gamma_\\mu x) - \\mu B'^y(\\gamma_\\mu x) ],\n\\end{align}\nand the velocity is given by $$\\mathbf{v} = \\frac{\\mathbf{E} \\times \\mathbf{B}}{B^2}$$ in flat spacetime. Additionally, $f(x)=1+\\sin (5\\pi x)$, $-1<\\mu<1$ is the wave speed relative to the grid frame and $\\gamma_\\mu = (1-\\mu^2)^{-1/2}$, and $g(x) = \\cos (5\\pi \\gamma_\\mu x)/\\pi$.\n\nFor the eventual purpose of testing convergence, any quantity $Q$ evolves as $Q(t,x) = Q(0,x-\\mu t)$\n\nSee the [Tutorial-GiRaFFEfood_NRPy](Tutorial-GiRaFFEfood_NRPy.ipynb) tutorial notebook for more general detail on how this is used.\n\n\n\n\n# Table of Contents:\n$$\\label{toc}$$\n\nThis notebook is organized as follows\n\n1. [Step 1](#initializenrpy): Import core NRPy+ modules and set NRPy+ parameters \n1. [Step 2](#vector_ak): Set the vector $A_k$\n1. [Step 3](#vectors_for_velocity): Set the vectors $B^i$ and $E^i$ for the velocity\n1. [Step 4](#vi): Calculate $v^i$\n1. [Step 5](#code_validation): Code Validation against `GiRaFFEfood_NRPy/GiRaFFEfood_NRPy_1D_tests` NRPy+ module\n1. [Step 6](#latex_pdf_output): Output this notebook to $\\LaTeX$-formatted PDF file\n\n\n\n# Step 1: Import core NRPy+ modules and set NRPy+ parameters \\[Back to [top](#toc)\\]\n$$\\label{initializenrpy}$$\n\nHere, we will import the NRPy+ core modules and set the reference metric to Cartesian, set commonly used NRPy+ parameters, and set C parameters that will be set from outside the code eventually generated from these expressions. We will also set up a parameter to determine what initial data is set up, although it won't do much yet.\n\n\n```python\n# Step 0: Add NRPy's directory to the path\n# https://stackoverflow.com/questions/16780014/import-file-from-parent-directory\nimport os,sys\nnrpy_dir_path = os.path.join(\"..\")\nif nrpy_dir_path not in sys.path:\n sys.path.append(nrpy_dir_path)\n\n# Step 0.a: Import the NRPy+ core modules and set the reference metric to Cartesian\nfrom outputC import * # NRPy+: Core C code output module\nimport finite_difference as fin # NRPy+: Finite difference C code generation module\nimport NRPy_param_funcs as par # NRPy+: Parameter interface\nimport grid as gri # NRPy+: Functions having to do with numerical grids\nimport indexedexp as ixp # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support\n\nimport reference_metric as rfm # NRPy+: Reference metric support\npar.set_parval_from_str(\"reference_metric::CoordSystem\",\"Cartesian\")\nrfm.reference_metric()\n\n# Step 1a: Set commonly used parameters.\nthismodule = \"GiRaFFEfood_NRPy_1D\"\n# Set the spatial dimension parameter to 3.\npar.set_parval_from_str(\"grid::DIM\", 3)\nDIM = par.parval_from_str(\"grid::DIM\")\n\n```\n\n##### \n\n# Step 2: Set the vector $A_k$ \\[Back to [top](#toc)\\]\n$$\\label{vector_ak}$$\n\nThe vector potential is given as\n\\begin{align}\nA_x &= 0 \\\\\nA_y &= \\left \\{ \\begin{array}{lll}\\gamma_\\mu x - 0.015 & \\mbox{if} & x \\leq -0.1/\\gamma_\\mu \\\\\n1.15 \\gamma_\\mu x - 0.03g(x) & \\mbox{if} & -0.1/\\gamma_\\mu \\leq x \\leq 0.1/\\gamma_\\mu \\\\ \n1.3 \\gamma_\\mu x - 0.015 & \\mbox{if} & x \\geq 0.1/\\gamma_\\mu \\end{array} \\right. , \\\\\nA_z &= y - \\gamma_\\mu (1-\\mu)x .\n\\end{align}\n\nHowever, to take full advantage of NRPy+'s automated function generation capabilities, we want to write this without the `if` statements, replacing them with calls to `fabs()`. To do so, we will use rewrite these functions in terms of maxima and minima. Critically, we will define these functions in the following way: \n\\begin{align}\n\\min(a,b) &= \\tfrac{1}{2} \\left( a+b - \\lvert a-b \\rvert \\right) \\\\\n\\max(a,b) &= \\tfrac{1}{2} \\left( a+b + \\lvert a-b \\rvert \\right). \\\\\n\\end{align}\nFor real numbers, these operate exactly as expected. In the case $a>b$,\n\\begin{align}\n\\min(a,b) &= \\tfrac{1}{2} \\left( a+b - (a-b) \\right) = b \\\\\n\\max(a,b) &= \\tfrac{1}{2} \\left( a+b + (a-b) \\right) = a, \\\\\n\\end{align}\nand in the case $a\n\n# Step 3: Set the vectors $B^i$ and $E^i$ for the velocity \\[Back to [top](#toc)\\]\n$$\\label{vectors_for_velocity}$$\n\nNow, we will set the magnetic and electric fields that we will need to define the initial velocities. First, we need to define $$f(x)=1+\\sin (5\\pi x);$$ note that in the definition of $B^i$, we need $f(x')$ where $x'=\\gamma_\\mu x$.\n\n\n\n```python\nxprime = gammamu*x\nf_AW = 1.0 + sp.sin(5.0*M_PI*xprime)\nprint(f_AW)\n```\n\n sin(5.0*M_PI*xx0/sqrt(1 - mu_AW**2)) + 1.0\n\n\nWe will now set the magnetic field in the wave frame:\n\\begin{align}\nB'^{x'}(x') = &\\ 1.0,\\ B'^y(x') = 1.0, \\\\\nB'^z(x') = &\\ \\left \\{ \\begin{array}{lll} 1.0 & \\mbox{if} & x' \\leq -0.1 \\\\\n\t\t\t\t1.0+0.15 f(x') & \\mbox{if} & -0.1 \\leq x' \\leq 0.1 \\\\\n\t\t\t\t1.3 & \\mbox{if} & x' \\geq 0.1 \\end{array} \\right. .\n\\end{align}\n\n\n\n```python\nBleftpU = ixp.zerorank1()\nBleftpU[0] = sp.sympify(1.0)\nBleftpU[1] = sp.sympify(1.0)\nBleftpU[2] = sp.sympify(1.0)\n\nBcenterpU = ixp.zerorank1()\nBcenterpU[0] = sp.sympify(1.0)\nBcenterpU[1] = sp.sympify(1.0)\nBcenterpU[2] = 1.0 + 0.15*f_AW\n\nBrightpU = ixp.zerorank1()\nBrightpU[0] = sp.sympify(1.0)\nBrightpU[1] = sp.sympify(1.0)\nBrightpU[2] = sp.sympify(1.3)\n\n```\n\nNow, we will set the electric field in the wave frame:\n\\begin{align}\nE'^{x'}(x') &= -B'^z(0,x'), \\\\ \nE'^y(x') &= 0.0, \\\\ \nE'^z(x') &= 1.0 .\n\\end{align}\n\n\n```python\nEleftpU = ixp.zerorank1()\nEleftpU[0] = -BleftpU[2]\nEleftpU[1] = sp.sympify(0.0)\nEleftpU[2] = sp.sympify(1.0)\n\nEcenterpU = ixp.zerorank1()\nEcenterpU[0] = -BcenterpU[2]\nEcenterpU[1] = sp.sympify(0.0)\nEcenterpU[2] = sp.sympify(1.0)\n\nErightpU = ixp.zerorank1()\nErightpU[0] = -BrightpU[2]\nErightpU[1] = sp.sympify(0.0)\nErightpU[2] = sp.sympify(1.0)\n\n```\n\nNext, we must transform the fields into the grid frame. We'll do the magnetic fields first.\n\\begin{align}\n B^x(0,x) = &\\ B'^{x'}(\\gamma_\\mu x) , \\\\\n B^y(0,x) = &\\ \\gamma_\\mu [ B'^y(\\gamma_\\mu x) - \\mu E'^z(\\gamma_\\mu x) ] , \\\\ \n B^z(0,x) = &\\ \\gamma_\\mu [ B'^z(\\gamma_\\mu x) + \\mu E'^y(\\gamma_\\mu x) ] , \n\\end{align}\n\n\n\n```python\nBleftU = ixp.zerorank1()\nBleftU[0] = BleftpU[0]\nBleftU[1] = gammamu*(BleftpU[1]-mu_AW*EleftpU[2])\nBleftU[2] = gammamu*(BleftpU[2]+mu_AW*EleftpU[1])\n\nBcenterU = ixp.zerorank1()\nBcenterU[0] = BcenterpU[0]\nBcenterU[1] = gammamu*(BcenterpU[1]-mu_AW*EcenterpU[2])\nBcenterU[2] = gammamu*(BcenterpU[2]+mu_AW*EcenterpU[1])\n\nBrightU = ixp.zerorank1()\nBrightU[0] = BrightpU[0]\nBrightU[1] = gammamu*(BrightpU[1]-mu_AW*ErightpU[2])\nBrightU[2] = gammamu*(BrightpU[2]+mu_AW*ErightpU[1])\n\n```\n\nAnd now the electric fields:\n\\begin{align}\n E^x(0,x) = &\\ E'^{x'}(\\gamma_\\mu x) , \\\\ \n E^y(0,x) = &\\ \\gamma_\\mu [ E'^y(\\gamma_\\mu x) + \\mu B'^z(\\gamma_\\mu x) ] ,\\\\ \n E^z(0,x) = &\\ \\gamma_\\mu [ E'^z(\\gamma_\\mu x) - \\mu B'^y(\\gamma_\\mu x) ],\n\\end{align}\n\n\n\n```python\nEleftU = ixp.zerorank1()\nEleftU[0] = EleftpU[0]\nEleftU[1] = gammamu*(EleftpU[1]+mu_AW*BleftpU[2])\nEleftU[2] = gammamu*(EleftpU[2]-mu_AW*BleftpU[1])\n\nEcenterU = ixp.zerorank1()\nEcenterU[0] = EcenterpU[0]\nEcenterU[1] = gammamu*(EcenterpU[1]+mu_AW*BcenterpU[2])\nEcenterU[2] = gammamu*(EcenterpU[2]-mu_AW*BcenterpU[1])\n\nErightU = ixp.zerorank1()\nErightU[0] = ErightpU[0]\nErightU[1] = gammamu*(ErightpU[1]+mu_AW*BrightpU[2])\nErightU[2] = gammamu*(ErightpU[2]-mu_AW*BrightpU[1])\n\n```\n\n\n\n# Step 4: Calculate $v^i$ \\[Back to [top](#toc)\\]\n$$\\label{vi}$$\n\nNow, we calculate $$\\mathbf{v} = \\frac{\\mathbf{E} \\times \\mathbf{B}}{B^2},$$ which is equivalent to $$v^i = [ijk] \\frac{E^j B^k}{B^2},$$ where $[ijk]$ is the Levi-Civita symbol and $B^2 = \\gamma_{ij} B^i B^j$ is a trivial dot product in flat space.\n\n\n\n```python\nimport WeylScal4NRPy.WeylScalars_Cartesian as weyl\nLeviCivitaSymbolDDD = weyl.define_LeviCivitaSymbol_rank3()\n\nBleft2 = BleftU[0]*BleftU[0] + BleftU[1]*BleftU[1] + BleftU[2]*BleftU[2]\nBcenter2 = BcenterU[0]*BcenterU[0] + BcenterU[1]*BcenterU[1] + BcenterU[2]*BcenterU[2]\nBright2 = BrightU[0]*BrightU[0] + BrightU[1]*BrightU[1] + BrightU[2]*BrightU[2]\n\nValenciavU = ixp.register_gridfunctions_for_single_rank1(\"AUX\",\"ValenciavU\")\n\nValenciavleftU = ixp.zerorank1()\nfor i in range(DIM):\n for j in range(DIM):\n for k in range(DIM):\n ValenciavleftU[i] += LeviCivitaSymbolDDD[i][j][k] * EleftU[j] * BleftU[k] / Bleft2\n \nValenciavcenterU = ixp.zerorank1()\nfor i in range(DIM):\n for j in range(DIM):\n for k in range(DIM):\n ValenciavcenterU[i] += LeviCivitaSymbolDDD[i][j][k] * EcenterU[j] * BcenterU[k] / Bcenter2\n \nValenciavrightU = ixp.zerorank1()\nfor i in range(DIM):\n for j in range(DIM):\n for k in range(DIM):\n ValenciavrightU[i] += LeviCivitaSymbolDDD[i][j][k] * ErightU[j] * BrightU[k] / Bright2\n```\n\n\n\n# Step 5: Code Validation against `GiRaFFEfood_NRPy/GiRaFFEfood_NRPy_1D_tests` NRPy+ module \\[Back to [top](#toc)\\]\n$$\\label{code_validation}$$\n\nHere, as a code validation check, we verify agreement in the SymPy expressions for the `GiRaFFE` Aligned Rotator initial data equations we intend to use between\n1. this tutorial and \n2. the NRPy+ [`GiRaFFEfood_NRPy/GiRaFFEfood_NRPy_1D_tests.py`](../edit/GiRaFFEfood_NRPy/GiRaFFEfood_NRPy_1D_tests.py) module.\n\n\n\n\n```python\n# Reset the list of gridfunctions, as registering a gridfunction\n# twice will spawn an error.\ngri.glb_gridfcs_list = []\n\nimport GiRaFFEfood_NRPy.GiRaFFEfood_NRPy_1D_tests as gfho\ngfho.GiRaFFEfood_NRPy_1D_tests()\n\nprint(\"Consistency check between GiRaFFEfood_NRPy tutorial and NRPy+ module: ALL SHOULD BE ZERO.\")\n\nfor i in range(DIM):\n\n print(\"ValenciavleftU[\"+str(i)+\"] - gfho.ValenciavleftU[\"+str(i)+\"] = \" + str(ValenciavleftU[i] - gfho.ValenciavleftU[i]))\n print(\"AleftD[\"+str(i)+\"] - gfho.AleftD[\"+str(i)+\"] = \" + str(AleftD[i] - gfho.AleftD[i]))\n print(\"ValenciavcenterU[\"+str(i)+\"] - gfho.ValenciavcenterU[\"+str(i)+\"] = \" + str(ValenciavcenterU[i] - gfho.ValenciavcenterU[i]))\n print(\"AcenterD[\"+str(i)+\"] - gfho.AcenterD[\"+str(i)+\"] = \" + str(AcenterD[i] - gfho.AcenterD[i]))\n print(\"ValenciavrightU[\"+str(i)+\"] - gfho.ValenciavrightU[\"+str(i)+\"] = \" + str(ValenciavrightU[i] - gfho.ValenciavrightU[i]))\n print(\"ArightD[\"+str(i)+\"] - gfho.ArightD[\"+str(i)+\"] = \" + str(ArightD[i] - gfho.ArightD[i]))\n```\n\n Consistency check between GiRaFFEfood_NRPy tutorial and NRPy+ module: ALL SHOULD BE ZERO.\n ValenciavleftU[0] - gfho.ValenciavleftU[0] = 0\n AleftD[0] - gfho.AleftD[0] = 0\n ValenciavcenterU[0] - gfho.ValenciavcenterU[0] = 0\n AcenterD[0] - gfho.AcenterD[0] = 0\n ValenciavrightU[0] - gfho.ValenciavrightU[0] = 0\n ArightD[0] - gfho.ArightD[0] = 0\n ValenciavleftU[1] - gfho.ValenciavleftU[1] = 0\n AleftD[1] - gfho.AleftD[1] = 0\n ValenciavcenterU[1] - gfho.ValenciavcenterU[1] = 0\n AcenterD[1] - gfho.AcenterD[1] = 0\n ValenciavrightU[1] - gfho.ValenciavrightU[1] = 0\n ArightD[1] - gfho.ArightD[1] = 0\n ValenciavleftU[2] - gfho.ValenciavleftU[2] = 0\n AleftD[2] - gfho.AleftD[2] = 0\n ValenciavcenterU[2] - gfho.ValenciavcenterU[2] = 0\n AcenterD[2] - gfho.AcenterD[2] = 0\n ValenciavrightU[2] - gfho.ValenciavrightU[2] = 0\n ArightD[2] - gfho.ArightD[2] = 0\n\n\n\n\n# Step 6: Output this notebook to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n$$\\label{latex_pdf_output}$$\n\nThe following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n[Tutorial-GiRaFFEfood_NRPy_1D_tests.pdf](Tutorial-GiRaFFEfood_NRPy_1D_tests.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means.)\n\n\n```python\n!jupyter nbconvert --to latex --template latex_nrpy_style.tplx --log-level='WARN' Tutorial-GiRaFFEfood_NRPy_1D_tests.ipynb\n!pdflatex -interaction=batchmode Tutorial-GiRaFFEfood_NRPy_1D_tests.tex\n!pdflatex -interaction=batchmode Tutorial-GiRaFFEfood_NRPy_1D_tests.tex\n!pdflatex -interaction=batchmode Tutorial-GiRaFFEfood_NRPy_1D_tests.tex\n!rm -f Tut*.out Tut*.aux Tut*.log\n```\n\n This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex)\n restricted \\write18 enabled.\n entering extended mode\n This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex)\n restricted \\write18 enabled.\n entering extended mode\n This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex)\n restricted \\write18 enabled.\n entering extended mode\n\n", "meta": {"hexsha": "0bfc2fdaee79302bb2c9dd58b867b7281b79d3ee", "size": 24284, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "in_progress/Tutorial-GiRaFFEfood_NRPy_1D_tests.ipynb", "max_stars_repo_name": "KAClough/nrpytutorial", "max_stars_repo_head_hexsha": "2cc3b22cb1092bc10890237dd8ee3b6881c36b52", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-12-23T05:31:25.000Z", "max_stars_repo_stars_event_max_datetime": "2019-12-23T05:31:25.000Z", "max_issues_repo_path": "in_progress/Tutorial-GiRaFFEfood_NRPy_1D_tests.ipynb", "max_issues_repo_name": "KAClough/nrpytutorial", "max_issues_repo_head_hexsha": "2cc3b22cb1092bc10890237dd8ee3b6881c36b52", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "in_progress/Tutorial-GiRaFFEfood_NRPy_1D_tests.ipynb", "max_forks_repo_name": "KAClough/nrpytutorial", "max_forks_repo_head_hexsha": "2cc3b22cb1092bc10890237dd8ee3b6881c36b52", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 39.1046698873, "max_line_length": 353, "alphanum_fraction": 0.5441031132, "converted": true, "num_tokens": 6559, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7490872243177517, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.44395897830543346}} {"text": "```python\nimport itertools\nimport pathlib\n\nimport matplotlib.pyplot as plt\nimport pandas as pd\nfrom myst_nb import glue\n\nfrom src.commons import NUM_EXPERIMENTS\nfrom src.decorators import repeat, get_from_cache_or_run\nfrom src.utils import build_plots_dir_path, build_cache_dir_path\nfrom src.visualization import PLOT_DPI\n\nplt.ioff() # turn off interactive plotting\nplt.style.use('../../../src/phd.mplstyle')\n\nroot_dir = pathlib.Path().cwd().parent.parent.parent\ncwd_dir = pathlib.Path().cwd()\n\nplot_dir = build_plots_dir_path(root_dir) / cwd_dir.name\ncache_dir = build_cache_dir_path(root_dir) / cwd_dir.name\n\nglue('num_experiments', NUM_EXPERIMENTS, display=False)\n```\n\n\n\n# Statistical verification of results\n\nIn order to assess the significance and performance of obtained results, two statistical approaches were used. The majority of metrics were compared using the Bayesian estimation (BEST) method {cite}`kruschke2013bayesian`, while the other straightforward metrics were just averaged.\n\n## Bayesian analysis\nThe Bayesian approach towards comparing data from multiple groups was used instead of traditional methods of _null hypothesis significance testing_ (NHST). They are more intuitive than the calculation and interpretation of _p-value_ scores, provides complete information about credible parameter values and allow more coherent inferences from data {cite}`dienes2011bayesian`.\n\nBenavoli depicted the perils of the frequentist NHST approach when comparing machine learning classifiers in {cite}`benavoli2017time`, which is particularly suited for this work. He points out the following reasons against using the NHST methods:\n- it does not estimate the probability of hypotheses,\n- point-wise null hypotheses are practically always false,\n- the _p-value_ does not separate between the effect size and the sample size,\n- it ignores magnitude and uncertainty,\n- it yields no information about the null hypothesis,\n- there is no principled way to decide the $\\alpha$ level.\n\nAdditionally, in 2016 the _[American Statistical Association](https://www.amstat.org/)_ made a statement against _p-values_ {cite}`wasserstein2016asa` which might be a motivation for other disciplines to pursue the Bayesian approach.\n\n````{margin}\n```{admonition} T-distribution\nLike the normal distribution, the T-distribution is bell-shaped and symmetric, but it has heavier tails, which means it tends to produce values that fall far from its mean.\n\nTail heaviness is determined by a parameter called _degrees of freedom_ $\\nu$ with smaller values giving heavier tails and higher values, making the T-distribution resembles a standard normal distribution with a mean of 0 and a standard deviation of 1.\n```\n````\n\nIn this work, we focus on establishing a descriptive mathematical model of the data $D$ using the Bayes' theorem deriving the posterior probability as a consequence of two antecedents: a prior probability and a \"likelihood function\" derived from a statistical model for the observed data {eq}`bayesian_approach`.\n\n```{math}\n:label: bayesian_approach\n\\underbrace{p(\\mu, \\sigma, \\nu|D)}_{\\text{posterior}} = \\underbrace{p(D|\\mu, \\sigma, \\nu)}_{\\text{likelihood}} \\times \\underbrace{p(\\mu, \\sigma, \\nu)}_{\\text{prior}} \\big/ \\underbrace{p(D)}_{\\text{evidence}}\n```\n\nEach experiment is performed {glue:}`num_experiments` times, generating independent samples, which according to the Central Limit Theorem, should be enough to consider it is approximating the normal distribution {cite}`kwak2017central` {cite}`islam2018sample`. To further provide a robust solution towards dealing with potential outliers, the _Student t-distribution_ is chosen. The prior distribution, described with three parameters - $\\mu$ (expected mean value), $\\sigma$ (standard deviation) and $\\nu$ (degrees of freedom) is presented using the Equation {eq}`bayesian_prior`. The standard deviation $\\sigma$ parameter uniformly covers a vast possible parameter space. The degrees of freedom follows a shifted exponential distribution controlling the normality of the data. When $\\nu>30$, the Student-t distribution is close to a normal distribution. However, if $\\nu$ is small, Student t-distributions have a heavy tail. Therefore, value of $\\nu \\sim \\mathrm{Exp}(\\frac{1}{29})$ allows the model to be more tolerant for potential outliers.\n\n```{math}\n:label: bayesian_prior\n\\begin{align}\n \\mu &\\sim \\mathrm{N}(\\mu_D, \\sigma^2_D) \\\\\n \\sigma &\\sim \\mathrm{U}(\\frac{1}{100}, 1000) \\\\\n \\nu &\\sim \\mathrm{Exp}(\\frac{1}{29})\n\\end{align}\n```\nThe posterior distribution is approximated arbitrarily high accuracy by generating a large representative sample using _Markov chain Monte Carlo_ (MCMC) methods. Its sample provides thousands of combinations of parameter values $<\\mu, \\sigma, \\nu>$. Each such combination of values is representative of credible parameter values that simultaneously accommodate the observed data and the prior distribution. From the MCMC sample, one can infer credible parameter values like the mean or standard deviation.\n\n### Example\nTo show an example of this technique, the performance of an ACS2 algorithm operating in a multistep, toy-problem - Simple Maze environment {cite}`gerard2000yacs` (see Figure {numref}`{number} `) will be discussed using three methods - the summary statistics, frequentist approach and the Bayesian estimation.\n\n:::{figure-md} simple-maze-env\n\n\nThe Simple Maze environment is a _Partially Observable Markov Decision Problem_ where the agent is placed in the starting location (denoted as \"S\"), at the beginning of each new trial, and the goal is to reach the final state \"F\" by executing four possible actions \u2013 moving north, east, south or west. The bolded lines represent the walls, and the goal can be reached optimally in seven successive steps.\n:::\n\n\n```python\nimport gym\nimport lcs.agents.acs2 as acs2\n\nSIMPLE_MAZE_TRIALS = 500\nLEARNING_RATE = 0.1\nDISCOUNT_FACTOR = 0.8\n\n\ndef simple_maze_env_provider():\n import gym_yacs_simple_maze # noqa: F401\n return gym.make('SimpleMaze-v0')\n\n\ndef custom_metrics(agent, env):\n population = agent.population\n return {\n 'pop': len(population),\n 'reliable': len([cl for cl in population if cl.is_reliable()])\n }\n\n\n@get_from_cache_or_run(cache_path=f'{cache_dir}/simple-maze/acs2.dill')\n@repeat(num_times=NUM_EXPERIMENTS, use_ray=True)\ndef run_acs2(env_provider):\n simple_maze_env = env_provider()\n\n cfg = acs2.Configuration(\n classifier_length=4,\n number_of_possible_actions=4,\n beta=LEARNING_RATE,\n gamma=DISCOUNT_FACTOR,\n do_ga=False,\n metrics_trial_frequency=1,\n user_metrics_collector_fcn=custom_metrics)\n\n agent = acs2.ACS2(cfg)\n return agent.explore(simple_maze_env, SIMPLE_MAZE_TRIALS)\n\n\n@get_from_cache_or_run(cache_path=f'{cache_dir}/simple-maze/acs2ga.dill')\n@repeat(num_times=NUM_EXPERIMENTS, use_ray=True)\ndef run_acs2ga(env_provider):\n simple_maze_env = env_provider()\n\n cfg = acs2.Configuration(\n classifier_length=4,\n number_of_possible_actions=4,\n beta=LEARNING_RATE,\n gamma=DISCOUNT_FACTOR,\n do_ga=True,\n metrics_trial_frequency=1,\n user_metrics_collector_fcn=custom_metrics)\n\n agent = acs2.ACS2(cfg)\n return agent.explore(simple_maze_env, SIMPLE_MAZE_TRIALS)\n\n\n# run computations\nacs2_runs_metrics = run_acs2(simple_maze_env_provider)\nacs2ga_runs_metrics = run_acs2ga(simple_maze_env_provider)\n\n# build dataframes\nacs2_metrics_df = pd.DataFrame(itertools.chain(*acs2_runs_metrics))\nacs2ga_metrics_df = pd.DataFrame(itertools.chain(*acs2ga_runs_metrics))\n\n# extract population size in last trial\nacs2_population_counts = acs2_metrics_df.query(f'trial == {SIMPLE_MAZE_TRIALS - 1}')['reliable']\nacs2ga_population_counts = acs2ga_metrics_df.query(f'trial == {SIMPLE_MAZE_TRIALS - 1}')['reliable']\n\n# glue variables\nglue('stats_simple_maze_trials', SIMPLE_MAZE_TRIALS, display=False)\n```\n\n\n\nThe ACS2 agent will be tested in two variants - with and without the genetic algorithm modification. All other settings are identical. Both agents in each experiment will execute {glue:}`stats_simple_maze_trials` trials, each time randomly selecting an action. Finally, each such experiment will be independently repeated {glue:}`num_experiments` times.\n\n````{admonition} Null hypothesis\nThe classifier population sizes obtained by ACS2 and ACS2 GA in the last trial $x$ are equal.\n````\n\n**Descriptive statistics**\n\nThe first option is to understand the data and extract basic summary statistics. Figure {numref}`{number} ` presents a whisker plot showing basic data aggregations like the minimum, first quartile, median, third quartile, maximum values alongside the histogram visualization. While the values look similar, it is still unclear whether they can be considered the same.\n\n\n\n```python\ndef plot_summary_statistics(data, labels, plot_filename=None):\n fig = plt.figure(figsize=(14, 8))\n colors = ['lightblue', 'lightgreen']\n\n # boxplot\n ax1 = fig.add_subplot(121)\n\n bp = ax1.boxplot(data,\n patch_artist=True,\n labels=labels)\n\n for patch, color in zip(bp['boxes'], colors):\n patch.set_facecolor(color)\n\n ax1.set_title('Whisker plot', pad=20)\n ax1.set_ylabel('Observed population size values')\n ax1.grid(which='major', axis='y', linestyle='--')\n\n # histogram\n ax2 = fig.add_subplot(122)\n for d, l, c in zip(data, labels, colors):\n ax2.hist(d, bins=10, label=l, color=c, edgecolor='black', alpha=.8, linewidth=1)\n\n ax2.set_title('Histogram', pad=20)\n ax2.grid(which='major', axis='y', linestyle='--')\n ax2.legend()\n\n # common settings\n fig.suptitle('ACS2 and ACS2 GA Comparison', fontweight='bold')\n fig.subplots_adjust(top=0.85)\n\n if plot_filename:\n fig.savefig(plot_filename, dpi=PLOT_DPI)\n\n return fig\n\n\nsummary_plot_fig = plot_summary_statistics(\n [acs2_population_counts.to_numpy(), acs2ga_population_counts.to_numpy()],\n ['ACS2', 'ACS2 GA'],\n plot_filename=f'{plot_dir}/summary-stats.png')\n```\n\n:::{figure-md} summary-stats\n:class: full-width\n\n\nDescriptive statistics depicting population size obtained after executing two versions of the ACS2 agent in the Simple Maze environment.\n:::\n\n**Frequentist approach**\n\nFor the frequentist approach first two hypotheses about data distribution are formed:\n\n```{math}\n\\begin{align*}\n&H_0: \\text{The classifier population sizes obtained by ACS2 and ACS2 GA are equal}\\\\\n&H_1: \\text{The classifier population sizes obtained by ACS2 and ACS2 GA are different}\n\\end{align*}\n```\n\nIn traditional NHST workflow, the first step would be to apply normality tests to verify whether the data is normally distributed. However, looking at the histograms from the Figure {numref}`{number} ` we might assume that the data follow the Gaussian distribution and skip it.\n\nIf the $H_0$ hypothesis is rejected, there is no significant difference between the means. To do so, a _p-value_ will be calculated and compared with a certain threshold $\\alpha \\leq 0.05$. If the _p-value_ falls below the threshold, it means that the null hypothesis $H_0$ can be rejected and **there is 95% confidence that both means are significantly different**.\n\n\n```python\nfrom scipy.stats import ttest_ind\n\n# don't assume equal variances\nvalue, pvalue = ttest_ind(acs2_population_counts, acs2ga_population_counts, equal_var=False)\n\nglue('stats-pvalue', round(pvalue, 2), display=False)\n```\n\n\n\nIn this case, after running the two-sided t-Test, the calculated _p-value_ is {glue:}`stats-pvalue`, which indicates strong evidence for the $H_0$ hypothesis, meaning that it is retained. Remember that NHST does not accept the null hypothesis; it can be only rejected or failed to be rejected. Stating that it is correct implies 100% certainty, which is not valid in this methodology.\n\n**Bayesian estimation**\n\nThroughout this work a [PyMC3](https://docs.pymc.io/en/v3/) open-sourced Python framework is used for the probabilistic programming. Figure {numref}`{number} ` depicts two separate Student-t posterior distribution hyper-parameters estimated by using the MCMC method.\n\n\n```python\nfrom src.bayes_estimation import bayes_estimate\n\n\n@get_from_cache_or_run(cache_path=f'{cache_dir}/simple-maze/bayes-acs2.dill')\ndef bayes_estimate_acs2(data):\n return bayes_estimate(data)\n\n\n@get_from_cache_or_run(cache_path=f'{cache_dir}/simple-maze/bayes-acs2ga.dill')\ndef bayes_estimate_acs2ga(data):\n return bayes_estimate(data)\n\n\n# run estimations\nbayes_acs2_trace = bayes_estimate_acs2(acs2_population_counts.to_numpy())\nbayes_acs2ga_trace = bayes_estimate_acs2ga(acs2ga_population_counts.to_numpy())\n\n\ndef plot_bayes_parameters(data, labels, plot_filename=None):\n colors = ['lightblue', 'lightgreen']\n hist_kwargs = {'bins': 20, 'edgecolor': 'black', 'alpha': .8, 'linewidth': 1}\n\n fig, axs = plt.subplots(2, 3, figsize=(16, 8))\n\n for i, (d, l, c) in enumerate(zip(data, labels, colors)):\n axs[i][0].hist(d['mu'], color=c, label=l, **hist_kwargs)\n axs[i][0].set_title(f'$\\mu_{{{l}}}$')\n\n axs[i][1].hist(d['std'], color=c, label=l, **hist_kwargs)\n axs[i][1].set_title(f'$\\sigma_{{{l}}}$')\n\n axs[i][2].hist(d['nu'], color=c, label=l, **hist_kwargs)\n axs[i][2].set_title(fr'$\\nu_{{{l}}}$')\n\n # share axes\n axs[0][0].sharex(axs[1][0])\n axs[0][1].sharex(axs[1][1])\n axs[0][2].sharex(axs[1][2])\n\n # common\n handles, labels = [(a + b) for a, b in zip(axs[0][0].get_legend_handles_labels(), axs[1][0].get_legend_handles_labels())]\n fig.suptitle('Posterior distribution of ACS2 and ACS2 GA Student-t parameters', fontweight='bold')\n fig.legend(handles, labels, loc='lower center', ncol=2, bbox_to_anchor=(0.5, -0.1))\n fig.tight_layout()\n\n if plot_filename:\n fig.savefig(plot_filename, dpi=PLOT_DPI)\n\n return fig\n\n\nbayes_parameters_fig = plot_bayes_parameters(\n [bayes_acs2_trace, bayes_acs2ga_trace],\n ['ACS2', 'ACS2 GA'],\n plot_filename=f'{plot_dir}/stats-bayes-parameters.png')\n\nglue('stats_bayes_params_fig', bayes_parameters_fig, display=False)\n```\n\n\n\n```{glue:figure} stats_bayes_params_fig\n:name: \"stats_bayes_params_fig\"\nParameter distributions of Student-t estimation of population size data for two agents.\n```\n\n````{margin}\n```{admonition} Credible values\nThe credible values are defined by the _highest density interval_ **HDI**, which is a useful summary of where the bulk of the most credible values falls.\n\nBy definition, every value inside the HDI has a higher probability density than any value outside the HDI, and the total mass of points inside the 95% HDI is 95% of the distribution.\n```\n````\n\nThe posterior distribution allows assessing the null value more credibly. Contrary to NHST, it can also accept $H_0$. To do so, a researcher defines a _region of practical equivalence_ (ROPE) {cite}`kruschke2011bayesian`, enclosing the values of the parameter that are deemed to be negligibly different from the null values for practical purposes. When nearly all the credible values fall within the ROPE, the null value is said to be accepted for practical purposes.\n\nFigure {numref}`{number} ` shows the graphical example of verifying the null hypothesis using the difference between two posterior distributions of population means $\\mu_{ACS2}-\\mu_{ACS2GA}$. The ROPE region (red sector) was assumed to be $[-1, 1]$, which means that the difference of one classifier is acceptable to consider two populations equal. Blue lines represent the area of credible values of the previously calculated difference. Since the credible value region does not fall into the ROPE region, the $H_0$ is rejected (while the NHST approach retained it).\n\n\n```python\nimport arviz\nimport numpy as np\nfrom scipy.stats import gaussian_kde\n\ndef plot_kde(data_diff, rope=None, hdi_prob=0.95, plot_filename=None):\n if rope is None:\n rope = [-1, 1]\n\n fig = plt.figure(figsize=(16, 8))\n ax = fig.add_subplot(111)\n\n x = np.linspace(min(data_diff), max(data_diff), 1000)\n gkde = gaussian_kde(data_diff)\n ax.plot(x, gkde(x), color='orange', label='$\\mu_{ACS2}-\\mu_{ACS2GA}$')\n ax.fill_between(x, gkde(x), alpha=0.1, color='orange')\n\n # no difference vertical line\n ax.axvline(0, color='black', alpha=0.5, linestyle='--')\n\n # high density interval\n hdi = arviz.hdi(data_diff, hdi_prob=0.95)\n ax.axvline(x=hdi[0], linestyle='--', color='blue', label=f'{int(hdi_prob * 100)}% HDI')\n ax.axvline(x=hdi[1], linestyle='--', color='blue')\n\n # ROPE\n ax.axvline(x=rope[0], linestyle='--', color='red', label='ROPE')\n ax.axvline(x=rope[1], linestyle='--', color='red')\n ax.fill_between(np.arange(rope[0], rope[1], 0.01), 0.5, color='red', alpha=0.05)\n\n ax.set_ylim(0, 0.5)\n fig.suptitle(r'Assessing $H_0$ using Bayes estimation', fontweight='bold')\n\n plt.legend()\n\n if plot_filename:\n fig.savefig(plot_filename, dpi=PLOT_DPI)\n\n return fig\n\n\ndata_diff = bayes_acs2_trace['mu'] - bayes_acs2ga_trace['mu']\n\nbayes_delta_fig = plot_kde(data_diff, plot_filename=f'{plot_dir}/stats-bayes-delta.png')\n\nglue('stats_bayes_delta_fig', bayes_delta_fig, display=False)\n```\n\n```{glue:figure} stats_bayes_delta_fig\n:name: \"stats_bayes_delta_fig\"\n\nAssessing $H_0$ using the probabilistic density function of $\\mu$ parameter difference of two examined populations. The ROPE interval was chosen to $[-1, 1]$ meaning that a difference of one classifier can be neglected in terms of population equality. Even though the mean difference is less than one, the $H_0$ is rejected because the HDI region is not contained within the ROPE.\n```\n", "meta": {"hexsha": "a32cac37ea0cb2c8b2b7f4d16f5b7af3d657a71c", "size": 125566, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "book/chapters/2_selected_topics/25_stats.ipynb", "max_stars_repo_name": "khozzy/phd", "max_stars_repo_head_hexsha": "9a05572a6960d948320669c51e0c80bb9d037d4a", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "book/chapters/2_selected_topics/25_stats.ipynb", "max_issues_repo_name": "khozzy/phd", "max_issues_repo_head_hexsha": "9a05572a6960d948320669c51e0c80bb9d037d4a", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "book/chapters/2_selected_topics/25_stats.ipynb", "max_forks_repo_name": "khozzy/phd", "max_forks_repo_head_hexsha": "9a05572a6960d948320669c51e0c80bb9d037d4a", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 207.204620462, "max_line_length": 53592, "alphanum_fraction": 0.8979899017, "converted": true, "num_tokens": 4480, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.7490872131147276, "lm_q1q2_score": 0.44395897166577525}} {"text": "# Problem set 1: Financial Frictions, Liquidity and the Business Cycle.\n\nThis notebook sums up relevant information on the setup for the exercises. If further suggests what to think about, before going to exercise classes. The solution follows from \"PS1.ipynb\".\n\n# Exercise 3.5 in JT (The Theory of Corporate Finance)\n\nConsider the continuous the continuous investment model with decreasing returns to scale outlined in chapter 3 of JT. The core of the model is as follows:\n- Let $I\\in[0, \\infty)$ be the level of investment in a given project.\n- Entrepreneur proposes a project. If successful the project generates income $R(I)$; if not the project generates $0$.\n- The probability of success depends on the behavior of the entrepreneur. If E behaves $(b)$ then probability of success is $p_H\\in(0,1)$. If E does not behave $(nb)$ then the probability of success is $p_L$ where $0\\leq p_L0, R''(I)<0$. \n 2. *Regularity condition 1:* Under perfect information a positive investment level would be optimal, i.e. $R'(0)>1/p_H$. \n 3. *Regularity condition 2:* Under perfect information a finite level of investment is optiaml, i.e. $ lim_{I\\rightarrow \\infty} R'(I) < 1/p_H$. \n- Assume perfect competition between lenders.\n- We will consider loan agreements where the entrepreneur *pledges income* $R(I)-R_b(I)$, to the lender. This leaves the entrepreneur with $R_b(I)$ if the project is successful.\n\n\n### Suggestions before exercise class:\n\n1. Write up the utility for an entrepreneur, in the case where he behaves $(u_b)$ and in the case where he does not $(u_{nb})$.\n2. Write up the *incentive compatibility constraint* (IC) stating what level of $R_b(I)$ is needed, for the entrepreneur to choose to behave.\n3. Write up the *individual rationality constraint* (IR) for the lender, ensuring that he will agree to the loan contract.\n4. What does the *perfect competition* amongst lenders imply, for the contract the entrepreneur will offer? \n5. Given that lenders are profit maximizing, what does this imply for the level $R_b(I)$ in the loan contract? (Hint: Think about the (IC) constraint).\n\n# Exercise 6.1 in JT: Privately known private benefit and market breakdown\n\nLet us start with a brief outline of the setup (from section 6.2), compared to exercise 3.5 (a lot of is the same, will not be repeated here):\n\n* Two types of entrepreneurs: Good and bad types with private benefits of not behaving $B_H>B_L$. (good type has $B_L$)\n* No equity (A=0),\n* Investment is not continuous, but either 0 or I.\n* Investment is either successfull (return R) or not (return 0).\n* Capital markets put probability $\\alpha\\in(0,1)$ on type 'good' and $1-\\alpha$ on type 'bad'. \n* Regularity conditions:\n\n$$ \\begin{align}\n p_H\\left(R-\\dfrac{B_H}{\\Delta p}\\right)A$) yields $R$ in return w. probability $p$. Otherwise $0$ return.\n* Agents differ in the probability of success $p$. Is distributed on $[0,1]$ according to the pdf function $f(p_i)$. (As usual $F(p)$ is the share of agents with $p_i\\leq p$. $f(p)$ is the marginal change in that probability around $p_i=p$.)\n* Total supply of loanable funds is simply presented by the increasing function $S(r)$. \n* This $S(r)$ is supplied by a competetive market (e.g. banks).\n* Assume perfect information: This means that loan contracts **can depend on the probability of sucess p**.\n* Compared to before: No assumption of private benefits from not behaving.\n\n### Suggestions before exercise class:\n\nThe same basic things: \n1. What is the (IR) constraint of an entrepreneur that has the option of (1) entering into agreement recceiving $R-R_b(p)$ if the project suceeds (zero otherwise) or (2) depositing the wealth $A$ and getting a *normal return* of r?\n2. What is the zero profits condition for a bank that receives the $pR_b(p)$ if the project is successful (0 otherwise), by investing $I-A$?\n", "meta": {"hexsha": "a9223b201119da277918e7f6c72185238e38841a", "size": 94457, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "PS1/PS1_setup.ipynb", "max_stars_repo_name": "ChampionApe/FinancialFrictions2019", "max_stars_repo_head_hexsha": "7d5b91a88c2077d6df6d03ccb3b5bc072f150a5a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "PS1/PS1_setup.ipynb", "max_issues_repo_name": "ChampionApe/FinancialFrictions2019", "max_issues_repo_head_hexsha": "7d5b91a88c2077d6df6d03ccb3b5bc072f150a5a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "PS1/PS1_setup.ipynb", "max_forks_repo_name": "ChampionApe/FinancialFrictions2019", "max_forks_repo_head_hexsha": "7d5b91a88c2077d6df6d03ccb3b5bc072f150a5a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 300.8184713376, "max_line_length": 47052, "alphanum_fraction": 0.9198894735, "converted": true, "num_tokens": 2435, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.7490872075132152, "lm_q1q2_score": 0.443958968345946}} {"text": "# Muscle modeling\n\nMarcos Duarte\n\nThere are two major classes of muscle models that have been used in biomechanics and motor control: the Hill-type and Huxley-type models. They differ mainly on how the contractile element is modeled. In Hill-type models, the modeling of the contractile element is phenomenological; arbitrary mathematical functions are used to reproduce experimental observations relating muscle characteristics (such as excitation/activation, muscle length and velocity) with the muscle force. In Huxley-type models, the modeling of the contractile element is mechanistic; the mathematical functions used represent the hypothesized mechanisms for the cross-bridge dynamics (Tsianos and Loeb, 2013). Huxley-type models tend to produce more realistic results than Hill-type models for certain conditions but they have a higher computational demand. For this reason, Hill-type models are more often employed in musculoskeletal modeling and simulation. \n\nHill-type muscle models are presented in several texts (e.g., Erdermir et al. 2007; He et al., 1991; McMahon, 1984; Nigg and Herzog, 2007; Robertson et al., 2013, Thelen, 2003; Tsianos and Loeb, 2013, Winters, 1990; Zajac, 1989; Zatsiorsky and Prilutsky, 2012) and implemented in many software for modeling and simulation of the musculoskeletal dynamics of human movement (e.g., the free and open source software [OpenSim](https://simtk.org/home/opensim)). \n\nNext, let's see a brief overview of a Hill-type muscle model and a basic implementation in Python. \n\n## Hill-type muscle model\n\nHill-type models are developed to reproduce the dependence of force with the length and velocity of the muscle-tendon unit and parameters are lumped and made dimensionless in order to represent different muscles with few changes in these parameters. A Hill-type model is complemented with the modeling of the activation dynamics (i.e., the temporal pattern of muscle activation and deactivation as a function of the neural excitation) to produce more realistic results. As a result, the force generated will be a function of three factors: the length and velocity of the muscle-tendon unit and its activation level $a$. \n\nA Hill-type muscle model has three components (see figure below): two for the muscle, an active contractile element (CE) and a passive elastic element (PE) in parallel with the CE, and one component for the tendon, an elastic element (SE) in series with the muscle. In some variations, a damping component is added parallel to the CE as a fourth element. A [pennation angle](http://en.wikipedia.org/wiki/Muscle_architecture) (angle of the pennate fibers with respect to the force-generating axis) is also included in the model. In a simpler approach, the muscle and tendon are assumed massless.\n\n
Figure. A Hill-type muscle model with three components: two for the muscle, an active contractile element, $\\mathsf{CE}$, and a passive elastic element in parallel, $\\mathsf{PE}$, with the $\\mathsf{CE}$, and one component for the tendon, an elastic element in series, $\\mathsf{SE}$, with the muscle. $\\mathsf{L_{MT}}$: muscle\u2013tendon length, $\\mathsf{L_T}$: tendon length, $\\mathsf{L_M}$: muscle fiber length, $\\mathsf{F_T}$: tendon force, $\\mathsf{F_M}$: muscle force, and $\u03b1$: pennation angle.
\n\nLet's now revise the models of a Hill-type muscle with three components and activation dynamics by two references: \n 1. [Thelen (2003)](http://simtk-confluence.stanford.edu:8080/display/OpenSim/Thelen+2003+Muscle+Model) with some of the adjustments described in Millard et al. (2013). Hereafter, Thelen2003Muscle or T03.\n 2. [McLean, Su, van den Bogert (2003)](http://www.ncbi.nlm.nih.gov/pubmed/14986412). Hereafter, McLean2003Muscle or M03.\n \nFirst, let's import the necessary Python libraries and customize the environment:\n\n\n```python\nimport numpy as np\nfrom scipy.integrate import ode, odeint\nimport matplotlib.pyplot as plt\n%matplotlib inline\n#%matplotlib nbagg \nimport matplotlib\nmatplotlib.rcParams['lines.linewidth'] = 3\nmatplotlib.rcParams['font.size'] = 13\nmatplotlib.rcParams['lines.markersize'] = 5\nmatplotlib.rc('axes', grid=True, labelsize=14, titlesize=16, ymargin=0.05)\nmatplotlib.rc('legend', numpoints=1, fontsize=11)\n```\n\n### Force-length relationship\n\nIn a Hill-type model, the force a muscle can generate depends on its length due to two factors: \n\n1. The active force of the contractile element (CE), which in turn depends on the spatial superposition of the actin and myosin molecules to form cross-bridges at the sarcomere. A maximum number of cross-bridges will be formed at an optimal fiber length, generating a maximum force. When a fiber is too stretched or too shortened, fewer cross-bridges will be formed, decreasing the force generated. \n2. The passive and parallel elastic element (PE), which behaves as a nonlinear spring where no force is generated below a certain length (the slack length) and force increases with the muscle elongation.\n\n#### Force-length relationship of the contractile element\n\nThelen2003Muscle represented the normalized force-length relationship of the contractile element by a Gaussian function:\n\n$$ \\bar{f}_{l,CE} = exp\\left[-(\\bar{L}_M-1)^2/\\gamma\\right] $$\n\nwhere $\\gamma$ is a shape factor and $\\bar{L}_M$ is the muscle fiber length normalized by the optimal muscle fiber length at which maximal force can be produced, $L_{Mopt}$:\n\n$$\\bar{L}_M=\\frac{L_M}{L_{Mopt}}$$\n\nThelen2003Muscle adopted $\\gamma=0.45$. The actual force produced is obtained multiplying $\\bar{f}_{l,CE}$ by the maximum isometric muscle force, $F_{M0}$. Thelen2003Muscle assumed that the maximum isometric muscle forces for old adults were 30% lower than those used for young adults.\n\nMcLean2003Muscle represented the force-length relationship of the contractile element (not normalized) as a function of muscle length (not normalized) by a quadratic function:\n\n$$ \nf_{l,CE} = max \\left\\{ \n \\begin{array}{l l}\n F_{Mmin} \\\\\n F_{M0}\\left[1 - \\left(\\frac{L_M-L_{Mopt}}{WL_{Mopt}}\\right)^2\\right]\n\\end{array} \\right.\n$$\n\nwhere $W$ is a dimensionless parameter describing the width of the force-length relationship. A minimum force level $F_{Mmin}$ is employed for numerical stability. \nMcLean2003Muscle adopted $W=1$ and $F_{Mmin}=10 N$. \n\nThe corresponding Python functions are:\n\n\n```python\ndef flce_T03(lm=1, gammal=0.45):\n \"\"\"Thelen (2003) force of the contractile element as function of muscle length.\n \n Parameters\n ----------\n lm : float, optional (default=1)\n normalized muscle fiber length\n gammal : float, optional (default=0.45)\n shape factor\n\n Returns\n -------\n fl : float\n normalized force of the muscle contractile element\n \"\"\"\n \n fl = np.exp(-(lm-1)**2/gammal)\n \n return fl\n```\n\n\n```python\ndef flce_M03(lm=1, lmopt=1, fm0=1, fmmin=0.001, wl=1):\n \"\"\"McLean (2003) force of the contractile element as function of muscle length.\n \n Parameters\n ----------\n lm : float, optional (default=1)\n muscle (fiber) length\n lmopt : float, optional (default=1)\n optimal muscle fiber length\n fm0 : float, optional (default=1)\n maximum isometric muscle force\n fmmin : float, optional (default=0.001)\n minimum muscle force\n wl : float, optional (default=1)\n shape factor of the contractile element force-length curve\n\n Returns\n -------\n fl : float\n force of the muscle contractile element\n \"\"\"\n \n fl = np.max([fmmin, fm0*(1 - ((lm - lmopt)/(wl*lmopt))**2)])\n \n return fl\n```\n\nAnd plots of these functions:\n\n\n```python\nlm = np.arange(0, 2.02, .02)\nfce_T03 = np.zeros(lm.size)\nfce_M03 = np.zeros(lm.size)\nfor i in range(len(lm)):\n fce_T03[i] = flce_T03(lm[i])\n fce_M03[i] = flce_M03(lm[i])\n```\n\n\n```python\nplt.figure(figsize=(7, 4))\nplt.plot(lm, fce_T03, 'b', label='T03')\nplt.plot(lm, fce_M03, 'g', label='M03')\nplt.xlabel('Normalized length')\nplt.ylabel('Normalized force')\nplt.legend(loc='best')\nplt.suptitle('Force-length relationship of the contractile element', y=1, fontsize=16)\nplt.show()\n```\n\nSimilar results when the same parameters are used.\n\n#### Force-length relationship of the parallel element\n\nThelen2003Muscle represents the normalized force of the parallel (passive) element of the muscle as a function of muscle length (normalized by the optimal muscle fiber length) by an exponential function:\n\n$$ \\bar{F}_{PE}(\\bar{L}_M) = \\frac{exp\\left[k_{PE}(\\bar{L}_M-1)/\\epsilon_{M0}\\right]-1}{exp(k_{PE})-1} $$\n\nwhere $k_{PE}$ is an exponential shape factor and $\\epsilon_{M0}$ is the passive muscle strain due to maximum isometric force:\n\n$$\\epsilon_{M0}=\\frac{L_M(F_{M0})-L_{Mslack}}{L_{Mslack}}$$\n\nwhere $L_{Mslack}$ is the muscle slack length. Thelen2003Muscle adopted $L_{Mslack} = L_{Mopt}$. \nThelen2003Muscle adopted $k_{PE}=5$ and $\\epsilon_{M0}=0.6$ for young adults ($\\epsilon_{M0}=0.5$ for old adults). The actual force produced is obtained multiplying $\\bar{F}_{PE}$ by the maximum isometric muscle force, $F_{M0}$.\n\nMcLean2003Muscle represents the force of the parallel (passive) element of the muscle (not normalized) as a function of muscle length (not normalized) by a quadratic function:\n\n$$ \nF_{PE}(L_M) = \\left\\{ \n \\begin{array}{l l}\n 0 \\quad & \\text{if} \\quad L_M \\leq L_{Mslack} \\\\\n k_{PE}(L_M - L_{Mslack})^2 \\quad & \\text{if} \\quad L_M > L_{Mslack}\n\\end{array} \\right.\n$$\n\nwhere $k_{PE}$ is a stiffness parameter of the parallel element such that the passive muscle force is equal to the normalized maximum isometric force of the muscle when the CE is stretched to its maximal length for active force production:\n\n$$ k_{PE} = \\frac{F_{M0}}{(WL_{Mopt})^2} $$\n\nMcLean2003Muscle adopted $L_{Mslack} = L_{Mopt}$.\n\nThe corresponding Python functions are:\n\n\n```python\ndef fpelm_T03(lm=1, kpe=5, epsm0=0.6):\n \"\"\"Thelen (2003) force of the muscle parallel element as function of muscle length.\n \n Parameters\n ----------\n lm : float, optional (default=1)\n normalized muscle fiber length\n kpe : float, optional (default=5)\n exponential shape factor\n epsm0 : float, optional (default=0.6)\n passive muscle strain due to maximum isometric force\n\n Returns\n -------\n fpe : float\n normalized force of the muscle parallel (passive) element\n \"\"\"\n \n if lm < 1:\n fpe = 0\n else:\n fpe = (np.exp(kpe*(lm-1)/epsm0)-1)/(np.exp(kpe)-1)\n \n return fpe\n```\n\n\n```python\ndef fpelm_M03(lm=1, lmopt=1, fm0=1, lmslack=1, wp=1):\n \"\"\"McLean (2003) force of the muscle parallel element as function of muscle length.\n \n Parameters\n ----------\n lm : float, optional (default=1)\n muscle fiber length\n lmopt : float, optional (default=1)\n optimal muscle (fiber) length\n fm0 : float, optional (default=1)\n maximum isometric muscle force\n lmslack : float, optional (default=1)\n muscle slack length\n wp : float, optional (default=1)\n shape factor of the parallel element force-length curve\n\n Returns\n -------\n fpe : float\n force of the muscle parallel (passive) element\n \"\"\"\n \n kpe = fm0/(wp*lmopt)**2\n if lm <= lmslack:\n fpe = 0\n else:\n fpe = kpe*(lm-lmslack)**2\n \n return fpe\n```\n\nAnd plots of these functions:\n\n\n```python\nlm = np.arange(0, 2.02, .02)\nfpe_T03 = np.zeros(lm.size)\nfpe_M03 = np.zeros(lm.size)\nfor i in range(len(lm)):\n fpe_T03[i] = fpelm_T03(lm[i])\n fpe_M03[i] = fpelm_M03(lm[i])\n```\n\n\n```python\nfig, (ax1, ax2) = plt.subplots(nrows=1, ncols=2, sharex=True, sharey=True, figsize=(10, 4))\nax1.plot(lm[:86], fce_T03[:86], 'b', label='Active')\nax1.plot(lm[:86], fpe_T03[:86], 'r', label='Passive')\nax1.plot(lm[:86], fce_T03[:86] + fpe_T03[:86], 'g', label='Total')\nax1.text(0.1, 2.6, 'T03')\nax1.set_xlim([0, 1.7])\nax1.set_xlabel('Normalized length')\nax1.set_ylabel('Normalized force')\n#ax1.legend(loc='best')\nax2.plot(lm[:86], fce_M03[:86], 'b', label='Active')\nax2.plot(lm[:86], fpe_M03[:86], 'r', label='Passive')\nax2.plot(lm[:86], fce_M03[:86] + fpe_M03[:86], 'g', label='Total')\nax2.text(0.1, 2.6, 'M03')\nax2.set_xlim([0, 1.7])\nax2.set_xlabel('Normalized length')\nax2.legend(loc='best')\nplt.suptitle('Muscle force-length relationship', y=1, fontsize=16)\nplt.tight_layout()\nplt.show()\n```\n\nThe results are different at the maximum stretching because Thelen2003Muscle and McLean2003Muscle model differently the passive component. \nThese results were simulated for a maximum muscle activation (an activation level, $a$, of 1, where 0 is no activation). The effect of different activation levels on the total muscle force (but only the active force is affected) is shown in the next figure:\n\n\n```python\nlm = np.arange(0, 2.02, .02)\nfce_T03_als = np.zeros((lm.size, 5))\nals = [0, 0.25, 0.50, 0.75, 1.0]\nfor j, al in enumerate(als):\n for i in range(len(lm)):\n fce_T03_als[i, j] = flce_T03(lm[i])*al\n```\n\n\n```python\nfig, ax = plt.subplots(nrows=1, ncols=1, sharex=True, sharey=True, figsize=(6, 5))\nfor j, al in enumerate(als):\n ax.plot(lm[:86], fce_T03_als[:86, j] + fpe_T03[:86], label='%.2f'%al)\nax.text(0.1, 2.6, 'T03')\nax.set_xlim([0, 1.7])\nax.set_xlabel('Normalized length')\nax.set_ylabel('Normalized force')\nax.legend(loc='best', title='Activation level')\nax.set_title('Muscle force-length relationship', y=1, fontsize=16)\nplt.tight_layout()\nplt.show()\n```\n\n#### Force-length relationship of the series element (tendon)\n\nThelen2003Muscle represented the tendon force of the series element as a function of the normalized tendon length (in fact, tendon strain) by an exponential function during an initial nonlinear toe region and by a linear function thereafter:\n\n$$ \n\\bar{F}_{SE}(\\bar{L}_T) = \\left\\{ \n \\begin{array}{l l}\n \\frac{\\bar{F}_{Ttoe}}{exp(k_{Ttoe})-1}\\left[exp(k_{Ttoe}\\epsilon_T/\\epsilon_{Ttoe})-1\\right] \\quad & \\text{if} \\quad \\epsilon_T \\leq \\epsilon_{Ttoe} \\\\\n k_{Tlin}(\\epsilon_T - \\epsilon_{Ttoe}) + \\bar{F}_{Ttoe} \\quad & \\text{if} \\quad \\epsilon_T > \\epsilon_{Ttoe}\n\\end{array} \\right.\n$$\n\nwhere $\\epsilon_{T}$ is the tendon strain:\n\n$$\\epsilon_{T} = \\frac{L_T-L_{Tslack}}{L_{Tslack}}$$\n\n$L_{Tslack}$ is the tendon slack length, $\\epsilon_{Ttoe}$ is the tendon strain above which the tendon exhibits linear behavior, $k_{Ttoe}$ is an exponential shape factor, and $k_{Tlin}$ is a linear scale factor. The parameters are chosen such that the tendon elongation at the normalized maximal isometric force of the muscle is 4% of the tendon length ($\\epsilon_{T0}=0.04$). \nThelen2003Muscle adopted $k_{Ttoe}=3$ and the transition from nonlinear to linear behavior occurs for normalized tendon forces greater than $\\bar{F}_{Ttoe}=0.33$. For continuity of slopes at the transition, $\\epsilon_{Ttoe}=0.609\\epsilon_{T0}$ and $k_{Tlin}=1.712/\\epsilon_{T0}$. The actual force produced is obtained multiplying $\\bar{F}_{SE}$ by the maximum isometric muscle force, $F_{M0}$.\n\nMcLean2003Muscle represented the tendon force (not normalized) of the series element as a function of the tendon length (not normalized) by the same quadratic function used for the force of the muscle passive element:\n\n$$ \nF_{SE}(L_T) = \\left\\{ \n \\begin{array}{l l}\n 0 \\quad & \\text{if} \\quad L_T \\leq L_{Tslack} \\\\\n k_T(L_T - L_{Tslack})^2 \\quad & \\text{if} \\quad L_T > L_{Tslack}\n\\end{array} \\right.\n$$\n\nwhere $k_T$ is the tendon stiffness. The stiffness parameter $k_T$ is chosen such that the tendon elongation is 4% at the maximum isometric force, $k_T=(1/\\epsilon_{T0})^2=625$ for $F_{M0}=1$.\n\nThe corresponding Python functions are:\n\n\n```python\ndef fselt_T03(lt=1, ltslack=1, epst0=0.04, kttoe=3):\n \"\"\"Thelen (2003) force-length relationship of tendon as function of tendon length.\n \n Parameters\n ----------\n lt : float, optional (default=1)\n normalized tendon length\n ltslack : float, optional (default=1)\n normalized tendon slack length\n epst0 : float, optional (default=0.04)\n tendon strain at the maximal isometric muscle force\n kttoe : float, optional (default=3)\n linear scale factor\n\n Returns\n -------\n fse : float\n normalized force of the tendon series element\n \"\"\"\n\n epst = (lt-ltslack)/ltslack\n fttoe = 0.33\n # values from OpenSim Thelen2003Muscle\n epsttoe = .99*epst0*np.e**3/(1.66*np.e**3 - .67)\n ktlin = .67/(epst0 - epsttoe)\n #\n if epst <= 0:\n fse = 0\n elif epst <= epsttoe:\n fse = fttoe/(np.exp(kttoe)-1)*(np.exp(kttoe*epst/epsttoe)-1)\n else:\n fse = ktlin*(epst-epsttoe) + fttoe\n \n return fse\n```\n\n\n```python\ndef fselt_M03(lt, ltslack=1, fm0=1, epst0=0.04):\n \"\"\"McLean (2003) force-length relationship of tendon as function of tendon length.\n \n Parameters\n ----------\n lt : float, optional (default=1)\n tendon length\n ltslack : float, optional (default=1)\n tendon slack length\n fm0 : float, optional (default=1)\n maximum isometric muscle force\n epst0 : float, optional (default=0.04)\n tendon strain at the maximal isometric muscle force\n\n Returns\n -------\n fse : float\n force of the tendon series element\n \"\"\"\n\n kt = fm0/epst0**2\n if lt <= ltslack:\n fse = 0\n else:\n fse = kt*(lt-ltslack)**2\n \n return fse\n```\n\nAnd plots of these functions:\n\n\n```python\nlt = np.arange(1, 1.051, .001)\nfse_T03 = np.zeros(lt.size)\nfse_M03 = np.zeros(lt.size)\nfor i in range(len(lt)):\n fse_T03[i] = fselt_T03(lt[i])\n fse_M03[i] = fselt_M03(lt[i])\n```\n\n\n```python\nplt.figure(figsize=(7, 4))\nplt.plot(lt-1, fse_T03, 'b', label='T03')\nplt.plot(lt-1, fse_M03, 'g', label='M03')\nplt.plot(0.04, 1, 'ro', markersize=8)\nplt.text(0.04, 0.7, '$\\epsilon_{T0}$', fontsize=22)\nplt.xlabel('Tendon strain')\nplt.ylabel('Normalized force')\nplt.legend(loc='upper left')\nplt.suptitle('Tendon force-length relationship (series element)', y=1, fontsize=16)\nplt.show()\n```\n\nSimilar results when the same parameters are used.\n\n### Force-velocity relationship of the contractile element\n\nThe force-velocity relation of the contractile element for shortening (concentric activation) is based on the well known Hill's equation of a hyperbola describing that the product between force $F$ and velocity $V$ of the contractile element is constant (Winters, 1990; Winters, 1995):\n\n$$ (F+a')(V+b') = (F_{0}+a')b' $$\n\nwhere $a'$, $b'$, and $F_{0}$ are constants. \n\nWe can rewrite the equation above with constants more meaningful to our modeling: \n\n$$ (F_{M}+A_f F_{Mlen})(V_M+A_f V_{Mmax}) = A_f F_{Mlen}V_{Mmax}(1+A_f) $$\n\nwhere $F_{M}$ and $V_M$ are the contractile element force and velocity, respectively, and the three constants are: $V_{Mmax}$, the maximum unloaded velocity (when $F_{M}=0$), $F_{Mlen}$, the maximum isometric force (when $V_M=0$), and $A_f$, a shape factor which specifies the concavity of the hyperbola.\n\nBased on the equation above for the shortening phase and in Winters (1990, 1995) for the lengthening phase, Thelen2003Muscle employed the following force-velocity equation:\n\n$$ V_M = (0.25+0.75a)\\,V_{Mmax}\\frac{\\bar{F}_M-a\\bar{f}_{l,CE}}{b} $$\n\nwhere\n\n$$ \nb = \\left\\{ \n \\begin{array}{l l l}\n a\\bar{f}_{l,CE} + \\bar{F}_M/A_f \\quad & \\text{if} \\quad \\bar{F}_M \\leq a\\bar{f}_{l,CE} & \\text{(shortening)} \\\\\n \\\\\n \\frac{(2+2/A_f)(a\\bar{f}_{l,CE}\\bar{f}_{Mlen} - \\bar{F}_M)}{\\bar{f}_{Mlen}-1} \\quad & \\text{if} \\quad \\bar{F}_M > a\\bar{f}_{l,CE} & \\text{(lengthening)} \n\\end{array} \\right.\n$$ \n\nwhere $a$ is the activation level and $\\bar{f}_{Mlen}$ is a constant for the maximum force generated at the lengthening phase (normalized by the maximum isometric force). \nThelen2003Muscle adopted $A_f=0.25$, $V_{Mmax}=10L_{Mopt}/s$, $\\bar{f}_{Mlen}=1.4$ for young adults ($V_{Mmax}=8L_{Mopt}/s$ and $\\bar{f}_{Mlen}=1.8$ for old adults). Note that the dependences of the force with the activation level and with the muscle length are already incorporated in the expression above. \n\nMcLean2013Muscle employed:\n\n$$ \n\\bar{f}_{v,CE} = \\left\\{ \n \\begin{array}{l l l}\n \\frac{\\lambda(a)V_{Mmax} + V_M}{\\lambda(a)V_{Mmax} - V_M/A_f} \\quad & \\text{if} \\quad V_M \\leq 0 & \\text{(shortening)} \\\\\n \\\\\n \\frac{\\bar{f}_{Mlen}V_M + d_1}{V_M + d_1} \\quad & \\text{if} \\quad 0 < V_M \\leq \\gamma d_1 & \\text{(slow lengthening)} \\\\\n \\\\\n d_3 + d_2V_M \\quad & \\text{if} \\quad V_M > \\gamma d_1 & \\text{(fast lengthening)} \n\\end{array} \\right.\n$$ \n\nwhere\n\n$$ \\begin{array}{l l}\n \\lambda(a) = 1-e^{-3.82a} + a\\:e^{-3.82} \\\\\n \\\\\n d_1 = \\frac{V_{Mmax}A_f(\\bar{f}_{Mlen}-1)}{S(A_f+1)} \\\\\n \\\\\n d_2 = \\frac{S(A_f+1)}{V_{Mmax}A_f(\\gamma+1)^2} \\\\\n \\\\\n d_3 = \\frac{(\\bar{f}_{Mlen}-1)\\gamma^2}{(\\gamma+1)^2} + 1\n\\end{array} $$\n\nwhere $\\lambda(a)$ is a scaling factor to account for the influence of the activation level $a$ on the force-velocity relationship, $\\bar{f}_{Mlen}$ is the asymptotic (maximum) value of $\\bar{F}_M$, $S$ is a parameter to double the slope of the force-velocity curve at zero velocity, and $\\gamma$ is a dimensionless parameter to ensure the transition between the hyperbolic and linear parts of the lengthening phase. \nMcLean2013Muscle adopted $A_f=0.25$, $V_{Mmax}=10L_{Mopt}/s$, $\\bar{f}_{Mlen}=1.5$, $S=2.0$, and $\\gamma=5.67$.\n\nLet's write these expressions as Python code and visualize them:\n\n\n```python\ndef vmfce_T03(fm, flce=1, lmopt=1, a=1, vmmax=1, fmlen=1.4, af=0.25):\n \"\"\"Thelen (2003) velocity of the force-velocity relationship as function of CE force.\n \n Parameters\n ----------\n fm : float\n normalized muscle force\n flce : float, optional (default=1)\n normalized muscle force due to the force-length relationship\n lmopt : float, optional (default=1)\n optimal muscle fiber length\n a : float, optional (default=1)\n muscle activation level\n vmmax : float, optional (default=1)\n maximum muscle velocity for concentric activation\n fmlen : float, optional (default=1.4)\n normalized maximum force generated at the lengthening phase\n af : float, optional (default=0.25)\n shape factor\n\n Returns\n -------\n vm : float\n velocity of the muscle\n \"\"\"\n \n vmmax = vmmax*lmopt\n if fm <= a*flce: # isometric and concentric activation\n b = a*flce + fm/af\n else: # eccentric activation\n b = (2 + 2/af)*(a*flce*fmlen - fm)/(fmlen - 1) \n vm = (0.25 + 0.75*a)*vmmax*(fm - a*flce)/b\n \n return vm\n```\n\nLet's find an expression for contractile element force as function of muscle velocity given the equation above, i.e. we want to invert the equation. For that, let's use [Sympy](http://www.sympy.org/):\n\n\n```python\ndef fvce_T03_symb():\n # Thelen (2003) velocity of the force-velocity relationship as function of CE force\n \n from sympy import symbols, solve, collect, Eq\n a, flce, fm, af, fmlen, vmmax = symbols('a, flce, fm, af, fmlen, vmmax', positive=True)\n vm = symbols('vm', real=True)\n \n b = a*flce + fm/af\n vm_eq = Eq(vm - (0.25 + 0.75*a)*vmmax*(fm - a*flce)/b)\n sol = solve(vm_eq, fm)\n print('fm <= a*flce:\\n', collect(sol[0], vmmax),'\\n')\n \n b = (2 + 2/af)*(a*flce*fmlen - fm)/(fmlen - 1)\n vm_eq = Eq(vm - (0.25 + 0.75*a)*vmmax*(fm - a*flce)/b)\n sol = solve(vm_eq, fm)\n print('fm > a*flce:\\n', collect(sol[0], (vmmax*af, fmlen, vm)))\n\nfvce_T03_symb()\n```\n\n fm <= a*flce:\n a*af*flce*(4.0*vm + vmmax*(3.0*a + 1))/(-4.0*vm + vmmax*(3.0*a*af + af)) \n \n fm > a*flce:\n a*flce*(af*vmmax*(3.0*a*fcemax - 3.0*a + fcemax - 1) + fcemax*(8.0*af*vm + 8.0*vm))/(af*vmmax*(3.0*a*fcemax - 3.0*a + fcemax - 1) + vm*(8.0*af + 8.0))\n\n\nAnd here is the function we need to compute contractile element force as function of muscle velocity:\n\n\n```python\ndef fvce_T03(vm=0, flce=1, lmopt=1, a=1, vmmax=1, fmlen=1.4, af=0.25):\n \"\"\"Thelen (2003) force of the contractile element as function of muscle velocity.\n \n Parameters\n ----------\n vm : float, optional (default=0)\n muscle velocity\n flce : float, optional (default=1)\n normalized muscle force due to the force-length relationship\n lmopt : float, optional (default=1)\n optimal muscle fiber length\n a : float, optional (default=1)\n muscle activation level\n vmmax : float, optional (default=1)\n maximum muscle velocity for concentric activation\n fmlen : float, optional (default=1.4)\n normalized maximum force generated at the lengthening phase\n af : float, optional (default=0.25)\n shape factor\n\n Returns\n -------\n fvce : float\n normalized force of the muscle contractile element\n \"\"\"\n\n vmmax = vmmax*lmopt\n if vm <= 0: # isometric and concentric activation\n fvce = af*a*flce*(4*vm + vmmax*(3*a + 1))/(-4*vm + vmmax*af*(3*a + 1))\n else: # eccentric activation\n fvce = a*flce*(af*vmmax*(3*a*fmlen - 3*a + fmlen - 1) + 8*vm*fmlen*(af + 1))/\\\n (af*vmmax*(3*a*fmlen - 3*a + fmlen - 1) + 8*vm*(af + 1))\n \n return fvce\n```\n\nHere is the Python function for the McLean (2003) model:\n\n\n```python\ndef fvce_M03(vm=0, lmopt=1, a=1, vmmax=1, fmlen=1.5, af=0.25, s=2, gammav=5.67):\n \"\"\"McLean (2003) contractile element force as function of muscle velocity.\n \n Parameters\n ----------\n vm : float, optional (default=0)\n muscle velocity\n lmopt : float, optional (default=1)\n optimal muscle fiber length\n a : float, optional (default=1)\n muscle activation level\n vmmax : float, optional (default=1)\n maximum muscle velocity for concentric activation\n fmlen : float, optional (default=1.5)\n normalized maximum force generated at the lengthening phase\n af : float, optional (default=0.25)\n shape factor\n s : float, optional (default=2)\n to double the slope of the force-velocity curve at zero velocity\n gammav : float, optional (default=5.67)\n to ensure the smooth transition of the lengthening phase\n\n Returns\n -------\n fvce : float\n normalized force of the muscle contractile element\n \"\"\"\n\n vmmax = vmmax*lmopt\n d1 = vmmax*af*(fmlen - 1)/(s*(af + 1))\n d2 = s*(af + 1)/(vmmax*af*(gammav + 1)**2)\n d3 = (fmlen - 1)*gammav**2/(gammav + 1)**2 + 1\n lbd = 1 - np.exp(-3.82*a) + a*np.exp(-3.82)\n if vm <= 0: # isometric and concentric activation\n fvce = (lbd*vmmax + vm)/(lbd*vmmax - vm/af)\n elif 0 < vm <= gammav*d1: # slow lengthening\n fvce = (fmlen*vm + d1)/(vm + d1)\n elif vm > gammav*d1: # fast lengthening\n fvce = d3 + d2*vm\n \n return fvce\n```\n\nWe can invert this equation to get an expression for muscle velocity as function of the contractile element force:\n\n\n```python\ndef vmfce_M03(fvce=1, lmopt=1, a=1, vmmax=1, fmlen=1.5, af=0.25, s=2, gammav=5.67):\n \"\"\"McLean (2003) contractile element velocity as function of CE force.\n \n Parameters\n ----------\n fvce : float, optional (default=1)\n normalized muscle force\n lmopt : float, optional (default=1)\n optimal muscle fiber length\n a : float, optional (default=1)\n muscle activation level\n vmmax : float, optional (default=1)\n maximum muscle velocity for concentric activation\n fmlen : float, optional (default=1.5)\n normalized maximum force generated at the lengthening phase\n af : float, optional (default=0.25)\n shape factor\n s : float, optional (default=2)\n to double the slope of the force-velocity curve at zero velocity\n gammav : float, optional (default=5.67)\n to ensure the smooth transition of the lengthening phase\n\n Returns\n -------\n fvce : float\n muscle velocity\n \"\"\"\n \n vmmax = vmmax*lmopt\n d1 = vmmax*af*(fmlen - 1)/(s*(af + 1))\n d2 = s*(af + 1)/(vmmax*af*(gammav + 1)**2)\n d3 = (fmlen - 1)*gammav**2/(gammav + 1)**2 + 1\n lbd = 1 - np.exp(-3.82*a) + a*np.exp(-3.82)\n if 0 <= fvce <= 1: # isometric and concentric activation\n vm = (lbd*vmmax*(1 - fvce))/(1 + fvce/af)\n elif 1 < fvce <= gammav*d1*d2 + d3: # slow lengthening\n vm = d1*(fvce - 1)/(fmlen - fvce)\n elif fvce > gammav*d1*d2 + d3: # fast lengthening\n vm = (fvce - d3)/d2\n \n return vm\n```\n\nLet's use these functions to compute muscle force as a function of the muscle velocity considering two levels of activation:\n\n\n```python\nvm1_T03 = np.linspace(-1, 1, 201)\nfce1_T03 = np.zeros(vm1_T03.size)\nvm2_T03 = np.linspace(-.63, .63, 201)\nfce2_T03 = np.zeros(vm2_T03.size)\nfor i in range(len(vm1_T03)):\n fce1_T03[i] = fvce_T03(vm=vm1_T03[i])\n fce2_T03[i] = fvce_T03(vm=vm2_T03[i], a=0.5)\n```\n\n\n```python\nvm1_M03 = np.linspace(-1, 1, 201)\nfce1_M03 = np.zeros(vm1_M03.size)\nvm2_M03 = np.linspace(-.63, .63, 201)\nfce2_M03 = np.zeros(vm2_M03.size)\nfor i in range(len(vm1_M03)):\n fce1_M03[i] = fvce_M03(vm=vm1_M03[i])\n fce2_M03[i] = fvce_M03(vm=vm2_M03[i], a=0.5)\nfce2_M03 = fce2_M03*0.5\n```\n\n\n```python\nfig, (ax1, ax2) = plt.subplots(nrows=1, ncols=2, sharex=True, sharey=True, figsize=(10, 4))\nax1.plot(vm1_T03, fce1_T03, 'b', label='T03)')\nax1.plot(vm1_M03, fce1_M03, 'g', label='M03)')\nax1.set_ylabel('Normalized force')\nax1.set_xlabel('Normalized velocity')\nax1.text(-.9, 1.7, 'Activation = 1.0')\nax2.plot(vm2_T03, fce2_T03, 'b', label='T03')\nax2.plot(vm2_M03, fce2_M03, 'g', label='M03')\nax2.text(-.9, 1.7, 'Activation = 0.5')\nax2.set_xlabel('Normalized velocity')\nax2.legend(loc='best')\nplt.suptitle('Force-velocity relationship of the contractile element', y=1, fontsize=16)\nplt.tight_layout()\nplt.show()\n```\n\nIdentical results for the shortening phase when $a=1$ and similar results for the lengthening phase when the same parameters are used.\n\n#### Muscle power\n\nThe muscle power is the product between force and velocity:\n\n\n```python\nP_T03 = np.abs(fce1_T03*vm1_T03)\n```\n\nLet's visualize the muscle power only for the concentric phase (muscle shortening):\n\n\n```python\nplt.figure(figsize=(7, 4))\nplt.plot(vm1_T03[:101], fce1_T03[:101], 'b', label='Force')\nplt.xlabel('Normalized velocity')\nplt.ylabel('Normalized force', color='b')\n#plt.legend(loc='upper left')\nplt.gca().invert_xaxis()\nplt.gca().twinx()\nplt.plot(vm1_T03[:101], P_T03[:101], 'g', label='Power')\nplt.ylabel('Normalized power', color='g')\n#plt.legend(loc='upper right')\nplt.suptitle('Muscle power', y=1, fontsize=16)\nplt.show()\n```\n\n#### Force-length-velocity relationship\n\nLet's visualize the effects of the length and velocity on the total (active plus passive) muscle force:\n\n\n```python\nlms = np.linspace(0, 1.65, 101)\nvms = np.linspace(-1, .76, 101)\nfce_T03 = np.zeros(lms.size)\nfpe_T03 = np.zeros(lms.size)\nfm_T03 = np.zeros((lms.size, vms.size))\nfor i in range(len(lms)):\n fce_T03[i] = flce_T03(lm=lms[i])\n fpe_T03[i] = fpelm_T03(lm=lms[i]) \n for j in range(len(vms)):\n fm_T03[j, i] = fvce_T03(vm=vms[j], flce=fce_T03[i]) + fpe_T03[i]\n```\n\n\n```python\nlms = np.linspace(0, 1.65, 101)\nvms = np.linspace(-1, .76, 101)\nfce_M03 = np.zeros(lms.size)\nfpe_M03 = np.zeros(lms.size)\nfm_M03 = np.zeros((lms.size, vms.size))\nfor i in range(len(lms)):\n fce_M03[i] = flce_M03(lm=lms[i])\n fpe_M03[i] = fpelm_M03(lm=lms[i]) \n for j in range(len(vms)):\n fm_M03[j, i] = fvce_M03(vm=vms[j])*fce_M03[i] + fpe_M03[i]\n```\n\n\n```python\nfrom mpl_toolkits.mplot3d import Axes3D\n\ndef flv3dplot(ax, lm, vm, fm, model):\n # 3d plot\n lm2, vm2 = np.meshgrid(lm, vm)\n ax.plot_surface(lm2, vm2, fm, rstride=2, cstride=2, cmap=plt.cm.coolwarm,\n linewidth=.5, antialiased=True)\n ax.plot(np.ones(vms.size), vms, fm[:, np.argmax(lm>=1)], 'w', linewidth=4)\n ax.plot(lm, np.zeros(lm.size), fm[np.argmax(vm>=0),:], 'w', linewidth=4)\n ax.set_xlim3d(lm[0], lm[-1])\n ax.set_ylim3d(vm[0], vm[-1])\n #ax.set_zlim3d(np.min(fm), np.max(fm))\n ax.set_zlim3d(0, 2)\n ax.set_xlabel('Normalized length')\n ax.set_ylabel('Normalized velocity')\n ax.set_zlabel('Normalized force')\n ax.view_init(20, 225)\n ax.locator_params(nbins=6)\n ax.text(-0.4, 0.7, 2.5, model, fontsize=14)\n \nfig = plt.figure(figsize=(12, 6))\nax1 = fig.add_subplot(1, 2, 1, projection='3d')\nflv3dplot(ax1, lms, vms, fm_T03, 'T03')\nax2 = fig.add_subplot(1, 2, 2, projection='3d')\nflv3dplot(ax2, lms, vms, fm_M03, 'M03')\nplt.suptitle('Force-length-velocity relationship', y=1, fontsize=16)\nplt.tight_layout()\nplt.show()\n```\n\n### Activation dynamics\n\nActivation dynamics represents the fact that a muscle cannot instantly activate or deactivate because of the electrical and chemical processes involved and it is usually integrated with a Hill-type model. In its simplest form, the activation dynamics is generally represented as a first-order ODE. \n\nThelen2003Muscle employed the following first-order [ordinary differential equation (ODE)](http://en.wikipedia.org/wiki/Ordinary_differential_equation):\n\n$$ \\frac{\\mathrm{d}a}{\\mathrm{d}t} = \\frac{u-a}{\\tau(a, u)} $$\n\nwith a lower activation bound to both activation and excitation.\n\nwhere $u$ and $a$ are the muscle excitation and activation, respectively (both are function of time), and $\\tau$ is a variable time constant to represent the activation and deactivation times, given by:\n\n$$ \n\\tau(a, u) = \\left\\{ \n \\begin{array}{l l}\n t_{act}(0.5+1.5a) \\quad & \\text{if} \\quad u > a\\\\\n \\frac{t_{deact}}{(0.5+1.5a)} \\quad & \\text{if} \\quad u \\leq a\n\\end{array} \\right.\n$$\n\nThelen2003Muscle adopted activation, $t_{act}$, and deactivation, $t_{deact}$, time constants for young adults equal to 15 and 50 ms, respectively (for old adults, Thelen2003Muscle adopted 15 and 60 ms, respectively).\n\nMcLean2003Muscle expressed the activation dynamics as the following first-order ODE:\n\n$$ \\frac{\\mathrm{d}a}{\\mathrm{d}t} = (u - a)(c_1u + c_2) $$\n\nwith a lower activation bound to both activation and excitation.\n\nwhere $c_1 + c_2$ is the activation rate constant (when $u = 1$), the inverse of $t_{act}$, and $c_2$ is the deactivation rate constant (when $u = 0$), the inverse of $t_{deact}$. \nMcLean2003Muscle adopted $c_1=3.3 s^{-1}$ and $c_2=16.7 s^{-1}$, resulting in time constants of 50 ms and 60 ms for activation and deactivation, respectively.\n\nIn Python, the numeric first-order ODE for the activation dynamics presented in Thelen2003Muscle can be expressed as:\n\n\n```python\ndef actdyn_T03(t, a, t_act, t_deact, u_max, u_min, t0=0, t1=1):\n \"\"\"Thelen (2003) activation dynamics, the derivative of `a` at `t`.\n\n Parameters\n ----------\n t : float\n time instant [s]\n a : float (0 <= a <= 1)\n muscle activation\n t_act : float\n activation time constant [s]\n t_deact : float\n deactivation time constant [s]\n u_max : float (0 < u_max <= 1), optional (default=1)\n maximum value for muscle excitation\n u_min : float (0 < u_min < 1), optional (default=0.01)\n minimum value for muscle excitation\n t0 : float [s], optional (default=0)\n initial time instant for muscle excitation equals to u_max\n t1 : float [s], optional (default=1)\n final time instant for muscle excitation equals to u_max\n\n Returns\n -------\n adot : float \n derivative of `a` at `t`\n \"\"\"\n\n u = excitation(t, u_max, u_min)\n if u > a:\n adot = (u - a)/(t_act*(0.5 + 1.5*a))\n else:\n adot = (u - a)/(t_deact/(0.5 + 1.5*a))\n\n return adot\n```\n\nIn Python, the numeric first-order ODE for the activation dynamics presented in McLean2003Muscle can be expressed as:\n\n\n```python\ndef actdyn_M03(t, a, t_act, t_deact, u_max=1, u_min=0.01, t0=0, t1=1):\n \"\"\"McLean (2003) activation dynamics, the derivative of `a` at `t`.\n\n Parameters\n ----------\n t : float\n time instant [s]\n a : float (0 <= a <= 1)\n muscle activation\n t_act : float\n activation time constant [s]\n t_deact : float\n deactivation time constant [s]\n u_max : float (0 < u_max <= 1), optional (default=1)\n maximum value for muscle excitation\n u_min : float (0 < u_min < 1), optional (default=0.01)\n minimum value for muscle excitation\n t0 : float [s], optional (default=0)\n initial time instant for muscle excitation equals to u_max\n t1 : float [s], optional (default=1)\n final time instant for muscle excitation equals to u_max\n\n Returns\n -------\n adot : float \n derivative of `a` at `t`\n \"\"\"\n \n c2 = 1/t_deact\n c1 = 1/t_act - c2\n u = excitation(t, u_max, u_min)\n adot = (u - a)*(c1*u + c2)\n \n return adot\n```\n\nLet's simulate the activation signal for a rectangular function as excitation signal:\n\n\n```python\ndef excitation(t, u_max=1, u_min=0.01, t0=0.1, t1=0.4):\n \"\"\"Excitation signal, a square wave.\n \n Parameters\n ----------\n t : float\n time instant [s]\n u_max : float (0 < u_max <= 1), optional (default=1)\n maximum value for muscle excitation\n u_min : float (0 < u_min < 1), optional (default=0.01)\n minimum value for muscle excitation\n t0 : float [s], optional (default=0.1)\n initial time instant for muscle excitation equals to u_max\n t1 : float [s], optional (default=0.4)\n final time instant for muscle excitation equals to u_max\n\n Returns\n -------\n u : float (0 < u <= 1)\n excitation signal\n \"\"\"\n \n u = u_min\n if t >= t0 and t <= t1:\n u = u_max\n \n return u\n```\n\nWe will solve the equation for $a$ by numerical integration using the [`scipy.integrate.ode`](http://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.integrate.ode.html) class of numeric integrators, particularly the `dopri5`, an explicit runge-kutta method of order (4)5 due to Dormand and Prince (a.k.a. ode45 in Matlab):\n\n\n```python\nimport warnings\ndef actdyn_ode45(fun, t0=0, t1=1, a0=0, t_act=0.015, t_deact=0.050, u_max=1, u_min=0.01):\n # Runge-Kutta (4)5 due to Dormand & Prince with variable stepsize ODE solver\n \n f = ode(fun).set_integrator('dopri5', nsteps=1, max_step=0.01, atol=1e-8) \n f.set_initial_value(a0, t0).set_f_params(t_act, t_deact, u_max, u_min)\n # suppress Fortran warning\n warnings.filterwarnings(\"ignore\", category=UserWarning)\n data = []\n while f.t < t1:\n f.integrate(t1, step=True)\n data.append([f.t, excitation(f.t, u_max, u_min), np.max([f.y, u_min])])\n warnings.resetwarnings()\n data = np.array(data)\n \n return data\n```\n\nSolving the problem for two different maximum excitation levels:\n\n\n```python\n# using the values for t_act and t_deact from Thelen2003Muscle for both models\nact1_T03 = actdyn_ode45(fun=actdyn_T03, u_max=1.0)\nact2_T03 = actdyn_ode45(fun=actdyn_T03, u_max=0.5)\nact1_M03 = actdyn_ode45(fun=actdyn_M03, u_max=1.0)\nact2_M03 = actdyn_ode45(fun=actdyn_M03, u_max=0.5)\n# using the values for t_act and t_deact from McLean2003Muscle\nact3_M03 = actdyn_ode45(fun=actdyn_M03, u_max=1.0, t_act=0.050, t_deact=0.060)\nact4_M03 = actdyn_ode45(fun=actdyn_M03, u_max=0.5, t_act=0.050, t_deact=0.060)\n```\n\nAnd the results:\n\n\n```python\nfig, axs = plt.subplots(nrows=2, ncols=2, sharex=True, sharey=True, figsize=(10, 6))\naxs[0, 0].plot(act1_T03[:, 0], act1_T03[:, 1], 'r:', label='Excitation')\naxs[0, 0].plot(act1_T03[:, 0], act1_T03[:, 2], 'b', label='T03 [15, 50] ms')\naxs[0, 0].plot(act1_M03[:, 0], act1_M03[:, 2], 'g', label='M03 [15, 50] ms')\naxs[0, 0].set_ylabel('Level')\naxs[0, 1].plot(act2_T03[:, 0], act2_T03[:, 1], 'r:', label='Excitation')\naxs[0, 1].plot(act2_T03[:, 0], act2_T03[:, 2], 'b', label='T03 [15, 50] ms')\naxs[0, 1].plot(act2_M03[:, 0], act2_M03[:, 2], 'g', label='M03 [15, 50] ms')\naxs[1, 1].set_xlabel('Time (s)')\naxs[0, 1].legend()\naxs[1, 0].plot(act1_T03[:, 0], act1_T03[:, 1], 'r:', label='Excitation')\naxs[1, 0].plot(act1_T03[:, 0], act1_T03[:, 2], 'b', label='T03 [15, 50] ms')\naxs[1, 0].plot(act3_M03[:, 0], act3_M03[:, 2], 'g', label='M03 [50, 60] ms')\naxs[1, 0].set_xlabel('Time (s)')\naxs[1, 0].set_ylabel('Level')\naxs[1, 1].plot(act2_T03[:, 0], act2_T03[:, 1], 'r:', label='Excitation')\naxs[1, 1].plot(act2_T03[:, 0], act2_T03[:, 2], 'b', label='T03 [15, 50] ms')\naxs[1, 1].plot(act4_M03[:, 0], act4_M03[:, 2], 'g', label='M03 [50, 60] ms')\naxs[1, 1].set_xlabel('Time (s)')\naxs[1, 1].legend()\nplt.suptitle('Activation dynamics', y=1, fontsize=16)\nplt.tight_layout()\nplt.show()\n```\n\nSimilar results when the same parameters are used (first row), but different bahavior when the typical values of each study are compared (second row).\n\n### Muscle modeling parameters\n\nWe have seen two types of parameters in the muscle modeling: parameters related to the mathematical functions used to model the muscle and tendon behavior and parameters related to the properties of specific muscles and tendons (e.g., maximum isometric force, optimal fiber length, pennation angle, and tendon slack). In general the first type of parameters are independent of the muscle-tendon unit being modeled (but dependent of the model!) while the second type of parameters is changed for each muscle-tendon unit (for instance, see http://isbweb.org/data/delp/ for some of these parameters).\n\n### Limitations of Hill-type muscle models\n\nAs with any modeling, Hill-type muscle models are a simplification of the reality. For instance, a typical Hill-type muscle model (as implemented here) does not capture time-dependent muscle behavior, such as force depression after quick muscle shortening, force enhancement after quick muscle lengthening, viscoelastic properties (creep and relaxation), and muscle fatigue (Zatsiorsky and Prilutsky, 2012). There are enhanced models that capture these properties but it seems their complexity are not worthy for the most common applications of human movement simulation.\n\n## Exercises\n\n1. The results presented in this text depend on the parameters used in the model. These parameters may vary because of different properties of the muscle and tendon but also because different mathematical functions may be used. \n a. Change some of the parameters and reproduce the plots shown here and discuss these results (e.g., use the parameters for different muscles from OpenSim or the data from [http://isbweb.org/data/delp/](http://isbweb.org/data/delp/)). \n b. Select another reference (e.g., Anderson, 2007) about muscle modeling that uses different mathematical functions and repeat the previous item.\n\n## References\n\n- Anderson C (2007) [Equations for Modeling the Forces Generated by Muscles and Tendons](https://docs.google.com/viewer?url=https%3A%2F%2Fsimtk.org%2Fdocman%2Fview.php%2F124%2F604%2FMuscleAndTendonForcesClayAnderson20070521.doc) ([PDF](https://drive.google.com/open?id=0BxbW72zV7WmUVUh0MldGOGZ6aHc&authuser=0)). BioE215 Physics-based Simulation of Biological Structures. \n- Erdemir A, McLean S, Herzog W, van den Bogert AJ (2007) [Model-based estimation of muscle forces exerted during movements](http://www.ncbi.nlm.nih.gov/pubmed/17070969). Clinical Biomechanics, 22, 131\u2013154. \n- He J, Levine WS, Loeb GE (1991) [Feedback gains for correcting small perturbations to standing posture](https://drive.google.com/open?id=0BxbW72zV7WmUekRXY09GSEhUVlE&authuser=0). IEEE Transactions on Automatic Control, 36, 322\u2013332. \n- McLean SG, Su A, van den Bogert AJ (2003) [Development and validation of a 3-D model to predict knee joint loading during dynamic movement](http://www.ncbi.nlm.nih.gov/pubmed/14986412). Journal of Biomechanical Engineering, 125, 864-74. \n- McMahon TA (1984) [Muscles, Reflexes, and Locomotion](https://archive.org/details/McMahonTAMusclesReflexesAndLocomotionPrincetonUniversityPress1984). Princeton University Press, Princeton, New Jersey. \n- Millard M, Uchida T, Seth A, Delp SL (2013) [Flexing computational muscle: modeling and simulation of musculotendon dynamics](http://www.ncbi.nlm.nih.gov/pubmed/23445050). Journal of Biomechanical Engineering, 135, 021005. \n- Nigg BM and Herzog W (2006) [Biomechanics of the Musculo-skeletal System](https://books.google.com.br/books?id=hOIeAQAAIAAJ&dq=editions:ISBN0470017678). 3rd Edition. Wiley. \n- Robertson G, Caldwell G, Hamill J, Kamen G (2013) [Research Methods in Biomechanics](http://books.google.com.br/books?id=gRn8AAAAQBAJ). 2nd Edition. Human Kinetics. \n- Thelen DG (2003) [Adjustment of muscle mechanics model parameters to simulate dynamic contractions in older adults](http://homepages.cae.wisc.edu/~thelen/pubs/jbme03.pdf). Journal of Biomechanical Engineering, 125(1):70\u201377.\n- Tsianos GA and Loeb GE (2013) [Muscle Physiology and Modeling](http://www.scholarpedia.org/article/Muscle_Physiology_and_Modeling). Scholarpedia, 8(10):12388. \n- Winters JM (1990) [Hill-based muscle models: a systems engineering perspective](http://link.springer.com/chapter/10.1007%2F978-1-4613-9030-5_5). In [Multiple Muscle Systems: Biomechanics and Movement Organization](http://link.springer.com/book/10.1007/978-1-4613-9030-5), edited by JM Winters and SL Woo, Springer-Verlag, New York. \n- Winters JM (1995) [An Improved Muscle-Reflex Actuator for Use in Large-Scale Neuromusculoskeletal Models](http://www.ncbi.nlm.nih.gov/pubmed/7486344). Annals of Biomedical Engineering, 23, 359\u2013374. \n- Zajac FE (1989) [Muscle and tendon: properties, models, scaling and application to biomechanics and motor control](http://www.ncbi.nlm.nih.gov/pubmed/2676342). Critical Reviews in Biomedical Engineering 17:359-411. \n- Zatsiorsky V and Prilutsky B (2012) [Biomechanics of Skeletal Muscles](http://books.google.com.br/books?id=THXfHT8L5MEC). Human Kinetics. \n", "meta": {"hexsha": "020e372ea96efb35564dab0aee96388d6e7641df", "size": 820252, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/MuscleModeling.ipynb", "max_stars_repo_name": "modenaxe/BMC", "max_stars_repo_head_hexsha": "b6f6e473878ab7b0c19430d1b66b6dba09059c63", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-06-23T20:09:07.000Z", "max_stars_repo_stars_event_max_datetime": "2018-06-23T20:09:07.000Z", "max_issues_repo_path": "notebooks/MuscleModeling.ipynb", "max_issues_repo_name": "modenaxe/BMC", "max_issues_repo_head_hexsha": "b6f6e473878ab7b0c19430d1b66b6dba09059c63", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/MuscleModeling.ipynb", "max_forks_repo_name": "modenaxe/BMC", "max_forks_repo_head_hexsha": "b6f6e473878ab7b0c19430d1b66b6dba09059c63", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2019-01-02T23:17:40.000Z", "max_forks_repo_forks_event_max_datetime": "2019-01-02T23:17:40.000Z", "avg_line_length": 519.8048162231, "max_line_length": 466410, "alphanum_fraction": 0.9244878403, "converted": true, "num_tokens": 14166, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.44393548099487945}} {"text": "```\nCopyright (c) Gradient Institute. All rights reserved.\nLicensed under the Apache 2.0 License.\n```\n\n\nThis notebook tests out the two stage ridge regression model again OLS and other ridge models for estimating treatment effects.\n\n\n\nHahn, P.R., Carvalho, C.M., Puelz, D., He, J., 2018. Regularization and Confounding in Linear Regression for Treatment Effect Estimation. Bayesian Anal. 13. https://doi.org/10.1214/16-BA1044\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nfrom sklearn.datasets import load_boston\nfrom sklearn.linear_model import LinearRegression, Ridge\nfrom sklearn.metrics import r2_score, mean_squared_error\nfrom sklearn.kernel_approximation import RBFSampler\nfrom sklearn.model_selection import train_test_split\n\nfrom twostageridge import TwoStageRidge, ridge_weights\n\nnp.random.seed(42)\n\n%matplotlib inline\n```\n\n# Generate the data\n\n\n```python\ndata = load_boston()\nX = data.data\nN, _ = X.shape\nD = 300\nregulariser_range = (-2, 2)\nrbf_gamma = 1.\n\n\n# Standardise the controls\nX -= X.mean(axis=0)\nX /= X.std(axis=0)\n\n\n# Make controls high dimensional\nX = RBFSampler(n_components=D, gamma=rbf_gamma).fit_transform(X)\n\n\n# Generate a treatment column\nepsilon = .5\ngamma = np.random.randn(D)\nZ = X @ gamma + np.random.randn(N) * epsilon\n\n\n# Generate the target\nalpha = 0.5\nnu = .6\nbeta = np.random.randn(D)\nY = alpha * Z + X @ beta + np.random.randn(N) * nu\n\n\n# Split this into training and testing data to get an idea of how the models overfit\nW = np.hstack((Z[:, np.newaxis], X))\nW, W_test, Y, Y_test = train_test_split(W, Y, test_size=0.1, shuffle=True)\nN_train, N_test = len(Y), len(Y_test)\n\n\n# \"treatment\" index, we need to give this to the two stage model.\ntreatment_ind = 0\n```\n\n# Ridge regression test\n\nRegularisation on alpha and beta \"out of the box\"\n\n$$\\mathcal{L}_Y = \\frac{1}{N} \\sum^N_{i=1} \\|Y_i - (\\alpha Z_i + X_i \\beta) \\|^2_2 + \\lambda \\alpha^2 + \\lambda \\|\\beta\\|_2^2$$\n\nAnd \"confounded\"\n\n$$\\mathcal{L}_Y = \\frac{1}{N} \\sum^N_{i=1} \\|Y_i - (\\alpha Z_i + X_i \\beta) \\|^2_2 + \\lambda \\|\\beta\\|_2^2$$\n\n\n```python\nols = LinearRegression().fit(W, Y)\nalpha_ols = ols.coef_[0]\nscore_ols = ols.score(W_test, Y_test)\n\n#ts = TwoStageRidge(treatment_index=0, regulariser1=.1, regulariser2=.1).fit(W, Y)\n#alpha_ts = ts.alpha_\n\nregularisers = np.logspace(*regulariser_range, 20)\nalpha_ridge = np.zeros_like(regularisers)\nalpha_conf = np.zeros_like(regularisers)\nalpha_ts = np.zeros_like(regularisers)\nscores_ridge = np.zeros_like(regularisers)\nscores_conf = np.zeros_like(regularisers)\nscores_ts = np.zeros_like(regularisers)\n\nfor i, regulariser in enumerate(regularisers):\n ts = TwoStageRidge(treatment_index=0, regulariser1=regulariser, regulariser2=regulariser).fit(W, Y)\n alpha_ts[i] = ts.alpha_\n scores_ts[i] = r2_score(Y_test, ts.predict(W_test)) \n \n reg = Ridge(alpha=regulariser).fit(W, Y) \n alpha_ridge[i] = reg.coef_[0]\n scores_ridge[i] = r2_score(Y_test, reg.predict(W_test))\n \n Wint = np.hstack((W, np.ones((N_train, 1))))\n Wint_test= np.hstack((W_test, np.ones((N_test, 1)))) \n reg = np.ones(Wint.shape[1]) * regulariser\n reg[0] = 0.\n reg_conf_weights = ridge_weights(Wint, Y, gamma=reg)\n alpha_conf[i] = reg_conf_weights[0]\n scores_conf[i] = r2_score(Y_test, Wint_test @ reg_conf_weights)\n\nprint(f\"R^2 OLS = {score_ols:.4f}\")\n \nfig, (ax1, ax2) = plt.subplots(2, 1, figsize=(8, 6), dpi=150)\n\nax1.semilogx(regularisers, alpha_ridge, label=\"Ridge\")\nax1.semilogx(regularisers, alpha_conf, label=\"Ridge confounded\")\nax1.semilogx(regularisers, alpha_ts, label=\"Two Stage $\\\\lambda_c = \\\\lambda_d = \\\\lambda$\")\nax1.semilogx([regularisers[0], regularisers[-1]], [alpha_ols, alpha_ols], label=\"OLS\")\nax1.semilogx([regularisers[0], regularisers[-1]], [alpha, alpha], label=\"True\")\nax1.set_ylabel(\"$\\\\alpha$\")\nax1.grid()\nax1.legend()\n\nax2.semilogx(regularisers, scores_ridge, label=\"Ridge\")\nax2.semilogx(regularisers, scores_conf, label=\"Ridge confounded\")\nax2.semilogx(regularisers, scores_ts, label=\"Two Stage\")\nax2.grid()\nax2.set_ylabel(\"Held-out $R^2$\")\nax2.set_xlabel(\"$\\\\lambda$\")\nax2.legend()\n\nplt.show()\n```\n\n# Two-stage test\n\n## Vary $\\lambda_d$\n\n\\begin{align}\n\\mathcal{L}_Z &= \\frac{1}{N} \\sum^N_{i=1} \\|Z_i - X_i \\beta_c \\|^2_2 + \\lambda_c \\|\\beta_c\\|_2^2 \\\\\n\\mathcal{L}_Y &= \\frac{1}{N} \\sum^N_{i=1} \\|Y_i - [\\alpha(Z_i - X_i \\beta_c) + X_i \\beta_d] \\|^2_2 + \\lambda \\|\\beta_d\\|_2^2\n\\end{align}\n\n\n\n\n\n```python\n#regularisers = np.logspace(*regulariser_range, 20)\nalpha_ts_d = np.zeros_like(regularisers)\ny_scores = np.zeros_like(regularisers)\nz_scores = np.zeros_like(regularisers)\n\n\nfor i, regulariser in enumerate(regularisers):\n ts_d = TwoStageRidge(treatment_index=0, regulariser1=1., regulariser2=regulariser).fit(W, Y)\n alpha_ts_d[i] = ts_d.alpha_\n y_scores[i] = r2_score(Y_test, ts_d.predict(W_test))\n z_scores[i] = ts_d.score_stage1(W_test)\n\n\nfig, (ax1, ax2) = plt.subplots(2, 1, figsize=(8, 6), dpi=150)\nax1.semilogx(regularisers, alpha_ts_d, label=\"Two-stage\")\nax1.semilogx([regularisers[0], regularisers[-1]], [alpha_ols, alpha_ols], label=\"OLS\")\nax1.semilogx([regularisers[0], regularisers[-1]], [alpha, alpha], label=\"True\")\nax1.set_ylabel(\"$\\\\alpha$\")\nax1.grid()\nax1.legend()\nax2.semilogx(regularisers, y_scores, label=\"Second stage: $Y$\")\nax2.semilogx(regularisers, z_scores, label=\"First stage: $Z$\")\nax2.grid()\nax2.legend()\nax2.set_ylabel(\"Held-out $R^2$\")\nax2.set_xlabel(\"$\\\\lambda_d$\")\nplt.show()\n```\n\n## Vary $\\lambda_c$\n\n\\begin{align}\n\\mathcal{L}_Z &= \\frac{1}{N} \\sum^N_{i=1} \\|Z_i - X_i \\beta_c \\|^2_2 + \\lambda_c \\|\\beta_c\\|_2^2 \\\\\n\\mathcal{L}_Y &= \\frac{1}{N} \\sum^N_{i=1} \\|Y_i - [\\alpha(Z_i - X_i \\beta_c) + X_i \\beta_d] \\|^2_2 + \\lambda \\|\\beta_d\\|_2^2\n\\end{align}\n\n\n\n\n```python\n#regularisers = np.logspace(-2, 3, 50)\nalpha_ts_c = np.zeros_like(regularisers)\ny_scores = np.zeros_like(regularisers)\nz_scores = np.zeros_like(regularisers)\n\n\nfor i, regulariser in enumerate(regularisers):\n ts_c = TwoStageRidge(treatment_index=0, regulariser1=regulariser, regulariser2=1.).fit(W, Y)\n alpha_ts_c[i] = ts_c.alpha_\n y_scores[i] = r2_score(Y_test, ts_c.predict(W_test))\n z_scores[i] = ts_c.score_stage1(W_test)\n\n\nfig, (ax1, ax2) = plt.subplots(2, 1, figsize=(8, 6), dpi=150)\nax1.semilogx(regularisers, alpha_ts_c, label=\"Two-stage\")\nax1.semilogx([regularisers[0], regularisers[-1]], [alpha_ols, alpha_ols], label=\"OLS\")\nax1.semilogx([regularisers[0], regularisers[-1]], [alpha, alpha], label=\"True\")\nax1.set_ylabel(\"$\\\\alpha$\")\nax1.grid()\nax1.legend()\nax2.semilogx(regularisers, y_scores, label=\"Second stage: $Y$\")\nax2.semilogx(regularisers, z_scores, label=\"First stage: $Z$\")\nax2.grid()\nax2.legend()\nax2.set_ylabel(\"Held-out $R^2$\")\nax2.set_xlabel(\"$\\\\lambda_c$\")\nplt.show()\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "1772db102df1c5e6c371942d83e97af7fc3b1d2e", "size": 306657, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/regularisation_bias_exploration.ipynb", "max_stars_repo_name": "gradientinstitute/twostageridge", "max_stars_repo_head_hexsha": "f19ccf30c0520a3b7e1523b091b2652057dbb18f", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/regularisation_bias_exploration.ipynb", "max_issues_repo_name": "gradientinstitute/twostageridge", "max_issues_repo_head_hexsha": "f19ccf30c0520a3b7e1523b091b2652057dbb18f", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/regularisation_bias_exploration.ipynb", "max_forks_repo_name": "gradientinstitute/twostageridge", "max_forks_repo_head_hexsha": "f19ccf30c0520a3b7e1523b091b2652057dbb18f", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 837.8606557377, "max_line_length": 132088, "alphanum_fraction": 0.9496962404, "converted": true, "num_tokens": 2195, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878414043816, "lm_q2_score": 0.7217432062975978, "lm_q1q2_score": 0.4439354708098667}} {"text": "```python\nfrom IPython.display import display, Math, Latex\nimport pystan\nimport matplotlib.pyplot as plt\nimport numpy as np\n```\n\n\n\n\n## Dr Peter Hurley & Dr Phil Rooney\n\n## Data Science\n\n$$Data + Model = Prediction$$\n* *Extracting information from Data*\n* *Using information to make predictions, decisions etc*\n\n

What is Probability?

\n
\n\n

Idea: Probability is the Long Run Expected Number of Occurances of an Event

\n
\n
\n \n

When you roll a fair dice once it will come up with only one answer, but after doing this many times each outcome will occur 1/6 of the time. So the probability of an outcome is 1/6.

\n
\n

Another way of saying this: The probability of an event A is equal to the number of ways in which A can occur divided by the number of ways in which any event can happen.

\n
\n

This then helps us answer questions such as, what is the probability of drawing three aces in a row from a shuffled deck of cards?

\n
\n

There are four ways of picking the first ace, out of 52 possible outcomes, three ways of picking the first ace, out of 51 possible outcomes, and two ways of picking the first ace, out of 50 possible outcomes. Therefore:

\n\n\n\\begin{equation}\nP(3\\ Aces) = \\frac{4}{52} * \\frac{3}{51} * \\frac{2}{50} \n\\end{equation}\n\n

We can also ask more nuanced questions, such as what is the probability of event A given event B. For example, what is the probability of a card is a King given it is a face card?

\n
\n\n\n\\begin{equation}\nP(A\\ |\\ B) = \\frac{P(A \\cap\tB)}{P(B)}\n\\end{equation}\n\n\\begin{equation}\nP(King\\ |\\ Face\\ Card) = \\frac{4}{12} = \\frac{1}{3}\n\\end{equation}\n\n\n

Suppose a family has two children and suppose one of the children is a boy. What is the probability that both children are boys?

\n \n\n

Total number of ways in which a family can have two children:

\n \n
\n\n

{BB}, {BG}, {GB}, {GG}

\n \n
\n\n

Of these three involve at least one boy, and of these only one pairing is two boys so:

\n \n
\n\n\n\\begin{equation}\nP(Two\\ Boys\\ |\\ One\\ child\\ definitely\\ a\\ boy) = \\frac{1}{3}\n\\end{equation}\n\n
\n\n

A nice extension to this problem is: Suppose a family has two children and suppose one of the children is a boy born on a Tuesday. What is the probability that both children are boys?

\n\n\n

The Monte Hall Problem:Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, \"Do you want to pick door No. 2?\" Is it to your advantage to switch your choice?

\n
\n\n\\begin{equation}\nP(Prize\\ behind\\ any\\ random\\ door) = \\frac{1}{3}\n\\end{equation}\n\n
\n\n\\begin{equation}\nP(Keep\\ Door\\ 1\\ and\\ Win) = \\frac{P(Prize\\ Door\\ 1\\ and\\ Host\\ Opens\\ Door\\ 3)}{Host\\ Opens\\ Door\\ 3} \n\\end{equation}\n\n
\n\n\\begin{equation}\nP(Prize\\ Door\\ 1\\ and\\ Host\\ Opens\\ Door\\ 3) = \\frac{1}{3} * \\frac{1}{2} = \\frac{1}{6}\n\\end{equation}\n\n
\n\n

The probability the host opens door 3 depends on which door the prize is behind.

\n\n\\begin{equation}\nP(Host\\ Door\\ 3) = P(Host\\ Door\\ 3\\ |\\ Prize\\ Door\\ 1) * P(Prize\\ Door\\ 1) + P(Host\\ Door\\ 3\\ |\\ Prize\\ Door\\ 2) * P(Prize\\ Door\\ 3)+ P(Host\\ Door\\ 3\\ |\\ Prize\\ Door\\ 2) * P(Prize\\ Door\\ 3) \n\\end{equation}\n\n
\n\n\\begin{equation}\nP(Host\\ Door\\ 3) = \\frac{1}{2} * \\frac{1}{3} + 1 * \\frac{1}{3} + 0*\\frac{1}{3} = \\frac{1}{2}\n\\end{equation}\n\n
\n\n

So the probability of winning if we keep our door is:

\n\n
\n\n\\begin{equation}\nP(Keep\\ Door\\ 1\\ and\\ Win) = \\frac{\\frac{1}{6}}{\\frac{1}{2}} = \\frac{1}{3}\n\\end{equation}\n\n
\n\n

And so the probability of winning if we swap is: <\\p>\n \n\\begin{equation}\nP(Swap\\ Door\\ 2\\ and\\ Win) = 1 - \\frac{1}{3} = \\frac{2}{3}\n\\end{equation}\n\n

We can manipulate these equations as in other areas of maths as we did above

\n
\n\n\\begin{equation}\nP(A\\ |\\ B) = \\frac{P(A \\cap\tB)}{P(B)}\n\\end{equation}\n\n\\begin{equation}\nP(A \\cap\tB) = P(A) * P(B | A)\n\\end{equation}\n\n\\begin{equation}\nP(A \\cap\tB) = P(B) * P(A | B)\n\\end{equation}\n\n\\begin{equation}\nP(A) * P(B | A) = P(B) * P(A | B)\n\\end{equation}\n\n\\begin{equation}\nP(B | A) = \\frac{P(B) * P(A | B)}{P(A)}\n\\end{equation}\n\n

This last equation is Bayes Theorem, a source of heated debate among statisticians throughout the 20th century.

\n\n
\n\n

You may wonder what the fuss is as surely everyone accepts the above?

\n\n
\n\n

The issue comes in when we consider how far we can apply this equation.

\n\n## Probabilistic models \n* A model describes data that one could observe from a system\n* Uses probability theory to express all forms of uncertainty and noise associated with our model\n* Inverse probability (i.e. Bayes rule) to infer parameters, adapt models, make predictions and learn from data.\n\n\n
\n\nA good example of how Bayesian probability is more intuitive is to think of a situation such as the probability of rain, given the sky is cloudy, or that there is a sprinkler nearby. We use prior information to decide if it has rained, rather than just looking at the number of times it has rained in past.\n\n
\n\nAlternatively there can also be cases where a frequentist model breaks down. Suppose you are an astronomer looking at Saturn with an old telescope. Combining many measurements of the shape of the planet will not improve your result on resolving the shape. However given a model of how a planet with a ring or without you can test how well each model describes Saturn and infer the correct shape. \n\n
\n\nOther key cases are those in which you can never repeat the experiment, e.g. modelling how the universe grew since the big bang or the diversification of songbirds in the Andes. \n\n\n\n\\begin{equation}\nP(B | A) = \\frac{P(B) * P(A | B)}{P(A)}\n\\end{equation}\n\n

P(B | A) = Posterior
\n P(B) = Prior
\n P(A | B) = Likelihood
\n P(A) = Marginal Likelihood
\n

\n\n
\n\n

Transparent way of including model prior information
\nGives full probability distribution\n

\n
\n\n

As the area under a posterior must equal one the marginal likelihood can be thought of as a normalistion value, and so the equation is often written as:

\n\n
\n\n\\begin{equation}\nP(B | A) \\propto P(B) * P(A | B)\n\\end{equation}\n\n
\n\n

In that case your posterior is your prior state of belief being updated by the new data.

\n\n
\n\n

More precisely however you P(A) is actually an integral which is often difficult to solve, and must be done numerically, motivates the need for MCMC type sampling. MCMC-like algorithms will generate points proportionally to the posterior values in each region. A simplistic way of thinking about this is a sampler generates some initial starting parameter values, calculates how well a model with these parameters explains the data and then generates a new set of parameter values nearby. If these new values are an improvement we always accept the new point and move there, otherwise we have a small chance we move there. The exact way in which this works varies by method.

\n\n
\n\n

So far so good, but there is a lot of debate and discomfort with how you choose an appropriate prior. However disagreements over what prior is used can at least be transparent, and the correct description of the current state of knowledge before new data is an important and necessary part of debate.

\n\n### Why bother with all this Bayesian probabilistic stuff?\n* Prior information help us extract more information from data\n\n### Prob. model of straight line\n$\\mathbf{y}=m\\mathbf{x}+c + \\epsilon$\n\nlikelihood = $p(\\mathbf{y}|m,c,\\mathbf{x}) = \\mathcal{N}(m\\mathbf{x}+c,\\sigma^2)$\n\nprior = $p(m,c) = \\mathcal{N}(0,\\Sigma_p)$\n\nposterior (after a bit of maths)\n\n$p(m,c|\\mathbf{y},\\mathbf{x}) \\propto \\mathcal{N}(\\sigma^{-2}(\\sigma^{-2}\\mathbf{x}\\mathbf{x}^T + \\Sigma_{p}^{-1})^{-1}\\mathbf{x}\\mathbf{y},$\n$\\sigma^{-2}\\mathbf{x}\\mathbf{x}^T + \\Sigma_{p}^{-1})$\n\nSimple example straight line, error only in y, simple gaussian prior\n\nWhat happens if intercept is positive? gets complicated fast\n\n\n\n\n\n\n\nHave prior information on intercept\n\n$c \\sim \\mathcal{N}(0.3,0.1)$\n\n\n\n\n\n\n\n\n\n* The uncertainty in our inferred model parameters contains information.\n\n* We do not want to just know the best answer, we\nwant to know how uncertain we are.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n### Using Uncertianty in decision making means making better decisions\n\n* Business consider risk when making decisions\n* Uncertianty should be considered when making decisions based on predictions\n\nSee [Blog post on Bayesian Decision making](https://pdh21.github.io/jekyll/update/2019/06/06/FACYNation_BDT.html)\n\n\n\n\n\n# Stan \n\n\n* Language for building probabilistic models\n* Algorithms for fitting probabilistic models\n* Interfaces for using probabilistic models \n\n\n\n## Language\nSpecify probabilisitic model: the joint probability distribution function,\n\n$$p(\\theta,x)$$\n\n* $\\theta$ = parameters\n* $x$ = data\n* $p$ = model\n\n\nStan is a language:\n* Statically typed, imperative\n* User can define programs using parameters, data and $\\log p(\\theta,x)$\n* Any program can be written as long as $\\log p(\\theta,x)$ is differentiable\n\n\nStatically typed like C, C++. i.e. types declared at compile time rather than run time like Python.\n\nImperative, you tell the compiler what you want to happen, step by step\n\n\n\n## Algorithms\n* Bayesian Inference: No U-Turn Hamiltonian Monte Carlo,\n - $p(\\theta|x)$ approximated with $[\\theta^1,\\theta^2,..\\theta^N]$\n* Approximate Bayesian Inference: Variational Inference,\n - $\\hat{p}(\\theta|x) \\approx q(\\hat{\\phi})$, where $\\phi = argmin_{\\phi} D_{KL}(q(\\theta|\\phi)||p(\\theta,x))$\n* Optimisation: Max \n - $\\hat{\\theta} = argmax_{\\theta} p(\\theta,x)$\n \n\n\n\n## Interfaces\n* CmdStan,PyStan,RStan\n* C++ API\n* C++ auto-diff library\n* Software built with Stan: RStanArm, brms, prophet\n\n

As an example lets consider a question of if a coin is fair or not. We begin with a flat prior and update it as we flip a coin.

\n\n\n\n\n\n\n```python\ncoin_model =\"\"\"// Inferring a Rate\ndata { \n int n; \n int h;\n} \nparameters {\n real theta;\n} \nmodel {\n // Prior Distribution for Rate Theta\n theta ~ normal(3, 0.5);\n \n // Observed Counts\n h ~ binomial(n, theta);\n}\"\"\"\n\nsm=pystan.StanModel(model_code=coin_model)\n```\n\n INFO:pystan:COMPILING THE C++ CODE FOR MODEL anon_model_f5e1c19b675a739c99f9dc1c95646755 NOW.\n\n\n\n```python\nmodel_data={\n 'n':100,\n 'h':55,\n}\n\n```\n\n\n```python\nfit=sm.sampling(data=model_data,chains=4,iter=1000,seed=12329)\n```\n\n\n```python\nprint(fit)\n\nplt.plot(np.arange(0, fit['theta'].size), fit['theta'])\nplt.xlabel('Sample')\nplt.ylabel('Theta')\nplt.show()\n\nplt.hist(fit['theta'])\nplt.xlabel('Theta')\nplt.ylabel('Number of Samples')\nplt.show\n```\n\n\n```python\nmodel_data={\n 'n':10,\n 'h':7,\n}\n\nfit=sm.sampling(data=model_data,chains=4,iter=1000,seed=12329)\n\nprint(fit)\n\nplt.hist(fit['theta'])\nplt.xlabel('Theta')\nplt.ylabel('Number of Samples')\nplt.show\n```\n\n\n```python\nmodel_data={\n 'n':100,\n 'h':54,\n}\n\nfit=sm.sampling(data=model_data,chains=4,iter=1000,seed=12329)\n\nprint(fit)\n\nplt.hist(fit['theta'])\nplt.xlabel('Theta')\nplt.ylabel('Number of Samples')\nplt.show\n```\n\n

Sometimes our data may come in a more complex form. For instance imagine if you have a game that involves coin tosses. You filp a coin until you flip a heads and a tails in a row. Your opponent flips a coin until they flip two heads in a row. All the information you have is who won each game. From this you want to understand if one coin was biased or not.

\n\n

As HT and HH in two coin flips are equally likely we may run the code above and see how close our theta value is to 0.5

\n \n

After 2,000 flips your opponent has won 747 times but you have won 1,253.

\n \n \n\n\n```python\nmodel_data={\n 'n':2000,\n 'h':747,\n}\n\nfit=sm.sampling(data=model_data,chains=4,iter=1000,seed=12329)\n\nprint(fit)\n\nplt.hist(fit['theta'])\nplt.xlabel('Theta')\nplt.ylabel('Number of Samples')\nplt.show\n```\n\n

So can we conclusively say that our coin was biased?

\n \n

Once again this is a case where we should have questioned our underlying assumptions, i.e. that both opponents had a 50% chance of winning. In future this will be covered in part by prior predictive checks which should give us intuition on how our model behaves. For now let us first generate data in the way described assuming both players have a fair coin

\n\n\n```python\nbob_wins = 0\nalice_wins = 0\n\nfor i in range(2000):\n bob_seq = np.array([])\n alice_seq = np.array([])\n bob_poss_win = 0\n alice_poss_win = 0\n while bob_wins + alice_wins < 2000:\n bob_seq = np.append(bob_seq, np.random.randint(2,size=1))\n alice_seq = np.append(alice_seq, np.random.randint(2,size=1))\n if np.prod((bob_seq.size > 1) and (bob_seq[-2:] == [0,0])):\n bob_poss_win = 1\n if np.prod((alice_seq.size > 1) and (alice_seq[-2:] == [0,1])):\n alice_poss_win = 1\n if (bob_poss_win == 1) and (alice_poss_win == 1):\n bob_seq = np.array([])\n alice_seq = np.array([])\n bob_poss_win = 0\n alice_poss_win = 0\n elif bob_poss_win == 1:\n bob_wins += 1 \n bob_seq = np.array([])\n alice_seq = np.array([])\n bob_poss_win = 0\n elif alice_poss_win == 1:\n alice_wins += 1 \n bob_seq = np.array([])\n alice_seq = np.array([])\n alice_poss_win = 0\n \nprint(bob_wins)\nprint(alice_wins)\n\n```\n\n 752\n 1248\n\n\n

Understanding a problem deeply is important when building a model. Misunderstanding a model can lead to misinterpretation. In this case there is a subtle difference, if you need to flip HT and first flip a H you will not win if on the next coin toss you flip another H - but you are still only one flip away. If you need to flip HH and after one H toss flip a T, you are again two flips away from winning.

\n\n
\n\n## Probabilistic Programming\n*Define probability models & solve automatically*\n\n\n\n\n\n\n```python\nrate_diff = \"\"\"// Difference Between Two Rates\ndata { \n int n1; \n int n2; \n int k1;\n int k2;\n} \nparameters {\n real theta1;\n real theta2;\n} \ntransformed parameters {\n real delta;\n delta <- theta1 - theta2;\n}\nmodel {\n // Prior Distribution for Rate Theta\n theta1 ~ beta(1, 1);\n theta2 ~ beta(1, 1);\n // Observed Counts\n k1 ~ binomial(n1, theta1);\n k2 ~ binomial(n2, theta2);\n}\"\"\"\n\nsm=pystan.StanModel(model_code=rate_diff)\n```\n\n INFO:pystan:COMPILING THE C++ CODE FOR MODEL anon_model_b712d3d58d6ce64ce339b400e820a652 NOW.\n\n\n\n```python\nmodel_data = {\n 'k1': 50,\n 'k2': 62,\n 'n1': 100,\n 'n2': 100,\n}\n\nfit=sm.sampling(data=model_data,chains=4,iter=1000,seed=12329)\n\nplt.hist(fit['delta'])\nplt.show()\n```\n\n\n

So let us now consider a model where there are more than one unknown and interacting parameters. In this case of five sets of eager volunteers distributed an equal number of surveys to randomly selected houses. Unfortunately nobody recorded the number of surveys in each bundle and now we need to estimate the rate of return. Assuming there was no more than 500 surveys in a bundle we can attempt to calculate this. But first let us discuss how we think the posterior will look.

\n
\n \n \n\n\n```python\nplt.plot(np.arange(0, 20), 1./np.arange(0, 20))\n```\n\n\n

# Inferring Return Rate and Numbers of Surveys from Observed Returns\n
\n# Observed Returns\n
\n\nfor (i in 1:m){\n
\n\nk[i] ~ dbin(theta,n)\n
\n\n}\n
\n\n\\# Priors on Rate Theta and Number n\n
\n\ntheta ~ dbeta(1,1)\n
\nn ~ dcat(p[])\n
\nfor (i in 1:nmax){\n
\np[i] <- 1/nmax\n
\n}

\n
\n\n\n\n```python\nsurvey_model = \"\"\"data { \n int nmax;\n int m;\n int k[m];\n}\ntransformed data {\n int nmin; // Minimal possible n\n \n nmin <- max(k);\n}\nparameters {\n real theta;\n}\ntransformed parameters {\n vector[nmax] lp_parts; // Log probability for each n\n\n // First part of the trick for mixture model\n for (n in 1:nmax)\n if (n < nmin)\n lp_parts[n] <- log(1.0 / nmax) + negative_infinity(); // Zero probability\n else\n lp_parts[n] <- log(1.0 / nmax) + binomial_log(k, n, theta); \n}\nmodel {\n // Second part of the trick for mixture model\n increment_log_prob(log_sum_exp(lp_parts));\n}\ngenerated quantities {\n int n;\n simplex[nmax] prob_n;\n \n // Transforming lp_parts to probabilities of each n\n prob_n <- softmax(lp_parts);\n n <- categorical_rng(prob_n);\n}\"\"\"\n\nsm=pystan.StanModel(model_code=survey_model)\n```\n\n INFO:pystan:COMPILING THE C++ CODE FOR MODEL anon_model_78f7cfc1daf3a7ccdfbee88500ff4157 NOW.\n\n\n!!! DISCLAIMER !!!\n
\nStan is not the correct tool to use if you are working with discrete parameters - in this case n. Expect errors and do this using a more suitable tool for real world applications. \n\n
\n\n\n```python\nmodel_data={\n 'k': np.array([16, 18, 22, 25, 27]),\n 'm': 5,\n 'nmax': 500\n}\n\nfit=sm.sampling(data=model_data,chains=4,iter=1000,seed=12329)\n\nprint(fit)\n\n```\n\n WARNING:pystan:Maximum (flat) parameter count (1000) exceeded: skipping diagnostic tests for n_eff and Rhat.\n To run all diagnostics call pystan.check_hmc_diagnostics(fit)\n WARNING:pystan:2 of 2000 iterations ended with a divergence (0.1 %).\n WARNING:pystan:Try running with adapt_delta larger than 0.8 to remove the divergences.\n /opt/conda/lib/python3.7/site-packages/numpy/core/_methods.py:117: RuntimeWarning: invalid value encountered in subtract\n x = asanyarray(arr - arrmean)\n\n\n Inference for Stan model: anon_model_78f7cfc1daf3a7ccdfbee88500ff4157.\n 4 chains, each with iter=1000; warmup=500; thin=1; \n post-warmup draws per chain=500, total post-warmup draws=2000.\n \n mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat\n theta 0.19 6.7e-3 0.14 0.05 0.08 0.15 0.26 0.55 429 1.0\n lp_parts[1] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[2] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[3] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[4] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[5] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[6] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[7] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[8] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[9] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[10] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[11] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[12] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[13] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[14] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[15] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[16] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[17] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[18] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[19] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[20] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[21] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[22] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[23] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[24] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[25] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[26] -inf nan nan -inf -inf -inf -inf -inf nan nan\n lp_parts[27] -170.3 3.79 73.24 -291.3 -232.4 -168.5 -109.9 -44.1 373 1.0\n lp_parts[28] -162.5 3.75 72.36 -282.5 -223.9 -160.3 -102.4 -39.05 372 1.0\n lp_parts[29] -155.8 3.71 71.48 -274.9 -216.4 -153.3 -96.12 -35.15 372 1.0\n lp_parts[30] -149.9 3.66 70.6 -268.1 -209.8 -147.0 -90.55 -31.99 371 1.0\n lp_parts[31] -144.5 3.62 69.73 -261.8 -203.6 -141.2 -85.55 -29.41 371 1.0\n lp_parts[32] -139.6 3.58 68.86 -255.9 -198.0 -136.0 -81.01 -27.62 370 1.0\n lp_parts[33] -135.1 3.54 68.0 -250.5 -192.7 -131.1 -76.86 -26.13 370 1.0\n lp_parts[34] -130.9 3.49 67.14 -245.4 -187.8 -126.6 -73.05 -25.08 369 1.0\n lp_parts[35] -127.0 3.45 66.28 -240.6 -183.2 -122.3 -69.52 -23.99 369 1.0\n lp_parts[36] -123.3 3.41 65.43 -236.0 -178.8 -118.3 -66.25 -23.6 368 1.0\n lp_parts[37] -119.9 3.37 64.59 -231.7 -174.7 -114.5 -63.2 -22.69 368 1.0\n lp_parts[38] -116.7 3.33 63.75 -227.6 -170.7 -111.0 -60.36 -22.16 367 1.0\n lp_parts[39] -113.6 3.28 62.92 -223.6 -166.9 -107.6 -57.7 -21.81 367 1.0\n lp_parts[40] -110.8 3.24 62.09 -219.8 -163.3 -104.4 -55.21 -21.62 366 1.0\n lp_parts[41] -108.1 3.2 61.27 -216.2 -159.9 -101.3 -52.92 -21.56 366 1.0\n lp_parts[42] -105.5 3.16 60.45 -212.7 -156.6 -98.42 -50.8 -21.37 365 1.0\n lp_parts[43] -103.0 3.12 59.65 -209.4 -153.4 -95.63 -48.84 -21.3 365 1.0\n lp_parts[44] -100.7 3.08 58.85 -206.2 -150.4 -92.96 -47.07 -21.19 365 1.0\n lp_parts[45] -98.57 3.04 58.05 -203.0 -147.5 -90.41 -45.4 -21.05 364 1.0\n lp_parts[46] -96.48 3.0 57.27 -200.0 -144.6 -87.97 -43.88 -20.95 364 1.0\n lp_parts[47] -94.48 2.96 56.49 -197.1 -141.9 -85.62 -42.46 -20.91 363 1.0\n lp_parts[48] -92.58 2.92 55.72 -194.3 -139.3 -83.4 -41.04 -20.88 363 1.0\n lp_parts[49] -90.76 2.89 54.96 -191.6 -136.7 -81.29 -39.67 -20.84 363 1.0\n lp_parts[50] -89.03 2.85 54.2 -188.9 -134.3 -79.32 -38.37 -20.77 363 1.0\n lp_parts[51] -87.37 2.81 53.46 -186.4 -131.9 -77.31 -37.44 -20.71 362 1.0\n lp_parts[52] -85.79 2.77 52.73 -183.9 -129.6 -75.45 -36.6 -20.65 362 1.0\n lp_parts[53] -84.27 2.73 52.0 -181.4 -127.3 -73.66 -36.17 -20.62 362 1.0\n lp_parts[54] -82.83 2.7 51.29 -179.1 -125.1 -71.94 -35.14 -20.57 362 1.0\n lp_parts[55] -81.44 2.66 50.59 -176.8 -123.1 -70.35 -34.18 -20.59 361 1.0\n lp_parts[56] -80.11 2.63 49.89 -174.6 -121.0 -68.75 -33.51 -20.56 361 1.0\n lp_parts[57] -78.85 2.59 49.21 -172.4 -119.1 -67.18 -32.85 -20.55 361 1.0\n lp_parts[58] -77.63 2.56 48.55 -170.2 -117.6 -65.94 -32.38 -20.55 361 1.0\n lp_parts[59] -76.47 2.52 47.89 -168.2 -115.7 -64.78 -31.8 -20.54 361 1.01\n lp_parts[60] -75.35 2.49 47.25 -166.1 -114.1 -63.41 -31.5 -20.52 361 1.01\n lp_parts[61] -74.29 2.46 46.63 -164.2 -112.3 -61.97 -30.96 -20.52 361 1.01\n lp_parts[62] -73.27 2.42 46.01 -162.5 -110.8 -60.84 -30.65 -20.53 361 1.01\n lp_parts[63] -72.29 2.39 45.42 -160.6 -109.0 -59.86 -30.15 -20.55 361 1.01\n lp_parts[64] -71.36 2.36 44.83 -158.7 -107.5 -59.2 -29.98 -20.57 361 1.01\n lp_parts[65] -70.46 2.33 44.27 -157.2 -106.0 -58.1 -29.62 -20.55 361 1.01\n lp_parts[66] -69.6 2.3 43.72 -155.7 -104.7 -57.73 -29.31 -20.53 361 1.01\n lp_parts[67] -68.78 2.27 43.19 -153.9 -103.2 -56.63 -28.88 -20.54 362 1.01\n lp_parts[68] -68.0 2.24 42.68 -152.3 -101.7 -55.75 -28.91 -20.52 362 1.01\n lp_parts[69] -67.25 2.21 42.18 -150.6 -100.4 -54.86 -28.67 -20.49 363 1.01\n lp_parts[70] -66.53 2.19 41.71 -149.1 -99.17 -54.42 -28.41 -20.51 363 1.01\n lp_parts[71] -65.85 2.16 41.25 -147.5 -97.94 -53.46 -28.47 -20.5 364 1.01\n lp_parts[72] -65.2 2.14 40.82 -146.0 -96.55 -53.29 -28.36 -20.49 365 1.01\n lp_parts[73] -64.57 2.11 40.41 -144.5 -95.18 -52.82 -28.29 -20.49 366 1.01\n lp_parts[74] -63.98 2.09 40.02 -143.3 -94.38 -52.27 -28.13 -20.49 367 1.01\n lp_parts[75] -63.41 2.07 39.66 -143.0 -93.31 -51.7 -28.0 -20.49 368 1.01\n lp_parts[76] -62.87 2.04 39.31 -141.5 -92.29 -50.98 -27.97 -20.49 370 1.01\n lp_parts[77] -62.36 2.02 38.99 -140.1 -91.91 -50.76 -27.93 -20.49 371 1.01\n lp_parts[78] -61.87 2.0 38.7 -139.2 -90.97 -49.86 -27.83 -20.47 373 1.01\n lp_parts[79] -61.41 1.99 38.43 -138.1 -90.0 -49.65 -27.7 -20.46 374 1.01\n lp_parts[80] -60.97 1.97 38.19 -136.9 -89.07 -49.07 -27.52 -20.46 376 1.01\n lp_parts[81] -60.56 1.95 37.97 -135.7 -88.03 -49.1 -27.44 -20.46 378 1.01\n lp_parts[82] -60.16 1.94 37.78 -134.8 -87.37 -49.07 -27.49 -20.46 380 1.01\n lp_parts[83] -59.79 1.92 37.62 -133.7 -86.91 -49.03 -27.3 -20.48 382 1.01\n lp_parts[84] -59.44 1.91 37.49 -132.4 -85.94 -48.53 -27.11 -20.46 385 1.01\n lp_parts[85] -59.11 1.9 37.38 -131.1 -85.16 -48.08 -27.13 -20.46 387 1.01\n lp_parts[86] -58.8 1.89 37.3 -130.3 -84.14 -47.72 -26.94 -20.46 390 1.01\n lp_parts[87] -58.51 1.88 37.25 -129.3 -83.41 -47.13 -26.94 -20.46 393 1.01\n lp_parts[88] -58.24 1.87 37.23 -129.7 -82.52 -46.89 -26.67 -20.45 395 1.01\n lp_parts[89] -57.99 1.87 37.24 -128.6 -82.42 -47.01 -26.8 -20.46 398 1.01\n lp_parts[90] -57.76 1.86 37.27 -128.7 -82.41 -46.67 -26.72 -20.47 401 1.01\n lp_parts[91] -57.54 1.86 37.34 -129.9 -82.05 -46.67 -26.42 -20.48 404 1.01\n lp_parts[92] -57.34 1.86 37.43 -130.5 -81.14 -46.3 -26.39 -20.46 407 1.01\n lp_parts[93] -57.16 1.85 37.55 -130.9 -80.64 -46.11 -26.57 -20.45 410 1.01\n lp_parts[94] -56.99 1.85 37.7 -133.8 -80.36 -46.03 -26.51 -20.46 413 1.01\n lp_parts[95] -56.84 1.86 37.87 -136.7 -79.59 -45.79 -26.41 -20.47 416 1.01\n lp_parts[96] -56.7 1.86 38.07 -138.3 -78.93 -45.78 -26.42 -20.47 419 1.01\n lp_parts[97] -56.58 1.87 38.3 -137.2 -78.75 -45.77 -26.45 -20.45 422 1.01\n lp_parts[98] -56.48 1.87 38.56 -136.1 -78.36 -45.5 -26.48 -20.47 425 1.01\n lp_parts[99] -56.39 1.88 38.84 -135.1 -78.23 -45.28 -26.27 -20.47 427 1.01\n lp_parts[100] -56.31 1.89 39.14 -134.9 -77.88 -45.14 -26.29 -20.47 430 1.01\n lp_parts[101] -56.25 1.9 39.47 -136.6 -77.86 -45.19 -26.21 -20.45 433 1.01\n lp_parts[102] -56.2 1.91 39.83 -139.3 -77.35 -45.09 -26.08 -20.45 436 1.01\n lp_parts[103] -56.16 1.92 40.2 -142.1 -76.9 -45.3 -26.08 -20.45 438 1.01\n lp_parts[104] -56.14 1.93 40.6 -144.8 -77.38 -45.2 -26.08 -20.45 441 1.01\n lp_parts[105] -56.13 1.95 41.03 -147.6 -76.63 -44.94 -26.15 -20.45 443 1.01\n lp_parts[106] -56.13 1.96 41.47 -150.4 -76.56 -44.48 -26.29 -20.45 446 1.01\n lp_parts[107] -56.14 1.98 41.93 -153.2 -75.81 -44.24 -26.12 -20.45 448 1.01\n lp_parts[108] -56.17 2.0 42.42 -156.0 -75.52 -44.26 -26.01 -20.45 450 1.01\n lp_parts[109] -56.21 2.02 42.92 -158.9 -75.28 -44.0 -25.87 -20.45 451 1.01\n lp_parts[110] -56.26 2.04 43.44 -161.7 -75.69 -43.74 -25.94 -20.44 453 1.01\n lp_parts[111] -56.32 2.06 43.98 -164.6 -75.17 -43.55 -25.89 -20.44 454 1.01\n lp_parts[112] -56.39 2.09 44.54 -167.4 -75.05 -43.67 -25.84 -20.44 456 1.01\n lp_parts[113] -56.47 2.11 45.11 -170.3 -74.64 -43.73 -25.92 -20.44 457 1.01\n lp_parts[114] -56.56 2.14 45.7 -173.2 -74.64 -43.56 -25.96 -20.44 458 1.01\n lp_parts[115] -56.66 2.16 46.3 -176.1 -74.43 -43.22 -26.08 -20.45 459 1.01\n lp_parts[116] -56.77 2.19 46.92 -179.0 -74.32 -43.1 -25.93 -20.45 460 1.01\n lp_parts[117] -56.89 2.22 47.56 -181.9 -73.87 -43.1 -26.08 -20.45 460 1.01\n lp_parts[118] -57.03 2.25 48.2 -184.8 -73.91 -43.14 -26.04 -20.45 461 1.01\n lp_parts[119] -57.17 2.28 48.86 -187.8 -74.02 -43.16 -26.03 -20.45 461 1.01\n lp_parts[120] -57.32 2.31 49.54 -190.7 -73.7 -43.07 -26.05 -20.46 462 1.01\n lp_parts[121] -57.48 2.34 50.22 -193.6 -73.26 -43.33 -26.02 -20.45 462 1.01\n lp_parts[122] -57.65 2.37 50.92 -196.6 -73.13 -43.19 -25.98 -20.45 463 1.01\n lp_parts[123] -57.82 2.4 51.63 -199.6 -73.09 -43.3 -25.88 -20.45 463 1.01\n lp_parts[124] -58.01 2.43 52.35 -202.6 -73.1 -43.22 -25.95 -20.46 463 1.01\n lp_parts[125] -58.2 2.47 53.08 -205.5 -72.88 -43.05 -26.11 -20.46 463 1.01\n lp_parts[126] -58.41 2.5 53.81 -208.5 -73.02 -42.9 -26.22 -20.46 464 1.01\n lp_parts[127] -58.62 2.53 54.56 -211.5 -72.65 -42.83 -26.37 -20.46 464 1.01\n lp_parts[128] -58.84 2.57 55.32 -214.6 -72.46 -42.77 -26.34 -20.46 464 1.01\n lp_parts[129] -59.06 2.6 56.08 -217.6 -72.2 -42.64 -26.11 -20.47 464 1.01\n lp_parts[130] -59.3 2.64 56.86 -220.6 -72.3 -42.42 -26.04 -20.48 464 1.01\n lp_parts[131] -59.54 2.68 57.64 -223.6 -72.39 -42.22 -26.01 -20.47 464 1.01\n lp_parts[132] -59.79 2.71 58.43 -226.7 -71.95 -42.07 -26.09 -20.45 464 1.01\n lp_parts[133] -60.05 2.75 59.23 -229.7 -71.53 -42.17 -26.03 -20.46 464 1.01\n lp_parts[134] -60.31 2.79 60.03 -232.8 -71.38 -42.5 -25.97 -20.46 464 1.01\n lp_parts[135] -60.59 2.83 60.84 -235.9 -71.09 -42.76 -25.9 -20.46 464 1.01\n lp_parts[136] -60.86 2.86 61.66 -238.9 -71.15 -42.9 -25.89 -20.45 464 1.01\n lp_parts[137] -61.15 2.9 62.48 -242.0 -71.14 -42.63 -25.92 -20.45 464 1.01\n lp_parts[138] -61.44 2.94 63.31 -245.1 -70.82 -42.63 -25.92 -20.44 463 1.01\n lp_parts[139] -61.74 2.98 64.15 -248.2 -70.33 -42.66 -25.93 -20.45 463 1.01\n lp_parts[140] -62.05 3.02 64.99 -251.3 -70.33 -42.58 -25.92 -20.46 463 1.01\n lp_parts[141] -62.36 3.06 65.84 -254.4 -70.09 -42.61 -25.86 -20.46 463 1.01\n lp_parts[142] -62.68 3.1 66.69 -257.5 -69.69 -42.65 -25.83 -20.46 463 1.01\n lp_parts[143] -63.01 3.14 67.55 -260.6 -69.49 -42.48 -25.78 -20.45 463 1.01\n lp_parts[144] -63.34 3.18 68.41 -263.7 -69.4 -42.32 -25.89 -20.45 462 1.01\n lp_parts[145] -63.68 3.22 69.27 -266.9 -69.04 -42.46 -25.84 -20.46 462 1.01\n lp_parts[146] -64.02 3.26 70.14 -270.0 -69.3 -42.6 -25.84 -20.45 462 1.01\n lp_parts[147] -64.37 3.3 71.02 -273.2 -69.47 -42.73 -25.84 -20.45 462 1.01\n lp_parts[148] -64.72 3.35 71.9 -276.3 -69.08 -42.7 -25.91 -20.45 462 1.01\n lp_parts[149] -65.09 3.39 72.78 -279.5 -68.93 -42.48 -26.06 -20.45 461 1.01\n lp_parts[150] -65.45 3.43 73.67 -282.6 -68.59 -42.3 -25.93 -20.45 461 1.01\n lp_parts[151] -65.83 3.47 74.56 -285.8 -68.62 -42.45 -26.12 -20.46 461 1.01\n lp_parts[152] -66.2 3.51 75.45 -288.9 -68.93 -42.51 -26.07 -20.47 461 1.01\n lp_parts[153] -66.59 3.56 76.35 -292.1 -68.75 -42.72 -26.01 -20.47 461 1.01\n lp_parts[154] -66.98 3.6 77.25 -295.3 -68.7 -42.71 -26.0 -20.47 460 1.01\n lp_parts[155] -67.37 3.64 78.15 -298.5 -68.53 -42.8 -25.96 -20.48 460 1.01\n lp_parts[156] -67.77 3.69 79.06 -301.7 -68.56 -42.6 -25.85 -20.49 460 1.01\n lp_parts[157] -68.17 3.73 79.97 -304.9 -68.31 -42.49 -25.9 -20.48 460 1.01\n lp_parts[158] -68.58 3.77 80.88 -308.1 -68.6 -42.43 -25.91 -20.48 459 1.01\n lp_parts[159] -69.0 3.82 81.8 -311.3 -68.45 -42.59 -25.83 -20.47 459 1.01\n lp_parts[160] -69.42 3.86 82.71 -314.5 -68.9 -42.65 -25.84 -20.48 459 1.01\n lp_parts[161] -69.84 3.91 83.64 -317.7 -68.86 -42.76 -25.91 -20.48 459 1.01\n lp_parts[162] -70.27 3.95 84.56 -320.9 -68.7 -42.86 -25.91 -20.48 458 1.01\n lp_parts[163] -70.71 3.99 85.48 -324.2 -68.69 -43.01 -25.94 -20.47 458 1.01\n lp_parts[164] -71.15 4.04 86.41 -327.4 -69.13 -43.01 -25.89 -20.48 458 1.01\n lp_parts[165] -71.59 4.08 87.34 -330.6 -69.09 -43.03 -25.95 -20.47 458 1.01\n lp_parts[166] -72.04 4.13 88.28 -333.9 -69.45 -43.09 -25.86 -20.47 458 1.01\n lp_parts[167] -72.49 4.17 89.21 -337.1 -69.38 -42.77 -25.92 -20.48 457 1.01\n lp_parts[168] -72.95 4.22 90.15 -340.4 -70.1 -42.94 -25.93 -20.49 457 1.01\n lp_parts[169] -73.41 4.26 91.08 -343.6 -70.25 -42.92 -25.82 -20.5 457 1.01\n lp_parts[170] -73.87 4.31 92.03 -346.9 -70.78 -42.9 -25.86 -20.5 457 1.01\n lp_parts[171] -74.34 4.35 92.97 -350.1 -71.56 -43.03 -25.82 -20.49 456 1.01\n lp_parts[172] -74.82 4.4 93.91 -353.4 -72.35 -42.96 -25.89 -20.49 456 1.01\n lp_parts[173] -75.3 4.44 94.86 -356.7 -73.24 -42.83 -26.03 -20.48 456 1.01\n lp_parts[174] -75.78 4.49 95.81 -360.0 -74.02 -42.73 -26.0 -20.49 456 1.01\n lp_parts[175] -76.26 4.53 96.76 -363.2 -74.91 -42.66 -26.03 -20.49 456 1.01\n lp_parts[176] -76.76 4.58 97.71 -366.5 -75.74 -42.57 -25.99 -20.48 455 1.01\n lp_parts[177] -77.25 4.62 98.66 -369.8 -76.63 -42.43 -25.96 -20.48 455 1.01\n lp_parts[178] -77.75 4.67 99.61 -373.1 -77.54 -42.64 -25.93 -20.46 455 1.01\n lp_parts[179] -78.25 4.72 100.57 -376.4 -78.46 -42.63 -25.93 -20.45 455 1.01\n lp_parts[180] -78.76 4.76 101.53 -379.7 -79.22 -42.71 -25.94 -20.45 454 1.01\n lp_parts[181] -79.27 4.81 102.49 -383.0 -78.76 -42.64 -25.97 -20.45 454 1.01\n lp_parts[182] -79.78 4.85 103.45 -386.3 -78.31 -42.49 -25.97 -20.46 454 1.01\n lp_parts[183] -80.3 4.9 104.41 -389.6 -78.6 -42.55 -25.91 -20.46 454 1.0\n lp_parts[184] -80.82 4.95 105.37 -392.9 -79.5 -42.61 -25.99 -20.46 454 1.0\n lp_parts[185] -81.34 4.99 106.33 -396.2 -80.4 -42.51 -25.94 -20.46 453 1.0\n lp_parts[186] -81.87 5.04 107.3 -399.6 -81.31 -42.47 -25.82 -20.45 453 1.0\n lp_parts[187] -82.4 5.09 108.26 -402.9 -82.22 -42.34 -25.77 -20.45 453 1.0\n lp_parts[188] -82.94 5.13 109.23 -406.2 -83.13 -42.21 -25.76 -20.45 453 1.0\n lp_parts[189] -83.48 5.18 110.2 -409.6 -84.04 -42.31 -25.72 -20.46 453 1.0\n lp_parts[190] -84.02 5.23 111.17 -412.9 -84.96 -42.5 -25.87 -20.46 453 1.0\n lp_parts[191] -84.57 5.27 112.14 -416.2 -85.89 -42.55 -25.89 -20.46 452 1.0\n lp_parts[192] -85.11 5.32 113.11 -419.6 -86.81 -42.65 -26.02 -20.46 452 1.0\n lp_parts[193] -85.67 5.37 114.08 -422.9 -87.74 -42.75 -25.94 -20.46 452 1.0\n lp_parts[194] -86.22 5.41 115.06 -426.3 -88.67 -42.52 -25.95 -20.46 452 1.0\n lp_parts[195] -86.78 5.46 116.03 -429.6 -89.61 -42.44 -26.03 -20.46 452 1.0\n lp_parts[196] -87.34 5.51 117.01 -433.0 -90.55 -42.35 -26.09 -20.46 451 1.0\n lp_parts[197] -87.91 5.55 117.99 -436.3 -91.49 -42.35 -26.18 -20.46 451 1.0\n lp_parts[198] -88.47 5.6 118.96 -439.7 -92.43 -42.45 -26.21 -20.47 451 1.0\n lp_parts[199] -89.05 5.65 119.94 -443.0 -93.38 -42.75 -26.12 -20.47 451 1.0\n lp_parts[200] -89.62 5.7 120.92 -446.4 -94.33 -42.7 -26.13 -20.46 451 1.0\n lp_parts[201] -90.2 5.74 121.9 -449.8 -95.29 -42.45 -26.19 -20.47 451 1.0\n lp_parts[202] -90.78 5.79 122.88 -453.2 -96.25 -42.42 -26.15 -20.48 450 1.0\n lp_parts[203] -91.36 5.84 123.86 -456.5 -97.21 -42.34 -26.13 -20.47 450 1.0\n lp_parts[204] -91.95 5.88 124.84 -459.9 -98.17 -42.42 -26.24 -20.47 450 1.0\n lp_parts[205] -92.54 5.93 125.83 -463.3 -99.13 -42.51 -26.16 -20.47 450 1.0\n lp_parts[206] -93.13 5.98 126.81 -466.7 -100.1 -42.6 -26.08 -20.47 450 1.0\n lp_parts[207] -93.72 6.03 127.8 -470.1 -101.0 -42.52 -26.09 -20.47 450 1.0\n lp_parts[208] -94.32 6.07 128.78 -473.5 -102.0 -42.26 -26.18 -20.46 450 1.0\n lp_parts[209] -94.92 6.12 129.77 -476.8 -103.0 -42.29 -26.2 -20.47 449 1.0\n lp_parts[210] -95.52 6.17 130.75 -480.2 -104.0 -42.14 -26.09 -20.47 449 1.0\n lp_parts[211] -96.13 6.22 131.74 -483.6 -104.9 -42.16 -26.11 -20.46 449 1.0\n lp_parts[212] -96.74 6.26 132.73 -487.0 -105.9 -41.99 -26.02 -20.46 449 1.0\n lp_parts[213] -97.35 6.31 133.72 -490.4 -106.9 -42.15 -25.92 -20.45 449 1.0\n lp_parts[214] -97.96 6.36 134.71 -493.9 -107.9 -42.03 -25.97 -20.45 449 1.0\n lp_parts[215] -98.58 6.41 135.7 -497.3 -108.9 -41.86 -25.96 -20.46 449 1.0\n lp_parts[216] -99.2 6.46 136.69 -500.7 -109.9 -41.73 -25.99 -20.46 448 1.0\n lp_parts[217] -99.82 6.5 137.68 -504.1 -110.9 -41.72 -26.11 -20.46 448 1.0\n lp_parts[218] -100.4 6.55 138.67 -507.5 -111.9 -41.8 -26.14 -20.47 448 1.0\n lp_parts[219] -101.0 6.6 139.66 -510.9 -112.9 -41.93 -26.11 -20.48 448 1.0\n lp_parts[220] -101.7 6.65 140.65 -514.3 -113.9 -41.77 -26.06 -20.48 448 1.0\n lp_parts[221] -102.3 6.69 141.65 -517.8 -114.9 -41.74 -26.18 -20.47 448 1.0\n lp_parts[222] -102.9 6.74 142.64 -521.2 -115.9 -41.73 -26.13 -20.46 448 1.0\n lp_parts[223] -103.6 6.79 143.64 -524.6 -116.9 -41.68 -26.08 -20.47 447 1.0\n lp_parts[224] -104.2 6.84 144.63 -528.1 -118.0 -41.62 -26.11 -20.47 447 1.0\n lp_parts[225] -104.8 6.89 145.63 -531.5 -119.0 -41.59 -26.19 -20.48 447 1.0\n lp_parts[226] -105.5 6.93 146.62 -534.9 -120.0 -41.75 -26.16 -20.49 447 1.0\n lp_parts[227] -106.1 6.98 147.62 -538.4 -121.0 -41.68 -26.1 -20.48 447 1.0\n lp_parts[228] -106.8 7.03 148.61 -541.8 -122.0 -41.64 -26.15 -20.47 447 1.0\n lp_parts[229] -107.4 7.08 149.61 -545.3 -123.1 -41.53 -26.12 -20.47 447 1.0\n lp_parts[230] -108.1 7.13 150.61 -548.7 -124.1 -41.6 -26.04 -20.48 447 1.0\n lp_parts[231] -108.7 7.17 151.61 -552.1 -125.1 -41.78 -26.03 -20.49 446 1.0\n lp_parts[232] -109.4 7.22 152.6 -555.6 -126.2 -41.72 -25.97 -20.49 446 1.0\n lp_parts[233] -110.1 7.27 153.6 -559.1 -127.2 -41.58 -25.98 -20.49 446 1.0\n lp_parts[234] -110.7 7.32 154.6 -562.5 -128.2 -41.82 -25.96 -20.5 446 1.0\n lp_parts[235] -111.4 7.37 155.6 -566.0 -129.3 -41.88 -25.97 -20.5 446 1.0\n lp_parts[236] -112.0 7.42 156.6 -569.4 -130.3 -42.15 -25.94 -20.5 446 1.0\n lp_parts[237] -112.7 7.46 157.6 -572.9 -131.4 -42.22 -26.09 -20.51 446 1.0\n lp_parts[238] -113.4 7.51 158.6 -576.3 -132.4 -42.31 -26.17 -20.5 446 1.0\n lp_parts[239] -114.1 7.56 159.6 -579.8 -133.5 -42.23 -26.19 -20.49 446 1.0\n lp_parts[240] -114.7 7.61 160.6 -583.3 -134.5 -42.35 -26.2 -20.49 445 1.0\n lp_parts[241] -115.4 7.66 161.61 -586.7 -135.6 -42.26 -26.25 -20.49 445 1.0\n lp_parts[242] -116.1 7.71 162.61 -590.2 -136.6 -42.6 -26.17 -20.49 445 1.0\n lp_parts[243] -116.8 7.75 163.61 -593.7 -137.7 -42.71 -26.16 -20.49 445 1.0\n lp_parts[244] -117.4 7.8 164.61 -597.2 -138.7 -42.91 -26.19 -20.48 445 1.0\n lp_parts[245] -118.1 7.85 165.62 -600.6 -139.8 -42.85 -26.3 -20.48 445 1.0\n lp_parts[246] -118.8 7.9 166.62 -604.1 -140.9 -42.94 -26.25 -20.47 445 1.0\n lp_parts[247] -119.5 7.95 167.62 -607.6 -141.9 -43.13 -26.22 -20.47 445 1.0\n lp_parts[248] -120.2 8.0 168.63 -611.1 -143.0 -43.13 -26.24 -20.46 445 1.0\n lp_parts[249] -120.9 8.04 169.63 -614.6 -144.1 -43.14 -26.25 -20.46 445 1.0\n lp_parts[250] -121.6 8.09 170.64 -618.1 -145.1 -43.28 -26.16 -20.46 445 1.0\n lp_parts[251] -122.3 8.14 171.64 -621.5 -146.2 -43.46 -26.17 -20.45 444 1.0\n lp_parts[252] -123.0 8.19 172.65 -625.0 -147.3 -43.78 -26.06 -20.45 444 1.0\n lp_parts[253] -123.7 8.24 173.65 -628.5 -148.4 -44.14 -26.15 -20.46 444 1.0\n lp_parts[254] -124.4 8.29 174.66 -632.0 -149.5 -44.34 -26.22 -20.47 444 1.0\n lp_parts[255] -125.1 8.34 175.66 -635.5 -150.5 -44.71 -26.24 -20.47 444 1.0\n lp_parts[256] -125.8 8.38 176.67 -639.0 -151.6 -45.07 -26.25 -20.47 444 1.0\n lp_parts[257] -126.5 8.43 177.68 -642.5 -152.7 -45.41 -26.19 -20.48 444 1.0\n lp_parts[258] -127.2 8.48 178.68 -646.0 -153.8 -45.45 -26.16 -20.48 444 1.0\n lp_parts[259] -127.9 8.53 179.69 -649.5 -154.9 -45.48 -26.19 -20.48 444 1.0\n lp_parts[260] -128.6 8.58 180.7 -653.0 -156.0 -45.8 -26.25 -20.49 444 1.0\n lp_parts[261] -129.3 8.63 181.71 -656.5 -157.1 -46.17 -26.24 -20.49 444 1.0\n lp_parts[262] -130.1 8.68 182.71 -660.0 -158.1 -46.54 -26.24 -20.49 443 1.0\n lp_parts[263] -130.8 8.73 183.72 -663.6 -159.2 -46.83 -26.3 -20.5 443 1.0\n lp_parts[264] -131.5 8.77 184.73 -667.1 -160.3 -47.16 -26.21 -20.5 443 1.0\n lp_parts[265] -132.2 8.82 185.74 -670.6 -161.4 -47.53 -26.11 -20.51 443 1.0\n lp_parts[266] -132.9 8.87 186.75 -674.1 -162.5 -47.91 -26.09 -20.51 443 1.0\n lp_parts[267] -133.7 8.92 187.76 -677.6 -163.6 -48.29 -26.04 -20.52 443 1.0\n lp_parts[268] -134.4 8.97 188.77 -681.1 -164.7 -48.67 -26.12 -20.52 443 1.0\n lp_parts[269] -135.1 9.02 189.78 -684.7 -165.8 -49.05 -26.17 -20.52 443 1.0\n lp_parts[270] -135.9 9.07 190.79 -688.2 -166.9 -49.41 -26.21 -20.51 443 1.0\n lp_parts[271] -136.6 9.12 191.8 -691.7 -168.1 -49.18 -26.24 -20.5 443 1.0\n lp_parts[272] -137.3 9.16 192.81 -695.2 -169.2 -49.52 -26.18 -20.5 443 1.0\n lp_parts[273] -138.1 9.21 193.82 -698.8 -170.3 -49.91 -26.18 -20.51 443 1.0\n lp_parts[274] -138.8 9.26 194.83 -702.3 -171.4 -50.29 -26.12 -20.51 442 1.0\n lp_parts[275] -139.5 9.31 195.84 -705.8 -172.5 -50.68 -26.08 -20.49 442 1.0\n lp_parts[276] -140.3 9.36 196.85 -709.3 -173.6 -51.07 -26.17 -20.49 442 1.0\n lp_parts[277] -141.0 9.41 197.86 -712.9 -174.7 -51.46 -26.14 -20.48 442 1.0\n lp_parts[278] -141.7 9.46 198.87 -716.4 -175.9 -51.85 -26.04 -20.48 442 1.0\n lp_parts[279] -142.5 9.51 199.88 -719.9 -177.0 -52.24 -25.97 -20.49 442 1.0\n lp_parts[280] -143.2 9.55 200.89 -723.5 -178.1 -52.63 -25.97 -20.49 442 1.0\n lp_parts[281] -144.0 9.6 201.91 -727.0 -179.2 -53.03 -25.94 -20.49 442 1.0\n lp_parts[282] -144.7 9.65 202.92 -730.6 -180.3 -53.43 -25.95 -20.49 442 1.0\n lp_parts[283] -145.5 9.7 203.93 -734.1 -181.5 -53.82 -25.98 -20.49 442 1.0\n lp_parts[284] -146.2 9.75 204.94 -737.7 -182.6 -54.22 -26.13 -20.48 442 1.0\n lp_parts[285] -147.0 9.8 205.95 -741.2 -183.7 -54.62 -26.19 -20.48 442 1.0\n lp_parts[286] -147.7 9.85 206.97 -744.7 -184.9 -55.03 -26.15 -20.47 442 1.0\n lp_parts[287] -148.5 9.9 207.98 -748.3 -186.0 -55.43 -26.18 -20.48 442 1.0\n lp_parts[288] -149.2 9.95 208.99 -751.8 -187.1 -55.84 -26.16 -20.48 442 1.0\n lp_parts[289] -150.0 10.0 210.01 -755.4 -188.3 -56.24 -26.08 -20.48 441 1.0\n lp_parts[290] -150.8 10.04 211.02 -758.9 -189.4 -56.65 -26.04 -20.49 441 1.0\n lp_parts[291] -151.5 10.09 212.03 -762.5 -190.5 -57.06 -26.1 -20.5 441 1.0\n lp_parts[292] -152.3 10.14 213.05 -766.1 -191.7 -57.47 -26.13 -20.5 441 1.0\n lp_parts[293] -153.0 10.19 214.06 -769.6 -192.8 -57.89 -26.24 -20.49 441 1.0\n lp_parts[294] -153.8 10.24 215.07 -773.2 -194.0 -58.3 -26.27 -20.49 441 1.0\n lp_parts[295] -154.6 10.29 216.09 -776.7 -195.1 -58.72 -26.18 -20.49 441 1.0\n lp_parts[296] -155.3 10.34 217.1 -780.3 -196.2 -59.13 -26.08 -20.51 441 1.0\n lp_parts[297] -156.1 10.39 218.12 -783.8 -197.4 -59.55 -26.0 -20.5 441 1.0\n lp_parts[298] -156.9 10.44 219.13 -787.4 -198.5 -59.97 -26.03 -20.5 441 1.0\n lp_parts[299] -157.7 10.49 220.15 -791.0 -199.7 -60.39 -26.0 -20.49 441 1.0\n lp_parts[300] -158.4 10.53 221.16 -794.5 -200.8 -60.81 -25.97 -20.49 441 1.0\n lp_parts[301] -159.2 10.58 222.18 -798.1 -202.0 -61.24 -25.95 -20.49 441 1.0\n lp_parts[302] -160.0 10.63 223.19 -801.7 -203.1 -61.66 -25.9 -20.5 441 1.0\n lp_parts[303] -160.8 10.68 224.21 -805.2 -204.3 -62.09 -25.93 -20.51 441 1.0\n lp_parts[304] -161.5 10.73 225.22 -808.8 -205.4 -62.51 -25.93 -20.51 441 1.0\n lp_parts[305] -162.3 10.78 226.24 -812.4 -206.6 -62.94 -25.95 -20.52 440 1.0\n lp_parts[306] -163.1 10.83 227.25 -816.0 -207.8 -63.37 -25.85 -20.52 440 1.0\n lp_parts[307] -163.9 10.88 228.27 -819.5 -208.9 -63.8 -25.81 -20.51 440 1.0\n lp_parts[308] -164.7 10.93 229.28 -823.1 -210.1 -64.23 -25.8 -20.51 440 1.0\n lp_parts[309] -165.4 10.98 230.3 -826.7 -211.2 -64.67 -25.84 -20.51 440 1.0\n lp_parts[310] -166.2 11.02 231.32 -830.3 -212.4 -65.1 -25.88 -20.51 440 1.0\n lp_parts[311] -167.0 11.07 232.33 -833.8 -213.6 -65.54 -25.9 -20.51 440 1.0\n lp_parts[312] -167.8 11.12 233.35 -837.4 -214.7 -65.97 -25.92 -20.5 440 1.0\n lp_parts[313] -168.6 11.17 234.37 -841.0 -215.9 -66.41 -25.86 -20.5 440 1.0\n lp_parts[314] -169.4 11.22 235.38 -844.6 -217.1 -66.85 -25.83 -20.5 440 1.0\n lp_parts[315] -170.2 11.27 236.4 -848.2 -218.2 -67.29 -25.78 -20.5 440 1.0\n lp_parts[316] -171.0 11.32 237.42 -851.8 -219.4 -67.73 -25.75 -20.51 440 1.0\n lp_parts[317] -171.8 11.37 238.43 -855.3 -220.6 -68.17 -25.67 -20.52 440 1.0\n lp_parts[318] -172.6 11.42 239.45 -858.9 -221.7 -68.62 -25.63 -20.53 440 1.0\n lp_parts[319] -173.4 11.47 240.47 -862.5 -222.9 -69.06 -25.62 -20.52 440 1.0\n lp_parts[320] -174.1 11.52 241.48 -866.1 -224.1 -69.51 -25.6 -20.52 440 1.0\n lp_parts[321] -174.9 11.57 242.5 -869.7 -225.3 -69.96 -25.57 -20.53 440 1.0\n lp_parts[322] -175.7 11.61 243.52 -873.3 -226.4 -70.41 -25.61 -20.52 440 1.0\n lp_parts[323] -176.5 11.66 244.54 -876.9 -227.6 -70.85 -25.7 -20.52 440 1.0\n lp_parts[324] -177.3 11.71 245.55 -880.5 -228.8 -71.31 -25.68 -20.52 439 1.0\n lp_parts[325] -178.2 11.76 246.57 -884.1 -230.0 -71.76 -25.67 -20.51 439 1.0\n lp_parts[326] -179.0 11.81 247.59 -887.7 -231.2 -72.21 -25.63 -20.51 439 1.0\n lp_parts[327] -179.8 11.86 248.61 -891.3 -232.3 -72.66 -25.66 -20.51 439 1.0\n lp_parts[328] -180.6 11.91 249.63 -894.9 -233.5 -73.12 -25.69 -20.51 439 1.0\n lp_parts[329] -181.4 11.96 250.64 -898.5 -234.7 -73.57 -25.75 -20.52 439 1.0\n lp_parts[330] -182.2 12.01 251.66 -902.1 -235.9 -74.03 -25.74 -20.52 439 1.0\n lp_parts[331] -183.0 12.06 252.68 -905.7 -237.1 -74.49 -25.71 -20.51 439 1.0\n lp_parts[332] -183.8 12.11 253.7 -909.3 -238.3 -74.95 -25.67 -20.5 439 1.0\n lp_parts[333] -184.6 12.16 254.72 -912.9 -239.5 -75.41 -25.7 -20.5 439 1.0\n lp_parts[334] -185.4 12.21 255.74 -916.5 -240.6 -75.87 -25.74 -20.49 439 1.0\n lp_parts[335] -186.2 12.25 256.75 -920.1 -241.8 -76.33 -25.74 -20.5 439 1.0\n lp_parts[336] -187.1 12.3 257.77 -923.7 -243.0 -76.8 -25.84 -20.5 439 1.0\n lp_parts[337] -187.9 12.35 258.79 -927.3 -244.2 -77.26 -25.95 -20.49 439 1.0\n lp_parts[338] -188.7 12.4 259.81 -930.9 -245.4 -77.73 -25.93 -20.48 439 1.0\n lp_parts[339] -189.5 12.45 260.83 -934.5 -246.6 -78.19 -25.93 -20.48 439 1.0\n lp_parts[340] -190.3 12.5 261.85 -938.1 -247.8 -78.66 -25.96 -20.47 439 1.0\n lp_parts[341] -191.1 12.55 262.87 -941.7 -249.0 -79.13 -26.01 -20.47 439 1.0\n lp_parts[342] -192.0 12.6 263.89 -945.4 -250.2 -79.6 -26.02 -20.46 439 1.0\n lp_parts[343] -192.8 12.65 264.91 -949.0 -251.4 -80.07 -26.12 -20.47 439 1.0\n lp_parts[344] -193.6 12.7 265.93 -952.6 -252.6 -80.54 -26.22 -20.48 439 1.0\n lp_parts[345] -194.4 12.75 266.95 -956.2 -253.8 -81.01 -26.34 -20.48 439 1.0\n lp_parts[346] -195.3 12.8 267.97 -959.8 -255.0 -81.49 -26.35 -20.48 439 1.0\n lp_parts[347] -196.1 12.85 268.99 -963.4 -256.2 -81.96 -26.43 -20.48 438 1.0\n lp_parts[348] -196.9 12.89 270.0 -967.1 -257.4 -82.44 -26.5 -20.48 438 1.0\n lp_parts[349] -197.7 12.94 271.02 -970.7 -258.6 -82.91 -26.5 -20.48 438 1.0\n lp_parts[350] -198.6 12.99 272.04 -974.3 -259.8 -83.39 -26.52 -20.47 438 1.0\n lp_parts[351] -199.4 13.04 273.06 -977.9 -261.0 -83.87 -26.63 -20.47 438 1.0\n lp_parts[352] -200.2 13.09 274.08 -981.5 -262.2 -84.35 -26.67 -20.47 438 1.0\n lp_parts[353] -201.0 13.14 275.1 -985.2 -263.4 -84.83 -26.75 -20.47 438 1.0\n lp_parts[354] -201.9 13.19 276.12 -988.8 -264.6 -85.31 -26.68 -20.48 438 1.0\n lp_parts[355] -202.7 13.24 277.14 -992.4 -265.8 -85.79 -26.58 -20.48 438 1.0\n lp_parts[356] -203.5 13.29 278.17 -996.0 -267.1 -86.27 -26.56 -20.48 438 1.0\n lp_parts[357] -204.4 13.34 279.19 -999.7 -268.3 -86.75 -26.56 -20.48 438 1.0\n lp_parts[358] -205.2 13.39 280.21 -1003 -269.5 -87.24 -26.6 -20.48 438 1.0\n lp_parts[359] -206.1 13.44 281.23 -1006 -270.7 -87.72 -26.69 -20.48 438 1.0\n lp_parts[360] -206.9 13.49 282.25 -1010 -271.9 -88.21 -26.73 -20.48 438 1.0\n lp_parts[361] -207.7 13.54 283.27 -1014 -273.1 -88.7 -26.85 -20.48 438 1.0\n lp_parts[362] -208.6 13.59 284.29 -1017 -274.3 -89.19 -26.97 -20.47 438 1.0\n lp_parts[363] -209.4 13.63 285.31 -1021 -275.6 -89.67 -27.04 -20.47 438 1.0\n lp_parts[364] -210.2 13.68 286.33 -1025 -276.8 -90.16 -26.95 -20.47 438 1.0\n lp_parts[365] -211.1 13.73 287.35 -1028 -278.0 -90.66 -27.0 -20.47 438 1.0\n lp_parts[366] -211.9 13.78 288.37 -1032 -279.2 -91.15 -27.12 -20.47 438 1.0\n lp_parts[367] -212.8 13.83 289.39 -1036 -280.4 -91.64 -27.24 -20.47 438 1.0\n lp_parts[368] -213.6 13.88 290.41 -1039 -281.7 -92.13 -27.36 -20.48 438 1.0\n lp_parts[369] -214.5 13.93 291.44 -1043 -282.9 -92.63 -27.49 -20.48 438 1.0\n lp_parts[370] -215.3 13.98 292.46 -1046 -284.1 -93.12 -27.61 -20.48 438 1.0\n lp_parts[371] -216.2 14.03 293.48 -1050 -285.3 -93.62 -27.66 -20.48 438 1.0\n lp_parts[372] -217.0 14.08 294.5 -1054 -286.5 -94.11 -27.73 -20.49 438 1.0\n lp_parts[373] -217.9 14.13 295.52 -1057 -287.8 -94.61 -27.79 -20.49 438 1.0\n lp_parts[374] -218.7 14.18 296.54 -1061 -289.0 -95.11 -27.92 -20.48 437 1.0\n lp_parts[375] -219.6 14.23 297.56 -1065 -290.2 -95.61 -28.04 -20.48 437 1.0\n lp_parts[376] -220.4 14.28 298.58 -1068 -291.5 -96.11 -28.17 -20.48 437 1.0\n lp_parts[377] -221.3 14.33 299.61 -1072 -292.7 -96.61 -28.3 -20.47 437 1.0\n lp_parts[378] -222.1 14.38 300.63 -1076 -293.9 -97.11 -28.42 -20.47 437 1.0\n lp_parts[379] -223.0 14.42 301.65 -1079 -295.1 -97.61 -28.5 -20.48 437 1.0\n lp_parts[380] -223.8 14.47 302.67 -1083 -296.4 -98.12 -28.58 -20.48 437 1.0\n lp_parts[381] -224.7 14.52 303.69 -1087 -297.6 -98.62 -28.71 -20.48 437 1.0\n lp_parts[382] -225.5 14.57 304.71 -1090 -298.8 -99.12 -28.84 -20.48 437 1.0\n lp_parts[383] -226.4 14.62 305.74 -1094 -300.1 -99.63 -28.97 -20.49 437 1.0\n lp_parts[384] -227.2 14.67 306.76 -1097 -301.3 -100.1 -29.1 -20.5 437 1.0\n lp_parts[385] -228.1 14.72 307.78 -1101 -302.5 -100.6 -29.23 -20.5 437 1.0\n lp_parts[386] -229.0 14.77 308.8 -1105 -303.8 -101.1 -29.37 -20.5 437 1.0\n lp_parts[387] -229.8 14.82 309.82 -1108 -305.0 -101.6 -29.5 -20.49 437 1.0\n lp_parts[388] -230.7 14.87 310.85 -1112 -306.2 -102.1 -29.64 -20.48 437 1.0\n lp_parts[389] -231.5 14.92 311.87 -1116 -307.5 -102.6 -29.77 -20.48 437 1.0\n lp_parts[390] -232.4 14.97 312.89 -1119 -308.7 -103.1 -29.91 -20.48 437 1.0\n lp_parts[391] -233.3 15.02 313.91 -1123 -310.0 -103.7 -29.94 -20.48 437 1.0\n lp_parts[392] -234.1 15.07 314.94 -1127 -311.2 -104.2 -29.83 -20.48 437 1.0\n lp_parts[393] -235.0 15.12 315.96 -1130 -312.4 -104.7 -29.74 -20.48 437 1.0\n lp_parts[394] -235.9 15.17 316.98 -1134 -313.7 -105.2 -29.83 -20.47 437 1.0\n lp_parts[395] -236.7 15.22 318.0 -1138 -314.9 -105.7 -29.96 -20.48 437 1.0\n lp_parts[396] -237.6 15.26 319.03 -1141 -316.2 -106.2 -30.1 -20.48 437 1.0\n lp_parts[397] -238.5 15.31 320.05 -1145 -317.4 -106.7 -30.23 -20.49 437 1.0\n lp_parts[398] -239.3 15.36 321.07 -1149 -318.7 -107.3 -30.37 -20.49 437 1.0\n lp_parts[399] -240.2 15.41 322.09 -1152 -319.9 -107.8 -30.51 -20.5 437 1.0\n lp_parts[400] -241.1 15.46 323.12 -1156 -321.2 -108.3 -30.65 -20.49 437 1.0\n lp_parts[401] -241.9 15.51 324.14 -1160 -322.4 -108.8 -30.79 -20.48 437 1.0\n lp_parts[402] -242.8 15.56 325.16 -1163 -323.6 -109.3 -30.93 -20.48 437 1.0\n lp_parts[403] -243.7 15.61 326.19 -1167 -324.9 -109.9 -31.07 -20.49 437 1.0\n lp_parts[404] -244.5 15.66 327.21 -1171 -326.1 -110.4 -31.21 -20.48 437 1.0\n lp_parts[405] -245.4 15.71 328.23 -1174 -327.4 -110.9 -31.35 -20.48 436 1.0\n lp_parts[406] -246.3 15.76 329.25 -1178 -328.6 -111.4 -31.49 -20.47 436 1.0\n lp_parts[407] -247.2 15.81 330.28 -1182 -329.9 -111.9 -31.64 -20.47 436 1.0\n lp_parts[408] -248.0 15.86 331.3 -1185 -331.1 -112.5 -31.78 -20.47 436 1.0\n lp_parts[409] -248.9 15.91 332.32 -1189 -332.4 -113.0 -31.92 -20.47 436 1.0\n lp_parts[410] -249.8 15.96 333.35 -1193 -333.7 -113.5 -32.07 -20.48 436 1.0\n lp_parts[411] -250.7 16.01 334.37 -1196 -334.9 -114.0 -32.22 -20.47 436 1.0\n lp_parts[412] -251.5 16.06 335.39 -1200 -336.2 -114.6 -32.36 -20.47 436 1.0\n lp_parts[413] -252.4 16.11 336.42 -1204 -337.4 -115.1 -32.51 -20.47 436 1.0\n lp_parts[414] -253.3 16.16 337.44 -1207 -338.7 -115.6 -32.66 -20.47 436 1.0\n lp_parts[415] -254.2 16.21 338.46 -1211 -339.9 -116.2 -32.81 -20.46 436 1.0\n lp_parts[416] -255.1 16.26 339.49 -1215 -341.2 -116.7 -32.96 -20.46 436 1.0\n lp_parts[417] -255.9 16.31 340.51 -1218 -342.4 -117.2 -33.11 -20.46 436 1.0\n lp_parts[418] -256.8 16.36 341.53 -1222 -343.7 -117.7 -33.26 -20.46 436 1.0\n lp_parts[419] -257.7 16.4 342.56 -1226 -345.0 -118.3 -33.41 -20.46 436 1.0\n lp_parts[420] -258.6 16.45 343.58 -1229 -346.2 -118.8 -33.56 -20.47 436 1.0\n lp_parts[421] -259.5 16.5 344.6 -1233 -347.5 -119.3 -33.71 -20.47 436 1.0\n lp_parts[422] -260.4 16.55 345.63 -1237 -348.7 -119.9 -33.87 -20.47 436 1.0\n lp_parts[423] -261.2 16.6 346.65 -1240 -350.0 -120.4 -34.02 -20.48 436 1.0\n lp_parts[424] -262.1 16.65 347.67 -1244 -351.3 -120.9 -34.18 -20.47 436 1.0\n lp_parts[425] -263.0 16.7 348.7 -1248 -352.5 -121.5 -34.33 -20.47 436 1.0\n lp_parts[426] -263.9 16.75 349.72 -1251 -353.8 -122.0 -34.49 -20.47 436 1.0\n lp_parts[427] -264.8 16.8 350.75 -1255 -355.1 -122.5 -34.64 -20.47 436 1.0\n lp_parts[428] -265.7 16.85 351.77 -1259 -356.3 -123.1 -34.8 -20.47 436 1.0\n lp_parts[429] -266.6 16.9 352.79 -1262 -357.6 -123.6 -34.96 -20.48 436 1.0\n lp_parts[430] -267.4 16.95 353.82 -1266 -358.9 -124.2 -35.12 -20.48 436 1.0\n lp_parts[431] -268.3 17.0 354.84 -1270 -360.1 -124.7 -35.28 -20.49 436 1.0\n lp_parts[432] -269.2 17.05 355.86 -1274 -361.4 -125.2 -35.44 -20.49 436 1.0\n lp_parts[433] -270.1 17.1 356.89 -1277 -362.7 -125.8 -35.6 -20.48 436 1.0\n lp_parts[434] -271.0 17.15 357.91 -1281 -363.9 -126.3 -35.76 -20.49 436 1.0\n lp_parts[435] -271.9 17.2 358.94 -1285 -365.2 -126.9 -35.92 -20.49 436 1.0\n lp_parts[436] -272.8 17.25 359.96 -1288 -366.5 -127.4 -36.08 -20.49 436 1.0\n lp_parts[437] -273.7 17.3 360.98 -1292 -367.7 -127.9 -36.24 -20.5 436 1.0\n lp_parts[438] -274.6 17.35 362.01 -1296 -369.0 -128.5 -36.41 -20.5 435 1.0\n lp_parts[439] -275.5 17.4 363.03 -1299 -370.3 -129.0 -36.57 -20.5 435 1.0\n lp_parts[440] -276.4 17.45 364.06 -1303 -371.5 -129.6 -36.74 -20.5 435 1.0\n lp_parts[441] -277.3 17.5 365.08 -1307 -372.8 -130.1 -36.9 -20.49 435 1.0\n lp_parts[442] -278.2 17.55 366.11 -1310 -374.1 -130.7 -37.07 -20.49 435 1.0\n lp_parts[443] -279.1 17.6 367.13 -1314 -375.4 -131.2 -37.23 -20.48 435 1.0\n lp_parts[444] -280.0 17.65 368.15 -1318 -376.6 -131.8 -37.4 -20.48 435 1.0\n lp_parts[445] -280.9 17.69 369.18 -1321 -377.9 -132.3 -37.57 -20.48 435 1.0\n lp_parts[446] -281.8 17.74 370.2 -1325 -379.2 -132.8 -37.73 -20.49 435 1.0\n lp_parts[447] -282.7 17.79 371.23 -1329 -380.5 -133.4 -37.9 -20.5 435 1.0\n lp_parts[448] -283.6 17.84 372.25 -1333 -381.7 -133.9 -38.07 -20.49 435 1.0\n lp_parts[449] -284.5 17.89 373.28 -1336 -383.0 -134.5 -38.24 -20.5 435 1.0\n lp_parts[450] -285.4 17.94 374.3 -1340 -384.3 -135.0 -38.41 -20.51 435 1.0\n lp_parts[451] -286.3 17.99 375.32 -1344 -385.6 -135.6 -38.58 -20.52 435 1.0\n lp_parts[452] -287.2 18.04 376.35 -1347 -386.9 -136.1 -38.75 -20.52 435 1.0\n lp_parts[453] -288.1 18.09 377.37 -1351 -388.1 -136.7 -38.93 -20.52 435 1.0\n lp_parts[454] -289.0 18.14 378.4 -1355 -389.4 -137.2 -39.1 -20.53 435 1.0\n lp_parts[455] -289.9 18.19 379.42 -1358 -390.7 -137.8 -39.27 -20.53 435 1.0\n lp_parts[456] -290.8 18.24 380.45 -1362 -392.0 -138.4 -39.44 -20.53 435 1.0\n lp_parts[457] -291.7 18.29 381.47 -1366 -393.3 -138.9 -39.62 -20.52 435 1.0\n lp_parts[458] -292.6 18.34 382.5 -1369 -394.5 -139.5 -39.79 -20.53 435 1.0\n lp_parts[459] -293.5 18.39 383.52 -1373 -395.8 -140.0 -39.97 -20.54 435 1.0\n lp_parts[460] -294.4 18.44 384.55 -1377 -397.1 -140.6 -40.14 -20.53 435 1.0\n lp_parts[461] -295.3 18.49 385.57 -1381 -398.4 -141.1 -40.32 -20.52 435 1.0\n lp_parts[462] -296.2 18.54 386.6 -1384 -399.7 -141.7 -40.5 -20.52 435 1.0\n lp_parts[463] -297.1 18.59 387.62 -1388 -401.0 -142.2 -40.67 -20.53 435 1.0\n lp_parts[464] -298.0 18.64 388.65 -1392 -402.2 -142.8 -40.85 -20.54 435 1.0\n lp_parts[465] -298.9 18.69 389.67 -1395 -403.5 -143.4 -41.03 -20.55 435 1.0\n lp_parts[466] -299.8 18.74 390.7 -1399 -404.8 -143.9 -41.21 -20.55 435 1.0\n lp_parts[467] -300.8 18.79 391.72 -1403 -406.1 -144.5 -41.39 -20.55 435 1.0\n lp_parts[468] -301.7 18.84 392.75 -1406 -407.4 -145.0 -41.57 -20.55 435 1.0\n lp_parts[469] -302.6 18.89 393.77 -1410 -408.7 -145.6 -41.75 -20.56 435 1.0\n lp_parts[470] -303.5 18.94 394.8 -1414 -410.0 -146.2 -41.93 -20.57 435 1.0\n lp_parts[471] -304.4 18.99 395.82 -1418 -411.3 -146.7 -42.11 -20.56 435 1.0\n lp_parts[472] -305.3 19.04 396.85 -1421 -412.5 -147.3 -42.29 -20.57 435 1.0\n lp_parts[473] -306.2 19.09 397.87 -1425 -413.8 -147.8 -42.48 -20.58 435 1.0\n lp_parts[474] -307.1 19.14 398.9 -1429 -415.1 -148.4 -42.66 -20.58 435 1.0\n lp_parts[475] -308.1 19.18 399.92 -1432 -416.4 -149.0 -42.84 -20.59 435 1.0\n lp_parts[476] -309.0 19.23 400.95 -1436 -417.7 -149.5 -43.03 -20.6 435 1.0\n lp_parts[477] -309.9 19.28 401.97 -1440 -419.0 -150.1 -43.21 -20.6 435 1.0\n lp_parts[478] -310.8 19.33 403.0 -1444 -420.3 -150.7 -43.39 -20.59 434 1.0\n lp_parts[479] -311.7 19.38 404.02 -1447 -421.6 -151.2 -43.58 -20.59 434 1.0\n lp_parts[480] -312.6 19.43 405.05 -1451 -422.9 -151.8 -43.77 -20.6 434 1.0\n lp_parts[481] -313.5 19.48 406.07 -1455 -424.2 -152.4 -43.95 -20.61 434 1.0\n lp_parts[482] -314.5 19.53 407.1 -1458 -425.5 -152.9 -44.14 -20.62 434 1.0\n lp_parts[483] -315.4 19.58 408.12 -1462 -426.8 -153.5 -44.33 -20.62 434 1.0\n lp_parts[484] -316.3 19.63 409.15 -1466 -428.1 -154.1 -44.52 -20.64 434 1.0\n lp_parts[485] -317.2 19.68 410.17 -1470 -429.4 -154.6 -44.7 -20.65 434 1.0\n lp_parts[486] -318.1 19.73 411.2 -1473 -430.7 -155.2 -44.89 -20.67 434 1.0\n lp_parts[487] -319.1 19.78 412.22 -1477 -431.9 -155.8 -45.08 -20.68 434 1.0\n lp_parts[488] -320.0 19.83 413.25 -1481 -433.2 -156.3 -45.27 -20.69 434 1.0\n lp_parts[489] -320.9 19.88 414.27 -1484 -434.5 -156.9 -45.46 -20.71 434 1.0\n lp_parts[490] -321.8 19.93 415.3 -1488 -435.8 -157.5 -45.65 -20.73 434 1.0\n lp_parts[491] -322.7 19.98 416.33 -1492 -437.1 -158.0 -45.84 -20.74 434 1.0\n lp_parts[492] -323.7 20.03 417.35 -1496 -438.4 -158.6 -46.04 -20.76 434 1.0\n lp_parts[493] -324.6 20.08 418.38 -1499 -439.7 -159.2 -46.23 -20.78 434 1.0\n lp_parts[494] -325.5 20.13 419.4 -1503 -441.0 -159.8 -46.42 -20.8 434 1.0\n lp_parts[495] -326.4 20.18 420.43 -1507 -442.3 -160.3 -46.61 -20.82 434 1.0\n lp_parts[496] -327.4 20.23 421.45 -1510 -443.6 -160.9 -46.81 -20.84 434 1.0\n lp_parts[497] -328.3 20.28 422.48 -1514 -444.9 -161.5 -47.0 -20.85 434 1.0\n lp_parts[498] -329.2 20.33 423.5 -1518 -446.2 -162.1 -47.2 -20.85 434 1.0\n lp_parts[499] -330.1 20.38 424.53 -1522 -447.5 -162.6 -47.39 -20.86 434 1.0\n lp_parts[500] -331.1 20.43 425.56 -1525 -448.9 -163.2 -47.59 -20.86 434 1.0\n n 186.53 6.66 125.54 39.0 83.0 147.0 267.55 463.6 356 1.0\n prob_n[1] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[2] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[3] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[4] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[5] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[6] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[7] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[8] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[9] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[10] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[11] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[12] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[13] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[14] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[15] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[16] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[17] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[18] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[19] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[20] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[21] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[22] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[23] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[24] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[25] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[26] 0.0 nan 0.0 0.0 0.0 0.0 0.0 0.0 nan nan\n prob_n[27] 1.2e-4 1.1e-4 3.2e-3 2.8e-120 1.3e-94 1.4e-66 7.9e-41 1.3e-11 844 1.0\n prob_n[28] 5.6e-4 4.7e-4 0.01 1.8e-116 6.8e-91 5.1e-63 1.4e-37 2.0e-9 529 1.01\n prob_n[29] 1.1e-3 8.4e-4 0.02 3.6e-113 1.2e-87 5.8e-60 7.7e-35 10.0e-8 387 1.01\n prob_n[30] 1.4e-3 9.4e-4 0.02 3.4e-110 9.2e-85 3.2e-57 2.0e-32 2.3e-6 354 1.01\n prob_n[31] 1.4e-3 8.1e-4 0.02 1.9e-107 4.1e-82 9.7e-55 3.0e-30 3.0e-5 414 1.01\n prob_n[32] 1.4e-3 6.2e-4 0.02 6.3e-105 1.2e-79 1.9e-52 2.8e-28 2.5e-4 630 1.01\n prob_n[33] 1.5e-3 4.9e-4 0.01 1.4e-102 2.2e-77 2.5e-50 1.8e-26 1.0e-3 909 1.0\n prob_n[34] 1.8e-3 5.1e-4 0.01 2.4e-100 3.1e-75 2.3e-48 8.0e-25 2.5e-3 833 1.0\n prob_n[35] 2.2e-3 6.3e-4 0.02 2.9e-98 3.2e-73 1.6e-46 2.7e-23 6.6e-3 678 1.0\n prob_n[36] 2.6e-3 7.6e-4 0.02 2.8e-96 2.5e-71 8.9e-45 7.2e-22 0.02 604 1.0\n prob_n[37] 3.1e-3 8.3e-4 0.02 2.1e-94 1.6e-69 3.9e-43 1.5e-20 0.02 593 1.0\n prob_n[38] 3.3e-3 8.3e-4 0.02 1.3e-92 8.3e-68 1.4e-41 2.6e-19 0.04 617 1.0\n prob_n[39] 3.5e-3 7.5e-4 0.02 6.9e-91 3.6e-66 4.1e-40 3.7e-18 0.06 698 1.0\n prob_n[40] 3.6e-3 6.5e-4 0.02 3.0e-89 1.3e-64 1.0e-38 4.5e-17 0.07 862 1.0\n prob_n[41] 3.7e-3 5.9e-4 0.02 1.1e-87 4.1e-63 2.2e-37 4.6e-16 0.07 1036 1.0\n prob_n[42] 3.9e-3 5.8e-4 0.02 3.7e-86 1.1e-61 4.0e-36 4.0e-15 0.07 1105 1.0\n prob_n[43] 4.0e-3 5.9e-4 0.02 1.0e-84 2.6e-60 6.5e-35 2.8e-14 0.07 1102 1.0\n prob_n[44] 4.2e-3 5.9e-4 0.02 2.6e-83 5.5e-59 9.4e-34 1.7e-13 0.08 1086 1.0\n prob_n[45] 4.4e-3 5.9e-4 0.02 5.9e-82 1.0e-57 1.2e-32 8.1e-13 0.09 1089 1.0\n prob_n[46] 4.6e-3 5.8e-4 0.02 1.2e-80 1.8e-56 1.4e-31 4.3e-12 0.09 1114 1.0\n prob_n[47] 4.7e-3 5.7e-4 0.02 2.2e-79 2.7e-55 1.5e-30 1.5e-11 0.08 1146 1.0\n prob_n[48] 4.9e-3 5.7e-4 0.02 3.7e-78 3.7e-54 1.3e-29 7.2e-11 0.09 1170 1.0\n prob_n[49] 5.2e-3 5.7e-4 0.02 5.6e-77 4.8e-53 1.2e-28 2.6e-10 0.09 1155 1.0\n prob_n[50] 5.4e-3 5.7e-4 0.02 7.9e-76 5.6e-52 8.9e-28 9.0e-10 0.08 1175 1.0\n prob_n[51] 5.6e-3 5.8e-4 0.02 1.0e-74 6.1e-51 5.9e-27 3.3e-9 0.09 1151 1.0\n prob_n[52] 5.9e-3 6.0e-4 0.02 1.3e-73 6.2e-50 3.9e-26 7.6e-9 0.09 1124 1.0\n prob_n[53] 6.1e-3 6.3e-4 0.02 1.4e-72 5.8e-49 2.4e-25 1.4e-8 0.09 1025 1.0\n prob_n[54] 6.2e-3 6.6e-4 0.02 1.5e-71 5.2e-48 1.4e-24 2.4e-8 0.09 950 1.0\n prob_n[55] 6.4e-3 6.8e-4 0.02 1.5e-70 4.3e-47 7.2e-24 6.0e-8 0.09 892 1.0\n prob_n[56] 6.4e-3 6.9e-4 0.02 1.4e-69 3.1e-46 3.1e-23 1.4e-7 0.09 867 1.0\n prob_n[57] 6.4e-3 6.6e-4 0.02 1.2e-68 2.3e-45 1.6e-22 2.9e-7 0.08 905 1.0\n prob_n[58] 6.4e-3 6.3e-4 0.02 1.0e-67 1.5e-44 7.1e-22 4.4e-7 0.08 960 1.0\n prob_n[59] 6.3e-3 6.0e-4 0.02 8.3e-67 6.2e-44 2.2e-21 7.3e-7 0.08 1019 1.0\n prob_n[60] 6.1e-3 5.8e-4 0.02 6.3e-66 3.3e-43 6.4e-21 1.0e-6 0.08 1026 1.0\n prob_n[61] 6.0e-3 5.6e-4 0.02 4.5e-65 1.8e-42 2.7e-20 1.6e-6 0.08 1043 1.0\n prob_n[62] 5.9e-3 5.4e-4 0.02 3.1e-64 1.0e-41 1.0e-19 2.8e-6 0.08 1067 1.0\n prob_n[63] 5.7e-3 5.2e-4 0.02 1.6e-63 5.0e-41 2.9e-19 3.8e-6 0.07 1095 1.0\n prob_n[64] 5.6e-3 5.0e-4 0.02 1.0e-62 2.2e-40 7.4e-19 5.1e-6 0.07 1123 1.0\n prob_n[65] 5.5e-3 4.9e-4 0.02 6.4e-62 1.1e-39 1.3e-18 5.8e-6 0.07 1111 1.0\n prob_n[66] 5.4e-3 4.8e-4 0.02 2.8e-61 3.7e-39 2.7e-18 8.9e-6 0.07 1091 1.0\n prob_n[67] 5.3e-3 4.8e-4 0.02 1.3e-60 1.7e-38 5.9e-18 1.2e-5 0.07 1060 1.0\n prob_n[68] 5.3e-3 4.9e-4 0.02 6.6e-60 8.0e-38 1.8e-17 1.4e-5 0.07 1019 1.0\n prob_n[69] 5.2e-3 4.9e-4 0.02 3.5e-59 3.3e-37 3.4e-17 1.5e-5 0.07 975 1.0\n prob_n[70] 5.2e-3 5.0e-4 0.02 1.8e-58 1.1e-36 7.9e-17 2.1e-5 0.07 942 1.0\n prob_n[71] 5.2e-3 5.0e-4 0.02 7.9e-58 4.1e-36 1.3e-16 2.2e-5 0.06 910 1.0\n prob_n[72] 5.2e-3 5.0e-4 0.02 3.8e-57 1.4e-35 2.4e-16 2.2e-5 0.06 894 1.0\n prob_n[73] 5.2e-3 5.0e-4 0.01 1.6e-56 5.3e-35 3.6e-16 2.1e-5 0.06 885 1.0\n prob_n[74] 5.2e-3 5.0e-4 0.01 7.2e-56 2.0e-34 4.6e-16 2.6e-5 0.06 886 1.0\n prob_n[75] 5.2e-3 4.9e-4 0.01 1.8e-55 4.4e-34 9.3e-16 2.8e-5 0.06 891 1.0\n prob_n[76] 5.1e-3 4.8e-4 0.01 4.0e-55 1.0e-33 1.8e-15 3.4e-5 0.06 899 1.0\n prob_n[77] 5.1e-3 4.7e-4 0.01 1.7e-54 3.3e-33 2.8e-15 3.2e-5 0.06 919 1.0\n prob_n[78] 5.1e-3 4.6e-4 0.01 6.8e-54 3.9e-33 4.5e-15 3.4e-5 0.06 952 1.0\n prob_n[79] 5.1e-3 4.5e-4 0.01 1.4e-53 1.2e-32 9.2e-15 3.4e-5 0.06 995 1.0\n prob_n[80] 5.1e-3 4.3e-4 0.01 4.5e-53 2.8e-32 1.3e-14 4.1e-5 0.06 1047 1.0\n prob_n[81] 5.0e-3 4.2e-4 0.01 1.4e-52 7.5e-32 1.4e-14 4.7e-5 0.06 1104 1.0\n prob_n[82] 5.0e-3 4.0e-4 0.01 4.7e-52 2.1e-31 1.8e-14 5.1e-5 0.06 1157 1.0\n prob_n[83] 5.0e-3 3.9e-4 0.01 1.0e-51 4.0e-31 1.4e-14 4.8e-5 0.05 1192 1.0\n prob_n[84] 4.9e-3 3.8e-4 0.01 3.6e-51 6.0e-31 2.0e-14 5.7e-5 0.05 1222 1.0\n prob_n[85] 4.9e-3 3.7e-4 0.01 1.2e-50 1.5e-30 2.7e-14 6.4e-5 0.05 1246 1.0\n prob_n[86] 4.8e-3 3.7e-4 0.01 3.6e-50 3.3e-30 4.9e-14 6.2e-5 0.05 1265 1.0\n prob_n[87] 4.8e-3 3.6e-4 0.01 1.0e-49 8.6e-30 7.6e-14 8.0e-5 0.05 1280 1.0\n prob_n[88] 4.7e-3 3.5e-4 0.01 1.2e-49 1.6e-29 1.3e-13 8.2e-5 0.05 1293 1.0\n prob_n[89] 4.7e-3 3.5e-4 0.01 2.1e-49 3.3e-29 1.2e-13 9.1e-5 0.05 1306 1.0\n prob_n[90] 4.6e-3 3.4e-4 0.01 3.7e-49 3.9e-29 1.3e-13 8.7e-5 0.05 1321 1.0\n prob_n[91] 4.6e-3 3.3e-4 0.01 3.1e-49 3.9e-29 1.8e-13 9.4e-5 0.05 1339 1.0\n prob_n[92] 4.5e-3 3.3e-4 0.01 1.4e-49 6.5e-29 1.9e-13 1.2e-4 0.05 1360 1.0\n prob_n[93] 4.5e-3 3.2e-4 0.01 1.3e-49 1.4e-28 2.3e-13 1.2e-4 0.05 1384 1.0\n prob_n[94] 4.5e-3 3.1e-4 0.01 1.5e-50 2.1e-28 2.6e-13 9.8e-5 0.05 1408 1.0\n prob_n[95] 4.4e-3 3.0e-4 0.01 8.9e-52 3.3e-28 3.2e-13 1.1e-4 0.05 1451 1.0\n prob_n[96] 4.4e-3 2.9e-4 0.01 2.3e-52 7.1e-28 4.1e-13 1.1e-4 0.05 1507 1.0\n prob_n[97] 4.3e-3 2.9e-4 0.01 6.7e-52 1.4e-27 4.1e-13 1.1e-4 0.05 1554 1.0\n prob_n[98] 4.3e-3 2.8e-4 0.01 2.0e-51 1.6e-27 4.2e-13 1.1e-4 0.05 1557 1.0\n prob_n[99] 4.3e-3 2.8e-4 0.01 5.6e-51 1.9e-27 4.5e-13 1.1e-4 0.04 1559 1.0\n prob_n[100] 4.3e-3 2.8e-4 0.01 7.9e-51 2.4e-27 6.4e-13 1.3e-4 0.04 1560 1.0\n prob_n[101] 4.2e-3 2.8e-4 0.01 3.8e-51 3.2e-27 7.6e-13 1.2e-4 0.04 1561 1.0\n prob_n[102] 4.2e-3 2.8e-4 0.01 2.6e-52 3.5e-27 6.4e-13 1.4e-4 0.04 1560 1.0\n prob_n[103] 4.2e-3 2.7e-4 0.01 1.7e-53 5.9e-27 7.5e-13 1.5e-4 0.04 1559 1.0\n prob_n[104] 4.2e-3 2.7e-4 0.01 1.1e-54 8.0e-27 5.8e-13 1.5e-4 0.04 1557 1.0\n prob_n[105] 4.2e-3 2.7e-4 0.01 7.1e-56 5.4e-27 6.5e-13 1.5e-4 0.04 1555 1.0\n prob_n[106] 4.1e-3 2.7e-4 0.01 4.5e-57 1.1e-26 8.2e-13 1.3e-4 0.04 1552 1.0\n prob_n[107] 4.1e-3 2.7e-4 0.01 2.8e-58 1.2e-26 1.3e-12 1.1e-4 0.04 1550 1.0\n prob_n[108] 4.1e-3 2.7e-4 0.01 1.8e-59 2.3e-26 1.6e-12 1.4e-4 0.04 1548 1.0\n prob_n[109] 4.1e-3 2.7e-4 0.01 1.1e-60 3.3e-26 1.6e-12 1.5e-4 0.04 1546 1.0\n prob_n[110] 4.1e-3 2.6e-4 0.01 6.6e-62 3.9e-26 2.0e-12 1.7e-4 0.04 1544 1.0\n prob_n[111] 4.0e-3 2.6e-4 0.01 3.9e-63 2.7e-26 2.6e-12 1.6e-4 0.04 1544 1.0\n prob_n[112] 4.0e-3 2.6e-4 0.01 2.3e-64 4.0e-26 3.1e-12 1.7e-4 0.04 1543 1.0\n prob_n[113] 4.0e-3 2.6e-4 0.01 1.4e-65 4.6e-26 2.7e-12 1.7e-4 0.04 1520 1.0\n prob_n[114] 3.9e-3 2.6e-4 10.0e-3 8.0e-67 7.3e-26 2.5e-12 1.5e-4 0.04 1455 1.0\n prob_n[115] 3.9e-3 2.6e-4 9.9e-3 4.6e-68 7.1e-26 2.9e-12 1.5e-4 0.04 1394 1.0\n prob_n[116] 3.8e-3 2.7e-4 9.8e-3 2.6e-69 8.7e-26 4.1e-12 1.3e-4 0.04 1338 1.0\n prob_n[117] 3.8e-3 2.7e-4 9.6e-3 1.5e-70 9.6e-26 4.6e-12 1.5e-4 0.04 1286 1.0\n prob_n[118] 3.8e-3 2.7e-4 9.5e-3 8.2e-72 1.5e-25 4.5e-12 1.3e-4 0.04 1239 1.0\n prob_n[119] 3.7e-3 2.7e-4 9.4e-3 4.6e-73 1.4e-25 4.2e-12 1.3e-4 0.04 1199 1.0\n prob_n[120] 3.7e-3 2.7e-4 9.3e-3 2.5e-74 1.3e-25 4.1e-12 1.3e-4 0.04 1172 1.0\n prob_n[121] 3.6e-3 2.7e-4 9.2e-3 1.4e-75 1.7e-25 4.5e-12 1.3e-4 0.04 1149 1.0\n prob_n[122] 3.6e-3 2.7e-4 9.1e-3 7.3e-77 2.6e-25 3.4e-12 1.3e-4 0.04 1136 1.0\n prob_n[123] 3.6e-3 2.7e-4 9.0e-3 3.9e-78 3.0e-25 3.9e-12 1.3e-4 0.04 1137 1.0\n prob_n[124] 3.5e-3 2.6e-4 8.9e-3 2.1e-79 3.2e-25 3.5e-12 1.5e-4 0.04 1142 1.0\n prob_n[125] 3.5e-3 2.6e-4 8.8e-3 1.1e-80 3.0e-25 3.6e-12 1.4e-4 0.03 1149 1.0\n prob_n[126] 3.4e-3 2.5e-4 8.7e-3 5.7e-82 3.6e-25 4.4e-12 1.2e-4 0.03 1159 1.0\n prob_n[127] 3.4e-3 2.5e-4 8.6e-3 2.9e-83 3.2e-25 4.9e-12 1.0e-4 0.03 1176 1.0\n prob_n[128] 3.4e-3 2.5e-4 8.5e-3 1.5e-84 4.4e-25 5.2e-12 8.8e-5 0.03 1195 1.0\n prob_n[129] 3.3e-3 2.4e-4 8.4e-3 7.6e-86 5.7e-25 5.6e-12 8.7e-5 0.03 1214 1.0\n prob_n[130] 3.3e-3 2.4e-4 8.4e-3 3.8e-87 6.9e-25 6.3e-12 1.1e-4 0.03 1234 1.0\n prob_n[131] 3.3e-3 2.3e-4 8.3e-3 1.9e-88 7.4e-25 7.6e-12 1.2e-4 0.03 1253 1.0\n prob_n[132] 3.2e-3 2.3e-4 8.2e-3 9.5e-90 5.9e-25 9.2e-12 1.2e-4 0.03 1271 1.0\n prob_n[133] 3.2e-3 2.3e-4 8.2e-3 4.7e-91 8.5e-25 1.1e-11 1.1e-4 0.03 1288 1.0\n prob_n[134] 3.2e-3 2.3e-4 8.1e-3 2.3e-92 1.3e-24 1.0e-11 1.2e-4 0.03 1302 1.0\n prob_n[135] 3.2e-3 2.2e-4 8.1e-3 1.1e-93 1.5e-24 7.2e-12 1.2e-4 0.03 1314 1.0\n prob_n[136] 3.1e-3 2.2e-4 8.0e-3 5.4e-95 2.0e-24 5.4e-12 1.3e-4 0.03 1326 1.0\n prob_n[137] 3.1e-3 2.2e-4 8.0e-3 2.6e-96 1.9e-24 4.7e-12 1.3e-4 0.03 1337 1.0\n prob_n[138] 3.1e-3 2.2e-4 7.9e-3 1.2e-97 1.8e-24 5.8e-12 1.3e-4 0.03 1347 1.0\n prob_n[139] 3.1e-3 2.1e-4 7.9e-3 5.8e-99 2.6e-24 6.3e-12 1.2e-4 0.03 1357 1.0\n prob_n[140] 3.0e-3 2.1e-4 7.8e-3 2.7e-100 4.2e-24 6.0e-12 1.2e-4 0.03 1367 1.0\n prob_n[141] 3.0e-3 2.1e-4 7.8e-3 1.3e-101 4.2e-24 5.7e-12 1.2e-4 0.03 1377 1.0\n prob_n[142] 3.0e-3 2.1e-4 7.7e-3 5.9e-103 5.0e-24 5.7e-12 1.2e-4 0.03 1387 1.0\n prob_n[143] 2.9e-3 2.0e-4 7.7e-3 2.7e-104 7.4e-24 5.9e-12 1.3e-4 0.03 1397 1.0\n prob_n[144] 2.9e-3 2.0e-4 7.6e-3 1.2e-105 1.1e-23 6.7e-12 1.4e-4 0.03 1408 1.0\n prob_n[145] 2.9e-3 2.0e-4 7.5e-3 5.6e-107 1.1e-23 7.3e-12 1.3e-4 0.03 1418 1.0\n prob_n[146] 2.9e-3 2.0e-4 7.5e-3 2.5e-108 1.6e-23 6.7e-12 1.2e-4 0.03 1427 1.0\n prob_n[147] 2.8e-3 1.9e-4 7.4e-3 1.1e-109 1.4e-23 5.9e-12 1.2e-4 0.03 1437 1.0\n prob_n[148] 2.8e-3 1.9e-4 7.3e-3 5.1e-111 9.3e-24 5.2e-12 1.3e-4 0.03 1446 1.0\n prob_n[149] 2.8e-3 1.9e-4 7.2e-3 2.3e-112 1.5e-23 4.9e-12 1.2e-4 0.03 1457 1.0\n prob_n[150] 2.7e-3 1.9e-4 7.2e-3 1.0e-113 1.8e-23 6.1e-12 1.0e-4 0.03 1467 1.0\n prob_n[151] 2.7e-3 1.8e-4 7.1e-3 4.4e-115 2.2e-23 7.1e-12 1.1e-4 0.03 1478 1.0\n prob_n[152] 2.7e-3 1.8e-4 7.0e-3 1.9e-116 2.3e-23 5.9e-12 10.0e-5 0.03 1507 1.0\n prob_n[153] 2.7e-3 1.8e-4 6.9e-3 8.4e-118 1.9e-23 6.8e-12 9.6e-5 0.03 1542 1.0\n prob_n[154] 2.6e-3 1.7e-4 6.9e-3 3.6e-119 2.0e-23 5.1e-12 9.9e-5 0.03 1572 1.0\n prob_n[155] 2.6e-3 1.7e-4 6.8e-3 1.6e-120 2.4e-23 5.2e-12 9.9e-5 0.03 1597 1.0\n prob_n[156] 2.6e-3 1.7e-4 6.7e-3 6.6e-122 2.6e-23 4.3e-12 1.1e-4 0.03 1616 1.0\n prob_n[157] 2.6e-3 1.6e-4 6.7e-3 2.8e-123 2.8e-23 5.0e-12 1.2e-4 0.03 1638 1.0\n prob_n[158] 2.5e-3 1.6e-4 6.6e-3 1.2e-124 3.2e-23 5.9e-12 1.1e-4 0.03 1658 1.0\n prob_n[159] 2.5e-3 1.6e-4 6.5e-3 5.0e-126 2.8e-23 5.8e-12 1.1e-4 0.03 1675 1.0\n prob_n[160] 2.5e-3 1.6e-4 6.5e-3 2.1e-127 3.2e-23 6.0e-12 1.1e-4 0.03 1681 1.0\n prob_n[161] 2.5e-3 1.6e-4 6.4e-3 8.8e-129 2.0e-23 5.6e-12 1.1e-4 0.03 1672 1.0\n prob_n[162] 2.5e-3 1.6e-4 6.3e-3 3.7e-130 2.1e-23 4.3e-12 1.1e-4 0.03 1660 1.0\n prob_n[163] 2.4e-3 1.6e-4 6.3e-3 1.5e-131 2.5e-23 4.2e-12 1.1e-4 0.03 1644 1.0\n prob_n[164] 2.4e-3 1.5e-4 6.2e-3 6.2e-133 2.7e-23 3.5e-12 1.1e-4 0.03 1633 1.0\n prob_n[165] 2.4e-3 1.5e-4 6.2e-3 2.5e-134 1.8e-23 3.3e-12 1.1e-4 0.03 1625 1.0\n prob_n[166] 2.4e-3 1.5e-4 6.2e-3 1.0e-135 1.8e-23 3.5e-12 1.1e-4 0.03 1614 1.0\n prob_n[167] 2.4e-3 1.5e-4 6.1e-3 4.2e-137 1.3e-23 2.7e-12 1.1e-4 0.03 1603 1.0\n prob_n[168] 2.4e-3 1.5e-4 6.1e-3 1.7e-138 1.4e-23 3.7e-12 1.0e-4 0.02 1590 1.0\n prob_n[169] 2.4e-3 1.5e-4 6.1e-3 6.8e-140 7.2e-24 3.7e-12 10.0e-5 0.02 1576 1.0\n prob_n[170] 2.3e-3 1.5e-4 6.0e-3 2.7e-141 6.5e-24 3.8e-12 1.1e-4 0.02 1561 1.0\n prob_n[171] 2.3e-3 1.5e-4 6.0e-3 1.1e-142 3.3e-24 4.0e-12 1.2e-4 0.02 1546 1.0\n prob_n[172] 2.3e-3 1.5e-4 6.0e-3 4.3e-144 1.7e-24 3.9e-12 1.1e-4 0.02 1530 1.0\n prob_n[173] 2.3e-3 1.5e-4 6.0e-3 1.7e-145 6.8e-25 3.1e-12 1.0e-4 0.02 1515 1.0\n prob_n[174] 2.3e-3 1.5e-4 6.0e-3 6.7e-147 3.1e-25 3.6e-12 9.7e-5 0.02 1499 1.0\n prob_n[175] 2.3e-3 1.5e-4 5.9e-3 2.6e-148 1.3e-25 4.3e-12 8.8e-5 0.02 1484 1.0\n prob_n[176] 2.3e-3 1.5e-4 5.9e-3 1.0e-149 5.7e-26 4.7e-12 8.5e-5 0.02 1470 1.0\n prob_n[177] 2.3e-3 1.6e-4 5.9e-3 4.0e-151 2.3e-26 4.8e-12 8.6e-5 0.02 1457 1.0\n prob_n[178] 2.3e-3 1.6e-4 5.9e-3 1.5e-152 9.2e-27 5.0e-12 8.8e-5 0.02 1444 1.0\n prob_n[179] 2.3e-3 1.6e-4 5.9e-3 6.0e-154 6.6e-27 5.4e-12 9.6e-5 0.02 1433 1.0\n prob_n[180] 2.3e-3 1.6e-4 5.9e-3 2.3e-155 1.0e-26 4.4e-12 9.5e-5 0.02 1422 1.0\n prob_n[181] 2.3e-3 1.6e-4 5.9e-3 8.8e-157 1.6e-26 4.4e-12 9.4e-5 0.02 1413 1.0\n prob_n[182] 2.3e-3 1.6e-4 5.9e-3 3.4e-158 7.6e-27 4.3e-12 8.6e-5 0.02 1405 1.0\n prob_n[183] 2.3e-3 1.6e-4 5.9e-3 1.3e-159 3.1e-27 4.7e-12 9.2e-5 0.02 1398 1.0\n prob_n[184] 2.3e-3 1.6e-4 5.9e-3 4.8e-161 1.3e-27 5.1e-12 8.9e-5 0.02 1393 1.0\n prob_n[185] 2.3e-3 1.6e-4 5.8e-3 1.8e-162 5.1e-28 5.2e-12 8.9e-5 0.02 1388 1.0\n prob_n[186] 2.3e-3 1.6e-4 5.8e-3 6.9e-164 2.1e-28 4.5e-12 9.2e-5 0.02 1386 1.0\n prob_n[187] 2.3e-3 1.6e-4 5.8e-3 2.6e-165 8.4e-29 5.1e-12 9.5e-5 0.02 1384 1.0\n prob_n[188] 2.3e-3 1.6e-4 5.8e-3 9.6e-167 3.4e-29 5.6e-12 1.0e-4 0.02 1384 1.0\n prob_n[189] 2.3e-3 1.6e-4 5.8e-3 3.6e-168 1.3e-29 6.2e-12 1.1e-4 0.02 1385 1.0\n prob_n[190] 2.3e-3 1.5e-4 5.7e-3 1.3e-169 5.4e-30 6.4e-12 1.1e-4 0.02 1387 1.0\n prob_n[191] 2.2e-3 1.5e-4 5.7e-3 4.9e-171 2.1e-30 5.6e-12 1.0e-4 0.02 1390 1.0\n prob_n[192] 2.2e-3 1.5e-4 5.7e-3 1.8e-172 8.5e-31 4.8e-12 9.5e-5 0.02 1394 1.0\n prob_n[193] 2.2e-3 1.5e-4 5.7e-3 6.6e-174 3.3e-31 4.6e-12 8.9e-5 0.02 1400 1.0\n prob_n[194] 2.2e-3 1.5e-4 5.6e-3 2.4e-175 1.3e-31 4.0e-12 8.2e-5 0.02 1406 1.0\n prob_n[195] 2.2e-3 1.5e-4 5.6e-3 8.8e-177 5.2e-32 4.0e-12 8.6e-5 0.02 1413 1.0\n prob_n[196] 2.2e-3 1.5e-4 5.6e-3 3.2e-178 2.0e-32 4.5e-12 8.0e-5 0.02 1420 1.0\n prob_n[197] 2.2e-3 1.5e-4 5.5e-3 1.2e-179 7.9e-33 5.2e-12 7.7e-5 0.02 1428 1.0\n prob_n[198] 2.2e-3 1.5e-4 5.5e-3 4.2e-181 3.1e-33 5.4e-12 6.9e-5 0.02 1437 1.0\n prob_n[199] 2.2e-3 1.4e-4 5.5e-3 1.5e-182 1.2e-33 5.4e-12 6.6e-5 0.02 1445 1.0\n prob_n[200] 2.2e-3 1.4e-4 5.4e-3 5.4e-184 4.6e-34 4.2e-12 6.6e-5 0.02 1454 1.0\n prob_n[201] 2.2e-3 1.4e-4 5.4e-3 1.9e-185 1.8e-34 3.7e-12 6.9e-5 0.02 1463 1.0\n prob_n[202] 2.1e-3 1.4e-4 5.4e-3 6.9e-187 6.8e-35 4.1e-12 6.6e-5 0.02 1471 1.0\n prob_n[203] 2.1e-3 1.4e-4 5.4e-3 2.5e-188 2.6e-35 5.0e-12 6.6e-5 0.02 1479 1.0\n prob_n[204] 2.1e-3 1.4e-4 5.3e-3 8.8e-190 9.9e-36 5.0e-12 6.7e-5 0.02 1486 1.0\n prob_n[205] 2.1e-3 1.4e-4 5.3e-3 3.1e-191 3.8e-36 5.5e-12 6.6e-5 0.02 1492 1.0\n prob_n[206] 2.1e-3 1.4e-4 5.3e-3 1.1e-192 1.4e-36 4.7e-12 6.1e-5 0.02 1498 1.0\n prob_n[207] 2.1e-3 1.3e-4 5.2e-3 3.9e-194 5.4e-37 4.1e-12 6.7e-5 0.02 1503 1.0\n prob_n[208] 2.1e-3 1.3e-4 5.2e-3 1.4e-195 2.0e-37 4.4e-12 6.8e-5 0.02 1506 1.0\n prob_n[209] 2.1e-3 1.3e-4 5.2e-3 4.7e-197 7.7e-38 4.7e-12 7.0e-5 0.02 1509 1.0\n prob_n[210] 2.0e-3 1.3e-4 5.1e-3 1.6e-198 2.9e-38 5.9e-12 6.4e-5 0.02 1510 1.0\n prob_n[211] 2.0e-3 1.3e-4 5.1e-3 5.7e-200 1.1e-38 5.6e-12 6.1e-5 0.02 1510 1.0\n prob_n[212] 2.0e-3 1.3e-4 5.1e-3 2.0e-201 4.0e-39 6.5e-12 6.6e-5 0.02 1508 1.0\n prob_n[213] 2.0e-3 1.3e-4 5.0e-3 6.9e-203 1.5e-39 6.6e-12 6.6e-5 0.02 1506 1.0\n prob_n[214] 2.0e-3 1.3e-4 5.0e-3 2.4e-204 5.6e-40 6.9e-12 7.4e-5 0.02 1502 1.0\n prob_n[215] 2.0e-3 1.3e-4 5.0e-3 8.2e-206 2.1e-40 6.6e-12 8.1e-5 0.02 1497 1.0\n prob_n[216] 2.0e-3 1.3e-4 4.9e-3 2.8e-207 7.6e-41 7.1e-12 7.2e-5 0.02 1491 1.0\n prob_n[217] 1.9e-3 1.3e-4 4.9e-3 9.6e-209 2.8e-41 8.1e-12 7.7e-5 0.02 1484 1.0\n prob_n[218] 1.9e-3 1.3e-4 4.9e-3 3.3e-210 1.0e-41 9.4e-12 7.0e-5 0.02 1476 1.0\n prob_n[219] 1.9e-3 1.3e-4 4.9e-3 1.1e-211 3.8e-42 9.5e-12 6.3e-5 0.02 1467 1.0\n prob_n[220] 1.9e-3 1.3e-4 4.8e-3 3.8e-213 1.4e-42 9.4e-12 6.2e-5 0.02 1458 1.0\n prob_n[221] 1.9e-3 1.3e-4 4.8e-3 1.3e-214 5.0e-43 7.8e-12 6.4e-5 0.02 1447 1.0\n prob_n[222] 1.9e-3 1.3e-4 4.8e-3 4.4e-216 1.8e-43 8.7e-12 6.6e-5 0.02 1437 1.0\n prob_n[223] 1.9e-3 1.3e-4 4.7e-3 1.5e-217 6.7e-44 9.0e-12 5.9e-5 0.02 1425 1.0\n prob_n[224] 1.9e-3 1.3e-4 4.7e-3 5.0e-219 2.4e-44 9.9e-12 6.1e-5 0.02 1412 1.0\n prob_n[225] 1.8e-3 1.3e-4 4.7e-3 1.7e-220 8.7e-45 9.4e-12 6.5e-5 0.02 1398 1.0\n prob_n[226] 1.8e-3 1.3e-4 4.7e-3 5.6e-222 3.1e-45 1.1e-11 6.2e-5 0.02 1384 1.0\n prob_n[227] 1.8e-3 1.2e-4 4.6e-3 1.9e-223 1.1e-45 1.0e-11 5.7e-5 0.02 1370 1.0\n prob_n[228] 1.8e-3 1.2e-4 4.6e-3 6.2e-225 4.0e-46 9.3e-12 5.8e-5 0.02 1356 1.0\n prob_n[229] 1.8e-3 1.2e-4 4.6e-3 2.1e-226 1.4e-46 1.0e-11 6.2e-5 0.02 1343 1.0\n prob_n[230] 1.8e-3 1.2e-4 4.5e-3 6.9e-228 5.2e-47 1.1e-11 5.8e-5 0.02 1330 1.0\n prob_n[231] 1.8e-3 1.2e-4 4.5e-3 2.3e-229 1.8e-47 1.1e-11 6.0e-5 0.02 1317 1.0\n prob_n[232] 1.8e-3 1.2e-4 4.5e-3 7.5e-231 6.5e-48 1.0e-11 6.5e-5 0.02 1305 1.0\n prob_n[233] 1.8e-3 1.2e-4 4.5e-3 2.5e-232 2.3e-48 9.8e-12 6.6e-5 0.02 1293 1.0\n prob_n[234] 1.7e-3 1.2e-4 4.4e-3 8.1e-234 8.2e-49 1.0e-11 6.9e-5 0.02 1282 1.0\n prob_n[235] 1.7e-3 1.2e-4 4.4e-3 2.7e-235 2.9e-49 1.0e-11 6.8e-5 0.02 1272 1.0\n prob_n[236] 1.7e-3 1.2e-4 4.4e-3 8.8e-237 1.0e-49 9.3e-12 6.9e-5 0.02 1263 1.0\n prob_n[237] 1.7e-3 1.2e-4 4.4e-3 2.9e-238 3.6e-50 7.6e-12 6.9e-5 0.02 1254 1.0\n prob_n[238] 1.7e-3 1.2e-4 4.4e-3 9.3e-240 1.3e-50 6.9e-12 7.0e-5 0.02 1246 1.0\n prob_n[239] 1.7e-3 1.2e-4 4.3e-3 3.0e-241 4.4e-51 5.4e-12 6.0e-5 0.02 1240 1.0\n prob_n[240] 1.7e-3 1.2e-4 4.3e-3 9.9e-243 1.5e-51 6.1e-12 5.5e-5 0.02 1234 1.0\n prob_n[241] 1.7e-3 1.2e-4 4.3e-3 3.2e-244 5.4e-52 5.8e-12 5.4e-5 0.02 1228 1.0\n prob_n[242] 1.7e-3 1.2e-4 4.3e-3 1.0e-245 1.9e-52 5.9e-12 5.2e-5 0.02 1224 1.0\n prob_n[243] 1.7e-3 1.2e-4 4.3e-3 3.3e-247 6.5e-53 5.1e-12 5.0e-5 0.02 1221 1.0\n prob_n[244] 1.6e-3 1.2e-4 4.2e-3 1.1e-248 2.3e-53 4.3e-12 5.4e-5 0.02 1218 1.0\n prob_n[245] 1.6e-3 1.2e-4 4.2e-3 3.5e-250 7.8e-54 3.6e-12 5.4e-5 0.02 1217 1.0\n prob_n[246] 1.6e-3 1.2e-4 4.2e-3 1.1e-251 2.7e-54 3.8e-12 5.4e-5 0.02 1216 1.0\n prob_n[247] 1.6e-3 1.2e-4 4.2e-3 3.6e-253 9.3e-55 3.5e-12 4.8e-5 0.02 1216 1.0\n prob_n[248] 1.6e-3 1.2e-4 4.2e-3 1.1e-254 3.2e-55 3.0e-12 5.0e-5 0.02 1217 1.0\n prob_n[249] 1.6e-3 1.2e-4 4.2e-3 3.6e-256 1.1e-55 3.0e-12 5.0e-5 0.02 1218 1.0\n prob_n[250] 1.6e-3 1.2e-4 4.1e-3 1.2e-257 3.8e-56 3.1e-12 5.0e-5 0.02 1220 1.0\n prob_n[251] 1.6e-3 1.2e-4 4.1e-3 3.7e-259 1.3e-56 2.7e-12 4.9e-5 0.02 1223 1.0\n prob_n[252] 1.6e-3 1.2e-4 4.1e-3 1.2e-260 4.4e-57 2.2e-12 5.3e-5 0.02 1226 1.0\n prob_n[253] 1.6e-3 1.2e-4 4.1e-3 3.7e-262 1.5e-57 1.5e-12 5.2e-5 0.02 1229 1.0\n prob_n[254] 1.6e-3 1.2e-4 4.1e-3 1.2e-263 5.1e-58 1.3e-12 5.7e-5 0.02 1234 1.0\n prob_n[255] 1.5e-3 1.2e-4 4.0e-3 3.7e-265 1.7e-58 8.7e-13 5.3e-5 0.02 1233 1.0\n prob_n[256] 1.5e-3 1.1e-4 4.0e-3 1.2e-266 5.8e-59 6.0e-13 4.9e-5 0.02 1241 1.0\n prob_n[257] 1.5e-3 1.1e-4 4.0e-3 3.6e-268 2.0e-59 5.1e-13 4.8e-5 0.02 1248 1.0\n prob_n[258] 1.5e-3 1.1e-4 4.0e-3 1.1e-269 6.7e-60 5.3e-13 4.7e-5 0.02 1251 1.0\n prob_n[259] 1.5e-3 1.1e-4 4.0e-3 3.6e-271 2.2e-60 4.2e-13 5.0e-5 0.02 1255 1.0\n prob_n[260] 1.5e-3 1.1e-4 4.0e-3 1.1e-272 7.5e-61 2.9e-13 5.0e-5 0.02 1259 1.0\n prob_n[261] 1.5e-3 1.1e-4 3.9e-3 3.5e-274 2.5e-61 2.0e-13 5.0e-5 0.02 1263 1.0\n prob_n[262] 1.5e-3 1.1e-4 3.9e-3 1.1e-275 8.5e-62 1.6e-13 4.5e-5 0.02 1267 1.0\n prob_n[263] 1.5e-3 1.1e-4 3.9e-3 3.4e-277 2.8e-62 1.1e-13 4.7e-5 0.02 1271 1.0\n prob_n[264] 1.5e-3 1.1e-4 3.9e-3 1.1e-278 9.5e-63 7.4e-14 4.6e-5 0.02 1275 1.0\n prob_n[265] 1.5e-3 1.1e-4 3.9e-3 3.3e-280 3.2e-63 5.1e-14 4.5e-5 0.02 1278 1.0\n prob_n[266] 1.5e-3 1.1e-4 3.8e-3 1.0e-281 1.1e-63 3.5e-14 4.6e-5 0.02 1281 1.0\n prob_n[267] 1.4e-3 1.1e-4 3.8e-3 3.1e-283 3.5e-64 4.5e-14 5.1e-5 0.01 1283 1.0\n prob_n[268] 1.4e-3 1.1e-4 3.8e-3 9.6e-285 1.2e-64 3.1e-14 5.2e-5 0.01 1285 1.0\n prob_n[269] 1.4e-3 1.1e-4 3.8e-3 3.0e-286 3.9e-65 2.2e-14 5.7e-5 0.01 1287 1.0\n prob_n[270] 1.4e-3 1.0e-4 3.8e-3 9.1e-288 1.3e-65 1.5e-14 5.2e-5 0.01 1288 1.0\n prob_n[271] 1.4e-3 1.0e-4 3.7e-3 2.8e-289 4.2e-66 1.0e-14 5.0e-5 0.01 1289 1.0\n prob_n[272] 1.4e-3 1.0e-4 3.7e-3 8.6e-291 1.4e-66 6.9e-15 4.6e-5 0.01 1289 1.0\n prob_n[273] 1.4e-3 1.0e-4 3.7e-3 2.6e-292 4.6e-67 4.7e-15 4.6e-5 0.01 1289 1.0\n prob_n[274] 1.4e-3 1.0e-4 3.7e-3 8.0e-294 1.5e-67 3.2e-15 4.7e-5 0.01 1288 1.0\n prob_n[275] 1.4e-3 1.0e-4 3.7e-3 2.4e-295 4.9e-68 2.2e-15 4.6e-5 0.01 1286 1.0\n prob_n[276] 1.4e-3 1.0e-4 3.6e-3 7.5e-297 1.6e-68 1.5e-15 5.1e-5 0.01 1284 1.0\n prob_n[277] 1.4e-3 1.0e-4 3.6e-3 2.3e-298 5.3e-69 1.0e-15 5.2e-5 0.01 1281 1.0\n prob_n[278] 1.4e-3 1.0e-4 3.6e-3 6.9e-300 1.7e-69 6.8e-16 4.8e-5 0.01 1278 1.0\n prob_n[279] 1.4e-3 1.0e-4 3.6e-3 2.1e-301 5.6e-70 4.6e-16 4.7e-5 0.01 1275 1.0\n prob_n[280] 1.3e-3 1.0e-4 3.6e-3 6.3e-303 1.8e-70 3.1e-16 5.1e-5 0.01 1271 1.0\n prob_n[281] 1.3e-3 1.0e-4 3.6e-3 1.9e-304 6.0e-71 2.1e-16 5.7e-5 0.01 1266 1.0\n prob_n[282] 1.3e-3 1.0e-4 3.6e-3 5.8e-306 1.9e-71 1.4e-16 6.0e-5 0.01 1261 1.0\n prob_n[283] 1.3e-3 10.0e-5 3.5e-3 1.7e-307 6.3e-72 9.4e-17 5.9e-5 0.01 1256 1.0\n prob_n[284] 1.3e-3 10.0e-5 3.5e-3 5.2e-309 2.0e-72 6.3e-17 5.6e-5 0.01 1250 1.0\n prob_n[285] 1.3e-3 9.9e-5 3.5e-3 0.0 6.6e-73 4.2e-17 5.7e-5 0.01 1244 1.0\n prob_n[286] 1.3e-3 9.9e-5 3.5e-3 0.0 2.1e-73 2.8e-17 5.1e-5 0.01 1238 1.0\n prob_n[287] 1.3e-3 9.9e-5 3.5e-3 0.0 6.8e-74 1.9e-17 4.6e-5 0.01 1231 1.0\n prob_n[288] 1.3e-3 9.9e-5 3.5e-3 0.0 2.2e-74 1.3e-17 4.5e-5 0.01 1225 1.0\n prob_n[289] 1.3e-3 9.9e-5 3.4e-3 0.0 7.1e-75 8.3e-18 4.7e-5 0.01 1217 1.0\n prob_n[290] 1.3e-3 9.9e-5 3.4e-3 0.0 2.3e-75 5.5e-18 4.6e-5 0.01 1209 1.0\n prob_n[291] 1.3e-3 9.9e-5 3.4e-3 0.0 7.3e-76 3.7e-18 4.8e-5 0.01 1200 1.0\n prob_n[292] 1.3e-3 9.9e-5 3.4e-3 0.0 2.3e-76 2.4e-18 5.1e-5 0.01 1192 1.0\n prob_n[293] 1.3e-3 9.9e-5 3.4e-3 0.0 7.4e-77 1.6e-18 5.0e-5 0.01 1184 1.0\n prob_n[294] 1.3e-3 9.8e-5 3.4e-3 0.0 2.4e-77 1.1e-18 4.5e-5 0.01 1176 1.0\n prob_n[295] 1.3e-3 9.8e-5 3.4e-3 0.0 7.6e-78 7.0e-19 4.4e-5 0.01 1167 1.0\n prob_n[296] 1.3e-3 9.8e-5 3.3e-3 0.0 2.4e-78 4.6e-19 4.2e-5 0.01 1159 1.0\n prob_n[297] 1.3e-3 9.8e-5 3.3e-3 0.0 7.7e-79 3.1e-19 4.2e-5 0.01 1152 1.0\n prob_n[298] 1.2e-3 9.8e-5 3.3e-3 0.0 2.4e-79 2.0e-19 4.7e-5 0.01 1144 1.0\n prob_n[299] 1.2e-3 9.8e-5 3.3e-3 0.0 7.7e-80 1.3e-19 5.1e-5 0.01 1136 1.0\n prob_n[300] 1.2e-3 9.8e-5 3.3e-3 0.0 2.4e-80 8.6e-20 5.2e-5 0.01 1129 1.0\n prob_n[301] 1.2e-3 9.8e-5 3.3e-3 0.0 7.7e-81 5.7e-20 5.1e-5 0.01 1121 1.0\n prob_n[302] 1.2e-3 9.8e-5 3.3e-3 0.0 2.4e-81 3.7e-20 5.5e-5 0.01 1114 1.0\n prob_n[303] 1.2e-3 9.8e-5 3.3e-3 0.0 7.7e-82 2.4e-20 5.2e-5 0.01 1106 1.0\n prob_n[304] 1.2e-3 9.8e-5 3.2e-3 0.0 2.4e-82 1.6e-20 5.7e-5 0.01 1097 1.0\n prob_n[305] 1.2e-3 9.8e-5 3.2e-3 0.0 7.7e-83 1.0e-20 5.6e-5 0.01 1088 1.0\n prob_n[306] 1.2e-3 9.8e-5 3.2e-3 0.0 2.4e-83 6.7e-21 5.4e-5 0.01 1079 1.0\n prob_n[307] 1.2e-3 9.8e-5 3.2e-3 0.0 7.6e-84 4.4e-21 5.4e-5 0.01 1070 1.0\n prob_n[308] 1.2e-3 9.8e-5 3.2e-3 0.0 2.4e-84 2.8e-21 5.7e-5 0.01 1060 1.0\n prob_n[309] 1.2e-3 9.8e-5 3.2e-3 0.0 7.4e-85 1.8e-21 6.1e-5 0.01 1051 1.0\n prob_n[310] 1.2e-3 9.8e-5 3.2e-3 0.0 2.3e-85 1.2e-21 6.6e-5 0.01 1041 1.0\n prob_n[311] 1.2e-3 9.8e-5 3.2e-3 0.0 7.2e-86 7.7e-22 6.1e-5 0.01 1031 1.0\n prob_n[312] 1.2e-3 9.8e-5 3.1e-3 0.0 2.3e-86 5.0e-22 5.8e-5 0.01 1022 1.0\n prob_n[313] 1.2e-3 9.9e-5 3.1e-3 0.0 7.0e-87 3.2e-22 5.8e-5 0.01 1012 1.0\n prob_n[314] 1.2e-3 9.9e-5 3.1e-3 0.0 2.2e-87 2.1e-22 5.7e-5 0.01 1002 1.0\n prob_n[315] 1.2e-3 9.9e-5 3.1e-3 0.0 6.8e-88 1.3e-22 5.9e-5 0.01 992 1.0\n prob_n[316] 1.2e-3 9.9e-5 3.1e-3 0.0 2.1e-88 8.5e-23 5.9e-5 0.01 982 1.0\n prob_n[317] 1.2e-3 9.9e-5 3.1e-3 0.0 6.6e-89 5.5e-23 6.2e-5 0.01 972 1.0\n prob_n[318] 1.2e-3 10.0e-5 3.1e-3 0.0 2.0e-89 3.5e-23 6.4e-5 0.01 962 1.0\n prob_n[319] 1.2e-3 1.0e-4 3.1e-3 0.0 6.3e-90 2.3e-23 6.9e-5 0.01 952 1.0\n prob_n[320] 1.2e-3 1.0e-4 3.1e-3 0.0 1.9e-90 1.4e-23 7.5e-5 0.01 943 1.0\n prob_n[321] 1.2e-3 1.0e-4 3.1e-3 0.0 6.0e-91 9.2e-24 7.4e-5 0.01 933 1.0\n prob_n[322] 1.2e-3 1.0e-4 3.1e-3 0.0 1.9e-91 5.9e-24 7.5e-5 0.01 924 1.0\n prob_n[323] 1.2e-3 1.0e-4 3.1e-3 0.0 5.7e-92 3.8e-24 7.8e-5 0.01 913 1.0\n prob_n[324] 1.2e-3 1.0e-4 3.1e-3 0.0 1.8e-92 2.4e-24 7.9e-5 0.01 902 1.0\n prob_n[325] 1.2e-3 1.0e-4 3.1e-3 0.0 5.4e-93 1.5e-24 7.0e-5 0.01 890 1.01\n prob_n[326] 1.2e-3 1.0e-4 3.0e-3 0.0 1.7e-93 9.7e-25 7.2e-5 0.01 880 1.01\n prob_n[327] 1.2e-3 1.0e-4 3.0e-3 0.0 5.1e-94 6.2e-25 7.1e-5 0.01 869 1.01\n prob_n[328] 1.2e-3 1.0e-4 3.0e-3 0.0 1.6e-94 3.9e-25 7.3e-5 0.01 858 1.01\n prob_n[329] 1.2e-3 1.0e-4 3.0e-3 0.0 4.8e-95 2.5e-25 7.2e-5 0.01 848 1.01\n prob_n[330] 1.2e-3 1.0e-4 3.0e-3 0.0 1.5e-95 1.6e-25 7.1e-5 0.01 839 1.01\n prob_n[331] 1.2e-3 1.1e-4 3.0e-3 0.0 4.5e-96 9.9e-26 6.7e-5 0.01 826 1.01\n prob_n[332] 1.2e-3 1.1e-4 3.0e-3 0.0 1.4e-96 6.3e-26 6.8e-5 0.01 813 1.01\n prob_n[333] 1.2e-3 1.1e-4 3.0e-3 0.0 4.1e-97 4.0e-26 7.0e-5 0.01 802 1.01\n prob_n[334] 1.2e-3 1.1e-4 3.0e-3 0.0 1.3e-97 2.5e-26 7.3e-5 0.01 790 1.01\n prob_n[335] 1.2e-3 1.1e-4 3.0e-3 0.0 3.8e-98 1.6e-26 7.1e-5 0.01 780 1.01\n prob_n[336] 1.2e-3 1.1e-4 3.0e-3 0.0 1.2e-98 9.9e-27 6.8e-5 0.01 770 1.01\n prob_n[337] 1.2e-3 1.1e-4 3.0e-3 0.0 3.5e-99 6.2e-27 6.7e-5 0.01 760 1.01\n prob_n[338] 1.2e-3 1.1e-4 3.0e-3 0.0 1.1e-99 3.9e-27 6.1e-5 0.01 751 1.01\n prob_n[339] 1.2e-3 1.1e-4 3.0e-3 0.0 3.3e-100 2.4e-27 5.8e-5 0.01 743 1.01\n prob_n[340] 1.2e-3 1.1e-4 3.0e-3 0.0 9.8e-101 1.5e-27 5.8e-5 0.01 735 1.01\n prob_n[341] 1.2e-3 1.1e-4 3.0e-3 0.0 3.0e-101 9.6e-28 5.8e-5 0.01 727 1.01\n prob_n[342] 1.2e-3 1.1e-4 3.0e-3 0.0 9.0e-102 6.0e-28 5.4e-5 0.01 720 1.01\n prob_n[343] 1.2e-3 1.1e-4 3.0e-3 0.0 2.7e-102 3.7e-28 5.6e-5 0.01 714 1.01\n prob_n[344] 1.2e-3 1.1e-4 3.0e-3 0.0 8.2e-103 2.3e-28 5.1e-5 0.01 708 1.01\n prob_n[345] 1.2e-3 1.1e-4 3.0e-3 0.0 2.5e-103 1.5e-28 4.6e-5 0.01 702 1.01\n prob_n[346] 1.2e-3 1.1e-4 3.0e-3 0.0 7.4e-104 9.1e-29 4.2e-5 0.01 697 1.01\n prob_n[347] 1.2e-3 1.1e-4 3.0e-3 0.0 2.2e-104 5.7e-29 4.1e-5 0.01 692 1.01\n prob_n[348] 1.2e-3 1.1e-4 3.0e-3 0.0 6.7e-105 3.5e-29 3.8e-5 0.01 687 1.01\n prob_n[349] 1.2e-3 1.1e-4 3.0e-3 0.0 2.0e-105 2.2e-29 3.7e-5 0.01 683 1.01\n prob_n[350] 1.2e-3 1.2e-4 3.0e-3 0.0 6.0e-106 1.4e-29 3.8e-5 0.01 679 1.01\n prob_n[351] 1.2e-3 1.2e-4 3.0e-3 0.0 1.8e-106 8.4e-30 3.4e-5 0.01 676 1.01\n prob_n[352] 1.2e-3 1.2e-4 3.0e-3 0.0 5.4e-107 5.2e-30 3.3e-5 0.01 673 1.01\n prob_n[353] 1.2e-3 1.2e-4 3.0e-3 0.0 1.6e-107 3.2e-30 3.3e-5 0.01 669 1.01\n prob_n[354] 1.2e-3 1.2e-4 3.0e-3 0.0 4.8e-108 2.0e-30 3.6e-5 0.01 666 1.01\n prob_n[355] 1.2e-3 1.2e-4 3.0e-3 0.0 1.4e-108 1.2e-30 3.9e-5 0.01 664 1.01\n prob_n[356] 1.2e-3 1.2e-4 3.0e-3 0.0 4.3e-109 7.6e-31 4.0e-5 0.01 661 1.01\n prob_n[357] 1.2e-3 1.2e-4 3.0e-3 0.0 1.3e-109 4.7e-31 3.8e-5 0.01 658 1.01\n prob_n[358] 1.2e-3 1.2e-4 3.0e-3 0.0 3.8e-110 2.9e-31 3.7e-5 0.01 656 1.01\n prob_n[359] 1.2e-3 1.2e-4 3.0e-3 0.0 1.1e-110 1.8e-31 3.4e-5 0.01 654 1.01\n prob_n[360] 1.2e-3 1.2e-4 3.0e-3 0.0 3.3e-111 1.1e-31 3.0e-5 0.01 652 1.01\n prob_n[361] 1.2e-3 1.2e-4 3.0e-3 0.0 9.9e-112 6.7e-32 2.9e-5 0.01 650 1.01\n prob_n[362] 1.2e-3 1.2e-4 3.0e-3 0.0 2.9e-112 4.1e-32 3.2e-5 0.01 649 1.01\n prob_n[363] 1.2e-3 1.2e-4 3.0e-3 0.0 8.7e-113 2.5e-32 2.9e-5 0.01 648 1.01\n prob_n[364] 1.2e-3 1.2e-4 3.0e-3 0.0 2.6e-113 1.5e-32 2.6e-5 0.01 647 1.01\n prob_n[365] 1.2e-3 1.2e-4 2.9e-3 0.0 7.6e-114 9.5e-33 2.3e-5 0.01 646 1.01\n prob_n[366] 1.2e-3 1.2e-4 2.9e-3 0.0 2.2e-114 5.8e-33 2.0e-5 0.01 646 1.01\n prob_n[367] 1.2e-3 1.2e-4 2.9e-3 0.0 6.6e-115 3.5e-33 2.0e-5 0.01 646 1.01\n prob_n[368] 1.2e-3 1.2e-4 2.9e-3 0.0 2.0e-115 2.2e-33 1.9e-5 0.01 645 1.01\n prob_n[369] 1.2e-3 1.2e-4 2.9e-3 0.0 5.8e-116 1.3e-33 1.7e-5 0.01 644 1.01\n prob_n[370] 1.2e-3 1.2e-4 2.9e-3 0.0 1.7e-116 8.0e-34 1.5e-5 0.01 642 1.01\n prob_n[371] 1.2e-3 1.2e-4 2.9e-3 0.0 5.0e-117 4.9e-34 1.3e-5 0.01 640 1.0\n prob_n[372] 1.2e-3 1.2e-4 2.9e-3 0.0 1.5e-117 3.0e-34 1.3e-5 0.01 638 1.0\n prob_n[373] 1.2e-3 1.2e-4 2.9e-3 0.0 4.3e-118 1.8e-34 1.1e-5 0.01 636 1.0\n prob_n[374] 1.2e-3 1.2e-4 2.9e-3 0.0 1.3e-118 1.1e-34 10.0e-6 0.01 633 1.0\n prob_n[375] 1.2e-3 1.2e-4 2.9e-3 0.0 3.7e-119 6.7e-35 8.8e-6 0.01 631 1.0\n prob_n[376] 1.2e-3 1.2e-4 2.9e-3 0.0 1.1e-119 4.1e-35 7.8e-6 0.01 629 1.0\n prob_n[377] 1.2e-3 1.2e-4 2.9e-3 0.0 3.2e-120 2.5e-35 6.8e-6 0.01 627 1.0\n prob_n[378] 1.2e-3 1.2e-4 2.9e-3 0.0 9.3e-121 1.5e-35 6.5e-6 0.01 624 1.0\n prob_n[379] 1.2e-3 1.2e-4 2.9e-3 0.0 2.7e-121 9.0e-36 7.2e-6 0.01 622 1.0\n prob_n[380] 1.2e-3 1.2e-4 2.9e-3 0.0 7.9e-122 5.4e-36 7.9e-6 0.01 620 1.0\n prob_n[381] 1.2e-3 1.2e-4 2.9e-3 0.0 2.3e-122 3.3e-36 7.1e-6 0.01 618 1.0\n prob_n[382] 1.2e-3 1.2e-4 2.9e-3 0.0 6.8e-123 2.0e-36 6.3e-6 0.01 616 1.0\n prob_n[383] 1.2e-3 1.2e-4 2.9e-3 0.0 2.0e-123 1.2e-36 5.6e-6 0.01 614 1.0\n prob_n[384] 1.2e-3 1.2e-4 2.8e-3 0.0 5.7e-124 7.2e-37 4.9e-6 0.01 612 1.0\n prob_n[385] 1.2e-3 1.2e-4 2.8e-3 0.0 1.7e-124 4.4e-37 4.3e-6 0.01 610 1.0\n prob_n[386] 1.2e-3 1.2e-4 2.8e-3 0.0 4.8e-125 2.6e-37 3.8e-6 0.01 607 1.0\n prob_n[387] 1.2e-3 1.2e-4 2.8e-3 0.0 1.4e-125 1.6e-37 3.3e-6 0.01 605 1.0\n prob_n[388] 1.2e-3 1.2e-4 2.8e-3 0.0 4.1e-126 9.5e-38 2.9e-6 0.01 603 1.0\n prob_n[389] 1.2e-3 1.2e-4 2.8e-3 0.0 1.2e-126 5.7e-38 2.6e-6 0.01 602 1.0\n prob_n[390] 1.2e-3 1.2e-4 2.8e-3 0.0 3.4e-127 3.4e-38 2.3e-6 0.01 600 1.0\n prob_n[391] 1.2e-3 1.2e-4 2.8e-3 0.0 9.9e-128 2.0e-38 2.0e-6 0.01 598 1.0\n prob_n[392] 1.2e-3 1.2e-4 2.8e-3 0.0 2.9e-128 1.2e-38 1.7e-6 0.01 596 1.0\n prob_n[393] 1.2e-3 1.2e-4 2.8e-3 0.0 8.3e-129 7.3e-39 1.5e-6 0.01 594 1.0\n prob_n[394] 1.2e-3 1.2e-4 2.8e-3 0.0 2.4e-129 4.4e-39 1.3e-6 0.01 592 1.0\n prob_n[395] 1.2e-3 1.2e-4 2.8e-3 0.0 6.9e-130 2.6e-39 1.2e-6 0.01 591 1.0\n prob_n[396] 1.1e-3 1.2e-4 2.8e-3 0.0 2.0e-130 1.6e-39 1.0e-6 0.01 589 1.0\n prob_n[397] 1.1e-3 1.2e-4 2.8e-3 0.0 5.8e-131 9.3e-40 8.8e-7 0.01 587 1.0\n prob_n[398] 1.1e-3 1.2e-4 2.8e-3 0.0 1.7e-131 5.6e-40 7.7e-7 0.01 586 1.0\n prob_n[399] 1.1e-3 1.2e-4 2.8e-3 0.0 4.8e-132 3.3e-40 6.7e-7 0.01 584 1.0\n prob_n[400] 1.1e-3 1.2e-4 2.8e-3 0.0 1.4e-132 2.0e-40 5.8e-7 0.01 583 1.0\n prob_n[401] 1.1e-3 1.2e-4 2.8e-3 0.0 4.0e-133 1.2e-40 5.1e-7 0.01 581 1.0\n prob_n[402] 1.1e-3 1.2e-4 2.8e-3 0.0 1.1e-133 7.0e-41 4.4e-7 0.01 580 1.0\n prob_n[403] 1.1e-3 1.2e-4 2.8e-3 0.0 3.3e-134 4.2e-41 3.8e-7 0.01 579 1.0\n prob_n[404] 1.1e-3 1.2e-4 2.8e-3 0.0 9.3e-135 2.5e-41 3.3e-7 0.01 577 1.0\n prob_n[405] 1.1e-3 1.2e-4 2.8e-3 0.0 2.7e-135 1.5e-41 2.9e-7 0.01 576 1.0\n prob_n[406] 1.1e-3 1.2e-4 2.8e-3 0.0 7.7e-136 8.7e-42 2.5e-7 0.01 575 1.0\n prob_n[407] 1.1e-3 1.2e-4 2.8e-3 0.0 2.2e-136 5.2e-42 2.2e-7 0.01 573 1.0\n prob_n[408] 1.1e-3 1.2e-4 2.8e-3 0.0 6.3e-137 3.1e-42 1.9e-7 0.01 572 1.0\n prob_n[409] 1.1e-3 1.2e-4 2.8e-3 0.0 1.8e-137 1.8e-42 1.6e-7 0.01 571 1.0\n prob_n[410] 1.1e-3 1.2e-4 2.8e-3 0.0 5.1e-138 1.1e-42 1.4e-7 0.01 570 1.0\n prob_n[411] 1.1e-3 1.2e-4 2.7e-3 0.0 1.5e-138 6.3e-43 1.2e-7 0.01 569 1.0\n prob_n[412] 1.1e-3 1.2e-4 2.7e-3 0.0 4.2e-139 3.7e-43 1.0e-7 0.01 568 1.0\n prob_n[413] 1.1e-3 1.2e-4 2.7e-3 0.0 1.2e-139 2.2e-43 9.0e-8 0.01 567 1.0\n prob_n[414] 1.1e-3 1.2e-4 2.7e-3 0.0 3.4e-140 1.3e-43 7.8e-8 0.01 566 1.0\n prob_n[415] 1.1e-3 1.2e-4 2.7e-3 0.0 9.7e-141 7.7e-44 6.7e-8 0.01 565 1.0\n prob_n[416] 1.1e-3 1.2e-4 2.7e-3 0.0 2.7e-141 4.5e-44 5.8e-8 0.01 564 1.0\n prob_n[417] 1.1e-3 1.2e-4 2.7e-3 0.0 7.8e-142 2.7e-44 5.0e-8 0.01 563 1.0\n prob_n[418] 1.1e-3 1.2e-4 2.7e-3 0.0 2.2e-142 1.6e-44 4.3e-8 0.01 562 1.0\n prob_n[419] 1.1e-3 1.2e-4 2.7e-3 0.0 6.3e-143 9.2e-45 3.7e-8 0.01 561 1.0\n prob_n[420] 1.1e-3 1.2e-4 2.7e-3 0.0 1.8e-143 5.4e-45 3.2e-8 0.01 560 1.0\n prob_n[421] 1.1e-3 1.2e-4 2.7e-3 0.0 5.1e-144 3.2e-45 2.7e-8 0.01 559 1.0\n prob_n[422] 1.1e-3 1.2e-4 2.7e-3 0.0 1.4e-144 1.9e-45 2.3e-8 0.01 558 1.0\n prob_n[423] 1.1e-3 1.1e-4 2.7e-3 0.0 4.1e-145 1.1e-45 2.0e-8 0.01 557 1.0\n prob_n[424] 1.1e-3 1.1e-4 2.7e-3 0.0 1.2e-145 6.4e-46 1.7e-8 0.01 556 1.0\n prob_n[425] 1.0e-3 1.1e-4 2.7e-3 0.0 3.3e-146 3.7e-46 1.5e-8 0.01 555 1.0\n prob_n[426] 1.0e-3 1.1e-4 2.7e-3 0.0 9.2e-147 2.2e-46 1.3e-8 0.01 554 1.0\n prob_n[427] 1.0e-3 1.1e-4 2.7e-3 0.0 2.6e-147 1.3e-46 1.1e-8 0.01 553 1.0\n prob_n[428] 1.0e-3 1.1e-4 2.7e-3 0.0 7.3e-148 7.5e-47 9.1e-9 0.01 552 1.0\n prob_n[429] 1.0e-3 1.1e-4 2.7e-3 0.0 2.1e-148 4.4e-47 7.8e-9 0.01 551 1.0\n prob_n[430] 1.0e-3 1.1e-4 2.7e-3 0.0 5.8e-149 2.6e-47 6.7e-9 0.01 550 1.0\n prob_n[431] 1.0e-3 1.1e-4 2.7e-3 0.0 1.6e-149 1.5e-47 5.7e-9 0.01 549 1.0\n prob_n[432] 1.0e-3 1.1e-4 2.7e-3 0.0 4.6e-150 8.7e-48 4.8e-9 0.01 548 1.0\n prob_n[433] 1.0e-3 1.1e-4 2.7e-3 0.0 1.3e-150 5.1e-48 4.1e-9 0.01 547 1.0\n prob_n[434] 1.0e-3 1.1e-4 2.7e-3 0.0 3.7e-151 3.0e-48 3.5e-9 0.01 546 1.0\n prob_n[435] 10.0e-4 1.1e-4 2.7e-3 0.0 1.0e-151 1.7e-48 3.0e-9 0.01 545 1.0\n prob_n[436] 9.9e-4 1.1e-4 2.7e-3 0.0 2.9e-152 1.0e-48 2.5e-9 0.01 544 1.0\n prob_n[437] 9.9e-4 1.1e-4 2.7e-3 0.0 8.1e-153 5.8e-49 2.2e-9 0.01 543 1.0\n prob_n[438] 9.8e-4 1.1e-4 2.7e-3 0.0 2.3e-153 3.4e-49 1.8e-9 0.01 542 1.0\n prob_n[439] 9.8e-4 1.1e-4 2.7e-3 0.0 6.4e-154 2.0e-49 1.6e-9 0.01 541 1.0\n prob_n[440] 9.7e-4 1.1e-4 2.7e-3 0.0 1.8e-154 1.1e-49 1.3e-9 0.01 540 1.0\n prob_n[441] 9.7e-4 1.1e-4 2.7e-3 0.0 5.0e-155 6.6e-50 1.1e-9 0.01 539 1.0\n prob_n[442] 9.6e-4 1.1e-4 2.7e-3 0.0 1.4e-155 3.8e-50 9.5e-10 0.01 538 1.0\n prob_n[443] 9.6e-4 1.1e-4 2.7e-3 0.0 3.9e-156 2.2e-50 8.0e-10 0.01 537 1.0\n prob_n[444] 9.5e-4 1.1e-4 2.7e-3 0.0 1.1e-156 1.3e-50 6.8e-10 0.01 536 1.0\n prob_n[445] 9.5e-4 1.1e-4 2.7e-3 0.0 3.1e-157 7.4e-51 5.8e-10 0.01 535 1.0\n prob_n[446] 9.4e-4 1.1e-4 2.7e-3 0.0 8.6e-158 4.3e-51 4.9e-10 0.01 534 1.0\n prob_n[447] 9.4e-4 1.1e-4 2.6e-3 0.0 2.4e-158 2.5e-51 4.1e-10 0.01 533 1.0\n prob_n[448] 9.3e-4 1.1e-4 2.6e-3 0.0 6.7e-159 1.4e-51 3.5e-10 0.01 532 1.0\n prob_n[449] 9.3e-4 1.1e-4 2.6e-3 0.0 1.9e-159 8.3e-52 2.9e-10 0.01 531 1.0\n prob_n[450] 9.2e-4 1.1e-4 2.6e-3 0.0 5.2e-160 4.8e-52 2.5e-10 0.01 530 1.0\n prob_n[451] 9.2e-4 1.1e-4 2.6e-3 0.0 1.5e-160 2.8e-52 2.1e-10 0.01 529 1.0\n prob_n[452] 9.1e-4 1.2e-4 2.6e-3 0.0 4.0e-161 1.6e-52 1.8e-10 0.01 528 1.0\n prob_n[453] 9.1e-4 1.2e-4 2.6e-3 0.0 1.1e-161 9.2e-53 1.5e-10 0.01 527 1.0\n prob_n[454] 9.0e-4 1.2e-4 2.6e-3 0.0 3.1e-162 5.3e-53 1.2e-10 0.01 525 1.0\n prob_n[455] 9.0e-4 1.2e-4 2.6e-3 0.0 8.7e-163 3.0e-53 1.0e-10 0.01 522 1.0\n prob_n[456] 8.9e-4 1.2e-4 2.6e-3 0.0 2.4e-163 1.7e-53 8.8e-11 0.01 518 1.0\n prob_n[457] 8.9e-4 1.2e-4 2.6e-3 0.0 6.7e-164 1.0e-53 7.4e-11 0.01 515 1.0\n prob_n[458] 8.8e-4 1.2e-4 2.6e-3 0.0 1.9e-164 5.8e-54 6.2e-11 0.01 512 1.0\n prob_n[459] 8.8e-4 1.2e-4 2.7e-3 0.0 5.2e-165 3.3e-54 5.2e-11 0.01 508 1.0\n prob_n[460] 8.7e-4 1.2e-4 2.7e-3 0.0 1.4e-165 1.9e-54 4.4e-11 0.01 505 1.0\n prob_n[461] 8.7e-4 1.2e-4 2.7e-3 0.0 4.0e-166 1.1e-54 3.7e-11 0.01 501 1.0\n prob_n[462] 8.6e-4 1.2e-4 2.7e-3 0.0 1.1e-166 6.2e-55 3.1e-11 0.01 497 1.0\n prob_n[463] 8.6e-4 1.2e-4 2.7e-3 0.0 3.0e-167 3.6e-55 2.6e-11 0.01 493 1.0\n prob_n[464] 8.5e-4 1.2e-4 2.7e-3 0.0 8.4e-168 2.1e-55 2.2e-11 0.01 488 1.0\n prob_n[465] 8.5e-4 1.2e-4 2.7e-3 0.0 2.3e-168 1.2e-55 1.8e-11 0.01 484 1.0\n prob_n[466] 8.4e-4 1.2e-4 2.7e-3 0.0 6.4e-169 6.7e-56 1.5e-11 0.01 479 1.0\n prob_n[467] 8.4e-4 1.2e-4 2.7e-3 0.0 1.8e-169 3.8e-56 1.3e-11 0.01 474 1.0\n prob_n[468] 8.3e-4 1.2e-4 2.7e-3 0.0 4.9e-170 2.2e-56 1.1e-11 0.01 469 1.0\n prob_n[469] 8.3e-4 1.3e-4 2.7e-3 0.0 1.3e-170 1.3e-56 8.8e-12 0.01 464 1.0\n prob_n[470] 8.3e-4 1.3e-4 2.7e-3 0.0 3.7e-171 7.2e-57 7.3e-12 0.01 458 1.0\n prob_n[471] 8.2e-4 1.3e-4 2.7e-3 0.0 1.0e-171 4.1e-57 6.1e-12 0.01 453 1.0\n prob_n[472] 8.2e-4 1.3e-4 2.8e-3 0.0 2.8e-172 2.3e-57 5.1e-12 0.01 447 1.0\n prob_n[473] 8.2e-4 1.3e-4 2.8e-3 0.0 7.7e-173 1.3e-57 4.3e-12 0.01 441 1.0\n prob_n[474] 8.1e-4 1.3e-4 2.8e-3 0.0 2.1e-173 7.6e-58 3.5e-12 0.01 436 1.01\n prob_n[475] 8.1e-4 1.4e-4 2.8e-3 0.0 5.9e-174 4.3e-58 2.9e-12 0.01 430 1.01\n prob_n[476] 8.0e-4 1.4e-4 2.8e-3 0.0 1.6e-174 2.4e-58 2.5e-12 0.01 424 1.01\n prob_n[477] 8.0e-4 1.4e-4 2.8e-3 0.0 4.4e-175 1.4e-58 2.0e-12 0.01 417 1.01\n prob_n[478] 8.0e-4 1.4e-4 2.9e-3 0.0 1.2e-175 7.9e-59 1.7e-12 0.01 411 1.01\n prob_n[479] 8.0e-4 1.4e-4 2.9e-3 0.0 3.3e-176 4.5e-59 1.4e-12 0.01 405 1.01\n prob_n[480] 7.9e-4 1.5e-4 2.9e-3 0.0 9.2e-177 2.6e-59 1.2e-12 0.01 399 1.01\n prob_n[481] 7.9e-4 1.5e-4 2.9e-3 0.0 2.5e-177 1.5e-59 9.7e-13 0.01 392 1.01\n prob_n[482] 7.9e-4 1.5e-4 3.0e-3 0.0 6.9e-178 8.2e-60 8.1e-13 0.01 386 1.01\n prob_n[483] 7.9e-4 1.5e-4 3.0e-3 0.0 1.9e-178 4.7e-60 6.7e-13 0.01 380 1.01\n prob_n[484] 7.8e-4 1.6e-4 3.1e-3 0.0 5.2e-179 2.6e-60 5.5e-13 0.01 373 1.01\n prob_n[485] 7.8e-4 1.6e-4 3.1e-3 0.0 1.4e-179 1.5e-60 4.6e-13 0.01 367 1.01\n prob_n[486] 7.8e-4 1.7e-4 3.1e-3 0.0 3.9e-180 8.5e-61 3.8e-13 0.01 361 1.01\n prob_n[487] 7.8e-4 1.7e-4 3.2e-3 0.0 1.1e-180 4.8e-61 3.1e-13 0.01 355 1.01\n prob_n[488] 7.8e-4 1.7e-4 3.2e-3 0.0 2.9e-181 2.7e-61 2.6e-13 0.01 349 1.01\n prob_n[489] 7.8e-4 1.8e-4 3.3e-3 0.0 7.9e-182 1.5e-61 2.1e-13 0.01 343 1.01\n prob_n[490] 7.8e-4 1.8e-4 3.3e-3 0.0 2.2e-182 8.7e-62 1.8e-13 0.01 338 1.01\n prob_n[491] 7.8e-4 1.9e-4 3.4e-3 0.0 5.9e-183 4.9e-62 1.5e-13 0.01 332 1.01\n prob_n[492] 7.8e-4 1.9e-4 3.5e-3 0.0 1.6e-183 2.8e-62 1.2e-13 0.01 327 1.01\n prob_n[493] 7.8e-4 2.0e-4 3.5e-3 0.0 4.4e-184 1.6e-62 10.0e-14 0.01 322 1.01\n prob_n[494] 7.8e-4 2.0e-4 3.6e-3 0.0 1.2e-184 8.8e-63 8.2e-14 0.01 317 1.01\n prob_n[495] 7.8e-4 2.1e-4 3.7e-3 0.0 3.2e-185 5.0e-63 6.8e-14 0.01 312 1.01\n prob_n[496] 7.8e-4 2.1e-4 3.8e-3 0.0 8.8e-186 2.8e-63 5.6e-14 0.01 307 1.01\n prob_n[497] 7.8e-4 2.2e-4 3.8e-3 0.0 2.4e-186 1.6e-63 4.6e-14 0.01 303 1.01\n prob_n[498] 7.8e-4 2.3e-4 3.9e-3 0.0 6.5e-187 8.9e-64 3.8e-14 0.01 299 1.01\n prob_n[499] 7.8e-4 2.3e-4 4.0e-3 0.0 1.8e-187 5.0e-64 3.1e-14 0.01 294 1.01\n prob_n[500] 7.8e-4 2.4e-4 4.1e-3 0.0 4.8e-188 2.8e-64 2.6e-14 10.0e-3 291 1.01\n lp__ -19.19 0.03 0.53 -20.55 -19.24 -19.01 -18.92 -18.87 369 1.01\n \n Samples were drawn using NUTS at Tue Feb 25 10:53:16 2020.\n For each parameter, n_eff is a crude measure of effective sample size,\n and Rhat is the potential scale reduction factor on split chains (at \n convergence, Rhat=1).\n\n\n\n```python\n\nplt.hist(fit['theta'])\nplt.xlabel('Theta')\nplt.ylabel('Number of Samples')\nplt.show\n\n\n```\n\n\n```python\n\nplt.plot(fit['theta'], fit['n'], '.')\nplt.xlabel('Theta')\nplt.ylabel('Number of Samples')\nplt.show\n```\n\n

Finally let us consider a coin tossing example from earlier, but in this case we will have run the experiment twice. We will then use data from both experiments to fit our head rates.

\n \n
\n\n\n```python\ntwo_exp_rate = \"\"\"// Inferring a Common Rate, With Posterior Predictive\ndata { \n int n1; \n int n2; \n int k1;\n int k2;\n}\nparameters {\n real theta;\n} \nmodel {\n // Prior on Single Rate Theta\n theta ~ beta(1, 1);\n \n // Observed Counts\n k1 ~ binomial(n1, theta);\n k2 ~ binomial(n2, theta);\n}\"\"\"\n\nsm=pystan.StanModel(model_code=two_exp_rate)\n```\n\n INFO:pystan:COMPILING THE C++ CODE FOR MODEL anon_model_bd8f6a5f874a46897ced59e716bf395f NOW.\n\n\n\n```python\ndata_1 = {\n'k1': 4,\n'k2': 6,\n'n1': 10,\n'n2': 10}\n\ndata_2 = {\n'k1': 0,\n'k2': 10,\n'n1': 10,\n'n2': 10}\n```\n\n\n```python\nfit1=sm.sampling(data=data_1,chains=4,iter=10000,seed=12329)\n\n\nplt.hist(fit1['theta'])\nplt.show()\n\nfit2=sm.sampling(data=data_2,chains=4,iter=10000,seed=12329)\n\n\nplt.hist(fit2['theta'])\nplt.show()\n\n\n\n```\n\n\n```python\ntwo_exp_rate_gen = \"\"\"// Inferring a Common Rate, With Posterior Predictive\ndata { \n int n1; \n int n2; \n int k1;\n int k2;\n}\nparameters {\n real theta;\n} \nmodel {\n // Prior on Single Rate Theta\n theta ~ beta(1, 1);\n \n // Observed Counts\n k1 ~ binomial(n1, theta);\n k2 ~ binomial(n2, theta);\n}\ngenerated quantities {\n int postpredk1;\n int postpredk2;\n \n // Posterior Predictive\n postpredk1 <- binomial_rng(n1, theta);\n postpredk2 <- binomial_rng(n2, theta);\n}\"\"\"\n\nsm=pystan.StanModel(model_code=two_exp_rate_gen)\n```\n\n INFO:pystan:COMPILING THE C++ CODE FOR MODEL anon_model_1070d695aa6003acbae0397bae8b9e8b NOW.\n\n\n\n```python\nfit1=sm.sampling(data=data_1,chains=4,iter=10000,seed=12329)\n\n\nplt.hist(fit1['theta'])\nplt.show()\n\nfit2=sm.sampling(data=data_2,chains=4,iter=10000,seed=12329)\n\n\nplt.hist(fit2['theta'])\nplt.show()\n\n```\n\n\n```python\nplt.hist(fit1['postpredk1'])\n```\n\n\n```python\nheatmap = np.zeros([11,11])\n\nfor x in np.arange(2000):\n heatmap[int(fit2['postpredk1'][x]), int(fit2['postpredk2'][x])] += 1\n```\n\n\n```python\nplt.imshow(heatmap, cmap='hot', interpolation='nearest')\nplt.show()\n```\n\n

Not only is it useful to understand your model before you fit it, but to understand how well your data is explained by your model. Part of this process is creative but there are also tools that exist that can help.

\n\n\n```python\n\n```\n\n\n```python\nx=np.random.normal(3,1,10)\n```\n\n\n```python\nx\n```\n\n\n\n\n array([4.48581462, 3.935212 , 2.13096659, 2.43710064, 5.40903363,\n 2.33496308, 1.64093622, 2.86598126, 3.23251945, 2.31631115])\n\n\n\n\n```python\nmodel=\"\"\"\ndata {\nint N; //number of datapoints\nreal x[N]; //x values\n}\n\n\nparameters {\nreal mu; //mean of normal distribution\nreal sigma;\n}\n\nmodel {\n\nsigma ~ cauchy(0,1);\nx ~ normal(mu,sigma);\n}\n\"\"\"\n```\n\n\n```python\nsm=pystan.StanModel(model_code=model)\n```\n\n INFO:pystan:COMPILING THE C++ CODE FOR MODEL anon_model_23840a14d34f49990979ae476b61c011 NOW.\n\n\n\n```python\ndata={\n 'N':10,\n 'x':x\n}\n```\n\n\n```python\nfit=sm.sampling(data=data,chains=4)\n```\n\n WARNING:pystan:1 of 4000 iterations saturated the maximum tree depth of 10 (0.025 %)\n WARNING:pystan:Run again with max_treedepth larger than 10 to avoid saturation\n\n\n\n```python\nfit\n```\n\n\n\n\n Inference for Stan model: anon_model_23840a14d34f49990979ae476b61c011.\n 4 chains, each with iter=2000; warmup=1000; thin=1; \n post-warmup draws per chain=1000, total post-warmup draws=4000.\n \n mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat\n mu 3.07 9.2e-3 0.42 2.24 2.79 3.07 3.34 3.93 2136 1.0\n sigma 1.29 7.5e-3 0.34 0.82 1.06 1.22 1.45 2.13 1993 1.0\n lp__ -8.02 0.03 1.1 -11.03 -8.45 -7.7 -7.23 -6.93 1308 1.0\n \n Samples were drawn using NUTS at Tue Feb 25 12:30:50 2020.\n For each parameter, n_eff is a crude measure of effective sample size,\n and Rhat is the potential scale reduction factor on split chains (at \n convergence, Rhat=1).\n\n\n\n\n```python\nplt.hist(fit['sigma'])\n```\n\n\n```python\nplt.scatter(fit['mu'],fit['sigma'])\n```\n\n\n```python\nimport seaborn as sns\nimport pandas as pd\n```\n\n\n```python\ndf=pd.DataFrame(np.vstack((fit['mu'],fit['sigma'])).T,columns=[r'$\\mu$',r'$\\sigma$'])\n\n```\n\n\n```python\ndf[0:5]\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
$\\mu$$\\sigma$
02.7210631.731366
12.9501221.320691
23.2106561.188615
33.2306991.271445
43.5946631.501862
\n
\n\n\n\n\n```python\ng=sns.PairGrid(df)\ng.map_diag(plt.hist)\ng.map_lower(sns.kdeplot)\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "fa7a73cef243ab28d2951727843c4354dcac594d", "size": 330002, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Mindlab_Course_PartI.ipynb", "max_stars_repo_name": "DataJavelin/MindLab_training", "max_stars_repo_head_hexsha": "d09fb753db994df5de253aa28bf9386c87a2504c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Mindlab_Course_PartI.ipynb", "max_issues_repo_name": "DataJavelin/MindLab_training", "max_issues_repo_head_hexsha": "d09fb753db994df5de253aa28bf9386c87a2504c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Mindlab_Course_PartI.ipynb", "max_forks_repo_name": "DataJavelin/MindLab_training", "max_forks_repo_head_hexsha": "d09fb753db994df5de253aa28bf9386c87a2504c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 114.6636553162, "max_line_length": 24620, "alphanum_fraction": 0.7315743541, "converted": true, "num_tokens": 70216, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.7090191337850933, "lm_q1q2_score": 0.4439340217803159}} {"text": "```python\nimport numpy as np\nimport numba\nimport matplotlib.pyplot as plt\nimport sympy as sym\nplt.style.use('presentation.mplstyle')\n%matplotlib notebook\n\ndef d2np(d):\n \n names = []\n numbers = ()\n dtypes = []\n for item in d:\n names += item \n if type(d[item]) == float:\n numbers += (d[item],)\n dtypes += [(item,float)]\n if type(d[item]) == int:\n numbers += (d[item],)\n dtypes += [(item,int)]\n if type(d[item]) == np.ndarray:\n numbers += (d[item],)\n dtypes += [(item,np.float64,d[item].shape)]\n return np.array([numbers],dtype=dtypes)\n```\n\n\n```python\ni_1d,i_1q,i_2d,i_2q,v_cd,v_cq = sym.symbols('i_1d,i_1q,i_2d,i_2q,v_cd,v_cq') \ndi_1d,di_1q,di_2d,di_2q,dv_cd,dv_cq = sym.symbols('di_1d,di_1q,di_2d,di_2q,dv_cd,dv_cq') \nL_1,L_2,C_ac,C_dc = sym.symbols('L_1,L_2,C_ac,C_dc') \nR_1,R_2 = sym.symbols('R_1,R_2') \nomega,dummy = sym.symbols('omega,dummy') \nv_sd,v_sq,v_dc= sym.symbols('v_sd,v_sq,v_dc') \neta_d,eta_q = sym.symbols('eta_d,eta_q') \np_ref,q_ref = sym.symbols('p_ref,q_ref') \ni_2d_ref,i_2q_ref = sym.symbols('i_2d_ref,i_2q_ref') \n\n#i_2d = i_2d_ref\n#i_2q = i_2q_ref\n\ndi_1d = 1/L_1*(0.5*eta_d*v_dc - R_1*i_1d + L_1*omega*i_1q - v_cd)\ndi_1q = 1/L_1*(0.5*eta_q*v_dc - R_1*i_1q - L_1*omega*i_1d - v_cq)\ndv_cd = 1/C_ac*( i_1d + C_ac*omega*v_cq - i_2d)\ndv_cq = 1/C_ac*( i_1q - C_ac*omega*v_cd - i_2q) \ndi_2d = 1/L_2*(v_cd - R_2*i_2d + L_2*omega*i_2q - v_sd)\ndi_2q = 1/L_2*(v_cq - R_2*i_2q - L_2*omega*i_2d - v_sq)\n\n\n''' \n'''\n\n\n```\n\n\n\n\n ' \\n'\n\n\n\n\n```python\ns = sym.solve([ di_1d, di_1q, dv_cd, dv_cq, di_2d, di_2q],\n [ i_1d, i_1q, v_cd, v_cq, i_2d, i_2q])\n\nfor item in s:\n print(item, '=', sym.simplify(s[item]))\n```\n\n i_1d = (0.5*eta_d*v_dc*(C_ac**2*L_2**2*R_1*omega**4 + C_ac**2*R_1*R_2**2*omega**2 - 2.0*C_ac*L_2*R_1*omega**2 + R_1 + R_2) + 0.5*eta_q*omega*v_dc*(C_ac**2*L_1*L_2**2*omega**4 + C_ac**2*L_1*R_2**2*omega**2 - 2.0*C_ac*L_1*L_2*omega**2 - C_ac*L_2**2*omega**2 - C_ac*R_2**2 + L_1 + L_2) - omega*v_sq*(-C_ac*L_1*L_2*omega**2 + C_ac*R_1*R_2 + L_1 + L_2) + v_sd*(C_ac*L_1*R_2*omega**2 + C_ac*L_2*R_1*omega**2 - R_1 - R_2))/(C_ac**2*L_1**2*L_2**2*omega**6 + C_ac**2*L_1**2*R_2**2*omega**4 + C_ac**2*L_2**2*R_1**2*omega**4 + C_ac**2*R_1**2*R_2**2*omega**2 - 2.0*C_ac*L_1**2*L_2*omega**4 - 2.0*C_ac*L_1*L_2**2*omega**4 - 2.0*C_ac*L_1*R_2**2*omega**2 - 2.0*C_ac*L_2*R_1**2*omega**2 + L_1**2*omega**2 + 2.0*L_1*L_2*omega**2 + L_2**2*omega**2 + R_1**2 + 2.0*R_1*R_2 + R_2**2)\n i_1q = (-0.5*eta_d*omega*v_dc*(C_ac**2*L_1*L_2**2*omega**4 + C_ac**2*L_1*R_2**2*omega**2 - 2.0*C_ac*L_1*L_2*omega**2 - C_ac*L_2**2*omega**2 - C_ac*R_2**2 + L_1 + L_2) + 0.5*eta_q*v_dc*(C_ac**2*L_2**2*R_1*omega**4 + C_ac**2*R_1*R_2**2*omega**2 - 2.0*C_ac*L_2*R_1*omega**2 + R_1 + R_2) + omega*v_sd*(-C_ac*L_1*L_2*omega**2 + C_ac*R_1*R_2 + L_1 + L_2) + v_sq*(C_ac*L_1*R_2*omega**2 + C_ac*L_2*R_1*omega**2 - R_1 - R_2))/(C_ac**2*L_1**2*L_2**2*omega**6 + C_ac**2*L_1**2*R_2**2*omega**4 + C_ac**2*L_2**2*R_1**2*omega**4 + C_ac**2*R_1**2*R_2**2*omega**2 - 2.0*C_ac*L_1**2*L_2*omega**4 - 2.0*C_ac*L_1*L_2**2*omega**4 - 2.0*C_ac*L_1*R_2**2*omega**2 - 2.0*C_ac*L_2*R_1**2*omega**2 + L_1**2*omega**2 + 2.0*L_1*L_2*omega**2 + L_2**2*omega**2 + R_1**2 + 2.0*R_1*R_2 + R_2**2)\n v_cd = (0.5*eta_d*v_dc*(-C_ac*L_1*L_2**2*omega**4 - C_ac*L_1*R_2**2*omega**2 + L_1*L_2*omega**2 + L_2**2*omega**2 + R_1*R_2 + R_2**2) + 0.5*eta_q*omega*v_dc*(C_ac*L_2**2*R_1*omega**2 + C_ac*R_1*R_2**2 + L_1*R_2 - L_2*R_1) + omega*v_sq*(C_ac*L_1**2*R_2*omega**2 + C_ac*R_1**2*R_2 - L_1*R_2 + L_2*R_1) + v_sd*(-C_ac*L_1**2*L_2*omega**4 - C_ac*L_2*R_1**2*omega**2 + L_1**2*omega**2 + L_1*L_2*omega**2 + R_1**2 + R_1*R_2))/(C_ac**2*L_1**2*L_2**2*omega**6 + C_ac**2*L_1**2*R_2**2*omega**4 + C_ac**2*L_2**2*R_1**2*omega**4 + C_ac**2*R_1**2*R_2**2*omega**2 - 2.0*C_ac*L_1**2*L_2*omega**4 - 2.0*C_ac*L_1*L_2**2*omega**4 - 2.0*C_ac*L_1*R_2**2*omega**2 - 2.0*C_ac*L_2*R_1**2*omega**2 + L_1**2*omega**2 + 2.0*L_1*L_2*omega**2 + L_2**2*omega**2 + R_1**2 + 2.0*R_1*R_2 + R_2**2)\n v_cq = (-0.5*eta_d*omega*v_dc*(C_ac*L_2**2*R_1*omega**2 + C_ac*R_1*R_2**2 + L_1*R_2 - L_2*R_1) + 0.5*eta_q*v_dc*(-C_ac*L_1*L_2**2*omega**4 - C_ac*L_1*R_2**2*omega**2 + L_1*L_2*omega**2 + L_2**2*omega**2 + R_1*R_2 + R_2**2) - omega*v_sd*(C_ac*L_1**2*R_2*omega**2 + C_ac*R_1**2*R_2 - L_1*R_2 + L_2*R_1) + v_sq*(-C_ac*L_1**2*L_2*omega**4 - C_ac*L_2*R_1**2*omega**2 + L_1**2*omega**2 + L_1*L_2*omega**2 + R_1**2 + R_1*R_2))/(C_ac**2*L_1**2*L_2**2*omega**6 + C_ac**2*L_1**2*R_2**2*omega**4 + C_ac**2*L_2**2*R_1**2*omega**4 + C_ac**2*R_1**2*R_2**2*omega**2 - 2.0*C_ac*L_1**2*L_2*omega**4 - 2.0*C_ac*L_1*L_2**2*omega**4 - 2.0*C_ac*L_1*R_2**2*omega**2 - 2.0*C_ac*L_2*R_1**2*omega**2 + L_1**2*omega**2 + 2.0*L_1*L_2*omega**2 + L_2**2*omega**2 + R_1**2 + 2.0*R_1*R_2 + R_2**2)\n i_2d = (-0.5*eta_d*v_dc*(C_ac*L_1*R_2*omega**2 + C_ac*L_2*R_1*omega**2 - R_1 - R_2) + 0.5*eta_q*omega*v_dc*(-C_ac*L_1*L_2*omega**2 + C_ac*R_1*R_2 + L_1 + L_2) - omega*v_sq*(C_ac**2*L_1**2*L_2*omega**4 + C_ac**2*L_2*R_1**2*omega**2 - C_ac*L_1**2*omega**2 - 2.0*C_ac*L_1*L_2*omega**2 - C_ac*R_1**2 + L_1 + L_2) - v_sd*(C_ac**2*L_1**2*R_2*omega**4 + C_ac**2*R_1**2*R_2*omega**2 - 2.0*C_ac*L_1*R_2*omega**2 + R_1 + R_2))/(C_ac**2*L_1**2*L_2**2*omega**6 + C_ac**2*L_1**2*R_2**2*omega**4 + C_ac**2*L_2**2*R_1**2*omega**4 + C_ac**2*R_1**2*R_2**2*omega**2 - 2.0*C_ac*L_1**2*L_2*omega**4 - 2.0*C_ac*L_1*L_2**2*omega**4 - 2.0*C_ac*L_1*R_2**2*omega**2 - 2.0*C_ac*L_2*R_1**2*omega**2 + L_1**2*omega**2 + 2.0*L_1*L_2*omega**2 + L_2**2*omega**2 + R_1**2 + 2.0*R_1*R_2 + R_2**2)\n i_2q = (-0.5*eta_d*omega*v_dc*(-C_ac*L_1*L_2*omega**2 + C_ac*R_1*R_2 + L_1 + L_2) - 0.5*eta_q*v_dc*(C_ac*L_1*R_2*omega**2 + C_ac*L_2*R_1*omega**2 - R_1 - R_2) + omega*v_sd*(C_ac**2*L_1**2*L_2*omega**4 + C_ac**2*L_2*R_1**2*omega**2 - C_ac*L_1**2*omega**2 - 2.0*C_ac*L_1*L_2*omega**2 - C_ac*R_1**2 + L_1 + L_2) - v_sq*(C_ac**2*L_1**2*R_2*omega**4 + C_ac**2*R_1**2*R_2*omega**2 - 2.0*C_ac*L_1*R_2*omega**2 + R_1 + R_2))/(C_ac**2*L_1**2*L_2**2*omega**6 + C_ac**2*L_1**2*R_2**2*omega**4 + C_ac**2*L_2**2*R_1**2*omega**4 + C_ac**2*R_1**2*R_2**2*omega**2 - 2.0*C_ac*L_1**2*L_2*omega**4 - 2.0*C_ac*L_1*L_2**2*omega**4 - 2.0*C_ac*L_1*R_2**2*omega**2 - 2.0*C_ac*L_2*R_1**2*omega**2 + L_1**2*omega**2 + 2.0*L_1*L_2*omega**2 + L_2**2*omega**2 + R_1**2 + 2.0*R_1*R_2 + R_2**2)\n\n\n\n```python\ns = sym.solve([ di_1d, di_1q, dv_cd, dv_cq, di_2d, di_2q],\n [ eta_d, eta_q, i_1d, i_1q, v_cd, v_cq])\n\nfor item in s:\n print(item, '=', sym.simplify(s[item]))\n```\n\n eta_d = 2.0*(-C_ac*R_1*omega*(L_2*i_2d*omega + R_2*i_2q + v_sq) - L_1*i_2q*omega + R_1*i_2d - (C_ac*L_1*omega**2 - 1.0)*(-L_2*i_2q*omega + R_2*i_2d + v_sd))/v_dc\n eta_q = 2.0*(C_ac*R_1*omega*(-L_2*i_2q*omega + R_2*i_2d + v_sd) + L_1*i_2d*omega + R_1*i_2q - (C_ac*L_1*omega**2 - 1.0)*(L_2*i_2d*omega + R_2*i_2q + v_sq))/v_dc\n i_1d = -C_ac*omega*(L_2*i_2d*omega + R_2*i_2q + v_sq) + i_2d\n i_1q = C_ac*omega*(-L_2*i_2q*omega + R_2*i_2d + v_sd) + i_2q\n v_cd = -L_2*i_2q*omega + R_2*i_2d + v_sd\n v_cq = L_2*i_2d*omega + R_2*i_2q + v_sq\n\n\n\n```python\ns = sym.solve([ dv_cd, dv_cq, di_2d, di_2q],\n [ v_cd, v_cq, i_2d, i_2q])\n\nfor item in s:\n print(item, '=', sym.simplify(s[item]))\n```\n\n v_cd = (C_ac*R_2*omega*v_sq + R_2*i_1d + i_1q*omega*(C_ac*L_2**2*omega**2 + C_ac*R_2**2 - L_2) - v_sd*(C_ac*L_2*omega**2 - 1))/(C_ac**2*L_2**2*omega**4 + C_ac**2*R_2**2*omega**2 - 2*C_ac*L_2*omega**2 + 1)\n v_cq = (-C_ac*R_2*omega*v_sd + R_2*i_1q - i_1d*omega*(C_ac*L_2**2*omega**2 + C_ac*R_2**2 - L_2) - v_sq*(C_ac*L_2*omega**2 - 1))/(C_ac**2*L_2**2*omega**4 + C_ac**2*R_2**2*omega**2 - 2*C_ac*L_2*omega**2 + 1)\n i_2d = (-C_ac**2*R_2*omega**2*v_sd + C_ac*R_2*i_1q*omega - C_ac*omega*v_sq*(C_ac*L_2*omega**2 - 1) - i_1d*(C_ac*L_2*omega**2 - 1))/(C_ac**2*L_2**2*omega**4 + C_ac**2*R_2**2*omega**2 - 2*C_ac*L_2*omega**2 + 1)\n i_2q = (-C_ac**2*R_2*omega**2*v_sq - C_ac*R_2*i_1d*omega + C_ac*omega*v_sd*(C_ac*L_2*omega**2 - 1) - i_1q*(C_ac*L_2*omega**2 - 1))/(C_ac**2*L_2**2*omega**4 + C_ac**2*R_2**2*omega**2 - 2*C_ac*L_2*omega**2 + 1)\n\n\n\n```python\ns = sym.solve([ di_1d, di_1q],\n [ i_1d, i_1q])\n\nfor item in s:\n print(item, '=', sym.simplify(s[item]))\n```\n\n i_1d = 0.5*(L_1*omega*(eta_q*v_dc - 2.0*v_cq) + R_1*(eta_d*v_dc - 2.0*v_cd))/(L_1**2*omega**2 + R_1**2)\n i_1q = 0.5*(-L_1*omega*(eta_d*v_dc - 2.0*v_cd) + R_1*(eta_q*v_dc - 2.0*v_cq))/(L_1**2*omega**2 + R_1**2)\n\n\n\n```python\neq_p = p_ref - 3.0/2.0*(v_sd*i_2d + v_sq*i_2q)\neq_q = q_ref - 3.0/2.0*(v_sd*i_2q - v_sq*i_2d)\n\ns = sym.solve([ eq_p, eq_q],\n [i_2d, i_2q])\n\n\nfor item in s:\n print(item, '=', sym.simplify(s[item]))\n```\n\n i_2d = 0.666666666666667*(p_ref*v_sd - q_ref*v_sq)/(v_sd**2 + v_sq**2)\n i_2q = 0.666666666666667*(p_ref*v_sq + q_ref*v_sd)/(v_sd**2 + v_sq**2)\n\n\n\n```python\nv_dc = 700.0\nv_sq = 325.0\nv_sd = 0.0\np_ref = 10.0e3\nq_ref = 900.0e3\nomega = 2.0*np.pi*50.0\nL_1 = 200.0e-6\nL_2 = 200.0e-6\nR_1 = 0.1\nR_2 = 0.1\nC_ac = 100e-6\n\ni_2d_ref = 2.0/3.0*(p_ref*v_sd - q_ref*v_sq)/(v_sd**2 + v_sq**2)\ni_2q_ref = 2.0/3.0*(p_ref*v_sq + q_ref*v_sd)/(v_sd**2 + v_sq**2)\n\neta_d = 2.0*(-C_ac*R_1*omega*(L_2*i_2d_ref*omega + R_2*i_2q_ref + v_sq) - L_1*i_2q_ref*omega + R_1*i_2d_ref - (C_ac*L_1*omega**2 - 1.0)*(-L_2*i_2q_ref*omega + R_2*i_2d_ref + v_sd))/v_dc\neta_q = 2.0*(C_ac*R_1*omega*(-L_2*i_2q_ref*omega + R_2*i_2d_ref + v_sd) + L_1*i_2d_ref*omega + R_1*i_2q_ref - (C_ac*L_1*omega**2 - 1.0)*(L_2*i_2d_ref*omega + R_2*i_2q_ref + v_sq))/v_dc\nprint(i_2d_ref)\nprint(i_2q_ref)\n\nprint('eta_d = ',eta_d)\nprint(eta_q)\n\neta = eta_d + 1j*eta_q\nangle = np.angle(eta)\neta_m = np.abs(eta)\n\nif eta_m > 1.0:\n eta_m = 1.0\n\neta = eta_m*np.exp(1j*angle)\neta_d = eta.real\neta_q = eta.imag\ni_2d = (-0.5*eta_d*v_dc*(C_ac*L_1*R_2*omega**2 + C_ac*L_2*R_1*omega**2 - R_1 - R_2) + 0.5*eta_q*omega*v_dc*(-C_ac*L_1*L_2*omega**2 + C_ac*R_1*R_2 + L_1 + L_2) - omega*v_sq*(C_ac**2*L_1**2*L_2*omega**4 + C_ac**2*L_2*R_1**2*omega**2 - C_ac*L_1**2*omega**2 - 2.0*C_ac*L_1*L_2*omega**2 - C_ac*R_1**2 + L_1 + L_2) - v_sd*(C_ac**2*L_1**2*R_2*omega**4 + C_ac**2*R_1**2*R_2*omega**2 - 2.0*C_ac*L_1*R_2*omega**2 + R_1 + R_2))/(C_ac**2*L_1**2*L_2**2*omega**6 + C_ac**2*L_1**2*R_2**2*omega**4 + C_ac**2*L_2**2*R_1**2*omega**4 + C_ac**2*R_1**2*R_2**2*omega**2 - 2.0*C_ac*L_1**2*L_2*omega**4 - 2.0*C_ac*L_1*L_2**2*omega**4 - 2.0*C_ac*L_1*R_2**2*omega**2 - 2.0*C_ac*L_2*R_1**2*omega**2 + L_1**2*omega**2 + 2.0*L_1*L_2*omega**2 + L_2**2*omega**2 + R_1**2 + 2.0*R_1*R_2 + R_2**2)\ni_2q = (-0.5*eta_d*omega*v_dc*(-C_ac*L_1*L_2*omega**2 + C_ac*R_1*R_2 + L_1 + L_2) - 0.5*eta_q*v_dc*(C_ac*L_1*R_2*omega**2 + C_ac*L_2*R_1*omega**2 - R_1 - R_2) + omega*v_sd*(C_ac**2*L_1**2*L_2*omega**4 + C_ac**2*L_2*R_1**2*omega**2 - C_ac*L_1**2*omega**2 - 2.0*C_ac*L_1*L_2*omega**2 - C_ac*R_1**2 + L_1 + L_2) - v_sq*(C_ac**2*L_1**2*R_2*omega**4 + C_ac**2*R_1**2*R_2*omega**2 - 2.0*C_ac*L_1*R_2*omega**2 + R_1 + R_2))/(C_ac**2*L_1**2*L_2**2*omega**6 + C_ac**2*L_1**2*R_2**2*omega**4 + C_ac**2*L_2**2*R_1**2*omega**4 + C_ac**2*R_1**2*R_2**2*omega**2 - 2.0*C_ac*L_1**2*L_2*omega**4 - 2.0*C_ac*L_1*L_2**2*omega**4 - 2.0*C_ac*L_1*R_2**2*omega**2 - 2.0*C_ac*L_2*R_1**2*omega**2 + L_1**2*omega**2 + 2.0*L_1*L_2*omega**2 + L_2**2*omega**2 + R_1**2 + 2.0*R_1*R_2 + R_2**2)\n\nprint(i_2d)\nprint(i_2q)\n\np = 3.0/2.0*(v_sd*i_2d + v_sq*i_2q)\nq = 3.0/2.0*(v_sd*i_2q - v_sq*i_2d)\n\nprint('p = ', p/1000)\nprint('q = ', q/1000)\n\nprint('p = ', p/1000)\nprint('q = ', q/1000)\n\n```\n\n -1846.1538461538462\n 20.51282051282051\n eta_d = -1.063155919300432\n 0.2745925438638739\n -1746.44514159\n -85.3469189756\n p = -41.6066230006\n q = 851.392006525\n p = -41.6066230006\n q = 851.392006525\n\n\n\n```python\n\n```\n\n\n```python\n\n@numba.jit(nopython=True, cache=True)\ndef b2b_ctrl1(struct,i,m):\n '''\n Doubly Fed Induction Machine in with neglected dynamics and\n rotor side converter and control level 1 already implemented.\n i_rd = i_rd_ref and i_rq = i_rq_ref without dynamics \n '''\n\n x_idx = struct[i]['b2b_idx']\n v_dc = float(struct[i]['x'][x_idx+0,0])\n \n L_1 = struct[i]['L_1']\n L_2 = struct[i]['L_2']\n R_1 = struct[i]['R_1']\n R_2 = struct[i]['R_2']\n C_dc = struct[i]['C_dc']\n\n omega_1 = struct[i]['omega_1']\n omega_2 = struct[i]['omega_2']\n \n i_1d_ref = struct[i]['i_1d_ref'] \n i_1q_ref = struct[i]['i_1q_ref'] \n i_2d_ref = struct[i]['i_2d_ref'] \n i_2q_ref = struct[i]['i_2q_ref'] \n \n i_1d = i_1d_ref\n i_1q = i_1q_ref\n i_2d = i_2d_ref\n i_2q = i_2q_ref\n \n v_1d = struct[i]['v_1d']\n v_1q = struct[i]['v_1q']\n v_2d = struct[i]['v_2d']\n v_2q = struct[i]['v_2q']\n \n \n eta_1d = 2.0*(R_1*i_1d - L_1*i_1q*omega_1 + v_1d)/v_dc\n eta_1q = 2.0*(R_1*i_1q + L_1*i_1d*omega_1 + v_1q)/v_dc\n eta_2d = 2.0*(R_2*i_2d - L_2*i_2q*omega_2 + v_2d)/v_dc\n eta_2q = 2.0*(R_2*i_2q + L_2*i_2d*omega_2 + v_2q)/v_dc\n \n i_dc_1 = 3.0/4.0*(eta_1d*i_1d + eta_1q*i_1q)\n i_dc_2 = 3.0/4.0*(eta_2d*i_2d + eta_2q*i_2q) \n dv_dc = 1.0/C_dc*(-i_dc_1 - i_dc_2)\n \n struct[i]['eta_1d'] = eta_1d\n struct[i]['eta_1q'] = eta_1q\n struct[i]['eta_2d'] = eta_2d\n struct[i]['eta_2q'] = eta_2q\n struct[i]['i_dc_1'] = i_dc_1\n struct[i]['i_dc_2'] = i_dc_2\n \n struct[i]['p_1'] = 3.0/2.0*(v_1d*i_1d + v_1q*i_1q)\n struct[i]['q_1'] = 3.0/2.0*(v_1d*i_1q - v_1q*i_1d)\n\n struct[i]['p_2'] = 3.0/2.0*(v_2d*i_2d + v_2q*i_2q)\n struct[i]['q_2'] = 3.0/2.0*(v_2d*i_2q - v_2q*i_2d)\n \n struct[i]['f'][x_idx+0,0] = dv_dc\n \n return 0\n\n```\n\n\n```python\n\n```\n\n\n```python\n@numba.jit(nopython=True, cache=True)\ndef b2b_ctrl2(struct,i,m):\n '''\n Control level 2 for DC Voltage\n \n '''\n \n x_idx = struct[i]['b2b_ctrl_idx']\n xi_v_dc = float(struct[i]['x'][x_idx+0,0]) \n\n S_b = struct[i]['S_b']\n V_dc_b = struct[i]['V_dc_b']\n \n K_v_p = struct[i]['K_v_p']\n K_v_i = struct[i]['K_v_i'] \n\n v_dc = struct[i]['v_dc']\n \n v_dc_ref = struct[i]['v_dc_ref']\n p_1_ref = struct[i]['p_1_ref']\n q_1_ref = struct[i]['q_1_ref'] \n p_2_ref = struct[i]['p_2_ref']\n q_2_ref = struct[i]['q_2_ref'] \n\n v_1d = struct[i]['v_1d']\n v_1q = struct[i]['v_1q']\n v_2d = struct[i]['v_2d']\n v_2q = struct[i]['v_2q']\n \n error_v_dc = (v_dc - v_dc_ref)/V_dc_b\n \n p_ref = (K_v_p * error_v_dc + K_v_i*xi_v_dc)*S_b\n \n if struct[i]['vdc_ctrl'] == 1:\n p_ref_1 = p_ref\n\n if struct[i]['vdc_ctrl'] == 2:\n p_ref_2 = p_ref\n \n den = (v_1d**2 + v_1q**2)\n \n den_1 = 0.001\n if den_1 > 0.0:\n den_1 = (v_1d**2 + v_1q**2)\n\n den_2 = 0.001\n if den_2 > 0.0:\n den_2 = (v_2d**2 + v_2q**2)\n \n i_1d_ref = 2.0/3.0*(p_1_ref*v_1d - q_1_ref*v_1q)/den_1\n i_1q_ref = 2.0/3.0*(p_1_ref*v_1q + q_1_ref*v_1d)/den_1\n \n i_2d_ref = 2.0/3.0*(p_2_ref*v_2d - q_2_ref*v_1q)/den_2\n i_2q_ref = 2.0/3.0*(p_2_ref*v_2q + q_2_ref*v_1d)/den_2\n \n struct[i]['i_1d_ref'] = i_1d_ref\n struct[i]['i_1q_ref'] = i_1q_ref\n struct[i]['i_2d_ref'] = i_2d_ref\n struct[i]['i_2q_ref'] = i_2q_ref\n \n dxi_v_dc = error_v_dc\n \n struct[i]['f'][x_idx+0,0] = dxi_v_dc\n \n return 0\n\n```\n\n\n```python\n\n```\n\n\n```python\nR_1 = R_2 = 0.1\nL_1 = L_2 = 0.5e-3\nOmega_b = 2.0*np.pi*50.0\nC_dc = 2200.0e-6\nomega_1 = omega_2 = Omega_b\n\nd =dict(R_1 = R_1,\n R_2 = R_2,\n L_1 = L_1,\n L_2 = L_2,\n C_dc = C_dc,\n b2b_idx = 0,\n b2b_ctrl_idx = 1,\n v_dc = 800.0,\n omega_1 = omega_1,\n omega_2 = omega_2, \n i_1d_ref = 0.0,\n i_1q_ref = 100.0, \n i_2d_ref = 0.0, \n i_2q_ref = -100.0, \n i_dc_1 = 0.0,\n i_dc_2 = 0.0,\n eta_1d = 0.0,\n eta_1q = 0.0, \n eta_2d = 0.0, \n eta_2q = 0.0, \n v_1d = 0.0,\n v_1q = 325.0,\n v_2d = 0.0,\n v_2q = 325.0,\n p_1 = 0.0,\n q_1 = 0.0,\n p_2 = 0.0,\n q_2 = 0.0, \n x_idx = 0,\n xi_v_dc = 0.0, \n S_b = 0.5e6,\n V_dc_b = 800.0,\n K_v_p = 0.1,\n K_v_i = 0.0, \n v_dc_ref = 750.0,\n p_1_ref = 0.0,\n q_1_ref = 0.0,\n p_2_ref = 0.0,\n q_2_ref = 0.0,\n vdc_ctrl = 1,\n x = np.array([[800.0],[0.0]]),\n f = np.array([[0.0],[0.0]]) \n )\n\n\n\n\n\n\n\n\nstruct = d2np(d)\n\ni=0\nm=2\nb2b_ctrl1(struct,i,m)\nb2b_ctrl2(struct,i,m)\nprint(struct[i]['p_1'])\nprint(struct[i]['p_2'])\nprint(struct[i]['i_dc_1'])\nprint(struct[i]['i_dc_2'])\nprint(struct[i]['f'])\n```\n\n 48750.0\n -48750.0\n 62.8125\n -59.0625\n [[ -1.70454545e+03]\n [ 6.25000000e-02]]\n\n\n\n```python\nstruct = d2np(d)\n\nsys_d = dict(x = np.array([[800.0],[0.0]]),\n f = np.zeros((2,1)))\n\nsys_struct = d2np(sys_d)\n\n@numba.jit(nopython=True, cache=True)\ndef f_eval(sys_struct,struct):\n N_states = 2\n for i in range(1):\n struct[i]['x'][:,0] = sys_struct[0]['x'][N_states*i:N_states*(i+1),0]\n b2b_ctrl1(struct,i,m)\n b2b_ctrl2(struct,i,m)\n sys_struct[0]['f'][N_states*i:N_states*(i+1),:] = struct[i]['f']\n return 0\n\n```\n\n\n```python\n@numba.jit(nopython=True, cache=True)\ndef run(sys_struct,struct): \n N_steps = 1000\n N_states = 2\n\n Dt = 10.0e-3\n Omega_r = np.zeros((N_steps,1))\n Omega_t = np.zeros((N_steps,1))\n P_1 = np.zeros((N_steps,1))\n Q_1 = np.zeros((N_steps,1))\n P_2 = np.zeros((N_steps,1))\n Q_2 = np.zeros((N_steps,1))\n V_dr = np.zeros((N_steps,1))\n V_qr = np.zeros((N_steps,1))\n I_dr = np.zeros((N_steps,1))\n I_qr = np.zeros((N_steps,1))\n Tau_e = np.zeros((N_steps,1))\n T = np.zeros((N_steps,1))\n X = np.zeros((N_steps,N_states))\n\n V_dc = np.zeros((N_steps,1))\n p_ref = 0.0\n q_ref = 0.0 \n xi_p = 0.0\n xi_q = 0.0\n \n struct[0]['x'][:,0] = np.copy(sys_struct[0]['x'][0:2,0])\n \n for it in range(N_steps):\n t = Dt*float(it)\n \n # perturbations and references\n\n struct[0]['p_1_ref'] = 0.0\n struct[0]['p_2_ref'] = 0.0\n struct[0]['q_1_ref'] = 0.0\n struct[0]['q_2_ref'] = 0.0 \n \n if t>2.0:\n struct[0]['p_1_ref'] = 1.0e6\n if t>3.0:\n struct[0]['p_2_ref'] = 0.1e6\n \n ## solver \n\n f_eval(sys_struct,struct)\n f1 = np.copy(sys_struct[0]['f'])\n x1 = np.copy(sys_struct[0]['x'])\n \n sys_struct[0]['x'][:]= np.copy(x1 + Dt*f1)\n f_eval(sys_struct,struct)\n f2 = np.copy(sys_struct[0]['f'])\n \n sys_struct[0]['x'][:]= np.copy(x1 + 0.5*Dt*(f1 + f2)) \n\n for i in range(1):\n struct[i]['x'][:,0] = sys_struct[0]['x'][2*i:2*(i+1),0]\n\n \n T[it,0] = t\n V_dc[it,0] = float(struct[0]['v_dc'])\n X[it,:] = sys_struct[0]['x'][:].T\n return T,X,V_dc\n%timeit run(sys_struct, struct)\n```\n\n 724 \u00b5s \u00b1 41.2 \u00b5s per loop (mean \u00b1 std. dev. of 7 runs, 1000 loops each)\n\n\n\n```python\nrun(sys_struct, struct)\n```\n\n\n\n\n (array([[ 0. ],\n [ 0.01],\n [ 0.02],\n [ 0.03],\n [ 0.04],\n [ 0.05],\n [ 0.06],\n [ 0.07],\n [ 0.08],\n [ 0.09],\n [ 0.1 ],\n [ 0.11],\n [ 0.12],\n [ 0.13],\n [ 0.14],\n [ 0.15],\n [ 0.16],\n [ 0.17],\n [ 0.18],\n [ 0.19],\n [ 0.2 ],\n [ 0.21],\n [ 0.22],\n [ 0.23],\n [ 0.24],\n [ 0.25],\n [ 0.26],\n [ 0.27],\n [ 0.28],\n [ 0.29],\n [ 0.3 ],\n [ 0.31],\n [ 0.32],\n [ 0.33],\n [ 0.34],\n [ 0.35],\n [ 0.36],\n [ 0.37],\n [ 0.38],\n [ 0.39],\n [ 0.4 ],\n [ 0.41],\n [ 0.42],\n [ 0.43],\n [ 0.44],\n [ 0.45],\n [ 0.46],\n [ 0.47],\n [ 0.48],\n [ 0.49],\n [ 0.5 ],\n [ 0.51],\n [ 0.52],\n [ 0.53],\n [ 0.54],\n [ 0.55],\n [ 0.56],\n [ 0.57],\n [ 0.58],\n [ 0.59],\n [ 0.6 ],\n [ 0.61],\n [ 0.62],\n [ 0.63],\n [ 0.64],\n [ 0.65],\n [ 0.66],\n [ 0.67],\n [ 0.68],\n [ 0.69],\n [ 0.7 ],\n [ 0.71],\n [ 0.72],\n [ 0.73],\n [ 0.74],\n [ 0.75],\n [ 0.76],\n [ 0.77],\n [ 0.78],\n [ 0.79],\n [ 0.8 ],\n [ 0.81],\n [ 0.82],\n [ 0.83],\n [ 0.84],\n [ 0.85],\n [ 0.86],\n [ 0.87],\n [ 0.88],\n [ 0.89],\n [ 0.9 ],\n [ 0.91],\n [ 0.92],\n [ 0.93],\n [ 0.94],\n [ 0.95],\n [ 0.96],\n [ 0.97],\n [ 0.98],\n [ 0.99],\n [ 1. ],\n [ 1.01],\n [ 1.02],\n [ 1.03],\n [ 1.04],\n [ 1.05],\n [ 1.06],\n [ 1.07],\n [ 1.08],\n [ 1.09],\n [ 1.1 ],\n [ 1.11],\n [ 1.12],\n [ 1.13],\n [ 1.14],\n [ 1.15],\n [ 1.16],\n [ 1.17],\n [ 1.18],\n [ 1.19],\n [ 1.2 ],\n [ 1.21],\n [ 1.22],\n [ 1.23],\n [ 1.24],\n [ 1.25],\n [ 1.26],\n [ 1.27],\n [ 1.28],\n [ 1.29],\n [ 1.3 ],\n [ 1.31],\n [ 1.32],\n [ 1.33],\n [ 1.34],\n [ 1.35],\n [ 1.36],\n [ 1.37],\n [ 1.38],\n [ 1.39],\n [ 1.4 ],\n [ 1.41],\n [ 1.42],\n [ 1.43],\n [ 1.44],\n [ 1.45],\n [ 1.46],\n [ 1.47],\n [ 1.48],\n [ 1.49],\n [ 1.5 ],\n [ 1.51],\n [ 1.52],\n [ 1.53],\n [ 1.54],\n [ 1.55],\n [ 1.56],\n [ 1.57],\n [ 1.58],\n [ 1.59],\n [ 1.6 ],\n [ 1.61],\n [ 1.62],\n [ 1.63],\n [ 1.64],\n [ 1.65],\n [ 1.66],\n [ 1.67],\n [ 1.68],\n [ 1.69],\n [ 1.7 ],\n [ 1.71],\n [ 1.72],\n [ 1.73],\n [ 1.74],\n [ 1.75],\n [ 1.76],\n [ 1.77],\n [ 1.78],\n [ 1.79],\n [ 1.8 ],\n [ 1.81],\n [ 1.82],\n [ 1.83],\n [ 1.84],\n [ 1.85],\n [ 1.86],\n [ 1.87],\n [ 1.88],\n [ 1.89],\n [ 1.9 ],\n [ 1.91],\n [ 1.92],\n [ 1.93],\n [ 1.94],\n [ 1.95],\n [ 1.96],\n [ 1.97],\n [ 1.98],\n [ 1.99],\n [ 2. ],\n [ 2.01],\n [ 2.02],\n [ 2.03],\n [ 2.04],\n [ 2.05],\n [ 2.06],\n [ 2.07],\n [ 2.08],\n [ 2.09],\n [ 2.1 ],\n [ 2.11],\n [ 2.12],\n [ 2.13],\n [ 2.14],\n [ 2.15],\n [ 2.16],\n [ 2.17],\n [ 2.18],\n [ 2.19],\n [ 2.2 ],\n [ 2.21],\n [ 2.22],\n [ 2.23],\n [ 2.24],\n [ 2.25],\n [ 2.26],\n [ 2.27],\n [ 2.28],\n [ 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3.2 ],\n [ 3.21],\n [ 3.22],\n [ 3.23],\n [ 3.24],\n [ 3.25],\n [ 3.26],\n [ 3.27],\n [ 3.28],\n [ 3.29],\n [ 3.3 ],\n [ 3.31],\n [ 3.32],\n [ 3.33],\n [ 3.34],\n [ 3.35],\n [ 3.36],\n [ 3.37],\n [ 3.38],\n [ 3.39],\n [ 3.4 ],\n [ 3.41],\n [ 3.42],\n [ 3.43],\n [ 3.44],\n [ 3.45],\n [ 3.46],\n [ 3.47],\n [ 3.48],\n [ 3.49],\n [ 3.5 ],\n [ 3.51],\n [ 3.52],\n [ 3.53],\n [ 3.54],\n [ 3.55],\n [ 3.56],\n [ 3.57],\n [ 3.58],\n [ 3.59],\n [ 3.6 ],\n [ 3.61],\n [ 3.62],\n [ 3.63],\n [ 3.64],\n [ 3.65],\n [ 3.66],\n [ 3.67],\n [ 3.68],\n [ 3.69],\n [ 3.7 ],\n [ 3.71],\n [ 3.72],\n [ 3.73],\n [ 3.74],\n [ 3.75],\n [ 3.76],\n [ 3.77],\n [ 3.78],\n [ 3.79],\n [ 3.8 ],\n [ 3.81],\n [ 3.82],\n [ 3.83],\n [ 3.84],\n [ 3.85],\n [ 3.86],\n [ 3.87],\n [ 3.88],\n [ 3.89],\n [ 3.9 ],\n [ 3.91],\n [ 3.92],\n [ 3.93],\n [ 3.94],\n [ 3.95],\n [ 3.96],\n [ 3.97],\n [ 3.98],\n [ 3.99],\n [ 4. ],\n [ 4.01],\n [ 4.02],\n [ 4.03],\n [ 4.04],\n [ 4.05],\n [ 4.06],\n [ 4.07],\n [ 4.08],\n [ 4.09],\n [ 4.1 ],\n [ 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[ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.],\n [ 800.]]))\n\n\n\n\n```python\nsys_struct['x'][:]= np.zeros((6,1))\nstruct['v_qs'] = 0.0\nstruct['v_ds'] = 690.0*np.sqrt(2.0/3.0)\nstruct['tau_t'] = 0.0\nsys_struct[0]['x'][0,0] = Omega_b*0.9/struct[0]['N_tr']/struct[0]['N_pp']\nsys_struct[0]['x'][3,0] = Omega_b*1.1/struct[1]['N_tr']/struct[0]['N_pp']\nT,X,Tau_e,P_s_1,Q_s_1,P_r_1,Q_r_1,P_s_2,Q_s_2,P_r_2,Q_r_2,V_dr,V_qr,Omega_r,Omega_t,I_dr,I_qr = run(sys_struct, struct)\n```\n\n\n```python\nfig, axes = plt.subplots(nrows=1, ncols=1, figsize=(8, 5), sharex = True)\naxes.plot(T,Tau_e)\nfig.savefig('dfim_tau_e.svg', bbox_inches='tight')\n```\n\n\n```python\nfig, axes = plt.subplots(nrows=2, ncols=1, figsize=(8, 8), sharex = True)\naxes[0].plot(T,P_s_1/1e6, label='$\\sf p_{s1}$')\naxes[0].plot(T,Q_s_1/1e6, label='$\\sf q_{s1}$')\naxes[0].plot(T,P_s_2/1e6, label='$\\sf p_{s2}$')\naxes[0].plot(T,Q_s_2/1e6, label='$\\sf q_{s2}$')\n\naxes[1].plot(T,P_r_1/1e6, label='$\\sf p_{r1}$')\naxes[1].plot(T,Q_r_1/1e6, label='$\\sf q_{r1}$')\naxes[1].plot(T,P_r_2/1e6, label='$\\sf p_{r2}$')\naxes[1].plot(T,Q_r_2/1e6, label='$\\sf q_{r2}$')\n\naxes[0].legend()\naxes[1].legend()\nfig.savefig('dfim_tau_e.svg', bbox_inches='tight')\n\n```\n\n\n```python\nfig, axes = plt.subplots(nrows=1, ncols=1, figsize=(8, 5), sharex = True)\naxes.plot(T,Omega_t)\n```\n\n\n```python\nfig, axes = plt.subplots(nrows=2, ncols=1, figsize=(8, 8), sharex = True)\naxes[0].plot(T,V_dr, label='$\\sf v_{dr}$')\naxes[0].plot(T,V_qr, label='$\\sf v_{qr}$')\n```\n\n\n```python\nfig, axes = plt.subplots(nrows=2, ncols=1, figsize=(8, 8), sharex = True)\naxes[0].plot(T,Omega_t, label='$\\sf v_{dr}$')\n\n```\n\n\n```python\nOmega_t[0]\n```\n\n\n```python\nfig, axes = plt.subplots(nrows=2, ncols=1, figsize=(8, 8), sharex = True)\naxes[0].plot(T,I_dr, label='$\\sf i_{dr}$')\naxes[0].plot(T,I_qr, label='$\\sf i_{qr}$')\n```\n\n\n```python\nfig, axes = plt.subplots(nrows=2, ncols=1, figsize=(8, 8), sharex = True)\naxes[0].plot(T,X[:,5], label='$\\sf x$')\n\n```\n\n\n```python\nnp.random.normal(500e3,100e3)\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "0b62bc35f3d80bb6d38870c6ca77cc45db5a6b04", "size": 90936, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "models/vsc_ctrl2.ipynb", "max_stars_repo_name": "pydgrid/pydgrid", "max_stars_repo_head_hexsha": "c56073c385f42883c79333533f7cfb8383a173aa", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 15, "max_stars_repo_stars_event_min_datetime": "2019-01-29T08:22:39.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-13T20:41:32.000Z", "max_issues_repo_path": "models/vsc_ctrl2.ipynb", "max_issues_repo_name": "pydgrid/pydgrid", "max_issues_repo_head_hexsha": "c56073c385f42883c79333533f7cfb8383a173aa", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2017-11-28T21:34:52.000Z", "max_issues_repo_issues_event_max_datetime": "2017-11-28T21:34:52.000Z", "max_forks_repo_path": "models/vsc_ctrl2.ipynb", "max_forks_repo_name": "pydgrid/pydgrid", "max_forks_repo_head_hexsha": "c56073c385f42883c79333533f7cfb8383a173aa", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2018-02-15T02:12:47.000Z", "max_forks_repo_forks_event_max_datetime": "2020-02-16T17:52:15.000Z", "avg_line_length": 31.9073684211, "max_line_length": 2196, "alphanum_fraction": 0.2689144013, "converted": true, "num_tokens": 23377, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256472515683, "lm_q2_score": 0.5273165233795672, "lm_q1q2_score": 0.44390857360045094}} {"text": "# Decoding Board from Data\n\n\n```python\n#necessary imports\nimport pandas\nimport numpy as np\nimport os\nimport tensorflow.keras as keras\nfrom keras.models import Model\nfrom keras.layers import Dense, Input\nfrom IPython.display import display\nimport sympy as sp\nsp.init_printing(use_latex = True)\nimport math\n\nimport matplotlib.pyplot as plt\n%matplotlib inline\n\n%run Drivers.ipynb\n\nEMPTY = 1;\nCOLOR = 0;\n\nBLACK = -1;\nWHITE = 1;\n\nWIDTH = 9;\n```\n\n# Data Categorization and Assignment\n\n\n```python\nBoards = []\nMoves = []\ndef Main():\n path = \"./go9\"\n for entry in os.scandir(path): #I changed my mind i love python\n Go = True\n Board = createEmptyBoard() # 0 - 80 = [color, empty], 81 = [turn, turn]\n with open(entry) as f:\n if Go:\n for line in f:\n if line[0] == ';': # this is the line with all the moves.\n Go = False\n copy = \"\"\n for c in line:\n if c != \"[\" and c != \"]\" and c != \")\":\n copy += c\n arr = copy[1:].split(';')\n for a in arr:\n move = Decode_Move(a[1:])\n if Decode_Move(a[1:]) == -1:\n print(entry)\n return\n else:\n color = 1\n if(a[0] == 'B'):\n color = -1\n Boards.append(Board)\n Moves.append(move)\n if(move != 81):\n Board = Move(Board, move, color)[1]\nMain()\nBoards = np.array(Boards)\nMoves = np.array(Moves)\n```\n\n\n```python\nprint(Boards.shape)\n```\n\n (414124, 82, 2)\n\n\n\n```python\n# Example Position:\nprintBoard(Boards[25])\n```\n\n # A B C D E F G H I\n 1 . . . . . . . . .\n 2 . . . . @ . . . .\n 3 . . O . @ O O . .\n 4 . . . O O @ . @ .\n 5 . . @ @ @ @ @ O .\n 6 . . . @ O O O O .\n 7 . . @ O O . . . .\n 8 . . . @ @ . . . .\n 9 . . . . . . . . .\n \n\n\n\n```python\nX = Boards\nY = keras.utils.to_categorical(Moves, 82)\n\ntraining_samples = int(0.9 * X.shape[0])\nX_train, X_test = X[:training_samples], X[training_samples:] # Inputs\nY_train, Y_test = Y[:training_samples], Y[training_samples:] # Outputs\n\nprint(X.shape)\nprint(Y.shape)\n```\n\n (414124, 82, 2)\n (414124, 82)\n\n\n# Building the Model\n\n\n```python\ninput_shape = (82, 2)\n\nmodel = keras.models.Sequential()\nmodel.add(keras.layers.Dense(2, input_shape = input_shape))\nmodel.add(keras.layers.Flatten())\nmodel.add(keras.layers.Dense(82, activation = 'relu'))\nmodel.add(keras.layers.Dense(82, activation = 'relu'))\nmodel.add(keras.layers.Dense(82, activation = 'relu'))\nmodel.add(keras.layers.Dense(82, activation = 'relu'))\nmodel.add(keras.layers.Dropout(.1))\nmodel.add(keras.layers.Dense(82, activation = 'relu'))\n\nmodel.compile(loss=keras.losses.CategoricalCrossentropy(), optimizer = keras.optimizers.Adam(), metrics = [keras.metrics.CategoricalAccuracy()])\n\nmodel.summary()\n```\n\n Model: \"sequential_1\"\n _________________________________________________________________\n Layer (type) Output Shape Param # \n =================================================================\n dense_6 (Dense) (None, 82, 2) 6 \n _________________________________________________________________\n flatten_1 (Flatten) (None, 164) 0 \n _________________________________________________________________\n dense_7 (Dense) (None, 82) 13530 \n _________________________________________________________________\n dense_8 (Dense) (None, 82) 6806 \n _________________________________________________________________\n dense_9 (Dense) (None, 82) 6806 \n _________________________________________________________________\n dense_10 (Dense) (None, 82) 6806 \n _________________________________________________________________\n dropout_1 (Dropout) (None, 82) 0 \n _________________________________________________________________\n dense_11 (Dense) (None, 82) 6806 \n =================================================================\n Total params: 40,760\n Trainable params: 40,760\n Non-trainable params: 0\n _________________________________________________________________\n\n\n# Training\n\n\n```python\n# Load Weights\n#model.load_weights('mini_weights.h5')\n```\n\n\n```python\n#Train the model\nprint(X_train.shape)\nprint(X_test.shape)\nprint(Y_train.shape)\nprint(Y_test.shape)\nhistory = model.fit(X_train, Y_train, batch_size = 32, epochs = 128, workers = 10, verbose = 1, validation_data = (X_test, Y_test))\n```\n\n\n```python\n# Save Weights\nmodel.save_weights('mini_weights.h5')\n```\n\n\n```python\nplt.figure(1)\n\nplt.subplot(211)\nplt.plot(history.history['categorical_accuracy'])\nplt.plot(history.history['val_categorical_accuracy'])\nplt.title('model accuracy')\nplt.ylabel('accuracy')\nplt.xlabel('epoch')\nplt.legend(['train', 'val'], loc='upper left')\n\nplt.subplot(212)\nplt.plot(history.history['loss'])\nplt.plot(history.history['val_loss'])\nplt.title('model loss') \n\nplt.ylabel('loss')\nplt.xlabel('epoch')\nplt.legend(['train', 'val'], loc='upper left')\nplt.tight_layout()\nplt.show()\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "46f6d8551f063044a78a9824e79e8aa5cd5bc767", "size": 52208, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "GOnet.ipynb", "max_stars_repo_name": "CSCI4850/s21-team6-project", "max_stars_repo_head_hexsha": "2d3f6e759a303e819aae73a098975360480a1355", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "GOnet.ipynb", "max_issues_repo_name": "CSCI4850/s21-team6-project", "max_issues_repo_head_hexsha": "2d3f6e759a303e819aae73a098975360480a1355", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "GOnet.ipynb", "max_forks_repo_name": "CSCI4850/s21-team6-project", "max_forks_repo_head_hexsha": "2d3f6e759a303e819aae73a098975360480a1355", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 135.9583333333, "max_line_length": 28372, "alphanum_fraction": 0.8080370824, "converted": true, "num_tokens": 1321, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744939732855, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.44383117403643735}} {"text": "# 1. Recall\n\n### 1.1 Recall: Transformer\n\n\n\n### 1.2 Recall: BERT\n\n\n\n### 1.3 Recall: Language Model\n\n- For the sequence $\\color{MediumOrchid}{w_{1}, w_{2}, \\ldots, w_{n}}$, using the chain rule, we have: \n\n$$\nP\\left(w_{1}, \\ldots, w_{n}\\right)=P\\left(w_{n} \\mid w_{1}, \\ldots, w_{n-1}\\right) P\\left(w_{n-1} \\mid w_{1}, \\ldots, w_{n-2}\\right) \\ldots P\\left(w_{2} \\mid w_{1}\\right) P\\left(w_{1}\\right)\n$$\n\n- N-Gram Approximation: $\\color{MediumOrchid}{P\\left(w_{1}, \\ldots, w_{n}\\right)=\\prod_{i=1}^{n} P\\left(w_{i} \\mid w_{i-N+1}, \\ldots, w_{i-1}\\right)}$\n\n- Applications:\n - Machine Translation: $\\color{MediumOrchid}{P(\\text{the cat is small })>P(\\text{ small is the cat} )}$\n - Grammar Checking: $\\color{MediumOrchid}{P(\\text{ He graduated from SJTU. )>P(He graduated on SJTU.)}}$\n\n# 2. GPT\n\nGPT: Generative Pre-Training\n\n\u76f8\u5173\u8bba\u6587:\n\n1. Radford, A., & Narasimhan, K. (2018). Improving Language Understanding by Generative Pre-Training.\n\n2. Radford, A., Wu, J., Child, R., Luan, D., Amodei, D., & Sutskever, I. (2019). Language Models are Unsupervised Multitask Learners.\n\n3. Brown, T. et al. \u201cLanguage Models are Few-Shot Learners.\u201d ArXiv abs/2005.14165 (2020): n. pag.\n\n### 2.1 Introduction\n\n1. In this paper, we explore a semi-supervised approach for language understanding tasks using a combination of unsupervised pre-training and supervised fine-tuning.\n\n\n2. Our goal is to learn a universal representation that transfers with little adaptation to a wide range of tasks.\n\n\n3. We employ a two-stage training procedure. \n - First, we use a language modeling objective on the unlabeled data to learn the initial parameters of a neural network model. \n - Subsequently, we adapt these parameters to a target task using the corresponding supervised objective.\n \n \n4. For our model architecture, we use the Transformer, This model choice provides us with a more structured memory for handling long-term dependencies in text, compared to alternatives like recurrent networks, resulting in robust transfer performance across diverse tasks. \n\n\n5. We evaluate our approach on four types of language understanding tasks:\n - natural language inference;\n - question answering;\n - semantic similarity;\n - text classification.\n\n### 2.2 Related Work\n\n#### Semi-supervised learning for NLP\n\nOver the last few years, researchers have demonstrated the benefits of using word embeddings, which are trained on unlabeled corpora, to improve performance on a variety of tasks. These approaches, however, mainly transfer word-level information, whereas we aim to capture higher-level semantics.\n\n\n#### Unsupervised pre-training\n\n1. Unsupervised pre-training is a special case of semi-supervised learning where the goal is to find a good initialization point instead of modifying the supervised learning objective.\n\n\n2. Subsequent research demonstrated that pre-training acts as a regularization scheme, enabling better generalization in deep neural networks.\n\n\n3. The closest line of work to ours involves pre-training a neural network using a language modeling objective and then fine-tuning it on a target task with supervision. \n\n\n4. Our choice of **transformer** networks allows us to **capture longer-range linguistic structure**, as demonstrated in our experiments.\n\n\n5. Other approaches use **hidden representations** from a pre-trained language or machine translation model as auxiliary features while training a supervised model on the target task. \n\n\n#### Auxiliary training objectives\n\n1. Adding auxiliary unsupervised training objectives is an alternative form of semi-supervised learning. \n\n\n2. Our experiments also use an auxiliary objective, but as we show, unsupervised pre-training already learns several linguistic aspects relevant to target tasks.\n\n### 2.3 Framework\n\n#### 2.3.1 Unsupervised pre-training\n\n\n \n \n
\n\n\u7ed9\u5b9a\u8bed\u6599\u5e93\u7684token\u96c6\u5408 $\\color{MediumOrchid}{\\mathcal{U}=\\left\\{u_{1}, \\ldots, u_{n}\\right\\}}$\uff0c\u4f7f\u7528\u6807\u51c6\u7684\u8bed\u8a00\u6a21\u578b\u6781\u5927\u5316\u4f3c\u7136\u51fd\u6570\uff1a\n\n$$\n\\color{MediumOrchid}{L_{1}(\\mathcal{U})=\\sum_{i} \\log P\\left(u_{i} \\mid u_{i-k}, \\ldots, u_{i-1} ; \\Theta\\right)} \\tag{1}\n$$\n\n- $k:$ size of the context window\n\nGPT\u4f7f\u7528Multi-layer Transformer Decoder\u6765\u8bad\u7ec3\u8bed\u8a00\u6a21\u578b: \n\n$$\n\\begin{eqnarray}\n\\color{red}{h_{0}} &=& \\color{red}{UW_{e} + W_{p}~~ ???} \\\\\nh_{l} &=& \\text{transformer_block}(h_{l-1}) \\quad \\forall l \\in [1, n] \\tag{2}\\\\\nP(u) &=& \\text{softmax}(h_{n}W_{e}^{\\mathrm{T}})\n\\end{eqnarray}\n$$\n\n- $U=(u_{-k}, \\cdots, u_{-1}):$ context vector of tokens;\n- $n:$ number of layers;\n- $W_{e}:$ token embedding matrix;\n- $W_{p}:$ position embedding matrix.\n\n#### 2.3.2 Supervised fine-tuning\n\n\u8bbe\u6709\u6807\u6ce8\u6570\u636e\u96c6 $\\mathcal{C}$\uff0c\u6bcf\u4e00\u4e2a\u6837\u672c\u7531\u8f93\u5165\u5e8f\u5217 $x^{1}, \\cdots, x^{m}$ \u548c\u6807\u7b7e $y$ \u7ec4\u6210\u3002\n\n\u5c06\u8f93\u5165\u5e8f\u5217\u4f20\u5165 pre-trained \u6a21\u578b\u4e2d\uff0c\u53d6\u6700\u540e\u4e00\u5c42transformer block\u7684\u6fc0\u6d3b\u9879 $h_{l}^{m}$ \uff0c\u9001\u8fdb\u4e00\u4e2a\u7ebf\u6027\u8f93\u51fa\u5c42\uff0c\u9884\u6d4b $y$:\n\n$$\nP\\left(y \\mid x^{1}, \\ldots, x^{m}\\right)=\\operatorname{softmax}\\left(h_{l}^{m} W_{y}\\right) \\tag{3}\n$$\n\n\u76ee\u6807\u662f\u6700\u5927\u5316\u5982\u4e0b\u76ee\u6807\u51fd\u6570: \n\n$$\nL_{2}(\\mathcal{C})=\\sum_{(x, y)} \\log P\\left(y \\mid x^{1}, \\ldots, x^{m}\\right) \\tag{4}\n$$\n\n\u6700\u7ec8\u548c$L_{1}$\u4e00\u8d77\u4f18\u5316\uff0c\u76ee\u6807\u51fd\u6570\u4e3a:\n\n$$\nL_{3}(\\mathcal{C})=L_{2}(\\mathcal{C})+\\lambda * L_{1}(\\mathcal{C}) \\tag{5}\n$$\n\n\n\u52a0\u4e0afine-tuning\u7684\u76ee\u6807\u51fd\u6570\u4e00\u8d77\u4f18\u5316\uff0c\u53ef\u4ee5:\n- improving generalization of the supervised model;\n- accelerating convergence. \n\n# 3. Transformer-XL\n\n- Segment-level Recurrence\n- Relative Positional Embedding\n\n\u5728NLP\u9886\u57df\uff0c\u5904\u7406\u8bed\u8a00\u5efa\u6a21\u95ee\u9898\u6709\u4e24\u79cd\u6700\u5148\u8fdb\u7684\u67b6\u6784\uff1aRNN\u548cTransformer\u3002RNN\u6309\u7167\u5e8f\u5217\u987a\u5e8f\u9010\u4e2a\u5b66\u4e60\u8f93\u5165\u7684\u5355\u8bcd\u6216\u5b57\u7b26\u4e4b\u95f4\u7684\u5173\u7cfb\uff1b\u800cTransformer\u5219\u63a5\u6536\u4e00\u6574\u6bb5\u5e8f\u5217\uff0c\u7136\u540e\u4f7f\u7528self-Attention\u673a\u5236\u6765\u5b66\u4e60\u5b83\u4eec\u4e4b\u95f4\u7684\u4f9d\u8d56\u5173\u7cfb\u3002\n\n\u4f46\u5b83\u4eec\u90fd\u6709\u4e00\u4e2a\u5171\u540c\u4e0d\u8db3\u4e4b\u5904: unable to model dependencies longer than a fixed length.\n\nTransformer-XL \u540c\u65f6\u7ed3\u5408\u4e86RNN\u5e8f\u5217\u5efa\u6a21\u548cTransformer\u81ea\u6ce8\u610f\u529b\u673a\u5236\u7684\u4f18\u70b9\uff0c\u5728\u8f93\u5165\u6570\u636e\u7684\u6bcf\u4e2a\u7247\u6bb5\u4e0a\u4f7f\u7528Transformer\u7684Self-Attention\u6a21\u5757\uff0c\u5e76\u4f7f\u7528\u5faa\u73af\u673a\u5236\u6765\u5b66\u4e60\u8fde\u7eed\u6bb5\u4e4b\u95f4\u7684\u4f9d\u8d56\u5173\u7cfb\u3002\n\n### 3.1 vanilla Transformer\n\n\n\nAl-Rfou\u7b49\u4eba\u57fa\u4e8eTransformer\u63d0\u51fa\u4e86vanilla model\uff0c\u5b83\u6839\u636e\u4e4b\u524d\u7684\u5b57\u7b26\u9884\u6d4b\u7247\u6bb5\u4e2d\u7684\u4e0b\u4e00\u4e2a\u5b57\u7b26\u3002\u4f8b\u5982\uff0c\u5b83\u4f7f\u7528$x_{1}, x_{2}, \\cdots, x_{n-1}$\u9884\u6d4b\u5b57\u7b26$x_{n}$\uff0c\u800c\u5728$x_{n}$\u4e4b\u540e\u7684\u5e8f\u5217\u5219\u88abmask\u6389\u3002\u8bba\u6587\u4e2d\u4f7f\u752864\u5c42\u6a21\u578b\uff0c\u5e76\u4ec5\u9650\u4e8e\u5904\u7406512\u4e2a\u5b57\u7b26\u8fd9\u79cd\u76f8\u5bf9\u8f83\u77ed\u7684\u8f93\u5165\uff0c\u56e0\u6b64\u5b83\u5c06\u8f93\u5165\u5206\u6210\u6bb5\uff0c\u5e76\u5206\u522b\u4ece\u6bcf\u4e2a\u6bb5\u4e2d\u8fdb\u884c\u5b66\u4e60\uff0c\u5982\u4e0a\u56fe\u6240\u793a\u3002\u5728Evaluation\u9636\u6bb5\uff0c\u8be5\u6a21\u578b\u4f1a\u5728\u6bcf\u4e00\u6b65\u4e2d\u5c06\u8f93\u5165\u5411\u53f3\u79fb\u52a8\u4e00\u4e2a\u5b57\u7b26\uff0c\u4ee5\u6b64\u5b9e\u73b0\u5bf9\u5355\u4e2a\u5b57\u7b26\u7684\u9884\u6d4b\u3002\n\n\n\u4f46vanilla model\u4ecd\u6709\u4e9b\u7f3a\u70b9:\n\n1. \u56e0\u4e3asegments\u4e4b\u95f4\u72ec\u7acb\u8bad\u7ec3\uff0c\u6240\u4ee5\u4e0d\u540c\u7684token\u4e4b\u95f4\uff0c\u6700\u957f\u7684\u4f9d\u8d56\u5173\u7cfb\uff0c\u5c31\u53d6\u51b3\u4e8esegment\u7684\u957f\u5ea6\uff1b\n\n\n2. \u51fa\u4e8e\u6548\u7387\u8003\u8651\uff0c\u5728\u5212\u5206segments\u7684\u65f6\u5019\uff0c\u4e0d\u8003\u8651\u53e5\u5b50\u7684\u81ea\u7136\u8fb9\u754c\uff0c\u800c\u662f\u6839\u636e\u56fa\u5b9a\u7684\u957f\u5ea6\u6765\u5212\u5206\u5e8f\u5217\uff0c\u5bfc\u81f4\u5206\u5272\u51fa\u6765\u7684segments\u5728\u8bed\u4e49\u4e0a\u662f\u4e0d\u5b8c\u6574\u7684\u3002\uff08context fragmentation problem\uff09\uff1b\n\n\n3. \u63a8\u7406\u901f\u5ea6\u6162: \u5728Evaluation\u9636\u6bb5\uff0c\u4e00\u822c\u53d6\u6700\u540e\u4e00\u4e2a\u4f4d\u7f6e\u7684\u9690\u5411\u91cf\u4f5c\u4e3a\u8f93\u51fa\u3002\u4e3a\u4e86\u5145\u5206\u5229\u7528\u4e0a\u4e0b\u6587\u5173\u7cfb\uff0c\u5728\u6bcf\u505a\u5b8c\u4e00\u6b21\u9884\u6d4b\u4e4b\u540e\uff0c\u5c31\u5bf9\u6574\u4e2a\u5e8f\u5217\u5411\u53f3\u79fb\u52a8\u4e00\u4e2a\u4f4d\u7f6e\uff0c\u518d\u505a\u4e00\u6b21\u8ba1\u7b97\uff0c\u5982\u4e0a\u56fe\uff08b\uff09\u6240\u793a\uff0c\u5219\u5bfc\u81f4\u8ba1\u7b97\u6548\u7387\u975e\u5e38\u4f4e\u3002\n\n### 3.2 Segment-Level Recurrence with State Reuse\n\n\n\nTransformer-XL\u5728\u5bf9\u5f53\u524dsegment\u8fdb\u884c\u5904\u7406\u7684\u65f6\u5019\uff0c\u7f13\u5b58\u5e76\u5229\u7528\u4e0a\u4e00\u4e2asegment\u4e2d\u6240\u6709layer\u7684\u9690\u5411\u91cf\u5e8f\u5217\uff0c\u800c\u4e14\u4e0a\u4e00\u4e2asegment\u7684\u6240\u6709\u9690\u5411\u91cf\u5e8f\u5217\u53ea\u53c2\u4e0e\u524d\u5411\u8ba1\u7b97\uff0c\u4e0d\u518d\u8fdb\u884c\u53cd\u5411\u4f20\u64ad\uff0c\u8fd9\u5c31\u662f\u6240\u8c13\u7684segment-level recurrence\u3002\n\n#### \u7b26\u53f7\u8bf4\u660e:\n\n\u4e24\u4e2a\u8fde\u7eed\u7684segments\u8868\u793a\u4e3a $s_{\\tau}=\\left[x_{\\tau, 1}, x_{\\tau, 2}, \\ldots, x_{\\tau, L}\\right],\\ s_{\\tau+1}=\\left[x_{\\tau+1, 1}, x_{\\tau+1, 2}, \\ldots, x_{\\tau+1, L}\\right], \\ L$\u662f\u5e8f\u5217\u957f\u5ea6\uff1b\n\n\n\u5047\u8bbe\u6574\u4e2a\u6a21\u578b\u4e2d\uff0c\u5305\u542b$~N~$\u5c42Transformer-block\uff0c\u90a3\u4e48\u6bcf\u4e2asegment\u4e2d\u5c31\u6709$~N~$\u7ec4\u957f\u5ea6\u4e3a$L$\u7684\u9690\u5411\u91cf\u5e8f\u5217\uff1b\n\n\n$\\mathbf{h}_{\\tau}^{n} \\in \\mathbb{R}^{L \\times d}$\u2014\u2014\u8868\u793a\u7b2c$~\\tau~$\u4e2asegment\u7684\u7b2c$~n~$\u5c42\u9690\u5411\u91cf\u5e8f\u5217\uff1b\n\n\n$\\text{SG}$\u662fstop-gradient\uff0c\u4e0d\u5728\u5bf9$~s_{\\tau}$ \u7684\u9690\u5411\u91cf\u505a\u53cd\u5411\u4f20\u64ad\uff1b\n\n\n$\\widetilde{\\mathbf{h}}_{\\tau+1}^{n-1}$ \u662f\u5bf9\u4e24\u4e2a\u9690\u5411\u91cf\u5e8f\u5217\u6cbf\u957f\u5ea6\u65b9\u5411\u7684\u62fc\u63a5\uff0c\\[\\]\u5185\u4e24\u4e2a\u9690\u5411\u91cf\u7684\u7ef4\u5ea6\u90fd\u662f$L \\times d$\uff0c\u62fc\u63a5\u4e4b\u540e\u7684\u5411\u91cf\u7ef4\u5ea6\u662f $2L \\times d$;\n\n\n$\\mathbf{q}$ \u7684\u8ba1\u7b97\u65b9\u5f0f\u4e0d\u53d8\uff0c\u53ea\u4f7f\u7528\u5f53\u524dsegment\u4e2d\u9690\u5411\u91cf\uff0c\u8ba1\u7b97\u5f97\u5230\u7684$\\mathbf{q}$\u5e8f\u5217\u957f\u5ea6\u4ecd\u662f$L$\uff1b\n\n\n$\\mathbf{k}, \\mathbf{v}$\u91c7\u7528\u62fc\u63a5\u4e4b\u540e\u7684$\\widetilde{\\mathbf{h}}$\u6765\u8ba1\u7b97\uff0c\u8ba1\u7b97\u51fa\u6765\u7684\u5e8f\u5217\u957f\u5ea6\u662f$2L$;\n\n\nTransformer\u7684\u8f93\u51fa\u9690\u5411\u91cf\u5e8f\u5217\u957f\u5ea6\u53d6\u51b3\u4e8equery\u7684\u5e8f\u5217\u957f\u5ea6\uff0c\u800c\u4e0d\u662fkey\u548cvalue.\n\n$$\n\\begin{array}{l}\n\\widetilde{\\mathbf{h}}_{\\tau+1}^{n-1}=\\left[\\mathrm{SG}\\left(\\mathbf{h}_{\\tau}^{n-1}\\right) \\circ \\mathbf{h}_{\\tau+1}^{n-1}\\right] \\\\\n\\mathbf{q}_{\\tau+1}^{n}, \\mathbf{k}_{\\tau+1}^{n}, \\mathbf{v}_{\\tau+1}^{n}=\\mathbf{h}_{\\tau+1}^{n-1} \\mathbf{W}_{q}^{\\top}, \\widetilde{\\mathbf{h}}_{\\tau+1}^{n-1} \\mathbf{W}_{k}^{\\top}, \\widetilde{\\mathbf{h}}_{\\tau+1}^{n-1} \\mathbf{W}_{v}^{\\top} \\\\\n\\mathbf{h}_{\\tau+1}^{n}= \\text{Transformer-Layer} \\left(\\mathbf{q}_{\\tau+1}^{n}, \\mathbf{k}_{\\tau+1}^{n}, \\mathbf{v}_{\\tau+1}^{n}\\right)\n\\end{array}\n$$\n\n\u8bad\u7ec3\u548c\u9884\u6d4b\u8fc7\u7a0b\u5982Fig2\u6240\u793a\u3002\u9700\u6ce8\u610f\u7684\u4e00\u70b9: \u5728\u5f53\u524dsegment\u4e2d\uff0c\u7b2c$~n~$\u5c42\u7684\u6bcf\u4e2a\u9690\u5411\u91cf\u7684\u8ba1\u7b97\uff0c\u9664\u4e86\u4f9d\u8d56\u5f53\u524d\u4f4d\u7f6e\u7684\u4e0b\u4e00\u5c42\u9690\u5411\u91cf\uff0c\u8fd8\u4e0e\u524d$L-1$\u4e2a\u4f4d\u7f6e\u7684\u9690\u5411\u91cf\u5b58\u5728\u4f9d\u8d56\u5173\u7cfb\uff0c\u800c\u4e14\u6bcf\u5f80\u4e0b\u8d70\u4e00\u5c42\uff0c\u4f9d\u8d56\u5173\u7cfb\u957f\u5ea6\u90fd\u4f1a\u589e\u52a0$(L-1)$\uff0c\u6240\u4ee5\u6700\u957f\u7684\u4f9d\u8d56\u5173\u7cfb\u662f$N(L-1)$\u3002\u5728\u5bf9\u957f\u6587\u672c\u8fdb\u884c\u8ba1\u7b97\u7684\u65f6\u5019\uff0c\u53ef\u4ee5\u7f13\u5b58\u4e0a\u4e00\u4e2asegment\u7684\u9690\u5411\u91cf\u7684\u7ed3\u679c\uff0c\u4e0d\u5fc5\u91cd\u590d\u8ba1\u7b97\uff0c\u5927\u5e45\u63d0\u9ad8\u8ba1\u7b97\u6548\u7387\u3002\n\n\n\u4e0a\u6587\u4e2d\uff0c\u6211\u4eec\u53ea\u4fdd\u5b58\u4e86\u4e0a\u4e00\u4e2asegment\uff0c\u5b9e\u9645\u64cd\u4f5c\u7684\u65f6\u5019\uff0c\u53ef\u4ee5\u4fdd\u5b58\u5c3d\u53ef\u80fd\u591a\u7684segments\uff0c\u53ea\u8981\u5185\u5b58\u6216\u8005\u663e\u5b58\u653e\u5f97\u4e0b\u3002\u8bba\u6587\u4e2d\u7684\u8bd5\u9a8c\u5728\u8bad\u7ec3\u7684\u65f6\u5019\uff0c\u53ea\u7f13\u5b58\u4e00\u4e2asegment\uff0c\u5728\u9884\u6d4b\u7684\u65f6\u5019\uff0c\u4f1a\u7f13\u5b58\u591a\u4e2asegments\u3002\n\n### 3.3 Relative Positional Encodings\n\nTransformer-XL\u653e\u5f03\u4e86vanilla transformer\u7edd\u5bf9\u4f4d\u7f6e\u7f16\u7801\uff0c\u800c\u91c7\u7528\u76f8\u5bf9\u4f4d\u7f6e\u7f16\u7801\u3002\u5177\u4f53\u5730\uff0c\u5728\u8ba1\u7b97Attention Score\u7684\u65f6\u5019\uff0c\u53ea\u8003\u8651query\u5411\u91cf\u4e0ekey\u5411\u91cf\u7684\u76f8\u5bf9\u4f4d\u7f6e\u5173\u7cfb\uff0c\u5e76\u4e14\u5c06\u8fd9\u79cd\u76f8\u5bf9\u4f4d\u7f6e\u5173\u7cfb\uff0c\u52a0\u5165\u5230\u6bcf\u4e00\u5c42Transformer\u7684Attention\u7684\u8ba1\u7b97\u4e2d\u3002\n\n\nVanilla Transformer\u4e2d\u4f4d\u7f6eEmbedding\u7684\u8ba1\u7b97\u516c\u5f0f\u5982\u4e0b: \n\n$$\n\\begin{aligned} \n\\mathbf { A } _ { i , j } ^ { \\mathrm { abs } } \n& = \\left\\{ \\mathbf { W }_{ q } \\left( \\mathbf { E }_{ x_{ i } } + \\mathbf { U }_{ i } \\right) \\right\\}^{\\top} \n\\left\\{ \\mathbf { W }_{ k } \\left( \\mathbf { E }_{ x_{ j } } + \\mathbf { U }_{ j } \\right) \\right\\}\\\\\n\\quad \\\\\n& = \\underbrace { \\mathbf { E } _ { x _ { i } } ^ { \\top } \\mathbf { W } _ { q } ^ { \\top } \\mathbf { W } _ { k } \\mathbf { E } _ { x _ { j } } } _ { ( a ) } + \\underbrace { \\mathbf { E } _ { x _ { i } } ^ { \\top } \\mathbf { W } _ { q } ^ { \\top } \\mathbf { W } _ { k } \\mathbf { U } _ { j } } _ { ( b ) } + \\underbrace { \\mathbf { U } _ { i } ^ { \\top } \\mathbf { W } _ { q } ^ { \\top } \\mathbf { W } _ { k } \\mathbf { E } _ { x _ { j } } } _ { ( c ) } + \\underbrace { \\mathbf { U } _ { i } ^ { \\top } \\mathbf { W } _ { q } ^ { \\top } \\mathbf { W } _ { k } \\mathbf { U } _ { j } } _ { ( d ) } \n\\end{aligned}\n$$\n\n\u800cTransformer-XL\u4e2d\u4f7f\u7528\u76f8\u5bf9\u4f4d\u7f6e\u8ba1\u7b97attention score\u7684\u516c\u5f0f\u5982\u4e0b: \n\n$$\n\\begin{aligned} \n\\mathbf { A } _ { i , j } ^ { \\mathrm { rel } } \n& = \\underbrace { \\mathbf { E } _ { x _ { i } } ^ { \\top } \\mathbf { W } _ { q } ^ { \\top } \\mathbf { W } _ { k,E } \\mathbf { E } _ { x _ { j } } } _ { ( a ) } + \\underbrace { \\mathbf { E } _ { x _ { i } } ^ { \\top } \\mathbf { W } _ { q } ^ { \\top } \\mathbf { W } _ { k,R } \\color{DeepSkyBlue}{ \\mathbf { R } _ { i-j }} } _ { ( b ) } + \\underbrace { \\color{red}{u ^ { \\top }} \\mathbf { W } _ { k,E } \\mathbf { E } _ { x _ { j } } } _ { ( c ) } + \\underbrace { \\color{red}{ v ^ { \\top }} \\mathbf { W } _ { k,R } \\color{DeepSkyBlue}{ \\mathbf { R } _ { i-j }} } _ { ( d ) } \n\\end{aligned}\n$$\n\n\u5176\u4e2d: \n- $\\color{red}{u,v}$ \u662ftrainable parameters; \n\n\n- $\\mathbf{W}_{k,E}$ \u7528\u4e8e\u751f\u6210\u57fa\u4e8e\u5185\u5bb9\u7684key\u5411\u91cf\uff1b\n\n\n- $\\mathbf{W}_{k,R}$ \u7528\u4e8e\u751f\u6210\u57fa\u4e8e\u4f4d\u7f6e\u7684key\u5411\u91cf\uff1b\n\n\n- $\\mathbf{R} \\in \\mathbb{R}^{L_{max} \\ \\ \\times d}$\uff0c\u7b2c$~i~$\u884c\u8868\u793a\u76f8\u5bf9\u4f4d\u7f6e\u95f4\u9694\u4e3a$~i~$\u7684\u4f4d\u7f6e\u5411\u91cf\u3002\u8bba\u6587\u4e2d\u5f3a\u8c03$\\mathbf{R}$\u91c7\u7528\u6b63\u5f26\u51fd\u6570\u751f\u6210\uff0c\u800c\u4e0d\u662f\u901a\u8fc7\u5b66\u4e60\u5f97\u5230\u7684\u3002\n\n\u6700\u540e\uff0c\u5bf9\u4e8e\u4e00\u4e2a$N$ \u5c42\u7684 single attention head \u7684Transformer-XL\u7684\u8ba1\u7b97\u516c\u5f0f\u5982\u4e0b: \n\n$\\text{For} \\quad n=1,\\cdots, N:$ \n\n\n\n# 3. XLNet\n\n*XLNet: Generalized Autoregressive Pretraining for Language Understanding*\n\nPart of Advances in Neural Information Processing Systems 32 (NeurIPS 2019)\n\n**Authors:**\n\nZhilin Yang, Zihang Dai, Yiming Yang, Jaime Carbonell, Russ R. Salakhutdinov, Quoc V. Le\n\n## 3.1 AR vs. AE\n\n\n\n### 3.1.1 Auto-Regression LM\n\nELMo(2018.03)/BERT(2018.10.11)\u51fa\u6765\u4e4b\u524d\uff0c\u5927\u5bb6\u901a\u5e38\u8bb2\u7684Language Model\u5176\u5b9e\u662f\u6839\u636e\u4e0a\u6587\u5185\u5bb9\u9884\u6d4b\u4e0b\u4e00\u4e2a\u53ef\u80fd\u8ddf\u968f\u7684\u5355\u8bcd\uff0c\u5c31\u662f\u5e38\u8bf4\u7684\u81ea\u5de6\u5411\u53f3\u7684Language Model\uff0c\u6216\u8005\u53cd\u8fc7\u6765\u4e5f\u884c\u3002\u8fd9\u79cdLanguage Model\u88ab\u79f0\u4e3a\u81ea\u56de\u5f52\u8bed\u8a00\u6a21\u578b\uff08Auto-Regression LM\uff09\u3002GPT\u662f\u5178\u578b\u7684AR LM\u3002ELMo\u5c3d\u7ba1\u770b\u4e0a\u53bb\u662f\u5229\u7528\u4e86\u4e0a\u6587\uff0c\u4e5f\u5229\u7528\u4e86\u4e0b\u6587\uff0c\u4f46\u662f\u672c\u8d28\u4e0a\u662fAuto-Regression LM\u3002ELMo\u662f\u5206\u522b\u505a\u4e86\u4e24\u4e2a\u65b9\u5411\u7684\u81ea\u56de\u5f52LM\uff0c\u7136\u540e\u628aLSTM\u7684\u4e24\u4e2a\u65b9\u5411\u7684\u9690\u72b6\u6001\u62fc\u63a5\u5230\u4e00\u8d77\uff0c\u6765\u4f53\u73b0\u53cc\u5411\u8bed\u8a00\u6a21\u578b\u8fd9\u4e2a\u4e8b\u60c5\u3002\u6240\u4ee5\u672c\u8d28\u4e0a\u8ba4\u8bc6Auto-Regression LM\n\n\u7ed9\u5b9a\u6587\u672c\u5e8f\u5217 $\\pmb{x}=\\left[x_{1}, \\ldots, x_{T}\\right]$\uff0cLanguage Model\u7684\u76ee\u6807\u662f\u8c03\u6574\u53c2\u6570\u4f7f\u5f97\u8bad\u7ec3\u6570\u636e\u4e0a\u7684\u4f3c\u7136\u51fd\u6570\u6700\u5927: \n\n$$\n\\max _{\\theta} \\log p_{\\theta}(\\pmb{x}) \\color{red}{=} \\sum_{t=1}^{T} \\log p_{\\theta}\\left(x_{t} \\mid \\pmb{x}_{\u662f\u65e0\u6cd5\u540c\u65f6\u5229\u7528\u4e0a\u4e0b\u6587\u7684\u4fe1\u606f\u3002\n\n### 3.1.2 Auto-Encoder LM\n\nBERT\u662f\u4e00\u79cd\u5178\u578b\u7684\u81ea\u7f16\u7801\u8bed\u8a00\u6a21\u578b\uff08Auto-Encoder LM\uff09\u3002\u5b83\u901a\u8fc7\u5c06\u5e8f\u5217$~\\pmb{x}~$\u4e2d\u968f\u673a\u6311\u900915%\u7684Token\u53d8\u6210\\[MASK\\]\u5f97\u5230\u5e26\u566a\u58f0\u7248\u672c\u7684 $\\hat{\\pmb{x}}$\u3002\u5047\u8bbe\u88abMask\u7684\u539f\u59cb\u503c\u4e3a$\\bar{\\pmb{x}}$\uff0c\u90a3\u4e48BERT\u5e0c\u671b\u5c3d\u91cf\u6839\u636e\u4e0a\u4e0b\u6587\u6062\u590d\uff08\u731c\u6d4b\uff09\u51fa\u539f\u59cb\u503c\uff0c\u4e5f\u5c31\u662f: \n\n$$\n\\max _{\\theta} \\log p_{\\theta}(\\overline{\\mathbf{x}} \\mid \\hat{\\mathbf{x}}) \\color{red}{ \\approx } \\sum_{t=1}^{T} m_{t} \\log p_{\\theta}\\left(x_{t} \\mid \\hat{\\mathbf{x}}\\right)=\\sum_{t=1}^{T} m_{t} \\log \\frac{\\exp \\left(H_{\\theta}(\\mathbf{x})_{t}^{T} e\\left(x_{t}\\right)\\right)}{\\sum_{x^{\\prime}} \\exp \\left(H_{\\theta}(\\mathbf{x})_{t}^{T} e\\left(x^{\\prime}\\right)\\right)}\n$$\n\n- $m_{t}=1$ \u8868\u793a$t$\u65f6\u523b\u662f\u4e00\u4e2aMASK\uff0c\u9700\u8981\u6062\u590d\uff1b\n- $H_{\\theta}$ \u662f\u4e00\u4e2aTransformer\uff0c\u5b83\u628a\u957f\u5ea6\u4e3a$T$\u7684\u5e8f\u5217$\\pmb{x}$\u6620\u5c04\u4e3a\u9690\u72b6\u6001\u7684\u5e8f\u5217 $H_{\\theta}(\\mathbf{x})=\\left[H_{\\theta}(\\mathbf{x})_{1}, H_{\\theta}(\\mathbf{x})_{2}, \\ldots, H_{\\theta}(\\mathbf{x})_{T}\\right]$\n- '$\\color{red}{\\approx}$' \uff0c\u662f\u56e0\u4e3a\u5f15\u5165\u4e86\u6761\u4ef6\u72ec\u7acb\u7684\u5047\u8bbe(Independent Assumption)\uff0c$P(New ~ York|is,a,city) \\color{red}{\\approx} P(New|is,a,city)\\cdot P(York|is,a,city)$\n\n\nAuto-Encoder LM\u80fd\u6bd4\u8f83\u81ea\u7136\u5730\u878d\u5165\u53cc\u5411\u8bed\u8a00\u6a21\u578b\uff0c\u540c\u65f6\u770b\u5230\u88ab\u9884\u6d4b\u5355\u8bcd\u7684\u4e0a\u6587\u548c\u4e0b\u6587\uff0c\u8fd9\u662f\u4f18\u70b9\u3002\u4f46\u662f\uff0c\u5728\u8f93\u5165\u4fa7\u5f15\u5165\\[MASK\\]\u6807\u8bb0\uff0c\u5bfc\u81f4Pre-training\u548cFine-tuning\u9636\u6bb5\u4e0d\u4e00\u81f4\u7684\u95ee\u9898\uff0c\u56e0\u4e3aFine-tuning\u9636\u6bb5\u662f\u770b\u4e0d\u5230\\[MASK\\]\u6807\u8bb0\u7684\u3002\n\nXLNet \u7684\u51fa\u53d1\u70b9\u5c31\u662f\uff1a\u80fd\u5426\u878d\u5408\u81ea\u56de\u5f52 LM \u548c DAE LM \u4e24\u8005\u7684\u4f18\u70b9\u3002\u5177\u4f53\u6765\u8bf4\u5c31\u662f\uff0c\u7ad9\u5728 AR \u7684\u89d2\u5ea6\uff0c\u5982\u4f55\u5f15\u5165\u548c\u53cc\u5411\u8bed\u8a00\u6a21\u578b\u7b49\u4ef7\u7684\u6548\u679c.\n\n## 3.2 Permutation Language Model\n\n\n\n#### \u8981\u70b9\uff1a\n- Sample a factorization order; \n\n- Determine the attention masks based on the order;\n\n- Optimize a standard language modeling objective:\n\n$$\n\\max _{\\theta} \\mathbb{E}_{\\mathbf{z} \\sim \\mathcal{Z}_{T}}\\left[\\sum_{t=1}^{T} \\log p_{\\theta}\\left(x_{z_{t}} \\mid \\mathbf{x}_{\\mathbf{z}_{\u901a\u8fc7\u968f\u673a\u53d6\u4e00\u53e5\u8bdd\u6392\u5217\u7684\u4e00\u79cd\uff0c\u7136\u540e\u5c06\u672b\u5c3e\u4e00\u5b9a\u91cf\u7684\u8bcd\u7ed9\u201c\u906e\u63a9\u201d\uff08\u548cBERT\u7684[MASK]\u6709\u4e9b\u4e0d\u540c\uff09\u6389\uff0c\u6700\u540e\u7528Auto-Regression\u7684\u65b9\u5f0f\u6765\u6309\u7167\u8fd9\u79cd\u6392\u5217\u65b9\u5f0f\u4f9d\u6b21\u9884\u6d4b\u88ab\u201c\u906e\u63a9\u201d\u6389\u7684\u8bcd\u3002\n\n\n\n\n\u6700\u540e\u201c\u906e\u63a9\u201d\u7684token\u957f\u5ea6\u600e\u4e48\u9009\u62e9\u5462\uff1f\n\u4f5c\u8005\u8bbe\u4e86\u4e00\u4e2a\u8d85\u53c2\u6570$K$\uff0c$K$\u7b49\u4e8e\u603b\u957f\u5ea6\u9664\u4ee5\u9700\u8981\u9884\u6d4b\u7684\u4e2a\u6570\u3002\u4ee5\u4e0a\u56fe\u4e3a\u4f8b\uff0c\u4e2d\u957f\u4e3a7\uff0c\u9700\u8981\u9884\u6d4b\u7684\u957f\u5ea6\u4e3a2\uff0c\u4e8e\u662f$K=7/2$\u3002\u8bba\u6587\u4e2d\u5b9e\u9a8c\u5f97\u51fa\u7684\u6700\u4f73$K$\u503c\u4ecb\u4e8e6\u548c7\u4e4b\u95f4\u3002\u5982\u679c\u53bb$K$\u7684\u5bfc\u6570\uff08\u5373$\\frac{1}{6}, \\frac{1}{7}$\uff09\uff0c\u8f6c\u5316\u4e3a\u767e\u5206\u6bd4\u4e3a\uff0814.3%\uff0c16.7%\uff09\u4e4b\u95f4\u3002\u800cBERT\u4e2d\u5c06Token\u66ff\u6362\u4e3a\\[MASK\\]\u7684\u6bd4\u5217\u5c31\u662f15%\uff0c\u4e8c\u8005\u4e4b\u95f4\u5e94\u8be5\u6709\u672c\u8d28\u4e0a\u7684\u8054\u7cfb\u3002\n\n\n\u5173\u4e8e\u53e5\u5b50\u6392\u5217\u7684\u91c7\u6837\uff1a\u5bf9\u4e8e\u4e00\u4e2a\u957f\u5ea6\u4e3a$T$\u7684\u53e5\u5b50\uff0c\u6709$T!$\u4e2d\u6392\u5217\uff0c\u5982\u679c\u904d\u5386\u6bcf\u79cd\u6392\u5217\uff0c\u662f\u4e0d\u73b0\u5b9e\u7684\u3002\u7528 $\\mathcal{Z}_{T}$ \u8868\u793a\u6240\u6709\u6392\u5217\u7ec4\u6210\u7684\u96c6\u5408\uff0c$\\mathcal{z}$ \u8868\u793a\u4ece$\\mathcal{Z}_{T}$\u91c7\u6837\u5f97\u5230\u4e00\u79cd\u6392\u5e8f\uff0c\u8bb0\u4e3a$\\mathcal{z} \\sim \\mathcal{Z}_{T}$\n\nXLNet\u5e76\u4e0d\u662f\u6253\u4e71\u8f93\u5165\u53e5\u5b50\u7684\u987a\u5e8f\uff0c\u800c\u662f\u901a\u8fc7Transformer\u7684Attention Masks\u6765\u5de7\u5999\u5b9e\u73b0\u7684\u3002\n\n\n\n## 3.3 Reparameterization\n\nPermutation Language Model\u7684\u601d\u60f3\u5f88\u7b80\u5355\uff0c\u4f46\u5982\u679c\u8fd8\u662f\u7528standard Transformer parameterization\uff0c\u5c31\u4f1a\u6709\u95ee\u9898\uff0cstandard Transformer parameterization\u516c\u5f0f\u4e3a: \n\n$$\n\\max _{\\theta} \\ \\log p_{\\theta}(\\pmb{x}) \\color{red}{=} \\sum_{t=1}^{T} \\log p_{\\theta}\\left(x_{t} \\mid \\pmb{x}_{ \\[3,4,5,1,2\\]\n2. is a city York New \u2014\u2014> \\[3,4,5,2,1\\]\n\n\u5bf9\u4e8e\u7b2c1\u79cd\u6392\u5217\uff0c\u5047\u8bbe\u6211\u4eec\u8981\u9884\u6d4b$z_{4} = New$\uff0c\u5219\u6709: \n\n$$\np_{\\theta}\\left( \\text{New} \\mid \\text{is, a, city} \\right) = \\frac{\\exp \n\\left \\{ h_{\\theta} \\left( \\text{is, a, city} \\right)^{\\mathrm{T}} \\cdot e(\\text{New}) \\right\\} }{ \\sum_{x^{\\prime}} \\exp \n\\left \\{ h_{\\theta} \\left( \\text{is, a, city} \\right)^{\\mathrm{T}} \\cdot e(x^{\\prime}) \\right\\} }\n$$\n\n\u540c\u7406\uff0c\u5bf9\u4e8e\u7b2c2\u4e2d\u6392\u5217\uff0c\u5047\u8bbe\u6211\u4eec\u8981\u9884\u6d4b$z_{4} = New$\uff0c\u540c\u6837\u6709: \n\n$$\np_{\\theta}\\left( \\text{New} \\mid \\text{is, a, city} \\right) = \\frac{\\exp \n\\left \\{ h_{\\theta} \\left( \\text{is, a, city} \\right)^{\\mathrm{T}} \\cdot e(\\text{New}) \\right\\} }{ \\sum_{x^{\\prime}} \\exp \n\\left \\{ h_{\\theta} \\left( \\text{is, a, city} \\right)^{\\mathrm{T}} \\cdot e(x^{\\prime}) \\right\\} }\n$$\n\n\u4e0a\u9762\u4e24\u4e2a\u516c\u5f0f\u5f97\u5230\u7684\u6982\u7387\u662f\u76f8\u7b49\u7684\uff0c\u4f46\u662f\u5bf9\u4e8e\u4e24\u79cd\u6392\u5217\uff0c\u5b83\u4eec\u7684\u6982\u7387\u5e94\u8be5\u662f\u4e0d\u76f8\u7b49\u7684\uff0c\u800c\u95ee\u9898\u7684\u539f\u56e0\u5728\u4e8e$h_{\\theta}\\left(\\pmb{x}_{1: t-1}\\right)$\u6ca1\u6709\u5efa\u6a21\u4f4d\u7f6e\u4fe1\u606f\u3002\n\n\u4e3a\u4e86\u89e3\u51b3\u4e0a\u8ff0\u95ee\u9898\uff0cXLNet\u63d0\u51fa\u4e86\u4e00\u79cd\u65b0\u7684\u53c2\u6570\u5316\u8868\u793a\u65b9\u6cd5: \n\n$$\np_{\\theta}\\left(X_{z_{t}}=x \\mid \\mathbf{x}_{z_{ t)$\u65f6\uff0c\u9700\u8981\u5305\u542b\u5185\u5bb9\u4fe1\u606f$x_{z_{t}}$\n\n\u5bf9\u4e8e\u4e0a\u9762\u7684\u4e24\u70b9\u8981\u6c42\uff0c\u666e\u901a\u7684Transformer Self-Attention\u662f\u4e0d\u80fd\u6ee1\u8db3\u7684\uff0c\u4e3e\u4f8b\u8bf4\u660e: \n\n\n\n\u4e3a\u89e3\u51b3\u4e0a\u8ff0\u95ee\u9898\uff0cXLNet\u5f15\u5165\u4e86Two-Stream Self-Attention\u7684\u8bbe\u8ba1: \n\n\n\n\n\u4ece\u4e0a\u56fe\u53ef\u4ee5\u770b\u5230\uff0c\u5728\u8ba1\u7b97Attention\u65f6\uff0c\u5f15\u5165\u4e86\u4e24\u4e2aStream\uff0c\u4e5f\u5c31\u662f\u4e24\u4e2a\u9690\u72b6\u6001: \n\n- \u5185\u5bb9\u9690\u72b6\u6001 $h_{\\theta}(\\mathbf{x}_{z\n\n\n# Week 47: Support Vector Machines and Summary of Course\n**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n\nDate: **Dec 8, 2021**\n\nCopyright 1999-2021, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n\n## Overview of week 47\n\n* **Thursday**: Support Vector Machines, classification and regression.\n\n * [Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h21/forelesningsvideoer/LectureNovember25.mp4?vrtx=view-as-webpage)\n\n* **Friday**: Support Vector Machines and Summary of Course\n\n * [Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK3155/h21/forelesningsvideoer/LectureNovember26.mp4?vrtx=view-as-webpage)\n\n_Reading recommendations:\n1. Geron's chapter 5.\n\n2. Hastie et al Chapter 12 (sections 12.1-12.3 are the most relevant ones)\n\n3. Bishop chapter 7, with sections 7.1 and 7.2 as the essential ones\n\n[See overview video on Support Vector Machines](https://www.youtube.com/watch?v=efR1C6CvhmE&ab_channel=StatQuestwithJoshStarmer). See also [this video](https://www.youtube.com/watch?v=N1vOgolbjSc&ab_channel=AliceZhao).\n\n## Support Vector Machines, overarching aims\n\nA Support Vector Machine (SVM) is a very powerful and versatile\nMachine Learning method, capable of performing linear or nonlinear\nclassification, regression, and even outlier detection. It is one of\nthe most popular models in Machine Learning, and anyone interested in\nMachine Learning should have it in their toolbox. SVMs are\nparticularly well suited for classification of complex but small-sized or\nmedium-sized datasets. \n\nThe case with two well-separated classes only can be understood in an\nintuitive way in terms of lines in a two-dimensional space separating\nthe two classes (see figure below).\n\nThe basic mathematics behind the SVM is however less familiar to most of us. \nIt relies on the definition of hyperplanes and the\ndefinition of a **margin** which separates classes (in case of\nclassification problems) of variables. It is also used for regression\nproblems.\n\nWith SVMs we distinguish between hard margin and soft margins. The\nlatter introduces a so-called softening parameter to be discussed\nbelow. We distinguish also between linear and non-linear\napproaches. The latter are the most frequent ones since it is rather\nunlikely that we can separate classes easily by say straight lines.\n\n## Hyperplanes and all that\n\nThe theory behind support vector machines (SVM hereafter) is based on\nthe mathematical description of so-called hyperplanes. Let us start\nwith a two-dimensional case. This will also allow us to introduce our\nfirst SVM examples. These will be tailored to the case of two specific\nclasses, as displayed in the figure here based on the usage of the petal data.\n\nWe assume here that our data set can be well separated into two\ndomains, where a straight line does the job in the separating the two\nclasses. Here the two classes are represented by either squares or\ncircles.\n\n\n```\n%matplotlib inline\n\nfrom sklearn import datasets\nfrom sklearn.svm import SVC, LinearSVC\nfrom sklearn.linear_model import SGDClassifier\nfrom sklearn.preprocessing import StandardScaler\nimport matplotlib\nimport matplotlib.pyplot as plt\nplt.rcParams['axes.labelsize'] = 14\nplt.rcParams['xtick.labelsize'] = 12\nplt.rcParams['ytick.labelsize'] = 12\n\n\niris = datasets.load_iris()\nX = iris[\"data\"][:, (2, 3)] # petal length, petal width\ny = iris[\"target\"]\n\nsetosa_or_versicolor = (y == 0) | (y == 1)\nX = X[setosa_or_versicolor]\ny = y[setosa_or_versicolor]\n\n\n\nC = 5\nalpha = 1 / (C * len(X))\n\nlin_clf = LinearSVC(loss=\"hinge\", C=C, random_state=42)\nsvm_clf = SVC(kernel=\"linear\", C=C)\nsgd_clf = SGDClassifier(loss=\"hinge\", learning_rate=\"constant\", eta0=0.001, alpha=alpha,\n max_iter=100000, random_state=42)\n\nscaler = StandardScaler()\nX_scaled = scaler.fit_transform(X)\n\nlin_clf.fit(X_scaled, y)\nsvm_clf.fit(X_scaled, y)\nsgd_clf.fit(X_scaled, y)\n\nprint(\"LinearSVC: \", lin_clf.intercept_, lin_clf.coef_)\nprint(\"SVC: \", svm_clf.intercept_, svm_clf.coef_)\nprint(\"SGDClassifier(alpha={:.5f}):\".format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_)\n\n# Compute the slope and bias of each decision boundary\nw1 = -lin_clf.coef_[0, 0]/lin_clf.coef_[0, 1]\nb1 = -lin_clf.intercept_[0]/lin_clf.coef_[0, 1]\nw2 = -svm_clf.coef_[0, 0]/svm_clf.coef_[0, 1]\nb2 = -svm_clf.intercept_[0]/svm_clf.coef_[0, 1]\nw3 = -sgd_clf.coef_[0, 0]/sgd_clf.coef_[0, 1]\nb3 = -sgd_clf.intercept_[0]/sgd_clf.coef_[0, 1]\n\n# Transform the decision boundary lines back to the original scale\nline1 = scaler.inverse_transform([[-10, -10 * w1 + b1], [10, 10 * w1 + b1]])\nline2 = scaler.inverse_transform([[-10, -10 * w2 + b2], [10, 10 * w2 + b2]])\nline3 = scaler.inverse_transform([[-10, -10 * w3 + b3], [10, 10 * w3 + b3]])\n\n# Plot all three decision boundaries\nplt.figure(figsize=(11, 4))\nplt.plot(line1[:, 0], line1[:, 1], \"k:\", label=\"LinearSVC\")\nplt.plot(line2[:, 0], line2[:, 1], \"b--\", linewidth=2, label=\"SVC\")\nplt.plot(line3[:, 0], line3[:, 1], \"r-\", label=\"SGDClassifier\")\nplt.plot(X[:, 0][y==1], X[:, 1][y==1], \"bs\") # label=\"Iris-Versicolor\"\nplt.plot(X[:, 0][y==0], X[:, 1][y==0], \"yo\") # label=\"Iris-Setosa\"\nplt.xlabel(\"Petal length\", fontsize=14)\nplt.ylabel(\"Petal width\", fontsize=14)\nplt.legend(loc=\"upper center\", fontsize=14)\nplt.axis([0, 5.5, 0, 2])\n\nplt.show()\n```\n\n## What is a hyperplane?\n\nThe aim of the SVM algorithm is to find a hyperplane in a\n$p$-dimensional space, where $p$ is the number of features that\ndistinctly classifies the data points.\n\nIn a $p$-dimensional space, a hyperplane is what we call an affine subspace of dimension of $p-1$.\nAs an example, in two dimension, a hyperplane is simply as straight line while in three dimensions it is \na two-dimensional subspace, or stated simply, a plane. \n\nIn two dimensions, with the variables $x_1$ and $x_2$, the hyperplane is defined as\n\n$$\nb+w_1x_1+w_2x_2=0,\n$$\n\nwhere $b$ is the intercept and $w_1$ and $w_2$ define the elements of a vector orthogonal to the line \n$b+w_1x_1+w_2x_2=0$. \nIn two dimensions we define the vectors $\\boldsymbol{x} =[x1,x2]$ and $\\boldsymbol{w}=[w1,w2]$. \nWe can then rewrite the above equation as\n\n$$\n\\boldsymbol{x}^T\\boldsymbol{w}+b=0.\n$$\n\nFor figures, see [handwritten notes](https://github.com/CompPhysics/MachineLearning/tree/master/doc/HandWrittenNotes/2021) for Thursday November 25.\n\n## A $p$-dimensional space of features\n\nWe limit ourselves to two classes of outputs $y_i$ and assign these classes the values $y_i = \\pm 1$. \nIn a $p$-dimensional space of say $p$ features we have a hyperplane defines as\n\n$$\nb+wx_1+w_2x_2+\\dots +w_px_p=0.\n$$\n\nIf we define a \nmatrix $\\boldsymbol{X}=\\left[\\boldsymbol{x}_1,\\boldsymbol{x}_2,\\dots, \\boldsymbol{x}_p\\right]$\nof dimension $n\\times p$, where $n$ represents the observations for each feature and each vector $x_i$ is a column vector of the matrix $\\boldsymbol{X}$,\n\n$$\n\\boldsymbol{x}_i = \\begin{bmatrix} x_{i1} \\\\ x_{i2} \\\\ \\dots \\\\ \\dots \\\\ x_{ip} \\end{bmatrix}.\n$$\n\nIf the above condition is not met for a given vector $\\boldsymbol{x}_i$ we have\n\n$$\nb+w_1x_{i1}+w_2x_{i2}+\\dots +w_px_{ip} >0,\n$$\n\nif our output $y_i=1$.\nIn this case we say that $\\boldsymbol{x}_i$ lies on one of the sides of the hyperplane and if\n\n$$\nb+w_1x_{i1}+w_2x_{i2}+\\dots +w_px_{ip} < 0,\n$$\n\nfor the class of observations $y_i=-1$, \nthen $\\boldsymbol{x}_i$ lies on the other side. \n\nEquivalently, for the two classes of observations we have\n\n$$\ny_i\\left(b+w_1x_{i1}+w_2x_{i2}+\\dots +w_px_{ip}\\right) > 0.\n$$\n\nWhen we try to separate hyperplanes, if it exists, we can use it to construct a natural classifier: a test observation is assigned a given class depending on which side of the hyperplane it is located.\n\n## The two-dimensional case\n\nLet us try to develop our intuition about SVMs by limiting ourselves to a two-dimensional\nplane. To separate the two classes of data points, there are many\npossible lines (hyperplanes if you prefer a more strict naming) \nthat could be chosen. Our objective is to find a\nplane that has the maximum margin, i.e the maximum distance between\ndata points of both classes. Maximizing the margin distance provides\nsome reinforcement so that future data points can be classified with\nmore confidence.\n\nWhat a linear classifier attempts to accomplish is to split the\nfeature space into two half spaces by placing a hyperplane between the\ndata points. This hyperplane will be our decision boundary. All\npoints on one side of the plane will belong to class one and all points\non the other side of the plane will belong to the second class two.\n\nUnfortunately there are many ways in which we can place a hyperplane\nto divide the data. Below is an example of two candidate hyperplanes\nfor our data sample.\n\n## Getting into the details\n\nLet us define the function\n\n$$\nf(x) = \\boldsymbol{w}^T\\boldsymbol{x}+b = 0,\n$$\n\nas the function that determines the line $L$ that separates two classes (our two features), see the figures in the [handwritten notes](https://github.com/CompPhysics/MachineLearning/tree/master/doc/HandWrittenNotes/2021) for Thursday November 25.\n. \n\nAny point defined by $\\boldsymbol{x}_i$ and $\\boldsymbol{x}_2$ on the line $L$ will satisfy $\\boldsymbol{w}^T(\\boldsymbol{x}_1-\\boldsymbol{x}_2)=0$. \n\nThe signed distance $\\delta$ from any point defined by a vector $\\boldsymbol{x}$ and a point $\\boldsymbol{x}_0$ on the line $L$ is then\n\n$$\n\\delta = \\frac{1}{\\vert\\vert \\boldsymbol{w}\\vert\\vert}(\\boldsymbol{w}^T\\boldsymbol{x}+b).\n$$\n\n## First attempt at a minimization approach\n\nHow do we find the parameter $b$ and the vector $\\boldsymbol{w}$? What we could\ndo is to define a cost function which now contains the set of all\nmisclassified points $M$ and attempt to minimize this function\n\n$$\nC(\\boldsymbol{w},b) = -\\sum_{i\\in M} y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b).\n$$\n\nWe could now for example define all values $y_i =1$ as misclassified in case we have $\\boldsymbol{w}^T\\boldsymbol{x}_i+b < 0$ and the opposite if we have $y_i=-1$. Taking the derivatives gives us\n\n$$\n\\frac{\\partial C}{\\partial b} = -\\sum_{i\\in M} y_i,\n$$\n\nand\n\n$$\n\\frac{\\partial C}{\\partial \\boldsymbol{w}} = -\\sum_{i\\in M} y_ix_i.\n$$\n\n## Solving the equations\n\nWe can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations\n\n$$\nb \\leftarrow b +\\eta \\frac{\\partial C}{\\partial b},\n$$\n\nand\n\n$$\n\\boldsymbol{w} \\leftarrow \\boldsymbol{w} +\\eta \\frac{\\partial C}{\\partial \\boldsymbol{w}},\n$$\n\nwhere $\\eta$ is our by now well-known learning rate.\n\n## Problems with the Simpler Approach\n\nThe equations we discussed above can be coded rather easily (the\nframework is similar to what we developed for logistic\nregression). \n\nThere are however problems with this approach, although it looks\npretty straightforward to implement. When running such a calculation, we can easily end up with many diffeent lines which separate the two classes.\n\nFor small\ngaps between the entries, we may also end up needing many iterations\nbefore the solutions converge and if the data cannot be separated\nproperly into two distinct classes, we may not experience a converge\nat all.\n\n## A better approach\n\nA better approach is rather to try to define a large margin between\nthe two classes (if they are well separated from the beginning).\n\nThus, we wish to find a margin $M$ with $\\boldsymbol{w}$ normalized to\n$\\vert\\vert \\boldsymbol{w}\\vert\\vert =1$ subject to the condition\n\n$$\ny_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) \\geq M \\hspace{0.1cm}\\forall i=1,2,\\dots, n.\n$$\n\nAll points are thus at a signed distance from the decision boundary defined by the line $L$. The parameters $b$ and $w_1$ and $w_2$ define this line. \n\nWe seek thus the largest value $M$ defined by\n\n$$\n\\frac{1}{\\vert \\vert \\boldsymbol{w}\\vert\\vert}y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) \\geq M \\hspace{0.1cm}\\forall i=1,2,\\dots, n,\n$$\n\nor just\n\n$$\ny_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) \\geq M\\vert \\vert \\boldsymbol{w}\\vert\\vert \\hspace{0.1cm}\\forall i=1,2,\\dots,n.\n$$\n\nIf we scale the equation so that $\\vert \\vert \\boldsymbol{w}\\vert\\vert = 1/M$, we have to find the minimum of \n$\\boldsymbol{w}^T\\boldsymbol{w}=\\vert \\vert \\boldsymbol{w}\\vert\\vert_2^2$ (the norm) subject to the condition\n\n$$\ny_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) \\geq 1 \\hspace{0.1cm}\\forall i=1,2,\\dots,n.\n$$\n\nWe have thus defined our margin as the invers of the norm of\n$\\boldsymbol{w}$. We want to minimize the norm in order to have a as large as\npossible margin $M$. Before we proceed, we need to remind ourselves\nabout Lagrangian multipliers.\n\n## A quick Reminder on Lagrangian Multipliers\n\nConsider a function of three independent variables $f(x,y,z)$ . For the function $f$ to be an\nextreme we have\n\n$$\ndf=0.\n$$\n\nA necessary and sufficient condition is\n\n$$\n\\frac{\\partial f}{\\partial x} =\\frac{\\partial f}{\\partial y}=\\frac{\\partial f}{\\partial z}=0,\n$$\n\ndue to\n\n$$\ndf = \\frac{\\partial f}{\\partial x}dx+\\frac{\\partial f}{\\partial y}dy+\\frac{\\partial f}{\\partial z}dz.\n$$\n\nIn many problems the variables $x,y,z$ are often subject to constraints (such as those above for the margin)\nso that they are no longer all independent. It is possible at least in principle to use each \nconstraint to eliminate one variable\nand to proceed with a new and smaller set of independent varables.\n\nThe use of so-called Lagrangian multipliers is an alternative technique when the elimination\nof variables is incovenient or undesirable. Assume that we have an equation of constraint on \nthe variables $x,y,z$\n\n$$\n\\phi(x,y,z) = 0,\n$$\n\nresulting in\n\n$$\nd\\phi = \\frac{\\partial \\phi}{\\partial x}dx+\\frac{\\partial \\phi}{\\partial y}dy+\\frac{\\partial \\phi}{\\partial z}dz =0.\n$$\n\nNow we cannot set anymore\n\n$$\n\\frac{\\partial f}{\\partial x} =\\frac{\\partial f}{\\partial y}=\\frac{\\partial f}{\\partial z}=0,\n$$\n\nif $df=0$ is wanted\nbecause there are now only two independent variables! Assume $x$ and $y$ are the independent \nvariables.\nThen $dz$ is no longer arbitrary.\n\n## Adding the Multiplier\n\nHowever, we can add to\n\n$$\ndf = \\frac{\\partial f}{\\partial x}dx+\\frac{\\partial f}{\\partial y}dy+\\frac{\\partial f}{\\partial z}dz,\n$$\n\na multiplum of $d\\phi$, viz. $\\lambda d\\phi$, resulting in\n\n$$\ndf+\\lambda d\\phi = (\\frac{\\partial f}{\\partial z}+\\lambda\n\\frac{\\partial \\phi}{\\partial x})dx+(\\frac{\\partial f}{\\partial y}+\\lambda\\frac{\\partial \\phi}{\\partial y})dy+\n(\\frac{\\partial f}{\\partial z}+\\lambda\\frac{\\partial \\phi}{\\partial z})dz =0.\n$$\n\nOur multiplier is chosen so that\n\n$$\n\\frac{\\partial f}{\\partial z}+\\lambda\\frac{\\partial \\phi}{\\partial z} =0.\n$$\n\nWe need to remember that we took $dx$ and $dy$ to be arbitrary and thus we must have\n\n$$\n\\frac{\\partial f}{\\partial x}+\\lambda\\frac{\\partial \\phi}{\\partial x} =0,\n$$\n\nand\n\n$$\n\\frac{\\partial f}{\\partial y}+\\lambda\\frac{\\partial \\phi}{\\partial y} =0.\n$$\n\nWhen all these equations are satisfied, $df=0$. We have four unknowns, $x,y,z$ and\n$\\lambda$. Actually we want only $x,y,z$, $\\lambda$ needs not to be determined, \nit is therefore often called\nLagrange's undetermined multiplier.\nIf we have a set of constraints $\\phi_k$ we have the equations\n\n$$\n\\frac{\\partial f}{\\partial x_i}+\\sum_k\\lambda_k\\frac{\\partial \\phi_k}{\\partial x_i} =0.\n$$\n\n## Setting up the Problem\n\nIn order to solve the above problem, we define the following Lagrangian function to be minimized\n\n$$\n{\\cal L}(\\lambda,b,\\boldsymbol{w})=\\frac{1}{2}\\boldsymbol{w}^T\\boldsymbol{w}-\\sum_{i=1}^n\\lambda_i\\left[y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)-1\\right],\n$$\n\nwhere $\\lambda_i$ is a so-called Lagrange multiplier subject to the condition $\\lambda_i \\geq 0$.\n\nTaking the derivatives with respect to $b$ and $\\boldsymbol{w}$ we obtain\n\n$$\n\\frac{\\partial {\\cal L}}{\\partial b} = -\\sum_{i} \\lambda_iy_i=0,\n$$\n\nand\n\n$$\n\\frac{\\partial {\\cal L}}{\\partial \\boldsymbol{w}} = 0 = \\boldsymbol{w}-\\sum_{i} \\lambda_iy_i\\boldsymbol{x}_i.\n$$\n\nInserting these constraints into the equation for ${\\cal L}$ we obtain\n\n$$\n{\\cal L}=\\sum_i\\lambda_i-\\frac{1}{2}\\sum_{ij}^n\\lambda_i\\lambda_jy_iy_j\\boldsymbol{x}_i^T\\boldsymbol{x}_j,\n$$\n\nsubject to the constraints $\\lambda_i\\geq 0$ and $\\sum_i\\lambda_iy_i=0$. \nWe must in addition satisfy the [Karush-Kuhn-Tucker](https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions) (KKT) condition\n\n$$\n\\lambda_i\\left[y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) -1\\right] \\hspace{0.1cm}\\forall i.\n$$\n\n1. If $\\lambda_i > 0$, then $y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)=1$ and we say that $x_i$ is on the boundary.\n\n2. If $y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)> 1$, we say $x_i$ is not on the boundary and we set $\\lambda_i=0$. \n\nWhen $\\lambda_i > 0$, the vectors $\\boldsymbol{x}_i$ are called support vectors. They are the vectors closest to the line (or hyperplane) and define the margin $M$.\n\n## The problem to solve\n\nWe can rewrite\n\n$$\n{\\cal L}=\\sum_i\\lambda_i-\\frac{1}{2}\\sum_{ij}^n\\lambda_i\\lambda_jy_iy_j\\boldsymbol{x}_i^T\\boldsymbol{x}_j,\n$$\n\nand its constraints in terms of a matrix-vector problem where we minimize w.r.t. $\\lambda$ the following problem\n\n$$\n\\frac{1}{2} \\boldsymbol{\\lambda}^T\\begin{bmatrix} y_1y_1\\boldsymbol{x}_1^T\\boldsymbol{x}_1 & y_1y_2\\boldsymbol{x}_1^T\\boldsymbol{x}_2 & \\dots & \\dots & y_1y_n\\boldsymbol{x}_1^T\\boldsymbol{x}_n \\\\\ny_2y_1\\boldsymbol{x}_2^T\\boldsymbol{x}_1 & y_2y_2\\boldsymbol{x}_2^T\\boldsymbol{x}_2 & \\dots & \\dots & y_1y_n\\boldsymbol{x}_2^T\\boldsymbol{x}_n \\\\\n\\dots & \\dots & \\dots & \\dots & \\dots \\\\\n\\dots & \\dots & \\dots & \\dots & \\dots \\\\\ny_ny_1\\boldsymbol{x}_n^T\\boldsymbol{x}_1 & y_ny_2\\boldsymbol{x}_n^T\\boldsymbol{x}_2 & \\dots & \\dots & y_ny_n\\boldsymbol{x}_n^T\\boldsymbol{x}_n \\\\\n\\end{bmatrix}\\boldsymbol{\\lambda}-\\mathbb{1}\\boldsymbol{\\lambda},\n$$\n\nsubject to $\\boldsymbol{y}^T\\boldsymbol{\\lambda}=0$. Here we defined the vectors $\\boldsymbol{\\lambda}^T =[\\lambda_1,\\lambda_2,\\dots,\\lambda_n]$ and \n$\\boldsymbol{y}^T=[y_1,y_2,\\dots,y_n]$.\n\n## The last steps\n\nSolving the above problem, yields the values of $\\lambda_i$.\nTo find the coefficients of your hyperplane we need simply to compute\n\n$$\n\\boldsymbol{w}=\\sum_{i} \\lambda_iy_i\\boldsymbol{x}_i.\n$$\n\nWith our vector $\\boldsymbol{w}$ we can in turn find the value of the intercept $b$ (here in two dimensions) via\n\n$$\ny_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)=1,\n$$\n\nresulting in\n\n$$\nb = \\frac{1}{y_i}-\\boldsymbol{w}^T\\boldsymbol{x}_i,\n$$\n\nor if we write it out in terms of the support vectors only, with $N_s$ being their number, we have\n\n$$\nb = \\frac{1}{N_s}\\sum_{j\\in N_s}\\left(y_j-\\sum_{i=1}^n\\lambda_iy_i\\boldsymbol{x}_i^T\\boldsymbol{x}_j\\right).\n$$\n\nWith our hyperplane coefficients we can use our classifier to assign any observation by simply using\n\n$$\ny_i = \\mathrm{sign}(\\boldsymbol{w}^T\\boldsymbol{x}_i+b).\n$$\n\nBelow we discuss how to find the optimal values of $\\lambda_i$. Before we proceed however, we discuss now the so-called soft classifier.\n\n## A soft classifier\n\nTill now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined.\n\nSuppose now that classes overlap in feature space, as shown in the\nfigure in the [handwritten notes](https://github.com/CompPhysics/MachineLearning/tree/master/doc/HandWrittenNotes/2021) for Thursday November 25.\n\nOne way to deal with this problem before we define the\nso-called **kernel approach**, is to allow a kind of slack in the sense\nthat we allow some points to be on the wrong side of the margin.\n\nWe introduce thus the so-called **slack** variables $\\boldsymbol{\\xi} =[\\xi_1,x_2,\\dots,x_n]$ and \nmodify our previous equation\n\n$$\ny_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)=1,\n$$\n\nto\n\n$$\ny_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)=1-\\xi_i,\n$$\n\nwith the requirement $\\xi_i\\geq 0$. The total violation is now $\\sum_i\\xi$. \nThe value $\\xi_i$ in the constraint the last constraint corresponds to the amount by which the prediction\n$y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)=1$ is on the wrong side of its margin. Hence by bounding the sum $\\sum_i \\xi_i$,\nwe bound the total amount by which predictions fall on the wrong side of their margins.\n\nMisclassifications occur when $\\xi_i > 1$. Thus bounding the total sum by some value $C$ bounds in turn the total number of\nmisclassifications.\n\n## Soft optmization problem\n\nThis has in turn the consequences that we change our optmization problem to finding the minimum of\n\n$$\n{\\cal L}=\\frac{1}{2}\\boldsymbol{w}^T\\boldsymbol{w}-\\sum_{i=1}^n\\lambda_i\\left[y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)-(1-\\xi_)\\right]+C\\sum_{i=1}^n\\xi_i-\\sum_{i=1}^n\\gamma_i\\xi_i,\n$$\n\nsubject to\n\n$$\ny_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)=1-\\xi_i \\hspace{0.1cm}\\forall i,\n$$\n\nwith the requirement $\\xi_i\\geq 0$.\n\nTaking the derivatives with respect to $b$ and $\\boldsymbol{w}$ we obtain\n\n$$\n\\frac{\\partial {\\cal L}}{\\partial b} = -\\sum_{i} \\lambda_iy_i=0,\n$$\n\nand\n\n$$\n\\frac{\\partial {\\cal L}}{\\partial \\boldsymbol{w}} = 0 = \\boldsymbol{w}-\\sum_{i} \\lambda_iy_i\\boldsymbol{x}_i,\n$$\n\nand\n\n$$\n\\lambda_i = C-\\gamma_i \\hspace{0.1cm}\\forall i.\n$$\n\nInserting these constraints into the equation for ${\\cal L}$ we obtain the same equation as before\n\n$$\n{\\cal L}=\\sum_i\\lambda_i-\\frac{1}{2}\\sum_{ij}^n\\lambda_i\\lambda_jy_iy_j\\boldsymbol{x}_i^T\\boldsymbol{x}_j,\n$$\n\nbut now subject to the constraints $\\lambda_i\\geq 0$, $\\sum_i\\lambda_iy_i=0$ and $0\\leq\\lambda_i \\leq C$. \nWe must in addition satisfy the Karush-Kuhn-Tucker condition which now reads\n\n$$\n\\lambda_i\\left[y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) -(1-\\xi_)\\right]=0 \\hspace{0.1cm}\\forall i,\n$$\n\n$$\n\\gamma_i\\xi_i = 0,\n$$\n\nand\n\n$$\ny_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) -(1-\\xi_) \\geq 0 \\hspace{0.1cm}\\forall i.\n$$\n\n## Kernels and non-linearity\n\nThe cases we have studied till now, were all characterized by two classes\nwith a close to linear separability. The classifiers we have described\nso far find linear boundaries in our input feature space. It is\npossible to make our procedure more flexible by exploring the feature\nspace using other basis expansions such as higher-order polynomials,\nwavelets, splines etc.\n\nIf our feature space is not easy to separate, as shown in the figure\nin the [handwritten notes](https://github.com/CompPhysics/MachineLearning/tree/master/doc/HandWrittenNotes/2021) for Thursday November 25.\n, we can achieve a better separation by introducing more complex\nbasis functions. The ideal would be to, via a specific transformation to \nobtain a separation between the classes which is almost linear. \n\nThe change of basis, from $x\\rightarrow z=\\phi(x)$ leads to the same type of equations to be solved, except that\nwe need to introduce for example a polynomial transformation to a two-dimensional training set.\n\n\n```\nimport numpy as np\nimport os\n\nnp.random.seed(42)\n\n# To plot pretty figures\nimport matplotlib\nimport matplotlib.pyplot as plt\nplt.rcParams['axes.labelsize'] = 14\nplt.rcParams['xtick.labelsize'] = 12\nplt.rcParams['ytick.labelsize'] = 12\n\n\nfrom sklearn.svm import SVC\nfrom sklearn import datasets\n\n\n\nX1D = np.linspace(-4, 4, 9).reshape(-1, 1)\nX2D = np.c_[X1D, X1D**2]\ny = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])\n\nplt.figure(figsize=(11, 4))\n\nplt.subplot(121)\nplt.grid(True, which='both')\nplt.axhline(y=0, color='k')\nplt.plot(X1D[:, 0][y==0], np.zeros(4), \"bs\")\nplt.plot(X1D[:, 0][y==1], np.zeros(5), \"g^\")\nplt.gca().get_yaxis().set_ticks([])\nplt.xlabel(r\"$x_1$\", fontsize=20)\nplt.axis([-4.5, 4.5, -0.2, 0.2])\n\nplt.subplot(122)\nplt.grid(True, which='both')\nplt.axhline(y=0, color='k')\nplt.axvline(x=0, color='k')\nplt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], \"bs\")\nplt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], \"g^\")\nplt.xlabel(r\"$x_1$\", fontsize=20)\nplt.ylabel(r\"$x_2$\", fontsize=20, rotation=0)\nplt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16])\nplt.plot([-4.5, 4.5], [6.5, 6.5], \"r--\", linewidth=3)\nplt.axis([-4.5, 4.5, -1, 17])\nplt.subplots_adjust(right=1)\nplt.show()\n```\n\n## The equations\n\nSuppose we define a polynomial transformation of degree two only (we continue to live in a plane with $x_i$ and $y_i$ as variables)\n\n$$\nz = \\phi(x_i) =\\left(x_i^2, y_i^2, \\sqrt{2}x_iy_i\\right).\n$$\n\nWith our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity)\n\n$$\n{\\cal L}=\\sum_i\\lambda_i-\\frac{1}{2}\\sum_{ij}^n\\lambda_i\\lambda_jy_iy_j\\boldsymbol{z}_i^T\\boldsymbol{z}_j,\n$$\n\nsubject to the constraints $\\lambda_i\\geq 0$, $\\sum_i\\lambda_iy_i=0$, and for the support vectors\n\n$$\ny_i(\\boldsymbol{w}^T\\boldsymbol{z}_i+b)= 1 \\hspace{0.1cm}\\forall i,\n$$\n\nfrom which we also find $b$.\nTo compute $\\boldsymbol{z}_i^T\\boldsymbol{z}_j$ we define the kernel $K(\\boldsymbol{x}_i,\\boldsymbol{x}_j)$ as\n\n$$\nK(\\boldsymbol{x}_i,\\boldsymbol{x}_j)=\\boldsymbol{z}_i^T\\boldsymbol{z}_j= \\phi(\\boldsymbol{x}_i)^T\\phi(\\boldsymbol{x}_j).\n$$\n\nFor the above example, the kernel reads\n\n$$\nK(\\boldsymbol{x}_i,\\boldsymbol{x}_j)=[x_i^2, y_i^2, \\sqrt{2}x_iy_i]^T\\begin{bmatrix} x_j^2 \\\\ y_j^2 \\\\ \\sqrt{2}x_jy_j \\end{bmatrix}=x_i^2x_j^2+2x_ix_jy_iy_j+y_i^2y_j^2.\n$$\n\nWe note that this is nothing but the dot product of the two original\nvectors $(\\boldsymbol{x}_i^T\\boldsymbol{x}_j)^2$. Instead of thus computing the\nproduct in the Lagrangian of $\\boldsymbol{z}_i^T\\boldsymbol{z}_j$ we simply compute\nthe dot product $(\\boldsymbol{x}_i^T\\boldsymbol{x}_j)^2$.\n\nThis leads to the so-called\nkernel trick and the result leads to the same as if we went through\nthe trouble of performing the transformation\n$\\phi(\\boldsymbol{x}_i)^T\\phi(\\boldsymbol{x}_j)$ during the SVM calculations.\n\n## The problem to solve\nUsing our definition of the kernel We can rewrite again the Lagrangian\n\n$$\n{\\cal L}=\\sum_i\\lambda_i-\\frac{1}{2}\\sum_{ij}^n\\lambda_i\\lambda_jy_iy_j\\boldsymbol{x}_i^T\\boldsymbol{z}_j,\n$$\n\nsubject to the constraints $\\lambda_i\\geq 0$, $\\sum_i\\lambda_iy_i=0$ in terms of a convex optimization problem\n\n$$\n\\frac{1}{2} \\boldsymbol{\\lambda}^T\\begin{bmatrix} y_1y_1K(\\boldsymbol{x}_1,\\boldsymbol{x}_1) & y_1y_2K(\\boldsymbol{x}_1,\\boldsymbol{x}_2) & \\dots & \\dots & y_1y_nK(\\boldsymbol{x}_1,\\boldsymbol{x}_n) \\\\\ny_2y_1K(\\boldsymbol{x}_2,\\boldsymbol{x}_1) & y_2y_2(\\boldsymbol{x}_2,\\boldsymbol{x}_2) & \\dots & \\dots & y_1y_nK(\\boldsymbol{x}_2,\\boldsymbol{x}_n) \\\\\n\\dots & \\dots & \\dots & \\dots & \\dots \\\\\n\\dots & \\dots & \\dots & \\dots & \\dots \\\\\ny_ny_1K(\\boldsymbol{x}_n,\\boldsymbol{x}_1) & y_ny_2K(\\boldsymbol{x}_n\\boldsymbol{x}_2) & \\dots & \\dots & y_ny_nK(\\boldsymbol{x}_n,\\boldsymbol{x}_n) \\\\\n\\end{bmatrix}\\boldsymbol{\\lambda}-\\mathbb{1}\\boldsymbol{\\lambda},\n$$\n\nsubject to $\\boldsymbol{y}^T\\boldsymbol{\\lambda}=0$. Here we defined the vectors $\\boldsymbol{\\lambda} =[\\lambda_1,\\lambda_2,\\dots,\\lambda_n]$ and \n$\\boldsymbol{y}=[y_1,y_2,\\dots,y_n]$. \nIf we add the slack constants this leads to the additional constraint $0\\leq \\lambda_i \\leq C$.\n\nWe can rewrite this (see the solutions below) in terms of a convex optimization problem of the type\n\n$$\n\\begin{align*}\n &\\mathrm{min}_{\\lambda}\\hspace{0.2cm} \\frac{1}{2}\\boldsymbol{\\lambda}^T\\boldsymbol{P}\\boldsymbol{\\lambda}+\\boldsymbol{q}^T\\boldsymbol{\\lambda},\\\\ \\nonumber\n &\\mathrm{subject\\hspace{0.1cm}to} \\hspace{0.2cm} \\boldsymbol{G}\\boldsymbol{\\lambda} \\preceq \\boldsymbol{h} \\hspace{0.2cm} \\wedge \\boldsymbol{A}\\boldsymbol{\\lambda}=f.\n\\end{align*}\n$$\n\nBelow we discuss how to solve these equations. Here we note that the matrix $\\boldsymbol{P}$ has matrix elements $p_{ij}=y_iy_jK(\\boldsymbol{x}_i,\\boldsymbol{x}_j)$.\nGiven a kernel $K$ and the targets $y_i$ this matrix is easy to set up. The constraint $\\boldsymbol{y}^T\\boldsymbol{\\lambda}=0$ leads to $f=0$ and $\\boldsymbol{A}=\\boldsymbol{y}$. How to set up the matrix $\\boldsymbol{G}$ is discussed later. Here note that the inequalities $0\\leq \\lambda_i \\leq C$ can be split up into\n$0\\leq \\lambda_i$ and $\\lambda_i \\leq C$. These two inequalities define then the matrix $\\boldsymbol{G}$ and the vector $\\boldsymbol{h}$.\n\n## Different kernels and Mercer's theorem\n\nThere are several popular kernels being used. These are\n1. Linear: $K(\\boldsymbol{x},\\boldsymbol{y})=\\boldsymbol{x}^T\\boldsymbol{y}$,\n\n2. Polynomial: $K(\\boldsymbol{x},\\boldsymbol{y})=(\\boldsymbol{x}^T\\boldsymbol{y}+\\gamma)^d$,\n\n3. Gaussian Radial Basis Function: $K(\\boldsymbol{x},\\boldsymbol{y})=\\exp{\\left(-\\gamma\\vert\\vert\\boldsymbol{x}-\\boldsymbol{y}\\vert\\vert^2\\right)}$,\n\n4. Tanh: $K(\\boldsymbol{x},\\boldsymbol{y})=\\tanh{(\\boldsymbol{x}^T\\boldsymbol{y}+\\gamma)}$,\n\nand many other ones.\n\nAn important theorem for us is [Mercer's\ntheorem](https://en.wikipedia.org/wiki/Mercer%27s_theorem). The\ntheorem states that if a kernel function $K$ is symmetric, continuous\nand leads to a positive semi-definite matrix $\\boldsymbol{P}$ then there\nexists a function $\\phi$ that maps $\\boldsymbol{x}_i$ and $\\boldsymbol{x}_j$ into\nanother space (possibly with much higher dimensions) such that\n\n$$\nK(\\boldsymbol{x}_i,\\boldsymbol{x}_j)=\\phi(\\boldsymbol{x}_i)^T\\phi(\\boldsymbol{x}_j).\n$$\n\nSo you can use $K$ as a kernel since you know $\\phi$ exists, even if\nyou don\u2019t know what $\\phi$ is. \n\nNote that some frequently used kernels (such as the Sigmoid kernel)\ndon\u2019t respect all of Mercer\u2019s conditions, yet they generally work well\nin practice.\n\n## The moons example\n\n\n```\nfrom __future__ import division, print_function, unicode_literals\n\nimport numpy as np\nnp.random.seed(42)\n\nimport matplotlib\nimport matplotlib.pyplot as plt\nplt.rcParams['axes.labelsize'] = 14\nplt.rcParams['xtick.labelsize'] = 12\nplt.rcParams['ytick.labelsize'] = 12\n\n\nfrom sklearn.svm import SVC\nfrom sklearn import datasets\n\n\n\nfrom sklearn.pipeline import Pipeline\nfrom sklearn.preprocessing import StandardScaler\nfrom sklearn.svm import LinearSVC\n\n\nfrom sklearn.datasets import make_moons\nX, y = make_moons(n_samples=100, noise=0.15, random_state=42)\n\ndef plot_dataset(X, y, axes):\n plt.plot(X[:, 0][y==0], X[:, 1][y==0], \"bs\")\n plt.plot(X[:, 0][y==1], X[:, 1][y==1], \"g^\")\n plt.axis(axes)\n plt.grid(True, which='both')\n plt.xlabel(r\"$x_1$\", fontsize=20)\n plt.ylabel(r\"$x_2$\", fontsize=20, rotation=0)\n\nplot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\nplt.show()\n\nfrom sklearn.datasets import make_moons\nfrom sklearn.pipeline import Pipeline\nfrom sklearn.preprocessing import PolynomialFeatures\n\npolynomial_svm_clf = Pipeline([\n (\"poly_features\", PolynomialFeatures(degree=3)),\n (\"scaler\", StandardScaler()),\n (\"svm_clf\", LinearSVC(C=10, loss=\"hinge\", random_state=42))\n ])\n\npolynomial_svm_clf.fit(X, y)\n\ndef plot_predictions(clf, axes):\n x0s = np.linspace(axes[0], axes[1], 100)\n x1s = np.linspace(axes[2], axes[3], 100)\n x0, x1 = np.meshgrid(x0s, x1s)\n X = np.c_[x0.ravel(), x1.ravel()]\n y_pred = clf.predict(X).reshape(x0.shape)\n y_decision = clf.decision_function(X).reshape(x0.shape)\n plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2)\n plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1)\n\nplot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5])\nplot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n\nplt.show()\n\n\nfrom sklearn.svm import SVC\n\npoly_kernel_svm_clf = Pipeline([\n (\"scaler\", StandardScaler()),\n (\"svm_clf\", SVC(kernel=\"poly\", degree=3, coef0=1, C=5))\n ])\npoly_kernel_svm_clf.fit(X, y)\n\npoly100_kernel_svm_clf = Pipeline([\n (\"scaler\", StandardScaler()),\n (\"svm_clf\", SVC(kernel=\"poly\", degree=10, coef0=100, C=5))\n ])\npoly100_kernel_svm_clf.fit(X, y)\n\nplt.figure(figsize=(11, 4))\n\nplt.subplot(121)\nplot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])\nplot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\nplt.title(r\"$d=3, r=1, C=5$\", fontsize=18)\n\nplt.subplot(122)\nplot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])\nplot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\nplt.title(r\"$d=10, r=100, C=5$\", fontsize=18)\n\nplt.show()\n\ndef gaussian_rbf(x, landmark, gamma):\n return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2)\n\ngamma = 0.3\n\nx1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1)\nx2s = gaussian_rbf(x1s, -2, gamma)\nx3s = gaussian_rbf(x1s, 1, gamma)\n\nXK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)]\nyk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])\n\nplt.figure(figsize=(11, 4))\n\nplt.subplot(121)\nplt.grid(True, which='both')\nplt.axhline(y=0, color='k')\nplt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c=\"red\")\nplt.plot(X1D[:, 0][yk==0], np.zeros(4), \"bs\")\nplt.plot(X1D[:, 0][yk==1], np.zeros(5), \"g^\")\nplt.plot(x1s, x2s, \"g--\")\nplt.plot(x1s, x3s, \"b:\")\nplt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1])\nplt.xlabel(r\"$x_1$\", fontsize=20)\nplt.ylabel(r\"Similarity\", fontsize=14)\nplt.annotate(r'$\\mathbf{x}$',\n xy=(X1D[3, 0], 0),\n xytext=(-0.5, 0.20),\n ha=\"center\",\n arrowprops=dict(facecolor='black', shrink=0.1),\n fontsize=18,\n )\nplt.text(-2, 0.9, \"$x_2$\", ha=\"center\", fontsize=20)\nplt.text(1, 0.9, \"$x_3$\", ha=\"center\", fontsize=20)\nplt.axis([-4.5, 4.5, -0.1, 1.1])\n\nplt.subplot(122)\nplt.grid(True, which='both')\nplt.axhline(y=0, color='k')\nplt.axvline(x=0, color='k')\nplt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], \"bs\")\nplt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], \"g^\")\nplt.xlabel(r\"$x_2$\", fontsize=20)\nplt.ylabel(r\"$x_3$ \", fontsize=20, rotation=0)\nplt.annotate(r'$\\phi\\left(\\mathbf{x}\\right)$',\n xy=(XK[3, 0], XK[3, 1]),\n xytext=(0.65, 0.50),\n ha=\"center\",\n arrowprops=dict(facecolor='black', shrink=0.1),\n fontsize=18,\n )\nplt.plot([-0.1, 1.1], [0.57, -0.1], \"r--\", linewidth=3)\nplt.axis([-0.1, 1.1, -0.1, 1.1])\n \nplt.subplots_adjust(right=1)\n\nplt.show()\n\n\nx1_example = X1D[3, 0]\nfor landmark in (-2, 1):\n k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)\n print(\"Phi({}, {}) = {}\".format(x1_example, landmark, k))\n\nrbf_kernel_svm_clf = Pipeline([\n (\"scaler\", StandardScaler()),\n (\"svm_clf\", SVC(kernel=\"rbf\", gamma=5, C=0.001))\n ])\nrbf_kernel_svm_clf.fit(X, y)\n\n\nfrom sklearn.svm import SVC\n\ngamma1, gamma2 = 0.1, 5\nC1, C2 = 0.001, 1000\nhyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)\n\nsvm_clfs = []\nfor gamma, C in hyperparams:\n rbf_kernel_svm_clf = Pipeline([\n (\"scaler\", StandardScaler()),\n (\"svm_clf\", SVC(kernel=\"rbf\", gamma=gamma, C=C))\n ])\n rbf_kernel_svm_clf.fit(X, y)\n svm_clfs.append(rbf_kernel_svm_clf)\n\nplt.figure(figsize=(11, 7))\n\nfor i, svm_clf in enumerate(svm_clfs):\n plt.subplot(221 + i)\n plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5])\n plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n gamma, C = hyperparams[i]\n plt.title(r\"$\\gamma = {}, C = {}$\".format(gamma, C), fontsize=16)\n\nplt.show()\n```\n\n## Mathematical optimization of convex functions\n\nA mathematical (quadratic) optimization problem, or just optimization problem, has the form\n\n$$\n\\begin{align*}\n &\\mathrm{min}_{\\lambda}\\hspace{0.2cm} \\frac{1}{2}\\boldsymbol{\\lambda}^T\\boldsymbol{P}\\boldsymbol{\\lambda}+\\boldsymbol{q}^T\\boldsymbol{\\lambda},\\\\ \\nonumber\n &\\mathrm{subject\\hspace{0.1cm}to} \\hspace{0.2cm} \\boldsymbol{G}\\boldsymbol{\\lambda} \\preceq \\boldsymbol{h} \\wedge \\boldsymbol{A}\\boldsymbol{\\lambda}=f.\n\\end{align*}\n$$\n\nsubject to some constraints for say a selected set $i=1,2,\\dots, n$.\nIn our case we are optimizing with respect to the Lagrangian multipliers $\\lambda_i$, and the\nvector $\\boldsymbol{\\lambda}=[\\lambda_1, \\lambda_2,\\dots, \\lambda_n]$ is the optimization variable we are dealing with.\n\nIn our case we are particularly interested in a class of optimization problems called convex optmization problems. \nIn our discussion on gradient descent methods we discussed at length the definition of a convex function. \n\nConvex optimization problems play a central role in applied mathematics and we recommend strongly [Boyd and Vandenberghe's text on the topics](http://web.stanford.edu/~boyd/cvxbook/).\n\n## How do we solve these problems?\n\nIf we use Python as programming language and wish to venture beyond\n**scikit-learn**, **tensorflow** and similar software which makes our\nlives so much easier, we need to dive into the wonderful world of\nquadratic programming. We can, if we wish, solve the minimization\nproblem using say standard gradient methods or conjugate gradient\nmethods. However, these methods tend to exhibit a rather slow\nconverge. So, welcome to the promised land of quadratic programming.\n\nThe functions we need are contained in the quadratic programming package **CVXOPT** and we need to import it together with **numpy** as\n\n\n```\nimport numpy\nimport cvxopt\n```\n\nThis will make our life much easier. You don't need t write your own optimizer.\n\n## A simple example\n\nWe remind ourselves about the general problem we want to solve\n\n$$\n\\begin{align*}\n &\\mathrm{min}_{x}\\hspace{0.2cm} \\frac{1}{2}\\boldsymbol{x}^T\\boldsymbol{P}\\boldsymbol{x}+\\boldsymbol{q}^T\\boldsymbol{x},\\\\ \\nonumber\n &\\mathrm{subject\\hspace{0.1cm} to} \\hspace{0.2cm} \\boldsymbol{G}\\boldsymbol{x} \\preceq \\boldsymbol{h} \\wedge \\boldsymbol{A}\\boldsymbol{x}=f.\n\\end{align*}\n$$\n\nLet us show how to perform the optmization using a simple case. Assume we want to optimize the following problem\n\n$$\n\\begin{align*}\n &\\mathrm{min}_{x}\\hspace{0.2cm} \\frac{1}{2}x^2+5x+3y \\\\ \\nonumber\n &\\mathrm{subject to} \\\\ \\nonumber\n &x, y \\geq 0 \\\\ \\nonumber\n &x+3y \\geq 15 \\\\ \\nonumber\n &2x+5y \\leq 100 \\\\ \\nonumber\n &3x+4y \\leq 80. \\\\ \\nonumber\n\\end{align*}\n$$\n\nThe minimization problem can be rewritten in terms of vectors and matrices as (with $x$ and $y$ being the unknowns)\n\n$$\n\\frac{1}{2}\\begin{bmatrix} x\\\\ y \\end{bmatrix}^T \\begin{bmatrix} 1 & 0\\\\ 0 & 0 \\end{bmatrix} \\begin{bmatrix} x \\\\ y \\end{bmatrix} + \\begin{bmatrix}3\\\\ 4 \\end{bmatrix}^T \\begin{bmatrix}x \\\\ y \\end{bmatrix}.\n$$\n\nSimilarly, we can now set up the inequalities (we need to change $\\geq$ to $\\leq$ by multiplying with $-1$ on bot sides) as the following matrix-vector equation\n\n$$\n\\begin{bmatrix} -1 & 0 \\\\ 0 & -1 \\\\ -1 & -3 \\\\ 2 & 5 \\\\ 3 & 4\\end{bmatrix}\\begin{bmatrix} x \\\\ y\\end{bmatrix} \\preceq \\begin{bmatrix}0 \\\\ 0\\\\ -15 \\\\ 100 \\\\ 80\\end{bmatrix}.\n$$\n\nWe have collapsed all the inequalities into a single matrix $\\boldsymbol{G}$. We see also that our matrix\n\n$$\n\\boldsymbol{P} =\\begin{bmatrix} 1 & 0\\\\ 0 & 0 \\end{bmatrix}\n$$\n\nis clearly positive semi-definite (all eigenvalues larger or equal zero). \nFinally, the vector $\\boldsymbol{h}$ is defined as\n\n$$\n\\boldsymbol{h} = \\begin{bmatrix}0 \\\\ 0\\\\ -15 \\\\ 100 \\\\ 80\\end{bmatrix}.\n$$\n\nSince we don't have any equalities the matrix $\\boldsymbol{A}$ is set to zero\nThe following code solves the equations for us\n\n\n```\n# Import the necessary packages\nimport numpy\nfrom cvxopt import matrix\nfrom cvxopt import solvers\nP = matrix(numpy.diag([1,0]), tc=\u2019d\u2019)\nq = matrix(numpy.array([3,4]), tc=\u2019d\u2019)\nG = matrix(numpy.array([[-1,0],[0,-1],[-1,-3],[2,5],[3,4]]), tc=\u2019d\u2019)\nh = matrix(numpy.array([0,0,-15,100,80]), tc=\u2019d\u2019)\n# Construct the QP, invoke solver\nsol = solvers.qp(P,q,G,h)\n# Extract optimal value and solution\nsol[\u2019x\u2019] \nsol[\u2019primal objective\u2019]\n```\n\n## Support Vector Machines and Regression\n\nMaterial may be added if of interest. See Bishop chapter 7.1 for a discussion.\n\n## Summary of course\n\n## What? Me worry? No final exam in this course!\n\n\n\n

Figure 1:

\n\n\n## What is the link between Artificial Intelligence and Machine Learning and some general Remarks\n\nArtificial intelligence is built upon integrated machine learning\nalgorithms as discussed in this course, which in turn are fundamentally rooted in optimization and\nstatistical learning.\n\nCan we have Artificial Intelligence without Machine Learning? See [this post for inspiration](https://www.linkedin.com/pulse/what-artificial-intelligence-without-machine-learning-claudia-pohlink).\n\n## Going back to the beginning of the semester\n\nTraditionally the field of machine learning has had its main focus on\npredictions and correlations. These concepts outline in some sense\nthe difference between machine learning and what is normally called\nBayesian statistics or Bayesian inference.\n\nIn machine learning and prediction based tasks, we are often\ninterested in developing algorithms that are capable of learning\npatterns from given data in an automated fashion, and then using these\nlearned patterns to make predictions or assessments of newly given\ndata. In many cases, our primary concern is the quality of the\npredictions or assessments, and we are less concerned with the\nunderlying patterns that were learned in order to make these\npredictions. This leads to what normally has been labeled as a\nfrequentist approach.\n\n## Not so sharp distinctions\n\nYou should keep in mind that the division between a traditional\nfrequentist approach with focus on predictions and correlations only\nand a Bayesian approach with an emphasis on estimations and\ncausations, is not that sharp. Machine learning can be frequentist\nwith ensemble methods (EMB) as examples and Bayesian with Gaussian\nProcesses as examples.\n\nIf one views ML from a statistical learning\nperspective, one is then equally interested in estimating errors as\none is in finding correlations and making predictions. It is important\nto keep in mind that the frequentist and Bayesian approaches differ\nmainly in their interpretations of probability. In the frequentist\nworld, we can only assign probabilities to repeated random\nphenomena. From the observations of these phenomena, we can infer the\nprobability of occurrence of a specific event. In Bayesian\nstatistics, we assign probabilities to specific events and the\nprobability represents the measure of belief/confidence for that\nevent. The belief can be updated in the light of new evidence.\n\n## Topics we have covered this year\n\nThe course has two central parts\n\n1. Statistical analysis and optimization of data\n\n2. Machine learning\n\n## Statistical analysis and optimization of data\n\nThe following topics have been discussed:\n1. Basic concepts, expectation values, variance, covariance, correlation functions and errors;\n\n2. Simpler models, binomial distribution, the Poisson distribution, simple and multivariate normal distributions;\n\n3. Central elements from linear algebra, matrix inversion and SVD\n\n4. Gradient methods for data optimization\n\n5. Estimation of errors using cross-validation, bootstrapping and jackknife methods;\n\n6. Practical optimization using Singular-value decomposition and least squares for parameterizing data.\n\n7. Principal Component Analysis to reduce the number of features.\n\n## Machine learning\n\nThe following topics will be covered\n1. Linear methods for regression and classification:\n\na. Ordinary Least Squares\n\nb. Ridge regression\n\nc. Lasso regression\n\nd. Logistic regression\n\n5. Neural networks and deep learning:\n\na. Feed Forward Neural Networks\n\nb. Convolutional Neural Networks\n\nc. Recurrent Neural Networks\n\n4. Decisions trees and ensemble methods:\n\na. Decision trees\n\nb. Bagging and voting\n\nc. Random forests\n\nd. Boosting and gradient boosting\n\n5. Support vector machines\n\na. Binary classification and multiclass classification\n\nb. Kernel methods\n\nc. Regression\n\n## Learning outcomes and overarching aims of this course\n\nThe course introduces a variety of central algorithms and methods\nessential for studies of data analysis and machine learning. The\ncourse is project based and through the various projects, normally\nthree, you will be exposed to fundamental research problems\nin these fields, with the aim to reproduce state of the art scientific\nresults. The students will learn to develop and structure large codes\nfor studying these systems, get acquainted with computing facilities\nand learn to handle large scientific projects. A good scientific and\nethical conduct is emphasized throughout the course. \n\n* Understand linear methods for regression and classification;\n\n* Learn about neural network;\n\n* Learn about bagging, boosting and trees\n\n* Support vector machines\n\n* Learn about basic data analysis;\n\n* Be capable of extending the acquired knowledge to other systems and cases;\n\n* Have an understanding of central algorithms used in data analysis and machine learning;\n\n* Work on numerical projects to illustrate the theory. The projects play a central role and you are expected to know modern programming languages like Python or C++.\n\n## Perspective on Machine Learning\n\n1. Rapidly emerging application area\n\n2. Experiment AND theory are evolving in many many fields. Still many low-hanging fruits.\n\n3. Requires education/retraining for more widespread adoption\n\n4. A lot of \u201cword-of-mouth\u201d development methods\n\nHuge amounts of data sets require automation, classical analysis tools often inadequate. \nHigh energy physics hit this wall in the 90\u2019s.\nIn 2009 single top quark production was determined via [Boosted decision trees, Bayesian\nNeural Networks, etc.](https://arxiv.org/pdf/0903.0850.pdf). Similarly, the search for Higgs was a statistical learning tour de force. See this link on [Kaggle.com](https://www.kaggle.com/c/higgs-boson).\n\n## Machine Learning Research\n\nWhere to find recent results:\n1. Conference proceedings, arXiv and blog posts!\n\n2. **NIPS**: [Neural Information Processing Systems](https://papers.nips.cc)\n\n3. **ICLR**: [International Conference on Learning Representations](https://openreview.net/group?id=ICLR.cc/2018/Conference#accepted-oral-papers)\n\n4. **ICML**: International Conference on Machine Learning\n\n5. [Journal of Machine Learning Research](http://www.jmlr.org/papers/v19/) \n\n6. [Follow ML on ArXiv](https://arxiv.org/list/cs.LG/recent)\n\n## Starting your Machine Learning Project\n\n1. Identify problem type: classification, regression\n\n2. Consider your data carefully\n\n3. Choose a simple model that fits 1. and 2.\n\n4. Consider your data carefully again! Think of data representation more carefully.\n\n5. Based on your results, feedback loop to earliest possible point\n\n## Choose a Model and Algorithm\n\n1. Supervised?\n\n2. Start with the simplest model that fits your problem\n\n3. Start with minimal processing of data\n\n## Preparing Your Data\n\n1. Shuffle your data\n\n2. Mean center your data\n\n * Why?\n\n3. Normalize the variance\n\n * Why?\n\n4. [Whitening](https://multivariatestatsjl.readthedocs.io/en/latest/whiten.html)\n\n * Decorrelates data\n\n * Can be hit or miss\n\n5. When to do train/test split?\n\nWhitening is a decorrelation transformation that transforms a set of\nrandom variables into a set of new random variables with identity\ncovariance (uncorrelated with unit variances).\n\n## Which Activation and Weights to Choose in Neural Networks\n\n1. RELU? ELU?\n\n2. Sigmoid or Tanh?\n\n3. Set all weights to 0?\n\n * Terrible idea\n\n4. Set all weights to random values?\n\n * Small random values\n\n## Optimization Methods and Hyperparameters\n1. Stochastic gradient descent\n\na. Stochastic gradient descent + momentum\n\n2. State-of-the-art approaches:\n\n * RMSProp\n\n * Adam\n\n * and more\n\nWhich regularization and hyperparameters? $L_1$ or $L_2$, soft\nclassifiers, depths of trees and many other. Need to explore a large\nset of hyperparameters and regularization methods.\n\n## Resampling\n\nWhen do we resample?\n\n1. [Bootstrap](https://www.cambridge.org/core/books/bootstrap-methods-and-their-application/ED2FD043579F27952363566DC09CBD6A)\n\n2. [Cross-validation](https://www.youtube.com/watch?v=fSytzGwwBVw&ab_channel=StatQuestwithJoshStarmer)\n\n3. Jackknife and many other\n\n## Other courses on Data science and Machine Learning at UiO\n\nThe link here gives an excellent overview of courses on Machine learning at UiO.\n\n1. [STK2100 Machine learning and statistical methods for prediction and classification](http://www.uio.no/studier/emner/matnat/math/STK2100/index-eng.html). \n\n2. [IN3050/IN4050 Introduction to Artificial Intelligence and Machine Learning](https://www.uio.no/studier/emner/matnat/ifi/IN3050/index-eng.html). Introductory course in machine learning and AI with an algorithmic approach. \n\n3. [STK-INF3000/4000 Selected Topics in Data Science](http://www.uio.no/studier/emner/matnat/math/STK-INF3000/index-eng.html). The course provides insight into selected contemporary relevant topics within Data Science. \n\n4. [IN4080 Natural Language Processing](https://www.uio.no/studier/emner/matnat/ifi/IN4080/index.html). Probabilistic and machine learning techniques applied to natural language processing. \n\n5. [STK-IN4300 \u2013 Statistical learning methods in Data Science](https://www.uio.no/studier/emner/matnat/math/STK-IN4300/index-eng.html). An advanced introduction to statistical and machine learning. For students with a good mathematics and statistics background.\n\n6. [IN-STK5000 Adaptive Methods for Data-Based Decision Making](https://www.uio.no/studier/emner/matnat/ifi/IN-STK5000/index-eng.html). Methods for adaptive collection and processing of data based on machine learning techniques. \n\n7. [IN5400/INF5860 \u2013 Machine Learning for Image Analysis](https://www.uio.no/studier/emner/matnat/ifi/IN5400/). An introduction to deep learning with particular emphasis on applications within Image analysis, but useful for other application areas too.\n\n8. [TEK5040 \u2013 Dyp l\u00e6ring for autonome systemer](https://www.uio.no/studier/emner/matnat/its/TEK5040/). The course addresses advanced algorithms and architectures for deep learning with neural networks. The course provides an introduction to how deep-learning techniques can be used in the construction of key parts of advanced autonomous systems that exist in physical environments and cyber environments.\n\n## Additional courses of interest\n\n1. [STK4051 Computational Statistics](https://www.uio.no/studier/emner/matnat/math/STK4051/index-eng.html)\n\n2. [STK4021 Applied Bayesian Analysis and Numerical Methods](https://www.uio.no/studier/emner/matnat/math/STK4021/index-eng.html)\n\n## What's the future like?\n\nBased on multi-layer nonlinear neural networks, deep learning can\nlearn directly from raw data, automatically extract and abstract\nfeatures from layer to layer, and then achieve the goal of regression,\nclassification, or ranking. Deep learning has made breakthroughs in\ncomputer vision, speech processing and natural language, and reached\nor even surpassed human level. The success of deep learning is mainly\ndue to the three factors: big data, big model, and big computing.\n\nIn the past few decades, many different architectures of deep neural\nnetworks have been proposed, such as\n1. Convolutional neural networks, which are mostly used in image and video data processing, and have also been applied to sequential data such as text processing;\n\n2. Recurrent neural networks, which can process sequential data of variable length and have been widely used in natural language understanding and speech processing;\n\n3. Encoder-decoder framework, which is mostly used for image or sequence generation, such as machine translation, text summarization, and image captioning.\n\n## Types of Machine Learning, a repetition\n\nThe approaches to machine learning are many, but are often split into two main categories. \nIn *supervised learning* we know the answer to a problem,\nand let the computer deduce the logic behind it. On the other hand, *unsupervised learning*\nis a method for finding patterns and relationship in data sets without any prior knowledge of the system.\nSome authours also operate with a third category, namely *reinforcement learning*. This is a paradigm \nof learning inspired by behavioural psychology, where learning is achieved by trial-and-error, \nsolely from rewards and punishment.\n\nAnother way to categorize machine learning tasks is to consider the desired output of a system.\nSome of the most common tasks are:\n\n * Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning.\n\n * Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.\n\n * Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.\n\n * Other unsupervised learning algortihms like **Boltzmann machines**\n\n## Why Boltzmann machines?\n\nWhat is known as restricted Boltzmann Machines (RMB) have received a lot of attention lately. \nOne of the major reasons is that they can be stacked layer-wise to build deep neural networks that capture complicated statistics.\n\nThe original RBMs had just one visible layer and a hidden layer, but recently so-called Gaussian-binary RBMs have gained quite some popularity in imaging since they are capable of modeling continuous data that are common to natural images. \n\nFurthermore, they have been used to solve complicated [quantum mechanical many-particle problems or classical statistical physics problems like the Ising and Potts classes of models](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.91.045002).\n\n## Boltzmann Machines\n\nWhy use a generative model rather than the more well known discriminative deep neural networks (DNN)? \n\n* Discriminitave methods have several limitations: They are mainly supervised learning methods, thus requiring labeled data. And there are tasks they cannot accomplish, like drawing new examples from an unknown probability distribution.\n\n* A generative model can learn to represent and sample from a probability distribution. The core idea is to learn a parametric model of the probability distribution from which the training data was drawn. As an example\n\na. A model for images could learn to draw new examples of cats and dogs, given a training dataset of images of cats and dogs.\n\nb. Generate a sample of an ordered or disordered phase, having been given samples of such phases.\n\nc. Model the trial function for [Monte Carlo calculations](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.91.045002).\n\n## Some similarities and differences from DNNs\n\n1. Both use gradient-descent based learning procedures for minimizing cost functions\n\n2. Energy based models don't use backpropagation and automatic differentiation for computing gradients, instead turning to Markov Chain Monte Carlo methods.\n\n3. DNNs often have several hidden layers. A restricted Boltzmann machine has only one hidden layer, however several RBMs can be stacked to make up Deep Belief Networks, of which they constitute the building blocks.\n\nHistory: The RBM was developed by amongst others [Geoffrey Hinton](https://en.wikipedia.org/wiki/Geoffrey_Hinton), called by some the \"Godfather of Deep Learning\", working with the University of Toronto and Google.\n\n## Boltzmann machines (BM)\n\nA BM is what we would call an undirected probabilistic graphical model\nwith stochastic continuous or discrete units.\n\nIt is interpreted as a stochastic recurrent neural network where the\nstate of each unit(neurons/nodes) depends on the units it is connected\nto. The weights in the network represent thus the strength of the\ninteraction between various units/nodes.\n\nIt turns into a Hopfield network if we choose deterministic rather\nthan stochastic units. In contrast to a Hopfield network, a BM is a\nso-called generative model. It allows us to generate new samples from\nthe learned distribution.\n\n## A standard BM setup\n\nA standard BM network is divided into a set of observable and visible units $\\hat{x}$ and a set of unknown hidden units/nodes $\\hat{h}$.\n\nAdditionally there can be bias nodes for the hidden and visible layers. These biases are normally set to $1$.\n\nBMs are stackable, meaning they cwe can train a BM which serves as input to another BM. We can construct deep networks for learning complex PDFs. The layers can be trained one after another, a feature which makes them popular in deep learning\n\nHowever, they are often hard to train. This leads to the introduction of so-called restricted BMs, or RBMS.\nHere we take away all lateral connections between nodes in the visible layer as well as connections between nodes in the hidden layer. The network is illustrated in the figure below.\n\n## The structure of the RBM network\n\n\n\n\n

Figure 1:

\n\n\n## The network\n\n**The network layers**:\n1. A function $\\mathbf{x}$ that represents the visible layer, a vector of $M$ elements (nodes). This layer represents both what the RBM might be given as training input, and what we want it to be able to reconstruct. This might for example be given by the pixels of an image or coefficients representing speech, or the coordinates of a quantum mechanical state function.\n\n2. The function $\\mathbf{h}$ represents the hidden, or latent, layer. A vector of $N$ elements (nodes). Also called \"feature detectors\".\n\n## Goals\n\nThe goal of the hidden layer is to increase the model's expressive\npower. We encode complex interactions between visible variables by\nintroducing additional, hidden variables that interact with visible\ndegrees of freedom in a simple manner, yet still reproduce the complex\ncorrelations between visible degrees in the data once marginalized\nover (integrated out).\n\n**The network parameters, to be optimized/learned**:\n1. $\\mathbf{a}$ represents the visible bias, a vector of same length as $\\mathbf{x}$.\n\n2. $\\mathbf{b}$ represents the hidden bias, a vector of same lenght as $\\mathbf{h}$.\n\n3. $W$ represents the interaction weights, a matrix of size $M\\times N$.\n\n## Joint distribution\n\nThe restricted Boltzmann machine is described by a Boltzmann distribution\n\n\n
\n\n$$\n\\begin{equation}\n\tP_{rbm}(\\mathbf{x},\\mathbf{h}) = \\frac{1}{Z} e^{-\\frac{1}{T_0}E(\\mathbf{x},\\mathbf{h})},\n\\label{_auto1} \\tag{1}\n\\end{equation}\n$$\n\nwhere $Z$ is the normalization constant or partition function, defined as\n\n\n
\n\n$$\n\\begin{equation}\n\tZ = \\int \\int e^{-\\frac{1}{T_0}E(\\mathbf{x},\\mathbf{h})} d\\mathbf{x} d\\mathbf{h}.\n\\label{_auto2} \\tag{2}\n\\end{equation}\n$$\n\nIt is common to ignore $T_0$ by setting it to one.\n\n## Network Elements, the energy function\n\nThe function $E(\\mathbf{x},\\mathbf{h})$ gives the **energy** of a\nconfiguration (pair of vectors) $(\\mathbf{x}, \\mathbf{h})$. The lower\nthe energy of a configuration, the higher the probability of it. This\nfunction also depends on the parameters $\\mathbf{a}$, $\\mathbf{b}$ and\n$W$. Thus, when we adjust them during the learning procedure, we are\nadjusting the energy function to best fit our problem.\n\nAn expression for the energy function is\n\n$$\nE(\\hat{x},\\hat{h}) = -\\sum_{ia}^{NA}b_i^a \\alpha_i^a(x_i)-\\sum_{jd}^{MD}c_j^d \\beta_j^d(h_j)-\\sum_{ijad}^{NAMD}b_i^a \\alpha_i^a(x_i)c_j^d \\beta_j^d(h_j)w_{ij}^{ad}.\n$$\n\nHere $\\beta_j^d(h_j)$ and $\\alpha_i^a(x_j)$ are so-called transfer functions that map a given input value to a desired feature value. The labels $a$ and $d$ denote that there can be multiple transfer functions per variable. The first sum depends only on the visible units. The second on the hidden ones. **Note** that there is no connection between nodes in a layer.\n\nThe quantities $b$ and $c$ can be interpreted as the visible and hidden biases, respectively.\n\nThe connection between the nodes in the two layers is given by the weights $w_{ij}$.\n\n## Defining different types of RBMs\nThere are different variants of RBMs, and the differences lie in the types of visible and hidden units we choose as well as in the implementation of the energy function $E(\\mathbf{x},\\mathbf{h})$. \n\n**Binary-Binary RBM:**\n\nRBMs were first developed using binary units in both the visible and hidden layer. The corresponding energy function is defined as follows:\n\n\n
\n\n$$\n\\begin{equation}\n\tE(\\mathbf{x}, \\mathbf{h}) = - \\sum_i^M x_i a_i- \\sum_j^N b_j h_j - \\sum_{i,j}^{M,N} x_i w_{ij} h_j,\n\\label{_auto3} \\tag{3}\n\\end{equation}\n$$\n\nwhere the binary values taken on by the nodes are most commonly 0 and 1.\n\n**Gaussian-Binary RBM:**\n\nAnother varient is the RBM where the visible units are Gaussian while the hidden units remain binary:\n\n\n
\n\n$$\n\\begin{equation}\n\tE(\\mathbf{x}, \\mathbf{h}) = \\sum_i^M \\frac{(x_i - a_i)^2}{2\\sigma_i^2} - \\sum_j^N b_j h_j - \\sum_{i,j}^{M,N} \\frac{x_i w_{ij} h_j}{\\sigma_i^2}. \n\\label{_auto4} \\tag{4}\n\\end{equation}\n$$\n\n## More about RBMs\n1. Useful when we model continuous data (i.e., we wish $\\mathbf{x}$ to be continuous)\n\n2. Requires a smaller learning rate, since there's no upper bound to the value a component might take in the reconstruction\n\nOther types of units include:\n1. Softmax and multinomial units\n\n2. Gaussian visible and hidden units\n\n3. Binomial units\n\n4. Rectified linear units\n\nTo read more, see [Lectures on Boltzmann machines in Physics](https://github.com/CompPhysics/ComputationalPhysics2/blob/gh-pages/doc/pub/notebook2/ipynb/notebook2.ipynb).\n\n## Autoencoders: Overarching view\n\nAutoencoders are artificial neural networks capable of learning\nefficient representations of the input data (these representations are called codings) without\nany supervision (i.e., the training set is unlabeled). These codings\ntypically have a much lower dimensionality than the input data, making\nautoencoders useful for dimensionality reduction. \n\nMore importantly, autoencoders act as powerful feature detectors, and\nthey can be used for unsupervised pretraining of deep neural networks.\n\nLastly, they are capable of randomly generating new data that looks\nvery similar to the training data; this is called a generative\nmodel. For example, you could train an autoencoder on pictures of\nfaces, and it would then be able to generate new faces. Surprisingly,\nautoencoders work by simply learning to copy their inputs to their\noutputs. This may sound like a trivial task, but we will see that\nconstraining the network in various ways can make it rather\ndifficult. For example, you can limit the size of the internal\nrepresentation, or you can add noise to the inputs and train the\nnetwork to recover the original inputs. These constraints prevent the\nautoencoder from trivially copying the inputs directly to the outputs,\nwhich forces it to learn efficient ways of representing the data. In\nshort, the codings are byproducts of the autoencoder\u2019s attempt to\nlearn the identity function under some constraints.\n\n[Video on autoencoders](https://www.coursera.org/lecture/building-deep-learning-models-with-tensorflow/autoencoders-1U4L3)\n\nSee also A. Geron's textbook, chapter 15.\n\n## Bayesian Machine Learning\n\nThis is an important topic if we aim at extracting a probability\ndistribution. This gives us also a confidence interval and error\nestimates.\n\nBayesian machine learning allows us to encode our prior beliefs about\nwhat those models should look like, independent of what the data tells\nus. This is especially useful when we don\u2019t have a ton of data to\nconfidently learn our model.\n\n[Video on Bayesian deep learning](https://www.youtube.com/watch?v=E1qhGw8QxqY&ab_channel=AndrewGordonWilson)\n\nSee also the [slides here](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Articles/lec03.pdf).\n\n## Reinforcement Learning\n\nReinforcement Learning (RL) is one of the most exciting fields of\nMachine Learning today, and also one of the oldest. It has been around\nsince the 1950s, producing many interesting applications over the\nyears.\n\nIt studies\nhow agents take actions based on trial and error, so as to maximize\nsome notion of cumulative reward in a dynamic system or\nenvironment. Due to its generality, the problem has also been studied\nin many other disciplines, such as game theory, control theory,\noperations research, information theory, multi-agent systems, swarm\nintelligence, statistics, and genetic algorithms.\n\nIn March 2016, AlphaGo, a computer program that plays the board game\nGo, beat Lee Sedol in a five-game match. This was the first time a\ncomputer Go program had beaten a 9-dan (highest rank) professional\nwithout handicaps. AlphaGo is based on deep convolutional neural\nnetworks and reinforcement learning. AlphaGo\u2019s victory was a major\nmilestone in artificial intelligence and it has also made\nreinforcement learning a hot research area in the field of machine\nlearning.\n\n[Lecture on Reinforcement Learning](https://www.youtube.com/watch?v=FgzM3zpZ55o&ab_channel=stanfordonline).\n\nSee also A. Geron's textbook, chapter 16.\n\n## Transfer learning\n\nThe goal of transfer learning is to transfer the model or knowledge\nobtained from a source task to the target task, in order to resolve\nthe issues of insufficient training data in the target task. The\nrationality of doing so lies in that usually the source and target\ntasks have inter-correlations, and therefore either the features,\nsamples, or models in the source task might provide useful information\nfor us to better solve the target task. Transfer learning is a hot\nresearch topic in recent years, with many problems still waiting to be studied.\n\n[Lecture on transfer learning](https://www.ias.edu/video/machinelearning/2020/0331-SamoryKpotufe).\n\n## Adversarial learning\n\nThe conventional deep generative model has a potential problem: the\nmodel tends to generate extreme instances to maximize the\nprobabilistic likelihood, which will hurt its performance. Adversarial\nlearning utilizes the adversarial behaviors (e.g., generating\nadversarial instances or training an adversarial model) to enhance the\nrobustness of the model and improve the quality of the generated\ndata. In recent years, one of the most promising unsupervised learning\ntechnologies, generative adversarial networks (GAN), has already been\nsuccessfully applied to image, speech, and text.\n\n[Lecture on adversial learning](https://www.youtube.com/watch?v=CIfsB_EYsVI&ab_channel=StanfordUniversitySchoolofEngineering).\n\n## Dual learning\n\nDual learning is a new learning paradigm, the basic idea of which is\nto use the primal-dual structure between machine learning tasks to\nobtain effective feedback/regularization, and guide and strengthen the\nlearning process, thus reducing the requirement of large-scale labeled\ndata for deep learning. The idea of dual learning has been applied to\nmany problems in machine learning, including machine translation,\nimage style conversion, question answering and generation, image\nclassification and generation, text classification and generation,\nimage-to-text, and text-to-image.\n\n## Distributed machine learning\n\nDistributed computation will speed up machine learning algorithms,\nsignificantly improve their efficiency, and thus enlarge their\napplication. When distributed meets machine learning, more than just\nimplementing the machine learning algorithms in parallel is required.\n\n## Meta learning\n\nMeta learning is an emerging research direction in machine\nlearning. Roughly speaking, meta learning concerns learning how to\nlearn, and focuses on the understanding and adaptation of the learning\nitself, instead of just completing a specific learning task. That is,\na meta learner needs to be able to evaluate its own learning methods\nand adjust its own learning methods according to specific learning\ntasks.\n\n## The Challenges Facing Machine Learning\n\nWhile there has been much progress in machine learning, there are also challenges.\n\nFor example, the mainstream machine learning technologies are\nblack-box approaches, making us concerned about their potential\nrisks. To tackle this challenge, we may want to make machine learning\nmore explainable and controllable. As another example, the\ncomputational complexity of machine learning algorithms is usually\nvery high and we may want to invent lightweight algorithms or\nimplementations. Furthermore, in many domains such as physics,\nchemistry, biology, and social sciences, people usually seek elegantly\nsimple equations (e.g., the Schr\u00f6dinger equation) to uncover the\nunderlying laws behind various phenomena. In the field of machine\nlearning, can we reveal simple laws instead of designing more complex\nmodels for data fitting? Although there are many challenges, we are\nstill very optimistic about the future of machine learning. As we look\nforward to the future, here are what we think the research hotspots in\nthe next ten years will be.\n\nSee the article on [Discovery of Physics From Data: Universal Laws and Discrepancies](https://www.frontiersin.org/articles/10.3389/frai.2020.00025/full)\n\n## Explainable machine learning\n\nMachine learning, especially deep learning, evolves rapidly. The\nability gap between machine and human on many complex cognitive tasks\nbecomes narrower and narrower. However, we are still in the very early\nstage in terms of explaining why those effective models work and how\nthey work.\n\n**What is missing: the gap between correlation and causation**. Standard Machine Learning is based on what e have called a frequentist approach. \n\nMost\nmachine learning techniques, especially the statistical ones, depend\nhighly on correlations in data sets to make predictions and analyses. In\ncontrast, rational humans tend to reply on clear and trustworthy\ncausality relations obtained via logical reasoning on real and clear\nfacts. It is one of the core goals of explainable machine learning to\ntransition from solving problems by data correlation to solving\nproblems by logical reasoning.\n\n**Bayesian Machine Learning is one of the exciting research directions in this field**.\n\n## Quantum machine learning\n\nQuantum machine learning is an emerging interdisciplinary research\narea at the intersection of quantum computing and machine learning.\n\nQuantum computers use effects such as quantum coherence and quantum\nentanglement to process information, which is fundamentally different\nfrom classical computers. Quantum algorithms have surpassed the best\nclassical algorithms in several problems (e.g., searching for an\nunsorted database, inverting a sparse matrix), which we call quantum\nacceleration.\n\nWhen quantum computing meets machine learning, it can be a mutually\nbeneficial and reinforcing process, as it allows us to take advantage\nof quantum computing to improve the performance of classical machine\nlearning algorithms. In addition, we can also use the machine learning\nalgorithms (on classic computers) to analyze and improve quantum\ncomputing systems.\n\n[Lecture on Quantum ML](https://www.youtube.com/watch?v=Xh9pUu3-WxM&ab_channel=InstituteforPure%26AppliedMathematics%28IPAM%29).\n\n[Read interview with Maria Schuld on her work on Quantum Machine Learning](https://physics.aps.org/articles/v13/179?utm_campaign=weekly&utm_medium=email&utm_source=emailalert). See also [her recent textbook](https://www.springer.com/gp/book/9783319964232).\n\n## Quantum machine learning algorithms based on linear algebra\n\nMany quantum machine learning algorithms are based on variants of\nquantum algorithms for solving linear equations, which can efficiently\nsolve N-variable linear equations with complexity of O(log2 N) under\ncertain conditions. The quantum matrix inversion algorithm can\naccelerate many machine learning methods, such as least square linear\nregression, least square version of support vector machine, Gaussian\nprocess, and more. The training of these algorithms can be simplified\nto solve linear equations. The key bottleneck of this type of quantum\nmachine learning algorithms is data input\u2014that is, how to initialize\nthe quantum system with the entire data set. Although efficient\ndata-input algorithms exist for certain situations, how to efficiently\ninput data into a quantum system is as yet unknown for most cases.\n\n## Quantum reinforcement learning\n\nIn quantum reinforcement learning, a quantum agent interacts with the\nclassical environment to obtain rewards from the environment, so as to\nadjust and improve its behavioral strategies. In some cases, it\nachieves quantum acceleration by the quantum processing capabilities\nof the agent or the possibility of exploring the environment through\nquantum superposition. Such algorithms have been proposed in\nsuperconducting circuits and systems of trapped ions.\n\n## Quantum deep learning\n\nDedicated quantum information processors, such as quantum annealers\nand programmable photonic circuits, are well suited for building deep\nquantum networks. The simplest deep quantum network is the Boltzmann\nmachine. The classical Boltzmann machine consists of bits with tunable\ninteractions and is trained by adjusting the interaction of these bits\nso that the distribution of its expression conforms to the statistics\nof the data. To quantize the Boltzmann machine, the neural network can\nsimply be represented as a set of interacting quantum spins that\ncorrespond to an adjustable Ising model. Then, by initializing the\ninput neurons in the Boltzmann machine to a fixed state and allowing\nthe system to heat up, we can read out the output qubits to get the\nresult.\n\n## Social machine learning\n\nMachine learning aims to imitate how humans\nlearn. While we have developed successful machine learning algorithms,\nuntil now we have ignored one important fact: humans are social. Each\nof us is one part of the total society and it is difficult for us to\nlive, learn, and improve ourselves, alone and isolated. Therefore, we\nshould design machines with social properties. Can we let machines\nevolve by imitating human society so as to achieve more effective,\nintelligent, interpretable \u201csocial machine learning\u201d?\n\nAnd much more.\n\n## The last words?\n\nEarly computer scientist Alan Kay said, **The best way to predict the\nfuture is to create it**. Therefore, all machine learning\npractitioners, whether scholars or engineers, professors or students,\nneed to work together to advance these important research\ntopics. Together, we will not just predict the future, but create it.\n\n## Best wishes to you all and thanks so much for your heroic efforts this semester\n\n\n\n\n

Figure 1:

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YES\n2. YES", "lm_q1_score": 0.5851011686727231, "lm_q2_score": 0.7577943767446202, "lm_q1q2_score": 0.44338637544689513}} {"text": "# ClimateMARGO.jl tutorial\n\nThe following schematic shows the full formulation of the MARGO model, which is described in detail in the accompanying (but not yet peer-reviewed) manuscript, available for free at [EarthArXiv.org/5bgyc](https://eartharxiv.org/5bgyc). No-policy baseline emissions $q(t)$ are prescribed, leading to changing greenhouse gas (CO$_{2e}$) concentrations, radiative forcing, temperatures, and climate damages. Emissions can be decreased by **M**itigation; concentrations can be decreased by carbon dioxide **R**emoval; forcing can be decreased by solar **G**eo-engineering; and the \"adapted\" temperature that leads to climate damages can be reduced by **A**dapting to the changed climate.\n\n\n### Import software\n\nThe MARGO model has been implemented and registered as the Julia package `ClimateMARGO.jl`, which can be installed with the command `Pkg.add(\"ClimateMARGO\")`.\n\nIf you are running this tutorial via Binder, there is no need to install the package; just import it using the command below.\n\n\n```julia\nusing Revise\n```\n\n\n```julia\nusing ClimateMARGO # Julia implementation of the MARGO model\nusing PyPlot # A basic plotting package\n```\n\n \u250c Info: Precompiling ClimateMARGO [d3f62095-a717-45bf-aadc-ac9dfc258fa6]\n \u2514 @ Base loading.jl:1260\n\n\n\n```julia\nusing ClimateMARGO.Models\nusing ClimateMARGO.Utils\nusing ClimateMARGO.Diagnostics\n```\n\n## Model configuration\n\nAn instance of MARGO `ClimateModelParameters` requires specifying a `name` and the parameters for three model subcomponents:\n1. `Domain`: determines the time period of interest, as well as the timestep and\u2013 in the policy response mode\u2013 the \"present\" decision-making year.\n2. `Economics`: determines exogenous economic growth, the cliamte damage cost function, and the cost function for each of the four controls, and the discount rate.\n3. `Physics`: determines the parameters that govern the aiborn fraction of emissions that remain in the atmosphere, the strength of climate feedbacks, and the rate of ocean heat uptake.\n\nAn instance of the MARGO `ClimateModel` is determined by combining the prescribed set of `ClimateModelParameters` with timeseries of the four `Controls`, which will act as the control variables in the optimization framework:\n1. **M**itigation of baseline emissions\n2. **R**emoval of carbon dioxide from the atmopshere\n3. **G**eongineering by solar radiation management, and\n4. **A**daptating to the changed climate to reduce the damages of climate change impacts.\n\nHere, we run through the full construction of the model with its \"default\" parameter values.\n\n\n```julia\nname = \"default\";\n```\n\n#### 1. Setting up the temporal grid\nFirst, we need to set up a time-frame for our experiment. Let's begin our simulation in the year 2020\u2013 present day\u2013 and consider out to 2200, with a 5-year timestep for computational efficiency.\n\n\n```julia\ninitial_year = 2020. # [yr]\nfinal_year = 2200. # [yr]\ndt = 5. # [yr]\nt_arr = t(initial_year, final_year, dt);\n```\n\nWhile the model allows for shifting the \"present\" year forward or backward in time to simulate past and future policy decision-making processes, we will only consider the simplest case in which we take the perspective of a policy decision maker in the year 2020, which is also the initial year of our simulation.\n\n\n```julia\npresent_year = initial_year\ndom = Domain(dt, initial_year, initial_year, final_year);\n```\n\n#### 2. Configuring the carbon cycle and energy balance models\n\nCO$_{2e}$ concentrations are given by:\n\\begin{equation}\nc_{M,R}(t) = c_{0} + \\int_{t_{0}}^{t} rq(t')(1-M(t')) \\text{ d}t' - q_{0} \\int_{t_{0}}^{t} R(t')\\text{ d}t'.\n\\end{equation}\nwhere $q(t)$ is a specific timeseries of no-climate-policy baseline emissions (with $q(t_{0}) = q_{0}$), $c_{0}$ is the initial CO$_{2e}$ concentration, and $r$ is the fraction of emitted CO$_{2e}$ that remains in the atmosphere net of short-term uptake by the ocean and biosphere. $M(t')$ and $R(t')$ represent the effects emissions mitigation and carbon dioxide removal, respectively, on CO$_{2e}$ concentrations relative to the baseline; they will be initially set to zero and later determined by solving an optimization problem.\n\nThe default no-policy scenario is one of rapid, fossil-fueled growth which leads to an accumulation of $c(t) = 1400$ ppm of CO$_{2e}$ in the atmosphere by 2150, when emissions are assumed to finally reach net-zero.\n\n\n```julia\nc0 = 460. # [ppm]\nr = 0.5; # [1] fraction of emissions remaining after biosphere and ocean uptake (Solomon 2009)\n\nq0 = 7.5\nq0mult = 3.\nt_peak = 2100.\nt_zero = 2150.\nq = ramp_emissions(t_arr, q0, q0mult, t_peak, t_zero);\n```\n\n\n```julia\nfigure(figsize=(9,3.5))\n\nsubplot(1,2,1)\nplot(t_arr, ppm_to_GtCO2(q))\nxlabel(\"year\")\nylabel(L\"baseline emissions $q(t)$ [GtCO2 / year]\")\nxlim([2020, 2200])\ngrid(true)\n\nsubplot(1,2,2)\nq_effective = effective_emissions(r, q, 0., 0.) # No mitigation, no carbon removal\nc_baseline = c(c0, q_effective, dt)\nplot(t_arr, c_baseline)\nxlabel(\"year\")\nylabel(L\"baseline concentrations $c(t)$ [ppm]\")\nxlim([2020, 2200])\ntight_layout();\ngrid(true)\n```\n\nThese CO$_{2e}$ concentrations drive an anomalous greenhouse effect, which is represented by the radiative forcing\n\\begin{equation}\nF_{M,R,G} = a \\ln\\left(\\frac{c_{M,R}(t)}{c_{0}}\\right) - G(t)F_{\\infty},\n\\end{equation}\nwhere $a$ is an empirically determined coefficient, $G(t)$ represents the effects of geoengineering, and $F_{\\infty} = 8.5$ W/m$^2$ is a scaling factor for the effects of geoengineering by Solar Radiation Modification (SRM).\n\n\n```julia\na = (6.9/2.)/log(2.); # F4xCO2/2 / log(2) [W m^-2]\nFinf = 8.5;\n```\n\n\n```julia\nfigure(figsize=(4.5,3.2))\nF0 = 3.0\nF_baseline = F(a, c0, Finf, c_baseline, 0.)\nplot(t_arr, F_baseline .+ F0)\nxlabel(\"year\")\nylabel(L\"baseline radiative forcing $F(t)$ [W/m$^2$]\")\nxlim([2020, 2200])\ngrid(true)\nylim([0,10.]);\n```\n\nNext, we configure MARGO's energy balance model, which is forced by the controlled forcing $F_{M,R,G}$. The two-layer energy balance model can be solved, approximately, as:\n\\begin{equation}\n T_{M,R,G}(t) - T_{0} = \\frac{F_{M,R,G}(t)}{B + \\kappa} + \\frac{\\kappa}{B} \\int_{t_{0}}^{t} \\frac{e^{\\frac{t'-t}{\\tau_{D}}}}{\\tau_{D}} \\frac{F_{M,R,G}(t')}{B+\\kappa} \\, \\text{d}t',\n\\end{equation}\nwhere $T_{0}$ is the initial temperature change relative to preindustrial, $B$ is the feedback parameter, $\\kappa$ is , and the timescale $\\tau_{D} = \\dfrac{C_{D}}{B} \\dfrac{B + \\kappa}{\\kappa}$ is a more convenient expression for the deep ocean heat capacity $C_{D}$.\n\nDefault parameter values are taken from [Geoffroy et al. 2013](https://journals.ametsoc.org/doi/full/10.1175/JCLI-D-12-00195.1)'s physically-based calibration of the two-layer model to CMIP5.\n\n\n```julia\n# Two-layer EBM parameters\nB = 1.13; # Feedback parameter [J yr^-1 m^-2 K^-1]\nCd = 106.; # Deep ocean heat capacity [J m^-2 K^-1]\n\u03ba = 0.73; # Heat exchange coefficient [J yr^-1 m^2 K^-1]\n\n# Initial condition: present-day temperature, relative to pre-industrial\nT0 = 1.1; # [degC] Berkeley Earth Surface Temperature (Rohde 2013)\n\nprint(\"\u03c4D = \", Int64(round(\u03c4d(Cd, B, \u03ba))), \" years\")\n```\n\n \u03c4D = 239 years\n\nThese physical parameters can be used to diagnose the climate sensitivity to a doubling of CO$_{2}$ ($ECS$).\n\n\n```julia\nprint(\"ECS = \", round(ECS(a, B),digits=1) ,\"\u00baC\")\n```\n\n ECS = 3.1\u00baC\n\nThese parameters define the physical model, which is instantiated by the calling the `Physics` constructor method:\n\n\n```julia\nPhys = Physics(c0, T0, a, B, Cd, \u03ba, r);\n```\n\n#### 3. Configuring the simple economic model\n\nEconomic growth in MARGO (in terms of Gross World Product, GWP) is exogenous $E(t) = E_{0} (1 + \\gamma)^{(t-t_{0})}$ and is entirely determined by the growth rate $\\gamma$. By default, we set $\\gamma = 2\\%$.\n\n\n```julia\nE0 = 100. # Gross World Product at t0 [10^12$ yr^-1]\n\u03b3 = 0.02 # economic growth rate\n\nfigure(figsize=(4, 2.5))\nplot(t_arr, E(t_arr, E0, \u03b3))\nxlabel(\"year\")\nylabel(\"GWP [trillion USD]\")\nxlim([2020, 2200])\nylim([0, 3000]);\ngrid(true)\n```\n\nEconomic damages $D_{M,R,G,A} = \\tilde{\\beta}E(t) (\\Delta T_{M,R,G})^{2} (1 - A(t))$, expressed as a fraction $\\tilde{\\beta}(\\Delta T_{M,R,G})^{2}$ of the instantaneous Gross World Product, increase quadratically with temperature change relative to preindustrial. They can be decrease by adaptation controls $A(t)$. The default value of the damage parameter $\\tilde{\\beta}$ corresponds to damages of 2\\% of GWP at 3\u00baC of warming.\n\n\n```julia\n\u03b2tilde = 0.02/(3.0)^2; # damages [%GWP / celsius^2]\n```\n\nThe total cost of climate controls is simply the sum of their individual costs, each of which follows a parabolic cost function:\n\\begin{equation}\n \\mathcal{C}_{M,R,G,A} = \\mathcal{C}_{M}M^2 + \\mathcal{C}_{R}R^2 + \\mathcal{C}_{G}G^2 + \\mathcal{C}_{A}A^2\n\\end{equation}\n\nThe calculation of the reference control costs $\\mathcal{C}_{M}, \\mathcal{C}_{R}, \\mathcal{C}_{G}, \\mathcal{C}_{A}$ are somewhat more complicated; see our Methods in [the preprint](https://eartharxiv.org/5bgyc/) and `defaults.jl` for details. Here, we simply provide their default numerical values, where the costs of mitigation $\\mathcal{C}_{M} = \\tilde{\\mathcal{C}}_{M} E(t)$ and geoengineering $\\mathcal{C}_{G} = \\tilde{\\mathcal{C}}_{G} E(t)$ grow with the size of the global economy and are thus specified as a fraction of GWP, while adaptaiton and removal costs are in trillions of USD per year.\n\n\n```julia\nmitigate_cost = 0.034; # [trillion USD / year / GtCO2]\n\nremove_cost = 13.; # [trillion USD / year]\nadapt_cost = 4.5; # [trillion USD / year]\n\n#mitigate_cost = 0.02; # [% GWP]\ngeoeng_cost = 0.046; # [% GWP]\n```\n\nClimate damages and control costs are discounted at the relatively low rate of $\\rho = 1\\%$, such that future damages and costs are reduced by a multiplicative discount factor $(1 - \\rho)^{(t-t_{0})}$.\n\n\n```julia\n\u03c1 = 0.01;\n\nfigure(figsize=(4, 2.5))\nplot(t_arr, discount(t_arr, \u03c1, present_year)*100)\nxlabel(\"year\")\nylabel(\"discount factor (%)\")\nxlim([2020, 2200])\nylim([0, 100]);\ngrid(true)\n```\n\nThese parameters, in addition to a no-policy baseline emissions time-series and present-day control values, define the economic model.\n\n\n```julia\nEcon = Economics(\n E0, \u03b3, \u03b2tilde, \u03c1, Finf,\n mitigate_cost, remove_cost, geoeng_cost, adapt_cost,\n 0.1, 0., 0., nothing, # Initial condition on control deployments at t[1]\n q\n);\n```\n\n#### A MARGO model configuration is uniquely determined by the parameters defined above\n\n\n```julia\nparams = ClimateModelParameters(\n name,\n dom,\n Econ,\n Phys\n);\n```\n\n## Instanciating the MARGO climate model\n\n\n\nAlong with economic and physical model components, the timeseries for each of the four controls must be specified. By default, we simply set these to zero.\n\n\n```julia\nCont = Controls(\n zeros(size(t_arr)), # mitigate\n zeros(size(t_arr)), # remove\n zeros(size(t_arr)), # geoeng\n zeros(size(t_arr)) # adapt\n);\n```\n\nThe above parameters determine the full configuration of the MARGO model, which is instanced using the `ClimateModel` constructor method:\n\n\n```julia\nm = ClimateModel(\n params,\n Cont\n);\n```\n\n## Model optimization\n\n#### Formulating the optimization problem\nBy default, the optimization problem we solve is for the most cost-effective combination of controls, as determined by minimization of the discounted net present value,\n\n\\begin{equation}\n \\min\\left\\{\\int_{t_{0}}^{t_{f}} \\mathcal{C}_{M,R,G,A} (1 + \\rho)^{-(t-t_{0})} \\text{ d}t\\right\\}\n\\end{equation}\n\nwhich keep controlled damages below the level corresponding to a chosen temperature threshold $T^{\\star}$,\n\n\\begin{equation}\n \\tilde{\\beta} E(t) (T_{M,R,G})^{2} (1 - A(t)) < \\tilde{\\beta} E(t) (T^{\\star})^{2}.\n\\end{equation}\n\n\n```julia\ntemp_goal = 2.0;\n```\n\n#### Additional policy / inertia constraints\nThe optimization is also subject to the constraints described below.\n\nFirst, we set a upper bounds on the maximum plausible deployment of each control.\n\n\n```julia\nmax_deployment = Dict(\"mitigate\"=>1., \"remove\"=>1., \"geoeng\"=>1., \"adapt\"=>0.4);\n```\n\nSecond, we set upper limits on how quickly each control can be ramped up or down.\n\n(Adaptation is treated differently since we it interpret it as buying insurance against future climate damages, although the financing is spread evenly over the entire period.)\n\n\n```julia\nmax_slope = Dict(\"mitigate\"=>1. /40., \"remove\"=>1. /40., \"geoeng\"=>1. /30., \"adapt\"=>0.);\n```\n\nThird, we impose restrictions on when controls can be first deployed. In particular, since carbon dioxide removal and solar radiation modification do not yet exist at scale, we delay these until 2030 and 2050, respectively, at the earliest.\n\n\n```julia\ndelay_deployment = Dict(\n \"mitigate\"=>0.,\n \"remove\"=>10.,\n \"geoeng\"=>30.,\n \"adapt\"=>0.\n);\n```\n\n#### Running the optimization\n\nThe optimization takes about ~40 seconds the first time it is run as the code compiles, but runs virtually instantly afterwards, even if model parameter values are changed.\n\n\n```julia\nusing ClimateMARGO.Optimization\n```\n\n\n```julia\n@time msolve = optimize_controls!(\n m,\n obj_option = \"temp\", temp_goal = temp_goal,\n max_deployment = max_deployment, max_slope = max_slope, delay_deployment = delay_deployment\n);\n```\n\n \n ******************************************************************************\n This program contains Ipopt, a library for large-scale nonlinear optimization.\n Ipopt is released as open source code under the Eclipse Public License (EPL).\n For more information visit http://projects.coin-or.org/Ipopt\n ******************************************************************************\n \n Solve_Succeeded\n 42.754889 seconds (122.36 M allocations: 6.838 GiB, 5.84% gc time)\n\n\nIt is always good to check that the solver actually converged on a valid solution!\n\n## Plotting MARGO results\n\nWe provide some convenient functions for plotting basic model results.\n\n\n```julia\nusing ClimateMARGO.Plotting\n```\n\n\n```julia\nplot_state(m)\n```\n\n## Customizing MARGO\n\nTry changing some of the parameters above and re-running the model!\n\nFor example, in the simulation below we set a more stringent temperature goal of $T^{\\star} = 1.5$ and omit solar geoengineering and adaptation completely (by setting their maximum deployment to zero).\n\n\n```julia\ntemp_goal = 1.5\nmax_deployment[\"geoeng\"] = 0.\nmax_deployment[\"adapt\"] = 0.\n\n@time optimize_controls!(\n m,\n obj_option = \"temp\", temp_goal = temp_goal,\n max_deployment = max_deployment, max_slope = max_slope, delay_deployment = delay_deployment\n);\n```\n\n Solve_Succeeded\n 0.065928 seconds (95.70 k allocations: 4.138 MiB)\n\n\n\n```julia\nplot_state(m)\n```\n\n## Saving (and loading) MARGO simulations (or parameter configurations)\n\n\n```julia\nusing ClimateMARGO.IO\nexport_path = tempname() * \".json\"\n```\n\n\n\n\n \"/var/folders/dr/t79w5jxj7xs_l_mklgcl1xt40000gn/T/jl_odZ3ea.json\"\n\n\n\n\n```julia\nexport_parameters(export_path, params);\n```\n\n\n```julia\nnew_params = import_parameters(export_path)\n```\n\n\n\n\n ClimateModelParameters(\"default\", Domain(5.0, 2020.0, 2020.0, 2200.0), Economics(100.0, 0.02, 0.0022222222222222222, 0.01, 8.5, 0.034, 13.0, 0.046, 4.5, 0.1, 0, 0, nothing, [7.5, 8.4375, 9.375, 10.3125, 11.25, 12.1875, 13.125, 14.0625, 15.0, 15.9375 \u2026 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 \u2026 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]), Physics(460.0, 1.1, 4.977297891066924, 1.13, 106.0, 0.73, 0.5))\n\n\n\nLet's say we want to see what happens if climate feedbacks $B$ are twice as strong as they are now (which means a lower feedback parameter).\n\n\n```julia\nnew_params.name = \"stronger_feedbacks\"\nnew_params.physics.B /= 2.;\n```\n\n\n```julia\nnew_m = ClimateModel(new_params, Cont)\n```\n\n\n\n\n ClimateModel(\"stronger_feedbacks\", Domain(5.0, 2020.0, 2020.0, 2200.0), Economics(100.0, 0.02, 0.0022222222222222222, 0.01, 8.5, 0.034, 13.0, 0.046, 4.5, 0.1, 0, 0, nothing, [7.5, 8.4375, 9.375, 10.3125, 11.25, 12.1875, 13.125, 14.0625, 15.0, 15.9375 \u2026 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 \u2026 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]), Physics(460.0, 1.1, 4.977297891066924, 0.565, 106.0, 0.73, 0.5), Controls([0.1, 0.22500004999104373, 0.350000099979977, 0.47500014996564954, 0.6000001999457527, 0.7250002499144974, 0.850000299848677, 0.9477994934106518, 0.9500529028429148, 0.9520860716881722 \u2026 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.1250000498613665, 0.18812839182026328, 0.19581990939457214, 0.2037938618329738, 0.21204567993938628, 0.14519923744826585, 0.1455444508241079, 0.14585592447748327 \u2026 0.02323950104723206, 0.022436294233423786, 0.021661454988364037, 0.02091394120850451, 0.02019275167609972, 0.01949692432666522, 0.018825534558218938, 0.018177693618293595, 0.017552548132442072, 0.016949238880767884], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 \u2026 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 \u2026 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]))\n\n\n\nThe new equilibrium climate sensitivity (recall $ECS = F_{2x}/B$) is now:\n\n\n```julia\nprint(\"ECS \u2248 $(round(ECS(new_m), digits=1))\u00b0C\")\n```\n\n ECS \u2248 6.1\u00b0C\n\n\n```julia\n\n```\n\n\n```julia\n\n```\n\n\n```julia\n\n```\n", "meta": 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"2c631ca92bc5159cf3b8772ea239d02aa9ea448b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 781.9431939979, "max_line_length": 297754, "alphanum_fraction": 0.9509713482, "converted": true, "num_tokens": 5624, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943712746406, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.44338636128150405}} {"text": "\n\n\n\n\n
Created: Tuesday 31 January 2017github.com/rhyswhitley/fire_limitation
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Quantifying the uncertainity of a global fire limitation model using Bayesian inference

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Part 2: Bayesian inference

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\n1,* Douglas Kelley, \n2 Ioannis Bistinas, \n3, 4 Chantelle Burton, \n1 Tobias Marthews, \n5 Rhys Whitley\n
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\n1 Centre for Ecology and Hydrology, Maclean Building, Crowmarsh Gifford, Wallingford, Oxfordshire, United Kingdom\n
\n2 Vrije Universiteit Amsterdam, Faculty of Earth and Life Sciences, Amsterdam, Netherlands\n
\n3 Met Office United Kingdom, Exeter, United Kingdom\n
\n4 Geography, University of Exeter, Exeter, United Kingdom\n
\n5 Natural Perils Pricing, Commercial & Consumer Portfolio & Pricing, Suncorp Group, Sydney, Australia\n
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Summary

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\nThis notebook aims to quantify the model parameters of a global fire model (defined below). The model is driven by a number of covariates (Xi=1, 2, ... M) that describe: cropland, pasture and urban area footprints; frequency of lightening ignitions, population density, net primary productivity (NPP) and Alpha, a proxy measure of available soil moisture in the root zone. The model attempts to predict the impact of fire through burnt area and is thus the model target (Y).\n

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\nPython code and calculations below\n
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\n\n### Model description\nThe model considers percentage of burnt area to be the joint product of a set of conditions that modulate fire through fuel load, ignitions, moisture and supression. Each function assumes some equilibrium point that desribes the optimal conditions for fire, that may be proportionally modified through some empirical relationship. These are briefly outlined below for the sake of comprehension in this notebook, but can be referred to in more detail in the model protocol located in the docs/ folder (model protocol).\n\n\\begin{eqnarray}\n F_{burn} &=& \\prod_{i}S(x_{i}) \\\\[1em]\n\\end{eqnarray}\n\nWhere $S(x_{i})$ representes some measure of fire conditions by $i =$ fuel, moisture, ignitions and anthropagenic supression, and is describe by a sigmoid:\n\n\\begin{equation}\n S(x_{i=fuel, moist, ignite, suppr}) = \\frac{1}{1 + \\exp\\{-b\\cdot(x_i-a)\\}}\n\\end{equation}\n\nThe $fuel$ and $moist$ sigmoid considers only NPP and fuel moisture and therefore have no hyper-parameters. Sigmoids $ignite$ and $suppr$ describe an aggregation of other climate and land-use covariates. Because these sigmoids are influenced by an aggregation of different drivers, they are influenced in turn by different sets of hyper-parameters; these are now described below. \n\n#### Fuel load covariate (no hyper-parameters)\n\\begin{equation}\n x_{fuel} = NPP \n\\end{equation}\n\n#### Moisture covariate\n\\begin{equation}\n x_{moist} = \\alpha\n\\end{equation}\n\n\n#### Ignition covariate \n\\begin{equation}\n x_{ignite} = Lightn + k_p\\cdot A_{pasture} + k_{d1}\\cdot\\rho_{population}\n\\end{equation}\n\nWhere $Lightn$ is the number of cloud-to-ground lightning strikes, modified as per Kelley et al. 2014.\n\n#### Supression covariate \n\\begin{equation}\n x_{supress} = A_{urban} + k_C\\cdot A_{Crop} + k_{d2}\\cdot\\rho_{population} \n\\end{equation}\n\nThis leaves 12 free parameters that need to be optimised against observations of burnt area. \n\n### Load libraries\n\n\n```python\nimport os\nimport numpy as np\nimport pandas as pd\n\nimport pymc3 as pm3 \nfrom pymc3.backends import SQLite\nfrom scipy import optimize\nfrom theano import tensor as tt\n\nimport matplotlib.pyplot as plt\n\n# setup nice plotting\nplt.style.use('ggplot')\n%matplotlib inline\n\n# paths and parameters\noutPath = \"../data/globfire.csv.gz\"\n```\n\n## 2.1 Fire limitation model definition\n\nCould possibly contain this in a class object, but I'm not sure theano can instantiate the object to be used by the GPU. If I've made absolutely no sense just then, then I would leave the following as is.\n\n\n```python\ndef fuel_load(NPP):\n \"\"\"\n Definition to describe fuel load: while return the input; capability to be modified later.\n \"\"\"\n return NPP\n\ndef moisture(alpha):\n \"\"\"\n Definition to describe moisture: currently equals alpha; capability to be modified later.\n \"\"\"\n return (alpha) \n\ndef ignition(lightning, pasture_area, pop_density, kp, kd1):\n \"\"\"\n Definition for the measure of ignition\n \"\"\"\n return lightning + kp*pasture_area + kd1*pop_density\n\ndef supression(crop_area, pop_density, kd2):\n \"\"\"\n Definition for the measure of fire supression\n \"\"\"\n return crop_area + kd2*pop_density\n\ndef tt_sigmoid(x, a, b):\n \"\"\"\n Sigmoid function to describe limitation using tensor\n \"\"\"\n return 1.0/(1.0 + tt.exp(a*x + b))\n```\n\n## 2.2 Import data\n\nLoad data and do any necessary transformation needed for the Bayesian modelling framework. For testing purposes I've limited the number of rows I'm importing (here about 100K). If you plan to do the full dataset, then delete the `nrows` variabe in the cell below. \n\n\n```python\nDATAPATH = os.path.expanduser(outPath)\n\nfd = pd.read_csv(DATAPATH, nrows=1e5)\n```\n\nDo a sanity check to make sure our data has imported correctly.\n\n\n```python\nfd.info()\n```\n\n \n RangeIndex: 100000 entries, 0 to 99999\n Data columns (total 8 columns):\n alpha 100000 non-null float64\n cropland 100000 non-null float64\n fire 100000 non-null float64\n lightning_ignitions 100000 non-null float64\n NPP 100000 non-null float64\n pasture 100000 non-null float64\n population_density 100000 non-null float64\n urban_area 100000 non-null float64\n dtypes: float64(8)\n memory usage: 6.1 MB\n\n\n## 2.3 Baysian framework\n\nA simple explanation of Baye's law is:\n\n\\begin{equation}\n P(\\beta|X) \\propto P(\\beta)\\cdot P(X|\\beta)\n\\end{equation}\n\nwhere $X$ is our data (observations of some arbitrary system), and $\\beta$ our set of unexplained parameters that describe the reponse of our _proposed understanding_ of this system as it varies with $X$.\n\n### 2.3.1 Prior definitions\nBecause I have no idea what the uncertainty on the hyper parameters should look like (beyond $\\beta> 0$), I've set them all as uniform. Some of them can possibly be describe as exponential or half-normal, due to the physical nature of $\\beta$, but we can play around with that later.\n\n\\begin{eqnarray}\n P(\\beta) &=& \\prod_{i=1}^{4}P(a_i)\\prod_{i=1}^{4}P(b_i)\\cdot P(\\sigma)\\cdot P(k_c)P(k_p)P(k_{d,1})P(k_{d,2}) \\\\[1.5em]\n P(a) = P(b) = P(\\sigma) &=& \\mathcal{N}(0, 1) \\\\[1em]\n P(k_c) = P(k_p) = P(k_{d,1}) = P(k_{d,2}) &=& \\mathcal{U}(\\beta_{\\min}, \\beta_{\\max}) \\\\[1.5em]\n\\end{eqnarray}\n\nI'm not totally sure about the maths above being right, but it's just to show that _full_ prior is normal. Important, because we'll also describe the error (likelihood) as normal, such that the posterior is therefore normal (conjugate); i.e. $\\mathcal{N}\\times\\mathcal{N}=\\mathcal{N}$ (expansion happens in the mean of the exponent). \n\nBack to the code.., `pymc3` is quite funky in that it allows me to create an empty `Model()` object and just add things to it as I need them using a `with` statement. I've called our Bayesian model `fire_error` as that is what we are trying to Quantify.\n\n\n\n\n```python\nwith pm3.Model() as fire_error:\n \n# first for the sigmoids (the 4's at the end state that I want 4 of them)\n a_s4 = pm3.Normal('sig_a', mu=0, sd=100, shape=4)\n b_s4 = pm3.Normal('sig_b', mu=0, sd=100, shape=4)\n \n# now for the hyper-parameters that describe the independent fire condition covariates\n\n kp = pm3.Normal('kp', mu=500, sd=100)\n kd1 = pm3.Normal('kd1', mu=500, sd=100)\n kd2 = pm3.Normal('kd2', mu=500, sd=100)\n\n# kp = pm3.Uniform('kp', 0, 1e3)\n# kd1 = pm3.Uniform('kd1', 0, 1e3)\n# kd2 = pm3.Uniform('kd2', 0, 1e3)\n \n# describe the standard deviation in the error term\n sigma = pm3.HalfNormal('sigma', sd=1)\n```\n\n### 2.3.2 Likelihood definition\n\nFor the sake of simplicity (and because I don't really know any better), we define the model error as normally distributed (i.i.d.) although it most likely isn't. We could make this more complicated later by defining the error as heteroscedastic, but I wouldn't bother with that until we have some idea of the convergence. We're describing the error (observations minus model predictions) as follows:\n\n\\begin{eqnarray}\n P(X|\\beta) &=& \\mathcal{N}(F_{burn}, \\sigma) \\\\[1em]\n \\mathcal{N}(F_{burn}, \\sigma) &=& \\frac{N}{\\sigma\\sqrt{2\\pi}}\\exp\\left\\{\\sum_{i=1}^{N}\\left(\\frac{y_i - F_{burn, i}}{\\sigma_i}\\right)^2\\right\\}\n\\end{eqnarray}\n\nwhere $y_i$ is a set of observations we're attempting to optimise on. Below is the code that describes the above:\n\n\n```python\nwith fire_error:\n \n # transform hyper-covariates \n x_1 = fd[\"NPP\"].values\n \n x_2 = fd[\"alpha\"].values\n \n x_3 = ignition(fd[\"lightning_ignitions\"].values, \\\n fd[\"pasture\"].values, \\\n fd[\"population_density\"].values, \\\n kp, kd1)\n \n x_4 = supression(fd[\"cropland\"].values, \\\n fd[\"population_density\"].values, \\\n kd2)\n \n # list of primary covariates\n xs = [x_1, x_2, x_3, x_4]\n Nx = len(xs)\n \n # burnt area is assumed to be the product of the 4 sigmoids\n prediction = np.product([tt_sigmoid(xs[i], a_s4[i], b_s4[i]) for i in range(Nx)])\n \n # calculate the error between observed and predicted burnt area\n error = pm3.Normal('error', mu=prediction, sd=sigma, observed=fd['fire'].values)\n```\n\n### 2.3.3 Posterior sampling\n\nBecause it is nigh impossible to determine the posterior solution analytically we will instead sample the information space to **infer** the posterior solutions for each of the model parameters. In this case we are using a Metropolis-Hasting step MCMC.\n\nI've tried using No-U-Turn (NUTS) sampling (which is the new kid on the block), but there are issues with it's current implementation in pymc3 (see github repo issues). Can use it once problems are ironed out - but TBH it doesn't matter if we're getting a reasonable convergence.\n\n\n```python\nwith fire_error:\n \n # help the sampling out by quickly finding an optimal start position\n start = pm3.find_MAP(model=fire_error.model, fmin=optimize.fmin_powell)\n \n # set the step-method (criteria algorithm for moving around information space)\n step = pm3.Metropolis()\n \n # save our sampling to disk so we can access it later\n db_save = SQLite('../data/firemodel_trace.db')\n \n # do the sampling\n mcmc_traces = pm3.sample(5e3, step=step, start=start, njobs=-1, trace=db_save)\n```\n\n Optimization terminated successfully.\n Current function value: -293612.186103\n Iterations: 8\n Function evaluations: 1426\n\n\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 5000/5000.0 [13:37<00:00, 6.00it/s]\n\n\n\n```python\npm3.traceplot(mcmc_traces);\n```\n\n\n```python\ntype(mcmc_traces)\n```\n\n\n\n\n pymc3.backends.base.MultiTrace\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "f78c9bd91a913aba7c60b067a2ad1a7d2da10ad6", "size": 153283, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/bayesian_inference.ipynb", "max_stars_repo_name": "rhyswhitley/fire_limitation", "max_stars_repo_head_hexsha": "3a55aa14afe5899c2e41ee89fdb4095073830a8a", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2019-10-07T14:26:02.000Z", "max_stars_repo_stars_event_max_datetime": "2019-11-20T23:24:01.000Z", "max_issues_repo_path": "notebooks/bayesian_inference.ipynb", "max_issues_repo_name": "rhyswhitley/fire_limitation", "max_issues_repo_head_hexsha": "3a55aa14afe5899c2e41ee89fdb4095073830a8a", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/bayesian_inference.ipynb", "max_forks_repo_name": "rhyswhitley/fire_limitation", "max_forks_repo_head_hexsha": "3a55aa14afe5899c2e41ee89fdb4095073830a8a", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 5, "max_forks_repo_forks_event_min_datetime": "2017-03-20T10:11:44.000Z", "max_forks_repo_forks_event_max_datetime": "2019-12-18T07:39:49.000Z", "avg_line_length": 316.0474226804, "max_line_length": 136284, "alphanum_fraction": 0.9109164095, "converted": true, "num_tokens": 3230, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.6224593452091672, "lm_q1q2_score": 0.4433355790382698}} {"text": "# Aim of this notebook\n\n* To construct the singular curve of universal type to finalize the solution of the optimal control problem\n\n# Preamble\n\n\n```python\nfrom sympy import *\ninit_printing(use_latex='mathjax')\n\n# Plotting\n%matplotlib inline\n## Make inline plots raster graphics\nfrom IPython.display import set_matplotlib_formats\n## Import modules for plotting and data analysis\nimport matplotlib.pyplot as plt\nfrom matplotlib import gridspec,rc,colors\nimport matplotlib.ticker as plticker\n# Parameters for seaborn plots\nimport seaborn as sns\nclrs = sns.color_palette(\"Spectral\", 6)\ndef set_plot_style(usetex=False):\n sns.set_style('white', {'axes.linewidth': 0.5})\n sns.set(style='white', font_scale=1.1,#context='paper',\n rc={'xtick.major.size': 6, 'ytick.major.size': 6, 'legend.fontsize': 14,\n 'text.usetex': usetex, 'font.family': 'serif', 'font.serif': ['Verdana'],\n 'text.latex.preamble': r\"\\usepackage{type1cm}\"}) \n plt.rcParams['xtick.major.size'] = 6\n plt.rcParams['xtick.major.width'] = 1\n plt.rcParams['ytick.major.size'] = 6\n plt.rcParams['ytick.major.width'] = 1\n plt.rcParams['xtick.bottom'] = True\n plt.rcParams['ytick.left'] = True\n \nset_plot_style(True)\n\nimport pandas as pd\npd.set_option('mode.chained_assignment',None)\n\nimport numpy as np\nfrom scipy.optimize import fsolve, root\nfrom scipy.integrate import ode\nbackend = 'dopri5'\nimport warnings\n\n# Timer\nimport time\n\nfrom copy import deepcopy\n\nfrom itertools import cycle\npalette_size = 10;\nclrs = sns.color_palette(\"Reds\",palette_size)\niclrs = cycle(clrs) # iterated colors\n\nclrs0 = sns.color_palette(\"Set1\",palette_size)\n\n# Suppress warnings\nimport warnings\nwarnings.filterwarnings(\"ignore\")\n```\n\n# Parameter values\n\n* Birth rate and const of downregulation are defined below in order to fit some experim. data\n\n\n```python\nd = .13 # death rate\n\u03b1 = .3 # low equilibrium point at expression of the main pathway (high equilibrium is at one)\n\u03b8 = .45 # threshold value for the expression of the main pathway\n\u03ba = 40 # robustness parameter\n```\n\n* Symbolic variables - the list insludes \u03bc & \u03bcbar, because they will be varied later\n\n\n```python\n\u03c3, \u03c60, \u03c6, x, \u03bc, \u03bcbar = symbols('sigma, phi0, phi, x, mu, mubar')\n```\n\n* Main functions\n\n\n```python\nA = 1-\u03c3*(1-\u03b8)\nEminus = (\u03b1*A-\u03b8)**2/2\n\u0394E = A*(1-\u03b1)*((1+\u03b1)*A/2-\u03b8)\n\u0394Ef = lambdify(\u03c3,\u0394E)\n```\n\n* Birth rate and cost of downregulation\n\n\n```python\nb = (0.1*(exp(\u03ba*(\u0394Ef(1)))+1)-0.14*(exp(\u03ba*\u0394Ef(0))+1))/(exp(\u03ba*\u0394Ef(1))-exp(\u03ba*\u0394Ef(0))) # birth rate\n\u03c7 = 1-(0.14*(exp(\u03ba*\u0394Ef(0))+1)-b*exp(\u03ba*\u0394Ef(0)))/b\nb, \u03c7\n```\n\n\n\n\n$$\\left ( 0.140168330860362, \\quad 0.325961223954473\\right )$$\n\n\n\n\n```python\nc_relative = 0.1\nc = c_relative*(b-d)/b+(1-c_relative)*\u03c7/(exp(\u03ba*\u0394Ef(0))+1) # cost of resistance\nc\n```\n\n\n\n\n$$0.00833519849448376$$\n\n\n\n* Hamiltonian *H* and a part of it \u03c1 that includes the control variable \u03c3\n\n\n```python\nh = b*(\u03c7/(exp(\u03ba*\u0394E)+1)*(1-x)+c*x)\nH = -\u03c60 + \u03c6*(b*(\u03c7/(exp(\u03ba*\u0394E)+1)-c)*x*(1-x)+\u03bc*(1-x)/(exp(\u03ba*\u0394E)+1)-\u03bcbar*exp(-\u03ba*Eminus)*x) + h\n\u03c1 = (\u03c6*(b*\u03c7*x+\u03bc)+b*\u03c7)/(exp(\u03ba*\u0394E)+1)*(1-x)-\u03c6*\u03bcbar*exp(-\u03ba*Eminus)*x\n\u03c11 = (\u03c6*(b*\u03c7*x+\u03bc)+b*\u03c7)/(exp(\u03ba*\u0394E)+1)*(1-x)\n\u03c12 = \u03c6*\u03bcbar*exp(-\u03ba*Eminus)*x\nn = b*(1-\u03c7*(1-x)/(exp(\u03ba*\u0394E)+1)-c*x)-d\nH, \u03c1, n\n```\n\n\n\n\n$$\\left ( \\phi \\left(\\frac{\\mu \\left(- x + 1\\right)}{e^{40 \\left(- 0.385 \\sigma + 0.7\\right) \\left(- 0.3575 \\sigma + 0.2\\right)} + 1} - \\bar{\\mu} x e^{- 20 \\left(- 0.165 \\sigma - 0.15\\right)^{2}} + x \\left(-0.00116833086036159 + \\frac{0.045689440686899}{e^{40 \\left(- 0.385 \\sigma + 0.7\\right) \\left(- 0.3575 \\sigma + 0.2\\right)} + 1}\\right) \\left(- x + 1\\right)\\right) - \\phi_{0} + 0.00116833086036159 x + \\frac{0.045689440686899 \\left(- x + 1\\right)}{e^{40 \\left(- 0.385 \\sigma + 0.7\\right) \\left(- 0.3575 \\sigma + 0.2\\right)} + 1}, \\quad - \\bar{\\mu} \\phi x e^{- 20 \\left(- 0.165 \\sigma - 0.15\\right)^{2}} + \\frac{\\left(- x + 1\\right) \\left(\\phi \\left(\\mu + 0.045689440686899 x\\right) + 0.045689440686899\\right)}{e^{40 \\left(- 0.385 \\sigma + 0.7\\right) \\left(- 0.3575 \\sigma + 0.2\\right)} + 1}, \\quad - 0.00116833086036159 x - \\frac{0.140168330860362 \\left(- 0.325961223954473 x + 0.325961223954473\\right)}{e^{40 \\left(- 0.385 \\sigma + 0.7\\right) \\left(- 0.3575 \\sigma + 0.2\\right)} + 1} + 0.0101683308603616\\right )$$\n\n\n\n* Same but for no treatment (\u03c3 = 0)\n\n\n```python\nh0 = h.subs(\u03c3,0)\nH0 = H.subs(\u03c3,0)\n\u03c10 = \u03c1.subs(\u03c3,0)\nH0, \u03c10\n```\n\n\n\n\n$$\\left ( \\phi \\left(0.00368423989943599 \\mu \\left(- x + 1\\right) - 0.637628151621773 \\bar{\\mu} x - 0.001 x \\left(- x + 1\\right)\\right) - \\phi_{0} + 0.001 x + 0.000168330860361587, \\quad - 0.637628151621773 \\bar{\\mu} \\phi x + 0.00368423989943599 \\left(- x + 1\\right) \\left(\\phi \\left(\\mu + 0.045689440686899 x\\right) + 0.045689440686899\\right)\\right )$$\n\n\n\n* Machinery: definition of the Poisson brackets\n\n\n```python\nPoissonBrackets = lambda H1, H2: diff(H1,x)*diff(H2,\u03c6)-diff(H1,\u03c6)*diff(H2,x)\n```\n\n* Necessary functions and defining the right hand side of dynamical equations\n\n\n```python\n\u03c1f = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),\u03c1)\n\u03c11f = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),\u03c11)\n\u03c12f = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),\u03c12)\n\u03c10f = lambdify((x,\u03c6,\u03bc,\u03bcbar),\u03c10)\ndxd\u03c4 = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),-diff(H,\u03c6))\nd\u03c6d\u03c4 = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),diff(H,x))\ndVd\u03c4 = lambdify((x,\u03c3),h)\nd\u03c1d\u03c3 = lambdify((\u03c3,x,\u03c6,\u03bc,\u03bcbar),diff(\u03c1,\u03c3))\nd\u03b4\u03c1d\u03c4 = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),-PoissonBrackets(\u03c10-\u03c1,H))\ndef ode_rhs(t,state,\u03bc,\u03bcbar):\n x, \u03c6, V, \u03b4\u03c1 = state\n \u03c3s = [0,1]\n if (d\u03c1d\u03c3(1.,x,\u03c6,\u03bc,\u03bcbar)<0) and (d\u03c1d\u03c3(\u03b8,x,\u03c6,\u03bc,\u03bcbar)>0):\n \u03c3star = fsolve(d\u03c1d\u03c3,.8,args=(x,\u03c6,\u03bc,\u03bcbar,))[0]\n else:\n \u03c3star = 1.;\n if \u03c1f(x,\u03c6,\u03c3star,\u03bc,\u03bcbar) < \u03c10f(x,\u03c6,\u03bc,\u03bcbar):\n sgm = 0\n else:\n sgm = \u03c3star\n return [dxd\u03c4(x,\u03c6,sgm,\u03bc,\u03bcbar),d\u03c6d\u03c4(x,\u03c6,sgm,\u03bc,\u03bcbar),dVd\u03c4(x,sgm),d\u03b4\u03c1d\u03c4(x,\u03c6,\u03c3star,\u03bc,\u03bcbar)]\ndef \u03c3starf(x,\u03c6,\u03bc,\u03bcbar):\n if (d\u03c1d\u03c3(1.,x,\u03c6,\u03bc,\u03bcbar)<0) and (d\u03c1d\u03c3(\u03b8,x,\u03c6,\u03bc,\u03bcbar)>0):\n \u03c3star = fsolve(d\u03c1d\u03c3,.8,args=(x,\u03c6,\u03bc,\u03bcbar,))[0]\n else:\n \u03c3star = 1.;\n if \u03c1f(x,\u03c6,\u03c3star,\u03bc,\u03bcbar) < \u03c10f(x,\u03c6,\u03bc,\u03bcbar):\n sgm = 0\n else:\n sgm = \u03c3star\n return sgm\n```\n\n\n```python\ndef get_primary_field(name, experiment,\u03bc,\u03bcbar):\n solutions = {}\n solver = ode(ode_rhs).set_integrator(backend)\n \u03c40 = experiment['\u03c40']\n tms = np.linspace(\u03c40,experiment['T_end'],1e3+1)\n for x0 in experiment['x0']:\n \u03b4\u03c10 = \u03c10.subs(x,x0).subs(\u03c6,0)-\u03c1.subs(x,x0).subs(\u03c6,0).subs(\u03c3,1.)\n solver.set_initial_value([x0,0,0,\u03b4\u03c10],0.).set_f_params(\u03bc,\u03bcbar)\n sol = []; k = 0;\n while (solver.t < experiment['T_end']) and (solver.y[0]<=1.) and (solver.y[0]>=0.):\n solver.integrate(tms[k])\n sol.append([solver.t]+list(solver.y))\n k += 1\n solutions[x0] = {'solution': sol}\n for x0, entry in solutions.items():\n entry['\u03c4'] = [entry['solution'][j][0] for j in range(len(entry['solution']))]\n entry['x'] = [entry['solution'][j][1] for j in range(len(entry['solution']))]\n entry['\u03c6'] = [entry['solution'][j][2] for j in range(len(entry['solution']))]\n entry['V'] = [entry['solution'][j][3] for j in range(len(entry['solution']))]\n entry['\u03b4\u03c1'] = [entry['solution'][j][4] for j in range(len(entry['solution']))]\n return solutions\ndef get_\u03b4\u03c1_value(tme,x0,\u03bc,\u03bcbar):\n solver = ode(ode_rhs).set_integrator(backend)\n \u03b4\u03c10 = \u03c10.subs(x,x0).subs(\u03c6,0)-\u03c1.subs(x,x0).subs(\u03c6,0).subs(\u03c3,1.)\n solver.set_initial_value([x0,0,0,\u03b4\u03c10],0.).set_f_params(\u03bc,\u03bcbar)\n while (solver.t < tme) and (solver.y[0]<=1.) and (solver.y[0]>=0.):\n solver.integrate(tme)\n sol = [solver.t]+list(solver.y)\n return solver.y[3]\ndef get_\u03b4\u03c1_ending(params,\u03bc,\u03bcbar):\n tme, x0 = params\n solver = ode(ode_rhs).set_integrator(backend)\n \u03b4\u03c10 = \u03c10.subs(x,x0).subs(\u03c6,0)-\u03c1.subs(x,x0).subs(\u03c6,0).subs(\u03c3,1.)\n solver.set_initial_value([x0,0,0,\u03b4\u03c10],0.).set_f_params(\u03bc,\u03bcbar)\n \u03b4\u03c4 = 1.0e-8; tms = [tme,tme+\u03b4\u03c4]\n _k = 0; sol = []\n while (_k=0.):\n solver.integrate(tms[_k])\n sol.append(solver.y)\n _k += 1\n #print(sol)\n return(sol[0][3],(sol[1][3]-sol[0][3])/\u03b4\u03c4)\ndef get_state(tme,x0,\u03bc,\u03bcbar):\n solver = ode(ode_rhs).set_integrator(backend)\n \u03b4\u03c10 = \u03c10.subs(x,x0).subs(\u03c6,0)-\u03c1.subs(x,x0).subs(\u03c6,0).subs(\u03c3,1.)\n solver.set_initial_value([x0,0,0,\u03b4\u03c10],0.).set_f_params(\u03bc,\u03bcbar)\n \u03b4\u03c4 = 1.0e-8; tms = [tme,tme+\u03b4\u03c4]\n _k = 0; sol = []\n while (solver.t < tms[-1]) and (solver.y[0]<=1.) and (solver.y[0]>=0.):\n solver.integrate(tms[_k])\n sol.append(solver.y)\n _k += 1\n return(list(sol[0])+[(sol[1][3]-sol[0][3])/\u03b4\u03c4])\n```\n\n# Machinery for the universal line\n\n* To find the universal singular curve we need to define two parameters\n\n\n```python\n\u03b30 = PoissonBrackets(PoissonBrackets(H,H0),H)\n\u03b31 = PoissonBrackets(PoissonBrackets(H0,H),H0)\n```\n\n* The dynamics\n\n\n```python\ndxd\u03c4SingExpr = -(\u03b30*diff(H0,\u03c6)+\u03b31*diff(H,\u03c6))/(\u03b30+\u03b31)\nd\u03c6d\u03c4SingExpr = (\u03b30*diff(H0,x)+\u03b31*diff(H,x))/(\u03b30+\u03b31)\ndVd\u03c4SingExpr = (\u03b30*h0+\u03b31*h)/(\u03b30+\u03b31)\n\u03c3SingExpr = \u03b31*\u03c3/(\u03b30+\u03b31)\n```\n\n* Machinery for Python: lambdify the functions above\n\n\n```python\ndxd\u03c4Sing = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),dxd\u03c4SingExpr)\nd\u03c6d\u03c4Sing = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),d\u03c6d\u03c4SingExpr)\ndVd\u03c4Sing = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),dVd\u03c4SingExpr)\n\u03c3Sing = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),\u03c3SingExpr)\n```\n\n\n```python\ndef ode_rhs_Sing(t,state,\u03bc,\u03bcbar):\n x, \u03c6, V = state\n if (d\u03c1d\u03c3(1.,x,\u03c6,\u03bc,\u03bcbar)<0) and (d\u03c1d\u03c3(\u03b8,x,\u03c6,\u03bc,\u03bcbar)>0):\n \u03c3star = fsolve(d\u03c1d\u03c3,.8,args=(x,\u03c6,\u03bc,\u03bcbar,))[0]\n else:\n \u03c3star = 1.;\n return [dxd\u03c4Sing(x,\u03c6,\u03c3star,\u03bc,\u03bcbar),d\u03c6d\u03c4Sing(x,\u03c6,\u03c3star,\u03bc,\u03bcbar),dVd\u03c4Sing(x,\u03c6,\u03c3star,\u03bc,\u03bcbar)]\ndef get_universal_curve(end_point,tmax,Nsteps,\u03bc,\u03bcbar):\n tms = np.linspace(end_point[0],tmax,Nsteps);\n solver = ode(ode_rhs_Sing).set_integrator(backend)\n solver.set_initial_value(end_point[1:4],end_point[0]).set_f_params(\u03bc,\u03bcbar)\n _k = 0; sol = []\n while (solver.t < tms[-1]):\n solver.integrate(tms[_k])\n sol.append([solver.t]+list(solver.y))\n _k += 1\n return sol\ndef get_\u03c3_universal(tme,end_point,\u03bc,\u03bcbar):\n \u03b4\u03c4 = 1.0e-8; tms = [tme,tme+\u03b4\u03c4]\n solver = ode(ode_rhs_Sing).set_integrator(backend)\n solver.set_initial_value(end_point[1:4],end_point[0]).set_f_params(\u03bc,\u03bcbar)\n _k = 0; sol = []\n while (solver.t < tme+\u03b4\u03c4):\n solver.integrate(tms[_k])\n sol.append([solver.t]+list(solver.y))\n _k += 1\n x, \u03c6 = sol[0][:2]\n sgm = fsolve(lambda \u03c3: dxd\u03c4(x,\u03c6,\u03c3,\u03bc,\u03bcbar)-(sol[1][0]-sol[0][0])/\u03b4\u03c4,\u03b8/2)[0]\n return sgm\ndef get_state_universal(tme,end_point,\u03bc,\u03bcbar):\n solver = ode(ode_rhs_Sing).set_integrator(backend)\n solver.set_initial_value(end_point[1:4],end_point[0]).set_f_params(\u03bc,\u03bcbar)\n solver.integrate(tme)\n return [solver.t]+list(solver.y)\n```\n\n\n```python\ndef ode_rhs_with_\u03c3star(t,state,\u03bc,\u03bcbar):\n x, \u03c6, V = state\n if (d\u03c1d\u03c3(1.,x,\u03c6,\u03bc,\u03bcbar)<0) and (d\u03c1d\u03c3(\u03b8,x,\u03c6,\u03bc,\u03bcbar)>0):\n \u03c3 = fsolve(d\u03c1d\u03c3,.8,args=(x,\u03c6,\u03bc,\u03bcbar,))[0]\n else:\n \u03c3 = 1.;\n return [dxd\u03c4(x,\u03c6,\u03c3,\u03bc,\u03bcbar),d\u03c6d\u03c4(x,\u03c6,\u03c3,\u03bc,\u03bcbar),dVd\u03c4(x,\u03c3)]\ndef ode_rhs_with_given_\u03c3(t,state,\u03c3,\u03bc,\u03bcbar):\n x, \u03c6, V = state\n return [dxd\u03c4(x,\u03c6,\u03c3,\u03bc,\u03bcbar),d\u03c6d\u03c4(x,\u03c6,\u03c3,\u03bc,\u03bcbar),dVd\u03c4(x,\u03c3)]\ndef get_trajectory_with_\u03c3star(starting_point,tmax,Nsteps,\u03bc,\u03bcbar):\n tms = np.linspace(starting_point[0],tmax,Nsteps)\n solver = ode(ode_rhs_with_\u03c3star).set_integrator(backend)\n solver.set_initial_value(starting_point[1:],starting_point[0]).set_f_params(\u03bc,\u03bcbar)\n sol = []; _k = 0;\n while solver.t < max(tms) and (solver.y[0]<=1.) and (solver.y[0]>=0.):\n solver.integrate(tms[_k])\n sol.append([solver.t]+list(solver.y))\n _k += 1\n return sol\ndef get_trajectory_with_given_\u03c3(starting_point,tmax,Nsteps,\u03c3,\u03bc,\u03bcbar):\n tms = np.linspace(starting_point[0],tmax,100)\n solver = ode(ode_rhs_with_given_\u03c3).set_integrator(backend)\n solver.set_initial_value(starting_point[1:],starting_point[0]).set_f_params(\u03c3,\u03bc,\u03bcbar)\n sol = []; _k = 0;\n while solver.t < max(tms) and (solver.y[0]<=1.) and (solver.y[0]>=0.):\n solver.integrate(tms[_k])\n sol.append([solver.t]+list(solver.y))\n _k += 1\n return sol\ndef get_state_with_\u03c3star(tme,starting_point,\u03bc,\u03bcbar):\n solver = ode(ode_rhs_with_\u03c3star).set_integrator(backend)\n solver.set_initial_value(starting_point[1:4],starting_point[0]).set_f_params(\u03bc,\u03bcbar)\n solver.integrate(tme)\n return [solver.t]+list(solver.y)\ndef get_finalizing_point_from_universal_curve(tme,tmx,end_point,\u03bc,\u03bcbar):\n unv_point = get_state_universal(tme,end_point,\u03bc,\u03bcbar)\n return get_state_with_\u03c3star(tmx,unv_point,\u03bc,\u03bcbar)[1]\n```\n\n# Field of optimal trajectories as the solution of the Bellman equation\n\n* \u03bc & \u03bcbar are varied by *T* and *T*bar ($\\mu=1/T$ and $\\bar\\mu=1/\\bar{T}$)\n\n\n```python\ntmx = 180.\nend_switching_curve = {'t': 19., 'x': .7} \n# for \u03a4, \u03a4bar in zip([28]*5,[14,21,28,35,60]):\n\u03a4 = 10.5; \u03a4bar = 14.\n\u03bc = 1./\u03a4; \u03bcbar = 1./\u03a4bar\nprint(\"Parameters: \u03bc = %.5f, \u03bcbar = %.5f\"%(\u03bc,\u03bcbar))\nend_switching_curve['t'], end_switching_curve['x'] = fsolve(get_\u03b4\u03c1_ending,(end_switching_curve['t'],end_switching_curve['x']),args=(\u03bc,\u03bcbar),xtol=1.0e-12)\nend_point = [end_switching_curve['t']]+get_state(end_switching_curve['t'],end_switching_curve['x'],\u03bc,\u03bcbar)\nprint(\"Ending point for the switching line: \u03c4 = %.1f days, x = %.1f%%\" % (end_point[0], end_point[1]*100))\nprint(\"Checking the solution - should give zero values: \")\nprint(get_\u03b4\u03c1_ending([end_switching_curve['t'],end_switching_curve['x']],\u03bc,\u03bcbar))\nprint(\"* Constructing the primary field\")\nprimary_field1 = []\nexperiments = {\n 'sol1': { 'T_end': tmx, '\u03c40': 0., 'x0': list(np.linspace(0,end_switching_curve['x']-(1e-3),7)) } }\nfor name, values in experiments.items():\n primary_field1.append(get_primary_field(name,values,\u03bc,\u03bcbar))\nprimary_field2 = []\nexperiments = {\n 'sol1': { 'T_end': tmx, '\u03c40': 0., 'x0': list(np.linspace(end_switching_curve['x']+(3e-6),1.,7)) } }\nfor name, values in experiments.items():\n primary_field2.append(get_primary_field(name,values,\u03bc,\u03bcbar))\nprint(\"* Constructing the switching curve\")\nswitching_curve = []\nx0s = np.linspace(end_switching_curve['x'],1,21); _y = end_switching_curve['t']\n\nfor x0 in x0s:\n tme = fsolve(get_\u03b4\u03c1_value,_y,args=(x0,\u03bc,\u03bcbar))[0]\n if (tme>0):\n switching_curve = switching_curve+[[tme,get_state(tme,x0,\u03bc,\u03bcbar)[0]]]\n _y = tme\nprint(\"* Constructing the universal curve\")\nuniversal_curve = get_universal_curve(end_point,tmx,25,\u03bc,\u03bcbar)\nprint(\"* Finding the last characteristic\")\n#time0 = time.time()\ntuniv = fsolve(get_finalizing_point_from_universal_curve,tmx-40.,args=(tmx,end_point,\u03bc,\u03bcbar,))[0]\n#print(\"The proccess to find the last characteristic took %0.1f minutes\" % ((time.time()-time0)/60.))\nuniv_point = get_state_universal(tuniv,end_point,\u03bc,\u03bcbar)\nprint(\"The last point on the universal line:\")\nprint(univ_point)\nlast_trajectory = get_trajectory_with_\u03c3star(univ_point,tmx,50,\u03bc,\u03bcbar)\nprint(\"Final state:\")\nfinal_state = get_state_with_\u03c3star(tmx,univ_point,\u03bc,\u03bcbar)\nprint(final_state)\n```\n\n Parameters: \u03bc = 0.09524, \u03bcbar = 0.07143\n Ending point for the switching line: \u03c4 = 11.1 days, x = 62.9%\n Checking the solution - should give zero values: \n (-1.8766899605925048e-10, 9.321241741773813e-10)\n * Constructing the primary field\n * Constructing the switching curve\n * Constructing the universal curve\n * Finding the last characteristic\n The last point on the universal line:\n [168.21633500053056, 0.6285637714632423, -0.24007763436787447, 1.3270312549851486]\n Final state:\n [180.0, -8.373857163235243e-14, -0.3878381029647683, 1.6191478880823515]\n\n\n\n```python\n# Plotting\nplt.rcParams['figure.figsize'] = (4.5, 3.2)\n_k = 0\nfor solutions in primary_field1:\n for x0, entry in solutions.items():\n plt.plot(entry['\u03c4'], entry['x'], '-', linewidth=1, color=clrs0[1])\n _k += 1\n_k = 0\nfor solutions in primary_field2:\n for x0, entry in solutions.items():\n plt.plot(entry['\u03c4'], entry['x'], '-', linewidth=1, color=clrs0[1])\n _k += 1\nplt.plot([x[0] for x in switching_curve],[x[1] for x in switching_curve],linewidth=3,color=\"k\",zorder=4,linestyle=\"dashed\")\nplt.plot([end_point[0]],[end_point[1]],marker='o',color=\"black\",zorder=4)\n\nplt.xlim([0,120]); plt.ylim([0,1]);\nytks = np.arange(0,1.2,.2)\nplt.yticks(ytks,[int(y*100) for y in ytks])\nplt.xlabel(\"Backward time (days)\"); plt.ylabel(\"Fraction of resistant cells (\\%)\")\n# plt.show()\nplt.savefig(\"../figures/draft/Fig2-0.pdf\",format='pdf',bbox_inches='tight')\n```\n\n\n```python\n# Plotting\nplt.rcParams['figure.figsize'] = (4.5, 3.2)\n_k = 0\nfor solutions in primary_field1:\n for x0, entry in solutions.items():\n plt.plot(entry['\u03c4'], entry['x'], '-', linewidth=1, color=clrs0[1])\n _k += 1\n_k = 0\nfor solutions in primary_field2:\n for x0, entry in solutions.items():\n plt.plot(entry['\u03c4'], entry['x'], '-', linewidth=1, color=clrs0[1])\n _k += 1\nplt.plot([x[0] for x in switching_curve],[x[1] for x in switching_curve],linewidth=3,color=\"k\",zorder=4,linestyle=\"dashed\")\n# plt.plot([end_point[0]],[end_point[1]],marker='o',color=\"black\",zorder=4)\nplt.plot([x[0] for x in universal_curve],[x[1] for x in universal_curve],linewidth=3,color=\"k\",zorder=3)\nfor tend in [42,64,86,110]:\n tuniv = fsolve(get_finalizing_point_from_universal_curve,tend-20.,args=(tend,end_point,\u03bc,\u03bcbar,))[0]\n univ_point = get_state_universal(tuniv,end_point,\u03bc,\u03bcbar)\n trajectory = get_trajectory_with_\u03c3star(univ_point,tend,50,\u03bc,\u03bcbar)\n plt.plot([x[0] for x in trajectory],[x[1] for x in trajectory],linewidth=1,color=clrs0[4])\n trajectory = get_trajectory_with_given_\u03c3(univ_point,tend+20,100,0,\u03bc,\u03bcbar)\n plt.plot([x[0] for x in trajectory],[x[1] for x in trajectory],linewidth=1,color=clrs0[4])\nplt.xlim([0,120]); plt.ylim([0,1]);\nytks = np.arange(0,1.2,.2)\nplt.yticks(ytks,[int(y*100) for y in ytks])\nplt.xlabel(\"Backward time (days)\"); plt.ylabel(\"Fraction of resistant cells (\\%)\")\n# plt.show()\nplt.savefig(\"../figures/draft/Fig2-1.pdf\",format='pdf',bbox_inches='tight')\n```\n\n\n```python\n# Plotting\nplt.rcParams['figure.figsize'] = (3.5, 2.5)\n\n\u03c3s = np.linspace(0,1,1001)\nplt.plot(\u03c3s,[\u03c1f(.9,0,\u03c3,\u03bc,\u03bcbar) for \u03c3 in \u03c3s],linewidth=2,color=\"k\")\n\u03c3imx = np.argmax([\u03c1f(.9,0,\u03c3,\u03bc,\u03bcbar) for \u03c3 in \u03c3s])\nprint(\u03c3imx)\nplt.plot(\u03c3s[\u03c3imx],[\u03c1f(.9,0,\u03c3s[\u03c3imx],\u03bc,\u03bcbar)],'ro')\nplt.plot(\u03c3s,[\u03c11f(.9,0,\u03c3,\u03bc,\u03bcbar) for \u03c3 in \u03c3s],'g--',linewidth=1,zorder=-5)\nplt.plot(\u03c3s,[\u03c12f(.9,0,\u03c3,\u03bc,\u03bcbar) for \u03c3 in \u03c3s],'b--',linewidth=1,zorder=-5)\nplt.ylim([-.0002,.0082]);\nplt.savefig(\"../figures/draft/Fig3-A.pdf\",format='pdf',bbox_inches='tight')\n```\n\n\n```python\nfig, ax = plt.subplots()\nfor solution in primary_field2:\n k = 0\n for x0, entry in solution.items():\n if (k==0):\n print(\"Terminal point: \",entry['\u03c4'][0],entry['x'][0],entry['\u03c6'][0])\n kk = 16\n print(entry['\u03c4'][kk],entry['x'][kk],entry['\u03c6'][kk])\n \u03c1yy = [\u03c1f(entry['x'][kk],entry['\u03c6'][kk],\u03c3,\u03bc,\u03bcbar) for \u03c3 in \u03c3s]\n plt.plot(\u03c3s,\u03c1yy,linewidth=2,color=\"k\")\n \u03c3imx = np.argmax(\u03c1yy)\n print(\u03c3imx)\n plt.plot(\u03c3s[\u03c3imx],\u03c1yy[\u03c3imx],'ro')\n plt.plot(\u03c3s,[\u03c11f(entry['x'][kk],entry['\u03c6'][kk],\u03c3,\u03bc,\u03bcbar) for \u03c3 in \u03c3s],'g--',linewidth=1,zorder=-5)\n plt.plot(\u03c3s,[-\u03c12f(entry['x'][kk],entry['\u03c6'][kk],\u03c3,\u03bc,\u03bcbar) for \u03c3 in \u03c3s],'b--',linewidth=1,zorder=-5)\n plt.ylim([-.0002,.0082]);\n break\n k = k + 1\nplt.savefig(\"../figures/draft/Fig3-B.pdf\",format='pdf',bbox_inches='tight')\n```\n\n\n```python\nfig, ax = plt.subplots()\nfor solution in primary_field2:\n k = 0\n for x0, entry in solution.items():\n if (k==0):\n print(\"Terminal point: \",entry['\u03c4'][0],entry['x'][0],entry['\u03c6'][0])\n kk = 70\n print(entry['\u03c4'][kk],entry['x'][kk],entry['\u03c6'][kk])\n \u03c1yy = [\u03c1f(entry['x'][kk],entry['\u03c6'][kk],\u03c3,\u03bc,\u03bcbar) for \u03c3 in \u03c3s]\n plt.plot(\u03c3s,\u03c1yy,linewidth=2,color=\"k\")\n \u03c3imx = np.argmax(\u03c1yy)\n print(\u03c3imx)\n plt.plot(\u03c3s[\u03c3imx],\u03c1yy[\u03c3imx],'ro')\n plt.plot(\u03c3s,[\u03c11f(entry['x'][kk],entry['\u03c6'][kk],\u03c3,\u03bc,\u03bcbar) for \u03c3 in \u03c3s],'g--',linewidth=1,zorder=-5)\n plt.plot(\u03c3s,[-\u03c12f(entry['x'][kk],entry['\u03c6'][kk],\u03c3,\u03bc,\u03bcbar) for \u03c3 in \u03c3s],'b--',linewidth=1,zorder=-5)\n plt.ylim([-.0002,.0082]);\n break\n k = k + 1\nplt.savefig(\"../figures/draft/Fig3-C.pdf\",format='pdf',bbox_inches='tight')\n```\n\n\n```python\nfig, ax = plt.subplots()\nkk = 10\nxu = universal_curve[kk][1]\n\u03c6u = universal_curve[kk][2]\nprint(\"Point on the universal curve: \",universal_curve[kk][0],xu,\u03c6u)\n\n\u03c1yy = [\u03c1f(xu,\u03c6u,\u03c3,\u03bc,\u03bcbar) for \u03c3 in \u03c3s]\nplt.plot(\u03c3s,\u03c1yy,linewidth=2,color=\"k\")\n\u03c3imx = np.argmax(\u03c1yy[1:]) #except zero\nprint(\u03c3imx)\nplt.plot(\u03c3s[\u03c3imx],\u03c1yy[\u03c3imx],'ro')\nplt.plot([0],\u03c1yy[\u03c3imx],'ro')\nplt.plot(\u03c3s,[\u03c11f(xu,\u03c6u,\u03c3,\u03bc,\u03bcbar) for \u03c3 in \u03c3s],'g--',linewidth=1,zorder=-5)\nplt.plot(\u03c3s,[-\u03c12f(xu,\u03c6u,\u03c3,\u03bc,\u03bcbar) for \u03c3 in \u03c3s],'b--',linewidth=1,zorder=-5)\nplt.ylim([-.0002,.0082]);\n# ax.yaxis.set_major_formatter(plt.NullFormatter())\nplt.savefig(\"../figures/draft/Fig3-D.pdf\",format='pdf',bbox_inches='tight')\n```\n\n# Preparation for second figure\n\n\n```python\n# Plotting\nplt.rcParams['figure.figsize'] = (6.75, 4.5)\n_k = 0\nfor solutions in primary_field1:\n for x0, entry in solutions.items():\n plt.plot(entry['\u03c4'], entry['x'], '-', linewidth=1, color=clrs0[1])\n _k += 1\n_k = 0\nfor solutions in primary_field2:\n for x0, entry in solutions.items():\n plt.plot(entry['\u03c4'], entry['x'], '-', linewidth=1, color=clrs0[1])\n _k += 1\nplt.plot([x[0] for x in switching_curve],[x[1] for x in switching_curve],linewidth=3,color=clrs0[0],zorder=4,linestyle=\"dashed\")\nplt.plot([end_point[0]],[end_point[1]],marker='o',color=\"black\",zorder=4)\nplt.plot([x[0] for x in universal_curve],[x[1] for x in universal_curve],linewidth=3,color=clrs0[0],zorder=3)\nfor tend in [80,110,140]:\n tuniv = fsolve(get_finalizing_point_from_universal_curve,tend-20.,args=(tend,end_point,\u03bc,\u03bcbar,))[0]\n univ_point = get_state_universal(tuniv,end_point,\u03bc,\u03bcbar)\n trajectory = get_trajectory_with_\u03c3star(univ_point,tend,50,\u03bc,\u03bcbar)\n plt.plot([x[0] for x in trajectory],[x[1] for x in trajectory],linewidth=1,color=clrs0[4])\n trajectory = get_trajectory_with_given_\u03c3(univ_point,tend+20,100,0,\u03bc,\u03bcbar)\n plt.plot([x[0] for x in trajectory],[x[1] for x in trajectory],linewidth=1,color=clrs0[4])\nplt.xlim([0,120]); plt.ylim([0,1]);\nplt.xlabel(\"Backward time (days)\"); plt.ylabel(\"Fraction of resistant cells (\\%)\")\nplt.show()\n```\n\n\n```python\nplt.rcParams['figure.figsize'] = (6.75, 4.5)\n\n_k = 0\nfor solutions in primary_field1:\n for x0, entry in solutions.items():\n if _k==5:\n sol = [[1,\u03c4,\u03c3starf(x,\u03c6,\u03bc,\u03bcbar),x,exp((b-d)*\u03c4-V)] for \u03c4,x,\u03c6,V in zip(entry['\u03c4'],entry['x'],entry['\u03c6'],entry['V'])]\n if _k==6:\n trajectory_thr = [[\u03c4,x,\u03c6,V] for \u03c4,x,\u03c6,V in zip(entry['\u03c4'],entry['x'],entry['\u03c6'],entry['V'])]\n sol += [[0,\u03c4,\u03c3starf(x,\u03c6,\u03bc,\u03bcbar),x,exp((b-d)*\u03c4-V)] for \u03c4,x,\u03c6,V in trajectory_thr]\n T0 = max([x[0] for x in trajectory_thr]) \n _k += 1\n#plt.plot(\u03c41, x1, '-', linewidth=1, color=clrs0[1])\n#plt.plot(\u03c4thr, xthr, '--', linewidth=1, color=clrs0[1])\nprint(T0/30.)\n\nplt.plot([end_point[0]],[end_point[1]],marker='o',color=\"black\",zorder=4)\nfor tend in [180]:\n tuniv = fsolve(get_finalizing_point_from_universal_curve,tend-20.,args=(tend,end_point,\u03bc,\u03bcbar,))[0]\n univ_point = get_state_universal(tuniv,end_point,\u03bc,\u03bcbar)\n trajectory = get_trajectory_with_\u03c3star(univ_point,tend,50,\u03bc,\u03bcbar)\n plt.plot([x[0] for x in trajectory],[x[1] for x in trajectory],linewidth=1,color=clrs0[4])\n sol += [[3,\u03c4,\u03c3starf(x,\u03c6,\u03bc,\u03bcbar),x,exp((b-d)*\u03c4-V)] for \u03c4,x,\u03c6,V in trajectory]\n universal_curve = get_universal_curve(end_point,univ_point[0],50,\u03bc,\u03bcbar)\n plt.plot([x[0] for x in universal_curve],[x[1] for x in universal_curve],linewidth=3,color=clrs0[0],zorder=3)\n sol = [[3,\u03c4,get_\u03c3_universal(\u03c4,end_point,\u03bc,\u03bcbar),x,exp((b-d)*\u03c4-V)] for \u03c4,x,\u03c6,V in universal_curve] + sol\n trajectory = get_trajectory_with_\u03c3star([0,end_switching_curve['x'],0,0],end_point[0],50,\u03bc,\u03bcbar)\n sol = [[3,\u03c4,\u03c3starf(x,\u03c6,\u03bc,\u03bcbar),x,exp((b-d)*\u03c4-V)] for \u03c4,x,\u03c6,V in trajectory] + sol\nfor tend in [124]:\n tuniv = fsolve(get_finalizing_point_from_universal_curve,tend-20.,args=(tend,end_point,\u03bc,\u03bcbar,))[0]\n univ_point = get_state_universal(tuniv,end_point,\u03bc,\u03bcbar)\n# trajectory = get_trajectory_with_\u03c3star(univ_point,tend,50,\u03bc,\u03bcbar)\n trajectory = get_trajectory_with_given_\u03c3(univ_point,tend+20,200,0,\u03bc,\u03bcbar)\n plt.plot([x[0] for x in trajectory],[x[1] for x in trajectory],linewidth=1,color=clrs0[4])\n sol += [[2,\u03c4,\u03c3starf(x,\u03c6,\u03bc,\u03bcbar),x,exp((b-d)*\u03c4-V)] for \u03c4,x,\u03c6,V in trajectory]\n universal_curve = get_universal_curve(end_point,univ_point[0],50,\u03bc,\u03bcbar)\n plt.plot([x[0] for x in universal_curve],[x[1] for x in universal_curve],linewidth=3,color=clrs0[0],zorder=3)\n sol = [[2,\u03c4,get_\u03c3_universal(\u03c4,end_point,\u03bc,\u03bcbar),x,exp((b-d)*\u03c4-V)] for \u03c4,x,\u03c6,V in universal_curve] + sol\n trajectory = get_trajectory_with_\u03c3star([0,end_switching_curve['x'],0,0],end_point[0],150,\u03bc,\u03bcbar)\n sol = [[2,\u03c4,\u03c3starf(x,\u03c6,\u03bc,\u03bcbar),x,exp((b-d)*\u03c4-V)] for \u03c4,x,\u03c6,V in trajectory] + sol\nplt.xlim([0,180]); plt.ylim([0,1]);\nplt.xlabel(\"Backward time (days)\"); plt.ylabel(\"Fraction of resistant cells (\\%)\")\nplt.show()\n```\n\n\n```python\ndf_res = pd.DataFrame(sol,columns=['part','time','sigma','resistance','fold_change'])\ndf_res[:10]\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
parttimesigmaresistancefold_change
020.0000001.00.8326631
120.0747771.00.8318161.00018377186716
220.1495551.00.8309621.00036509677104
320.2243321.00.8301001.00054395111730
420.2991101.00.8292311.00072031112691
520.3738871.00.8283531.00089415283525
620.4486651.00.8274681.00106545209120
720.5234421.00.8265751.00123418455620
820.5982191.00.8256741.00140032570334
920.6729971.00.8247651.00156385081649
\n
\n\n\n\n\n```python\npd.DataFrame(sol).to_csv('../figures/draft/Fig6-trjs_optimal.csv',index=False,header=False)\n```\n\n\n```python\ndf_res_ = df_res.loc[lambda df: df.part==3].sort_values('time').drop('part',axis=1)\ndf_res_[:10]\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
timesigmaresistancefold_change
2000.0000001.00.8326631
2010.2273841.00.8300651.00055119847425
2020.4547691.00.8273961.00107932271479
2030.6821531.00.8246531.00158369331340
2040.9095381.00.8218351.00206361475324
2051.1369221.00.8189411.00251837517607
2061.3643071.00.8159671.00294724615568
2071.5916911.00.8129121.00334948247831
2081.8190751.00.8097751.00372432193087
2092.0464601.00.8065521.00407098509788
\n
\n\n\n\n\n```python\ndf_res_.to_csv('../figures/draft/Fig7-trj_optimal.csv',index=False,header=False)\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "1e9948a180041b3b3658845476e7d682bcf0b3d7", "size": 181513, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "scripts/D. Field of optimal trajectories [Python].ipynb", "max_stars_repo_name": "aakhmetz/AkhmKim2019Scripts", "max_stars_repo_head_hexsha": "c348f6702a135e30aea5fc1eb3d8f4ca18b146e3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-11-04T00:10:17.000Z", "max_stars_repo_stars_event_max_datetime": "2019-11-04T00:10:17.000Z", "max_issues_repo_path": "scripts/D. Field of optimal trajectories [Python].ipynb", "max_issues_repo_name": "aakhmetz/AkhmKim2019Scripts", "max_issues_repo_head_hexsha": "c348f6702a135e30aea5fc1eb3d8f4ca18b146e3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "scripts/D. Field of optimal trajectories [Python].ipynb", "max_forks_repo_name": "aakhmetz/AkhmKim2019Scripts", "max_forks_repo_head_hexsha": "c348f6702a135e30aea5fc1eb3d8f4ca18b146e3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2019-11-04T00:10:01.000Z", "max_forks_repo_forks_event_max_datetime": "2019-11-04T00:10:01.000Z", "avg_line_length": 133.1716801174, "max_line_length": 29664, "alphanum_fraction": 0.8344856842, "converted": true, "num_tokens": 11061, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.712232184238947, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.443335569061811}} {"text": "+ This notebook is part of *Quiz review 3* in the OCW MIT course 18.06 by Prof Gilbert Strang [1]\n+ Created by me, Dr Juan H Klopper\n + Head of Acute Care Surgery\n + Groote Schuur Hospital\n + University Cape Town\n + Email me with your thoughts, comments, suggestions and corrections \n
Linear Algebra OCW MIT18.06 IPython notebook [2] study notes by Dr Juan H Klopper is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.\n\n+ [1] OCW MIT 18.06\n+ [2] Fernando P\u00e9rez, Brian E. Granger, IPython: A System for Interactive Scientific Computing, Computing in Science and Engineering, vol. 9, no. 3, pp. 21-29, May/June 2007, doi:10.1109/MCSE.2007.53. URL: http://ipython.org\n\n\n```python\nfrom IPython.core.display import HTML, Image\ncss_file = 'style.css'\nHTML(open(css_file, 'r').read())\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n```python\nfrom sympy import init_printing, Matrix, symbols, sqrt, Rational\nfrom numpy import matrix, transpose, sqrt, eye\nfrom numpy.linalg import pinv, inv, det, svd, norm, eig\nfrom scipy.linalg import pinv2\nfrom warnings import filterwarnings\n```\n\n\n```python\ninit_printing(use_latex = 'mathjax')\nfilterwarnings('ignore')\n```\n\n# Review of select topics\n\n## Quick facts\n\n+ Eigenvalues\n + There are shortcuts that we can sometimes employ to calculate them\n+ In symmetric matrices A = AT\n + Their eigenvalues are always real\n + There are always enough eigenvectors and we can choose them to be orthogonal\n + They van be diagonalized and factorized as Q&Lambda'QT\n+ Similar matrices are any square matrices that are related by A = M-1BM\n + They have the same eigenvalues (not eigenvectors)\n + As one grows / decays so does the other Ak = M-1BkM\n\n## Exercise problems\n\n### Differential equation matrix\n\n+ Consider the following and solve for a general solution and solve for *e*At\n$$ \\frac{du}{dt}={A}{u}=\\begin{bmatrix}0&-1&0\\\\1&0&-1\\\\0&1&0\\end{bmatrix}{u} $$\n\n+ There are no initial condition, so we need the general solutions\n + They will be in the form\n $$ u\\left(t\\right)={c}_{1}{e}^{{\\lambda}_{1}{t}}{x}_{1}+{c}_{2}{e}^{{\\lambda}_{2}{t}}{x}_{2}+{c}_{3}{e}^{{\\lambda}_{3}{t}}{x}_{3} $$\n\n\n```python\nA = Matrix([[0, -1, 0], [1, 0, -1], [0, 1, 0]])\nA, A.det()\n```\n\n\n\n\n$$\\begin{pmatrix}\\left[\\begin{matrix}0 & -1 & 0\\\\1 & 0 & -1\\\\0 & 1 & 0\\end{matrix}\\right], & 0\\end{pmatrix}$$\n\n\n\n+ It is clearly singular (dependent rows and columns)\n\n\n```python\nA.transpose()\n```\n\n\n\n\n$$\\left[\\begin{matrix}0 & 1 & 0\\\\-1 & 0 & 1\\\\0 & -1 & 0\\end{matrix}\\right]$$\n\n\n\n+ It is skew-symmetric and therefor eigenvalues are purely complex numbers (including 0*i*)\n\n\n```python\nA.eigenvals()\n```\n\n\n\n\n$$\\begin{Bmatrix}0 : 1, & - \\sqrt{2} i : 1, & \\sqrt{2} i : 1\\end{Bmatrix}$$\n\n\n\n+ The solution is thus as follows\n$$ u\\left(t\\right)={c}_{1}{x}_{1}+{c}_{2}{e}^{{\\sqrt{2}i}{t}}{x}_{2}+{c}_{3}{e}^{{-\\sqrt{2}i}{t}}{x}_{3} $$\n+ The solution moves around the unit circle (doesn't grow / decay)\n+ It returns to the same value (it's periodic) after a certain time *t*\n$$ \\sqrt{2}{i}{T}=2{\\pi}{i};\\quad{e}^{0}=1 \\\\ \\sqrt{2}{T}=2{\\pi} \\\\ {T}={\\pi}\\sqrt{2} $$\n\n+ Finding *u*(*t*) allows the following\n$$ u\\left(t\\right)={e}^{At}u\\left(0\\right) $$\n\n+ If A is diagonalizable (A = SΛS-1) then we have the following\n$$ {e}^{At}=S{e}^{\\Lambda{t}}{S}^{-1} \\\\ { e }^{ \\Lambda t }=\\begin{bmatrix} { e }^{ { \\lambda }_{ 1 }t } & 0 & 0 \\\\ 0 & \\ddots & 0 \\\\ 0 & 0 & { e }^{ { \\lambda }_{ n }t } \\end{bmatrix} $$\n\n### Orthogonal eigenvalues\n\n+ Which matrices have orthogonal eigenvectors?\n\n+ The following\n + Symmetric matrices\n + When AT = -A (skew-symmetric)\n + Orthogonal matrices\n + In general these are all when AAT=ATA\n\n### Definitions\n\n+ Given the following\n$$ { \\lambda }_{ 1 }=0;\\quad { \\lambda }_{ 2 }=c;\\quad { \\lambda }_{ 3 }=2\\\\ { x }_{ 1 }=\\begin{bmatrix} 1 \\\\ 1 \\\\ 1 \\end{bmatrix},\\quad { x }_{ 2 }=\\begin{bmatrix} 1 \\\\ -1 \\\\ 0 \\end{bmatrix},\\quad { x }_{ 3 }=\\begin{bmatrix} 1 \\\\ 1 \\\\ 2 \\end{bmatrix} $$\n\n+ Is the matrix A diagonalizable and of so for which value(s) of *c*?\n + So we need enough enough eigenvectors and they should be independent\n + They are indeed\n + More so they are orthogonal\n + So, for all *c* the matrix is diagonalizable\n\n+ Is A symmetric and if so, for which value(s) of *s*?\n + The eigenvalues all have to be real values\n + Thus all real values for *c*\n\n+ Is A positive definite and if so, for which values of *c*?\n + This is a sub-case of symmetric matrices\n + There are a lot of tests for positive definite matrices\n + One of the eigenvalues are zero, so it can, at best, be semi-definite, for *c* ≥ 0\n\n+ Is this a Markov matrix and if so, for which values of *c*?\n + One of the eigenvalues must be 1 and the others must be smaller\n + So, no\n\n+ Could ½A be a projection matrix?\n + They are symmetric and eigenvalues must be real\n + Any projection matrix eigenvalues must be 0 and 1\n $$ P=\\frac{A}{2} \\\\ {P}^{2}=P \\\\ \\therefore {\\lambda}^{2}=\\lambda \\\\ \\therefore \\lambda = 0; \\quad \\lambda = 1 $$\n + Thus *c* = 0 or *c* = 2 will work (for ½A we will have ½λ)\n\n### Singular value decomposition\n\n+ In SVD we have the following\n$$ A=U\\Sigma {V}^{T} $$\n+ Where U and V are orthogonal matrices and Σ a diagonal matrix\n\n+ Every matrix has a SVD\n$$ {A}^{T}A=\\left( V {\\Sigma}^{T} {U}^{T} \\right) \\left( U {\\Sigma} {V}^{T} \\right) \\\\ {A}^{T} A = V \\left( {\\Sigma}^{T} {\\Sigma} \\right) {V}^{T} $$\n+ V is the eigenvector matrix for AT\n+ &Sigma: has along its main diagonal the square roots of the eigenvalues\n+ U is similarly calculated as the eigenvector matrix of AAT\n+ There is always, though, as sign issue when choosing V and U\n + For whichever signs are chosen for V, this forces the signs for U which can be checked against the following\n $$ A{v}_{i} = {\\sigma}_{i} {u}_{i} \\\\ AV = \\Sigma U $$\n+ Σ can tell us a lot about A\n + All values must be ≥ 0\n + If it contains a 0 along the main diagonal, A is singular\n\n### Symmetric AND orthogonal matrices (matrices that are both)\n+ A = AT = A-1\n\n+ What can be said about the eigenvalues of these?\n + Symmetric matrices have real eigenvalues and the orthogonal matrix eigenvalues must have length 1; ||λ|| = 1\n\n+ Is A sure to be positive definite?\n + No, as λ can be -1\n\n+ Does it have repeated eigenvalues?\n + Yes (if *n* ≥ 2, some eigenvalues must be repeated)\n\n+ Is it diagonalizable?\n + Most definitely\n\n+ Is it non-singular\n + Yes (no zero eigenvalues)\n\n+ Prove that the following is a projection matrix\n$$ \\frac{1}{2} \\left( A+I \\right) $$ \n + Squaring it should result in the same\n $$ \\frac{1}{4} \\left( {A}^{2} + 2A + I \\right) $$\n + This begs the question, what is A2?\n + Well, if A equals its inverse, A2 = I\n + As an aside the eigenvalues of A + I will be twice the eigenvalues of A\n\n## Example problems\n\n### Example problem 1\n\n+ Find the eigenvalues and eigenvectors of the following\n + 1: The projection matrix P\n $$ {P} = \\frac{{a}{a}^{T}}{{a}^{T}{a}}; \\quad a= \\begin{bmatrix}3\\\\4\\end{bmatrix} $$\n + 2: The matrix Q\n $$ Q = \\begin{bmatrix}0.6&-0.8\\\\0.8&0.6\\end{bmatrix} $$\n + 3: The matrix R\n $$ R = 2P - I $$\n\n#### Solution\n\n+ 1:\n\n\n```python\na = matrix([[3], [4]]) # Not using sympy\nP = (a * transpose(a)) / (transpose(a) * a)\nP\n```\n\n\n\n\n matrix([[ 0.36, 0.48],\n [ 0.48, 0.64]])\n\n\n\n+ The eigenvalues of a projection matrix are either 0 or 1\n\n\n```python\neig(P) # eig() gives the eigenvalues and eigenvector matrix\n```\n\n\n\n\n (array([ 0., 1.]), matrix([[-0.8, -0.6],\n [ 0.6, -0.8]]))\n\n\n\n+ 2:\n\n\n```python\nQ = matrix([[0.6, -0.8], [0.8, 0.6]])\nQ\n```\n\n\n\n\n matrix([[ 0.6, -0.8],\n [ 0.8, 0.6]])\n\n\n\n+ Note that Q is a projection matrix\n\n\n```python\neig(Q) # eigenvalues come in complex conjugate pairs\n```\n\n\n\n\n (array([ 0.6+0.8j, 0.6-0.8j]),\n matrix([[ 0.70710678+0.j , 0.70710678-0.j ],\n [ 0.00000000-0.70710678j, 0.00000000+0.70710678j]]))\n\n\n\n+ 3:\n\n+ R will have the same eigenvectors, but (shifted) eigenvalues\n\n\n```python\nR = 2 * P - eye(2)\nR\n```\n\n\n\n\n matrix([[-0.28, 0.96],\n [ 0.96, 0.28]])\n\n\n\n\n```python\neig(R)\n```\n\n\n\n\n (array([-1., 1.]), matrix([[-0.8, -0.6],\n [ 0.6, -0.8]]))\n\n\n\n+ The eigenvalues of P was 0 and 1\n + 2 × 0 - 1 = -1\n + 2 × 1 - 1 = 1\n\n\n```python\n\n```\n", "meta": {"hexsha": "63580ffe67eeb5f6905badf847a592c79f948f8f", "size": 21115, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "forks/MIT_OCW_Linear_Algebra_18_06-master/III_09_Quiz_review.ipynb", "max_stars_repo_name": "solomonxie/jupyter-notebooks", "max_stars_repo_head_hexsha": "65999f179e037242138de72f512dda4bf00c7379", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-02-13T05:52:05.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-08T09:52:35.000Z", "max_issues_repo_path": "forks/MIT_OCW_Linear_Algebra_18_06-master/III_09_Quiz_review.ipynb", "max_issues_repo_name": "solomonxie/jupyter-notebooks", "max_issues_repo_head_hexsha": "65999f179e037242138de72f512dda4bf00c7379", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "forks/MIT_OCW_Linear_Algebra_18_06-master/III_09_Quiz_review.ipynb", "max_forks_repo_name": "solomonxie/jupyter-notebooks", "max_forks_repo_head_hexsha": "65999f179e037242138de72f512dda4bf00c7379", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 26.9323979592, "max_line_length": 708, "alphanum_fraction": 0.4928723656, "converted": true, "num_tokens": 3743, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5156199157230156, "lm_q2_score": 0.8596637487122112, "lm_q1q2_score": 0.44325974966112197}} {"text": "# Enzyme Kinetics\nWe now study common reaction mechanisms that describe enzyme catalysis. Enzymes can dramatically accelerate the rate of biochemical reactions inside and outside living cells. The absolute rates of biochemical reactions are key biological design variables because they can evolve from a very low rate as determined by the mass action kinetics based on collision frequencies, to a very high and specific reaction rate as determined by appropriately-evolved enzyme properties. We first describe the procedure used to derive enzymatic rate laws, which we then apply to the Michaelis-Menten reaction mechanism, then to the Hill model, and finally to the symmetry model. The first is used to describe plain chemical transformations, while the latter two are used to describe regulatory effects. \n\n**MASSpy** will be used to demonstrate some of the topics in this chapter. \n\n\n```python\nfrom mass import (\n MassModel, MassMetabolite, MassReaction, Simulation, MassSolution)\nfrom mass.visualization import plot_time_profile, plot_phase_portrait\n```\n\nOther useful packages are also imported at this time.\n\n\n```python\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nfrom scipy.integrate import solve_ivp\n```\n\n## Enzyme Catalysis\nEnzymes are catalysts that accelerate biochemical transformations in cells. Almost all enzymes are proteins. There are also catalytically active ribonucleic acids, called \"ribozymes.\" The fundamental properties of enzyme catalysis are described in this section. \n\n### Enzymatic activity\nThe activity of an enzyme is measured by determining the increase in the reaction rate relative to the absence of the enzyme. In other words we compare the reaction rate of the un-catalyzed reaction to the catalyzed rate. The ratio can be thought of as an acceleration factor and this number can be quite high, sometimes by many million-fold. \n\n### Reaction and substrate specificity\nEnzymes are usually very specific both with respect to the type of reaction being catalyzed (reaction specificity) and with respect to the reactants (the \"substrates\") that they act on. Highly specific enzymes catalyze the cleavage of only one type of a chemical bond, and only in one substrate. Other enzymes may have a narrow reaction specificity, but broad substrate specificity, i.e., they act on a number of chemically similar substrates. Rare enzymes exist that have both low reaction specificity and low substrate specificity. \n\n\n\n**Figure 5.1:** Basic principles of enzyme catalysis. From (Koolman, 2005).\n\n### Catalysis\nAs discussed in Chapter 2 (Figure 2.4), two molecules can only react with each other if they collide in a favorable orientation. Such collisions may be rare, and thus the reaction rate is slow. An un-catalyzed reaction starts with a favorable collision as shown in Figure 5.1a. Before the products are formed, the collision complex A-B has to pass through what is called a _transition state_. Its formation requires _activation energy_. Since activation energies can be quite high, only a few A-B complexes have this amount of energy, and thus a productive transition state arises only for a fraction of favorable collisions. As a result, conversion only happens occasionally even when the reaction is thermodynamically feasible; i.e., when the net change in Gibbs free energy is negative ($\\Delta G < 0$). \n\nEnzymes can facilitate the probability of a favorable collision and lower the activation energy barrier, see Figure 5.1b,c. Enzymes are able to bind their substrates in the catalytic site. As a result, the substrates are favorably oriented relative to one another, greatly enhancing the probability that productive A-B complexes form. The transition state is stabilized leading to a lowered activation energy barrier. \n\n### Information on enzymes \nDetailed information is available on a large number of enzymes. This include structural information, the organism source, and other characteristics. An example is shown in Figure 5.2. Many online sources of such information exist. \n\n\n\n**Figure 5.2:** Detailed information on enzymes is available. From PDB.\n\n## Deriving Enzymatic Rate Laws\nThe chemical events underlying the catalytic activities of enzymes are described by a reaction mechanism. A reaction mechanism is comprised of the underlying elementary reactions that are believed to take place. A rate law is then formulated to describe the rate of reaction. \n\nA rate law describes the conversion of a substrate $(x_1)$ by an enzyme into a product $(x_2)$: \n\n$$\\begin{equation} x_1 \\stackrel{v}{\\rightarrow} x_2 \\tag{5.1} \\end{equation}$$\n\nwhere $v$ is a function of the concentrations of the chemical species involved in the reaction. The steps involved in the development and analysis of enzymatic rate laws are illustrated in Figure 5.3 and they are as follows: \n\n\n\n**Figure 5.3:** The process of formulating enzymatic rate laws. QSSA represents the quasi-steady state assumption and QEA represents the quasi-equilibrium assumption.\n\n* Formulate the dynamic mass balances based on the elementary reactions in the postulated reaction mechanism, \n\n* Identify time invariants, or conservation relationships, \n\n* Reduce the dynamic dimension of the reaction mechanism by eliminating dynamically dependent variables using the conservation relationships, \n\n* Apply commonly used simplifying kinetic assumptions to formulate a rate law, representing a reduction in the dynamic dimension of the kinetic model, \n\n* Apply mathematical and numerical analysis to determine when the simplifying assumptions are valid and the reaction rate law can be used; and \n\n* Identify key dimensionless parameter ratios. This last step is optional and used by those interested in deeper mathematical analysis of the properties of the rate laws. \n\nThe use of enzymatic rate laws in dynamic network models is hampered by their applicability in vivo based on in vitro measurements. From a practical standpoint, with the numerical simulation capacity that is now routinely available, applying simplifying assumptions may no longer be needed for computational simplification and convenience. However, it is useful to help understand the historical origin of enzymatic rate laws, the simplifications on which they are based, and when it may be desirable to use them. \n\n## Michaelis-Menten Kinetics\nThe simplest enzymatic reaction mechanism, first proposed by Henri (Henri, 1903) but named after Michaelis and Menten (Michaelis, 1913) is; \n\n$$\\begin{equation} S + E \\underset{k_{-1}}{\\stackrel{k_1}{\\rightleftharpoons}} X \\stackrel{k_2}{\\rightarrow} E + P \\tag{5.2} \\end{equation}$$\n\nwhere a substrate, $S$, binds reversibly to the enzyme, $E$, to form the intermediate, $X$, which can break down to give the product, $P$, and regenerate the enzyme. Note that it is similar to the reaction mechanism of two connected reversible bi-linear reactions (Eq. (4.38)) with $x_5 = x_2$, as one of the original reactants $(E)$ is regained in the second step. Historically speaking, the Michaelis-Menten scheme is the most important enzymatic reaction mechanism. A detailed account of the early history of Michaelis-Menten kinetics is found in (Segal, 1959).\n\n### Step 1: Dynamic mass balances for Michaelis-Menten kinetics\nApplying the law of mass action to the Michaelis-Menten reaction mechanism, one obtains four differential equations that describe the dynamics of the concentrations of the four chemical species involved in the reaction mechanism: \n\n$$\\begin{align} \\frac{ds}{dt} &= -k_1es + k_{-1}x, & s(t = 0) = s_0 \\ \\ &\\tag{5.3a} \\\\ \\frac{dx}{dt} &= k_1es - (k_{-1} + k_2)x, & x(t = 0) = 0 \\ \\ &\\tag{5.3b} \\\\ \\frac{de}{dt} &= -k_1es + (k_{-1} + k_2)x, & e(t = 0) = e_0 \\ \\ &\\tag{5.3c} \\\\ \\frac{dp}{dt} &= k_2x, & p(t = 0) = 0 \\ \\ &\\tag{5.3d}\\\\ \\end{align}$$\n\nwhere the lower case letters denote the concentrations of the corresponding chemical species. The initial conditions shown are for typical initial rate experiments where substrate and free enzyme are mixed together at time $t=0$. $e_0$ and $s_0$ denote the initial concentration of enzyme and substrate, respectively. No mass exchange occurs with the environment. \n\n### Step 2: Finding the time invariants for Michaelis-Menten kinetics\nUsing $\\textbf{x} = (s, e, x, p)$ and $\\textbf{v} = (k_1es, \\ k_{-1}x, \\ k_2x)$ the stoichiometrix matrix is \n\n$$\\begin{equation} \\textbf{S} = \\begin{pmatrix} {-1} & {1} & {0} \\\\ {-1} & {1} & {1} \\\\ {1} & {-1} & {-1} \\\\ {0} & {0} & {1} \\\\ \\end{pmatrix} \\end{equation}$$\n$$\\tag{5.4}$$\n\nIt has a rank of 2 and thus there are two conservation quantities. They are the total concentration of the enzyme and total concentration of the substrate: \n\n$$\\begin{align} e_0 & = e + x \\tag{5.5} \\\\ s_0 &= s + x + p \\tag{5.6} \\end{align}$$\n\n### Step 3: Reducing the dynamic description for Michaelis-Menten kinetics\nAs a consequence of the two conservation relationships, only two of equations 5.3 are dynamically independent. Choosing the substrate, $s$, and the intermediate complex, $x$, concentrations as the two independent variables, the reaction dynamics are described by: \n\n$$\\begin{align} \\frac{ds}{dt} &= -k_1e_0s + (k_1s + k_{-1})x, \\ &s(t = 0)=s_0 \\ \\ &\\tag{5.7} \\\\ \\frac{dx}{dt} &= k_1e_0s - (k_1s + k_{-1} + k_2)x, \\ &x(t = 0) = 0 \\ \\ &\\tag{5.8} \\\\ \\end{align}$$\n\nThe major problem with this mass action kinetic model is that it is mathematically intractable (Hommes, 1962). Equations 5.7 and 5.8 can be reduced to an Abel type differential equation whose solution cannot be obtained in a closed form.\n\n### Step 4: Applying kinetic assumptions for Michaelis-Menten kinetics\nA closed form analytical solution to the mass action kinetic equations, 5.7 and 5.8, is only attainable by using simplifying kinetic assumptions. Two assumptions are used: the _quasi-steady state assumption_ (QSSA) and the _quasi-equilibrium assumption_ (QEA). \n\n#### The quasi-steady state assumption:\nThe rationale behind the quasi-steady state assumption (Briggs, 1925) is that, after a rapid transient phase, the intermediate, $X$, reaches a quasi-stationary state in which its concentration does not change appreciably with time. Applying this assumption to Eq. (5.8) (i.e., $dx/dt=0$) gives the concentration of the intermediate complex as: \n\n$$\\begin{equation} x_{qss} = \\frac{e_0s}{K_m + s} \\tag{5.9} \\end{equation}$$\n\nwhere $K_m = (k_{-1} + k_2)/k_1$ is the well-known Michaelis constant. Substituting $x_{qss}$ into the differential equation for the substrate (Eq. (5.7)) gives the rate law \n\n$$\\begin{equation} \\frac{ds}{dt} = \\frac{-k_2e_0s}{K_m + s} \\tag{5.10} \\end{equation}$$\n\nwhich is the well-known Michaelis-Menten equation, where $v_m$ is the maximum reaction rate (or reaction velocity). \n\nInitially, the quasi-steady state assumption was justified based on physical intuition, but justification for its applicability is actually found within the theory of singular perturbations (Bowen, 1963) Eq. (5.10) can be shown to be the first term in an asymptotic series solution derived from singular perturbation theory (Heineken,1967), (Meiske, 1978); see review in (Palsson, 1984). \n\n#### The quasi-equilibrium assumption:\nHere, one assumes that the binding step quickly reaches a quasi-equilibrium state (Henri, 1903), (Michaelis, 1913) where \n\n$$\\begin{equation} \\frac{se}{x} = \\frac{s(e_0 - x)}{x} = \\frac{k_{-1}}{k_1} = K_d, \\ \\text{or} \\ x_{qe} = \\frac{e_0s}{K_d + s} \\tag{5.11} \\end{equation}$$\n\nholds. $K_d$ is the disassociation equilibrium constant. Note the similarity to Eq. (5.9). Hence, one obtains the rate law \n\n$$\\begin{equation} \\frac{dp}{dt} = \\frac{k_2e_0s}{K_d + s} \\tag{5.12} \\end{equation}$$\n\nby using Eq. (5.11) in the differential equation for the product $P$. \n\n### Step 5: Numerical solutions for Michaelis-Menten kinetics\nThe full dynamic description of the kinetics of the reaction (Eq. (5.7) and (5.8)) can be obtained by direct numerical integration. The results are most conveniently shown on a phase portrait along with the transient response of the concentrations on both the fast and slow time scales, see Figure 5.4. \n\n#### QSSA Solution for Michaelis-Menten kinetics\n\n\n```python\nt0 = 0\ntf = 1e3\n# QSSA Assumption\n# Define function to integrate\ndef qssa(t, s, *params):\n k2, e0, Km = params\n dsdt = (-k2*e0*s)/(Km + s)\n return dsdt\n\n# Define initial conditions and parameters for integration\ns0 = 1\n\ne0 = (1/100)\nk2 = 1\nKm = 1\nparams = [k2, e0, Km]\n\n# Obtain numerical solutions\nsol_obj = solve_ivp(fun=lambda t, s: qssa(t, s, *params),\n t_span=(t0, tf), y0=[s0])\nt, s_sol = (sol_obj.t, sol_obj.y)\nx_sol = np.array([(e0 * val)/(Km + val) for val in s_sol])\n# Store solutions into Solution Objects\nqssa_sol = MassSolution(\n \"QSSA\", solution_type=\"Conc\", \n data_dict={\"s\": s_sol[0], \"x\": x_sol[0]},\n time=t, interpolate=False)\n```\n\n#### Numerical Solution for Michaelis-Menten kinetics\n\n\n```python\nmodel = MassModel('Michaeli_Menten')\n## Define metabolites\ns = MassMetabolite(\"s\")\ne = MassMetabolite(\"e\")\nx = MassMetabolite(\"x\")\np = MassMetabolite(\"p\")\n# Define reactions\nv1 = MassReaction(\"v1\")\nv2 = MassReaction(\"v2\", reversible=False)\nv1.add_metabolites({s: -1, e: -1, x: 1})\nv2.add_metabolites({x: -1, e: 1, p: 1})\nmodel.add_reactions([v1, v2])\n## Define parameters\nv1.kf = 2\nv1.Keq = 2\nv2.kf = 1\n# Define initial conditions\nmodel.update_initial_conditions({s: s0, e: e0, x: 0, p: 0})\n\n# Solve\nMM_simulation = Simulation(model, verbose=True)\nconc_sol, flux_sol = MM_simulation.simulate(model, (t0, tf))\n```\n\n Set parameter Username\n\n\n \u001b[93mWARNING:\u001b[0m \u001b[93mNo compartments found in model. Therefore creating compartment 'compartment' for entire model.\u001b[0m\n\n\n Successfully loaded MassModel 'Michaeli_Menten' into RoadRunner.\n\n\n\n```python\nfig_5_4 = plt.figure(figsize=(9, 7))\ngs = fig_5_4.add_gridspec(nrows=2, ncols=2, width_ratios=[1, 1.5],\n height_ratios=[1, 1])\n\nax1 = fig_5_4.add_subplot(gs[0, 0])\nax2 = fig_5_4.add_subplot(gs[0, 1])\nax3 = fig_5_4.add_subplot(gs[1, 1])\n# Phase portrait of both solutions' substrate vs. intermediate\nplot_phase_portrait(\n conc_sol, x=s, y=x, ax=ax1, legend=[\"Numerical Solution\", \"lower outside\"],\n annotate_time_points=\"endpoints\",\n annotate_time_points_color=[\"r\", \"b\"],\n annotate_time_points_labels=True);\nplot_phase_portrait(\n qssa_sol, x=s, y=x, ax=ax1, legend=[\"QSSA\", \"lower outside\"], \n xlabel=s.id, ylabel=x.id, linestyle=[\"--\"],\n title=(\"(a) Phase Portrait\", {\"size\": \"x-large\"}),\n annotate_time_points=\"endpoints\",\n annotate_time_points_labels=True);\n\n# Time profile of solutions' substrate concentration\nplot_time_profile(conc_sol, observable=s, ax=ax2);\nplot_time_profile(\n qssa_sol, observable=s, ax=ax2, \n xlabel=\"Time\", ylabel=\"Concentration\",\n title=(\"(b) Substrate Concentration\", {\"size\": \"x-large\"}),\n linestyle=[\"--\"]);\n# Time profile of solutions' intermediate concentration\nplot_time_profile(conc_sol, observable=x, ax=ax3);\nplot_time_profile(\n qssa_sol, observable=x, ax=ax3,\n xlabel=\"Time\", ylabel=\"Concentration\",\n title=(\"(c) Intermediate Concentration\", {\"size\": \"x-large\"}),\n linestyle=[\"--\"]);\nfig_5_4.tight_layout()\n```\n\n**Figure 5.4:** The transient response of the Michaelis-Menten reaction mechanism, for $k_2 = k_{-1}, \\ 100e_0 = K_m \\ \\text{and} \\ s_0 = K_m$. (a) The phase portrait . (b) The substrate concentrations. (c) The intermediate concentrations. The solid and dashed line represent the quasi-steady state and the full numerical solution respectively.\n\n* _The phase portrait_. The phase portrait is shown in of Figure 5.4a and it shows how the reaction rapidly approaches the quasi-steady state line and then moves along that line towards the equilibrium in the origin where the reaction has gone to completion. \n\n* _The fast motion_. With the slider on moved to the left, Figure 5.4 shows the changes in the concentrations during the faster time scale. The intermediate concentration exhibits a significant fast motion, while the substrate does not move far from its initial value. \n\n* _The slow motion_. The changes in the concentrations during the slower time scale are shown when the slider is on the right of Figure 5.4. Both the substrate and the intermediate complex decay towards zero. During the decay process, the complex is in a quasi-stationary state and the motion of the substrate drives the reaction dynamics. The quasi-steady state solution gives a good description of the motion on the slower time scale. \n\n### Step 6: Identification of dimensionless parameters for Michaelis-Menten kinetics\nSimulation studies suggests that there are three dimensionless parameters of interest: \n\n$$\\begin{equation} a = k_2/k_{-1}, \\ b = e_0/K_m, \\ c = s_0/K_m \\tag{5.13} \\end{equation}$$\n\nThis result is also found by rigorous mathematical analysis (Palsson, 1984). The dynamic behavior of the reaction is determined by three dimensionless groups: a ratio of kinetic constants and the two initial conditions scaled to $K_m$. \n\n1. The first dimensionless group, a, is a ratio consisting only of kinetic constants, $k_2/k_{-1}$. This ratio has been called the 'stickiness number' (Palsson, 1984), (Palsson, 1984a), since a substrate is said to stick well to an enzyme if $k_2 > k_{-1}$. Once $X$ is formed it is more likely to break down to yield the product than to revert back to substrate. \n\n2. The second dimensionless number, $e_0/K_m$, is a dimensionless concentration parameter - the total enzyme concentration relative to the Michaelis constant. This quantity varies from one situation to another and takes particularly different values under _in vitro_ and _in vivo_ conditions. The enzyme concentrations used _in vitro_ are several orders of magnitude lower than the $K_m$ values (Masters, 1977), (Srere, 1967), (Srere, 1970). In vivo enzyme concentrations can approach the same order of magnitude as $K_m$. \n\n3. The third dimensionless ratio, $s_0/K_m$, is the initial condition for the substrate concentration. Typical values for this ratio _in vivo_ is on the order of unity. \n\n\n```python\n# Define new initial conditions and parameters for integration\ns0 = (1/100)\n\ne0 = (1/100)\nk2 = 1\nKm = 1\nparams = [k2, e0, Km]\n\n# Obtain numerical solutions\nsol_obj = solve_ivp(fun=lambda t, s: qssa(t, s, *params),\n t_span=(t0, tf), y0=[s0])\ns_sol = sol_obj.y\nx_sol = np.array([(e0 * val)/(Km + val) for val in s_sol])\n# Store solutions into MassSolution Objects\nqssa_sol = MassSolution(\n \"QSSA\", solution_type=\"Conc\", \n data_dict={\"s\": s_sol[0], \"x\": x_sol[0]},\n time=sol_obj.t, interpolate=False)\n\n# Update initial conditions for MassModel\nmodel.update_initial_conditions({s: s0})\n\n# Solve\nMM_simulation = Simulation(model, verbose=True)\nconc_sol, flux_sol = MM_simulation.simulate(model, (t0, tf))\n```\n\n \u001b[93mWARNING:\u001b[0m \u001b[93mNo compartments found in model. Therefore creating compartment 'compartment' for entire model.\u001b[0m\n\n\n Successfully loaded MassModel 'Michaeli_Menten' into RoadRunner.\n\n\n\n```python\nfig_5_5 = plt.figure(figsize=(9, 7))\ngs = fig_5_5.add_gridspec(nrows=2, ncols=2, width_ratios=[1, 1.5],\n height_ratios=[1, 1])\n\nax1 = fig_5_5.add_subplot(gs[0, 0])\nax2 = fig_5_5.add_subplot(gs[0, 1])\nax3 = fig_5_5.add_subplot(gs[1, 1])\n# Phase portrait of both solutions' substrate vs. intermediate\nplot_phase_portrait(\n conc_sol, x=s, y=x, ax=ax1, legend=[\"Numerical Solution\", \"lower outside\"],\n annotate_time_points=\"endpoints\",\n annotate_time_points_color=[\"r\", \"b\"],\n annotate_time_points_labels=True);\nplot_phase_portrait(\n qssa_sol, x=s, y=x, ax=ax1, legend=[\"QSSA\", \"lower outside\"], \n xlabel=s.id, ylabel=x.id, linestyle=[\"--\"],\n title=(\"(a) Phase Portrait\", {\"size\": \"x-large\"}),\n annotate_time_points=\"endpoints\",\n annotate_time_points_labels=True);\n\n# Time profile of solutions' substrate concentration\nplot_time_profile(conc_sol, observable=s, ax=ax2);\nplot_time_profile(qssa_sol, observable=s, ax=ax2, \n xlabel=\"Time\", ylabel=\"Concentration\",\n title=(\"(b) Substrate Concentration\", {\"size\": \"x-large\"}),\n linestyle=[\"--\"]);\n# Time profile of solutions' intermediate concentration\nplot_time_profile(conc_sol, observable=x, ax=ax3);\nplot_time_profile(qssa_sol, observable=x, ax=ax3,\n xlabel=\"Time\", ylabel=\"Concentration\",\n title=(\"(c) Intermediate Concentration\", {\"size\": \"x-large\"}),\n linestyle=[\"--\"]);\nfig_5_5.tight_layout()\n```\n\n**Figure 5.5:** The transient response of the Michaelis-Menten reaction mechanism, for $k_2 = k_{-1},\\ 100e_0 = K_m \\ \\text{and} \\ s_0 = K_m$. (a) The phase portrait . (b) The substrate concentrations. (c) The slow transients. The solid and dashed line represent the quasi-steady state and the full numerical solution respectively.\n\n### Comment on the criterion $e_0 << s_0$\nHistorically, the commonly accepted criterion for the applicability of the quasi-steady state assumption is that the initial concentration of the enzyme must be much smaller than that of the substrate. The actual criterion is $e_0 << K_m, \\ \\text{or} \\ b << 1$ (Palsson, 1984). Figure 5.5 shows the reaction dynamics for $e_0 = K_m, \\ e_0 = 100s_0, \\ 100k_2 = k_{-1}$, which is analogous to Figure 5.4, except the initial substrate concentration is now a hundred times smaller than $K_m$. In other words, we have $e_0 = s_0 << K_m$ and, as demonstrated in Figure 5.5, the quasi-steady state assumption is applicable. \n\n## Hill-kinetics for Enzyme Regulation\n### Regulated enzymes\nEnzyme activity is regulated by the binding of small molecules to the enzyme resulting in an altered enzymatic activity. Such binding can inhibit or activate the catalytic activities of the enzyme. The regulation of enzymes such regulators represents a 'tug of war' between the functional states of the enzyme, see Figure 5.6. A simple extension of the oldest reaction mechanisms for ligand binding to oligomeric protein, i.e., oxygen binding to hemoglobin, is commonly used to obtain simple rate laws for regulated enzymes (Hill, 1910).\n\n\n\n**Figure 5.6:** An example of a regulated multimeric enzyme. The T form of the enzyme created by inhibitor binding is inactive, where as the R form, where no inhibitor is bound, is catalytically active. From (Koolman, 2005) (reprinted with permission).\n\n### The reaction mechanism for Hill-kinetics\nThe Hill reaction mechanism is based on two reactions: a catalytic conversion and the sequestration of the enzyme in an inactive form. It assumes that the catalyzed reaction is an irreversible bi-molecular reaction between the substrate, $S$, and the enzyme, $E$, to form the product,$P$, and the free enzyme in a single elementary reaction: \n\n$$\\begin{equation} S + E \\stackrel{k}{\\rightarrow} E + P \\tag{5.14} \\end{equation}$$\n\nThe enzyme in turn can be put into a catalytically inactive state, $X$, through binding simultaneously and reversibly to $\\nu$ molecules of an inhibitor, $I$: \n\n$$\\begin{equation} E + {\\nu}I \\underset{k_{-i}^-}{\\stackrel{k_{i}^+}{\\rightleftharpoons}} X \\tag{5.15} \\end{equation}$$\n\nNumerical values for $\\nu$ often exceed unity. Thus, the regulatory action of $I$ is said to be lumped in the simple $E$ to $X$ transformation, as values of $\\nu$ greater than 1 are chemically unrealistic. Numerical values estimated from data show that the best fit values for $\\nu$ are not integers; for instance $\\nu$ is found to be around 2.3 to 2.6 for $O_2$ binding to hemoglobin. Section 5.5 describes more realistic reaction mechanisms of serial binding of an inhibitor to a regulated enzyme to sequester it in an inactive form. \n\n### Step 1: Dynamic mass balances for Hill-kinetics\nThe mass action kinetic equations are \n\n$$\\begin{equation} \\frac{ds}{dt} = -v_1, \\ \\frac{de}{dt} = -v_2 + v_3, \\ \\frac{dp}{dt} = v_1, \\ \\frac{di}{dt} = -\\nu (v_2 - v_3), \\ \\frac{ds}{dt} = v_2 - v_3 \\end{equation}$$\n\nwhere the reaction rates are \n\n$$\\begin{equation} v_1 = kse, \\ v_2 = k_i^+i^{\\nu}e, \\ v_3 = k_i^-x \\tag{5.16} \\end{equation}$$\n\n### Step 2: Finding the time invariants for Hill-kinetics\nWe define $\\textbf{x} = (s, e, p, i, x) \\ \\text{and} \\ \\textbf{v} = (ks, k_i^+i^{\\nu}e, k_i^-x)$. The stoichiometric matrix is then \n\n$$\\begin{equation} \\textbf{S} = \\begin{pmatrix} {-1} & {0} & {0} \\\\ {0} & {-1} & {1} \\\\ {1} & {0} & {0} \\\\ {0} & {-\\nu} & {\\nu} \\\\ {0} & {1} & {-1} \\\\ \\end{pmatrix} \\end{equation}$$\n$$\\tag{5.17}$$\n\nand has a rank of two. The conservation quantities are a balance on the substrate, the enzyme, and the inhibitor: \n\n$$\\begin{equation} s_0 = s + p, \\ e_0 = e + x, \\ i_0 = i + \\nu x \\tag{5.18} \\end{equation}$$\n\n### Step 3: Reducing the dynamic description for Hill-kinetics\nWe need two differential equations to simulate the dynamic response and then the remaining three variables can be computed from the conservation relationships. We can choose the substrate, $s$, and the concentration of the enzyme, $e$: \n\n$$\\begin{equation} \\frac{ds}{dt} = kse, \\ \\frac{de}{dt} = -k_i^+i^{\\nu}e + k_i^-x \\tag{5.19} \\end{equation}$$\n\nthen $p$, $x$ and $i$ are computed from Eq. (5.18). \n\n### Step 4: Applying simplifying kinetic assumptions for Hill-kinetics\n\nIf we assume that the binding of the inhibitor is fast, so that a quasi-equilibrium forms for the reaction of Eq. (5.15), we have \n\n$$\\begin{equation} v_2 = v_3, \\ \\text{thus} \\ x = (k_i^+/k_i^-)i^{\\nu}e = (i/K_i)^{\\nu}e, \\ \\text{and} \\ \\frac{de}{dt} = \\frac{dx}{dt} = \\frac{di}{dt} = 0 \\tag{5.20} \\end{equation}$$\n\nwhere $K_i$ is a \"per-site\" dissociation constant for Eq. (5.15). The enzyme is in one of two states, so that we have the mass balance \n\n$$\\begin{equation} e_0 = e + x = (1 + (i/K_i)^{\\nu})e \\ \\text{or} \\ e(i) = \\frac{e_0}{1 + (i/K_i)^{\\nu}} \\tag{5.21} \\end{equation}$$\n\nwhere $e_0$ is the total concentration of the enzyme. Using the mass balance and the quasi-equilibrium assumption gives the flux through the regulated reaction as \n\n$$\\begin{equation} v(i) = ke(i)s = \\frac{ke_0s}{1 + (i/K_i)^{\\nu}} = \\frac{v_m}{1 + (i/K_i)^{\\nu}} \\tag{5.22} \\end{equation}$$\n\nwith $v_m = ke_0s$. The Hill model has three parameters: 1) $\\nu$, the degree of cooperativity, 2) $K_i$, the dissociation constant for the inhibitor and, 3) $v_m$, the maximum reaction rate or the capacity of the enzyme. We note that \n\n$$\\begin{equation} f_e = \\frac{e(i)}{e_0} = \\frac{1}{1 + (i/K_i)^{\\nu}} \\tag{5.23} \\end{equation}$$\n\nrepresents the fraction of the enzyme that is in the active state. Note that $f_e \\lt 1$ for any finite concentration of the inhibitor. \n\n\n```python\nt0 = 0\ntf = 10\n\ndef hill(t, state_vars, *params):\n s, p, e, i, x = state_vars \n k1, k_plus,k_minus, nu = params\n # Reaction Rates\n v1 = k1 * s * e\n v2 = k_plus * i**nu * e\n v3 = k_minus * x\n # Differential equations\n diffeqs =[-v1, # ds/dt\n v1, # dp/dt\n -v2 + v3, # de/dt\n -nu*(v2 - v3), # di/dt\n v2 - v3] # dx/dt\n return diffeqs\n\n# Define initial conditions\ns0, p0, e0, i0, x0 = (1, 0, 1, 1, 0)\n\n# Define paramters\nk1 = 1\nk_plus, k_minus = (100, 100)\nnu = 2\nparams = [k1, k_plus, k_minus, nu]\n\n# Obtain numerical solutions\nsol_obj = solve_ivp(\n fun=lambda t, state_vars: hill(t, state_vars, *params),\n t_span=(t0, tf), y0=[s0, p0, e0, i0, x0])\n# Store solutions into Solution Objects\nsol_dict = dict(zip([\"s\", \"p\", \"e\", \"i\", \"x\"], sol_obj.y))\nhill_sol = MassSolution(\n \"Hill\", solution_type=\"Conc\", data_dict=sol_dict,\n time=sol_obj.t, interpolate=False)\n```\n\n\n```python\nfig_5_7 = plt.figure(figsize=(9, 8))\ngs = fig_5_7.add_gridspec(nrows=2, ncols=2, width_ratios=[1, 1],\n height_ratios=[1, 1])\n\nax1 = fig_5_7.add_subplot(gs[0, 0])\nax2 = fig_5_7.add_subplot(gs[0, 1])\nax3 = fig_5_7.add_subplot(gs[1, 0])\nax4 = fig_5_7.add_subplot(gs[1, 1])\n\nplot_phase_portrait(\n hill_sol, x=\"s\", y=\"e\", ax=ax1, xlabel=\"s\", ylabel=\"e\",\n xlim=(-0.05, 1.05), ylim=(-0.05, 1.05),\n title=(\"(a) Phase Portrait of s vs. e\", {\"size\": \"x-large\"}),\n annotate_time_points=\"endpoints\",\n annotate_time_points_labels=True);\nplot_phase_portrait(\n hill_sol, x=\"e\", y=\"x\", ax=ax2, xlabel=\"e\", ylabel=\"x\",\n xlim=(-0.05, 1.05), ylim=(-0.05, 1.05),\n title=(\"(b) Phase Portrait of e vs. x\", {\"size\": \"x-large\"}),\n annotate_time_points=\"endpoints\",\n annotate_time_points_labels=True);\n\nplot_phase_portrait(\n hill_sol, x=\"i\", y=\"x\", ax=ax3, xlabel=\"i\", ylabel=\"x\",\n xlim=(-0.05, 1.05), ylim=(-0.05, 1.05),\n title=(\"(a) Phase Portrait of i vs. x\", {\"size\": \"x-large\"}),\n annotate_time_points=\"endpoints\",\n annotate_time_points_labels=True);\nplot_time_profile(\n hill_sol, ax=ax4, legend=\"right outside\", \n title=(\"(d) Concentration Profiles\", {\"size\": \"x-large\"}));\nfig_5_7.tight_layout()\n```\n\n**Figure 5.7:** The transient response of the Hill reaction mechanism, for $k_i^+ = k_i^- = 100$, $k = 1$, $\\nu = 2$, $x_0 = 0$ and $e_0 = s_0 = i_0 = 1$. (a) The phase portraits of $s$ and $e$. (b) The phase portraits of $e$ and $x$. (c) The phase portraits of $i$ and $x$. (d) The concentration profiles.\n\n### Step 5: Numerical solutions for Hill-kinetics\nThe dynamic response of the Hill reaction mechanism is shown in Figure 5.7. The trace in the $s$ vs. $e$ phase portrait is L-shaped, showing a rapid initial equilibration of the enzyme to the inhibitor (the vertical line), followed by the slower conversion of the product (the horizontal line). These two reactions are naturally (stoichiometrically) decoupled and separated in time for the numerical values of the kinetic constants used. \n\nThe phase portraits for $e$ vs. $x$ and $i$ vs. $x$ are straight lines as given by the conservation Eq. (5.18), see Figure 5.7b,c. The two phase transient responses in Figure 5.7d shows the rapid equilibration of the enzyme and the slow conversion of substrate. Under these parameter conditions, the QEA should give good results. \n\n### Step 6: Estimating key parameters \nThere are two features of the Hill rate law that are of interest: \n\n#### Applicability of the quasi-equilibrium assumption. \nGiven the fact that the two reactions have characteristic times scales, their relative magnitude is of key concern when it comes to the justification of the QEA: \n\n$$\\begin{equation} a = (\\frac{\\text{characteristic binding time of the inhibitor}}{\\text{characteristic turnover time of the substrate}}) = \\frac{k}{k_i^+} \\tag{5.24} \\end{equation}$$\n\nIf $a$ is much smaller than unity, we would expect the QEA to be valid. In Figure 5.7, $a$ is 0.01. \n\n#### Regulatory characteristics \nThe Hill rate law has a sigmoidal shape with sensitivity of the reaction rate to the end product concentration as \n\n$$\\begin{equation} v_i = \\frac{\\partial v}{\\partial i} = \\frac{-\\nu v_m}{i} \\frac{(i/K_i)^{\\nu}}{[1 + (i/K_i)^{\\nu}]^2} \\tag{5.25} \\end{equation}$$\n\nwhich has a maximum \n\n$$\\begin{equation} v_i^* = -\\frac{v_m}{K_i}N(\\nu) \\ \\text{where} \\ N(\\nu) = \\frac{1}{4\\nu}(\\nu - 1)^{1 - 1/\\nu}(\\nu + 1)^{1 + 1/\\nu} \\tag{5.26} \\end{equation}$$\n\nat the inflection point \n\n$$\\begin{equation} i^* = K_i(\\frac{\\nu - 1}{\\nu + 1})^{1/\\nu} \\tag{5.27} \\end{equation}$$\n\nFor plausible values of $\\nu$, the function $N(\\nu)$ is on the order of unity (Table 5.1), and hence the maximum sensitivity $v_i^*$ is on the order of $(-v_m/K_i)$. The ratio $(K_i/v_m)$ can be interpreted as a time constant characterizing the inhibition process; \n\n$$\\begin{equation} t_i = \\frac{K_i}{v_m} = [\\frac{\\text{concentration}}{\\text{concentration/time}}]\\tag{5.28} \\end{equation}$$\n\nThis estimate represents an upper bound since the steady state concentration of $i$ can be different from $i^*$. The turnover of the substrate happens on a time scale defined by the rate constant $t_s = 1/k$. Thus, a key dimensionless property is \n\n$$\\begin{equation} b = \\frac{t_s}{t_i} = \\frac{1/k}{K_i/v_m} = \\frac{v_m}{kK_i} = \\frac{e_t}{K_i} \\tag{5.29} \\end{equation}$$\n\nTherefore, the dimensionless parameter $b$ can be interpreted as a ratio of time constants or as a ratio of concentration ranges. \n\n**Table 5.1:** The values of the function $N(\\nu)$ and $i^*/K_i$ at the inflection point.\n\n\n```python\ndef N(nu): # N(v)\n return (1/(4*nu))*((nu-1)**(1-1/nu))*((nu+1)**(1+1/nu))\n\ndef i_Ki(nu): # i*/Ki\n return ((nu-1)/(nu+1))**(1/nu)\n\ncols = [nu for nu in np.linspace(2, 5, 4)]\ntab_5_1 = pd.DataFrame([[round(N(nu), 2) for nu in cols], \n [round(i_Ki(nu), 2) for nu in cols]],\n index=['N($\\\\nu$)', '$i^{*}$$/K_{i}$'], columns=cols)\ntab_5_1.index.rename('$\\\\nu$', inplace=True)\ntab_5_1\n```\n\n\n\n\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
2.03.04.05.0
$\\nu$
N($\\nu$)0.650.841.071.30
$i^{*}$$/K_{i}$0.580.790.880.92
\n
\n\n\n\n## The Symmetry Model\nThe regulatory molecules are often chemically quite different than the substrate molecule. They thus often have a different binding site on the protein molecule than the catalytic site. It called an _allosteric site_. One of the earliest enzyme kinetic models that accounted for allosterism was the symmetry model (Monod, 1965), named after certain assumed symmetry properties of the subunits of the enzyme. It is a mechanistically realistic description of regulatory enzymes. An example of a multimeric regulatory enzyme is given in Figure 5.6.\n\n### The reaction mechanism for the symmetry model\nThe main chemical conversion in the symmetry model is as before and is described by Equation (5.14). The symmetry model postulates that the regulated enzyme lies naturally in two forms, $E$ and $X$, and is converted between the two states simply as \n\n$$\\begin{equation} E \\underset{k_{-}}{\\stackrel{k_+}{\\rightleftharpoons}} X \\tag{5.30} \\end{equation}$$\n\nThe equilibrium constant for this reaction, \n\n$$\\begin{equation} L = k_+/k_- = x/e \\tag{5.31} \\end{equation}$$\n\nhas a special name, the allosteric constant. Then $\\nu$ molecules of an inhibitor, $I$, can bind sequentially to $X$ as \n\n$$\\begin{equation} \\begin{matrix} {X} & {+} & {I} & {\\underset{k_i^-}{\\stackrel{\\nu k_i^+}{\\rightleftharpoons}}} & {X_1} \\\\ {X_1} & {+} & {I} & {\\underset{2 k_i^-}{\\stackrel{(\\nu-1) k_i^+}{\\rightleftharpoons}}} & {X_2} \\\\ {\\vdots} & {} & {} & {} & {\\vdots} \\\\ {X_{\\nu - 1}} & {+} & {I} & {\\underset{\\nu k_i^-}{\\stackrel{k_i^+}{\\rightleftharpoons}}} & {X_{\\nu}} \\\\ \\end{matrix}\\end{equation}$$\n$$\\tag{5.32}$$\n\nwhere the binding steps have the same dissociation constant, $K_i = k_i^- / k_i^+$. We will discuss the most common case of a tetramer here, i.e., $\\nu = 4$, see Figure 5.8.\n\n\n\n**Figure 5.8:** The reaction mechanisms for the symmetry model. The enzyme has four binding sites for the inhibitor.\n\n### Step 1: Dynamic mass balances for the symmetry model\nThe conversion rate of the substrate is \n\n$$\\begin{equation} v = kse \\tag{5.33} \\end{equation}$$\n\nwhereas the enzyme sequestration is characterized by the reaction rates \n\n$$\\begin{equation} \\begin{matrix} {v_1 = k^+e,} & {v_2 = k^-x,} & {v_3 = 4k_i^+xi,} \\\\ {v_4 = k_i^-x_1,} & {v_5 = 3 k_i^+x_1i,} & {v_6 = 2k_i^-x_2,} \\\\ {v_7 = k_i^+x_2i,} & {v_8 = k_i^-x_3,} & {v_9 = k_i^+x_3i,} \\\\ {} & {v_{10} = 4k_i^-x_4} & {} \\\\ \\end{matrix} \\end{equation}$$\n$$\\tag{5.34}$$\n\nThe dynamic mass balances on the various states of the enzyme are: \n\n$$\\begin{align} \\frac{de}{dt} &= -v_1 + v_2\\ \\ &\\tag{5.35a} \\\\ \\frac{dx}{dt} &= v_1 - v_2 - v_3 + v_4\\ \\ &\\tag{5.35b} \\\\ \\frac{di}{dt} &= -v_3 + v_4 - v_5 + v_6 - v_7 + v_8 - v_9 + v_{10}\\ \\ &\\tag{5.35c} \\\\ \\frac{dx_1}{dt} &= v_3 - v_4 - v_5 + v_6\\ \\ &\\tag{5.35d} \\\\ \\frac{dx_2}{dt} &= v_5 - v_6 - v_7 + v_8\\ \\ &\\tag{5.35e} \\\\ \\frac{dx_3}{dt} &= v_7 - v_8 - v_9 + v_{10}\\ \\ &\\tag{5.35f} \\\\ \\frac{dx_4}{dt} &= v_9 - v_{10}\\ \\ &\\tag{5.35g}\\\\ \\end{align}$$\n\n### Step 2: Finding the time invariants for the symmetry model\nThe stoichiometric matrix for $\\textbf{x} = (e, x, i, x_1, x_2, x_3, x_4)$ is a 7x10 matrix:\n\n$$\\begin{equation} \\textbf{S} = \\begin{pmatrix} {-1} & {1} & {0} & {0} & {0} & {0} & {0} & {0} & {0} & {0} \\\\ {1} & {-1} & {-1} & {1} & {0} & {0} & {0} & {0} & {0} & {0} \\\\ {0} & {0} & {-1} & {1} & {-1} & {1} & {-1} & {1} & {-1} & {1} \\\\ {0} & {0} & {1} & {-1} & {-1} & {1} & {0} & {0} & {0} & {0} \\\\ {0} & {0} & {0} & {0} & {1} & {-1} & {-1} & {1} & {0} & {0} \\\\ {0} & {0} & {0} & {0} & {0} & {0} & {1} & {-1} & {-1} & {1} \\\\ {0} & {0} & {0} & {0} & {0} & {0} & {0} & {0} & {1} & {-1} \\\\ \\end{pmatrix} \\end{equation}$$\n$$\\tag{5.36}$$\n\nthat has a rank of 5. Thus, there are two conservation relationships, for the enzyme: $e_0 = e + x + x_1 + x_2 + x_3 + x_4$;, and, for the inhibitor: $i_0 = i + x_1 + 2x_2 + 3x_3 + 4x_4$. If the dynamic mass balances on the substrate and product are taken into account, a third conservation $s_0 = s + p$ appears. \n\n### Step 3: Reducing the dynamic description for the symmetry model\nWe leave it to the reader to pick two dynamic variables from the full kinetic model as the dependent variables and then eliminate them from the dynamic description using the conservation relationships. The impetus for doing so algebraically becomes smaller as the number of differential equations grows. Most standard software packages will integrate a dynamically redundant set of differential equations and such substitution is not necessary to obtain the numerical solutions. \n\n### Step 4: Using simplifying kinetic assumptions to derive a rate law for the symmetry model\nThe serial binding of an inhibitor to X that has four binding sites is shown in Figure 5.8. The derivation of the rate law is comprised of four basic steps: \n\n1. Mass balance on enzyme: \n\n$$\\begin{equation} e_0 = e + x + x_1 + x_2 + x_3 + x_4 \\tag{5.37} \\end{equation}$$\n\n2. QEA for binding steps: \n\n$$\\begin{align} 4k_i^+ix = k_i^-x_1 \\ &\\Rightarrow \\ x_1 = \\frac{4}{1}x (i/K_i) = 4x(i/K_i)\\ \\ &\\tag{5.38a} \\\\ 3k_i^+ix_1 = 2k_i^-x_2 \\ &\\Rightarrow \\ x_2 = \\frac{3}{2}x_1(i/K_i) = 6x(i/K_i)^2 \\ \\ &\\tag{5.38b} \\\\ 2k_i^+ix_2 = 3k_i^-x_3 \\ &\\Rightarrow \\ x_3 = \\frac{2}{3}x_2(i/K_i) = 4x(i/K_i)^3 \\ \\ &\\tag{5.38ac} \\\\ k_i^+ix_3 = 4k_i^-x_4 \\ &\\Rightarrow \\ x_4 = \\frac{1}{4}x_3(i/K_i) = x(i/K_i)^4 \\ \\ &\\tag{5.38d} \\\\ \\end{align}$$\n\n3. Combine 1 and 2: \n\n$$\\begin{align} e_0 &= e + x + 4x(i/K_i) + 6x(i/K_i)^2 + 4x(i/K_i)^3 + x(i/K_i)^4 \\\\ &= e + x(1 + (i/K_i))^4 \\ \\text{where} \\ x=Le \\\\ &= e(1 + L(1 + (i/K_i)))^4 \\end{align}$$\n$$\\tag{5.39}$$\n\n4. Form the rate law: The reaction rate is given by: $v = kse$. We can rewrite the last part of Eq. (5.39) as: \n\n$$\\begin{equation} e = \\frac{e_0}{1 + L(1 + i/K_i)^4} \\tag{5.40} \\end{equation}$$\n\nleading to the rate law: \n\n$$\\begin{equation} v(s, i) = \\frac{ke_0s}{1 + L(1 + i/K_i)^4} \\tag{5.41} \\end{equation}$$\n\nThis rate law generalizes to: \n\n$$\\begin{equation} v(s, i) = \\frac{ke_0s}{1 + L(1 + i/K_i)^{\\nu}} = \\frac{v_m}{1 + L(1 + i/K_i)^{\\nu}} \\tag{5.42} \\end{equation}$$\n\nfor any $\\nu$. The reader can find the same key dimensionless groups as for the Hill rate law. Note again the fraction \n\n$$\\begin{equation} f_e = \\frac{e}{e_0} = \\frac{1}{1 + L(1 + i/K_i)^{\\nu}} \\tag{5.43} \\end{equation}$$\n\nthat describes the what fraction of the enzyme is in the catalytically active state. \n\n\n```python\nt0 = 0\ntf = 15\n\ndef symmetry(t, state_vars, *params):\n s, p, e, i, x, x1, x2, x3, x4 = state_vars\n k1, k_plus, k_minus, ki_plus, ki_minus = params\n # Enzyme Reaction Rates\n v1 = k_plus * e; v2 = k_minus * x;\n v3 = 4 * ki_plus * i * x; v4 = ki_minus * x1;\n v5 = 3 * ki_plus * i * x1; v6 = 2 * ki_minus * x2;\n v7 = 2 * ki_plus * i * x2; v8 = 3 * ki_minus * x3;\n v9 = ki_plus * i * x3; v10 = 4 * ki_minus * x4;\n # Differential equations to integrate\n diffeqs = [-k1 * s * e, # ds/dt\n k1 * s * e, # dp/dt\n -v1 + v2, # de/dt\n -v3 + v4 - v5 + v6 - v7 + v8 - v9 + v10, # di/dt\n v1 - v2 - v3 + v4, # dx/dt\n v3 - v4 - v5 + v6, # dx1/dt\n v5 - v6 - v7 + v8, # dx2/dt\n v7 - v8 - v9 + v10, # dx3/dt\n v9 - v10] # dx4/dt\n return diffeqs\n\n# Define initial conditions\ns0, p0, e0, i0, x0 = (1, 0, 1, 1, 0)\nx1_0, x2_0, x3_0, x4_0 = (0, 0, 0, 0)\n\n# Define paramters\nk1 = 1; \nk_plus, k_minus = (100, 100)\nki_plus, ki_minus = (2, 2)\nparams = [k1, k_plus,k_minus, ki_plus, ki_minus]\n\n# Obtain numerical solutions\nsol_obj = solve_ivp(fun=lambda t, state_vars: symmetry(t, state_vars, *params),\n t_span=(t0, tf), y0=[s0, p0, e0, i0, x0, x1_0, x2_0, x3_0, x4_0])\n# Store solutions into Solution Objects\nsol_dict = dict(zip([\"s\", \"p\", \"e\", \"i\", \"x\", \"x1\", \"x2\", \"x3\", \"x4\"], sol_obj.y))\n\nx_total = sum(sol_dict[k] for k in [\"x\", \"x1\", \"x2\", \"x3\", \"x4\"])\ni_bound = sum(i*sol_dict[k] for i, k in zip([1, 2, 3, 4], [\"x1\", \"x2\", \"x3\", \"x4\"]))\n\nsol_dict.update({\"x_total\": x_total, \"i_bound\": i_bound})\n\nsymmetry_sol = MassSolution(\n \"Symmetry\", solution_type=\"Conc\", data_dict=sol_dict,\n time=sol_obj.t, interpolate=False)\n```\n\n\n```python\nfig_5_9 = plt.figure(figsize=(10, 8))\ngs = fig_5_9.add_gridspec(nrows=2, ncols=2, width_ratios=[1, 1],\n height_ratios=[1, 1])\n\nax1 = fig_5_9.add_subplot(gs[0, 0])\nax2 = fig_5_9.add_subplot(gs[0, 1])\nax3 = fig_5_9.add_subplot(gs[1, 0])\nax4 = fig_5_9.add_subplot(gs[1, 1])\n\nplot_phase_portrait(\n symmetry_sol, x=\"s\", y=\"e\", ax=ax1, xlabel=\"s\", ylabel=\"e\",\n xlim=(-0.05, 1.05), ylim=(-0.05, 1.05),\n title=(\"(a) Phase Portrait of s vs. e\", {\"size\": \"x-large\"}),\n annotate_time_points=\"endpoints\",\n annotate_time_points_labels=True);\n\nplot_phase_portrait(\n symmetry_sol, x=\"e\", y=\"x_total\", ax=ax2, \n xlabel=\"e\", ylabel='x + x1 + x2 + x3 + x4',\n xlim=(-0.05, 1.05), ylim=(-0.05, 1.05),\n title=(\"(b) Phase Portrait of e vs. x_total\", {\"size\": \"x-large\"}),\n annotate_time_points=\"endpoints\",\n annotate_time_points_labels=True);\n\nplot_phase_portrait(\n symmetry_sol, x=\"i\", y=\"i_bound\", ax=ax3, \n xlabel=\"i\", ylabel='1*x1 + 2*x2 + 3*x3 + 4*x4',\n xlim=(-0.05, 1.05), ylim=(-0.05, 1.05),\n title=(\"(a) Phase Portrait of i vs. x\", {\"size\": \"x-large\"}),\n annotate_time_points=\"endpoints\",\n annotate_time_points_labels=True);\n\nplot_time_profile(\n symmetry_sol, observable=list(\n k for k in symmetry_sol.keys() if k not in [\n \"x\", \"x1\", \"x2\", \"x3\", \"x4\"]),\n ax=ax4, legend=\"right outside\",\n title=(\"(d) Concentration Profiles\", {\"size\": \"x-large\"}));\nfig_5_9.tight_layout()\n```\n\n**Figure 5.9:** The transient response of the symmetry model, for $k^+ = k^- = 100$, $k_i^+ = k_i^- = 2$, $k = 1$, $\\nu = 4$, $x_0 = x_{1, 0} = x_{2, 0} = x_{3, 0} = x_{4, 0} = 0$ and $e_0 = s_0 = i_0 = 1$. (a) The phase portraits of $s$ and $e$. (b) The phase portraits of $e$ and $(x + x_1 + x_2 + x_3 + x_4)$. (c) The phase portraits of $i$ and $(x_1 + 2x_2 + 3x_3 + 4x_4)$. (d) Concentration and pool profiles.\n\n### Step 5: Numerical solutions for the symmetry model\nThese equations can be simulated. Typically the conformational changes between $E$ and $X$ are fast as are the inhibitor binding steps relative to the catalysis rate. Numerical simulations were carried out for this situation and the results are plotted in Figure 5.9. \n\n* Figure 5.9a shows how the substrate-enzyme phase portrait is L-shaped showing that the sequestration of the enzyme in the inhibited form (the vertical line) is faster than the conversion of the substrate (the horizontal line). \n\n* Figure 5.9b shows the redistribution of the total enzyme among the active and inactive forms, that is, $e$ vs. $(x + x_1 + x_2 + x_3 + x_4)$. The fraction of the enzyme in the inactive form is about 0.29. \n\n* Figure 5.9c shows the redistribution of the inhibitor between the free and bound form; $i$ vs. $(x_1 + 2x_2 + 3x_3 + 4x_4)$. This panel shows that the fraction the inhibitor that is bound is high, 0.70. \n\n* Finally, Figure 5.9d show the transient changes in the concentrations and pools on the fast and slow time scales. Note that two natural aggregation variables appear: the total enzyme in the inactive form, and the total number of inhibitor molecules bound to the enzyme. \n\n## Scaling Dynamic Descriptions \nThe analysis of simple equations requires the \"proper frame of mind.\" In step 6 of the process of formulating rate laws, this notion is translated into quantitative measures. We need to scale the variables with respect to intrinsic reference scales and thereby cast our mathematical descriptions into appropriate coordinate systems. All parameters then aggregate into dimensionless property ratios that, if properly interpreted, have a clear physical significance. \n\n### The scaling process: \nThe examples above illustrate the decisive role of time constants and their use to analyze simple situations and to elucidate intrinsic reference scales. Identification of unimportant terms is sometimes more difficult and familiarity with a formal scaling procedure is useful. This procedure basically consists of four steps: \n\n1. Identify logical reference scales. This step is perhaps the most difficult. It relies partly on physical intuition, and the use of time constants is surprisingly powerful even when analyzing steady situations. \n\n2. Introduce reference scales into the equations and make the variables dimensionless. \n\n3. Collect the parameters into dimensionless property ratios. The number of dimensionless parameters is always the same and it is given by the well-known Buckingham Pi theorem. \n \n4. Interpret the results. The dimensionless groups that appear can normally be interpreted as ratios of the time constants, such as those discussed above. \n\nScaling of equations is typically only practiced for small models and for analysis purposes only. Numerical simulations of complex models are essentially always performed with absolute values of the variables. \n\n### The importance of intrinsic reference scales \nThe process by which the equations are made dimensionless is not unique. The 'correct' way of putting the equations into a dimensionless form, where judgments of relative orders of magnitude can be made, is called _scaling_. The scaling process is defined by Lin and Segel (Segel, 1974) as: \n\n\"...select intrinsic reference quantities so that each term in the dimensional equations transforms into a product of a constant dimensional factor which closely estimates the term's order of magnitude and a dimensionless factor of unit order of magnitude.\"\n\nIn other words, if one has an equation which is a sum of terms $T_i$ as: \n\n$$\\begin{equation} T_1 + T_2 + \\dots = 0 \\tag{5.44} \\end{equation}$$\n\none tries to scale the _variables_ involved so that they are of unit order of magnitude or \n\n$$\\begin{equation} t_i = \\frac{\\text{variable}_i}{\\text{intrinistic reference scale}_i} \\approx \\text{unit order of magnitude} \\tag{5.45} \\end{equation}$$\n\nIntroducing these dimensionless variables into equation (5.44) results in the dimensionless form: \n\n$$\\begin{equation} \\pi_1 t_1 + \\pi_2 t_2 + \\dots = 0 \\tag{5.44} \\end{equation}$$\n\nwhere the dimensionless multipliers, $\\pi_i$ are the dimensionless groups and they will indicate the order of magnitude of the product, $\\pi_it_i$. Once the equations are in this form, order of magnitude judgements can be made based on the dimensionless groups. \n\n## Summary \n\n* Enzymes are highly specialized catalysts that can dramatically accelerate the rates of biochemical reactions. \n\n* Reaction mechanisms are formulated for the chemical conversions carried out by enzymes in terms of elementary reactions. \n\n* Rate laws for enzyme reaction mechanisms are derived based on simplifying assumptions. \n\n* Two simplifying assumptions are commonly used: the quasi-steady state (QSSA) and the quasi-equilibrium assumptions (QEA). \n\n* The validity of the simplifying assumptions can be determined using scaling of the equations followed by mathematical and numerical analysis. \n\n* A number of rate laws have been developed for enzyme catalysis and for the regulation of enzymes. Only three reaction mechanisms were described in this chapter. \n\n$\\tiny{\\text{\u00a9 B. \u00d8. Palsson 2011;}\\ \\text{This publication is in copyright.}\\\\ \\text{Subject to statutory exception and to the provisions of relevant collective licensing agreements,}\\\\ \\text{no reproduction of any part may take place without the written permission of Cambridge University Press.}}$\n", "meta": {"hexsha": "02160b72e93130b376d6c4e4eb505a1c973648fd", "size": 275612, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/education/sb2/chapters/sb2_chapter5.ipynb", "max_stars_repo_name": "z-haiman/MASSpy", "max_stars_repo_head_hexsha": "aeeed1e3f9d1058e9485247a86f85cb94eeecbc9", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/education/sb2/chapters/sb2_chapter5.ipynb", "max_issues_repo_name": "z-haiman/MASSpy", "max_issues_repo_head_hexsha": "aeeed1e3f9d1058e9485247a86f85cb94eeecbc9", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/education/sb2/chapters/sb2_chapter5.ipynb", "max_forks_repo_name": "z-haiman/MASSpy", "max_forks_repo_head_hexsha": "aeeed1e3f9d1058e9485247a86f85cb94eeecbc9", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 234.1648258284, "max_line_length": 62120, "alphanum_fraction": 0.8780423204, "converted": true, "num_tokens": 14771, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.724870282120402, "lm_q1q2_score": 0.4431726234650594}} {"text": "# Lab 01 - Local Optimization\n## Tasks\n- Introduction to PyTorch - will be used throughout the course\n- Introduction to autograd\n- Implement gradient descent using autograd framework\n- Optimize a simple quadrupole triplet\n\n## Set up environment\n\n\n```python\n!pip install git+https://github.com/uspas/2021_optimization_and_ml --quiet\n```\n\n\n```python\n%reset -f\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n#matplotlib graphs will be included in your notebook, next to the code:\n%matplotlib inline\n\n#import toy accelerator package\nfrom uspas_ml.accelerator_toy_models import simple_lattices\n\nimport torch\n```\n\n## Introduction to PyTorch\nPyTorch is a machine learning library which implements autograd functionality (more on this in a bit). It will be useful for creating and training surrogate models. Many function implemented in numpy for `ndarrays` are also implemented in PyTorch with the `torch.Tensor` class. Here we show some examples of using `torch.Tensor`.\n\n\n```python\ndata = [[1, 2],[3, 4]]\nx_data = torch.tensor(data)\n```\n\n\n```python\nnp_array = np.array(data)\nx_np = torch.from_numpy(np_array)\n```\n\n\n```python\nx_ones = torch.ones_like(x_data) # retains the properties of x_data\nprint(f\"Ones Tensor: \\n {x_ones} \\n\")\n\nx_rand = torch.rand_like(x_data, dtype=torch.float) # overrides the datatype of x_data\nprint(f\"Random Tensor: \\n {x_rand} \\n\")\n```\n\n\n```python\nshape = (2,3,)\nrand_tensor = torch.rand(shape)\nones_tensor = torch.ones(shape)\nzeros_tensor = torch.zeros(shape)\n\nprint(f\"Random Tensor: \\n {rand_tensor} \\n\")\nprint(f\"Ones Tensor: \\n {ones_tensor} \\n\")\nprint(f\"Zeros Tensor: \\n {zeros_tensor}\")\n```\n\n\n```python\ntensor = torch.rand(3,4)\n\nprint(f\"Shape of tensor: {tensor.shape}\")\nprint(f\"Datatype of tensor: {tensor.dtype}\")\nprint(f\"Device tensor is stored on: {tensor.device}\")\n```\n\n\n```python\ntensor = torch.ones(4, 4)\nprint('First row: ',tensor[0])\nprint('First column: ', tensor[:, 0])\nprint('Last column:', tensor[..., -1])\ntensor[:,1] = 0\nprint(tensor)\n```\n\n\n```python\nt1 = torch.cat([tensor, tensor, tensor], dim=1)\nprint(t1)\n```\n\n\n```python\n# This computes the matrix multiplication between two tensors. y1, y2, y3 will have the same value\ny1 = tensor @ tensor.T\ny2 = tensor.matmul(tensor.T)\n\ny3 = torch.rand_like(tensor)\ntorch.matmul(tensor, tensor.T, out=y3)\n\n\n# This computes the element-wise product. z1, z2, z3 will have the same value\nz1 = tensor * tensor\nz2 = tensor.mul(tensor)\n\nz3 = torch.rand_like(tensor)\ntorch.mul(tensor, tensor, out=z3)\n```\n\n\n```python\n# This sums all the elements of a tensor\nagg = tensor.sum()\nagg_item = agg.item()\nprint(agg_item, type(agg_item))\n```\n\n\n```python\n# Add a scalar to all the elements of a tensor\nprint(tensor, \"\\n\")\ntensor.add_(5)\nprint(tensor)\ntensor += 5\nprint(tensor)\n```\n\n## A note regarding tensor shapes\nPaying attention to the shape of your tensors/numpy arrays is important, especially if you don't have a lot of experience programming. In general ML algorithms accept 2D arrays where the shape is given by `(number_of_samples, dimension_of_samples)`. Even if the input parameter is 1D the input shape should be `(number_of_samples, 1)`. \n\nFurthermore it is important to remember that slicing along a single axis returns a 1D tensor/array ie. `x[:,0]` returns an array shape of `(n_samples,)`.\n\nFunctions that will be useful for manipulating the shape of a tensor/array:\n- `.reshape()`\n- `.unsqueeze()`\n- `.squeeze()`\n- `.flatten()`\n\n\n## GPU Integration\nTo speed up many of the computations that we will do in this course we will move our data/models to a GPU. This works best when doing large matrix calculations or batch optimization routines. Keep in mind the time cost associated with moving data back and forth between a CPU and a GPU, this can slow down smaller scale calculations. \n\n\n```python\n# We move our tensor to the GPU if available\nif torch.cuda.is_available():\n tensor = tensor.to('cuda')\n```\n\n## Autograd with PyTorch\nIts hard to overstate the importance of Automatic Differentiation (commonly referred to Autograd) to the machine learning community. It forms the basis for training/optimizing surrogate models such as Gaussian Processes and Neural Networks.\n\nAutograd is a framework for doing calculations while keeping track of the derivatives associated with each calculation step. We will then be able to calculate the total derivative of say, the beam size vs. a quadrupole strength parameter. This will be used to train surrogate models with derivative based optimization methods (gradient descent and its descendants).\n\nAutograd works by creating what is known by a computational graph (an example of which is below). The graph has a set of what are known as leaf nodes (or tensors) $a,w_1,w_2,w_3,w_4$ which represent input variables that we choose. We wish to compute how changing these variables will effect the result $L$. Each of the lines on the computation graph represent a single calculation which we will need the derivate of to calculate the total derivative.\n\n\nOur calculation will involve a number of discrete steps, each which create intermediate variables seen below:\n\n$\n\\begin{align}\n b & = w_1 * a\\\\\n c & = w_2 * a\\\\\n d & = w_3*b + w_4*c\\\\\n L & = 10 -d\n\\end{align}\n$\n\nIn order to calculate the total derivative of L with respect to say $w_1$ we use the chain rule\n\n$$\n\\frac{dL}{dw_1} = \\frac{\\partial L}{\\partial d}\\frac{\\partial d}{\\partial b}\\frac{\\partial b}{\\partial w_1}\n$$\n\nUsing PyTorch the partial derivatives for each calculation are easily calculated for a wide variety of calculations and variable manipulations that we need for model building and programming. Furthermore, this is easily done with thousands of variables and/or manipulations at a time! Thanks PyTorch!\n\nLets take a look at how this can be done in practice. We start with defining some variables that we will need the derivatives with respect to by specifying `requires_grad = True` during creation.\n\n\n```python\n#define variables (the chosen values here are arbitrary)\na = torch.tensor([2.], requires_grad=True)\nw1 = torch.tensor([3.], requires_grad=True)\nw2 = torch.tensor([1.], requires_grad=True)\nw3 = torch.tensor([0.5], requires_grad=True)\nw4 = torch.tensor([-2.], requires_grad=True)\n\n#do the calculation\nb = w1*a\nc = w2*a\nd = w3*b + w4*c\nL = 10 - d\n```\n\nTo calculate the derivatives we call `L.backward()`. After calling `backward()` the gradients are stored in the `.grad` attribute of each variable.\n\n\n```python\nL.backward()\nprint(a.grad)\nprint(w4.grad)\n```\n\nNote that we are calculating the gradient of $a$ and $w_4$ evaluated at their currently set values. In other words the grad attribute gives us\n\n$$\n\\frac{dL}{da}\\Big\\rvert_{a = 2}\n$$\n\n
\n \n**Task:** \n Calculate (by hand) the quantity $\\frac{dL}{dw_1}$, and check that this is consistent with the result given by `w1.grad`.\n \n
\n\n\n```python\nprint(w1.grad) # Check that this is consistent with your analytical calculation\n```\n\nNote that, if you redo the calculation a second time, the gradient will **accumulate**:\n\n\n```python\n# redo the calculation a second time\nb = w1*a\nc = w2*a\nd = w3*b + w4*c\nL = 10 - d\n# call backward to calculate the gradients a second time\nL.backward()\n# print gradient\nprint(w1.grad)\n```\n\nIf you want to avoid this, you can call `.grad.zero_()`\n\n\n```python\n# zero the gradient\nw1.grad.zero_()\n# redo the calculation a third time\nb = w1*a\nc = w2*a\nd = w3*b + w4*c\nL = 10 - d\n# call backward to calculate the gradients a third time\nL.backward()\n# print gradient\nprint(w1.grad)\n```\n\nNote that you can also ask `pytorch` to temporarily stop tracking the gradients for certain operations:\n\n\n```python\nwith torch.no_grad():\n w1 = 2*w1\n```\n\n## Gradient descent\n\n
\n \n**Task:** \n Now its your turn! Implement a gradient descent algortihm to minimize a simple test function below using autograd. Complete the implementation of the function `gradient_descent` below, to perform 100 iterations of gradient descent.\n\n- implement the gradient descent algorithm using the above-mentioned features of pytorch\n- record the values of `X0` as you go and generate a plot that shows how gradient descent travels through input space\n- given the form of `test_function`, which value do you expect `X0` to converge to?\n- redo the plot for 3 different values of the step size `alpha` [0.005, 0.1, 1.0 ]\n \n
\n\n\n```python\ndef test_function(X):\n \"\"\"\n Function to be optimized.\n \n Input: X, a pytorch tensor of shape (2,)\n Output: a single scalar\n \"\"\"\n return (X[0] - 0.5)**2 + X[1]**2\n\ndef gradient_descent(X0, function, alpha=0.1, n_iterations=100):\n \"\"\"\n Performs 100 iterations of gradient descent\n \n Inputs:\n X0: a pytorch tensor to be modified in-place by the gradient descent algorithm\n function: a callable, that takes X0 as input\n alpha: step size for the gradient descent algorithm\n n_iterations: number of iterations of gradient descent\n \n Returns:\n history_X: array of values of X\n history_f: array of values of f\n \"\"\"\n history_X = []\n history_f = []\n for iteration in range(n_iterations): # iterations of gradient descent\n f = function(X0)\n \n # Your code that modifies X0 in-place here\n # ...\n \n # Keep a copy in history (do not modify these lines)\n history_X.append( X0.detach().numpy().copy() ) \n history_f.append( f.detach().numpy().copy() )\n \n return np.array(history_X), np.array(history_f)\n```\n\n\n```python\n# Test your code by executing the code below\nX0 = torch.tensor([0.9,0.9], requires_grad=True)\nalpha = 0.1\n\nhistory_X, history_f = gradient_descent( X0, test_function, alpha)\n\n# Plot the trajectory of the points\nplt.scatter( history_X[:,0], history_X[:,1], c=np.arange(len(history_X)))\nplt.xlim(-1, 1)\nplt.ylim(-1, 1)\n```\n\n## Accelerator example\nNow we will use gradient descent to optimize the strengths of quadrupoles in a triplet configuration. \n\n\n\n\nThe function below creates a quadrupole triplet as defined in simple_lattices.py (see the accelerator toy models) and returns $\\sqrt{\\sigma_x^2 + \\sigma_y^2}$ for an initial beam matrix defined in the function below.\n\nRemember that the beam matrix takes the form of \n\n$\n\\Sigma = \n\\begin{bmatrix}\n\\sigma_{11} & \\sigma_{12} & \\sigma_{13} & \\sigma_{14} & \\sigma_{15} & \\sigma_{16} \\\\\n\\sigma_{21} & \\sigma_{22} & \\sigma_{23} & \\sigma_{24} & \\sigma_{25} & \\sigma_{26} \\\\\n\\sigma_{31} & \\sigma_{32} & \\sigma_{33} & \\sigma_{34} & \\sigma_{35} & \\sigma_{36} \\\\\n\\sigma_{41} & \\sigma_{42} & \\sigma_{43} & \\sigma_{44} & \\sigma_{45} & \\sigma_{46} \\\\\n\\sigma_{51} & \\sigma_{52} & \\sigma_{53} & \\sigma_{54} & \\sigma_{55} & \\sigma_{56} \\\\\n\\sigma_{61} & \\sigma_{62} & \\sigma_{63} & \\sigma_{64} & \\sigma_{65} & \\sigma_{66} \\\\\n\\end{bmatrix}\n$\n\nwhere for example $\\sigma_{11} = \\sigma_x^2, \\sigma_{12} = \\sigma_{x,x'}$ etc.\n\nWe start with the following initial beam matrix.\n$\n\\Sigma = \n\\begin{bmatrix}\n1e-6 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 1e-8 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 1e-6 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1e-8 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1e-8 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1e-8 \\\\\n\\end{bmatrix}\n$\n\n\n```python\ndef beamsize(K , noise = 0.0):\n '''\n calculate sqrt(sigma_x^2 + sigma_y^2)\n \n K : magnetic strength of each quad magnet, torch.tensor, shape (3,)\n noise : rms noise level of measurement, float (default 0.0)\n \n Returns: the beam size in mm\n '''\n \n #generate initial beam matrix\n # - x/y/z geometric emittance of 1.0e-8 m-rad\n init_beam_matrix = torch.eye(6) * 1.0e-8\n\n #set x_rms beam size to 1 mm and rms divergence to 0.1 mrad\n init_beam_matrix[0,0] = 1.0e-3 ** 2 \n init_beam_matrix[1,1] = 1.0e-4 ** 2 \n init_beam_matrix[2,2] = 1.0e-3 ** 2 \n init_beam_matrix[3,3] = 1.0e-4 ** 2 \n \n #create accelerator lattice object with one quad and a drift\n line = simple_lattices.create_triplet(K)\n \n #propagate beam matrix\n final_beam_matrix = line.propagate_beam_matrix(init_beam_matrix, noise)\n return 1.e3*torch.sqrt(final_beam_matrix[0,0] + final_beam_matrix[2,2]) #Convert to mm\n```\n\n\n```python\n#example usage\nK = torch.tensor([1.0, -1.0, 0.5], requires_grad = True)\nprint(K.requires_grad)\nsize = beamsize(K)\nprint(size)\nsize.backward()\nprint(K.grad)\n```\n\n## Gradient Descent on the Accelerator Example\n\n
\n \n**Task:**\n Now apply gradient descent to optimizing the quadrupole strengths to minimize the total final beam size using the above function. Write the corresponding code below. (Feel free to reuse the function `gradient_descent` and to tune `alpha` and `n_iterations` to reach convergence - which you can assess by plotting the returned history ; do not hesitate to use a large value for `alpha`.)\n \nWhat is the value of the beamsize that is obtained at the end of the optimization?\n \n
\n\n\n```python\nK = torch.tensor([1.0, -1.0, 0.5], requires_grad = True)\n# Your code here\n# ...\n```\n\n**Your answer here:** (What is the value of the beamsize that is obtained at the end of the optimization?)\n\n## Gradient descent with numerical differentiation\n\n
\n \n**Homework:**\n Try to minimize again the beam size, but using numerical differentiation instead of autograd. In this case, you will **not** call the `.backward` and `.grad` functions, but instead you will need to calculate the gradient numerically: vary in the input `K` by `h=1.e-4` in each direction, in order to compute the gradient of `beam_size`.\n \n(Note that, because we do not call `.backward`, we also do not need to set `requires_grad = True`.)\n \nDoes the algorithm reach the same final value, for the beamsize?\n
\n\n\n```python\ndef gradient_descent_nd(X0, function, alpha=0.1, n_iterations=100):\n \"\"\"\n Performs n_iterations iterations of gradient descent, \n with numerical differentiation\n \n Inputs:\n X0: a pytorch tensor to be modified in-place by the gradient descent algorithm\n function: a callable, that takes X0 as input\n alpha: step size for the gradient descent algorithm\n n_iterations: number of iterations of gradient descent\n \n Returns:\n history_X: array of values of X\n history_f: array of values of f\n \"\"\"\n history_X = []\n history_f = []\n for iteration in range(n_iterations): # iterations of gradient descent\n f = function(X0)\n \n # Your code that modifies X0 in-place here\n # ...\n \n # Keep a copy in history (do not modify these lines)\n history_X.append( X0.detach().numpy().copy() ) \n history_f.append( f.detach().numpy().copy() )\n \n return np.array(history_X), np.array(history_f)\n```\n\n\n```python\n# Your code here: call `gradient_descent_nd` and plot the corresponding `history_f`\n```\n\n**Your answer here:** (Does the algorithm reach the same value of `K`?)\n\n## Gradient descent with noise\n\nThe function below emulates noise in the beam size measurement. \nCall this function several times for a given value of `K`. \n\n\n```python\ndef beamsize_with_noise(K):\n return beamsize(K, noise=2.e-5)\n```\n\n\n```python\n# Call this function several times to get a sense of how much \n# the noise is changing the result\nbeamsize_with_noise(K)\n```\n\n
\n \n**Homework:**\n Now perform the same optimization as before, but on the function `beam_size_with_noise` instead of `beam_size`. Does the analytical gradient descent reach approximately the same final value of the beamsize as before? Does the version of gradient descent with finite difference reach the same value too? Why? (Plot the history of the beamsize in both cases.) \n
\n\n\n```python\n# Your code here\n# ...\n```\n\n**Your answer here:** (Does the analytical gradient descent reach approximately the same final value of the beamsize as before? Does the version of gradient descent with finite difference reach the same value too? Why?)\n", "meta": {"hexsha": "31df7ee06c0c47de2e7b9512bc1311963c55c7d9", "size": 127893, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "labs/lab_01/lab_01.ipynb", "max_stars_repo_name": "uspas/2021_optimization_and_ml", "max_stars_repo_head_hexsha": "7280d3c1bbed47058f9fdaaefd8f29d28f4d6898", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2021-06-21T19:45:49.000Z", "max_stars_repo_stars_event_max_datetime": "2021-07-15T00:58:56.000Z", "max_issues_repo_path": "labs/lab_01/lab_01.ipynb", "max_issues_repo_name": "uspas/2021_optimization_and_ml", "max_issues_repo_head_hexsha": "7280d3c1bbed47058f9fdaaefd8f29d28f4d6898", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "labs/lab_01/lab_01.ipynb", "max_forks_repo_name": "uspas/2021_optimization_and_ml", "max_forks_repo_head_hexsha": "7280d3c1bbed47058f9fdaaefd8f29d28f4d6898", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2021-06-25T19:21:44.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-03T03:58:38.000Z", "avg_line_length": 166.9621409922, "max_line_length": 82600, "alphanum_fraction": 0.8908071591, "converted": true, "num_tokens": 4206, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819591324418, "lm_q2_score": 0.7248702880639791, "lm_q1q2_score": 0.443172616833453}} {"text": "# Homework and bake-off: word-level entailment with neural networks\n\n\n```python\n__author__ = \"Christopher Potts\"\n__version__ = \"CS224u, Stanford, Fall 2020\"\n```\n\n## Contents\n\n1. [Overview](#Overview)\n1. [Set-up](#Set-up)\n1. [Data](#Data)\n1. [Baseline](#Baseline)\n 1. [Representing words: vector_func](#Representing-words:-vector_func)\n 1. [Combining words into inputs: vector_combo_func](#Combining-words-into-inputs:-vector_combo_func)\n 1. [Classifier model](#Classifier-model)\n 1. [Baseline results](#Baseline-results)\n1. [Homework questions](#Homework-questions)\n 1. [Hypothesis-only baseline [2 points]](#Hypothesis-only-baseline-[2-points])\n 1. [Alternatives to concatenation [2 points]](#Alternatives-to-concatenation-[2-points])\n 1. [A deeper network [2 points]](#A-deeper-network-[2-points])\n 1. [Your original system [3 points]](#Your-original-system-[3-points])\n1. [Bake-off [1 point]](#Bake-off-[1-point])\n\n## Overview\n\nThe general problem is word-level natural language inference. Training examples are pairs of words $(w_{L}, w_{R}), y$ with $y = 1$ if $w_{L}$ entails $w_{R}$, otherwise $0$.\n\nThe homework questions below ask you to define baseline models for this and develop your own system for entry in the bake-off, which will take place on a held-out test-set distributed at the start of the bake-off. (Thus, all the data you have available for development is available for training your final system before the bake-off begins.)\n\n## Set-up\n\nSee [the first notebook in this unit](nli_01_task_and_data.ipynb) for set-up instructions.\n\n\n```python\nfrom collections import defaultdict\nimport json\nimport numpy as np\nimport os\nimport pandas as pd\nfrom torch_shallow_neural_classifier import TorchShallowNeuralClassifier\nimport nli\nimport utils\n```\n\n\n```python\nDATA_HOME = 'data'\n\nNLIDATA_HOME = os.path.join(DATA_HOME, 'nlidata')\n\nwordentail_filename = os.path.join(\n NLIDATA_HOME, 'nli_wordentail_bakeoff_data.json')\n\nGLOVE_HOME = os.path.join(DATA_HOME, 'glove.6B')\n```\n\n## Data\n\nI've processed the data into a train/dev split that is designed to put some pressure on our models to actually learn these semantic relations, as opposed to exploiting regularities in the sample. \n\nThe defining feature of the dataset is that the `train` and `dev` __vocabularies__ are disjoint. That is, if a word `w` appears in a training pair, it does not occur in any text pair. It follows from this that there are also no word-pairs shared between train and dev, as you would expect. This should require your models to learn abstract relationships, as opposed to memorizing incidental properties of individual words in the dataset.\n\n\n```python\nwith open(wordentail_filename) as f:\n wordentail_data = json.load(f)\n```\n\nThe keys are the splits plus a list giving the vocabulary for the entire dataset:\n\n\n```python\nwordentail_data.keys()\n```\n\n\n\n\n dict_keys(['dev', 'train', 'vocab'])\n\n\n\n\n```python\nwordentail_data['train'][: 5]\n```\n\n\n\n\n [[['abode', 'house'], 1],\n [['abortion', 'anaemia'], 0],\n [['abortion', 'aneurysm'], 0],\n [['abortion', 'blindness'], 0],\n [['abortion', 'deafness'], 0]]\n\n\n\n\n```python\nnli.get_vocab_overlap_size(wordentail_data)\n```\n\n\n\n\n 0\n\n\n\nBecause no words are shared between `train` and `dev`, no pairs are either:\n\n\n```python\nnli.get_pair_overlap_size(wordentail_data)\n```\n\n\n\n\n 0\n\n\n\nHere is the label distribution:\n\n\n```python\npd.DataFrame(wordentail_data['train'])[1].value_counts()\n```\n\n\n\n\n 0 7000\n 1 1283\n Name: 1, dtype: int64\n\n\n\nThis is a challenging label distribution \u2013 there are more than 5 times as more non-entailment cases as entailment cases.\n\n## Baseline\n\nEven in deep learning, __feature representation is vital and requires care!__ For our task, feature representation has two parts: representing the individual words and combining those representations into a single network input.\n\n### Representing words: vector_func\n\nLet's consider two baseline word representations methods:\n\n1. Random vectors (as returned by `utils.randvec`).\n1. 50-dimensional GloVe representations.\n\n\n```python\ndef randvec(w, n=50, lower=-1.0, upper=1.0):\n \"\"\"Returns a random vector of length `n`. `w` is ignored.\"\"\"\n return utils.randvec(n=n, lower=lower, upper=upper)\n```\n\n\n```python\ndef load_glove50():\n glove_src = os.path.join(GLOVE_HOME, 'glove.6B.50d.txt')\n # Creates a dict mapping strings (words) to GloVe vectors:\n GLOVE = utils.glove2dict(glove_src)\n return GLOVE\n\nGLOVE = load_glove50()\n\ndef glove_vec(w):\n \"\"\"Return `w`'s GloVe representation if available, else return\n a random vector.\"\"\"\n return GLOVE.get(w, randvec(w, n=50))\n```\n\n### Combining words into inputs: vector_combo_func\n\nHere we decide how to combine the two word vectors into a single representation. In more detail, where `u` is a vector representation of the left word and `v` is a vector representation of the right word, we need a function `vector_combo_func` such that `vector_combo_func(u, v)` returns a new input vector `z` of dimension `m`. A simple example is concatenation:\n\n\n```python\ndef vec_concatenate(u, v):\n \"\"\"Concatenate np.array instances `u` and `v` into a new np.array\"\"\"\n return np.concatenate((u, v))\n```\n\n`vector_combo_func` could instead be vector average, vector difference, etc. (even combinations of those) \u2013 there's lots of space for experimentation here; [homework question 2](#Alternatives-to-concatenation-[2-points]) below pushes you to do some exploration.\n\n### Classifier model\n\nFor a baseline model, I chose `TorchShallowNeuralClassifier`:\n\n\n```python\nnet = TorchShallowNeuralClassifier(early_stopping=True)\n```\n\n### Baseline results\n\nThe following puts the above pieces together, using `vector_func=glove_vec`, since `vector_func=randvec` seems so hopelessly misguided for our problem!\n\n\n```python\nbaseline_experiment = nli.wordentail_experiment(\n train_data=wordentail_data['train'],\n assess_data=wordentail_data['dev'],\n model=net,\n vector_func=glove_vec,\n vector_combo_func=vec_concatenate)\n```\n\n Stopping after epoch 140. Validation score did not improve by tol=1e-05 for more than 10 epochs. Final error is 1.023185983300209\n\n precision recall f1-score support\n \n 0 0.870 0.950 0.908 1732\n 1 0.506 0.263 0.346 334\n \n accuracy 0.839 2066\n macro avg 0.688 0.607 0.627 2066\n weighted avg 0.811 0.839 0.818 2066\n \n\n\n## Homework questions\n\nPlease embed your homework responses in this notebook, and do not delete any cells from the notebook. (You are free to add as many cells as you like as part of your responses.)\n\n### Hypothesis-only baseline [2 points]\n\nDuring our discussion of SNLI and MultiNLI, we noted that a number of research teams have shown that hypothesis-only baselines for NLI tasks can be remarkably robust. This question asks you to explore briefly how this baseline affects our task.\n\nFor this problem, submit two functions:\n\n1. A `vector_combo_func` function called `hypothesis_only` that simply throws away the premise, using the unmodified hypothesis (second) vector as its representation of the example.\n\n1. A function called `run_hypothesis_only_evaluation` that does the following:\n 1. Loops over the two `vector_combo_func` values `vec_concatenate` and `hypothesis_only`, calling `nli.wordentail_experiment` to train on the 'train' portion and assess on the 'dev' portion, with `glove_vec` as the `vector_func`. So that the results are consistent, use an `sklearn.linear_model.LogisticRegression` with default parameters as the model.\n 1. Returns a `dict` mapping `function_name` strings to the 'macro-F1' score for that pair, as returned by the call to `nli.wordentail_experiment`. (Tip: you can get the `str` name of, e.g., `hypothesis_only` with `hypothesis_only.__name__`.)\n \nThe functions `test_hypothesis_only` and `test_run_hypothesis_only_evaluation` will help ensure that your functions have the desired logic.\n\n\n```python\ndef hypothesis_only(u, v):\n ##### YOUR CODE HERE\n return v\n\ndef run_hypothesis_only_evaluation():\n ##### YOUR CODE HERE\n from sklearn.linear_model import LogisticRegression\n result = dict()\n lr = LogisticRegression()\n for combo in [hypothesis_only, vec_concatenate]:\n experiment = nli.wordentail_experiment(\n train_data=wordentail_data['train'],\n assess_data=wordentail_data['dev'],\n model=lr,\n vector_func=glove_vec,\n vector_combo_func=combo)\n result[combo.__name__] = experiment['macro-F1']\n return result\n\n```\n\n\n```python\ndef test_hypothesis_only(hypothesis_only):\n v = hypothesis_only(1, 2)\n assert v == 2\n```\n\n\n```python\ntest_hypothesis_only(hypothesis_only)\n```\n\n\n```python\ndef test_run_hypothesis_only_evaluation(run_hypothesis_only_evaluation):\n results = run_hypothesis_only_evaluation()\n assert all(x in results for x in ('hypothesis_only', 'vec_concatenate')), \\\n (\"The return value of `run_hypothesis_only_evaluation` does not \"\n \"have the intended kind of keys.\")\n assert isinstance(results['vec_concatenate'], float), \\\n (\"The values of the `run_hypothesis_only_evaluation` result \"\n \"should be floats.\")\n```\n\n\n```python\ntest_run_hypothesis_only_evaluation(run_hypothesis_only_evaluation)\n```\n\n precision recall f1-score support\n \n 0 0.853 0.971 0.908 1732\n 1 0.462 0.129 0.201 334\n \n accuracy 0.835 2066\n macro avg 0.657 0.550 0.555 2066\n weighted avg 0.789 0.835 0.794 2066\n \n precision recall f1-score support\n \n 0 0.863 0.954 0.906 1732\n 1 0.473 0.213 0.293 334\n \n accuracy 0.834 2066\n macro avg 0.668 0.583 0.600 2066\n weighted avg 0.800 0.834 0.807 2066\n \n\n\n### Alternatives to concatenation [2 points]\n\nWe've so far just used vector concatenation to represent the premise and hypothesis words. This question asks you to explore two simple alternative:\n\n1. Write a function `vec_diff` that, for a given pair of vector inputs `u` and `v`, returns the element-wise difference between `u` and `v`.\n\n1. Write a function `vec_max` that, for a given pair of vector inputs `u` and `v`, returns the element-wise max values between `u` and `v`.\n\nYou needn't include your uses of `nli.wordentail_experiment` with these functions, but we assume you'll be curious to see how they do!\n\n\n```python\ndef vec_diff(u, v):\n ##### YOUR CODE HERE\n return u - v\n\n\ndef vec_max(u, v):\n ##### YOUR CODE HERE\n return np.maximum(u, v)\n\n\n```\n\n\n```python\ndef test_vec_diff(vec_diff):\n u = np.array([10.2, 8.1])\n v = np.array([1.2, -7.1])\n result = vec_diff(u, v)\n expected = np.array([9.0, 15.2])\n assert np.array_equal(result, expected), \\\n \"Expected {}; got {}\".format(expected, result)\n```\n\n\n```python\ntest_vec_diff(vec_diff)\n```\n\n\n```python\ndef test_vec_max(vec_max):\n u = np.array([1.2, 8.1])\n v = np.array([10.2, -7.1])\n result = vec_max(u, v)\n expected = np.array([10.2, 8.1])\n assert np.array_equal(result, expected), \\\n \"Expected {}; got {}\".format(expected, result)\n```\n\n\n```python\ntest_vec_max(vec_max)\n```\n\n### A deeper network [2 points]\n\nIt is very easy to subclass `TorchShallowNeuralClassifier` if all you want to do is change the network graph: all you have to do is write a new `build_graph`. If your graph has new arguments that the user might want to set, then you should also redefine `__init__` so that these values are accepted and set as attributes.\n\nFor this question, please subclass `TorchShallowNeuralClassifier` so that it defines the following graph:\n\n$$\\begin{align}\nh_{1} &= xW_{1} + b_{1} \\\\\nr_{1} &= \\textbf{Bernoulli}(1 - \\textbf{dropout_prob}, n) \\\\\nd_{1} &= r_1 * h_{1} \\\\\nh_{2} &= f(d_{1}) \\\\\nh_{3} &= h_{2}W_{2} + b_{2}\n\\end{align}$$\n\nHere, $r_{1}$ and $d_{1}$ define a dropout layer: $r_{1}$ is a random binary vector of dimension $n$, where the probability of a value being $1$ is given by $1 - \\textbf{dropout_prob}$. $r_{1}$ is multiplied element-wise by our first hidden representation, thereby zeroing out some of the values. The result is fed to the user's activation function $f$, and the result of that is fed through another linear layer to produce $h_{3}$. (Inside `TorchShallowNeuralClassifier`, $h_{3}$ is the basis for a softmax classifier; no activation function is applied to it because the softmax scaling is handled internally by the loss function.)\n\nFor your implementation, please use `nn.Sequential`, `nn.Linear`, and `nn.Dropout` to define the required layers.\n\nFor comparison, using this notation, `TorchShallowNeuralClassifier` defines the following graph:\n\n$$\\begin{align}\nh_{1} &= xW_{1} + b_{1} \\\\\nh_{2} &= f(h_{1}) \\\\\nh_{3} &= h_{2}W_{2} + b_{2}\n\\end{align}$$\n\nThe following code starts this sub-class for you, so that you can concentrate on `build_graph`. Be sure to make use of `self.dropout_prob`.\n\nFor this problem, submit just your completed `TorchDeepNeuralClassifier`. You needn't evaluate it, though we assume you will be keen to do that!\n\nYou can use `test_TorchDeepNeuralClassifier` to ensure that your network has the intended structure.\n\n\n```python\nimport torch.nn as nn\n\nclass TorchDeepNeuralClassifier(TorchShallowNeuralClassifier):\n def __init__(self, dropout_prob=0.7, **kwargs):\n self.dropout_prob = dropout_prob\n super().__init__(**kwargs)\n\n def build_graph(self):\n \"\"\"Complete this method!\n\n Returns\n -------\n an `nn.Module` instance, which can be a free-standing class you\n write yourself, as in `torch_rnn_classifier`, or the outpiut of\n `nn.Sequential`, as in `torch_shallow_neural_classifier`.\n\n \"\"\"\n ##### YOUR CODE HERE\n return nn.Sequential(\n nn.Linear(self.input_dim, self.hidden_dim),\n nn.Dropout(p=self.dropout_prob),\n self.hidden_activation,\n nn.Linear(self.hidden_dim, self.n_classes_))\n\n\n```\n\n\n```python\ndef test_TorchDeepNeuralClassifier(TorchDeepNeuralClassifier):\n dropout_prob = 0.55\n assert hasattr(TorchDeepNeuralClassifier(), \"dropout_prob\"), \\\n \"TorchDeepNeuralClassifier must have an attribute `dropout_prob`.\"\n try:\n inst = TorchDeepNeuralClassifier(dropout_prob=dropout_prob)\n except TypeError:\n raise TypeError(\"TorchDeepNeuralClassifier must allow the user \"\n \"to set `dropout_prob` on initialization\")\n inst.input_dim = 10\n inst.n_classes_ = 5\n graph = inst.build_graph()\n assert len(graph) == 4, \\\n \"The graph should have 4 layers; yours has {}\".format(len(graph))\n expected = {\n 0: 'Linear',\n 1: 'Dropout',\n 2: 'Tanh',\n 3: 'Linear'}\n for i, label in expected.items():\n name = graph[i].__class__.__name__\n assert label in name, \\\n (\"The {} layer of the graph should be a {} layer; \"\n \"yours is {}\".format(i, label, name))\n assert graph[1].p == dropout_prob, \\\n (\"The user's value for `dropout_prob` should be the value of \"\n \"`p` for the Dropout layer.\")\n```\n\n\n```python\ntest_TorchDeepNeuralClassifier(TorchDeepNeuralClassifier)\n```\n\n### Your original system [3 points]\n\nThis is a simple dataset, but its \"word-disjoint\" nature ensures that it's a challenging one, and there are lots of modeling strategies one might adopt. \n\nYou are free to do whatever you like. We require only that your system differ in some way from those defined in the preceding questions. They don't have to be completely different, though. For example, you might want to stick with the model but represent examples differently, or the reverse.\n\nYou are free to use different pretrained word vectors and the like.\n\nPlease embed your code in this notebook so that we can rerun it.\n\nIn the cell below, please provide a brief technical description of your original system, so that the teaching team can gain an understanding of what it does. This will help us to understand your code and analyze all the submissions to identify patterns and strategies. We also ask that you report the best score your system got during development, just to help us understand how systems performed overall.\n\n\n```python\n# PLEASE MAKE SURE TO INCLUDE THE FOLLOWING BETWEEN THE START AND STOP COMMENTS:\n# 1) Textual description of your system.\n# 2) The code for your original system.\n# 3) The score achieved by your system in place of MY_NUMBER.\n# With no other changes to that line.\n# You should report your score as a decimal value <=1.0\n# PLEASE MAKE SURE NOT TO DELETE OR EDIT THE START AND STOP COMMENTS\n\n# NOTE: MODULES, CODE AND DATASETS REQUIRED FOR YOUR ORIGINAL SYSTEM \n# SHOULD BE ADDED BELOW THE 'IS_GRADESCOPE_ENV' CHECK CONDITION. DOING\n# SO ABOVE THE CHECK MAY CAUSE THE AUTOGRADER TO FAIL.\n\n# START COMMENT: Enter your system description in this cell.\n# The system uses:\n# The Glove 50 embedded space used previously as we didn't find any significant improvement in using other \n# embedded spaces. \n#\n# vec_concatenate to combine the 2 vector words in a single representation. If the hypothesis belongs to the \n# premise hypernyms, hyponym, member_holonyms, antonyms, derivationally_related_forms, pertainyms as defined \n# by Wordnet, we translate the relevant representation by a value which is halfway between the \n# mean of the sums of the rows of the Glove embedded space and its max. For that purpose we split evenly a 50 array \n# between these features. So that we have a 150 array, 50 for word1, 50 for word 2 and 50 for the different wordnet \n# features.\n#\n# Finally, we use a 3 hidden layers neural network classifier defined in the class TorchDeepNCMultipleLayers. \n#\n# My peak score was: 0.679\n\nif 'IS_GRADESCOPE_ENV' not in os.environ:\n from nltk.corpus import wordnet as wn\n import torch.nn as nn\n \n utils.fix_random_seeds()\n \n class TorchDeepNCMultipleLayers(TorchShallowNeuralClassifier):\n\n def __init__(self, nhidden=1, **kwargs):\n \"\"\"\n Generalisation of TorchShallowNeuralClassifier with multiple hidden layers. Each hidden layers keeps the\n dimension of the hidden layer has defined in the super class.\n :param nhidden: number of hidden layers\n \"\"\"\n super().__init__(**kwargs)\n self.nhidden = nhidden\n\n def build_graph(self):\n \"\"\"\n Returns\n -------\n an `nn.Module` instance, which can be a free-standing class you\n write yourself, as in `torch_rnn_classifier`, or the output of\n `nn.Sequential`, as in `torch_shallow_neural_classifier`.\n\n \"\"\"\n graph_list = [nn.Linear(self.input_dim, self.hidden_dim)]\n graph_list.append(self.hidden_activation)\n for i in range(self.nhidden - 1):\n graph_list.append(nn.Linear(self.hidden_dim, self.hidden_dim))\n graph_list.append(self.hidden_activation)\n graph_list.append(nn.Linear(self.hidden_dim, self.n_classes_))\n graph = nn.Sequential(*tuple(graph_list))\n return graph\n \n def wordnet_features(word1, word2, methodname):\n \"\"\"\n Returns 1 if the synsets extracted from the hierarchy methodname of the premise intersect the synsets of the \n hypothesis. Otherwise returns -1\n :param word1: word in premise\n :param word2: word in hypothesis\n :param methodname: wordnet hierarchy considered (hypernym or hyponym)\n :return:\n \"\"\"\n try:\n hyps = [h for ss in wn.synsets(word1) for h in getattr(ss, methodname)()]\n except AttributeError:\n hyps = list()\n for ss in wn.synsets(word1):\n try:\n for h in getattr(ss.lemmas()[0], methodname)():\n hyps.append(h)\n except TypeError:\n pass\n syns = wn.synsets(word2)\n\n output = 1 if set(hyps) & set(syns) else -1\n return output\n \n def word_entail_featurize_wordnet_1(data, vector_func, vector_combo_func, hierarchy, adjustment_param):\n \"\"\"\n Modified version of the featurize function used in nli.wordentail_experiment. Uses WordNet hierarchy.\n :param data:\n :param vector_func:\n :param vector_combo_func:\n :param hierarchy: hierarchy in wordnet considered, 'hypernyms' or 'hyponyms'\n :param adjustment_param: Parameters by which we translate the vector output from vector_combo_func when\n the hypothesis is included in the wordnet hierarchy of word1 (hypernyms or hyponyms)\n :return:\n \"\"\"\n X = []\n y = []\n for (w1, w2), label in data:\n is_related = list()\n for h in hierarchy:\n is_related.append(wordnet_features(w1, w2, h))\n rep = vector_combo_func(vector_func(w1), vector_func(w2))\n\n # build a vector of length vector_func(w1) with info from wordnet\n vec_size = vector_func(w1).shape[0]\n hierarchy_size = vec_size // len(hierarchy)\n coeff_vec = np.array([x for item in is_related for x in [item] * hierarchy_size])\n remaining_items = vec_size % len(hierarchy)\n if remaining_items > 0:\n # makes sure that coeff_vec has the right size\n coeff_vec = np.concatenate((coeff_vec, np.zeros(remaining_items)))\n adj_vec = coeff_vec * adjustment_param\n\n rep = np.concatenate((rep, adj_vec))\n X.append(rep)\n y.append(label)\n return X, y\n\n def word_entail_featurize_wordnet(hierarchy, param):\n \"\"\"\n Wrapper of word_entail_featurize_wordnet_1, allowing to define parameters and adjustment_param\n in the feature function.\n \"\"\"\n return lambda *args, **kwargs: word_entail_featurize_wordnet_1(*args, hierarchy=hierarchy,\n adjustment_param=param, **kwargs)\n \n def nli_model():\n \"\"\"\n Original system\n \"\"\"\n glove_dic = GLOVE\n df = pd.DataFrame.from_dict(glove_dic, orient='index')\n sum_rows = df.sum(axis=1)\n\n # calculate param, average distance between the mean and the max/min sum of row values of the Glove embedded space.\n # will be used in word_entail_featurize_wordnet to adjust vectors when they share an hypernym.\n n = df.shape[1]\n mean, max1, min1 = sum_rows.mean() / n, sum_rows.max() / n, sum_rows.min() / n\n param = 0.25 * (abs(min1) + abs(max1)) + mean\n\n featurize_func = word_entail_featurize_wordnet(['hypernyms', 'hyponyms', 'member_holonyms', 'antonyms', 'antonyms',\n 'derivationally_related_forms', 'pertainyms'], param)\n model = TorchDeepNCMultipleLayers(hidden_dim=150, nhidden=3, batch_size=300, eta=0.001, hidden_activation=nn.Tanh())\n \n experiment = nli.wordentail_experiment(\n train_data=wordentail_data['train'],\n assess_data=wordentail_data['dev'],\n model=model,\n vector_func=glove_vec,\n vector_combo_func=vec_concatenate,\n featurize_func=featurize_func)\n return experiment\n \n model_result = nli_model()\n\n# STOP COMMENT: Please do not remove this comment.\n```\n\n Stopping after epoch 73. Training loss did not improve more than tol=1e-05. Final error is 0.15200649778125808.\n\n precision recall f1-score support\n \n 0 0.888 0.931 0.909 1732\n 1 0.524 0.392 0.449 334\n \n accuracy 0.844 2066\n macro avg 0.706 0.662 0.679 2066\n weighted avg 0.829 0.844 0.835 2066\n \n\n\n## Bake-off [1 point]\n\nThe goal of the bake-off is to achieve the highest __macro-average F1__ score on a test set that we will make available at the start of the bake-off. The announcement will go out on the discussion forum. To enter, you'll be asked to run `nli.bake_off_evaluation` on the output of your chosen `nli.wordentail_experiment` run. \n\nThe cells below this one constitute your bake-off entry.\n\nThe rules described in the [Your original system](#Your-original-system-[3-points]) homework question are also in effect for the bake-off.\n\nSystems that enter will receive the additional homework point, and systems that achieve the top score will receive an additional 0.5 points. We will test the top-performing systems ourselves, and only systems for which we can reproduce the reported results will win the extra 0.5 points.\n\nLate entries will be accepted, but they cannot earn the extra 0.5 points. Similarly, you cannot win the bake-off unless your homework is submitted on time.\n\nThe announcement will include the details on where to submit your entry.\n\n\n```python\n# Enter your bake-off assessment code into this cell.\n# Please do not remove this comment.\nif 'IS_GRADESCOPE_ENV' not in os.environ:\n pass\n # Please enter your code in the scope of the above conditional.\n ##### YOUR CODE HERE\n\n```\n\n\n```python\n# On an otherwise blank line in this cell, please enter\n# your macro-avg f1 value as reported by the code above.\n# Please enter only a number between 0 and 1 inclusive.\n# Please do not remove this comment.\nif 'IS_GRADESCOPE_ENV' not in os.environ:\n pass\n # Please enter your score in the scope of the above conditional.\n ##### YOUR CODE HERE\n\n```\n", "meta": {"hexsha": "38fba62c8f6afb66b00c29fb8b56d282dc17169e", "size": 36991, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "hw_wordentail-Copy1.ipynb", "max_stars_repo_name": "Geodego/cs224u", "max_stars_repo_head_hexsha": "7b3de9d94d7b5ec0d96a49e67e2a65734e085e50", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "hw_wordentail-Copy1.ipynb", "max_issues_repo_name": "Geodego/cs224u", "max_issues_repo_head_hexsha": "7b3de9d94d7b5ec0d96a49e67e2a65734e085e50", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "hw_wordentail-Copy1.ipynb", "max_forks_repo_name": "Geodego/cs224u", "max_forks_repo_head_hexsha": "7b3de9d94d7b5ec0d96a49e67e2a65734e085e50", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.5162882527, "max_line_length": 642, "alphanum_fraction": 0.5658944067, "converted": true, "num_tokens": 6414, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.7772998663336158, "lm_q1q2_score": 0.44294638977408085}} {"text": "

Wiki Game Bot


\nThis notebook steps through a program designed to find an optimal solution for The Wiki Game, where players compete to find the shortest path between 2 random wikipedia articles by clicking links within the page. \nOur strategy is to use GloVe word vectors to pick the links that match the target link the best. To measure similarity between links we are using the cosine similarity formula.\n
\n\\begin{equation}\ncos(a, b) = \\frac{ab}{\\left \\| a \\right \\|\\left \\| b \\right \\|}\n\\end{equation}\n
\nThe perfect solution could be computed by checking every link on every page for the target link, however this is exponentially computationally expensive. To find a balance between computing speed and the depth of search we are using a proccess inspired by Beam Search. By only searching the top n links (with the highest cosine similarity) the program is generally able to find a solution quickly. The program has certain limitations due to its use of the word embeddings. If the words of an article name aren't in the embedding dictionary then it's unable to converge on the target. If an article is a person, the embeddings are limited with names and may only converge on celebrities and famous names.\n\n

Import Statements

\n
  • Numpy is used to handle the formatting & multiplication of the GloVe word embedding
    \n
  • urllib is used for opening and retrieving data from Wikipedia URLs
    \n
  • re (Regular Expressions) is used for searching and replacing parts of strings (links)
    \n
  • BeautifulSoup is used for parsing the HTML anchors of from Wikipedia Pages

    \n\n\n\n```python\nimport numpy as np\nimport re \nfrom bs4 import BeautifulSoup\nfrom urllib.request import urlopen\n```\n\n\n```python\ndef create_embeddings():\n global embeddings_index\n embeddings_index = {}\n\n #Open the 300 dimensional embedding as a readable file \n with open('glove.6B.300d.txt', 'r', encoding='utf-8') as f:\n #Loop Through Each Line\n for line in f:\n #Lower & Strip the line, then split into a list\n glove_parsing = line.lower().strip().split()\n\n #Take the first element (which is a word)\n embedding_word = glove_parsing[0]\n\n #Then take the embedding weights and place them into an array\n embedding_weights = np.array(glove_parsing[1:])\n\n #Create a dictionary where keys correspond to a given word and the values are the given weights \n embeddings_index[embedding_word] = embedding_weights\n return embeddings_index\n```\n\n\n```python\ndef retrive_wiki_links(url):\n \"\"\"Given a string input, url, returns a list of all blue article links\"\"\"\n \n #Retrive the Entire html of the Wiki Page and Convert to a String from Bytes \n html = str(urlopen(url).read())\n\n #Find the Start of the Content Page on the html\n index_start = html.index('id=\"mw-content-text\"')\n \n #Find Where the End of the Content Page is, this is Usually Where the Reference Header Begins\n #By Doing this we also Avoid Picking up the Links in the Reference Section \n index_end = None\n if ('id=\"References\"') in html:\n index_end = html.index('id=\"References\"')\n #Sometimes a Note Secection Exists, Which Comes Before the Reference Section, which also Contains Links we Wish to Avoid\n if ('id=\"Notes\"') in html:\n index_end = html.index('id=\"Notes\"')\n\n #Concatenate the html to only include the Content Section \n html_clean = html[:index_start+1] + html[:index_end]\n\n #Process the html data using the Beautiful Soup library \n cleaned_data = BeautifulSoup(html_clean, 'html.parser')\n\n #Retrive a list of all anchor tags \n tags = cleaned_data('a')\n\n #Retrive href links \n tag_list = []\n for tag in tags:\n tag = tag.get('href')\n tag_list.append(tag)\n\n #Remvoe None type elements\n tag_list = [x for x in tag_list if x != None]\n\n #Loop through the tags and filter\n wiki_links = []\n for tag in tag_list:\n #Blue links have no unique identifiers compared to other links, therefore other link types must be removed\n removed_terms = ['.jpg', '.jpeg', '/User:', '/wiki/Talk', '.png', '/wiki/Wikipedia',\n '/wiki/Help', '/wiki/File', '/wiki/Template', '/wiki/Special', 'https://',\n 'disambiguation']\n if ('/wiki/') in tag and not any(substring in tag for substring in removed_terms):\n wiki_links.append(tag)\n return wiki_links\n```\n\n\n```python\ndef wiki_link_cleaner(link):\n \"\"\"Clean the raw link into words readable by the encodings\"\"\"\n \n link = re.sub('^(.*?)\\/wiki\\/', '', link) \n link = link.replace('_', ' ').split('#')[0]\n link = re.sub(r'[^\\w\\s]',' ', link)\n return link \n```\n\n\n```python\ndef words_to_embeddings(wiki_links):\n \"\"\"given a list of links(wiki_links), return them as the embedding of the words\"\"\"\n \n #Strip the links down to the keywords\n clean_links = []\n for link in wiki_links:\n link = wiki_link_cleaner(link)\n clean_links.append(link)\n \n #Loop through the links to make embeddings\n embedding_tags = []\n for link in clean_links:\n words = link.split()\n\n #If the word isn't within the embeddings, it is removed\n for word in words:\n if embeddings_index.get(word.lower()) is None:\n words.remove(word)\n \n #Create a 2d array (length of words by the dimesnion of the word embedding) \n word_vecs = np.zeros((len(words), 300))\n\n #Add the respective weights to the array\n for i in range(len(words)):\n word_vecs[i] = embeddings_index.get(words[i].lower())\n\n #Take the mean of the weights (used for multi word links)\n #It's possible to use a more complex method of multi-word embeddings\n embedding_tags.append(np.mean(word_vecs, axis=0))\n\n embedding_tags = np.array(embedding_tags)\n return embedding_tags\n```\n\n\n```python\n#Function used to determine the similarity between words \ndef cosine_similarity(a, b):\n \"\"\"Calulates the cosine similarity between vectors a and b\"\"\"\n \n numerator = np.dot(a, b)\n denominator = np.dot(np.linalg.norm(a), np.linalg.norm(b))\n return numerator / denominator\n\n```\n\n\n```python\n\ndef beam_search(target, embedding_tags, BEAM_VALUE):\n \"\"\"Uses the principles of a beam search algorithm to find the best path to a range of new links\"\"\"\n \n tag_values = []\n target = target.lower()\n words = target.split()\n\n #If the word isn't within the embeddings, it is removed\n for word in words:\n if embeddings_index.get(word) is None:\n words.remove(word)\n \n #Create a 2d array (length of words by the dimesnion of the word embedding) \n word_vecs = np.zeros((len(words), 300))\n\n #Add the respective weights to the array\n for i in range(len(words)):\n word_vecs[i] = embeddings_index.get(words[i])\n\n #Take the mean of the weights (used for multi word links)\n target_embedding = np.mean(word_vecs, axis=0)\n\n #Calculate the cosine similarities\n for i in range(len(embedding_tags)):\n tag_values.append(cosine_similarity(embedding_tags[i], target_embedding))\n\n for j in range(len(tag_values)):\n if np.isnan(tag_values[j]):\n tag_values[j] = 0\n\n #Beam value adjusted if is larger than the number of values\n temp = 0\n if len(tag_values) < BEAM_VALUE:\n temp = BEAM_VALUE\n BEAM_VALUE = len(tag_values)\n\n #Output the top BEAM_VALUE number of links\n output = np.argpartition(np.array(tag_values), -BEAM_VALUE)[-BEAM_VALUE:]\n\n BEAM_VALUE = temp\n return output\n\n```\n\n\n```python\ndef dirty_to_searchable(index, word_list, return_link = False):\n \"\"\"takes a half link and makes it a full wikipedia link\"\"\"\n \n index = int(index)\n link = word_list[index]\n if return_link == True:\n link = 'https://en.wikipedia.org'+link\n return link\n```\n\n\n```python\ndef searchable_to_dirty(link):\n \"\"\"removes the wikipedia part of the link\"\"\"\n return link.split(\".wikipedia.org\")[-1]\n```\n\n\n```python\ndef wiki_game(starting_link, target_link):\n target = wiki_link_cleaner(target_link)\n blacklist = []\n link_position = {}\n number_of_searches = 0\n searches = []\n searches_history = []\n beam = 8\n\n #initial page search\n wiki_links = retrive_wiki_links(starting_link)\n\n #Create embeddings of the links\n embedding_tags = words_to_embeddings(wiki_links)\n \n #Run beam search to find the best embeddings (best 12 links on the first page)\n beam_initial_positions = beam_search(target, embedding_tags, 12)\n\n #Format new links from the beam search\n for i in beam_initial_positions:\n searches.append(dirty_to_searchable(i, wiki_links, True))\n searches_history.append(searches)\n \n #Keep checking new links until the target_link is found\n print('This may take a minute...')\n while target_link not in searches:\n positions_beam_1 = []\n total_links_beam_1 = []\n total_links_beam_2 = []\n number_of_searches += 1 \n \n #Limit the number of searches to prevent an infinite loop\n if number_of_searches > 9:\n print('Max search limit exceeded')\n is_successful = False\n break\n \n #Filter out links that have already been used to prevent looping\n for i in searches:\n wiki_links = retrive_wiki_links(i)\n for wiki_link in wiki_links:\n while wiki_links.count(wiki_link) != 1:\n wiki_links.remove(wiki_link)\n if wiki_link in blacklist:\n wiki_links.remove(wiki_link)\n else:\n total_links_beam_1.append(wiki_link)\n blacklist.append(wiki_link)\n\n for x in total_links_beam_1:\n while total_links_beam_1.count(x) != 1:\n total_links_beam_1.remove(x)\n\n #Beam search starts by picking top 8 links, and increases its beam by 3 each iteration\n embedding_tags = words_to_embeddings(total_links_beam_1)\n positions_beam_1 = beam_search(target, embedding_tags, beam)\n beam += 3\n \n #format the new links and add them to searches_history for the next iteration\n searches = []\n for position in positions_beam_1:\n link = dirty_to_searchable(position, total_links_beam_1, True)\n searches.append(link) \n searches_history.append(searches)\n print('searching...')\n \n return searches_history, starting_link, target_link\n```\n\n\n```python\n\ndef track_link_path(searches_history, starting_link, target_link):\n \"\"\"Works backwards from the solution, using searches_history to find a path to the starting_link\"\"\"\n \n path = []\n term = searchable_to_dirty(target_link)\n path.append(target_link)\n for j in range(len(searches_history))[:-1][::-1]:\n for i in searches_history[j]:\n wiki_links = retrive_wiki_links(i)\n for link in wiki_links:\n if link == term:\n path.append(i)\n term = searchable_to_dirty(i)\n break\n else:\n continue\n break\n path.append(starting_link)\n return path[::-1]\n\n```\n\n\n```python\nembeddings_index = create_embeddings() #Takes a minute to run\n```\n\n\n```python\nstarting_link = input('Starting Link --- ')\ntarget_link = input('Target Link --- ')\n\nsearches_history, starting_link, target_link = wiki_game(starting_link, target_link)\npath = track_link_path(searches_history, starting_link, target_link)\n\nfor i in range(len(path)-1):\n print(path[i], end = \" ---> \")\nprint(path[-1])\n\n```\n\n Starting Link --- https://en.wikipedia.org/wiki/Apple_pie\n Target Link --- https://en.wikipedia.org/wiki/Shrek\n This may take a minute...\n searching...\n searching...\n searching...\n https://en.wikipedia.org/wiki/Apple_pie ---> https://en.wikipedia.org/wiki/Ginger ---> https://en.wikipedia.org/wiki/Gingerbread ---> https://en.wikipedia.org/wiki/Gingerbread_man ---> https://en.wikipedia.org/wiki/Shrek\n\n\n\n```python\nstarting_link = 'https://en.wikipedia.org/wiki/Interference_(communication)'\ntarget_link = 'https://en.wikipedia.org/wiki/Popcorn'\n\nsearches_history, starting_link, target_link = wiki_game(starting_link, target_link)\npath = track_link_path(searches_history, starting_link, target_link)\n\nfor i in range(len(path)-1):\n print(path[i], end = \" ---> \")\nprint(path[-1])\n\n```\n\n This may take a minute...\n searching...\n searching...\n searching...\n https://en.wikipedia.org/wiki/Interference_(communication) ---> https://en.wikipedia.org/wiki/Wireless_networks ---> https://en.wikipedia.org/wiki/Wi-Fi ---> https://en.wikipedia.org/wiki/Microwave_oven ---> https://en.wikipedia.org/wiki/Popcorn\n\n\n\n```python\nstarting_link = 'https://en.wikipedia.org/wiki/Early_Christianity'\ntarget_link = 'https://en.wikipedia.org/wiki/Eminem'\n\nsearches_history, starting_link, target_link = wiki_game(starting_link, target_link)\npath = track_link_path(searches_history, starting_link, target_link)\n\nfor i in range(len(path)-1):\n print(path[i], end = \" ---> \")\nprint(path[-1])\n```\n\n This may take a minute...\n searching...\n searching...\n https://en.wikipedia.org/wiki/Early_Christianity ---> https://en.wikipedia.org/wiki/Christian_music ---> https://en.wikipedia.org/wiki/Music_download ---> https://en.wikipedia.org/wiki/Eminem\n\n\n\n```python\nstarting_link = 'https://en.wikipedia.org/wiki/Apple_pie'\ntarget_link = 'https://en.wikipedia.org/wiki/Shrek'\n\nsearches_history, starting_link, target_link = wiki_game(starting_link, target_link)\npath = track_link_path(searches_history, starting_link, target_link)\n\nfor i in range(len(path)-1):\n print(path[i], end = \" ---> \")\nprint(path[-1])\n```\n\n This may take a minute...\n searching...\n searching...\n searching...\n https://en.wikipedia.org/wiki/Apple_pie ---> https://en.wikipedia.org/wiki/Ginger ---> https://en.wikipedia.org/wiki/Gingerbread ---> https://en.wikipedia.org/wiki/Gingerbread_man ---> https://en.wikipedia.org/wiki/Shrek\n\n", "meta": {"hexsha": "e6b6b39cb72a0e0fefbe5dff2f823fd151672787", "size": 20799, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "WikiBot.ipynb", 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"avg_line_length": 36.3618881119, "max_line_length": 774, "alphanum_fraction": 0.5544016539, "converted": true, "num_tokens": 3298, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7772998611746912, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.44294638683425397}} {"text": "\n \n \n \n \n
    \n \n \n
    \n\n# Orthogonal Random Forest: Use Cases and Examples\n\nOrthogonal Random Forest (ORF) combines orthogonalization,\na technique that effectively removes the confounding effect in two-stage estimation,\nwith generalized random forests, a flexible method for estimating treatment effect heterogeneity. Due to the orthogonalization aspect of this method, the ORF performs especially well in the presence of high-dimensional confounders. For more details, see [this paper](https://arxiv.org/abs/1806.03467).\n\nThe EconML SDK implements the following OrthoForest variants:\n\n* ContinuousTreatmentOrthoForest: suitable for continuous treatments\n\n* DiscreteTreatmentOrthoForest: suitable for discrete treatments\n\nIn this notebook, we show the performance of the ORF on synthetic data.\n\n**Notebook contents:**\n\n1. Example usage with continuous treatment synthetic data\n\n2. Example usage with binary treatment synthetic data\n\n3. Example usage with multiple discrete treatment synthetic data\n\n4. Example usage with real continuous treatment observational data\n\n\n```python\nimport econml\n```\n\n\n```python\n# Main imports\nfrom econml.ortho_forest import ContinuousTreatmentOrthoForest, DiscreteTreatmentOrthoForest\nfrom econml.sklearn_extensions.linear_model import WeightedLassoCVWrapper, WeightedLasso, WeightedLassoCV\n\n# Helper imports\nimport numpy as np\nfrom itertools import product\nfrom sklearn.linear_model import Lasso, LassoCV, LogisticRegression, LogisticRegressionCV\nimport matplotlib.pyplot as plt\n\n%matplotlib inline\n```\n\n## 1. Example Usage with Continuous Treatment Synthetic Data\n\n### 1.1. DGP \nWe use the data generating process (DGP) from [here](https://arxiv.org/abs/1806.03467). The DGP is described by the following equations:\n\n\\begin{align}\nT =& \\langle W, \\beta\\rangle + \\eta, & \\;\\eta \\sim \\text{Uniform}(-1, 1)\\\\\nY =& T\\cdot \\theta(X) + \\langle W, \\gamma\\rangle + \\epsilon, &\\; \\epsilon \\sim \\text{Uniform}(-1, 1)\\\\\nW \\sim& \\text{Normal}(0,\\, I_{n_w})\\\\\nX \\sim& \\text{Uniform}(0,1)^{n_x}\n\\end{align}\n\nwhere $W$ is a matrix of high-dimensional confounders and $\\beta, \\gamma$ have high sparsity.\n\nFor this DGP, \n\\begin{align}\n\\theta(x) = \\exp(2\\cdot x_1).\n\\end{align}\n\n\n```python\n# Treatment effect function\ndef exp_te(x):\n return np.exp(2*x[0])\n```\n\n\n```python\n# DGP constants\nnp.random.seed(123)\nn = 1000\nn_w = 30\nsupport_size = 5\nn_x = 1\n# Outcome support\nsupport_Y = np.random.choice(range(n_w), size=support_size, replace=False)\ncoefs_Y = np.random.uniform(0, 1, size=support_size)\nepsilon_sample = lambda n: np.random.uniform(-1, 1, size=n)\n# Treatment support \nsupport_T = support_Y\ncoefs_T = np.random.uniform(0, 1, size=support_size)\neta_sample = lambda n: np.random.uniform(-1, 1, size=n) \n\n# Generate controls, covariates, treatments and outcomes\nW = np.random.normal(0, 1, size=(n, n_w))\nX = np.random.uniform(0, 1, size=(n, n_x))\n# Heterogeneous treatment effects\nTE = np.array([exp_te(x_i) for x_i in X])\nT = np.dot(W[:, support_T], coefs_T) + eta_sample(n)\nY = TE * T + np.dot(W[:, support_Y], coefs_Y) + epsilon_sample(n)\n\n# ORF parameters and test data\n# The following parameters are set according to theory\nsubsample_power = 0.88\nsubsample_ratio = ((n/np.log(n_w))**(subsample_power)) / n\nlambda_reg = np.sqrt(np.log(n_w) / (10 * subsample_ratio * n))\nX_test = np.array(list(product(np.arange(0, 1, 0.01), repeat=n_x)))\n```\n\n### 1.2. Train Estimator\n\n**Note:** The models in the final stage of the estimation (``model_T_final``, ``model_Y_final``) need to support sample weighting. \n\nIf the models of choice do not support sample weights (e.g. ``sklearn.linear_model.LassoCV``), the ``econml`` packages provides a convenient wrapper for these models ``WeightedModelWrapper`` in order to allow sample weights. \n\nIf the model of choice is a linear (regression) model such as Lasso, you should set ``sample_type=\"weighted\"``. Otherwise, set ``sample_type=\"sampled\"``.\n\n\n```python\nest = ContinuousTreatmentOrthoForest(\n n_trees=200, min_leaf_size=5,\n max_depth=50, subsample_ratio=2*subsample_ratio, bootstrap=False, \n model_T=Lasso(alpha=lambda_reg),\n model_Y=Lasso(alpha=lambda_reg),\n model_T_final=WeightedLasso(alpha=lambda_reg), \n model_Y_final=WeightedLasso(alpha=lambda_reg),\n random_state=123)\n```\n\n\n```python\nest.fit(Y, T, X, W)\n```\n\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 4.3s\n [Parallel(n_jobs=-1)]: Done 185 out of 200 | elapsed: 5.8s remaining: 0.4s\n [Parallel(n_jobs=-1)]: Done 200 out of 200 | elapsed: 5.8s finished\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.1s\n [Parallel(n_jobs=-1)]: Done 118 out of 200 | elapsed: 1.0s remaining: 0.7s\n [Parallel(n_jobs=-1)]: Done 200 out of 200 | elapsed: 1.4s finished\n\n\n\n\n\n \n\n\n\n\n```python\ntreatment_effects = est.effect(X_test)\n```\n\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.4s\n [Parallel(n_jobs=-1)]: Done 100 out of 100 | elapsed: 2.2s finished\n\n\n### 1.3. Performance Visualization\n\n\n```python\ny = treatment_effects[:, 0]\nplt.plot(X_test, y, label='ORF estimate')\nexpected_te = np.array([exp_te(x_i) for x_i in X_test])\nplt.plot(X_test[:, 0], expected_te, 'b--', label='True effect')\nplt.ylabel(\"Treatment Effect\")\nplt.xlabel(\"x\")\nplt.legend()\nplt.show()\n```\n\n## 2. Example Usage with Binary Treatment Synthetic Data\n\n### 2.1. DGP \nWe use the following DGP:\n\n\\begin{align}\nT \\sim & \\text{Bernoulli}\\left(f(W)\\right), &\\; f(W)=\\sigma(\\langle W, \\beta\\rangle + \\eta), \\;\\eta \\sim \\text{Uniform}(-1, 1)\\\\\nY = & T\\cdot \\theta(X) + \\langle W, \\gamma\\rangle + \\epsilon, & \\; \\epsilon \\sim \\text{Uniform}(-1, 1)\\\\\nW \\sim & \\text{Normal}(0,\\, I_{n_w}) & \\\\\nX \\sim & \\text{Uniform}(0,\\, 1)^{n_x}\n\\end{align}\n\nwhere $W$ is a matrix of high-dimensional confounders, $\\beta, \\gamma$ have high sparsity and $\\sigma$ is the sigmoid function.\n\nFor this DGP, \n\\begin{align}\n\\theta(x) = \\exp( 2\\cdot x_1 ).\n\\end{align}\n\n\n```python\n# DGP constants\nnp.random.seed(1234)\nn = 1000\nn_w = 30\nsupport_size = 5\nn_x = 1\n# Outcome support\nsupport_Y = np.random.choice(range(n_w), size=support_size, replace=False)\ncoefs_Y = np.random.uniform(0, 1, size=support_size)\nepsilon_sample = lambda n: np.random.uniform(-1, 1, size=n)\n# Treatment support \nsupport_T = support_Y\ncoefs_T = np.random.uniform(0, 1, size=support_size)\neta_sample = lambda n: np.random.uniform(-1, 1, size=n) \n\n# Generate controls, covariates, treatments and outcomes\nW = np.random.normal(0, 1, size=(n, n_w))\nX = np.random.uniform(0, 1, size=(n, n_x))\n# Heterogeneous treatment effects\nTE = np.array([exp_te(x_i) for x_i in X])\n# Define treatment\nlog_odds = np.dot(W[:, support_T], coefs_T) + eta_sample(n)\nT_sigmoid = 1/(1 + np.exp(-log_odds))\nT = np.array([np.random.binomial(1, p) for p in T_sigmoid])\n# Define the outcome\nY = TE * T + np.dot(W[:, support_Y], coefs_Y) + epsilon_sample(n)\n\n# ORF parameters and test data\n# The following parameters are set according to theory\nsubsample_power = 0.88\nsubsample_ratio = ((n/np.log(n_w))**(subsample_power)) / n\nlambda_reg = np.sqrt(np.log(n_w) / (10 * subsample_ratio * n))\nX_test = np.array(list(product(np.arange(0, 1, 0.01), repeat=n_x)))\n```\n\n### 2.2. Train Estimator \n\n\n```python\nest = DiscreteTreatmentOrthoForest(\n n_trees=200, min_leaf_size=10,\n max_depth=30, subsample_ratio=2*subsample_ratio, bootstrap=False,\n propensity_model = LogisticRegression(C=1/(X.shape[0]*lambda_reg), penalty='l1', solver='saga'),\n model_Y = Lasso(alpha=lambda_reg),\n propensity_model_final=LogisticRegression(C=1/(X.shape[0]*lambda_reg), penalty='l1', solver='saga'), \n model_Y_final=WeightedLasso(alpha=lambda_reg)\n)\n```\n\n\n```python\nest.fit(Y, T, X, W)\n```\n\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.1s\n [Parallel(n_jobs=-1)]: Done 118 out of 200 | elapsed: 0.8s remaining: 0.5s\n [Parallel(n_jobs=-1)]: Done 200 out of 200 | elapsed: 1.2s finished\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.1s\n [Parallel(n_jobs=-1)]: Done 200 out of 200 | elapsed: 1.1s finished\n\n\n\n\n\n \n\n\n\n\n```python\ntreatment_effects = est.effect(X_test)\n```\n\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.4s\n [Parallel(n_jobs=-1)]: Done 100 out of 100 | elapsed: 3.2s finished\n\n\n### 2.3. Performance Visualization\n\n\n```python\ny = treatment_effects\nplt.plot(X_test, y, label='ORF estimate')\nexpected_te = np.array([exp_te(x_i) for x_i in X_test])\nplt.plot(X_test[:, 0], expected_te, 'b--', label='True effect')\nplt.ylabel(\"Treatment Effect\")\nplt.xlabel(\"x\")\nplt.legend()\nplt.show()\n```\n\n## 3. Example Usage with Multiple Treatment Synthetic Data\n\n### 3.1 DGP \nWe use the following DGP:\n\n\\begin{align}\nY = & \\sum_{t=1}^{n_{\\text{treatments}}} 1\\{T=t\\}\\cdot \\theta_{T}(X) + \\langle W, \\gamma\\rangle + \\epsilon, \\; \\epsilon \\sim \\text{Unif}(-1, 1), \\\\\n\\text{Pr}[T=t \\mid W] \\propto & \\exp\\{\\langle W, \\beta_t \\rangle\\}, \\;\\;\\;\\; \\forall t\\in \\{0, 1, \\ldots, n_{\\text{treatments}}\\} \n\\end{align}\n\nwhere $W$ is a matrix of high-dimensional confounders, $\\beta_t, \\gamma$ are sparse.\n\nFor this particular example DGP we used $n_{\\text{treatments}}=3$ and \n\\begin{align}\n\\theta_1(x) = & \\exp( 2 x_1 ),\\\\\n\\theta_2(x) = & 3 \\cdot \\sigma(100\\cdot (x_1 - .5)),\\\\\n\\theta_3(x) = & -2 \\cdot \\sigma(100\\cdot (x_1 - .25)),\n\\end{align}\nwhere $\\sigma$ is the sigmoid function.\n\n\n```python\ndef get_test_train_data(n, n_w, support_size, n_x, te_func, n_treatments):\n # Outcome support\n support_Y = np.random.choice(range(n_w), size=support_size, replace=False)\n coefs_Y = np.random.uniform(0, 1, size=support_size)\n epsilon_sample = lambda n: np.random.uniform(-1, 1, size=n)\n # Treatment support \n support_T = support_Y\n coefs_T = np.random.uniform(0, 1, size=(support_size, n_treatments))\n eta_sample = lambda n: np.random.uniform(-1, 1, size=n) \n # Generate controls, covariates, treatments and outcomes\n W = np.random.normal(0, 1, size=(n, n_w))\n X = np.random.uniform(0, 1, size=(n, n_x))\n # Heterogeneous treatment effects\n TE = np.array([te_func(x_i, n_treatments) for x_i in X])\n log_odds = np.dot(W[:, support_T], coefs_T)\n T_sigmoid = np.exp(log_odds)\n T_sigmoid = T_sigmoid/np.sum(T_sigmoid, axis=1, keepdims=True)\n T = np.array([np.random.choice(n_treatments, p=p) for p in T_sigmoid])\n TE = np.concatenate((np.zeros((n,1)), TE), axis=1)\n Y = TE[np.arange(n), T] + np.dot(W[:, support_Y], coefs_Y) + epsilon_sample(n)\n X_test = np.array(list(product(np.arange(0, 1, 0.01), repeat=n_x)))\n\n return (Y, T, X, W), (X_test, np.array([te_func(x, n_treatments) for x in X_test]))\n```\n\n\n```python\nimport scipy.special\ndef te_func(x, n_treatments):\n return [np.exp(2*x[0]), 3*scipy.special.expit(100*(x[0] - .5)) - 1, -2*scipy.special.expit(100*(x[0] - .25))]\n\nnp.random.seed(123)\n(Y, T, X, W), (X_test, te_test) = get_test_train_data(1000, 3, 3, 1, te_func, 4)\n```\n\n### 3.2 Train Estimator\n\n\n```python\nest = DiscreteTreatmentOrthoForest(n_trees=500,\n model_Y = WeightedLasso(alpha=lambda_reg))\n```\n\n\n```python\nest.fit(Y, T, X, W)\n```\n\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.3s\n [Parallel(n_jobs=-1)]: Done 208 tasks | elapsed: 3.7s\n [Parallel(n_jobs=-1)]: Done 500 out of 500 | elapsed: 9.1s finished\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.4s\n [Parallel(n_jobs=-1)]: Done 112 tasks | elapsed: 2.2s\n [Parallel(n_jobs=-1)]: Done 272 tasks | elapsed: 5.4s\n [Parallel(n_jobs=-1)]: Done 500 out of 500 | elapsed: 9.9s finished\n\n\n\n\n\n \n\n\n\n\n```python\ntreatment_effects = est.const_marginal_effect(X_test)\n```\n\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.9s\n [Parallel(n_jobs=-1)]: Done 100 out of 100 | elapsed: 5.1s finished\n\n\n### 3.3 Performance Visualization\n\n\n```python\ny = treatment_effects\nfor it in range(y.shape[1]):\n plt.plot(X_test, y[:, it], label='ORF estimate T={}'.format(it))\n plt.plot(X_test[:, 0], te_test[:, it], '--', label='True effect T={}'.format(it))\nplt.ylabel(\"Treatment Effect\")\nplt.xlabel(\"x\")\nplt.legend()\nplt.show()\n```\n\n## 4. Example usage with real continuous treatment observational data\n\nWe applied our technique to Dominick\u2019s dataset, a popular historical dataset of store-level orange juice prices and sales provided by University of Chicago Booth School of Business. \n\nThe dataset is comprised of a large number of covariates $W$, but researchers might only be interested in learning the elasticity of demand as a function of a few variables $x$ such\nas income or education. \n\nWe applied the `ContinuousTreatmentOrthoForest` to estimate orange juice price elasticity\nas a function of income, and our results, unveil the natural phenomenon that lower income consumers are more price-sensitive.\n\n### 4.1. Data\n\n\n```python\n# A few more imports\nimport os\nimport pandas as pd\nimport urllib.request\nfrom sklearn.preprocessing import StandardScaler\n```\n\n\n```python\n# Import the data\nfile_name = \"oj_large.csv\"\n\nif not os.path.isfile(file_name):\n print(\"Downloading file (this might take a few seconds)...\")\n urllib.request.urlretrieve(\"https://msalicedatapublic.blob.core.windows.net/datasets/OrangeJuice/oj_large.csv\", file_name)\noj_data = pd.read_csv(file_name)\noj_data.head()\n```\n\n Downloading file (this might take a few seconds)...\n\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    storebrandweeklogmovefeatpriceAGE60EDUCETHNICINCOMEHHLARGEWORKWOMHVAL150SSTRDISTSSTRVOLCPDIST5CPWVOL5
    02tropicana409.01869503.870.2328650.2489350.1142810.5532050.1039530.3035850.4638872.1101221.1428571.927280.376927
    12tropicana468.72323103.870.2328650.2489350.1142810.5532050.1039530.3035850.4638872.1101221.1428571.927280.376927
    22tropicana478.25322803.870.2328650.2489350.1142810.5532050.1039530.3035850.4638872.1101221.1428571.927280.376927
    32tropicana488.98719703.870.2328650.2489350.1142810.5532050.1039530.3035850.4638872.1101221.1428571.927280.376927
    42tropicana509.09335703.870.2328650.2489350.1142810.5532050.1039530.3035850.4638872.1101221.1428571.927280.376927
    \n
    \n\n\n\n\n```python\n# Prepare data\nY = oj_data['logmove'].values\nT = np.log(oj_data[\"price\"]).values\nscaler = StandardScaler()\nW1 = scaler.fit_transform(oj_data[[c for c in oj_data.columns if c not in ['price', 'logmove', 'brand', 'week', 'store']]].values)\nW2 = pd.get_dummies(oj_data[['brand']]).values\nW = np.concatenate([W1, W2], axis=1)\nX = oj_data[['INCOME']].values\n```\n\n### 4.2. Train Estimator\n\n\n```python\n# Define some parameters\nn_trees = 2000\nmin_leaf_size = 50\nmax_depth = 20\nsubsample_ratio = 0.02\nbootstrap = False\n```\n\n\n```python\nest = ContinuousTreatmentOrthoForest(\n n_trees=n_trees, min_leaf_size=min_leaf_size, max_depth=max_depth, \n subsample_ratio=subsample_ratio, bootstrap=bootstrap, \n model_T=Lasso(alpha=0.1),\n model_Y=Lasso(alpha=0.1),\n model_T_final=WeightedLassoCVWrapper(cv=3), \n model_Y_final=WeightedLassoCVWrapper(cv=3)\n )\n```\n\n\n```python\nest.fit(Y, T, X, W)\n```\n\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 296 tasks | elapsed: 0.5s\n [Parallel(n_jobs=-1)]: Done 2000 out of 2000 | elapsed: 2.2s finished\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.1s\n [Parallel(n_jobs=-1)]: Done 2000 out of 2000 | elapsed: 2.3s finished\n\n\n\n\n\n \n\n\n\n\n```python\nmin_income = 10.0 \nmax_income = 11.1\ndelta = (max_income - min_income) / 100\nX_test = np.arange(min_income, max_income + delta - 0.001, delta).reshape(-1, 1)\n```\n\n\n```python\nimport time\nt0 = time.time()\nte_pred = est.const_marginal_effect(X_test)\nprint(time.time() - t0)\n```\n\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 41.0s\n\n\n 285.2490632534027\n\n\n [Parallel(n_jobs=-1)]: Done 101 out of 101 | elapsed: 4.8min finished\n\n\n### 4.3. Performance Visualization\n\n\n```python\n# Plot Orange Juice elasticity as a function of income\nplt.plot(np.ndarray.flatten(X_test), te_pred[:, 0], label=\"OJ Elasticity\")\nplt.xlabel(r'$\\log$(Income)')\nplt.ylabel('Orange Juice Elasticity')\nplt.legend()\nplt.title(\"Orange Juice Elasticity vs Income\")\nplt.show()\n```\n\n### 4.4 Bootstrap Confidence Intervals\n\nWe can also use a bootstrap estimator to generate confidence intervals by changing how we call `fit`; in order to return results in a few minutes we're limiting the number of trees to 100 and the number of bootstrap samples to 10 in the code below, but for better estimates these numbers can be increased at the cost of increased runtime.\n\n\n```python\nfrom econml.inference import BootstrapInference\nest = ContinuousTreatmentOrthoForest(\n n_trees=100, min_leaf_size=min_leaf_size, max_depth=max_depth, \n subsample_ratio=subsample_ratio, bootstrap=bootstrap, \n model_T=Lasso(alpha=0.1),\n model_Y=Lasso(alpha=0.1),\n model_T_final=WeightedLassoCVWrapper(cv=3), \n model_Y_final=WeightedLassoCVWrapper(cv=3)\n)\n```\n\n\n```python\nest.fit(Y, T, X, W, inference=BootstrapInference(n_bootstrap_samples=10, n_jobs=-1))\nte_pred_interval = est.const_marginal_effect_interval(X_test, alpha=0.02)\n```\n\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 100 out of 100 | elapsed: 0.2s finished\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 100 out of 100 | elapsed: 0.1s finished\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.3s\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.6s\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.6s\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.6s\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.7s\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.6s\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.7s\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.6s\n [Parallel(n_jobs=-1)]: Done 100 out of 100 | elapsed: 3.2s finished\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 100 out of 100 | elapsed: 3.3s finished\n 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finished\n [Parallel(n_jobs=-1)]: Done 100 out of 100 | elapsed: 3.5s finished\n [Parallel(n_jobs=-1)]: Done 100 out of 100 | elapsed: 3.5s finished\n [Parallel(n_jobs=-1)]: Done 100 out of 100 | elapsed: 3.5s finished\n [Parallel(n_jobs=-1)]: Done 7 out of 10 | elapsed: 8.0s remaining: 3.4s\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 100 out of 100 | elapsed: 1.9s finished\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.1s\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.1s\n [Parallel(n_jobs=-1)]: Done 100 out of 100 | elapsed: 0.5s finished\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 16 tasks | elapsed: 0.2s\n [Parallel(n_jobs=-1)]: Done 100 out of 100 | elapsed: 0.8s finished\n [Parallel(n_jobs=-1)]: Done 100 out of 100 | elapsed: 0.6s finished\n [Parallel(n_jobs=-1)]: Done 10 out of 10 | elapsed: 9.6s finished\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n [Parallel(n_jobs=-1)]: Using backend ThreadingBackend with 8 concurrent workers.\n\n\n\n```python\nplt.plot(np.ndarray.flatten(X_test), te_pred[:, 0], label=\"OJ Elasticity\")\nplt.fill_between(np.ndarray.flatten(X_test), \n te_pred_interval[0][:, 0], \n te_pred_interval[1][:, 0], alpha=.5, label=\"1-99% CI\")\nplt.xlabel(r'$\\log$(Income)')\nplt.ylabel('Orange Juice Elasticity')\nplt.title(\"Orange Juice Elasticity vs Income\")\nplt.legend()\nplt.show()\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "695234ee0b34980bef167ff12b1819fc607269d5", "size": 134310, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/Orthogonal Random Forest Examples.ipynb", "max_stars_repo_name": "LeihuaYe/EconML", "max_stars_repo_head_hexsha": "83706947980b9e772f8fefaac4e30c195a572c45", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-04-14T19:53:08.000Z", "max_stars_repo_stars_event_max_datetime": "2020-04-14T19:53:08.000Z", "max_issues_repo_path": "notebooks/Orthogonal Random Forest Examples.ipynb", "max_issues_repo_name": "LeihuaYe/EconML", "max_issues_repo_head_hexsha": "83706947980b9e772f8fefaac4e30c195a572c45", "max_issues_repo_licenses": ["MIT"], 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YES\n2. YES", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.63341027059799, "lm_q1q2_score": 0.44291493999928827}} {"text": "\n\n# Tutorial 2: Learning Hyperparameters\n**Week 1, Day 2: Linear Deep Learning**\n\n**By Neuromatch Academy**\n\n__Content creators:__ Saeed Salehi, Andrew Saxe\n\n__Content reviewers:__ Polina Turishcheva, Antoine De Comite, Kelson Shilling-Scrivo\n\n__Content editors:__ Anoop Kulkarni\n\n__Production editors:__ Khalid Almubarak, Spiros Chavlis\n\n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n

    \n\n---\n# Tutorial Objectives\n\n* Training landscape\n* The effect of depth\n* Choosing a learning rate\n* Initialization matters\n\n\n\n```python\n# @title Tutorial slides\n\n# @markdown These are the slides for the videos in the tutorial\nfrom IPython.display import IFrame\nIFrame(src=f\"https://mfr.ca-1.osf.io/render?url=https://osf.io/sne2m/?direct%26mode=render%26action=download%26mode=render\", width=854, height=480)\n```\n\n---\n# Setup\n\nThis a GPU-Free tutorial!\n\n\n```python\n# @title Install dependencies\n!pip install git+https://github.com/NeuromatchAcademy/evaltools --quiet\n\nfrom evaltools.airtable import AirtableForm\n```\n\n\n```python\n# Imports\nimport time\nimport numpy as np\nimport matplotlib\nimport matplotlib.pyplot as plt\n```\n\n\n```python\n# @title Figure settings\n\nfrom ipywidgets import interact, IntSlider, FloatSlider, fixed\nfrom ipywidgets import HBox, interactive_output, ToggleButton, Layout\nfrom mpl_toolkits.axes_grid1 import make_axes_locatable\n\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/content-creation/main/nma.mplstyle\")\n```\n\n\n```python\n# @title Plotting functions\n\ndef plot_x_y_(x_t_, y_t_, x_ev_, y_ev_, loss_log_, weight_log_):\n \"\"\"\n \"\"\"\n plt.figure(figsize=(12, 4))\n plt.subplot(1, 3, 1)\n plt.scatter(x_t_, y_t_, c='r', label='training data')\n plt.plot(x_ev_, y_ev_, c='b', label='test results', linewidth=2)\n plt.xlabel('x')\n plt.ylabel('y')\n plt.legend()\n plt.subplot(1, 3, 2)\n plt.plot(loss_log_, c='r')\n plt.xlabel('epochs')\n plt.ylabel('mean squared error')\n plt.subplot(1, 3, 3)\n plt.plot(weight_log_)\n plt.xlabel('epochs')\n plt.ylabel('weights')\n plt.show()\n\n\ndef plot_vector_field(what, init_weights=None):\n \"\"\"\n \"\"\"\n n_epochs=40\n lr=0.15\n x_pos = np.linspace(2.0, 0.5, 100, endpoint=True)\n y_pos = 1. / x_pos\n xx, yy = np.mgrid[-1.9:2.0:0.2, -1.9:2.0:0.2]\n zz = np.empty_like(xx)\n x, y = xx[:, 0], yy[0]\n\n x_temp, y_temp = gen_samples(10, 1.0, 0.0)\n\n cmap = matplotlib.cm.plasma\n plt.figure(figsize=(8, 7))\n ax = plt.gca()\n\n if what == 'all' or what == 'vectors':\n for i, a in enumerate(x):\n for j, b in enumerate(y):\n temp_model = ShallowNarrowLNN([a, b])\n da, db = temp_model.dloss_dw(x_temp, y_temp)\n zz[i, j] = temp_model.loss(temp_model.forward(x_temp), y_temp)\n scale = min(40 * np.sqrt(da**2 + db**2), 50)\n ax.quiver(a, b, - da, - db, scale=scale, color=cmap(np.sqrt(da**2 + db**2)))\n\n if what == 'all' or what == 'trajectory':\n if init_weights is None:\n for init_weights in [[0.5, -0.5], [0.55, -0.45], [-1.8, 1.7]]:\n temp_model = ShallowNarrowLNN(init_weights)\n _, temp_records = temp_model.train(x_temp, y_temp, lr, n_epochs)\n ax.scatter(temp_records[:, 0], temp_records[:, 1],\n c=np.arange(len(temp_records)), cmap='Greys')\n ax.scatter(temp_records[0, 0], temp_records[0, 1], c='blue', zorder=9)\n ax.scatter(temp_records[-1, 0], temp_records[-1, 1], c='red', marker='X', s=100, zorder=9)\n else:\n temp_model = ShallowNarrowLNN(init_weights)\n _, temp_records = temp_model.train(x_temp, y_temp, lr, n_epochs)\n ax.scatter(temp_records[:, 0], temp_records[:, 1],\n c=np.arange(len(temp_records)), cmap='Greys')\n ax.scatter(temp_records[0, 0], temp_records[0, 1], c='blue', zorder=9)\n ax.scatter(temp_records[-1, 0], temp_records[-1, 1], c='red', marker='X', s=100, zorder=9)\n\n if what == 'all' or what == 'loss':\n contplt = ax.contourf(x, y, np.log(zz+0.001), zorder=-1, cmap='coolwarm', levels=100)\n divider = make_axes_locatable(ax)\n cax = divider.append_axes(\"right\", size=\"5%\", pad=0.05)\n cbar = plt.colorbar(contplt, cax=cax)\n cbar.set_label('log (Loss)')\n\n ax.set_xlabel(\"$w_1$\")\n ax.set_ylabel(\"$w_2$\")\n ax.set_xlim(-1.9, 1.9)\n ax.set_ylim(-1.9, 1.9)\n\n plt.show()\n\n\ndef plot_loss_landscape():\n \"\"\"\n \"\"\"\n x_temp, y_temp = gen_samples(10, 1.0, 0.0)\n\n xx, yy = np.mgrid[-1.9:2.0:0.2, -1.9:2.0:0.2]\n zz = np.empty_like(xx)\n x, y = xx[:, 0], yy[0]\n\n for i, a in enumerate(x):\n for j, b in enumerate(y):\n temp_model = ShallowNarrowLNN([a, b])\n zz[i, j] = temp_model.loss(temp_model.forward(x_temp), y_temp)\n\n temp_model = ShallowNarrowLNN([-1.8, 1.7])\n loss_rec_1, w_rec_1 = temp_model.train(x_temp, y_temp, 0.02, 240)\n\n temp_model = ShallowNarrowLNN([1.5, -1.5])\n loss_rec_2, w_rec_2 = temp_model.train(x_temp, y_temp, 0.02, 240)\n\n plt.figure(figsize=(12, 8))\n ax = plt.subplot(1, 1, 1, projection='3d')\n ax.plot_surface(xx, yy, np.log(zz+0.5), cmap='coolwarm', alpha=0.5)\n ax.scatter3D(w_rec_1[:, 0], w_rec_1[:, 1], np.log(loss_rec_1+0.5),\n c='k', s=50, zorder=9)\n ax.scatter3D(w_rec_2[:, 0], w_rec_2[:, 1], np.log(loss_rec_2+0.5),\n c='k', s=50, zorder=9)\n plt.axis(\"off\")\n ax.view_init(45, 260)\n\n plt.show()\n\n\ndef depth_widget(depth):\n if depth == 0:\n depth_lr_init_interplay(depth, 0.02, 0.9)\n else:\n depth_lr_init_interplay(depth, 0.01, 0.9)\n\n\ndef lr_widget(lr):\n depth_lr_init_interplay(50, lr, 0.9)\n\n\ndef depth_lr_interplay(depth, lr):\n depth_lr_init_interplay(depth, lr, 0.9)\n\n\ndef depth_lr_init_interplay(depth, lr, init_weights):\n n_epochs = 600\n\n x_train, y_train = gen_samples(100, 2.0, 0.1)\n model = DeepNarrowLNN(np.full((1, depth+1), init_weights))\n\n plt.figure(figsize=(10, 5))\n plt.plot(model.train(x_train, y_train, lr, n_epochs),\n linewidth=3.0, c='m')\n\n plt.title(\"Training a {}-layer LNN with\"\n \" $\\eta=${} initialized with $w_i=${}\".format(depth, lr, init_weights), pad=15)\n plt.yscale('log')\n plt.xlabel('epochs')\n plt.ylabel('Log mean squared error')\n plt.ylim(0.001, 1.0)\n plt.show()\n\n\ndef plot_init_effect():\n depth = 15\n n_epochs = 250\n lr = 0.02\n\n x_train, y_train = gen_samples(100, 2.0, 0.1)\n\n plt.figure(figsize=(12, 6))\n for init_w in np.arange(0.7, 1.09, 0.05):\n model = DeepNarrowLNN(np.full((1, depth), init_w))\n plt.plot(model.train(x_train, y_train, lr, n_epochs),\n linewidth=3.0, label=\"initial weights {:.2f}\".format(init_w))\n plt.title(\"Training a {}-layer narrow LNN with $\\eta=${}\".format(depth, lr), pad=15)\n plt.yscale('log')\n plt.xlabel('epochs')\n plt.ylabel('Log mean squared error')\n plt.legend(loc='lower left', ncol=4)\n plt.ylim(0.001, 1.0)\n plt.show()\n\n\nclass InterPlay:\n def __init__(self):\n self.lr = [None]\n self.depth = [None]\n self.success = [None]\n self.min_depth, self.max_depth = 5, 65\n self.depth_list = np.arange(10, 61, 10)\n self.i_depth = 0\n self.min_lr, self.max_lr = 0.001, 0.105\n self.n_epochs = 600\n self.x_train, self.y_train = gen_samples(100, 2.0, 0.1)\n self.converged = False\n self.button = None\n self.slider = None\n\n def train(self, lr, update=False, init_weights=0.9):\n if update and self.converged and self.i_depth < len(self.depth_list):\n depth = self.depth_list[self.i_depth]\n self.plot(depth, lr)\n self.i_depth += 1\n self.lr.append(None)\n self.depth.append(None)\n self.success.append(None)\n self.converged = False\n self.slider.value = 0.005\n if self.i_depth < len(self.depth_list):\n self.button.value = False\n self.button.description = 'Explore!'\n self.button.disabled = True\n self.button.button_style = 'danger'\n else:\n self.button.value = False\n self.button.button_style = ''\n self.button.disabled = True\n self.button.description = 'Done!'\n time.sleep(1.0)\n\n elif self.i_depth < len(self.depth_list):\n depth = self.depth_list[self.i_depth]\n # assert self.min_depth <= depth <= self.max_depth\n assert self.min_lr <= lr <= self.max_lr\n self.converged = False\n\n model = DeepNarrowLNN(np.full((1, depth), init_weights))\n self.losses = np.array(model.train(self.x_train, self.y_train, lr, self.n_epochs))\n if np.any(self.losses < 1e-2):\n success = np.argwhere(self.losses < 1e-2)[0][0]\n if np.all((self.losses[success:] < 1e-2)):\n self.converged = True\n self.success[-1] = success\n self.lr[-1] = lr\n self.depth[-1] = depth\n self.button.disabled = False\n self.button.button_style = 'success'\n self.button.description = 'Register!'\n else:\n self.button.disabled = True\n self.button.button_style = 'danger'\n self.button.description = 'Explore!'\n else:\n self.button.disabled = True\n self.button.button_style = 'danger'\n self.button.description = 'Explore!'\n self.plot(depth, lr)\n\n def plot(self, depth, lr):\n fig = plt.figure(constrained_layout=False, figsize=(10, 8))\n gs = fig.add_gridspec(2, 2)\n ax1 = fig.add_subplot(gs[0, :])\n ax2 = fig.add_subplot(gs[1, 0])\n ax3 = fig.add_subplot(gs[1, 1])\n\n ax1.plot(self.losses, linewidth=3.0, c='m')\n ax1.set_title(\"Training a {}-layer LNN with\"\n \" $\\eta=${}\".format(depth, lr), pad=15, fontsize=16)\n ax1.set_yscale('log')\n ax1.set_xlabel('epochs')\n ax1.set_ylabel('Log mean squared error')\n ax1.set_ylim(0.001, 1.0)\n\n ax2.set_xlim(self.min_depth, self.max_depth)\n ax2.set_ylim(-10, self.n_epochs)\n ax2.set_xlabel('Depth')\n ax2.set_ylabel('Learning time (Epochs)')\n ax2.set_title(\"Learning time vs depth\", fontsize=14)\n ax2.scatter(np.array(self.depth), np.array(self.success), c='r')\n\n # ax3.set_yscale('log')\n ax3.set_xlim(self.min_depth, self.max_depth)\n ax3.set_ylim(self.min_lr, self.max_lr)\n ax3.set_xlabel('Depth')\n ax3.set_ylabel('Optimial learning rate')\n ax3.set_title(\"Empirically optimal $\\eta$ vs depth\", fontsize=14)\n ax3.scatter(np.array(self.depth), np.array(self.lr), c='r')\n\n plt.show()\n```\n\n\n```python\n# @title Helper functions\n\natform = AirtableForm('appn7VdPRseSoMXEG','W1D2_T2','https://portal.neuromatchacademy.org/api/redirect/to/9c55f6cb-cdf9-4429-ac1c-ec44fe64c303')\n\n\ndef gen_samples(n, a, sigma):\n \"\"\"\n Generates `n` samples with `y = z * x + noise(sgma)` linear relation.\n\n Args:\n n : int\n a : float\n sigma : float\n Retutns:\n x : np.array\n y : np.array\n \"\"\"\n assert n > 0\n assert sigma >= 0\n\n if sigma > 0:\n x = np.random.rand(n)\n noise = np.random.normal(scale=sigma, size=(n))\n y = a * x + noise\n else:\n x = np.linspace(0.0, 1.0, n, endpoint=True)\n y = a * x\n return x, y\n\n\nclass ShallowNarrowLNN:\n \"\"\"\n Shallow and narrow (one neuron per layer) linear neural network\n \"\"\"\n def __init__(self, init_ws):\n \"\"\"\n init_ws: initial weights as a list\n \"\"\"\n assert isinstance(init_ws, list)\n assert len(init_ws) == 2\n self.w1 = init_ws[0]\n self.w2 = init_ws[1]\n\n def forward(self, x):\n \"\"\"\n The forward pass through netwrok y = x * w1 * w2\n \"\"\"\n y = x * self.w1 * self.w2\n return y\n\n def loss(self, y_p, y_t):\n \"\"\"\n Mean squared error (L2) with 1/2 for convenience\n \"\"\"\n assert y_p.shape == y_t.shape\n mse = ((y_t - y_p)**2).mean()\n return mse\n\n def dloss_dw(self, x, y_t):\n \"\"\"\n partial derivative of loss with respect to weights\n\n Args:\n x : np.array\n y_t : np.array\n \"\"\"\n assert x.shape == y_t.shape\n Error = y_t - self.w1 * self.w2 * x\n dloss_dw1 = - (2 * self.w2 * x * Error).mean()\n dloss_dw2 = - (2 * self.w1 * x * Error).mean()\n return dloss_dw1, dloss_dw2\n\n def train(self, x, y_t, eta, n_ep):\n \"\"\"\n Gradient descent algorithm\n\n Args:\n x : np.array\n y_t : np.array\n eta: float\n n_ep : int\n \"\"\"\n assert x.shape == y_t.shape\n\n loss_records = np.empty(n_ep) # pre allocation of loss records\n weight_records = np.empty((n_ep, 2)) # pre allocation of weight records\n\n for i in range(n_ep):\n y_p = self.forward(x)\n loss_records[i] = self.loss(y_p, y_t)\n dloss_dw1, dloss_dw2 = self.dloss_dw(x, y_t)\n self.w1 -= eta * dloss_dw1\n self.w2 -= eta * dloss_dw2\n weight_records[i] = [self.w1, self.w2]\n\n return loss_records, weight_records\n\n\nclass DeepNarrowLNN:\n \"\"\"\n Deep but thin (one neuron per layer) linear neural network\n \"\"\"\n def __init__(self, init_ws):\n \"\"\"\n init_ws: initial weights as a numpy array\n \"\"\"\n self.n = init_ws.size\n self.W = init_ws.reshape(1, -1)\n\n def forward(self, x):\n \"\"\"\n x : np.array\n input features\n \"\"\"\n y = np.prod(self.W) * x\n return y\n\n def loss(self, y_t, y_p):\n \"\"\"\n mean squared error (L2 loss)\n\n Args:\n y_t : np.array\n y_p : np.array\n \"\"\"\n assert y_p.shape == y_t.shape\n mse = ((y_t - y_p)**2 / 2).mean()\n return mse\n\n def dloss_dw(self, x, y_t, y_p):\n \"\"\"\n analytical gradient of weights\n\n Args:\n x : np.array\n y_t : np.array\n y_p : np.array\n \"\"\"\n E = y_t - y_p # = y_t - x * np.prod(self.W)\n Ex = np.multiply(x, E).mean()\n Wp = np.prod(self.W) / (self.W + 1e-9)\n dW = - Ex * Wp\n return dW\n\n def train(self, x, y_t, eta, n_epochs):\n \"\"\"\n training using gradient descent\n\n Args:\n x : np.array\n y_t : np.array\n eta: float\n n_epochs : int\n \"\"\"\n loss_records = np.empty(n_epochs)\n loss_records[:] = np.nan\n for i in range(n_epochs):\n y_p = self.forward(x)\n loss_records[i] = self.loss(y_t, y_p).mean()\n dloss_dw = self.dloss_dw(x, y_t, y_p)\n if np.isnan(dloss_dw).any() or np.isinf(dloss_dw).any():\n return loss_records\n self.W -= eta * dloss_dw\n return loss_records\n```\n\n\n```python\n#@title Set random seed\n\n#@markdown Executing `set_seed(seed=seed)` you are setting the seed\n\n# for DL its critical to set the random seed so that students can have a\n# baseline to compare their results to expected results.\n# Read more here: https://pytorch.org/docs/stable/notes/randomness.html\n\n# Call `set_seed` function in the exercises to ensure reproducibility.\nimport random\nimport torch\n\ndef set_seed(seed=None, seed_torch=True):\n if seed is None:\n seed = np.random.choice(2 ** 32)\n random.seed(seed)\n np.random.seed(seed)\n if seed_torch:\n torch.manual_seed(seed)\n torch.cuda.manual_seed_all(seed)\n torch.cuda.manual_seed(seed)\n torch.backends.cudnn.benchmark = False\n torch.backends.cudnn.deterministic = True\n\n print(f'Random seed {seed} has been set.')\n\n\n# In case that `DataLoader` is used\ndef seed_worker(worker_id):\n worker_seed = torch.initial_seed() % 2**32\n np.random.seed(worker_seed)\n random.seed(worker_seed)\n```\n\n\n```python\n#@title Set device (GPU or CPU). Execute `set_device()`\n# especially if torch modules used.\n\n# inform the user if the notebook uses GPU or CPU.\n\ndef set_device():\n device = \"cuda\" if torch.cuda.is_available() else \"cpu\"\n if device != \"cuda\":\n print(\"GPU is not enabled in this notebook. \\n\"\n \"If you want to enable it, in the menu under `Runtime` -> \\n\"\n \"`Hardware accelerator.` and select `GPU` from the dropdown menu\")\n else:\n print(\"GPU is enabled in this notebook. \\n\"\n \"If you want to disable it, in the menu under `Runtime` -> \\n\"\n \"`Hardware accelerator.` and select `None` from the dropdown menu\")\n\n return device\n```\n\n\n```python\nSEED = 2021\nset_seed(seed=SEED)\nDEVICE = set_device()\n```\n\n Random seed 2021 has been set.\n GPU is enabled in this notebook. \n If you want to disable it, in the menu under `Runtime` -> \n `Hardware accelerator.` and select `None` from the dropdown menu\n\n\n---\n# Section 1: A Shallow Narrow Linear Neural Network\n\n*Time estimate: ~30 mins*\n\n\n```python\n# @title Video 1: Shallow Narrow Linear Net\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1F44y117ot\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"6e5JIYsqVvU\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('video 1: Shallow Narrow Linear Net')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n## Section 1.1: A Shallow Narrow Linear Net\n\nTo better understand the behavior of neural network training with gradient descent, we start with the incredibly simple case of a shallow narrow linear neural net, since state-of-the-art models are impossible to dissect and comprehend with our current mathematical tools.\n\nThe model we use has one hidden layer, with only one neuron, and two weights. We consider the squared error (or L2 loss) as the cost function. As you may have already guessed, we can visualize the model as a neural network:\n\n
    \n\n
    \n\nor by its computation graph:\n\n
    \n\nor on a rare occasion, even as a reasonably compact mapping:\n\n$$ loss = (y - w_1 \\cdot w_2 \\cdot x)^2 $$\n\n
    \n\nImplementing a neural network from scratch without using any Automatic Differentiation tool is rarely necessary. The following two exercises are therefore **Bonus** (optional) exercises. Please ignore them if you have any time-limits or pressure and continue to Section 1.2.\n\n### Analytical Exercise 1.1: Loss Gradients (Optional)\n\nOnce again, we ask you to calculate the network gradients analytically, since you will need them for the next exercise. We understand how annoying this is.\n\n$\\dfrac{\\partial{loss}}{\\partial{w_1}} = ?$\n\n$\\dfrac{\\partial{loss}}{\\partial{w_2}} = ?$\n\n
    \n\n---\n#### Solution\n\n$\\dfrac{\\partial{loss}}{\\partial{w_1}} = -2 \\cdot w_2 \\cdot x \\cdot (y - w_1 \\cdot w_2 \\cdot x)$\n\n$\\dfrac{\\partial{loss}}{\\partial{w_2}} = -2 \\cdot w_1 \\cdot x \\cdot (y - w_1 \\cdot w_2 \\cdot x)$\n\n---\n\n\n### Coding Exercise 1.1: Implement simple narrow LNN (Optional)\n\nNext, we ask you to implement the `forward` pass for our model from scratch without using PyTorch.\n\nAlso, although our model gets a single input feature and outputs a single prediction, we could calculate the loss and perform training for multiple samples at once. This is the common practice for neural networks, since computers are incredibly fast doing matrix (or tensor) operations on batches of data, rather than processing samples one at a time through `for` loops. Therefore, for the `loss` function, please implement the **mean** squared error (MSE), and adjust your analytical gradients accordingly when implementing the `dloss_dw` function.\n\nFinally, complete the `train` function for the gradient descent algorithm:\n\n\\begin{equation}\n\\mathbf{w}^{(t+1)} = \\mathbf{w}^{(t)} - \\eta \\nabla loss (\\mathbf{w}^{(t)})\n\\end{equation}\n\n\n```python\nclass ShallowNarrowExercise:\n \"\"\"Shallow and narrow (one neuron per layer) linear neural network\n \"\"\"\n def __init__(self, init_weights):\n \"\"\"\n Args:\n init_weights (list): initial weights\n \"\"\"\n assert isinstance(init_weights, (list, np.ndarray, tuple))\n assert len(init_weights) == 2\n self.w1 = init_weights[0]\n self.w2 = init_weights[1]\n\n\n def forward(self, x):\n \"\"\"The forward pass through netwrok y = x * w1 * w2\n\n Args:\n x (np.ndarray): features (inputs) to neural net\n\n returns:\n (np.ndarray): neural network output (prediction)\n \"\"\"\n #################################################\n ## Implement the forward pass to calculate prediction\n ## Note that prediction is not the loss\n # Complete the function and remove or comment the line below\n #raise NotImplementedError(\"Forward Pass `forward`\")\n #################################################\n y = self.w1 * self.w2 * x \n return y\n\n\n def dloss_dw(self, x, y_true):\n \"\"\"Gradient of loss with respect to weights\n\n Args:\n x (np.ndarray): features (inputs) to neural net\n y_true (np.ndarray): true labels\n\n returns:\n (float): mean gradient of loss with respect to w1\n (float): mean gradient of loss with respect to w2\n \"\"\"\n assert x.shape == y_true.shape\n #################################################\n ## Implement the gradient computation function\n # Complete the function and remove or comment the line below\n #raise NotImplementedError(\"Gradient of Loss `dloss_dw`\")\n #################################################\n dloss_dw1 = np.mean(-2 * self.w2 * x * (y_true - self.w1 * self.w2 * x))\n dloss_dw2 = np.mean(-2 * self.w1 * x * (y_true - self.w1 * self.w2 * x))\n return dloss_dw1, dloss_dw2\n\n\n def train(self, x, y_true, lr, n_ep):\n \"\"\"Training with Gradient descent algorithm\n\n Args:\n x (np.ndarray): features (inputs) to neural net\n y_true (np.ndarray): true labels\n lr (float): learning rate\n n_ep (int): number of epochs (training iterations)\n\n returns:\n (list): training loss records\n (list): training weight records (evolution of weights)\n \"\"\"\n assert x.shape == y_true.shape\n\n loss_records = np.empty(n_ep) # pre allocation of loss records\n weight_records = np.empty((n_ep, 2)) # pre allocation of weight records\n\n for i in range(n_ep):\n y_prediction = self.forward(x)\n loss_records[i] = loss(y_prediction, y_true)\n dloss_dw1, dloss_dw2 = self.dloss_dw(x, y_true)\n #################################################\n ## Implement the gradient descent step\n # Complete the function and remove or comment the line below\n #raise NotImplementedError(\"Training loop `train`\")\n #################################################\n self.w1 -= lr * dloss_dw1\n self.w2 -= lr * dloss_dw2 \n weight_records[i] = [self.w1, self.w2]\n\n return loss_records, weight_records\n\n\ndef loss(y_prediction, y_true):\n \"\"\"Mean squared error\n\n Args:\n y_prediction (np.ndarray): model output (prediction)\n y_true (np.ndarray): true label\n\n returns:\n (np.ndarray): mean squared error loss\n \"\"\"\n assert y_prediction.shape == y_true.shape\n #################################################\n ## Implement the MEAN squared error\n # Complete the function and remove or comment the line below\n #raise NotImplementedError(\"Loss function `loss`\")\n #################################################\n mse = np.mean((y_prediction - y_true) ** 2)\n return mse\n\n\n#add event to airtable\natform.add_event('Coding Exercise 1.1: Implement simple narrow LNN')\n\nset_seed(seed=SEED)\nn_epochs = 211\nlearning_rate = 0.02\ninitial_weights = [1.4, -1.6]\nx_train, y_train = gen_samples(n=73, a=2.0, sigma=0.2)\nx_eval = np.linspace(0.0, 1.0, 37, endpoint=True)\n## Uncomment to run\nsn_model = ShallowNarrowExercise(initial_weights)\nloss_log, weight_log = sn_model.train(x_train, y_train, learning_rate, n_epochs)\ny_eval = sn_model.forward(x_eval)\nplot_x_y_(x_train, y_train, x_eval, y_eval, loss_log, weight_log)\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content-dl/tree/main//tutorials/W1D2_LinearDeepLearning/solutions/W1D2_Tutorial2_Solution_46492cd6.py)\n\n*Example output:*\n\n\n\n\n\n## Section 1.2: Learning landscapes\n\n\n```python\n# @title Video 2: Training Landscape\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1Nv411J71X\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"k28bnNAcOEg\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 2: Training Landscape')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nAs you may have already asked yourself, we can analytically find $w_1$ and $w_2$ without using gradient descent:\n\n\\begin{equation}\nw_1 \\cdot w_2 = \\dfrac{y}{x}\n\\end{equation}\n\nIn fact, we can plot the gradients, the loss function and all the possible solutions in one figure. In this example, we use the $y = 1x$ mapping:\n\n**Blue ribbon**: shows all possible solutions: $~ w_1 w_2 = \\dfrac{y}{x} = \\dfrac{x}{x} = 1 \\Rightarrow w_1 = \\dfrac{1}{w_2}$\n\n**Contour background**: Shows the loss values, red being higher loss\n\n**Vector field (arrows)**: shows the gradient vector field. The larger yellow arrows show larger gradients, which correspond to bigger steps by gradient descent.\n\n**Scatter circles**: the trajectory (evolution) of weights during training for three different initializations, with blue dots marking the start of training and red crosses ( **x** ) marking the end of training. You can also try your own initializations (keep the initial values between `-2.0` and `2.0`) as shown here:\n```python\nplot_vector_field('all', [1.0, -1.0])\n```\n\nFinally, if the plot is too crowded, feel free to pass one of the following strings as argument:\n\n```python\nplot_vector_field('vectors') # for vector field\nplot_vector_field('trajectory') # for training trajectory\nplot_vector_field('loss') # for loss contour\n```\n\n**Think!**\n\nExplore the next two plots. Try different initial values. Can you find the saddle point? Why does training slow down near the minima?\n\n\n```python\nplot_vector_field('all') # for training trajectory\n```\n\nHere, we also visualize the loss landscape in a 3-D plot, with two training trajectories for different initial conditions.\nNote: the trajectories from the 3D plot and the previous plot are independent and different.\n\n\n```python\nplot_loss_landscape()\n```\n\n\n```python\n# @title Student Response\nfrom ipywidgets import widgets\n\n\ntext=widgets.Textarea(\n value='Type your here and Push submit',\n placeholder='Type something',\n description='',\n disabled=False\n)\n\nbutton = widgets.Button(description=\"Submit!\")\n\ndisplay(text,button)\n\ndef on_button_clicked(b):\n atform.add_answer('q1', text.value)\n print(\"Submission successful!\")\n\nbutton.on_click(on_button_clicked)\n```\n\n\n Textarea(value='Type your here and Push submit', placeholder='Type something')\n\n\n\n Button(description='Submit!', style=ButtonStyle())\n\n\n\n```python\n# @title Video 3: Training Landscape - Discussion\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1py4y1j7cv\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"0EcUGgxOdkI\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 3: Training Landscape - Discussiond')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n---\n# Section 2: Depth, Learning rate, and initialization\n*Time estimate: ~45 mins*\n\nSuccessful deep learning models are often developed by a team of very clever people, spending many many hours \"tuning\" learning hyperparameters, and finding effective initializations. In this section, we look at three basic (but often not simple) hyperparameters: depth, learning rate, and initialization.\n\n## Section 2.1: The effect of depth\n\n\n```python\n# @title Video 4: Effect of Depth\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1z341167di\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"Ii_As9cRR5Q\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 4: Effect of Depth')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nWhy might depth be useful? What makes a network or learning system \"deep\"? The reality is that shallow neural nets are often incapable of learning complex functions due to data limitations. On the other hand, depth seems like magic. Depth can change the functions a network can represent, the way a network learns, and how a network generalizes to unseen data. \n\nSo let's look at the challenges that depth poses in training a neural network. Imagine a single input, single output linear network with 50 hidden layers and only one neuron per layer (i.e. a narrow deep neural network). The output of the network is easy to calculate:\n\n$$ prediction = x \\cdot w_1 \\cdot w_2 \\cdot \\cdot \\cdot w_{50} $$\n\nIf the initial value for all the weights is $w_i = 2$, the prediction for $x=1$ would be **exploding**: $y_p = 2^{50} \\approx 1.1256 \\times 10^{15}$. On the other hand, for weights initialized to $w_i = 0.5$, the output is **vanishing**: $y_p = 0.5^{50} \\approx 8.88 \\times 10^{-16}$. Similarly, if we recall the chain rule, as the graph gets deeper, the number of elements in the chain multiplication increases, which could lead to exploding or vanishing gradients. To avoid such numerical vulnerablities that could impair our training algorithm, we need to understand the effect of depth.\n\n\n### Interactive Demo 2.1: Depth widget\n\nUse the widget to explore the impact of depth on the training curve (loss evolution) of a deep but narrow neural network.\n\n**Think!**\n\nWhich networks trained the fastest? Did all networks eventually \"work\" (converge)? What is the shape of their learning trajectory?\n\n\n```python\n# @markdown Make sure you execute this cell to enable the widget!\n\n_ = interact(depth_widget,\n depth = IntSlider(min=1, max=100,\n step=5, value=0,\n continuous_update=False))\n```\n\n\n interactive(children=(IntSlider(value=1, continuous_update=False, description='depth', min=1, step=5), Output(\u2026\n\n\n\n```python\n# @title Video 5: Effect of Depth - Discussion\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1Qq4y1H7uk\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"EqSDkwmSruk\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 5: Effect of Depth - Discussion')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n## Section 2.2: Choosing a learning rate\n\nThe learning rate is a common hyperparameter for most optimization algorithms. How should we set it? Sometimes the only option is to try all the possibilities, but sometimes knowing some key trade-offs will help guide our search for good hyperparameters.\n\n\n```python\n# @title Video 6: Learning Rate\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV11f4y157MT\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"w_GrCVM-_Qo\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 6: Learning Rate')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n### Interactive Demo 2.2: Learning rate widget\n\nHere, we fix the network depth to 50 layers. Use the widget to explore the impact of learning rate $\\eta$ on the training curve (loss evolution) of a deep but narrow neural network.\n\n**Think!**\n\nCan we say that larger learning rates always lead to faster learning? Why not? \n\n\n```python\n# @markdown Make sure you execute this cell to enable the widget!\n\n_ = interact(lr_widget,\n lr = FloatSlider(min=0.005, max=0.045, step=0.005, value=0.005,\n continuous_update=False, readout_format='.3f',\n description='eta'))\n```\n\n\n interactive(children=(FloatSlider(value=0.005, continuous_update=False, description='eta', max=0.045, min=0.00\u2026\n\n\n\n```python\n# @title Video 7: Learning Rate - Discussion\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1Aq4y1p7bh\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"cmS0yqImz2E\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 7: Learning Rate - Discussion')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n## Section 2.3: Depth vs Learning Rate\n\n\n```python\n# @title Video 8: Depth and Learning Rate\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1V44y1177e\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"J30phrux_3k\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 8: Depth and Learning Rate')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n### Interactive Demo 2.3: Depth and Learning-Rate\n\n\n**Important instruction**\nThe exercise starts with 10 hidden layers. Your task is to find the learning rate that delivers fast but robust convergence (learning). When you are confident about the learning rate, you can **Register** the optimal learning rate for the given depth. Once you press register, a deeper model is instantiated, so you can find the next optimal learning rate. The Register button turns green only when the training converges, but does not imply the fastest convergence. Finally, be patient :) the widgets are slow.\n\n\n**Think!**\n\nCan you explain the relationship between the depth and optimal learning rate?\n\n\n```python\n# @markdown Make sure you execute this cell to enable the widget!\nintpl_obj = InterPlay()\n\nintpl_obj.slider = FloatSlider(min=0.005, max=0.105, step=0.005, value=0.005,\n layout=Layout(width='500px'),\n continuous_update=False,\n readout_format='.3f',\n description='eta')\n\nintpl_obj.button = ToggleButton(value=intpl_obj.converged, description='Register')\n\nwidgets_ui = HBox([intpl_obj.slider, intpl_obj.button])\nwidgets_out = interactive_output(intpl_obj.train,\n {'lr': intpl_obj.slider,\n 'update': intpl_obj.button,\n 'init_weights': fixed(0.9)})\n\ndisplay(widgets_ui, widgets_out)\n```\n\n\n HBox(children=(FloatSlider(value=0.005, continuous_update=False, description='eta', layout=Layout(width='500px\u2026\n\n\n\n Output()\n\n\n\n```python\n# @title Video 9: Depth and Learning Rate - Discussion\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV15q4y1p7Uq\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"7Fl8vH7cgco\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 9: Depth and Learning Rate - Discussion')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n## Section 2.4: Why initialization is important\n\n\n```python\n# @title Video 10: Initialization Matters\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1UL411J7vu\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"KmqCz95AMzY\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 10: Initialization Matters')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nWe\u2019ve seen, even in the simplest of cases, that depth can slow learning. Why? From the chain rule, gradients are multiplied by the current weight at each layer, so the product can vanish or explode. Therefore, weight initialization is a fundamentally important hyperparameter.\n\nAlthough in practice initial values for learnable parameters are often sampled from different $\\mathcal{Uniform}$ or $\\mathcal{Normal}$ probability distribution, here we use a single value for all the parameters.\n\nThe figure below shows the effect of initialization on the speed of learning for the deep but narrow LNN. We have excluded initializations that lead to numerical errors such as `nan` or `inf`, which are the consequence of smaller or larger initializations.\n\n\n```python\n# @markdown Make sure you execute this cell to see the figure!\n\nplot_init_effect()\n```\n\n\n```python\n# @title Video 11: Initialization Matters Explained\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1hM4y1T7gJ\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"vKktGdiQDsE\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 11: Initialization Matters Explained')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n---\n# Summary\n\nIn the second tutorial, we have learned what is the training landscape, and also we have see in depth the effect of the depth of the network and the learning rate, and their interplay. Finally, we have seen that initialization matters and why we need smart ways of initialization.\n\n\n```python\n# @title Video 12: Tutorial 2 Wrap-up\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1P44y117Pd\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"r3K8gtak3wA\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 12: Tutorial 2 Wrap-up')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n\n```python\n\n# @title Airtable Submission Link\nfrom IPython import display as IPydisplay\nIPydisplay.HTML(\n f\"\"\"\n
    \n \n \n
    \"\"\" )\n```\n\n\n\n\n\n
    \n \n \n
    \n\n\n\n---\n# Bonus\n\n## Hyperparameter interaction\n\nFinally, let's put everything we learned together and find best initial weights and learning rate for a given depth. By now you should have learned the interactions and know how to find the optimal values quickly. If you get `numerical overflow` warnings, don't be discouraged! They are often caused by \"exploding\" or \"vanishing\" gradients.\n\n**Think!**\n\nDid you experience any surprising behaviour \nor difficulty finding the optimal parameters?\n\n\n```python\n# @markdown Make sure you execute this cell to enable the widget!\n\n_ = interact(depth_lr_init_interplay,\n depth = IntSlider(min=10, max=51, step=5, value=25,\n continuous_update=False),\n lr = FloatSlider(min=0.001, max=0.1,\n step=0.005, value=0.005,\n continuous_update=False,\n readout_format='.3f',\n description='eta'),\n init_weights = FloatSlider(min=0.1, max=3.0,\n step=0.1, value=0.9,\n continuous_update=False,\n readout_format='.3f',\n description='initial weights'))\n```\n\n\n interactive(children=(IntSlider(value=25, continuous_update=False, description='depth', max=51, min=10, step=5\u2026\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "d100992ea47400175733e28cdc427879a3bf52d3", "size": 931037, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W1D2_LinearDeepLearning/student/ED_W1D2_Tutorial2.ipynb", "max_stars_repo_name": "eduardojdiniz/course-content-dl", "max_stars_repo_head_hexsha": "8d66641683651bce7b0179b6d890aef5a048a8b9", "max_stars_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tutorials/W1D2_LinearDeepLearning/student/ED_W1D2_Tutorial2.ipynb", "max_issues_repo_name": "eduardojdiniz/course-content-dl", "max_issues_repo_head_hexsha": "8d66641683651bce7b0179b6d890aef5a048a8b9", "max_issues_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/W1D2_LinearDeepLearning/student/ED_W1D2_Tutorial2.ipynb", "max_forks_repo_name": "eduardojdiniz/course-content-dl", "max_forks_repo_head_hexsha": "8d66641683651bce7b0179b6d890aef5a048a8b9", "max_forks_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 410.872462489, "max_line_length": 460156, "alphanum_fraction": 0.9323474792, "converted": true, "num_tokens": 15085, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544085240401, "lm_q2_score": 0.63341027751814, "lm_q1q2_score": 0.4429149289589951}} {"text": "```python\nimport numpy as np\nimport math\nfrom random import random\nimport matplotlib.pyplot as plt\nimport sympy\nfrom random import seed\nimport math \n```\n\n# Question 1 (2 points)\nUse online, library and other resources to answer the following questions. You can use Internet references, library resources, your prior knowledge, and any other resource. For each question, provide references that you used.\n\nProvide two applications in computer science for each of the following methodologies.\n\n\na. Monte Carlo Simulation\\\nb. Markov Decision Process\\\nc. Random Walks\\\nd. Resampling method\n\n\n## Random Walks\nThe Random Walks theory applying in the investment. According to random walk theory, market price variations have the same distribution and are unrelated. As a result, it is concluded that a curent stock price and past movement cannot be used to forecast its future movement. In light of this, random walk theory states that stocks follow an unpredictable and random direction, rendering all methods of stock price prediction useless in the long run. Thus, we consider using random walks model to predict the stock price in the furture.\n\nReference: Phung, A. (2020, August 28). How can random walk theory be applied to investing https://www.investopedia.com/ask/answers/08/random-walk-theory.asp. \n\n## Monte Carlo Simulation\nAccording to this article, Monte Carlo simulation improves project management. Graves (2001) spoke about the various forms of probability distributions that can be used to estimate project durations. The merit of Monte Carlo Simulation model is that it can simplify time management and cost management, and that the PM team can know the consequences of the project without having a huge cost. They will be able to predict project schedules and budgets that are more practical.\n\nReference: Kwak, Y., Ingall, L. Exploring Monte Carlo Simulation Applications for Project Management. *Risk Manag 9, 44*\u201357 (2007). https://doi.org/10.1057/palgrave.rm.8250017\n\n\n\n\n\n\n \n\n# Question 2 (4 points)\nA gambler has the chance to enter a gamble using a fair coin. **The gambler needs to invest $100 to enter the game.** If the coin is tossed a head (p = 0.5), the gambler wins one dollar otherwise losses one dollar. The game stops after N=100 trials (100 times tossing the coin).\n\na) Model this problem and run a simulation model to determine if the gambler is better off getting into this game. Repeat the simulation when the number of trials N = 1000. \n\nBecause the gambler needs to pay \\$100 to join the game at least, so we set the first round if the gambler win, the full balance will be \\$101. Conversely, if the gambler lose, the full balance will be \\$99.\n\nAccording to the question, the minimum rounds to play this game is 100 trals. After we run 100 times, the full balance is \\$106. If we run 1000 times, the full balance is \\$154. Therefore, I highly suggest gambler join the game as soon as possible(p = 0.5).\n\n\n\n\n```python\ndef gamblerCoin(N, p):\n if N < 100:\n print(\"You need to invest $100 to enter the game\")\n return\n seed(1)\n res = []\n res.append(99 if random() < (1-p) else 101) # first time tossing a coin, and needs to invest $100 at the beginning\n for i in range(1, N):\n tossed = -1 if random() < (1-p) else 1 \n tmp = res[i-1] + tossed # previous result + new result\n res.append(tmp) \n plt.plot(res)\n plt.xlabel(\"trials\")\n plt.ylabel(\"profits\")\n plt.show()\n return res[-1]\n\ngamblerCoin(100, 0.5)\n```\n\n\n```python\ngamblerCoin(1000, 0.5)\n```\n\nb) If the dealer uses an unfair coin (p = 0.49), does the gambler enters the game using N=100? Repeat the game when the number of trails N =1000.\n\n- Compared to p = 0.5, the gambler will earn less money if p = 0.49\n- N = 100 does not mean gambler will lose the money (the full balance is still 100)\n- After the model run 1000 trials, the full balance is \\$128. It's not caused a profunded effect for gambler \n\n\n\n```python\ngamblerCoin(100, 0.49)\n```\n\n\n```python\ngamblerCoin(1000, 0.49)\n```\n\nNow, the gambler is given a chance to exit the game only if the remaining dollar in the middle of the game is \\$100 (for example the gambler can exit when 25 tosses left and he owns \\$100). Let us call this situation State100.\n\nc) Model this problem and run a simulation model to estimate the number of times the gambler can exit the game, when N= 100 and N=1000.\n - N = 100, 7 times\n - N = 1000, 44 times \n - The default seed is `1`\n\n\n\n```python\ndef stateOneHundred(N, p):\n State100 = 100\n seed(1)\n count = 0\n res = []\n res.append(99 if random() < (1-p) else 101) # first time tossing a coin, and needs to invest $100 at the beginning\n for i in range(1, N):\n tossed = -1 if random() < (1-p) else 1 \n tmp = res[i-1] + tossed # previous result + new result\n res.append(tmp)\n if tmp == State100:\n count += 1\n return count\n\nprint(stateOneHundred(100, 0.5))\nprint(stateOneHundred(1000, 0.5))\n```\n\n 7\n 44\n\n\nd) Is there any difference between different State100s? For example, do you see the gambler makes different decision when there are 20 tossing coin left compared with 2 tossing coin left.\n\nYes. for example, if 20 tossing coins left maybe the gambler is able to earn \\$20. However the gambler will also have a chance to lose \\$20 correspondingly. On the other hand, if the situation is 2 tossing coins left, the gambler only has a chance to earn \\$2 or lose \\$2 at most. Thus, the expected values are different, meaning that the gambler has a different choice to get the final result.\n\n# Question 3 (5 points)\nWe want to simulate a dollar slot machine that has three reels (spinning cylinders). To play the game, we first insert \\$1 and pull the handle. All three reels then spin independently, and each reel will stop randomly on one of the five pictures it contains. Once the reels stop, we examine the three pictures and determine our winnings. If we have exactly 2 cherries, we win \\$1 (so we break even). If we have 3 bells, we win \\$2. If we have 3 cherries, we win \\$3. For all other outcomes, we win nothing. The composition of each reel is summarized below.\n\na. Provide details for a strategy to determine what picture each reel stops on, given a U (0,1) number. Detailed strategy means that you need to divide the uniform variable range U(0,1) into sub-intervals and assign pictures. For example, if the random variable is between 0 and x (x is between 0 and 1) then it is a cherry, etc.\n\n\n\n---\nb. Complete the following table to simulate 5 replications of this system. What is the average profit over these five games?\n- The average profit over these five games is negative 0.4.\n\n![3b.png](data:image/png;base64,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\n\nc. Assume that the gambler starts with \\$100. Write a Python code to determine the expected value of the player after 50 rounds of the game.\n- The `n` parameter specifies the number of rounds in the game, while the `profits` parameter specifies how much the gambler wants to start with. The result then shows your how much money do you have after game. The probability of a slot machine is used to create this model.\n- please run the model to see the expected value below. \n\n\n```python\ndef slotMachine(n, profits):\n if profits <= 0:\n print(\"You cannot play the game\") # must have money to enter the game\n return\n reelOne, reelTwo, reelThree = [], [], [] # three reels\n first = np.random.uniform(0, 1, n) # random selected between 0 to 1\n # set the probability for each picture\n for i in first:\n if i <= 0.4:\n reelOne.append(\"Lemon\")\n elif i > 0.4 and i <= 0.6:\n reelOne.append(\"Bell\")\n else:\n reelOne.append(\"Cherry\")\n second = np.random.uniform(0, 1, n) # random selected between 0 to 1\n for i in second:\n if i <= 0.6:\n reelTwo.append(\"Lemon\")\n elif i > 0.6 and i <= 0.8:\n reelTwo.append(\"Bell\")\n else:\n reelTwo.append(\"Cherry\")\n third = np.random.uniform(0, 1, n) # random selected between 0 to 1\n for i in third:\n if i <= 0.4:\n reelThree.append(\"Lemon\")\n elif i > 0.4 and i <= 0.8:\n reelThree.append(\"Bell\")\n else:\n reelThree.append(\"Cherry\")\n zipped = zip(reelOne, reelTwo, reelThree) # zipped the results\n rounds = list(zipped)\n \n values = 0 # earn or lose the money per time (insert $1 for play game so needs minus 1 each time)\n for i in range(len(rounds)):\n if rounds[i][0] and rounds[i][1] and rounds[i][2] == \"Cherry\":\n values += 3\n values -= 1\n elif rounds[i][0] and rounds[i][1] or rounds[i][0] and rounds[i][2] or rounds[i][1] and rounds[i][2] == \"Cherry\":\n values += 1\n values -= 1\n elif rounds[i][0] and rounds[i][1] and rounds[i][2] == \"Bell\":\n values += 2\n values -= 1\n else:\n values += 0\n values -= 1\n return values+profits # net profits + your money\n\nslotMachine(50, 100)\n```\n\n\n\n\n 120\n\n\n\n# Question 4 (5 points)\nConsider the integral\na. What distribution function this probability distribution function represents? Use Normal distribution table to determine the \u201cexact\u201d value of I? (Show your work to receive credit. I\u2019m not testing whether your calculator can integrate this numerically!)\n\nb. Evaluate I using Monte Carlo integration using a Python code and U(0,1) random numbers. (Provide the details of the Python code.)\n\n- The blue points that is under the curve.\n- The yellow points that is above the function.\n\n- Reference: https://stackoverflow.com/questions/30030659/in-python-what-is-the-difference-between-random-uniform-and-random-random\n\n\n```python\nimport random \nimport numpy as np\nimport matplotlib.pyplot as plt\n\nrandom.seed(2)\nf = lambda x:(1/math.sqrt(2*np.pi))*np.exp(-x**2/2) # equation\na = 1\nb = 1.5\nNumSteps = 1000000 \nXIntegral=[] \nYIntegral=[]\nXRectangle=[] \nYRectangle=[]\n\nymin = f(a)\nymax = ymin\n# extract min and max without repeated min/max\nfor i in range(NumSteps):\n x = a + (b - a) * float(i) / NumSteps # calculate the rectangle area\n y = f(x)\n if y < ymin: ymin = y\n if y > ymax: ymax = y\n\n# Monte Carlo method\nA = (b - a) * (ymax - ymin)\nN = 1000000 \nM = 0\nfor k in range(N):\n x = a + (b - a) * random.uniform(0, 1)\n y = ymin + (ymax - ymin) * random.uniform(0, 1)\n if y <= f(x):\n M += 1 \n XIntegral.append(x)\n YIntegral.append(y) \n else:\n XRectangle.append(x) \n YRectangle.append(y) \nNumericalIntegral = M / N * A\nprint (\"Numerical integration I = \" + str(NumericalIntegral))\n\nXLin=np.linspace(a,b)\nYLin=[]\nfor x in XLin:\n YLin.append(f(x))\n\n# plt.axis ([a, b, 0, f(b)]) \nplt.plot (XLin,YLin, color=\"red\" , linewidth=\"4\") \nplt.scatter(XIntegral, YIntegral, color=\"blue\", marker =\".\") # visual the points\nplt.scatter(XRectangle, YRectangle, color=\"yellow\", marker =\".\") # visual the points\nplt.title (\"Numerical Integration using Monte Carlo method\")\nplt.show()\n```\n\n# Question 5 (4 points)\nUsing the Gradient Descent algorithm and the Newton-Raphson method in Python to calculate the maximum/minimum value of\n\n$$\n\ud835\udc66 = 3\ud835\udc65^4 \u2212 5\ud835\udc65+2\n$$\nUse Python codes to determine the possible local maximum/minimum of this function.\n\n- According to the plot, we need to find the **minimum value** of the equation.\n\n\n```python\ndef gradientDescent(Xvalue):\n x = np.linspace(0, 1.2, 100) \n y= 3*x**4-5*x+2\n\n # visualize the equation in order to confirm we need to find the maximum or the minimum value\n fig = plt.figure()\n axdef = fig.add_subplot(1, 1, 1)\n axdef.spines['left'].set_position('center')\n axdef.spines['bottom'].set_position('zero')\n axdef.spines['right'].set_color('none')\n axdef.spines['top'].set_color('none')\n axdef.xaxis.set_ticks_position('bottom')\n axdef.yaxis.set_ticks_position('left')\n plt.plot(x,y, 'r')\n \n # according to the plot, we need to find the minimum value of the equation.\n\n Gradf = lambda x: 12*x**3-5 # define the gradient function (The derivative of y equation)\n\n ActualX = Xvalue\n LearningRate = 0.01 # learn rate variable, 0.01 not too many iterations and can keep the optimal solution\n PrecisionValue = 0.000001 # defines the degree of precision \n PreviousStepSize = 1 \n MaxIteration = 10000 # will be used to stop the procedure if it does not converge\n IterationCounter = 0 # counter\n\n xValues = [] \n while PreviousStepSize > PrecisionValue and IterationCounter < MaxIteration: # Define the gradient descent\n PreviousX = ActualX\n ActualX = ActualX - LearningRate * Gradf(PreviousX) # find the max \n xValues.append(ActualX)\n PreviousStepSize = abs(ActualX - PreviousX) \n IterationCounter = IterationCounter+1 \n # print(\"Number of iterations = \",IterationCounter,\"\\nActual value of x is = \", ActualX) \n\n # draw the maximum point of the function \n plt.scatter(ActualX, 3*ActualX**4-5*ActualX+2)\n plt.title(\"The optimal coordinate\")\n plt.show() \n\n print('The last one X values of f(x) minimum is', xValues[-1])\n xValues = np.array(xValues) # change the array type to numpt array in order to compute smoothly\n y = (3*xValues**4)-5*xValues+2\n print(\"The minimum value of y in the iterations is\", y[-1])\n\ngradientDescent(1.2) # based on the plot, the far-right point value is 1.2, so let we start with 1.2\n```\n\n\n```python\ndef NewtonRaphsonMethod(xValue):\n x = np.linspace(0, 1.2, num = 100) # creates numerical sequences(start, stop, num: Number of samples to generate. Default is 50. Must be non-negative.)\n y = 3*x**4-5*x+2\n\n fig = plt.figure()\n axdef = fig.add_subplot(1, 1, 1)\n axdef.spines['left'].set_position('center')\n axdef.spines['bottom'].set_position('zero')\n axdef.spines['right'].set_color('none')\n axdef.spines['top'].set_color('none')\n axdef.xaxis.set_ticks_position('bottom')\n axdef.yaxis.set_ticks_position('left')\n\n # print('Value of x at the minimum of the function', x[np.argmin(y)]) # np.argmin(): returns the indices of the minimum values along an axis\n\n plt.plot(x,y, 'r')\n\n\n\n FirstDerivative = lambda x: 12*x**3-5\n SecondDerivative = lambda x: 36*x**2 \n\n ActualX = xValue\n PrecisionValue = 0.000001 \n PreviousStepSize = 1 \n MaxIteration = 10000 \n IterationCounter = 0 \n points = []\n while PreviousStepSize > PrecisionValue and IterationCounter < MaxIteration:\n PreviousX = ActualX\n ActualX = ActualX - FirstDerivative(PreviousX)/ SecondDerivative(PreviousX) # actual value - first derivative/ second derivative\n points.append(ActualX) # the iteration add in array\n PreviousStepSize = abs(ActualX - PreviousX) # diff between current and previous\n IterationCounter = IterationCounter+1 \n # print(\"Number of iterations = \",IterationCounter,\"\\nActual value of x is = \",ActualX) \n\n # draw the maximum point of the function \n plt.scatter(ActualX, 3*ActualX**4-5*ActualX+2)\n plt.title(\"The optimal coordinate\")\n plt.show() \n print(\"X value of f(x) minimum is \", ActualX)\n print(\"The minimum value of y is\", 3*ActualX**4-5*ActualX+2) \n\n\nNewtonRaphsonMethod(1.2)\n\n```\n\n\n```python\n!apt-get install texlive texlive-xetex texlive-latex-extra pandoc\n!pip install pypandoc\nfrom google.colab import drive\ndrive.mount('/content/drive')\n!cp drive/My Drive/Colab Notebooks/test.ipynb ./\n!jupyter nbconvert --to PDF \"mynotebook.ipynb\"\n```\n\n\n```python\n\"\"\"\nfrom google.colab import drive\ndrive.mount('/content/drive')\n!cp drive/My Drive/Colab Notebooks/Untitled.ipynb ./\n!jupyter nbconvert --to pdf fileName.ipynb\n\"\"\"\n```\n", "meta": {"hexsha": "344a99d8a5edf05b93fd99c67047bf5be57841fe", "size": 477364, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "exam1/.ipynb_checkpoints/Chiu_Exam 1-checkpoint.ipynb", "max_stars_repo_name": "twyunting/CSC-632_Intro-to-Simulation-and-Modeling", "max_stars_repo_head_hexsha": "4287c2d049929b3bf702980749e662b60f50a58a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "exam1/.ipynb_checkpoints/Chiu_Exam 1-checkpoint.ipynb", "max_issues_repo_name": "twyunting/CSC-632_Intro-to-Simulation-and-Modeling", "max_issues_repo_head_hexsha": "4287c2d049929b3bf702980749e662b60f50a58a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "exam1/.ipynb_checkpoints/Chiu_Exam 1-checkpoint.ipynb", "max_forks_repo_name": "twyunting/CSC-632_Intro-to-Simulation-and-Modeling", "max_forks_repo_head_hexsha": "4287c2d049929b3bf702980749e662b60f50a58a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 526.3109151047, "max_line_length": 190872, "alphanum_fraction": 0.9410617474, "converted": true, "num_tokens": 95672, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.7520125682019722, "lm_q1q2_score": 0.44285202761494435}} {"text": "###### Text provided under a Creative Commons Attribution license, CC-BY. All code is made available under the FSF-approved MIT license. (c) Daniel Koehn based on Jupyter notebooks by Marc Spiegelman [Dynamical Systems APMA 4101](https://github.com/mspieg/dynamical-systems) and Kyle Mandli from his course [Introduction to numerical methods](https://github.com/mandli/intro-numerical-methods), notebook style sheet by L.A. Barba, N.C. Clementi [Engineering Computations](https://github.com/engineersCode)\n\n\n```python\n# Execute this cell to load the notebook's style sheet, then ignore it\nfrom IPython.core.display import HTML\ncss_file = '../style/custom.css'\nHTML(open(css_file, \"r\").read())\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n# Exploring the Lorenz Equations\n\nThe Lorenz Equations are a 3-D dynamical system that is a simplified model of Rayleigh-Benard thermal convection. They are derived and described in detail in Edward Lorenz' 1963 paper [Deterministic Nonperiodic Flow](http://journals.ametsoc.org/doi/pdf/10.1175/1520-0469%281963%29020%3C0130%3ADNF%3E2.0.CO%3B2) in the Journal of Atmospheric Science. In their classical form they can be written\n\n\\begin{equation}\n\\begin{split}\n\\frac{\\partial X}{\\partial t} &= \\sigma( Y - X)\\\\\n\\frac{\\partial Y}{\\partial t} &= rX - Y - XZ \\\\\n\\frac{\\partial Z}{\\partial t} &= XY -b Z\n\\end{split}\n\\tag{1}\n\\end{equation}\n\nwhere $\\sigma$ is the \"Prandtl number\", $r = \\mathrm{Ra}/\\mathrm{Ra}_c$ is a scaled \"Rayleigh number\" and $b$ is a parameter that is related to the the aspect ratio of a convecting cell in the original derivation.\n\nHere, $X(t)$, $Y(t)$ and $Z(t)$ are the time dependent amplitudes of the streamfunction and temperature fields, expanded in a highly truncated Fourier Series where the streamfunction contains one cellular mode\n\n$$\n \\psi(x,z,t) = X(t)\\sin(a\\pi x)\\sin(\\pi z)\n$$\n\nand temperature has two modes\n\n$$\n \\theta(x,z,t) = Y(t)\\cos(a\\pi x)\\sin(\\pi z) - Z(t)\\sin(2\\pi z)\n$$\n\nThis Jupyter notebook, will provide some simple python routines for numerical integration and visualization of the Lorenz Equations.\n\n## Numerical solution of the Lorenz Equations\n\nWe have to solve the uncoupled ordinary differential equations (1) using the finite difference method introduced in [this lecture](https://nbviewer.jupyter.org/github/daniel-koehn/Differential-equations-earth-system/blob/master/02_finite_difference_intro/1_fd_intro.ipynb).\n\nThe approach is similar to the one used in [Exercise: How to sail without wind](https://nbviewer.jupyter.org/github/daniel-koehn/Differential-equations-earth-system/blob/master/02_finite_difference_intro/3_fd_ODE_example_sailing_wo_wind.ipynb), except that eqs.(1) are coupled ordinary differential equations, we have an additional differential equation and the RHS are more complex. \n\nApproximating the temporal derivatives in eqs. (1) using the **backward FD operator** \n\n\\begin{equation}\n\\frac{df}{dt} = \\frac{f(t)-f(t-dt)}{dt} \\notag\n\\end{equation}\n\nwith the time sample interval $dt$ leads to \n\n\\begin{equation}\n\\begin{split}\n\\frac{X(t)-X(t-dt)}{dt} &= \\sigma(Y - X)\\\\\n\\frac{Y(t)-Y(t-dt)}{dt} &= rX - Y - XZ\\\\\n\\frac{Y(t)-Y(t-dt)}{dt} &= XY -b Z\\\\\n\\end{split}\n\\notag\n\\end{equation}\n\nAfter solving for $X(t), Y(t), Z(t)$, we get the **explicit time integration scheme** for the Lorenz equations:\n\n\\begin{equation}\n\\begin{split}\nX(t) &= X(t-dt) + dt\\; \\sigma(Y - X)\\\\\nY(t) &= Y(t-dt) + dt\\; (rX - Y - XZ)\\\\\nZ(t) &= Z(t-dt) + dt\\; (XY -b Z)\\\\\n\\end{split}\n\\notag\n\\end{equation}\n\nand by introducing a temporal dicretization $t^n = n * dt$ with $n \\in [0,1,...,nt]$, where $nt$ denotes the maximum time steps, the final FD code becomes:\n\n\\begin{equation}\n\\begin{split}\nX^{n} &= X^{n-1} + dt\\; \\sigma(Y^{n-1} - X^{n-1})\\\\\nY^{n} &= Y^{n-1} + dt\\; (rX^{n-1} - Y^{n-1} - X^{n-1}Z^{n-1})\\\\\nZ^{n} &= Z^{n-1} + dt\\; (X^{n-1}Y^{n-1} - b Z^{n-1})\\\\\n\\end{split}\n\\tag{2}\n\\end{equation}\n\nThe Python implementation is quite straightforward, because we can reuse some old codes ...\n\n##### Exercise 1\n\nFinish the function `Lorenz`, which computes and returns the RHS of eqs. (1) for a given $X$, $Y$, $Z$.\n\n\n```python\n%matplotlib inline\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom mpl_toolkits.mplot3d import Axes3D\n```\n\n\n```python\ndef Lorenz(X,Y,Z,sigma,r,b):\n \n '''\n Returns the RHS of the Lorenz equations\n '''\n\n # ADD RHS OF LORENZ EQUATIONS (1) HERE!\n X_dot_rhs =\n Y_dot_rhs =\n Z_dot_rhs =\n\n # return the state derivatives\n return X_dot_rhs, Y_dot_rhs, Z_dot_rhs\n```\n\nNext, we write the function to solve the Lorenz equation `SolveLorenz` based on the `sailing_boring` code from the [Exercise: How to sail without wind](https://nbviewer.jupyter.org/github/daniel-koehn/Differential-equations-earth-system/blob/master/02_finite_difference_intro/3_fd_ODE_example_sailing_wo_wind.ipynb)\n\n##### Exercise 2\n\nFinish the FD-code implementation `SolveLorenz`\n\n\n```python\ndef SolveLorenz(tmax, dt, X0, Y0, Z0, sigma=10.,r=28.,b=8./3.0):\n \n '''\n Integrate the Lorenz equations from initial condition (X0,Y0,Z0)^T at t=0 \n for parameters sigma, r, b\n \n Returns: X, Y, Z, time\n '''\n \n # Compute number of time steps based on tmax and dt\n nt = (int)(tmax/dt)\n \n # vectors for storage of X, Y, Z positions and time t\n X = np.zeros(nt + 1)\n Y = np.zeros(nt + 1)\n Z = np.zeros(nt + 1)\n t = np.zeros(nt + 1)\n \n # define initial position and time\n X[0] = X0\n Y[0] = Y0\n Z[0] = Z0\n \n # start time stepping over time samples n\n for n in range(1,nt + 1):\n \n # compute RHS of Lorenz eqs. (1) at current position (X,Y,Z)^T\n X_dot_rhs, Y_dot_rhs, Z_dot_rhs = Lorenz(X[n-1],Y[n-1],Z[n-1],sigma,r,b)\n \n # compute new position using FD approximation of time derivative\n # ADD FD SCHEME OF THE LORENZ EQS. HERE!\n X[n] = \n Y[n] = \n Z[n] =\n t[n] = n * dt\n\n return X, Y, Z, t\n```\n\nFinally, we create a function to plot the solution (X,Y,Z)^T of the Lorenz eqs. ...\n\n\n```python\ndef PlotLorenzXvT(X,Y,Z,t,sigma,r,b):\n \n '''\n Create time series plots of solutions of the Lorenz equations X(t),Y(t),Z(t)\n '''\n\n plt.figure()\n ax = plt.subplot(111)\n ax.plot(t,X,'r',label='X')\n ax.plot(t,Y,'g',label='Y')\n ax.plot(t,Z,'b',label='Z')\n ax.set_xlabel('time t')\n plt.title('Lorenz Equations: $\\sigma=${}, $r=${}, $b=${}'.format(sigma,r,b))\n # Shrink current axis's height by 10% on the bottom\n box = ax.get_position()\n ax.set_position([box.x0, box.y0 + box.height * 0.1,\n box.width, box.height * 0.9])\n\n # Put a legend below current axis\n ax.legend(loc='upper center', bbox_to_anchor=(0.5, -0.05),ncol=3)\n plt.show()\n```\n\n... and a function to plot the trajectory in the **phase space portrait**:\n\n\n```python\ndef PlotLorenz3D(X,Y,Z,sigma,r,b):\n '''\n Show 3-D Phase portrait using mplot3D\n '''\n # do some fancy 3D plotting\n fig = plt.figure()\n ax = fig.gca(projection='3d')\n ax.plot(X,Y,Z)\n ax.set_xlabel('X')\n ax.set_ylabel('Y')\n ax.set_zlabel('Z')\n plt.title('Lorenz Equations: $\\sigma=${}, $r=${}, $b=${}'.format(sigma,r,b))\n plt.show()\n```\n\n##### Exercise 3\n\nSolve the Lorenz equations for a Prandtl number $\\sigma=10$, $b=8/3$ and a scaled Rayleigh number $r=0.5$, starting from the initial condition ${\\bf{X_0}}=(X_0,Y_0,Z_0)^T=(2,3,4)^T$. Plot the temporal evolution and 3D phase potrait of the solution $(X(t),Y(t),Z(t))^T$. Mark the fix points, you derived in [Stationary Solutions of Time-Dependent Problems](http://nbviewer.ipython.org/urls/github.com/daniel-koehn/Differential-equations-earth-system/tree/master/03_Lorenz_equations/02_Stationary_solutions_of_DE.ipynb) in the 3D phase portrait. Describe and interpret the results.\n\n\n```python\n# SET THE PARAMETERS HERE!\nsigma= \nb = \n\n# SET THE INITIAL CONDITIONS HERE!\nX0 = \nY0 = \nZ0 =\n\n# Set maximum integration time and sample interval dt\ntmax = 30\ndt = 0.01\n\n# SET THE RAYLEIGH NUMBER HERE!\nr =\n\n# Solve the Equations\nX, Y, Z, t = SolveLorenz(tmax, dt, X0, Y0, Z0, sigma,r,b)\n\n# and Visualize as a time series\nPlotLorenzXvT(X,Y,Z,t,sigma,r,b)\n\n# and as a 3-D phase portrait\nPlotLorenz3D(X,Y,Z,sigma,r,b)\n```\n\n##### Exercise 4\n\nSolve the Lorenz equations for a Prandtl number $\\sigma=10$, $b=8/3$ and a scaled Rayleigh number $r=10$, starting from the initial condition ${\\bf{X_0}}=(X_0,Y_0,Z_0)^T=(2,3,4)^T$. Plot the temporal evolution and 3D phase potrait of the solution $(X(t),Y(t),Z(t))^T$. Mark the fix points, you derived in [Stationary Solutions of Time-Dependent Problems](http://nbviewer.ipython.org/urls/github.com/daniel-koehn/Differential-equations-earth-system/tree/master/03_Lorenz_equations/02_Stationary_solutions_of_DE.ipynb) in the 3D phase portrait. Describe and interpret the results.\n\n\n```python\n# SET THE PARAMETERS HERE!\nsigma= \nb = \n\n# SET THE INITIAL CONDITIONS HERE!\nX0 = \nY0 = \nZ0 =\n\n# Set maximum integration time and sample interval dt\ntmax = 30\ndt = 0.01\n\n# SET THE RAYLEIGH NUMBER HERE!\nr =\n\n# Solve the Equations\nX, Y, Z, t = SolveLorenz(tmax, dt, X0, Y0, Z0, sigma,r,b)\n\n# and Visualize as a time series\nPlotLorenzXvT(X,Y,Z,t,sigma,r,b)\n\n# and as a 3-D phase portrait\nPlotLorenz3D(X,Y,Z,sigma,r,b)\n```\n\n##### Exercise 5\n\nSolve the Lorenz equations again for a Prandtl number $\\sigma=10$, $b=8/3$ and a scaled Rayleigh number $r=10$. However, starting from the initial condition ${\\bf{X_0}}=(X_0,Y_0,Z_0)^T=(-2,-3,4)^T$. Plot the temporal evolution and 3D phase potrait of the solution $(X(t),Y(t),Z(t))^T$. Mark the fix points, you derived in [Stationary Solutions of Time-Dependent Problems](http://nbviewer.ipython.org/urls/github.com/daniel-koehn/Differential-equations-earth-system/tree/master/03_Lorenz_equations/02_Stationary_solutions_of_DE.ipynb) in the 3D phase portrait. Describe and interpret the results. How does the solution change compared to exercise 4?\n\n\n```python\n# SET THE PARAMETERS HERE!\nsigma= \nb = \n\n# SET THE INITIAL CONDITIONS HERE!\nX0 = \nY0 = \nZ0 =\n\n# Set maximum integration time and sample interval dt\ntmax = 30\ndt = 0.01\n\n# SET THE RAYLEIGH NUMBER HERE!\nr =\n\n# Solve the Equations\nX, Y, Z, t = SolveLorenz(tmax, dt, X0, Y0, Z0, sigma,r,b)\n\n# and Visualize as a time series\nPlotLorenzXvT(X,Y,Z,t,sigma,r,b)\n\n# and as a 3-D phase portrait\nPlotLorenz3D(X,Y,Z,sigma,r,b)\n```\n\n##### Exercise 6\n\nSolve the Lorenz equations for a Prandtl number $\\sigma=10$, $b=8/3$ and a scaled Rayleigh number $r=28$, starting from the initial condition ${\\bf{X_0}}=(X_0,Y_0,Z_0)^T=(2,3,4)^T$. Plot the temporal evolution and 3D phase potrait of the solution $(X(t),Y(t),Z(t))^T$. Mark the fix points, you derived in [Stationary Solutions of Time-Dependent Problems](http://nbviewer.ipython.org/urls/github.com/daniel-koehn/Differential-equations-earth-system/tree/master/03_Lorenz_equations/02_Stationary_solutions_of_DE.ipynb) in the 3D phase portrait. Describe and interpret the results. Compare with the previous results.\n\n\n```python\n# SET THE PARAMETERS HERE!\nsigma= \nb = \n\n# SET THE INITIAL CONDITIONS HERE!\nX0 = \nY0 = \nZ0 =\n\n# Set maximum integration time and sample interval dt\ntmax = 30\ndt = 5e-4\n\n# SET THE RAYLEIGH NUMBER HERE!\nr =\n\n# Solve the Equations\nX, Y, Z, t = SolveLorenz(tmax, dt, X0, Y0, Z0, sigma,r,b)\n\n# and Visualize as a time series\nPlotLorenzXvT(X,Y,Z,t,sigma,r,b)\n\n# and as a 3-D phase portrait\nPlotLorenz3D(X,Y,Z,sigma,r,b)\n```\n\n##### Exercise 7\n\nIn his 1963 paper Lorenz also investigated the influence of small changes of the initial conditions on the long-term evolution of the thermal convection problem for large Rayleigh numbers. \n\nSolve the Lorenz equations for a Prandtl number $\\sigma=10$, $b=8/3$ and a scaled Rayleigh number $r=28$, however starting from the initial condition ${\\bf{X_0}}=(X_0,Y_0,Z_0)^T=(2,3.001,4)^T$. Plot the temporal evolution and compare with the solution of exercise 6. Describe and interpret the results.\n\nExplain why Lorenz introduced the term **Butterfly effect** based on your results.\n\n\n```python\n# SET THE PARAMETERS HERE!\nsigma= \nb = \n\n# SET THE INITIAL CONDITIONS HERE!\nX0 = \nY0 = \nZ0 =\n\n# Set maximum integration time and sample interval dt\ntmax = 30\ndt = 5e-4\n\n# SET THE RAYLEIGH NUMBER HERE!\nr =\n\n# Solve the Equations\nX1, Y1, Z1, t = SolveLorenz(tmax, dt, X0, Y0, Z0, sigma,r,b)\n\n# and Visualize differences as a time series\nPlotLorenzXvT(X-X1,Y-Y1,Z-Z1,t,sigma,r,b)\n\n# and Visualize as a time series\nPlotLorenzXvT(X1,Y1,Z1,t,sigma,r,b)\n\n# and Visualize as a time series\nPlotLorenzXvT(X,Y,Z,t,sigma,r,b)\n```\n\n##### Exercise 8\n\nSolve the Lorenz equations for a Prandtl number $\\sigma=10$, $b=8/3$ and a scaled Rayleigh number $r=350$, starting from the initial condition ${\\bf{X_0}}=(X_0,Y_0,Z_0)^T=(2,3,4)^T$. Plot the temporal evolution and 3D phase potrait of the solution $(X(t),Y(t),Z(t))^T$. Mark the fix points, you derived in [Stationary Solutions of Time-Dependent Problems](http://nbviewer.ipython.org/urls/github.com/daniel-koehn/Differential-equations-earth-system/tree/master/03_Lorenz_equations/02_Stationary_solutions_of_DE.ipynb) in the 3D phase portrait. Describe and interpret the results. Compare with the previous result from exercise 8.\n\n\n```python\n# SET THE PARAMETERS HERE!\nsigma= \nb = \n\n# SET THE INITIAL CONDITIONS HERE!\nX0 = \nY0 = \nZ0 =\n\n# Set maximum integration time and sample interval dt\ntmax = 8.\ndt = 5e-4\n\n# SET THE RAYLEIGH NUMBER HERE!\nr =\n\n# Solve the Equations\nX, Y, Z, t = SolveLorenz(tmax, dt, X0, Y0, Z0, sigma,r,b)\n\n# and Visualize as a time series\nPlotLorenzXvT(X,Y,Z,t,sigma,r,b)\n\n# and as a 3-D phase portrait\nPlotLorenz3D(X,Y,Z,sigma,r,b)\n```\n\n## What we learned:\n\n- How to solve the Lorenz equations using a simple finite-difference scheme. \n\n- How to visualize the solution of ordinary differential equations using the temporal evolution and phase portrait.\n\n- Exporing the dynamic of non-linear differential equations and the sensitivity of small changes of the initial conditions to the long term evolution of the system.\n\n- Why physicists can only predict the time evolution of complex dynamical systems to some extent.\n", "meta": {"hexsha": "cb3448d410839139690980c2784f852304c60f8c", "size": 25777, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "03_Lorenz_equations/03_LorenzEquations_fdsolve.ipynb", "max_stars_repo_name": "daniel-koehn/Differential-equations-earth-system", "max_stars_repo_head_hexsha": "3916cbc968da43d0971b7139476350c1dd798746", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 30, "max_stars_repo_stars_event_min_datetime": "2019-10-16T19:07:36.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-10T03:48:44.000Z", "max_issues_repo_path": "03_Lorenz_equations/03_LorenzEquations_fdsolve.ipynb", "max_issues_repo_name": "daniel-koehn/Differential-equations-earth-system", "max_issues_repo_head_hexsha": "3916cbc968da43d0971b7139476350c1dd798746", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "03_Lorenz_equations/03_LorenzEquations_fdsolve.ipynb", "max_forks_repo_name": "daniel-koehn/Differential-equations-earth-system", "max_forks_repo_head_hexsha": "3916cbc968da43d0971b7139476350c1dd798746", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2020-11-19T08:21:55.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-10T09:33:37.000Z", "avg_line_length": 35.4566712517, "max_line_length": 656, "alphanum_fraction": 0.5357101292, "converted": true, "num_tokens": 5266, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.8397339596505965, "lm_q1q2_score": 0.4428055921667102}} {"text": "\n\n# Lambda School Data Science Module 132\n## Sampling, Confidence Intervals, and Hypothesis Testing\n\n## Prepare - examine other available hypothesis tests\n\nIf you had to pick a single hypothesis test in your toolbox, t-test would probably be the best choice - but the good news is you don't have to pick just one! Here's some of the others to be aware of:\n\n\n```\nimport numpy as np\nfrom scipy.stats import chisquare # One-way chi square test\n\n# Chi square can take any crosstab/table and test the independence of rows/cols\n# The null hypothesis is that the rows/cols are independent -> low chi square\n# The alternative is that there is a dependence -> high chi square\n# Be aware! Chi square does *not* tell you direction/causation\n\nind_obs = np.array([[1, 1], [2, 2]]).T\nprint(ind_obs)\nprint(chisquare(ind_obs, axis=None))\n\ndep_obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T\nprint(dep_obs)\nprint(chisquare(dep_obs, axis=None))\n```\n\n [[1 2]\n [1 2]]\n Power_divergenceResult(statistic=0.6666666666666666, pvalue=0.8810148425137847)\n [[16 32]\n [18 24]\n [16 16]\n [14 28]\n [12 20]\n [12 24]]\n Power_divergenceResult(statistic=23.31034482758621, pvalue=0.015975692534127565)\n\n\n\n```\n# Distribution tests:\n# We often assume that something is normal, but it can be important to *check*\n\n# For example, later on with predictive modeling, a typical assumption is that\n# residuals (prediction errors) are normal - checking is a good diagnostic\n\nfrom scipy.stats import normaltest\n# Poisson models arrival times and is related to the binomial (coinflip)\nsample = np.random.poisson(5, 1000)\nprint(normaltest(sample)) # Pretty clearly not normal\n```\n\n NormaltestResult(statistic=28.280826942833155, pvalue=7.225975047337626e-07)\n\n\n\n```\n# Kruskal-Wallis H-test - compare the median rank between 2+ groups\n# Can be applied to ranking decisions/outcomes/recommendations\n# The underlying math comes from chi-square distribution, and is best for n>5\nfrom scipy.stats import kruskal\n\nx1 = [1, 3, 5, 7, 9]\ny1 = [2, 4, 6, 8, 10]\nprint(kruskal(x1, y1)) # x1 is a little better, but not \"significantly\" so\n\nx2 = [1, 1, 1]\ny2 = [2, 2, 2]\nz = [2, 2] # Hey, a third group, and of different size!\nprint(kruskal(x2, y2, z)) # x clearly dominates\n```\n\n KruskalResult(statistic=0.2727272727272734, pvalue=0.6015081344405895)\n KruskalResult(statistic=7.0, pvalue=0.0301973834223185)\n\n\nAnd there's many more! `scipy.stats` is fairly comprehensive, though there are even more available if you delve into the extended world of statistics packages. As tests get increasingly obscure and specialized, the importance of knowing them by heart becomes small - but being able to look them up and figure them out when they *are* relevant is still important.\n\n## T-test Assumptions\n\n\n\n- Independence of means\n\nAre the means of our voting data independent (do not affect the outcome of one another)?\n \nThe best way to increase thel likelihood of our means being independent is to randomly sample (which we did not do).\n\n\n\n```\nfrom scipy.stats import ttest_ind\n\n?ttest_ind\n```\n\n- \"Homogeneity\" of Variance? \n\nIs the magnitude of the variance between the two roughly the same?\n\nI think we're OK on this one for the voting data, although it probably could be better, one party was larger than the other.\n\nIf we suspect this to be a problem then we can use Welch's T-test\n\n\n```\n?ttest_ind\n```\n\n- \"Dependent Variable\" (sample means) are Distributed Normally\n\n\n\nLots of statistical tests depend on normal distributions. We can test for normality using Scipy as was shown above.\n\nThis assumption is often assumed even if the assumption is a weak one. If you strongly suspect that things are not normally distributed, you can transform your data to get it looking more normal and then run your test. This problem typically goes away for large sample sizes (yay Central Limit Theorem) and is often why you don't hear it brought up. People declare the assumption to be satisfied either way. \n\n\n\n## Central Limit Theorem\n\n\n\n\n```\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\n\nsample_means = []\nfor x in range(0,3000):\n coinflips = np.random.binomial(n=1, p=.5, size=30)\n one_sample = coinflips\n sample_means.append(coinflips.mean())\n\nprint(len(sample_means))\nprint(sample_means)\n```\n\n 3000\n [0.5, 0.36666666666666664, 0.43333333333333335, 0.6, 0.3333333333333333, 0.6, 0.6, 0.4666666666666667, 0.3333333333333333, 0.5333333333333333, 0.5666666666666667, 0.6333333333333333, 0.5, 0.36666666666666664, 0.43333333333333335, 0.5333333333333333, 0.4666666666666667, 0.4, 0.6333333333333333, 0.5333333333333333, 0.6666666666666666, 0.5333333333333333, 0.5333333333333333, 0.5333333333333333, 0.3333333333333333, 0.6, 0.5, 0.5666666666666667, 0.43333333333333335, 0.5666666666666667, 0.5333333333333333, 0.4666666666666667, 0.4666666666666667, 0.5666666666666667, 0.5666666666666667, 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0.4, 0.5, 0.5, 0.5, 0.5, 0.5, 0.36666666666666664, 0.43333333333333335, 0.5, 0.5333333333333333, 0.5, 0.5, 0.6, 0.5666666666666667, 0.6, 0.5333333333333333, 0.6, 0.5, 0.4, 0.6333333333333333, 0.3333333333333333, 0.5666666666666667, 0.43333333333333335, 0.6666666666666666, 0.4666666666666667, 0.4666666666666667, 0.5666666666666667, 0.5, 0.3333333333333333, 0.43333333333333335, 0.5, 0.5666666666666667, 0.43333333333333335, 0.5666666666666667, 0.6333333333333333, 0.3333333333333333, 0.5333333333333333, 0.4666666666666667, 0.5666666666666667, 0.4, 0.6, 0.43333333333333335, 0.5666666666666667, 0.5333333333333333, 0.5666666666666667, 0.43333333333333335, 0.5333333333333333, 0.3, 0.6, 0.5666666666666667, 0.43333333333333335, 0.4666666666666667, 0.4666666666666667, 0.5666666666666667, 0.6333333333333333, 0.5333333333333333, 0.5333333333333333, 0.4, 0.4, 0.43333333333333335, 0.43333333333333335, 0.6333333333333333, 0.4, 0.5333333333333333, 0.36666666666666664, 0.3, 0.5666666666666667, 0.5, 0.4, 0.5, 0.6, 0.43333333333333335, 0.5, 0.3333333333333333, 0.5666666666666667, 0.5333333333333333, 0.5666666666666667, 0.4, 0.5333333333333333, 0.4, 0.5666666666666667, 0.5333333333333333, 0.5666666666666667, 0.7333333333333333, 0.5333333333333333, 0.43333333333333335, 0.43333333333333335, 0.7, 0.5, 0.5333333333333333, 0.6, 0.43333333333333335, 0.5666666666666667, 0.5, 0.6, 0.43333333333333335, 0.5333333333333333, 0.4666666666666667, 0.43333333333333335, 0.5, 0.6666666666666666, 0.3333333333333333, 0.5, 0.4666666666666667, 0.5, 0.5333333333333333, 0.5333333333333333, 0.5333333333333333, 0.5, 0.5666666666666667, 0.43333333333333335, 0.6, 0.5, 0.7333333333333333, 0.6, 0.5333333333333333, 0.6, 0.5333333333333333, 0.5666666666666667, 0.5666666666666667, 0.4666666666666667, 0.5, 0.4, 0.36666666666666664, 0.5333333333333333, 0.43333333333333335, 0.4666666666666667, 0.5, 0.5, 0.43333333333333335, 0.4666666666666667, 0.5666666666666667, 0.4, 0.6333333333333333, 0.36666666666666664, 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0.5333333333333333, 0.6333333333333333, 0.3, 0.4666666666666667, 0.6, 0.5, 0.5333333333333333, 0.5333333333333333, 0.3333333333333333, 0.36666666666666664, 0.5333333333333333, 0.6666666666666666, 0.4, 0.5666666666666667, 0.4, 0.6666666666666666, 0.26666666666666666, 0.6333333333333333, 0.43333333333333335, 0.5, 0.3, 0.5, 0.5333333333333333, 0.43333333333333335, 0.43333333333333335, 0.5666666666666667, 0.5333333333333333, 0.6666666666666666, 0.4666666666666667, 0.5, 0.5, 0.36666666666666664, 0.5, 0.3333333333333333, 0.3, 0.5666666666666667, 0.5, 0.4666666666666667, 0.5, 0.43333333333333335, 0.4, 0.4666666666666667, 0.6, 0.4666666666666667, 0.6333333333333333, 0.5, 0.4, 0.43333333333333335, 0.6, 0.4, 0.4666666666666667, 0.4666666666666667, 0.6666666666666666, 0.4666666666666667, 0.5, 0.6666666666666666, 0.4, 0.43333333333333335, 0.4666666666666667, 0.5]\n\n\n\n```\ndf = pd.DataFrame({'a': one_sample})\ndf.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    a
    00
    11
    21
    31
    40
    \n
    \n\n\n\n\n```\ndf.a.hist();\n```\n\n\n```\nax = plt.hist(sample_means, bins=30)\nplt.title('Distribution of 3000 sample means \\n (of 30 coinflips each)');\n```\n\nWhat does the Central Limit Theorem State? That no matter the initial distribution of the population, the distribution of sample means taken will approximate a normal distribution as $n \\rightarrow \\infty$.\n\nThis has very important implications for hypothesis testing and is precisely the reason why the t-distribution begins to approximate the normal distribution as our sample size increases. \n\n## Standard Error of the Mean\n\nWhat does it mean to \"estimate\"? the Population mean?\n\n\n```\nimport numpy as np\nimport pandas as pd\n\nlambda_heights = np.random.uniform(4,6.5, size=2000)\nprint(len(lambda_heights))\nlambda_heights\n```\n\n 2000\n\n\n\n\n\n array([4.51572466, 6.28356165, 5.74660261, ..., 4.02088121, 5.68035051,\n 4.58423192])\n\n\n\n\n```\nprint(\"Population Mean:\", lambda_heights.mean())\nprint(\"Population Standard Deviation:\", lambda_heights.std())\n```\n\n Population Mean: 5.265856045117739\n Population Standard Deviation: 0.7158986145354737\n\n\n\n```\npopulation = pd.DataFrame({'heights': lambda_heights})\nprint(population.shape)\npopulation.head()\n```\n\n (2000, 1)\n\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    heights
    04.515725
    16.283562
    25.746603
    34.647367
    45.055953
    \n
    \n\n\n\n\n```\nsample = population.sample(100)\nprint(sample.shape)\nsample.head()\n```\n\n (100, 1)\n\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    heights
    695.397222
    5445.582838
    12855.448180
    5245.178870
    2804.684905
    \n
    \n\n\n\n\n```\nprint(\"Sample Mean 1:\", sample['heights'].mean())\n```\n\n Sample Mean 1: 5.328376748609253\n\n\n\n```\nsample = population.sample(100)\nprint(sample.shape)\nsample.head()\n```\n\n (100, 1)\n\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    heights
    11224.531799
    15025.109466
    14235.608430
    1205.754977
    6864.754457
    \n
    \n\n\n\n\n```\nprint(\"Sample Mean 2:\", sample['heights'].mean())\n```\n\n Sample Mean 2: 5.230470707607443\n\n\n## Build and Interpret a Confidence Interval\n\n\n\n\n```\ncoinflips_100 = np.random.binomial(n=1, p=.5, size=100)\n\nsample_std = np.std(coinflips_100)\nprint(\"sample standard deviation:\", sample_std)\nsample_size = len(coinflips_100)\nprint(\"sample size:\", sample_size)\n```\n\n sample standard deviation: 0.4950757517794625\n sample size: 100\n\n\n\n```\nstandard_error = sample_std / (sample_size**(.5))\n\nprint(\"standard error:\", standard_error)\n```\n\n standard error: 0.04950757517794625\n\n\n\n```\nfrom scipy import stats\n\nstderr = stats.sem(coinflips_100, ddof=0)\nstderr\n```\n\n\n\n\n 0.04950757517794625\n\n\n\n### What confidence level do we want our confidence interval to represent?\n\n95% confidence Interval? 99% confidence interval? \n\n\n```\nt = stats.t.ppf(.975 , sample_size-1)\n```\n\n\n```\nsample_mean = coinflips_100.mean()\n```\n\n\n```\nconfidence_interval = (sample_mean - t*stderr, sample_mean + t*stderr)\n\nmargin_of_error = t*stderr\n\nprint(\"Sample Mean\", sample_mean)\nprint(\"Margin of Error:\", margin_of_error)\nprint(\"Confidence Interval:\", confidence_interval)\n```\n\n Sample Mean 0.43\n Margin of Error: 0.09823376989617144\n Confidence Interval: (0.33176623010382855, 0.5282337698961714)\n\n\n\n```\nconfidence_interval[0]\n```\n\n\n\n\n 0.33176623010382855\n\n\n\n\n```\nconfidence_interval[1]\n```\n\n\n\n\n 0.5282337698961714\n\n\n\n## Graphically Represent a Confidence Interval\n\n\n```\nimport seaborn as sns\n\nsns.kdeplot(coinflips_100)\nplt.axvline(x=confidence_interval[0], color='red')\nplt.axvline(x=confidence_interval[1], color='red')\nplt.axvline(x=sample_mean, color='k');\n```\n\n## Relationship between Confidence Intervals and T-tests\n\nConfidence Interval == Bounds of statistical significance for our t-test\n\nA sample mean that falls inside of our confidence interval will \"FAIL TO REJECT\" our null hypothesis\n\nA sample mean that falls outside of our confidence interval will \"REJECT\" our null hypothesis\n\n\n```\nfrom scipy.stats import t, ttest_1samp\n```\n\n\n```\nimport numpy as np\n\ncoinflip_means = []\nfor x in range(0,100):\n coinflips = np.random.binomial(n=1, p=.5, size=30)\n coinflip_means.append(coinflips.mean())\n\nprint(coinflip_means)\n```\n\n [0.36666666666666664, 0.6, 0.4666666666666667, 0.4, 0.5, 0.5666666666666667, 0.4, 0.5666666666666667, 0.4666666666666667, 0.5666666666666667, 0.5333333333333333, 0.4666666666666667, 0.5, 0.7333333333333333, 0.36666666666666664, 0.6333333333333333, 0.5666666666666667, 0.5, 0.6, 0.36666666666666664, 0.5333333333333333, 0.5666666666666667, 0.5, 0.43333333333333335, 0.6333333333333333, 0.4, 0.6, 0.23333333333333334, 0.6333333333333333, 0.5666666666666667, 0.3, 0.43333333333333335, 0.36666666666666664, 0.6333333333333333, 0.4, 0.4666666666666667, 0.43333333333333335, 0.43333333333333335, 0.6, 0.5333333333333333, 0.5666666666666667, 0.4666666666666667, 0.6333333333333333, 0.5666666666666667, 0.5666666666666667, 0.6333333333333333, 0.5, 0.6333333333333333, 0.43333333333333335, 0.6, 0.43333333333333335, 0.43333333333333335, 0.43333333333333335, 0.5333333333333333, 0.36666666666666664, 0.6666666666666666, 0.6, 0.5666666666666667, 0.4666666666666667, 0.5, 0.5, 0.7333333333333333, 0.4666666666666667, 0.4666666666666667, 0.5666666666666667, 0.5666666666666667, 0.4666666666666667, 0.36666666666666664, 0.4666666666666667, 0.4, 0.43333333333333335, 0.4666666666666667, 0.3333333333333333, 0.4666666666666667, 0.6666666666666666, 0.5666666666666667, 0.5, 0.6, 0.5, 0.5333333333333333, 0.36666666666666664, 0.6666666666666666, 0.4, 0.43333333333333335, 0.5, 0.4, 0.4, 0.5333333333333333, 0.5333333333333333, 0.5333333333333333, 0.6, 0.4666666666666667, 0.6666666666666666, 0.6333333333333333, 0.4666666666666667, 0.4666666666666667, 0.5333333333333333, 0.4, 0.36666666666666664, 0.5666666666666667]\n\n\n\n```\n# Sample Size\nn = len(coinflip_means)\n# Degrees of Freedom\ndof = n-1\n# The Mean of Means:\nmean = np.mean(coinflip_means)\n# Sample Standard Deviation\nsample_std = np.std(coinflip_means, ddof=1)\n# Standard Error\nstd_err = sample_std/n**.5\n\nCI = t.interval(.95, dof, loc=mean, scale=std_err)\nprint(\"95% Confidence Interval: \", CI)\n```\n\n 95% Confidence Interval: (0.4852859432008737, 0.5240473901324597)\n\n\n\n```\n'''You can roll your own CI calculation pretty easily. \nThe only thing that's a little bit challenging \nis understanding the t stat lookup'''\n\n# 95% confidence interval\nt_stat = t.ppf(.975, dof)\nprint(\"t Statistic:\", t_stat)\n\nCI = (mean-(t_stat*std_err), mean+(t_stat*std_err))\nprint(\"Confidence Interval\", CI)\n```\n\n t Statistic: 1.9842169515086827\n Confidence Interval (0.4852859432008737, 0.5240473901324597)\n\n\nA null hypothesis that's just inside of our confidence interval == fail to reject\n\n\n\n\n```\nttest_1samp(coinflip_means, .471714)\n```\n\n\n\n\n Ttest_1sampResult(statistic=3.3737254397556526, pvalue=0.0010597598838256272)\n\n\n\nA null hypothesis that's just outside of our confidence interval == reject\n\n\n\n\n```\nttest_1samp(coinflip_means, .471713)\n```\n\n\n\n\n Ttest_1sampResult(statistic=3.373827820709339, pvalue=0.0010594068333503433)\n\n\n\n\n```\ndef confidence_interval(data, confidence=0.95):\n \"\"\"\n Calculate a confidence interval around a sample mean for given data.\n Using t-distribution and two-tailed test, default 95% confidence. \n \n Arguments:\n data - iterable (list or numpy array) of sample observations\n confidence - level of confidence for the interval\n \n Returns:\n tuple of (mean, lower bound, upper bound)\n \"\"\"\n data = np.array(data)\n mean = np.mean(data)\n n = len(data)\n stderr = stats.sem(data)\n interval = stderr * stats.t.ppf((1 + confidence) / 2.0, n - 1)\n return (mean, mean - interval, mean + interval)\n```\n\n## Run a $\\chi^{2}$ Test \"by hand\" (Using Numpy)\n\n\n```\ndf = pd.read_csv('https://raw.githubusercontent.com/ryanleeallred/datasets/master/adult.csv', na_values=\" ?\")\nprint(df.shape)\ndf.head()\n```\n\n (32561, 15)\n\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    ageworkclassfnlwgteducationeducation-nummarital-statusoccupationrelationshipracesexcapital-gaincapital-losshours-per-weekcountrysalary
    039State-gov77516Bachelors13Never-marriedAdm-clericalNot-in-familyWhiteMale2174040United-States<=50K
    150Self-emp-not-inc83311Bachelors13Married-civ-spouseExec-managerialHusbandWhiteMale0013United-States<=50K
    238Private215646HS-grad9DivorcedHandlers-cleanersNot-in-familyWhiteMale0040United-States<=50K
    353Private23472111th7Married-civ-spouseHandlers-cleanersHusbandBlackMale0040United-States<=50K
    428Private338409Bachelors13Married-civ-spouseProf-specialtyWifeBlackFemale0040Cuba<=50K
    \n
    \n\n\n\n\n```\ndf.columns\n\n```\n\n\n\n\n Index(['age', 'workclass', 'fnlwgt', 'education', 'education-num',\n 'marital-status', 'occupation', 'relationship', 'race', 'sex',\n 'capital-gain', 'capital-loss', 'hours-per-week', 'country', 'salary'],\n dtype='object')\n\n\n\n\n```\ndf.describe()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    agefnlwgteducation-numcapital-gaincapital-losshours-per-week
    count32561.0000003.256100e+0432561.00000032561.00000032561.00000032561.000000
    mean38.5816471.897784e+0510.0806791077.64884487.30383040.437456
    std13.6404331.055500e+052.5727207385.292085402.96021912.347429
    min17.0000001.228500e+041.0000000.0000000.0000001.000000
    25%28.0000001.178270e+059.0000000.0000000.00000040.000000
    50%37.0000001.783560e+0510.0000000.0000000.00000040.000000
    75%48.0000002.370510e+0512.0000000.0000000.00000045.000000
    max90.0000001.484705e+0616.00000099999.0000004356.00000099.000000
    \n
    \n\n\n\n\n```\ndf.describe(exclude='number')\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    workclasseducationmarital-statusoccupationrelationshipracesexcountrysalary
    count307253256132561307183256132561325613197832561
    unique816714652412
    topPrivateHS-gradMarried-civ-spouseProf-specialtyHusbandWhiteMaleUnited-States<=50K
    freq22696105011497641401319327816217902917024720
    \n
    \n\n\n\n\n```\ncut_points = [0, 9, 19, 29, 39, 49, 1000]\nlabel_names = ['0-9', '10-19', '20-29', '30-39', '40-49', '50+']\ndf['hours_per_week_categories'] = pd.cut(df['hours-per-week'], cut_points, labels=label_names)\n\ndf.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    ageworkclassfnlwgteducationeducation-nummarital-statusoccupationrelationshipracesexcapital-gaincapital-losshours-per-weekcountrysalaryhours_per_week_categories
    039State-gov77516Bachelors13Never-marriedAdm-clericalNot-in-familyWhiteMale2174040United-States<=50K40-49
    150Self-emp-not-inc83311Bachelors13Married-civ-spouseExec-managerialHusbandWhiteMale0013United-States<=50K10-19
    238Private215646HS-grad9DivorcedHandlers-cleanersNot-in-familyWhiteMale0040United-States<=50K40-49
    353Private23472111th7Married-civ-spouseHandlers-cleanersHusbandBlackMale0040United-States<=50K40-49
    428Private338409Bachelors13Married-civ-spouseProf-specialtyWifeBlackFemale0040Cuba<=50K40-49
    \n
    \n\n\n\n\n```\ndf['sex'].value_counts()\n```\n\n\n\n\n Male 21790\n Female 10771\n Name: sex, dtype: int64\n\n\n\n\n```\ndf['hours_per_week_categories'].value_counts()\n```\n\n\n\n\n 40-49 18336\n 50+ 6462\n 30-39 3667\n 20-29 2392\n 10-19 1246\n 0-9 458\n Name: hours_per_week_categories, dtype: int64\n\n\n\n\n```\ndf = df.sort_values(by='hours_per_week_categories', ascending=True)\n\ndf.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    ageworkclassfnlwgteducationeducation-nummarital-statusoccupationrelationshipracesexcapital-gaincapital-losshours-per-weekcountrysalaryhours_per_week_categories
    3129055Self-emp-not-inc41938Bachelors13Married-civ-spouseProf-specialtyWifeWhiteFemale008United-States<=50K0-9
    517232NaN134886HS-grad9Married-civ-spouseNaNWifeWhiteFemale002United-States>50K0-9
    2292817NaN33266610th6Never-marriedNaNOwn-childWhiteFemale004United-States<=50K0-9
    790235Private359131Bachelors13Married-civ-spouseProf-specialtyWifeWhiteFemale729808NaN>50K0-9
    660441Private406603HS-grad9Never-marriedOther-serviceNot-in-familyWhiteMale006Iran<=50K0-9
    \n
    \n\n\n\n\n```\ncontingency_table = pd.crosstab(df['sex'], df['hours_per_week_categories'], margins=True)\n\ncontingency_table\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    hours_per_week_categories0-910-1920-2930-3940-4950+All
    sex
    Female235671128719145636102810771
    Male2235751105175312700543421790
    All45812462392366718336646232561
    \n
    \n\n\n\n\n```\nfemalecount = contingency_table.iloc[0][0:6].values\nfemalecount\n```\n\n\n\n\n array([ 235, 671, 1287, 1914, 5636, 1028])\n\n\n\n\n```\nmalecount = contingency_table.iloc[1][0:6].values\nmalecount\n```\n\n\n\n\n array([ 223, 575, 1105, 1753, 12700, 5434])\n\n\n\n\n```\nimport matplotlib.pyplot as plt\nimport seaborn as sns\n\n#Plots the bar chart\nfig = plt.figure(figsize=(10, 5))\nsns.set(font_scale=1.8)\ncategories = [\"0-9\",\"10-19\",\"20-29\",\"30-39\",\"40-49\",\"50+\"]\np1 = plt.bar(categories, malecount, 0.55, color='#d62728')\np2 = plt.bar(categories, femalecount, 0.55, bottom=malecount)\nplt.legend((p2[0], p1[0]), ('Female', 'Male'))\nplt.xlabel('Hours per Week Worked')\nplt.ylabel('Count')\nplt.show()\n```\n\n## Expected Value Calculation\n\\begin{align}\nexpected_{i,j} =\\frac{(row_{i} \\text{total})(column_{j} \\text{total}) }{(\\text{total observations})} \n\\end{align}\n\n\n```\n# Get Row Sums\nrow_sums = contingency_table.iloc[0:2, 6].values\ncol_sums = contingency_table.iloc[2, 0:6].values\n\nprint(row_sums)\nprint(col_sums)\n```\n\n [10771 21790]\n [ 458 1246 2392 3667 18336 6462]\n\n\n\n```\ntotal = contingency_table.loc['All','All']\ntotal\n```\n\n\n\n\n 32561\n\n\n\n\n```\nexpected = []\nfor i in range(len(row_sums)):\n expected_row = []\n for column in col_sums:\n expected_val = column*row_sums[i]/total\n expected_row.append(expected_val)\n expected.append(expected_row)\n \n\nexpected = np.array(expected)\nprint(expected.shape) \nprint(expected)\n```\n\n (2, 6)\n [[ 151.50388502 412.16995793 791.26046497 1213.02346365\n 6065.44811277 2137.59411566]\n [ 306.49611498 833.83004207 1600.73953503 2453.97653635\n 12270.55188723 4324.40588434]]\n\n\n\n```\nobserved = pd.crosstab(df['sex'], df['hours_per_week_categories']).values\nprint(observed.shape)\nobserved\n```\n\n (2, 6)\n\n\n\n\n\n array([[ 235, 671, 1287, 1914, 5636, 1028],\n [ 223, 575, 1105, 1753, 12700, 5434]])\n\n\n\n## Chi-Squared Statistic with Numpy\n\n\\begin{align}\n\\chi^2 = \\sum \\frac{(observed_{i}-expected_{i})^2}{(expected_{i})}\n\\end{align}\n\nFor the $observed$ values we will just use a version of our contingency table without the margins as a numpy array. In this way, if our observed values array and our expected values array are the same shape, then we can subtract them and divide them directly which makes the calculations a lot cleaner. No for loops!\n\n\n```\nchi_squared = ((observed - expected)**2/(expected)).sum()\nprint(f\"Chi-Squared: {chi_squared}\")\n```\n\n Chi-Squared: 2287.190943926107\n\n\n\n```\n# Calculate Degrees of Freedom\ndof = (len(row_sums)-1)*(len(col_sums)-1)\nprint(f\"Degrees of Freedom: {dof}\") \n```\n\n Degrees of Freedom: 5\n\n\n## Run a $\\chi^{2}$ Test using Scipy\n\n\n```\nchi_squared, p_value, dof, expected = stats.chi2_contingency(observed)\n\nprint(f\"Chi-Squared: {chi_squared}\")\nprint(f\"P-value: {p_value}\")\nprint(f\"Degrees of Freedom: {dof}\") \nprint(\"Expected: \\n\", np.array(expected))\n```\n\n Chi-Squared: 2287.190943926107\n P-value: 0.0\n Degrees of Freedom: 5\n Expected: \n [[ 151.50388502 412.16995793 791.26046497 1213.02346365\n 6065.44811277 2137.59411566]\n [ 306.49611498 833.83004207 1600.73953503 2453.97653635\n 12270.55188723 4324.40588434]]\n\n\nNull Hypothesis: Hours worked per week bins is **independent** of sex. \n\nDue to a p-value of 0, we REJECT the null hypothesis that hours worked per week and sex are independent, and conclude that there is an association between hours worked per week and sex. \n\n## Assignment - Build a confidence interval\n\nA confidence interval refers to a neighborhood around some point estimate, the size of which is determined by the desired p-value. For instance, we might say that 52% of Americans prefer tacos to burritos, with a 95% confidence interval of +/- 5%.\n\n52% (0.52) is the point estimate, and +/- 5% (the interval $[0.47, 0.57]$) is the confidence interval. \"95% confidence\" means a p-value $\\leq 1 - 0.95 = 0.05$.\n\nIn this case, the confidence interval includes $0.5$ - which is the natural null hypothesis (that half of Americans prefer tacos and half burritos, thus there is no clear favorite). So in this case, we could use the confidence interval to report that we've failed to reject the null hypothesis.\n\nBut providing the full analysis with a confidence interval, including a graphical representation of it, can be a helpful and powerful way to tell your story. Done well, it is also more intuitive to a layperson than simply saying \"fail to reject the null hypothesis\" - it shows that in fact the data does *not* give a single clear result (the point estimate) but a whole range of possibilities.\n\nHow is a confidence interval built, and how should it be interpreted? It does *not* mean that 95% of the data lies in that interval - instead, the frequentist interpretation is \"if we were to repeat this experiment 100 times, we would expect the average result to lie in this interval ~95 times.\"\n\nFor a 95% confidence interval and a normal(-ish) distribution, you can simply remember that +/-2 standard deviations contains 95% of the probability mass, and so the 95% confidence interval based on a given sample is centered at the mean (point estimate) and has a range of +/- 2 (or technically 1.96) standard deviations.\n\nDifferent distributions/assumptions (90% confidence, 99% confidence) will require different math, but the overall process and interpretation (with a frequentist approach) will be the same.\n\nYour assignment - using the data from the prior module ([congressional voting records](https://archive.ics.uci.edu/ml/datasets/Congressional+Voting+Records)):\n\n\n### Confidence Intervals:\n1. Generate and numerically represent a confidence interval\n2. Graphically (with a plot) represent the confidence interval\n3. Interpret the confidence interval - what does it tell you about the data and its distribution?\n\n### Chi-squared tests:\n4. Take a dataset that we have used in the past in class that has **categorical** variables. Pick two of those categorical variables and run a chi-squared tests on that data\n - By hand using Numpy\n - In a single line using Scipy\n\nStretch goals:\n\n1. Write a summary of your findings, mixing prose and math/code/results. *Note* - yes, this is by definition a political topic. It is challenging but important to keep your writing voice *neutral* and stick to the facts of the data. Data science often involves considering controversial issues, so it's important to be sensitive about them (especially if you want to publish).\n2. Apply the techniques you learned today to your project data or other data of your choice, and write/discuss your findings here.\n3. Refactor your code so it is elegant, readable, and can be easily run for all issues.\n\n\n\n---\n\n### Confidence Intervals:\n1. Generate and numerically represent a confidence interval\n2. Graphically (with a plot) represent the confidence interval\n3. Interpret the confidence interval - what does it tell you about the data and its distribution?\n\n\n```\ndf1 = pd.read_csv('https://archive.ics.uci.edu/ml/machine-learning-databases/voting-records/house-votes-84.data', header=None)\n```\n\n\n```\ndf = df.replace('n',0)\n```\n\n\n```\ndf = df.replace('y',1)\n```\n\n\n\n---\n\n### Chi-squared tests:\n4. Take a dataset that we have used in the past in class that has **categorical** variables. Pick two of those categorical variables and run a chi-squared tests on that data\n - By hand using Numpy\n - In a single line using Scipy\n\n\n```\ndf = pd.read_csv('https://raw.githubusercontent.com/ryanleeallred/datasets/master/adult.csv', na_values=\" ?\")\nprint(df.shape)\ndf.head()\n```\n\n (32561, 15)\n\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    ageworkclassfnlwgteducationeducation-nummarital-statusoccupationrelationshipracesexcapital-gaincapital-losshours-per-weekcountrysalary
    039State-gov77516Bachelors13Never-marriedAdm-clericalNot-in-familyWhiteMale2174040United-States<=50K
    150Self-emp-not-inc83311Bachelors13Married-civ-spouseExec-managerialHusbandWhiteMale0013United-States<=50K
    238Private215646HS-grad9DivorcedHandlers-cleanersNot-in-familyWhiteMale0040United-States<=50K
    353Private23472111th7Married-civ-spouseHandlers-cleanersHusbandBlackMale0040United-States<=50K
    428Private338409Bachelors13Married-civ-spouseProf-specialtyWifeBlackFemale0040Cuba<=50K
    \n
    \n\n\n\n\n```\ndf.describe()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    agefnlwgteducation-numcapital-gaincapital-losshours-per-week
    count32561.0000003.256100e+0432561.00000032561.00000032561.00000032561.000000
    mean38.5816471.897784e+0510.0806791077.64884487.30383040.437456
    std13.6404331.055500e+052.5727207385.292085402.96021912.347429
    min17.0000001.228500e+041.0000000.0000000.0000001.000000
    25%28.0000001.178270e+059.0000000.0000000.00000040.000000
    50%37.0000001.783560e+0510.0000000.0000000.00000040.000000
    75%48.0000002.370510e+0512.0000000.0000000.00000045.000000
    max90.0000001.484705e+0616.00000099999.0000004356.00000099.000000
    \n
    \n\n\n\n\n```\ndf.describe(exclude='number')\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    workclasseducationmarital-statusoccupationrelationshipracesexcountrysalary
    count307253256132561307183256132561325613197832561
    unique816714652412
    topPrivateHS-gradMarried-civ-spouseProf-specialtyHusbandWhiteMaleUnited-States<=50K
    freq22696105011497641401319327816217902917024720
    \n
    \n\n\n\n\n```\ncut_points = [0, 9, 19, 29, 39, 49, 100]\nlabel_names = ['0-9', '10-19', '20-29', '30-39', '40-49', '50+']\ndf['hours_per_week_catergories'] = pd.cut(df['hours-per-week'], cut_points, labels=label_names)\n\n```\n\n\n```\ndf.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    ageworkclassfnlwgteducationeducation-nummarital-statusoccupationrelationshipracesexcapital-gaincapital-losshours-per-weekcountrysalaryhours_per_week_catergories
    039State-gov77516Bachelors13Never-marriedAdm-clericalNot-in-familyWhiteMale2174040United-States<=50K40-49
    150Self-emp-not-inc83311Bachelors13Married-civ-spouseExec-managerialHusbandWhiteMale0013United-States<=50K10-19
    238Private215646HS-grad9DivorcedHandlers-cleanersNot-in-familyWhiteMale0040United-States<=50K40-49
    353Private23472111th7Married-civ-spouseHandlers-cleanersHusbandBlackMale0040United-States<=50K40-49
    428Private338409Bachelors13Married-civ-spouseProf-specialtyWifeBlackFemale0040Cuba<=50K40-49
    \n
    \n\n\n\n\n```\ndf['education'].value_counts()\n\n```\n\n\n\n\n HS-grad 10501\n Some-college 7291\n Bachelors 5355\n Masters 1723\n Assoc-voc 1382\n 11th 1175\n Assoc-acdm 1067\n 10th 933\n 7th-8th 646\n Prof-school 576\n 9th 514\n 12th 433\n Doctorate 413\n 5th-6th 333\n 1st-4th 168\n Preschool 51\n Name: education, dtype: int64\n\n\n\n\n```\ndf['hours_per_week_catergories'].value_counts()\n```\n\n\n\n\n 40-49 18336\n 50+ 6462\n 30-39 3667\n 20-29 2392\n 10-19 1246\n 0-9 458\n Name: hours_per_week_catergories, dtype: int64\n\n\n\n\n```\ndf = df.sort_values(by='hours_per_week_catergories', ascending=True)\ndf.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    ageworkclassfnlwgteducationeducation-nummarital-statusoccupationrelationshipracesexcapital-gaincapital-losshours-per-weekcountrysalaryhours_per_week_catergories
    3129055Self-emp-not-inc41938Bachelors13Married-civ-spouseProf-specialtyWifeWhiteFemale008United-States<=50K0-9
    2090977Self-emp-not-inc71676Some-college10WidowedAdm-clericalNot-in-familyWhiteFemale019441United-States<=50K0-9
    2812728Local-gov207213Assoc-acdm12Never-marriedCraft-repairOwn-childWhiteMale005United-States<=50K0-9
    847275NaN173064Bachelors13Married-civ-spouseNaNHusbandWhiteMale006United-States<=50K0-9
    1961640NaN170649Some-college10Married-civ-spouseNaNHusbandWhiteMale008United-States<=50K0-9
    \n
    \n\n\n\n\n```\n\n```\n\n\n```\ncontingency_table = pd.crosstab(df['education'], df['hours_per_week_catergories'], margins=True)\n```\n\n\n```\ncontingency_table\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    hours_per_week_catergories0-910-1920-2930-3940-4950+All
    education
    10th227496134500107933
    11th441471871375391211175
    12th637655522545433
    1st-4th2721268923168
    5th-6th314223822036333
    7th-8th1431489236497646
    9th922506131656514
    Assoc-acdm2426761216062141067
    Assoc-voc1522551598732581382
    Bachelors67120237479299214605355
    Doctorate861730162190413
    HS-grad11627565812296487173610501
    Masters2137641628805591723
    Preschool0461026551
    Prof-school7151955193287576
    Some-college100409771879386412687291
    All45812462392366718336646232561
    \n
    \n\n\n\n\n```\nhsgrad_count = contingency_table.iloc[11][0:6].values\nhsgrad_count\n```\n\n\n\n\n array([ 116, 275, 658, 1229, 6487, 1736])\n\n\n\n\n```\nbachelors_count = contingency_table.iloc[9][0:6].array\nbachelors_count\n```\n\n\n\n\n \n [67, 120, 237, 479, 2992, 1460]\n Length: 6, dtype: int64\n\n\n\n\n```\nimport matplotlib.pyplot as plt\nimport seaborn as sns\nfig = plt.figure(figsize=(10,5))\nsns.set(font_scale=1.8)\ncategories = [\"0-9\",\"10-19\",\"20-29\",\"30-39\",\"40-49\",\"50+\"]\np1 = plt.bar(categories, hsgrad_count, .55, color='#d62728')\np2 = plt.bar(categories, bachelors_count, .55, bottom=hsgrad_count)\nplt.legend((p2[0], p1[0]), ('High School Grad', 'Bachelors Degree'))\nplt.xlabel('Hours per Week Worked')\nplt.ylabel('Count')\nplt.show()\n```\n\n# Run a \ud835\udf122 Test using Scipy\n\n\n```\n \n```\n\n\n```\n\n```\n\n## Resources\n\n- [Interactive visualize the Chi-Squared test](https://homepage.divms.uiowa.edu/~mbognar/applets/chisq.html)\n- [Calculation of Chi-Squared test statistic](https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test)\n- [Visualization of a confidence interval generated by R code](https://commons.wikimedia.org/wiki/File:Confidence-interval.svg)\n- [Expected value of a squared standard normal](https://math.stackexchange.com/questions/264061/expected-value-calculation-for-squared-normal-distribution) (it's 1 - which is why the expected value of a Chi-Squared with $n$ degrees of freedom is $n$, as it's the sum of $n$ squared standard normals)\n\n\n```\n\n```\n", "meta": {"hexsha": "96c9cc0eabace6f182bd16a1b7fc3668e3e5c8e7", "size": 314791, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Pierre_nelson_Copy_of_LS_DS6_132_Sampling_Confidence_Intervals_and_Hypothesis_Testing.ipynb", "max_stars_repo_name": "alxanderpierre/DS-Unit-1-Sprint-3-Statistical-Tests-and-Experiments", "max_stars_repo_head_hexsha": "e7824ca938d776c4ea7062f3e1936df73841d9d3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Pierre_nelson_Copy_of_LS_DS6_132_Sampling_Confidence_Intervals_and_Hypothesis_Testing.ipynb", "max_issues_repo_name": "alxanderpierre/DS-Unit-1-Sprint-3-Statistical-Tests-and-Experiments", "max_issues_repo_head_hexsha": "e7824ca938d776c4ea7062f3e1936df73841d9d3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Pierre_nelson_Copy_of_LS_DS6_132_Sampling_Confidence_Intervals_and_Hypothesis_Testing.ipynb", "max_forks_repo_name": "alxanderpierre/DS-Unit-1-Sprint-3-Statistical-Tests-and-Experiments", "max_forks_repo_head_hexsha": "e7824ca938d776c4ea7062f3e1936df73841d9d3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 74.0336312324, "max_line_length": 45128, "alphanum_fraction": 0.6636498502, "converted": true, "num_tokens": 44135, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736783928749126, "lm_q2_score": 0.7718435083355187, "lm_q1q2_score": 0.4427899434128546}} {"text": "```python\n# import numpy as np\n\n# # !/usr/bin/env python3\n# # -*- coding: utf-8 -*-\n# \"\"\"\n# Created on 20181219\n\n# @author: zhangji\n\n# Trajection of a ellipse, Jeffery equation. \n# \"\"\"\n\n# %pylab inline\n# pylab.rcParams['figure.figsize'] = (25, 11)\n# fontsize = 40\n\n# import numpy as np\n# import scipy as sp\n# from scipy.optimize import leastsq, curve_fit\n# from scipy import interpolate\n# from scipy.interpolate import interp1d\n# from scipy.io import loadmat, savemat\n# # import scipy.misc\n\n# import matplotlib\n# from matplotlib import pyplot as plt\n# from matplotlib import animation, rc\n# import matplotlib.ticker as mtick\n# from mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\n# from mpl_toolkits.mplot3d import Axes3D, axes3d\n\n# from sympy import symbols, simplify, series, exp\n# from sympy.matrices import Matrix\n# from sympy.solvers import solve\n\n# from IPython.display import display, HTML\n# from tqdm import tqdm_notebook as tqdm\n# import pandas as pd\n# import re\n# from scanf import scanf\n# import os\n# import glob\n\n# from codeStore import support_fun as spf\n# from src.support_class import *\n# from src import stokes_flow as sf\n\n# rc('animation', html='html5')\n# PWD = os.getcwd()\n# font = {'size': 20}\n# matplotlib.rc('font', **font)\n# np.set_printoptions(linewidth=90, precision=5)\n\nimport os\nimport glob\nimport re\nimport pandas as pd\nfrom scanf import scanf\nimport natsort \nimport numpy as np\nimport scipy as sp\nfrom scipy.optimize import leastsq, curve_fit\nfrom scipy import interpolate\nfrom scipy import spatial\n# from scipy.interpolate import interp1d\nfrom scipy.io import loadmat, savemat\n# import scipy.misc\nimport importlib\nfrom IPython.display import display, HTML\nimport pickle\n\nimport matplotlib\nfrom matplotlib import pyplot as plt\nfrom matplotlib import colors as mcolors\nfrom matplotlib import animation, rc\nimport matplotlib.ticker as mtick\nfrom mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\nfrom mpl_toolkits.mplot3d import Axes3D, axes3d\nfrom matplotlib import cm\nfrom mpl_toolkits.axes_grid1 import make_axes_locatable\n\nfrom time import time\nfrom src.support_class import *\nfrom src import jeffery_model as jm\nfrom codeStore import support_fun as spf\n\n# %matplotlib notebook\n\nrc('animation', html='html5')\nfontsize = 40\nfigsize = (30, 16)\nPWD = os.getcwd()\n```\n\n\n```python\n# make average lookup table form existing date\njob_dir = 'planeShearRatex_0.25c'\n\nwith open('%s.pickle' % job_dir, 'rb') as handle:\n table_data = pickle.load(handle)\nn_case = len(table_data)\n \nU_avr = [table_data[0][1][0][2] * 0 for i0 in range(6)]\nt_lmt = np.zeros(6)\nfor tpsi, table_psi_data in table_data:\n for (ty, tx, tU), i0 in zip(table_psi_data, range(len(U_avr))):\n U_avr[i0] = U_avr[i0] + tU / n_case\n t1 = np.nanmax(np.abs(tU.values))\n t_lmt[i0] = np.max((t_lmt[i0], t1))\n \n# interpolate over 1d, psi, for ellipse case psi is fake, and it just copy the data. \ntable_data = []\nfor tU in U_avr:\n tx = tU.columns.values # norm_phi\n ty = tU.index.values # norm_theta\n table_data.append((ty, tx, tU))\nt_name = job_dir + '_avr'\nwith open('%s.pickle' % t_name, 'wb') as handle:\n pickle.dump(table_data, handle, protocol=pickle.HIGHEST_PROTOCOL)\nprint('save table_data to %s.pickle' % t_name)\n```\n\n save table_data to planeShearRatex_0.25c_avr.pickle\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "1763ae3baef46865a08e449fe6c81ee389c5be89", "size": 5286, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "head_Force/loop_table/make_Avr_Table.ipynb", "max_stars_repo_name": "pcmagic/stokes_flow", "max_stars_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-11-11T05:00:53.000Z", "max_stars_repo_stars_event_max_datetime": "2018-11-11T05:00:53.000Z", "max_issues_repo_path": "head_Force/loop_table/make_Avr_Table.ipynb", "max_issues_repo_name": "pcmagic/stokes_flow", "max_issues_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "head_Force/loop_table/make_Avr_Table.ipynb", "max_forks_repo_name": "pcmagic/stokes_flow", "max_forks_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.3793103448, "max_line_length": 94, "alphanum_fraction": 0.5732122588, "converted": true, "num_tokens": 951, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.6654105521116443, "lm_q1q2_score": 0.44277120727663527}} {"text": "# Machine learning for medicine\n## Using ML\n\nIn this notebook we're going to apply ML to a dataset and contrast it with what you may do in a standard Discovery project.\nThe goal of this notebook is to empower you to perform parallel, ML-style analysis on your data and maybe pick up on some cool patterns that you otherwise would have missed.\n\n### Imports\n\n\n```python\nimport pandas as pds\nimport numpy as np\nfrom ipywidgets import interact, interactive, fixed, interact_manual\nimport scipy.stats as stats\nimport ipywidgets as widgets\nimport matplotlib.pyplot as plt\n\nimport networkx as nx\n```\n\n## Diabetes\n\nWe're going to work with a straight forward static system in this experiment.\nWe've got four variables: $x,y,z,w$ that we're trying to study.\n\n\\begin{equation}\nx = \\text{diabetes} \\in {\\{\\text{yes},\\text{no}\\}}\\\\\ny = \\text{bloodsugar} \\\\\nz = \\text{insulin} \\\\ \nw = \\text{potassium}\n\\end{equation}\n\nThanks to the first year and a half of med school, we have some ideas of how these variables relate to each other.\n\n\n```python\nhealthy_patient = nx.Graph()\nhealthy_patient.add_nodes_from([0,1,2,3])\nhealthy_patient.add_edge([2,1])\nhealthy_patient.add_edge([])\n```\n\n## Data from an Experiment\n### Experiment 1\n\n### Experiment 2\nWe've got a whole bunch of people with anemia.\nWe're going to see if a new drug, Awesomumab, increases the Hemoglobin in patients.\nWe recruit about 500 patients for this study, 250 in the drug arm and 250 in the placebo arm.\nFor each of the patients we have a pre-study hemoglobin and a post-study hemoglobin.\nWe want to know if the patients that received Awesomumab had an elevated hemoglobin compared to those that received only placebo.\n\nLet's take a look at our data.\n\n\n```python\ndef sys(mod_strength,unknown_val):\n ins_level = np.random.randint(0,10,size=(100,))\n dz_state = np.zeros(100)\n dz_state[60:] = 1\n np.random.shuffle(dz_state)\n \n unknown_state = np.ones(100)\n unknown_state[50:] = unknown_val\n np.random.shuffle(unknown_state)\n \n blood_glucose = -(dz_state - 1)*(unknown_state) * 100 + (dz_state)*(unknown_state)*mod_strength*ins_level + np.random.normal(0,10,size=dz_state.shape)\n #200*(-dz_state + 1) + (1-dz_state)*(10*ins_level - 100) + np.random.normal(0,10,size=dz_state.shape)\n\n x = ins_level\n y = blood_glucose\n \n #Plotting\n fig = plt.figure()\n ax = fig.add_axes([0,0,1,1])\n #ax.axis('off')\n ax.scatter(ins_level,blood_glucose)\n #plt.xlim((-10,10))\n #plt.ylim((-10,10))\n pears = stats.pearsonr(x,y)\n spear = stats.spearmanr(x,y)\n plt.title(pears)\n plt.show()\n \nw = interactive(sys,mod_strength=(-10,10,1),unknown_val=(0,1,1))\ndisplay(w)\n```\n\n\n interactive(children=(IntSlider(value=0, description='mod_strength', max=10, min=-10), IntSlider(value=0, desc\u2026\n\n\n## The example\n\nWe'll work in the context of Diabetes.\nSpecifically, we're going to study how Pancreas $\\beta$ cells, insulin, blood glucose, and potassium all interact.\n\nThe core of the example is developed and described [elsewhere]() as it is out of the scope of the discussion here.\n\n### \n", "meta": {"hexsha": "2fce3b9e91223b80d8c5a7043368ac2b971f23bb", "size": 7054, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ML_Med_useML.ipynb", "max_stars_repo_name": "amurugan19/Med_ML", "max_stars_repo_head_hexsha": "961c667b5cdf8e6ecf7cdbe25deb5eb132fc0857", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "ML_Med_useML.ipynb", "max_issues_repo_name": "amurugan19/Med_ML", "max_issues_repo_head_hexsha": "961c667b5cdf8e6ecf7cdbe25deb5eb132fc0857", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ML_Med_useML.ipynb", "max_forks_repo_name": "amurugan19/Med_ML", "max_forks_repo_head_hexsha": "961c667b5cdf8e6ecf7cdbe25deb5eb132fc0857", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-02-17T21:26:15.000Z", "max_forks_repo_forks_event_max_datetime": "2020-02-17T21:26:15.000Z", "avg_line_length": 37.7219251337, "max_line_length": 1347, "alphanum_fraction": 0.5963992061, "converted": true, "num_tokens": 825, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.6688802603710086, "lm_q1q2_score": 0.4427471455440709}} {"text": "# \u7b2c\u5341\u4e03\u8bb2\uff1a\u89e3\u8fde\u7eed\u72b6\u6001\u7684MDP\n\n\u5728\u4e0a\u4e00\u8bb2\uff0c\u6211\u4eec\u7ed9\u51fa\u4e86\u5f3a\u5316\u5b66\u4e60\u548c\u9a6c\u5c14\u53ef\u592b\u51b3\u7b56\u8fc7\u7a0b\u7684\u5b9a\u4e49\u3002\n\nMDP\u662f\u4e00\u4e2a\u4e94\u5143\u7ec4$\\left(S,A,\\{P_{sa}\\},\\gamma,R\\right)$\uff0c\u4e0a\u4e00\u8bb2\u6211\u4eec\u4e00\u76f4\u4f7f\u7528\u7684\u4e00\u4e2a\u4f4e\u9636MDP\u7684\u4f8b\u5b50\u2014\u2014\u673a\u5668\u4eba\u8eb2\u907f\u969c\u788d\u7269\u5230\u8fbe\u76ee\u7684\u5730\uff08$+1$\uff09\uff1a\n\n$$\n\\begin{array}\n{|c|c|c|c|}\n\\hline\n?&?&?&+1\\\\\n\\hline\n?&\\textrm{\u5899}&?&-1\\\\\n\\hline\n?&?&?&?\\\\\n\\hline\n\\end{array}\n$$\n\n* \u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c\u5171\u6709$11$\u4e2a\u72b6\u6001\u2014\u2014$\\lvert S\\rvert=11$\uff1b\n* \u673a\u5668\u4eba\u80fd\u591f\u505a\u51fa\u7684\u52a8\u4f5c\u4e3a$A=\\{\\textrm{N, S, E, W}\\}$\uff0c\u5373\u5411\u5317\u3001\u5357\u3001\u4e1c\u3001\u897f\u79fb\u52a8\uff1b\n* \u72b6\u6001\u8f6c\u6362\u6982\u7387$P_{sa}$\u4e3a\u5904\u4e8e\u72b6\u6001$s$\u65f6\uff0c\u505a\u51fa\u52a8\u4f5c$a$\u540e\u72b6\u6001\u53d8\u4e3a$s'$\u7684\u6982\u7387\uff0c\u6bd4\u5982\u4ece\u72b6\u6001$(3,1)$\u5f00\u59cb\uff0c\u505a\u51fa\u52a8\u4f5c\u201c\u5411\u5317\u79fb\u52a8\u201d\uff0c\u5219\u673a\u5668\u4eba\u6a21\u578b\u53ef\u80fd\u4f1a\u56e0\u4e3a\u673a\u68b0\u4f20\u52a8\u566a\u97f3\u6709$0.1$\u7684\u6982\u7387\u201c\u5411\u4e1c\u79fb\u52a8\u201d\u3001\u6709$0.1$\u7684\u6982\u7387\u7684\u6982\u7387\u201c\u5411\u897f\u79fb\u52a8\u201d\u3001\u6709$0.8$\u7684\u6982\u7387\u201c\u5411\u5317\u79fb\u52a8\u201d\uff0c\u5219\u771f\u6b63\u843d\u5728\u72b6\u6001$(3.2)$\u7684\u6982\u7387\u4e3a$0.8$\u5982\u56fe\uff1a\n\n$$\n\\begin{array}\n{|c|c|c|c|}\n\\hline\n?&?&?&+1\\\\\n\\hline\n?&\\textrm{\u5899}&0.8\\uparrow&-1\\\\\n\\hline\n?&0.1\\leftarrow&s&0.1\\rightarrow\\\\\n\\hline\n\\end{array}\n$$\n\n* \u5956\u52b1\u51fd\u6570$R=\\begin{cases}\\pm1&\\textrm{absorbing state}\\\\-0.02&\\textrm{else where}\\end{cases}$\uff0c\u5373\u5728$(4,2)$\u65f6\u5956\u52b1\u4e3a$-1$\uff0c\u4ee3\u8868\u4efb\u52a1\u5931\u8d25\u5e76\u7ed3\u675f\uff1b\u800c\u5728$(4,3)$\u72b6\u6001\u4ee3\u8868\u5956\u52b1$+1$\uff0c\u4ee3\u8868\u4efb\u52a1\u6210\u529f\u5e76\u7ed3\u675f\uff1b\u5176\u4ed6\u72b6\u6001\u5956\u52b1$-0.02$\uff08**\u5438\u6536\u72b6\u6001\uff08absorbing state\uff09**\u8868\u793a\u8fdb\u5165\u8be5\u72b6\u6001\u5373\u65f6\u5956\u52b1$+1$\uff0c\u5e76\u4e14\u5176\u4ed6\u72b6\u6001\u5956\u52b1\u5747\u5f52\u96f6\uff0c\u4e5f\u5c31\u662f\u4efb\u52a1\u7ed3\u675f\uff09\u3002\n\n* \u6298\u6263\u56e0\u5b50$\\gamma=0.99$\uff0c\u8fd9\u4e2a\u503c\u4ee3\u8868\u5f53\u524d\u5956\u52b1\u4e0e\u672a\u6765\u5956\u52b1\u95f4\u7684\u6743\u91cd\uff08\u5f53\u524d\u4e3a$\\gamma^0$\uff0c\u4e0b\u4e00\u6b65\u4e3a$\\gamma^1$\uff0c\u518d\u4e0b\u4e00\u6b65\u4e3a$\\gamma^2$\u7b49\u7b49\uff0c\u8fd9\u4e00\u9879\u9f13\u52b1\u6a21\u578b\u5c3d\u65e9\u83b7\u5f97\u5956\u52b1\uff09\u3002\n\n\u800c\u7b56\u7565$\\pi:S\\to A$\uff0c\u5bf9\u672c\u4f8b\u6765\u8bf4\u5c31\u662f\u4e00\u4e2a\u673a\u5668\u4eba\u63a7\u5236\u7b56\u7565\uff0c\u5b83\u662f\u4e00\u4e2a\u4ece\u72b6\u6001\u5230\u52a8\u4f5c\u7684\u6620\u5c04\uff0c\u4e5f\u5c31\u662f\u544a\u8bc9\u673a\u5668\u4eba\u5728\u6bcf\u4e00\u79cd\u72b6\u6001\u4e0b\u8be5\u6267\u884c\u4ec0\u4e48\u52a8\u4f5c\u3002\u6211\u4eec\u7684\u76ee\u6807\u5c31\u662f\u627e\u5230\u4e00\u4e2a\u80fd\u591f\u6700\u5927\u5316\u6211\u4eec\u9884\u671f\u603b\u6536\u76ca\u7684\u7b56\u7565\uff08\u5373\u5c3d\u53ef\u80fd\u591a\u7684\u83b7\u5f97\u5956\u52b1\uff09\u3002\n\n\u6211\u4eec\u5b9a\u4e49\u4e86\u7b56\u7565$\\pi$\u7684\u4ef7\u503c\u51fd\u6570$V^\\pi(s)=\\mathrm E\\left[R(s_0)+\\gamma R(s_1)+\\gamma^2R(s_2)+\\cdots\\right]$\uff0c\u89e3\u91ca\u4e3a\u201c\u5728\u72b6\u6001$s$\u4e0b\u542f\u52a8\u673a\u5668\u4eba\u5e76\u4f7f\u5176\u5728\u7b56\u7565$\\pi$\u4e0b\u8fd0\u884c\uff0c\u6240\u5f97\u5230\u7684\u6298\u6263\u5956\u52b1\u4e4b\u548c\u7684\u671f\u671b\u201d\u3002\u5bf9\u4e8e\u76ee\u6807\u800c\u8a00\uff0c\u5176\u4e2d\u4e00\u79cd\u65b9\u6cd5\u662f\u627e\u5230\u6700\u5927\u7684\u4ef7\u503c\u51fd\u6570\uff1a\n\n1. \u6211\u4eec\u5148\u8981\u8ba1\u7b97$\\displaystyle V^*(s)=\\max_\\pi V^\\pi(s)$\u2014\u2014\u8fd9\u662f\u6267\u884c\u4efb\u610f\u7b56\u7565\u80fd\u591f\u5f97\u5230\u7684\u6700\u5927\u6536\u76ca\uff0c\u6bd4\u5982\u8fd9\u4e2a\u6700\u4f18\u4ef7\u503c\u51fd\u6570\u770b\u4e0a\u53bb\u5982\u4e0b\u56fe\u6240\u793a\uff1a\n $$\n \\begin{array}\n {|c|c|c|c|}\n \\hline\n .86&.90&.95&+1\\\\\n \\hline\n .82&\\textrm{\u5899}&.69&-1\\\\\n \\hline\n .78&.75&.71&.49\\\\\n \\hline\n \\end{array}\n $$\n \u8fd9\u5c31\u662f$V^*$\uff0c\u5b83\u8868\u793a\u4ece\u4efb\u610f\u72b6\u6001\u5f00\u59cb\uff0c\u80fd\u591f\u5f97\u5230\u7684\u6298\u6263\u5956\u52b1\u4e4b\u548c\u7684\u9884\u671f\u503c\u3002\n\n2. \u4e00\u65e6\u5f97\u5230\u4e86$V^*$\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u4f7f\u7528$\\displaystyle\\pi^*(s)=\\arg\\max_{a\\in A}\\sum_{s'\\in S}P_{sa}\\left(s'\\right)V^*\\left(s'\\right)$\u8ba1\u7b97\u51fa\u6700\u4f73\u7b56\u7565$\\pi^*$\u3002\n $$\n \\begin{array}\n {|c|c|c|c|}\n \\hline\n \\rightarrow&\\rightarrow&\\rightarrow&+1\\\\\n \\hline\n \\uparrow&\\textrm{\u5899}&\\uparrow&-1\\\\\n \\hline\n \\uparrow&\\leftarrow&\\leftarrow&\\leftarrow\\\\\n \\hline\n \\end{array}\n $$\n3. \u7b97\u6cd5\u7684\u5b9e\u73b0\u5c31\u662f\u5229\u7528\u8d1d\u5c14\u66fc\u65b9\u7a0b$\\displaystyle V^*(s)=R(s)+\\max_{a\\in A}\\gamma\\sum_{s'\\in S}P_{s\\pi(s)}\\left(s'\\right)V^*\\left(s'\\right)$\uff0c\u5728\u72b6\u6001$s$\u65f6\u7684\u6700\u4f18\u6298\u6263\u5956\u52b1\u4e4b\u548c\u4e3a\u201c\u5728$s$\u5f00\u59cb\u65f6\u7684\u5373\u65f6\u5956\u52b1\u201d\u52a0\u4e0a\u201c\u6240\u6709\u52a8\u4f5c\u4e2d\u80fd\u591f\u6700\u5927\u5316\u672a\u6765\u6298\u6263\u5956\u52b1\u603b\u548c\u52a8\u4f5c\u5e8f\u5217\u5bf9\u5e94\u7684\u672a\u6765\u5956\u52b1\u201d\u3002\u4e8e\u662f\u6211\u4eec\u53ef\u4ee5\u5e94\u7528\u4e00\u4e2a\u503c\u8fed\u4ee3\u7b97\u6cd5\uff0c\u4e5f\u5c31\u662f\u5229\u7528\u8d1d\u5c14\u66fc\u65b9\u7a0b\u4e0d\u505c\u7684\u8fed\u4ee3$\\displaystyle V(s):=R(s)+\\max_{a\\in A}\\gamma\\sum_{s'\\in S}P_{sa}\\left(s'\\right)V\\left(s'\\right)$\u3002\u8fd9\u6837$V(s)$\u5c31\u4f1a\u6536\u655b\u4e8e$V^*(s)$\uff0c\u800c\u6709\u4e86$V^*(s)$\u5c31\u53ef\u4ee5\u5f97\u5230$\\pi^*(s)$\u3002\n\n\u6700\u540e\uff0c\u6211\u4eec\u8fd8\u4ecb\u7ecd\u4e86\u7b56\u7565\u8fed\u4ee3\u7b97\u6cd5\uff1a\u6839\u636e\u968f\u673a\u521d\u59cb\u5316\u7684\u7b56\u7565$\\pi$\u8ba1\u7b97$V^\\pi$\u5e76\u66f4\u65b0$V$\uff08\u5373\u4e3a\u7279\u5b9a\u7684\u7b56\u7565\u8ba1\u7b97\u51fa\u4ef7\u503c\u51fd\u6570\uff0c\u4f7f\u7528\u8d1d\u5c14\u66fc\u65b9\u7a0b\u7ec4\u8ba1\u7b97\u6bcf\u4e2a\u72b6\u6001\u7684\u6298\u6263\u5956\u52b1\u603b\u548c\uff09\uff0c\u7136\u540e\u5047\u8bbe\u6211\u4eec\u5df2\u7ecf\u627e\u5230\u7684\u7b56\u7565\u662f\u6700\u4f73\u7b56\u7565\uff0c\u6839\u636e$\\displaystyle\\pi(s):=\\arg\\max_{a\\in A}\\sum_{s'\\in S}P_{sa}\\left(s'\\right)V\\left(s'\\right)$\u5e26\u5165\u5047\u8bbe\u7684\u6700\u4f18\u4ef7\u503c\u51fd\u6570$V$\u627e\u5230\u65b0\u7684\u7b56\u7565\u518d\u66f4\u65b0$\\pi$\u3002\u91cd\u590d\u8fd9\u4e24\u6b65\u6700\u7ec8\u4f1a\u4f7f\u5f97$V$\u6536\u655b\u4e8e$V^*$\uff0c$\\pi$\u6536\u655b\u4e8e$\\pi^*$\u3002\u5982\u56fe\uff1a\n\n1. \u968f\u673a\u521d\u59cb\u5316\u4e00\u4e2a\u7b56\u7565\uff1a\n $$\n \\begin{array}\n {|c|c|c|c|}\n \\hline\n \\downarrow&\\rightarrow&\\rightarrow&+1\\\\\n \\hline\n \\rightarrow&\\textrm{\u5899}&\\rightarrow&-1\\\\\n \\hline\n \\uparrow&\\rightarrow&\\uparrow&\\leftarrow\\\\\n \\hline\n \\end{array}\n $$\n2. \u89e3\u8d1d\u5c14\u66fc\u65b9\u7a0b\u7ec4$\\displaystyle V^\\pi(s)=R(s)+\\gamma\\sum_{s'\\in S}P_{s\\pi(s)}\\left(s'\\right)V^\\pi\\left(s'\\right)$\uff0c\u5728\u672c\u4f8b\u4e2d\u6709$11$\u79cd\u72b6\u6001\uff0c\u6240\u4ee5\u5c31\u6709\u7531$11$\u4e2a\u4ef7\u503c\u51fd\u6570\u7ec4\u6210\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff0c\u89e3\u8fd9\u4e2a\u65b9\u7a0b\u7ec4\u5c31\u53ef\u4ee5\u5f97\u5230\u4e2a\u72b6\u6001\u5bf9\u5e94\u7684\u4ef7\u503c\u51fd\u6570\u7684\u503c\uff0c\u518d\u586b\u5728\u56fe\u4e2d\u5373\u53ef\u5f97\u5230\u4e00\u4e2a\u65b0\u7684\u7b56\u7565\u3002\n\n\u4ee5\u4e0a\u5c31\u662f\u4e0a\u4e00\u8bb2\u7684\u4e3b\u8981\u5185\u5bb9\uff0c\u4e0b\u9762\u6211\u4eec\u4f1a\u63a5\u7740\u8fd9\u4e24\u79cd\u7b97\u6cd5\uff08\u503c\u8fed\u4ee3\u4e0e\u7b56\u7565\u8fed\u4ee3\uff09\u7ee7\u7eedMDP\u7684\u4ecb\u7ecd\u3002\n\n## 4. \u5177\u6709\u8fde\u7eed\u72b6\u6001\u7684MDP\n\n\u5230\u76ee\u524d\u4e3a\u6b62\uff0c\u6211\u4eec\u4ecb\u7ecd\u7684\u90fd\u662f\u5177\u6709\u6709\u9650\u4e2a\u72b6\u6001\u7684MDP\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u8981\u8ba8\u8bba\u5177\u6709\u65e0\u9650\u4e2a\u72b6\u6001\u7684MDP\u3002\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5bf9\u4e8e\u4e00\u8f86\u8f66\uff0c\u6211\u4eec\u4f1a\u7528\u72b6\u6001$(x,y,\\theta,\\dot x,\\dot y,\\dot\\theta)$\u6765\u8868\u793a\uff0c\u5176\u4e2d$(x,y)$\u8868\u793a\u8f66\u5b50\u7684\u4f4d\u7f6e\uff1b$\\theta$\u8868\u793a\u8f66\u5b50\u7684\u65b9\u5411\uff1b\u8f66\u5b50\u5728$x,y$\u65b9\u5411\u4e0a\u7684\u901f\u5ea6\u5206\u91cf$\\dot x,\\dot y$\uff1b\u4ee5\u53ca\u89d2\u901f\u5ea6$\\dot\\theta$\u3002\u56e0\u6b64\uff0c$S=\\mathbb R^6$\u662f\u4e00\u4e2a\u65e0\u9650\u72b6\u6001\u96c6\uff0c\u56e0\u4e3a\u5bf9\u4e8e\u4e00\u8f86\u8f66\u6765\u8bf4\uff0c\u6709\u65e0\u9650\u4e2a\u53ef\u80fd\u7684\u4f4d\u7f6e\u548c\u65b9\u5411\u3002\uff08\u7406\u8bba\u4e0a\u8bb2\uff0c\u8f66\u5b50\u7684\u65b9\u5411$\\theta$\u5e94\u8be5\u5728$\\theta\\in[-\\pi,\\pi)$\u53d6\u503c\uff0c\u800c\u4e0d\u662f$\\theta\\in\\mathbb R$\uff0c\u4f46\u5728\u6211\u4eec\u7684\u4f8b\u5b50\u4e2d\uff0c\u8fd9\u4e00\u70b9\u5e76\u4e0d\u91cd\u8981\u3002\uff09\u7c7b\u4f3c\u7684\uff0c\u5728[\u95ee\u9898\u96c64](http://cs229.stanford.edu/materials/ps4.pdf)\u4e2d\u7684\u5012\u7f6e\u949f\u6446\u95ee\u9898\uff0c\u949f\u6446\u7684\u72b6\u6001\u4e3a$(x,\\theta,\\dot x,\\dot\\theta)$\uff0c\u800c$\\theta$\u5c31\u662f\u949f\u6446\u7684\u89d2\u5ea6\u3002\u518d\u6bd4\u5982\u6211\u4eec\u7684\u9065\u63a7\u76f4\u5347\u673a\uff0c\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u98de\u884c\uff0c\u5b83\u7684\u72b6\u6001\u4e3a$(x,y,z,\\phi,\\theta,\\psi,\\dot x,\\dot y,\\dot z,\\dot\\phi,\\dot\\theta,\\dot\\psi)$\uff0c\u5176\u4e2d$\\phi$\u4e3a\u6a2a\u6eda\u89d2\u3001$\\theta$\u4e3a\u4fef\u4ef0\u89d2\u3001$\\psi$\u4e3a\u822a\u5411\u89d2\u3002\n\n\u5728\u672c\u8282\u6211\u4eec\u5c06\u8003\u8651\u72b6\u6001\u7a7a\u95f4\u4e3a$S=\\mathbb R^n$\u7684\u60c5\u5f62\uff0c\u5e76\u4ecb\u7ecd\u5982\u4f55\u89e3\u51fa\u8fd9\u79cdMDP\u3002\n### 4.1 \u79bb\u6563\u5316\uff08Discretization\uff09\n\n\u63cf\u8ff0\u4e00\u4e2a\u8fde\u7eed\u72b6\u6001MDP\u7684\u6700\u7b80\u5355\u7684\u65b9\u6cd5\u5c31\u662f\u5c06\u5b83\u7684\u72b6\u6001\u7a7a\u95f4\u79bb\u6563\u5316\uff0c\u7136\u540e\u5728\u5bf9\u5176\u5e94\u7528\u524d\u9762\u4ecb\u7ecd\u7684\u503c\u8fed\u4ee3\u6216\u7b56\u7565\u8fed\u4ee3\u8fdb\u884c\u6c42\u89e3\u3002\n\n\u6bd4\u5982\u6211\u4eec\u6709\u4e00\u4e2a\u4e8c\u7ef4\u72b6\u6001$(s_1,s_2)$\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u4f7f\u7528\u7f51\u683c\u5c06\u5176\u72b6\u6001\u7a7a\u95f4\u79bb\u6563\u5316\uff1a\n\n\n\n\u56fe\u4e2d\u7684\u6bcf\u4e00\u683c\u90fd\u4ee3\u8868\u4e00\u4e2a\u72ec\u7acb\u7684\u72b6\u6001$\\bar s$\u3002\u4e8e\u662f\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u79bb\u6563\u72b6\u6001\u7684MDP$\\left(\\bar S,A,\\left\\{P_{\\bar sa}\\right\\},\\gamma,R\\right)$\u53bb\u8fd1\u4f3c\u539f\u6765\u7684\u8fde\u7eed\u72b6\u6001MDP\uff0c\u800c$\\left\\{P_{\\bar sa}\\right\\}$\u662f\u5173\u4e8e\u79bb\u6563\u72b6\u6001\u7684\u72b6\u6001\u8f6c\u6362\u6982\u7387\u3002\u7136\u540e\u5c31\u53ef\u4ee5\u5e94\u7528\u524d\u9762\u4ecb\u7ecd\u8fc7\u7684\u503c\u8fed\u4ee3\u6216\u7b56\u7565\u8fed\u4ee3\u89e3\u51faMDP$\\left(\\bar S,A,\\left\\{P_{\\bar sa}\\right\\},\\gamma,R\\right)$\u7684$V^*\\left(\\bar s\\right)$\u548c$\\pi^*\\left(\\bar s\\right)$\u3002\u5f53\u6211\u4eec\u7684\u5b9e\u9645\u7cfb\u7edf\u5904\u4e8e\u72b6\u6001$s\\in S$\u65f6\uff0c\u6211\u4eec\u9700\u8981\u9009\u62e9\u4e00\u4e2a\u52a8\u4f5c\u5e76\u6267\u884c\uff0c\u6b64\u65f6\u6211\u4eec\u5c31\u5e94\u8be5\u8ba1\u7b97\u8fde\u7eed\u72b6\u6001$s$\u76f8\u5bf9\u5e94\u7684\u79bb\u6563\u72b6\u6001$\\bar s$\uff0c\u7136\u540e\u6267\u884c\u52a8\u4f5c$\\pi^*(\\bar s)$\u3002\n\n\u8fd9\u79cd\u57fa\u4e8e\u79bb\u6563\u5316\u7684\u5b9e\u73b0\u65b9\u6cd5\u53ef\u4ee5\u5e94\u7528\u4e0e\u5f88\u591aMDP\u95ee\u9898\u7684\u89e3\u51b3\uff0c\u4f46\u662f\u5b83\u6709\u4e24\u4e2a\u7f3a\u70b9\uff1a\n\n* \u9996\u5148\u662f\u8be5\u65b9\u6cd5\u5bf9\u4e8e$V^*$\uff08\u4ee5\u53ca$\\pi^*$\uff09\u7684\u8868\u8fbe\u8fc7\u4e8e\u7eaf\u7cb9\uff0c\u65b9\u6cd5\u5047\u8bbe\u4ef7\u503c\u51fd\u6570\u5728\u6bcf\u4e00\u4e2a\u79bb\u6563\u5316\u7684\u533a\u95f4\u4e0a\u53d6\u5e38\u6570\u503c\uff08\u5373\u4ef7\u503c\u51fd\u6570\u5bf9\u4e8e\u6bcf\u4e2a\u7f51\u683c\u6765\u8bf4\u662f\u4e00\u4e2a\u5206\u6bb5\u5e38\u6570\uff09\u3002\n\n \u4e3a\u4e86\u66f4\u52a0\u5f62\u8c61\u5316\u7684\u7406\u89e3\u8fd9\u79cd\u8868\u8fbe\u7684\u5c40\u9650\uff0c\u8003\u8651\u76d1\u7763\u5b66\u4e60\u95ee\u9898\u4e2d\u4f7f\u7528\u51fd\u6570\u62df\u5408\u6570\u636e\u96c6\u7684\u8fc7\u7a0b\uff1a\n\n \n\n \u5f88\u5bb9\u6613\u770b\u51fa\uff0c\u505a\u7ebf\u6027\u56de\u5f52\u5c31\u53ef\u4ee5\u5f88\u597d\u7684\u62df\u5408\u6570\u636e\u3002\u7136\u800c\u5982\u679c\u6211\u4eec\u5c06$x$\u5750\u6807\u79bb\u6563\u5316\uff0c\u7136\u540e\u4f7f\u7528\u5206\u6bb5\u5e38\u6570\u6765\u8868\u793a\u79bb\u6563\u533a\u95f4\uff0c\u5219\u5bf9\u6570\u636e\u7684\u62df\u5408\u5c06\u53d8\u6210\u4e0b\u56fe\u4e2d\u7684\u6837\u5b50\uff1a\n\n \n\n \u76f8\u6bd4\u4e8e\u5e73\u6ed1\u7684\u51fd\u6570\uff0c\u8fd9\u79cd\u5206\u6bb5\u5e38\u6570\u8868\u793a\u6cd5\u6709\u4e00\u4e9b\u7f3a\u9677\u3002\u5bf9\u4e8e\u8f93\u5165\u6765\u8bf4\u5b83\u5f97\u5230\u7684\u8f93\u51fa\u4e0d\u591f\u5e73\u6ed1\uff0c\u800c\u4e14\u65e0\u6cd5\u5c06\u4e0d\u540c\u7684\u79bb\u6563\u533a\u95f4\u5f52\u7eb3\u5728\u540c\u4e00\u4e2a\u8868\u8fbe\u5f0f\u4e2d\u3002\u5bf9\u4e8e\u8fd9\u79cd\u6570\u636e\u8868\u793a\u6cd5\uff0c\u6211\u4eec\u9700\u8981\u4e00\u4e2a\u5f88\u7ec6\u7684\u79bb\u6563\u5316\uff08\u7f51\u683c\u975e\u5e38\u7ec6\uff09\u624d\u80fd\u505a\u51fa\u8f83\u597d\u7684\u8fd1\u4f3c\u3002\n\n* \u53e6\u4e00\u4e2a\u7f3a\u9677\u5219\u662f**\u7ef4\u6570\u707e\u96be\uff08curse of dimensionality\uff09**\u3002\u8bbe$S=\\mathbb R^n$\uff0c\u6211\u4eec\u5c06$n$\u7ef4\u4e2d\u7684\u6bcf\u4e00\u4e2a\u7ef4\u5ea6\u79bb\u6563\u5316\u6210$k$\u4e2a\u503c\uff0c\u90a3\u4e48\u79bb\u6563\u7684\u72b6\u6001\u7684\u603b\u6570\u5c31\u6709$k^n$\u4e2a\u4e4b\u591a\u3002\u5bf9\u4e8e\u72b6\u6001\u7a7a\u95f4\u7684\u7ef4\u6570$n$\u6765\u8bf4\uff0c\u8fd9\u4e2a\u79bb\u6563\u72b6\u6001\u603b\u6570\u5448\u6307\u6570\u589e\u52a0\uff0c\u6240\u4ee5\u5e76\u4e0d\u9002\u7528\u4e8e\u5927\u578b\u95ee\u9898\u7684\u89e3\u51b3\u3002\u5bf9\u4e8e\u4e00\u4e2a$10$\u7ef4\u72b6\u6001\u7a7a\u95f4\uff0c\u5982\u679c\u6211\u4eec\u5c06\u6bcf\u4e2a\u7ef4\u5ea6\u79bb\u6563\u5316\u4e3a$100$\u4e2a\u503c\uff0c\u90a3\u4e48\u5c31\u4f1a\u5f97\u5230$100^{10}=10^{20}$\u4e2a\u79bb\u6563\u72b6\u6001\uff0c\u8fd9\u5bf9\u76ee\u524d\u7684\u8ba1\u7b97\u673a\u6765\u8bf4\u592a\u5927\u4e86\u3002\n\n \u6309\u7167\u5b9e\u8df5\u7ecf\u9a8c\uff0c\u79bb\u6563\u5316\u5728\u89e3\u51b3$1$\u7ef4\u6216$2$\u4e3a\u95ee\u9898\u65f6\u6548\u679c\u975e\u5e38\u597d\uff08\u4e14\u5b9e\u73b0\u8d77\u6765\u65b9\u4fbf\u5feb\u6377\uff09\u3002\u4e5f\u8bb8\uff0c\u51ed\u501f\u4e00\u4e9b\u667a\u6167\u5e76\u5728\u79bb\u6563\u5316\u65b9\u6cd5\u9009\u62e9\u4e0a\u5c0f\u5fc3\u8c28\u614e\uff0c\u90a3\u4e48\u8fd9\u79cd\u65b9\u6cd5\u4e5f\u80fd\u591f\u5904\u7406$4$\u7ef4\u95ee\u9898\u3002\u5982\u679c\u6211\u4eec\u7279\u522b\u667a\u6167\u53c8\u975e\u5e38\u5e78\u8fd0\u7684\u8bdd\uff0c\u53ef\u80fd\u80fd\u591f\u8ba9\u8fd9\u79cd\u65b9\u6cd5\u6491\u5230\u89e3\u51b3$6$\u7ef4\u95ee\u9898\u3002\u4f46\u662f\u5982\u679c\u95ee\u9898\u590d\u6742\u5ea6\u518d\u589e\u52a0\uff0c\u8fd9\u79cd\u65b9\u6cd5\u5c31\u65e0\u80fd\u4e3a\u529b\u4e86\u3002\n\n### 4.2 \u4ef7\u503c\u51fd\u6570\u8fd1\u4f3c\n\n\u6211\u4eec\u518d\u6765\u4ecb\u7ecd\u53e6\u4e00\u79cd\u80fd\u591f\u6c42\u89e3\u5177\u6709\u8fde\u7eed\u72b6\u6001\u7684MDP\u7684\u65b9\u6cd5\uff0c\u8fd9\u6b21\u6211\u4eec\u76f4\u63a5\u5bf9$V^*$\u505a\u8fd1\u4f3c\uff0c\u4e5f\u5c31\u7701\u53bb\u4e86\u79bb\u6563\u5316\u7684\u6b65\u9aa4\u3002\u8fd9\u79cd\u5b9e\u73b0\u79f0\u4e3a**\u4ef7\u503c\u51fd\u6570\u8fd1\u4f3c\uff08value function approximation\uff09**\uff0c\u5df2\u7ecf\u5e94\u7528\u5728\u5f88\u591a\u5f3a\u5316\u5b66\u4e60\u95ee\u9898\u4e2d\u4e86\u3002\n\n#### 4.2.1 \u4f7f\u7528\u6a21\u578b\u6216\u6a21\u62df\u5668\n\n\u4e3a\u4e86\u5b9e\u73b0\u4ef7\u503c\u51fd\u6570\u8fd1\u4f3c\uff0c\u6211\u4eec\u9700\u8981\u4e3aMDP\u5047\u8bbe\u4e00\u4e2a**\u6a21\u578b\uff08model\uff09**\u6216**\u6a21\u62df\u5668\uff08simulator\uff09**\u3002\u975e\u6b63\u5f0f\u7684\u5b9a\u4e49\uff0c\u6a21\u62df\u5668\u662f\u4e00\u4e2a\u9ed1\u76d2\uff0c\u80fd\u591f\u63a5\u53d7\u4efb\u610f\u7684\uff08\u8fde\u7eed\u503c\uff09\u72b6\u6001$s_t$\u548c\u52a8\u4f5c$a_t$\u4f5c\u4e3a\u8f93\u5165\uff0c\u5e76\u6309\u7167\u72b6\u6001\u8f6c\u6362\u6982\u7387$P_{s_ta_t}$\u8f93\u51fa\u4e0b\u4e00\u4e2a\u72b6\u6001$s_{t+1}$\uff1a\n\n\n\n\u6211\u4eec\u6709\u51e0\u79cd\u5f97\u5230\u6a21\u578b\u7684\u65b9\u6cd5\uff0c\u5176\u4e2d\u4e00\u79cd\u5c31\u662f\u7269\u7406\u6a21\u62df\u3002\u6bd4\u5982[\u95ee\u9898\u96c64](cs229.stanford.edu/materials/ps4.pdf)\u4e2d\u5012\u7f6e\u949f\u6446\u7684\u4f8b\u5b50\uff0c\u6211\u4eec\u5047\u8bbe\u7cfb\u7edf\u7684\u6240\u6709\u53c2\u6570\u5df2\u77e5\uff0c\u6bd4\u5982\u6746\u7684\u957f\u5ea6\u3001\u6746\u7684\u8d28\u91cf\u7b49\u7b49\uff0c\u6839\u636e\u8fd9\u4e9b\u8f93\u5165\u53c2\u6570\uff0c\u4f7f\u7528\u5404\u79cd\u7269\u7406\u5b9a\u5f8b\u8ba1\u7b97\u4f4d\u4e8e$t$\u65f6\u523b\u7684\u5f53\u524d\u72b6\u6001\u5728\u6267\u884c\u52a8\u4f5c$a$\u4e4b\u540e\uff0c\u7cfb\u7edf\u5728$t+1$\u65f6\u523b\u7cfb\u7edf\u7684\u72b6\u6001\uff08\u5305\u62ec\u5c0f\u8f66\u7684\u3001\u6746\u7684\u89d2\u5ea6\u7b49\uff09\u3002\u6211\u4eec\u4e5f\u53ef\u4ee5\u4f7f\u7528\u73b0\u6210\u7684\u7269\u7406\u6a21\u62df\u8f6f\u4ef6\uff0c\u6211\u4eec\u53ea\u9700\u8981\u5728\u8f6f\u4ef6\u4e2d\u5bf9\u8fd9\u4e2a\u673a\u68b0\u7cfb\u7edf\u8fdb\u884c\u7269\u7406\u5efa\u6a21\uff0c\u5e76\u7ed9\u51fa\u5f53\u524d\u72b6\u6001$s_t$\u548c\u8981\u6267\u884c\u7684\u52a8\u4f5c$a_t$\uff0c\u5c31\u53ef\u4ee5\u8ba1\u6a21\u62df\u51fa\u672a\u6765\u4e00\u5c0f\u6bb5\u65f6\u95f4\u5185\uff0c\u5904\u4e8e$t+1$\u65f6\u523b\u7cfb\u7edf\u7684\u72b6\u6001\u3002\uff08\u6bd4\u5982[Open Dynamics Engine](http://www.ode.com)\u5c31\u662f\u4e00\u6b3e\u514d\u8d39\u7684\u5f00\u6e90\u7269\u7406\u6a21\u62df\u5668\uff0c\u5b83\u5b8c\u5168\u53ef\u4ee5\u7528\u6765\u6a21\u62df\u201c\u5012\u7f6e\u949f\u6446\u201d\u95ee\u9898\uff0c\u5e76\u5df2\u7ecf\u5728\u5f3a\u5316\u5b66\u4e60\u7814\u7a76\u8005\u4e2d\u5efa\u7acb\u4e86\u826f\u597d\u7684\u53e3\u7891\u3002\uff09\n\n\u53e6\u4e00\u79cd\u65b9\u6cd5\u662f\u4eceMDP\u6536\u96c6\u5230\u7684\u6570\u636e\u4e2d\u201c\u5b66\u5230\u201d\u4e00\u4e2a\u6a21\u578b\u3002\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5047\u8bbe\u6211\u4eec\u901a\u8fc7\u4e0d\u505c\u7684\u6267\u884c\u52a8\u4f5c\uff0c\u5728MDP\u4e2d\u505a\u4e86$m$\u6b21**\u8bd5\u9a8c\uff08trials\uff09**\uff0c\u6bcf\u4e00\u6b21\u8bd5\u9a8c\u90fd\u8fdb\u884c\u4e86$T$\u6b65\u3002\u8fd9\u6837\u7684\u8bd5\u9a8c\u53ef\u4ee5\u901a\u8fc7\u968f\u673a\u9009\u62e9\u52a8\u4f5c\u3001\u4f9d\u7167\u7279\u5b9a\u7684\u7b56\u7565\u6216\u4f7f\u7528\u5176\u4ed6\u9009\u62e9\u52a8\u4f5c\u7684\u65b9\u5f0f\u3002\u4e8e\u662f\u6211\u4eec\u53ef\u4ee5\u5f97\u5230$m$\u4e2a\u5982\u4e0b\u7684\u72b6\u6001\u5e8f\u5217\uff1a\n\n$$\n\\require{AMScd}\n\\begin{CD}\ns_0^{(1)} @>{a_0^{(1)}}>> s_1^{(1)} @>{a_1^{(1)}}>> s_2^{(1)} @>{a_2^{(1)}}>> \\cdots @>{a_{T-1}^{(1)}}>> s_T^{(1)}\n\\end{CD}\\\\\n\\begin{CD}\ns_0^{(2)} @>{a_0^{(2)}}>> s_1^{(2)} @>{a_1^{(2)}}>> s_2^{(2)} @>{a_2^{(2)}}>> \\cdots @>{a_{T-1}^{(2)}}>> s_T^{(2)}\n\\end{CD}\\\\\n\\vdots\\\\\n\\begin{CD}\ns_0^{(m)} @>{a_0^{(m)}}>> s_1^{(m)} @>{a_1^{(m)}}>> s_2^{(m)} @>{a_2^{(m)}}>> \\cdots @>{a_{T-1}^{(m)}}>> s_T^{(m)}\n\\end{CD}\\\\\n$$\n\n\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5e94\u7528\u4e00\u79cd\u5b66\u4e60\u7b97\u6cd5\uff0c\u7c7b\u4f3c\u4e00\u4e2a\u51fd\u6570\uff0c\u8f93\u5165\u662f$s_t$\u548c$a_t$\uff0c\u800c\u8f93\u51fa\u662f\u9884\u6d4b\u7684\u72b6\u6001$s_{t+1}$\u3002\n\n\u6bd4\u5982\u6211\u4eec\u53ef\u80fd\u4f1a\u5efa\u7acb\u4e00\u4e2a\u7ebf\u6027\u6a21\u578b\uff1a\n\n$$\ns_{t+1}=As_t+Ba_t\\tag{5}\n$$\n\n\u4f7f\u7528\u7c7b\u4f3c\u4e8e\u7ebf\u6027\u56de\u5f52\u7684\u7b97\u6cd5\u3002\u6b64\u5904\u6a21\u578b\u7684\u53c2\u6570\u662f\u77e9\u9635$A$\u548c$B$\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u6536\u96c6\u5230\u7684$m$\u6b21\u6d4b\u8bd5\u6570\u636e\u6765\u4f30\u8ba1\u8fd9\u4e24\u4e2a\u77e9\u9635\uff1a\n\n$$\n\\arg\\min_{A,B}\\sum_{i=1}^m\\sum_{t=0}^{T-1}\\left\\lVert s_{t+1}^{(i)}-\\left(As_t^{(i)}+Ba_t^{(i)}\\right)\\right\\rVert^2\n$$\n\n\uff08\u4e5f\u5c31\u662f\u6c42\u53c2\u6570\u7684\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u7684\u6b65\u9aa4\u3002\uff09\n\n\u5728\u7b97\u6cd5\u627e\u51fa$A,B$\u4e4b\u540e\uff0c\u5176\u4e2d\u4e00\u79cd\u9009\u62e9\u662f\u5efa\u7acb\u4e00\u4e2a**\u786e\u5b9a\u6027\uff08deterministic\uff09**\u6a21\u578b\uff0c\u5b83\u63a5\u53d7$s_t,a_t$\u4f5c\u4e3a\u8f93\u5165\uff0c\u5e76\u8f93\u51fa\u4e00\u4e2a\u786e\u5b9a\u7684$s_{t+1}$\u3002\u6bd4\u5982\u6211\u4eec\u603b\u662f\u4f7f\u7528$(5)$\u5f0f\u8ba1\u7b97$s_{t+1}$\u3002\u53e6\u4e00\u79cd\u9009\u62e9\u662f\u5efa\u7acb\u4e00\u4e2a**\u968f\u673a\u6027\uff08stochastic\uff09**\u6a21\u578b\uff0c\u5728\u8fd9\u79cd\u6a21\u578b\u4e2d\uff0c$s_{t+1}$\u662f\u4e00\u4e2a\u5173\u4e8e\u8f93\u5165$s_t,a_t$\u7684\u968f\u673a\u51fd\u6570\uff1a$\\displaystyle s_{t+1}=As_t+Ba_t+\\epsilon_t$\uff0c\u800c$\\epsilon_t$\u662f\u4e00\u4e2a\u566a\u97f3\u9879\uff0c\u6211\u4eec\u901a\u5e38\u4ee4\u5176\u670d\u4ece$\\epsilon_t\\sim\\mathcal N(0,\\varSigma)$\u3002\uff08\u534f\u65b9\u5dee\u77e9\u9635$\\varSigma$\u4e5f\u53ef\u4ee5\u901a\u8fc7\u6d4b\u8bd5\u6570\u636e\u76f4\u63a5\u4f30\u8ba1\u51fa\u6765\u3002\uff09\n\n\u8fd9\u91cc\u4e3e\u51fa\u7684\u4f8b\u5b50\u662f\u5c06$s_{t+1}$\u770b\u505a\u662f\u5173\u4e8e\u5f53\u524d\u72b6\u6001\u548c\u52a8\u4f5c\u7684\u7ebf\u6027\u51fd\u6570\uff0c\u6211\u4eec\u5f53\u7136\u4e5f\u53ef\u4ee5\u4f7f\u7528\u975e\u7ebf\u6027\u51fd\u6570\u5bf9\u5176\u5efa\u6a21\u3002\u6bd4\u5982\u8bf4\u7528\u51fd\u6570$s_{t+1}=A\\phi_s(s_t)+B\\phi_a(a_t)$\u5efa\u6a21\uff0c\u5176\u4e2d$\\phi_s,\\phi_a$\u662f\u5173\u4e8e\u72b6\u6001\u548c\u52a8\u4f5c\u7684\u67d0\u4e9b\u975e\u7ebf\u6027\u7279\u5f81\u6620\u5c04\u3002\u6216\u8005\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e9b\u975e\u7ebf\u6027\u5b66\u4e60\u7b97\u6cd5\u2014\u2014\u6bd4\u5982\u5c40\u90e8\u52a0\u6743\u56de\u5f52\uff08\u53c2\u89c1[\u7b2c\u4e09\u8bb2](chapter02.ipynb)\uff09\u2014\u2014\u5c06$s_{t+1}$\u62df\u5408\u4e3a\u5173\u4e8e$s_t,a_t$\u7684\u51fd\u6570\u3002\u8fd9\u4e9b\u5b9e\u73b0\u7684\u65b9\u6cd5\u65e2\u53ef\u4ee5\u7528\u6765\u6784\u5efa\u786e\u5b9a\u6027\u6a21\u62df\u5668\uff0c\u4e5f\u53ef\u4ee5\u7528\u6765\u6784\u5efa\u968f\u673a\u6027\u6a21\u62df\u5668\u3002\n\n#### 4.2.2 \u62df\u5408\u503c\u8fed\u4ee3\uff08Fitted value iteration\uff09\n\n\u73b0\u5728\u6765\u4ecb\u7ecd**\u62df\u5408\u503c\u8fed\u4ee3\uff08fitted value iteration\uff09**\u7b97\u6cd5\uff0c\u7528\u4e8e\u5728\u5177\u6709\u8fde\u7eed\u72b6\u6001\u7684MDP\u4e0a\u8fd1\u4f3c\u503c\u51fd\u6570\u3002\u5728\u540e\u9762\u7684\u8ba8\u8bba\u4e2d\u6211\u4eec\u5047\u8bbeMDP\u5177\u6709\u8fde\u7eed\u7684\u72b6\u6001\u7a7a\u95f4$S=\\mathbb R^n$\uff0c\u4f46\u662f\u52a8\u4f5c\u7a7a\u95f4$A$\u89c4\u6a21\u8f83\u5c0f\u5e76\u4e14\u662f\u79bb\u6563\u7684\uff08\u56e0\u4e3a\u5728\u5b9e\u8df5\u4e2d\uff0c\u901a\u5e38\u72b6\u6001\u7a7a\u95f4\u7684\u7ef4\u5ea6\u8981\u6bd4\u52a8\u4f5c\u7a7a\u95f4\u5927\u5f88\u591a\uff0c\u6240\u4ee5\u52a8\u4f5c\u7a7a\u95f4\u901a\u5e38\u6bd4\u8f83\u5bb9\u6613\u79bb\u6563\u5316\uff09\u3002\n\n\u56de\u5fc6\u5728\u503c\u8fed\u4ee3\u4e2d\u7684\u66f4\u65b0\u89c4\u5219\uff1a\n\n$$\n\\begin{align}\nV(s)&:=R(s)+\\gamma R\\int_{s'}P_{sa}\\left(s'\\right)V\\left(s'\\right)\\mathrm ds'\\tag{6}\\\\\n&=R(s)+\\gamma\\max_a\\mathrm E_{s'\\sim P_{sa}}\\left[V\\left(s'\\right)\\right]\\tag{7}\n\\end{align}\n$$\n\n\uff08\u5728[\u4e0a\u4e00\u8bb2](chapter16.ipynb)\u7b2c\u4e8c\u8282\u4e2d\uff0c\u6211\u4eec\u5f97\u5230\u7684\u503c\u8fed\u4ee3\u66f4\u65b0\u89c4\u5219$\\displaystyle V(s):=R(s)+\\max_{a\\in A}\\gamma\\sum_{s'\\in S}P_{sa}\\left(s'\\right)V\\left(s'\\right)$\u4e2d\u5bf9\u72b6\u6001\u4f7f\u7528\u4e86\u6c42\u548c\u8fd0\u7b97\uff0c\u800c\u5728\u8fd9\u91cc\u6211\u4eec\u5bf9\u72b6\u6001\u4f7f\u7528\u7684\u662f\u79ef\u5206\u8fd0\u7b97\uff0c\u8fd9\u8bf4\u660e\u6211\u4eec\u73b0\u5728\u662f\u5728\u8fde\u7eed\u72b6\u6001\u4e0b\uff08\u800c\u4e0d\u662f\u5728\u79bb\u6563\u72b6\u6001\u4e0b\uff09\u6c42\u89e3MDP\u3002\uff09\n\n\u62df\u5408\u503c\u8fed\u4ee3\u7684\u6838\u5fc3\u601d\u60f3\u5c31\u662f\u5728$s^{(1)},\\cdots,s^{(m)}$\u7684\u6709\u9650\u62bd\u6837\u4e0a\u8fdb\u884c\u8fd1\u4f3c\u7684\u8fed\u4ee3\u6b65\u9aa4\u3002\u6211\u4eec\u5c06\u4f7f\u7528\u76d1\u7763\u5b66\u4e60\u7b97\u6cd5\uff08\u5728\u4e0b\u9762\u7684\u8ba8\u8bba\u4e2d\u5c06\u4f7f\u7528\u7ebf\u6027\u56de\u5f52\uff09\uff0c\u5c06\u503c\u51fd\u6570\u770b\u505a\u4e00\u4e2a\u5173\u4e8e\u72b6\u6001\u7684\u7ebf\u6027\u6216\u975e\u7ebf\u6027\u51fd\u6570\uff0c\u7136\u540e\u518d\u6765\u8fd1\u4f3c\u8fd9\u4e2a\u51fd\u6570\uff1a\n\n$$\nV(s)=\\theta^T\\phi(s)\n$$\n\n\u8fd9\u91cc\u7684$\\phi$\u662f\u67d0\u4e2a\u5173\u4e8e\u72b6\u6001\u7684\u6620\u5c04\u3002\n\n\u5728$m$\u4e2a\u72b6\u6001\u7684\u6709\u9650\u6837\u672c\u4e2d\uff0c\u5bf9\u4e8e\u6bcf\u4e00\u4e2a\u72b6\u6001$s$\uff0c\u7b97\u6cd5\u5c06\u5148\u8ba1\u7b97\u4e00\u4e2a\u8bb0\u4e3a$y^{(i)}$\u7684\u91cf\uff08\u6211\u4eec\u5728\u540e\u9762\u4f1a\u628a\u8fd9\u4e2a\u91cf\u5f53\u505a$R(s)+\\gamma\\max_a\\mathrm E_{s'\\sim P_{sa}}\\left[V\\left(s'\\right)\\right]$\u7684\u8fd1\u4f3c\u503c\uff0c\u5373$(7)$\u5f0f\uff09\uff1b\u7136\u540e\u4f1a\u5e94\u7528\u4e00\u4e2a\u76d1\u7763\u5b66\u4e60\u7b97\u6cd5\uff0c\u5c1d\u8bd5\u4f7f$V(s)$\u9760\u8fd1$R(s)+\\gamma\\max_a\\mathrm E_{s'\\sim P_{sa}}\\left[V\\left(s'\\right)\\right]$\uff08\u6362\u53e5\u8bdd\u8bf4\uff0c\u5c1d\u8bd5\u4f7f$V\\left(s^{(i)}\\right)$\u5c3d\u91cf\u9760\u8fd1$y^{(i)}$\uff09\u3002\u7b97\u6cd5\u7684\u6b65\u9aa4\u5982\u4e0b\uff1a\n\n\u91cd\u590d\uff1a`{`\n1. \u968f\u673a\u62bd\u6837$m$\u4e2a\u72b6\u6001$s^{(1)},s^{(2)},\\cdots,s^{(m)}\\in S$\uff1b\n2. \u521d\u59cb\u5316$\\theta:=0$\uff1b\n3. \u5bf9\u6bcf\u4e2a\u72b6\u6001 $i=1,\\cdots,m$ `{`\n * \u5bf9\u6bcf\u4e2a\u52a8\u4f5c$a\\in A$ `{`\n * \u62bd\u6837$s_1',\\cdots,s_k'\\sim P_{s^{(i)}a}$\uff08\u4f7f\u7528MDP\u6a21\u578b\u6839\u636e$s,a$\u9884\u6d4b$s'$\uff09\uff1b\n * \u4ee4$\\displaystyle q(a)=\\frac{1}{k}\\sum_{j=1}^kR\\left(s^{(i)}\\right)+\\gamma V\\left(s_j'\\right)$ // \u56e0\u6b64\uff0c$q(a)$\u5c31\u662f\u4e00\u4e2a\u5bf9$\\displaystyle R(s^{(i)})+\\gamma\\mathrm E_{s'\\sim P_{s^{(i)}a}}\\left[V\\left(s'\\right)\\right]$\u7684\u4f30\u8ba1\uff1b\n\n `}`\n \n * \u4ee4$\\displaystyle y^{(i)}=\\max_aq(a)$ // \u56e0\u6b64\uff0c$y^{(i)}$\u5c31\u662f\u4e00\u4e2a\u5bf9$\\displaystyle R(s^{(i)})+\\gamma\\max_a\\mathrm E_{s'\\sim P_{s^{(i)}a}}\\left[V\\left(s'\\right)\\right]$\u7684\u4f30\u8ba1\uff1b\n \n `}`\n \n // \u5728\u539f\u6765\u7684\uff08\u7528\u4e8e\u89e3\u79bb\u6563\u72b6\u6001\u7684\uff09\u503c\u8fed\u4ee3\u7b97\u6cd5\u4e2d\uff0c\u6211\u4eec\u6309\u7167$V\\left(s^{(i)}\\right):=y^{(i)}$\u66f4\u65b0\u503c\u51fd\u6570\uff1b\n \n // \u800c\u5728\u8fd9\u4e2a\u7b97\u6cd5\u4e2d\uff0c\u6211\u4eec\u5c06\u4f7f\u7528\u76d1\u7763\u5b66\u4e60\u7b97\u6cd5\uff08\u7ebf\u6027\u56de\u5f52\uff09\u6765\u5b9e\u73b0$V\\left(s^{(i)}\\right)\\approx y^{(i)}$\uff0c\u5373\u4f7f$V\\left(s^{(i)}\\right)$\u5c3d\u91cf\u9760\u8fd1$y^{(i)}$\uff1b\n * \u4ee4$\\displaystyle\\theta:=\\arg\\min_\\theta\\frac{1}{2}\\sum_{i=1}^m\\left(\\theta^T\\phi\\left(s^{(i)}\\right)-y^{(i)}\\right)^2$\n\n`}`\n\n\u4ee5\u4e0a\u5c31\u662f\u4f7f\u7528\u7ebf\u6027\u56de\u5f52\u4f7f$V\\left(s^{(i)}\\right)$\u9760\u8fd1$y^{(i)}$\u7684\u62df\u5408\u503c\u8fed\u4ee3\u3002\u7c7b\u6bd4\u6807\u51c6\u76d1\u7763\u5b66\u4e60\uff08\u56de\u5f52\uff09\u7b97\u6cd5\uff0c\u5728\u56de\u5f52\u7b97\u6cd5\u4e2d\u6211\u4eec\u5df2\u77e5\u8bad\u7ec3\u96c6$\\left(x^{(1)},y^{(1)}\\right),\\left(x^{(2)},y^{(2)}\\right),\\cdots,\\left(x^{(m)},y^{(m)}\\right)$\uff0c\u60f3\u8981\u5b66\u4e60\u5230\u4e00\u4e2a\u4ece$x$\u5230$y$\u7684\u51fd\u6570\u6620\u5c04\uff0c\u8fd9\u4e2a\u7b97\u6cd5\u4e0e\u56de\u5f52\u552f\u4e00\u4e0d\u540c\u7684\u662f$s$\u4ee3\u66ff\u4e86$y$\u7684\u4f4d\u7f6e\u3002\u867d\u7136\u4e0a\u9762\u7684\u4f8b\u5b50\u4e2d\u4f7f\u7528\u7684\u662f\u7ebf\u6027\u56de\u5f52\uff0c\u4e0d\u8fc7\u663e\u7136\u4e5f\u53ef\u4ee5\u4f7f\u7528\u5176\u4ed6\u56de\u5f52\u7b97\u6cd5\uff08\u5982\u5c40\u90e8\u52a0\u6743\u56de\u5f52\u7b49\uff09\u3002\n\n\u4e0e\u79bb\u6563\u72b6\u6001\u4e0a\u7684\u503c\u8fed\u4ee3\u4e0d\u540c\u7684\u662f\uff0c\u4e0d\u80fd\u88ab\u8bc1\u660e\u6536\u655b\u3002\u4e0d\u8fc7\u4e0d\u7528\u62c5\u5fc3\uff0c\u5728\u5b9e\u8df5\u4e2d\u8be5\u7b97\u6cd5\u901a\u5e38\u6536\u655b\uff08\u6216\u8fd1\u8fbe\u5230\u4f3c\u6536\u655b\u7684\u7a0b\u5ea6\uff09\uff0c\u800c\u4e14\u5bf9\u5f88\u591a\u95ee\u9898\u90fd\u975e\u5e38\u6709\u6548\u3002\u6ce8\u610f\u5230\u5982\u679c\u6211\u4eec\u4f7f\u7528MDP\u7684\u786e\u5b9a\u6027\u6a21\u62df\u5668/\u6a21\u578b\uff0c\u5219\u5c31\u53ef\u7528$k=1$\u7b80\u5316\u7b97\u6cd5\u3002\u8fd9\u662f\u56e0\u4e3a$(7)$\u5f0f\u4e2d\u7684\u671f\u671b\u53d8\u6210\u4e86\u5173\u4e8e\u4e00\u4e2a\u786e\u5b9a\u6027\u5206\u5e03\u7684\u671f\u671b\uff08\u5728\u786e\u5b9a\u6027\u6a21\u578b\u4e2d\uff0c\u6211\u4eec\u5728\u7ed9\u5b9a\u8f93\u5165\u4e0b\u505a\u7684\u6bcf\u6b21\u62bd\u6837\u5c06\u5f97\u5230\u540c\u6837\u7684\u8f93\u51fa\uff09\uff0c\u4e8e\u662f\u4f7f\u7528\u4e00\u4e2a\u6837\u672c\u5c31\u53ef\u4ee5\u8ba1\u7b97\u76f8\u5e94\u7684\u671f\u671b\u503c\u4e86\u3002\u5982\u679c\u4e0d\u662f\u786e\u5b9a\u6027\u6a21\u578b\uff0c\u90a3\u4e48\u6211\u4eec\u5fc5\u987b\u62bd\u53d6$k$\u4e2a\u6837\u672c\uff0c\u7136\u540e\u518d\u7528\u5747\u503c\u53bb\u8fd1\u4f3c\u76f8\u5e94\u7684\u671f\u671b\uff08\u89c1\u7b97\u6cd5\u4f2a\u4ee3\u7801\u4e2d\u5173\u4e8e$q(a)$\u7684\u5b9a\u4e49\uff09\u3002\n\n\u6700\u540e\uff0c\u62df\u5408\u503c\u8fed\u4ee3\u7684\u8f93\u51fa$V$\u662f\u4e00\u4e2a\u5173\u4e8e$V^*$\u7684\u8fd1\u4f3c\uff0c\u8fd9\u540c\u65f6\u6697\u793a\u4e86\u7b56\u7565\u7684\u5b9a\u4e49\u3002\u5f53\u7cfb\u7edf\u5904\u4e8e\u72b6\u6001$s$\u65f6\uff0c\u9700\u8981\u9009\u62e9\u4e00\u4e2a\u52a8\u4f5c\uff0c\u4e8e\u662f\u6211\u4eec\u4f7f\u7528\uff1a\n\n$$\n\\arg\\max_a\\mathrm E_{s'\\sim P_{sa}}\\left[V^*\\left(s'\\right)\\right]\\tag{8}\n$$\n\n\uff08\u7531\u4e8e\u72b6\u6001\u662f\u8fde\u7eed\u7684\uff0c\u6211\u4eec\u5c31\u65e0\u6cd5\u8ba1\u7b97\u6bcf\u4e00\u4e2a\u72b6\u6001\u7684\u6700\u4f18\u4ef7\u503c\u51fd\u6570\u4e86\u3002\u6240\u4ee5\u6211\u4eec\u6bcf\u6b21\u5728\u9700\u8981\u505a\u4e0b\u4e00\u4e2a\u52a8\u4f5c\u65f6\u8fdb\u884c\u8ba1\u7b97\uff1a\u5728\u7cfb\u7edf\u5904\u4e8e\u7279\u5b9a\u72b6\u6001\u65f6\uff0c\u6267\u884c\u67d0\u52a8\u4f5c\u4f1a\u8fdb\u5165\u4e0b\u4e00\u4e2a\u72b6\u6001$s'$\uff0c\u5f0f\u5b50\u4f1a\u627e\u5230\u80fd\u591f\u4f7f$\\displaystyle\\mathrm E_{s'\\sim P_{sa}}\\left[V^*\\left(s'\\right)\\right]$\u53d6\u5230\u6700\u5927\u7684\u52a8\u4f5c\u3002\uff09\n\n\u8fd9\u4e00\u6b65\u8ba1\u7b97/\u8fd1\u4f3c\u7c7b\u4f3c\u4e8e\u62df\u5408\u503c\u8fed\u4ee3\u91cc\u5c42\u7684\u5faa\u73af\u4e2d\u7528\u62bd\u6837$s_1',\\cdots,s_k'\\sim P_{sa}$\u8fd1\u4f3c\u671f\u671b\u7684\u6b65\u9aa4\uff08\u540c\u6837\u7684\uff0c\u5982\u679c\u6a21\u62df\u5668\u662f\u786e\u5b9a\u6027\u7684\uff0c\u5219\u53d6$k=1$\u5373\u53ef\uff09\u3002\n\n\u5728\u5b9e\u8df5\u4e2d\uff0c\u6211\u4eec\u4e5f\u7ecf\u5e38\u4f7f\u7528\u5176\u4ed6\u65b9\u6cd5\u6765\u8fd1\u4f3c\u8fd9\u4e2a\u6b65\u9aa4\u3002\u4e3e\u4e2a\u4f8b\u5b50\uff1a\n* \u5982\u679c\u6211\u4eec\u6b63\u5728\u4f7f\u7528\u4e00\u4e2a\u786e\u5b9a\u6027\u6a21\u578b\uff0c\u5219\u6709$s_{t+1}=f(s_t,a_t)$\uff08\u5373\u6709\u4e86\u5173\u4e8e\u5f53\u524d\u72b6\u6001\u548c\u52a8\u4f5c\u7684\u51fd\u6570\uff09\u3002\u4e8e\u662f\u4e0a\u9762\u7684\u5f0f\u5b50\u5c31\u53ef\u4ee5\u76f4\u63a5\u7b80\u5316\u4e3a$\\displaystyle\\arg\\max_aV^*\\left(f(s,a)\\right)$\uff08\u56e0\u4e3a$s'=f(s,a)$\uff09\uff1b\n* \u53e6\u4e00\u79cd\u5e38\u89c1\u7684\u60c5\u5f62\u662f\u5728\u968f\u673a\u6027\u6a21\u578b\u4e2d\uff0c\u5f53\u6a21\u62df\u5668\u91c7\u7528$s_{t+1}=f(s_t,a_t)+\\epsilon_t$\uff0c\u5176\u4e2d$f$\u662f\u5173\u4e8e\u72b6\u6001\u7684\u786e\u5b9a\u6027\u51fd\u6570\uff08\u6bd4\u5982$f(s_t,a_t)=As_t+Ba_t$\uff09\uff0c$\\epsilon$\u662f\u4e00\u4e2a\u670d\u4ece\u96f6\u671f\u671b\u9ad8\u65af\u5206\u5e03\u7684\u566a\u97f3\u9879\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\n\n$$\n\\arg\\max_aV^*\\left(f(s,a)\\right)\n$$\n\n\u9009\u62e9\u52a8\u4f5c\u3002\u4e5f\u5c31\u662f\u8bf4\u4ee4$\\epsilon=1$\uff08\u5373\u5ffd\u7565\u6a21\u62df\u5668\u7684\u566a\u97f3\uff09\u5e76\u4ee4$k=1$\u3002\u7b49\u4ef7\u7684\uff0c\u8fd9\u4e5f\u53ef\u4ee5\u4ece$(8)$\u5f0f\u4e2d\u5229\u7528\u8fd1\u4f3c\u63a8\u5bfc\u51fa\u6765\uff1a\n\n$$\n\\begin{align}\n\\mathrm E_{s'}\\left[V\\left(s'\\right)\\right]&\\approx V\\left(\\mathrm E_{s'}\\left[s'\\right]\\right)\\tag{9}\\\\\n&=V\\left(f(s,a)\\right)\\tag{10}\n\\end{align}\n$$\n\n\u6b64\u5904\u7684\u9884\u671f\u503c\u662f\u5728\u968f\u673a\u7684$s'\\sim P_{sa}$\u4e0a\u7684\u3002\u53ea\u8981\u566a\u97f3\u9879$\\epsilon$\u8f83\u5c0f\uff0c\u5219\u8fd9\u4e2a\u8fd1\u4f3c\u901a\u5e38\u90fd\u662f\u5408\u7406\u7684\u3002\n\n\u4e0d\u8fc7\uff0c\u5bf9\u4e8e\u4e0d\u9002\u5408\u505a\u8fd9\u79cd\u8fd1\u4f3c\u7684\u95ee\u9898\uff0c\u4f7f\u7528\u6a21\u578b\u62bd\u6837$k\\lvert A\\rvert$\u4e2a\u72b6\u6001\u4ee5\u8fd1\u4f3c\u4e0a\u9762\u7684\u671f\u671b\uff0c\u53ef\u80fd\u4f1a\u4f7f\u8ba1\u7b97\u7684\u4ee3\u4ef7\u975e\u5e38\u5927\u3002\n", "meta": {"hexsha": "0381dfb4c39d692d774b3e3be3084d769ea3ef21", "size": 14444, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "8-HMM/note/LSJU-chapter17.ipynb", "max_stars_repo_name": "PeterChenYijie/MachineLearningZeroToALL", "max_stars_repo_head_hexsha": "b14005c3e0b5a39a0ba82db5c9791f682b5effd5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 8, "max_stars_repo_stars_event_min_datetime": "2018-04-20T09:10:20.000Z", "max_stars_repo_stars_event_max_datetime": "2019-02-16T07:50:32.000Z", "max_issues_repo_path": "8-HMM/note/LSJU-chapter17.ipynb", "max_issues_repo_name": "DeepInDeeper/MachineLearningZeroToALL", "max_issues_repo_head_hexsha": "b14005c3e0b5a39a0ba82db5c9791f682b5effd5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "8-HMM/note/LSJU-chapter17.ipynb", "max_forks_repo_name": "DeepInDeeper/MachineLearningZeroToALL", "max_forks_repo_head_hexsha": "b14005c3e0b5a39a0ba82db5c9791f682b5effd5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2020-01-27T00:55:59.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-25T00:07:56.000Z", "avg_line_length": 50.3275261324, "max_line_length": 634, "alphanum_fraction": 0.566809748, "converted": true, "num_tokens": 7713, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804478040617, "lm_q2_score": 0.7154239957834733, "lm_q1q2_score": 0.4426903804806687}} {"text": "# Power Generation Emission Calculation Methods\n\nThis notebook contains the methodologies in calculating the emission from Power Generation in order to develop Life Cycle Inventories for Power Generation.\n\nThe different methodologies are from:\n\n No. | Country | Pub. Yr |\n -- | ------- | --------- |\n 1.0 | Japan | 2000 |\n 2. | Canada | 2001 |\n 3. | Brazil | 2003 |\n 4. | Indonesia | 2003 |\n 5. | Korea | 2004 |\n 6. | China | 2007 |\n 7. | Thailand | 2008 |\n 8. | Singapore | 2010 |\n\n\n## 1. Japan\nCalculating Direct Emission\n\n\n```python\nimport sympy as sp\nimport numpy as np\n```\n\n\n```python\nfrom sympy.interactive import printing\nprinting.init_printing(use_latex=True)\nfrom sympy import *\nfrom numpy import linalg\n```\n\n\n```python\nx,y = sp.symbols('x y')\n```\n\n\n```python\nlatex(Integral(cos(x)**2, (x,0,pi)))\n```\n\n\n\n\n '\\\\int\\\\limits_{0}^{\\\\pi} \\\\cos^{2}{\\\\left(x \\\\right)}\\\\, dx'\n\n\n\n### 1.1 $CO_2$\n\n$$CO_2 em_{int} = CO_2 em_{coeff} * f$$\n\n$CO_2 em_{int}$ = CO_2 emission intensity [kg-CO_2/kWh]
    \n$CO_2 em_{coeff}$ = CO_2 emission coefficient [kg-CO_2/kg-fuel]
    \n$f$ = amount of fuel consumed to produce 1 kWh of electricity [kg-fuel/kWh]
    \n\n\n$$Coal CO_2 em_{coeff} = 0.201 + 0.087 * Coal_{NCV}$$\n\n$Coal CO_2 em_{coeff}$ = CO_2 emission coefficient of coal [kg-CO_2/kg-coal]
    \n$Coal_{NCV}$ = net calorific value of coal [MJ/kg-coal]\n\n### 1.2 $SO_2$\n\n$$SO_2 em_{int} = \\frac{TFC * TSC * (M_{SO2}/M_s)*(1- E_{t,SO2}*(P_{equipped}/P_{total})}{(NEP}$$\n\nTFC = Total fuel consumption in each power station, 1997 [kg]
    \nTSC =
    \n$M_{SO2}$ =
    \nM_s =
    \n$E_{t,SO2}$ =
    \n$P_{equipped}$ =
    \n$P_{total}$ =
    \nNEP =
    \n\n### 1.3 $NO_x$\n\n$$ NO_x em_{int} = \\frac{TLHF * (NO*FC) * (1- E_{t_NOx} * P_{equipped}/P_{total})}{NEP}$$\n\n$NO_x em_{int}$ =
    \n$TLHF$
    \n$NO$ =
    \n$FC$ =
    \n$E_{t_NOx}$ =
    \n$P_{equipped}$ =
    \n$P_{total}$ =
    \n$NEP$=
    \n\n### 1.4 Non-methane volatile organic compound (NMVOC), CH_4, CO\n\nfrom literatures [14 - 15]\n\n$$ em_{int} = \\frac{em_f * NHV_{total}}{NEP}$$\n\n$em_{int}$ =
    \n$em_f$ =
    \n$NHV_{total}$ =
    \n$NEP$
    \n\n### 2.5 Dust (all particulates)\n\n#### 2.5.a Dust in Oil Powered Stations\n\n$$dust = 0.38 + 1.25 * S_{oil}$$\n\n$dust$ = dust formation [g/liter-oil consumed in oil power station]
    \n$S_{oil}$ = sulfur content in oil [wt%]
    \nThis paper was calculated using the average vaule of $S_{oil}$ 0.81 wt% based on Ministry of International Trade and Industry, Agency of Natural Resources and Energy (1999).\n\n#### 2.5.a.1 Desulfurizer in Power Station\n\nWhen not equipped with desulfurizer, using this equation:\n\n$$dust = \\frac{1.25 * SCO * TOC * (1- 0.8)}{NEP}$$\n\nWhen equipped with desulfurizer, using this equation:\n\n$$dust = \\frac{1.25 * SCO * TOC * (1- 0.8) * (1 - 0.9)}{NEP}$$\n\n\n#### 2.5.b Dust in Coal Powered Stations \n\nNot equipped with desulfurizer\n$$dust = \\frac{AshC * TCC * 0.5 * (1- 0.995)}{NEP}$$\n\nEquipped with desulfurizer\n$$dust = \\frac{AshC * TCC * 0.5 * (1- 0.995) * (1-0.9)}{NEP}$$\n\n### 1.6 Heavy Metals is taken from [3,7,8]\n\nCountry | As | Cd | Or | Hg | Ni | Pb | V | Zn |\n--------| -- | -- | -- | -- | -- | -- | - | -- |\nAustralia| 2 | 0.1 | 10 | 0.1 | 10 | 10 | 20 | 20 |\nCanada | 10 | 0.5 | 20 | 0.1 | 20 | 30 | 40 | 50 |\nChina | 5 |0.5 |30 |0.1 |50 |40 |50 |50|\nIndonesia | 10 |0.5| 20| 0.1| 20 |40 |40| 50|\nJapan | 10 |0.5| 20 |0.1| 20 |40 |40 |50|\nUSA | 15 |0.5| 20| 0.2| 20 |15| 35| 20|\nSouth Africa |3 |0.01 |100 |0.2| 20| 10| 50| 30|\nRussia | 10| 0.5| 20| 0.1| 20| 40| 40| 50|\n\n\n```python\n\n```\n\n\n```python\n\n\n\n\n```\n\n\n```python\n\n```\n\n## Development of LCI_s for Electricity Grid Mixes in Each Electric Company\n\n$$ EID = \\frac {EIP * NEP}{(NEP - LP - LT) * (1-(LD/100))} $$\n\nEID = emission intensity of each substance based on net electricity distributed to customers by each electric company [kg/kWh]
    \nEIP = emission intensity of each substance based on net electriciy production [kg/kWH]
    \nNEP = net electricity production in each power station [kWh]
    \nLP = loss of electricity in pumping [kWh]
    \nLT = loss of electricity in transformer [kWh]
    \nLD = distribution loss [%]
    \n\n\n```python\n\n```\n", "meta": {"hexsha": "eae55657c8e8ddac4998c7771e0d8568639dced6", "size": 7285, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "LCI/1_calc_methods_JP.ipynb", "max_stars_repo_name": "msuherma/LifeCycleAssessment_LCA", "max_stars_repo_head_hexsha": "1f77b4c0d2b9829dba2c736ff78213dbf6504be5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-12-04T21:36:53.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-10T09:35:06.000Z", "max_issues_repo_path": "LCI/1_calc_methods_JP.ipynb", "max_issues_repo_name": "msuherma/LifeCycleAssessment_LCA", "max_issues_repo_head_hexsha": "1f77b4c0d2b9829dba2c736ff78213dbf6504be5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "LCI/1_calc_methods_JP.ipynb", "max_forks_repo_name": "msuherma/LifeCycleAssessment_LCA", "max_forks_repo_head_hexsha": "1f77b4c0d2b9829dba2c736ff78213dbf6504be5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 26.2050359712, "max_line_length": 182, "alphanum_fraction": 0.4908716541, "converted": true, "num_tokens": 1567, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.44269037042165593}} {"text": "```python\n\"\"\"\n================================\n Data pre-processing\n================================\n\n(See the project documentation for more info)\n\nThe goal is to process data before using it to train ML algorithms :\n- Extraction of accelerations for activity 1 (rest activity)\n- Transitory regime suppression on activity 1\n- Calculation of theta angle between Z' and Z (ground's normal axis)\n- System rotation towards Z earth axis\n- Offset removal\n- Data augmentation for less represented activities\n\"\"\"\nprint(__doc__)\n```\n\n \n ================================\n Data pre-processing\n ================================\n \n (See the project documentation for more info)\n \n The goal is to process data before using it to train ML algorithms :\n - Extraction of accelerations for activity 1 (rest activity)\n - Transitory regime suppression on activity 1\n - Calculation of theta angle between Z' and Z (ground's normal axis)\n - System rotation towards Z earth axis\n - Offset removal\n - Data augmentation for less represented activities\n \n\n\n\n```python\n# Imports statements\n\nimport pandas as pd\nimport numpy as np\n# from math import cos, sin\nfrom utils.colorpalette import black, red, blue, green, yellow, pink, brown, violet\nfrom utils.activities import activities_labels\nimport matplotlib.pyplot as plt\nfrom matplotlib.lines import Line2D \n```\n\n\n```python\n# Import data into memory\n\nraw_data = pd.read_csv('../data/1.csv',header=None,delimiter=',')\nraw_data.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    01234
    00.01502221521531
    11.01667207220471
    22.01611195719061
    33.01601193918311
    44.01643196518791
    \n
    \n\n\n\n\n```python\n# Prepare further plotting activities\ncolor_map = np.array([black, red, blue, green, yellow, pink, brown, violet])\naxe_name = ['X', 'Y', 'Z']\nactivities = np.array(raw_data[4])\nx_min, x_max = raw_data[0].min() - 1, raw_data[0].max() + 1\n```\n\n\n```python\n# Show data before processing\ny_min, y_max, xx, yy, fig, subplot = [], [], [], [], [], []\nlegend = []\nfor activity, color in zip(activities_labels, color_map):\n legend.append(Line2D([0], [0], marker='o', label=activity, ls='None', markerfacecolor=color, markeredgecolor='k'))\nfor k in range(0,3):\n y_min.append(raw_data[k+1].min() - 1)\n y_max.append(raw_data[k+1].max() + 1)\n xx_tmp, yy_tmp = np.meshgrid(np.arange(x_min, x_max, 1000),np.arange(y_min[k], y_max[k], 100))\n xx.append(xx_tmp)\n yy.append(yy_tmp)\n fig.append(plt.figure())\n subplot.append(fig[k].add_subplot(111))\n subplot[k].scatter(raw_data[0], raw_data[k+1], s=1,c=color_map[activities])\n subplot[k].set_title('Acceleration on ' + axe_name[k])\nlegend = plt.legend(handles=legend, loc='upper center', bbox_to_anchor=(1, 2), title='Activities')\nplt.show()\n```\n\n\n```python\n#Prepare for processing\nclean_data = []\nclean_data.append(raw_data[0])\n```\n\n\n```python\n# Transitory regime suppression on activity 1\nnp_raw_data = np.array(raw_data, dtype=object)\nbool_mask_on_act_1 = np_raw_data[:, 4] == 1 # Boolean mask to only select rows concerning activity 1\n#activity_1_data = np_raw_data[bool_mask_on_act_1]\nbool_mask_on_permanent_regime = (np_raw_data[:, 0] >= 3200) & (np_raw_data[:, 0] <= 16000)\nact_1_data_permanent_regime = np_raw_data[bool_mask_on_act_1 & bool_mask_on_permanent_regime]\n```\n\n\n```python\n# Show activity 1 data after transitory regime suppression\nactivities = np.array(act_1_data_permanent_regime[:,4], dtype=int)\n\nx_min, x_max = act_1_data_permanent_regime[0].min() - 1, act_1_data_permanent_regime[0].max() + 1\ny_min, y_max, xx, yy, fig, subplot = [], [], [], [], [], []\nlegend = []\nfor activity, color in zip(activities_labels, color_map):\n legend.append(Line2D([0], [0], marker='o', label=activity, ls='None', markerfacecolor=color, markeredgecolor='k'))\nfor k in range(0,3):\n y_min.append(act_1_data_permanent_regime[k+1].min() - 1)\n y_max.append(act_1_data_permanent_regime[k+1].max() + 1)\n xx_tmp, yy_tmp = np.meshgrid(np.arange(x_min, x_max, 1000),np.arange(y_min[k], y_max[k], 100))\n xx.append(xx_tmp)\n yy.append(yy_tmp)\n fig.append(plt.figure())\n subplot.append(fig[k].add_subplot(111))\n subplot[k].scatter(act_1_data_permanent_regime[:,0], act_1_data_permanent_regime[:,k+1], s=1,c=color_map[activities])\n subplot[k].set_title('Acceleration on ' + axe_name[k])\nlegend = plt.legend(handles=legend, loc='upper center', bbox_to_anchor=(1, 2), title='Activities')\nplt.show()\n```\n\n\n```python\n\n```\n\n\n```python\n# Look for theta value : \nfrom sympy.solvers import solve\nfrom sympy import Symbol, sin, cos\nfrom math import sqrt\ntheta = Symbol('theta')\nindex_mean, xp_mean, yp_mean, zp_mean, activity_mean = act_1_data_permanent_regime.mean(axis=0)\nabs_gamma_mean = sqrt(xp_mean**2+yp_mean**2+zp_mean**2)\ntheta = solve(sin(theta)*yp_mean+cos(theta)*zp_mean, abs_gamma_mean, dict=True)\n# TODO : Find a way that this equation returns results !\n```\n\n\n```python\n# System rotation towards Z earth axis - see the report for documentation\nrotation_matrix = np.array([[1, 0, 0],\n [0, cos(theta), -sin(theta)]\n [0, sin(theta), cos(theta)]])\n\nfor row_index in clean_data:\n Gamma_xp_yp_zp = clean_data.iloc[row_index][1, 2, 3]\n Gamma_x_y_z = np.matmul(rotation_matrix, Gamma_xp_yp_zp)\n clean_data.iloc[row_index][1, 2, 3] = Gamma_x_y_z\n```\n\n\n```python\n# Offset suppression\n\n# TODO :\n# Should we really delete the offset though? Maybe it just corresponds to gravity, so change of system first !\n# At rest, Gamma is expected to be 1g, but is calculated to be around 3,7g.\n# So there might be offsets, indeed, but in which direction?\n\nmean_acc_by_act = raw_data[[1, 2, 3, 4]].groupby([4], as_index=False).mean().sort_values(by=4, ascending=True)\nmean_acc_at_act_1 = mean_acc_by_act.iloc[0] # Offset is calculated at rest (activity 1)\n\nfor k in range(1,4):\n clean_data.append(raw_data[k] - mean_acc_at_act_1[k])\n```\n\n\n```python\n# Show changes after offset suppression\nlegend = []\nfor activity, color in zip(activities_labels, color_map):\n legend.append(Line2D([0], [0], marker='o', label=activity, ls='None', markerfacecolor=color, markeredgecolor='k'))\n\ny_min, y_max, xx, yy, fig, subplot = [], [], [], [], [], []\nfor k in range(0,3):\n y_min.append(clean_data[k+1].min() - 1)\n y_max.append(clean_data[k+1].max() + 1)\n xx_tmp, yy_tmp = np.meshgrid(np.arange(x_min, x_max, 1000),np.arange(y_min[k], y_max[k], 100))\n xx.append(xx_tmp)\n yy.append(yy_tmp)\n fig.append(plt.figure())\n subplot.append(fig[k].add_subplot(111))\n subplot[k].scatter(clean_data[0], clean_data[k+1], s=1,c=color_map[activities])\n subplot[k].set_title('Acceleration on ' + axe_name[k])\nlegend = plt.legend(handles=legend, loc='upper center', bbox_to_anchor=(1, 2), title='Activities')\nplt.show()\n```\n\n\n```python\n# Data augmentation\n```\n\n\n```python\n# Push data changes into new csv file\nclean_data.append(raw_data[4])\n```\n\n\n```python\n# Garbage snippet : backup if we want to plot subplots on a same figure\n\n\n# x_min, x_max = data[0].min() - 1, data[0].max() + 1\n# y_min, y_max = (min(data[1].min(), data[2].min(), data[3].min()) - 1,\n# max(data[1].max(), data[2].max(), data[3].max()) + 1)\n# xx, yy = np.meshgrid(np.arange(x_min, x_max, 1000),np.arange(y_min, y_max, 100))\n\n# fig = plt.figure()\n\n# subplot1 = fig.add_subplot(311)\n# subplot1.scatter(data[0], data[1], s=1,c=color_map[activities])\n# subplot1.set_title('Acceleration on X')\n\n# subplot2 = fig.add_subplot(312)\n# subplot2.scatter(data[0], data[2], s=1,c=color_map[activities])\n# subplot2.set_title('Acceleration on Y')\n\n# subplot3 = fig.add_subplot(313)\n# subplot3.scatter(data[0], data[3], s=1,c=color_map[activities])\n# subplot3.set_title('Acceleration on Z')\n```\n", "meta": {"hexsha": "231124cae19198759a7af39c3085b7fd100b5857", "size": 308081, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "src/preprocessing.ipynb", "max_stars_repo_name": "Caojerem/Caojerem", "max_stars_repo_head_hexsha": "06c6fd56b1022c9f24fe4b3a1189b970be622959", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/preprocessing.ipynb", "max_issues_repo_name": "Caojerem/Caojerem", "max_issues_repo_head_hexsha": "06c6fd56b1022c9f24fe4b3a1189b970be622959", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/preprocessing.ipynb", "max_forks_repo_name": "Caojerem/Caojerem", "max_forks_repo_head_hexsha": "06c6fd56b1022c9f24fe4b3a1189b970be622959", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 626.1808943089, "max_line_length": 85742, "alphanum_fraction": 0.9420412164, "converted": true, "num_tokens": 2599, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723316991792861, "lm_q2_score": 0.658417500561683, "lm_q1q2_score": 0.4426749569220149}} {"text": "## Configurations for Colab\n\n\n```python\nimport sys\nIN_COLAB = \"google.colab\" in sys.modules\n\nif IN_COLAB:\n !apt install python-opengl\n !apt install ffmpeg\n !apt install xvfb\n !pip install pyvirtualdisplay\n !pip install gym\n from pyvirtualdisplay import Display\n \n # Start virtual display\n dis = Display(visible=0, size=(400, 400))\n dis.start()\n```\n\n# 05. Noisy Networks for Exploration\n\n[M. Fortunato et al., \"Noisy Networks for Exploration.\" arXiv preprint arXiv:1706.10295, 2017.](https://arxiv.org/pdf/1706.10295.pdf)\n\n\nNoisyNet is an exploration method that learns perturbations of the network weights to drive exploration. The key insight is that a single change to the weight vector can induce a consistent, and potentially very complex, state-dependent change in policy over multiple time steps.\n\nFirstly, let's take a look into a linear layer of a neural network with $p$ inputs and $q$ outputs, represented by\n\n$$\ny = wx + b,\n$$\n\nwhere $x \\in \\mathbb{R}^p$ is the layer input, $w \\in \\mathbb{R}^{q \\times p}$, and $b \\in \\mathbb{R}$ the bias.\n\nThe corresponding noisy linear layer is defined as:\n\n$$\ny = (\\mu^w + \\sigma^w \\odot \\epsilon^w) x + \\mu^b + \\sigma^b \\odot \\epsilon^b,\n$$\n\nwhere $\\mu^w + \\sigma^w \\odot \\epsilon^w$ and $\\mu^b + \\sigma^b \\odot \\epsilon^b$ replace $w$ and $b$ in the first linear layer equation. The parameters $\\mu^w \\in \\mathbb{R}^{q \\times p}, \\mu^b \\in \\mathbb{R}^q, \\sigma^w \\in \\mathbb{R}^{q \\times p}$ and $\\sigma^b \\in \\mathbb{R}^q$ are learnable, whereas $\\epsilon^w \\in \\mathbb{R}^{q \\times p}$ and $\\epsilon^b \\in \\mathbb{R}^q$ are noise random variables which can be generated by one of the following two ways:\n\n1. **Independent Gaussian noise**: the noise applied to each weight and bias is independent, where each random noise entry is drawn from a unit Gaussian distribution. This means that for each noisy linear layer, there are $pq + q$ noise variables (for $p$ inputs to the layer and $q$ outputs).\n2. **Factorised Gaussian noise:** This is a more computationally efficient way. It produces 2 random Gaussian noise vectors ($p, q$) and makes $pq + q$ noise entries by outer product as follows:\n\n$$\n\\begin{align}\n\\epsilon_{i,j}^w &= f(\\epsilon_i) f(\\epsilon_j),\\\\\n\\epsilon_{j}^b &= f(\\epsilon_i),\\\\\n\\text{where } f(x) &= sgn(x) \\sqrt{|x|}.\n\\end{align}\n$$\n\nIn all experiements of the paper, the authors used Factorised Gaussian noise, so we will go for it as well.\n\n\n```python\nimport math\nimport os\nfrom typing import Dict, List, Tuple\n\nimport gym\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport torch.optim as optim\nfrom IPython.display import clear_output\n```\n\n## Replay buffer\n\nPlease see *01.dqn.ipynb* for detailed description.\n\n\n```python\nclass ReplayBuffer:\n \"\"\"A simple numpy replay buffer.\"\"\"\n\n def __init__(self, obs_dim: int, size: int, batch_size: int = 32):\n self.obs_buf = np.zeros([size, obs_dim], dtype=np.float32)\n self.next_obs_buf = np.zeros([size, obs_dim], dtype=np.float32)\n self.acts_buf = np.zeros([size], dtype=np.float32)\n self.rews_buf = np.zeros([size], dtype=np.float32)\n self.done_buf = np.zeros(size, dtype=np.float32)\n self.max_size, self.batch_size = size, batch_size\n self.ptr, self.size, = 0, 0\n\n def store(\n self,\n obs: np.ndarray,\n act: np.ndarray, \n rew: float, \n next_obs: np.ndarray, \n done: bool,\n ):\n self.obs_buf[self.ptr] = obs\n self.next_obs_buf[self.ptr] = next_obs\n self.acts_buf[self.ptr] = act\n self.rews_buf[self.ptr] = rew\n self.done_buf[self.ptr] = done\n self.ptr = (self.ptr + 1) % self.max_size\n self.size = min(self.size + 1, self.max_size)\n\n def sample_batch(self) -> Dict[str, np.ndarray]:\n idxs = np.random.choice(self.size, size=self.batch_size, replace=False)\n return dict(obs=self.obs_buf[idxs],\n next_obs=self.next_obs_buf[idxs],\n acts=self.acts_buf[idxs],\n rews=self.rews_buf[idxs],\n done=self.done_buf[idxs])\n\n def __len__(self) -> int:\n return self.size\n```\n\n## Noisy Layer\n\n**References:**\n- https://github.com/higgsfield/RL-Adventure/blob/master/5.noisy%20dqn.ipynb\n- https://github.com/Kaixhin/Rainbow/blob/master/model.py\n\n\n```python\nclass NoisyLinear(nn.Module):\n \"\"\"Noisy linear module for NoisyNet.\n \n Attributes:\n in_features (int): input size of linear module\n out_features (int): output size of linear module\n std_init (float): initial std value\n weight_mu (nn.Parameter): mean value weight parameter\n weight_sigma (nn.Parameter): std value weight parameter\n bias_mu (nn.Parameter): mean value bias parameter\n bias_sigma (nn.Parameter): std value bias parameter\n \n \"\"\"\n\n def __init__(self, in_features: int, out_features: int, std_init: float = 0.5):\n \"\"\"Initialization.\"\"\"\n super(NoisyLinear, self).__init__()\n \n self.in_features = in_features\n self.out_features = out_features\n self.std_init = std_init\n\n self.weight_mu = nn.Parameter(torch.Tensor(out_features, in_features))\n self.weight_sigma = nn.Parameter(\n torch.Tensor(out_features, in_features)\n )\n self.register_buffer(\n \"weight_epsilon\", torch.Tensor(out_features, in_features)\n )\n\n self.bias_mu = nn.Parameter(torch.Tensor(out_features))\n self.bias_sigma = nn.Parameter(torch.Tensor(out_features))\n self.register_buffer(\"bias_epsilon\", torch.Tensor(out_features))\n\n self.reset_parameters()\n self.reset_noise()\n\n def reset_parameters(self):\n \"\"\"Reset trainable network parameters (factorized gaussian noise).\"\"\"\n mu_range = 1 / math.sqrt(self.in_features)\n self.weight_mu.data.uniform_(-mu_range, mu_range)\n self.weight_sigma.data.fill_(\n self.std_init / math.sqrt(self.in_features)\n )\n self.bias_mu.data.uniform_(-mu_range, mu_range)\n self.bias_sigma.data.fill_(\n self.std_init / math.sqrt(self.out_features)\n )\n\n def reset_noise(self):\n \"\"\"Make new noise.\"\"\"\n epsilon_in = self.scale_noise(self.in_features)\n epsilon_out = self.scale_noise(self.out_features)\n\n # outer product\n self.weight_epsilon.copy_(epsilon_out.ger(epsilon_in))\n self.bias_epsilon.copy_(epsilon_out)\n\n def forward(self, x: torch.Tensor) -> torch.Tensor:\n \"\"\"Forward method implementation.\n \n We don't use separate statements on train / eval mode.\n It doesn't show remarkable difference of performance.\n \"\"\"\n return F.linear(\n x,\n self.weight_mu + self.weight_sigma * self.weight_epsilon,\n self.bias_mu + self.bias_sigma * self.bias_epsilon,\n )\n \n @staticmethod\n def scale_noise(size: int) -> torch.Tensor:\n \"\"\"Set scale to make noise (factorized gaussian noise).\"\"\"\n x = torch.FloatTensor(np.random.normal(loc=0.0, scale=1.0, size=size))\n\n return x.sign().mul(x.abs().sqrt())\n```\n\n## Noisy Network\n\nWe use NoisyLinear for the last two FC layers, and there is a method to reset noise at every step.\nThese are the only differences from the example of *01.dqn.ipynb*.\n\n\n```python\nclass Network(nn.Module):\n def __init__(self, in_dim: int, out_dim: int):\n \"\"\"Initialization.\"\"\"\n super(Network, self).__init__()\n\n self.feature = nn.Linear(in_dim, 128)\n self.noisy_layer1 = NoisyLinear(128, 128)\n self.noisy_layer2 = NoisyLinear(128, out_dim)\n\n def forward(self, x: torch.Tensor) -> torch.Tensor:\n \"\"\"Forward method implementation.\"\"\"\n feature = F.relu(self.feature(x))\n hidden = F.relu(self.noisy_layer1(feature))\n out = self.noisy_layer2(hidden)\n \n return out\n \n def reset_noise(self):\n \"\"\"Reset all noisy layers.\"\"\"\n self.noisy_layer1.reset_noise()\n self.noisy_layer2.reset_noise()\n```\n\n## DQN + NoisyNet Agent (w/o DuelingNet)\n\nHere is a summary of DQNAgent class.\n\n| Method | Note |\n| --- | --- |\n|select_action | select an action from the input state. |\n|step | take an action and return the response of the env. |\n|compute_dqn_loss | return dqn loss. |\n|update_model | update the model by gradient descent. |\n|target_hard_update| hard update from the local model to the target model.|\n|train | train the agent during num_frames. |\n|test | test the agent (1 episode). |\n|plot | plot the training progresses. |\n\nIn the paper, NoisyNet is used as a component of the Dueling Network Architecture, which includes Double-DQN and Prioritized Experience Replay. However, we don't implement them to simplify the tutorial. One thing to note is that NoisyNet is an alternertive to $\\epsilon$-greedy method, so all $\\epsilon$ related lines are removed. Please check all comments with *NoisyNet*.\n\n\n```python\nclass DQNAgent:\n \"\"\"DQN Agent interacting with environment.\n \n Attribute:\n env (gym.Env): openAI Gym environment\n memory (ReplayBuffer): replay memory to store transitions\n batch_size (int): batch size for sampling\n target_update (int): period for target model's hard update\n gamma (float): discount factor\n dqn (Network): model to train and select actions\n dqn_target (Network): target model to update\n optimizer (torch.optim): optimizer for training dqn\n transition (list): transition information including\n state, action, reward, next_state, done\n \"\"\"\n\n def __init__(\n self, \n env: gym.Env,\n memory_size: int,\n batch_size: int,\n target_update: int,\n gamma: float = 0.99,\n ):\n \"\"\"Initialization.\n \n Args:\n env (gym.Env): openAI Gym environment\n memory_size (int): length of memory\n batch_size (int): batch size for sampling\n target_update (int): period for target model's hard update\n gamma (float): discount factor\n \"\"\"\n # NoisyNet: All attributes related to epsilon are removed\n obs_dim = env.observation_space.shape[0]\n action_dim = env.action_space.n\n \n self.env = env\n self.memory = ReplayBuffer(obs_dim, memory_size, batch_size)\n self.batch_size = batch_size\n self.target_update = target_update\n self.gamma = gamma\n \n # device: cpu / gpu\n self.device = torch.device(\n \"cuda\" if torch.cuda.is_available() else \"cpu\"\n )\n print(self.device)\n\n # networks: dqn, dqn_target\n self.dqn = Network(obs_dim, action_dim).to(self.device)\n self.dqn_target = Network(obs_dim, action_dim).to(self.device)\n self.dqn_target.load_state_dict(self.dqn.state_dict())\n self.dqn_target.eval()\n \n # optimizer\n self.optimizer = optim.Adam(self.dqn.parameters())\n\n # transition to store in memory\n self.transition = list()\n \n # mode: train / test\n self.is_test = False\n\n def select_action(self, state: np.ndarray) -> np.ndarray:\n \"\"\"Select an action from the input state.\"\"\"\n # NoisyNet: no epsilon greedy action selection\n selected_action = self.dqn(\n torch.FloatTensor(state).to(self.device)\n ).argmax()\n selected_action = selected_action.detach().cpu().numpy()\n \n if not self.is_test:\n self.transition = [state, selected_action]\n \n return selected_action\n\n def step(self, action: np.ndarray) -> Tuple[np.ndarray, np.float64, bool]:\n \"\"\"Take an action and return the response of the env.\"\"\"\n next_state, reward, done, _ = self.env.step(action)\n\n if not self.is_test:\n self.transition += [reward, next_state, done]\n self.memory.store(*self.transition)\n \n return next_state, reward, done\n\n def update_model(self) -> torch.Tensor:\n \"\"\"Update the model by gradient descent.\"\"\"\n samples = self.memory.sample_batch()\n\n loss = self._compute_dqn_loss(samples)\n\n self.optimizer.zero_grad()\n loss.backward()\n self.optimizer.step()\n \n # NoisyNet: reset noise\n self.dqn.reset_noise()\n self.dqn_target.reset_noise()\n\n return loss.item()\n \n def train(self, num_frames: int, plotting_interval: int = 200):\n \"\"\"Train the agent.\"\"\"\n self.is_test = False\n \n state = self.env.reset()\n update_cnt = 0\n losses = []\n scores = []\n score = 0\n\n for frame_idx in range(1, num_frames + 1):\n action = self.select_action(state)\n next_state, reward, done = self.step(action)\n\n state = next_state\n score += reward\n \n # NoisyNet: removed decrease of epsilon\n\n # if episode ends\n if done:\n state = self.env.reset()\n scores.append(score)\n score = 0\n\n # if training is ready\n if len(self.memory) >= self.batch_size:\n loss = self.update_model()\n losses.append(loss)\n update_cnt += 1\n \n # if hard update is needed\n if update_cnt % self.target_update == 0:\n self._target_hard_update()\n\n # plotting\n if frame_idx % plotting_interval == 0:\n self._plot(frame_idx, scores, losses)\n \n self.env.close()\n \n def test(self) -> List[np.ndarray]:\n \"\"\"Test the agent.\"\"\"\n self.is_test = True\n \n state = self.env.reset()\n done = False\n score = 0\n \n frames = []\n while not done:\n frames.append(self.env.render(mode=\"rgb_array\"))\n action = self.select_action(state)\n next_state, reward, done = self.step(action)\n\n state = next_state\n score += reward\n \n print(\"score: \", score)\n self.env.close()\n \n return frames\n\n def _compute_dqn_loss(self, samples: Dict[str, np.ndarray]) -> torch.Tensor:\n \"\"\"Return dqn loss.\"\"\"\n device = self.device # for shortening the following lines\n state = torch.FloatTensor(samples[\"obs\"]).to(device)\n next_state = torch.FloatTensor(samples[\"next_obs\"]).to(device)\n action = torch.LongTensor(samples[\"acts\"].reshape(-1, 1)).to(device)\n reward = torch.FloatTensor(samples[\"rews\"].reshape(-1, 1)).to(device)\n done = torch.FloatTensor(samples[\"done\"].reshape(-1, 1)).to(device)\n \n # G_t = r + gamma * v(s_{t+1}) if state != Terminal\n # = r otherwise\n curr_q_value = self.dqn(state).gather(1, action)\n next_q_value = self.dqn_target(next_state).max(\n dim=1, keepdim=True\n )[0].detach()\n mask = 1 - done\n target = (reward + self.gamma * next_q_value * mask).to(self.device)\n\n # calculate dqn loss\n loss = F.smooth_l1_loss(curr_q_value, target)\n\n return loss\n\n def _target_hard_update(self):\n \"\"\"Hard update: target <- local.\"\"\"\n self.dqn_target.load_state_dict(self.dqn.state_dict())\n \n def _plot(\n self, \n frame_idx: int, \n scores: List[float], \n losses: List[float], \n ):\n \"\"\"Plot the training progresses.\"\"\"\n clear_output(True)\n plt.figure(figsize=(20, 5))\n plt.subplot(131)\n plt.title('frame %s. score: %s' % (frame_idx, np.mean(scores[-10:])))\n plt.plot(scores)\n plt.subplot(132)\n plt.title('loss')\n plt.plot(losses)\n plt.show()\n```\n\n## Environment\n\nYou can see the [code](https://github.com/openai/gym/blob/master/gym/envs/classic_control/cartpole.py) and [configurations](https://github.com/openai/gym/blob/master/gym/envs/__init__.py#L53) of CartPole-v0 from OpenAI's repository.\n\n\n```python\n# environment\nenv_id = \"CartPole-v0\"\nenv = gym.make(env_id)\nif IN_COLAB:\n env = gym.wrappers.Monitor(env, \"videos\", force=True)\n```\n\n## Set random seed\n\n\n```python\nseed = 777\n\ndef seed_torch(seed):\n torch.manual_seed(seed)\n if torch.backends.cudnn.enabled:\n torch.backends.cudnn.benchmark = False\n torch.backends.cudnn.deterministic = True\n\nnp.random.seed(seed)\nseed_torch(seed)\nenv.seed(seed)\n```\n\n\n\n\n [777]\n\n\n\n## Initialize\n\n\n```python\n# parameters\nnum_frames = 20000\nmemory_size = 2000\nbatch_size = 64\ntarget_update = 150\n\n# train\nagent = DQNAgent(env, memory_size, batch_size, target_update)\n```\n\n cpu\n\n\n## Train\n\n\n```python\nagent.train(num_frames)\n```\n\n## Test\n\nRun the trained agent (1 episode).\n\n\n```python\nframes = agent.test()\n```\n\n score: 200.0\n\n\n## Render\n\n\n```python\nif IN_COLAB: # for colab\n import base64\n import glob\n import io\n import os\n\n from IPython.display import HTML, display\n\n\n def ipython_show_video(path: str) -> None:\n \"\"\"Show a video at `path` within IPython Notebook.\"\"\"\n if not os.path.isfile(path):\n raise NameError(\"Cannot access: {}\".format(path))\n\n video = io.open(path, \"r+b\").read()\n encoded = base64.b64encode(video)\n\n display(HTML(\n data=\"\"\"\n \n \"\"\".format(encoded.decode(\"ascii\"))\n ))\n\n list_of_files = glob.glob(\"videos/*.mp4\")\n latest_file = max(list_of_files, key=os.path.getctime)\n print(latest_file)\n ipython_show_video(latest_file)\n \nelse: # for jupyter\n from matplotlib import animation\n from JSAnimation.IPython_display import display_animation\n from IPython.display import display\n\n\n def display_frames_as_gif(frames: List[np.ndarray]) -> None:\n \"\"\"Displays a list of frames as a gif, with controls.\"\"\"\n patch = plt.imshow(frames[0])\n plt.axis('off')\n\n def animate(i):\n patch.set_data(frames[i])\n\n anim = animation.FuncAnimation(\n plt.gcf(), animate, frames = len(frames), interval=50\n )\n display(display_animation(anim, default_mode='loop'))\n\n\n # display \n display_frames_as_gif(frames)\n```\n\n\n\n\n\n
    \n \n
    \n \n
    \n \n \n \n \n \n \n \n \n \n
    \n Once \n Loop \n Reflect \n
    \n
    \n\n\n\n\n\n", "meta": {"hexsha": "ecb065b1867a87a8edaef4a65c28540d8c551894", "size": 485921, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "05.noisy_net.ipynb", "max_stars_repo_name": "leeyaf/rainbow-is-all-you-need", "max_stars_repo_head_hexsha": "692796a9515cc40154aab5672bf06e08ba9f8b8b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "05.noisy_net.ipynb", "max_issues_repo_name": "leeyaf/rainbow-is-all-you-need", "max_issues_repo_head_hexsha": "692796a9515cc40154aab5672bf06e08ba9f8b8b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "05.noisy_net.ipynb", "max_forks_repo_name": "leeyaf/rainbow-is-all-you-need", "max_forks_repo_head_hexsha": "692796a9515cc40154aab5672bf06e08ba9f8b8b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 424.3851528384, "max_line_length": 62192, "alphanum_fraction": 0.9414699097, "converted": true, "num_tokens": 4780, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723317123102956, "lm_q2_score": 0.658417487156366, "lm_q1q2_score": 0.4426749565548817}} {"text": "Last year [Duolingo](http://duolingo.com) published [a paper](https://aclweb.org/anthology/P/P16/P16-1174.pdf) describing a learner model which can be used to describe learners' learning and forgetting. Together with this paper authors published also a [data set containing 13 million learning traces](https://s3.amazonaws.com/duolingo-papers/publications/settles.acl16.learning_traces.13m.csv.gz) allowing other peaple to reproduce the original results. As I've [already written](https://papousek.github.io/analysis-of-half-life-regression-model-made-by-duolingo.html), the original model is realy bad. \n\nOur [research group](http://www.fi.muni.cz/adaptivelearning/) use learner modeling techniques quite often and my research is focused on [adaptive practice of factual knowledge in domains with varied prior knowledge](http://www.fi.muni.cz/~xpelanek/publications/EDM14-adaptive-facts.pdf). Learning a new language is a good example of this kind of domain. A lot of peaple try to learn a new languge many times (at school, private lessons, ...) and I assume that when a learner starts using Duolingo, she has already some kind of prior knowledge. Our approach to model this phenomenon is to seperate learners' prior knowledge on one side and the process of learning and forgetting on the other side. Of course, we use a learner's prior knowledge as an input to the model of learning.\n\nIn the following text, I describe how we usually model prior knowledge and I use data set from Duolingo as an example. The full code of this analysis is available on [GitHub](https://github.com/papousek/duolingo-halflife-regression) and is based on the fork of the original repository to minimize the probability of my error.\n\nIf you want to run this Jupyter notebook on your own, please download the [original data set](https://s3.amazonaws.com/duolingo-papers/publications/settles.acl16.learning_traces.13m.csv.gz) to the **data** directory. To be more familiar with the data, please read the [previous analysis](https://papousek.github.io/analysis-of-half-life-regression-model-made-by-duolingo.html).\n\n\n```python\n%matplotlib inline\n%load_ext autoreload\n%autoreload 2\n```\n\n\n```python\nfrom evaluation import plot_model_stats\nfrom models import ItemAverage, Elo\nfrom proposal import load_train_test_set, train_model, grid_search\nimport matplotlib.pylab as plt\nimport numpy as np\nimport pandas as pd\nimport seaborn as sns\nimport spiderpig as sp\n\nsns.set(style=\"white\")\n```\n\nTo cache intermidiate results I use [spiderpig](https://github.com/papousek/spiderpig), a library I have written on my own for this purpose. Spiderpig has to be initialized and I already decleare some global parameters.\n\n\n```python\nsp.init(\n directory='.spiderpig',\n traces_filename='./data/settles.acl16.learning_traces.13m.csv.gz',\n traces_nrows=1000000,\n max_in_memory_entries=10\n)\n```\n\nSince I need to fit some parameters I divide the data to train/test set.\n\n\n```python\ntrainset, testset = load_train_test_set()\n```\n\n## Baseline: Item Average\n\nAs a baseline I use \"item average\" model. This model computes the average probability of recall for each item (lexeme) and uses this average for the estimation of the future probability of recall. Although this model is realy stupid, based on our experience it serves as a good baseline. Your model should definitely be better than this.\n\n\n```python\nitem_average, _ = train_model('ItemAverage')\n```\n\nTo evaluate quality of predictions, we can use a lot of metrics\n([AUC](https://en.wikipedia.org/wiki/Receiver_operating_characteristic#Area_under_the_curve),\n[MAE](https://en.wikipedia.org/wiki/Mean_absolute_error), [RMSE](https://en.wikipedia.org/wiki/Root-mean-square_deviation), ...), see [this paper](http://www.fi.muni.cz/~xpelanek/publications/metrics.pdf) for more info Although these metrics are usually very simple, they can have big negative impact on your evaluation. Based on our experience, RMSE serves well in the most cases. To look at predictions more closely, we use \"calibration\" curve and histogram of predictions. In the optimal case, the calibration curve is aligned with diagonal.\n\n\n```python\npredicted = np.zeros(len(testset))\nfor i, lexeme_id in enumerate(testset['lexeme_id'].values):\n predicted[i] = item_average.predict(lexeme_id)\nplot_model_stats(predicted, testset['p_recall'], 50)\n```\n\n## Elo\n\nFor each learner s we track her skill $\\theta_s$ and for each item (lexeme) i we track its difficulty $b_i$. When we want to predict the probality of correct answer, we substract item difficulty from a learner's skill and transform it by a logistic function. \n\n$$\n\\begin{align}\nP(correct_{si} = 1) &= \\frac{1}{1 + e^{-(\\theta_s - b_i)}}\n\\end{align}\n$$\n\nThis model is known as [one-parameter IRT model](https://en.wikipedia.org/wiki/Item_response_theory) (or Rasch model). We can use joint maximum likelihood to fit skills and difficulties, but this method is not applicable online, becuase it needs all data for its computation. To estimate parameters online, we can inspire from [Elo rating system](https://en.wikipedia.org/wiki/Elo_rating_system) originally designed to rate chess players. This method can easily work with a stream of answers. After a learner s solves an item i, we update the learner's skill and the item difficulty as follows (constant K stands for sensitivity of the estimate to the last update and $correct_{si}\\in[0, 1]$):\n\n$$\n\\begin{align}\n\\theta_s &= \\theta_s + K\\cdot(correct_{si}- P(correct_{si} = 1)) \\\\\nb_i &= b_i - K\\cdot(correct_{si} - P(correct_{si} = 1))\n\\end{align}\n$$\n\nInitially, values of skills $\\theta_s$ and difficulties $b_i$ are set to 0. To make the estimate stable and converging we need to replace the sensitivity constant K by an uncertainty function ensuring that later changes have less weight. Without that we could easily lose information from past updates.\n\n$$\nK \\sim U(n) = \\frac{\\alpha}{1 + \\beta n}\n$$\n\nVariable n stands for a number of past updates and a, b are meta-parameters fitted to data.\n\n**Important note**: Since this model does not handle learning, it is necessary to update skills and difficulties only once per each learner and item (we ignore repeated answers).\n\nTo fit meta-parameters we use simple grid search.\n\n\n```python\nelo_search = grid_search('Elo', ('alpha', (0.5, 1.5)), ('beta', (0.01, 0.11)))\n```\n\n\n```python\nsns.heatmap(\n elo_search.pivot_table(columns='alpha', index='beta', values='rmse', dropna=False).iloc[::-1],\n cmap='viridis_r',\n rasterized=True,\n linewidths=0.0\n)\nplt.xlabel('$\\\\alpha$')\nplt.ylabel('$\\\\beta$')\nelo_best_params = elo_search.ix[elo_search['rmse'].idxmin()]\nprint('Best meta-parameters:')\nprint(elo_best_params)\nplt.show()\n```\n\nBased on RMSE, it seems that the improvement in comparison to the baseline is very small, but calibration curve looks much better.\n\n\n```python\nelo, _ = train_model('Elo', alpha=elo_best_params['alpha'], beta=elo_best_params['beta'])\npredicted = np.zeros(len(testset))\nfor i, (user_id, lexeme_id, p_recall) in enumerate(testset[['user_id', 'lexeme_id', 'p_recall']].values):\n predicted[i] = elo.predict(user_id, lexeme_id, p_recall)\nplot_model_stats(predicted, testset['p_recall'], 50)\n```\n\nThe model also provides us with information about items and learners. Here are histograms of item difficulties and learners' prior skills.\n\n\n```python\nplt.hist(list(elo._difficulty.values()), bins=20)\nplt.xlabel('difficulty')\nplt.ylabel('count')\nplt.show()\n```\n\n\n```python\nplt.hist(list(elo._skill.values()), bins=20)\nplt.xlabel('prior skill')\nplt.ylabel('count')\nplt.show()\n```\n\n## Conclusion\n\nUsing Elo to estimate difficulties of items and learners' skills improves quality of predictions. Still, this model can't provide information shown by skill meters, because the estimated knowledge does not degrade in time. On the other side, estimation from this model can be used as an input for another model handling learning and forgetting, e.g., [PFAE with staircase time shift function](http://www.fi.muni.cz/~xpelanek/publications/memory-geography.pdf).\n", "meta": {"hexsha": "ce8fee3050935ad62ecc15ea3c2e18214a3d560f", "size": 157638, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Prior Knowledge.ipynb", "max_stars_repo_name": "ihsgnef/duolingo-halflife-regression", "max_stars_repo_head_hexsha": "01c7895eee0450462b5277a055d2ae1de58f1be5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Prior Knowledge.ipynb", "max_issues_repo_name": "ihsgnef/duolingo-halflife-regression", "max_issues_repo_head_hexsha": "01c7895eee0450462b5277a055d2ae1de58f1be5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Prior Knowledge.ipynb", "max_forks_repo_name": "ihsgnef/duolingo-halflife-regression", "max_forks_repo_head_hexsha": "01c7895eee0450462b5277a055d2ae1de58f1be5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 404.2, "max_line_length": 57534, "alphanum_fraction": 0.9272002943, "converted": true, "num_tokens": 1952, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723316860482763, "lm_q2_score": 0.6584175005616829, "lm_q1q2_score": 0.4426749482763282}} {"text": "###### Content under Creative Commons Attribution license CC-BY 4.0, code under BSD 3-Clause License \u00a9 2018 parts of this notebook are from ([this Jupyter notebook](https://nbviewer.jupyter.org/github/heinerigel/coursera/blob/master/Notebooks4Coursera/W2/W2_P1.ipynb)) by Heiner Igel ([@heinerigel](https://github.com/heinerigel)) which is a supplemenatry material to his Coursera lecture [Computers, Waves, Simulations: A Practical Introduction to Numerical Methods using Python](https://www.coursera.org/learn/computers-waves-simulations), additional modifications by D. Koehn, notebook style sheet by L.A. Barba, N.C. Clementi\n\n\n```python\n# Execute this cell to load the notebook's style sheet, then ignore it\nfrom IPython.core.display import HTML\ncss_file = '../style/custom.css'\nHTML(open(css_file, \"r\").read())\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n# What is an \"optimum\" grid point distance?\n\nAfter the introduction to the finite-difference method in the last class, you might think: how do I choose the \"optimum\" spatial or temporal grid point distance $dx$ or $dt$ for a given FD-scheme in order to find the sweet spot between numerical accuracy of the solution and computation time? To achieve this goal, I introduce the concept of \"gridpoints \nper wavelength\" and demonstrate the accuracy for an under- and oversampled computation of the first derivative of the sine function.\n\n\n```python\n# Import Libraries\nimport numpy as np\nfrom math import *\nimport matplotlib.pyplot as plt\n```\n\nWe initialize a space-dependent sine function\n\n\\begin{equation}\nf(x)= \\sin (k x) \\notag\n\\end{equation}\n\nwhere the wavenumber k is\n\n\\begin{equation}\nk = \\dfrac{2 \\pi}{\\lambda} \\notag\n\\end{equation}\n\nand $\\lambda$ is wavelength.\n\n\n```python\n# Initial parameters\nxmax = 10.0 # maximum extension of physical domain (m)\nnx = 200 # number of gridpoints \ndx = xmax/(nx-1) # grid increment dx (m)\nx = np.linspace(0,xmax,nx) # space coordinates\n\n# Initialization of sine function\nl = 20*dx # wavelength\nk = 2*pi/l # wavenumber\nf = np.sin(k*x)\n```\n\n\n```python\n# Define figure size\nplt.figure(figsize=(10,5))\n\n# Plot sine function\nplt.plot(x, f)\nplt.title('Sine function')\nplt.xlabel('x, m')\nplt.ylabel('Amplitude')\nplt.xlim((0, xmax))\nplt.grid()\nplt.show()\n```\n\nIn the cell below we calculate the central finite-difference derivative of f(x) using two points\n\n\\begin{equation} \nf^{\\prime}(x)=\\dfrac{f(x+dx)-f(x-dx)}{2dx}\\notag\n\\end{equation} \n\nand compare with the analytical derivative\n\n\\begin{equation} \nf^{\\prime}(x) = k \\cos(k x)\\notag\n\\end{equation} \n\n\n```python\n# First derivative with central difference operator\n\n# Initiation of numerical and analytical derivatives \nnder=np.zeros(nx) # numerical derivative\nader=np.zeros(nx) # analytical derivative\n\n# Numerical derivative of the given function\nfor i in range (1, nx-1):\n nder[i]=(f[i+1]-f[i-1])/(2*dx)\n\n# Analytical derivative of the given function\nader= k * np.cos(k*x)\n\n# Exclude boundaries\nader[0]=0.\nader[nx-1]=0.\n\n# Error (rms) \nrms = np.sqrt(np.mean(nder-ader)**2)\n```\n\n\n```python\n# Define figure size\nplt.figure(figsize=(10,5))\n\n# Plotting numerical & analytical solution and their difference\nplt.plot (x, nder,label=\"Numerical Derivative, 2-point central FD\", lw=2, ls='-', color=\"blue\")\nplt.plot (x, ader, label=\"Analytical Derivative\", lw=2, ls=\"--\",color=\"red\")\nplt.plot (x, nder-ader, label=\"Difference\", lw=2, ls=\":\")\nplt.title(\"First derivative, Err (rms) = %.6f \" % (rms) )\nplt.xlabel('x, m')\nplt.ylabel('Amplitude')\nplt.legend(loc='lower left')\nplt.grid()\nplt.show()\n```\n\n### The concept of number of points per wavelength\n\n\\begin{equation}\nn_\\lambda = \\dfrac{\\lambda}{dx} \\notag\n\\end{equation}\n\nHow does the error of the numerical derivative change with the number of points per wavelength?\n\n\n```python\n# Define figure size\nplt.figure(figsize=(10,5))\n\n# Plotting number of points per wavelength\n# ----------------------------------------\nplt.plot (x, nder,label=\"Numerical Derivative, 2-point central FD\", marker='o', color=\"blue\")\nplt.title(\"First derivative, Error = %.6f, $n_\\lambda$ = %.2f \" % ( rms, l/dx) )\nplt.xlabel('x, m')\nplt.ylabel('Amplitude')\nplt.legend(loc='lower left')\nplt.xlim((xmax/2-l,xmax/2+l))\nplt.grid()\nplt.show()\n```\n\n### Investigate the error as a function of grid points per wavelength\n\nNext, we investigate how the error of the FD solution changes as a function of grid points per wavelength\n\n\n```python\n# Define a range of number of points per wavelength, e.g. [nmin=3,5,6 ... ,nmax=15]\n# Loop over points, calculate corresponding wavelength and calculate error\n\n# Initialize vectors\nnmin=1\nnmax=15\nna = np.zeros(nmax-nmin+1) # Vector with number of points per wavelength\nerr = np.zeros(nmax-nmin+1) # Vector with error\n\nj = -1 # array index\n\n# Loop through finite-difference derivative calculation\nfor n in range (nmin,nmax+1):\n \n j = j+1 # array index\n na[j] = n\n\n \n # Initialize sin function\n l = na[j]*dx # wavelength\n k = 2*pi/l # wavenumber\n f = np.sin(k*x)\n\n # Numerical derivative of the sin function\n for i in range (1, nx-1):\n nder[i]=(f[i+1]-f[i-1])/(2*dx)\n\n # Analytical derivative of the sin function\n ader= k * np.cos(k*x)\n \n # Exclude boundaries\n ader[0]=0.\n ader[nx-1]=0.\n\n i0 = np.int(nx/2)\n \n # Error (rms) \n err[j] = (nder[i0]-ader[i0])**2/ader[i0]**2 * 100\n```\n\n\n```python\n# Define figure size\nplt.figure(figsize=(12,5))\n\n# Plotting error as function of number of points per wavelength\n# -------------------------------------------------------------\nplt.semilogy(na,err, lw=2, ls='-', marker='o', color=\"blue\")\nplt.title('Error as a function of $n_\\lambda$ ')\nplt.xlabel('n$_\\lambda$')\nplt.ylabel('rms ')\nplt.grid()\nplt.show()\n```\n\n## We learned\n\n* 2-point finite-difference approximations can provide estimates of the 1st derivative of a function\n* The accuracy depends on the \"number of points per wavelength\", i.e., how well we sample the original function\n* The more points we use the more accurate is the derivative approximation\n", "meta": {"hexsha": "f958f15486ae0319123ea883c06bc5692cf59030", "size": 205981, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "02_finite_difference_intro/2_fd_optimum_gridpoint_dist.ipynb", "max_stars_repo_name": "daniel-koehn/Differential-equations-earth-system", "max_stars_repo_head_hexsha": "3916cbc968da43d0971b7139476350c1dd798746", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 30, "max_stars_repo_stars_event_min_datetime": "2019-10-16T19:07:36.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-10T03:48:44.000Z", "max_issues_repo_path": "02_finite_difference_intro/2_fd_optimum_gridpoint_dist.ipynb", "max_issues_repo_name": "daniel-koehn/Differential-equations-earth-system", "max_issues_repo_head_hexsha": "3916cbc968da43d0971b7139476350c1dd798746", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "02_finite_difference_intro/2_fd_optimum_gridpoint_dist.ipynb", "max_forks_repo_name": "daniel-koehn/Differential-equations-earth-system", "max_forks_repo_head_hexsha": "3916cbc968da43d0971b7139476350c1dd798746", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2020-11-19T08:21:55.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-10T09:33:37.000Z", "avg_line_length": 242.9021226415, "max_line_length": 86700, "alphanum_fraction": 0.9097392478, "converted": true, "num_tokens": 2567, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.8519527982093666, "lm_q1q2_score": 0.44260764396651486}} {"text": "\n\n# Lambda School Data Science Module 132\n## Sampling, Confidence Intervals, and Hypothesis Testing\n\n## Prepare - examine other available hypothesis tests\n\nIf you had to pick a single hypothesis test in your toolbox, t-test would probably be the best choice - but the good news is you don't have to pick just one! Here's some of the others to be aware of:\n\n\n```\nimport numpy as np\nfrom scipy.stats import chisquare # One-way chi square test\n\n# Chi square can take any crosstab/table and test the independence of rows/cols\n# The null hypothesis is that the rows/cols are independent -> low chi square\n# The alternative is that there is a dependence -> high chi square\n# Be aware! Chi square does *not* tell you direction/causation\n\nind_obs = np.array([[1, 1], [2, 2]]).T\nprint(ind_obs)\nprint(chisquare(ind_obs, axis=None))\n\ndep_obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T\nprint(dep_obs)\nprint(chisquare(dep_obs, axis=None))\n```\n\n [[1 2]\n [1 2]]\n Power_divergenceResult(statistic=0.6666666666666666, pvalue=0.8810148425137847)\n [[16 32]\n [18 24]\n [16 16]\n [14 28]\n [12 20]\n [12 24]]\n Power_divergenceResult(statistic=23.31034482758621, pvalue=0.015975692534127565)\n\n\n\n```\n# Distribution tests:\n# We often assume that something is normal, but it can be important to *check*\n\n# For example, later on with predictive modeling, a typical assumption is that\n# residuals (prediction errors) are normal - checking is a good diagnostic\n\nfrom scipy.stats import normaltest\n# Poisson models arrival times and is related to the binomial (coinflip)\nsample = np.random.poisson(5, 1000)\nprint(normaltest(sample)) # Pretty clearly not normal\n```\n\n NormaltestResult(statistic=22.273646023922907, pvalue=1.4565963588062096e-05)\n\n\n\n```\n# Kruskal-Wallis H-test - compare the median rank between 2+ groups\n# Can be applied to ranking decisions/outcomes/recommendations\n# The underlying math comes from chi-square distribution, and is best for n>5\nfrom scipy.stats import kruskal\n\nx1 = [1, 3, 5, 7, 9]\ny1 = [2, 4, 6, 8, 10]\nprint(kruskal(x1, y1)) # x1 is a little better, but not \"significantly\" so\n\nx2 = [1, 1, 1]\ny2 = [2, 2, 2]\nz = [2, 2] # Hey, a third group, and of different size!\nprint(kruskal(x2, y2, z)) # x clearly dominates\n```\n\n KruskalResult(statistic=0.2727272727272734, pvalue=0.6015081344405895)\n KruskalResult(statistic=7.0, pvalue=0.0301973834223185)\n\n\nAnd there's many more! `scipy.stats` is fairly comprehensive, though there are even more available if you delve into the extended world of statistics packages. As tests get increasingly obscure and specialized, the importance of knowing them by heart becomes small - but being able to look them up and figure them out when they *are* relevant is still important.\n\n## T-test Assumptions\n\n\n```\nfrom scipy.stats import ttest_ind\n\n?ttest_ind\n```\n\n\n\n- Independence of means\n\nAre the means of our voting data independent (do not affect the outcome of one another)?\n \nThe best way to increase thel likelihood of our means being independent is to randomly sample (which we did not do).\n\n- \"Homogeneity\" of Variance? \n\nIs the magnitude of the variance between the two roughly the same?\n\nI think we're OK on this one for the voting data, although it probably could be better, one party was larger than the other.\n\nIf we suspect this to be a problem then we can use Welch's T-test\n\n- \"Dependent Variable\" (sample means) are Distributed Normally\n\n\n\nLots of statistical tests depend on normal distributions. We can test for normality using Scipy as was shown above.\n\nThis assumption is often assumed even if the assumption is a weak one. If you strongly suspect that things are not normally distributed, you can transform your data to get it looking more normal and then run your test. This problem typically goes away for large sample sizes (yay Central Limit Theorem) and is often why you don't hear it brought up. People declare the assumption to be satisfied either way. \n\n\n\n## Central Limit Theorem\n\n\n\n\n```\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\n\nsample_means = []\nfor x in range(0,3000):\n coinflips = np.random.binomial(n=1, p=.5, size=30)\n one_sample = coinflips\n sample_means.append(coinflips.mean())\n\nprint(len(sample_means))\nprint(sample_means)\n```\n\n 3000\n [0.4666666666666667, 0.6, 0.5, 0.6, 0.36666666666666664, 0.5, 0.5666666666666667, 0.4, 0.5666666666666667, 0.5666666666666667, 0.7333333333333333, 0.5, 0.6, 0.43333333333333335, 0.5333333333333333, 0.5, 0.5, 0.36666666666666664, 0.5666666666666667, 0.4666666666666667, 0.36666666666666664, 0.5333333333333333, 0.3, 0.43333333333333335, 0.5666666666666667, 0.43333333333333335, 0.5, 0.43333333333333335, 0.43333333333333335, 0.5666666666666667, 0.5333333333333333, 0.6333333333333333, 0.43333333333333335, 0.6, 0.5666666666666667, 0.43333333333333335, 0.5333333333333333, 0.6, 0.4666666666666667, 0.5333333333333333, 0.5, 0.43333333333333335, 0.43333333333333335, 0.5, 0.5666666666666667, 0.3333333333333333, 0.4666666666666667, 0.36666666666666664, 0.5666666666666667, 0.6, 0.5, 0.6, 0.43333333333333335, 0.4666666666666667, 0.5333333333333333, 0.5333333333333333, 0.4666666666666667, 0.4666666666666667, 0.43333333333333335, 0.4666666666666667, 0.4, 0.6, 0.43333333333333335, 0.3333333333333333, 0.4666666666666667, 0.26666666666666666, 0.43333333333333335, 0.43333333333333335, 0.5333333333333333, 0.5, 0.6, 0.3, 0.5666666666666667, 0.4, 0.6666666666666666, 0.4666666666666667, 0.5333333333333333, 0.5, 0.43333333333333335, 0.43333333333333335, 0.5333333333333333, 0.5, 0.43333333333333335, 0.5666666666666667, 0.4666666666666667, 0.36666666666666664, 0.3, 0.4, 0.5666666666666667, 0.4, 0.6333333333333333, 0.5333333333333333, 0.4, 0.5333333333333333, 0.36666666666666664, 0.5333333333333333, 0.5, 0.5666666666666667, 0.43333333333333335, 0.5666666666666667, 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0.4, 0.43333333333333335, 0.5, 0.6, 0.4666666666666667, 0.43333333333333335, 0.5, 0.5666666666666667, 0.36666666666666664, 0.4666666666666667, 0.5333333333333333, 0.6, 0.5, 0.6333333333333333, 0.5333333333333333, 0.5333333333333333, 0.6, 0.6666666666666666, 0.6333333333333333, 0.4, 0.4666666666666667, 0.43333333333333335, 0.5333333333333333, 0.6666666666666666, 0.6, 0.4666666666666667, 0.6, 0.5, 0.5, 0.5, 0.4666666666666667, 0.4666666666666667, 0.5666666666666667, 0.43333333333333335, 0.5, 0.36666666666666664, 0.5666666666666667, 0.7, 0.36666666666666664, 0.5333333333333333, 0.6, 0.3, 0.5333333333333333, 0.5, 0.43333333333333335, 0.6, 0.4666666666666667, 0.5]\n\n\n\n```\n# Create dataframe with single coin flip\n```\n\n\n```\n# Plot histogram to look at distribution of a single coin flip \n```\n\n\n```\n# Plot histogram to look at distribution of all coin flips\n```\n\nWhat does the Central Limit Theorem State? That no matter the initial distribution of the population, the distribution of sample means taken will approximate a normal distribution as $n \\rightarrow \\infty$.\n\nThis has very important implications for hypothesis testing and is precisely the reason why the t-distribution begins to approximate the normal distribution as our sample size increases. \n\n## Standard Error of the Mean\n\nWhat does it mean to \"estimate\"? the Population mean?\n\n\n```\nimport numpy as np\nimport pandas as pd\n\n# Average Height\nmu = 70\nsigma = 3\n\nlambda_heights = np.random.normal(mu, sigma, 2000)\nprint(len(lambda_heights))\nlambda_heights\n```\n\n 2000\n\n\n\n\n\n array([64.52980773, 75.16776072, 70.69565154, ..., 69.35265898,\n 72.82941193, 71.11879291])\n\n\n\n\n```\nimport seaborn as sns\n\nsns.distplot(lambda_heights)\nplt.title('Distribution of Heights (in inches)');\n```\n\n\n```\nprint(\"Population Mean:\", lambda_heights.mean())\nprint(\"Population Standard Deviation:\", lambda_heights.std())\n```\n\n Population Mean: 69.9296448140553\n Population Standard Deviation: 2.9611799110435335\n\n\n\n```\npopulation = pd.DataFrame({'heights': lambda_heights})\nprint(population.shape)\npopulation.head()\n```\n\n (2000, 1)\n\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    heights
    064.529808
    175.167761
    270.695652
    369.006209
    472.512828
    \n
    \n\n\n\n\n```\n# Take a random sample and print sample mean\n```\n\n\n```\n# Take a different random sample and print sample mean\n```\n\n## Build and Interpret a Confidence Interval\n\n\n\n\n```\n\n```\n\n### What confidence level do we want our confidence interval to represent?\n\n95% confidence Interval? 99% confidence interval? \n\n\n```\n\n```\n\n## Graphically Represent a Confidence Interval\n\n\n```\n\n```\n\n## Relationship between Confidence Intervals and T-tests\n\nConfidence Interval == Bounds of statistical significance for our t-test\n\nA sample mean that falls inside of our confidence interval will \"FAIL TO REJECT\" our null hypothesis\n\nA sample mean that falls outside of our confidence interval will \"REJECT\" our null hypothesis\n\n\n```\nfrom scipy.stats import t, ttest_1samp\n```\n\n\n```\nimport numpy as np\n\ncoinflip_means = []\nfor x in range(0,100):\n coinflips = np.random.binomial(n=1, p=.5, size=30)\n coinflip_means.append(coinflips.mean())\n\nprint(coinflip_means)\n```\n\n [0.5, 0.6, 0.3333333333333333, 0.43333333333333335, 0.5666666666666667, 0.5, 0.5, 0.43333333333333335, 0.6, 0.36666666666666664, 0.5, 0.6333333333333333, 0.4666666666666667, 0.5333333333333333, 0.5333333333333333, 0.4, 0.4666666666666667, 0.6, 0.5666666666666667, 0.5333333333333333, 0.4, 0.5, 0.5333333333333333, 0.5333333333333333, 0.36666666666666664, 0.6333333333333333, 0.3, 0.5, 0.5, 0.4666666666666667, 0.4666666666666667, 0.6, 0.6333333333333333, 0.5333333333333333, 0.36666666666666664, 0.5, 0.6, 0.5, 0.5, 0.4666666666666667, 0.5333333333333333, 0.43333333333333335, 0.36666666666666664, 0.6666666666666666, 0.5333333333333333, 0.5333333333333333, 0.5333333333333333, 0.5, 0.4, 0.4666666666666667, 0.43333333333333335, 0.5, 0.6, 0.4666666666666667, 0.5666666666666667, 0.4, 0.4666666666666667, 0.43333333333333335, 0.4666666666666667, 0.4666666666666667, 0.5333333333333333, 0.7, 0.6666666666666666, 0.5, 0.6333333333333333, 0.43333333333333335, 0.5, 0.36666666666666664, 0.5333333333333333, 0.43333333333333335, 0.4666666666666667, 0.36666666666666664, 0.4666666666666667, 0.5, 0.36666666666666664, 0.36666666666666664, 0.6333333333333333, 0.5333333333333333, 0.5, 0.6, 0.4666666666666667, 0.5666666666666667, 0.4666666666666667, 0.5666666666666667, 0.4666666666666667, 0.4666666666666667, 0.5, 0.5333333333333333, 0.6, 0.4, 0.43333333333333335, 0.4, 0.5666666666666667, 0.5, 0.5333333333333333, 0.8, 0.5333333333333333, 0.3333333333333333, 0.5, 0.43333333333333335]\n\n\nA null hypothesis that's just inside of our confidence interval == fail to reject\n\n\n\n\n```\n\n```\n\nA null hypothesis that's just outside of our confidence interval == reject\n\n\n\n\n```\n\n```\n\n\n```\ndef confidence_interval(data, confidence=0.95):\n \"\"\"\n Calculate a confidence interval around a sample mean for given data.\n Using t-distribution and two-tailed test, default 95% confidence. \n \n Arguments:\n data - iterable (list or numpy array) of sample observations\n confidence - level of confidence for the interval\n \n Returns:\n tuple of (mean, lower bound, upper bound)\n \"\"\"\n data = np.array(data)\n mean = np.mean(data)\n n = len(data)\n stderr = stats.sem(data)\n interval = stderr * stats.t.ppf((1 + confidence) / 2.0, n - 1)\n return (mean, mean - interval, mean + interval)\n```\n\n## Run a $\\chi^{2}$ Test \"by hand\" (Using Numpy)\n\n\n```\ndf = pd.read_csv('https://raw.githubusercontent.com/ryanleeallred/datasets/master/adult.csv', na_values=\" ?\")\nprint(df.shape)\ndf.head()\n```\n\n (32561, 15)\n\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    ageworkclassfnlwgteducationeducation-nummarital-statusoccupationrelationshipracesexcapital-gaincapital-losshours-per-weekcountrysalary
    039State-gov77516Bachelors13Never-marriedAdm-clericalNot-in-familyWhiteMale2174040United-States<=50K
    150Self-emp-not-inc83311Bachelors13Married-civ-spouseExec-managerialHusbandWhiteMale0013United-States<=50K
    238Private215646HS-grad9DivorcedHandlers-cleanersNot-in-familyWhiteMale0040United-States<=50K
    353Private23472111th7Married-civ-spouseHandlers-cleanersHusbandBlackMale0040United-States<=50K
    428Private338409Bachelors13Married-civ-spouseProf-specialtyWifeBlackFemale0040Cuba<=50K
    \n
    \n\n\n\n## Expected Value Calculation\n\\begin{align}\nexpected_{i,j} =\\frac{(row_{i} \\text{total})(column_{j} \\text{total}) }{(\\text{total observations})} \n\\end{align}\n\n\n```\n\n```\n\n## Chi-Squared Statistic with Numpy\n\n\\begin{align}\n\\chi^2 = \\sum \\frac{(observed_{i}-expected_{i})^2}{(expected_{i})}\n\\end{align}\n\nFor the $observed$ values we will just use a version of our contingency table without the margins as a numpy array. In this way, if our observed values array and our expected values array are the same shape, then we can subtract them and divide them directly which makes the calculations a lot cleaner. No for loops!\n\n\n```\n\n```\n\n## Run a $\\chi^{2}$ Test using Scipy\n\n\n```\n\n```\n\nNull Hypothesis: Hours worked per week bins is **independent** of sex. \n\nDue to a p-value of 0, we REJECT the null hypothesis that hours worked per week and sex are independent, and conclude that there is an association between hours worked per week and sex. \n\n## Assignment - Build a confidence interval\n\nA confidence interval refers to a neighborhood around some point estimate, the size of which is determined by the desired p-value. For instance, we might say that 52% of Americans prefer tacos to burritos, with a 95% confidence interval of +/- 5%.\n\n52% (0.52) is the point estimate, and +/- 5% (the interval $[0.47, 0.57]$) is the confidence interval. \"95% confidence\" means a p-value $\\leq 1 - 0.95 = 0.05$.\n\nIn this case, the confidence interval includes $0.5$ - which is the natural null hypothesis (that half of Americans prefer tacos and half burritos, thus there is no clear favorite). So in this case, we could use the confidence interval to report that we've failed to reject the null hypothesis.\n\nBut providing the full analysis with a confidence interval, including a graphical representation of it, can be a helpful and powerful way to tell your story. Done well, it is also more intuitive to a layperson than simply saying \"fail to reject the null hypothesis\" - it shows that in fact the data does *not* give a single clear result (the point estimate) but a whole range of possibilities.\n\nHow is a confidence interval built, and how should it be interpreted? It does *not* mean that 95% of the data lies in that interval - instead, the frequentist interpretation is \"if we were to repeat this experiment 100 times, we would expect the average result to lie in this interval ~95 times.\"\n\nFor a 95% confidence interval and a normal(-ish) distribution, you can simply remember that +/-2 standard deviations contains 95% of the probability mass, and so the 95% confidence interval based on a given sample is centered at the mean (point estimate) and has a range of +/- 2 (or technically 1.96) standard deviations.\n\nDifferent distributions/assumptions (90% confidence, 99% confidence) will require different math, but the overall process and interpretation (with a frequentist approach) will be the same.\n\nYour assignment - using the data from the prior module ([congressional voting records](https://archive.ics.uci.edu/ml/datasets/Congressional+Voting+Records)):\n\n\n### Confidence Intervals:\n1. Generate and numerically represent a confidence interval\n2. Graphically (with a plot) represent the confidence interval\n3. Interpret the confidence interval - what does it tell you about the data and its distribution?\n\n### Chi-squared tests:\n4. Take a dataset that we have used in the past in class that has **categorical** variables. Pick two of those categorical variables and run a chi-squared tests on that data\n - By hand using Numpy\n - In a single line using Scipy\n\nStretch goals:\n\n1. Write a summary of your findings, mixing prose and math/code/results. *Note* - yes, this is by definition a political topic. It is challenging but important to keep your writing voice *neutral* and stick to the facts of the data. Data science often involves considering controversial issues, so it's important to be sensitive about them (especially if you want to publish).\n2. Apply the techniques you learned today to your project data or other data of your choice, and write/discuss your findings here.\n3. Refactor your code so it is elegant, readable, and can be easily run for all issues.\n\n\n```\n# TODO - your code!\n```\n\n## Resources\n\n- [Interactive visualize the Chi-Squared test](https://homepage.divms.uiowa.edu/~mbognar/applets/chisq.html)\n- [Calculation of Chi-Squared test statistic](https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test)\n- [Visualization of a confidence interval generated by R code](https://commons.wikimedia.org/wiki/File:Confidence-interval.svg)\n- [Expected value of a squared standard normal](https://math.stackexchange.com/questions/264061/expected-value-calculation-for-squared-normal-distribution) (it's 1 - which is why the expected value of a Chi-Squared with $n$ degrees of freedom is $n$, as it's the sum of $n$ squared standard normals)\n", "meta": {"hexsha": "839a56a0c5fa44e030ca33b05125cd5415f04ce7", "size": 102746, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Copy_of_LS_DS6_132_Sampling_Confidence_Intervals_and_Hypothesis_Testing_updated.ipynb", "max_stars_repo_name": "ngriggs/DS-Unit-1-Sprint-3-Statistical-Tests-and-Experiments", "max_stars_repo_head_hexsha": "fd1eab8d8d86bdebce319bdaf18741a48241a5cc", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Copy_of_LS_DS6_132_Sampling_Confidence_Intervals_and_Hypothesis_Testing_updated.ipynb", "max_issues_repo_name": "ngriggs/DS-Unit-1-Sprint-3-Statistical-Tests-and-Experiments", "max_issues_repo_head_hexsha": "fd1eab8d8d86bdebce319bdaf18741a48241a5cc", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Copy_of_LS_DS6_132_Sampling_Confidence_Intervals_and_Hypothesis_Testing_updated.ipynb", "max_forks_repo_name": "ngriggs/DS-Unit-1-Sprint-3-Statistical-Tests-and-Experiments", "max_forks_repo_head_hexsha": "fd1eab8d8d86bdebce319bdaf18741a48241a5cc", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 89.8915135608, "max_line_length": 45946, "alphanum_fraction": 0.7064897904, "converted": true, "num_tokens": 30473, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.7981867825403177, "lm_q1q2_score": 0.44257099644039744}} {"text": "# Plotting with Matplotlib\n\n## Prepare for action\n\n\n```\nimport numpy as np\nimport scipy as sp\nimport sympy\n\n# Pylab combines the pyplot functionality (for plotting) with the numpy\n# functionality (for mathematics and for working with arrays) in a single namespace\n# aims to provide a closer MATLAB feel (the easy way). Note that his approach\n# should only be used when doing some interactive quick and dirty data inspection.\n# DO NOT USE THIS FOR SCRIPTS\n#from pylab import *\n\n# the convienient Matplotib plotting interface pyplot (the tidy/right way)\n# use this for building scripts. The examples here will all use pyplot.\nimport matplotlib.pyplot as plt\n\n# for using the matplotlib API directly (the hard and verbose way)\n# use this when building applications, and/or backends\nimport matplotlib as mpl\n```\n\nHow would you like the IPython notebook show your plots? In order to use the\nmatplotlib IPython magic youre IPython notebook should be launched as\n\n ipython notebook --matplotlib=inline\n\nMake plots appear as a pop up window, chose the backend: 'gtk', 'inline', 'osx', 'qt', 'qt4', 'tk', 'wx'\n \n %matplotlib qt\n \nor inline the notebook (no panning, zooming through the plot). Not working in IPython 0.x\n \n %matplotib inline\n \n\n\n```\n# activate pop up plots\n#%matplotlib qt\n# or change to inline plots\n%matplotlib inline\n```\n\n### Matplotlib documentation\n\nFinding your own way (aka RTFM). Hint: there is search box available!\n\n* http://matplotlib.org/contents.html\n\nThe Matplotlib API docs:\n\n* http://matplotlib.org/api/index.html\n\nPyplot, object oriented plotting:\n\n* http://matplotlib.org/api/pyplot_api.html\n* http://matplotlib.org/api/pyplot_summary.html\n\nExtensive gallery with examples:\n\n* http://matplotlib.org/gallery.html\n\n### Tutorials for those who want to start playing\n\nIf reading manuals is too much for you, there is a very good tutorial available here:\n\n* http://nbviewer.ipython.org/github/jrjohansson/scientific-python-lectures/blob/master/Lecture-4-Matplotlib.ipynb\n\nNote that this tutorial uses\n\n from pylab import *\n\nwhich is usually not adviced in more advanced script environments. When using\n \n import matplotlib.pyplot as plt\n\nyou need to preceed all plotting commands as used in the above tutorial with\n \n plt.\n\n\nGive me more!\n\n[EuroScipy 2012 Matlotlib tutorial](http://www.loria.fr/~rougier/teaching/matplotlib/). Note that here the author uses ```from pylab import * ```. When using ```import matplotliblib.pyplot as plt``` the plotting commands need to be proceeded with ```plt.```\n\n\n## Plotting template starting point\n\n\n```\n# some sample data\nx = np.arange(-10,10,0.1)\n```\n\nTo change the default plot configuration values.\n\n\n```\npage_width_cm = 13\ndpi = 200\ninch = 2.54 # inch in cm\n# setting global plot configuration using the RC configuration style\nplt.rc('font', family='serif')\nplt.rc('xtick', labelsize=12) # tick labels\nplt.rc('ytick', labelsize=20) # tick labels\nplt.rc('axes', labelsize=20) # axes labels\n# If you don\u2019t need LaTeX, don\u2019t use it. It is slower to plot, and text\n# looks just fine without. If you need it, e.g. for symbols, then use it.\n#plt.rc('text', usetex=True) #<- P-E: Doesn't work on my Mac\n```\n\n\n```\n# create a figure instance, note that figure size is given in inches!\nfig, ax = plt.subplots(nrows=1, ncols=1, figsize=(8,6))\n# set the big title (note aligment relative to figure)\nfig.suptitle(\"suptitle 16, figure alignment\", fontsize=16)\n\n# actual plotting\nax.plot(x, x**2, label=\"label 12\")\n\n\n# set axes title (note aligment relative to axes)\nax.set_title(\"title 14, axes alignment\", fontsize=14)\n\n# axes labels\nax.set_xlabel('xlabel 12')\nax.set_ylabel(r'$y_{\\alpha}$ 12', fontsize=8)\n\n# legend\nax.legend(fontsize=12, loc=\"best\")\n\n# saving the figure in different formats\nfig.savefig('figure-%03i.png' % dpi, dpi=dpi)\nfig.savefig('figure.svg')\nfig.savefig('figure.eps')\n```\n\n\n```\n# following steps are only relevant when using figures as pop up windows (with %matplotlib qt)\n# to update a figure with has been modified\nfig.canvas.draw()\n# show a figure\nfig.show()\n```\n\n C:\\Anaconda\\lib\\site-packages\\matplotlib\\figure.py:371: UserWarning: matplotlib is currently using a non-GUI backend, so cannot show the figure\n \"matplotlib is currently using a non-GUI backend, \"\n\n\n## Exercise\n\nThe current section is about you trying to figure out how to do several plotting features. You should use the previously mentioned resources to find how to do that. In many cases, google is your friend!\n\n* add a grid to the plot\n\n\n\n```\nplt.plot(x,x**2)\n#Write code to show grid in plot here\naxes = plt.gca() \naxes.grid(True)\n```\n\n* change the location of the legend to different places\n\n\n\n```\nplt.plot(x,x**2, label=\"label 12\")\nplt.legend(fontsize=12, loc=4)\n```\n\n* find a way to control the line type and color, marker type and color, control the frequency of the marks (`markevery`). See plot options at: http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot \n\n\n\n```\nplt.plot(x,x**2, 'o-', markevery=5, linewidth = 2, markerfacecolor='r')\n```\n\n* add different sub-plots\n\n\n\n```\nfig, ax = plt.subplots(nrows=2, ncols=1)\nax[0].plot(x,x**2)\nax[1].plot(x, x**3)\n```\n\n* size the figure such that when included on an A4 page the fonts are given in their true size\n\n\n\n```\n\n```\n\n* make a contour plot\n\n\n\n```\nX, Y = np.meshgrid(x,x)\nZ = X**2+Y**2\n\ncnt = plt.contour(Z, vmin=abs(Z).min(), vmax=abs(Z).max(), extent=[0, 1, 0, 1])\n\n```\n\n* use twinx() to create a second axis on the right for the second plot\n\n\n\n```\nfig, ax1 = plt.subplots()\nax1.plot(x,x**2)\nax2 = ax1.twinx()\nax2.plot(x,x**4, 'r')\n\nfor label in ax2.get_yticklabels():\n label.set_color(\"red\")\n```\n\n* add horizontal and vertical lines using axvline(), axhline()\n\n\n\n```\nplt.plot(x,x**2)\nplt.axvline(x=0, ymin=0, ymax=1)\nplt.axhline(y=50, xmin=-10, xmax=10)\n```\n\n* autoformat dates for nice printing on the x-axis using fig.autofmt_xdate()\n\n\n```\nimport datetime\ndates = np.array([datetime.datetime.now() + datetime.timedelta(days=i) for i in xrange(24)])\nfig, ax = plt.subplots(nrows=1, ncols=1)\nax.bar(dates, np.random.rand(24))\nfig.autofmt_xdate(bottom=0.2, rotation=30, ha='right')\n```\n\n## Advanced exercises\n\nWe are going to play a bit with regression\n\n* Create a vector x of equally spaced number between $x \\in [0, 5\\pi]$ of 1000 points (keyword: linspace)\n\n\n```\nx = np.linspace(0, 5*np.pi, 1000)\n```\n\n* create a vector y, so that y=sin(x) with some random noise\n\n\n```\ny = np.sin(x)+(np.random.random(1000)-0.5)\n```\n\n* plot it like this: \n\n\n```\nfig, ax = plt.subplots(1, 1)\nax.plot(x, y, '.')\nax.plot(x, np.sin(x), 'b--', linewidth=3, label='y=sin(x)')\n\nax.legend(loc =1)\n```\n\nTry to do a polynomial fit on y(x) with different polynomial degree (Use numpy.polyfit to obtain coefficients)\n\nPlot it like this (use np.poly1d(coef)(x) to plot polynomials) \n\n\n\n```\nfig, ax = plt.subplots(1, 1)\nax.plot(x, y, '.')\nax.plot(x, np.sin(x), 'b--', linewidth=3, label='y=sin(x)')\n\nfor i in range(0, 10):\n coeff = np.polyfit(x, y, i)\n ax.plot(x, np.polyval(coeff, x), label='deg=%i'%i)\n\nax.legend(loc=7, bbox_to_anchor=(1.32, 0.5))\n```\n\n\n```\n\n```\n", "meta": {"hexsha": "ec9f6249f38f080a9374114d3396f1ac5a6a2cf2", "size": 359638, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lesson 3/results/tlbl.ipynb", "max_stars_repo_name": "gtpedrosa/Python4WindEnergy", "max_stars_repo_head_hexsha": "f8ad09018420cfb3a419173f97b129de7118d814", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 48, "max_stars_repo_stars_event_min_datetime": "2015-01-19T18:21:10.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-27T22:41:06.000Z", "max_issues_repo_path": "lesson 3/results/tlbl.ipynb", "max_issues_repo_name": "arash7444/Python4WindEnergy", "max_issues_repo_head_hexsha": "8f97a5f86e81ce01d80dafb6f8104165fd3ad397", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2016-05-24T06:07:07.000Z", "max_issues_repo_issues_event_max_datetime": "2016-05-24T08:26:29.000Z", "max_forks_repo_path": "lesson 3/results/tlbl.ipynb", "max_forks_repo_name": "arash7444/Python4WindEnergy", "max_forks_repo_head_hexsha": "8f97a5f86e81ce01d80dafb6f8104165fd3ad397", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 24, "max_forks_repo_forks_event_min_datetime": "2015-06-26T14:44:07.000Z", "max_forks_repo_forks_event_max_datetime": "2021-06-07T18:36:52.000Z", "avg_line_length": 502.2877094972, "max_line_length": 77779, "alphanum_fraction": 0.9322401971, "converted": true, "num_tokens": 1965, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.7981867825403177, "lm_q1q2_score": 0.44257099644039744}} {"text": "
    \n \n
    \n\n# Getting started with Python and Jupyter\n*Roberto Di Remigio*, *Luca Frediani*\n\nDuring the course we will use [Python] to solve some of the exercises and explore concepts in quantum mechanics.\nInstead of just using the command line, we will exploit [Jupyter notebooks].\nYou will be able to look at the exercises and run them on your browser. We recommend you install Python on your machine anyway: we think you will find knowledge of Python useful also in the future.\nThe easiest way to get a reasonably complete Python distribution working is *via* [Anaconda], detailed instructions for Windows, Max OS X and Linux are [here].\n\n[Python]: https://en.wikipedia.org/wiki/Python_(programming_language)\n[Jupyter notebooks]: https://jupyter.readthedocs.org\n[Anaconda]: https://www.continuum.io/why-anaconda\n[here]: http://docs.continuum.io/anaconda/install\n\n## Why Python?\n\nPython is an interpreted scripting language that has become more and more popular in the past years.\nIt can be used (and it **is used**) to do pretty much anything. It can be your pocket calculator, you can use it to [plot data], [calculate derivatives and integrals], perform tasks in [linear algebra] and much, much more!\n\n[plot data]: http://matplotlib.org/\n[calculate derivatives and integrals]: http://www.sympy.org/en/index.html\n[linear algebra]: http://www.numpy.org/\n\nThere are a lot of resources out there to learn Python.\n[Mark Bakker]'s collection of Jupyter notebooks is really great. It covers a lot of ground, plus you have screencasts for the notebooks covering the basics of Python.\nWe also recommend the book [Learning IPython for Interactive Computing and Data Visualization] by Cyrille Roussant.\nThe ebook is available from the University's library. You will have to be on campus to access it *via* [this link],\nclick the __Fulltekst tilgjengelig p\u00e5: ebrary Academic Complete Non-US__ link to go to the ebook.\nFinally, to learn more about how to fully exploit a Jupyter notebook we recommend you read the tutorials in this [page]\n\n[Mark Bakker]: http://mbakker7.github.io/exploratory_computing_with_python/\n[Learning IPython for Interactive Computing and Data Visualization]: http://ipython-books.github.io/minibook/\n[this link]: http://bibsys-almaprimo.hosted.exlibrisgroup.com/primo_library/libweb/action/display.do?tabs=viewOnlineTab&ct=display&fn=search&doc=BIBSYS_ILS71524751750002201&indx=2&recIds=BIBSYS_ILS71524751750002201&recIdxs=1&elementId=1&renderMode=poppedOut&displayMode=full&frbrVersion=9&frbg=&&dscnt=0&scp.scps=scope%3A%28SC_OPEN_ACCESS%29%2Cscope%3A%28%22UBTO%22%29%2Cprimo_central_multiple_fe&tb=t&mode=Basic&vid=UBTO&srt=rank&tab=default_tab&dum=true&vl(freeText0)=ipython&dstmp=1454313553096\n[page]: https://www.datacamp.com/community/tutorials/tutorial-jupyter-notebook#gs.svdmCKs\n\n## First steps\n\nLet's start off by using Python as a calculator. There are different types of **cells** in Jupyter notebooks. We will be mostly using **code cells** to write and execute code and **Markdown cells** to write comments to the code.\nA Markdown cell is just text. You can add images and also equations, using $\\LaTeX$:\n\n\\begin{equation}\n f(x) = ax^2 + bx + c\n\\end{equation}\n\nCode cells contain Python code. Each code cell has an input and an output section. The input section contains code you write, the ouput section contains the results of executing your code.\nMove your cursor to the code cell right below this Markdown cell and press [shift][Enter].\nThis triggers evaluation of the cell and you will see the output of the computation right below.\n\n\n```python\n6 * 2\n```\n\n\n\n\n 12\n\n\n\nWhen programming, you can (and should) use variables to hold values, so that you can reference them later in your code. Pressing [shift][Enter] will always trigger evaluation of a code cell and print out the results.\n\n\n```python\na = 6\nb = 2\na * b\n```\n\n\n\n\n 12\n\n\n\nPython provides the `print` function to write output to the screen.\n\n\n```python\na = 6\nb = 2\nc = a * b\nprint(a)\nprint(b)\nprint(c)\n```\n\n 6\n 2\n 12\n\n\nOf course, we can add more information to a print statement! A line starting with `#` is a comment, the intepreter will just ignore it. It is useful to add comments, to remind both yourself and others what you were trying to do when writing the code ;)\n\n\n```python\na = 6 # Assign value to a\nb = 2 # Assign value to b\nc = a * b # Perform operation\n# Now print everything out!\nprint('a =', a)\nprint('b =', b)\nprint('c =', c)\n```\n\n a = 6\n b = 2\n c = 12\n\n\n## Python modules: basic plotting with `matplotlib`\n\nWhen you first install Python and launch an instance of the interpreter (the program running the commands) you cannot do very much. Luckily, Python was designed to be highly extensible using independent packages called **modules**. Much of the functionality that makes Python appealing is provided *via* modules.\nA module is a collection of functions designed to perform a very specific set of tasks.\n\nFor example, the [`matplotlib`] module provides the functions needed to perform plotting in Python.\nTo use it we need to **import** it. There are many different ways to import a module, for the moment we import only the plottig part of `matplotlib` and call it `plt`. The second command ensures that the plots are generated in this Notebook and not in a separate window.\n\n[`matplotlib`]: http://matplotlib.org/\n\n\n```python\nimport matplotlib.pyplot as plt\n# make sure we see it on this notebook\n%matplotlib inline\n```\n\nModules only have to be imported once in a Jupyter session. After the import, any plotting function may be called from any code cell as `plt.function`. In the following we are plotting the data table:\n\n x | y\n---|---\n 0 | 1 \n 1 | 2 \n 2 | 3 \n 3 | 5 \n 4 | 2 \n 5 | 6 \n\n\n```python\nimport matplotlib.pyplot as plt\n# make sure we see it on this notebook\n%matplotlib inline\nplt.plot([1, 2, 3, 5, 2, 6])\nplt.show()\n```\n\n## Python modules: arrays with `numpy`\n\nOK, let's plot something more interesting than just a bunch of points.\nThe function we want to plot is:\n\\begin{equation}\n f(x) = 10x^3\\mathrm{e}^{-x^2} + \\frac{\\sin(x^5)}{\\cos(x^3)}\n\\end{equation}\nHow do we do it? We could of course create a table of $x$,$y$ pairs and feed it to the `plt.plot` function as in the example above, but that's too tedious.\nThe module [`numpy`] comes to the rescue. First of all, we import `numpy` and give it the name `np`:\n\n[`numpy`]: http://www.numpy.org/\n\n\n```python\nimport numpy as np\n```\n\nNext, we create an **array** of, for example, 10 equally spaced points between -4 and +4 with the `linspace` command. This array will contain the values of the `x` variables where we will evaluate the function $f(x)$.\n\n\n```python\nx = np.linspace(-4, 4, 10)\nprint(x)\n```\n\n [-4. -3.11111111 -2.22222222 -1.33333333 -0.44444444 0.44444444\n 1.33333333 2.22222222 3.11111111 4. ]\n\n\nThe next step is to evaluate the function at all the $x$ points listed in the `x` array and save the results in another array, which we call `y`:\n\n\n```python\ny = 10 * x**3 * np.exp(-x**2) + np.sin(x**5) / np.cos(x**3)\nprint(y)\n```\n\n [ 0.40449721 -2.48343478 -33.46262352 -5.23117497 -0.73796062\n 0.73796062 5.23117497 33.46262352 2.48343478 -0.40449721]\n\n\nNotice that:\n1. Raising to power has the syntax `x**3` and **not** `x^3`\n2. The exponential, sine and cosine are included in `numpy` and you have to invoke them as `np.exp`, `np.sin` and `np.cos` since you are giving `numpy` arrays as arguments.\nWe are now ready to plot! \n\n\n```python\nimport matplotlib.pyplot as plt\n# make sure we see it on this notebook\n%matplotlib inline\nplt.plot(x, y)\n```\n\nDone! OK, it doesn't look that great. That's because we are using only 10 sampling points. Try changing the number of points generated by `np.linspace` from 10 to another, higher value. Re-run all the cells and see what changes.\n\nWe can also apply other changes. First of all, we can define a Python function to calculate $f(x)$. We can then reuse this function wherever we want instead of typing it down everytime!\nA Python function:\n1. Starts with the `def` statement\n2. Followed by the name of the function\n3. In parentheses, a comma-separated list of **function arguments**. In this case the `numpy` array with the $x$ points.\n4. A colon\n5. The body of the function\nOur function first calculates $f(x)$ and stores the values in the `y` array, then **returns** the array.\n\n\n```python\ndef my_first_function(x):\n import numpy as np\n y = 10 * x**3 * np.exp(-x**2) + np.sin(x**5) / np.cos(x**3)\n return y\n```\n\n\n```python\nfunction = my_first_function(x)\nprint(function)\n```\n\n [ 0.40449721 -2.48343478 -33.46262352 -5.23117497 -0.73796062\n 0.73796062 5.23117497 33.46262352 2.48343478 -0.40449721]\n\n\nAs you can see, calling the function gives the same results as calculated above.\nAs further improvements we will add a legend to the plot, labels to the $x$ and $y$ axes and a title. Python functions in `matplotlib` and `numpy` take many arguments. Pressing [shift][Tab] when typing in a code cell will open an help box with a lot of useful information on the function arguments.\n\n\n```python\nplt.plot(x, my_first_function(x), 'r--')\n# Write label on x axis\nplt.xlabel('x-axis')\n# Write label on y axis\nplt.ylabel('y-axis')\n# Write title\nplt.title('First Python Figure')\n# Add legend\nplt.legend('f')\n```\n", "meta": {"hexsha": "cf98d1a31768d67523a22441e000f77ccaf39104", "size": 48920, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "00_getting_started.ipynb", "max_stars_repo_name": "ilfreddy/seminars", "max_stars_repo_head_hexsha": "c7e13874b41cc906a45b672e5b85c57d6880473e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2017-02-04T01:34:33.000Z", "max_stars_repo_stars_event_max_datetime": "2021-06-12T12:27:37.000Z", "max_issues_repo_path": "00_getting_started.ipynb", "max_issues_repo_name": "ilfreddy/seminars", "max_issues_repo_head_hexsha": "c7e13874b41cc906a45b672e5b85c57d6880473e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2020-03-30T11:00:35.000Z", "max_issues_repo_issues_event_max_datetime": "2020-05-12T05:42:24.000Z", "max_forks_repo_path": "00_getting_started.ipynb", "max_forks_repo_name": "ilfreddy/seminars", "max_forks_repo_head_hexsha": "c7e13874b41cc906a45b672e5b85c57d6880473e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 7, "max_forks_repo_forks_event_min_datetime": "2016-04-26T20:42:43.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-06T11:12:57.000Z", "avg_line_length": 86.891651865, "max_line_length": 13278, "alphanum_fraction": 0.8281275552, "converted": true, "num_tokens": 2574, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.7981867777396212, "lm_q1q2_score": 0.442570993778553}} {"text": "# KVLCC2 Ikeda method\n\n# Purpose\nHow good is original Ikeda method for this ship?\n\n# Methodology\nRun PyScoresII and calculate Ikeda\n\n# Setup\n\n\n```python\n# %load imports.py\n\"\"\"\nThese is the standard setup for the notebooks.\n\"\"\"\n\n%matplotlib inline\n%load_ext autoreload\n%autoreload 2\n\nimport sys\nsys.path.append(\"../../\")\n\nimport pandas as pd\npd.options.display.max_rows = 999\npd.options.display.max_columns = 999\npd.set_option(\"display.max_columns\", None)\nimport numpy as np\nimport os\nimport matplotlib.pyplot as plt\nfrom collections import OrderedDict\nimport copy\nfrom sklearn.pipeline import Pipeline\nfrom rolldecayestimators.transformers import CutTransformer, LowpassFilterDerivatorTransformer, ScaleFactorTransformer, OffsetTransformer\nfrom rolldecayestimators.direct_estimator_cubic import EstimatorQuadraticB, EstimatorCubic\nfrom rolldecayestimators.ikeda_estimator import IkedaQuadraticEstimator\nimport src.equations as equations\nimport rolldecayestimators.lambdas as lambdas\nfrom rolldecayestimators.substitute_dynamic_symbols import lambdify\nimport rolldecayestimators.symbols as symbols\nimport sympy as sp\n\nfrom sympy.physics.vector.printing import vpprint, vlatex\nfrom IPython.display import display, Math, Latex\n\nfrom sklearn.metrics import r2_score\nimport shipflowmotionshelpers.shipflowmotionshelpers as helpers\nimport src.visualization.visualize as visualize\nimport scipy\nfrom copy import deepcopy\nimport joblib\n```\n\n Duplicate key in file WindowsPath('C:/Users/maa/.matplotlib/stylelib/paper.mplstyle'), line 462 ('figure.figsize : 5, 3 ## figure size in inches')\n Duplicate key in file WindowsPath('C:/Users/maa/.matplotlib/stylelib/paper.mplstyle'), line 463 ('figure.dpi : 100 ## figure dots per inch')\n\n\n\n```python\nimport pyscores2\nimport pyscores2.runScores2\nimport pyscores2.xml_hydrostatics\nfrom pyscores2.output import OutputFile\nfrom rolldecayestimators.ikeda import Ikeda, IkedaR\n\nfrom rolldecayestimators.simplified_ikeda_class import SimplifiedIkeda, SimplifiedIkedaABS\nfrom rolldecayestimators.simplified_ikeda import limits_kawahara\nfrom pyscores2.runScores2 import Calculation\nimport shutil\nfrom reports import mdl_results\n```\n\n\n```python\ndf_rolldecays=mdl_results.df_rolldecays\n```\n\n\n```python\nrow = df_rolldecays.iloc[2]\n```\n\n## Run ScoresII\n\n\n```python\nxml_parser = pyscores2.xml_hydrostatics.Parser(fileName='../../data/external/KVLCC2m_kbk_final_ScoresData.xml')\nindata = xml_parser.convertToScores2Indata(conditionName='Design')\nindata.runOptions[\"IJ\"].set_value(1)\nindata.runOptions[\"IK\"].set_value(2)\n```\n\n\n```python\nindata.kxx = row.KXX*0.78 # To get correct natural frequency\nindata.kyy = row.KZZ\nindata.speedMax=15.5\nindata.speedIncrement=15\nindata.waveFrequenciesMax = 1.0\nindata.waveFrequenciesMin = 0.3\nindata.waveFrequenciesIncrement = 0.015\n#indata.zcg = run.loading_condition.kg\n```\n\n\n```python\nindata.save('../../models/KVLCC2_speed.IN')\n```\n\n\n```python\nsave_dir_name = 'scores'\nif not os.path.exists(save_dir_name):\n os.mkdir(save_dir_name)\n \ncalculation = Calculation(outDataDirectory='scores')\n```\n\n\n```python\nif os.name == 'nt':\n calculation.run(indata=indata) # (This only works on windows)\n shutil.copyfile(calculation.outDataPath,'../../data/interim/KVLCC2_speed.out')\nelse:\n calculation.outDataPath = '../../data/interim/KVLCC2_speed.out'\n```\n\n Running Scores2 for KVLCC2m_kbk_final Design\n\n\n C:\\python36-64\\lib\\re.py:212: FutureWarning: split() requires a non-empty pattern match.\n return _compile(pattern, flags).split(string, maxsplit)\n\n\n## Load ScoresII results\n\n\n```python\noutput_file = OutputFile(filePath=calculation.outDataPath)\noutput_file.results\n```\n\n\n\n\n {0.0: {0.0: ,\n 30.0: ,\n 60.0: ,\n 90.0: ,\n 120.0: ,\n 150.0: ,\n 180.0: },\n 15.0: {0.0: ,\n 30.0: ,\n 60.0: ,\n 90.0: ,\n 120.0: ,\n 150.0: ,\n 180.0: }}\n\n\n\n\n```python\ndf = output_file.get_result()\n```\n\n\n```python\ndf.describe()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    frequenciesencounterFrequencieswaveLengthsheaveAmplitudeheavePhasepitchAmplitudepitchPhasesurgeAmplitudesurgePhaseforcesmomentsspeedwave directionswayAmplitudeswayPhaseyawAmplitudeyawPhaserollAmplituderollPhase
    count658.000000658.000000658.000000658.000000658.000000658.000000658.000000658.000000658.000000658.000000658.000000658.000000658.000000470.000000470.000000470.000000470.000000470.000000470.000000
    mean0.4680330.468033368.6700000.552034103.7357140.298860-14.2442250.2341764.96945341.159458198.4570577.50000090.0000000.34720964.3778720.14129750.9772340.96208641.333617
    std0.1677760.214466181.2259730.377948105.1128460.19835085.1814210.23536191.02052743.8755151409.3879367.50570660.0456450.25989660.2041300.099550103.0942351.634676100.136149
    min0.3019700.21313061.6200000.000200-179.8000000.001000-176.4000000.000000-175.4000000.990810-4040.0000000.0000000.0000000.000900-166.5000000.000500-179.1000000.000400-179.600000
    25%0.3413300.328962208.4700000.15885060.7250000.082275-85.1750000.021250-87.6000006.707588-168.7500000.00000030.0000000.08580079.6750000.030850-10.9000000.117025-57.725000
    50%0.4088200.398135368.6700000.602900163.5500000.344500-49.1500000.19095032.10000026.2417500.0000007.50000090.0000000.30670089.5500000.1567503.6000000.43780079.750000
    75%0.5436600.533963528.8700000.902550174.6750000.46535068.1750000.36460090.70000063.642375490.50000015.000000150.0000000.58732590.9000000.218425172.5750001.039675115.825000
    max0.9999801.786820675.7200001.191500179.4000000.650000176.7000001.047400166.100000200.3692507580.00000015.000000180.0000000.833100171.2000000.351600179.50000015.787100177.600000
    \n
    \n\n\n\n\n```python\ndf.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    frequenciesencounterFrequencieswaveLengthsheaveAmplitudeheavePhasepitchAmplitudepitchPhasesurgeAmplitudesurgePhaseforcesmomentsspeedwave directionswayAmplitudeswayPhaseyawAmplitudeyawPhaserollAmplituderollPhase
    00.999980.9999861.620.0035-115.20.0041-116.60.0032-31.628.449000.00.00.0NaNNaNNaNNaNNaNNaN
    10.906580.9065874.970.007561.10.009064.80.0036110.232.863500.00.00.0NaNNaNNaNNaNNaNNaN
    20.835260.8352688.320.0116-76.10.0207-51.10.0125136.238.749500.00.00.0NaNNaNNaNNaNNaNNaN
    30.778490.77849101.670.0213-140.80.0327-151.50.0140-72.244.880750.00.00.0NaNNaNNaNNaNNaNNaN
    40.731920.73192115.020.0272137.90.0636159.50.0326-64.253.955000.00.00.0NaNNaNNaNNaNNaNNaN
    \n
    \n\n\n\n\n```python\ndf[r'lambda/lpp'] = df['waveLengths']/row.lpp \n\n\nfig,ax=plt.subplots()\nfor index, group in df.groupby(by=['speed','wave direction']):\n group.plot(x=r'lambda/lpp', y='heaveAmplitude', style='o-', label=index, ax=ax)\n \nax.grid(True)\nax.legend();\nax.set_ylabel('Heave');\n```\n\n\n```python\nRAO_15_0 = df.groupby(by=['speed','wave direction']).get_group((15,180))\n```\n\n\n```python\nfig,ax=plt.subplots()\nRAO_15_0.plot(x=r'lambda/lpp', y='heaveAmplitude', style='o-', ax=ax)\nax.set_xlim((0,1.8))\n\nfig,ax=plt.subplots()\nRAO_15_0.plot(x=r'lambda/lpp', y='pitchAmplitude', style='o-', ax=ax)\n#ax.set_xlim((0,1.8))\n```\n\n\n```python\ndf_roll_damping = output_file.get_roll_damping()\ndf_roll_damping\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    calculated_wave_damping_in_rollcritical_wave_damping_in_rollnatural_roll_frequencyroll_damping_ratiospeedwave_direction
    09054.07738000.00.298380.0261500.00.0
    19054.07738000.00.298380.0371800.030.0
    29054.07738000.00.298380.0414600.060.0
    39054.07738000.00.298380.0371800.090.0
    49054.07738000.00.298380.0261200.0120.0
    5NaNNaNNaNNaN0.0150.0
    6NaNNaNNaNNaN0.0180.0
    79054.07738000.00.298380.00432215.00.0
    89054.07738000.00.298380.01269015.030.0
    99054.07738000.00.298380.04164015.060.0
    109054.07738000.00.298380.01343015.090.0
    119054.07738000.00.298380.00612915.0120.0
    12NaNNaNNaNNaN15.0150.0
    13NaNNaNNaNNaN15.0180.0
    \n
    \n\n\n\n## Run Ikeda\n\n\n```python\nw = 2.462149630662348\n\nscale_factor=row.scale_factor\nV = row.ship_speed*1.852/3.6/np.sqrt(scale_factor)\n\nif not row.BKL:\n BKL=0\nelse:\n BKL=row.BKL/scale_factor\n\nif not row.BKB:\n BKB = 0\nelse:\n BKB=row.BKB/scale_factor\n\nkg = row.kg/scale_factor\n \n#fi_as = np.deg2rad([1,10])\nfi_as = np.deg2rad(10)\n\nikeda = Ikeda.load_scoresII(V=V, w=w, fi_a=fi_as, indata=indata, output_file=output_file, \n scale_factor=scale_factor, BKL=BKL, BKB=BKB, kg=kg)\n\nR = 0.05*row.beam/scale_factor # Just guessing...\nikeda.R = R\n```\n\n\n```python\ndef calculate_ikeda(ikeda):\n\n output = pd.DataFrame()\n output['B_44_hat'] = ikeda.calculate_B44()\n output['B_W0_hat'] = float(ikeda.calculate_B_W0())\n output['B_W_hat'] = float(ikeda.calculate_B_W())\n output['B_F_hat'] = ikeda.calculate_B_F()\n output['B_E_hat'] = ikeda.calculate_B_E()\n output['B_BK_hat'] = ikeda.calculate_B_BK()\n output['B_L_hat'] = float(ikeda.calculate_B_L())\n output['Bw_div_Bw0'] = float(ikeda.calculate_Bw_div_Bw0())\n return output\n```\n\n\n```python\nresult_datas = calculate_ikeda(ikeda) # DataFrame with two roll amplitudes\n```\n\n c:\\dev\\prediction-of-roll-damping-using-fully-nonlinear-potential-flow-and-ikedas-method\\reports\\venv\\lib\\site-packages\\rolldecayestimators\\ikeda_speed.py:595: RuntimeWarning: invalid value encountered in sqrt\n gamma=sqrt(pi)*f3*(rmax+2*M/H*sqrt(B0**2*A0**2))/((2*Ts*(1-OG/Ts)*sqrt(H0_prim*sigma_prim)))\n\n\n\n```python\nresult_datas\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    B_44_hatB_W0_hatB_W_hatB_F_hatB_E_hatB_BK_hatB_L_hatBw_div_Bw0
    00.0022640.0000640.0003030.0004980.0001180.00.0013454.754679
    \n
    \n\n\n\n## Simplified Ikeda also...\n\n\n```python\nlpp = row.lpp/scale_factor\nbeam = row.beam/scale_factor\nkg = row.kg/scale_factor\nvolume = row.Volume/(scale_factor**3)\ndraught = (row.TA + row.TF)/2/scale_factor\nA0 = row.A0\n\nif not row.BKL:\n BKL=0\nelse:\n BKL = row.BKL\n\nif not row.BKB:\n BKB = 0\nelse:\n BKB = row.BKB\n\nsi = SimplifiedIkeda(V=V, w=w, fi_a=fi_as, beam=beam, lpp=lpp, kg = kg, volume=volume, draught=draught, A0=A0, BKL=BKL, BKB=BKB)\n```\n\n\n```python\ndef calculate_SI(si):\n \n output = pd.DataFrame()\n output['B_44_hat'] = si.calculate_B44()\n output['B_W0_hat'] =si.calculate_B_W0()\n output['B_W_hat'] =si.calculate_B_W()\n output['B_F_hat'] =si.calculate_B_F()\n output['B_E_hat'] =si.calculate_B_E()\n output['B_BK_hat'] =si.calculate_B_BK()\n output['B_L_hat'] =si.calculate_B_L()\n output['Bw_div_Bw0'] =si.calculate_Bw_div_Bw0()\n \n return output\n```\n\n\n```python\nresult_datas_SI = calculate_SI(si=si)\n```\n\n\n```python\nresult_datas_SI\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    B_44_hatB_W0_hatB_W_hatB_F_hatB_E_hatB_BK_hatB_L_hatBw_div_Bw0
    0NaN0.0007310.0034760.0002960.001962NaN0.001374.754679
    \n
    \n\n\n\n\n```python\ndf_results = pd.DataFrame(columns=result_datas.columns)\ndf_results.loc['ikeda']=result_datas.iloc[0]\ndf_results.loc['SI']=result_datas_SI.iloc[0]\n\n```\n\n\n\n\n```python\ninteresting = ['B_W_hat','B_F_hat','B_E_hat']\ndf_results[interesting].plot(kind='bar',stacked=True)\n```\n\n\n```python\nlimits_kawahara\n```\n\n\n\n\n {'CB': (0.5, 0.85),\n 'B/d': (2.5, 4.5),\n 'OG/d': (-1.5, 0.2),\n 'CMID': (0.9, 0.99),\n 'bBk/B': (0.01, 0.06),\n 'lBk/LPP': (0.05, 0.4),\n 'OMEGA_hat': (0, 1.0)}\n\n\n\n\n```python\ndf_limits = pd.DataFrame(data = limits_kawahara, index = ['min','max']).transpose()\ndf_limits\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    minmax
    CB0.500.85
    B/d2.504.50
    OG/d-1.500.20
    CMID0.900.99
    bBk/B0.010.06
    lBk/LPP0.050.40
    OMEGA_hat0.001.00
    \n
    \n\n\n\n\n```python\ng=9.81\nomega_hat = lambdas.omega_hat(beam=beam, g=g, omega0=w)\n```\n\n\n```python\nCb = volume/(lpp*beam*draught)\nOG = draught-kg\nship_limits = {\n 'CB': Cb,\n 'B/d': beam/draught,\n 'OG/d': OG/draught,\n 'CMID': A0,\n 'bBk/B': BKB/beam,\n 'lBk/LPP': BKL/beam,\n 'OMEGA_hat': omega_hat}\nship_limits = pd.Series(ship_limits,name='ship')\n```\n\n\n```python\ndf_limits['ship'] = ship_limits\n```\n\n\n```python\ndf_limits_clean = df_limits.copy()\nif df_limits.loc['bBk/B','ship']==0:\n df_limits_clean.drop('bBk/B', inplace=True)\n \nif df_limits.loc['lBk/LPP','ship']==0:\n df_limits_clean.drop('lBk/LPP', inplace=True)\n```\n\n\n```python\nfig,ax=plt.subplots()\nax.errorbar(df_limits_clean.index,df_limits_clean['ship'],yerr=[df_limits_clean['ship']-df_limits_clean['min'],df_limits_clean['max']-df_limits_clean['ship']], \n fmt='ok', lw=1, ecolor='gray', capsize=20)\nax.set_title('Ship vs. SI limits')\n```\n\n\n```python\ndf_ = df_limits_clean.sub(df_limits['min'],axis=0)\ndf_limits_normalized = df_.div(df_['max'], axis=0)\n```\n\n\n\n\n```python\nfig,ax=plt.subplots()\nax.errorbar(df_limits_normalized.index,df_limits_normalized['ship'],yerr=[df_limits_normalized['ship']-df_limits_normalized['min'],\n df_limits_normalized['max']-df_limits_normalized['ship']], fmt='ok', lw=1, ecolor='gray', capsize=20)\nax.set_title('Ship vs. SI limits')\nax.set_ylabel('Norlimized limit')\n```\n", "meta": {"hexsha": "d4e2c75c39365b7abe16e7e536b96bdbe1b382a2", "size": 180788, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "reports/ISOPE_outline/00.4_KVLCC2_Ikeda_method_speed.ipynb", "max_stars_repo_name": "rddaz2013/Prediction-of-roll-motion-using-fully-nonlinear-potential-flow-and-Ikedas-method", "max_stars_repo_head_hexsha": "ac0a27e31d64edc8ae8912b6ed10005029868c90", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "reports/ISOPE_outline/00.4_KVLCC2_Ikeda_method_speed.ipynb", "max_issues_repo_name": "rddaz2013/Prediction-of-roll-motion-using-fully-nonlinear-potential-flow-and-Ikedas-method", "max_issues_repo_head_hexsha": "ac0a27e31d64edc8ae8912b6ed10005029868c90", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "reports/ISOPE_outline/00.4_KVLCC2_Ikeda_method_speed.ipynb", "max_forks_repo_name": "rddaz2013/Prediction-of-roll-motion-using-fully-nonlinear-potential-flow-and-Ikedas-method", "max_forks_repo_head_hexsha": "ac0a27e31d64edc8ae8912b6ed10005029868c90", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-06-05T15:38:54.000Z", "max_forks_repo_forks_event_max_datetime": "2021-06-05T15:38:54.000Z", "avg_line_length": 108.5813813814, "max_line_length": 72864, "alphanum_fraction": 0.8100537646, "converted": true, "num_tokens": 8705, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947155710233, "lm_q2_score": 0.6757645944891559, "lm_q1q2_score": 0.44255466190094367}} {"text": "# Implementation\n
    \n\nThis section presents the complete computational algorithm, its\nimplementation in Python code, animation of the solution, and\nverification of the implementation.\n\nA real implementation of the basic computational algorithm from\nthe sections [Formulating a recursive algorithm](wave1D_fd1.ipynb#wave:string:alg) and [Sketch of an implementation](wave1D_fd1.ipynb#wave:string:impl) can be\nencapsulated in a function, taking all the input data for the problem\nas arguments. The physical input data consists of $c$, $I(x)$,\n$V(x)$, $f(x,t)$, $L$, and $T$. The numerical input is the mesh\nparameters $\\Delta t$ and $\\Delta x$.\n\nInstead of specifying $\\Delta t$ *and* $\\Delta x$, we can specify one\nof them and the Courant number $C$ instead, since having explicit\ncontrol of the Courant number is convenient when investigating the\nnumerical method. Many find it natural to prescribe the resolution of\nthe spatial grid and set $N_x$. The solver function can then compute\n$\\Delta t = CL/(cN_x)$. However, for comparing $u(x,t)$ curves (as\nfunctions of $x$) for various Courant numbers\nit is more convenient to keep $\\Delta t$ fixed for\nall $C$ and let $\\Delta x$ vary according to $\\Delta x = c\\Delta t/C$.\nWith $\\Delta t$ fixed, all frames correspond to the same time $t$,\nand this simplifies animations that compare simulations with different\nmesh resolutions. Plotting functions of $x$\nwith different spatial resolution is trivial,\nso it is easier to let $\\Delta x$ vary in the simulations than $\\Delta t$.\n\n## Callback function for user-specific actions\n
    \n\n\nThe solution at all spatial points at a new time level is stored in an\narray `u` of length $N_x+1$. We need to decide what to do with\nthis solution, e.g., visualize the curve, analyze the values, or write\nthe array to file for later use. The decision about what to do is left to\nthe user in the form of a user-supplied function \n`user_action(u, x, t, n)`, where `u` is the solution at the spatial points `x` at time `t[n]`.\nThe `user_action` function is called from the solver at each time level `n`.\n\nIf the user wants to plot the solution or store the solution at a\ntime point, she needs to write such a function and take appropriate\nactions inside it. We will show examples on many such `user_action`\nfunctions.\n\nSince the solver function makes calls back to the user's code\nvia such a function, this type of function is called a *callback function*.\nWhen writing general software, like our solver function, which also needs\nto carry out special problem- or solution-dependent actions\n(like visualization),\nit is a common technique to leave those actions to user-supplied\ncallback functions.\n\nThe callback function can be used to terminate the solution process\nif the user returns `True`. For example,\n\n\n```python\ndef my_user_action_function(u, x, t, n):\n return np.abs(u).max() > 10\n```\n\nis a callback function that will terminate the solver function (given below) of the\namplitude of the waves exceed 10, which is here considered as a numerical\ninstability.\n\n\n## The solver function\n
    \n\n\nA first attempt at a solver function is listed below.\n\n\n```python\n#\u00a0NBVAL_IGNORE_OUTPUT\nimport numpy as np\nimport time as time\nfrom devito import Constant, Grid, TimeFunction, SparseTimeFunction, Function, Eq, solve, Operator, Buffer\n```\n\n\n```python\n# %load -s solver, src-wave/wave1D/wave1D_u0.py\ndef solver(I, V, f, c, L, dt, C, T, user_action=None):\n \"\"\"Solve u_tt=c^2*u_xx + f on (0,L)x(0,T].\"\"\"\n Nt = int(round(T/dt))\n t = np.linspace(0, Nt*dt, Nt+1) # Mesh points in time\n dx = dt*c/float(C)\n Nx = int(round(L/dx))\n x = np.linspace(0, L, Nx+1) # Mesh points in space\n C2 = C**2 # Help variable in the scheme\n \n # Make sure dx and dt are compatible with x and t\n dx = x[1] - x[0]\n dt = t[1] - t[0]\n\n # Initialising functions f and V if not provided\n if f is None or f == 0 :\n f = lambda x, t: 0\n if V is None or V == 0:\n V = lambda x: 0\n \n t0 = time.perf_counter() # Measure CPU time\n\n # Set up grid\n grid = Grid(shape=(Nx+1), extent=(L))\n t_s = grid.stepping_dim\n \n # Create and initialise u\n u = TimeFunction(name='u', grid=grid, time_order=2, space_order=2)\n u.data[:,:] = I(x[:])\n\n x_dim = grid.dimensions[0]\n t_dim = grid.time_dim\n \n # The wave equation we are trying to solve\n pde = (1/c**2)*u.dt2-u.dx2\n \n # Source term and injection into equation\n dt_symbolic = grid.time_dim.spacing \n src = SparseTimeFunction(name='f', grid=grid, npoint=Nx+1, nt=Nt+1)\n \n for i in range(Nt):\n src.data[i] = f(x, t[i])\n \n src.coordinates.data[:, 0] = x\n src_term = src.inject(field=u.forward, expr=src * (dt_symbolic**2))\n stencil = Eq(u.forward, solve(pde, u.forward))\n\n # Set up special stencil for initial timestep with substitution for u.backward\n v = Function(name='v', grid=grid, npoint=Nx+1, nt=1)\n v.data[:] = V(x[:])\n stencil_init = stencil.subs(u.backward, u.forward - dt_symbolic*v)\n\n # Boundary conditions\n bc = [Eq(u[t_s+1, 0], 0)]\n bc += [Eq(u[t_s+1, Nx], 0)]\n\n # Create and apply operators\n op_init = Operator([stencil_init]+src_term+bc)\n op = Operator([stencil]+src_term+bc)\n \n op_init.apply(time_M=1, dt=dt)\n op.apply(time_m=1, time_M=Nt, dt=dt)\n \n cpu_time = time.perf_counter() - t0\n \n return u.data[-1], x, t, cpu_time\n\n```\n\nA couple of remarks about the above code is perhaps necessary:\n\n * Although we give `dt` and compute `dx` via `C` and `c`, the resulting\n `t` and `x` meshes do not necessarily correspond exactly to these values\n because of rounding errors. To explicitly ensure that `dx` and `dt`\n correspond to the cell sizes in `x` and `t`, we recompute the values.\n\n * According to the particular choice made in the section [Callback function for user-specific actions](#wave:pde1:impl:useraction), a true value returned from `user_action` should terminate the simulation.\n\n\n```python\n#\u00a0NBVAL_IGNORE_OUTPUT\nimport matplotlib.pyplot as plt\n\ndef u_exact(x, t):\n return x*(L-x)*(1 + 0.5*t)\n\ndef I(x):\n return u_exact(x, 0)\n\ndef V(x):\n return 0.5*u_exact(x, 0)\n\ndef f(x, t):\n return 2*(1 + 0.5*t)*c**2\n\nc = 1.5\nL = 2.5\nNx = 100\nC = 0.75\ndt = C*(L/Nx)/c\nT = 18\nu, x, t, cpu_time = solver(I, V, f, c, L, dt, C, T)\n\nplt.plot(x, u)\nplt.xlabel('x')\nplt.ylabel('u')\nplt.legend(loc='best')\nplt.show()\n```\n\n## Verification: exact quadratic solution\n
    \n\n\nWe use the test problem derived in the section [A slightly generalized model problem](wave1D_fd1.ipynb#wave:pde2:fd) for\nverification. Below is a unit test based on this test problem\nand realized as a proper *test function* compatible with the unit test\nframeworks nose or pytest.\n\n\n```python\n# %load -s test_quadratic, src-wave/wave1D/wave1D_u0.py\ndef test_quadratic():\n \"\"\"Check that u(x,t)=x(L-x)(1+t/2) is exactly reproduced.\"\"\"\n\n def u_exact(x, t):\n return x*(L-x)*(1 + 0.5*t)\n\n def I(x):\n return u_exact(x, 0)\n\n def V(x):\n return 0.5*u_exact(x, 0)\n\n def f(x, t):\n return 2*(1 + 0.5*t)*c**2\n\n L = 2.5\n c = 1.5\n C = 0.75\n Nx = 6 # Very coarse mesh for this exact test\n dt = C*(L/Nx)/c\n T = 18\n\n def assert_no_error(u, x, t, n):\n u_e = u_exact(x, t[n])\n print(np.abs(u- u_e).min())\n diff = np.abs(u - u_e).max()\n tol = 1E-7\n assert diff < tol\n\n solver(I, V, f, c, L, dt, C, T,\n user_action=assert_no_error)\n\n```\n\nWhen this function resides in the file [`wave1D_u0.py`](https://github.com/devitocodes/devito_book/blob/master/fdm-devito-notebooks/02_wave/src-wave/wave1D/wave1D_u0.py), one can run\npytest to check that all test functions with names `test_*()`\nin this file work:\n\n Terminal> py.test -s -v wave1D_u0.py\n\n\n## Verification: convergence rates\n
    \n\n\nA more general method, but not so reliable as a verification method,\nis to compute the convergence rates and see if they coincide with\ntheoretical estimates. Here we expect a rate of 2 according to\nthe various results in the section [Analysis of the difference equations\n](wave_analysis.ipynb#wave:pde1:analysis).\nA general function for computing convergence rates can be written like\nthis:\n\n\n```python\n# %load -s convergence_rates, src-wave/wave1D/wave1D_u0.py\ndef convergence_rates(\n u_exact, # Python function for exact solution\n I, V, f, c, L, # physical parameters\n dt0, num_meshes, C, T): # numerical parameters\n \"\"\"\n Half the time step and estimate convergence rates for\n for num_meshes simulations.\n \"\"\"\n # First define an appropriate user action function\n global error\n error = 0 # error computed in the user action function\n\n def compute_error(u, x, t, n):\n global error # must be global to be altered here\n # (otherwise error is a local variable, different\n # from error defined in the parent function)\n if n == 0:\n error = 0\n else:\n error = max(error, np.abs(u - u_exact(x, t[n])).max())\n\n # Run finer and finer resolutions and compute true errors\n E = []\n h = [] # dt, devito_solver adjusts dx such that C=dt*c/dx\n dt = dt0\n for i in range(num_meshes):\n solver(I, V, f, c, L, dt, C, T,\n user_action=compute_error)\n # error is computed in the final call to compute_error\n E.append(error)\n h.append(dt)\n dt /= 2 # halve the time step for next simulation\n print('E:')\n print(E)\n print('h:')\n print(h)\n # Convergence rates for two consecutive experiments\n r = [np.log(E[i]/E[i-1])/np.log(h[i]/h[i-1])\n for i in range(1,num_meshes) if h[i-1] != 0 and h[i] != h[i-1] and E[i-1] != 0]\n return r\n\n```\n\nUsing the analytical solution from the section [Using an analytical solution of physical significance](wave1D_fd1.ipynb#wave:pde2:fd:standing:waves), we can call `convergence_rates` to\nsee if we get a convergence rate that approaches 2 and use the final\nestimate of the rate in an `assert` statement such that this function becomes\na proper test function:\n\n\n```python\n# %load -s test_convrate_sincos, src-wave/wave1D/wave1D_u0.py\ndef test_convrate_sincos():\n n = m = 2\n L = 1.0\n u_exact = lambda x, t: np.cos(m*np.pi/L*t)*np.sin(m*np.pi/L*x)\n\n r = convergence_rates(\n u_exact=u_exact,\n I=lambda x: u_exact(x, 0),\n V=lambda x: 0,\n f=0,\n c=1,\n L=L,\n dt0=0.1,\n num_meshes=6,\n C=0.9,\n T=1)\n print('rates sin(x)*cos(t) solution:')\n print([round(r_,2) for r_ in r])\n assert abs(r[-1] - 2) < 0.002\n\n```\n\nDoing `py.test -s -v wave1D_u0.py` will run also this test function and\nshow the rates 2.05, 1.98, 2.00, 2.00, and 2.00 (to two decimals).\n\n\n## Visualization: animating the solution\n
    \n\nNow that we have verified the implementation it is time to do a\nreal computation where we also display evolution of the waves\non the screen. Since the `solver` function knows nothing about\nwhat type of visualizations we may want, it calls the callback function\n`user_action(u, x, t, n)`. We must therefore write this function and\nfind the proper statements for plotting the solution.\n\n### Function for administering the simulation\n\nThe following `viz` function\n\n1. defines a `user_action` callback function\n for plotting the solution at each time level,\n\n2. calls the `solver` function, and\n\n3. combines all the plots (in files) to video in different formats.\n\n\n```python\n# %load -s viz, src-wave/wave1D/wave1D_u0.py\ndef viz(\n I, V, f, c, L, dt, C, T, # PDE parameters\n umin, umax, # Interval for u in plots\n animate=True, # Simulation with animation?\n tool='matplotlib', # 'matplotlib' or 'scitools'\n devito_solver_function=solver, # Function with numerical algorithm\n ):\n \"\"\"Run devito_solver and visualize u at each time level.\"\"\"\n\n def plot_u_st(u, x, t, n):\n \"\"\"user_action function for devito_solver.\"\"\"\n plt.plot(x, u, 'r-',\n xlabel='x', ylabel='u',\n axis=[0, L, umin, umax],\n title='t=%f' % t[n], show=True)\n # Let the initial condition stay on the screen for 2\n # seconds, else insert a pause of 0.2 s between each plot\n time.sleep(2) if t[n] == 0 else time.sleep(0.2)\n plt.savefig('frame_%04d.png' % n) # for movie making\n\n class PlotMatplotlib:\n def __call__(self, u, x, t, n):\n \"\"\"user_action function for devito_solver.\"\"\"\n if n == 0:\n plt.ion()\n self.lines = plt.plot(x, u, 'r-')\n plt.xlabel('x'); plt.ylabel('u')\n plt.axis([0, L, umin, umax])\n plt.legend(['t=%f' % t[n]], loc='lower left')\n else:\n self.lines[0].set_ydata(u)\n plt.legend(['t=%f' % t[n]], loc='lower left')\n plt.draw()\n time.sleep(2) if t[n] == 0 else time.sleep(0.2)\n plt.savefig('tmp_%04d.png' % n) # for movie making\n\n if tool == 'matplotlib':\n import matplotlib.pyplot as plt\n plot_u = PlotMatplotlib()\n elif tool == 'scitools':\n import scitools.std as plt # scitools.easyviz interface\n plot_u = plot_u_st\n import time, glob, os\n\n # Clean up old movie frames\n for filename in glob.glob('tmp_*.png'):\n os.remove(filename)\n\n # Call devito_solver and do the simulaton\n user_action = plot_u if animate else None\n u, x, t, cpu = devito_solver_function(\n I, V, f, c, L, dt, C, T, user_action)\n\n # Make video files\n fps = 4 # frames per second\n codec2ext = dict(flv='flv', libx264='mp4', libvpx='webm',\n libtheora='ogg') # video formats\n filespec = 'tmp_%04d.png'\n movie_program = 'ffmpeg'\n for codec in codec2ext:\n ext = codec2ext[codec]\n cmd = '%(movie_program)s -r %(fps)d -i %(filespec)s '\\\n '-vcodec %(codec)s movie.%(ext)s' % vars()\n os.system(cmd)\n\n if tool == 'scitools':\n # Make an HTML play for showing the animation in a browser\n plt.movie('tmp_*.png', encoder='html', fps=fps,\n output_file='movie.html')\n return cpu\n\n```\n\n### Dissection of the code\n\nThe `viz` function can either use SciTools or Matplotlib for\nvisualizing the solution. The `user_action` function based on SciTools\nis called `plot_u_st`, while the `user_action` function based on\nMatplotlib is a bit more complicated as it is realized as a class and\nneeds statements that differ from those for making static plots.\nSciTools can utilize both Matplotlib and Gnuplot (and many other\nplotting programs) for doing the graphics, but Gnuplot is a relevant\nchoice for large $N_x$ or in two-dimensional problems\nas Gnuplot is significantly faster than\nMatplotlib for screen animations.\n\n\nA function inside another function, like `plot_u_st` in the above code\nsegment, has access to *and remembers* all the local variables in the\nsurrounding code inside the `viz` function (!). This is known in\ncomputer science as a *closure* and is very convenient to program\nwith. For example, the `plt` and `time` modules defined outside\n`plot_u` are accessible for `plot_u_st` when the function is called\n(as `user_action`) in the `solver` function. Some may think, however,\nthat a class instead of a closure is a cleaner and\neasier-to-understand implementation of the user action function, see\nthe section [Building a general 1D wave equation solver](wave1D_fd2.ipynb#wave:pde2:software).\n\nThe `plot_u_st` function just makes a standard SciTools `plot` command\nfor plotting `u` as a function of `x` at time `t[n]`. To achieve a\nsmooth animation, the `plot` command should take keyword arguments\ninstead of being broken into separate calls to `xlabel`, `ylabel`,\n`axis`, `time`, and `show`. Several `plot` calls will automatically\ncause an animation on the screen. In addition, we want to save each\nframe in the animation to file. We then need a filename where the\nframe number is padded with zeros, here `tmp_0000.png`,\n`tmp_0001.png`, and so on. The proper printf construction is then\n`tmp_%04d.png`. The section [Making animations](../01_vib/vib_undamped.ipynb#vib:ode1:anim) contains more basic\ninformation on making animations.\n\nThe solver is called with an argument `plot_u` as `user_function`.\nIf the user chooses to use SciTools, `plot_u` is the `plot_u_st`\ncallback function, but for Matplotlib it is an instance of the\nclass `PlotMatplotlib`. Also this class makes use of variables\ndefined in the `viz` function: `plt` and `time`.\nWith Matplotlib, one has to make the first plot the standard way, and\nthen update the $y$ data in the plot at every time level. The update\nrequires active use of the returned value from `plt.plot` in the first\nplot. This value would need to be stored in a local variable if we\nwere to use a closure for the `user_action` function when doing the\nanimation with Matplotlib. mathcal{I}_t is much easier to store the\nvariable as a class attribute `self.lines`. Since the class is essentially a\nfunction, we implement the function as the special method `__call__`\nsuch that the instance `plot_u(u, x, t, n)` can be called as a standard\ncallback function from `solver`.\n\n### Making movie files\n\nFrom the\n`frame_*.png` files containing the frames in the animation we can\nmake video files.\n% if BOOK == \"book\":\nthe section [vib:ode1:anim](#vib:ode1:anim) presents basic information on how to\nuse the `ffmpeg` program for producing video files\nin different modern formats: Flash, MP4, Webm, and Ogg.\n% else:\nWe use the `ffmpeg` program to combine individual\nplot files to movies in modern formats: Flash, MP4, Webm, and Ogg.\nA typical `ffmpeg` command for creating a movie file\nin Ogg format\nwith 4 frames per second built from a collection of\nplot files with names generated by `frame_%04d.png`,\nlook like\n\n Terminal> ffmpeg -r 25 -i frame_%04d.png -c:v libtheora movie.ogg\n\n\nThe different formats require different video encoders (`-c:v`) to\nbe installed: Flash applies `flv`, WebM applies `libvpx`, and MP4\napplies `libx264`:\n\n Terminal> ffmpeg -r 25 -i frame_%04d.png -c:v flv movie.flv\n Terminal> ffmpeg -r 25 -i frame_%04d.png -c:v libvpx movie.webm\n Terminal> ffmpeg -r 25 -i frame_%04d.png -c:v libx264 movie.mp4\n\n\nPlayers like `vlc`, `mplayer`, `gxine`, and `totem`\ncan be used to play these movie files.\n\nNote that padding the frame counter with zeros in the `frame_*.png`\nfiles, as specified by the `%04d` format, is essential so that the wildcard\nnotation `frame_*.png` expands to the correct set of files.\n% endif\n\n\nThe `viz` function creates an `ffmpeg` command\nwith the proper arguments for each of the formats Flash, MP4, WebM,\nand Ogg. The task is greatly simplified by having a\n`codec2ext` dictionary for mapping\nvideo codec names to filename extensions.\n% if BOOK == \"book\":\nAs mentioned in the section [vib:ode1:anim](#vib:ode1:anim), only\n% else:\nOnly\n% endif\ntwo formats are actually needed to ensure that all browsers can\nsuccessfully play the video: MP4 and WebM.\n\nSome animations having a large number of plot files may not\nbe properly combined into a video using `ffmpeg`.\nA method that always works is to play the PNG files as an animation\nin a browser using JavaScript code in an HTML file.\nThe SciTools package has a function `movie` (or a stand-alone command\n`scitools movie`) for creating such an HTML player. The `plt.movie`\ncall in the `viz` function shows how the function is used.\nThe file `movie.html` can be loaded into a browser and features\na user interface where the speed of the animation can be controlled.\nNote that the movie in this case consists of the `movie.html` file\nand all the frame files `tmp_*.png`.\n\n\n### Skipping frames for animation speed\n\nSometimes the time step is small and $T$ is large, leading to an\ninconveniently large number of plot files and a slow animation on the\nscreen. The solution to such a problem is to decide on a total number\nof frames in the animation, `num_frames`, and plot the solution only for\nevery `skip_frame` frames. For example, setting `skip_frame=5` leads\nto plots of every 5 frames. The default value `skip_frame=1` plots\nevery frame.\nThe total number of time levels (i.e., maximum\npossible number of frames) is the length of `t`, `t.size` (or `len(t)`),\nso if we want `num_frames` frames in the animation,\nwe need to plot every `t.size/num_frames` frames:\n\n```python\ndef plot_u(t):\n skip_frame = int(t.size/float(num_frames))\n if n % skip_frame == 0 or n == t.size-1:\n st.plot(x, u, 'r-', ...)\n```\n\nThe initial condition (`n=0`) is included by `n % skip_frame == 0`,\nas well as every `skip_frame`-th frame.\nAs `n % skip_frame == 0` will very seldom be true for the\nvery final frame, we must also check if `n == t.size-1` to\nget the final frame included.\n\nA simple choice of numbers may illustrate the formulas: say we have\n801 frames in total (`t.size`) and we allow only 60 frames to be\nplotted. As `n` then runs from 801 to 0, we need to plot every 801/60\nframe, which with integer division yields 13 as `skip_frame`. Using\nthe mod function, `n % skip_frame`, this operation is zero every time\n`n` can be divided by 13 without a remainder. That is, the `if` test\nis true when `n` equals $0, 13, 26, 39, ..., 780, 801$. The associated\ncode is included in the `plot_u` function, inside the `viz` function,\nin the file [`wave1D_u0.py`](https://github.com/devitocodes/devito_book/blob/master/fdm-devito-notebooks/02_wave/src-wave/wave1D/wave1D_u0.py).\n\n## Running a case\n
    \n\nThe first demo of our 1D wave equation solver concerns vibrations of a\nstring that is initially deformed to a triangular shape, like when picking\na guitar string:\n\n\n
    \n\n$$\n\\begin{equation}\nI(x) = \\left\\lbrace\n\\begin{array}{ll}\nax/x_0, & x < x_0,\\\\ \na(L-x)/(L-x_0), & \\hbox{otherwise}\n\\end{array}\\right.\n\\label{wave:pde1:guitar:I} \\tag{1}\n\\end{equation}\n$$\n\nWe choose $L=75$ cm, $x_0=0.8L$, $a=5$ mm, and a time frequency\n$\\nu = 440$ Hz. The relation between the wave speed $c$ and $\\nu$ is\n$c=\\nu\\lambda$, where $\\lambda$ is the wavelength, taken as $2L$ because\nthe longest wave on the string forms frac{1}{2} a wavelength. There is no\nexternal force, so $f=0$ (meaning we can neglect gravity),\nand the string is at rest initially, implying $V=0$.\n\nRegarding numerical parameters, we need to specify a $\\Delta t$.\nSometimes it is more natural to think of a spatial resolution instead\nof a time step. A natural semi-coarse spatial resolution in the present\nproblem is $N_x=50$. We can then choose the associated $\\Delta t$ (as required\nby the `viz` and `solver` functions) as the stability limit:\n$\\Delta t = L/(N_xc)$. This is the $\\Delta t$ to be specified,\nbut notice that if $C<1$, the actual $\\Delta x$ computed in `solver` gets\nlarger than $L/N_x$: $\\Delta x = c\\Delta t/C = L/(N_xC)$. (The reason\nis that we fix $\\Delta t$ and adjust $\\Delta x$, so if $C$ gets\nsmaller, the code implements this effect in terms of a larger $\\Delta x$.)\n\nA function for setting the physical and numerical parameters and\ncalling `viz` in this application goes as follows:\n\n\n```python\n# %load -s guitar, src-wave/wave1D/wave1D_u0.py\ndef guitar(C):\n \"\"\"Triangular wave (pulled guitar string).\"\"\"\n L = 0.75\n x0 = 0.8*L\n a = 0.005\n freq = 440\n wavelength = 2*L\n c = freq*wavelength\n omega = 2*pi*freq\n num_periods = 1\n T = 2*pi/omega*num_periods\n # Choose dt the same as the stability limit for Nx=50\n dt = L/50./c\n\n def I(x):\n return a*x/x0 if x < x0 else a/(L-x0)*(L-x)\n\n umin = -1.2*a; umax = -umin\n cpu = viz(I, 0, 0, c, L, dt, C, T, umin, umax,\n animate=True, tool='scitools')\n\n```\n\nThe associated program has the name [`wave1D_u0.py`](https://github.com/devitocodes/devito_book/blob/master/fdm-devito-notebooks/02_wave/src-wave/wave1D/wave1D_u0.py). Run\nthe program and watch the [movie of the vibrating string](mov-wave/guitar_C0.8/movie.html).\nThe string should ideally consist of straight segments, but these are\nsomewhat wavy due to numerical approximation. Run the case with the\n`wave1D_u0.py` code and $C=1$ to see the exact solution.\n\n\n## Working with a scaled PDE model\n\n\nDepending on the model, it may be a substantial job to establish\nconsistent and relevant physical parameter values for a case. The\nguitar string example illustrates the point. However, by *scaling*\nthe mathematical problem we can often reduce the need to estimate\nphysical parameters dramatically. The scaling technique consists of\nintroducing new independent and dependent variables, with the aim that\nthe absolute values of these lie in $[0,1]$. We introduce the\ndimensionless variables (details are found in Section 3.1.1 in [[Langtangen_scaling]](#Langtangen_scaling))\n\n$$\n\\bar x = \\frac{x}{L},\\quad \\bar t = \\frac{c}{L}t,\\quad\n\\bar u = \\frac{u}{a}\n$$\n\nHere, $L$ is a typical length scale, e.g., the length of the domain,\nand $a$ is a typical size of $u$, e.g., determined from the\ninitial condition: $a=\\max_x|I(x)|$.\n\nWe get by the chain rule that\n\n$$\n\\frac{\\partial u}{\\partial t} =\n\\frac{\\partial}{\\partial\\bar t}\\left(a\\bar u\\right)\n\\frac{d\\bar t}{dt} =\n\\frac{ac}{L}\\frac{\\partial\\bar u}{\\partial\\bar t}\n$$\n\nSimilarly,\n\n$$\n\\frac{\\partial u}{\\partial x}\n= \\frac{a}{L}\\frac{\\partial\\bar u}{\\partial\\bar x}\n$$\n\nInserting the dimensionless variables in the PDE gives, in case $f=0$,\n\n$$\n\\frac{a^2c^2}{L^2}\\frac{\\partial^2\\bar u}{\\partial\\bar t^2}\n= \\frac{a^2c^2}{L^2}\\frac{\\partial^2\\bar u}{\\partial\\bar x^2}\n$$\n\nDropping the bars, we arrive at the scaled PDE\n\n\n
    \n\n$$\n\\begin{equation}\n\\frac{\\partial^2 u}{\\partial t^2} = \\frac{\\partial^2 u}{\\partial x^2},\n\\quad x\\in (0,1),\\ t\\in (0,cT/L),\n\\label{_auto1} \\tag{2}\n\\end{equation}\n$$\n\nwhich has no parameter $c^2$ anymore. The initial conditions are scaled\nas\n\n$$\na\\bar u(\\bar x, 0) = I(L\\bar x)\n$$\n\nand\n\n$$\n\\frac{a}{L/c}\\frac{\\partial\\bar u}{\\partial\\bar t}(\\bar x,0) = V(L\\bar x),\n$$\n\nresulting in\n\n$$\n\\bar u(\\bar x, 0) = \\frac{I(L\\bar x)}{\\max_x |I(x)|},\\quad\n\\frac{\\partial\\bar u}{\\partial\\bar t}(\\bar x,0) = \\frac{L}{ac}V(L\\bar x)\n$$\n\nIn the common case $V=0$ we see that there are no physical parameters to be\nestimated in the PDE model!\n\nIf we have a program implemented for the physical wave equation with\ndimensions, we can obtain the dimensionless, scaled version by\nsetting $c=1$. The initial condition of a guitar string,\ngiven in ([1](#wave:pde1:guitar:I)), gets its scaled form by choosing\n$a=1$, $L=1$, and $x_0\\in [0,1]$. This means that we only need to\ndecide on the $x_0$ value as a fraction of unity, because\nthe scaled problem corresponds to setting all\nother parameters to unity. In the code we can just set\n`a=c=L=1`, `x0=0.8`, and there is no need to calculate with\nwavelengths and frequencies to estimate $c$!\n\nThe only non-trivial parameter to estimate in the scaled problem\nis the final end time of the simulation, or more precisely, how it relates\nto periods in periodic solutions in time, since we often want to\nexpress the end time as a certain number of periods.\nThe period in the dimensionless problem is 2, so the end time can be\nset to the desired number of periods times 2.\n\nWhy the dimensionless period is 2 can be explained by the following\nreasoning.\nSuppose that $u$ behaves as $\\cos (\\omega t)$ in time in the original\nproblem with dimensions. The corresponding period is then $P=2\\pi/\\omega$, but\nwe need to estimate $\\omega$. A typical solution of the wave\nequation is $u(x,t)=A\\cos(kx)\\cos(\\omega t)$, where $A$ is an amplitude\nand $k$ is related to the wave length $\\lambda$ in space: $\\lambda = 2\\pi/k$.\nBoth $\\lambda$ and $A$ will be given by the initial condition $I(x)$.\nInserting this $u(x,t)$ in the PDE yields $-\\omega^2 = -c^2k^2$, i.e.,\n$\\omega = kc$. The period is therefore $P=2\\pi/(kc)$.\nIf the boundary conditions are $u(0,t)=u(L,t)$, we need to have\n$kL = n\\pi$ for integer $n$. The period becomes $P=2L/nc$. The longest\nperiod is $P=2L/c$. The dimensionless period $\\tilde P$ is obtained\nby dividing $P$ by the time scale $L/c$, which results in $\\tilde P=2$.\nShorter waves in the initial condition will have a dimensionless\nshorter period $\\tilde P=2/n$ ($n>1$).\n\n\n# Exercises\n\n\n\n\n\n## Exercise 1: Simulate a standing wave\n
    \n\nThe purpose of this exercise is to simulate standing waves on $[0,L]$\nand illustrate the error in the simulation.\nStanding waves arise from an initial condition\n\n$$\nu(x,0)= A \\sin\\left(\\frac{\\pi}{L}mx\\right),\n$$\n\nwhere $m$ is an integer and $A$ is a freely chosen amplitude.\nThe corresponding exact solution can be computed and reads\n\n$$\nu(x,t) = A\\sin\\left(\\frac{\\pi}{L}mx\\right)\n\\cos\\left(\\frac{\\pi}{L}mct\\right)\n$$\n\n**a)**\nExplain that for a function $\\sin kx\\cos \\omega t$ the wave length\nin space is $\\lambda = 2\\pi /k$ and the period in time is $P=2\\pi/\\omega$.\nUse these expressions to find the wave length in space and period in\ntime of $u$ above.\n\n\n\n**Solution.**\nSince the sin and cos functions depend on $x$ and $t$, respectively,\nthe sin function will run through one period as $x$ increases by $\\frac{2\\pi}{k}$, while the cos function starts repeating as $t$ increases by $\\frac{2\\pi}{\\omega}$.\n\nThe wave length in space becomes\n\n$$\n\\lambda = \\frac{2\\pi}{\\frac{\\pi}{L}m} = \\frac{2L}{m}\n$$\n\nThe period in time becomes\n\n$$\nP = \\frac{2\\pi}{\\frac{\\pi}{L}mc} = \\frac{2L}{mc}\n$$\n\n\n\n**b)**\nImport the `solver` function from `wave1D_u0.py` into a new file\nwhere the `viz` function is reimplemented such that it\nplots either the numerical *and* the exact solution, *or* the error.\n\n\n\n**Solution.**\nSee code below.\n\n\n\n**c)**\nMake animations where you illustrate how the error\n$e^n_i =u(x_i, t_n)- u^n_i$\ndevelops and increases in time. Also make animations of\n$u$ and $u$ simultaneously.\n\n\n\n**Hint 1.**\nQuite long time simulations are needed in order to display significant\ndiscrepancies between the numerical and exact solution.\n\n\n\n\n\n**Hint 2.**\nA possible set of parameters is $L=12$, $m=9$, $c=2$, $A=1$, $N_x=80$,\n$C=0.8$. The error mesh function $e^n$ can be simulated for 10 periods,\nwhile 20-30 periods are needed to show significant differences between\nthe curves for the numerical and exact solution.\n\n\n\n\n\n**Solution.**\nThe code:\n\n```python\nimport sys\nsys.path.insert(1, \"src-wave/wave1D\")\n\nfrom wave1D_u0 import devito_solver\n#from wave1D_u0v import devito_solver # allows faster vectorized operations\nimport numpy as np\n\ndef viz(I, V, f, c, L, dt, C, T,\n ymax, # y axis: [-ymax, ymax]\n u_exact, # u_exact(x, t)\n animate='u and u_exact', # or 'error'\n movie_filename='movie',\n ):\n \"\"\"Run devito_solver and visualize u at each time level.\"\"\"\n import matplotlib as plt\n import time, glob, os\n\n class Plot:\n def __init__(self, ymax, frame_name='frame'):\n self.max_error = [] # hold max amplitude errors\n self.max_error_t = [] # time points corresp. to max_error\n self.frame_name = frame_name\n self.ymax = ymax\n\n def __call__(self, u, x, t, n):\n \"\"\"user_action function for devito_solver.\"\"\"\n if animate == 'u and u_exact':\n plt.plot(x, u, 'r-',\n x, u_exact(x, t[n]), 'b--',\n xlabel='x', ylabel='u',\n axis=[0, L, -self.ymax, self.ymax],\n title='t=%f' % t[n], show=True)\n else:\n error = u_exact(x, t[n]) - u\n local_max_error = np.abs(error).max()\n # self.max_error holds the increasing amplitude error\n if self.max_error == [] or \\\n local_max_error > max(self.max_error):\n self.max_error.append(local_max_error)\n self.max_error_t.append(t[n])\n # Use user's ymax until the error exceeds that value.\n # This gives a growing max value of the yaxis (but\n # never shrinking)\n self.ymax = max(self.ymax, max(self.max_error))\n plt.plot(x, error, 'r-',\n xlabel='x', ylabel='error',\n axis=[0, L, -self.ymax, self.ymax],\n title='t=%f' % t[n], show=True)\n plt.savefig('%s_%04d.png' % (self.frame_name, n))\n\n # Clean up old movie frames\n for filename in glob.glob('frame_*.png'):\n os.remove(filename)\n\n plot = Plot(ymax)\n u, x, t, cpu = devito_solver(I, V, f, c, L, dt, C, T, plot)\n\n # Make plot of max error versus time\n plt.figure()\n plt.plot(plot.max_error_t, plot.max_error)\n plt.xlabel('time'); plt.ylabel('max abs(error)')\n plt.savefig('error.png')\n plt.savefig('error.pdf')\n\n # Make .flv movie file\n fps = 4 # Frames per second\n codec2ext = dict(flv='flv') #, libx64='mp4', \n #libvpx='webm', libtheora='ogg')\n\n filespec = 'frame_%04d.png'\n movie_program = 'ffmpeg'\n for codec in codec2ext:\n ext = codec2ext[codec]\n cmd = '%(movie_program)s -r %(fps)d -i %(filespec)s '\\\n '-vcodec %(codec)s %(movie_filename)s.%(ext)s' % vars()\n os.system(cmd)\n\ndef simulations():\n from numpy import sin, cos, pi\n L = 12 # length of domain\n m = 9 # 2L/m: wave length or period in space (2*pi/k, k=pi*m/L)\n c = 2 # wave velocity\n A = 1 # amplitude\n Nx = 80\n C = 0.8\n P = 2*pi/(pi*m*c/L) # 1 period in time\n #T = 10*P\n # Choose dt the same as the stability limit for Nx=50\n dt = L/50./c \n\n def u_exact(x, t):\n return A*sin(pi*m*x/L)*cos(pi*m*c*t/L)\n\n def I(x):\n return u_exact(x, 0)\n\n V = 0\n f = 0\n\n viz(I, V, f, c, L, dt, C, 10.5*P,\n 0.1, u_exact,\n animate='error',\n movie_filename='error')\n\n # Very long simulation to demonstrate different curves\n viz(I, V, f, c, L, dt, C, 30*P,\n 1.2*A, u_exact,\n animate='u and u_exact',\n movie_filename='solution')\n\nif __name__ == '__main__':\n simulations()\n```\n\n\n\n\nFilename: `wave_standing`.\n\n\n\n### Remarks\n\nThe important\nparameters for numerical quality are $C$ and $k\\Delta x$, where\n$C=c\\Delta t/\\Delta x$ is the Courant number and $k$ is defined above\n($k\\Delta x$ is proportional to how many mesh points we have per wave length\nin space, see the [Numerical dispersion relation](wave_analysis.ipynb#wave:pde1:num:dispersion) section for explanation).\n\n\n\n\n\n\n\n\n\n## Exercise 2: Add storage of solution in a user action function\n
    \n\nExtend the `plot_u` function in the file `wave1D_u0.py` to also store\nthe solutions `u` in a list.\nTo this end, declare `all_u` as\nan empty list in the `viz` function, outside `plot_u`, and perform\nan append operation inside the `plot_u` function. Note that a\nfunction, like `plot_u`, inside another function, like `viz`,\nremembers all local variables in `viz` function, including `all_u`,\neven when `plot_u` is called (as `user_action`) in the `solver` function.\nTest both `all_u.append(u)` and `all_u.append(u.copy())`.\nWhy does one of these constructions fail to store the solution correctly?\nLet the `viz` function return the `all_u` list\nconverted to a two-dimensional `numpy` array.\n\n\n\n**Solution.**\nWe have to explicitly use a copy of u, i.e. as `all_u.append(u.copy())`, otherwise we just get a reference to `u`, which goes on changing with the computations.\n\n```python\n#!/usr/bin/env python\n\"\"\"\n1D wave equation with u=0 at the boundary.\nSimplest possible implementation.\n\nThe key function is::\n\n u, x, t, cpu = solver(I, V, f, c, L, dt, C, T, user_action)\n\nwhich solves the wave equation u_tt = c**2*u_xx on (0,L) with u=0\non x=0,L, for t in (0,T]. Initial conditions: u=I(x), u_t=V(x).\n\nT is the stop time for the simulation.\ndt is the desired time step.\nC is the Courant number (=c*dt/dx), which specifies dx.\nf(x,t) is a function for the source term (can be 0 or None).\nI and V are functions of x.\n\nuser_action is a function of (u, x, t, n) where the calling\ncode can add visualization, error computations, etc.\n\"\"\"\n\nimport numpy as np\n\ndef solver(I, V, f, c, L, dt, C, T, user_action=None):\n \"\"\"Solve u_tt=c^2*u_xx + f on (0,L)x(0,T].\"\"\"\n Nt = int(round(T/dt))\n t = np.linspace(0, Nt*dt, Nt+1) # Mesh points in time\n dx = dt*c/float(C)\n Nx = int(round(L/dx))\n x = np.linspace(0, L, Nx+1) # Mesh points in space\n C2 = C**2 # Help variable in the scheme\n # Make sure dx and dt are compatible with x and t\n dx = x[1] - x[0]\n dt = t[1] - t[0]\n\n if f is None or f == 0 :\n f = lambda x, t: 0\n if V is None or V == 0:\n V = lambda x: 0\n\n u = np.zeros(Nx+1) # Solution array at new time level\n u_1 = np.zeros(Nx+1) # Solution at 1 time level back\n u_2 = np.zeros(Nx+1) # Solution at 2 time levels back\n\n import time; t0 = time.clock() # for measuring CPU time\n\n # Load initial condition into u_1\n for i in range(0,Nx+1):\n u_1[i] = I(x[i])\n\n if user_action is not None:\n user_action(u_1, x, t, 0)\n\n # Special formula for first time step\n n = 0\n for i in range(1, Nx):\n u[i] = u_1[i] + dt*V(x[i]) + \\\n 0.5*C2*(u_1[i-1] - 2*u_1[i] + u_1[i+1]) + \\\n 0.5*dt**2*f(x[i], t[n])\n u[0] = 0; u[Nx] = 0\n\n if user_action is not None:\n user_action(u, x, t, 1)\n\n # Switch variables before next step\n u_2[:] = u_1; u_1[:] = u\n\n for n in range(1, Nt):\n # Update all inner points at time t[n+1]\n for i in range(1, Nx):\n u[i] = - u_2[i] + 2*u_1[i] + \\\n C2*(u_1[i-1] - 2*u_1[i] + u_1[i+1]) + \\\n dt**2*f(x[i], t[n])\n\n # Insert boundary conditions\n u[0] = 0; u[Nx] = 0\n if user_action is not None:\n if user_action(u, x, t, n+1):\n break\n\n # Switch variables before next step\n u_2[:] = u_1; u_1[:] = u\n\n cpu_time = t0 - time.clock()\n return u, x, t, cpu_time\n\ndef test_quadratic():\n \"\"\"Check that u(x,t)=x(L-x)(1+t/2) is exactly reproduced.\"\"\"\n\n def u_exact(x, t):\n return x*(L-x)*(1 + 0.5*t)\n\n def I(x):\n return u_exact(x, 0)\n\n def V(x):\n return 0.5*u_exact(x, 0)\n\n def f(x, t):\n return 2*(1 + 0.5*t)*c**2\n\n L = 2.5\n c = 1.5\n C = 0.75\n Nx = 6 # Very coarse mesh for this exact test\n dt = C*(L/Nx)/c\n T = 18\n\n def assert_no_error(u, x, t, n):\n u_e = u_exact(x, t[n])\n diff = np.abs(u - u_e).max()\n tol = 1E-13\n assert diff < tol\n\n solver(I, V, f, c, L, dt, C, T,\n user_action=assert_no_error)\n\ndef test_constant():\n \"\"\"Check that u(x,t)=Q=0 is exactly reproduced.\"\"\"\n u_const = 0 # Require 0 because of the boundary conditions\n C = 0.75\n dt = C # Very coarse mesh\n u, x, t, cpu = solver(I=lambda x:\n 0, V=0, f=0, c=1.5, L=2.5,\n dt=dt, C=C, T=18)\n tol = 1E-14\n assert np.abs(u - u_const).max() < tol\n\ndef viz(\n I, V, f, c, L, dt, C, T, # PDE parameters\n umin, umax, # Interval for u in plots\n animate=True, # Simulation with animation?\n tool='matplotlib', # 'matplotlib' or 'scitools'\n solver_function=solver, # Function with numerical algorithm\n ):\n \"\"\"Run solver, store and visualize u at each time level.\"\"\"\n\n all_u = []\n def plot_u_st(u, x, t, n):\n \"\"\"user_action function for solver.\"\"\"\n plt.plot(x, u, 'r-',\n xlabel='x', ylabel='u',\n axis=[0, L, umin, umax],\n title='t=%f' % t[n], show=True)\n # Let the initial condition stay on the screen for 2\n # seconds, else insert a pause of 0.2 s between each plot\n time.sleep(2) if t[n] == 0 else time.sleep(0.2)\n #plt.savefig('frame_%04d.png' % n) # for movie making\n plt.savefig('tmp_%04d.png' % n) # for movie making \n all_u.append(u.copy()) \n\n class PlotMatplotlib:\n def __call__(self, u, x, t, n):\n \"\"\"user_action function for solver.\"\"\"\n if n == 0:\n plt.ion()\n self.lines = plt.plot(x, u, 'r-')\n plt.xlabel('x'); plt.ylabel('u')\n plt.axis([0, L, umin, umax])\n plt.legend(['t=%f' % t[n]], loc='lower left')\n else:\n self.lines[0].set_ydata(u)\n plt.legend(['t=%f' % t[n]], loc='lower left')\n plt.draw()\n time.sleep(2) if t[n] == 0 else time.sleep(0.2)\n plt.savefig('tmp_%04d.png' % n) # for movie making\n\n if tool == 'matplotlib':\n import matplotlib.pyplot as plt\n plot_u = PlotMatplotlib()\n elif tool == 'scitools':\n import scitools.std as plt # scitools.easyviz interface\n plot_u = plot_u_st\n import time, glob, os\n\n # Clean up old movie frames\n for filename in glob.glob('tmp_*.png'):\n os.remove(filename)\n\n # Call solver and do the simulaton\n user_action = plot_u if animate else None\n u, x, t, cpu = solver_function(\n I, V, f, c, L, dt, C, T, user_action)\n\n # Make video files\n fps = 4 # frames per second\n codec2ext = dict(flv='flv', libx264='mp4', libvpx='webm',\n libtheora='ogg') # video formats\n filespec = 'tmp_%04d.png'\n movie_program = 'ffmpeg'\n for codec in codec2ext:\n ext = codec2ext[codec]\n cmd = '%(movie_program)s -r %(fps)d -i %(filespec)s '\\\n '-vcodec %(codec)s movie.%(ext)s' % vars()\n os.system(cmd)\n\n if tool == 'scitools':\n # Make an HTML play for showing the animation in a browser\n plt.movie('tmp_*.png', encoder='html', fps=fps,\n output_file='movie.html')\n return cpu, np.array(all_u)\n\ndef guitar(C):\n \"\"\"Triangular wave (pulled guitar string).\"\"\"\n L = 0.75\n x0 = 0.8*L\n a = 0.005\n freq = 440\n wavelength = 2*L\n c = freq*wavelength\n from math import pi\n w = 2*pi*freq\n num_periods = 1\n T = 2*pi/w*num_periods\n # Choose dt the same as the stability limit for Nx=50\n dt = L/50./c\n\n def I(x):\n return a*x/x0 if x < x0 else a/(L-x0)*(L-x)\n\n umin = -1.2*a; umax = -umin\n cpu, all_u = viz(I, 0, 0, c, L, dt, C, T, umin, umax,\n animate=True, tool='scitools')\n # checking\n #for e in all_u:\n # print e[int(len(all_u[1])/2)]\n\nif __name__ == '__main__':\n #test_quadratic()\n import sys\n try:\n C = float(sys.argv[1])\n print('C=%g' % C)\n except IndexError:\n C = 0.85\n print('Courant number: %.2f' % C)\n guitar(C)\n```\n\n\nFilename: `wave1D_u0_s_store`.\n\n\n\n\n\n\n\n\n## Exercise 3: Use a class for the user action function\n
    \n\nRedo [Exercise 2: Add storage of solution in a user action function](#wave:exer:store:list) using a class for the user\naction function. Let the `all_u` list be an attribute in this class\nand implement the user action function as a method (the special method\n`__call__` is a natural choice). The class versions avoid that the\nuser action function depends on parameters defined outside the\nfunction (such as `all_u` in [Exercise 2: Add storage of solution in a user action function](#wave:exer:store:list)).\n\n\n\n**Solution.**\nUsing a class, we get\n\n```python\n#!/usr/bin/env python\n\"\"\"\n1D wave equation with u=0 at the boundary.\nSimplest possible implementation.\n\nThe key function is::\n\n u, x, t, cpu = solver(I, V, f, c, L, dt, C, T, user_action)\n\nwhich solves the wave equation u_tt = c**2*u_xx on (0,L) with u=0\non x=0,L, for t in (0,T]. Initial conditions: u=I(x), u_t=V(x).\n\nT is the stop time for the simulation.\ndt is the desired time step.\nC is the Courant number (=c*dt/dx), which specifies dx.\nf(x,t) is a function for the source term (can be 0 or None).\nI and V are functions of x.\n\nuser_action is a function of (u, x, t, n) where the calling\ncode can add visualization, error computations, etc.\n\"\"\"\n\nimport numpy as np\n\ndef solver(I, V, f, c, L, dt, C, T, user_action=None):\n \"\"\"Solve u_tt=c^2*u_xx + f on (0,L)x(0,T].\"\"\"\n Nt = int(round(T/dt))\n t = np.linspace(0, Nt*dt, Nt+1) # Mesh points in time\n dx = dt*c/float(C)\n Nx = int(round(L/dx))\n x = np.linspace(0, L, Nx+1) # Mesh points in space\n C2 = C**2 # Help variable in the scheme\n # Make sure dx and dt are compatible with x and t\n dx = x[1] - x[0]\n dt = t[1] - t[0]\n\n if f is None or f == 0 :\n f = lambda x, t: 0\n if V is None or V == 0:\n V = lambda x: 0\n\n u = np.zeros(Nx+1) # Solution array at new time level\n u_1 = np.zeros(Nx+1) # Solution at 1 time level back\n u_2 = np.zeros(Nx+1) # Solution at 2 time levels back\n\n import time; t0 = time.clock() # for measuring CPU time\n\n # Load initial condition into u_1\n for i in range(0,Nx+1):\n u_1[i] = I(x[i])\n\n if user_action is not None:\n user_action(u_1, x, t, 0)\n\n # Special formula for first time step\n n = 0\n for i in range(1, Nx):\n u[i] = u_1[i] + dt*V(x[i]) + \\\n 0.5*C2*(u_1[i-1] - 2*u_1[i] + u_1[i+1]) + \\\n 0.5*dt**2*f(x[i], t[n])\n u[0] = 0; u[Nx] = 0\n\n if user_action is not None:\n user_action(u, x, t, 1)\n\n # Switch variables before next step\n u_2[:] = u_1; u_1[:] = u\n\n for n in range(1, Nt):\n # Update all inner points at time t[n+1]\n for i in range(1, Nx):\n u[i] = - u_2[i] + 2*u_1[i] + \\\n C2*(u_1[i-1] - 2*u_1[i] + u_1[i+1]) + \\\n dt**2*f(x[i], t[n])\n\n # Insert boundary conditions\n u[0] = 0; u[Nx] = 0\n if user_action is not None:\n if user_action(u, x, t, n+1):\n break\n\n # Switch variables before next step\n u_2[:] = u_1; u_1[:] = u\n\n cpu_time = t0 - time.clock()\n return u, x, t, cpu_time\n\ndef test_quadratic():\n \"\"\"Check that u(x,t)=x(L-x)(1+t/2) is exactly reproduced.\"\"\"\n\n def u_exact(x, t):\n return x*(L-x)*(1 + 0.5*t)\n\n def I(x):\n return u_exact(x, 0)\n\n def V(x):\n return 0.5*u_exact(x, 0)\n\n def f(x, t):\n return 2*(1 + 0.5*t)*c**2\n\n L = 2.5\n c = 1.5\n C = 0.75\n Nx = 6 # Very coarse mesh for this exact test\n dt = C*(L/Nx)/c\n T = 18\n\n def assert_no_error(u, x, t, n):\n u_e = u_exact(x, t[n])\n diff = np.abs(u - u_e).max()\n tol = 1E-13\n assert diff < tol\n\n solver(I, V, f, c, L, dt, C, T,\n user_action=assert_no_error)\n\ndef test_constant():\n \"\"\"Check that u(x,t)=Q=0 is exactly reproduced.\"\"\"\n u_const = 0 # Require 0 because of the boundary conditions\n C = 0.75\n dt = C # Very coarse mesh\n u, x, t, cpu = solver(I=lambda x:\n 0, V=0, f=0, c=1.5, L=2.5,\n dt=dt, C=C, T=18)\n tol = 1E-14\n assert np.abs(u - u_const).max() < tol\n\ndef viz(\n I, V, f, c, L, dt, C, T, # PDE parameters\n umin, umax, # Interval for u in plots\n animate=True, # Simulation with animation?\n tool='matplotlib', # 'matplotlib' or 'scitools'\n solver_function=solver, # Function with numerical algorithm\n ):\n \"\"\"Run solver, store and visualize u at each time level.\"\"\"\n\n class PlotUst:\n def __init__(self):\n self.all_u = []\n def __call__(self, u, x, t, n):\n \"\"\"user_action function for solver.\"\"\"\n plt.plot(x, u, 'r-',\n xlabel='x', ylabel='u',\n axis=[0, L, umin, umax],\n title='t=%f' % t[n], show=True)\n # Let the initial condition stay on the screen for 2\n # seconds, else insert a pause of 0.2 s between each plot\n time.sleep(2) if t[n] == 0 else time.sleep(0.2)\n plt.savefig('tmp_%04d.png' % n) # for movie making \n self.all_u.append(u.copy())\n\n class PlotMatplotlib:\n def __call__(self, u, x, t, n):\n \"\"\"user_action function for solver.\"\"\"\n if n == 0:\n plt.ion()\n self.lines = plt.plot(x, u, 'r-')\n plt.xlabel('x'); plt.ylabel('u')\n plt.axis([0, L, umin, umax])\n plt.legend(['t=%f' % t[n]], loc='lower left')\n else:\n self.lines[0].set_ydata(u)\n plt.legend(['t=%f' % t[n]], loc='lower left')\n plt.draw()\n time.sleep(2) if t[n] == 0 else time.sleep(0.2)\n plt.savefig('tmp_%04d.png' % n) # for movie making\n\n if tool == 'matplotlib':\n import matplotlib.pyplot as plt\n plot_u = PlotMatplotlib()\n elif tool == 'scitools':\n import scitools.std as plt # scitools.easyviz interface\n plot_u = PlotUst()\n import time, glob, os\n\n # Clean up old movie frames\n for filename in glob.glob('tmp_*.png'):\n os.remove(filename)\n\n # Call solver and do the simulaton\n user_action = plot_u if animate else None\n u, x, t, cpu = solver_function(\n I, V, f, c, L, dt, C, T, user_action)\n\n # Make video files\n fps = 4 # frames per second\n codec2ext = dict(flv='flv', libx264='mp4', libvpx='webm',\n libtheora='ogg') # video formats\n filespec = 'tmp_%04d.png'\n movie_program = 'ffmpeg'\n for codec in codec2ext:\n ext = codec2ext[codec]\n cmd = '%(movie_program)s -r %(fps)d -i %(filespec)s '\\\n '-vcodec %(codec)s movie.%(ext)s' % vars()\n os.system(cmd)\n\n if tool == 'scitools':\n # Make an HTML play for showing the animation in a browser\n plt.movie('tmp_*.png', encoder='html', fps=fps,\n output_file='movie.html')\n return cpu, np.array(plot_u.all_u)\n\ndef guitar(C):\n \"\"\"Triangular wave (pulled guitar string).\"\"\"\n L = 0.75\n x0 = 0.8*L\n a = 0.005\n freq = 440\n wavelength = 2*L\n c = freq*wavelength\n from math import pi\n w = 2*pi*freq\n num_periods = 1\n T = 2*pi/w*num_periods\n # Choose dt the same as the stability limit for Nx=50\n dt = L/50./c\n\n def I(x):\n return a*x/x0 if x < x0 else a/(L-x0)*(L-x)\n\n umin = -1.2*a; umax = -umin\n cpu, all_u = viz(I, 0, 0, c, L, dt, C, T, umin, umax,\n animate=True, tool='scitools')\n # checking\n #for e in all_u:\n # print e[int(len(all_u[1])/2)]\n\nif __name__ == '__main__':\n #test_quadratic()\n import sys\n try:\n C = float(sys.argv[1])\n print('C=%g' % C)\n except IndexError:\n C = 0.85\n print('Courant number: %.2f' % C)\n guitar(C)\n```\n\n\nFilename: `wave1D_u0_s2c`.\n\n\n\n\n\n\n\n\n## Exercise 4: Compare several Courant numbers in one movie\n
    \n\nThe goal of this exercise is to make movies where several curves,\ncorresponding to different Courant numbers, are visualized. Write a\nprogram that resembles `wave1D_u0_s2c.py` in [Exercise 3: Use a class for the user action function](#wave:exer:store:list:class), but with a `viz` function that\ncan take a list of `C` values as argument and create a movie with\nsolutions corresponding to the given `C` values. The `plot_u` function\nmust be changed to store the solution in an array (see [Exercise 2: Add storage of solution in a user action function](#wave:exer:store:list) or [Exercise 3: Use a class for the user action function](#wave:exer:store:list:class) for\ndetails), `solver` must be computed for each value of the Courant\nnumber, and finally one must run through each time step and plot all\nthe spatial solution curves in one figure and store it in a file.\n\nThe challenge in such a visualization is to ensure that the curves in\none plot correspond to the same time point. The easiest remedy is to\nkeep the time resolution constant and change the space resolution\nto change the Courant number. Note that each spatial grid is needed for\nthe final plotting, so it is an option to store those grids too.\n\n\n\n**Solution.**\nModifying the code to store all solutions for each $C$ value and also each corresponding spatial grid (needed for final plotting), we get\n\n```python\n#!/usr/bin/env python\n\"\"\"\n1D wave equation with u=0 at the boundary.\nSimplest possible implementation.\n\nThe key function is::\n\n u, x, t, cpu = solver(I, V, f, c, L, dt, C, T, user_action)\n\nwhich solves the wave equation u_tt = c**2*u_xx on (0,L) with u=0\non x=0,L, for t in (0,T]. Initial conditions: u=I(x), u_t=V(x).\n\nT is the stop time for the simulation.\ndt is the desired time step.\nC is the Courant number (=c*dt/dx), which specifies dx.\nf(x,t) is a function for the source term (can be 0 or None).\nI and V are functions of x.\n\nuser_action is a function of (u, x, t, n) where the calling\ncode can add visualization, error computations, etc.\n\"\"\"\n\nimport numpy as np\n\ndef solver(I, V, f, c, L, dt, C, T, user_action=None):\n \"\"\"Solve u_tt=c^2*u_xx + f on (0,L)x(0,T].\"\"\"\n Nt = int(round(T/dt))\n t = np.linspace(0, Nt*dt, Nt+1) # Mesh points in time\n dx = c*dt/C\n Nx = int(round(L/dx))\n x = np.linspace(0, L, Nx+1) # Mesh points in space\n C2 = C**2 # Help variable in the scheme\n # Recompute to make sure dx and dt are compatible with x and t\n dx = x[1] - x[0]\n dt = t[1] - t[0]\n\n if f is None or f == 0 :\n f = lambda x, t: 0\n if V is None or V == 0:\n V = lambda x: 0\n\n u = np.zeros(Nx+1) # Solution array at new time level\n u_1 = np.zeros(Nx+1) # Solution at 1 time level back\n u_2 = np.zeros(Nx+1) # Solution at 2 time levels back\n\n import time; t0 = time.clock() # for measuring CPU time\n\n # Load initial condition into u_1\n for i in range(0,Nx+1):\n u_1[i] = I(x[i])\n\n if user_action is not None:\n user_action(u_1, x, t, 0)\n\n # Special formula for first time step\n n = 0\n for i in range(1, Nx):\n u[i] = u_1[i] + dt*V(x[i]) + \\\n 0.5*C2*(u_1[i-1] - 2*u_1[i] + u_1[i+1]) + \\\n 0.5*dt**2*f(x[i], t[n])\n u[0] = 0; u[Nx] = 0\n\n if user_action is not None:\n user_action(u, x, t, 1)\n\n # Switch variables before next step\n u_2[:] = u_1; u_1[:] = u\n\n for n in range(1, Nt):\n # Update all inner points at time t[n+1]\n for i in range(1, Nx):\n u[i] = - u_2[i] + 2*u_1[i] + \\\n C2*(u_1[i-1] - 2*u_1[i] + u_1[i+1]) + \\\n dt**2*f(x[i], t[n])\n\n # Insert boundary conditions\n u[0] = 0; u[Nx] = 0\n if user_action is not None:\n if user_action(u, x, t, n+1):\n break\n\n # Switch variables before next step\n u_2[:] = u_1; u_1[:] = u\n\n cpu_time = time.clock() - t0\n return u, x, t, cpu_time\n\ndef viz(\n I, V, f, c, L, dt, C, T, # PDE parameters\n umin, umax, # Interval for u in plots\n animate=True, # Simulation with animation?\n tool='matplotlib', # 'matplotlib' or 'scitools'\n solver_function=solver, # Function with numerical algorithm\n ):\n \"\"\"\n Run solver, store and viz. u at each time level with all C values.\n \"\"\"\n \n class PlotUst:\n def __init__(self):\n self.all_u = []\n self.all_u_for_all_C = []\n self.x_mesh = [] # need each mesh for final plots\n def __call__(self, u, x, t, n):\n \"\"\"user_action function for solver.\"\"\"\n self.all_u.append(u.copy())\n if t[n] == T: # i.e., whole time interv. done for this C\n self.x_mesh.append(x)\n self.all_u_for_all_C.append(self.all_u)\n self.all_u = [] # reset to empty list\n \n if len(self.all_u_for_all_C) == len(C): # all C done\n print('Finished all C. Proceed with plots...')\n # note: n will here be the last index in t[n]\n for n_ in range(0, n+1): # for each tn\n plt.plot(self.x_mesh[0], \n self.all_u_for_all_C[0][n_],\n axis=[0, L, umin, umax],\n title='Solutions for all \\\n C at t=%f' % t[n_])\n plt.hold('on')\n \n for j in range(1, len(C)):\n # build plot at this tn with each \n # sol. from the different C values\n plt.plot(self.x_mesh[j], \n self.all_u_for_all_C[j][n_],\n axis=[0, L, umin, umax])\n plt.xlabel('x'); plt.ylabel('u')\n plt.hold('off')\n plt.show()\n # Let the init. cond. stay on the screen for\n # 2 sec, else insert a pause of 0.2 s \n # between each plot \n time.sleep(2) if t[n_] == 0 else \\\n time.sleep(0.2) \n plt.savefig('tmp_%04d.png' % n_) # for movie\n \n\n class PlotMatplotlib:\n def __init__(self):\n self.all_u = []\n self.all_u_for_all_C = []\n def __call__(self, u, x, t, n):\n \"\"\"user_action function for solver.\"\"\"\n self.all_u.append(u.copy())\n if t[n] == T: # i.e., whole time interv. done for this C\n self.all_u_for_all_C.append(self.all_u)\n self.all_u = [] # reset to empty list\n \n if len(self.all_u_for_all_C) == len(C): # all C done\n print('Finished all C. Proceed with plots...')\n plt.ion()\n # note: n will here be the last index in t[n]\n for n_ in range(0, n+1): # for each tn\n plt.plot(x, self.all_u_for_all_C[0][n_])\n plt.axis([0, L, umin, umax])\n plt.hold(True)\n for j in range(1, len(C)):\n # build plot at this tn with each \n # sol. from the different C values\n plt.plot(x, self.all_u_for_all_C[j][n_])\n plt.axis([0, L, umin, umax])\n plt.xlabel('x'); plt.ylabel('u')\n plt.title('Solutions for all \\\n C at t=%f' % t[n_])\n plt.hold(False)\n plt.draw()\n # Let the init. cond. stay on the screen for\n # 2 sec, else insert a pause of 0.2 s \n # between each plot \n time.sleep(2) if t[n_] == 0 else \\\n time.sleep(0.2) \n plt.savefig('tmp_%04d.png' % n_) # for movie\n\n if tool == 'matplotlib':\n import matplotlib.pyplot as plt\n plot_u = PlotMatplotlib()\n elif tool == 'scitools':\n import scitools.std as plt # scitools.easyviz interface\n plot_u = PlotUst()\n import time, glob, os\n\n # Clean up old movie frames\n for filename in glob.glob('tmp_*.png'):\n os.remove(filename)\n\n # Call solver and do the simulaton\n user_action = plot_u if animate else None\n for C_value in C:\n print('C_value --------------------------------- ')\n print(C_value)\n u, x, t, cpu = solver_function(\n I, V, f, c, L, dt, C_value, T, user_action)\n\n # Make video files\n fps = 4 # frames per second\n codec2ext = dict(flv='flv', libx264='mp4', libvpx='webm',\n libtheora='ogg') # video formats\n filespec = 'tmp_%04d.png'\n movie_program = 'ffmpeg'\n for codec in codec2ext:\n ext = codec2ext[codec]\n cmd = '%(movie_program)s -r %(fps)d -i %(filespec)s '\\\n '-vcodec %(codec)s movie.%(ext)s' % vars()\n os.system(cmd)\n\n if tool == 'scitools':\n # Make an HTML play for showing the animation in a browser\n plt.movie('tmp_*.png', encoder='html', fps=fps,\n output_file='movie.html')\n return cpu\n\ndef guitar(C):\n \"\"\"Triangular wave (pulled guitar string).\"\"\"\n L = 0.75\n x0 = 0.8*L\n a = 0.005\n freq = 440\n wavelength = 2*L\n c = freq*wavelength\n from math import pi\n w = 2*pi*freq\n num_periods = 1\n T = 2*pi/w*num_periods\n # Choose dt the same as the stability limit for Nx=50\n dt = L/50./c\n dx = dt*c/float(C)\n # Now dt is considered fixed and a list of C \n # values is made by reducing increasing the dx value \n # in steps of 10%. \n all_C = [C]\n all_C.append(c*dt/(1.1*dx))\n all_C.append(c*dt/(1.2*dx))\n\n def I(x):\n return a*x/x0 if x < x0 else a/(L-x0)*(L-x)\n\n umin = -1.2*a; umax = -umin\n cpu = viz(I, 0, 0, c, L, dt, all_C, T, umin, umax,\n animate=True, tool='scitools')\n #cpu = viz(I, 0, 0, c, L, dt, all_C, T, umin, umax,\n # animate=True, tool='matplotlib')\n print('cpu = ')\n print(cpu)\n\nif __name__ == '__main__':\n import sys\n try:\n C = float(sys.argv[1])\n print('C=%g' % C)\n except IndexError:\n C = 0.85\n print('Courant number: %.2f' % C)\n # The list of C values will be generated from this C value\n guitar(C)\n```\n\n\nFilename: `wave_numerics_comparison`.\n\n\n\n\n\n\n\n\n## Exercise 5: Implementing the solver function as a generator\n
    \n\nThe callback function `user_action(u, x, t, n)` is called from the\n`solver` function (in, e.g., `wave1D_u0.py`) at every time level and lets\nthe user work perform desired actions with the solution, like plotting it\non the screen. We have implemented the callback function in the typical\nway it would have been done in C and Fortran. Specifically, the code looks\nlike\n\n```python\nif user_action is not None:\n if user_action(u, x, t, n):\n break\n```\n\nMany Python programmers, however, may claim that `solver` is an iterative\nprocess, and that iterative processes with callbacks to the user code is\nmore elegantly implemented as *generators*. The rest of the text has little\nmeaning unless you are familiar with Python generators and the `yield`\nstatement.\n\nInstead of calling `user_action`, the `solver` function\nissues a `yield` statement, which is a kind of `return` statement:\n\n```python\nyield u, x, t, n\n```\n\nThe program control is directed back to the calling code:\n\n```python\nfor u, x, t, n in solver(...):\n # Do something with u at t[n]\n```\n\nWhen the block is done, `solver` continues with the statement after `yield`.\nNote that the functionality of terminating the solution process if\n`user_action` returns a `True` value is not possible to implement in the\ngenerator case.\n\nImplement the `solver` function as a generator, and plot the solution\nat each time step.\nFilename: `wave1D_u0_generator`.\n\n\n\n\n\n\n\n\n## Project 6: Calculus with 1D mesh functions\n
    \n\nThis project explores integration and differentiation of\nmesh functions, both with scalar and vectorized implementations.\nWe are given a mesh function $f_i$ on a spatial one-dimensional\nmesh $x_i=i\\Delta x$, $i=0,\\ldots,N_x$, over the interval $[a,b]$.\n\n\n**a)**\nDefine the discrete derivative of $f_i$ by using centered\ndifferences at internal mesh points and one-sided differences\nat the end points. Implement a scalar version of\nthe computation in a Python function and write an associated unit test\nfor the linear case $f(x)=4x-2.5$ where the discrete derivative should\nbe exact.\n\n\n\n**Solution.**\nSee code below.\n\n\n\n**b)**\nVectorize the implementation of the discrete derivative.\nExtend the unit test to check the validity of the implementation.\n\n\n\n**Solution.**\nSee code below.\n\n\n\n**c)**\nTo compute the discrete integral $F_i$ of $f_i$, we assume that\nthe mesh function $f_i$ varies linearly between the mesh points.\nLet $f(x)$ be such a linear interpolant of $f_i$. We then\nhave\n\n$$\nF_i = \\int_{x_0}^{x_i} f(x) dx\n$$\n\nThe exact integral of a piecewise linear function $f(x)$ is\ngiven by the Trapezoidal rule. Show\nthat if $F_{i}$ is already computed, we can find $F_{i+1}$\nfrom\n\n$$\nF_{i+1} = F_i + \\frac{1}{2}(f_i + f_{i+1})\\Delta x\n$$\n\nMake a function for the scalar implementation of the discrete integral\nas a mesh function. That is, the function should return\n$F_i$ for $i=0,\\ldots,N_x$.\nFor a unit test one can use the fact that the above defined\ndiscrete integral of a linear\nfunction (say $f(x)=4x-2.5$) is exact.\n\n\n\n**Solution.**\nWe know that the difference $F_{i+1} - F_i$ must amount to the area\nof a trapezoid, which is exactly what $\\frac{1}{2}(f_i + f_{i+1})\\Delta x$ is.\nTo show the relation above, we may start with the Trapezoidal rule:\n\n$$\nF_{i+1} = \\Delta x \\left[\\frac{1}{2}f(x_0) + \\sum_{j=1}^{n-1}f(x_j) + \\frac{1}{2}f(x_n) \\right] \\thinspace . \\nonumber\n$$\n\nSince $n = i+1$, and since the final term in the sum may be separated out from the sum and split in two, this may be written as\n\n$$\nF_{i+1} = \\Delta x \\left[\\frac{1}{2}f(x_0) + \\sum_{j=1}^{i-1}f(x_j) + \\frac{1}{2}f(x_i) + \\frac{1}{2}f(x_i) + \\frac{1}{2}f(x_{i+1}) \\right] \\thinspace . \\nonumber\n$$\n\nThis may further be written as\n\n$$\nF_{i+1} = \\Delta x \\left[\\frac{1}{2}f(x_0) + \\sum_{j=1}^{i-1}f(x_j) + \\frac{1}{2}f(x_i)\\right] + \\Delta x \\left[\\frac{1}{2}f(x_i) + \\frac{1}{2}f(x_{i+1}) \\right] \\thinspace . \\nonumber\n$$\n\nFinally, this gives\n\n$$\nF_{i+1} = F_i + \\frac{1}{2}(f_i + f_{i+1})\\Delta x\n$$\n\nSee code below for implementation.\n\n\n\n**d)**\nVectorize the implementation of the discrete integral.\nExtend the unit test to check the validity of the implementation.\n\n\n\n**Hint.**\nInterpret the recursive formula for $F_{i+1}$ as a sum.\nMake an array with each element of the sum and use the \"cumsum\"\n(`numpy.cumsum`) operation to compute the accumulative sum:\n`numpy.cumsum([1,3,5])` is `[1,4,9]`.\n\n\n\n\n\n**Solution.**\nSee code below.\n\n\n\n**e)**\nCreate a class `MeshCalculus` that can integrate and differentiate\nmesh functions. The class can just define some methods that call\nthe previously implemented Python functions. Here is an example\non the usage:\n\n```python\nimport numpy as np\ncalc = MeshCalculus(vectorized=True)\nx = np.linspace(0, 1, 11) # mesh\nf = np.exp(x) # mesh function\ndf = calc.differentiate(f, x) # discrete derivative\nF = calc.integrate(f, x) # discrete anti-derivative\n```\n\n\n**Solution.**\nSee code below.\n\n\n\n\n\n\n**Solution.**\nThe final version of the code reads\n\n```python\n# -*- coding: utf-8 -*-\n\"\"\"\nCalculus with a 1D mesh function.\n\"\"\"\nimport numpy as np\n\nclass MeshCalculus:\n def __init__(self, vectorized=True):\n self.vectorized = vectorized\n\n def differentiate(self, f, x):\n '''\n Computes the derivative of f by centered differences, but \n forw and back difference at the start and end, respectively.\n '''\n dx = x[1] - x[0]\n Nx = len(x) - 1 # number of spatial steps\n num_dfdx = np.zeros(Nx+1)\n # Compute approximate derivatives at end-points first\n num_dfdx[0] = (f(x[1]) - f(x[0]))/dx # FD approx.\n num_dfdx[Nx] = (f(x[Nx]) - f(x[Nx-1]))/dx # BD approx.\n # proceed with approximate derivatives for inner mesh points\n if self.vectorized:\n num_dfdx[1:-1] = (f(x[2:]) - f(x[:-2]))/(2*dx)\n else: # scalar version\n for i in range(1, Nx):\n num_dfdx[i] = (f(x[i+1]) - f(x[i-1]))/(2*dx)\n return num_dfdx\n \n def integrate(self, f, x):\n '''\n Computes the integral of f(x) over the interval \n covered by x. \n '''\n dx = x[1] - x[0]\n F = np.zeros(len(x)) \n F[0] = 0 # starting value for iterative scheme\n if self.vectorized:\n all_trapezoids = np.zeros(len(x)-1) \n all_trapezoids[:] = 0.5*(f(x[:-1]) + f(x[1:]))*dx \n F[1:] = np.cumsum(all_trapezoids)\n else: # scalar version\n for i in range(0, len(x)-1):\n F[i+1] = F[i] + 0.5*(f(x[i]) + f(x[i+1]))*dx \n return F\n \ndef test_differentiate():\n def f(x):\n return 4*x - 2.5\n def dfdx(x):\n derivatives = np.zeros(len(x))\n derivatives[:] = 4\n return derivatives\n \n a = 0; b = 1; Nx = 10\n x = np.linspace(a, b, Nx+1) \n exact_dfdx = dfdx(x) # compute exact derivatives\n # test vectorized version\n calc_v = MeshCalculus(vectorized=True)\n num_dfdx = calc_v.differentiate(f, x)\n print np.abs(num_dfdx - exact_dfdx)\n diff = np.abs(num_dfdx - exact_dfdx).max()\n tol = 1E-14\n assert diff < tol\n # test scalar version\n calc = MeshCalculus(vectorized=False)\n num_dfdx = calc.differentiate(f, x)\n print np.abs(num_dfdx - exact_dfdx)\n diff = np.abs(num_dfdx - exact_dfdx).max()\n assert diff < tol\n \ndef test_integrate():\n def f(x):\n return 4*x - 2.5\n a = 0; b = 1; Nx = 10\n# a = 2.5/4; b = 10; Nx = 2\n x = np.linspace(a, b, Nx+1) \n # The exact integral amounts to the total area of two triangles\n I_exact = 0.5*abs(2.5/4 - a)*f(a) + 0.5*abs(b - 2.5/4)*f(b)\n # test vectorized version \n calc_v = MeshCalculus(vectorized=True) \n F = calc_v.integrate(f, x)\n print F, I_exact\n diff = np.abs(F[-1] - I_exact)\n print diff\n tol = 1E-14\n assert diff < tol \n # test scalar version\n calc = MeshCalculus(vectorized=False) \n F = calc.integrate(f, x)\n print F, I_exact\n diff = np.abs(F[-1] - I_exact)\n print diff\n assert diff < tol\n \n \nif __name__ == '__main__':\n test_differentiate()\n test_integrate()\n```\n\n\n\nFilename: `mesh_calculus_1D`.\n\n\n", "meta": {"hexsha": "07b646ecc1f5e4234d1da21f1fcdd27ea3a297f2", "size": 113179, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "fdm-devito-notebooks/02_wave/wave1D_prog.ipynb", "max_stars_repo_name": "devitocodes/devito_book", "max_stars_repo_head_hexsha": "30405c3d440a1f89df69594fd0704f69650c1ded", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2020-07-17T13:19:15.000Z", "max_stars_repo_stars_event_max_datetime": "2021-03-27T05:21:09.000Z", "max_issues_repo_path": "fdm-jupyter-book/notebooks/02_wave/wave1D_prog.ipynb", "max_issues_repo_name": "devitocodes/devito_book", "max_issues_repo_head_hexsha": "30405c3d440a1f89df69594fd0704f69650c1ded", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 73, "max_issues_repo_issues_event_min_datetime": "2020-07-14T15:38:52.000Z", "max_issues_repo_issues_event_max_datetime": "2020-09-25T11:54:59.000Z", "max_forks_repo_path": "fdm-jupyter-book/notebooks/02_wave/wave1D_prog.ipynb", "max_forks_repo_name": "devitocodes/devito_book", "max_forks_repo_head_hexsha": "30405c3d440a1f89df69594fd0704f69650c1ded", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-03-27T05:21:14.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-27T05:21:14.000Z", "avg_line_length": 43.6142581888, "max_line_length": 14912, "alphanum_fraction": 0.5641859356, "converted": true, "num_tokens": 20788, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.7371581626286834, "lm_q1q2_score": 0.4424337481433768}} {"text": "# The Ising model and phase transitions\n\n### Remarks on completing the module\n\n\nThis assignment is summatively assessed.\nIt is imperative that you submit the notebook on time.\n\n\n\n```python\n##### from ipywidgets import widgets, interact, interactive, fixed\nfrom ipywidgets import widgets, interact, interactive, fixed\nfrom ipywidgets import Button, HBox, VBox\nimport shelve\nassessmentName=\"ID\";\nimport os\n\ndef get_last_value(key):\n if os.path.isfile('.choices.shelve') or os.path.isfile('.choices.shelve.dir'):\n s=shelve.open('.choices.shelve')\n return s.get(key,None)\n return None\n\ndef make_value_change_fn(assessmentName,name):\n def fn(change):\n s=shelve.open('.choices.shelve')\n key='{0}_{1}'.format(assessmentName,name)\n s[key]=change['new']\n s.close()\n return fn\n \nclass myFloatBox:\n def __init__(self,name,description,long_description):\n self.name=name\n self.description=description\n self.long_description=long_description\n def getWidget(self):\n self.widgets=[ \n widgets.FloatText(\n description=self.description,\n disabled=False,\n value=get_last_value('{0}_{1}'.format(assessmentName,self.name))\n )]\n \n txt=widgets.HTMLMath(\n value=self.long_description,\n placeholder='',\n description='',\n )\n \n self.widget=VBox([txt]+self.widgets)\n self.widgets[0].observe(make_value_change_fn(assessmentName,self.name), names='value')\n\n return self.widget\n```\n\n### Ising model\nThe task for this assignment is to implement the Ising Model introduced in the lecture. The structure in terms of a code skeleton provided below needs to be followed. Otherwise the automatic tests, which allow you to test different parts of you implementation, will not work.\n\nWe consider an Ising model, in which the interaction energy $E_i=E(s_i)$ of spin $i$ is calculated from\n\n\\begin{align}\nE(s_{i}) = \\frac{J}{2} \\sum\\limits_{j} (1-s_{i} s_{j})\n\\end{align}\nwhere the sum is over the 4 nearest neighbours of $i$.\n\nWe will restrict the calculation to a 2 dimensional grid throughout, hence 4 nearest neighbours, $x=\\pm 1$, $y=\\pm 1$. Notice that the expression for $E$ is different from the form considered in the lecture. \n\n\n\nTo simplify the calculations, we will use the dimensionless interaction energy \n$$\n\\mathcal{E}(s_\\mathrm{i}) \\equiv \\frac{\\beta}{2} \\sum\\limits_{j} (1-s_{i} s_{j}),\n$$\nwhere\n$$\n\\beta = \\frac{J}{kT},\n$$\nin the following. Here, $k$ is Boltzmann's constant, and $T$ is the temperature.\n\nGiven all $N$ spin states, we calculate\nthe ensenble-averaged macroscopic magnetization $\\bar{M}$, as\n\\begin{align}\n\\bar{M} = \\left\\langle\\left|\\frac{1}{N}\\sum_{i=1}^N s_{i}\\right|\\right\\rangle\n\\end{align}\n\nThe $\\langle\\rangle$ brackets denote the ensemble average. The parameter $J>0$ has the dimensions of energy, and consequently $\\beta$ is dimensionless.\n\nFollow the numbered steps in the following cells.\n\n\n\nThe cells below describe how to proceed, step-by-step. Begin by reading through all steps, comparing the instructions\nto the Ising model discussed in the lecture. Complete the assignment using the cells below. Several cells allow you to test your implementation step by step.\n\n\n#### 1. Set up the regular grid\n\nSet up a 2D grid in the form of a **python class**, called `Grid`. The class should contain\n - the spin array\n - the value of $J$\n \nThe spin array should be a 2D array of dimension $L^2$, with $L=32$. We will address a particular spin with its 2D Cartesian coordinates $(x,y)$, where $x=0\\cdots L-1$ and $y=1\\cdots L-1$ are the indices in the 2D array. So, for example, spin $s_{xy}$ refers to the spin located at vertex $(x,y)$ of the grid.\n\nInitialize the spins on the grid randomly, meaning each spin can be either up, $s_{xy}=1$, or down,\n$s_{xy}=-1$, with equal probability.\n\nWhen performing calculations on the grid, we will assume **periodic boundary conditions**\n\n** no marks **\n\n#### 2. Calculate the energy\n\nWrite a method, `energy`, as part of the class `Grid`, which calculates the interaction energy of a given spin, $s_{xy}$, by summing over its four nearest neighbours. The function should take the grid array, $\\beta$, and the cell indices $x$ and $y$ as parameters. It should **return a python tuple** containing two dimensionless energies corresponding to the energy of the current spin state of cell $xy$, $\\mathcal{E}_\\mathrm{c} \\equiv \\mathcal{E}\\left(s_{xy}^\\mathrm{current}\\right)$ and the energy of the flipped spin state $\\mathcal{E}_\\mathrm{f} \\equiv \\mathcal{E}\\left(s_{xy}^\\mathrm{flipped}\\right)$.\n\nThis means that for a cell with spin state $s_{xy} = 1$, the method should return $\\left(\\mathcal{E}\\left(s_{xy} = 1\\right), \\mathcal{E}\\left(s_{xy} = -1\\right)\\right)$ and vice versa.\n\n** Remember to account for periodic boudnary conditions on the grid.**\n\nYou can test the implementation of this method using the test cells provded. What are the interaction energies of cells (6,6), (15,0) and (31, 17) of the assignment grid given below (please include the answer to this question in the PDF you hand in). \n\n** 2 marks**\n\n#### 3. Calculate the probability of flipping a spin\n\nThe probability that the spin at vertex $(x,y)$ is flipped depends on the spin states of its neighbours and the value of $\\beta$ as explained in the lecture.\n\nWrite a method `prob_flip` which calculates the probability that spin $s_{xy}$ is flipped, given the (dimensionless) interaction energies for the current state $\\mathcal{E}_\\mathrm{c}$ and the flipped state $\\mathcal{E}_\\mathrm{f}$. \n\nThe probability for a flip is given by\n\\begin{align}\n\\mathcal{P}_\\mathrm{flip} = \n\\begin{cases}\n\\exp\\left(-\\left[\\mathcal{E}_\\mathrm{f} - \\mathcal{E}_\\mathrm{c}\\right]\\right) & \\text{if } \\mathcal{E}_\\mathrm{f} > \\mathcal{E}_\\mathrm{c}, \\\\\n1 & \\text{if } \\mathcal{E}_\\mathrm{f} \\leq \\mathcal{E}_\\mathrm{c}.\n\\end{cases}\n\\end{align}\n\nYou can test the implementation of this method using the test cells provided. What are the probabilities for cells (12, 12), (18,0) and (31, 12) of the assignment grid given below (please include the answer to this question in the PDF you hand in)?\n\n** 2 marks **\n\n#### 4. Calculate the macroscopic magnetisation, $M$\n\nWrite a method which calculates the current macroscopic magnetisation of a given grid, and add it to the `Grid` class. The function should take the grid-array as a parameter and return the mean, macroscopic magnetisation,\n\n$$ M=\\frac{1}{N}\\sum_{i=1}^N s_i\\,.$$\n\nYou can test the implementation of this method using the test cells provded. Calculate the magnetisation of the assignment grid. State the answer to 3 significant digits on the PDF file you hand in. \n\n**2 marks**\n\n#### 5. Red-black sweep\n\nWrite a method to sweep over all spins in turn, first in $x$ (say) and then in $y$ in a **red-black pattern**. \nRed-black means, first loop over all the red cells on a red-black chessboard pattern (looping over them $x$ first, then $y$). Once all the red cells are done,\nthen update all the black cells. Add this method to the `Grid` class.\n\nFor each spin in turn, flip the spin or not following the criterion discussed in the lecture. This means that the spin in each cell in turn should be flipped with a probability $\\mathcal{P}_\\mathrm{flip}$ (as discussed in step **3**). \n\nYou can use the methods implemented in step **2** and **3**.\n\n** no marks **\n\n\n#### 6. Thermalisation and magnetisation measurement\nStarting from a random configuration, the system needs to be evolved over a number of full red-black sweeps in order to reach thermal equilibrium. This *thermalization* is part of the method you develop in this step.\n\nWrite a method that starts by sweeping the grid $N_\\mathrm{therm}$ times to allow for the system to thermalize before you carry out any measurements. \n\nNext, the method should perform a further $N_\\mathrm{measure}$ sweeps, while in addition computing and recording the value of $M$ after every sweep. Use the method you developed in step **4** and the sweep implementaton of step **5**. \n\n$N_\\mathrm{therm}$ and $N_\\mathrm{measure}$ are input parameters of the method. The method should return a numpy array of length $N_\\mathrm{measure}$, containing the magentisations measured after each measurement sweep. \n\nAdd this method to the `Grid` class.\n\n**no marks**\n\n#### 7. Thermalisation \n\nPlot the magnetisation over time for 1000 full mesh sweeps for $\\beta = 0.1, 0.8$ and $1.6$ (include the thermalisation period in the plot). Include this plot in the PDF you hand in. Save the plot to a file \n'Thermalization.pdf'\n\n**4 marks**\n\n#### 8. Ensemble-averaged magnetisation, $\\bar M$, as a function of $\\beta$\n\nOnce the system has thermalized (a good choice is $N_\\mathrm{therm} =400$ thermalisation sweeps), measure the time-averaged magnetisation over 1000 sweeps. From this, estimate the ensemble-averaged magnetisation, $\\bar M$. Plot $\\bar M$ as a function of $1/\\beta$ for $\\beta = 1.6, 1.3, 1.1, 1.0, 0.9, 0.8, 0.7, 0.6, 0.4, 0.1$. Include this plot in the PDF you hand in. Save the plot to a file 'Magnetisation.pdf'.\n\n**4 marks**\n\n#### 9. Critical temperature\n\nThe critical temperature, $T_c$, in the mean field approximation, is given by\n$$\nT_c=\\frac{h J}{2k_B},\n$$\nwhere h is the number of nearest-neighbours, as discussed in the lecture. Use this to calculate the critical value $\\beta_c$ which corresponds to the critical temperature and mark it in the plot produced in step **8** (If you could not produce the plot in step **8** you may also mention the numerical value of $\\beta_c$ on the solution you hand in).\n\n**2 marks**\n\n#### 10. Mean field approximation\n\nThe ensemble-averaged value of the mean magnitization in the mean field approximation, is the solution of\n$$\n\\bar M - \\tanh\\left( \\frac{T_c\\bar M}{T} \\right) = 0\\,,\n$$\nas discussed in the lecture. This equation can not be solved analytically for $\\bar M$, for given\n$T_c/T$.\n\nRewrite the equation in terms of $\\beta$ using the relation between $T$ and $\\beta$ derived before, and solve the resulting formula numerically for $\\beta = 1.6, 1.3, 1.1, 1.0, 0.9, 0.8, 0.7, 0.6, 0.4, 0.1$. You may want to use the root finding methods implemented in previous exercises. Redo the plot of step **8**, but on top of the numerical result, over plot the mean field approximation. Add a legend to the plot which allows to distinguish the solution obtained in step **8** from the mean field approximation.\n\nSave the plot as 'MeanField.pdf'\n\n**4 marks**\n\n\n## Solution \n\nUse the cells below to complete the assignment. Use the test cells provided to make sure you are on the right track.\n\n\n```python\n%matplotlib inline\nimport numpy as np\nimport matplotlib.pyplot as plt \nimport sys\nimport os\nsys.path.append(os.getcwd())\nfrom scipy.interpolate import CubicSpline\nimport pickle\n```\n\n\n```python\n# This cell is hidden from the student's notebook. It generates the buttons used in the answers.\nfrom ipywidgets import widgets, interact, interactive, fixed\nfrom ipywidgets import Button, HBox, VBox\nimport shelve\nassessmentName=\"test\";\nimport os\n\ndef get_last_value(key):\n if os.path.isfile('.choices.shelve') or os.path.isfile('.choices.shelve.dir'):\n s=shelve.open('.choices.shelve')\n return s.get(key,None)\n return None\n\n\nclass myRadioButton:\n def __init__(self,name,description,options):\n self.name=name\n self.options=options\n self.description=description\n def getWidget(self):\n def on_value_change(change):\n s=shelve.open('.choices.shelve')\n key=self.getKey()\n s[key]=change['new']\n s.close()\n\n self.widget=widgets.RadioButtons(\n options=self.options,\n value=get_last_value(self.getKey()),\n description=self.description,\n disabled=False\n )\n self.widget.observe(on_value_change, names='value')\n\n return self.widget\n def getKey(self):\n return '{0}_{1}'.format(assessmentName,self.name)\n \n \ndef on_value_change(change):\n s=shelve.open('.choices.shelve')\n key='{0}_{1}_{2}'.format(assessmentName,self.name,i)\n s[key]=change['new']\n s.close()\n\ndef make_value_change_fn(assessmentName,name,i):\n def fn(change):\n s=shelve.open('.choices.shelve')\n key='{0}_{1}_{2}'.format(assessmentName,name,i)\n s[key]=change['new']\n s.close()\n return fn\n\nclass myCheckBoxSet:\n def __init__(self,name,description,options):\n self.name=name\n self.options=options\n self.description=description\n def getWidget(self):\n keys=['{0}_{1}_{2}'.format(assessmentName,self.name,i) for i in range(len(self.options))] \n self.widgets=[ widgets.Checkbox(value=get_last_value(key),\n description=o,\n disabled=False\n ) for key,o in zip(keys,self.options)]\n \n txt=widgets.HTMLMath(\n value=self.description,\n placeholder='',\n description='',\n )\n\n \n self.widget=VBox([txt]+self.widgets)\n for i,w in enumerate(self.widgets):\n w.observe(make_value_change_fn(assessmentName,self.name,i), names='value')\n\n return self.widget\nimport mywidgets \n```\n\n\n```python\nclass Grid:\n def __init__(self, size, beta):\n '''This function initialises the grid, i.e. it sets the \n grid size, the value of beta and initialises the cells of the \n grid with randomly chosen 'plus' (1) or 'minus' (-1) states.'''\n # set self.size, self.beta, and self.cells\n # self.cells is a 2D array, so that self.cells[i,j]=+1 or -1, the spin in grid location (i,j)\n # YOUR CODE HERE\n self.size = size\n self.beta = beta\n self.cells = []\n for i in range(size):\n tmp = []\n for j in range(size):\n if (np.random.random() > 0.5):\n tmp.append(1)\n else:\n tmp.append(-1)\n self.cells.append(tmp)\n# raise NotImplementedError()\n \n def energy(self, i, j, beta, grid):\n '''This function calculates the energies 'e_plus' and 'e_minus' \n corresponding to the two possible states 'plus' and 'minus' for the spin at location (i, j)\n of a given grid with a given value of beta.\n returns: the two energy states 'e_current' and 'e_flip' as a tuple. \n 'e_current' is the energy of the spin (i,j) in its current spin state\n 'e_flip' is the energy of the spin (i,j) if you were to flip its spin\n '''\n # YOUR CODE HERE\n e_current = 0\n if i + 1 >= len(grid):\n e_current += 1 - grid[0][j] * grid[i][j]\n e_current += 1 - grid[i - 1][j] * grid[i][j]\n else:\n e_current += 1 - grid[i - 1][j] * grid[i][j]\n e_current += 1 - grid[i + 1][j] * grid[i][j]\n \n if j + 1 >= len(grid[0]):\n e_current += 1 - grid[i][0] * grid[i][j]\n e_current += 1 - grid[i][j - 1] * grid[i][j]\n else:\n e_current += 1 - grid[i][j + 1] * grid[i][j]\n e_current += 1 - grid[i][j - 1] * grid[i][j]\n e_current *= beta / 2\n e_flip = 4 * beta - e_current\n return e_current, e_flip\n \n def prob_flip(self, e_current, e_flip):\n '''This function calculates the probability of a spin flip \n for a given spin, given the energies e_current and e_flip \n of the current and the flipped state for the cell.\n returns: the probability for the flip'''\n # YOUR CODE HERE\n# raise NotImplementedError()\n probability_for_flip = 0\n if e_current > e_flip:\n probability_for_flip = 1\n else:\n probability_for_flip = np.exp(-1 * abs(e_flip - e_current))\n return probability_for_flip\n \n def sweep(self):\n '''This function carries out a single red-black sweep. \n returns: nothing. For each spin in turn, it compute the probability for \n flipping the spin, using the prob_flip function. Comparing the probablity,\n it draws a random number to decide whether or not the flip the spin '''\n # YOUR CODE HERE\n new_grid = self.cells.copy()\n for i in range(0,len(new_grid),2):\n for j in range(0,len(new_grid[0]),2):\n e_current, e_flip = self.energy(i, j, self.beta, self.cells)\n probability_for_flip = self.prob_flip(e_current, e_flip)\n if (np.random.random() < probability_for_flip):\n new_grid[i][j] *= -1\n for i in range(1,len(new_grid),2):\n for j in range(1,len(new_grid[0]),2):\n e_current, e_flip = self.energy(i, j, self.beta, self.cells)\n probability_for_flip = self.prob_flip(e_current, e_flip)\n if (np.random.random() < probability_for_flip):\n new_grid[i][j] *= -1\n self.cells = new_grid\n new_grid = self.cells.copy()\n for i in range(0,len(new_grid),2):\n for j in range(1,len(new_grid[0]),2):\n e_current, e_flip = self.energy(i, j, self.beta, self.cells)\n probability_for_flip = self.prob_flip(e_current, e_flip)\n if (np.random.random() < probability_for_flip):\n new_grid[i][j] *= -1\n for i in range(1,len(new_grid),2):\n for j in range(0,len(new_grid[0]),2):\n e_current, e_flip = self.energy(i, j, self.beta, self.cells)\n probability_for_flip = self.prob_flip(e_current, e_flip)\n if (np.random.random() < probability_for_flip):\n new_grid[i][j] *= -1\n new_grid = self.cells.copy()\n# raise NotImplementedError()\n \n \n def magnetisation(self, grid):\n '''This function calculates the mean magnetisation of all the spin in the grid\n returns: the mean magnetisation M'''\n # YOUR CODE HERE\n count = 0\n for i in grid:\n for j in i:\n count += j\n M = count / len(grid) / len(grid[0])\n# raise NotImplementedError()\n return M\n \n def do_sweeps(self, n_therm, n_measure):\n '''This function carries out n_therm thermalisation sweeps and n_measure measurement sweeps.\n At the end of each measurement sweep the average magnetisation is computed and recorded.\n returns: an array of length 'n_measure' containing the recorded magnetisations for each measurement sweep.\n It uses the sweep function, and the magnitization function'''\n # YOUR CODE HERE\n for i in range(n_therm):\n self.sweep()\n magnetisation = []\n for i in range(n_measure):\n self.sweep()\n magnetisation.append(self.magnetisation(self.cells))\n return magnetisation\n \n```\n\nYou can use the cells below to test your implementation\n\n### Test and assignment grids\n\nThe cell below loads the test grids, `test_grid_1` and `test_grid_2`, and their corresponding values for $\\beta$, `test_beta_1` and `test_beta_2`.\n\nIn addition, it loads `assignement_grid` and `assignment_beta`.\n\nUse the first two grids to test your implementation. Use the third grid to answer the assignment questions.\n\n\n```python\nfilename = 'test_grid_1.pickle'\nf = open(filename, 'rb')\n(test_grid_1, test_beta_1) = pickle.load(f)\nf.close()\n\nfilename = 'test_grid_2.pickle'\nf = open(filename, 'rb')\n(test_grid_2, test_beta_2) = pickle.load(f)\nf.close()\n\nfilename = 'assignment_grid.pickle'\nf = open(filename, 'rb')\n(assignment_grid, assignment_beta) = pickle.load(f)\nf.close()\n\nprint(\" grid 1 loaded, size=\", len(test_grid_1),\" , beta= \", test_beta_1)\nprint(\" grid 2 loaded, size=\", len(test_grid_2),\" , beta= \", test_beta_2)\nprint(\" grid 3 loaded, size=\", len(assignment_grid),\" , beta= \", assignment_beta)\n```\n\n grid 1 loaded, size= 32 , beta= 1.6\n grid 2 loaded, size= 32 , beta= 0.8\n grid 3 loaded, size= 32 , beta= 0.1\n\n\n#### 2. Interaction energy calculation\n\nThe cell below allows you to test your interaction energy calculation method. If it does not return an error your implementation might be correct.\n\n\n\n```python\ng1 = Grid(len(test_grid_1), test_beta_1)\ng2 = Grid(len(test_grid_2), test_beta_2)\ncells = [(6,6), (15,0), (31,17)]\nenergies_1 = [(0.0, 6.4), (0.0, 6.4), (0.0, 6.4)]\nenergies_2 = [(0.8, 2.4000000000000004), (1.6, 1.6), (2.4, 0.8)]\n\nfor c, cell in enumerate(cells):\n i = cell[0]\n j = cell[1]\n e_1 = g1.energy(i, j, test_beta_1, test_grid_1)\n e_2 = g2.energy(i, j, test_beta_2, test_grid_2)\n assert(np.isclose(e_1, energies_1[c]).all())\n assert(np.isclose(e_2, energies_2[c]).all())\n\nprint(\"Your implementation might be correct!\")\n```\n\n Your implementation might be correct!\n\n\n#### 3. Probability calculation\n\nThe cell below allows you to test your probability calculation method. If it does not return an error your implementation might be correct.\n\n\n\n```python\ng1 = Grid(len(test_grid_1), test_beta_1)\nenergies = [(0.1, 0.3), (0.2, 0.2), (0.3, 0.1), (1.5, 1.6), (0.1, 1.6), (0.8, 2.4)]\nprobabilities = [0.8187307530779818, 1, 1, 0.9048374180359595, 0.22313016014842982, 0.20189651799465544]\nthis_prob = []\n\nfor i, e in enumerate(energies):\n this_prob.append(g1.prob_flip(e[0], e[1]))\nassert(np.isclose(this_prob, probabilities).all())\n\nprint(\"Your implementation might be correct!\")\n```\n\n Your implementation might be correct!\n\n\n#### 4. Magnetisation calculation\n\nThe cell below allows you to test your magnetisation method. If it does not return an error your implementation might be correct.\n\n\n\n```python\ng1 = Grid(len(test_grid_1), test_beta_1)\nassert(np.isclose(g1.magnetisation(test_grid_1),0.193359375))\ng2 = Grid(len(test_grid_2), test_beta_2)\nassert(np.isclose(g2.magnetisation(test_grid_2),-0.3203125))\n\nprint(\"Your implementation might be correct!\")\n```\n\n Your implementation might be correct!\n\n\n#### The following hidden cell uses the assignment grid to test\n - the calculation of energies ** 2 marks**\n - the calculation of the probabilities ** 2 marks **\n - the calculation of the magnetization ** 2 marks **\n \n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n\n#### 5 Implement the red-black sweep\n\n\n\n#### 6 Implement the thermalization step\n\n#### 7. Thermalisation \n\nPlot the magnetisation over time for 1000 full mesh sweeps for $\\beta = 0.1, 0.8$ and $1.6$ (include the thermalisation period in the plot). Include this plot in the PDF you hand in. Save the plot as file \n'Thermalisation.pdf'\n\n**4 marks**\n\n\n```python\n# for each value of beta=[0.1, 0.8, 1.6]\n# generate a grid of size=32, with the given value of beta\n# perform N=1000 red-black sweeps (each sweep runs over the full 32x32 grid)\n# calculate the mean magnetization, M for each sweep\n# Plot M as a function of sweep number.\n# You may want to use some of the plotting commands below.\nbetas = [0.1, 0.8, 1.6]\nsize = 32\ngrids = []\nmags = []\ntmp = []\nfor beta in betas:\n grids.append(Grid(size, beta))\nfor grid in grids:\n for i in range(1000):\n grid.sweep()\n tmp.append(grid.magnetisation(grid.cells))\n mags.append(tmp)\n tmp = []\n# set-up the figure\nprint(\"Calculation finished\")\nfig, ax = plt.subplots(1,1, figsize = (8, 5))\nfile = \"Thermalisation.pdf\"\n# caculate mag, the average magnetization, for N=1000 sweeps\n# # YOUR CODE HERE\n# raise NotImplementedError()\n# plot the result, annotate the file, and save the file\nax.set_xlabel(r'$N_{steps}$')\nax.set_ylabel(r'$M$')\nax.set_ylim([-1.05, 1.05])\ni = 0\nfor mag in mags:\n # pass the value of beta into the plot command to generate the label, as in\n ax.plot(np.arange(len(mag)), mag, label='beta=%.2f'%betas[i])\n i += 1\nax.legend()\nplt.savefig(file)\nfig.show()\n\n\n\n```\n\n#### 8. Ensemble-averaged magnetisation, $\\bar M$, as a function of $\\beta$\n\nOnce the system has thermalized (a good choice is $N_\\mathrm{therm} =400$ thermalisation sweeps), measure the time-averaged magnetisation over 1000 sweeps. From this, estimate the ensemble-averaged magnetisation, $\\bar M$. Plot $\\bar M$ as a function of $1/\\beta$ for $\\beta = 1.6, 1.3, 1.1, 1.0, 0.9, 0.8, 0.7, 0.6, 0.4, 0.1$. Include this plot in the PDF you hand in. Save the plot to a file 'Magnetisation.pdf'.\n\n\nPerform the following steps:\n - for each of the listed values of beta:\n - create a $32\\times32$ random grid\n - sweep the for N=400 initial 'thermalization' steps\n - **then** sweep for another 1000 steps, calculating $M$ after every sweep\n - use this to compute the ensemble-averaged magnitization, $\\bar M$ for that value of $\\beta$\n - plot $\\bar M$ as a function of $\\beta^{-1}$\n\n**4 marks**\n\n\n\n```python\n# Step 8: Magnetisation\n\n\n# set-up the figure\nfile = \"Magnetisation.pdf\" \nfig, ax = plt.subplots(1,1, figsize = (8, 5))\n\n# the range of values of beta\nsize = 32\nbetas = [1.6,1.3,1.1,1.0,0.9,0.8,0.7,0.6,0.4,0.1]\nmean_mags = []\n# Loop over values of beta, computing the ensemble-averaged M for each beta\n# name the resulting ensemble-averaged M mean_mags\n# It is an array with the same dimension as betas\nfor beta in betas:\n grid = Grid(size, beta)\n mean_mags.append(np.mean(grid.do_sweeps(400, 1000)))\n# YOUR CODE HERE\n\n# make the plot\nax.set_xlabel(r'$\\beta^{-1}$')\nax.set_ylabel(r'$\\bar M$')\nax.set_ylim([-1.05, 1.05])\nax.set_xlim(0,4)\nax.set_title(r'$\\beta^{-1}-\\bar M$')\nplt.plot(1./np.array(betas), mean_mags)\nplt.savefig(file)\nfig.show()\n\n```\n\n#### 9. Critical temperature\n\nThe critical temperature, $T_c$, in the mean field approximation, is given by\n$$\nT_c=\\frac{h J}{2k_B},\n$$\nwhere h is the number of nearest-neighbours, as discussed in the lecture. Use this to calculate the critical value $\\beta_c$ which corresponds to the critical temperature and mark it in the plot produced in step **8** (If you could not produce the plot in step **8** you may also mention the numerical value of $\\beta_c$ on the solution you hand in).\n\nEnter your answer in the box below. If you don't see a box, execute the hidden cell below.\n\n**2 marks**\n\n\n```python\nbeta_crit=mywidgets.myFloatBox('Phasetransition','P1','beta_c=', 'Enter your analytically calculated value of beta_c to 3 sig figs')\nbeta_crit.getWidget()\n```\n\n\n VBox(children=(HTMLMath(value='Enter your analytically calculated value of beta_c to 3 sig figs', placeholder=\u2026\n\n\n\n```python\n\n```\n\n#### 10. Mean field approximation\n\nThe ensemble-averaged value of the mean magnitization in the mean field approximation, is the solution of\n$$\n\\bar M - \\tanh\\left( \\frac{T_c\\bar M}{T} \\right) = 0\\,,\n$$\nas discussed in the lecture. This equation can not be solved analytically for $\\bar M$, for given\n$T_c/T$.\n\nRewrite the equation in terms of $\\beta$ using the relation between $T$ and $\\beta$ derived before, and solve the resulting formula numerically for $\\beta = 1.6, 1.3, 1.1, 1.0, 0.9, 0.8, 0.7, 0.6, 0.4, 0.1$. You may want to use the root finding methods implemented in previous exercises. Redo the plot of step **8**, but on top of the numerical result, over plot the mean field approximation. Add a legend to the plot which allows to distinguish the solution obtained in step **8** from the mean field approximation.\n\nThe numerical value of $\\bar M$ versus $\\beta$ shows that the transition from $\\bar M=0$ at hight $T$ to $\\bar M>0$ at low $T$ is not infinitely sharp. To quantify where the transition occurs, it might be useful to compute\n$\\beta_{1/2}$, the value of $\\beta$ where $\\bar M=1/2$. Calculate this for both the numerical and mean field approximation, and indicate the point $(\\beta,\\bar M)=(\\beta_{1/2}, 1/2)$ on the plot.\n\nSave the plot as 'MeanField.pdf'\n\n\n**4 marks**\n\n\n```python\n# Implement the mean field calculation here: calculate mean_mag_MF, the mean field approximation to the magnetisation\n# for a given value of beta. Also implement the calculation of the critical value, beta_c, according to the\n# MFA\n\nfrom scipy.optimize import fsolve\n# YOUR CODE HERE\ndef solve_m(M, beta):\n return M - np.math.tanh(M * 2 * beta)\n\ndef solve_beta(beta, M):\n return M - np.math.tanh(M * 2 * beta)\n\nbetas = [1.6,1.3,1.1,1.0,0.9,0.8,0.7,0.6,0.4,0.1]\nbetas_tmp = []\nfor beta in betas:\n for i in range(3):\n betas_tmp.append(beta)\nmean_mag_MF = []\nfor beta in betas:\n mean_mag_MF.append(fsolve(solve_m, 1, beta)[0])\n mean_mag_MF.append(fsolve(solve_m, 0, beta)[0])\n mean_mag_MF.append(fsolve(solve_m, -1, beta)[0])\nbeta_half = fsolve(solve_beta, 0, 0.5)[0]\nT = 1 / beta_half\n# raise NotImplementedError()\n```\n\n\n```python\n# YOUR CODE HERE\nfig, ax = plt.subplots(1,1, figsize = (8, 5))\nfile = \"MeanField.pdf\" \nax.set_xlabel(r'$\\beta$')\nax.set_ylabel(r'$\\bar M$')\nax.set_ylim([-1.05, 1.05])\nplt.plot(np.array(betas), mean_mags, label=\"experiment data line\")\nplt.plot(np.array(betas_tmp), mean_mag_MF, '*', label=\"numerical solution for given points\")\nplt.plot(1/T, 0.5, 'x', label=\"beta value when M_mean=1/2\")\nplt.title(r'$\\beta$-$\\bar M$')\nplt.legend()\nplt.savefig(file)\nfig.show()\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "ab0e8736687b0a5155bafaa4e723436e95eda814", "size": 151713, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Core 1/scientific computing/ipython notebook/Phase Transition 19/phase_transition.ipynb", "max_stars_repo_name": "hwhv66/CourseworkDurham", "max_stars_repo_head_hexsha": "eec812d38776cf61c6fd7925e5706c8698926fa3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Core 1/scientific computing/ipython notebook/Phase Transition 19/phase_transition.ipynb", "max_issues_repo_name": "hwhv66/CourseworkDurham", "max_issues_repo_head_hexsha": "eec812d38776cf61c6fd7925e5706c8698926fa3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Core 1/scientific computing/ipython notebook/Phase Transition 19/phase_transition.ipynb", "max_forks_repo_name": "hwhv66/CourseworkDurham", "max_forks_repo_head_hexsha": "eec812d38776cf61c6fd7925e5706c8698926fa3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-11-24T06:35:40.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-24T06:35:40.000Z", "avg_line_length": 97.7532216495, "max_line_length": 51868, "alphanum_fraction": 0.8256378821, "converted": true, "num_tokens": 7800, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573376, "lm_q2_score": 0.7371581568543044, "lm_q1q2_score": 0.44243373413418646}} {"text": "```python\n# %load imports.py\n## Local packages:\n\n%matplotlib inline\n%load_ext autoreload\n%autoreload 2\n%config Completer.use_jedi = False ## (To fix autocomplete)\n\n## External packages:\nimport pandas as pd\npd.options.display.max_rows = 999\npd.options.display.max_columns = 999\npd.set_option(\"display.max_columns\", None)\nimport numpy as np\nnp.set_printoptions(linewidth=150)\n\nimport numpy as np\nimport os\nimport matplotlib.pyplot as plt\n#if os.name == 'nt':\n# plt.style.use('presentation.mplstyle') # Windows\n\nimport plotly.express as px \nimport plotly.graph_objects as go\n\nimport seaborn as sns\nimport sympy as sp\nfrom sympy.physics.mechanics import (dynamicsymbols, ReferenceFrame,\n Particle, Point)\nfrom sympy.physics.vector.printing import vpprint, vlatex\nfrom IPython.display import display, Math, Latex\nfrom src.substitute_dynamic_symbols import run, lambdify\n\nimport pyro\n\nimport sklearn\nimport pykalman\nfrom statsmodels.sandbox.regression.predstd import wls_prediction_std\nimport statsmodels.api as sm\n\nfrom scipy.integrate import solve_ivp\n\n## Local packages:\nfrom src.data import mdl\n\nfrom src.symbols import *\nfrom src.parameters import *\nimport src.symbols as symbols\nfrom src import prime_system\nfrom src.models import regression\nfrom src.visualization.regression import show_pred\nfrom src.visualization.plot import track_plot\n\n## Load models:\n# (Uncomment these for faster loading):\nimport src.models.vmm_martin as vmm \nfrom src.models.vmm import ModelSimulator\nfrom example_ship2 import ship_parameters, df_parameters\n\n\nif os.name == 'nt':\n plt.style.use('../docs/book/book.mplstyle') # Windows\n \nfrom src.visualization.plot import track_plot, plot\n```\n\n\n```python\nfrom src.extended_kalman_vmm import ExtendedKalman\nfrom src.extended_kalman_filter import loglikelihood\n```\n\n## Load test\n\n\n```python\nid=22773\ndf, units, meta_data = mdl.load(dir_path = '../data/raw', id=id)\ndf.index = df.index.total_seconds()\ndf.index-=df.index[0]\ndf['x0']-=df.iloc[0]['x0']\ndf['y0']-=df.iloc[0]['y0']\ndf['psi']-=df.iloc[0]['psi']\n\n```\n\n\n```python\nfig,ax=plt.subplots()\nfig.set_size_inches(10,10)\ntrack_plot(df=df, lpp=meta_data.lpp, x_dataset='x0', y_dataset='y0', psi_dataset='psi', beam=meta_data.beam, ax=ax);\n```\n\n## Filtering\n\n### Lowpass filter\n\n\n```python\nfrom numpy import cos as cos\nfrom numpy import sin as sin\nfrom src.data.lowpass_filter import lowpass_filter\n\ndf_lowpass = df.copy()\nt = df_lowpass.index\nts = np.mean(np.diff(t))\nfs = 1/ts\n\nposition_keys = ['x0','y0','psi']\nfor key in position_keys:\n df_lowpass[key] = lowpass_filter(data=df_lowpass[key], fs=fs, cutoff=1, order=1)\n\ndf_lowpass['x01d_gradient'] = x1d_ = np.gradient(df_lowpass['x0'], t)\ndf_lowpass['y01d_gradient'] = y1d_ = np.gradient(df_lowpass['y0'], t)\ndf_lowpass['r'] = r_ = np.gradient(df_lowpass['psi'], t)\n\npsi_ = df_lowpass['psi']\n\ndf_lowpass['u'] = x1d_*cos(psi_) + y1d_*sin(psi_)\ndf_lowpass['v'] = -x1d_*sin(psi_) + y1d_*cos(psi_)\n\nvelocity_keys = ['u','v','r']\nfor key in velocity_keys:\n df_lowpass[key] = lowpass_filter(data=df_lowpass[key], fs=fs, cutoff=1, order=1)\n```\n\n\n```python\ndata = df.copy()\ndata['thrust'] = data['Prop/PS/Thrust'] + data['Prop/SB/Thrust']\n\ndata['u'] = df_lowpass['u']\ndata['v'] = df_lowpass['v']\ndata['r'] = df_lowpass['r']\ndata=data.iloc[200:-100]\ndata.index-=data.index[0]\n```\n\n### Extended Kalman filter\n\n\n```python\nparameters = df_parameters[\"prime\"].copy()\nek = ExtendedKalman(vmm=vmm, \n parameters=parameters, \n ship_parameters=ship_parameters)\n\n```\n\n#### Simulate with system model\n\n\n```python\nE = np.array([\n [0,0,0],\n [0,0,0],\n [0,0,0],\n [1,0,0],\n [0,1,0],\n [0,0,1],\n ],\n)\n\nt = np.linspace(0,10,100)\ndata_ = pd.DataFrame(index=t)\ndata_['delta'] = 0.0\ndata_['thrust'] = 20\ndata_['x0'] = 0\ndata_['y0'] = 0\ndata_['psi'] = 0\ndata_['u'] = 2\ndata_['v'] = 0\ndata_['r'] = 0\n\nek.simulate(data=data, input_columns=['delta','thrust'], \n E=E).tail()\n```\n\n\n```python\ndata_frames = {'data':data, 'sim':ek.df_simulation}\n\nfig,ax=plt.subplots()\nstyles = {\n 'Mesurement' : {\n 'linestyle' : '',\n 'marker' : '.',\n 'ms' : 1,\n 'zorder':-10,\n },\n \n 'Kalman filter' : {\n 'lw' : 2,\n },\n \n \n}\n\nfor label,df_ in data_frames.items():\n track_plot(\n df=df_,\n lpp=ship_parameters[\"L\"],\n beam=ship_parameters[\"B\"],\n ax=ax,\n label=label,\n plot_boats=True,\n **styles.get(label,{})\n );\nax.legend()\n\nplot(data_frames, keys=ek.df_simulation.columns);\n```\n\n### Extended Kalman filter and RTS smoother\n\n\n```python\nCd = np.array([\n [1, 0, 0, 0, 0, 0],\n [0, 1, 0, 0, 0, 0],\n [0, 0, 1, 0, 0, 0],\n])\n\nE = np.array([\n [0,0,0],\n [0,0,0],\n [0,0,0],\n [1,0,0],\n [0,1,0],\n [0,0,1],\n ],\n)\n\nP_prd = np.diag([0.1, 0.1, np.deg2rad(0.01), 0.01, 0.01, np.deg2rad(0.01)])\nQd = np.diag([0.01, 0.01, np.deg2rad(0.01)]) #process variances: u,v,r\n\nerror_max_pos = 0.1\nsigma_pos = error_max_pos/3\nvariance_pos = sigma_pos**2\n\nerror_max_psi = np.deg2rad(0.1)\nsigma_psi = error_max_psi/3\nvariance_psi = sigma_psi**2\n\nRd = np.diag([variance_pos, variance_pos, variance_psi])\n\nek.filter(\n data=data, P_prd=P_prd, Qd=Qd, Rd=Rd, E=E, Cd=Cd, \n input_columns=['delta','thrust'],\n )\n\n\nek.smoother();\n```\n\n\n```python\ndataframes = {\n 'Mesurement' : data,\n #'Kalman filter' : ek.df_kalman,\n 'RTS': ek.df_smooth,\n}\n\nstyles = {\n 'Mesurement' : {\n 'linestyle' : '',\n 'marker' : '.',\n 'ms' : 1,\n 'zorder':-10,\n },\n \n 'Kalman filter' : {\n 'lw' : 2,\n },\n \n \n}\n\nplot(dataframes = dataframes, \n fig_size=(10,10), \n styles = ['r-','g-','b-'],\n keys=['x0','y0','psi','u','v','r','u1d','v1d','r1d']);\n```\n\n\n```python\nek2 = ek.copy()\n```\n\n\n```python\nloglikelihood(ek2.time_steps_smooth)\n```\n\n\n```python\nek2.parameters\n```\n\n\n```python\ndef vary(parameters):\n \n ek2.parameters = parameters\n \n ek2.filter(\n data=data, P_prd=P_prd, Qd=Qd, Rd=Rd, E=E, Cd=Cd, \n input_columns=['delta','thrust'],\n )\n\n\n ek2.smoother();\n \n return ek2\n \n```\n\n\n```python\nfrom scipy.optimize import minimize\n\n\ndef _vary(x, keys:list):\n parameters = deepcopy(ek.parameters)\n parameters[keys] = x\n \n ek_ = vary(parameters)\n return ek_\n \ndef fun(x,keys:list):\n ek_ = _vary(x, keys)\n return -loglikelihood(ek_.time_steps_smooth)\n \n```\n\n\n```python\nfrom copy import deepcopy\n\neks = []\nkey = 'Ndelta'\nkeys = [key]\nvalues = np.linspace(0.5*ek.parameters[key], 1.5*ek.parameters[key], 5)\nfor value in values:\n \n ek_ = _vary(value, keys)\n eks.append(ek_.copy())\n \n```\n\n\n```python\nloglikelihoods = []\n\nfor ek_ in eks:\n \n loglikelihoods.append(loglikelihood(ek_.time_steps_smooth))\n \n```\n\n\n```python\nfig,ax=plt.subplots()\n\nax.plot(values, loglikelihoods)\nax.set_xlabel(key)\nax.set_ylabel('loglikelihood')\n```\n\n\n```python\nek.parameters.values\n```\n\n\n```python\n%%time\n#keys = ek.parameters.keys()\n#keys = ['Ndelta']\n\nkeys = [\n'Xu', \n'Ndelta', \n'Nv',\n'Nr',\n]\n\nparameters = ek.parameters[keys]\nx0 = parameters.values\n#bounds = [(-np.inf,0.0001) if value < 0 else (0.0001,np.inf) for value in parameters.values]\n\nsmall=0.5\nlarge=3\nbounds = [(large*value,small*value) if value < 0 else (small*value,large*value) for value in parameters.values]\n\n#res = minimize(fun, x0=x0, bounds=bounds, tol=10**6, args=(keys,))\n\nres = minimize(fun, x0=x0, tol=10**6, args=(keys,))\n```\n\n\n```python\nres\n```\n\n\n```python\nx0\n```\n\n\n```python\nek_ = _vary(res.x, keys=keys)\nek_.simulate(data=data, input_columns=['delta','thrust'], \n E=E).tail()\n```\n\n\n```python\n#keys = [\n#'Xu', \n#'Xvr', \n#'Ydelta', \n#'Yv', \n#'Yur', \n#'Ndelta', \n#'Nv',\n#]\n#\n#x = [-0.003 , -0.006 , 0.003 , -0.01231641, 0.00413222, -0.0015 , -0.00318395]\n#ek_ = _vary(x, keys=keys)\n#ek_.simulate(data=data, input_columns=['delta','thrust'], \n# E=E, x0=data.iloc[0][[\"x0\", \"y0\", \"psi\", \"u\", \"v\", \"r\"]].values).tail()\n```\n\n\n```python\ndata_frames = {'data':data, 'sim':ek_.df_simulation}\n\nfig,ax=plt.subplots()\nstyles = {\n 'Mesurement' : {\n 'linestyle' : '',\n 'marker' : '.',\n 'ms' : 1,\n 'zorder':-10,\n },\n \n 'Kalman filter' : {\n 'lw' : 2,\n },\n \n \n}\n\nfor label,df_ in data_frames.items():\n track_plot(\n df=df_,\n lpp=ship_parameters[\"L\"],\n beam=ship_parameters[\"B\"],\n ax=ax,\n label=label,\n plot_boats=True,\n **styles.get(label,{})\n );\nax.legend()\n\nplot(data_frames, keys=ek.df_simulation.columns);\n```\n\n\n```python\n%%time\ninput = {'delta':0,'thrust':1.0}\npsi = 0\nu = 2\nv = 0\nr = 0\nx_dot = run(\n ek._lambda_f,\n **ek.parameters,\n **ek.ship_parameters,\n **input,\n psi=psi,\n u=u,\n v=v,\n r=r,\n )\n```\n\n\n```python\nek._lambda_f\n```\n\n\n```python\nparameters = deepcopy(ek.parameters)\nship_parameters = deepcopy(ek.ship_parameters)\n\nfrom inspect import signature\ns = signature(ek._lambda_f)\nkeys = list(set(ek.ship_parameters) & set(s.parameters.keys()))\nship_parameters = {key:value for key,value in ek.ship_parameters.items() if key in keys}\n \n\n```\n\n\n```python\n%%time\n\ninput = {'delta':0,'thrust':1.0}\npsi = 0\nu = 2\nv = 0\nr = 0\n\nek._lambda_f(**parameters,\n **input,\n **ship_parameters,\n psi=psi,\n u=u,\n v=v,\n r=r,)\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "0c3a9265281af92015e7fc60311bf03a5b8d86a0", "size": 19451, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/15.49_EK_smoother_3DOF_thrust.ipynb", "max_stars_repo_name": "martinlarsalbert/wPCC", "max_stars_repo_head_hexsha": "16e0d4cc850d503247916c9f5bd9f0ddb07f8930", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 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YES\n2. YES", "lm_q1_score": 0.6513548646660543, "lm_q2_score": 0.6791786926816161, "lm_q1q2_score": 0.44238634545570177}} {"text": "\n\n\n# Tutorial 1 Part 1 - Sequential Probability Ratio Test\n\nPlease execute the cell below to initialize the notebook environment\n\n\n```python\nimport numpy as np # import numpy\nimport scipy as sp # import scipy\nfrom scipy import stats\n\nimport matplotlib.pyplot as plt # import matplotlib\n```\n\n\n```python\n#@title Figure Settings\n#@ Figure Settings\n%matplotlib inline\nfig_w, fig_h = (8, 6)\nplt.rcParams.update({'figure.figsize': (fig_w, fig_h)})\n# %config InlineBackend.figure_format = 'retina'\n```\n\n\n```python\n#@title Helper functions\ndef simulate_and_plot_SPRT_fixedtime(sigma, stop_time, num_samples):\n \"\"\"Helper function for Exercise 1\"\"\"\n\n evidence_history_list = []\n print(\"#Trial\\tTotal_Evidence\\tDecision\")\n for i in range(num_sample):\n evidence_history, decision, data = simulate_SPRT_fixedtime(sigma, stop_time)\n print(\"{}\\t{:f}\\t{}\".format(i,evidence_history[-1], decision))\n evidence_history_list.append(evidence_history)\n\n fig, ax = plt.subplots()\n maxlen_evidence = np.max(list(map(len,evidence_history_list)))\n ax.plot(np.zeros(maxlen_evidence),'--',c='red',alpha=1.0)\n for evidences in evidence_history_list:\n ax.plot(np.arange(len(evidences)), evidences)\n ax.set_xlabel(\"Time\")\n ax.set_ylabel(\"Cumulated log likelihood ratio\")\n ax.set_title(\"Log likelihood ratio trajectories under the fixed-time stopping rule\")\n \n plt.show(fig)\n\ndef simulate_and_plot_accuracy_vs_stoptime(sigma, stop_time_list, num_sample):\n \"\"\"Helper function for Exercise 2\"\"\"\n accuracy_list, _ = simulate_accuracy_vs_stoptime(sigma, stop_time_list, num_sample=num_sample)\n \n fig, ax = plt.subplots()\n ax.plot(stop_time_list,accuracy_list)\n ax.set_xlabel('Stop Time')\n ax.set_ylabel('Average Accuracy')\n\n plt.show(fig)\n\ndef simulate_and_plot_SPRT_fixedthreshold(sigma, num_sample, alpha):\n \"\"\"Helper function for Exercise 3\"\"\"\n # calculate evidence threshold from error rate\n threshold = threshold_from_errorrate(alpha)\n\n # run simulation\n evidence_history_list = []\n\n print(\"#Trial\\tTime\\tCumulated Evidence\\tDecision\")\n for i in range(num_sample):\n evidence_history, decision, data = simulate_SPRT_threshold(sigma, threshold, batch_size=10)\n print(\"{}\\t{}\\t{:f}\\t{}\".format(i,len(data), evidence_history[-1], decision))\n evidence_history_list.append(evidence_history)\n\n fig, ax = plt.subplots()\n maxlen_evidence = np.max(list(map(len,evidence_history_list)))\n ax.plot(np.repeat(threshold,maxlen_evidence + 1),c=\"red\")\n ax.plot(-np.repeat(threshold,maxlen_evidence + 1),c=\"red\")\n ax.plot(np.zeros(maxlen_evidence + 1),'--',c='red',alpha=0.5)\n\n for evidences in evidence_history_list:\n ax.plot(np.arange(len(evidences) + 1), np.concatenate([[0],evidences]))\n \n ax.set_xlabel(\"Time\")\n ax.set_ylabel(\"Cumulated log likelihood ratio\")\n ax.set_title(\"Log likelihood ratio trajectories under the threshold rule\")\n \n plt.show(fig)\n\ndef simulate_and_plot_speed_vs_accuracy(sigma, threshold_list, num_sample):\n \"\"\"Helper function for Exercise 4\"\"\"\n accuracy_list, decision_speed_list = simulate_accuracy_vs_speed(sigma, threshold_list, num_sample=num_sample)\n\n fig, ax = plt.subplots()\n ax.plot(decision_speed_list, accuracy_list, linestyle=\"--\", marker=\"o\")\n ax.plot([0.0, 1.0], [0.5, 0.5], c='red') # plot baseline for random choice\n ax.set_xlabel(\"Average Decision Speed\")\n ax.set_ylabel('Average Accuracy')\n ax.set_title(\"Speed/Accuracy Tradeoff\")\n ax.set_ylim(bottom=0.45)\n \n plt.show(fig)\n```\n\n# Introduction \n\n\n```python\n#@title Video 1 \n# Insert the ID of the corresponding youtube video\nfrom IPython.display import YouTubeVideo\nvideo = YouTubeVideo(id=\"3CG92WsTk4Q\", width=854, height=480, fs=1)\nprint(\"Video available at https://youtu.be/\" + video.id)\nvideo\n```\n\n Video available at https://youtu.be/3CG92WsTk4Q\n\n\n\n\n\n\n\n\n\n\n\n### Tutorial objectives\nIn this tutorial, we will consider a simplified random dot motion task. On each trial $i$, we are shown a dot moving at velocity $v_i$, either in either a leftward ($v<0$) or rightward ($v>0$) direction. Although the dots' velocity varies from trial to trial, the set of all $v_i$ are generated by a fixed probability distribution, which we know to be either:\n$$\n\\\\\n\\begin{eqnarray}\np_L &=& \\mathcal{N}(-1,\\; \\sigma^2) \\\\\n&&\\textrm{or} \\\\\np_R &=& \\mathcal{N}(+1,\\; \\sigma^2) \\\\\n\\end{eqnarray} \n\\\\\n$$\nWe want to determine which of these two possibilities is the true data generating distribution. \n\nIn W2D1, we learned how to choose between possibilities based on the relative weight of the evidence. Depending on the sensory evidence and our prior experience, we learned to choose between these *two* options: accepting hypothesis $H_L$, that the data comes from the $p_L$ distribution, or accepting $H_R$, that it comes from $p_R$. \n\nHere, we add a *third* option: choose to collect more evidence before making a decision.\n\n---\n\nIn this notebook we will perform a *Sequential Probability Ratio Test* between two hypotheses $H_L$ and $H_R$ by running simulations of a *Drift Diffusion Model (DDM)*. \n\nAs independent and identically distributed(*i.i.d*) samples from the true data-generating distribution coming in, we accumulate our evidence linearly until a certain criterion is met before deciding which hypothesis to accept. Two types of stopping criterion/stopping rule will be implemented: after seeing a fixed amount of data, and after the likelihood ratio passes a pre-defined threshold. Due to the noisy nature of observations, there will be a *drifting* term governed by expected mean output and a *diffusion* term governed by observation noise.\n\nIn this tutorial, you will\n\n* Simulate Drift-Diffusion Model with different stopping rules.\n* Observe the relation between accuracy and reaction time, get an intuition about the speed/accuracy tradeoff.\n\n\n---\n\n### Sequential Probability Ratio Test(SPRT)\n\n\n\n\n\nSuppose we receive a sequence of independent samples from distribution $p$. We know that $p$ is from $\\{p_0,p_1\\}$ determined by a binary latent variable $x$ and need to test between the two hyptheses:\n\n$H_L: p=p_L \\text{ or } x=0$\n\n$H_R: p=p_R \\text{ or } x=1$\n\nWhen we see $n$ samples $\\{x_{1}...x_n\\}$, we want to calculate the total log likelihood ratio as our evidence for decision:\n\n$$S_n = \\log \\frac{\\prod_{i=1}^n p_R(x_i)}{\\prod_{i=1}^n p_L(x_i)} = \\sum_{i=1}^n \\log p_R(x_i) - \\sum_{i=}^n \\log p_L(x_i) \\tag{1}$$\n\nDue to the independence of samples, this can be calculated in a incremental way when new data points come in sequentially:\n\n$$ S_n = S_{n-1} + \\log \\frac{p_R(x_n)}{p_L(x_n)} = S_{n-1} + \\log \\Lambda_n \\tag{2}$$\n\nThe stopping rule can be implemented in two ways:\n\n\n\n1. Fixed time \n\nMake a decision based on $S_n$ immediately when we have collected $n$ samples. That is, accept $H_R$ if $S_n > 0$, accept $H_L$ if $S_n < 0$, and accept $H_R$ with probability $\\frac{1}{2}$ if $S_n = 0$. The significance level or desired error rate $\\alpha$ can then be determined as \n\n$$\\alpha = \\frac{1}{1+\\exp(|S_n|)} \\tag{4}$$\n\n2. Thresholding \n\nWe assume equal probability to make a false positive decision and to make a false negative decision, and denote it with $\\alpha$. Then we accept hypothesis $H_R$ if $S_n \\ge b$ or accept hypothesis $H_L$ if $S_n \\le a$ where the thresholds are determined by \n\n$$a=\\log \\frac{\\alpha}{1-\\alpha},b=\\log \\frac{1-\\alpha}{\\alpha} \\tag{3}$$\n\n\n# SPRT as a Drift Diffusion Model (DDM)\n\n\n```python\n#@title Video 2\nfrom IPython.display import YouTubeVideo\nvideo = YouTubeVideo(id=\"a-O5n_d5tOA\", width=854, height=480, fs=1, start=0, end=40)\nprint(\"Video available at https://youtu.be/\" + video.id)\nvideo\n```\n\n Video available at https://youtu.be/a-O5n_d5tOA\n\n\n\n\n\n\n\n\n\n\n\n\n\nSPRT as a Drift Diffusion Model (DDM)\n\nLet's assume two different Gaussian observation models conditioned on discrete latent variable $z$ \n\n$$p_L(x|z=0) = \\mathcal{N}(\\mu_L,\\sigma_L^2)$$\n\n$$p_R(x|z=1) = \\mathcal{N}(\\mu_R,\\sigma_R^2)$$\n\nThen the log likelihood ratio for a single data point $x_i$ is \n\n$$ \\log \\Lambda_i = \\log \\frac{\\sigma_L}{\\sigma_R} -0.5 \\biggr[\\frac{(x_i-\\mu_1)^2}{\\sigma_R^2} - \\frac{(x_i-\\mu_L)^2}{\\sigma_L^2}\\biggl] \\tag{5}$$\n\nWithout loss of generality, let's further assume the true data generating distribution is $p_R$. In this case $x_i$ can be expressed as $x_i = \\mu_R + \\sigma_R \\epsilon$ where $\\epsilon$ comes from a standard Gaussian. The foregoing formula can then be rewritten as \n\n$$\n\\log \\Lambda_i = \\bigg( \\log \\frac{\\sigma_L}{\\sigma_R} + 0.5 \\frac{(\\mu_R-\\mu_L)^2}{\\sigma_L^2} \\bigg) + \\bigg( \\frac{\\mu_R-\\mu_L}{\\sigma_L^2}\\epsilon -0.5\\big[1-(\\frac{\\sigma_R}{\\sigma_L})^2\\big]\\epsilon^2 \\bigg) \\tag{6}\n$$\n\nwhere the first two constant terms serve as the *drifting* part and the last terms are the *diffusion* part. If we further let $\\sigma_L=\\sigma_R$, we can get rid of the quadratic term and this reduces to the classical discrete drift-diffusion equation where we have analytical solutions for mean and expected auto-covariance:\n\n$$\n\\log \\Lambda_i = 0.5 \\frac{(\\mu_R-\\mu_L)^2}{\\sigma_L^2} + \\frac{\\mu_R-\\mu_L}{\\sigma_L^2}\\epsilon, \\text{where } \\epsilon \\sim \\mathcal{N}(0,1)\n$$\n\n\n\n# 1. Simulating DDM with fixed-time stopping rule\n\n\n\n```python\n#@title Video 3a\nfrom IPython.display import YouTubeVideo\nvideo = YouTubeVideo(id=\"TpqMzI0iZJ8\", width=854, height=480, fs=1, start=0, end=40)\nprint(\"Video available at https://youtu.be/\" + video.id)\nvideo\n```\n\n Video available at https://youtu.be/TpqMzI0iZJ8\n\n\n\n\n\n\n\n\n\n\n\nExercise 1\n---\n\nAssume we are performing a random dot motion task and at each time we see a moving dot with velocity $x_t$. All data points are sampled from the same distribution $p$, which is either $p_L=\\mathcal{N}(-\\mu,\\sigma^2)$ or $p_R=\\mathcal{N}(\\mu,\\sigma^2)$. Let's now generate some simulated data under this setting and perform SPRT using the fixed time stopping rule. \n\nIn this exercise, without loss of generality, we assume the true data-generating model is $p_R$.\n\n* Complete the code in function `simulate_SPRT_fixedtime` to create two Gaussian random variables to represent our observation models.\n* Complete the function `log_likelihood_ratio` to calculate log likelihood ratio for a sequence of data points\n* Complete the code in function `simulate_SPRT_fixedtime` to calculate cumulated evidence given a list of individual evidences\n* Run 10 simulations and plot the DDM traces by commenting out our provided code\n\n\n\n\n```python\ndef log_likelihood_ratio(xvec,p0,p1):\n \"\"\"\n Given a sequence(vector) of observed data, calculate the log of likelihood ratio of p1 and p0\n\n Args:\n xvec (numpy vector): a vector of scalar measurements \n p0 (Gaussian random variable): a normal random variable with `logpdf` method \n p1 (Gaussian random variable): a normal random variable with `logpdf` method \n\n Returns:\n llvec: a vector of log likelihood ratios for each input data point\n \"\"\"\n ################################################################################\n ## Insert your code here to:\n ## Calculate the log of likelihood ratios of `p1` and `p0` given a vector of measurements\n ## Hint: using `p1.logpdf`\n ################################################################################\n raise NotImplementedError(\"Function `log_likelihood_ratio` incomplete\")\n\n\ndef simulate_SPRT_fixedtime(sigma, stop_time, true_dist=1):\n \"\"\"\n Simulate a Sequential Probability Ratio Test with fixed time stopping rule.\n Two observation models are 1D Gaussian distributions N(1,sigma^2) and N(-1,sigma^2).\n\n Args:\n sigma (float): standard deviation \n stop_time (int): number of samples to take before stopping\n\n Returns:\n evidence_history (numpy vector): the history of cumulated evidence given generated data\n decision (int): 1 for pR, 0 for pL\n data (numpy vector): the generated sequences of data in this trial \n \"\"\"\n muL = -1.0\n muR = 1.0 \n ################################################################################\n ## Insert your code here to:\n ## Create two Gaussian variables `pL` and `pR` with mean `muL` and `muR` respectively and same std. `sigma`\n ## Hint: using `stats.norm(...)` to construct an instance of 1D Gaussian distribution\n ################################################################################\n\n\n if 'pL' not in locals() or 'pR' not in locals():\n raise NotImplementedError(\"Function `simulate_SPRT_fixedtime` incomplete\")\n\n # Generate a random sequence of data \n if true_dist == 1:\n data = pR.rvs(size=stop_time)\n else: \n data = pL.rvs(size=stop_time)\n \n # Calculate cumulated evidence\n ll_vec = log_likelihood_ratio(data, pL, pR)\n ################################################################################\n ## Insert your code here to:\n ## Calculate cumulated evidence given a vector of individual evidences \n ## Hint: use `np.cumsum`\n ################################################################################\n evidence_history = None\n\n if evidence_history is None:\n raise NotImplementedError(\"Function `simulate_SPRT_fixedtime` incomplete\")\n\n # Make decision\n if evidence_history[-1] > 0:\n decision = 1\n elif evidence_history[-1] < 0:\n decision = 0 \n else: \n decision = np.random.randint(2)\n \n return evidence_history, decision, data \n\nsigma = 3.5 # standard deviation for pL and pR\nnum_sample = 10 # number of simulations to run for each stopping time\nstop_time = 150 # stopping time\n\n\n#######p#########################################################################\n## Un-comment the following code block after completing this exercise\n################################################################################\n#plot_and_simulate_SPRT_fixedtime(sigma, stop_time, num_sample)\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W2D3_DecisionMaking/solutions/W2D3_Tutorial1_Solution_a06a0721.py)\n\n*Example output:*\n\n\n\n\n\nNow let's look at how the dynamics change if you change the noise level and stopping time. \n\n\n\n* Play with different noise levels and stopping times and observe the corresponding trajectories. Once you have completed exercise 1, check the box to enable the demo.\n\n*Hint*: you can click `...`>`Form`>`Hide code` to hide the code section if the slider is too far from the figure\n\n\n```python\n#@title # Interactive Demo: Noise level and stopping time { run : \"auto\" }\n\nsigma = 4 #@param {type:\"slider\", min:0.05, max:10.0, step:0.05}\nnum_sample = 10 # number of simulations to run for each stopping time\nstop_time = 394 #@param {type:\"slider\", min:5, max:500, step:1}\n################################################################################\n## Un-comment the following code block after complete the exercise above\n################################################################################\ntry:\n simulate_and_plot_SPRT_fixedtime(sigma, stop_time, num_sample)\nexcept:\n print(\"Complete Exercise 1 first\")\n```\n\n# 2. Accuracy vs. Stopping time\n\n\n\n```python\n#@title Video 3b\nfrom IPython.display import YouTubeVideo\nvideo = YouTubeVideo(id=\"TpqMzI0iZJ8\", width=854, height=480, fs=1, start=41)\nprint(\"Video available at https://youtu.be/\" + video.id)\nvideo\n```\n\n Video available at https://youtu.be/TpqMzI0iZJ8\n\n\n\n\n\n\n\n\n\n\n\nExercise 2\n---\n\nIf you stop taking samples too early, e.g., make a decision after only seeing 5 samples, or there's a huge amount of observation noise that buries the signal, you are likely to be driven by observation noise to a negative cumulated log likelihood ratio and thus make a wrong decision. You could get a sense of this by increasing noise level or decreasing stopping time in the last exercise.\n\nNow let's look at how decision accuracy varies with the number of samples we see quantitatively. First we'll fix our observation noise level. In this exercise you will run several repeated simulations for a certain stopping time to calculate the average decision accuracy. Do this for a range of stopping times and plot the relation between average decision accuracy and stopping time. You should get a positive correlation between these two quantities.\n\n* Choose a noise level. For example, $\\sigma=3$ \n* Complete the function `simulate_accuracy_vs_stoptime` to simulate and compute corresponding average accuracies for a list of stopping times.\n* Plot accuracy versus stopping time using the pre-written codes \n\n\n\n\n\n```python\nsigma = 4.65 # standard deviation for observation noise\nnum_sample = 200 # number of simulations to run for each stopping time\n# list of stopping times to play with\nstop_time_list = list(range(1,150,10))\n\ndef simulate_accuracy_vs_stoptime(sigma, stop_time_list, num_sample=100):\n \"\"\"\n Calculate the average decision accuracy vs. stopping time by running repeated SPRT simulations\n for each stop time \n\n Args:\n sigma (float): standard deviation for observation model \n stop_list_list (list-like object): a list of stopping times to run over\n num_sample (int): number of simulations to run per stopping time \n\n Returns:\n accuracy_list: a list of average accuracies corresponding to input `stop_time_list`\n decisions_list: a list of decisions made in all trials\n \"\"\"\n accuracy_list = []\n decisions_list = []\n for stop_time in stop_time_list:\n decision_list = []\n ################################################################################\n ## Insert your code here to:\n ## * Run `num_sample` repeated simulations, collect decision into `decision_list`\n ## * Calculate average decision accuracy as `accuracy`\n ################################################################################\n for i in range(num_sample):\n # _, decision,_= ...\n # append to container\n pass \n # accuracy = ...\n\n if not 'accuracy' in locals():\n raise NotImplementedError(\"function `simulate_accuracy_vs_stoptime` incomplete\")\n\n accuracy_list.append(accuracy)\n decisions_list.append(decision_list)\n\n return accuracy_list, decisions_list\n\n################################################################################\n## Un-comment the following code after completing this exercise\n################################################################################\n# simulate_and_plot_accuracy_vs_stoptime(sigma, stop_time_list, num_sample)\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W2D3_DecisionMaking/solutions/W2D3_Tutorial1_Solution_a5a89cc2.py)\n\n*Example output:*\n\n\n\n\n\n\n**Suggestions** \n\n* Play with difference values of noise level `sigma` and observe how that affects the curve. What does that mean? *Hint*: you can click `...`>`Form`>`Hide code` to hide the code section if the slider is too far from the figure\n\n\n\n```python\n#@title Interactive Demo: Accuracy vs. Stopping Time { run : \"auto\"}\n\nsigma = 5.5 #@param {type:\"slider\", min:0.05, max:10.0, step:0.05}\nnum_sample = 200 # number of simulations to run for each stopping time\n# list of stopping times to play with\nstop_time_list = list(range(1,150,10))\n\ntry:\n simulate_and_plot_accuracy_vs_stoptime(sigma, stop_time_list, num_sample)\nexcept NotImplementedError:\n print(\"Complete Exercise 2 first\")\n```\n\n# Simulate DDM with fixed thresholds\n\n\nExercise 3\n---\n\nIn this exercise, we will use thresholding as our stopping rule and observe the behavior of the DDM. \n\nWith thresholding stopping rule, we define a desired error rate and will continue making measurements until that error rate is reached. Experimental evidence suggested that evidence accumulation and thresholding stopping strategy happens at neuronal level (see [this article](https://www.annualreviews.org/doi/full/10.1146/annurev.neuro.29.051605.113038) for further reading).\n\n* Complete the function `threshold_from_errorrate` to calculate the evidence threshold from desired error rate $\\alpha$ as described in the formulas below. The evidence thresholds $th_L$ and $th_R$ for $p_L$ and $p_R$ are opposite of each other as shown below, so you can just return the absolute value.\n$$\n\\begin{align}\n th_{L} &= \\log \\frac{\\alpha}{1-\\alpha} &= -th_{R} \\\\\n th_{R} &= \\log \\frac{1-\\alpha}{\\alpha} &= -th{_L}\\\\\n \\end{align}\n $$\n\n* Complete the function `simulate_SPRT_threshold` to simulate an SPRT with thresholding stopping rule given noise level and desired threshold \n\n* Run repeated simulations for a given noise level and a desired error rate visualize the DDM traces using our provided code \n\n\n\n```python\ndef threshold_from_errorrate(alpha):\n \"\"\"\n Calculate log likelihood ratio threshold from desired error rate `alpha`\n\n Args:\n alpha (float): in (0,1), the desired error rate\n\n Return:\n threshold: corresponding evidence threshold\n \"\"\"\n ################################################################################\n ## Insert your code here to: \n ## * calculate the evidence threshold from desired error rate\n ################################################################################\n raise NotImplementedError(\"function `threshold_from_errorrate` incomplete\")\n\n\ndef simulate_SPRT_threshold(sigma, threshold , true_dist=1, batch_size=100):\n \"\"\"\n Simulate a Sequential Probability Ratio Test with thresholding stopping rule.\n Two observation models are 1D Gaussian distributions N(1,sigma^2) and N(-1,sigma^2).\n\n Args:\n sigma (float): standard deviation \n threshold (float): desired log likelihood ratio threshold to achieve before making decision\n batch_size (int): generate and process data in batches instead of serially for speed. The size of each batch\n\n Returns:\n evidence_history (numpy vector): the history of cumulated evidence given generated data\n decision (int): 1 for pR, 0 for pL \n data (numpy vector): the generated sequences of data in this trial \n \"\"\"\n muL = -1.0\n muR = 1.0 \n \n pL = stats.norm(muL, sigma) \n pR = stats.norm(muR, sigma)\n\n has_enough_data = False \n \n data = np.zeros(0) \n evidence_history = np.zeros(0)\n current_evidence = 0.0\n while not has_enough_data:\n # generate a batch of data \n if true_dist == 1:\n data_batch = pR.rvs(size=batch_size)\n else: \n data_batch = pL.rvs(size=batch_size)\n\n # individual log likelihood ratios\n ll_batch = log_likelihood_ratio(data_batch, pL, pR)\n # cumulated evidence for this batch\n evidence_history_batch = np.cumsum(ll_batch) + current_evidence\n # update the collection of all data\n data = np.concatenate([ data, data_batch])\n # update the collection of all cumulated evidence history\n evidence_history = np.concatenate([evidence_history, evidence_history_batch])\n current_evidence = evidence_history[-1]\n \n # check if we've got enough data\n if np.abs(current_evidence) > threshold:\n has_enough_data = True \n\n # find earliest time to cross threshold\n ################################################################################\n ## Insert your code here to: \n ## * Find the index of first data that crosses one of the thresholds\n ## and assign that to `id_crossing`\n ## Hint: use `np.argmax` on a vector of Bool numbers will give you the index of the \n ## first non-zero value\n ################################################################################\n \n if id_crossing not in locals():\n raise NotImplementedError(\"function `simulate_SPRT_threshold` incomplete\")\n\n # discard redundant data because of batching\n if len(data)-1 > id_crossing:\n data = data[:id_crossing + 1]\n evidence_history = evidence_history[:id_crossing+1]\n\n # Make decision \n if evidence_history[-1] >0:\n decision = 1\n elif evidence_history[-1] <0:\n decision = 0 \n else: \n decision = np.random.randint(2)\n\n return evidence_history, decision, data\n\n\nsigma = 2.8 \nnum_sample = 10\nlog10_alpha = -6.5 #log10(alpha)\nalpha = np.power(10.0,log10_alpha)\n\n\n# Uncomment the line below to run the simulation and visualize the results\n# simulate_and_plot_SPRT_fixedthreshold(sigma, num_sample, alpha)\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W2D3_DecisionMaking/solutions/W2D3_Tutorial1_Solution_490660b2.py)\n\n*Example output:*\n\n\n\n\n\n**Suggestion**\n\n* Play with difference values of `alpha` and `sigma` and observe how that affects the dynamics of Drift-Diffusion Model. *Hint*: you can click `...`>`Form`>`Hide code` to hide the code section if the slider is too far from the figure\n\n\n```python\n#@title Interactive Demo: Noise levels and thresholds { run : \"auto\"}\nsigma = 5 #@param {type:\"slider\", min:0.05, max:5.0, step:0.05}\nnum_sample = 10\nlog10_alpha = -5.9 #@param {type:\"slider\", min:-8, max:-0.1, step:0.1}\nalpha = np.power(10.0, log10_alpha)\nthreshold = threshold_from_errorrate(alpha)\n# construct a simulation model\nnum_sample = 10\n\n################################################################################\n## Un-comment the following code block after completing the exercise above\n################################################################################\ntry:\n simulate_and_plot_SPRT_fixedthreshold(sigma, num_sample, alpha)\nexcept NotImplementedError:\n print(\"Finish exercise 3 first\")\n```\n\n# Speed/Accuracy Tradeoff\n\n---\n### EXERCISE 4: \n\nThe faster you make a decision (by allowing higher error rate $\\alpha$), the lower your accuracy will be. This is known as the speed/accuracy tradeoff. To illustrate the speed/accuracy under thresholding stopping rule, let's run some simulations under differenct thresholds and look at how average decision speed changes with average decision accuracy. \n\n* Complete the function `simulate_accuracy_vs_speed` to simulate and compute average accuracies vs. average decision speeds for a list of error rates.\n\n* We've set up a list of error rates. Run repeated simulations and collect average accuracy with average speed for each error rate in this list, and use our provided code to visualize the speed/accuracy tradeoff. You should see a negative correlation between speed and accuracy.\n\n\n\n```python\nsigma = 3.95 \nnum_repeats = 500 # number of simulations to run for each error rate\nalpha_list = np.power(10,list(range(-5,0)) + np.linspace(-0.9,-0.1,9).tolist()) # list of error rates\n\nthreshold_list = list(map(threshold_from_errorrate, alpha_list))\n\ndef simulate_accuracy_vs_speed(sigma, threshold_list, num_sample=100):\n \"\"\"\n Calculate the average decision accuracy vs. average decision speed by running repeated SPRT simulations\n with thresholding stopping rule for each threshold\n\n Args:\n sigma (float): standard deviation for observation model \n threshold_list (list-like object): a list of evidence thresholds to run over\n num_sample (int): number of simulations to run per stopping time \n\n Returns:\n accuracy_list: a list of average accuracies corresponding to input `stop_time_list`\n decision_speed_list: a list of average decision speeds\n \"\"\"\n decision_speed_list = [] # container for average decision speed for each alpha\n accuracy_list = [] # container for decision accuracy for each alpha\n for threshold in threshold_list:\n decision_time_list = [] # container for decision time for each simulation\n decision_list = [] # container for decision for every simulation\n for i in range(num_repeats):\n # run simulation and get decision of current simulation\n ################################################################################\n ## Insert your code here to: \n ## * run a simulation of SPRT with thresholding using function `simulate_SPRT_threshold`\n ## * calculate decision time(number of data seen) from return values and assign that to `decision_time`\n ################################################################################\n \n if 'decision_time' not in locals():\n raise NotImplementedError(\"function `simulate_accuracy_vs_speed` incomplete\")\n decision_list.append(decision)\n decision_time_list.append(decision_time)\n ################################################################################\n ## Insert your code here to: \n ## * Calculate decision speed given a list of decision times \n ## and assign that to variable `decision_speed`\n ################################################################################\n\n if 'decision_speed' not in locals():\n raise NotImplementedError(\"function `simulate_accuracy_vs_speed` incomplete\")\n decision_accuracy = sum(decision_list) / len(decision_list)\n decision_speed_list.append(decision_speed)\n accuracy_list.append(decision_accuracy)\n\n return accuracy_list, decision_speed_list\n\n\n################################################################################\n## Un-comment the following code block after completing this exercise\n################################################################################\n# simulate_and_plot_speed_vs_accuracy(sigma, threshold_list, num_sample)\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W2D3_DecisionMaking/solutions/W2D3_Tutorial1_Solution_be044152.py)\n\n*Example output:*\n\n\n\n\n\n**Suggestions**\n\n* Play with difference values of noise level `sigma` and observe how that affects the speed/accuracy tradeoff. *Hint*: you can click `...`>`Form`>`Hide code` to hide the code section if the slider is too far from the figure\n\n\n```python\n#@title Interactive Cell { run : \"auto\" }\nsigma = 3.75 #@param {type:\"slider\", min:0.05, max:5.0, step:0.05}\nnum_repeats = 500 # number of simulations to run for each error rate\nalpha_list = np.power(10,list(range(-5,0)) + np.linspace(-0.9,-0.1,9).tolist()) # list of error rates\n\n################################################################################\n## Un-comment the following code block after completing the exercise above\n################################################################################\ntry:\n threshold_list = list(map(threshold_from_errorrate, alpha_list))\n accuracy_list, decision_speed_list = simulate_accuracy_vs_speed(sigma, threshold_list, num_sample=num_sample)\n# # Plotting\n \n fig, ax = plt.subplots()\n ax.plot(decision_speed_list, accuracy_list, linestyle=\"--\", marker=\"o\")\n ax.plot([0.0, 1.0], [0.5, 0.5], c='red') # plot baseline for random choice\n ax.set_xlabel(\"Average Decision Speed\")\n ax.set_ylabel('Average Accuracy')\n ax.set_title(\"Speed/Accuracy Tradeoff\")\n ax.set_ylim(bottom=0.45)\n plt.show()\n \nexcept NotImplementedError:\n print(\"Finish the functions above first.\")\n\n```\n\n# Summary\n\nGood job! By simulating Drift Diffusion Models to perform decision making, you have learnt how to \n\n1. Calculate individual sample evidence as the log likelihood ratio of two candidate models, accumulate evidence from new data points, and make decision based on current evidence in `Exercise 1`\n2. Run repeated simulations to get an estimate of decision accuraries in `Exercise 2`\n3. Implement the thresholding stopping rule where we can control our error rate by taking adequate amounts of data, and calculate the evidence threshold from desired error rate in `Exercise 3`\n4. Explore and gain intuition about the speed/accuracy tradeoff for perceptual decision making in `Exercise 4`\n", "meta": {"hexsha": "a20848e2fab1137c952e964b0883638900d93c7e", "size": 406020, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W2D3_DecisionMaking/student/W2D3_Tutorial1.ipynb", "max_stars_repo_name": "liuxiaomiao123/NeuroMathAcademy", "max_stars_repo_head_hexsha": "16a7969604a300bf9fbb86f8a5b26050ebd14c65", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-07-03T04:39:09.000Z", "max_stars_repo_stars_event_max_datetime": "2020-07-12T02:08:31.000Z", "max_issues_repo_path": "tutorials/W2D3_DecisionMaking/student/W2D3_Tutorial1.ipynb", "max_issues_repo_name": "NinaHKivanani/course-content", "max_issues_repo_head_hexsha": "3c91dd1a669cebce892486ba4f8086b1ef2e1e49", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-06-22T22:57:03.000Z", "max_issues_repo_issues_event_max_datetime": "2020-06-22T22:57:03.000Z", "max_forks_repo_path": "tutorials/W2D3_DecisionMaking/student/W2D3_Tutorial1.ipynb", "max_forks_repo_name": "NinaHKivanani/course-content", "max_forks_repo_head_hexsha": "3c91dd1a669cebce892486ba4f8086b1ef2e1e49", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-01-23T20:16:15.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-23T20:16:15.000Z", "avg_line_length": 294.0043446778, "max_line_length": 101688, "alphanum_fraction": 0.9150066499, "converted": true, "num_tokens": 7418, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.727975460709318, "lm_q1q2_score": 0.4423638760151838}} {"text": "```python\nimport sys\nimport os\nimport PIL\nimport numpy as np\nfrom numpy.linalg import norm\nfrom math import *\nfrom scipy import ndimage\nfrom scipy import misc\nimport skimage\n\nfrom sympy import *\n\n%matplotlib inline \nimport matplotlib.pyplot as plt\n\n```\n\n\n```python\ndef estimateModelAttempt1():\n r,g = symbols('r g')\n a0,r0,g0 = symbols('a0 r0 g0')\n a1,r1,g1 = symbols('a1 r1 g1')\n \n print solveset(Eq(r0, a0*r + (1-a0)*255), r)\n \n # r as a function of a0, r0\n # Eq(r, rFromA0R0)\n rFromA0R0 = (255*a0 + r0 - 255)/a0\n \n print Eq(r1, a1*r + (1-a1)*255).subs(r, rFromA0R0)\n # r1 as a function of a0, r0, a1\n # Eq(r1, r1FromA0R0A1)\n r1FromA0R0A1 = -255*a1 + 255 + a1*(255*a0 + r0 - 255)/a0\n \n # Similarly, g1 as a function of a0, g0, a1\n # Eq(g1, g1FromA0R0A1)\n g1FromA0G0A1 = -255*a1 + 255 + a1*(255*a0 + g0 - 255)/a0\n \n print solveset(Eq(r1, r1FromA0R0A1), a1)\n a1FromA0R0R1 = a0*(r1 - 255)/(r0 - 255)\n \n g1FromA0G0R0R1 = simplify(g1FromA0G0A1.subs(a1, a1FromA0R0R1))\n print g1FromA0G0R0R1\n # a0 disappears!\n g1FromA0G0R0R1 = (g0*r1 - 255*g0 + 255*r0 - 255*r1)/(r0 - 255)\n \n print '==='\n \nestimateModelAttempt1()\n\n```\n\n {(255*a0 + r0 - 255)/a0}\n Eq(r1, -255*a1 + 255 + a1*(255*a0 + r0 - 255)/a0)\n {a0*(r1 - 255)/(r0 - 255)}\n (g0*r1 - 255*g0 + 255*r0 - 255*r1)/(r0 - 255)\n ===\n\n\n\n```python\ndef testModel1():\n r0,g0,b0 = (157.,212.,244.)\n r1,g1,b1 = (218.,239.,251.)\n print 'g1FromR = ', (g0*r1 - 255*g0 + 255*r0 - 255*r1)/(r0 - 255)\n print 'g1FromB = ', (g0*b1 - 255*g0 + 255*b0 - 255*b1)/(b0 - 255)\n print 'b1FromR = ', (b0*r1 - 255*b0 + 255*r0 - 255*r1)/(r0 - 255)\n print 'b1FromG = ', (b0*g1 - 255*b0 + 255*g0 - 255*g1)/(g0 - 255)\n print 'r1FromG = ', (r0*g1 - 255*r0 + 255*g0 - 255*g1)/(g0 - 255)\n print 'r1FromB = ', (r0*b1 - 255*r0 + 255*b0 - 255*b1)/(b0 - 255) \n \ntestModel1()\n```\n\n g1FromR = 238.765306122\n g1FromB = 239.363636364\n b1FromR = 250.846938776\n b1FromG = 250.906976744\n r1FromG = 218.534883721\n r1FromB = 219.363636364\n\n\n\n```python\ndef testModel1_2():\n r0,g0,b0 = (127.,255.,255.)\n r1,g1,b1 = (255.,255.,255.)\n print 'g1FromR = ', (g0*r1 - 255*g0 + 255*r0 - 255*r1)/(r0 - 255)\n print 'b1FromR = ', (b0*r1 - 255*b0 + 255*r0 - 255*r1)/(r0 - 255)\n print 'b1FromG = ', (b0*g1 - 255*b0 + 255*g0 - 255*g1)/(g0 - 255)\n print 'r1FromG = ', (r0*g1 - 255*r0 + 255*g0 - 255*g1)/(g0 - 255)\n \ntestModel1_2()\n```\n\n\n```python\ndef estimateSimplerModel():\n r,g = symbols('r g')\n a0,r0,g0 = symbols('a0 r0 g0')\n a1,r1,g1 = symbols('a1 r1 g1')\n \n r0def = a0*r + (1-a0)*255\n g0def = a0*g + (1-a0)*255\n\n # Does not depend on a0, getting (r - 255)/(g - 255)\n print simplify((r0def-255)/(g0def-255))\n \n # Conclusion 1: (r0-255)/(g0-255) = (r1-255)/(g1-255) if they come from the same color\n # If the background was black, we'd get r0/g0 = r1/g1\n \n # Conclusion 2: we can't recover the true color and alpha values if the background\n # is grayscale. \n \nestimateSimplerModel()\n\n```\n\n (r - 255)/(g - 255)\n\n\n\n```python\ndef testSimpleModel():\n r0,g0,b0 = (157.,212.,244.)\n r1,g1,b1 = (218.,239.,251.)\n \n # Should be the same\n print (r0-255.)/(g0-255.)\n print (r1-255.)/(g1-255.)\n \n # Should be the same\n print (r0-255.)/(b0-255.)\n print (r1-255.)/(b1-255.)\n \ntestSimpleModel()\n```\n\n 2.27906976744\n 2.3125\n 8.90909090909\n 9.25\n\n\n\n```python\ndef testColor0():\n ref = np.array([81.,179.,235.])\n v0 = np.array([157.,212.,244.])\n v1 = np.array([218.,239.,251.])\n v2 = np.array([110.,191.,238.])\n \n alphas0 = (v0 - 255) / (ref - 255)\n alphas1 = (v1 - 255) / (ref - 255)\n alphas2 = (v2 - 255) / (ref - 255)\n print 'alphas0', alphas0\n print 'alphas1', alphas1\n print 'alphas2', alphas2\n \n \ntestColor0()\n\n```\n\n alphas0 [ 0.56321839 0.56578947 0.55 ]\n alphas1 [ 0.21264368 0.21052632 0.2 ]\n alphas2 [ 0.83333333 0.84210526 0.85 ]\n\n\n\n```python\ndef linearToSrgb(L):\n if L <= 0.0031308:\n return L * 12.92 * 255.0\n else: \n return 255.0 * ((1.055 * L**0.41667) - 0.055)\nlinearToSrgb = np.vectorize(linearToSrgb)\n\ndef sRgbToLinearRgb(S):\n S = S/255.0\n if (S <= 0.04045):\n return S/12.92\n else: \n return ((S+0.055)/1.055)**2.4\nsRgbToLinearRgb = np.vectorize(sRgbToLinearRgb)\n\ndef testSrgbToLinear():\n srgb0 = np.array([157.,212.,244.])\n linear0 = sRgbToLinearRgb(srgb0)\n srgba1 = linearToSrgb(linear0)\n \ntestSrgbToLinear()\n\ndef testWithGamma1():\n ref = np.array([241.,230.,42.])\n v0 = np.array([243.,232.,66.])\n v1 = np.array([245.,239.,127.])\n \n ref = v0\n \n #ref = sRgbToLinearRgb(ref)\n #v0 = sRgbToLinearRgb(v0)\n #v1 = sRgbToLinearRgb(v1)\n \n alphas0 = (v0 - 255) / (ref - 255)\n alphas1 = (v1 - 255) / (ref - 255)\n print 'alphas0', alphas0\n print 'alphas1', alphas1\n \n r,g,b = ref\n r0,g0,b0 = v0\n r1,g1,b1 = v1\n print 'g1FromR = ', (g*r1 - 255*g + 255*r - 255*r1)/(r - 255)\n print 'g1FromB = ', (g*b1 - 255*g + 255*b - 255*b1)/(b - 255)\n print 'b1FromR = ', (b*r1 - 255*b + 255*r - 255*r1)/(r - 255)\n print 'b1FromG = ', (b*g1 - 255*b + 255*g - 255*g1)/(g - 255)\n print 'r1FromG = ', (r*g1 - 255*r + 255*g - 255*g1)/(g - 255)\n print 'r1FromB = ', (r*b1 - 255*r + 255*b - 255*b1)/(b - 255) \n \ntestWithGamma1()\n\n```\n\n alphas0 [ 1. 1. 1.]\n alphas1 [ 0.83333333 0.69565217 0.67724868]\n g1FromR = 235.833333333\n g1FromB = 239.423280423\n b1FromR = 97.5\n b1FromG = 123.52173913\n r1FromG = 246.652173913\n r1FromB = 246.873015873\n\n\n\n```python\ndef testWithGamma2():\n ref = np.array([81.,179.,235.])\n v0 = np.array([126.,198.,241.])\n v1 = np.array([84.,180.,235.])\n \n ref = sRgbToLinearRgb(ref)\n v0 = sRgbToLinearRgb(v0)\n v1 = sRgbToLinearRgb(v1)\n \n alphas0 = (v0 - 255) / (ref - 255)\n alphas1 = (v1 - 255) / (ref - 255)\n print 'alphas0', alphas0\n print 'alphas1', alphas1\n \n r,g,b = ref\n r0,g0,b0 = v0\n r1,g1,b1 = v1\n print 'g1FromR = ', (g*r1 - 255*g + 255*r - 255*r1)/(r - 255)\n print 'g1FromB = ', (g*b1 - 255*g + 255*b - 255*b1)/(b - 255)\n print 'b1FromR = ', (b*r1 - 255*b + 255*r - 255*r1)/(r - 255)\n print 'b1FromG = ', (b*g1 - 255*b + 255*g - 255*g1)/(g - 255)\n print 'r1FromG = ', (r*g1 - 255*r + 255*g - 255*g1)/(g - 255)\n print 'r1FromB = ', (r*b1 - 255*r + 255*b - 255*b1)/(b - 255) \n \ntestWithGamma2()\n\n```\n\n alphas0 [ 0.99950433 0.99955244 0.9998078 ]\n alphas1 [ 0.999975 0.9999779 1. ]\n g1FromR = 0.457149449509\n g1FromB = 0.450785782838\n b1FromR = 0.837124043939\n b1FromG = 0.836386719912\n r1FromG = 0.0879160909596\n r1FromB = 0.0822827071298\n\n\n\n```python\ndef testPlotImage():\n # image = ndimage.io.imread('../DaltonLensTests/gnuplotLt5CairoCropped.png')\n # image = ndimage.io.imread('../DaltonLensTests/gnuplotLt5Cropped.png')\n # image = ndimage.io.imread('../DaltonLensTests/gnuplotLt5ScreenCaptureCropped.png')\n # image = ndimage.io.imread('../DaltonLensTests/ComplexPlotCropped2.png')\n image = ndimage.io.imread('../DaltonLensTests/RandomPlotsCropped.png')\n # image = ndimage.io.imread('../DaltonLensTests/XcodeBackground.png')\n float_image = skimage.util.dtype.img_as_float(image[:,:,0:3])\n npix = float_image.shape[0]*float_image.shape[1]\n float_image = float_image.reshape((npix,3))\n print float_image.shape\n plt.plot (255.0 - float_image[:,0]*255., 255.0 - float_image[:,1]*255., '.')\n plt.axis([0, 255, 0, 255]) \n plt.figure()\n \n plt.plot (255.0 - float_image[:,0]*255., 255.0 - float_image[:,2]*255., '.')\n plt.axis([0, 255, 0, 255]) \n plt.figure()\n \n plt.plot (255.0 - float_image[:,1]*255., 255.0 - float_image[:,2]*255., '.')\n plt.axis([0, 255, 0, 255]) \n plt.figure()\n \n # Algorithm IDEA:\n # compute points (R-255, G-255), (R-255, B-255), (G-255, B-255)\n # see if they are in the same line as the reference points (fit a line, distance to line < k)\n # count the number of discriminant points (max 3). Values very close to 255 for every channel\n # are not informative. Informative if min(R,G), min(R,B) or min(G,B) < e.g. 100\n # if compatible and informative, mark as definitely a match\n # if compatible and one neighbor is a match, accept it too\n \n # (b0*r1 - 255*b0 + 255*r0 - 255*r1)/(r0 - 255)\n # ratios = (float_image[:,0]*255. - 255.0001)/(float_image[:,2]*255. - 255.0001)\n # plt.plot (np.arange(0, npix, 1), ratios, '.')\n # plt.figure()\n \ntestPlotImage()\n\n```\n\n\n```python\nfrom mpl_toolkits.mplot3d import Axes3D\ndef plotUncertainty():\n fig = plt.figure()\n ax = fig.add_subplot(111, projection='3d')\n \n R = np.arange(0.0, 254.0, 1.0)\n G = np.arange(0.0, 254.0, 1.0)\n \n R, G = np.meshgrid(R, G)\n gamma = ((256.0-R)/(255.0-G)) - ((255.0-R)/(256.0-G))\n ax.set_xlabel('R')\n ax.set_ylabel('G')\n ax.plot_surface(R, G, gamma, color='b')\n \nplotUncertainty()\n```\n\n\n```python\ndef gammaOf(R,G):\n return ((256.0-R)/(255.0-G)) - ((255.0-R)/(256.0-G))\ngammaOf_ufunc = np.frompyfunc(gammaOf, 2, 1)\n\ndef distFromOrigin(R,G):\n return (255.0-R)/(255.0-G)**2\n\ndef plotUncertaintyFromRatio():\n R = np.arange(1.0, 254.0, 1.0)\n G = np.arange(1.0, 254.0, 1.0)\n \n gamma = gammaOf_ufunc.outer(R,G).flatten()\n print np.shape(gamma)\n \n rOverGValues = np.frompyfunc(distFromOrigin, 2, 1).outer(R,G).flatten()\n print rOverGValues\n print gamma\n \n plt.axis('auto')\n plt.plot(rOverGValues.flatten(), gamma.flatten())\n plt.figure()\n \nplotUncertaintyFromRatio()\n```\n", "meta": {"hexsha": "18a32f6d7b017e57e77578bd7a37e151dae90dc4", "size": 160264, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "python/AntiAliasing.ipynb", "max_stars_repo_name": "Waitsnake/DaltonLens", "max_stars_repo_head_hexsha": "c310acc8a4158f5d9d27c0add52a4eddcf8e730a", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2018-02-05T23:29:13.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-23T08:53:12.000Z", "max_issues_repo_path": "python/AntiAliasing.ipynb", "max_issues_repo_name": "Waitsnake/DaltonLens", "max_issues_repo_head_hexsha": "c310acc8a4158f5d9d27c0add52a4eddcf8e730a", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": 9, "max_issues_repo_issues_event_min_datetime": "2018-05-01T18:27:52.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-19T00:32:33.000Z", "max_forks_repo_path": "python/AntiAliasing.ipynb", "max_forks_repo_name": "Waitsnake/DaltonLens", "max_forks_repo_head_hexsha": "c310acc8a4158f5d9d27c0add52a4eddcf8e730a", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-04-04T17:02:17.000Z", "max_forks_repo_forks_event_max_datetime": "2020-04-04T17:02:17.000Z", "avg_line_length": 266.6622296173, "max_line_length": 87958, "alphanum_fraction": 0.8985673638, "converted": true, "num_tokens": 3923, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754607093178, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.44236387601518373}} {"text": "# The Deconfounder in Action\n\nIn this notebook, we are going to see **the deconfounder in action**.\n\nWe will perform **causal inference** with the deconfounder on a **breast cancer** dataset.\n\n**Goal:**\nTo convince all of us that the deconfounder is **easy** to use!\n\n\nThe **deconfounder** operates in three steps:\n\n1. **Fit** a factor model to the assigned causes; it leads to a candidate substitute confounder.\n2. **Check** the factor model with a predictive check.\n3. **Correct** for the substitute confounder in a causal inference.\n\n\nLet's get started!\n\n\n# Getting ready to work!\n\n\n```python\n!pip install tensorflow_probability\n```\n\n Requirement already satisfied: tensorflow_probability in /usr/local/lib/python3.6/dist-packages (0.9.0)\n Requirement already satisfied: six>=1.10.0 in /usr/local/lib/python3.6/dist-packages (from tensorflow_probability) (1.12.0)\n Requirement already satisfied: gast>=0.2 in /usr/local/lib/python3.6/dist-packages (from tensorflow_probability) (0.3.3)\n Requirement already satisfied: cloudpickle>=1.2.2 in /usr/local/lib/python3.6/dist-packages (from tensorflow_probability) (1.3.0)\n Requirement already satisfied: numpy>=1.13.3 in /usr/local/lib/python3.6/dist-packages (from tensorflow_probability) (1.18.2)\n Requirement already satisfied: decorator in /usr/local/lib/python3.6/dist-packages (from tensorflow_probability) (4.4.2)\n\n\n\n```python\n%tensorflow_version 1.x\nimport tensorflow as tf\nimport numpy as np\nimport numpy.random as npr\nimport pandas as pd\nimport tensorflow as tf\nimport tensorflow_probability as tfp\nimport statsmodels.api as sm\n\nfrom tensorflow_probability import edward2 as ed\nfrom sklearn.datasets import load_breast_cancer\nfrom pandas.plotting import scatter_matrix\nfrom scipy import sparse, stats\nfrom sklearn.model_selection import train_test_split\nfrom sklearn.linear_model import LogisticRegression \nfrom sklearn.metrics import confusion_matrix, classification_report, roc_auc_score, roc_curve\n\nimport matplotlib\nmatplotlib.rcParams.update({'font.sans-serif' : 'Helvetica',\n 'axes.labelsize': 10,\n 'xtick.labelsize' : 6,\n 'ytick.labelsize' : 6,\n 'axes.titlesize' : 10})\nimport matplotlib.pyplot as plt\n\nimport seaborn as sns\ncolor_names = [\"windows blue\",\n \"amber\",\n \"crimson\",\n \"faded green\",\n \"dusty purple\",\n \"greyish\"]\ncolors = sns.xkcd_palette(color_names)\nsns.set(style=\"white\", palette=sns.xkcd_palette(color_names), color_codes = False)\n```\n\n TensorFlow 1.x selected.\n\n\n /usr/local/lib/python3.6/dist-packages/statsmodels/tools/_testing.py:19: FutureWarning: pandas.util.testing is deprecated. Use the functions in the public API at pandas.testing instead.\n import pandas.util.testing as tm\n\n\n\n```python\n!pip show tensorflow\n```\n\n Name: tensorflow\n Version: 1.15.2\n Summary: TensorFlow is an open source machine learning framework for everyone.\n Home-page: https://www.tensorflow.org/\n Author: Google Inc.\n Author-email: packages@tensorflow.org\n License: Apache 2.0\n Location: /tensorflow-1.15.2/python3.6\n Requires: absl-py, keras-preprocessing, astor, tensorboard, tensorflow-estimator, wrapt, six, google-pasta, keras-applications, numpy, opt-einsum, termcolor, grpcio, protobuf, wheel, gast\n Required-by: stable-baselines, magenta, fancyimpute\n\n\n\n```python\n!pip show tensorflow_probability\n```\n\n Name: tensorflow-probability\n Version: 0.7.0\n Summary: Probabilistic modeling and statistical inference in TensorFlow\n Home-page: http://github.com/tensorflow/probability\n Author: Google LLC\n Author-email: no-reply@google.com\n License: Apache 2.0\n Location: /tensorflow-1.15.2/python3.6\n Requires: decorator, cloudpickle, six, numpy\n Required-by: tensorflow-gan, tensor2tensor, magenta, kfac, dm-sonnet\n\n\n\n```python\n# set random seed so everyone gets the same number\nimport random\nrandseed = 123\nprint(\"random seed: \", randseed)\nrandom.seed(randseed)\nnp.random.seed(randseed)\ntf.set_random_seed(randseed)\n```\n\n random seed: 123\n\n\n## The scikit-learn breast cancer dataset\n\n* It is a data set about **breast cancer**. \n* We are interested in how tumor properties **affect** cancer diagnosis. \n* The **(multiple) causes** are tumor properties, e.g. sizes, compactness, symmetry, texture. \n* The **outcome** is tumor diagnosis, whether the breast cancer is diagnosed as malignant or benign.\n\n\n\n\n```python\ndata = load_breast_cancer()\n```\n\n\n```python\nprint(data['DESCR'])\n```\n\n .. _breast_cancer_dataset:\n \n Breast cancer wisconsin (diagnostic) dataset\n --------------------------------------------\n \n **Data Set Characteristics:**\n \n :Number of Instances: 569\n \n :Number of Attributes: 30 numeric, predictive attributes and the class\n \n :Attribute Information:\n - radius (mean of distances from center to points on the perimeter)\n - texture (standard deviation of gray-scale values)\n - perimeter\n - area\n - smoothness (local variation in radius lengths)\n - compactness (perimeter^2 / area - 1.0)\n - concavity (severity of concave portions of the contour)\n - concave points (number of concave portions of the contour)\n - symmetry \n - fractal dimension (\"coastline approximation\" - 1)\n \n The mean, standard error, and \"worst\" or largest (mean of the three\n largest values) of these features were computed for each image,\n resulting in 30 features. For instance, field 3 is Mean Radius, field\n 13 is Radius SE, field 23 is Worst Radius.\n \n - class:\n - WDBC-Malignant\n - WDBC-Benign\n \n :Summary Statistics:\n \n ===================================== ====== ======\n Min Max\n ===================================== ====== ======\n radius (mean): 6.981 28.11\n texture (mean): 9.71 39.28\n perimeter (mean): 43.79 188.5\n area (mean): 143.5 2501.0\n smoothness (mean): 0.053 0.163\n compactness (mean): 0.019 0.345\n concavity (mean): 0.0 0.427\n concave points (mean): 0.0 0.201\n symmetry (mean): 0.106 0.304\n fractal dimension (mean): 0.05 0.097\n radius (standard error): 0.112 2.873\n texture (standard error): 0.36 4.885\n perimeter (standard error): 0.757 21.98\n area (standard error): 6.802 542.2\n smoothness (standard error): 0.002 0.031\n compactness (standard error): 0.002 0.135\n concavity (standard error): 0.0 0.396\n concave points (standard error): 0.0 0.053\n symmetry (standard error): 0.008 0.079\n fractal dimension (standard error): 0.001 0.03\n radius (worst): 7.93 36.04\n texture (worst): 12.02 49.54\n perimeter (worst): 50.41 251.2\n area (worst): 185.2 4254.0\n smoothness (worst): 0.071 0.223\n compactness (worst): 0.027 1.058\n concavity (worst): 0.0 1.252\n concave points (worst): 0.0 0.291\n symmetry (worst): 0.156 0.664\n fractal dimension (worst): 0.055 0.208\n ===================================== ====== ======\n \n :Missing Attribute Values: None\n \n :Class Distribution: 212 - Malignant, 357 - Benign\n \n :Creator: Dr. William H. Wolberg, W. Nick Street, Olvi L. Mangasarian\n \n :Donor: Nick Street\n \n :Date: November, 1995\n \n This is a copy of UCI ML Breast Cancer Wisconsin (Diagnostic) datasets.\n https://goo.gl/U2Uwz2\n \n Features are computed from a digitized image of a fine needle\n aspirate (FNA) of a breast mass. They describe\n characteristics of the cell nuclei present in the image.\n \n Separating plane described above was obtained using\n Multisurface Method-Tree (MSM-T) [K. P. Bennett, \"Decision Tree\n Construction Via Linear Programming.\" Proceedings of the 4th\n Midwest Artificial Intelligence and Cognitive Science Society,\n pp. 97-101, 1992], a classification method which uses linear\n programming to construct a decision tree. Relevant features\n were selected using an exhaustive search in the space of 1-4\n features and 1-3 separating planes.\n \n The actual linear program used to obtain the separating plane\n in the 3-dimensional space is that described in:\n [K. P. Bennett and O. L. Mangasarian: \"Robust Linear\n Programming Discrimination of Two Linearly Inseparable Sets\",\n Optimization Methods and Software 1, 1992, 23-34].\n \n This database is also available through the UW CS ftp server:\n \n ftp ftp.cs.wisc.edu\n cd math-prog/cpo-dataset/machine-learn/WDBC/\n \n .. topic:: References\n \n - W.N. Street, W.H. Wolberg and O.L. Mangasarian. Nuclear feature extraction \n for breast tumor diagnosis. IS&T/SPIE 1993 International Symposium on \n Electronic Imaging: Science and Technology, volume 1905, pages 861-870,\n San Jose, CA, 1993.\n - O.L. Mangasarian, W.N. Street and W.H. Wolberg. Breast cancer diagnosis and \n prognosis via linear programming. Operations Research, 43(4), pages 570-577, \n July-August 1995.\n - W.H. Wolberg, W.N. Street, and O.L. Mangasarian. Machine learning techniques\n to diagnose breast cancer from fine-needle aspirates. Cancer Letters 77 (1994) \n 163-171.\n\n\n***For simplicity, we will work with the first 10 features, i.e. the mean radius/texture/perimeter***/....\n\n\n```python\nnum_fea = 10\ndf = pd.DataFrame(data[\"data\"][:,:num_fea], columns=data[\"feature_names\"][:num_fea])\n```\n\n\n```python\ndf.shape\n```\n\n\n\n\n (569, 10)\n\n\n\n\n```python\ndf.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    mean radiusmean texturemean perimetermean areamean smoothnessmean compactnessmean concavitymean concave pointsmean symmetrymean fractal dimension
    017.9910.38122.801001.00.118400.277600.30010.147100.24190.07871
    120.5717.77132.901326.00.084740.078640.08690.070170.18120.05667
    219.6921.25130.001203.00.109600.159900.19740.127900.20690.05999
    311.4220.3877.58386.10.142500.283900.24140.105200.25970.09744
    420.2914.34135.101297.00.100300.132800.19800.104300.18090.05883
    \n
    \n\n\n\n\n```python\ndfy = data[\"target\"]\n```\n\n\n```python\ndfy.shape, dfy[:100] # binary outcomes\n```\n\n\n\n\n ((569,),\n array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1,\n 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,\n 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0,\n 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0,\n 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0]))\n\n\n\n## Preparing the dataset for the deconfounder\n\n### Only one step of preprocessing needed!\n\n### We need to get rid of the highly correlated causes\n\n**Why** do we need to get rid of highly correlated causes?\n\nIf two causes are **highly correlated**, a valid substitute confounder will largely **inflate the variance** of causal estimates downstream.\n\nThis phenomenon is **closely related to** the variance inflation phenomenon in linear regression.\n\n\n\n\n\n***A more technical explanation (ignorable)***\n\nThink of the extreme case where two causes are perfectly collinear $A_1 = 5A_2$. The only random variable Z that \n$$A_1 \\perp A_2 | Z,$$\n\n$(A_1, A_2)$ **must** be a **deterministic function** of Z. For example, $Z = A_1$ or $Z = A_2$.\n\nSuch a substitute confounder Z **breaks one of the conditions** the deconfounder requires. See \"***A note on overlap***\" in the theory section of the paper.\n\n\n**How** do we get rid of highly correlated causes?\n\n* We first make a **scatter plot** of **all pairs** of the causes.\n\n* It reveals which causes are **highly correlated**.\n\n* We will **exclude** these highly correlated causes by hand.\n\n\n\n\n```python\nsns.pairplot(df, size=1.5)\n```\n\n\n```python\n# perimeter and area are highly correlated with radius\nfea_cols = df.columns[[(not df.columns[i].endswith(\"perimeter\")) \\\n and (not df.columns[i].endswith(\"area\")) \\\n for i in range(df.shape[1])]]\n```\n\n\n```python\ndfX = pd.DataFrame(df[fea_cols])\n\nprint(dfX.shape, dfy.shape)\n```\n\n (569, 8) (569,)\n\n\n### How does the dataset look like after preprocessing?\n\n\n```python\n# The causes\ndfX.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    mean radiusmean texturemean smoothnessmean compactnessmean concavitymean concave pointsmean symmetrymean fractal dimension
    017.9910.380.118400.277600.30010.147100.24190.07871
    120.5717.770.084740.078640.08690.070170.18120.05667
    219.6921.250.109600.159900.19740.127900.20690.05999
    311.4220.380.142500.283900.24140.105200.25970.09744
    420.2914.340.100300.132800.19800.104300.18090.05883
    \n
    \n\n\n\n\n```python\n# The outcome\ndfy[:25]\n```\n\n\n\n\n array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1,\n 0, 0, 0])\n\n\n\n# The dataset is ready. Let's do causal inference with the deconfounder!\n\n## Step 1: Fit a factor model to the assigned causes; it leads to a substitute confounder.\n\n\n### We start with trying out a random factor model. How about a probabilistic PCA model?\n\nThe matrix of assigned causes $X$\n\n* It has N=569 rows and D=8 columns. \n* N is the number of subjects/data points.\n* D is the number of causes/data dimension.\n\n### Step 1.1: Some chores first...\n\n#### Standardize the data\nThis step is optional to the deconfounder. \n\nIt only makes finding a good probabilistic PCA model easier.\n\n\n```python\n# dfX.std()\n```\n\n\n```python\n# standardize the data for PPCA\nX = np.array((dfX - dfX.mean())/dfX.std())\n```\n\n#### Then holdout some data!\n\nWe will later need to check the factor model with some heldout data.\nSo let's holdout some now.\n\n\n```python\n# randomly holdout some entries of X\nnum_datapoints, data_dim = X.shape\n\nholdout_portion = 0.2\nn_holdout = int(holdout_portion * num_datapoints * data_dim)\n\nholdout_row = np.random.randint(num_datapoints, size=n_holdout)\nholdout_col = np.random.randint(data_dim, size=n_holdout)\nholdout_mask = (sparse.coo_matrix((np.ones(n_holdout), \\\n (holdout_row, holdout_col)), \\\n shape = X.shape)).toarray()\n\nholdout_subjects = np.unique(holdout_row)\nholdout_mask = np.minimum(1, holdout_mask)\n\nx_train = np.multiply(1-holdout_mask, X)\nx_vad = np.multiply(holdout_mask, X)\n```\n\n### Step 1.2: We are ready to fit a probabilistic PCA model to x_train.\n\nThis step of \"**fitting** a factor model\" involves **inferring latent variables** in probability models. \n\nWe will rely on **Tensorflow Probability**, a library for probabilistic reasoning and statistical analysis in TensorFlow.\n\nThere are many **other probabilistic programming toolboxes** for fitting factor models, e.g. Pyro, Stan. \n\nSome of the latent variable models can also be fit with **scikit-learn**. \n\nWe are free to use any of these with the deconfounder!\n\n\n\n\n\n**What does a probabilistic PCA model look like?**\n\n* Probabilistic PCA is a dimensionality reduction technique. It models data with a lower dimensional latent space.\n\n* We consider the assigned causes of the $n$th subject. We write it as $\\mathbf{x}_n$, which is a $D=8$ dimensional vector.\n\n* The probabilistic PCA assumes the following data generating process for each $\\mathbf{x}_n$, $n = 1, ..., N$:\n\n\\begin{equation*}\n\\mathbf{z}_{n} \\stackrel{iid}{\\sim} N(\\mathbf{0}, \\mathbf{I}_K),\n\\end{equation*}\n\n\\begin{equation*}\n\\mathbf{x}_n \\mid \\mathbf{z}_n\n\\sim N(\\mathbf{z}_n\\mathbf{W}, \\sigma^2\\mathbf{I}_D).\n\\end{equation*}\n\n\n* We construct a $K$-dimensional substitute confounder $\\mathbf{z}_{n}$ for each subject $n$, $n = 1, ..., N$. \n* Each $\\mathbf{z}_{n}$ is a $K$-dimensional latent vector, $n = 1, ..., N$. \n\n\n\n\n\n```python\n# we allow both linear and quadratic model\n# for linear model x_n has mean z_n * W\n# for quadratic model x_n has mean b + z_n * W + (z_n**2) * W_2\n# quadractice model needs to change the checking step accordingly\n\ndef ppca_model(data_dim, latent_dim, num_datapoints, stddv_datapoints, mask, form=\"linear\"):\n w = ed.Normal(loc=tf.zeros([latent_dim, data_dim]),\n scale=tf.ones([latent_dim, data_dim]),\n name=\"w\") # parameter\n z = ed.Normal(loc=tf.zeros([num_datapoints, latent_dim]),\n scale=tf.ones([num_datapoints, latent_dim]), \n name=\"z\") # local latent variable / substitute confounder\n if form == \"linear\":\n x = ed.Normal(loc=tf.multiply(tf.matmul(z, w), mask),\n scale=stddv_datapoints * tf.ones([num_datapoints, data_dim]),\n name=\"x\") # (modeled) data\n elif form == \"quadratic\":\n b = ed.Normal(loc=tf.zeros([1, data_dim]),\n scale=tf.ones([1, data_dim]),\n name=\"b\") # intercept\n w2 = ed.Normal(loc=tf.zeros([latent_dim, data_dim]),\n scale=tf.ones([latent_dim, data_dim]),\n name=\"w2\") # quadratic parameter\n x = ed.Normal(loc=tf.multiply(b + tf.matmul(z, w) + tf.matmul(tf.square(z), w2), mask),\n scale=stddv_datapoints * tf.ones([num_datapoints, data_dim]),\n name=\"x\") # (modeled) data\n return x, (w, z)\n\nlog_joint = ed.make_log_joint_fn(ppca_model)\n```\n\n**Let's fit a probabilistic PCA model.**\n\n\n```python\nlatent_dim = 2\nstddv_datapoints = 0.1\n\nmodel = ppca_model(data_dim=data_dim,\n latent_dim=latent_dim,\n num_datapoints=num_datapoints,\n stddv_datapoints=stddv_datapoints, \n mask=1-holdout_mask)\n```\n\nThe cell below implements **variational inference** for probabilistic PCA in tensorflow probability.\n\nYou are free to fit the probabilistic PCA in your favourite ways with your favourite package. \n\nNote: approximate inference is perfectly fine!\n\nIt is orthogonal to our discussion around the deconfounder.\n\nLet's **ignore** that for now (and forever).\n\n\n\n```python\ndef variational_model(qb_mean, qb_stddv, qw_mean, qw_stddv, \n qw2_mean, qw2_stddv, qz_mean, qz_stddv):\n qb = ed.Normal(loc=qb_mean, scale=qb_stddv, name=\"qb\")\n qw = ed.Normal(loc=qw_mean, scale=qw_stddv, name=\"qw\")\n qw2 = ed.Normal(loc=qw2_mean, scale=qw2_stddv, name=\"qw2\")\n qz = ed.Normal(loc=qz_mean, scale=qz_stddv, name=\"qz\")\n return qb, qw, qw2, qz\n\n\nlog_q = ed.make_log_joint_fn(variational_model)\n\ndef target(b, w, w2, z):\n \"\"\"Unnormalized target density as a function of the parameters.\"\"\"\n return log_joint(data_dim=data_dim,\n latent_dim=latent_dim,\n num_datapoints=num_datapoints,\n stddv_datapoints=stddv_datapoints,\n mask=1-holdout_mask,\n w=w, z=z, w2=w2, b=b, x=x_train)\n\ndef target_q(qb, qw, qw2, qz):\n return log_q(qb_mean=qb_mean, qb_stddv=qb_stddv,\n qw_mean=qw_mean, qw_stddv=qw_stddv,\n qw2_mean=qw2_mean, qw2_stddv=qw2_stddv,\n qz_mean=qz_mean, qz_stddv=qz_stddv,\n qw=qw, qz=qz, qw2=qw2, qb=qb)\n\nqb_mean = tf.Variable(np.ones([1, data_dim]), dtype=tf.float32)\nqw_mean = tf.Variable(np.ones([latent_dim, data_dim]), dtype=tf.float32)\nqw2_mean = tf.Variable(np.ones([latent_dim, data_dim]), dtype=tf.float32)\nqz_mean = tf.Variable(np.ones([num_datapoints, latent_dim]), dtype=tf.float32)\nqb_stddv = tf.nn.softplus(tf.Variable(0 * np.ones([1, data_dim]), dtype=tf.float32))\nqw_stddv = tf.nn.softplus(tf.Variable(-4 * np.ones([latent_dim, data_dim]), dtype=tf.float32))\nqw2_stddv = tf.nn.softplus(tf.Variable(-4 * np.ones([latent_dim, data_dim]), dtype=tf.float32))\nqz_stddv = tf.nn.softplus(tf.Variable(-4 * np.ones([num_datapoints, latent_dim]), dtype=tf.float32))\n\nqb, qw, qw2, qz = variational_model(qb_mean=qb_mean, qb_stddv=qb_stddv,\n qw_mean=qw_mean, qw_stddv=qw_stddv,\n qw2_mean=qw2_mean, qw2_stddv=qw2_stddv,\n qz_mean=qz_mean, qz_stddv=qz_stddv)\n\n\nenergy = target(qb, qw, qw2, qz)\nentropy = -target_q(qb, qw, qw2, qz)\n\nelbo = energy + entropy\n\n\noptimizer = tf.train.AdamOptimizer(learning_rate = 0.05)\ntrain = optimizer.minimize(-elbo)\n\ninit = tf.global_variables_initializer()\n\nt = []\n\nnum_epochs = 500\n\nwith tf.Session() as sess:\n sess.run(init)\n\n for i in range(num_epochs):\n sess.run(train)\n if i % 5 == 0:\n t.append(sess.run([elbo]))\n\n b_mean_inferred = sess.run(qb_mean)\n b_stddv_inferred = sess.run(qb_stddv)\n w_mean_inferred = sess.run(qw_mean)\n w_stddv_inferred = sess.run(qw_stddv)\n w2_mean_inferred = sess.run(qw2_mean)\n w2_stddv_inferred = sess.run(qw2_stddv)\n z_mean_inferred = sess.run(qz_mean)\n z_stddv_inferred = sess.run(qz_stddv)\n \nprint(\"Inferred axes:\")\nprint(w_mean_inferred)\nprint(\"Standard Deviation:\")\nprint(w_stddv_inferred)\n\nplt.plot(range(1, num_epochs, 5), t)\nplt.show()\n\ndef replace_latents(b, w, w2, z):\n\n def interceptor(rv_constructor, *rv_args, **rv_kwargs):\n \"\"\"Replaces the priors with actual values to generate samples from.\"\"\"\n name = rv_kwargs.pop(\"name\")\n if name == \"b\":\n rv_kwargs[\"value\"] = b\n elif name == \"w\":\n rv_kwargs[\"value\"] = w\n elif name == \"w\":\n rv_kwargs[\"value\"] = w2\n elif name == \"z\":\n rv_kwargs[\"value\"] = z\n return rv_constructor(*rv_args, **rv_kwargs)\n\n return interceptor\n```\n\nSo we just played some **magic** to **fit the probabilistic PCA** to the matrix of assigned causes $\\mathbf{X}$.\n\n\n**The only important thing here is: **\n\nWe have **inferred** the latent variables $\\mathbf{z}_n, n=1, ..., N$ and the parameters $\\mathbf{W}$.\n\nSpecifically, we have obtained from this step\n\n```\nw_mean_inferred,\nw_stddv_inferred,\nz_mean_inferred,\nz_stddv_inferred.\n```\n\n\n\n## Step 2: Check the factor model with a predictive check.\n\n\nNow we are ready to **check** the probabilistic PCA model.\n\nThe checking step is **very important** to the deconfounder. \n\nPleeeeeze **always** check the factor model!\n\n**How** do we perform the predictive check?\n\n\n1. We will **generate** some replicated datasets for the heldout entries.\n2. And then **compare** the replicated datasets with the original dataset on the heldout entries.\n3. If they **look similar**, then we are good to go.\n\n\n\n#### Step 2.1: We generate some replicated datasets first.\n\n* We will start with generating some **replicated datasets** from the predictive distribution of the assigned causes $X$:\n\\begin{align}\n p(\\mathbf{X^{rep}_{n,heldout}} \\,|\\, \\mathbf{X_{n, obs}}) =\n \\int p(\\mathbf{X_{n, heldout}} \\,|\\, \\mathbf{z}_n) p(\\mathbf{z_n} \\,|\\, \\mathbf{X}_{n, obs}) \\mathrm{d} \\mathbf{z_n}.\n\\end{align}\n\n* That is, we generate these datasets from a probabilistic PCA model given the **inferred** latent variables $\\hat{p}(\\mathbf{z}_n)$ and $\\hat{p}(\\mathbf{W})$:\n\n\\begin{equation*}\n\\mathbf{z}_{n} \\sim \\hat{p}(\\mathbf{z}_n),\n\\end{equation*}\n\n\\begin{equation*}\n\\mathbf{W} \\sim \\hat{p}(\\mathbf{W}),\n\\end{equation*}\n\n\\begin{equation*}\n\\mathbf{x}_n \\mid \\mathbf{z}_n\n\\sim N(\\mathbf{z}_n\\mathbf{W}, \\sigma^2\\mathbf{I}_D).\n\\end{equation*}\n\n\n* These replicated datasets tell us what the assigned causes $X$ **should look like** if it is indeed generated by the fitted probabilistic PCA model.\n\n\n\n```python\nn_rep = 100 # number of replicated datasets we generate\nholdout_gen = np.zeros((n_rep,*(x_train.shape)))\n\nfor i in range(n_rep):\n b_sample = npr.normal(b_mean_inferred, b_stddv_inferred)\n w_sample = npr.normal(w_mean_inferred, w_stddv_inferred)\n w2_sample = npr.normal(w2_mean_inferred, w2_stddv_inferred)\n z_sample = npr.normal(z_mean_inferred, z_stddv_inferred)\n\n with ed.interception(replace_latents(b_sample, w_sample, w2_sample, z_sample)):\n generate = ppca_model(\n data_dim=data_dim, latent_dim=latent_dim,\n num_datapoints=num_datapoints, stddv_datapoints=stddv_datapoints,\n mask=np.ones(x_train.shape))\n\n with tf.Session() as sess:\n x_generated, _ = sess.run(generate)\n\n # look only at the heldout entries\n holdout_gen[i] = np.multiply(x_generated, holdout_mask)\n```\n\n#### Step 2.2: Then we compute the test statistic on both the original and the replicated dataset.\n\n\n\n* We use the **test statistic** of **expected heldout log likelihood**:\n\\begin{align}\n t(\\mathbf{X_{n,heldout}}) = \\mathbb{E}_{\\mathbf{Z}, \\mathbf{W}}[{\\log p(\\mathbf{X_{n,heldout}} \\,|\\, \\mathbf{Z}, \\mathbf{W}) \\,|\\,\n \\mathbf{X_{n,obs}}}].\n\\end{align}\n\n* We calculate this test statistic **for each $n$** and for **both** the **original** dataset $\\mathbf{X_{n,heldout}}$ and the **replicated** dataset $\\mathbf{X^{rep}_{n,heldout}}$.\n\n\n\n\n```python\nn_eval = 100 # we draw samples from the inferred Z and W\nobs_ll = []\nrep_ll = []\nfor j in range(n_eval):\n w_sample = npr.normal(w_mean_inferred, w_stddv_inferred)\n z_sample = npr.normal(z_mean_inferred, z_stddv_inferred)\n \n holdoutmean_sample = np.multiply(z_sample.dot(w_sample), holdout_mask)\n obs_ll.append(np.mean(stats.norm(holdoutmean_sample, \\\n stddv_datapoints).logpdf(x_vad), axis=1))\n\n rep_ll.append(np.mean(stats.norm(holdoutmean_sample, \\\n stddv_datapoints).logpdf(holdout_gen),axis=2))\n \nobs_ll_per_zi, rep_ll_per_zi = np.mean(np.array(obs_ll), axis=0), np.mean(np.array(rep_ll), axis=0)\n```\n\n#### Step 2.3: Finally we compare the test statistic of the original and the replicated dataset.\n\n\n* We compare the test statistics via the $p$-values.\n\\begin{equation*}\n \\text{$p$-value} = p\\left(t(\\mathbf{X_{n,heldout}^{rep}}) < t(\\mathbf{X_{n, heldout}})\\right).\n\\end{equation*}\n\n* The **smaller** the $p$-value is, the **more different** the original dataset is from the replicated dataset.\n\n* We **fail** the check if the $p$-value is **small**.\n\n* Note this goes in the opposite direction to the conventional usage of $p$-values.\n\n\n\nWe compute a $p$-value for each $n$ and output the average $p$-values.\n\n\n```python\npvals = np.array([np.mean(rep_ll_per_zi[:,i] < obs_ll_per_zi[i]) for i in range(num_datapoints)])\nholdout_subjects = np.unique(holdout_row)\noverall_pval = np.mean(pvals[holdout_subjects])\nprint(\"Predictive check p-values\", overall_pval)\n```\n\n Predictive check p-values 0.10207589285714287\n\n\n**We passed the check!**\n\nThe substitute confounder $\\mathbf{z}_n$ constructed in Step 1 is valid. We are ready to move on!\n\n#### An optional step\n\nWe can also peak at **the predictive check of individual subjects**.\n\nThis step is just for fun. It is how we generate Figure 2 of the paper.\n\n\n\n* We randomly choose a subject.\n* Plot the kernel density estimate of the test statistic on the replicated datasets.\n* Plot the test statistic on the original dataset (the dashed vertical line).\n\n\n\n\n\n```python\nsubject_no = npr.choice(holdout_subjects) \nsns.kdeplot(rep_ll_per_zi[:,subject_no]).set_title(\"Predictive check for subject \"+str(subject_no))\nplt.axvline(x=obs_ll_per_zi[subject_no], linestyle='--')\n```\n\n## Step 3: Correct for the substitute confounder in a causal inference.\n\n**How** to estimate causal effects?\n\n* For simplicity, we fit a logistic regression as an outcome model here. \n\n* The target is the observed outcome $y_n$, $n=1,\\ldots, N$.\n\n* The regressor is the multiple causes $\\mathbf{X}_n$, $n=1,\\ldots, N$.\n\n**How** to correct for the substitute confounder?\n\n* We include the substitute confounder $\\mathbf{Z}_n$, $n=1,\\ldots, N$, into the regressors.\n\n\n```python\n# approximate the (random variable) substitute confounders with their inferred mean.\nZ_hat = z_mean_inferred \n# augment the regressors to be both the assigned causes X and the substitute confounder Z\nX_aug = np.column_stack([X, Z_hat])\n```\n\n\n```python\n# holdout some data from prediction later\nX_train, X_test, y_train, y_test = train_test_split(X_aug, dfy, test_size=0.2, random_state=0)\n```\n\n\n```python\ndcfX_train = sm.add_constant(X_train)\ndcflogit_model = sm.Logit(y_train, dcfX_train)\ndcfresult = dcflogit_model.fit_regularized(maxiter=5000)\nprint(dcfresult.summary())\n```\n\n Optimization terminated successfully. (Exit mode 0)\n Current function value: 0.13286976182886223\n Iterations: 85\n Function evaluations: 85\n Gradient evaluations: 85\n Logit Regression Results \n ==============================================================================\n Dep. Variable: y No. Observations: 455\n Model: Logit Df Residuals: 444\n Method: MLE Df Model: 10\n Date: Fri, 03 Apr 2020 Pseudo R-squ.: 0.7971\n Time: 23:28:08 Log-Likelihood: -60.456\n converged: True LL-Null: -297.99\n Covariance Type: nonrobust LLR p-value: 9.348e-96\n ==============================================================================\n coef std err z P>|z| [0.025 0.975]\n ------------------------------------------------------------------------------\n const 0.7997 0.260 3.074 0.002 0.290 1.309\n x1 -2.9778 0.964 -3.088 0.002 -4.868 -1.088\n x2 -1.4487 0.381 -3.804 0.000 -2.195 -0.702\n x3 -1.4633 0.744 -1.968 0.049 -2.921 -0.006\n x4 0.3681 0.872 0.422 0.673 -1.342 2.078\n x5 -1.1352 0.900 -1.261 0.207 -2.900 0.630\n x6 -1.6511 1.240 -1.332 0.183 -4.081 0.779\n x7 -0.5368 0.506 -1.060 0.289 -1.530 0.456\n x8 0.4164 0.765 0.544 0.586 -1.083 1.916\n x9 0.3544 1.734 0.204 0.838 -3.044 3.753\n x10 -1.0729 1.360 -0.789 0.430 -3.739 1.593\n ==============================================================================\n \n Possibly complete quasi-separation: A fraction 0.17 of observations can be\n perfectly predicted. This might indicate that there is complete\n quasi-separation. In this case some parameters will not be identified.\n\n\n\n```python\nres = pd.DataFrame({\"causal_mean\": dcfresult.params[:data_dim+1], \\\n \"causal_std\": dcfresult.bse[:data_dim+1], \\\n \"causal_025\": dcfresult.conf_int()[:data_dim+1,0], \\\n \"causal_975\": dcfresult.conf_int()[:data_dim+1,1], \\\n \"causal_pval\": dcfresult.pvalues[:data_dim+1]})\nres[\"causal_sig\"] = (res[\"causal_pval\"] < 0.05)\nres = res.T\nres.columns = np.concatenate([[\"intercept\"], np.array(dfX.columns)])\nres = res.T\n```\n\n\n```python\nres\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    causal_meancausal_stdcausal_025causal_975causal_pvalcausal_sig
    intercept0.7996570.2601160.2898391.309470.00211043True
    mean radius-2.977830.964373-4.86796-1.087690.0020162True
    mean texture-1.448670.380799-2.19502-0.7023130.000142218True
    mean smoothness-1.463260.743651-2.92079-0.005734250.0491055True
    mean compactness0.3680690.87232-1.341652.077780.673067False
    mean concavity-1.135240.90044-2.900070.6295880.207394False
    mean concave points-1.651081.23964-4.080720.7785730.182893False
    mean symmetry-0.5368260.506479-1.529510.4558550.289182False
    mean fractal dimension0.4163580.765082-1.083181.915890.586304False
    \n
    \n\n\n\nWe check the predictions to see if the logistic outcome model is a good outcome model.\n\n\n```python\n# make predictions with the causal model \ndcfX_test = X_test\ndcfy_predprob = dcfresult.predict(sm.add_constant(dcfX_test))\ndcfy_pred = (dcfy_predprob > 0.5)\nprint(classification_report(y_test, dcfy_pred))\n```\n\n precision recall f1-score support\n \n 0 0.96 0.91 0.93 47\n 1 0.94 0.97 0.96 67\n \n accuracy 0.95 114\n macro avg 0.95 0.94 0.95 114\n weighted avg 0.95 0.95 0.95 114\n \n\n\n# We are done!\n\nWe have computed the average causal effect of raising the causes by one unit (see the \"causal mean\" column above).\n\n# Is the deconfounder worth the effort?\n\nWe finally compare the **causal** estimation (with the deconfounder) with the **noncausal** estimation (with vanilla regression).\n\n## The classical logistic regression! Note it is noncausal :-(\n\n\n```python\n# regress the outcome against the causes only (no substitute confounders)\nnodcfX_train = sm.add_constant(X_train[:,:X.shape[1]])\nnodcflogit_model = sm.Logit(y_train, nodcfX_train)\nnodcfresult = nodcflogit_model.fit_regularized(maxiter=5000)\nprint(nodcfresult.summary())\n```\n\n Optimization terminated successfully. (Exit mode 0)\n Current function value: 0.13451539390572878\n Iterations: 71\n Function evaluations: 71\n Gradient evaluations: 71\n Logit Regression Results \n ==============================================================================\n Dep. Variable: y No. Observations: 455\n Model: Logit Df Residuals: 446\n Method: MLE Df Model: 8\n Date: Fri, 03 Apr 2020 Pseudo R-squ.: 0.7946\n Time: 23:28:08 Log-Likelihood: -61.205\n converged: True LL-Null: -297.99\n Covariance Type: nonrobust LLR p-value: 3.283e-97\n ==============================================================================\n coef std err z P>|z| [0.025 0.975]\n ------------------------------------------------------------------------------\n const 0.7632 0.255 2.998 0.003 0.264 1.262\n x1 -3.2642 0.904 -3.611 0.000 -5.036 -1.492\n x2 -1.7189 0.307 -5.602 0.000 -2.320 -1.117\n x3 -1.2257 0.498 -2.462 0.014 -2.202 -0.250\n x4 0.3115 0.741 0.420 0.674 -1.142 1.765\n x5 -1.1317 0.717 -1.578 0.115 -2.538 0.274\n x6 -2.0079 1.172 -1.714 0.087 -4.304 0.289\n x7 -0.5413 0.324 -1.673 0.094 -1.176 0.093\n x8 0.5775 0.677 0.853 0.394 -0.750 1.905\n ==============================================================================\n \n Possibly complete quasi-separation: A fraction 0.15 of observations can be\n perfectly predicted. This might indicate that there is complete\n quasi-separation. In this case some parameters will not be identified.\n\n\n\n```python\nres[\"noncausal_mean\"] = np.array(nodcfresult.params)\nres[\"noncausal_std\"] = np.array(nodcfresult.bse)\nres[\"noncausal_025\"] = np.array(nodcfresult.conf_int()[:,0])\nres[\"noncausal_975\"] = np.array(nodcfresult.conf_int()[:,1])\nres[\"noncausal_pval\"] = np.array(nodcfresult.pvalues)\nres[\"noncausal_sig\"] = (res[\"noncausal_pval\"] < 0.05)\n```\n\n\n```python\nres[\"diff\"] = res[\"causal_mean\"] - res[\"noncausal_mean\"]\nres[\"pval_diff\"] = res[\"causal_pval\"] - res[\"noncausal_pval\"]\n```\n\n\n```python\nnodcfX_test = sm.add_constant(X_test[:,:X.shape[1]])\nnodcfy_predprob = nodcfresult.predict(nodcfX_test)\nnodcfy_pred = (nodcfy_predprob > 0.5)\n```\n\n**Causal models do not hurt predictions here!**\n\n\n```python\ndcflogit_roc_auc = roc_auc_score(y_test, dcfy_pred)\ndcffpr, dcftpr, dcfthresholds = roc_curve(y_test, dcfy_predprob)\nnodcflogit_roc_auc = roc_auc_score(y_test, nodcfy_pred)\nnodcffpr, nodcftpr, nodcfthresholds = roc_curve(y_test, nodcfy_predprob)\nplt.figure()\nplt.plot(nodcffpr, nodcftpr, label='Noncausal Logistic Regression (area = %0.9f)' % nodcflogit_roc_auc)\nplt.plot(dcffpr, dcftpr, label='Causal Logistic Regression (area = %0.9f)' % dcflogit_roc_auc)\nplt.plot([0, 1], [0, 1],'r--')\nplt.xlim([0.0, 1.0])\nplt.ylim([0.0, 1.05])\nplt.xlabel('False Positive Rate')\nplt.ylabel('True Positive Rate')\nplt.title('Receiver operating characteristic')\nplt.legend(loc=\"lower right\")\nplt.savefig('Log_ROC')\nplt.show()\n```\n\n**But causal models do change the regression coefficients and which features are significant.**\n\n* The mean smoothness is a feature **significantly correlated** with the cancer diagnosis.\n\n* But it does **not significantly** **causally affect** the cancer diagnosis.\n\n* The effect of all features are **over-estimated** with the noncausal model except the \"mean compactness\".\n\n\n```python\nres.sort_values(\"pval_diff\", ascending=True)[[\"pval_diff\", \"causal_pval\", \"noncausal_pval\", \"causal_sig\", \"noncausal_sig\", \"causal_mean\", \"noncausal_mean\"]]\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    pval_diffcausal_pvalnoncausal_pvalcausal_signoncausal_sigcausal_meannoncausal_mean
    mean compactness-0.001309450.6730676.743764e-01FalseFalse0.3680690.311494
    intercept-0.000606160.002110432.716595e-03TrueTrue0.7996570.763172
    mean texture0.0001421970.0001422182.123746e-08TrueTrue-1.44867-1.718860
    mean radius0.001711020.00201623.051830e-04TrueTrue-2.97783-3.264181
    mean smoothness0.03527010.04910551.383540e-02TrueTrue-1.46326-1.225668
    mean concavity0.09272530.2073941.146687e-01FalseFalse-1.13524-1.131677
    mean concave points0.09630940.1828938.658376e-02FalseFalse-1.65108-2.007921
    mean fractal dimension0.1925210.5863043.937831e-01FalseFalse0.4163580.577489
    mean symmetry0.1947750.2891829.440683e-02FalseFalse-0.536826-0.541292
    \n
    \n\n\n\n* We include causes into the regression **one-by-one**.\n* The deconfounder coefficients **does not** flip signs.\n* But classical logistic regression coefficients **does** flip signs.\n* This suggests that **the deconfounder is causal**. \n* It is because **causal** coefficients **do not change** as we include more variables into the system; causal estimation already controls for confounders so that it is causal.\n* However, **correlation** coefficients **can change** as we include more variables into the system; if the added variable is a confounder, than the regression coefficients change to account for the confounding effects.\n\n\n```python\n# The deconfounder with causes added one-by-one\n# The first i coefficient is the causal coefficient of the first i causes.\n# i = 1, ..., 8.\nfor i in range(X.shape[1]):\n print(i, \"causes included\")\n # augment the regressors to be both the assigned causes X and the substitute confounder Z\n X_aug = np.column_stack([X[:,:i], Z_hat])\n # holdout some data from prediction later\n X_train, X_test, y_train, y_test = train_test_split(X_aug, dfy, test_size=0.2, random_state=0)\n dcfX_train = sm.add_constant(X_train)\n dcflogit_model = sm.Logit(y_train, dcfX_train)\n dcfresult = dcflogit_model.fit_regularized(maxiter=5000)\n print(dcfresult.summary()) \n```\n\n 0 causes included\n Optimization terminated successfully. (Exit mode 0)\n Current function value: 0.18480230713099138\n Iterations: 22\n Function evaluations: 22\n Gradient evaluations: 22\n Logit Regression Results \n ==============================================================================\n Dep. Variable: y No. Observations: 455\n Model: Logit Df Residuals: 452\n Method: MLE Df Model: 2\n Date: Fri, 03 Apr 2020 Pseudo R-squ.: 0.7178\n Time: 23:28:09 Log-Likelihood: -84.085\n converged: True LL-Null: -297.99\n Covariance Type: nonrobust LLR p-value: 1.267e-93\n ==============================================================================\n coef std err z P>|z| [0.025 0.975]\n ------------------------------------------------------------------------------\n const 0.9124 0.206 4.436 0.000 0.509 1.315\n x1 -0.7687 0.211 -3.643 0.000 -1.182 -0.355\n x2 -5.0965 0.543 -9.394 0.000 -6.160 -4.033\n ==============================================================================\n 1 causes included\n Optimization terminated successfully. (Exit mode 0)\n Current function value: 0.1627111817680479\n Iterations: 30\n Function evaluations: 30\n Gradient evaluations: 30\n Logit Regression Results \n ==============================================================================\n Dep. Variable: y No. Observations: 455\n Model: Logit Df Residuals: 451\n Method: MLE Df Model: 3\n Date: Fri, 03 Apr 2020 Pseudo R-squ.: 0.7516\n Time: 23:28:09 Log-Likelihood: -74.034\n converged: True LL-Null: -297.99\n Covariance Type: nonrobust LLR p-value: 9.246e-97\n ==============================================================================\n coef std err z P>|z| [0.025 0.975]\n ------------------------------------------------------------------------------\n const 0.7397 0.217 3.410 0.001 0.315 1.165\n x1 -2.1235 0.513 -4.143 0.000 -3.128 -1.119\n x2 -1.2129 0.253 -4.800 0.000 -1.708 -0.718\n x3 -3.8954 0.621 -6.276 0.000 -5.112 -2.679\n ==============================================================================\n 2 causes included\n Optimization terminated successfully. (Exit mode 0)\n Current function value: 0.15261356019021724\n Iterations: 37\n Function evaluations: 37\n Gradient evaluations: 37\n Logit Regression Results \n ==============================================================================\n Dep. Variable: y No. Observations: 455\n Model: Logit Df Residuals: 450\n Method: MLE Df Model: 4\n Date: Fri, 03 Apr 2020 Pseudo R-squ.: 0.7670\n Time: 23:28:09 Log-Likelihood: -69.439\n converged: True LL-Null: -297.99\n Covariance Type: nonrobust LLR p-value: 1.268e-97\n ==============================================================================\n coef std err z P>|z| [0.025 0.975]\n ------------------------------------------------------------------------------\n const 0.7502 0.227 3.309 0.001 0.306 1.195\n x1 -3.3034 0.701 -4.710 0.000 -4.678 -1.929\n x2 -1.0262 0.344 -2.985 0.003 -1.700 -0.352\n x3 -1.7916 0.355 -5.040 0.000 -2.488 -1.095\n x4 -2.4318 0.785 -3.097 0.002 -3.971 -0.893\n ==============================================================================\n \n Possibly complete quasi-separation: A fraction 0.10 of observations can be\n perfectly predicted. This might indicate that there is complete\n quasi-separation. In this case some parameters will not be identified.\n 3 causes included\n Optimization terminated successfully. (Exit mode 0)\n Current function value: 0.14196497629747212\n Iterations: 44\n Function evaluations: 44\n Gradient evaluations: 44\n Logit Regression Results \n ==============================================================================\n Dep. Variable: y No. Observations: 455\n Model: Logit Df Residuals: 449\n Method: MLE Df Model: 5\n Date: Fri, 03 Apr 2020 Pseudo R-squ.: 0.7832\n Time: 23:28:09 Log-Likelihood: -64.594\n converged: True LL-Null: -297.99\n Covariance Type: nonrobust LLR p-value: 1.173e-98\n ==============================================================================\n coef std err z P>|z| [0.025 0.975]\n ------------------------------------------------------------------------------\n const 0.9137 0.247 3.700 0.000 0.430 1.398\n x1 -3.4320 0.752 -4.564 0.000 -4.906 -1.958\n x2 -1.1337 0.363 -3.119 0.002 -1.846 -0.421\n x3 -1.3748 0.462 -2.976 0.003 -2.280 -0.469\n x4 -0.7584 0.495 -1.533 0.125 -1.728 0.211\n x5 -2.6149 0.804 -3.251 0.001 -4.191 -1.039\n ==============================================================================\n \n Possibly complete quasi-separation: A fraction 0.15 of observations can be\n perfectly predicted. This might indicate that there is complete\n quasi-separation. In this case some parameters will not be identified.\n 4 causes included\n Optimization terminated successfully. (Exit mode 0)\n Current function value: 0.1402052748348448\n Iterations: 57\n Function evaluations: 57\n Gradient evaluations: 57\n Logit Regression Results \n ==============================================================================\n Dep. Variable: y No. Observations: 455\n Model: Logit Df Residuals: 448\n Method: MLE Df Model: 6\n Date: Fri, 03 Apr 2020 Pseudo R-squ.: 0.7859\n Time: 23:28:09 Log-Likelihood: -63.793\n converged: True LL-Null: -297.99\n Covariance Type: nonrobust LLR p-value: 5.396e-98\n ==============================================================================\n coef std err z P>|z| [0.025 0.975]\n ------------------------------------------------------------------------------\n const 0.8689 0.248 3.504 0.000 0.383 1.355\n x1 -3.4797 0.755 -4.608 0.000 -4.960 -2.000\n x2 -1.1444 0.363 -3.153 0.002 -1.856 -0.433\n x3 -1.1184 0.466 -2.399 0.016 -2.032 -0.205\n x4 0.9297 0.722 1.288 0.198 -0.485 2.345\n x5 -1.6538 0.827 -2.000 0.045 -3.274 -0.033\n x6 -3.1811 0.932 -3.413 0.001 -5.008 -1.354\n ==============================================================================\n \n Possibly complete quasi-separation: A fraction 0.15 of observations can be\n perfectly predicted. This might indicate that there is complete\n quasi-separation. In this case some parameters will not be identified.\n 5 causes included\n Optimization terminated successfully. (Exit mode 0)\n Current function value: 0.13737900142942064\n Iterations: 60\n Function evaluations: 60\n Gradient evaluations: 60\n Logit Regression Results \n ==============================================================================\n Dep. Variable: y No. Observations: 455\n Model: Logit Df Residuals: 447\n Method: MLE Df Model: 7\n Date: Fri, 03 Apr 2020 Pseudo R-squ.: 0.7902\n Time: 23:28:09 Log-Likelihood: -62.507\n converged: True LL-Null: -297.99\n Covariance Type: nonrobust LLR p-value: 1.395e-97\n ==============================================================================\n coef std err z P>|z| [0.025 0.975]\n ------------------------------------------------------------------------------\n const 0.8515 0.251 3.393 0.001 0.360 1.343\n x1 -3.7237 0.770 -4.836 0.000 -5.233 -2.215\n x2 -1.2666 0.367 -3.451 0.001 -1.986 -0.547\n x3 -1.7016 0.617 -2.756 0.006 -2.912 -0.492\n x4 0.8248 0.694 1.189 0.235 -0.535 2.185\n x5 -1.1228 0.704 -1.594 0.111 -2.503 0.257\n x6 -0.5378 1.062 -0.506 0.613 -2.620 1.544\n x7 -2.2185 1.087 -2.041 0.041 -4.349 -0.088\n ==============================================================================\n \n Possibly complete quasi-separation: A fraction 0.18 of observations can be\n perfectly predicted. This might indicate that there is complete\n quasi-separation. In this case some parameters will not be identified.\n 6 causes included\n Optimization terminated successfully. (Exit mode 0)\n Current function value: 0.13516533343345713\n Iterations: 71\n Function evaluations: 71\n Gradient evaluations: 71\n Logit Regression Results \n ==============================================================================\n Dep. Variable: y No. Observations: 455\n Model: Logit Df Residuals: 446\n Method: MLE Df Model: 8\n Date: Fri, 03 Apr 2020 Pseudo R-squ.: 0.7936\n Time: 23:28:09 Log-Likelihood: -61.500\n converged: True LL-Null: -297.99\n Covariance Type: nonrobust LLR p-value: 4.396e-97\n ==============================================================================\n coef std err z P>|z| [0.025 0.975]\n ------------------------------------------------------------------------------\n const 0.7814 0.256 3.049 0.002 0.279 1.284\n x1 -3.1186 0.854 -3.652 0.000 -4.792 -1.445\n x2 -1.3462 0.367 -3.664 0.000 -2.066 -0.626\n x3 -1.1913 0.691 -1.724 0.085 -2.546 0.163\n x4 1.0127 0.716 1.415 0.157 -0.390 2.415\n x5 -0.5867 0.792 -0.741 0.459 -2.139 0.966\n x6 -1.6608 1.187 -1.399 0.162 -3.987 0.665\n x7 -0.6526 1.062 -0.615 0.539 -2.734 1.428\n x8 -1.8720 1.099 -1.703 0.089 -4.026 0.282\n ==============================================================================\n \n Possibly complete quasi-separation: A fraction 0.16 of observations can be\n perfectly predicted. This might indicate that there is complete\n quasi-separation. In this case some parameters will not be identified.\n 7 causes included\n Optimization terminated successfully. (Exit mode 0)\n Current function value: 0.1331994800829468\n Iterations: 77\n Function evaluations: 77\n Gradient evaluations: 77\n Logit Regression Results \n ==============================================================================\n Dep. Variable: y No. Observations: 455\n Model: Logit Df Residuals: 445\n Method: MLE Df Model: 9\n Date: Fri, 03 Apr 2020 Pseudo R-squ.: 0.7966\n Time: 23:28:09 Log-Likelihood: -60.606\n converged: True LL-Null: -297.99\n Covariance Type: nonrobust LLR p-value: 1.448e-96\n ==============================================================================\n coef std err z P>|z| [0.025 0.975]\n ------------------------------------------------------------------------------\n const 0.7813 0.258 3.033 0.002 0.276 1.286\n x1 -3.2282 0.861 -3.748 0.000 -4.916 -1.540\n x2 -1.4509 0.382 -3.798 0.000 -2.200 -0.702\n x3 -1.5318 0.738 -2.075 0.038 -2.978 -0.085\n x4 0.5140 0.829 0.620 0.535 -1.111 2.139\n x5 -1.0940 0.894 -1.224 0.221 -2.846 0.658\n x6 -1.8328 1.201 -1.527 0.127 -4.186 0.520\n x7 -0.6254 0.483 -1.294 0.196 -1.572 0.322\n x8 0.7810 1.558 0.501 0.616 -2.273 3.835\n x9 -0.8963 1.326 -0.676 0.499 -3.496 1.703\n ==============================================================================\n \n Possibly complete quasi-separation: A fraction 0.17 of observations can be\n perfectly predicted. This might indicate that there is complete\n quasi-separation. In this case some parameters will not be identified.\n\n\n\n```python\n# Logistic regression with causes added one-by-one\n# The first i coefficient is the causal coefficient of the first i causes.\n# i = 1, ..., 8.\nfor i in range(X.shape[1]):\n print(i, \"causes included\")\n # augment the regressors to be both the assigned causes X and the substitute confounder Z\n X_aug = np.column_stack([X[:,:i]])\n # holdout some data from prediction later\n X_train, X_test, y_train, y_test = train_test_split(X_aug, dfy, test_size=0.2, random_state=0)\n dcfX_train = sm.add_constant(X_train)\n dcflogit_model = sm.Logit(y_train, dcfX_train)\n dcfresult = dcflogit_model.fit_regularized(maxiter=5000)\n print(dcfresult.summary()) \n```\n\n 0 causes included\n Optimization terminated successfully. (Exit mode 0)\n Current function value: 0.6549205599225008\n Iterations: 6\n Function evaluations: 6\n Gradient evaluations: 6\n Logit Regression Results \n ==============================================================================\n Dep. Variable: y No. Observations: 455\n Model: Logit Df Residuals: 454\n Method: MLE Df Model: 0\n Date: Fri, 03 Apr 2020 Pseudo R-squ.: 7.357e-13\n Time: 23:28:09 Log-Likelihood: -297.99\n converged: True LL-Null: -297.99\n Covariance Type: nonrobust LLR p-value: nan\n ==============================================================================\n coef std err z P>|z| [0.025 0.975]\n ------------------------------------------------------------------------------\n const 0.5639 0.098 5.783 0.000 0.373 0.755\n ==============================================================================\n 1 causes included\n Optimization terminated successfully. (Exit mode 0)\n Current function value: 0.2953829271568602\n Iterations: 15\n Function evaluations: 15\n Gradient evaluations: 15\n Logit Regression Results \n ==============================================================================\n Dep. Variable: y No. Observations: 455\n Model: Logit Df Residuals: 453\n Method: MLE Df Model: 1\n Date: Fri, 03 Apr 2020 Pseudo R-squ.: 0.5490\n Time: 23:28:09 Log-Likelihood: -134.40\n converged: True LL-Null: -297.99\n Covariance Type: nonrobust LLR p-value: 3.955e-73\n ==============================================================================\n coef std err z P>|z| [0.025 0.975]\n ------------------------------------------------------------------------------\n const 0.6566 0.155 4.225 0.000 0.352 0.961\n x1 -3.5061 0.353 -9.934 0.000 -4.198 -2.814\n ==============================================================================\n 2 causes included\n Optimization terminated successfully. (Exit mode 0)\n Current function value: 0.2592096971350075\n Iterations: 20\n Function evaluations: 20\n Gradient evaluations: 20\n Logit Regression Results \n ==============================================================================\n Dep. Variable: y No. Observations: 455\n Model: Logit Df Residuals: 452\n Method: MLE Df Model: 2\n Date: Fri, 03 Apr 2020 Pseudo R-squ.: 0.6042\n Time: 23:28:09 Log-Likelihood: -117.94\n converged: True LL-Null: -297.99\n Covariance Type: nonrobust LLR p-value: 6.397e-79\n ==============================================================================\n coef std err z P>|z| [0.025 0.975]\n ------------------------------------------------------------------------------\n const 0.6744 0.168 4.018 0.000 0.345 1.003\n x1 -3.6479 0.390 -9.347 0.000 -4.413 -2.883\n x2 -0.9815 0.181 -5.424 0.000 -1.336 -0.627\n ==============================================================================\n 3 causes included\n Optimization terminated successfully. (Exit mode 0)\n Current function value: 0.162465976738943\n Iterations: 28\n Function evaluations: 29\n Gradient evaluations: 28\n Logit Regression Results \n ==============================================================================\n Dep. Variable: y No. Observations: 455\n Model: Logit Df Residuals: 451\n Method: MLE Df Model: 3\n Date: Fri, 03 Apr 2020 Pseudo R-squ.: 0.7519\n Time: 23:28:09 Log-Likelihood: -73.922\n converged: True LL-Null: -297.99\n Covariance Type: nonrobust LLR p-value: 8.272e-97\n ==============================================================================\n coef std err z P>|z| [0.025 0.975]\n ------------------------------------------------------------------------------\n const 0.9963 0.231 4.306 0.000 0.543 1.450\n x1 -4.9102 0.601 -8.170 0.000 -6.088 -3.732\n x2 -1.7394 0.284 -6.115 0.000 -2.297 -1.182\n x3 -2.2080 0.336 -6.578 0.000 -2.866 -1.550\n ==============================================================================\n \n Possibly complete quasi-separation: A fraction 0.12 of observations can be\n perfectly predicted. This might indicate that there is complete\n quasi-separation. In this case some parameters will not be identified.\n 4 causes included\n Optimization terminated successfully. (Exit mode 0)\n Current function value: 0.15485253636195323\n Iterations: 35\n Function evaluations: 35\n Gradient evaluations: 35\n Logit Regression Results \n ==============================================================================\n Dep. Variable: y No. Observations: 455\n Model: Logit Df Residuals: 450\n Method: MLE Df Model: 4\n Date: Fri, 03 Apr 2020 Pseudo R-squ.: 0.7636\n Time: 23:28:10 Log-Likelihood: -70.458\n converged: True LL-Null: -297.99\n Covariance Type: nonrobust LLR p-value: 3.495e-97\n ==============================================================================\n coef std err z P>|z| [0.025 0.975]\n ------------------------------------------------------------------------------\n const 0.9646 0.233 4.146 0.000 0.509 1.421\n x1 -4.6125 0.614 -7.509 0.000 -5.816 -3.409\n x2 -1.6563 0.295 -5.610 0.000 -2.235 -1.078\n x3 -1.7342 0.372 -4.667 0.000 -2.463 -1.006\n x4 -0.9279 0.353 -2.631 0.009 -1.619 -0.237\n ==============================================================================\n \n Possibly complete quasi-separation: A fraction 0.13 of observations can be\n perfectly predicted. This might indicate that there is complete\n quasi-separation. In this case some parameters will not be identified.\n 5 causes included\n Optimization terminated successfully. (Exit mode 0)\n Current function value: 0.14242454080229494\n Iterations: 44\n Function evaluations: 45\n Gradient evaluations: 44\n Logit Regression Results \n ==============================================================================\n Dep. Variable: y No. Observations: 455\n Model: Logit Df Residuals: 449\n Method: MLE Df Model: 5\n Date: Fri, 03 Apr 2020 Pseudo R-squ.: 0.7825\n Time: 23:28:10 Log-Likelihood: -64.803\n converged: True LL-Null: -297.99\n Covariance Type: nonrobust LLR p-value: 1.444e-98\n ==============================================================================\n coef std err z P>|z| [0.025 0.975]\n ------------------------------------------------------------------------------\n const 0.8234 0.242 3.396 0.001 0.348 1.299\n x1 -4.6125 0.636 -7.252 0.000 -5.859 -3.366\n x2 -1.7131 0.303 -5.648 0.000 -2.308 -1.119\n x3 -1.9515 0.395 -4.941 0.000 -2.726 -1.177\n x4 0.4077 0.512 0.797 0.425 -0.595 1.410\n x5 -1.7514 0.464 -3.777 0.000 -2.660 -0.843\n ==============================================================================\n \n Possibly complete quasi-separation: A fraction 0.16 of observations can be\n perfectly predicted. This might indicate that there is complete\n quasi-separation. In this case some parameters will not be identified.\n 6 causes included\n Optimization terminated successfully. (Exit mode 0)\n Current function value: 0.13837349335376822\n Iterations: 60\n Function evaluations: 60\n Gradient evaluations: 60\n Logit Regression Results \n ==============================================================================\n Dep. Variable: y No. Observations: 455\n Model: Logit Df Residuals: 448\n Method: MLE Df Model: 6\n Date: Fri, 03 Apr 2020 Pseudo R-squ.: 0.7887\n Time: 23:28:10 Log-Likelihood: -62.960\n converged: True LL-Null: -297.99\n Covariance Type: nonrobust LLR p-value: 2.361e-98\n ==============================================================================\n coef std err z P>|z| [0.025 0.975]\n ------------------------------------------------------------------------------\n const 0.7376 0.248 2.975 0.003 0.252 1.223\n x1 -3.5398 0.807 -4.384 0.000 -5.122 -1.957\n x2 -1.6821 0.302 -5.563 0.000 -2.275 -1.089\n x3 -1.3522 0.491 -2.756 0.006 -2.314 -0.391\n x4 0.6207 0.527 1.177 0.239 -0.413 1.655\n x5 -1.0073 0.623 -1.617 0.106 -2.228 0.214\n x6 -2.1317 1.138 -1.873 0.061 -4.362 0.098\n ==============================================================================\n \n Possibly complete quasi-separation: A fraction 0.14 of observations can be\n perfectly predicted. This might indicate that there is complete\n quasi-separation. In this case some parameters will not be identified.\n 7 causes included\n Optimization terminated successfully. (Exit mode 0)\n Current function value: 0.135339334890979\n Iterations: 62\n Function evaluations: 62\n Gradient evaluations: 62\n Logit Regression Results \n ==============================================================================\n Dep. Variable: y No. Observations: 455\n Model: Logit Df Residuals: 447\n Method: MLE Df Model: 7\n Date: Fri, 03 Apr 2020 Pseudo R-squ.: 0.7933\n Time: 23:28:10 Log-Likelihood: -61.579\n converged: True LL-Null: -297.99\n Covariance Type: nonrobust LLR p-value: 5.570e-98\n ==============================================================================\n coef std err z P>|z| [0.025 0.975]\n ------------------------------------------------------------------------------\n const 0.7287 0.250 2.916 0.004 0.239 1.219\n x1 -3.6556 0.796 -4.592 0.000 -5.216 -2.095\n x2 -1.7323 0.307 -5.644 0.000 -2.334 -1.131\n x3 -1.1122 0.483 -2.304 0.021 -2.058 -0.166\n x4 0.7408 0.542 1.367 0.172 -0.321 1.803\n x5 -0.8364 0.614 -1.363 0.173 -2.039 0.366\n x6 -2.2682 1.139 -1.992 0.046 -4.500 -0.037\n x7 -0.5397 0.324 -1.666 0.096 -1.175 0.095\n ==============================================================================\n \n Possibly complete quasi-separation: A fraction 0.15 of observations can be\n perfectly predicted. This might indicate that there is complete\n quasi-separation. In this case some parameters will not be identified.\n\n\n**We note that the causal coefficient of x4 is stable with the (causal) deconfounder but the correlation coefficent of x4 flips sign with the (noncausal) logistic regression.**\n\n# Takeaways\n\n\n\n* The deconfounder is **not hard** to use.\n* We simply **fit** a factor model, **check** it, and **infer** causal effects with the substitute confounder.\n* Please **always check** the factor model.\n* The deconfounder **makes a difference**.\n* The deconfounder **deconfounds**.\n\n\n\n# Acknowledgements\n\nWe thank Suresh Naidu for suggesting the adding-causes-one-by-one idea.\n", "meta": {"hexsha": "0fcd0e42cf3fdd02085ad7d63e9a8dcde0f4e984", "size": 629116, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "deconfounder_tutorial.ipynb", "max_stars_repo_name": "blei-lab/deconfounder_tutorial", "max_stars_repo_head_hexsha": "54e2a6bd73736f688313751ccfbc9119d6060d4d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 84, "max_stars_repo_stars_event_min_datetime": "2018-09-20T19:04:57.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-30T18:21:13.000Z", "max_issues_repo_path": "deconfounder_tutorial.ipynb", "max_issues_repo_name": "cleeway/deconfounder_tutorial", "max_issues_repo_head_hexsha": "54e2a6bd73736f688313751ccfbc9119d6060d4d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2020-04-02T13:02:43.000Z", "max_issues_repo_issues_event_max_datetime": "2020-06-03T12:28:16.000Z", "max_forks_repo_path": "deconfounder_tutorial.ipynb", "max_forks_repo_name": "cleeway/deconfounder_tutorial", "max_forks_repo_head_hexsha": "54e2a6bd73736f688313751ccfbc9119d6060d4d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 21, "max_forks_repo_forks_event_min_datetime": "2018-10-08T13:48:16.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-06T13:28:58.000Z", "avg_line_length": 200.7389917039, "max_line_length": 433438, "alphanum_fraction": 0.8345996605, "converted": true, "num_tokens": 21559, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754489059775, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.44236386884272855}} {"text": "\n\n\n# Tutorial-IllinoisGRMHD: outer_boundaries.C\n\n## Authors: Leo Werneck & Zach Etienne\n\n**This module is currently under development**\n\n## In this tutorial module we explain the outer boundary conditions imposed on the quantities evolved within `IllinoisGRMHD`\n\n### Required and recommended citations:\n\n* **(Required)** Etienne, Z. B., Paschalidis, V., Haas R., M\u00f6sta P., and Shapiro, S. L. IllinoisGRMHD: an open-source, user-friendly GRMHD code for dynamical spacetimes. Class. Quantum Grav. 32 (2015) 175009. ([arxiv:1501.07276](http://arxiv.org/abs/1501.07276)).\n* **(Required)** Noble, S. C., Gammie, C. F., McKinney, J. C., Del Zanna, L. Primitive Variable Solvers for Conservative General Relativistic Magnetohydrodynamics. Astrophysical Journal, 641, 626 (2006) ([astro-ph/0512420](https://arxiv.org/abs/astro-ph/0512420)).\n* **(Recommended)** Del Zanna, L., Bucciantini N., Londrillo, P. An efficient shock-capturing central-type scheme for multidimensional relativistic flows - II. Magnetohydrodynamics. A&A 400 (2) 397-413 (2003). DOI: 10.1051/0004-6361:20021641 ([astro-ph/0210618](https://arxiv.org/abs/astro-ph/0210618)).\n\n\n\n# Table of Contents\n$$\\label{toc}$$\n\nThis module is organized as follows\n\n0. [Step 0](#src_dir): **Source directory creation**\n1. [Step 1](#introduction): **Introduction**\n1. [Step 2](#outer_boundaries__c): **`outer_boundaries.C`**\n 1. [Step 2.a](#outer_boundaries__amu): *The vector potential variables*\n 1. [Step 2.a.i](#outer_boundaries__amu__linear_extrapolation): Defining the linear extrapolation operators\n 1. [Step 2.a.ii](#outer_boundaries__amu__applying_bcs): Applying outer boundary conditions to $A_{\\mu}$\n 1. [Step 2.b](#outer_boundaries__hydro_vars): *The hydrodynamic variables*\n 1. [Step 2.b.i](#outer_boundaries__hydro_vars__zero_deriv_outflow): Defining the zero derivative, outflow operators\n 1. [Step 2.b.ii](#outer_boundaries__hydro_vars__applying_bcs): Applying boundary conditions to $\\left\\{P,\\rho_{b},v^{i}\\right\\}$\n 1. [Step 2.c](#outer_boundaries__conservatives): *The conservative variables*\n1. [Step 3](#code_validation): **Code validation**\n1. [Step 4](#latex_pdf_output): **Output this notebook to $\\LaTeX$-formatted PDF file**\n\n\n\n# Step 0: Source directory creation \\[Back to [top](#toc)\\]\n$$\\label{src_dir}$$\n\nWe will now use the [cmdline_helper.py NRPy+ module](Tutorial-Tutorial-cmdline_helper.ipynb) to create the source directory within the `IllinoisGRMHD` NRPy+ directory, if it does not exist yet.\n\n\n```python\n# Step 0: Creation of the IllinoisGRMHD source directory\n# Step 0a: Add NRPy's directory to the path\n# https://stackoverflow.com/questions/16780014/import-file-from-parent-directory\nimport os,sys\nnrpy_dir_path = os.path.join(\"..\",\"..\")\nif nrpy_dir_path not in sys.path:\n sys.path.append(nrpy_dir_path)\n\n# Step 0b: Load up cmdline_helper and create the directory\nimport cmdline_helper as cmd\nIGM_src_dir_path = os.path.join(\"..\",\"src\")\ncmd.mkdir(IGM_src_dir_path)\n\n# Step 0c: Create the output file path\noutfile_path__outer_boundaries__C = os.path.join(IGM_src_dir_path,\"outer_boundaries.C\")\n```\n\n\n\n# Step 1: Introduction \\[Back to [top](#toc)\\]\n$$\\label{introduction}$$\n\n\n\n# Step 2: `outer_boundaries.C` \\[Back to [top](#toc)\\]\n$$\\label{outer_boundaries__c}$$\n\nThe strategy used to set outer boundary for the primitives, $\\left\\{P,\\rho_{b},v^{i}\\right\\}$, and for the scalar and vector potentials, $\\left\\{\\left[\\sqrt{\\gamma}\\Phi\\right],A_{i}\\right\\}$, follows eqs. (39) and (40) of the [original release paper of IllinoisGRMHD](https://arxiv.org/pdf/1501.07276.pdf). For example, if we are trying to apply boundary condition along the $x$-direction, we would have\n\n$$\n\\boxed{\nE_{i+1}\n=\n\\left\\{\n\\begin{align}\nE_{i}\\ , &{\\rm\\ if\\ } E\\in\\left\\{P,\\rho_{b},v^{y},v^{z}\\right\\},{\\rm\\ or\\ } E=v^{x}\\ {\\rm and\\ } v^{x}\\geq0\\\\\n0\\ , &{\\rm\\ if\\ } E=v^{x}\\ {\\rm and\\ } v^{x}<0\\\\\n2E_{i} - E_{i-1}\\ , &{\\rm\\ if\\ } E\\in\\left\\{\\left[\\sqrt{\\gamma}\\Phi\\right],A_{x},A_{y},A_{z}\\right\\}\n\\end{align}\n\\right.\n}\\ ,\n$$\n\nfor the ghostzone points along the *positive* $x$-direction, and\n\n$$\n\\boxed{\nE_{i-1}\n=\n\\left\\{\n\\begin{align}\nE_{i}\\ , &{\\rm\\ if\\ } E\\in\\left\\{P,\\rho_{b},v^{y},v^{z}\\right\\},{\\rm\\ or\\ } E=v^{x}\\ {\\rm and\\ } v^{x}\\geq0\\\\\n0\\ , &{\\rm\\ if\\ } E=v^{x}\\ {\\rm and\\ } v^{x}<0\\\\\n2E_{i} - E_{i+1}\\ , &{\\rm\\ if\\ } E\\in\\left\\{\\left[\\sqrt{\\gamma}\\Phi\\right],A_{x},A_{y},A_{z}\\right\\}\n\\end{align}\n\\right.\n}\\ ,\n$$\n\nfor the ghostzone points along the *negative* $x$-direction.In this way, linear extrapolation outer boundary conditions are applied to the vector potential variables $\\left\\{\\left[\\sqrt{\\gamma}\\Phi\\right],A_{i}\\right\\}$ and zero-derivative, outflow outer boundary conditions are applied to the hydrodynamic variables $\\left\\{P,\\rho_{b},v^{i}\\right\\}$.\n\n\n\n## Step 2.a: The vector potential variables \\[Back to [top](#toc)\\]\n$$\\label{outer_boundaries__amu}$$\n\n\n\n### Step 2.a.i: Defining the linear extrapolation operators \\[Back to [top](#toc)\\]\n$$\\label{outer_boundaries__amu__linear_extrapolation}$$\n\nWe start by applying outer boundary conditions to $\\left\\{\\left[\\sqrt{\\gamma}\\Phi\\right],A_{i}\\right\\}$. We follow the prescription described above:\n\n$$\n\\boxed{\n\\begin{align}\n\\text{Positive direction: }E_{i+1} = 2E_{i} - E_{i-1}\\ , &{\\rm\\ if\\ } E\\in\\left\\{\\left[\\sqrt{\\gamma}\\Phi\\right],A_{x},A_{y},A_{z}\\right\\}\\\\\n\\text{Negative direction: }E_{i-1} = 2E_{i} - E_{i+1}\\ , &{\\rm\\ if\\ } E\\in\\left\\{\\left[\\sqrt{\\gamma}\\Phi\\right],A_{x},A_{y},A_{z}\\right\\}\n\\end{align}\n}\\ ,\n$$\n\nwhich uses a linear extrapolation outer boundary condition.\n\n\n```python\n%%writefile $outfile_path__outer_boundaries__C\n/*******************************************************\n * Outer boundaries are handled as follows:\n * (-1) Update RHS quantities, leave RHS quantities zero on all outer ghostzones (including outer AMR refinement, processor, and outer boundaries)\n * ( 0) Let MoL update all evolution variables\n * ( 1) Apply outer boundary conditions (BCs) on A_{\\mu}\n * ( 2) Compute B^i from A_i everywhere, synchronize B^i\n * ( 3) Call con2prim to get primitives on interior pts\n * ( 4) Apply outer BCs on {P,rho_b,vx,vy,vz}.\n * ( 5) (optional) set conservatives on outer boundary.\n *******************************************************/\n\n#include \"cctk.h\"\n#include \n#include \n#include \n#include \"cctk_Arguments.h\"\n#include \"cctk_Parameters.h\"\n\n#include \"IllinoisGRMHD_headers.h\"\n#include \"IllinoisGRMHD_EoS_lowlevel_functs.C\"\n#include \"inlined_functions.C\"\n\n#define IDX(i,j,k) CCTK_GFINDEX3D(cctkGH,(i),(j),(k))\n\n#define XMAX_OB_LINEAR_EXTRAP(FUNC,imax) for(int k=0;k\n\n### Step 2.a.ii: Applying outer boundary conditions to $A_{\\mu}$ \\[Back to [top](#toc)\\]\n$$\\label{outer_boundaries__amu__applying_bcs}$$\n\nNow we apply boundary conditions to $A_{\\mu}$. The code below is pretty straightforward, but it is useful to understand the following `cctk` variables (refer to the **Cactus Variables** paragraph of section C1.6.2 of the [Einstein Toolkit UserGuide](https://einsteintoolkit.org/usersguide/UsersGuidech9.html#x13-81000C1.6) for further details):\n\n1. `cctk_lsh[i]`: the number of *total* number of grid points along direction $x^{i}$, used *by each processor*.\n2. `cctk_bbox[i]`: an array of integers that tell if the boundary gridpoints used by each processor are *internal* (i.e. artificial) or *physical* (i.e. actual boundary points). The variable follows the pattern:\n 1. `cctk_bbox[0]`: **Direction**: $x$ | **Orientation**: $+$ | Returns $\\color{red}{0}$ if the boundary is $\\color{red}{\\text{artificial}}$ and $\\color{blue}{1}$ if it is $\\color{blue}{\\text{physical}}$\n 1. `cctk_bbox[1]`: **Direction**: $x$ | **Orientation**: $-$ | Returns $\\color{red}{0}$ if the boundary is $\\color{red}{\\text{artificial}}$ and $\\color{blue}{1}$ if it is $\\color{blue}{\\text{physical}}$\n 1. `cctk_bbox[2]`: **Direction**: $y$ | **Orientation**: $+$ | Returns $\\color{red}{0}$ if the boundary is $\\color{red}{\\text{artificial}}$ and $\\color{blue}{1}$ if it is $\\color{blue}{\\text{physical}}$\n 1. `cctk_bbox[3]`: **Direction**: $y$ | **Orientation**: $-$ | Returns $\\color{red}{0}$ if the boundary is $\\color{red}{\\text{artificial}}$ and $\\color{blue}{1}$ if it is $\\color{blue}{\\text{physical}}$\n 1. `cctk_bbox[4]`: **Direction**: $z$ | **Orientation**: $+$ | Returns $\\color{red}{0}$ if the boundary is $\\color{red}{\\text{artificial}}$ and $\\color{blue}{1}$ if it is $\\color{blue}{\\text{physical}}$\n 1. `cctk_bbox[5]`: **Direction**: $z$ | **Orientation**: $-$ | Returns $\\color{red}{0}$ if the boundary is $\\color{red}{\\text{artificial}}$ and $\\color{blue}{1}$ if it is $\\color{blue}{\\text{physical}}$\n\n\n```python\n%%writefile -a $outfile_path__outer_boundaries__C\n\n\n/*********************************************\n * Apply outer boundary conditions on A_{\\mu}\n ********************************************/\nextern \"C\" void IllinoisGRMHD_outer_boundaries_on_A_mu(CCTK_ARGUMENTS) {\n DECLARE_CCTK_ARGUMENTS;\n DECLARE_CCTK_PARAMETERS;\n\n if(CCTK_EQUALS(EM_BC,\"frozen\")) return;\n\n bool Symmetry_none=false; if(CCTK_EQUALS(Symmetry,\"none\")) Symmetry_none=true;\n\n int levelnumber = GetRefinementLevel(cctkGH);\n\n IllinoisGRMHD_convert_ADM_to_BSSN__enforce_detgtij_eq_1__and_compute_gtupij(cctkGH,cctk_lsh, gxx,gxy,gxz,gyy,gyz,gzz,alp,\n gtxx,gtxy,gtxz,gtyy,gtyz,gtzz,\n gtupxx,gtupxy,gtupxz,gtupyy,gtupyz,gtupzz,\n phi_bssn,psi_bssn,lapm1);\n\n // Don't apply approximate outer boundary conditions on initial data, which should be defined everywhere, or on levels != [coarsest level].\n if(cctk_iteration==0 || levelnumber!=0) return;\n\n if(cctk_nghostzones[0]!=cctk_nghostzones[1] || cctk_nghostzones[0]!=cctk_nghostzones[2])\n CCTK_VError(VERR_DEF_PARAMS,\"ERROR: IllinoisGRMHD outer BC driver does not support unequal number of ghostzones in different directions!\");\n for(int which_bdry_pt=0;which_bdry_pt\n\n## Step 2.b: The hydrodynamic variables \\[Back to [top](#toc)\\]\n$$\\label{outer_boundaries__hydro_vars}$$\n\n\n\n### Step 2.b.i: Defining the zero derivative, outflow operators \\[Back to [top](#toc)\\]\n$$\\label{outer_boundaries__hydro_vars__zero_deriv_outflow}$$\n\nWe now apply outer boundary conditions to $\\left\\{P,\\rho_{b},v^{i}\\right\\}$, imposing zero derivative, outflow boundary conditions. We follow the prescription described above:\n\n$$\n\\boxed{\n\\begin{matrix}\n\\text{Positive direction: }E_{i+1}\n=\n\\left\\{\n\\begin{matrix}\nE_{i}\\ , &{\\rm\\ if\\ } E\\in\\left\\{P,\\rho_{b},v^{y},v^{z}\\right\\},{\\rm\\ or\\ } E=v^{x}\\ {\\rm and\\ } v^{x}\\geq0\\\\\n0\\ , &{\\rm\\ if\\ } E=v^{x}\\ {\\rm and\\ } v^{x}<0\n\\end{matrix}\n\\right.\\\\\n\\text{Negative direction: }E_{i-1}\n=\n\\left\\{\n\\begin{matrix}\nE_{i}\\ , &{\\rm\\ if\\ } E\\in\\left\\{P,\\rho_{b},v^{y},v^{z}\\right\\},{\\rm\\ or\\ } E=v^{x}\\ {\\rm and\\ } v^{x}\\geq0\\\\\n0\\ , &{\\rm\\ if\\ } E=v^{x}\\ {\\rm and\\ } v^{x}<0\n\\end{matrix}\n\\right.\n\\end{matrix}\n}\\ .\n$$\n\n\n```python\n%%writefile -a $outfile_path__outer_boundaries__C\n\n\n#define XMAX_OB_SIMPLE_COPY(FUNC,imax) for(int k=0;k0.) vx[IDX(imin,j,k)]=0.;\n#define YMIN_INFLOW_CHECK(vy,jmin) for(int k=0;k0.) vy[IDX(i,jmin,k)]=0.;\n#define ZMIN_INFLOW_CHECK(vz,kmin) for(int j=0;j0.) vz[IDX(i,j,kmin)]=0.;\n```\n\n Appending to ../src/outer_boundaries.C\n\n\n\n\n### Step 2.b.ii: Applying boundary conditions to $\\left\\{P,\\rho_{b},v^{i}\\right\\}$ \\[Back to [top](#toc)\\]\n$$\\label{outer_boundaries__hydro_vars__applying_bcs}$$\n\nAs with the previous case, applying the boundary conditions is a straightforward procedure. We refer the reader to the `cctk` quantities discussed in [Step 2.a.ii](#outer_boundaries__amu__applying_bcs), in case clarifications are needed.\n\n\n```python\n%%writefile -a $outfile_path__outer_boundaries__C\n\n\n\n/*******************************************************\n * Apply outer boundary conditions on {P,rho_b,vx,vy,vz}\n * It is better to apply BCs on primitives than conservs,\n * because small errors in conservs can be greatly\n * amplified in con2prim, sometimes leading to unphysical\n * primitives & unnecessary fixes.\n *******************************************************/\nextern \"C\" void IllinoisGRMHD_outer_boundaries_on_P_rho_b_vx_vy_vz(CCTK_ARGUMENTS) {\n DECLARE_CCTK_ARGUMENTS;\n DECLARE_CCTK_PARAMETERS;\n\n if(CCTK_EQUALS(Matter_BC,\"frozen\")) return;\n\n bool Symmetry_none=false; if(CCTK_EQUALS(Symmetry,\"none\")) Symmetry_none=true;\n\n int levelnumber = GetRefinementLevel(cctkGH);\n\n // Don't apply approximate outer boundary conditions on initial data, which should be defined everywhere, or on levels != [coarsest level].\n if(cctk_iteration==0 || levelnumber!=0) return;\n\n int ENABLE=1;\n\n IllinoisGRMHD_convert_ADM_to_BSSN__enforce_detgtij_eq_1__and_compute_gtupij(cctkGH,cctk_lsh, gxx,gxy,gxz,gyy,gyz,gzz,alp,\n gtxx,gtxy,gtxz,gtyy,gtyz,gtzz,\n gtupxx,gtupxy,gtupxz,gtupyy,gtupyz,gtupzz,\n phi_bssn,psi_bssn,lapm1);\n\n //if(levelnumber<=11110) {\n if(cctk_nghostzones[0]!=cctk_nghostzones[1] || cctk_nghostzones[0]!=cctk_nghostzones[2])\n CCTK_VError(VERR_DEF_PARAMS,\"ERROR: IllinoisGRMHD outer BC driver does not support unequal number of ghostzones in different directions!\");\n for(int which_bdry_pt=0;which_bdry_pt\n\n## Step 2.c: The conservative variables \\[Back to [top](#toc)\\]\n$$\\label{outer_boundaries__conservatives}$$\n\nAfter we have applied boundary conditions to our primitives (i.e. hydrodynamics) variables, we [make sure their values lie within the physical range and then recompute the conservatives](Tutorial-IllinoisGRMHD__apply_tau_floor__enforce_limits_on_primitives_and_recompute_conservs.ipynb). Notice that the boundary conditions are then not applied directly to the conservative variables. The reason why the code is structured in this way is because small variations in the values of the conservative variables can cause the conservative-to-primitive algorithm to fail.\n\n\n```python\n%%writefile -a $outfile_path__outer_boundaries__C\n\n\n /**********************************\n * Piecewise Polytropic EOS Patch *\n * Setting up the EOS struct *\n **********************************/\n /*\n * The short piece of code below takes care\n * of initializing the EOS parameters.\n * Please refer to the \"inlined_functions.C\"\n * source file for the documentation on the\n * function.\n */\n eos_struct eos;\n initialize_EOS_struct_from_input(eos);\n\n#pragma omp parallel for\n for(int k=0;k=cctk_lsh[0]-cctk_nghostzones[0]) ||\n ((cctk_bbox[2]) && j=cctk_lsh[1]-cctk_nghostzones[1]) ||\n ((cctk_bbox[4]) && k=cctk_lsh[2]-cctk_nghostzones[2])) {\n int index = CCTK_GFINDEX3D(cctkGH,i,j,k);\n int ww;\n\n CCTK_REAL METRIC[NUMVARS_FOR_METRIC],dummy=-1e100; // Set dummy to insane value, to ensure it isn't being used.\n ww=0;\n //psi[index] = exp(phi[index]);\n METRIC[ww] = phi_bssn[index];ww++;\n METRIC[ww] = dummy; ww++; // Don't need to set psi.\n METRIC[ww] = gtxx[index]; ww++;\n METRIC[ww] = gtxy[index]; ww++;\n METRIC[ww] = gtxz[index]; ww++;\n METRIC[ww] = gtyy[index]; ww++;\n METRIC[ww] = gtyz[index]; ww++;\n METRIC[ww] = gtzz[index]; ww++;\n METRIC[ww] = lapm1[index]; ww++;\n METRIC[ww] = betax[index]; ww++;\n METRIC[ww] = betay[index]; ww++;\n METRIC[ww] = betaz[index]; ww++;\n METRIC[ww] = gtupxx[index]; ww++;\n METRIC[ww] = gtupyy[index]; ww++;\n METRIC[ww] = gtupzz[index]; ww++;\n METRIC[ww] = gtupxy[index]; ww++;\n METRIC[ww] = gtupxz[index]; ww++;\n METRIC[ww] = gtupyz[index]; ww++;\n\n CCTK_REAL U[MAXNUMVARS];\n ww=0;\n U[ww] = rho_b[index]; ww++;\n U[ww] = P[index]; ww++;\n U[ww] = vx[index]; ww++;\n U[ww] = vy[index]; ww++;\n U[ww] = vz[index]; ww++;\n U[ww] = Bx[index]; ww++;\n U[ww] = By[index]; ww++;\n U[ww] = Bz[index]; ww++;\n\n struct output_stats stats;\n CCTK_REAL CONSERVS[NUM_CONSERVS],TUPMUNU[10],TDNMUNU[10];\n\n const int already_computed_physical_metric_and_inverse=0;\n CCTK_REAL g4dn[4][4],g4up[4][4];\n IllinoisGRMHD_enforce_limits_on_primitives_and_recompute_conservs(already_computed_physical_metric_and_inverse,U,stats,eos,METRIC,g4dn,g4up, TUPMUNU,TDNMUNU,CONSERVS);\n\n rho_b[index] = U[RHOB];\n P[index] = U[PRESSURE];\n vx[index] = U[VX];\n vy[index] = U[VY];\n vz[index] = U[VZ];\n\n rho_star[index]=CONSERVS[RHOSTAR];\n tau[index] =CONSERVS[TAUENERGY];\n mhd_st_x[index]=CONSERVS[STILDEX];\n mhd_st_y[index]=CONSERVS[STILDEY];\n mhd_st_z[index]=CONSERVS[STILDEZ];\n\n if(update_Tmunu) {\n ww=0;\n eTtt[index] = TDNMUNU[ww]; ww++;\n eTtx[index] = TDNMUNU[ww]; ww++;\n eTty[index] = TDNMUNU[ww]; ww++;\n eTtz[index] = TDNMUNU[ww]; ww++;\n eTxx[index] = TDNMUNU[ww]; ww++;\n eTxy[index] = TDNMUNU[ww]; ww++;\n eTxz[index] = TDNMUNU[ww]; ww++;\n eTyy[index] = TDNMUNU[ww]; ww++;\n eTyz[index] = TDNMUNU[ww]; ww++;\n eTzz[index] = TDNMUNU[ww];\n }\n //if(i==5 && j==5 && k==5) CCTK_VInfo(CCTK_THORNSTRING,\"%e %e %e %e\",eTtt[index],eTtx[index],eTty[index],eTxy[index]);\n //CCTK_VInfo(CCTK_THORNSTRING,\"YAY: \"); for(ww=0;ww<10;ww++) CCTK_VInfo(CCTK_THORNSTRING,\"%e \",TDNMUNU[ww]); CCTK_VInfo(CCTK_THORNSTRING,\"\");\n }\n }\n}\n\n\n```\n\n Appending to ../src/outer_boundaries.C\n\n\n\n\n# Step 3: Code validation \\[Back to [top](#toc)\\]\n$$\\label{code_validation}$$\n\nFirst we download the original `IllinoisGRMHD` source code and then compare it to the source code generated by this tutorial notebook.\n\n\n```python\n# Verify if the code generated by this tutorial module\n# matches the original IllinoisGRMHD source code\n\n# First download the original IllinoisGRMHD source code\nimport urllib\nfrom os import path\n\noriginal_IGM_file_url = \"https://bitbucket.org/zach_etienne/wvuthorns/raw/5611b2f0b17135538c9d9d17c7da062abe0401b6/IllinoisGRMHD/src/outer_boundaries.C\"\noriginal_IGM_file_name = \"outer_boundaries-original.C\"\noriginal_IGM_file_path = os.path.join(IGM_src_dir_path,original_IGM_file_name)\n\n# Then download the original IllinoisGRMHD source code\n# We try it here in a couple of ways in an attempt to keep\n# the code more portable\ntry:\n original_IGM_file_code = urllib.request.urlopen(original_IGM_file_url).read().decode(\"utf-8\")\n # Write down the file the original IllinoisGRMHD source code\n with open(original_IGM_file_path,\"w\") as file:\n file.write(original_IGM_file_code)\nexcept:\n try:\n original_IGM_file_code = urllib.urlopen(original_IGM_file_url).read().decode(\"utf-8\")\n # Write down the file the original IllinoisGRMHD source code\n with open(original_IGM_file_path,\"w\") as file:\n file.write(original_IGM_file_code)\n except:\n # If all else fails, hope wget does the job\n !wget -O $original_IGM_file_path $original_IGM_file_url\n\n# Perform validation\nValidation__outer_boundaries__C = !diff $original_IGM_file_path $outfile_path__outer_boundaries__C\n\nif Validation__outer_boundaries__C == []:\n # If the validation passes, we do not need to store the original IGM source code file\n !rm $original_IGM_file_path\n print(\"Validation test for outer_boundaries.C: PASSED!\")\nelse:\n # If the validation fails, we keep the original IGM source code file\n print(\"Validation test for outer_boundaries.C: FAILED!\")\n # We also print out the difference between the code generated\n # in this tutorial module and the original IGM source code\n print(\"Diff:\")\n for diff_line in Validation__outer_boundaries__C:\n print(diff_line)\n```\n\n Validation test for outer_boundaries.C: FAILED!\n Diff:\n 19a20\n > #include \"IllinoisGRMHD_EoS_lowlevel_functs.C\"\n 31a33\n > \n 73a76\n > \n 91a95\n > \n 155c159,170\n < // FIXME: only for single gamma-law EOS.\n ---\n > \n > /**********************************\n > * Piecewise Polytropic EOS Patch *\n > * Setting up the EOS struct *\n > **********************************/\n > /*\n > * The short piece of code below takes care\n > * of initializing the EOS parameters.\n > * Please refer to the \"inlined_functions.C\"\n > * source file for the documentation on the\n > * function.\n > */\n 157,165c172,173\n < eos.neos=neos;\n < eos.K_poly=K_poly;\n < eos.rho_tab[0]=rho_tab[0];\n < eos.P_tab[0]=P_tab[0];\n < eos.gamma_th=gamma_th;\n < eos.eps_tab[0]=eps_tab[0];\n < eos.k_tab[0]=k_tab[0]; eos.k_tab[1]=k_tab[1];\n < eos.gamma_tab[0]=gamma_tab[0]; eos.gamma_tab[1]=gamma_tab[1];\n < \n ---\n > initialize_EOS_struct_from_input(eos);\n > \n 246a255\n > \n\n\n\n\n# Step 4: Output this notebook to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n$$\\label{latex_pdf_output}$$\n\nThe following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n[Tutorial-IllinoisGRMHD__outer_boundaries.pdf](Tutorial-IllinoisGRMHD__outer_boundaries.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means).\n\n\n```python\nlatex_nrpy_style_path = os.path.join(nrpy_dir_path,\"latex_nrpy_style.tplx\")\n#!jupyter nbconvert --to latex --template $latex_nrpy_style_path --log-level='WARN' Tutorial-IllinoisGRMHD__outer_boundaries.ipynb\n#!pdflatex -interaction=batchmode Tutorial-IllinoisGRMHD__outer_boundaries.tex\n#!pdflatex -interaction=batchmode Tutorial-IllinoisGRMHD__outer_boundaries.tex\n#!pdflatex -interaction=batchmode Tutorial-IllinoisGRMHD__outer_boundaries.tex\n!rm -f Tut*.out Tut*.aux Tut*.log\n```\n", "meta": {"hexsha": "44502e675a0da30d701691355375370bbdfee904", "size": 37702, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "IllinoisGRMHD/doc/Tutorial-IllinoisGRMHD__outer_boundaries.ipynb", "max_stars_repo_name": "ksible/nrpytutorial", "max_stars_repo_head_hexsha": "4ca6e9da22def2a9c9bcbcad75847fd1db159f4b", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "IllinoisGRMHD/doc/Tutorial-IllinoisGRMHD__outer_boundaries.ipynb", "max_issues_repo_name": "ksible/nrpytutorial", "max_issues_repo_head_hexsha": "4ca6e9da22def2a9c9bcbcad75847fd1db159f4b", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "IllinoisGRMHD/doc/Tutorial-IllinoisGRMHD__outer_boundaries.ipynb", "max_forks_repo_name": "ksible/nrpytutorial", "max_forks_repo_head_hexsha": "4ca6e9da22def2a9c9bcbcad75847fd1db159f4b", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 49.6732542819, "max_line_length": 571, "alphanum_fraction": 0.5717468569, "converted": true, "num_tokens": 9674, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754489059774, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.44236386884272844}} {"text": "$$ \\LaTeX \\text{ command declarations here.}\n\\newcommand{\\R}{\\mathbb{R}}\n\\renewcommand{\\vec}[1]{\\mathbf{#1}}\n\\newcommand{\\X}{\\mathcal{X}}\n\\newcommand{\\D}{\\mathcal{D}}\n\\newcommand{\\G}{\\mathcal{G}}\n\\newcommand{\\Parents}{\\mathrm{Parents}}\n\\newcommand{\\NonDesc}{\\mathrm{NonDesc}}\n\\newcommand{\\I}{\\mathcal{I}}\n$$\n\n\n```python\nfrom __future__ import division\n\n# scientific\n%matplotlib inline\nfrom matplotlib import pyplot as plt;\nimport numpy as np;\n\n# ipython\nimport IPython;\n\n# python\nimport os;\n\n#####################################################\n\n# image processing\nimport PIL;\n\n# trim and scale images\ndef trim(im, percent=100):\n print(\"trim:\", percent);\n bg = PIL.Image.new(im.mode, im.size, im.getpixel((0,0)))\n diff = PIL.ImageChops.difference(im, bg)\n diff = PIL.ImageChops.add(diff, diff, 2.0, -100)\n bbox = diff.getbbox()\n if bbox:\n x = im.crop(bbox)\n return x.resize(((x.size[0]*percent)//100,\n (x.size[1]*percent)//100),\n PIL.Image.ANTIALIAS);\n\n\n#####################################################\n\n# daft (rendering PGMs)\nimport daft;\n\n# set to FALSE to load PGMs from static images\nRENDER_PGMS = False;\n\n# decorator for pgm rendering\ndef pgm_render(pgm_func):\n def render_func(path, percent=100, render=None, *args, **kwargs):\n print(\"render_func:\", percent);\n # render\n render = render if (render is not None) else RENDER_PGMS;\n \n if render:\n print(\"rendering\");\n # render\n pgm = pgm_func(*args, **kwargs);\n pgm.render();\n pgm.figure.savefig(path, dpi=300);\n \n # trim\n img = trim(PIL.Image.open(path), percent);\n img.save(path, 'PNG');\n else:\n print(\"not rendering\");\n \n # error\n if not os.path.isfile(path):\n raise(\"Error: Graphical model image %s not found.\" \n + \"You may need to set RENDER_PGMS=True.\");\n \n # display\n return IPython.display.Image(filename=path);#trim(PIL.Image.open(path), percent);\n \n return render_func;\n\n######################################################\n```\n\n# EECS 445: Machine Learning\n## Lecture 15: Exponential Families & Bayesian Networks\n* Instructor: **Jacob Abernethy**\n* Date: November 7, 2016\n\n*Lecture Exposition Credit:* Benjamin Bray & Valliappa Chockalingam\n\n## References\n\n- **[MLAPP]** Murphy, Kevin. [*Machine Learning: A Probabilistic Perspective*](https://mitpress.mit.edu/books/machine-learning-0). 2012.\n- **[Koller & Friedman 2009]** Koller, Daphne and Nir Friedman. [*Probabilistic Graphical Models*](https://mitpress.mit.edu/books/probabilistic-graphical-models). 2009.\n- **[Hero 2008]** Hero, Alfred O.. [*Statistical Methods for Signal Processing*](http://web.eecs.umich.edu/~hero/Preprints/main_564_08_new.pdf). 2008.\n- **[Blei 2011]** Blei, David. [*Notes on Exponential Families*](https://www.cs.princeton.edu/courses/archive/fall11/cos597C/lectures/exponential-families.pdf). 2011.\n- **[Jordan 2010]** Jordan, Michael I.. [*The Exponential Family: Basics*](http://www.cs.berkeley.edu/~jordan/courses/260-spring10/other-readings/chapter8.pdf). 2008.\n\n## Outline\n\n- Exponential Families\n - Sufficient Statistics & Pitman-Koopman-Darmois Theorem\n - Mean and natural parameters\n - Maximum Likelihood estimation\n- Probabilistic Graphical Models\n - Directed Models (Bayesian Networks)\n - Conditional Independence & Factorization\n - Examples\n\n# Exponential Families\n\n> Uses material from **[MLAPP]** \u00a79.2 and **[Hero 2008]** \u00a73.5, \u00a74.4.2\n\n### Exponential Family: Introduction\n\nWe have seen many distributions.\n* Bernoulli\n* Gaussian\n* Exponential\n* Gamma \n \nMany of these belong to a more general class called the **exponential family**.\n\n### Exponential Family: Introduction\n\nWhy do we care?\n* only family of distributions with finite-dimensional **sufficient statistics**\n* only family of distributions for which **conjugate priors** exist\n* makes the least set of assumptions subject to some user-chosen constraints (**Maximum Entropy**)\n* core of generalized linear models and **variational inference**\n\n### Sufficient Statistics\n\n**Recall:** A **statistic** $T(\\D)$ is a function of the observed data $\\D$.\n- Mean, $T(x_1, \\dots, x_n) = \\frac{1}{n}\\sum_{k=1}^n x_k$\n- Variance, maximum, mode, etc.\n\n### Sufficient Statistics: Definition\n\nSuppose we have a model $P$ with parameters $\\theta$. Then,\n\n> A statistic $T(\\D)$ is **sufficient** for $\\theta$ if no other statistic calculated from the same sample provides any additional information about the parameter.\n\nThat is, if $T(\\D_1) = T(\\D_2)$, our estimate of $\\theta$ given $\\D_1$ or $\\D_2$ will be the same.\n- Mathematically, $P(\\theta | T(\\D), \\D) = P(\\theta | T(\\D))$ independently of $\\D$\n\n### Sufficient Statistics: Example\n\nSuppose $X \\sim \\mathcal{N}(\\mu, \\sigma^2)$ and we observe $\\mathcal{D} = (x_1, \\dots, x_n)$. Let\n- $\\hat\\mu$ be the sample mean\n- $\\hat{\\sigma}^2$ be the sample variance\n\nThen $T(\\mathcal{D}) = (\\hat\\mu, \\hat{\\sigma}^2)$ is sufficient for $\\theta=(\\mu, \\sigma^2)$.\n- Two samples $\\D_1$ and $\\D_2$ with the same mean and variance give the same estimate of $\\theta$\n\n(we are sweeping some details under the rug)\n\n### Exponential Family: Definition\n\n**INTUITION**: $p()$ has **exp family form** when density $p(x | \\theta)$ can be written as $$\\exp(\\text{Linear combination of }\\theta\\text{ and features of }x)$$\n\n\n\n**DEF:** $p(x | \\theta)$ has **exponential family form** if:\n$$\n\\begin{align}\np(x | \\theta)\n&= \\frac{1}{Z(\\theta)} h(x) \\exp\\left[ \\eta(\\theta)^T \\phi(x) \\right] \\\\\n&= h(x) \\exp\\left[ \\eta(\\theta)^T \\phi(x) - A(\\theta) \\right]\n\\end{align}\n$$\n\n\n- $Z(\\theta)$ is the **partition function** for normalization\n- $A(\\theta) = \\log Z(\\theta)$ is the **log partition function**\n- $\\phi(x) \\in \\R^d$ is a vector of **sufficient statistics**\n- $\\eta(\\theta)$ maps $\\theta$ to a set of **natural parameters**\n- $h(x)$ is a scaling constant, usually $h(x)=1$\n\n### Example: Bernoulli\n\nThe Bernoulli distribution can be written as\n$$\n\\begin{align}\n\\mathrm{Ber}(x | \\mu)\n&= \\mu^x (1-\\mu)^{1-x} \\\\\n&= \\exp\\left[ x \\log \\mu + (1-x) \\log (1-\\mu) \\right] \\\\\n&= \\exp\\left[ \\eta(\\mu)^T \\phi(x) \\right]\n\\end{align}\n$$\n\nwhere $\\eta(\\mu) = (\\log\\mu, \\log(1-\\mu))$ and $\\phi(x) = (x, 1-x)$\n- There is a linear dependence between features $\\phi(x)$\n- This representation is **overcomplete**\n- $\\eta$ is not uniquely determined\n\n### Example: Bernoulli\n\nInstead, we can find a **minimal** parameterization:\n$$\n\\begin{align}\n\\mathrm{Ber}(x | \\mu) \n&= (1-\\mu) \\exp\\left[ x \\log\\frac{\\mu}{1-\\mu} \\right]\n\\end{align}\n$$\n\nThis gives **natural parameters** $\\eta = \\log \\frac{\\mu}{1-\\mu}$.\n- Now, $\\eta$ is unique\n\n### Other Examples\n\nExponential Family Distributions:\n- Multivariate normal\n- Exponential\n- Dirichlet\n\nNon-examples:\n- Student t-distribution can't be written in exponential form\n- Uniform distribution support depends on the parameters $\\theta$\n\n### Log-Partition Function\n\nDerivatives of the **log-partition function** $A(\\theta)$ yield **cumulants** of the sufficient statistics *(Exercise!)*\n- $\\nabla_\\theta A(\\theta) = E[\\phi(x)]$\n- $\\nabla^2_\\theta A(\\theta) = \\text{Cov}[ \\phi(x) ]$\n\nThis guarantees that $A(\\theta)$ is convex!\n- Its Hessian is the covariance matrix of $X$, which is positive-definite.\n- Later, this will guarantee a unique global maximum of the likelihood! \n\n#### Proof of Convexity: First Derivative\n\n$$\n\\begin{align}\n\\frac{dA}{d\\theta}\n&= \\frac{d}{d\\theta} \\left[ \\log \\int exp(\\theta\\phi(x))h(x)dx \\right] \\\\\n&= \\frac{\\frac{d}{d\\theta} \\int exp(\\theta\\phi(x))h(x)dx)}{\\int exp(\\theta\\phi(x))h(x)dx)} \\\\\n&= \\frac{\\int \\phi(x)exp(\\theta\\phi(x))h(x)dx}{exp(A(\\theta))} \\\\\n&= \\int \\phi(x) \\exp[\\theta\\phi(x)-A(\\theta)] h(x) dx \\\\\n&= \\int \\phi(x) p(x) dx \\\\\n&= E[\\phi(x)]\n\\end{align}\n$$\n\n#### Proof of Convexity: Second Derivative\n\n$$\n\\begin{align}\n\\frac{d^2A}{d\\theta^2}\n& = \\int \\phi(x)\\exp[\\theta \\phi(x) - A(\\theta)] h(x) (\\phi(x) - A'(\\theta)) dx \\\\\n& = \\int \\phi(x) p(x) (\\phi(x) - A'(\\theta))dx \\\\\n& = \\int \\phi^2(x) p(X) dx - A'(\\theta) \\int \\phi(x)p(x)dx \\\\\n& = E[\\phi^2(x)] - E[\\phi(x)]^2 \\hspace{2em} (\\because A'(\\theta) = E[\\phi(x)]) \\\\ \n& = Var[\\phi(x)]\n\\end{align}\n$$\n\n#### Proof of Convexity: Second Derivative\n\nFor multi-variate case, we have \n\n$$ \\frac{\\partial^2A}{\\partial\\theta_i \\partial\\theta_j} = E[\\phi_i(x)\\phi_j(x)] - E[\\phi_i(x)] E[\\phi_j(x)]$$\n\nand hence,\n$$ \\nabla^2A(\\theta) = Cov[\\phi(x)] $$\n\nSince covariance is positive definite, we have $A(\\theta)$ convex as required.\n\n### Exponential Family: Likelihood for a *Set* of Data\n\nFor data $\\D = (x_1, \\dots, x_N)$, the likelihood is\n$$\np(\\D|\\theta)\n= \\left[ \\prod_{k=1}^N h(x_k) \\right] Z(\\theta)^{-N} \\exp\\left[ \\eta(\\theta)^T \\left(\\sum_{k=1}^N \\phi(x_k) \\right) \\right]\n$$\n\nThe sufficient statistics are now $\\phi(\\D) = \\sum_{k=1}^N \\phi(x_k)$.\n- **Bernoulli:** $\\phi = \\# Heads$\n- **Normal:** $\\phi = [ \\sum_k x_k, \\sum_k x_k^2 ]$\n\n### Exponential Family: MLE\n\nFor natural parameters $\\theta$ and data $\\D = (x_1, \\dots, x_N)$, \n$$\n\\log p(\\D|\\theta) = \\theta^T \\phi(\\D) - N A(\\theta)\n$$\n\nSince $-A(\\theta)$ is concave and $\\theta^T\\phi(\\D)$ linear,\n- the log-likelihood is concave\n- under many conditions, there is a unique global maximum!\n> Strict convexity of the log-partition function requires that we are working with a \"regular exponential family\". More on this can be found in [These Notes](http://pages.cs.wisc.edu/~jerryzhu/cs731/expfamily.pdf). \n\n### Exponential Family: MLE\n\nTo find the maximum, recall $\\nabla_\\theta A(\\theta) = E_\\theta[\\phi(x)]$, so\n\\begin{align*}\n\\nabla_\\theta \\log p(\\D | \\theta) & =\n\\nabla_\\theta(\\theta^T \\phi(\\D) - N A(\\theta)) \\\\\n& = \\phi(\\D) - N E_\\theta[\\phi(X)] = 0\n\\end{align*}\nWhich gives\n$$E_\\theta[\\phi(X)] = \\frac{\\phi(\\D)}{N} = \\frac{1}{N} \\sum_{k=1}^N \\phi(x_k)$$\n\nAt the MLE $\\hat\\theta_{MLE}$, the empirical average of sufficient statistics equals their expected value.\n- this is called **moment matching**\n\n### Exponential Family: MLE for the Bernoulli\n\nAs an example, consider the Bernoulli distribution\n- Sufficient statistic $N$, $\\phi(\\D) = \\# Heads$\n\n$$\n\\hat\\mu_{MLE} = \\frac{\\# Heads}{N}\n$$\n\n### Bayes for Exponential Family\n\nExact Bayesian analysis is considerably simplified if the prior is **conjugate** to the likelihood.\n- Simply, this means that prior $p(\\theta)$ has the same form as the posterior $p(\\theta|\\mathcal{D})$.\n\nThis requires likelihood to have finite sufficient statistics\n* Exponential family to the rescue!\n\n**Note**: We will release some notes on cojugate priors + exponential families. It's hard to learn from slides and needs a bit more description. \n\n### Likelihood for exponential family\n\nLikelihood: \n$$ p(\\mathcal{D}|\\theta) \\propto g(\\theta)^N \\exp[\\eta(\\theta)^T s_N]\\\\\ns_N = \\sum_{i=1}^{N}\\phi(x_i)$$\n\nIn terms of canonical parameters:\n$$ p(\\mathcal{D}|\\eta) \\propto \\exp[N\\eta^T \\bar{s} -N A(\\eta)] \\\\\n\\bar s = \\frac{1}{N}s_N $$\n\n### Conjugate prior for exponential family\n\n* The prior and posterior for an exponential family involve two parameters, $\\tau$ and $\\nu$, initially set to $\\tau_0, \\nu_0$\n\n$$ p(\\theta| \\nu_0, \\tau_0) \\propto g(\\theta)^{\\nu_0} \\exp[\\eta(\\theta)^T \\tau_0] $$\n\n* Denote $ \\tau_0 = \\nu_0 \\bar{\\tau}_0$ to separate out the size of the **prior pseudo-data**, $\\nu_0$ , from the mean of the sufficient statistics on this pseudo-data, $\\tau_0$ . Hence,\n\n$$ p(\\theta| \\nu_0, \\bar \\tau_0) \\propto \\exp[\\nu_0\\eta^T \\bar \\tau_0 - \\nu_0 A(\\eta)] $$\n\n* Think of $\\tau_0$ as a \"guess\" of the future sufficient statistics, and $\\nu_0$ as the strength of this guess\n\n### Prior: Example\n\n$$\n\\begin{align}\np(\\theta| \\nu_0, \\tau_0) \n&\\propto (1-\\theta)^{\\nu_0} \\exp[\\tau_0\\log(\\frac{\\theta}{1-\\theta})] \\\\\n&= \\theta^{\\tau_0}(1-\\theta)^{\\nu_0 - \\tau_0}\n\\end{align}\n$$\n\nDefine $\\alpha = \\tau_0 +1 $ and $\\beta = \\nu_0 - \\tau_0 +1$ to see that this is a **beta distribution**.\n\n### Posterior\n\nPosterior: \n$$ p(\\theta|\\mathcal{D}) = p(\\theta|\\nu_N, \\tau_N) = p(\\theta| \\nu_0 +N, \\tau_0 +s_N) $$\n\nNote that we obtain **hyper-parameters** by adding. Hence,\n\n$$ \\begin{align}\np(\\eta|\\mathcal{D})\n&\\propto \\exp[\\eta^T (\\nu_0 \\bar\\tau_0 + N \\bar s) - (\\nu_0 + N) A(\\eta) ] \\\\\n&= p(\\eta|\\nu_0 + N, \\frac{\\nu_0 \\bar\\tau_0 + N \\bar s}{\\nu_0 + N})\n\\end{align}$$\nwhere $\\bar s = \\frac 1 N \\sum_{i=1}^{N}\\phi(x_i)$.\n\n* *posterior hyper-parameters are a convex combination of the prior mean hyper-parameters and the average of the sufficient statistics.*\n\n## Break time!\n\n\n\n# Probabilistic Graphical Models\n\n> Uses material from **[MLAPP]** \u00a710.1, 10.2 and **[Koller & Friedman 2009]**.\n\n> \"I basically know of two principles for treating complicated systems in simple ways: the first is the principle of modularity and the second is the principle of abstraction. I am an apologist for computational probability in machine learning because I believe that probability theory implements these two principles in deep and intriguing ways — nameley through factorization and through averaging. Exploiting these two mechanisms as fully as possible seems to me to be the way forward in machine learning\"  – Michael Jordan (qtd. in MLAPP)\n\n### Graphical Models: Motivation\n\nSuppose we observe multiple correlated variables $x=(x_1, \\dots, x_n)$.\n- Words in a document\n- Pixels in an image\n\nHow can we compactly represent the **joint distribution** $p(x|\\theta)$?\n- How can we tractably *infer* one set of variables given another?\n- How can we efficiently *learn* the parameters?\n\n### Joint Probability Tables\n\nOne (bad) choice is to write down a **Joint Probability Table**.\n- For $n$ binary variables, we must specify $2^n - 1$ probabilities!\n- Expensive to store and manipulate\n- Impossible to learn so many parameters\n- Very hard to interpret!\n\n> Can we be more concise?\n\n### Motivating Example: Coin Flips\n\nWhat is the joint distribution of three independent coin flips?\n- Explicitly specifying the JPT requires $2^3-1=7$ parameters.\n\nAssuming independence, $P(X_1, X_2, X_3) = P(X_1)P(X_2)P(X_3)$\n- Each marginal $P(X_k)$ only requires one parameter, the bias\n- This gives a total of $3$ parameters, compared to $8$.\n\n> Exploiting the **independence structure** of a joint distribution leads to more concise representations.\n\n### Motivating Example: Naive Bayes\n\nIn Naive Bayes, we assumed the features $X_1, \\dots, X_N$ were independent given the class label $C$:\n$$\nP(x_1, \\dots, x_N, c) = P(c) \\prod_{k=1}^N P(x_k | c)\n$$\n\nThis greatly simplified the learning procedure:\n- Allowed us to look at each feature individually\n- Only need to learn $O(CN)$ probabilities, for $C$ classes and $N$ features\n\n### Conditional Independence\n\nThe key to efficiently representing large joint distributions is to make **conditional independence** assumptions of the form\n$$\nX \\perp Y \\mid Z\n\\iff\np(X,Y|Z) = p(X|Z)p(Y|Z)\n$$\n\n> Once $z$ is known, information about $x$ does not tell us any information about $y$ and vice versa.\n\nAn effective way to represent these assumptions is with a **graph**.\n\n### Bayesian Networks: Definition\n\nA **Bayesian Network** $\\mathcal{G}$ is a directed acyclic graph whose nodes represent random variables $X_1, \\dots, X_n$.\n- Let $\\Parents_\\G(X_k)$ denote the parents of $X_k$ in $\\G$\n- Let $\\NonDesc_\\G(X_k)$ denote the variables in $\\G$ who are not descendants of $X_k$.\n\n> Examples will come shortly...\n\n### Bayesian Networks: Local Independencies\n\nEvery Bayesian Network $\\G$ encodes a set $\\I_\\ell(\\G)$ of **local independence assumptions**:\n\n> For each variable $X_k$, we have $(X_k \\perp \\NonDesc_\\G(X_k) \\mid \\Parents_\\G(X_k))$\n\nEvery node $X_k$ is conditionally independent of its nondescendants given its parents.\n\n\n### Example: Naive Bayes\n\nThe graphical model for Naive Bayes is shown below:\n- $\\Parents_\\G(X_k) = \\{ C \\}$, $\\NonDesc_\\G(X_k) = \\{ X_j \\}_{j\\neq k} \\cup \\{ C \\}$\n- Therefore $X_j \\perp X_k \\mid C$ for any $j \\neq k$\n\n\n```python\n@pgm_render\ndef pgm_naive_bayes():\n pgm = daft.PGM([4,3], origin=[-2,0], node_unit=0.8, grid_unit=2.0);\n # nodes\n pgm.add_node(daft.Node(\"c\", r\"$C$\", -0.25, 2));\n pgm.add_node(daft.Node(\"x1\", r\"$X_1$\", -1, 1));\n pgm.add_node(daft.Node(\"x2\", r\"$X_2$\", -0.5, 1));\n pgm.add_node(daft.Node(\"dots\", r\"$\\cdots$\", 0, 1, plot_params={ 'ec' : 'none' }));\n pgm.add_node(daft.Node(\"xN\", r\"$X_N$\", 0.5, 1));\n \n # edges\n pgm.add_edge(\"c\", \"x1\", head_length=0.08);\n pgm.add_edge(\"c\", \"x2\", head_length=0.08);\n pgm.add_edge(\"c\", \"xN\", head_length=0.08);\n \n return pgm;\n```\n\n\n```python\n%%capture\npgm_naive_bayes(\"images/naive-bayes.png\");\n```\n\n### Subtle Point: Graphs & Distributions\n\nA Bayesian network $\\G$ over variables $X_1, \\dots, X_N$ encodes a set of **conditional independencies**.\n- Shows independence structure, nothing more.\n- Does **not** tell us how to assign probabilities to a configuration $(x_1, \\dots x_N)$ of the variables.\n\nThere are **many** distributions $P$ satisfying the independencies in $\\G$.\n- Many joint distributions share a common structure, which we exploit in algorithms.\n- The distribution $P$ may satisfy other independencies **not** encoded in $\\G$.\n\n### Subtle Point: Graphs & Distributions\n\nIf $P$ satisfies the independence assertions made by $\\G$, we say that\n- $\\G$ is an **I-Map** for $P$\n- or that $P$ **satisfies** $\\G$.\n\nAny distribution satisfying $\\G$ shares common structure.\n- We will exploit this structure in our algorithms\n- This is what makes graphical models so **powerful**!\n\n### Review: Chain Rule for Probability\n\nWe can factorize any joint distribution via the **Chain Rule for Probability**:\n$$\n\\begin{align}\nP(X_1, \\dots, X_N)\n&= P(X_1) P(X_2, \\dots, X_N | X_1) \\\\\n&= P(X_1) P(X_2 | X_1) P(X_3, \\dots, X_N | X_1, X_2) \\\\\n&= \\prod_{k=1}^N P(X_k | X_1, \\dots, X_{k-1})\n\\end{align}\n$$\n\n> Here, the ordering of variables is arbitrary. This works for any permutation.\n\n### Bayesian Networks: Topological Ordering\n\nEvery network $\\G$ induces a **topological (partial) ordering** on its nodes:\n> Parents assigned a lower index than their children\n\n\n```python\n@pgm_render\ndef pgm_topological_order():\n pgm = daft.PGM([4, 4], origin=[-4, 0])\n\n # Nodes\n pgm.add_node(daft.Node(\"x1\", r\"$1$\", -3.5, 2))\n pgm.add_node(daft.Node(\"x2\", r\"$2$\", -2.5, 1.3))\n pgm.add_node(daft.Node(\"x3\", r\"$3$\", -2.5, 2.7))\n pgm.add_node(daft.Node(\"x4\", r\"$4$\", -1.5, 1.6))\n pgm.add_node(daft.Node(\"x5\", r\"$5$\", -1.5, 2.3))\n pgm.add_node(daft.Node(\"x6\", r\"$6$\", -0.5, 1.3))\n pgm.add_node(daft.Node(\"x7\", r\"$7$\", -0.5, 2.7))\n\n # Add in the edges.\n pgm.add_edge(\"x1\", \"x4\", head_length=0.08)\n pgm.add_edge(\"x1\", \"x5\", head_length=0.08)\n pgm.add_edge(\"x2\", \"x4\", head_length=0.08)\n pgm.add_edge(\"x3\", \"x4\", head_length=0.08)\n pgm.add_edge(\"x3\", \"x5\", head_length=0.08)\n pgm.add_edge(\"x4\", \"x6\", head_length=0.08)\n pgm.add_edge(\"x4\", \"x7\", head_length=0.08)\n pgm.add_edge(\"x5\", \"x7\", head_length=0.08)\n \n return pgm;\n```\n\n\n```python\n%%capture\npgm_topological_order(\"images/topological-order.png\")\n```\n\n### Factorization Theorem: Statement\n\n**Theorem:** *(Koller & Friedman 3.1)* If $\\G$ is an I-map for $P$, then $P$ **factorizes** as follows:\n$$\nP(X_1, \\dots, X_N) = \\prod_{k=1}^N P(X_k \\mid \\Parents_\\G(X_k))\n$$\n\n> Let's prove it together!\n\n### Factorization Theorem: Proof\n\nFirst, apply the chain rule to any topological ordering:\n$$\nP(X_1, \\dots, X_N) = \\prod_{k=1}^N P(X_k \\mid X_1, \\dots, X_{k-1})\n$$\n\nConsider one of the factors $P(X_k \\mid X_1, \\dots, X_{k-1})$. \n\n### Factorization Theorem: Proof\n\nSince our variables $X_1,\\dots,X_N$ are in topological order,\n- $\\Parents_\\G(X_k) \\subseteq \\{ X_1, \\dots, X_{k-1} \\}$\n- None of $X_k$'s descendants can possibly lie in $\\{ X_1, \\dots, X_{k-1} \\}$\n\nTherefore, $\\{ X_1, \\dots, X_{k-1} \\} = \\Parents_\\G(X_k) \\cup \\mathcal{Z}$\n- for some $\\mathcal{Z} \\subseteq \\NonDesc_\\G(X_k)$.\n\n### Factorization Theorem: Proof\n\nRecall the following property of conditional independence:\n$$\n( X \\perp Y, W \\mid Z ) \\implies (X \\perp Y \\mid Z)\n$$\n\nSince $\\G$ is an I-map for $P$ and $\\mathcal{Z} \\subseteq \\NonDesc_\\G(X_k)$, we have\n$$\\begin{align}\n& (X_k \\perp \\NonDesc_\\G(X_k) \\mid \\Parents_\\G(X_k)) \\\\\n\\implies & (X_k \\perp \\mathcal{Z} \\mid \\Parents_\\G(X_k))\n\\end{align}\n$$\n\n### Factorization Theorem: Proof\n\nWe have just shown $(X_k \\perp \\mathcal{Z} \\mid \\Parents_\\G(X_k))$, therefore\n$$\nP(X_k \\mid X_1, \\dots, X_{k-1}) = P(X_k \\mid \\Parents_\\G(X_k))\n$$\n\n- Recall $\\{ X_1, \\dots, X_{k-1} \\} = \\Parents_\\G(X_k) \\cup \\mathcal{Z}$.\n\n> **Remember:** $X_k$ is conditionally independent of its nondescendants given its parents!\n\n### Factorization Theorem: End of Proof\n\nApplying this to every factor, we see that\n$$\n\\begin{align}\nP(X_1, \\dots, X_N)\n&= \\prod_{k=1}^N P(X_k \\mid X_1, \\dots, X_{k-1}) \\\\\n&= \\prod_{k=1}^N P(X_k \\mid \\Parents_\\G(X_k))\n\\end{align}\n$$\n\n### Factorization Theorem: Consequences\n\nWe just proved that for any $P$ satisfying $\\G$,\n$$\nP(X_1, \\dots, X_N) = \\prod_{k=1}^N P(X_k \\mid \\Parents_\\G(X_k))\n$$\n\nIt suffices to store **conditional probability tables** $P(X_k | \\Parents_\\G(X_k))$!\n- Requires $O(N2^k)$ features if each node has $\\leq k$ parents\n- Substantially more compact than **JPTs** for $N$ large, $\\G$ sparse\n- We can also specify that a CPD is Gaussian, Dirichlet, etc.\n\n### Example: Fully Connected Graph\n\nA **fully connected graph** makes no independence assumptions.\n$$\nP(A,B,C) = P(A) P(B|A) P(C|A,B)\n$$\n\n\n```python\n@pgm_render\ndef pgm_fully_connected_a():\n pgm = daft.PGM([4, 4], origin=[0, 0])\n\n # nodes\n pgm.add_node(daft.Node(\"a\", r\"$A$\", 2, 3.5))\n pgm.add_node(daft.Node(\"b\", r\"$B$\", 1.3, 2.5))\n pgm.add_node(daft.Node(\"c\", r\"$C$\", 2.7, 2.5))\n\n # add in the edges\n pgm.add_edge(\"a\", \"b\", head_length=0.08)\n pgm.add_edge(\"a\", \"c\", head_length=0.08)\n pgm.add_edge(\"b\", \"c\", head_length=0.08)\n \n return pgm;\n```\n\n\n```python\n%%capture\npgm_fully_connected_a(\"images/fully-connected-a.png\")\n```\n\n### Example: Fully Connected Graph\n\nThere are many possible fully connected graphs:\n$$\n\\begin{align}\nP(A,B,C) \n&= P(A) P(B|A) P(C|A,B) \\\\\n&= P(B) P(C|B) P(A|B,C)\n\\end{align}\n$$\n\n\n```python\n@pgm_render\ndef pgm_fully_connected_b():\n pgm = daft.PGM([8, 4], origin=[0, 0])\n\n # nodes\n pgm.add_node(daft.Node(\"a1\", r\"$A$\", 2, 3.5))\n pgm.add_node(daft.Node(\"b1\", r\"$B$\", 1.5, 2.8))\n pgm.add_node(daft.Node(\"c1\", r\"$C$\", 2.5, 2.8))\n\n # add in the edges\n pgm.add_edge(\"a1\", \"b1\", head_length=0.08)\n pgm.add_edge(\"a1\", \"c1\", head_length=0.08)\n pgm.add_edge(\"b1\", \"c1\", head_length=0.08)\n \n # nodes\n pgm.add_node(daft.Node(\"a2\", r\"$A$\", 4, 3.5))\n pgm.add_node(daft.Node(\"b2\", r\"$B$\", 3.5, 2.8))\n pgm.add_node(daft.Node(\"c2\", r\"$C$\", 4.5, 2.8))\n\n # add in the edges\n pgm.add_edge(\"b2\", \"c2\", head_length=0.08)\n pgm.add_edge(\"b2\", \"a2\", head_length=0.08)\n pgm.add_edge(\"c2\", \"a2\", head_length=0.08)\n \n return pgm;\n```\n\n\n```python\n%%capture\npgm_fully_connected_b(\"images/fully-connected-b.png\")\n```\n\n### Bayesian Networks & Causality\n\nThe fully-connected example brings up a crucial point:\n\n> Directed edges do **not** necessarily represent causality.\n\nBayesian networks encode **independence assumptions** only.\n- This representation is not unique.\n\n### Example: Markov Chain\n\nState at time $t$ depends only on state at time $t-1$.\n$$\nP(X_0, X_1, \\dots, X_N) = P(X_0) \\prod_{t=1}^N P(X_t \\mid X_{t-1})\n$$\n\n\n```python\n@pgm_render\ndef pgm_markov_chain():\n pgm = daft.PGM([6, 6], origin=[0, 0])\n\n # Nodes\n pgm.add_node(daft.Node(\"x1\", r\"$\\mathbf{x}_1$\", 2, 2.5))\n pgm.add_node(daft.Node(\"x2\", r\"$\\mathbf{x}_2$\", 3, 2.5))\n pgm.add_node(daft.Node(\"ellipsis\", r\" . . . \", 3.7, 2.5, offset=(0, 0), plot_params={\"ec\" : \"none\"}))\n pgm.add_node(daft.Node(\"ellipsis_end\", r\"\", 3.7, 2.5, offset=(0, 0), plot_params={\"ec\" : \"none\"}))\n pgm.add_node(daft.Node(\"xN\", r\"$\\mathbf{x}_N$\", 4.5, 2.5))\n\n # Add in the edges.\n pgm.add_edge(\"x1\", \"x2\", head_length=0.08)\n pgm.add_edge(\"x2\", \"ellipsis\", head_length=0.08)\n pgm.add_edge(\"ellipsis_end\", \"xN\", head_length=0.08)\n \n return pgm;\n```\n\n\n```python\n%%capture\npgm_markov_chain(\"images/markov-chain.png\")\n```\n\n### Example: Hidden Markov Model\n\nNoisy observations $X_k$ generated from hidden Markov chain $Y_k$.\n$$\nP(\\vec{X}, \\vec{Y}) = P(Y_1) P(X_1 \\mid Y_1) \\prod_{k=2}^N \\left(P(Y_k \\mid Y_{k-1}) P(X_k \\mid Y_k)\\right)\n$$\n\n\n```python\n@pgm_render\ndef pgm_hmm():\n pgm = daft.PGM([7, 7], origin=[0, 0])\n\n # Nodes\n pgm.add_node(daft.Node(\"Y1\", r\"$Y_1$\", 1, 3.5))\n pgm.add_node(daft.Node(\"Y2\", r\"$Y_2$\", 2, 3.5))\n pgm.add_node(daft.Node(\"Y3\", r\"$\\dots$\", 3, 3.5, plot_params={'ec':'none'}))\n pgm.add_node(daft.Node(\"Y4\", r\"$Y_N$\", 4, 3.5))\n\n pgm.add_node(daft.Node(\"x1\", r\"$X_1$\", 1, 2.5, observed=True))\n pgm.add_node(daft.Node(\"x2\", r\"$X_2$\", 2, 2.5, observed=True))\n pgm.add_node(daft.Node(\"x3\", r\"$\\dots$\", 3, 2.5, plot_params={'ec':'none'}))\n pgm.add_node(daft.Node(\"x4\", r\"$X_N$\", 4, 2.5, observed=True))\n\n\n # Add in the edges.\n pgm.add_edge(\"Y1\", \"Y2\", head_length=0.08)\n pgm.add_edge(\"Y2\", \"Y3\", head_length=0.08)\n pgm.add_edge(\"Y3\", \"Y4\", head_length=0.08)\n\n pgm.add_edge(\"Y1\", \"x1\", head_length=0.08)\n pgm.add_edge(\"Y2\", \"x2\", head_length=0.08)\n pgm.add_edge(\"Y4\", \"x4\", head_length=0.08)\n \n return pgm;\n```\n\n\n```python\n%%capture\npgm_hmm(\"images/hmm.png\")\n```\n\n### Example: Plate Notation\n\nWe can represent (conditionally) iid variables using **plate notation**.\n\n\n```python\n@pgm_render\ndef pgm_plate_example():\n pgm = daft.PGM([4,3], origin=[-2,0], node_unit=0.8, grid_unit=2.0);\n # nodes\n pgm.add_node(daft.Node(\"lambda\", r\"$\\lambda$\", -0.25, 2));\n \n pgm.add_node(daft.Node(\"t1\", r\"$\\theta_1$\", -1, 1.3));\n pgm.add_node(daft.Node(\"t2\", r\"$\\theta_2$\", -0.5, 1.3));\n pgm.add_node(daft.Node(\"dots1\", r\"$\\cdots$\", 0, 1.3, plot_params={ 'ec' : 'none' }));\n pgm.add_node(daft.Node(\"tN\", r\"$\\theta_N$\", 0.5, 1.3));\n \n pgm.add_node(daft.Node(\"x1\", r\"$X_1$\", -1, 0.6));\n pgm.add_node(daft.Node(\"x2\", r\"$X_2$\", -0.5, 0.6));\n pgm.add_node(daft.Node(\"dots2\", r\"$\\cdots$\", 0, 0.6, plot_params={ 'ec' : 'none' }));\n pgm.add_node(daft.Node(\"xN\", r\"$X_N$\", 0.5, 0.6));\n \n \n pgm.add_node(daft.Node(\"LAMBDA\", r\"$\\lambda$\", 1.5, 2));\n pgm.add_node(daft.Node(\"THETA\", r\"$\\theta_k$\", 1.5,1.3));\n pgm.add_node(daft.Node(\"XX\", r\"$X_k$\", 1.5,0.6));\n \n # edges\n pgm.add_edge(\"lambda\", \"t1\", head_length=0.08);\n pgm.add_edge(\"lambda\", \"t2\", head_length=0.08);\n pgm.add_edge(\"lambda\", \"tN\", head_length=0.08);\n pgm.add_edge(\"t1\", \"x1\", head_length=0.08);\n pgm.add_edge(\"t2\", \"x2\", head_length=0.08);\n pgm.add_edge(\"tN\", \"xN\", head_length=0.08);\n \n \n pgm.add_edge(\"LAMBDA\", \"THETA\", head_length=0.08);\n pgm.add_edge(\"THETA\", \"XX\", head_length=0.08);\n \n pgm.add_plate(daft.Plate([1.1,0.4,0.8,1.2], label=r\"$\\qquad\\quad\\; K$\",\n shift=-0.1))\n \n return pgm;\n```\n\n\n```python\n%%capture\npgm_plate_example(\"images/plate-example.png\")\n```\n", "meta": {"hexsha": "533ea18fa9370c9646359ee8ede9f31dba2b9205", "size": 353134, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lecture15_exp_families_bayesian_networks/lecture15_exp_families_bayesian_networks.ipynb", "max_stars_repo_name": "xipengwang/umich-eecs445-f16", "max_stars_repo_head_hexsha": "298407af9fd417c1b6daa6127b17cb2c34c2c772", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 97, "max_stars_repo_stars_event_min_datetime": "2016-09-11T23:15:35.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-22T08:03:24.000Z", "max_issues_repo_path": "lecture15_exp_families_bayesian_networks/lecture15_exp_families_bayesian_networks.ipynb", "max_issues_repo_name": "eecs445-f16/umich-eecs445-f16", "max_issues_repo_head_hexsha": "298407af9fd417c1b6daa6127b17cb2c34c2c772", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "lecture15_exp_families_bayesian_networks/lecture15_exp_families_bayesian_networks.ipynb", "max_forks_repo_name": "eecs445-f16/umich-eecs445-f16", "max_forks_repo_head_hexsha": "298407af9fd417c1b6daa6127b17cb2c34c2c772", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 77, "max_forks_repo_forks_event_min_datetime": "2016-09-12T20:50:46.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-03T14:41:23.000Z", "avg_line_length": 216.381127451, "max_line_length": 68736, "alphanum_fraction": 0.8939127923, "converted": true, "num_tokens": 9168, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073802837477, "lm_q2_score": 0.8031737987125612, "lm_q1q2_score": 0.4422334212216694}} {"text": "# Markdown Cell\n\n\n\n## Sub heading this is big text\n\n#### Smaller subheadings\n\nThis is just a _paragraph_ and this is **bolded**.\n\n\n```python\n# code cell\nname = \"Jonathan\"\n```\n\n# Tips and Tricks\n\nA markdown cell lets you do many _useful things_.\n\n\n```python\nimport numpy as np\n\n# don't do:\n# from numpy import *\n```\n\n\n```python\nmax(\"a\")\n```\n\n\n```python\nnp.max(\"a\")\n```\n\n## Imports\n\n\n```python\n# %matplotlib inline\n# %config InlineBackend.figure_format='retina'\n\nimport pandas as pd\nimport matplotlib.pyplot as plt\nimport seaborn as sns\nimport os\nfrom pivottablejs import pivot_ui\nimport sys\nimport warnings\n\nwarnings.filterwarnings(\"ignore\")\n```\n\n# Keyboard shortcuts\n\nPractice doing these a few times and try to force yourself to use them whenever you can. \n\n## Help (h)\nFor help in the Notebook, `ESC` + `h`\n(doesn't work in Lab)\n\n## Select multiple cells\n\nj, k, and arrow keys while holding shift selects multiple cells.\n\n\n```python\nfirst = 1\n```\n\n\n```python\nsecond = 2\n```\n\n\n```python\nthird = 3\n```\n\n## Split a cell with -\n\nWell, `ctrl + shift + -`, but remember it by horizontal.\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n## Add a new cell (a)bove\n\n## Add a new cell (b)elow\n\n\n```python\n\n```\n\n## (d)(d)elete \n## unD(z)o \n\n## (c)opy cells\n\n## (v)aste cells\n\n# Pivot Tables w/ pandas\n\nA library and example from: http://nicolas.kruchten.com/content/2015/09/jupyter_pivottablejs/\n\n\n```python\ncanadian_politics = pd.read_csv(\"../data/mps2.csv\")\n```\n\n\n```python\n# recommend using .head()\ncanadian_politics.head(10)\n```\n\n\n```python\nsns.distplot(canadian_politics[\"Age\"].dropna());\n```\n\n\n```python\nsns.set_context(\"poster\", font_scale=1.3)\n```\n\n\n```python\nfig, ax = plt.subplots(figsize=(12, 8))\nsns.distplot(canadian_politics[\"Age\"].dropna())\nfig.tight_layout()\n```\n\n# Enhanced Pandas Dataframe Display\n\n\n```python\n# Province, Party, (deselect) Average, Age, Heatmap\n```\n\n\n```python\npivot_ui(canadian_politics)\n```\n\n\n```python\nnewdf = pd.read_clipboard(sep='\\t')\nnewdf.fillna(\"\")\n```\n\n\n```python\n\n```\n\n\n```python\ncanadian_politics['Age-bin'] = pd.cut(canadian_politics['Age'], [x for x in range(10, 100, 5)])\n# pd.qcut# neat!\n```\n\n# Tab -- Your Friend\n\n\n```python\nimport numpy as np\n```\n\n\n```python\nfrom numpy.random import chisquare, choice\n```\n\n\n```python\nnp.random.chisquare()\n```\n\n\n```python\n# pure tab right \u2193 less useful\nnp.random.choice()\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n## shift-tab\n\n\n```python\n# shift-tab right \u2193 more useful\nnp.linspace(start=50, stop=100, endpoint=False)\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n## shift-tab-tab\n\n\n```python\nnp.linspace(start=50, end=120)\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n## shift-tab-tab-tab\n\n\n```python\nnp.linspace(start=50, stop=150, num=100, endpoint=False)\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n## shift-tab-tab-tab-tab\n\n\n```python\nplt.plot(np.linspace(start, stop, num=50, ))\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n## DO NOT TRY shift-tab-tab-tab-tab-tab\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n## ?\n\n\n```python\nnp.linspace?\n```\n\n\n```python\n?np.linspace\n```\n\n## ??\n\n(Lab can scroll if you click)\n\n\n```python\nnp.linspace??\n```\n\n\n```python\n!code ~/miniconda3/envs/dspy3/lib/python3.6/site-packages/numpy/core/function_base.py\n```\n\n# Random stuff\n\n\n```python\nimport textwrap\ndef example_function():\n \"\"\"Docstring for example function\"\"\"\n \n print(textwrap.dedent(\"\"\"\n This is a multi-lined string\n that I want to write inside of a function.\n Notice what happens when I print this.\n And when something is indented more.\"\"\"))\n\n\nexample_function()\n```\n\n\n```python\n# python3.6+\n```\n\n\n```python\nname\n```\n\n\n```python\nf\"{name}'s name is not Alex.\"\n```\n\n\n```python\nage = 37\nf\"{age} plus 2 = {age + 2}\"\n\n# Note: \n# f\"\"\"{example_dictionary[\"key\"]}\"\"\"\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n## Inspect _everything_ and Find and Replace\n\n\n```python\n# But first find and replace\ndef silly_function(xval):\n \"\"\"Takes a value and returns the value.\"\"\"\n xval_sq = xval ** 2.0\n 3 + 1\n xval_abs = np.sqrt(xval_sq)\n return xval_abs\n```\n\n\n```python\nsilly_function(-2,)\n```\n\n\n```python\nsilly_function?\n```\n\n\n```python\nsilly_function??\n```\n\n\n```python\n!ls ../data/\n```\n\n\n```python\ncoal_years = !ls ../data/coal_prod_20*.csv\n```\n\n\n```python\ncoal_years\n```\n\n\n```python\nfrom glob import glob \n```\n\n\n```python\nfor filename in glob(\"../data/coal_prod_20*.csv\"):\n print(filename)\n```\n\n## Line numbers (lowercase \"L\")\n\nType lowercase \"L\" to have line numbers; shift-L for line numbers notebook-wide.\n\n## Move blocks of code around\n\n\n### Indent/dedent\n\n Cmd + [\n Cmd + ]\n\n### Comment\n\n Cmd + /\n\n\n```python\nex_dictionary = {}\n\n# Indent/dedent/comment\n\nfor index in range(5):\n ex_dictionary[\"float_one\"] = 1\n ex_dictionary[\"float_two\"] = 2\n ex_dictionary[\"float_three\"] = 3\n ex_dictionary[\"float_four\"] = 4\n```\n\n\n```python\nex_dictionary\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n## Multicursor magic\n\nHold down `option`, click and drag (for big cursor).\n\ncmd + click == wherever you click. \n\nShift command P -- command palette Classic Notebook\nShift command C -- command palette in JupyterLab\n\nHide left bar (CMD + B)\n\nFull screen CMD + Shift + D (but CTRL + CMD F for browser fullscreen)\n\nMove cells around mouse\n\n\n```python\nexample[\"one_better_neat\"] = 1\nexample[\"two_better_neat\"] = 2\nexample[\"three_better_neat\"] = 3\nexample[\"four_better_neat\"] = 4\n```\n\n## Monospace\n\nYou can also get `monospaced` fonts by indenting 4 spaces:\n\n mkdir toc\n cd toc\n\n## Syntax Highlighting\n\nWrap with triple-backticks and language:\n\n```bash\nmkdir toc\ncd toc\nwget https://repo.continuum.io/miniconda/Miniconda3-latest-MacOSX-x86_64.sh\n```\n\n```SQL\nSELECT first_name,\n last_name,\n year_of_birth\nFROM presidents\nWHERE year_of_birth > 1800;\n```\n\n## Headings and LaTeX\n\nWith text and $\\LaTeX$ support.\n\n$$\\begin{align}\n B'&=-\\nabla \\times E,\\\\\n E'&=\\nabla \\times B - 4\\pi j\n\\end{align}$$\n\n# Magics\n\n - `%` $\\equiv$ inline magic\n - `%%` $\\equiv$ cell magic\n \n## Read through the documentation: https://ipython.readthedocs.io/en/stable/interactive/magics.html\n\n\n```latex\n%%latex\n\nIf you want to get crazier$\\ldots$\n\n\\begin{equation}\n\\oint_S {E_n dA = \\frac{1}{{\\varepsilon _0 }}} Q_\\textrm{inside}\n\\end{equation}\n```\n\n\n```python\n%%python2\nprint \"hi\"\n```\n\n\n```bash\n%%bash\nwget http://www.ast.cam.ac.uk/%7Erfc/vpfit12.2.tar.gz\nmkdir -p vpfit12\ncd vpfit12\ntar -xvzf ../vpfit12.2.tar.gz\n```\n\n## Scripting\n\n\n```python\nnormal_argument = 12.4\nsecond_argument = 98.4\n\narg_with_spaces = \"the secret to life\"\n```\n\n\n```bash\n%%bash -s {normal_argument} {second_argument}\necho \"This script knows the value of the argument: $1\"\necho \"It also has no trouble with the second argument: $2\"\n```\n\n\n```bash\n%%bash -s \"$arg_with_spaces\"\necho \"This bash script knows $1.\"\n```\n\n\n```python\n# %%R -i df -o df2\n# df2 <- \n```\n\n\n```python\nls vpfit10/\n```\n\n\n```python\ntailthing = \"*.ipynb\"\n```\n\n\n```python\ntailthing\n```\n\n\n```python\n!ls {tailthing}\n```\n\n\n```python\noutput = !ls \n```\n\n\n```python\noutput\n```\n\n## Need to set or change environment variables\n\n\n```python\n%env \n```\n\n## Danger zone\n\n _, _N and _iN\n\n\n```python\n!pwd\n```\n\n\n```python\na = 3\na\n```\n\n\n```python\nprint(canadian_politics.head().to_latex())\n```\n\n\n```python\n5 * 83\n```\n\n\n```python\n_\n```\n\n\n```python\n3 + 7\n```\n\n\n```python\n_\n```\n\n\n```python\n\n```\n\n\n```python\nprint(_81)\n```\n\n\n```python\n\n```\n\n\n```python\nsaved = _25\n```\n\n\n```python\nsaved\n```\n\n\n```python\n%history \n```\n\n\n```python\n%history -opf alex.txt\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "be1a7bf6e3dccb4690ca3789439405c09f5bc3fe", "size": 852652, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/01-Tips-and-tricks.ipynb", "max_stars_repo_name": "jbwhit/jupyter-tips-and-tricks", "max_stars_repo_head_hexsha": "c0b9f6cf6c422146743ebb53fba8914ade5611f5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 260, "max_stars_repo_stars_event_min_datetime": "2015-09-11T15:57:35.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-13T23:42:13.000Z", "max_issues_repo_path": "notebooks/01-Tips-and-tricks.ipynb", "max_issues_repo_name": "Lukematic/jupyter-tips-and-tricks", "max_issues_repo_head_hexsha": "c0b9f6cf6c422146743ebb53fba8914ade5611f5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, 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YES\n2. YES", "lm_q1_score": 0.5506073802837478, "lm_q2_score": 0.8031737963569014, "lm_q1q2_score": 0.44223341992462584}} {"text": "# Ikeda $B_e$ assumtion.\n\n\n```python\nfrom rolldecayestimators import equations\n```\n\n Duplicate key in file WindowsPath('C:/Users/maa/.matplotlib/stylelib/paper.mplstyle'), line 461 ('figure.figsize : 5, 3 ## figure size in inches')\n Duplicate key in file WindowsPath('C:/Users/maa/.matplotlib/stylelib/paper.mplstyle'), line 462 ('figure.dpi : 100 ## figure dots per inch')\n\n\n# Purpose\nThe quadratic or cubic model can be expressed using the linearized equivalent damping ($B_e$) according to .:\n\n\n```python\nequations.B_e_equation\n```\n\n\n\n\n$\\displaystyle B_{e} = B_{1} + \\frac{8 B_{2} \\omega_{0} \\phi_{a}}{3 \\pi}$\n\n\n\n\n```python\nequations.B_e_equation_cubic\n```\n\n\n\n\n$\\displaystyle B_{e} = B_{1} + \\frac{8 B_{2} \\omega_{0} \\phi_{a}}{3 \\pi} + 0.75 B_{3} \\omega_{0}^{2} \\phi_{a}^{2}$\n\n\n\nBut I have some doubt about the validity of this, which will be investigated in this notebook.\n\n# Methodology\nA quadratic and cubic model from Simplified Ikeda will be used to calculate $B_e$. $B_e$ will also be obtained from Roll-decay simulations with these models, will the value be the same?\n\n# WIP - improvements\n(WORK IN PROGRESS)\nUse this section only if the notebook is not final.\n\nNotable TODOs:\n* todo 1\n* todo 2\n* todo 3\n\n## Results\nDescribe and comment the most important results.\n\n# Suggested next steps\nState suggested next steps, based on results obtained in this notebook.\n\n# Setup\n\n\n```python\n# %load imports.py\n\"\"\"\nThese is the standard setup for the notebooks.\n\"\"\"\n\n%matplotlib inline\n%load_ext autoreload\n%autoreload 2\n\n#from jupyterthemes import jtplot\n#jtplot.style(theme='onedork', context='notebook', ticks=True, grid=False)\n\nimport pandas as pd\npd.options.display.max_rows = 999\npd.options.display.max_columns = 999\npd.set_option(\"display.max_columns\", None)\nimport numpy as np\nimport os\nimport matplotlib.pyplot as plt\nfrom collections import OrderedDict\n#plt.style.use('paper')\nfrom reports.paper_writing import save_fig\n\n#import data\nimport copy\nfrom mdldb.run import Run\n\nfrom sklearn.pipeline import Pipeline\nfrom rolldecayestimators.transformers import CutTransformer, LowpassFilterDerivatorTransformer, ScaleFactorTransformer, OffsetTransformer\nfrom rolldecayestimators.direct_estimator_cubic import EstimatorQuadraticB, EstimatorCubic\nfrom rolldecayestimators.ikeda_estimator import IkedaQuadraticEstimator\nimport rolldecayestimators.equations as equations\nimport rolldecayestimators.lambdas as lambdas\nfrom rolldecayestimators.substitute_dynamic_symbols import lambdify\nimport rolldecayestimators.symbols as symbols\nimport sympy as sp\n\nfrom sympy.physics.vector.printing import vpprint, vlatex\nfrom IPython.display import display, Math, Latex\n\nfrom sklearn.metrics import r2_score\nfrom src.data import database\nfrom mdldb import tables\nimport shipflowmotionshelpers.shipflowmotionshelpers as helpers\nimport src.visualization.visualize as visualize\n\n```\n\n\n```python\nfrom rolldecayestimators.simplified_ikeda_class import SimplifiedIkeda\nimport rolldecayestimators\nfrom scipy.integrate import solve_ivp\n```\n\n\n```python\nzeta_lambda = lambdify(sp.solve(equations.extinction_equation,symbols.zeta)[0])\n\ndef calculate_B_n(X_amplitudes):\n B_ns=[None,]\n for i in range(len(X_amplitudes)-1):\n \n row1 = X_amplitudes.iloc[i]\n row2 = X_amplitudes.iloc[i+1]\n t_ = row2.name - row1.name\n B_n = zeta_lambda(omega0=row1['omega0'],phi_0=row1['phi_a'], phi_a=row2['phi_a'],\n t=t_)\n B_ns.append(B_n)\n \n return B_ns\n```\n\n## Linear\n\n\n```python\nMath(vlatex(equations.roll_decay_equation_himeno_linear))\n```\n\n\n\n\n$\\displaystyle A_{44} \\ddot{\\phi} + B_{1} \\dot{\\phi} + C_{1} \\phi = 0$\n\n\n\n\n```python\neq_acceleration_linear = sp.Eq(symbols.phi_dot_dot,\n sp.solve(equations.roll_decay_equation_himeno_linear,symbols.phi_dot_dot)[0])\nMath(vlatex(eq_acceleration_linear))\n```\n\n\n\n\n$\\displaystyle \\ddot{\\phi} = - \\frac{B_{1} \\dot{\\phi} + C_{1} \\phi}{A_{44}}$\n\n\n\n\n```python\naccelaration_linear_lambda = lambdify(sp.solve(equations.roll_decay_equation_himeno_linear,symbols.phi_dot_dot)[0])\n```\n\n## Quadratic\n\n\n```python\nMath(vlatex(equations.roll_decay_equation_himeno_quadratic_b))\n```\n\n\n\n\n$\\displaystyle A_{44} \\ddot{\\phi} + C_{1} \\phi + \\left(B_{1} + B_{2} \\left|{\\dot{\\phi}}\\right|\\right) \\dot{\\phi} = 0$\n\n\n\n\n```python\neq_acceleration_quadratic = sp.Eq(symbols.phi_dot_dot,\n sp.solve(equations.roll_decay_equation_himeno_quadratic_b,symbols.phi_dot_dot)[0])\n\naccelaration_quadratic_lambda = lambdify(sp.solve(equations.roll_decay_equation_himeno_quadratic_b,symbols.phi_dot_dot)[0])\n\nMath(vlatex(eq_acceleration_quadratic))\n```\n\n\n\n\n$\\displaystyle \\ddot{\\phi} = - \\frac{B_{1} \\dot{\\phi} + B_{2} \\left|{\\dot{\\phi}}\\right| \\dot{\\phi} + C_{1} \\phi}{A_{44}}$\n\n\n\n\n```python\nclass RollDecayLinear():\n \n def __init__(self,A_44, B_1, C_1):\n self.parameters = {\n 'A_44':A_44,\n 'B_1':B_1,\n 'C_1':C_1,\n }\n \n def time_step(self,t,states):\n \n phi = states[0]\n phi1d = states[1]\n phi2d = accelaration_linear_lambda(**self.parameters, phi=phi, phi1d=phi1d)\n \n d_states_dt = np.array([phi1d, phi2d])\n return d_states_dt\n \n def simulate(self,t,phi0=np.deg2rad(10),phi1d0=0):\n \n initial_state = [phi0,phi1d0]\n \n t_span = [t[0], t[-1]]\n \n result = solve_ivp(fun=simulation.time_step, t_span=t_span, y0=initial_state, t_eval=t)\n assert result.success\n df_result = pd.DataFrame(index=result.t, data=result.y.T, columns = ['phi','phi1d'])\n return df_result\n \nclass RollDecayQuadratic(RollDecayLinear):\n \n def __init__(self,A_44, B_1, B_2, C_1):\n self.parameters = {\n 'A_44':A_44,\n 'B_1':B_1,\n 'B_2':B_2,\n 'C_1':C_1,\n }\n \n def time_step(self,t,states):\n \n phi = states[0]\n phi1d = states[1]\n phi2d = accelaration_quadratic_lambda(**self.parameters, phi=phi, phi1d=phi1d)\n \n d_states_dt = np.array([phi1d, phi2d])\n return d_states_dt\n```\n\n\n```python\nN=100000\nA_44 = 2.2\nB_1 = 0.10\nB_2 = 1.5\nC_1 = 0.5\n\nt = np.linspace(0,200,N)\nphi0=np.deg2rad(10)\nphi1d0 = 0\ninitial_state = [phi0,phi1d0]\n\nsimulations = {\n 'linear':RollDecayLinear(A_44=A_44, B_1=B_1, C_1=C_1),\n 'quadratic':RollDecayQuadratic(A_44=A_44, B_1=B_1, B_2=B_2, C_1=C_1),\n}\n```\n\n\n```python\nequations.C_equation_linear\n```\n\n\n\n\n$\\displaystyle C = GM g m$\n\n\n\n\n```python\nA_44_eq = sp.Eq(symbols.A_44, equations.A44)\nA_44_eq\n```\n\n\n\n\n$\\displaystyle A_{44} = \\frac{GM g m}{\\omega_{0}^{2}}$\n\n\n\n\n```python\neqs = [\n A_44_eq,\n equations.C_equation_linear,\n\n]\nomega0_eq = sp.Eq(symbols.omega0,sp.solve(eqs, symbols.omega0, symbols.GM)[1][0])\nomega0_eq\n```\n\n\n\n\n$\\displaystyle \\omega_{0} = \\sqrt{\\frac{C}{A_{44}}}$\n\n\n\n\n```python\nomega0 = np.sqrt(C_1/A_44)\n```\n\n\n```python\nt_span = [t[0], t[-1]]\n\nresults = {}\nX_amplitudes = {}\nfor name,simulation in simulations.items():\n \n df_result = simulation.simulate(t=t, phi0=phi0, phi1d0=phi1d0)\n \n results[name]=df_result\n X_amplitudes[name]=rolldecayestimators.measure.calculate_amplitudes_and_damping(X=df_result)\n```\n\n\n```python\nfor name in results.keys():\n fig,ax=plt.subplots()\n df_result = results[name]\n amplitudes = X_amplitudes[name]\n df_result.plot(y='phi',ax=ax)\n amplitudes.plot(y='phi_a', ax=ax)\n ax.grid(True)\n ax.set_title(name)\n```\n\n\n```python\nequations.extinction_equation\n```\n\n\n\n\n$\\displaystyle \\phi_{a} = \\phi_{0}{\\left(t \\right)} e^{- \\omega_{0} t \\zeta}$\n\n\n\n\n```python\nsp.Eq(symbols.zeta,sp.solve(equations.extinction_equation,symbols.zeta)[0])\n```\n\n\n\n\n$\\displaystyle \\zeta = \\frac{\\log{\\left(\\frac{\\phi_{0}{\\left(t \\right)}}{\\phi_{a}} \\right)}}{\\omega_{0} t}$\n\n\n\n\n```python\nequations.B_e_equation\n```\n\n\n\n\n$\\displaystyle B_{e} = B_{1} + \\frac{8 B_{2} \\omega_{0} \\phi_{a}}{3 \\pi}$\n\n\n\n\n```python\nfor name in results.keys():\n amplitudes = X_amplitudes[name]\n amplitudes['B_n2'] = calculate_B_n(amplitudes)\n \n omega0=amplitudes['omega0']\n phi_a=amplitudes['phi_a']\n amplitudes['B_e'] = B_1 + B_2*8/(3*np.pi)*omega0*phi_a\n \n amplitudes['B_1/2omega0'] = B_1/(2*omega0*A_44)\n amplitudes['B_e/2omega0'] = amplitudes['B_e']/(2*omega0*A_44)\n \n```\n\n\n\n\n```python\nfor name in results.keys():\n amplitudes = X_amplitudes[name]\n fig,ax=plt.subplots()\n amplitudes.plot(x='phi_a', y='B_n2', style='-', ax=ax)\n amplitudes.plot(x='phi_a', y='B_1/2omega0', style='--', ax=ax)\n amplitudes.plot(x='phi_a', y='B_e/2omega0', style='--',ax=ax)\n \n y_lim = ax.get_xlim()\n #ax.set_ylim(0,y_lim[1])\n ax.set_title(name)\n```\n\n\n```python\ndef align_yaxis(ax1, v1, ax2, v2):\n \"\"\"adjust ax2 ylimit so that v2 in ax2 is aligned to v1 in ax1\"\"\"\n _, y1 = ax1.transData.transform((0, v1))\n _, y2 = ax2.transData.transform((0, v2))\n inv = ax2.transData.inverted()\n _, dy = inv.transform((0, 0)) - inv.transform((0, y1-y2))\n miny, maxy = ax2.get_ylim()\n ax2.set_ylim(miny+dy, maxy+dy)\n```\n\n\n```python\nfrom scipy.integrate import cumtrapz\n\ndf_results = results['linear'].copy()\nphi = df_results['phi']\nphi1d = df_results['phi1d']\n\ndf_results['B'] = phi1d*B_1\n\nfig,ax=plt.subplots()\ndf_results.plot(y='phi', ax=ax, lw=2, alpha=1)\nax.legend(loc='upper left')\nax.set_ylabel('$\\phi$ [rad]')\n\nax_damping = ax.twinx()\ndf_results.plot(y='B', style='r-', lw=2, ax=ax_damping)\nax_damping.set_ylabel('Damping B [Nm]')\n\nalign_yaxis(ax, 0, ax_damping, 0)\n```\n\n\n```python\ndf_results['E_kin'] = 1/2*A_44*phi1d**2\nE_loss = cumtrapz(df_results['B'],x=phi)\nE_loss = np.concatenate([[0],E_loss])\ndf_results['E_loss'] = E_loss\ndf_results['E_pot'] = C_1*phi**2/2\ndf_results['E_sys'] = df_results['E_kin'] + df_results['E_pot']\ndf_results['E_tot'] = df_results['E_loss'] + df_results['E_sys']\n```\n\n\n\n\n```python\n#with plt.style.context('paper'):\nfig,ax=plt.subplots()\n#fig.set_size_inches(15,10)\n#fig.set_dpi(300)\ndf_results.plot.area(y = ['E_kin','E_pot','E_loss'], label=[r'$E_{kin}$',r'$E_{pot}$',r'$E_{loss}$'], color=['r','g','b'], ax=ax)\nax.set_xlabel('Time [s]')\nax.set_ylabel('Energy [kNm]')\n\nsave_fig(fig, name='energy_transfer')\n```\n\n\n```python\nE_loss2 = cumtrapz(phi1d,x=phi)\nE_loss2 = np.concatenate([[0],E_loss2])\nB_es = df_results['E_loss']/E_loss2\n\n```\n\n\n```python\n\n```\n\n\n```python\nfrom scipy.integrate import cumtrapz\n\ndf_results = results['quadratic'].copy()\nphi = df_results['phi']\nphi1d = df_results['phi1d']\n\ndf_results['B'] = (B_1 + B_2*np.abs(phi1d))*phi1d \n\nfig,ax=plt.subplots()\ndf_results.plot(y='phi', ax=ax, lw=2, alpha=1)\nax.legend(loc='upper left')\nax.set_ylabel('$\\phi$ [rad]')\n\nax_damping = ax.twinx()\ndf_results.plot(y='B', style='r-', lw=2, ax=ax_damping)\nax_damping.set_ylabel('Damping B [Nm]')\n\nalign_yaxis(ax, 0, ax_damping, 0)\n```\n\n\n```python\ndf_results['E_kin'] = 1/2*A_44*phi1d**2\nE_loss = cumtrapz(df_results['B'],x=phi)\nE_loss = np.concatenate([[0],E_loss])\ndf_results['E_loss'] = E_loss\ndf_results['E_pot'] = C_1*phi**2/2\ndf_results['E_sys'] = df_results['E_kin'] + df_results['E_pot']\ndf_results['E_tot'] = df_results['E_loss'] + df_results['E_sys']\n```\n\n\n```python\nE_loss2 = cumtrapz(phi1d,x=phi)\nE_loss2 = np.concatenate([[1],E_loss2])\n\ndf_results['B_e'] = df_results['E_loss']/E_loss2\n```\n\n\n```python\nfig,ax=plt.subplots()\ndf_results.plot(y='B_e', ax=ax)\n```\n\n\n```python\nfig,ax=plt.subplots()\ndf_results.plot(y='B_e', ax=ax)\nax.set_xlim(3,200)\nax.set_ylim(0.16,0.20)\n```\n\n\n\n\n```python\namplitudes = X_amplitudes['quadratic']\ndf_results['phi_a'] = np.interp(df_results.index,amplitudes.index,amplitudes['phi_a'])\nomega0 = np.sqrt(C_1/A_44)\ndf_results['B_e_formula'] = B_1 + B_2*8/(3*np.pi)*omega0*df_results['phi_a']\n\nfig,ax=plt.subplots()\nmask = df_results['phi_a']<0.11\ndf_results.loc[mask].plot(x='phi_a', y=['B_e','B_e_formula'],ax=ax)\n```\n\n\n```python\nomega0 = np.sqrt(C_1/A_44)\ndf_results['zeta'] = df_results['B_e']/(2*omega0*A_44)\n```\n\n\n```python\nfig,ax=plt.subplots()\nmask = df_results['phi_a']<0.11\ndf_results.loc[mask].plot(x='phi_a', y='zeta',ax=ax)\n```\n\n\n```python\nB_e = df_results['B_e'].iloc[-1]\nB_e\n```\n\n\n\n\n 0.16615505838874878\n\n\n\n\n```python\nsimulation = RollDecayLinear(A_44=A_44, B_1=B_e, C_1=C_1)\ndf_result = simulation.simulate(t=t, phi0=phi0, phi1d0=phi1d0)\n```\n\n\n```python\nfig,ax=plt.subplots()\nresults['quadratic'].plot(y='phi', ax=ax, label='quadratic')\nresults['linear'].plot(y='phi', ax=ax, style='--', label='linear')\ndf_result.plot(y='phi', ax=ax, style='--', label='linearized')\n\n```\n\n\n```python\nfig,ax=plt.subplots()\n\n\ndf_error = pd.DataFrame(index=df_result.index)\nref = results['quadratic']['phi']\ndf_error['linear'] = results['linear']['phi'] - ref\ndf_error['linearized'] = df_result['phi'] - ref\n\ndf_error.plot(ax=ax)\n```\n\n\n```python\n\n```\n\n\n```python\ndamping = equations.roll_decay_equation_himeno_quadratic_b.lhs.subs([(symbols.A_44,0),(symbols.C_1,0)])\nMath(vlatex(damping))\n```\n\n\n\n\n$\\displaystyle \\left(B_{1} + B_{2} \\left|{\\dot{\\phi}}\\right|\\right) \\dot{\\phi}$\n\n\n\n\n```python\nx = sp.symbols('x')\ndamping2 = symbols.B_1*x + symbols.B_2*x**2\ndamping2\n```\n\n\n\n\n$\\displaystyle B_{1} x + B_{2} x^{2}$\n\n\n\n\n```python\nx = sp.symbols('x')\ndamping3 = symbols.B_1*x\ndamping3\n```\n\n\n\n\n$\\displaystyle B_{1} x$\n\n\n\n\n```python\ns = sp.fourier_series(damping3, (x, -symbols.omega0/2*symbols.t, symbols.omega0/2*symbols.t))\ns\n```\n\n\n\n\n$\\displaystyle \\frac{B_{1} \\omega_{0} t \\sin{\\left(\\frac{2 \\pi x}{\\omega_{0} t} \\right)}}{\\pi} - \\frac{B_{1} \\omega_{0} t \\sin{\\left(\\frac{4 \\pi x}{\\omega_{0} t} \\right)}}{2 \\pi} + \\frac{B_{1} \\omega_{0} t \\sin{\\left(\\frac{6 \\pi x}{\\omega_{0} t} \\right)}}{3 \\pi} + \\ldots$\n\n\n\n\n```python\ns_truncate = s.truncate(n=2)\ns_truncate\n```\n\n\n\n\n$\\displaystyle \\frac{B_{1} \\omega_{0} t \\sin{\\left(\\frac{2 \\pi x}{\\omega_{0} t} \\right)}}{\\pi} - \\frac{B_{1} \\omega_{0} t \\sin{\\left(\\frac{4 \\pi x}{\\omega_{0} t} \\right)}}{2 \\pi}$\n\n\n\n\n```python\nsp.integrate(s_truncate,(symbols.t, -symbols.omega0/2*symbols.t, symbols.omega0/2*symbols.t))\n```\n\n\n\n\n$\\displaystyle - \\frac{B_{1} \\omega_{0} \\left(- \\frac{\\omega_{0}^{2} t^{2} \\sin{\\left(\\frac{4 \\pi x}{\\omega_{0}^{2} t} \\right)}}{8} - \\frac{\\pi t x \\cos{\\left(\\frac{4 \\pi x}{\\omega_{0}^{2} t} \\right)}}{2} - \\frac{2 \\pi^{2} x^{2} \\operatorname{Si}{\\left(\\frac{4 \\pi x}{\\omega_{0}^{2} t} \\right)}}{\\omega_{0}^{2}}\\right)}{\\pi} + \\frac{B_{1} \\omega_{0} \\left(\\frac{\\omega_{0}^{2} t^{2} \\sin{\\left(\\frac{4 \\pi x}{\\omega_{0}^{2} t} \\right)}}{8} + \\frac{\\pi t x \\cos{\\left(\\frac{4 \\pi x}{\\omega_{0}^{2} t} \\right)}}{2} + \\frac{2 \\pi^{2} x^{2} \\operatorname{Si}{\\left(\\frac{4 \\pi x}{\\omega_{0}^{2} t} \\right)}}{\\omega_{0}^{2}}\\right)}{\\pi} + \\frac{B_{1} \\omega_{0} \\left(- \\frac{\\omega_{0}^{2} t^{2} \\sin{\\left(\\frac{8 \\pi x}{\\omega_{0}^{2} t} \\right)}}{8} - \\pi t x \\cos{\\left(\\frac{8 \\pi x}{\\omega_{0}^{2} t} \\right)} - \\frac{8 \\pi^{2} x^{2} \\operatorname{Si}{\\left(\\frac{8 \\pi x}{\\omega_{0}^{2} t} \\right)}}{\\omega_{0}^{2}}\\right)}{2 \\pi} - \\frac{B_{1} \\omega_{0} \\left(\\frac{\\omega_{0}^{2} t^{2} \\sin{\\left(\\frac{8 \\pi x}{\\omega_{0}^{2} t} \\right)}}{8} + \\pi t x \\cos{\\left(\\frac{8 \\pi x}{\\omega_{0}^{2} t} \\right)} + \\frac{8 \\pi^{2} x^{2} \\operatorname{Si}{\\left(\\frac{8 \\pi x}{\\omega_{0}^{2} t} \\right)}}{\\omega_{0}^{2}}\\right)}{2 \\pi}$\n\n\n\n\n```python\n\n```\n\n\n```python\nfunc = 2*x\ns = sp.fourier_series(func, (x, -10, 10))\ns\n```\n\n\n\n\n$\\displaystyle \\frac{40 \\sin{\\left(\\frac{\\pi x}{10} \\right)}}{\\pi} - \\frac{20 \\sin{\\left(\\frac{\\pi x}{5} \\right)}}{\\pi} + \\frac{40 \\sin{\\left(\\frac{3 \\pi x}{10} \\right)}}{3 \\pi} + \\ldots$\n\n\n\n\n```python\ns_trunate = s.truncate(n=10)\ns_lambda = lambdify(s_trunate)\nfunc_lambda = lambdify(func)\n\nx_ = np.linspace(-10,20,100)\ny_ = s_lambda(x=x_)\ny = func_lambda(x=x_)\n\nfig,ax=plt.subplots()\nax.plot(x_,y, label='function')\nax.plot(x_,y_, label='fourier series n=1')\n```\n\n\n```python\nsp.integrate(x,(x, 0, 1))\n```\n\n\n\n\n$\\displaystyle \\frac{1}{2}$\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "d129449bc581c05968d5a1096f2559d5d18d5eec", "size": 377079, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/02.2_ikeda_Be_assumption.ipynb", "max_stars_repo_name": "rddaz2013/Prediction-of-roll-motion-using-fully-nonlinear-potential-flow-and-Ikedas-method", "max_stars_repo_head_hexsha": "ac0a27e31d64edc8ae8912b6ed10005029868c90", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/02.2_ikeda_Be_assumption.ipynb", "max_issues_repo_name": "rddaz2013/Prediction-of-roll-motion-using-fully-nonlinear-potential-flow-and-Ikedas-method", "max_issues_repo_head_hexsha": "ac0a27e31d64edc8ae8912b6ed10005029868c90", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/02.2_ikeda_Be_assumption.ipynb", "max_forks_repo_name": "rddaz2013/Prediction-of-roll-motion-using-fully-nonlinear-potential-flow-and-Ikedas-method", "max_forks_repo_head_hexsha": "ac0a27e31d64edc8ae8912b6ed10005029868c90", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-06-05T15:38:54.000Z", "max_forks_repo_forks_event_max_datetime": "2021-06-05T15:38:54.000Z", "avg_line_length": 257.0408997955, "max_line_length": 38512, "alphanum_fraction": 0.9217591009, "converted": true, "num_tokens": 5446, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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Doing Math With Python PDF ( http://index-of.es/Varios-2/Doing%20Math%20with%20Python.pdf)\n1. Doing Math With Python site (http://www.nostarch.com/doingmathwithpython/ )\n\n# \u306f\u3058\u3081\u306b\n\n\u6570\u5b66\u3068\u8a00\u3063\u3066\u3082\u5e83\u3044\u8a71\u3067\u3001\u8abf\u3079\u306a\u304c\u3089\u3044\u308d\u3093\u306a\u8a71\u984c\u306b\u9032\u3093\u3067\u3057\u307e\u3046\u3068\u306f\u601d\u3046\u3082\u306e\u306e\u3001\u3068\u308a\u3042\u3048\u305a\u601d\u3044\u3064\u3044\u305f\u3053\u3068\u3092\u66f8\u3044\u3066\u304a\u304f\u3068\u3001\u8208\u5473\u306e\u5bfe\u8c61\u304c\u308f\u304b\u3063\u3066\u3044\u3044\u3068\u601d\u3046\u306e\u3067\u66f8\u3044\u3066\u304a\u304f\u3002\n\n1. \u6570\u5f0f\u3002\u6570\u5b66\u306b\u304a\u3044\u3066\u6570\u5f0f\u3092\u6570\u5b66\u306e\u4e16\u754c\u3067\u666e\u901a\u306a\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u306f\u5927\u4e8b\u3060\u3068\u601d\u3046\u3002$x^2$\u3092 `x**2` \u3068\u66f8\u304f\u3068\u304d\u3001\u4f55\u304c\u9055\u3046\u304b\u3068\u3044\u3046\u3068\u30d5\u30a9\u30f3\u30c8\u304c\u9055\u3046\u3001\u8a18\u53f7\u304c\u9055\u3046\u3002\u3044\u307e\u306f$ \\LaTeX$ \u3067\u66f8\u3044\u305f\u3002\n1. \u6570\u5f0f\u6f14\u7b97\u3002$x$ \u306b\u6570\u3092\u4ee3\u5165\u3059\u308b\u3053\u3068\u306f\u30d7\u30ed\u30b0\u30e9\u30df\u30f3\u30b0\u3067\u3067\u304d\u3066\u3001\u624b\u7d9a\u304d\u7684\u306b\u3044\u308d\u3044\u308d\u8907\u96d1\u306a\u8a08\u7b97\u3082\u3067\u304d\u308b\u304c\u3001\u6570\u5f0f\u3092\u6570\u5f0f\u81ea\u4f53\u3067\u64cd\u4f5c\u3057\u3066\u3001\u6982\u5ff5\u3092\u62bd\u8c61\u5316\u3057\u3066\u8003\u3048\u308b\u3053\u3068\u304c\u6570\u5b66\u306e\u672c\u8cea\u306e\u4e00\u90e8\u3067\u3042\u308b\u3002\n1. sympy \u3068\u5225\u306b python \u306b\u306f numpy \u30e2\u30b8\u30e5\u30fc\u30eb\u304c\u3042\u3063\u3066\u8a08\u7b97\u306b\u4fbf\u5229\u306b\u4f7f\u308f\u308c\u308b\u3002 \u5fc5\u8981\u306b\u5fdc\u3058\u3066\u4f7f\u3046\u3002\n1. \u56f3\u306f matplotlib \u3092\u4f7f\u3046\u3002\n\n\n\n# \u6570\u5217\u306e\u5408\u8a08 \u30b7\u30b0\u30de $\\Sigma$\n\n\n```python\n# python \u3067\u6570\u306e\u5408\u8a08\u306f\u6b21\u306e\u3088\u3046\u306b\u66f8\u304f \r\nsum ([12,34])\n```\n\n\n\n\n 46\n\n\n\n\n```python\n# sympy \u306b\u6570\u306e\u5408\u8a08\u306e\u8a18\u53f7\u30b7\u30b0\u30de\u304c\u3042\u308b\r\nfrom sympy import *\r\nfrom sympy.abc import *\r\nSum(k, (k, 1, m))\n```\n\n\n\n\n$\\displaystyle \\sum_{k=1}^{m} k$\n\n\n\n\n```python\nSum(k, (k, 1, m)).doit()\n```\n\n\n\n\n$\\displaystyle \\frac{m^{2}}{2} + \\frac{m}{2}$\n\n\n\n\n```python\n# \u4e0a\u8a18\u306e\u51fa\u529b\u3092\u56e0\u6570\u5206\u89e3\u3057\u3066\u6b21\u306e\u3088\u3046\u306b\u3057\u305f\u3044\r\n# cf. factoring, expand, simplify\r\n%%latex\r\n\\displaystyle\r\n\\frac{m(m+1)}{2}\n```\n\n\n\\displaystyle\n\n\\frac{m(m+1)}{2}\n\n\n\n```python\n# \u56e0\u6570\u5206\u89e3\u306f factor \u3092\u4f7f\u3046\r\nfactor(Sum(k, (k, 1, m)).doit())\n```\n\n\n```python\n# \u5b8c\u6210\u54c1\r\nEq(Sum(k, (k, 1, m)), factor(Sum(k, (k, 1, m)).doit()))\n```\n\n# SymPy \u306e\u6b69\u304d\u65b9\r\n\r\n\u4f8b\u3048\u3070\u3001\u6570\u5217\u306e\u5408\u8a08\u30b7\u30b0\u30de $\\Sigma$ \u3092 sympy \u3067\u3069\u3046\u66f8\u304f\u304b\u3001\u305d\u3082\u305d\u3082\u66f8\u3051\u308b\u306e\u304b\u3001\u540d\u524d\u306f\u306a\u3093\u3068\u3044\u3046\u306e\u304b, Sigma \u306a\u306e\u304b Sum \u306a\u306e\u304b\u3001\u95a2\u6570\u3068\u3057\u3066\u4f7f\u3048\u308b\u306e\u304b\u305f\u3060\u306e\u8a18\u53f7\u306a\u306e\u304b\u3001\u3068\u3044\u3046\u3068\u304d\u3002\r\n\r\n`from sympy import *` \u3057\u3066\u3001 dir() \u3092\u51fa\u529b\u3059\u308b\u3002 \r\n\u51fa\u529b\u3092\u30a8\u30c7\u30a3\u30bf\u30fc\u306b\u30b3\u30d4\u30da\u3057\u3066\u3001\u305d\u308c\u3089\u3057\u304d\u547d\u4ee4\u3092\u63a2\u3059\u3002 \r\nSum \u3068\u3044\u3046\u306e\u304c\u3042\u3063\u305f\u3002\r\n\u30b3\u30fc\u30c9\u30bb\u30eb\u3067 Sum \u3092\u5b9f\u884c\u3059\u308b\u3068 `sympy.concrete.summations.Sum` \u3068\u51fa\u529b\u3055\u308c\u308b\u3002 \u3053\u308c\u3092 google \u3067\u691c\u7d22\u3059\u308b\u3068\u30c9\u30ad\u30e5\u30e1\u30f3\u30c8\u304c\u898b\u3089\u308c\u308b\u3002 \r\nhelp(Sum) \u3067 help \u304c\u898b\u3089\u308c\u308b\u3002\r\n\r\npython \u306e\u95a2\u6570\u3084\u30e1\u30bd\u30c3\u30c9\u306f\u5c0f\u6587\u5b57 lower caracter \u3067\u59cb\u307e\u308b\u306e\u304c\u539f\u5247\u306a\u306e\u3067\u3001\u5927\u6587\u5b57\u3067\u59cb\u307e\u308b\u306e\u306f sympy \u306e\u51fa\u529b\u306e\u304b\u3089\u3093\u3060\u547d\u4ee4\u3067\u3042\u308b\u3002\r\n\r\n\u3042\u3068\u306f\u5b9f\u9a13\u5b66\u7fd2\r\n\n\n\n```python\nfrom sympy import *\r\nSum\n```\n\n\n\n\n sympy.concrete.summations.Sum\n\n\n\n# Abs \u3092\u4f8b\u306b\u5b9f\u9a13\n\n\n```python\n# Abs \u306b\u3064\u3044\u3066\u5b9f\u9a13\r\n# sympy \u3067\u30b7\u30f3\u30dc\u30eb\u306b\u5024\u304c\u5165\u3063\u3066\u3044\u308b\u5834\u5408\u3001Abs \u306f\u95a2\u6570\u3068\u3057\u3066\u50cd\u304d\u3001\u5f15\u6570\u304c\u30b7\u30f3\u30dc\u30eb\u306e\u5834\u5408\u3001\u6570\u5f0f\u51e6\u7406\u30b7\u30b9\u30c6\u30e0\u3068\u3057\u3066\u50cd\u304f\r\nfrom sympy import *\r\nfrom sympy.abc import *\r\ndisplay(Abs(x))\r\nx = -3\r\ndisplay(Abs(x))\r\nx = symbols('x')\r\ndisplay(Abs(x))\r\ndisplay(latex(Abs(x)))\n```\n\n\n```latex\n%%latex\r\n\\left|{x}\\right|\n```\n\n\n\\left|{x}\\right|\n\n\n\n```python\n# help(abs) \u306b\u3088\u308b\u3068\u3001Abs \u304c\u3042\u308b\u5834\u5408\u3001\u901a\u5e38\u306e\u7d44\u307f\u8fbc\u307f\u95a2\u6570 abs \u306f Abs \u306b\u306a\u308b\u3001\u3068\u306e\u3053\u3068\u306a\u306e\u3067\u5b9f\u9a13\r\nx = symbols('x')\r\nabs(x) # => \\left|{x}\\right|\n```\n\n\n\n\n$\\displaystyle \\left|{x}\\right|$\n\n\n\n# \u7a4d\u5206\u8a18\u53f7 Integral \u3068\u5fae\u5206\u8a18\u53f7 Derivative\r\n\r\n\u7a4d\u5206\u3059\u308b\u306f integrate \u5fae\u5206\u3059\u308b\u306f diff\r\n\n\n\n```python\nfrom sympy import *\ninit_printing()\n\nx = symbols ('x')\na = Integral (cos(x) * exp(x), x)\ndisplay(a)\ndisplay(a.doit())\ndisplay(Eq (a, (factor(a.doit()))))\n```\n\n\u4eca\u307e\u3067\u306e\u4f8b\u304b\u3089\u308f\u304b\u3063\u305f\u3053\u3068\u3002\n\n1. `from sympy import *` \u3067\u3053\u308c\u4ee5\u964d\u306a\u306b\u3082\u3053\u3068\u308f\u3089\u305a\u306b\u56db\u5247\u6f14\u7b97\u3092\u3075\u304f\u3081\u6570\u5b66\u8a18\u53f7\u304c sympy \u306e\u6f14\u7b97\u5b50\u306b\u306a\u3063\u3066\u3044\u308b\u3002\n1. `x = symbols('x')` \u306f\u3001python \u3068\u5171\u5b58\u3059\u308b\u306e\u3067\u3001\u306a\u3093\u3089\u304b\u306e\u5ba3\u8a00\u304c\u5fc5\u8981\u306a\u306e\u306f\u308f\u304b\u308b\u3002`x = 3`\u3068\u5165\u308c\u305f\u3089\u305d\u308c\u4ee5\u964d\u306f `x` \u306f 3 \u306b\u306a\u3063\u3066\u3001\u307e\u305f `x = symbols('x')` \u3068\u3057\u305f\u3089 `x` \u306b\u623b\u3063\u305f\u3002 `from sympy.abc import *` \u3068\u3044\u3046\u4fbf\u5229\u306a\u9053\u5177\u3082\u3042\u308b\u3002\n3. \u5f0f\u3067\u9805\u306e\u9806\u5e8f\u306f\u7dad\u6301\u3055\u308c\u306a\u3044\u3002` Integral (cos(x) * exp(x), x)` \u306e `cos(x)`\u3068`exp(x)`\u306f\u6570\u5f0f\u8868\u73fe\u306b\u306a\u3063\u305f\u3068\u304d\u306b\u9806\u5e8f\u304c\u9006\u306b\u306a\u3063\u3066\u3044\u308b\u3002\n4. `doit()`\u306f\u6570\u5f0f\u30ec\u30d9\u30eb\u306e\u8a08\u7b97\u3068\u3044\u3046\u304b\u3001\u8a55\u4fa1\u3092\u3059\u308b\u304c\u3001\u3044\u307e `((x + 2) * (x - 2)).doit()` \u3068\u3057\u305f\u3089\u5c55\u958b\u3057\u306a\u304b\u3063\u305f\u3002\u30e1\u30bd\u30c3\u30c9 `factor()`\u3001`expand()` \u306a\u3069\u3092\u4f7f\u3046\u3002\n\n\n\n```python\nfrom sympy import *\ninit_printing()\n\nx = symbols ('x')\n((x + 2) * (x - 2)).expand()\n```\n\n\n```python\n((x + 2) * (x - 2))\n```\n\n\n```python\nexpand((x + 2) * (x - 2))\n```\n\n\n```python\nfrom sympy import *\r\ninit_printing()\r\nx = symbols ('x')\r\nfactor(x**2 - 4)\n```\n\n\n```python\nintegrate (cos(x) * exp(x), x)\n```\n\n# sympy \u306e\u51fa\u529b\u306f\u5fc5\u305a\u3057\u3082\u7406\u60f3\u7684\u3067\u306f\u306a\u3044\u306e\u3067\u3001latex \u3067\u6574\u5f62\u3059\u308b\r\n\n\n\n```python\nfrom sympy import *\r\ninit_printing()\r\nx = symbols ('x')\r\na = Integral (cos(x) * exp(x), x)\r\ndisplay(Eq (a, a.doit()))\r\nlatex(Eq (a, factor(a.doit())))\n```\n\n\n```latex\n%%latex\n\\displaystyle\n\\int e^{x} \\cos{\\left(x \\right)}\\, dx = \\frac{e^{x} \\sin{\\left(x \\right)}}{2} + \\frac{e^{x} \\cos{\\left(x \\right)}}{2} \\\\\n\\int{e^x \\cos (x)}dx = \\frac{e^x}{2} \\sin(x) + \\frac{e^x}{2} \\cos(x) \\\\\n\\int e^{x} \\cos{\\left(x \\right)}\\, dx = \\frac{\\left(\\sin{\\left(x \\right)} + \\cos{\\left(x \\right)}\\right) e^{x}}{2} \\\\\n\\int e^{x} \\cos{\\left(x \\right)}\\, dx = \\frac{e^{x}}{2}\\left(\\sin{\\left(x \\right)} + \\cos{\\left(x \\right)}\\right) \n```\n\n\n\\displaystyle\n\\int e^{x} \\cos{\\left(x \\right)}\\, dx = \\frac{e^{x} \\sin{\\left(x \\right)}}{2} + \\frac{e^{x} \\cos{\\left(x \\right)}}{2} \\\\\n\\int{e^x \\cos (x)}dx = \\frac{e^x}{2} \\sin(x) + \\frac{e^x}{2} \\cos(x) \\\\\n\\int e^{x} \\cos{\\left(x \\right)}\\, dx = \\frac{\\left(\\sin{\\left(x \\right)} + \\cos{\\left(x \\right)}\\right) e^{x}}{2} \\\\\n\\int e^{x} \\cos{\\left(x \\right)}\\, dx = \\frac{e^{x}}{2}\\left(\\sin{\\left(x \\right)} + \\cos{\\left(x \\right)}\\right) \n\n\n# \u3044\u307e\u3053\u3053\n\n`doit()`\u306e\u30d8\u30eb\u30d7\u3092\u8aad\u3093\u3067\u898b\u308b\u3002\n\n\n```python\n# `doit()`\u306e\u30d8\u30eb\u30d7\u3092\u8aad\u3093\u3067\u898b\u308b\u3002\nfrom sympy import *\nx = symbols('x')\nhelp(x.doit)\n```\n\n\n```python\n# doit() \u306e\u4f7f\u3044\u65b9\u3002 doit(a) \u3067\u306f\u30c0\u30e1\u3002doit(deep=False) \u3068\u304b\u306b\u4f7f\u3046\u3002\nfrom sympy import *\ninit_printing()\nx = symbols ('x')\na = 2 * Integral (x, x)\ndisplay(a)\nprint()\ndisplay(a.doit())\nprint()\ndisplay(Eq(a, a.doit()))\n```\n\n`init_printing()`\u3068\u3059\u308b\u3068\u3001\u30a2\u30a6\u30c8\u30d7\u30c3\u30c8\u304c\u6570\u5f0f\u8868\u793a\u306b\u306a\u308b\u3002\n\n# \u5fae\u5206\n\n\u5fae\u5206\u306e\u5b9f\u9a13\n\n\n\n\n```python\n# Derivative \u5fae\u5206\u306e\u5b9f\u9a13\r\nfrom sympy import *\r\nfrom sympy.abc import *\r\ninit_printing()\r\nexpr = x**2\r\ndisplay(Eq(Derivative(expr), expr.diff()))\r\nprint()\r\ndisplay(Eq(Derivative(expr), diff(expr)))\r\nprint()\r\ndisplay(Eq(Derivative(expr), (Derivative(expr)).doit()))\n```\n\n\n```python\nEq(Derivative(exp(x)), diff(exp(x)))\n```\n\n# \u56e0\u6570\u5206\u89e3\u3068\u5f0f\u306e\u5c55\u958b\n\n`factor()` \u3068 `expand()` \u3092\u4f7f\u3063\u3066\u307f\u308b\u3002\n\n\n```python\nfrom sympy import *\nfrom sympy.abc import *\ninit_printing()\na = (x + 2) * (x - 2)\nb = (x**2 - 4)\nc = (x**2 - 4) / (x - 2)\ndisplay(a, b, c)\n```\n\n\n```python\nfactor((x**2 - 4))\n```\n\n\n```python\ndisplay(factor (b))\ndisplay(factor (c))\n```\n\n\n```python\ndisplay(expand (a))\ndisplay(expand (b))\ndisplay(expand (c))\n```\n\n\n```python\nd = (x - 2)\nb / d\n```\n\n\u5f0f\u306e\u56e0\u6570\u5206\u89e3\u3084\u5c55\u958b\u304c\u5168\u90e8\u81ea\u52d5\u5316\u3067\u304d\u308b\u308f\u3051\u3067\u306f\u306a\u3044\u3002\n\n\u3042\u305f\u308a\u307e\u3048\u3060\u306d\u3002\n\n\n```python\n# \u591a\u9805\u5f0f\u306f\u305d\u308c\u306a\u308a\u306b\u4e26\u3079\u66ff\u3048\u305f\u308a\u6574\u7406\u306f\u3057\u3066\u304f\u308c\u308b\r\ndisplay(x**2 + y**3 + x + x**3 + x**2 + y**4)\r\ndisplay(x * (x+1) ** 2 + y**3 *(y + 1))\r\ndisplay(expand(x * (x+1) ** 2 + y**3 *(y + 1)))\r\n\n```\n\nRational() \u3068\u304b\n\n\n```python\n# Rational() \u3068\u304b\nfrom sympy import *\nRational (3, 2)\n```\n\n\n```python\nRational(8, 4)\n```\n\n\n```python\nsqrt(8)\n```\n\n\u6570\u5f0f\u306e\u30b7\u30f3\u30dc\u30eb\u3092\u4e8b\u524d\u306b\u5ba3\u8a00\u3059\u308b\u5834\u5408\u3002\n\n\n\n```python\n# \u30b7\u30f3\u30dc\u30eb\u306b\u3064\u3044\u3066\u8003\u5bdf\u3001\u5b9f\u9a13\nfrom sympy import *\nx, y, z = symbols(\"x y z\")\nk, m, n = symbols(\"k m n\", integer=True)\nf, g, h = symbols('f g h', cls=Function)\n```\n\n\n\u610f\u5473\u306f\u591a\u5206\u3001`x, y, z` \u306f\u4efb\u610f\u306e\u5909\u6570\u3001`f, m, n` \u306f\u6574\u6570\u3001`f, g, h` \u306f\u95a2\u6570\u3068\u3044\u3046\u3053\u3068\u3060\u3068\u601d\u3046\u304c\u3001`i, j`\u304c\u306a\u3044\u3002\n\n\n```python\nfrom sympy import *\r\nfrom sympy.abc import *\r\ninit_printing()\r\ndisplay(alpha, beta, chi, delta, epsilon, eta, gamma, iota, kappa, lamda, mu, nu, omega, omicron, phi, pi, psi, rho, sigma, tau, theta, upsilon, xi, zeta)\n```\n\nRational()\u306e\u4e2d\u3067\u5909\u6570\u306f\u4f7f\u3048\u306a\u304b\u3063\u305f\n\n\n```python\n# Rational(x, y) # => TypeError\r\ndisplay(x / y)\n```\n\n\u6570\u5f0f\u306e\u5272\u308a\u7b97\u3068\u304b\n\n\n```python\n(x**2 - 2 * x + 1) / (x - 1)\n```\n\n\n```python\nfactor((x**2 - 2 * x + 1) / (x - 1))\n```\n\n\n```python\nfactor ((x**2 - 2*x + 1)) / (x - 1)\n```\n\n\n```python\ndisplay(pi, i, e)\n```\n\n\n```python\npi.evalf()\n```\n\n\n```python\nexp(1).evalf()\n```\n\n# \u30d8\u30eb\u30d7\u306e\u6c42\u3081\u65b9\n\n`evalf()`\u306e\u4f8b\u3067\u306f`evalf?`\u3068\u304b`evalf??`\u3068\u304b\u304c\u7c21\u5358\u3067\u3088\u3044\u3002\n\n`help(evalf)`\u3082\u3042\u308b\u3002`help(x.evalf)`\u3082\u3042\u308b\u3002\n\n\n```python\nevalf?\n# Adaptive numerical evaluation of SymPy expressions, using mpmath\n# for mathematical functions.\n```\n\n\n```python\nevalf??\n```\n\n\n```python\nexp(1).evalf()\n```\n\n\n```python\n-oo.evalf()\n```\n\n\n```python\n-oo\n```\n\n# \u5f0f\u306e\u7c21\u7d20\u5316\n\n`simplification`\u3068\u3044\u3046\u306e\u304b\u3002\u81ea\u52d5\u3067\u3084\u3089\u308c\u308b\u306e\u306f\u3054\u304f\u5358\u7d14\u306a\u5834\u5408\u3060\u3051\u3002\n\n\n\n```python\nfrom sympy import *\ninit_printing()\nx, y = symbols('x y')\nexpr = x + 2 * y\ndisplay((x * expr))\ndisplay(expand (x * expr))\ndisplay(factor(expand (x * expr)))\n```\n\n\n```python\nfrom sympy import *\ninit_printing()\nx, t, z, nu = symbols ('x t z nu')\n\nexpr = sin (x) * exp(x)\ndisplay(Eq(Derivative(expr), diff(expr)))\n```\n\n\n```python\nexpr2=exp(x)*sin(x) + exp(x)*cos(x)\r\ndisplay(Eq(Integral(expr2), integrate (expr2)))\n```\n\n\n```python\ndisplay(integrate(sin(x**2), (x, -oo, oo)))\r\ndisplay(Integral(sin(x**2), (x, -oo, oo)))\r\n# \u6b21\u306e\u5f0f\u306f\u5b9f\u884c\u3059\u308b\u3068\u7d42\u308f\u3089\u306a\u3044\r\n# display(Eq(Integral(sin(x**2), (x, -oo, oo)), integrate(sin(x**2), (x, -oo, oo))))\n```\n\n\n```python\n# \u6975\u9650 limit\r\nlimit (sin(x)/x, x, 0)\n```\n\n# \u3044\u307e\u3053\u3053\n\n\n```python\n# \\sqrt 2 $ \u3092100\u6841\u307e\u3067\u8a08\u7b97\u3059\u308b\r\nsqrt(2).evalf(100)\n```\n\n\n```python\npi.evalf(100)\n```\n\n\n```python\n2**(1/2)\n```\n\n\n```python\nval = 2**(1/2)\nprint(\"{0:.100f}\".format(val))\n```\n\n 1.4142135623730951454746218587388284504413604736328125000000000000000000000000000000000000000000000000\n\n\n\n```python\n# \u3053\u308c\u306f\u30a8\u30e9\u30fc\u306b\u306a\u308b\n# val = sqrt(2)\n# print(\"{0:.100f}\".format(val))\n```\n\n\n```python\nval = 2**(1/2)\n```\n\n\n```python\n# 2**(1/2)\u306ffloat\u306a\u306e\u3067evalf()\u304c\u4f7f\u3048\u306a\u3044\u3002exp\u8868\u793a\u306b\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\nexp(ln(2) / 2).evalf(100)\n```\n\n\n```python\n###### 1/2+1/3\u3092\u6709\u7406\u6570\u3068\u3057\u3066\u8a08\u7b97\u3059\u308b\nRational(1,2)+Rational(1,3)\n```\n\n\n```python\nx + x + y + x\n```\n\n\n```python\nexpand((x+y)**5)\n```\n\n###### \u5909\u6570\u5165\u308a\u306e\u6570\u5f0f\u306f\u30d9\u30ad\u304c\u5927\u304d\u3044\u306e\u304b\u3089\u5c0f\u3055\u3044\u306e\u306b\u4e26\u3079\u3089\u308c\u308b\u306e\u306f\u30a8\u30e9\u3044\n\n\n```python\n3*x*y**2 + 3*y*x**2 + x**3 + y**3\n```\n\n\n```python\nexpand(x + y, complex = True)\n```\n\n###### \u4e0a\u306e\u306f\u8907\u7d20\u6570\u8868\u8a18\u3067\u30c9\u30a4\u30c4\u6587\u5b57\u306e$ R $\u3068$ I $\u3092\u4f7f\u3063\u3066\u3044\u308b\n\u30d5\u30e9\u30af\u30c8\u30a5\u30fc\u30eb\u3068\u3044\u3046\u3089\u3057\u3044\n$$\n\\mathfrak R (x) + \\mathfrak R (y) + i \\mathfrak I x + \\mathfrak I y\n$$\n\u3060\u3044\u305f\u3044\u5408\u3063\u3066\u3044\u308b\u304b\u306a\u3002\n\n\n```python\n%%script false\n'''\nUnit converter: Miles and Kilometers\n'''\ndef print_menu():\n print('1. Kilometers to Miles')\n print('2. Miles to Kilometers')\n\ndef km_miles():\n km = float(input('Enter distance in kilometers: '))\n miles = km / 1.609\n\n print('Distance in miles: {0}'.format(miles))\n\ndef miles_km():\n miles = float(input('Enter distance in miles: '))\n km = miles * 1.609\n print('Distance in kilometers: {0}'.format(km))\n\nif __name__ == '__main__':\n print_menu()\n choice = input('Which conversion would you like to do?: ')\nif choice == '1':\n km_miles()\nif choice == '2':\n miles_km()\n```\n\n### if __name__ == '__main__':\n\u306f\u30ce\u30fc\u30c8\u30d6\u30c3\u30af\u306b\u305d\u3050\u308f\u306a\u3044\u307f\u305f\u3044\u3002 \n\u53d6\u3063\u3066\u3057\u307e\u3063\u305f\u65b9\u304c\u3044\u3044\u307f\u305f\u3044\u3002 \n\n\u305d\u3093\u306a\u3053\u3068\u3092\u8a00\u3048\u3070\u3001input()\u3082\u30ce\u30fc\u30c8\u30d6\u30c3\u30af\u306b\u305d\u3050\u308f\u306a\u3044\u307f\u305f\u3044\u3002 \ninput()\u304c\u3042\u308b\u3068\u8a55\u4fa1\u304c\u305d\u306e\u30bb\u30eb\u3067\u6b62\u307e\u3063\u3066\u3057\u307e\u3046\u3002 \n\u306a\u308b\u307b\u3069\u3002\n\n\n```python\n'''\nQuadratic equation root calculator\n'''\ndef roots(a, b, c):\n D = (b*b - 4*a*c)**0.5\n x_1 = (-b + D)/(2*a)\n x_2 = (-b - D)/(2*a)\n print('x1: {0}'.format(x_1))\n print('x2: {0}'.format(x_2))\n\na, b, c = 1, 2, 1\nroots(float(a), float(b), float(c))\n```\n\n x1: -1.0\n x2: -1.0\n\n\n###### (x+y)6 \u306e\u5c55\u958b\u5f62\u3092\u8a08\u7b97\u3059\u308b\u3002\n\n\n```python\nfrom sympy import *\ninit_printing()\nx, y = symbols(\"x,y\")\n\nexpand((x + y)**6)\n```\n\n\n```python\nsimplify(x**6 + 6*x**5*y)\n```\n\n###### python\u8868\u8a18\u306b\u623b\u3059 init_printing()\u306b\u306f\u3069\u3046\u3057\u305f\u3089\u3088\u3044\u304b\n\u30a2\u30a4\u30c7\u30a2\u306f \ninit_printing(use_pythonformat: True) \nx**2 \n\u3068\u304b\n\ninit_printing(pretty_print=False)\n\n\u3067\u3057\u305f\u3002\n\n\n```python\ninit_printing(pretty_print=False)\nx**2\n```\n\n\n\n\n x**2\n\n\n\n\n```python\nexpand((x + y)**6)\n```\n\n\n\n\n x**6 + 6*x**5*y + 15*x**4*y**2 + 20*x**3*y**3 + 15*x**2*y**4 + 6*x*y**5 + y**6\n\n\n\n\n```python\n# sin(x)/cos(x)\u3092\u7c21\u5358\u5316\u3059\u308b\nfrom sympy import *\ninit_printing()\nx = symbols('x')\nsimplify(sin(x)/cos(x))\n```\n\n### simplify\u306e\u4ee3\u66ff\n\n\u7c21\u5358\u5316\u3068\u306f\u3044\u304f\u3076\u3093\u66d6\u6627\u306a\u7528\u8a9e\u306e\u305f\u3081\u3001\u3088\u308a\u76ee\u7684\u3092\u660e\u78ba\u306b\u3057\u305f simplify \u306e\u4ee3\u66ff\u304c\u5b58\u5728\u3059\u308b: powsimp (\u6307\u6570\u306e\u7c21\u5358\u5316), trigsimp (\u4e09\u89d2\u95a2\u6570\u3092\u542b\u3080\u6570\u5f0f), logcombine, radsimp, togeter.\n\n\n```python\nfactor(x**2 + 2*x + 1)\n```\n\n\n```python\nsimplify(x**2 + 2*x + 2)\n```\n\n\n```python\ntrigsimp(sin(x)**4 - 2*cos(x)**2*sin(x)**2 + cos(x)**4)\n```\n\n\n```python\nexpand_trig(sin(x+y))\n```\n\n\n```python\ntrigsimp(sin(x)*cos(y)+sin(y)*cos(x))\n```\n\n\n```python\nexpand_trig(tan(2*x))\n```\n\n\n```python\nsqrt(x*y)\n```\n\n\n```python\nln(x)\n```\n\n\n```python\nx, y = symbols('x y')\nexpand_log(log(x*y))\n```\n\n\n```python\nx, y = symbols('x y', positive=True)\nexpand_log(log(x*y))\n```\n\n\n```python\nx, y = symbols('x y')\nexpand_log(log(x*y), force=True)\n```\n\n\n```python\nlogcombine(log(x) + log(y), force=True)\n```\n\n\n```python\nn=symbols('n')\nlogcombine(n*ln(x),force=True)\n```\n\n\n```python\nx,y,z=symbols('x y z')\nk, m, n = symbols('k m n')\nfactorial(n)\n```\n\n\n```python\nbinomial(n,k)\n```\n\n\n```python\nbinomial(5,3)\n```\n\n\n```python\ngamma(z)\n```\n\n\n```python\ngamma(10)\n```\n\n\n```python\nfactorial(9)\n```\n\n\u3053\u308c\u306f`LaTeX`\n

    \n\n$$\n\\Gamma(z) = \\int_0^\\infty t^{z - 1}e^{-t}\\,dt\n$$\n\n\n\n```latex\n%%latex\n\\displaystyle\n\n\\Gamma(z) = \\int_0^\\infty t^{z - 1}e^{-t},dt\n```\n\n\n\\displaystyle\n\n\\Gamma(z) = \\int_0^\\infty t^{z - 1}e^{-t},dt\n\n\n\n```python\n# hyper([a_1, ..., a_p], [b_1, ..., b_q], z)\n# hypergeometric function\nhyper([1,2],[3],z)\n```\n\n\n\n\n$\\displaystyle {{}_{2}F_{1}\\left(\\begin{matrix} 1, 2 \\\\ 3 \\end{matrix}\\middle| {z} \\right)}$\n\n\n\n\n```python\n# rewrite\ntan(x).rewrite(sin)\n```\n\n\n```python\nfactorial(x).rewrite(gamma)\n# For some tips on applying more targeted rewriting, see the Advanced Expression Manipulation section\n```\n\n\n```python\nexpand_func(gamma(x + 3))\n```\n\n\n```python\nhyperexpand(hyper([1,1],[2],z))\n```\n\n\n```python\nexpr = meijerg([[1],[1]], [[1],[]],-z)\nexpr\n```\n\n\n\n\n$\\displaystyle {G_{2, 1}^{1, 1}\\left(\\begin{matrix} 1 & 1 \\\\1 & \\end{matrix} \\middle| {- z} \\right)}$\n\n\n\n\n```python\nhyperexpand(expr)\n```\n\n\n```python\nn , k = symbols('n k', integer=True)\ncombsimp(factorial(n)/factorial(n - 3))\n```\n\n\n```python\ncombsimp(binomial(n+1, k+1)/binomial(n, k))\n```\n\n\n```python\ngammasimp(gamma(x)*gamma(1 - x))\n```\n\n### sympy\u306etutorial\n\n\n```python\nfrom sympy import *\ninit_printing()\nx = symbols('x')\na = Integral(cos(x)*exp(x),x)\nEq(a, a.doit())\n```\n\n\n```python\na.doit()\n```\n\n\n```python\nb = Integral(x**3)\nEq(b, b.doit())\n```\n\n### Sympy Tutorial\u306b\u79fb\u308a\u307e\u3057\u305f\n\n\n```python\nfrom sympy import *\ninit_printing()\nx, t, z, nu, y = symbols('x t z nu y')\n\ndiff(sin('x')*exp(x),x)\n```\n\n\n```python\ndiff(x**2)\n```\n\n\n```python\ndiff(sin('x')*exp(x))\n```\n\n\n```python\nexp(x)\n```\n\n\n```python\nexp(1)\n```\n\n\n```python\ndiff(exp(x))\n```\n\n\n```python\nexp(1).evalf()\n```\n\n\n```python\nexpand_trig(sin(x + y))\n```\n\n\n```python\nexpand_trig(sin(2*x))\n```\n\n\n```python\nexpand_trig(cos(x + y))\n```\n\n\n```python\nexpand_trig(cos(2*x))\n```\n\n\n```python\nexpand_trig(sin(3*x))\n```\n\n\n```python\nintegrate(exp(x)*sin(x) + exp(x)*cos(x))\n```\n\n\n```python\nintegrate(sin(x**2),(x,-oo, oo))\n```\n\n\n```python\nlimit(sin(x)/x, x, 0)\n```\n\n\n```python\nfrom sympy import *\ninit_printing()\nx = symbols('x')\ndisplay(factor(x**2 - 4))\ndisplay(factor(x**2 - 2))\n```\n\n\n```python\nsolve(x**2 - 2, x)\n```\n\n\n```python\n# solve the differential equation y'' - y = e^t\nfrom sympy import *\ninit_printing()\nx, y ,z, t = symbols('x y z t')\ny = Function('y')\ndsolve(Eq(y(t).diff(t, t) - y(t), exp(t)), y(t))\n```\n\n\n```python\ny(t).diff(t,t)\n```\n\n### eigenvals()\n\n\n```python\nMatrix([[1,2],[2,2]]).eigenvals()\n```\n\n\n```python\nfrom sympy import *\ninit_printing()\nx, y ,z, nu, t = symbols('x y z nu t')\n\nbesselj(nu,z).rewrite(jn)\n```\n\n\n```python\nbesselj(nu, z)\n```\n\n\n```python\nIntegral(cos(x)**2,(x,0,pi))\n```\n\n\n```python\nintegrate(cos(x)**2,(x,0,pi))\n```\n\n\n```python\nfrom sympy import*\ninit_printing()\nx = symbols('x')\nexpr = x + 1\nexpr.subs(x,2)\n```\n\n\n```python\nx\n```\n\n\n```python\nEq((x + 1)**2, x**2 + 2*x + 1)\n```\n\n\n```python\nsolve(Eq((x + 1)**2, 2*x))\n```\n\n\n```python\n(x + 1)**2 == (x**2 + 2*x + 1)\n```\n\n\n\n\n False\n\n\n\n\n```python\nexpand((x + 1)**2) == (x**2 + 2*x + 1)\n```\n\n\n\n\n True\n\n\n\n\n```python\na = (x + 1)**2\nb = x**2 + 2*x + 1\na\n```\n\n\n```python\na - b\n```\n\n\n```python\nsimplify(a - b)\n```\n\n\n```python\nif simplify(a - b) == 0:\n print (\"right\")\nelse:\n print (\"no\")\n```\n\n right\n\n\n\n```python\na.equals(b)\n```\n\n\n\n\n True\n\n\n\n\n```python\na = cos(x)**2 - sin(x)**2\nb = cos(2*x)\na.equals(b)\n```\n\n\n\n\n True\n\n\n\n\n```python\nx^y\n```\n\n\n```python\ntype(Integer(1) + 1)\n```\n\n\n\n\n sympy.core.numbers.Integer\n\n\n\n\n```python\nInteger(3)\n```\n\n\n```python\ntype(1 + 1)\n```\n\n\n\n\n int\n\n\n\n\n```python\nInteger(3) / Integer(2)\n```\n\n\n```python\n3/2\n```\n\n\n```python\n# from __future__ import division\n1/2\n```\n\n\n```python\nRational(1,2)\n```\n\n\n```python\nx + 1/2\n```\n\n### \u4ee3\u5165\n\n\n```python\nfrom sympy import *\nx, y, z = symbols('x y z')\nexpr = cos(x) + 1\nexpr.subs(x, y)\n```\n\n\n```python\nexpr.subs(x, 0)\n```\n\n\n```python\nexpr = x ** y\nexpr\n```\n\n\n```python\nexpr = expr.subs(y, x**y)\nexpr\n\n```\n\n\n```python\nexpr = sin(2*x) + cos(2*x)\nexpand_trig(expr)\n```\n\n\n```python\nexpr.subs(sin(2*x), 2*sin(x)*cos(x))\n```\n\n### \u591a\u91cd\u4ee3\u5165\n\n\n```python\nfrom sympy import*\ninit_printing()\nx,y,z,t = symbols('x y z t')\nexpr = x**3 + 4*x*y - z\nexpr.subs([(x,2),(y,4),(z,0)])\n```\n\n\n```python\nexpr = x**4 - 4*x**3 + 4*x**2 - 2*x + 3\nexpr\n```\n\n\n```python\nreplacements = [(x**i, y**i) for i in range(5) if i % 2 ==0]\nprint (replacements)\ndisplay(expr.subs(replacements))\n```\n\n### sympify\n\n\n```python\nstr_expr = \"x**2+ 3*x - 1/2\"\nexpr = sympify(str_expr)\nexpr\n```\n\n\n```python\nexpr.subs(x,2)\n```\n\n### evalf\n\n\n```python\nsqrt(8).evalf()\n```\n\n\n```python\nE.evalf()\n```\n\n\n```python\npi.evalf(100)\n```\n\n\n```python\nexpr = cos(2*x)\nexpr.evalf(subs={x: 2.4})\n```\n\n\n```python\nexpr.subs(x,2.4).evalf()\n```\n\n\n```python\none = cos(1)**2 + sin(1)**2\n(one - 1).evalf(chop=True)\n```\n\n\n```python\none = cos(1)**2 + sin(1)**2\n(one - 1).evalf()\n```\n\n### lambdify\n`lambdify` \u3069\u3093\u306a\u3068\u304d\u306b\u5f79\u306b\u305f\u3064\u306e\u3060\u308d\u3046\u304b\u3002\n\n\u307e\u3042\u3044\u3044\u304b\u3002\u9032\u3082\u3046\u3002\n\n\n```python\nimport numpy\na = numpy.arange(10)\na\n```\n\n\n\n\n array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])\n\n\n\n\n```python\nrange (10)\n```\n\n\n\n\n range(0, 10)\n\n\n\n\n```python\nrange(0, 10)[3]\n```\n\n\n```python\na[3]\n```\n\n\n\n\n 3\n\n\n\n\n```python\nlist(a)\n```\n\n\n\n\n [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]\n\n\n\n\n```python\nlist(range(0, 10))\n```\n\n\n```python\nexpr = sin(x)\nf = lambdify(x, expr, \"numpy\")\nf(a)\n```\n\n\n\n\n array([ 0. , 0.84147098, 0.90929743, 0.14112001, -0.7568025 ,\n -0.95892427, -0.2794155 , 0.6569866 , 0.98935825, 0.41211849])\n\n\n\n\n```python\nf = lambdify(x, expr, \"math\")\nf(0.1)\n```\n\n\n```python\nlambdify?\n```\n\n### sympy_name:numerical_function pairs\n\n\n```python\ndef mysin(x):\n \"\"\"\n My sine. Note that this is only accurate for small x.\n \"\"\"\n return x\nf = lambdify(x, expr, {\"sin\":mysin})\nf(0.1)\n```\n\n### printing\n\nstr, srepr, ASCII pretty printer, Unicode pretty printer, LaTeX, MathML, Dot\n\ninit_session() works\n1. frm sympy import *\n2. init_printing()\n3. common symbols\n\n\n\n```python\nfrom sympy import *\ninit_session()\n```\n\n IPython console for SymPy 1.7.1 (Python 3.7.10-64-bit) (ground types: python)\n \n These commands were executed:\n >>> from __future__ import division\n >>> from sympy import *\n >>> x, y, z, t = symbols('x y z t')\n >>> k, m, n = symbols('k m n', integer=True)\n >>> f, g, h = symbols('f g h', cls=Function)\n >>> init_printing()\n \n Documentation can be found at https://docs.sympy.org/1.7.1/\n \n\n\n### continued fractions\n\n\n\n```python\nfrom sympy import *\ninit_printing()\nx,y,z = symbols('x y z')\n\ndef list_to_frac(l):\n expr = Integer(0)\n for i in reversed(l[1:]):\n expr += i\n expr = 1 / expr\n return l[0] + expr\nfrac = list_to_frac([x,y,z])\nfrac\n```\n\n\n```python\nlist_to_frac([1,2,3,4])\n```\n\n\n```python\nsyms = symbols('a0:5')\nsyms\n```\n\n\n```python\na0, a1, a2, a3, a4 = syms\n\nfrac = list_to_frac(syms)\nfrac\n```\n\n\n```python\n# \u5b9f\u9a13\r\nfrom sympy import *\r\ninit_printing()\r\na11 = symbols('a11')\r\ndisplay (a11)\r\ndisplay(symbols('a_11'))\n```\n\n\n```python\n# \u5b9f\u9a13\r\nfrom sympy import *\r\ninit_printing()\r\na_seq = [-1, 3, 23, 8]\r\nn, r = symbols('n, r')\r\na_n = Function('a')(n)\r\nterms = 4\r\nshort_expr = Sum(a_n * r**n, (n, 0, terms - 1))\r\ndisplay(short_expr)\r\n# coeffed_short_expr = short_expr.doit().subs(\r\n# (a_n.subs(n, i), a_seq[i]) for i in range(terms)) # 8*r**3 + 23*r**2 + 3*r - 1\r\n# func_short_expr = lambdify(r, coeffed_short_expr, 'numpy')\n```\n\n\n```python\n# \u5b9f\u9a13\r\nfrom sympy import *\r\ninit_printing()\r\na = symbols('a', shape=(3,3))\r\ndisplay(a)\n```\n\n### cancel\n\n\n```python\nfrac=cancel(frac)\nfrac\n```\n\n\n\n\n frac\n\n\n\n\n```python\n%%script false\nl=[]\nfrac = apart(frac, a0)\nfrac\n```\n\n\n```python\n%%script false\nl.append(a0)\nfrac = 1/(frac - a0)\nfrac\n```\n\n\n```python\n%%script false\nfrac = apart(frac, a1)\nfrac\n```\n\n\n```python\n%%script false\nl.append(a1)\nfrac = 1/(frac - a1)\nfrac = apart(frac, a2)\nfrac\n```\n\n\n```python\n%%script false\nl.append(a2)\nfrac = 1/(frac - a2)\nfrac = apart(frac, a3)\nfrac\n```\n\n\n```python\n%%script false\nl.append(a3)\nfrac = 1/(frac - a3)\nfrac = apart(frac, a4)\nfrac\n```\n\n\n```python\n%%script false\nl.append(a4)\nlist_to_frac(l)\n```\n\n\n```python\n%%script false\r\nl\n```\n\n### random\n\n\n```python\nimport random\nl = list(symbols('a0:5'))\nrandom.shuffle(l)\norig_frac = frac = cancel(list_to_frac(l))\ndel l\n```\n\n\n```python\nfrac\n```\n\n\n```python\nl=[]\nfrac = apart(frac, a1)\nfrac\n```\n\n\n```python\nl.append(a1)\nfrac = 1/(frac - a1)\nfrac\n```\n\n\n```python\nfrac = apart(frac, a3)\nfrac\n```\n\n\n```python\nl.append(a3)\nfrac = 1/(frac - a3)\nfrac\n```\n\n\n```python\nfrac = apart(frac, a0)\nfrac\n```\n\n\n```python\nl.append(a0)\nfrac = 1/(frac - a0)\nfrac\n```\n\n\n```python\nfrac = apart(frac, a2)\nfrac\n```\n\n\n```python\nl.append(a2)\nfrac = 1/(frac - a2)\nfrac\n```\n\n\n```python\nfrac = apart(frac, a4)\nfrac\n```\n\n\n```python\nl.append(a4)\nlist_to_frac(l)\n```\n\n\n```python\nl\n```\n\n\n```python\norig_frac\n```\n\n\n```python\ncancel(list_to_frac(l))\n```\n", "meta": {"hexsha": "04cf9ca393af91825c5cae311d22363d6e5a7069", "size": 496822, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "mathwithpython01.ipynb", "max_stars_repo_name": "kalz2q/-yjupyternotebooks", "max_stars_repo_head_hexsha": "ba37ac7822543b830fe8602b3f611bb617943463", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-09-16T03:45:19.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-16T03:45:19.000Z", "max_issues_repo_path": "mathwithpython01.ipynb", "max_issues_repo_name": "kalz2q/-yjupyternotebooks", "max_issues_repo_head_hexsha": "ba37ac7822543b830fe8602b3f611bb617943463", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "mathwithpython01.ipynb", "max_forks_repo_name": "kalz2q/-yjupyternotebooks", "max_forks_repo_head_hexsha": "ba37ac7822543b830fe8602b3f611bb617943463", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 66.0579710145, "max_line_length": 7998, "alphanum_fraction": 0.7152622066, "converted": true, "num_tokens": 8732, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982315512489, "lm_q2_score": 0.6825737473266735, "lm_q1q2_score": 0.4421700664215281}} {"text": "# Columns and nullspaces in a matrix\n\n+ This notebook is part of lecture 6 *Columnspace and nullspace* in the OCW MIT course 18.06 by Prof Gilbert Strang [1]\n+ Created by me, Dr Juan H Klopper\n + Head of Acute Care Surgery\n + Groote Schuur Hospital\n + University Cape Town\n + Email me with your thoughts, comments, suggestions and corrections \n
    Linear Algebra OCW MIT18.06 IPython notebook [2] study notes by Dr Juan H Klopper is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.\n\n+ [1] OCW MIT 18.06\n+ [2] Fernando P\u00e9rez, Brian E. Granger, IPython: A System for Interactive Scientific Computing, Computing in Science and Engineering, vol. 9, no. 3, pp. 21-29, May/June 2007, doi:10.1109/MCSE.2007.53. URL: http://ipython.org\n\n\n```python\nfrom IPython.core.display import HTML, Image\ncss_file = 'style.css'\nHTML(open(css_file, 'r').read())\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n```python\n#import numpy as np\nfrom sympy import init_printing, Matrix, symbols\n#import matplotlib.pyplot as plt\n#import seaborn as sns\n#from IPython.display import Image\nfrom warnings import filterwarnings\n\ninit_printing(use_latex = 'mathjax')\n%matplotlib inline\nfilterwarnings('ignore')\n```\n\n# Columnspace and nullspace of a matrix\n\n## Columnspaces of matrices\n\n* We saw in the previous lecture that columns of a matrix can form vectors\n* Consider now the LU-decomposition of *A*\n$$ PA = PLU $$\n\n* The union P∪L (all vectors in P or L or both) is NOT a subspace\n* The intersection P∩L (or vectors in P and L) is a subspace (because their intersection is only the zero vector)\n\n* The intersection of any two subspaces is a subspace\n\n* Consider the following example matrix\n\n\n```python\nA = Matrix([[1, 1, 2], [2, 1, 3], [3, 1, 4], [4, 1, 5]])\nA\n```\n\n\n\n\n$$\\left[\\begin{matrix}1 & 1 & 2\\\\2 & 1 & 3\\\\3 & 1 & 4\\\\4 & 1 & 5\\end{matrix}\\right]$$\n\n\n\n* Each of the column spaces are vectors (column space) in ℝ4\n\n* The linear combinations of all the column vectors form a subspace\n* Is it the whole *V* = ℝ4, though?\n\n* The reason why we ask is because we want to bring it back to a system of linear equations and ask the question: Is there (always) a solution to the following:\n$$ {A} \\overline {x}= \\overline {b} $$\n\n\n* Thus, which right-hand sides *b* are allowed?\n* In our example above we are in ℝ4 and we ask if linear combination of all of them fill ℝ4\n\n* From our example above some right-hand sides will be allowed (they form a subspace)\n* Let's look at an example for **b**\n\n\n```python\nx1, x2, x3 = symbols('x1, x2, x3')\nvec_x = Matrix([x1, x2, x3])\nb = Matrix([1, 2, 3, 4])\nA, vec_x, b\n```\n\n\n\n\n$$\\left ( \\left[\\begin{matrix}1 & 1 & 2\\\\2 & 1 & 3\\\\3 & 1 & 4\\\\4 & 1 & 5\\end{matrix}\\right], \\quad \\left[\\begin{matrix}x_{1}\\\\x_{2}\\\\x_{3}\\end{matrix}\\right], \\quad \\left[\\begin{matrix}1\\\\2\\\\3\\\\4\\end{matrix}\\right]\\right )$$\n\n\n\n\n```python\nA * vec_x\n```\n\n\n\n\n$$\\left[\\begin{matrix}x_{1} + x_{2} + 2 x_{3}\\\\2 x_{1} + x_{2} + 3 x_{3}\\\\3 x_{1} + x_{2} + 4 x_{3}\\\\4 x_{1} + x_{2} + 5 x_{3}\\end{matrix}\\right]$$\n\n\n\n* You can do the row multiplication, but it's easy to see from above we are asking about linear combinations of the columns, i.e. how many (*x*1) of column 1 plus how many (*x*2) of column 2 plus how many (*x*3) of column 3 equals **b**?\n* Well, since **b** is the same as the first column, **x** would be\n$$ \\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \\end{bmatrix} $$\n\n* So we can solve for all values of **b** if **b** is in the column space\n\n### Linear independence\n\n* We really need to know if the columns above are linearly independent\n* We note that column three above is a linear combination of the first two, so adds nothing new\n* Actually, we could also throw away the first one because it is column 3 plus -1 times column 2\n* Same for column 2\n* We thus have two columns left and we say that the column space is of dimension 2 (a 2-dimensional subspace of ℝ4)\n\n## The nullspace\n\n* It contains all solutions **x** for A**x**=0\n* This solution(s) is in ℝ3\n\n\n```python\nzero_b = Matrix([0, 0, 0, 0])\nA, vec_x, zero_b\n```\n\n\n\n\n$$\\left ( \\left[\\begin{matrix}1 & 1 & 2\\\\2 & 1 & 3\\\\3 & 1 & 4\\\\4 & 1 & 5\\end{matrix}\\right], \\quad \\left[\\begin{matrix}x_{1}\\\\x_{2}\\\\x_{3}\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0\\\\0\\\\0\\\\0\\end{matrix}\\right]\\right )$$\n\n\n\n* Some solutions would be\n$$ \\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\end{bmatrix} $$\n$$ \\begin{bmatrix} 1 \\\\ 1 \\\\ -1 \\end{bmatrix} $$\n$$ \\begin{bmatrix} 2 \\\\ 2 \\\\ -2 \\end{bmatrix} $$\n* In fact, we have:\n$$ {c} \\begin{bmatrix} 1 \\\\ 1 \\\\ -1 \\end{bmatrix} $$\n* It is thus a line\n* The nullspace is a line in ℝ3\n\n* **PLEASE** remember, for any space the rules of addition and scalar multiplication must hold for vectors to remain in that space\n\n\n```python\n\n```\n", "meta": {"hexsha": "5dc6e612f67fad9f59800798959b4a66e80af9a0", "size": 13729, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Beginners_Guide_Math_LinAlg/Math/I_07_Column_and_null_spaces.ipynb", "max_stars_repo_name": "DanielMabadeje/Artificial-Intelligence-Deep-Learning-Machine-Learning-Tutorials", "max_stars_repo_head_hexsha": "7adab3877fc1d3f1d5f57e6c1743dae8f76f72c5", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 3266, "max_stars_repo_stars_event_min_datetime": "2017-08-06T16:51:46.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-30T07:34:24.000Z", "max_issues_repo_path": "Beginners_Guide_Math_LinAlg/Math/I_07_Column_and_null_spaces.ipynb", "max_issues_repo_name": "hashDanChibueze/Artificial-Intelligence-Deep-Learning-Machine-Learning-Tutorials", "max_issues_repo_head_hexsha": "bef2c415d154a052c00e99a05f0870af7a5819ac", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 150, "max_issues_repo_issues_event_min_datetime": "2017-08-28T14:59:36.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-11T23:21:35.000Z", "max_forks_repo_path": "Beginners_Guide_Math_LinAlg/Math/I_07_Column_and_null_spaces.ipynb", "max_forks_repo_name": "hashDanChibueze/Artificial-Intelligence-Deep-Learning-Machine-Learning-Tutorials", "max_forks_repo_head_hexsha": "bef2c415d154a052c00e99a05f0870af7a5819ac", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 1449, "max_forks_repo_forks_event_min_datetime": "2017-08-06T17:40:59.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-31T12:03:24.000Z", "avg_line_length": 30.5768374165, "max_line_length": 708, "alphanum_fraction": 0.5009104815, "converted": true, "num_tokens": 2365, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123243, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.44217004877322075}} {"text": " \n#### Procesamiento Digital de Se\u00f1ales\n\n# Trabajo Pr\u00e1ctico N\u00ba0\n#### Gisela Farace\n\n\n# Introducci\u00f3n\nJupyter Notebook es una herramienta para la confecci\u00f3n de reportes t\u00e9cnicos, dado que permite la interacci\u00f3n en el mismo ambiente de: \n1. un procesador de texto elemental (formato Markdown) que permite resaltar texto, en forma de *it\u00e1lica* o **negrita** de manera muy legible (haciendo doble click en este texto podr\u00e1s ver el c\u00f3digo fuente estilo Markdown). Cuenta con estilos predefinidos:\n\n# T\u00edtulo 1\n## T\u00edtulo 2\n### T\u00edtulo 3\n\ny tambi\u00e9n la capacidad de incluir enlaces a otras p\u00e1ginas, como por ejemplo [esta p\u00e1gina](https://medium.com/ibm-data-science-experience/markdown-for-jupyter-notebooks-cheatsheet-386c05aeebed) donde encontrar\u00e1s m\u00e1s funcionalidades del lenguaje **Markdown**\n\n2. capacidad para incluir lenguaje matem\u00e1tico estilo LaTex, tanto de forma presentada\n\n\\begin{equation}\nT(z) = \\frac{Y(z)}{X(z)} = \\frac{ b_2 \\, z^{-2} + b_1 \\, z^{-1} + b_0 }\n{a_2 \\, z^{-2} + a_1 \\, z^{-1} + a_0}\n\\end{equation}\n\ncomo *inline* en el propio p\u00e1rrafo $y[k] = \\frac{1}{a_0} \\left( \\sum_{m=0}^{M} b_m \\; x[k-m] - \\sum_{n=1}^{N} a_n \\; y[k-n] \\right) $\n\n3. La posibilidad de incluir scripts en Python, como los que usaremos para las simulaciones en los TPs de la materia. En este caso usaremos el *testbench0.py* como ejemplo. Una vez que lo probamos y estamos seguros que funciona de forma esperada en *Spyder*, podemos incluir los resultados de la simulaci\u00f3n de manera casi transparente. Solo tenemos que agregar una celda de c\u00f3digo donde incluimos el c\u00f3digo, y los resultados directamente quedan incluidos en este documento.\n\n\n```python\n# M\u00f3dulos para Jupyter\nimport warnings\nwarnings.filterwarnings('ignore')\n\nimport numpy as np\nimport matplotlib as mpl\n#%% Inicializaci\u00f3n de librer\u00edas\n# Setup inline graphics: Esto lo hacemos para que el tama\u00f1o de la salida, \n# sea un poco m\u00e1s adecuada al tama\u00f1o del documento\nmpl.rcParams['figure.figsize'] = (50,20)\n\nimport matplotlib.pyplot as plt\nimport pdsmodulos as pds\n\n#%% Esto tiene que ver con cuestiones de presentaci\u00f3n de los gr\u00e1ficos,\n# NO ES IMPORTANTE\nfig_sz_x = 14\nfig_sz_y = 13\nfig_dpi = 80 # dpi\n\nfig_font_family = 'Ubuntu'\nfig_font_size = 16\n\nplt.rcParams.update({'font.size':fig_font_size})\nplt.rcParams.update({'font.family':fig_font_family})\n\n##############################################\n#%% A partir de aqu\u00ed comienza lo IMPORTANTE #\n#############################################\n\ndef my_testbench( sig_type ):\n \n # Datos generales de la simulaci\u00f3n\n fs = 1000.0 # frecuencia de muestreo (Hz)\n N = 1000 # cantidad de muestras\n \n ts = 1/fs # tiempo de muestreo\n df = fs/N # resoluci\u00f3n espectral\n \n # grilla de sampleo temporal\n tt = np.linspace(0, (N-1)*ts, N).flatten()\n \n # grilla de sampleo frecuencial\n ff = np.linspace(0, (N-1)*df, N).flatten()\n\n # Concatenaci\u00f3n de matrices:\n # guardaremos las se\u00f1ales creadas al ir poblando la siguiente matriz vac\u00eda\n x = np.array([], dtype=np.float).reshape(N,0)\n ii = 0\n \n # estructuras de control de flujo\n if sig_type['tipo'] == 'senoidal':\n \n \n # calculo cada senoidal de acuerdo a sus par\u00e1metros\n for this_freq in sig_type['frecuencia']:\n # prestar atenci\u00f3n que las tuplas dentro de los diccionarios tambi\u00e9n pueden direccionarse mediante \"ii\"\n aux = sig_type['amplitud'][ii] * np.sin( 2*np.pi*this_freq*tt + sig_type['fase'][ii] )\n # para concatenar horizontalmente es necesario cuidar que tengan iguales FILAS\n x = np.hstack([x, aux.reshape(N,1)] )\n ii += 1\n \n elif sig_type['tipo'] == 'ruido':\n \n # calculo cada se\u00f1al de ruido incorrelado (blanco), Gausiano de acuerdo a sus par\u00e1metros\n # de varianza\n for this_var in sig_type['varianza']:\n aux = np.sqrt(this_var) * np.random.randn(N,1)\n # para concatenar horizontalmente es necesario cuidar que tengan iguales FILAS\n x = np.hstack([x, aux] )\n \n # Podemos agregar alg\u00fan dato extra a la descripci\u00f3n de forma program\u00e1tica\n # {0:.3f} significa 0: primer argunmento de format\n # .3f formato flotante, con 3 decimales\n # $ ... $ indicamos que incluiremos sintaxis LaTex: $\\hat{{\\sigma}}^2$\n sig_props['descripcion'] = [ sig_props['descripcion'][ii] + ' - $\\hat{{\\sigma}}^2$ :{0:.3f}'.format( np.var(x[:,ii])) for ii in range(0,len(sig_props['descripcion'])) ]\n \n else:\n \n print(\"Tipo de se\u00f1al no implementado.\") \n return\n \n #%% Presentaci\u00f3n gr\u00e1fica de los resultados\n \n plt.figure(1)\n line_hdls = plt.plot(tt, x)\n plt.title('Se\u00f1al: ' + sig_type['tipo'] )\n plt.xlabel('tiempo [segundos]')\n plt.ylabel('Amplitud [V]')\n # plt.grid(which='both', axis='both')\n \n # presentar una leyenda para cada tipo de se\u00f1al\n axes_hdl = plt.gca()\n \n # este tipo de sintaxis es *MUY* de Python\n axes_hdl.legend(line_hdls, sig_type['descripcion'], loc='upper right' )\n \n plt.show()\n\n```\n\nDado que nuestro *testbench* ha sido desarrollado de manera funcional, llamando a la funci\u00f3n *my_testbench()* con diferentes par\u00e1metros, podemos lograr funcionalidades diferentes, como mostramos a continuaci\u00f3n primero con una senoidal:\n\n\n```python\nf0 = 500\nsig_props = { 'tipo': 'senoidal', \n 'frecuencia': (f0, f0, f0), # Uso de tuplas para las frecuencias \n 'amplitud': (1, 1, 1),\n 'fase': (0, 0, 0)\n } \n# Como tambi\u00e9n puedo agregar un campo descripci\u00f3n de manera program\u00e1tica\n# este tipo de sintaxis es *MUY* de Python\nsig_props['descripcion'] = [ str(a_freq) + ' Hz' for a_freq in sig_props['frecuencia'] ]\n \n# Invocamos a nuestro testbench exclusivamente: \nmy_testbench( sig_props )\n```\n\nY ahora con una se\u00f1al aleatoria, en este caso ruido blanco Gaussiano incorrelado de varianza $\\sigma^2$:\n\n\n```python\n# Usar CTRL+1 para comentar o descomentar el bloque de abajo.\nsig_props = { 'tipo': 'ruido', \n 'varianza': (0.5, 1, 1) # Uso de tuplas para las frecuencias \n } \nsig_props['descripcion'] = [ '$\\sigma^2$ = ' + str(a_var) for a_var in sig_props['varianza'] ]\n \n# Invocamos a nuestro testbench exclusivamente: \nmy_testbench( sig_props )\n\n```\n\nComo puede verse en la figura anterior, al samplear una distribuci\u00f3n estad\u00edstica de media nula y varianza $\\sigma^2=1$, obtenemos realizaciones cuyo par\u00e1metro $\\sigma^2$ estimado, es decir $\\hat\\sigma^2$, tienen una desviaci\u00f3n respecto al verdadero valor (sesgo). Nos ocuparemos de estudiar el sesgo y la varianza de algunos estimadores cuando veamos **Estimaci\u00f3n Espectral**.\n\n# Una vez terminado ...\nUna vez que hayas termiando con la confecci\u00f3n del documento, podemos utilizar una ventaja muy importante de este tipo de documentos que es la posibilidad de compartirlos *online* mediante la [p\u00e1gina de nbviewer](http://nbviewer.jupyter.org/). Para ello es necesario que tu notebook y todos los recursos asociados est\u00e9n alojados en un repositorio de [Github](https://github.com/). Como ejemplo, pod\u00e9s ver este mismo documento disponible [online](http://nbviewer.jupyter.org/github/marianux/pdstestbench/blob/master/notebook0.ipynb).\n\n# Un gatito\n\n\n\n\n\n# Modificaciones\n- Cambie la frecuencia y amplitud del 1er gr\u00e1fico\n- Cambie la varianza del 2do gr\u00e1fico\n- Inclu\u00ed la imagen de un gatito\n\n", "meta": {"hexsha": "06956d5a290646cb1034be98f80c33e6532fb108", "size": 901755, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebook0.ipynb", "max_stars_repo_name": "gfarace/prueba_jupyter", "max_stars_repo_head_hexsha": "4ea8cd499566612afdfc2ee48aa539e32c02e605", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebook0.ipynb", "max_issues_repo_name": "gfarace/prueba_jupyter", "max_issues_repo_head_hexsha": "4ea8cd499566612afdfc2ee48aa539e32c02e605", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebook0.ipynb", "max_forks_repo_name": "gfarace/prueba_jupyter", "max_forks_repo_head_hexsha": "4ea8cd499566612afdfc2ee48aa539e32c02e605", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 3142.0034843206, "max_line_length": 841648, "alphanum_fraction": 0.9610093651, "converted": true, "num_tokens": 2080, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.7549149813536518, "lm_q1q2_score": 0.4417016269153324}} {"text": "# CAM Methods Benchmark\n\n**Goal:** to compare CAM methods using regular explaining metrics. \n**Author:** lucas.david@ic.unicamp.br\n\nUse GPUs if you are running *Score-CAM* or *Quantitative Results* sections.\n\n\n```python\n#@title\n \nfrom google.colab import drive\ndrive.mount('/content/drive')\n```\n\n\n```python\nimport tensorflow as tf\n\n# base_dir = '/content/drive/MyDrive/'\nbase_dir = '/home/ldavid/Workspace'\n# data_dir = '/root/tensorflow_datasets'\ndata_dir = '/home/ldavid/Workspace/datasets/'\n\nclass Config:\n seed = 218402\n\n class data:\n path = '/root/tensorflow_datasets/amazon-from-space'\n size = (256, 256)\n shape = (*size, 3)\n batch_size = 32\n shuffle_buffer_size = 8 * batch_size\n prefetch_buffer_size = tf.data.experimental.AUTOTUNE\n train_shuffle_seed = 120391\n shuffle = False\n\n class model:\n backbone = tf.keras.applications.ResNet101V2\n last_spatial_layer = 'post_relu'\n # backbone = tf.keras.applications.EfficientNetB6\n # last_spatial_layer = 'eb6'\n # backbone = tf.keras.applications.VGG16\n # last_spatial_layer = 'block5_pool'\n\n gap_layer_name = 'avg_pool'\n include_top = False\n classifier_activation = None\n\n custom = True\n fine_tune_layers = 0.6\n freeze_batch_norm = False\n\n weights = f'{base_dir}/logs/amazon-from-space/resnet101-sw-ce-fine-tune/weights.h5'\n \n class training:\n valid_size = 0.3\n\n class explaining:\n noise = tf.constant(.2)\n repetitions = tf.constant(8)\n\n score_cam_activations = 'all'\n\n \u03bb_pos = tf.constant(1.)\n \u03bb_neg = tf.constant(1.)\n \u03bb_bg = tf.constant(1.)\n\n report = f'{base_dir}/logs/amazon-from-space/resnet101-sw-ce-fine-tune/cam-score.txt'\n```\n\n\n```python\npreprocess = tf.keras.applications.resnet_v2.preprocess_input\ndeprocess = lambda x: (x + 1) * 127.5\n\n# preprocess = tf.keras.applications.res.preprocess_input\n# deprocess = lambda x: x\n\n# preprocess = tf.keras.applications.vgg16.preprocess_input\n# deprocess = lambda x: x[..., ::-1] + [103.939, 116.779, 123.68]\n\nto_image = lambda x: tf.cast(tf.clip_by_value(deprocess(x), 0, 255), tf.uint8)\nmasked = lambda x, maps: x * tf.image.resize(maps, Config.data.size)\n```\n\n## Setup\n\n\n```python\n! pip -qq install tensorflow_addons\n```\n\n\n```python\nimport os\nimport shutil\nfrom time import time\nfrom math import ceil\n\nimport numpy as np\nimport pandas as pd\nimport tensorflow_addons as tfa\nimport tensorflow_datasets as tfds\nimport matplotlib.pyplot as plt\nimport seaborn as sns\n\nfrom tensorflow.keras import callbacks\n```\n\n\n```python\nfor d in tf.config.list_physical_devices('GPU'):\n print(d)\n print(f'Setting device {d} to memory-growth mode.')\n try:\n tf.config.experimental.set_memory_growth(d, True)\n except Exception as e:\n print(e)\n```\n\n PhysicalDevice(name='/physical_device:GPU:0', device_type='GPU')\n Setting device PhysicalDevice(name='/physical_device:GPU:0', device_type='GPU') to memory-growth mode.\n Physical devices cannot be modified after being initialized\n\n\n\n```python\nR = tf.random.Generator.from_seed(Config.seed, alg='philox')\nC = np.asarray(sns.color_palette(\"Set1\", 21))\nCMAP = sns.color_palette(\"Set1\", 21, as_cmap=True)\n\nsns.set_style(\"whitegrid\", {'axes.grid' : False})\n```\n\n\n```python\nnp.set_printoptions(linewidth=120)\n```\n\n\n```python\ndef normalize(x, reduce_min=True, reduce_max=True):\n if reduce_min: x -= tf.reduce_min(x, axis=(-3, -2), keepdims=True)\n if reduce_max: x = tf.math.divide_no_nan(x, tf.reduce_max(x, axis=(-3, -2), keepdims=True))\n\n return x\n\ndef visualize(\n image,\n title=None,\n rows=2,\n cols=None,\n i0=0,\n figsize=(9, 4),\n cmap=None,\n full=True\n):\n if image is not None:\n if isinstance(image, (list, tuple)) or len(image.shape) > 3: # many images\n if full: plt.figure(figsize=figsize)\n cols = cols or ceil(len(image) / rows)\n for ix in range(len(image)):\n plt.subplot(rows, cols, i0+ix+1)\n visualize(image[ix],\n cmap=cmap,\n title=title[ix] if title is not None and len(title) > ix else None)\n if full: plt.tight_layout()\n return\n\n if isinstance(image, tf.Tensor): image = image.numpy()\n if image.shape[-1] == 1: image = image[..., 0]\n plt.imshow(image, cmap=cmap)\n \n if title is not None: plt.title(title)\n plt.axis('off')\n```\n\n\n```python\ndef observe_labels(probs, labels, ix):\n p = probs[ix]\n l = labels[ix]\n s = tf.argsort(p, direction='DESCENDING')\n\n d = pd.DataFrame({\n 'idx': s,\n 'label': tf.gather(CLASSES, s).numpy().astype(str),\n 'predicted': tf.gather(p, s).numpy().round(2),\n 'ground-truth': tf.gather(l, s).numpy()\n })\n\n return d[(d['ground-truth']==1) | (d['predicted'] > 0.05)]\n\n\ndef plot_heatmap(i, m):\n plt.imshow(i)\n plt.imshow(m, cmap='jet', alpha=0.5)\n plt.axis('off')\n```\n\n## Related Work\n\n### Summary\n\n\n```python\n#@title\n\nd = [\n ['1512.04150', 'CAM', 'GoogleNet', 'ILSVRC-15 val', 56.4, 43.00, None, None, None, None],\n ['1512.04150', 'CAM', 'VGG-16', 'ILSVRC-15 val', 57.2, 45.14, None, None, None, None],\n ['1512.04150', 'Backprop', 'GoogleNet', 'ILSVRC-15 val', 61.31, 50.55, None, None, None, None],\n ['1512.04150', 'Backprop', 'GoogleNet', 'ILSVRC-15 test', None, 37.10, None, None, None, None],\n \n ['1610.02391', 'CAM', 'VGG-16', 'ILSVRC-15 val', 57.2, 45.14, None, None, None, None, None],\n ['1610.02391', 'Grad-CAM', 'VGG-16', 'ILSVRC-15 val', 56.51, 46.41, None, None, None, None, None],\n ['1610.02391', 'CAM', 'GoogleNet', 'ILSVRC-15 val', 60.09, 49.34, None, None, None, None, None],\n ['1610.02391', 'Grad-CAM', 'GoogleNet', 'ILSVRC-15 val', 60.09, 49.34, None, None, None, None, None],\n\n ['1710.11063', 'Grad-CAM', 'VGG-16', 'ILSVRC-2012 val', None, None, 46.56, 13.42, 29.28, None, None],\n ['1710.11063', 'Grad-CAM++', 'VGG-16', 'ILSVRC-2012 val', None, None, 36.84, 17.05, 70.72, None, None],\n ['1710.11063', 'Grad-CAM', 'AlexNet', 'ILSVRC-2012 val', None, None, 82.86, 3.16, 13.44, None, None],\n ['1710.11063', 'Grad-CAM++', 'AlexNet', 'ILSVRC-2012 val', None, None, 62.75, 8.24, 86.56, None, None],\n ['1710.11063', 'Grad-CAM', 'VGG-16', 'Pascal 2007 val', None, None, 28.54, 21.43, 39.44, None, None],\n ['1710.11063', 'Grad-CAM++', 'VGG-16', 'Pascal 2007 val', None, None, 19.53, 18.96, 61.47, None, None],\n ['1710.11063', 'Grad-CAM', 'AlexNet', 'Pascal 2007 val', None, None, 45.82, 14.38, 27.21, None, None],\n ['1710.11063', 'Grad-CAM++', 'AlexNet', 'Pascal 2007 val', None, None, 29.16, 19.76, 72.79, None, None],\n ['1710.11063', 'Grad-CAM', 'ResNet-50', 'ILSVRC-2012 val', None, None, 30.36, 22.11, 39.49, None, None],\n ['1710.11063', 'Grad-CAM++', 'ResNet-50', 'ILSVRC-2012 val', None, None, 28.90, 22.16, 60.51, None, None],\n ['1710.11063', 'Grad-CAM', 'ResNet-50', 'Pascal 2007 val', None, None, 20.86, 21.99, 41.39, None, None],\n ['1710.11063', 'Grad-CAM++', 'ResNet-50', 'Pascal 2007 val', None, None, 16.19, 19.52, 58.61, None, None],\n\n ['1710.11063', 'Grad-CAM', 'VGG-16', 'Pascal 2012 val', None, None, None, None, None, 0.33, None],\n ['1710.11063', 'Grad-CAM++', 'VGG-16', 'Pascal 2012 val', None, None, None, None, None, 0.34, None],\n\n ['1910.01279', 'Backprop Vanilla', 'VGG-16', 'ILSVRC-2012 val', None, None, None, None, None, None, 41.3],\n ['1910.01279', 'Backprop Smooth', 'VGG-16', 'ILSVRC-2012 val', None, None, None, None, None, None, 42.4],\n ['1910.01279', 'Backprop Integraded', 'VGG-16', 'ILSVRC-2012 val', None, None, None, None, None, None, 44.7],\n\n ['1910.01279', 'Grad-CAM', 'VGG-16', 'ILSVRC-2012 val', None, None, 47.80, 19.60, None, None, 48.1],\n ['1910.01279', 'Grad-CAM++', 'VGG-16', 'ILSVRC-2012 val', None, None, 45.50, 18.90, None, None, 49.3],\n ['1910.01279', 'Score-CAM', 'VGG-16', 'ILSVRC-2012 val', None, None, 31.50, 30.60, None, None, 63.7],\n]\n\nd = pd.DataFrame(\n d,\n columns=[\n 'Source', 'Method', 'Arch', 'Dataset',\n 'Loc_Error_T-1', 'Loc_Error_T-5',\n 'Avg_Drop', 'Incr_in_confidence', 'Win_%', 'mLoc_I^c(s=0)',\n 'E-Pointing_Game'\n ]\n)\n```\n\n\n```python\n#@title\n\n(d.groupby(['Dataset', 'Method'], as_index=False)\n .mean()\n .replace(np.nan, '', regex=True))\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    DatasetMethodLoc_Error_T-1Loc_Error_T-5Avg_DropIncr_in_confidenceWin_%mLoc_I^c(s=0)E-Pointing_Game
    0ILSVRC-15 testBackprop37.1
    1ILSVRC-15 valBackprop61.3150.55
    2ILSVRC-15 valCAM57.722545.655
    3ILSVRC-15 valGrad-CAM58.347.875
    4ILSVRC-2012 valBackprop Integraded44.7
    5ILSVRC-2012 valBackprop Smooth42.4
    6ILSVRC-2012 valBackprop Vanilla41.3
    7ILSVRC-2012 valGrad-CAM51.89514.572527.40333348.1
    8ILSVRC-2012 valGrad-CAM++43.497516.587572.59666749.3
    9ILSVRC-2012 valScore-CAM31.530.663.7
    10Pascal 2007 valGrad-CAM31.7419.26666736.013333
    11Pascal 2007 valGrad-CAM++21.62666719.41333364.29
    12Pascal 2012 valGrad-CAM0.33
    13Pascal 2012 valGrad-CAM++0.34
    \n
    \n\n\n\n\n```python\n#@title Full Report\n\nd.replace(np.nan, '', regex=True)\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    SourceMethodArchDatasetLoc_Error_T-1Loc_Error_T-5Avg_DropIncr_in_confidenceWin_%mLoc_I^c(s=0)E-Pointing_Game
    01512.04150CAMGoogleNetILSVRC-15 val56.443.0
    11512.04150CAMVGG-16ILSVRC-15 val57.245.14
    21512.04150BackpropGoogleNetILSVRC-15 val61.3150.55
    31512.04150BackpropGoogleNetILSVRC-15 test37.1
    41610.02391CAMVGG-16ILSVRC-15 val57.245.14
    51610.02391Grad-CAMVGG-16ILSVRC-15 val56.5146.41
    61610.02391CAMGoogleNetILSVRC-15 val60.0949.34
    71610.02391Grad-CAMGoogleNetILSVRC-15 val60.0949.34
    81710.11063Grad-CAMVGG-16ILSVRC-2012 val46.5613.4229.28
    91710.11063Grad-CAM++VGG-16ILSVRC-2012 val36.8417.0570.72
    101710.11063Grad-CAMAlexNetILSVRC-2012 val82.863.1613.44
    111710.11063Grad-CAM++AlexNetILSVRC-2012 val62.758.2486.56
    121710.11063Grad-CAMVGG-16Pascal 2007 val28.5421.4339.44
    131710.11063Grad-CAM++VGG-16Pascal 2007 val19.5318.9661.47
    141710.11063Grad-CAMAlexNetPascal 2007 val45.8214.3827.21
    151710.11063Grad-CAM++AlexNetPascal 2007 val29.1619.7672.79
    161710.11063Grad-CAMResNet-50ILSVRC-2012 val30.3622.1139.49
    171710.11063Grad-CAM++ResNet-50ILSVRC-2012 val28.922.1660.51
    181710.11063Grad-CAMResNet-50Pascal 2007 val20.8621.9941.39
    191710.11063Grad-CAM++ResNet-50Pascal 2007 val16.1919.5258.61
    201710.11063Grad-CAMVGG-16Pascal 2012 val0.33
    211710.11063Grad-CAM++VGG-16Pascal 2012 val0.34
    221910.01279Backprop VanillaVGG-16ILSVRC-2012 val41.3
    231910.01279Backprop SmoothVGG-16ILSVRC-2012 val42.4
    241910.01279Backprop IntegradedVGG-16ILSVRC-2012 val44.7
    251910.01279Grad-CAMVGG-16ILSVRC-2012 val47.819.648.1
    261910.01279Grad-CAM++VGG-16ILSVRC-2012 val45.518.949.3
    271910.01279Score-CAMVGG-16ILSVRC-2012 val31.530.663.7
    \n
    \n\n\n\n**Localization Error**\n\nAs defined in http://image-net.org/challenges/LSVRC/2015/index#maincomp:\n\nLet $d(c_i,C_k)=0$ if $c_i=C_k$ and 1 otherwise. Let $f(b_i,B_k)=0$ if $b_i$ and $B_k$ have more than 50% overlap, and 1 otherwise. The error of the algorithm on an individual image will be computed using:\n\n$$e=\\frac{1}{n} \\cdot \\sum_k min_{i} min_{m} max \\{d(c_i,C_k), f(b_i,B_{km}) \\}$$\n\n**Pixel Perturbation** (Full-Gradient)\n\nFirst form: remove $k$ most salient pixels from the image and measure impact on output confidence (high impact expected for good saliency methods, similar to Avg. Drop %). This might add artifacts (edges).\n\nSecond form: remove $k$ least salient pixels from the image and measure output confidence (low impact expected for good saliency methods).\n\n**Average Drop %** (Grad-CAM++, Score-CAM)\n\nThe avg. percentual drop in the confidence of a model for a particular image $x_i$ and class $c$, when only the highlighted region is provided ($M_i\\circ x_i$):\n\n$$\u2211_i^N \\frac{max(0, Y_i^c \u2212 O_i^c)}{Y_i^c} 100$$\n\n* $Y_i^c = f(x_i)^c$\n* $O_i^c = f(M_i\\circ x_i)^c$\n\n**Increase in confidence %** (Grad-CAM++, Score-CAM)\n\nMeasures scenarios where removing background noise must improve classification confidence.\n\n$$\u2211^N_i \\frac{Y^c_i < O^c_i}{N}100$$\n\n\n### CAM\n\n$M(f, x)^u_{ij} = \\text{relu}(\\sum_k w^u_k A_{ij}^k)$\n\n\n```python\n@tf.function\ndef sigmoid_cam(x, y):\n print(f'CAM tracing x:{x.shape} y:{y.shape}')\n\n l, a = nn_s(x, training=False)\n y = tf.einsum('bhwk,ku->buhw', a, sW)\n\n return l, y[..., tf.newaxis]\n```\n\n### Grad-CAM\n\n$M(f, x)^u_{ij} = \\text{relu}(\\sum_k \\sum_{lm}\\frac{\\partial S_u}{\\partial A_{lm}^k} A_{ij}^k)$\n\n\n```python\n@tf.function\ndef sigmoid_gradcam(x, y):\n print(f'Grad-CAM tracing x:{x.shape} y:{y.shape}')\n\n with tf.GradientTape(watch_accessed_variables=False) as t:\n t.watch(x)\n l, a = nn_s(x, training=False)\n\n dlda = t.batch_jacobian(l, a)\n\n weights = tf.reduce_sum(dlda, axis=(-3, -2)) # bc(hw)k -> bck\n maps = tf.einsum('bhwc,buc->buhw', a, weights)\n\n return l, maps[..., tf.newaxis]\n```\n\n\nNote that for fully-convolutional network, with a single densely-connected softmax classifier at its end:\n\n$S_u = \\sum_k w^k_u [\\frac{1}{hw}\\sum_{lm}^{hw} A^k_{lm}]$\n\nThen:\n\n\\begin{align}\n\\frac{\\partial S_u}{\\partial A_{lm}^k} &= \\frac{\\partial}{\\partial A_{lm}^k} \\sum_k w^k_u [\\frac{1}{hw}\\sum_{lm}^{hw} A^k_{lm}] \\\\\n &= w^k_u [\\frac{1}{hw}\\frac{\\partial A^k_{lm}}{\\partial A^k_{lm}}] \\\\\n &= \\frac{w^k_u}{hw}\n\\end{align}\n\nAll constants (including $\\frac{1}{hw}$) are erased when we apply normalization.\nTherefore, in these conditions, this this method is equivalent to `CAM`.\n\n### Grad-CAM++\n\n$M(f, x)^u_{ij} = \\sum_k \\sum_{lm} \\alpha_{lm}^{ku} \\text{relu}(\\frac{\\partial S_u}{\\partial A_{lm}^k}) A_{ij}^k$\n\nWhere\n\n$\\alpha_{lm}^{ku} = \\frac{(\\frac{\\partial S_u}{\\partial A_{lm}^k})^2}{2(\\frac{\\partial S_u}{\\partial A_{lm}^k})^2 + \\sum_{ab} A_{ab}^k(\\frac{\\partial S_u}{\\partial A_{lm}^k})^3}$\n\n\n```python\n@tf.function\ndef sigmoid_gradcampp(x, y):\n print(f'Grad-CAM++ tracing x:{x.shape} y:{y.shape}')\n\n with tf.GradientTape(watch_accessed_variables=False) as tape:\n tape.watch(x)\n s, a = nn_s(x, training=False)\n\n dsda = tape.batch_jacobian(s, a)\n\n dyda = tf.einsum('bu,buhwk->buhwk', tf.exp(s), dsda)\n d2 = dsda**2\n d3 = dsda**3\n aab = tf.reduce_sum(a, axis=(1, 2)) # (BK)\n akc = tf.math.divide_no_nan(\n d2,\n 2.*d2 + tf.einsum('bk,buhwk->buhwk', aab, d3)) # (2*(BUHWK) + (BK)*BUHWK)\n\n weights = tf.einsum('buhwk,buhwk->buk', akc, tf.nn.relu(dyda)) # w: buk\n maps = tf.einsum('buk,bhwk->buhw', weights, a) # a:bhwk, m: buhw\n\n return s, maps[..., tf.newaxis]\n```\n\n### Score CAM\n\n$M(f, x)^u_{ij} = \\text{relu}(\u2211_k \\text{softmax}(C(A_l^k)_u) A_l^k)$\n\nWhere\n\n$\nC(A_l^k)_u = f(X_b \\circ \\psi(\\text{up}(A_l^k)))_u - f(X_b)_u\n$\n\n\nThis algorithm has gone through updated. I followed the most recent implementation in [haofanwang/Score-CAM](https://github.com/haofanwang/Score-CAM/blob/master/cam/scorecam.py#L47).\n\n\n```python\ndef sigmoid_scorecam(x, y):\n acts_used = Config.explaining.score_cam_activations\n\n l, a = nn_s(x, training=False)\n\n if acts_used == 'all' or acts_used is None:\n acts_used = a.shape[-1]\n\n # Sorting kernels from highest to lowest variance.\n std = tf.math.reduce_std(a, axis=(1, 2))\n a_high_std = tf.argsort(std, axis=-1, direction='DESCENDING')[:, :acts_used]\n a = tf.gather(a, a_high_std, axis=3, batch_dims=1)\n\n s = tf.Variable(tf.zeros((x.shape[0], y.shape[1], *Config.data.size)), name='sc_maps')\n\n for ix in range(acts_used):\n a_ix = a[..., ix:ix+1]\n\n if tf.reduce_min(a_ix) == tf.reduce_max(a_ix):\n break\n\n s.assign_add(_scorecam_feed(x, a_ix))\n \n return l, s[..., tf.newaxis]\n\n@tf.function\ndef _scorecam_feed(x, a_ix):\n print('Score-CAM feed tracing')\n a_ix = tf.image.resize(a_ix, Config.data.size)\n\n b = normalize(a_ix)\n fm = nn(x * b, training=False)\n fm = tf.nn.sigmoid(fm)\n fm = tf.einsum('bc,bhw->bchw', fm, a_ix[..., 0])\n\n return fm\n```\n\nVectorized implementation:\n\nThe number of activating kernels used was defined in `Config.explaining.score_cam_activations`. \nFor each batch of 16 samples (`100 MB = 16 \u00d7 512\u00d7512\u00d73 \u00d7 8\u00f71000\u00f71000`):\n\n1. 16 samples are feed-forwarded, generating the logits `(16, 20)` and activations `(16, 16, 16, score_cam_activations)`\n2. The masked input `(16 \u00d7 score_cam_activations, 512, 512, 3)` is created (`score_cam_activations \u00d7 100 MB`).\n3. The masked input is feed-forwarded (batching used to prevent the GPU from blowing up).\n\n\n```python\ndef sigmoid_scorecam(x, y):\n acts_used = Config.explaining.score_cam_activations\n\n l, a = nn_s(x, training=False)\n\n std = tf.math.reduce_std(a, axis=(1, 2))\n a_high_std = tf.argsort(std, axis=-1, direction='DESCENDING')[:, :acts_used]\n a = tf.gather(a, a_high_std, axis=-1, batch_dims=-1)\n\n a = tf.image.resize(a, Config.data.size)\n b = normalize(a)\n\n b = tf.einsum('bhwc,bhwk->bkhwc', x, b) # outer product over 2 ranks\n b = tf.reshape(b, (-1, *Config.data.shape)) # batchify (B*A, H, W, C)\n\n fm = nn.predict(b, batch_size=Config.data.batch_size)\n fm = tf.nn.sigmoid(fm)\n fm = tf.reshape(fm, (x.shape[0], acts_used, fm.shape[1])) # unbatchify\n\n s = tf.einsum('bhwk,bkc->bchw', a, fm)\n s = tf.nn.relu(s)\n s = s[..., tf.newaxis]\n\n return l, s\n```\n\n## Dataset\n\n### Augmentation Policy\n\n\n```python\ndef default_policy_fn(image):\n # image = tf.image.resize_with_crop_or_pad(image, *Config.data.size)\n # mask = tf.image.resize_with_crop_or_pad(mask, *Config.data.size)\n\n return image\n```\n\n### Preparing and Performance Settings\n\n\n```python\ndata_path = Config.data.path\n```\n\n\n```bash\n%%bash\n\nif [ ! -d /root/tensorflow_datasets/amazon-from-space/ ]; then\n mkdir -p /root/tensorflow_datasets/amazon-from-space/\n\n # gdown --id 12wCmah0FFPIjI78YJ2g_YWFy97gaA5S9 --output /root/tensorflow_datasets/amazon-from-space/train-jpg.tfrecords\n\n cp /content/drive/MyDrive/datasets/amazon-from-space/train-jpg.tfrecords \\\n /root/tensorflow_datasets/amazon-from-space/\nelse\n echo \"Dir $data_path found. Skipping.\"\nfi\n```\n\n\n```python\nclass AmazonFromSpace:\n num_train_samples = 40479\n num_test_samples = 61191\n\n classes_ = np.asarray(\n ['agriculture', 'artisinal_mine', 'bare_ground', 'blooming', 'blow_down',\n 'clear', 'cloudy', 'conventional_mine', 'cultivation', 'habitation', 'haze',\n 'partly_cloudy', 'primary', 'road', 'selective_logging', 'slash_burn', 'water'])\n\n @classmethod\n def int2str(cls, indices):\n return cls.classes_[indices]\n \n @staticmethod\n def _bytes_feature(value):\n return tf.train.Feature(bytes_list=tf.train.BytesList(value=[value.tobytes()]))\n\n @staticmethod\n def _int64_feature(value):\n return tf.train.Feature(int64_list=tf.train.Int64List(value=[value]))\n \n @staticmethod\n def decode_fn(record_bytes):\n r = tf.io.parse_single_example(record_bytes, {\n 'filename': tf.io.FixedLenFeature([], tf.string),\n 'image': tf.io.FixedLenFeature([], tf.string),\n 'height': tf.io.FixedLenFeature([], tf.int64, default_value=[256]),\n 'width': tf.io.FixedLenFeature([], tf.int64, default_value=[256]),\n 'channels': tf.io.FixedLenFeature([], tf.int64, default_value=[3]),\n 'label': tf.io.VarLenFeature(tf.int64),\n })\n \n r['image'] = tf.reshape(tf.io.decode_raw(r['image'], tf.uint8),\n (r['height'], r['width'], r['channels']))\n r['label'] = tf.sparse.to_dense(r['label'])\n\n return r\n \n @classmethod\n def load(cls, tfrecords_path):\n return tf.data.TFRecordDataset(tfrecords_path).map(cls.decode_fn, num_parallel_calls=tf.data.AUTOTUNE)\n```\n\n\n```python\nCLASSES = AmazonFromSpace.classes_\nint2str = AmazonFromSpace.int2str\nnum_samples = AmazonFromSpace.num_train_samples\nnum_train_samples = int((1-Config.training.valid_size)*num_samples)\nnum_valid_samples = int(Config.training.valid_size*num_samples)\n```\n\n\n```python\nfrom functools import partial\n\n\n@tf.function\ndef load_fn(d, augment=False):\n image = d['image']\n labels = d['label']\n\n image = tf.cast(image, tf.float32)\n image = tf.ensure_shape(image, Config.data.shape)\n\n # image, _ = adjust_resolution(image)\n image = (augment_policy_fn(image)\n if augment\n else default_policy_fn(image))\n \n image = preprocess(image)\n\n return image, labels_to_one_hot(labels)\n\n\ndef adjust_resolution(image):\n es = tf.constant(Config.data.size, tf.float32)\n xs = tf.cast(tf.shape(image)[:2], tf.float32)\n\n ratio = tf.reduce_min(es / xs)\n xsn = tf.cast(tf.math.ceil(ratio * xs), tf.int32)\n\n image = tf.image.resize(image, xsn, preserve_aspect_ratio=True, method='nearest')\n\n return image, ratio\n\n\ndef labels_to_one_hot(labels):\n return tf.reduce_max(\n tf.one_hot(labels, depth=CLASSES.shape[0]),\n axis=0)\n\n\ndef prepare(ds, batch_size, cache=False, shuffle=False, augment=False):\n if cache: ds = ds.cache()\n if shuffle: ds = ds.shuffle(Config.data.shuffle_buffer_size, reshuffle_each_iteration=True, seed=Config.data.train_shuffle_seed)\n\n return (ds.map(partial(load_fn, augment=augment), num_parallel_calls=tf.data.AUTOTUNE)\n .batch(batch_size, drop_remainder=True)\n .prefetch(Config.data.prefetch_buffer_size))\n```\n\n\n```python\ntrain_dataset = AmazonFromSpace.load(f'{data_dir}/amazon-from-space/train-jpg.tfrecords')\nvalid_dataset = train_dataset.take(num_valid_samples)\ntrain_dataset = train_dataset.skip(num_valid_samples)\n\n# train = prepare(train_dataset, Config.data.batch_size)\nvalid = prepare(valid_dataset, Config.data.batch_size)\n```\n\n### Examples in The Dataset\n\n\n```python\n#@title\n\nfor stage, batches, samples in zip(('validation',),\n (valid,),\n (num_valid_samples,)):\n print(stage)\n print(f' {batches}')\n print(f' samples: {samples}')\n print(f' steps : {samples // Config.data.batch_size}')\n print()\n```\n\n validation\n \n samples: 12143\n steps : 379\n \n\n\n\n```python\n#@title\n\nfor images, labels in valid.take(1):\n gt = [' '.join((e[:3] for e in CLASSES[l].astype(str))) for l in labels.numpy().astype(bool)]\n visualize(to_image(images[:16]), gt, rows=4, figsize=(12, 10));\n```\n\n## Network\n\n\n```python\nprint(f'Loading {Config.model.backbone.__name__}')\n\nbackbone = Config.model.backbone(\n input_shape=Config.data.shape,\n include_top=Config.model.include_top,\n classifier_activation=Config.model.classifier_activation,\n)\n```\n\n Loading ResNet101V2\n\n\n\n```python\nfrom tensorflow.keras.layers import Conv2D, Dense, Dropout, GlobalAveragePooling2D\n\nclass DenseKur(Dense):\n \"\"\"Dense with Softmax Weights.\n \"\"\"\n def call(self, inputs):\n kernel = self.kernel\n ag = kernel # tf.abs(kernel)\n ag = ag - tf.reduce_max(ag, axis=-1, keepdims=True)\n ag = tf.nn.softmax(ag)\n\n outputs = inputs @ (ag*kernel)\n\n if self.use_bias:\n outputs = tf.nn.bias_add(outputs, self.bias)\n\n if self.activation is not None:\n outputs = self.activation(outputs)\n\n return outputs\n```\n\n\n```python\ndef build_specific_classifier(\n backbone,\n classes,\n dropout_rate=0.5,\n name=None,\n gpl='avg_pool',\n):\n x = tf.keras.Input(Config.data.shape, name='images')\n y = backbone(x)\n y = GlobalAveragePooling2D(name='avg_pool')(y)\n # y = Dense(classes, name='predictions')(y)\n y = DenseKur(classes, name='predictions')(y)\n\n return tf.keras.Model(x, y, name=name)\n\nbackbone.trainable = False\nnn = build_specific_classifier(backbone, len(CLASSES), name='resnet101_afs')\n```\n\n\n```python\nif Config.model.fine_tune_layers:\n print(f'Unfreezing {Config.model.fine_tune_layers:.0%} layers.')\n\n backbone.trainable = True\n frozen_layer_ix = int((1-Config.model.fine_tune_layers) * len(backbone.layers))\n\n for ix, l in enumerate(backbone.layers):\n l.trainable = (ix > frozen_layer_ix and\n (not isinstance(l, tf.keras.layers.BatchNormalization) or\n not Config.model.freeze_batch_norm))\n \nprint(f'Loading weights from {Config.model.weights}')\nnn.load_weights(Config.model.weights)\n```\n\n Unfreezing 60% layers.\n Loading weights from /home/ldavid/Workspace/logs/amazon-from-space/resnet101-sw-ce-fine-tune/weights.h5\n\n\n\n```python\nbackbone.trainable = False\n```\n\n\n```python\nnn_s = tf.keras.Model(\n inputs=nn.inputs,\n outputs=[nn.output, nn.get_layer(Config.model.gap_layer_name).input],\n name='nn_spatial')\n```\n\n\n```python\nsW, sb = nn_s.get_layer('predictions').weights\nsW = sW * tf.nn.softmax(sW - tf.reduce_max(sW, axis=-1, keepdims=True))\n```\n\n## Saliency Methods\n\n\n```python\nDT = 0.5\n\n\u03bb_pos = Config.explaining.\u03bb_pos\n\u03bb_neg = Config.explaining.\u03bb_neg\n\u03bb_bg = Config.explaining.\u03bb_bg\n```\n\n### Min-Max CAM (ours, test ongoing)\n\n$M(f, x)^u_{ij} = \\sum_k [w^u_k - \\frac{1}{|N_x|} \\sum_{n\\in N_x} w_k^n] A_{i,j}^k$\n\nWhere\n\n$N_x = C_x\\setminus \\{u\\}$\n\n\n```python\n@tf.function\ndef min_max_sigmoid_cam(x, y):\n print(f'Min-Max CAM (tracing x:{x.shape} p:{y.shape})')\n\n l, a = nn_s(x, training=False)\n\n c = len(CLASSES)\n s_n = tf.reduce_sum(sW, axis=-1, keepdims=True)\n s_n = s_n - sW\n\n w = \u03bb_pos*sW - \u03bb_neg*s_n/(c-1)\n\n maps = tf.einsum('bhwk,ku->buhw', a, w)\n return l, maps[..., tf.newaxis]\n```\n\n\n```python\n@tf.function\ndef contextual_min_max_sigmoid_cam(x, y):\n print(f'Contextual Min-Max CAM (tracing x:{x.shape} p:{y.shape})')\n\n l, a = nn_s(x, training=False)\n p = tf.nn.sigmoid(l)\n\n d = tf.cast(p > DT, tf.float32)\n c = tf.reduce_sum(d, axis=-1)\n c = tf.reshape(c, (-1, 1, 1)) # detections (b, 1)\n\n w = d[:, tf.newaxis, :] * sW[tf.newaxis, ...] # expand kernels in d and batches in sW\n w_n = tf.reduce_sum(w, axis=-1, keepdims=True)\n w_n = w_n - w\n\n w = \u03bb_pos*sW - \u03bb_neg*w_n / tf.maximum(c-1, 1)\n\n maps = tf.einsum('bhwk,bku->buhw', a, w)\n return l, maps[..., tf.newaxis]\n```\n\n#### Contextual ReLU MinMax\n\n$M(f, x)^u_{ij} = \\sum_k [w^{u+}_k - \\frac{1}{|N_x|} \\sum_{n\\in N_x} w^{n+}_k +\\frac{1}{|C_i|}\\sum_{n\\in C_i} w^{n-}_k] A_{i,j}^k$\n\nWhere\n\n$N_x = C_x\\setminus \\{u\\}$\n\n\n```python\n@tf.function\ndef contextual_relu_min_max_sigmoid_cam(x, y):\n print(f'Contextual ReLU Min-Max CAM (tracing x:{x.shape} p:{y.shape})')\n\n l, a = nn_s(x, training=False)\n p = tf.nn.sigmoid(l)\n\n d = tf.cast(p > DT, tf.float32)\n c = tf.reshape(tf.reduce_sum(d, axis=-1), (-1, 1, 1)) # select only detected\n\n w = d[:, tf.newaxis, :] * sW[tf.newaxis, ...] # expand kernels in d and batches in sW\n wa = tf.reduce_sum(w, axis=-1, keepdims=True)\n wn = wa - w\n\n w = ( \u03bb_pos * tf.nn.relu(sW)\n - \u03bb_neg * tf.nn.relu(wn) / tf.maximum(c-1, 1)\n + \u03bb_bg * tf.minimum(0., wa) / tf.maximum(c, 1))\n\n maps = tf.einsum('bhwk,bku->buhw', a, w)\n return l, maps[..., tf.newaxis]\n```\n\n\n```python\n@tf.function\ndef contextual_relu_min_max_sigmoid_cam_2(x, y):\n l, a = nn_s(x, training=False)\n p = tf.nn.sigmoid(l)\n\n aw = tf.einsum('bhwk,ku->buhw', a, sW)\n\n d = p > DT\n c = tf.reduce_sum(tf.cast(d, tf.float32), axis=-1)\n c = tf.reshape(c, (-1, 1, 1, 1))\n\n e = tf.repeat(d[:, tf.newaxis, ...], d.shape[1], axis=1)\n e = tf.linalg.set_diag(e, tf.fill(e.shape[:-1], False))\n\n z = tf.fill(aw.shape, -np.inf)\n\n an = tf.where(e[..., tf.newaxis, tf.newaxis], aw[:, tf.newaxis, ...], z[:, tf.newaxis, ...])\n ab = tf.where(d[..., tf.newaxis, tf.newaxis], aw, z)\n\n an = tf.reduce_max(an, axis=2)\n ab = tf.reduce_max(ab, axis=2, keepdims=True)\n\n maps = (\n tf.maximum(0., aw)\n - tf.maximum(0., an)\n + tf.minimum(0., ab)\n )\n\n return l, maps[..., tf.newaxis]\n```\n\n### Min-Max Grad-CAM (ours, test ongoing)\n\n$M(f, x)^u_{ij} = \\text{relu}(\\sum_k \\sum_{l,m}\\frac{\\partial J_u}{\\partial A_{l,m}^k} A_{i,j}^k)$\n\nWhere\n\n$J_u = S_u - \\frac{1}{|N|} \\sum_{n\\in N_x} S_n$ \n$N_x = C_x\\setminus \\{u\\}$\n\n\n\n```python\ndef min_max_activation_gain(y, s):\n c = len(CLASSES)\n # shape(s) == (b, c)\n s_n = tf.reduce_sum(s, axis=-1, keepdims=True) # shape(s_n) == (b)\n s_n = s_n-s # shape(s_n) == (b, c)\n\n return \u03bb_pos*s - \u03bb_neg*s_n / (c-1)\n\n@tf.function\ndef min_max_sigmoid_gradcam(x, y):\n print(f'Min-Max Grad-CAM (tracing x:{x.shape} p:{y.shape})')\n\n with tf.GradientTape(watch_accessed_variables=False) as t:\n t.watch(x)\n l, a = nn_s(x, training=False)\n loss = min_max_activation_gain(y, l)\n\n dlda = t.batch_jacobian(loss, a)\n \n weights = tf.reduce_sum(dlda, axis=(-3, -2))\n maps = tf.einsum('bhwc,buc->buhw', a, weights)\n\n return l, maps[..., tf.newaxis]\n```\n\n\n```python\ndef contextual_min_max_activation_gain(y, s, p):\n d = tf.cast(p > DT, tf.float32)\n c = tf.reduce_sum(d, axis=-1, keepdims=True) # only detections\n\n sd = s*d\n s_n = tf.reduce_sum(sd, axis=-1, keepdims=True) # sum logits detected (b, 1)\n\n return \u03bb_pos*s - \u03bb_neg*(s_n - sd)/tf.maximum(c-1, 1)\n\n@tf.function\ndef contextual_min_max_sigmoid_gradcam(x, y):\n print(f'Contextual Min-Max Grad-CAM (tracing x:{x.shape} p:{y.shape})')\n\n with tf.GradientTape(watch_accessed_variables=False) as t:\n t.watch(x)\n l, a = nn_s(x, training=False)\n p = tf.nn.sigmoid(l)\n loss = contextual_min_max_activation_gain(y, l, p)\n\n dlda = t.batch_jacobian(loss, a)\n\n weights = tf.reduce_sum(dlda, axis=(-3, -2))\n maps = tf.einsum('bhwc,buc->buhw', a, weights) # a*weights\n\n return l, maps[..., tf.newaxis]\n```\n\n\n```python\ndef contextual_relu_min_max_activation_gain(y, s):\n p = tf.nn.sigmoid(s)\n d = tf.cast(p > DT, tf.float32)\n c = tf.reduce_sum(d, axis=-1, keepdims=True)\n\n sd = s*d # only detections\n sa = tf.reduce_sum(sd, axis=-1, keepdims=True) # sum logits detected (b, 1)\n sn = sa - sd\n\n return tf.stack((\n \u03bb_pos * s,\n \u03bb_neg * sn / tf.maximum(c-1, 1),\n \u03bb_bg * (sn+sd) / tf.maximum(c, 1)\n ), axis=1)\n\n@tf.function\ndef contextual_relu_min_max_sigmoid_gradcam(x, y):\n print(f'Contextual ReLU Min-Max Grad-CAM (tracing x:{x.shape} y:{y.shape})')\n\n with tf.GradientTape(watch_accessed_variables=False) as t:\n t.watch(x)\n \n l, a = nn_s(x, training=False)\n loss = contextual_relu_min_max_activation_gain(y, l)\n\n dlda = t.batch_jacobian(loss, a)\n\n w, wn, wa = dlda[:, 0], dlda[:, 1], dlda[:, 2]\n \n w = ( tf.nn.relu(w)\n - tf.nn.relu(wn)\n + tf.minimum(0., wa))\n\n weights = tf.reduce_sum(w, axis=(-3, -2))\n maps = tf.einsum('bhwc,buc->buhw', a, weights)\n\n return l, maps[..., tf.newaxis]\n```\n\n## Qualitative Analysis\n\n\n```python\n#@title\n\ndef visualize_explaining_many(\n x,\n y,\n p,\n maps,\n N=None,\n max_detections=3\n):\n N = N or len(x)\n plt.figure(figsize=(16, 2*N))\n rows = N\n cols = 1+2*max_detections\n\n actual = [','.join(CLASSES[_y]) for _y in y.numpy().astype(bool)]\n for ix in range(N):\n detections = p[ix] > DT\n\n visualize_explaining(\n x[ix],\n actual[ix],\n detections,\n p[ix],\n maps[ix],\n i0=cols*ix,\n rows=rows,\n cols=cols,\n full=False,\n max_detections=max_detections\n )\n \n plt.tight_layout()\n\ndef visualize_explaining(image,\n labels,\n detections,\n probs,\n cams,\n full=True,\n i0=0,\n rows=2,\n cols=None,\n max_detections=3):\n detections = detections.numpy()\n \n im = to_image(image)\n _maps = tf.boolean_mask(cams, detections)\n _maps = tf.image.resize(_maps, Config.data.size)\n _masked = to_image(masked(image, _maps))\n\n plots = [im, *_maps[:max_detections]]\n title = [labels] + [f'{d} {p:.0%}' for d, p in zip(CLASSES[detections], probs.numpy()[detections])]\n\n visualize(plots, title, full=full, rows=rows, cols=cols, i0=i0)\n\n for ix, s in enumerate(_maps[:max_detections]):\n plt.subplot(rows, cols, i0+len(plots)+ix+1)\n plot_heatmap(im, s[..., 0])\n```\n\n\n```python\ncams = {}\n\nfor x, y in valid.take(1):\n l = tf.convert_to_tensor(nn.predict(x))\n p = tf.nn.sigmoid(l)\n\n # Only samples with two or more objects:\n s = tf.reduce_sum(tf.cast(p > DT, tf.int32), axis=1) > 1\n x, y, l, p = x[s], y[s], l[s], p[s]\n```\n\n#### CAM\n\n\n```python\n_, maps = sigmoid_cam(x, y)\n\nmaps = tf.nn.relu(maps)\nmaps = normalize(maps)\n\ncams['cam'] = maps\n```\n\n\n```python\nvisualize_explaining_many(x, y, p, maps)\n```\n\n#### Grad-CAM\n\n\n```python\n_, maps = sigmoid_gradcam(x, y)\n\nmaps = tf.nn.relu(maps)\nmaps = normalize(maps)\n# visualize_explaining_many(x, y, p, maps)\n\ncams['gradcam'] = maps\n```\n\n#### Grad-CAM++\n\n\n```python\n_, maps = sigmoid_gradcampp(x, y)\n\nmaps = tf.nn.relu(maps)\nmaps = normalize(maps)\n# visualize_explaining_many(x, y, p, maps)\n\ncams['gradcampp'] = maps\n```\n\n### Score-CAM\n\n\n```python\n_, maps = sigmoid_scorecam(x, y)\n\nmaps = tf.nn.relu(maps)\nmaps = normalize(maps)\nvisualize_explaining_many(x, y, p, maps)\n\ncams['scorecam'] = maps\n```\n\n### Min-Max CAM\n\n#### Vanilla\n\n\n```python\n%%time\n\n_, maps = min_max_sigmoid_cam(x, y)\n\nmaps = tf.nn.relu(maps)\nmaps = normalize(maps)\n# visualize_explaining_many(x, y, p, maps)\n\ncams['minmax_cam'] = maps\n```\n\n#### Contextual\n\n\n```python\n%%time\n\n_, maps = contextual_min_max_sigmoid_cam(x, y)\n\nmaps = tf.nn.relu(maps)\nmaps = normalize(maps)\n# visualize_explaining_many(x, y, p, maps)\n\ncams['contextual_minmax_cam'] = maps\n```\n\n#### Contextual ReLU\n\n\n```python\n%%time\n\n_, maps = contextual_relu_min_max_sigmoid_cam(x, y)\n\nmaps = tf.nn.relu(maps)\nmaps = normalize(maps)\n# visualize_explaining_many(x, y, p, maps)\n\ncams['contextual_relu_minmax_cam'] = maps\n```\n\n\n```python\n%%time\n\n_, maps = contextual_relu_min_max_sigmoid_cam_2(x, y)\n\nmaps = tf.nn.relu(maps)\nmaps = normalize(maps)\n# visualize_explaining_many(x, y, p, maps)\n\ncams['contextual_relu_minmax_cam_2'] = maps\n```\n\n### Min-Max Grad-CAM\n\n#### Vanilla\n\n\n```python\n_, maps = min_max_sigmoid_gradcam(x, y)\n\nmaps = tf.nn.relu(maps)\nmaps = normalize(maps)\nvisualize_explaining_many(x, y, p, maps)\n\ncams['minmax_gradcam'] = maps\n```\n\n#### Contextual\n\n\n```python\n_, maps = contextual_min_max_sigmoid_gradcam(x, y)\n\nmaps = tf.nn.relu(maps)\nmaps = normalize(maps)\nvisualize_explaining_many(x, y, p, maps)\n\ncams['contextual_minmax_gradcam'] = maps\n```\n\n#### Contextual ReLU\n\n\n```python\n_, maps = contextual_relu_min_max_sigmoid_gradcam(x, y)\n\nmaps = tf.nn.relu(maps)\nmaps = normalize(maps)\nvisualize_explaining_many(x, y, p, maps)\n\ncams['contextual_relu_minmax_gradcam'] = maps\n```\n\n### Summary\n\n\n```python\nobserving = 'cam minmax_cam contextual_minmax_cam contextual_relu_minmax_cam'.split()\ntitles = 'Input CAM MM C-MM CG-MM'.split()\n\nprint('Results selected for vis:', *observing, sep='\\n ')\n```\n\n\n```python\n#@title\n\ndetections = p > DT\nindices = tf.where(detections)\nsample_ix, label_ix = indices[:, 0], indices[:, 1]\n\nvisualize(\n sum(zip(to_image(tf.gather(x, sample_ix)[:48]).numpy(),\n *(tf.image.resize(cams[c][detections][:48], Config.data.size).numpy()\n for c in observing)),\n ()),\n title=titles,\n rows=sample_ix[:48].shape[0],\n figsize=(12, 80)\n);\n```\n\n\n```python\n#@title\n\nplt.figure(figsize=(12, 80))\n\nselected_images = to_image(tf.gather(x, sample_ix)[:48]).numpy()\n\nrows = len(selected_images)\ncols = len(observing) + 1\n\nfor ix, im in enumerate(selected_images):\n plt.subplot(rows, cols, ix*cols + 1)\n plt.imshow(im)\n plt.axis('off')\n\n for j, method in enumerate(observing):\n map = cams[method][detections][ix]\n map = tf.image.resize(map, Config.data.size)\n \n plt.subplot(rows, cols, ix*cols + j + 2)\n plot_heatmap(im, map[..., 0])\n\n if ix == 0:\n plt.title(titles[j+1])\n\nplt.tight_layout()\n```\n\n## Quantitative Analysis\n\n#### Metrics\n\n##### **Increase in Confidence %**\n> *Removing background noise should improve confidence (higher=better)*\n\n$\\frac{1}{\u2211 |C_i|} \u2211^N_i\u2211^{C_i}_c [Y^c_i < O_{ic}^c] 100$\n\nThought: this probably works better for *softmax* classifiers.\n\n\n```python\ndef increase_in_confidence(\n p, # f(x) (batch, classes)\n y, # f(x)[f(x) > 0.5] (detections, 1)\n o, # f(x*mask(x, m)) (detections, classes)\n samples_ix,\n units_ix\n):\n oc = tf.gather(o, units_ix, axis=1, batch_dims=1) # (detections, 1)\n\n incr = np.zeros(p.shape, np.uint32)\n incr[samples_ix, units_ix] = tf.cast(y < oc, tf.uint32).numpy()\n\n return incr.sum(axis=0)\n```\n\n##### **Average Drop %**\n> *Masking with an accurate mask should not decrease confidence (lower=better)*\n\n$\\frac{1}{\u2211 |C_i|} \u2211_i^N \u2211_c^{C_i} \\frac{max(0, Y_i^c \u2212 O_{ic}^c)}{Y_i^c} 100$\n\nMeasures if your mask is correctly positioned on top of the important regions that determine the class of interest.\n\n\n```python\ndef average_drop(p, y, o, samples_ix, units_ix):\n oc = tf.gather(o, units_ix, axis=1, batch_dims=1)\n\n drop = np.zeros(p.shape)\n drop[samples_ix, units_ix] = (tf.nn.relu(y - oc) / y).numpy()\n\n return drop.sum(axis=0)\n```\n\n##### **Average Retention % (ours, testing ongoing)**\n> *Masking the input with an accurate complement mask for class $c$ should decrease confidence in class $c$ (higher=better)*\n\n$\\frac{1}{\u2211 |C_i|} \u2211_i^N \u2211_c^{C_i} \\frac{max(0, Y_i^c \u2212 \\bar{O}_{ic}^c)}{Y_i^c} 100$\n\nWhere $\\bar{O}_{ic}^c = f(x_i \\circ (1-\\psi(M(f, x_i)_{hw}^c))^c$\n\nMasking the input $x_i$ for all classes except $c$ should cause the model's confidence in $c$ to drop.\n\n###### **Average Drop & Retention % (ours)**\n\n\\begin{align}\n\\frac{\\text{drop} + (1-\\text{retention})}{2}\n&= \\frac{1}{2 \u2211 |C_i|} \u2211_i^N \u2211_c^{C_i} [\\frac{max(0, Y_i^c \u2212 O_{ic}^c)}{Y_i^c} + (1-\\frac{max(0, Y_i^c \u2212 \\bar{O}_{ic}^c)}{Y_i^c})] 100 \\\\\n&= \\frac{1}{2 \u2211 |C_i|} \u2211_i^N \u2211_c^{C_i} [1 + \\frac{Y_i^c \u2212 O_{ic}^c - (Y_i^c \u2212 \\bar{O}_{ic}^c)}{Y_i^c}] 100 \\\\\n&= \\frac{1}{2 \u2211 |C_i|} \u2211_i^N \u2211_c^{C_i} [1 + \\frac{\\bar{O}_{ic}^c \u2212 O_{ic}^c}{Y_i^c}] 100\n\\end{align}\n\nWhere\n\n* $O_{ic}^c = f(x_i \\circ \\psi(M(f, x_i)_{hw}^c)^c$\n* $\\bar{O}_{ic}^c = f(x_i \\circ (1-\\psi(M(f, x_i)_{hw}^c))^c$\n\n\n##### **Average Drop of Others % (ours, testing ongoing)**\n\n> *An ideal mask for class $c$ is not retaining any objects of other classes (higher=better)*\n\n$\\frac{1}{\u2211 |C_i|} \u2211_i^N \u2211_c^{C_i} \\frac{1}{|D_i|} \u2211_d^{D_i} \\frac{max(0, Y_i^d \u2212 O_{ic}^d)}{Y_i^d} 100$\n\nMasking the input $x_i$ for a given class $c$ should cause the confidence in other classes to drop.\n\nI.e., $f(x_i\\circ \\psi(M(f, x_i)^c_{hw}))^d \\sim 0, \\forall d\\in D_i = C_i\\setminus \\{c\\}$.\n\nFor single-label problems, $D_i = \\emptyset$ and Average Drop of Others is not defined. How to solve this?\n\n\n```python\ndef average_drop_of_others(p, s, y, o, samples_ix, units_ix):\n # Drop of all units, for all detections\n d = tf.gather(p, samples_ix)\n d = tf.nn.relu(d - o) / d\n\n # Remove drop of class `c` and non-detected classes\n detected = tf.cast(tf.gather(s, samples_ix), tf.float32)\n d = d*detected\n d = tf.reduce_sum(d, axis=-1) - tf.gather(d, units_ix, axis=1, batch_dims=1)\n c = tf.reduce_sum(detected, axis=-1)\n\n # Normalize by the number of peer labels for detection `c`\n d = d / tf.maximum(1., c -1)\n\n drop = np.zeros(p.shape)\n drop[samples_ix, units_ix] = d.numpy()\n\n return drop.sum(axis=0)\n```\n\n##### **Average Retention of Others % (ours, testing ongoing)**\n\n> *An ideal mask complement for class $c$ should cover all objects of the other classes (lower=better)*\n\n$\\frac{1}{\u2211 |D_i|} \u2211_i^N \u2211_c^{C_i} \\frac{1}{|D_i|} \u2211_d^{D_i} \\frac{max(0, Y_i^d \u2212 \\bar{O}_{ic}^d)}{Y_i^d} 100$\n\nMasking the input $x_i$ for all classes except $c$ should cause the confidence in other classes to stay the same or increase.\n\nI.e., $f(x_i\\circ (1-\\psi(M(f, x_i)^c_{ij})))^d \\approx f(x_i)^d, \\forall d\\in D_i = C_i\\setminus \\{c\\}$.\n\n#### Experiments\n\n\n```python\n#@title Testing Loop\n\ndef experiment_with(dataset, cam_method, cam_modifier):\n print(f'Testing {cam_method.__name__}')\n t = time()\n r = cam_evaluation(nn, dataset, cam_method=cam_method, cam_modifier=cam_modifier)\n print(f'elapsed: {(time() - t)/60:.1f} minutes', end='\\n\\n')\n\n return r.assign(method=cam_method.__name__)\n\n\ndef cam_evaluation(nn, dataset, cam_method, cam_modifier):\n metric_names = ('increase %', 'avg drop %', 'avg retention %',\n 'avg drop of others %', 'avg retention of others %',\n 'detections')\n metrics = (np.zeros(len(CLASSES), np.uint16),\n np.zeros(len(CLASSES)),\n np.zeros(len(CLASSES)),\n np.zeros(len(CLASSES)),\n np.zeros(len(CLASSES)),\n np.zeros(len(CLASSES), np.uint16))\n\n try:\n for step, (x, y) in enumerate(dataset):\n p, maps = cam_method(x, y)\n p = tf.nn.sigmoid(p)\n maps = cam_modifier(maps)\n\n for e, f in zip(metrics, cam_evaluation_step(nn, x, p, maps)):\n e += f\n\n print('.', end='' if (step+1) % 80 else '\\n')\n print()\n\n except KeyboardInterrupt:\n print('interrupted')\n\n metrics, detections = metrics[:-1], metrics[-1]\n\n results = {n: 100*m/detections for n, m in zip(metric_names, metrics)}\n results['label'] = CLASSES\n results['detections'] = detections\n results = pd.DataFrame(results)\n\n print(f'Average Drop %: {results[\"avg drop %\"].mean():.4}%')\n print(f'Average Increase %: {results[\"increase %\"].mean():.4}%')\n\n return results\n\n\ndef cam_evaluation_step(nn, x, p, m):\n s = p > DT\n w = tf.where(s)\n samples_ix, units_ix = w[:, 0], w[:, 1]\n\n md = tf.image.resize(m[s], Config.data.size)\n\n detections = tf.reduce_sum(tf.cast(s, tf.uint32), axis=0)\n\n y = p[s] # (batch, c) --> (detections)\n xs = tf.gather(x, samples_ix) # (batch, 300, 300, 3) --> (detections, 300, 300, 3)\n\n o = nn.predict(masked(xs, md), batch_size=Config.data.batch_size)\n o = tf.nn.sigmoid(o)\n \n co = nn.predict(masked(xs, 1 -md), batch_size=Config.data.batch_size)\n co = tf.nn.sigmoid(co)\n \n samples_ix, units_ix = samples_ix.numpy(), units_ix.numpy()\n\n incr = increase_in_confidence(p, y, o, samples_ix, units_ix)\n drop = average_drop(p, y, o, samples_ix, units_ix)\n rete = average_drop(p, y, co, samples_ix, units_ix)\n\n drop_of_others = average_drop_of_others(p, s, y, o, samples_ix, units_ix)\n rete_of_others = average_drop_of_others(p, s, y, co, samples_ix, units_ix)\n\n return incr, drop, rete, drop_of_others, rete_of_others, detections.numpy()\n```\n\n\n```python\nmethods_being_tested = (\n # Baseline\n # sigmoid_cam,\n # sigmoid_gradcam,\n\n # Best solutions\n # sigmoid_gradcampp,\n sigmoid_scorecam,\n\n # Ours\n # min_max_sigmoid_cam,\n # contextual_min_max_sigmoid_cam,\n # contextual_relu_min_max_sigmoid_cam,\n # contextual_relu_min_max_sigmoid_cam_2,\n)\n\nrelu_and_normalize = lambda c: normalize(tf.nn.relu(c))\n```\n\n\n```python\nresults = pd.concat(\n [\n experiment_with(valid, m, relu_and_normalize)\n for m in methods_being_tested\n ]\n)\n```\n\n Testing sigmoid_scorecam\n Score-CAM feed tracing\n ................................................................................\n ................................................................................\n ................................................................................\n ................................................................................\n ...........................................................\n Average Drop %: 39.5%\n Average Increase %: 14.13%\n elapsed: 1745.5 minutes\n \n\n\n :40: RuntimeWarning: invalid value encountered in true_divide\n results = {n: 100*m/detections for n, m in zip(metric_names, metrics)}\n\n\n\n```python\n# if os.path.exists(Config.report):\n# raise FileExistsError('You are asking me to override a report file.')\n\nresults.to_csv(Config.report, index=False)\n```\n\n#### Report\n\n\n```python\nresults = pd.read_csv(Config.report)\n```\n\n\n```python\nmethods = (\n 'sigmoid_cam',\n 'sigmoid_gradcampp',\n 'sigmoid_scorecam',\n 'min_max_sigmoid_cam',\n 'contextual_min_max_sigmoid_cam',\n 'contextual_relu_min_max_sigmoid_cam',\n)\n\nmethods_detailed = (\n 'sigmoid_cam',\n 'sigmoid_scorecam',\n 'contextual_min_max_sigmoid_cam',\n 'contextual_relu_min_max_sigmoid_cam'\n)\n\nmetric_names = (\n 'increase %',\n 'avg drop %',\n 'avg retention %',\n 'avg drop of others %',\n 'avg retention of others %',\n 'f1 score',\n 'f1 score negatives'\n)\n\nminimizing_metrics = {'avg drop %', 'avg retention of others %', 'f1 score negatives'}\n```\n\n\n```python\ndef fb_score(a, b, beta=1):\n beta2 = beta**2\n denom = (beta2 * b + a)\n denom[denom == 0.] = 1\n return (1+beta2) * a * b / denom\n\nresults['f1/2 score'] = fb_score(results['avg retention %'], results['avg drop of others %'], beta=1/2)\nresults['f1 score'] = fb_score(results['avg retention %'], results['avg drop of others %'], beta=1)\nresults['f2 score'] = fb_score(results['avg retention %'], results['avg drop of others %'], beta=2)\n\nresults['f1/2 score negatives'] = fb_score(results['avg drop %'], results['avg retention of others %'], beta=1/2)\nresults['f1 score negatives'] = fb_score(results['avg drop %'], results['avg retention of others %'], beta=1)\nresults['f2 score negatives'] = fb_score(results['avg drop %'], results['avg retention of others %'], beta=2)\n```\n\n\n```python\n(results\n .drop('detections', axis=1)\n .groupby('method')\n .mean()\n .round(4)/100)\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    increase %avg drop %avg retention %avg drop of others %avg retention of others %f1/2 scoref1 scoref2 scoref1/2 score negativesf1 score negativesf2 score negatives
    method
    sigmoid_scorecam0.1412590.3950450.4361190.3521020.2679540.0647940.1723190.3099530.0507520.1330860.247012
    \n
    \n\n\n\n\n```python\n#@title Macro Average (Class-Balanced)\n\nmacro_avg = (\n results\n .groupby('method')\n .mean()\n .reindex(methods)[list(metric_names)]\n) / 100\n\nmacro_avg_hm = macro_avg.copy()\nfor m in minimizing_metrics:\n macro_avg_hm[m] = 1-macro_avg_hm[m]\nmacro_avg_hm -= macro_avg_hm.min(axis=0)\nmacro_avg_hm /= macro_avg_hm.max(axis=0) + 1e-07\n\nplt.figure(figsize=(6, 6))\nsns.heatmap(\n macro_avg_hm,\n fmt='.2%',\n annot=macro_avg,\n cmap='RdPu',\n cbar=False,\n xticklabels=[c.replace('_', ' ') for c in macro_avg.columns],\n yticklabels=[i.replace('_', ' ') for i in macro_avg.index],\n);\n```\n\n\n```python\n#@title Weighted Average (Class Frequency Weighted)\n\ntotal_detections = (\n results\n .groupby('method')\n .agg({'detections': 'sum'})\n .rename(columns={'detections': 'total_detections'})\n)\n\nw_avg = results.merge(total_detections, how='left', left_on='method', right_index=True)\n\nmetric_results = {\n m: w_avg[m] * w_avg.detections / w_avg.total_detections\n for m in metric_names\n}\nmetric_results['method'] = w_avg.method\nmetric_results['label'] = w_avg.label\n\nw_avg = (\n pd.DataFrame(metric_results)\n .groupby('method')\n .sum()\n .reindex(methods) / 100\n)\n\nhm = w_avg.copy()\nfor m in minimizing_metrics:\n hm[m] = 1-hm[m]\nhm -= hm.min(axis=0)\nhm /= hm.max(axis=0) + 1e-07\n\nplt.figure(figsize=(6, 6))\nsns.heatmap(\n hm,\n fmt='.2%',\n annot=w_avg,\n cmap='RdPu',\n cbar=False,\n xticklabels=[c.replace('_', ' ') for c in w_avg.columns],\n yticklabels=[i.replace('_', ' ') for i in w_avg.index]\n);\n```\n\n\n```python\n#@title Detailed Results per Class\n\nplt.figure(figsize=(16, 6))\nsns.boxplot(\n data=results[results.method.isin(methods_detailed)]\n .melt(('method', 'label'), metric_names, 'metric'),\n hue='method',\n x='metric',\n y='value'\n);\n```\n\n\n```python\n#@title F1 Score by Label and CAM Method\n\nplt.figure(figsize=(16, 6))\nsns.barplot(\n data=results.sort_values('f1 score', ascending=False),\n hue='method',\n y='f1 score',\n x='label'\n)\nplt.xticks(rotation=-45);\n```\n", "meta": {"hexsha": "c5e0bcb978787665450b992c5b4addd7e14d3e6e", "size": 968003, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/experiments/2-benchmarks/cam-benchmarks-amazon-from-space.ipynb", "max_stars_repo_name": "lucasdavid/minmax-cam", "max_stars_repo_head_hexsha": "be94495a86d2c810f5522177b76be1959e47df5e", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-12-05T16:40:56.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-10T12:43:38.000Z", "max_issues_repo_path": "notebooks/experiments/2-benchmarks/cam-benchmarks-amazon-from-space.ipynb", "max_issues_repo_name": "lucasdavid/minmax-cam", "max_issues_repo_head_hexsha": "be94495a86d2c810f5522177b76be1959e47df5e", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/experiments/2-benchmarks/cam-benchmarks-amazon-from-space.ipynb", "max_forks_repo_name": "lucasdavid/minmax-cam", "max_forks_repo_head_hexsha": "be94495a86d2c810f5522177b76be1959e47df5e", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-02-11T11:48:37.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-11T11:48:37.000Z", "avg_line_length": 484001.5, "max_line_length": 968002, "alphanum_fraction": 0.9290983602, "converted": true, "num_tokens": 19419, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.6893056040203135, "lm_q1q2_score": 0.4415938848774858}} {"text": "```python\n# Imports\nimport sys\n\nfrom tqdm import tqdm\n\nimport torch\nimport numpy as np\nimport matplotlib.pyplot as plt\n\ncuda = torch.cuda.is_available()\n%matplotlib inline\n```\n\n# Variational Autoencoder\n\nThe variational autoencoder (VAE) was described in its current form in [Kingma 2013](https://arxiv.org/abs/1312.6114). The model consists of an encoder/inference network $q_{\\phi}(z|x)$ and a decoder/generative network $p_{\\theta}(x|z)$. The main idea is that it is possible to both reconstruct and generate samples from from some input distribution by learning a variational distribution over the latent variable $z$.\n\n\n\nThe VAE therefore has a bottleneck structure, where the input $x$ is encoded into a latent variable $z$. New data can then be generated by feeding a latent code into the generator network - $\\widehat{x} \\sim p_{\\theta}(z|x)$. The diagram above shows the generative model (right) and how the latent variable $z$ is inferred from $x$ (left).\n\nBelow we will instantiate a new variational autoencoder with this bottleneck structure consisting of a 2-layer encoder network turning an input MNIST image into a latent code: $784 \\to 256 \\to 128 \\to 32$. We also have a decoder that performs the operation in reverse: $32 \\to 128 \\to 256 \\to 784$.\n\n\n```python\nfrom semi.models import VariationalAutoencoder\nfrom semi.layers import GaussianSample\nmodel = VariationalAutoencoder([784, 32, [256, 128]])\nmodel\n```\n\n /Users/jakobhavtorn/repos/vae/semi/models/vae.py:113: UserWarning: nn.init.xavier_normal is now deprecated in favor of nn.init.xavier_normal_.\n init.xavier_normal(m.weight.data)\n\n\n\n\n\n VariationalAutoencoder(\n (encoder): Encoder(\n (hidden): ModuleList(\n (0): Linear(in_features=784, out_features=256, bias=True)\n (1): Linear(in_features=256, out_features=128, bias=True)\n )\n (sample): GaussianSample(\n (mu): Linear(in_features=128, out_features=32, bias=True)\n (log_var): Linear(in_features=128, out_features=32, bias=True)\n )\n )\n (decoder): Decoder(\n (hidden): ModuleList(\n (0): Linear(in_features=32, out_features=128, bias=True)\n (1): Linear(in_features=128, out_features=256, bias=True)\n )\n (reconstruction): Linear(in_features=256, out_features=784, bias=True)\n (output_activation): Sigmoid()\n )\n )\n\n\n\nNotice how the middle most layer consists of a `GaussianSample` layer, in which we turn the input digit into the parameters of a Normal distribution with parameters $\\mu$ and $\\sigma$. This allows us to use the *reparametrization trick* to sample from this distribution to introduce stochasticity into the network.\n\n\n```python\nfrom torch.autograd import Variable\n\ngaussian = GaussianSample(10, 1)\nz, mu, log_var = gaussian(Variable(torch.ones(1, 10)))\n\nprint(f\"sample {float(z.data):.2f} drawn from N({float(mu.data):.2f}, {float(log_var.exp().data):.2f})\")\n```\n\n sample 2.28 drawn from N(0.68, 2.05)\n\n\n## Training\n\nHow do we go about training a variational autoencoder then? We want to model the data distribution $p(x)$, from here we can introduce a variational distribution $q(z|x)$ by multiplying and dividing by this distribution. Now we employ Jensen's inequality to move the logarithm inside the integral, we can do this because $\\log$ is concave and because $q(z|x)$ is a probability distribution. From here on we just rearrange and we see that a lower bound on the marginal probability of the data $p(x)$ is just an expectation over the likelihood of the data minus the KL-divergence between the variational distribution and a prior $p(z)$.\n\n\\begin{align}\n\\log p(x) &= \\log \\int p(x, z) \\ dz = \\log \\int q(z|x) \\frac{p(x, z)}{q(z|x)} \\ dz\\\\\n &\\geq \\int q(z|x) \\log \\frac{p(x, z)}{q(z|x)} \\ dz = \\int q(z|x) \\log p(x|z) + \\log \\frac{p(z)}{q(z|x)} \\ dz\\\\\n &= \\int q(z|x) \\log p(x|z) \\ dz + \\int q(z|x) \\log \\frac{p(z)}{q(z|x)} \\ dz\\\\\n &= \\mathbb{E}_{q(z|x)} [\\log p(x|z)] - KL(q(z|x)||p(z)) = \\mathcal{L}(x)\n\\end{align}\n\nTo make things even more concrete, we show how we can go from this equation to an actual algorithm. Recall that the expectation is just an arithmetic mean, which can be approximated using Monte-Carlo samples. In fact for most applications we can do with just a single sample, even though this provides infinite variance.\n\n$$\\mathbb{E}_{q(z|x)} [\\log p(x|z)] = \\lim_{N \\to \\infty} \\frac{1}{N} \\sum_{i=1}^{N} \\log p(x_i|z_i) \\approx \\frac{1}{M} \\sum_{i=1}^{M} \\log p(x_i|z_i)$$\n\nAs you can see the likelihood is just the log probability of the data given the latent variable, but the latent variable is itself derived from the data - we can just use the reconstruction error! In the MNIST case, it is most fitting to use the Bernoulli / binary cross entropy.\n\nFinally, the second term is the Kullback-Leibler divergence. It states that whatever distribution we learn over $q(z|x)$ can never be very far from the prior $p(z)$. This is both good and bad news. Ideally we want a reconstruction that is as good as possible, i.e. only relying on the likelihood, which will only occur if $q(z|x) = p(z)$. This will never happen in practice as the q-distribution will be very complex in order to produce latent variables that result convincing samples. On the plus side, the KL-term acts as a regularizer that pulls the distribution towards the prior, which is the whole reason why we can create samples.\n\nThis term can either be computed analytically, or by sampling similarly to the way we did it in the expectation, which you can see in the printed doc string below.\n\n\n```python\nprint(model._kld.__doc__)\n```\n\n \n Computes the KL-divergence of\n some element z.\n \n KL(q||p) = -\u222b q(z) log [ p(z) / q(z) ]\n = -E[log p(z) - log q(z)]\n \n :param z: sample from q-distribuion\n :param q_param: (mu, log_var) of the q-distribution\n :param p_param: (mu, log_var) of the p-distribution\n :return: KL(q||p)\n \n\n\n\n```python\nfrom semi.data import get_mnist\n\n_, loader_train, loader_validation = get_mnist(location=\"./\", batch_size=64, num_workers=0)\n\n# We use this custom BCE function until PyTorch implements reduce=False\ndef binary_cross_entropy(r, x):\n return -torch.sum(x * torch.log(r + 1e-8) + (1 - x) * torch.log(1 - r + 1e-8), dim=-1)\n\noptimizer = torch.optim.Adam(model.parameters(), lr=3e-4, betas=(0.9, 0.999))\n```\n\n\n```python\nfor epoch in range(50):\n model.train()\n total_loss, m = 0, 0\n for (u, _) in tqdm(loader_train):\n u = Variable(u)\n\n if cuda:\n u = u.cuda(device=0)\n\n reconstruction = model(u)\n \n likelihood = -binary_cross_entropy(reconstruction, u)\n kl_divergence = model.kl_divergence\n\n elbo = likelihood - kl_divergence\n \n loss = -torch.mean(elbo)\n\n loss.backward()\n optimizer.step()\n optimizer.zero_grad()\n\n total_loss += loss.item()\n\n m += 1\n\n print(f\"Epoch: {epoch}\\t ELBO: {total_loss / m:.2f}\\tLL: {likelihood.mean():.2f}\\tKL: {kl_divergence.mean():.2f}\")\n```\n\n \n \n 0%| | 0/938 [00:00 0.5*rmax)[0]\nFWHMmin=wavefilter[nnn[0]]\nFWHMmax=wavefilter[nnn[-1]]\nfilterwid=FWHMmax-FWHMmin #nm, FWHM\n```\n\n ../obs/filters/bessell_V.par\n\n\n# define wavelength array,\n- cover the range of 350nm to 1050nm, depend on the spectral resolution wanted. \n\n\n```python\n#specr0=2000 ; no unit\n#if(keyword_set(specr)) then specr0=specr\nsampling=2.0 #pixels per spectral resolution element ?1D or 2D/linear or area?\n\n#if(keyword_set(specsample)) then sampling=specsample\n\n#delta_lambda=500.0/specr0/sampling ; in the same unit as lambda0\ndelta_lambda=0.1755555\n#if(keyword_set(deltal)) then delta_lambda=deltal # has to be in unit of nm\nnarray=int((1000.0-350.0)/delta_lambda) \n#figure out the wavelength array length, from 350nm to 1000nm, spacing at delta_lambda\nwavearr=350.0+delta_lambda*pl.frange(narray-1)\n# select out the array of V band filter\nii=np.logical_and(wavearr >= vmin, wavearr <= vmax)\nwavetmp2=wavearr[ii]\nx=np.interp(wavetmp2,wavefilter,fluxfilter)\nintegratef4=x*wavetmp2\n```\n\n /share/soft/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:12: MatplotlibDeprecationWarning: numpy.arange\n if sys.path[0] == '':\n\n\n\n```python\ndef integral(x,y):\n nn=len(x)\n \n dx=x[1:]-x[:-1]\n yy=0.5*(y[1:]+y[:-1])\n \n return np.sum(dx*yy)\n \n```\n\n\n```python\nintegconst=integral(wavetmp2, integratef4) # int(lambda*Rlambda*dlambda)\n```\n\n\n```python\nintegconst\n```\n\n\n\n\n 44654.97566743437\n\n\n\n# some less basic parameters, may change, but not often\n- qsys=0.10\t; throughput of the whole system, should be a function of lambda\n\n\n```python\nthroughput=pd.read_csv('../obs/IFU_throughput.dat',sep='\\s+',header=None,skiprows=1)\nlambdaq=np.array(throughput[8])\nqtot=np.array(throughput[9]) #; throughput of the whole system,\n#if(not keyword_set(qinput)) then qinput=1.0\nqinput=1.0\nqe=0.8 \n#;assuming the total throughput cannot reach the theory value, 0.3 is the upper limit. \nqtot[qtot>=0.3]=0.3 \nq=qtot*qinput #*qe ;qtot of CSST already includes the CCD efficiency \nfovsp=0.2 # diameter of fiber (or spaxel) in arcsec ?\n#fov2=3.14159/4.0*(0.2)^2 ; fiber area in (arcsec)^2\nfov2=(fovsp)**2 #*3.14159/4.0 ; fiber (or spaxel) area in (arcsec)^2\n#if(keyword_set(fovp)) then fov2=(fovp)^2;*3.14159/4.0\n# for slit (point source)\n#if(keyword_set(slitwidth)) then fov2=1 \nslitunit=0.074 # arcsec. the length of slit which conresponds to a pixel length on IFU CCD \n```\n\n# SKY\n - define V band sky brightness\n\n\n```python\niskyv0=22.5 # in Johnson V mag/arcsec^2 unit\n#if(keyword_set(skyv)) then iskyv0=skyv\nlambdav=filtereff #in nm\n\n#sky brightness corresponding to this sky magnitude\niskyv0_jy=3631.0*10**(-iskyv0/2.5+3.0) # sky flux in V in mJy/arcsec^2 unit\niskyv0_nm=iskyv0_jy*3.0/(lambdav/100.0)**2 #sky flux in V in 10^(-13)erg/s/cm^2/nm (/arcsec^2 ?)\n\n#readin the ground sky spectrum \nskybg_50=pd.read_csv('../obs/skybg_50_10.dat',sep='\\s+',header=None,skiprows=14)\nwavesky=np.array(skybg_50[0])\nfluxsky1=np.array(skybg_50[1])\nfluxsky2=fluxsky1/wavesky*1.98 #change the sky flux unit to 10^(-13)erg/s/cm^2/nm/arcsec^2\n\n#This fluxsky is in unit of phot/s/nm/arcsec^2/m^2, to convert it to F_lambda/arcsec^2, \n#need to do fluxsky(phot/s/nm/arcsec^2/m^2)*h(6.625*10^{-27}erg.s)*nu(1/s)*10{-4}(m^2/cm^2)\n#=fluxsky*c(3.0*10^{17}nm/s)/lambda(nm)*6.6*10{-31} erg/s/cm^2/nm/arcsec^2\n#=fluxsky/lambda*1.98*10^{-13}erg/s/cm^2/nm/arcsec^2 \n\n#find out the normalization of the sky,\n#normalization=iskyv0_nm*(integrate(bandpass*lambda*dlambda)/integrate(bandpass*lambda*F_sky_lambda*dlambda))\nii=np.logical_and(wavesky >= vmin, wavesky <= vmax)\nwavetmp=wavesky[ii]\nfluxtmp=fluxsky1[ii]\n\nx=np.interp(wavetmp,wavefilter,fluxfilter)\nvfluxtmp=x*fluxtmp*1.98 #bandpass*lambda*F_sky_lambda(fluxsky2)=bandpass*fluxsky*1.98, x10^(-13)\nskyintegrate=integral(wavetmp, vfluxtmp)\nskynorm=iskyv0_nm*integconst/skyintegrate \nfluxsky3=np.interp(wavearr,wavesky,fluxsky2)\nfluxsky=fluxsky3*skynorm # get the sky spectrum in wavearr grid, the unit should now be the same as fluxvega: 10^(-13) erg/s/nm/cm^2 (/arcsec^2 ?)\n\nfluxskypp=fluxsky\n\nprint(fluxskypp)\n#a second way of estimating the Sky, if know the sky electron number per pixel\n\n'''\n if(keyword_set(skyperpixel)) then begin\n ;since the numbers are given by the main survey, our detected Sky electron will be less, so scale a rough factor of 0.9\n scaletemp=0.9\n ii=where(wavearr ge 255 and wavearr le 400, counta)\n fluxskypp[ii]=0.028/counta \n ii=where(wavearr ge 400 and wavearr le 600, countb)\n fluxskypp[ii]=0.229/countb \n ii=where(wavearr ge 600 and wavearr le 900, countc)\n fluxskypp[ii]=0.301/countc \n ii=where(wavearr ge 900, countd)\n fluxskypp[ii]=0.301/countd \n fluxskypp=fluxskypp/0.074^2*fov2*scaletemp\n end \n'''\n```\n\n [3.26635242e-05 5.87930256e-04 5.87513766e-04 ... 9.80157011e-04\n 1.01348412e-03 1.06961878e-03]\n\n\n\n\n\n '\\n if(keyword_set(skyperpixel)) then begin\\n ;since the numbers are given by the main survey, our detected Sky electron will be less, so scale a rough factor of 0.9\\n scaletemp=0.9\\n ii=where(wavearr ge 255 and wavearr le 400, counta)\\n fluxskypp[ii]=0.028/counta \\n ii=where(wavearr ge 400 and wavearr le 600, countb)\\n fluxskypp[ii]=0.229/countb \\n ii=where(wavearr ge 600 and wavearr le 900, countc)\\n fluxskypp[ii]=0.301/countc \\n ii=where(wavearr ge 900, countd)\\n fluxskypp[ii]=0.301/countd \\n fluxskypp=fluxskypp/0.074^2*fov2*scaletemp\\n end \\n'\n\n\n\n# define basic target brightness, parameters constantly change\n\n\n```python\nitarget=22. # in Johnson V mag/arcsec^2 unit\n#if(keyword_set(targetmag)) then itarget=targetmag\nitarget_jy=3631.0*10**(-itarget/2.5+3.0) # target flux in V in mJy/arcsec^2 unit\nitarget_nm=itarget_jy*3.0/(lambdav/100.0)**2 #target flux in V in 10^(-13)erg/s/cm^2/nm (/arcsec^2 ?)\n\n#readin the galaxy spectrum\n'''\n ; readcol, '../obs/allgalaxy.dat', wavegal, eflux, s0f, saf, sbf, scf, /silent\n ; wavegal=wavegal*1000.0\n ; spectype0=4\t;default use Sb galaxy template spectrum\n ; if(keyword_set(spectype)) then spectype0=spectype ;unless specified\n ; if(spectype0 eq 1) then galflux1=eflux\n ; if(spectype0 eq 2) then galflux1=s0f\n ; if(spectype0 eq 3) then galflux1=saf\n ; if(spectype0 eq 4) then galflux1=sbf\n ; if(spectype0 eq 5) then galflux1=scf\n'''\n\ntplfile='../obs/SFgal_tpl/SFgal_texp_FeH0_tau5_Ew10.fits'\n# if(keyword_set(galtpl)) then tplfile='../obs/SFgal_tpl/'+galtpl\nsfgal=fits.open(tplfile)\nwavegal=sfgal[1].data['wave']/10. #change A to nm\ngalflux2=sfgal[1].data['flux']\ngalflux1=np.interp(wavearr,wavegal,galflux2)\n\n#;normalize the galaxy spectrum to the V band magnitude specified.\nii=np.logical_and(wavegal >= vmin, wavegal <= vmax)\nwavetmp=wavegal[ii]\nfluxtmp=galflux2[ii]\nx=np.interp(wavetmp,wavefilter,fluxfilter)\nvfluxtmp=x*wavetmp*fluxtmp #bandpass*lambda*F_gal_lambda\ngalintegrate=integral(wavetmp,vfluxtmp)\ngalnorm=itarget_nm*integconst/galintegrate\ngalflux=galnorm*galflux1 # the unit should now be in 10^(-13)erg/s/nm/cm^2 (/arcsec^2 ?)\n```\n\n\n```python\nsfgal.info()\n```\n\n Filename: ../obs/SFgal_tpl/SFgal_texp_FeH0_tau5_Ew10.fits\n No. Name Ver Type Cards Dimensions Format\n 0 PRIMARY 1 PrimaryHDU 4 () \n 1 1 BinTableHDU 16 5138R x 4C ['D', 'D', 'D', 'D'] \n\n\n\n```python\nsfgal[1].columns\n```\n\n\n\n\n ColDefs(\n name = 'wave'; format = 'D'\n name = 'flux'; format = 'D'\n name = 'stellar'; format = 'D'\n name = 'gas_emission'; format = 'D'\n )\n\n\n\n\n```python\nsfgal[1].header\n```\n\n\n\n\n XTENSION= 'BINTABLE' / binary table extension \n BITPIX = 8 / array data type \n NAXIS = 2 / number of array dimensions \n NAXIS1 = 32 / length of dimension 1 \n NAXIS2 = 5138 / length of dimension 2 \n PCOUNT = 0 / number of group parameters \n GCOUNT = 1 / number of groups \n TFIELDS = 4 / number of table fields \n TTYPE1 = 'wave ' \n TFORM1 = 'D ' \n TTYPE2 = 'flux ' \n TFORM2 = 'D ' \n TTYPE3 = 'stellar ' \n TFORM3 = 'D ' \n TTYPE4 = 'gas_emission' \n TFORM4 = 'D ' \n\n\n\n# define observation information, parameters constantly change\n\n\n```python\nobst=300.0 # in seconds, single integration time\n#if(keyword_set(obstime)) then obst=obstime\nrepn=1.0 # repeating time\n#if(keyword_set(repeatnum)) then repn=repeatnum\nnpixw=3.0\n#if(keyword_set(npixel_width)) then npixw=npixel_width\n# sky of slit area (slitwidth*npixw*slitlength) will go into the CCD\n#if(keyword_set(slitwidth)) then fluxskypp=fluxskypp*slitwidth*npixw*slitunit \n \nexpf2=np.zeros(narray)\nexpfemi=np.zeros(narray)\nsnarray=np.zeros(narray)\nmockgal=np.zeros(narray)\ntmp=np.zeros(narray)\nlista=np.zeros(narray*10).reshape(narray,10)\n```\n\n\n```python\nfor i in range(narray):\n lambda0=wavearr[i]\n qlambda=np.interp(lambda0,lambdaq,q)\n hv=planckh*cc/lambda0 #;10^{-10}erg\n delta_hz=cc*delta_lambda/lambda0/lambda0*sampling #;10^17 1/s\n \n #now that many fluxes are in 10^(-13)erg/s/nm/cm^2, to convert it to Jy, need to multiple: \n #lambda0^2/c(in nm)=lambda0^2(nm)/(3.*10^(17))*10^(-13)erg/s/Hz/cm^2\n #=lambda^2(nm)*3.33*10^(-31)erg/s/Hz/cm^2=lambda^2(nm)*3.33*10^(-8)Jy\n \n #find out sky value at lambda0 \n #calculate n_sky/pixel\n isky=fluxsky[i]*lambda0**2*0.0333*fov2 #in uJy/spaxel unit\n iskyall=isky*telarea/1000.0 #in 10-26 erg/s/Hz /spaxel\n fsky=qlambda*iskyall*delta_hz #10^{-9} erg/s /spaxel\n nsky=fsky/hv*10.0 #in unit of #e/s /spaxel\n \n '''\n if(keyword_set(skyperpixel)) then begin\n nsky=fluxskypp[i]*sampling ; #e/s in npixw*sampling pixels \n endif\n ;print, \"Sky electron counts\", nsky, nsky0, fluxskypp[i]\n '''\n \n #calculate n_source/pixel\n isource=galflux[i]*lambda0**2*0.0333*fov2 #in uJy/spaxel unit\n isall=isource*telarea/1000.0 #in 10-26 erg/s/Hz /spaxel\n fs=qlambda*isall*delta_hz #10^{-9} erg/s /spaxel\n ns=fs/hv*10.0 #in unit of #e/s /spaxel\n #print, \"Source electron counts\", ns\n \n darkn=(darkc*repn*obst*npixw*sampling)\n rnn2=rn**2*(repn*npixw*sampling)\n sourcenn=(ns*repn*obst)\n skynn=(nsky*repn*obst)\n tmp[i]=skynn\n \n #nn1=sqrt(2.0*rnn^2+2.0*darkn^2+sourcenn^2+2.0*skynn^2)\n #nn1=sqrt(rnn^2+darkn^2+sourcenn^2+skynn^2)\n nn1=np.sqrt(rnn2+darkn+skynn+sourcenn) #total noise\n sn1=repn*ns*obst/nn1 #S/N\n snarray[i]=sn1\n #nn=sqrt(2.0*rnn^2+2.0*darkn^2+2.0*skynn^2)\n nn=np.sqrt(rnn2+darkn+skynn) #system noise\n #print, \"total noise, system noise, sn, readnoise, dark, source, sky\", nn1, nn, sn1, rnn, darkn, sourcenn, skynn \n \n #set the detection limit \n detlimit=1.0\n #if(keyword_set(snlimit)) then detlimit=snlimit\n #N_{source}/sqrt(N_{source}+nn^2)=detlimit, ==m\n #N_{source}^2-m^2*N_{source}-m^2*nn^2=0 (solve the equation)\n #N_{source}=(m^2+sqrt(m^4+4m^2*nn^2))/2.0\n nntmp=detlimit**2+np.sqrt(detlimit**4+4.0*detlimit**2*nn**2)\n nntmp=nntmp/2.0\n\n #calculate detection limit in uJy and mag\n \n fnn=nntmp\n f1=fnn/obst/repn #in e/s\n f2=f1*hv #in 10^{-10} erg/s\n f3=f2/delta_lambda #in 10^{-10} erg/s/nm\n f1=f3/telarea #in 10^{-10} erg/s/nm/cm^2\n f2=f1/qlambda #in 10^{-10} erg/s/nm/cm^2\n expf2[i]=f2/fov2*100000. # in 10^{-15} erg/s/nm/cm^2/arcsec^2\n #expfemi[i]=expf2[i]*delta_lambda*sampling # in 10^{-15} erg/s/cm^2/arcsec^2, the above multiplied by the spectral resolution\n #print, \"detection limit is\", f2,\"microJy/arcsec^2\"\n #print, \"detection limit is\", magf2, \"AB mag / arcsec^2\"\n mockgal[i]=galflux[i]+galflux[i]/snarray[i]*np.random.randn(1,1)[0][0] #in 10^{-13} erg/s/nm/cm^2\n\n lista[i,:]=[lambda0, sn1, galflux[i], nn1,\\\n np.sqrt(sourcenn), nn, np.sqrt(rnn2),np.sqrt(darkn), \\\n np.sqrt(skynn), mockgal[i]]\n \n```\n\n\n```python\n'''\n ;mockgal=galflux+galflux/snarray*randomn(seed,narray)\n ; plot,wavearr,galflux,xrange=[350,1000],xs=1,xtitle='lambda (nm)',ytitle='galaxy flux'\n ; oplot,wavearr,mockgal,color=250\n ; oplot,wavearr,galflux,thick=3\n ; label=strcompress(string(obst))+'s *'+strcompress(string(repn))+' times'\n ; xyouts,800,max(galflux),label,charsize=2\n ; plot,wavearr,expf2*0.01,xrange=[350,1000],xs=1,xtitle='lambda (nm)',ytitle='expected galaxy flux'\n ; oplot,wavearr,galflux,color=250\n \n ; plot,wavearr,fluxsky,xrange=[350,1000],xs=1,xtitle='lambda (nm)',ytitle='sky flux'\n ; plot,wavearr,tmp,xrange=[350,1000],xs=1,xtitle='lambda (nm)',ytitle='#photon'\n \n \n ; ii=where(wavearr ge vmin and wavearr le vmax)\n ; wavetmp=wavearr(ii)\n ; fluxtmp=expf2(ii)/100000. ;10^{-10} erg/s/cm^2/arcsec^2\n ; x=interpol(fluxfilter, wavefilter, wavetmp)\n ; vfluxtmp=x*wavetmp*fluxtmp ;bandpass*lambda*F_gal_lambda\n ; gexpintegrate=integral(wavetmp, vfluxtmp)\n ; magf2=-2.5*(alog10((gexpintegrate*(lambdav/100.0)^2)/(integconst*3631.0*3.0)))\n ; print,'magf2=',magf2\n ; plot,wavetmp,fluxtmp,xrange=[350,1000],xs=1,xtitle='lambda',ytitle='expected galaxy flux'\n'''\n```\n\n\n\n\n \"\\n ;mockgal=galflux+galflux/snarray*randomn(seed,narray)\\n ; plot,wavearr,galflux,xrange=[350,1000],xs=1,xtitle='lambda (nm)',ytitle='galaxy flux'\\n ; oplot,wavearr,mockgal,color=250\\n ; oplot,wavearr,galflux,thick=3\\n ; label=strcompress(string(obst))+'s *'+strcompress(string(repn))+' times'\\n ; xyouts,800,max(galflux),label,charsize=2\\n ; plot,wavearr,expf2*0.01,xrange=[350,1000],xs=1,xtitle='lambda (nm)',ytitle='expected galaxy flux'\\n ; oplot,wavearr,galflux,color=250\\n \\n ; plot,wavearr,fluxsky,xrange=[350,1000],xs=1,xtitle='lambda (nm)',ytitle='sky flux'\\n ; plot,wavearr,tmp,xrange=[350,1000],xs=1,xtitle='lambda (nm)',ytitle='#photon'\\n \\n \\n ; ii=where(wavearr ge vmin and wavearr le vmax)\\n ; wavetmp=wavearr(ii)\\n ; fluxtmp=expf2(ii)/100000. ;10^{-10} erg/s/cm^2/arcsec^2\\n ; x=interpol(fluxfilter, wavefilter, wavetmp)\\n ; vfluxtmp=x*wavetmp*fluxtmp ;bandpass*lambda*F_gal_lambda\\n ; gexpintegrate=integral(wavetmp, vfluxtmp)\\n ; magf2=-2.5*(alog10((gexpintegrate*(lambdav/100.0)^2)/(integconst*3631.0*3.0)))\\n ; print,'magf2=',magf2\\n ; plot,wavetmp,fluxtmp,xrange=[350,1000],xs=1,xtitle='lambda',ytitle='expected galaxy flux'\\n\"\n\n\n\n\n```python\nii=np.logical_and(wavearr >= FWHMmin , wavearr <= FWHMmax)\nwavetmp=wavearr[ii]\nif len(snarray[ii]) % 2 ==0:\n snm=sorted(list(snarray[ii]))[int(0.5*len(snarray[ii]))]\nelse:\n snm=np.median(snarray[ii]) #the median SN of FWHM range to acchieve the sn limit\nim=np.where(snarray[ii] == snm)[0]\nprint(snarra)\nfact=np.reshape(expf2[ii][im]*0.01/galflux[ii][im],1)\nprint(fact)\nfact=fact[0]\nprint(fact)\nlimitmag=-2.5*np.log10(fact)+itarget \n\n#print,limitmag\n#oplot,wavetmp, fluxtmp, color=250\nsnmean=np.median(snarray)\nprint(snmean)\n```\n\n [37.63560326]\n 37.635603262237744\n 0.05607035207145872\n\n\n\n```python\nz=0.0\n#if(keyword_set(redshift)) then z=redshift\nwaveha=656.3*(1.0+z)\nii=np.logical_and(wavearr >= (waveha-0.5) , wavearr < (waveha+0.5)) #1nm ,10A\nnii=len(np.where(ii==1)[0])\nii_1=np.logical_and(wavearr >= (waveha-10) , wavearr <= (waveha-5))\nii_2=np.logical_and(wavearr <= (waveha+10) , wavearr >= (waveha+5))\nicont=np.logical_or(ii_1==1,ii_2==1)\n\ncontrms=np.sqrt(np.sum(mockgal[icont]**2)/len(mockgal[icont]))\nprint(contrms)\nh=3*contrms*np.sqrt(nii) # hight>3 con\nw=1. # width of 10A\nlimitemif=np.sqrt(2*np.pi)*h*w #h=3*cont, w=10A \n#print,limtmag,limitemif\n```\n\n 0.008009696733561928\n\n\n\n```python\nprint('The END!')\n```\n\n The END!\n\n\n\n```python\nprint('d:', d)\nprint('darkc:', darkc)\nprint('rn:', rn)\nprint(filterfile)\nprint('sampling:', sampling)\nprint('delta_lambda:', delta_lambda)\nprint('qinput:', qinput)\nprint('fov2:', fov2)\nprint('iskyv0:', iskyv0)\nprint('fluxskypp:', fluxskypp)\nprint('itarget:', itarget)\nprint('tplfile:',tplfile)\nprint('obst:',obst)\nprint('repn:',repn)\nprint('npixw:',npixw)\nprint('fluxskypp:',fluxskypp)\nprint('z:',z)\nprint(integconst)\n```\n\n d: 200.0\n darkc: 0.017\n rn: 4\n ../obs/filters/bessell_V.par\n sampling: 2.0\n delta_lambda: 0.1755555\n qinput: 1.0\n fov2: 0.04000000000000001\n iskyv0: 22.5\n fluxskypp: [3.26635242e-05 5.87930256e-04 5.87513766e-04 ... 9.80157011e-04\n 1.01348412e-03 1.06961878e-03]\n itarget: 22.0\n tplfile: ../obs/SFgal_tpl/SFgal_texp_FeH0_tau5_Ew10.fits\n obst: 300.0\n repn: 1.0\n npixw: 3.0\n fluxskypp: [3.26635242e-05 5.87930256e-04 5.87513766e-04 ... 9.80157011e-04\n 1.01348412e-03 1.06961878e-03]\n z: 0.0\n 44654.97566743437\n\n\n# write dat\n\n\n```python\niddat=np.array(list(range(10)))\nnamedat=np.array(['lambda','S/N','tar_flux','tot_noise','sc_noise', \\\n 'sys_noise', 'readnoise','dark_noise', 'sky_noise', 'mockgal'])\nunit=np.array(['nm', ' ','1e-13 erg/s/cm2/nm',\\\n '#e','#e','#e','#e','#e','#e', '1e-13 erg/s/cm2/nm'])\n```\n\n\n```python\n[fits.Column(name=namedat[i],array=np.array(lista[:,i]),format='k') for i in range(len(namedat))]\n```\n\n\n\n\n [name = 'lambda'; format = 'k',\n name = 'S/N'; format = 'k',\n name = 'tar_flux'; format = 'k',\n name = 'tot_noise'; format = 'k',\n name = 'sc_noise'; format = 'k',\n name = 'sys_noise'; format = 'k',\n name = 'readnoise'; format = 'k',\n name = 'dark_noise'; format = 'k',\n name = 'sky_noise'; format = 'k',\n name = 'mockgal'; format = 'k']\n\n\n\n\n```python\nhdr=fits.Header()\nfor i in range(len(namedat)):\n hdr[str(i)]=unit[i]\nhun1=fits.PrimaryHDU(header=hdr)\n\nhun2=fits.BinTableHDU.from_columns([fits.Column(name=namedat[i],array=np.array(lista[:,i]),format='1E') for i in range(len(namedat))])\nhdulist = fits.HDUList([hun1,hun2])\nhdulist.writeto('noise.fits')\n```\n\n# read fits\n\n\n```python\naaa=fits.open('noise.fits')\n```\n\n\n```python\naaa.info()\n```\n\n Filename: noise.fits\n No. Name Ver Type Cards Dimensions Format\n 0 PRIMARY 1 PrimaryHDU 14 () \n 1 1 BinTableHDU 28 3702R x 10C [1E, 1E, 1E, 1E, 1E, 1E, 1E, 1E, 1E, 1E] \n\n\n\n```python\naaa[0].header\n```\n\n\n\n\n SIMPLE = T / conforms to FITS standard \n BITPIX = 8 / array data type \n NAXIS = 0 / number of array dimensions \n EXTEND = T \n 0 = 'nm ' \n 1 = '' \n 2 = '1e-13 erg/s/cm2/nm' \n 3 = '#e ' \n 4 = '#e ' \n 5 = '#e ' \n 6 = '#e ' \n 7 = '#e ' \n 8 = '#e ' \n 9 = '1e-13 erg/s/cm2/nm' \n\n\n\n\n```python\naaa[1].data\n```\n\n\n\n\n FITS_rec([(350. , 0.00696866, 0.00048873, 11.255384 , 0.2800625 , 11.2519 , 9.797959, 5.531727, 0.07240252, -0.02861151),\n (350.17557, 0.00651509, 0.00045201, 11.259164 , 0.27084038, 11.255906 , 9.797959, 5.531727, 0.30888805, 0.04023228),\n (350.3511 , 0.00598966, 0.000411 , 11.2589445, 0.25968695, 11.25595 , 9.797959, 5.531727, 0.31048384, 0.05475378),\n ...,\n (999.3798 , 0.0174631 , 0.00034563, 11.285209 , 0.44393095, 11.276474 , 9.797959, 5.531727, 0.7475748 , 0.01462729),\n (999.55536, 0.01739325, 0.00034563, 11.285915 , 0.44305614, 11.277216 , 9.797959, 5.531727, 0.75868005, 0.00467519),\n (999.7309 , 0.01732252, 0.00034563, 11.287188 , 0.44217926, 11.2785225, 9.797959, 5.531727, 0.7778652 , 0.02839942)],\n dtype=(numpy.record, [('lambda', '>f4'), ('S/N', '>f4'), ('tar_flux', '>f4'), ('tot_noise', '>f4'), ('sc_noise', '>f4'), ('sys_noise', '>f4'), ('readnoise', '>f4'), ('dark_noise', '>f4'), ('sky_noise', '>f4'), ('mockgal', '>f4')]))\n\n\n\n\n```python\n# if this file has existed, I need remove it\nif(os.path.exists('../results/noise_'+str(filtersel)+'.fits'))==1:\n os.remove('../results/noise_'+str(filtersel)+'.fits')\n```\n\n\n```python\nbbb=fits.open('../results/noise_bessell_V.par_SFgal_texp_FeH0_tau5_Ew10.fits.fits')\n```\n\n\n```python\nbbb.info()\n```\n\n Filename: ../results/noise_bessell_V.par_SFgal_texp_FeH0_tau5_Ew10.fits.fits\n No. Name Ver Type Cards Dimensions Format\n 0 PRIMARY 1 PrimaryHDU 14 () \n 1 1 BinTableHDU 28 3702R x 10C [1E, 1E, 1E, 1E, 1E, 1E, 1E, 1E, 1E, 1E] \n\n\n\n```python\nbbb[0].header\n```\n\n\n\n\n SIMPLE = T / conforms to FITS standard \n BITPIX = 8 / array data type \n NAXIS = 0 / number of array dimensions \n EXTEND = T \n 0 = 'nm ' \n 1 = '' \n 2 = '1e-13 erg/s/cm2/nm' \n 3 = '#e ' \n 4 = '#e ' \n 5 = '#e ' \n 6 = '#e ' \n 7 = '#e ' \n 8 = '#e ' \n 9 = '1e-13 erg/s/cm2/nm' \n\n\n\n\n```python\nbbb[1].columns\n```\n\n\n\n\n ColDefs(\n name = 'lambda'; format = '1E'\n name = 'S/N'; format = '1E'\n name = 'tar_flux'; format = '1E'\n name = 'tot_noise'; format = '1E'\n name = 'sc_noise'; format = '1E'\n name = 'sys_noise'; format = '1E'\n name = 'readnoise'; format = '1E'\n name = 'dark_noise'; format = '1E'\n name = 'sky_noise'; format = '1E'\n name = 'mockgal'; format = '1E'\n )\n\n\n\n\n```python\nplt.plot(bbb[1].data['lambda'],bbb[1].data['mockgal'])\n```\n\n\n```python\nplt.plot(wavearr,galflux)\n#plt.plot(wavearr,mockgal)\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "186280753eb108f0f8b050e7e15d02119cecc4d6", "size": 80324, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "prog/snppv1.0.ipynb", "max_stars_repo_name": "nanamomoko/snpp", "max_stars_repo_head_hexsha": "05c9601df9e5f6b58f533376d748e22bde56a066", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "prog/snppv1.0.ipynb", "max_issues_repo_name": "nanamomoko/snpp", "max_issues_repo_head_hexsha": "05c9601df9e5f6b58f533376d748e22bde56a066", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "prog/snppv1.0.ipynb", "max_forks_repo_name": "nanamomoko/snpp", "max_forks_repo_head_hexsha": "05c9601df9e5f6b58f533376d748e22bde56a066", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 72.298829883, "max_line_length": 23449, "alphanum_fraction": 0.728935312, "converted": true, "num_tokens": 8852, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.785308580887758, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.4414820301190715}} {"text": "# Introduction\n\nIn this tutorial, we will develop a model of semantic knowledge based on the same model from the simulation-based ACT-R [tutorial](http://act-r.psy.cmu.edu/software/). As explained below, the model organizes knowledge into a semantic network in which concepts are represented by nodes and relationships are represented by connections between nodes within the network. \n\n## Task\n\nIn the task, participants must verify whether one category is a member of another category by responding \"yes\" or \"no\". For example, a person might recieve a question such as \"Is a bird a canary?\" and must respond \"yes\" or \"no\" with the appropriate keys. \n\n# Semantic Model\n\nAs illustrated in the figure below, category relationships are represented as a semantic network in which nodes correspond to concepts and directed edges correspond to membership. The model can answer questions about category membership directly through a process called direct verification, or indirectly through a process called category chaining. For example, the model can answer the question \"Is a bird an animal?\" through direct verification by traversing from the bird node to the animal node using a single memory retrieval. By contrast, the question \"is a canary an animal?\" is answered indirectly through category chaining, which involves traversing from the canary node to the bird node and then from the bird node to the animal node using successive memory retrievals. \n\n\n\n\n## Declarative Memory\n\nDeclarative memory $M$ consists of 20 chunks with the slots, (i.e., domain), $Q = \\{\\textrm{object, attribute, value}\\}$. The object slot refers to the entity being evaluated (e.g. canary); the attribute slot refers to the particular property on which the object is being evaluated (e.g. category, flies); and the value slot refers to the particular value of the attribute (e.g. bird, true). For example, chunk 20 is $c_{20}(\\rm{object}) = \\textrm{bird}$, $c_{20}(\\textrm{attribute}) =\\textrm{category}$, $c_{20}(\\textrm{value}) = \\textrm{animal}$, which indicates that the object bird belongs to the category animal. \n\n\n### Activation\n\nMemory activation for chunk $m$ is defined as \n\n\\begin{equation}\\label{eq:semantic_activation}\na_m = \\textrm{blc} + \\rho_m + \\epsilon_m\n\\end{equation}\n\nwhere $\\textrm{blc}$ is the base-level constant and $\\rho_m$ is partial matching. We will denote the expected value as $E[a_m] = \\mu_m$. The base-level constant simply scales memory activation up or down. Partial matching allows chunks to be retrieved as a function of dissimilarity to the retrieval request values, $\\mathbf{r}$. For simplicity, partial matching is defined as a weighted count of mismatching slot-values:\n\n\\begin{equation}\\label{eq:penalty_function}\n\\rho_m = -\\delta \\sum_{q \\in Q_r} I^c\\left(r(q), c_{m}(q)\\right)\n\\end{equation}\n\nwhere $Q_r = \\{\\textrm{object},\\textrm{attribute}\\}$ is the set of slots in the retrieval request, the mismatch penalty parameter $\\delta$ controls the steepness of the dissimilarity gradient, and $I^c$ is an indicator function: \n\n$$ I^c(x,y) =\n \\begin{cases}\n 1 & x \\neq y\\\\\n 0 & x = y\n \\end{cases}\n$$\n\nIn cases where a chunk does not have a slot corresponding to a slot in the retrieval request, we treat it as a mismatch, i.e., $I^c(x, \\emptyset)=1.$ Thus, chunks that are more similar to the retrieval request are more likely to be retrieved.\n\n## Markov Process\n\nWe will represent the Semantic model as a discrete time Markov process. In a Markov process, a system transitions from state to state, such that the next state depends only on the current state. This constraint on state transitions is known as the Markov property. As is often the case, procedural knowledge is deterministic in the model. We will omit these states, which includes stimulus encoding, to simplify the model, as they do not change the predictions. Instead, we focus on states that are the product of stochastic declarative memory processes. Thus, the state space for the Semantic model is \n\n$S = \\{s_{\\rm ir}, s_{\\rm cc1}, s_{\\rm cc2}, s_{\\rm yes}, s_{\\rm no}\\}$\n\nThe states are defined as follows:\n- $s_{\\rm ir}$ is the initial retrieval state\n- $s_{\\rm cc1}$ is the first category chain retrieval state\n- $s_{\\rm ccs}$ is the second category chain retrieval state\n- $s_{\\rm yes}$ is the state in which the model responds \"yes\"\n- $s_{\\rm no}$ is the state in which model responds \"no\". \n\nA convenient way to visualize a Markov process is with a directed graph in which nodes represent states and edges (e.g. arrows) that indicate possible transitions. In the graph below, the process starts at $s_{\\rm ir}$ from which point it can transition to states $s_{\\rm cc1}$, $s_{\\rm yes}$, or $s_{\\rm no}$. The process terminates when a response is given, Thus, $s_{\\rm yes}$, and $s_{\\rm no}$ are absorbing states. If the model transitions from $s_{\\rm ir}$, to $s_{\\rm cc1}$, it performs second retrieval based on category chaining. From $s_{\\rm cc1}$, the model can transition to $s_{\\rm yes}$ or $s_{\\rm no}$ where a response is made, or to $s_{\\rm cc2}$ where a third memory retrieval is performed through category chainining. From $s_{\\rm cc2}$, the model can only transition to $s_{\\rm yes}$ or $s_{\\rm no}$. In principle, there is no limit to the number of category chain states, but for the stimuli consider here, the maximum is two.\n\n\n\n\n## Retrieval Probability\n\nAlthough many transitions in the model are deterministic based on the assumption of well-established procedural knowledge, other transitions are stochastic due to noise in declarative memory processes. In those cases, the probability of retrieving chunk $m$ given retrieval request $\\mathbf{r}$ is computed from the approximation reported in Weaver (2008):\n\n$\\Pr(\\mathbf{c}_m; \\mathbf{r}) = \\frac{e^{\\mu_m/\\sigma}}{\\sum_{k|\\mathbf{c}_k \\in M} e^{\\mu_k/\\sigma} + e^{\\mu_{m^\\prime}/\\sigma}},$\n\nwhere $\\sigma = s\\sqrt{2}$ controls activation noise and $s$ is the logistic scale parameter. Typically, we are interested in the probability of transitioning into a state rather than the probability of retrieving a particular chunk. Let $R_{s_i, s_j}$ be the set of chunks that would result in a transition from a state $s_i$ to a state, $s_j$. The probability of transitioning from $s_i$ to $s_j$ is:\n\n$\\Pr(s_i\\rightarrow s_j; \\mathbf{r}) = \\frac{ \\sum_{m|\\mathbf{c}_m \\in R_{s_i, s_j}}e^{\\mu_m/\\sigma}}{\\sum_{k|\\mathbf{c}_k \\in M} e^{\\mu_k/\\sigma} + e^{\\mu_{m^\\prime}/\\sigma}}.$\n\n\n## Initial State Vector\n\nAfter encoding the stimulus into chunk $\\mathbf{c}_{s,\\textrm{imaginal}}$, the Markov process begins in state $s_{\\rm ir}$ with probability 1. Formally, this is represented with the following initial state vector:\n\n\n \nUpon entering the state $s_{\\rm ir}$, the initial retrieval request $\\mathbf{r} = \\{(\\rm object, \\mathbf{c}_{s, \\textrm{imaginal}}(\\rm object), (attribute, category)\\}$ is issued to declarative memory.\n\n## Transition Matrix\n\nThe probabilities of transiting from state to state is organized into a matrix. Let $\\mathbf{P}$ be the transition probability matrix, defined as: \n\n\n\nWe use $p_{r,c}$ to denote the probability of transitioning from state $r$ to state $c$. Note that $s_{\\rm yes}$ and $s_{\\rm no}$ are absorbing states, so their self-transitions occur with probability 1 and the probabilities of transitioning to other states are zero. After encoding the stimulus into chunk $\\mathbf{c}_{s,\\rm imaginal}$, the Markov process begins in state $s_{\\rm ir}$ with probability 1, where the initial retrieval request $\\mathbf{r} = \\{(\\rm object, c_{s,\\rm imaginal}(\\rm object)), (attribute, category)\\}$ is issued to declarative memory. Depending on the outcome of the initial retrieval request, the model can transition to state $s_{\\rm cc1}$ with probability $p_{1,2}$, state $s_{\\rm yes}$ with probability $p_{1,4}$, or state $s_{\\rm no}$ with probability $p_{1,5}$. The mapping between the results of the retrieval request and $s_{\\rm yes}$ is given by:\n\\begin{equation}\nR_{{\\rm ir},{\\rm yes}} = \\{\\mathbf{c}_m \\in M: \\forall {q\\in Q}, p_{j}(q) = c_m(q) \\},\n\\label{eq:map_yes}\n\\end{equation}\nwhere $\\mathbf{p}_j = \\{(\\rm object, c_{s,\\rm imaginal}(\\rm object)), (\\rm attribute, category),(value, c_{s,\\rm imaginal}(category))\\}$. $R_{{\\rm ir},{\\rm yes}}$ contains a single chunk because $\\mathbf{p}_j$ requires an exact match. The mapping between the result of the retrieval request and a category chain state is given by: \n\\begin{equation}\nR_{{\\rm ir},{\\rm cci}} = \\{\\mathbf{c}_m \\in M: \\forall {q\\in Q}, p_k(q) = c_m(q) \\},\n\\label{eq:map_cc}\n\\end{equation}\nwhere $\\mathbf{p}_k = \\{(\\rm object, c_{s,\\rm imaginal}(\\rm object)), (\\rm attribute, category), (\\rm value, \\neg c_{s,\\rm imaginal}(\\rm category)\\}$ denotes the conditions for production rule $k$. The mapping between the result of the retrieval request and $s_{\\rm no}$ is given by:\n\\begin{equation}\nR_{{\\rm ir},{\\rm no}} = \\{\\mathbf{c}_m \\in M: \\exists {q \\in Q_{j}} \\textrm{ s.t. } p_j(q) \\neq c_m(q)\\} \\cup \\mathbf{c}_{m^\\prime},\n\\label{eq:map_no}\n\\end{equation}\nwhere $Q_{j} = \\{\\rm object, attribute\\}$. If the model transitions into absorbing states $s_{\\rm yes}$ or $s_{\\rm no}$, a response is emitted and the model terminates. However, if the model transitions to state $s_{\\rm cc1}$, a new retrieval request $\\mathbf{r}_{\\rm cc1}$ is formed by assigning the value of the value slot of retrieved chunk $\\mathbf{c}_{r1} \\in R_{{\\rm ir}, {\\rm cc1}}$, which we denote $c_{r1}({\\rm value})$, to the value of the object slot in the new retrieval request: $\\mathbf{r}_{\\rm cc1} = \\{\\rm (object, c_{r1}(value)), (attribute, category)\\}$. \n In addition, $\\mathbf{c}_{s, \\rm imaginal}$ is modified after each category chain $i \\in \\{1,2\\}$ as follows: $c_{s,\\rm imaginal}(\\rm object) = c_{ri}(value)$.\n \n For some stimuli considered here it is possible to transition to $s_{\\rm cc2}$ where a second category chain is performed. The mapping between the results of retrieval request $\\mathbf{r}_{cc1}$ and state $s_{\\rm cc2}$ is $R_{\\rm cc1,cc2}$. In state $s_{\\rm cc2}$, a new retrieval retrieval request $\\mathbf{r}_{cc2} = \\{\\rm (object, c_{r2}), (attribute,category) \\}$ is formed from retrieved chunk $c_{r2} \\in R_{\\rm cc1,cc2}$. \n\n### A worked example \n\nAs a concrete example, consider the question for canary-animal in which the model arrives at the correct answer \"yes\" through the process of category chaining. The model encodes the stimulus into chunk $\\mathbf{c}_{s,\\rm imaginal} = \\{\\rm (object, canary), (category, animal)\\}$ before issuing retrieval request $\\mathbf{r} = \\{(\\textrm{object},\\textrm{canary}), (\\textrm{attribute},\\textrm{category})\\}$. Let $\\mathbf{p}_{j} = \\{(\\textrm{object},\\textrm{canary}), (\\textrm{attribute},\\textrm{category}), (\\textrm{value}, \\textrm{animal})\\}$ be the conditions for the \"yes\" production rule, and $\\mathbf{p}_{k} = \\{(\\textrm{object},\\textrm{canary}), (\\textrm{attribute},\\textrm{category}), (\\textrm{value}, \\lnot \\textrm{animal})\\}$ be the conditions for the category chaining production rule. Using the mappings defined above, the probability of transitioning from the initial retrieval state, $s_{\\rm ir},$ to the first category chain retrieval state, $s_{\\rm cc1},$ is given by,\n$p_{1,2} = \\frac{\\sum_{m|\\mathbf{c}_m \\in R_{\\rm ir, cc1}} e^{\\mu_m/\\sigma}}{\\sum_{k|\\mathbf{c}_k \\in M} e^{\\mu_k/\\sigma} + e^{\\mu_{m^\\prime}/\\sigma}}.$ \n\nThe probability of transitioning from the initial retrieval state, $s_{\\rm ir}$, to a no response, $s_{\\rm no},$ is given by,\n\n$p_{1,5} = \\frac{\\sum_{m|\\mathbf{c}_m \\in R_{\\rm ir,no}} e^{\\mu_m/\\sigma}}{\\sum_{k|\\mathbf{c}_k \\in M} e^{\\mu_k/\\sigma} + e^{\\mu_{m^\\prime}/\\sigma}}.$\n\nThe model cannot transition from $s_{\\rm ir}$ to $s_{\\rm yes}$ to verify that a canary is an animal upon the first retrieval because $R_{\\rm ir,yes} = \\emptyset$ at this point in the process. Thus, $p_{1,4} = 0$, which implies $p_{1,5} = 1 - p_{1,2}$. \n\nLet us assume that the model transitions from $s_{\\rm ir}$ to $s_{\\rm cc1}$ by retrieving $\\mathbf{c}_{r1} = \\{(\\rm object, canary),(\\rm attribute, category), (\\rm value, bird)\\}$. A new retrieval request $\\mathbf{r}_{\\rm cc1} = \\{(\\rm object, bird),(\\rm attribute, category) \\}$ is used to perform a retrieval for category chaining and the chunk in the imaginal buffer becomes $\\mathbf{c}_{s,\\rm imaginal} = \\{\\rm (object, bird), (category, animal)\\}$. From $s_{\\rm cc1}$, the model can transition to $s_{\\rm cc2}$ with probability $p_{2,3}$, to $s_{\\rm yes}$ with probability $p_{2,4}$, or to $s_{\\rm no}$ with probability $p_{2,5}$ depending on the result of the retrieval request $\\mathbf{r}_{cc1}$. The probability of $p_{2,3} = 0$ because $R_{\\rm cc1,cc2} = \\emptyset$, i.e., there are no chunks in memory for which a bird is not an animal. The probability of transitioning to $s_{\\rm yes}$ is,\n\n$p_{2,4} = \\frac{\\sum_{m|\\mathbf{c}_m \\in R_{\\rm cc1,yes}} e^{\\mu_m/\\sigma}}{\\sum_{k|\\mathbf{c}_k \\in M} e^{\\mu_k/\\sigma} + e^{\\mu_{m^\\prime}/\\sigma}},$\n\nwhich is obtained by retrieving $\\mathbf{c}_{r2} = \\{(\\rm object, bird), (\\rm attribute, category), (\\rm value, animal)\\}$. The complementary probability of transitioning to $s_{\\rm no}$ is obtained by:\n\n$p_{2,5} = \\frac{\\sum_{m|\\mathbf{c}_m \\in R_{\\rm cc1, no}} e^{\\mu_m/\\sigma}}{\\sum_{k|\\mathbf{c}_k \\in M} e^{\\mu_k/\\sigma} + e^{\\mu_{m^\\prime}/\\sigma}} = 1-p_{2,4}$\n\n#### Generating the Transition Matrix\n\nIn the code block below, we will generate the transition matrix for the question \"Is a canary an animal?\". Before doing so, we will first load the required dependencies.\n\n\n```julia\ncd(@__DIR__)\nusing Pkg\nPkg.activate(\"../../\")\nusing StatsPlots, ACTRModels, Distributions, Turing, NamedArrays\ninclude(\"Semantic_Model.jl\")\nRandom.seed!(354301);\n```\n\n \u001b[32m\u001b[1m Activating\u001b[22m\u001b[39m environment at `~/.julia/dev/ACTRFundamentalTools/Project.toml`\n\n\nIn the following example, the transition matrix is generated with the function `transition_matrix`. We begin by defining the `stimulus` and the model parameters in `parm`. Next we create an ACT-R model object and pass it to to `transition_matrix`. Lastly, we add labels to the transition matrix with `NamedArray`.\n\n\n```julia\nblc = 1.0\nparms = (noise = true, \u03c4 = 0.0, s = 0.2, mmp = true, \u03b4 = 1.0)\nstimulus = (object = :canary, category = :animal, ans = :yes)\n# populate declarative memory\nchunks = populate_memory()\n# create declarative memory object\nmemory = Declarative(;memory=chunks)\n# create act-r object\nactr = ACTR(;declarative=memory, parms..., blc)\n# generate transition matrix\ntmat = transition_matrix(actr, stimulus, blc)\n# states: initial retrieval, category chain 1, category chain 2, respond yes and respond no\nstates = [\"Sir\", \"Scc1\", \"Scc2\", \"Syes\", \"Sno\"]\n# name the rows and columns\nNamedArray(tmat, (states, states), (\"t\",\"t+1\")) |> x->round.(x, digits=2)\n```\n\n\n\n\n 5\u00d75 Named Matrix{Float64}\n t \u2572 t+1 \u2502 Sir Scc1 Scc2 Syes Sno\n \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n Sir \u2502 0.0 0.8 0.0 0.0 0.2\n Scc1 \u2502 0.0 0.0 0.0 0.8 0.2\n Scc2 \u2502 0.0 0.0 1.0 0.0 0.0\n Syes \u2502 0.0 0.0 0.0 1.0 0.0\n Sno \u2502 0.0 0.0 0.0 0.0 1.0\n\n\n\n#### Response Probabilities\n\nBased on the transition matrix above, there is one path to a \"yes\" response and two paths to a \"no\" response. The probability of responding yes is \n\n$\\rm Pr(\\rm Response = yes) = p_{1,2}\\times p_{2,4} = .80 \\times .80 = .64$\n\nThe probability of responding \"no\" is \n\n$\\rm Pr(\\rm Response = no) = \\underbrace{p_{1,5}}_{\\textrm{ no on initial retrieval}} + \\underbrace{p_{1,2}\\times p_{2,5}}_{\\textrm{no on category chain}} = .20 + .80 \\times .20 = .36$\n\nAs expected, $\\rm Pr(\\rm Response = no) = 1 - \\rm Pr(\\rm Response = yes)$.\n\n# Generate Data\n\nThe data generation process uses four primary functions, shown below. Each function is annotated. The function `simulate` is the top-level function that initializes the data generation process. It accepts the following inputs\n\n- parms: a `NamedTuple` of fixed parameters\n- stimulus: a `NamedTuple` of stimulus slot-value pairs\n- n_reps: the number of trials or repetitions for a given stimulus\n- blc: keyword argument for the base-level constant parameter.\n\n`simulate` performs two primary functions. First, it initializes the ACT-R model with the desired parameters. Second, it iterates through simulated trials with the same stimulus and counts the number of correct responses `k`. `simulate` returns a `NamedTuple` containing the stimulus, the number of simulated trials, and the number of correct responses. \n\nInside the for loop of `simulate`, the function `simulate_trial` generates data for a single simulated trial. `simulate_trial` requires two inputs:\n\n- `actr`: an ACT-R model object\n- `stimulus`: a stimulus containing values for object and category\n\n`simulate_trial` performs chained retrievals until a yes or no state is researched in the markov process. On each iteration of the while loop the model performs the following actions:\n\n1. retrieve a chunk\n2. evaluate whether to respond yes, respond no, or to chain the category in a subsequent memory retrieval\n\nIf the a retrieval failure occurs, `k` remains zero indicating a no response, `retrieving` is set to false and the simulation terminates for the present trial. If the function `direct_verify` returns true, the model responds \"yes\". `direct_verify` corresponds to production rule $p_j$ described above. In this case, `k` = 1 and `retrieving` is set to false to terminate the simulation. If `chain_category` returns true, the model chains the category and attempts an additional memory retrieval by seting the object slot of the probe to the value of the object slot in the retrieved chunk. The remaining condition is that a mismatching chunk was retrieved, in which case `retrieving` is set to false, and `k` remains zero, indicating a \"no\" response. \n\n\n```julia\nfunction simulate(parms, stimulus, n_reps; blc)\n # populate declarative memory\n chunks = populate_memory()\n # generate declarative memory object\n memory = Declarative(;memory=chunks)\n # generate ACTR object\n actr = ACTR(;declarative=memory, parms..., blc)\n # the number of correct responses\n k = 0\n # count the number of correct answers, k\n for rep in 1:n_reps\n k += simulate_trial(actr, stimulus)\n end\n # return data that constains stimulus information, number of trials, \n # and correct answers\n return (stimulus..., N = n_reps, k = k)\nend\n\nfunction simulate_trial(actr, stimulus)\n retrieving = true\n # create memory probe or retrieval request\n probe = stimulus\n chunks = actr.declarative.memory\n # k = 1 if answer is \"yes\", 0 otherwise\n k = 0\n while retrieving\n # generate retrieval probabilities\n p,_ = retrieval_probs(actr; object=probe.object, attribute=:category)\n # sample a chunk index proportional to retrieval probabilities\n idx = sample(1:length(p), weights(p))\n # Last element corresponds to a retrieval failure\n # stop retrieval processes\n if idx == length(p)\n retrieving = false\n # retrieved chunk matches retrieval request, stop retrieving\n # and set k = 1 for \"yes\" response\n elseif direct_verify(chunks[idx], probe)\n retrieving = false\n k += 1\n # perform another retrieval with category chaining\n # modify the retrieval request based on the retrieved chunk\n elseif chain_category(chunks[idx], probe)\n probe = delete(probe, :object)\n probe = (object = chunks[idx].slots.value, probe...)\n # no chunks match, stop retrieving and respond \"no\" with k = 0\n else\n retrieving = false\n end\n end\n return k\nend\n\n\"\"\"\nAnswer yes via direct verification if retrieved chunk matches\nprobe on the object slot, the attribute slot equals category and the \nvalue slot matches the value of the probe's category slot\n\"\"\"\nfunction direct_verify(chunk, probe)\n return match(chunk, object=probe.object,\n value=probe.category, attribute=:category)\nend\n\n\"\"\"\nChain category if retrieved chunk matches\nprobe on the object slot, the attribute slot equals category and the \nvalue slot does not match the value of the probe's category slot\n\"\"\"\nfunction chain_category(chunk, probe)\n return match(chunk, ==, !=, ==, object=probe.object,\n value=probe.category, attribute=:category)\nend\n```\n\n\n\n\n chain_category\n\n\n\nWe will use the questions \"Is a canary a bird\" and \"Is a canary an animal\" as the stimuli for the simulation. In the code below, 10 trials are generated for each question. The data are formated as an array of `NamedTuples` where\n\n- object: is the object slot\n\n- category: the category slot\n\n- ans: the correct answer\n\n- N: the number of trials\n\n- k: the number of trials in which yes is given as a response\n\n\n```julia\n# true blc value\nblc = 1.0\n# fixed parameters\nparms = (noise = true, \u03c4 = 0.0, s = .2, mmp = true, \u03b4 = 1.0)\nstimuli = get_stimuli()\nn_reps = 10\ndata = map(x -> simulate(parms, x, n_reps; blc), stimuli)\n```\n\n\n\n\n 2-element Vector{NamedTuple{(:object, :category, :ans, :N, :k), Tuple{Symbol, Symbol, Symbol, Int64, Int64}}}:\n (object = :canary, category = :bird, ans = :yes, N = 10, k = 8)\n (object = :canary, category = :animal, ans = :yes, N = 10, k = 7)\n\n\n\n## Define Likelihood Function\n\nThe probability of responding \"Yes\" is obtained from the stationary distribution vector, $\\boldsymbol{\\pi} \\in \\mathbb{R}^{1XN}$. The stationary distribution vector describes the limiting behavior of the Markov process, such that $\\boldsymbol{\\pi} = \\lim_{n \\to \\infty} \\mathbf{s} \\mathbf{P}^n$ and $\\boldsymbol{\\pi} = \\boldsymbol{\\pi}{P}$. The probability of responding \"Yes\" is given by $\\theta = \\pi_4$. The likelihood of responding \"Yes\" on $k$ trials out of a total of $n_t$ trials is given by the binomial likelihood function:\n\n$\\mathcal{L}(\\theta ; k, n_t) = {n_t \\choose k} \\theta^k (1-\\theta)^{n_t-k} $\n\n\nThe annotated code below impliments the likelihood function defined mathematically above. The function `computeLL` is the top-level function that computes the log likelihood across all data. The function `initial_state` generates the initial state vector. `transition_matrix` generates the transition matrix and the function `probability_yes` computes the log likelihood of $k$ \"yes\" responses out of $N$ trials using the binomial log PDF function. The most complex part of the code is the function `transition_matrix`. At a high level, the code matches the formal discription above, which involves the following steps\n\n1. compute a vector p that represents the retrieval probability of each chunk\n2. define the mapping between states by dividing chunk indices into sets that correspond to direct verification (yes response), category chain, and mismatching chunks (no response)\n3. request chunk that matches category chain production rule\n4. modify probe/retrieval request for the next category chain\n5. continue steps 1-4 while there is a chunk that matches the category chain production rule\n\n\n\n```julia\nusing Parameters, StatsBase, NamedTupleTools\n\nimport Distributions: logpdf, loglikelihood\n\nstruct Semantic{T1,T2} <: ContinuousUnivariateDistribution\n blc::T1\n parms::T2\nend\n\nloglikelihood(d::Semantic, data::Array{<:NamedTuple,1}) = logpdf(d, data)\n\nSemantic(;blc, parms) = Semantic(blc, parms)\n\nfunction initial_state(blc)\n s0 = zeros(typeof(blc), 5)\n s0[1] = 1\n return s0\nend\n\nfunction computeLL(parms, data; blc)\n act = zero(typeof(blc))\n # populate declarative memory\n chunks = populate_memory(act)\n # create declarative memory object\n memory = Declarative(;memory=chunks)\n # create act-r object\n actr = ACTR(;declarative=memory, parms..., blc)\n # create initial state vector\n s0 = initial_state(blc)\n LL = 0.0\n for d in data\n # create transition matrix\n tmat = transition_matrix(actr, d, blc)\n # compute probability of \"yes\"\n LL += probability_yes(tmat, s0, d)\n end\n return LL\nend\n\nfunction probability_yes(tmat, s0, d)\n z = s0' * tmat^3; \u03b8 = z[4]\n # sometimes \u03b8 is nan because of exponentiation of activation\n return isnan(\u03b8) ? (return -Inf) : logpdf(Binomial(d.N, \u03b8), d.k)\nend\n\n\"\"\"\npopulatates transition matrix consisting of 5 states:\n* `s1`: initial retrieval\n* `s2 `: chain category 1\n* `s3`: chain category 2\n* `s4`: respond yes\n* `s5`: respond no\n\"\"\"\nfunction transition_matrix(actr, stim, blc)\n chunks = actr.declarative.memory\n Nc = length(chunks) + 1\n probe::typeof(stim) = stim\n probe = stim\n N = 5\n # populate transition matrix\n tmat = zeros(typeof(blc), N, N)\n # compute retrieval probabilities, p\n p,_ = retrieval_probs(actr; object=get_object(probe), attribute=:category)\n # find indices of chunks associated with direct verification, category chaining and mismatching conditions\n direct_indices = find_indices(actr, object=get_object(probe), value=get_category(probe))\n chain_indices = find_indices(actr, ==, !=, ==, object=get_object(probe), value=get_category(probe), attribute=:category)\n mismatch_indices = setdiff(1:Nc, direct_indices, chain_indices)\n # use indices to compute probability of category chain, direct verification (yes), and mismatch (no)\n tmat[1,2] = sum(p[chain_indices])\n tmat[1,4] = sum(p[direct_indices])\n tmat[1,5] = sum(p[mismatch_indices])\n # attempt to extract chunk associated with category chaining\n chain_chunk = get_chunks(actr,==,!=,==, object = get_object(probe),\n value = get_category(probe), attribute=:category)\n cnt = 1\n # continue the process above as long as category chaining can be performed.\n while !isempty(chain_chunk)\n cnt += 1\n probe = (object = get_chunk_value(chain_chunk[1]), delete(probe, :object)...)\n p,_ = retrieval_probs(actr; object=get_object(probe), attribute=:category)\n direct_indices = find_indices(actr, object=get_object(probe), value=get_category(probe))\n chain_indices = find_indices(actr, ==, !=, ==, object=get_object(probe), value=get_category(probe), attribute=:category)\n mismatch_indices = setdiff(1:Nc, direct_indices, chain_indices)\n tmat[cnt,2] = sum(p[chain_indices])\n tmat[cnt,4] = sum(p[direct_indices])\n tmat[cnt,5] = sum(p[mismatch_indices])\n chain_chunk = get_chunks(actr,==,!=,==, object = get_object(probe),\n value = get_category(probe), attribute=:category)\n end\n # set self-transitions to 1 if row i sums to 0.0\n map(i -> sum(tmat[i,:]) == 0.0 ? (tmat[i,i] = 1.0) : nothing, 1:size(tmat, 2))\n return tmat\nend\n```\n\n\n\n\n transition_matrix\n\n\n\n## Define Model\n\nThe prior distributions and model is summarized as follows:\n\n\\begin{align}\n\\rm blc \\sim Normal(1,1)\n\\end{align}\n\n\\begin{align}\n\\theta = \\mathbf{\\pi_4}\n\\end{align}\n\n\\begin{align}\nk \\sim \\rm Binomial(\\theta, N)\n\\end{align}\n\nwhere $ \\mathbf{\\pi}$ is the stationary distribution and index 4 corresponds to state $s_{\\rm yes}$. In computer code, the model is specified as follows:\n\n\n```julia\n@model model(data, parms) = begin\n blc ~ Normal(1.0, 1.0)\n data ~ Semantic(blc, parms)\nend\n```\n\n\n\n\n model (generic function with 1 method)\n\n\n\n## Estimate Parameters\n\nNow that the priors, likelihood and Turing model have been specified, we can now estimate the parameters. In the following code, we will run four MCMC chains with the NUTS sample for 2,000 iterations and omit the first 1,000 warmup samples. \n\n\n```julia\n# Settings of the NUTS sampler.\nn_samples = 1500\nn_adapt = 1500\nspecs = NUTS(n_adapt, 0.65)\nn_chains = 4\nchain = sample(model(data, parms), specs, MCMCThreads(), n_samples, n_chains, progress=true)\n```\n\n \u250c Info: Found initial step size\n \u2502 \u03f5 = 0.41250000000000003\n \u2514 @ Turing.Inference /home/dfish/.julia/packages/Turing/y0DW3/src/inference/hmc.jl:188\n \u250c Info: Found initial step size\n \u2502 \u03f5 = 0.4\n \u2514 @ Turing.Inference /home/dfish/.julia/packages/Turing/y0DW3/src/inference/hmc.jl:188\n \u250c Info: Found initial step size\n \u2502 \u03f5 = 0.4\n \u2514 @ Turing.Inference /home/dfish/.julia/packages/Turing/y0DW3/src/inference/hmc.jl:188\n \u250c Info: Found initial step size\n \u2502 \u03f5 = 0.4\n \u2514 @ Turing.Inference /home/dfish/.julia/packages/Turing/y0DW3/src/inference/hmc.jl:188\n \u250c Warning: The current proposal will be rejected due to numerical error(s).\n \u2502 isfinite.((\u03b8, r, \u2113\u03c0, \u2113\u03ba)) = (true, false, false, false)\n \u2514 @ AdvancedHMC /home/dfish/.julia/packages/AdvancedHMC/yd6UP/src/hamiltonian.jl:47\n \u250c Warning: The current proposal will be rejected due to numerical error(s).\n \u2502 isfinite.((\u03b8, r, \u2113\u03c0, \u2113\u03ba)) = (true, true, false, true)\n \u2514 @ AdvancedHMC /home/dfish/.julia/packages/AdvancedHMC/yd6UP/src/hamiltonian.jl:47\n \u250c Warning: The current proposal will be rejected due to numerical error(s).\n \u2502 isfinite.((\u03b8, r, \u2113\u03c0, \u2113\u03ba)) = (true, false, false, false)\n \u2514 @ AdvancedHMC /home/dfish/.julia/packages/AdvancedHMC/yd6UP/src/hamiltonian.jl:47\n \u250c Warning: The current proposal will be rejected due to numerical error(s).\n \u2502 isfinite.((\u03b8, r, \u2113\u03c0, \u2113\u03ba)) = (true, false, false, false)\n \u2514 @ AdvancedHMC /home/dfish/.julia/packages/AdvancedHMC/yd6UP/src/hamiltonian.jl:47\n \u250c Warning: The current proposal will be rejected due to numerical error(s).\n \u2502 isfinite.((\u03b8, r, \u2113\u03c0, \u2113\u03ba)) = (true, false, false, false)\n \u2514 @ AdvancedHMC /home/dfish/.julia/packages/AdvancedHMC/yd6UP/src/hamiltonian.jl:47\n \u250c Warning: The current proposal will be rejected due to numerical error(s).\n \u2502 isfinite.((\u03b8, r, \u2113\u03c0, \u2113\u03ba)) = (true, false, false, false)\n \u2514 @ AdvancedHMC /home/dfish/.julia/packages/AdvancedHMC/yd6UP/src/hamiltonian.jl:47\n \u250c Warning: The current proposal will be rejected due to numerical error(s).\n \u2502 isfinite.((\u03b8, r, \u2113\u03c0, \u2113\u03ba)) = (true, true, false, true)\n \u2514 @ AdvancedHMC /home/dfish/.julia/packages/AdvancedHMC/yd6UP/src/hamiltonian.jl:47\n \u250c Warning: The current proposal will be rejected due to numerical error(s).\n \u2502 isfinite.((\u03b8, r, \u2113\u03c0, \u2113\u03ba)) = (true, false, false, false)\n \u2514 @ AdvancedHMC /home/dfish/.julia/packages/AdvancedHMC/yd6UP/src/hamiltonian.jl:47\n\n\n\n\n\n Chains MCMC chain (1500\u00d713\u00d74 Array{Float64, 3}):\n \n Iterations = 1501:1:3000\n Number of chains = 4\n Samples per chain = 1500\n Wall duration = 13.72 seconds\n Compute duration = 52.29 seconds\n parameters = blc\n internals = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size\n \n Summary Statistics\n \u001b[1m parameters \u001b[0m \u001b[1m mean \u001b[0m \u001b[1m std \u001b[0m \u001b[1m naive_se \u001b[0m \u001b[1m mcse \u001b[0m \u001b[1m ess \u001b[0m \u001b[1m rhat \u001b[0m \u001b[1m \u001b[0m \u22ef\n \u001b[90m Symbol \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m \u001b[0m \u22ef\n \n blc 1.5328 0.7085 0.0091 0.0204 1166.2415 1.0023 \u22ef\n \u001b[36m 1 column omitted\u001b[0m\n \n Quantiles\n \u001b[1m parameters \u001b[0m \u001b[1m 2.5% \u001b[0m \u001b[1m 25.0% \u001b[0m \u001b[1m 50.0% \u001b[0m \u001b[1m 75.0% \u001b[0m \u001b[1m 97.5% \u001b[0m\n \u001b[90m Symbol \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m\n \n blc 0.4439 0.9933 1.4321 1.9695 3.1418\n\n\n\n\n## Results\n\nIn the summary output above, rhat is very close to 1, indicating convergence of the MCMC chains.The trace plot below also shows good mixing: the trace of each chain fluctuates randomly, and the traces are plotted on top of each other. \n\nThe autocorrelation plot in the second pannel shows low autocorrelation, indicating efficient sampling of the posterior distribution. In the third panel, the density plot shows that the posterior distribution of blc encompasses the data generating value of blc = 1 and is highly skewed.\n\n\n```julia\npyplot()\nch = group(chain, :blc)\nfont_size = 12\np1 = plot(ch, xaxis=font(font_size), yaxis=font(font_size), seriestype=(:traceplot),\n grid=false, size=(250,100), titlefont=font(font_size))\np2 = plot(ch, xaxis=font(font_size), yaxis=font(font_size), seriestype=(:autocorplot),\n grid=false, size=(250,100), titlefont=font(font_size))\np3 = plot(ch, xaxis=font(font_size), yaxis=font(font_size), seriestype=(:mixeddensity),\n grid=false, size=(250,100), titlefont=font(font_size))\npc\u03c4 = plot(p1, p2, p3, layout=(3,1), size=(600,600))\n```\n\n### Posterior Predictive Distribution\n\nThe plots below show the posterior predictive distributions for percent correct for the question \"is a canary a bird\", which requires no category chains and the question \"is a canary an animal?\", which requires 1 category chain. Comparing the distributions, we see that, on average, category chaining reduces accuracy. The reduction in accuracy can be attributed to the increase in opportunities to incorrectly respond \"no\": one opportunity following the initial retrieval and a second opportunity following the category chain retrieval.\n\n\n```julia\nfont_size = 12\nhit_rates(s) = posterior_predictive(x -> hit_rate(parms, s, n_reps; x...), chain, 1000)\npreds = map(s -> hit_rates(s), stimuli)\npredictive_plot = histogram(preds, xlabel=\"% Correct\" ,ylabel=\"Probability\", xaxis=font(font_size), yaxis=font(font_size),\n grid=false, color=:grey, leg=false, titlefont=font(font_size), xlims=(0,1.1),\n layout=(2,1), ylims=(0,0.4), normalize=:probability, size=(600,600), title=[\"Is a canary a bird?\" \"Is a canary an animal?\"])\n```\n\n# References\n\nWeaver, R. (2008). Parameters, predictions, and evidence in computational modeling: A statistical view informed by ACT\u2013R. Cognitive Science, 32(8), 1349-1375.\n", "meta": {"hexsha": "35255661bf421103f2010809b6aee8fb05579f67", "size": 214875, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "models/Semantic/.ipynb_checkpoints/Semantic_Model-checkpoint.ipynb", "max_stars_repo_name": "itsdfish/ACTRFundamentalTools.jl", "max_stars_repo_head_hexsha": "0314b4a79c71712d36052a549f577dd4b4bfd065", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-03-26T09:17:29.000Z", "max_stars_repo_stars_event_max_datetime": "2021-03-26T09:17:29.000Z", "max_issues_repo_path": "models/Semantic/Semantic_Model.ipynb", "max_issues_repo_name": "itsdfish/ACTRFundamentalTools.jl", "max_issues_repo_head_hexsha": "0314b4a79c71712d36052a549f577dd4b4bfd065", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "models/Semantic/Semantic_Model.ipynb", "max_forks_repo_name": "itsdfish/ACTRFundamentalTools.jl", "max_forks_repo_head_hexsha": "0314b4a79c71712d36052a549f577dd4b4bfd065", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 277.6162790698, "max_line_length": 138161, "alphanum_fraction": 0.8955578825, "converted": true, "num_tokens": 9801, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.4414645235353425}} {"text": "```python\nimport numpy as np\nimport sympy as sym\nimport numba\nimport pydae.build as db\n```\n\n\n```python\n\n```\n\n## Formulation\n\nBackward solution:\n\n$$\n\\begin{split} \n\\mathbf {\\dot x} & = \\mathbf {f (x,y^{ini},u^{ini}) } \\\\\n\\mathbf 0 & = \\mathbf {g (x,y^{ini},u^{ini}) } \n\\end{split}\n$$\n\nForeward solution:\n\n$$\n\\begin{split} \n\\mathbf {\\dot x} & = \\mathbf {f (x,y^{run},u^{run}) } \\\\\n\\mathbf 0 & = \\mathbf {g (x,y^{run},u^{run}) } \n\\end{split}\n$$\n\n### Auxiliar equations\n\n\\begin{eqnarray}\nv_d &=& v_1\\sin(\\delta - \\theta_1) \\\\\nv_q &=& v_1\\cos(\\delta - \\theta_1) \n\\end{eqnarray}\n\n\\begin{eqnarray}\np_e = i_d \\left(v_d + R_a i_d\\right) + i_q \\left(v_q + R_a i_q\\right) \n\\end{eqnarray}\n\n\n### Differential equations\n\n$$\n\\begin{eqnarray}\n f_1 = \\dot \\delta &=& \\Omega_b \\left( \\omega -1 \\right) \\\\\n f_2 = \\dot \\omega &=& \\frac{1}{2H} \\left( p_m - p_e - D \\left( \\omega - 1 \\right) \\right) \\\\\n f_3 = \\dot e_q &=& \\frac{1}{T'_{d0}} \\left( -e'_q - \\left(X_d - X'_d \\right) i_d + v_f^\\star \\right) \\\\\n f_4 = \\dot e_d &=& \\frac{1}{T'_{q0}} \\left( -e'_d - \\left(X_q - X'_q \\right) i_q \\right) \n\\end{eqnarray}\n$$\n\n### Algebraic equations\n\n$$\n\\begin{eqnarray}\ng_1 &=& v_q + R_a i_q + X_d' i_d - e_q \\\\\ng_2 &=& v_d + R_a i_d - X_q' i_q - e_d \\\\\ng_3 &=& p_t - \\left(v_1 v_0 \\sin \\left(\\theta_1 - \\theta_0 \\right) \\right)/X_l \\\\\ng_4 &=& q_t + \\left(v_1 v_0 \\cos \\left(\\theta_1 - \\theta_0 \\right) \\right)/X_l - v_1^2/X_l \\\\\ng_5 &=& i_d v_d + i_q v_q - p_t \\\\\ng_6 &=& i_d v_q - i_q v_d - q_t \n\\end{eqnarray}\n$$ \n\n\n\n\n\nThe dynamic states are:\n\n$$\n\\mathbf{f} =\n\\left[\n\\begin{array}{c}\nf_1\\\\\nf_2\\\\\nf_3\\\\\nf_4\n\\end{array}\n\\right]\n\\;\\;\\;\\;\\;\\;\n\\mathbf{g} =\n\\left[\n\\begin{array}{c}\ng_1\\\\\ng_2\\\\\ng_3\\\\\ng_4\\\\\ng_5\\\\\ng_6\n\\end{array}\n\\right]\n\\;\\;\\;\\;\\;\\;\n$$\n\n$$\n\\mathbf x = \\left[\n\\begin{array}{c} \n\\delta \\\\ \n\\omega \\\\ \n e_q'\\\\ \n e_d' \n\\end{array} \\right]\n\\;\\;\\;\\;\n\\mathbf {y^{ini}} = \\left[\n\\begin{array}{c} \ni_d\\\\\ni_q\\\\\nv_1\\\\\n\\theta_1\\\\\np_m\\\\\nv_f \n\\end{array} \\right] \n\\;\\;\\;\\;\n\\mathbf {y^{run}} = \\left[\n\\begin{array}{c} \ni_d\\\\\ni_q\\\\\nv_1\\\\\n\\theta_1\\\\\np_t\\\\\nq_t \n\\end{array} \\right] \n\\;\\;\\;\\;\n\\mathbf {u^{ini}} = \\left[\n\\begin{array}{c} \n p_t\\\\\n q_t \n\\end{array} \\right] \n\\;\\;\\;\\;\n\\mathbf {u^{run}} = \\left[\n\\begin{array}{c} \np_m \\\\\nv_f\n\\end{array} \\right]\n$$\n\n### Outputs\n\n\n$$\n\\mathbf{h} =\n\\left[\n\\begin{array}{c}\np_e \\\\\np_m\n\\end{array}\n\\right]\n\\;\\;\\;\\;\\;\\;\n\\mathbf{z} =\n\\left[\n\\begin{array}{c}\np_e\\\\\np_m\n\\end{array}\n\\right]\n$$\n \n \n\n\n \n \n\n## System definition \n\n\n```python\nparams_dict = {'X_d':1.81,'X1d':0.3,'T1d0':8.0,\n 'X_q':1.76,'X1q':0.65,'T1q0':1.0,\n 'R_a':0.003,'X_line': 0.05, \n 'H':3.5,'D':1.0,\n 'Omega_b':2*np.pi*50,'omega_s':1.0,\n 'v_0':1.0,'theta_0':0.0, 'B_shunt':0.0}\n\n\nu_ini_dict = {'p_t':0.8, 'q_t':0.2} # for the initialization problem\nu_run_dict = {'p_m':0.8,'v_f':1.0} # for the running problem (here initialization and running problem are the same)\n\n\nx_list = ['delta','omega','e1q','e1d'] # [inductor current, PI integrator]\ny_ini_list = ['i_d','i_q','v_1','theta_1','p_m','v_f'] # for the initialization problem\ny_run_list = ['i_d','i_q','v_1','theta_1','p_t','q_t'] # for the running problem (here initialization and running problem are the same)\n\nsys_vars = {'params_dict':params_dict,\n 'u_list':u_run_dict,\n 'x_list':x_list,\n 'y_list':y_run_list}\n\nexec(db.sym_gen_str()) # exec to generate the required symbolic varables and constants\n```\n\n\n```python\n# Auxiliar equations\nv_d = v_1*sin(delta - theta_1) \nv_q = v_1*cos(delta - theta_1) \n\np_e = i_d*(v_d + R_a*i_d) + i_q*(v_q + R_a*i_q) \n\n# Differential equations\nddelta = Omega_b*(omega - omega_s)\ndomega = 1/(2*H)*(p_m - p_e - D*(omega - omega_s))\nde1q = 1/T1d0*(-e1q - (X_d - X1d)*i_d + v_f)\nde1d = 1/T1q0*(-e1d + (X_q - X1q)*i_q)\n\n# Algebraic equations\ng_1 = v_q + R_a*i_q + X1d*i_d - e1q\ng_2 = v_d + R_a*i_d - X1q*i_q - e1d\ng_3 = p_t - (v_1*v_0*sin(theta_1 - theta_0))/X_line\ng_4 = q_t + (v_1*v_0*cos(theta_1 - theta_0))/X_line - v_1**2/X_line - v_1**2*B_shunt\ng_5 = i_d*v_d + i_q*v_q - p_t\ng_6 = i_d*v_q - i_q*v_d - q_t\n\n# Outputs \nh_1 = p_m\nh_2 = p_e\n\n# System dictionary\nsys = {'name':'smib_milano_ex8p1_4ord',\n 'params_dict':params,\n 'f_list':[ddelta,domega,de1q,de1d],\n 'g_list':[g_1,g_2,g_3,g_4,g_5,g_6],\n 'x_list':x_list,\n 'y_ini_list':y_ini_list,\n 'y_run_list':y_run_list,\n 'u_run_dict':u_run_dict,\n 'u_ini_dict':u_ini_dict,\n 'h_dict':{'p_m':h_1,'p_e':h_2}}\n\n\nsys = db.system(sys)\ndb.sys2num(sys) # building system module\n \n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "7070577bc7ce8ae1b60fe6e098ec563b93fb6932", "size": 7915, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "_build/html/_sources/smib_milano_ex8p1/smib_milano_ex8p1_4ord_builder.ipynb", "max_stars_repo_name": "pydae/pydae_doc", "max_stars_repo_head_hexsha": "5e0b0252de02b3264af94542e26ab5ac38b62e50", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "_build/html/_sources/smib_milano_ex8p1/smib_milano_ex8p1_4ord_builder.ipynb", "max_issues_repo_name": "pydae/pydae_doc", "max_issues_repo_head_hexsha": "5e0b0252de02b3264af94542e26ab5ac38b62e50", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "_build/html/_sources/smib_milano_ex8p1/smib_milano_ex8p1_4ord_builder.ipynb", "max_forks_repo_name": "pydae/pydae_doc", "max_forks_repo_head_hexsha": "5e0b0252de02b3264af94542e26ab5ac38b62e50", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 26.5604026846, "max_line_length": 144, "alphanum_fraction": 0.4271636134, "converted": true, "num_tokens": 1956, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.44146452353534243}} {"text": "# Continuous-Time Markov Chains\n## Introduction\n***Pure birth process*** and ***birth and death model***.\n## Continuous-Time Markov Chains\n\n$\\DeclareMathOperator*{\\argmin}{argmin}\n\\DeclareMathOperator*{\\argmax}{argmax}\n\\DeclareMathOperator*{\\plim}{plim}\n\\newcommand{\\ffrac}{\\displaystyle \\frac}\n\\newcommand{\\d}[1]{\\displaystyle{#1}}\n\\newcommand{\\space}{\\text{ }}\n\\newcommand{\\bspace}{\\;\\;\\;\\;}\n\\newcommand{\\bbspace}{\\;\\;\\;\\;\\;\\;\\;\\;}\n\\newcommand{\\QQQ}{\\boxed{?\\:}}\n\\newcommand{\\void}{\\left.\\right.}\n\\newcommand{\\Tran}[1]{{#1}^{\\mathrm{T}}}\n\\newcommand{\\CB}[1]{\\left\\{ #1 \\right\\}}\n\\newcommand{\\SB}[1]{\\left[ #1 \\right]}\n\\newcommand{\\P}[1]{\\left( #1 \\right)}\n\\newcommand{\\abs}[1]{\\left| #1 \\right|}\n\\newcommand{\\norm}[1]{\\left\\| #1 \\right\\|}\n\\newcommand{\\given}[1]{\\left. #1 \\right|}\n\\newcommand{\\using}[1]{\\stackrel{\\mathrm{#1}}{=}}\n\\newcommand{\\asim}{\\overset{\\text{a}}{\\sim}}\n\\newcommand{\\RR}{\\mathbb{R}}\n\\newcommand{\\EE}{\\mathbb{E}}\n\\newcommand{\\II}{\\mathbb{I}}\n\\newcommand{\\NN}{\\mathbb{N}}\n\\newcommand{\\ZZ}{\\mathbb{Z}}\n\\newcommand{\\QQ}{\\mathbb{Q}}\n\\newcommand{\\PP}{\\mathbb{P}}\n\\newcommand{\\AcA}{\\mathcal{A}}\n\\newcommand{\\FcF}{\\mathcal{F}}\n\\newcommand{\\AsA}{\\mathscr{A}}\n\\newcommand{\\FsF}{\\mathscr{F}}\n\\newcommand{\\dd}{\\mathrm{d}}\n\\newcommand{\\I}[1]{\\mathrm{I}\\left( #1 \\right)}\n\\newcommand{\\N}[1]{\\mathcal{N}\\left( #1 \\right)}\n\\newcommand{\\Exp}[1]{\\mathrm{E}\\left[ #1 \\right]}\n\\newcommand{\\Var}[1]{\\mathrm{Var}\\left[ #1 \\right]}\n\\newcommand{\\Avar}[1]{\\mathrm{Avar}\\left[ #1 \\right]}\n\\newcommand{\\Cov}[1]{\\mathrm{Cov}\\left( #1 \\right)}\n\\newcommand{\\Corr}[1]{\\mathrm{Corr}\\left( #1 \\right)}\n\\newcommand{\\ExpH}{\\mathrm{E}}\n\\newcommand{\\VarH}{\\mathrm{Var}}\n\\newcommand{\\AVarH}{\\mathrm{Avar}}\n\\newcommand{\\CovH}{\\mathrm{Cov}}\n\\newcommand{\\CorrH}{\\mathrm{Corr}}\n\\newcommand{\\ow}{\\text{otherwise}}\n\\newcommand{\\FSD}{\\text{FSD}}\n\\newcommand{\\SSD}{\\text{SSD}}\\CB{X\\P t,t\\geq 0}$ is a ***continuous-time Markov chain*** if $\\forall\\; s,t \\geq 0$ and *nonnegative* integers $i,j,x\\P u$, $0\\leqq u < s$, we have\n\n$\\bspace P\\CB{X\\P{t+s} = j \\mid X\\P s = i,X\\P u = x\\P u,0\\leq u s+t\\mid T_i > s} = P\\CB{T_i > t}$, $\\forall\\; s,t\\geq 0$. Hence, the random variable $T_i$ is memoryless and must thus be **exponenetially distributed**.\n\n## Birth and Death Processes\nA system, with states are represented by the number of people in the system at that time. Suppose that whenever there're $n$ people in the system,\n\n1. new arrivals enter the system at an exponential rate $\\lambda_n$\n2. people leave the system at an exponential rate $\\mu_n$\n\nthen we have the state transition rates and probabilities\n\n$\\bspace\\begin{align}\nv_0 &= \\lambda_0,\\\\\nv_i &= \\lambda_i + \\mu_i,\\bspace i>0 \\\\\nP_{01} &= 1,\\\\\nP_{i,i+1} &= \\ffrac{\\lambda_i}{\\lambda_i + \\mu_i},\\bspace i>0\\\\\nP_{i,i-1} &= \\ffrac{\\mu_i}{\\lambda_i + \\mu_i},\\bspace i>0\n\\end{align}$\n\nNote that $v_i$ represents the rate of time that either a birth or a death occurs, which is exponentially distributed with rate $\\lambda_i + \\mu_i$\n\n**e.g.2** Poisson Process\n\nA birth and death process for which $\\forall\\; n \\geq 0$, we have\n\n$\\bspace\\begin{align}\n\\mu_n &= 0\\\\\n\\lambda_n &= \\lambda\n\\end{align}\\newcommand{\\wp}{\\text{with probability }}$\n\n**e.g.3** Yule process\n\nConsider a population whose members can only give birth to new members but cannot die, with each member acting independently and the time for each member giving birth exponentially distributed with rate $\\lambda$. Let $X\\P t$ be the population size at time $t$, then $\\CB{X\\P t,t\\geq0}$ is a pure birth process with $\\lambda_n = n\\lambda$, $n\\geq 0$.\n\n**e.g.4** A Linear Growth Model with Immigration\n\n$\\bspace\\begin{align}\n\\mu_n &= n\\mu, &n\\geq 1\\\\\n\\lambda_n &= n\\lambda + \\theta,& n\\geq 0\n\\end{align}$\n\nThis is the ***linear growth process with immigration***. Suppose that $X\\P 0 = i$ and let $M\\P t = \\Exp{X\\P t}$. Now solve $M\\P t$.\n\n>$M\\P{t+h} = \\Exp{X\\P{t+h}} = \\Exp{\\Exp{X\\P{t+h}\\mid X\\P t}}$\n>\n>So given the population at time $t$, the population at time $t+h$ could be the following with certain probabilities.\n>\n>$\\bspace X\\P{t+h} = \\begin{cases}\nX\\P t + 1,&\\wp \\P{\\theta + X\\P t \\lambda}\\cdot h + o\\P h\\\\\nX\\P t - 1,&\\wp \\P{X\\P t \\mu} \\cdot h + o\\P h\\\\\nX\\P t,& \\wp 1 - \\P{\\theta + X\\P t \\lambda + X\\P t +\\mu}\\cdot h + o\\P h\n\\end{cases}$\n>\n>Therefore, $\\Exp{X\\P{t+h}\\mid X\\P t} = X\\P t + \\P{\\theta + X\\P t \\lambda - X\\P t \\mu}\\cdot h + o\\P h$ and taking the expectation yields $M\\P{t+h} = M\\P t + \\P{\\lambda - \\mu}M\\P t \\cdot h + \\theta \\cdot h + o\\P h$\n>\n>Letting $h\\to0$ we have $M'\\P t = \\P{\\lambda - \\mu}M\\P t + \\theta$. Solve the equation, we have\n>\n>$\\bspace M\\P t = \\ffrac{\\theta}{\\lambda - \\mu} \\P{e^{\\P{\\lambda - \\mu}t} - 1} +ie^{\\P{\\lambda - \\mu}t}$\n>\n>Finally by the fact that $M\\P 0 = i$ we reach the final solution that \n>\n>$\\bspace M\\P t = \\theta t + i$\n***\n\n**e.g.5** The Queueing System, $M/M/1$\n\nConsider a single-server station. Customers arrives in accordance with a Poisson process having rate $\\lambda$. Upon arrival, each customer goes directly into service if the server is free; if not, then the customer joins the queue. The successive service times are assumed to be independent exponential random variables having mean $1/\\mu$ (or rate $\\mu$).\n\nThe preceding is known as the $M/M/1$ queueing system. The first $M$ refers to the fact that the interarrival process is ***Markovian***, since it's a Poisson process and the second to the fact that the service distribution is **exponential**, hence, **Markovian**. The $1$ refers to the fact that there's only one server.\n\nAnd we can write this system as a birth and death process with\n\n$\\bspace\\begin{align}\n\\mu_n &= \\mu,&n\\geq 1\\\\\n\\lambda_n &= \\lambda,& n\\geq 0\n\\end{align}$\n***\n\n**e.g.6** A Multiserver Exponential Queueing System\n\nNow there're $s$ servers and thus the birth and death process is with parameters\n\n$\\bspace\\begin{align}\n\\mu_n &= \\begin{cases}\nn\\mu,& 1\\leq n \\leq s\\\\\ns\\mu,& n > s\\\\\n\\end{cases}\\\\\n\\lambda_n &= \\lambda, \\bbspace n\\geq 0\n\\end{align}$\n\nAnd this is known as $M/M/s$ queueing model.\n***\n\nNow we implement a deeper discussion on the mean and variance of the process. Consider a general birth and death process with birth rates $\\CB{\\lambda_n}$ and death rates $\\CB{\\mu_n}$ where $\\mu_0 = 0$. And let $T_i$ denote the time, starting from state $i$, it takes for the process to enter state $i+1$, $i\\geq0$. Recursively, we can compute $\\Exp{T_i}$, $i\\geq 0$. Firstly\n\n$\\bspace \\Exp{T_0 } = \\ffrac{1}{\\lambda_0}$\n\nThen for $i>0$, we condition whether the first transition takes the process into state $i-1$ or $i+1$. Here's the indicator variable and the corresponding conditional expectation\n\n$\\bspace I_i = \\begin{cases}\n1,&\\text{from state } i \\text{ to state } i+1\\\\ \n0,&\\text{from state } i \\text{ to state } i-1\\\\ \n\\end{cases} \\bspace\\Longrightarrow\\bspace \\left\\{\\begin{align}\n\\Exp{T_i \\mid I_i = 1} &= \\ffrac{1}{\\lambda_i + \\mu_i}\\\\\n\\Exp{T_i \\mid I_i = 0} &= \\ffrac{1}{\\lambda_i + \\mu_i} + \\Exp{T_{i-1}} + \\Exp{T_i}\\\\\n\\end{align}\\right.\\bspace\\Longrightarrow$\n\n$\\bspace\\begin{align}\\Exp{T_i} &= \\ffrac{1}{\\lambda_i + \\mu_i} \\cdot \\ffrac{\\lambda_i}{\\lambda_i + \\mu_i} + \\P{\\ffrac{1}{\\lambda_i + \\mu_i} + \\Exp{T_{i-1}} + \\Exp{T_i}} \\cdot \\ffrac{\\mu_i}{\\lambda_i + \\mu_i} \\\\\n&= \\ffrac{1}{\\lambda_i + \\mu_i} + \\ffrac{\\mu_i}{\\lambda_i + \\mu_i} \\cdot\\P{\\Exp{T_{i-1}} + \\Exp{T_i}}\\\\\n&= \\ffrac{1}{\\lambda_i} + \\ffrac{\\mu_i}{\\lambda_i}\\cdot\\Exp{T_{i-1}},\\bspace i\\geq 1\n\\end{align}$\n\nSuppose now that we wanted to determine the expected time to go from state $i$ to state $j$ where $i t$\n\nTherefore, for $i t}\\\\\n&= \\sum_{k=i}^{j-1}\\P{ e^{-\\lambda_k \\cdot t} \\prod_{r\\neq k,r=i}^{j-1} \\ffrac{\\lambda_r}{\\lambda_r - \\lambda_k}}\\\\\nP\\CB{X\\P t = j \\mid X\\P 0 = i} &= P\\CB{X\\P t < j+1 \\mid X\\P 0 = i} - P\\CB{X\\P t < j \\mid X\\P 0 = i} \\\\\n&= \\sum_{k=i}^{j}\\P{ e^{-\\lambda_k \\cdot t} \\prod_{r\\neq k,r=i}^{j} \\ffrac{\\lambda_r}{\\lambda_r - \\lambda_k}} - \\sum_{k=i}^{j-1} \\P{e^{-\\lambda_k \\cdot t} \\prod_{r\\neq k,r=i}^{j-1} \\ffrac{\\lambda_r}{\\lambda_r - \\lambda_k}}\\\\ \n&= P_{ij}\\P t \\\\\nP_{ii}\\P t &= P\\CB{X_i > t} = e^{-\\lambda_i \\cdot t}\n\\end{align}$\n\n***\n\nFor the general cases, considering the corresponding ***Chapman\u2013Kolmogorov equations*** we have\n\n$Lemma.3$\n\n$\\bspace\\begin{align}\nP_{ij}\\P{t+s} &= P\\CB{X\\P{t+s} = j \\mid X\\P0 = i}\\\\\n&= \\sum_{k=0}^\\infty P\\CB{X\\P{t+s} = j,X\\P t = k \\mid X\\P0 = i}\\\\\n&= \\sum_{k=0}^\\infty P\\CB{X\\P{t+s} = j \\mid X\\P0 = i,X\\P t = k}\\cdot P\\CB{X\\P t = k \\mid X\\P0 = i}\\\\\n&= \\sum_{k=0}^\\infty P\\CB{X\\P{t+s} = j \\mid X\\P t = k}\\cdot P\\CB{X\\P t = k \\mid X\\P0 = i}\\\\\n&= \\sum_{k=0}^\\infty P_{kj}\\P s \\cdot P_{ik}\\P t = \\sum_{k=0}^\\infty P_{ik}\\P t P_{kj}\\P s\n\\end{align}$\n\nWe then derive \n\n$\\bspace\\begin{align}\nP_{ij}'\\P t &= \\lim_{h \\to 0} \\ffrac{P_{ij}\\P{t+h} - P_{ij}\\P t}{h}= \\lim_{h \\to 0} \\ffrac{ \\P{\\sum\\limits_{k\\neq i} P_{ik}\\P h \\cdot P_{kj} \\P t} - \\P{1-P_{ii}\\P h} P_{ij}\\P t}{h} \\\\\n&= \\lim_{h \\to 0}\\P{ \\sum_{k\\neq i} \\ffrac{P_{ik}\\P h}{h}P_{kj} \\P t - \\P{\\ffrac{1-P_{ii}\\P h}{h}}P_{ij}\\P t }\n\\end{align}$\n\nTo find the limit, we assume that we can interchange the limit and the summation in the preceding, and obtain\n\n$\\bspace \\d{P_{ij}'\\P t = \\sum_{k\\neq i}\\P{ \\lim_{h \\to 0} \\ffrac{P_{ik}\\P h}{h}}P_{kj} \\P t - \\lim_{h \\to 0}\\P{\\ffrac{1-P_{ii}\\P h}{h}}P_{ij}\\P t}$\n\nNow to find $\\lim_{h \\to 0} \\ffrac{P_{ik}\\P h}{h}$ and $\\lim_{h \\to 0}\\ffrac{1-P_{ii}\\P h}{h}$, we need to introduce $q_{ij}$, which is defined as (***different from the textbook!!! Frecking!!!***)\n\n$\\bspace q_{ij} = \\begin{cases}\nv_i P_{ij}, & j\\neq i\\\\\n-\\sum\\limits_{i\\neq j} q_{ij} = -v_i,& j = i\n\\end{cases}, \\bspace \\d{\\forall \\: i, \\sum_{j}q_{ij} = 0}$\n\nSince $v_i$ is the rate at which the process makes a transition when in state $i$ and $P_{ij}$ (from $\\mathbf P$) is the probability that this transition is into state $j$, it follows that $q_{ij}$ is the rate, when in state $i$, at which the process makes a transition into state $j$, called the ***instantaneous transition rates***.\n\nAfter this we have\n\n$Lemma.2$\n\n>$\\bspace\\d{\\lim_{h \\to 0}}\\ffrac{1-P_{ii}\\P h}{h} = v_i,\\bspace \\d{\\lim_{h \\to 0}} \\ffrac{P_{ij}\\P h}{h} = q_{ij}, i\\neq j$\n\n$Proof$\n\n>Skipped...\n\nThen after so many efforts, we finally make it to\n\n$Theorem.1$ Kolmogorov\u2019s Backward Equations\n\n>For all states $i$, $j$, and times $t\\geq 0$,\n>\n>$\\bspace \\d{P_{ij}'\\P t = \\sum_{k\\neq i} q_{ik}P_{kj}\\P t - v_i P_{ij}\\P t}$\n\n$Remark$\n\n>We also write $q_{ii} = -v_i = -q_i$ where $q_i = \\abs{q_{ii}}>0$, thus, $P_{ii} = 0$ and $P_{ij} = \\ffrac{q_{ij}}{q_{i}} = \\ffrac{q_{ij}}{\\sum_{j\\neq i} q_{ij}}$\n\n**e.g.9** Pure Birth Process\n\nThe backward equations for the pure birth process\n\n>$\\bspace\\begin{align}\nP_{ij}'\\P{t} &= \\sum_{k\\neq i,i+1} 0\\cdot P_{kj}\\P t + q_{i,i+1} P_{i+1,j} \\P t - v_i P_{ij}\\P t\\\\\n&= v_i \\cdot P_{i,i+1}P_{i+1,j}\\P t - \\P{\\lambda_i + 0} P_{ij}\\P t\\\\\n&= \\lambda_i \\ffrac{\\lambda_i}{\\lambda_i + 0}P_{i+1,j}\\P t - \\lambda_i P_{ij}\\P t\\\\\n&= \\lambda_i P_{i+1,j}\\P t - \\lambda_i P_{ij}\\P t\n\\end{align}$\n***\n\n**e.g.10** Birth and Death Process\n\nThe backward equations for the birth and death process\n\n>$\\bspace\\begin{align}\nP_{0j}'\\P{t} &= v_0 P_{01}P_{1j}\\P t - v_0 P_{0j}\\P t = \\lambda_0 P_{1j}\\P t - \\lambda_0 P_{0j} \\P t = \\lambda_0\\P{P_{1j}\\P t - P_{0j}\\P t}\\\\\nP_{1j}'\\P t &= v_i\\P{P_{i,i+1}P_{i+1,j}\\P t + P_{i,i-1}P_{i-1,j}\\P t} - v_i P_{ij}\\P t\\\\\n&= \\P{\\lambda_i + \\mu_i}\\P{\\ffrac{\\lambda_i}{\\lambda_i + \\mu_i} P_{i+1,j}\\P t + \\ffrac{\\mu_i}{\\lambda_i + \\mu_i} P_{i-1,j}\\P t} - \\P{\\lambda_i + \\mu_i}P_{ij}\\P t\\\\\n&= \\lambda_i P_{i+1,j}\\P t + \\mu_i P_{i-1,j}\\P t - \\P{\\lambda_i + \\mu_i}P_{ij}\\P t,\\bspace i > 0\n\\end{align}$\n***\n\n\nAnother set of equations, similarly, are \n\n$\\bspace\\begin{align}\nP_{ij}'\\P t &= \\lim_{h \\to 0} \\ffrac{P_{ij}\\P{t+h} - P_{ij}\\P t}{h}= \\lim_{h \\to 0} \\ffrac{ \\P{\\sum\\limits_{k\\neq j} P_{ik}\\P t \\cdot P_{kj} \\P h} - \\P{1-P_{jj}\\P h} P_{ij}\\P t}{h} \\\\\n&= \\lim_{h \\to 0}\\P{ \\sum_{k\\neq j} P_{ik} \\P t \\ffrac{P_{kj}\\P h}{h} - \\P{\\ffrac{1-P_{jj}\\P h}{h}}P_{ij}\\P t }\\\\\n\\end{align}$\n\nStill we assume that we can interchange limit with summation, we obtain, from $Lemma.2$\n\n$Theorem.2$ Kolmogorov\u2019s Forward Equations\n\n>Under suitable regularity conditions, say, models with finite states, or all birth and death processes,\n>\n>$\\bspace \\d{P_{ij}'\\P t = \\sum_{k\\neq j} P_{ik}\\P t q_{kj}- v_j P_{ij}\\P t}$\n\nFor the pure birth process, the forward equation is written as \n\n$\\bspace \\begin{align}\nP_{ij}'\\P t &= \\sum_{k\\neq j,j-1} P_{ik}\\P t q_{kj} + P_{i,j-1}\\P t q_{j-1,j} - v_j P_{ij}\\P t\\\\\n&= 0 + P_{i,j-1}\\P t \\cdot \\lambda_{j-1} \\ffrac{\\lambda_{j-1}}{\\lambda_{j-1} + 0} - \\lambda_j P_{ij}\\P t \\\\\n&= \\lambda_{j-1} P_{i,j-1}\\P t - \\lambda_j P_{ij}\\P t\\\\\n&=\\begin{cases}\n-\\lambda_i P_{ii}\\P t,& j=i\\\\\n\\lambda_{j-1} P_{i,j-1}\\P t - \\lambda_j P_{ij}\\P t, & j\\geq i+1\n\\end{cases}\n\\end{align}$\n\nThen we solve the ODE and obtain\n\n$Proposition.4$ Pure Birth Process\n\n>$\\bspace \\begin{align}\nP_{ii}\\P t &= e^{-\\lambda_i t}, && i \\geq 0\\\\\nP_{ij}\\P t &= \\lambda_{j-1} e^{-\\lambda_j t}\\int_0^t e^{-\\lambda_j s}P_{i,j-1} \\P s \\;\\dd s, && j \\geq i+1\n\\end{align}$\n\n$Proof$\n\n>Just keep in mind that $P_{ii}\\P 0 = 1$ and $P_{ij}\\P 0 = 0$ and it can't be more obvious\n\n**e.g.12** Birth and Death Process\n\nThe forward equation now is\n\n>$\\bspace\\begin{align}\nP_{0j}'\\P{t} &= \\sum_{k\\neq 0} P_{ik}\\P t q_{k0} - v_0 P_{i0}\\P t\\\\\n&= \\sum_{k\\neq 0,1} P_{ik}\\P t q_{k0} + P_{i1}\\P t q_{10} - \\lambda_0 P_{i0}\\P t\\\\\n&= 0 + P_{i1}\\P t v_1 P_{10} - \\lambda_0 P_{i0}\\P t\\\\\n&= P_{i1}\\P t \\P{\\lambda_1 + \\mu_1}\\ffrac{\\mu_1}{\\lambda_1 + \\mu_1}- \\lambda_0 P_{i0}\\P t\\\\\n&= \\mu_1 P_{i1}\\P t- \\lambda_0 P_{i0}\\P t\\\\\nP_{1j}'\\P t &= P_{i,j-1}\\P t q_{j-1,j} + P_{i,j+1}\\P t q_{j+1,j} - v_j P_{ij}\\P t\\\\\n&= P_{i,j-1}\\P t \\P{\\lambda_{j-1} + \\mu_{j-1}}\\ffrac{\\lambda_{j-1}}{\\lambda_{j-1} + \\mu_{j-1}} + P_{i,j+1}\\P t \\P{\\lambda_{j+1} + \\mu_{j+1}} \\ffrac{\\mu_{j+1}}{\\lambda_{j+1} + \\mu_{j+1}} - \\P{\\lambda_j + \\mu_j}P_{ij}\\P t\\\\\n&= \\lambda_{j-1}P_{i,j-1}\\P t + \\mu_{j+1}P_{i,j+1}\\P t \\P{\\lambda_j + \\mu_j}P_{ij}\\P t\n\\end{align}$\n\n## Limiting Probabilities\nIn analogy with discrete-time Markov chains, the probability that a continuous-time Markov chain will be in state $j$ at time $t$ often converges to a limiting value that is independent of the initial state.\n\n$\\bspace P_j \\equiv \\d{\\lim_{t\\to\\infty} P_{ij}\\P t}$\n\nwhere we are assuming that the limit *exists* and is *independent of the initial state* $i$.\n\nTo find $P_{j}$, consider the **forward equations**.\n\n$\\bspace P_{ij}'\\P t = \\d{\\sum_{k \\neq j} P_{ik}\\P t q_{kj} - v_j P_{ij}\\P t \\Longrightarrow \\lim_{t\\to\\infty} P_{ij}'\\P t = \\sum_{k \\neq j} P_k \\cdot q_{kj} - v_j \\cdot P_j}$\n\nfor all states $i$ and $\\sum_j P_j = 1$ will do the problem. One last thing is to note that $\\d{\\lim_{t\\to\\infty} P_{ij}'\\P t = 0}$, since if not, then with $t\\to \\infty$, $P_{ij}\\P t$ will move to either $+\\infty$ or $-\\infty$. So in short, we have\n\n$\\bspace \\begin{cases}\n\\sum\\limits_{k \\neq j} P_k \\cdot q_{kj} - v_j \\cdot P_j = 0, & \\forall\\: j\\\\\n\\sum\\limits_j P_j = 1\n\\end{cases}$\n\n$Remark$\n\n>**backword equations** won't do, since\n>\n>$\\bspace P_{ij}'\\P t = \\d{\\sum_{k \\neq i} q_{ik} P_{kj}\\P t - v_i P_{ij}\\P t \\Longrightarrow \\lim_{t\\to\\infty} P_{ij}'\\P t = \\sum_{k \\neq i} q_{ik} P_j - v_i \\cdot P_j =P_j\\sum_k q_{ik} = P_j \\cdot 0= 0}$\n>\n>And the two sufficient conditions for the existence $P_j$ are\n>\n>- ***irreducible***: all states of the Markov chain ***communicate***, meaning that there's positive probability of starting in state $i$ and ever being in state $j$, for all $i$, $j$. Or in mathematical form: *Let $S$ be the state space*. $\\forall \\: i,j\\in S$, $\\exists\\: t > 0$, $s.t.$ $P_{ij}\\P t > 0$. And if not, the two states are in different classes.\n>- ***positive recurrrent***: *starting in any state, the mean time to return to that state is finite*\n>\n>Unfortunately to decide whether a continuous-time Markov chain is **positive recurrent** is not a easy job. Write $\\mathbf Q = \\P{q_{ij}}$ and still we have $-q_{ii} = v_i = q_i$. If $\\sup q_i < \\infty$ and $\\inf q_i > 0$, whether state $i$ is **positive recurrent** can be determined exactly the same way as if we're dealing with a discrete-time Markov Chain. Later we'll show that **time reversible** $\\Rightarrow$ **positive recurrent**: $\\d{\\lim_{t\\to\\infty} P_{ii}\\P t > 0}$\n>\n>And more about **recurrent**. In discrete-time Markov Chain, beinig **recurrent** means $f_{ii} = \\sum_{n=1}^\\infty f_{ii}^\\P{n} = 1 \\iff \\sum_{n=1}^\\infty P_{ii}^\\P{n} = \\infty$. Now in continuous-time Markov Chain, we change the summation to integral, $\\d{\\int_{0}^\\infty P_{ii}\\P t\\;\\dd t = \\infty}$.\n>\n>**Irreducibility** is an easier job now. First we update some notation. $\\mathbf Q = \\P{q_{ij}}$, $\\mathbf P\\P t = \\P{P_{ij}\\P{t}}$, and the **jump chain** or the **embedded chain** (ref [Time Reversibility](#Time-Reversibility)), $\\mathbf P = \\P{P_{ij}}$.\n>\n>And from $\\mathbf Q$ we can easily write $\\mathbf P$ like\n>\n>$\\bspace \\mathbf Q = \\begin{pmatrix}\n-v_1 & q_{12} & q_{13}\\\\ \nq_{21} & -v_2 & q_{23}\\\\ \nq_{31} & q_{32} & -v_3\n\\end{pmatrix} \\iff \\mathbf P = \\begin{Vmatrix}\n0 & \\ffrac{q_{12}}{v_1} & \\ffrac{q_{13}}{v_1}\\\\ \n\\ffrac{q_{21}}{v_2} & 0 & \\ffrac{q_{23}}{v_2}\\\\ \n\\ffrac{q_{31}}{v_3} & \\ffrac{q_{32}}{v_3} & 0\n\\end{Vmatrix}$\n>\n>Since it's too hard to derive $\\mathbf P\\P t = \\P{P_{ij}\\P{t}}$, although you can do so that you can directly see whether such $t$ exists so that $P_{ij}\\P t > 0$, we suggest handling $\\mathbf P$ as if it were a transition probabilities matrix from a discrete-time Markov Chain.\n>\n>And since through working on $\\mathbf P$ we can decide whether the Markov Chain is **irreducible**, and $\\mathbf P$ is equivalent to $\\mathbf Q$, could we obtain the conclusion by only working on $\\mathbf Q$. **YES YOU CAN!**\n>\n>*Our object is to find whether there's an $n$ such that $q_{ij}^\\P{n} > 0$, where $q_{ij}^\\P{n}$ have the similar definitions like $\\d{P_{ij}^{n} = \\sum_{k=0}^\\infty P_{ik}^{m} P_{kj}^{n-m}}$*.\n\n$Remark$ \n\n>Up to this point, back to solving the equation to obtain the **limiting probabilities**, remember we **CANNOT** first obtain $\\mathbf P$ and solve $\\begin{cases}\n\\pi = \\pi \\mathbf P\\\\\n\\pi \\cdot \\mathbf 1 = 1\n\\end{cases}$, whose solution is the **limiting distribution** of the **jump chain**...\n>\n>So, anyway, if we also define $\\pi_i = \\d{\\lim_{t \\to \\infty} P_{ii}\\P t}$, the equation could be simplied as\n\n>$\\bspace\\begin{cases}\n\\sum\\limits_{k \\neq j} P_k \\cdot q_{kj} - v_j \\cdot P_j = \\pi \\mathbf Q = 0, & \\forall\\: j\\\\\n\\sum\\limits_j \\pi_j =\\pi\\cdot \\mathbf 1 = 1\n\\end{cases}$\n>\n>And more **importantly**, **this equation set has a solution** $\\iff$ **the limiting probabilities or the stationary probabilities exist**.\n>\n>And if it exists, we call this Markov Chain ***ergodic***.\n>\n>And we do this because it's hard to find the limiting probabilities in **Continuous Markov Chain** while at that time, **limiting probabilities** $\\Leftrightarrow$ **stationary probabilities**. Thus we let the **stationary probabilities** be a row vector $\\pi$ with $\\pi = \\pi P\\P{t}$.\n>\n>And this requires that $\\d{\\lim_{t\\to 0} \\dfrac{P\\P t - I}{t} = P'\\P 0 \\equiv Q}$ or, $\\d{\\lim_{t\\to 0} \\dfrac{P_{ij}\\P t - \\delta_{ij}}{t} = q_{ij}}$.\n>\n>Here $Q$, is a matrix with zero row sum, negative diagram entries and all other positive probabilities. Also we see that $P\\P 0 = I$, $i.e.$, $P_{ij}\\P0 = \\delta_{ij}$\n\n$Remark$\n\n>$\\sum\\limits_{k \\neq j} P_k \\cdot q_{kj} - v_j \\cdot P_j =0$, this equation (also known as the ***balance equations***) has one more intepretation:\n>\n>- $v_j \\cdot P_j$ is the *rate at which the process **leaves** state $j$*: When the process is in state $j$, it leaves at rate $v_j$; and $P_j$ is the *proportion of time it is in state $j$*\n>- $\\sum\\limits_{k \\neq j} P_k \\cdot q_{kj}$ is the *rate at which the process **enters** state $j$*: for all states named $k$ other than $j$, the *process enters $j$ at a rate $q_{kj}$*; and $P_k$ is the *proportion of time it is in state $k$*\n\nLet us now determine the limiting probabilities for a birth and death process. For each state, from $0$ to $n\\geq1$, we have a table\n\n$$\\begin{array}{cc}\\hline\n\\text{State} & \\text{Rate at which leave} = \\text{rate at which enter}\\\\ \\hline\n0 & \\lambda_0 P_0 = \\mu_1 P_1 \\\\\n1 & \\P{\\lambda_1+\\mu_1}P_1 = \\mu_2P_2 + \\lambda_0 P_0 \\\\\n2 & \\P{\\lambda_2+\\mu_2}P_2 = \\mu_3P_3 + \\lambda_1 P_1 \\\\\nn,n\\geq 1 & \\P{\\lambda_n+\\mu_n}P_n = \\mu_{n-1}P_{n-1} + \\lambda_{n+1} P_{n+1}\\\\ \\hline\n\\end{array}\n$$\n\nAdding these equations together, we have $\\forall\\: n\\geq 0$, $\\lambda_n P_n = \\mu_{n+1}P_{n+1}$. Solving these inductively, we have, in terms of $P_0$\n\n$\\bspace\\begin{align}\nP_1 &= \\ffrac{\\lambda_0}{\\mu_1}P_0 \\\\\nP_2 &= \\ffrac{\\lambda_1}{\\mu_2}P_1 = \\ffrac{\\lambda_1\\lambda_0}{\\mu_2\\mu_1}P_0 \\\\\nP_3 &= \\ffrac{\\lambda_2}{\\mu_3}P_2 = \\ffrac{\\lambda_2\\lambda_1\\lambda_0}{\\mu_3\\mu_2\\mu_1}P_0 \\\\\n&\\vdots\\\\\nP_n &= \\ffrac{\\lambda_{n-1}}{\\mu_n}P_{n-1} = \\ffrac{\\lambda_{n-1}\\lambda_{n-2}\\cdots\\lambda_2\\lambda_1\\lambda_0}{\\mu_n\\mu_{n-1}\\cdots\\mu_2\\mu_1}P_0\n\\end{align}$\n\nAnd by using the fact that $\\sum_{n=0}^\\infty P_n = 1 $, we can solve for $P_0$ that\n\n$\\bspace P_0 = \\ffrac{1}{1 + \\sum\\limits_{n=1}^{\\infty}\\ffrac{\\lambda_{n-1}\\lambda_{n-2}\\cdots\\lambda_2\\lambda_1\\lambda_0}{\\mu_n\\mu_{n-1}\\cdots\\mu_2\\mu_1}} \\Rightarrow P_n = \\ffrac{\\lambda_0 \\lambda_1 \\cdots \\lambda_{n-1}}{\\mu_1 \\mu_2\\cdots \\mu_n\\P{1 + \\sum\\limits_{n=1}^{\\infty}\\ffrac{\\lambda_{n-1}\\lambda_{n-2}\\cdots\\lambda_2\\lambda_1\\lambda_0}{\\mu_n\\mu_{n-1}\\cdots\\mu_2\\mu_1}}},\\bspace n\\geq1$\n\nThe foregoing equations also show us to the *necessary* condition that the limiting probabilities exist:\n\n$\\bspace \\d{\\sum\\limits_{n=1}^{\\infty}\\ffrac{\\lambda_{n-1}\\lambda_{n-2}\\cdots\\lambda_2\\lambda_1\\lambda_0}{\\mu_n\\mu_{n-1}\\cdots\\mu_2\\mu_1} < \\infty}$\n\n(and this condition is also *sufficient*.)\n\nAnd in **e.g.6**, the multiserver exponential queueing system, we have the condition reduced to\n\n$\\bspace \\d{\\sum_{n=s+1}^\\infty \\ffrac{\\lambda^n}{\\P{s\\mu}^n}<\\infty \\iff \\ffrac{\\lambda}{s\\mu}<1}$\n\nIn **e.g.4**, the linear growth model with immigration, the condition is reduced to\n\n$\\bspace \\d{\\sum_{n=1}^\\infty \\ffrac{\\theta\\P{\\theta+\\lambda}\\cdots\\P{\\theta+\\P{n-1}\\lambda}}{n!\\mu^n}<\\infty \\iff \\lim_{n\\to\\infty} \\ffrac{\\ffrac{\\theta\\P{\\theta+\\lambda}\\cdots\\P{\\theta+n\\lambda}}{\\P{n+1}!\\mu^{n+1}}}{\\ffrac{\\theta\\P{\\theta+\\lambda}\\cdots\\P{\\theta+\\P{n-1}\\lambda}}{n!\\mu^n}} = \\lim_{n\\to\\infty}\\ffrac{\\theta+n\\lambda}{\\P{n+1}\\mu} = \\ffrac{\\lambda}{\\mu}<1}$\n\n$Remark$\n\n>The previous method to find the limit of a summationis called the ***ratio test***.\n\n**e.g.13** Machine Repair Model\n\nThere're $M$ machines and $1$ serviceman. Suppose that the amount of time each machine runs(survives) before breaking down is *exponentially distributed with mean* $\\ffrac{1}{\\lambda}$, and suppose that the amount of time that it takes for the serviceman to fix a machine is exponentially distributed with mean $\\ffrac{1}{\\mu}$. Show time!\n\n>Usually, we'll focus on those that are **NOT** in use. We say the system is in state $n$ whenever $n$ machines are not in use, then the preceding is a *birth and death process* having parameters\n>\n>$\\bspace\\begin{align}\n\\mu_n &= \\mu,\\bbspace\\bbspace\\:\\!\\; n\\geq 1\\\\\n\\lambda_n &= \\begin{cases}\n\\P{M-n}\\lambda, & n\\leq M\\\\\n0, & n>M\n\\end{cases}\n\\end{align}$\n>\n>where a *failing* machine is regarded as an *arrival* and a *fixed* machine as a *departure*. With this we have the $P_0$\n>\n>$\\bspace\\begin{align}\nP_0 &= \\ffrac{1}{1+\\sum\\limits_{n=1}^M\\P{M\\lambda \\P{M-1}\\lambda \\cdots \\P{M-n+1}\\lambda \\cdot \\mu^{-n}}} = \\ffrac{1}{1+\\sum\\limits_{n=1}^M \\P{\\ffrac{M!}{\\P{M-n}!}\\P{\\ffrac{\\lambda}{\\mu}}^n}}\\\\\nP_n &= \\ffrac{\\ffrac{M!}{\\P{M-n}!}\\P{\\ffrac{\\lambda}{\\mu}}^n}{1+\\sum\\limits_{n=1}^M \\P{\\ffrac{M!}{\\P{M-n}!}\\P{\\ffrac{\\lambda}{\\mu}}^n}},\\bspace n=0,1,\\dots,M\n\\end{align}$\n>\n>We then find the average number of machines not in use\n\n>$\\bspace \\d{\\sum_{n=0}^MnP_n = \\ffrac{\\sum\\limits_{n=0}^M\\P{n\\cdot\\ffrac{M!}{\\P{M-n}!}\\P{\\ffrac{\\lambda}{\\mu}}^n}}{1+\\sum\\limits_{n=1}^M \\P{\\ffrac{M!}{\\P{M-n}!}\\P{\\ffrac{\\lambda}{\\mu}}^n}}}$\n>\n>And the long-run proportion of time that a given machine is working, which is equivalent to the limiting probability of the machine working\n>\n>$\\bspace\\begin{align}\nP\\CB{\\text{machine } i\\text{ is working}} &= \\sum_{n=0}^M P\\CB{\\text{machine } i\\text{ is working} \\mid \\text{state }n}\\cdot P_n\\\\\n&= \\sum_{n=0}^M P\\CB{\\text{machine } i\\text{ is not one of the }n\\text{ broken machines}}\\cdot P_n\\\\\n&= \\sum_{n=0}^M \\ffrac{M-n}{M}P_n = 1 - \\sum_{n=0}^M\\ffrac{nP_n}{M}\n\\end{align}$\n\n## Time Reversibility\n\nFor an **ergodic** **continuous-time** Markov chain, with limiting probablities $P_i$, ignoring the amount of time spent in each state, then this sequence constitutes a **discrete-time** Markov chain with transition probabilities $P_{ij}$, called ***jump chain*** or ***embedded chain***. This is also **ergodic** and we denote its limiting probabilities as $\\pi_i$. That is, $\\pi_i$ is the unique solution of \n\n$\\bspace \\left\\{\\begin{align}\n\\pi_i &= \\sum\\nolimits_j \\pi_j P_{ji}, & \\forall\\: i\\\\\n\\sum\\nolimits_i \\pi_i &= 1\n\\end{align}\\right. \\iff $$\\left\\{\\begin{align}\n\\pi &= \\pi \\mathbf P\\\\\n\\pi \\cdot \\mathbf 1 &= 1\n\\end{align}\\right.$\n\nThus, intuitively, $P_i = \\ffrac{\\pi_i/v_i}{\\sum\\limits_j\\pi_j/v_j}$. Alternatively (or to check the intuition\ud83d\ude06), since for the limiting probabilities, the equality $\\forall\\: i$, $v_i P_i = \\sum\\limits_{j\\neq i} P_j \\, q_{ji}$ must hold, or equivalently since $P_{ii} = 0$, $v_i P_i = \\sum\\limits_{j} P_j v_j P_{ji}$, after plug it in,\n\n$\\bspace \\pi_i = \\d{\\sum_j \\pi_j P_{ji}},\\bspace \\forall\\: i$\n\n***\n\nNow consider the reversed process. Suppose now that the **continuous-time** Markov chain has been in operation for a long time and we trace the process going back from time $T$. First note that\n\n$\\bspace\\begin{align}\nP&\\CB{\\text{process is in state }i\\text{throughout }\\SB{t-s,t}\\mid X\\P t = i}\\\\\n&= \\ffrac{\\text{process is in state }i\\text{throughout }\\SB{t-s,t}}{P\\CB{X\\P t = i}}\\\\\n&= \\ffrac{P\\CB{X\\P{t-s} = i}e^{-v_i\\cdot s}}{P\\CB{X\\P t = i}}\\\\\n&\\using{\\text{large }t}e^{-v_i\\cdot s}\n\\end{align}$\n\nIn other words, *going backward in time, the amount of time the process spends in state $i$ is also exponentially distributed with rate $v_i$*. And our conclusion is that\n\n$\\bspace$*The continuous-time Markov chain will be **time reversible**, in the sense that the process reversed in time has the same probabilistic structure as the original process, if the **embedded chain** is **time reversible***. That is:\n\n$\\bspace\\forall\\: i,j\\bspace \\pi_i P_{ij} = \\pi_j P_{ji}$\n\nAnd further using the fact that $P_i = \\ffrac{\\pi_i/v_i}{\\sum\\limits_j\\pi_j/v_j}$, the preceding condition is equivalent to\n\n$\\bspace\\forall\\: i,j\\bspace P_i \\cdot q_{ij} = P_j \\cdot q_{ji}$\n\nwhich has the interpretation: *the rate at which the process goes directly from state $i$ to state $j$ is equal to the rate at which it goes directly from $j$ to $i$*.\n\n$Remark$\n\n>**Time reversible** $\\Rightarrow$ **POSITIVE recurrent** (From Instructor GAO Wujun)\n\n$Proposition.5$\n\n>An ergodic birth and death process is time reversible.\n\nRecall that a process is time reversible $iff$\n\n$\\bspace\\forall\\: i,j\\bspace P_i \\cdot q_{ij} = P_j \\cdot q_{ji}$\n\nThus, if, as lucky as we are, able to find a set of numbers $P_i$ such that the foregoing is satisfied, then the Markov chain is **time reversible** and the $P_i$s are the **long-run probabilities**. That is,\n\n$Proposition.7$\n\n>If for some set $\\CB{P_i}$, $\\d{\\sum_i P_i = 1}$, $P_i \\geq 0$ and $\\forall\\: i\\neq j$, $P_i \\cdot q_{ij} = P_j \\cdot q_{ji}$, then the continuous-time Markov chain is **time reversible** and $P_i$ represents the **limiting probability** of being in state $i$.\n\n$Proof$\n\n>For fixed $i$ we obtain upon summing $P_i \\cdot q_{ij} = P_j \\cdot q_{ji}$ over all $j \\neq i$\n>\n>$\\bspace \\d{\\sum_{j\\neq i}P_i \\cdot q_{ij} = \\sum_{j\\neq i}P_j \\cdot q_{ji}}$\n>\n>since $\\sum_{j\\neq i} q_{ij} = v_i$, the preceding is just equivalent to\n>\n>$\\bspace \\d{v_i P_i = \\sum_{j\\neq i}P_j \\cdot q_{ji}}$\n>\n>That's just the **balance equations**! Thus, $P_i$s are the **limiting probabilities**.\n\nConsider a **continuous-time** Markov chain with state space $S$, the ***truncated*** to the set $A \\subset S$ one is obtained by further define $q_{ij} = 0$, $\\forall\\: i\\in A$, $j\\notin A$. That is:\n\n$\\bspace$*Transitions out of the class $A$ are no longer allowed, whereas ones in $A$ continue at the same rates as before*. A useful result is that: *if the chain is **time reversible**, then so is the **truncated** one.\n\n$Proposition.8$\n\n>A time reversible chain with limiting probabilities $P_j$, $j\\in S$ that is truncated to the set $A\\in S$ and remains *irreducible*, is also time reversible and has limiting probabilities $P_j^A$ given by\n>\n>$\\bspace P_j^A = \\ffrac{P_j}{\\sum\\limits_{i\\in A}P_i},\\bspace j\\in A$\n\n$Proof$\n\n>By $Proposition\\,6.7$, we need to show that, with $P_j^A = \\ffrac{P_j}{\\sum\\limits_{i\\in A}P_i}$,\n>\n>$\\bspace P_i^A q_{ij} = P_j^A q_{ji},\\bspace $for $i\\in A$, $j\\in A$\n>\n>Or equivalently, $P_i^A q_{ij} = P_j^A q_{ji}$ for $i\\in A$, $j\\in A$. But this follows since the original chain is, by assumption, time reversible.\n\n$Proposition.9$\n\n>If $\\CB{X_i\\P t, t\\geq 0}$ are, for $i=1,\\dots,n$, *independent* **time reversible** **continuous-time** Markov chains, then the vector process $\\CB{\\P{X_i\\P t,\\dots,X_n\\P t},t\\geq 0}$ is also a **time reversible continuous-time** Markov chain.\n\n## The Reversed Chain\nSkipped\n## Uniformization\nSkipped\n## Computing the Transition Probabilities\nNow the backward equations is rewritten as (here we use a slightly different notation from the textbook...)\n\n$\\d{\\bspace P_{ij}'\\P t = \\sum_{k\\neq i}q_{ik}P_{kj}\\P t - v_i P_{ij}\\P t = \\sum_{k} q_{ik}P_{kj}\\P t } \\Rightarrow \\mathbf P'\\P t= \\mathbf Q \\mathbf P\\P t$\n\nand similarly the forward ond, $\\mathbf P'\\P t= \\mathbf P\\P t \\mathbf Q$. We then solve the equation and obtain\n\n$\\bbspace\\mathbf P\\P t = \\mathbf P\\P 0 \\cdot e^{\\mathbf Q \\cdot t}$\n\nSince $\\mathbf P\\P 0 =\\mathbf I$, this yields that $\\mathbf P\\P t = e^{\\mathbf Q \\cdot t} = \\sum\\limits_{n=0}^\\infty \\mathbf Q^n\\ffrac{t^n}{n!}$\n\nAnd that's not for practical uses, so we need to introduce two approximation method:\n\n- $\\d{e^{\\mathbf R t} = \\lim_{n\\to\\infty} \\P{\\mathbf I + \\mathbf R\\ffrac{t}{n}}^n }$\n- for large $n$, $\\d{e^{-\\mathbf R t} = \\lim_{n\\to\\infty} \\P{\\mathbf I - \\mathbf R\\ffrac{t}{n}}^n}\\Rightarrow \\mathbf P\\P t = \\P{\\mathbf I - \\mathbf R\\ffrac{t}{n}}^{-n}$\n\n***\n", "meta": {"hexsha": "bb70716f2f6bd0be2cc405f862d3c633828dd00e", "size": 45024, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Probability and Statistics/Applied Random Process/Chap_06_Continuous-Time Markov Chains.ipynb", "max_stars_repo_name": "XavierOwen/Notes", "max_stars_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2018-11-27T10:31:08.000Z", "max_stars_repo_stars_event_max_datetime": "2019-01-20T03:11:58.000Z", "max_issues_repo_path": "Probability and Statistics/Applied Random Process/Chap_06_Continuous-Time Markov Chains.ipynb", "max_issues_repo_name": "XavierOwen/Notes", "max_issues_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Probability and Statistics/Applied Random Process/Chap_06_Continuous-Time Markov Chains.ipynb", "max_forks_repo_name": "XavierOwen/Notes", "max_forks_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-07-14T19:57:23.000Z", "max_forks_repo_forks_event_max_datetime": "2020-07-14T19:57:23.000Z", "avg_line_length": 56.0697384807, "max_line_length": 500, "alphanum_fraction": 0.5413113006, "converted": true, "num_tokens": 13349, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.7905303162021596, "lm_q1q2_score": 0.44137441537614497}} {"text": "```python\n#mapping some polygons - \nimport pandas as pd\nimport geopandas as gpd\nfrom shapely.geometry.polygon import Polygon\nimport cartopy.crs as ccrs\nimport cartopy.feature as cfeature\nimport matplotlib.pyplot as plt\nfrom matplotlib import cm\nimport folium\nimport numpy as np\n%matplotlib inline\n```\n\n\n```python\n#USE ONCE\n# Lat = np.linspace(36.01,40.01,200)\n# Lon = np.linspace(-84.01, -75.01,200)\n# vapoly = {'Lat':Lat,'Lon':Lon}\n# vapolydf = pd.DataFrame(vapoly)\n# vapolydf.head()\n# vapolydf.to_csv(r'vapoly.csv')\n```\n\n\n```python\n#dataframes for csv files - Buffers\ndfbuffer = pd.read_csv('bufferlatlongV2.csv', delimiter = ',').astype(float)\ndfhurdat = pd.read_csv('hurdatcleanva.csv', delimiter = ',')\ndfcoast = pd.read_csv('coastlatlongV2.csv', delimiter = ',').astype(float)\ndfbasin = pd.read_csv('jamesbasinlatlongV2.csv', delimiter = ',').astype(float)\ndfva = pd.read_csv('vapoly.csv',delimiter = ',').astype(float)\n\n```\n\n\n```python\ndfhurdat.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    Storm NumberStorm NameStorm StatusLatLonTimeMaxspeed
    0AL041851UNNAMEDTS36.8-75.11851-08-25 18:00:0040.0
    1AL031854UNNAMEDTS36.8-75.91854-09-10 06:00:0040.0
    2AL031856UNNAMEDTS37.0-76.01856-08-20 00:00:0050.0
    3AL031856UNNAMEDTS38.0-75.31856-08-20 06:00:0050.0
    4AL021857UNNAMEDTS36.3-75.81857-09-14 06:00:0050.0
    \n
    \n\n\n\n\n```python\ndfbuffer.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    LatLon
    037.3-75.4
    137.3-75.3
    237.3-75.3
    337.0-75.5
    436.4-75.3
    \n
    \n\n\n\n\n```python\n# # import Point, Polygon \n# from sympy import Point, Polygon, Line \n# #isolate the storm and get a list of points.\n# #dfhurdat.head()\n# points = dfhurdat[dfhurdat['Storm Number'] == \"AL031856\"]\n# datachop = points[['Lat','Lon']].copy().values #conversion to an array\n# #create a line object\n# # using intersection() \n# isIntersection = poly1.intersection(Line(p1, Point(3, 2)))\n\n```\n\n\n```python\nimport scipy as sp\nimport scipy.interpolate\n```\n\n\n```python\nimport numpy as np\nlat = dfhurdat[['Lat']].copy().values\nlatmin = min(lat)\nlatmax = max(lat)\nlon = dfhurdat[['Lon']].copy().values\nlonmin = min(lon)\nlonmax = max(lon)\nlength = 100000\n# londf = pd.DataFrame(lat)\n# latmin = min(latdf['Lat'])\n# latmax = max(latdf['Lat'])\n# lonmin = min(londf['Lat']) \n# lonmax = max(londf['Lat'])\nupsample_lat = np.linspace(latmin,latmax,length)\n#print(upsample_lat)\nupsample_lon = np.linspace(lonmin,lonmax,length)\n#print(upsample_lon)\ndf1 = pd.DataFrame(upsample_lat)\ndf2 = pd.DataFrame(upsample_lon)\n# Place the DataFrames side by side\ndfupsample = pd.concat([df1, df2], axis=1)\ndfupsample.columns = ['Lat', 'Lon']\ndfupsample\nlen(dfupsample)\n```\n\n\n\n\n 100000\n\n\n\n\n```python\n#create the polygons\n#polygon to reach beyond state a little\n#listPoint = [[13.415449261665342, 52.502674590782519],[13.416039347648621, 52.50250152147968],[13.415787220001221, 52.501845158120446],[13.416162729263306, 52.502201097675766],[13.415406346321104, 52.502334982450677],[13.415111303329468,52.50204435400651]]\n#polygon = {'type': 'Polygon', 'coordinates': [listPoint]}\n\nbuffer_geom =Polygon(zip(dfbuffer['Lon'],dfbuffer['Lat']))\nbasin_geom = Polygon(zip(dfbasin['Lon'],dfbasin['Lat']))\ncrs = {'init' : 'epsg:4326'}\nbufferpoly = gpd.GeoDataFrame(index = [0], crs = crs, geometry = [buffer_geom])\nbufferpoly.to_file(filename = 'buffer.geojson', driver = 'GeoJSON')\nbasinpoly = gpd.GeoDataFrame(index = [0], crs = crs, geometry = [basin_geom])\nbasinpoly.to_file(filename = 'basin.geojson', driver = 'GeoJSON')\n```\n\n /Users/williampc/opt/anaconda3/envs/geop/lib/python3.9/site-packages/pyproj/crs/crs.py:53: FutureWarning: '+init=:' syntax is deprecated. ':' is the preferred initialization method. When making the change, be mindful of axis order changes: https://pyproj4.github.io/pyproj/stable/gotchas.html#axis-order-changes-in-proj-6\n return _prepare_from_string(\" \".join(pjargs))\n\n\n\n```python\n### for use with original data from nhc ###\n# import pandas as pd\n# from datetime import datetime\n# def lat_lon_to_float (v):\n# \"\"\"Convert strings from NHC to float locations\"\"\"\n# if (v[-1] == 'S') or (v[-1] == 'W'):\n# multiplier = -1\n# else:\n# multiplier = 1\n# return float(v[:-1])*multiplier\n```\n\n\n```python\n### for use with original data from nhc ###\n# hurdata = []\n# with open ('hurdat2.txt', 'r') as f:\n# for line in f.readlines():\n# if line.startswith('AL'):\n# storm_id = line.split(',')\n# storm_number = storm_id[0].strip()\n# storm_name = storm_id[1].strip()\n# else:\n# location_line = line.split(',')\n# dt = datetime.strptime(location_line[0] + location_line[1],\"%Y%m%d %H%M\")\n# storm_status = location_line[3].strip()\n# storm_lat = lat_lon_to_float(location_line[4].strip())\n# storm_lon = lat_lon_to_float(location_line[5].strip())\n# max_speed = float(location_line[6].strip())\n# hurdata.append([storm_number,storm_name,storm_status,storm_lat,storm_lon,dt,max_speed])\n#df = pd.DataFrame(hurdata, columns = ['Storm Number','Storm Name', 'Storm Status', 'Lat', 'Lon','Time', 'Max Speed'])\n\n```\n\n\n```python\ndfhurdat.head()\n\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    Storm NumberStorm NameStorm StatusLatLonTimeMaxspeed
    0AL041851UNNAMEDTS36.8-75.11851-08-25 18:00:0040.0
    1AL031854UNNAMEDTS36.8-75.91854-09-10 06:00:0040.0
    2AL031856UNNAMEDTS37.0-76.01856-08-20 00:00:0050.0
    3AL031856UNNAMEDTS38.0-75.31856-08-20 06:00:0050.0
    4AL021857UNNAMEDTS36.3-75.81857-09-14 06:00:0050.0
    \n
    \n\n\n\n\n```python\ndatachop = dfhurdat[['Lat','Lon']].copy().values\ndfhd = pd.DataFrame(datachop).astype(float)\ndfhd.columns = ['Lat', 'Lon']\ndfhd.values.tolist()\nprint(dfhd)\n```\n\n Lat Lon\n 0 36.8 -75.1\n 1 36.8 -75.9\n 2 37.0 -76.0\n 3 38.0 -75.3\n 4 36.3 -75.8\n .. ... ...\n 285 37.8 -82.0\n 286 38.8 -82.0\n 287 39.5 -80.5\n 288 36.5 -77.7\n 289 37.0 -76.7\n \n [290 rows x 2 columns]\n\n\n\n```python\n# # #changing to a GeoDataFrame to create geometry series\n# hurdatgdf = gpd.GeoDataFrame(df, geometry=gpd.points_from_xy(df.Lon,df.Lat))\n# # hurdatgdf.head()\n# # len(hurdatgdf)\n# T = pd.merge(dfbuffer, dfhurdat, how='inner', on=['Lat', 'Lon'])\n# print(T)\n# len(T)\n# T\n#dfinbuffer = pd.merge(dfhurdat,dfupsample,on=['Lat','Lon'])\n#dfinbuffer = pd.merge(left=dfhurdat, right=dfupsample, left_on='Lat', right_on='Lon')\n#dfinbuffer = pd.DatFrame.merge(left = dfhurdat, right = dfupsample, left_on = ['Lat','Lon'], right_on = ['Lat','Lon'], how = 'right')\n\n# isIntersection = bufferpoly.intersection(Line(dfhd))\n#dfinbuffer = pd.merge(dfupsample.astype(float), dfva.astype(float), on=[\"Lat\",\"Lon\"])\n#pd.set_option(\"display.max_rows\", None, \"display.max_columns\", None)\n#print(dfinbuffer)\n# len(dfinbuffer)\n# print(dfinbuffer)\n```\n\n\n```python\nfrom sympy import Point, Polygon, Line \nfrom sympy import Point, Polygon\nimport pandas as pd\nimport geopandas as gpd\nfrom shapely.geometry import *\n```\n\n\n```python\n# total map with all storms and buffers\nimport matplotlib.pyplot as plt\nimport cartopy.crs as ccrs\nimport cartopy.feature as cfeature\n```\n\n\n```python\nplot_crs = ccrs.LambertConformal(central_longitude =-100., central_latitude = 45)\ndata_crs = ccrs.PlateCarree()\n```\n\n\n```python\nimport matplotlib.patches as mpatches\nimport matplotlib.pyplot as plt\nimport shapely.geometry as sgeom\nimport cartopy.crs as ccrs\nimport cartopy.io.shapereader as shpreader\n\n\ndef basincoords():\n \"\"\"\n Return a list of latitudes and a list of longitudes (lons, lats)\n for James River Basin \n\n \"\"\"\n basinlon = dfbasin['Lon']\n basinlat = dfbasin['Lat']\n\n return basinlon, basinlat\n\ndef buffercoords():\n \"\"\"\n Return a list of latitudes and a list of longitudes (lons, lats)\n for James River Basin \n\n \"\"\"\n bufferlon = dfbuffer['Lon']\n bufferlat = dfbuffer['Lat']\n \n\n return bufferlon, bufferlat\nbufferlon, bufferlat = buffercoords()\nbasinlon, basinlat = basincoords()\n\n#ax.set_title('James River Basin and Buffer - Virginia, USA')\n\n# turn the lons and lats into a shapely LineString\nbuffer = sgeom.LineString(zip(bufferlon, bufferlat))\nbasin = sgeom.LineString(zip(basinlon, basinlat))\n\n\nfig = plt.figure(figsize = (7,7))\nax = plt.subplot(1,1,1,projection = plot_crs)\n\nax.set_extent([-85,-70,32,40],data_crs)\nax.coastlines('50m', edgecolor = 'k', linewidth = 0.75)\nax.add_feature(cfeature.STATES, linewidth = 0.5)\nax.add_feature(cfeature.RIVERS, linewidth = 0.85)\nax.add_feature(cfeature.OCEAN)\nax.add_geometries([buffer], ccrs.PlateCarree(),facecolor='#C8A2C8', alpha=0.5)\nax.add_geometries([basin], ccrs.PlateCarree(),facecolor='rgb', edgecolor='k')\nax.plot(dfupsample['Lon'], dfupsample['Lat'],transform = data_crs)\n\n#for storm_number in vahurc['Storm Number'].unique():\n #data = df[jameshurc['Storm Number'] == storm_number]\n #print(vahurc)\nax.plot(dfupsample['Lon'], dfupsample['Lat'],dfbuffer['Lon'],dfbuffer['Lat'],dfbasin['Lat'],dfbasin['Lon'], transform = data_crs)\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "7d9fcf320805a1bc56f94a94b5725ee5e2001ab2", "size": 93980, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "buffer.ipynb", "max_stars_repo_name": "williampc8985/VT-JamesRiver", "max_stars_repo_head_hexsha": "6bacd10f4fd6158db74973ddc1abd89b650efc9f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "buffer.ipynb", "max_issues_repo_name": "williampc8985/VT-JamesRiver", "max_issues_repo_head_hexsha": "6bacd10f4fd6158db74973ddc1abd89b650efc9f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "buffer.ipynb", "max_forks_repo_name": "williampc8985/VT-JamesRiver", "max_forks_repo_head_hexsha": "6bacd10f4fd6158db74973ddc1abd89b650efc9f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 127.3441734417, "max_line_length": 71536, "alphanum_fraction": 0.8452117472, "converted": true, "num_tokens": 4094, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593452091672, "lm_q2_score": 0.7090191276365462, "lm_q1q2_score": 0.4413355819294195}} {"text": "## Background\n\n### Nitrogen fertilizer recommendations\nThe concept behind the **Economic Optimum Nitrogen Rate** approach (also referred to as the **Maximum Return to Nitrogen** approach) is to make the most favorable nitrogen fertilizer recommendation considering three variables:\n\n* Grain price (\\$ per kg)\n* Fertilizer cost (\\$ per kg)\n* Grain yield response to nitrogen fertilizer (modeled from input data)\n\n#### Why *nitrogen* and not other nutrients?\nMany crops, (e.g., corn, wheat, potatoes) depend on nitrogen fertilizer to achieve profitable yields. Although the same is true for other nutrients (e.g., phosphorus, potassium, sulfur, etc.), many would probably agree that nitrogen deserves special attention because it is usually required in high quantities, it can cause environmental harm, and it tends to be more difficult to manage because of its elusive behavior in the soil. A diagram describing the fundamental pathways among the different pools in the nitrogen cycle illustrates its complexity (**Figure 1**). When nitrogen is introduced to agricultural soils, either naturally or synthetically, it is immediately vulnerable to be lost. Some losses may be temporary, while others are permanent. When nitrogen is lost, it can contribute to environmental pollution and cause other damages related to health and quality of living. Nitrogen pollution can be present in both water (including groundwater, lakes, rivers, etc.) and in the air (as particulate matter or as nitrous oxides, which are strong greenhouse gases).\n\n\n\n**Figure 1**: The nitrogen cycle\n\n**With so many potential loss pathways, it is no wonder why farmers care so much about the nitrogen they apply to their crops. The** `EONR` **tool is an attempt to support them, through economics-driven research.**\n\n### EONR? MRTN? ..what?\nAs its name suggests, the **Maximum Return to Nitrogen** (MRTN) is the maximum achievable profit expected after accounting for the cost of nitrogen fertilizer. The economic optimum nitrogen rate (EONR) is the nitrogen rate where the MRTN is reached. In statistical terms, the EONR is the point where the monetary return from yield is equal to the cost of the increase in nitrogen fertilizer input. The approach uses a best-fit statistical model that utilizes input data from replicated field trials with several nitrogen rate treatments. **Figure 2** is a typical nitrogen rate response curve for corn.\n\n\n\n**Figure 2:** Plot generated from the `EONR` package. The blue points are observed experimental data (nitrogen rate vs yield return to nitrogen). Yield is expressed as monetary return, and is simply the grain price multiplied by grain yield. The blue line is the best-fit quadratic-plateau model representing gross return to nitrogen, the red line is the cost of nitrogen fertilizer, and the green line is the difference between the two and represents the net return to nitrogen.\n\nA model is fit to the experimental data to describe the return to nitrogen for *each* particular dataset (e.g., quadratic, quadratic-plateau, etc.). Differences among replications from these trials can contribute to uncertainty in estimating the EONR, and this is illustrated by the rather wide confidence intervals on either side of the 162 kg ha$^{-1}$ estimated EONR value (90% confidence bounds are 130 kg ha$^{-1}$ and 208$^{-1}$). Although there is perhaps a lot of uncertainty in the EONR estimate, it is a good baseline for making nitrogen fertilizer recommendations for a given location or soil.\n\nTo generate an EONR/MRTN plot as in **Figure 2**, it is necessary to conduct a nitrogen response field experiment (**Figure 3**). Experiments can be \"small-plot\", where it is more feasible to include more nitrogen rates and replications, or experiments can be \"strip trials\", where experiments are generally easier to establish.\n\n\n\n**Figure 3:** Aerial photo of a nitrogen rate response experiment for corn (photo captured in July when the crop is about shoulder-high).\n\nNitrogen deficiencies are visible in the small-plot experiment in **Figure 3** (plot boundaries were rendered over the photo to visualize the individual plots more easily). The experiment in this photo had nine (9) nitrogen rates applied at three (3) different times during the season, and was replicated four (4) times.\n\n### The quadratic-plateau model\nThe quadratic-plateau model is often identified as the most appropriate model for nitrogen response in corn. It is a piecewise function that can be described as:\n\n\\begin{equation}\ny_i =\\begin{cases}\\beta_0 + \\beta_1x_i + \\beta_2x_i^2 + \\epsilon_i & if\\ x_i < \\frac{-\\beta_1}{2\\beta_2}\n\\\\{\\beta_0} - \\frac{{\\beta_1^2}}{4{\\beta_2}} + \\epsilon_i & if\\ x_i \\geq \\frac{-\\beta_1}{2\\beta_2}\\end{cases}\n\\label{qp}\n\\end{equation}\n\nwhere $y_{i}$ can represent grain yield (kg ha$^{-1}$) or monetary return (\\\\$ ha$^{-1}$) from grain yield, $x_{i}$ represents quantity of nitrogen fertilizer applied, and $\\beta_{0}$, $\\beta_{1}$, and $\\beta_{2}$ are the coefficients estimated from the experimental data assuming identically and independently distributed errors, $\\epsilon_{i}$.\n\nThe point on the x-axis where the net return curve (green) reaches the maximum return is the **EONR/MRTN**. The profile-likelihood 90\\% CIs are illustrated as a transparent grey box surrounding the EONR/MRTN point.\n\n\n\n### The quadratic model\nThe quadratic model may be desireable if grain yield reaches a maximum, then begins to decline with increasing rates of nitrogen fertilizer (i.e., it exhibits a sort of toxicity effect). Compared to the quadratic-plateau model, the estimated ONR generally tends to be greater when using the quadratic model. It can be described as:\n\n\\begin{equation}\ny_i =\\beta_0 + \\beta_1x_i + \\beta_2x_i^2 + \\epsilon_i\n\\label{q}\n\\end{equation}\n\nwhere $y_{i}$ can represent grain yield (kg ha$^{-1}$) or monetary return (\\\\$ ha$^{-1}$) from grain yield, $x_{i}$ represents quantity of nitrogen fertilizer applied, and $\\beta_{0}$, $\\beta_{1}$, and $\\beta_{2}$ are the coefficients estimated from the experimental data assuming identically and independently distributed errors, $\\epsilon_{i}$.\n\nQuadratic models tend to be most popular in the literature, at least for describing how to calculate confidence intervals (CIs) for the EONR ([Hernandez & Mulla, 2008](https://dl.sciencesocieties.org/publications/aj/abstracts/100/5/1221); [Sela et al., 2017](https://dl.sciencesocieties.org/publications/jeq/abstracts/46/2/311)). However, even in cases where a quadratic model happens to fit the observed data best, [Cerrato & Blackmer (1990)](https://www.agronomy.org/publications/aj/abstracts/82/1/AJ0820010138) imply that the idea of using the quadratic model for maize is absurd because it predicts rapid decreases in yields when fertilizer is applied at higher than optimal rates, a trend that is not generally supported by evidence for maize. Furthermore, the quadratic model produces a systematic bias of overestimated maximum grain yield and optimum nitrogen fertilizer rate [(Bullock & Bullock, 1994)](https://www.agronomy.org/publications/aj/abstracts/86/1/AJ0860010191).\n\n### Confidence intervals\nOne of the major novelties of the `EONR` package is that it calculates profile-likelihood CIs (as well as Wald and bootstrap CIs) from data fit by the model.\n\nFrom a scientific perspective, it is widely recognized that large uncertainties exist around the estimated EONR computed from yield data and that it is essential to report CIs. Still, few examples exist in the agronomic literature where CIs are actually estimated ([Bachmaier & Gandorfer, 2009](https://www.researchgate.net/publication/225680100_A_conceptual_framework_for_judging_the_precision_agriculture_hypothesis_with_regard_to_site-specific_nitrogen_application); [Hernandez & Mulla, 2008](https://dl.sciencesocieties.org/publications/aj/abstracts/100/5/1221); [Jaynes, 2011](https://link.springer.com/article/10.1007/s11119-010-9168-3); [Sela et al., 2017](https://dl.sciencesocieties.org/publications/jeq/abstracts/46/2/311); [Qin et al., 2018](https://dl.sciencesocieties.org/publications/aj/articles/110/6/2596). Of these examples, only Jaynes (2011) calculated CIs for the quadratic-plateau response function (which has generally been recognized as the most appropriate model for describing yield response to nitrogen in corn). [Hernandez & Mulla, 2008](https://dl.sciencesocieties.org/publications/aj/abstracts/100/5/1221) describe three general methods that can be used for estimating CIs about the EONR:\n\n* the profile-likelihood based CI\n* a bootstrap-derived CI\n* the Wald CI\n\n#### Profile-likelihood\nThe profile-likelihood is the most accurate of any of the approaches (requires reparameterization). The profile-likelihood confidence intervals are computed according to the following general steps:\n\n1. Fit a model (e.g., quadratic-plateau, quadratic, etc.) to the observed data and calculate the sum of squared errors (refered to as the sum of squared errors of the \"full\" model, $SSE(\\hat{\\theta_{2}})$).\n2. Calculate the $SSE$ of the model subject to the constraint that $\\theta_{2}$ (the parameter representing the optimum nitrogen rate) has a fixed value (refered to as the sum of squared errors of the \"reduced\" model, $SSE(\\tilde{\\theta_{2}})$).\n3. The likelihood ratio statistic succinctly expresses how much more likely the fit of the full model is than reduced model. Calculate the likelihood ratio statistic to quantify the difference between the full and reduced model:\n\n\\begin{equation}\n\\tau(\\theta_{2}) = \\frac{{(SSE(\\tilde{\\theta_{2}})-SSE(\\hat{\\theta_{2}}))}}{SSE(\\hat{\\theta_{2}})/(n-p)}\n\\label{tau_lr}\n\\end{equation}\n\nwhere $n$ is the number of observations and $p$ is the total number of parameters in the model (i.e., $p$ = 3 for the quadratic-plateau model).\n\n4. Invert the likelihood ratio statistic to obtain a confidence interval about the full model for $\\theta_{2}$. That is, for a given $\\alpha$ level of significance, the profile-likelihood CI for $\\theta_{2}$ is the set of all $\\tilde{\\theta_{2}}$ for which:\n\n\\begin{equation}\n\\tau(\\theta_{2}) \\leq Q(t_{d}, f)\n\\label{tau_inv}\n\\end{equation}\n\nwhere $d$ is the degrees of freedom and $Q({t_{d}, f})$ is the ${f}$th quantile of the ${t}$-value distribution ($t_{d}$).\n\nBecause $\\tilde{\\theta_{2}}$ was intentionally set away from $\\hat{\\theta_{2}}$ (step 2), any increase in $SSE(\\tilde{\\theta_{2}})$ compared to $SSE(\\hat{\\theta_{2}})$ is derived from $\\theta_{2}$ instead of any of the other model parameters. Because $SSE(\\tilde{\\theta_{2}})$ should never be less than $SSE(\\hat{\\theta_{2}})$, $\\tau(\\theta_{2})$ is a positive value for both the lower and upper confidence interval.\n\nThe algorithm for computing the profile-likelihood confidence intervals iteratively checks all $\\tau(\\theta_{2})$ that are less than or equal to the test statistic for the given $\\alpha$ value. The `EONR` package uses the Nelder-Mead optimization algorithm ([Nelder & Mead, 1965](https://academic.oup.com/comjnl/article-abstract/7/4/308/354237?redirectedFrom=fulltext)) to efficiently find the confidence interval.\n\n#### Bootstrapping\nBootstrapping involves sampling residuals of the original data with replacement (requires reparameterization similar to the profile-likelihood approach). This is a worthy alternative to the profile-likelihood approach, but it is not always perfect.\n\n#### Wald-type (+/- 1 standard error)\nThe Wald-type approach is the simplest, but it has poor performance with small sample sizes and nonlinear models.\n\\begin{equation}\n\\hat{\\theta} = Q(t_{d}, 1-\\alpha/2)\\:{SE}(\\hat{\\theta})\n\\label{wald}\n\\end{equation}\n\nWhere $\\hat{\\theta}$ is the parameter estimate, ${d}$ is the degrees of freedom, ${d=n-k,n}$ is the number of observations, $Q({t_{d}, f})$ is the ${f}$th quantile of the ${t}$-value distribution ($t_{d}$) with ${k}$ treatments, and ${SE}(\\hat{\\theta})$ is the standard error of the parameter estimate ([Cook & Weisberg, 1990](https://www.tandfonline.com/doi/abs/10.1080/01621459.1990.10476233); [Hernandez & Mulla, 2008](https://dl.sciencesocieties.org/publications/aj/abstracts/100/5/1221)).\n\n### The social cost of nitrogen\nThe `EONR` package also allows the user to define a __social cost of nitrogen__, which is then used in the optimum nitrogen rate calculations based on residual soil nitrogen (nitrogen fertilizer not taken up by the crop at the end of the season).\n\nThe traditional approach for calculating the EONR considers only the cost of the nitrogen fertilizer product, and does not consider other unintended costs of nitrogen application. The social cost of nitrogen, defined as the present value of monetary damages caused by an incremental increase in nitrogen, has been suggested as a method to place a value on pollution and other damages (e.g., health, quality of living, etc.) caused by nitrogen fertilizer application ([Keeler et al., 2016](http://stacks.iop.org/1748-9326/9/i=7/a=074002?key=crossref.75a91c07d59a4043a07280d01299d0d8)). Because of the complexity of the nitrogen cycle and the spatial and temporal variability associated with it, the social cost of nitrogen is extremely difficult to quantify and is fraught with uncertainty. Additionally, the basis for what might be considered a healthy environment or an acceptable quality of living is highly subjective and may change abruptly depending on many factors. The social cost of nitrogen is a straightforward concept, however, and it can be a useful method to assess the value of economic gain versus damages from agricultural production.\n\nWhen total nitrogen uptake is measured in field experiments, there is an opportunity to calculate the quantity of nitrogen that we know was not utilized by the crop (residual nitrogen). `EONR` uses crop nitrogen uptake (and optionally, nitrogen in the soil at the beginning of the season) to model end-of-season residual nitrogen as a function of crop-available nitrogen at the beginning of the season (including nitrogen applied as fertilizer). This residual nitrogen value can be multiplied by a social cost (per unit residual nitrogen) to determine the monetary damages that a given experimental treatment might be contributing to pollution or other damages caused by nitrogen fertilizer application. `EONR` then adjusts the optimum nitrogen rate after also considering these social costs derived from experimental data. This is meaningful because it provides a basis for analyzing the costs of pollution and other damages caused by nitrogen fertilizer at the field scale. Although analysis at the regional scale is worthwhile, results oftentimes to not translate to the field scale.\n\nDepending on the economic scenario defined by the user `EONR` will calculate one of the following:\n\n1. **Agronomic Optimum Nitrogen Rate** *(AONR)*: both the cost of nitrogen fertilizer and social cost of nitrogen are ignored\n2. **Economic Optimum Nitrogen Rate** *(EONR)*: cost of nitrogen fertilizer is considered but social cost is ignored\n3. **Socially Optimum Nitrogen Rate** *(SONR)*: both cost of nitrogen fertilizer and social cost of nitrogen are considered\n\nThe `EONR` package is able to calculate the **AONR**, **EONR**, or **SONR** for your dataset by simply adjusting the economic scenario (i.e., by adjusting `price_grain`, `cost_n_fert`, `cost_n_social`, and even `costs_fixed`).\n", "meta": {"hexsha": "d2a65953f2a5d5ad6f3ef5a420e3f108f6969bf6", "size": 17553, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/_build/background.ipynb", "max_stars_repo_name": "tnigon/eonr", "max_stars_repo_head_hexsha": "fbbbf4fa8b0ab46851c6c43b2714ae461eb13f05", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2019-05-21T00:03:46.000Z", "max_stars_repo_stars_event_max_datetime": "2021-04-17T04:50:15.000Z", "max_issues_repo_path": "docs/source/background.ipynb", "max_issues_repo_name": "tnigon/eonr", "max_issues_repo_head_hexsha": "fbbbf4fa8b0ab46851c6c43b2714ae461eb13f05", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2019-03-29T21:56:50.000Z", "max_issues_repo_issues_event_max_datetime": "2021-04-13T18:02:15.000Z", "max_forks_repo_path": "docs/source/background.ipynb", "max_forks_repo_name": "tnigon/eonr", "max_forks_repo_head_hexsha": "fbbbf4fa8b0ab46851c6c43b2714ae461eb13f05", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 97.5166666667, "max_line_length": 1226, "alphanum_fraction": 0.7162308437, "converted": true, "num_tokens": 3677, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850933, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.44133557582518745}} {"text": "# Online Drift Detection on the Wine Quality Dataset\n\nIn the context of deployed models, data (model queries) usually arrive sequentially and we wish to detect it as soon as possible after its occurence. One approach is to perform a test for drift every $W$ time-steps, using the $W$ samples that have arrived since the last test. Such a strategy could be implemented using any of the offline detectors implemented in `alibi-detect`, but being both sensitive to slight drift and responsive to severe drift is difficult. If the window size $W$ is too small then slight drift will be undetectable. If it is too large then the delay between test-points hampers responsiveness to severe drift.\n\nAn alternative strategy is to perform a test each time data arrives. However the usual offline methods are not applicable because the process for computing p-values is too expensive and doesn't account for correlated test outcomes when using overlapping windows of test data. \n\nOnline detectors instead work by computing the test-statistic once using the first $W$ data points and then updating the test-statistic sequentially at low cost. When no drift has occured the test-statistic fluctuates around its expected value and once drift occurs the test-statistic starts to drift upwards. When it exceeds some preconfigured threshold value, drift is detected.\n\nUnlike offline detectors which require the specification of a threshold p-value (a false positive rate), the online detectors in `alibi-detect` require the specification of an expected run-time (ERT) (an inverted FPR). This is the number of time-steps that we insist our detectors, on average, should run for in the absense of drift before making a false detection. Usually we would like the ERT to be large, however this results in insensitive detectors which are slow to respond when drift does occur. There is a tradeoff between the expected run time and the expected detection delay. \n\nTo target the desired ERT, thresholds are configured during an initial configuration phase via simulation. This configuration process is only suitable when the amount reference data (most likely the training data of the model of interest) is relatively large (ideally around an order of magnitude larger than the desired ERT). Configuration can be expensive (less so with a GPU) but allows the detector to operate at low-cost during deployment. \n\nThis notebook demonstrates online drift detection using two different two-sample distance metrics for the test-statistic, the maximum mean discrepency (MMD) and least-squared density difference (LSDD), both of which can be updated sequentially at low cost. \n\n### Backend\n\nThe online detectors are implemented in both the *PyTorch* and *TensorFlow* frameworks with support for CPU and GPU. Various preprocessing steps are also supported out-of-the box in Alibi Detect for both frameworks and an example will be given in this notebook. Alibi Detect does however not install PyTorch for you. Check the [PyTorch docs](https://pytorch.org/) how to do this. \n\n### Dataset\n\nThe [Wine Quality Data Set](https://archive.ics.uci.edu/ml/datasets/wine+quality) consists of 4898 and 1599 samples of white and red wine respectively. Each sample has an associated quality (as determined by experts) and 11 numeric features indicating its acidity, density, pH etc. We consider the regression problem of tring to predict the quality of white wine samples given these features. We will then consider whether the model remains suitable for predicting the quality of red wine samples or whether the associated change in the underlying distribution should be considered as drift.\n\n## Online detection with MMD and Pytorch\n\nThe Maximum Mean Discepency (MMD) is a distance-based measure between 2 distributions *p* and *q* based on the mean embeddings $\\mu_{p}$ and $\\mu_{q}$ in a reproducing kernel Hilbert space $F$:\n\n\\begin{align}\nMMD(F, p, q) & = || \\mu_{p} - \\mu_{q} ||^2_{F} \\\\\n\\end{align}\n\nGiven reference samples $\\{X_i\\}_{i=1}^{N}$ and test samples $\\{Y_i\\}_{i=t}^{t+W}$ we may compute an unbiased estimate $\\widehat{MMD}^2(F, \\{X_i\\}_{i=1}^N, \\{Y_i\\}_{i=t}^{t+W})$ of the squared MMD between the two underlying distributions. Depending on the size of the reference and test windows, $N$ and $W$ respectively, this can be relatively expensive. However, once computed it is possible to update the statistic to estimate to the squared MMD between the distributions underlying $\\{X_i\\}_{i=1}^{N}$ and $\\{Y_i\\}_{i=t+1}^{t+1+W}$ at a very low cost, making it suitable for online drift detection.\n\nBy default we use a [radial basis function kernel](https://en.wikipedia.org/wiki/Radial_basis_function_kernel), but users are free to pass their own kernel of preference to the detector.\n\n\n```python\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport torch\nimport tensorflow as tf\nimport pandas as pd\nimport scipy\nfrom sklearn.decomposition import PCA\n\nnp.random.seed(0)\ntorch.manual_seed(0)\ntf.random.set_seed(0)\n```\n\n### Load data\n\nFirst we load in the data:\n\n\n```python\nred = pd.read_csv(\n \"https://storage.googleapis.com/seldon-datasets/wine_quality/winequality-red.csv\", sep=';'\n)\nwhite = pd.read_csv(\n \"https://storage.googleapis.com/seldon-datasets/wine_quality/winequality-white.csv\", sep=';'\n)\nwhite.describe()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    fixed acidityvolatile aciditycitric acidresidual sugarchloridesfree sulfur dioxidetotal sulfur dioxidedensitypHsulphatesalcoholquality
    count4898.0000004898.0000004898.0000004898.0000004898.0000004898.0000004898.0000004898.0000004898.0000004898.0000004898.0000004898.000000
    mean6.8547880.2782410.3341926.3914150.04577235.308085138.3606570.9940273.1882670.48984710.5142675.877909
    std0.8438680.1007950.1210205.0720580.02184817.00713742.4980650.0029910.1510010.1141261.2306210.885639
    min3.8000000.0800000.0000000.6000000.0090002.0000009.0000000.9871102.7200000.2200008.0000003.000000
    25%6.3000000.2100000.2700001.7000000.03600023.000000108.0000000.9917233.0900000.4100009.5000005.000000
    50%6.8000000.2600000.3200005.2000000.04300034.000000134.0000000.9937403.1800000.47000010.4000006.000000
    75%7.3000000.3200000.3900009.9000000.05000046.000000167.0000000.9961003.2800000.55000011.4000006.000000
    max14.2000001.1000001.66000065.8000000.346000289.000000440.0000001.0389803.8200001.08000014.2000009.000000
    \n
    \n\n\n\nWe can see that the data for both red and white wine samples take the same format.\n\n\n```python\nred.describe()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    fixed acidityvolatile aciditycitric acidresidual sugarchloridesfree sulfur dioxidetotal sulfur dioxidedensitypHsulphatesalcoholquality
    count1599.0000001599.0000001599.0000001599.0000001599.0000001599.0000001599.0000001599.0000001599.0000001599.0000001599.0000001599.000000
    mean8.3196370.5278210.2709762.5388060.08746715.87492246.4677920.9967473.3111130.65814910.4229835.636023
    std1.7410960.1790600.1948011.4099280.04706510.46015732.8953240.0018870.1543860.1695071.0656680.807569
    min4.6000000.1200000.0000000.9000000.0120001.0000006.0000000.9900702.7400000.3300008.4000003.000000
    25%7.1000000.3900000.0900001.9000000.0700007.00000022.0000000.9956003.2100000.5500009.5000005.000000
    50%7.9000000.5200000.2600002.2000000.07900014.00000038.0000000.9967503.3100000.62000010.2000006.000000
    75%9.2000000.6400000.4200002.6000000.09000021.00000062.0000000.9978353.4000000.73000011.1000006.000000
    max15.9000001.5800001.00000015.5000000.61100072.000000289.0000001.0036904.0100002.00000014.9000008.000000
    \n
    \n\n\n\nWe shuffle and normalise the data such that each feature takes a value in \\[0,1\\], as does the quality we seek to predict. We assue that our model was trained on white wine samples, which therefore forms the reference distribution, and that red wine samples can be considered to be drawn from a drifted distribution.\n\n\n```python\nwhite, red = np.asarray(white, np.float32), np.asarray(red, np.float32)\nn_white, n_red = white.shape[0], red.shape[0]\n\ncol_maxes = white.max(axis=0)\nwhite, red = white / col_maxes, red / col_maxes\nwhite, red = white[np.random.permutation(n_white)], red[np.random.permutation(n_red)]\nX = white[:, :-1]\nX_corr = red[:, :-1]\n```\n\nAlthough it may not be necessary on this relatively low-dimensional data for which individual features are semantically meaningful, we demonstrate how [principle component analysis (PCA)](https://en.wikipedia.org/wiki/Principal_component_analysis) can be performed as a preprocessing stage to project raw data onto a lower dimensional representation which more concisely captures the factors of variation in the data. As not to bias the detector it is necessary to fit the projection using a split of the data which isn't then passed as reference data. We additionally split off some white wine samples to act as undrifted data during deployment.\n\n\n```python\nX_train = X[:(n_white//2)]\nX_ref = X[(n_white//2):(3*n_white//4)]\nX_h0 = X[(3*n_white//4):]\n```\n\nNow we define a PCA object to be used as a preprocessing function to project the 11-D data onto a 2-D representation. We learn the first 2 principal components on the training split of the reference data.\n\n\n```python\npca = PCA(2)\npca.fit(X_train)\n```\n\n\n\n\n PCA(n_components=2)\n\n\n\nHopefully the learned preprocessing step has learned a projection such that in the lower dimensional space the two samples are distinguishable.\n\n\n```python\nenc_h0 = pca.transform(X_h0)\nenc_h1 = pca.transform(X_corr)\n\nplt.scatter(enc_h0[:,0], enc_h0[:,1], alpha=0.2, color='green', label='white wine')\nplt.scatter(enc_h1[:,0], enc_h1[:,1], alpha=0.2, color='red', label='red wine')\nplt.legend(loc='upper right')\nplt.show()\n```\n\nNow we can define our online drift detector. We specify an expected run-time (in the absence of drift) of 50 time-steps, and a window size of 10 time-steps. Upon initialising the detector thresholds will be computed using 2500 boostrap samples. These values of `ert`, `window_size` and `n_bootstraps` are lower than a typical use-case in order to demonstrate the average behaviour of the detector over a large number of runs in a reasonable time. \n\n\n```python\nfrom alibi_detect.cd import MMDDriftOnline\n\nert = 50\nwindow_size = 10\n\ncd = MMDDriftOnline(\n X_ref, ert, window_size, backend='pytorch', preprocess_fn=pca.transform, n_bootstraps=2500\n)\n```\n\n No GPU detected, fall back on CPU.\n\n\n 3%|\u258e | 74/2500 [00:00<00:03, 736.80it/s]\n\n Generating permutations of kernel matrix..\n\n\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 2500/2500 [00:04<00:00, 599.06it/s]\n Computing thresholds: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 10/10 [00:02<00:00, 3.59it/s]\n\n\nWe now define a function which will simulate a single run and return the run-time. Note how the detector acts on single instances at a time, the run-time is considered as the time elapsed after the test-window has been filled, and that the detector is stateful and must be reset between detections.\n\n\n```python\ndef time_run(cd, X, window_size):\n n = X.shape[0]\n perm = np.random.permutation(n)\n t = 0\n cd.reset()\n while True:\n pred = cd.predict(X[perm[t%n]])\n if pred['data']['is_drift'] == 1:\n return t\n else:\n t += 1\n```\n\nNow we look at the distribution of run-times when operating on the held-out data from the reference distribution of white wine samples. We report the average run-time, however note that the targeted run-time distribution, a Geometric distribution with mean `ert`, is very high variance so the empirical average may not be that close to `ert` over a relatively small number of runs. We can see that the detector accurately targets the desired Geometric distribution however by inspecting the linearity of a [Q-Q plot](https://en.wikipedia.org/wiki/Q%E2%80%93Q_plot).\n\n\n```python\nn_runs = 250\ntimes_h0 = [time_run(cd, X_h0, window_size) for _ in range(n_runs)]\nprint(f\"Average run-time under no-drift: {np.mean(times_h0)}\")\n_ = scipy.stats.probplot(np.array(times_h0), dist=scipy.stats.geom, sparams=1/ert, plot=plt)\n```\n\nIf we run the detector in an identical manner but on data from the drifted distribution of red wine samples the average run-time is much lower.\n\n\n```python\nn_runs = 250\ntimes_h1 = [time_run(cd, X_corr, window_size) for _ in range(n_runs)]\nprint(f\"Average run-time under drift: {np.mean(times_h1)}\")\n```\n\n Average run-time under drift: 6.004\n\n\n## Online detection with LSDD and TensorFlow\n\nHere we address the same problem but using the least squares density difference (LSDD) as the two-sample distance in a manner similar to [Bu et al. (2017)](https://ieeexplore.ieee.org/abstract/document/7890493). The LSDD between two distributions $p$ and $q$ on $\\mathcal{X}$ is defined as $$LSDD(p,q) = \\int_{\\mathcal{X}} (p(x)-q(x))^2 \\,dx$$ and also has an empirical estimate $\\widehat{LSDD}(\\{X_i\\}_{i=1}^N, \\{Y_i\\}_{i=t}^{t+W})$ that can be updated at low cost as the test window is updated to $\\{Y_i\\}_{i=t+1}^{t+1+W}$.\n\nWe additionally show that TensorFlow can also be used as the backend and that sometimes it is not necessary to perform preprocessing, making definition of the drift detector simpler. Moreover, in the absence of a learned preprocessing stage we may use all of the reference data available.\n\n\n```python\nX_ref = np.concatenate([X_train, X_ref], axis=0)\n```\n\nAnd now we define the LSDD-based online drift detector, again with an `ert` of 50 and `window_size` of 10.\n\n\n```python\nfrom alibi_detect.cd import LSDDDriftOnline\n\ncd = LSDDDriftOnline(\n X_ref, ert, window_size, backend='tensorflow', n_bootstraps=2500,\n)\n```\n\n Computing thresholds: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 9/9 [00:09<00:00, 1.00s/it]\n\n\nWe run this new detector on the held out reference data and again see that in the absence of drift the distribution of run-times follows a Geometric distribution with mean `ert`.\n\n\n```python\nn_runs = 250\ntimes_h0 = [time_run(cd, X_h0, window_size) for _ in range(n_runs)]\nprint(f\"Average run-time under no-drift: {np.mean(times_h0)}\")\n_ = scipy.stats.probplot(np.array(times_h0), dist=scipy.stats.geom, sparams=1/ert, plot=plt)\n```\n\nAnd when drift has occured the detector is very fast to respond.\n\n\n```python\nn_runs = 250\ntimes_h1 = [time_run(cd, X_corr, window_size) for _ in range(n_runs)]\nprint(f\"Average run-time under drift: {np.mean(times_h1)}\")\n```\n\n Average run-time under drift: 4.328\n\n", "meta": {"hexsha": "3e4b982b58faaf7c42d97e902b07b4e97b9ca4d2", "size": 121794, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/cd_online_wine.ipynb", "max_stars_repo_name": "ojcobb/alibi-detect", "max_stars_repo_head_hexsha": "436e99efb88c922dc04cec97b0b52d88bf439d65", "max_stars_repo_licenses": ["ECL-2.0", "Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "examples/cd_online_wine.ipynb", "max_issues_repo_name": "ojcobb/alibi-detect", "max_issues_repo_head_hexsha": "436e99efb88c922dc04cec97b0b52d88bf439d65", "max_issues_repo_licenses": ["ECL-2.0", "Apache-2.0"], "max_issues_count": 40, "max_issues_repo_issues_event_min_datetime": "2021-03-21T15:59:53.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-31T22:10:24.000Z", "max_forks_repo_path": "examples/cd_online_wine.ipynb", "max_forks_repo_name": "ojcobb/alibi-detect", "max_forks_repo_head_hexsha": "436e99efb88c922dc04cec97b0b52d88bf439d65", "max_forks_repo_licenses": ["ECL-2.0", "Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 136.8471910112, "max_line_length": 53446, "alphanum_fraction": 0.8395076933, "converted": true, "num_tokens": 5846, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.44133557582518734}} {"text": "\n\n\n# Start-to-Finish Example: Unit Testing `GiRaFFE_NRPy`: Interpolating Metric Face-Values\n\n## Author: Patrick Nelson\n\n## This module Validates the `FCVAL` routine for `GiRaFFE`.\n\n**Notebook Status:** Validated\n\n**Validation Notes:** This module will validate the routines in [Tutorial-GiRaFFE_NRPy-Metric_Face_Values](Tutorial-GiRaFFE_NRPy-Metric_Face_Values.ipynb).\n\n### NRPy+ Source Code for this module: \n* [GiRaFFE_NRPy/GiRaFFE_NRPy_Metric_Face_Values.py](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_Metric_Face_Values.py) [\\[**tutorial**\\]](Tutorial-GiRaFFE_NRPy-FCVAL.ipynb) Generates the C code to interpolate gridfunctions to cell faces.\n\n## Introduction:\n\nThis notebook validates the code that will interpolate the metric gridfunctions on cell faces. These values, along with the reconstruction of primitive variables on the faces, are necessary for the Riemann solvers to compute the fluxes through the cell faces.\n\nIt is, in general, good coding practice to unit test functions individually to verify that they produce the expected and intended output. We will generate test data with arbitrarily-chosen analytic functions and calculate gridfunctions at the cell centers on a small numeric grid. We will then compute the values on the cell faces in two ways: first, with our interpolator, then second, we will shift the grid and compute them analytically. Then, we will rerun the function at a finer resolution. Finally, we will compare the results of the two runs to show third-order convergence.\n\nWhen this notebook is run, the difference between the approximate and exact metric gridfunctions will be output to text files that can be found in the same directory as this notebook. These will be read in in [Step 3](#convergence), and used there to confirm convergence order of the algorithm. \n\n\n\n# Table of Contents\n$$\\label{toc}$$\n\nThis notebook is organized as follows\n\n1. [Step 1](#setup): Set up core functions and parameters for unit testing the FCVAL algorithm\n 1. [Step 1.a](#expressions) Write expressions for the metric gridfunctions\n 1. [Step 1.b](#ccodekernels) Generate C functions to calculate the gridfunctions\n 1. [Step 1.c](#free_parameters) Set free parameters in the code\n1. [Step 2](#mainc): `FCVAL_unit_test.c`: The Main C Code\n 1. [Step 2.a](#compile_run): Compile and run the code\n1. [Step 3](#convergence): Code validation: Verify that relative error in numerical solution converges to zero at the expected order\n1. [Step 4](#latex_pdf_output): Output this notebook to $\\LaTeX$-formatted PDF file\n\n\n\n# Step 1: Set up core functions and parameters for unit testing the FCVAL algorithm \\[Back to [top](#toc)\\]\n$$\\label{setup}$$\n\nWe'll start by appending the relevant paths to `sys.path` so that we can access sympy modules in other places. Then, we'll import NRPy+ core functionality and set up a directory in which to carry out our test. \n\n\n```python\nimport os, sys # Standard Python modules for multiplatform OS-level functions\n# First, we'll add the parent directory to the list of directories Python will check for modules.\nnrpy_dir_path = os.path.join(\"..\")\nif nrpy_dir_path not in sys.path:\n sys.path.append(nrpy_dir_path)\nnrpy_dir_path = os.path.join(\"..\",\"..\")\nif nrpy_dir_path not in sys.path:\n sys.path.append(nrpy_dir_path)\n\nfrom outputC import outCfunction, lhrh # NRPy+: Core C code output module\nimport sympy as sp # SymPy: The Python computer algebra package upon which NRPy+ depends\nimport finite_difference as fin # NRPy+: Finite difference C code generation module\nimport NRPy_param_funcs as par # NRPy+: Parameter interface\nimport grid as gri # NRPy+: Functions having to do with numerical grids\nimport indexedexp as ixp # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support\nimport reference_metric as rfm # NRPy+: Reference metric support\nimport cmdline_helper as cmd # NRPy+: Multi-platform Python command-line interface\n\nout_dir = \"Validation/\"\ncmd.mkdir(out_dir)\nsubdir = \"FCVAL\"\ncmd.mkdir(os.path.join(out_dir,subdir))\n\nthismodule = \"Start_to_Finish_UnitTest-GiRaFFE_NRPy-Metric_Face_Values\"\n\n# Set the finite-differencing order to 2\npar.set_parval_from_str(\"finite_difference::FD_CENTDERIVS_ORDER\", 2)\n\n```\n\n\n\n## Step 1.a: Write expressions for the metric gridfunctions \\[Back to [top](#toc)\\]\n$$\\label{expressions}$$\n\nNow, we'll choose some functions with arbitrary forms to generate test data. We'll need to set ten gridfunctions, so expressions are being pulled from several previously written unit tests.\n\n\\begin{align}\n\\gamma_{xx} &= ax^3 + by^3 + cz^3 + dy^2 + ez^2 + f \\\\\n\\gamma_{yy} &= gx^3 + hy^3 + lz^3 + mx^2 + nz^2 + p \\\\\n\\gamma_{zz} &= px^3 + qy^3 + rz^3 + sx^2 + ty^2 + u. \\\\\n\\gamma_{xy} &= a \\exp\\left(-\\left((x-b)^2+(y-c)^2+(z-d)^2\\right)\\right) \\\\\n\\gamma_{xz} &= f \\exp\\left(-\\left((x-g)^2+(y-h)^2+(z-l)^2\\right)\\right) \\\\\n\\gamma_{yz} &= m \\exp\\left(-\\left((x-n)^2+(y-o)^2+(z-p)^2\\right)\\right), \\\\\n\\beta^x &= \\frac{2}{\\pi} \\arctan(ax + by + cz) \\\\\n\\beta^y &= \\frac{2}{\\pi} \\arctan(bx + cy + az) \\\\\n\\beta^z &= \\frac{2}{\\pi} \\arctan(cx + ay + bz) \\\\\n\\alpha &= 1 - \\frac{1}{2+x^2+y^2+z^2} \\\\\n\\end{align}\n\n\n\n```python\na,b,c,d,e,f,g,h,l,m,n,o,p,q,r,s,t,u = par.Cparameters(\"REAL\",thismodule,[\"a\",\"b\",\"c\",\"d\",\"e\",\"f\",\"g\",\"h\",\"l\",\"m\",\"n\",\"o\",\"p\",\"q\",\"r\",\"s\",\"t\",\"u\"],1e300)\nM_PI = par.Cparameters(\"#define\",thismodule,[\"M_PI\"], \"\")\n\ngammaDD = ixp.register_gridfunctions_for_single_rank2(\"AUXEVOL\",\"gammaDD\",\"sym01\",DIM=3)\nbetaU = ixp.register_gridfunctions_for_single_rank1(\"AUXEVOL\",\"betaU\",DIM=3)\nalpha = gri.register_gridfunctions(\"AUXEVOL\",\"alpha\")\n\npar.set_parval_from_str(\"reference_metric::CoordSystem\",\"Cartesian\")\nrfm.reference_metric()\nx = rfm.xxCart[0]\ny = rfm.xxCart[1]\nz = rfm.xxCart[2]\n\ngammaDD[0][0] = a*x**3 + b*y**3 + c*z**3 + d*y**2 + e*z**2 + f\ngammaDD[1][1] = g*x**3 + h*y**3 + l*z**3 + m*x**2 + n*z**2 + o\ngammaDD[2][2] = p*x**3 + q*y**3 + r*z**3 + s*x**2 + t*y**2 + u\ngammaDD[0][1] = a * sp.exp(-((x-b)**2 + (y-c)**2 + (z-d)**2))\ngammaDD[0][2] = f * sp.exp(-((x-g)**2 + (y-h)**2 + (z-l)**2))\ngammaDD[1][2] = m * sp.exp(-((x-n)**2 + (y-o)**2 + (z-p)**2))\n\nbetaU[0] = (sp.sympify(2)/M_PI) * sp.atan(a*x + b*y + c*z)\nbetaU[1] = (sp.sympify(2)/M_PI) * sp.atan(b*x + c*y + a*z)\nbetaU[2] = (sp.sympify(2)/M_PI) * sp.atan(c*x + a*y + b*z)\n\nalpha = sp.sympify(1) - sp.sympify(1) / (sp.sympify(2) + x**2 + y**2 + z**2)\n```\n\n\n\n## Step 1.b: Generate C functions to calculate the gridfunctions \\[Back to [top](#toc)\\]\n$$\\label{ccodekernels}$$\n\nHere, we will use the NRPy+ function `outCfunction()` to generate C code that will calculate our metric gridfunctions over an entire grid. We will also call the function to generate the function we are testing. \n\n\n```python\nmetric_gfs_to_print = [\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"gammaDD00\"),rhs=gammaDD[0][0]),\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"gammaDD01\"),rhs=gammaDD[0][1]),\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"gammaDD02\"),rhs=gammaDD[0][2]),\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"gammaDD11\"),rhs=gammaDD[1][1]),\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"gammaDD12\"),rhs=gammaDD[1][2]),\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"gammaDD22\"),rhs=gammaDD[2][2]),\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"betaU0\"),rhs=betaU[0]),\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"betaU1\"),rhs=betaU[1]),\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"betaU2\"),rhs=betaU[2]),\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"alpha\"),rhs=alpha),\\\n ]\n\ndesc = \"Calculate the metric gridfunctions\"\nname = \"calculate_metric_gfs\"\noutCfunction(\n outfile = os.path.join(out_dir,name+\".h\"), desc=desc, name=name,\n params =\"const paramstruct *restrict params,REAL *restrict xx[3],REAL *restrict auxevol_gfs\",\n body = fin.FD_outputC(\"returnstring\",metric_gfs_to_print,params=\"outCverbose=False\").replace(\"IDX4\",\"IDX4S\"),\n loopopts=\"AllPoints,Read_xxs\")\n\nimport GiRaFFE_NRPy.GiRaFFE_NRPy_Metric_Face_Values as FCVAL\nFCVAL.GiRaFFE_NRPy_FCVAL(os.path.join(out_dir,subdir))\n\n```\n\n Output C function calculate_metric_gfs() to file Validation/calculate_metric_gfs.h\n\n\n\n\n## Step 1.c: Set free parameters in the code \\[Back to [top](#toc)\\]\n$$\\label{free_parameters}$$\n\nWe also need to create the files that interact with NRPy's C parameter interface. \n\n\n```python\n# Step 3.d.i: Generate declare_Cparameters_struct.h, set_Cparameters_default.h, and set_Cparameters[-SIMD].h\n# par.generate_Cparameters_Ccodes(os.path.join(out_dir))\n\n# Step 3.d.ii: Set free_parameters.h\nwith open(os.path.join(out_dir,\"free_parameters.h\"),\"w\") as file:\n file.write(\"\"\"\n// Override parameter defaults with values based on command line arguments and NGHOSTS.\nparams.Nxx0 = atoi(argv[1]);\nparams.Nxx1 = atoi(argv[2]);\nparams.Nxx2 = atoi(argv[3]);\nparams.Nxx_plus_2NGHOSTS0 = params.Nxx0 + 2*NGHOSTS;\nparams.Nxx_plus_2NGHOSTS1 = params.Nxx1 + 2*NGHOSTS;\nparams.Nxx_plus_2NGHOSTS2 = params.Nxx2 + 2*NGHOSTS;\n// Step 0d: Set up space and time coordinates\n// Step 0d.i: Declare \\Delta x^i=dxx{0,1,2} and invdxx{0,1,2}, as well as xxmin[3] and xxmax[3]:\nconst REAL xxmin[3] = {-1.0,-1.0,-1.0};\nconst REAL xxmax[3] = { 1.0, 1.0, 1.0};\n\nparams.dxx0 = (xxmax[0] - xxmin[0]) / ((REAL)params.Nxx0);\nparams.dxx1 = (xxmax[1] - xxmin[1]) / ((REAL)params.Nxx1);\nparams.dxx2 = (xxmax[2] - xxmin[2]) / ((REAL)params.Nxx2);\nprintf(\"dxx0,dxx1,dxx2 = %.5e,%.5e,%.5e\\\\n\",params.dxx0,params.dxx1,params.dxx2);\nparams.invdx0 = 1.0 / params.dxx0;\nparams.invdx1 = 1.0 / params.dxx1;\nparams.invdx2 = 1.0 / params.dxx2;\n\\n\"\"\")\n\n# Generates declare_Cparameters_struct.h, set_Cparameters_default.h, and set_Cparameters[-SIMD].h\npar.generate_Cparameters_Ccodes(os.path.join(out_dir))\n```\n\n\n\n# Step 2: `FCVAL_unit_test.c`: The Main C Code \\[Back to [top](#toc)\\]\n$$\\label{mainc}$$\n\n\n\n\n```python\n%%writefile $out_dir/FCVAL_unit_test.c\n// These are common packages that we are likely to need.\n#include \"stdio.h\"\n#include \"stdlib.h\"\n#include \"math.h\"\n#include \"string.h\" // Needed for strncmp, etc.\n#include \"stdint.h\" // Needed for Windows GCC 6.x compatibility\n#include // Needed to set a random seed.\n\n#define REAL double\n#include \"declare_Cparameters_struct.h\"\n\nconst int NGHOSTS = 3;\n\nREAL a,b,c,d,e,f,g,h,l,m,n,o,p,q,r,s,t,u;\n\n// Standard NRPy+ memory access:\n#define IDX4S(g,i,j,k) \\\n( (i) + Nxx_plus_2NGHOSTS0 * ( (j) + Nxx_plus_2NGHOSTS1 * ( (k) + Nxx_plus_2NGHOSTS2 * (g) ) ) )\n\nconst int kronecker_delta[4][3] = { { 0,0,0 },\n { 1,0,0 },\n { 0,1,0 },\n { 0,0,1 } };\n\n// Give gridfunctions their names:\n#define GAMMADD00GF 0\n#define GAMMADD01GF 1\n#define GAMMADD02GF 2\n#define GAMMADD11GF 3\n#define GAMMADD12GF 4\n#define GAMMADD22GF 5\n#define BETAU0GF 6\n#define BETAU1GF 7\n#define BETAU2GF 8\n#define ALPHAGF 9\n#define GAMMA_FACEDD00GF 10\n#define GAMMA_FACEDD01GF 11\n#define GAMMA_FACEDD02GF 12\n#define GAMMA_FACEDD11GF 13\n#define GAMMA_FACEDD12GF 14\n#define GAMMA_FACEDD22GF 15\n#define BETA_FACEU0GF 16\n#define BETA_FACEU1GF 17\n#define BETA_FACEU2GF 18\n#define ALPHA_FACEGF 19\n#define NUM_AUXEVOL_GFS 20\n\n#include \"calculate_metric_gfs.h\"\n#include \"FCVAL/interpolate_metric_gfs_to_cell_faces.h\"\n\nint main(int argc, const char *argv[]) {\n paramstruct params;\n#include \"set_Cparameters_default.h\"\n\n // Step 0c: Set free parameters, overwriting Cparameters defaults\n // by hand or with command-line input, as desired.\n#include \"free_parameters.h\"\n#include \"set_Cparameters-nopointer.h\"\n\n // Step 0e: Set up cell-centered Cartesian coordinate grids\n REAL *xx[3];\n xx[0] = (REAL *)malloc(sizeof(REAL)*Nxx_plus_2NGHOSTS0);\n xx[1] = (REAL *)malloc(sizeof(REAL)*Nxx_plus_2NGHOSTS1);\n xx[2] = (REAL *)malloc(sizeof(REAL)*Nxx_plus_2NGHOSTS2);\n for(int j=0;j\n\n## Step 2.a: Compile and run the code \\[Back to [top](#toc)\\]\n$$\\label{compile_run}$$\n\nNow that we have our file, we can compile it and run the executable.\n\n\n```python\nimport time\n\nprint(\"Now compiling, should take ~2 seconds...\\n\")\nstart = time.time()\ncmd.C_compile(os.path.join(out_dir,\"FCVAL_unit_test.c\"), os.path.join(out_dir,\"FCVAL_unit_test\"))\nend = time.time()\nprint(\"Finished in \"+str(end-start)+\" seconds.\\n\\n\")\n\nprint(\"Now running...\\n\")\nstart = time.time()\ncmd.Execute(os.path.join(\"Validation\",\"FCVAL_unit_test\"), \"10 10 10\",\"out.txt\")\n# To do a convergence test, we'll also need a second grid with twice the resolution.\ncmd.Execute(os.path.join(\"Validation\",\"FCVAL_unit_test\"), \"20 20 20\",\"out.txt\")\nend = time.time()\nprint(\"Finished in \"+str(end-start)+\" seconds.\\n\\n\")\n\n```\n\n Now compiling, should take ~2 seconds...\n \n Compiling executable...\n (EXEC): Executing `gcc -Ofast -fopenmp -march=native -funroll-loops Validation/FCVAL_unit_test.c -o Validation/FCVAL_unit_test.exe -lm`...\n (BENCH): Finished executing in 1.0330591201782227 seconds.\n Finished compilation.\n Finished in 1.18806791305542 seconds.\n \n \n Now running...\n \n (EXEC): Executing `cmd /c Validation\\FCVAL_unit_test 10 10 10`...\n (BENCH): Finished executing in 0.23601365089416504 seconds.\n (EXEC): Executing `cmd /c Validation\\FCVAL_unit_test 20 20 20`...\n (BENCH): Finished executing in 0.2540144920349121 seconds.\n Finished in 0.7620437145233154 seconds.\n \n \n\n\n\n\n# Step 3: Code validation: Verify that relative error in numerical solution converges to zero at the expected order \\[Back to [top](#toc)\\]\n$$\\label{convergence}$$\n\nHere, we import the data at two resolutions and wrote to text files. This data consists of the absolute error of a metric gridfunction at each in the grid. We'll plot a portion of this data along the axis at the lower resolution along with that same data at the higher resolution scaled to demonstrate that this error converges to 0 at the expected rate. Since our algorithm uses a third-order polynomial, we expect fourth-order convergence here.\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nplt.rcParams.update({'font.size': 12})\n\nData1 = np.loadtxt(\"out10-numer.txt\")\nData2 = np.loadtxt(\"out20-numer.txt\")\n\ndef IDX4(i,j,k,Nxx_plus_2NGHOSTS0,Nxx_plus_2NGHOSTS1,Nxx_plus_2NGHOSTS2):\n return (i) + Nxx_plus_2NGHOSTS0 * ( (j) + Nxx_plus_2NGHOSTS1 * ( (k) + Nxx_plus_2NGHOSTS2 * (0) ) )\n\nx1 = np.zeros(10)\na1 = np.zeros(10)\nfor i in range(10):\n x1[i] = Data1[IDX4(i+3,8,8,16,16,16),1]\n a1[i] = Data1[IDX4(i+3,8,8,16,16,16),0]\nx2 = np.zeros(20)\na2 = np.zeros(20)\nfor i in range(20):\n x2[i] = Data2[IDX4(i+3,13,13,26,26,26),1]\n a2[i] = Data2[IDX4(i+3,13,13,26,26,26),0]\n\nplt.figure()\na = plt.plot(x1,a1,'.',label=\"dx\")\nb = plt.plot(x2,a2*(2**4),label=\"dx/2, times (20/10)^4\")\nplt.legend()\nplt.xlabel(\"x\")\nplt.ylabel(\"alpha\")\nplt.show()\n\nconvergence_satisfactory = np.log2(a1/a2[0:-1:2])>3\nif not convergence_satisfactory.all():\n sys.exit(1)\n```\n\n\n\n# Step 4: Output this notebook to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n$$\\label{latex_pdf_output}$$\n\nThe following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n[Tutorial-Start_to_Finish_UnitTest-GiRaFFE_NRPy-Metric_Face_Values.pdf](Tutorial-Start_to_Finish_UnitTest-GiRaFFE_NRPy-Metric_Face_Values.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means.)\n\n\n```python\nimport cmdline_helper as cmd # NRPy+: Multi-platform Python command-line interface\ncmd.output_Jupyter_notebook_to_LaTeXed_PDF(\"Tutorial-Start_to_Finish_UnitTest-GiRaFFE_NRPy-Metric_Face_Values\")\n```\n\n Notebook output to PDF is only supported on Linux systems, with pdflatex installed.\n\n", "meta": {"hexsha": "a58d1b33abd33b046cb515bf0f36de709ae08256", "size": 42443, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "in_progress/Tutorial-Start_to_Finish_UnitTest-GiRaFFE_NRPy-Metric_Face_Values.ipynb", "max_stars_repo_name": "goncalo-andrade/nrpytutorial", "max_stars_repo_head_hexsha": "4fbcb51c936864b442daefd176bd6a5277c00116", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-06-09T16:16:21.000Z", "max_stars_repo_stars_event_max_datetime": "2020-06-09T16:16:21.000Z", "max_issues_repo_path": "in_progress/Tutorial-Start_to_Finish_UnitTest-GiRaFFE_NRPy-Metric_Face_Values.ipynb", "max_issues_repo_name": "goncalo-andrade/nrpytutorial", "max_issues_repo_head_hexsha": "4fbcb51c936864b442daefd176bd6a5277c00116", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "in_progress/Tutorial-Start_to_Finish_UnitTest-GiRaFFE_NRPy-Metric_Face_Values.ipynb", "max_forks_repo_name": "goncalo-andrade/nrpytutorial", "max_forks_repo_head_hexsha": "4fbcb51c936864b442daefd176bd6a5277c00116", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 71.5733558179, "max_line_length": 17144, "alphanum_fraction": 0.7307212026, "converted": true, "num_tokens": 5910, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191214879992, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.44133556817074643}} {"text": "# FastEMRIWaveforms Tutorial\n## ICERM Workshop: Waveform acceleration with machine learning and GPUs\n\n### Michael Katz, Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Lead developer for FastEMRIWaveforms\n\nIn this tutorial, you will learn the basics of building an accelereated EMRI waveform. We encourage participants to see our paper ([arxiv.org/2008.06071](https://arxiv.org/abs/2008.06071)) and the FastEMRIWaveforms [package documentation](https://bhptoolkit.org/FastEMRIWaveforms/) for more information, as well as our forthcoming paper that will describe the waveform in much more detail than our PRL submission. \n\nImport packages:\n\n\n```python\nimport matplotlib.pyplot as plt\n%matplotlib inline\nimport numpy as np\n\nimport h5py\n\n\nfrom few.amplitude import romannet\nfrom few.utils.utility import check_for_file_download, p_to_y\nfrom few.trajectory.flux import RunSchwarzEccFluxInspiral\nfrom few.amplitude.romannet import RomanAmplitude\nfrom few.amplitude.interp2dcubicspline import Interp2DAmplitude\nfrom few.waveform import FastSchwarzschildEccentricFlux\n```\n\n## A Quick Introduction to FastEMRIWaveforms\n\n### Collaborators: Alvin Chua, Niels Warburton, Scott Hughes, Lorenzo Speri\n\n\n```python\ngenerate = FastSchwarzschildEccentricFlux()\n\nM = 1e6 # large mass\nmu = 1e1 # small mass\np0 = 12.0 # separation \ne0 = 0.4 # eccentricity\ntheta = np.pi/3. # polar viewing angle\nphi = np.pi/4. # azimuthal viewing angle\ndist = 1.0 # distance in Gpc\n\nT = 1/365. # in years\ndt = 10.0 # time spacing of data stream\n\nwave = generate(M, mu, p0, e0, theta, phi, dist, T=T, dt=dt)\n\nplt.plot(wave.real)\nplt.plot(wave.imag)\n```\n\nFastEMRIWaveforms is the first fully relativistic template generation tool for extreme mass ratio inspirals. We think of it more as a framework than any specific waveform. In this tutorial, we will discuss the main pieces of this framework and how they relate to our first fully relativistic waveform model shown above. \n\nAs a basic primer, our five key points we want to achieve with this framework are:\n\n* Accuracy: Our fast waveforms must be accurate when compared to slow and accurate waveforms generated by the waveform modeling community.\n\n* Modularity: These waveforms are to be built out of a set of modules. These modules are to be easily interchangeable, as well as stand alone tools available for more in-depth analysis.\n\n* Flexibility: This framework must be easily adaptable to new computational methods and/or improvements in EMRI physics. \n\n* Easy User Interface: All modules and complete waveforms are to have a front-facing python interface with clear and extensive documentation, as well as many examples.\n\n* Parallelization: These waveforms must take advantage of parallelization techniques such as OpenMP, as well as accelerator hardware such as GPUs. \n\n## Basics of an EMRI Waveform\n\nThe guiding equation for building an EMRI waveform is given by\n\n\\begin{equation}\nh_+-ih_x = \\frac{1}{r}\\sum_{lmkn}\\left(-\\frac{Z_{lmkn}}{\\omega_{mkn}^2}\\right)\\left(S_{lmkn}(\\theta)e^{-im\\phi}\\right)e^{i\\Phi_{mkn}} = \\frac{1}{r}\\sum_{lmkn}A_{lmkn}\\Theta_{lmkn}e^{i\\Phi_{mkn}}.\n\\end{equation}\n\n\n\nHere we are concerned with generating fast and accurate waveforms. These waveforms are created with a sequence of modules. We will discuss the three main modules used to produce these waveforms. The first module is the Trajectory module which takes initial parameters and produces arrays for all of the evolving quantities of concern. These include the phase evolution, {$\\Phi_\\varphi,\\Phi_\\theta, \\Phi_r$}, from the start to the end of the EMRI orbit. With these quantities, we can produce:\n\n\\begin{equation}\n\\Phi_{mkn} = m\\Phi_\\varphi + k\\Phi_\\theta + n\\Phi_r.\n\\end{equation}\n\nThe trajectory also produces orbital quantities over time. These include the separation ($p$), eccentricity ($e$), and the inclination angle of the orbit ($\\iota$). With these arrays containing the orbital evolution of these values, we generate the amplitudes, $A_{lmkn}$. While the evolution is over time, the various phasing and amplitude computations are produced within a frequency decomposition into $(l,m,k,n)$ modes.\n\nWith phases and amplitdues in hand, we combine this with the angular harmonic information ($\\Theta_{lmkn}$) to produce the final waveform. In this step, we calculate the sum of all modes at each time point in the template data stream. \n\n## Overall Waveform Strategy\n\n* We calculate our trajectories and amplitudes as sparse arrays in time by using a large adaptive stepping integrator. This produces arrays with $\\sim100$ points. All of these quantities vary slowly and smoothly. After these calculations are complete, we scale this up to the actual data stream cadence. \n\n* The speed of EMRI waveforms is strongly determined by the amount of harmonic content. Higher eccentricities require more modes to produce a high fidelity waveform. In order to make our waveforms as efficient as possible, we perform an online mode content calculation that removes modes from consideration if they do not contribute to the waveform power determined by a user-defined threshold. \n\n## Current Waveform Model: Schwarzschild Eccentric\n\n* No $k$ modes\n* Orbit is equatorial\n* $S_{lmkn}(\\theta)e^{-im\\phi}$ reduces to $_{-2}Y_{lm}(\\theta,\\phi)$\n* $l:\\{2,10\\}$, $m:\\{-l,l\\}$, $n:\\{-30,30\\}$ $\\rightarrow$ 3843 modes. \n\n## Fast Trajectories: $\\{p, e, \\Phi_\\varphi, \\Phi_r\\}$\n\nWe are not going to spend too much time on the trajectory part. However, we need to generate it in order to build the rest of our waveform. To build the trajectory, we integrate with large steps using an RK8 integrator. \n\n\n```python\ntraj = RunSchwarzEccFluxInspiral()\n```\n\n\n```python\np0 = 16.0 # initial separation\ne0 = 0.4 # initial eccentricity\nmu = 180. # iniital small mass in solar masses, produces approximately 1 yr waveform\nM = 1e6 # initial large mass in solar masses\ndt = 10.0 # sets initial step size\nT = 1.0 # in years\n\nt, p, e, Phi_phi, Phi_r, flux = traj(M, mu, p0, e0, T=T, dt=dt)\nprint(\"length:\", len(t), \"duration:\", t[-1])\n```\n\n\n```python\nfig, axes = plt.subplots(2, 3)\nplt.subplots_adjust(wspace=0.3)\nfig.set_size_inches(14, 8)\naxes = axes.ravel()\n\nylabels = [r'$e$', r'$p$', r'$e$', r'$\\Phi_\\phi$', r'$\\Phi_r$', r'Flux']\nxlabels = [r'$p$', r'$t$', r'$t$', r'$t$', r'$t$', r'$t$', r'$t$', r'$t$']\nys = [e, p, e, Phi_phi, Phi_r, flux]\nxs = [p, t, t, t, t, t]\n\nfor i, (ax, x, y, xlab, ylab) in enumerate(zip(axes, xs, ys, xlabels, ylabels)):\n ax.plot(x, y, lw=0.5)\n ax.scatter(x, y, s=5)\n ax.set_xlabel(xlab, fontsize=16)\n ax.set_ylabel(ylab, fontsize=16)\n```\n\n## RomanNet Amplitudes: $A_{lmn}$\n\nTo generate the amplitudes, we use a RomanNet ([arXiv:1811.05491](https://arxiv.org/abs/1811.05491)). Roman stands for Reduced Order Modelling with Artificial Neurons. When training a neural network, it can be crucial to hand it data that is distilled to highlight the most pertinent information. A nice tool for doing this is Reduced Order Modeling. Reduced order modelling projects the information with lossless compression down to a lower dimensional space. We take our complex mode amplitude vectors containing 3843 modes and project this down to a real-valued space with 198 values. The neural network is then trained with inputs given by the $p$ and $e$ values and outputs given by the reduced order coefficients ($\\alpha_i$):\n\n\\begin{equation}\nA_{lmn}\\in\\mathbb{C}^{3843}\\xrightarrow{\\mathit{ROM}}\\alpha_i\\in\\mathbb{R}^{198}\\xrightarrow{\\mathit{train}} f(p,e)=\\alpha_i\n\\end{equation}\n\nThe neural network itself is extremely simple. It is a fully connected network with a LeakyReLU activation on all layers but the final layer. This means the neural network can be built simply with a sequence of linear matrix multiplications followed by a pass through the activation function. Once the neural network is trained, we will have a set of weights. This is where we will start in this tutorial. \n\nDuring online evaluation of the waveform, we perform the reverse process:\n\n\\begin{equation}\nf(p,e)\\xrightarrow{\\mathit{eval}}\\alpha_i\\in\\mathbb{R}^{198}\\xrightarrow{\\mathit{project}}A_{lmn}\\in\\mathbb{C}^{3843}\n\\end{equation}\n\nThis method has pros and cons:\n\nPros:\n\n* This is more of a global fit, rather than individual fits to given modes. This generally means storage of less information in memory. If we do individual fits of every mode, the memory necessary to store this information would scale badly with mode content. We are currently working in Schwarzschild Eccentric. As we go to generic Kerr, we expect the number of modes to increase by a factor of $\\sim10$.\n\n* Due to its global fit nature, this method is generally faster to evaluate than individual interpolants. \n\n* Since it is a neural network and a linear projection, this is very suitable to GPUs.\n\n* We expect this method, or methods similar to this, to scale better with dimensionality as we move towards the end goal of generic Kerr orbits. \n\nCons:\n\n* For extremely quiet modes, this method can be less accurate. However, as these modes are quiet, this does not result in a significant loss of accuracy in the final waveform. Caution must be taken when using this method to analyze individual mode amplitudes. There is an approximate floor in the amplitude values at $\\sim10^{-5}$. For reference, the loudest modes at a given $p$ and $e$ value are usually $\\sim0.1-1.0$. (These amplitudes are not scaled for distance.|) \n\n* Training these neural networks can be more of an art than a science. It takes a lot of trial and error to get this right. \n\n### Constructing the neural network from trained weights\n\n\n```python\n\n\n# prepare to load the weights\n\npath_to_few_dir = romannet.__file__[:-25]\n\nweight_file = fp = \"SchwarzschildEccentricInput.hdf5\"\ncheck_for_file_download(fp, path_to_few_dir)\n\nweights = []\nbias = []\ndim1 = []\ndim2 = []\n\n# get highest layer number\nnum_layers = 0\n\n# extract all necessary information from the file\nwith h5py.File(path_to_few_dir + \"few/files/\" + weight_file, \"r\") as fp:\n \n # get basic information\n num_teuk_modes = fp.attrs[\"num_teuk_modes\"]\n transform_factor = fp.attrs[\"transform_factor\"]\n break_index = fp.attrs[\"break_index\"]\n \n # determine layer arrangement \n for key, value in fp.items():\n if key == \"reduced_basis\":\n continue\n\n layer_num = int(key[1:])\n\n if layer_num > num_layers:\n num_layers = layer_num\n\n # get weights and bias\n for i in range(1, num_layers + 1):\n temp = {}\n for let in [\"w\", \"b\"]:\n mat = fp.get(let + str(i))[:]\n temp[let] = np.asarray(mat)\n\n weights.append(temp[\"w\"])\n bias.append(temp[\"b\"])\n dim1.append(temp[\"w\"].shape[0])\n dim2.append(temp[\"w\"].shape[1])\n\n # get the post network transform matrix\n transform_matrix = np.asarray(fp[\"reduced_basis\"])\n\n# activation function\n# we use a factor of 0.2 for negative values\ndef LeakyReLU(x):\n out = (x >= 0.0) * x + (x < 0.0) * 0.2*x\n return out\n\n# build the neural network\ndef RomanNet(p, e):\n \n p = np.atleast_1d(p)\n e = np.atleast_1d(e)\n \n # convert from the p coordinate to a special y coordinate\n # see the documentation for more details\n y = p_to_y(p, e)\n \n # prepare input\n x = np.array([y, e])\n \n # basic fully connected network\n for layer_i in range(num_layers):\n \n # linear transformation\n x = np.dot(weights[layer_i].T, x) + bias[layer_i][:, np.newaxis]\n \n # do not want to activate last layer\n if layer_i < num_layers - 1:\n # non-linear activatation\n x = LeakyReLU(x)\n \n # separate real and imaginary\n x = x[:break_index] + 1j * x[break_index:]\n \n # project back to amplitude basis\n out = np.dot(transform_matrix.T, x)/transform_factor\n return out.T\n \n \n```\n\n\n```python\n# test it\np_test = np.array([11.0, 10.0, 10.0])\ne_test = np.array([0.2, 0.3, 0.1])\n\nRomanNet(p_test, e_test)\n```\n\n### Produce amplitudes associated with our trajectories\n\n\n```python\n# generate amplitudes with roman net\namps = RomanNet(p, e)\n\n# check against actual code\nRomanNetTrue = RomanAmplitude()\namps_check_1 = RomanNetTrue(p, e)\n\nassert np.allclose(amps, amps_check_1)\n```\n\n### Check our RomanNet global fit against accurate values\n\n\n```python\n# get accurate values \n# each mode is fitted with a bicubic spline\nBicubicAmps = Interp2DAmplitude()\namps_check_2 = BicubicAmps(p, e)\n```\n\n#### Print vectors\n\n\n```python\nprint(\"romannet:\", amps_check_1[0][0:10], \"\\nBicubic Spline:\", amps_check_2[0][0:10])\n```\n\nWe see if we take a quick examination of our results (a small subset) we see the problem with the global fit. It cannot handle modes with a very small amplitude. Therefore, you might naively think that this method might not work.\n\n#### Compare via the cosine between the vectors\n\n\n```python\ncos = (np.dot(amps_check_2.conj()[0].T, amps_check_1[0]) /\n np.sqrt(np.dot(amps_check_2.conj()[0], amps_check_2[0]) \n * np.dot(amps_check_1.conj()[0], amps_check_1[0]))).real\n\nprint('Cos:', cos)\n```\n\nWe now see that if we compare the results as a whole, the results match very well. If we look at individual modes that have high power, they are likely to strongly match when the bicubic spline is compared to the RomanNet method. \n\n## GPU-accelerated Waveform Build: $\\sum_{lmn}$\n\n### Quick primer on GPUs\n\n\n\n\n\nGPUs run code in parallel in a configuration of grids, blocks, and threads.\n\n* **Threads** are the the actual software units that run the code. Threads run independently of one another. Threads are referenced in code using `threadIdx.x`.\n* **Blocks** are an array of threads. This array can be 1, 2, or 3 dimensions. In more than one dimension, thread indices are referenced using `x`, `y`, and `z` (e.g. `threadIdx.z`). Blocks, similar to threads, are referenced using `blockIdx.x`, `blockIdx.y`, or `blockIdx.z`. The size of a block, i.e. the number of threads along a given dimension is given as `blockDim.x`. **Note**: In applications I have worked on, I rarely ever use more than 1 dimension of threads. \n* **Grids** are an array of blocks. This array can be 1, 2, or 3 dimensions. There is no reference for grids as they efffectively represent the entire GPU kernel. The size of a grid, i.e. the number of blocks along a dimension, is determined with `gridDim.x`. \n\nOrganizing your code properly into blocks and threads is a key component of maximizing your efficiency. \n\nAnother equally important aspect to the maximization of efficiency is the proper use of the GPU memory structures. There are three main GPU memory structures to consider (there are a few more but they are for more specific uses). These are global memory, shared memory, and local memory.\n\n* **Global memory** is contained on the host (or off-chip). Global memory is accessible by every block and thread in the entire grid. It contains many GBs of RAM. (The GPUs I use currently, which are Tesla V100s, have ~16GB. The new A100s have 40-80GB.) Because this memory is contained off-chip, it is slow to access. A key to accessing global memory efficiently is to use so-called \"memory coalescence.\" This effectively means that neighboring threads access neighboring addresses in memory. This allows the compiler to make up to 32 memory reads at the same time, rather than 32 separate memory reads. We will see this in a simple case below. \n\n* **Local memory** is also contained on the host. Local memory consists of any arrays allocated within the kernel specific to each thread. Therefore, this memory is only accessible by the thread it is created on. Usually, around 512 KB are available for local memory for each thread. Since, this memory is off-chip, it is also slow to access. Local memory is, however, always accessed in a memory coalescing manner. \n\n* **Shared memory** is different. Shared memory is located on-chip. It is, therefore, much faster to read from (~100x faster). Shared memory is accessible by all the threads on a given block. The catch is that only ~48 KB are available for shared memory. Therefore, leveraging shared memory effectively is key to the efficiency of GPU code. \n\nTo sum up, there are two main points to consider when beginning to program on GPUs:\n\n* Layout your grid effectively for your given problem.\n* Use memory effectively: leverage the availability of shared memory and read from global memory in a coalescing fashion. \n\nSome parting thoughts on GPUs:\n\n* For maximal efficiency and stability, I usually code in C++/CUDA so that everything is precompiled and tested. \n* There are python libraries that leverage the power of GPUs. Check out numba, CuPy, PyTorch, Tensorflow, PyCUDA. \n* Lately, I have really focused on writing CPU/GPU agnostic code. This means that the source code is ~99% the same between the two. On the python side I usually sub in CuPy for Numpy. We will see some basic pointers on this below. In C++, I use short compiler directives to make minimal changes.\n* Generally speaking it is optimal to store all quantities in a 1 dimensional array when working with GPUs. Let's say you have a two dimensional array that has dimensions (dim1, dim2) and is referenced with (i, j). You can turn this into a 1D array that references each value with (i * dim2 + j). \n\n### Why are GPUs important for EMRI Waveforms?\n\nThe waveform summation is the key bottleneck. This is an operation that is uniquely suited to GPUs. You can see the improvement below.\n\n\n\n### Basic example coded in python\n\nWe are going to write code in python that will reflect what we will do on the GPU. It will obviously still be on the CPU, but will give you a chance to see how to write some basic GPU code and understand how it works. **Make sure to read the comments in the code.**\n\nWe will do two examples. \n\n#### Multiply two arrays\n\nHere we will multiply two arrays as we would on the GPU. \n\n\n```python\n# setup our GPU quantities that would come in if we were actually on GPUs. \n\nNUM_THREADS = 64 # needs to be a multiple of 32\nblockDim = NUM_THREADS # blockDim.x\n```\n\n\n```python\n\n# this initial piece is a basic of CPU/GPU agnostic code in python\ntry:\n import cupy as xp\n \nexcept ModuleNotFoundError:\n import numpy as xp\n\n# __global__\ndef multiply_arrays(array_out, array1, array2, n):\n \"\"\"\n // what this would like like in CUDA\n \n // if (threadIdx.x + blockDim.x * blockIdx.x >= n) return;\n for (int i = threadIdx.x + blockDim.x * blockIdx.x; i < n; i += blockDim.x * gridDim.x)\n {\n array_out[i] = array1[i] * array2[i];\n }\n \"\"\"\n # The loops here are to simulate the GPU. \n # In reality the GPU will run all threads and blocks in parallel\n \n # begin simulation\n for block in range(num_blocks):\n for thread in range(NUM_THREADS):\n # end simulation\n \n # get the overall index in the grid\n # based on the thread and block\n i = thread + block * blockDim\n \n # since the GPU runs threads in parallel\n # we need to make sure the GPU does not index a value\n # on the last block that goes over the array length\n if i >= n:\n continue\n \n # use our index to get values out of the array\n # notice this simple statement uses memory coalescence\n # neighboring threads will have consecutive indices\n # therefore, they will access consecutive addresses in each array\n array_out[i] = array1[i] * array2[i]\n \n\nlength = int(2 ** 14)\n\n# initialize arrays\narray1 = xp.random.rand(length)\narray2 = xp.random.rand(length)\n \n# prepare output array\narray_out = xp.zeros_like(array1)\n\n# get the number of blocks\n# the number of blocks multiplied by the number of threads per block\n# must be greater than the length of the array \nnum_blocks = int(np.ceil((length + NUM_THREADS -1)/NUM_THREADS))\n\n# this would actually be called as\n\"\"\"\nmultiply_arrays<<>>(array_out, array1, array2, length);\ncudaDeviceSynchronize();\n\"\"\"\n\nmultiply_arrays(array_out, array1, array2, length)\n\n# confirm it\nassert np.allclose(array_out, array1 * array2)\n```\n\n#### Linear Interpolation to scale up an array \n\nFor our second example it will be a bit more complicated. This way we can see how to use shared memory. \n\nHere we will scale up the size of an array using linear interpolation. The key here is that we need every block to read in the original array to shared memory. From there, we perform the interpolation. This allows us to achieve memory coalescence with all global reads and ensure that all reads when actually interpolating are done from shared memory. \n\n\n```python\n# in CUDA, when you statically allocate shared arrays,\n# you cannot use a variable length\n# therefore, in C++/CUDA you need to declare a max value up front\n\n#define MAX_INPUT 1000\n\ndef linear_interpolation(array_out, array_in, n_out, n_in, dx_out, dx_in):\n \n # we want to read in to shared memory\n # want to do this with memory coalescence\n # since this is only on a specific block\n # we need to use only the threads on this block\n \n \"\"\"\n // the below simulated code block in CUDA would look like this:\n \n // in CUDA, you have to declare shared arrays\n \n __shared__ double shared_array_in[MAX_INPUT];\n \n \n for (int i = threadIdx.x; i < n_in; i += blockDim.x)\n {\n // memory coalescence only needed from array_in (global memory)\n shared_array_in[i] = array_in[i];\n }\n \n // we need to make sure the threads all finish this operation before we move on\n // Therefore, we force the threads to synchronize\n \n __syncthreads();\n \n \"\"\"\n \n \n # we will just do the following to simulate in python easily\n shared_array_in = xp.zeros(n_in)\n \n for thread in range(NUM_THREADS):\n i = thread \n \n # this while statement simulates the above commented loop\n while (i < n_in):\n # here is the memory coalesced reads\n shared_array_in[i] = array_in[i]\n \n # just for simulation\n i += blockDim\n \n \n \"\"\"\n // this is what the below would really like it in CUDA\n // if (threadIdx.x + blockDim.x * blockIdx.x >= n_out) return;\n for (int i = threadIdx.x + blockDim.x * blockIdx.x; i < n_out; i += blockDim.x * gridDim.x)\n {\n // get the new out value assuming equal spacing\n double x_new = dx_out * i;\n \n // get index of the point in the original array below the new point\n int ind_in = (int) (x_new / dx_in);\n \n // get the below x value\n double x_old = ind_in * dx_in;\n\n // slope of segment\n double m = (shared_array_in[ind_in + 1] - shared_array_in[ind_in])/dx_in;\n\n // interpolate\n double new_value = m * (x_new - x_old) + shared_array_in[ind_in];\n array_out[i] = new_value;\n }\n \n \"\"\"\n # The loops here are to simulate the GPU. \n # In reality the GPU will run all threads and blocks in parallel\n \n # begin simulation\n for block in range(num_blocks):\n for thread in range(NUM_THREADS):\n # end simulation\n \n # get the overall index in the grid\n # based on the thread and block\n i = thread + block * blockDim\n \n # since the GPU runs threads in parallel\n # we need to make sure the GPU does not index a value\n # on the last block that goes over the array length\n if i >= n_out:\n continue\n \n x_new = dx_out * i\n \n ind_in = int(x_new / dx_in)\n x_old = ind_in * dx_in\n \n m = (shared_array_in[ind_in + 1] - shared_array_in[ind_in])/dx_in\n \n new_value = m * (x_new - x_old) + shared_array_in[ind_in]\n array_out[i] = new_value\n \n\n# original length\nlength_in = 100\n\n# prepare input arrays\nx_in = xp.arange(length_in)\ny_in = (x_in ** 2).astype(xp.float64)\n\n# set length out\nlength_out = int(2 ** 14)\n\n# setup the new x_values\nx_new = xp.linspace(x_in[0], x_in[-1], length_out + 1)[:-1]\n\n# change in original x values\ndx_in = 1.0\n\n# we will actually only use the spacing to find new points\ndx_out = x_new[1] - x_new[0]\n \ny_out = xp.zeros(length_out)\n\nnum_blocks = int(np.ceil((length_out + NUM_THREADS -1)/NUM_THREADS))\n\n# this would actually be called as\n# linear_interpolation<<>>(y_out, y_in, length_out, length_in, dx_out, dx_in);\n# cudaDeviceSynchronize();\nlinear_interpolation(y_out, y_in, length_out, length_in, dx_out, dx_in)\n\n# confirm it\nplt.plot(x_new, y_out, lw=6, label='out')\nplt.plot(x_in, y_in, '.', label='in')\n```\n\n### Comment on CPU/GPU Agnostic code in C++/CUDA and Python\n\nMaking CPU/GPU agnostic code consists of 3 main parts: \n\n* Sub CuPy for NumPy\n* Compiler directives in C++/CUDA\n* Easy transition from Python to C++ through an augmented Cython process\n\nAbove, we saw an example of how to deal with the Python side. The Cython functionality is effectively a decorator function that gets the pointer of a Numpy or Cupy array and sends that into the C++ code. Below is a basic example of how to use basic compiler directives to make your code more CPU/GPU agnostic.\n\n**Note**: I generally argue that we should build codes for GPUs and then adapt them to CPUs, not the other way around which is the typical direction. In my experience, the CPU codes adapted from GPU codes are just as fast or within a small percentage of the speed of CPU designed codes. And, generally, it is much harder to optimize going from CPU->GPU rather than GPU->CPU. \n\n\n\n## Future Plans\n\n* Build fast trajectories in Kerr under the NIT framework\n* Generate amplitudes in Kerr regime\n* Further analyze how to determine best methods for mode content inclusion\n\nWe need your help! If anyone is interested in working on these types of issues, please let myself or any of my collaborators (listed above) know!\n", "meta": {"hexsha": "d89c6352018277c8b0a370882e22a0e24b57b5a5", "size": 35594, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ICERM_tutorial_GPUs_ML/.ipynb_checkpoints/ICERM_tutorial-checkpoint.ipynb", "max_stars_repo_name": "basuparth/ICERM_Workshop", "max_stars_repo_head_hexsha": "ebabce680fc87e90ff1de30246dcda9beb384bb4", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "ICERM_tutorial_GPUs_ML/.ipynb_checkpoints/ICERM_tutorial-checkpoint.ipynb", "max_issues_repo_name": "basuparth/ICERM_Workshop", "max_issues_repo_head_hexsha": "ebabce680fc87e90ff1de30246dcda9beb384bb4", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ICERM_tutorial_GPUs_ML/.ipynb_checkpoints/ICERM_tutorial-checkpoint.ipynb", "max_forks_repo_name": "basuparth/ICERM_Workshop", "max_forks_repo_head_hexsha": "ebabce680fc87e90ff1de30246dcda9beb384bb4", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 40.401816118, "max_line_length": 742, "alphanum_fraction": 0.5949879193, "converted": true, "num_tokens": 6516, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8128673269042767, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.4412758258669137}} {"text": "\n\n# \ucc38\uace0 \ubb38\uc11c\nhttps://pytorch.org/tutorials/beginner/blitz/tensor_tutorial.html#sphx-glr-beginner-blitz-tensor-tutorial-py \nhttps://pytorch.org/tutorials/beginner/blitz/autograd_tutorial.html \n\n\nDeep Learning with PyTorch\n\n\nWhat is PyTorch?\n================\n\nIt\u2019s a Python-based scientific computing package targeted at two sets of\naudiences:\n\n- A replacement for NumPy to use the power of GPUs\n- a deep learning research platform that provides maximum flexibility\n and speed\n\nGetting Started\n---------------\n\nTensors\n^^^^^^^\n\nTensors are similar to NumPy\u2019s ndarrays, with the addition being that\nTensors can also be used on a GPU to accelerate computing.\n\n\n\n\n```python\nfrom __future__ import print_function\nimport torch\n```\n\n\n```python\nprint(torch.cuda.is_available())\n```\n\n True\n\n\n **\uc8fc\uc758) \uc704 \uacb0\uacfc\uac00 False\uc774\uba74 Colab \uba54\ub274\uc758 \ub7f0\ud0c0\uc784->\ub7f0\ud0c0\uc784 \uc720\ud615\ubcc0\uacbd\uc5d0\uc11c \ud558\ub4dc\uc6e8\uc5b4 \uac00\uc18d\uae30\ub97c GPU\ub85c \uc120\ud0dd\ud558\uace0 \ub2e4\uc2dc \uc2dc\uc791**\n\n\n```python\nx = torch.empty(5, 3) # 5\ud589, 3\uc5f4\uc758 \ud150\uc11c. \uac12\uc740 \ubaa8\ub450 garbage\nprint(x)\n```\n\n tensor([[-3.0226e+35, 3.0672e-41, 3.3631e-44],\n [ 0.0000e+00, nan, 0.0000e+00],\n [ 1.1578e+27, 1.1362e+30, 7.1547e+22],\n [ 4.5828e+30, 1.2121e+04, 7.1846e+22],\n [ 9.2198e-39, 7.0374e+22, 0.0000e+00]])\n\n\n\n```python\nprint(x.size()) # size, shape\n```\n\n torch.Size([5, 3])\n\n\n\n```python\nprint(x.shape)\n```\n\n torch.Size([5, 3])\n\n\n

    Uninitialized vs random

    An uninitialized matrix is declared,\n but does not contain definite known\n values before it is used. When an\n uninitialized matrix is created,\n whatever values were in the allocated\n memory at the time will appear as the initial values.

    \n\n\n---\n\n\n\n\ntorch.empty\ub294 \ucd08\uae30\ud654\ud558\uc9c0 \uc54a\uc740 \uac12(\ud560\ub2f9\ubc1b\uc740 \uba54\ubaa8\ub9ac\uc5d0 \uc788\ub294 \uac12)\uc774 \ub4e4\uc5b4\uc788\ub2e4\n\nConstruct a 5x3 matrix, uninitialized:\n\n\uc989 \uac12\uc774 garbage\uc784. random\uc774 \uc544\ub2d8.\n\n\n\n\n```python\nx = torch.empty(5, 3)\nprint(x)\n```\n\n tensor([[-3.0226e+35, 3.0672e-41, 3.3631e-44],\n [ 0.0000e+00, nan, 0.0000e+00],\n [ 4.4721e+21, 1.5956e+25, 4.7399e+16],\n [ 3.7293e-08, 3.9664e+28, 6.9397e+22],\n [ 1.7260e+25, 2.2856e+20, 5.0948e-14]])\n\n\n\n```python\nprint(torch.get_default_dtype())\n```\n\n torch.float32\n\n\n\n```python\nprint(x.dtype)\n```\n\n torch.float32\n\n\ntorch.empty\ub294 torch\uc758 global default dtype \uc774\uba74\uc11c garbage \uac12\uc744 \uac00\uc9c0\ub294 \uc6d0\uc18c\uac00 \ucc44\uc6cc\uc9c4 \ud150\uc11c\ub97c \uc0dd\uc131.\n\n\uc989 \ub514\ud3f4\ud2b8 \ud0c0\uc785\uc744 \uac16\ub294 \ube48 \ud150\uc11c\ub97c \uc0dd\uc131.\n\n\n```python\nx = torch.FloatTensor(5,3)\nprint(x)\n```\n\n tensor([[-3.0226e+35, 3.0672e-41, 3.7835e-44],\n [ 0.0000e+00, nan, 7.1547e+22],\n [ 1.3733e-14, 6.4069e+02, 4.3066e+21],\n [ 1.1824e+22, 4.3066e+21, 6.3828e+28],\n [ 3.8016e-39, 0.0000e+00, 0.0000e+00]])\n\n\n

    \uc8fc\uc758) torch.Tensor\uc640 torch.tensor \uc0ac\uc6a9 \uc608.

    \n\ntorch.Tensor\ub294 Tensor \ud074\ub798\uc2a4\ub97c instantiation. \uc989 torch.empty\uc640 \uac19\uc740 \ub3d9\uc791.\n\ntorch.tensor\ub294 \uc778\uc790\ub85c \ubc1b\ub294 \ub370\uc774\ud130 \uac12\uc744 \ubcf5\uc81c. \uc778\uc790\ub294 \ub9ac\uc2a4\ud2b8, \ud29c\ud50c, ndarray, \uc2a4\uce7c\ub77c \uac12 \ub4f1 \ub2e4\uc591\ud55c \ud615\ud0dc. \ud150\uc11c\ub97c \ub9cc\ub4dc\ub294 \uac00\uc7a5 \uc77c\ubc18\uc801 \ud615\ud0dc\uc784.\n\n\n```python\nx = torch.Tensor(5,3)\nprint(x, x.dtype) # \uc758\ubbf8\uc5c6\ub294 \uac12.\n```\n\n tensor([[-3.0226e+35, 3.0672e-41, 2.3694e-38],\n [ 9.2196e-41, -3.0226e+35, 3.0672e-41],\n [-3.0478e+35, 3.0672e-41, -3.1655e+35],\n [ 3.0672e-41, -3.1656e+35, 3.0672e-41],\n [-3.1655e+35, 3.0672e-41, 0.0000e+00]]) torch.float32\n\n\n\n```python\nx = torch.tensor([[0.1, 1.2], [2.2, 3.1], [4.9, 5.2]])\nprint(x, x.dtype)\n```\n\n tensor([[0.1000, 1.2000],\n [2.2000, 3.1000],\n [4.9000, 5.2000]]) torch.float32\n\n\n\n```python\nx = torch.tensor([0, 1]) # dtype\uc744 \uc778\uc790\uc758 \uac12\uc5d0\uc11c \uc720\ucd94\ud558\uc5ec \uc124\uc815\nprint(x, x.dtype) \n```\n\n tensor([0, 1]) torch.int64\n\n\n\n```python\ny = torch.tensor([0.0, 1.0])\nprint(y, y.dtype)\n```\n\n tensor([0., 1.]) torch.float32\n\n\n\n```python\ntry:\n z = torch.tensor([[0.11111, 0.222222, 0.3333333]], dtype=torch.float64, device=torch.device('cuda:0')) # creates a torch.cuda.DoubleTensor\n print(z, z.dtype)\nexcept Exception as e:\n print('Error:', e)\n```\n\n tensor([[0.1111, 0.2222, 0.3333]], device='cuda:0', dtype=torch.float64) torch.float64\n\n\nCPU \ud658\uacbd\uc5d0\uc120 \uc704 \ubb38\uc7a5\uc774 \uc624\ub958 \ubc1c\uc0dd.\n\nGPU \ud658\uacbd\uc73c\ub85c \ub7f0\ud0c0\uc784 \ubcc0\uacbd \ud6c4 \uc2e4\ud589.\n\n\n\n\n```python\nx = torch.tensor(3.14159) # Create a scalar (zero-dimensional tensor)\nprint(x, x.dtype)\n```\n\n tensor(3.1416) torch.float32\n\n\n\n```python\nx = torch.tensor([]) # Create an empty tensor (of size (0,))\nprint(x, x.dtype)\n```\n\n tensor([]) torch.float32\n\n\n\n```python\ntry:\n x = torch.tensor(3,2) # \uc778\uc790\uac00 \ub9e4\uce58\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c \uc624\ub958\uac00 \ub098\uc57c \ud568.\n print(x, x.dtype)\nexcept Exception as e:\n print('Error:', e)\n```\n\n Error: tensor() takes 1 positional argument but 2 were given\n\n\n\n\n---\n\n\n\n

    Random

    \nConstruct a randomly initialized matrix:\n\n\n\n\n```python\nx = torch.rand(5, 3)\nprint(x)\n```\n\n tensor([[0.6106, 0.1358, 0.2298],\n [0.4580, 0.9274, 0.1602],\n [0.1059, 0.0834, 0.8331],\n [0.0217, 0.5579, 0.4679],\n [0.8884, 0.5640, 0.5379]])\n\n\nConstruct a matrix filled zeros and of dtype long:\n\n\n\n\n```python\nx = torch.zeros(5, 3, dtype=torch.long)\nprint(x)\n```\n\n tensor([[0, 0, 0],\n [0, 0, 0],\n [0, 0, 0],\n [0, 0, 0],\n [0, 0, 0]])\n\n\n

    \ub370\uc774\ud130\ub97c \uc778\uc790\ub85c \uc8fc\uc5b4 \ud150\uc11c \uc0dd\uc131

    \nConstruct a tensor directly from data:\n\n\n\n\n```python\nx = torch.tensor([5.5, 3])\nprint(x, x.dtype)\n```\n\n tensor([5.5000, 3.0000]) torch.float32\n\n\nor create a tensor based on an existing tensor. These methods\nwill reuse properties of the input tensor, e.g. dtype, unless\nnew values are provided by user\n\n\n\nx\uc640 \uac19\uc740 \ud0c0\uc785\uc758 \uc0c8 \ud150\uc11c \uc0dd\uc131\n\n\n```python\nx = x.new_ones(5, 3) # new_* methods take in sizes\nprint(x, x.dtype)\n```\n\n tensor([[1., 1., 1.],\n [1., 1., 1.],\n [1., 1., 1.],\n [1., 1., 1.],\n [1., 1., 1.]]) torch.float32\n\n\n\n```python\nx = x.new_ones(5, 3, dtype=torch.double) # new_* methods take in sizes\nprint(x, x.dtype)\n```\n\n tensor([[1., 1., 1.],\n [1., 1., 1.],\n [1., 1., 1.],\n [1., 1., 1.],\n [1., 1., 1.]], dtype=torch.float64) torch.float64\n\n\n\n```python\nx = torch.randn_like(x, dtype=torch.float) # override dtype!\nprint(x, x.dtype) # result has the same size\n```\n\n tensor([[ 0.8815, -0.4184, 0.0153],\n [-0.1281, -0.1178, 0.4411],\n [ 1.3856, 0.2023, -0.7654],\n [-1.6897, 0.8962, 0.8546],\n [ 0.8858, -1.1012, 0.8380]]) torch.float32\n\n\nGet its size: size() or shape\n\n\n\n\n```python\nprint(x.size())\nprint(x.shape)\n```\n\n torch.Size([5, 3])\n torch.Size([5, 3])\n\n\n

    Note

    ``torch.Size`` is in fact a tuple, so it supports all tuple operations.

    \n\nOperations\n^^^^^^^^^^\nThere are multiple syntaxes for operations. In the following\nexample, we will take a look at the addition operation.\n\n

    \ud150\uc11c \ub354\ud558\uae30

    \n\nAddition: syntax 1\n\n\n\n\n```python\ny = torch.rand(5, 3)\nprint('x\\n', x)\nprint('y\\n', y)\nprint('x+y\\n', x + y)\n```\n\n x\n tensor([[ 0.8815, -0.4184, 0.0153],\n [-0.1281, -0.1178, 0.4411],\n [ 1.3856, 0.2023, -0.7654],\n [-1.6897, 0.8962, 0.8546],\n [ 0.8858, -1.1012, 0.8380]])\n y\n tensor([[0.2691, 0.3013, 0.0288],\n [0.3602, 0.8997, 0.1451],\n [0.0238, 0.5176, 0.7644],\n [0.1169, 0.9153, 0.7058],\n [0.3358, 0.9711, 0.0666]])\n x+y\n tensor([[ 1.1506e+00, -1.1708e-01, 4.4037e-02],\n [ 2.3207e-01, 7.8194e-01, 5.8625e-01],\n [ 1.4095e+00, 7.1991e-01, -1.0388e-03],\n [-1.5728e+00, 1.8116e+00, 1.5604e+00],\n [ 1.2216e+00, -1.3014e-01, 9.0465e-01]])\n\n\nAddition: syntax 2\n\n\n\n\n```python\nprint(torch.add(x, y))\n```\n\n tensor([[ 1.1506e+00, -1.1708e-01, 4.4037e-02],\n [ 2.3207e-01, 7.8194e-01, 5.8625e-01],\n [ 1.4095e+00, 7.1991e-01, -1.0388e-03],\n [-1.5728e+00, 1.8116e+00, 1.5604e+00],\n [ 1.2216e+00, -1.3014e-01, 9.0465e-01]])\n\n\nAddition: providing an output tensor as argument\n\n\n\n\n```python\nresult = torch.empty(5, 3)\nprint(result)\n```\n\n tensor([[ 2.8641e+08, 3.0673e-41, 4.4037e-02],\n [ 2.3207e-01, 7.8194e-01, 5.8625e-01],\n [ 1.4095e+00, 7.1991e-01, -1.0388e-03],\n [-1.5728e+00, 1.8116e+00, 1.5604e+00],\n [ 1.2216e+00, -1.3014e-01, 9.0465e-01]])\n\n\n\n```python\ntorch.add(x, y, out=result)\nprint(result)\n```\n\n tensor([[ 1.1506e+00, -1.1708e-01, 4.4037e-02],\n [ 2.3207e-01, 7.8194e-01, 5.8625e-01],\n [ 1.4095e+00, 7.1991e-01, -1.0388e-03],\n [-1.5728e+00, 1.8116e+00, 1.5604e+00],\n [ 1.2216e+00, -1.3014e-01, 9.0465e-01]])\n\n\n\n```python\nres = x + y\nprint(res)\n```\n\n tensor([[ 1.1506e+00, -1.1708e-01, 4.4037e-02],\n [ 2.3207e-01, 7.8194e-01, 5.8625e-01],\n [ 1.4095e+00, 7.1991e-01, -1.0388e-03],\n [-1.5728e+00, 1.8116e+00, 1.5604e+00],\n [ 1.2216e+00, -1.3014e-01, 9.0465e-01]])\n\n\nAddition: in-place\n\n\n\n\n```python\n# adds x to y\ny.add_(x)\nprint(y)\n```\n\n tensor([[ 1.1506e+00, -1.1708e-01, 4.4037e-02],\n [ 2.3207e-01, 7.8194e-01, 5.8625e-01],\n [ 1.4095e+00, 7.1991e-01, -1.0388e-03],\n [-1.5728e+00, 1.8116e+00, 1.5604e+00],\n [ 1.2216e+00, -1.3014e-01, 9.0465e-01]])\n\n\n# **_\ub85c \ub05d\ub098\ub294 operation\uc740 in-place**\n

    Note

    Any operation that mutates a tensor in-place is post-fixed with an ``_``.\n For example: ``x.copy_(y)``, ``x.t_()``, will change ``x``.

    \n\nYou can use standard NumPy-like indexing with all bells and whistles!\n\n\n\n\n\n\n```python\n\n```\n\n# autograd\n\n\ud559\uc2b5\uc5d0\uc11c error backpropagation\uc744 \uc704\ud55c \uadf8\ub798\ub514\uc5b8\ud2b8 \uacc4\uc0b0\uc744 \uc790\ub3d9\ud654\ud55c PyTorch \uae30\ub2a5\n\n\n```python\nimport torch\n\na = torch.tensor([2., 3.], requires_grad=True)\nb = torch.tensor([6., 4.], requires_grad=True)\n```\n\n\n```python\na\n```\n\n\n\n\n tensor([2., 3.], requires_grad=True)\n\n\n\n\n```python\nb\n```\n\n\n\n\n tensor([6., 4.], requires_grad=True)\n\n\n\n\n```python\nQ = 3*a**3 - b**2\n```\n\n\\begin{align}Q=3a^{3}-b^{3}\\end{align}\n\n\n```python\nQ\n```\n\n\n\n\n tensor([-12., 65.], grad_fn=)\n\n\n\n\\begin{align}\\frac{\\partial Q}{\\partial a} = 9a^2\\end{align}\n\n\\begin{align}\\frac{\\partial Q}{\\partial b} = -2b\\end{align}\n\n\\begin{align}\\frac{dQ}{dQ} = 1\\end{align}\n\n\n```python\nexternal_grad = torch.tensor([1., 1.])\nQ.backward(gradient=external_grad)\n```\n\n\n```python\nQ\n```\n\n\n\n\n tensor([-12., 65.], grad_fn=)\n\n\n\n\n```python\na.grad, b.grad\n```\n\n\n\n\n (tensor([36., 81.]), tensor([-12., -8.]))\n\n\n\n\n```python\n# check if collected gradients are correct\nprint(9*a**2 == a.grad)\nprint(-2*b == b.grad)\n```\n\n tensor([True, True])\n tensor([True, True])\n\n\n\n```python\n\n```\n\n# \uc77c\ubc18\uc801\uc73c\ub85c\ub294 Numpy \ubc30\uc5f4\uacfc \uc720\uc0ac\ud55c \uc2e0\ud0dd\uc2a4\ub97c \uac00\uc9c4\ub2e4\n\n# Indexing tensors\n\n## Python slicing \uacfc \uc720\uc0ac\n\nhttps://drive.google.com/file/d/1qYbFGElZiXcB4VgUiolBp0f2YZS1pzGw/view?usp=sharing\n\n\n\n\n```python\nsome_list = list(range(6))\nsome_list[:]\n```\n\n\n\n\n [0, 1, 2, 3, 4, 5]\n\n\n\n\n```python\nprint(some_list[1:4])\nprint(some_list[1:])\nprint(some_list[:4])\nprint(some_list[:-1])\nprint(some_list[1:4:2]) # 1 to 3 ([1,2,3]), step 2 ([1,3])\n```\n\n [1, 2, 3]\n [1, 2, 3, 4, 5]\n [0, 1, 2, 3]\n [0, 1, 2, 3, 4]\n [1, 3]\n\n\n\n```python\npoints = torch.tensor([[4.0, 1.0], [5.0, 3.0], [2.0, 1.0]])\nprint(points)\nprint(points.shape)\n\n```\n\n tensor([[4., 1.],\n [5., 3.],\n [2., 1.]])\n torch.Size([3, 2])\n\n\n\n```python\nprint(points[1:])\n```\n\n tensor([[5., 3.],\n [2., 1.]])\n\n\n\n```python\nprint(points[1:, :])\n```\n\n tensor([[5., 3.],\n [2., 1.]])\n\n\n\n```python\nprint(points[1:, 0])\n```\n\n tensor([5., 2.])\n\n\n\n```python\nprint(points[None])\n```\n\n tensor([[[4., 1.],\n [5., 3.],\n [2., 1.]]])\n\n\n\n```python\nprint(points[None].size())\n```\n\n torch.Size([1, 3, 2])\n\n\n\n```python\npoints.shape\n```\n\n\n\n\n torch.Size([3, 2])\n\n\n\n\n```python\npoints\n```\n\n\n\n\n tensor([[4., 1.],\n [5., 3.],\n [2., 1.]])\n\n\n\n# unsqueeze - dimension \ud655\uc7a5. dim\uc73c\ub85c \uc9c0\uc815\ub41c \ub514\uba58\uc158\uc744 \ud558\ub098 \ud655\uc7a5\n\n\n```python\nprint(points.unsqueeze(dim=0))\nprint(points.unsqueeze(dim=0).size())\n```\n\n tensor([[[4., 1.],\n [5., 3.],\n [2., 1.]]])\n torch.Size([1, 3, 2])\n\n\n\n```python\nunsq_points = points.unsqueeze(dim=0)\nprint(unsq_points)\nprint(unsq_points.size())\n```\n\n tensor([[[4., 1.],\n [5., 3.],\n [2., 1.]]])\n torch.Size([1, 3, 2])\n\n\n# squeeze - dimension \ucd95\uc18c\n### \uc6d0\uc18c 1\uac1c\uc9dc\ub9ac dimension\uc744 \uc81c\uac70\n\n\n```python\nsq_points = unsq_points.squeeze()\nprint(sq_points)\nprint(sq_points.size())\n```\n\n tensor([[4., 1.],\n [5., 3.],\n [2., 1.]])\n torch.Size([3, 2])\n\n\n\n\n---\n\nResizing: If you want to resize/reshape tensor, you can use ``torch.view``:\n\n\n\nx\ub294 (4,4), y\ub294 (16,), z\ub294 (inferred, 8)\uc778\ub370 \uc804\uccb4\uac00 16 \uc6d0\uc18c\uc774\ubbc0\ub85c inferred\ub294 2\uac00 \ub418\uc5b4 (2,8)\n\n\n```python\nx = torch.randn(4, 4)\ny = x.view(16)\nz = x.view(-1, 8) # the size -1 is inferred from other dimensions\nprint('x.size()=', x.size(), '\\ny.size()=', y.size(), '\\nz.size()=', z.size())\nprint(x, '\\n', y, '\\n', z)\n```\n\n x.size()= torch.Size([4, 4]) \n y.size()= torch.Size([16]) \n z.size()= torch.Size([2, 8])\n tensor([[-0.3112, -0.8315, 2.0023, -0.8920],\n [-0.7311, -0.1806, -1.3087, -0.4823],\n [ 1.4226, -2.2820, -0.0773, -2.7146],\n [ 1.2870, 1.0097, 0.1546, 1.3778]]) \n tensor([-0.3112, -0.8315, 2.0023, -0.8920, -0.7311, -0.1806, -1.3087, -0.4823,\n 1.4226, -2.2820, -0.0773, -2.7146, 1.2870, 1.0097, 0.1546, 1.3778]) \n tensor([[-0.3112, -0.8315, 2.0023, -0.8920, -0.7311, -0.1806, -1.3087, -0.4823],\n [ 1.4226, -2.2820, -0.0773, -2.7146, 1.2870, 1.0097, 0.1546, 1.3778]])\n\n\n---\n# transpose\n\n\n```python\npoints = torch.tensor([[3.0, 1.0, 2.0], [4.0, 1.0, 7.0]])\nprint(points)\n```\n\n tensor([[3., 1., 2.],\n [4., 1., 7.]])\n\n\n## \uc11c\ub85c \ubc14\uafc0 \ub450 \ub514\uba58\uc158\uc744 \uc9c0\uc815\n\n\n```python\npoints_tr = points.transpose(0, 1)\n```\n\n\n```python\npoints_tr\n```\n\n\n\n\n tensor([[3., 4.],\n [1., 1.],\n [2., 7.]])\n\n\n\n\n```python\npoints\n```\n\n\n\n\n tensor([[3., 1., 2.],\n [4., 1., 7.]])\n\n\n\n\n```python\npoints_t = points.t() # 2D or less dimension\nprint(points_t)\n```\n\n tensor([[3., 4.],\n [1., 1.],\n [2., 7.]])\n\n\n## Multidimensional\n\n\n```python\nsome_t = torch.ones(3, 4, 5)\ntranspose_t = some_t.transpose(0, 2) # 0\ubc88\uc9f8\uc640 2\ubc88\uc9f8 \ub514\uba58\uc158\uc744 transpose\nsome_t.shape\n```\n\n\n\n\n torch.Size([3, 4, 5])\n\n\n\n\n```python\nsome_t\n```\n\n\n\n\n tensor([[[1., 1., 1., 1., 1.],\n [1., 1., 1., 1., 1.],\n [1., 1., 1., 1., 1.],\n [1., 1., 1., 1., 1.]],\n \n [[1., 1., 1., 1., 1.],\n [1., 1., 1., 1., 1.],\n [1., 1., 1., 1., 1.],\n [1., 1., 1., 1., 1.]],\n \n [[1., 1., 1., 1., 1.],\n [1., 1., 1., 1., 1.],\n [1., 1., 1., 1., 1.],\n [1., 1., 1., 1., 1.]]])\n\n\n\n\n```python\ntranspose_t.shape\n```\n\n\n\n\n torch.Size([5, 4, 3])\n\n\n\n# Contiguous\n\n\uac00\uc7a5 \uc624\ub978\ucabd \ucc28\uc6d0\uc774 \uc21c\ucc28\uc801\uc73c\ub85c \uc800\uc7a5\ub41c \ud150\uc11c\n\n\n```python\nsome_t.is_contiguous()\n```\n\n\n\n\n True\n\n\n\ncontinous\ud55c \ud150\uc11c\uc758 transpose\ub294 \uc2e4\uc81c \ub370\uc774\ud130\ub97c \uc62e\uae30\ub294 \uac83\uc774 \uc544\ub2c8\ub77c \uc811\uadfc\ud558\ub294 \ubc29\ubc95\ub9cc \ub2e4\ub974\uac8c \ud558\uae30 \ub54c\ubb38\uc5d0 continous\ud558\uc9c0 \uc54a\uc74c\n\n\n```python\ntranspose_t.is_contiguous()\n```\n\n\n\n\n False\n\n\n\n## Contiguous\ub85c \ubcc0\ud658\ud558\uae30\n\n\n```python\npoints = torch.tensor([[4.0, 1.0], [5.0, 3.0], [2.0, 1.0]])\n```\n\n\n```python\npoints_t = points.t()\n```\n\n\n```python\npoints_t\n```\n\n\n\n\n tensor([[4., 5., 2.],\n [1., 3., 1.]])\n\n\n\n\n```python\npoints_t.is_contiguous()\n```\n\n\n\n\n False\n\n\n\n\uc2e4\uc81c \ub370\uc774\ud130\ub97c contiguous\ud558\uac8c \uc7ac\ubc30\uce58\n\n\n```python\npoints_t_cont = points_t.contiguous()\n```\n\n\n```python\npoints_t_cont\n```\n\n\n\n\n tensor([[4., 5., 2.],\n [1., 3., 1.]])\n\n\n\n\n```python\npoints_t_cont.contiguous()\n```\n\n\n\n\n tensor([[4., 5., 2.],\n [1., 3., 1.]])\n\n\n\n## tensor view()\n\n\uac19\uc740 storage \uc5d0\uc11c \ubcf4\ub294 \uad00\uc810 (shape)\ub9cc \ub2e4\ub974\uac8c \ud568\n\n\n```python\nt = torch.rand(4, 4)\nb = t.view(2, 8)\nprint(t, t.shape)\nprint(b, b.shape)\n```\n\n tensor([[0.4407, 0.9571, 0.6890, 0.5465],\n [0.2858, 0.9516, 0.4148, 0.6201],\n [0.9849, 0.9962, 0.1261, 0.0170],\n [0.2510, 0.7078, 0.5342, 0.2367]]) torch.Size([4, 4])\n tensor([[0.4407, 0.9571, 0.6890, 0.5465, 0.2858, 0.9516, 0.4148, 0.6201],\n [0.9849, 0.9962, 0.1261, 0.0170, 0.2510, 0.7078, 0.5342, 0.2367]]) torch.Size([2, 8])\n\n\n\n```python\nt.is_contiguous()\n```\n\n\n\n\n True\n\n\n\n\n```python\nb.is_contiguous()\n```\n\n\n\n\n True\n\n\n\n## view \uac19\uc740 operation\uc740 contiguous tensor\uc5d0\uc11c\ub9cc \ub3d9\uc791\n\n\n```python\nx = torch.tensor([[1,2,3], [4,5,6]])\nx\n```\n\n\n\n\n tensor([[1, 2, 3],\n [4, 5, 6]])\n\n\n\n\n```python\nx.shape\n```\n\n\n\n\n torch.Size([2, 3])\n\n\n\n\n```python\nx_t = x.t()\nx_t\n```\n\n\n\n\n tensor([[1, 4],\n [2, 5],\n [3, 6]])\n\n\n\n\n```python\ny = x.view(3,2) # transpose\uac00 \uc544\ub2d8\ny\n```\n\n\n\n\n tensor([[1, 2],\n [3, 4],\n [5, 6]])\n\n\n\n\n```python\ny.is_contiguous()\n```\n\n\n\n\n True\n\n\n\n\n```python\nx_t.is_contiguous()\n```\n\n\n\n\n False\n\n\n\ncontiguous\ud558\uc9c0 \uc54a\uc740 \ud150\uc11c\ub85c view\ub97c \ubd80\ub974\uba74 \uc624\ub958\uac00 \ubc1c\uc0dd\n\n\n```python\ntry:\n x_t.view(2, 3) # error\nexcept Exception as e:\n print('Error:', e)\n```\n\n Error: view size is not compatible with input tensor's size and stride (at least one dimension spans across two contiguous subspaces). Use .reshape(...) instead.\n\n\n\n```python\nx_t_cont = x_t.contiguous()\nx_t_cont\n```\n\n\n\n\n tensor([[1, 4],\n [2, 5],\n [3, 6]])\n\n\n\n\n```python\nx_t_cont.view(2, 3)\n```\n\n\n\n\n tensor([[1, 4, 2],\n [5, 3, 6]])\n\n\n\n\n```python\nx_t\n```\n\n\n\n\n tensor([[1, 4],\n [2, 5],\n [3, 6]])\n\n\n\nx_t\uc640 x_t_cont\ub294 \ud45c\uc2dc\ub418\ub294 \uac12\uc740 \uac19\uc73c\ub098 \ub0b4\ubd80\uc758 \ub370\uc774\ud130 \uc21c\uc11c\uac00 \ub2e4\ub974\uac8c \ubc30\uce58\ub418\uc5b4 \uc788\uc74c\n\n\n```python\nx = torch.randn(2,3,4)\nx_t = x.transpose(0, 2)\nprint(x.shape)\nprint(x_t.shape)\n\n```\n\n torch.Size([2, 3, 4])\n torch.Size([4, 3, 2])\n\n\n\n```python\nx\n```\n\n\n\n\n tensor([[[-0.0203, 1.4557, 0.6741, -0.9552],\n [ 0.5372, 1.0151, 1.3597, 0.4710],\n [-1.3028, -0.1712, -0.4442, -0.3842]],\n \n [[ 0.8422, -0.2642, 0.1464, -0.7503],\n [-0.7967, 1.5664, -1.2226, -1.3853],\n [ 0.3374, -1.1332, -0.7648, 0.3682]]])\n\n\n\n\n```python\nx.view(2, 12)\n```\n\n\n\n\n tensor([[-0.0203, 1.4557, 0.6741, -0.9552, 0.5372, 1.0151, 1.3597, 0.4710,\n -1.3028, -0.1712, -0.4442, -0.3842],\n [ 0.8422, -0.2642, 0.1464, -0.7503, -0.7967, 1.5664, -1.2226, -1.3853,\n 0.3374, -1.1332, -0.7648, 0.3682]])\n\n\n\n\n```python\ntry:\n x_t.view(4,6) # Error\nexcept Exception as e:\n print('Error:', e)\n```\n\n Error: view size is not compatible with input tensor's size and stride (at least one dimension spans across two contiguous subspaces). Use .reshape(...) instead.\n\n\n## reshape()\n\n\n```python\ny = x.reshape((2, 12))\n```\n\n\n```python\nprint(y.size())\ny\n```\n\n torch.Size([2, 12])\n\n\n\n\n\n tensor([[-0.0203, 1.4557, 0.6741, -0.9552, 0.5372, 1.0151, 1.3597, 0.4710,\n -1.3028, -0.1712, -0.4442, -0.3842],\n [ 0.8422, -0.2642, 0.1464, -0.7503, -0.7967, 1.5664, -1.2226, -1.3853,\n 0.3374, -1.1332, -0.7648, 0.3682]])\n\n\n\n\n```python\nz = x.reshape((6, -1)) # -1 \uc740 \uc54c\uc544\uc11c \ub9de\ucd94\ub77c\ub294 \uc758\ubbf8\n```\n\n\n```python\nz\n```\n\n\n\n\n tensor([[-0.0203, 1.4557, 0.6741, -0.9552],\n [ 0.5372, 1.0151, 1.3597, 0.4710],\n [-1.3028, -0.1712, -0.4442, -0.3842],\n [ 0.8422, -0.2642, 0.1464, -0.7503],\n [-0.7967, 1.5664, -1.2226, -1.3853],\n [ 0.3374, -1.1332, -0.7648, 0.3682]])\n\n\n\n\n```python\nz = x.reshape((3, -1))\n```\n\n\n```python\nz\n```\n\n\n\n\n tensor([[-0.0203, 1.4557, 0.6741, -0.9552, 0.5372, 1.0151, 1.3597, 0.4710],\n [-1.3028, -0.1712, -0.4442, -0.3842, 0.8422, -0.2642, 0.1464, -0.7503],\n [-0.7967, 1.5664, -1.2226, -1.3853, 0.3374, -1.1332, -0.7648, 0.3682]])\n\n\n\n\n```python\ny = x.reshape((-1, 2))\n```\n\n\n```python\ny\n```\n\n\n\n\n tensor([[-0.0203, 1.4557],\n [ 0.6741, -0.9552],\n [ 0.5372, 1.0151],\n [ 1.3597, 0.4710],\n [-1.3028, -0.1712],\n [-0.4442, -0.3842],\n [ 0.8422, -0.2642],\n [ 0.1464, -0.7503],\n [-0.7967, 1.5664],\n [-1.2226, -1.3853],\n [ 0.3374, -1.1332],\n [-0.7648, 0.3682]])\n\n\n\n\nNumPy Bridge\n------------\n\nConverting a Torch Tensor to a NumPy array and vice versa is a breeze.\n\n## The Torch Tensor and NumPy array will share their underlying memory locations (if the Torch Tensor is on CPU), and changing one will change the other.\n\nConverting a Torch Tensor to a NumPy Array\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n\n\n\n\n```python\na = torch.ones(5)\nprint(a)\nprint(type(a))\n```\n\n tensor([1., 1., 1., 1., 1.])\n \n\n\n\n```python\nb = a.numpy()\nprint(b)\nprint(type(b))\n```\n\n [1. 1. 1. 1. 1.]\n \n\n\nConverting NumPy Array to Torch Tensor\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\nSee how changing the np array changed the Torch Tensor automatically\n\n\n\n\n```python\nimport numpy as np\na = np.ones(5)\nb = torch.from_numpy(a)\nnp.add(a, 1, out=a)\nprint(a)\nprint(b)\n```\n\n [2. 2. 2. 2. 2.]\n tensor([2., 2., 2., 2., 2.], dtype=torch.float64)\n\n\nAll the Tensors on the CPU except a CharTensor support converting to\nNumPy and back.\n\nCUDA Tensors\n------------\n\nTensors can be moved onto any device using the ``.to`` method.\n\n\n\n\n```python\n# let us run this cell only if CUDA is available\n# We will use ``torch.device`` objects to move tensors in and out of GPU\nif torch.cuda.is_available():\n device = torch.device(\"cuda\") # a CUDA device object\n y = torch.ones_like(x, device=device) # directly create a tensor on GPU\n x = x.to(device) # or just use strings ``.to(\"cuda\")``\n z = x + y\n print(z)\n print(z.dtype, z.device)\n print(z.to(\"cpu\", torch.double)) # ``.to`` can also change dtype together!\n print(z.device)\n```\n\n tensor([[[ 0.9797, 2.4557, 1.6741, 0.0448],\n [ 1.5372, 2.0151, 2.3597, 1.4710],\n [-0.3028, 0.8288, 0.5558, 0.6158]],\n \n [[ 1.8422, 0.7358, 1.1464, 0.2497],\n [ 0.2033, 2.5664, -0.2226, -0.3853],\n [ 1.3374, -0.1332, 0.2352, 1.3682]]], device='cuda:0')\n torch.float32 cuda:0\n tensor([[[ 0.9797, 2.4557, 1.6741, 0.0448],\n [ 1.5372, 2.0151, 2.3597, 1.4710],\n [-0.3028, 0.8288, 0.5558, 0.6158]],\n \n [[ 1.8422, 0.7358, 1.1464, 0.2497],\n [ 0.2033, 2.5664, -0.2226, -0.3853],\n [ 1.3374, -0.1332, 0.2352, 1.3682]]], dtype=torch.float64)\n cuda:0\n\n\n\n```python\nprint(z.to(\"cpu\").dtype)\n```\n\n torch.float32\n\n\n\n```python\nprint(z.to(\"cpu\").device)\n```\n\n cpu\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "3679981dcd9bccd49001f4c8e252c72252eac939", "size": 79661, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tensor_intro.ipynb", "max_stars_repo_name": "skimaza/assist_ai", "max_stars_repo_head_hexsha": "912401e8cd0026cfc572041d074710f3a3946bd0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tensor_intro.ipynb", "max_issues_repo_name": "skimaza/assist_ai", "max_issues_repo_head_hexsha": "912401e8cd0026cfc572041d074710f3a3946bd0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tensor_intro.ipynb", "max_forks_repo_name": "skimaza/assist_ai", "max_forks_repo_head_hexsha": "912401e8cd0026cfc572041d074710f3a3946bd0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.9642745221, "max_line_length": 226, "alphanum_fraction": 0.389927317, "converted": true, "num_tokens": 9320, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370308082623217, "lm_q2_score": 0.6926419831347361, "lm_q1q2_score": 0.44123428235273837}} {"text": "# Data Space Report\n\n\n\n\n\n## Pittsburgh Bridges Data Set\n\n\n\n Andy Warhol Bridge - Pittsburgh.\n\nReport created by Student Francesco Maria Chiarlo s253666, for A.A 2019/2020.\n\n**Abstract**:The aim of this report is to evaluate the effectiveness of distinct, different statistical learning approaches, in particular focusing on their characteristics as well as on their advantages and backwards when applied on a relatively small dataset as the one employed within this report, that is Pittsburgh Bridgesdataset.\n\n**Key words**:Statistical Learning, Machine Learning, Bridge Design.\n\n## TOC:\n* [Imports Section](#imports-section)\n* [Dataset's Attributes Description](#attributes-description)\n* [Data Preparation and Investigation](#data-preparation)\n* [Learning Models](#learning-models)\n* [Improvements and Conclusions](#improvements-and-conclusions)\n* [References](#references)\n\n### Imports Section \n\n\n```python\nfrom utils.all_imports import *;\n%matplotlib inline\n```\n\n None\n\n\n\n```python\n# Set seed for notebook repeatability\nnp.random.seed(0)\n```\n\n### Dataset's Attributes Description \n\nThe analyses that I aim at accomplishing while using as means the methods or approaches provided by both Statistical Learning and Machine Learning fields, concern the dataset Pittsburgh Bridges, and what follows is a overview and brief description of the main characteristics, as well as, basic information about this precise dataset.\n\nThe Pittsburgh Bridges dataset is a dataset available from the web site called mainly *\"UCI Machine Learing Repository\"*, which is one of the well known web site that let a large amount of different datasets, from different domains or fields, to be used for machine-learning research and which have been cited in peer-reviewed academic journals.\n\nIn particular, the dataset I'm going to treat and analyze, which is Pittsburgh Bridges dataset, has been made freely available from the Western Pennsylvania Regional Data Center (WPRDC), which is a project led by the University Center of Social and Urban Research (UCSUR) at the University of Pittsburgh (\"University\") in collaboration with City of Pittsburgh and The County of Allegheny in Pennsylvania. The WPRDC and the WPRDC Project is supported by a grant from the Richard King Mellon Foundation.\n\nIn order to be more precise, from the official and dedicated web page, within UCI Machine Learning cite, Pittsburgh Bridges dataset is a dataset that has been created after the works of some co-authors which are:\n- Yoram Reich & Steven J. Fenves from Department of Civil Engineering and Engineering Design Research Center Carnegie Mellon University Pittsburgh, PA 15213\n\nThe Pittsburgh Bridges dataset is made of up to 108 distinct observations and each of that data sample is made of 12 attributes or features where some of them are considered to be continuous properties and other to be categorical or nominal properties. Those variables are the following:\n\n- **RIVER**: which is a nominal type variable that can assume the subsequent possible discrete values which are: A, M, O. Where A stands for Allegheny river, while M stands for Monongahela river and lastly O stands for Ohio river.\n\n\n\n\n- **LOCATION**: which represents a nominal type variable too, and assume a positive integer value from 1 up to 52 used as categorical attribute.\n- **ERECTED**: which might be either a numerical or categorical variable, depending on the fact that we want to aggregate a bunch of value under a categorical quantity. What this means is that, basically such attribute is made of date starting from 1818 up to 1986, but we may imagine to aggregate somehow these data within a given category among those suggested, that are CRAFTS, EMERGENING, MATURE, MODERN.\n- **PURPOSE**: which is a categorical attribute and represents the reason why a particular bridge has been built, which means that this attribute represents what kind of vehicle can cross the bridge or if the bridge has been made just for people. For this reasons the allowd values for this attributes are the following: WALK, AQUEDUCT, RR, HIGHWAY. Three out of four are self explained values, while RR value that might be tricky at first glance, it just stands for railroad.\n- **LENGTH**: which represents the bridge's length, is a numerical attribute if we just look at the real number values that go from 804 up to 4558, but we can again decide to handle or arrange such values so that they can be grouped into range of values mapped into SHORT, MEDIUM, LONG so that we can refer to a bridge's length by means of these new categorical values.\n- **LANES**: which is a categorical variable which is represented by numerical values, that are 1, 2, 4, 6 which indicate the number of distinct lanes that a bridge in Pittsburgh city may have. The larger the value the wider the bridge.\n- **CLEAR-G**: specifies whether a vertical navigation clearance requirement was enforced in the design or not.\n- **T-OR-D**: which is a nominal attribute, in other words, a categorical attribute that can assume THROUGH, DECK values. In order to be more precise, this samples attribute deals with structural elements of a bridge. In fact, a deck is the surface of a bridge and this structural element, of bridge's superstructure, may be constructed of concrete, steel, open grating, or wood. On the other hand, a through arch bridge, also known as a half-through arch bridge or a through-type arch bridge, is a bridge that is made from materials such as steel or reinforced concrete, in which the base of an arch structure is below the deck but the top rises above it.\n\n\n\n\n- **MATERIAL**: which is a categorical or nominal variable and is used to describe the bridge telling which is the main or core material used to build it.\n This attribute can assume one of the possible, following values which are: WOOD, IRON, STEEL. Furthermore, we expect to see somehow a bit of correlation between the values assumed by the pairs represented by T-OR-D and MATERIAL columns, when looking just to them.\n- **SPAN**: which is a categorical or nominal value and has been recorded by means of three possible values for each sample, that are SHORT, MEDIUM, LONG. This attribute, within the field of Structural Engineering, is the distance between two intermediate supports for a structure, e.g. a beam or a bridge. A span can be closed by a solid beam or by a rope. The first kind is used for bridges, the second one for power lines, overhead telecommunication lines, some type of antennas or for aerial tramways. \n\n\n\n- **REL-L**: which is a categorical or nominal variable and stands for relative length of the main span of the bridge to the total crossing length, it can assume three possible values that are S, S-F, F.\n- Lastly, **TYPE** which indicates as a categorical or nominal attributes what type of bridge each record represents, among the possible 6 distinct classes or types of bridges that are: WOOD, SUSPEN, SIMPLE-T, ARCH, CANTILEV, CONT-T.\n\n\n```python\n# Show TYPE of Bridges\n# ------------------------\n# show_bridges_types_images()\n```\n\n## Data Investigation \n\nThe aim of this chapter is to get in the data, that are available within Pittsburgh Bridge Dataset, in order to investigate a bit more in to detail and generally speaking deeper the main or high level statistics quantities, such as mean, median, standard deviation of each attribute, as well as displaying somehow data distribution for each attribute by means of histogram plots. This phase allows or enables us to decide which should be the best feature to be selected as the target variable, in other word the attribute that will represent the dependent variable with respect to the remaining attributes that instead will play the role of predictors and independent variables, as well.\n\nIn order to investigate and explore our data we make usage of *Pandas library*. We recall mainly that, in computer programming, Pandas is a software library written for the Python programming language* for *data manipulation and analysis*. In particular, it offers data structures and operations for manipulating numerical tables and time series. It is free software and a interesting and funny things about such tool is that the name is derived from the term \"panel data\", an econometrics term for data sets that include observations over multiple time periods for the same individuals.\nWe also note that as the analysis proceeds we will introduce other computer programming as well as programming libraries that allow or enable us to fulfill our goals.\n\nInitially, once I have downloaded from the provided web page the dataset with the data samples about Pittsburgh Bridge we load the data by means of functions available using python library's pandas. We notice that the overall set of data points is large up to 108 records or rows, which are sorted by Erected attributes, so this means that are sorted in decreasing order from the oldest bridge which has been built in 1818 up to the most modern bridge that has been erected in 1986. Then we display the first 5 rows to get an overview and have a first idea about what is inside the overall dataset, and the result we obtain by means of head() function applied onto the fetched dataset is equals to what follows:\n\n### Read Input Data\n\n\n```python\n# Some global script variables\n# --------------------------------------------------------------------------- #\ndataset_path, dataset_name, column_names, TARGET_COL = \\\n get_dataset_location() # Info Data to be fetched\nestimators_list, estimators_names = get_estimators() # Estimator to be trained\n\n# variables used for pass through arrays used to store results\npos_gs = 0; pos_cv = 0\n\n# Array used for storing graphs\nplots_names = list(map(lambda xi: f\"{xi}_learning_curve.png\", estimators_names))\npca_kernels_list = ['linear', 'poly', 'rbf', 'cosine', 'sigmoid']\ncv_list = list(range(10, 1, -1))\n```\n\n\n```python\n# Parameters to be tested for Cross-Validation Approach\n# -----------------------------------------------------\nparam_grids = []\nparmas_logreg = {\n 'penalty': ('l1', 'l2', 'elastic', None),\n 'solver': ('newton-cg', 'lbfgs', 'liblinear', 'sag', 'saga'),\n 'fit_intercept': (True, False),\n 'tol': (1e-4, 1e-3, 1e-2),\n 'class_weight': (None, 'balanced'),\n 'C': (10.0, 1.0, .1, .01, .001, .0001),\n # 'random_state': (0,),\n}; param_grids.append(parmas_logreg)\n\nparmas_knn_clf = {\n 'n_neighbors': (2,3,4,5,6,7,8,9,10),\n 'weights': ('uniform', 'distance'),\n 'metric': ('euclidean', 'minkowski', 'manhattan'),\n 'leaf_size': (5, 10, 15, 30),\n 'algorithm': ('ball_tree', 'kd_tree', 'brute'),\n}; param_grids.append(parmas_knn_clf)\n\nparams_sgd_clf = {\n 'loss': ('log', 'modified_huber'), # ('hinge', 'log', 'modified_huber', 'squared_hinge', 'perceptron')\n 'penalty': ('l2', 'l1', 'elasticnet'),\n 'alpha': (1e-1, 1e-2, 1e-3, 1e-4),\n 'max_iter': (50, 100, 150, 200, 500, 1000, 1500, 2000, 2500),\n 'class_weight': (None, 'balanced'),\n 'learning_rate': ('optimal',),\n 'tol': (None, 1e-2, 1e-4, 1e-5, 1e-6),\n # 'random_state': (0,),\n}; param_grids.append(params_sgd_clf)\n\nkernel_type = 'svm-rbf-kernel'\nparams_svm_clf = {\n # 'gamma': (1e-7, 1e-4, 1e-3, 1e-2, 0.1, 1.0, 10, 1e+2, 1e+3, 1e+5, 1e+7),\n 'gamma': (1e-5, 1e-3, 1e-2, 0.1, 1.0, 10, 1e+2, 1e+3, 1e+5),\n 'max_iter':(1e+2, 1e+3, 2 * 1e+3, 5 * 1e+3, 1e+4, 1.5 * 1e+3),\n 'degree': (1,2,4,8),\n 'coef0': (.001, .01, .1, 0.0, 1.0, 10.0),\n 'shrinking': (True, False),\n 'kernel': ['linear', 'poly', 'rbf', 'sigmoid',],\n 'class_weight': (None, 'balanced'),\n 'C': (1e-4, 1e-3, 1e-2, 0.1, 1.0, 10, 1e+2, 1e+3),\n 'probability': (True,),\n}; param_grids.append(params_svm_clf)\n\nparmas_tree = {\n 'splitter': ('random', 'best'),\n 'criterion':('gini', 'entropy'),\n 'max_features': (None, 'sqrt', 'log2'),\n 'max_depth': (None, 3, 5, 7, 10,),\n 'splitter': ('best', 'random',),\n 'class_weight': (None, 'balanced'),\n}; param_grids.append(parmas_tree)\n\nparmas_random_forest = {\n 'n_estimators': (3, 5, 7, 10, 30, 50, 70, 100, 150, 200),\n 'criterion':('gini', 'entropy'),\n 'bootstrap': (True, False),\n 'min_samples_leaf': (1,2,3,4,5),\n 'max_features': (None, 'sqrt', 'log2'),\n 'max_depth': (None, 3, 5, 7, 10,),\n 'class_weight': (None, 'balanced', 'balanced_subsample'),\n}; param_grids.append(parmas_random_forest)\n\n# Some variables to perform different tasks\n# -----------------------------------------------------\nN_CV, N_KERNEL, N_GS = 9, 5, 6;\nnrows = N_KERNEL // 2 if N_KERNEL % 2 == 0 else N_KERNEL // 2 + 1;\nncols = 2; grid_size = [nrows, ncols]\n```\n\n\n```python\n# READ INPUT DATASET\n# --------------------------------------------------------------------------- #\ndataset = pd.read_csv(os.path.join(dataset_path, dataset_name), names=column_names, index_col=0)\n```\n\n\n```python\n# SHOW SOME STANDARD DATASET INFOS\n# --------------------------------------------------------------------------- #\n# print('Dataset shape: {}'.format(dataset.shape)); print(dataset.info())\n```\n\n\n```python\n# SHOWING FIRSTS N-ROWS AS THEY ARE STORED WITHIN DATASET\n# --------------------------------------------------------------------------- #\n# dataset.head(5)\n```\n\nWhat we can notice from just the table above is that there are some attributes that are characterized by a special character that is '?' which stands for a missing value, so by chance there was not possibility to get the value for this attribute, such as for LENGTH and SPAN attributes. Analyzing in more details the dataset we discover that there are up to 6 different attributes, in the majority attributes with categorical or nominal nature such as CLEAR-G, T-OR-D, MATERIAL, SPAN, REL-L, and TYPE that contain at list one row characterized by the fact that one of its attributes is set to assuming '?' value that stands, as we already know for a missing value.\n\nHere, we can follow different strategies that depends onto the level of complexity as well as accuracy we want to obtain or achieve for models we are going to fit to the data after having correctly pre-processed them, speaking about what we could do with missing values. In fact one can follow the simplest way and can decide to simply discard those rows that contain at least one attribute with a missing value represented by the '?' symbol. Otherwise one may alos decide to follow a different strategy that aims at keeping also those rows that have some missing values by means of some kind of technique that allows to establish a potential substituting value for the missing one.\n\nSo, in this setting, that is our analyses, we start by just leaving out those rows that at least contain one attribute that has a missing value, this choice leads us to reduce the size of our dataset from 108 records to 70 remaining samples, with a drop of 38 data examples, which may affect the final results, since we left out more or less the 46\\% of the data because of missing values.\n\n\n```python\n# INVESTIGATING DATASET IN ORDER TO DETECT NULL VALUES\n# --------------------------------------------------------------------------- #\n# print('Before preprocessing dataset and handling null values')\nresult = dataset.isnull().values.any(); # print('There are any null values ? Response: {}'.format(result))\nresult = dataset.isnull().sum(); # print('Number of null values for each predictor:\\n{}'.format(result))\n```\n\n\n```python\n# DISCOVERING VALUES WITHIN EACH PREDICTOR DOMAIN\n# --------------------------------------------------------------------------- #\ncolumns_2_avoid = ['ERECTED', 'LENGTH', 'LOCATION', 'LANES']\nlist_columns_2_fix = show_categorical_predictor_values(dataset, columns_2_avoid)\n```\n\n\n```python\n# FIXING, UPDATING NULL VALUES CODED AS '?' SYMBOL\n# WITHIN EACH CATEGORICAL VARIABLE, IF DETECTED ANY\n# --------------------------------------------------------------------------- #\n# print('\"Before\" removing \\'?\\' rows, Dataset dim:', dataset.shape)\nfor _, predictor in enumerate(list_columns_2_fix):\n dataset = dataset[dataset[predictor] != '?']\n# print('\"After\" removing \\'?\\' rows, Dataset dim: ', dataset.shape); print('-' * 50)\n_ = show_categorical_predictor_values(dataset, columns_2_avoid)\n```\n\n\n```python\n# INTERMEDIATE RESULT FOUND\n# --------------------------------------------------------------------------- #\nfeatures_vs_values = preprocess_categorical_variables(dataset, columns_2_avoid); print(dataset.info())\n```\n\n \n Index: 88 entries, E1 to E90\n Data columns (total 12 columns):\n # Column Non-Null Count Dtype \n --- ------ -------------- ----- \n 0 RIVER 88 non-null int64 \n 1 LOCATION 88 non-null object\n 2 ERECTED 88 non-null int64 \n 3 PURPOSE 88 non-null int64 \n 4 LENGTH 88 non-null object\n 5 LANES 88 non-null object\n 6 CLEAR-G 88 non-null int64 \n 7 T-OR-D 88 non-null int64 \n 8 MATERIAL 88 non-null int64 \n 9 SPAN 88 non-null int64 \n 10 REL-L 88 non-null int64 \n 11 TYPE 88 non-null int64 \n dtypes: int64(9), object(3)\n memory usage: 8.9+ KB\n None\n\n\n\n```python\n# dataset.head(5)\n```\n\nThe next step is represented by the effort of mapping categorical variables into numerical variables, so that them are comparable with the already existing numerical or continuous variables, and also by mapping the categorical variables into numerical variables we allow or enable us to perform some kind of normalization or just transformation onto the entire dataset in order to let some machine learning algorithm to work better or to take advantage of normalized data within our pre-processed dataset. Furthermore, by transforming first the categorical attributes into a continuous version we are also able to calculate the \\textit{heatmap}, which is a very useful way of representing a correlation matrix calculated on the whole dataset. Moreover we have displayed data distribution for each attribute by means of histogram representation to take some useful information about the number of occurrences for each possible value, in particular for those attributes that have a categorical nature.\n\n\n```python\n# MAP NUMERICAL VALUES TO INTEGER VALUES\n# --------------------------------------------------------------------------- #\n# print('Before', dataset.shape)\ncolumns_2_map = ['ERECTED', 'LANES']\nfor _, predictor in enumerate(columns_2_map):\n dataset = dataset[dataset[predictor] != '?']\n dataset[predictor] = np.array(list(map(lambda x: int(x), dataset[predictor].values)))\n# print('After', dataset.shape); print(dataset.info())\n```\n\n\n```python\n# dataset.head(5)\n```\n\n\n```python\n# MAP NUMERICAL VALUES TO FLOAT VALUES\n# --------------------------------------------------------------------------- #\n# print('Before', dataset.shape)\ncolumns_2_map = ['LOCATION', 'LANES', 'LENGTH'] \nfor _, predictor in enumerate(columns_2_map):\n dataset = dataset[dataset[predictor] != '?']\n dataset[predictor] = np.array(list(map(lambda x: float(x), dataset[predictor].values)))\n```\n\n\n```python\nresult = dataset.isnull().values.any() # print('After handling null values\\nThere are any null values ? Response: {}'.format(result))\nresult = dataset.isnull().sum() # print('Number of null values for each predictor:\\n{}'.format(result))\n# dataset.head(5)\n```\n\n\n```python\n# dataset.describe(include='all')\n```\n\n### Descriptive Statistics\nAfter having performed the inital preprocessing phase about Pittsburgh Bridge Dataset, where we have clined the dataset from missing values as well as properly coded all the features, that are attributes and variables by which our dataset is made of, in order to reflect their own nature, wheter categorical or numerical, so continuous, we go ahead doing another step further, consisting in describing features properties by means of some usefull and well know tools coming from Descriptive Statistics area, that is a branch of the Statistics id considered as a whole.\n\nIn particular we are going to exploit some features that made up statistician's toolbox such as histograms, pie charts and the like, describing also their advantages as well as some of their backwards.\n\n\n```python\n# sns.pairplot(dataset, hue='T-OR-D', size=1.5)\n```\n\n__Histograms__:\nthe main advantage of using such a chart is that i can be employed to describe the frequencies with which single values or a subset of distict values wihtin a range occurs for a given sample of observations, independently whether such a sample is representing a part of an entire population of examples and measurements, or the population itself, reminding that usually we deal with subsets or samples obtained or randomly sampled by an entire popualtion which might be real or just hypothetical and supposed. In particualr the advantages that Histogram graphs allow us to observe looking at a sampel of records and measurements are the following:\n- If the variable taken into account is a continuous variable we may decide dto discretize the range of possible values into a number of subintervals, that are also referred to as bins, and observe how the data is distributed into the different subintervals.\n- In particular, continuing from above, the histogram we might see can suggest us if the sample has one or more picks, describing which are the most occuring values or the most populated subintervalls, as well as the histogram follows a bell-like shape in order to spot also whether the graphs shows greater upper-tail or lower-tail, or a positive skew and altertnatively a negaitve skew. Where, in the former case usually we knwo that that the sample of data shows a greater probability of observing data measurements from the upper side of the bell-shaped graph, otherwise from the lower side of again a bell-shapef graph.\n- It is also important to say that generally speaking all these kind of observations and analyses are well suited for variables and featrues that assumes values that are continuous in nature, such as height, weight, or as in our dataset LOCATION variable\n- Instead, if the variable under investigation is discrete or categorical in nature, the histogram graph is better called a bar graph and is a suitable choice for describing occourrencies or frequencies of different categories and classes, since someitmes there is not a natural order among the values such as Colors, evne if we might find a natural order as dress's sizes.\n\nHere, what follows is the sequence of several histograms created and illustrated to describe some other characteristics of the variables by which the dataset is made as well as to show the level or type of relationship of the frequency or the occurency of each value for a given attribute with the values assumed by the target variable we have selected amongst the overall variables.\n\n\n```python\ncolumns_2_avoid = ['ERECTED', 'LENGTH', 'LOCATION']; show_frequency_distribution_predictor(dataset, predictor_name='RIVER', columns_2_avoid=columns_2_avoid, features_vs_values=features_vs_values, hue=TARGET_COL, verbose=1)\n```\n\nThe Histogram related to the frequency, in other sense the occurency, of RIVER datset's feature shows us that:\n- Among the three main rivers that cross the pittsburgh town, that are *Allegheny*, *Monongahela*, and *Ohio*, the one with the highest number of bridges the first Allegheny, followed by Monongahela, and finally the Ohio river whcih is also the converging river of the former two preciding rivers.\n- Instead, if we depict and illustrate the occurency, of RIVER datset's feature over our target variable T-OR-D dataset's feature we can understand that among the two binary values, that are DECK and THROUGH, the second seems the most exploited floor system for building bridges between the opposite edges of the rivers. Furthermore, speaking about bridges built around Ohio river just THROUGH structural element is the only technique adopted for those bridges.\n- What we can also sau about RIVER feature is that Allegheny and Monongahela show more or less the same number of bridges made from THROUGH surface, while for DECK surface Allegheny bits all the other rivers and Ohio does not figure among the rivers where there are bridges with DECK like structure at all.\n\n\n```python\n# show_frequency_distribution_predictors(dataset, columns_2_avoid)\nshow_frequency_distribution_predictor(dataset, predictor_name='T-OR-D', columns_2_avoid=columns_2_avoid, features_vs_values=features_vs_values)\n```\n\n\n```python\n# show_frequency_distribution_predictors(dataset, columns_2_avoid)\nshow_frequency_distribution_predictor(dataset, predictor_name='CLEAR-G', columns_2_avoid=columns_2_avoid, features_vs_values=features_vs_values, hue=TARGET_COL)\n```\n\nInstead looking at CLEAR-G feature we can notice that the *Vertical Clearance for navigation* is allowed for the majority of the bridges and when looing at the relationship of such a feature with the T-OR-D target variable we can see that THROUGH technology is the most adopted amongst the both the bridges that have or not gained the vertical Clearance for navigation, and in particular the THROUCH system is far more popular than DECK surface system in both G and N bridges, recalling us how the THROUGH technique become so important and widely spread across time and space while speaking about bridge constructing.\n\n\n```python\n# show_frequency_distribution_predictors(dataset, columns_2_avoid)\nshow_frequency_distribution_predictor(dataset, predictor_name='SPAN', columns_2_avoid=columns_2_avoid, features_vs_values=features_vs_values, hue=TARGET_COL)\n```\n\nSpan is the distance between two intermediate supports for a structure, e.g. a beam or a bridge. A span can be closed by a solid beam or by a rope. The first kind is used for bridges, the second one for power lines, overhead telecommunication lines, some type of antennas or for aerial tramways. With such a definition kept in mind what we can undestand is that:\n- looking at the histogram graph about Occurency distribution of Bridge SPAN feature is that, since the three rivers are conisdered to be large riversa long most of their length considering the portion of them that cross the city of Pittsburgh, it becomes natural to observe that MEDIUM Span samples are the most occurring examples, while SHORT Span samples are the least frequent records and LONG span samples range in the between but seems to reach more closely the MEDIUM Span records.\n- As usual, also here, we continue to observe that THROUGH brdiges are the ind of bridges analysing the T-OR-D feature that collect the majority of samples, while DECK bridges are just characterized by brdiges with LONG or MEDIUM Span feature and no SHORT span.\n- Moreover, the ration between DECK and THROUGH reaches more or less 1 over 5, that is every 5 THROUGH bridges we find a DECK bridge with either LONG or MEDIUM Span elements.\n\n\n```python\n# show_frequency_distribution_predictors(dataset, columns_2_avoid)\nshow_frequency_distribution_predictor(dataset, predictor_name='MATERIAL', columns_2_avoid=columns_2_avoid, features_vs_values=features_vs_values, hue=TARGET_COL)\n```\n\nIn the histohram graph illustrated above, we can clearly understand that:\n- the STELL element is the most frequently exploited material for building bridges due to its strengthen and its restinance to the corrosion caused by the surrounding environment, while WOOD-like bridges are still present they are far less frequent than STEEL-like bridges but still have better properties than IRON bridges that are least frequent bridges since Iron leads to heavier bridges and Iron requires more extensive maintanance than Steel bridges and have less elastic properties than Wood brdiges.\n- However, THROUGH-like bridges are the kind of bridges that present istances that have all examples of bridges with all the available materials that the dataset has shown, while DECK-like bridges exploit just Steel material for building bridges.\n\n\n```python\n# show_frequency_distribution_predictors(dataset, columns_2_avoid)\nshow_frequency_distribution_predictor(dataset, predictor_name='REL-L', columns_2_avoid=columns_2_avoid, features_vs_values=features_vs_values, hue=TARGET_COL)\n```\n\nWe knwo that The REL-L property is the relative length of the main span to the total crossing length. With this short notion about such a feature in mind, what we can suggest observing ther first histogram depicted above is that:\n- FULL kind of Bridge, shortly *F*, is the most frequent example of feature of the Pittsburgh bridges, also, the Through system is the bridge system that the most exploit or is characterized by FULL REL-L property\n- The, SMALL kind of Bridge, shortlu *S* is the second property in number of instances that show such a feature among the bridges and it is only a kind of feature shown only from THROUGH-like bridges, this means that we do not find bridges that show such a property amongs the DECK-like bridges.\n- Lastly, an intermediate solution, represented by SMALL-FULL property, shortly *S-F*, is more or less present in both type of bridges that are classified with DECK or THROUGH system speaking about T-OR-D attribute.\n\n\n```python\n# show_frequency_distribution_predictors(dataset, columns_2_avoid)\nshow_frequency_distribution_predictor(dataset, predictor_name='TYPE', columns_2_avoid=columns_2_avoid, features_vs_values=features_vs_values, hue=TARGET_COL)\n```\n\nLastly, we have to speak about the TYPE feature, which is an attribute that refers to the kind of architecture or strucutre used for building the final shape of the bridges. Looking at the first picture above, that is the first Histogram, what we can notice is that:\n- SIMPLE-T architecture is the most frequent kind of shape or strucutre adopted to build brdiges amongs the pittsburgh bridges, than it is folowed by ARCH-like brideges.\n- Howevere, starting from the ARCH-like brideges and going ahead considering the other remaining kind of technique for giving a strucuture to a bridge, what we can undestand is that these attribute are more or less distrubuted equally, instead SIMPLE-T shows the highest value to refer to the number of instances characterized by SIMPLE-T value for this attribute within the datast.\n- Furthermore, DECK-like Birdges are just characterized by up to 4 out of 7 possible values for TYPE attribute, while THROUGH-like bridges show examples of istances from all of the possible kind of architectures for building a bridge.\n\n### Correlation Matrix Analysis\n\nIn fields of statistics as well as statistical learning, where the latter comes partly from the former, _correlation matrix_ is a table showing correlation coefficients between variables. Each cell in the table shows the correlation between two variables. A correlation matrix is used to summarize data, as an input into a more advanced analysis, and as a diagnostic for advanced analyses.\nKey decisions to be made when creating a correlation matrix include: choice of correlation statistic, coding of the variables, treatment of missing data, and presentation.\nTypically, a correlation matrix is square, with the same variables shown in the rows and columns.\n\n__Applications of a correlation matrix__:\nthere are three broad reasons for computing a correlation matrix:\n\n- To summarize a large amount of data where the goal is to see patterns. In our example above, the observable pattern is that all the variables highly correlate with each other.\n- To input into other analyses. For example, people commonly use correlation matrixes as inputs for exploratory factor analysis, confirmatory factor analysis, structural equation models, and linear regression when excluding missing values pairwise.\n- As a diagnostic when checking other analyses. For example, with linear regression, a high amount of correlations suggests that the linear regression estimates will be unreliable.\n\n__Treatment of missing values__:\nthe data that we use to compute correlations often contain missing values. This can either be because we did not collect this data or don\u2019t know the responses. Various strategies exist for dealing with missing values when computing correlation matrixes. A best practice is usually to use _multiple imputation_. However, people more commonly use _pairwise missing values_ (sometimes known as partial correlations). This involves computing correlation using all the non-missing data for the two variables. Alternatively, some use _listwise deletion_, also known as case-wise deletion, which only uses observations with no missing data. Both pairwise and case-wise deletion assume that data is missing completely at random.\n\n\n```python\n# corr_matrix = dataset.corr()\n```\n\n__Coding of the variables__: if you also have data from a survey, you'll need to decide how to code the data before computing the correlations. Changes in codings tend to have little effect, except when extreme.\n\n__Presentation__:\nwhen presenting a correlation matrix, you'll need to consider various options including:\n\n- Whether to show the whole matrix, as above or just the non-redundant bits, as below (arguably the 1.00 values in the main diagonal should also be removed).\n- How to format the numbers (for example, best practice is to remove the 0s prior to the decimal places and decimal-align the numbers, as above, but this can be difficult to do in most software).\n- Whether to show statistical significance (e.g., by color-coding cells red).\n- Whether to color-code the values according to the correlation statistics (as shown below).\n- Rearranging the rows and columns to make patterns clearer.\n\nThis shows correlations between the stated importance of various things of attributes used to describe records, examples, and samples within bridge dataset. The line of 1.00s going from the top left to the bottom right is the main diagonal, which shows that each variable always perfectly correlates with itself. This matrix is symmetrical, with the same correlation is shown above the main diagonal being a mirror image of those below the main diagonal.\n\n\n```python\n# display_heatmap(corr_matrix)\n```\n\nThis kind of presentation allows us, once we choose a row number and column number let's say _ith-row_ and _jth-column_, we can check locally the value assigned by the correlection factor to the pair made from to distinct features if _\"i\"_ strctly different from _\"j\"_. As an instance, what we can suggest observing the correlation matrix depicted just above, as well as exploting some common properties of symmetic and square matrices, is that:\n- along the area that spread near the main diagonal the resulting features pairs seemes to either positively moderately correlate or positively waekly correlate.\n- Convercely, along the area that spreads near the anti-diagonal matrix the resulting features pairs seems to instead either negatively moderately correlate or negatively waekly correlate.\n\nFinally, as examples, in order to exploit the fact that we can access directly to the correlaction valuecomputed by means the math formula provided by the expression of correlation facotr, we can say that:\n- the pair represented by ERECTED and LANES (3d-row, 6th col) features seems to moderately positively correlate, with a value equals to 0.65. This is also reasoable since that by the time while the Pittsburgh city was growing in size also the need of more infrastructures and building for working and let the people in town to leave, lead to increase in winde the different bridges to manage the traffic from and to the city.\n- on the other hand, still speaking about ERECTED feature, when it is coupled with TYPE feature (3d-row, 12th col) we see in contrast that them are characterized by a negative value of correlation that lead to interprete the pair as negatively moderately correlated. The principal reason about such a behavior may be imputed by the observation that through the different years the building techniques and technologies have been employed to constrcut better, stronger bridges abandoning oldest techniques which instead imply the exploitation of less technological materials sucha s wood that requires instead more frequently maintenance tasks.\n\n### Pie chart as a continuation of Correlation Matrix Analysis\n\nHere, within this subsequent section, I'm going to discuss and analyse the usefulness of exploiting piec-chart like graphs for describing some features or beahvior of correlation matrix values.\n\nThe first pie chart followed by also a realted histogram both aim at explaining and depicting how the features pairs are distributed among the three main subintervals that are named weak, moderate and strong as facotrs to label the kind of correlation the value of correlation referrend to a given pair represent, which interval are the following:\n- weak, if $p_{ith}$: $p_{ith} \\leq .5$\n- moderate, if $p_{ith}$: $.5 < p_{ith} < .8$\n- strong, if $p_{ith}$: $p_{ith} \\geq .8$\n\n\n```python\n# show_pie_hist_charts_abs_corr(corr_matrix, figsize=(2, 2), gridshape=None)\n```\n\nThe insight that we can understand is that up to nearly 90.90% of pairs of features is weakly correlated and just 9.09% is moderately correlated, without performing any finer distinction among positively or negatively correlated in each group. While we can notice that no pair is strongly correlated. Moreover, looking at the related histogram, just illustrated below the pie chart, we can clearly understand that just 5 over 12 features are showing also a moderate correlation patterns, which are: CLER-G, ERECTED, LANES; RIVER, and SPAN. Furthermore, we can end up saying that among those 5 features just ERECTED and SPAN shw the larger number of pairs in which them moderately correlate. instead in the majority of cases the possible pairs of features in large measure seem to weakly correlate.\n\nThe other two graphs, the first a pie chart and the second a histogram are a kind of zoom in of the other two graphs illustrated in the paragraphs above. In particular those two subsequents graphs aim at exploring in a deeper way the difference about correlation factor, taking into account also the positivness, or convercely the negativness of the kind of correlation, and so not just the information about the strongness in absolute value of the correlation factor.\n\n\n```python\n# show_pie_hist_charts_corr(corr_matrix, figsize=(10, 10), gridshape=None)\n```\n\nWhat we can understand analysing the two graphs is that the pie chart allows us to see that the features seems to be more negatively weakly correlated than positively weakly correlated, and following from the prior pie chart, also here the weak correlation dominates over the other kinds or strengthen of correlations, that are the remaining moderate and strong ones. In particualr negative weak correlation rises up to nearly 50% and positive weak correlation reaaches up to 40%, in other words, the latter is more or less 10 percentage point less than the former.\n\nWhile from the bar chart obtained from a histogram partly related to the pie chart that is above, we can undestand that, considering the same set of featrues which also show a bit of moderate correlation - that are CLER-G, ERECTED, LANES; RIVER, and SPAN - only three up to five of them show also somehow a little positive moderate correlation with some features, while the others mostly show a positive moderate correlation\n\n## Prepare data for Training Step\n\n\n```python\n# Make distinction between Target Variable and Predictors\n# --------------------------------------------------------------------------- #\nrescaledX, y, columns = prepare_data_for_train(dataset, target_col=TARGET_COL)\n```\n\n Summary about Target Variable {target_col}\n --------------------------------------------------\n 2 57\n 1 13\n Name: T-OR-D, dtype: int64\n shape features matrix X, after normalizing: (70, 11)\n\n\n\n```python\n# sns.pairplot(dataset, hue=TARGET_COL, height=2.5);\n```\n\n### Pricipal Component Analysis\n\nAfter having investigate the data points inside the dataset, I move one to another section of my report where I decide to explore examples that made up the entire dataset using a particular technique in the field of statistical analysis that corresponds, precisely, to so called Principal Component Analysis. In fact, the major objective of this section is understand whether it is possible to transform, by means of some kind of linear transformation given by a mathematical calculation, the original data examples into reprojected representation that allows me to retrieve most useful information to be later exploited at training time. So, lets dive a bit whitin what is and which are main concepts, pros and cons about Principal Component Analysis.\n\nFirstly, we know that **Principal Component Analysis**, more shortly PCA, is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called *principal components*. This transformation is defined in such a way that:\n- the first principal component has the largest possible variance (that is, accounts for as much of the variability in the data as possible),\n- and each succeeding component in turn has the highest variance possible under the constraint that it is orthogonal to the preceding components.\n\nThe resulting vectors, each being a linear combination of the variables and containing n observations, are an uncorrelated orthogonal basis set. PCA is sensitive to the relative scaling of the original variables.\n\nPCA is mostly used as a tool in *exploratory data analysis* and for making predictive models, for that reasons I used such a technique here, before going through the different learning technique for producing my models.\n\n#### Several Different Implementation\n\nFrom the theory and the filed of research in statistics, we know that out there, there are several different implementation and way of computing principal component analysis, and each adopted technique has different performance as well as numerical stability. The three major derivations are:\n- PCA by means of an iterative based procedure of extraing pricipal components one after the other selecting each time the one that account for the most of variance along its own axis, within the remainig subspace to be derived.\n- The second possible way of performing PCA is done via calculation of *Covariance Matrix* applied to attributes, that are our independent predictive variables, used to represent data points.\n- Lastly, it is used the technique known as *Singular Valued Decomposition* applied to the overall data points within our dataset.\n\nReading scikit-learn documentation, I discovered that PCA's derivation uses the *LAPACK implementation* of the *full SVD* or a *randomized truncated SVD* by the method of *Halko et al. 2009*, depending on the shape of the input data and the number of components to extract. Therefore I will descrive mainly that way of deriving the method with respect to the others that, instead, will be described more briefly and roughly.\n\n#### PCA's Iterative based Method\nGoing in order, as depicted briefly above, I start describing PCA obtained by means of iterative based procedure to extract one at a time a new principal componet explointing the data points at hand.\n\nWe begin, recalling that, PCA is defined as an orthogonal linear transformation that transforms the data to a new coordinate system such that the greatest variance by some scalar projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on.\n\nWe suppose to deal with a data matrix X, with column-wise zero empirical mean, where each of the n rows represents a different repetition of the experiment, and each of the p columns gives a particular kind of feature.\n\nFrom a math poitn of view, the transformation is defined by a set of p-dimensional vectors of weights or coefficients $\\mathbf {w} _{(k)}=(w_{1},\\dots ,w_{p})_{(k)}$ that map each row vector $\\mathbf{x}_{(i)}$ of X to a new vector of principal component scores ${\\displaystyle \\mathbf {t} _{(i)}=(t_{1},\\dots ,t_{l})_{(i)}}$, given by: ${\\displaystyle {t_{k}}_{(i)}=\\mathbf {x} _{(i)}\\cdot \\mathbf {w} _{(k)}\\qquad \\mathrm {for} \\qquad i=1,\\dots ,n\\qquad k=1,\\dots ,l}$.\n\nIn this way all the individual variables ${\\displaystyle t_{1},\\dots ,t_{l}}$ of t considered over the data set successively inherit the maximum possible variance from X, with each coefficient vector w constrained to be a unit vector.\n\nMore precisely, the first component In order to maximize variance has to satisfy the following expression:\n\n${\\displaystyle \\mathbf {w} _{(1)}={\\underset {\\Vert \\mathbf {w} \\Vert =1}{\\operatorname {\\arg \\,max} }}\\,\\left\\{\\sum _{i}\\left(t_{1}\\right)_{(i)}^{2}\\right\\}={\\underset {\\Vert \\mathbf {w} \\Vert =1}{\\operatorname {\\arg \\,max} }}\\,\\left\\{\\sum _{i}\\left(\\mathbf {x} _{(i)}\\cdot \\mathbf {w} \\right)^{2}\\right\\}}$\n\nSo, with $w_{1}$ found, the first principal component of a data vector $x_{1}$ can then be given as a score $t_{1(i)} = x_{1} \u22c5 w_{1}$ in the transformed co-ordinates, or as the corresponding vector in the original variables, $(x_{1} \u22c5 w_{1})w_{1}$.\n\nThe others remainig components are computed as folloes. The kth component can be found by subtracting the first k \u2212 1 principal components from X, as in the following expression:\n\n- ${\\displaystyle \\mathbf {\\hat {X}} _{k}=\\mathbf {X} -\\sum _{s=1}^{k-1}\\mathbf {X} \\mathbf {w} _{(s)}\\mathbf {w} _{(s)}^{\\rm {T}}}$\n\n- and then finding the weight vector which extracts the maximum variance from this new data matrix ${\\mathbf {w}}_{{(k)}}={\\underset {\\Vert {\\mathbf {w}}\\Vert =1}{\\operatorname {arg\\,max}}}\\left\\{\\Vert {\\mathbf {{\\hat {X}}}}_{{k}}{\\mathbf {w}}\\Vert ^{2}\\right\\}={\\operatorname {\\arg \\,max}}\\,\\left\\{{\\tfrac {{\\mathbf {w}}^{T}{\\mathbf {{\\hat {X}}}}_{{k}}^{T}{\\mathbf {{\\hat {X}}}}_{{k}}{\\mathbf {w}}}{{\\mathbf {w}}^{T}{\\mathbf {w}}}}\\right\\}$\n\nIt turns out that:\n- from the formulas depicted above me get the remaining eigenvectors of $X^{T}X$, with the maximum values for the quantity in brackets given by their corresponding eigenvalues. Thus the weight vectors are eigenvectors of $X^{T}X$.\n- The kth principal component of a data vector $x_(i)$ can therefore be given as a score $t_{k(i)} = x_{(i)} \u22c5 w_(k)$ in the transformed co-ordinates, or as the corresponding vector in the space of the original variables, $(x_{(i)} \u22c5 w_{(k)}) w_{(k)}$, where $w_{(k)}$ is the kth eigenvector of $X^{T}X$.\n- The full principal components decomposition of X can therefore be given as: ${\\displaystyle \\mathbf {T} =\\mathbf {X} \\mathbf {W}}$, where W is a p-by-p matrix of weights whose columns are the eigenvectors of $X^{T}X$.\n\n#### Covariance Matrix for PCA analysis\n\nPCA made from covarian matrix computation requires the calculation of sample covariance matrix of the dataset as follows: $\\mathbf{Q} \\propto \\mathbf{X}^T \\mathbf{X} = \\mathbf{W} \\mathbf{\\Lambda} \\mathbf{W}^T$.\n\nThe empirical covariance matrix between the principal components becomes ${\\displaystyle \\mathbf {W} ^{T}\\mathbf {Q} \\mathbf {W} \\propto \\mathbf {W} ^{T}\\mathbf {W} \\,\\mathbf {\\Lambda } \\,\\mathbf {W} ^{T}\\mathbf {W} =\\mathbf {\\Lambda } }$.\n\n\n#### Singular Value Decomposition for PCA analysis\n\nFinally, the principal components transformation can also be associated with another matrix factorization, the singular value decomposition (SVD) of X, ${\\displaystyle \\mathbf {X} =\\mathbf {U} \\mathbf {\\Sigma } \\mathbf {W} ^{T}}$, where more precisely:\n- \u03a3 is an n-by-p rectangular diagonal matrix of positive numbers $\u03c3_{(k)}$, called the singular values of X;\n- instead U is an n-by-n matrix, the columns of which are orthogonal unit vectors of length n called the left singular vectors of X;\n- Then, W is a p-by-p whose columns are orthogonal unit vectors of length p and called the right singular vectors of X.\n\nfactorizingn the matrix ${X^{T}X}$, it can be written as:\n\n${\\begin{aligned}\\mathbf {X} ^{T}\\mathbf {X} &=\\mathbf {W} \\mathbf {\\Sigma } ^{T}\\mathbf {U} ^{T}\\mathbf {U} \\mathbf {\\Sigma } \\mathbf {W} ^{T}\\\\&=\\mathbf {W} \\mathbf {\\Sigma } ^{T}\\mathbf {\\Sigma } \\mathbf {W} ^{T}\\\\&=\\mathbf {W} \\mathbf {\\hat {\\Sigma }} ^{2}\\mathbf {W} ^{T}\\end{aligned}}$\n\nWhere we recall that ${\\displaystyle \\mathbf {\\hat {\\Sigma }} }$ is the square diagonal matrix with the singular values of X and the excess zeros chopped off that satisfies ${\\displaystyle \\mathbf {{\\hat {\\Sigma }}^{2}} =\\mathbf {\\Sigma } ^{T}\\mathbf {\\Sigma } } {\\displaystyle \\mathbf {{\\hat {\\Sigma }}^{2}} =\\mathbf {\\Sigma } ^{T}\\mathbf {\\Sigma } }$. Comparison with the eigenvector factorization of $X^{T}X$ establishes that the right singular vectors W of X are equivalent to the eigenvectors of $X^{T}X$ , while the singular values $\u03c3_{(k)}$ of X are equal to the square-root of the eigenvalues $\u03bb_{(k)}$ of $X^{T}X$ . \n\nAt this point we understand that using the singular value decomposition the score matrix T can be written as:\n\n$\\begin{align} \\mathbf{T} & = \\mathbf{X} \\mathbf{W} \\\\ & = \\mathbf{U}\\mathbf{\\Sigma}\\mathbf{W}^T \\mathbf{W} \\\\ & = \\mathbf{U}\\mathbf{\\Sigma} \\end{align}$\n\nso each column of T is given by one of the left singular vectors of X multiplied by the corresponding singular value. This form is also the polar decomposition of T.\n\nEfficient algorithms exist to calculate the SVD, as in scikit-learn package, of X without having to form the matrix $X^{T}X$, so computing the SVD is now the standard way to calculate a principal components analysis from a data matrix\n\n\n```python\nshow_table_pc_analysis(X=rescaledX)\n```\n\n Cumulative varation explained(percentage) up to given number of pcs:\n\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    # PCSCumulative Varation Explained (percentage)
    0247.738342
    1575.856460
    2682.615768
    3788.413903
    4892.661938
    5995.976841
    61098.432807
    \n
    \n\n\n\n#### Major Pros & Cons of PCA\n\n\n\n## Learning Process \n\nHere in this section we are going to partly describy and in the remaining to test and evaluate performance of various machine learning models that we selected and adopted to built up learning models for accomplishing supervised machine learning tasks related to classification problems. More precisely, we focused on binary classification problemsa, since the target variables, that is T-OR-D feature, amongst the 12 features from which the dataset is made of, and from which the more or less hundered of records are described, is a binary categorical feature which can assume the two values represented by labels DECK and THROUGH describing in two distinct manner a property about each bridge within the dataset, that property refers to the system used for constructing the bridge surface, that is commonly called deck, for let veicheles or trains or whatevert to cross the rivers that are three distinct rivers: A, M, O. Where A stands for Allegheny river, while M stands for Monongahela river and lastly O stands for Ohio river.\n\nBefore describing the fine tuning process applied to the different models accordingly to their own major properties, characteristics and features, we have decided and established to test each model performance by looking at how well all of them is going just exploiting the default setting and runnng cross-validation protocol, in other word also referred to as policy, to check the accuracy level, as ana instance and some other metrics.\n\nTo be more detailed we follow the common machine leaning working flow, that requires to split the data set, after having preprocessed it properly and in a suitable manner to meet mahcine learning models needs, into subsets commonly referred to as training set and test set. Where, the former is exploited to build up a inference model and the latter is used to check model's performance as weel as behavior up on held out sample of istances never seen before and so those examples that the learning model wasn't provided with to learn weights and select right hyper-params to plug in back into model at the end of training procedure, in order to later test the model with the found weights as well as hyper-parameters and if it meet our requirements in terms of performance values reached at test time, then it will be ready for deployments.\n\n\n\nAs we can see from the image just above what we have to do, following the purposed machine learning scheme or work flow are the subsequent steps:\n\n__(1) Split Dataset into Train and Test sets__:\nAfter having done with preprocessing phase we separate or divide the dataset into training set and test set, where usually the training set is bigger than the test set, and in some cases the test set, for recording some kind of performance measures, can be made of just a single instance against which the model will be tested and checked.\n\n__(2) Split Train set into smaller Train set and a Validation set__:\nOnce have made the first split then we hold out a part for later checking the test set and we focus on training set. In fact once we have selected a machine learning model amongst those we want to adopt to fit a model and compare the results obtained we try to indentify the best fit parameters to setting the model with them. To reach such a goal we can adopt different aproaches to split further the training set into a validation set and a smaller train set in order to emulate before doing test phase a possible beahvior of the trainng model once we think it is ready for the following phase that is test step.\nThere are several procedure for establishing how to divide training set into two new subsets, that are validation and a little bit smaller train set, where all of them are also connected or reffered to a learning algorithm called cross-validation which consists roughly speaking into testing the model against a smaller portion of training set in order to record and measure the performance of a model before saying we are ready to proceed with test phase. Among the existing Cross-Validation Procedure we adopt and so describe briefly the following:\n- **K-fold Cross-Validation**: which is a cross validation protocol exploited so that we split the training set into K-folds, in other words K-subsets all of the same size, a part eventually the last one which will be the remainder set of samples. One at a time the K-fold are left out and the model is firstly trained against K-1 subsets(folds) and then test against the left out fold for recording the performanc. At the end of the k-times we have trained the model we have recorded the performance measures and we so can average the results and understand how model in average is well doing. In oder words we can either assume the mean value as the driving value to assume the model as satisfying our constraint on performance measures or adopt the best result amongst the k-trains as the settings for hyper-params to adopt. This procedure is feasible and suitable if we do not carea botu the fact that, in cases of classification tasks, the categories might be balanced or not in terms of number of instances, as well as, it can be adopted also if we want to show a learning curve and some other performance graphics or schemes as Confusion matrtices and ROC or recall-precision curves. Latly usual values for K are: 5, 10, 20.\n\n- **Leave One Out Cross-Validation**: It is a special case of the K-fold Cross-Validation. In fact, instead of adopting the usual values as 5, 10, 20, for K as the number of possible subsets, we establish to identify each single instance as a possible fold. It's clear that this algorithm requires more time to be completed and does not allow to show up the graphics cited just above, since we do not enough data for confusion matrix and ROC or recall-precision curves.\n- **Stratified Cross-Validation**: It is a good compromise when the dataset is still large enough to be exploited for training purposes but we decide to split into K-folds the traning set such that each subset have the same proportion of samples coming from the differn classes. This operation reveals to be necessary or even mandatory when we detect that the dataset does not show the same number of samples, more or less, for each class, in other word when the dataset is unbalanced for a given attribute that is the target attribute. This means that, while trying to mitigate the issue about the unbalanced dataset we thunk as well as hope that this management let the model to fit a model which will not be affected heavily just from the most numerous class, bus still learn how to classify the samples coming from the other less numerous classes, without too mcuh misclassifying errors. As usual also with Stratified Cross-Validation we are able to show same graphics as with plain K-fold Cross-Validation, the difference is that the folds are not randomly sampled from the original training set, but yet are sampled in the same proportion per each class in order to have the same number of samples for each class inside each fold.\n\nWe try out all of the three described cross-validation techniques to measure how weel default settings for different models are doing to gaina a base line against wich compare later results coming from fine tuning process carried out exploiting grid search technique for selecting the best combinatin of purposed values for each candidate machine learning technique.\n\n__(3) Grid Search approach__: is the technique adopted when taking into account one at a time all the machine learning algorithm for selecting the best set of hyper-parameters. It consists into defining for each model a grid of possible values for different hyper-parameters that in some sense represent our degree of freedom referred to some of the properties that characterize all the different models. The grid of values might be real number ranging within a more or less interval, or a string value used to trigger a certain feature of a model combined with other related aspects of the machine learning algorithm of the given model. We recall that the standard grid search will proceed until all possible combination of provided values have been test and training with such settings have been carried out. Opposite to classic grid search it is another technique called Random Grid Search, which implies instead to let a model to choose or sample randomly the hyper-parameters within the ranges or intervals related to each hyper-params. The latter technique can be potentially less expensive since we test a reduced number of combination but might be sub-optimal even if the results can be still acceptable and meaningful.\n\n\n\n\n__(4) Metrics and model performance's evaluation tools__: before deploying we have to test our model against a test set, that is a subset created or obtained from overall dataset which was, after preprocessing phase that turns feature values somehow into numbers, devided into two distinct sets generally of different size. The test set evaluation implies exploiting some metrics such as Accuracy, but yet there exist several others that partly are derived from confusion matrix evaluation tool, such as Precsio, Recal F1-score, and the like.\n\nSo what we understand is that we can make use of a bouch of metrics but rather than using directly those metrics we can explore model's performance by means of more useful tools such as Confusion Matrix, Roc curve in order to better understanding model's behavior when fed with unobserved new samples, as well as, how to set a threshold for determing when a target variable will suggest us that the classified sample belong to one class or the other. So, here we briefly describe which instruments we exploit to measure model's peformance starting from confusion matrix and moving ahead toward ROC curve.\n\n**Confusion Matrix**: in statistics a confusion matrix it's a grid or matrix of numbers and in the simplest scenario, correspoding to a binary classification task, it aims at showing how well a model was going when applied onto unknow or preiviously unseen data points and samples, in the following manner. Arbitrary we establish that along the rows we have the fraction of samples that the model has calssified or has assigned to agiven class, and so the rows account for *Predicted valuers*. Vice versa, along the columns we have the total number of samples for each class that all together resemble the so called *Actual Values* as we illustrate in the picture just below:\n\nSo, such a table of numbers allows us to measure the fraction of correctly classified examples belonging to the Positive class, also referred to as *True Positive(TP)* or to the Negative class, also named *True Negative(TN)* and at the same time we can derive also the fractions of wrongly classified Positive samples and Negative ones, respectively knwon as *False Positve(FP)* and *False Negative (FN)*. It is clear that looking at the diagonal matrix we can undestand that the larger the value along the diagonal the better the model was able to correctly identify the samples accordingly to their own actual class.\nFrom the four statistics measure depiceted above, that are TP,TN, FP, and FN, by the time have been dirved other useful metrics that can be exploited when the most used and well knwon measurement performance of accuracy is not enough due to the fact as an instance we wanto to analyze deeper our model behavior toward optimization of some goal or because we have to deal with dataset which are not balanced through classes to be predicted and so we wanto some other metrics to ensure the goodness of our solutions.\n\n**Roc Curve**: for the reason cited some lines earlier by exploiting the four basis metrics have been developed other useful tools for ensuring the goodness of a solution, among those tools we decide to adopt the following one, knwon as Roc Curve. It is a curve, whose acronim stands for Receiver Operating Curve, largely employed in the field of *Decison Theory* and aims at finding, siggesting or showing how moedl's performance varywhen we are going to set different thresholds for a simple scenario in which we are going to solve a binary classification problem. Such a curve require to show on the x-axis the fraction of samples corresponding to *False Positve Rate (FPR)* at a different values of the model's hyper-params corresponding to threshold set for classifying items at inference time, as well as on the x-axis the fraction of samples corresponding to *True Positive Rate(TRP)* in order to plot a curve that originates at coordinates (0,0) and terminates at coordinates (1,1), varying in the middle accordingly to the pairs of values recorded at a given threshold for (FPR,TPR) pair. We are also reporting two driving curves that are respectvely the curve related to the *Random Classifier* which corresponds to a classifier that for each instance is just randomly selecting the predicted class, and the *Perfetct Classifier* that always correctly classifyes all the samples. Our goal by analysing such a graphics is to identify the threshold value such that we are near the points on the curve that are not so far from the upper-left corner so that to maximise the TPR as well as minimizing the FPR. Another useful quantity related to Roc Curve is represented by the so called amount Area Under the Curve (AUC) that suggests us how much area under the rco curve whas accounted while varying the threshold for classiffying the test samples, in particular the higher the value the better the classifier is doing varying the threshlds. Lastly we notice that the Random Sample accounts for an AUC equls to 0.5 %, while the Perfect Classifier for 1.0 % so we aim at obtaing a value for AUC that is in at least in the between but that approaches the unity. Here, below is presented an example of Roc Curve example:\n\n\n\nLastly, before moving ahead with describing Machine Learning Models we provide a brief list of other useful metrics that can be exploited if necessary during our analyses:\n\n\n\n**Learning Curve**:\nlearning curves constitute a great tool to diagnose bias and variance in any supervised learning algorithm. It comes in handt when we have to face against the so called **Bias-Variance Trade-Off**. In order to explain what that trade-off implies we have to say briefly what follows:\n- In supervised learning, we assume there's a real relationship between feature(s) and target and estimate this unknown relationship with a model. Provided the assumption is true, there really is a model, which we'll call $f$, which describes perfectly the relationship between features and target.\n- In practice, $f$ is almost always completely unknown, and we try to estimate it with a model $\\hat{f}$. We use a certain training set and get a certain $\\hat{f}$. If we use a different training set, we are very likely to get a different $\\hat{f}$. As we keep changing training sets, we get different outputs for $\\hat{f}$. The amount by which $\\hat{f}$ varies as we change training sets is called **variance**.\n- While, for most real-life scenarios, however, the true relationship between features and target is complicated and far from linear. Simplifying assumptions give **bias** to a model. The more erroneous the assumptions with respect to the true relationship, the higher the bias, and vice-versa.\n- In practice, however, we need to accept a trade-off. We can\u2019t have both low bias and low variance, so we want to aim for something in the middle, knowing that:\n\\begin{equation}\n\\begin{cases}\nY = f(X) + \\quad irreducible \\quad error \\\\\nf(X) = \\hat{f(X)} + \\quad reducible \\quad error \\\\\nY = \\hat{f(X)} + \\quad reducible \\quad error + \\quad irreducible \\quad error \\\\\n \\end{cases}\n\\end{equation}\n\n\n \n \n
    \n\n\n \n \n
    \n\n\n__(5)Deployment__: The last step for building a machine learning system comprise the deployment of one or more machine learning trained models that fit and satisfy constraints that was intially fixed before doing any kind of analysis. The main goal of deployemnt is to employ such statistics models for predicting, in other words making inference, against new data, unknown obeservatons for which we do not knwo their target class values.\n\n\n```python\ntry_comparing_various_online_solvers_by_kernel(X=rescaledX, y=y, kernels=None, heldout=[.75, .7, .65, .6, .55, .5, .45, .4, .35, .33, .3, .25, .2], rounds=20, n_classes=-1, n_components=9, estimators=estimators_list, cv=StratifiedKFold(2), verbose=0, show_fig=True, save_fig=False, stratified_flag=True, n_splits=3, random_state=42, gridshape=(3, 2), figsize=(15, 15), title=\"try comparing various online solvers by kernel\", fig_name=\"try_comparing_various_online_solvers_by_kernel.png\")\n```\n\n## Learning Models \n\n### Learning Curve\nPerforming learning curve for a certain number of machine learning methods applied using the default configurations provided by the scikit-learn API, as illustrated looking at the online documentation. The result obtained is the following, in particular we have established to focus on just a kind of kernel trick instead of analysing all the beahviors from different kernel modes for kernelPca unsupervised learning technique, since we have notice, when carrying out several trials with different kenrel tricks the results found seem to be nearly the same, with some minor differences, so that we select the first kernel trick as a representative for the model:\n\n\n```python\nX = rescaledX; n_components=9\n# learning_curves_by_components( # train_sizes=list(range(5, 50)), # figsize=(20,5)\n# learning_curves_by_kernels(estimators_list[:], estimators_names[:], X, y, train_sizes=np.linspace(.1, 1.0, 10), n_components=n_components, pca_kernels_list=pca_kernels_list[0], verbose=0, by_pairs=True, savefigs=True, figs_dest=os.path.join('figures', 'learning_curve', f\"Pcs_{n_components}\"))\n```\n\nWhat we can learn observig or looking at the bounch of graphics we have displayed just above is that not all of the different machine learning techiniques applied on dataset pre-processed by means of a precise kernel trick adopted for KernelPca show the same behavior, and even some plots suggest us that a machine learning algorithm may seem not suitable for such a small dataset for determining the well-known trend of a learning curve while exploiting a learning technique. To be more precise, we are going to speak roughly and briefly a little bit about each graphics, in the order as they have been shown here before, recalling that we have adopted up to 9 Principal components that account for nearly 95% of comulative explained variance related to the dataset at hand:\n- The first general observation or thought that we can explain is that for all the pictures the initial gap between the two curves is most of the time wide, large, in other words, important, more or less about 20% in percentage.\n- Speaking about both learning curve obtained from applying Gaussian Niave Bayes, before, and K-Nearest Neighbor, after, what we can say is that the curve related to Training Score is decreasing more or less of 10 point percentage, reaching 90% from the top, while the Cross-alidation curve seems to be constant for the first 6 trials and then to improve a little bit, but for the former model we can see increasing and descreasing that do not allow to fix a trend for measuring a gap between the two curves, instead for the latter model seems that the two curve converge at a certain point once the traingin set is more or less the overall dataset, which means we are going to exploit the most of information.\n- Instead, if we focus on Logistic Regression technique and SGD Classifier, we have observed that before a given numebr of trials in both picitures the two curves, that are Trainign and Cross-validation score curves, seem to follow an independent trend increasing and descrising independently, but then the two curves start to follow a decreasing trend so we can hipotize that after that precise trials we may have reached the desired training size.\n- The learning curve associated with SVC classifier suggests us that it seems to behave looking at both Traingin and Cross-Validation curve more or less as the previous graphics, but while the Training curve as decreases loosing accuracy score value such a curve tightenes the varaince, instead the Cross-validation curve keeps a wide varaince stretching it while increasing the training set size, meaaning that the models seem to be more unsure about the results.\n- The last two graphics are the most problematics and questionable ones, since while enalarging the training size we do not see either improvements or worsening referring to the Training curve, while the Cross-lvalidation curve seems at the very beginning to show a samew behavior as the Training curve, but then after a sequence of trials with increasing training size we get an improvement in terms of accuracy scores. But, since we normally expect to observe a learning curve where Training curve as well as Cross-validation curve respectively decreases and increases their performance at least at a given point where we hope the gap existing between them will be as small as possible wecan think that such those two graphics are nto explaining directly the real issues we face while dealing with Decsion Tree and Radom Forest Classifiers with such a small dataset.\n\n### Cross Validation\n\nPerforming cross validation for all the Machine Learning Models to be fine tuned, later once the cross validation results are available to be plotted and to be considered as a base line, since those results are computed when the models have been prepared and configured with default hyper-parameter values.\n\n\n```python\n# Performe all Cross-Validations\n# -----------------------------------------------------------------\n# naive_bayes_classifier_grid_search(rescaledX, y)\nplot_dest = os.path.join(\"figures\", \"n_comp_2_analysis\", \"cross_validation\"); plots_names = list(map(lambda xi: f\"{xi}_learning_curve.png\", estimators_names))\n\nn = len(estimators_list); dfs_list, df_strfd = fit_all_by_n_components(estimators_list=estimators_list[:n], estimators_names=estimators_names[:n], X=X, y=y, random_state=0, test_size=.33, n_components=9, cv_list=cv_list[:N_CV], show_plots=False, pca_kernels_list=pca_kernels_list[:N_KERNEL-2], verbose=0, plot_dest=plot_dest)\nshow_df_with_mean_at_bottom(df_strfd) # show_df_with_mean_at_bottom(df_strfd) # df_strfd.head(df_strfd.shape[0])\n```\n\n## Naive Bayes Classification\n\nNaive Bayes models are a group of extremely fast and simple classification algorithms that are often suitable for very high-dimensional datasets. Because they are so fast and have so few tunable parameters, they end up being very useful as a quick-and-dirty baseline for a classification problem. Here I will provide an intuitive and brief explanation of how naive Bayes classifiers work, followed by its exploitation onto my datasets.\n\nI start saying that Naive Bayes classifiers are built on Bayesian classification methods. These rely on Bayes's theorem, which is an equation describing the relationship of conditional probabilities of statistical quantities.\n\nn Bayesian classification, we're interested in finding the probability of a label given some observed features, which we can write as P(L | features). Bayes's theorem tells us how to express this in terms of quantities we can compute more directly:\n\n$P(L|features)=\\frac{P(features|L)}{P(L)P(features)}$\n\nIf we are trying to decide between two labels, and we call them L1 and L2, then one way to make this decision is to compute the ratio of the posterior probabilities for each label:\n\n$\\frac{P(L1 | features)}{P(L2 | features)}=\\frac{P(features | L1)P(L1)}{P(features | L2)P(L2)}$\n\nAll we need now is some model by which we can compute P(features | $L_{i}$)\n\nfor each label. Such a model is called a generative model because it specifies the hypothetical random process that generates the data. Specifying this generative model for each label is the main piece of the training of such a Bayesian classifier. The general version of such a training step is a very difficult task, but we can make it simpler through the use of some simplifying assumptions about the form of this model.\n\nThis is where the \"naive\" in \"naive Bayes\" comes in: if we make very naive assumptions about the generative model for each label, we can find a rough approximation of the generative model for each class, and then proceed with the Bayesian classification. Different types of naive Bayes classifiers rest on different naive assumptions about the data, and we will examine a few of these in the following sections.\n\n#### Gaussian Naive Bayes\n\nPerhaps the easiest naive Bayes classifier to understand is Gaussian naive Bayes. In this classifier, the assumption is that data from each label is drawn from a simple Gaussian distribution. In fact, one extremely fast way to create a simple model is to assume that the data is described by a Gaussian distribution with no covariance between dimensions. This model can be fit by simply finding the mean and standard deviation of the points within each label, which is all you need to define such a distribution.\n\n$P(x_i \\mid y) = \\frac{1}{\\sqrt{2\\pi\\sigma^2_y}} \\exp\\left(-\\frac{(x_i - \\mu_y)^2}{2\\sigma^2_y}\\right)$\n\nThe parameters $\\sigma_{y}$ and $\\mu_{y}$ are estimated usually using maximum likelihood.\n\n\n```python\n# GaussianNB\n# -----------------------------------\nshow_df_with_mean_at_bottom(dfs_list[pos_cv]) # dfs_list[pos_cv].head(dfs_list[pos_cv].shape[0])\n```\n\n\n```python\nplot_name = plots_names[pos_cv]\nshow_learning_curve(dfs_list[pos_cv], n=len(cv_list[:N_CV]), figsize=(15, 7), plot_dest=plot_dest, grid_size=grid_size, plot_name=plot_name)\n```\n\n#### When to Use Naive Bayes\n\nBecause naive Bayesian classifiers make such stringent assumptions about data, they will generally not perform as well as a more complicated model. That said, they have several advantages:\n\n- They are extremely fast for both training and prediction\n- They provide straightforward probabilistic prediction\n- They are often very easily interpretable\n- They have very few (if any) tunable parameters\n\n\nThese advantages mean a naive Bayesian classifier is often a good choice as an initial baseline classification. If it performs suitably, then congratulations: you have a very fast, very interpretable classifier for your problem. If it does not perform well, then you can begin exploring more sophisticated models, with some baseline knowledge of how well they should perform.\n\nNaive Bayes classifiers tend to perform especially well in one of the following situations:\n\n- When the naive assumptions actually match the data (very rare in practice)\n- For very well-separated categories, when model complexity is less important\n- For very high-dimensional data, when model complexity is less important\n\nThe last two points seem distinct, but they actually are related: as the dimension of a dataset grows, it is much less likely for any two points to be found close together (after all, they must be close in every single dimension to be close overall). This means that clusters in high dimensions tend to be more separated, on average, than clusters in low dimensions, assuming the new dimensions actually add information. For this reason, simplistic classifiers like naive Bayes tend to work as well or better than more complicated classifiers as the dimensionality grows: once you have enough data, even a simple model can be very powerful.\n\n## Logistic Regression\n| Learning Technique | Type of Learner | Type of Learning | Classification | Regression |\n| --- | --- | --- | --- | --- |\n| *Logistic Regression* | *Linear Model* | *Supervised Learning* | *Supported* | *Not-Supported* |\n\nLogistic regression, is a linear model for classification rather than regression. Logistic regression is also known in the literature as logit regression, maximum-entropy classification (MaxEnt) or the log-linear classifier. In this model, the probabilities describing the possible outcomes of a single trial are modeled using a logistic function.\n\nLogistic regression is implemented in LogisticRegression. This implementation can fit binary, One-vs-Rest, or multinomial logistic regression with optional $l_{1}$, $l_{2}$ or Elastic-Net regularization.\n\nAs an optimization problem, binary class $l_{2}$ penalized logistic regression minimizes the following cost function:\n\n- $\\min_{w, c} \\frac{1}{2}w^T w + C \\sum_{i=1}^n \\log(\\exp(- y_i (X_i^T w + c)) + 1)$\n\nSimilarly, $l_{1}$ regularized logistic regression solves the following optimization problem:\n\n- $\\min_{w, c} \\|w\\|_1 + C \\sum_{i=1}^n \\log(\\exp(- y_i (X_i^T w + c)) + 1)$\n\nElastic-Net regularization is a combination of $l_{1}$ and $l_{2}$, so minimizes the following cost function:\n\n- $\\min_{w, c} \\frac{1 - \\rho}{2}w^T w + \\rho \\|w\\|_1 + C \\sum_{i=1}^n \\log(\\exp(- y_i (X_i^T w + c)) + 1)$\n- where $\\rho$ controls the strength of $l_{1}$ regularization vs. $l_{2}$ regularization\n\n### Cross-Validation Results\n\n\n```python\n# LogisticRegression\n# -----------------------------------\npos_cv = pos_cv + 1; show_df_with_mean_at_bottom(dfs_list[pos_cv]) # dfs_list[pos_cv].head(dfs_list[pos_cv].shape[0])\n```\n\n\n```python\nplot_name = plots_names[pos_cv]\nshow_learning_curve(dfs_list[pos_cv], n=len(cv_list[:N_CV]), figsize=(15, 7),plot_dest=plot_dest, grid_size=grid_size, plot_name=plot_name)\n```\n\n### Grdi-Search Results\n\n\n```javascript\n%%javascript\nIPython.OutputArea.prototype._should_scroll = function(lines) { return false; }\n```\n\n\n```python\nplot_dest = os.path.join(\"figures\", \"n_comp_9_analysis\", \"grid_search\"); X = copy.deepcopy(rescaledX)\n\ndf_gs, df_auc_gs, df_pvalue = grid_search_all_by_n_components(\n estimators_list=estimators_list[pos_gs+1], \\\n param_grids=param_grids[pos_gs],\n estimators_names=estimators_names[pos_gs+1], \\\n X=X, y=y,\n n_components=9,\n random_state=0, show_plots=False, show_errors=False, verbose=1, plot_dest=plot_dest, debug_var=False)\ndf_9, df_9_auc = df_gs, df_auc_gs\n```\n\nLooking at the results obtained running *Logistic Regression Classifier* against our dataset splitted into training set and test set and adopting a different kernel trick applied to *kernel-Pca* unsupervised preprocessing method we can state that generally speaking all the methods shonw a very high *Train Accuracy Score* which reaches in the most of the case a value greater than *90%*. However only one trial out of five, that is trial in which we adopted *Cosine Trick* we was able to account for *74%* of accuracy than the other cases where instead we do not reach a *train accuracy score* greater than *50%*. So, we can end up saying that the other models either overfit to the *train set* and wasn't able to generalize well on *test set*, or the fact that our dataset is not a balanced one leads to models and estimators that were able to correctly predict one among the two classes and more specifically, the models seems to recognize better the *class 0*, that is *Deck Bridges* than *class 1*, that is *Through Bridges*. In other words, usually working with unbalanced dataset we expect that the most frequent classes or most numerous classes were advantaged against the less numerous, but here emploping Logisti Regression Classifier we obtained models that were more able to correctly classify the less numerous class and to worngly predict the more numerous class. More precisely:\n- speaking about __Linear kernel Pca based Logisti Classifier__, we can notice that such a model exploiting a default threshold of *.5* reaches a poor test accuracy score of just *41%* with respect to a train accuracy score of *97%*. The model indeed overfits to the overall train set and tends to better predict the less numerous class, so the model gains weight parameters suitable to identify class 0 samples. Looking at *recall and precision scores*, the model was really precise when predicting class 1 examples and was able to correcly predict labels for class 0, so maximzes recall of negative class. But we cannot say it is also precise when predicting class 0 this means that it wrongly inferes the true label for positive class. Lastly the model obtained high and low weighted average precision and recall, such that weigthed *F1-score* was low as well. Speaking about *Roc curve and Auc Score*, we can unbderstand that the model obtains a intermediate Auc score, of *.64* than the Random Classifier, and the relationship between *FPR and TPR* is linear most of the time changing the threshold value for classification.\n- observing __Polynomial kernel Pca based Logisti Classifier__, we can notice that such a model exploiting a default threshold of *.5* reaches even a lower test accuracy of *32%* than a still high train accuracy of *91%*. So also here for this trial the resulting model overfit to the train set and becaus of both lower accuracy scores we can can state that the model wrongly predicts a larger number of samples from class 1. In fact the model's precision and recall of class 0 and class 1 lowered than the values seens for the previous trial, while the precision and recall of class 1 and class 0 still remain the same, so this model preidcts with high precision samples from class 1 but with great uncertainty about class 0, even if most of the sample from such a classes were correctly labeled. Looking at *Roc Curve and Auc Score* we can observe that the best found model with this configuration indeed is going slightly bettere than random classifier, in fact has a auc score equals just to *.59* and the *TPR and FPR* behaviors is that them grows linearly while modifying the default threshold value most of the time.\n- review __Rbf kernel Pca based Logisti Classifier__, we can stricly and briefly saying that as the two preivous models also here discussing this estimator performance we do not obtain satisfying results in fact the model behaves more or less as the first reviewed, and more precisely the model obtained a slighlty better accuracy on test set of *.47* and a weighted F1-Score of *.5*, that allow for a Auc score that reaches a value of *.68* However also this model overfit to the train set with an accuracy score of *92%*, and is more able to correcly predict class 1 instances with high precision and class 0 instances with more uncertainty, even if has a high recall related to class 0.\n- __Cosine kernel Pca based Logisti Classifier__ results to be the best solution found while performing grid search algorithm for Logisti Regression method, when it is fixed a defualt threshold of *.5* for classification. The rationale is that this trial retrieves a model that does not overfit to the train set, since test set accuracy is *74%*, just nearly 20 percent points than train accuracy score of *92%*. Moreover, we obtain high values for both *averaged precions, recalla and F1-Score metrics*, where the latter more precisely was even gretaer than test accuracy score, reachiung a value of *77%*. However, this model as others is less precise when predicting labels for class 0, than when inferring class 1 lables, this is mostly due to the fact that the dataset is not balanced. So we still remain more confident and precise when predicting class labels for class 1 examples. Looking at Roc Curve and Auc Score, we can say that for this model we have a curve which is able to account for up to *77%* of auc, and that this model works fine for many thresholds, in particular we can imagine to lower down a little bit the default threshold so that we can improve *TPR* reducing slightly *FPR* scores.\n- lastly, also __Sigmoid kernel Pca based Logisti Classifier__ as other previous trials except the one represented by the moedl trained fixing Cosine Trick as kernel Pca method, gains poor and lower performance, due to overfit issue and as other more or less same low performance models generally speaking obtains high and low weighted average precision and recall scores, meaning that while the few instances predicted as belonging to class 1 was done with high precision instead of samples from class 0 which was prediceted with high uncertainty, even if most of the time the model correctly predicts instances that indeed belongs to class 0. The Roc Curve and Auc Score of *62%* show that also this run leads to a model which TPR and FPR are most of the time growing linearly across the thresholds.\n\n__Significance Analysis__: finally, when looking at the different graphics related to the test which aims at investigating the diagnostic power of our different models we have fine tuned, picking the best one for such a test we can notice that beacues of the *signficance level* $\\alpha$ set equal to *0.05 that is 5% of chance to reject the Null-Hypothesis $H_{0}$*, we have obtained not grdi search result from training set that was able to overcome such cutt-off value of *%%* and therefore the different models are not uncertain enough to be adopted and configured with those hyper-parameters and model's weights for describing the underling model related to the data.\n\n#### Table Fine Tuned Hyper-Params (Logisti Regression)\n\n\n```python\nshow_table_summary_grid_search(df_gs, df_auc_gs, df_pvalue)\n```\n\nLooking at the table dispalyed just above that shows the details about the selected values for hyper-parameters specified during grid search, in the different situations accordingly to the fixed kernel-trick for kernel Pca unsupervised method we can state that, referring to the first two columns of *Train and Test Accuracy*, we can recognize which trials lead to more overfit results such as for *Linear, Polynomial, Rbf, and Sigmoid Tricks* or less overfit solution such as in the case of *Cosine Trick*. Speaking about the hyper-parameters, we can say what follows:\n- speaking about the *hyper-param C*, that is inverse of regularization strength where smaller values specify stronger regularization, we observe that except the Cosine kernel trick case all other kernel-Pca tricks adopted have preferred to exploit a very low value for *C parameter* equals to *0.001* and accounts for a very strtong regularization, but such a choice does not lead to models that obtained a high generalization capability, instead the *Cosine based kernel-Pca* model opted for a default value for such a parameter.\n- instead referring to *class_weight parameter*, we knwo that it can be set with balanced strategy which stends for a strategy where values of y to automatically adjust weights inversely proportional to class frequencies in the input data as *n_samples / (n_classes * np.bincount(y))*, we have been surprised that all the method that obtained worst performance choose a balanced strategy than the best model which was fine even with a default strategy that does not require to use a balanced mode.\n- instead *fit_intercept parameter* refers to the fact that we specify if a constant (a.k.a. bias or intercept) should be added to the decision function, and allows for modeling a certain behavior and a certain response different from zero even when the input sample is mostly made of zero components, we can understand that in all the cases the models obtained best results enabling such strategy and so the models are fitted taking into account also a intercept weight or parameter, increasing model complexity.\n- model's *penalty parameter* allows to specify the norm used in the penalization, among the folowing list of possible choices *l1, l2, elasticnet*. In all the models the best choice was for *l2 regularization*, this means that all the models opted for a kind of regularization that do not consider at all the *l1 normalization* as a regularization technique, so we avoid to obtain models that instead may lead weights to zero values, in other words sparse models.\n- model's *solver parameter* which is the algorithm to use in the optimization problem. It is curios to notice that almost all the models except cosine based kernel-Pca which adopted *liblinear* solver. What we can understand is that for all the overfitted models the choice of *sag* solver does not lead to significant results in term of performance, and we can say instead that we correctly except that for such a small dataset a *liblinear* choice is the most suitable and the best model found here is coherent with such a suggestion from theory field.\n- lastly, looking at *tol parameter*, which stends for tolerance for stopping criteria, we can clearly see that the first two models adopted a lowe tolerance value instead the last three preferred a lower value of tolerance, so the first two methods accordingly with the kind of kernel trick technique adopted for kernel-Pca seem to go well when a tolerance value is not so small as the last three methods, furthermore the first two methods request less time than the last three because of the lareger tolerance set for training convergence.\n\n## Knn\n\n| Learning Technique | Type of Learner | Type of Learning |Classification | Regression | Clustering |\n| --- | --- | --- | --- | --- | --- |\n| *K-Nearest Neighbor* | *Instance-based or Non-generalizing* | *Supervised and Usupervised Learning* | *Supported* | *Supported* | *Supported*|\n\nIn *Pattern Recognition*, the *K-Nearest Neighbors Algorithm (k-NN)* is a __non-parametric method__ used for classification and regression. In both cases, the input consists of the k closest training examples in the feature space. The output depends on whether k-NN is used for classification or regression:\n\n- In *k-NN classification*, the output is a class membership. An object is classified by a plurality vote of its neighbors, with the object being assigned to the class most common among its k nearest neighbors (k is a positive integer, typically small). If k = 1, then the object is simply assigned to the class of that single nearest neighbor.\n- In *k-NN regression*, the output is the property value for the object. This value is the average of the values of k nearest neighbors.\n\nWhat follows is a briefly explanation of Knn:\n\nThe training examples are vectors in a multidimensional feature space, each with a class label. The training phase of the algorithm consists only of storing the feature vectors and class labels of the training samples. In the classification phase, k is a user-defined constant, and an unlabeled vector (a query or test point) is classified by assigning the label which is most frequent among the k training samples nearest to that query point. A commonly used distance metric for continuous variables is *Euclidean distance*.\n\n\n\nExample of k-NN classification. The test sample (green dot) should be classified either to blue squares or to red triangles. If k = 3 (solid line circle) it is assigned to the red triangles because there are 2 triangles and only 1 square inside the inner circle. If k = 5 (dashed line circle) it is assigned to the blue squares (3 squares vs. 2 triangles inside the outer circle).\n\n__Choice of Nearest Neighbors Algorithm__: The optimal algorithm for a given dataset is a complicated choice, and depends on a number of factors, number of samples *N* (i.e. n_samples) and dimensionality *D* (i.e. n_features):\n- *Brute force* query time grows as $O[D N]$\n- *Ball tree* query time grows as approximately $O[D \\log(N)]$\n- *KD tree* query time changes with *D* in a way that is difficult to precisely characterise. For small *D* (less than 20 or so) the cost is approximately $O[D \\log(N)]$, and the KD tree query can be very efficient. For larger *D*, the cost increases to nearly O[DN], and the overhead due to the tree structure can lead to queries which are slower than brute force.\n\nTherefore, we end up saying that, for small data sets (*N* less than 30 or so), $\\log(N)$ is comparable to *N*, and brute force algorithms can be more efficient than a tree-based approach. Both KDTree and BallTree address this through providing a leaf size parameter: this controls the number of samples at which a query switches to brute-force. This allows both algorithms to approach the efficiency of a brute-force computation for small *N*.\n\n### Cross-Validation Results\n\n\n```python\n# Knn\n# -----------------------------------\npos_cv = pos_cv + 1; show_df_with_mean_at_bottom(dfs_list[pos_cv]) # dfs_list[pos_cv].head(dfs_list[pos_cv].shape[0])\n```\n\n\n```python\nplot_name = plots_names[pos_cv]\nshow_learning_curve(dfs_list[pos_cv], n=len(cv_list[:N_CV]), figsize=(15, 7), plot_dest=plot_dest, grid_size=grid_size, plot_name=plot_name)\n```\n\n### Grid-Search Results\n\n\n```python\npos_gs = pos_gs + 1; plot_dest = os.path.join(\"figures\", \"n_comp_9_analysis\", \"grid_search\"); X = copy.deepcopy(rescaledX)\n\ndf_gs, df_auc_gs, df_pvalue = grid_search_all_by_n_components(\n estimators_list=estimators_list[pos_gs+1], \\\n param_grids=param_grids[pos_gs],\n estimators_names=estimators_names[pos_gs+1], \\\n X=X, y=y,\n n_components=9,\n random_state=0, show_plots=False, show_errors=False, verbose=1, plot_dest=plot_dest, debug_var=False)\ndf_9 = merge_dfs_by_common_columns(df_9, df_gs); df_9_auc = merge_dfs_by_common_columns(df_9_auc, df_auc_gs) # # df_9, df_9_auc = pd.concat([df_9, df_gs], axis=0), pd.concat([df_9_auc, df_auc_gs], axis=0)\n```\n\nLooking at the results obtained running *Knn Classifier* against our dataset splitted into training set and test set and adopting a different kernel trick applied to *kernel-Pca* unsupervised preprocessing method we can state generally speaking that all the such a *Statistical Learning technique* leads to a sequence of results that on average are more than appreciable beacues of the accuracy scores obtained at test time which compared against the same score but related to train phase allows us to understand that during the model creation the resulting classifiers do not overfit to the data, and even when the training score was high it was still lower than the scores in terms of accuracy obtained from the Logisti Regression Models which overfit to the data. Moreover, looking at the weighted values of *Recall, Precision, and F1-Scores* we can notably claim that the classifiers based on Knn obtained good performance and and except for one trial where we got lower and worst results, when *Sigmoid Trick* is selected, in the remaning cases have gotten remarkable results. More precisely we can say what follows:\n- speaking about __Linear kernel Pca based Knn Classifier__, when adoping the default threshold of *.5* for classification purposes we have a model that reaches an accuracy of *71%* at test time against an accuracy of *92%* at train step, while the Auc score reaches a value of *76%* with a Roc Curve that shows a behavior for which the model for a first set of thresholds let *TPR* grows faster than *FPR*, and only when we take into account larger thresholds we can understand that the trend is reversed. Looking at classification report we can see that the model has high precision and recall for class 1, so this means that the classifier has high confidence when predicting class 1 labels, instead it is less certain when predicting class 0 instances because has low precision, even if the model was able to predict correctly all the samples from class 0, leading to high recall.\n- observing __Polynomial kernel Pca based Knn Estimator__, we can notice that such a model exploiting a default threshold of *.5* reaches an accuracy of *74%* at test time against an accuracy of *89%* at train step, while the Auc score reaches a value of *77%*. What we can immediately understand is that the second model we have trained for Knn classifier is able to better generalize because obtained a higher accuracy score for test set which is also less far from train accuracy score, moreover the model has a slightly greater precision and recall when referring to class 1, while the precision and recall about class 0 seem to be more or less the same.\n- review __Rbf kernel Pca based Knn Classifier__, we can notice that such a model exploiting a default threshold of *.5* reaches an accuracy of *79%* at test time against an accuracy of *95%* at train step, while the Auc score reaches a value of *74%*. We can understand that even if this model, having selected *Rbf kernel Trick for kernel-Pca*, corresponds to the estimator that among Knn classifiers is the one with the best performance in terms of accuracy we notice that the corresponding auc score is less than the other first two analyzed trials where we end up saying that such classifiers lead to acceptable results. However, this method set with the hyper-params found by the grid-search algorithm reveals a higher value of precision related to class 0, meaning that *Rbf kernel Pca based Logisti Classifier* has a higher precision than previous models when classifying instances as belonging to class 0, while precisin and recall metrics for class 1 was more or less the same. This classifier is the one that we should select since it has higher precision values for better classifyng new instances.\n- looking at __Cosine kernel Pca based Knn Classifier__, we can notice that such a model exploiting a default threshold of *.5* reaches an accuracy of *59%* at test time against an accuracy of *92%* at train step, while the Auc score reaches a value of *62%*. We can clearly see that that such a model corresponds to the worst solution amongst the models we have trained exploiting *Knn classifier*, because we can state that due to a mostly lower accuracy score obtained at test time than the accuracy score referred to training time the classifier seems to have overfit to the data. In particular speaking about Precision and Recall scores about class 1, from classification report, the model seems to be mostly precise when predicting class 1 as label for the instances to be classified, however was misclassifying nearly half of the samples from class 1. Furthermore, the model also do not obtain fine results in terms of metrics when we focus on class 0 precision and recall. This model is the oen we should avoid, and do not exploit.\n- finally, referring to __Sigmoid kernel Pca based Knn Model__, we can notice that such a model exploiting a default threshold of *.5* reaches an accuracy of *65%* at test time against an accuracy of *92%* at train step, while the Auc score reaches a value of *72%*. It has values for performance metrics such as *precisio, recall, and F1-Score* which are more or less quyite similar to those of the first models trained for Knn classifier, that are those which are exploiting lienar and polynomial tricks, however this model misclassifyes larger number of class 1 instances lowering down the precision related to class 0, as well as lowering down recall from class 1. Also such a classifier with respect the first three trial is not sufficiently fine to be accepted so that we can exclude it from our choice.\n\n__Significance Analysis__: finally, when looking at the different graphics related to the test which aims at investigating the diagnostic power of our different models we have fine tuned, picking the best one for such a test we can notice that beacues of the *signficance level* $\\alpha$ set equal to *0.05 that is 5% of chance to reject the Null-Hypothesis $H_{0}$*, we have obtained following results. Two classifier out of five, which are *Linear- and Poly-kernel Pca based Knn Classifier* have a p-value that widely exceeds the value set for significance level, so for those two cases rejecting Null-Hypothesis will causes a *Type I error*. Also looking at *Cosine- and Sigmoid-kernel Pca based Knn Classifier* we can state same conclusions as just above for the two previous models. While just *Rbf-kernel Pca based Knn Classifier* seems the right classifier that obtained a p-value lower than the pre-defined p-value of *5%*, so we have to select this method as the one which allows us to reject the Null-Hypothesis and adopting such a classifier for describing the data behavior.\n\n#### Table Fine Tuned Hyper-Params(Knn Classifier)\n\n\n```python\nshow_table_summary_grid_search(df_gs, df_auc_gs, df_pvalue)\n```\n\nLooking at the table dispalyed just above that shows the details about the selected values for hyper-parameters specified during grid search, in the different situations accordingly to the fixed kernel-trick for kernel Pca unsupervised method we can state that, referring to the first two columns of *Train and Test Accuracy*, we can recognize which trials lead to more overfit results such as for *Cosine, and Sigmoid Tricks* or less overfit solution such as in the case of *Linear, Polynomial, and Rbf Trick*. Speaking about the hyper-parameters, we can say what follows:\n- looking at the *algorithm paramter*, which can be set alternatively as *brute, kd-tree, and ball-tree* where each algorithm represents an differen strategy for implementing neighbor-based learning with pros and cons in terms of requested training time, memory usage and inference performance in terms of elapsed time, we can clearly understand that the choice of the kind of kernel trick for performing kernel-Pca does not care since all the trials preferred and selectd *ball-tree* strategy to solve the problem. It means that the grid search algorithm, when forced to try all the possible combination recorded such a choice as the best hyper-param which leads to building an expensive data strucuture which aims at integrating somehow distance information to achieve better performance scores. So, this should make us reason about the fact that it is still a good choice or we should re-run the procedure excluding such algorithm. In fact the answer to such a issue depend on the forecast about the number of queryes we aim to solve. If it will be huge in future than ball-tree algorthm was a good choice and a goo parameter included amongst the hyper-params grid of values, otherwise we should get rid of it.\n- referring to *leaf_size parameter*, we can notice that also here the choice of a specific kernel trick for performing kernel-Pca algorithm does not affect the value tha such a parameter has assumed amongst those proposed. However, recalling that leaf size hyper-param is used to monitor and control the tree-like structure of our solutions we can understand that since the value is pretty low the obtained trees were allowed to grow toward maximum depth.\n- speaking about *distance parameter*, the best solution through different trials was *Euclidean distance*, which also corresponds to the default choice, furthermore the choice of a kernel trick in the context of the other grid values was not affecting the choice of *distance parameter*.\n- *n_neighbors parameter* is the one which is most affected and influenced by the choice of kernel trick for performing the kernel-Pca preprocessing method, since three out of five trials found 3 as the best value which are *Linear, Poly and Cosine tricks*, however only the first two examples using such a low number of neighbors still obtained fine results, instead the best trial which is the classifier characterized from Rbf kernel trick for kernel-Pca has selected 7 as the best value for the number of neighbors meanning that such a classifier required a greater number of neighbor before estimating class label and also that the query time t solve the inference would be longer.\n- lastly, the *weights param* is involved when we want to assign a certain weight to examples used during classification, where usually farway points will have less effect and nearby point grow their importance. The most frequent choice was represented by the *distance strategy*, which assign a weigh value to each sample of train set involved during classification a value proportional to the inverse of the distance of that sample from the query point. Only the Sigmoid kernel trick case instead adopted a weights strategy which corresponds to the default choice which is the uniform strategy.\n\nIf we imagine to build up an *Ensemble Classifier* from the family of *Average Methods*, which state that the underlying principle leading their creation requires to build separate and single classifiers than averaging their prediction in regression context or adopting a majority vote strategy for the classification context, we can claim that amongst the purposed Knn classifier, for sure, we could employ the classifier foudn from the first three trials because of their performance metrics and also because Ensemble Methods such as Bagging Classifier, usually work fine exploiting an ensemble of independent and fine tuned classifier differently from Boosting Methods which instead are based on weak learners.\n\n## Stocastic Gradient Descent\n| Learning Technique | Type of Learner | Type of Learning | Classification | Regression | Clustering |\n| --- | --- | --- | --- | --- | --- |\n| *Stochastic Gradient Descent (SGD)* | *Linear Model* | *Supervised Learning*| *Supported* | *Supported* | *Not-Supported*|\n\nStochastic Gradient Descent (SGD) is a simple yet very efficient approach to fitting linear classifiers and regressors under convex loss functions such as (linear) Support Vector Machines and Logistic Regression. Even though SGD has been around in the machine learning community for a long time, it has received a considerable amount of attention just recently in the context of large-scale learning.\n\nSGD has been successfully applied to large-scale and sparse machine learning problems often encountered in *text classification * and *natural language processing*. Given that the data is sparse, the classifiers in this module easily scale to problems with more than 10^5 training examples and more than 10^5 features.\n\n__Mathematical formulation__: we describe here the mathematical details of the SGD procedure.\n\nGiven a set of training examples $(x_1, y_1), \\ldots, (x_n, y_n)$ where $x_i \\in \\mathbf{R}^m$ and $y_i \\in \\mathcal{R}$ ($y_i \\in {-1, 1}$ for classification), our goal is to learn a linear scoring function $f(x) = w^T x + b$ with model parameters \n$w \\in \\mathbf{R}^m$ and intercept $b \\in \\mathbf{R}$. In order to make predictions for binary classification, we simply look at the sign of $f(x)$. To find the model parameters, we minimize the regularized training error given by:\n- $E(w,b) = \\frac{1}{n}\\sum_{i=1}^{n} L(y_i, f(x_i)) + \\alpha R(w)$\n- where **L** is a loss function that measures model (mis)fit and **R** is a regularization term (aka penalty) that penalizes model complexity; $\\alpha > 0$ is a non-negative hyperparameter that controls the regularization stength.\n\nDifferent choices for **L** entail different classifiers or regressors:\n\n- Hinge (soft-margin): equivalent to Support Vector Classification: $L(y_i, f(x_i)) = \\max(0, 1 - y_i f(x_i))$.\n- Perceptron: $L(y_i, f(x_i)) = \\max(0, - y_i f(x_i))$\n- Modified Huber: $L(y_i, f(x_i)) = \\max(0, 1 - y_i f(x_i))^2$ if $y_i f(x_i) >1$, otherwise in $L(y_i, f(x_i)) = -4 y_i f(x_i)$\n- Log: equivalent to Logistic Regression: $L(y_i, f(x_i)) = \\log(1 + \\exp (-y_i f(x_i)))$\n- Least-Squares: Linear regression (Ridge or Lasso depending on **R**): $L(y_i, f(x_i)) = \\frac{1}{2}(y_i - f(x_i))^2$.\n- Huber: less sensitive to outliers than least-squares. It is equivalent to least squares when $|y_i - f(x_i)| \\leq \\varepsilon$, and $L(y_i, f(x_i)) = \\varepsilon |y_i - f(x_i)| - \\frac{1}{2} \\varepsilon^2$ otherwise.\n- Epsilon-Insensitive: (soft-margin) equivalent to Support Vector Regression: $L(y_i, f(x_i)) = \\max(0, |y_i - f(x_i)| - \\varepsilon)$\n\nFinally, popular choices for the regularization term (the penalty parameter) include:\n- L2 norm: $R(w) := \\frac{1}{2} \\sum_{j=1}^{m} w_j^2 = ||w||_2^2$\n- L1 norm: $R(w) := \\sum_{j=1}^{m} |w_j|$ ,which leads to sparse solutions.\n- Elastic Net: $R(w) := \\frac{\\rho}{2} \\sum_{j=1}^{n} w_j^2 +(1-\\rho) \\sum_{j=1}^{m} |w_j|$, a convex combination of L2 and L1, where $\\rho$ is given by $1 - l1_{ratio}$.\n\n__Adavantages and Backwards__: the advantages of Stochastic Gradient Descent are:\n- Efficiency.\n- Ease of implementation (lots of opportunities for code tuning).\n- Complexity: The major advantage of SGD is its efficiency, which is basically linear in the number of training examples. If X is a matrix of size (n, p) training has a cost of $O(k n \\bar p)$, where k is the number of iterations (epochs) and $\\bar p$ is the average number of non-zero attributes per sample.\n\nThe disadvantages of Stochastic Gradient Descent include:\n- SGD requires a number of hyperparameters such as the regularization parameter and the number of iterations.\n- SGD is sensitive to feature scaling.\n\n### Cross-Validation Result\n\n\n```python\n# SGDClassifier\n# -----------------------------------\npos_cv = pos_cv + 1; show_df_with_mean_at_bottom(dfs_list[pos_cv]) # dfs_list[pos_cv].head(dfs_list[pos_cv].shape[0])\n```\n\n\n```python\nplot_name = plots_names[pos_cv]\nshow_learning_curve(dfs_list[pos_cv], n=len(cv_list[:N_CV]), figsize=(15, 7), plot_dest=plot_dest, grid_size=grid_size, plot_name=plot_name)\n```\n\n### Grid-Search Result\n\n\n```python\npos_gs = pos_gs + 1; plot_dest = os.path.join(\"figures\", \"n_comp_9_analysis\", \"grid_search\"); X = copy.deepcopy(rescaledX)\n\ndf_gs, df_auc_gs, df_pvalue = grid_search_all_by_n_components(\n estimators_list=estimators_list[pos_gs+1], \\\n param_grids=param_grids[pos_gs],\n estimators_names=estimators_names[pos_gs+1], \\\n X=X, y=y,\n n_components=9,\n random_state=0, show_plots=False, show_errors=False, verbose=1, plot_dest=plot_dest, debug_var=False)\ndf_9 = merge_dfs_by_common_columns(df_9, df_gs); df_9_auc = merge_dfs_by_common_columns(df_9_auc, df_auc_gs) # df_9, df_9_auc = pd.concat([df_9, df_gs], axis=0), pd.concat([df_9_auc, df_auc_gs], axis=0)\n```\n\nLooking at the results obtained running *Sgd Classifier* against our dataset splitted into training set and test set and adopting a different kernel trick applied to *kernel-Pca* unsupervised preprocessing method we can state generally speaking that looking at the weighted values of *Recall, Precision, and F1-Scores* we obtained good performance and and except for one trial where we got lower and worst results, when *Polynomial and Rbf Tricks* is selected, in the remaning cases have gotten remarkable results. More precisely we can say what follows:\n\n- speaking about __Linear kernel Pca based Sgd Classifier__, when adoping the default threshold of *.5* for classification purposes we have a model that reaches an accuracy of *65%* at test time against an accuracy of *92%* at train step, while the Auc score reaches a value of *79%* with a Roc Curve that shows a behavior for which the model increases its *TPR* without affecting the *FPR* score, however at a given point the Roc Curve trend turns so that the two cited scores begin to increase linearly and with a slope lower than that of Random Classifier so that FPR increases faster. The model is very precise when predicting class 1 instances but it has a recall of just *54%* so misclassified more or less half of samples from class 1 and this fact influenced instead the precision of class 0 that is a bit low, just *32%*, while class 0 recall is very high. Since the test accuracy score loses nearly 30 percent points we can assume that sucha model quite overfit to train data, we are not really encouraged to adopt it except we decied to exploit it for including it in an ensemble classifier, more boosting like than bagging one.\n\n- observing __Polynomial kernel Pca based Sgd Estimator__, we can notice that such a model exploiting a default threshold of *.5* reaches an accuracy of *76%* at test time against an accuracy of *92%* at train step, while the Auc score reaches a value of *73%*. It represents the best result obtained running th SGD based Training Algorithm upon our input dataset, in particular it obtained high precision and high recall for class 1, in other words such a model is able to recognize and correctly classify most of the data examples whose true label is indeed class 1. However, even if the model has high recall related to class 0, since the dataset is unbalanced we cannot say the same things for precision score about the class 0. So the model is somewhat uncertain when predicting class 0 as label value for new observations.\n\n- review __Rbf kernel Pca based Sgd Classifier__, we can notice that such a model exploiting a default threshold of *.5* reaches an accuracy of *82%* at test time against an accuracy of *92%* at train step, while the Auc score reaches a value of *57%*. In particular such a trial along with the *PCosine kernel Pca based Sgd Classifier* are the two attempts that lead to worts results, since the model overfit against the data employed at training time, but also the model gained weights that tend to predict every thing as class 1 instance. So, the resulting scores tell us that the model is highly precise and obtained high recall related to class 1, convercely has very low performance for precision and recall referred to class 0. Since such a model is performing just a little bit better than random classifier, can be largely adopted along other similar models for building voting classifier, following boosting like classifier policy and behavior.\n\n- looking at __Cosine kernel Pca based Sgd Classifier__, we can notice that such a model exploiting a default threshold of *.5* reaches an accuracy of *32%* at test time against an accuracy of *95%* at train step, while the Auc score reaches a value of just *59%*. Here the fine tuned model obtained from grid-search approach tells us that we are able to classify with high precision a few data examples from class 1, and even if we correctly classify all instances from class 0, we also wrongly predict class labels for most of instances,. whose true label is class 1. This means that the model is highly uncertain when predicting class 0 as the output target label. Moreover, the model's ROC Curve performs slighltly better than the random classifier, and we end up saying that such a model has gained weights and hyper-params that tend to predict the unknown instances as belonging to class 0 most of the time. We cannot neither say that switching the class labels between the two classes will allow us to obtain a better result since the roc curve trend is just a little bit better than the random classifier.\n\n- finally, referring to __Sigmoid kernel Pca based Sgd Model__, we can notice that such a model exploiting a default threshold of *.5* reaches an accuracy of *44%* at test time against an accuracy of *92%* at train step, while the Auc score reaches a value of *66%*. This model behaves more or less as the model obtained from the first trial performed for Sgd-based classifier, so as the first model is slightly worst than the best model found here when adopting as classifier Sgd technique, that is the *Cosine kernel Pca based Sgd Classifier*.\n\n__Significance Analysis__: finally, when looking at the different graphics related to the test which aims at investigating the diagnostic power of our different models we have fine tuned for *SGD Classifier*, picking the best one for such a test we can notice that beacues of the *signficance level* $\\alpha$ set equal to *0.05 that is 5% of chance to reject the Null-Hypothesis $H_{0}$*, we have obtained following results. Adopting the SGD statistical learning technique for classification fine tuned as above with hyper-params selectd also depending on the kind of *kernel-trick adopted for kernel-Pca unsupervised technique*, we can calim that only two out of five trials lead to a *p-vlaue* worst than *selected significance level equal to 5%*, which are *Linear- and Cosine-kernel Pca based Sgd Classifier*, so rejecting the *Null-Hypotesis* for those two cases will results into a *Type I Error*. While the remaining three cases, that are *Poly-, Rbf- and Sigmoid-kernel Pca based Sgd Classifier* have obtained a p-value over the range $[.9, 3]$ *in percet points*, so we are satisfyed for the results obtained in terms of significance scores, however, only *Poly-, and Rbf-kernel Pca based Sgd Classifier* really matter or are worth models since they do not overfit too much and do not go worstly as *Sigmoid-kernel Pca based Sgd Classifier* at test time.\n\n#### Table Fine Tuned Hyper-Params(SGD Classifier)\n\n\n```python\nshow_table_summary_grid_search(df_gs, df_auc_gs, df_pvalue)\n```\n\nLooking at the table dispalyed just above that shows the details about the selected values for hyper-parameters specified during grid search, in the different situations accordingly to the fixed kernel-trick for kernel Pca unsupervised method we can state that, referring to the first two columns of *Train and Test Accuracy*, we can recognize which trials lead to more overfit results such as for *Rbfd Trick* or less overfit solution such as in the case of *Linear, Polynomial, Cosine, and Sigmoid Tricks*. Speaking about the hyper-parameters, we can say what follows:\n\n- looking at __alpha hyper-parameter__, that is constant that multiplies the regularization term. The higher the value, the stronger the regularization. Also used to compute the learning rate when set to *learning_rate* is set to *'optimal'*, as was here, we can notice that the final choice through the different trials was more or less tha same, meanning that the adopted kernel trick for performing kernel-Pca does not affected appreciably such a hyper-param, which three cases out of five was set to *0.1*, and the remaining case adopted *0.0001*, *0.001* for respectively Cosine and Sigmoid based *kernel-Pca*. This also remind us that while training the classifiers was not necessary to force a high regularization contribute for reducing the overfit as well as the learning process, even if we know that *Rbf kernel Pca based Sgd Classifier* overfits mostly against train data, and gained weights that encourages predicting all samples as belonging to class 1.\n\n- reviewing __class_weight hyper-param__, what we can state about such a parameter is that it represents weights associated with classes. If not given, all classes are supposed to have weight one. The *\u201cbalanced\u201d mode* uses the values of y to automatically adjust weights inversely proportional to class frequencies in the input data as __n_samples / (n_classes * np.bincount(y))__. In particular we can notice that three out five models that were fine tuned accepted or selected *balanced weights*, which are *Linear-, Sigomoid-, Cosine-kernel Pca based Sgd Classifier*, while the remaining obtain better, when setting uniform weights which are models *Polynomial-, Rbf-kernel Pca based Sgd Classifier*. So the choiche of the right *kernel-trick* affected the subsequent selection at fine tuning time of the *class_weight hyper-param*. What we can further notice is that *Polynomial-, Rbf-kernel Pca based Sgd Classifier* more or less adopted same kind of values for hyper-params, as an instance for penalty hyper-param, however Polynomial model got worst performance in terms of accuracy but considering the other metrics simultaneously we can understand that the Poly model overfits less than Rbf one and so get better performance in general.\n\n- speaking of __learning_rate hyper-param__, since we force this as the unique available choice it was just report for completeness.\n\n- interesting it is the discussion about __loss parameter__, if fact we know that the possible options are *\u2018hinge\u2019, \u2018log\u2019, \u2018modified_huber\u2019, \u2018squared_hinge\u2019, \u2018perceptron\u2019*, where the *\u2018log\u2019 loss* gives logistic regression, a probabilistic classifier. *\u2018modified_huber\u2019* is another smooth loss that brings tolerance to outliers as well as probability estimates. *\u2018squared_hinge\u2019* is like hinge but is quadratically penalized. \u2018perceptron\u2019 is the linear loss used by the perceptron algorithm. Here, speaking about loss parameter we can clearly understand that the choice of a particular kernel trick does not affect the following choice of the loss function to be optimized, in fact uniformly all the models adopted or tend to prefer *modified_huber* loss function, allowing the models to fit to the data taking into account the fact that such a loss function is less sensitive to outliers, recalling inn fact that the Huber loss function is used in robust statistics, M-estimation and additive modelling. This loss is so cllaed beacues it derives from the plain version normally exploited for regression problems.\n\n- also when referring to __max iteration parameter__, we can easily say that thte models evenly adopted somewhat small number of iteration before stopping the learning procedure, this might be also becaues we work with a small dataset and so a set of data points that is small tend to overfit quickly and this migth be the reason for which in order to avoid too much overfit the training procedure performed employing grid-search technique for fine-tuning tend to prefer tiny number of iterations set for training the model.\n\n- __penalty parameter__, we recall here that it represents regularization term to be used. More precisely, defaults to *\u2018l2\u2019* which is the standard regularizer for linear SVM models. *\u2018l1\u2019* and *\u2018elasticnet\u2019* might bring *sparsity* to the model (feature selection) not achievable with *\u2018l2\u2019*. Also for such a hyper-param the choice of a particular *kernel-trick* to be used for *kernel-Pca* was affecting the subsequent selection of penalty contribute to regularize learning task, as was for *class weight hyper-param*. Here three over five models that are *Linear-, Sigomoid-, Cosine-kernel Pca based Sgd Classifier* adopted *l1-norm* as regularization term so the models's weights tend to be more sparse, while the remaining *Polynomial-, Rbf-kernel Pca based Sgd Classifier* models adopted *l2-nrom*. For the trials we have done, the models with *l1-regularization term* seem to get worst performance, more precisely *Sigomoid-, Cosine-kernel Pca based Sgd Classifier* even were worser than random classifier, while the *Linear-kernel Pca based Sgd Classifier* was slightly worst than Polynomial one, so does not overfit too much however we can say it can be exploited for ensemble method that follows a Boosting Policy.\n\nIf we imagine to build up an *Ensemble Classifier* from the family of *Average Methods*, which state that the underlying principle leading their creation requires to build separate and single classifiers than averaging their prediction in regression context or adopting a majority vote strategy for the classification context, we can claim that amongst the purposed *Sgd classifier*, for sure, we could employ the classifier found from all the trials, except for *Rbf, Cosine and Sigmoid kernel Pca based Sgd Classifiers*, since the first model is overly overfitting to the data used at train time and more precisely most of the time predicted correctly just samples from class 1 and misclassifyes instances from class 0, the others instead assumed the opposite behavior. Also, because of their performance metrics and also because Ensemble Methods such as Bagging Classifier, usually work fine exploiting an ensemble of independent and fine tuned classifier differently from Boosting Methods which instead are based on weak learners.\n\n## Support Vector Machines Classifier\n\n\n\n| Learning Technique | Type of Learner | Type of Learning | Classification | Regression | Clustering | Outlier Detection \n| --- | --- | --- | --- | --- | --- | --- |\n| *Support Vector Machines(SVMs)* | *Discriminative Model* | *Supervised Learning*| *Supported* | *Supported* | *Not-Supported* | *Supported* |\n\nHere, in this section I'm going to exploit a machine learning techinique known as Support Vectors Machines in order to detect and so select the best model I can produce throughout the usage of data points contained within the dataset at hand. So let discuss a bit those kind of classifiers.\n\nIn machine learning, **support-vector machines**, shortly SVMs, are *supervised learning models* with associated learning algorithms that analyze data used for classification and regression analysis. Given a set of training examples, each marked as belonging to one or the other of two categories, an SVM training algorithm builds a model that assigns new examples to one category or the other, making it a *non-probabilistic binary linear classifier*. An SVM model is a representation of the examples as points in space, mapped so that the examples of the separate categories are divided by a clear gap that is as wide as possible. New examples are then mapped into that same space and predicted to belong to a category based on the side of the gap on which they fall.\n\nMore formally, a support-vector machine constructs a hyperplane or set of hyperplanes in a high-dimensional space, which can be used for classification, regression. Intuitively, a good separation is achieved by the hyperplane that has the largest distance to the nearest training-data point of any class, so-called *functional margin*, since in general the larger the margin, the lower the *generalization error* of the classifier.\n\n#### Mathematical formulation of SVMs\nHere, I'm going to describe the main mathematical properties and characteristics used to derive from a math point of view the algorithm derived and proven by researches when they have studied SVMs classifiers.\n\nI start saying and recalling that A support vector machine constructs a hyper-plane or set of hyper-planes in a high or infinite dimensional space, which can be used for classification, regression or other tasks. Intuitively, a good separation is achieved by the hyper-plane that has the largest distance to the nearest training data points of any class so-called functional margin, since in general the larger the margin the lower the generalization error of the classifier.\n\nWhen demonstrating SVMs classifier algorithm I suppose that We are given a training dataset of *n*n points of the form:\n\n\\begin{align}\n(\\vec{x_1} y_1),..,(\\vec{x_n},y_n)\n\\end{align}\n\nwhere the $y_{1}$ are either 1 or \u22121, each indicating the class to which the point $\\vec{x}_{i}$ belongs. Each $\\vec{x}_{i}$is a *p-dimensional real vector*. We want to find the \"maximum-margin hyperplane\" that divides the group of points $\\vec{x}_{i}$ for which $y_{1}$ = 1from the group of points for which $y_{1}$ = \u2212 1, which is defined so that the distance between the hyperplane and the nearest point $\\vec{x}_{i}$ from either group is maximized.\n\nAny hyperplane can be written as the set of points $\\vec{x}_{i}$ satisfying : $\\vec{w}_{i}\\cdot{\\vec{x}_{i}} - b = 0$; where $\\vec{w}_{i}$ is the, even if not necessarily, normal vector to the hyperplane. The parameter $\\tfrac {b}{\\|\\vec{w}\\|}$ determines the offset of the hyperplane from the origin along the normal vector $\\vec{x}_{i}$.\n\nArrived so far, I have to distiguish between two distinct cases which both depende on the nature of data points that generally made up a given dataset. Those two different cases are called *Hard-Margin* and *Soft Margin* and, respectively.\n\nThe first case, so the ***Hard-Margin*** case, happens just for really optimistics datasets. In fact it is the case when the training data is linearly separable, hence, we can select two parallel hyperplanes that separate the two classes of data, so that the distance between them is as large as possible. The region bounded by these two hyperplanes is called the \"margin\", and the maximum-margin hyperplane is the hyperplane that lies halfway between them. With a normalized or standardized dataset, these hyperplanes can be described by the equations:\n- $\\vec{w}_{i}\\cdot{\\vec{x}_{i}} - b = 1$, that is anything on or above this boundary is of one class, with label 1;\n- and, $\\vec{w}_{i}\\cdot{\\vec{x}_{i}} - b = -1$, that is anything on or above this boundary is of one class, with label -1.\n\nWe can notice also that the distance between these two hyperplanes is ${\\displaystyle {\\tfrac {2}{\\|{\\vec {w}}\\|}}}$ so to maximize the distance between the planes we want to minimize \u2016 w \u2192 \u2016 {\\displaystyle \\|{\\vec {w}}\\|} \\|{\\vec {w}}\\|. The distance is computed using the distance from a point to a plane equation. We also have to prevent data points from falling into the margin, we add the following constraint: for each *i*:\n- either, ${\\displaystyle {\\vec {w}}\\cdot {\\vec {x}}_{i}-b\\geq 1}$, ${\\displaystyle y_{i}=1}$;\n- or, ${\\displaystyle {\\vec {w}}\\cdot {\\vec {x}}_{i}-b\\leq -1}$, if ${\\displaystyle y_{i}=-1}$.\n\nThese constraints state that each data point must lie on the correct side of the margin.\nFinally, we collect all the previous observations and define the following optimization problem:\n- from $y_{i}(\\vec{w}_{i}\\cdot{\\vec{x}_{i}} - b) \\geq 1$, for all $1 \\leq i \\leq n$;\n- to minimize ${\\displaystyle y_{i}({\\vec {w}}\\cdot {\\vec {x}}_{i}-b)\\geq 1}$ ${\\displaystyle i=1,\\ldots ,n}$.\n\nThe classifier we obtain is made from ${\\vec {w}}$ and ${\\displaystyle b}$ that solve this problem, and he max-margin hyperplane is completely determined by those ${\\vec {x}}_{i}$ that lie nearest to it. These $\\vec{x}_{i}$ are called *support vectors*.\n\nThe other case, so the ***Soft-Margin*** case, convercely happens when the training data is not linearly separable. To deal with such situation, as well as, to extend SVM to cases in which the data are not linearly separable, we introduce the hinge loss function, that is: $max(y_{i}(\\vec{w}_{i}\\cdot{\\vec{x}_{i}} - b))$.\nOnce we have provided the new loss function we go ahead with the new optimization problem that we aim at minimizing that is:\n\n\\begin{align}\n{\\displaystyle \\left[{\\frac {1}{n}}\\sum _{i=1}^{n}\\max \\left(0,1-y_{i}({\\vec {w}}\\cdot {\\vec {x}}_{i}-b)\\right)\\right]+\\lambda \\lVert {\\vec {w}}\\rVert ^{2},}\n\\end{align}\n\nwhere the parameter \\lambda determines the trade-off between increasing the margin size and ensuring that the ${\\vec {x}}_{i}$ lie on the correct side of the margin. Thus, for sufficiently small values of\\lambda , the second term in the loss function will become negligible, hence, it will behave similar to the hard-margin SVM, if the input data are linearly classifiable, but will still learn if a classification rule is viable or not.\n\nWhat we notice from last equations written just above is that we deal with a quadratic programming problem, and its solution is provided, detailed below.\n\nWe start defining a *Primal Problem* as follow:\n- For each $\\{1,\\,\\ldots ,\\,n\\}$ we introduce a variable ${\\displaystyle \\zeta _{i}=\\max \\left(0,1-y_{i}(w\\cdot x_{i}-b)\\right)}$. Note that ${\\displaystyle \\zeta _{i}}$ is the smallest nonnegative number satisfying ${\\displaystyle y_{i}(w\\cdot x_{i}-b)\\geq 1-\\zeta _{i}}$;\n- we can rewrite the optimization problem as follows: ${\\displaystyle {\\text{minimize }}{\\frac {1}{n}}\\sum _{i=1}^{n}\\zeta _{i}+\\lambda \\|w\\|^{2}}$, ${\\displaystyle {\\text{subject to }}y_{i}(w\\cdot x_{i}-b)\\geq 1-\\zeta _{i}\\,{\\text{ and }}\\,\\zeta _{i}\\geq 0,\\,{\\text{for all }}i.}$\n\nHowever, by solving for the *Lagrangian dual* of the above problem, one obtains the simplified problem:\n\\begin{align}\n {\\displaystyle {\\text{maximize}}\\,\\,f(c_{1}\\ldots c_{n})=\\sum _{i=1}^{n}c_{i}-{\\frac {1}{2}}\\sum _{i=1}^{n}\\sum _{j=1}^{n}y_{i}c_{i}(x_{i}\\cdot x_{j})y_{j}c_{j},} \\\\\n{\\displaystyle {\\text{subject to }}\\sum _{i=1}^{n}c_{i}y_{i}=0,\\,{\\text{and }}0\\leq c_{i}\\leq {\\frac {1}{2n\\lambda }}\\;{\\text{for all }}i.} \n\\end{align}\n- moreover, the variables $c_i$ are defined as ${\\displaystyle {\\vec {w}}=\\sum _{i=1}^{n}c_{i}y_{i}{\\vec {x}}_{i}}$. Where, ${\\displaystyle c_{i}=0}$ exactly when ${\\displaystyle {\\vec {x}}_{i}}$ lies on the correct side of the margin, and ${\\displaystyle 00}$ and ${\\displaystyle c<0}$.\n\nWhat follows is the application or use of SVMs classifier for learning a model that best fit the training data in order to be able to classify new instance in a reliable way, selecting the most promising model trained.\n\n### Cross-Validation Result\n\n\n```python\n# RandomForestClassifier\n# -----------------------------------\npos_cv = pos_cv + 1; show_df_with_mean_at_bottom(dfs_list[pos_cv]) # dfs_list[pos_cv].head(dfs_list[pos_cv].shape[0])\n```\n\n\n```python\nplot_name = plots_names[pos_cv]\nshow_learning_curve(dfs_list[pos_cv], n=len(cv_list[:N_CV]), figsize=(15, 7), plot_dest=plot_dest, grid_size=grid_size, plot_name=plot_name)\n```\n\n### Grid-Search Result\n\n\n```python\npos_gs = pos_gs + 1; plot_dest = os.path.join(\"figures\", \"n_comp_9_analysis\", \"grid_search\"); X = copy.deepcopy(rescaledX)\n\ndf_gs, df_auc_gs, df_pvalue = grid_search_all_by_n_components(\n estimators_list=estimators_list[pos_gs+1], \\\n param_grids=param_grids[pos_gs],\n estimators_names=estimators_names[pos_gs+1], \\\n X=X, y=y,\n n_components=9,\n random_state=0, show_plots=False, show_errors=False, verbose=1, plot_dest=plot_dest, debug_var=False)\ndf_9 = merge_dfs_by_common_columns(df_9, df_gs); df_9_auc = merge_dfs_by_common_columns(df_9_auc, df_auc_gs) # # df_9, df_9_auc = pd.concat([df_9, df_gs], axis=0), pd.concat([df_9_auc, df_auc_gs], axis=0)\n```\n\nGenerally speaking, looking at results coming from the trials we have carried out with grid search algorithm for the purpose of fine tuning the SVM classifier adopted to fit models against the dataset at hand, we can claim that only three models out of five gives us important, significant performance, while oen model even if corresponds to that with the best Auc score is not satisfying our main goal we have in mind, that is correctly classify the majority instances from both classes, lastly the remaing classifier that corresponds to poly-tirkc kernel Pca based classifier is that with worst performance in terms of measurement metrics. To be more precise we can say what follows:\n- looking at __Linear-trick kenrle-Pca based Svm Model__, we can state that it belongs to the set of three classifier we consider satisfying results for such a classifier, since it reaches an Auc score equals to .74, however discussing the Roc Curve we can observe that it starts with a higher slope and then ends with a lower slope for describing the relation between the Sensitivity and 1-Specificity scores. Moreover, the model obtain high values for both precision and recall related to class 1, but the recall of class 0 was higher than the precision of the same class, so we get troubles when classifying as class 0 new instances since we are less sure about the result.\n- focusing on __Poly-trick kenrle-Pca based Svm Classifier__, this trials corresponds to the worst classifier gathered from grid-search approach when fine tuning such model because the setting adopted for the model leads to a Auc score lower than the random class, so this model should be discarded, even if seems to get good performance when looking at calculated classification report.\n- speaking of __Rbf-trick kenrle-Pca based Svm Classifier__, here we can state that the fianl model leads to a very well performing roc curve, in fact the Auc score is even .73, one of the highes found, and also the interval of thresholds before the TPR and FPR scores begin to linearly increas is very wide. However the classifier with a default threshold of .5, seems not to perform adequately, since it classify correctly most of the samples from class 1 but wrongly predicts class label for samples belonging to class 0, meaning it was a model with high recall but low precision for for class 1 and very poor performance generally speaking for class 0 related metrics.\n- also the __Cosine-trick kenrle-Pca based Svm Classifier__ shows the same issues more or less as the previous classifier, in other words, the model was well performing for class 1 with high recall and precision but really bad performing for precision and recall related to class 0. Thus, looking to Roc Curve and Auc Score, we can say that the shape of Roc Curve shows a model which in the first part has a higher slope that describes how the TPR and FPR increases with the different thresholds, while int the second half the steepness of the slope reduces, however the Auc score is not soo good, just .67.\n- Finally, discussing the __Sigmoid-trick kenrle-Pca based Svm Classifier__, we can observe that with a default threshold of 0.5 the model was able to correctly classify all the instances belonging to class 0 leading to high recall as well as high precisoin for class 0 and class 1 respectively, however, both precision and recall of class 0 and class 1 was convercely very low, this means tha the model is very sure when classifyng instances as belonging to class 1 but is almost unsure when facing an instance that virtually belongs to class 0. Focusing on Roc Curve the model has a tiny range of thresholds where the model increases the Sensitivity without changing the 1-Specificity, but, at a given point the remaing part of roc curve seems to follow a linear curve for describing the relation beween TPR and FPR fractions, leading the model to record a Auc score of .7.\n\n__Significance Analysis__: finally, when looking at the different graphics related to the test which aims at investigating the diagnostic power of our different models we have fine tuned for *SGD Classifier*, picking the best one for such a test we can notice that beacues of the *signficance level* $\\alpha$ set equal to *0.05 that is 5% of chance to reject the Null-Hypothesis $H_{0}$*, we have obtained following results.\n\n#### Table Fine Tuned Hyper-Params(SVMs Classifier)\n\n\n```python\nshow_table_summary_grid_search(df_gs, df_auc_gs, df_pvalue)\n```\n\nLooking at the table dispalyed just above, referred to the found hyper-params values identifyed by grid search procedure that attempt to fine tuning the Svm Classifier combining it with different possbile kernel tricks for kernel-Pca unsupervised learning technique, we can clearly understand that the accuracy results gained by the best models retrieved by grid-search procedures corresponds to value near 90 percent of accuracy.However, the choice of kenrel trick affected importantly and widely the identification of proper values for hyper-params such as *C, gamma, and kernel-trick related to Svm Classifier*.\n\nMore precisely, we can say that for *C hyper-param* we go across a wide range of values from 0.0001 up to 10 and except to Rbf and Cosine tricks adopted for kernel-Pca, there are no other models that share the same value of hyper-param C. What we can further note is that as the kernel-trick adopted to performe kernel-Pca becomes more fancy and complicated from the viewpoint of math formulation the value assumend by *C hyper-param* seems to become greater and greater, where we recall that the *parameter C* controls the trade off between errors of the SVM on training data and margin maximization. So as we increase the complexity of kernel-trick for kernel-Pca we observe that the resulting marigns becomes more and more hard.\n\nInstead, if we think intuitively of the *gamma parameter* as that parameter which defines how far the influence of a single training example reaches, with low values meaning *'far'* and high values meaning *'close'*, then we can see a trend for which as the margin becomes harder the influence of the single seems to be closer.\n\nFinally, looking at kernel trick chosen from different models we observe that in the most of the cases the polynomial kernel was the best choice and only sigmoid trick was adopted when also the kernel-Pca technique was performed exploiting the same kind of trick, as well as the linear trick was selected as the best choice combining with a kernel-Pca that instead adopted a Rbf-kernel.\n\n#### Advantages and Backwards of SVMs\n\nFinally, I conclude this section providing a description of major advantages and backwards of such a machine learning technique, that have been noticed by researches who studied SVMs properties. The advantages of support vector machines are:\n\n- Effective in high dimensional spaces.\n- Still effective in cases where number of dimensions is greater than the number of samples.\n- Uses a subset of training points in the decision function (called support vectors), so it is also memory efficient.\n- Versatile: different Kernel functions can be specified for the decision function. Common kernels are provided, but it is also possible to specify custom kernels.\n\nOn the other and, the disadvantages of support vector machines include:\n\n- If the number of features is much greater than the number of samples, avoid over-fitting in choosing Kernel functions and regularization term is crucial.\n- SVMs do not directly provide probability estimates, these are calculated using an expensive five-fold cross-validation (see Scores and probabilities, below).\n\n## Decision Tree Models\n| Learning Technique | Type of Learner | Type of Learning | Classification | Regression | Clustering | Outlier Detection \n| --- | --- | --- | --- | --- | --- | --- |\n| *Decision Trees* | *Non-parametric Model* | *Supervised Learning*| *Supported* | *Supported* | *Not-Supported* | *Not-Supported* |\n\nDecision Trees, for short DTs, are a *non-parametric supervised learning method* used for classification and regression. The goal is to create a model that predicts the value of a target variable by learning simple decision rules inferred from the data features.\n\nTheir mathematical formulation is generally provided as follows: Given training vectors $x_{i} \\in R^{n}$, $i=1,\u2026, l$ and a label vector $y \\in R^{l}$, a decision tree recursively partitions the space such that the samples with the same labels are grouped together.\n\nLet the data at node $m$ be represented by $Q$. For each candidate split $\\theta = (j, t_{m})$\nconsisting of a feature $j$ and threshold $t_{m}$, partition the data into $Q_{left}(\\theta)$ and $Q_{right}(\\theta)$ subsets as:\n\n\\begin{align}\\begin{aligned}Q_{left}(\\theta) = {(x, y) | x_j <= t_m}\\\\Q_{right}(\\theta) = Q \\setminus Q_{left}(\\theta)\\end{aligned}\\end{align}\n\nThe impurity at $m$ is computed using an impurity function $H()$, the choice of which depends on the task being solved (classification or regression) like:\n\n\\begin{align}\nG(Q, \\theta) = \\frac{n_{left}}{N_m} H(Q_{left}(\\theta)) + \\frac{n_{right}}{N_m} H(Q_{right}(\\theta))\n\\end{align}\n\nSelect the parameters that minimises the impurity: $\\theta^* = \\operatorname{argmin}_\\theta G(Q, \\theta)$.\n\nRecurse for subsets $Q_{left}(\\theta^*)$ and $Q_{right}(\\theta^*)$ until the maximum allowable depth is reached,\n$N_m < \\min_{samples}$ or N_m = 1.\n\nSpeaking about *Classification Criteria* referred to the procedure used for learining or fit to the data a decision tree we can state what follows: If a target is a classification outcome taking on values $0,1,\u2026,K-1$, for node $m$, representing a region $R_{m}$ with $N_{m}$ observations, let $p_{mk} = 1/ N_m \\sum_{x_i \\in R_m} I(y_i = k)$ be the proportion of class $k$ observations in node $m$.\n\nSo, Common measures of impurity are:\n- Gini, specified as $H(X_m) = \\sum_k p_{mk} (1 - p_{mk})$\n\n\n- Entropy, definead as $(X_m) = - \\sum_k p_{mk} \\log(p_{mk})$\n\nwhere, we recall that $X_{m}$ is the training data in node $m$.\n\n\n```python\npos_cv = pos_cv +1; show_df_with_mean_at_bottom(dfs_list[pos_cv]) # dfs_list[pos_cv].head(dfs_list[pos_cv].shape[0])\n```\n\n\n```python\nplot_name = plots_names[pos_cv]\nshow_learning_curve(dfs_list[pos_cv], n=len(cv_list[:N_CV]), figsize=(15, 7), plot_dest=plot_dest, grid_size=grid_size, plot_name=plot_name)\n```\n\n### Grid Search Result\n\n\n```python\npos_gs = pos_gs + 1; plot_dest = os.path.join(\"figures\", \"n_comp_9_analysis\", \"grid_search\"); X = copy.deepcopy(rescaledX)\n\ndf_gs, df_auc_gs, df_pvalue = grid_search_all_by_n_components(\n estimators_list=estimators_list[pos_gs+1], \\\n param_grids=param_grids[pos_gs],\n estimators_names=estimators_names[pos_gs+1], \\\n X=X, y=y,\n n_components=9,\n random_state=0, show_plots=False, show_errors=False, verbose=1, plot_dest=plot_dest, debug_var=False)\ndf_9 = merge_dfs_by_common_columns(df_9, df_gs); df_9_auc = merge_dfs_by_common_columns(df_9_auc, df_auc_gs) # df_9, df_9_auc = pd.concat([df_9, df_gs], axis=0), pd.concat([df_9_auc, df_auc_gs], axis=0)\n```\n\nLooking at the results obtained running grid-search algorithm applied to Decision Tree Classifier we can rougly saying that the different models will obtain performances with default classification thresholds that are not as good as the results obtained from previous models, also because the models do not show Roc Curve along with their corresponding Auc scores that enable us saying that such models can perform well during inference also varying the default threshold. In particular we can say what follows:\n- looking at __Linear kernel-Pca based Decision Tree Classifier__, we notice that with the default threshold the model obtain high precision and recall for class 1 examples, meaning it is able to correctly classify most of the samples from class 1 as well as few examples from class 0 are exchanged as belonging to class 1. However speaking about class 0 we notice that we have obtained a very low precision and a 50 percent of recall that means that the model with default threshold misclassifyes half of the samples from class 0 and we are not really sure that what we have classified as the class 0 instance it really belong to class 0. Finally looking at Roc Curve we can observe that even from the very beginning the Sensitivity and 1-Specificity grow linearly with a slope value sligthly bigger than the slope of referring curve represented by Random Classifier, however at a given point the slope changes and as the thresholds approaches to higher values and the curve approaches to the top the slope decreases importantly, instead the value of Auc score account for 0.62.\n\n- while looking at __Poly kernel-Pca based Decision Tree Model__, we can clearly understand that such a classifier is not good enough to be exploited for further inferences since it accounts for just .58 value of Auc score and observing Roc Curve we can conclude that it goes slightly better than the curve provided by random classifier. Moreover, the model when we adopt the default threshold seems to correctly classify most of the istances from class 0, but wrongly predict the class for instances of the opposite categories, in fact it is characterized from law precision referred to class 0, and since we want to correctly predict labels for both categories, here with such a classifier we are not able to satisfy such a constraint.\n\n- the classifier corresponding to a __Rbf kernel-Pca based Decision Tree__, here is the model which leads to the worst performances, since the roc curve graphics is even worst than the random classifier and the roc curve accounts for a Auc score that is less than .5, more precisely just .44. So this result will be discarded, even if the model seems to correctly recognize samples from class 1 but wrongly predict labels for the class 0 samples, and again also here we have to state that we are not able to meet the constraint of correctly classify most of the data examples as we expect from a well defined classifier.\n\n- referring to __Cosine kernel-Pca based Decision Tree Classifier__, when adopting a default threshold we notice that the model even if correctly classifyes all samples from class 0 leading to high recall for such a category, we can say also that the model has a low precision for class 0, meaning that the model confuses many samples from class 1 in fact is characterized from low value of recalll for the class 1, however when predicts a label equals to category one it is almost always sure about the choice. Thus, looking at Roc Curve and Auc score we can note that the model is characterized from a firt phase in which the Sensitivityu and 1-Specificity are not growing linearly following a line with a slope even lower than the one of Random Classifier as the preivous models, accounting for a AUC score equals to *45%*, we cannot neither decided to switch the labels for trying to train another classifier with such a new configuration, since the AUC does not suggest us to follow this new available and well-knwon strategy.\n\n- Lastly the __Sigmoid kernel-Pca based Decision Tree Classifier__, with a default threshold of .5 for classification shows peformance scores that are more or less analog to those seen previously for *Rbf and Cosine kernel-Pca based Decision Tree Models*. The only difference is that the model get a AUC score much closer to that of Random Classifier. Again, also this model is able to predict with high recall the samples belonging to class 1, instead wrongly predict class labels for those samples whcih come from class 0.\n\n__Significance Analysis__: finally, when looking at the different graphics related to the test which aims at investigating the diagnostic power of our different models we have fine tuned for *SGD Classifier*, picking the best one for such a test we can notice that beacues of the *signficance level* $\\alpha$ set equal to *0.05 that is 5% of chance to reject the Null-Hypothesis $H_{0}$*, we have obtained following results. Adopting the Decision Trees statistical learning technique for classification fine tuned as above with hyper-params selectd also depending on the kind of *kernel-trick adopted for kernel-Pca unsupervised technique*, we can claim that only two out of five trials lead to a *p-vlaue* worst than *selected significance level equal to 5%*, which are *Rbf- and Cosine-kernel Pca based Sgd Classifier*, so rejecting the *Null-Hypotesis* for those two cases will results into a *Type I Error*. While the remaining three cases, that are *Linear Poly-, and Sigmoid-kernel Pca based Sgd Classifier* have obtained a p-value over the range $[15, 90] \\in R$ *in percet points*. Holthough we have at least two out of five classifier fine tuned that seem to allow us reject the *Null-Hypothesis* in order to justify the usae of such a models so fine-tuned and that accept the weights and hyper-parameters values we have discovered we end up saying that there is no one of the previous models that for sure we will accept to employ for inference and classification tasks due to their worst performance, rougly speaking. In other words, it seems that *Decision Tree Classification technique* does not work fine with such a small, unbalanced dataset.\n\n#### Table Fine Tuned Hyper-Params(Decision Trees Classifier)\n\n\n```python\nshow_table_summary_grid_search(df_gs, df_auc_gs, df_pvalue)\n```\n\nLooking at the table shown above and obtained running grid-search algorithm applied to Decision Tree Classifier, only three out of five classifier get significant accuracy values, and those are Decision Tree created by means of *Linear, Cosine Sigmoid tricks adopted for kernel-Pca*, since we reaches accuracy up to *71%, 74%, and 74%* respectively. However, the worst classifier reveals to be *Polynomial based kernel Pca Decision Tree* with an accuracy on test set that reaches just *41%*, lastly the *Sigmoid based kernel Pca Decision Tree* was slightly worst than the previous three classifier that we have told were the best models, since the latter *Sigmoid based kernel Pca Decision Tree* has reached an accuracy score just 10 percent points less than the previous three, that is *62%*, however, not enough to consider it a good enough classifier. Looking at the hyper-parameters results shown from the summary table included in the report just above for DecisionTree Classification algorithm we can explain those results as follows:\n\n- referring to the **class_weight hyper-param**, which represents weights associated with classes and if we disable such a setting so that it is valued to None, all classes are supposed to have weight one. Instead the *\u201cbalanced\u201d* mode uses the values of y to automatically adjust weights inversely proportional to class frequencies in the input data as n_samples / (n_classes * np.bincount(y)). We can notice that the choice of whether adopting a weighted strategy as balanced for performing training phase was chosen as best hyper-param value for *class_weight param* from just two out of five resulting fine tuned classifiers which were *Rbf, Sigmoid kernel-Pca based Decision Tree Classifier*, while the remaining adopted a uniform strategy. However discarding the worst classifier such a parameter half of the time was chose to be balanced and the remaining time to be uniform. In other words the choice of the kernel-trick at preprocessing time affected later the choice at grid-search training for *class_weight hyper-param*.\n\n- reviewing **criterion decision trees' param**, which stands for the function to measure the quality of a split, where supported criteria are *\u201cgini\u201d for the Gini impurity and \u201centropy\u201d for the information gain*. The most chosen *measure the quality* was the *gini for the Gini impurity*, in fact *Linear, Rbf, Sigmoid kernel-Pca based Decision Tree Classifier* selected this technique, while the remaining fine tuned classifiers *Poly and Cosine kernel-Pca based Decision Tree Classifiers* take advantages from exploiting *\u201centropy\u201d for the information gain*. Again the choice at pre-processing time of a specific kernel-trick amongst those available for kernel-Pca unsupervised procedure leads to a particular *criterion* adopted from the trees based estimator when building the trees strucuture.\n\n- looking at **max_depth trees' param**, which stands for the maximum depth of the tree. If None, then nodes are expanded until all leaves are pure or until all leaves contain less than *min_samples_split samples*, we clearly understand that for such a unbalanced and small dataset the best strategy while building the trees data strucutre was to keep growing the trees until all leaves are pure or until all leaves contain less than *min_samples_split samples*. In other words the hyper-param was set with a None value suggesting not to stop growing the trees at a given point, but rather to expand as much as possible. Only for *Cosine kernel-Pca based Decision Tree Classifiers* the best max_depth value for the hyper-parameter involved in this analysis was set to a really small size, just three nodes, which means that the preprocessed data via Cosine kernel-trick, does need to* much attributes to be taken into account before arriving into a leaf node.\n\n- also describing **max_features hyper-parameter**, that is, the number of features to consider when looking for the best split, where if *\u201csqrt\u201d, then max_features=sqrt(n_features)*, *if \u201clog2\u201d, then max_features=log2(n_features)*, and *if None, then max_features=n_features,* we can notice that the initial choice of the kernel-trick for pre-processing the data points within the dataset do not affect the final selection about the right technique to be adopted for calculating the number of features to be considered when looking for the best split. Moreover, since we are dealing with a small dataset, unbalanced with respect the class labels and also with a small number of features, we can reasonable understand that in most of the cases the classifier get better performance scores just considering a number of features equal to the available features after having performed the kernel-Pca algorithm. Only *Rbf kernel-Pca based Decision Tree Classifiers* selected a strategy tgat corresponds to *\u201csqrt\u201d, then max_features=sqrt(n_features)* for deciding the features to be identifyed for the best split.\n\n- lastly, when speaking about the strategy used to choose the split at each node, where supported strategies are *\u201cbest\u201d to choose the best split and \u201crandom\u201d to choose the best random split*, we are saying that we are referring to **splitter hyper-parameter**. Here, for the trails we have carryed out, what we can say about such a hyper-param, is that since we are dealing with a small dataset, unbalanced and with not so much large number of features, also after having preprocessed it and discarded some useless features, is that the diverese fine tuned models in most of the cases adopted a best strategy, which requires more trainign time, while just for a single case corresponding to *Cosine kernel-Pca based Decision Tree Classifier*, the retrieved model seems to go better when adopint a random strategy.\n\nThe choice of kernel-Pca with a specific kernel-trick was decisive and affects the hyper-partameters set for building the If we imagine to build up an *Ensemble Classifier* from the family of *Average Methods*, which state that the underlying principle leading their creation requires to build separate and single classifiers than averaging their prediction in regression context or adopting a majority vote strategy for the classification context, we can claim that amongst the purposed decision trees classifier, for sure, we could employ the classifiers found from the __Linear, Rbf, Cosine and Sigmoid kernel-Pca based Decision Tree Classifier__ because of their performance metrics and also because Ensemble Methods such as Bagging Classifier, usually work fine exploiting an ensemble of independent and fine tuned classifier differently from Boosting Methods which instead are based on weak learners.\n\n#### Decision Tree's Advantages & Bacwards\n\nSome advantages of decision trees are:\n\n- Simple to understand and to interpret. Trees can be visualised.\n- Requires little data preparation. Other techniques often require data normalisation, dummy variables need to be created and blank values to be removed. Note however that this module does not support missing values.\n- The cost of using the tree (i.e., predicting data) is logarithmic in the number of data points used to train the tree.\n- Able to handle both numerical and categorical data. Other techniques are usually specialised in analysing datasets that have only one type of variable. See algorithms for more information.\n- Able to handle multi-output problems.\n\n- Uses a white box model. If a given situation is observable in a model, the explanation for the condition is easily explained by boolean logic. By contrast, in a black box model (e.g., in an artificial neural network), results may be more difficult to interpret.\n- Possible to validate a model using statistical tests. That makes it possible to account for the reliability of the model.\n- Performs well even if its assumptions are somewhat violated by the true model from which the data were generated.\n\nThe disadvantages of decision trees include:\n\n- Decision-tree learners can create over-complex trees that do not generalise the data well. This is called overfitting. Mechanisms such as pruning (not currently supported), setting the minimum number of samples required at a leaf node or setting the maximum depth of the tree are necessary to avoid this problem.\n- Decision trees can be unstable because small variations in the data might result in a completely different tree being generated. This problem is mitigated by using decision trees within an ensemble.\n- The problem of learning an optimal decision tree is known to be NP-complete under several aspects of optimality and even for simple concepts. Consequently, practical decision-tree learning algorithms are based on heuristic algorithms such as the greedy algorithm where locally optimal decisions are made at each node. Such algorithms cannot guarantee to return the globally optimal decision tree. This can be mitigated by training multiple trees in an ensemble learner, where the features and samples are randomly sampled with replacement.\n- There are concepts that are hard to learn because decision trees do not express them easily, such as XOR, parity or multiplexer problems.\n- Decision tree learners create biased trees if some classes dominate. It is therefore recommended to balance the dataset prior to fitting with the decision tree.\n\n\n\n## Ensemble methods\n\nThe goal of ensemble methods is to combine the predictions of several base estimators built with a given learning algorithm in order to improve generalizability / robustness over a single estimator.\n\nTwo families of ensemble methods are usually distinguished:\n- In averaging methods, the driving principle is to build several estimators independently and then to average their predictions. On average, the combined estimator is usually better than any of the single base estimator because its variance is reduced. So, some examples are: Bagging methods, Forests of randomized trees, but still exist more classifiers;\n- Instead, in boosting methods, base estimators are built sequentially and one tries to reduce the bias of the combined estimator. The motivation is to combine several weak models to produce a powerful ensemble. Hence, some examples are: AdaBoost, Gradient Tree Boosting,but still exist more options.\n\n## Random Forests\n\n| Learning Technique | Type of Learner | Type of Learning | Classification | Regression | Ensemble Family |\n| --- | --- | --- | --- | --- | --- |\n| *RandomForest* | *Ensemble Method (Meta-Estimator)* | *Supervised Learning* | *Supported* | *Supported* | *Averaging Methods* |\n\nThe **sklearn.ensemble module** includes two averaging algorithms based on randomized decision trees: the RandomForest algorithm and the Extra-Trees method. Both algorithms are perturb-and-combine techniques, specifically designed for trees. This means a diverse set of classifiers is created by introducing randomness in the classifier construction. The prediction of the ensemble is given as the averaged prediction of the individual classifiers.\n\nIn random forests (see RandomForestClassifier and RandomForestRegressor classes), each tree in the ensemble is built from a sample drawn with replacement (i.e., a bootstrap sample) from the training set.\n\nThe main parameters to adjust when using these methods is *number of estimators* and *maxima features*. The former is the number of trees in the forest. The larger the better, but also the longer it will take to compute. In addition, note that results will stop getting significantly better beyond a critical number of trees. The latter is the size of the random subsets of features to consider when splitting a node. The lower the greater the reduction of variance, but also the greater the increase in bias.\n\nEmpirical good default values are maxima features equals to null, that means always considering all features instead of a random subset, for regression problems, and maxima features equals to \"sqrt\", using a random subset of size sqrt(number of features)) for classification tasks, where number of features is the number of features in the data. The best parameter values should always be cross-validated.\n\nWe note that the size of the model with the default parameters is $O( M * N * log (N) )$, where $M$ is the number of trees and $N$ is the number of samples.\n\n### Cross-Validation Result\n\n\n```python\npos_cv = pos_cv + 1; show_df_with_mean_at_bottom(dfs_list[pos_cv]) # dfs_list[pos_cv].head(dfs_list[pos_cv].shape[0])\n```\n\n\n```python\nplot_name = plots_names[pos_cv]\nshow_learning_curve(dfs_list[pos_cv], figsize=(15, 7), n=len(cv_list[:N_CV]), plot_dest=plot_dest, grid_size=grid_size, plot_name=plot_name)\n```\n\n### Grid-Search Result\n\n\n```python\npos_gs = pos_gs + 1; plot_dest = os.path.join(\"figures\", \"n_comp_9_analysis\", \"grid_search\"); X = copy.deepcopy(rescaledX)\n\ndf_gs, df_auc_gs, df_pvalue = grid_search_all_by_n_components(\n estimators_list=estimators_list[pos_gs+1], \\\n param_grids=param_grids[pos_gs],\n estimators_names=estimators_names[pos_gs+1], \\\n X=X, y=y,\n n_components=9,\n random_state=0, show_plots=False, show_errors=False, verbose=1, plot_dest=plot_dest, debug_var=False)\ndf_9 = merge_dfs_by_common_columns(df_9, df_gs); df_9_auc = merge_dfs_by_common_columns(df_9_auc, df_auc_gs) # df_9, df_9_auc = pd.concat([df_9, df_gs], axis=0), pd.concat([df_9_auc, df_auc_gs], axis=0)\n```\n\nThe results collected when running Random Forest Classifier Algorithm as the ensemble method of choice here, by means of grid search approach wer can say, widely speaking, that only two out of five models lead to classifiers which are able to satisy our constrain of correctly classify the most of the samples coming from different classes that are class 0 and class 1. More precisely we can say what follows:\n- Looking at __Linear kenrel-Pca baed Random Forest Classifier__, with a default threshld the model correctly classifyes all instances from class 1 but wrongly predicts labesl for instances from class 0 and so results into a model with a high recall and low precision for class 1, and a low recall as welll as low precision for class 0. Moreover modifying the default threshold we dono not observe looking at Roc Curve an improvement in the classification capability of the model, the evidence is also suggested by the fact that we do not obtainm a Auc score higher than .5 which is the referring score related to the Random Classifier, this means that such a model should be ignored and not employed.\n- instead lookinga at __Poly kernel-Pca based Random Forest Classifier__, with a default threshold of .5 the model allows to correctly classify most of the samples from class 1 but we correctly classify just half of the samples that instead belon to the class 0, so the model has high recall and preicision for class 1 but low precision and a recal of 50 percent for class 0. While lookng at Rocu Curve it shows us that the model for a first range of thresholds has both TPR and FPR metrics that grow linearly with a slope higher than the slope used by lien of Random Classifier, however after a given point the slope changes and decreaes to a value lower than 0.5, meaing that in the first range chanign the threshold the model seems to gain grower TPR than FPR, after the the FPR is increasing more significantly than the TPR and so we do not have to select thresholds too much high. However the model get a Auc Score of .61 that is not so greater than .5 and so also such a model is not really well performing.\n- while, speaking about __Rbf kernel-Pca based Random Forest Classifier___, with a default thresold of .5 allows for a model wich reaches a Auc Score of .78 which is the highest score amongs the Random Forest Classifiers built here in this section. Morevore, it allows us to obtain a model wich correctly classify most of the class 1 instances in fact we get high precision and higfh recall so we also have a model that misclassify few instances from class 0. Alos looking at the Roc Curve we can notice that for a very large set of thresholds the TPR metric grows faster than the FPR metric than the model for high thresholds seems to let FPR to grows much faster than TPR, however this trial leads to a classifier that could be righltly selected for predicting the samples from different classes, because shows us really good performance.\n- referring to the __Cosine kernel-Pca based Random Classifier__, we can quickly say that sucha a classifier is slighlty better than the one obtained still for Random Forest Classifier but with a linear trick for kernel-Pca, in fact the Auc score is higher than the latter model but is just slighly higher also than the Random Classifier, so we end up saying that such a configuration do not lead to a model that we want to exploit for classification. The main reason is that we have wrongly predicted the class label for half of class 0 examples as well as we have wrognly assigned labels to class 1 samples in huge number so that the model results with a low value of precision related to class 0.\n- Lastly, __Sigmoid kernel-Pca based Random Classifier__ with a default .5 threshold leads to model with a very high precision and a middle recall for class 1, instead we obtain a middle recall and low precision for class 0, so we correcly classify more or less a little bit more samples from class 0, but even if the number of wrongly classified examples from class 1 is not huge it is a value comparable to the number of correctly classified claas 0 exmaples so that the precision goes down. Howqever the model allows for a roc curve that obtain a score of .67 the second score for Random Forest Classifier even if the accuracy obtained from the model is only the third amongst Random Forest classifier.\n\n\n```python\nshow_table_summary_grid_search(df_gs, df_auc_gs, df_pvalue)\n```\n\nLooking at the table reported above for Random Forest Classifiers we can noticed that the three out of five classffiers that are linear, polyomial and sigmoid-trick kernel-Pca based models do not adopt a boostrap approach, while Rbf and Cosine trick kernel-Pca based Random Forests decided to adopt such a strategy to get better results during inference. While the gini index criterion was mostly preferred than entropy, in fact the latter was adopted just from sigmoid kernel-Pca based model. Finally the number of estimators adopted from the different models varyes from model to model, in fact we can see that three out of five cases the number of estimators was lower than 10 estimators, while there are two cases in whihc we exploit 50 and even hundered of estiamtors to be able to correctly classify samples.\n\n## Summary Results\n\n### Summary Tables about Analyses done by means of different number of included Pricipal Components\n\n\n```python\n# df_9_, df_12_ = reshape_dfs_acc([df_9, df_12], num_col=N_KERNEL, n_cp_list=[9, 11])\n\n# res = create_widget_list_df_vertical([df_9_, df_9_auc]); display.display(res)\n# res = create_widget_list_df_vertical([df_12_, df_12_auc]); display.display(res)\n```\n\n### Summary Test\n\nHere, in the following section I'm going to emulate a test in which I will test the different possible kinds of kernel trick, in other sense techniques, available for a Principal Component Analysis, shortly PCA, unsupervised statistical learning technique in order to remap the original features into a new N-dimensional reference system by means of the kernel approach adopted during the computation.\n\nOnce the new N-dimensional feature space is available and ready, I will experiment a bounch of selected machine learning methods and procedures applied directly on the first two most informative principal components, that is, also referred to as PCA1 and PCA2, respectively, in order to display a sequence of decision boundaries and contours retrieved after having runned each method on the selected dataset, which has been divided into halves, ofd the same size, and with the same proportion of the two classes of the target variable.\n\nWhat follows is the related code, to the desciption given just above, and the results are also available through several rows of images that represent the contour and decision boundaries obtained thank to the several combinations of PCA's kernel trick and machine learning method for fitting a classifier:\n\n\n```python\nkernel_pca = ['linear', 'poly', 'rbf', 'cosine', 'sigmoid'] # linear, poly, rbf, sigmoid, cosine, precomputed\nscaler_techniques = ['StandardScaler', 'Normalize', 'MinMaxScaler']\n\n# Trying only StandardScaler approach\nerr_list = classifier_comparison_by_pca_kernels(X, y, start_clf=0, stop_clf=10, scaler_technique=scaler_techniques[0], straitified_flag=True, kernels_pca_list=kernel_pca[:], figsize=(27, 9), by_pairs=False, singles=False, verbose=0, record_errors=True, avoid_func=False,)\n```\n\nDescribing the several pictures we have obtained throughout the combination of kernel tricks available for kernelPCA technique together with different supervised machine learning techniques for building classifiers and models, we can end up saying what follows.\n\nLooking at the first picture of each row of graphs, that is those pictures showing just data points witout any kind of decision regions as well as decision boundaries, what we understand is that the data points that are the data examples will group creating different shapes accordingly to the kind of kernel trick adopted for kernelPCA method, in particular, we can immediately see that the two categories, where blue points stands for THROUGH-like bridges while red points for DECK-like bridges, are not equally in numbers, but blue points are the greater among the two, moreover the two categories does not seem to seaprate very well but the picture is crowded with both types of categories that are strctlu closed one another. More precisely we can see that:\n- using *linear* kernel trick for kernelPCA procedure data points seem to be widely spread along vertical axis, and group mostly near the center of the picutre;\n- using *poly* kernel trick for kernelPCA procedure instead data points are mostly clusterd near the left bottom corner where seem to form a straight line and there are few examples on the upper side and some less points on the right side of the sdame picture;\n- while using *rbf* kernel trick for kernelPCA procedure data points seem to spread as the data points represented in the first picture so in the middle of the area but are tightly related so that are less spread along the horizontal axis;\n- when exploiting *cosine* kernel trick for kernelPCA method the data points are widely spread and tend to reach the top of the picture;\n- fianlly, when adopting *sigmoid* kernel trick for kernelPCA we can see that data points are mostly clusterd in the center of the graph.\n\nSpeaking about decision boundaries and decision regions about the selected and fitted to the data machine learning methods, what we can say is the following:\n- Looking at **Nearest-Neighbor Method** graphs for describing decision boundaries and decision regions we notice that in the majority of cases the decision boundaries and decision regions are prominent for the THROUGH-like bridges, sometimes the area referring to DECK-like samples are sourranded by the decision regions of the other class and the transition to the two decision boundaries is very sharp, not easilly describable.\n- While, looking at **Linear SVM Classifier**, and knowing the fact that we are fitting a linear classifier to the data, we are aware and so it's clear that the expected decision regions follow a pattern made from several strips of shifting shades of colors from dark red to dark blue. More precisely, three out of five Linear Svm classifiers, in paritcular those that correspond to classifiers fitted when kernel trick for Kernel PCA was set to *'rbf', 'cosine', 'sigmoid'* respectively and one at a time, show more or less the same pattern, so this classification technique combined with these kernel tricks for KernelPCA seems to behave more or less at the same way. Instead Liner Svm combined with poly kernel trick seems to lead the classifier and the resulting decisin boundaries to follow a symmetric patter with respect to the vertical axis. Finally the first combination of KernelPca and Linear Svm technique, that is linear kernel trick plus linear svm, leads to a less aggressive or finer slope of the linear decision regions. We can end up saying that in the majority of cases the transition from one extreme or edge to the other of the shade of color is smoother and continue with respect to the Nearest-Neighbor Approach.\n- Speaking about **RBF kernel SVM** combined with a preprocessed datset with the various kernel tricks for kernelPca Procedure we can observe that the attempt of finding decision regions on one side advantages the more numerous class that is the class corresponding to those data points classified as THROUGH-like bridges, while penalize the other which is referred to smaller region. However it seems that the classifieris able to correctly classify the data points corresponding to the less numerous class while the data points of the other class sometimes are misclassified more frequently.\n- Looking at classifiers trained by means of **Gaussian Process technique** we can ascertain that decision boundaries and decision regions seem to follow a straight pattern where the data points are mixing the most, while far from the bigger cluster of points that come from both categories the decison boundaries are assuming higher order so that resemble smooth nonlinear curves. In particular while in all other cases the blue region seems to occupy the left side of the graph, sometimes near the bottom and other times near the top-right, for Gaussian Process technique combined with sigmoid kernel trick for kernelPca procedure we observe that the pattern observed above is the opposite.\n- Even if these three methods have different characteristics they seems to lead to or provide more or less, and somehow, resulting decision boundaries and decision regions that follow a similar nature that is regions obtained dividing the available two-dimensional plane into subregions that corresponds to square regioons or alternatively irregular regions that are not corresponding to some kind of curve but rather to segmentation of the available area. These methods are respectively **Deciosn Trees, Random Forests, and Adabosts**. Where the two latter can be seen as a improvement of Decsion Tree because often the two latter are based on the decision tree classifier as unit of the overall classifier as are generally described Random Forests, and AdaBost. However Adabost and Random Forests seem to beahve more or less in the same way, in the sense that both show a predominance of reagins and subregions linked to the THROUGH class, even if the transition from one region to the other is mcuh smoother than the transition of the Decision Tree based models.\n- The **Niave Bayes Classifier**, when applied to the data points once preprocessed using one at a time all the suggested kernel tricks for kernelPca method as a classifier technique, leads to a results in terms of decision boundaries and regions that vary the most from one kernel trick to the other. In particular using the first three proposed kernel tricks that are 'linear', 'poly', 'rbf' the decison regions connected to the Deck-like bridges are concentric with respect to the surrounding area that instead is widely associated to the other class that is THROUGH-like bridges. More Precisely for 'linear' kernel the resulting decision boundaries are wide and spread along the vertical axis instead for 'poly', 'rbf' tend to be narrowe and to be located near the bottom of the graphic. Instead looking at the graphic that referes to the data points when preprocessed by means of cosine kenrel trick for kernelPca method we notice that it seems to lead to a opposite or simmetryc graphic with respect to the horizontal axis when compared with the graphic obtained by means of linear kernel trick. Lastly the sigmoid kernel trick leads to a graphic that seems to classifies data points from THROUGH class associating them to the left and right sides of the piciture while the top and bottom centered horizontal strip seems to be associated with data points from DECK class and more precisely the dark red areas are spotted mosty near either the top or bottom areas.\n- The last classifier proposed for this thiny and rough experiment is he one known as **Quadratic Discriminant Analysis**, or more shortly *QDA*. The resulting graphics suggest us that by means of such technique we observe that the DECK class is the class among the two which affects mostly the models capabilites, since the decision regions are mostly represented by shades of colors that range in the majority of case around the red color, enabling us to summarize that the DECK class differently from other preceding models will be the most frequently predicted class with respect to the other class that is the THROUGH class.\n\nHaving performed the analyses discussed just above, employing graphics and so qualitaty approach for investigating some of the most known and exploited methods we can summarize that since we adopt jsut two PCS out of eleven possible components for predicting classes among DECK and THROUGH for T-OR-D dependent variable as our predictive or target variable, is is reallyu difficult to correctly classify all the majority of the data samples since the decisoin bundaries vary heavily from one method to the other also due to the fact that we exploit few information and knwoledge and we cannot find patterns that lead to a more precise classification. We need to exploit more features to reach better performance at classification time and find better decision boundaries that allow to separate the data points without mixing them.\n\n\n```python\nplot_dest = os.path.join(\"figures\", \"n_comp_12_analysis\", \"grid_search\"); X = rescaledX; pos = pos + 1\ndf_gs, df_auc_gs = grid_search_all_by_n_components(estimators_list=estimators_list[pos_gs+1], param_grids=param_grids[pos_gs], estimators_names=estimators_names[pos_gs+1], X=X, y=y, n_components=12, random_state=0, show_plots=False, show_errors=False, verbose=1, plot_dest=plot_dest, debug_var=False)\ndf_12, df_12_auc = df_gs, df_auc_gs\n```\n\n\n```python\ncreate_widget_list_df([df_gs, df_auc_gs]) #print(df_gs); print(df_auc_gs)\n```\n\n\n```python\nplot_dest = os.path.join(\"figures\", \"n_comp_12_analysis\", \"grid_search\"); X = rescaledX; pos = pos + 1\ndf_gs, df_auc_gs = grid_search_all_by_n_components(estimators_list=estimators_list[pos_gs+1], param_grids=param_grids[pos_gs], estimators_names=estimators_names[pos_gs+1], X=X, y=y, n_components=12, random_state=0, show_plots=False, show_errors=False, verbose=1, plot_dest=plot_dest, debug_var=False)\ndf_12, df_12_auc = pd.concat([df_12, df_gs], axis=0), pd.concat([df_12_auc, df_auc_gs], axis=0)\n```\n\n\n```python\ncreate_widget_list_df([df_gs, df_auc_gs]) #print(df_gs); print(df_auc_gs)\n```\n\n\n```python\nplot_dest = os.path.join(\"figures\", \"n_comp_12_analysis\", \"grid_search\"); X = rescaledX; pos = pos + 1\ndf_gs, df_auc_gs = grid_search_all_by_n_components(estimators_list=estimators_list[pos_gs+1], param_grids=param_grids[pos_gs], estimators_names=estimators_names[pos_gs+1], X=X, y=y, n_components=12, random_state=0, show_plots=False, show_errors=False, verbose=1, plot_dest=plot_dest, debug_var=False)\ndf_12, df_12_auc = pd.concat([df_12, df_gs], axis=0), pd.concat([df_12_auc, df_auc_gs], axis=0)\n```\n\n\n```python\ncreate_widget_list_df([df_gs, df_auc_gs]) #print(df_gs); print(df_auc_gs)\n```\n\n\n```python\nplot_dest = os.path.join(\"figures\", \"n_comp_12_analysis\", \"grid_search\"); X = rescaledX; pos = pos + 1\ndf_gs, df_auc_gs = grid_search_all_by_n_components(estimators_list=estimators_list[pos_gs+1], param_grids=param_grids[pos_gs], estimators_names=estimators_names[pos_gs+1], X=X, y=y, n_components=12, random_state=0, show_plots=False, show_errors=False, verbose=1, plot_dest=plot_dest, debug_var=False)\ndf_12, df_12_auc = pd.concat([df_12, df_gs], axis=0), pd.concat([df_12_auc, df_auc_gs], axis=0)\n```\n\n\n```python\ncreate_widget_list_df([df_gs, df_auc_gs]) #print(df_gs); print(df_auc_gs)\n```\n\n\n```python\nplot_dest = os.path.join(\"figures\", \"n_comp_12_analysis\", \"grid_search\"); X = rescaledX; pos = pos + 1\ndf_gs, df_auc_gs = grid_search_all_by_n_components(estimators_list=estimators_list[pos_gs+1], param_grids=param_grids[pos_gs], estimators_names=estimators_names[pos_gs+1], X=X, y=y, n_components=12, random_state=0, show_plots=False, show_errors=False, verbose=1, plot_dest=plot_dest, debug_var=False)\ndf_12, df_12_auc = pd.concat([df_12, df_gs], axis=0), pd.concat([df_12_auc, df_auc_gs], axis=0)\n```\n\n\n```python\ncreate_widget_list_df([df_gs, df_auc_gs]) #print(df_gs); print(df_auc_gs)\n```\n\n\n```python\nplot_dest = os.path.join(\"figures\", \"n_comp_12_analysis\", \"grid_search\"); X = rescaledX; pos = pos + 1\ndf_gs, df_auc_gs = grid_search_all_by_n_components(estimators_list=estimators_list[pos_gs+1], param_grids=param_grids[pos_gs], estimators_names=estimators_names[pos_gs+1], X=X, y=y, n_components=12, random_state=0, show_plots=False, show_errors=False, verbose=1, plot_dest=plot_dest, debug_var=False)\ndf_12, df_12_auc = pd.concat([df_12, df_gs], axis=0), pd.concat([df_12_auc, df_auc_gs], axis=0)\n```\n\n\n```python\ncreate_widget_list_df([df_gs, df_auc_gs]) #print(df_gs); print(df_auc_gs)\n```\n\n### Improvements and Conclusions \n\nExtension that we can think of to better improve the analyses we can perform on such a relative tiny dataset many include, for preprocessing phases:\n- Selecting different *Feature Extraction ant Dimensionality Reduction Techniques* other than Pca or kernel Pca such as: \n*linear discriminant analysis (LDA)*, or *canonical correlation analysis (CCA) techniques* as a pre-processing step.\n\nExtension that we can think of to better improve the analyses we can perform on such a relative tiny dataset many include, for training phases:\n\n- Selecting different *Ensemble Methods, investigating both Average based and Boosting based Statistical Learning Methods*.\n\nExtension that we can think of to better improve the analyses we can perform on such a relative tiny dataset many include, for diagnostic analyses after having performed train and test phases:\n\n- Using other measures, indicators and ghraphical plots such as the *Total Operating Characteristic (TOC)*, since also such a measure characterizes diagnostic ability while revealing more information than the ROC. In fact for each threshold, ROC reveals two ratios, TP/(TP + FN) and FP/(FP + TN). In other words, ROC reveals hits/(hits + misses) and false alarms/(false alarms + correct rejections). On the other hand, TOC shows the total information in the contingency table for each threshold. Lastly, the TOC method reveals all of the information that the ROC method provides, plus additional important information that ROC does not reveal, i.e. the size of every entry in the contingency table for each threshold.\n\n## References section \n### Main References\n- Data Domain Information part:\n - (Deck) https://en.wikipedia.org/wiki/Deck_(bridge)\n - (Cantilever bridge) https://en.wikipedia.org/wiki/Cantilever_bridge\n - (Arch bridge) https://en.wikipedia.org/wiki/Deck_(bridge)\n- Machine Learning part:\n - (Theory Book) https://jakevdp.github.io/PythonDataScienceHandbook/\n - (Feature Extraction: PCA) https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.PCA.html\n - (Linear Model: Logistic Regression) https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n - (Neighbor-based Learning: Knn) https://scikit-learn.org/stable/modules/neighbors.html\n - (Stochastc Learning: SGD Classifier) https://scikit-learn.org/stable/modules/sgd.html#sgd\n - (Discriminative Model: SVM) https://scikit-learn.org/stable/modules/svm.html\n - (Non-Parametric Learning: Decsion Trees) https://scikit-learn.org/stable/modules/tree.html#tree\n - (Ensemble, Non-Parametric Learning: RandomForest) https://scikit-learn.org/stable/modules/ensemble.html#forest\n- Metrics:\n - (F1-Accuracy-Precision-Recall) https://towardsdatascience.com/beyond-accuracy-precision-and-recall-3da06bea9f6c\n- Statistics:\n - (Correlation and dependence) https://en.wikipedia.org/wiki/Correlation_and_dependence\n - (KDE) https://jakevdp.github.io/blog/2013/12/01/kernel-density-estimation/\n- Chart part:\n - (Seaborn Charts) https://acadgild.com/blog/data-visualization-using-matplotlib-and-seaborn\n- Third Party Library:\n - (sklearn) https://scikit-learn.org/stable/index.html\n - (statsmodels) https://www.statsmodels.org/stable/index.html#\n\n \n### Others References\n- Plots:\n - (Python Plot) https://www.datacamp.com/community/tutorials/matplotlib-tutorial-python?utm_source=adwords_ppc&utm_campaignid=898687156&utm_adgroupid=48947256715&utm_device=c&utm_keyword=&utm_matchtype=b&utm_network=g&utm_adpostion=&utm_creative=255798340456&utm_targetid=aud-299261629574:dsa-473406587955&utm_loc_interest_ms=&utm_loc_physical_ms=1008025&gclid=Cj0KCQjw-_j1BRDkARIsAJcfmTFu4LAUDhRGK2D027PHiqIPSlxK3ud87Ek_lwOu8rt8A8YLrjFiHqsaAoLDEALw_wcB\n- Markdown Math part:\n - (Math Symbols Latex) https://oeis.org/wiki/List_of_LaTeX_mathematical_symbols\n - (CheatSheet) https://www.ibm.com/support/knowledgecenter/SSHGWL_1.2.3/analyze-data/markd-jupyter.html\n - (Tutorial 1) https://share.cocalc.com/share/b4a30ed038ee41d868dad094193ac462ccd228e2/Homework%20/HW%201.2%20-%20Markdown%20and%20LaTeX%20Cheatsheet.ipynb?viewer=share\n - (Tutorial 2) https://jupyter-notebook.readthedocs.io/en/stable/examples/Notebook/Typesetting%20Equations.html\n\n\n```python\n\n```\n", "meta": {"hexsha": "6dd19309ec4bcdab3ac88d035e60c35b89b7f688", "size": 785101, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "pittsburgh-bridges-data-set-analysis/models-analyses/complete_analyses/.ipynb_checkpoints/Data Space Report (Official)-v2.0.4-checkpoint.ipynb", "max_stars_repo_name": "franec94/Pittsburgh-Bridge-Dataset", "max_stars_repo_head_hexsha": "682ff0e3979ca565637e858cc36dc07c2aeda7d6", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "pittsburgh-bridges-data-set-analysis/models-analyses/complete_analyses/.ipynb_checkpoints/Data Space Report (Official)-v2.0.4-checkpoint.ipynb", "max_issues_repo_name": "franec94/Pittsburgh-Bridge-Dataset", "max_issues_repo_head_hexsha": "682ff0e3979ca565637e858cc36dc07c2aeda7d6", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 7, "max_issues_repo_issues_event_min_datetime": "2021-02-02T22:51:40.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-12T00:39:08.000Z", "max_forks_repo_path": "pittsburgh-bridges-data-set-analysis/models-analyses/complete_analyses/.ipynb_checkpoints/Data Space Report (Official)-v2.0.4-checkpoint.ipynb", "max_forks_repo_name": "franec94/Pittsburgh-Bridge-Dataset", "max_forks_repo_head_hexsha": "682ff0e3979ca565637e858cc36dc07c2aeda7d6", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 317.7260218535, "max_line_length": 356368, "alphanum_fraction": 0.8923170395, "converted": true, "num_tokens": 44790, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.6926419831347361, "lm_q1q2_score": 0.4412342728067892}} {"text": "###### Content under Creative Commons Attribution license CC-BY 4.0, code under MIT license (c) 2019 Daniel Koehn, based on (c)2014 L.A. Barba, G.F. Forsyth, C.D. Cooper [CFD Python](https://github.com/barbagroup/CFDPython), (c)2013 L.A. Barba, also under CC-BY.\n\n\n```python\nfrom IPython.core.display import HTML\ncss_file = '../style/custom.css'\nHTML(open(css_file, 'r').read())\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n# 1D (non)linear Advection\n\nIn the [last lecture](https://github.com/daniel-koehn/Differential-equations-earth-system/tree/master/03_Lorenz_equations#ordinary-differential-equations-in-earth-sciences-the-lorenz-equations), we looked into numerical integration methods for the solution of ordinary differential equations (ODEs), using a simplified thermal convection problem - the Lorenz equations. In this module, we will study the numerical solution of **partial differential equations (PDEs)**, where the unknown is a multi-variate function. The problem could depend on time, $t$, and one spatial dimension $x$ (or more), which means we need to build a discretization grid with each independent variable.\n\nWe will start our discussion of numerical PDEs with 1-D linear and non-linear advection equations.\n\n## 1D Linear Advection\n\nThe *one-dimensional linear advection equation* is the simplest, most basic model that can be used to learn something about numerical solution of PDEs. It's surprising that this little equation can teach us so much! Here it is:\n\n$$\n\\begin{equation}\n\\frac{\\partial u}{\\partial t} + c \\frac{\\partial u}{\\partial x} = 0 \\tag{1}\n\\end{equation}\n$$\n\nThe equation represents a *wave* propagating with speed $c$ in the $x$ direction, without change of shape. For that reason, it's sometimes called the *one-way wave equation*.\n\nWith an initial condition $u(x,0)=u_0(x)$, the equation has an exact solution given by:\n\n$$\n\\begin{equation}\nu(x,t)=u_0(x-ct) \\tag{2}\n\\end{equation}\n$$\n\n##### Exercise 1\n\nProve that eq. (2) solves the 1D linear convection equation (1). Take the time and space derivative and stick them into the equation to see that it holds!\n\nLook at the exact solution for a moment ... we know two things about it: \n\n1. its shape does not change, being always the same as the initial waveform $u_0$, only shifted in the $x$-direction; and \n2. it's constant along so-called **characteristic curves**, $x-ct=$constant. This means that for any point in space and time, you can move back along the characteristic curve to $t=0$ to know the value of the solution.\n\n\n#### Characteristic curves for positive wave speed.\n\nWhy do we call the equations *linear*? PDEs can be either linear or non-linear. In a linear equation, the unknown function $u$ and its derivatives appear only in linear terms, in other words, there are no products, powers, or transcendental functions applied on them. \n\nWhat is the most important feature of linear equations? Do you remember? In case you forgot: solutions can be superposed to generate new solutions that still satisfy the original equation. This is super useful!\n\n## Finite-differences\n\nIn the previous lessons, we discretized time derivatives; now we have derivatives in both space *and* time, so we need to discretize with respect to *both* these variables. \n\nImagine a *space-time* plot, where the coordinates in the vertical direction represent advancing in time\u2014for example, from $t^n$ to $t^{n+1}$\u2014and the coordinates in the horizontal direction move in space: consecutive points are $x_{i-1}$, $x_i$, and $x_{i+1}$. This creates a grid where a point has both a temporal and spatial index. Here is a graphical representation of the space-time grid:\n\n$$\n\\begin{matrix}\nt^{n+1} & \\rightarrow & \\bullet && \\bullet && \\bullet \\\\\nt^n & \\rightarrow & \\bullet && \\bullet && \\bullet \\\\\n& & x_{i-1} && x_i && x_{i+1}\n\\end{matrix}\n$$\n\nFor the numerical solution of $u(x,t)$, we'll use subscripts to denote the spatial position, like $u_i$, and superscripts to denote the temporal instant, like $u^n$. We would then label the solution at the top-middle point in the grid above as follows:\n$u^{n+1}_{i}$.\n\nEach grid point below has an index $i$, corresponding to the spatial position and increasing to the right, and an index $n$, corresponding to the time instant and increasing upwards. A small grid segment would have the following values of the numerical solution at each point:\n\n$$\n\\begin{matrix}\n& &\\bullet & & \\bullet & & \\bullet \\\\\n& &u^{n+1}_{i-1} & & u^{n+1}_i & & u^{n+1}_{i+1} \\\\\n& &\\bullet & & \\bullet & & \\bullet \\\\\n& &u^n_{i-1} & & u^n_i & & u^n_{i+1} \\\\\n& &\\bullet & & \\bullet & & \\bullet \\\\\n& &u^{n-1}_{i-1} & & u^{n-1}_i & & u^{n-1}_{i+1} \\\\\n\\end{matrix}\n$$\n\nAnother way to explain our discretization grid is to say that it is built with constant steps in time and space, $\\Delta t$ and $\\Delta x$, as follows:\n\n$$\n\\begin{eqnarray}\nx_i &=& i\\, \\Delta x \\quad \\text{and} \\quad t^n= n\\, \\Delta t \\nonumber \\\\\nu_i^n &=& u(i\\, \\Delta x, n\\, \\Delta t) \\notag\n\\end{eqnarray}\n$$\n\n### Discretizing our model equation\n\nLet's see how to discretize the 1-D linear advection equation in both space and time. By definition, the partial derivative with respect to time changes only with time and not with space; its discretized form changes only the $n$ indices. Similarly, the partial derivative with respect to $x$ changes with space not time, and only the $i$ indices are affected. \n\nWe'll discretize the spatial coordinate $x$ into points indexed from $i=0$ to $N$, and then step in discrete time intervals of size $\\Delta t$.\n\nFrom the definition of a derivative (and simply removing the limit), we know that for $\\Delta x$ sufficiently small:\n\n$$\n\\begin{equation}\n\\frac{\\partial u}{\\partial x}\\approx \\frac{u(x+\\Delta x)-u(x)}{\\Delta x} \\notag\n\\end{equation}\n$$\n\nThis formula could be applied at any point $x_i$. But note that it's not the only way that we can estimate the derivative. The geometrical interpretation of the first derivative $\\partial u/ \\partial x$ at any point is that it represents the slope of the tangent to the curve $u(x)$. In the sketch below, we show a slope line at $x_i$ and mark it as \"exact.\" If the formula written above is applied at $x_i$, it approximates the derivative using the next spatial grid point: it is then called a _forward difference_ formula. \n\nBut as shown in the sketch below, we could also estimate the spatial derivative using the point behind $x_i$, in which case it is called a _backward difference_. We could even use the two points on each side of $x_i$, and obtain what's called a _central difference_ (but in that case the denominator would be $2\\Delta x$).\n\n\n#### Three finite-difference approximations at $x_i$.\n\nWe have three possible ways to represent a discrete form of $\\partial u/ \\partial x$:\n\n* Forward difference: uses $x_i$ and $x_i + \\Delta x$,\n* Backward difference: uses $x_i$ and $x_i- \\Delta x$,\n* Central difference: uses two points on either side of $x_i$.\n\nThe sketch above also suggests that some finite-difference formulas might be better than others: it looks like the *central difference* approximation is closer to the slope of the \"exact\" derivative. Curious if this is just an effect of our exaggerated picture? We'll show you later how to make this observation rigorous!\n\nThe three formulas are:\n\n$$\n\\begin{eqnarray}\n\\frac{\\partial u}{\\partial x} & \\approx & \\frac{u(x_{i+1})-u(x_i)}{\\Delta x} \\quad\\text{Forward}\\notag\\\\\n\\frac{\\partial u}{\\partial x} & \\approx & \\frac{u(x_i)-u(x_{i-1})}{\\Delta x} \\quad\\text{Backward}\\notag\\\\\n\\frac{\\partial u}{\\partial x} & \\approx & \\frac{u(x_{i+1})-u(x_{i-1})}{2\\Delta x} \\quad\\text{Central}\\notag\n\\end{eqnarray}\n$$\n\n**Euler's method** is equivalent to using a forward-difference scheme for the time derivative. Let's stick with that, and choose the backward-difference scheme for the space derivative (forward in time & backward in space **FTBS Method**). Our discrete equation is then:\n\n$$\n\\begin{equation}\n\\frac{u_i^{n+1}-u_i^n}{\\Delta t} + c \\frac{u_i^n - u_{i-1}^n}{\\Delta x} = 0 \\notag\n\\end{equation}\n$$\n\nwhere $n$ and $n+1$ are two consecutive steps in time, while $i-1$ and $i$ are two neighboring points of the discretized $x$ coordinate. With given initial conditions, the only unknown in this discretization is $u_i^{n+1}$. We solve for this unknown to get an equation that lets us step in time, as follows:\n\n$$\n\\begin{equation}\nu_i^{n+1} = u_i^n - c \\frac{\\Delta t}{\\Delta x}(u_i^n-u_{i-1}^n) \\notag\n\\end{equation}\n$$\n\nWe like to make drawings of a grid segment, showing the grid points that influence our numerical solution. This is called a **stencil**. Below is the stencil for solving our model equation with the finite-difference formula we wrote above.\n\n\n#### Stencil for the \"forward-time/backward-space\" scheme.\n\n## And compute!\n\nAlright. Let's get a little Python on the road. First: we need to load our array and plotting libraries, as usual.\n\n\n```python\nimport numpy\nfrom matplotlib import pyplot\n%matplotlib inline\n```\n\nWe also set notebook-wide plotting parameters for the font family and the font size by modifying entries of the `rcParams` dictionary.\n\n\n```python\n# Set the font family and size to use for Matplotlib figures.\npyplot.rcParams['font.family'] = 'serif'\npyplot.rcParams['font.size'] = 16\n```\n\nAs a first exercise, we'll solve the 1D linear advection equation with a *square wave* initial condition, defined as follows:\n\n$$\n\\begin{equation}\nu(x,0)=\\begin{cases}2 & \\text{where } 0.5\\leq x \\leq 1,\\\\\n1 & \\text{everywhere else in } (0, 2)\n\\end{cases}\n\\notag\n\\end{equation}\n$$\n\nWe also need a boundary condition on $x$: let $u=1$ at $x=0$. Our spatial domain for the numerical solution will only cover the range $x\\in (0, 2)$.\n\n\n#### Square wave initial condition.\n\nNow let's define a few variables; we want to make an evenly spaced grid of points within our spatial domain. In the code below, we define a variable called `nx` that will be the number of spatial grid points, and a variable `dx` that will be the distance between any pair of adjacent grid points. We also can define a step in time, `dt`, a number of steps, `nt`, and a value for the wave speed: we like to keep things simple and make $c=1$. \n\n\n```python\n# Set parameters.\nnx = 41 # number of spatial discrete points\nL = 2.0 # length of the 1D domain\ndx = L / (nx - 1) # spatial grid size\nnt = 25 # number of time steps\ndt = 0.02 # time-step size\nc = 1.0 # advection speed\n\n# Define the grid point coordinates.\nx = numpy.linspace(0.0, L, num=nx)\n```\n\nWe also need to set up our initial conditions. Here, we use the NumPy function `numpy.ones()` defining an array which is `nx`-element long with every value equal to $1$. How useful! We then *change a slice* of that array to the value $u=2$, to get the square wave, and we print out the initial array just to admire it. But which values should we change? The problem states that we need to change the indices of `u` such that the square wave begins at $x = 0.5$ and ends at $x = 1$.\n\nWe can use the [`numpy.where()`](https://docs.scipy.org/doc/numpy/reference/generated/numpy.where.html) function to return a list of indices where the vector $x$ meets some conditions.\nThe function [`numpy.logical_and()`](https://docs.scipy.org/doc/numpy/reference/generated/numpy.logical_and.html) computes the truth value of `x >= 0.5` **and** `x <= 1.0`, element-wise.\n\n\n```python\n# Set initial conditions with 1.0 everywhere (for now).\nu0 = numpy.ones(nx)\n\n# Get a list of indices where 0.5 <= x <= 1.0.\nmask = numpy.where(numpy.logical_and(x >= 0.5, x <= 1.0))\nprint(mask)\n```\n\n (array([10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], dtype=int64),)\n\n\nWith the list of indices, we can now update our initial conditions to get a square-wave shape.\n\n\n```python\n# Set initial condition u = 2.0 where 0.5 <= x <= 1.0.\nu0[mask] = 2.0\nprint(u0)\n```\n\n [1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 1. 1. 1.\n 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]\n\n\nNow let's take a look at those initial conditions we've built with a handy plot.\n\n\n```python\n# Plot the initial conditions.\npyplot.figure(figsize=(4.0, 4.0))\npyplot.title('Initial conditions')\npyplot.xlabel('x')\npyplot.ylabel('u')\npyplot.grid()\npyplot.plot(x, u0, color='C0', linestyle='--', linewidth=2)\npyplot.xlim(0.0, L)\npyplot.ylim(0.0, 2.5);\n```\n\nIt does look pretty close to what we expected. But it looks like the sides of the square wave are not perfectly vertical. Is that right? Think for a bit.\n\nNow it's time to write some code for the discrete form of the advection equation using our chosen FTBS finite-difference scheme. \n\nFor every element of our array `u`, we need to perform the operation: \n\n$$\nu_i^{n+1} = u_i^n - c \\frac{\\Delta t}{\\Delta x}(u_i^n-u_{i-1}^n)\n$$\n\nWe'll store the result in a new (temporary) array `un`, which will be the solution $u$ for the next time-step. We will repeat this operation for as many time-steps as we specify and then we can see how far the wave has traveled. \n\nWe first initialize the placeholder array `un` to hold the values we calculate for the $n+1$ time step, using once again the NumPy function `ones()`.\n\nThen, we may think we have two iterative operations: one in space and one in time, so we may start by nesting a spatial loop inside the time loop, as shown below. You see that the code for the finite-difference scheme is a direct expression of the discrete equation: \n\n\n```python\nu = u0.copy()\nfor n in range(1, nt):\n un = u.copy()\n for i in range(1, nx):\n u[i] = un[i] - c * dt / dx * (un[i] - un[i - 1])\n```\n\n**Note 1**\u2014We stressed above that our physical problem needs a boundary condition at $x=0$. Here we do not need to impose it at every iteration because our discretization does not change the value of u[0]: it remains equal to one and our boundary condition is therefore satisfied during the whole computation!\n\nNow let's inspect our solution array after advancing in time with a line plot.\n\n\n```python\n# Plot the solution after nt time steps\n# along with the initial conditions.\npyplot.figure(figsize=(4.0, 4.0))\npyplot.xlabel('x')\npyplot.ylabel('u')\npyplot.grid()\npyplot.plot(x, u0, label='Initial',\n color='C0', linestyle='--', linewidth=2)\npyplot.plot(x, u, label='nt = {}'.format(nt),\n color='C1', linestyle='-', linewidth=2)\npyplot.legend()\npyplot.xlim(0.0, L)\npyplot.ylim(0.0, 2.5);\n```\n\nThat's funny. Our square wave has definitely moved to the right, but it's no longer in the shape of a top-hat. **What's going on?**\n\n##### Dig deeper\n\nThe solution differs from the expected square wave because the discretized equation is an approximation of the continuous differential equation that we want to solve. There are errors: we knew that. But the modified shape of the initial wave is something curious. Maybe it can be improved by making the grid spacing finer. Why don't you try it? Does it help?\n\n## Spatial truncation error\n\nRecall the finite-difference approximation we are using for the spatial derivative:\n\n$$\n\\begin{equation}\n\\frac{\\partial u}{\\partial x}\\approx \\frac{u(x+\\Delta x)-u(x)}{\\Delta x}\\notag\n\\end{equation}\n$$\n\nWe obtain it by using the definition of the derivative at a point, and simply removing the limit, in the assumption that $\\Delta x$ is very small. But we already learned with Euler's method that this introduces an error, called the *truncation error*.\n\nUsing a Taylor series expansion for the spatial terms now, we see that the backward-difference scheme produces a first-order method, in space.\n\n$$\n\\begin{equation}\n\\frac{\\partial u}{\\partial x}(x_i) = \\frac{u(x_i)-u(x_{i-1})}{\\Delta x} + \\frac{\\Delta x}{2} \\frac{\\partial^2 u}{\\partial x^2}(x_i) - \\frac{\\Delta x^2}{6} \\frac{\\partial^3 u}{\\partial x^3}(x_i)+ \\cdots\n\\notag\n\\end{equation}\n$$\n\nThe dominant term that is neglected in the finite-difference approximation is of $\\mathcal{O}(\\Delta x)$. We also see that the approximation *converges* to the exact derivative as $\\Delta x \\rightarrow 0$. That's good news!\n\nIn summary, the chosen \"forward-time/backward space\" difference scheme is first-order in both space and time: the truncation errors are $\\mathcal{O}(\\Delta t, \\Delta x)$. We'll come back to this!\n\n## Non-linear advection\n\nLet's move on to the non-linear advection equation, using the same methods as before. The 1-D convection equation is:\n\n$$\n\\begin{equation}\n\\frac{\\partial u}{\\partial t} + u \\frac{\\partial u}{\\partial x} = 0 \\tag{3}\n\\end{equation}\n$$\n\nThe only difference with the linear case is that we've replaced the constant wave speed $c$ by the variable speed $u$. The equation is non-linear because now we have a product of the solution and one of its derivatives: the product $u\\,\\partial u/\\partial x$. This changes everything!\n\nWe're going to use the same discretization as for linear convection: forward difference in time and backward difference in space. Here is the discretized equation:\n\n$$\n\\begin{equation}\n\\frac{u_i^{n+1}-u_i^n}{\\Delta t} + u_i^n \\frac{u_i^n-u_{i-1}^n}{\\Delta x} = 0 \\notag\n\\end{equation}\n$$\n\nSolving for the only unknown term, $u_i^{n+1}$, gives an equation that can be used to advance in time:\n\n$$\n\\begin{equation}\nu_i^{n+1} = u_i^n - u_i^n \\frac{\\Delta t}{\\Delta x} (u_i^n - u_{i-1}^n) \\tag{4}\n\\end{equation}\n$$\n\nThere is very little that needs to change from the code written so far. In fact, we'll even use the same square-wave initial condition. But let's re-initialize the variable `u` with the initial values, and re-enter the numerical parameters here\n\n\n```python\n# Set parameters.\nnx = 41 # number of spatial discrete points\nL = 2.0 # length of the 1D domain\ndx = L / (nx - 1) # spatial grid size\nnt = 10 # number of time steps\ndt = 0.02 # time-step size\n\nx = numpy.linspace(0.0, L, num=nx)\n\n# Re-initialization of u0\nu0 = numpy.ones(nx)\nmask = numpy.where(numpy.logical_and(x >= 0.5, x <= 1.0))\nu0[mask] = 2.0\n```\n\n How does it look?\n\n\n```python\n# Plot the initial conditions.\npyplot.figure(figsize=(4.0, 4.0))\npyplot.title('Initial conditions')\npyplot.xlabel('x')\npyplot.ylabel('u')\npyplot.grid()\npyplot.plot(x, u0, color='C0', linestyle='--', linewidth=2)\npyplot.xlim(0.0, L)\npyplot.ylim(0.0, 2.5);\n```\n\n##### Exercise 2\n\nBy changing just one line of code in the solution of the linear advection, we are able to get the non-linear solution: the line that corresponds to the discrete equation now has `un[i]`, replacing the constant velocity `c`. Implement the FTBS-FD approach of the non-linear advection problem eq.(4) in the cell below, run the code ...\n\n\n```python\n# MODIFY THE LINEAR ADVECTION CODE BELOW TO SOLVE THE NONLINEAR ADVECTION PROBLEM!\nu = u0.copy()\nfor n in range(1, nt):\n un = u.copy()\n for i in range(1, nx):\n u[i] = un[i] - c * dt / dx * (un[i] - un[i - 1])\n```\n\n... and plot the solution along the initial condition\n\n\n```python\n# Plot the solution after nt time steps\n# along with the initial conditions.\npyplot.figure(figsize=(4.0, 4.0))\npyplot.xlabel('x')\npyplot.ylabel('u')\npyplot.grid()\npyplot.plot(x, u0, label='Initial',\n color='C0', linestyle='--', linewidth=2)\npyplot.plot(x, u, label='nt = {}'.format(nt),\n color='C1', linestyle='-', linewidth=2)\npyplot.legend()\npyplot.xlim(0.0, L)\npyplot.ylim(0.0, 2.5);\n```\n\nHmm. That's quite interesting: like in the linear case, we see that we have lost the sharp sides of our initial square wave, but there's more. Now, the wave has also lost symmetry! It seems to be lagging on the rear side, while the front of the wave is steepening. Is this another form of numerical error, do you ask? No! It's physics!\n\n##### Dig deeper\n\nThink about the effect of having replaced the constant wave speed $c$ by the variable speed given by the solution $u$. It means that different parts of the wave move at different speeds. Make a sketch of an initial wave and think about where the speed is higher and where it is lower ...\n\n## What we learned:\n\n- How to solve the 1D (non)linear advection problem by the FTBS finite difference method\n\n- While the analytical solution of the linear advection problem should only shift the initial waveform to the right, the FD solution on a coarse spatial grid also changes the shape of the initial waveform\n\n- In case of the 1D non-linear advection problem we lose the symmetry of the initial waveform\n", "meta": {"hexsha": "78b493182ed58409104cdb6349c82c13a36494f9", "size": 76074, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "04_Advection_1D/01_Advection_1D.ipynb", "max_stars_repo_name": "daniel-koehn/Differential-equations-earth-system", "max_stars_repo_head_hexsha": "3916cbc968da43d0971b7139476350c1dd798746", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 30, "max_stars_repo_stars_event_min_datetime": "2019-10-16T19:07:36.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-10T03:48:44.000Z", "max_issues_repo_path": "04_Advection_1D/01_Advection_1D.ipynb", "max_issues_repo_name": "daniel-koehn/Differential-equations-earth-system", "max_issues_repo_head_hexsha": "3916cbc968da43d0971b7139476350c1dd798746", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "04_Advection_1D/01_Advection_1D.ipynb", "max_forks_repo_name": "daniel-koehn/Differential-equations-earth-system", "max_forks_repo_head_hexsha": "3916cbc968da43d0971b7139476350c1dd798746", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2020-11-19T08:21:55.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-10T09:33:37.000Z", "avg_line_length": 81.9762931034, "max_line_length": 16476, "alphanum_fraction": 0.7825801194, "converted": true, "num_tokens": 6555, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5078118642792044, "lm_q2_score": 0.8688267796346599, "lm_q1q2_score": 0.44120054670197417}} {"text": "# Teorema de Bayes, distribuciones previas y previas conjugadas\n\n\n\n> Justo como ilustra la imagen de arriba, dadas dos VA $A$ y $B$, el Teorema de Bayes se puede escribir como:\n> \n> $$\n P(A | B) = \\frac{P(B | A) P(A)}{P(B)}\n $$\n \n> En este notebook recapitularemos qu\u00e9 significa cada uno de esos elementos, estudiaremos porqu\u00e9 es complejo el c\u00e1lculo exacto de la distribuci\u00f3n posterior, y veremos un m\u00e9todo anal\u00edtico para hacer este c\u00e1lculo exacto.\n\n> **Objetivos:**\n> - Explicar porqu\u00e9 el c\u00e1lculo exacto de la distribuci\u00f3n posterior es una tarea intratable en muchas ocasiones.\n> - Principio de Maximum Aposteriori (MAP).\n> - Comprender el concepto de distribuci\u00f3n previa conjugada.\n\n> **Referencias:**\n> - Bayesian Methods for Machine Learning course, HSE University, Coursera.\n> - Statistical Rethinking, Richard McElreath, 2018.\n\n## 1. Teorema de Bayes\n\nComo ya vimos, el enfoque Bayesiano considera que los par\u00e1metros $\\theta$ son los que se consideran aleatorios, y los datos $X$ est\u00e1n fijos como evidencia.\n\nEn ese sentido, nos interesa modelar la distribuci\u00f3n de los par\u00e1metros dada la evidencia que estamos observando $P(\\theta | X)$, lo cual, por el Teorema de Bayes podemos escribir como:\n\n$$\nP(\\theta | X) = \\frac{P(X | \\theta) P(\\theta)}{P(X)},\n$$\n\ndonde:\n- $P(\\theta | X)$ se conoce como **distribuci\u00f3n posterior** (en ingl\u00e9s *posterior*). La posterior nos dice la probabilidad de los par\u00e1metros despu\u00e9s de haber observado los datos.\n- $P(X | \\theta)$ se conoce como **funci\u00f3n de verosimilitud** (en ingl\u00e9s *likelihood*). La verosimilitud nos dice que tan bien los par\u00e1metros explican los datos.\n- $P(\\theta)$ se conoce como **distribuci\u00f3n previa** (en ingl\u00e9s *prior*). En la previa, incluimos todo el conocimiento que podamos tener acerca de los par\u00e1metros.\n- $P(X)$ se conoce como **distribuci\u00f3n de evidencia** (en ingl\u00e9s *evidence*).\n\n#### \u00bfQu\u00e9 es la (distribuci\u00f3n de) evidencia $P(X)$?\n\nEjemplos:\n\n1. Imaginemos que estamos trabajando con visi\u00f3n por computadora de im\u00e1genes de diferentes artistas (por ejemplo Van Gogh, DaVinci, Monet). \u00bfQu\u00e9 es $P(X)$?\n\n **Respuesta**\n\n2. Imaginemos que estamos estimando la probabilidad de que un cliente compre en nuestra e-commerce en las siguientes dos semanas (clasificador binario). En este caso:\n\n $$\n P(\\theta | X, y) = \\frac{P(y | X, \\theta) P(\\theta | X)}{P(y | X)},\n $$\n\n \u00bfQu\u00e9 es $P(y | X)$?\n \n **Respuesta**\n\nPor tanto, modelar la distribuci\u00f3n de evidencia es una tarea bastante compleja que en la mayor\u00eda de los casos no somos capaces de realizar. \n\nPor tanto, en esta clase, estaremos estudiando metodolog\u00edas anal\u00edticas para evitar el c\u00e1lculo de la distribucion de evicencia.\n\n## 2. Maximum Aposteriori (MAP)\n\nRecordemos la clase de [m\u00ednimos cuadrados desde una perspectiva probabil\u00edstica](../tema3/3_regresion_lineal.ipynb).\n\nEn ciertas ocasiones no nos interesa modelar toda la distribuci\u00f3n de los par\u00e1metros, sino simplemente encontrar el valor *m\u00e1s probable* de los par\u00e1metros dadas las observaciones. Esto da lugar al estimador MAP:\n\n$$\n\\theta_{MAP} = \\arg \\max_{\\theta} P(\\theta | X),\n$$\n\nel cual, usando el Teorema de Bayes se puede escribir como:\n\n$$\n\\theta_{MAP} = \\arg \\max_{\\theta} \\frac{P(X | \\theta) P(\\theta)}{P(X)}.\n$$\n\nSin embargo, como la evidencia $P(X)$ no depende de los par\u00e1metros $\\theta$:\n\n$$\n\\theta_{MAP} = \\arg \\max_{\\theta} P(X | \\theta) P(\\theta).\n$$\n\n\n\nDe esta manera, observamos que para el c\u00e1lculo del estimador MAP, evitamos completamente el c\u00e1lculo de la distribuci\u00f3n de evidencia. El problema de estimaci\u00f3n resultante lo podemos resolver de manera eficiente usando m\u00e9todos num\u00e9ricos.\n\n#### Y si siempre nos quedamos con el estimador MAP, \u00bfCu\u00e1l es el problema?\n\nSi solamente nos qued\u00e1ramos con el estimador MAP, el enfoque Bayesiano pierde el sentido. Recordamos que bajo el enfoque Bayesiano nos interesa modelar la incertidumbre de los par\u00e1metros ante las observaciones que hacemos del mundo, y con un estimador MAP obtenemos los valores **fijos** m\u00e1s probables de los par\u00e1metros.\n\nEsto tiene varias implicaciones:\n\n1. Hab\u00edamos dicho que una de las ventajas del enfoque Bayesiano era que nos permit\u00eda hacer aprendizaje on-line, solo haciendo los c\u00e1lculos relativos a cada paso:\n \n $$\n P_k(\\theta) = P(\\theta | x_k) = \\frac{P(x_k | \\theta) P_{k-1}(\\theta)}{P(x_k)}.\n $$\n \n Sin embargo, no podemos usar la estimaci\u00f3n MAP como nueva previa en el pr\u00f3ximo paso, debido a que ser\u00eda una funci\u00f3n impulso:\n \n $$\n P_{k-1}(\\theta) = \\delta(\\theta - \\theta_{MAP}) = \\left\\{\\begin{array}{cc}\n 1 & \\text{si } \\theta=\\theta_{MAP} \\\\\n 0 & \\text{en otro caso}\n \\end{array}\\right.,\n $$\n \n y esto no nos aportar\u00eda informaci\u00f3n al siguiente paso:\n \n $$\n P_k(\\theta) = P(\\theta | x_k) = \\frac{P(x_k | \\theta) \\delta(\\theta - \\theta_{MAP})}{P(x_k)} = \\delta(\\theta - \\theta_{MAP}).\n $$\n\n2. Podr\u00edamos caer en un caso donde $\\theta_{MAP}$ sea un valor at\u00edpico, y aunque sea el \"m\u00e1s probable\", la mayor\u00eda de los datos est\u00e9n concentrados en otra regi\u00f3n. Por ejemplo\n\n\n```python\n# Importar matplotlib.pyplot\nfrom matplotlib import pyplot as plt\n# Importar scipy.stats.norm\nfrom scipy.stats import norm\n# Importar numpy\nimport numpy as np\n```\n\n\n```python\n# Definir VA X, Y\nX = norm(loc=0.5, scale=0.1)\nY = norm(loc=0.7, scale=0.01)\n```\n\n\n```python\n# Suma ponderada de densidades\nx = np.linspace(0, 1, 1000)\nfz = 0.8 * X.pdf(x) + 0.2 * Y.pdf(x)\n```\n\n\n```python\n# Graficar densidad de Z\nplt.plot(x, fz)\n```\n\n3. Si solo estimamos un punto, no podr\u00edamos estimar regiones de credibildiad (intervalos de confianza).\n\n\n\n## 3. Distribuciones conjugadas\n\nDe manera que los estimadores MAP son bastante \u00fatiles en ciertas aplicaciones, pero si queremos a\u00fan m\u00e1s flexibilidad en cuanto a la estimaci\u00f3n de la distribuci\u00f3n de los par\u00e1metros, no nos es \u00fatil.\n\nSi queremos estimar esta distribucion dadas las observaciones debemos buscar otros caminos. Uno de ellos son las **distribuciones conjugadas**.\n\nRecapitulando, la distribuci\u00f3n posterior la podemos escribir como:\n\n$$\nP(\\theta | X) = \\frac{P(X | \\theta) P(\\theta)}{P(X)},\n$$\n\ndonde:\n\n- La funci\u00f3n de verosimilitud $P(X | \\theta)$ la fija el modelo.\n- La distribuci\u00f3n de evidencia $P(X)$ la fijan los datos.\n\nPeeeero, la distribuci\u00f3n previa es algo que podemos elegir de acuerdo a **nuestra experiencia, conocimiento previo o simplemente para acomodar los c\u00e1lculos**.\n\nEs decir, podr\u00edamos elegir la distribuci\u00f3n previa $P(\\theta)$ con el \u00fanico fin de evitar el c\u00e1lculo de la la distribuci\u00f3n de evidencia $P(X)$.\n\n> *Definici\u00f3n.* (**Previa conjugada**) Decimos que la distribuci\u00f3n previa $P(\\theta)$ es conjugada a la funci\u00f3n de verosimilitud $P(X | \\theta)$, si la distribuci\u00f3n posterior pertenece a la misma familia de distribuciones que la distribuci\u00f3n previa.\n\n\u00bfC\u00f3mo es esto posible?\n\nSupongamos que tenemos dos dos densidades normales distintas.\n\n\n```python\n# Definir VA X, Y\nX = norm(loc=0.5, scale=0.1)\nY = norm(loc=0.7, scale=0.15)\n```\n\n\n```python\n# Graficar densidades\nx = np.linspace(0, 1.4, 1000)\nplt.plot(x, X.pdf(x), label=\"Densidad 1\")\nplt.plot(x, Y.pdf(x), label=\"Densidad 2\")\nplt.legend()\n```\n\nSi multiplicamos estas dos densidades, y renormalizamos para que la integral de $1$, obtenemos de nuevo una distribuci\u00f3n normal:\n\n\n```python\nfrom scipy.integrate import quad\n```\n\n\n```python\ndef pdf_z(x):\n return X.pdf(x) * Y.pdf(x)\n```\n\n\n```python\nval, err = quad(pdf_z, -10, 10)\nval, err\n```\n\n\n\n\n (1.1959423430718399, 1.4430475689561403e-11)\n\n\n\n\n```python\n# Graficar densidad de Z\nx = np.linspace(0, 1.4, 1000)\nplt.plot(x, X.pdf(x), label=\"Densidad 1\")\nplt.plot(x, Y.pdf(x), label=\"Densidad 2\")\nplt.plot(x, X.pdf(x) * Y.pdf(x) / val, label=\"Densidad del producto\")\nplt.legend()\n```\n\nMatem\u00e1ticamente, sean $X_1 \\sim \\mathcal{N}(\\mu_1, \\sigma_1^2)$ y $X_2 \\sim \\mathcal{N}(\\mu_2, \\sigma_2^2)$. Entonces:\n\n\\begin{align}\n\\mathcal{N}(x | \\mu_1, \\sigma_1^2) \\mathcal{N}(x | \\mu_2, \\sigma_2^2) & \\propto \\exp\\left\\{-\\frac{(x - \\mu_1)^2}{2 \\sigma_1^2}\\right\\} \\exp\\left\\{-\\frac{(x - \\mu_2)^2}{2 \\sigma_2^2}\\right\\} \\\\\n& = \\exp\\left\\{-\\frac{(x - \\mu_1)^2}{2 \\sigma_1^2} -\\frac{(x - \\mu_2)^2}{2 \\sigma_2^2} \\right\\} \\\\\n& = \\exp\\left\\{-\\frac{\\sigma_2^2(x - \\mu_1)^2 + \\sigma_1^2(x - \\mu_2)^2}{2 \\sigma_1^2 \\sigma_2^2}\\right\\} \\\\\n& = \\exp\\left\\{-\\frac{(x - \\bar{\\mu})^2}{2 \\bar{\\sigma}^2} + const\\right\\} \\\\\n& \\propto \\exp\\left\\{-\\frac{(x - \\bar{\\mu})^2}{2 \\bar{\\sigma}^2}\\right\\}\n\\end{align}\n\nDe modo que el producto vuelve a ser una densidad normal. Ver [el siguiente enlace](https://www.johndcook.com/blog/2012/10/29/product-of-normal-pdfs/).\n\nCon base en lo anterior, tomando la posterior:\n\n$$\nP(\\theta | X) = \\frac{P(X | \\theta) P(\\theta)}{P(X)},\n$$\n\nSi la funci\u00f3n de verosimilitud $P(X | \\theta) = \\mathcal{N}(X | \\theta, \\sigma^2)$, y la previa se selecciona como:\n\n$$\nP(\\theta) = \\mathcal{N}(\\theta | m, s^2),\n$$\n\nobtenemos:\n\n$$\nP(\\theta | X) = \\mathcal{N}(\\theta | a, b^2).\n$$\n\nEs decir, para una distribuci\u00f3n normal sobre los datos con media $\\theta$, la previa conjugada es una distribuci\u00f3n normal sobre $\\theta$.\n\n#### \u00bfY qu\u00e9 pasa con la evidencia?\n\nRecordemos que la distribuci\u00f3n posterior es una distribuci\u00f3n **sobre los par\u00e1metros**. Debido a que la distribuci\u00f3n de evidencia no depende de los par\u00e1metros, la podemos pensar como una constante de normalizaci\u00f3n para que la distribuci\u00f3n resultante integre (sume) uno.\n\nEsto es una buena noticia, porque operacionalmente, no nos debemos preocupar mucho por las constantes que hacen que la distribuci\u00f3n integre a uno. Debemos tener especial cuidado con que **la densidad resultante tenga la misma forma funcional que la previa**.\n\n**Ejercicio:** supongamos que la funci\u00f3n de verosimilitud y la distribuci\u00f3n previa son:\n\n$$\np(x | \\theta) = \\mathcal{N}(x |\\theta, 1) \\qquad p(\\theta) = \\mathcal{N}(\\theta |0, 1)\n$$\n\nEncontrar completamente la densidad de probabilidad posterior\n\n$$\np(\\theta | x) = \\frac{p(x | \\theta) p(\\theta)}{p(x)}.\n$$\n\nEn clase ...\n\nVeremos algunos ejemplos adicionales, pero antes debemos aprender un par de distribuciones m\u00e1s.\n\n## 4. Algunas distribuciones de probabilidad importantes\n\n### 4.1. Distribuci\u00f3n Gamma\n\nEs una distribuci\u00f3n continua, cuyo soporte son los reales positivos ($x\\in\\mathbb{R}_+$), de dos par\u00e1metros $a, b >0$, cuya funci\u00f3n de densidad de probabilidad es:\n\n$$\n\\Gamma(x | a, b) = \\frac{b^a}{\\Gamma(a)} x^{a - 1} \\exp\\{-b x\\}\n$$\n\n\n```python\n# Importar scipy.stats.gamma\nfrom scipy.stats import gamma\n```\n\n\n```python\ngamma?\n```\n\n\n```python\n# Graficar para diferentes valores de a y b\nx = np.linspace(0, 5, 100)\nplt.plot(x, gamma.pdf(x, a=1, scale=1 / 2), label=\"$a=1, b=2$\")\nplt.plot(x, gamma.pdf(x, a=2, scale=1 / 2), label=\"$a=2, b=2$\")\nplt.plot(x, gamma.pdf(x, a=3, scale=1 / 2), label=\"$a=3, b=2$\")\nplt.plot(x, gamma.pdf(x, a=0.5, scale=1 / 1), label=\"$a=0.5, b=1$\")\nplt.legend()\n```\n\nSi $X \\sim \\Gamma(a, b)$, algunos estad\u00edsticos importantes son:\n\n$$\nE[X] = \\frac{a}{b}\n$$\n\n$$\nMode[X] = \\frac{a - 1}{b}, \\quad \\text{para } a > 1\n$$\n\n$$\nVar[X] = \\frac{a}{b^2}\n$$\n\nLa distribuci\u00f3n Gamma es importante porque es la previa conjugada para la **precisi\u00f3n** en una verosimilitud normal.\n\n\u00bfQu\u00e9 es la precisi\u00f3n? Es el rec\u00edproco de la varianza. Es decir, la densidad\n\n$$\n\\mathcal{N}(x | \\mu, \\sigma^2) = \\frac{1}{\\sqrt{2 \\pi \\sigma^2}} \\exp\\left\\{-\\frac{(x - \\mu)^2}{2 \\sigma^2}\\right\\},\n$$\n\nen t\u00e9rminos de la precisi\u00f3n $\\gamma = \\frac{1}{\\sigma^2}$ es\n\n$$\n\\mathcal{N}\\left(x | \\mu, \\frac{1}{\\gamma}\\right) = \\frac{\\sqrt{\\gamma}}{\\sqrt{2 \\pi}} \\exp\\left\\{-\\gamma\\frac{(x - \\mu)^2}{2}\\right\\}.\n$$\n\n\u00bfCu\u00e1l es la previa conjugada respecto a la precisi\u00f3n? La respuesta es la distribuci\u00f3n Gamma. Veamos\n\n$$\np(\\gamma | x) = \\frac{p(x | \\gamma) p(\\gamma)}{p(x)} \\propto p(x | \\gamma) p(\\gamma)\n$$\n\nSi la verosimilitud es normal con par\u00e1metro la precisi\u00f3n: $p(x | \\gamma) = \\mathcal{N}\\left(x | \\mu, \\frac{1}{\\gamma}\\right) \\propto \\gamma^{1/2} \\exp\\left\\{-\\gamma s\\right\\}$, con $s=\\frac{(x - \\mu)^2}{2}$, y la previa es una distribuci\u00f3n gamma: $p(\\gamma) = \\Gamma(\\gamma|a,b) \\propto \\gamma^{a-1} \\exp\\{-b \\gamma\\}$, entonces\n\n$$\np(\\gamma | x) \\propto \\gamma^{1/2} \\exp\\left\\{-\\gamma s\\right\\} \\gamma^{a-1} \\exp\\{-b \\gamma\\} = \\gamma^{a - 1/2} \\exp\\{-(s+b) \\gamma\\},\n$$\n\nesto es:\n\n$$\np(\\gamma | x) = \\Gamma(\\gamma|a+1/2,b+s) \n$$\n\n### 4.2. Distribuci\u00f3n Beta\n\nEs una distribuci\u00f3n continua, cuyo soporte es el intervalo $[0, 1]$, de dos par\u00e1metros $a, b>0$, cuya funci\u00f3n de densidad de probabilidad es:\n\n$$\nB(x | a, b) = \\frac{\\Gamma(a + b)}{\\Gamma(a)\\Gamma(b)} x^{a - 1} (1 - x)^{b - 1}\n$$\n\n\n```python\n# Importar scipy.stats.gamma\nfrom scipy.stats import beta\n```\n\n\n```python\n# Graficar para diferentes valores de a y b\nx = np.linspace(0, 1, 100)\nplt.plot(x, beta.pdf(x, a=0.5, b=0.5), label=\"$a=0.5, b=0.5$\")\nplt.plot(x, beta.pdf(x, a=1, b=1), label=\"$a=1, b=1$\")\nplt.plot(x, beta.pdf(x, a=2, b=2), label=\"$a=2, b=2$\")\nplt.plot(x, beta.pdf(x, a=2, b=5), label=\"$a=2, b=5$\")\nplt.plot(x, beta.pdf(x, a=10, b=5), label=\"$a=10, b=5$\")\nplt.legend(loc=\"upper left\", bbox_to_anchor=(1.05, 1))\n```\n\nSi $X \\sim Beta(a, b)$, algunos estad\u00edsticos importantes son:\n\n$$\nE[X] = \\frac{a}{a + b}\n$$\n\n$$\nMode[X] = \\frac{a - 1}{a + b - 2}, \\quad \\text{para } a > 1, \\text{ y } b > 1\n$$\n\n$$\nVar[X] = \\frac{ab}{(a + b)^2(a + b - 1)}\n$$\n\nLa distribuci\u00f3n Beta es importante porque es la previa conjugada para la verosimilitud de Bernoulli.\n\n___\n#### Distribuci\u00f3n de Bernoulli\n\nSupongamos que tenemos el experimento trillado de tirar una moneda al aire, y sabemos que cae cara con probabilidad $\\theta$ y sello con probabilidad $1 - \\theta$.\n\nAsignamos la VA $X$ de la siguiente manera:\n\n$$\nX = \\left\\{\\begin{array}{cc}\n1 & \\text{si cara} \\\\\n0 & \\text{si sello} \\\\\n\\end{array}\\right.\n$$\n\nUna suposici\u00f3n plausible es que cada tiro de la moneda es independiente.\n\nSupongamos que hemos tirado $5$ veces la moneda obteniendo los siguientes datos:\n\n$$\nD = \\{ 1, 1, 0, 1, 0\\}.\n$$\n\nEn este sentido, la verosimilitud de los datos es:\n\n$$\nP(D | \\theta) = \\prod_{i=1}^{5} P(X_i | \\theta) = \\theta^{3}(1 - \\theta)^{2}.\n$$\n\nEn general, si la moneda cae $N_1$ veces cara y $N_0$ veces sello, la verosimilitud ser\u00eda:\n\n$$\nP(D | \\theta) = \\theta^{N_1}(1 - \\theta)^{N_0},\n$$\n\ndonde $N=N_0+N_1$ es el n\u00famero total de tiros.\n___\n\nYa que tenemos todo claro, veamos que la distribuci\u00f3n Beta es la previa conjugada para la verosimilitud de Bernoulli:\n\n$$\np(\\theta | X) = \\frac{p(X | \\theta) p(\\theta)}{p(X)} \\propto p(X | \\theta) p(\\theta)\n$$\n\nSi la verosimilitud es de Bernoulli con probabilidad $\\theta$: $p(X | \\theta) = \\theta^{N_1}(1 - \\theta)^{N_0}$, y la previa es una distribuci\u00f3n Beta: $p(\\theta) = B(\\theta | a, b) \\propto \\theta^{a - 1} (1 - \\theta)^{b - 1}$, entonces\n\n$$\np(\\theta | X) \\propto \\theta^{N_1}(1 - \\theta)^{N_0} \\theta^{a - 1} (1 - \\theta)^{b - 1} = \\theta^{a + N_1 - 1} (1 - \\theta)^{b + N_0 -1},\n$$\n\nesto es:\n\n$$\np(\\theta | X) = B(\\theta | a + N_1, b + N_0).\n$$\n\n**Ilustraci\u00f3n:**\n\nDefinimos la variable aleatoria (discreta) binomial:\n\n$$\nX = \\left\\{ \\begin{array}{ccc} 1 & \\text{si} & \\text{la moneda cae cara} \\\\\n 0 & \\text{si} & \\text{la moneda cae sello} \\end{array} \\right.\n$$\n\ncon funci\u00f3n de masa de probabilidad dada por:\n\n- $P(X=1) = P(X=1 | \\theta) = \\theta$;\n- $P(X=0) = P(X=0 | \\theta) = 1 - P(X=1) = 1 - \\theta$.\n\nDe esta manera, la funci\u00f3n de verosimilitud satisface:\n\n$$\nP(\\mathcal{D} | \\theta) = \\mathcal{L}(\\theta) = \\theta^{N_1} (1 - \\theta)^{N_0}.\n$$\n\nAcabamos de demostrar que la distribuci\u00f3n Beta sobre el par\u00e1metro $\\theta$ es la previa conjugada:\n\n$$\nP(\\theta) = Beta(\\theta | a, b) \\propto \\theta^{a-1} (1 - \\theta)^{b - 1}\n$$\n\n\n```python\nfrom scipy.stats import bernoulli\n```\n\n\n```python\n# VA real\ntheta_true = 0.7\nX = bernoulli(p=theta_true)\n```\n\n\n```python\n# Datos\ndata = X.rvs(size=100)\ndata\n```\n\n\n\n\n array([1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0,\n 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1,\n 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1,\n 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1,\n 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0])\n\n\n\n\n```python\n# Verosimilitud\nN1 = (data == 1).sum()\nN0 = (data == 0).sum()\ntheta = np.linspace(0, 1, 101)\nlikelihood = theta**N1 * (1 - theta)**N0\n```\n\n\n```python\nplt.plot(theta, likelihood)\nplt.legend(loc=\"upper left\", bbox_to_anchor=(1.05, 1))\n```\n\n\n```python\n# Previa\na0 = 5\nb0 = 5\nprior = beta(a=a0, b=b0)\n```\n\n\n```python\nplt.plot(theta, prior.pdf(theta))\n```\n\n\n```python\n# Posterior\na1 = a0+N1\nb1 = b0+N0\nposterior = beta(a=a0+N1, b=b0+N0)\n```\n\n\n```python\nplt.plot(theta, posterior.pdf(theta))\n```\n\n\n```python\nposterior.mean()\n```\n\n\n\n\n 0.7\n\n\n\n**Recibimos datos de otros 100 tiros de la moneda**\n\n\n```python\nfrom copy import copy\n```\n\n\n```python\nprior = copy(posterior)\n```\n\n\n```python\n# Datos nuevos\ndata_new = X.rvs(size=100)\ndata_new\n```\n\n\n\n\n array([1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1,\n 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1,\n 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0,\n 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1,\n 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0])\n\n\n\n\n```python\nN1 = (data_new == 1).sum()\nN0 = (data_new == 0).sum()\n```\n\n\n```python\n# Posterior con los nuevos datos\na2 = a1 + N1\nb2 = b1 + N0\nposterior = beta(a=a2, b=b2)\n```\n\n\n```python\nplt.plot(theta, posterior.pdf(theta))\nplt.plot(theta, prior.pdf(theta))\n```\n\n\n```python\nprior.a\n```\n\n\n\n\n 0.0\n\n\n\n## Conclusi\u00f3n\n\nPara evitar el c\u00e1lculo de la distribuci\u00f3n de evidencia vimos dos posibilidades:\n\n1. Estimador MAP:\n - Obtenemos el par\u00e1metro m\u00e1s probable dada la evidencia.\n - No se modela la distribuci\u00f3n; obtenemos un valor fijo.\n - No se puede usar para aprendizaje on-line.\n - Podr\u00edamos estimar un valor at\u00edpico.\n \n2. Previas conjugadas:\n - Seleccionamos la distribuci\u00f3n previa para evitar el c\u00e1lculo de la evidencia.\n - Modelamos la distribuci\u00f3n posterior.\n - Se puede usar para aprendizaje on-line.\n - Al selecci\u00f3nar la previa conjugada con el fin de evitar el c\u00e1lculo de la evidencia, es posible que esto vaya en contra de la intuici\u00f3n o conocimiento previo que tengamos acerca del problema. No siempre la previa conjugada es adecuada.\n\n\n\n
    \nCreated with Jupyter by Esteban Jim\u00e9nez Rodr\u00edguez.\n
    \n", "meta": {"hexsha": "61a54a653fc34494e349e79e1eb0ccc8afcd87de", "size": 201444, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "modulo1/tema4/1_bayes_previa_conjugadas.ipynb", "max_stars_repo_name": "G4ll4rd0/mebo2021", "max_stars_repo_head_hexsha": "11096e6bb9c897c38ae02f9c30f90e1c878ee619", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "modulo1/tema4/1_bayes_previa_conjugadas.ipynb", "max_issues_repo_name": "G4ll4rd0/mebo2021", "max_issues_repo_head_hexsha": "11096e6bb9c897c38ae02f9c30f90e1c878ee619", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "modulo1/tema4/1_bayes_previa_conjugadas.ipynb", "max_forks_repo_name": "G4ll4rd0/mebo2021", "max_forks_repo_head_hexsha": "11096e6bb9c897c38ae02f9c30f90e1c878ee619", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-02-03T22:50:53.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-03T22:50:53.000Z", "avg_line_length": 164.4440816327, "max_line_length": 39396, "alphanum_fraction": 0.8899992057, "converted": true, "num_tokens": 6644, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5078118642792044, "lm_q2_score": 0.8688267643505193, "lm_q1q2_score": 0.44120053894050626}} {"text": "```python\nfrom IPython.core.display import HTML\nfrom IPython.display import Image\nimport ipywidgets as widgets\nimport IPython.display as display\n\nimport sympy as sp\nimport numpy as np\nfrom utils import symplot, symdisp, round_expr\n\nHTML(\"\"\"\n\n\"\"\")\n```\n\n# *Circuitos El\u00e9tricos I - Semana 8*\n\n### Circuitos RL e RC de primeira ordem\n\nOs quatro tipos poss\u00edveis de circuitos de primeira ordem est\u00e3o ilustrados na figura abaixo.\n\n\n\nLembrando que um circuito de primeira ordem qualquer, com v\u00e1rios resistores e fontes, por exemplo, pode ser reduzido a um dos quatro circuitos acima fazendo $R=R_{th}$, $v_s=v_{th}$ e $i_s=i_{N}$. Logo, desde que o circuito contenha apenas um elemento indutor ou capacitor, a an\u00e1lise de um circuito de primeira ordem deve ser feita primeiramente determinando-se o circuito equivalente de Th\u00e9venin ou de Norton conectado aos terminais do elemento em quest\u00e3o.\n\n### EDO linear homog\u00eanea\n\nSeja para a corrente passando pelo indutor (circuito RL) ou para a tens\u00e3o aplicada aos terminais do capacitor (circuito RC), a aplica\u00e7\u00e3o das leis de Kirchhoff e da **conven\u00e7\u00e3o passiva** nos circuitos RL e RC sempre levar\u00e3o a uma EDO homog\u00eanea separ\u00e1vel de primeira ordem do tipo\n\n$$ \\begin{equation}\\label{eq1} \\large \\frac{dx(t)}{dt} = -\\frac{1}{\\tau}x(t), \\end{equation}$$ \n\ncom $\\tau$ sendo a constante de tempo do circuito. A solu\u00e7\u00e3o de (\\ref{eq1}) pode ser obtida via integra\u00e7\u00e3o fazendo\n\n$$ \\begin{equation} \\large \\int_{x\\left(t_{0}^+\\right)}^{x(t)} \\frac{d u}{u}=-\\frac{1}{\\tau} \\int_{t_{0}^+}^{t} d v \\end{equation}$$.\n\nLogo, a solu\u00e7\u00e3o da EDO homog\u00eanea ser\u00e1 dada por $$ \\begin{equation} \\large x(t) = x(t_0^+)e^{-\\frac{(t-t_0^+)}{\\tau}}. \\end{equation} $$\n\n### Resposta natural\n\nA resposta natural de circuitos RL/RC corresponder\u00e1 a solu\u00e7\u00e3o da EDO homog\u00eanea, ou seja,\n\n$$ \\begin{equation} \\large i_L(t) = i_L(t_0^+)e^{-\\frac{(t-t_0^+)}{\\tau}}, \\end{equation}$$\n\ncom $\\tau = L/R$ para o circuito RL e\n\n$$\\begin{equation} \\large v_C(t) = v_C(t_0^+)e^{-\\frac{(t-t_0^+)}{\\tau}}, \\end{equation} $$\n\ncom $\\tau = RC$ para o circuito RC.\n\n### Resposta ao degrau\n\nA resposta ao degrau de circuitos RL/RC corresponder\u00e1 a solu\u00e7\u00e3o da EDO homog\u00eanea adicionada da solu\u00e7\u00e3o particular (ou solu\u00e7\u00e3o de regime estacion\u00e1rio). Logo,\n\n$$\\large i_L(t) = i_L(\\infty) + A_0e^{-\\frac{(t-t_0^+)}{\\tau}}, $$\n\ncom $\\tau = L/R$ para o circuito RL e\n\n$$\\large v_C(t) = v_C(\\infty) + A_0e^{-\\frac{(t-t_0^+)}{\\tau}}, $$\n\ncom $\\tau = RC$ para o circuito RC.\n\nA constante $A_o$ da resposta ao degrau pode ser determinada utilizando as condi\u00e7\u00f5es iniciais de corrente no indutor ou tens\u00e3o no capacitor. \n\nDesse modo, para o circuito RL\n\n$$\\large{\\begin{align} i_L(t_0^+) &= i_L(\\infty) + A_0e^{-\\frac{(t_0^+-t_0^+)}{\\tau}}\\nonumber\\\\ &= i_L(\\infty) + A_0 \\nonumber\\\\ \\Rightarrow A_0 &= i_L(t_0^+)- i_L(\\infty)\\nonumber\n\\end{align} }$$\n\ne para o circuito RC\n\n$$\\large{\\begin{align} v_C(t_0^+) &= v_C(\\infty) + A_0e^{-\\frac{(t_0^+-t_0^+)}{\\tau}}\\nonumber\\\\ &= v_C(\\infty) + A_0 \\nonumber\\\\ \\Rightarrow A_0 &= v_C(t_0^+)- v_C(\\infty)\\nonumber\n\\end{align} }$$\n\n### Resposta geral\n\nA resposta geral de circuitos RL/RC corresponder\u00e1 \u00e0s express\u00f5es,\n\n$$ \\begin{equation} \\large i_L(t) = i_L(\\infty) + \\left[i_L(t_0^+)- i_L(\\infty)\\right]e^{-\\frac{(t-t_0^+)}{\\tau}},\\end{equation} $$\n\ncom $\\tau = L/R$ para o circuito RL e\n\n$$ \\begin{equation} \\large v_C(t) = v_C(\\infty) + \\left[v_C(t_0^+)- v_C(\\infty)\\right]e^{-\\frac{(t-t_0^+)}{\\tau}},\\end{equation} $$\n\ncom $\\tau = RC$ para o circuito RC.\n\nFinalmente, para qualquer circuito de primeira ordem,\n\n$$\\begin{equation} \\large x(t) = x(\\infty) + \\left[x(t_0^+)-x(\\infty)\\right]e^{-\\frac{(t-t_0^+)}{\\tau}},\\end{equation} $$\n\ncom $\\tau$ sendo a constante de tempo associada a este circuito.\n\n### Problema 1\n\nPara circuito da figura abaixo, a chave encontra-se conectada ao terminal $a$ h\u00e1 muito tempo. Em $t=0s$, posi\u00e7\u00e3o da chave muda do ponto $a$ para o ponto $b$. Em $t=20 ms$, a chave \u00e9 desconectada do ponto $b$, permanecendo aberta. Determine:\n\na. A corrente no indutor para $0^+s\\leq t \\leq 20^-ms$.\\\nb. A tens\u00e3o nos terminais indutor em $t=0^-s$ e $t=0^+s$.\\\nc. A tens\u00e3o nos terminais do resistor de 40 $\\Omega$ em $t=0^-s$ e $t=0^+s$.\\\nd. A corrente no indutor para $t\\geq20^+ms$.\\\ne. A tens\u00e3o nos terminais do indutor em $t=20^+ms$.\n\n\n\n\nSimula\u00e7\u00e3o do circuito dispon\u00edvel no link: https://tinyurl.com/yfs69qqu\n\n#### Visualiza\u00e7\u00e3o das curvas $i_L(t)$, $v_L(t)$ e $p_L(t)$\n\n\n```python\n# par\u00e2metros do circuito\nL = 1\nvth = -80e-3\nRth = 18\n\n# informa\u00e7\u00f5es obtidas pela an\u00e1lise do circuito\niL_inf = vth/Rth # iL(infinito)\niL_t0 = 2.9e-3 # iL(t0+)\n\n\u03c4 = L/Rth # constante de tempo do circuito RL\nt0 = 0 # instante inicial\n\nt = sp.symbols('t', real=True) # define a vari\u00e1vel tempo\n\niL = sp.Piecewise( (iL_t0, t<0), # iL(t) para t=0) )*1000 # iL(t) para t>t0+\n\nsymdisp('i_L(t) =', round_expr(iL.simplify(), 3), 'mA')\n\nintervalo = np.linspace(t0-2*\u03c4, t0+8*\u03c4, 400)\nsymplot(t, iL, intervalo, funLabel = '$i_L(t)$ [mA]')\n```\n\n\n```python\nvL = L*sp.diff(iL,t)\nvL = vL.simplify()\n\nsymdisp('v_L(t) = ', vL, ' mV')\n\nsymplot(t, vL, intervalo, funLabel = '$v_L(t)$ [mV]')\n```\n\n\n```python\npL = vL*iL\npL = pL.simplify()\n\nsymdisp('p_L(t) = ', round_expr(pL,3), ' \u03bcW')\n\nsymplot(t, pL, intervalo, funLabel = '$p_L(t)$ [\u03bcW]')\n```\n\n\n```python\nimport ipywidgets as widgets\nimport IPython.display as display\n\nimg1 = open('./figures/J11C2a.gif', 'rb').read()\nimg2 = open('./figures/J11C2b.gif', 'rb').read()\n\nwi1 = widgets.Image(value=img1, format='gif', width=600, height=400)\nwi2 = widgets.Image(value=img2, format='gif', width=600, height=400)\n\nsidebyside = widgets.HBox([wi1, wi2])\ndisplay.display(sidebyside)\n```\n\n### Problema 2\n\nNo circuito da figura abaixo, antes de conectado ao circuito, o capacitor possui uma tens\u00e3o inicial de $v_C=10~V$. Em $t=0s$, o capacitor \u00e9 conectado ao circuito. Determine:\n\na. O circuito equivalente de Th\u00e9venin do ponto de vista dos terminais do capacitor.\\\nb. A tens\u00e3o no capacitor $v_C(t)$ para $t\\geq 0^+s$.\\\nc. A tens\u00e3o $v_x(t)$ para $t\\geq 0^+s$. \n\n\n\nSimula\u00e7\u00e3o do circuito dispon\u00edvel no link: https://tinyurl.com/yzhty8w3\n", "meta": {"hexsha": "2a0c9e168fad9f4364d4a052666252e2ec5f3c73", "size": 54458, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Jupyter notebooks/Circuitos Eletricos I - Semana 8.1.ipynb", "max_stars_repo_name": "Jefferson-Lopes/ElectricCircuits", "max_stars_repo_head_hexsha": "bf2075dc0731cacece75f7b0b378c180630bdf85", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Jupyter notebooks/Circuitos Eletricos I - Semana 8.1.ipynb", "max_issues_repo_name": "Jefferson-Lopes/ElectricCircuits", "max_issues_repo_head_hexsha": "bf2075dc0731cacece75f7b0b378c180630bdf85", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Jupyter notebooks/Circuitos Eletricos I - Semana 8.1.ipynb", "max_forks_repo_name": "Jefferson-Lopes/ElectricCircuits", "max_forks_repo_head_hexsha": "bf2075dc0731cacece75f7b0b378c180630bdf85", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 124.6178489703, "max_line_length": 14364, "alphanum_fraction": 0.8552829704, "converted": true, "num_tokens": 2262, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.7956580927949806, "lm_q1q2_score": 0.44116891266155966}} {"text": "\n\n## Data-driven Design and Analyses of Structures and Materials (3dasm)\n\n## Lecture 9\n\n### Miguel A. Bessa | M.A.Bessa@tudelft.nl | Associate Professor\n\n**What:** A lecture of the \"3dasm\" course\n\n**Where:** This notebook comes from this [repository](https://github.com/bessagroup/3dasm_course)\n\n**Reference for entire course:** Murphy, Kevin P. *Probabilistic machine learning: an introduction*. MIT press, 2022. Available online [here](https://probml.github.io/pml-book/book1.html)\n\n**How:** We try to follow Murphy's book closely, but the sequence of Chapters and Sections is different. The intention is to use notebooks as an introduction to the topic and Murphy's book as a resource.\n* If working offline: Go through this notebook and read the book.\n* If attending class in person: listen to me (!) but also go through the notebook in your laptop at the same time. Read the book.\n* If attending lectures remotely: listen to me (!) via Zoom and (ideally) use two screens where you have the notebook open in 1 screen and you see the lectures on the other. Read the book.\n\n**Optional reference (the \"bible\" by the \"bishop\"... pun intended \ud83d\ude06) :** Bishop, Christopher M. *Pattern recognition and machine learning*. Springer Verlag, 2006.\n\n**References/resources to create this notebook:**\n* [Car figure](https://korkortonline.se/en/theory/reaction-braking-stopping/)\n\nApologies in advance if I missed some reference used in this notebook. Please contact me if that is the case, and I will gladly include it here.\n\n## **OPTION 1**. Run this notebook **locally in your computer**:\n1. Confirm that you have the 3dasm conda environment (see Lecture 1).\n\n2. Go to the 3dasm_course folder in your computer and pull the last updates of the [repository](https://github.com/bessagroup/3dasm_course):\n```\ngit pull\n```\n3. Open command window and load jupyter notebook (it will open in your internet browser):\n```\nconda activate 3dasm\njupyter notebook\n```\n4. Open notebook of this Lecture.\n\n## **OPTION 2**. Use **Google's Colab** (no installation required, but times out if idle):\n\n1. go to https://colab.research.google.com\n2. login\n3. File > Open notebook\n4. click on Github (no need to login or authorize anything)\n5. paste the git link: https://github.com/bessagroup/3dasm_course\n6. click search and then click on the notebook for this Lecture.\n\n\n```python\n# Basic plotting tools needed in Python.\n\nimport matplotlib.pyplot as plt # import plotting tools to create figures\nimport numpy as np # import numpy to handle a lot of things!\nfrom IPython.display import display, Math # to print with Latex math\n\n%config InlineBackend.figure_format = \"retina\" # render higher resolution images in the notebook\nplt.style.use(\"seaborn\") # style for plotting that comes from seaborn\nplt.rcParams[\"figure.figsize\"] = (8,4) # rescale figure size appropriately for slides\n```\n\n## Outline for today\n\n* Linear models for regression\n - A practical session on how to train linear regression models\n\n**Reading material**: This notebook + Chapter 11\n\n## Today's lecture is going to be more practical\n\nSince we covered the fundamentals of Bayesian and non-Bayesian machine learning...\n\n* Today we will focus on how to train **linear regression models** using [scikit-learn](https://scikit-learn.org)\n\n* In a later lecture we will derive the models that we are going to cover today.\n\nAs we learned in Lecture 2, let's load the pandas dataframe that is in the \"docs\" folder of Lecture 9:\n\n\n```python\nimport pandas as pd\n# read csv data provided by someone else (this time I also specify that the first column provides the indices)\ncar_prob_df = pd.read_csv(\"docs/data_for_car_prob.csv\", index_col=0)\nprint(car_prob_df)\n```\n\n x y\n 0 9.516939 29.749036\n 1 72.398757 642.132203\n 2 17.950326 36.648484\n 3 9.440853 18.604106\n 4 78.791008 769.656168\n 5 16.961121 57.971010\n 6 65.410368 559.093313\n 7 58.671099 463.686613\n 8 21.550603 92.242676\n 9 36.866913 197.688573\n 10 15.728748 56.885233\n 11 58.511494 388.753795\n 12 57.419190 399.807488\n 13 38.459157 213.181519\n 14 8.841742 20.387384\n 15 60.733051 516.341724\n 16 49.256663 307.931956\n 17 35.895121 181.123049\n 18 79.195652 750.178284\n 19 69.156669 553.153541\n 20 77.634896 746.031880\n 21 9.254011 20.810698\n 22 15.451468 39.872527\n 23 14.438247 42.118771\n 24 13.410999 44.775122\n 25 53.747057 375.013937\n 26 10.283719 19.438868\n 27 82.005477 742.336845\n 28 81.805562 706.620282\n 29 51.837742 345.212876\n 30 20.283785 65.303165\n 31 28.359647 155.185137\n 32 74.993715 676.628982\n 33 21.827564 81.150935\n 34 70.519111 700.520033\n 35 74.208532 622.453560\n 36 14.518958 40.927570\n 37 13.357644 39.770922\n 38 75.346253 707.973754\n 39 44.923956 251.300805\n 40 26.801159 124.098654\n 41 29.906265 118.100900\n 42 40.226356 215.082100\n 43 66.282662 537.845048\n 44 47.342777 308.558833\n 45 3.087674 5.947997\n 46 21.254611 101.295276\n 47 46.939484 345.778352\n 48 38.875692 219.095582\n 49 76.705452 742.720134\n\n\nAs before, we can separate the data into inputs (features) $x$ and outputs $y$ (targets)\n\n\n```python\nData_x = car_prob_df['x'].values # select the input VALUES from your dataframe into Data_x\nData_y = car_prob_df['y'].values # select the output VALUES from your dataframe inta Data_y\nprint(\"Data_x is:\\n\",Data_x)\nprint(\"\\nData_y is:\\n\",Data_y)\n```\n\n Data_x is:\n [ 9.51693942 72.39875748 17.95032583 9.44085299 78.79100778 16.96112056\n 65.4103675 58.67109927 21.55060313 36.86691294 15.72874781 58.51149357\n 57.41918959 38.45915667 8.84174221 60.73305107 49.25666345 35.89512052\n 79.19565172 69.15666925 77.63489641 9.25401128 15.45146824 14.43824684\n 13.41099874 53.74705712 10.28371886 82.00547705 81.80556249 51.8377421\n 20.28378484 28.35964692 74.99371524 21.82756352 70.51911096 74.20853195\n 14.51895792 13.35764354 75.34625316 44.92395642 26.80115926 29.90626522\n 40.22635624 66.28266205 47.34277718 3.08767411 21.25461134 46.93948443\n 38.87569199 76.70545196]\n \n Data_y is:\n [ 29.74903647 642.13220315 36.64848446 18.60410602 769.65616843\n 57.97101034 559.09331318 463.68661322 92.24267632 197.68857288\n 56.88523327 388.75379474 399.80748803 213.18151905 20.38738432\n 516.34172363 307.93195589 181.12304936 750.17828361 553.15354059\n 746.03187971 20.81069833 39.87252654 42.11877078 44.77512244\n 375.01393668 19.43886782 742.33684483 706.62028237 345.21287569\n 65.30316533 155.18513747 676.62898211 81.15093549 700.52003305\n 622.45356019 40.92757044 39.77092163 707.97375405 251.30080489\n 124.09865438 118.10089977 215.08209978 537.84504756 308.55883254\n 5.94799685 101.29527607 345.77835213 219.09558165 742.72013356]\n\n\nAnd we can plot the data:\n\n\n```python\nfig_car_data, ax_car_data = plt.subplots() # create a plot\nax_car_data.plot(Data_x, Data_y, 'b.')\nax_car_data.set_xlabel(\"x (m/s)\", fontsize=20) # create x-axis label with font size 20\nax_car_data.set_ylabel(\"y (m)\", fontsize=20) # create y-axis label with font size 20\nax_car_data.set_title(\"Car stopping distance problem\", fontsize=20); # create title with font size 20\n```\n\n## Supervised learning: regression models\n\nAs we have been discussing, when we do regression via supervised learning we want to:\n\n* create a machine learning model\n* train it on known data (known inputs $x$ and outputs $y$)\n* predict for new (unseen) data points, i.e. predict $y^*$ for a new value of $x$.\n\nToday we will talk about the simplest models: **linear regression**.\n\n## Linear regression models\n\nLinear regression models encompass a class of machine learning methods that is larger than you might think...\n\nAs we will see, despite being called \"linear\" these models can do more than fitting a simple \"line\" to our data.\n\nFor now, let's consider 1d datasets, i.e. where we have one input $x$ and one output $y$.\n\n### Simplest 1d linear regression model: fitting a line to your data\n\n1. Observation distribution:\n\nUsually, assumed as a Gaussian distribution,\n\n$$\np(y|x, \\mathbf{z}) = \\mathcal{N}(y| \\mu_{y|z} = \\mathbf{w}^T \\boldsymbol{\\phi}(x), \\sigma_{y|z}^2 = \\sigma^2)\n$$\n\nwhere $\\mathbf{z} = (\\mathbf{w}, \\sigma)$ are all the hidden rv's of the model, i.e. the model parameters.\n* the vector $\\mathbf{w} = [w_0, w_1]^T$ includes the **bias** term $w_0$ and the **weight** $w_1$.\n* the vector $\\boldsymbol{\\phi}(x) = [1, x]^T$ includes the **basis functions**.\n\n2. A chosen Prior distribution on each hidden rv of $\\mathbf{z}$:\n\nUsually, the prior on $w_0$ and $\\sigma$ is the Uniform distribution.\n\nHowever, the prior on the weight $w_1$ is often chosen as something else (but it can also be Uniform).\n\nDoes this model remind you of something we did?\n\n* Car stopping distance problem when we knew one of the rv's and fixed $x$!\n\nNote that,\n\n$$\n\\begin{align}\np(y|x, \\mathbf{z}) &= \\mathcal{N}(y| \\mathbf{w}^T \\boldsymbol{\\phi}(x), \\sigma^2) \\\\\n&= \\mathcal{N}(y| w_0 + w_1 x, \\sigma^2) \\\\\n\\end{align}\n$$\n\nwhere we previously called $w_0 \\equiv b$, $w_1 \\equiv z$ and $\\sigma^2 \\equiv \\sigma_{y|z}^2$.\n\nTherefore, the only difference is that we now start to consider more than one rv, so we group them into vector $\\mathbf{z} = (\\mathbf{w}, \\sigma)$.\n\nAbout notation:\n\n* We can also write $\\mathbf{z}^T = [\\mathbf{w}^T, \\sigma]$.\n\n### 1d linear regression models with different basis functions: fitting a polynomial to your data\n\nHowever, we know that usually a straight line does not provide a good fit to most data sets.\n\nA very important realization:\n\n* The basis functions vector $\\boldsymbol{\\phi}(x)$ does not need to be a *linear transformation*.\n\n* It could be a polynomial or any other **nonlinear** transformation of the feature (input) $x$.\n\nAs long as the parameters of the basis functions vector $\\boldsymbol{\\phi}(x)$ are **fixed**, the model remains **linear in the parameters**, even if is not linear in the input (feature). That's why we still call this a **linear model**.\n\nHere's how our **linear** regression model looks like for a polynomial basis functions vector $\\boldsymbol{\\phi}(x)$.\n\n1. Observation distribution:\n\n$$\np(y|x, \\mathbf{z}) = \\mathcal{N}(y| \\mu_{y|z} = \\mathbf{w}^T \\boldsymbol{\\phi}(x), \\sigma_{y|z}^2 = \\sigma^2)\n$$\n\nwhere $\\mathbf{z} = (\\mathbf{w}, \\sigma)$ are all the hidden rv's of the model, i.e. the model parameters.\n* the vector $\\mathbf{w} = [w_0, w_1, w_2 ..., w_d]^T$ includes the **bias** term $w_0$ and the remaining **weights** $w_i$ with $i=1,..., d$.\n* the vector $\\boldsymbol{\\phi}(x) = [1, x, x^2, ..., x^d]^T$ includes the **basis functions**, which now correspond to a polynomial of degree $d$.\n\n2. A chosen Prior distribution for each hidden rv of $\\mathbf{z}$, as mentioned previously.\n\n## Linear regression models from a Bayesian perspective\n\nThe choice of likelihood and prior determines what is the linear regression model that you are choosing!\n\n| Likelihood | Prior (on the weights) | Posterior | Name of the model | Book section |\n|--- |--- |--- |--- |--- |\n| Gaussian | Uniform | Point estimate | Least Squares regression | 11.2.2 |\n| Gaussian | Gaussian | Point estimate | Ridge regression | 11.3 |\n| Gaussian | Laplace | Point estimate | Lasso regression | 11.4 |\n| Student-$t$ | Uniform | Point estimate | Robust regression | 11.6.1 |\n| Laplace | Uniform | Point estimate | Robust regression | 11.6.2 |\n| Gaussian | Gaussian | Gaussian | Bayesian linear regression | 11.7 |\n\nWe will derive some of these models in a later class. Today, we focus on the practical aspects!\n\n## Training (fitting) a linear model with Scikit-learn\n\nLet's see how to use [scikit-learn](https://scikit-learn.org) to train linear regression models.\n\n* [scikit-learn](https://scikit-learn.org) is a well-documented and user-friendly library that is great for introducing machine learning.\n\n* You should really read the documentation.\n - It includes many useful examples.\n - It provides a short introduction to common machine learning algorithms.\n\n## Example 1: training a linear model for the car stopping distance problem\n\n\n\nLet's start by importing from scikit-learn the simplest linear regression model:\n\n* Least Squares Regression.\n\nAnd let's consider the simplest basis function:\n\n* A line (polynomial of degree 1).\n\n\n```python\nfrom sklearn.linear_model import LinearRegression # For Least Squares Regression\nfrom sklearn.preprocessing import PolynomialFeatures # For Polynomial basis functions\nfrom sklearn.pipeline import make_pipeline # to link different objects\n```\n\nNow let's define the model (Least Squares Regression + polynomial of degree 1)\n\n\n```python\n# We start by defining the model (polynomial basis + Least Squares Regression)\ndegree = 1 # degree of polynomial we want to fit\npoly_model = make_pipeline(PolynomialFeatures(degree),LinearRegression())\n```\n\nThen we train the model for our data (input Data_x and output Data_y that were loaded with pandas)\n\n\n```python\n# Uncomment line below (this is just for students to understand: don't panic when encountering an error!)\n#poly_model.fit(Data_x,Data_y) # but it gives an ERROR!\n```\n\nThis gives an error! Fortunately, scikit-learn tells us what happened...\n\nScikit-learn expects the inputs to be formatted as a 2D array (matrix), instead of a 1D array (vector).\n\nThis happens because usually we fit machine learning models for multidimensional inputs.\n\n\n```python\n# Reshape the input vector into a 2D array:\nData_X = np.reshape(Data_x, (-1, 1)) # we use capital letters for matrices and lower case for vectors\n```\n\n\n```python\nprint(Data_X)\n```\n\n [[ 9.51693942]\n [72.39875748]\n [17.95032583]\n [ 9.44085299]\n [78.79100778]\n [16.96112056]\n [65.4103675 ]\n [58.67109927]\n [21.55060313]\n [36.86691294]\n [15.72874781]\n [58.51149357]\n [57.41918959]\n [38.45915667]\n [ 8.84174221]\n [60.73305107]\n [49.25666345]\n [35.89512052]\n [79.19565172]\n [69.15666925]\n [77.63489641]\n [ 9.25401128]\n [15.45146824]\n [14.43824684]\n [13.41099874]\n [53.74705712]\n [10.28371886]\n [82.00547705]\n [81.80556249]\n [51.8377421 ]\n [20.28378484]\n [28.35964692]\n [74.99371524]\n [21.82756352]\n [70.51911096]\n [74.20853195]\n [14.51895792]\n [13.35764354]\n [75.34625316]\n [44.92395642]\n [26.80115926]\n [29.90626522]\n [40.22635624]\n [66.28266205]\n [47.34277718]\n [ 3.08767411]\n [21.25461134]\n [46.93948443]\n [38.87569199]\n [76.70545196]]\n\n\nAfter reshaping the input as a 2D array, we see that we can fit the model!\n\n\n```python\npoly_model.fit(Data_X,Data_y) # now we were able to train (fit) our linear model to the data!\n```\n\n\n\n\n Pipeline(steps=[('polynomialfeatures', PolynomialFeatures(degree=1)),\n ('linearregression', LinearRegression())])\n\n\n\nThat's it! Here's your first ML model: fitting a straight line \ud83d\ude06\n\nNow that we have a model, we can predict the output $y^*$ for any new input point $x^*$.\n\nIn particular, we can predict the output for each of the input points $x$ that we used for training the model (i.e. at Data_X).\n\n\n```python\ny_pred = poly_model.predict(Data_X) # In scikit-learn, predicting from a model is a one-liner\n```\n\nDone! These are the predictions for all your training points.\n\nBut we can also predict the output $y^*$ for other points.\n\nThis enables us to visualize the model by predicting the output for a uniformly spaced set of points.\n\n\n```python\n# Now create linearly spaced points for plotting our linear model\nx_plot = np.linspace(0, 90, 200) # 200 points uniformly spaced\ny_plot = poly_model.predict(np.reshape(x_plot, (-1, 1))) # prediction of those points (note the reshape again)\n```\n\nFinally, we can just plot the data, the predictions for each input training point, and the linear model.\n\n\n```python\n# The usual plotting style for this model and the data.\nfig_poly, ax_poly = plt.subplots() # create a plot\nax_poly.plot(Data_x, Data_y, 'b.', markersize=12,label=\"Data\") # Markers locating data points)\nax_poly.plot(Data_x, y_pred, 'm*', markersize=12,label=\"Predictions\") # Markers locating prediction points)\nlegend_str = \"Linear regression with Polynomial of degree \" + str(degree)\nax_poly.plot(x_plot, y_plot, 'm-', linewidth=2,label=legend_str) # polynomial interpolation plotted\nax_poly.set_xlabel(\"x (m/s)\", fontsize=20) # create x-axis label with font size 20\nax_poly.set_ylabel(\"y (m)\", fontsize=20) # create y-axis label with font size 20\nax_poly.set_title(\"Car stopping distance problem\", fontsize=20) # create title with font size 20\nax_poly.legend(loc='upper left', fontsize=15) # replot legend\nplt.close(fig_poly) # do not plot the figure now. We will show it in a later cell\n```\n\n\n```python\nfig_poly # just call the figure that we created in the previous cell.\n```\n\nNow we can see all the things that we have done in a single plot!\n\n## Exercise 1\n\n1. Put it all together and create a linear model using a polynomial of degree 2 for the same data.\n\n\n2. Compare your plot with the one obtained for the straight line (polynomial of degree 1).\n\n\n3. Play a bit with your code by changing the degree of the polynomial. What happens?\n\n\n```python\n# Write your code for Exercise 1:\n\n# until here.\n```\n\nIf you keep increasing the polynomial degree, you see that the prediction gets worse!\n\nThis is called **overfitting**.\n\n- Overfitting is a natural consequence of having a model that is more complex than it should be!\n\n- As we know, the mean of the data that originates from the car stopping distance problem is generated with a quadratic model: $y = z_1 x + z_2 x^2$\n * Therefore, we don't need additional complexity to describe the *mean* of the data!\n\nAlso, the fit is better **within the domain** that we used for training ($x\\in [3, 83]$) than away from it.\n\nIn other words, we **interpolate** better than we **extrapolate**.\n\n* Nevertheless, overfitting is also an issue when we are interpolating...\n - See this by making the degree very high, e.g. 30, and plot for $x\\in [5, 80]$.\n\nIt might be surprising, but this is an important issue in ML.\n\nIt is very common to use models that are complicated (have many parameters) and that perform poorly even when interpolating, but especially when extrapolating!\n\n## Example 2: linear model for a noiseless problem\n\nConsider now a problem not governed by a polynomial law and without uncertainty.\n\nLet's consider the function $x\\sin(x)$ in the domain $x\\in[0,10]$. We did this in Lecture 2!\n\n\n```python\n# 1. Define the function f(x) = x sin(x)\ndef f(x):\n return x * np.sin(x)\n# 2. Create a vector of 50 points that are uniformly spaced between 0 and 10\nn_data = 50 # number of points for plotting the function\nx_data = np.linspace(0, 10, n_data) # uniformly spaced points\n# 3. Compute the output vector:\ny_data = f(x_data)\n# 4. Plot the function and the data\nfig1, ax1 = plt.subplots() # This opens a new figure\n# Plot points and interpolate them:\nax1.plot(x_data, y_data, 'ko:', markersize=6, linewidth=2,label=u'ground truth: $f(x) = x\\,\\sin(x)$')\nax1.set_xlabel('$x$', fontsize=20) # label of the x axis\nax1.set_ylabel('$f(x)$', fontsize=20) # label of the y axis\nax1.legend(loc='upper left', fontsize=15) # plot legend in the upper left corner\n```\n\nWith lots of data, even linear interpolation between the points can approximate the function well.\n\nHowever, what if we use just a few points from our dataset x_data?\n\n\n```python\nn_train = 5 # points to train the algorithm\nx_train = np.linspace(0, 10, n_train) # 5 points uniformly distributed\ny_train = f(x_train)\nax1.plot(x_train, y_train, 'r*', markersize=12,label=\"Training points\") # Markers locating training points\nax1.plot(x_train, y_train, 'g-', linewidth=2, label=u'local linear interpolation') # linear interpolation\nax1.legend(loc='upper left', fontsize=15) # replot legend\nfig1 # replot fig1 now overlaying the plot in the previous cell\n```\n\nThis is called local interpolation because each line only depends on the two points it is connecting (not on the other points).\n\n* This is not an ML model! It's different from the linear model we created for the car stopping distance problem.\n\nSo, let's train a linear ML model using a polynomial of degree 4 as basis function for this example.\n\n\n```python\ndegree = 4 # degree of polynomial\n\npoly_model = make_pipeline(PolynomialFeatures(degree), LinearRegression()) # model\n\nX_train = np.reshape(x_train, (-1, 1)) # convert input vector into 2d array\n\npoly_model.fit(X_train,y_train) # fit the polynomial to our 5 training points\n#y_pred = poly_model.predict(X_train) # prediction of our polynomial\n\nX_data = np.reshape(x_data, (-1, 1)) # Don't forget to convert to a 2d array\ny_pred = poly_model.predict(X_data) # prediction our model for all 50 data points\n\n# Plot x_data and prediction as a blue line:\nax1.plot(x_data, y_pred, 'b-', linewidth=2, label=\"Polynomial of degree %d prediction\" % degree)\n\n# Replot figure and legend:\nax1.legend(loc='upper left', fontsize=15)\nfig1\n```\n\nOur polynomial (blue) is clearly different to the function that we want to \"learn\", i.e. $x \\sin(x)$.\n\nHow do we evaluate the quality of our approximation?\n\n* By evaluating the error of our polynomial model in the points that we didn't use in the fit.\n\nTwo common metrics are $R^2$ and $\\text{MSE}$ (you will have to search for them and explain them!)\n\n\n```python\n# Import error metrics:\nfrom sklearn.metrics import mean_squared_error, r2_score\n\n# Compute MSE and R2 for the polynomial model we fitted\nmse_value = mean_squared_error(y_data, y_pred)\nr2_value = r2_score(y_data, y_pred)\n\nprint('MSE for polynomial = ', mse_value)\nprint('R2 score for polynomial = ', r2_value)\n```\n\n MSE for polynomial = 5.826961574280784\n R2 score for polynomial = 0.578691860751741\n\n\nAs expected, these predictions are not great because:\n\n* We want $\\text{MSE}$ to be as low as possible\n\n* The closer $R^2$ is to 1.0 the better\n\nYou will dive deeper into this when solving the **Midterm Project**.\n\n## Example 3: linear model for noisy datasets\n\nLet's consider one last case where we perturb the function of Example 2 $f(x) = x\\sin(x)$ with noise.\n\n* This is similar to what happens in the car stopping distance problem.\n\n* Here, the difference is that at every point the noise is random (not as predictable as before)\n\nLet's \"fabricate\" such dataset.\n\n\n```python\nseed = 1987 # set a random seed so that everyone gets the same result\nnp.random.seed(seed)\n\n# Let's perturb every y_data point with Gaussian noise\nrandom_std = 0.5 + 1.0 * np.random.random(y_data.shape)\n\n# Then, take the random value for STD from 0.5 to 1.5 for each\n# data point and create noise following a Gaussian distribution with\n# that STD at that point:\nnoise = np.random.normal(0, random_std)\n\n# The perturbed data becomes:\ny_noisy_data = y_data + noise\n```\n\nFor comparison, we plot the noisy data with the noiseless function that we would like to discover $x \\sin(x)$:\n\n\n```python\nfig2, ax2 = plt.subplots() # This opens a new figure\n\n# Plot the noiseless function (\"the ground thruth\")\nax2.plot(x_data, y_data, 'r:', linewidth=2,\n label=u'ground truth: $f(x) = x\\,\\sin(x)$')\n\n# Plot the noisy dataset that we are given:\nplt.errorbar(x_data, y_noisy_data, random_std, fmt='kX',\n markersize=6, label=u'noisy dataset')\n\nax2.set_xlabel('$x$', fontsize=20) # label of the x axis\nax2.set_ylabel('$f(x)$', fontsize=20) # label of the y axis\nax2.legend(loc='upper left', fontsize=15) # plot legend in the upper left corner\n```\n\nNote a couple of things:\n\n* The black \"x\" marks the actual measured value.\n\n* The black bars indicate the noise in each data point (each data point has a different noise value). Formally, we call this aleatoric uncertainty.\n - In the plot, I centered the error bars around the measured value. Note that the bars do not have the same length (different standard deviation).\n\n### A note on data preprocessing\n\nUsually, when we are given a dataset we need to find a way to **train** and **test** our model using that data.\n\nHowever, to test our model we have to use data that we have not used in training, otherwise we would be cheating!\n\nThis is done by splitting the dataset (in this case x_data) into two sets:\n\n1. **Training** set (for example: 75% of the dataset)\n\n\n2. **Test** set with the remaining points of the dataset\n\nScikit-learn has a very easy way of doing this:\n\n\n```python\nfrom sklearn.model_selection import train_test_split\n\nX_data = np.reshape(x_data,(-1,1)) # a 2D array that scikit-learn likes\n# Let's split the data points into 10% for the training set (5 points)\n# and the rest for the test set:\nX_train, X_test, y_train, y_test = train_test_split(X_data,\n y_noisy_data, test_size=0.90,\n random_state=seed)\n```\n\nNote that the train_test_split module of scikit-learn picks points pseudo-randomly according to the random_state seed value.\n\nLet's visualize the training and testing sets:\n\n\n```python\nx_train = X_train[:] # converting back to 1D array for plotting\nx_test = X_test[:] # converting back to 1D array for plotting\n# Plot the noisy training dataset:\nax2.plot(x_train, y_train, 'g*', markersize=18, label=\"Training points\") # Markers locating training points\nax2.plot(x_test, y_test, 'bs', markersize=6,label=\"Testing points\") # Markers locating training points\nax2.set_xlabel('$x$', fontsize=20) # label of the x axis\nax2.set_ylabel('$f(x)$', fontsize=20) # label of the y axis\nax2.legend(loc='upper left', fontsize=15) # plot legend in the upper left corner\nfig2\n```\n\nLet's create a new figure with less clutter by just plotting the ground truth function and the training points.\n\n\n```python\nfig3, ax3 = plt.subplots() # This opens a new figure\n\n# Plot the noiseless function (\"the ground thruth\")\nax3.plot(x_data, y_data, 'r:', linewidth=2, label=u'ground truth: $f(x) = x\\,\\sin(x)$')\nax3.plot(x_train, y_train, 'g*', markersize=18, label=\"Training points\") # Markers locating training points\nax3.set_xlabel('$x$', fontsize=20) # label of the x axis\nax3.set_ylabel('$f(x)$', fontsize=20) # label of the y axis\nax3.legend(loc='upper left', fontsize=15) # plot legend in the upper left corner\n```\n\n## Exercise 2\n\nFit a polynomial of degree 4 to this training data and calculate the $R^2$ and $\\text{MSE}$ metrics for the testing data.\n\n\n```python\n# Write your code for Exercise 2:\n\n# until here.\n```\n\nWell done...\n\nYet, this does not seem like a great result, does it?\n\nThe $R^2$ value is so bad that it is even negative!\n\n* What explains this result?\n\n* Can we do something to fix this while still using polynomials?\n\n* If we used more points would that help?\n\n* What if we increased the degree of the polynomial?\n\n### You will explore these and other things in Part 1 of the Midterm Project...\n\nHave fun!\n\n## Solution for Exercise 1\n\n```python\ndegree = 2\npoly_model = make_pipeline(PolynomialFeatures(degree),LinearRegression()) # model\npoly_model.fit(np.reshape(Data_x, (-1, 1)),Data_y) # training (fitting)\ny_pred = poly_model.predict(Data_X) # predict y for Data_x\nx_plot = np.linspace(0, 90, 200) # 200 points uniformly spaced\ny_plot = poly_model.predict(np.reshape(x_plot, (-1, 1))) # prediction for those points\nfig_poly, ax_poly = plt.subplots() # create a plot\nax_poly.plot(Data_x, Data_y, 'b.', markersize=12,label=\"Data\") # Markers locating data points)\nax_poly.plot(Data_x, y_pred, 'm*', markersize=12,label=\"Predictions\") # Markers locating prediction points)\nlegend_str = \"Linear regression with Polynomial of degree \" + str(degree)\nax_poly.plot(x_plot, y_plot, 'm-', linewidth=2,label=legend_str) # polynomial interpolation plotted\nax_poly.set_xlabel(\"x (m/s)\", fontsize=20) # create x-axis label with font size 20\nax_poly.set_ylabel(\"y (m)\", fontsize=20) # create y-axis label with font size 20\nax_poly.set_title(\"Car stopping distance problem\", fontsize=20) # create title with font size 20\nax_poly.legend(loc='upper left') # replot legend\n```\n\n### Solution to Exercise 2\n\n``` python\ndegree = 4 # degree of polynomial we want to fit\npoly_model2 = make_pipeline(PolynomialFeatures(degree),LinearRegression())\npoly_model2.fit(X_train,y_train) # fit the polynomial to our 5 points in x_train\ny_pred = poly_model2.predict(X_data) # prediction of our polynomial\n# Compute MSE and R2 for the polynomial model we fitted\nmse_value = mean_squared_error(y_noisy_data, y_pred)\nr2_value = r2_score(y_noisy_data, y_pred)\nprint('MSE for polynomial = ', mse_value)\nprint('R2 score for polynomial = ', r2_value)\n# Then, plot the polynomial prediction on top of fig1:\nax3.plot(x_data, y_pred, 'bo-', markersize=6, linewidth=2, label=\"Polynomial of degree %d prediction\" % degree)\n# Replot figure and legend:\nax3.legend(loc='upper right')\nfig3\n```\n", "meta": {"hexsha": "a69b1ce15d4a15c4917522f075e7c418c00eb3a2", "size": 665365, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Lectures/Lecture9/3dasm_Lecture9.ipynb", "max_stars_repo_name": "shushu-qin/3dasm_course", "max_stars_repo_head_hexsha": "a53ce9f8d7c692a9b1356946ec11e60b35b7bbcd", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2022-02-07T18:45:48.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-23T21:45:27.000Z", "max_issues_repo_path": "Lectures/Lecture9/3dasm_Lecture9.ipynb", "max_issues_repo_name": "shushu-qin/3dasm_course", "max_issues_repo_head_hexsha": "a53ce9f8d7c692a9b1356946ec11e60b35b7bbcd", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Lectures/Lecture9/3dasm_Lecture9.ipynb", "max_forks_repo_name": "shushu-qin/3dasm_course", "max_forks_repo_head_hexsha": "a53ce9f8d7c692a9b1356946ec11e60b35b7bbcd", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 6, "max_forks_repo_forks_event_min_datetime": "2022-02-07T18:45:49.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-25T19:30:17.000Z", "avg_line_length": 368.0116150442, "max_line_length": 126356, "alphanum_fraction": 0.9339760883, "converted": true, "num_tokens": 8411, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.824461932846258, "lm_q1q2_score": 0.44116828472849157}} {"text": "\n\n\n# Javascript Interface Code for Scalar Field Collapse\n\n## Author: Lucas Graham\n### Based on the previous simulation notebooks by Karinne Summers, and Tutorial-Start_to_Finish-BSSNCurvilinear-ScalarField_Collapse by Leonardo Werneck & Zach Etienne\n\n## This module implements a basic numerical relativity code to determine the evolution of a massless scalar field, and see if it collapses.\n\n## Introduction:\nHere we use NRPy+ to generate the C source code necessary to simulate a scalar field collapsing into a black hole, based on initial conditions generated in Javascript.\n\n### OLD\nHere we use NRPy+ to generate the C source code necessary to set up initial data for two black holes (Brill-Lindquist, [Brill & Lindquist, Phys. Rev. 131, 471, 1963](https://journals.aps.org/pr/abstract/10.1103/PhysRev.131.471); see also Eq. 1 of [Brandt & Br\u00fcgmann, arXiv:gr-qc/9711015v1](https://arxiv.org/pdf/gr-qc/9711015v1.pdf)). Then we use it to generate the RHS expressions for [Method of Lines](https://reference.wolfram.com/language/tutorial/NDSolveMethodOfLines.html) time integration based on the [explicit Runge-Kutta fourth-order scheme](https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods) (RK4).\n### OLD\n\nThe entire algorithm is outlined as follows, with links to the relevant NRPy+ tutorial notebooks listed at each step:\n\n1. Allocate memory for gridfunctions, including temporary storage for the Method of Lines time integration\n * [**NRPy+ tutorial on Method of Lines algorithm**](Tutorial-Method_of_Lines-C_Code_Generation.ipynb).\n1. Set gridfunction values to initial data \n * [**NRPy+ tutorial on Brill-Lindquist initial data**](Tutorial-ADM_Initial_Data-Brill-Lindquist.ipynb)\n * [**NRPy+ tutorial on validating Brill-Lindquist initial data**](Tutorial-Start_to_Finish-BSSNCurvilinear-Setting_up_Exact_Initial_Data.ipynb).\n1. Next, integrate the initial data forward in time using the Method of Lines coupled to a Runge-Kutta explicit timestepping algorithm:\n 1. At each RK time substep, do the following:\n 1. Evaluate BSSN RHS expressions \n * [**NRPy+ tutorial on BSSN right-hand sides**](Tutorial-BSSN_time_evolution-BSSN_RHSs.ipynb)\n * [**NRPy+ tutorial on BSSN gauge condition right-hand sides**](Tutorial-BSSN_time_evolution-BSSN_gauge_RHSs.ipynb) \n 1. Apply singular, curvilinear coordinate boundary conditions [*a la* the SENR/NRPy+ paper](https://arxiv.org/abs/1712.07658)\n * [**NRPy+ tutorial on setting up singular, curvilinear boundary conditions**](Tutorial-Start_to_Finish-Curvilinear_BCs.ipynb)\n 1. Enforce constraint on conformal 3-metric: $\\det{\\bar{\\gamma}_{ij}}=\\det{\\hat{\\gamma}_{ij}}$ \n * [**NRPy+ tutorial on enforcing $\\det{\\bar{\\gamma}_{ij}}=\\det{\\hat{\\gamma}_{ij}}$ constraint**](Tutorial-BSSN-Enforcing_Determinant_gammabar_equals_gammahat_Constraint.ipynb). \n 1. At the end of each iteration in time, output the Hamiltonian constraint violation \n * [**NRPy+ tutorial on BSSN constraints**](Tutorial-BSSN_constraints.ipynb).\n \n\n\n\n# Table of Contents\n$$\\label{toc}$$\n\nThis notebook is organized as follows\n\n1. [Step 1](#initializenrpy): Set core NRPy+ parameters for numerical grids and reference metric\n1. [Step 2](#initial_data) Set up ADM initial data for the Scalar Field\n1. [Step 3](#ADMtoBSSN) Convert ADM initial data to BSSN-in-curvilinear coordinates\n1. [Step 4](#bssn): Output C code for BSSN spacetime solve\n 1. [Step 4.a](#bssnrhs): Output C code for BSSN RHS expressions, and add the *rescaled* $T^{\\mu\\nu}$ source terms\n 1. [Step 4.b](#hamconstraint): Output C code for Hamiltonian constraint\n 1. [Step 4.c](#enforce3metric): Enforce conformal 3-metric $\\det{\\bar{\\gamma}_{ij}}=\\det{\\hat{\\gamma}_{ij}}$ constraint\n 1. [Step 4.d](#ccodegen) Generate C code kernels for BSSN expressions, in parallel if possible\n 1. [Step 4.e](#cparams_rfm_and_domainsize) Output C codes needed for declaring and setting Cparameters; also set `free_parameters.h`\n\n1. [Step 5](#bc_functs): Set up boundary condition functions for chosen singular, curvilinear coordinate system\n1. [Step 6](#main_ccode) The main C code: `ScalarFieldCollapse_Playground.c`\n1. [Step 7](#compileexec): Compile generated C codes & perform the scalar field collapse calculation\n1. [Step 8](#visualization) Visualization\n 1. [Step 8.a](#movie_dynamics) Dynamics of the solution\n 1. [Step 8.a.i](#genimages) Generate images for visualization animation\n 1. [Step 8.a.ii](#gemnvideo) Generate visualization animation\n 1. [Step 8.b](#convergence) Convergence of constraint violation\n\n\n```python\n# nrpytutorial should be in a subdirectory, so, do the following.\nimport os,sys\nnrpy_dir_path = os.path.join(\"nrpytutorial\")\nif nrpy_dir_path not in sys.path:\n sys.path.append(nrpy_dir_path)\n```\n\n\n\n# Step 1: Set core NRPy+ parameters for numerical grids and reference metric \\[Back to [top](#toc)\\]\n$$\\label{initializenrpy}$$\n\n\n```python\n# Step P1: Import needed NRPy+ core modules:\nfrom outputC import lhrh,outputC,outCfunction # NRPy+: Core C code output module\nimport NRPy_param_funcs as par # NRPy+: Parameter interface\nimport sympy as sp # SymPy: The Python computer algebra package upon which NRPy+ depends\nimport finite_difference as fin # NRPy+: Finite difference C code generation module\nimport grid as gri # NRPy+: Functions having to do with numerical grids\nimport indexedexp as ixp # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support\nimport reference_metric as rfm # NRPy+: Reference metric support\nimport cmdline_helper as cmd # NRPy+: Multi-platform Python command-line interface\nimport shutil\n\n# Step P2: Create C code output directory:\nCcodesdir = os.path.join(\"BSSN_Scalar_Collapse/\")\n# First remove C code output directory if it exists\n# Courtesy https://stackoverflow.com/questions/303200/how-do-i-remove-delete-a-folder-that-is-not-empty\nshutil.rmtree(Ccodesdir, ignore_errors=True)\n# Then create a fresh directory\ncmd.mkdir(Ccodesdir)\n\n# Step P3: Create executable output directory:\noutdir = os.path.join(Ccodesdir,\"output/\")\ncmd.mkdir(outdir)\n\n# Step 1: Set the spatial dimension parameter\n# to three (BSSN is a 3+1 decomposition\n# of Einstein's equations), and then read\n# the parameter as DIM.\nDIM = 3\npar.set_parval_from_str(\"grid::DIM\",DIM)\n\n# Step 1.a: Enable SIMD-optimized code?\n# I.e., generate BSSN and Ricci C code kernels using SIMD-vectorized\n# compiler intrinsics, which *greatly improve the code's performance*,\n# though at the expense of making the C-code kernels less\n# human-readable.\n# * Important note in case you wish to modify the BSSN/Ricci kernels\n# here by adding expressions containing transcendental functions\n# (e.g., certain scalar fields):\n# Note that SIMD-based transcendental function intrinsics are not\n# supported by the default installation of gcc or clang (you will\n# need to use e.g., the SLEEF library from sleef.org, for this\n# purpose). The Intel compiler suite does support these intrinsics\n# however without the need for external libraries.\n#enable_SIMD = True\n\n# Step 2: Set some core parameters, including CoordSystem MoL timestepping algorithm,\n# FD order, floating point precision, and CFL factor:\n# Choices are: Spherical, SinhSpherical, SinhSphericalv2, Cylindrical, SinhCylindrical,\n# SymTP, SinhSymTP\nCoordSystem = \"Spherical\"\n\n# domain_size sets the default value for:\n# * Spherical's params.RMAX\n# * SinhSpherical*'s params.AMAX\n# * Cartesians*'s -params.{x,y,z}min & .{x,y,z}max\n# * Cylindrical's -params.ZMIN & .{Z,RHO}MAX\n# * SinhCylindrical's params.AMPL{RHO,Z}\n# * *SymTP's params.AMAX\ndomain_size = 10 # Length scale of computational domain\nFD_order = 8 # Finite difference order: even numbers only, starting with 2. 12 is generally unstable\n\n# sinh_width sets the default value for:\n# * SinhSpherical's params.SINHW\n# * SinhCylindrical's params.SINHW{RHO,Z}\n# * SinhSymTP's params.SINHWAA\nsinh_width = 0.2 # If Sinh* coordinates chosen\n\n# sinhv2_const_dr sets the default value for:\n# * SinhSphericalv2's params.const_dr\n# * SinhCylindricalv2's params.const_d{rho,z}\nsinhv2_const_dr = 0.05 # If Sinh*v2 coordinates chosen\n\n# SymTP_bScale sets the default value for:\n# * SinhSymTP's params.bScale\nSymTP_bScale = 0.5 # If SymTP chosen\n\n# Step 2.b: Set the timestepping order,\n# the core data type, and the CFL factor.\n# Step 2.b: Set the order of spatial and temporal derivatives;\n# the core data type, and the CFL factor.\n# RK_method choices include: Euler, \"RK2 Heun\", \"RK2 MP\", \"RK2 Ralston\", RK3, \"RK3 Heun\", \"RK3 Ralston\",\n# SSPRK3, RK4, DP5, DP5alt, CK5, DP6, L6, DP8\nRK_method = \"RK4\"\nREAL = \"double\" # Best to use double here.\nCFL_FACTOR = 0.1 # (GETS OVERWRITTEN WHEN EXECUTED.) In pure axisymmetry (symmetry_axes = 2 below) 1.0 works fine. Otherwise 0.5 or lower.\n\n# Set the lapse & shift conditions\nLapseCondition = \"OnePlusLog\"\nShiftCondition = \"GammaDriving2ndOrder_Covariant\"\n\n# Step 3: Generate Runge-Kutta-based (RK-based) timestepping code.\n# As described above the Table of Contents, this is a 3-step process:\n# 3.A: Evaluate RHSs (RHS_string)\n# 3.B: Apply boundary conditions (post_RHS_string, pt 1)\n# 3.C: Enforce det(gammahat) = det(gammahat) constraint (post_RHS_string, pt 2)\nimport MoLtimestepping.C_Code_Generation as MoL\nfrom MoLtimestepping.RK_Butcher_Table_Dictionary import Butcher_dict\nRK_order = Butcher_dict[RK_method][1]\ncmd.mkdir(os.path.join(Ccodesdir,\"MoLtimestepping/\"))\nMoL.MoL_C_Code_Generation(RK_method,\n RHS_string = \"\"\"\nRicci_eval(&rfmstruct, ¶ms, RK_INPUT_GFS, auxevol_gfs);\nrhs_eval(&rfmstruct, ¶ms, auxevol_gfs, RK_INPUT_GFS, RK_OUTPUT_GFS);\"\"\",\n post_RHS_string = \"\"\"\napply_bcs_curvilinear(¶ms, &bcstruct, NUM_EVOL_GFS, evol_gf_parity, RK_OUTPUT_GFS);\nenforce_detgammahat_constraint(&rfmstruct, ¶ms, RK_OUTPUT_GFS);\\n\"\"\",\n outdir = os.path.join(Ccodesdir,\"MoLtimestepping/\"))\n\n# Step 4: Set the coordinate system for the numerical grid\npar.set_parval_from_str(\"reference_metric::CoordSystem\",CoordSystem)\nrfm.reference_metric() # Create ReU, ReDD needed for rescaling B-L initial data, generating BSSN RHSs, etc.\n\n# Step 5: Set the finite differencing order to FD_order (set above).\npar.set_parval_from_str(\"finite_difference::FD_CENTDERIVS_ORDER\", FD_order)\n\n# Step 6: Copy SIMD/SIMD_intrinsics.h to $Ccodesdir/SIMD/SIMD_intrinsics.h\ncmd.mkdir(os.path.join(Ccodesdir,\"SIMD\"))\nshutil.copy(os.path.join(\"nrpytutorial/SIMD/\")+\"SIMD_intrinsics.h\",os.path.join(Ccodesdir,\"SIMD/\"))\n\n# Step 7: Impose spherical symmetry by demanding that all\n# derivatives in the angular directions vanish\npar.set_parval_from_str(\"indexedexp::symmetry_axes\",\"12\")\n```\n\n\n\n## Step 1.c: Output needed C code for finding the minimum proper distance between grid points, needed for [CFL](https://en.wikipedia.org/w/index.php?title=Courant%E2%80%93Friedrichs%E2%80%93Lewy_condition&oldid=806430673)-limited timestep \\[Back to [top](#toc)\\]\n$$\\label{cfl}$$\n\nIn order for our explicit-timestepping numerical solution to the scalar wave equation to be stable, it must satisfy the [CFL](https://en.wikipedia.org/w/index.php?title=Courant%E2%80%93Friedrichs%E2%80%93Lewy_condition&oldid=806430673) condition:\n$$\n\\Delta t \\le \\frac{\\min(ds_i)}{c},\n$$\nwhere $c$ is the wavespeed, and\n$$ds_i = h_i \\Delta x^i$$ \nis the proper distance between neighboring gridpoints in the $i$th direction (in 3D, there are 3 directions), $h_i$ is the $i$th reference metric scale factor, and $\\Delta x^i$ is the uniform grid spacing in the $i$th direction:\n\n\n```python\n# Output the find_timestep() function to a C file.\nrfm.out_timestep_func_to_file(os.path.join(Ccodesdir,\"find_timestep.h\"))\n```\n\n\n\n# Step 2: Set up ADM Initial Conditions for the Scalar Field \\[Back to [top](#toc)\\]\n$$\\label{initial_data}$$\n\nAs documented [in the scalar field Gaussian pulse initial data NRPy+ tutorial notebook](nrpytutorial/Tutorial-ADM_Initial_Data-ScalarField.ipynb), we will now set up the scalar field initial data, storing the densely-sampled result to file.\n\nThe initial data function `ScalarField_InitialData` requires `SciPy`, so let's make sure it's installed.\n\n\n```python\n# Step 2.a: Import necessary Python and NRPy+ modules\nimport ScalarField.ScalarField_InitialData as sfid\n\n# Step 2.b: Set the initial data parameters\noutputfilename = os.path.join(outdir,\"SFID.txt\")\nID_Family = \"Gaussian_pulse\"\npulse_amplitude = 0.4\npulse_center = 0\npulse_width = 1\nNr = 30000\nrmax = domain_size*1.1\n\n# Step 2.c: Generate the initial data #Generates a text file so it won't work with emscripten. Leo is making C code.\nsfid.ScalarField_InitialData(outputfilename,ID_Family,\n pulse_amplitude,pulse_center,pulse_width,Nr,rmax)\n\n# Step 2.d: Generate the needed C code\nsfid.NRPy_param_funcs_register_C_functions_and_NRPy_basic_defines(Ccodesdir=Ccodesdir)\n```\n\n Generated the ADM initial data for the gravitational collapse \n of a massless scalar field in Spherical coordinates.\n \n Type of initial condition: Scalar field: \"Gaussian\" Shell\n ADM quantities: Time-symmetric\n Lapse condition: Pre-collapsed\n Parameters: amplitude = 0.4,\n center = 0,\n width = 1,\n domain size = 11.0,\n number of points = 30000,\n Initial data file = BSSN_Scalar_Collapse/output/SFID.txt.\n \n Output C function ID_scalarfield_ADM_quantities() to file BSSN_Scalar_Collapse/ID_scalarfield_ADM_quantities.h\n Output C function ID_scalarfield_spherical() to file BSSN_Scalar_Collapse/ID_scalarfield_spherical.h\n Output C function ID_scalarfield_xx0xx1xx2_to_BSSN_xx0xx1xx2() to file BSSN_Scalar_Collapse/ID_scalarfield_xx0xx1xx2_to_BSSN_xx0xx1xx2.h\n Output C function ID_scalarfield() to file BSSN_Scalar_Collapse/ID_scalarfield.h\n\n\n\n\n# Step 3: Convert ADM to BSSN Coordinates \\[Back to [top](#toc)\\]\n$$\\label{ADMtoBSSN}$$\n\n\n```python\nimport BSSN.ADM_Numerical_Spherical_or_Cartesian_to_BSSNCurvilinear as AtoBnum\nAtoBnum.Convert_Spherical_or_Cartesian_ADM_to_BSSN_curvilinear(\"Spherical\",\"ID_scalarfield_ADM_quantities\",\n Ccodesdir=Ccodesdir,loopopts=\"\")\n```\n\n Output C function ID_BSSN_lambdas() to file BSSN_Scalar_Collapse/ID_BSSN_lambdas.h\n Output C function ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs() to file BSSN_Scalar_Collapse/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h\n Output C function ID_BSSN__ALL_BUT_LAMBDAs() to file BSSN_Scalar_Collapse/ID_BSSN__ALL_BUT_LAMBDAs.h\n\n\n\n\n# Step 4: Output C code for BSSN spacetime solve \\[Back to [top](#toc)\\]\n$$\\label{bssn}$$\n\n\n\n## Step 4.a: Set up the BSSN and ScalarField right-hand-side (RHS) expressions, and add the *rescaled* $T^{\\mu\\nu}$ source terms \\[Back to [top](#toc)\\]\n$$\\label{bssnrhs}$$\n\n`BSSN.BSSN_RHSs()` sets up the RHSs assuming a spacetime vacuum: $T^{\\mu\\nu}=0$. (This might seem weird, but remember that, for example, *spacetimes containing only single or binary black holes are vacuum spacetimes*.) Here, using the [`BSSN.BSSN_stress_energy_source_terms`](../edit/BSSN/BSSN_stress_energy_source_terms.py) ([**tutorial**](Tutorial-BSSN_stress_energy_source_terms.ipynb)) NRPy+ module, we add the $T^{\\mu\\nu}$ source terms to these equations.\n\n\n```python\nimport time\nimport BSSN.BSSN_RHSs as rhs\nimport BSSN.BSSN_gauge_RHSs as gaugerhs\npar.set_parval_from_str(\"BSSN.BSSN_gauge_RHSs::LapseEvolutionOption\", LapseCondition)\npar.set_parval_from_str(\"BSSN.BSSN_gauge_RHSs::ShiftEvolutionOption\", ShiftCondition)\n\nprint(\"Generating symbolic expressions for BSSN RHSs...\")\nstart = time.time()\n# Enable rfm_precompute infrastructure, which results in\n# BSSN RHSs that are free of transcendental functions,\n# even in curvilinear coordinates, so long as\n# ConformalFactor is set to \"W\" (default).\ncmd.mkdir(os.path.join(Ccodesdir,\"rfm_files/\"))\npar.set_parval_from_str(\"reference_metric::enable_rfm_precompute\",\"True\")\npar.set_parval_from_str(\"reference_metric::rfm_precompute_Ccode_outdir\",os.path.join(Ccodesdir,\"rfm_files/\"))\n\n# Evaluate BSSN + BSSN gauge RHSs with rfm_precompute enabled:\nimport BSSN.BSSN_quantities as Bq\npar.set_parval_from_str(\"BSSN.BSSN_quantities::LeaveRicciSymbolic\",\"True\")\n\nrhs.BSSN_RHSs()\n\n# Evaluate the Scalar Field RHSs\nimport ScalarField.ScalarField_RHSs as sfrhs\nsfrhs.ScalarField_RHSs()\n\n# Compute ScalarField T^{\\mu\\nu}\n# Compute the scalar field energy-momentum tensor\nimport ScalarField.ScalarField_Tmunu as sfTmunu\nsfTmunu.ScalarField_Tmunu()\nT4UU = sfTmunu.T4UU\n\nimport BSSN.BSSN_stress_energy_source_terms as Bsest\nBsest.BSSN_source_terms_for_BSSN_RHSs(T4UU)\nrhs.trK_rhs += Bsest.sourceterm_trK_rhs\nfor i in range(DIM):\n # Needed for Gamma-driving shift RHSs:\n rhs.Lambdabar_rhsU[i] += Bsest.sourceterm_Lambdabar_rhsU[i]\n # Needed for BSSN RHSs:\n rhs.lambda_rhsU[i] += Bsest.sourceterm_lambda_rhsU[i]\n for j in range(DIM):\n rhs.a_rhsDD[i][j] += Bsest.sourceterm_a_rhsDD[i][j]\n \ngaugerhs.BSSN_gauge_RHSs()\n\n# We use betaU as our upwinding control vector:\nBq.BSSN_basic_tensors()\nbetaU = Bq.betaU\n\nimport BSSN.Enforce_Detgammahat_Constraint as EGC\nenforce_detg_constraint_symb_expressions = EGC.Enforce_Detgammahat_Constraint_symb_expressions()\n\n# Next compute Ricci tensor\npar.set_parval_from_str(\"BSSN.BSSN_quantities::LeaveRicciSymbolic\",\"False\")\nBq.RicciBar__gammabarDD_dHatD__DGammaUDD__DGammaU()\n\n# Now register the Hamiltonian as a gridfunction.\nH = gri.register_gridfunctions(\"AUX\",\"H\")\n\n# Then define the Hamiltonian constraint and output the optimized C code.\nimport BSSN.BSSN_constraints as bssncon\nbssncon.BSSN_constraints(add_T4UUmunu_source_terms=False)\nBsest.BSSN_source_terms_for_BSSN_constraints(T4UU)\nbssncon.H += Bsest.sourceterm_H\n\n# Add Kreiss-Oliger dissipation\ndiss_strength = par.Cparameters(\"REAL\",\"ScalarFieldCollapse\",[\"diss_strength\"],0.1)\n\nalpha_dKOD = ixp.declarerank1(\"alpha_dKOD\")\ncf_dKOD = ixp.declarerank1(\"cf_dKOD\")\ntrK_dKOD = ixp.declarerank1(\"trK_dKOD\")\nsf_dKOD = ixp.declarerank1(\"sf_dKOD\")\nsfM_dKOD = ixp.declarerank1(\"sfM_dKOD\")\nbetU_dKOD = ixp.declarerank2(\"betU_dKOD\",\"nosym\")\nvetU_dKOD = ixp.declarerank2(\"vetU_dKOD\",\"nosym\")\nlambdaU_dKOD = ixp.declarerank2(\"lambdaU_dKOD\",\"nosym\")\naDD_dKOD = ixp.declarerank3(\"aDD_dKOD\",\"sym01\")\nhDD_dKOD = ixp.declarerank3(\"hDD_dKOD\",\"sym01\")\n\nfor k in range(3):\n gaugerhs.alpha_rhs += diss_strength*alpha_dKOD[k]*rfm.ReU[k]\n rhs.cf_rhs += diss_strength* cf_dKOD[k]*rfm.ReU[k]\n rhs.trK_rhs += diss_strength* trK_dKOD[k]*rfm.ReU[k]\n sfrhs.sf_rhs += diss_strength* sf_dKOD[k]*rfm.ReU[k]\n sfrhs.sfM_rhs += diss_strength* sfM_dKOD[k]*rfm.ReU[k]\n for i in range(3):\n if \"2ndOrder\" in ShiftCondition:\n gaugerhs.bet_rhsU[i] += diss_strength* betU_dKOD[i][k]*rfm.ReU[k]\n gaugerhs.vet_rhsU[i] += diss_strength* vetU_dKOD[i][k]*rfm.ReU[k]\n rhs.lambda_rhsU[i] += diss_strength*lambdaU_dKOD[i][k]*rfm.ReU[k]\n for j in range(3):\n rhs.a_rhsDD[i][j] += diss_strength*aDD_dKOD[i][j][k]*rfm.ReU[k]\n rhs.h_rhsDD[i][j] += diss_strength*hDD_dKOD[i][j][k]*rfm.ReU[k]\n \n# Now that we are finished with all the rfm hatted\n# quantities in generic precomputed functional\n# form, let's restore them to their closed-\n# form expressions.\npar.set_parval_from_str(\"reference_metric::enable_rfm_precompute\",\"False\") # Reset to False to disable rfm_precompute.\nrfm.ref_metric__hatted_quantities()\nend = time.time()\nprint(\"(BENCH) Finished BSSN symbolic expressions in \"+str(end-start)+\" seconds.\")\n\ndef BSSN_plus_ScalarField_RHSs():\n print(\"Generating C code for BSSN RHSs in \"+par.parval_from_str(\"reference_metric::CoordSystem\")+\" coordinates.\")\n start = time.time()\n\n # Construct the left-hand sides and right-hand-side expressions for all BSSN RHSs\n lhs_names = [ \"alpha\", \"cf\", \"trK\", \"sf\", \"sfM\" ]\n rhs_exprs = [gaugerhs.alpha_rhs, rhs.cf_rhs, rhs.trK_rhs, sfrhs.sf_rhs, sfrhs.sfM_rhs]\n for i in range(3):\n lhs_names.append( \"betU\"+str(i))\n rhs_exprs.append(gaugerhs.bet_rhsU[i])\n lhs_names.append( \"lambdaU\"+str(i))\n rhs_exprs.append(rhs.lambda_rhsU[i])\n lhs_names.append( \"vetU\"+str(i))\n rhs_exprs.append(gaugerhs.vet_rhsU[i])\n for j in range(i,3):\n lhs_names.append( \"aDD\"+str(i)+str(j))\n rhs_exprs.append(rhs.a_rhsDD[i][j])\n lhs_names.append( \"hDD\"+str(i)+str(j))\n rhs_exprs.append(rhs.h_rhsDD[i][j])\n\n # Sort the lhss list alphabetically, and rhss to match.\n # This ensures the RHSs are evaluated in the same order\n # they're allocated in memory:\n lhs_names,rhs_exprs = [list(x) for x in zip(*sorted(zip(lhs_names,rhs_exprs), key=lambda pair: pair[0]))]\n\n # Declare the list of lhrh's\n BSSN_evol_rhss = []\n for var in range(len(lhs_names)):\n BSSN_evol_rhss.append(lhrh(lhs=gri.gfaccess(\"rhs_gfs\",lhs_names[var]),rhs=rhs_exprs[var]))\n\n # Set up the C function for the BSSN RHSs\n desc=\"Evaluate the BSSN RHSs\"\n name=\"rhs_eval\"\n outCfunction(\n outfile = os.path.join(Ccodesdir,name+\".h\"), desc=desc, name=name,\n params = \"\"\"rfm_struct *restrict rfmstruct,const paramstruct *restrict params,\n const REAL *restrict auxevol_gfs,const REAL *restrict in_gfs,REAL *restrict rhs_gfs\"\"\",\n body = fin.FD_outputC(\"returnstring\",BSSN_evol_rhss, params=\"outCverbose=False,enable_SIMD=True\",\n upwindcontrolvec=betaU),\n loopopts = \"InteriorPoints,enable_SIMD,enable_rfm_precompute\")\n end = time.time()\n print(\"(BENCH) Finished BSSN_RHS C codegen in \" + str(end - start) + \" seconds.\")\n\ndef Ricci():\n print(\"Generating C code for Ricci tensor in \"+par.parval_from_str(\"reference_metric::CoordSystem\")+\" coordinates.\")\n start = time.time()\n desc=\"Evaluate the Ricci tensor\"\n name=\"Ricci_eval\"\n outCfunction(\n outfile = os.path.join(Ccodesdir,name+\".h\"), desc=desc, name=name,\n params = \"\"\"rfm_struct *restrict rfmstruct,const paramstruct *restrict params,\n const REAL *restrict in_gfs,REAL *restrict auxevol_gfs\"\"\",\n body = fin.FD_outputC(\"returnstring\",\n [lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"RbarDD00\"),rhs=Bq.RbarDD[0][0]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"RbarDD01\"),rhs=Bq.RbarDD[0][1]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"RbarDD02\"),rhs=Bq.RbarDD[0][2]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"RbarDD11\"),rhs=Bq.RbarDD[1][1]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"RbarDD12\"),rhs=Bq.RbarDD[1][2]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"RbarDD22\"),rhs=Bq.RbarDD[2][2])],\n params=\"outCverbose=False,enable_SIMD=True\"),\n loopopts = \"InteriorPoints,enable_SIMD,enable_rfm_precompute\")\n end = time.time()\n print(\"(BENCH) Finished Ricci C codegen in \" + str(end - start) + \" seconds.\")\n```\n\n Generating symbolic expressions for BSSN RHSs...\n (BENCH) Finished BSSN symbolic expressions in 17.095833778381348 seconds.\n\n\n\n\n## Step 4.b: Output C code for Hamiltonian constraint \\[Back to [top](#toc)\\]\n$$\\label{hamconstraint}$$\n\nNext output the C code for evaluating the Hamiltonian constraint [(**Tutorial**)](Tutorial-BSSN_constraints.ipynb). In the absence of numerical error, this constraint should evaluate to zero. However it does not due to numerical (typically truncation and roundoff) error. We will therefore measure the Hamiltonian constraint violation to gauge the accuracy of our simulation, and, ultimately determine whether errors are dominated by numerical finite differencing (truncation) error as expected.\n\n\n```python\ndef Hamiltonian():\n start = time.time()\n print(\"Generating optimized C code for Hamiltonian constraint. May take a while, depending on CoordSystem.\")\n # Set up the C function for the Hamiltonian RHS\n desc=\"Evaluate the Hamiltonian constraint\"\n name=\"Hamiltonian_constraint\"\n outCfunction(\n outfile = os.path.join(Ccodesdir,name+\".h\"), desc=desc, name=name,\n params = \"\"\"rfm_struct *restrict rfmstruct,const paramstruct *restrict params,\n REAL *restrict in_gfs, REAL *restrict auxevol_gfs, REAL *restrict aux_gfs\"\"\",\n body = fin.FD_outputC(\"returnstring\",lhrh(lhs=gri.gfaccess(\"aux_gfs\", \"H\"), rhs=bssncon.H),\n params=\"outCverbose=False\"),\n loopopts = \"InteriorPoints,enable_rfm_precompute\")\n\n end = time.time()\n print(\"(BENCH) Finished Hamiltonian C codegen in \" + str(end - start) + \" seconds.\")\n```\n\n\n\n## Step 4.c: Enforce conformal 3-metric $\\det{\\bar{\\gamma}_{ij}}=\\det{\\hat{\\gamma}_{ij}}$ constraint \\[Back to [top](#toc)\\]\n$$\\label{enforce3metric}$$\n\nThen enforce conformal 3-metric $\\det{\\bar{\\gamma}_{ij}}=\\det{\\hat{\\gamma}_{ij}}$ constraint (Eq. 53 of [Ruchlin, Etienne, and Baumgarte (2018)](https://arxiv.org/abs/1712.07658)), as [documented in the corresponding NRPy+ tutorial notebook](nrpytutorial/Tutorial-BSSN_enforcing_determinant_gammabar_equals_gammahat_constraint.ipynb)\n\nApplying curvilinear boundary conditions should affect the initial data at the outer boundary, and will in general cause the $\\det{\\bar{\\gamma}_{ij}}=\\det{\\hat{\\gamma}_{ij}}$ constraint to be violated there. Thus after we apply these boundary conditions, we must always call the routine for enforcing the $\\det{\\bar{\\gamma}_{ij}}=\\det{\\hat{\\gamma}_{ij}}$ constraint:\n\n\n```python\ndef gammadet():\n start = time.time()\n print(\"Generating optimized C code for gamma constraint. May take a while, depending on CoordSystem.\")\n\n # Set up the C function for the det(gammahat) = det(gammahat)\n EGC.output_Enforce_Detgammahat_Constraint_Ccode(Ccodesdir,exprs=enforce_detg_constraint_symb_expressions)\n end = time.time()\n print(\"(BENCH) Finished gamma constraint C codegen in \" + str(end - start) + \" seconds.\")\n```\n\n\n\n## Step 4.d: Generate C code kernels for BSSN expressions, in parallel if possible \\[Back to [top](#toc)\\]\n$$\\label{ccodegen}$$\n\n\n```python\n# Step 4.d: C code kernel generation\n# Step 4.d.i: Create a list of functions we wish to evaluate in parallel\nfuncs = [BSSN_plus_ScalarField_RHSs,Ricci,Hamiltonian,gammadet]\n\ntry:\n if os.name == 'nt':\n # It's a mess to get working in Windows, so we don't bother. :/\n # https://medium.com/@grvsinghal/speed-up-your-python-code-using-multiprocessing-on-windows-and-jupyter-or-ipython-2714b49d6fac\n raise Exception(\"Parallel codegen currently not available in Windows\")\n # Step 4.d.ii: Import the multiprocessing module.\n import multiprocess as multiprocessing\n\n # Step 4.d.iii: Define master function for parallelization.\n # Note that lambdifying this doesn't work in Python 3\n def master_func(arg):\n funcs[arg]()\n\n # Step 4.d.iv: Evaluate list of functions in parallel if possible;\n # otherwise fallback to serial evaluation:\n pool = multiprocessing.Pool()\n pool.map(master_func,range(len(funcs)))\nexcept:\n # Steps 4.d.iii-4.d.v, alternate: As fallback, evaluate functions in serial.\n for func in funcs:\n func()\n```\n\n Generating C code for BSSN RHSs in Spherical coordinates.\n Output C function rhs_eval() to file BSSN_Scalar_Collapse/rhs_eval.h\n (BENCH) Finished BSSN_RHS C codegen in 58.68921256065369 seconds.\n Generating C code for Ricci tensor in Spherical coordinates.\n Output C function Ricci_eval() to file BSSN_Scalar_Collapse/Ricci_eval.h\n (BENCH) Finished Ricci C codegen in 43.368809938430786 seconds.\n Generating optimized C code for Hamiltonian constraint. May take a while, depending on CoordSystem.\n Output C function Hamiltonian_constraint() to file BSSN_Scalar_Collapse/Hamiltonian_constraint.h\n (BENCH) Finished Hamiltonian C codegen in 88.60661959648132 seconds.\n Generating optimized C code for gamma constraint. May take a while, depending on CoordSystem.\n Output C function enforce_detgammahat_constraint() to file BSSN_Scalar_Collapse/enforce_detgammahat_constraint.h\n (BENCH) Finished gamma constraint C codegen in 0.25756216049194336 seconds.\n\n\n\n\n## Step 4.e: Output C codes needed for declaring and setting Cparameters; also set `free_parameters.h` \\[Back to [top](#toc)\\]\n$$\\label{cparams_rfm_and_domainsize}$$\n\nBased on declared NRPy+ Cparameters, first we generate `declare_Cparameters_struct.h`, `set_Cparameters_default.h`, and `set_Cparameters[-SIMD].h`.\n\nThen we output `free_parameters.h`, which sets initial data parameters, as well as grid domain & reference metric parameters, applying `domain_size` and `sinh_width`/`SymTP_bScale` (if applicable) as set above\n\n\n```python\n# Step 4.e.i: Generate declare_Cparameters_struct.h, set_Cparameters_default.h, and set_Cparameters[-SIMD].h\npar.generate_Cparameters_Ccodes(os.path.join(Ccodesdir))\n\n# Step 4.e.ii: Set free_parameters.h\n# Output to $Ccodesdir/free_parameters.h reference metric parameters based on generic\n# domain_size,sinh_width,sinhv2_const_dr,SymTP_bScale,\n# parameters set above.\nrfm.out_default_free_parameters_for_rfm(os.path.join(Ccodesdir,\"free_parameters.h\"),\n domain_size,sinh_width,sinhv2_const_dr,SymTP_bScale)\n\n# Step 4.e.iii: Generate set_Nxx_dxx_invdx_params__and__xx.h:\nrfm.set_Nxx_dxx_invdx_params__and__xx_h(Ccodesdir)\n\n# Step 4.e.iv: Generate xx_to_Cart.h, which contains xx_to_Cart() for\n# (the mapping from xx->Cartesian) for the chosen\n# CoordSystem:\nrfm.xx_to_Cart_h(\"xx_to_Cart\",\"./set_Cparameters.h\",os.path.join(Ccodesdir,\"xx_to_Cart.h\"))\n\n# Step 4.e.v: Generate declare_Cparameters_struct.h, set_Cparameters_default.h, and set_Cparameters[-SIMD].h\npar.generate_Cparameters_Ccodes(os.path.join(Ccodesdir))\n```\n\n\n\n# Step 5: Set up boundary condition functions for chosen singular, curvilinear coordinate system \\[Back to [top](#toc)\\]\n$$\\label{bc_functs}$$\n\nNext apply singular, curvilinear coordinate boundary conditions [as documented in the corresponding NRPy+ tutorial notebook](Tutorial-Start_to_Finish-Curvilinear_BCs.ipynb)\n\n\n```python\nimport CurviBoundaryConditions.CurviBoundaryConditions as cbcs\ncbcs.Set_up_CurviBoundaryConditions(os.path.join(Ccodesdir,\"boundary_conditions/\"),\n Cparamspath=os.path.join(\"../\"),path_prefix=\"nrpytutorial/\") \n```\n\n Wrote to file \"BSSN_Scalar_Collapse/boundary_conditions/parity_conditions_symbolic_dot_products.h\"\n Evolved parity: ( aDD00:4, aDD01:5, aDD02:6, aDD11:7, aDD12:8, aDD22:9,\n alpha:0, betU0:1, betU1:2, betU2:3, cf:0, hDD00:4, hDD01:5, hDD02:6,\n hDD11:7, hDD12:8, hDD22:9, lambdaU0:1, lambdaU1:2, lambdaU2:3, sf:0,\n sfM:0, trK:0, vetU0:1, vetU1:2, vetU2:3 )\n Auxiliary parity: ( H:0 )\n AuxEvol parity: ( RbarDD00:4, RbarDD01:5, RbarDD02:6, RbarDD11:7,\n RbarDD12:8, RbarDD22:9 )\n Wrote to file \"BSSN_Scalar_Collapse/boundary_conditions/EigenCoord_Cart_to_xx.h\"\n\n\n\n\n# Step 6: `ScalarFieldCollapse_Playground.c`: The Main C Code \\[Back to [top](#toc)\\]\n\n$$\\label{main_ccode}$$\n\n## Main c files\n\n\n```python\n# Part P0: Define REAL, set the number of ghost cells NGHOSTS (from NRPy+'s FD_CENTDERIVS_ORDER),\n# and set the CFL_FACTOR (which can be overwritten at the command line)\n\nwith open(os.path.join(Ccodesdir,\"ScalarFieldCollapse_Playground_REAL__NGHOSTS__CFL_FACTOR.h\"), \"w\") as file:\n file.write(\"\"\"\n// Part P0.a: Set the number of ghost cells, from NRPy+'s FD_CENTDERIVS_ORDER\n#define NGHOSTS \"\"\"+str(int(FD_order/2)+1)+\"\"\"\n// Part P0.b: Set the numerical precision (REAL) to double, ensuring all floating point\n// numbers are stored to at least ~16 significant digits\n#define REAL \"\"\"+REAL+\"\"\"\n// Part P0.c: Set the number of ghost cells, from NRPy+'s FD_CENTDERIVS_ORDER\nREAL CFL_FACTOR = \"\"\"+str(CFL_FACTOR)+\"\"\"; // Set the CFL Factor. Can be overwritten at command line.\\n\"\"\")\n```\n\n\n```python\n%%writefile $Ccodesdir/ScalarFieldCollapse_Playground.c\n//This is being written into a C-file \n//Emscripten compiles the C code\n// Step P0: Define REAL and NGHOSTS; and declare CFL_FACTOR. This header is generated in NRPy+.\n#include \"ScalarFieldCollapse_Playground_REAL__NGHOSTS__CFL_FACTOR.h\"\n\n#include \"rfm_files/rfm_struct__declare.h\"\n\n#include \"declare_Cparameters_struct.h\"\n\n#include \"emscripten.h\"\n\n// All SIMD intrinsics used in SIMD-enabled C code loops are defined here:\n#include \"SIMD/SIMD_intrinsics.h\"\n\n// Step P1: Import needed header files\n#include \"stdio.h\"\n#include \"stdlib.h\"\n#include \"math.h\"\n#include \"time.h\"\n#include \"stdint.h\" // Needed for Windows GCC 6.x compatibility\n#ifndef M_PI\n#define M_PI 3.141592653589793238462643383279502884L\n#endif\n#ifndef M_SQRT1_2\n#define M_SQRT1_2 0.707106781186547524400844362104849039L\n#endif\n#define wavespeed 1.0 // Set CFL-based \"wavespeed\" to 1.0.\n#define alpha_threshold (2e-3) // Value below which we rule gravitational collapse has happened\n\n// Step P2: Declare the IDX4S(gf,i,j,k) macro, which enables us to store 4-dimensions of\n// data in a 1D array. In this case, consecutive values of \"i\"\n// (all other indices held to a fixed value) are consecutive in memory, where\n// consecutive values of \"j\" (fixing all other indices) are separated by\n// Nxx_plus_2NGHOSTS0 elements in memory. Similarly, consecutive values of\n// \"k\" are separated by Nxx_plus_2NGHOSTS0*Nxx_plus_2NGHOSTS1 in memory, etc.\n#define IDX4S(g,i,j,k) \\\n( (i) + Nxx_plus_2NGHOSTS0 * ( (j) + Nxx_plus_2NGHOSTS1 * ( (k) + Nxx_plus_2NGHOSTS2 * (g) ) ) )\n#define IDX4ptS(g,idx) ( (idx) + (Nxx_plus_2NGHOSTS0*Nxx_plus_2NGHOSTS1*Nxx_plus_2NGHOSTS2) * (g) )\n#define IDX3S(i,j,k) ( (i) + Nxx_plus_2NGHOSTS0 * ( (j) + Nxx_plus_2NGHOSTS1 * ( (k) ) ) )\n#define LOOP_REGION(i0min,i0max, i1min,i1max, i2min,i2max) \\\n for(int i2=i2min;i2Cartesian via\n// {xx[0][i0],xx[1][i1],xx[2][i2]}->{xCart[0],xCart[1],xCart[2]}\n#include \"xx_to_Cart.h\"\n\n// Step P5: Defines set_Nxx_dxx_invdx_params__and__xx(const int EigenCoord, const int Nxx[3],\n// paramstruct *restrict params, REAL *restrict xx[3]),\n// which sets params Nxx,Nxx_plus_2NGHOSTS,dxx,invdx, and xx[] for\n// the chosen Eigen-CoordSystem if EigenCoord==1, or\n// CoordSystem if EigenCoord==0.\n#include \"set_Nxx_dxx_invdx_params__and__xx.h\"\n\n// Step P6: Include basic functions needed to impose curvilinear\n// parity and boundary conditions.\n#include \"boundary_conditions/CurviBC_include_Cfunctions.h\"\n\n// Step P7: Implement the algorithm for upwinding.\n// *NOTE*: This upwinding is backwards from\n// usual upwinding algorithms, because the\n// upwinding control vector in BSSN (the shift)\n// acts like a *negative* velocity.\n//#define UPWIND_ALG(UpwindVecU) UpwindVecU > 0.0 ? 1.0 : 0.0\n\n// Step P8: Include function for enforcing detgammahat constraint.\n#include \"enforce_detgammahat_constraint.h\"\n\n// Step P9: Find the CFL-constrained timestep\n#include \"find_timestep.h\"\n\n// Step P10: Declare initial data input struct:\n// stores data from initial data solver,\n// so they can be put on the numerical grid.\ntypedef struct __ID_inputs {\n int interp_stencil_size;\n int numlines_in_file;\n REAL *r_arr,*sf_arr,*psi4_arr,*alpha_arr;\n} ID_inputs;\n\n// Part P11: Declare all functions for setting up ScalarField initial data.\n/* Routines to interpolate the ScalarField solution and convert to ADM & T^{munu}: */\n#include \"../nrpytutorial/ScalarField/ScalarField_interp.h\"\n#include \"ID_scalarfield_ADM_quantities.h\"\n#include \"ID_scalarfield_spherical.h\"\n#include \"ID_scalarfield_xx0xx1xx2_to_BSSN_xx0xx1xx2.h\"\n#include \"ID_scalarfield.h\"\n\n/* Next perform the basis conversion and compute all needed BSSN quantities */\n#include \"ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h\"\n#include \"ID_BSSN__ALL_BUT_LAMBDAs.h\"\n#include \"ID_BSSN_lambdas.h\"\n\n\n\nfloat fr[30000];\nfloat fsf[30000];\nfloat fpsi4[30000];\nfloat falpha[30000];\n\n\n// Step P12: Set the generic driver function for setting up BSSN initial data\nvoid initial_data(const paramstruct *restrict params,const bc_struct *restrict bcstruct,\n const rfm_struct *restrict rfmstruct,\n REAL *restrict xx[3], REAL *restrict auxevol_gfs, REAL *restrict in_gfs) {\n#include \"set_Cparameters.h\"\n//We will change this to getter functions to grab the initial data from Leos initial conditions\n // Step 1: Set up ScalarField initial data\n // Step 1.a: Read ScalarField initial data from data file\n // Open the data file:\n // [Deleted to hard code in initial conditions]\n // Count the number of lines in the data file:\n \n \n // Allocate space for all data arrays:\n int numlines_in_file = 30000;\n REAL *r_arr = (REAL *)malloc(sizeof(REAL)*numlines_in_file);\n REAL *sf_arr = (REAL *)malloc(sizeof(REAL)*numlines_in_file);\n REAL *psi4_arr = (REAL *)malloc(sizeof(REAL)*numlines_in_file);\n REAL *alpha_arr = (REAL *)malloc(sizeof(REAL)*numlines_in_file);\n\n // Read from the data file, filling in arrays\n // read_datafile__set_arrays() may be found in ScalarField/ScalarField_interp.h\n #include \"../SFID.h\"\n //REAL *r_arr = r_arrjs;\n //REAL *sf_arr = sf_arrjs;\n //REAL *psi4_arr = psi4_arrjs;\n //REAL *alpha_arr = alpha_arrjs;\n \n //fprintf(stderr,\"R_arr %e\");\n \n for(int ln =0; ln NUM_EVOL_GFS) {\n fprintf(stderr,\"Error: NUM_AUX_GFS > NUM_EVOL_GFS. Either reduce the number of auxiliary gridfunctions,\\n\");\n fprintf(stderr,\" or allocate (malloc) by hand storage for *diagnostic_output_gfs. \\n\");\n exit(1);\n }\n\n // Step 0j: Allocate memory for gridfunctions\n#include \"MoLtimestepping/RK_Allocate_Memory.h\"\n REAL *restrict auxevol_gfs = (REAL *)malloc(sizeof(REAL) * NUM_AUXEVOL_GFS * Nxx_plus_2NGHOSTS_tot);\n\n // Step 0k: Set up precomputed reference metric arrays\n // Step 0k.i: Allocate space for precomputed reference metric arrays.\n#include \"rfm_files/rfm_struct__malloc.h\"\n\n // Step 0k.ii: Define precomputed reference metric arrays.\n {\n #include \"set_Cparameters-nopointer.h\"\n #include \"rfm_files/rfm_struct__define.h\"\n }\n\n // Step 1a: Set up initial data to an exact solution\n initial_data(¶ms,&bcstruct, &rfmstruct, xx, auxevol_gfs, y_n_gfs);\n\n // Step 1b: Apply boundary conditions, as initial data\n // are sometimes ill-defined in ghost zones.\n // E.g., spherical initial data might not be\n // properly defined at points where r=-1.\n apply_bcs_curvilinear(¶ms, &bcstruct, NUM_EVOL_GFS,evol_gf_parity, y_n_gfs);\n enforce_detgammahat_constraint(&rfmstruct, ¶ms, y_n_gfs);\n\n //Step 2: Assign pointers/Initialize global variables\n arrNGHOSTS[0] = NGHOSTS;\n arrNGHOSTS[1] = Nxx_plus_2NGHOSTS0;\n arrNGHOSTS[2] = Nxx_plus_2NGHOSTS1;\n arrNGHOSTS[3] = Nxx_plus_2NGHOSTS2;\n dt_p = dt; \n //These appear to be all the pointers needed \n params_p = params;\n rfmstruct_p = rfmstruct;\n bcstruct_p = bcstruct;\n y_n_gfs_p = y_n_gfs;\n auxevol_gfs_p = auxevol_gfs;\n k_odd_gfs_p = k_odd_gfs;\n k_even_gfs_p = k_even_gfs;\n y_nplus1_running_total_gfs_p = y_nplus1_running_total_gfs;\n xx_p[0]=xx[0];\n xx_p[1]=xx[1];\n xx_p[2]=xx[2];\n //printf(\"%f\",xx_p[0]);\n\n return 0;\n}\n\n// stepForward() function:\n// Step 1: Define and initialize variables from initialize() function so they can be used in the RK-like Method\n// of Lines timestepping algorithm\n// Step 2: Step forward one timestep (t -> t+dt) in time using chosen RK-like MoL timestepping algorithm\n// Second half of main, does one single timescript that Javascript loops\nvoid EMSCRIPTEN_KEEPALIVE stepForward(j){\n\n // Step 1: Redefine and initialize variables. In order to call each time-step one by one, we need to redefine\n // some variables used in the MoL timestepping algorithm with the saved values from the initialization\n // step\n int Nxx_plus_2NGHOSTS0 = arrNGHOSTS[1];\n int Nxx_plus_2NGHOSTS1 = arrNGHOSTS[2];\n int Nxx_plus_2NGHOSTS2 = arrNGHOSTS[3];\n int Nxx_plus_2NGHOSTS_tot = Nxx_plus_2NGHOSTS0*Nxx_plus_2NGHOSTS1*Nxx_plus_2NGHOSTS2;\n int N_final = N_final_p;\n int output_every_N = output_every_N_p;\n //REAL *restrict diagnostic_output_gfs = diagnostic_output_gfs_p;\n REAL dt = dt_p;\n paramstruct params = params_p;\n rfm_struct rfmstruct = rfmstruct_p;\n bc_struct bcstruct = bcstruct_p;\n REAL *y_n_gfs = y_n_gfs_p;\n REAL *restrict auxevol_gfs = auxevol_gfs_p;\n REAL *k_odd_gfs = k_odd_gfs_p;\n REAL *k_even_gfs = k_even_gfs_p;\n REAL *restrict y_nplus1_running_total_gfs = y_nplus1_running_total_gfs_p;\n REAL *xx[3];\n xx[0]=xx_p[0];\n xx[1]=xx_p[1];\n xx[2]=xx_p[2];\n //if(j){\n // size_t len = sizeof(y_n_gfs_p)/sizeof(y_n_gfs_p[0]);\n // fprintf(stderr, \"The length of the array is: %zu\\n\", len);\n //}\n#include \"boundary_conditions/driver_bcstruct.h\"\n\n // Step 2: Step forward one timestep (t -> t+dt) in time using\n // chosen RK-like MoL timestepping algorithm\n // Step 3.a: Output 2D data file periodically, for visualization\n // if(n%output_every_N == 0) {\n // Evaluate Hamiltonian constraint violation\n // Hamiltonian_constraint(&rfmstruct, ¶ms, y_n_gfs,auxevol_gfs, diagnostic_output_gfs);\n\n //char filename[100];\n //sprintf(filename,\"out%d-%08d.txt\",Nxx[0],n);\n //const int i1mid=Nxx_plus_2NGHOSTS1/2;\n //const int i2mid=Nxx_plus_2NGHOSTS2/2;\n //FILE *fp = fopen(filename, \"w\");\n //for( int i0=NGHOSTS;i0\n\n# Step 7: Compile\n\n\\[Back to [top](#toc)\\]\n$$\\label{compileexec}$$\n\nUse emscripten to generate compilied html, wasm and js files. Copy the wasm and js files to the main javascript directory.\n\n\n```python\nmain_c_file = os.path.join(Ccodesdir,\"ScalarFieldCollapse_Playground.c\")\nmain_output_file = os.path.join(outdir,\"ScalarFieldCollapse.html\")\nprint(\"Attempting to compile\\n\", main_c_file, \"\\nto\\n\", main_output_file,\"\\n\")\ncmd.C_compile(main_c_file, main_output_file, compile_mode=\"emscripten\")\n\nprint(\"\\nFiles in output directory are:\\n\", os.listdir(outdir))\n```\n\n Attempting to compile\n BSSN_Scalar_Collapse/ScalarFieldCollapse_Playground.c \n to\n BSSN_Scalar_Collapse/output/ScalarFieldCollapse.html \n \n Compiling executable...\n (EXEC): Executing `emcc -std=gnu99 -s -O3 -march=native -funroll-loops -s ALLOW_MEMORY_GROWTH=1 BSSN_Scalar_Collapse/ScalarFieldCollapse_Playground.c -o BSSN_Scalar_Collapse/output/ScalarFieldCollapse.html -lm `...\n (BENCH): Finished executing in 11.24559497833252 seconds.\n Finished compilation.\n \n Files in output directory are:\n ['SFID.txt', 'ScalarFieldCollapse.wasm', 'ScalarFieldCollapse.js', 'ScalarFieldCollapse.html']\n\n\n\n```bash\n%%bash\ncp BSSN_Scalar_Collapse/output/ScalarFieldCollapse.wasm ./Scalar_Collapse_WebMaterials\ncp BSSN_Scalar_Collapse/output/ScalarFieldCollapse.js ./Scalar_Collapse_WebMaterials\ncp BSSN_Scalar_Collapse/output/ScalarFieldCollapse.html ./Scalar_Collapse_WebMaterials\n```\n\n\n\n# Step 8: Visualization\n\n\\[Back to [top](#toc)\\]\n$$\\label{visualization}$$\n\n\n\n### Step 8a: Movie Dynamics\n\n$$\\label{movie_dynamics}$$\nDynamics of the solution\n\n\n\n### Step 8a: Movie Dynamics\nCheck that the Hamiltonian violations are constrained.\n$$\\label{convergence}$$\n\n\n```python\n\n```\n", "meta": {"hexsha": "746d72800bc3303380148c2a3343c852ebc68a0a", "size": 69638, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "New_Initials_EMCC-BSSNCurvilinear-Scalar_Collapse.ipynb", "max_stars_repo_name": "wucap/NRPyJS", "max_stars_repo_head_hexsha": "2029b169f05011b686cc7d270849ff18af4115d4", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "New_Initials_EMCC-BSSNCurvilinear-Scalar_Collapse.ipynb", "max_issues_repo_name": "wucap/NRPyJS", "max_issues_repo_head_hexsha": "2029b169f05011b686cc7d270849ff18af4115d4", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "New_Initials_EMCC-BSSNCurvilinear-Scalar_Collapse.ipynb", "max_forks_repo_name": "wucap/NRPyJS", "max_forks_repo_head_hexsha": "2029b169f05011b686cc7d270849ff18af4115d4", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 48.2591822592, "max_line_length": 626, "alphanum_fraction": 0.6018265889, "converted": true, "num_tokens": 15810, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7634837635542924, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.4409083348472241}} {"text": "```python\n# import numpy as np\n\n# # !/usr/bin/env python3\n# # -*- coding: utf-8 -*-\n# \"\"\"\n# Created on 20181219\n\n# @author: zhangji\n\n# Trajection of a ellipse, Jeffery equation. \n# \"\"\"\n\n# %pylab inline\n# pylab.rcParams['figure.figsize'] = (25, 11)\n# fontsize = 40\n\n# import numpy as np\n# import scipy as sp\n# from scipy.optimize import leastsq, curve_fit\n# from scipy import interpolate\n# from scipy.interpolate import interp1d\n# from scipy.io import loadmat, savemat\n# # import scipy.misc\n\n# import matplotlib\n# from matplotlib import pyplot as plt\n# from matplotlib import animation, rc\n# import matplotlib.ticker as mtick\n# from mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\n# from mpl_toolkits.mplot3d import Axes3D, axes3d\n\n# from sympy import symbols, simplify, series, exp\n# from sympy.matrices import Matrix\n# from sympy.solvers import solve\n\n# from IPython.display import display, HTML\n# from tqdm import tqdm_notebook as tqdm\n# import pandas as pd\n# import re\n# from scanf import scanf\n# import os\n# import glob\n\n# from codeStore import support_fun as spf\n# from src.support_class import *\n# from src import stokes_flow as sf\n\n# rc('animation', html='html5')\n# PWD = os.getcwd()\n# font = {'size': 20}\n# matplotlib.rc('font', **font)\n# np.set_printoptions(linewidth=90, precision=5)\n\n%load_ext autoreload\n%autoreload 2\n\nfrom tqdm import tqdm_notebook\nimport os\nimport glob\nimport natsort \nimport numpy as np\nimport scipy as sp\nfrom scipy.optimize import leastsq, curve_fit\nfrom scipy import interpolate, integrate\nfrom scipy import spatial\n# from scipy.interpolate import interp1d\nfrom scipy.io import loadmat, savemat\n# import scipy.misc\nimport importlib\nfrom IPython.display import display, HTML\nimport pandas as pd\nimport pickle\nimport re\nfrom scanf import scanf\n\nimport matplotlib\nfrom matplotlib import pyplot as plt\nimport matplotlib.colors as colors\nfrom matplotlib import animation, rc\nimport matplotlib.ticker as mtick\nfrom mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\nfrom mpl_toolkits.mplot3d import Axes3D, axes3d\nfrom mpl_toolkits.axes_grid1.axes_divider import make_axes_locatable\nfrom mpl_toolkits.mplot3d.art3d import Line3DCollection\nfrom matplotlib import cm\n\nfrom time import time\nfrom datetime import datetime\nfrom src.support_class import *\nfrom src.objComposite import *\nfrom src import jeffery_model as jm\nfrom codeStore import support_fun as spf\nfrom codeStore import support_fun_table as spf_tb\n# %matplotlib notebook\n\nrc('animation', html='html5')\nfontsize = 40\nPWD = os.getcwd()\n```\n\n\n```python\n# load (u_i^{Ej}, \\omega_i^{Ej}) and (u_i^a, \\omega_i^a) from dir, standard version.\n# see the method of base flow for detail\n\n# calculate \\omega_1^{E4} and \\omega_2^{E4} at large spin limit. \ndef load_wE4_list(job_dir):\n t_dir = os.path.join(PWD, job_dir, '*.pickle')\n pickle_names = glob.glob(t_dir)\n problem_kwarg_list = []\n wE41_list = []\n wE42_list = []\n\n for pickle_name in pickle_names:\n# print(pickle_name)\n with open(pickle_name, 'rb') as myinput:\n pickle_dict = pickle.load(myinput)\n ta1 = (pickle_dict['uw_Base_list'][4][3] + pickle_dict['uw_Base_list'][5][4]) / 2\n ta2 = (pickle_dict['uw_Base_list'][4][4] - pickle_dict['uw_Base_list'][5][3]) / 2\n wE41_list.append(ta1)\n wE42_list.append(ta2)\n problem_kwarg_list.append(pickle_dict['problem_kwargs'])\n \n wE41_list = np.array(wE41_list)\n wE42_list = np.array(wE42_list)\n problem_kwarg_list = np.array(problem_kwarg_list)\n return wE41_list, wE42_list, problem_kwarg_list\n```\n\n\n```python\njob_dir_list = ['omega_E42_zero_a', \n 'omega_E42_zero_b', \n 'omega_E42_zero_c', ]\n\ntdata_list = []\nfor job_dir in job_dir_list:\n wE41_list, wE42_list, problem_kwarg_list = load_wE4_list(job_dir)\n ch_list = np.array([i0['ch'] for i0 in problem_kwarg_list])\n tdata_list.append((wE41_list, wE42_list, ch_list))\n```\n\n\n```python\n# %matplotlib notebook\n%matplotlib inline\n\nfigsize=np.array((16, 9))*0.5\ndpi = 500 if 'inline' in matplotlib.get_backend() else 100\n\nfig, axs = plt.subplots(1, 2, figsize=figsize, dpi=dpi)\nfig.patch.set_facecolor('white')\n\nfor wE41_list, wE42_list, ch_list in tdata_list:\n axi = axs[0]\n axi.plot(ch_list, wE41_list, '.')\n axi.set_xscale('log')\n axi.set_xlabel('$n$')\n axi.set_ylabel('$\\\\bar \\\\omega_1^{E4}$')\n\n axi = axs[1]\n axi.plot(ch_list, wE42_list, '.')\n axi.set_xscale('log')\n axi.set_xlabel('$n$')\n axi.set_ylabel('$\\\\bar \\\\omega_2^{E4}$')\nplt.tight_layout()\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "a8364501747d2df5c114dc7fd11b00177bd9b29e", "size": 243529, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "baseFlow/toyModel_attractor/omegaE412.ipynb", "max_stars_repo_name": "pcmagic/stokes_flow", "max_stars_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-11-11T05:00:53.000Z", "max_stars_repo_stars_event_max_datetime": "2018-11-11T05:00:53.000Z", "max_issues_repo_path": "baseFlow/toyModel_attractor/omegaE412.ipynb", "max_issues_repo_name": "pcmagic/stokes_flow", "max_issues_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "baseFlow/toyModel_attractor/omegaE412.ipynb", "max_forks_repo_name": "pcmagic/stokes_flow", "max_forks_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 1049.6939655172, "max_line_length": 236384, "alphanum_fraction": 0.9534593416, "converted": true, "num_tokens": 1316, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.6959583313396339, "lm_q1q2_score": 0.44082715016262713}} {"text": "# Calculate original Ikeda for one ship\n\n\n```python\n# %load ../../imports.py\n\"\"\"\nThese is the standard setup for the notebooks.\n\"\"\"\n\n%matplotlib inline\n%load_ext autoreload\n%autoreload 2\n\nfrom jupyterthemes import jtplot\n#jtplot.style(theme='onedork', context='notebook', ticks=True, grid=False)\n\nimport warnings\nwarnings.filterwarnings(\"ignore\")\nfrom pylab import rcParams\nrcParams['figure.figsize'] = 15, 8\n\n\nimport pandas as pd\npd.options.display.max_rows = 999\npd.options.display.max_columns = 999\npd.set_option(\"display.max_columns\", None)\nimport numpy as np\nimport os\nimport matplotlib.pyplot as plt\n#plt.style.use('paper')\n\n#import data\nimport copy\nfrom rolldecay.bis_system import BisSystem\nfrom rolldecay import database\nfrom mdldb.tables import Run\n\nfrom sklearn.pipeline import Pipeline\nfrom rolldecayestimators.transformers import CutTransformer, LowpassFilterDerivatorTransformer, ScaleFactorTransformer, OffsetTransformer\nfrom rolldecayestimators.direct_estimator_cubic import EstimatorQuadraticB, EstimatorCubic\nfrom rolldecayestimators.ikeda_estimator import IkedaQuadraticEstimator\n\n```\n\n\n```python\ndb = database.get_db()\n\ndf_simplified_ikeda = database.load(rolldecay_table_name='rolldecay_simplified_ikeda', limit_score=0.5, \n exclude_table_name='rolldecay_exclude')\n\ndf_rolldecay = database.load(rolldecay_table_name='rolldecay_quadratic_b', limit_score=0.9, \n exclude_table_name='rolldecay_exclude')\n\ndf_rolldecay_cubic = database.load(rolldecay_table_name='rolldecay_cubic_b', limit_score=0.9, \n exclude_table_name='rolldecay_exclude')\n```\n\n\n```python\ndf_rolldecay.head()\n```\n\n### Selecting one ship\n\n\n```python\nproject_number=20157375\n```\n\n\n```python\nmask = (df_rolldecay['project_number'] == project_number)\ndf_rolldecay.loc[mask]\n```\n\n### Ballast loading condition\n\n\n```python\nTF=9.4\n#TF=11.7\nmask = ((df_rolldecay['project_number'] == project_number) &\n (df_rolldecay['TF'] == TF)\n )\ndf_ballast = df_rolldecay.loc[mask].copy()\ndf_ballast\n```\n\n\n```python\nmask = ((df_rolldecay_cubic['project_number'] == project_number) &\n (df_rolldecay_cubic['TF'] == TF)\n )\ndf_ballast_cubic = df_rolldecay_cubic.loc[mask].copy()\ndf_ballast_cubic\n```\n\n\n```python\ndf_ballast_simplified_ikeda = df_simplified_ikeda.loc[df_ballast.index].copy()\ndf_ballast_simplified_ikeda\n```\n\n\n```python\nmeta_data=df_ballast.iloc[0]\nmeta_data_cubic=df_ballast_cubic.loc[meta_data.name]\nmeta_data_ikeda=df_ballast_simplified_ikeda.loc[meta_data.name]\n\ndb_run = db.session.query(Run).get(int(meta_data.name))\ndf = database.load_run(db_run)\n```\n\n\n```python\nlowpass_filter = LowpassFilterDerivatorTransformer(cutoff=2, minimum_score=0.99)\ncutter = CutTransformer(phi_max=np.deg2rad(9), phi_min=np.deg2rad(0.25), phi1d_start_tolerance=0.015)\noffset_transformer = OffsetTransformer()\n\nsteps = [\n ('filter',lowpass_filter),\n# ('scaler',scaler), # Is froude scaling a good idea??\n ('cutter', cutter), \n ('offset_transformer',offset_transformer)\n]\n \npreprocessor = Pipeline(steps) # define the pipeline object.\npreprocessor.fit(df)\nX = preprocessor.transform(df)\n```\n\n\n```python\ndirect_estimator = EstimatorQuadraticB.load(data=meta_data, X=X)\ndirect_estimator_cubic = EstimatorCubic.load(data=meta_data_cubic, X=X)\nikeda_estimator = IkedaQuadraticEstimator.load(data=meta_data_ikeda, X=X)\n\nfig,ax=plt.subplots()\ndirect_estimator.plot_fit(ax=ax)\nfig,ax=plt.subplots()\ndirect_estimator_cubic.plot_fit(ax=ax)\nfig,ax=plt.subplots()\nikeda_estimator.plot_fit(ax=ax)\n```\n\n\n```python\nmeta_data_ikeda['B_e']\n```\n\n\n```python\nmeta_data['B_1']\n```\n\n\n```python\nmeta_data['B_2']\n```\n\n## Run Scores2\n\n\n```python\nfrom pyscores2.indata import Indata\nfrom pyscores2.runScores2 import Calculation\nfrom pyscores2.output import OutputFile\n```\n\n\n```python\nindata_file_name = 'data207_name%s_loading_condition_id%s' % (meta_data.name, meta_data.loading_condition_id)\nindata_path = os.path.join('scores2', indata_file_name)\nindata = Indata()\nindata.open(indata_path)\n\nindata.kxx=meta_data['KXX']\nindata.displacement=meta_data['Volume']\nindata.rho=1000\nindata.zcg=meta_data['kg']-meta_data['TA'] #(important)\nindata.lcb=meta_data['lcg']\n\nindata.waveDirectionMin=0\nindata.waveDirectionMax=180\nindata.waveDirectionIncrement=90\nindata.speedMin=0\nindata.speedMax=18.6\nindata.speedIncrement=18.6\n\n#indata.waveFrequenciesMin=\n#indata.waveFrequenciesMax=\n#indata.waveFrequenciesIncrement=\n\n```\n\n\n```python\noutdata_directory=r'scores2/result'\nif not os.path.exists(outdata_directory):\n os.mkdir(outdata_directory)\n\ncalculation = Calculation(outDataDirectory=outdata_directory)\ncalculation.run(indata=indata)\n```\n\n\n```python\noutput_file = OutputFile(filePath=calculation.outDataPath)\ndf = output_file.get_result()\n\nfig,ax=plt.subplots()\nfor index, group in df.groupby(by=['speed','wave direction'], sort=False):\n group.plot(x='frequencies', y='rollAmplitude', style='o-', label=index, ax=ax)\n \nax.grid(True)\nax.legend();\nax.set_ylabel('Roll');\n```\n\n\n```python\ndf_roll_damping = output_file.get_roll_damping()\ndf_roll_damping\n```\n\n\n```python\ndf_roll_damping.set_index(['speed','wave_direction'], inplace=True)\n```\n\n\n```python\nscores_damping = df_roll_damping.loc[0,0]\nscores_damping\n```\n\n\n```python\nBW_scores = scores_damping['calculated_wave_damping_in_roll']\n```\n\n\n```python\nmeta_data['omega0']/np.sqrt(meta_data['scale_factor'])\n```\n\n\n```python\nimport rolldecayestimators.simplified_ikeda as si\n```\n\n\n```python\nscale_factor = meta_data['scale_factor']\nLPP=meta_data['lpp']\nBeam=meta_data['beam']\nDRAFT=(meta_data['TA']+meta_data['TF'])/2\nVolume=meta_data['Volume']\nCB=Volume/(Beam*DRAFT*LPP)\nCMID=meta_data['A0']\nOG=-meta_data['kg'] + DRAFT\nPHI=5\nlBK=meta_data['BKL']\nbBK=meta_data['BKB']\nOMEGA=meta_data['omega0']\nV=meta_data['ship_speed']*1.852/3.6\n\n#Froude scale:\nLPP/=scale_factor\nBeam/=scale_factor\nDRAFT/=scale_factor\nVolume/=scale_factor**3\nOG/=scale_factor\nlBK/=scale_factor\nbBK/=scale_factor\n#OMEGA=meta_data['omega0']\nV/=np.sqrt(scale_factor)\n\n\nB44HAT, BFHAT, BWHAT, BEHAT, BBKHAT, BLHAT = si.calculate_roll_damping(LPP=LPP,Beam=Beam,CB=CB,\n CMID=CMID,OG=OG,PHI=PHI,lBK=lBK,bBK=bBK,OMEGA=OMEGA,\n DRAFT=DRAFT, V=V, verify_input=False, limit_inputs=True)\n```\n\n\n```python\nsi_result = pd.Series(name='SI-method')\nsi_result['B44HAT']=B44HAT\nsi_result['BFHAT']=BFHAT\nsi_result['BWHAT']=BWHAT\nsi_result['BEHAT']=BEHAT\nsi_result['BBKHAT']=BBKHAT\nsi_result['BLHAT']=BLHAT\n```\n\n\n```python\nsi_result\n```\n\n\n```python\nBWHAT\n```\n\n\n```python\nfrom rolldecayestimators import equations\nfrom rolldecayestimators import symbols, lambdas\nfrom rolldecayestimators.substitute_dynamic_symbols import lambdify\nimport sympy as sp\n```\n\n\n```python\nlambdas.B44_hat_equation\n```\n\n\n```python\nlhs=symbols.B_44_hat\nrhs=sp.sqrt(symbols.beam/(2*symbols.g))/(symbols.rho*symbols.Disp*symbols.beam**2) \nsp.Eq(lhs, rhs)\n```\n\n\n```python\nequations.omega_hat_equation\n```\n\n\n```python\ng=9.81\nrho=1000\nBW_hat_scores=lambdas.B_hat_lambda(B=BW_scores, Disp=Volume*scale_factor**3, beam=Beam*scale_factor, \n g=g, rho=rho)\nBW_hat_scores\n```\n\n\n```python\nBFHAT+BW_hat_scores+BBKHAT+BLHAT\n```\n\n\n```python\nPHI=6 # roll amplitude [deg]\nB_e = lambdas.B_e_lambda(B_1=df_ballast['B_1'], B_2=df_ballast['B_2'], phi_a=np.deg2rad(PHI), \n omega0=df_ballast['omega0'])\n\n\ndf_ballast['B_e_hat'] = lambdas.B_e_hat_lambda(B_e=B_e, Disp=Volume, beam=Beam, \n g=g, rho=rho)\ndf_ballast[['ship_speed','B_e_hat']]\ndf_ballast['V']=df_ballast['ship_speed']*1.852/3.6/np.sqrt(scale_factor)\n```\n\n\n```python\nB44HAT\n```\n\n## Ikeda\n\n\n```python\nfrom rolldecayestimators import ikeda_speed\n```\n\n\n```python\ninputs = pd.DataFrame()\nN=50\nspeeds=np.linspace(df_ballast['ship_speed'].min(),df_ballast['ship_speed'].max(),N)*1.852/3.6/np.sqrt(scale_factor)\ninputs['w'] = w = np.linspace(df_ballast['omega0'].min(),df_ballast['omega0'].max(),N) # roll frequency [rad/s]\n\ninputs['V'] = speeds # Ship speed [m/s]\ninputs['d'] = DRAFT # Draught of hull [m]\n#BW_scores_model=BW_scores/(scale_factor**4/np.sqrt(scale_factor))\n#inputs['Bw0'] = BW_scores_model # Zero speed wave damping [Nm*s/rad]\nws,B_W0 = output_file.calculate_B_W0()\n#B_W0/=(scale_factor**4/np.sqrt(scale_factor))\n#ws/=np.sqrt(scale_factor)\n#ws*=np.sqrt(scale_factor)\n\nB_W0/=(scale_factor**3.5)\nB_W0/=100\nws*=np.sqrt(scale_factor)\n\ninputs['Bw0']=np.interp(w,ws,B_W0) # Zero speed wave damping [Nm*s/rad]\ninputs['fi_a'] = np.deg2rad(PHI) # Roll amplitude [rad]\ninputs['B'] = Beam # Breadth of hull [m]\ninputs['A'] = CMID*Beam*DRAFT # Area of cross section of hull [m2]\ninputs['bBK'] = bBK #breadth of Bilge keel [m] !!(=height???)\ninputs['R'] = 0.1 # Bilge Radis [m]\ninputs['OG'] = OG # distance from roll axis to still water level [m]\ninputs['Ho'] = Beam/(2*DRAFT) # half breadth to draft ratio B/(2*d) [-]\ninputs['ra'] = ra = rho # density of water (1025) [kg/m3]\ninputs['Cb'] =CB # Block coeff [-]\ninputs['L']=LPP # Ship length [m]\ninputs['LBK']=lBK # Bilge keel length [m]\ninputs['Bw_div_Bw0_max'] = np.inf # maxmum allowed difference between Bw0 and at speed.\ninputs['visc']=1.15*10**-6\ninputs['g']=g=9.81\nresults_ikeda = inputs.apply(ikeda_speed.calculate_B44_series,axis=1)\n```\n\n\n```python\nresults_ikeda_hat = results_ikeda.apply(func=lambdas.B_hat_lambda,Disp=Volume, beam=Beam, \n g=g, rho=rho)\nresults_ikeda_hat = pd.concat((inputs, results_ikeda_hat), axis=1)\n\n```\n\n\n```python\ninputs_si = pd.DataFrame()\ninputs_si['V']=speeds\n\ninputs_si['LPP']=LPP\ninputs_si['Beam']=Beam\ninputs_si['DRAFT']=DRAFT\ninputs_si['CB']=Volume/(Beam*DRAFT*LPP)\ninputs_si['CMID']=CMID\ninputs_si['OG']=OG\ninputs_si['PHI']=PHI\ninputs_si['lBK']=lBK\ninputs_si['bBK']=bBK\ninputs_si['OMEGA']=inputs['w'] \n```\n\n\n```python\ndef si_method(row):\n \n B44HAT, BFHAT, BWHAT, BEHAT, BBKHAT, BLHAT = si.calculate_roll_damping(**row, verify_input=False, \n limit_inputs=True, \n BWHAT_lim=np.inf)\n si_result = pd.Series(name=row.name)\n si_result['B44HAT']=B44HAT\n si_result['BFHAT']=BFHAT\n si_result['BWHAT']=BWHAT\n si_result['BEHAT']=BEHAT\n si_result['BBKHAT']=BBKHAT\n si_result['BLHAT']=BLHAT \n \n return si_result\n```\n\n\n```python\nresults_si_hat=inputs_si.apply(func=si_method, axis=1)\nresults_si_hat = pd.concat((inputs_si, results_si_hat), axis=1)\n```\n\n\n```python\nrenamings = {\n 'B44HAT':'B_44',\n 'BFHAT':'B_F',\n 'BWHAT':'B_W',\n 'BEHAT':'B_E',\n 'BBKHAT':'B_BK',\n 'BLHAT':'B_L',\n}\n\nresults_si_hat.rename(columns=renamings, inplace=True)\n```\n\n\n\n\n```python\nfig,ax=plt.subplots()\nresults_ikeda_hat.plot.area(x='V', y = ['B_BK','B_F','B_L','B_W',], ax=ax)\ndf_ballast.plot(x='V', y='B_e_hat', style='ro', label='model test', ax=ax)\nax.legend()\n#ax.set_ylim(0,0.014)\nax.set_title('Original Ikeda compared to model tests')\n\nfig,ax=plt.subplots()\nresults_si_hat.plot.area(x='V', y = ['B_BK','B_F','B_E','B_L','B_W',], ax=ax)\ndf_ballast.plot(x='V', y='B_e_hat', style='ro', label='model test', ax=ax)\nax.legend()\nax.set_ylim(0,0.014)\nax.set_title('Simplified Ikeda compared to model tests')\n```\n\n\n```python\nresults_ikeda_hat['B_E'] = results_si_hat['B_E'].copy()\nresults_si_hat['B_BK'] = results_ikeda_hat['B_BK'].copy()\n\n```\n\n\n\n\n```python\nfig,ax=plt.subplots()\nresults_ikeda_hat.plot.area(x='V', y = ['B_BK','B_F','B_E','B_L','B_W',], ax=ax)\ndf_ballast.plot(x='V', y='B_e_hat', style='ro', label='model test', ax=ax)\nax.legend()\nax.set_ylim(0,0.014)\nax.set_title('Original Ikeda compared to model tests')\n\nfig,ax=plt.subplots()\nresults_si_hat.plot.area(x='V', y = ['B_BK','B_F','B_E','B_L','B_W',], ax=ax)\ndf_ballast.plot(x='V', y='B_e_hat', style='ro', label='model test', ax=ax)\nax.legend()\nax.set_ylim(0,0.014)\nax.set_title('Simplified Ikeda compared to model tests')\n\n```\n\n\n## Sectional Lewis coefficients\n$$a=B / 2\\left(1+a_{1}+a_{3}\\right)$$\n$$a_{1}=\\left(1+a_{3}\\right)(H-1) /(H+1)$$\n$$a_{3}=\\frac{\\left.-C_{1}+3+\\sqrt{(} 9-2 C_{1}\\right)}{C_{1}}$$\n$$C_{1}=\\left(3+\\frac{4 \\sigma_{s}}{\\pi}\\right)+\\left(1-\\frac{4 \\sigma_{s}}{\\pi}\\right)\\left(\\frac{H-1}{H+1}\\right)^{2}$$\n$$\\sigma_{\\mathrm{s}}=\\frac{S}{B T}$$\n$$H=\\frac{B}{2 T}$$\n\n\n```python\nTs=indata.ts/scale_factor\nbwl=indata.bs/scale_factor\n\nareas=indata.cScores*Ts*bwl\na, a_1, a_3, sigma_s, H = ikeda_speed.calculate_sectional_lewis(B_s=bwl, T_s=Ts, S_s=areas)\n \n```\n\n\n\n\n```python\nxs=np.linspace(0,LPP,21)\nR=Beam*0.05 # Guessing\nd=DRAFT\nwE=inputs.iloc[0]['w']\nfi_a = inputs.iloc[0]['fi_a']\nB_E = ikeda_speed.eddy(bwl=bwl, a_1=a_1, a_3=a_3, sigma=sigma_s, xs=xs, H0=H, Ts=Ts, OG=OG, \n R=R, d=d, wE=wE, fi_a=fi_a)\n```\n\n\n```python\nB_E_hat=lambdas.B_hat_lambda(B_E,Disp=Volume, beam=Beam, \n g=g, rho=rho)\nB_E_hat\n```\n\n## Using Ikeda class\n\n\n```python\nfrom rolldecayestimators.ikeda import Ikeda, IkedaR\nfi_a = inputs['fi_a']\nw = inputs['w']\nV = inputs['V']\nikeda = Ikeda.load_scoresII(V=V, w=w, fi_a=fi_a, indata=indata, output_file=output_file, \n scale_factor=scale_factor, lBK=inputs['LBK'], bBK=inputs['bBK'])\n\nikeda.R = inputs['R'].min()\n\n```\n\n\n```python\ndef calculate(inputs, ikeda):\n\n output = inputs.copy()\n output['B_44_hat'] = ikeda.calculate_B44()\n output['B_W0'] =ikeda.calculate_B_W0()\n output['B_W'] =ikeda.calculate_B_W()\n output['B_F'] =ikeda.calculate_B_F()\n output['B_E'] =ikeda.calculate_B_E()\n output['B_BK'] =ikeda.calculate_B_BK()\n output['B_L'] =ikeda.calculate_B_L()\n output['Bw_div_Bw0'] =ikeda.calculate_Bw_div_Bw0()\n \n return output\n```\n\n\n\n\n```python\noutput = calculate(inputs=inputs, ikeda=ikeda)\n\nfig,ax=plt.subplots()\noutput.plot.area(x='V', y = ['B_BK','B_F','B_E','B_L','B_W',], ax=ax)\nax.legend()\nax.set_ylim(0,0.014)\nax.set_title('Original Ikeda compared to model tests');\n```\n\n\n```python\nikeda_r = IkedaR.load_scoresII(V=V, w=w, fi_a=fi_a, indata=indata, output_file=output_file, \n scale_factor=scale_factor, lBK=inputs['LBK'], bBK=inputs['bBK'])\n\n```\n\n\n\n\n```python\noutput = calculate(inputs=inputs, ikeda=ikeda_r)\n\nfig,ax=plt.subplots()\noutput.plot.area(x='V', y = ['B_BK','B_F','B_E','B_L','B_W',], ax=ax)\ndf_ballast.plot(x='V', y='B_e_hat', style='ro', label='model test', ax=ax)\nax.legend()\nax.set_ylim(0,0.014)\nax.set_title('Original Ikeda compared to model tests');\n```\n\n\n```python\ndef calculate_one(ikeda):\n\n output = pd.Series()\n output['B_44_hat'] = ikeda.calculate_B44()[0]\n output['B_W0'] =ikeda.calculate_B_W0()[0]\n output['B_W'] =ikeda.calculate_B_W()[0]\n output['B_F'] =ikeda.calculate_B_F()[0]\n output['B_E'] =ikeda.calculate_B_E()[0]\n output['B_BK'] =ikeda.calculate_B_BK()[0]\n output['B_L'] =ikeda.calculate_B_L()[0]\n output['Bw_div_Bw0'] =ikeda.calculate_Bw_div_Bw0()[0]\n \n return output\n```\n\n\n\n\n```python\nfrom rolldecayestimators.ikeda import Ikeda, IkedaR\nfi_a = inputs['fi_a'].mean()\nw = inputs['w'].mean()\nV = 0\nRs = np.linspace(0.01,0.2,50)*inputs.iloc[0]['B']\nikeda_R = Ikeda.load_scoresII(V=V, w=w, fi_a=fi_a, indata=indata, output_file=output_file, \n scale_factor=scale_factor, lBK=inputs['LBK'].iloc[0], \n bBK=inputs['bBK'].iloc[0])\n\noutputs=pd.DataFrame()\nfor R_ in Rs:\n ikeda_R.R=R_\n output = calculate_one(ikeda=ikeda_R)\n outputs=outputs.append(output, ignore_index=True)\n \nfig,ax=plt.subplots()\noutputs['R']=Rs\noutputs[r'R/beam']=Rs/inputs.iloc[0]['B']\noutputs.plot.area(x=r'R/beam', y = ['B_BK','B_F','B_E','B_L','B_W',], ax=ax);\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "25219bac4a6c5477de5cb388ea1e8f5d50409e3f", "size": 29876, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/rolldecay/06_ikeda/01.01_ikeda_one_ship.ipynb", "max_stars_repo_name": "martinlarsalbert/rolldecay", "max_stars_repo_head_hexsha": "bee335c27e6519bd32ed9488751bc13e7f6fa9b3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/rolldecay/06_ikeda/01.01_ikeda_one_ship.ipynb", "max_issues_repo_name": "martinlarsalbert/rolldecay", "max_issues_repo_head_hexsha": "bee335c27e6519bd32ed9488751bc13e7f6fa9b3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2021-02-02T23:07:50.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-13T03:27:41.000Z", "max_forks_repo_path": "notebooks/rolldecay/06_ikeda/01.01_ikeda_one_ship.ipynb", "max_forks_repo_name": "martinlarsalbert/rolldecay", "max_forks_repo_head_hexsha": "bee335c27e6519bd32ed9488751bc13e7f6fa9b3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.8139534884, "max_line_length": 146, "alphanum_fraction": 0.534844022, "converted": true, "num_tokens": 4990, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.6334102567576901, "lm_q1q2_score": 0.44082714135209117}} {"text": "(section-topics-real)=\n# Real-valued input challenge\n\nMost real-world environments or datasets are represented using exclusively continue-valued inputs or alongside discrete attributes. Unfortunately, the described ALCS implementations were not initially designed for such representations. The fundamental modification required to overcome the limitation above is the adaptation of rule representation within the system. Despite the variety of possible approaches listed in this chapter, each comes with its own merits and perils.\n\nTwo significant concepts involving the discretization and interval encoding using dedicated alphabets will be pursued later, while the hindmost ones involving the neural network and fuzzy logic will be only mentioned. There are two main reasons for this decision:\n\n1. drastic changes in rule representation requires significant modifications in existing and tightly coupled components. In some cases, some of them are not usable in their proposed form - like the ALP component in ACS systems needs a complete redefinition for other alphabet encodings,\n2. rule interpretability is relevantly blemished.\n\nTherefore, even if some approaches are more appealing for the problem, their usage violates principal ALCS virtue of transparency (creating a compact set of human-readable rules) or does not formally specify the behaviour of related inner mechanisms fulfilling the evolution and assessing each classifier. However, more detailed research is highly advised but will not be pursued herein.\n\n(section-topics-real-discretization)=\n## Discretization\n````{margin}\n```{admonition} Ternary representation\nA typical representation of a rule in LCS systems comprising three symbols - $\\{0, 1, \\#\\}$\n```\n````\n\nOne of the approaches for handling continuous attributes is to partition them into a number of sub-ranges and treat each sub-range as a category. This process is known as _discretization_ and is used across many families of ML algorithms in order to create better models {cite}`elomaa2004efficient`. In the XCS family, the modification of XCSI {cite}`wilson2001compact` adopted the algorithm for the integer domain. The modification was evaluated on the _Wisconsin Breast Cancer_ dataset and turned out to be competitive with other ML algorithms. Minding the nature of ALCS, the usage of such nominal values would also be the most straightforward approach. ALCS systems by design are not limited by _ternary representation_, therefore creating an arbitrary number of potential states is achievable. The first such implementation called rACS was proposed by Unold and Mianowski in 2016 {cite}`unold2016real` and tested successfully on [](section-topics-environments-corridor) and [](section-topics-environments-grid) environments. Both have a regular structure, meaning that the intervals are evenly distributed throughout the investigated range.\n\nUnfortunately, the number of ways to discretize a continuous attribute is infinite. Kotsiantis in {cite}`kotsiantis2006discretization` takes a survey of possible discretization techniques, but in this work, the preferred method is to divide the search-space into $n$ equally spaced intervals, referred later as _bins_ or _buckets_. When the $n$ is large, system precision is increased, resulting obviously in the growth of the classifier population. Such a population can be further optimized by compacting the classifiers operating in the neighbouring space regions {cite}`wyatt2004building` {cite}`wilson2001compact`. On the contrary, when the $n$ is low, the system under-performs due to the inability of creating accurate classifiers.\n\nThe process of assigning a discrete value for each consecutive interval region within $[0, 1]$ range is depicted on Figure {numref}`{number} `.\n\n:::{figure-md} discretization-fig\n\n\nRepresentation of allele in rACS as a natural number of the partition. Figure taken from {cite}`unold2016real`.\n:::\n\nSuch a solution for real-value representation in a simple scenario does not require significant changes to any components and retain the human readability and interpretability of created rules. However, when an arbitrary number of bins for each observation attribute is used, certain modifications might be necessary (for example, additional restrictions for the GA cross-over operator).\n\n(section-topics-real-interval)=\n## Interval predicates\nThe discretization approach assumed that a distinct value of the condition attribute represents a fixed-length interval part of an input range. Another approach is to encode the input using custom hyper-rectangular boundaries described by half-open interval $[p_i, q_i)$, which matches the environment signal $x_i$ if $p_i \\leq x_i < q_i$.\n\nSuch approach facilitates creation of more general and compact population size, due to arbitrarily interval ranges. In 1999 Wilson took the approach to adapt the XCS to automatically search for optimally decisive thresholds {cite}`wilson1999get`. His approach representation named _center-spread representation_ (CSR) used to represent an interval tuple $(c_i, s_i) \\ \\text{where} \\ c_i, s_i \\in \\mathbb{R}$ where $c_i$ is the center of the interval, and $s_i$ its spread. The interval is therefore described in Equation {math:numref}`csr-eq`:\n\n```{math}\n:label: csr-eq\n\n\\begin{align}\np_i &= \\text{min}(p_{min}, c_i - s_i) \\\\\nq_i &= \\text{max}(q_{max}, c_i + s_i)\n\\end{align}\n```\nHis system, called XCSR, differs only at the input interface, mutation and covering operators. Preliminary tests revealed the weakness of the crude mutation operator. Moreover, the testing environment, which is very regular and therefore did not challenge the agent with interesting problems like noise or data contradiction.\n\nIn 2003 Stone and Bull thoroughly addressed those problems in {cite}`stone2003real` by introducing two new representations - ordered-bounded (OBR) and unordered-bounded (UBR). The OBR is an extension of Wilson XCSI {cite}`wilson2000mining` but enhanced with real-valued interval tuple $(l_i, u_i) \\ \\text{where} \\ l_i, u_i \\in \\mathbb{R}$. Here $l_i$ and $u_i$ relates to lower and upper bounds, respectively. The encoding imposes $l_i < u_i$ ordering; therefore, all system components that might inadvertently change must be recognized and taken care of. This requirement was relaxed in UBR {cite}`wilson2001function`, thus the same interval can be encoded by both $(p_i, q_i)$ and $(q_i, p_i)$. The advantage is that no operator constraints are needed when the ordering restriction is violated, which constitutes a form of epistasis between $l_i$ and $u_i$, as their values are mutually dependent. The authors showed that CSR and OBR are biased in interval generation in bounded solution spaces. The UBR obviating limitations of OBR was assumed to yield better results in one of the testing problems and therefore considered superior to the other ones.\n\nFinally, in 2005 Dam and Abbass {cite}`dam2005real` recognized that UBR changes the semantics of the chromosome by alternating between min and max genes. This discrepancy is challenging for the XCS because it disturbs the genetic process evolving the population of classifiers {cite}`holland1975adaptation` {cite}`goldberg1989`. They present the Min-Percentage representation of $(m_i, k_i) \\ \\text{where} \\ m_i, k_i \\in \\mathbb{R}$ tuple, where the interval is determined by the Equation {math:numref}`mp-eq`, which compared with UBR did not provide any substantial improvements.\n\n```{math}\n:label: mp-eq\n\n\\begin{align}\np_i &= m_i \\\\\ns_i &= k_i \\cdot (k_{max} - p_i) \\\\\nq_i &= m_i + s_i\n\\end{align}\n```\n\nIt is also worth mentioning an important, subtly different, family of learning classifier systems handling real-valued input natively by _approximating functions_. The most popular implementation of XCSF {cite}`wilson2001function` {cite}`hamzeh2005evolutionary` computes the payoff value locally instead of learning its prediction through a gradient-descent type update. The classifier's condition covers the continuous input by an OBR interval. Enhanced with additional weights vector, updated by regression techniques, the output prediction can be calculated. The classifier structure was further simplified by eliminating the proposed action. Because of the lack of explicit state anticipation capabilities, the function approximation learning classifiers systems are not considered in this work.\n\n(section-topics-real-neural)=\n## Neural networks\nO\u2019Hara and Bull experimented with representing the rule structure of an XCS classifier with two artificial neural networks, introducing the system named X-NCS {cite}`ohara2004prediction` {cite}`o2005building`. While the first one, called the _action-network_, determines the application condition of the classifiers replacing the condition-action part, the latter one - _anticipation network_ - forms the description of the predicted next state.\n\nBoth networks are fully connected multi-layered perceptrons (MLP) with the same nodes in their hidden layer. The input layer in both cases matches the observation state vector provided by the environment. In the action network, the size of an output layer is equal to the number of possible actions incremented with an extra node signifying a non-matching situation. Hence, the anticipation network is responsible for predicting the next state; the size of its output layer is equal to the input one. Figures {numref}`{number} ` and {numref}`{number} ` visualize both topologies.\n\n````{tabbed} Action Network\n```{figure} ../../_static/graphs/xncs_action_network.gv.png\n:name: xncs-action-network\n\nThe topology of the fully connected MLP of the network determines the agent's action based on the observed state (input layer). The number of output nodes equals the number of possible actions with an extra state representing a non-matching case. Figure adapted from {cite}`o2005building`.\n```\n````\n\n````{tabbed} Anticipation Network\n```{figure} ../../_static/graphs/xncs_anticipation_network.gv.png\n:name: xncs-anticipation-network\n\nThe topology of fully connected MLP of the network determines the anticipated states using the observed environmental state (input layer). Specific output nodes refer to particular input nodes. Figure adapted from {cite}`o2005building`.\n```\n````\n\nThe system starts with an initial random population, containing the maximum number of classifiers considered in a particular experiment, as opposed to standard XCS {cite}`wilson1995classifier`. The network weights are randomly initialized in the range $[-1.0, 1.0]$ and are updated throughout the experiment using two search techniques - the local search performed by backpropagation complemented by the global sampling performed by GA.\n\nIn order to assess the general system error, all of the internal mechanisms remained unchanged except for the calculation of the rule\u2019s absolute error, which is defined as the sum of prediction and lookahead error (measuring the correctness of the prediction).\n\nThe critical difference between the classic ternary representation is the lack of explicit wildcard symbol and hence no explicit pass-through of input to anticipations. A concept of a reliable classifier cannot be applied in X-NCS - the anticipation accuracy is based on a percentage of accurate anticipations per presentation.\n\nThe authors tested the extension in various configurations showing promising results, especially on discrete multistep problems. Due to novel rule representation, the X-NCS system is suited for real-valued data representation, but the conceptual differences make it difficult to compare it with other systems. Aspects like vague generalization metric, constant population size or the anticipation accuracy computation would require dedicated research solely on this implementation.\n\nInterestingly, two years later, a similar system evolved. The authors take advantage of the idea of both the function mapping {cite}`wilson2002classifiers` and the neural anticipation {cite}`o2005building`. The classifier structure was extended with a parametrized anticipatory function $an_f$, which is trained using supervised learning based on the current state $\\sigma_t$ and the next state $\\sigma_{t+1}$. The proposed XCS with anticipation mappings (XCSAM) system {cite}`bull2007anticipation` does not use the lookahead error and focuses on the value of actions. The authors claim that the anticipatory capabilities are obtained as the \"side effects\". Because of the lack of any dedicated prediction components, the anticipatory burden falls on the function $an_f$. Preliminary experiments showed that both X-NCS and XCSAM obtain comparable results on discrete multistep environments, however both of the systems can be generalized to problems involving continuous inputs {cite}`lanzi2005xcs`.\n\n(section-topics-real-fuzzy)=\n## Fuzzy representation\nThe Michigan-style genetic fuzzy rule-based system {cite}`cord2001genetic` is a machine learning system that employs linguistic rules and fuzzy sets in its representation and an evolutionary algorithm for rule discovery. Casillas proposed relevant modifications to the XCS and introduced a new modification called Fuzzy-XCS {cite}`casillas2004fuzzy`. The newly created system was capable of dealing with continuous states and actions while maintaining maximal generalization. A similar solution was also proposed later by Bonarini in {cite}`bonarini2007fixcs`.\n\nKondziela undertook an approach to create a fuzzy variant of ACS in 2021 {cite}`kondziela2021facs`. The presented idea modified the system comprising four major elements {cite}`rudnik2011koncepcja`:\n\n- _fuzzification_ - assigns set membership of current environmental perception,\n- _inference_ - aggregates the results by comparing the membership functions with collected knowledge,\n- _knowledge-store_ - stores population of classifiers where representing rules using the _Fuzzy Inference System_ (Mamandi) of `IF..AND..THEN` statements {cite}`zadeh1996fuzzy`,\n- _defuzzification_ - provides a single value response obtained from the inference phase determining the final action value.\n\nAs the first step the vector of environment signal determines set memberships using predefined functions {cite}`casillas2007fuzzy` {cite}`bonarini2007fixcs`. Then using the rule representation described by Equation {math:numref}`fuzzy-eq` the match set is formed. Each input value $X_i$ is equal to a linguistic set of $\\tilde{A_i} = \\{ A_{i1} \\lor \\dots \\lor A_{il} \\}$ meaning that classifier internally consists of a rule described with $\\{0, 1, \\# \\}$ symbols (ternary alphabet).\n\n```{math}\n:label: fuzzy-eq\n\n\\mathbf{IF}\\ X_1\\ \\text{is}\\ \\tilde{A_1}\\ \\text{and}\\ \\dots \\ X_n\\ \\text{is}\\ \\tilde{A_n}\\ \\mathbf{THEN}\\ Y\\ \\text{is}\\ B\n```\n\nIn the next step, the action set $[A]$ is formed in the same way as in traditional ACS, but the final action selection procedure differs - it is proposed by taking advantage of each rule's membership function values, and the _Center of Gravity_ method for defuzzification {cite}`kondziela2021facs`.\n\nPreliminary tests made on multistep, discrete maze environments showed that fuzzy ACS implementation successfully predicted more than 75% of encountered situations and maintained a limited number of reliable classifiers (although both metrics were highly oscillating). The author did not report any other operational performance indicators.\n\nThe usage of fuzzy logic enables the system to handle the real-valued input naturally. The apparent impediment is the requirement to specify membership functions for each environmental perception upfront. The selection of optimal values is complicated {cite}`bonarini1999fuzzy`, and further increases the number of overall system tunable parameters. The other identified flaw is the GA phase which is not suited for new representation. Both the mutation and cross-over operators should be reviewed accordingly.\n\n", "meta": {"hexsha": "282383c809251b9a1b7191ff5f72813bb15adfc1", "size": 18290, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "book/chapters/2_selected_topics/23_real_value_challenge.ipynb", "max_stars_repo_name": "khozzy/phd", "max_stars_repo_head_hexsha": "9a05572a6960d948320669c51e0c80bb9d037d4a", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "book/chapters/2_selected_topics/23_real_value_challenge.ipynb", "max_issues_repo_name": "khozzy/phd", "max_issues_repo_head_hexsha": "9a05572a6960d948320669c51e0c80bb9d037d4a", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "book/chapters/2_selected_topics/23_real_value_challenge.ipynb", "max_forks_repo_name": "khozzy/phd", "max_forks_repo_head_hexsha": "9a05572a6960d948320669c51e0c80bb9d037d4a", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 92.3737373737, "max_line_length": 1168, "alphanum_fraction": 0.7388190268, "converted": true, "num_tokens": 3403, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.7341195385342971, "lm_q1q2_score": 0.4406100013337586}} {"text": "- [Otto Group Product Classification](#Otto-Group-Product-Classification)\n - [Contexto](#Contexto)\n - [Dados](#Dados)\n- [Benchmark](#Benchmark)\n- [Cross Validation](#Cross-Validation)\n- [Tratamento](#Tratamento)\n - [Correla\u00e7\u00e3o](#Correla\u00e7\u00e3o)\n - [Filtrando colunas](#Filtrando-colunas)\n - [Resultado](#Resultado)\n- [Train/Test split](#Train/Test-split)\n - [Feature Scaling](#Feature-Scaling)\n - [Confusion Matrix](#Confusion-Matrix)\n- [Modelo Dummy Classifier](#Modelo-Dummy-Classifier)\n- [Boosting](#Boosting)\n- [Gradient Descent](#Gradient-Descent)\n- [XGBoost](#XGBoost)\n - [Model](#Model)\n - [Objective Function:](#Objective-Function:)\n - [CART](#CART)\n - [Training](#Training)\n - [Additive](#Additive)\n- [GridSearchCV](#GridSearchCV)\n - [Aplicando GridSearchCV ao XGBClassifier:](#Aplicando-GridSearchCV-ao-XGBClassifier:)\n - [Aplicando GridSearchCV ao Decision Tree Classifier:](#Aplicando-GridSearchCV-ao-Decision-Tree-Classifier:)\n- [Trees](#Trees)\n - [Decision Tree](#Decision-Tree)\n - [Distribui\u00e7\u00e3o dos dados](#Distribui\u00e7\u00e3o-dos-dados)\n - [Filtrar dados](#Filtrar-dados)\n - [Verificando resultado](#Verificando-resultado)\n- [Random Forest](#Random-Forest)\n - [Utilizando o algoritmo](#Utilizando-o-algoritmo)\n - [Verificando com Cross Validation](#Verificando-com-Cross-Validation)\n - [Importancia das features para a RF](#Importancia-das-features-para-a-RF)\n - [Gini](#Gini)\n - [O indice](#O-indice)\n - [Para Decisions Trees](#Para-Decisions-Trees)\n - [Decrease Mean Importance](#Decrease-Mean-Importance)\n - [ExtraTrees](#ExtraTrees)\n- [Neur\u00f4nio Artificial](#Neur\u00f4nio-Artificial)\n - [Entrada](#Entrada)\n - [Fun\u00e7\u00e3o agregadora](#Fun\u00e7\u00e3o-agregadora)\n - [Neur\u00f4nio](#Neur\u00f4nio)\n - [Formula](#Formula)\n - [MLP Classifier](#MLP-Classifier)\n- [Conclus\u00e3o](#Conclus\u00e3o)\n- [Refer\u00eancias Bibliogr\u00e1ficas](#Refer\u00eancias-Bibliogr\u00e1ficas)\n\n# Otto Group Product Classification\n\nEste notebook \u00e9 uma proposta de solu\u00e7\u00e3o utilizando t\u00e9cnicas de data-mining e machine learn para o problema de classifica\u00e7\u00e3o de produtos da companhia Otto dispon\u00edveis em: [Kaggle (challenge): Otto group product classification](https://www.kaggle.com/c/otto-group-product-\nclassification-challenge)\n\n## Contexto\n\nRetirado da descri\u00e7\u00e3o do problema, temos que o grupo Otto \u00e9 uma das maiores companhias de *e-commerce* do mundo, e possui s filiais em mais de 20 paises. Vendem milh\u00f5es de produtos ao redor do mundo todos os dias com centezas de produtos sendo adicionados constantemente.\n\nA an\u00e1lise de consist\u00eancia da performance dos produtos deles \u00e9 crucial, entretando, com a infraestrutura de escala global que possuem, produtos identicos s\u00e3o classifidados de maneira diferenciada. Entretanto a an\u00e1lise da qualidade dos produtos depende fortemente da acur\u00e1cia na habilidade de agrupar produtos semelhantes. Quanto melhor for a classifica\u00e7\u00e3o, mais intuitivamente eles ter um maior alcance com seus produtos.\n\n## Dados\n\nForam disponibilizados 2 bases de dados separadas. A primeira delas cont\u00e9m 61878 registros com r\u00f3tulo da classifica\u00e7\u00e3o do produto e 144368 de registros sem o r\u00f3tulo.\n\nS\u00e3o um total de 93 caracter\u00edsticas na qual n\u00e3o h\u00e1 a descri\u00e7\u00e3o do que significa cada uma delas. Sendo que n\u00e3o h\u00e1 dados faltando. O range dos dados v\u00e3o de 0 a 352.\n\n\n```python\n# Configure to show multiples outputs from a single cell\nfrom IPython.core.interactiveshell import InteractiveShell\nInteractiveShell.ast_node_interactivity = \"all\"\nimport zipfile\nimport pandas as pd\nimport matplotlib.pyplot as plt\nfrom pandas.plotting import scatter_matrix\nimport numpy as np\nimport math\nfrom sklearn.metrics import classification_report, confusion_matrix\nfrom sklearn.neural_network import MLPClassifier\n\n```\n\n\n```python\nwith zipfile.ZipFile('Datasets.zip') as ziped_file:\n with ziped_file.open('Datasets/train.csv') as train_file:\n df_train = pd.read_csv(train_file, header=0).set_index('id')\n with ziped_file.open('Datasets/test.csv') as test_file:\n df_test = pd.read_csv(test_file, header=0).set_index('id')\ndf_target = pd.DataFrame(df_train.pop('target')) # Get the target\ndf_target.target = pd.Categorical(df_target.target) # Transform target in Categorical type\ndf_target['categories'] = df_target.target.cat.codes # Add the codes in a columns\ndf_target.head() # Show target classes\ndf_train.head() # The train dataset\ndf_test.head() # It hasn't target\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    targetcategories
    id
    1Class_10
    2Class_10
    3Class_10
    4Class_10
    5Class_10
    \n
    \n\n\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    feat_1feat_2feat_3feat_4feat_5feat_6feat_7feat_8feat_9feat_10...feat_84feat_85feat_86feat_87feat_88feat_89feat_90feat_91feat_92feat_93
    id
    11000000000...0100000000
    20000000100...0000000000
    30000000100...0000000000
    41001615001...22012000000
    50000000000...0100001000
    \n

    5 rows \u00d7 93 columns

    \n
    \n\n\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    feat_1feat_2feat_3feat_4feat_5feat_6feat_7feat_8feat_9feat_10...feat_84feat_85feat_86feat_87feat_88feat_89feat_90feat_91feat_92feat_93
    id
    10000000003...001112000000
    2221416000000...0000040020
    301121000000...0000200001
    40001000000...0310000000
    51001001203...0000000900
    \n

    5 rows \u00d7 93 columns

    \n
    \n\n\n\n# Benchmark\n\nA vari\u00e1vel results \u00e9 um acumulador para salvar os resultados na base de treino e teste de cada um dos modelos e compar\u00e1-los ao final.\n\nSegue a estrutura:\n\n`\n 'modelo':\n 'teste': value\n 'treino': value\n`\n\n\n```python\nfrom sklearn.model_selection import train_test_split\n\nresults = {}\ndef add_results(model, train, test):\n results[model] = {\n 'train': train*100,\n 'test': test*100,\n }\n```\n\n# Cross Validation\n\nA abordagem para a Valida\u00e7\u00e3o Cruzada \u00e9 a utiliza\u00e7\u00e3o do m\u00e9todo de k-parti\u00e7\u00f5es. Neste m\u00e9todo, o conjunto de dados \u00e9 dividido em k parti\u00e7\u00f5es [(WITTEN e FRANK, 2000)](ftp://ftp.ingv.it/pub/manuela.sbarra/Data%20Mining%20Practical%20Machine%20Learning%20Tools%20and%20Techniques%20-%20WEKA.pdf), testes extensivos em diversas bases de dados, utilizando diversos algoritmos, identificaram o valor de k para identificar a melhor margem de erro como sendo 10, tamb\u00e9m de forma rand\u00f4mica. Ent\u00e3o, o conjunto de dados de treinamento \u00e9 criado com k \u2013 1 parti\u00e7\u00f5es, e apenas uma parti\u00e7\u00e3o \u00e9 utilizada para testes. S\u00e3o realizadas k itera\u00e7\u00f5es, aonde cada parti\u00e7\u00e3o \u00e9 utilizada uma vez para testes enquanto as outras s\u00e3o utilizadas para treinamento. Ap\u00f3s todas as parti\u00e7\u00f5es terem sido utilizadas para teste, a margem de erro de cada itera\u00e7\u00e3o \u00e9 somada e a m\u00e9dia das k itera\u00e7\u00f5es se torna a margem de erro do modelo.\n\n \n
    Representa\u00e7\u00e3o do m\u00e9todo Cross Validation com k = 10.\n**Fonte**: BABATUNDE et al., 2015.
    \n\n# Tratamento\n\nSer\u00e1 realizada as etapas de feature selection e feature engineering. Correla\u00e7\u00e3o entre features Ser\u00e1 realizada uma an\u00e1lise da correla\u00e7\u00e3o entre as features. Visto que h\u00e1 um total de 93 colunas que n\u00e3o foi disponibilizada nenhuma informa\u00e7\u00e3o sobre o que s\u00e3o elas e o que representam e portanto, esta an\u00e1lize ajudar\u00e1 a identificar as rela\u00e7\u00f5es entre as features.\n\n## Correla\u00e7\u00e3o\n\n\nA correla\u00e7\u00e3o entre duas vari\u00e1veis \u00e9 quando existe algum la\u00e7o matem\u00e1tico que envolve o valor de duas vari\u00e1veis de alguma forma [ESTAT\u00cdSTICA II - CORRELA\u00c7\u00c3O E REGRESS\u00c3O](http://www.ctec.ufal.br/professor/mgn/05CorrelacaoERegressao.pdf).\n\nUma das maneiras mais simples de se identificar a correla\u00e7\u00e3o entre duas vari\u00e1veis \u00e9 plotando-as em um gr\u00e1fico, para tentar identificar alguma rela\u00e7\u00e3o entre elas, entretanto, como s\u00e3o um total de 93 features, dificulta visualizar a correla\u00e7\u00e3o em forma gr\u00e1fica.\n\nA correla\u00e7\u00e3o de [Pearson](https://pt.wikipedia.org/wiki/Coeficiente_de_correla%C3%A7%C3%A3o_de%0A_Pearson) mede o grau da correla\u00e7\u00e3o (e a direc\u00e7\u00e3o dessa correla\u00e7\u00e3o - se positiva ou negativa) entre duas vari\u00e1veis de escala m\u00e9trica (intervalar ou de r\u00e1cio/raz\u00e3o).\n\nJ\u00e1 a correla\u00e7\u00e3o de [Spearman](https://pt.wikipedia.org/wiki/Coeficiente_de_correla%C3%A7%C3%A3o_de_postos_de_Spearman) entre duas vari\u00e1veis \u00e9 igual \u00e0 correla\u00e7\u00e3o de Pearson entre os valores de postos daquelas duas vari\u00e1veis. Enquanto a correla\u00e7\u00e3o de Pearson avalia rela\u00e7\u00f5es lineares, a correla\u00e7\u00e3o de Spearman avalia rela\u00e7\u00f5es mon\u00f3tonas, sejam elas lineares ou n\u00e3o.\n\nVisto ambos os tipos de correla\u00e7\u00e3o, utilizaremos a de Pearson para avaliar se h\u00e1 alguma correla\u00e7\u00e3o linear crescente ou decrescente entre as vari\u00e1veis, pois esta rela\u00e7\u00e3o nos possibilita remover uma delas sem prejuizos aos modelos de machine learn.\n\n\n```python\nshape = (df_train.shape[1], df_train.shape[1])\nupper_matrix = np.tril(np.ones(shape)).astype(np.bool)\nnp.fill_diagonal(upper_matrix, False)\ncorrelation = df_train.corr('pearson').abs().where(upper_matrix)\ncorrelation\n```\n\n\n\n\n
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    feat_1feat_2feat_3feat_4feat_5feat_6feat_7feat_8feat_9feat_10...feat_84feat_85feat_86feat_87feat_88feat_89feat_90feat_91feat_92feat_93
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    feat_780.2792020.0942560.0219791.449856e-020.0707090.0612500.5670840.0796230.0159220.091760...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
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    feat_890.0968510.1055270.1747811.837145e-010.1199510.0350420.0635110.0079740.0191470.061498...0.1036430.0535820.0118220.0660080.022552NaNNaNNaNNaNNaN
    feat_900.0103100.5150220.0150689.454061e-030.0048420.0540340.1295780.0268070.0206980.049908...0.0060130.0039310.0198030.0146960.0316790.027764NaNNaNNaNNaN
    feat_910.0372640.0263830.0124171.031241e-020.0120120.0124650.0685060.0959900.0147420.024025...0.0034440.0230910.0240050.0288500.0336530.0159170.014812NaNNaNNaN
    feat_920.0547770.0082190.0669218.763105e-020.0653310.0154790.0322610.0136080.0697070.006869...0.0484310.0434840.0493930.0014240.0701200.1296220.0353110.104226NaNNaN
    feat_930.0817830.0545930.0068141.574563e-020.0020380.0085210.0349120.0051310.0060380.041316...0.0037230.0233900.0290350.4999900.0086310.0306500.0398640.0000450.003653NaN
    \n

    93 rows \u00d7 93 columns

    \n
    \n\n\n\n## Filtrando colunas\n\nA partir da matriz de correla\u00e7\u00e3o assima, buscamos agora identificar quais das colunas possuem uma forte correla\u00e7\u00e3o de acordo com a tabela a seguir.\n\nComo sugerido por [Makuka,2012](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3576830/)\n\n
    Interpreta\u00e7\u00e3o do resultado de correla\u00e7\u00e3o
    \n\n|Valor absoluto|Significado|\n|---|---|\n|0.9 < v | Muito forte |\n|0.7 < v <= 0.9 | Forte |\n|0.5 < v <= 0.7 | Moderada |\n|0.3 < v <= 0.5 | Fraca |\n|0.0 < v <= 0.3 | Desprez\u00edvel |\n\n\n```python\nstrong_correlation = correlation.where(correlation > 0.8)\nstrong_correlation = strong_correlation.dropna(how='all', axis=(0,1))\ncorr_features = strong_correlation[strong_correlation.notnull()].stack().index\ncorr_features_size = len(corr_features)\nif corr_features_size:\n col = math.floor(math.log2(corr_features_size)) or 1\n row = math.ceil(corr_features_size/col)\n figure, axis = plt.subplots(row, col, figsize=[15,2*row])\n figure.tight_layout()\n for idx, (feature1, feature2) in enumerate(corr_features):\n if row == 1: # Has a single element\n plot = axis.scatter(df_train[feature1],df_train[feature2])\n plot = axis.set_xlabel(feature1)\n plot = axis.set_ylabel(feature2)\n plot = axis.annotate(strong_correlation[feature2][feature1],xy=(0,0))\n elif col == 1: # Has multiples elements, but is a array\n plot = axis[idx].scatter(df_train[feature1], df_train[feature2])\n plot = axis[idx].set_xlabel(feature1)\n plot = axis[idx].set_ylabel(feature2)\n plot = axis[idx].annotate(strong_correlation[feature2][feature1],xy=(0,0))\n else: # Multitle elements and is a matrix\n plot = axis[int(idx/col), idx%col].scatter(df_train[feature1], df_train[feature2])\n plot = axis[int(idx/col), idx%col].set_xlabel(feature1)\n plot = axis[int(idx/col), idx%col].set_ylabel(feature2)\n plot = axis[int(idx/col), idx%col].annotate(strong_correlation[feature2][feature1],xy=(0,0))\n plt.show()\n```\n\n## Resultado\n\nA correla\u00e7\u00e3o mostra que n\u00e3o h\u00e1 uma fort\u00edssima correla\u00e7\u00e3o entre as features, entretanto, h\u00e1 10 colunas que est\u00e3o fortemente correlacionadas. Porem buscamos uma correla\u00e7\u00e3o fort\u00edssima para n\u00e3o remover features com comportamentos diferentes.\n\n# Train/Test split\n\nUtilizaremos 80% da base de treino para efetivamente treinar o modelo e 20% para averiguar a performance do modelo.\n\n\n```python\nfrom sklearn.model_selection import cross_val_score\nfrom sklearn.metrics import confusion_matrix\nfrom sklearn.preprocessing import StandardScaler\n\nX = df_train\ny = df_target.categories\nX_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n```\n\n## Feature Scaling\n\nTrata-se do processo de transformar todos os dados da amostra para uma unidade padr\u00e3o, neste problema utilizaremos a t\u00e9cnica de padroniza\u00e7\u00e3o que consiste em remover a m\u00e9dia dos dados e colocar todos na escala do desvio padr\u00e3o [Wikipedia](https://en.wikipedia.org/wiki/Feature_scaling). Em que $\\bar{x}$ \u00e9 a m\u00e9dia e $\\sigma$ \u00e9 o desvio padr\u00e3o.\n\n\\begin{equation}\n x' = \\frac{x - \\bar{x}}{\\sigma}\n\\end{equation}\n\n\n```python\nsc_X = StandardScaler()\nsc_X_train = sc_X.fit_transform(X_train)\nsc_X_test = sc_X.transform(X_test)\n```\n\nFeature scaling foi aplicado nos dataframes de **features** e utilizado nos modelos, mas o resultado n\u00e3o apresentou mudan\u00e7a. Os modelos continuaram com exatamente as mesmas performances.\n\n## Confusion Matrix\n\nA matriz de confu\u00e7\u00e3o \u00e9 uma m\u00e9trica para algor\u00edtmos supervisionados em que \u00e9 poss\u00edvel estabelecer uma rela\u00e7\u00e3o entre os acertos e erros durante a classifica\u00e7\u00e3o do conjunto de amostras. Basicamente elabora-se uma matriz em que nas colunas e linhas s\u00e3o as poss\u00edveis classes. Cada c\u00e9lula traz a contagem de amostras que eram da Label X (coluna) e foram classificadas na Label Y (linha). Dessa forma, na matriz, a diagonal principal trar\u00e1 os acertos do classificador [Microsoft](https://docs.microsoft.com/pt-br/sql/analysis-services/data- mining/classification-matrix-analysis-services-data-mining).\n\n Veja o exemplo a seguir:\n\n|Classificador\\Real|Label 1|Label 2|Label 3|\n|---|-------|-------|-------|\n|**Label 1**|10|10|0|\n|**Label 2**|1|10|1|\n|**Label 3**|0|0|3|\n\nPlot para matriz de confus\u00e3o encontrado em [Scikit](http://scikit- learn.org/stable/auto_examples/model_selection/plot_confusion_matrix.html#sphx- glr-auto-examples-model-selection-plot-confusion-matrix-py) e adaptado para o problema\n\n\n```python\nimport itertools\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nfrom sklearn.metrics import confusion_matrix\n\ndef plot_confusion_matrix(cm, classes,\n normalize=False,\n title='Confusion matrix',\n cmap=plt.cm.Blues):\n \"\"\"\n This function prints and plots the confusion matrix.\n Normalization can be applied by setting `normalize=True`.\n \"\"\"\n if normalize:\n cm = cm.astype('float') / cm.sum(axis=1)[:, np.newaxis]\n print(\"Normalized confusion matrix\")\n else:\n print('Confusion matrix, without normalization')\n\n plt.figure(figsize=(11, 7))\n plt.imshow(cm, interpolation='nearest', cmap=cmap)\n plt.title(title)\n plt.colorbar()\n\n fmt = '.2f' if normalize else 'd'\n thresh = cm.max() / 2.\n for i, j in itertools.product(range(cm.shape[0]), range(cm.shape[1])):\n plt.text(j, i, format(cm[i, j], fmt),\n horizontalalignment=\"center\",\n color=\"white\" if cm[i, j] > thresh else \"black\")\n\n plt.tight_layout()\n plt.ylabel('True label')\n plt.xlabel('Predicted label')\n plt.show()\n```\n\n## Modelo Dummy Classifier\n\nDummy Classifier \u00e9 um modelo que faz predi\u00e7\u00f5es usando regras simples.\n\nO dummy \u00e9 importante para termos como par\u00e2metro de compara\u00e7\u00e3o com outros modelos.N\u00e3o pode ser utilizado em problemas reais porque ele \u00e9 apenas para realizar compara\u00e7\u00f5es e trabalha com aleatoriedade e frequencia de repeti\u00e7\u00f5es para realizar as predi\u00e7\u00f5es.\n\n\nUsamos dois tipos de estrat\u00e9gia:\n\n* **Stratified**: realiza predi\u00e7\u00f5es baseadas na distribui\u00e7\u00e3o das classes da base de treino. (Ex.: 10% A, 20% B, 50% C, 20% D)\n* **Most Frequent**: sempre prediz com a classe mais frequente na base de treino\n\n\n```python\nfrom sklearn.dummy import DummyClassifier\n\ndef dummies(X_train, y_train, X_test, y_test):\n models = ['most_frequent', 'stratified']\n\n for model in models:\n clf = DummyClassifier(strategy=model)\n clf.fit(X_train, y_train)\n score = clf.score(X_train, y_train)\n y_pred = clf.predict(X_test)\n cm = confusion_matrix(y_test, y_pred)\n\n plot_confusion_matrix(cm, classes=model)\n # Cross validation\n accuracies = cross_val_score(estimator=clf, X=X_train, y=y_train, cv=10)\n \n add_results(model, clf.score(X_train, y_train), clf.score(X_test, y_test))\n print(model, 'train dataset score: %.2f' % score)\n print('M\u00e9dia: %.2f' % accuracies.mean())\n print('Desvio padr\u00e3o: %.4f' % accuracies.std())\n\ndummies(X_train, y_train, X_test, y_test)\n```\n\n## Boosting\n\nA defini\u00e7\u00e3o de boosting \u00e9 que at\u00e9 mesmo algor\u00edtmos fracos de machine larning podem se tornar potentes [(KEARNS, 1988)](https://www.cis.upenn.edu/~mkearns/papers/boostnote.pdf).\n\nUm algor\u00edtmo fraco de aprendizagem pode ser definido como modelos ou regras que n\u00e3o possuem boa acur\u00e1cia ou aparentam ser ineficientes, tais como modelos *dummy*: mais frequente, estratificado, rand\u00f4mico. J\u00e1 algor\u00edtmos de aprendizagem forte, s\u00e3o aqueles que apresentam uma boa taxa de acertos [(FREUND e SCHAPIRE)](http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=4BF3325D8222B3234BB95971FCAD8759?doi=10.1.1.56.9855&rep=rep1&type=pdf).\n\n**Exemplo - Corrida de cavalos**[(FREUND e SCHAPIRE)](http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=4BF3325D8222B3234BB95971FCAD8759?doi=10.1.1.56.9855&rep=rep1&type=pdf):\n\n Como determinar em qual cavalor apostar, considerando um conjunto de dados dispon\u00edveis tais como informa\u00e7\u00f5es do cavalo, do dono, das corridas anteriores e etc. Ao perguntar para especialistas cada um deles ir\u00e1 falar coisas distintas e ainda assim muito imprecisas (modelos fracos)! Mas seria poss\u00edvel utilizar as regras de aposta de cada especialista e gerar uma \u00fanica regra que seja capaz de predizer o cavalor vencedor da corrida utilizando boost \n\n## Gradient Descent\n\n\n\n\nUm algor\u00edtmo de gradient descendent \u00e9 uma forma de minimizar o valor de uma fun\u00e7\u00e3o interativamente, na qual s\u00e3o dados um conjunto de parametros e ela busca a partir da\u00ed o menor valor[(TOUSSAINT, 2012)](https://ipvs.informatik.uni- stuttgart.de/mlr/marc/notes/gradientDescent.pdf). De forma que:\n\n\\begin{equation}\n y_{min} = F(x_1) > F(x_2) > F(x_3) > ... > F(x_n),\\ onde:\\ F(x_n) < precis\u00e3o\n\\end{equation}\n\nUm pseudo algor\u00edtmo que pode ser proposto para um problema de gradient \u00e9:\n\n x = inital_value\n step = 0.01\n repita\n xprev=x\n x = xperv - step * F(xprev)\n enquanto abs(x - xprev) > precisao\n\n# XGBoost\n\nXGBoost \u00e9 um algoritmo que implementa *gradient boosting* de Decision Trees de forma r\u00e1pida e com alta performance. **Gradient Boosting** \u00e9 uma t\u00e9cnica de *machine learning* para problemas de regress\u00e3o e classifica\u00e7\u00e3o que produz um modelo de predi\u00e7\u00e3o na forma de *ensemble* de modelos de predi\u00e7\u00f5es fracas, normalmente \u00e1rvores de decis\u00f5es. Boosting \u00e9 um processo sequencial, mas como o `XGBoost` consegue implement\u00e1-lo de forma paralela? Sabemos que cada \u00e1rvore pode ser produzida apenas depois que produzida a \u00e1rvore anterior, mas o processo de criar as \u00e1rvores pode ser paralelizado utilizando todos os n\u00facleos a disposi\u00e7\u00e3o.\n\n## Model\n\n### Objective Function:\n\n\\begin{equation}\n \\text{obj}(\\theta) = L(\\theta) + \\Omega(\\theta)\n\\end{equation}\n\n**L- Training Loss function**: Mede predi\u00e7\u00e3o do modelo na base de treino. (M\u00e9trica: *Mean Squared Error*(MSE)) \n**Omega- Regularization function **: Controla a complexidade do modelo (Ajuda a evitar o *Overfitting*)\n\nnota: As *objective functions* devem sempre possuir *training loss* e *regularization*\n\n\n\n### CART\n\nUso de *CARTs* (Classification And Regression Trees) no ensemble das \u00e1rvores \n\n\n\n\nModelo de ensemble de \u00e1rvores IGUAL ao modelo Random Forest, mas onde est\u00e1 ent\u00e3o a diferen\u00e7a?\n\n## Training\n\n### Additive\n\nTraining:\n\nPrecisamos agora melhorar os param\u00eatros da fun\u00e7\u00e3o de **Regularization**, mas como fazer isso? Fazer isso aqui \u00e9 muito mais dif\u00edcil do que em problemas de otimiza\u00e7\u00e3o tradicionais, onde voc\u00ea pode usar o gradiente para isso. N\u00e3o \u00e9 f\u00e1cil treinar todas as \u00e1rvores ao mesmo tempo. Em vez disso, usamos uma **estrat\u00e9gia aditiva**: consertamos o que aprendemos e adicionamos uma nova \u00e1rvore de cada vez.\n\n\n\\begin{split}\\hat{y}_i^{(0)} &= 0\\\\\n \\hat{y}_i^{(1)} &= f_1(x_i) = \\hat{y}_i^{(0)} + f_1(x_i)\\\\\n \\hat{y}_i^{(2)} &= f_1(x_i) + f_2(x_i)= \\hat{y}_i^{(1)} + f_2(x_i)\\\\\n &\\dots\\\\\n \\hat{y}_i^{(t)} &= \\sum_{k=1}^t f_k(x_i)= \\hat{y}_i^{(t-1)} + f_t(x_i)\n\\end{split}\n\n\n```python\n%%time\nfrom xgboost import XGBClassifier\n\ndef xgboost(X_train, y_train, X_test, y_test): \n xgbclf = XGBClassifier(\n learning_rate=0.01,\n n_estimators=140,\n max_depth=6,\n min_child_weight=6,\n gamma=0,\n subsample=0.8,\n colsample_bytree=0.8,\n nthread=8,\n scale_pos_weight=1\n )\n print('XGBoost fit')\n xgbclf.fit(X_train, y_train)\n print('XGBoost train score')\n train_score = xgbclf.score(X_train, y_train)\n print('XGBoost test score')\n y_pred = xgbclf.predict(X_test)\n\n print('XGBoost confusion matrix')\n cm = confusion_matrix(y_test, y_pred)\n\n print('XGBoost cross validation')\n accuracies = cross_val_score(estimator=xgbclf, X=X_train, y=y_train, cv=10)\n \n print('XGBoost results')\n add_results('xgboost', xgbclf.score(X_train, y_train), xgbclf.score(X_test, y_test))\n \n plot_confusion_matrix(cm, classes=xgbclf)\n print('Resultado na base de treino %.2f' % train_score)\n print('Resultado M\u00e9dio na base de teste: %.2f' % accuracies.mean())\n print('Desvio padr\u00e3o: %.4f' % accuracies.std())\n \n\nxgboost(X_train, y_train, X_test, y_test)\n```\n\n# GridSearchCV\n\nA ferramenta GridSearch disponibilizada pelo Scikit, gera de forma exaustiva candidatos a partir de um grid de par\u00e2metros especificados com o atributo param_grid.\n\n\n```python\ndt_params = [{\n 'max_depth': [40, 50, 60, 80, 100, 120],\n 'max_features': [70, 80, 90, 92],\n 'min_samples_leaf': [2, 5, 10, 20, 30, 40]\n}]\n\nxgb_params = [{\n 'max_depth': [4, 5, 6],\n 'min_child_weight': [4, 5, 6]\n}]\n```\n\n\n```python\n%%time\nfrom sklearn.model_selection import GridSearchCV\nfrom sklearn.tree import DecisionTreeClassifier\n\ndef search_params(classifier, params):\n clf = classifier()\n grid_search = GridSearchCV(estimator=clf,\n param_grid=params,\n cv = 10,\n n_jobs=-1)\n\n grid_search = grid_search.fit(X_train, y_train)\n print(grid_search.best_score_, grid_search.best_params_)\n return grid_search.best_score_\n```\n\n CPU times: user 20 ms, sys: 10 ms, total: 30 ms\n Wall time: 371 ms\n\n\n## Aplicando GridSearchCV ao XGBClassifier:\n\n\n```python\n%%time\nfrom xgboost import XGBClassifier\n\n# Takes long time to run\nsearch_params(XGBClassifier, xgb_params)\n```\n\n 0.797705143227 {'max_depth': 6, 'min_child_weight': 6}\n CPU times: user 2min 44s, sys: 667 ms, total: 2min 45s\n Wall time: 1h 30min 54s\n\n\n## Aplicando GridSearchCV ao Decision Tree Classifier:\n\n\n```python\nsearch_params(DecisionTreeClassifier, dt_params)\n```\n\n 0.724374772736 {'max_depth': 50, 'max_features': 70, 'min_samples_leaf': 10}\n\n\n\n\n\n 0.72437477273645512\n\n\n\n# Trees\n\n## Decision Tree\n\nOs dados s\u00e3o separados recursivamente formando uma \u00e1rvore de decis\u00e3o baseada nas features.Pode-se definir uma \u00e1rvore de decis\u00e3o, conforme diz (MITCHELL, 1997), como um m\u00e9todo para aproximar valores discretos em fun\u00e7\u00f5es, onde a fun\u00e7\u00e3o de aprendizagem \u00e9 representada por uma \u00e1rvore de decis\u00e3o. Tais \u00e1rvores aprendidas podem ser representadas - a n\u00edvel de c\u00f3digo fonte - como conjuntos de estruturas condicionais \"se-ent\u00e3o\" para melhorar a leitura e entendimento humano, de acordo com (MITCHELL, 1997).\n\nEstes algoritmos s\u00e3o muito utilizados, segundo (MITCHELL, 1997), na \u00e1rea de algoritmos de infer\u00eancia indutiva, e dentre as aplica\u00e7\u00f5es de tais algoritmos, tem-se m\u00e1quinas que aprenderam a diagnosticar casos da medicina, ou ainda, para avaliar o risco de inadimpl\u00eancia dos requerentes de cr\u00e9ditos em bancos.\n\nPara visualizar de forma mais f\u00e1cil a representa\u00e7\u00e3o de uma \u00e1rvore, a figura 3, representada abaixo, caracteriza uma \u00e1rvore de decis\u00e3o em que a m\u00e1quina deve decidir com base nas vari\u00e1veis do tempo (ensolarado, nublado ou chuvoso), se pode ou n\u00e3o ocorrer uma partida de t\u00eanis. Al\u00e9m das vari\u00e1veis de tempo, tem-se outras vari\u00e1veis que podem ser levadas em conta dependendo da condi\u00e7\u00e3o clim\u00e1tica local, como umidade (alta ou normal) e o vento (forte ou fraco).\n\n\n\nO algoritmo de \u00e1rvores de decis\u00e3o classifica inst\u00e2ncias ou dados, ordenando-os apartir da raiz da \u00e1rvore, para os n\u00f3s de suas folhas. Cada n\u00f3 da \u00e1rvore exemplifica uma pergunta (teste) de alguns - atributos - de inst\u00e2ncia, e cada ramo descendente de um n\u00f3 corresponde para um dos poss\u00edveis valores de tal atributo (MITCHELL, 1997). Vale a pena citar: O algoritmo ID3 (QUINLAN, 1986) aprende sobre \u00e1rvores de decis\u00e3o construindo-as de cima para baixo (n\u00f3 raiz para as ramifica\u00e7\u00f5es) tentando buscar respostas para a pergunta \"Qual atributo devemos testar na raiz da \u00e1rvore?\", sendo assim, cada atributo instanciado \u00e9 calculado por meio de testes estat\u00edsticos, para determinar o qu\u00e3o bem (\u00f3timo) tal atributo, isolado dos demais, classifica os exemplos de treinamento.\n\nQuando o melhor atributo \u00e9 selecionado e utilizado como teste no n\u00f3 principal da \u00e1rvore, cria-se um descendente para cada valor admiss\u00edvel deste atributo e os exemplos de treinamento s\u00e3o sorteados para o n\u00f3 filho mais apropriado. O processo inteiro \u00e9 ent\u00e3o repetido utilizando treinamentos associados a cada descendente para selecionar o melhor atributo para testar na \u00e1rvore. Quando realizado dessa forma, o algoritmo tenta de forma \u201cgulosa\u201c3.4. O modelo 49 Figura 3 \u2013 Exemplo de \u00e1rvore de decis\u00e3o, sobre condi\u00e7\u00f5es para realiza\u00e7\u00e3o de um jogo de t\u00eanis.\n\n\n```python\nfrom sklearn.model_selection import cross_val_score\n\ndef fit_tree(X_train, y_train, X_test, y_test, tree_description='decision_tree'):\n tree_clf = DecisionTreeClassifier(max_features=70, min_samples_leaf=10, max_depth=40)\n tree_clf.fit(X_train, y_train)\n\n inner_score = tree_clf.score(X_train, y_train)\n tree_fit = cross_val_score(tree_clf, X_train, y_train)\n \n add_results(tree_description, tree_clf.score(X_train, y_train), tree_clf.score(X_test, y_test))\n \n return inner_score, tree_fit.mean(), tree_fit.std()\n\n\"inner: {:.2f} cross: {:.2f} +/- {:.2f}\".format(*fit_tree(X_train, y_train, X_test, y_test))\n```\n\n\n\n\n 'inner: 0.81 cross: 0.71 +/- 0.01'\n\n\n\n## Distribui\u00e7\u00e3o dos dados\n\nUm dos modelos a ser utilizado ser\u00e1 o decision tree no m\u00e9todo de montagem random forest. Este modelo de predi\u00e7\u00e3o possui um problema de vi\u00e9s quando uma das classes na base de treino \u00e9 mais predominante do que outra, ou seja, a distribui\u00e7\u00e3o das classes na base de treino devem ser semelhantes para evitar problemas de [overfiting](http://docs.aws.amazon.com/machine-learning/latest/dg/model-fit-underfitting-vs-overfitting.html).\n\nPara tanto, precisa-se descobrir qual a contagem de cada classe dispon\u00edvel na base de treino, montaremos um histograma para verificar a diferen\u00e7a entre elas.\n\n\n```python\ncounts = [0] *len(df_target.target.cat.categories)\n\ndef reduce(target):\n counts[target.categories] += 1\n return counts[target.categories]\n\ndf_target['increase_count'] = df_target.apply(reduce, axis=1)\ndf_target.groupby('target').count()\ndf_target.groupby('target')['increase_count'].max().sum() == df_target.target.count()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    categoriesincrease_count
    target
    Class_119291929
    Class_21612216122
    Class_380048004
    Class_426912691
    Class_527392739
    Class_61413514135
    Class_728392839
    Class_884648464
    Class_949554955
    \n
    \n\n\n\n\n\n\n True\n\n\n\n### Filtrar dados\n\nAgora, iremos filtrar os dados deixando apenas os primeiros registros. O crit\u00e9rio de filtrar os dados ser\u00e1 pegar a classe que possue o menor n\u00famero e utilizar ele como base para remover os demais, considerando um tamanho m\u00e1ximo de at\u00e9 2x o da menor classe\n\n\n```python\ndistance_percent = 2\nminimum_value = df_target.groupby('target')['increase_count'].max().min()\ndf_rtarget = df_target[ df_target.increase_count < minimum_value*distance_percent ]\ndf_rtarget.groupby('target').count()\ndf_rtrain = df_train.drop( df_target[df_target.increase_count >= minimum_value * distance_percent].index )\ndf_rtrain.shape[0] == df_rtarget.shape[0]\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    categoriesincrease_count
    target
    Class_119291929
    Class_238573857
    Class_338573857
    Class_426912691
    Class_527392739
    Class_638573857
    Class_728392839
    Class_838573857
    Class_938573857
    \n
    \n\n\n\n\n\n\n True\n\n\n\n### Verificando resultado\n\nAp\u00f3s aplicar uma melhor distribui\u00e7\u00e3o nos dados, rodou-se novamene o algor\u00edtmo da decision tree e percebeu-se que a acur\u00e1cia do modelo diminuiu, e portanto, n\u00e3o ser\u00e1 utilizado.\n\n\n```python\nX_tr, X_te, y_tr, y_te = train_test_split(df_rtrain, df_rtarget.target, test_size=0.2)\n\"inner: {:.2f} cross: {:.2f} +/- {:.2f}\".format(*fit_tree(X_tr, y_tr, X_te, y_te))\n```\n\n\n\n\n 'inner: 0.77 cross: 0.65 +/- 0.00'\n\n\n\n# Random Forest\n\nBreiman breiman, 2001, descreve Random Forests como uma evolu\u00e7\u00e3o das decisions trees, onde v\u00e1rias \u00e1vores s\u00e3o formadas para criar um modelo com maior precis\u00e3o. Isto \u00e9 feito a partir da separa\u00e7\u00e3o dos Dados em conjutos de dados menores e aleat\u00f3rios. Cada \u00e1rvore \u00e9 contruida a partir de um peda\u00e7o aleat\u00f3rio dos dados. Quando um novo dado chega, a predi\u00e7\u00e3o \u00e9 feita por todas as \u00c1rvores e ao fim \u00e9 feita uma vota\u00e7\u00e3o por maioria, ou seja, a categoria com mais votos ganha e o resultado \u00e9 dado.\n\n\n\nDe acordo com breiman, 2001, as RFs corrigem a maior parte dos problemas de Overfitting que as \u00c1rvores de decis\u00e3o apresentam. Tudo depende do quanto as DT contidas dentro da Random Forest. Isto \u00e9, o quanto elas representam os dados.\n\n## Utilizando o algoritmo\n\n\n```python\nfrom sklearn.ensemble import RandomForestClassifier\n\ndef test_random(params, X_train, y_train, X_test, y_test, name='random_forest'):\n rfclf = RandomForestClassifier(**params)\n rfclf = rfclf.fit(X_train, y_train)\n \n train_score = rfclf.score(X_train, y_train)\n test_score = rfclf.score(X_test, y_test)\n\n add_results(name, train_score, test_score)\n return name, train_score, test_score\nparams = {'n_estimators': 10, 'max_features': 70, 'min_samples_leaf': 10, 'max_depth': 40}\ntest_random({}, X_train, y_train, X_test, y_test)\ntest_random(params, X_train, y_train, X_test, y_test, 'random_forest_otimized')\n```\n\n\n\n\n ('random_forest', 0.9920003232192639, 0.77957336780866193)\n\n\n\n\n\n\n ('random_forest_otimized', 0.84002666558926908, 0.7690691661279897)\n\n\n\n## Verificando com Cross Validation\n\nCross validation ir\u00e1 predizer um peda\u00e7o do dataset utilizando o modelo treinado com o resto dos dados que n\u00e3o fazem parte deste dataset.\n\n\n```python\nrfclf = RandomForestClassifier(**params)\nrfclf.fit(X_train, y_train)\nrfscores = cross_val_score(rfclf, X_train, y_train)\nprint (\"{} de precis\u00e3o\".format(rfscores.mean() * 100))\n\n```\n\n\n\n\n RandomForestClassifier(bootstrap=True, class_weight=None, criterion='gini',\n max_depth=40, max_features=70, max_leaf_nodes=None,\n min_impurity_decrease=0.0, min_impurity_split=None,\n min_samples_leaf=10, min_samples_split=2,\n min_weight_fraction_leaf=0.0, n_estimators=10, n_jobs=1,\n oob_score=False, random_state=None, verbose=0,\n warm_start=False)\n\n\n\n 76.09793654862037 de precis\u00e3o\n\n\n## Importancia das features para a RF\n\nA seguir vemos quais as influ\u00eancias de cada uma das features para o uso no random forest. Quanto maior no gr\u00e1fico, maior \u00e9 a import\u00e2ncia da feature.\n\n### Gini\n\nO m\u00e9todo utilizado para gerar a import\u00e2ncia das features no modelo \u00e9 a Decrease Mean Importance, que utiliza em seus c\u00e1lculos um indicador de impureza no sistema. No caso do random forest implementado [(LOUPPE et al.,2013)](https://pdfs.semanticscholar.org/2635/19c5a43fbf981da5ba873062219c50fdf56d.pdf), este indicador \u00e9 o Gini Impurity que pode ser entendido como uma redu\u00e7\u00e3o da probabilidade de errar a classifica\u00e7\u00e3o de uma categoria dentro de um algor\u00edtmo de \u00e1rvore [(Sebastian Raschaka)](https://sebastianraschka.com/faq/docs/decision-tree-binary.html).\n\n#### O indice\n\nO indice de Gini pode ser calculado utilizando a seguinte f\u00f3rmula[(TEKIMONO,2009)](http://people.revoledu.com/kardi/tutorial/DecisionTree/how-\nto-measure-impurity.htm):\n\n\\begin{equation}\n Gini = 1- \\sum_{i=1} p_i^2\n\\end{equation}\n\nEm que $p_i$ \u00e9 a probabilidade da ocorr\u00eancia de uma determinada classe, desconsiderando os atributos. Ou seja $N_i$ \u00e9 o n\u00famero de ocorr\u00eancias da classe i e N \u00e9 o total de elementos das classes:\n\n\\begin{equation}\n p_i = \\frac{N_i}{N}\n\\end{equation}\n\n#### Para Decisions Trees\n\nPara Classification and Regression Trees (CART), utiliza-se o indice de Gini modificado, isto \u00e9, calcula-se ainda as probabilidades em $p_i$, mas agora utiliza-se do indice de Gini nos filhos da esquerda $t_l$ e direita $t_r$. Recalcula-se as probabilidades para ambos os n\u00f3s tamb\u00e9m em $p_l$ e $p_r$ utilizando como base as poss\u00edveis classes reduzidas a $N_t$ [(LOUPPE et al.,2013)](https://pdfs.semanticscholar.org/2635/19c5a43fbf981da5ba873062219c50fdf56d.pdf).\n\n\\begin{equation}\n i(s, t) = Gini(t) - p_l Gini(t_l) - p_r Gini(t_r) \\\\\n p(t) = \\frac{N_{l|r}}{N_t}\n\\end{equation}\n\n#### Decrease Mean Importance\n\nPara calcular a import\u00e2ncia de uma feature X ao tentar predizer uma label Y, utiliza- se os indices de impureza com a propor\u00e7\u00e3o de $N_f$ amostras em rela\u00e7\u00e3o ao total $N$. $N_T$ \u00e9 o total de \u00e1rvores na floresta. Assim, para uma Random Forest a conta \u00e9:\n\n\\begin{equation}\n I(X_m) = \\frac{1}{N_T} \\sum_{T} \\sum_{t \\epsilon T:v(s)=X_m} pf(t)i(s,t) \\\\\n pf(f) = \\frac{N_f}{N}\n\\end{equation}\n\n\n```python\nfig, axis = plt.subplots(figsize=(15, 5))\nplot = axis.bar(df_train.columns, rfclf.feature_importances_)\nplot = axis.set_xticklabels(df_train.columns.values, rotation='vertical')\nplot = axis.set_xlabel('feature')\nplot = axis.set_ylabel('importance')\nplt.show()\n```\n\n## ExtraTrees\n\nO [Scikit Learn](http://scikit- learn.org/stable/modules/generated/sklearn.ensemble.ExtraTreesClassifier.html) nos apresenta um tipo diferente de random forest que pode apresentar resultados melhores que o [RandomForestClassifier](http://scikit- learn.org/stable/modules/generated/sklearn.ensemble.RandomForestClassifier.html). Assim como afirma que as extra tree devem ser utilizadas apenas em algor\u00edtmos de montagem Como o Extra Trees Classifier e Regressor.\n\nO que diferencia uma extra tree de uma decision tree \u00e9 a forma que \u00e9 feita a constru\u00e7\u00e3o da \u00e1rvore. Enquanto uma decision tree utiliza c\u00f3pia dos dados e sub amostras para realizar as divis\u00f5es de cada n\u00f3. Uma extra tree utiliza um ponto de divis\u00e3o randomico e utiliza toda a base de treino para crescer a \u00e1rvore [(GEURTS, ERNST e WEHENKEL, 2005)](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.65.7485&rep=rep1&type=pdf).\n\n\n```python\nfrom sklearn.ensemble import ExtraTreesClassifier\n\netc = ExtraTreesClassifier()\netscores = cross_val_score(etc, X_train, y_train)\nprint (\"{} de precis\u00e3o\".format((etscores.mean() * 100)))\n\netc = etc.fit(X_train, y_train)\nadd_results('extra_trees', etc.score(X_train, y_train), etc.score(X_test, y_test))\nprint(\"Inner score\", etc.score(X_train, y_train))\n```\n\n 76.8898405821295 de precis\u00e3o\n Inner score 1.0\n\n\n## Neur\u00f4nio Artificial\n\n\n\n#### Entrada\n\n- Sinais de entrada {x1,x2,...,xn}.\n- Cada sinal de entrada e ponderado por 1 peso.{w1,w2,...,wn}.\n- O peso \u00e9 adquirido a partir do treino.\n\n#### Fun\u00e7\u00e3o agregadora\n\n- Recebe todos os sinais e realiza a soma dos produtos dos sinais.\n\n#### Neur\u00f4nio\n\n- Tem a fun\u00e7\u00e3o de deixar, passar ou inibir um sinal de saida de acordo com a entrada.\n- Teta \u00e9 a limiar de ativacao(ponderado),'u' \u00e9 o potencial de ativa\u00e7\u00e3o que \u00e9 passado para a fun\u00e7\u00e3o (g(u)), que \u00e9 a fun\u00e7\u00e3o de ativa\u00e7\u00e3o que \u00e9 responsavel pela saida que permite o sinal passar ou n\u00e3o ou at\u00e9 mesmo modificalo.\n\n#### Formula\n\n- Potencial de ativa\u00e7\u00e3o\n\n\n\n### MLP Classifier\n\nEsse algoritmo \u00e9 um classificador Perceptron de Multicamadas usado para fazer o treinamento de modelos, e \u00e9 uma biblioteca do Scikit-Learn.\n\n\n```python\n%%time\n\nfrom sklearn.neural_network import MLPClassifier\n\nmlp = MLPClassifier(solver='adam',activation='relu',max_iter=250)\nmlp.fit(X_train, y_train)\nsaidas = mlp.predict(X_test)\nscoreTreino = mlp.score(X_train, y_train)\nscoreTeste = mlp.score(X_test, y_test)\n\nprint('Score treino: ', scoreTreino)\nprint('Score teste: ', scoreTeste)\n\nmlpscores = cross_val_score(mlp, X_train, y_train)\n\nprint('Score: {} +/- {}'.format(mlpscores.mean(), mlpscores.std()))\n\nadd_results('multi_layer_perceptron', scoreTreino, scoreTeste)\n```\n\n Score treino: 0.87135873298\n Score teste: 0.785956690368\n Score: 0.7825550109810641 +/- 0.007188504494548886\n CPU times: user 4min 13s, sys: 4min 39s, total: 8min 52s\n Wall time: 2min 17s\n\n\n\n```python\nfrom sklearn.neural_network import MLPClassifier\n\nmp = confusion_matrix(y_test,saidas);\nplot_confusion_matrix(mp, classes=mlp)\n```\n\n## Preprocessamento de dados\n\nA rede neural pode ter dificuldade em convergir antes de atingir o n\u00famero m\u00e1ximo\nde itera\u00e7\u00f5es permitido se os dados n\u00e3o forem normalizados. Multi-layer\nPerceptron \u00e9 sens\u00edvel ao dimensionamento de features, portanto, \u00e9 altamente\nrecomend\u00e1vel dimensionar seus dados. Usaremos o StandardScaler incorporado para\npadroniza\u00e7\u00e3o.\n\n\n```python\n%%time\nfrom sklearn.preprocessing import StandardScaler\nscaler = StandardScaler()\n# Fit only to the training data\nscaler.fit(X_train)\n\n# Now apply the transformations to the data:\nX_train = scaler.transform(X_train)\nX_test = scaler.transform(X_test)\n```\n\n CPU times: user 107 ms, sys: 3.33 ms, total: 110 ms\n Wall time: 109 ms\n\n\n\n```python\n%%time\n\nmlp = MLPClassifier(hidden_layer_sizes=(30,30,30))\nmlp.fit(X_train,y_train)\npredictions = mlp.predict(X_test)\n```\n\n CPU times: user 1min 21s, sys: 2min 14s, total: 3min 35s\n Wall time: 57.6 s\n\n\n\n```python\nfrom sklearn.metrics import classification_report,confusion_matrix\nprint(classification_report(y_test,predictions))\nmp = confusion_matrix(y_test,predictions)\nplot_confusion_matrix(mp, classes=mlp)\nadd_results('MultiLayerPerceptron parametrized', X_train, y_train, X_test, y_test)\n```\n\ncoefs\\_ \u00e9 uma lista de matrizes de peso, onde a matriz de peso no \u00edndice i\nrepresenta os pesos entre a camada i e a camada i + 1.\n\nintercepts\\_ \u00e9 uma lista de vetores de polariza\u00e7\u00e3o, onde o vetor no \u00edndice i\nrepresenta os valores de polariza\u00e7\u00e3o adicionados \u00e0 camada i + 1.\n\n\n```python\nprint(len(mlp.coefs_), len(mlp.coefs_[0]), len(mlp.intercepts_[0]))\n```\n\n 4 93 30\n\n\n# Conclus\u00e3o\n\nComo conclus\u00e3o, tivemos a utiliza\u00e7\u00e3o do modelo Random Forest e Extreme Gradient Boosting otimizados. Mas o gr\u00e1fico a seguir ir\u00e1 mostrar os resultados com a base de treino e base de teste.\n\n\n```python\ncolumns = [x.replace('_',' ') for x in results.keys()]\ntrain = []\ntest = []\nwidth=0.4\nbase = np.arange(len(columns))\nfor key in results:\n train.append(results[key]['train'])\n test.append(results[key]['test'])\nfig, ax=plt.subplots(figsize=[10,10])\nfig = ax.bar(base, train, width)\nfig = ax.bar(base+width, test, width)\nfig = ax.set_xticks(base+width/2)\nfig = ax.set_xticklabels(columns, rotation='45')\nfig = ax.legend(['Base de treino', 'Base de teste'])\nplt.show()\n```\n\n# Refer\u00eancias Bibliogr\u00e1ficas\n\nhttp://scikit- learn.org/stable/modules/generated/sklearn.dummy.DummyClassifier.html#sklearn.dummy.DummyClassifier\n\nhttps://www.analyticsvidhya.com/blog/2016/03/complete-guide-parameter-tuning- xgboost-with-codes-python/\n\nhttp://scikit- learn.org/stable/modules/generated/sklearn.neural_network.MLPClassifier.html\n\nftp://ftp.sas.com/pub/neural/FAQ3.html#A_hu\n\n[MITCHELL](https://dl.acm.org/citation.cfm?id=505283), Tom M. Machine learning. 1997. Burr Ridge, IL: McGraw Hill, v. 45, n. 37, p. 870-877, 1997.\n[QUINLAN](http://hunch.net/~coms-4771/quinlan.pdf), J.. Ross . Induction of decision trees. Machine learning, v. 1, n. 1, p. 81-106, 1986.\n[BREIMAN](https://www.stat.berkeley.edu/users/breiman/randomforest2001.pdf), Leo. Random forests. Machine learning, v. 45, n. 1, p. 5-32, 2001.\n\nBABATUNDE, Oluleye, ARMSTRONG, Leisa, DIEPEVEEN, Dean e LENG, J. Comparative analysis of Genetic Algorithm and Particle Swam Optimization: An application in precision agriculture. 2015. **Asian Journal of Computer and Information Systems**. 3. 1-12.\n", "meta": {"hexsha": "4fd85bd330176b69b476c2396fd5fca96a25b66f", "size": 404621, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "product-solution.ipynb", "max_stars_repo_name": "Wall-eSociety/Product-Classification", "max_stars_repo_head_hexsha": "b7194bf0043f41e0c9a0d205a5ed49b81bc4dc94", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2017-11-20T13:42:36.000Z", "max_stars_repo_stars_event_max_datetime": "2018-09-12T11:20:20.000Z", "max_issues_repo_path": "product-solution.ipynb", "max_issues_repo_name": "pwaila/Product-Classification", "max_issues_repo_head_hexsha": "93739918bd9ed62d982aad35321f71034c8ff3c3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "product-solution.ipynb", "max_forks_repo_name": "pwaila/Product-Classification", "max_forks_repo_head_hexsha": "93739918bd9ed62d982aad35321f71034c8ff3c3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2018-03-14T19:33:02.000Z", "max_forks_repo_forks_event_max_datetime": "2018-09-12T11:20:22.000Z", "avg_line_length": 102.0481715006, "max_line_length": 51314, "alphanum_fraction": 0.7597109394, "converted": true, "num_tokens": 28356, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7341195269001831, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.4406099943510988}} {"text": "\u6700\u521d\u306b\u5fc5\u8981\u306a\u30e9\u30a4\u30d6\u30e9\u30ea\u3092\u8aad\u307f\u8fbc\u307f\u307e\u3059\u3002\n\n\n```python\nfrom sympy import *\nfrom sympy.physics.quantum import *\nfrom sympy.physics.quantum.qubit import Qubit, QubitBra, measure_all, measure_all_oneshot\nfrom sympy.physics.quantum.gate import H,X,Y,Z,S,T,CPHASE,CNOT,SWAP,UGate,CGateS,gate_simp\nfrom sympy.physics.quantum.gate import IdentityGate as _I\nfrom sympy.physics.quantum.qft import *\n\nfrom sympy.printing.dot import dotprint\ninit_printing()\n\n%matplotlib inline\nimport matplotlib.pyplot as plt\nfrom sympy.physics.quantum.circuitplot import CircuitPlot,labeller, Mz,CreateOneQubitGate\n\n```\n\n## \uff08\u72ed\u7fa9\u306e\uff09\u91cf\u5b50\u30d7\u30ed\u30b0\u30e9\u30df\u30f3\u30b0\u306e\u624b\u9806\n\n1. \u8a08\u7b97\u306b\u5fc5\u8981\u306a\u91cf\u5b50\u30d3\u30c3\u30c8\uff08\u91cf\u5b50\u30ec\u30b8\u30b9\u30bf\uff09\u3092\u6e96\u5099\u3057\u3066\u3001\u305d\u306e\u5024\u3092\u521d\u671f\u5316\u3059\u308b\n\n2. \u91cf\u5b50\u8a08\u7b97\u3092\u30e6\u30cb\u30bf\u30ea\u884c\u5217\uff08\u30b2\u30fc\u30c8\u6f14\u7b97\u5b50\uff09\u3067\u8a18\u8ff0\u3059\u308b\n\n3. \u30e6\u30cb\u30bf\u30ea\u884c\u5217\u3092\u91cf\u5b50\u30d3\u30c3\u30c8\u306b\u4f5c\u7528\u3059\u308b\n\n4. \u6e2c\u5b9a\u3059\u308b\n\n\n#### \uff08\uff11\u306e\u4f8b\uff09\u8a08\u7b97\u306b\u5fc5\u8981\u306a\u91cf\u5b50\u30d3\u30c3\u30c8\uff08\u91cf\u5b50\u30ec\u30b8\u30b9\u30bf\uff09\u3092\u6e96\u5099\u3057\u3066\u3001\u305d\u306e\u5024\u3092\u521d\u671f\u5316\u3059\u308b\n\n\n```python\n# \u5168\u3066 0 \u306e\uff13\u91cf\u5b50\u30d3\u30c3\u30c8\u3092\u6e96\u5099\nQubit('000')\n```\n\n#### \uff08\uff12\u306e\u4f8b\uff09\u91cf\u5b50\u8a08\u7b97\u3092\u30e6\u30cb\u30bf\u30ea\u884c\u5217\uff08\u30b2\u30fc\u30c8\u6f14\u7b97\u5b50\uff09\u3067\u8a18\u8ff0\u3059\u308b\n\n\n```python\n# \u57fa\u672c\u7684\u306a\u30e6\u30cb\u30bf\u30ea\u6f14\u7b97\u5b50\npprint(represent(X(0),nqubits=1))\npprint(represent(Y(0),nqubits=1))\npprint(represent(Z(0),nqubits=1))\npprint(represent(H(0),nqubits=1))\npprint(represent(S(0),nqubits=1))\npprint(represent(S(0)**(-1),nqubits=1))\npprint(represent(T(0),nqubits=1))\npprint(represent(T(0)**(-1),nqubits=1))\npprint(represent(CNOT(1,0),nqubits=2))\n```\n\n \u23a10 1\u23a4\n \u23a2 \u23a5\n \u23a31 0\u23a6\n \u23a10 -\u2148\u23a4\n \u23a2 \u23a5\n \u23a3\u2148 0 \u23a6\n \u23a11 0 \u23a4\n \u23a2 \u23a5\n \u23a30 -1\u23a6\n \u23a11 1 \u23a4\n \u23a2\u2500\u2500 \u2500\u2500 \u23a5\n \u23a2\u221a2 \u221a2 \u23a5\n \u23a2 \u23a5\n \u23a21 -\u221a2 \u23a5\n \u23a2\u2500\u2500 \u2500\u2500\u2500\u2500\u23a5\n \u23a3\u221a2 2 \u23a6\n \u23a11 0\u23a4\n \u23a2 \u23a5\n \u23a30 \u2148\u23a6\n \u23a11 0 \u23a4\n \u23a2 \u23a5\n \u23a30 -\u2148\u23a6\n \u23a11 0 \u23a4\n \u23a2 \u23a5\n \u23a2 \u2148\u22c5\u03c0\u23a5\n \u23a2 \u2500\u2500\u2500\u23a5\n \u23a2 4 \u23a5\n \u23a30 \u212f \u23a6\n \u23a11 0 \u23a4\n \u23a2 \u23a5\n \u23a2 -\u2148\u22c5\u03c0 \u23a5\n \u23a2 \u2500\u2500\u2500\u2500\u2500\u23a5\n \u23a2 4 \u23a5\n \u23a30 \u212f \u23a6\n \u23a11 0 0 0\u23a4\n \u23a2 \u23a5\n \u23a20 1 0 0\u23a5\n \u23a2 \u23a5\n \u23a20 0 0 1\u23a5\n \u23a2 \u23a5\n \u23a30 0 1 0\u23a6\n\n\n#### \uff08\uff13\u306e\u4f8b\uff09\u30e6\u30cb\u30bf\u30ea\u884c\u5217\u3092\u91cf\u5b50\u30d3\u30c3\u30c8\u306b\u4f5c\u7528\u3059\u308b\n\n\n```python\n# \u30e6\u30cb\u30bf\u30ea\u884c\u5217\u3092\u91cf\u5b50\u30d3\u30c3\u30c8\u306b\u4f5c\u7528\u3059\u308b\u306b\u306f\u3001qapply() \u3092\u4f7f\u3044\u307e\u3059\u3002\nhadamard3 = H(2)*H(1)*H(0)\nqapply(hadamard3*Qubit('000'))\n```\n\n#### \uff08\uff14\u306e\u4f8b\uff09\u6e2c\u5b9a\u3059\u308b\n\n\n```python\n# \u6e2c\u5b9a\u306f\u3001qapply() \u3057\u305f\u91cf\u5b50\u72b6\u614b\u306b\u5bfe\u3057\u3066\u3001measure_all_oneshot() \u3067\u78ba\u7387\u7684\u306a\u7d50\u679c\u3092\u5f97\u307e\u3059\u3002\nfor i in range(10):\n pprint(measure_all_oneshot(qapply(hadamard3*Qubit('000'))))\n```\n\n \u2758011\u27e9\n \u2758000\u27e9\n \u2758000\u27e9\n \u2758100\u27e9\n \u2758100\u27e9\n \u2758100\u27e9\n \u2758011\u27e9\n \u2758001\u27e9\n \u2758100\u27e9\n \u2758000\u27e9\n\n\n\n```python\n# SymPy\u306e\u91cf\u5b50\u30b7\u30df\u30e5\u30ec\u30fc\u30bf\u30fc\u3067\u306f\u3001\u5185\u90e8\u3067\u91cf\u5b50\u72b6\u614b\u3092\u53b3\u5bc6\u306b\u8a08\u7b97\u3057\u3066\u3001\u3059\u3079\u3066\u306e\u72b6\u614b\u3092\u4fdd\u6301\u3057\u3066\u3044\u307e\u3059\u3002\n# \u305d\u306e\u305f\u3081\u3002measure_all() \u3067\u306f\u3001\u5168\u3066\u306e\u91cf\u5b50\u72b6\u614b\u306e\u78ba\u7387\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\nmeasure_all(qapply(hadamard3*Qubit('000')))\n```\n\n## \u3010\u7df4\u7fd2\u554f\u984c\u3011\u3044\u3064\u3082\u306e\u8aac\u660e\u8cc7\u6599\u306e\u91cf\u5b50\u56de\u8def\u3092\u30d7\u30ed\u30b0\u30e9\u30df\u30f3\u30b0\u624b\u9806\u306b\u305d\u3063\u3066\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002\n\n\n\n\n```python\n### 1. \u8a08\u7b97\u306b\u5fc5\u8981\u306a\u91cf\u5b50\u30d3\u30c3\u30c8\uff08\u91cf\u5b50\u30ec\u30b8\u30b9\u30bf\uff09\u3092\u6e96\u5099\u3057\u3066\u3001\u305d\u306e\u5024\u3092\u521d\u671f\u5316\u3059\u308b\n## \uff12\u91cf\u5b50\u30d3\u30c3\u30c8\u3092 0 \u3067\u521d\u671f\u5316\u3057\u3066\u304f\u3060\u3055\u3044\u3002\nQubit('00')\n```\n\n\n```python\n### 2. \u91cf\u5b50\u8a08\u7b97\u3092\u30e6\u30cb\u30bf\u30ea\u884c\u5217\uff08\u30b2\u30fc\u30c8\u6f14\u7b97\u5b50\uff09\u3067\u8a18\u8ff0\u3059\u308b\n## Hadamard \u306e\u30c6\u30f3\u30bd\u30eb\u7a4d \u306e\u884c\u5217\u8868\u73fe\u3092\u8868\u793a\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\nrepresent(H(1)*H(0),nqubits=2)\n```\n\n\n```python\n## CNOT \u3092 Hadamard \u3067\u631f\u3093\u3060\u30b2\u30fc\u30c8\u64cd\u4f5c \u306e\u884c\u5217\u8868\u73fe\u3092\u8868\u793a\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\nrepresent(H(0)*CNOT(1,0)*H(0),nqubits=2)\n```\n\n\n```python\n### 3. \u30e6\u30cb\u30bf\u30ea\u884c\u5217\u3092\u91cf\u5b50\u30d3\u30c3\u30c8\u306b\u4f5c\u7528\u3059\u308b\n## Hadamard \u306e\u30c6\u30f3\u30bd\u30eb\u7a4d \u3092 `Qubit('00')` \u306b\u4f5c\u7528\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\nqapply(H(1)*H(0)*Qubit('00'))\n```\n\n\n```python\n## \u6b21\u306b\u3001CNOT \u3092 Hadamard \u3067\u631f\u3093\u3060\u30b2\u30fc\u30c8\u64cd\u4f5c \u3092 \u524d\u306e\u72b6\u614b\u306b\u4f5c\u7528\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\n\nqapply(H(0)*CNOT(1,0)*H(0)*H(1)*H(0)*Qubit('00'))\n```\n\n\n```python\n### 4. \u6e2c\u5b9a\u3059\u308b\n## measure_all() \u3092\u4f7f\u3063\u3066\u3001\u305d\u308c\u305e\u308c\u306e\u72b6\u614b\u304c\u6e2c\u5b9a\u3055\u308c\u308b\u78ba\u7387\u3092\u8868\u793a\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\n\nmeasure_all(qapply(H(0)*CNOT(1,0)*H(0)*H(1)*H(0)*Qubit('00')))\n```\n\n## \u3010\u8ab2\u984c\uff11\u3011\u30b0\u30ed\u30fc\u30d0\u30fc\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\n\n\u554f\uff11\uff09 \n\uff11. \u6b21\u306e\u300c\u554f\uff11\u306e\u521d\u671f\u72b6\u614b\u300d quest_state \u3092\u5165\u529b\u3068\u3057\u3066\u3001\u3053\u306e\u91cf\u5b50\u72b6\u614b\u306b $\\lvert 111 \\rangle $ \u304c\u542b\u307e\u308c\u308b\u304b \n\u3000\u30b0\u30ed\u30fc\u30d0\u30fc\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u4f7f\u3063\u3066\u8abf\u3079\u3066\u304f\u3060\u3055\u3044\u3002 \n\n\uff12. \u4e0a\u306e\u6761\u4ef6\u3067\u3001\u3053\u306e\u91cf\u5b50\u72b6\u614b\u306b $\\lvert 101 \\rangle $ \u304c\u542b\u307e\u308c\u308b\u304b\u3092\u30b0\u30ed\u30fc\u30d0\u30fc\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092 \n\u3000\u4f7f\u3063\u3066\u8abf\u3079\u308b\u8003\u5bdf\u3092\u3057\u307e\u3059\u3002\uff08\u3046\u307e\u304f\u3044\u304b\u306a\u3044\u4f8b\u3092\u898b\u307e\u3059\uff09 \n\u3000 \n\u3000\u30fb\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f5c\u308a\u3001\u5b9f\u969b\u306f\u3001$\\lvert 101 \\rangle $ \u304c\u9ad8\u78ba\u7387\u3067\u691c\u51fa\u3055\u308c\u308b\u3053\u3068\u3092\u8abf\u3079\u3066\u304f\u3060\u3055\u3044\u3002 \n\u3000\u30fb\u306a\u305c\u3001\u521d\u671f\u72b6\u614b\u306b\u542b\u307e\u308c\u3066\u3044\u306a\u3044\u72b6\u614b\u304c\u691c\u51fa\u3055\u308c\u308b\u304b\u7406\u7531\u3092\u8003\u3048\u307e\u3057\u3087\u3046\u3002\uff08\u89e3\u7b54\u306f\u53e3\u982d\u3067\u3088\u3044\uff09 \n\u3000 \n\u3000 \n\u554f\uff12\uff09 \n\uff11. \u4e0b\u306e\u300c\u554f\uff12\u306e\u521d\u671f\u72b6\u614b\u300dquest2_state \u3092\u5165\u529b\u3068\u3057\u3066\u3001\u554f\uff11\u3068\u540c\u69d8\u3001 \n\u3000$\\lvert 111 \\rangle $ \u3068 $\\lvert 101 \\rangle $ \u306e\u72b6\u614b\u306b\u306e\u691c\u77e5\u306b\u3064\u3044\u3066 \u30b0\u30ed\u30fc\u30d0\u30fc\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u9069\u7528\u3057\u3066\u3001 \n\u3000\u305d\u306e\u72b6\u6cc1\u3092\u8003\u5bdf\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\u3000 \n\u3000 \n\n\n**\u4ee5\u964d\u3001\u3010\u8ab2\u984c\uff11\u3011\u554f\uff11\u2212\uff11\uff09\u306e\u56de\u7b54\u6b04\uff1a** \n\n\n```python\n# \u554f\uff11\u306e\u521d\u671f\u72b6\u614b\nquest_state = CNOT(1,0)*CNOT(2,1)*H(2)*H(0)*Qubit('000')\nCircuitPlot(quest_state,nqubits=3)\n```\n\n\n```python\n# \u8a08\u7b97\u3057\u305f\u521d\u671f\u72b6\u614b\u3092 init_state \u3068\u3059\u308b\ninit_state = qapply(quest_state)\ninit_state\n```\n\n\n```python\n# \u4ee5\u964d\u3067\u5f79\u7acb\u3061\u305d\u3046\u306a\u95a2\u6570\u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002\ndef CCX(c1,c2,t): return CGateS((c1,c2),X(t))\ndef hadamard(s,n):\n h = H(s)\n for i in range(s+1,n+s):\n h = H(i)*h\n return h\ndef CCZ(c1,c2,t): return (H(t)*CCX(c1,c2,t)*H(t)) # \uff23\uff23\uff3a\u6f14\u7b97\u5b50\u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002\ndef DOp(n): return (Qubit('0'*n)*QubitBra('0'*n)*2-_I(0)) # \u30b2\u30fc\u30c8\u64cd\u4f5c\u3067\u8a08\u7b97\u3059\u308b\u306b\u306f\u3001\u4e0a\u8a18\u30b3\u30e1\u30f3\u30c8\u306e\u3088\u3046\u306a\u6f14\u7b97\u306b\u306a\u308a\u307e\u3059\u3002\nh_3 = hadamard(0,3)\nd_3 = h_3 * DOp(3) * h_3 # \u5e73\u5747\u5024\u5468\u308a\u306e\u53cd\u8ee2\u64cd\u4f5c\n# represent(d_3,nqubits=3)\n```\n\n\n```python\n# | 111 > \u306e\u691c\u7d22\u3059\u308b\u91cf\u5b50\u56de\u8def\u3092\u4f5c\u6210\u3059\u308b\u3002\n\n\nmark_7 = CCZ(1,2,0)\ngrover_7 = gate_simp(d_3*mark_7*d_3*mark_7)\nstate1_7 = qapply(d_3*mark_7*init_state)\npprint(state1_7)\nqapply(d_3*mark_7*state1_7)\n```\n\n\n```python\n# \u4e0a\u3067\u4f5c\u3063\u305f\u91cf\u5b50\u56de\u8def\u3092\u521d\u671f\u72b6\u614b\u3068\u4f5c\u7528\u3055\u305b\u3066 measure_all_oneshot() \u3067\u4f55\u56de\u304b\u8a66\u884c\u3057\u3066\u3001\u7d50\u679c\u3092\u307f\u308b\u3002\n\n\nfor i in range(10):\n pprint(measure_all_oneshot(qapply(grover_7*init_state)))\n```\n\n \u2758011\u27e9\n \u2758111\u27e9\n \u2758100\u27e9\n \u2758111\u27e9\n \u2758111\u27e9\n \u2758010\u27e9\n \u2758001\u27e9\n \u2758100\u27e9\n \u2758100\u27e9\n \u2758100\u27e9\n\n\n**\u4ee5\u964d\u3001\u3010\u8ab2\u984c\uff11\u3011\u554f\uff11\u2212\uff12\uff09\u306e\u56de\u7b54\u6b04\uff1a** \n\n\n```python\n# | 101 > \u306e\u691c\u7d22\u3059\u308b\u91cf\u5b50\u56de\u8def\u3092\u4f5c\u6210\u3059\u308b\u3002\n\n\nmark_5 = X(1)*CCZ(1,2,0)*X(1)\ngrover_5 = gate_simp(d_3*mark_5*d_3*mark_5)\nstate1_5 = qapply(d_3*mark_5*init_state)\npprint(state1_5)\nqapply(d_3*mark_5*state1_5)\n```\n\n\n```python\n# \u4e0a\u3067\u4f5c\u3063\u305f\u91cf\u5b50\u56de\u8def\u3092\u521d\u671f\u72b6\u614b\u3068\u4f5c\u7528\u3055\u305b\u3066 measure_all() \u3067\u304b\u304f\u72b6\u614b\u306e\u78ba\u7387\u3092\u307f\u3066\u3001\u8003\u5bdf\u3059\u308b\u3002\n\n\nmeasure_all(qapply(grover_5*init_state))\n```\n\n**\u4ee5\u964d\u3001\u3010\u8ab2\u984c\uff11\u3011\u554f\uff12\u2212\uff11\uff09\u306e\u56de\u7b54\u6b04\uff1a** \n\n\n```python\n# \u554f\uff12\u306e\u521d\u671f\u72b6\u614b\nquest2_state = CNOT(2,1)*H(2)*X(2)*CNOT(2,1)*CNOT(2,0)*H(2)*X(2)*Qubit('000')\nCircuitPlot(quest2_state,nqubits=3)\n```\n\n\n```python\n# \u554f\uff12\u306e\u56de\u7b54\u6b04\uff08\uff11\uff09\n\n\ninit2_state = qapply(quest2_state)\ninit2_state\n```\n\n\n```python\n# \u554f\uff12\u306e\u56de\u7b54\u6b04\uff08\uff12\uff09\n\n\nfor i in range(10):\n pprint(measure_all_oneshot(qapply(grover_7*init2_state)))\n```\n\n \u2758101\u27e9\n \u2758110\u27e9\n \u2758111\u27e9\n \u2758101\u27e9\n \u2758001\u27e9\n \u2758111\u27e9\n \u2758110\u27e9\n \u2758101\u27e9\n \u2758101\u27e9\n \u2758000\u27e9\n\n\n\n```python\n# \u554f\uff12\u306e\u56de\u7b54\u6b04\uff08\uff13\uff09\n\n\nmeasure_all(qapply(grover_5*init2_state))\n```\n\n## \u3010\u8ab2\u984c\uff12\u3011\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\n\n\u554f\uff11\uff09 \n\uff11. \uff13\u91cf\u5b50\u30d3\u30c3\u30c8\u3092\u5bfe\u8c61\u306b\u3057\u305f\u3001\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3092\u884c\u3044\u307e\u3059\u3002 \n\u3000|000>, |001>, ..., |110>, |111> \u306e\u5168\u3066\u306e\u72b6\u614b\u306e\u305d\u308c\u305e\u308c\u306e QFT \u306e\u7d50\u679c\u3092\u51fa\u3057\u3066\u304f\u3060\u3055\u3044\u3002 \n\u3000 \n\u3000\u3000\u3000\u30d2\u30f3\u30c8\uff09sympy.physics.quantum.qft \u306e QFT \u95a2\u6570\u3092\u4f7f\u3044\u307e\u3059\u3002\n\n\uff12. QFT(0,3) \u306e\u91cf\u5b50\u56de\u8def\u56f3\u3092 CircuitPlot() \u3067\u4f5c\u56f3\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\u3000 \n\u3000 \n\u554f\uff12\uff09 \n\uff11. \uff13\u91cf\u5b50\u30d3\u30c3\u30c8\u3092\u5bfe\u8c61\u306b\u3057\u305f\u3001\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3092\u57fa\u672c\u7684\u306a\u91cf\u5b50\u30b2\u30fc\u30c8\u3060\u3051\u3067\u8868\u3057\u3066\u304f\u3060\u3055\u3044\u3002 \n\u3000 $\\sqrt{T}$\u30b2\u30fc\u30c8\u3067\u3042\u308b Rk(n,4) \u306f\u5229\u7528\u3057\u3066\u3082\u3088\u3044\u3002\n\u3000 \n\u3000\u30fb\u6f14\u7b97\u3092\u30c6\u30f3\u30bd\u30eb\u7a4d\u3067\u8868\u3057\u3066\u304f\u3060\u3055\u3044\u3002 \n\u3000\u30fb\uff08\u3053\u306e\u5834\u5408\u306e\u91cf\u5b50\u56de\u8def\u56f3\u306f\u3001\u3046\u307e\u304f\u63cf\u3051\u307e\u305b\u3093\u3002\uff09 \n\u3000 \n\u3000 \n\n\n**\u4ee5\u964d\u3001\u3010\u8ab2\u984c\uff12\u3011\u554f\uff11\u2212\uff11\uff09\u306e\u56de\u7b54\u6b04\uff1a**\n\n\n```python\n## QFT(0,3) \u306e\u884c\u5217\u8868\u73fe\u3092\u8868\u793a\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\n\nqft3=QFT(0,3)\nrepresent(qft3,nqubits=3)\n```\n\n\n```python\n# |000> \u3092\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\n\nqapply(qft3*Qubit('000'))\n```\n\n\n```python\n# |001> \u3092\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\n\nqapply(qft3*Qubit('001'))\n```\n\n\n```python\n# |010> \u3092\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\n\nqapply(qft3*Qubit('010'))\n```\n\n\n```python\n# |011> \u3092\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\n\nqapply(qft3*Qubit('011'))\n```\n\n\n```python\n# |100> \u3092\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\n\nqapply(qft3*Qubit('100'))\n```\n\n\n```python\n# |101> \u3092\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\n\nqapply(qft3*Qubit('101'))\n```\n\n\n```python\n# |110> \u3092\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\n\nqapply(qft3*Qubit('110'))\n```\n\n\n```python\n# |111> \u3092\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\n\nqapply(qft3*Qubit('111'))\n```\n\n**\u4ee5\u964d\u3001\u3010\u8ab2\u984c\uff12\u3011\u554f\uff11\u2212\uff12\uff09\u306e\u56de\u7b54\u6b04\uff1a**\n\n\n```python\n### QFT(0,3) \u306f\u3001SymPy \u3067\u306f\u3072\u3068\u584a\u308a\u306e\u307e\u3068\u307e\u3063\u305f\u30aa\u30da\u30ec\u30fc\u30bf\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u307e\u3059\u3002\n### \u57fa\u672c\u30b2\u30fc\u30c8\u3092\u77e5\u308b\u305f\u3081\u306b\u306f\u3001decompose() \u3092\u4f7f\u3044\u307e\u3059\u3002\nQFT(0,3).decompose()\n```\n\n\n```python\n# QFT(0,3) \u306e\u91cf\u5b50\u56de\u8def\u56f3\u3092 CircuitPlot() \u3067\u4f5c\u56f3\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\n\nCircuitPlot(QFT(0,3).decompose(), nqubits=3)\n```\n\n\n```python\n# decompose() \u3057\u305f\u4e0a\u8a18\u306e\u56de\u8def\u3092\u6539\u3081\u3066\u3001\u5b9a\u7fa9\u3057\u306a\u304a\u3057\u307e\u3059\u3002\nqft3_decomp = SWAP(0,2)*H(0)*CGateS((0,),S(1))*H(1)*CGateS((0,),T(2))*CGateS((1,),S(2))*H(2)\nqft3_decomp\n```\n\n\n```python\n# \u4e0a\u8a18\u3067\u5b9a\u7fa9\u3057\u306a\u304a\u3057\u305f QFT \u306e\u91cf\u5b50\u56de\u8def\u56f3\u3092 CircuitPlot() \u3067\u4f5c\u56f3\u3057\u307e\u3059\u3002\n# QFT(0,3).decompose() \u306e\u91cf\u5b50\u56de\u8def\u56f3\u3068\u6bd4\u8f03\u3057\u3066\u304f\u3060\u3055\u3044\u3002\nCircuitPlot(qft3_decomp,nqubits=3)\n```\n\n**\u4ee5\u964d\u3001\u3010\u8ab2\u984c\uff12\u3011\u554f\uff12\u2212\uff11\uff09\u306e\u89e3\u7b54\u6b04\uff1a** \n\uff08\u30d2\u30f3\u30c8\uff09$c_{g}$ \u3092\u30b0\u30ed\u30fc\u30d0\u30eb\u4f4d\u76f8\u3068\u3057\u3066\u3001Z\u8ef8\u56de\u8ee2\n$ R_{z\\theta} = c_{g} X \\cdot R_{z\\theta/2}^{\\dagger} \\cdot X \\cdot R_{z\\theta/2} $\n\u3068\u8868\u305b\u308b\u3053\u3068\u3092\u4f7f\u3044\u307e\u3059\u3002\n\n\n```python\n# S = c\u30fbX\u30fbT\u2020\u30fbX\u30fbT \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3057\u307e\u3059\u3002\npprint(represent(S(0),nqubits=1))\nrepresent(exp(I*pi/4)*X(0)*T(0)**(-1)*X(0)*T(0),nqubits=1)\n```\n\n\n```python\n# T = c\u30fbX\u30fbsqrt(T)\u2020\u30fbX\u30fbsqrt(T) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3057\u307e\u3059\u3002\npprint(represent(T(0),nqubits=1))\nrepresent(exp(I*pi/8)*X(0)*Rk(0,4)**(-1)*X(0)*Rk(0,4),nqubits=1)\n```\n\n\n```python\n# qft3_decomp = SWAP(0,2)*H(0)*CGateS((0,),S(1))*H(1)*CGateS((0,),T(2))*CGateS((1,),S(2))*H(2)\n# qft3_decomp \u3092\u898b\u306a\u304c\u3089\u3001\u5236\u5fa1\uff33\u30b2\u30fc\u30c8\u3092\u7f6e\u304d\u63db\u3048\u3066\u3001qft3_decomp2 \u3078\u4ee3\u5165\u3057\u307e\u3059\u3002\n\n\nqft3_decomp2 = SWAP(0,2)*H(0)*CNOT(0,1)*T(1)**(-1)*CNOT(0,1)*T(1)*H(1)*CGateS((0,),T(2))*CNOT(1,2)*T(2)**(-1)*CNOT(1,2)*T(2)*H(2)\nqft3_decomp2\n```\n\n\n```python\n# qft3_decomp2 = SWAP(0,2)*H(0)*CNOT(0,1)*T(1)**(-1)*CNOT(0,1)*T(1)*H(1)*CGateS((0,),T(2))*CNOT(1,2)*T(2)**(-1)*CNOT(1,2)*T(2)*H(2)\n# qft3_decomp \u3092\u898b\u306a\u304c\u3089\u3001\u5236\u5fa1\uff34\u30b2\u30fc\u30c8\u3092\u7f6e\u304d\u63db\u3048\u3066\u3001qft3_decomp3 \u3078\u4ee3\u5165\u3057\u307e\u3059\u3002\n\nqft3_decomp3 = SWAP(0,2)*H(0)*CNOT(0,1)*T(1)**(-1)*CNOT(0,1)*T(1)*H(1)*CNOT(0,2)*Rk(2,4)**(-1)*CNOT(0,2)*Rk(2,4)*CNOT(1,2)*T(2)**(-1)*CNOT(1,2)*T(2)*H(2)\nqft3_decomp3\n```\n\n\n```python\n# |000> \u306e\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306e\u7d50\u679c\u3092\u307f\u307e\u3059\u3002\n### \u30b2\u30fc\u30c8\u64cd\u4f5c\u304c\u5c11\u3057\u8907\u96d1\u306b\u306a\u308b\u305f\u3081\u3001SymPy\u304c\u3046\u307e\u304f\u5224\u65ad\u3067\u304d\u307e\u305b\u3093\u3002\n### represent()\u3067\u8a08\u7b97\u3057\u307e\u3059\u3002\u89e3\u7b54\u4f8b\u3067\u306f\u3001\u7d50\u679c\u304c\u7e26\u30d9\u30af\u30c8\u30eb\u3067\u884c\u6570\u304c\u9577\u304f\u306a\u308b\u306e\u3092\u5acc\u3044\u3001transpose()\u3057\u307e\u3059\u3002\n\n# \uff08\u89e3\u7b54\u4f8b\uff09transpose(represent(qft3_decomp2*Qubit('000'), nqubits=3))\n\n\ntranspose(represent(qft3_decomp2*Qubit('000'), nqubits=3))\n```\n\n\n```python\n# |001> \u306e\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306e\u7d50\u679c\u3092\u307f\u307e\u3059\u3002\n### \u30b0\u30ed\u30fc\u30d0\u30eb\u4f4d\u76f8 exp(I*pi/4) \u3092\u304b\u3051\u308b\u3068\u540c\u3058\u306b\u306a\u308a\u307e\u3059\u3002\n\n\nexp(I*pi/4)*transpose(represent(qft3_decomp2*Qubit('001'), nqubits=3))\n```\n\n\n```python\n# |010> \u306e\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306e\u7d50\u679c\u3092\u307f\u307e\u3059\u3002\n### \u30b0\u30ed\u30fc\u30d0\u30eb\u4f4d\u76f8 exp(I*pi/4) \u3092\u304b\u3051\u308b\u3068\u540c\u3058\u306b\u306a\u308a\u307e\u3059\u3002\n\n\nexp(I*pi/4)*transpose(represent(qft3_decomp2*Qubit('010'), nqubits=3))\n```\n\n\n```python\n# |011> \u306e\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306e\u7d50\u679c\u3092\u307f\u307e\u3059\u3002\n### \u30b0\u30ed\u30fc\u30d0\u30eb\u4f4d\u76f8 exp(I*pi/2) \u3092\u304b\u3051\u308b\u3068\u540c\u3058\u306b\u306a\u308a\u307e\u3059\u3002\n\n\nexp(I*pi/2)*transpose(represent(qft3_decomp2*Qubit('011'), nqubits=3))\n```\n\n\n```python\n# |100> \u306e\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306e\u7d50\u679c\u3092\u307f\u307e\u3059\u3002\n\n\ntranspose(represent(qft3_decomp2*Qubit('100'), nqubits=3))\n```\n\n\n```python\n# |101> \u306e\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306e\u7d50\u679c\u3092\u307f\u307e\u3059\u3002\n### \u30b0\u30ed\u30fc\u30d0\u30eb\u4f4d\u76f8 exp(I*pi/4) \u3092\u304b\u3051\u308b\u3068\u540c\u3058\u306b\u306a\u308a\u307e\u3059\u3002\n\n\nexp(I*pi/4)*transpose(represent(qft3_decomp2*Qubit('101'), nqubits=3))\n```\n\n\n```python\n# |110> \u306e\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306e\u7d50\u679c\u3092\u307f\u307e\u3059\u3002\n### \u30b0\u30ed\u30fc\u30d0\u30eb\u4f4d\u76f8 exp(I*pi/4) \u3092\u304b\u3051\u308b\u3068\u540c\u3058\u306b\u306a\u308a\u307e\u3059\u3002\n\n\nexp(I*pi/4)*transpose(represent(qft3_decomp2*Qubit('110'), nqubits=3))\n```\n\n\n```python\n# |111> \u306e\u91cf\u5b50\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306e\u7d50\u679c\u3092\u307f\u307e\u3059\u3002\n### \u30b0\u30ed\u30fc\u30d0\u30eb\u4f4d\u76f8 exp(I*pi/2) \u3092\u304b\u3051\u308b\u3068\u540c\u3058\u306b\u306a\u308a\u307e\u3059\u3002\n\n\nexp(I*pi/2)*transpose(represent(qft3_decomp2*Qubit('111'), nqubits=3))\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "181d9852404732816c735fcf0d2d403108570c01", 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YES\n2. YES", "lm_q1_score": 0.8354835411997897, "lm_q2_score": 0.5273165233795672, "lm_q1q2_score": 0.44056427628632255}} {"text": ">*This is an interactive `jupyter` notebook version of the paper [arXiv:1906.08263](https://arxiv.org/abs/1906.08263). Run the notebook to make the plots interactive. Google colab version is available [here]( https://colab.research.google.com/drive/10jw5T07mhvdP0hp60HqVqNqWK49Cfzby).*\n\n##### Pre-loading packages and codes\n\nIf you can see this sentence, click the black triangle on the left to collapse this section.\n\n\n```python\n!pip install lenstronomy\n```\n\n Collecting lenstronomy\n \u001b[?25l Downloading https://files.pythonhosted.org/packages/b0/47/a50157a9b34135debb75655201887a42d71c3a6ee6fa8406eb4a9e59b50d/lenstronomy-0.8.2.tar.gz (5.8MB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 5.8MB 4.3MB/s \n \u001b[?25hRequirement already satisfied: numpy>=1.13 in /usr/local/lib/python3.6/dist-packages (from lenstronomy) (1.16.4)\n Requirement already satisfied: scipy>=0.14.0 in /usr/local/lib/python3.6/dist-packages (from lenstronomy) (1.3.0)\n Collecting configparser (from lenstronomy)\n Downloading https://files.pythonhosted.org/packages/ba/05/6c96328e92e625fc31445d24d75a2c92ef9ba34fc5b037fe69693c362a0d/configparser-3.7.4-py2.py3-none-any.whl\n Building wheels for collected packages: lenstronomy\n Building wheel for lenstronomy (setup.py) ... \u001b[?25l\u001b[?25hdone\n Stored in directory: /root/.cache/pip/wheels/58/ba/a1/983a88d921a559e52db01095b532a4113c9e545226a879d4ab\n Successfully built lenstronomy\n Installing collected packages: configparser, lenstronomy\n Successfully installed configparser-3.7.4 lenstronomy-0.8.2\n\n\n\n```python\n!pip install mgefit\n```\n\n Collecting mgefit\n \u001b[?25l Downloading https://files.pythonhosted.org/packages/d8/8c/73f1bc792b947457c26bf568f87b8d977654f4643585315b43ecb4e25f45/mgefit-5.0.12.tar.gz (11.5MB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 11.5MB 4.7MB/s \n \u001b[?25hRequirement already satisfied: numpy in /usr/local/lib/python3.6/dist-packages (from mgefit) (1.16.4)\n Requirement already satisfied: scipy in /usr/local/lib/python3.6/dist-packages (from mgefit) (1.3.0)\n Requirement already satisfied: matplotlib in /usr/local/lib/python3.6/dist-packages (from mgefit) (3.0.3)\n Requirement already satisfied: kiwisolver>=1.0.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib->mgefit) (1.1.0)\n Requirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.6/dist-packages (from matplotlib->mgefit) (0.10.0)\n Requirement already satisfied: python-dateutil>=2.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib->mgefit) (2.5.3)\n Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib->mgefit) (2.4.0)\n Requirement already satisfied: setuptools in /usr/local/lib/python3.6/dist-packages (from kiwisolver>=1.0.1->matplotlib->mgefit) (41.0.1)\n Requirement already satisfied: six in /usr/local/lib/python3.6/dist-packages (from cycler>=0.10->matplotlib->mgefit) (1.12.0)\n Building wheels for collected packages: mgefit\n Building wheel for mgefit (setup.py) ... \u001b[?25l\u001b[?25hdone\n Stored in directory: /root/.cache/pip/wheels/b6/ad/08/a25b9fd4e4419d304787593dda44651238bec609ca340e1187\n Successfully built mgefit\n Installing collected packages: mgefit\n Successfully installed mgefit-5.0.12\n\n\n\n```python\n## plot settings\n\nimport matplotlib.pyplot as plt\nimport seaborn as sns\n\nsns.set(style='ticks', context='paper', #font='Helvetica', \n font_scale=1.5)\nsns.set_style({\"xtick.direction\": \"in\",\"ytick.direction\": \"in\", \n \"axes.linewidth\": 1.5})\n\ndef set_font_scale(font_scale):\n sns.set(style='ticks', context='paper', font='Times New Roman', \n font_scale=font_scale)\n sns.set_style({\"xtick.direction\": \"in\",\"ytick.direction\": \"in\", \n \"axes.linewidth\": 1.5,})\n \n# from cb2\nemerald = '#66c2a5' #sns.xkcd_rgb['emerald']\norange = '#fc8d62' #sns.xkcd_rgb['bright orange']\nblue = '#8da0cb' #sns.xkcd_rgb['light purple'] \npink = '#e78ac3'\n\n\ncmap = sns.cubehelix_palette(start=0.5, rot=-1.5, gamma=1, hue=1, light=-0., \n dark=0.8, reverse=False, as_cmap=True)\n```\n\n\n```python\nimport lenstronomy\n\nlenstronomy.__version__\n```\n\n\n\n\n '0.8.2'\n\n\n\n\n```python\nimport numpy as np\nfrom ipywidgets import interact\nfrom ipywidgets import FloatSlider\nfrom ipywidgets import IntSlider\n\nfrom lenstronomy.LensModel.Profiles.nfw_ellipse import NFW_ELLIPSE\nfrom lenstronomy.LensModel.Profiles.sersic_ellipse_potential import SersicEllipse\nfrom lenstronomy.Util.param_util import phi_q2_ellipticity\n\nimport matplotlib.gridspec as gridspec\nimport matplotlib.patches\n\nsersic = SersicEllipse()\n\nn = 50\nwidth = 2\nx = np.linspace(-width, width, n)\ny = np.linspace(-width, width, n)\n\nX, Y = np.meshgrid(x, y)\nXf = X.flatten()\nYf = Y.flatten()\n \ndef plot_potential_to_density_contours(q, n_sersic):\n fig = plt.figure(figsize=(8, 6))\n gs1 = gridspec.GridSpec(1, 2)\n gs1.update(left=0.0, right=1., wspace=0.0, hspace=0.05)\n axes = []\n axes.append(fig.add_subplot(gs1[0, 0]))\n axes.append(fig.add_subplot(gs1[0, 1]))\n\n r_sersic = 1\n k_eff = 5\n pa = 0\n\n e1, e2 = phi_q2_ellipticity(pa, q)\n\n f = sersic.function(Xf, Yf, n_sersic, r_sersic, k_eff, e1, e2)\n f = f.reshape((n, -1))\n\n f_xx, f_yy, _ = sersic.hessian(Xf, Yf, n_sersic, r_sersic, k_eff, e1, e2)\n kappa = (f_xx + f_yy).reshape((n, -1))\n\n levels = np.array([0, width-0.2]) #np.linspace(0, w-0.2, 5)\n pot_lines = sersic.function(levels, levels*0, n_sersic, r_sersic, k_eff, \n e1, e2)\n\n p = .7\n pot_lines = np.linspace(pot_lines[0]**p, pot_lines[1]**p, 5)**(1/p)\n\n reds = [ '#fd8d3c', '#fc4e2a', '#e31a1c', '#b10026']\n reds.reverse()\n blues = ['#0c2c84', '#225ea8', '#1d91c0', '#41b6c4', '#7fcdbb']\n\n axes[0].contour(X, Y, f, levels=pot_lines[1:], linestyles=['--'], \n linewidths=[1.3], colors=blues)\n axes[0].axis('equal')\n axes[0].set_xticks([])\n axes[0].set_yticks([])\n axes[0].set_xlim(-width, width)\n axes[0].set_ylim(-width, width)\n\n axes[0].annotate(r'$q=${:.2f}'.format(q), (-width*0.8, width*0.8))\n\n _f_xx, _f_yy, _ = sersic.hessian(levels, levels*0, n_sersic, r_sersic,\n k_eff, e1, e2)\n kappa_lines = _f_xx + _f_yy\n\n kappa_lines = 1/np.linspace(1/kappa_lines[1]**p, \n 1/kappa_lines[0]**p, 5)**(1/p)\n\n axes[1].contour(X, Y, kappa, linestyles=['-'], levels=kappa_lines[:-1], \n linewidths=[1.3], colors=reds)\n axes[1].axis('equal')\n axes[1].set_xticks([])\n axes[1].set_yticks([])\n axes[1].set_xlim(-width, width)\n axes[1].set_ylim(-width, width) \n\n # 1. Get transformation operators for axis and figure\n ax0tr = axes[0].transData # Axis 0 -> Display\n ax1tr = axes[1].transData # Axis 1 -> Display\n figtr = fig.transFigure.inverted() # Display -> Figure\n # 2. Transform arrow start point from axis 0 to figure coordinates\n ptB = figtr.transform(ax0tr.transform((width-0.15, 0.)))\n # 3. Transform arrow end point from axis 1 to figure coordinates\n ptE = figtr.transform(ax1tr.transform((-width+0.18, 0.)))\n\n arrow = matplotlib.patches.FancyArrowPatch(\n ptB, ptE, transform=fig.transFigure, # Place arrow in figure coord system\n fc = \"white\", edgecolor='k', connectionstyle=\"arc3,rad=0.0\", \n arrowstyle='simple,head_length=0.45,head_width=.8,tail_width=0.35', \n alpha = 1., mutation_scale = 15., linewidth=1.1)\n # 5. Add patch to list of objects to draw onto the figure\n fig.patches.append(arrow)\n\n\n axes[0].set_title(r'Deflection potential')\n axes[1].set_title(r'Surface density')\n\n plt.show()\n```\n\n\n```python\nfrom scipy.special import comb\nimport cmath\n\nP = 10\n\n## Euler nodes and weights\nkes = np.arange(2*P+1)\nbetas = np.sqrt(2*P * np.log(10) / 3. + 1j * 2*np.pi * kes)\nepsilons = np.zeros(2*P+1)\n\nepsilons[0] = 0.5\nepsilons[1:P+1] = 1.\nepsilons[-1] = 1/2**P\n\nfor k in range(1, P):\n epsilons[2*P-k] = epsilons[2*P-k+1] + 1/2**P * comb(P, k)\n\netas = (-1)**kes * epsilons * 2 * np.sqrt(2*np.pi) * 10**(P/3)\n\n\ndef transform(func, sigmas, **kwargs):\n \"\"\"\n \"\"\"\n f_sigmas = np.zeros_like(sigmas)\n\n f_sigmas = np.sum(etas * func(sigmas[:,np.newaxis]*betas[np.newaxis, :],\n **kwargs).real, \n axis=1\n )\n\n return f_sigmas\n\n\ndef decompose(func, sigma_start=0.02, sigma_end=15, N_sigma=15, **kwargs):\n \"\"\"\n Compute the amplitudes and sigmas of Gaussian components using the\n integral transform with Gaussian kernel. The returned values are in the \n convention of eq. (2.13).\n \"\"\"\n sigmas = np.logspace(np.log10(sigma_start), np.log10(sigma_end), N_sigma)\n\n f_sigmas = transform(func, sigmas, **kwargs)\n\n # weighting for trapezoid method integral\n f_sigmas[0] *= 0.5\n f_sigmas[-1] *= 0.5\n\n del_log_sigma = np.abs(np.diff(np.log(sigmas)).mean())\n\n f_sigmas *= del_log_sigma / np.sqrt(2.*np.pi)\n\n return f_sigmas, sigmas\n \n\ndef sersic_func(R, n_sersic=1):\n \"\"\"\n \"\"\"\n b_n = 1.9992 * n_sersic - 0.3271\n\n return np.exp(-b_n*R**(1./n_sersic) + b_n)\n\n\ndef nfw_func(R, r_s=5, amp=1./1.39552425):\n \"\"\"\n \"\"\"\n shape = R.shape\n R = R.flatten()\n x = R/r_s\n \n f = np.empty(shape=x.shape, dtype=x.dtype)\n \n range1 = (x > 1.)\n if np.any(range1):\n s = x[range1]\n f[range1] = (1 - np.arccos(1/s) / np.sqrt(s*s - 1)) / (s*s - 1)\n \n range2 = (x < 1.)\n if np.any(range2):\n s = x[range2]\n f[range2] = (1 - np.arccosh(1/s) / np.sqrt(1 - s*s)) / (s*s - 1)\n \n range3 = np.logical_and(np.logical_not(range1), np.logical_not(range2))\n if np.any(range3):\n s = x[range3]\n f[range3] = 1./3.\n\n return amp * f.reshape(shape)\n\n\ndef sum_gauss_components(R, A_sigmas, sigmas):\n \"\"\"\n \"\"\"\n total = np.zeros_like(R)\n\n for i, r in enumerate(R):\n total[i] = np.sum(A_sigmas * np.exp(-r*r/2/sigmas**2))\n\n return total\n```\n\n\n```python\ndef plot_gauss_decompose(NFW=True, n_sersic=4, N_sigma=15,\n sigma_start=0.02, sigma_end=15):\n \"\"\"\n \"\"\"\n blue = '#4575b4'\n red = '#d73027'\n \n r_s = 5\n \n sigs = np.logspace(-2, 2, 1000)\n f_sigs_sersic = transform(sersic_func, sigs, n_sersic=n_sersic)\n f_sigs_nfw = transform(nfw_func, sigs, r_s=r_s)\n\n amps_sersic, sigmas_sersic = decompose(sersic_func, sigma_start=sigma_start, \n sigma_end=sigma_end, N_sigma=N_sigma,\n n_sersic=n_sersic)\n \n amps_nfw, sigmas_nfw = decompose(nfw_func, sigma_start=0.005*r_s, \n sigma_end=50*r_s, N_sigma=15,\n r_s=r_s)\n\n Rs = np.logspace(-1, 1, 500)\n sersic_approx = sum_gauss_components(Rs, amps_sersic, sigmas_sersic)\n nfw_approx = sum_gauss_components(Rs, amps_nfw, sigmas_nfw)\n\n sersic_true = sersic_func(Rs, n_sersic=n_sersic)\n nfw_true = nfw_func(Rs, r_s=r_s)\n \n fig = plt.figure(figsize=(12, 4))\n\n ax1 = fig.add_subplot(131)\n ax1.loglog(sigs, f_sigs_sersic, color=blue, lw=2., ls='-')\n\n for s in sigmas_sersic:\n ax1.axvline(s, .97, 1, color=blue)\n \n if NFW:\n ax1.loglog(sigs, f_sigs_nfw, color=red, lw=2., ls='--')\n \n for s in sigmas_nfw:\n ax1.axvline(s, .97, 1, color=red)\n \n ax1.set_ylim(1e-4, 1e3)\n ax1.set_xlim(1e-2, 1e2)\n #ax1.axis('equal')\n #ax1.legend()\n ax1.set_xlabel(r'$\\sigma$')\n ax1.set_ylabel(r'$f\\ (\\sigma)$')\n\n ax2 = fig.add_subplot(132)\n #ax2.loglog(Rs, sersic_true, color='k', lw=2., ls='-')\n ax2.loglog(Rs, sersic_approx, color=blue, lw=2., ls='-')\n if NFW:\n ax2.loglog(Rs, nfw_approx, color=red, lw=2., ls='--')\n ax2.set_ylim(1e-4, 1e3)\n ax2.set_xlim(1e-1, 1e1)\n #ax2.axis('equal')\n #ax1.legend()\n ax2.set_xlabel(r'$R/R_{\\rm eff}$')\n ax2.set_ylabel(r'$\\Sigma_{\\rm approx}$')\n\n ax3 = fig.add_subplot(133)\n ax3.plot(Rs, (sersic_true - sersic_approx)/np.sqrt(sersic_true)*100, \n color=blue, lw=2., ls='-', \n label=r'$n_{\\rm Sersic}=$'+'{}'.format(n_sersic))\n if NFW:\n ax3.plot(Rs, (nfw_true - nfw_approx)/np.sqrt(nfw_true)*100, \n color=red, lw=2., ls='--', label=r'Projected NFW')\n ax3.set_xscale('log')\n ax3.set_ylim(-2, 2)\n ax3.set_xlim(1e-1, 1e1)\n ax3.set_xlabel(r'$R/R_{\\rm eff}$')\n ax3.set_ylabel(r'Residual / Noise')\n ax3.legend()\n\n fig.tight_layout()\n plt.show()\n```\n\n\n```python\nfrom mgefit.mge_fit_1d import mge_fit_1d\n\ndef plot_mge_comparison(n_sersic=1, N_sigma=15):\n blue = '#4575b4'\n\n R = np.logspace(-1, 1, 200)\n amps, sigs = decompose(sersic_func, N_sigma=N_sigma, n_sersic=n_sersic)\n\n sersic_approx = sum_gauss_components(R, amps, sigs)\n\n sersic_R = sersic_func(R, n_sersic=n_sersic)\n mge = mge_fit_1d(R, sersic_R, ngauss=N_sigma, plot=False, quiet=True)\n\n fig = plt.figure(figsize=(12, 4))\n\n ax1 = fig.add_subplot(131)\n ax1.clear()\n\n for a, s in zip(amps, sigs):\n ax1.loglog(R, a*np.exp(-R**2/2/s**2), color='#91bfdb', \n ls=':', lw=1.5)\n\n ax1.loglog(R, sersic_approx, color=blue, lw=2, \n label=r'$n_{\\rm Sersic}=$'+'{}'.format(n_sersic))\n ax1.set_xlim(1e-1, 1e1)\n ax1.set_ylim(1e-4, 1e3)\n ax1.set_xlabel(r'$R/R_{\\rm eff}$')\n ax1.set_ylabel(r'Our method $\\Sigma_{\\rm approx}$')\n ax1.legend()\n\n ax2 = fig.add_subplot(132)\n ax2.clear()\n ax2.loglog(mge.x, mge.gauss*mge.weights[None, :], 'lightgrey', \n lw=1.5, ls='-.')\n\n ax2.loglog(mge.x, mge.yfit, 'grey', lw=2, ls='--', \n label=r'$n_{\\rm Sersic}=$'+'{}'.format(n_sersic))\n\n ax2.set_ylim(1e-4, 1e3)\n ax2.set_xlim(1e-1, 1e1)\n ax2.set_xlabel(r'$R/R_{\\rm eff}$')\n ax2.set_ylabel(r\"Cappellari (2002) $\\Sigma_{\\rm approx}$\")\n ax2.legend()\n\n ax3 = fig.add_subplot(133)\n ax3.clear()\n ax3.plot(R, (sersic_R - sersic_approx)/np.sqrt(sersic_R)*100, color=blue, \n lw=2., ls='-', label='Our method')\n ax3.plot(mge.x, (mge.err)/np.sqrt(mge.y)*100, color='grey', lw=2, ls='--', \n label='Cappellari (2002)', zorder=-2)\n ax3.set_xscale('log')\n ax3.set_ylim(-2, 2)\n ax3.set_xlim(1e-1, 1e1)\n ax3.set_xlabel(r'$R/R_{\\rm eff}$')\n ax3.set_ylabel(r'Residual / Noise')\n ax3.legend()\n\n fig.tight_layout()\n plt.show()\n```\n\n\n```python\nfrom scipy.special import wofz\nfrom copy import deepcopy\n\n\nclass EllipticalGaussianLens(object):\n \"\"\"\n \"\"\"\n def __init__(self, min_ellipticity=1e-5):\n \"\"\"\n \"\"\"\n self.min_ellipticity = 1e-5\n \n def quarantine(self, q):\n \"\"\"\n \"\"\"\n if 1 - q < self.min_ellipticity:\n q = 1 - self.min_ellipticity\n \n return q\n \n def get_deflection(self, x, y, amp, sigma, q):\n \"\"\"\n \"\"\"\n q = self.quarantine(q)\n\n _p = q / sigma / np.sqrt(2 * (1. - q**2))\n\n sig_func_re, sig_func_im = self.sigma_function(_p * x, _p * y, q)\n\n alpha_x = amp * sigma * np.sqrt(2*np.pi/(\n 1.-q**2)) * sig_func_re\n alpha_y = - amp * sigma * np.sqrt(\n 2 * np.pi / (1. - q ** 2)) * sig_func_im\n\n return alpha_x, alpha_y\n \n def get_convergence(self, x, y, amp, sigma, q):\n \"\"\"\n \"\"\"\n return amp * np.exp(-(q*q*x*x + y*y)/2./sigma/sigma)\n \n def get_shear(self, x, y, amp, sigma, q):\n \"\"\"\n \"\"\"\n q = self.quarantine(q)\n\n _p = q / sigma / np.sqrt(2 * (1. - q**2))\n sig_func_re, sig_func_im = self.sigma_function(_p * x, _p * y, q)\n\n kappa = self.get_convergence(x, y, amp, sigma, q)\n\n shear = - 1/(1-q*q) * ((1+q**2)*kappa - 2*q*amp + np.sqrt(\n 2*np.pi) * q*q * amp * (x - 1j*y) / sigma / np.sqrt(1-q*q) * (\n sig_func_re - 1j*sig_func_im))\n \n return shear.real, -shear.imag\n \n def get_magnification(self, x, y, amp, sigma, q):\n \"\"\"\n \n \"\"\"\n kappa = self.get_convergence(x, y, amp, sigma, q)\n g1, g2 = self.get_shear(x, y, amp, sigma, q)\n\n return 1. / ((1.-kappa)**2 - (g1**2+g2**2))\n\n def sigma_function(self, x, y, q):\n \"\"\"\n Compute the function varsigma(z; q) from equation (4.12).\n\n :param x: Real part of complex variable, x = Re(z)\n :type x: float or numpy.array\n :param y: Imaginary part of complex variable, y = Re(z)\n :type y: float or numpy.array\n :param q: Axis ratio.\n :type q: float\n :return: value of Sigma function, equation (4.12) from Shajib (2019)\n :rtype: tuple (type, type) with type being type(x) or type(y)\n \"\"\"\n y_sign = np.sign(y)\n y_ = deepcopy(y) * y_sign\n z = x + 1j * y_\n zq = q * x + 1j * y_ / q\n\n w = wofz(z)\n wq = wofz(zq)\n\n # exponential factor in the 2nd term of eqn. (4.15) of Shajib (2019)\n exp_factor = np.exp(-x * x * (1 - q * q) - y_ * y_ * (1 / q / q - 1))\n\n sigma_func_real = w.imag - exp_factor * wq.imag\n sigma_func_imag = (- w.real + exp_factor * wq.real) * y_sign \n \n return sigma_func_real, sigma_func_imag\n\n def get_critical_curves(self, amp, sigma, q, compute_window=4, \n start_scale=0.05, max_order=10):\n \"\"\"\n This method is modified from a `lenstronomy` function written by \n Simon Birrer.\n \n :param amp:\n :param sigma:\n :param q:\n :param compute_window:\n :param tiling_scale:\n :return:\n \"\"\"\n\n numPix = int(2 * compute_window / start_scale)\n \n arr = np.linspace(-compute_window, compute_window, numPix)\n x_grid_init, y_grid_init = np.meshgrid(arr, arr)\n\n mag_init = self.get_magnification(x_grid_init, y_grid_init, \n amp, sigma, q) \n\n x_crit_list = []\n y_crit_list = []\n \n # iterate through original triangles and return ra_crit, dec_crit list\n for i in range(numPix-1):\n for j in range(numPix-1):\n edge1 = [x_grid_init[i, j], y_grid_init[i, j], mag_init[i, j]]\n edge2 = [x_grid_init[i+1, j+1], y_grid_init[i+1, j+1], \n mag_init[i+1, j+1]]\n edge_90_1 = [x_grid_init[i, j+1], y_grid_init[i, j+1], \n mag_init[i, j+1]]\n edge_90_2 = [x_grid_init[i+1, j], y_grid_init[i+1, j], \n mag_init[i+1, j]]\n x_crit, y_crit = self._tiling_crit(edge1, edge2, edge_90_1, \n max_order, amp, sigma, q)\n x_crit_list += x_crit # list addition\n y_crit_list += y_crit # list addition\n x_crit, y_crit = self._tiling_crit(edge1, edge2, edge_90_2, \n max_order, amp, sigma, q)\n x_crit_list += x_crit # list addition\n y_crit_list += y_crit # list addition\n \n return np.array(x_crit_list), np.array(y_crit_list)\n\n def _tiling_crit(self, edge1, edge2, edge_90, max_order, amp, sigma, q):\n \"\"\"\n Tile a rectangular triangle and compares the signs of the magnification. \n This method is modified from a `lenstronomy` function written by Simon Birrer.\n\n :param edge1: [x_coordinate, y_coordinate, magnification]\n :param edge2: [x_coordinate, y_coordinate, magnification]\n :param edge_90: [x_coordinate, y_coordinate, magnification]\n :param max_order: maximal order to fold triangle\n :return:\n \"\"\"\n x_1, y_1, mag_1 = edge1\n x_2, y_2, mag_2 = edge2\n x_3, y_3, mag_3 = edge_90\n sign_list = np.sign([mag_1, mag_2, mag_3])\n \n # if all signs are the same\n if sign_list[0] == sign_list[1] and sign_list[0] == sign_list[2]:\n return [], []\n else:\n # Split triangle along the long axis\n # execute tiling twice\n # add ra_crit and dec_crit together\n # if max depth has been reached, return the mean value in the triangle\n max_order -= 1\n \n if max_order <= 0:\n return [(x_1 + x_2 + x_3)/3], [(y_1 + y_2 + y_3)/3]\n else:\n # split triangle\n # find point in the middle of the long axis to split triangle\n x_90_ = (x_1 + x_2)/2 \n y_90_ = (y_1 + y_2)/2\n mag_90_ = self.get_magnification(x_90_, y_90_, amp, sigma, q)\n edge_90_ = [x_90_, y_90_, mag_90_]\n x_crit, y_crit = self._tiling_crit(edge_90, edge1, edge_90_, \n max_order, amp, sigma, q)\n x_crit_2, y_crit_2 = self._tiling_crit(edge_90, edge2, edge_90_, \n max_order, amp, sigma, q)\n x_crit += x_crit_2\n y_crit += y_crit_2\n \n return x_crit, y_crit\n \n def get_critical_curves_and_caustics(self, amp, sigma, q, **kwargs):\n \"\"\"\n \"\"\"\n x_crit, y_crit = self.get_critical_curves(amp, sigma, q, **kwargs)\n \n alpha_x, alpha_y = self.get_deflection(x_crit, y_crit, amp, sigma, q)\n \n x_caus, y_caus = x_crit - alpha_x, y_crit - alpha_y\n \n return x_crit, y_crit, x_caus, y_caus\n```\n\n\n```python\nfrom matplotlib.colors import LinearSegmentedColormap\nfrom matplotlib.patches import Rectangle\nfrom matplotlib.patches import Circle\nfrom matplotlib.patches import FancyArrow\nfrom matplotlib.legend_handler import HandlerPatch\n\n\nclass HandlerEllipse(HandlerPatch):\n \"\"\"\n \"\"\"\n def __init__(self):\n pass\n \n def create_artists(self, legend, orig_handle,\n xdescent, ydescent, width, height, fontsize, trans):\n \"\"\"\n \"\"\"\n center = 0.5 * width - 0.5 * xdescent, 0.5 * height - 0.5 * ydescent\n p = mpatches.Ellipse(xy=center, width=width + xdescent,\n height=height + ydescent)\n self.update_prop(p, orig_handle, legend)\n p.set_transform(trans)\n \n return [p]\n \n \ndef make_legend_arrow(legend, orig_handle,\n xdescent, ydescent,\n width, height, fontsize):\n \"\"\"\"\"\"\n p = FancyArrow(0, 0.5*height, width, 0, width=1.5, \n length_includes_head=True, head_width=0.5*height, \n overhang=.1 )\n return p\n \n \ndef plot_gaussian_lens(amplitude=2., \u03c3=1., q=0.5):\n \"\"\"\n \"\"\"\n amp = amplitude\n sigma = \u03c3\n w = 4\n n = 11\n\n emerald = '#66c2a5' #sns.xkcd_rgb['emerald']\n orange = '#fc8d62' #sns.xkcd_rgb['bright orange']\n blue = '#8da0cb' #sns.xkcd_rgb['light purple'] \n\n xs = np.linspace(-w, w, n)\n ys = np.linspace(-w, w, n)\n\n X, Y = np.meshgrid(xs, ys)\n \n gaussian_lens = EllipticalGaussianLens()\n \n kappa = gaussian_lens.get_convergence(X, Y, amp, sigma, q)\n alpha_x, alpha_y = gaussian_lens.get_deflection(X, Y, amp, sigma, q)\n\n x_01 = 0\n y_01 = w/4\n\n x_02 = 0\n y_02 = w/2\n\n x_03 = 0\n y_03 = w*3/4\n \n dx = 0.01\n dy = 0.01\n\n xs1 = []\n ys1 = []\n\n xs2 = []\n ys2 = []\n\n xs3 = []\n ys3 = []\n\n while y_01 >= 0.:\n xs1.append(x_01)\n ys1.append(y_01)\n\n alpha_x1, alpha_y1 = gaussian_lens.get_deflection(x_01, y_01, amp, sigma, q)\n\n y_01 += - alpha_x1 / alpha_y1 * dx\n x_01 += dx\n\n while y_02 >= 0.:\n xs2.append(x_02)\n ys2.append(y_02)\n\n alpha_x2, alpha_y2 = gaussian_lens.get_deflection(x_02, y_02, amp, sigma, q)\n\n y_02 += - alpha_x2 / alpha_y2 * dx\n x_02 += dx\n\n while y_03 >= 0.:\n xs3.append(x_03)\n ys3.append(y_03)\n\n alpha_x3, alpha_y3 = gaussian_lens.get_deflection(x_03, y_03, amp, sigma, q)\n y_03 += - alpha_x3 / alpha_y3 * dx\n x_03 += dx\n\n xs1 = np.array(xs1)\n ys1 = np.array(ys1)\n xs2 = np.array(xs2)\n ys2 = np.array(ys2)\n xs3 = np.array(xs3)\n ys3 = np.array(ys3)\n\n fullxs1 = np.append(xs1, xs1[::-1])\n fullys1 = np.append(ys1, -ys1[::-1])\n fullxs2 = np.append(xs2, xs2[::-1])\n fullys2 = np.append(ys2, -ys2[::-1])\n fullxs3 = np.append(xs3, xs3[::-1])\n fullys3 = np.append(ys3, -ys3[::-1])\n\n fullxs1 = np.append(fullxs1, -xs1)\n fullys1 = np.append(fullys1, -ys1)\n fullxs2 = np.append(fullxs2, -xs2)\n fullys2 = np.append(fullys2, -ys2)\n fullxs3 = np.append(fullxs3, -xs3)\n fullys3 = np.append(fullys3, -ys3)\n\n\n fullxs1 = np.append(fullxs1, -xs1[::-1])\n fullys1 = np.append(fullys1, ys1[::-1])\n fullxs2 = np.append(fullxs2, -xs2[::-1])\n fullys2 = np.append(fullys2, ys2[::-1])\n fullxs3 = np.append(fullxs3, -xs3[::-1])\n fullys3 = np.append(fullys3, ys3[::-1])\n\n x_crit, y_crit, x_caus, y_caus = gaussian_lens.get_critical_curves_and_caustics(amp, sigma, q)\n\n fig = plt.figure(figsize=(10, 5))\n ax = fig.add_subplot(121)\n\n cmap = LinearSegmentedColormap.from_list('mycmap', ['white', '#fdcdac'])\n\n\n ep1, = ax.plot(fullxs1, fullys1, color='#7570b3', ls='--', zorder=1, lw=1.5)\n ep2, = ax.plot(fullxs2, fullys2, color='#7570b3', ls='--', zorder=1, lw=1.5)\n ep3, = ax.plot(fullxs3, fullys3, color='#7570b3', ls='--', zorder=1, lw=1.5)\n\n\n ax.quiver(xs, ys, -alpha_x, -alpha_y, color=emerald, zorder=2, \n angles='xy', scale_units='xy', scale=4, label='deflection')\n ax.axis('equal')\n\n convergence = ax.imshow(np.arcsinh(np.arcsinh(np.arcsinh(kappa))), \n interpolation='bicubic', cmap=cmap, \n label='convergence', extent=(-w, w, -w, w))\n ax.set_xlabel(r'$x/\\sigma$')\n ax.set_ylabel(r'$y/\\sigma$')\n ax.set_yticks([-2, 0, 2])\n ax.set_xticks([-2, 0, 2])\n ax.set_xlim(-w, w)\n ax.set_ylim(-w, w)\n ax.set_aspect('auto')\n\n height = 40\n width = 1\n\n arrow = ax.arrow(100, 1000, 2.5, 0.6, label='My label', color=emerald)\n c = Circle((0.5, 0.5), 0.25, facecolor='#fdcdac',\n edgecolor=\"none\", linewidth=0.5)\n\n ax.legend([c, arrow, ep1], \n ['convergence', 'deflection', 'isopotential'], \n handler_map={\n Rectangle: HandlerPatch(patch_func=HandlerEllipse),\n FancyArrow : HandlerPatch(patch_func=make_legend_arrow),\n }, loc='upper right')\n\n\n ax1 = fig.add_subplot(122)\n \n _line = np.array([500, 600])\n \n ax1.plot(x_crit, y_crit, 'o', markersize=0.05, c='#222222')\n ax1.plot(_line, _line, c='#222222', label='critical curve')\n\n ax1.plot(x_caus, y_caus, 'o', c='#e78ac3', markersize=0.05)\n ax1.plot(_line, _line, c='#e78ac3', label='caustic')\n\n ax1.axis('equal')\n ax1.set_xlim(-w, w)\n ax1.set_ylim(-w, w)\n ax1.legend(loc='upper right')\n ax1.set_xlabel(r'$x/\\sigma$')\n ax1.set_yticks([])\n ax1.set_xticks([-2, 0, 2])\n\n\n fig.subplots_adjust(wspace=0)\n \n plt.show()\n```\n\n# Unified lensing and kinematic analysis for _any_ elliptical mass profile\n\n### Anowar J. Shajib \n\n_Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547, US_\n\nEmail: ajshajib@astro.ucla.edu\n\n>>## Abstract\n \n>>We demonstrate an efficient method to compute the strong-gravitational-lensing deflection angle and magnification for any elliptical surface-density profile. This method solves a numerical hurdle in lens modelling that has lacked a general solution for nearly three decades. The hurdle emerges because it is prohibitive to derive analytic expressions of the lensing quantities for most elliptical mass profiles. In our method, we first decompose an elliptical mass profile into concentric Gaussian components. We introduce an integral transform that provides us with a fast and accurate algorithm for this Gaussian decomposition. We derive analytic expressions of the lensing quantities for a Gaussian component. As a result, we can compute these quantities for the total mass profile by adding up the contributions from the individual components. This lensing analysis self-consistently completes the kinematic description in terms of Gaussian components presented by Cappellari ([2008](http://adsabs.harvard.edu/abs/2008MNRAS.390...71C)). Our method is general without extra computational burden unlike other methods currently in use.\n\n>>**Key words:** gravitational lensing: strong \u2013 galaxies: kinematics and dynamics \u2013 methods:analytical \u2013 methods: numerical \u2013 methods: data analysis\n\n\n\n\n## 1 Introduction: precise computation made easy\n\nGravitational lensing (hereafter, lensing) has versatile applications in astrophysics and cosmology. Lensing is the effect when light bends while passing by a massive object. If two galaxies sit along the same line-of-sight of an observer, the background galaxy appears multiple times due to lensing. This system is called a galaxy-scale strong-lensing system (hereafter, lens). Lenses are useful to measure the Hubble constant $H_0$, dark matter subhalo mass function, dust characteristics in galaxies, mass of super-massive black holes, stellar initial mass function, etc. (e.g., Falco et al. [1999](http://adsabs.harvard.edu/abs/1999ApJ...523..617F), Peng et al. [2006](http://adsabs.harvard.edu/abs/2006ApJ...649..616P), Suyu et al. [2013](http://adsabs.harvard.edu/abs/2013ApJ...766...70S), Vegetti et al. [2014](http://adsabs.harvard.edu/abs/2014MNRAS.442.2017V), Schechter et al. [2014](http://adsabs.harvard.edu/abs/2014ApJ...793...96S)).\n\nIn several applications, stellar kinematics play a complementary role to lensing. Lensing-only observables suffer from mass-sheet degeneracy (MSD)—if a mass profile is appropriately rescaled after an infinite mass-sheet is added on top of it, then all the lensing observables stay invariant except the time delay (Falco et al. [1985](http://adsabs.harvard.edu/abs/1985ApJ...289L...1F), Schneider & Sluse [2014](http://adsabs.harvard.edu/abs/2014A%26A...564A.103S)). When a measured quantity depends on the mass profile, the MSD adds uncertainty to the measurement. Kinematics help break this degeneracy. Lensing observables probe the projected mass; kinematic observables probe the three-dimensional potential. Thus, the lensing–kinematics combination tightly constrains the mass profile, and enables us to robustly measure astrophysical and cosmological quantities (e.g., Treu et al. [2004](http://adsabs.harvard.edu/abs/2004ApJ...611..739T), Barnabe et al. [2011](http://adsabs.harvard.edu/abs/2011MNRAS.415.2215B)).

    \n\nIn practice, it is difficult to compute lensing and kinematic quantities for elliptical mass profiles, which are required to describe the most common kind of lenses, i.e., elliptical galaxies. We need to integrate over density profiles while fitting a model to either the lensing or the kinematic data. For example, in lensing, the deflection angle is an integral over the surface density profile; in kinematics, the line-of-sight velocity dispersion is a double integral over three-dimensional mass and light profiles. For most elliptical density profiles, we are unable to express these integrals with elementary or special functions. Special functions are numerically well-studied, hence fast algorithms to compute them are usually available. Thus, these functions can be numerically convenient for evaluating model-predicted observables in large numbers (e.g., $\\sim10^6$) when sampling the lens model posterior with Markov chain Monte Carlo (MCMC) methods, or even when only searching for the best-fit model. Otherwise, it would be inefficient to numerically compute lensing and kinematic integrals for general elliptical profiles.\n\nUsually, the numerical difficulties are circumvented through simplifying assumptions or approximations. We now outline common assumptions and approximations in kinematic and lensing analyses, noting that these can limit the accuracy and precision of the inferences.\n\nEither axisymmetry or spherical symmetry is usually assumed for kinematic analysis. If we start with a surface density profile for lensing analysis, we need to deproject this profile along the line of sight to compute the kinematics. This deprojection has an infinite degeneracy (Contopoulos [1956](http://adsabs.harvard.edu/abs/1956ZA.....39..126C)). Therefore, it is necessary to choose a line-of-sight symmetry when deprojecting, for which either axisymmetry or spherical symmetry is often a convenient choice.\n\n\nIn lensing analysis, spherical symmetry is rarely sufficient and we need to consider ellipticity to achieve the required precision. All the lensing quantities are related to the deflection potential or its derivatives. The time delay depends on the deflection potential difference. The gradient of the deflection potential gives the deflection angle. The Hessian of the deflection potential relates to the surface density, the shear, and the magnification. To efficiently compute these quantities for the elliptical case, we can find the following three approximations in the literature:\n\n1. **Ellipticity in deflection potential:** The gradient and the Hessian of an elliptical deflection potential can be easily computed through numerical differentiation (e.g., Kovner [1987](http://adsabs.harvard.edu/abs/1987Natur.325..507K), Golse & Kneib [2002](http://adsabs.harvard.edu/abs/2002A%26A...390..821G)). However, this solution is not general, as the surface density becomes dumbbell-shaped for an elliptical deflection potential with axis ratio $q \\lesssim 0.6$ (Fig. [1](#scrollTo=YpMX6aGH9BWl), Kassiola & Kovner [1993](http://adsabs.harvard.edu/abs/1993ApJ...417..450K)). This oddly shaped surface-density is unphysical.\n\n2. **Elliptical power-law profile:** We can efficiently compute the lensing quantities for the elliptical power-law profile using numerical approximation or analytical expressions (for the isothermal case, Korman [1994](http://adsabs.harvard.edu/abs/1994A%26A...284..285K); and for the general case, Barkana [1998](http://adsabs.harvard.edu/abs/1998ApJ...502..531B), Tessore & Metcalf [2015](http://adsabs.harvard.edu/abs/2015A%26A...580A..79T)). This profile can be sufficient to use in statistical studies that do not require detailed modelling of individual lenses (e.g., Koopmans et al. [2009](http://adsabs.harvard.edu/abs/2009ApJ...703L..51K), Sonnenfeld et al. [2013](http://adsabs.harvard.edu/abs/2013ApJ...777...97S)). However, adopting a power-law profile artificially breaks the MSD and it could potentially bias the $H_0$ measurement (e.g., Schneider & Sluse [2013](http://adsabs.harvard.edu/abs/2013A%26A...559A..37S), Sonnenfeld [2018](http://adsabs.harvard.edu/abs/2018MNRAS.474.4648S)). Therefore, we need to explore different, physically-motivated mass models, such as a _composite model_ that explicitly accounts for the luminous and the dark components (Suyu et al. [2014](http://adsabs.harvard.edu/abs/2014ApJ...788L..35S), Y\u0131ld\u0131r\u0131m et al. [2019](https://ui.adsabs.harvard.edu/abs/2019arXiv190407237Y)).\n\n3. **Chameleon profile:** The Sersic profile well describes the surface brightness of a galaxy (Sersic [1968](http://adsabs.harvard.edu/abs/1968adga.book.....S)). The Chameleon profile approximates the Sersic profile within a few per cent (Dutton et al. [2011](http://adsabs.harvard.edu/abs/2011MNRAS.417.1621D)). We can efficiently compute lensing quantities for the Chameleon profile using analytic expressions. However, this profile only describes the baryonic component. The precise lensing analysis of elliptical Navarro–Frenk–White (NFW) profile for the dark component still lacks a general solution (Navarro et al. [1997](http://adsabs.harvard.edu/abs/1996ApJ...462..563N)).\n\nIn a nutshell, these approximations for elliptical lensing analysis are only applicable in restricted regimes as described above.\n\n\n\n\n```python\ninteract(plot_potential_to_density_contours, \n q=FloatSlider(min=0.1, max=0.95, step=0.05, value=0.6, continuous_update=False), \n n_sersic=FloatSlider(min=1, max=6, step=0.5, value=4, continuous_update=False));\n\n#plot_potential_to_density_contours(q=0.6, n_sersic=4)\n```\n\n\n\n>**Figure 1:** Elliptical deflection potential (left column) producing dumbbell-shaped surface-density (right column). The dashed contours in the left column are isopotential curves for Sersic profile, and the solid contours in the right column are corresponding isodensity curves. The dumbbell shape in surface density is unphysical and it gets more pronounced for higher ellipticity in the deflection potential. Hence, we can not use elliptical deflection potential to simplify lensing analysis of moderately elliptical galaxies. We need to treat ellipticity in the surface density, not deflection potential, to make our lensing analysis generally consistent with our physical priors.\n\nIn this paper, we present a general method to precisely compute the gradient and the Hessian of the deflection potential for any elliptical surface-density profile. The method follows a ''divide and conquer'' strategy. We can approximately divide, or decompose, an elliptical profile into concentric Gaussian components (e.g., Bendinelli [1991](http://adsabs.harvard.edu/abs/1991ApJ...366..599B)). For this Gaussian decomposition, we devise a fast and accurate algorithm by introducing an integral transform. For each Gaussian component, we derive analytic expressions of the gradient and the Hessian of the deflection potential. For the deprojected Gaussian component, Cappellari ([2008]((http://adsabs.harvard.edu/abs/2008MNRAS.390...71C))) derives the line-of-sight velocity dispersion. We can combine the computed quantities from each Gaussian component back together to obtain these quantities for the total density profile. In this way, the lensing and kinematic descriptions are self-consistently unified. At the same time, this method is general, as we can apply it to lensing with any elliptical surface-density and to kinematics with either axisymmetry or spherical symmetry. Our method is more efficient than numerical integration to compute lensing quantities for an elliptical surface-density profile.\n\nWe organize this paper as follows. In Section [2](#scrollTo=DvRslW9qN1SL), we motivate the ''divide and conquer'' strategy behind our method and introduce an integral transform that provides a fast algorithm to decompose any elliptical surface-density profile into concentric Gaussian components. In Section [3](#scrollTo=alI-uL1t9HSR), we summarize the kinematic description of the Gaussian components from Cappellari ([2008](http://adsabs.harvard.edu/abs/2008MNRAS.390...71C)). In Section [4](#scrollTo=TzkbBKWhcIbn), we derive the gradient and the Hessian of the deflection potential for elliptical Gaussian surface-density. Next in Section [5](#scrollTo=uHG1Plf5apG9), we demonstrate a proof-of-concept for our method using simulated data. Then, we summarize the paper in Section [6](#scrollTo=oout9ImHc58C). Additionally in Appendix [A](#scrollTo=9qdjVclOfmVk), we prove some fundamental theorems for the integral transform introduced in Section [2](#scrollTo=DvRslW9qN1SL). This integral transform with a Gaussian kernel is the continuous case of concentric Gaussian decomposition. The theorems in Appendix [A](#scrollTo=9qdjVclOfmVk) establish the existence, uniqueness, and invertibility of this transform.\n\n\n## 2 Decomposing an elliptical profile into concentric Gaussian components\n\nWe aim to decompose an elliptical surface-density profile into simpler functions to make the computation tractable. This function should be simple enough so that we can both \n\n1. express the deflection angle in terms of elementary or special functions, and\n2. easily deproject it into three-dimension and compute the enclosed mass for kinematic analysis.\n\nThe Gaussian function meets both of these criteria. We validate criterion (1) in Section [4.2](#scrollTo=slP9xat7i8YZ), where we express the deflection angle for an elliptical Gaussian surface-density profile with the complex error function. For criterion (2), deprojecting a two-dimensional Gaussian into three-dimension is straightforward, as the Abel inversion of a two-dimensional Gaussian is a three-dimensional Gaussian. The enclosed mass for a three-dimensional Gaussian has the form of the error function, which we can efficiently compute without integrating numerically. Given these points, we approximately decompose an elliptical surface-density profile as\n\n\\begin{align} \\tag{2.1}\\label{21}\n\t\\Sigma(x, y) \\approx \\sum_{j=0}^J {\\Sigma_0}_{j} \\exp\\left(-\\frac{q^2x^2+y^2}{2 \\sigma_j^2}\\right),\t\n\\end{align}\n\nwhere ${\\Sigma_0}_ j$ is the amplitude of the $j$-th Gaussian, and all the components have a common axis ratio $q$. Similar decomposition into concentric Gaussian components has been used in the literature to fit the surface brightness profile of galaxies [called as the multi-Gaussian expansion (MGE) by Emsellem et al. ([1994](http://adsabs.harvard.edu/abs/1994A%26A...285..739E)), and the mixture-of-Gaussians by Hogg & Lang ([2013](http://adsabs.harvard.edu/abs/2013PASP..125..719H))]. The lensing and kinematic quantities of our interest—namely the gradient and the Hessian of the deflection potential, and the line-of-sight velocity dispersion—follow the principle of superposition. As a result, we can compute these quantities separately for each Gaussian component and then add them together to recover these quantities for the total surface-density profile (see Fig. [2](#scrollTo=M0C0Fo4jsRPe&line=5&uniqifier=1)).\n\n
    \n\n>**Figure 2.** \"Divide and conquer'' strategy to compute lensing quantities for elliptical surface-density profile. In this figure, we choose the deflection field as the quantity of interest to illustrate the method. However, this method works equally well for other quantities such as the lensing shear and the line-of-sight velocity dispersion. The steps in the strategy are organized into columns and the arrows show the progression of the steps. We explain each step in the text at the top of the corresponding column.\n\nWe describe the kinematics and lensing analyses for the Gaussian components in Sections [3](#scrollTo=alI-uL1t9HSR) and [4](#scrollTo=TzkbBKWhcIbn), but first, we need a fast method to decompose an elliptical profile into concentric Gaussian components. Cappellari ([2002](http://adsabs.harvard.edu/abs/2002MNRAS.333..400C)) presents a method that uses non-linear optimization. However, this non-linear optimization method is computationally too expensive to implement within MCMC. Although, this method has been implemented to compute the kinematic observable while sampling from the lens model posterior (e.g., Birrer et al. [2019](http://adsabs.harvard.edu/abs/2018arXiv180901274B)). Computing the kinematic observable can involve at least one numerical integration, which is the main bottleneck in efficiency, not the non-linear optimization method for Gaussian decomposition. To make our lens-modelling method efficient, we require a **(i)** general, **(ii)** precise, and **(iii)** fast technique to decompose a function into concentric Gaussian components. In Sections [2.1](#scrollTo=bw1Bl8DWOSNv) and [2.2](#scrollTo=U-DGfxIw6q0g), we provide a technique that satisfies these three requirements.\n\n### 2.1 An integral transform for fast Gaussian decomposition\n\nNow, we introduce an integral transform with a Gaussian kernel. Using this transform, we obtain an algorithm to efficiently decompose an elliptical surface-density profile into concentric Gaussian components.\n\nWe start with the simple one-dimensional case of the integral transform. We aim to approximate a function $F(x)$ as a sum of Gaussian components as\n\n\\begin{equation} \\tag{2.2}\\label{22}\n\tF(x) \\approx \\sum_{n=0}^{N} A_n \\exp \\left( -\\frac{x^2}{2\\sigma_n^2} \\right),\n\\end{equation}\n\nwhere $A_n$ and $\\sigma_n$ are respectively the amplitude and the standard deviation of the $n$-th Gaussian component. We can convert this discrete summation into a continuous integral by taking $N \\to \\infty$. Accordingly, we define the following integral transform:\n\n\\begin{equation} \\tag{2.3} \\label{23}\n\tF(x) \\equiv \\int_0^\\infty \\frac{f(\\sigma)}{\\sqrt{2 \u03c0} \\sigma} \\exp\\left( -\\frac{x^2}{2\\sigma^2}\\right) \\mathrm{d} \\sigma.\n\\end{equation}\n\nHere, the amplitude $A_n$ is converted into a function $f(\\sigma)/\\sqrt{2 \u03c0}\\sigma$. We call $F(x)$ as the _transform_ of $f(\\sigma)$. We prove three fundamental properties of this integral transform in Appendix A. These three properties tell us that\n\n\n1. this integral transform exists for most mass and light profiles of practical use,\n2. the transform is unique for these functions, and\n3. the integral transform is invertible.\n\nWe call $f(\\sigma)$ as the _inverse transform_ of $F(x)$. The inverse transform is given by\n\n\\begin{equation} \\tag{2.4}\\label{24}\n\tf(\\sigma) = \\frac{1}{\\mathrm{i} \\sigma^2} \\sqrt{\\frac{2}{\u03c0}} \\int_{C} z F(z) \\exp \\left( \\frac{z^2}{2 \\sigma^2} \\right) \\mathrm{d} z.\n\\end{equation}\n\nHere, $\\mathrm{i}$ is the imaginary unit as $\\mathrm{i} = \\sqrt{-1}$. Also, we have extended $F(x)$ to some region on the complex plane and wrote it as $F(z)$, where $z$ is a complex variable. The contour $C$ for the integral lies within the region where $F(z)$ is defined (for details, see Appendix [A](#scrollTo=9qdjVclOfmVk) and Fig. [A1](#scrollTo=zBNYTOeevKkJ)). In Section [2.1.1](#scrollTo=vzcmlzwNAVdE), we provide an algorithm that does not require $C$ to be explicitly specified for computing the inverse transform $f(\\sigma)$.\n\nWe can use the inverse transform to decompose a function into concentric Gaussian components and the forward transform to recover the original function by combining the Gaussian components. First in Section [2.1.1](#scrollTo=vzcmlzwNAVdE), we first provide an efficient algorithm to compute the inverse transform from equation ([2.4](#mjx-eqn-24)); then in Section [2.1.2](#scrollTo=QSdpwV_JB_45), we discuss a method for computing the forward transform from equation ([2.3](#mjx-eqn-23)).\n\n#### 2.1.1 Computing the inverse transform\n\nThe integral transform in equation ([2.3](#mjx-eqn-23)) can be converted into a Laplace transform by suitable change of variables (Remark [A.9](#scrollTo=cmm1SJ9lvRYH)). Therefore, we can use any of the several algorithms available for inverse Laplace transform by appropriately changing the variables (for a simple overview of the algorithms, see Abate & Whitt [2006](http://pubsonline.informs.org/doi/10.1287/ijoc.1050.0137)). In this paper, we modify the Euler algorithm to approximate equation ([2.4](#mjx-eqn-24)) as\n\n\\begin{equation} \\tag{2.5}\\label{25}\n\tf(\\sigma) \\approx \\sum_{n=0}^{2P} \\eta_n \\textrm{Re} \\left[ F \\left( \\sigma \\chi_n \\right) \\right].\n\\end{equation}\n\n(Abate et al. [2000](https://doi.org/10.1007/978-1-4757-4828-4_8)). Here, the weights $\\eta_n$ and nodes $\\chi_n$ can be complex-valued and they are independent of $f(\\sigma)$. The weights and the nodes are given by\n\n\\begin{equation} \\tag{2.6}\\label{26}\n\\chi_n = \\left[\\frac{2P \\log (10)}{3} + 2 \u03c0 \\mathrm{i} n \\right]^{1/2}, \\\\\n\\eta_n = (-1)^n\\ 2\\sqrt{2 \u03c0}\\ 10^{P/3} \\xi_p, \\\\\n\\xi_0 = \\frac{1}{2},\\quad \\xi_n = 1,\\ 1\\leq n \\leq P,\\quad \\xi_{2P} = \\frac{1}{2^P}, \\\\\n\\xi_{2P-n} = \\xi_{2P-n+1} + 2^{-P} \\binom Pn,\\ 0 < n < P. \n\\end{equation}\n \n We can precompute the weights and the nodes just once before the MCMC sampling. In that way, computing them does not add any extra burden in computing the likelihood. The precision of the inverse transform is $\\sim \\mathcal{O}(10^{-0.6P})$ (Abate & Whitt [2006](http://pubsonline.informs.org/doi/10.1287/ijoc.1050.0137)). Therefore, the value of $P$ can be appropriately chosen to achieve a required precision. Note that the decimal precision of the machine sets an effective upper limit for $P$. For example, the precision will not improve with increasing $P$ when $P \\gtrsim 12$ for 32-bit floating point number, and when $P \\gtrsim 27$ for 64-bit floating point number. Thus, equation ([2.5](#mjx-eqn-25)) gives a straightforward, fast, and precise algorithm to compute the inverse transform.\n\n\n\n#### 2.1.2 Computing the forward transform\n\nLet us approximate the forward transform integral such that we can recover $F(x)$ from only a finite number of $f(\\sigma)$ values computed at fixed $\\sigma$'s. This finite number should be on the order of tens to keep the lensing analysis computationally feasible, as we have to compute lensing quantities for each Gaussian component individually. We write equation ([2.3](#mjx-eqn-23)) as\n\n\\begin{equation} \\tag{2.7}\\label{27}\n\tF(x) = \\frac{1}{\\sqrt{2 \u03c0}} \\int_0^{\\infty} f(\\sigma) \\exp \\left(-\\frac{x^2}{2 \\sigma^2} \\right) \\mathrm{d} (\\log \\sigma) \\\\\n\t\\Rightarrow F(x) = \\lim_{N \\to \\infty} \\sum_{n=1}^N \\frac{f(\\sigma_n)}{\\sqrt{2 \u03c0}} \\exp \\left(-\\frac{x^2}{2\\sigma_n^2} \\right) \\Delta(\\log\\sigma)_n \\\\\n\t\\Rightarrow F(x) \\approx \\sum_{n=1}^{N} A_n \\exp \\left(-\\frac{x^2}{2\\sigma_n^2} \\right).\n\\end{equation}\n\nWe have recovered the form of equation ([2.2](#mjx-eqn-22)) by taking logarithmically spaced $\\sigma_n$. Here, the amplitudes are $A_n = w_n f(\\sigma_n) \\Delta (\\log \\sigma)_n/\\sqrt{2\\pi}$. The weights $w_n$ depend on the choice of the numerical integration method. We use the trapezoidal method with weights $w_1 = 0.5$, $w_n = 1$ for $1 r_{\\rm s}), \\\\\n\t\t\t\\frac{2}{3} \\rho_{\\rm s} r_{\\rm s} & (R = r_{\\rm s}), \\\\\n\t\t\t\\frac{2 \\rho_{\\rm s} r_{\\rm s}}{(R/r_{\\rm s})^2 - 1}\n\\left[ 1- \\frac{ {\\rm sech}^{-1} {(R/r_{\\rm s}) } } {\\sqrt{1-(R/r_{\\rm s})^2}} \\right] & (R < r_{\\rm s})\n\t\t\\end{array}\n\t\\right.\n\\end{equation}\n\n(Bartelmann [1996](http://adsabs.harvard.edu/abs/1996A%26A...313..697B)). Here, $\\rho_{\\rm s}$ is the three-dimensional density normalization, and $r_{\\rm s}$ is the scale radius. The Sersic profile is given by\n\n\\begin{equation} \\tag{2.9}\\label{29}\n\t\\Sigma_{\\rm \\text{Sersic}}(R) = \\Sigma_{\\rm eff} \\exp\\left[ - b_n \\left\\{ (R/R_{\\rm eff})^{1/n_{\\rm \\text{Sersic}}} - 1 \\right\\} \\right]\n\\end{equation}\n\n(Sersic [1968](http://adsabs.harvard.edu/abs/1968adga.book.....S)). Here, the normalizing factor $b_n$ ensures that half of the total projected mass is contained within the effective radius $R_{\\rm eff}$. We only need to compute as many $f(\\sigma)$ values as the number of Gaussian components. We can appropriately choose this number to achieve the required precision for approximating the original function within a given range of $R$. In this example, we set $r_{\\rm s}=5R_{\\rm eff}$ and assume 1 per cent Poisson noise at $R = R_{\\rm eff}$. Then, we can approximate both the projected NFW profile and the Sersic function within the noise level with only 15 Gaussian components in the range $0.1R_{\\rm eff} \\leq R \\leq 10R_{\\rm eff}$. The standard deviations $\\sigma_n$ of the 15 Gaussians are logarithmically spaced between $0.005r_{\\rm s}$ and $50r_{\\rm s}$ for the NFW profile, and between $0.02R_{\\rm eff}$ and $15R_{\\rm eff}$ for the Sersic profile. Thus using the integral transform method, we can decompose a function into concentric Gaussian components within any required precision by appropriately choosing the component number $N$.\n\n\n```python\ninteract(plot_gauss_decompose, \n n_sersic=FloatSlider(min=1., max=6., step=0.5, value=1., continuous_update=False),\n N_sigma=IntSlider(min=10, max=20, value=15, continuous_update=False), \n sigma_start=FloatSlider(min=0.01, max=0.05, step=0.005, value=0.02, continuous_update=False), \n sigma_end=FloatSlider(min=10, max=20, step=1, value=15, continuous_update=False));\n\n#plot_gauss_decompose(n_sersic=1)\n```\n\n>**Figure 3.** Decomposing the Sersic profile and the projected NFW profile into concentric Gaussian components using the integral transform with a Gaussian kernel. The solid blue lines correspond to the Sersic profiles. The red, dashed lines correspond to the two-dimensional projected NFW profile. Both profiles are normalized to have $\\Sigma (R_{\\rm eff}) = 1$. **Left:** the inverse transform $f(\\sigma)$ of the Sersic profiles and the projected NFW profile. Here, we choose the NFW scale radius $r_{\\rm s} = 5R_{\\rm eff}$. To decompose a function into 15 Gaussian components, we only need to compute $f(\\sigma)$ at 15 points. These points are marked along the top border as blue ticks for the Sersic profiles and as red ticks for the NFW profile. **Center:** recovering the original profile as $\\Sigma_{\\rm approx}(R)$ using the forward transform by combining the 15 Gaussian components. We do not plot the true form of $\\Sigma_{\\rm \\text{Sersic}}(R)$ or $\\Sigma_{\\rm \\text{NFW}}(R)$, because they are visually almost indistinguishable from $\\Sigma_{\\rm approx}(R)$. **Right:** noise-normalized difference between the recovered profile $\\Sigma_{\\rm approx}(R)$ and the true form of $\\Sigma_{\\rm \\text{Sersic}}(R)$ or $\\Sigma_{\\rm \\text{NFW}}(R)$. We assume 1 per cent Poisson noise at $R = R_{\\rm eff}$ to obtain the noise level for normalizing the residual. Our method approximates the NFW profile and the Sersic profile as a sum of 15 Gaussians within the noise level for $0.1R_{\\rm eff} \\leq R \\leq 10R_{\\rm eff}$.\n\n>*(Interactive options are set only for the Sersic profile.)*\n\n#### 2.1.3 The integral transform method is more efficient than the MGE method.\n\nWe compare our method to decompose a one-dimensional function into concentric Gaussian components with the MGE method (Fig. [4](#scrollTo=a2aIUCNQnzqG)). We use both methods to decompose the Sersic function into 15 Gaussian components. The MGE method approximates the Sersic function within the noise level up to $\\sim 6R_{\\rm eff}$, whereas our method approximates the Sersic function within the noise level up to 10$R_{\\rm eff}$ with the same number of components. Albeit, we can increase the number of Gaussian components in the MGE method to reach the desired precision. In our method, the precision of the decomposition can be affected by both $P$ in equation ([2.5](#mjx-eqn-25)) and the number of Gaussians $N$ in equation ([2.7](#mjx-eqn-27)). However, if $P$ is appropriately chosen so that $10^{-0.6P}$ is sufficiently (e.g., by a factor of $\\sim 10^{-2}$–$10^{-4}$) smaller than the required precision, then the precision predominantly depends on $N$. For lensing and kinematic analyses, increasing the number of Gaussian components $N$ introduces more computational burden than increasing $P$. Therefore, it is advisable to first choose a sufficiently large $P$ and then adjust the number of Gaussians to achieve the required precision. Note that the real power of our method is in its efficiency. A `python` implementation of our method is $\\sim 10^3$ times faster than the MGE method to decompose a one-dimensional function into concentric Gaussian components with similar or better precision.\n\n\n```python\n# runtime of the integral transform method\n\n%%timeit -n100 -r5\n\n_ = decompose(sersic_func, N_sigma=15)\n```\n\n 100 loops, best of 5: 104 \u00b5s per loop\n\n\n\n```python\nfrom mgefit.mge_fit_1d import mge_fit_1d\n\nR = np.logspace(-1, 1, 200)\nsersic_R = sersic_func(R, n_sersic=1)\n```\n\n\n```python\n# runtime of the MGE method\n\n%%timeit -n10 -r5\n\n_ = mge_fit_1d(R, sersic_R, ngauss=15, plot=False, quiet=True)\n```\n\n 10 loops, best of 5: 716 ms per loop\n\n\n\n```python\ninteract(plot_mge_comparison, \n n_sersic=FloatSlider(min=1, max=6, step=0.5, value=1, continuous_update=False), \n N_sigma=IntSlider(min=10, max=20, value=15, continuous_update=False));\n\n#plot_mge_comparison()\n```\n\n>**Figure 4.** Comparison between our method and the multi-Gaussian expansion (MGE) method from Cappellari ([2002](http://adsabs.harvard.edu/abs/2002MNRAS.333..400C)) to decompose a one-dimensional Sersic function into concentric Gaussian components. **Left:** the Sersic function (solid, blue line) approximated with 15 Gaussian components using our Gaussian decomposition method. The dotted, lighter-blue lines show the individual Gaussian components. Some of the Gaussian components are out of the figure range. **Center:** same as the left figure but using the MGE method with 15 Gaussian components. The dashed, grey line shows the Sersic function approximated by MGE and the dot-dashed, lighter-grey lines show individual Gaussian components. **Right:** comparison of the noise-normalized residual for the two methods. We assume 1 per cent Poisson noise at effective radius $R_{\\rm eff}$ to obtain the noise level for normalizing the residual. The MGE method approximates the Sersic function within the noise level up to $\\sim 6 R_{\\rm eff}$, whereas our method approximates the Sersic function within the noise level up to $10R_{\\rm eff}$. More importantly, our method is $\\sim 10^3$ times faster than the MGE method to decompose a one-dimensional function into concentric Gaussian components.\n\n\n### 2.2 Decomposing a two-dimensional elliptical profile with the one-dimensional transform\n\nSo far, we have discussed the one-dimensional case of the integral transform; now we show that the one-dimensional transform is sufficient to decompose a two-dimensional elliptical profile. We can extend the one-dimensional integral transform from equation ([2.3](#mjx-eqn-23)) into a two-dimensional integral transform for a function $f(\\sigma_1, \\sigma_2)$ as\n\n\\begin{equation} \\tag{2.10}\\label{210}\n\tF(x, y) = \\int_0^\\infty \\mathrm{d} \\sigma_1 \\int_0^\\infty \\mathrm{d} \\sigma_2 \\ \\frac{f(\\sigma_1, \\sigma_2)}{2 \u03c0 \\sigma_1 \\sigma_2}\\ \\exp\\left( -\\frac{x^2}{2\\sigma_1^2} - \\frac{y^2}{2\\sigma_2^2} \\right).\n\\end{equation}\n\nIf $F(x,y)$ is elliptically symmetric, then we can express it as $F(R)$ in terms of the _elliptical radius_ $R=\\sqrt{q^2x^2+y^2}$ and axis ratio $q$. Then, we can write\n\n\\begin{equation} \\tag{2.11}\\label{211}\n\t\tF(x,y) = F\\left(R(q)\\right) = \\int_0^{\\infty} F\\left(R(\\varrho)\\right)\\ \\delta(\\varrho-q)\\ \\mathrm{d} \\varrho \\\\\n\t\t= \\int_0^{\\infty} \\mathrm{d} \\varrho \\ \\delta(\\varrho-q \\mathcal{}) \\int_0^{\\infty} \\mathrm{d} \\sigma\\ \\frac{f_y(\\sigma)}{\\sqrt{2\u03c0}\\sigma}\\ \\exp\\left(-\\frac{R(\\varrho)^2}{2\\sigma^2} \\right),\n\\end{equation}\n\nwhere $f_y(\\sigma)$ is the inverse transform of $F(0,y)$. If we make the change of variables $\\sigma_1 = \\sigma/\\varrho$, $\\sigma_2=\\sigma$, this integral becomes\n\n\\begin{equation} \\tag{2.12}\\label{212}\n\t\tF(x,y) = \\frac{1}{2\\pi} \\int_0^{\\infty} \\mathrm{d} \\sigma_1 \\int_0^{\\infty} \\mathrm{d} \\sigma_2 \\frac{\\sqrt{2\u03c0}q \\ \\delta(\\sigma_2/\\sigma_1-q)\\ f_y(\\sigma_2)}{\\sigma_1 \\sigma_2} \\\\\n\t\t\\qquad\\qquad\\qquad\\qquad \\times \\exp \\left(-\\frac{x^2}{2\\sigma_1^2}-\\frac{y^2}{2\\sigma_2^2} \\right).\n\\end{equation} \n\nBecause of the uniqueness property (Theorem [A.7](#scrollTo=ls5kM1XABtfj)), comparing equations ($\\ref{210}$) and ($\\ref{212}$) we can write\n\n\\begin{equation} \\tag{2.13}\\label{213}\n\tf(\\sigma_1, \\sigma_2) = \t\\sqrt{2\u03c0}q \\ \\delta\\left(\\frac{\\sigma_2}{\\sigma_1}-q \\right)f_y(\\sigma_2).\n\\end{equation}\n\nTherefore, for an elliptically symmetric function, it is sufficient to numerically compute the one-dimensional inverse transform $f_y(\\sigma_2)$ along the $y$-axis. As a result, we can express a two-dimensional elliptical function as a sum of elliptical Gaussian components as\n\n\\begin{equation} \\tag{2.14}\\label{214}\n\tF(x, y) \\approx\t\\sum_{n=1}^{N} A_n \\exp \\left( -\\frac{q^2 x^2 + y^2}{2\\sigma_n^2} \\right).\n\\end{equation}\n\nBy now, we have shown that the integral transform method meets all of our three requirements for decomposing a function into concentric Gaussian components:\n\n- **generality:** the method applies to most mass and light profiles of practical use,\n- **precision:** the method achieves any required precision over a given range by appropriately choosing the number of Gaussian components, and\n- **efficiency:** the method runs approximately $\\sim 10^3$ times faster than the previously available method.\n\nWith these three requirements met, our lensing analysis method has cleared the first hurdle to be feasible in practice. In Section [4.2](#scrollTo=slP9xat7i8YZ), we show that we can efficiently compute the gradient and the Hessian of the deflection potential for an elliptical Gaussian surface-density. With that, our method also clears the final hurdle to be efficient. Next in Section [3](#scrollTo=alI-uL1t9HSR), we summarize the kinematic analysis for the Gaussian components from Cappellari ([2008](http://adsabs.harvard.edu/abs/2008MNRAS.390...71C)). Then in Section [4](#scrollTo=TzkbBKWhcIbn), we describe the lensing analysis for the Gaussian components and complete the unification of lensing and kinematic descriptions.\n\n\n## 3 Kinematics of Gaussian components\n\nCappellari ([2008](http://adsabs.harvard.edu/abs/2008MNRAS.390...71C)) presents the Jeans anisotropic modelling of kinematics for a mass profile decomposed into concentric Gaussian components. We summarize the analysis here to complete our unified framework. The kinematic observable is the luminosity-weighted, line-of-sight velocity dispersion. The velocity dispersion can be an integrated measurement within a single aperture or it can be spatially resolved on the plane of the sky. To compute this quantity for a combination of mass and light profiles, we need to solve the Jeans equations. We can decompose the surface mass-density profile into concentric Gaussian components as\n\n\\begin{equation} \\tag{3.1}\\label{31}\n\t\\Sigma (x, y) \\approx \\sum_{j=1}^J\t{\\Sigma_{0}}_{j} \\exp \\left( - \\frac{q_j^2 x^2 + y^2}{2 \\sigma_j^2} \\right),\n\\end{equation}\n\nand decompose the surface brightness profile into concentric Gaussian components as\n\n\\begin{equation} \\tag{3.2}\\label{32}\n\tI (x, y) \\approx \\sum_{k=1}^K\t{I_{0}}_{k} \\exp \\left( - \\frac{q_k^2 x^2 + y^2}{2 \\sigma_k^2} \\right).\n\\end{equation}\n\nHere, we use different subscript letters to make the context of the Gaussian decomposition clear: we use the subscript $j$ for a component of the mass profile and the subscript $k$ for a component of the light profile. We have also allowed different ellipticity for each Gaussian component represented by $q_j$ or $q_k$. This is the most general case, for example, when structures with different ellipticities constitute the total mass or light distribution. We first need to deproject these two-dimensional profiles into three-dimension as the kinematic quantities depend on the three-dimensional distributions of mass and light. We assume axisymmetry or spherical symmetry for the deprojected three-dimensional structure to circumvent the infinite degeneracy in deprojection. First, we provide the kinematic analysis for the axisymmetric case in Section [3.1](#scrollTo=1rgbrAIw-HmT); then we do the same for the simpler case of spherical symmetry in Section [3.2](#scrollTo=orW-XSQB_wa0).\n\n\n### 3.1 Axisymmetric case\n\nFor an axisymmetric system, the cylindrical coordinates $(R^{\\prime}, z^{\\prime}, \\phi^{\\prime})$ are the most suitable to express the Jeans equations. We use the prime symbol to denote the coordinates in the system's _symmetry-frame_, where the $z^{\\prime}$-axis aligns with the axis of symmetry. We assign $(x, y, z)$ coordinates to the _sky frame_, where the $z$-axis aligns with the line of sight and the $x$-axis aligns with the projected major axis. If the galaxy is inclined by an angle $\\iota$, then the $(x^{\\prime}, y^{\\prime}, z^{\\prime})$ coordinates in the symmetry frame relate to the the $(x, y, z)$ coordinates in the sky frame as\n\n\\begin{equation} \\tag{3.3}\\label{33}\n\t\\begin{pmatrix} x^{\\prime} \\\\ y^{\\prime} \\\\ z^{\\prime}\t\\end{pmatrix} \n\t= \n\t\\begin{pmatrix}\n\t\t1 &0 &0 \\\\ 0 &\\cos \\iota &-\\sin \\iota \\\\ 0 &\\sin \\iota & \\cos \\iota \n\t\\end{pmatrix}\n\t\\begin{pmatrix} x \\\\ y \\\\ z\t\\end{pmatrix} .\n\\end{equation}\n\nIn the symmetry frame, equation ([3.1](#mjx-eqn-31)) deprojects into the mass density profile $\\rho$ as\n\n\\begin{equation} \\tag{3.4}\\label{34}\n\t\\rho(R^{\\prime}, z^{\\prime}) = \\sum_{j=1}^J \t\\frac{{q_j^{\\prime}}^2 {\\Sigma_{0}}_{j}}{\\sqrt{2\u03c0} \\sigma_j q_j} \\exp \\left( - \\frac{{q_j^{\\prime}}^2 {R^{\\prime}}^2 + {z^{\\prime}}^2}{2 \\sigma_j^2} \\right),\n\\end{equation}\n\nand equation ([3.2](#mjx-eqn-32)) deprojects into the light density profile $l$ as \n\n\\begin{equation} \\tag{3.5}\\label{35}\n\tl (R^{\\prime}, z^{\\prime}) = \\sum_{k=1}^K\t\\frac{{q_k^{\\prime}}^2 {I_0}_{k}}{\\sqrt{2\u03c0} \\sigma_k q_k} \\exp \\left( - \\frac{{q_k^{\\prime}}^2 {R^{\\prime}}^2 + {z^{\\prime}}^2}{2 \\sigma_k^2} \\right).\n\\end{equation}\n\nHere, the intrinsic axis ratio $q^{\\prime}$ relates to the projected axis ratio $q$ as\n\n\\begin{equation} \\tag{3.6}\\label{36}\n\tq^{\\prime} = \\frac{\\sqrt{q^2 - \\cos^2 \\iota}}{\\sin \\iota}.\t\n\\end{equation}\n\nWe can first solve the Jeans equations for these mass and light density profiles in the symmetry frame to get the intrinsic velocity dispersions and then integrate along the line of sight to obtain the line-of-sight velocity dispersion.\n \nThe Jeans equations to solve for an axisymmetric system are \n\n\\begin{equation} \\tag{3.7}\\label{37}\n\t\\frac{b\\ l\\ \\overline{v_{z^{\\prime}}^2} - l\\ \\overline{v_{\\phi^{\\prime}}^2}}{R^{\\prime}} + \\frac{\\partial \\left( b\\ l\\ \\overline{v_{z^{\\prime}}^2} \\right)}{\\partial R^{\\prime}} = - l \\frac{\\partial \\Phi}{\\partial R^{\\prime}}, \\\\\n\t\\frac{\\partial \\left( l\\ \\overline{v_{z^{\\prime}}^2} \\right)}{\\partial z^{\\prime}} = - l \\frac{\\partial \\Phi}{\\partial z^{\\prime}}.\n\\end{equation}\n\nHere, the gravitational potential $\\Phi$ relates to the three-dimensional mass density $\\rho$ by $\\nabla^2 \\Phi = \\rho$, and $b$ represents the anisotropy as in $\\overline{v_{R^{\\prime}}^2} = b \\overline{v_{z^{\\prime}}^2}$. We can let $b_k$ for each luminous Gaussian component have different values to approximate the luminosity-weighted anisotropy parameter as\n\n\\begin{equation} \\tag{3.8}\\label{38}\n\t\\beta_{z^{\\prime}} (R^{\\prime}, z^{\\prime}) \\equiv 1 - \\frac{\\overline{v_{z^{\\prime}}^2}}{\\overline{v_{R^{\\prime}}^2}} \\approx 1 - \\frac{\\sum_k l_k}{\\sum_k b_k l_k}\n\\end{equation}\n\n(Binney & Mamon [1982](http://adsabs.harvard.edu/abs/1982MNRAS.200..361B), Cappllari [2008](http://adsabs.harvard.edu/abs/2008MNRAS.390...71C)). The line-of-sight second velocity moment for total mass profile obtained from solving the Jeans equations is given by\n\n\\begin{equation} \\tag{3.9}\\label{39}\n\t\\overline{v_{\\rm los}^2}\t (x, y) = 2 \\sqrt{\u03c0} G \\int_0^1 \\sum_{j=1}^J \\sum_{k=1}^K \\frac{{q_j^{\\prime}}^3 {q_k^{\\prime}}^2 {\\Sigma_0}_j {I_0}_k u^2}{\\sigma_j q_j \\sigma_k q_k} \\\\\n\t\\times \\frac{\\sigma_k^2\\left( \\cos^2 \\iota + b_k \\sin^2 \\iota \\right) + \\mathcal{D}x^2 \\sin^2 \\iota}{\\left(1-\\mathcal{C}u^2 \\right)\\sqrt{\\left(\\mathcal{A} + \\mathcal{B} \\cos^2 \\iota \\right) \\left[1 - \\left(1-{q_j^{\\prime}}^2\\right)u^2 \\right]}} \\\\\n\t \\times \\exp \\left(-\\mathcal{A} \\left[x^2 + \\frac{(\\mathcal{A}+\\mathcal{B})y^2}{\\mathcal{A}+\\mathcal{B}\\cos^2 \\iota} \\right] \\right) \\mathrm{d} u,\n\\end{equation}\n\nwhere $G$ is the gravitational constant and\n\n\\begin{equation} \\tag{3.10}\\label{310}\n\t\t\\mathcal{A} = \\frac{1}{2} \\left( \\frac{u^2 {q_j^{\\prime}}^2}{\\sigma_j^2} + \\frac{{q_k^{\\prime}}^2}{\\sigma_k^2} \\right), \\\\\n\t\t\\mathcal{B} = \\frac{1}{2}\\left\\{ \\frac{1-{q_k^{\\prime}}^2}{\\sigma_k^2} + \\frac{{q_j^{\\prime}}^2\\left(1-{q_j^{\\prime}}^2\\right)u^4}{\\sigma_j^2 \\left[ 1- \\left(1-{q_j^{\\prime}}^2\\right) u^2\\right] } \\right\\},\\\\\n\t\t\\mathcal{C} = 1 - {q_j^{\\prime}}^2 - \\frac{{q_j^{\\prime}}^2\\sigma_k^2}{\\sigma_j^2},\\\\\n\t\t\\mathcal{D} = 1 - b_k{q_k^{\\prime}}^2 - \\left[(1-b_k)\\mathcal{C} + (1-{q_j^{\\prime}}^2)b_k \\right] u^2\n\\end{equation}\n\n(Cappellari [2008](http://adsabs.harvard.edu/abs/2008MNRAS.390...71C)). The line-of-sight velocity dispersion $\\sigma_{\\rm los}$ relates to the second velocity moment by $\\overline{v^2_{\\rm los}} = v_{\\rm mean}^2 + \\sigma^2_{\\rm los}$, where $v_{\\rm mean}$ is the stellar mean velocity.\n\n### 3.2 Spherical case\n\nIf we assume the system is spherically symmetric, then the spherical coordinates $(r, \\phi, \\theta)$ are the most suitable to express the Jeans equations. In this coordinate system, the mass density profile deprojected from equation ([3.1](#mjx-eqn-31)) takes the form \n\n\\begin{equation} \\tag{3.11}\\label{311}\n\t\\rho(r) = \\sum_{j=1}^J \\frac{{\\Sigma_{0}}_j }{\\sqrt{2 \u03c0} \\sigma_j q_j} \\exp \\left( -\\frac{r^2}{2 \\sigma_j^2}\\right),\n\\end{equation}\n\nand the light density profile deprojected from equation ([3.2](#mjx-eqn-32)) turns into\n\n\\begin{equation} \\tag{3.12}\\label{312}\n\tl(r) = \\sum_{k=1}^K \\frac{{I_{0}}_k}{\\sqrt{2 \u03c0} \\sigma_k q_k} \\exp \\left( -\\frac{r^2}{2 \\sigma_k^2}\\right).\n\\end{equation}\n\nThe projected axis ratio $q$ shows up in these equations to keep the total mass and luminosity conserved. We can express the three-dimensional enclosed mass for this density profile as\n\n\\begin{equation} \\tag{3.13}\\label{313}\n\tM(r) = \\sum_{j=1}^J \\frac{2\u03c0 \\sigma_j^2 {\\Sigma_0}_j}{q_j} \\left[ \\mathrm{erf} \\left( \\frac{r}{\\sqrt{2}\\sigma_j} \\right) - \\sqrt{\\frac{2}{\u03c0}} \\frac{r}{\\sigma_j} \\exp \\left( - \\frac{r^2}{2 \\sigma_j^2} \\right) \\right],\n\\end{equation}\n\nwhere $\\mathrm{erf}\\ (x)$ is the error function. The spherical Jeans equation is\n\n\\begin{equation} \\tag{3.14}\\label{314}\n\t\\frac{\\mathrm{d} \\left( l\\ \\overline{v_r^2}\\right)}{\\mathrm{d} r} + \\frac{2 \\beta\\ l\\ \\overline{v_r^2}}{r} = - l \\frac{\\mathrm{d} \\Phi}{\\mathrm{d} r},\t\n\\end{equation}\n\nwhere $\\beta(r)$ is the anisotropy parameter given by \n\n\\begin{equation} \\tag{3.15}\\label{315}\n\t\\beta(r) = 1 - {\\overline{v_{\\theta}^2}}/{\\overline{v_{r}^2}}.\n\\end{equation}\n\nSpherical symmetry imposes that $\\overline{v_{\\theta}^2} = \\overline{v_{\\phi}^2}$. By solving the Jeans equation for the spherically symmetric case, we can obtain the line-of-sight second velocity moment as\n\n\\begin{equation} \\tag{3.16}\\label{316}\n\t\\overline{v_{\\rm los}^2} (x, y) = \\frac{2G}{I(x, y)} \\int_{\\sqrt{x^2+y^2}}^\\infty \\mathcal{K}_\\beta \\left(\\frac{r}{\\sqrt{x^2+y^2}} \\right)\\ l(r)\\ M(r)\\ \\frac{\\mathrm{d} r}{r}\n\\end{equation}\n\n(Mamon & Lokas [2005](http://adsabs.harvard.edu/abs/2005MNRAS.363..705M)). Here, the function $\\mathcal{K}_\\beta (\\upsilon)$ depends on the form of the anisotropy parameter $\\beta(r)$. For the isotropic case with $\\beta = 0$, the function $\\mathcal{K}_\\beta$ shapes into\n\n\\begin{equation} \\tag{3.17}\\label{317}\n\t\\mathcal{K}_{\\beta} (\\upsilon) = \\sqrt{1 - \\frac{1}{\\upsilon^2}}.\n\\end{equation}\n\nFor the Osipkov-Merritt parameterization $\\beta(r) = r^2/(r^2 + r_{\\rm ani}^2)$, where $r_{\\rm ani}$ is a scale radius, the function $\\mathcal{K}_{\\beta}$ takes the form\n\n\\begin{equation} \\tag{3.18}\\label{318}\n\t\\mathcal{K}_{\\beta} \\left( \\upsilon \\right) = \\frac{\\upsilon_{\\rm ani}^2 + 1/2}{(\\upsilon_{\\rm ani}+1)^{3/2}} \\left( \\frac{\\upsilon^2 + \\upsilon^2_{\\rm ani}}{\\upsilon} \\right) \\tan^{-1} \\left(\\sqrt{\\frac{\\upsilon^2 - 1}{\\upsilon_{\\rm ani}^2 + 1}}\\right) \\\\\n\t \\qquad\\qquad\\qquad\\qquad - \\frac{1/2}{\\upsilon_{\\rm ani}^2 + 1} \\sqrt{1 - \\frac{1}{\\upsilon^2}}\n\\end{equation}\n\nwith $\\upsilon_{\\rm ani} = r_{\\rm ani}/\\sqrt{x^2 + y^2}$ (Osipkov [1979](http://adsabs.harvard.edu/abs/1979PAZh....5...77O), Merritt [1985a](http://adsabs.harvard.edu/abs/1985MNRAS.214P..25M), [b](http://adsabs.harvard.edu/abs/1985AJ.....90.1027M)). See equation (A16) of Mamon & Lokas ([2005](http://adsabs.harvard.edu/abs/2005MNRAS.363..705M)) for the form of $\\mathcal{K}_{\\beta}$ corresponding to other parameterizations of $\\beta(r)$. When assuming spherical symmetry is sufficient, we can use equation ([3.16](#mjx-eqn-316)) to compute the line-of-sight velocity dispersion in a much simpler way than the axisymmetric case \\[cf. equation ([3.9](#mjx-eqn-3.9))\\].\n\nThe kinematic description of an elliptical mass distribution by decomposing it into concentric Gaussian components is thus well developed in the literature. In the next section, we unify the lensing description with the kinematic description under the same framework.\n\n## 4 Lensing by Gaussian components\n\nIn this section, we present the lensing analysis for an elliptical surface-density profile decomposed into concentric Gaussian components. In Section 2, we introduced an integral transform that efficiently decomposes an elliptical surface-density profile into Gaussian components as\n\n\\begin{equation} \\tag{4.1}\\label{41}\n\t\\Sigma (x, y) \\approx \\sum_{j=1}^J\t{\\Sigma_{0}}_{j} \\exp \\left( - \\frac{q_j^2 x^2 + y^2}{2 \\sigma_j^2} \\right)\t.\n\\end{equation}\n\nWe can compute a lensing quantity for each individual Gaussian component, and then linearly add the contributions from all the components to obtain the total lensing quantity. For example, if $\\mathbf{\u03b1}_j(x, y)$ is the deflection at position $(x, y)$ for the $j$-th Gaussian component, then the total deflection is simply given by $\\mathbf{\u03b1} (x, y) = \\sum_{j=1}^J \\mathbf{\u03b1}_j (x, y)$. Therefore, it is sufficient to analyze the lensing properties of one elliptical Gaussian component. We use the complex formulation of lensing to solve the deflection integral for an elliptical Gaussian surface-density profile. Below, we first lay out the complex formalism of lensing in Section [4.1](#scrollTo=wACcjB-yg4s3); then we study the lensing properties of an elliptical Gaussian surface-density profile in Section [4.2](#scrollTo=slP9xat7i8YZ).\n\n### 4.1 Complex formulation of lensing\n\nThe strong lensing effect is usually described using the vector formulation on the two-dimensional image plane. We first define the lensing quantities in the familiar vector formulation, then we translate them to the complex formulation. The convergence $\\kappa$ is a dimensionless surface-density defined as $\\kappa \\equiv \\Sigma/\\Sigma_{\\rm crit}$, where the critical density $\\Sigma_{\\rm crit}$ is given by\n\n\\begin{equation} \\tag{4.2}\\label{42}\n\t\\Sigma_{\\rm crit} = \\frac{c^2 D_{\\rm s}}{4 \u03c0 G D_{\\rm ds} D_{\\rm d}}.\t\n\\end{equation}\n\nHere, $c$ is the speed of light. The three angular diameter distances are $D_{\\rm d}$: between the observer and the deflector, $D_{\\rm s}$: between the observer and the source, and $D_{\\rm ds}$: between the deflector and the source. The convergence $\\kappa$ relates to the vector deflection angle ${\\mathbf{\u03b1}}$ as $\\kappa = \\nabla \\cdot \\mathbf{\u03b1} / 2$. The deflection angle $\\mathbf{\u03b1}$ is the gradient of the deflection potential as $\\mathbf{\u03b1} = \\nabla \\psi$, thus the convergence $\\kappa$ relates to the deflection potential $\\psi$ as $\\kappa = \\nabla^2 \\psi / 2$. The Hessian matrix of the deflection potential is\n\n\n\\begin{equation} \\tag{4.3}\\label{43}\n\t\\mathsf{H} = \\begin{pmatrix}\n \t\t\t\\dfrac{\\partial^2 \\psi}{\\partial^2 x} &\\dfrac{\\partial^2 \\psi}{\\partial x \\partial y} \\\\\n \t\t\t\\dfrac{\\partial^2 \\psi}{\\partial x \\partial y} &\\dfrac{\\partial^2 \\psi}{\\partial^2 y}\n \t\t\\end{pmatrix}.\n\\end{equation}\n\nThe convergence $\\kappa$, the shear parameters $(\\gamma_1, \\gamma_2)$, and the magnification $\\mu$ relate to the Hessian, since we can express them as\n\n\\begin{equation} \\tag{4.4}\\label{44}\n\t\t\\kappa = \\frac{1}{2}\\left(\\frac{\\partial^2 \\psi}{\\partial^2 x} + \\frac{\\partial^2 \\psi}{\\partial^2 y} \\right), \\\\\n\t\t\\gamma_1 = \\frac{1}{2} \\left(\\frac{\\partial^2 \\psi}{\\partial^2 x} - \\frac{\\partial^2 \\psi}{\\partial^2 y} \\right), \\\\\n\t\t\\gamma_2 = \\frac{\\partial^2 \\psi}{\\partial x \\partial y}, \\\\\n \\mu = \\frac{1}{\\det\\ ({\\mathsf{I}} - {\\mathsf{H}})},\n\\end{equation}\n\nwhere $\\mathsf{I}$ is the identity matrix. Thus, if we start with a convergence $\\kappa$ and derive the deflection $\\mathbf{\u03b1}$ and the shear parameters $(\\gamma_1, \\gamma_2)$, then we know the gradient and the Hessian of the deflection potential.\n\nNow we reformulate the lensing quantities on the complex plane. Following Bourassa et al. ([1973](http://adsabs.harvard.edu/abs/1973ApJ...185..747B)), we can express the deflection vector $\\mathbf{\u03b1}$ as a complex quantity\n\n\\begin{equation} \\tag{4.5}\\label{45}\n\t\\alpha(z) \\equiv \\alpha_x + \\mathrm{i} \\alpha_y,\t\n\\end{equation}\n\nwhere the complex quantity $z = x + \\mathrm{i}y$ corresponds to the position vector $\\mathbf{r} = (x, y)$. We can define a complex deflection potential $\\psi(z)$ with its real part equal to the usual deflection potential (Schramm [1990](http://adsabs.harvard.edu/abs/1990A%26A...231...19S)). Then, the complex deflection angle is the Wirtinger derivative of the deflection potential as\n\n\\begin{equation} \\tag{4.6}\\label{46}\n\t\\alpha (z) = \\frac{\\partial \\psi}{\\partial x} + \\mathrm{i} \\frac{\\partial \\psi}{\\partial y} = 2 \\frac{\\partial \\psi}{\\partial z^*}.\n\\end{equation}\n\nWe can express the convergence $\\kappa$ as\n\n\\begin{equation} \\tag{4.7}\\label{47}\n\t\\kappa = \\frac{\\partial \\alpha^*}{\\partial z^*}.\n\\end{equation}\n\nFurthermore, the complex shear $\\gamma \\equiv \\gamma_1 + \\mathrm{i} \\gamma_2$ satisfies the relation\n\n\\begin{equation} \\tag{4.8}\\label{48}\n\t\\gamma^* = \\frac{\\partial \\alpha^*}{\\partial z}.\n\\end{equation}\n\nUsing this complex formulation, we analyze the lensing properties of an elliptical Gaussian convergence next in Section [4.2](#scrollTo=slP9xat7i8YZ). \n\n\n### 4.2 Lensing by elliptical Gaussian convergence\n\nWe derive the deflection angle and shear for the elliptical Gaussian convergence\n\n\\begin{equation} \\tag{4.9}\\label{49}\n\t\\kappa(R) = \\kappa_0 \\exp \\left( -\\frac{R^2}{2 \\sigma^2}\\right),\n\\end{equation}\n\nwhere $R=\\sqrt{q^2x^2 + y^2}$ is the elliptical radius. Using the complex formulation, the deflection angle for the elliptical convergence can be obtained from\n\n\\begin{equation} \\tag{4.10}\\label{410}\n\t\\alpha^*(z) = 2\\ \\rm{sgn}(z) \\ \\int_0^{R(z)} \\mathrm{d} \\zeta \\frac{\\zeta \\kappa(\\zeta)}{\\sqrt{q^2 z^2 - (1-q^2) \\zeta^2}} \\\\ \n = \\frac{2 \\kappa_0 \\ }{ q z} \\int_0^{R(z)} \\mathrm{d} \\zeta \\frac{\\zeta \\exp (-\\zeta^2/2\\sigma^2)}{\\sqrt{1 - {(1-q^2)\\zeta^2}/{q^2 z^2}}},\n\\end{equation}\n\nwhere $\\textrm{sgn}(z) \\equiv \\sqrt{z^2}/z$ is the complex sign function, and $R(z) = \\sqrt{q^2x^2 + y^2}$ is the semi-minor axis length for the ellipse with axis-ratio $q$ that goes through the point $z=x+iy$ (Bourassa & Kantowski [1975](http://adsabs.harvard.edu/abs/1975ApJ...195...13B), Bray [1984](http://adsabs.harvard.edu/abs/1984MNRAS.208..511B)). With changes of variables $s=1/2\\sigma^2$, $t = (1-q^2)/q^2 z^2$, $\\tau = \\sqrt{1 - t\\zeta^2}$, we can express equation ([4.10](#mjx-eqn-410)) as\n\n\\begin{equation} \\tag{4.11}\\label{411}\n\\alpha^*(z) = \\frac{2 \\kappa_0 \\ \\mathrm{e}^{-s/t}}{q z t} \\int_{\\sqrt{1-tR(z)^2}}^{1} \\mathrm{d} \\tau \\exp \\left( \\frac{s}{t} \\tau^2 \\right) \\\\\n = \\frac{2\\kappa_0 \\ \\mathrm{e}^{-s/t}}{ q z t} \\left[\\frac{1}{2} \\sqrt{\\frac{\u03c0 t}{s}}\\ \\mathrm{erfi} \\left( \\sqrt{\\frac{s}{t}} \\tau \\right) \\right]_{\\sqrt{1-t(q^2x^2 + y^2)}}^{1} \\\\\n= \\kappa_0 \\sigma \\sqrt{\\frac{2 \u03c0}{1-q^2}} \\exp \\left( -\\frac{q^2 z^2}{2\\sigma^2(1 - q^2)}\\right) \\left[ \\mathrm{erfi} \\left( \\frac{q z}{\\sigma \\sqrt{2(1-q^2)}} \\right) \\right. \\\\\n\t\t \\qquad \\qquad \\qquad \\qquad - \\left. \\mathrm{erfi} \\left( \\frac{ q^2 x + i y}{\\sigma \\sqrt{2(1-q^2)}} \\right) \\right] \\\\\n\t\t= \\kappa_0 \\sigma \\sqrt{\\frac{2\u03c0}{1-q^2}}\\ \\varsigma(z;q),\n\\end{equation}\n\nwhere $\\mathrm{erfi} \\ (z) \\equiv -\\mathrm{i}\\ \\mathrm{erf} \\ (\\mathrm{i}z)$ and we have defined the function\n\n\\begin{equation} \\tag{4.12}\\label{412}\n\t\\varsigma (z; q) \\equiv \\exp \\left( -\\frac{q^2 z^2}{2\\sigma^2(1 - q^2)}\\right) \\left[ \\mathrm{erfi} \\left( \\frac{q z}{\\sigma \\sqrt{2(1-q^2)}} \\right) \\right. \\\\\n\t\t\\qquad \\qquad \\qquad \\qquad - \\left. \\mathrm{erfi} \\left( \\frac{ q^2 x + \\mathrm{i} y}{\\sigma \\sqrt{2(1-q^2)}} \\right) \\right].\n\\end{equation}\n\nWe obtain the complex conjugate of the complex shear from equation ([4.8](#mjx-eqn-48)) as\n\n\\begin{equation} \\tag{4.13}\\label{413}\n\t\\gamma^*(z) = -\\frac{\\kappa_0} {1-q^2} \\left[ (1+q^2) \\exp \\left( - \\frac{q^2x^2+y^2} {2\\sigma^2} \\right) \\right. - 2q \\\\ \n\t\\quad\\ + \\frac{\\sqrt{2 \u03c0} q^2 z}{\\sigma\\sqrt{1-q^2}} \\exp \\left(-\\frac{q^2 z^2}{2\\sigma^2(1-q^2)}\\right) \\left\\{ \\mathrm{erfi} \\left( \\frac{qz} {\\sigma\\sqrt{2(1-q^2)}}\\right) \\right. \\\\\n\t\\quad \\left. \\left. - \\mathrm{erfi} \\left( \\frac{q^2x + \\mathrm{i} y} {\\sigma\\sqrt{2(1-q^2)}}\\right) \\right\\} \\right] \\\\\n\t= -\\frac{1}{1-q^2} \\left[ (1+q^2) \\kappa(x, y) - 2q\\kappa_0 + \\frac{\\sqrt{2\u03c0} q^2\\kappa_0 z}{\\sigma \\sqrt{1-q^2}}\\ \\varsigma(z;q) \\right] .\n\\end{equation}\n\nBoth the deflection angle and the shear contain the function $\\varsigma(z;q)$. This function relates to the Faddeeva function $w_{\\rm F}(z)$. First, we write the function $\\varsigma(z;q)$ as\n\n\\begin{equation} \\tag{4.14}\\label{414}\n\t\\varsigma(z;q) = \\varpi \\left(\\frac{q z}{\\sigma \\sqrt{2(1-q^2)}}; 1\\right) - \\varpi \\left(\\frac{q z}{\\sigma \\sqrt{2(1-q^2)}}; q \\right)\n\\end{equation}\n\nwhere $\\varpi (z;q) = \\exp\\left(-z^2 \\right)\\ \\rm{erfi} \\left(qx + iy/q \\right)$. We can express $\\varpi(z;q)$ using the Faddeeva function $w_{\\rm F}(z)$ as\n\n\\begin{equation} \\tag{4.15}\\label{415}\n\t\\varpi (z;q) = \t\\mathrm{e}^{-x^2 - 2\\mathrm{i}xy} \\mathrm{e}^{y^2} - \\mathrm{i} \\exp\\left[ -x^2 (1-q^2) - y^2 (1/q^2 -1) \\right] \\\\\n\t \\qquad \\times w_{\\rm F}(q x + \\mathrm{i} y/q).\n\\end{equation}\n\nThus, we can compute the deflection angle and the shear using the Faddeeva function (Fig. [5](#scrollTo=MtnsSzoDteie)). Faddeeva function is a well-studied special function for its various applications in physics, for example, in radiative transfer and in plasma physics (Armstrong [1967](http://adsabs.harvard.edu/abs/1967JQSRT...7...61A), Jimenez-Dominguez et al. [1989](http://adsabs.harvard.edu/abs/1989NIMPA.278..625J)). We can readily compute $w_{\\rm F}(z)$ in `python` using the `scipy.special.wofz` function. For some other popular programming languages, code-packages to compute this function are available at the web-address [http://ab-initio.mit.edu/Faddeeva](http://ab-initio.mit.edu/Faddeeva). In this paper, we use the algorithm outlined by Zaghloul ([2017](http://doi.acm.org/10.1145/3119904)) to compute $w_{\\rm F}(z)$ with relative error less than $4\\times10^{-5}$ over the whole complex plane. We state this algorithm in Appendix [B](#scrollTo=1LZhVuG8ft0R). A `python` implementation of this algorithm is approximately twice as fast as the function provided by `scipy`. As a result, we can efficiently compute the gradient and Hessian of the deflection potential for an elliptical Gaussian convergence using equations ([4.11](#mjx-eqn-411)) and ([4.12](#mjx-eqn-412)).\n\n\n```python\ninteract(plot_gaussian_lens, \n amplitude=FloatSlider(value=2., min=1., max=5, step=.5, continuous_update=False), \n \u03c3=FloatSlider(value=1., min=.5, max=2, step=.1, continuous_update=False),\n q=FloatSlider(value=.5, min=.05, max=1., step=.05, continuous_update=False));\n\n#plot_gaussian_lens()\n```\n\n>**Figure 5.** Lensing quantities for an elliptical Gaussian convergence profile. **Left:** convergence (orange shade), deflection field (green arrows), isopotential contours (blue, dashed contours). The arrow directions are for the negative of the deflection angles and the lengths are shrunk by a factor of 4 for nicer visualization. **Right:** Critical curves (black lines) and corresponding caustics (pink lines). The solid-contour caustic corresponds to the solid-contour critical curve, similarly dot-dashed contours correspond to each other. We express the gradient and the Hessian of the deflection potential for an elliptical Gaussian convergence using the complex error function, as a result we can efficiently compute them.\n\nNow, we turn our attention to computing the deflection potential. The deflection potential $\\psi (z)$ is given by\n\n\\begin{equation} \\tag{4.16} \\label{416}\n\t\\psi(z) = \\mathrm{Re} \\left(\\int^z_{0} \\alpha^*(z^\\prime)\\ \\mathrm{d} z^\\prime \\right),\n\\end{equation}\n\nwhere we set $\\psi(0) = 0$. Often times we are interested in the potential difference between two points $z_1$ and $z_2$ given by\n\n\\begin{equation} \\tag{4.17}\\label{417}\n\t\\Delta \\psi = \\psi (z_2) - \\psi (z_1) = \\mathrm{Re} \\left( \\int_{z_1}^{z_2} \\alpha^*(z^\\prime)\\ \\mathrm{d} z^\\prime \\right).\n\\end{equation}\n\nThis integral is independent of the choice of a contour. We have to carry out this integral numerically. However, the number of times we need to compute it in most applications, e.g., for computing time delays, is much fewer than that for $\\alpha(z)$. We can also numerically solve the Poisson equation $\\nabla^2 \\psi = 2\\kappa$ using the Fourier transform of the deflection potential $\\hat{\\psi} \\equiv \\mathcal{F}[\\psi]$ (van de Ven [2009](http://adsabs.harvard.edu/abs/2009MNRAS.398..607V)). This equation turns into $k^2 \\hat{\\psi} = 2\\hat{\\kappa}$ in the Fourier domain. The solution of the Poisson equation is then $\\psi = \\mathcal{F}^{-1}(2\\hat{\\kappa}/k^2)$. We can analytically compute the forward Fourier transform because the convergence has the Gaussian form. Then, we need to compute only the inverse transform numerically. Although obtaining the deflection potential necessitates a numerical integration or a numerical Fourier transform, we can keep the computational burden under control in most applications by computing this quantity only for a feasible number of models sampled from the lens model posterior.\n\n\n## 5 Adding it all together: proof of concept\n\nIn this section, we demonstrate the feasibility of our method to model lenses. We first simulate synthetic data of a mock lens and then model the lens using our method. We use the publicly available lens-modelling software [`lenstronomy`](https://github.com/sibirrer/lenstronomy) to simulate the synthetic data and perform the model-fitting (Birrer & Amara [2018](http://adsabs.harvard.edu/abs/2018PDU....22..189B)). We added extra modules to `lenstronomy` to implement the lensing analysis presented in Section [4.2](#scrollTo=slP9xat7i8YZ).\n\nFor the mock strong lensing system, we adopt an elliptical NFW deflection potential for the dark component and an elliptical Chameleon convergence for the luminous component. We take realistic scale sizes and normalizations for these profiles. We choose the Chameleon profile for two reasons:\n\n1. we can analytically simulate the data with ellipticity in the convergence, and\n2. we know the Sersic-profile parameters that approximates the chosen Chameleon profile _a priori_, so we can check the fidelity of our method.\n\nWe parameterize the scaling of the NFW profile with two parameters: scale radius $r_{\\rm s}$ and the deflection angle $\\alpha_{\\rm s}$ at $r_{\\rm s}$. For a spherical NFW profile given by\n\n\\begin{equation} \\tag{5.1}\\label{51}\n\t\\rho_{\\rm NFW}(r) = \\frac{\\rho_{\\rm s}}{(r/r_{\\rm s})(1 + r/r_{\\rm s})^2},\n\\end{equation}\n\nthe normalization $\\rho_s$ relates to $\\alpha_s$ as\n\n\\begin{equation} \\tag{5.2}\\label{52}\n\t\\alpha_{\\rm s} = \\frac{4 \\rho_{\\rm s} r_{\\rm s}^2}{D_{\\rm d}\\Sigma_{\\rm crit}} \\left( 1 - \\ln 2 \\right).\n\\end{equation}\n\n(Meneghetti et al. [2003](http://adsabs.harvard.edu/abs/2003MNRAS.340..105M)). The elliptical Chameleon convergence is given by\n\n\\begin{equation} \\tag{5.3}\\label{53}\n\t\\kappa_{\\rm Chm} (x, y) = \\frac{\\kappa_0}{1 + q} \\left[ \\frac{1}{\\sqrt{x^2+y^2/q^2 + 4w_{\\rm c}^2/(1+q^2)}} \\right. \\\\\n\t \\quad \\quad \\quad \\quad \\left. - \\frac{1}{\\sqrt{x^2+y^2/q^2 + 4w_{\\rm t}^2/(1+q^2)}} \\right]\n\\end{equation}\n\n(Suyu et al. [2014](http://adsabs.harvard.edu/abs/2014ApJ...788L..35S)). We also add external shear to the mass profiles. Therefore, our fiducial lens mass profile has three components in total: elliptical NFW deflection potential, elliptical Chameleon convergence, and external shear.\n\nWe simulate data for this fiducial lens system with image quality similar to the _Hubble Space Telescope_ (_HST_) Wide-Field Camera 3 imaging in the F160W filter (see top-left panel of Fig. [6](#scrollTo=b4sIv2af2SHw)). We adopt 0.08 arcsec for the pixel size, 2197 s for exposure time, and a realistic point spread function (PSF) to achieve data quality similar to the lens sample presented by Shajib et al. ([2019](http://adsabs.harvard.edu/abs/2019MNRAS.483.5649S)).\n\nWe fit the synthetic data with a model composed of elliptical NFW deflection potential, elliptical Sersic convergence profile decomposed into concentric Gaussians, and external shear. Note that we take ellipticity in the deflection potential for the NFW profile due to a design restriction of `lenstronomy`. However, we can also extend an elliptical NFW convergence into Gaussians for lensing analysis in principle. For now, this limitation does not affect the point of this exercise to show that lensing analysis with Gaussian components is feasible. We take the PSF as known during model-fitting for simplicity. We also separately model the lens with the fiducial mass profiles for comparison. In both cases, the parameters for the light and the luminous mass profiles are joint except for the amplitudes. We fit the model to the data by using the MCMC method. For every sample point in the parameter space, we first decompose the elliptical Sersic profile into concentric Gaussian components using equations ([2.5](#mjx-eqn-25)) and ([2.7](#mjx-eqn-27)). Similar to the example in Section [2.1.2](#scrollTo=QSdpwV_JB_45), we take 15 Gaussian components with logarithmically spaced $\\sigma$'s between $0.02R_{\\rm eff}$ and $15 R_{\\rm eff}$. These 15 Gaussian components approximate the Sersic function well within the noise level for $0.1R_{\\rm eff} \\leq R \\leq 10R_{\\rm eff}$ (Fig. [3](#scrollTo=W3KtTXnnVNAy)). We compute the gradient and the Hessian of the deflection potential for each Gaussian component. Finally, we add the contributions from all the individual components together to obtain these quantities for the total mass profile. These total quantities are used to compute the likelihood in the MCMC method for fitting the model to the data. \n \n Our method fits the synthetic data very well (see the 'Normalized Residuals' plot in Fig. [6](#scrollTo=b4sIv2af2SHw)). The fiducial Sersic profile parameters are also recovered with reasonable to high accuracies at the same time (Table [1](#scrollTo=0PjFuyYK2WiS)). The total runtime is only approximately three times longer than using the fiducial model with the Chameleon profile. This loss in efficiency is a reasonable tradeoff for generality. Thus, we have demonstrated a feasible implementation of our lensing analysis in lens modelling.\n\n
    \n\n>**Figure 6.** Fitting synthetic lensing data with concentric Gaussian components of an elliptical Sersic profile for the luminous component. We fit the dark component with an elliptical NFW profile. The Sersic parameters for the lens light are joint with the luminous mass distribution except for the amplitudes letting the global mass-to-light ratio be a free parameter. We generated the synthetic data for a composite model with elliptical NFW and elliptical Chameleon profiles. In the 'Reconstructed source' plot, the pink contours outline the caustics. The blue star indicates the point source position. The green arrows in the 'Convergence and deflection' plot represent negative deflection angles and they are shrunk by a factor of 4 for nicer visualization. The red dots in the 'Magnification model' plot point out the image positions. Our method of computing lensing quantities for the Sersic profile with ellipticity in the convergence works well as evident from the 'Normalized Residual' plot. This method only takes approximately three times longer than using the fiducial model with the Chameleon profile. Unlike the Chameleon profile, however, our method is general.\n\n>>>**Table 1.** Fidelity of our lens modelling method. We simulate mock data with a fiducial model composed of elliptical Chameleon convergence, elliptical NFW deflection potential, and external shear. We test our method with the \"Gaussian''-model: elliptical Sersic convergence decomposed into concentric Gaussians, elliptical NFW deflection potential, and external shear. The 'True' rows contain the mock values of the fiducial model parameters. The ;Gaussian-fit' rows contain the parameters of the \"Gaussian''-model fit to the data. Similarly, the 'Fiducial-fit' rows contain the parameters of the fiducial model fit to the data. We do not provide uncertainty for the values that are accurate up to the displayed decimal point. The accuracy of our computational method with Gaussian components is comparable to that using the fiducial model.\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    Mass ProfileParameters
    Sersic$R_{\\rm eff}$ (arcsec)$n_{\\rm Sersic}$$q$
    		$\\phi$ (deg)
        Truea1.553.090.645
        Gaussian-fit1.51\u00b10.023.04\u00b10.030.6145.1\u00b10.3
    Chameleon$w_{\\rm t}$ (arcsec)$w_{\\rm c}$ (arcsec)$q$$\\phi$ (deg)
        True0.0381.70.645
        Fiducial-fit0.0381.70.645.0\u00b10.2
    NFW$r_{\\rm s}$ (arcsec)$\\alpha_{\\rm s}$ (arcsec)$q$$\\phi$ (deg)
        True510.945
        Gaussian-fit5.00.99\u00b10.030.87\u00b10.0244\u00b12
        Fiducial-fit5.01.00\u00b10.010.90\u00b10.0145\u00b12
    External shear$\\gamma$$\\phi$ (deg)––
        True0.0515.7––
        Gaussian-fit0.056\u00b10.0029\u00b13––
        Fiducial-fit0.051\u00b10.0016\u00b12––
    Notes. aThe true values of the Sersic-profile parameters correspond to the
    true values of the Chameleon-profile parameters.
    \n\n## 6 Conclusion: precision is feasible.\n\nIn this paper, we present a general method for precise lensing analysis of any elliptical convergence profile. Our method follows a \"divide and conquer'' strategy. In our method, we first decompose an elliptical convergence profile into concentric Gaussian components as\n\n\\begin{equation} \\tag{6.1}\\label{61}\n\t\\kappa(x, y) \\approx \\sum_{j=1}^J \\kappa_{0j} \\exp \\left( -\\frac{q^2 x^2 + y^2}{2 \\sigma_j^2} \\right)\n\\end{equation}\n\nWe then compute lensing quantities, e.g., the gradient and the Hessian of the deflection potential, for each Gaussian component. Finally, we add the lensing quantities from individual Gaussian components together to obtain these quantities for the total surface-density profile. Moreover, we can straightforwardly deproject a Gaussian component to obtain its corresponding three-dimensional density profile assuming either axisymmetry or spherical symmetry. Then, we can also compute the kinematic properties, such as the line-of-sight velocity dispersion, for each Gaussian component (Cappellari [2008](http://adsabs.harvard.edu/abs/2008MNRAS.390...71C)). We can then add the velocity dispersions from individual Gaussians together to obtain the total line-of-sight velocity dispersion. In this way, we self-consistently unify the lensing and kinematic descriptions of any elliptical mass profile.\n\nWe introduce an integral transform with a Gaussian kernel that leads us to a general, precise, and fast algorithm for decomposing a surface density profile into concentric Gaussian components. Without such an algorithm, decomposing into Gaussians would end up as a bottleneck in the lens modelling efficiency. We obtain the algorithm by first inverting the integral transform as\n\n\\begin{equation} \\tag{6.2}\\label{62}\n\tf(\\sigma) = \\frac{1}{\\mathrm{i}\\sigma^2} \\sqrt{\\frac{2}{\u03c0}} \\int_{C} z F(z) \\exp \\left( \\frac{z^2}{2 \\sigma^2} \\right)\\ \\mathrm{d} z.\n\\end{equation}\n\nAlthough this is an integral, we provide a straightforward formula to compute $f(\\sigma)$. The computed values of $f(\\sigma)$ then quantify the amplitudes $\\kappa_{0j}$ of the Gaussian components in equation ([6.1](#mjx-eqn-61)). As a result, this integral transform fulfills the three requirements for a decomposition algorithm to be **(i)** general, **(ii)** precise, and **(iii)** fast. To be specific, this decomposition algorithm is $\\sim 10^3$ times faster than the MGE algorithm from Cappellari ([2002](http://adsabs.harvard.edu/abs/2002MNRAS.333..400C)). Consequently, our lensing analysis requires the same order of CPU time as other methods currently in use to model a lens with a composite mass profile. Thus, the integral transform enables the lens modelling with the Gaussian components to be efficient and, in turn, makes our unified framework for lensing and kinematic analysis of an elliptical mass profile feasible.\n\nOur method enables precise lens modelling with an elliptical mass profile for several astrophysical applications. Specifically, our method gives an efficient method to model composite mass profiles with separate components for the baryonic matter and the dark matter. For example, the usual choices for these components are the Sersic and the NFW profiles; both are computationally difficult to directly implement in lens modelling for the elliptical case. Our method makes both of these profiles computationally tractable while achieving the required precision. Thus, our method will be useful in applications where a composite mass profile is essential for lens modelling, for example, in detecting dark-matter substructure, in measuring the Hubble constant, and in testing massive elliptical-galaxy formation theories (e.g., Vegetti et al. [2012](http://adsabs.harvard.edu/abs/2012Natur.481..341V), Wong et al. [2017](http://adsabs.harvard.edu/abs/2017MNRAS.465.4895W), Nightingale et al. [2019](http://adsabs.harvard.edu/abs/2019arXiv190107801N)).\n\n## Acknowledgements\n\nAJS thanks Simon Birrer and Shouman Das for helpful discussions. AJS also thanks the anonymous referee for very useful suggestions that improved this paper. AJS expresses gratitude to Adriano Agnello, Simon Birrer, Xuheng Ding, Xinnan Du, Abhimat K. Gautam, Briley Lewis, Michael Topping, and Tommaso Treu for providing feedbacks that greatly improved the writing and the presentation of this paper. AJS acknowledges support by National Aeronautics and Space Administration (NASA) through Space Telescope Science Institute grant HST-GO-15320.\n\nThis research made use of `lenstronomy` (Birrer & Amara [2018](http://adsabs.harvard.edu/abs/2018PDU....22..189B)), `numpy` (Oliphant [2015](http://web.mit.edu/dvp/Public/numpybook.pdf)), `scipy` (Jones et al. [2001](http://www.scipy.org/)), `jupyter` (Kluyver et al. [2016](https://eprints.soton.ac.uk/403913/)), and `matplotlib` (Hunter [2007](http://adsabs.harvard.edu/abs/2007CSE.....9...90H)).\n\n## Appendix A: Properties of the integral transform with a Gaussian kernel\n\nIn this appendix, we prove some fundamental properties of the integral transform with a Gaussian kernel. In three theorems, we prove that \n\n\n1. the integral transform exists for a function with certain characteristics,\n2. the transform is unique for a continuous function, and\n3. the transform is invertible.\n\nFirst, we define the integral transform.\n\n\n**Definition A.1.** Define an integral transform $\\mathcal{T}$ that takes a function $f(\\sigma):\\mathbb{R}_{\\geq 0} \\to \\mathbb{R}$ to a function $F(z):\\mathbb{C} \\to \\mathbb{C}$ as\n\n\\begin{equation} \\tag{\u03911}\\label{A1}\n\t\tF(z) \\equiv \\mathcal{T}[f] (z) \\equiv \\frac{1}{\\sqrt{2 \u03c0}} \\int_0^\\infty \\frac{f(\\sigma)}{\\sigma} \\exp\\left( -\\frac{z^2}{2\\sigma^2}\\right) \\mathrm{d} \\sigma.\n\\end{equation}\n\nNext, we define a _transformable_ function in the context of this paper.\n\n**Definition A.2.** A function $f(\\sigma):\\mathbb{R}_{\\geq 0} \\to \\mathbb{R}$ is said to be transformable, if it satisfies the following conditions:\n\n1. the function $f(\\sigma)$ is piecewise continuous,\n2. the function $f(\\sigma) = \\mathcal{O}(\\exp(c/2\\sigma^2))$ as $\\sigma \\to 0$, where $c \\in \\mathbb{R}_{\\geq 0}$,\n3. the function $f(\\sigma) = \\mathcal{O}(\\sigma^\\lambda)$ with $\\lambda < 0 $ as $\\sigma \\to \\infty$.\n\nWe refer to these three conditions as the _transformability conditions_. The namesake for the transformable function is made clear next in Theorem [A.3](#scrollTo=it_f3KX03WcS).\n\n**Theorem A.3. (Existence)** If $f(\\sigma)$ is transformable, then its integral transform $F(z)$ exists in the region of convergence (ROC) $\\mathrm{Re}(z^2) > c$.\n\n_Proof._ Divide the integral in equation ([A1](#mjx-eqn-A1)) as\n\n\\begin{equation} \\tag{A2}\\label{A2}\nF(z) = \\frac{1}{\\sqrt{2 \u03c0}} \\left( \\int_0^a \\mathrm{d} \\sigma + \\int_a^b \\mathrm{d} \\sigma + \\int_b^\\infty \\mathrm{d} \\sigma \\right) \\frac{f(\\sigma)}{\\sigma} \\exp\\left( -\\frac{z^2}{2\\sigma^2}\\right) \\\\\n= \\frac{1}{\\sqrt{2 \u03c0}} \\left( \\mathcal{I}_1 + \\mathcal{I}_2 + \\mathcal{I}_3 \\right),\n\\end{equation}\n\nwhere $00}$ such that $f(\\sigma) \\leq M_1 \\exp (c/2\\sigma^2)$ for $\\sigma \\leq a$. Then using Jensen's inequality, we have\n\n\\begin{equation} \\tag{A3}\\label{A3}\n\\newcommand{\\re}{{\\rm Re}}\n\\newcommand{\\im}{{\\rm Im}}\n\\newcommand{\\dd}{\\mathrm{d}}\n\\left|\\mathcal{I}_1\\right| \\leq \\int_0^a \\left|\\frac{f(\\sigma)}{\\sigma} \\exp\\left( -\\frac{z^2}{2\\sigma^2}\\right)\\right| \\mathrm{d} \\sigma \\\\\n\t\\Rightarrow \\left| \\mathcal{I}_1 \\right| \\leq \\int_0^a \\left| \\frac{M_1}{\\sigma} \\exp\\left( - \\frac{z^2 - c}{2\\sigma^2}\\right) \\right| \\mathrm{d} \\sigma \\\\\n\t \\Rightarrow \\left|\\mathcal{I}_1 \\right| \\leq M_1 \\int_0^a \\frac{1}{\\sigma} \\exp \\left(-\\frac{\\re (z^2) - c}{2 \\sigma^2} \\right) \\mathrm{d} \\sigma.\n\\end{equation}\n\nTherefore, the integral $\\mathcal{I}_1$ converges in the region $\\re (z^2) = x^2 - y^2 > c$, where $z = x + \\mathrm{i}y$, $x \\in \\mathbb{R}, y \\in \\mathbb{R}$.\n\n2. According to transformability condition 3, there exists $M_2 \\in \\mathbb{R}_{>0}$ such that $f(\\sigma) \\leq M_2 \\sigma^{\\lambda}$ for $\\sigma \\geq b$. Then, we have\n\n\\begin{equation} \\tag{A4}\\label{A4}\n \\newcommand{\\ra}{\\right|}\n \\newcommand{\\la}{\\left|}\n\t\t\\la \\mathcal{I}_3\\ra \\leq \\int_b^\\infty \\la \\frac{f(\\sigma)}{\\sigma} \\exp \\left( - \\frac{z^2}{2 \\sigma^2} \\right) \\ra \\dd \\sigma \\\\\n\t\t\\Rightarrow \\la \\mathcal{I}_3 \\ra \\leq \\int_b^\\infty M_2 \\sigma^{\\lambda-1} \\la \\exp \\left( - \\frac{z^2}{2 \\sigma^2}\\right) \\ra \\dd \\sigma \\\\\n\t\t\\Rightarrow \\la \\mathcal{I}_3 \\ra \\leq M_2 \\int_b^\\infty \\sigma^{\\lambda-1} \\dd \\sigma = -M_2\\frac{b^{\\lambda}}{\\lambda} < \\infty.\n\\end{equation}\n \nHere, we applied the inequality $\\left|\\exp (-z^2/2\\sigma^2)\\right| \\leq 1$ for $\\re (z^2) > c \\geq 0$. As a result, the integral $\\mathcal{I}_3$ converges.\n\nTherefore, the transform $F(z)$ exists in the ROC $\\re(z^2) > c$. $\\square$\n\nFig. [A1](#scrollTo=zBNYTOeevKkJ) shows the ROC for the integral in equation ([A1](#mjx-eqn-A1)). We can extend the ROC by the following two corollaries.\n\n
    \n\n>**Figure A1.** Region of convergence (shaded region) on the complex plane for the integral in equation ([A1](#mjx-eqn-A1)). The hyperbolic contour $C$ for the integral in equation ([A9](#mjx-eqn-A9)) is shown with solid black hyperbola.\n\n**Corollary A.4.** If a transformable function $f(\\sigma)$ additionally satisfies the condition $f(\\sigma) = \\mathcal{O} (\\sigma^\\beta \\exp(c/2\\sigma^2))$ with $\\beta \\geq 1$ as $\\sigma \\to 0$, then the integral in equation ([A1](#mjx-eqn-A1)) converges in the ROC $\\re (z^2) \\geq c$.\n\n_Proof._ According to the additional condition, there exists $ M_3 \\in \\mathbb{R}_{>0}$ such that $f(\\sigma) \\leq M_3 \\sigma^{\\beta} \\exp(c/2\\sigma^2)$ for $\\sigma \\leq a$. Then, we can rewrite equation ([A3](#mjx-eqn-A3)) as\n\n\\begin{equation} \\tag{A5}\\label{A5}\n\t\\la \\mathcal{I}_1 \\ra \\leq \\int_0^a \\la \\frac{f(\\sigma)}{\\sigma} \\exp\\left( -\\frac{z^2}{2\\sigma^2}\\right) \\ra \\dd \\sigma \\\\\n\t\\Rightarrow \\la \\mathcal{I}_1 \\ra \\leq \\int_0^a \\la M_3 \\sigma^{\\beta - 1} \\exp\\left( - \\frac{z^2 - c}{2\\sigma^2}\\right) \n \\ra \\dd \\sigma \\\\\n\t \\Rightarrow \\la \\mathcal{I}_1 \\ra \\leq M_3 \\int_0^a \\sigma^{\\beta-1} \\exp \\left(-\\frac{\\re (z^2) - c}{2 \\sigma^2} \\right) \\dd \\sigma.\n\\end{equation}\n\nFor $\\re (z^2) = c$, this becomes\n\n\\begin{equation} \\tag{A6}\\label{A6}\n\\la \\mathcal{I}_1 \\ra \\leq M_3 \\int_0^a \\sigma^{\\beta - 1} \\dd \\sigma = \\frac{M_3 a^{\\beta}}{\\beta} < \\infty.\n\\end{equation}\n\nTherefore, the ROC for the integral in equation ([A1](#mjx-eqn-A1)) extends to $\\re (z^2) \\geq c$. $\\square$\n\n**Corollary A.5.** If a transformable function $f(\\sigma)$ additionally satisfies $f(\\sigma) = \\mathcal{O} (\\sigma^\\beta)$ with $\\beta \\geq 1$ as $\\sigma \\to 0$, then the integral in equation ([A1](#mjx-eqn-A1)) converges in the ROC $\\re (z^2) \\geq 0$.\n\n_Proof._ The proof is trivial with the substitution $c = 0$ in Corollary [A.4](#scrollTo=TXZfV0aMKNY9). $\\square$\n\nNext we prove the uniqueness theorem for the transform. First, we state a well-known proof of the following lemma for completeness.\n\n**Lemma A.6.** If $f(x)$ is continuous in $[0, 1]$, and $\\int_0^1 x^n f(x)\\ \\dd x = 0$ for $n=0,1,2,\\dots$, then $f(x)=0$.\n\n_Proof._\tFrom the Weierstrass approximation theorem, for any $ \\epsilon>0$, there exists a polynomial $P_{\\epsilon}(x)$ such that $\\la{f(x)-P_{\\epsilon}(x)}\\ra< \\epsilon$ for all $x \\in [0,1]$. The hypothesis implies that $\\int_0^1 P_{\\epsilon}(x) f(x)\\ \\dd x = 0$. By taking the limit $\\epsilon \\to 0$, this equation becomes $\\int_0^1 f(x)f(x)\\ \\dd x = 0$. As $f(x)^2 \\geq 0$, we have $f(x)=0$. $\\square$\n\n**Theorem A.7. (Uniqueness)** If $f(\\sigma)$ and $g(\\sigma)$ are continuous, and $\\mathcal{T}[f](z) = \\mathcal{T}[g](z)$ for all $z$ in the ROC, then $f(\\sigma)=g(\\sigma)$.\n\n_Proof._\tDue to linearity, it is sufficient to prove that if $\\mathcal{T}[f](z) = 0$, then $f(\\sigma)=0$. Take $d$ such that the contour $\\re (z^2) = d$ lies in the ROC. By making the change of variables $s=\\exp(-1/2\\sigma^2)$, for $z^2=d+n+1$ with $n=0,1,2,\\dots$, we have\n\n\\begin{equation} \\tag{A7}\\label{A7}\n \\newcommand{\\uppi}{\u03c0}\n\t\t\t\\mathcal{T}[f](z) = \\frac{1}{\\sqrt{2\\uppi}} \\int_0^\\infty \\frac{f(\\sigma)}{\\sigma} \\exp \\left( -\\frac{d+n+1}{2\\sigma^2}\\right) \\dd \\sigma = 0 \\\\\n\t\t\t\\Rightarrow \\int_0^1 \\left[ - \\frac{s^d f(\\sqrt{-1/2\\log s})}{2 \\sqrt{2\\uppi} \\log s} \\right] s^n \\dd s = 0.\n\\end{equation}\n \n This integral exists as $s \\to 0$, because\n\t\\begin{equation} \\tag{A8}\\label{A8}\n\t\t\\lim_{s \\to 0}\t\\left[ - \\frac{s^d f(\\sqrt{-1/2\\log s})}{2 \\sqrt{2\\uppi} \\log s} \\right] = \\lim_{\\sigma \\to 0} \\left[ \\frac{\\sigma^2 f(\\sigma)}{\\sqrt{2 \\uppi}} \\exp \\left(- \\frac{d}{2\\sigma^2} \\right) \\right] = 0.\n\t\\end{equation}\n \nTherefore, according to Lemma [A.6](#scrollTo=-0XsGwfH39cf), we have $f(\\sigma)=0$. $\\square$\n\n**Theorem A.8. (Inversion)** If $F(z)$ is the transform of $f(\\sigma)$, then $f(\\sigma)$ is given by the inverse transform\n\n\\begin{equation} \\tag{A9}\\label{A9}\n\tf(\\sigma)= \\mathcal{T}^{-1}[F](\\sigma) = \\frac{1}{\\mathrm{i}\\sigma^2} \\sqrt{\\frac{2}{\\uppi}} \\int_{C} z F(z) \\exp \\left( \\frac{z^2}{2 \\sigma^2} \\right) \\dd z,\n\\end{equation}\n\nwhere the contour $C$ is the hyperbola $\\re (z^2) = d$ such that $C$ lies in ROC of $F(z)$.\n\n_Proof._ Write equation ([A1](#mjx-eqn-A1)) for $z^2 = d$ as\n\n\\begin{equation}\\tag{A10}\\label{A10}\n\tF\\left(\\sqrt{d}\\right) = \\frac{1}{\\sqrt{2 \\uppi}} \\int_0^\\infty \\frac{f(\\sigma)}{\\sigma} \\exp\\left( -\\frac{d}{2\\sigma^2}\\right) \\dd \\sigma .\n\\end{equation}\n\nWith the change of variables $p = 1/\\sigma^2$, this equation transforms into\n\n\\begin{equation} \\tag{A11} \\label{A11}\n\tF\\left(\\sqrt{d} \\right) = \\int_0^\\infty g(p) \\exp \\left( -\\frac{dp}{2} \\right) \\dd p,\n\\end{equation}\n\nwhere\n\\begin{equation} \\tag{A12}\\label{A12}\ng(p) = \\frac{\\sigma^2 f(\\sigma)}{2 \\sqrt{2 \\uppi}}.\n\\end{equation}\n\nDefine a new function\n\n\\begin{equation} \\tag{A13}\\label{A13}\nh(p) = \n \\begin{cases}\n g(p) \\exp \\left( -\\frac{dp}{2}\\right), & p \\geq 0,\\\\\n 0, &p < 0. \\\\ \n \\end{cases}\n\\end{equation}\n\nTheorem A.3 implies that $\\int_{-\\infty}^\\infty \\la{h(p)}\\ra \\ \\dd p < \\infty $, thus $h(p)$ belongs to the Lebesgue space $L^1 (\\mathbb{R})$. Therefore, we can take the Fourier transform of $h(p)$ as\n\n\\begin{equation} \\tag{A14}\\label{A14}\n\t\\hat{h}(\\nu) = \\frac{1}{\\sqrt{2 \\uppi}} \\int_{-\\infty}^{\\infty} h(p) \\mathrm{e}^{-\\mathrm{i}p\\nu} \\dd p \\\\\n\t=\\frac{1}{\\sqrt{2 \\uppi}} \\int_{0}^{\\infty} g(p) \\exp \\left( -\\frac{(d+2\\mathrm{i}\\nu)p}{2}\\right) \\dd p \\\\\n\t= \\frac{1}{\\sqrt{2 \\uppi}} F\\left( \\sqrt{d+2\\mathrm{i}\\nu} \\right),\n\\end{equation}\n\nwhere we used equation ([A11](#mjx-eqn-A11)) for the substitution in the last line. Now, take the inverse Fourier transform of $\\hat{h}(\\nu)$ as \n\n\\begin{equation} \\tag{A15}\\label{A15}\n\th(p) = \\frac{1}{\\sqrt{2\\uppi}} \\int_{-\\infty}^{\\infty} \\hat{h} (\\nu) e^{\\mathrm{i} p \\nu} \\dd \\nu \\\\\n\t\\Rightarrow g(p) \\exp \\left( - \\frac{dp}{2} \\right) = \\frac{1}{2 \\uppi} \\int_{-\\infty}^{\\infty} F\\left( \\sqrt{d + 2 \\mathrm{i} \\nu} \\right) e^{\\mathrm{i} p \\nu} \\dd \\nu \\\\\n\t\\Rightarrow \\frac{\\sigma^2 f(\\sigma)}{2 \\sqrt{2 \\uppi}} = \\frac{1}{2 \\uppi} \\int_{-\\infty}^{\\infty} F\\left( \\sqrt{d + 2 \\mathrm{i} \\nu} \\right) \\exp \\left(\\frac{d + 2 \\mathrm{i} \\nu}{2\\sigma^2} \\right) \\dd \\nu \\\\\n\t\\Rightarrow f(\\sigma) = \\frac{1}{\\mathrm{i}\\sigma^2} \\sqrt{\\frac{2}{\\uppi}} \\int_{C} z F(z) \\exp \\left( \\frac{z^2}{2 \\sigma^2} \\right) \\dd z.\n\\end{equation}\n\nHere, we used the substitution of variable $z^2 = d+2 \\mathrm{i} \\nu$ in the last line, which transforms the integral path to the hyperbolic contour $C$ given by $\\re (z^2) = d$. $\\square$\n\n**Remark A.9.** Equation ([A11](#mjx-eqn-A11)) has the form of a Laplace transform. Therefore, the integral transform with a Gaussian kernel for a transformable function can be converted into a Laplace transform by suitable change of variables.\n\n## Appendix B: Efficient algorithm to compute the Faddeeva function\n\nIn this appendix, we provide an efficient `python` [function](#scrollTo=FmSoVRxSnR1R) to compute the Faddeeva function $w_{\\rm F}(z)$ based on the algorithm from Zaghloul ([2017](http://doi.acm.org/10.1145/3119904)). The relative error of this algorithm is less than $4\\times 10^{-5}$ over the whole complex plane.\n\n\n```python\ndef wofz_approx(z):\n \"\"\"\n Compute the Faddeeva function w(z) using the approximation given in\n Zaghloul (2017).\n\n :param z: complex number or array\n :type z: complex\n :return: w_f\n :rtype: complex\n \"\"\"\n sqrt_pi = 1 / np.sqrt(np.pi)\n i_sqrt_pi = 1j * sqrt_pi\n\n wz = np.empty_like(z)\n\n z_imag2 = z.imag ** 2\n abs_z2 = z.real ** 2 + z_imag2\n\n reg1 = (abs_z2 >= 38000.)\n \n if np.any(reg1):\n wz[reg1] = i_sqrt_pi / z[reg1]\n\n reg2 = (256. <= abs_z2) & (abs_z2 < 38000.)\n if np.any(reg2):\n t = z[reg2]\n wz[reg2] = i_sqrt_pi * t / (t * t - 0.5)\n\n reg3 = (62. <= abs_z2) & (abs_z2 < 256.)\n if np.any(reg3):\n t = z[reg3]\n wz[reg3] = (i_sqrt_pi / t) * (1 + 0.5 / (t * t - 1.5))\n\n reg4 = (30. <= abs_z2) & (abs_z2 < 62.) & (z_imag2 >= 1e-13)\n if np.any(reg4):\n t = z[reg4]\n tt = t * t\n wz[reg4] = (i_sqrt_pi * t) * (tt - 2.5) / (tt * (tt - 3.) + 0.75)\n\n reg5 = (62. > abs_z2) & np.logical_not(reg4) & (abs_z2 > 2.5) & (\n z_imag2 < 0.072)\n if np.any(reg5):\n t = z[reg5]\n u = -t * t\n f1 = sqrt_pi\n f2 = 1\n s1 = [1.320522, 35.7668, 219.031, 1540.787, 3321.99, 36183.31]\n s2 = [1.841439, 61.57037, 364.2191, 2186.181, 9022.228, 24322.84,\n 32066.6]\n\n for s in s1:\n f1 = s - f1 * u\n for s in s2:\n f2 = s - f2 * u\n\n wz[reg5] = np.exp(u) + 1j * t * f1 / f2\n\n reg6 = (30.0 > abs_z2) & np.logical_not(reg5)\n if np.any(reg6):\n t3 = - 1j * z[reg6]\n\n f1 = sqrt_pi\n f2 = 1\n s1 = [5.9126262, 30.180142, 93.15558, 181.92853, 214.38239,\n 122.60793]\n s2 = [10.479857, 53.992907, 170.35400, 348.70392, 457.33448,\n 352.73063, 122.60793]\n\n for s in s1:\n f1 = f1 * t3 + s\n for s in s2:\n f2 = f2 * t3 + s\n\n wz[reg6] = f1 / f2\n\n return wz\n```\n", "meta": {"hexsha": "9012edc6bedb9d02312714cd891855d2700e7169", "size": 435783, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Unified lensing and kinematic analysis.ipynb", "max_stars_repo_name": "ajshajib/unified_lensing_and_kinematics", "max_stars_repo_head_hexsha": "f80a6a618dddb29ac71cb82b6ba03b19ef62c21e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Unified lensing and kinematic analysis.ipynb", "max_issues_repo_name": "ajshajib/unified_lensing_and_kinematics", "max_issues_repo_head_hexsha": "f80a6a618dddb29ac71cb82b6ba03b19ef62c21e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Unified lensing and kinematic analysis.ipynb", "max_forks_repo_name": "ajshajib/unified_lensing_and_kinematics", "max_forks_repo_head_hexsha": "f80a6a618dddb29ac71cb82b6ba03b19ef62c21e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 173.1358760429, "max_line_length": 105302, "alphanum_fraction": 0.8344910196, "converted": true, "num_tokens": 35982, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.7248702702332475, "lm_q1q2_score": 0.4404769661275599}} {"text": "\n\n# Tutorial 3: Synaptic transmission - Models of static and dynamic synapses\n**Week 2, Day 3: Real Neurons**\n\n**By Neuromatch Academy**\n\n__Content creators:__ Qinglong Gu, Songtin Li, John Murray, Richard Naud, Arvind Kumar\n\n__Content reviewers:__ Maryam Vaziri-Pashkam, Ella Batty, Lorenzo Fontolan, Richard Gao, Matthew Krause, Spiros Chavlis, Michael Waskom\n\n---\n# Tutorial Objectives\nSynapses connect neurons into neural networks or circuits. Specialized electrical synapses make direct, physical connections between neurons. In this tutorial, however, we will focus on **chemical synapses**, which are more common in the brain. These synapses do not physically join neurons. Instead, a spike in the presynaptic cell causes a chemical, or neurotransmitter, to be released into a small space between the neurons called the synaptic cleft. Once the chemical diffuses across that space, it changes the pearmeability of the postsynaptic membrane, which may result in a positive or negative change in the membrane voltage.\n\nIn this tutorial, we will model chemical synaptic transmission and study some interesting effects produced by **static synapses** and **dynamic synapses**.\n\nFirst, we will start by writing code to simulate static synapses -- whose weight is always fixed. \nNext, we will extend the model and model **dynamic synapses** -- whose synaptic strength is dependent on the recent spike history: synapses can either progressively increase or decrease the size of their effects on the post-synaptic neuron, based on the recent firing rate of its presynaptic partners. This feature of synapses in the brain is called **Short-Term Plasticity** and causes synapses to undergo *Facilitation* or *Depression*. \n\nOur goals for this tutorial are to:\n\n- simulate static synapses and study how excitation and inhibition affect the patterns in the neurons' spiking output\n- define mean- or fluctuation-driven regimes\n- simulate short-term dynamics of synapses (facilitation and depression)\n- study how a change in pre-synaptic firing history affects the synaptic weights (i.e., PSP amplitude)\n\n\n```python\n#@title Video 1: Static and dynamic synapses\nfrom IPython.display import YouTubeVideo\nvideo = YouTubeVideo(id='Hbz2lj2AO_0', width=854, height=480, fs=1)\nprint(\"Video available at https://youtube.com/watch?v=\" + video.id)\nvideo\n```\n\n---\n# Setup\n\n\n\n```python\n# Import libraries\nimport matplotlib.pyplot as plt\nimport numpy as np\n```\n\n\n```python\n# @title Figure Settings\nimport ipywidgets as widgets # interactive display\n%config InlineBackend.figure_format='retina'\n# use NMA plot style\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/course-content/master/nma.mplstyle\")\nmy_layout = widgets.Layout()\n```\n\n\n```python\n# @title Helper functions\n\n\ndef my_GWN(pars, mu, sig, myseed=False):\n \"\"\"\n Args:\n pars : parameter dictionary\n mu : noise baseline (mean)\n sig : noise amplitute (standard deviation)\n myseed : random seed. int or boolean\n the same seed will give the same random number sequence\n\n Returns:\n I : Gaussian White Noise (GWN) input\n \"\"\"\n\n # Retrieve simulation parameters\n dt, range_t = pars['dt'], pars['range_t']\n Lt = range_t.size\n\n # set random seed\n # you can fix the seed of the random number generator so that the results\n # are reliable. However, when you want to generate multiple realizations\n # make sure that you change the seed for each new realization.\n if myseed:\n np.random.seed(seed=myseed)\n else:\n np.random.seed()\n\n # generate GWN\n # we divide here by 1000 to convert units to seconds.\n I_GWN = mu + sig * np.random.randn(Lt) / np.sqrt(dt / 1000.)\n\n return I_GWN\n\n\ndef Poisson_generator(pars, rate, n, myseed=False):\n \"\"\"\n Generates poisson trains\n\n Args:\n pars : parameter dictionary\n rate : noise amplitute [Hz]\n n : number of Poisson trains\n myseed : random seed. int or boolean\n\n Returns:\n pre_spike_train : spike train matrix, ith row represents whether\n there is a spike in ith spike train over time\n (1 if spike, 0 otherwise)\n \"\"\"\n\n # Retrieve simulation parameters\n dt, range_t = pars['dt'], pars['range_t']\n Lt = range_t.size\n\n # set random seed\n if myseed:\n np.random.seed(seed=myseed)\n else:\n np.random.seed()\n\n # generate uniformly distributed random variables\n u_rand = np.random.rand(n, Lt)\n\n # generate Poisson train\n poisson_train = 1. * (u_rand < rate * (dt / 1000.))\n\n return poisson_train\n\n\ndef default_pars(**kwargs):\n pars = {}\n\n ### typical neuron parameters###\n pars['V_th'] = -55. # spike threshold [mV]\n pars['V_reset'] = -75. # reset potential [mV]\n pars['tau_m'] = 10. # membrane time constant [ms]\n pars['g_L'] = 10. # leak conductance [nS]\n pars['V_init'] = -65. # initial potential [mV]\n pars['E_L'] = -75. # leak reversal potential [mV]\n pars['tref'] = 2. # refractory time (ms)\n\n ### simulation parameters ###\n pars['T'] = 400. # Total duration of simulation [ms]\n pars['dt'] = .1 # Simulation time step [ms]\n\n ### external parameters if any ###\n for k in kwargs:\n pars[k] = kwargs[k]\n\n pars['range_t'] = np.arange(0, pars['T'], pars['dt']) # Vector of discretized time points [ms]\n\n return pars\n\n\ndef my_illus_LIFSYN(pars, v_fmp, v):\n \"\"\"\n Illustartion of FMP and membrane voltage\n\n Args:\n pars : parameters dictionary\n v_fmp : free membrane potential, mV\n v : membrane voltage, mV\n\n Returns:\n plot of membrane voltage and FMP, alongside with the spiking threshold\n and the mean FMP (dashed lines)\n \"\"\"\n\n plt.figure(figsize=(14, 5))\n plt.plot(pars['range_t'], v_fmp, 'r', lw=1.,\n label='Free mem. pot.', zorder=2)\n plt.plot(pars['range_t'], v, 'b', lw=1.,\n label='True mem. pot', zorder=1, alpha=0.7)\n plt.axhline(-55, 0, 1, color='k', lw=2., ls='--',\n label='Spike Threshold', zorder=1)\n plt.axhline(np.mean(v_fmp), 0, 1, color='r', lw=2., ls='--',\n label='Mean Free Mem. Pot.', zorder=1)\n plt.xlabel('Time (ms)')\n plt.ylabel('V (mV)')\n plt.legend(loc=[1.02, 0.68])\n plt.show()\n\n\ndef my_illus_STD(Poi_or_reg=False, rate=20., U0=0.5,\n tau_d=100., tau_f=50., plot_out=True):\n \"\"\"\n Only for one presynaptic train\n\n Args:\n Poi_or_reg : Poisson or regular input spiking trains\n rate : Rate of input spikes, Hz\n U0 : synaptic release probability at rest\n tau_d : synaptic depression time constant of x [ms]\n tau_f : synaptic facilitation time constantr of u [ms]\n plot_out : whether ot not to plot, True or False\n\n Returns:\n Nothing.\n \"\"\"\n\n T_simu = 10.0 * 1000 / (1.0 * rate) # 10 spikes in the time window\n pars = default_pars(T=T_simu)\n dt = pars['dt']\n\n if Poi_or_reg:\n # Poisson type spike train\n pre_spike_train = Poisson_generator(pars, rate, n=1)\n pre_spike_train = pre_spike_train.sum(axis=0)\n else:\n # Regular firing rate\n isi_num = int((1e3/rate)/dt) # number of dt\n pre_spike_train = np.zeros(len(pars['range_t']))\n pre_spike_train[::isi_num] = 1.\n\n u, R, g = dynamic_syn(g_bar=1.2, tau_syn=5., U0=U0,\n tau_d=tau_d, tau_f=tau_f,\n pre_spike_train=pre_spike_train,\n dt=pars['dt'])\n\n if plot_out:\n plt.figure(figsize=(12, 6))\n\n plt.subplot(221)\n plt.plot(pars['range_t'], R, 'b', label='R')\n plt.plot(pars['range_t'], u, 'r', label='u')\n plt.legend(loc='best')\n plt.xlim((0, pars['T']))\n plt.ylabel(r'$R$ or $u$ (a.u)')\n plt.subplot(223)\n spT = pre_spike_train > 0\n t_sp = pars['range_t'][spT] #spike times\n plt.plot(t_sp, 0. * np.ones(len(t_sp)), 'k|', ms=18, markeredgewidth=2)\n plt.xlabel('Time (ms)');\n plt.xlim((0, pars['T']))\n plt.yticks([])\n plt.title('Presynaptic spikes')\n\n plt.subplot(122)\n plt.plot(pars['range_t'], g, 'r', label='STP synapse')\n plt.xlabel('Time (ms)')\n plt.ylabel('g (nS)')\n plt.xlim((0, pars['T']))\n\n plt.tight_layout()\n\n if not Poi_or_reg:\n return g[isi_num], g[9*isi_num]\n\n\ndef plot_volt_trace(pars, v, sp):\n \"\"\"\n Plot trajetory of membrane potential for a single neuron\n\n Args:\n pars : parameter dictionary\n v : volt trajetory\n sp : spike train\n\n Returns:\n figure of the membrane potential trajetory for a single neuron\n \"\"\"\n\n V_th = pars['V_th']\n dt = pars['dt']\n if sp.size:\n sp_num = (sp/dt).astype(int) - 1\n v[sp_num] += 10\n\n plt.plot(pars['range_t'], v, 'b')\n plt.axhline(V_th, 0, 1, color='k', ls='--', lw=1.)\n plt.xlabel('Time (ms)')\n plt.ylabel('V (mV)')\n```\n\nIn the `Helper Function`:\n\n- Gaussian white noise generator: `my_GWN(pars, mu, sig, myseed=False)`\n- Poissonian spike train generator: `Poisson_generator(pars, rate, n, myseed=False)`\n- default parameter function (as before) and other plotting utilities\n\n---\n# Section 1: Static synapses\n\n## Section 1.1: Simulate synaptic conductance dynamics\n\nSynaptic input _in vivo_ consists of a mixture of **excitatory** neurotransmitters, which depolarizes the cell and drives it towards spike threshold, and **inhibitory** neurotransmitters that hyperpolarize it, driving it away from spike threshold. These chemicals cause specific ion channels on the postsynaptic neuron to open, resulting in a change in that neuron's conductance and, therefore, the flow of current in or out of the cell.\n\nThis process can be modelled by assuming that the presynaptic neuron's spiking activity produces transient changes in the postsynaptic neuron's conductance ($g_{\\rm syn}(t)$). Typically, the conductance transient is modeled as an exponential function. \n\nSuch conductance transients can be generated using a simple ordinary differential equation (ODE):\n\n\\\\\n\n\\begin{eqnarray}\n\\frac{dg_{\\rm syn}(t)}{dt} &=& \\bar{g}_{\\rm syn} \\sum_k \\delta(t-t_k) -g_{\\rm syn}(t)/\\tau_{\\rm syn}\n\\end{eqnarray}\n\n\\\\\n\nwhere $\\bar{g}_{\\rm syn}$ (often referred to as synaptic weight) is the maximum conductance elicited by each incoming spike, and $\\tau_{\\rm syn}$ is the synaptic time constant. Note that the summation runs over all spikes received by the neuron at time $t_k$.\n\nOhm's law allows us to convert conductance changes to the current as:\n\n\\\\\n\n\\begin{align}\nI_{\\rm syn}(t) = g_{\\rm syn}(t)(V(t)-E_{\\rm syn}) \\\\\n\\end{align}\n\n\\\\\n\nThe reversal potential $E_{\\rm syn}$ determines the direction of current flow and the excitatory or inhibitory nature of the synapse. \n\n**Thus, incoming spikes are filtered by an exponential-shaped kernel, effectively low-pass filtering the input. In other words, synaptic input is not white noise, but it is, in fact, colored noise, where the color (spectrum) of the noise is determined by the synaptic time constants of both excitatory and inhibitory synapses.**\n\nIn a neuronal network, the total synaptic input current $I_{\\rm syn}$ is the sum of both excitatory and inhibitory inputs. Assuming the total excitatory and inhibitory conductances received at time $t$ are $g_E(t)$ and $g_I(t)$, and their corresponding reversal potentials are $E_E$ and $E_I$, respectively, then the total synaptic current can be described as: \n\n\\\\\n\n\\begin{align}\nI_{\\rm syn}(V(t),t) = -g_E(t) (V-E_E) - g_I(t) (V-E_I)\n\\end{align}\n\n\\\\\n\nAccordingly, the membrane potential dynamics of the LIF neuron under synaptic current drive become:\n\n\\\\\n\n\\begin{eqnarray}\n\\tau_m\\frac{dV(t)}{dt} = -(V(t)-E_L) - \\frac{g_E(t)}{g_L} (V(t)-E_E) - \\frac{g_I(t)}{g_L} (V(t)-E_I) + \\frac{I_{\\rm inj}}{g_L}\\quad (2)\n\\end{eqnarray}\n\n\\\\\n\n$I_{\\rm inj}$ is an external current injected in the neuron, which is under experimental control; it can be GWN, DC, or anything else.\n\nWe will use Eq. (2) to simulate the conductance-based LIF neuron model below.\n\nIn the previous tutorials, we saw how the output of a single neuron (spike count/rate and spike time irregularity) changes when we stimulate the neuron with DC and GWN, respectively. Now, we are in a position to study how the neuron behaves when it is bombarded with both excitatory and inhibitory spikes trains -- as happens *in vivo*.\n\nWhat kind of input is a neuron receiving? When we do not know, we chose the simplest option. The simplest model of input spikes is given when every input spike arrives independently of other spikes, i.e., we assume that the input is Poissonian.\n\n## Section 1.2: Simulate LIF neuron with conductance-based synapses\n\nWe are now ready to simulate a LIF neuron with conductance-based synaptic inputs! The following code defines the LIF neuron with synaptic input modeled as conductance transients.\n\n\n```python\n# @markdown Execute this cell to get a function for conductance-based LIF neuron (run_LIF_cond)\n\ndef run_LIF_cond(pars, I_inj, pre_spike_train_ex, pre_spike_train_in):\n \"\"\"\n Conductance-based LIF dynamics\n\n Args:\n pars : parameter dictionary\n I_inj : injected current [pA]. The injected current here\n can be a value or an array\n pre_spike_train_ex : spike train input from presynaptic excitatory neuron\n pre_spike_train_in : spike train input from presynaptic inhibitory neuron\n\n Returns:\n rec_spikes : spike times\n rec_v : mebrane potential\n gE : postsynaptic excitatory conductance\n gI : postsynaptic inhibitory conductance\n\n \"\"\"\n\n # Retrieve parameters\n V_th, V_reset = pars['V_th'], pars['V_reset']\n tau_m, g_L = pars['tau_m'], pars['g_L']\n V_init, E_L = pars['V_init'], pars['E_L']\n gE_bar, gI_bar = pars['gE_bar'], pars['gI_bar']\n VE, VI = pars['VE'], pars['VI']\n tau_syn_E, tau_syn_I = pars['tau_syn_E'], pars['tau_syn_I']\n tref = pars['tref']\n dt, range_t = pars['dt'], pars['range_t']\n Lt = range_t.size\n\n # Initialize\n tr = 0.\n v = np.zeros(Lt)\n v[0] = V_init\n gE = np.zeros(Lt)\n gI = np.zeros(Lt)\n Iinj = I_inj * np.ones(Lt) # ensure Iinj has length Lt\n\n if pre_spike_train_ex.max() == 0:\n pre_spike_train_ex_total = np.zeros(Lt)\n else:\n pre_spike_train_ex_total = pre_spike_train_ex.sum(axis=0) * np.ones(Lt)\n\n if pre_spike_train_in.max() == 0:\n pre_spike_train_in_total = np.zeros(Lt)\n else:\n pre_spike_train_in_total = pre_spike_train_in.sum(axis=0) * np.ones(Lt)\n\n # simulation\n rec_spikes = [] # recording spike times\n for it in range(Lt - 1):\n if tr > 0:\n v[it] = V_reset\n tr = tr - 1\n elif v[it] >= V_th: # reset voltage and record spike event\n rec_spikes.append(it)\n v[it] = V_reset\n tr = tref / dt\n\n # update the synaptic conductance\n gE[it + 1] = gE[it] - (dt / tau_syn_E) * gE[it] + gE_bar * pre_spike_train_ex_total[it + 1]\n gI[it + 1] = gI[it] - (dt / tau_syn_I) * gI[it] + gI_bar * pre_spike_train_in_total[it + 1]\n\n # calculate the increment of the membrane potential\n dv = (dt / tau_m) * (-(v[it] - E_L) \\\n - (gE[it + 1] / g_L) * (v[it] - VE) \\\n - (gI[it + 1] / g_L) * (v[it] - VI) + Iinj[it] / g_L)\n\n # update membrane potential\n v[it+1] = v[it] + dv\n\n rec_spikes = np.array(rec_spikes) * dt\n\n return v, rec_spikes, gE, gI\n\n\nprint(help(run_LIF_cond))\n```\n\n### Exercise 1: Measure the mean free membrane potential\n\nLet's simulate the conductance-based LIF neuron with presynaptic spike trains generated by a `Poisson_generator` with rate 10 Hz for both excitatory and inhibitory inputs. Here, we choose 80 excitatory presynaptic spike trains and 20 inhibitory ones.\n\nPreviously, we've already learned that $CV_{\\rm ISI}$ can describe the irregularity of the output spike pattern. Now, we will introduce a new descriptor of the neuron membrane, i.e., the **Free Membrane Potential (FMP)** -- the membrane potential of the neuron when its spike threshold is removed. \n\nAlthough this is completely artificial, calculating this quantity allows us to get an idea of how strong the input is. We are mostly interested in knowing the mean and standard deviation (std.) of the FMP. In the exercise, you can visualize the FMP and membrane voltage with spike threshold.\n\n\n```python\n# To complete the exercise, uncomment the code and fill the missing parts (...)\npars = default_pars(T=1000.)\n# Add parameters\npars['gE_bar'] = 2.4 # [nS]\npars['VE'] = 0. # [mV] excitatory reversal potential\npars['tau_syn_E'] = 2. # [ms]\npars['gI_bar'] = 2.4 # [nS]\npars['VI'] = -80. # [mV] inhibitory reversal potential\npars['tau_syn_I'] = 5. # [ms]\n\n# generate presynaptic spike trains\npre_spike_train_ex = Poisson_generator(pars, rate=10, n=80)\npre_spike_train_in = Poisson_generator(pars, rate=10, n=20)\n\n# simulate conductance-based LIF model\nv, rec_spikes, gE, gI = run_LIF_cond(pars, 0, pre_spike_train_ex,\n pre_spike_train_in)\ndt, range_t = pars['dt'], pars['range_t']\nif rec_spikes.size:\n sp_num = (rec_spikes / dt).astype(int) - 1\n v[sp_num] = 10 # draw nicer spikes\n\n####################################################################\n## TODO for students: measure the free membrane potential\n# In order to measure the free membrane potential, first,\n# you should prevent the firing of the LIF neuron\n# How to prevent a LIF neuron from firing? Increse the threshold pars['V_th'].\n####################################################################\n# Change the threshold\n# pars['V_th'] = ...\n# Calculate FMP\n# v_fmp, _, _, _ = run_LIF_cond(pars, ..., ..., ...)\n\n# uncomment when you have filled the exercise\n# my_illus_LIFSYN(pars, v_fmp, v)\n```\n\n\n```python\n# to_remove solution\npars = default_pars(T=1000.)\n# Add parameters\npars['gE_bar'] = 2.4 # [nS]\npars['VE'] = 0. # [mV] excitatory reversal potential\npars['tau_syn_E'] = 2. # [ms]\npars['gI_bar'] = 2.4 # [nS]\npars['VI'] = -80. # [mV] inhibitory reversal potential\npars['tau_syn_I'] = 5. # [ms]\n\n# generate presynaptic spike trains\npre_spike_train_ex = Poisson_generator(pars, rate=10, n=80)\npre_spike_train_in = Poisson_generator(pars, rate=10, n=20)\n\n# simulate conductance-based LIF model\nv, rec_spikes, gE, gI = run_LIF_cond(pars, 0, pre_spike_train_ex,\n pre_spike_train_in)\ndt, range_t = pars['dt'], pars['range_t']\nif rec_spikes.size:\n sp_num = (rec_spikes / dt).astype(int) - 1\n v[sp_num] = 10 # draw nicer spikes\n\n# Change the threshold\npars['V_th'] = 1e3\n# Calculate FMP\nv_fmp, _, _, _ = run_LIF_cond(pars, 0, pre_spike_train_ex, pre_spike_train_in)\n\nwith plt.xkcd():\n my_illus_LIFSYN(pars, v_fmp, v)\n```\n\n### Interactive Demo: Conductance-based LIF Explorer with different E/I input\n\nIn the following, we can investigate how varying the ratio of excitatory to inhibitory inputs changes the firing rate and the spike time regularity (see the output text). \n\nTo change both the excitatory and inhibitory inputs, we will vary their firing rates. *However, if you wish, you can vary the strength and/or the number of these connections as well.* \n\nPay close attention to the mean free membrane potential (red dotted line) and its location with respect to the spike threshold (black dotted line). Try to develop a heuristic about the mean of the FMP and spike time irregularity ($CV_{\\rm ISI}$)\n\n\n```python\n# @title\n\n# @markdown Make sure you execute this cell to enable the widget!\n\nmy_layout.width = '450px'\n@widgets.interact(\n inh_rate=widgets.FloatSlider(20., min=10., max=60., step=5.,\n layout=my_layout),\n exc_rate=widgets.FloatSlider(10., min=2., max=20., step=2.,\n layout=my_layout)\n)\n\n\ndef EI_isi_regularity(exc_rate, inh_rate):\n\n pars = default_pars(T=1000.)\n # Add parameters\n pars['gE_bar'] = 3. # [nS]\n pars['VE'] = 0. # [mV] excitatory reversal potential\n pars['tau_syn_E'] = 2. # [ms]\n pars['gI_bar'] = 3. # [nS]\n pars['VI'] = -80. # [mV] inhibitory reversal potential\n pars['tau_syn_I'] = 5. # [ms]\n\n pre_spike_train_ex = Poisson_generator(pars, rate=exc_rate, n=80)\n pre_spike_train_in = Poisson_generator(pars, rate=inh_rate, n=20) # 4:1\n\n # Lets first simulate a neuron with identical input but with no spike\n # threshold by setting the threshold to a very high value\n # so that we can look at the free membrane potential\n pars['V_th'] = 1e3\n v_fmp, rec_spikes, gE, gI = run_LIF_cond(pars, 0, pre_spike_train_ex,\n pre_spike_train_in)\n\n # Now simulate a LIP with a regular spike threshold\n pars['V_th'] = -55.\n v, rec_spikes, gE, gI = run_LIF_cond(pars, 0, pre_spike_train_ex,\n pre_spike_train_in)\n dt, range_t = pars['dt'], pars['range_t']\n if rec_spikes.size:\n sp_num = (rec_spikes / dt).astype(int) - 1\n v[sp_num] = 10 # draw nicer spikes\n\n spike_rate = 1e3 * len(rec_spikes) / pars['T']\n\n cv_isi = 0.\n if len(rec_spikes) > 3:\n isi = np.diff(rec_spikes)\n cv_isi = np.std(isi) / np.mean(isi)\n\n print('\\n')\n plt.figure(figsize=(15, 10))\n plt.subplot(211)\n plt.text(500, -35, f'Spike rate = {spike_rate:.3f} (sp/s), Mean of Free Mem Pot = {np.mean(v_fmp):.3f}',\n fontsize=16, fontweight='bold', horizontalalignment='center',\n verticalalignment='bottom')\n plt.text(500, -38.5, f'CV ISI = {cv_isi:.3f}, STD of Free Mem Pot = {np.std(v_fmp):.3f}',\n fontsize=16, fontweight='bold', horizontalalignment='center',\n verticalalignment='bottom')\n\n plt.plot(pars['range_t'], v_fmp, 'r', lw=1.,\n label='Free mem. pot.', zorder=2)\n plt.plot(pars['range_t'], v, 'b', lw=1.,\n label='mem. pot with spk thr', zorder=1, alpha=0.7)\n plt.axhline(pars['V_th'], 0, 1, color='k', lw=1., ls='--',\n label='Spike Threshold', zorder=1)\n plt.axhline(np.mean(v_fmp),0, 1, color='r', lw=1., ls='--',\n label='Mean Free Mem. Pot.', zorder=1)\n plt.ylim(-76, -39)\n plt.xlabel('Time (ms)')\n plt.ylabel('V (mV)')\n plt.legend(loc=[1.02, 0.68])\n\n plt.subplot(223)\n plt.plot(pars['range_t'][::3], gE[::3], 'r', lw=1)\n plt.xlabel('Time (ms)')\n plt.ylabel(r'$g_E$ (nS)')\n\n plt.subplot(224)\n plt.plot(pars['range_t'][::3], gI[::3], 'b', lw=1)\n plt.xlabel('Time (ms)')\n plt.ylabel(r'$g_I$ (nS)')\n\n plt.tight_layout()\n```\n\n**Mean-driven and Fluctuation-driven regimes**\n\nIf we look at the figure above, we note that when the mean FMP is above spike threshold, the fluctuations in the FMP are rather small, and the neuron spikes in a fairly regular fashion. This regime, where the mean FMP is above the spike threshold, is called **mean-driven regime**. \n\n\nWhen the mean FMP is below the spike threshold, the fluctuations in the FMP are large, and the neuron's spikes are driven by these fluctuations. As a consequence, the neuron spikes in more Poisson-like fashion. This regime, where the mean FMP is below the spike threshold, and spikes are driven by the fluctuations, is called **fluctuation-driven regime**. \n\n## Think!\n\n- How much can you increase the spike pattern variability? Under what condition(s) might the neuron respond with Poisson-type spikes? Note that we injected Poisson-type spikes. (Think of the answer in terms of the ratio of the exc. and inh. input spike rates.)\n\n- Link to the balance of excitation and inhibition: one of the definitions of excitation and inhibition balance is that mean free membrane potential remains constant as excitatory and inhibitory input rates are increased. What do you think happens to the neuron firing rate as we change excitatory and inhibitory rates while keeping the neuron in balance? See [Kuhn, Aertsen, and Rotter (2004)](https://www.jneurosci.org/content/jneuro/24/10/2345.full.pdf) for much more on this.\n\n\n```python\n# to_remove explanation\n\n\"\"\"\nDiscussion:\n\n1. We can push the neuron to spike almost like a Poisson neuron. Of course given\nthat there is a refractoriness it will never spike completely like a Poisson process.\nPoisson type spike irregularity will be achieved when mean is small (far from the\nspike threshold) and fluctuations are large. This will achieved when excitatory\nand inhibitory rates are balanced -- i.e. ratio of exc and inh. spike rate is\nconstant as you vary the inout rate.\n\n2. Firing rate will increase because fluctuations will increase as we increase\nexc. and inh. rates. But if synapses are modelled as conductance as opposed to\ncurrents, fluctuations may start decrease at high input rates because neuron time\nconstant will drop.\n\n\"\"\";\n```\n\n---\n# Section 2: Short-term synaptic plasticity\nAbove, we modeled synapses with fixed weights. Now we will explore synapses whose weight change in some input conditions. \n\nShort-term plasticity (STP) is a phenomenon in which synaptic efficacy changes over time in a way that reflects the history of presynaptic activity. Two types of STP, with opposite effects on synaptic efficacy, have been experimentally observed. They are known as Short-Term Depression (STD) and Short-Term Facilitation (STF).\n\nThe mathematical model (_for more information see [here](http://www.scholarpedia.org/article/Short-term_synaptic_plasticity)_) of STP is based on the concept of a limited pool of synaptic resources available for transmission ($R$), such as, for example, the overall amount of synaptic vesicles at the presynaptic terminals. The amount of presynaptic resource changes in a dynamic fashion depending on the recent history of spikes. \n\nFollowing a presynaptic spike, (i) the fraction $u$ (release probability) of the available pool to be utilized increases due to spike-induced calcium influx to the presynaptic terminal, after which (ii) $u$ is consumed to increase the post-synaptic conductance. Between spikes, $u$ decays back to zero with time constant $\\tau_f$ and $R$ recovers to 1 with time constant $\\tau_d$. In summary, the dynamics of excitatory (subscript $E$) STP are given by:\n\n\\\\\n\n\\begin{eqnarray}\n&& \\frac{du_E}{dt} &=& -\\frac{u_E}{\\tau_f} + U_0(1-u_E^-)\\delta(t-t_{\\rm sp}) \\\\[.5mm]\n&& \\frac{dR_E}{dt} &=& \\frac{1-R_E}{\\tau_d} - u_E^+ R_E^- \\delta(t-t_{\\rm sp}) \\qquad (6) \\\\[.5mm] \n&& \\frac{dg_E(t)}{dt} &=& -\\frac{g_E}{\\tau_E} + \\bar{g}_E u_E^+ R_E^- \\delta(t-t_{\\rm sp})\n\\end{eqnarray}\n\n\\\\\n\nwhere $U_0$ is a constant determining the increment of $u$ produced by a spike. $u_E^-$ and $R_E^-$ denote the corresponding values just before the spike arrives, whereas $u_E^+$ refers to the moment right after the spike. $\\bar{g}_E$ denotes the maximum excitatory conductane, and $g_E(t)$ is calculated for all spiketimes $k$, and decays over time with a time constant $\\tau_{E}$. Similarly, one can obtain the dynamics of inhibitory STP (i.e., by replacing the subscript $E$ with $I$).\n\n\nThe interplay between the dynamics of $u$ and $R$ determines whether the joint effect of $uR$ is dominated by *depression* or *facilitation*. In the parameter regime of $\\tau_d \\gg \\tau_f$ and for large $U_0$, an initial spike incurs a large drop in $R$ that takes a long time to recover; therefore, the synapse is STD-dominated. In the regime of $\\tau_d \\ll \\tau_f$ and for small $U_0$, the synaptic efficacy is increased gradually by spikes, and consequently, the synapse is STF-dominated. This phenomenological model successfully reproduces the kinetic dynamics of depressed and facilitated synapses observed in many cortical areas.\n\n## Exercise 2: Compute $du$, $dR$ and $dg$\n\nAs we learned in several previous tutorials, the Euler numerical integration method involves the calculation of each derivative at step $n$:\n\n\\\\\n\n\\begin{eqnarray}\ndu_E &=& -\\frac{u_E[t]}{\\tau_f} dt + U_0(1-u_E[t])\\cdot \\text{sp_or_not[t+dt]} \\\\\ndR_E &=& \\frac{1-R_E[t]}{\\tau_d} dt - u_E[t+dt]R_E[t]\\cdot \\text{sp_or_not[t+dt]} \\\\\ndg_E &=& -\\frac{g_E[t]}{\\tau_{E}} dt + \\bar{g}_Eu_E[t+dt]R_E[t]\\cdot \\text{sp_or_not[t+dt]} \\\\\n\\end{eqnarray}\n\n\\\\\n\nwhere $\\text{sp_or_not}=1$ if there's a spike in the time window $dt$, and $\\text{sp_or_not}=0$ otherwise. In addition, note that any spike train generated by our `Poisson_generator` is binary. Then, the values are updated:\n\n\\\\\n\n\\begin{eqnarray}\n u_E[t+dt] &=& u_E[t] + du_E \\\\\n R_E[t+dt] &=& R_E[t] + dR_E \\\\\n g_E[t+dt] &=& g_E[t] + dg_E \\\\\n\\end{eqnarray}\n\n\\\\\n\nSimilarly, one can obtain the dynamics of inhibitory conductance.\n\n\n\n```python\ndef dynamic_syn(g_bar, tau_syn, U0, tau_d, tau_f, pre_spike_train, dt):\n \"\"\"\n Short-term synaptic plasticity\n\n Args:\n g_bar : synaptic conductance strength\n tau_syn : synaptic time constant [ms]\n U0 : synaptic release probability at rest\n tau_d : synaptic depression time constant of x [ms]\n tau_f : synaptic facilitation time constantr of u [ms]\n pre_spike_train : total spike train (number) input\n from presynaptic neuron\n dt : time step [ms]\n\n Returns:\n u : usage of releasable neurotransmitter\n R : fraction of synaptic neurotransmitter resources available\n g : postsynaptic conductance\n\n \"\"\"\n\n Lt = len(pre_spike_train)\n # Initialize\n u = np.zeros(Lt)\n R = np.zeros(Lt)\n R[0] = 1.\n g = np.zeros(Lt)\n\n # simulation\n for it in range(Lt - 1):\n\n #########################################################################\n ## TODO for students: compute du, dx and dg, remove NotImplementedError #\n # Note pre_spike_train[i] is binary, i.e., sp_or_not in the i-th timebin\n # Fill out function and remove\n raise NotImplementedError(\"Student excercise: compute the STP dynamics\")\n #########################################################################\n # Compute du\n du = ...\n u[it + 1] = u[it] + du\n # Compute dR\n dR = ...\n R[it + 1] = R[it] + dR\n # Compute dg\n dg = ...\n g[it + 1] = g[it] + dg\n\n return u, R, g\n\n\n# Uncomment this line after completing the dynamic_syn function\n# _ = my_illus_STD(Poi_or_reg=False, rate=20., U0=0.5, tau_d=100., tau_f=50.)\n```\n\n\n```python\n# to_remove solution\ndef dynamic_syn(g_bar, tau_syn, U0, tau_d, tau_f, pre_spike_train, dt):\n \"\"\"\n Short-term synaptic plasticity\n\n Args:\n g_bar : synaptic conductance strength\n tau_syn : synaptic time constant [ms]\n U0 : synaptic release probability at rest\n tau_d : synaptic depression time constant of x [ms]\n tau_f : synaptic facilitation time constantr of u [ms]\n pre_spike_train : total spike train (number) input\n from presynaptic neuron\n dt : time step [ms]\n\n Returns:\n u : usage of releasable neurotransmitter\n R : fraction of synaptic neurotransmitter resources available\n g : postsynaptic conductance\n\n \"\"\"\n\n Lt = len(pre_spike_train)\n # Initialize\n u = np.zeros(Lt)\n R = np.zeros(Lt)\n R[0] = 1.\n g = np.zeros(Lt)\n\n # simulation\n for it in range(Lt - 1):\n # Compute du\n du = -(dt / tau_f) * u[it] + U0 * (1.0 - u[it]) * pre_spike_train[it + 1]\n u[it + 1] = u[it] + du\n # Compute dR\n dR = (dt / tau_d) * (1.0 - R[it]) - u[it + 1] * R[it] * pre_spike_train[it + 1]\n R[it + 1] = R[it] + dR\n # Compute dg\n dg = -(dt / tau_syn) * g[it] + g_bar * R[it] * u[it + 1] * pre_spike_train[it + 1]\n g[it + 1] = g[it] + dg\n\n return u, R, g\n\n\nwith plt.xkcd():\n _ = my_illus_STD(Poi_or_reg=False, rate=20., U0=0.5, tau_d=100., tau_f=50.)\n```\n\n## Section 2.1: Short-term syaptic depression (STD)\n\n\n### Interactive Demo: STD Explorer with input rate\nBelow, an interactive demo that shows how Short-term synaptic depression (STD) changes for different firing rates of the presynaptic spike train and how the amplitude synaptic conductance $g$ changes with every incoming spike until it reaches its stationary state.\n\nDoes it matter if the neuron fires in a Poisson manner, rather than regularly?\n\n**Note:** `Poi_or_Reg=1`: for *Posisson type* and `Poi_or_Reg=0`: for *regular* presynaptic spikes.\n\n\n```python\n#@title\n\n#@markdown Make sure you execute this cell to enable the widget!\n\n\ndef my_STD_diff_rate(rate, Poi_or_Reg):\n _ = my_illus_STD(Poi_or_reg=Poi_or_Reg, rate=rate)\n\n\n_ = widgets.interact(my_STD_diff_rate, rate=(10., 100.1, 5.),\n Poi_or_Reg=(0, 1, 1))\n```\n\n\n```python\n# to_remove explanation\n\n\"\"\"\nDiscussion:\n\nIncreasing the input rate, we decrease the synaptic efficacy, i.e., the synaptic\nconductance decreases. This is the case for both Poisson or a regular spiking input.\nIn case of regular spiking, the synaptic conductance reaches a steady state. This\nwill not happen in the case of Poisson type spikes.\n\"\"\";\n```\n\n### Synaptic depression and presynaptic firing rate\nOnce, I asked an experimentalist about the experimental values of the PSP amplitude produced by a connection between two neocortical excitatory neurons. She asked: \"At what frequency?\" I was confused, but you will understand her question, now that you know that PSP amplitude depends on the spike history, and therefore on the spike rate of the presynaptic neuron. \n\nHere, we will study how the ratio of the synaptic conductance corresponding to the first and 10th spikes change as a function of the presynaptic firing rate (experimentalists often take the ratio of first and second PSPs). \n\nFor computational efficiency, we assume that the presynaptic spikes are regular. This assumption means that we do not have to run multiple trials. \n\n\n```python\n# @markdown STD conductance ratio with different input rate\n# Regular firing rate\ninput_rate = np.arange(5., 40.1, 5.)\ng_1 = np.zeros(len(input_rate)) # record the the PSP at 1st spike\ng_2 = np.zeros(len(input_rate)) # record the the PSP at 10th spike\n\nfor ii in range(len(input_rate)):\n g_1[ii], g_2[ii] = my_illus_STD(Poi_or_reg=False, rate=input_rate[ii],\n plot_out=False, U0=0.5, tau_d=100., tau_f=50)\n\nplt.figure(figsize=(11, 4.5))\n\nplt.subplot(121)\nplt.plot(input_rate, g_1, 'm-o', label='1st Spike')\nplt.plot(input_rate, g_2, 'c-o', label='10th Spike')\nplt.xlabel('Rate [Hz]')\nplt.ylabel('Conductance [nS]')\nplt.legend()\n\nplt.subplot(122)\nplt.plot(input_rate, g_2 / g_1, 'b-o')\nplt.xlabel('Rate [Hz]')\nplt.ylabel(r'Conductance ratio $g_{10}/g_{1}$')\nplt.tight_layout()\n```\n\nAs we increase the input rate the ratio of the first to tenth spike is increased, because the tenth spike conductance becomes smaller. This is a clear evidence of synaptic depression, as using the same amount of current has a smaller effect on the neuron.\n\n## Section 2.2: Short-term synaptic facilitation (STF)\n\n### Interactive Demo: STF explorer with input rate\nBelow, we see an illustration of a short-term facilitation example. Take note of the change in the synaptic variables: `U_0`, `tau_d`, and `tau_f`.\n\n- for STD, `U0=0.5, tau_d=100., tau_f=50.`\n\n- for STP, `U0=0.2, tau_d=100., tau_f=750.`\n \nHow does the synaptic conductance change as we change the input rate? What do you observe in the case of a regular input and a Poisson type one? \n\n\n```python\n# @title\n\n# @markdown Make sure you execute this cell to enable the widget!\n\n\ndef my_STD_diff_rate(rate, Poi_or_Reg):\n _ = my_illus_STD(Poi_or_reg=Poi_or_Reg, rate=rate, U0=0.2, tau_d=100., tau_f=750.)\n\n\n_ = widgets.interact(my_STD_diff_rate, rate=(4., 40.1, 2.),\n Poi_or_Reg=(0, 1, 1))\n```\n\n### Synaptic facilitation and presynaptic firing rate\n\nHere, we will study how the ratio of the synaptic conductance corresponding to the $1^{st}$ and $10^{th}$ spike changes as a function of the presynaptic rate. \n\n\n```python\n# @title STF conductance ratio with different input rates\n# Regular firing rate\ninput_rate = np.arange(2., 40.1, 2.)\ng_1 = np.zeros(len(input_rate)) # record the the PSP at 1st spike\ng_2 = np.zeros(len(input_rate)) # record the the PSP at 10th spike\n\nfor ii in range(len(input_rate)):\n g_1[ii], g_2[ii] = my_illus_STD(Poi_or_reg=False, rate=input_rate[ii],\n plot_out=False,\n U0=0.2, tau_d=100., tau_f=750.)\n\nplt.figure(figsize=(11, 4.5))\n\nplt.subplot(121)\nplt.plot(input_rate, g_1, 'm-o', label='1st Spike')\nplt.plot(input_rate, g_2, 'c-o', label='10th Spike')\nplt.xlabel('Rate [Hz]')\nplt.ylabel('Conductance [nS]')\nplt.legend()\n\nplt.subplot(122)\nplt.plot(input_rate, g_2 / g_1, 'b-o',)\nplt.xlabel('Rate [Hz]')\nplt.ylabel(r'Conductance ratio $g_{10}/g_{1}$')\nplt.tight_layout()\n```\n\n## Think!\n\nWhy does the ratio of the first and tenth spike conductance changes in a non-monotonic fashion for synapses with STF, even though it decreases monotonically for synapses with STD?\n\n\n```python\n# to_remove explanation\n\n\"\"\"\nDiscussion:\n\nBecause we have a facilitatory synapses, as the input rate increases synaptic\nresources released per spike also increase. Therefore, we expect that the synaptic\nconductance will increase with input rate. However, total synaptic resources are\nfinite. And they recover in a finite time. Therefore, at high frequency inputs\nsynaptic resources are rapidly deleted at a higher rate than their recovery, so\nafter first few spikes, only a small number of synaptic resources are left. This\nresults in decrease in the steady-state synaptic conductance at high frequency inputs.\n\"\"\";\n```\n\n---\n# Summary\n\nCongratulations! You have just finished the last tutorial of this day. Here, we saw how to model conductance-based synapses and also how to incorporate short-term dynamics in synaptic weights. \n\nWe covered the:\n\n- static synapses and how excitation and inhibition affect the neuronal output\n- mean- or fluctuation-driven regimes\n- short-term dynamics of synapses (both facilitation and depression)\n\nFinally, we incorporated all the aforementioned tools to study how a change in presynaptic firing history affects the synaptic weights!\n\nThere are many interesting things that you can try on your own to develop a deeper understanding of biological synapses. A couple of those are mentioned below in the optional boxes -- if you have time.\n\nBut now it is time to explore another important feature of biological synapses, i.e., spike timing dependent synaptic plasticity (go to the next tutorial).\n\n---\n# Bonus 1: Conductance-based LIF with STP\n\n\nPreviously, we looked only at how the presynaptic firing rate affects the presynaptic resource availability and thereby the synaptic conductance. It is straightforward to imagine that, while the synaptic conductances are changing, the output of the postsynaptic neuron will change as well. \n\nSo, let's put the STP on synapses impinging on an LIF neuron and see what happens. \n\n\n```python\n# @title Function for conductance-based LIF neuron with STP-synapses\n\ndef run_LIF_cond_STP(pars, I_inj, pre_spike_train_ex, pre_spike_train_in):\n \"\"\"\n conductance-based LIF dynamics\n\n Args:\n pars : parameter dictionary\n I_inj : injected current [pA]\n The injected current here can be a value or an array\n pre_spike_train_ex : spike train input from presynaptic excitatory neuron (binary)\n pre_spike_train_in : spike train input from presynaptic inhibitory neuron (binary)\n\n Returns:\n rec_spikes : spike times\n rec_v : mebrane potential\n gE : postsynaptic excitatory conductance\n gI : postsynaptic inhibitory conductance\n\n \"\"\"\n\n # Retrieve parameters\n V_th, V_reset = pars['V_th'], pars['V_reset']\n tau_m, g_L = pars['tau_m'], pars['g_L']\n V_init, V_L = pars['V_init'], pars['E_L']\n gE_bar, gI_bar = pars['gE_bar'], pars['gI_bar']\n U0E, tau_dE, tau_fE = pars['U0_E'], pars['tau_d_E'], pars['tau_f_E']\n U0I, tau_dI, tau_fI = pars['U0_I'], pars['tau_d_I'], pars['tau_f_I']\n VE, VI = pars['VE'], pars['VI']\n tau_syn_E, tau_syn_I = pars['tau_syn_E'], pars['tau_syn_I']\n tref = pars['tref']\n\n dt, range_t = pars['dt'], pars['range_t']\n Lt = range_t.size\n\n nE = pre_spike_train_ex.shape[0]\n nI = pre_spike_train_in.shape[0]\n\n # compute conductance Excitatory synapses\n uE = np.zeros((nE, Lt))\n RE = np.zeros((nE, Lt))\n gE = np.zeros((nE, Lt))\n for ie in range(nE):\n u, R, g = dynamic_syn(gE_bar, tau_syn_E, U0E, tau_dE, tau_fE,\n pre_spike_train_ex[ie, :], dt)\n\n uE[ie, :], RE[ie, :], gE[ie, :] = u, R, g\n\n gE_total = gE.sum(axis=0)\n\n # compute conductance Inhibitory synapses\n uI = np.zeros((nI, Lt))\n RI = np.zeros((nI, Lt))\n gI = np.zeros((nI, Lt))\n for ii in range(nI):\n u, R, g = dynamic_syn(gI_bar, tau_syn_I, U0I, tau_dI, tau_fI,\n pre_spike_train_in[ii, :], dt)\n\n uI[ii, :], RI[ii, :], gI[ii, :] = u, R, g\n\n gI_total = gI.sum(axis=0)\n\n # Initialize\n v = np.zeros(Lt)\n v[0] = V_init\n Iinj = I_inj * np.ones(Lt) # ensure I has length Lt\n\n # simulation\n rec_spikes = [] # recording spike times\n tr = 0.\n for it in range(Lt - 1):\n if tr > 0:\n v[it] = V_reset\n tr = tr - 1\n elif v[it] >= V_th: # reset voltage and record spike event\n rec_spikes.append(it)\n v[it] = V_reset\n tr = tref / dt\n\n # calculate the increment of the membrane potential\n dv = (dt / tau_m) * (-(v[it] - V_L) \\\n - (gE_total[it + 1] / g_L) * (v[it] - VE) \\\n - (gI_total[it + 1] / g_L) * (v[it] - VI) + Iinj[it] / g_L)\n\n # update membrane potential\n v[it+1] = v[it] + dv\n\n rec_spikes = np.array(rec_spikes) * dt\n\n return v, rec_spikes, uE, RE, gE, RI, RI, gI\n\n\nprint(help(run_LIF_cond_STP))\n```\n\n## Simulation of a postsynaptic neuron with STP synapses driven by Poisson type spike trains\n\nHere we have assumed that both excitatory and inhibitory synapses show short-term depression. Change the nature of synapses and study how spike pattern variability changes.\nIn the interactive demo, `tau_d = 500*tau_ratio (ms)` and `tau_f = 300*tau_ratio (ms)`.\n\nYou should compare the output of this neuron with what you observed in the previous tutorial when synapses were assumed to be static. \n\n_Note: it will take slightly longer time to run each case_\n\n### Interactive Demo: LIF with STP Explorer\n\n\n```python\n# @title\n\n# @markdown Make sure you execute this cell to enable the widget!\n\n\ndef LIF_STP(tau_ratio):\n pars = default_pars(T=1000)\n pars['gE_bar'] = 1.2 * 4 # [nS]\n pars['VE'] = 0. # [mV]\n pars['tau_syn_E'] = 5. # [ms]\n pars['gI_bar'] = 1.6 * 4 # [nS]\n pars['VI'] = -80. # [ms]\n pars['tau_syn_I'] = 10. # [ms]\n\n # here we assume that both Exc and Inh synapses have synaptic depression\n pars['U0_E'] = 0.45\n pars['tau_d_E'] = 500. * tau_ratio # [ms]\n pars['tau_f_E'] = 300. * tau_ratio # [ms]\n\n pars['U0_I'] = 0.45\n pars['tau_d_I'] = 500. * tau_ratio # [ms]\n pars['tau_f_I'] = 300. * tau_ratio # [ms]\n\n pre_spike_train_ex = Poisson_generator(pars, rate=15, n=80)\n pre_spike_train_in = Poisson_generator(pars, rate=15, n=20) # 4:1\n\n v, rec_spikes, uE, RE, gE, uI, RI, gI = run_LIF_cond_STP(pars, 0,\n pre_spike_train_ex,\n pre_spike_train_in)\n\n t_plot_range = pars['range_t'] > 200\n\n plt.figure(figsize=(11, 7))\n plt.subplot(211)\n plot_volt_trace(pars, v, rec_spikes)\n\n plt.subplot(223)\n plt.plot(pars['range_t'][t_plot_range], gE.sum(axis=0)[t_plot_range], 'r')\n plt.xlabel('Time (ms)')\n plt.ylabel(r'$g_E$ (nS)')\n\n plt.subplot(224)\n plt.plot(pars['range_t'][t_plot_range], gI.sum(axis=0)[t_plot_range], 'b')\n plt.xlabel('Time (ms)')\n plt.ylabel(r'$g_I$ (nS)')\n\n plt.tight_layout()\n\n\n_ = widgets.interact(LIF_STP, tau_ratio=(0.2, 1.1, 0.2))\n```\n\nWhen we vary the tau_ratio we are increasing `tau_f` and `tau_d` i.e. by increasing `tau_ratio` we are increasing the synaptic depression. The effect is same on both Exc and Inh conductances.\nThis is visible as a clear decrease in the firing rate of the neuron from 300-400ms onwards.\n\nNot much happens in the beginning because synaptic depression takes some time to become visible.\n\nIt is curious that while both excitatory and inhibitory conductances have depressed but output firing rate has still decreased.\n\nThere are two explanations of this:\n1. excitation has depressed more than the inhibition from their starting values.\n2. because synaptic conductances have depressed, membrane fluctuation size has decreased.\n\nWhich is more likely reason? Think.\n\n---\n# Bonus 2: STP Synapse Parameter Exploration\n\nVary the parameters of the above simulation and observe the spiking pattern of the postsynaptic neuron. \nWill the neuron show higher irregularity if the synapses have STP? If yes, what should be the nature of STP on static and dynamic synapses, respectively? \n\nCalculate the $CV_{\\rm ISI}$ for different `tau_ratio` after simulating the LIF neuron with STP (Hint:`run_LIF_cond_STP` help you understand the irregularity).\n\n\n## Functional implications of short-term dynamics of synapses\nAs you have seen above, if the firing rate is stationary, the synaptic conductance quickly reaches a fixed point. On the other hand, if the firing rate transiently changes, synaptic conductance will vary -- even if the change is as short as a single inter-spike-interval. Such small changes can be observed in a single neuron when input spikes are regular and periodic. If the input spikes are Poissonian, then one may have to perform an average over several neurons.\n\n_Come up with other functions that short-term dynamics of synapses can be used to implement and implement them._\n", "meta": {"hexsha": "0ae1a7f43dc9b046f70f4a6170751b682794ef19", "size": 65387, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W2D3_RealNeurons/W2D3_Tutorial3.ipynb", "max_stars_repo_name": "vasudev-sharma/course-content", "max_stars_repo_head_hexsha": "46fb9be49da52acb5df252dda43f11b6d1fe827f", "max_stars_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tutorials/W2D3_RealNeurons/W2D3_Tutorial3.ipynb", "max_issues_repo_name": "vasudev-sharma/course-content", "max_issues_repo_head_hexsha": "46fb9be49da52acb5df252dda43f11b6d1fe827f", "max_issues_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/W2D3_RealNeurons/W2D3_Tutorial3.ipynb", "max_forks_repo_name": "vasudev-sharma/course-content", "max_forks_repo_head_hexsha": "46fb9be49da52acb5df252dda43f11b6d1fe827f", "max_forks_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 41.4629042486, "max_line_length": 648, "alphanum_fraction": 0.5753590163, "converted": true, "num_tokens": 12713, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.66192288918838, "lm_q1q2_score": 0.4404504795421454}} {"text": "# \u7b2c\u5341\u4e5d\u8bb2\uff1a\u5fae\u5206\u52a8\u6001\u89c4\u5212\u53ca\u7ebf\u6027\u4e8c\u6b21\u578b\u9ad8\u65af\n\n\u5728\u7ee7\u7eed\u5b66\u4e60\u4e4b\u524d\uff0c\u6211\u4eec\u5148\u6765\u56de\u987e\u4e00\u4e0b[\u4e0a\u4e00\u8bb2](chapter18.ipynb)\u7684\u5185\u5bb9\uff1a\n* \u5176\u76ee\u6807\u662f$\\displaystyle\\max_a\\mathrm E\\left[R^{(0)}(s_0,a_0)+R^{(1)}(s_1,a_1)+\\cdots+R^{(T)}(s_T,a_T)\\right]$\uff1b\n* \u4e4b\u540e\u6211\u4eec\u5e94\u7528\u4e86\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\uff1a\n $$\n \\begin{align}\n V_T^*(s)&=\\max_{a_T}R(s_T,a_T)\\tag{1}\\\\\n V_t^*(s)&=\\max_a\\left(R(s,a)+\\sum_{s'}P_{sa}^{(t)}\\left(s'\\right)V_{t+1}^*\\left(s'\\right)\\right)\\tag{2}\\\\\n \\pi_t^*(s)&=\\arg\\max_a\\left(R(s,a)+\\sum_{s'}P_{sa}^{(t)}\\left(s'\\right)V_{t+1}^*\\left(s'\\right)\\right)\\tag{3}\n \\end{align}\n $$\n \u5176\u4e2d$(1)$\u5f0f\u662f\u6709\u9650\u65f6\u57dfMDP\u7684\u6700\u540e\u4e00\u6b65\uff1b\u5f97\u51fa\u6700\u540e\u4e00\u6b65\u540e\u5c31\u53ef\u4ee5\u4f7f\u7528$(2)$\u5f0f\u5411\u540e\u9012\u63a8\u6bcf\u4e00\u4e2a$V_{T-1},\\cdots,V_{0}$\uff1b\u6700\u540e\uff0c\u5728\u4f7f\u7528$\\arg\\max$\u5bf9\u6bcf\u4e00\u6b65\u4ef7\u503c\u51fd\u6570\u6700\u5927\u5316\u7684\u90e8\u5206\u505a\u8fd0\u7b97\uff0c\u5c31\u53ef\u4ee5\u5f97\u5230\u6bcf\u4e00\u6b65\u7684\u6700\u4f18\u7b56\u7565\u3002\n* \u63a5\u4e0b\u6765\u6211\u4eec\u4ecb\u7ecd\u4e86\u4e00\u4e2a\u5177\u4f53\u7684LQR\u95ee\u9898\uff1a\n * \u72b6\u6001\u4e3a$S\\in\\mathbb R^n$\uff1b\n * \u52a8\u4f5c\u4e3a$A\\in\\mathbb R^d$\uff1b\n * \u5c06\u4e0b\u4e00\u6b65\u4f5c\u4e3a\u5173\u4e8e\u5f53\u524d\u72b6\u6001\u53ca\u52a8\u4f5c\u7684\u51fd\u6570$s_{t+1}=A_ts_t+B_ta_t+w_t$\uff0c\u6b64\u5904\u7684$w_t\\sim\\mathcal N(0,\\varSigma_w)$\u662f\u4e00\u4e2a\u671f\u671b\u4e3a\u96f6\u3001\u534f\u65b9\u5dee\u4e3a$\\varSigma_w$\u7684\u9ad8\u65af\u566a\u97f3\u9879\u3002\n* \u4e0a\u9762\u5b9a\u4e49\u4e2d\u6709\u4e86\u5173\u4e8e\u5f53\u524d\u72b6\u6001\u53ca\u52a8\u4f5c\u7684\u51fd\u6570$s_{t+1}=f(s_t,a_t)$\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u9009\u62e9\u4e00\u4e2a\u7cfb\u7edf\u957f\u65f6\u95f4\u4fdd\u6301\u7684\u72b6\u6001\uff08\u7cfb\u7edf\u901a\u5e38\u90fd\u4f1a\u90fd\u5904\u4e8e\u8fd9\u79cd\u72b6\u6001\uff09$(\\bar s_t,\\bar a_t)$\uff0c\u5728\u8be5\u70b9\u4f7f\u7528\u7ebf\u6027\u51fd\u6570\u8fd1\u4f3c\u975e\u7ebf\u6027\u51fd\u6570$s_{t+1}=f(s_t,a_t)$\uff0c\u8fdb\u800c\u5f97\u5230$\\displaystyle s_{t+1}\\approx f(\\bar s_t,\\bar a_t)+(\\nabla_sf(\\bar s_t,\\bar a_t))^T(s_t-\\bar s_t)+(\\nabla_af(\\bar s_t,\\bar a_t))^T(a_t-\\bar a_t)$\u3002\u6700\u7ec8\u80fd\u591f\u5728$(\\bar s_t,\\bar a_t)$\u9644\u8fd1\u5f97\u5230\u4e00\u4e2a\u7ebf\u6027\u51fd\u6570$s_{t+1}=A_ts_t+B_ta_t$\u3002\n* LQR\u7684\u5956\u52b1\u51fd\u6570\u4e3a$\\displaystyle R^{(t)}(s_t,a_t)=-\\left(s_t^TU_ts_t+a^TV_ta_t\\right)$\uff0c\u5176\u4e2d\u77e9\u9635$U,V$\u662f\u534a\u6b63\u5b9a\u7684\uff0c\u6240\u4ee5\u8fd9\u4e2a\u5956\u52b1\u51fd\u6570\u603b\u662f\u975e\u6b63\u6570\uff1b\n* \u5bf9LQR\u4f7f\u7528\u4e0a\u9762\u7684\u52a8\u6001\u89c4\u5212\u5f97\u5230\u6c42\u89e3\u65b9\u6cd5\uff1a\n * \u4f7f\u7528$\\varPhi_t=-U_t,\\varPsi_t=0$\u521d\u59cb\u5316$V_T^*$\uff1b\n * \u5411\u540e\u9012\u63a8\u8ba1\u7b97$\\varPhi_t,\\varPsi_t$\uff0c\u4f7f\u7528\u79bb\u6563\u65f6\u95f4Riccati\u65b9\u7a0b\uff0c\u5c06\u5176\u5f53\u505a\u5173\u4e8e$\\varPhi_{t+1},\\varPsi_{t+1}$\u7684\u7ebf\u6027\u51fd\u6570\uff1b\uff08\u4f9d\u6b21\u6c42\u51fa$t=T-1,T-2,\\cdots,0$\u3002\uff09\n $$\n \\begin{align}\n \\varPhi_t&=A_t^T\\left(\\varPhi_{t+1}-\\varPhi_{t+1}B_t\\left(B_t^T\\varPhi_{t+1}B_t-V_t\\right)^{-1}B_t\\varPhi_{t+1}\\right)A_t-U_t\\\\\n \\varPsi_t&=-\\mathrm{tr}\\varSigma_w\\varPhi_{t+1}+\\varPsi_{t+1}\n \\end{align}\n $$\n * \u8ba1\u7b97$\\displaystyle L_t=\\left(B_t^T\\varPhi_{t+1}B_t-V_t\\right)^{-1}B_t^T\\varPhi_{t+1}A_t$\uff0c\u8fd9\u662f\u4e00\u4e2a\u5173\u4e8e$\\varPhi_{t+1},\\varPsi_{t+1}$\u7684\u51fd\u6570\uff1b\n * \u8ba1\u7b97$\\pi^*(s_t)=L_ts_t$\uff0c\u8fd9\u662f\u4e00\u4e2a\u5173\u4e8e$s_t$\u7684\u7ebf\u6027\u51fd\u6570\uff0c\u7cfb\u6570\u662f$L_t$\u3002\n \n \u8fd9\u4e2a\u7b97\u6cd5\u4e2d\u6709\u8da3\u7684\u4e00\u70b9\u662f\uff0c$L_t$\u4e0d\u4f9d\u9760$\\varPsi_t$\uff0c$\\varPhi_t$\u4e5f\u4e0d\u4f9d\u8d56\u4e8e$\\varPsi_t$\u3002\u6240\u4ee5\uff0c\u5373\u4f7f\u6211\u4eec\u4ece\u4e0d\u8003\u8651$\\varPsi$\u4e5f\u4e0d\u4f1a\u5f71\u54cd\u6700\u7ec8\u7684\u7ed3\u679c\u3002\u53e6\u4e00\u4e2a\u6709\u8da3\u7684\u5730\u65b9\u5728\u4e8e$\\varSigma_w$\u53ea\u5728$\\varPsi_t$\u4e2d\u51fa\u73b0\uff0c\u518d\u770bLQR\u5b9a\u4e49\u4e2d\u7684$s_{t+1}=A_ts_t+B_ta_t+w_t$\u53ef\u4ee5\u5f97\u51fa\uff0c\u5373\u4f7f\u5728\u4e0d\u77e5\u9053\u566a\u97f3\u9879\u534f\u65b9\u5dee\u7684\u524d\u63d0\u4e0b\uff0c\u4e5f\u53ef\u4ee5\u6b63\u786e\u7684\u6c42\u51fa\u6700\u4f18\u7b56\u7565\uff08\u4f46\u4e0d\u8981\u5fd8\u4e86\u6700\u4f18\u4ef7\u503c\u51fd\u6570\u662f\u53d7$\\varPsi$\u5f71\u54cd\u7684\uff09\u3002\u6700\u540e\uff0c\u8fd9\u4e24\u70b9\u90fd\u662f\u5bf9\u8fd9\u4e2a\u7279\u6b8aLQR\u6a21\u578b\uff08\u975e\u7ebf\u6027\u52a8\u529b\u7cfb\u7edf\uff09\u72ec\u6709\u7684\uff0c\u4e00\u65e6\u6211\u4eec\u5bf9\u6a21\u578b\u505a\u51fa\u6539\u53d8\uff0c\u5219\u8fd9\u4e24\u4e2a\u7279\u6027\u5c06\u4e0d\u590d\u5b58\u5728\uff0c\u4e5f\u5c31\u662f\u53ea\u6709\u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\u624d\u6709\u201c\u6700\u4f18\u7b56\u7565\u4e0d\u53d7\u566a\u97f3\u9879\u534f\u65b9\u5dee\u5f71\u54cd\u201d\u3002\u5728\u540e\u9762\u8ba8\u8bbaKalman\u6ee4\u6ce2\u7684\u65f6\u5019\u4f1a\u7528\u5230\u8fd9\u4e2a\u6027\u8d28\u3002\n\n## 8. \u5f3a\u5316\u5b66\u4e60\u7b97\u6cd5\u7684\u8c03\u8bd5\n\n\u6211\u4eec\u5728[\u7b2c\u5341\u4e00\u8bb2](chapter11.ipynb)\u4e2d\uff0c\u5728\u5173\u4e8e\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u7684\u8c03\u8bd5\u4ecb\u7ecd\u8fc7\u76f4\u5347\u673a\u6a21\u578b\u63d0\u5230\uff1a\n\n1. \u642d\u5efa\u4e00\u4e2a\u9065\u63a7\u76f4\u5347\u673a\u6a21\u62df\u5668\uff0c\u4e3b\u8981\u662f\u5bf9\u72b6\u6001\u8f6c\u6362\u6982\u7387$P_{sa}$\u8fdb\u884c\u5efa\u6a21\uff08\u6bd4\u5982\u901a\u8fc7\u5b66\u4e60\u7269\u7406\u77e5\u8bc6\uff0c\u4f9d\u7167\u7a7a\u6c14\u52a8\u529b\u5b66\u539f\u7406\u7f16\u5199\u6a21\u62df\u5668\uff1b\u6216\u8005\u4e5f\u53ef\u4ee5\u6536\u96c6\u5927\u91cf\u8bd5\u9a8c\u6570\u636e\uff0c\u901a\u8fc7\u62df\u5408\u8fd9\u4e9b\u6570\u636e\u4e00\u4e2a\u7ebf\u6027\u6216\u975e\u7ebf\u6027\u6a21\u578b\uff0c\u4ee5\u5f97\u5230\u4e00\u4e2a\u7528\u5f53\u524d\u72b6\u6001\u3001\u5f53\u524d\u64cd\u4f5c\u8868\u793a\u4e0b\u4e00\u6b65\u72b6\u6001\u7684\u51fd\u6570\uff09\uff1b\n2. \u9009\u62e9\u5956\u52b1\u51fd\u6570\uff0c\u6bd4\u5982$R(s)=\\left\\lVert s-s_\\mathrm{desired}\\right\\rVert^2$\uff08$s$\u4ee3\u8868\u76f4\u5347\u673a\u5f53\u524d\u7684\u4f4d\u7f6e\uff0c\u8fd9\u91cc\u793a\u4f8b\u7684\u51fd\u6570\u8868\u793a\u76f4\u5347\u673a\u5b9e\u9645\u4f4d\u7f6e\u4e0e\u671f\u671b\u4f4d\u7f6e\u7684\u5e73\u65b9\u8bef\u5dee\uff0c\u6211\u4eec\u5728[\u4e0a\u4e00\u8bb2](chapter18.ipynb)\u7684LQR\u4e2d\u5df2\u7ecf\u4f7f\u7528\u8fc7\u8fd9\u4e2a\u5956\u52b1\u51fd\u6570\u4e86\uff09\uff1b\n3. \u5728\u6a21\u62df\u5668\u4e2d\u8fd0\u884c\u5f3a\u5316\u5b66\u4e60\u7b97\u6cd5\u63a7\u5236\u76f4\u5347\u673a\u5e76\u5c1d\u8bd5\u6700\u5927\u5316\u5956\u52b1\u51fd\u6570\uff0c$\\mathrm E\\left[R(s_0)+R(s_1)+\\cdots+R(s_T)\\right]$\uff0c\u8fdb\u800c\u6c42\u5f97\u5bf9\u5e94\u7684\u7b56\u7565$\\pi_\\mathrm{RL}$\u3002\n\n\u73b0\u5728\uff0c\u5047\u8bbe\u6211\u4eec\u5df2\u7ecf\u6309\u7167\u4e0a\u8ff0\u6b65\u9aa4\u5f97\u5230\u4e86\u63a7\u5236\u6a21\u578b\uff0c\u4f46\u662f\u53d1\u73b0\u8be5\u6a21\u578b\u6bd4\u4eba\u7c7b\u76f4\u63a5\u64cd\u4f5c\u5dee\u5f88\u591a\uff0c\u63a5\u4e0b\u6765\u5e94\u8be5\u600e\u4e48\u529e\u5462\uff1f\u6211\u4eec\u53ef\u80fd\u4f1a\uff1a\n* \u6539\u8fdb\u6a21\u62df\u5668\uff08\u4e5f\u8bb8\u6a21\u62df\u5668\u6240\u7528\u7684\u6a21\u578b\u4e0d\u662f\u7ebf\u6027\u6216\u975e\u7ebf\u6027\u7684\uff1b\u4e5f\u8bb8\u9700\u8981\u6536\u96c6\u66f4\u591a\u6570\u636e\u6765\u62df\u5408\u6a21\u578b\uff1b\u4e5f\u8bb8\u9700\u8981\u8c03\u6574\u62df\u5408\u6a21\u578b\u65f6\u6240\u4f7f\u7528\u7684\u7279\u5f81\u503c\uff09\uff1b\n* \u4fee\u6539\u5956\u52b1\u51fd\u6570$R$\uff08\u4e5f\u8bb8\u5b83\u4e0d\u53ea\u662f\u4e00\u4e2a\u4e8c\u6b21\u51fd\u6570\uff09\uff1b\n* \u6539\u8fdb\u5b66\u4e60\u7b97\u6cd5\uff08\u4e5f\u8bb8RL\u5e76\u4e0d\u9002\u5408\u8fd9\u4e2a\u95ee\u9898\uff1b\u4e5f\u8bb8\u9700\u8981\u5bf9\u72b6\u6001\u8fdb\u884c\u66f4\u7cbe\u7ec6\u7684\u63cf\u8ff0\uff1b\u4e5f\u8bb8\u5728\u4ef7\u503c\u51fd\u6570\u8fd1\u4f3c\u8fc7\u7a0b\u4e2d\u4f7f\u7528\u7684\u7279\u5f81\u503c\u9700\u8981\u8c03\u6574\uff09\uff1b\n\n\u5bf9\u4e8e\u8c03\u6574\u7b97\u6cd5\u8fd9\u6837\u7684\u4efb\u52a1\uff0c\u6211\u4eec\u6709\u592a\u591a\u7684\u9009\u62e9\u65b9\u5411\uff0c\u5982\u679c\u9009\u62e9\u9519\u8bef\uff0c\u5219\u5c06\u4f1a\u6d6a\u8d39\u5927\u91cf\u7684\u65f6\u95f4\u7cbe\u529b\u3002\n\n\u63a5\u4e0b\u6765\u6211\u4eec\u5047\u8bbe\uff1a\n1. \u76f4\u5347\u673a\u6a21\u62df\u5668\u8db3\u591f\u7cbe\u786e\uff1b\n2. RL\u7b97\u6cd5\u80fd\u591f\u5728\u6a21\u62df\u5668\u4e2d\u6b63\u786e\u7684\u63a7\u5236\u98de\u673a\uff0c\u56e0\u6b64\u7b97\u6cd5\u6b63\u786e\u7684\u6700\u5927\u5316\u4e86\u9884\u671f\u603b\u6536\u76ca$V_t^{\\pi_\\mathrm{RL}}(s_0)=\\mathrm E\\left[R(s_0)+R(s_1)+\\cdots+R(s_T)\\mid\\pi_\\mathrm{RL},s_t=s_0\\right]$\uff1b\n3. \u6700\u5927\u5316\u9884\u671f\u603b\u6536\u76ca\u4e0e\u6a21\u578b\u6b63\u786e\u7684\u64cd\u4f5c\u98de\u673a\u5f3a\u76f8\u5173\u3002\n\n\u5982\u679c\u4e0a\u8ff0\u4e09\u6761\u5047\u8bbe\u5747\u6b63\u786e\uff0c\u5219\u53ef\u4ee5\u63a8\u51fa$\\pi_\\mathrm{RL}$\u80fd\u591f\u63a7\u5236\u597d\u771f\u6b63\u7684\u76f4\u5347\u673a\u3002\n\n\u4e8e\u662f\uff0c\u6211\u4eec\u53ef\u4ee5**\u4f9d\u6b21**\u505a\u51fa\u5982\u4e0b\u8bca\u65ad\uff1a\n1. \u9996\u5148\uff0c\u5982\u679c$\\pi_\\mathrm{RL}$\u5728\u6a21\u62df\u5668\u4e2d\u98de\u7684\u5f88\u597d\uff0c\u4f46\u662f\u5728\u5b9e\u9645\u98de\u673a\u4e0a\u5374\u8868\u73b0\u4e0d\u597d\uff0c\u5219\u8bf4\u660e\u662f\u6a21\u62df\u5668\u7684\u95ee\u9898\uff1b\n2. \u7136\u540e\uff0c\u7528$\\pi_\\mathrm{human}$\u8868\u793a\u4eba\u7c7b\u64cd\u7eb5\u8005\u5728\u64cd\u7eb5\u76f4\u5347\u673a\u65f6\u4f7f\u7528\u7684\u7b56\u7565\uff0c\u6bd4\u8f83\u4eba\u7c7b\u548c\u7b97\u6cd5\u7684\u7b56\u7565\u6240\u5bf9\u5e94\u7684\u4ef7\u503c\u51fd\u6570\uff0c\u5982\u679c$V^{\\pi_\\mathrm{RL}}\\lt V^{\\pi_\\mathrm{human}}$\uff0c\u5219\u8bf4\u660e\u662fRL\u7b97\u6cd5\u7684\u95ee\u9898\u3002\uff08\u7b97\u6cd5\u5e76\u6ca1\u6709\u6210\u529f\u7684\u6700\u5927\u5316\u9884\u671f\u603b\u6536\u76ca\u3002\uff09\n3. \u6700\u540e\uff0c\u5982\u679c\u7ecf\u6bd4\u8f83\u53d1\u73b0$V^{\\pi_\\mathrm{RL}}\\gt V^{\\pi_\\mathrm{human}}$\uff0c\u5219\u8bf4\u660e\u95ee\u9898\u51fa\u5728\u5956\u52b1\u51fd\u6570\u4e0a\u3002\uff08\u6700\u5927\u5316\u8fd9\u4e2a\u5956\u52b1\u51fd\u6570\u5e76\u6ca1\u6709\u4f7f\u5f97\u98de\u884c\u6c34\u5e73\u63d0\u9ad8\u3002\uff09\n\n\u4ee5\u4e0a\u4ec5\u662f\u4e00\u4e2a\u5f3a\u5316\u5b66\u4e60\u8c03\u8bd5\u7684\u4f8b\u5b50\uff0c\u56e0\u4e3a\u6211\u4eec\u6070\u597d\u627e\u5230\u4e86\u4e00\u4e2a\u6781\u4f18\u79c0\u7684\u76f4\u5347\u673a\u64cd\u7eb5\u8005\uff0c\u5982\u679c\u6ca1\u6709\u7684\u8bdd\u5219\u9700\u8981\u60f3\u51fa\u522b\u7684\u8c03\u8bd5\u65b9\u6cd5\u3002\u5728\u901a\u5e38\u7684\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u90fd\u9700\u8981\u81ea\u5df1\u627e\u51fa\u9488\u5bf9\u95ee\u9898\u7684\u6709\u6548\u7684\u8c03\u8bd5\u65b9\u6cd5\u3002\n\n## 9. \u5fae\u5206\u52a8\u6001\u89c4\u5212\uff08DDP: Differential Dynamic Programming\uff09\n\n\u7ee7\u7eed\u4f7f\u7528\u9065\u63a7\u76f4\u5347\u673a\u7684\u4f8b\u5b50\uff0c\u5047\u8bbe\u6211\u4eec\u5df2\u7ecf\u6709\u6a21\u62df\u5668\u5e76\u53ef\u4ee5\u901a\u8fc7\u6a21\u62df\u5668\u77e5\u9053$s_{t+1}=f(s_t,a_t)$\uff08\u5373\u4e0b\u4e00\u4e2a\u72b6\u6001\u662f\u4e00\u4e2a\u5173\u4e8e\u5f53\u524d\u72b6\u6001\u53ca\u52a8\u4f5c\u7684\u51fd\u6570\uff09\uff0c\u800c\u4e14\u8fd9\u4e2a\u6a21\u62df\u5668\u662f\u975e\u7ebf\u6027\u3001\u786e\u5b9a\u6027\u7684\uff0c\u566a\u97f3\u9879\u4e5f\u6bd4\u8f83\u5c0f\u3002\u6211\u4eec\u73b0\u5728\u60f3\u8981\u8ba9\u76f4\u5347\u673a\u98de\u51fa\u4e00\u4e9b\u7279\u5b9a\u8f68\u8ff9\uff0c\u4e8e\u662f\uff1a\n1. \u5148\u5199\u51fa\u8fd9\u4e2a\u8f68\u8ff9\uff1a$(\\bar s_0,\\bar a_0),(\\bar s_1,\\bar a_1),\\cdots,(\\bar s_T,\\bar a_T)$\uff0c\u8fd9\u4e5f\u79f0\u4e3a**\u6807\u51c6\u8f68\u8ff9\uff08nominal trajectory\uff09**\uff1b\uff08\u8fd9\u4e2a\u8f68\u8ff9\u4e5f\u53ef\u80fd\u6765\u81ea\u4e00\u4e2a\u5f88\u7c97\u7cd9\u7684\u63a7\u5236\u7b97\u6cd5\uff0c\u4f46\u662f\u867d\u7136\u7c97\u7cd9\uff0c\u5374\u4e5f\u662f\u63cf\u8ff0\u7684\u6211\u4eec\u60f3\u8981\u505a\u51fa\u7684\u52a8\u4f5c\uff0c\u6bd4\u5982\u5728\u7a7a\u4e2d\u505a$90^\\circ$\u8f6c\u5f2f\u4e4b\u7c7b\u7684\u52a8\u4f5c\u3002\uff09\n2. \u7136\u540e\uff0c\u6211\u4eec\u5728\u8fd9\u4e2a\u8f68\u8ff9\u9644\u8fd1\u7ebf\u6027\u5316$f$\u5f97\u5230\uff1a$\\displaystyle s_{t+1}\\approx f(\\bar s_t,\\bar a_t)+(\\nabla_sf(\\bar s_t,\\bar a_t))^T(s_t-\\bar s_t)+(\\nabla_af(\\bar s_t,\\bar a_t))^T(a_t-\\bar a_t)$\uff0c\u4e5f\u5c31\u662f\u8bf4\u80fd\u591f\u5f97\u5230\u4e00\u4e2a\u7ebf\u6027\u51fd\u6570$s_{t+1}=A_ts_t+B_ta_t$\u3002\u6211\u4eec\u8fd9\u662f\u5728\u8bfe\u7a0b\u4e2d\u7b2c\u4e00\u6b21\u4f7f\u7528LQR\u6765\u5904\u7406\u6709\u9650\u65f6\u57df\u4e0a\u7684\u975e\u5e73\u7a33\u7684\u52a8\u6001\u95ee\u9898\uff0c\u5c24\u5176\u8981\u6ce8\u610f\u7684\u662f\u8fd9\u91cc\u7684$A_t,B_t$\u662f\u4f9d\u8d56\u4e8e\u65f6\u95f4\u7684\uff08\u4e5f\u5c31\u662f\u8fd9\u662f\u4e00\u7cfb\u5217\u4e0d\u76f8\u7b49\u7684\u77e9\u9635\uff09\uff1b\uff08\u5373\u4f7f\u6807\u51c6\u8f68\u8ff9\u6765\u81ea\u4e00\u5957\u5f88\u7cdf\u7cd5\u7684\u63a7\u5236\uff0c\u4f46\u6211\u4eec\u4ecd\u7136\u5e0c\u671b\u7cfb\u7edf\u5728$t$\u65f6\u523b\u7684\u72b6\u6001\u548c\u52a8\u4f5c\u80fd\u591f\u8fd1\u4f3c\u4e8e\u8fd9\u4e2a\u7cdf\u7cd5\u7684\u8f68\u8ff9\u3002\u4e5f\u8bb8\u8fd9\u4e2a\u63a7\u5236\u7b97\u6cd5\u505a\u7684\u5de5\u4f5c\u5f88\u9a6c\u864e\uff0c\u4f46\u6bd5\u7adf\u8fd9\u4e2a\u8f68\u8ff9\u63cf\u8ff0\u4e86\u6211\u4eec\u60f3\u8981\u5f97\u5230\u7684\u52a8\u4f5c\uff0c\u4e5f\u5c31\u662f\u5e0c\u671b$(s_t,a_t)\\approx(\\bar s_t,\\bar a_t)$\u3002\uff09\n3. \u6709\u4e86\u7ebf\u6027\u6a21\u578b\uff0c\u518d\u4f7f\u7528LQR\u8ba1\u7b97\u6700\u4f18\u7b56\u7565$\\pi_t$\uff0c\u4e8e\u662f\u6211\u4eec\u5c31\u4f1a\u5f97\u5230\u4e00\u4e2a\u66f4\u597d\u7684\u7b56\u7565\uff1b\n4. \u5728\u6a21\u62df\u5668\u4e2d\u6309\u7167\u65b0\u7684\u7b56\u7565\u5b9e\u73b0\u4e00\u4e2a\u65b0\u7684\u8f68\u8ff9\uff0c\u4e5f\u5c31\u662f\uff1a\n * \u7528$\\bar s_0$\u521d\u59cb\u5316\u6a21\u62df\u5668\uff1b\n * \u7528\u521a\u5b66\u5230\u7684$\\pi_t$\u63a7\u5236\u6bcf\u4e00\u6b65\u7684\u52a8\u4f5c$a_t=\\pi_t(\\bar s_t)$\uff1b\n * \u6700\u7ec8\u5f97\u5230\u4e00\u5957\u65b0\u7684\u8f68\u8ff9$\\bar s_{t+1}=f(\\bar s_t,\\bar a_t)$\n5. \u5f97\u5230\u65b0\u7684\u8f68\u8ff9\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u7ee7\u7eed\u91cd\u590d\u4e0a\u9762\u7684\u64cd\u4f5c\uff0c\u4e0d\u505c\u7684\u4f18\u5316\u8fd9\u4e2a\u8f68\u8ff9\uff08\u4e5f\u5c31\u662f\u91cd\u590d\u6b65\u9aa42-5\uff09\u3002\n\n\u5728\u5b9e\u8df5\u4e2d\u80fd\u591f\u77e5\u9053\uff0c\u8fd9\u662f\u4e00\u4e2a\u975e\u5e38\u6709\u6548\u7684\u7b97\u6cd5\uff0cDDP\u5b9e\u9645\u4e0a\u662f\u4e00\u79cd\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5\uff0c\u5728\u6bcf\u6b21\u8fed\u4ee3\u4e2d\uff0c\u627e\u5230\u4e00\u4e2a\u7a0d\u597d\u7684\u70b9\u8fdb\u884c\u7ebf\u6027\u5316\uff0c\u8fdb\u800c\u5f97\u5230\u4e00\u4e2a\u7a0d\u597d\u7684\u7b56\u7565\uff0c\u91cd\u590d\u8fd9\u4e2a\u52a8\u4f5c\u6700\u7ec8\u5c31\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u6bd4\u8f83\u597d\u7684\u7b56\u7565\u4e86\u3002\uff08\u201c\u5fae\u5206\u201d\u6307\u7684\u5c31\u662f\u6bcf\u6b21\u8fed\u4ee3\u6211\u4eec\u90fd\u4f1a\u9009\u62e9\u4e00\u4e2a\u70b9\u5e76\u901a\u8fc7\u6c42\u5bfc\u505a\u7ebf\u6027\u5316\u3002\uff09\n\n## 10. \u7ebf\u6027\u4e8c\u6b21\u578b\u9ad8\u65af\uff08LQG: Linear Quadratic Gaussian\uff09\n\n### 10.1 Kalman\u6ee4\u6ce2\uff08Kalman Filter\uff09\n\n\u73b0\u5728\u4ecb\u7ecd\u53e6\u4e00\u79cd\u7c7b\u578b\u7684MDP\uff0c\u5728\u8fd9\u79cdMDP\u4e2d\uff0c\u6211\u4eec\u5e76\u4e0d\u80fd\u76f4\u63a5\u89c2\u5bdf\u5230\u6bcf\u4e00\u6b65\u7684\u72b6\u6001\uff08\u5728\u524d\u9762\u7684MDP\u4e2d\u6211\u4eec\u90fd\u4f1a\u5047\u8bbe\u7cfb\u7edf\u7684\u72b6\u6001\u5df2\u77e5\uff0c\u4e8e\u662f\u53ef\u4ee5\u8ba1\u7b97\u51fa\u7b56\u7565\uff0c\u5b83\u662f\u4e00\u4e2a\u5173\u4e8e\u72b6\u6001\u7684\u51fd\u6570\uff1b\u5728LQR\u4e2d\u6211\u4eec\u4e5f\u662f\u8ba1\u7b97$L_ts_t$\uff0c\u72b6\u6001\u90fd\u662f\u5df2\u77e5\u7684\uff09\u3002\n\n\u6211\u4eec\u6682\u65f6\u5148\u4e0d\u8c08\u63a7\u5236\uff0c\u6765\u8bf4\u8bf4\u666e\u901a\u7684\u4e0d\u80fd\u89c2\u5bdf\u5230\u72b6\u6001\u4fe1\u606f\u7684\u52a8\u6001\u7cfb\u7edf\u3002\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u6211\u4eec\u5bf9\u201c\u4f7f\u7528\u96f7\u8fbe\u8ffd\u8e2a\u98de\u884c\u5668\u201d\u7684\u8fc7\u7a0b\u8fdb\u884c\u6781\u5176\u7b80\u5316\u7684\u5efa\u6a21\uff1a\u7ebf\u6027\u6a21\u578b$s_{t+1}=As_t+w_t$\uff0c\u5176\u4e2d$s_t=\\begin{bmatrix}x_t\\\\\\dot x_t\\\\y_t\\\\\\dot y_t\\end{bmatrix},\\ A=\\begin{bmatrix}1&1&0&0\\\\0&0.9&0&0\\\\0&0&1&1\\\\0&0&0&0.9\\end{bmatrix}$\uff08\u8fd9\u662f\u4e00\u4e2a\u975e\u5e38\u7b80\u5316\u7684\u6a21\u578b\uff0c\u53ef\u4ee5\u7406\u89e3\u4e3a\u4e00\u67b6\u57282D\u7a7a\u95f4\u4e2d\u79fb\u52a8\u7684\u98de\u673a\uff09\u3002\n\n\u5728\u8fd9\u4e2a\u7b80\u5316\u6a21\u578b\u4e2d\u6211\u4eec\u65e0\u6cd5\u89c2\u5bdf\u5230\u6bcf\u4e00\u6b65\u7684\u72b6\u6001\uff0c\u4f46\u5047\u8bbe\u6211\u4eec\u53ef\u4ee5\u89c2\u5bdf\u5230\u53e6\u4e00\u4e2a\u91cf\u2014\u2014$y_t=Cs_t+v_t,\\ v_t\\sim\\mathcal N(0,\\varSigma_v)$\uff1a\u6bd4\u5982$C=\\begin{bmatrix}1&0&0&0\\\\0&0&1&0\\end{bmatrix}$\uff0c\u5219\u6709$Cs_t=\\begin{bmatrix}x_t\\\\y_t\\end{bmatrix}$\u3002\u8fd9\u4e2a\u91cf\u53ef\u80fd\u662f\u96f7\u8fbe\u8ffd\u8e2a\u5230\u7684\u98de\u884c\u5668\u7684\u4f4d\u7f6e\uff08\u53ea\u80fd\u5f97\u5230\u4f4d\u7f6e\u76f8\u5173\u7684\u72b6\u6001\uff0c\u800c\u4e0d\u80fd\u5f97\u5230\u52a0\u901f\u5ea6\u76f8\u5173\u7684\u72b6\u6001\uff09\u3002\u6bd4\u5982\u8bf4\u5728$\\mathrm{xOy}$\u5e73\u9762\u4e0a\uff0c\u6709\u4e00\u4e2a\u96f7\u8fbe\u4f4d\u4e8e$x$\u8f74\u67d0\u70b9\uff0c\u5176\u89c6\u89d2\u6cbf$y$\u8f74\u6b63\u65b9\u5411\u5927\u81f4\u5bf9\u7740\u98de\u884c\u5668\uff0c\u5e76\u4f9d\u6b21\u89c2\u5bdf\u5230\u4e00\u7cfb\u5217\u98de\u884c\u5668\u79fb\u52a8\u8fc7\u7a0b\u4e2d\u7684\u4e0d\u8fde\u7eed\u7684\u70b9\uff0c\u56e0\u4e3a\u96f7\u8fbe\u89c6\u89d2\u6cbf$y$\u8f74\u6b63\u534a\u8f74\u65b9\u5411\uff0c\u6240\u4ee5\u5f97\u5230\u7684\u70b9\u7684\u5750\u6807\u7684\u7eb5\u5750\u6807\u7684\u566a\u97f3\uff08\u65b9\u5dee\uff09\u5e94\u8be5\u5927\u4e8e\u6a2a\u5750\u6807\u7684\u65b9\u5dee\uff0c\u53ef\u4ee5\u8bf4\u8fd9\u4e9b\u70b9\u5206\u522b\u4f4d\u4e8e\u4e00\u4e2a\u4e2a\u5c0f\u7684\u9ad8\u65af\u5206\u5e03\u4e2d\uff0c\u800c\u8fd9\u4e9b\u9ad8\u65af\u5206\u5e03\u7684\u534f\u65b9\u5dee\u77e9\u9635\u5e94\u8be5\u578b\u4e3a$\\varSigma=\\begin{bmatrix}a&0\\\\0&b\\end{bmatrix},\\ a\\gt b$\uff08\u5373\u56e0\u4e3a\u5728$y$\u8f74\u65b9\u5411\u4e0a\u7684\u89c2\u6d4b\u7ed3\u679c\u66f4\u5bb9\u6613\u51fa\u73b0\u504f\u5dee\uff0c\u6240\u4ee5\u7b49\u9ad8\u7ebf\u56fe\u4e2d\u692d\u5706\u7684\u957f\u8f74\u4f1a\u6cbf\u7740$y$\u8f74\u65b9\u5411\u3002\uff09\u800c\u6211\u4eec\u60f3\u8981\u505a\u7684\u5c31\u662f\u901a\u8fc7\u89c2\u6d4b\u5230\u7684\u4e00\u7cfb\u5217\u5750\u6807\u4f30\u8ba1\u98de\u884c\u5668\u5728\u6bcf\u4e2a\u89c2\u6d4b\u70b9\u7684\u72b6\u6001$P(s_t\\mid y_1,\\cdots,y_t)$\u3002\n\n$s_0,\\cdots,s_t;\\ y_0,\\cdots,y_t$\u7ec4\u6210\u4e86\u4e00\u4e2a\u9ad8\u65af\u5206\u5e03\u7684\u8054\u5408\u5206\u5e03\uff08\u591a\u5143\u6b63\u6001\u5206\u5e03\uff09\uff0c\u53ef\u4ee5\u7528$z=\\begin{bmatrix}s_0\\\\\\vdots\\\\s_t\\\\y_0\\\\\\vdots\\\\y_t\\end{bmatrix},\\ z\\sim\\mathcal N(\\mu,\\varSigma)$\u8868\u793a\uff0c\u518d\u7ed3\u5408[\u7b2c\u5341\u4e09\u8bb2](chapter13.ipynb)\u4e2d\u4e86\u89e3\u7684\u591a\u5143\u6b63\u6001\u5206\u5e03\u7684\u8fb9\u7f18\u5206\u5e03\u4e0e\u6761\u4ef6\u5206\u5e03\u7684\u8ba1\u7b97\u65b9\u6cd5\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5bf9$P(s_t\\mid y_1,\\cdots,y_t)$\u8fdb\u884c\u6c42\u89e3\u3002\u867d\u7136\u8fd9\u4e2a\u601d\u8def\u5728\u6982\u5ff5\u4e0a\u662f\u53ef\u884c\u7684\uff0c\u4f46\u662f\u5728\u5b9e\u9645\u4e2d\u8ddf\u8e2a\u4e00\u4e2a\u98de\u884c\u5668\u4f1a\u5f97\u5230\u6210\u5343\u4e0a\u4e07\u4e2a\u4f4d\u7f6e\uff0c\u4e8e\u662f\u5c31\u4f1a\u6709\u4e00\u4e2a\u5de8\u5927\u7684\u534f\u65b9\u5dee\u77e9\u9635\uff08\u671f\u671b\u5411\u91cf\u548c\u534f\u65b9\u5dee\u77e9\u9635\u90fd\u4f1a\u968f\u7740\u65f6\u95f4\u3001\u6b65\u6570\u7684\u589e\u957f\u5448\u7ebf\u6027\u589e\u957f\uff0c\u5728\u8ba1\u7b97\u8fc7\u7a0b\u4e2d\u9700\u8981\u7528\u5230\u7684\u8fb9\u7f18\u5206\u5e03\u3001\u6761\u4ef6\u5206\u5e03\u90fd\u53ef\u80fd\u4f1a\u4f7f\u7528\u534f\u65b9\u5dee\u7684\u9006\uff0c\u800c\u8ba1\u7b97\u534f\u65b9\u5dee\u7684\u9006\u7684\u590d\u6742\u5ea6\u662f\u77e9\u9635\u89c4\u6a21\u7684\u5e73\u65b9\u7ea7\u522b\uff09\uff0c\u6240\u4ee5\u5bf9\u4e8e\u8ba1\u7b97\u6765\u8bf4\u53ef\u884c\u6027\u8f83\u4f4e\u3002\n\n\u4e8e\u662f\u6211\u4eec\u5f15\u5165**Kalman\u6ee4\u6ce2\uff08Kalman Filter\uff09**\u7b97\u6cd5\u6765\u89e3\u51b3\u8fd9\u4e2a\u6548\u7387\u95ee\u9898\u3002\u8fd9\u4e2a\u7b97\u6cd5\u5b9e\u9645\u4e0a\u662f\u4e00\u4e2a\u9690\u5f0f\u9a6c\u5c14\u53ef\u592b\u6a21\u578b\uff0c\u4f7f\u7528\u9012\u63a8\u7684\u65b9\u5f0f\u8ba1\u7b97\uff0c\u4e0b\u9762\u5199\u51fa\u7b97\u6cd5\u7684\u6b65\u9aa4\uff0c\u7b97\u6cd5\u7684\u601d\u8def\u53ef\u4ee5\u5728[Hidden Markov Models](http://cs229.stanford.edu/section/cs229-hmm.pdf)\u4e2d\u627e\u5230\uff1a\n* \u7b2c\u4e00\u6b65\u79f0\u4e3a**\u9884\u6d4b\uff08predict\uff09**\u6b65\u9aa4\uff0c\u4f7f\u7528$P(s_t\\mid y_1,\\cdots,y_t)$\u5bf9$P(s_{t+1}\\mid y_1,\\cdots,y_t)$\u8fdb\u884c\u9884\u6d4b\uff1b\n * \u6709$s_t\\mid y_0,\\cdots,y_t\\sim\\mathcal N\\left(s_{t\\mid t},\\varSigma_{t\\mid t}\\right)$\uff1b\n * \u5219\u6709$s_{t+1}\\mid y_0,\\cdots,y_t\\sim\\mathcal N\\left(s_{t+1\\mid t},\\varSigma_{t+1\\mid t}\\right)$\uff1b\n * \u5176\u4e2d$\\begin{cases}s_{t+1\\mid t}&=As_{t\\mid t}\\\\ \\varSigma_{t+1\\mid t}&=A\\varSigma_{t\\mid t}A^T+\\varSigma_t\\end{cases}$\uff1b\n * \u9700\u8981\u8bf4\u660e\u7684\u662f\uff0c\u6211\u4eec\u4f7f\u7528\u8bf8\u5982$s_t,y_t$\u8868\u793a\u6a21\u578b\u7684\u771f\u5b9e\u72b6\u6001\uff08\u5982\u4e3a\u89c2\u6d4b\u5230\u7684\u5b9e\u9645\u72b6\u6001\uff0c\u5df2\u89c2\u6d4b\u5230\u7684\u5b9e\u9645\u4f4d\u7f6e\uff09\uff0c\u800c\u4f7f\u7528$s_{t\\mid t},s_{t+1\\mid t},\\varSigma_{t\\mid t}$\u8868\u793a\u7531\u8ba1\u7b97\u5f97\u5230\u7684\u503c\uff1b\n* \u7b2c\u4e8c\u6b65\u79f0\u4e3a**\u66f4\u65b0\uff08update\uff09**\u6b65\u9aa4\uff0c\u4f7f\u7528\u4e0a\u4e00\u6b65\u9884\u6d4b\u7684$P(s_{t+1}\\mid y_1,\\cdots,y_t)$\u66f4\u65b0$P(s_{t+1}\\mid y_1,\\cdots,y_{t+1})$\uff0c\u4e5f\u5c31\u662f\u6709\u4e86\u9884\u6d4b\u7ed3\u679c\u4e4b\u540e\u518d\u5c06\u5bf9\u5e94\u7684\u6837\u672c\u7eb3\u5165\u6a21\u578b\uff1a\n * \u5176\u4e2d$s_{t+1\\mid t+1}=s_{t+1\\mid t}+K_{t+1}\\cdot\\left(y_{t+1}-Cs_{t+1\\mid t}\\right)$\uff1b\n * \u800c$K_{t+1}=\\varSigma_{t+1\\mid t}C^T\\left(C\\varSigma_{t+1\\mid t}C^T+\\varSigma_t\\right)^{-1}$\uff1b\n * $\\varSigma_{t+1\\mid t+1}=\\varSigma_{t+1\\mid t}+\\varSigma_{t+1\\mid t}\\cdot C^T\\left(C\\varSigma_{t+1\\mid t}C^T+\\varSigma_t\\right)^{-1}C\\varSigma_{t+1\\mid t}$\n\n Kalman\u6ee4\u6ce2\u7b97\u6cd5\u7684\u590d\u6742\u5ea6\u6bd4\u8ba1\u7b97\u534f\u65b9\u5dee\u77e9\u9635\u7684\u9006\u4f4e\u5f88\u591a\uff0c\u56e0\u4e3a\u91c7\u7528\u4e86\u9012\u63a8\u7684\u8ba1\u7b97\u8fc7\u7a0b\uff0c\u6bcf\u5f53\u6b65\u9aa4\u589e\u52a0\uff0c\u5219\u53ea\u9700\u591a\u6267\u884c\u4e00\u6b65\u8fed\u4ee3\uff0c\u800c\u4e14\u65e0\u9700\u4fdd\u7559\u592a\u591a\u6570\u636e\u5728\u5185\u5b58\u4e2d\uff0c\u76f8\u6bd4\u4e8e\u6c42\u4e00\u4e2a\u5de8\u5927\u77e9\u9635\u7684\u9006\u800c\u8a00\uff0c\u7b97\u6cd5\u590d\u6742\u5ea6\u4f4e\u4e86\u5f88\u591a\uff1a\n $$\n \\require{AMScd}\n \\begin{CD}\n y_1 @. y_2 @. y_3 \\\\\n @VVV @VVV @VVV \\\\\n P\\left(s_1\\mid y_1\\right) @>>> P\\left(s_2\\mid y_1,y_2\\right) @>>> P\\left(s_3\\mid y_1,y_2,y_3\\right) @>>> \\cdots\n \\end{CD}\n $$\n \u4ece\u8fd9\u4e2a\u8349\u56fe\u53ef\u4ee5\u770b\u51fa\u6765\uff0c\u5f53\u8ba1\u7b97$P\\left(s_3\\mid y_1,y_2,y_3\\right)$\u65f6\uff0c\u6211\u4eec\u5e76\u4e0d\u9700\u8981$y_1,y_2,P\\left(s_1\\mid y_1\\right)$\uff0c\u6bcf\u6b65\u8ba1\u7b97\u53ea\u9700\u8981\u77e5\u9053\u6700\u65b0\u7684\u89c2\u6d4b\u503c\u548c\u4e0a\u4e00\u6b65\u5f97\u51fa\u7684\u6982\u7387\u4f30\u8ba1\u5373\u53ef\u3002\n\n### 10.2 LQG\n\n\u5c06Kalman\u6ee4\u6ce2\u4e0eLQR\u7ed3\u5408\u8d77\u6765\u5c31\u53ef\u4ee5\u5f97\u5230**\u7ebf\u6027\u4e8c\u6b21\u9ad8\u65af\uff08LQG: Linear Quadratic Gaussian\uff09**\u7ebf\u6027\u4e8c\u6b21\u9ad8\u901f\u7b97\u6cd5\u3002\u5728LQR\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u5c06\u52a8\u4f5c$a_t$\u52a0\u56de\u5230\u6a21\u578b\u4e2d\uff1a\u8bbe\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u7cfb\u7edf\u4e3a$s_{t+1}=As_t+Ba_t+w_t,\\ w\\sim\\mathcal N(0,\\varSigma_w)$\uff0c\u800c\u4e14\u6211\u4eec\u65e0\u6cd5\u76f4\u63a5\u89c2\u6d4b\u5230\u72b6\u6001\uff0c\u53ea\u80fd\u89c2\u6d4b\u5230$y_t=Cs_t+v_t,\\ v\\sim\\mathcal N(0,\\varSigma_v)$\u3002\u73b0\u5728\u4f7f\u7528Kalman\u6ee4\u6ce2\u4f30\u8ba1\u72b6\u6001\uff1a\n* \u5047\u8bbe\u521d\u59cb\u72b6\u6001\u4e3a$s_{0\\mid0}=s_0,\\varSigma_{0\\mid0}=0$\uff1b\uff08\u5982\u679c\u4e0d\u77e5\u9053\u786e\u5207\u521d\u59cb\u72b6\u6001\uff0c\u6d4b\u53ef\u4ee5\u5927\u81f4\u4f30\u8ba1\u521d\u59cb\u72b6\u6001$s_0\\sim\\mathcal N\\left(s_{0\\mid0},\\varSigma_{0\\mid0}\\right)$\uff0c\u5373\u4f7f\u7528\u521d\u59cb\u72b6\u6001\u7684\u5e73\u5747\u503c\u548c\u534f\u65b9\u5dee\u7684\u4f30\u8ba1\u3002\uff09\n* \u9884\u6d4b\u6b65\u9aa4\uff0c$\\begin{cases}s_{t+1\\mid t}&=As_{t\\mid t}+Ba_t\\\\\\varSigma_{t+1\\mid t}&=A\\varSigma_{t\\mid t}A^T+\\varSigma_t\\end{cases}$\u3002\u5982\u6b64\uff0c\u6211\u4eec\u5c31\u201c\u89e3\u51b3\u201d\u4e3a\u4e86\u72b6\u6001\u89c2\u6d4b\u4e0d\u5230\u7684\u95ee\u9898\uff08\u8fd9\u91cc\u7684$A,B$\u90fd\u662f\u5df2\u77e5\u9879\uff09\uff1b\n* \u63a7\u5236\u6b65\u9aa4\uff0c\u6309\u7167LQR\u7b97\u6cd5\u4e2d\u7684\u6700\u4f18\u52a8\u4f5c$a_t=L_ts_t$\u8ba1\u7b97\u5176\u4e2d\u7684$L_t$\uff0c\u5e76\u6682\u65f6\u5ffd\u7565\u89c2\u6d4b\u4e0d\u5230\u72b6\u6001\u7684\u95ee\u9898\uff1b\u5f97\u5230$L_t$\u540e\uff0c\u9009\u62e9$a_t=L_ts_{t\\mid t}$\u4f5c\u4e3a\u6700\u4f18\u52a8\u4f5c\uff08\u6362\u53e5\u8bdd\u8bf4\uff0c\u56e0\u4e3a\u4e0d\u77e5\u9053\u72b6\u6001\uff0c\u6240\u4ee5\u6211\u4eec\u5bf9\u72b6\u6001\u7684\u6700\u4f73\u4f30\u8ba1\u5c31\u662f$s_{t\\mid t}$\uff09\u3002\n\n\u8fd9\u4e2a\u7b97\u6cd5\u5b9e\u9645\u4e0a\u662f\u5148\u8bbe\u8ba1\u4e86\u4e00\u4e2a\u72b6\u6001\u4f30\u8ba1\u7b97\u6cd5\uff0c\u7136\u540e\u518d\u4f7f\u7528\u4e86\u4e00\u4e2a\u63a7\u5236\u7b97\u6cd5\uff0c\u628a\u4e24\u4e2a\u7b97\u6cd5\u7ed3\u5408\u8d77\u6765\u5c31\u5f97\u5230\u4e86LQG\u3002\n", "meta": {"hexsha": "216988fad43da2b4f45e5b0bf6ac13a9396c1601", "size": 10545, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "other/note/LSJU-chapter19.ipynb", "max_stars_repo_name": "PeterChenYijie/MachineLearningZeroToALL", "max_stars_repo_head_hexsha": "b14005c3e0b5a39a0ba82db5c9791f682b5effd5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 8, "max_stars_repo_stars_event_min_datetime": "2018-04-20T09:10:20.000Z", "max_stars_repo_stars_event_max_datetime": "2019-02-16T07:50:32.000Z", "max_issues_repo_path": "other/note/LSJU-chapter19.ipynb", "max_issues_repo_name": "DeepInDeeper/MachineLearningZeroToALL", "max_issues_repo_head_hexsha": "b14005c3e0b5a39a0ba82db5c9791f682b5effd5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "other/note/LSJU-chapter19.ipynb", "max_forks_repo_name": "DeepInDeeper/MachineLearningZeroToALL", "max_forks_repo_head_hexsha": "b14005c3e0b5a39a0ba82db5c9791f682b5effd5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2020-01-27T00:55:59.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-25T00:07:56.000Z", "avg_line_length": 70.3, "max_line_length": 594, "alphanum_fraction": 0.6370791844, "converted": true, "num_tokens": 6279, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.4403726863921272}} {"text": "```python\n%matplotlib inline\n```\n\n\n```python\n!pip install pygments\nimport pygments\n!pip install flit\nimport flit\n!pip install scikit-learn\n!pip install scikit-aero\nfrom skaero.atmosphere import coesa\nfrom skaero.gasdynamics import isentropic, shocks\n!pip install numpy\nimport numpy as np\n```\n\n Requirement already satisfied: pygments in /srv/conda/envs/notebook/lib/python3.6/site-packages (2.10.0)\n Requirement already satisfied: flit in /srv/conda/envs/notebook/lib/python3.6/site-packages (3.4.0)\n Requirement already satisfied: requests in /srv/conda/envs/notebook/lib/python3.6/site-packages (from flit) (2.26.0)\n Requirement already satisfied: tomli in /srv/conda/envs/notebook/lib/python3.6/site-packages (from flit) (1.2.2)\n Requirement already satisfied: flit_core>=3.4.0 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from flit) (3.4.0)\n Requirement already satisfied: docutils in /srv/conda/envs/notebook/lib/python3.6/site-packages (from flit) (0.18)\n Requirement already satisfied: tomli-w in /srv/conda/envs/notebook/lib/python3.6/site-packages (from flit) (0.4.0)\n Requirement already satisfied: charset-normalizer~=2.0.0 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from requests->flit) (2.0.0)\n Requirement already satisfied: urllib3<1.27,>=1.21.1 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from requests->flit) (1.26.7)\n Requirement already satisfied: idna<4,>=2.5 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from requests->flit) (3.1)\n Requirement already satisfied: certifi>=2017.4.17 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from requests->flit) (2021.5.30)\n Requirement already satisfied: scikit-learn in /srv/conda/envs/notebook/lib/python3.6/site-packages (0.24.2)\n Requirement already satisfied: scipy>=0.19.1 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from scikit-learn) (1.5.3)\n Requirement already satisfied: joblib>=0.11 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from scikit-learn) (1.1.0)\n Requirement already satisfied: numpy>=1.13.3 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from scikit-learn) (1.19.5)\n Requirement already satisfied: threadpoolctl>=2.0.0 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from scikit-learn) (3.0.0)\n Requirement already satisfied: scikit-aero in /srv/conda/envs/notebook/lib/python3.6/site-packages (0.1)\n Requirement already satisfied: numpy in /srv/conda/envs/notebook/lib/python3.6/site-packages (1.19.5)\n\n\n\n```python\n!pip install numba\n```\n\n Requirement already satisfied: numba in /srv/conda/envs/notebook/lib/python3.6/site-packages (0.53.1)\r\n Requirement already satisfied: llvmlite<0.37,>=0.36.0rc1 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from numba) (0.36.0)\r\n Requirement already satisfied: numpy>=1.15 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from numba) (1.19.5)\r\n Requirement already satisfied: setuptools in /srv/conda/envs/notebook/lib/python3.6/site-packages (from numba) (58.0.4)\r\n\n\n\n```python\nA = np.array([1, 3, 7, 21])\noutput = np.std(A)\nprint(output)\nprint(output**4)\nprint(output/2)\nprint(output%2)\n```\n\n 7.810249675906654\n 3720.999999999999\n 3.905124837953327\n 1.810249675906654\n\n\n\n```python\nimport matplotlib.pyplot as plt\nfrom sklearn.covariance import EmpiricalCovariance, MinCovDet\n!pip install sims\nimport sims\n```\n\n Requirement already satisfied: sims in /srv/conda/envs/notebook/lib/python3.6/site-packages (2.0.0)\n Requirement already satisfied: scipy in /srv/conda/envs/notebook/lib/python3.6/site-packages (from sims) (1.5.3)\n Requirement already satisfied: scikit-image in /srv/conda/envs/notebook/lib/python3.6/site-packages (from sims) (0.17.2)\n Requirement already satisfied: xarray in /srv/conda/envs/notebook/lib/python3.6/site-packages (from sims) (0.16.2)\n Requirement already satisfied: matplotlib in /srv/conda/envs/notebook/lib/python3.6/site-packages (from sims) (3.3.4)\n Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.3 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from matplotlib->sims) (2.4.7)\n Requirement already satisfied: numpy>=1.15 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from matplotlib->sims) (1.19.5)\n Requirement already satisfied: python-dateutil>=2.1 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from matplotlib->sims) (2.8.2)\n Requirement already satisfied: kiwisolver>=1.0.1 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from matplotlib->sims) (1.3.1)\n Requirement already satisfied: cycler>=0.10 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from matplotlib->sims) (0.11.0)\n Requirement already satisfied: pillow>=6.2.0 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from matplotlib->sims) (8.3.2)\n Requirement already satisfied: six>=1.5 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from python-dateutil>=2.1->matplotlib->sims) (1.16.0)\n Requirement already satisfied: networkx>=2.0 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from scikit-image->sims) (2.6.3)\n Requirement already satisfied: imageio>=2.3.0 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from scikit-image->sims) (2.9.0)\n Requirement already satisfied: tifffile>=2019.7.26 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from scikit-image->sims) (2020.10.1)\n Requirement already satisfied: PyWavelets>=1.1.1 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from scikit-image->sims) (1.1.1)\n Requirement already satisfied: setuptools>=38.4 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from xarray->sims) (58.0.4)\n Requirement already satisfied: pandas>=0.25 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from xarray->sims) (1.1.5)\n Requirement already satisfied: pytz>=2017.2 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from pandas>=0.25->xarray->sims) (2021.3)\n\n\n\n```python\n!pip install geometer\nfrom geometer import *\n```\n\n Requirement already satisfied: geometer in /srv/conda/envs/notebook/lib/python3.6/site-packages (0.2.3)\r\n Requirement already satisfied: numpy<1.20,>=1.15 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from geometer) (1.19.5)\r\n Requirement already satisfied: sympy<=1.7,>=1.3 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from geometer) (1.7)\r\n Requirement already satisfied: mpmath>=0.19 in /srv/conda/envs/notebook/lib/python3.6/site-packages (from sympy<=1.7,>=1.3->geometer) (1.2.1)\r\n\n\n\n```python\nfrom numba import jit\nimport random\n```\n", "meta": {"hexsha": "4d7b7792383798ab61dfc1de0179768985d4de66", "size": 8804, "ext": "ipynb", "lang": "Jupyter Notebook", 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YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.6723316860482763, "lm_q1q2_score": 0.4403064773610227}} {"text": "## 9-1. 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(1-p)^{nt}$\u3067\u3042\u308b\u3002$n=10^{12}$\u3067$t=10^{3}$\u3068\u3057\u3001$q=0.99$\u3092\u5b9f\u73fe\u3059\u308b\u305f\u3081\u306b\u306f\u3001$p\\sim 10^{-18}$\u3067\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u3002\n\n\u4e2d\u6027\u5b50\u7dda\u30a8\u30e9\u30fc\u306e\u30ea\u30fc\u30af\u3068\u306e\u5927\u304d\u306a\u9055\u3044\u306f\u3001\u30ea\u30fc\u30af\u306f\u3059\u3050\u3055\u307e\u78ba\u8a8d\u3059\u308c\u30700,1\u306e\u4e2d\u9593\u7684\u306a\u72b6\u614b\u306b\u3042\u308b\u305f\u3081\u5fa9\u5e30\u304c\u53ef\u80fd\u306a\u3082\u306e\u306e\u3001\u4e2d\u6027\u5b50\u7dda\u306f\u751f\u3058\u305f\u5f8c\u306b\u306f\u3082\u3068\u306f\u3069\u306e\u72b6\u614b\u3067\u3042\u3063\u305f\u304b\u77e5\u308b\u3053\u3068\u304c\u3067\u304d\u306a\u3044\u70b9\u306b\u3042\u308b\u3002\n\u3053\u306e\u305f\u3081\u3001\u4e2d\u6027\u5b50\u7dda\u30a8\u30e9\u30fc\u304c\u554f\u984c\u3068\u306a\u308b\u9818\u57df\u3067\u306e\u5fdc\u7528\u3067\u306f\u3001\u8aa4\u308a\u8a02\u6b63\u6a5f\u80fd\u306e\u4ed8\u3044\u305fECC\u30e1\u30e2\u30ea\u304c\u5229\u7528\u3055\u308c\u308b\u3002ECC\u30e1\u30e2\u30ea\u3067\u306f\u5f8c\u8ff0\u306e\u8aa4\u308a\u8a02\u6b63\u3092\u884c\u3044\u30011bit\u307e\u3067\u306e\u30a8\u30e9\u30fc\u3092\u5c0f\u3055\u306a\u30aa\u30fc\u30d0\u30fc\u30d8\u30c3\u30c9\u3067\u8a02\u6b63\u3059\u308b\u3002\n\n\n\n### \u53e4\u5178\u30a8\u30e9\u30fc\u8a02\u6b63: \u591a\u6570\u6c7a\n\u6700\u3082\u5358\u7d14\u306a\u7b26\u53f7\u306f\u591a\u6570\u6c7a\u3067\u3042\u308b\u3002\u591a\u6570\u6c7a\u3067\u306f\u500b\u3005\u306e\u30d3\u30c3\u30c8\u3092$d$\u500d\u306b\u30b3\u30d4\u30fc\u3059\u308b\u3002\u3053\u306e\u6642\u3001$k$\u30d3\u30c3\u30c8\u306e\u60c5\u5831\u3092$n:=dk$\u30d3\u30c3\u30c8\u3067\u8868\u73fe\u3059\u308b\u3053\u3068\u306b\u306a\u308b\u3002\u591a\u6570\u6c7a\u306e\u5f15\u304d\u5206\u3051\u3092\u9632\u3050\u305f\u3081\u3001$d$\u306f\u5947\u6570\u3068\u3059\u308b\u3002\n\u73fe\u5b9f\u306b\u5b58\u5728\u3059\u308b$n$\u30d3\u30c3\u30c8\u306e\u3053\u3068\u3092\u7269\u7406\u30d3\u30c3\u30c8\u3001\u5b9f\u614b\u3068\u3057\u3066\u8868\u73fe\u3057\u3066\u3044\u308b$k$\u30d3\u30c3\u30c8\u306e\u3053\u3068\u3092\u8ad6\u7406\u30d3\u30c3\u30c8\u3068\u547c\u3076\u3002\n\n\u591a\u6570\u6c7a\u3067\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8aa4\u308a\u306e\u691c\u77e5\u3068\u8a02\u6b63\u3092\u884c\u3046\u3002\u500b\u3005\u306e\u8ad6\u7406\u30d3\u30c3\u30c8\u306b\u3064\u3044\u3066\u8907\u88fd\u3057\u305f$d$\u30d3\u30c3\u30c8\u306e\u60c5\u5831\u3092\u8aad\u307f\u51fa\u3057\u30010,1\u306e\u3069\u3061\u3089\u304c\u591a\u3044\u304b\u3092\u6570\u3048\u308b\u3002\u305d\u3057\u3066\u3001\u983b\u5ea6\u304c\u9ad8\u304b\u3063\u305f\u5024\u304c\u305d\u306e\u8ad6\u7406\u30d3\u30c3\u30c8\u306e\u5024\u3067\u3042\u3063\u305f\u3068\u5224\u5b9a\u3059\u308b\u3002\u521d\u671f\u306e\u5024\u304c\u3069\u3046\u3042\u308c\u3001\u7b26\u53f7\u5316\u3055\u308c\u305f\u72b6\u614b\u306f\u5168\u3066\u306e\u5024\u304c\u540c\u3058\u3067\u3042\u308b\u304b\u3089\u3001$d$\u30d3\u30c3\u30c8\u306e\u5024\u304c\u4e00\u3064\u3067\u3082\u4e00\u81f4\u3057\u3066\u3044\u306a\u3044\u5834\u5408\u3001\u4f55\u3089\u304b\u306e\u30a8\u30e9\u30fc\u304c\u751f\u3058\u3066\u3044\u308b\u3053\u3068\u304c\u78ba\u4fe1\u3067\u304d\u308b\u3002\u5f93\u3063\u3066$n$\u30d3\u30c3\u30c8\u304c\u4e38\u3054\u3068\u53cd\u8ee2\u3057\u306a\u3044\u9650\u308a\u5fc5\u305a\u8aa4\u308a\u3092\u691c\u77e5\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\n\n\u8aa4\u308a\u3092\u8a02\u6b63\u3057\u305f\u3044\u5834\u5408\u306f\u534a\u6570\u4ee5\u4e0a\u306e\u30d3\u30c3\u30c8\u304c\u6b63\u3057\u3044\u5024\u3092\u4fdd\u6301\u3057\u3066\u3044\u308c\u3070\u6b63\u3057\u3044\u7d50\u679c\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u305d\u308c\u305e\u308c\u306e\u30d3\u30c3\u30c8\u306b1\u3088\u308a\u5341\u5206\u5c0f\u3055\u3044\u78ba\u7387$p$\u3067\u30a8\u30e9\u30fc\u304c\u751f\u3058\u308b\u3068\u3059\u308b\u3068\u3001\u534a\u6570\u4ee5\u4e0a\u306e\u30d3\u30c3\u30c8\u304c\u30a8\u30e9\u30fc\u3092\u8d77\u3053\u3059\u78ba\u7387\u306f\u304a\u3088\u305d$p^{\\lfloor d/2 \\rfloor+1}$\u3067\u3042\u308b\u3002\n\u5f93\u3063\u3066\u3001$d$\u30922\u5897\u3084\u3059\u3054\u3068\u306b\u3001\u591a\u6570\u6c7a\u304c\u5931\u6557\u3059\u308b\u78ba\u7387\u306f\u30aa\u30fc\u30c0\u30fc\u3067$p$\u500d\u5c0f\u3055\u304f\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\n\u6700\u3082\u5358\u7d14\u306a$k=1$\u306e\u5834\u5408\u3092\u30b7\u30df\u30e5\u30ec\u30fc\u30c8\u3059\u308b\u30b3\u30fc\u30c9\u3092\u66f8\u304f\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002\n\n\n```python\nimport numpy as np\n\ndata = 1\nd = 31\np = 0.01\n\nprint(\"original bit: {}\".format(data))\nstate = np.repeat(data, d)\nis_flipped = (np.random.rand(d)Models and Pricing of Financial Derivativs HW_01\n\n**
    11510691 \u7a0b\u8fdc\u661f
    **\n\n\n\n## Question 1\n\nExplain carefully the difference between *selling* a *call* option and *buying* a *put* option.\n\n$Answer$\n\n>$\\DeclareMathOperator*{\\argmin}{argmin}\n\\DeclareMathOperator*{\\argmax}{argmax}\n\\newcommand{\\using}[1]{\\stackrel{\\mathrm{#1}}{=}}\n\\newcommand{\\ffrac}{\\displaystyle \\frac}\n\\newcommand{\\space}{\\text{ }}\n\\newcommand{\\bspace}{\\;\\;\\;\\;}\n\\newcommand{\\QQQ}{\\boxed{?\\:}}\n\\newcommand{\\void}{\\left.\\right.}\n\\newcommand{\\CB}[1]{\\left\\{ #1 \\right\\}}\n\\newcommand{\\SB}[1]{\\left[ #1 \\right]}\n\\newcommand{\\P}[1]{\\left( #1 \\right)}\n\\newcommand{\\dd}{\\mathrm{d}}\n\\newcommand{\\Tran}[1]{{#1}^{\\mathrm{T}}}\n\\newcommand{\\d}[1]{\\displaystyle{#1}}\n\\newcommand{\\EE}[2][\\,\\!]{\\mathbb{E}_{#1}\\left[#2\\right]}\n\\newcommand{\\Var}[2][\\,\\!]{\\mathrm{Var}_{#1}\\left[#2\\right]}\n\\newcommand{\\Cov}[2][\\,\\!]{\\mathrm{Cov}_{#1}\\left(#2\\right)}\n\\newcommand{\\Corr}[2][\\,\\!]{\\mathrm{Corr}_{#1}\\left(#2\\right)}\n\\newcommand{\\I}[1]{\\mathrm{I}\\left( #1 \\right)}\n\\newcommand{\\N}[1]{\\mathrm{N} \\left( #1 \\right)}\n\\newcommand{\\ow}{\\text{otherwise}}$Selling a call option: As the writer of the call option, I give the holder the right to buy an asset at a specified time $T$ for a specified price $K$. My payoff would be $-\\max\\P{S_T - K,0} = \\min\\P{K-S_T,0}$ for european call options. If I sold an american call option, the holder can exercise at any time before $T$.\n>\n>Buying a put option: As the holder of the put option, I actually was granted the right to sell an asset at a specified time $T$ for a specified price $K$. My payoff would be $\\max\\P{K - S_T, 0}$ Or before $T$ if what I bought is an american put option.\n\n## Question 2\n\nThe current price of a stock is $\\$94$, and $3$-month European call option with a strike price of $\\$95$ currently sell for $\\$4.70$. An investor who feels that the price of the stock will increase is trying to decide between buying $100$ shares and buying $2, 000$ call options ($20$ contracts). Both strategies involve an investment of $\\$9, 400$. What advice would you give? How high does the stock price have to rise for the option strategy to be more profitable?\n\n$Answer$\n\n>We can write their profit function on the stock price $S_t$.\n>\n>- Stock: $100\\P{S_t - 94}$\n>- Option: $2000\\big(\\max\\P{S_t - 95,0} - 4.7\\big)$\n>\n>They intersect at two points, $\\P{0,0}$ and $\\P{100,600}$. It's generally acknowledge that it's of higher possibility that the stock price moves less than more, thus I personally think that holding the stocks rather than the options have a better chance to profit.\n>\n>As for the second question, since we've already acquire their intersection, we can say that when the stock price goes higher than $100$, options will win more.\n\n## Question 3\n\nA trader *buys* a *European call* option and *sells* a *European put* option. The options have the same underlying asset, strike price, and maturity. Describe the trader\u2019s ***position***. Under what circumstances does the price of the call equal the price of the put?\n\n$Answer$\n\n>For the call, he has paid $c$ to buy the right that he can use $K$ to buy the underlying asset as time $T$. For the put, he has received $p$ and given the right to someone else selling him the asset at price $K$ at time $T$. The terminal payoff of the trader is\n>\n>$$\\P{S_T - K}^+ - \\P{K-S_T}^+ = S_T - K$$\n>\n>To let the prices equal, by the **Put-call parity** (or to avoid proving it, we replicate the strategy by buying an asset and borrowing $Ke^{-rT}$ at time $0$), we have \n>\n>$$c-p = S_0 - Ke^{-rT}$$\n>\n>Thus when $S_0 = Ke^{-rT}$, the price of the call $c$ equals the price of the put $p$.\n\n## Question 4\n\nThe price of a stock is $\\$40$. The price of a $1$-year *European put* option on the stock with a strike price of $\\$30$ is quoted as $\\$7$ and the price of a $1$-year *European call* option on the stock with a strike price of $\\$50$ is quoted as $\\$5$. Suppose that an investor *buys* $100$ shares, *shorts* $100$ *call* options, and *buys* $100$ put options. Draw a diagram illustrating how the investor\u2019s ***profit or loss*** varies with the stock price over the next year. How does your answer change if the investor *buys* $100$ shares, *shorts* $200$ *call* options, and *buys* $200$ *put* options?\n\n$Answer$\n\n>We first write its payoff function on the stock price:\n>\n>$$\\begin{align}\np &= 100S_T -100 \\cdot\\max\\P{S_T - 50, 0} + 100\\cdot \\max\\P{30 - S_T,0}\\\\\n&= \\begin{cases}\n5000, &\\text{if } S_T \\geq 50 \\\\\n100S_T, &\\text{if } 50 \\geq S_T \\geq 30 \\\\\n3000, &\\text{if } 30 \\geq S_T \\geq 0\n\\end{cases}\n\\end{align}\n$$\n>\n>\n>\n>After that, the payoff would change to:\n\n>$$\\begin{align}\np &= 100\\P{S_T - 40} + 200\\SB{5 - \\max\\P{S_T - 50, 0}} + 200\\SB{\\max\\P{30 - S_T,0} - 7}\\\\\n&= \\begin{cases}\n10000 - 100S_T, &\\text{if } S_T \\geq 50 \\\\\n100S_T, &\\text{if } 50 \\geq S_T \\geq 30 \\\\\n6000 - 100S_T, &\\text{if } 30 \\geq S_T \\geq 0\n\\end{cases}\n\\end{align}\n$$\n>\n>\n>\n>Also note that the investor has paid $4200$ and $4400$ initially for the two portfolios, respectively.\n\n## Question 5\n\nWhat is a *lower bound* for the price of a $1$-month *European put* option on a non-dividend-paying stock when the stock price is $\\$12$, the strike price is $\\$15$, and the risk-free interest rate is $6\\%$ per annum?\n\n$Answer$\n\n>The lower bound of the option can be obtained using the formula\n>\n>$\\begin{align}\np &\\geq K e^{-rT} - S_0 \\\\\n&= 15 \\cdot e^{-6\\% \\times 1/12} - 12 \\\\\n&\\approx 2.93\n\\end{align}$\n>\n>To prove this formula (***inevitable***!): \n>\n>- we construct a arbitrage portfolio: if not, then $p0$. (**Positive Output**)\n>- consider two portolios at time $0$: $A$: a put and an asset $S_0$; $B$: cash $Ke^{-rT}$. At time $T$, $A$ worth $\\P{K-S_T}^+ + S_T = \\max\\P{K,S_T}$ however $B$ worth $K\\leq \\max\\P{K,S_T}$. Thus at time $0$ this relationship turns to $p \\geq K e^{-rT} - S_0$\n\n## Qustion 6\n\"The *early exercise* of an American *put* is a trade-off between the *time value of money* and the *insurance value of a put*.\" Explain this statement.\n\n$Answer$\n\n>The early exercise of an American put option is to sell the stock to the writer at the Strike price $K$ before the expiration date $T$. Suppose he exercised at time $t$ thus his time value of money is $Ke^{-r\\P{T-t}}$. But then he can not sell the stock at $K$ at time $T$ any more. In return, he is now able to earn interest on it between the time of the early exercise and the expiration date.\n\n## Question 7\n\nThe price of a non-dividend-paying stock is $19$ and the price of a $3$-month European call option on the stock with a strike price of $\\$20$ is $\\$1$. The risk-free rate is $4\\%$ per annum. What is the price of a $3$-month European put option with a strike price of $\\$20$?\n\n$Answer$\n\n>By the put-call parity, we have: $1 + 20 \\times e^{-4\\% \\times 0.25} = p + 19$ thus $p = 1.80$\n>\n>To prove this parity (***inevitable***!), consider two portfolios at time $0$\n>\n>- a call and bond worth $c + Ke^{-rT}$\n- a put and an asset worth $p + S_0$\n>\n>At time $T$ these two portfolios both worth $\\max\\P{S_T,K}$ thus at time $0$, we have $c + Ke^{-rT} = p + S_0$\n\n## Question 8\n\nA $1$-month European put option on a non-dividend-paying stock is currently selling for $\\$2.50$. The stock price is $\\$47$, the strike price is $\\$50$, and the risk-free interest rate is $6\\%$ per annum. What opportunities are there for an arbitrageur?\n\n$Answer$\n\n>Based on the put-call parity, $c + Ke^{-rT} = S_0 + p \\Longrightarrow c + 49.75 = 47+2.5$. Thus there's always a chance for arbitraging. He can buy a stock and a put option using the borrowed money, $49.5$ with interest rate $6\\%$.\n>\n>Then he can win $50 - 49.5e^{0.06\\times 1/12} \\approx 0.25 $ if the stock price goes lower than $50$ or more if the stock price goes higher than $50$. \n\n## Question 9\n\n$\\P{1}$ The price of an American call on a non-dividend-paying stock is $\\$4$. The stock price is $\\$31$, the strike price is $\\$30$, and the expiration date is in $3$ months. The risk-free interest rate is $8\\%$. Derive upper and lower bounds for the price of an American put on the same stock with the same strike price and expiration date.\n\n$Answer$\n\n>$P \\geq p = c + Ke^{-rT}-S_0 = C + Ke^{-rT} - S_0 = 4 + 30 e^{-8\\%\\times1/4} -31 \\approx 2.41$\n>\n>And to find something that keeps over an American Put, we can use $K$ cash at the beginning, thus $c + K \\geq P+ S_0$ always holds. Therefore, $P \\leq c + K - S_0 = C + K - S_0 = 4 + 30 - 31 = 3$\n\n$\\P{2}$ Explain carefully the arbitrage opportunities in Problem $9(1)$ if the American put price is greater than the calculated upper bound.\n\n$Answer$\n\n>If the American Put price is greater than $3$ that is to say that $P \\geq C + K - S_0$, then we write an American put option and sell it to somebody, then use the money to buy a American call option, to borrow a stock then sell it to gain $S_0$. Send $K$ cash to the bank. Then when the American put option holder want to exercise, we can instantly use $K = 30$ to buy the stock and return to the stock lender. Up to now, we start from nothing to a American call option and a positive payoff and some interest.\n\n## Question 10\n\n$\\P{1}$ Suppose that $c_1$ , $c_2$ , and $c_3$ are the prices of European call options with strike prices $K_1$, $K_2$, and $K_3$, respectively, where $K_3 > K_2 > K_1$ and $K_3 \u2212 K_2 = K_2 \u2212 K_1$. All options have the same maturity. Show that $c_2 \\leq 0.5\\P{c_1 + c_3}$\n\n$Answer$\n\n>If not, then $2c_2 > c_1 + c_3$. So that we have the arbitrage chance. First write two call option with strike price $K_2$ and then use the money gained to buy two call option with strike price $K_1$ and $K_3$. We already have some money left now.\n>\n>Then we the exercise time comes, we can exercise all three at the same time since $2K_2 = K_1 + K_3$, meaning that we gain money from nothing. Thus $2c_2 \\leq c_1 + c_3$.\n\n$\\P{2}$ What is the result corresponding to that in Problem $10(1)$ For European put options?\n\n$Answer$\n\n>$p_2 \\leq 0.5\\P{p_1 + p_3}$, the proof is similar to the preceding one.\n\n\n```python\nimport math as m\nimport numpy as np\nimport matplotlib.pyplot as plt\n```\n\n\n```python\nX = np.linspace(0,80,9)\nY = np.zeros(9)\nY[0:4] = 3000\nY[4:6]= 100 * X[4:6]\nY[6:9]= 5000\n```\n\n\n```python\nfig = plt.figure()\nfig.set_size_inches(10,5)\nax = fig.add_subplot(111)\nax.plot(X,Y)\nax.set_xlabel('Stock Price')\nax.set_ylabel('Payoff')\n```\n\n\n```python\nX = np.linspace(0,80,9)\nY = np.zeros(9)\nY[0:4] = 6000 - 100*X[0:4]\nY[4:6]= 100 * X[4:6]\nY[6:9]= 10000 - 100*X[6:9]\n```\n\n\n```python\nfig = plt.figure()\nfig.set_size_inches(10,5)\nax = fig.add_subplot(111)\nax.plot(X,Y)\nax.set_xlabel('Stock Price')\nax.set_ylabel('Payoff')\n```\n\n\n```python\n# another method\nfig, axs = plt.subplots()\nfig.set_size_inches(10,5)\naxs.plot(X,Y)\naxs.set_xlabel('Stock Price')\naxs.set_ylabel('Payoff')\n```\n\n\n```python\n15*m.exp(-0.005)-12\n```\n\n\n\n\n 2.9251871878902342\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "43ae69231c020b75e3aa9dbb00ee5d0f27044f9e", "size": 83586, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "FinMath/Models and Pricing of Financial Derivatives/HW/HW_01_fixed.ipynb", "max_stars_repo_name": "XavierOwen/Notes", "max_stars_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2018-11-27T10:31:08.000Z", "max_stars_repo_stars_event_max_datetime": "2019-01-20T03:11:58.000Z", "max_issues_repo_path": "FinMath/Models and Pricing of Financial Derivatives/HW/HW_01_fixed.ipynb", "max_issues_repo_name": "XavierOwen/Notes", "max_issues_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "FinMath/Models and Pricing of Financial Derivatives/HW/HW_01_fixed.ipynb", "max_forks_repo_name": "XavierOwen/Notes", "max_forks_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-07-14T19:57:23.000Z", "max_forks_repo_forks_event_max_datetime": "2020-07-14T19:57:23.000Z", "avg_line_length": 181.3145336226, "max_line_length": 24992, "alphanum_fraction": 0.8847534276, "converted": true, "num_tokens": 3666, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5156199157230157, "lm_q2_score": 0.8539127529517043, "lm_q1q2_score": 0.4402944217117661}} {"text": "Based on\nhttps://www.tensorflow.org/tutorials/images/classification\n\nUnfortunately it proved out, that both the fruit and Pokemon datasets I tested would have been\ntoo complex and would have contained too few images for any current quantum neural\nnetwork to classify them successfully.\nTherefore this path was abandoned.\n\n\n```python\n%matplotlib inline\n\nimport collections\n# import os.path\n\nimport cirq\nfrom cirq.contrib.svg import SVGCircuit\nimport matplotlib.pyplot as plt\nimport sympy\nimport tensorflow as tf\nfrom tensorflow.keras import layers\nimport tensorflow_quantum as tfq\n```\n\nLoading the dataset\n\n\n```python\n# data_dir = \"./fruits360\"\nsplit_seed = 123\nimage_size = (8, 8)\nvalidation_split=0.2\ncolor_mode=\"grayscale\"\n\ntrain_ds = tf.keras.preprocessing.image_dataset_from_directory(\n \"./fruits-360/Training\",\n # validation_split=validation_split,\n # subset=\"training\",\n seed=split_seed,\n image_size=image_size,\n color_mode=color_mode\n)\nval_ds = tf.keras.preprocessing.image_dataset_from_directory(\n \"./fruits-360/Test\",\n # validation_split=validation_split,\n # subset=\"validation\",\n seed=split_seed,\n image_size=image_size,\n color_mode=color_mode\n)\nclass_names = train_ds.class_names\nprint(class_names)\n\nplt.figure(figsize=(10, 10))\nfor images, labels in train_ds.take(1):\n for i in range(9):\n ax = plt.subplot(3, 3, i + 1)\n plt.imshow(images[i].numpy().astype(\"uint8\"))\n plt.title(class_names[labels[i]])\n plt.axis(\"off\")\n```\n\n\n```python\n# Performance tuning\nAUTOTUNE = tf.data.AUTOTUNE\ntrain_ds = train_ds.cache().shuffle(1000).prefetch(buffer_size=AUTOTUNE)\nval_ds = val_ds.cache().prefetch(buffer_size=AUTOTUNE)\n\nclassical_model = tf.keras.models.Sequential([\n layers.experimental.preprocessing.Rescaling(1/255, input_shape=image_size),\n layers.Conv2D(16, 3, padding=\"same\", activation=\"relu\"),\n layers.Flatten(),\n layers.Dense(8, activation=\"relu\"),\n layers.Dense(2)\n])\n\nclassical_model.compile(optimizer='adam',\n loss=tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True),\n metrics=['accuracy'])\n\nclassical_model.summary()\n```\n\n\n```python\nclass CircuitLayerBuilder:\n def __init__(self, data_qubits, readout):\n self.data_qubits = data_qubits\n self.readout = readout\n\n def add_layer(self, circuit, gate, prefix):\n for i, qubit in enumerate(self.data_qubits):\n symbol = sympy.Symbol(prefix + '-' + str(i))\n circuit.append(gate(qubit, self.readout)**symbol)\n\ndef create_quantum_model():\n \"\"\"Create a QNN model circuit and readout operation to go along with it.\"\"\"\n data_qubits = cirq.GridQubit.rect(*image_size)\n readout = cirq.GridQubit.rect(8, 1)\n circuit = cirq.Circuit()\n\n # TODO\n\n # Prepare the readout qubit.\n circuit.append(cirq.X(readout))\n circuit.append(cirq.H(readout))\n\n builder = CircuitLayerBuilder(\n data_qubits = data_qubits,\n readout=readout)\n\n # Then add layers (experiment by adding more).\n builder.add_layer(circuit, cirq.XX, \"xx1\")\n builder.add_layer(circuit, cirq.ZZ, \"zz1\")\n\n # Finally, prepare the readout qubit.\n for qubit in readout:\n circuit.append(cirq.H(qubit))\n\n return circuit, cirq.Z(readout)\n\nmodel_circuit, model_readout = create_quantum_model()\n\nSVGCircuit(model_circuit)\n```\n\n\n```python\nquantum_model = tf.keras.Sequential([\n # The input is the data-circuit, encoded as a tf.string\n tf.keras.layers.Input(shape=(), dtype=tf.string),\n # The PQC layer returns the expected value of the readout gate, range [-1,1].\n tfq.layers.PQC(model_circuit, model_readout),\n])\n```\n\n\n```python\nclassical_model.fit(train_ds.)\n\n# TODO\n```\n", "meta": {"hexsha": "66a8279afe6e47c0e486632cd0c28233663220ca", "size": 35756, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "old_experiments/fruits.ipynb", "max_stars_repo_name": "AgenttiX/fys2029-project", "max_stars_repo_head_hexsha": "26dc885064721f40db10fad4405f3366f2cfdf4a", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-05-21T14:39:07.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-21T14:39:07.000Z", "max_issues_repo_path": "old_experiments/fruits.ipynb", "max_issues_repo_name": "AgenttiX/fys2029-project", "max_issues_repo_head_hexsha": "26dc885064721f40db10fad4405f3366f2cfdf4a", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-05-25T12:52:49.000Z", "max_issues_repo_issues_event_max_datetime": "2021-05-25T12:52:49.000Z", "max_forks_repo_path": "old_experiments/fruits.ipynb", "max_forks_repo_name": "AgenttiX/fys2029-project", "max_forks_repo_head_hexsha": "26dc885064721f40db10fad4405f3366f2cfdf4a", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 124.1527777778, "max_line_length": 17971, "alphanum_fraction": 0.8054312563, "converted": true, "num_tokens": 916, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357701094303, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.44029105081602554}} {"text": "```python\n__author__ = 'kjoseph'\nfrom array import array\n\nfrom deflection import *\nfrom initialize_functions import *\nfrom functions import *\nimport scipy.spatial\nFUNDAMENTALS = ['ae','ap','aa','be','bp','ba','oe','op','oa']\n\nEQUATION_FILE = \"../data/sexes_avg_new.txt\"\n\n\nt,M = get_t_and_M_from_file(EQUATION_FILE, FUNDAMENTALS)\ndeflection_computation = get_coefficients(M,t)\nH = np.dot(np.vstack((np.identity(9),-M)), np.hstack((np.identity(9),-M.transpose())))\nS_a,g_a = compute_s_and_g(M,t,'a',EPA_LIST)\nS_b,g_b = compute_s_and_g(M,t,'b',EPA_LIST)\nS_o,g_o = compute_s_and_g(M,t,'o',EPA_LIST)\n##TEST DEFLECTION\n\npm1 = ParameterStruct('t', 1,3,3,[0.556179039209, 1],[0.748935820403, 1],[0.608249328496,1],1,1)\npm2 = ParameterStruct('t', 1,3,3,[-0.42344161868, 1],[-0.659712135792, 1],[0.192399427834,1],1,1)\nopt_v = get_optimal(M,t,H,S_b,g_b,['a','o'], pm1,pm2)\n\nfor v in [[-0.447532713413, -0.356447458267, 0.429316014051], [-0.069087468087,0.258912533522,0.0270347073674]]:\n print scipy.spatial.distance.cosine(opt_v,v)\n\ndeflection_computation = np.array(deflection_computation, dtype=np.object_)\n\n\nvals_list = array('f',[2.45, 1.75, 0.29,1.85, 1.65, 0.30,1.49,.31,.75])\nfor x in compute_deflection_bad(vals_list,deflection_computation):\n print 'x: ', x\n\n# np.random.shuffle(M)\n# deflection_computation = get_coefficients(M, t)\n# print 'shuffled'\n# for x in compute_deflection_bad(vals_list,deflection_computation):\n# print x\n#for fundamental_index in range(N_FUNDAMENTALS):\n# c0, c1, c2 = get_constants_for_fundamental(fundamental_index,deflection_computation,value_list)\n# print FUNDAMENTALS[fundamental_index], -c1/(2*c0), c1, c0\n```\n\n\n```python\ndef get_t_and_M_from_file(eq_filename, fundamentals,spl_char= \"\\t\"):\n M = []\n t = []\n equation_file = open(eq_filename)\n i = 0\n for line in equation_file:\n t.append(set())\n line_spl = [l.strip() for l in line.split(spl_char)]\n M.append([float(x) for x in line_spl[1:]])\n\n coeff = line_spl[0].replace(\"Z\",\"\")\n for j in range(len(coeff)):\n if coeff[j] == '1':\n t[i].add(fundamentals[j])\n i+=1\n\n equation_file.close()\n return t, np.array(M)\n\nFUNDAMENTALS = ['ae','ap','aa','be','bp','ba','oe','op','oa']\n\nt, M = get_t_and_M_from_file(\"../data/sexes_avg_new.txt\",FUNDAMENTALS,\"\\t\")\n\nfund_eq = [[] for i in range(len(FUNDAMENTALS))]\nfor j in range(len(FUNDAMENTALS)):\n for i,coef in enumerate(t):\n coef = \"*\".join(coef)\n l = M[i,j]\n app_str = \"\"\n if l > 0:\n app_str = \"+\"\n if l == 0:\n continue\n elif coef != '':\n fund_eq[j].append(app_str +str(l)+\"*\"+coef)\n else:\n fund_eq[j].append(app_str+str(l))\n\n\nFUND_EQ_STRS = [\"\".join(x) for x in fund_eq]\nprint FUND_EQ_STRS\n```\n\n ['-0.155+0.41*ae+0.42*be-0.02*bp-0.1*ba+0.03*oe+0.06*op+0.05*be*ae+0.03*ae*op+0.12*be*oe-0.05*be*op-0.05*oe*bp+0.03*be*oe*ae-0.02*be*ae*op', '-0.1+0.56*ap+0.06*aa-0.105*be+0.44*bp+0.04*oe-0.05*ap*bp+0.01*be*oe', '+0.14+0.05*ae+0.705*aa-0.06*be+0.29*ba-0.06*aa*ba', '-0.14+0.11*ae+0.555*be-0.12*ba+0.05*op+0.11*be*oe-0.05*be*op-0.02*oe*bp+0.02*be*oe*ae', '+0.06+0.16*ap-0.15*be+0.685*bp+0.03*oe-0.015*op+0.01*be*ae+0.02*be*oe', '+0.17+0.02*ae-0.06*ap+0.3*aa+0.04*be+0.64*ba', '+0.025+0.11*be+0.61*oe-0.01*oa+0.03*be*ae+0.04*be*oe-0.03*oe*bp', '-0.395+0.15*be-0.11*bp-0.115*oe+0.66*op+0.07*oa+0.03*be*oe+0.03*be*op-0.05*bp*op', '-0.035+0.02*be+0.03*oe-0.05*op+0.745*oa']\n\n\n\n```python\nfrom sympy.solvers import solve\nfrom sympy import Symbol\nfrom sympy import sympify\nfrom sympy.polys import Poly\nfrom math import sqrt \n\n\nidentities = [1, 2]\nsentiment_words = [1,4]\nconstraints = [SocialEventConstraint(actor=1, behavior=1, object=2),\n EqualityConstraint(identity=1, sentiment_word=1)]\n \nequation_str = ''\n\nfor starter, list_of_terms in [['i',identities], ['z',sentiment_words]]:\n for term in list_of_terms:\n for epa in ['e','p','a']:\n id = starter+str(term)+epa\n symbols[id] = Symbol(id)\n\neq_constr = [constraint.get_constraint_string() for constraint in constraints]\nequation_str = \"+\".join(eq_constr) \nexpr = sympify(equation_str)\n\ndat = {\"i1e\": 1.39,\n \"i1p\": .88,\n \"i1a\": 0.96,\n \"z1e\": -1.92,\n \"z1p\": 1.00,\n \"z1a\": 1.62,\n \"i2e\": 1.49,\n \"i2p\": .31,\n \"i2a\": 0.75,\n }\nsubstitutions = dat.items()\nexpr = expr.subs(substitutions).expand()\np = Poly(expr).coeffs()\nprint expr\nprint -p[1]/(2*p[0])\n```\n\n\n```python\nclass User:\n \n def __init__(self, n_identities, sentences):\n self.sentences = sentences\n self.sentences_for_identity = [list() * n_identities]\n for sent_it, sentence in enumerate(sentences):\n self.add_sentence(sentence,sent_it)\n \n def add_sentence(sentence,sent_it=None):\n self.sentences.append(sentence)\n if not sent_it:\n sent_it = len(sentence)\n for identity in sentence.identities_contained():\n self.sentences_for_identity[identity].append(sent_it)\n```\n\n\n```python\nd = read_grouped_file\n\nfor g in d:\n construct_sentence\n add_to_user\n```\n\n\n```python\n\n```\n\n\n```python\nfrom nltk.stem.wordnet import WordNetLemmatizer\nfrom nltk.corpus import wordnet as wn\n\n_wnl = WordNetLemmatizer()\n```\n\n\n```python\nsent_dict = {}\nfor x in codecs.open(\"../../../../thesis/thesis_work/lcss_study/data/all_epa_terms.txt\",\"r\",\"utf8\"):\n x_spl = x.split(\"\\t\")\n sent_dict[x_spl[0]] = [float(z) for z in x_spl[1:]]\n\n\nIDENTITY_LIST_FN = \"../../../../thesis/thesis_work/lcss_study/data/identities_for_study.txt\"\nidentity_set = {x.strip().lower() for x in open(IDENTITY_LIST_FN)}\nfull_set_of_interesting_terms = identity_set|set(sent_dict.keys())\n```\n\n\n```python\n_wnl.lemmatize(\"stuck\",wn.VERB)\n```\n\n\n\n\n u'stick'\n\n\n\n\n```python\nsent_dict['deal with']\n```\n\n\n```python\nidentity_set = {i : x.strip().lower() for i,x in enumerate(open(IDENTITY_LIST_FN))}\n```\n\n\n```python\nfrom gensim.models.word2vec import Word2Vec\nmodel = Word2Vec.load_word2vec_format(\"../../../../thesis/thesis_work/identity_extraction/python/gensim_model/glove_twitter_50_raw_model.txt.gz\", binary=False)\n```\n\n WARNING:gensim.models.word2vec:consider setting layer size to a multiple of 4 for greater performance\n\n\n\n```python\n'miracle' in sent_dict\n```\n\n\n\n\n True\n\n\n", "meta": {"hexsha": "c7c84e362b45bdad679fbd4ea7baa4b2caa8ee7d", "size": 15027, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "python/src/example_usage.ipynb", "max_stars_repo_name": "kennyjoseph/act_paper_public", "max_stars_repo_head_hexsha": "8d2fff033b653c815a07bf032b37b90422085f58", "max_stars_repo_licenses": ["MIT", "Unlicense"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "python/src/example_usage.ipynb", "max_issues_repo_name": "kennyjoseph/act_paper_public", "max_issues_repo_head_hexsha": "8d2fff033b653c815a07bf032b37b90422085f58", "max_issues_repo_licenses": ["MIT", "Unlicense"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "python/src/example_usage.ipynb", "max_forks_repo_name": "kennyjoseph/act_paper_public", "max_forks_repo_head_hexsha": "8d2fff033b653c815a07bf032b37b90422085f58", "max_forks_repo_licenses": ["MIT", "Unlicense"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 39.9654255319, "max_line_length": 1317, "alphanum_fraction": 0.5578625141, "converted": true, "num_tokens": 2179, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660542, "lm_q2_score": 0.6757646010190476, "lm_q1q2_score": 0.4401625602428718}} {"text": "# Pandas\n\n\n```python\nimport pandas as pd\n```\n\n\n```python\npd.DataFrame({'some_data': [1, 2, 3], 'more_data': ['x', 'y', 'z']})\n```\n\n# NetworkX\n\n\n```python\nimport networkx as nx\nimport matplotlib.pyplot as plt\n```\n\n\n```python\nnx.draw(nx.petersen_graph())\n```\n\n# Shapely\n\n\n```python\nfrom shapely.geometry import Polygon\n```\n\n\n```python\nPolygon([(0, 0), (0, 1), (1, 0)])\n```\n\n# SymPy\n\n\n```python\nimport sympy\n```\n\n\n```python\nx = sympy.Symbol('x')\nsympy.Integral(sympy.sin(x**2) / x)\n```\n\n# ipyleaflet\n\n\n```python\nfrom ipyleaflet import Map, basemaps, basemap_to_tiles\n```\n\n\n```python\nm = Map(\n basemap=basemap_to_tiles(basemaps.NASAGIBS.ModisTerraTrueColorCR, \"2017-04-08\"),\n center=(52.204793, 360.121558),\n zoom=4\n)\n\nm\n```\n\n# Appendix\n\nThere is more:\n* Iris\n* Logs\n* Xarray\n* Snakeviz\n* PyMC3\n* Bokeh\n* Altair\n* ...\n\n\n```python\n\n```\n", "meta": {"hexsha": "9690b26cd7a684026af541a4245e5ff87098c5a9", "size": 3562, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "02_rich_visualizations/01_libraries.ipynb", "max_stars_repo_name": "silvanmelchior/jupyter-tricks", "max_stars_repo_head_hexsha": "5effd591a71901c5e9ce8190f67962bdb9235073", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "02_rich_visualizations/01_libraries.ipynb", "max_issues_repo_name": "silvanmelchior/jupyter-tricks", "max_issues_repo_head_hexsha": "5effd591a71901c5e9ce8190f67962bdb9235073", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "02_rich_visualizations/01_libraries.ipynb", "max_forks_repo_name": "silvanmelchior/jupyter-tricks", "max_forks_repo_head_hexsha": "5effd591a71901c5e9ce8190f67962bdb9235073", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 17.2912621359, "max_line_length": 95, "alphanum_fraction": 0.4761370017, "converted": true, "num_tokens": 301, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757645879592641, "lm_q2_score": 0.6513548646660542, "lm_q1q2_score": 0.44016255173631835}} {"text": "# Realization of Recursive Filters\n\n*This jupyter notebook is part of a [collection of notebooks](../index.ipynb) on various topics of Digital Signal Processing. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*\n\n## Quantization of Filter Coefficients\n\nThe finite numerical resolution of digital number representations has impact on the properties of filters, as already discussed for [non-recursive filters](../nonrecursive_filters/quantization_effects.ipynb#Quantization-Effects). The quantization of coefficients, state variables, algebraic operations and signals plays an important role in the design of recursive filters. Compared to non-recursive filters, the impact of quantization is often more prominent due to the feedback. Severe degradations from the desired characteristics and instability are potential consequences of a finite word length in practical implementations.\n\nA recursive filter of order $N \\geq 2$ can be [decomposed into second-order sections (SOS)](../recursive_filters/cascaded_structures.ipynb). Due to the grouping of poles/zeros to filter coefficients with a limited amplitude range, a realization by cascaded SOS is favorable in practice. We therefore limit our investigation of quantization effects to SOS. The transfer function of a SOS is given as\n\n\\begin{equation}\nH(z) = \\frac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}}\n\\end{equation}\n\nThis can be [split into a non-recursive part and a recursive part](../recursive_filters/introduction.ipynb#Recursive-Filters). The quantization effects of non-recursive filters have already been discussed. We therefore focus here on the recursive part given by the transfer function\n\n\\begin{equation}\nH(z) = \\frac{1}{1 + a_1 z^{-1} + a_2 z^{-2}}\n\\end{equation}\n\nThis section investigates the consequences of quantization in recursive filters. As for non-recursive filters, we first take a look at the quantization of filter coefficients. The structure used for the realization of the filter has impact on the quantization effects. We begin with the direct form followed by the coupled form, as example for an alternative structure.\n\n### Direct Form\n\nAbove transfer function of the recursive part of a SOS can be rewritten in terms of its complex conjugate poles $z_{\\infty}$ and $z_{\\infty}^*$ as\n\n\\begin{equation}\nH(z) = \\frac{1}{(z-z_{\\infty}) (z-z_{\\infty}^*)} = \\frac{z^{-2}}{ 1 \\underbrace{- 2 r \\cos(\\varphi)}_{a_1} \\; z^{-1} + \\underbrace{r^2}_{a_2} \\; z^{-2} }\n\\end{equation}\n\nwhere $r = |z_{\\infty}|$ and $\\varphi = \\arg \\{z_{\\infty}\\}$ denote the absolute value and phase of the pole $z_{\\infty}$, respectively. Let's assume a [linear uniform quantization](../quantization/linear_uniform_quantization_error.ipynb#Quantization-Error-of-a-Linear-Uniform-Quantizer) of the coefficients $a_1$ and $a_2$ with quantization step $Q$. Discarding clipping, the following relations for the locations of the poles can be found\n\n\\begin{align}\nr_n &= \\sqrt{n \\cdot Q} \\\\\n\\varphi_{nm} &= \\arccos \\left( \\sqrt{\\frac{m^2 Q}{4 n}} \\right)\n\\end{align}\nfor $n \\in \\mathbb{N}_0$ and $m \\in \\mathbb{Z}$. Quantization of the filter coefficients $a_1$ and $a_2$ into a finite number of amplitude values leads to a finite number of pole locations. In the $z$-plane the possible pole locations are given by the intersections of\n\n* circles whose radii $r_n$ are given by $r_n = \\sqrt{n \\cdot Q}$ with\n* equidistant vertical lines which intersect the horizontal axis at $\\frac{1}{2} m \\cdot Q$.\n\nThe finite number of pole locations may lead to deviations from a desired filter characteristic since a desired pole location is moved to the next possible pole location. The filter may even get unstable, when poles are moved outside the unit circle. For illustration, the resulting pole locations for a SOS realized in direct form are computed and plotted.\n\n\n```python\n%matplotlib inline\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom matplotlib.patches import Circle\nimport scipy.signal as sig\nimport itertools\n\n\ndef compute_pole_locations(Q):\n a1 = np.arange(-2, 2+Q, Q)\n a2 = np.arange(0, 1+Q, Q)\n \n p = np.asarray([np.roots([1, n, m]) for (n,m) in itertools.product(a1, a2)])\n p = p[np.imag(p)!=0]\n\n return p\n\ndef plot_pole_locations(p, Q):\n ax = plt.gca()\n for n in np.arange(np.ceil(2/Q)+1):\n circle = Circle((0,0), radius=np.sqrt(n*Q), fill=False, color='black', ls='solid', alpha=0.05)\n ax.add_patch(circle)\n ax.axvline(.5*n*Q, color='0.95')\n ax.axvline(-.5*n*Q, color='0.95')\n\n unit_circle = Circle((0,0), radius=1, fill=False, color='red', ls='solid')\n ax.add_patch(unit_circle) \n\n plt.plot(np.real(p), np.imag(p), 'b.', ms = 4)\n plt.xlabel(r'Re{$z$}')\n plt.ylabel(r'Im{$z$}')\n plt.axis([-1.1, 1.1, -1.1, 1.1])\n\n# compute and plot pole locations\nfor w in [5,6]:\n Q = 2/(2**(w-1)) # quantization stepsize\n plt.figure(figsize=(5, 5))\n p = compute_pole_locations(Q)\n plot_pole_locations(p, Q)\n plt.title(r'Direct form coefficient quantization to $w=%d$ bits'%w)\n```\n\n**Exercise**\n\n* What consequences does the distribution of pole locations on the desired characteristics of a filter have for e.g. low/high frequencies?\n\nSolution: Quantization of the original filter coefficients leads to a limited number of possible pole and zero locations. These locations are not uniformly distributed over the $z$-plane, as can be observed from above illustrations. The density of potential locations is especially low for low frequencies and close to the Nyquist frequency. The properties of a designed filter having poles and/or zeros at low/high frequencies will potentially deviate more when quantizing its coefficients, as a consequence.\n\n### Coupled Form\n\nBesides the quantization step $Q$, the pole distribution depends also on the topology of the filter. In order to gain a different distribution of pole locations after quantization, one has to derive structures where the coefficients of the multipliers are given by other values than the direct form coefficients $a_1$ and $a_2$. \n\nOne of these alternative structures is the coupled form (also known as Gold & Rader structure)\n\n\n\nwhere $\\Re\\{z_\\infty\\} = r \\cdot \\cos \\varphi$ and $\\Im\\{z_\\infty\\} = r \\cdot \\sin \\varphi$ denote the real- and imaginary part of the complex pole $z_\\infty$, respectively. Analysis of the structure reveals its difference equation as\n\n\\begin{align}\nw[k] &= x[k] + \\Re\\{z_\\infty\\} \\, w[k-1] - \\Im\\{z_\\infty\\} \\, y[k-1] \\\\\ny[k] &= \\Im\\{z_\\infty\\} \\, w[k-1] + \\Re\\{z_\\infty\\} \\, y[k-1]\n\\end{align}\n\nand its transfer function as\n\n\\begin{equation}\nH(z) = \\frac{\\Im\\{z_\\infty\\} \\; z^{-1}}{ 1 - 2 \\Re\\{z_\\infty\\} \\; z^{-1} + (\\Re\\{z_\\infty\\}^2 + \\Im\\{z_\\infty\\}^2) \\; z^{-2} }\n\\end{equation}\n\nNote that the numerator of the transfer function differs from the recursive only SOS given above. However, this can be considered in the design of the transfer function of a general SOS.\n\nThe real- and imaginary part of the pole $z_\\infty$ occur directly as coefficients for the multipliers in the coupled form. Quantization of these coefficients results therefore in a Cartesian grid of possible pole locations in the $z$-plane. This is illustrated in the following.\n\n\n```python\ndef compute_pole_locations(w):\n Q = 1/(2**(w-1)) # quantization stepsize\n a1 = np.arange(-1, 1+Q, Q)\n a2 = np.arange(-1, 1+Q, Q)\n \n p = np.asarray([n+1j*m for (n,m) in itertools.product(a1, a2) if n**2+m**2 <= 1])\n\n return p\n\ndef plot_pole_locations(p):\n ax = plt.gca()\n \n unit_circle = Circle((0,0), radius=1, fill=False, color='red', ls='solid')\n ax.add_patch(unit_circle) \n\n plt.plot(np.real(p), np.imag(p), 'b.', ms = 4)\n plt.xlabel(r'Re{$z$}')\n plt.ylabel(r'Im{$z$}')\n plt.axis([-1.1, 1.1, -1.1, 1.1])\n\n \n# compute and plot pole locations\nfor w in [5,6]:\n plt.figure(figsize=(5, 5))\n p = compute_pole_locations(w)\n plot_pole_locations(p)\n plt.title(r'Coupled form coefficient quantization to $w=%d$ bits'%w)\n```\n\n**Excercise**\n\n* What is the benefit of this representation in comparison to the direct from discussed in the previous section?\n\nSolution: A befit of the coupled form is a uniform distribution of potential pole and zero locations in the $z$-plane. This holds especially for low frequencies and close to the Nyquist frequency.\n\n### Example - Influence of coefficient quantization\n\nThe following example illustrates the effects of coefficient quantization for a recursive [Butterworth filter](https://en.wikipedia.org/wiki/Butterworth_filter) realized in cascaded SOSs in transposed direct form II.\n\n\n```python\nw = 16 # wordlength of filter coefficients\nN = 7 # order of filter\n\ndef uniform_midtread_quantizer(x, w, xmin=1):\n # quantization step\n Q = xmin/(2**(w-1))\n # limiter\n x = np.copy(x)\n idx = np.where(x <= -xmin)\n x[idx] = -1\n idx = np.where(x > xmin - Q)\n x[idx] = 1 - Q\n # linear uniform quantization\n xQ = Q * np.floor(x/Q + 1/2)\n \n return xQ\n\n# coefficients of recursive filter\nb, a = sig.butter(N, 0.2, 'low')\n# decomposition into SOS\nsos = sig.tf2sos(b, a, pairing='nearest')\nsos = sos/np.amax(np.abs(sos))\n# quantization of SOS coefficients\nsosq = uniform_midtread_quantizer(sos, w, xmin=2)\n# compute overall transfer function of (quantized) filter\nH = np.ones(512)\nHq = np.ones(512)\nfor n in range(sos.shape[0]):\n Om, Hn = sig.freqz(sos[n, 0:3], sos[n, 3:6])\n H = H * Hn\n Om, Hn = sig.freqz(sosq[n, 0:3], sosq[n, 3:6])\n Hq = Hq * Hn\n\n\n# plot magnitude responses\nplt.figure(figsize=(10, 3))\nplt.plot(Om, 20 * np.log10(abs(H)), label='continuous')\nplt.plot(Om, 20 * np.log10(abs(Hq)), label='quantized')\nplt.title('Magnitude response')\nplt.xlabel(r'$\\Omega$')\nplt.ylabel(r'$|H(e^{j \\Omega})|$ in dB')\nplt.legend(loc=3)\nplt.grid()\n# plot phase responses\nplt.figure(figsize=(10, 3))\nplt.plot(Om, np.unwrap(np.angle(H)), label='continuous')\nplt.plot(Om, np.unwrap(np.angle(Hq)), label='quantized')\nplt.title('Phase')\nplt.xlabel(r'$\\Omega$')\nplt.ylabel(r'$\\varphi (\\Omega)$ in rad')\nplt.legend(loc=3)\nplt.grid()\n```\n\n**Exercise**\n\n* Decrease the word length `w` of the filter. What happens? At what word length does the filter become unstable?\n* Increase the order `N` of the filter for a fixed word length `w`. What happens?\n\nSolution: The deviations from the continuous (desired) realization of the filter increase with decreasing word length. The filter with order `N=5` becomes unstable for `w = 10`. Increasing the order `N` of the filter for a fixed word length results also in instabilities. Consequently, for a high order filter also a higher word length is required.\n\n**Copyright**\n\nThis notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Digital Signal Processing - Lecture notes featuring computational examples, 2016-2018*.\n", "meta": {"hexsha": "b68a23a0aad8e3d1e28e6fc63ba2a3048b664f2f", "size": 237163, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "recursive_filters/quantization_of_coefficients.ipynb", "max_stars_repo_name": "mgrubisic/digital-signal-processing-lecture", "max_stars_repo_head_hexsha": "7098b958639eb5cfcabd110d26ddd30ff8444e0a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2018-12-29T19:13:49.000Z", "max_stars_repo_stars_event_max_datetime": "2020-05-25T09:53:21.000Z", "max_issues_repo_path": "recursive_filters/quantization_of_coefficients.ipynb", "max_issues_repo_name": "mgrubisic/digital-signal-processing-lecture", "max_issues_repo_head_hexsha": "7098b958639eb5cfcabd110d26ddd30ff8444e0a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "recursive_filters/quantization_of_coefficients.ipynb", "max_forks_repo_name": "mgrubisic/digital-signal-processing-lecture", "max_forks_repo_head_hexsha": "7098b958639eb5cfcabd110d26ddd30ff8444e0a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2020-10-17T07:48:22.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-17T06:28:58.000Z", "avg_line_length": 620.8455497382, "max_line_length": 71604, "alphanum_fraction": 0.9405725176, "converted": true, "num_tokens": 3094, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160666, "lm_q2_score": 0.7154239957834733, "lm_q1q2_score": 0.44004861135119205}} {"text": "# \u935b\u934a\u4f60\u7684\u8cc7\u6599\u5206\u6790\u529b | Python \u7a0b\u5f0f\u8a2d\u8a08\n\n> \u6a21\u7d44\u8207\u5957\u4ef6\n\n\u90ed\u8000\u4ec1 from [DATAINPOINT](https://www.datainpoint.com/)\n\n> One feature of Python that makes it useful for a wide range of tasks is that it comes \"batteries included\"\u2014that is, the Python standard library contains useful tools for a wide range of tasks.\n\n## \u7a0b\u5f0f\u5c01\u88dd\u7684\u5c64\u7d1a\n\n- **\u5957\u4ef6\uff08Libraries\uff09**\n - **\u6a21\u7d44\uff08Modules\uff09**\n - \u985e\u5225\uff08Classes\uff09\n - \u51fd\u5f0f\uff08Functions\uff09\n\n## \u5927\u7db1\n\n- `import` \u6307\u4ee4\n- \u81ea\u8a02\u6a21\u7d44\n- \u81ea\u8a02\u5957\u4ef6\n\n## `import` \u6307\u4ee4\n\n## \u4f7f\u7528 `import` \u6307\u4ee4\u8f09\u5165\u6a19\u6e96\u8207\u7b2c\u4e09\u65b9\u6a21\u7d44\u5957\u4ef6\n\n- \u6a19\u6e96\u6a21\u7d44\u5957\u4ef6\uff08Standard modules/libraries\uff09\n- \u5916\u90e8\u6a21\u7d44\u5957\u4ef6\uff08Third-party modules/libraries\uff09\n\n## \u6a19\u6e96\u6a21\u7d44\u5957\u4ef6\uff1a\u4f34\u96a8 Python \u76f4\u8b6f\u5668\u800c\u4f86\uff0c\u7121\u9700\u984d\u5916\u5b89\u88dd\n\n- `re`\n- `datetime`\n- `random`\n- `math`\n- ...etc.\n\nSource: \n\n## \u4f7f\u7528 `import` \u6307\u4ee4\uff1a\u4ee5 `math` \u70ba\u4f8b\n\n\n```python\nimport math\n\nprint(math.pi)\n```\n\n 3.141592653589793\n\n\n## \u4f7f\u7528 `import` \u6307\u4ee4\u642d\u914d\u7e2e\u5beb\n\n\n```python\nimport math as m\n\nprint(m.pi)\n```\n\n 3.141592653589793\n\n\n## \u4f7f\u7528 `from` \u642d\u914d `import` \u6307\u4ee4\u8f09\u5165\u90e8\u5206\u529f\u80fd\n\n\n```python\nfrom math import sin, pi\n\nprint(sin(pi/2))\n```\n\n 1.0\n\n\n## \u7b2c\u4e09\u65b9\u5957\u4ef6\uff1a\u4f7f\u7528 `pip install` \u6307\u4ee4\u5b89\u88dd\n\n```bash\npip install LIBRARY_NAME\n```\n\n## \u4f7f\u7528 `pip` \u5b89\u88dd\u7b2c\u4e09\u65b9\u5957\u4ef6 `numpy`\n\n```bash\npip install numpy\n```\n\n## \u6aa2\u8996\u5957\u4ef6\u5b89\u88dd\u7684\u8def\u5f91\uff1a\u4f7f\u7528 `__file__` \u5c6c\u6027\n\n```python\nimport numpy as np\nprint(np.__file__)\n```\n\n## \u5957\u4ef6\u5b89\u88dd\u7684\u8def\u5f91\uff08\u4ee5\u6211\u5011\u4e0a\u8ab2\u6240\u4f7f\u7528\u7684\u74b0\u5883\u70ba\u4f8b\uff09\n\n- \u6a19\u6e96\u6a21\u7d44\u5957\u4ef6\uff1a`/srv/conda/envs/notebook/lib/python3.7`\n- \u5916\u90e8\u6a21\u7d44\u5957\u4ef6\uff1a`/srv/conda/envs/notebook/lib/python3.7/site-packages`\n\n## \u524d\u5f80\u89c0\u770b this\n\nNew > Terminal\n\n```bash\ncd /srv/conda/envs/notebook/lib/python3.7\nls | grep this\ncat this.py\n```\n\n## \u524d\u5f80\u89c0\u770b numpy\n\nNew > Terminal\n\n```bash\ncd /srv/conda/envs/notebook/lib/python3.7/site-packages\nls | grep numpy\n```\n\n## `pip` \u662f Python \u7684\u6a21\u7d44\u5957\u4ef6\u7ba1\u7406\u5de5\u5177\n\n- \u4e0d\u9700\u984d\u5916\u5b89\u88dd\n- \u5354\u52a9\u4f7f\u7528\u8005\u5f9e [Python Pacakge Index](https://pypi.org/) \u4e0b\u8f09\u3001\u5b89\u88dd\u6216\u66f4\u65b0\n\n## \u6aa2\u8996 `pip` \u7248\u672c\n\n```bash\npip --version\n```\n\n## \u66f4\u65b0 `pip`\n\n- Linux / macOS\n\n```bash\npip install --upgrade pip\n```\n\n- Windows\n\n```bash\npython -m pip install -U pip\n```\n\n## \u81ea\u8a02\u6a21\u7d44\n\n## \u4f55\u8b02\u6a21\u7d44\n\n- \u5c07\u51fd\u5f0f\u6216\u985e\u5225\u5c01\u88dd\u5728\u4e00\u500b `.py` \u6a94\u6848\u4e2d\n- `.py` \u7684\u6a94\u540d\u5c31\u662f\u6a21\u7d44\u540d\u7a31\n\n## \u5c07 `celsius_to_fahrenheit()` \u51fd\u5f0f\u5c01\u88dd\u5728 `temperature_scale.py` \u7684\u6a94\u6848\u4e2d\n\n\u6a21\u7d44\u540d\u7a31\uff1a`temperature_scale`\u3002\n\n```python\n# temperature_scale.py\ndef celsius_to_fahrenheit(x):\n return x * 9/5 + 32\n```\n\n## \u5c07 `temperature_scale.py` \u5132\u5b58\u5728 `/notebooks` \u76ee\u9304\u4e0b\uff08\u4ee5\u6211\u5011\u4e0a\u8ab2\u6240\u4f7f\u7528\u7684\u74b0\u5883\u70ba\u4f8b\uff09\n\n\n```python\nimport temperature_scale as ts\n\nts.celsius_to_fahrenheit(30)\n```\n\n\n\n\n 86.0\n\n\n\n\n```python\nfrom temperature_scale import celsius_to_fahrenheit\n\ncelsius_to_fahrenheit(30)\n```\n\n\n\n\n 86.0\n\n\n\n\n```python\nfrom temperature_scale import celsius_to_fahrenheit as c2f\n\nc2f(30)\n```\n\n\n\n\n 86.0\n\n\n\n## \u5c07 `CelsiusFahrenheit` \u985e\u5225\u52a0\u5165\u5230 `temperature_scale.py` \u7684\u6a94\u6848\u4e2d\n\n\u6a21\u7d44\u540d\u7a31\uff1a`temperature_scale`\u3002\n\n```python\n# temperature_scale.py\nclass CelsiusFahrenheit:\n def c2f(self, x):\n return x * 9/5 + 32\n def f2c(self, x):\n return (x - 32) * 5/9\n```\n\n## \u5c07 `temperature_scale.py` \u5132\u5b58\u5728 `/notebooks` \u76ee\u9304\u4e0b\uff08\u4ee5\u6211\u5011\u4e0a\u8ab2\u6240\u4f7f\u7528\u7684\u74b0\u5883\u70ba\u4f8b\uff09\n\n\n```python\nimport temperature_scale as ts\n\ncelsius_fahrenheit = ts.CelsiusFahrenheit()\nprint(celsius_fahrenheit.c2f(30))\nprint(celsius_fahrenheit.f2c(86))\n```\n\n 86.0\n 30.0\n\n\n\n```python\nfrom temperature_scale import CelsiusFahrenheit\n\ncelsius_fahrenheit = CelsiusFahrenheit()\nprint(celsius_fahrenheit.c2f(30))\nprint(celsius_fahrenheit.f2c(86))\n```\n\n 86.0\n 30.0\n\n\n\n```python\nfrom temperature_scale import CelsiusFahrenheit as CF\n\ncelsius_fahrenheit = CF()\nprint(celsius_fahrenheit.c2f(30))\nprint(celsius_fahrenheit.f2c(86))\n```\n\n 86.0\n 30.0\n\n\n## \u81ea\u8a02\u5957\u4ef6\n\n## \u4f55\u8b02\u5957\u4ef6\n\n- \u591a\u500b\u529f\u80fd\u76f8\u95dc\u7684\u6a21\u7d44\uff08.py \u6a94\u6848\uff09\u53ef\u4ee5\u7d44\u7e54\u6210\u4e00\u500b\u5957\u4ef6\uff08\u8cc7\u6599\u593e\uff09\n- \u5c07 `temperature_scale.py` \u8907\u88fd\u5230 `demo_scripts` \u8cc7\u6599\u593e\n- \u8cc7\u6599\u593e\u540d\u7a31\u5c31\u662f\u5957\u4ef6\u540d\u7a31\uff08`demo_scripts`\uff09\n\n\n```python\nfrom demo_scripts.temperature_scale import CelsiusFahrenheit\n\ncelsius_fahrenheit = CelsiusFahrenheit()\nprint(celsius_fahrenheit.c2f(30))\nprint(celsius_fahrenheit.f2c(86))\n```\n\n 86.0\n 30.0\n\n\n## \u96a8\u5802\u7df4\u7fd2\n\n## \u96a8\u5802\u7df4\u7fd2\uff1a\u5728 `/notebooks` \u81ea\u8a02\u4e00\u500b\u540d\u70ba `bmi_calculator` \u7684\u6a21\u7d44\n\n\\begin{equation}\nBMI = \\frac{weight_{kg}}{height_{m}^2}\n\\end{equation}\n\n\n\n- \u5728\u8a72\u6a21\u7d44\u4e2d\u5b9a\u7fa9\u4e00\u500b\u985e\u5225 `BMI`\n- `BMI` \u985e\u5225\u5177\u5099\u5169\u500b\u65b9\u6cd5\n - `get_bmi()`\n - `get_bmi_label()`\n\n```python\n# bmi_calculator.py\nclass BMI:\n \"\"\"\n >>> from bmi_calculator import BMI\n >>> bmi = BMI(198, 129) # Zion Williamson\n >>> bmi.get_bmi()\n 32.90480563207836\n >>> bmi.get_bmi_label()\n 'Obese'\n >>> bmi = BMI(206, 113) # LeBron James\n >>> bmi.get_bmi()\n 26.628334433028563\n >>> bmi.get_bmi_label()\n 'Overweight'\n \"\"\"\n```\n\n## \u96a8\u5802\u7df4\u7fd2\uff1a\u5728 `/notebooks` \u65b0\u589e\u4e00\u500b\u540d\u70ba `casino` \u7684\u6a21\u7d44\n\n- \u5728\u8a72\u6a21\u7d44\u4e2d\u5b9a\u7fa9\u4e00\u500b\u985e\u5225 `Casino`\n- `Casino` \u985e\u5225\u5177\u5099\u65b9\u6cd5\uff1a\n - `toss_a_coin()`\n - `roll_a_dice()`\n - `deal_a_card()`\n \n\u5099\u8a3b\uff1a\u4f7f\u7528\u6a19\u6e96\u6a21\u7d44\u5957\u4ef6 [random](https://docs.python.org/3/library/random.html) \u5be6\u73fe\u300c\u96a8\u6a5f\u6027\u300d\u9700\u6c42\n\n```python\n# casino.py\nclass Casino:\n \"\"\"\n >>> from casino import Casino\n >>> my_casino = Casino()\n >>> my_casino.toss_a_coin()\n 'Head' # \u6216 'Tail'\n >>> my_casino.roll_a_dice()\n 6 # \u6216 1, 2, 3, 4, 5\n >>> my_casino.deal_a_card()\n '2 of hearts' # \u6216\u5176\u4ed6 51 \u7a2e\u5927\u5c0f\u82b1\u8272 'king of spades', 'ace of diamonds', 'jack of clubs'...\n \"\"\"\n```\n\n## \u96a8\u5802\u7df4\u7fd2\uff1a\u5728 `/notebooks` \u65b0\u589e\u4e00\u500b\u540d\u70ba `stats` \u7684\u6a21\u7d44\n\n- \u5728\u8a72\u6a21\u7d44\u4e2d\u5b9a\u7fa9\u4e00\u500b\u985e\u5225 `Stats`\n- `Stats` \u985e\u5225\u5177\u5099\u65b9\u6cd5\uff1a\n - `mean()`\n - `median()`\uff1a\n - `modes()`\uff1a\n\n```python\n# stats.py\nclass Stats:\n \"\"\"\n >>> from stats import Stats\n >>> statistics = Stats(1, 2, 2, 3, 4, 7, 9)\n >>> statistics.mean()\n 4.0\n >>> statistics.median()\n 3\n >>> statistics.modes()\n [2]\n >>> statistics = Stats(1, 2, 2, 3, 4, 7, 9, 7)\n >>> statistics.mean()\n 4.375\n >>> statistics.median()\n 3.5\n >>> statistics.modes()\n [2, 7]\n \"\"\"\n```\n\n## \u57f7\u884c\u6e2c\u8a66\n\n\n```python\n%load ../test_cases/test_cases_09.py\n```\n", "meta": {"hexsha": "6a8335ec3b2127c8758db661b95557ee33cddc8e", "size": 15629, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/09-modules-and-libraries.ipynb", "max_stars_repo_name": "datainpoint/classroom-introduction-to-python", "max_stars_repo_head_hexsha": "a5d4036829eda3a0ed1a0a0af752f541e4e015e7", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/09-modules-and-libraries.ipynb", "max_issues_repo_name": "datainpoint/classroom-introduction-to-python", "max_issues_repo_head_hexsha": "a5d4036829eda3a0ed1a0a0af752f541e4e015e7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/09-modules-and-libraries.ipynb", "max_forks_repo_name": "datainpoint/classroom-introduction-to-python", "max_forks_repo_head_hexsha": "a5d4036829eda3a0ed1a0a0af752f541e4e015e7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 18.5617577197, "max_line_length": 202, "alphanum_fraction": 0.4624096231, "converted": true, "num_tokens": 2448, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.44004861135119194}} {"text": "# Chapter 10 Introducing Python Statements\n\nPython Program Structure Revisited\n\nAt its core, Python syntax is composed of statements and expressions. Expressions\nprocess objects and are embedded in statements. Statements code the larger logic of a\nprogram\u2019s operation\u2014they use and direct expressions to process the objects we studied\nin the preceding chapters. Moreover, statements are where objects spring into existence\n(e.g., in expressions within assignment statements), and some statements create entirely new kinds of objects (functions, classes, and so on). Statements always exist in\nmodules, which themselves are managed with statements.\n\nThis chapter climbs the hierarchy to the next level:\n1. Programs are composed of modules.\n2. Modules contain statements.\n3. Statements contain expressions.\n4. Expressions create and process objects.\n\n# A Few Special Cases\n\nAs mentioned previously, in Python\u2019s syntax model:\n\u2022 The end of a line terminates the statement on that line (without semicolons).\n\u2022 Nested statements are blocked and associated by their physical indentation (without braces).\n\n\n```python\na = 1; b = 2; print(a + b) # Three statements on one line\n```\n\n 3\n\n\n\n```python\nmlist = [111,\n 222,\n 333]\n```\n\n\n```python\nprint(mlist)\n```\n\n [111, 222, 333]\n\n\nParentheses are the catchall device\u2014because any expression can be wrapped up in\nthem, simply inserting a left parenthesis allows you to drop down to the next line and\ncontinue your statement:\n\n\n```python\nfrom sympy import *\n```\n\n\n```python\nA, B, C, D = symbols('A, B , C , D')\n```\n\n\n```python\nX = symbols('X')\n```\n\n\n```python\nX = A + B + C + D\n```\n\n\n```python\nif (A == 1 and \n B == 2 and\n C == 3):\n print('spam'*3)\n```\n\n spamspamspam\n\n\n\n```python\nA, B, C = 1, 2, 3\n```\n\n\n```python\nexec('print(\"spam\");'*3)\n```\n\n spam\n spam\n spam\n\n\nAn older rule also allows for continuation lines when the prior line ends in a backslash:\n\n\n```python\nX = A + B + \\\nC + D\n```\n\n\n```python\nX\n```\n\n\n\n\n$\\displaystyle D + 6$\n\n\n\nThis alternative technique is dated, though, and is frowned on today because it\u2019s difficult to notice and maintain the backslashes, and it\u2019s fairly brittle\u2014there can be no\nspaces after the backslash, and omitting it can have unexpected effects if the next line\nis mistaken to be a new statement. It\u2019s also another throwback to the C language, where\nit is commonly used in \u201c#define\u201d macros; again, when in Pythonland, do as Pythonistas\ndo, not as C programmers do.\n\n# Block rule special case\n\n\n```python\nx, y = 26,57\nif x > y: print(x)\n```\n\n# A Simple Interactive Loop\n\n# Doing Math on User Inputs\n\n\n```python\nwhile True:\n reply = input('Enter text:')\n if reply == 'stop': break\n print(int(reply) ** 2)\nprint('Bye')\n```\n\n Enter text: 4546\n\n\n 20666116\n\n\n Enter text: 2313213\n\n\n 5350954383369\n\n\n Enter text: 4545\n\n\n 20657025\n\n\n Enter text: 54545\n\n\n 2975157025\n\n\n Enter text: 4545\n\n\n 20657025\n\n\n Enter text: 4545\n\n\n 20657025\n\n\n Enter text: 48636\n\n\n 2365460496\n\n\n Enter text: stop\n\n\n Bye\n\n\n# Handling Errors by Testing Inputs\n\n\n```python\nS = '123'\nT = 'xxx'\nS.isdigit(), T.isdigit()\n```\n\n\n\n\n (True, False)\n\n\n\n\n```python\nwhile True:\n reply = input('Enter text:')\n if reply == 'stop':\n break\n elif not reply.isdigit():\n print('Bad!' * 8)\n else:\n print(int(reply) ** 2)\nprint('Bye')\n```\n\n Enter text: 24\n\n\n 576\n\n\n Enter text: 45\n\n\n 2025\n\n\n Enter text: asdf\n\n\n Bad!Bad!Bad!Bad!Bad!Bad!Bad!Bad!\n\n\n Enter text: stop\n\n\n Bye\n\n\n# Handling Errors with try Statements\n\n\n```python\nwhile True:\n reply = input('Enter text:')\n if reply == 'stop': break\n try:\n num = int(reply)\n except:\n print('Bad!' * 8)\n else:\n print(int(reply) ** 2)\nprint('Bye')\n```\n\n Enter text: 456\n\n\n 207936\n\n\n Enter text: dasd\n\n\n Bad!Bad!Bad!Bad!Bad!Bad!Bad!Bad!\n\n\n Enter text: '454'\n\n\n Bad!Bad!Bad!Bad!Bad!Bad!Bad!Bad!\n\n\n Enter text: stop\n\n\n Bye\n\n\n# Nesting Code Three Levels Deep\n\n\n```python\nwhile True:\n reply = input('Enter text:')\n if reply == 'stop':\n break\n elif not reply.isdigit():\n print('Bad!' * 8)\n else:\n num = int(reply)\n if num < 20:\n print('low')\n else:\n print(num ** 2)\nprint('Bye')\n```\n\n Enter text: 454\n\n\n 206116\n\n\n Enter text: 4\n\n\n low\n\n\n Enter text: stop\n\n\n Bye\n\n\n# Chapter Summary\n\nThat concludes our quick look at Python statement syntax. This chapter introduced\nthe general rules for coding statements and blocks of code. As you\u2019ve learned, in Python\nwe normally code one statement per line and indent all the statements in a nested block\nthe same amount (indentation is part of Python\u2019s syntax). However, we also looked at\na few exceptions to these rules, including continuation lines and single-line tests and\nloops. Finally, we put these ideas to work in an interactive script that demonstrated a\nhandful of statements and showed statement syntax in action.\nIn the next chapter, we\u2019ll start to dig deeper by going over each of Python\u2019s basic procedural statements in depth. As you\u2019ll see, though, all statements follow the same general rules introduced here.\n\n\n```python\nimport sys\nfrom tkinter import *\n\nclass HelloClass:\n def __init__(self):\n widget = Button(None, text = 'Hello event world', command = self.quit)\n widget.pack()\n \n def quit(self):\n print('Hello class method world')\n sys.exit()\n \nHelloClass()\nmainloop()\n```\n\n\n```python\n# coding:utf-8\nimport turtle as t\nimport time\n\n#\u76ae\u5361\u4e18\n\n#\u57fa\u7840\u8bbe\u7f6e\nt.screensize(800,600)\nt.pensize(2) # \u8bbe\u7f6e\u753b\u7b14\u7684\u5927\u5c0f\nt.speed(10) # \u8bbe\u7f6e\u753b\u7b14\u901f\u5ea6\u4e3a10\n\n\n#\u753b\u5de6\u504f\u66f2\u7ebf\u51fd\u6570\ndef radian_left(ang,dis,step,n):\n for i in range(n):\n dis+=step #dis\u589e\u5927step\n t.lt(ang) #\u5411\u5de6\u8f6cang\u5ea6\n t.fd(dis) #\u5411\u524d\u8d70dis\u7684\u6b65\u957f\n \ndef radian_right(ang,dis,step,n):\n for i in range(n):\n dis+=step\n t.rt(ang) #\u5411\u5de6\u8f6cang\u5ea6\n t.fd(dis) #\u5411\u524d\u8d70dis\u7684\u6b65\u957f\n \n\n#\u753b\u8033\u6735\ndef InitEars():\n \n t.color(\"black\",\"yellow\")\n #\u5de6\u8033\u6735\u66f2\u7ebf\n t.pu() # \u63d0\u7b14\n t.goto(-50,100) # \u7b14\u5934\u521d\u59cb\u4f4d\u7f6e\n t.pd() # \u4e0b\u7b14\n t.setheading(110)#\u753b\u7b14\u89d2\u5ea6\n\n t.begin_fill()\n\n radian_left(1.2,0.4,0.1,40)\n \n t.setheading(270) #\u753b\u7b14\u89d2\u5ea6\n radian_left(1.2,0.4,0.1,40)\n\n t.setheading(44) #\u753b\u7b14\u89d2\u5ea6\n t.forward(32)\n t.end_fill()\n\n #\u53f3\u8033\u6735\u66f2\u7ebf \n t.pu() # \u63d0\u7b14\n t.goto(50,100) # \u7b14\u5934\u521d\u59cb\u4f4d\u7f6e\n t.pd() # \u4e0b\u7b14\n t.setheading(70)#\u753b\u7b14\u89d2\u5ea6\n\n t.begin_fill()\n radian_right(1.2,0.4,0.1,40)\n \n t.setheading(270) #\u753b\u7b14\u89d2\u5ea6\n radian_right(1.2,0.4,0.1,40)\n \n t.setheading(136) #\u753b\u7b14\u89d2\u5ea6\n t.forward(32)\n t.end_fill()\n \n #\u8033\u6735\u9ed1\n\n t.begin_fill()\n t.fillcolor(\"black\")\n t.pu() # \u63d0\u7b14\n t.goto(88,141) # \u7b14\u5934\u521d\u59cb\u4f4d\u7f6e\n t.pd() # \u4e0b\u7b14\n t.setheading(35)#\u753b\u7b14\u89d2\u5ea6\n\n radian_right(1.2,1.6,0.1,16)\n \n t.setheading(270) #\u753b\u7b14\u89d2\u5ea6\n radian_right(1.2,0.4,0.1,25)\n\n t.setheading(132) #\u753b\u7b14\u89d2\u5ea6\n t.forward(31)\n t.end_fill()\n\n t.begin_fill()\n t.fillcolor(\"black\")\n t.pu() # \u63d0\u7b14\n t.goto(-88,141) # \u7b14\u5934\u521d\u59cb\u4f4d\u7f6e\n t.pd() # \u4e0b\u7b14\n t.setheading(145)#\u753b\u7b14\u89d2\u5ea6\n\n radian_left(1.2,1.6,0.1,16)\n \n t.setheading(270) #\u753b\u7b14\u89d2\u5ea6\n radian_left(1.2,0.4,0.1,25)\n\n t.setheading(48) #\u753b\u7b14\u89d2\u5ea6\n t.forward(31)\n t.end_fill()\n \n\n #\u753b\u5c3e\u5df4\ndef InitTail():\n #\u5c3e\u5df4\n t.begin_fill()\n t.fillcolor(\"yellow\")\n t.pu() # \u63d0\u7b14\n t.goto(64,-140) # \u7b14\u5934\u521d\u59cb\u4f4d\u7f6e\n t.pd() # \u4e0b\u7b14\n t.setheading(10) #\u753b\u7b14\u89d2\u5ea6\n t.forward(20)\n t.setheading(90) #\u753b\u7b14\u89d2\u5ea6\n t.forward(20)\n t.setheading(10) #\u753b\u7b14\u89d2\u5ea6\n t.forward(10)\n t.setheading(80) #\u753b\u7b14\u89d2\u5ea6\n t.forward(100)\n t.setheading(35) #\u753b\u7b14\u89d2\u5ea6\n t.forward(80)\n t.setheading(260) #\u753b\u7b14\u89d2\u5ea6\n t.forward(100)\n t.setheading(205) #\u753b\u7b14\u89d2\u5ea6\n t.forward(40)\n t.setheading(260) #\u753b\u7b14\u89d2\u5ea6\n t.forward(37)\n t.setheading(205) #\u753b\u7b14\u89d2\u5ea6\n t.forward(20)\n t.setheading(260) #\u753b\u7b14\u89d2\u5ea6\n t.forward(25)\n t.setheading(175) #\u753b\u7b14\u89d2\u5ea6\n t.forward(30)\n t.setheading(100) #\u753b\u7b14\u89d2\u5ea6\n t.forward(13)\n t.end_fill()\n\n\n#\u753b\u811a\ndef InitFoots():\n #\u811a\n t.begin_fill()\n t.fillcolor(\"yellow\")\n t.pensize(2)\n t.pu() # \u63d0\u7b14\n t.goto(-70,-200) # \u7b14\u5934\u521d\u59cb\u4f4d\u7f6e\n t.pd() # \u4e0b\u7b14\n t.setheading(225) #\u753b\u7b14\u89d2\u5ea6\n radian_left(0.5,1.2,0,12)\n radian_left(35,0.6,0,4)\n radian_left(1,1.2,0,18)\n t.setheading(160) #\u753b\u7b14\u89d2\u5ea6\n t.forward(13)\n t.end_fill()\n\n t.begin_fill()\n t.fillcolor(\"yellow\")\n t.pensize(2)\n t.pu() # \u63d0\u7b14\n t.goto(70,-200) # \u7b14\u5934\u521d\u59cb\u4f4d\u7f6e\n t.pd() # \u4e0b\u7b14\n t.setheading(315) #\u753b\u7b14\u89d2\u5ea6\n radian_right(0.5,1.2,0,12)\n radian_right(35,0.6,0,4)\n radian_right(1,1.2,0,18)\n t.setheading(20) #\u753b\u7b14\u89d2\u5ea6\n t.forward(13)\n t.end_fill()\n\n\n#\u753b\u8eab\u4f53\ndef InitBody():\n #\u5916\u5f62\u8f6e\u5ed3\n t.begin_fill()\n t.pu() # \u63d0\u7b14\n t.goto(112,0) # \u7b14\u5934\u521d\u59cb\u4f4d\u7f6e\n t.pd() # \u4e0b\u7b14\n t.setheading(90) #\u753b\u7b14\u89d2\u5ea6\n t.circle(112,180) \n\n t.setheading(250) #\u753b\u7b14\u89d2\u5ea6\n\n radian_left(1.6,1.3,0,50)\n \n radian_left(0.8,1.5,0,25)\n\n t.setheading(255) #\u753b\u7b14\u89d2\u5ea6\n radian_left(0.4,1.6,0.2,27)\n\n radian_left(2.8,1,0,45)\n radian_right(0.9,1.4,0,31)\n\n t.setheading(355) #\u753b\u7b14\u89d2\u5ea6\n radian_right(0.9,1.4,0,31)\n\n radian_left(2.8,1,0,45)\n\n radian_left(0.4,7.2,-0.2,27)\n\n t.setheading(10) #\u753b\u7b14\u89d2\u5ea6\n radian_left(0.8,1.5,0,25)\n\n radian_left(1.6,1.3,0,50)\n\n t.end_fill()\n\n \ndef InitEyes():\n #\u5de6\u773c\u775b\n t.begin_fill()\n t.fillcolor(\"black\")\n t.pu() # \u63d0\u7b14\n t.goto(-46,10) # \u7b14\u5934\u521d\u59cb\u4f4d\u7f6e\n t.pd() # \u4e0b\u7b14\n t.setheading(90) #\u753b\u7b14\u89d2\u5ea6\n t.circle(5,360) \n t.end_fill()\n\n #\u53f3\u773c\u775b\n t.begin_fill()\n t.fillcolor(\"black\")\n t.pu() # \u63d0\u7b14\n t.goto(46,10) # \u7b14\u5934\u521d\u59cb\u4f4d\u7f6e\n t.pd() # \u4e0b\u7b14\n t.setheading(-90) #\u753b\u7b14\u89d2\u5ea6\n t.circle(5,360) \n t.end_fill()\n \n\n \n \n#\u753b\u8138\ndef InitFace():\n #\u8138\u86cb\n t.begin_fill()\n t.fillcolor(\"red\")\n t.pu() # \u63d0\u7b14\n t.goto(-63,-10) # \u7b14\u5934\u521d\u59cb\u4f4d\u7f6e\n t.pd() # \u4e0b\u7b14\n t.setheading(90) #\u753b\u7b14\u89d2\u5ea6\n t.circle(10,360) \n t.end_fill()\n\n t.begin_fill()\n t.fillcolor(\"red\")\n t.pu() # \u63d0\u7b14\n t.goto(63,-10) # \u7b14\u5934\u521d\u59cb\u4f4d\u7f6e\n t.pd() # \u4e0b\u7b14\n t.setheading(-90) #\u753b\u7b14\u89d2\u5ea6\n t.circle(10,360) \n t.end_fill()\n\n\n #\u5634\u5df4\n t.pensize(2.2)\n t.pu() # \u63d0\u7b14\n t.goto(0,0) # \u7b14\u5934\u521d\u59cb\u4f4d\u7f6e\n t.pd() # \u4e0b\u7b14\n t.setheading(235) #\u753b\u7b14\u89d2\u5ea6\n radian_right(5,0.8,0,30)\n\n t.pu() # \u63d0\u7b14\n t.goto(0,0) # \u7b14\u5934\u521d\u59cb\u4f4d\u7f6e\n t.pd() # \u4e0b\u7b14\n t.setheading(305) #\u753b\u7b14\u89d2\u5ea6\n radian_left(5,0.8,0,30)\n\n\n#\u753b\u624b\ndef InitHands():\n #\u5de6\u624b\n t.pensize(2)\n t.pu() # \u63d0\u7b14\n t.goto(-46,-100) # \u7b14\u5934\u521d\u59cb\u4f4d\u7f6e\n t.pd() # \u4e0b\u7b14\n t.setheading(285) #\u753b\u7b14\u89d2\u5ea6\n radian_right(0.4,1.2,0,26)\n radian_right(5,0.35,0,26)\n radian_right(0.3,1.2,0,15)\n\n #\u53f3\u624b\n t.pu() # \u63d0\u7b14\n t.goto(46,-100) # \u7b14\u5934\u521d\u59cb\u4f4d\u7f6e\n t.pd() # \u4e0b\u7b14\n t.setheading(255) #\u753b\u7b14\u89d2\u5ea6\n radian_left(0.4,1.2,0,26)\n radian_left(5,0.35,0,26)\n radian_left(0.3,1.2,0,15)\n\n\ndef CloseEyes():\n #\u5de6\u773c\u775b\n t.pu() # \u63d0\u7b14\n t.goto(-46,12) # \u7b14\u5934\u521d\u59cb\u4f4d\u7f6e\n t.pd() # \u4e0b\u7b14\n t.setheading(180) #\u753b\u7b14\u89d2\u5ea6\n t.forward(10)\n\n #\u53f3\u773c\u775b\n t.pu() # \u63d0\u7b14\n t.goto(46,12) # \u7b14\u5934\u521d\u59cb\u4f4d\u7f6e\n t.pd() # \u4e0b\u7b14\n t.setheading(0) #\u753b\u7b14\u89d2\u5ea6\n t.forward(10)\n \n\n#\u521d\u59cb\u5316\ndef Init():\n InitEars()\n InitTail()\n InitFoots()\n InitBody()\n InitFace()\n InitHands()\n InitEyes()\n \n \n #\u7728\u773c\u775b\ndef Upgarde():\n InitEars()\n InitTail()\n InitFoots()\n InitBody()\n InitFace()\n InitHands()\n CloseEyes()\n \ndef Upgarde_Init():\n InitEars()\n InitTail()\n InitFoots()\n InitBody()\n InitFace()\n InitHands()\n InitEyes()\n \n\n \ndef main():\n\n Init()\n \n t.tracer(False)\n \n #\u7728\u773c\u775b\u52a8\u753b\n for i in range(30):\n if i%2==0:\n t.reset()\n t.hideturtle()\n Upgarde()\n t.update()\n time.sleep(0.3)\n else:\n t.reset()\n t.hideturtle()\n Upgarde_Init()\n t.update()\n time.sleep(1)\n \n \nmain()\n\n#\u7ed3\u675f\u753b\u7b14\nt.done()\n```\n\n\n```python\nimport turtle as T\nimport random\nimport time\n\n# \u753b\u6a31\u82b1\u7684\u8eaf\u5e72(60,t)\ndef Tree(branch, t):\n time.sleep(0.000005)\n if branch > 3:\n if 8 <= branch <= 12:\n if random.randint(0, 2) == 0:\n t.color('snow') # \u767d\n else:\n t.color('lightcoral') # \u6de1\u73ca\u745a\u8272\n t.pensize(branch / 3)\n elif branch < 8:\n if random.randint(0, 1) == 0:\n t.color('snow')\n else:\n t.color('lightcoral') # \u6de1\u73ca\u745a\u8272\n t.pensize(branch / 2)\n else:\n t.color('sienna') # \u8d6d(zh\u011b)\u8272\n t.pensize(branch / 10) # 6\n t.forward(branch)\n a = 1.5 * random.random()\n t.right(20 * a)\n b = 1.5 * random.random()\n Tree(branch - 10 * b, t)\n t.left(40 * a)\n Tree(branch - 10 * b, t)\n t.right(20 * a)\n t.up()\n t.backward(branch)\n t.down()\n\n# \u6389\u843d\u7684\u82b1\u74e3\ndef Petal(m, t):\n for i in range(m):\n a = 200 - 400 * random.random()\n b = 10 - 20 * random.random()\n t.up()\n t.forward(b)\n t.left(90)\n t.forward(a)\n t.down()\n t.color('lightcoral') # \u6de1\u73ca\u745a\u8272\n t.circle(1)\n t.up()\n t.backward(a)\n t.right(90)\n t.backward(b)\n\n# \u7ed8\u56fe\u533a\u57df\nt = T.Turtle()\n# \u753b\u5e03\u5927\u5c0f\nw = T.Screen()\nt.hideturtle() # \u9690\u85cf\u753b\u7b14\nt.getscreen().tracer(5, 0)\nw.screensize(bg='wheat') # wheat\u5c0f\u9ea6\nt.left(90)\nt.up()\nt.backward(150)\nt.down()\nt.color('sienna')\n\n# \u753b\u6a31\u82b1\u7684\u8eaf\u5e72\nTree(79, t)\n# \u6389\u843d\u7684\u82b1\u74e3\nPetal(200, t)\nw.exitonclick()\n```\n\n\n```python\nfrom turtle import *\nfrom random import *\nfrom math import *\n\ndef tree(n,l):\n pd()#\u4e0b\u7b14\n #\u9634\u5f71\u6548\u679c\n t = cos(radians(heading()+45))/8+0.25\n pencolor(t,t,t)\n pensize(n/3)\n forward(l)#\u753b\u6811\u679d\n\n if n>0:\n b = random()*15+10 #\u53f3\u5206\u652f\u504f\u8f6c\u89d2\u5ea6\n c = random()*15+10 #\u5de6\u5206\u652f\u504f\u8f6c\u89d2\u5ea6\n d = l*(random()*0.25+0.7) #\u4e0b\u4e00\u4e2a\u5206\u652f\u7684\u957f\u5ea6\n #\u53f3\u8f6c\u4e00\u5b9a\u89d2\u5ea6,\u753b\u53f3\u5206\u652f\n right(b)\n tree(n-1,d)\n #\u5de6\u8f6c\u4e00\u5b9a\u89d2\u5ea6\uff0c\u753b\u5de6\u5206\u652f\n left(b+c)\n tree(n-1,d)\n #\u8f6c\u56de\u6765\n right(c)\n else:\n #\u753b\u53f6\u5b50\n right(90)\n n=cos(radians(heading()-45))/4+0.5\n pencolor(n,n*0.8,n*0.8)\n circle(3)\n left(90)\n #\u6dfb\u52a00.3\u500d\u7684\u98d8\u843d\u53f6\u5b50\n if(random()>0.7):\n pu()\n #\u98d8\u843d\n t = heading()\n an = -40 +random()*40\n setheading(an)\n dis = int(800*random()*0.5 + 400*random()*0.3 + 200*random()*0.2)\n forward(dis)\n setheading(t)\n #\u753b\u53f6\u5b50\n pd()\n right(90)\n n = cos(radians(heading()-45))/4+0.5\n pencolor(n*0.5+0.5,0.4+n*0.4,0.4+n*0.4)\n circle(2)\n left(90)\n pu()\n #\u8fd4\u56de\n t=heading()\n setheading(an)\n backward(dis)\n setheading(t)\n pu()\n backward(l)#\u9000\u56de\n\nbgcolor(0.5,0.5,0.5)#\u80cc\u666f\u8272\nht()#\u9690\u85cfturtle\nspeed(2)#\u901f\u5ea6 1-10\u6e10\u8fdb\uff0c0 \u6700\u5feb\ntracer(0,0)\npu()#\u62ac\u7b14\nbackward(100)\nleft(90)#\u5de6\u8f6c90\u5ea6\npu()#\u62ac\u7b14\nbackward(300)#\u540e\u9000300\ntree(12,100)#\u9012\u5f527\u5c42\ndone()\n```\n\n\n```python\nfrom turtle import *\nfrom random import *\nfrom math import *\n\ndef tree(n, l):\n pd()\n t = cos(radians(heading() + 45)) / 8 + 0.25\n pencolor(t, t, t)\n pensize(n / 4)\n forward(l)\n if n > 0:\n b = random() * 15 + 10\n c = random() * 15 + 10\n d = l * (random() * 0.35 + 0.6)\n right(b)\n tree(n - 1, d)\n left(b + c)\n tree(n - 1, d)\n right(c)\n else:\n right(90)\n n = cos(radians(heading() - 45)) / 4 + 0.5\n pencolor(n, n, n)\n circle(2)\n left(90)\n pu()\n backward(l)\nbgcolor(0.5, 0.5, 0.5)\nht()\nspeed(5)\ntracer(0, 0)\nleft(90)\npu()\nbackward(300)\ntree(20, 100)\ndone()\n```\n\n\n```python\nfrom turtle import *\nimport random\nimport time\n\nn = 100.0\n\nspeed(10)\nscreensize(bg='seashell')\nleft(90)\nforward(3*n)\ncolor(\"orange\", \"yellow\")\nbegin_fill()\nleft(126)\n\nfor i in range(5):\n forward(n/5)\n right(144)\n forward(n/5)\n left(72)\nend_fill()\nright(126)\n\ncolor(\"dark green\")\nbackward(n*4.8)\ndef tree(d, s):\n if d <= 0: return\n forward(s)\n tree(d-1, s*.8)\n right(120)\n tree(d-3, s*.5)\n right(120)\n tree(d-3, s*.5)\n right(120)\n backward(s)\ntree(15, n)\nbackward(n/2)\n\nfor i in range(200):\n a = 200 - 400 * random.random()\n b = 10 - 20 * random.random()\n up()\n forward(b)\n left(90)\n forward(a)\n down()\n if random.randint(0, 1) == 0:\n color('tomato')\n else:\n color('wheat')\n circle(2)\n up()\n backward(a)\n right(90)\n backward(b)\n\n# time.sleep(60)\n```\n\n\n```python\n# \u9b54\u6cd5\u5c11\u5973\nimport turtle as te\nimport time\nWriteStep = 5000 # \u8d1d\u585e\u5c14\u51fd\u6570\u7684\u53d6\u6837\u6b21\u6570\nSpeed = 5\nWidth = 1800 # \u754c\u9762\u5bbd\u5ea6\nHeight = 1800 # \u754c\u9762\u9ad8\u5ea6\nXh = 0 # \u8bb0\u5f55\u524d\u4e00\u4e2a\u8d1d\u585e\u5c14\u51fd\u6570\u7684\u624b\u67c4\nYh = 0\n\n\ndef Bezier(p1, p2, t): # \u4e00\u9636\u8d1d\u585e\u5c14\u51fd\u6570\n return p1 * (1 - t) + p2 * t\n\n\ndef Bezier_2(x1, y1, x2, y2, x3, y3): # \u4e8c\u9636\u8d1d\u585e\u5c14\u51fd\u6570\n te.goto(x1, y1)\n te.pendown()\n for t in range(0, WriteStep + 1):\n x = Bezier(Bezier(x1, x2, t / WriteStep),\n Bezier(x2, x3, t / WriteStep), t / WriteStep)\n y = Bezier(Bezier(y1, y2, t / WriteStep),\n Bezier(y2, y3, t / WriteStep), t / WriteStep)\n te.goto(x, y)\n te.penup()\n\n\ndef Bezier_3(x1, y1, x2, y2, x3, y3, x4, y4): # \u4e09\u9636\u8d1d\u585e\u5c14\u51fd\u6570\n x1 = -Width / 2 + x1\n y1 = Height / 2 - y1\n x2 = -Width / 2 + x2\n y2 = Height / 2 - y2\n x3 = -Width / 2 + x3\n y3 = Height / 2 - y3\n x4 = -Width / 2 + x4\n y4 = Height / 2 - y4 # \u5750\u6807\u53d8\u6362\n te.goto(x1, y1)\n te.pendown()\n for t in range(0, WriteStep + 1):\n x = Bezier(Bezier(Bezier(x1, x2, t / WriteStep), Bezier(x2, x3, t / WriteStep), t / WriteStep),\n Bezier(Bezier(x2, x3, t / WriteStep), Bezier(x3, x4, t / WriteStep), t / WriteStep), t / WriteStep)\n y = Bezier(Bezier(Bezier(y1, y2, t / WriteStep), Bezier(y2, y3, t / WriteStep), t / WriteStep),\n Bezier(Bezier(y2, y3, t / WriteStep), Bezier(y3, y4, t / WriteStep), t / WriteStep), t / WriteStep)\n te.goto(x, y)\n te.penup()\n\n\ndef Moveto(x, y): # \u79fb\u52a8\u5230svg\u5750\u6807\u4e0b\uff08x\uff0cy\uff09\n te.penup()\n te.goto(-Width / 2 + x, Height / 2 - y)\n\n\ndef line(x1, y1, x2, y2): # \u8fde\u63a5svg\u5750\u6807\u4e0b\u4e24\u70b9\n te.penup()\n te.goto(-Width / 2 + x1, Height / 2 - y1)\n te.pendown()\n te.goto(-Width / 2 + x2, Height / 2 - y2)\n te.penup()\n\n\ndef lineto(dx, dy): # \u8fde\u63a5\u5f53\u524d\u70b9\u548c\u76f8\u5bf9\u5750\u6807\uff08dx\uff0cdy\uff09\u7684\u70b9\n te.pendown()\n te.goto(te.xcor() + dx, te.ycor() - dy)\n te.penup()\n\n\ndef Lineto(x, y): # \u8fde\u63a5\u5f53\u524d\u70b9\u548csvg\u5750\u6807\u4e0b\uff08x\uff0cy\uff09\n te.pendown()\n te.goto(-Width / 2 + x, Height / 2 - y)\n te.penup()\n\n\ndef Horizontal(x): # \u505a\u5230svg\u5750\u6807\u4e0b\u6a2a\u5750\u6807\u4e3ax\u7684\u6c34\u5e73\u7ebf\n te.pendown()\n te.setx(x - Width / 2)\n te.penup()\n\n\ndef horizontal(dx): # \u505a\u5230\u76f8\u5bf9\u6a2a\u5750\u6807\u4e3adx\u7684\u6c34\u5e73\u7ebf\n te.seth(0)\n te.pendown()\n te.fd(dx)\n te.penup()\n\n\ndef vertical(dy): # \u505a\u5230\u76f8\u5bf9\u7eb5\u5750\u6807\u4e3ady\u7684\u5782\u76f4\u7ebf\n te.seth(-90)\n te.pendown()\n te.fd(dy)\n te.penup()\n te.seth(0)\n\n\ndef polyline(x1, y1, x2, y2, x3, y3): # \u505asvg\u5750\u6807\u4e0b\u7684\u6298\u7ebf\n te.penup()\n te.goto(-Width / 2 + x1, Height / 2 - y1)\n te.pendown()\n te.goto(-Width / 2 + x2, Height / 2 - y2)\n te.goto(-Width / 2 + x3, Height / 2 - y3)\n te.penup()\n\n\ndef Curveto(x1, y1, x2, y2, x, y): # \u4e09\u9636\u8d1d\u585e\u5c14\u66f2\u7ebf\u5230\uff08x\uff0cy\uff09\n te.penup()\n X_now = te.xcor() + Width / 2\n Y_now = Height / 2 - te.ycor()\n Bezier_3(X_now, Y_now, x1, y1, x2, y2, x, y)\n global Xh\n global Yh\n Xh = x - x2\n Yh = y - y2\n\n\ndef curveto_r(x1, y1, x2, y2, x, y): # \u4e09\u9636\u8d1d\u585e\u5c14\u66f2\u7ebf\u5230\u76f8\u5bf9\u5750\u6807\uff08x\uff0cy\uff09\n te.penup()\n X_now = te.xcor() + Width / 2\n Y_now = Height / 2 - te.ycor()\n Bezier_3(X_now, Y_now, X_now + x1, Y_now + y1,\n X_now + x2, Y_now + y2, X_now + x, Y_now + y)\n global Xh\n global Yh\n Xh = x - x2\n Yh = y - y2\n\n\ndef Smooth(x2, y2, x, y): # \u5e73\u6ed1\u4e09\u9636\u8d1d\u585e\u5c14\u66f2\u7ebf\u5230\uff08x\uff0cy\uff09\n global Xh\n global Yh\n te.penup()\n X_now = te.xcor() + Width / 2\n Y_now = Height / 2 - te.ycor()\n Bezier_3(X_now, Y_now, X_now + Xh, Y_now + Yh, x2, y2, x, y)\n Xh = x - x2\n Yh = y - y2\n\n\ndef smooth_r(x2, y2, x, y): # \u5e73\u6ed1\u4e09\u9636\u8d1d\u585e\u5c14\u66f2\u7ebf\u5230\u76f8\u5bf9\u5750\u6807\uff08x\uff0cy\uff09\n global Xh\n global Yh\n te.penup()\n X_now = te.xcor() + Width / 2\n Y_now = Height / 2 - te.ycor()\n Bezier_3(X_now, Y_now, X_now + Xh, Y_now + Yh,\n X_now + x2, Y_now + y2, X_now + x, Y_now + y)\n Xh = x - x2\n Yh = y - y2\n\nte.tracer(10)\nte.setup(Width, Height, 0, 0)\nte.pensize(1)\nte.speed(Speed)\nte.penup()\n\n# \u56fe\u5c42_2\n# time.sleep(20)\nte.color(\"black\", \"#F2F2F2\") # \u5916\u5957\nMoveto(61, 462)\nte.begin_fill()\nsmooth_r(12, -41, 27, -58)\ncurveto_r(-6, -36, 6, -118, 9, -132)\ncurveto_r(-15, -27, -23, -51, -26, -74)\ncurveto_r(4, -66, 38, -105, 65, -149)\nHorizontal(486)\ncurveto_r(12, 24, 40, 99, 33, 114)\ncurveto_r(39, 82, 55, 129, 39, 144)\nsmooth_r(-31, 23, -39, 28)\nsmooth_r(-12, 37, -12, 37)\nlineto(50, 92)\nHorizontal(445)\nsmooth_r(-29, -38, -31, -46)\nsmooth_r(78, -107, 72, -119)\nSmooth(355, 178, 340, 176)\nSmooth(272, 63, 264, 64)\nsmooth_r(-29, 67, -27, 73)\nCurveto(99, 292, 174, 428, 173, 439)\nsmooth_r(-8, 23, -8, 23)\nLineto(61, 462)\nte.end_fill()\n\nMoveto(60.5, 461.5) # \u9634\u5f71\nte.color(\"black\", \"#D3DFF0\")\nte.begin_fill()\ncurveto_r(0, 0, 17, -42, 27, -59)\ncurveto_r(-6, -33, 6, -128, 10, -133)\ncurveto_r(-15, -10, -27, -66, -27.285, -75)\nte.pencolor(\"#D3DFF0\")\ncurveto_r(12.285, 11, 82.963, 156, 82.963, 156)\nte.pencolor(\"black\")\nsmooth_r(12.322, 75, 19.322, 86)\ncurveto_r(-1, 11, -8, 25, -8, 25)\nHorizontal(60.5)\nte.end_fill()\n\nMoveto(444.5, 464)\nte.begin_fill()\ncurveto_r(0, 0, -29, -36, -31, -46)\nsmooth_r(53.59, -82.337, 53.59, -82.337)\nte.pencolor(\"#D3DFF0\")\nsmooth_r(86.41, -47.663, 96.072, -54.85)\nCurveto(563.5, 297.5, 570.5, 299.5, 518.5, 334)\nte.pencolor(\"black\")\ncurveto_r(-2, 16, -12, 33, -12, 37)\nsmooth_r(50, 92, 50, 93)\nHorizontal(444.5)\nte.end_fill()\n\nMoveto(195, 49)\nte.begin_fill()\nte.pencolor(\"#D3DFF0\")\npolyline(195, 49, 175.5, 106.5, 202.522, 49)\nte.pencolor(\"black\")\nHorizontal(195)\nte.pencolor(\"#D3DFF0\")\nte.end_fill()\n\nMoveto(327.997, 49)\nte.begin_fill()\nte.pencolor(\"#D3DFF0\")\ncurveto_r(0, 0, 11.503, 121.087, 13.503, 128.087)\ncurveto_r(11, 2, 54, 37, 54, 37)\nlineto(-40, -165.087)\nte.pencolor(\"black\")\nHorizontal(327.997)\nte.pencolor(\"#D3DFF0\")\nte.end_fill()\n\nte.pencolor(\"black\")\nline(94.5, 397.5, 107.5, 373.5) # \u76b1\u7eb9\nline(122.5, 317.5, 95.875, 274.699)\nline(122.5, 341.5, 141.5, 402.5)\nline(141.5, 409.5, 153.5, 431.5)\n# line(328,47.712,344,175.977)\nline(340.023, 49, 360.5, 144)\n# line(353.5,47.5,395.5,208.5)\nline(478.5, 95.5, 518.5, 161.5)\nline(518.5, 332.5, 460.5, 359.5)\npolyline(506.5, 369.5, 493.5, 402.5, 502.5, 443.5)\nMoveto(530, 429)\ncurveto_r(4, 16, -5, 33, -5, 33)\n\n# \u56fe\u5c42_3\nte.color(\"black\", \"#2b1d2a\") # \u5916\u5957\u5185\u4fa7\nMoveto(225, 462)\nte.begin_fill()\nHorizontal(165)\nsmooth_r(9, -15, 8, -25)\ncurveto_r(-47, -126, 6, -212, 12, -225)\nCurveto(185, 305, 202, 428, 225, 462)\nLineto(225, 462)\nte.end_fill()\n\nMoveto(390, 462)\nte.begin_fill()\ncurveto_r(10, -23, 34, -180, 35, -222) # !!!227\ncurveto_r(7, 4, 54, 45, 61, 61) # 61\nsmooth_r(-73, 101, -72, 118)\ncurveto_r(5, 15, 31, 46, 31, 45)\nLineto(390, 462)\nte.end_fill()\n# \u56fe\u5c42_4\nte.color(\"black\", \"#2b1d29\") # \u5916\u5957\u5185\u4fa7\nMoveto(225, 462)\nte.begin_fill()\ncurveto_r(-28, -50, -40, -166, -40, -250)\ncurveto_r(6, 51, -6, 87, 45, 106)\nsmooth_r(64, 27, 89, 24)\nsmooth_r(49, -18, 56, -20)\nsmooth_r(50, -10, 51, -85)\ncurveto_r(0, 29, -25, 201, -36, 225)\nLineto(225, 462)\nte.end_fill()\n# \u56fe\u5c42_5\nte.color(\"black\", \"#3D3D3D\") # \u8863\u670d\nMoveto(225, 462)\nte.begin_fill()\ncurveto_r(-5, -5, -22, -53, -23, -70)\nlineto(32, -13)\ncurveto_r(3, -25, 6, -28, 12, -36)\nsmooth_r(13, -12, 16, -12)\nvertical(-2)\ncurveto_r(45, 20, 64, 14, 94, 1)\nvertical(2)\ncurveto_r(8, -2, 15, 2, 17, 4)\nsmooth_r(0, 6, -2, 9)\ncurveto_r(10, 10, 10, 29, 11, 33)\nsmooth_r(23, 4, 25, 6)\nsmooth_r(-17, 83, -17, 78)\nLineto(225, 462)\nte.end_fill()\n# \u56fe\u5c42_6\nte.color(\"black\", \"#968281\") # \u8116\u5b50\nMoveto(262, 329)\nte.begin_fill()\nvertical(17)\ncurveto_r(1, 2, 44, 14, 45, 15)\nsmooth_r(3, 12, 3, 12)\nhorizontal(3)\nvertical(-5)\ncurveto_r(1, -3, 4, -6, 5, -7)\nlineto(36, -14)\ncurveto_r(1, -1, 3, -16, 2, -17)\nCurveto(318, 348, 296, 344, 262, 329)\nte.end_fill()\n# \u56fe\u5c42_8\nte.color(\"black\", \"#E7F1FF\") # \u767d\u8272\u8936\u76b1\nMoveto(225, 462)\nte.begin_fill()\nlineto(-3, -5) # -3,-3,-3,-5\ncurveto_r(0, -2, 4, -4, 5, -6)\nsmooth_r(16, 3, 19, -8)\nsmooth_r(0, -7, 0, -11)\nsmooth_r(5, -8, 9, -5)\nsmooth_r(19, -8, 19, -11)\nsmooth_r(6, -7, 6, -7)\nsmooth_r(7, -2, 9, -4)\nlineto(41, -2)\nlineto(12, 9)\nsmooth_r(3, 15, 7, 18)\nsmooth_r(15, 4, 17, 4)\nsmooth_r(4, -4, 6, -4)\nsmooth_r(6, 4, 5, 9)\nsmooth_r(0, 9, 0, 9)\nsmooth_r(1, 7, 7, 6)\nsmooth_r(8, 0, 8, 0)\nlineto(-2, 8)\nLineto(225, 462)\nte.end_fill()\n\nte.pensize(2)\nMoveto(240, 450)\nsmooth_r(0, 9, 3, 12)\nMoveto(372, 462)\ncurveto_r(-2, -4, -5, -29, -7, -28)\nte.pensize(1)\n# \u56fe\u5c42_7\nte.color(\"black\", \"#A2B8D6\") # \u8863\u9886\nMoveto(262, 331)\nte.begin_fill()\ncurveto_r(0, 8, -1, 13, 0, 15)\nsmooth_r(43, 14, 45, 15)\nlineto(3, 12)\nhorizontal(3)\nsmooth_r(-1, -3, 0, -5)\nlineto(5, -7)\nlineto(36, -14)\ncurveto_r(1, -1, 2, -12, 2, -15)\nsmooth_r(25, -2, 15, 13)\ncurveto_r(-2, 4, -7, 29, -7, 32)\nsmooth_r(-35, 19, -41, 22)\nsmooth_r(-9, 14, -12, 14)\nsmooth_r(-7, -12, -14, -15)\ncurveto_r(-19, -2, -41, -25, -41, -25)\nsmooth_r(-10, -26, -10, -30)\nSmooth(255, 332, 262, 331)\nte.end_fill()\n\nMoveto(262, 346)\nlineto(-12, -6)\nMoveto(369, 333)\ncurveto_r(2, 4, -6, 10, -15, 14)\n# \u56fe\u5c42_9\nte.color(\"black\", \"#151515\") # \u9886\u7ed3\nMoveto(247, 358)\nte.begin_fill()\ncurveto_r(-5, 3, -8, 20, -6, 23)\ncurveto_r(25, 21, 50, 17, 50, 17)\nlineto(-23, 64)\nhorizontal(22)\nsmooth_r(1, -13, 2, -16)\nlineto(13, -50)\ncurveto_r(2, 2, 7, 3, 10, 1)\nsmooth_r(18, 65, 18, 65)\nhorizontal(19)\nlineto(-24, -65)\ncurveto_r(21, 5, 39, -10, 44, -13)\ncurveto_r(5, -20, 1, -21, 0, -24)\ncurveto_r(-18, -2, -49, 15, -52, 17)\nsmooth_r(-11, -3, -15, -1)\nSmooth(252, 356, 247, 358)\nte.end_fill()\n# \u56fe\u5c42_10\nte.color(\"black\", \"#A2B8D6\") # \u8863\u9886\uff08\u900f\u8fc7\u9886\u7ed3\uff09\nMoveto(297, 387)\nte.begin_fill()\nlineto(-11, 6)\ncurveto_r(-1, 0, -20, -7, -30, -19)\nCurveto(259, 373, 297, 385, 297, 387)\nte.end_fill()\n\nMoveto(323, 384)\nte.begin_fill()\nlineto(8, 7)\nlineto(30, -14)\ncurveto_r(1, -1, 5, -6, 4, -7)\nSmooth(329, 379, 323, 384)\nte.end_fill()\n# \u56fe\u5c42_11\nte.color(\"black\", \"#F3EEEB\") # \u8138\nMoveto(185, 212)\nte.begin_fill()\ncurveto_r(4, -9, 46, -77, 52, -75)\ncurveto_r(-2, -17, 19, -68, 27, -73)\ncurveto_r(16, 15, 71, 108, 76, 112)\nsmooth_r(76, 53, 86, 60)\ncurveto_r(0, 65, -27, 75, -31, 76)\ncurveto_r(-50, 28, -70, 30, -85, 30)\nsmooth_r(-77, -22, -86, -26)\nCurveto(180, 302, 186, 228, 185, 212)\nte.end_fill()\n# \u56fe\u5c42_12\nte.color(\"black\", \"#2B1D29\") # \u5934\u53d1\nMoveto(189, 202)\nte.begin_fill()\ncurveto_r(-1, 22, 19, 51, 19, 51)\nsmooth_r(-10, -42, 7, -92)\nCurveto(212, 168, 196, 189, 189, 202)\nte.end_fill()\n\nMoveto(221, 155)\nte.begin_fill()\ncurveto_r(-2, 6, 5, 48, 5, 48)\nsmooth_r(18, -28, 20, -48)\ncurveto_r(-5, 24, 4, 43, 7, 50)\ncurveto_r(-10, -49, 3, -72, 13, -106)\ncurveto_r(-2, -7, -3, -32, -3, -35)\ncurveto_r(-17, 18, -27, 71, -27, 71)\nLineto(221, 155)\nte.end_fill()\n\nMoveto(264, 64)\nte.begin_fill()\ncurveto_r(-4, 5, 14, 100, 14, 100)\nsmooth_r(-6, -79, -5, -85)\ncurveto_r(0, 98, 49, 139, 49, 139)\nsmooth_r(8, -50, 3, -65)\nSmooth(272, 64, 264, 64)\nte.end_fill()\n\nMoveto(342, 176)\nte.begin_fill()\ncurveto_r(-1, 27, -10, 57, -10, 57)\nsmooth_r(20, -33, 17, -54)\nLineto(342, 176)\nte.end_fill()\n\nte.penup()\nte.begin_fill()\npolyline(349, 180, 353, 203, 361, 203)\npolyline(361, 203, 362, 188, 349, 180)\nte.end_fill()\n# \u56fe\u5c42_13\nte.pensize(2)\nMoveto(210, 180) # \u7709\u6bdb\ncurveto_r(5, -4, 63, 9, 63, 14)\nMoveto(338, 193)\ncurveto_r(0, -3, 18, -6, 18, -6)\nte.pensize(1)\n# \u56fe\u5c42_14\nte.color(\"black\", \"#D1D1D1\") # \u773c\u775b1\nte.pensize(2)\nMoveto(206, 212)\nte.begin_fill()\nlineto(15, -7)\ncurveto_r(4, -1, 26, -2, 30, 0)\nsmooth_r(10, 3, 12, 7)\nte.pencolor(\"#D1D1D1\")\nte.pensize(1)\nsmooth_r(2, 27, -1, 30)\nsmooth_r(-39, 5, -44, 1)\nSmooth(206, 212, 206, 212)\nte.end_fill()\n\nMoveto(384, 204)\nte.begin_fill()\nte.pencolor(\"black\")\nte.pensize(2)\ncurveto_r(-3, -1, -18, -1, -28, 1)\nsmooth_r(-9, 6, -10, 9)\nte.pencolor(\"#D1D1D1\")\nte.pensize(1)\nsmooth_r(3, 18, 6, 23)\nsmooth_r(38, 6, 40, 4)\nsmooth_r(10, -9, 13, -22)\nte.pencolor(\"black\")\nte.pensize(2)\nLineto(384, 204)\nte.end_fill()\n# \u56fe\u5c42_15\nte.color(\"#0C1631\", \"#0C1631\") # \u773c\u775b2\nte.pensize(1)\nMoveto(216, 206)\nte.begin_fill()\ncurveto_r(-1, 5, 0, 26, 7, 35)\nsmooth_r(30, 2, 33, 0)\nsmooth_r(5, -31, 2, -34)\nSmooth(219, 203, 216, 206)\nte.end_fill()\n\nMoveto(354, 207)\nte.begin_fill()\ncurveto_r(-2, 1, 2, 29, 4, 31)\nsmooth_r(30, 3, 33, 1)\nsmooth_r(6, -24, 4, -27)\nlineto(-11, -8)\nCurveto(382, 204, 357, 206, 354, 207)\nte.end_fill()\n\n# \u56fe\u5c42_17\nte.color(\"#F5F5F5\", \"#F5F5F5\") # \u773c\u775b3\nMoveto(253, 211)\nte.begin_fill()\ncurveto_r(-3, 0, -8, 8, 1, 10)\nSmooth(258, 210, 253, 211)\nte.end_fill()\n\nMoveto(392, 209)\nte.begin_fill()\nlineto(4, 3)\nvertical(4)\nlineto(-4, 2)\nCurveto(386, 214, 392, 209, 392, 209)\nte.end_fill()\n# \u56fe\u5c42_18\nte.color(\"#352F53\", \"#352F53\") # \u773c\u775b4\nMoveto(219, 229)\nte.begin_fill()\nsmooth_r(2, -5, 6, -4)\nsmooth_r(18, 13, 27, 1)\ncurveto_r(3, 0, 5, 3, 5, 3)\nvertical(13)\nHorizontal(224)\nLineto(219, 229)\nte.end_fill()\n\nMoveto(357, 227)\nte.begin_fill()\nsmooth_r(4, -6, 10, -2)\nsmooth_r(10, 13, 19, 1)\ncurveto_r(6, 0, 8, 6, 8, 6)\nlineto(-2, 9)\ncurveto_r(-12, 3, -29, 0, -32, -2)\nSmooth(357, 227, 357, 227)\nte.end_fill()\n\n# \u56fe\u5c42_19\nte.color(\"#9A90CB\", \"#9A90CB\") # \u773c\u775b5\nMoveto(227, 231)\nte.begin_fill()\ncurveto_r(-6, 0, -5, 5, -3, 8)\nsmooth_r(24, 2, 27, 0)\nsmooth_r(0, -8, -1, -8)\nSmooth(234, 231, 227, 231)\nte.end_fill()\n\nMoveto(361, 227)\nte.begin_fill()\ncurveto_r(2, 18, 26, 14, 30, 6)\nsmooth_r(-1, -3, -2, -4)\nsmooth_r(-15, 9, -24, -4)\nCurveto(363, 224, 361, 225, 361, 227)\nte.end_fill()\n\n# \u56fe\u5c42_16\nte.pencolor(\"black\") # \u773c\u775b(\u7ebf\u6761)\nte.pensize(3)\n# Moveto(206,213)\n# lineto(14,-8)\n# curveto_r(3,-1,30,0,33,1)\n# lineto(10,6)\nMoveto(225, 215)\ncurveto_r(10, 28, 22, 16, 24, 6)\nMoveto(365, 219)\ncurveto_r(4, 14, 18, 24, 22, -3)\nte.pensize(2)\nline(240.5, 207.5, 227.5, 211.5)\nline(245.5, 209.5, 227.5, 214.5)\nline(247.5, 211.5, 227.5, 217.5)\nline(247.5, 214.5, 229.5, 220.5)\nline(247.5, 218.5, 230.5, 223.5)\nline(246.5, 222.5, 232.5, 226.5)\nline(244.5, 225.5, 234.5, 228.5)\n\nline(377.5, 207.5, 367.5, 210.5)\nline(384.5, 207.5, 366.5, 212.5)\nline(385.5, 210.5, 366.5, 215.5)\nline(384.5, 213.5, 366.5, 218.5)\nline(384.5, 215.5, 367.5, 220.5)\nline(384.5, 218.5, 368.5, 223.5)\n# line(383.5,220.5,368.5,225.5)\nline(382.5, 223.5, 370.5, 227.5)\n# line(381.5,226.5,373.5,229.5)\n# \u56fe\u5c42_20\nte.pencolor(\"black\")\nMoveto(309, 270) # \u9f3b\u5b50\u3001\u5634\ncurveto_r(0, 0, 4, 7, 1, 9)\nline(296.5, 307.5, 303.5, 307.5)\nMoveto(315, 307)\nsmooth_r(10, -1, 10, 2)\n\nte.penup()\nte.hideturtle()\nte.update()\nte.done()\n\n```\n\n\n```python\nimport turtle as T\nimport random\nimport time\n\n# \u753b\u6a31\u82b1\u7684\u8eaf\u5e72(60,t)\ndef Tree(branch, t):\n t.hideturtle()\n time.sleep(0.00009)\n if branch > 3:\n if 8 <= branch <= 12:\n if random.randint(0, 2) == 0:\n t.color('snow') # \u767d\n else:\n t.color('lightcoral') # \u6de1\u73ca\u745a\u8272\n t.pensize(branch / 3)\n elif branch < 8:\n if random.randint(0, 1) == 0:\n t.color('snow')\n else:\n t.color('lightcoral') # \u6de1\u73ca\u745a\u8272\n t.pensize(branch / 2)\n else:\n t.color('sienna') # \u8d6d(zh\u011b)\u8272\n t.pensize(branch / 10) # 6\n t.forward(branch)\n a = 1.5 * random.random()\n t.right(20 * a)\n b = 1.5 * random.random()\n Tree(branch - 10 * b, t)\n t.left(40 * a)\n Tree(branch - 10 * b, t)\n t.right(20 * a)\n t.up()\n t.backward(branch)\n t.down()\n T\n\n\n# \u6389\u843d\u7684\u82b1\u74e3\ndef Petal(m, t):\n T.speed(0)\n for i in range(m):\n a = 200 - 400 * random.random()\n b = 10 - 20 * random.random()\n t.up()\n t.forward(b)\n t.left(90)\n t.forward(a)\n t.down()\n t.color('lightcoral') # \u6de1\u73ca\u745a\u8272\n t.circle(1)\n t.up()\n t.backward(a)\n t.right(90)\n t.backward(b)\n\n\n# \u7ed8\u56fe\u533a\u57df\nt = T.Turtle()\n# \u753b\u5e03\u5927\u5c0f\nw = T.Screen()\nt.hideturtle() # \u9690\u85cf\u753b\u7b14\nT.showturtle()\nt.getscreen().tracer(5, 0)\nw.screensize(3000,3000,bg='wheat') # wheat\u5c0f\u9ea6\nt.left(90)\nt.up()\nt.backward(150)\nt.down()\nt.color('sienna')\nt.hideturtle()\nT.speed(0)\n\n\n# \u753b\u6a31\u82b1\u7684\u8eaf\u5e72\nTree(79, t)\n# \u6389\u843d\u7684\u82b1\u74e3\nPetal(200, t)\nw.exitonclick()\n\n```\n\n\n```python\nfrom turtle import *\ncolors = ['red', 'purple', 'blue', 'green', 'yellow', 'orange']\nfor x in range(360):\n pencolor(colors[x % 6])\n width(x / 100 + 1)\n forward(x)\n left(59)\n```\n\n\n```python\na,*b = 'spam'\n```\n\n\n```python\na,b\n```\n\n\n\n\n ('s', ['p', 'a', 'm'])\n\n\n\n\n```python\nspam = ham = 'lunch'\n```\n\n\n```python\nspam += str(42)\n```\n\n\n```python\nspam\n```\n\n\n\n\n 'lunch4242'\n\n\n\n\n```python\nnudge = 1\nwink = 2\nA, B = nudge, wink\nprint(A, B)\n[C, D] = [nudge, wink]\nprint(C,D)\n```\n\n 1 2\n 1 2\n\n\n# Advanced sequence assignment patterns\n\n\n```python\nstring = 'SPAM'\na, b, c, d = string\n```\n\n\n```python\na, d\n```\n\n\n\n\n ('S', 'M')\n\n\n\n\n```python\na, b, c, d\n```\n\n\n\n\n ('S', 'P', 'A', 'M')\n\n\n\n\n```python\na, b, c = string[0], string[1], string[2:] #Index and slice \n```\n\n\n```python\na, b, c\n```\n\n\n\n\n ('S', 'P', 'AM')\n\n\n\n\n```python\na, b, c = list(string[:2]) + [string[2:]]\n```\n\n\n```python\na, b, c\n```\n\n\n\n\n ('S', 'P', 'AM')\n\n\n\n\n```python\n(a, b), c = string[:2], string[2:]\n```\n\n\n```python\na, b, c\n```\n\n\n\n\n ('S', 'P', 'AM')\n\n\n\n\n```python\n((a, b), c) = ('SP',\"AM\") # Paired by shape and position\n```\n\n\n```python\na, b, c\n```\n\n\n\n\n ('S', 'P', 'AM')\n\n\n\n\n```python\nred, green, blue = range(3)\n```\n\n\n```python\nred, blue\n```\n\n\n\n\n (0, 2)\n\n\n\n\n```python\nL = [1,2,3,4]\nwhile L:\n front, L = L[0], L[1:] # See next section for 3.0 alternative\n print(front, L)\n```\n\n 1 [2, 3, 4]\n 2 [3, 4]\n 3 [4]\n 4 []\n\n\n\n```python\nL =[1,2,3,4]\nwhile L:\n front = L.pop(0)\n print(front, L)\n```\n\n 1 [2, 3, 4]\n 2 [3, 4]\n 3 [4]\n 4 []\n\n\n\n```python\nseq = [1,2,3,4]\na, b, c, d = seq\nprint(a, b, c, d)\n```\n\n 1 2 3 4\n\n\n\n```python\na, *b = seq\n```\n\n\n```python\na, b\n```\n\n\n\n\n (1, [2, 3, 4])\n\n\n\n\n```python\n*a, b = seq\n```\n\n\n```python\na, b\n```\n\n\n\n\n ([1, 2, 3], 4)\n\n\n\n\n```python\na, *b, c = seq\n```\n\n\n```python\na, b, c\n```\n\n\n\n\n (1, [2, 3], 4)\n\n\n\n\n```python\na, b, *c = seq\n```\n\n\n```python\na, *b = 'spam'\n```\n\n\n```python\nprint(a,b)\na, *b, c = 'spam'\na, b, c\n```\n\n s ['p', 'a', 'm']\n\n\n\n\n\n ('s', ['p', 'a'], 'm')\n\n\n\n# Boundary cases\n\n\n```python\nseq\n```\n\n\n\n\n [1, 2, 3, 4]\n\n\n\n\n```python\na, b, c, *d = seq\nprint(a, b, c, d)\n```\n\n 1 2 3 [4]\n\n\n\n```python\na, b, c, d, *e = seq\nprint(a,b,c,d,e)\n```\n\n 1 2 3 4 []\n\n\n\n```python\na, b, *e, c, d = seq\n```\n\n\n```python\nprint(a,b,c,d,e)\n```\n\n 1 2 3 4 []\n\n\n\n```python\n*a, = seq\n```\n\n\n```python\na\n```\n\n\n\n\n [1, 2, 3, 4]\n\n\n\n\n```python\nT = (seq[1]) # A one-item tuple (not an expression)\n```\n\n\n```python\ntype(T), type((seq[1],))\n```\n\n\n\n\n (int, tuple)\n\n\n\n\n```python\nseq\n```\n\n\n\n\n [1, 2, 3, 4]\n\n\n\n\n```python\na, *b = seq\n```\n\n\n```python\na, b\n```\n\n\n\n\n (1, [2, 3, 4])\n\n\n\n\n```python\na, b = seq[0], seq[1:] # First, rest : traditional\n```\n\n\n```python\na, b\n```\n\n\n\n\n (1, [2, 3, 4])\n\n\n\n\n```python\nfor (a, *b, c) in [(1,2,3,4),(5,6,7,8),(9,10,11,12)]:\n print(a,b,c)\n```\n\n 1 [2, 3] 4\n 5 [6, 7] 8\n 9 [10, 11] 12\n\n\n\n```python\nfor allnum in [(1,2,3,4),(5,6,7,8)]:\n a, b, c = allnum[0], allnum[1:3], allnum[3]\n```\n\n# Multiple-Target Assignments\n\n\n```python\na = b = c = 'spam'\na, b, c\n```\n\n\n\n\n ('spam', 'spam', 'spam')\n\n\n\n\n```python\nc = 'spam'\nb = c\na = b\n```\n\n\n```python\nT= []\nT.append([123, 'xyz', 'zara', 'abc'])\n```\n\n\n```python\nT\n```\n\n\n\n\n [[123, 'xyz', 'zara', 'abc']]\n\n\n\n# Multiple-target assignment and shared references\n\n\n```python\na = b = 0\nb = b + 1\na, b\n```\n\n\n\n\n (0, 1)\n\n\n\nHere, changing b only changes b because numbers do not support in-place changes. As\nlong as the object assigned is immutable, it\u2019s irrelevant if more than one name references\nit.\n\n\n```python\na = b = []\nb.append(42)\nprint(a,b)\n```\n\n [42] [42]\n\n\nAs usual, though, we have to be more cautious when initializing variables to an empty mutable object such as a list or dictionary:\n\nThis time, because a and b reference the same object, appending to it in-place through\nb will impact what we see through a as well. This is really just another example of the\nshared reference phenomenon we first met in Chapter 6. To avoid the issue, initialize\nmutable objects in separate statements instead, so that each creates a distinct empty\nobject by running a distinct literal expression:\n\n\n```python\na = []\nb = []\nb.append(42)\nprint(a,b)\n```\n\n [] [42]\n\n\n# Augmented Assignments\n\n\n```python\na = a + b\nprint(a)\na += b\nprint(a)\n```\n\n [42]\n [42, 42]\n\n\nTable 11-2. Augmented assignment statements\nX += Y X &= Y X -= Y X |= Y\nX *= Y X ^= Y X /= Y X >>= Y\nX %= Y X <<= Y X **= Y X //= Y\n\n\n```python\n# a &= b unsupported operand type(s) for &=: 'list' and 'list'\n```\n\n\n```python\na, b = 36, 42\na &= b\n```\n\n\n```python\nbin(36),bin(42)\n```\n\n\n\n\n ('0b100100', '0b101010')\n\n\n\n\n```python\nbin(32)\n```\n\n\n\n\n '0b100000'\n\n\n\n\n```python\n36 & 42, 1 & 1, 1 & 2, 1 & 3\n```\n\n\n\n\n (32, 1, 0, 1)\n\n\n\n\n```python\nfor i in range(40):\n print(1 & i, end = ',')\n```\n\n 0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,\n\n\n```python\nfor i in range(50):\n print(36 & i, end = ', ')\n print(bin(36), bin(i), bin(36 & i), sep=', ')\n```\n\n 0, 0b100100, 0b0, 0b0\n 0, 0b100100, 0b1, 0b0\n 0, 0b100100, 0b10, 0b0\n 0, 0b100100, 0b11, 0b0\n 4, 0b100100, 0b100, 0b100\n 4, 0b100100, 0b101, 0b100\n 4, 0b100100, 0b110, 0b100\n 4, 0b100100, 0b111, 0b100\n 0, 0b100100, 0b1000, 0b0\n 0, 0b100100, 0b1001, 0b0\n 0, 0b100100, 0b1010, 0b0\n 0, 0b100100, 0b1011, 0b0\n 4, 0b100100, 0b1100, 0b100\n 4, 0b100100, 0b1101, 0b100\n 4, 0b100100, 0b1110, 0b100\n 4, 0b100100, 0b1111, 0b100\n 0, 0b100100, 0b10000, 0b0\n 0, 0b100100, 0b10001, 0b0\n 0, 0b100100, 0b10010, 0b0\n 0, 0b100100, 0b10011, 0b0\n 4, 0b100100, 0b10100, 0b100\n 4, 0b100100, 0b10101, 0b100\n 4, 0b100100, 0b10110, 0b100\n 4, 0b100100, 0b10111, 0b100\n 0, 0b100100, 0b11000, 0b0\n 0, 0b100100, 0b11001, 0b0\n 0, 0b100100, 0b11010, 0b0\n 0, 0b100100, 0b11011, 0b0\n 4, 0b100100, 0b11100, 0b100\n 4, 0b100100, 0b11101, 0b100\n 4, 0b100100, 0b11110, 0b100\n 4, 0b100100, 0b11111, 0b100\n 32, 0b100100, 0b100000, 0b100000\n 32, 0b100100, 0b100001, 0b100000\n 32, 0b100100, 0b100010, 0b100000\n 32, 0b100100, 0b100011, 0b100000\n 36, 0b100100, 0b100100, 0b100100\n 36, 0b100100, 0b100101, 0b100100\n 36, 0b100100, 0b100110, 0b100100\n 36, 0b100100, 0b100111, 0b100100\n 32, 0b100100, 0b101000, 0b100000\n 32, 0b100100, 0b101001, 0b100000\n 32, 0b100100, 0b101010, 0b100000\n 32, 0b100100, 0b101011, 0b100000\n 36, 0b100100, 0b101100, 0b100100\n 36, 0b100100, 0b101101, 0b100100\n 36, 0b100100, 0b101110, 0b100100\n 36, 0b100100, 0b101111, 0b100100\n 32, 0b100100, 0b110000, 0b100000\n 32, 0b100100, 0b110001, 0b100000\n\n\n\n```python\na, b = 36, 42\na -= b\nprint(a)\na, b = 36, 42\na |= b\nprint(a)\n```\n\n -6\n 46\n\n\n\n```python\nbin(36), bin(42), bin(46)\n```\n\n\n\n\n ('0b100100', '0b101010', '0b101110')\n\n\n\n\n```python\nimport pandas as pd\ntarget_url = \"http://aima.cs.berkeley.edu/data/iris.csv\"\ndata = pd.read_csv(target_url, header=None)\ndata.describe()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    0123
    count150.000000150.000000150.000000150.000000
    mean5.8433333.0540003.7586671.198667
    std0.8280660.4335941.7644200.763161
    min4.3000002.0000001.0000000.100000
    25%5.1000002.8000001.6000000.300000
    50%5.8000003.0000004.3500001.300000
    75%6.4000003.3000005.1000001.800000
    max7.9000004.4000006.9000002.500000
    \n
    \n\n\n\n\n```python\na, b = 36, 42\na *= b\nprint(a)\nprint(36*42)\na = 36; b = 42; a^=b; print(a)\n```\n\n 1512\n 1512\n 14\n\n\n\n```python\n36 ^ 42; print(bin(36), bin(42),bin(14))\n```\n\n 0b100100 0b101010 0b1110\n\n\n\n```python\nimport pandas as pd\ndata = pd.read_csv('F:\\house-votes-84.data',header= None)\ndata.head(434)\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    012345678910111213141516
    0republicannynyyynnny?yyyny
    1republicannynyyynnnnnyyyn?
    2democrat?yy?yynnnnynyynn
    3democratnyyn?ynnnnynynny
    4democratyyynyynnnny?yyyy
    ......................................................
    429democratynyn?nyyyynyn?yy
    430republicannnyyyynnyynyyyny
    431democratnnynnnyyyynnnnny
    432republicann?nyyynnnnyyyyny
    433republicannnnyyy????nyyyny
    \n

    434 rows \u00d7 17 columns

    \n
    \n\n\n\n\n```python\n36 ^ 42; print(bin(36), bin(42),bin(14))\n```\n\n 0b100100 0b101010 0b1110\n\n\n\n```python\nL1 = []\nfor i in range(len(str(bin(36)))):\n L1.append(str(bin(36))[i])\nL2 = []\nfor j in range(len(str(bin(42)))):\n L2.append(str(bin(42))[j])\n```\n\n\n```python\nprint(L1); print(L2)\n```\n\n ['0', 'b', '1', '0', '0', '1', '0', '0']\n ['0', 'b', '1', '0', '1', '0', '1', '0']\n\n\n\n```python\nT = []\nprint(type(T))\nfor k in range(len(L1)):\n T.append((L1[k],L2[k]))\n```\n\n \n\n\n\n```python\nprint(T)\n```\n\n [('0', '0'), ('b', 'b'), ('1', '1'), ('0', '0'), ('0', '1'), ('1', '0'), ('0', '1'), ('0', '0')]\n\n\n\n```python\nS1 = ''\nfor m in range(2,len(L1)):\n if T[m][0] == T[m][1]:\n S1 +='0'\n else:\n S1 +='1'\n```\n\n\n```python\nS2 = ''\nfor m in range(2,len(L1)):\n S2 += str(eval(T[m][0]) & eval(T[m][1]))\n```\n\n\n```python\nprint(S1,S2); S1 = '0b'+ S1; S2 = '0b'+ S2; print(eval(S1)); print(eval(S2))\n```\n\n 001110 100000\n 14\n 32\n\n\n\n```python\na = 36; b = 42;\n```\n\n\n```python\na/=b\nprint(a)\n```\n\n 0.8571428571428571\n\n\n\n```python\nprint(36/42); a == 36/42\n```\n\n 0.8571428571428571\n\n\n\n\n\n True\n\n\n\n\n```python\na = 36; b =42; a>>=b; print(a); print(bin(36),bin(42))\n```\n\n 0\n 0b100100 0b101010\n\n\n\n```python\nprint(int(36/(2**42)) == 0); print(36<<42 == 36*2**42)\n```\n\n True\n True\n\n\n\n```python\nprint(8>>3 == 8/(2**3))\n```\n\n True\n\n\n\n```python\nprint(8%3); print(42%36); a = 42%36; print(a)\n```\n\n 2\n 6\n 6\n\n\n\n```python\na = 42; b = 36; a**=b; print(a, a==42**36);\n```\n\n 27351078395457093635779121514268739090397938600214219718656 True\n\n\n\n```python\na = 42; b = 36; a //=b; print(a,a==42//36)\n```\n\n 1 True\n\n\n\n```python\nimport requests\nfrom lxml import etree\nfrom bs4 import BeautifulSoup\n\n\nheaders = {'User-Agent': \"Mozilla/5.0 (Windows NT 10.0; WOW64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/80.0.3970.5 Safari/537.36\"}\nurl='https://www.cnblogs.com/JYNNO1/p/10525649.html' #\u8f93\u5165\u6211\u4eec\u7684url\nget = requests.get(url) # get(url) \u5f97\u5230\u6211\u4eec\u7684\u7f51\u9875, text\u5c06\u6e90\u7f51\u9875\u8f6c\u5316\u4e3a\u5b57\u7b26\u4e32\nsoup = BeautifulSoup(get.content, 'html.parser')\ntext = soup.find_all(text=True)\nset([t.parent.name for t in text])\noutput = ''\nblacklist = [\n '[document]',\n 'a',\n 'body',\n 'div',\n 'h1',\n 'h2',\n 'head',\n 'html',\n 'li',\n 'p',\n 'pre',\n 'script',\n 'span',\n 'title',\n 'ul'\n]\nfor t in text:\n if t.parent.name not in blacklist:\n output += '{} '.format(t)\n```\n\n\n```python\nprint(output)\n```\n\n \n\n\n\n```python\nimport requests\nfrom bs4 import BeautifulSoup\nurl = 'https://www.cnblogs.com/JYNNO1/p/10525649.html'\nres = requests.get(url)\nhtml_page = res.content\nsoup = BeautifulSoup(html_page, 'html.parser')\ntext = soup.find_all(text=True)\n\noutput = ''\nblacklist = [\n '[document]',\n 'noscript',\n 'header',\n 'html',\n 'meta',\n 'head', \n 'input',\n 'script',\n # there may be more elements you don't want, such as \"style\", etc.\n]\n\nfor t in text:\n if t.parent.name not in blacklist:\n output += '{} '.format(t)\n\nprint(output,end='; ')\n```\n\n Python-\u4f4d\u64cd\u4f5c( &\u3001 | \u3001^\u3001~ \u3001>>\u3001 <<) - \u5df2\u7ecf\u5d29\u76d8\u4e86 - \u535a\u5ba2\u56ed \n \n done \n \n \n \n \n done \n \u5df2\u7ecf\u5d29\u76d8\u4e86 \n \n \n \n end: blogTitle \u535a\u5ba2\u7684\u6807\u9898\u548c\u526f\u6807\u9898 \n end: header \u5934\u90e8 \n \n \n \n \n \n \n \u535a\u5ba2\u56ed \n \n \n \n \u9996\u9875 \n \n \n \n \u65b0\u968f\u7b14 \n \n \n \n \u8054\u7cfb \n \n \n \u7ba1\u7406 \n \n \n \n \u8ba2\u9605 \n \n \n \n \n \n done \n \u968f\u7b14- \n 17\u00a0\n \u6587\u7ae0- \n 0\u00a0\n \u8bc4\u8bba- \n 0\u00a0\n \n \n \n \t\t\t\t\n \t\t\t end: blogStats \n end: navigator \u535a\u5ba2\u5bfc\u822a\u680f \n \n done \n \n \n \n Python-\u4f4d\u64cd\u4f5c( &\u3001 | \u3001^\u3001~ \u3001>>\u3001 <<) \n \n \n \n \n \u7528\u4e8e\u63d0\u9ad8\u8fd0\u7b97\u901f\u5ea6\uff0c\u89c4\u907f\u7b97\u672f\u8fd0\u7b97\u7b26\u3002 \n \u5728\u4f4d\u64cd\u4f5c\u8fd0\u7b97\u4e2d\uff0c\u4e0d\u5e94\u8be5\u8bd5\u56fe\u8003\u8651\u5148\u5f97\u5230\u4e00\u4e2a\u6570\u7684\u4e8c\u8fdb\u5236\u7801\uff0c\u800c\u5e94\u8be5\u5c06\u8fd9\u4e2a\u6570\u770b\u4f5c\u662f\u4e00\u4e2a\u4e8c\u8fdb\u5236\u7801\uff0c\u4e8c\u8fdb\u5236\u8865\u7801\u4e0e\u6574\u6570\u4e4b\u95f4\u662f\u4e00\u4e00\u5bf9\u5e94\u7684\u3002\u8bda\u7136 Python\u8bed\u8a00\u4e2d\u6709\u5185\u7f6e\u51fd\u6570\u00a0 bin\u5c06\u4e00\u4e2a\u6574\u6570\u8f6c\u6362\u4e3a\u4e8c\u8fdb\u5236\uff0cPython\u4e2d\u4f7f\u7528\u8be5\u51fd\u6570\u8f6c\u6362\u4e3a\u8d1f\u6570\u5e76\u4e0d\u662f\u5176\u8865\u7801\u3002\u56e0\u6b64\u4e0d\u80fd\u5148\u5f97\u5230\u8be5\u6570\u7684\u4e8c\u8fdb\u5236\u7801\u3002\u540c\u65f6\u7ede\u5c3d\u8111\u6c41\u5f97\u5230\u4e00\u4e2a\u6570\u7684\u4e8c\u8fdb\u5236\u8865\u7801\u662f\u6ca1\u6709\u5fc5\u8981\u7684\u3002 \n &\uff1a\u6309\u4f4d\u4e0e\u64cd\u4f5c\uff0c\u53ea\u6709 1 &1 \u4e3a1\uff0c\u5176\u4ed6\u60c5\u51b5\u4e3a0\u3002 \u53ef\u7528\u4e8e\u8fdb\u4f4d\u8fd0\u7b97 \u3002 \n |\uff1a\u6309\u4f4d\u6216\u64cd\u4f5c\uff0c\u53ea\u6709 0|0\u4e3a0\uff0c\u5176\u4ed6\u60c5\u51b5\u4e3a1\u3002 \n ~\uff1a\u9010\u4f4d\u53d6\u53cd\u3002 \n ^\uff1a\u5f02\u6216\uff0c\u76f8\u540c\u4e3a0\uff0c\u76f8\u5f02\u4e3a1\u3002 \u53ef\u7528\u4e8e\u52a0\u64cd\u4f5c\uff08\u4e0d\u5305\u62ec\u8fdb\u4f4d\u9879\uff09\u3002 \n <<\uff1a\u5de6\u79fb\u64cd\u4f5c\uff0c2\u7684\u5e42\u76f8\u5173 \n >>\uff1a\u53f3\u79fb\u64cd\u4f5c\uff0c2\u7684\u5e42\u76f8\u5173 \n \u5224\u65ad\u4e00\u4e2a\u6574\u6570\u7684\u4e8c\u8fdb\u5236\u8865\u7801\u4e2d 1 \u7684\u4e2a\u6570\uff1a \n \u3000\u3000\u65e2\u7136\u4e00\u4e2a\u6574\u6570\u5728\u8ba1\u7b97\u673a\u5185\u90e8\u4e3a\u4e8c\u8fdb\u5236\u7684\u8865\u7801\uff0c\u90a3\u4e48\u76f4\u63a5\u5bf9\u6574\u6570\u8fdb\u884c\u4f4d\u64cd\u4f5c\u5373\u53ef\uff0c\u6ca1\u6709\u5fc5\u8981\u8fdb\u884c\u8fdb\u5236\u7684\u8f6c\u6362\u3002 \n \u3000\u30001\uff09\u5c06\u6574\u6570\u901a\u8fc7\u79fb\u4f4d\u5e76\u4e0e1\u8fdb\u884c\u4e0e\u64cd\u4f5c\uff0c\u5373\u53ef\u5224\u65ad\u5f53\u65f6\u672b\u5c3e\u662f\u5426\u4e3a1. \u4f46\u662f \u7531\u4e8e\u6574\u6570\u4ee5\u4e8c\u8fdb\u5236\u8865\u7801\u7684\u65b9\u5f0f\u5b58\u50a8\uff0c\u6b63\u6570\u53f3\u79fb\u4e0e\u8d1f\u6570\u53f3\u79fb\u5f97\u5230\u7684\u6548\u679c\u5e76\u4e0d\u76f8\u540c\uff0c\u8d1f\u6570\u5728\u53f3\u79fb\u8fc7\u7a0b\u4e2d\u4f1a\u81ea\u52a8\u8865 1 .\u7531\u4e8e\u5728c \u6216c++\u8fd9\u79cd\u8bed\u8a00\u4e2d\u6570\u636e\u7684\u7c7b\u578b\u8981\u5148\u58f0\u660e\uff0c\u5982 int\u4e3a32\u4f4d\uff0c\u800c\u5728python\u4e2d\uff0c\u7531\u4e8e\u52a8\u6001\u8bed\u8a00\u7684\u7279\u6027\uff0c\u6570\u636e\u7684\u4f4d\u6570\u7406\u60f3\u4e0a\u662f\u4e0d\u53d7\u9650\u5236\u7684\uff0c\u56e0\u6b64\u53ef\u901a\u8fc7 \u79fb\u4f4d\u7684\u6b21\u6570\u8fdb\u884c\u5224\u65ad\uff0cint\u578b\u6570\u636e\u79fb\u4f4d\u4e0d\u80fd\u8d85\u8fc732\u5373\u53ef\uff0c\u8fd9\u6837\u7a0b\u5e8f\u7684\u5faa\u73af\u6b21\u6570\u6052\u5b9a\u4e3a32. \n \n 1 class Solution:\n 2 def NumberOf1(self, n):\n 3 # write code here \n 4 m = 0\n 5 result = 0\n 6 \n 7 while m < 32 :\n 8 if n & 1 :\n 9 result += 1\n 10 n = n >> 1\n 11 m += 1\n 12 return result \n \n \u3000\u30002\uff09\uff1a\u540c\u6837\u7684\uff0c\u53ef\u4ee5\u901a\u8fc7\u5de6\u79fb 1 \u518d\u4e0e\u8f93\u5165\u503c\u8fdb\u884c\u4e0e\u64cd\u4f5c\u8fdb\u884c\u5224\u65ad\u3002 \u4ecd\u7136\u662f\u7531\u4e8ePython\u4e0d\u4f1a\u5b58\u5728\u6ea2\u51fa\u73b0\u8c61 \uff0c\u56e0\u6b64\u9700\u8981\u7528\u5230 \u6570\u636e\u7c7b\u578b\u7684\u4f4d\u6570\uff0c\u8fdb\u884c\u9650\u5236\u3002 \n \n 1 class Solution():\n 2 def getResult(self, n):\n 3 m = 1\n 4 result = 0\n 5 i = 0\n 6 \n 7 while i < 32 :\n 8 i += 1\n 9 if m& n :\n 10 result += 1\n 11 m = m << 1\n 12 return result \n \n \u3000\u30003\uff09\u4e00\u4e2a\u4e8c\u8fdb\u5236\u51cf\u4e00\u518d\u4e8e\u81ea\u8eab\u4e0e\u64cd\u4f5c\u80fd\u591f\u5c06\u6700\u540e\u4e00\u4f4d1\u7f6e\u96f6\u3002\u5982 1011-1 = 1010 1010&1011 = 1010\u30011010-1 = 1001 1001 & 1010 = 1000\u30011000-1 = 0111 0111 &1000 = 0000 \n \u4f46\u662f \uff0cPython\u4e0d\u4f1a\u6ea2\u51fa\uff0c\u56e0\u6b64\u8d1f\u6570\u5728\u8fdb\u884c\u6301\u7eed\u51cf\u4e00 \u7684\u8fd0\u7b97\uff0c \u82e5\u901a\u8fc7\u5f53\u524d\u503c\u662f\u5426\u4e3a0\u8fdb\u884c\u5faa\u73af\u7ec8\u6b62\u6761\u4ef6\u4f1a\u5bfc\u81f4\u6b7b\u5faa\u73af \u3002\u56e0\u6b64\u9700\u8981\u4e3a\u6b63\u8d1f\u6570\u5206\u522b\u8bbe\u5b9a\u7ec8\u6b62\u6761\u4ef6\u3002 \n \n 1 class Solution2():\n 2 def getResult(self, n):\n 3 result = 0\n 4 if n >= 0:\n 5 while n:\n 6 result += 1\n 7 n = (n - 1)& n\n 8 else :\n 9 while n >= -2147483648 :\n 10 result += 1\n 11 n = (n - 1)& n \n 12 return result \n \n \u3000\u30004\uff09\uff1a\u4e00\u4e2a\u8d1f\u6570\u5fc5\u7136\u5bf9\u5e94\u4e00\u4e2a\u6b63\u6570\uff0c\u5148\u901a\u8fc7\u52a0\u6cd5\u8fd0\u7b97\u5f97\u5230\u8d1f\u6570\u7684\u76f8\u53cd\u6570\uff0c\u8fd9\u6837\u53ef\u4ee5\u5c06\u8d1f\u6570\u5f53\u4f5c\u6b63\u6570\u8fdb\u884c\u64cd\u4f5c\uff0c\u552f\u4e00\u4e0d\u540c\u7684\u662f\uff0c\u8d1f\u6570\u7684\u7b26\u53f7\u4f4d\u8981\u591a\u4e00\u4e2a 1 \n \n 1 # -*- coding:utf-8 -*- \n 2 class Solution:\n 3 def NumberOf1(self, n):\n 4 # write code here \n 5 result = 0\n 6 \n 7 if n < 0:\n 8 n = n + (1<<31 )\n 9 result += 1\n 10 result += bin(n)[2:].count( ' 1 ' )\n 11 return result \n \n \u3000\u30005\uff09\uff1a\u7ed3\u5408\u65b9\u6cd5 4 \u4e0e 3\u4e4b\u524d\u7684\u65b9\u6cd5\uff1a \n \n 1 # -*- coding:utf-8 -*- \n 2 class Solution:\n 3 def NumberOf1(self, n):\n 4 # write code here \n 5 result = 0\n 6 \n 7 if n < 0:\n 8 n = n + (1<<31 )\n 9 result += 1\n 10 while n:\n 11 result += 1\n 12 n = n&(n-1 )\n 13 return result \n \n \u00a0 \n \n \n \n \n \n \n \n \n \n posted @ \n 2019-03-14 21:16 \u00a0\n \u5df2\u7ecf\u5d29\u76d8\u4e86 \u00a0\n \u9605\u8bfb( ... )\u00a0\n \u8bc4\u8bba( ... )\u00a0\n \u7f16\u8f91 \u00a0\n \u6536\u85cf \n \n end: topics \u6587\u7ae0\u3001\u8bc4\u8bba\u5bb9\u5668 \n \n \n \n \n \n \n \n \n \u5237\u65b0\u8bc4\u8bba \u5237\u65b0\u9875\u9762 \u8fd4\u56de\u9876\u90e8 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n end: forFlow \n end: mainContent \u4e3b\u4f53\u5185\u5bb9\u5bb9\u5668 \n \n \n \n \n \n \n \n \n \n \n end: sideBarMain \n end: sideBar \u4fa7\u8fb9\u680f\u5bb9\u5668 \n \n end: main \n \n \n done \n Copyright \u00a9 2020 \u5df2\u7ecf\u5d29\u76d8\u4e86\n Powered by .NET Core on Kubernetes \n end: footer \n end: home \u81ea\u5b9a\u4e49\u7684\u6700\u5927\u5bb9\u5668 \n ; \n\n\n```python\nimport requests\nfrom bs4 import BeautifulSoup\nheaders = {'User-Agent': \"Mozilla/5.0 (Windows NT 10.0; WOW64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/80.0.3970.5 Safari/537.36\"}\n\nurl = 'https://blog.csdn.net/Betty_betty_betty/article/details/80898798'\nres = requests.get(url)\nhtml_page = res.content\nsoup = BeautifulSoup(html_page, 'html.parser')\ntext = soup.find_all(text=True)\n\noutput = ''\nblacklist = [\n '[document]',\n 'noscript',\n 'header',\n 'html',\n 'meta',\n 'head', \n 'input',\n 'script',\n # there may be more elements you don't want, such as \"style\", etc.\n]\n\nfor t in text:\n if t.parent.name not in blacklist:\n output += '{} '.format(t)\n\nprint(output)\n```\n\n \n \u4f4d\u8fd0\u7b97\u7b26\u4e2d\u7684 (\u5de6\u79fb) (\u53f3\u79fb) (\u65e0\u7b26\u53f7\u53f3\u79fb)2_\u5927\u6570\u636e_Dimples.-CSDN\u535a\u5ba2 \n \n \n \n \u81ea\u5b9a\u4e49\u76ae\u80a4\u6837\u5f0f \n \n \n js\u5f15\u7528 \n \n \n \n .MathJax, .MathJax_Message, .MathJax_Preview{\n display: none\n }\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \u4f4d\u8fd0\u7b97\u7b26\u4e2d\u7684 (\u5de6\u79fb) (\u53f3\u79fb) (\u65e0\u7b26\u53f7\u53f3\u79fb)2 \n \n \n \n \u6587\u7ae0\u7c7b\u578b \n \u539f\u521b DimplesDimples. \n \u6700\u540e\u53d1\u5e03\u4e8e2018-07-03 15:05:49 \n \u9605\u8bfb\u6570 573 \n \n \n \n \n \u6536\u85cf\n \n \n \u53d1\u5e03\u4e8e2018-07-03 15:05:49 \n \n \n \u5206\u7c7b\u4e13\u680f\uff1a \n \n java \n \n \n \n \n \n \u7248\u6743\u58f0\u660e\uff1a\u672c\u6587\u4e3a\u535a\u4e3b\u539f\u521b\u6587\u7ae0\uff0c\u9075\u5faa CC 4.0 BY-SA \u7248\u6743\u534f\u8bae\uff0c\u8f6c\u8f7d\u8bf7\u9644\u4e0a\u539f\u6587\u51fa\u5904\u94fe\u63a5\u548c\u672c\u58f0\u660e\u3002 \n \n \u672c\u6587\u94fe\u63a5\uff1a https://blog.csdn.net/Betty_betty_betty/article/details/80898798 \n \n \n \n \n \n \u5c55\u5f00 \n \n \n \n \n \n python\u5b89\u88c5\u624b\u518c\u5f00\u59cb \n python\u5b89\u88c5\u624b\u518c\u7ed3\u675f \n ####\u4e13\u680f\u5e7f\u544a\u4f4d\u56fe\u6587\u5207\u6362\u5f00\u59cb \n ####\u4e13\u680f\u5e7f\u544a\u4f4d\u56fe\u6587\u5207\u6362\u7ed3\u675f \n \n \n \n \n \n <<:\u67d0\u6570\u636e\u5de6\u79fbn\u4f4d\u662f\u8be5\u6570\u636e\u4e58\u4ee52\u7684n\u6b21\u5e42\uff0c\u53ef\u4ee5\u5b8c\u62102\u7684\u6b21\u5e42\u8fd0\u7b97 \n >>:\u67d0\u6570\u636e\u53f3\u79fbn\u4f4d\u662f\u8be5\u6570\u636e\u9664\u4ee52\u7684n\u6b21\u5e42\uff0c\u5bf9\u4e8e\u9ad8\u4f4d\u51fa\u73b0\u7684\u7a7a\u4f4d\uff0c\u539f\u6765\u662f\u4ec0\u4e48\u5c31\u7528\u4ec0\u4e48\u6765\u8865\u5168\u3002 \n >>>:\u65e0\u7b26\u53f7\u53f3\u79fb\u662f\u6570\u636e\u8fdb\u884c\u53f3\u79fb\u65f6\uff0c\u9ad8\u4f4d\u51fa\u73b0\u7684\u7a7a\u4f4d\uff0c\u65e0\u8bba\u539f\u9ad8\u4f4d\u662f\u4ec0\u4e48\uff0c\u7a7a\u4f4d\u90fd\u75280\u8865\u3002 \n \n \n \n \n \n \n \n \u70b9\u8d5e \n 2 \n \n \n \n \u6536\u85cf \n \n \n \u5206\u4eab \n \u6253\u8d4f\u5f00\u59cb \n \u6253\u8d4f\u7ed3\u675f \n \n \n \n \n \n \u6587\u7ae0\u4e3e\u62a5 \n \n \n \n \n \n \n \n \n \n \n \n DimplesDimples. \n \n \u53d1\u5e03\u4e86127 \u7bc7\u539f\u521b\u6587\u7ae0 \u00b7 \u83b7\u8d5e 18 \u00b7 \u8bbf\u95ee\u91cf 4\u4e07+ \n \n \n \u79c1\u4fe1\n \n \u5173\u6ce8 \n \n \n \n \n \n \n \n \u5c55\u5f00\u9605\u8bfb\u5168\u6587\n \n \n \n \n \n \n \n \u591a\u6761\u5e7f\u544a\u5982\u4e0b\u811a\u672c\u53ea\u9700\u5f15\u5165\u4e00\u6b21 \n \n \n \n \n \n \n \n \n \n \n \n d-flex \n \n \n \n \u53d1\u8868\u8bc4\u8bba \n \n \u6dfb\u52a0\u4ee3\u7801\u7247 \n \n \n \n \n \n HTML/XML \n objective-c \n Ruby \n PHP \n C \n C++ \n JavaScript \n Python \n Java \n CSS \n SQL \n \u5176\u5b83 \n \n \n \u8fd8\u80fd\u8f93\u5165 1000 \u4e2a\u5b57\u7b26 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \t\t\t\tjava \u65e0\u7b26\u53f7\u53f3\u79fb>>> \u6709\u7b26\u53f7\u53f3\u79fb>>\t\t \n \n \n 10-13 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t2585 \n \n \n \n \n \n \u6709\u7b26\u53f7\u53f3\u79fb&amp;gt;&amp;gt;\u6709\u7b26\u53f7\u53f3\u79fb\u5c31\u662f\u53f3\u79fb\u4e4b\u540e\uff0c\u5de6\u8fb9\u7684\u8865\u4e0a\u7b26\u53f7\u4f4d\uff0c\u6b63\u6570\u88650\uff0c\u8d1f\u6570\u88651\u65e0\u7b26\u53f7\u53f3\u79fb&amp;gt;&amp... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t sunshine \n \n \n \n \n \n \n \n \n \t\t\t\t\u4f4d\u64cd\u4f5c\uff08\u5de6\u79fb\u548c\u53f3\u79fb\uff09\t\t \n \n \n 08-10 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t9910 \n \n \n \n \n \n \u4f4d\u64cd\u4f5c\u63d0\u9ad8\u7a0b\u5e8f\u8fd0\u884c\u6548\u7387 \u51cf\u5c11\u9664\u6cd5\u548c\u53d6\u6a21\u7684\u8fd0\u7b97\u3002\u5728\u8ba1\u7b97\u673a\u7a0b\u5e8f\u4e2d\u6570\u636e\u7684\u4f4d\u662f\u53ef\u4ee5\u64cd\u4f5c\u7684\u6700\u5c0f\u6570\u636e\u5355\u4f4d\uff0c\u7406\u8bba\u4e0a\u53ef\u4ee5\u7528\u201d\u4f4d\u8fd0\u7b97\u201d\u6765\u5b8c\u6210\u6240\u6709\u7684\u8fd0\u7b97\u548c\u64cd\u4f5c\u3002\u5de6\u79fb\uff0c\u540e\u7a7a\u7f3a\u81ea\u52a8\u88650\uff1b\u53f3\u79fb\uff0c\u5206\u4e3a\u903b\u8f91\u53f3\u79fb\u548c\u7b97\u6570\u53f3\u79fb1\uff09\u903b\u8f91\u53f3... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t \u535a\u5ba2 \n \n \n \n \n \n \n \n \n \t\t\t\t\u5173\u4e8e\uff08\u4e8c\u8fdb\u5236\uff09\u4f4d\u79fb\u8fd0\u7b97\uff1a\u5e26\u7b26\u53f7\u53f3\u79fb\u4f4d\uff1b\u5e26\u7b26\u53f7\u5de6\u79fb\u4f4d\uff1b\u65e0\u7b26\u53f7\u53f3\u79fb\u4f4d\t\t \n \n \n 06-04 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t3682 \n \n \n \n \n \n >> \u5e26\u7b26\u53f7\u53f3\u79fb\u4f4d\u5206\u6790\uff1anum>> n\uff081\uff09\u76f8\u5f53\u4e8enum\uff08num\u7684\u4e8c\u8fdb\u5236\uff09\u5411\u53f3\u79fb\u52a8n\u4f4d\u3002\uff082\uff09\u6b63\u6570\u79fb\u4f4d\uff1a\u53f3\u8fb9\uff08\u4f4e\u4f4d\uff09\u79fb\u51fa\u90e8\u5206\uff0c\u76f4\u63a5\u820d\u5f03\uff0c\u5de6\u8fb9\uff08\u9ad8... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t weixin_42315600\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \n \t\t\t\t\u5728java\u4e2d\u65e0\u7b26\u53f7\u6570-2\uff0c\u8fdb\u884c\u5411\u53f3\u79fb\u52a83\u4f4d\uff0c\u7ed3\u679c\u4e3a\u4ec0\u4e48\u662f8\uff1f(\u4e5f\u5c31\u662fa=-2\uff0ca>>>=8)\t\t \n \n \n 03-13 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t405 \n \n \n \n \n \n \u8fd9\u91cc\u5199\u81ea\u5b9a\u4e49\u76ee\u5f55\u6807\u9898\u6b22\u8fce\u4f7f\u7528Markdown\u7f16\u8f91\u5668\u65b0\u7684\u6539\u53d8\u529f\u80fd\u5feb\u6377\u952e\u5408\u7406\u7684\u521b\u5efa\u6807\u9898\uff0c\u6709\u52a9\u4e8e\u76ee\u5f55\u7684\u751f\u6210\u5982\u4f55\u6539\u53d8\u6587\u672c\u7684\u6837\u5f0f\u63d2\u5165\u94fe\u63a5\u4e0e\u56fe\u7247\u5982\u4f55\u63d2\u5165\u4e00\u6bb5\u6f02\u4eae\u7684\u4ee3\u7801\u7247\u751f\u6210\u4e00\u4e2a\u9002\u5408\u4f60\u7684\u5217\u8868\u521b\u5efa\u4e00\u4e2a\u8868\u683c\u8bbe\u5b9a\u5185\u5bb9\u5c45\u4e2d... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t L_world_\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \n \n \n \n \t\t\t\tJava_\u4f4d\u8fd0\u7b97\u7b26_\u5de6\u79fb\u53f3\u79fb\t\t \n \n \n 05-22 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t193 \n \n \n \n \n \n \u5de6\u79fb\u548c\u53f3\u79fb\u90fd\u662f\u5bf9\u6574\u6570\u7684\u8865\u7801\u8fdb\u884c\u8fd0\u7b97\u7684\u5de6\u79fb\u5c31\u662f\u8865\u7801\u5411\u5de6\u79fb\u4e24\u4f4d\uff0c\u6bcf\u5de6\u79fb\u4e00\u4f4d\u5c31\u76f8\u5f53\u4e8e\u8865\u7801\u6240\u5bf9\u5e94\u7684\u5341\u8fdb\u5236\u7684\u6574\u6570 *2\u53f3\u79fb\u5c31\u662f\u8865\u7801\u5411\u53f3\u79fb\u4e24\u4f4d\uff0c\u6bcf\u53f3\u79fb\u4e00\u4f4d\u5c31\u76f8\u5f53\u4e8e\u8865\u7801\u6240\u5bf9\u5e94\u7684\u5341\u8fdb\u5236\u7684\u6574\u6570 \\261\u8865\u7801\uff1a00..0... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t Song_MJ\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \n \t\t\t\t\u5173\u4e8e\u8d1f\u6570\u7684\u53f3\u79fb>>\u4e0e\u65e0\u7b26\u53f7\u53f3\u79fb>>>\u8fd0\u7b97\u5c0f\u7ed3\t\t \n \n \n 03-10 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t1474 \n \n \n \n \n \n       \u5bf9\u4e8e\u5e26\u7b26\u53f7\u53f3\u79fb,\u82e5\u4e3a\u8d1f\u6570,\u5219\u5728\u5b58\u50a8\u65f6\u9996\u4f4d\u8868\u793a\u7b26\u53f7\u4f4d,\u5176\u503c\u4e3a1... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t \u738b\u660c\u65f6\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \n \t\t\t\tpython \u6ca1\u6709\u65e0\u7b26\u53f7\u53f3\u79fb\uff0c\u5de6\u79fb\t\t \n \n \n 02-27 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t721 \n \n \n \n \n \n python\u6ca1\u6709>>>\u65e0\u9644\u540e\u53f3\u79fb \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t weixin_44088837\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \t\t\t\t\u65e0\u7b26\u53f7\u53f3\u79fb\u95ee\u9898\t\t \n \n 07-21 \n \n \n >>>\u4f5c\u4e3a\u65e0\u7b26\u53f7\u53f3\u79fb\uff0c\u53ef\u4ee5\u4f7f\u7b26\u53f7\u53f3\u79fb\uff0c\u4f46\u662f\u6211\u8fd0\u7b97\u65f6\u5374\u53d1\u73b0\u4e00\u4e2a\u602a\u5f02\u7684\u95ee\u9898 byte n = (byte)((byte)0x80>>>4); \u7ed3\u679c\u662f-8 \u5982\u679c\u6539\u6210byte n = (byte)(0x80> \n \u8bba\u575b \n \n \n \n \n \n \n \t\t\t\tpython \u600e\u6837\u5b9e\u73b0\u65e0\u7b26\u53f7\u53f3\u79fb\uff1f \t\t \n \n 03-05 \n \n \n java \u91cc\u7684\u65e0\u7b26\u53f7\u53f3\u79fb\u662f >>> python\u7684\u65e0\u7b26\u53f7\u53f3\u79fb\u600e\u6837\u5b9e\u73b0\uff1f\uff1f \u6c42\u89e3\uff01 \n \u8bba\u575b \n \n \n \n \n \u591a\u6761\u5e7f\u544a\u5982\u4e0b\u811a\u672c\u53ea\u9700\u5f15\u5165\u4e00\u6b21 \n \n \n \n \n \n \t\t\t\t\u79fb\u4f4d\uff08\u5de6\u79fb\uff0c\u53f3\u79fb\u548c\u65e0\u7b26\u53f7\u53f3\u79fb\uff09\t\t \n \n \n 04-04 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t1948 \n \n \n \n \n \n public class Shift {\u00a0public static void main(String[] args) {\u00a0\u00a0int x = -50;\u00a0\u00a0int a = 50;\u00a0\u00a0int b = 5 ... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t yshunb\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \n \t\t\t\tC++\u4e2d\u5b9e\u73b0\u65e0\u7b26\u53f7\u53f3\u79fb\t\t \n \n \n 09-06 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t1373 \n \n \n \n \n \n \u65e0\u7b26\u53f7\u53f3\u79fb\u7684\u610f\u601d\u662f\u53f3\u79fb\u65f6\u4e0d\u8003\u8651\u7b26\u53f7\u95ee\u9898\uff0c\u5373\u65e0\u8bba\u53f3\u79fb\u6b63\u6570\u8fd8\u662f\u8d1f\u6570\uff0c\u5176\u6700\u9ad8\u4f4d\u90fd\u662f\u88650\u3002\u4e0d\u540c\u4e8e\u666e\u901a\u7684\u201c>>\u201d\u79fb\u4f4d\u64cd\u4f5c\uff0c\u4f7f\u7528\u201c>>\u201d\u53f3\u79fb\u65f6\uff0c\u662f\u4ee5\u7b26\u53f7\u6269\u5c55\u539f\u5219\u8fdb\u884c\u53f3\u79fb\uff0c\u5373\u5728\u53f3\u79fb\u8fc7\u7a0b\u4e2d\u5b83\u5c06\u4fdd\u6301\u539f\u6709\u6570\u636e\u7684\u6b63\u8d1f\u53f7\u4e0d\u53d8... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t \u51b0\u6b87\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \t\t\t\tJava \u4f4d\u8fd0\u7b97\u7b26\uff1a\u5de6\u79fb\u3001\u53f3\u79fb\u3001\u65e0\u7b26\u53f7\u53f3\u79fb\t\t \n \n \n 09-17 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t702 \n \n \n \n \n \n \u6982\u8ff0\u4e0a\u4e00\u7bc7\u6587\u7ae0\u6211\u4eec\u8bf4\u5230\u4e86Java\u8fd0\u7b97\u7b26\u7684\u4f18\u5148\u7ea7\u7684\u95ee\u9898\uff0c\u4e5f\u7ed9\u5927\u5bb6\u63d0\u4f9b\u4e86\u4e00\u4e2a\u7b80\u5355\u6613\u8bb0\u7684\u53e3\u8bc0\uff0c\u90a3\u4e48\u8fd9\u4e00\u7bc7\u535a\u6587\uff0c\u6211\u4eec\u5c31\u6765\u770b\u4e00\u770bJava\u8fd0\u7b97\u7b26\u4e2d\u7684\u4f4d\u8fd0\u7b97\u7b26\uff1a\u5de6\u79fb\uff0c\u53f3\u79fb\uff0c\u65e0\u7b26\u53f7\u53f3\u79fb\u3002\u7a0b\u5e8f\u4e2d\u7684\u6240\u6709\u6570\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u90fd... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t \u67a3\u9762\u5305\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \n \t\t\t\t\u4f4d\u8fd0\u7b97\u7b26\u4e4b---\u5de6\u79fb\u53f3\u79fb\u8fd0\u7b97\u7b26(\u7b80\u5355\u6613\u61c2)\t\t \n \n \n 08-07 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t3218 \n \n \n \n \n \n \u524d\u8a00:\u4f4d\u8fd0\u7b97\u7b26\u662f\u7528\u6765\u5bf9\u4e8c\u8fdb\u5236\u4f4d\u8fdb\u884c\u64cd\u4f5c\u7684c\u8bed\u8a00\u4e2d\u67096\u79cd\u4f4d\u8fd0\u7b97\u7b26:&\u6309\u4f4d\u4e0e|\u6309\u4f4d\u6216^\u6309\u4f4d\u5f02\u6216~\u53d6\u53cd<<\u5de6\u79fb>>\u53f3\u79fb\u672c\u7bc7\u6587\u7ae0\u6211\u4eec\u53ea\u8bb2\u5de6\u79fb\u548c\u53f3\u79fb\u8fd0\u7b97\u7b26,\u5176\u4ed6\u7684\u4f1a\u9646\u7eed\u5728\u5176\u4ed6\u7bc7\u8bb2... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t \u6c88\u660e\u4e00\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \n \t\t\t\t\u6309\u4f4d\u4e0e\u8fd0\u7b97\u7b26&\u4e0e\u53f3\u79fb\u8fd0\u7b97\u7b26>>\t\t \n \n \n 04-25 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t341 \n \n \n \n \n \n \u6765\u81ea\uff1ahttps://zhidao.baidu.com/question/146401227.html&\u8868\u793a\u4e24\u79cd\u8fd0\u7b97\u7b26\uff0c\u5176\u4e2d\u4e00\u79cd\u8868\u793a\u53d6\u503c\u8fd0\u7b97\u7b26\uff0c\u4e00\u79cd\u662f\u6309\u4f4d\u4e0e\u53d6\u503c\u8fd0\u7b97\u7b26int a=1;int *... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t luo_yu_1106\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \n \n \n \n \t\t\t\tJava\u5b66\u4e60\u624b\u518c\uff1a\u65e0\u7b26\u53f7\u5de6\u79fb&\u65e0\u7b26\u53f7\u53f3\u79fb&\u6709\u7b26\u53f7\u53f3\u79fb\t\t \n \n \n 04-12 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t683 \n \n \n \n \n \n \u4e00\u3001\u65e0\u7b26\u53f7\u53f3\u79fb\u201c>>\u201d\u4e0e\u6709\u7b26\u53f7\u53f3\u79fb\u201c>>>\u201dJava\u63d0\u4f9b\u4e86\u4e24\u79cd\u53f3\u79fb\u8fd0\u7b97\u7b26\uff1a\u201c>>\u201d\u548c\">>>\"\u3002\u5176\u4e2d\uff0c\u201c>>\u201d\u88ab\u79f0\u4e3a\u6709\u7b26\u53f7\u53f3\u79fb\u8fd0... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t \u6d69\u6bd4\u7684\u4e13\u680f \n \n \n \n \n \n \n \n \n \t\t\t\tjavaSE (\u4e8c\uff09\u903b\u8f91/\u4f4d\u8fd0\u7b97\u7b26\u3001\u5de6\u79fb\u53f3\u79fb\u8fd0\u7b97\u7b26\u3001switch\u8bed\u53e5\t\t \n \n \n 11-25 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t360 \n \n \n \n \n \n \u903b\u8f91\u8fd0\u7b97\u7b26 &amp;amp;amp;amp;amp; | ^ !&amp;amp;amp;amp;amp;\uff1a \u4e0e \u6709false\u5219false| \uff1a \u6216 ... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t PizAn\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \n \t\t\t\t\u4f4d\u8fd0\u7b97\u4ee5\u53ca\u4f4d\u7684\u5de6\u79fb\u548c\u53f3\u79fb\t\t \n \n \n 03-29 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t442 \n \n \n \n \n \n 1\u3001\u5de6\u79fb\uff1am<>\u548c\u65e0\u7b26\u53f7\u53f3\u79fb\u8fd0\u7b97\u7b26>>>\u7684\u533a\u522b\t\t \n \n \n 02-07 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t1\u4e07+ \n \n \n \n \n \n \u5728\u505a\u4e00\u4e2apcm\u97f3\u9891\u65f6\u9047\u5230\u4e86\u8fd9\u4e2a\u7b26\uff0c\u4f46\u662f\u6211\u770b\u6709\u4e9b\u767e\u5ea6\u7684\u5730\u65b9\u7684\u89e3\u91ca\u90fd\u4e0d\u80fd\u4ee4\u6211\u5f88\u61c2\uff0c\u6240\u4ee5\u5c31\u6574\u7406\u4e0b\u3002-----------------------------------------------------\u66f4... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t cobbwho\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \n \t\t\t\t\u7b97\u6cd5\uff08\u4e00\uff09\uff1a\u5982\u4f55\u9ad8\u6548\u7684\u7b97\u51fa2*8\u7684\u503c\uff0c\u4f4d\u79fb\u7b97\u6cd5\u539f\u7406\u89e3\u91ca\uff0c\u4e3a\u4ec0\u4e488\u5de6\u79fb1\u4f4d\uff0c4\u5de6\u79fb2\u4f4d\uff0c2\u5de6\u79fb3\u4f4d\uff0c1\u5de6\u79fb4\u4f4d\u7684\u7ed3\u679c\u4e3a16\t\t \n \n \n 03-31 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t2\u4e07+ \n \n \n \n \n \n \u4f4d\u79fb\u7b97\u6cd5\uff0c\u5982\u4f55\u9ad8\u6548\u7684\u7b97\u51fa2*8\u7684\u503c\uff0c\u4e3a\u4ec0\u4e488&amp;amp;lt;&amp;amp;lt;1\uff0c4&amp;amp;lt;&amp;am... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t seesun2012\u7684\u4e13\u680f \n \n \n \n \n \n \n \n \n \t\t\t\t\u65e0\u7b26\u53f7\u53f3\u79fb\u8fd0\u7b97\u7b26 (>>>)\t\t \n \n \n 08-07 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t958 \n \n \n \n \n \n \u8f6c\u8f7d\u81ea\uff1ahttp://hi.baidu.com/wlred/item/68abdeebf0910d2c5a2d64cc\u65e0\u7b26\u53f7\u53f3\u79fb\u8fd0\u7b97\u7b26 (>>>)\u662f\u53f3\u79fb\u8868\u8fbe\u5f0f\u7684\u4f4d\uff0c\u4e0d\u4fdd\u7559\u7b26\u53f7\u3002\u4f7f\u7528\u793a\u4f8bresult =... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t hugoshine\u7684\u4e13\u680f \n \n \n \n \n \n \n \n \n \t\t\t\t1.\u53cc\u4e0e(\u6216)\u548c\u5355\u4e0e(\u6216)\u7684\u533a\u522b2.\u5de6\u79fb\u53f3\u79fb\u65e0\u7b26\u53f7\u53f3\u79fb3.\u5b57\u7b26\u4e32\u7684\u5e94\u75284.\u7c7b\u578b\u5f3a\u5236\u8f6c\u6362\t\t \n \n \n 09-28 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t197 \n \n \n \n \n \n 1.\u7c7b\u578b\u5f3a\u5236\u8f6c\u6362 \u00a0 byte y=2; \u00a0int x=3; \u00a0 y=(byte)(x+y);\u00a0 byte a=2;\u00a0 \u00a0a+=3; \u5bf9(\u505a\u4e86\u5f3a\u5236\u8f6c\u6362\u52a8\u4f5c) \u00a0a=a+3; \u9519(\u7cbe\u5ea6\u4e22\u5931 byte\u4e0d\u53ef... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t insiston0635\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \n \t\t\t\tC\u8bed\u8a00\u4e2d\u7684\u5de6\u79fb\u4e0e\u53f3\u79fb\t\t \n \n \n 12-20 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t26 \n \n \n \n \n \n C\u8bed\u8a00\u4e2d\u7684\u5de6\u79fb\u4e0e\u53f3\u79fb\u5148\u8bf4\u5de6\u79fb,\u5de6\u79fb\u5c31\u662f\u628a\u4e00\u4e2a\u6570\u7684\u6240\u6709\u4f4d\u90fd\u5411\u5de6\u79fb\u52a8\u82e5\u5e72\u4f4d,\u5728C\u4e2d\u7528<<\u8fd0\u7b97\u7b26.\u4f8b\u5982:int i = 1;i = i << 2; //\u628ai\u91cc\u7684\u503c\u5de6\u79fb2\u4f4d\u4e5f\u5c31\u662f\u8bf4,1... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t weixin_30414245\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \n \t\t\t\tC/C++\u4e2d\u7684\u6709\u7b26\u53f7\u548c\u65e0\u7b26\u53f7\u7684\u4f4d\u8fd0\u7b97\u95ee\u9898\t\t \n \n \n 08-11 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t413 \n \n \n \n \n \n \u662f\u5bf9C/C++\u4e0b\u9762\u7684\u7ed3\u679c\u611f\u5230\u56f0\u60d1\uff1fprintf(\"%d\\n\", ((unsigned char)~0 >> 1)); //\u7ed3\u679c\u4e3a127printf(\"%d\\n\", -((unsigned char)~... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t \u767e\u4ebf\u4e91 \n \n \n \n \n \n \n \n \n \t\t\t\t2018-1-20\uff1a\u5de6\u79fb<<\u548c\u53f3\u79fb>>\u8fd0\u7b97\u4ee5\u53ca\u548c>>>\uff08\u65e0\u7b26\u53f7\u53f3\u79fb\uff09\u7684\u533a\u522b\u4ee5\u53ca\u4f4d\u8fd0\u7b97\u7684\u5c0f\u9898\u76ee...\t\t \n \n \n 01-20 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t8 \n \n \n \n \n \n \u9996\u5148\uff1a\u5de6\u79fb\u548c\u53f3\u79fb\u4ee5\u53ca\u65e0\u7b26\u53f7\u53f3\u79fb\u90fd\u662f\u5e94\u7528\u4e8e2\u7684\u6b21\u5e42\u8fd0\u7b97\u5de6\u79fb\u51e0\u4f4d\u5c31\u76f8\u5f53\u4e8e\u8be5\u6570\u636e\u4e58\u4ee52\u7684\u51e0\u6b21\u65b9\uff1a\u5373\u5b8c\u62102\u7684\u6b21\u5e42\u8fd0\u7b97\u53f3\u79fb\u51e0\u4f4d\u5c31\u76f8\u5f53\u4e8e\u8be5\u6570\u636e\u9664\u4ee52\u7684\u51e0\u6b21\u65b9\uff1a\u5373\u5b8c\u62102\u7684\u6b21\u5e42\u8fd0\u7b97\uff0c\u53f3\u79fb\u6709\u4e2a\u5c0f\u7279\u70b9\uff1a\u53f3\u79fb\u4e4b\u540e\u7a7a\u51fa\u7684\u6700\u9ad8... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t weixin_33766805\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \n \t\t\t\t\u65e0\u7b26\u53f7\u53f3\u79fb\u8fd0\u7b97\t\t \n \n \n 07-30 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t27 \n \n \n \n \n \n 1.\u975e\u6570\u503c\u8f6c\u6362\u621002.\u6240\u4ee5\u5927\u4e8e\u7b49\u4e8e0\u7684\u6570\u53d6\u6574\u6570\u90e8\u5206\"a\" >>> 0012.34 >>> 012\u8f6c\u8f7d\u4e8e:https://www.cnblogs.com/bibiaf... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t diancui3160\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \n \t\t\t\t\u4f4d\u8fd0\u7b973\t\t \n \n \n 03-26 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t160 \n \n \n \n \n \n /* \u5de6\u8fb9\u6700\u9ad8\u4f4d\u4e22\u5f03\uff0c\u53f3\u8fb9\u8865\u9f500>>:\u53f3\u79fb \u6700\u9ad8\u4f4d\u662f0\uff0c\u5de6\u8fb9\u8865\u9f500\uff1b\u6700\u9ad8\u4e3a\u662f1\uff0c\u5de6\u8fb9\u8865\u9f501>>>:\u65e0\u7b26\u53f7\u53f3\u79fb \u65e0\u8bba\u6700\u9ad8\u4f4d\u662f0\u8fd8\u662f1\uff0c\u5de6\u8fb9\u8865\u9f500\u9762\u8bd5\u9898\uff1a\u8bf7\u7528\u6700\u6709\u6548\u7387\u7684\u65b9\u5f0f\u5199\u51fa\u8ba1\u7b972\u4e58\u4ee58\u7684\u7ed3\u679c?2 * ... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t \u5348\u591c\u7684\u884c\u4eba \n \n \n \n \n \n \n \n \n \t\t\t\t\u5de6\u79fb\u53f3\u79fb\u5c0f\u603b\u7ed3\t\t \n \n \n 12-11 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t1\u4e07+ \n \n \n \n \n \n \u5de6\u79fb\u53f3\u79fb\u5c0f\u603b\u7ed3@(\u7ec4\u6210\u539f\u7406)\u5173\u4e8e\u6570\u7684\u79fb\u4f4d\uff0c\u7279\u522b\u9700\u8981\u6ce8\u610f\u6b63\u6570\uff0c\u4e09\u7801\u76f8\u540c\uff0c\u6240\u4ee5\u65e0\u8bba\u5de6\u79fb\u8fd8\u662f\u53f3\u79fb\u90fd\u662f\u88650.\u800c\u8d1f\u6570\u7684\u8865\u7801\u5c31\u9700\u8981\u6ce8\u610f\uff0c\u5de6\u79fb\u5728\u53f3\u8fb9\u88650\uff0c\u53f3\u79fb\u9700\u8981\u5728\u5de6\u8fb9\u88651\uff0c\u6709\u4e00\u4e2a\u5f88\u6709\u8da3\u7684\u8bef\u533a\u662f\uff0c\u8ba4\u4e3a\u7b26\u53f7\u4f4d\u4fdd\u6301\u4e0d\u53d8... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t Bing's Blog \n \n \n \n \n \n \n \n \n \t\t\t\tC/C++\u4e2d\u7684\u903b\u8f91\u53f3\u79fb\u3001\u7b97\u6570\u53f3\u79fb\u3001\u5faa\u73af\u5de6\u79fb\u3001\u5faa\u73af\u53f3\u79fb\t\t \n \n \n 10-08 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t1\u4e07+ \n \n \n \n \n \n C/C++\u8bed\u8a00\u4e2d\u903b\u8f91\u53f3\u79fb\u548c\u7b97\u6570\u53f3\u79fb\u5171\u4eab\u540c\u4e00\u4e2a\u8fd0\u7b97\u7b26>>\uff0c\u7f16\u8bd1\u5668\u51b3\u5b9a\u4f7f\u7528\u903b\u8f91\u53f3\u79fb\u8fd8\u662f\u7b97\u6570\u53f3\u79fb\uff0c\u6839\u636e\u7684\u662f\u8fd0\u7b97\u6570\u7684\u7c7b\u578b\u3002\u5982\u679c\u662funsigned\u5219\u91c7\u7528\u903b\u8f91\u53f3\u79fb\uff0c\u5982\u679c\u662fsigned\u5219\u91c7\u7528\u7b97\u6570\u53f3\u79fb\u3002\u5bf9\u4e8esigne... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t huizhang0110 \n \n \n \n \n \n \n \n \n \t\t\t\tC\u8bed\u8a00\u7684\u903b\u8f91\u53f3\u79fb\u548c\u7b97\u672f\u53f3\u79fb\t\t \n \n \n 03-11 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t5501 \n \n \n \n \n \n \u9996\u5148\u8bf4\u660e\u4e00\u4e0b\u8fd9\u4e24\u4e2a\u6982\u5ff5\uff1a\u903b\u8f91\u53f3\u79fb\uff1a\u53f3\u79fb\u540e\u5de6\u8fb9\u6dfb\u52a00\u7b97\u672f\u53f3\u79fb\uff1a\u53f3\u79fb\u540e\u6dfb\u52a0\u7684\u4f4d\u4e0e\u539f\u6570\u7684\u7b26\u53f7\u4f4d\u76f8\u540c\u5728C\u8bed\u8a00\u4e2d\uff0c\u5bf9\u4e8e\u79fb\u4f4d\u64cd\u4f5c\u6267\u884c\u7684\u662f\u903b\u8f91\u5de6\u79fb\u548c\u7b97\u672f\u53f3\u79fb\uff0c\u4e0d\u8fc7\u5bf9\u4e8e\u65e0\u7b26\u53f7\u7c7b\u578b\uff0c\u6240\u6709\u7684\u79fb\u4f4d\u64cd\u4f5c\u90fd\u662f\u903b\u8f91\u7684\u3002\u6240\u4ee5\u8981\u76f8\u5bf9... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t \u8427\u5341\u4e00\u7684\u6280\u672f\u5c0f\u7ad9 \n \n \n \n \n \n \n \n \n \t\t\t\t\u5173\u4e8ejava\u4e2d\u4f4d\u8fd0\u7b97\u7684\u5de6\u79fb\u3001\u53f3\u79fb\u3001\u65e0\u7b26\u53f7\u53f3\u79fb\t\t \n \n \n 12-23 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t7934 \n \n \n \n \n \n \u4eca\u5929\u5b66\u5230\u4e86java\u4e2d\u7684\u4f4d\u8fd0\u7b97\uff0c\u548cC\u8bed\u8a00\u8fd8\u662f\u6709\u6240\u4e0d\u540c\u7684\uff1a>>:\u53f3\u79fb\u8fd0\u7b97\u7b26\uff0cnum>>1\uff0c\u76f8\u5f53\u4e8enum*2\uff1b>>>:\u65e0\u7b26\u53f7\u53f3\u79fb\uff0c\u5ffd\u7565\u7b26\u53f7\u4f4d\uff0c\u7a7a\u4f4d\u4ee50\u8865\u9f50\u3002\u4f4d\u8fd0\u7b97\u7684\u4e00\u4e9b\u89c4\u5219\uff1a1.byte\u3001short\u3001cha... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t \u5149\u8000\u5929\u4e0b \n \n \n \n \n \n \n \n \n \t\t\t\t\u6709\u7b26\u53f7\u53f3\u79fb>>\uff0c\u65e0\u7b26\u53f7\u53f3\u79fb>>>\t\t \n \n \n 02-27 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t9025 \n \n \n \n \n \n \u8ba1\u7b97\u673a\u8868\u793a\u6570\u5b57\u6b63\u8d1f\u4e0d\u662f\u7528+-\u52a0\u51cf\u53f7\u6765\u8868\u793a\uff0c\u800c\u662f\u7528\u6700\u9ad8\u4f4d\u6570\u5b57\u6765\u8868\u793a\uff0c0\u8868\u793a\u6b63\uff0c1\u8868\u793a\u8d1f1.\u6709\u7b26\u53f7\u53f3\u79fb>>\uff08\u82e5\u6b63\u6570,\u9ad8\u4f4d\u88650,\u8d1f\u6570,\u9ad8\u4f4d\u88651\uff09\u6b63\u6570:\u4f8b\u59824>&am... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t BushRo \n \n \n \n \n \n \n \n \n \t\t\t\t\u5206\u6790\u8f6e\u5b50\uff08\u4e8c\uff09- << \uff0c>>\uff0c>> \uff08\u5de6\u79fb\u3001\u53f3\u79fb\u3001\u65e0\u7b26\u53f7\u53f3\u79fb\uff09\t\t \n \n \n 09-06 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t79 \n \n \n \n \n \n \u524d\u8a00\uff1a\u5199\u00a0\u5206\u6790\u8f6e\u5b50\uff08\u4e00\uff09-ArrayList.java\u00a0\u7684\u65f6\u5019\u770b\u5230\u6e90\u7801\u4e2d\u6709\u00a0int newCapacity = oldCapacity + (oldCapacity >> ... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t weixin_33805992\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \t\t\t\t\u4e3a\u4ec0\u4e48\u8981\u53f3\u79fb8\u4f4d >>8\t\t \n \n 04-12 \n \n \n byte b2_groupno = new byte; int groupno = ((int)b2_groupno & 0xFF)|(((int)b2_groupno & 0xFF)<<8); b2 \n \u8bba\u575b \n \n \n \n \n \n \n \n \t\t\t\t\u6309\u4f4d\u5de6\u53f3\u79fb\u4f4d\u8fd0\u7b97\u7b26\t\t \n \n \n 05-11 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t568 \n \n \n \n \n \n \u6309\u4f4d\u5de6\u53f3\u79fb\u4f4d\u8fd0\u7b97\u7b26>> \u4eca\u5929\u5728\u7fa4\u91cc\u603b\u7b97\u957f\u89c1\u8bc6\u4e86\u3002\u3002\u867d\u7136\u4e0d\u5e38\u7528\u8bb0\u4e0b\u6765...>> 1 = / 2>> 2 = / 4>> 3 = / 8>> 4 = / 16...\u5931\u5fc6\u7684\u732a\u732a(3224897) 17:04... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t xiaoj13\u7684\u4e13\u680f \n \n \n \n \n \n \n \n \n \t\t\t\t\u4f4d\u8fd0\u7b97\u7b26\u5de6\u79fb\u53f3\u79fb\u7684\u7279\u70b9\t\t \n \n \n 10-16 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t468 \n \n \n \n \n \n /*\u00a0* \u4f4d\u8fd0\u7b97\u7b26:\u00a0* \u00a0* \u00a0* >>:\u53f3\u79fb,\u6700\u9ad8\u7b26\u53f7\u4f4d\u5982\u679c\u662f1,\u5de6\u8fb9\u8865\u9f501,\u6700\u9ad8\u7b26\u53f7\u4f4d\u5982\u679c\u662f0,\u5de6\u8fb9\u8865\u9f500\u00a0* //>>\u53f3\u79fb\u52a8\u7279\u70b9:\u8be5\u7b26\u53f7\u5de6\u8fb9\u7684\u6570\u636e\u9664\u4ee52\u7684\u79fb\u52a8\u6b21\u5e42:24/2^2 = 6\u00a0* >... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t qq_39345059\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \n \t\t\t\tjava \u4f4d\u8fd0\u7b97\u7b26\uff08\u5de6\u79fb\u4e0e\u53f3\u79fb\uff09\t\t \n \n \n 01-19 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t1608 \n \n \n \n \n \n \u5de6\u79fb\u4ee3\u8868\u4e58\uff0c\u5de6\u79fb\u4e00\u4f4d\u4ee3\u8868\u4e582,\u5de6\u79fb\u4e24\u4f4d\u4ee3\u8868\u4e584\uff0c\u4f9d\u6b21\u9012\u589e \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 12<<1=24 \u00a0\u00a0 12<<2=48\u00a0\u53f3\u79fb\u4ee3\u8868\u9664,\u00a0 \u53f3\u79fb\u4e00\u4f4d... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t shuoxi666\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \n \t\t\t\tSecureCRT\u7684\u4e0b\u8f7d\u3001\u5b89\u88c5\uff08 \u8fc7\u7a0b\u975e\u5e38\u8be6\u7ec6\uff01\uff01\u503c\u5f97\u67e5\u770b\uff09\t\t \n \n \n 03-03 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t1\u4e07+ \n \n \n \n \n \n SecureCRT\u7684\u4e0b\u8f7d\u3001\u5b89\u88c5\u548c\u7834\u89e3\uff08 \u8fc7\u7a0b\u975e\u5e38\u8be6\u7ec6\uff01\uff01\u503c\u5f97\u67e5\u770b\uff09\u7b80\u5355\u4ecb\u7ecd\u4e0bSecureCRT\u4e00\u3001SecureCRT\u7684\u4e0b\u8f7d\u4e8c\u3001SecureCRT\u7684\u5b89\u88c5\u7b80\u5355\u4ecb\u7ecd\u4e0bSecureCRTSecureCRT\u662f\u4e00... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t \u8d85Ren\u4e13\u5c5e \n \n \n \n \n \n \n \n \n \t\t\t\tJava\u5b66\u4e60\u7684\u6b63\u786e\u6253\u5f00\u65b9\u5f0f\t\t \n \n \n 01-08 \n \n \t\t\t\t\t\u9605\u8bfb\u6570 \n \t\t\t\t\t28\u4e07+ \n \n \n \n \n \n \u5728\u535a\u4e3b\u8ba4\u4e3a\uff0c\u5bf9\u4e8e\u5165\u95e8\u7ea7\u5b66\u4e60java\u7684\u6700\u4f73\u5b66\u4e60\u65b9\u6cd5\u83ab\u8fc7\u4e8e\u89c6\u9891+\u535a\u5ba2+\u4e66\u7c4d+\u603b\u7ed3\uff0c\u524d\u4e09\u8005\u535a\u4e3b\u5c06\u6dcb\u6f13\u5c3d\u81f4\u5730\u6325\u6beb\u4e8e\u8fd9\u7bc7\u535a\u5ba2\u6587\u7ae0\u4e2d\uff0c\u81f3\u4e8e\u603b\u7ed3\u5728\u4e8e\u4e2a\u4eba\uff0c\u5b9e\u9645\u4e0a\u8d8a\u5230\u540e\u9762\u4f60\u4f1a\u53d1\u73b0\u5b66\u4e60\u7684\u6700\u597d\u65b9\u5f0f\u5c31\u662f\u9605\u8bfb\u53c2\u8003\u5b98\u65b9\u6587\u6863\u5176\u6b21... \n \n \n \u535a\u6587 \n \u6765\u81ea\uff1a\t \u7a0b\u5e8f\u5458\u5b9c\u6625\u7684\u535a\u5ba2 \n \n \n \n \n \n \n \n \n \n python \n \n \n \n json \n \n \n \n java \n \n \n \n mysql \n \n \n \n pycharm \n \n \n \n android \n \n \n \n linux \n \n \n \n json\u683c\u5f0f \n \n \n \n c# temp \u76ee\u5f55 \n \n \n \n bytes\u521d\u59cb\u5316 c# \n \n \n \n c#\u663e\u793a\u65e0\u7126\u70b9\u7a97\u53e3 \n \n \n \n c# \u7c7b\u662f\u5426\u7ee7\u627f\u6307\u5b9a\u63a5\u53e3 \n \n \n \n c# \u68c0\u67e5\u6570\u636e\u66f4\u65b0 \n \n \n \n c#\u8fde\u63a5\u626b\u63cf\u67aa \n \n \n \n c# \u8ddfc++ \u8fdb\u7a0b\u540c\u6b65 \n \n \n \n c#\u4e2d\u7684arrylist \n \n \n \n c#\u5b50\u7a97\u4f53\u91cd\u590d\u6253\u5f00 \n \n \n \n c# \u6e90\u7801 \u7f51\u7ad9\u76d1\u63a7 \n \n \n \n \n \n \n \n \n \u6ca1\u6709\u66f4\u591a\u63a8\u8350\u4e86\uff0c \u8fd4\u56de\u9996\u9875 \n \n \n \n \u00a9\ufe0f2019 CSDN \n \u76ae\u80a4\u4e3b\u9898: \u4e66\u9999\u6c34\u58a8 \n \u8bbe\u8ba1\u5e08:\n CSDN\u5b98\u65b9\u535a\u5ba2 \n \n \n \n \u4e3b\u9875\u5f15\u5165 \n \n

    \u4e2a\u4eba\u8d44\u6599

    \n \n \n \n \n \n \n \n \n \n \n \n DimplesDimples. \n \n \n \n TA\u7684\u4e2a\u4eba\u4e3b\u9875 > \n \n \n \n \n \n \u539f\u521b \n 127 \n \n \n \u7c89\u4e1d \n 19 \n \n \n \u83b7\u8d5e \n 18 \n \n \n \u8bc4\u8bba \n 1 \n \n \n \u8bbf\u95ee \n 4\u4e07+ \n \n \n \n \n \u7b49\u7ea7: \n \n \n \n \n \n \n \n \n \n \u5468\u6392\u540d: \n \n \n 2\u4e07+ \n \n \n \n \u79ef\u5206: \n \n 1744 \n \n \n \u603b\u6392\u540d: \n \n \n 4\u4e07+ \n \n \n \n \n \n \u52cb\u7ae0: \n \n \n \n \n \n \n \n \n \n \n \u6301\u4e4b\u4ee5\u6052 \n \n \n \u6388\u4e88\u6bcf\u4e2a\u81ea\u7136\u6708\u5185\u53d1\u5e034\u7bc7\u62164\u7bc7\u4ee5\u4e0a\u539f\u521b\u6216\u7ffb\u8bd1IT\u535a\u6587\u7684\u7528\u6237\u3002\u4e0d\u79ef\u8dec\u6b65\u65e0\u4ee5\u81f3\u5343\u91cc\uff0c\u4e0d\u79ef\u5c0f\u6d41\u65e0\u4ee5\u6210\u6c5f\u6d77\uff0c\u7a0b\u5e8f\u4eba\u751f\u7684\u7cbe\u5f69\u9700\u8981\u575a\u6301\u4e0d\u61c8\u5730\u79ef\u7d2f\uff01 \n \n \n \n \n \n \n \n \n \n \n \n \n 1024\u52cb\u7ae0 \n \n \n #1024\u7a0b\u5e8f\u5458\u8282#\u6d3b\u52a8\u52cb\u7ae0\uff0c\u5f53\u65e5\u53d1\u5e03\u539f\u521b\u535a\u5ba2\u5373\u53ef\u83b7\u5f97 \n \n \n \n \n \n \n \n \n \n \u5173\u6ce8 \n \n \n \u79c1\u4fe1 \n \n \n \n \n \n \n \u535a\u5ba2\u5185\u9875\u5de6\u4e0a\u89c6\u7a97-20181120 \n \n \n \n \n \u6700\u65b0\u6587\u7ae0 \n \n \n \n \n Hadoop\u600e\u6837\u907f\u514d\u6587\u4ef6\u88ab\u5207\u5206\uff1f \n \n \n \n Flume\u662f\u4ec0\u4e48\uff0c\u6709\u4ec0\u4e48\u4f5c\u7528\uff0cflume\u7684\u4e09\u4e2a\u7ec4\u4ef6\u3002 \n \n \n \n HA\u67b6\u6784\u4e2d\u7684\u8111\u88c2\uff0c\u4ec0\u4e48\u662f\u8111\u88c2\uff0c\u600e\u6837\u9884\u9632\u8111\u88c2\uff1f \n \n \n \n \u6570\u636e\u5e93\u4e8b\u52a1\u7684\u56db\u4e2a\u7279\u6027\u53ca\u542b\u4e49 \n \n \n \n MapReduce\u4e2d\u6392\u5e8f\u53d1\u751f\u5728\u54ea\u51e0\u4e2a\u9636\u6bb5\uff1f\u8fd9\u4e9b\u6392\u5e8f\u662f\u5426\u53ef\u4ee5\u907f\u514d\uff1f\u4e3a\u4ec0\u4e48\uff1f \n \n \n \n \n \n \u5206\u7c7b\u4e13\u680f \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n Flume \n \n ####\u662f\u5426\u4ed8\u8d39 \n 1\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n java \n \n ####\u662f\u5426\u4ed8\u8d39 \n 71\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n \u6570\u636e\u5e93 \n \n ####\u662f\u5426\u4ed8\u8d39 \n 18\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n Linux \n \n ####\u662f\u5426\u4ed8\u8d39 \n 14\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n vim \n \n ####\u662f\u5426\u4ed8\u8d39 \n 1\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n Hadoop \n \n ####\u662f\u5426\u4ed8\u8d39 \n 21\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n \u8ba1\u7b97\u673a\u57fa\u7840 \n \n ####\u662f\u5426\u4ed8\u8d39 \n 1\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n JSP \n \n ####\u662f\u5426\u4ed8\u8d39 \n 1\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n Zookeeper \n \n ####\u662f\u5426\u4ed8\u8d39 \n 1\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n HBase \n \n ####\u662f\u5426\u4ed8\u8d39 \n 3\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n python \n \n ####\u662f\u5426\u4ed8\u8d39 \n 1\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n hive \n \n ####\u662f\u5426\u4ed8\u8d39 \n 1\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n Spring \n \n ####\u662f\u5426\u4ed8\u8d39 \n 1\u7bc7 \n \n \n \n \n \n \u5c55\u5f00 \n \n \n \n \u5f52\u6863 \n \n \n \u5f52\u6863\u7edf\u8ba1 \n \n \n 2020\u5e741\u6708 2\u7bc7 \n \n \n \u5f52\u6863\u7edf\u8ba1 \n \n \n 2019\u5e7412\u6708 5\u7bc7 \n \n \n \u5f52\u6863\u7edf\u8ba1 \n \n \n 2019\u5e743\u6708 1\u7bc7 \n \n \n \u5f52\u6863\u7edf\u8ba1 \n \n \n 2018\u5e7412\u6708 6\u7bc7 \n \n \n \u5f52\u6863\u7edf\u8ba1 \n \n \n 2018\u5e7411\u6708 23\u7bc7 \n \n \n \u5f52\u6863\u7edf\u8ba1 \n \n \n 2018\u5e7410\u6708 15\u7bc7 \n \n \n \u5f52\u6863\u7edf\u8ba1 \n \n \n 2018\u5e749\u6708 18\u7bc7 \n \n \n \u5f52\u6863\u7edf\u8ba1 \n \n \n 2018\u5e748\u6708 40\u7bc7 \n \n \n \u5f52\u6863\u7edf\u8ba1 \n \n \n 2018\u5e747\u6708 20\u7bc7 \n \n \n \u5f52\u6863\u7edf\u8ba1 \n \n \n 2018\u5e746\u6708 5\u7bc7 \n \n \n \n \n \n \u5c55\u5f00 \n \n \n \n \u70ed\u95e8\u6587\u7ae0 \n \n \n \n \n Hadoop\u5b8c\u5168\u5206\u5e03\u5f0f\u642d\u5efa\u6b65\u9aa4 \n \u9605\u8bfb\u6570 4651 \n \n \n \n \u5f55\u516510\u4e2a\u5b66\u751f\u7684\u6570\u5b66\u8bed\u6587\u82f1\u8bed\u7684\u4e09\u79d1\u6210\u7ee9 \u8ba1\u7b97\u6bcf\u79d1\u7684\u5e73\u5747\u6210\u7ee9\u8ba1\u7b97\u51fa\u6bcf\u4e2a\u4eba\u7684\u603b\u6210\u7ee9 \u5e76\u627e\u51fa\u6bcf\u79d1\u6210\u7ee9\u7684\u6700\u9ad8\u5206\u4e0e\u6700\u4f4e\u5206 \n \u9605\u8bfb\u6570 2963 \n \n \n \n MySQL\u8bed\u8a00\u4e4b\u7ea6\u675f(constraint) \n \u9605\u8bfb\u6570 2563 \n \n \n \n HBase\u6709\u4ec0\u4e48\u4f18\u70b9\u548c\u7f3a\u70b9 \n \u9605\u8bfb\u6570 2429 \n \n \n \n \u5728window\u4e0b\u8bbe\u7f6ehosts\u914d\u7f6e\u4e3b\u673a\u540d\u548cip\u7684\u6620\u5c04 \n \u9605\u8bfb\u6570 2261 \n \n \n \n \n \n \u6700\u65b0\u8bc4\u8bba \n \n \n \n \u6210\u5458\u53d8\u91cf\u4e0e\u5c40\u90e8\u53d8\u91cf\u7684\u533a\u522b ... \n \n qq_29951485\uff1a \u4f5c\u8005\u4f60\u597d\uff0c\u4e00\u4e2a\u95ee\u9898\uff0cGCROOT\u5305\u542b\u5c40\u90e8\u53d8\u91cf\u8868\u91cc\u7684\u5f15\u7528\u7684\u5bf9\u8c61\uff0c\u90a3\u4e48\u7b49\u5230\u5c40\u90e8\u53d8\u91cf\u8868\u4e2d\u7684\u503c\u7f6e\u4e3anull\u65f6\uff0c\u5bf9\u8c61\u88ab\u56de\u6536\uff0c\u53ef\u4ee5\u7406\u89e3\u3002\u4f46\u662f\u6210\u5458\u53d8\u91cf\u968f\u7740\u5bf9\u8c61\u4e00\u8d77\u518d\u5806\u4e2d\uff0c\u90a3\u4e48\u88ab\u56de\u6536\u7684\u65f6\u673a\u5c31\u548c\u5bf9\u8c61\u4e00\u81f4\uff0c\u4e0d\u80fd\u5355\u72ec\u88ab\u56de\u6536\uff0c\u662f\u8fd9\u6837\u5417\uff1f \n \n \n \n \n \n \n \n \u535a\u5ba2\u5185\u9875\u5de6\u4e0b\u89c6\u7a97-20181130 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \u76ee\u5f55 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \u591a\u6761\u5e7f\u544a\u5982\u4e0b\u811a\u672c\u53ea\u9700\u5f15\u5165\u4e00\u6b21 \n \n \n \u5206\u7c7b\u4e13\u680f \n \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n Flume \n \n ####\u662f\u5426\u4ed8\u8d39 \n 1\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n java \n \n ####\u662f\u5426\u4ed8\u8d39 \n 71\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n \u6570\u636e\u5e93 \n \n ####\u662f\u5426\u4ed8\u8d39 \n 18\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n Linux \n \n ####\u662f\u5426\u4ed8\u8d39 \n 14\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n vim \n \n ####\u662f\u5426\u4ed8\u8d39 \n 1\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n Hadoop \n \n ####\u662f\u5426\u4ed8\u8d39 \n 21\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n \u8ba1\u7b97\u673a\u57fa\u7840 \n \n ####\u662f\u5426\u4ed8\u8d39 \n 1\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n JSP \n \n ####\u662f\u5426\u4ed8\u8d39 \n 1\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n Zookeeper \n \n ####\u662f\u5426\u4ed8\u8d39 \n 1\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n HBase \n \n ####\u662f\u5426\u4ed8\u8d39 \n 3\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n python \n \n ####\u662f\u5426\u4ed8\u8d39 \n 1\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n hive \n \n ####\u662f\u5426\u4ed8\u8d39 \n 1\u7bc7 \n \n \n \n \n \n ####\u662f\u5426\u4ed8\u8d39 \n Spring \n \n ####\u662f\u5426\u4ed8\u8d39 \n 1\u7bc7 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \u70b9\u8d5e \n \u53d6\u6d88\u70b9\u8d5e \n \n 2 \n \n \n \n \n \n \n \n \u6d77\u62a5 \n \n \n \n \n \n \n \n \n \n \u8bc4\u8bba \n \n \n \n \n \n \n \n \n \n \u76ee\u5f55 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \u6536\u85cf \n \n \t\t\t\t\t\u6536\u85cf\n \t\t\t\t\t\u53d6\u6d88\u6536\u85cf\n \t\t\t\t \n \n \n \n \n \n \n \n \n \n \t\t\t\t\t\u624b\u673a\u770b\n \t\t\t\t \n \n \n \n \n \n \n \n \u4e0a\u4e00\u7bc7 \n \n \n \n \n \n \n \n \u4e0b\u4e00\u7bc7 \n \n \n \u5bbd\u5c4f\u66f4\u591a\u6309\u94ae \n \n \n \n \n \n \u66f4\u591a \n \n \n \n \n \n \n \n \u4e0a\u4e00\u7bc7 \n \n \n \n \n \n \n \n \u4e0b\u4e00\u7bc7 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \u6253\u8d4f \n \n \n \u6253\u8d4f \n \n \n \n DimplesDimples. \n \u201c\u4f60\u7684\u9f13\u52b1\u5c06\u662f\u6211\u521b\u4f5c\u7684\u6700\u5927\u52a8\u529b\u201d \n \n \n \n 5C\u5e01 \n 10C\u5e01 \n 20C\u5e01 \n 50C\u5e01 \n 100C\u5e01 \n 200C\u5e01 \n \n \n \u786e\u5b9a \n \n \n \n \n \n \n \n \n \n \n \n \n \n \u5206\u4eab\u5230\u5fae\u4fe1\u670b\u53cb\u5708 \n \n \u00d7 \n \n \n \u626b\u4e00\u626b\uff0c\u624b\u673a\u6d4f\u89c8 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nt = np.arange(0.0, 12*np.pi, 0.01)\nx = np.sin(t)*(np.e**np.cos(t) - 2*np.cos(4*t)-np.sin(t/12)**5)\ny = np.cos(t)*(np.e**np.cos(t) - 2*np.cos(4*t)-np.sin(t/12)**5)\nplt.figure(figsize=(8,6))\nplt.axis('off')\nplt.plot(x,y,color='blue',linewidth = '2')\nplt.show()\n# plt.savefig(\"butter.jpg\",dpi=400)\n```\n\n\n```python\nfrom mpl_toolkits.mplot3d import Axes3D\nfrom matplotlib import cm\nfrom matplotlib.ticker import LinearLocator\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig = plt.figure()\nax = fig.gca(projection='3d')\n[x, t] = np.meshgrid(np.array(range(25)) / 24.0, np.arange(0, 575.5, 0.5) / 575 * 17 * np.pi - 2 * np.pi)\np = (np.pi / 2) * np.exp(-t / (8 * np.pi))\nu = 1 - (1 - np.mod(3.6 * t, 2 * np.pi) / np.pi) ** 4 / 2\ny = 2 * (x ** 2 - x) ** 2 * np.sin(p)\nr = u * (x * np.sin(p) + y * np.cos(p))\nh = u * (x * np.cos(p) - y * np.sin(p))\nsurf = ax.plot_surface(r * np.cos(t), r * np.sin(t), h, rstride=1, cstride=1,\n cmap=cm.gist_rainbow_r, linewidth=0, antialiased=True)\nplt.show()\n```\n\n\n```python\nfrom mpl_toolkits.mplot3d import Axes3D\nfrom matplotlib import cm\nfrom matplotlib.ticker import LinearLocator\nimport matplotlib.pyplot as plt\nimport numpy as np\n\n# \u7701\u7565\u4e86\u5934\u6587\u4ef6\uff0c\u53ef\u4ee5\u5728\u4e4b\u524d\u7684\u535a\u5ba2\u91cc\u770b\u5230\nfig = plt.figure()\nax = fig.gca(projection='3d')\n# \u5c06\u76f8\u4f4d\u5411\u540e\u79fb\u52a8\u4e866*pi\n[x, t] = np.meshgrid(np.array(range(25)) / 24.0, np.arange(0, 575.5, 0.5) / 575 * 20 * np.pi + 4*np.pi)\np = (np.pi / 2) * np.exp(-t / (8 * np.pi))\n# \u6dfb\u52a0\u8fb9\u7f18\u6270\u52a8\nchange = np.sin(15*t)/150\n# \u5c06t\u7684\u53c2\u6570\u51cf\u5c11\uff0c\u4f7f\u82b1\u74e3\u7684\u89d2\u5ea6\u53d8\u5927\nu = 1 - (1 - np.mod(3.3 * t, 2 * np.pi) / np.pi) ** 4 / 2 + change\ny = 2 * (x ** 2 - x) ** 2 * np.sin(p)\nr = u * (x * np.sin(p) + y * np.cos(p))\nh = u * (x * np.cos(p) - y * np.sin(p))\nc= cm.get_cmap('Reds')\nsurf = ax.plot_surface(r * np.cos(t), r * np.sin(t), h, rstride=1, cstride=1,\n cmap= c, linewidth=0, antialiased=True)\nplt.show()\n```\n\n\n```python\nfrom mpl_toolkits.mplot3d import Axes3D\nfrom matplotlib import cm\nfrom matplotlib.ticker import LinearLocator\nimport matplotlib.pyplot as plt\nimport numpy as np\nfig = plt.figure()\nax = fig.gca(projection='3d')\n[x, t] = np.meshgrid(np.array(range(25)) / 24.0, np.arange(0, 575.5, 0.5) / 575 * 30 * np.pi - 4*np.pi)\np = (np.pi / 2) * np.exp(-t / (8 * np.pi))\nchange = np.sin(20*t)/50\nu = 1 - (1 - np.mod(3.3 * t, 2 * np.pi) / np.pi) ** 4 / 2 + change\ny = 2 * (x ** 2 - x) ** 2 * np.sin(p)\nr = u * (x * np.sin(p) + y * np.cos(p)) * 1.5\nh = u * (x * np.cos(p) - y * np.sin(p))\nc= cm.get_cmap('magma')\nsurf = ax.plot_surface(r * np.cos(t), r * np.sin(t), h, rstride=1, cstride=1,\n cmap= c, linewidth=0, antialiased=True)\nplt.show()\n```\n\n\n```python\n# \u6211\u91c7\u7528requests\u5e93\nimport requests\nimport time\n\n# \u7528\u6765\u83b7\u53d6 \u65f6\u95f4\u6233\ndef gettime():\n return int(round(time.time() * 1000))\n\nif __name__ == '__main__':\n # \u7528\u6765\u81ea\u5b9a\u4e49\u5934\u90e8\u7684\n headers = {}\n # \u7528\u6765\u4f20\u9012\u53c2\u6570\u7684\n keyvalue = {}\n # \u76ee\u6807\u7f51\u5740(\u95ee\u53f7\u524d\u9762\u7684\u4e1c\u897f)\n url = 'http://data.stats.gov.cn/easyquery.htm'\n\n # \u5934\u90e8\u7684\u586b\u5145\n headers['User-Agent'] = 'Mozilla/5.0 (Macintosh; Intel Mac OS X 10_14) ' \\\n 'AppleWebKit/605.1.15 (KHTML, like Gecko) ' \\\n 'Version/12.0 Safari/605.1.15'\n\n # \u4e0b\u9762\u662f\u53c2\u6570\u7684\u586b\u5145\uff0c\u53c2\u8003\u56fe10\n keyvalue['m'] = 'QueryData'\n keyvalue['dbcode'] = 'hgnd'\n keyvalue['rowcode'] = 'zb'\n keyvalue['colcode'] = 'sj'\n keyvalue['wds'] = '[]'\n keyvalue['dfwds'] = '[{\"wdcode\":\"zb\",\"valuecode\":\"A0301\"}]'\n keyvalue['k1'] = str(gettime())\n\n # \u53d1\u51fa\u8bf7\u6c42\uff0c\u4f7f\u7528get\u65b9\u6cd5\uff0c\u8fd9\u91cc\u4f7f\u7528\u6211\u4eec\u81ea\u5b9a\u4e49\u7684\u5934\u90e8\u548c\u53c2\u6570\n # r = requests.get(url, headers=headers, params=keyvalue)\n # \u5efa\u7acb\u4e00\u4e2aSession\n s = requests.session()\n # \u5728Session\u57fa\u7840\u4e0a\u8fdb\u884c\u4e00\u6b21\u8bf7\u6c42\n r = s.get(url, params=keyvalue, headers=headers)\n # \u6253\u5370\u8fd4\u56de\u8fc7\u6765\u7684\u72b6\u6001\u7801\n print(r.status_code)\n # \u4fee\u6539dfwds\u5b57\u6bb5\u5185\u5bb9\n keyvalue['dfwds'] = '[{\"wdcode\":\"sj\",\"valuecode\":\"2000\"}]'\n # \u518d\u6b21\u8fdb\u884c\u8bf7\u6c42\n r = s.get(url, params=keyvalue, headers=headers)\n # \u6b64\u65f6\u6211\u4eec\u5c31\u80fd\u83b7\u53d6\u5230\u6211\u4eec\u641c\u7d22\u5230\u7684\u6570\u636e\u4e86\n print(r.text)\n```\n\n 200\n {\"returncode\":200,\"returndata\":{\"datanodes\":[{\"code\":\"zb.A030101_sj.2000\",\"data\":{\"data\":126743,\"dotcount\":0,\"hasdata\":true,\"strdata\":\"126743\"},\"wds\":[{\"valuecode\":\"A030101\",\"wdcode\":\"zb\"},{\"valuecode\":\"2000\",\"wdcode\":\"sj\"}]},{\"code\":\"zb.A030102_sj.2000\",\"data\":{\"data\":65437,\"dotcount\":0,\"hasdata\":true,\"strdata\":\"65437\"},\"wds\":[{\"valuecode\":\"A030102\",\"wdcode\":\"zb\"},{\"valuecode\":\"2000\",\"wdcode\":\"sj\"}]},{\"code\":\"zb.A030103_sj.2000\",\"data\":{\"data\":61306,\"dotcount\":0,\"hasdata\":true,\"strdata\":\"61306\"},\"wds\":[{\"valuecode\":\"A030103\",\"wdcode\":\"zb\"},{\"valuecode\":\"2000\",\"wdcode\":\"sj\"}]},{\"code\":\"zb.A030104_sj.2000\",\"data\":{\"data\":45906,\"dotcount\":0,\"hasdata\":true,\"strdata\":\"45906\"},\"wds\":[{\"valuecode\":\"A030104\",\"wdcode\":\"zb\"},{\"valuecode\":\"2000\",\"wdcode\":\"sj\"}]},{\"code\":\"zb.A030105_sj.2000\",\"data\":{\"data\":80837,\"dotcount\":0,\"hasdata\":true,\"strdata\":\"80837\"},\"wds\":[{\"valuecode\":\"A030105\",\"wdcode\":\"zb\"},{\"valuecode\":\"2000\",\"wdcode\":\"sj\"}]}],\"freshsort\":0,\"hasdatacount\":5,\"wdnodes\":[{\"nodes\":[{\"cname\":\"\u5e74\u672b\u603b\u4eba\u53e3\",\"code\":\"A030101\",\"dotcount\":0,\"exp\":\"\u5e74\u672b\u4eba\u53e3\u6570\u6307\u6bcf\u5e7412\u670831\u65e524\u65f6\u7684\u4eba\u53e3\u6570\u3002\u5e74\u5ea6\u7edf\u8ba1\u7684\u5168\u56fd\u4eba\u53e3\u603b\u6570\u5185\u672a\u5305\u62ec\u9999\u6e2f\u3001\u6fb3\u95e8\u7279\u522b\u884c\u653f\u533a\u548c\u53f0\u6e7e\u7701\u4ee5\u53ca\u6d77\u5916\u534e\u4fa8\u4eba\u6570\u3002\",\"ifshowcode\":false,\"memo\":\"1981\u5e74\u53ca\u4ee5\u524d\u4eba\u53e3\u6570\u636e\u4e3a\u6237\u7c4d\u7edf\u8ba1\u6570\uff1b1982\u30011990\u30012000\u30012010\u5e74\u6570\u636e\u4e3a\u5f53\u5e74\u4eba\u53e3\u666e\u67e5\u6570\u636e\u63a8\u7b97\u6570\uff1b\u5176\u4f59\u5e74\u4efd\u6570\u636e\u4e3a\u5e74\u5ea6\u4eba\u53e3\u62bd\u6837\u8c03\u67e5\u63a8\u7b97\u6570\u636e\u3002\u603b\u4eba\u53e3\u548c\u6309\u6027\u522b\u5206\u4eba\u53e3\u4e2d\u5305\u62ec\u73b0\u5f79\u519b\u4eba\uff0c\u6309\u57ce\u4e61\u5206\u4eba\u53e3\u4e2d\u73b0\u5f79\u519b\u4eba\u8ba1\u5165\u57ce\u9547\u4eba\u53e3\u3002\",\"name\":\"\u5e74\u672b\u603b\u4eba\u53e3\",\"nodesort\":\"1\",\"sortcode\":2785,\"tag\":\"\",\"unit\":\"\u4e07\u4eba\"},{\"cname\":\"\u7537\u6027\u4eba\u53e3\",\"code\":\"A030102\",\"dotcount\":0,\"exp\":\"\",\"ifshowcode\":false,\"memo\":\"\",\"name\":\"\u7537\u6027\u4eba\u53e3\",\"nodesort\":\"1\",\"sortcode\":2786,\"tag\":\"\",\"unit\":\"\u4e07\u4eba\"},{\"cname\":\"\u5973\u6027\u4eba\u53e3\",\"code\":\"A030103\",\"dotcount\":0,\"exp\":\"\",\"ifshowcode\":false,\"memo\":\"\",\"name\":\"\u5973\u6027\u4eba\u53e3\",\"nodesort\":\"1\",\"sortcode\":2787,\"tag\":\"\",\"unit\":\"\u4e07\u4eba\"},{\"cname\":\"\u57ce\u9547\u4eba\u53e3\",\"code\":\"A030104\",\"dotcount\":0,\"exp\":\"\u57ce\u9547\u4eba\u53e3\u662f\u6307\u5c45\u4f4f\u5728\u57ce\u9547\u8303\u56f4\u5185\u7684\u5168\u90e8\u5e38\u4f4f\u4eba\u53e3\u3002\",\"ifshowcode\":false,\"memo\":\"\",\"name\":\"\u57ce\u9547\u4eba\u53e3\",\"nodesort\":\"1\",\"sortcode\":2788,\"tag\":\"\",\"unit\":\"\u4e07\u4eba\"},{\"cname\":\"\u4e61\u6751\u4eba\u53e3\",\"code\":\"A030105\",\"dotcount\":0,\"exp\":\"\u4e61\u6751\u4eba\u53e3\u662f\u9664\u57ce\u9547\u4eba\u53e3\u4ee5\u5916\u7684\u5168\u90e8\u4eba\u53e3\u3002\",\"ifshowcode\":false,\"memo\":\"\",\"name\":\"\u4e61\u6751\u4eba\u53e3\",\"nodesort\":\"1\",\"sortcode\":2789,\"tag\":\"\",\"unit\":\"\u4e07\u4eba\"}],\"wdcode\":\"zb\",\"wdname\":\"\u6307\u6807\"},{\"nodes\":[{\"cname\":\"2000\u5e74\",\"code\":\"2000\",\"dotcount\":4,\"exp\":\"\",\"ifshowcode\":false,\"memo\":\"\",\"name\":\"2000\u5e74\",\"nodesort\":\"1\",\"sortcode\":21,\"tag\":\"\",\"unit\":\"\"}],\"wdcode\":\"sj\",\"wdname\":\"\u65f6\u95f4\"}]}}\n\n\n\n```python\nimport pandas as pd\ndata = pd.read_excel(\"F:\\Course\\Ad Design Project\\Adanced design Project\\Layout phase\\VeryIMMaterials\\Dist_between_2_ASVs_Reservior_No_shutdown\\Dist_between_2_ASVs_2km_Reservior_No_Shutdown.xlsx\",sheet_name='Rupture Data')\n```\n\n\n```python\ndata.head(554)\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    Time (s)Discharge Rate (kg/s)Cumulative Mass (kg)Pexit 1 (Bara)Pexit 2 (Bara)Temp 1 (K)Temp 2 (K)Density 1 (kg/m3)Density 2 (kg/m3)Velocity 1 (m/s)Velocity 2 (m/s)
    00.00.00000.000039.995839.9958293.1447293.144767.885467.88540.26550.2655
    10.01180.05370.000022.592322.5923252.5346252.534642.252142.2521253.1873253.1873
    20.1628.180862.818113.798613.7726231.6360231.574025.605625.5871220.4379220.4254
    30.2609.6585123.783912.401312.3740235.4435235.357121.534621.4889241.2826241.2221
    40.2609.6585123.783912.401312.3740235.4435235.357121.534621.4889241.2826241.2221
    ....................................
    54992.042.742612303.50071.15702.0338293.0632272.23231.34322.56571.0638294.4158
    55094.041.771712387.04411.15701.9950293.0677272.46251.34312.51370.9680294.4916
    55196.040.821112468.68631.15701.9511293.0718272.55041.34312.45660.8925294.5369
    55298.039.886012548.45831.15701.9128293.0756272.76161.34312.40570.8300294.7477
    553100.038.970512626.39941.15701.8701293.0790272.83121.34312.35040.7771294.6870
    \n

    554 rows \u00d7 11 columns

    \n
    \n\n\n\n\n```python\ndata.describe()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    Time (s)Discharge Rate (kg/s)Cumulative Mass (kg)Pexit 1 (Bara)Pexit 2 (Bara)Temp 1 (K)Temp 2 (K)Density 1 (kg/m3)Density 2 (kg/m3)Velocity 1 (m/s)Velocity 2 (m/s)
    count554.000000554.000000554.000000554.000000554.000000554.000000554.000000554.000000554.000000554.000000554.000000
    mean23.610830220.2540325240.3081444.9986745.362190268.013234266.6150166.8955437.372436274.070756284.984168
    std20.498052120.3335373356.5821333.1047502.7409648.6833616.4529365.0812804.65687047.28779815.001453
    min0.0000000.0000000.0000001.1570001.870100231.636000231.5740001.3431002.3504000.2655000.265500
    25%5.525000142.7137002043.9404253.1982253.719800264.603300264.6254504.1021754.789725279.212600283.126175
    50%19.350000192.5116505168.6790004.5590004.553350268.843150268.8305505.9647005.956950287.629750288.838800
    75%36.350000281.2509508069.7597256.4936006.481800270.508525270.7208258.8234258.805525290.000425290.835725
    max100.0000001180.05370012626.39940039.99580039.995800293.144700293.14470067.88540067.885400296.231100294.747700
    \n
    \n\n\n\n\n```python\nimport numpy as np # linear algebra\nimport pandas as pd # data processing, CSV file I/O (e.g. pd.read_csv)\nimport matplotlib.pyplot as plt\nimport matplotlib.gridspec as gridspec\nfrom mpl_toolkits.mplot3d import Axes3D\nimport seaborn as sns\ncolor = sns.color_palette()\nfrom sklearn.linear_model import LinearRegression\nfrom sklearn.tree import DecisionTreeRegressor, export_graphviz\nfrom sklearn.ensemble import RandomForestRegressor\nfrom sklearn.ensemble import GradientBoostingRegressor\nfrom sklearn.model_selection import GridSearchCV, train_test_split, KFold\nfrom sklearn.preprocessing import StandardScaler\nfrom sklearn.ensemble.partial_dependence import plot_partial_dependence\nfrom sklearn.ensemble.partial_dependence import partial_dependence\nfrom scipy.stats.stats import pearsonr\n```\n\n\n```python\nfrom IPython.display import Image\nimport pydotplus\n```\n\n\n```python\nsns.set_style(\"dark\")\ndef plot_corr(predictors):\n predictors = predictors[:]\n mcorr = data[predictors].corr()\n mask = np.zeros_like(mcorr, dtype=np.bool)\n mask[np.triu_indices_from(mask)] = True\n cmap = sns.diverging_palette(220, 10, as_cmap=True)\n g = sns.heatmap(mcorr, mask=mask, cmap=cmap, square=True, annot=True, fmt='0.2f')\n g.set_xticklabels(predictors, rotation=90)\n g.set_yticklabels(reversed(predictors))\n plt.show()\nplot_corr([\"Discharge Rate (kg/s)\",\"Cumulative Mass (kg)\",\"Pexit 1 (Bara)\",\"Pexit 2 (Bara)\",\"Temp 1 (K)\",\"Temp 2 (K)\",\"Density 1 (kg/m3)\",\"Velocity 1 (m/s)\",\"Velocity 2 (m/s)\"])\n```\n\nAugmented assignments have three advantages:*\n\u2022 There\u2019s less for you to type. Need I say more?\n\u2022 The left side only has to be evaluated once. In X += Y, X may be a complicated object\nexpression. In the augmented form, it only has to be evaluated once. However, in\nthe long form, X = X + Y, X appears twice and must be run twice. Because of this,\naugmented assignments usually run faster.\n\u2022 The optimal technique is automatically chosen. That is, for objects that support\nin-place changes, the augmented forms automatically perform in-place change operations instead of slower copies.\n\n\n```python\nimport time\nstart =time.time()\n#\u4e2d\u95f4\u5199\u4e0a\u4ee3\u7801\u5757\nL = [1,2]\nL = L + [3] # Concatenation\nprint(L)\nend = time.time()\nprint('Running time: %s Seconds'%(end-start))\n```\n\n [1, 2, 3]\n Running time: 0.0 Seconds\n\n\n\n```python\nimport time\nstart =time.time()\n#\u4e2d\u95f4\u5199\u4e0a\u4ee3\u7801\u5757\nL = [1,2]\nL = L.append(3) # Concatenation\nprint(L)\nend = time.time()\nprint('Running time: %s Seconds'%(end-start))\n```\n\n None\n Running time: 0.0 Seconds\n\n\nC/C++ programmers take note: although Python now supports statements like X += Y, it still does not have\nC\u2019s auto-increment/decrement operators (e.g., X++, \u2212\u2212X). These don\u2019t quite map to the Python object model\nbecause Python has no notion of in-place changes to immutable objects like numbers.\n\n\n```python\nimport timeit\nstart=timeit.default_timer()\n\n#\u4e2d\u95f4\u5199\u4ee3\u7801\u5757\n\nL = [1,2]\nL = L.append(3) # Concatenation\n\nend=timeit.default_timer()\nprint('Running time: %s Seconds'%(end-start))\n```\n\n Running time: 7.229999937408138e-05 Seconds\n\n\n\n```python\nimport timeit\nstart=timeit.default_timer()\n\n#\u4e2d\u95f4\u5199\u4ee3\u7801\u5757\n\nL = [1,2]\nL = L + [3] # Concatenation\n\nend=timeit.default_timer()\nprint('Running time: %s Seconds'%(end-start))\n```\n\n Running time: 7.509999886678997e-05 Seconds\n\n\n# And to add a set of items to the end, we can either concatenate again or call the list extend method:\n\n\n```python\nimport timeit\nstart = timeit.default_timer()\n\nL= [1,2,3,4]\nL = L + [5,6]\nprint(L)\n\nend = timeit.default_timer()\nprint(\"Running time: %s Seconds\"%(end-start))\n```\n\n [1, 2, 3, 4, 5, 6]\n Running time: 0.00024709999888727907 Seconds\n\n\n\n```python\nimport timeit\nstart = timeit.default_timer()\n\nL= [1,2,3,4]\nL += [5,6]\nprint(L)\n\nend = timeit.default_timer()\nprint(\"Running time: %s Seconds\"%(end-start))\n```\n\n [1, 2, 3, 4, 5, 6]\n Running time: 0.00022110000099928584 Seconds\n\n\n\n```python\nimport timeit\nstart = timeit.default_timer()\n\nL = [1,2,3,4]\nL.extend([5,6])\nprint(L)\n\nend = timeit.default_timer()\nprint(\"Running time: %s Seconds\"%(end-start))\n```\n\n [1, 2, 3, 4, 5, 6]\n Running time: 0.00042180000127700623 Seconds\n\n\n# \uff1f When we use augmented assignment to extend a list, we can forget these details\u2014forexample, Python automatically calls the quicker extend method instead of using the slower concatenation operation implied by \n\n\n```python\nimport pandas as pd\ndf = pd.DataFrame([[\"\"\"def all_unique(lst):\n return len(lst) == len(set(lst))\n\n\nx = [1,1,2,2,3,2,3,4,5,6]\ny = [1,2,3,4,5]\nprint(all_unique(x)) # False\nprint(all_unique(y)) # True\nprint(x,y)\"\"\"],[\"\"\"from collections import Counter\n\ndef anagram(first, second):\n return Counter(first) == Counter(second)\n\n\nprint(anagram(\"abcd3\", \"3acdb\")) # True\"\"\"],[\"\"\"import sys \n\nvariable = 30 \nprint(sys.getsizeof(variable)) # 24\"\"\"],[\"\"\"def byte_size(string):\n return(len(string.encode( 'utf-8' )))\n\n\nprint(byte_size( 'KDA' )) # 4\nprint(byte_size( 'Hello World' )) # 11 \"\"\"],[\"\"\"n = 2; \ns =\"Programming\"; \n\nprint(s * n);\n# ProgrammingProgramming \"\"\"],[\"\"\"s = \"programming is awesome\"\n\nprint(s.title())\n# Programming Is Awesome\"\"\"],[\"\"\"\n\nfrom math import ceil\n\ndef chunk(lst, size):\n return list(\n map(lambda x: lst[x * size:x * size + size],\n list(range(0, ceil(len(lst) / size)))))\n\nprint(chunk([1,2,3,4,5],2))\n# [[1,2],[3,4],5]\"\"\"],[\"\"\"def compact(lst):\n return list(filter(bool, lst))\n\n\nprint(compact([0, 1, False, 2, 3, 5 , s , 34]))\n# [ 1, 2, 3, a , s , 34 ]\"\"\"],[\"\"\"array = [[ 'a' , 'b' ], [ 'c' , 'd' ], [ 'e' , 'f' ]]\ntransposed = zip(*array)\nprint(transposed)\n# [( a , c , e ), ( b , d , f )]\"\"\"],[\"\"\"a = 3\nprint( 2 < a < 8) # True\nprint(1 == a < 2) # False\"\"\"],[\"\"\"hobbies = [\"basketball\", \"football\", \"swimming\"]\n\nprint(\"My hobbies are: \" + \", \".join(hobbies))\n# My hobbies are: basketball, football, swimming\"\"\"],[\"\"\"import re\n\ndef count_vowels(str):\n return len(re.findall('[aeiou]', str, re.IGNORECASE))\n\nprint(count_vowels( 'foobar' )) # 3\nprint(count_vowels( 'gym' )) # 0\"\"\"],[\"\"\"def decapitalize(string):\n return string[:1].lower() + string[1:]\n\n\nprint(decapitalize( 'FooBar' )) # fooBar\nprint(decapitalize( 'FooBar' )) # fooBar\"\"\"],[\"\"\"def spread(arg):\n ret = []\n for i in arg:\n if isinstance(i, list):\n ret.extend(i)\n else:\n ret.append(i)\n return ret\n\ndef deep_flatten(lst):\n result = []\n result.extend(\n spread(list(map(lambda x: deep_flatten(x) if type(x) == list else x, lst))))\n return result\n\n\nprint(deep_flatten([1, [2], [[3], 4], 5])) # [1,2,3,4,5]\"\"\"],[\"\"\"def difference(a, b):\n set_a = set(a)\n set_b = set(b)\n comparison = set_a.difference(set_b)\n return list(comparison)\n\n\nprint(difference([1,2,3], [1,2,4])) # [3]\"\"\"],[\"\"\"def difference_by(a, b, fn):\n b = set(map(fn, b))\n return [item for item in a if fn(item) not in b]\n\n\nfrom math import floor\nprint(difference_by([2.1, 1.2], [2.3, 3.4],floor)) # [1.2]\nprint(difference_by([{ 'x' : 2 }, { 'x' : 1 }], [{ 'x' : 1 }], lambda v : v[ 'x' ]))\n# [ { x: 2 } ]\n\"\"\"],[\"\"\"def add(a, b):\n return a + b\n\ndef subtract(a, b):\n return a - b\n\na, b = 4, 5\nprint((subtract if a > b else add)(a, b)) # 9 \"\"\"],[\"\"\"def has_duplicates(lst):\n return len(lst) != len(set(lst))\n\n\nx = [1,2,3,4,5,5]\ny = [1,2,3,4,5]\nprint(has_duplicates(x)) # True\nprint(has_duplicates(y)) # False\"\"\"],[\"\"\"def merge_two_dicts(a, b):\n c = a.copy() # make a copy of a \n c.update(b) # modify keys and values of a with the ones from b\n return c\n\n\na = { 'x' : 1, 'y' : 2}\nb = { 'y' : 3, 'z' : 4}\nprint(merge_two_dicts(a, b))\n# { y : 3, x : 1, z : 4}\n\"\"\"],[\"\"\"def to_dictionary(keys, values):\n return dict(zip(keys, values))\n\n\nkeys = [\"a\", \"b\", \"c\"] \nvalues = [2, 3, 4]\nprint(to_dictionary(keys, values))\n# { a : 2, c : 4, b : 3}\"\"\"],[\"\"\"lst = [\"a\", \"b\", \"c\", \"d\"]\nfor index, element in enumerate(lst): \n print(\"Value\", element, \"Index \", index, )\n\n# ( Value , a , Index , 0)\n# ( Value , b , Index , 1)\n#( Value , c , Index , 2)\n# ( Value , d , Index , 3)\"\"\"],[\"\"\"import time\n\nstart_time = time.time()\n\na = 1\nb = 2\nc = a + b\nprint(c) #3\n\nend_time = time.time()\ntotal_time = end_time - start_time\nprint(\"Time: \", total_time)\n\n# ( Time: , 1.1205673217773438e-05)\"\"\"],[\"\"\"try:\n 2*3\nexcept TypeError:\n print(\"An exception was raised\")\nelse:\n print(\"Thank God, no exceptions were raised.\")\n\n#Thank God, no exceptions were raised.\"\"\"],[\"\"\"def most_frequent(list):\n return max(set(list), key = list.count)\n\n\nlist = [1,2,1,2,3,2,1,4,2]\nprint(most_frequent(list))\ndel list\"\"\"],[\"\"\"# def palindrome(string):\n # from re import sub\n # s = sub('\\w',[W_], string.lower())\n # return s == s[::-1]\n\n\n#print(palindrome( 'taco cat' )) # True\"\"\"],[\"\"\"import operator\naction = {\n \"+\": operator.add,\n \"-\": operator.sub,\n \"/\": operator.truediv,\n \"*\": operator.mul,\n \"**\": pow\n}\nprint(action[\"-\"](50, 25)) # 25\"\"\"],[\"\"\"from copy import deepcopy\nfrom random import randint\n\ndef shuffle(lst):\n temp_lst = deepcopy(lst)\n m = len(temp_lst)\n while (m):\n m -= 1\n i = randint(0, m)\n temp_lst[m], temp_lst[i] = temp_lst[i], temp_lst[m]\n return temp_lst\n\n\nfoo = [1,2,3]\nprint(shuffle(foo)) # [2,3,1] , foo = [1,2,3]\"\"\"],[\"\"\"def spread(arg):\n ret = []\n for i in arg:\n if isinstance(i, list):\n ret.extend(i)\n else:\n ret.append(i)\n return ret\n\n\nprint(spread([1,2,3,[4,5,6],[7],8,9])) # [1,2,3,4,5,6,7,8,9]\"\"\"],[\"\"\"def swap(a, b):\n return b, a\n\na, b = -1, 14\nprint(swap(a, b)) # (14, -1)\nprint(spread([1,2,3,[4,5,6],[7],8,9])) # [1,2,3,4,5,6,7,8,9]\"\"\"],[\"\"\"d = { a : 1, b : 2}\n\nprint(d.get( c , 3)) # 3\"\"\"]],index=['\u91cd\u590d\u5143\u7d20\u5224\u5b9a','\u5b57\u7b26\u5143\u7d20\u7ec4\u6210\u5224\u5b9a','\u5185\u5b58\u5360\u7528','\u5b57\u8282\u5360\u7528','\u6253\u5370 N \u6b21\u5b57\u7b26\u4e32','\u5927\u5199\u7b2c\u4e00\u4e2a\u5b57\u6bcd','\u5206\u5757','\u538b\u7f29','\u89e3\u5305','\u94fe\u5f0f\u5bf9\u6bd4','\u9017\u53f7\u8fde\u63a5','\u5143\u97f3\u7edf\u8ba1',\n '\u9996\u5b57\u6bcd\u5c0f\u5199','\u5c55\u5f00\u5217\u8868', '\u5217\u8868\u7684\u5dee','\u901a\u8fc7\u51fd\u6570\u53d6\u5dee','\u94fe\u5f0f\u51fd\u6570\u8c03\u7528','\u68c0\u67e5\u91cd\u590d\u9879','\u5408\u5e76\u4e24\u4e2a\u5b57\u5178','\u5c06\u4e24\u4e2a\u5217\u8868\u8f6c\u5316\u4e3a\u5b57\u5178','\u4f7f\u7528\u679a\u4e3e','\u6267\u884c\u65f6\u95f4',\n 'Try else','\u5143\u7d20\u9891\u7387','\u56de\u6587\u5e8f\u5217','\u4e0d\u4f7f\u7528 if-else \u7684\u8ba1\u7b97\u5b50','Shuffle','\u5c55\u5f00\u5217\u8868','\u4ea4\u6362\u503c','\u5b57\u5178\u9ed8\u8ba4\u503c'],columns=['Python codes'])\n```\n\n\n```python\ndf.head(31)\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    Python codes
    \u91cd\u590d\u5143\u7d20\u5224\u5b9adef all_unique(lst):\\n return len(lst) == l...
    \u5b57\u7b26\u5143\u7d20\u7ec4\u6210\u5224\u5b9afrom collections import Counter\\n\\ndef anagram...
    \u5185\u5b58\u5360\u7528import sys \\n\\nvariable = 30 \\nprint(sys.getsi...
    \u5b57\u8282\u5360\u7528def byte_size(string):\\n return(len(string....
    \u6253\u5370 N \u6b21\u5b57\u7b26\u4e32n = 2; \\ns =\"Programming\"; \\n\\nprint(s * n);\\n...
    \u5927\u5199\u7b2c\u4e00\u4e2a\u5b57\u6bcds = \"programming is awesome\"\\n\\nprint(s.title(...
    \u5206\u5757\\n\\nfrom math import ceil\\n\\ndef chunk(lst, si...
    \u538b\u7f29def compact(lst):\\n return list(filter(bool...
    \u89e3\u5305array = [[ 'a' , 'b' ], [ 'c' , 'd' ], [ 'e'...
    \u94fe\u5f0f\u5bf9\u6bd4a = 3\\nprint( 2 < a < 8) # True\\nprint(1 == a ...
    \u9017\u53f7\u8fde\u63a5hobbies = [\"basketball\", \"football\", \"swimming...
    \u5143\u97f3\u7edf\u8ba1import re\\n\\ndef count_vowels(str):\\n retur...
    \u9996\u5b57\u6bcd\u5c0f\u5199def decapitalize(string):\\n return string[:...
    \u5c55\u5f00\u5217\u8868def spread(arg):\\n ret = []\\n for i in a...
    \u5217\u8868\u7684\u5deedef difference(a, b):\\n set_a = set(a)\\n ...
    \u901a\u8fc7\u51fd\u6570\u53d6\u5deedef difference_by(a, b, fn):\\n b = set(map(...
    \u94fe\u5f0f\u51fd\u6570\u8c03\u7528def add(a, b):\\n return a + b\\n\\ndef subtra...
    \u68c0\u67e5\u91cd\u590d\u9879def has_duplicates(lst):\\n return len(lst) ...
    \u5408\u5e76\u4e24\u4e2a\u5b57\u5178def merge_two_dicts(a, b):\\n c = a.copy() ...
    \u5c06\u4e24\u4e2a\u5217\u8868\u8f6c\u5316\u4e3a\u5b57\u5178def to_dictionary(keys, values):\\n return d...
    \u4f7f\u7528\u679a\u4e3elst = [\"a\", \"b\", \"c\", \"d\"]\\nfor index, element...
    \u6267\u884c\u65f6\u95f4import time\\n\\nstart_time = time.time()\\n\\na =...
    Try elsetry:\\n 2*3\\nexcept TypeError:\\n print(\"A...
    \u5143\u7d20\u9891\u7387def most_frequent(list):\\n return max(set(l...
    \u56de\u6587\u5e8f\u5217# def palindrome(string):\\n # from re import...
    \u4e0d\u4f7f\u7528 if-else \u7684\u8ba1\u7b97\u5b50import operator\\naction = {\\n \"+\": operator...
    Shufflefrom copy import deepcopy\\nfrom random import ...
    \u5c55\u5f00\u5217\u8868def spread(arg):\\n ret = []\\n for i in a...
    \u4ea4\u6362\u503cdef swap(a, b):\\n return b, a\\n\\na, b = -1, 1...
    \u5b57\u5178\u9ed8\u8ba4\u503cd = { a : 1, b : 2}\\n\\nprint(d.get( c , 3)) # 3
    \n
    \n\n\n\n\n```python\nfor i in range(30):\n exec(str((df.iloc[i].values)[0]))\n```\n\n False\n True\n [1, 1, 2, 2, 3, 2, 3, 4, 5, 6] [1, 2, 3, 4, 5]\n True\n 28\n 3\n 11\n ProgrammingProgramming\n Programming Is Awesome\n [[1, 2], [3, 4], [5]]\n [1, 2, 3, 5, 'programming is awesome', 34]\n \n True\n False\n My hobbies are: basketball, football, swimming\n 3\n 0\n fooBar\n fooBar\n [1, 2, 3, 4, 5]\n [3]\n [1.2]\n [{'x': 2}]\n 9\n True\n False\n {'x': 1, 'y': 3, 'z': 4}\n {'a': 2, 'b': 3, 'c': 4}\n Value a Index 0\n Value b Index 1\n Value c Index 2\n Value d Index 3\n 3\n Time: 0.0\n Thank God, no exceptions were raised.\n 2\n 25\n [1, 2, 3]\n [1, 2, 3, 4, 5, 6, 7, 8, 9]\n (14, -1)\n [1, 2, 3, 4, 5, 6, 7, 8, 9]\n 3\n\n\n\n```python\nhelp(re.sub)\n```\n\n Help on function sub in module re:\n \n sub(pattern, repl, string, count=0, flags=0)\n Return the string obtained by replacing the leftmost\n non-overlapping occurrences of the pattern in string by the\n replacement repl. repl can be either a string or a callable;\n if a string, backslash escapes in it are processed. If it is\n a callable, it's passed the Match object and must return\n a replacement string to be used.\n \n\n\n\n```python\ndef difference_by(a, b, fn):\n b = set(map(fn, b))\n return [item for item in a if fn(item) not in b]\n\n\nfrom math import floor\ndifference_by([2.1, 1.2], [2.3, 3.4],floor) # [1.2]\ndifference_by([{ 'x' : 2 }, { 'x' : 1 }], [{ 'x' : 1 }], lambda v : v[ 'x' ])\n# [ { x: 2 } ]\n```\n\n\n\n\n [{'x': 2}]\n\n\n\n\n```python\nlines = df.iloc[0].values\nprint(lines)\n```\n\n ['def all_unique(lst):\\n return len(lst) == len(set(lst))\\n\\n\\nx = [1,1,2,2,3,2,3,4,5,6]\\ny = [1,2,3,4,5]\\nprint(all_unique(x)) # False\\nprint(all_unique(y)) # True\\nprint(x,y)']\n\n\n\n```python\nprint(str(lines[0]))\n```\n\n def all_unique(lst):\n return len(lst) == len(set(lst))\n \n \n x = [1,1,2,2,3,2,3,4,5,6]\n y = [1,2,3,4,5]\n print(all_unique(x)) # False\n print(all_unique(y)) # True\n print(x,y)\n\n\n\n```python\nx,y\n```\n\n\n\n\n ([1, 1, 2, 2, 3, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5])\n\n\n\n\n```python\nexec_code = compile(str(lines),'', 'exec')\nprint(exec_code)\n```\n\n at 0x000002E0B0EAF390, file \"\", line 1>\n\n\n\n```python\nprint(df.iloc[1].values)\n```\n\n ['from collections import Counter\\n\\ndef anagram(first, second):\\n return Counter(first) == Counter(second)\\n\\n\\nanagram(\"abcd3\", \"3acdb\") # True']\n\n\n\n```python\nexec('from collections import Counter\\n\\ndef anagram(first, second):\\n return Counter(first) == Counter(second)\\n\\n\\nanagram(\"abcd3\", \"3acdb\") # True')\n```\n\n\n```python\nanagram(\"abcd3\",\"333333333333333333acdb\")\n```\n\n\n\n\n False\n\n\n\n\n```python\nexec(str(lines[0]))\n```\n\n False\n True\n [1, 1, 2, 2, 3, 2, 3, 4, 5, 6] [1, 2, 3, 4, 5]\n\n\n\n```python\nstr((df.iloc[1].values)[0])\n```\n\n\n\n\n 'from collections import Counter\\n\\ndef anagram(first, second):\\n return Counter(first) == Counter(second)\\n\\n\\nprint(anagram(\"abcd3\", \"3acdb\")) # True'\n\n\n\n\n```python\nexec(str((df.iloc[1].values)[0]))\n```\n\n True\n\n\n\n```python\nprint(dir(str))\n```\n\n ['__add__', '__class__', '__contains__', '__delattr__', '__dir__', '__doc__', '__eq__', '__format__', '__ge__', '__getattribute__', '__getitem__', '__getnewargs__', '__gt__', '__hash__', '__init__', '__init_subclass__', '__iter__', '__le__', '__len__', '__lt__', '__mod__', '__mul__', '__ne__', '__new__', '__reduce__', '__reduce_ex__', '__repr__', '__rmod__', '__rmul__', '__setattr__', '__sizeof__', '__str__', '__subclasshook__', 'capitalize', 'casefold', 'center', 'count', 'encode', 'endswith', 'expandtabs', 'find', 'format', 'format_map', 'index', 'isalnum', 'isalpha', 'isascii', 'isdecimal', 'isdigit', 'isidentifier', 'islower', 'isnumeric', 'isprintable', 'isspace', 'istitle', 'isupper', 'join', 'ljust', 'lower', 'lstrip', 'maketrans', 'partition', 'replace', 'rfind', 'rindex', 'rjust', 'rpartition', 'rsplit', 'rstrip', 'split', 'splitlines', 'startswith', 'strip', 'swapcase', 'title', 'translate', 'upper', 'zfill']\n\n\n\n```python\nprint(dir(str.__str__))\n```\n\n ['__call__', '__class__', '__delattr__', '__dir__', '__doc__', '__eq__', '__format__', '__ge__', '__get__', '__getattribute__', '__gt__', '__hash__', '__init__', '__init_subclass__', '__le__', '__lt__', '__name__', '__ne__', '__new__', '__objclass__', '__qualname__', '__reduce__', '__reduce_ex__', '__repr__', '__setattr__', '__sizeof__', '__str__', '__subclasshook__', '__text_signature__']\n\n\n\n```python\nprint(dir((str.__class__).__call__))\n```\n\n ['__call__', '__class__', '__delattr__', '__dir__', '__doc__', '__eq__', '__format__', '__ge__', '__get__', '__getattribute__', '__gt__', '__hash__', '__init__', '__init_subclass__', '__le__', '__lt__', '__name__', '__ne__', '__new__', '__objclass__', '__qualname__', '__reduce__', '__reduce_ex__', '__repr__', '__setattr__', '__sizeof__', '__str__', '__subclasshook__', '__text_signature__']\n\n\n\n```python\nimport inspect\n```\n\n\n```python\nhelp((str.__class__).__call__)\n```\n\n Help on wrapper_descriptor:\n \n __call__(self, /, *args, **kwargs)\n Call self as a function.\n \n\n\n# Naming conventions\n\nBesides these rules, there is also a set of naming conventions\u2014rules that are not required\nbut are followed in normal practice. For instance, because names with two leading and\ntrailing underscores (e.g., __name__) generally have special meaning to the Python interpreter, you should avoid this pattern for your own names. Here is a list of the conventions Python follows:\n\n\u2022 Names that begin with a single underscore (_X) are not imported by a from module\nimport * statement (described in Chapter 22).\n\n\u2022 Names that have two leading and trailing underscores (__X__) are system-defined\nnames that have special meaning to the interpreter.\n\n\u2022 Names that begin with two underscores and do not end with two more (__X) are\nlocalized (\u201cmangled\u201d) to enclosing classes (see the discussion of pseudoprivate\nattributes in Chapter 30).\n\n\u2022 The name that is just a single underscore (_) retains the result of the last expression\nwhen working interactively.\n\n# The Python 3.0 print Function\n\n\n```python\nx = 'spam'\ny = 99\nz = ['eggs']\nprint(x,y,z)\n```\n\n spam 99 ['eggs']\n\n\n\n```python\nL = print(x,y,z,sep='',end='')\n```\n\n spam99['eggs']\n\n\n```python\nprint(L)\n```\n\n None\n\n\n\n```python\nprint(x,y,z,sep='')\n```\n\n spam99['eggs']\n\n\n\n```python\nprint(x,y,z,sep='');print(x,y,z) # Two prints, same output line\nprint(x,y,z,end='');print(x,y,z)\n```\n\n spam99['eggs']\n spam 99 ['eggs']\n spam 99 ['eggs']spam 99 ['eggs']\n\n\n\n```python\nimport pyprind\nimport time\nimport pandas as pd\n\nbar = pyprind.ProgBar(10, monitor=True)\nfor i in range(10):\n time.sleep(0.01) # your computation here\n data = pd.read_excel(\"F:\\Course\\Ad Design Project\\Adanced design Project\\Layout phase\\VeryIMMaterials\\Dist_between_2_ASVs_Reservior_No_shutdown\\Dist_between_2_ASVs_2km_Reservior_No_Shutdown.xlsx\",\n sheet_name='Rupture Data')\n bar.update()\nprint(bar)\n```\n\n 0% [##########] 100% | ETA: 00:00:00\n Total time elapsed: 00:06:46\n\n\n Title: \n Started: 03/08/2020 20:24:03\n Finished: 03/08/2020 20:30:50\n Total time elapsed: 00:06:46\n CPU %: 95.50\n Memory %: 3.60\n\n\n\n```python\nprint('x','y','z',end='...\\n')\n```\n\n x y z...\n\n\n\n```python\nprint('x','y','z',sep='...',end='!\\n') # Multiple keywords\nprint('x','y','z',end='\\n', sep='...')\n```\n\n x...y...z!\n x...y...z\n\n\nHere is how the file keyword argument is used\u2014it directs the printed text to an open\noutput file or other compatible object for the duration of the single print (this is really\na form of stream redirection, a topic we will revisit later in this section):\n\n\n```python\nimport requests\nfrom bs4 import BeautifulSoup\nurl = 'https://www.cnblogs.com/JYNNO1/p/10525649.html'\nres = requests.get(url)\nhtml_page = res.content\nsoup = BeautifulSoup(html_page, 'html.parser')\ntext = soup.find_all(text=True)\n\noutput = ''\nblacklist = [\n '[document]',\n 'noscript',\n 'header',\n 'html',\n 'meta',\n 'head', \n 'input',\n 'script',\n # there may be more elements you don't want, such as \"style\", etc.\n]\n\nfor t in text:\n if t.parent.name not in blacklist:\n output += '{} '.format(t)\n\nprint(output, file=open('data.txt','w',encoding='utf-8'))\n```\n\n\n```python\nprint('x','y','z',sep='...',file=open('data1.txt','w'))\n```\n\n\n```python\nprint('x','y','z',sep='...'); print(open('data1.txt').read())\n```\n\n x...y...z\n x...y...z\n \n\n\n\n```python\ntext = '%s: %-.4f, %05d' % ('Result',3.14159,42)\nprint(text)\n```\n\n Result: 3.1416, 00042\n\n\n\n```python\nprint('%s: %-.4f,%05d' % ('Result',3.14159,42))\n```\n\n Result: 3.1416,00042\n\n\n\n```python\nimport sys\nsys.stdout.write('hello world\\n')\n```\n\n hello world\n\n\nThis code explicitly calls the write method of sys.stdout\u2014an attribute preset when\nPython starts up to an open file object connected to the output stream. The print\noperation hides most of those details, providing a simple tool for simple printing tasks.\n\n\n```python\nS = \"\"\"Here, we reset sys.stdout to a manually opened file named log.txt, located in the script\u2019s\nworking directory and opened in append mode (so we add to its current content). After\nthe reset, every print operation anywhere in the program will write its text to the end\nof the file log.txt instead of to the original output stream. The print operations are\nhappy to keep calling sys.stdout\u2019s write method, no matter what sys.stdout happens\nto refer to. Because there is just one sys module in your process, assigning\nsys.stdout this way will redirect every print anywhere in your program.\"\"\"\n```\n\n\n```python\nL = S.split()\n```\n\n\n```python\nprint(L, end = ' ')\n```\n\n ['Here,', 'we', 'reset', 'sys.stdout', 'to', 'a', 'manually', 'opened', 'file', 'named', 'log.txt,', 'located', 'in', 'the', 'script\u2019s', 'working', 'directory', 'and', 'opened', 'in', 'append', 'mode', '(so', 'we', 'add', 'to', 'its', 'current', 'content).', 'After', 'the', 'reset,', 'every', 'print', 'operation', 'anywhere', 'in', 'the', 'program', 'will', 'write', 'its', 'text', 'to', 'the', 'end', 'of', 'the', 'file', 'log.txt', 'instead', 'of', 'to', 'the', 'original', 'output', 'stream.', 'The', 'print', 'operations', 'are', 'happy', 'to', 'keep', 'calling', 'sys.stdout\u2019s', 'write', 'method,', 'no', 'matter', 'what', 'sys.stdout', 'happens', 'to', 'refer', 'to.', 'Because', 'there', 'is', 'just', 'one', 'sys', 'module', 'in', 'your', 'process,', 'assigning', 'sys.stdout', 'this', 'way', 'will', 'redirect', 'every', 'print', 'anywhere', 'in', 'your', 'program.'] \n\n# Manual stream redirection\n\n\n```python\nprint('x','y') # Or, in 2.6: print X, Y\nimport sys\nsys.stdout.write(str('x')+' '+ str('y') + '\\n')\n```\n\n x y\n x y\n\n\n\n```python\nimport sys\ntemp = sys.stdout\nsys.stdout = open('log.txt','a')\n```\n\n\n```python\nS1 = \"\"\"In fact, as this chapter\u2019s upcoming sidebar about print and stdout will explain, you\ncan even reset sys.stdout to an object that isn\u2019t a file at all, as long as it has the expected\ninterface: a method named write to receive the printed text string argument. When that\nobject is a class, printed text can be routed and processed arbitrarily per a write method\nyou code yourself\"\"\"\n```\n\n\n```python\nprint(S1.split())\n```\n\n\n```python\nprint(open('log.txt').read())\n```\n\n ['In', 'fact,', 'as', 'this', 'chapter\u2019s', 'upcoming', 'sidebar', 'about', 'print', 'and', 'stdout', 'will', 'explain,', 'you', 'can', 'even', 'reset', 'sys.stdout', 'to', 'an', 'object', 'that', 'isn\u2019t', 'a', 'file', 'at', 'all,', 'as', 'long', 'as', 'it', 'has', 'the', 'expected', 'interface:', 'a', 'method', 'named', 'write', 'to', 'receive', 'the', 'printed', 'text', 'string', 'argument.', 'When', 'that', 'object', 'is', 'a', 'class,', 'printed', 'text', 'can', 'be', 'routed', 'and', 'processed', 'arbitrarily', 'per', 'a', 'write', 'method', 'you', 'code', 'yourself']\n ['In', 'fact,', 'as', 'this', 'chapter\u2019s', 'upcoming', 'sidebar', 'about', 'print', 'and', 'stdout', 'will', 'explain,', 'you', 'can', 'even', 'reset', 'sys.stdout', 'to', 'an', 'object', 'that', 'isn\u2019t', 'a', 'file', 'at', 'all,', 'as', 'long', 'as', 'it', 'has', 'the', 'expected', 'interface:', 'a', 'method', 'named', 'write', 'to', 'receive', 'the', 'printed', 'text', 'string', 'argument.', 'When', 'that', 'object', 'is', 'a', 'class,', 'printed', 'text', 'can', 'be', 'routed', 'and', 'processed', 'arbitrarily', 'per', 'a', 'write', 'method', 'you', 'code', 'yourself']\n ['In', 'fact,', 'as', 'this', 'chapter\u2019s', 'upcoming', 'sidebar', 'about', 'print', 'and', 'stdout', 'will', 'explain,', 'you', 'can', 'even', 'reset', 'sys.stdout', 'to', 'an', 'object', 'that', 'isn\u2019t', 'a', 'file', 'at', 'all,', 'as', 'long', 'as', 'it', 'has', 'the', 'expected', 'interface:', 'a', 'method', 'named', 'write', 'to', 'receive', 'the', 'printed', 'text', 'string', 'argument.', 'When', 'that', 'object', 'is', 'a', 'class,', 'printed', 'text', 'can', 'be', 'routed', 'and', 'processed', 'arbitrarily', 'per', 'a', 'write', 'method', 'you', 'code', 'yourself']\n ['In', 'fact,', 'as', 'this', 'chapter\u2019s', 'upcoming', 'sidebar', 'about', 'print', 'and', 'stdout', 'will', 'explain,', 'you', 'can', 'even', 'reset', 'sys.stdout', 'to', 'an', 'object', 'that', 'isn\u2019t', 'a', 'file', 'at', 'all,', 'as', 'long', 'as', 'it', 'has', 'the', 'expected', 'interface:', 'a', 'method', 'named', 'write', 'to', 'receive', 'the', 'printed', 'text', 'string', 'argument.', 'When', 'that', 'object', 'is', 'a', 'class,', 'printed', 'text', 'can', 'be', 'routed', 'and', 'processed', 'arbitrarily', 'per', 'a', 'write', 'method', 'you', 'code', 'yourself']\n \n\n\n\n```python\ndf.iloc[1].values\n```\n\n\n\n\n array(['from collections import Counter\\n\\ndef anagram(first, second):\\n return Counter(first) == Counter(second)\\n\\n\\nprint(anagram(\"abcd3\", \"3acdb\")) # True'],\n dtype=object)\n\n\n\n\n```python\nimport sys\ntemp = sys.stdin\nsys.stdin = open('myfile.txt','a')\n```\n\n\n```python\nprint('hello world')\n```\n\n hello world\n\n\n\n```python\nmyfile = open('log.txt','w+')\n```\n\n\n```python\nmyfile.write(\"\"\"Let\u2019s work through a simple example that demonstrates file-processing basics. The\nfollowing code begins by opening a new text file for output, writing two lines (strings\nterminated with a newline marker, \\n), and closing the file. Later, the example opens\nthe same file again in input mode and reads the lines back one at a time with\nreadline. Notice that the third readline call returns an empty string; this is how Python\nfile methods tell you that you\u2019ve reached the end of the file (empty lines in the file come\nback as strings containing just a newline character, not as empty strings). \"\"\")\n```\n\n\n\n\n 591\n\n\n\n\n```python\nmyfile.close()\n```\n\n\n```python\nopen('log.txt').read()\n```\n\n\n\n\n 'Let\u2019s work through a simple example that demonstrates file-processing basics. The\\nfollowing code begins by opening a new text file for output, writing two lines (strings\\nterminated with a newline marker, \\n), and closing the file. Later, the example opens\\nthe same file again in input mode and reads the lines back one at a time with\\nreadline. Notice that the third readline call returns an empty string; this is how Python\\nfile methods tell you that you\u2019ve reached the end of the file (empty lines in the file come\\nback as strings containing just a newline character, not as empty strings). '\n\n\n\n\n```python\nmyfile = open('log.txt','a+')\nmyfile.write(\"\"\"When coded this way, the temporary file object created by open will automatically read\nand return one line on each loop iteration. This form is usually easiest to code, good\non memory use, and may be faster than some other options (depending on many variables, of course). Since we haven\u2019t reached statements or iterators yet, though, you\u2019ll\nhave to wait until Chapter 14 for a more complete explanation of this code.\"\"\")\n```\n\n\n\n\n 417\n\n\n\n\n```python\nmyfile.flush()\nmyfile.readlines()\n```\n\n\n\n\n []\n\n\n\n\n```python\nopen('log.txt').read()\n```\n\n\n\n\n 'Let\u2019s work through a simple example that demonstrates file-processing basics. The\\nfollowing code begins by opening a new text file for output, writing two lines (strings\\nterminated with a newline marker, \\n), and closing the file. Later, the example opens\\nthe same file again in input mode and reads the lines back one at a time with\\nreadline. Notice that the third readline call returns an empty string; this is how Python\\nfile methods tell you that you\u2019ve reached the end of the file (empty lines in the file come\\nback as strings containing just a newline character, not as empty strings). When coded this way, the temporary file object created by open will automatically read\\nand return one line on each loop iteration. This form is usually easiest to code, good\\non memory use, and may be faster than some other options (depending on many variables, of course). Since we haven\u2019t reached statements or iterators yet, though, you\u2019ll\\nhave to wait until Chapter 14 for a more complete explanation of this code.When coded this way, the temporary file object created by open will automatically read\\nand return one line on each loop iteration. This form is usually easiest to code, good\\non memory use, and may be faster than some other options (depending on many variables, of course). Since we haven\u2019t reached statements or iterators yet, though, you\u2019ll\\nhave to wait until Chapter 14 for a more complete explanation of this code.'\n\n\n\n\n```python\nimport pubchempy as pcp\nc = pcp.Compound.from_cid(5090)\nprint(c.molecular_formula)\nprint(c.molecular_weight)\nprint(c.isomeric_smiles)\n```\n\n C17H14O4S\n 314.4\n CS(=O)(=O)C1=CC=C(C=C1)C2=C(C(=O)OC2)C3=CC=CC=C3\n\n\n\n```python\nimport sys\ndef test(x):\n if x==0:\n print(r'Error--> x: can\\'t be zero', file=sys.stderr)\n else:\n print(x)\n```\n\n\n```python\ntest(0), test(1)\n```\n\n 1\n\n\n Error--> x: can\\'t be zero\n\n\n\n\n\n (None, None)\n\n\n\n\n```python\ntest(0); test(1)\n```\n\n 1\n\n\n Error--> x: can\\'t be zero\n\n\nNow that you know all about print redirections, the equivalence between printing and\nfile write methods should be fairly obvious. The following interaction prints both ways\nin 3.0, then redirects the output to an external file to verify that the same text is printed:\n\n\n```python\nX = 1; Y = 2\nprint(X, Y)\nimport sys\nsys.stdout.write(str(X)+' ' + str(Y) + '\\n')\nprint\n```\n\n 1 2\n 1 2\n\n\n\n```python\nf = open('alice.txt', encoding = 'utf-8')\nwhile f.readline()!='':\n print(f.readline())\n```\n\n \u5982\u679c\u662f\u6211\u81ea\u5df1\u7684\uff0c\u6211\u5c31\u4f7f\u52b2\u6253\n \n \u771f\u6253\u91cd\u4e86\u4ed6\u4f1a\u54ed\n \n \u4ed6\u53c8\u6709\u70b9\u53ef\u7231\n \n \u558a\u6211\u4f84\u5b50\u56de\u6765\u505a\u4f5c\u4e1a\n \n \u4e0d\u662f\u8bf4\u8fc7\u4e86\n \n \u6211\u73b0\u5728\u5c31\u662f\u8bb2\u4e00\u4ef6\u4e8b\u7684\u5fc3\u60c5\u554a\uff0c\u6ca1\u4ec0\u4e48\u4e0d\u5f00\u5fc3\u7684\n \n \u6240\u4ee5\u5c31\u4e0d\u7ee7\u7eed\u5566\n \n \u6240\u4ee5\u5982\u679c\u4f60\u60f3\u8981\u672a\u6765\u7684\u59bb\u5b50\u4e5f\u53ea\u7167\u987e\u4f60\u7684\u8bdd\uff0c\u4fdd\u59c6\u5c31\u53ef\u4ee5\u5566\n \n \u4e0d\u60f3\u8bf4\n \n \u5c31\u89c1\u4e00\u6b21\u9762\u800c\u5df2\uff0c\u6211\u7684\u813e\u6027\u4f60\u90fd\u4e0d\u6e05\u695a\n \n \u4f60\u5f88\u559c\u6b22\u6211\uff1f\n \n \u98ce\u683c\u7a81\u53d8\u554a\n \n \u51c0\u51c0\uff1a\u8fd8\u6ca1\u8fc7\u5173\u5462\n \n \u6709\u4e2a\u5927\u4f84\u5b50\u5c31\u591f\u5566\n \n \u770b\u6765\u8ddf\u6211\u8bf4\u8bdd\u62c9\u4f4e\u4f60\u80fd\u529b\u554a\n \n \u53c8\u8ba9\u4f60\u5927\u5f00\u773c\u754c\u4e86\n \n \u4f60\u6709\u8fc7\u51e0\u6b21\u604b\u7231\n \n \u6ca1\u8054\u7cfb\u4e86\u5427\n \n \u7a7a\u6c14\n \n \u54ea\u6562\u6253\u4f60\n \n \u4e0d\u7136\u5bb6\u5ead\u5730\u4f4d\u4e0d\u9ad8\n \n \u6211\u4e0d\u4f1a\u6512\u94b1\u554a\n \n \u90a3\u600e\u4e48\u529e\n \n \u4f60\u5bf9\u81ea\u5df1\u4ee5\u540e\u6709\u4ec0\u4e48\u89c4\u5212\u5417\n \n \u6211\u4e5f\u7ed9\u4e0d\u4e86\u4ec0\u4e48\u597d\u7684\u5efa\u8bae\u554a\n \n \u6d77\u5357\u7684\u7ecf\u6d4e\u8d38\u6613\u6e2f\u91cd\u5fc3\u8fd8\u6ca1\u786e\u5b9a\u4e0b\u6765\u4e48\n \n \u4f60\u73b0\u5728\u5c31\u662f\u8981\u5728\u6df1\u5733\u5de5\u4f5c\uff0c\u662f\u5417\n \n \n\n\n\n```python\nimport os\nf.close()\nos.remove('alice.txt')\n```\n\n# Version-Neutral Printing\n\nFinally, if you cannot restrict your work to Python 3.0 but still want your prints to be\ncompatible with 3.0, you have some options. For one, you can code 2.6 print statements and let 3.0\u2019s 2to3 conversion script translate them to 3.0 function calls automatically. See the Python 3.0 documentation for more details about this script; it\nattempts to translate 2.X code to run under 3.0.\n\n\n```python\nfrom __future__ import print_function\n```\n\nC:\\misc> c:\\python30\\python\n< print('spam') # 3.0 print function call syntax\nspam\n\n< print('spam', 'ham', 'eggs') # These are mutiple argments\nspam ham eggs\n\nThe first of these works the same in 2.6, but the second generates a tuple in the output:\nC:\\misc> c:\\python26\\python\n\n< print('spam') # 2.6 print statement, enclosing parens\nspam\n\n< print('spam', 'ham', 'eggs') # This is really a tuple object!\n('spam', 'ham', 'eggs')\n\n\n```python\nprint('%s %s %s ' %('spam','ham','eggs'))\nprint('{0} {1} {2}'.format('spam','ham','eggs'))\n```\n\n spam ham eggs \n spam ham eggs\n\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nt = np.arange(0.0, 12*np.pi, 0.01)\nx = np.sin(t)*(np.e**np.cos(t) - 2*np.cos(4*t)-np.sin(t/12)**5)\ny = np.cos(t)*(np.e**np.cos(t) - 2*np.cos(4*t)-np.sin(t/12)**5)\nplt.figure(figsize=(8,6))\nplt.axis('off')\nplt.plot(x,y,color='blue',linewidth = '2')\nplt.show()\n# plt.savefig(\"butter.jpg\",dpi=400)\n```\n\n# Chapter Summary\n\nIn this chapter, we began our in-depth look at Python statements by exploring assignments, expressions, and print operations. Although these are generally simple to use,\nthey have some alternative forms that, while optional, are often convenient in practice:\naugmented assignment statements and the redirection form of print operations, for\nexample, allow us to avoid some manual coding work. Along the way, we also studied\nthe syntax of variable names, stream redirection techniques, and a variety of common\nmistakes to avoid, such as assigning the result of an append method call back to a\nvariable.\nIn the next chapter, we\u2019ll continue our statement tour by filling in details about the\nif statement, Python\u2019s main selection tool; there, we\u2019ll also revisit Python\u2019s syntax\nmodel in more depth and look at the behavior of Boolean expressions. Before we move\non, though, the end-of-chapter quiz will test your knowledge of what you\u2019ve learned\nhere.\n\n\n```python\n\n```\n", "meta": {"hexsha": "045562c844bd9ae154e3c7cd1b54143129b01f41", "size": 667622, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Chapter 11 Introducing Python Statements.ipynb", "max_stars_repo_name": "nickcafferry/Learning-Python-4th-Edition", "max_stars_repo_head_hexsha": "6684afc3a97d9719c3cdb2d451959ac15cc896cd", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-12-08T07:56:09.000Z", "max_stars_repo_stars_event_max_datetime": "2019-12-08T07:56:09.000Z", "max_issues_repo_path": "Chapter 11 Introducing Python Statements.ipynb", "max_issues_repo_name": "nickcafferry/Learning-Python-4th-Edition", "max_issues_repo_head_hexsha": "6684afc3a97d9719c3cdb2d451959ac15cc896cd", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Chapter 11 Introducing Python Statements.ipynb", "max_forks_repo_name": "nickcafferry/Learning-Python-4th-Edition", "max_forks_repo_head_hexsha": "6684afc3a97d9719c3cdb2d451959ac15cc896cd", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 72.2456444108, "max_line_length": 84884, "alphanum_fraction": 0.7526744176, "converted": true, "num_tokens": 48931, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.7520125737597972, "lm_q1q2_score": 0.4400034248821999}} {"text": "# Homework and bake-off: word-level entailment with neural networks\n\n\n```python\n__author__ = \"Christopher Potts\"\n__version__ = \"CS224u, Stanford, Fall 2020\"\n```\n\n## Contents\n\n1. [Overview](#Overview)\n1. [Set-up](#Set-up)\n1. [Data](#Data)\n1. [Baseline](#Baseline)\n 1. [Representing words: vector_func](#Representing-words:-vector_func)\n 1. [Combining words into inputs: vector_combo_func](#Combining-words-into-inputs:-vector_combo_func)\n 1. [Classifier model](#Classifier-model)\n 1. [Baseline results](#Baseline-results)\n1. [Homework questions](#Homework-questions)\n 1. [Hypothesis-only baseline [2 points]](#Hypothesis-only-baseline-[2-points])\n 1. [Alternatives to concatenation [2 points]](#Alternatives-to-concatenation-[2-points])\n 1. [A deeper network [2 points]](#A-deeper-network-[2-points])\n 1. [Your original system [3 points]](#Your-original-system-[3-points])\n1. [Bake-off [1 point]](#Bake-off-[1-point])\n\n## Overview\n\nThe general problem is word-level natural language inference. Training examples are pairs of words $(w_{L}, w_{R}), y$ with $y = 1$ if $w_{L}$ entails $w_{R}$, otherwise $0$.\n\nThe homework questions below ask you to define baseline models for this and develop your own system for entry in the bake-off, which will take place on a held-out test-set distributed at the start of the bake-off. (Thus, all the data you have available for development is available for training your final system before the bake-off begins.)\n\n## Set-up\n\nSee [the first notebook in this unit](nli_01_task_and_data.ipynb) for set-up instructions.\n\n\n```python\nfrom collections import defaultdict\nimport json\nimport numpy as np\nimport os\nimport pandas as pd\nfrom torch_shallow_neural_classifier import TorchShallowNeuralClassifier\nimport nli\nimport utils\n```\n\n\n```python\nDATA_HOME = 'data'\n\nNLIDATA_HOME = os.path.join(DATA_HOME, 'nlidata')\n\nwordentail_filename = os.path.join(\n NLIDATA_HOME, 'nli_wordentail_bakeoff_data.json')\n\nGLOVE_HOME = os.path.join(DATA_HOME, 'glove.6B')\n```\n\n## Data\n\nI've processed the data into a train/dev split that is designed to put some pressure on our models to actually learn these semantic relations, as opposed to exploiting regularities in the sample. \n\nThe defining feature of the dataset is that the `train` and `dev` __vocabularies__ are disjoint. That is, if a word `w` appears in a training pair, it does not occur in any text pair. It follows from this that there are also no word-pairs shared between train and dev, as you would expect. This should require your models to learn abstract relationships, as opposed to memorizing incidental properties of individual words in the dataset.\n\n\n```python\nwith open(wordentail_filename) as f:\n wordentail_data = json.load(f)\n```\n\nThe keys are the splits plus a list giving the vocabulary for the entire dataset:\n\n\n```python\nwordentail_data.keys()\n```\n\n\n\n\n dict_keys(['dev', 'train', 'vocab'])\n\n\n\n\n```python\nwordentail_data['train'][: 5]\n```\n\n\n\n\n [[['abode', 'house'], 1],\n [['abortion', 'anaemia'], 0],\n [['abortion', 'aneurysm'], 0],\n [['abortion', 'blindness'], 0],\n [['abortion', 'deafness'], 0]]\n\n\n\n\n```python\nnli.get_vocab_overlap_size(wordentail_data)\n```\n\n\n\n\n 0\n\n\n\nBecause no words are shared between `train` and `dev`, no pairs are either:\n\n\n```python\nnli.get_pair_overlap_size(wordentail_data)\n```\n\n\n\n\n 0\n\n\n\nHere is the label distribution:\n\n\n```python\npd.DataFrame(wordentail_data['train'])[1].value_counts()\n```\n\n\n\n\n 0 7000\n 1 1283\n Name: 1, dtype: int64\n\n\n\nThis is a challenging label distribution \u2013 there are more than 5 times as more non-entailment cases as entailment cases.\n\n## Baseline\n\nEven in deep learning, __feature representation is vital and requires care!__ For our task, feature representation has two parts: representing the individual words and combining those representations into a single network input.\n\n### Representing words: vector_func\n\nLet's consider two baseline word representations methods:\n\n1. Random vectors (as returned by `utils.randvec`).\n1. 50-dimensional GloVe representations.\n\n\n```python\ndef randvec(w, n=50, lower=-1.0, upper=1.0):\n \"\"\"Returns a random vector of length `n`. `w` is ignored.\"\"\"\n return utils.randvec(n=n, lower=lower, upper=upper)\n```\n\n\n```python\ndef load_glove50():\n glove_src = os.path.join(GLOVE_HOME, 'glove.6B.50d.txt')\n # Creates a dict mapping strings (words) to GloVe vectors:\n GLOVE = utils.glove2dict(glove_src)\n return GLOVE\n\nGLOVE = load_glove50()\n\ndef glove_vec(w):\n \"\"\"Return `w`'s GloVe representation if available, else return\n a random vector.\"\"\"\n return GLOVE.get(w, randvec(w, n=50))\n```\n\n### Combining words into inputs: vector_combo_func\n\nHere we decide how to combine the two word vectors into a single representation. In more detail, where `u` is a vector representation of the left word and `v` is a vector representation of the right word, we need a function `vector_combo_func` such that `vector_combo_func(u, v)` returns a new input vector `z` of dimension `m`. A simple example is concatenation:\n\n\n```python\ndef vec_concatenate(u, v):\n \"\"\"Concatenate np.array instances `u` and `v` into a new np.array\"\"\"\n return np.concatenate((u, v))\n```\n\n`vector_combo_func` could instead be vector average, vector difference, etc. (even combinations of those) \u2013 there's lots of space for experimentation here; [homework question 2](#Alternatives-to-concatenation-[2-points]) below pushes you to do some exploration.\n\n### Classifier model\n\nFor a baseline model, I chose `TorchShallowNeuralClassifier`:\n\n\n```python\nnet = TorchShallowNeuralClassifier(early_stopping=True)\n```\n\n### Baseline results\n\nThe following puts the above pieces together, using `vector_func=glove_vec`, since `vector_func=randvec` seems so hopelessly misguided for our problem!\n\n\n```python\nbaseline_experiment = nli.wordentail_experiment(\n train_data=wordentail_data['train'],\n assess_data=wordentail_data['dev'],\n model=net,\n vector_func=glove_vec,\n vector_combo_func=vec_concatenate)\n```\n\n Stopping after epoch 32. Validation score did not improve by tol=1e-05 for more than 10 epochs. Final error is 2.5276186764240265\n\n precision recall f1-score support\n \n 0 0.866 0.946 0.904 1732\n 1 0.463 0.243 0.318 334\n \n accuracy 0.832 2066\n macro avg 0.665 0.594 0.611 2066\n weighted avg 0.801 0.832 0.809 2066\n \n\n\n## Homework questions\n\nPlease embed your homework responses in this notebook, and do not delete any cells from the notebook. (You are free to add as many cells as you like as part of your responses.)\n\n### Hypothesis-only baseline [2 points]\n\nDuring our discussion of SNLI and MultiNLI, we noted that a number of research teams have shown that hypothesis-only baselines for NLI tasks can be remarkably robust. This question asks you to explore briefly how this baseline affects our task.\n\nFor this problem, submit two functions:\n\n1. A `vector_combo_func` function called `hypothesis_only` that simply throws away the premise, using the unmodified hypothesis (second) vector as its representation of the example.\n\n1. A function called `run_hypothesis_only_evaluation` that does the following:\n 1. Loops over the two `vector_combo_func` values `vec_concatenate` and `hypothesis_only`, calling `nli.wordentail_experiment` to train on the 'train' portion and assess on the 'dev' portion, with `glove_vec` as the `vector_func`. So that the results are consistent, use an `sklearn.linear_model.LogisticRegression` with default parameters as the model.\n 1. Returns a `dict` mapping `function_name` strings to the 'macro-F1' score for that pair, as returned by the call to `nli.wordentail_experiment`. (Tip: you can get the `str` name of, e.g., `hypothesis_only` with `hypothesis_only.__name__`.)\n \nThe functions `test_hypothesis_only` and `test_run_hypothesis_only_evaluation` will help ensure that your functions have the desired logic.\n\n\n```python\ndef hypothesis_only(u, v):\n pass\n ##### YOUR CODE HERE\n return v\n\n\ndef run_hypothesis_only_evaluation():\n pass\n ##### YOUR CODE HERE\n from sklearn.linear_model import LogisticRegression\n vec_combo_funcs = (vec_concatenate, hypothesis_only)\n res = {}\n for vec_combo_func in vec_combo_funcs:\n baseline_experiment = nli.wordentail_experiment(\n train_data=wordentail_data['train'],\n assess_data=wordentail_data['dev'],\n model=LogisticRegression(),\n vector_func=glove_vec,\n vector_combo_func=vec_combo_func)\n res[vec_combo_func.__name__] = baseline_experiment['macro-F1']\n\n return res\n\n#run_hypothesis_only_evaluation()\n```\n\n\n```python\ndef test_hypothesis_only(hypothesis_only):\n v = hypothesis_only(1, 2)\n assert v == 2\n```\n\n\n```python\ntest_hypothesis_only(hypothesis_only)\n```\n\n\n```python\ndef test_run_hypothesis_only_evaluation(run_hypothesis_only_evaluation):\n results = run_hypothesis_only_evaluation()\n assert all(x in results for x in ('hypothesis_only', 'vec_concatenate')), \\\n (\"The return value of `run_hypothesis_only_evaluation` does not \"\n \"have the intended kind of keys.\")\n assert isinstance(results['vec_concatenate'], float), \\\n (\"The values of the `run_hypothesis_only_evaluation` result \"\n \"should be floats.\")\n```\n\n\n```python\ntest_run_hypothesis_only_evaluation(run_hypothesis_only_evaluation)\n```\n\n precision recall f1-score support\n \n 0 0.865 0.953 0.907 1732\n 1 0.481 0.228 0.309 334\n \n accuracy 0.835 2066\n macro avg 0.673 0.590 0.608 2066\n weighted avg 0.803 0.835 0.810 2066\n \n precision recall f1-score support\n \n 0 0.855 0.971 0.909 1732\n 1 0.495 0.147 0.226 334\n \n accuracy 0.838 2066\n macro avg 0.675 0.559 0.568 2066\n weighted avg 0.797 0.838 0.799 2066\n \n\n\n### Alternatives to concatenation [2 points]\n\nWe've so far just used vector concatenation to represent the premise and hypothesis words. This question asks you to explore two simple alternative:\n\n1. Write a function `vec_diff` that, for a given pair of vector inputs `u` and `v`, returns the element-wise difference between `u` and `v`.\n\n1. Write a function `vec_max` that, for a given pair of vector inputs `u` and `v`, returns the element-wise max values between `u` and `v`.\n\nYou needn't include your uses of `nli.wordentail_experiment` with these functions, but we assume you'll be curious to see how they do!\n\n\n```python\ndef vec_diff(u, v):\n pass\n ##### YOUR CODE HERE\n return u - v\n\ndef vec_max(u, v):\n pass\n ##### YOUR CODE HERE\n return np.maximum(u, v)\n\nu = np.array([10.2, 8.1])\nv = np.array([1.2, -7.1])\nvec_max(u,v)\n```\n\n\n\n\n array([10.2, 8.1])\n\n\n\n\n```python\ndef test_vec_diff(vec_diff):\n u = np.array([10.2, 8.1])\n v = np.array([1.2, -7.1])\n result = vec_diff(u, v)\n expected = np.array([9.0, 15.2])\n assert np.array_equal(result, expected), \\\n \"Expected {}; got {}\".format(expected, result)\n```\n\n\n```python\ntest_vec_diff(vec_diff)\n```\n\n\n```python\ndef test_vec_max(vec_max):\n u = np.array([1.2, 8.1])\n v = np.array([10.2, -7.1])\n result = vec_max(u, v)\n expected = np.array([10.2, 8.1])\n assert np.array_equal(result, expected), \\\n \"Expected {}; got {}\".format(expected, result)\n```\n\n\n```python\ntest_vec_max(vec_max)\n```\n\n### A deeper network [2 points]\n\nIt is very easy to subclass `TorchShallowNeuralClassifier` if all you want to do is change the network graph: all you have to do is write a new `build_graph`. If your graph has new arguments that the user might want to set, then you should also redefine `__init__` so that these values are accepted and set as attributes.\n\nFor this question, please subclass `TorchShallowNeuralClassifier` so that it defines the following graph:\n\n$$\\begin{align}\nh_{1} &= xW_{1} + b_{1} \\\\\nr_{1} &= \\textbf{Bernoulli}(1 - \\textbf{dropout_prob}, n) \\\\\nd_{1} &= r_1 * h_{1} \\\\\nh_{2} &= f(d_{1}) \\\\\nh_{3} &= h_{2}W_{2} + b_{2}\n\\end{align}$$\n\nHere, $r_{1}$ and $d_{1}$ define a dropout layer: $r_{1}$ is a random binary vector of dimension $n$, where the probability of a value being $1$ is given by $1 - \\textbf{dropout_prob}$. $r_{1}$ is multiplied element-wise by our first hidden representation, thereby zeroing out some of the values. The result is fed to the user's activation function $f$, and the result of that is fed through another linear layer to produce $h_{3}$. (Inside `TorchShallowNeuralClassifier`, $h_{3}$ is the basis for a softmax classifier; no activation function is applied to it because the softmax scaling is handled internally by the loss function.)\n\nFor your implementation, please use `nn.Sequential`, `nn.Linear`, and `nn.Dropout` to define the required layers.\n\nFor comparison, using this notation, `TorchShallowNeuralClassifier` defines the following graph:\n\n$$\\begin{align}\nh_{1} &= xW_{1} + b_{1} \\\\\nh_{2} &= f(h_{1}) \\\\\nh_{3} &= h_{2}W_{2} + b_{2}\n\\end{align}$$\n\nThe following code starts this sub-class for you, so that you can concentrate on `build_graph`. Be sure to make use of `self.dropout_prob`.\n\nFor this problem, submit just your completed `TorchDeepNeuralClassifier`. You needn't evaluate it, though we assume you will be keen to do that!\n\nYou can use `test_TorchDeepNeuralClassifier` to ensure that your network has the intended structure.\n\n\n```python\nimport torch.nn as nn\n\nclass TorchDeepNeuralClassifier(TorchShallowNeuralClassifier):\n def __init__(self, dropout_prob=0.7, **kwargs):\n self.dropout_prob = dropout_prob\n super().__init__(**kwargs)\n\n def build_graph(self):\n \"\"\"Complete this method!\n\n Returns\n -------\n an `nn.Module` instance, which can be a free-standing class you\n write yourself, as in `torch_rnn_classifier`, or the outpiut of\n `nn.Sequential`, as in `torch_shallow_neural_classifier`.\n\n \"\"\"\n pass\n ##### YOUR CODE HERE\n dropout_layer = nn.Dropout(p=self.dropout_prob)\n\n return nn.Sequential(\n nn.Linear(self.input_dim, self.hidden_dim),\n dropout_layer,\n self.hidden_activation,\n nn.Linear(self.hidden_dim, self.n_classes_))\n\n```\n\n\n```python\ndef test_TorchDeepNeuralClassifier(TorchDeepNeuralClassifier):\n dropout_prob = 0.55\n assert hasattr(TorchDeepNeuralClassifier(), \"dropout_prob\"), \\\n \"TorchDeepNeuralClassifier must have an attribute `dropout_prob`.\"\n try:\n inst = TorchDeepNeuralClassifier(dropout_prob=dropout_prob)\n except TypeError:\n raise TypeError(\"TorchDeepNeuralClassifier must allow the user \"\n \"to set `dropout_prob` on initialization\")\n inst.input_dim = 10\n inst.n_classes_ = 5\n graph = inst.build_graph()\n assert len(graph) == 4, \\\n \"The graph should have 4 layers; yours has {}\".format(len(graph))\n expected = {\n 0: 'Linear',\n 1: 'Dropout',\n 2: 'Tanh',\n 3: 'Linear'}\n for i, label in expected.items():\n name = graph[i].__class__.__name__\n assert label in name, \\\n (\"The {} layer of the graph should be a {} layer; \"\n \"yours is {}\".format(i, label, name))\n assert graph[1].p == dropout_prob, \\\n (\"The user's value for `dropout_prob` should be the value of \"\n \"`p` for the Dropout layer.\")\n```\n\n\n```python\ntest_TorchDeepNeuralClassifier(TorchDeepNeuralClassifier)\n```\n\n### Your original system [3 points]\n\nThis is a simple dataset, but its \"word-disjoint\" nature ensures that it's a challenging one, and there are lots of modeling strategies one might adopt. \n\nYou are free to do whatever you like. We require only that your system differ in some way from those defined in the preceding questions. They don't have to be completely different, though. For example, you might want to stick with the model but represent examples differently, or the reverse.\n\nYou are free to use different pretrained word vectors and the like.\n\nPlease embed your code in this notebook so that we can rerun it.\n\nIn the cell below, please provide a brief technical description of your original system, so that the teaching team can gain an understanding of what it does. This will help us to understand your code and analyze all the submissions to identify patterns and strategies. We also ask that you report the best score your system got during development, just to help us understand how systems performed overall.\n\n\n```python\nfrom torch_rnn_classifier import TorchRNNClassifier, TorchRNNModel\nimport torch\nfrom sklearn.metrics import classification_report, accuracy_score, f1_score\nimport time\n\nclass TorchRNNSentenceEncoderDataset(torch.utils.data.Dataset):\n def __init__(self, prem_seqs, hyp_seqs, prem_lengths, hyp_lengths, y=None):\n self.prem_seqs = prem_seqs\n self.hyp_seqs = hyp_seqs\n self.prem_lengths = prem_lengths\n self.hyp_lengths = hyp_lengths\n self.y = y\n assert len(self.prem_seqs) == len(self.hyp_seqs)\n assert len(self.hyp_seqs) == len(self.prem_lengths)\n assert len(self.prem_lengths) == len(self.hyp_lengths)\n if self.y is not None:\n assert len(self.hyp_lengths) == len(self.y)\n\n @staticmethod\n def collate_fn(batch):\n batch = list(zip(*batch))\n X_prem = torch.nn.utils.rnn.pad_sequence(batch[0], batch_first=True)\n X_hyp = torch.nn.utils.rnn.pad_sequence(batch[1], batch_first=True)\n prem_lengths = torch.tensor(batch[2])\n hyp_lengths = torch.tensor(batch[3])\n if len(batch) == 5:\n y = torch.tensor(batch[4])\n return X_prem, X_hyp, prem_lengths, hyp_lengths, y\n else:\n return X_prem, X_hyp, prem_lengths, hyp_lengths\n\n def __len__(self):\n return len(self.prem_seqs)\n\n def __getitem__(self, idx):\n if self.y is None:\n return (self.prem_seqs[idx], self.hyp_seqs[idx],\n self.prem_lengths[idx], self.hyp_lengths[idx])\n else:\n return (self.prem_seqs[idx], self.hyp_seqs[idx],\n self.prem_lengths[idx], self.hyp_lengths[idx],\n self.y[idx])\n \nclass TorchRNNSentenceEncoderClassifierModel(nn.Module):\n def __init__(self, prem_rnn, hyp_rnn, output_dim):\n super().__init__()\n self.prem_rnn = prem_rnn\n self.hyp_rnn = hyp_rnn\n self.output_dim = output_dim\n self.bidirectional = self.prem_rnn.bidirectional\n # Doubled because we concatenate the final states of\n # the premise and hypothesis RNNs:\n self.classifier_dim = self.prem_rnn.hidden_dim * 2\n # Bidirectionality doubles it again:\n if self.bidirectional:\n self.classifier_dim *= 2\n self.classifier_layer = nn.Linear(\n self.classifier_dim, self.output_dim)\n\n def forward(self, X_prem, X_hyp, prem_lengths, hyp_lengths):\n # Premise:\n _, prem_state = self.prem_rnn(X_prem, prem_lengths)\n prem_state = self.get_batch_final_states(prem_state)\n # Hypothesis:\n _, hyp_state = self.hyp_rnn(X_hyp, hyp_lengths)\n hyp_state = self.get_batch_final_states(hyp_state)\n # Final combination:\n state = torch.cat((prem_state, hyp_state), dim=1)\n # Classifier layer:\n logits = self.classifier_layer(state)\n return logits\n\n def get_batch_final_states(self, state):\n if self.prem_rnn.rnn.__class__.__name__ == 'LSTM':\n state = state[0].squeeze(0)\n else:\n state = state.squeeze(0)\n if self.bidirectional:\n state = torch.cat((state[0], state[1]), dim=1)\n return state \n \n \nclass TorchRNNSentenceEncoderClassifier(TorchRNNClassifier):\n\n def build_dataset(self, X, y=None):\n X_prem, X_hyp = zip(*X)\n X_prem, prem_lengths = self._prepare_sequences(X_prem)\n X_hyp, hyp_lengths = self._prepare_sequences(X_hyp)\n if y is None:\n return TorchRNNSentenceEncoderDataset(\n X_prem, X_hyp, prem_lengths, hyp_lengths)\n else:\n self.classes_ = sorted(set(y))\n self.n_classes_ = len(self.classes_)\n class2index = dict(zip(self.classes_, range(self.n_classes_)))\n y = [class2index[label] for label in y]\n return TorchRNNSentenceEncoderDataset(\n X_prem, X_hyp, prem_lengths, hyp_lengths, y)\n\n def build_graph(self):\n prem_rnn = TorchRNNModel(\n vocab_size=len(self.vocab),\n embedding=self.embedding,\n use_embedding=self.use_embedding,\n embed_dim=self.embed_dim,\n rnn_cell_class=self.rnn_cell_class,\n hidden_dim=self.hidden_dim,\n bidirectional=self.bidirectional,\n freeze_embedding=self.freeze_embedding)\n\n model = TorchRNNSentenceEncoderClassifierModel(\n prem_rnn, prem_rnn, output_dim=self.n_classes_)\n\n self.embed_dim = prem_rnn.embed_dim\n\n return model\n\n```\n\n\n```python\nstart_token = ''\nend_token = ''\nsep_token = ''\n# Load in GloVe embeddings\nglove_lookup = utils.glove2dict(\n os.path.join(GLOVE_HOME, 'glove.6B.300d.txt'))\n```\n\n\n```python\n## WordNet & GloVe analysis\nfrom nltk.corpus import wordnet as wn\nprint(f\"Vocabulary size: {len(wordentail_data['vocab'])}\")\nctr = 0\ng_ctr = 0\nn_ctr = 0\nfor word in wordentail_data['vocab']:\n syn = wn.synsets(word)\n if len(syn) == 0:\n ctr +=1\n gv = glove_lookup.get(word)\n if gv is None:\n g_ctr += 1\n if len(syn) == 0 and gv is None:\n n_ctr += 1\n \nprint(f\"Words not in WordNet {ctr}\")\nprint(f\"Words not in GloVe {g_ctr}\")\nprint(f\"Words in neither WordNet nor GloVe {n_ctr}\")\n\nsyn = wn.synsets(\"apple\")\nprint(syn[0].lemma_names())\nprint(syn[1].lemma_names())\n\nsyn = wn.synsets(\"carrot\")\nprint(syn)\nprint(f\"{syn[0].lemma_names()} ... {syn[0].definition()}\")\nprint(f\"{syn[1].lemma_names()} ... {syn[1].definition()}\")\nprint(f\"{syn[2].lemma_names()} ... {syn[2].definition()}\")\nprint(f\"{syn[3].lemma_names()} ... {syn[3].definition()}\")\n\n\n```\n\n Vocabulary size: 8470\n Words not in WordNet 577\n Words not in GloVe 391\n Words in neither WordNet nor GloVe 198\n ['apple']\n ['apple', 'orchard_apple_tree', 'Malus_pumila']\n [Synset('carrot.n.01'), Synset('carrot.n.02'), Synset('carrot.n.03'), Synset('carrot.n.04')]\n ['carrot'] ... deep orange edible root of the cultivated carrot plant\n ['carrot', 'cultivated_carrot', 'Daucus_carota_sativa'] ... perennial plant widely cultivated as an annual in many varieties for its long conical orange edible roots; temperate and tropical regions\n ['carrot'] ... orange root; important source of carotene\n ['carrot'] ... promise of reward as in\n\n\n\n```python\ndef get_word_definitions(w):\n syns = wn.synsets(w)\n defn = start_token + ' ' + w + ' '\n #defn = w + ' '\n for s in syns:\n defn += s.definition()\n defn += ' ' + sep_token + ' '\n defn += ' ' + end_token\n return defn.split()\n\ndef get_word_synonyms(w):\n syns = wn.synsets(w)\n st = {w}\n for s in syns:\n st.update(s.lemma_names())\n return list(st)\n\ndef get_entailment_sentence(wordentail):\n defn1 = get_word_definitions(wordentail[0][0])\n defn2 = get_word_definitions(wordentail[0][1])\n #defn1 = get_word_synonyms(wordentail[0][0])\n #defn2 = get_word_synonyms(wordentail[0][1])\n return (defn1, defn2)\n\ndef fit_simple_sentence_encoding_rnn_with_hyperparameter_search(X, y):\n vocab = [start_token, sep_token, end_token]\n vocab += wordentail_data['vocab']\n glove_embedding, nli_glove_vocab = utils.create_pretrained_embedding(\n glove_lookup, vocab) \n\n model = TorchRNNSentenceEncoderClassifier(\n nli_glove_vocab,\n embedding=glove_embedding,\n bidirectional=True,\n hidden_dim=100,\n early_stopping=True,\n max_iter=3)\n\n param_grid = {\n 'batch_size': [32, 64, 128, 256],\n 'eta': [0.0001, 0.001, 0.01, 0.1]}\n# param_grid = {\n# 'batch_size': [64],\n# 'eta': [0.01]}\n\n bestmod = utils.fit_classifier_with_hyperparameter_search(\n X, y, model, cv=3, param_grid=param_grid)\n\n return bestmod\n \ndef simple_example():\n print(\"Running a simple example\")\n start = time.time()\n\n train = [[get_entailment_sentence(wordentail), wordentail[1]] for wordentail in wordentail_data['train'][:1000]]\n test = [[get_entailment_sentence(wordentail), wordentail[1]] for wordentail in wordentail_data['train'][10:100]]\n\n X_train, y_train = zip(*train)\n X_dev, y_dev = zip(*test)\n\n print(\"Searching for best hyperparameters\")\n model = fit_simple_sentence_encoding_rnn_with_hyperparameter_search(X_train, y_train)\n model.max_iter = 1000\n \n model.fit(X_train, y_train)\n\n predictions = model.predict(X_dev)\n\n # Report:\n print(classification_report(y_dev, predictions, digits=3))\n macrof1 = utils.safe_macro_f1(y_dev, predictions)\n print(time.time() - start)\n\n#simple_example()\n\n```\n\n\n```python\n# PLEASE MAKE SURE TO INCLUDE THE FOLLOWING BETWEEN THE START AND STOP COMMENTS:\n# 1) Textual description of your system.\n# 2) The code for your original system.\n# 3) The score achieved by your system in place of MY_NUMBER.\n# With no other changes to that line.\n# You should report your score as a decimal value <=1.0\n# PLEASE MAKE SURE NOT TO DELETE OR EDIT THE START AND STOP COMMENTS\n\n# IMPORT ANY MODULES BELOW THE 'IS_GRADESCOPE_ENV' CHECK CONDITION. DOING\n# SO ABOVE THE CHECK MAY CAUSE THE AUTOGRADER TO FAIL.\n\n# START COMMENT: Enter your system description in this cell.\n# My peak score was: MY_NUMBER\nif 'IS_GRADESCOPE_ENV' not in os.environ:\n pass\n start_time = time.time()\n \n train = [[get_entailment_sentence(wordentail), wordentail[1]] for wordentail in wordentail_data['train']]\n test = [[get_entailment_sentence(wordentail), wordentail[1]] for wordentail in wordentail_data['dev']]\n\n X_train, y_train = zip(*train)\n \n print(\"Searching for best hyperparameters\")\n model = fit_simple_sentence_encoding_rnn_with_hyperparameter_search(X_train, y_train)\n model.max_iter = 1000\n\n print(\"Fitting the full model\")\n model.fit(X_train, y_train)\n\n X_dev, y_dev = zip(*test)\n predictions = model.predict(X_dev)\n\n # Report:\n rep = classification_report(y_dev, predictions, digits=3)\n print(rep)\n macrof1 = utils.safe_macro_f1(y_dev, predictions)\n res = {\n 'macro-F1': macrof1,\n 'model': model,\n 'train_data': wordentail_data['train'],\n 'assess_data': wordentail_data['dev'],\n }\n \n with open('hw_wordentail_orig_sys.txt', 'w') as f:\n print(f'Took {time.time() - start_time} seconds', file=f)\n print(f'Classification report: {rep}', file=f)\n print(f'Result dictionary: {res}', file=f)\n \n# STOP COMMENT: Please do not remove this comment.\n```\n\n Searching for best hyperparameters\n\n\n Finished epoch 3 of 3; error is 36.265901450067768\n\n Best params: {'batch_size': 32, 'eta': 0.001}\n Best score: 0.767\n Fitting the full model\n\n\n Stopping after epoch 18. Validation score did not improve by tol=1e-05 for more than 10 epochs. Final error is 10.007551011745818\n\n precision recall f1-score support\n \n 0 0.878 0.849 0.863 1732\n 1 0.330 0.386 0.356 334\n \n accuracy 0.774 2066\n macro avg 0.604 0.617 0.609 2066\n weighted avg 0.789 0.774 0.781 2066\n \n\n\n## Bake-off [1 point]\n\nThe goal of the bake-off is to achieve the highest __macro-average F1__ score on a test set that we will make available at the start of the bake-off. The announcement will go out on the discussion forum. To enter, you'll be asked to run `nli.bake_off_evaluation` on the output of your chosen `nli.wordentail_experiment` run. \n\nThe cells below this one constitute your bake-off entry.\n\nThe rules described in the [Your original system](#Your-original-system-[3-points]) homework question are also in effect for the bake-off.\n\nSystems that enter will receive the additional homework point, and systems that achieve the top score will receive an additional 0.5 points. We will test the top-performing systems ourselves, and only systems for which we can reproduce the reported results will win the extra 0.5 points.\n\nLate entries will be accepted, but they cannot earn the extra 0.5 points. Similarly, you cannot win the bake-off unless your homework is submitted on time.\n\nThe announcement will include the details on where to submit your entry.\n\n\n```python\n# Enter your bake-off assessment code into this cell.\n# Please do not remove this comment.\nif 'IS_GRADESCOPE_ENV' not in os.environ:\n pass\n # Please enter your code in the scope of the above conditional.\n ##### YOUR CODE HERE\n\n```\n\n\n```python\n# On an otherwise blank line in this cell, please enter\n# your macro-avg f1 value as reported by the code above.\n# Please enter only a number between 0 and 1 inclusive.\n# Please do not remove this comment.\nif 'IS_GRADESCOPE_ENV' not in os.environ:\n pass\n # Please enter your score in the scope of the above conditional.\n ##### YOUR CODE HERE\n\n```\n", "meta": {"hexsha": "8a0a2c14de10843417f61ab1b52e1a516ae79295", "size": 42688, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "hw_wordentail_bkup.ipynb", "max_stars_repo_name": "ammarhusain/cs224u", "max_stars_repo_head_hexsha": "bbdb0aaa6b7437481e2e1fab8e12bbf1996eecd1", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "hw_wordentail_bkup.ipynb", "max_issues_repo_name": "ammarhusain/cs224u", "max_issues_repo_head_hexsha": "bbdb0aaa6b7437481e2e1fab8e12bbf1996eecd1", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "hw_wordentail_bkup.ipynb", "max_forks_repo_name": "ammarhusain/cs224u", "max_forks_repo_head_hexsha": "bbdb0aaa6b7437481e2e1fab8e12bbf1996eecd1", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 34.8189233279, "max_line_length": 642, "alphanum_fraction": 0.5621017616, "converted": true, "num_tokens": 7653, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.782662489091802, "lm_q1q2_score": 0.43999445949219834}} {"text": "# Objective\n\nIn this lab, we will investigate the effects of temperature on electronic devices -- resistors that change value with temperature and thermally-generated free charge carriers on the reverse-bias current through a diode.\n\n**NOTE:** Do not skim-read the procedure. The wording is precise and specific, skipping steps or missing things will *guarantee* you will not finish your measurements within the scheduled lab time. Many problems will be avoided if you do not try to rush through the procedure.\n\n**READ THE PROCEDURE CAREFULLY**\n\n**READ THE PROCEDURE CAREFULLY**\n\n**slow down...!**\n\n# Theory\n\nSkip this section for now and start with the Procedure section, read this while waiting for your measurements to stabilize.\n\n## Silicon *pn* diode\n\nThe silicon *pn* junction diode has an ideal current-voltage relationship of:\n\n\\begin{equation}\nI_D = I_s \\left(e^{V_D \\cdot q / (n k_B T)} - 1\\right)\n\\end{equation}\n\nThe term $k_B T /q$ is known as the *thermal voltage* $V_T$ and is approximately $26 \\mathrm{mV}$ at room temperature, $n$ is called the *ideality factor* and is 1 for an ideal diode but larger (1--2 range) for a real diode. $I_S$ is called the *saturation current*. For reverse-bias (negative) voltages larger than a few hundred $\\mathrm{mV}$, the diode current can be considered $\\approx -I_S$, to a first-approximation.\n\nBesides the $1/T$ term explicitly showing up in the equation's exponent, the saturation current is a strong function of temperature as well. It can be calculated from geometrical and material parameters as:\n\n\\begin{equation}\nI_S = A q n_i^2 \\left(\\dfrac{D_n}{N_A L_n} + \\dfrac{D_p}{N_D L_p}\\right)\n\\end{equation}\n\nwhere $A$ is the device's cross-sectional area, $L_n$ and $L_p$ are electron and hole \"diffusion lengths,\" respectively, $D_{n,p}$ are the diffusion constants, and $N_{A,D}$ are the doping for the *p* and *n* sides. These parameters can be considered constants here.\n\nThe Einstein Relation can be proven to relate the ratio of the diffusion constant to mobility in a material as:\n\n\\begin{equation}\n\\dfrac{D_{n,p}}{\\mu_{n,p}} = \\dfrac{k_B T}{q} = V_T\n\\end{equation}\n\nSolving for the diffusion constant, $D_{n,p} = \\mu_{n,p} \\dfrac{k_B T}{q}$, re-write the saturation current and factor out temperature:\n\n\\begin{equation}\nI_S = A q n_i^2 T \\left(\\dfrac{\\mu_n k_B}{q N_A L_n} + \\dfrac{\\mu_p k_B}{q N_D L_p}\\right)\n\\end{equation}\n\nIgnore the fact that mobility changes with temperature due to several effects, reaching a maximum at an intermediate temperature.\n\nThe intrinsic carrier density also varies with temperature:\n\n\\begin{equation}\nn_i^2 = B T^3 e^{-E_G/(n k_b T)}\n\\end{equation}\n\nso,\n\n\\begin{equation}\nI_S = A q B T^4 e^{-E_G/(n k_b T)} \\left(\\dfrac{\\mu_n k_B}{q N_A L_n} + \\dfrac{\\mu_p k_B}{q N_D L_p}\\right)\n\\end{equation}\n\nWe are interested in the temperature dependence of $I_S$ for a specific device, so collapse all the temperature-independent terms into a single constant:\n\n\\begin{equation}\nI_S(T) = C \\cdot T^4 e^{-E_G/(n k_b T)}\n\\end{equation}\n\n\n## Temperature-dependent resistors\n\nAll materials change resistivity with changes in temperature. The different types of resistor materials have varying sensitivity to temperature. This change in resistance versus temperature for \"normal\" resistors is specified as its temperature coefficient of resistance (TCR) and has units of percent or PPM per degree. Usually, $T_{ref}$ is room temperature.\n\n\\begin{equation}\nR(T) = R_{T_{ref}}\\left[ 1 + TCR\\left(\\frac{T}{T_{ref}}\\right) \\right]\n\\end{equation}\n\nSee (Temperature Coefficient of Copper)[http://www.cirris.com/learning-center/general-testing/special-topics/177-temperature-coefficient-of-copper] for more information.\n\nThermistors are devices which are *intended to* change their resistance with temperature. Their obvious application is to measure temperature. Other interesting applications include measurement of air speed and density in aircraft as the airstream cools the device. Their resistance is dependent on temperature as approximately:\n\n\\begin{equation}\nR(T) = R_R \\exp \\left[B\\left(\\dfrac{1}{T} - \\dfrac{1}{T_R}\\right)\\right]\n\\end{equation}\n\nwhere $B$ is a constant for the specific device, and $R_R$ and $T_R$ are the resistance at the rated temperature. For the thermistors used in the lab, they are rated $1\\mathrm{k\\Omega}$ at $25^\\circ C$ ($R_R=1k$, $T_R=25^\\circ C$) with constant $B = 3930 K$. Be mindful of the temperature units in your calculations ($0^\\circ C = 273.15 K$). See the file `NTC-general-technical-information.pdf` on Blackboard for more background information and `NTC_Leaded_disks_M891.pdf` for the specific values.\n\nA more accurate determination of resistance or temperature is obtained from using actual values and temperatures provided in tables in the device's datasheet and interpolating between the values. Page 2 of the `NTC-standardizedrt.pdf` file on Blackboard describes the calculations and gives an example. We are using the $1 \\mathrm{k\\Omega}$ rated thermistor, whose data table is number \"`1009`\" in the datasheet, file `NTC_Leaded_disks_M891.pdf` on Blackboard.\n\n\n\n\n# Procedure\n\n\n\nThe general format of these instructions is: the first 1-2 sentences are your task to do and the rest of the item is a short description of what the task is about.\n\n**Goal:** Setup your oscilloscope to precisely measure very small DC values in the presence of noise and interference.\n\n1. Turn down both outputs of the benchtop power supply to zero before turning it on.\n\n1. Connect the $+6\\,\\mathrm{V}$ power supply terminal to the node labeled `6V_or_0-20V` in **Figure 1**.\n\n1. Connect the $+20\\,\\mathrm{V}$ power supply terminal to the node labeled `0-20V` in **Figure 1**.\n\n1. Connect the `COM` terminal of the power supply to the node labeled `GND` in **Figure 1**.\n\n1. Connect oscilloscope channel 1's \"+\" terminal (the tip) to the triangle side of the diode and the black ground clip lead to the node labeled `GND`.\n\n * This makes it appear that channel 1 is connected across *nothing*, which is not necessarily intuitive. The currents we will be measuring will start at a few $\\mathrm{nA}$, a million times below the range of cheap ammeters (the nice multimeters can measure this, but we don't have enough of them for every lab station). With this connection scheme, the saturation current $I_S$ is forced to go through the $10 \\,\\mathrm{M\\Omega}$ input resistance of the oscilloscope+probe combination. It is operating as a *current shunt*. The voltage measured across this resistance by the scope is proportional to the current through it via Ohms \"Law\". This current $I_S$ is shown in **Figure 1**.\n\n1. Connect oscilloscope channel 2's \"+\" terminal to the node between the MiniChamber's thermistor and the $2.7 \\,\\mathrm{k\\Omega}$ resistor.\n\n1. Turn on the power supply. The outputs should already have been both set to zero.\n\n1. Raise the $+6\\,\\mathrm{V}$ power supply output to about $+6\\,\\mathrm{V}$ and record the actual value shown on the power supply's meter. It is not critical to be at exactly $6.00$, but it **is** important to use the true measured value for your data collection and calculations.\n\n1. Touch the \"Default Setup\" key on the scope then select the \"Factory Default\" option in the soft menu (confirm that you want to reset to factory settings). This completely resets your scope's settings. Default Setup alone still leaves some settings unchanged.\n\n1. Touch the \"Acquire\" key on the scope and change the Acquire Mode to \"High Res\". The manual says the scope displays the average of the many samples taken by the front-end for each displayed pixel in this mode. \"Normal\" mode merely displays the first sample value in the interval.\n\n1. Change the trigger system to **trigger** on the **edge** of the **\"Line\" source**. The signal you see is the superposition of the DC value to be measured, noise, and the $60 \\,\\mathrm{Hz}$ electric and magnetic fields in the room from the building power. The signals we are interested in are DC only, but the probe will pick up a large amount of powerline interference as you can see on the screen. This keeps the display stable so we can then average-out the display and make more precise measurements.\n\n1. Change the vertical scale and vertical position of both channels ($V_{thm}$ and $V_{I_S}$) to show both the entire signal waveform and the channel's $0 \\,\\mathrm{V}$ reference level. The goal is to maximize the amount of screen area the signal takes up while still keeping the 0 reference and signal on the screen without spilling above or below. This gives you the most accurate measurements.\n\n1. Change the horizontal scale to show at least 5 periods of the $60 \\,\\mathrm{Hz}$ interference waveform.\n\n1. Using the measurement functions, display the *full-screen averages* of both channel voltages.\n\n1. Observe these measurements and record them when they stop slowly changing up or down.\n\n1. Increase the $0-20 \\,\\mathrm{V}$ power supply output in $2.0 \\,\\mathrm{V}$ increments until it reaches $6.0 \\,\\mathrm{V}$ and then $1.0 \\,\\mathrm{V}$ increments. (0, 2, 4, 6, **7**, **8**, **9**, ...)\n\n * At each increment, wait for the temperature to reach equilibrium, indicated when both measurements stop changing. Calculate a time to wait between making a change and taking each measurement that allows you to finish all your measurements at 10 minutes before the lab session ends.\n \n * These changes will be slow and small at first. $V_{I_S}$ will be around $40 \\,\\mathrm{mV}$ at the beginning. **(You)** Calculate the expected value of $V_{thm}$ at room temperature, when the thermistor resistance is around $1 \\,\\mathrm{k\\Omega}$.\n \n * At the end of each increment's time interval:\n * Record **5** data values:\n * The $20 \\mathrm{V}$ supply voltage\n * The $20 \\mathrm{V}$ supply current\n * The voltage at the node labeled `6V_or_0-20V`.\n * $V_{I_S}$ (oscilloscope channel 1)\n * $V_{thm}$ (oscilloscope channel 2)\n \n * When the $20 \\,\\mathrm{V}$ supply increment reaches $6 \\,\\mathrm{V}$: Move the power supply connection to node labeled `6V_or_0-20V` from the $6\\,\\mathrm{V}$ output to the $20 \\,\\mathrm{V}$ output. The diode, heater, and $2.7 \\,\\mathrm{k\\Omega}$ resistor will therefore all be connected to the $20 \\,\\mathrm{V}$ output and the $6 \\mathrm{V}$ power supply will then be disconnected.\n \n * Stop the experiment when either the thermistor reaches the calculated resistance for $100 \\,\\mathrm{^\\circ C}$ **or** the MiniChamber appears to be becoming damaged (smoke, noise, etc.).\n \n\n\n1. While waiting for each measurement to stabilize: setup your calculations and plots, viewing each new data point as you enter them. It is useful to pre-define the \"plot range\" to be the full number of rows you expect to have, this way you do not need to update that range for each new data point.\n\n1. Calculate what the thermistor's resistance will be at $100 \\mathrm{^\\circ C}$ using **Equation 9**. This is the maximum temperature the MiniChamber should reach with the corresponding thermistor resistance.\n\\newpage\n## Analysis\n\nUse Excel or Google Sheets to perform these calculations.\n\n1. Using circuit analysis and the thermistor datasheet, determine the **thermistor resistance** and **temperature** for each of your $V_{thm}$ measurements.\n\n2. Calculate the **heat power** supplied to the system by the heater resistor at each measurement point.\n\n3. Using circuit analysis, calculate the **diode saturation current $I_S$** as measured by the oscilloscope probe $V_{Is}$. (Technically we are measuring the diode's leakage current instead.)\n\n4. From the heater resistor's applied voltage and resulting measured current, calculate the **heater's effective resistance**. It changes with temperature :(\n\nAt this point, your spreadsheet will have 5 columns of raw data, and 5 columns of calculated values from this data. Keep the columns in separate groups to visually differentiate from raw data measurements and derived numbers; this is good lab practice.\n\nFrom the numbers, generate the following plots:\n(remember: \"$y$ vs. $x$\")\n\n1. $I_S$ vs. temperature\n\n * On this figure, also plot the theoretical value of $I_S$ versus temperature. Choose the constants $C$ and $n$ which best fit your measured data. $n$ will be between $1$ and $2$.\n * Use a log scale for $I_S$ and linear scale for temperature.\n * Compare the measured and theoretical plots and comment on them.\n * Compare this plot to Figure 5 of the `1N4148_NXP.pdf` datasheet.\n\n2. Temperature vs. heat power\n\n * Also calculate the slope in units of $^\\circ \\mathrm{C} / W$.\n * This represents the thermal resistance of the heat transfer from the heater resistor to the room of the entire device. ECEs use this type of information in calculating the size of heatsinks.\n \n3. Heater resistance vs. temperature\n\n * Find the slope as $\\mathrm{PPM} / ^\\circ \\mathrm{C}$. Parts-per-million (PPM) is like percentage, but uses $1 / 10^6$ instead of the $1 / 100$ factor for percentage. PPM is a very common term for specifying small relative measurements.\n\n## Report\n\nRefer to the document `Lab Report Guidelines.pdf` on Blackboard for the format for your report.\n", "meta": {"hexsha": "8bdc4100a58c7d4066802eec8baf1dd8aa35eef3", "size": 16831, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lab2-temperature.ipynb", "max_stars_repo_name": "etihwnad/misc-notebooks", "max_stars_repo_head_hexsha": "e4d449382bd36732ae0cd5daf338f9e2dceb0bc1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-03-16T08:20:07.000Z", "max_stars_repo_stars_event_max_datetime": "2020-03-16T08:20:07.000Z", "max_issues_repo_path": "lab2-temperature.ipynb", "max_issues_repo_name": "etihwnad/misc-notebooks", "max_issues_repo_head_hexsha": "e4d449382bd36732ae0cd5daf338f9e2dceb0bc1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "lab2-temperature.ipynb", "max_forks_repo_name": "etihwnad/misc-notebooks", "max_forks_repo_head_hexsha": "e4d449382bd36732ae0cd5daf338f9e2dceb0bc1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 55.1836065574, "max_line_length": 707, "alphanum_fraction": 0.6498128453, "converted": true, "num_tokens": 3372, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.6791787056691698, "lm_q1q2_score": 0.4399707552035091}} {"text": "\\title{Digital Latches with myHDL}\n\\author{Steven K Armour}\n\\maketitle\n\n# Refs\n@book{brown_vranesic_2014, place={New York, NY}, edition={3}, title={Fundamentals of digital logic with Verilog design}, publisher={McGraw-Hill}, author={Brown, Stephen and Vranesic, Zvonko G}, year={2014} },\n@book{lameres_2017, title={Introduction to logic circuits & logic design with Verilog}, publisher={springer}, author={LaMeres, Brock J}, year={2017} }\n\n# Acknowledgments\n\nAuthor of **myHDL** [Jan Decaluwe](http://www.myhdl.org/users/jandecaluwe.html) and the author of the **myHDL Peeker** [XESS Corp.](https://github.com/xesscorp/myhdlpeek)\n\n[**Draw.io**](https://www.draw.io/)\n\n**Xilinx**\n\n# Python Libraries Utilized\n\n\n```python\nimport numpy as np\nimport pandas as pd\nfrom sympy import *\ninit_printing()\n\nfrom myhdl import *\nfrom myhdlpeek import *\nimport random\n\n#python file of convince tools. Should be located with this notebook\nfrom sympy_myhdl_tools import *\n```\n\n# Latches vs Flip-Flops\n\nLatches and Flip-Flops are both metastaple logic circuit tobologies in that once loaded with a state they hold that state information till that state is upset by a new state or a reset command. But the diffrance between the two is that Flip-Flops are clock controlled devices built upon Latches where as Latches are not clock dependent\n\n# SR-Latch\n\n## Symbol and Internals\nThe Symbol for a SR-Latch and one representation of it's internals is shown below\n\n\n## Definition\n\n## State Diagram\n\n## myHDL SR-Latch Gate and Testing\n\n\nNeed Help Getting this Latch via Combo Cirucits working geting AlwayCombError in using out signal as argument in out signals next state out\ndef LatchCombo(S_in, rst, Q_out):\n \n\n @always_comb\n def logic():\n Q_out.next=(not rst) and (S_in or Q_out)\n return logicPeeker.clear()\nS_in, rst, Q_out=[Signal(bool(0)) for _ in range(3)]\nPeeker(S_in, 'S_in'); Peeker(rst, 'rst')\nPeeker(Q_out, 'Q_out')\n\nDUT=LatchCombo(S_in=S_in, rst=rst, Q_out=Q_out)\n\ninputs=[S_in, rst]\nsim=Simulation(DUT, Combo_TB(inputs), *Peeker.instances()).run() \nPeeker.to_wavedrom(start_time=0, stop_time=2*2**len(inputs), tock=True,\n title='AND 2 gate simulation',\n caption=f'after clock cycle {2**len(inputs)-1} ->random input')\n\n```python\n\n```\n\n## myHDL SR-Latch Behavioral and Testing\n\n\n```python\ndef SRLatch(S_in, rst, Q_out, Qn_out):\n @always_comb\n def logic():\n if S_in and rst==0:\n Q_out.next=1\n Qn_out.next=0\n \n elif S_in==0 and rst:\n Q_out.next=0\n Qn_out.next=1\n\n elif S_in and rst:\n Q_out.next=0\n Qn_out.next=0\n\n return logic\n```\n\n\n```python\nS_in, rst, Q_out, Qn_out=[Signal(bool(0)) for _ in range(4)]\nPeeker.clear()\n\nPeeker(S_in, 'S_in'); Peeker(rst, 'rst')\nPeeker(Q_out, 'Q_out'); Peeker(Qn_out, 'Qn_out')\n\nDUT=SRLatch(S_in=S_in, rst=rst, Q_out=Q_out, Qn_out=Qn_out)\ninputs=[S_in, rst]\nsim=Simulation(DUT, Combo_TB(inputs), *Peeker.instances()).run() \nPeeker.to_wavedrom(start_time=0, stop_time=2*2**len(inputs), tock=True,\n title='SRLatch Behavioral simulation',\n caption=f'after clock cycle {2**len(inputs)-1} ->random input')\n\n```\n\n : No more events\n\n\n\n
    \n\n\n\n\n\n```python\nMakeDFfromPeeker(Peeker.to_wavejson(start_time=0, stop_time=2**len(inputs) -1))\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    Q_outQn_outS_inrst
    00000
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    \n\n\n\n## myHDL SR-Latch Behavioral HDL Synthesis\n\n\n```python\ntoVerilog(SRLatch, S_in, rst, Q_out, Qn_out)\n#toVHDL(SRLatch, S_in, rst, Q_out, Qn_out)\n_=VerilogTextReader('SRLatch')\n```\n\n ***Verilog modual from SRLatch.v***\n \n // File: SRLatch.v\n // Generated by MyHDL 0.9.0\n // Date: Thu Oct 26 00:40:43 2017\n \n \n `timescale 1ns/10ps\n \n module SRLatch (\n S_in,\n rst,\n Q_out,\n Qn_out\n );\n \n \n input S_in;\n input rst;\n output Q_out;\n reg Q_out;\n output Qn_out;\n reg Qn_out;\n \n \n \n \n \n \n always @(rst, S_in) begin: SRLATCH_LOGIC\n if ((S_in && (rst == 0))) begin\n Q_out = 1;\n Qn_out = 0;\n end\n else if (((S_in == 0) && rst)) begin\n Q_out = 0;\n Qn_out = 1;\n end\n else if ((S_in && rst)) begin\n Q_out = 0;\n Qn_out = 0;\n end\n end\n \n endmodule\n \n\n\nThe following shows the **Xilinx**'s _Vivado 2016.1_ RTL generated schematic of our Behaviorla SRLatch from the synthesised verilog code. We can see that the systhizied version is quite apstract from fig lakdfjkaj. \n\n\n# Gated SR-Latch\n\n## myHDL SR-Latch Behavioral and Testing\n\n\n```python\ndef GSRLatch(S_in, rst, ena, Q_out, Qn_out):\n @always_comb\n def logic():\n if ena:\n if S_in and rst==0:\n Q_out.next=1\n Qn_out.next=0\n\n elif S_in==0 and rst:\n Q_out.next=0\n Qn_out.next=1\n\n elif S_in and rst:\n Q_out.next=0\n Qn_out.next=0\n else:\n pass\n\n return logic\n```\n\n\n```python\nS_in, rst, ena, Q_out, Qn_out=[Signal(bool(0)) for _ in range(5)]\nPeeker.clear()\n\nPeeker(S_in, 'S_in'); Peeker(rst, 'rst'); Peeker(ena, 'ena')\nPeeker(Q_out, 'Q_out'); Peeker(Qn_out, 'Qn_out')\n\nDUT=GSRLatch(S_in=S_in, rst=rst, ena=ena, Q_out=Q_out, Qn_out=Qn_out)\ninputs=[S_in, rst, ena]\nsim=Simulation(DUT, Combo_TB(inputs), *Peeker.instances()).run() \nPeeker.to_wavedrom(start_time=0, stop_time=2*2**len(inputs), tock=True,\n title='GSRLatch Behavioral simulation',\n caption=f'after clock cycle {2**len(inputs)-1} ->random input')\n\n```\n\n : No more events\n\n\n\n
    \n\n\n\n\n\n```python\nMakeDFfromPeeker(Peeker.to_wavejson(start_time=0, stop_time=2**len(inputs) -1))\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    Q_outQn_outS_inenarst
    000000
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    \n
    \n\n\n\n## myHDL SR-Latch Behavioral HDL Synthesis\n\n\n```python\ntoVerilog(GSRLatch, S_in, rst, ena, Q_out, Qn_out)\n#toVHDL(GSRLatch, S_in, rst,ena, Q_out, Qn_out)\n_=VerilogTextReader('GSRLatch')\n```\n\n ***Verilog modual from GSRLatch.v***\n \n // File: GSRLatch.v\n // Generated by MyHDL 0.9.0\n // Date: Thu Oct 26 00:40:44 2017\n \n \n `timescale 1ns/10ps\n \n module GSRLatch (\n S_in,\n rst,\n ena,\n Q_out,\n Qn_out\n );\n \n \n input S_in;\n input rst;\n input ena;\n output Q_out;\n reg Q_out;\n output Qn_out;\n reg Qn_out;\n \n \n \n \n \n \n always @(ena, S_in, rst) begin: GSRLATCH_LOGIC\n if (ena) begin\n if ((S_in && (rst == 0))) begin\n Q_out = 1;\n Qn_out = 0;\n end\n else if (((S_in == 0) && rst)) begin\n Q_out = 0;\n Qn_out = 1;\n end\n else if ((S_in && rst)) begin\n Q_out = 0;\n Qn_out = 0;\n end\n end\n else begin\n // pass\n end\n end\n \n endmodule\n \n\n\nThe following shows the **Xilinx**'s _Vivado 2016.1_ RTL generated schematic of our Behaviorla Gated SRLatch from the synthesised verilog code. We can see that the systhizied version is quite apstract from fig lakdfjkaj. \n\n\n# D-Latch\n\n## myHDL Behavioral D-Latch and Testing\n\n\n```python\ndef DLatch(D_in, ena, Q_out, Qn_out):\n #Normal Qn_out is not specifed since a not gate is so easily implimented\n @always_comb\n def logic():\n if ena:\n Q_out.next=D_in\n Qn_out.next=not D_in\n return logic\n```\n\n\n```python\nD_in, ena, Q_out, Qn_out=[Signal(bool(0)) for _ in range(4)]\n\nPeeker.clear()\n\nPeeker(D_in, 'D_in'); Peeker(ena, 'ena')\nPeeker(Q_out, 'Q_out'); Peeker(Qn_out, 'Qn_out')\n\nDUT=DLatch(D_in=D_in, ena=ena, Q_out=Q_out, Qn_out=Qn_out)\ninputs=[D_in, ena]\nsim=Simulation(DUT, Combo_TB(inputs), *Peeker.instances()).run() \nPeeker.to_wavedrom(start_time=0, stop_time=2*2**len(inputs), tock=True,\n title='DLatch Behavioral simulation',\n caption=f'after clock cycle {2**len(inputs)-1} ->random input')\n\n```\n\n : No more events\n\n\n\n
    \n\n\n\n\n\n```python\nMakeDFfromPeeker(Peeker.to_wavejson(start_time=0, stop_time=2**len(inputs) -1))\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    D_inQ_outQn_outena
    00000
    10011
    21010
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    \n
    \n\n\n\n## myHDL DLatch Behavioral HDL Synthesis\n\n\n```python\ntoVerilog(DLatch, D_in, ena, Q_out, Qn_out)\n#toVHDL(DLatch,D_in, ena, Q_out, Qn_out)\n_=VerilogTextReader('DLatch')\n```\n\n ***Verilog modual from DLatch.v***\n \n // File: DLatch.v\n // Generated by MyHDL 0.9.0\n // Date: Thu Oct 26 00:40:46 2017\n \n \n `timescale 1ns/10ps\n \n module DLatch (\n D_in,\n ena,\n Q_out,\n Qn_out\n );\n \n \n input D_in;\n input ena;\n output Q_out;\n reg Q_out;\n output Qn_out;\n reg Qn_out;\n \n \n \n \n \n \n always @(ena, D_in) begin: DLATCH_LOGIC\n if (ena) begin\n Q_out = D_in;\n Qn_out = (!D_in);\n end\n end\n \n endmodule\n \n\n\nThe following shows the **Xilinx**'s _Vivado 2016.1_ RTL generated schematic of our myHDL Dlatch with a exsplisit $\\bar{Q}$ verilog code. Note that becouse $\\bar{Q}$ is not normal declared in HDL code Vivado produced two RTL DLatchs and used a NOT Gate to acount for the negated output\n\n\n# Examples\n", "meta": {"hexsha": "cb2526780817698f2a5dd865c3d701c15f818f26", "size": 27397, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "myHDL_DigLogicFundamentals/.ipynb_checkpoints/myHDL_Latches-checkpoint.ipynb", "max_stars_repo_name": "PyLCARS/PythonUberHDL", "max_stars_repo_head_hexsha": "f7ae2293d6efaca7986d62540798cdf061383d06", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 31, "max_stars_repo_stars_event_min_datetime": "2017-10-09T12:15:14.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-28T09:05:21.000Z", "max_issues_repo_path": "myHDL_DigLogicFundamentals/myHDL_Latches.ipynb", "max_issues_repo_name": "cfelton/PythonUberHDL", "max_issues_repo_head_hexsha": "f7ae2293d6efaca7986d62540798cdf061383d06", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "myHDL_DigLogicFundamentals/myHDL_Latches.ipynb", "max_forks_repo_name": "cfelton/PythonUberHDL", "max_forks_repo_head_hexsha": "f7ae2293d6efaca7986d62540798cdf061383d06", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 12, "max_forks_repo_forks_event_min_datetime": "2018-02-09T15:36:20.000Z", "max_forks_repo_forks_event_max_datetime": "2021-04-20T21:39:12.000Z", "avg_line_length": 27.0989119683, "max_line_length": 648, "alphanum_fraction": 0.432529109, "converted": true, "num_tokens": 4061, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.7025300573952054, "lm_q1q2_score": 0.4398710541647257}} {"text": "\n\n# **MAC023 - Mec\u00e2nica das Estruturas**\n\n# ME-03 - Terceira Avalia\u00e7\u00e3o de Conhecimentos\n\nAlunos: \n\nBrian Maia \\\\\nMathews Edwirds \\\\\n\n# Condi\u00e7\u00f5es Gerais\n\nEsta avalia\u00e7\u00e3o tem como objetivo avaliar os conhecimentos adquiridos na primeira parte da disciplina de Mec\u00e2nica das Estruturas.\n\n\n---\n\nAs condic\u00f5es abaixo devem ser observadas: \n\n1. Ser\u00e3o formadas equipes e cada uma delas com no m\u00ednimo **2** e no m\u00e1ximo **3** integrantes. \n\n2. A avalia\u00e7\u00e3o ser\u00e1 realizada por meio da entrega de uma c\u00f3pia deste notebook com as solu\u00e7\u00f5es desenvolvidas at\u00e9 a data estipulada de entrega.\n\n\n3. Da entrega da avalia\u00e7\u00e3o.\n * Os documentos necess\u00e1rios para a entrega do trabalho s\u00e3o (1) os c\u00f3digos desenvolvidos pela equipe e (2) v\u00eddeo com a descri\u00e7\u00e3o da solu\u00e7\u00e3o. \n * A equipe deve usar este modelo de notebook para desenvolver os c\u00f3digos. \n * Os c\u00f3digos podem ser desenvolvidos combinado a linguagem LaTeX e computa\u00e7\u00e3o simb\u00f3lica ou computa\u00e7\u00e3o num\u00e9rica quando necess\u00e1rio.\n * Os gr\u00e1ficos necess\u00e1rios para a apresenta\u00e7\u00e3o da solu\u00e7\u00e3o devem estar embutidos no notebook.\n\n4. Da distribui\u00e7\u00e3o das quest\u00f5es.\n * A quantidade de quest\u00f5es ser\u00e1 a mesma para cada grupo. \n * Ser\u00e3o atribu\u00eddas as mesmas quest\u00f5es para todos os grupos.\n * A pontuac\u00e3o referente a cada quest\u00e3o ser\u00e1 igualit\u00e1ria e o valor total da avalia\u00e7\u00e3o ser\u00e1 100 pontos.\n\n5. As equipes devem ser formadas at\u00e9 \u00e0s **23:59 horas o dia 22/02/2022** por meio do preenchimento da planilha [[MAC023] Forma\u00e7\u00e3o das Equipes](https://docs.google.com/spreadsheets/d/1Dlftymao970nnrE4mu958iP8nMqKqSuhHiiLH91BKpQ/edit#gid=986079240).\n\n6. A forma\u00e7\u00e3o das equipes pode ser acompanhada arquivo indicado acima. Cada equipe ser\u00e1 indentificada por uma letra em ordem alfab\u00e9tica seguida do n\u00famero 3 (A3, B3, C3, e assim por diante). O arquivo est\u00e1 aberto para edi\u00e7\u00e3o e pode ser alterado pelos alunos at\u00e9 a data estipulada.\n\n7. Equipes formadas ap\u00f3s a data estabelecida para a forma\u00e7\u00e3o das equipes ter\u00e3o a nota da avalia\u00e7\u00e3o multiplicada por um coeficiente de **0.80**.\n\n8. A equipe deve indicar no arquivo de indica\u00e7\u00e3o de equipes um respons\u00e1vel pela entrega do projeto. \n * Somente o respons\u00e1vel pela entrega deve fazer o upload do arquivo na plataforma\n\n9. A entrega dos projetos deve ocorrer at\u00e9 \u00e0s **23:59 do dia 25/02/2022** na plataforma da disciplina pelo respons\u00e1vel pela entrega. \n * Caso a entrega seja feita por outro integrante diferente daquele indicado pela pela equipe a avalia\u00e7\u00e3o ser\u00e1 desconsiderada e n\u00e3o ser\u00e1 corrigida at\u00e9 que a a condi\u00e7\u00e3o de entrega seja satisfeita.\n\n10. Quaisquer d\u00favidas ou esclarecimentos devem ser encaminhadas pela sala de aula virtual.\n\n\n\n## (Q1) Implementa\u00e7\u00e3o do m\u00e9todo dos deslocamentos\n\nConsidere a estrutura abaixo de se\u00e7\u00e3o transversal retangular com 8 cm de base por 15 cm de altura. O material possui m\u00f3dulo de elasticidade de 70 GPa. Caso necess\u00e1rio, considere o coeficiente de Poisson igual a 0,30.\n\n\n\nProceda como se pede:\n\n1. Enumere os graus de liberdade da estrutura\n2. Determine os graus de liberdade associados as deslocabilidades\n3. Determine a matriz de rigidez do membro AC\n4. Determine a matriz de rigidez do membro CD\n5. Determine a matriz de rigidez do membro DB\n6. Determine a matriz de rigidez global da estrutura\n7. Apresente os deslocamentos horizontal, vertical e a rota\u00e7\u00e3o do ponto C.\n\n\n\n\n\n```\nimport numpy as np\nimport sympy as sp\nsp.init_printing(use_latex=True)\n```\n\n# Leitura do arquivo de dados\n\n\n```\n# leitura do arquivo de entrada\nfrom io import BytesIO\nimport pandas as pd\nimport requests\n\n# acesso via link\nlink='https://docs.google.com/spreadsheet/ccc?key=1ZBDTypR4MOZqEiIXDqxt2JfW2Oy20SYoNsi2QHJjxKk&output=csv'\n\nr = requests.get(link)\ndata = r.content\n \ndf = pd.read_csv(BytesIO(data), header=0) #, index_col=0\ndf\n```\n\n\n\n\n\n
    \n
    \n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    MAC023 - Estrutura do trabalho 03Unnamed: 1Unnamed: 2Unnamed: 3Unnamed: 4Unnamed: 5Unnamed: 6
    04NaNNaNNaNNaNNaNNaN
    100.000000e+000.0000000.01.01.00.0
    216.000000e+000.0000000.00.00.00.0
    321.200000e+010.0000000.00.00.00.0
    431.500000e+010.0000000.00.01.00.0
    51NaNNaNNaNNaNNaNNaN
    607.000000e+070.300000NaNNaNNaNNaN
    71NaNNaNNaNNaNNaNNaN
    801.200000e-020.000023NaNNaNNaNNaN
    93NaNNaNNaNNaNNaNNaN
    1000.000000e+001.0000000.00.0NaNNaN
    1111.000000e+002.0000000.00.0NaNNaN
    1222.000000e+003.0000000.00.0NaNNaN
    132NaNNaNNaNNaNNaNNaN
    1410.000000e+00-20.0000000.0NaNNaNNaN
    1520.000000e+00-20.0000000.0NaNNaNNaN
    \n
    \n \n\n \n\n \n
    \n
    \n\n\n\n\n\n```\n#\n# Vari\u00e1veis do programa -- descreva os nomes das vari\u00e1veis\n#\n\n# numnp : n\u00famero de pontos nodais\n# neq : n\u00famero de equa\u00e7\u00f5es\n# coord : coordenadas nodais\n# idx : array com os \u00edndices das equa\u00e7\u00f5es\n# nummat : n\u00famero de materiais\n\n# E : m\u00f3dulo de elasticidade\n# nu : coeficiente de poisson\n# gp : matriz com os dados definidos para cada uma das se\u00e7\u00f5es transversais\n# nlcase : n\u00famero de casos de carregamento\n# f : vetor de for\u00e7as nodais (global)\n# matp : propriedades dos materiais (A, E)\n# conec : matriz de conectividades\n# geop : array com as propriedades geom\u00e9tricas (A, I)\n# numgp : n\u00famero de se\u00e7\u00f5es geom\u00e9tricas\n```\n\n# Informa\u00e7\u00e3o dos n\u00f3s, suas coordenadas e as condi\u00e7\u00f5es de deslocabilidade\n\n\n```\nk=0\nnumnp = int(df.iloc[k,0])\ncoord=np.zeros((numnp,3))\nidx=np.zeros((numnp,3), int)\nk+=1\n\nprint(\"N\u00famero de n\u00f3s da estrutura: {}\".format(numnp))\nprint(\"Matriz com coordenadas (x,y,z) de cada um dos n\u00f3s: \\n{}\".format(coord))\nprint(\"Matriz de poss\u00edveis deslocamentos em cada uma das dire\u00e7\u00f5es x,y,z: \\n{}\".format(idx))\n```\n\n N\u00famero de n\u00f3s da estrutura: 4\n Matriz com coordenadas (x,y,z) de cada um dos n\u00f3s: \n [[0. 0. 0.]\n [0. 0. 0.]\n [0. 0. 0.]\n [0. 0. 0.]]\n Matriz de poss\u00edveis deslocamentos em cada uma das dire\u00e7\u00f5es x,y,z: \n [[0 0 0]\n [0 0 0]\n [0 0 0]\n [0 0 0]]\n\n\n\n```\nfor i in range(numnp):\n print(df.iloc[k+i,:].values)\n v = df.iloc[k+i,:].values\n node=int(v[0])\n x,y,z = v[1:4]\n coord[node,:] = v[1:4]\n idx[node,:] = [ int(j) for j in v[4:7] ]\n```\n\n [0. 0. 0. 0. 1. 1. 0.]\n [1. 6. 0. 0. 0. 0. 0.]\n [ 2. 12. 0. 0. 0. 0. 0.]\n [ 3. 15. 0. 0. 0. 1. 0.]\n\n\n\n```\ndisplay(numnp, coord, idx)\n```\n\n# Leitura das propriedades dos materiais\n\n\n```\nk=1\nk=k+numnp\nnummat=df.iloc[k,0]\nEy=np.zeros(nummat)\nnu=np.zeros(nummat)\n\nprint(\"Posicionamento na linha correta do arquivo: k={}\".format(k))\nprint(\"N\u00famero de materiais definidos no arquivo de entrada de dados: {}\".format(nummat))\nprint(\"M\u00f3dulo de elasticidade para os materiais definidos: {}\".format(Ey))\nprint(\"Coeficiente de Poisson para os materiais definidos: {}\".format(nu))\n\nk+=1\n```\n\n Posicionamento na linha correta do arquivo: k=5\n N\u00famero de materiais definidos no arquivo de entrada de dados: 1\n M\u00f3dulo de elasticidade para os materiais definidos: [0.]\n Coeficiente de Poisson para os materiais definidos: [0.]\n\n\n\n```\nfor i in range(nummat):\n v=df.iloc[k+i].values # linha do arquivo referente a um material: cont\u00e9m o m\u00f3dulo de elasticidade e o coeficiente de Poisson definido para o mesmo\n print(v)\n j=int(v[0])-1 # redu\u00e7\u00e3o de uma unidade para aloca\u00e7\u00e3o dos dados em estrutura interna\n Ey[j]=v[1] # m\u00f3dulo de elasticidade do material correspondente\n nu[j]=v[2] \n```\n\n [0.e+00 7.e+07 3.e-01 nan nan nan nan]\n\n\n# Leitura das informa\u00e7\u00f5es das se\u00e7\u00f5es transversais \n\n\n```\nk=6\nk=k+nummat\nnumgp = df.iloc[k,0]\ngp=np.zeros((numgp,2))\nk+=1\n```\n\n\n```\nfor i in range(numgp):\n v=df.iloc[k+i].values\n print(v)\n j=int(v[0])-1\n gp[j]=v[1:3] \n```\n\n [0.00e+00 1.20e-02 2.25e-05 nan nan nan nan]\n\n\n# Leitura e armazenamento das informa\u00e7\u00f5es dos elementos\n\n\n```\nk=8\nk = k+numgp\nnume = df.iloc[k,0]\nconec=np.zeros((nume,2))\nmatp=np.zeros(nume)\ngeop=np.zeros(nume)\nk+=1\n\nprint(\"Vetor de n\u00f3s que conectam a barra: {}\".format(conec))\nprint(\"Vetor que associa material \u00e0 barra: {}\".format(matp))\nprint(\"Vetor que associa se\u00e7\u00e3o \u00e0 barra: {} \\n\\n\".format(geop))\n\nfor i in range(nume):\n v=df.iloc[k+i].values\n conec[i,0] = v[1]\n conec[i,1] = v[2]\n matp[i] = v[3]\n geop[i] = v[4]\n\nprint(\"Vetor de n\u00f3s que conectam a barra: \\n{}\".format(conec))\nprint(\"Vetor que associa material \u00e0 barra: \\n{}\".format(matp))\nprint(\"Vetor que associa se\u00e7\u00e3o \u00e0 barra: \\n{}\".format(geop))\n```\n\n Vetor de n\u00f3s que conectam a barra: [[0. 0.]\n [0. 0.]\n [0. 0.]]\n Vetor que associa material \u00e0 barra: [0. 0. 0.]\n Vetor que associa se\u00e7\u00e3o \u00e0 barra: [0. 0. 0.] \n \n \n Vetor de n\u00f3s que conectam a barra: \n [[0. 1.]\n [1. 2.]\n [2. 3.]]\n Vetor que associa material \u00e0 barra: \n [0. 0. 0.]\n Vetor que associa se\u00e7\u00e3o \u00e0 barra: \n [0. 0. 0.]\n\n\n# Contabiliza\u00e7\u00e3o do n\u00famero de graus de liberdade do problema e como resultado temos o n\u00famero de equa\u00e7\u00f5es do sistema linear associado\n\n\n```\nneq=0\nfor n in range(numnp):\n # Para cada um dos poss\u00edveis deslocamentos representados\n # O vetor 'idx' armazena, agora, os \u00edndices dos graus de \n # liberdade com rela\u00e7\u00e3o ao referencial global\n for i in range(3): \n if idx[n,i]==0:\n idx[n,i]=neq\n neq+=1\n else:\n idx[n,i]=-1\n```\n\n\n```\ndisplay(neq, idx)\n```\n\n# Leitura e armazenamento dos dados de carregamento nodal\n\n\n```\nk=10\nk = k + nume\nnload = df.iloc[k,0]\nnode = np.zeros(nload)\nidirn = np.zeros(nload)\nfload = np.zeros(nload)\n\nnlcase=1\nr=np.zeros((neq,nlcase))\nk+=1\n\nprint(\"N\u00f3s em que as cargas do caso correspondente ser\u00e3o aplicadas: \\n{}\".format(node))\nprint(\"Dire\u00e7\u00f5es de aplica\u00e7\u00e3o v\u00e1lidas de cada carga: \\n{}\".format(idirn))\nprint(\"Magnitude das cargas aplicadas em cada uma das dire\u00e7\u00f5es principais: \\n{}\".format(fload))\nprint(\"Vetor de cargas no referencial global: \\n{}\".format(r))\n```\n\n N\u00f3s em que as cargas do caso correspondente ser\u00e3o aplicadas: \n [0. 0.]\n Dire\u00e7\u00f5es de aplica\u00e7\u00e3o v\u00e1lidas de cada carga: \n [0. 0.]\n Magnitude das cargas aplicadas em cada uma das dire\u00e7\u00f5es principais: \n [0. 0.]\n Vetor de cargas no referencial global: \n [[0.]\n [0.]\n [0.]\n [0.]\n [0.]\n [0.]\n [0.]\n [0.]\n [0.]]\n\n\n# Processamento das carga nodais, transformando a informa\u00e7\u00e3o local em informa\u00e7\u00e3o global\n\n\n```\nk=14\nl=0\nr=np.zeros((neq,nlcase))\nfor i in range(nload):\n v=df.iloc[k+i].values # n\u00f3, Fx, Fy, Mz \n print(v)\n node = int(v[0]) # n\u00f3 em que a carga est\u00e1 sendo aplicada\n fload = np.array([float(s) for s in v[1:4]]) # atribui as componentes de carga Fx, Fy, Mz\n idirn = np.array(idx[node]) # vetor com os graus de liberdade do n\u00f3, -1 indica deslocamento impedido\n\n print (node,idirn,fload)\n for m in range(len(idirn)):\n if idirn[m] > -1: # Verifica se a forca aplicada ira de fato influenciar na estrutura, verificar se n\u00e3o esta em cima do apoio\n r[idirn[m],l]=r[idirn[m],l] + fload[m] # c\u00e1lculo de r, que \u00e9 um vetor das for\u00e7as resultantes associadas a cada grau de liberdade\n```\n\n [ 1. 0. -20. 0. nan nan nan]\n 1 [1 2 3] [ 0. -20. 0.]\n [ 2. 0. -20. 0. nan nan nan]\n 2 [4 5 6] [ 0. -20. 0.]\n\n\n\n```\ndisplay(r)\n```\n\n\n array([[ 0.],\n [ 0.],\n [-20.],\n [ 0.],\n [ 0.],\n [-20.],\n [ 0.],\n [ 0.],\n [ 0.]])\n\n\n# C\u00e1lculo de matriz de rigidez de cada elemento\n\n\n```\n# Fun\u00e7\u00e3o que monta a matriz de rigidez de um elemento qualquer\ndef localstiff(_E,_A,_I,_L,):\n x, u1, u2, u3, u4, u5, u6, L, E, I, A=sp.var ('x u1 u2 u3 u4 u5 u6 L E I A');\n\n # Deslocamentos axiais devido a algum carregamento\n u = [1-x/L, 0., 0., x/L, 0., 0.]\n \n # Deslocamentos verticais devido a algum carregamento em n\u00f3s sem r\u00f3tula\n v = [0., 1-3*(x/L)**2+2*(x/L)**3, x-2*((x**2)/L)+3*((x**3)/(L**2)), 0., 3*(x/L)**2 - 2*(x/L)**3, -(x**2)/L + (x**3)/(L**2)]\n \n # Do Princ\u00edpio dos Trabalhos Virtuais tem-se W=U e chega-se em [K]{u}={F}\n K = sp.zeros(6) # matriz de rigidez local \n for i in range(6):\n for j in range(6):\n K[i,j] = (sp.integrate(sp.diff(E*I*v[i],x,2)*sp.diff(v[j],x,2),(x,0,L)) + sp.integrate(sp.diff(E*A*u[i],x,1)*sp.diff(u[j],x,1),(x,0,L))) \n \n KK= K.subs({E:_E,I:_I,A:_A,L:_L})\n KK=KK.evalf()\n return KK \n```\n\n# Fun\u00e7\u00e3o com a matriz de rota\u00e7\u00e3o para barras inclinadas\n\n\n```\n# Fun\u00e7\u00e3o que monta matriz de rota\u00e7\u00e3o para um quadro\t\ndef rotation_frame(_xyz):\n verbose=0\n xyz=np.array(_xyz) # vetor com as coordenadas dos dois pontos do elemento\n l=0 # tamanho do elemento\n\n # Calcula tamanho do elemento calculando a dist\u00e2ncia ente os dois pontos\n for i in range(3):\n l+=(xyz[i+3]-xyz[i])**2.\n \n l = np.sqrt(l) # l = sqrt((x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2)\n c=np.zeros(3) # dire\u00e7\u00e3o do elemento\n\n # Calcula dire\u00e7\u00e3o unit\u00e1ria do elemento\n for i in range(3):\n c[i]=(xyz[i+3]-xyz[i])/l\n\n R=np.eye(6) # matriz de rota\u00e7\u00e3o\n\n # Preenchendo a matriz de rota\u00e7\u00e3o\n R[0,0]=R[1,1]=c[0]\n R[0,1]=c[1]\n R[1,0]=-c[1]\n R[3,3]=R[4,4]=c[0]\n R[3,4]=c[1]\n R[4,3]=-c[1]\n\n return(R) \n```\n\n# Montagem da matrix global do problema\n\n\n```\nK=np.zeros((neq,neq))\nlength=[]\nbarras=[\"AC\",\"CD\",\"DB\"]\ncontador = 0\n\nfor n in range(nume):\n \n length.append((sum(coord[int(conec[n,0])]-coord[int(conec[n,1])])**2)**0.5)\n \n # Matriz local\n KK=localstiff(Ey[int(matp[n])], gp[int(geop[n])][0], gp[int(geop[n]),1], length[n]) \n \n lm=[] # lista com as dire\u00e7\u00f5es globais dos n\u00f3s locais\n xyz=[] # lista com as coordenadas globais dos n\u00f3s locais\n \n for j in range(len(conec[n])):\n lm=lm+list(idx[int(conec[n,j])])\n xyz=xyz+list(coord[int(conec[n,j])])\n\n R=rotation_frame(xyz) # montagem da matriz local de rota\u00e7\u00e3o\n KL=np.zeros((6,6)) # cria a matriz local do elemento \n KL=R.T * KK * R # montagem da matriz local do elemento no sistema de refer\u00eancia global\n\n print (\"Matriz local\", barras[contador])\n display (sp.Matrix(KL))\n\n # Adicionamos as matrizes locais KL na global K\n for i in range(len(lm)):\n if lm[i]!=-1:\n for j in range(len(lm)):\n if lm[j]!=-1:\n K[lm[i],lm[j]] += KL[i,j]\n \n if n!=nume-1:\n print(\"\\n\\n\")\n contador+=1\n```\n\n Matriz local AC\n\n\n\n$\\displaystyle \\left[\\begin{matrix}140000.0 & 0 & 0 & -140000.0 & 0 & 0\\\\0 & 87.5 & 787.5 & 0 & -87.5 & 262.5\\\\0 & 787.5 & 13650.0 & 0 & -787.5 & 3675.0\\\\-140000.0 & 0 & 0 & 140000.0 & 0 & 0\\\\0 & -87.5 & -787.5 & 0 & 87.5 & -262.5\\\\0 & 262.5 & 3675.0 & 0 & -262.5 & 1050.0\\end{matrix}\\right]$\n\n\n \n \n \n Matriz local CD\n\n\n\n$\\displaystyle \\left[\\begin{matrix}140000.0 & 0 & 0 & -140000.0 & 0 & 0\\\\0 & 87.5 & 787.5 & 0 & -87.5 & 262.5\\\\0 & 787.5 & 13650.0 & 0 & -787.5 & 3675.0\\\\-140000.0 & 0 & 0 & 140000.0 & 0 & 0\\\\0 & -87.5 & -787.5 & 0 & 87.5 & -262.5\\\\0 & 262.5 & 3675.0 & 0 & -262.5 & 1050.0\\end{matrix}\\right]$\n\n\n \n \n \n Matriz local DB\n\n\n\n$\\displaystyle \\left[\\begin{matrix}280000.0 & 0 & 0 & -280000.0 & 0 & 0\\\\0 & 700.0 & 3150.0 & 0 & -700.0 & 1050.0\\\\0 & 3150.0 & 27300.0 & 0 & -3150.0 & 7350.0\\\\-280000.0 & 0 & 0 & 280000.0 & 0 & 0\\\\0 & -700.0 & -3150.0 & 0 & 700.0 & -1050.0\\\\0 & 1050.0 & 7350.0 & 0 & -1050.0 & 2100.0\\end{matrix}\\right]$\n\n\n\n```\nprint (\"Matriz Global\")\ndisplay (sp.Matrix(K))\n```\n\n Matriz Global\n\n\n\n$\\displaystyle \\left[\\begin{matrix}13650.0 & 0.0 & -787.5 & 3675.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 280000.0 & 0.0 & 0.0 & -140000.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\-787.5 & 0.0 & 175.0 & 525.0 & 0.0 & -87.5 & 262.5 & 0.0 & 0.0\\\\3675.0 & 0.0 & 525.0 & 14700.0 & 0.0 & -787.5 & 3675.0 & 0.0 & 0.0\\\\0.0 & -140000.0 & 0.0 & 0.0 & 420000.0 & 0.0 & 0.0 & -280000.0 & 0.0\\\\0.0 & 0.0 & -87.5 & -787.5 & 0.0 & 787.5 & 2887.5 & 0.0 & 1050.0\\\\0.0 & 0.0 & 262.5 & 3675.0 & 0.0 & 2887.5 & 28350.0 & 0.0 & 7350.0\\\\0.0 & 0.0 & 0.0 & 0.0 & -280000.0 & 0.0 & 0.0 & 280000.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1050.0 & 7350.0 & 0.0 & 2100.0\\end{matrix}\\right]$\n\n\n# Resolve o sistema de equa\u00e7\u00f5es para os **deslocamentos**\n\n\n```\nu = np.linalg.solve(K,r)\nu\n```\n\n\n\n\n array([[-0.09332674],\n [ 0. ],\n [-0.9459219 ],\n [ 0.14394464],\n [ 0. ],\n [-2.8045477 ],\n [-0.94829461],\n [ 0. ],\n [ 4.72130499]])\n\n\n\n\n```\nprint(\"Deslocamento horizontal do ponto C:\\n{}\".format(u[1][0]))\nprint(\"Deslocamento vertical do ponto C:\\n{}\".format(u[2][0]))\nprint(\"Rota\u00e7\u00e3o do ponto C: \\n{}\".format(u[3][0]))\n```\n\n Deslocamento horizontal do ponto C:\n 0.0\n Deslocamento vertical do ponto C:\n -0.9459218981710655\n Rota\u00e7\u00e3o do ponto C: \n 0.1439446366782069\n\n\n#Implementa\u00e7\u00e3o da biblioteca anastruct para verificar os resultados\n\n\n```\n# Instala\u00e7\u00e3o dos pacotes para representa\u00e7\u00e3o computacional \n!pip install anastruct\n```\n\n Collecting anastruct\n Downloading anastruct-1.2.0-py3-none-any.whl (69 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 69 kB 7.6 MB/s \n \u001b[?25hRequirement already satisfied: scipy>=1.1.0 in /usr/local/lib/python3.7/dist-packages (from anastruct) (1.4.1)\n Requirement already satisfied: matplotlib>=3.0 in /usr/local/lib/python3.7/dist-packages (from anastruct) (3.2.2)\n Requirement already satisfied: numpy>=1.15.4 in /usr/local/lib/python3.7/dist-packages (from anastruct) (1.21.5)\n Requirement already satisfied: kiwisolver>=1.0.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib>=3.0->anastruct) (1.3.2)\n Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib>=3.0->anastruct) (3.0.7)\n Requirement already satisfied: python-dateutil>=2.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib>=3.0->anastruct) (2.8.2)\n Requirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.7/dist-packages (from matplotlib>=3.0->anastruct) (0.11.0)\n Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.7/dist-packages (from python-dateutil>=2.1->matplotlib>=3.0->anastruct) (1.15.0)\n Installing collected packages: anastruct\n Successfully installed anastruct-1.2.0\n\n\n\n```\n# importando os pacotes necess\u00e1rios\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom anastruct import SystemElements\n\n#-------------------------------------------------------------------------------\n# Informa\u00e7\u00f5es da estrutura\n#-------------------------------------------------------------------------------\nnode={\"A\":(0, 0), \"C\":(6, 0), \"D\":(12, 0), \"B\":(15, 0)}\n\n# Define as conectividades entre os n\u00f3s\nconec=[('A', 'C'), ('C', 'D'), ('D', 'B')]\n\n#-------------------------------------------------------------------------------\n# Montagem do modelo estrutural\n#-------------------------------------------------------------------------------\n# Instancia o sistema estrutural\nEA = 70e+7 * 0.012 # kPa*m^2\nEI = 70e+7 * 2.25e-5 # kPa*m^4\nss = SystemElements(EA=EA, EI=EI) \n#ss = SystemElements() \n\n# Implementa a conectividade entre os n\u00f3s gerando os elementos ou \"membros\"\nfor e in conec:\n element = (node[e[0]], node[e[1]])\n ss.add_element(location = element) # Add um elemento comum entre cada n\u00f3 conectado\n\n# Define os carregamentos pontuais\nF = -20\nnode_C = ss.find_node_id(node['C'])\nnode_D = ss.find_node_id(node['D'])\nss.point_load(node_id=node_C, Fy=F) # Add um carregamento ao n\u00f3 \"C\" da estrutura\nss.point_load(node_id=node_D, Fy=F) # Add um carregamento ao n\u00f3 \"D\" da estrutura\n\n# Apoios\nnode_id = ss.find_node_id(node['A'])\nss.add_support_hinged(node_id=node_id)\nnode_id = ss.find_node_id(node['B'])\nss.add_support_roll(node_id=node_id)\n\n#-------------------------------------------------------------------------------\n# Solu\u00e7\u00e3o e resultados do sistema estrutural\n#-------------------------------------------------------------------------------\nss.solve()\n\nss.show_structure(scale=0.7, figsize=(8,5), offset=(0,0))\n#ss.show_reaction_force(scale=0.9, figsize=(10,7), offset=(0,1))\n#ss.show_axial_force(scale=0.7, figsize=(10,7), offset=(0,0))\n#ss.show_shear_force(scale=0.7, figsize=(10,7), offset=(0,0))\n#ss.show_bending_moment(scale=0.7, figsize=(8,5), offset=(0,0))\nss.show_displacement(scale=0.7, figsize=(8,5), offset=(0,0))\n```\n", "meta": {"hexsha": "7900640a86a0fde1719780c45e30815dbd5a5ec0", "size": 125607, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "[MAC023]_Trabalho_03.ipynb", "max_stars_repo_name": "MathewsJosh/mecanica-estruturas-ufjf", "max_stars_repo_head_hexsha": "7f94b9a7cdacdb5a22b8fc491959f309be65acfb", 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"lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141572, "lm_q2_score": 0.7718434873426302, "lm_q1q2_score": 0.4398370577389473}} {"text": "

    Gotchas

    \n\n

    From

    \n\n

    To begin, we should make something about SymPy clear. SymPy is nothing more than a Python library, like NumPy, Django, or even modules in the Python standard library sys or re. What this means is that SymPy does not add anything to the Python language. Limitations that are inherent in the Python language are also inherent in SymPy. It also means that SymPy tries to use Python idioms whenever possible, making programming with SymPy easy for those already familiar with programming with Python. As a simple example, SymPy uses Python syntax to build expressions. Implicit multiplication (like 3x or 3 x) is not allowed in Python, and thus not allowed in SymPy. To multiply 3 and x, you must type 3*x with the *.

    \n\n
    In Julia:
    \n\n
      \n

    • implicit multiplication by a literal is supported, unlike Python

      \n
      • \n
    \n\n
    \n\n

    Symbols

    \n\n

    One consequence of this fact is that SymPy can be used in any environment where Python is available. We just import it, like we would any other library:

    \n\n\n```julia\n >>> from sympy import *\n```\n\n
    In Julia:
    \n\n
      \n

    • the functions from the sympy module are loaded with the package:

      \n
      • \n
    \n\n\n```julia\nusing SymPy\n```\n\n
    \n\n

    This imports all the functions and classes from SymPy into our interactive Python session. Now, suppose we start to do a computation.

    \n\n\n```julia\n >>> x + 1\n Traceback (most recent call last):\n ...\n NameError: name 'x' is not defined\n```\n\n
    In Julia:
    \n\n
      \n

    • the error output may differ, but an UndefVarError is thrown

      \n
      • \n
    \n\n\n```julia\nx + 1\n```\n\n\n\n\n UndefVarError(:x)\n\n\n\n\n
    \n\n

    Oops! What happened here? We tried to use the variable x, but it tells us that x is not defined. In Python, variables have no meaning until they are defined. SymPy is no different. Unlike many symbolic manipulation systems you may have used, in SymPy, variables are not defined automatically. To define variables, we must use symbols.

    \n\n\n```julia\n >>> x = symbols('x')\n >>> x + 1\n x + 1\n```\n\n
    In Julia:
    \n\n\n```julia\nx = symbols(\"x\")\nx + 1\n```\n\n\n\n\n\\begin{equation*}x + 1\\end{equation*}\n\n\n\n
    \n\n

    symbols takes a string of variable names separated by spaces or commas, and creates Symbols out of them. We can then assign these to variable names. Later, we will investigate some convenient ways we can work around this issue. For now, let us just define the most common variable names, x, y, and z, for use through the rest of this section

    \n\n\n```julia\n >>> x, y, z = symbols('x y z')\n```\n\n
    In Julia:
    \n\n\n```julia\nx, y, z = symbols(\"x y z\")\n```\n\n\n\n\n (x, y, z)\n\n\n\n
    \n\n

    As a final note, we note that the name of a Symbol and the name of the variable it is assigned to need not have anything to do with one another.

    \n\n\n```julia\n >>> a, b = symbols('b a')\n >>> a\n b\n >>> b\n a\n```\n\n
    In Julia:
    \n\n\n```julia\na, b = symbols(\"b a\")\na\n```\n\n\n\n\n\\begin{equation*}b\\end{equation*}\n\n\n\n\n```julia\nb\n```\n\n\n\n\n\\begin{equation*}a\\end{equation*}\n\n\n\n
    \n\n

    Here we have done the very confusing thing of assigning a Symbol with the name a to the variable b, and a Symbol of the name b to the variable a. Now the Python variable named a points to the SymPy Symbol named b, and visa versa. How confusing. We could have also done something like

    \n\n\n```julia\n >>> crazy = symbols('unrelated')\n >>> crazy + 1\n unrelated + 1\n```\n\n
    In Julia:
    \n\n\n```julia\ncrazy = symbols(\"unrelated\")\ncrazy + 1\n```\n\n\n\n\n\\begin{equation*}unrelated + 1\\end{equation*}\n\n\n\n
    \n\n

    This also shows that Symbols can have names longer than one character if we want.

    \n\n

    Usually, the best practice is to assign Symbols to Python variables of the same name, although there are exceptions: Symbol names can contain characters that are not allowed in Python variable names, or may just want to avoid typing long names by assigning Symbols with long names to single letter Python variables.

    \n\n

    To avoid confusion, throughout this tutorial, Symbol names and Python variable names will always coincide. Furthermore, the word \"Symbol\" will refer to a SymPy Symbol and the word \"variable\" will refer to a Python variable.

    \n\n

    Finally, let us be sure we understand the difference between SymPy Symbols and Python variables. Consider the following::

    \n\n\n```julia\n x = symbols('x')\n expr = x + 1\n x = 2\n print(expr)\n```\n\n

    What do you think the output of this code will be? If you thought 3, you're wrong. Let's see what really happens

    \n\n\n```julia\n >>> x = symbols('x')\n >>> expr = x + 1\n >>> x = 2\n >>> print(expr)\n x + 1\n```\n\n
    In Julia:
    \n\n
      \n

    • we must change to double quotes (or use @vars x, say)

      \n
      • \n
    \n\n\n```julia\nx = symbols(\"x\")\nexpr = x + 1\nx = 2\nexpr\n```\n\n\n\n\n\\begin{equation*}x + 1\\end{equation*}\n\n\n\n
    \n\n

    Changing x to 2 had no effect on expr. This is because x = 2 changes the Python variable x to 2, but has no effect on the SymPy Symbol x, which was what we used in creating expr. When we created expr, the Python variable x was a Symbol. After we created, it, we changed the Python variable x to 2. But expr remains the same. This behavior is not unique to SymPy. All Python programs work this way: if a variable is changed, expressions that were already created with that variable do not change automatically. For example

    \n\n\n```julia\n >>> x = 'abc'\n >>> expr = x + 'def'\n >>> expr\n 'abcdef'\n >>> x = 'ABC'\n >>> expr\n 'abcdef'\n```\n\n
    In Julia:
    \n\n
      \n

    • The * infix operator is used for string concatenation

      \n
      • \n
    \n\n\n```julia\nx = \"abc\"\nexpr = x * \"def\"\nexpr\n```\n\n\n\n\nabcdef\n\n\n\n\n```julia\nx = \"ABC\"\nexpr\n```\n\n\n\n\nabcdef\n\n\n\n
    \n\n

    Quick Tip

    \n\n

    To change the value of a Symbol in an expression, use subs

    \n\n\n```julia\n >>> x = symbols('x')\n >>> expr = x + 1\n >>> expr.subs(x, 2)\n 3\n```\n\n
    In Julia:
    \n\n\n```julia\nx = symbols(\"x\")\nexpr = x + 1\nexpr.subs(x, 2)\n```\n\n\n\n\n\\begin{equation*}3\\end{equation*}\n\n\n\n

    Or use the \"call\" form of subs: expr(x => 2)

    \n\n
    \n\n

    In this example, if we want to know what expr is with the new value of x, we need to reevaluate the code that created expr, namely, expr = x + 1. This can be complicated if several lines created expr. One advantage of using a symbolic computation system like SymPy is that we can build a symbolic representation for expr, and then substitute x with values. The correct way to do this in SymPy is to use subs, which will be discussed in more detail later.

    \n\n\n```julia\n >>> x = symbols('x')\n >>> expr = x + 1\n >>> expr.subs(x, 2)\n 3\n```\n\n
    In Julia:
    \n\n\n```julia\nx = symbols(\"x\")\nexpr = x + 1\nexpr.subs(x, 2)\n```\n\n\n\n\n\\begin{equation*}3\\end{equation*}\n\n\n\n
    \n\n

    TODO

    Add link to basic operations section

    \n
    \n\n

    Equals signs

    \n\n

    Another very important consequence of the fact that SymPy does not extend Python syntax is that = does not represent equality in SymPy. Rather it is Python variable assignment. This is hard-coded into the Python language, and SymPy makes no attempts to change that.

    \n\n

    You may think, however, that ==, which is used for equality testing in Python, is used for SymPy as equality. This is not quite correct either. Let us see what happens when we use ==.

    \n\n\n```julia\n >>> x + 1 == 4\n False\n```\n\n
    In Julia:
    \n\n
      \n

    • == is similar as in Python:

      \n
      • \n
    \n\n\n```julia\nx + 1 == 4\n```\n\n\n\n\n false\n\n\n\n

    Recall == promotes values, so we have a Julia object may be \"equal\" to a SymPy one:

    \n\n\n```julia\n0 == zero(Sym) ## or Sym(0)\n```\n\n\n\n\n true\n\n\n\n
    \n\n

    Instead of treating x + 1 == 4 symbolically, we just got False. In SymPy, == represents exact structural equality testing. This means that a == b means that we are asking if a = b. We always get a bool as the result of ==. There is a separate object, called Eq, which can be used to create symbolic equalities

    \n\n\n```julia\n >>> Eq(x + 1, 4)\n Eq(x + 1, 4)\n```\n\n
    In Julia:
    \n\n\n```julia\nEq(x + 1, 4)\n```\n\n\n\n\n\\begin{equation*}x + 1 = 4\\end{equation*}\n\n\n\n
    \n\n

    There is one additional caveat about == as well. Suppose we want to know if $(x + 1)^2 = x^2 + 2x + 1$. We might try something like this:

    \n\n\n```julia\n >>> (x + 1)**2 == x**2 + 2*x + 1\n False\n```\n\n
    In Julia:
    \n\n\n```julia\n(x + 1)^2 == x^2 + 2*x + 1\n```\n\n\n\n\n false\n\n\n\n
    \n\n

    We got False again. However, $(x + 1)^2$ does equal $x^2 + 2x + 1$. What is going on here? Did we find a bug in SymPy, or is it just not powerful enough to recognize this basic algebraic fact?

    \n\n

    Recall from above that == represents exact structural equality testing. \"Exact\" here means that two expressions will compare equal with == only if they are exactly equal structurally. Here, $(x + 1)^2$ and $x^2 + 2x + 1$ are not the same symbolically. One is the power of an addition of two terms, and the other is the addition of three terms.

    \n\n

    It turns out that when using SymPy as a library, having == test for exact structural equality is far more useful than having it represent symbolic equality, or having it test for mathematical equality. However, as a new user, you will probably care more about the latter two. We have already seen an alternative to representing equalities symbolically, Eq. To test if two things are equal, it is best to recall the basic fact that if a = b, then a - b = 0. Thus, the best way to check if a = b is to take a - b and simplify it, and see if it goes to 0. We will learn :ref:later <tutorial-simplify> that the function to do this is called simplify. This method is not infallible\u2013-in fact, it can be theoretically proven <http://en.wikipedia.org/wiki/Richardson%27s_theorem>_ that it is impossible to determine if two symbolic expressions are identically equal in general\u2013-but for most common expressions, it works quite well.

    \n\n\n```julia\n >>> a = (x + 1)**2\n >>> b = x**2 + 2*x + 1\n >>> simplify(a - b)\n 0\n >>> c = x**2 - 2*x + 1\n >>> simplify(a - c)\n 4*x\n```\n\n
    In Julia:
    \n\n\n```julia\na = (x + 1)^2\nb = x^2 + 2*x + 1\nsimplify(a - b)\n```\n\n\n\n\n\\begin{equation*}0\\end{equation*}\n\n\n\n\n```julia\nc = x^2 - 2*x + 1\nsimplify(a - c)\n```\n\n\n\n\n\\begin{equation*}4 x\\end{equation*}\n\n\n\n
    \n\n

    There is also a method called equals that tests if two expressions are equal by evaluating them numerically at random points.

    \n\n\n```julia\n >>> a = cos(x)**2 - sin(x)**2\n >>> b = cos(2*x)\n >>> a.equals(b)\n True\n```\n\n
    In Julia:
    \n\n\n```julia\na = cos(x)^2 - sin(x)^2\nb = cos(2*x)\na.equals(b)\n```\n\n\n\n\n true\n\n\n\n
    \n\n

    Two Final Notes: ^ and /

    \n\n

    You may have noticed that we have been using ** for exponentiation instead of the standard ^. That's because SymPy follows Python's conventions. In Python, ^ represents logical exclusive or. SymPy follows this convention:

    \n\n\n```julia\n >>> True ^ False\n True\n >>> True ^ True\n False\n >>> x^y\n Xor(x, y)\n```\n\n
    In Julia:
    \n\n
      \n

    • we export True and False for symbolic Boolean values

      \n
      • \n

    • This does not apply, as we use ^ for exponentiation.

      \n
      • \n

    • Use the prefix Or for logical

      \n
      • \n
    \n\n\n```julia\nOr(True, False)\n```\n\n\n\n\n\\begin{equation*}\\mathrm{True}\\end{equation*}\n\n\n\n\n```julia\nOr(True, True)\n```\n\n\n\n\n\\begin{equation*}\\mathrm{True}\\end{equation*}\n\n\n\n\n```julia\nOr(x, y)\n```\n\n\n\n\n\\begin{equation*}x \\vee y\\end{equation*}\n\n\n\n
    \n\n

    Finally, a small technical discussion on how SymPy works is in order. When you type something like x + 1, the SymPy Symbol x is added to the Python int 1. Python's operator rules then allow SymPy to tell Python that SymPy objects know how to be added to Python ints, and so 1 is automatically converted to the SymPy Integer object.

    \n\n

    This sort of operator magic happens automatically behind the scenes, and you rarely need to even know that it is happening. However, there is one exception. Whenever you combine a SymPy object and a SymPy object, or a SymPy object and a Python object, you get a SymPy object, but whenever you combine two Python objects, SymPy never comes into play, and so you get a Python object.

    \n\n\n```julia\n >>> type(Integer(1) + 1)\n \n >>> type(1 + 1)\n <... 'int'>\n```\n\n
    In Julia:
    \n\n
      \n

    • In Julia, most operations between SymPy objects and Julia objects will promote to a SymPy objects, but of course Julia objects combined will produce Julia Objects:

      \n
      • \n
    \n\n\n```julia\ntypeof(sympy.Integer(1) + 1)\n```\n\n\n\n\n SymPy.Sym\n\n\n\n\n```julia\ntypeof(1 + 1)\n```\n\n\n\n\n Int64\n\n\n\n

    To convert a Julia object to a SymPy object, the Sym constructor may be useful:

    \n\n\n```julia\nSym(1)\n```\n\n\n\n\n\\begin{equation*}1\\end{equation*}\n\n\n\n

    To convert a SymPy object to a Julia object, the N function is useful for numbers and booleans:

    \n\n\n```julia\nN(Sym(1)), N(True)\n```\n\n\n\n\n (1, true)\n\n\n\n

    And the lambdify function can produce a function from an expression:

    \n\n\n```julia\nex = x^2 - 2x + 2\nfn = lambdify(ex)\nfn(1) - ex(1)\n```\n\n\n\n\n\\begin{equation*}0\\end{equation*}\n\n\n\n
    \n\n

    Note

    \n\n

    On running the example above in SymPy Live, (1+1) is wrapped by Integer, so it does not show the correct output.

    \n\n

    This is usually not a big deal. Python ints work much the same as SymPy Integers, but there is one important exception: division. In SymPy, the division of two Integers gives a Rational:

    \n\n\n```julia\n >>> Integer(1)/Integer(3)\n 1/3\n >>> type(Integer(1)/Integer(3))\n \n```\n\n
    In Julia:
    \n\n\n```julia\nsympy.Integer(1)/sympy.Integer(3)\n```\n\n\n\n\n\\begin{equation*}\\frac{1}{3}\\end{equation*}\n\n\n\n\n```julia\ntypeof(sympy.Integer(1)/sympy.Integer(3))\n```\n\n\n\n\n SymPy.Sym\n\n\n\n

    And to get the Python, type, we can use __class__:

    \n\n\n```julia\n(sympy.Integer(1)/sympy.Integer(3)).__class__\n```\n\n\n\n\n PyObject \n\n\n\n
    \n\n

    But in Python / represents either integer division or floating point division, depending on whether you are in Python 2 or Python 3, and depending on whether or not you have run from __future__ import division:

    \n\n\n```julia\n >>> from __future__ import division\n >>> 1/2 #doctest: +SKIP\n 0.5\n```\n\n
    In Julia:
    \n\n
      \n

    • This does not apply, as / is not integer division.

      \n
      • \n
    \n\n
    \n\n

    !! note

    \n\n

    On running the example above in SymPy Live, (1/2) is wrapped by Integer, so it does not show the correct output.

    \n\n

    To avoid this, we can construct the rational object explicitly

    \n\n\n```julia\n >>> Rational(1, 2)\n 1/2\n```\n\n
    In Julia:
    \n\n
      \n

    • Rational from sympy is not exported, it would conflict with Julia's Rational costructor. We must qualify it:

      \n
      • \n
    \n\n\n```julia\nRational(1, 2)\n```\n\n\n\n\n 1//2\n\n\n\n\n```julia\nsympy.Rational(1, 2)\n```\n\n\n\n\n\\begin{equation*}\\frac{1}{2}\\end{equation*}\n\n\n\n
    \n\n

    This problem also comes up whenever we have a larger symbolic expression with int/int in it. For example:

    \n\n\n```julia\n >>> x + 1/2 #doctest: +SKIP\n x + 0.5\n```\n\n
    In Julia:
    \n\n
      \n

    • Int/Int will produce a floating point value, whereas Int//Int will produce a rational, which can then be promoted without loss to a symbolic object:

      \n
      • \n
    \n\n\n```julia\nx + 1/2 #doctest: +SKIP\n```\n\n\n\n\n\\begin{equation*}x + 0.5\\end{equation*}\n\n\n\n
    \n\n

    Note

    \n\n

    On running the example above in SymPy Live, (1/2) is wrapped by Integer, so it does not show the correct output.

    \n\n

    This happens because Python first evaluates 1/2 into 0.5, and then that is cast into a SymPy type when it is added to x. Again, we can get around this by explicitly creating a Rational:

    \n\n\n```julia\n >>> x + Rational(1, 2)\n x + 1/2\n```\n\n
    In Julia:
    \n\n\n```julia\nx + 1//2\n```\n\n\n\n\n\\begin{equation*}x + \\frac{1}{2}\\end{equation*}\n\n\n\n
    \n\n

    There are several tips on avoiding this situation in the :ref:gotchas document.

    \n\n

    Further Reading

    \n\n

    For more discussion on the topics covered in this section, see :ref:gotchas.

    \n\n
    \n\n

    return to index

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YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.793105951184112, "lm_q1q2_score": 0.4397538255159365}} {"text": "# ENV/ATM 415: Climate Laboratory\n\n[Brian E. J. Rose](http://www.atmos.albany.edu/facstaff/brose/index.html), University at Albany\n\n# Lecture 7: Elementary greenhouse models\n\n## Contents\n\n1. [A single layer atmosphere](#section1)\n2. [Introducing the two-layer grey gas model](#section2)\n3. [Tuning the grey gas model to observations](#section3)\n4. [Level of emission](#section4)\n5. [Radiative forcing in the 2-layer grey gas model](#section5)\n6. [Summary](#section6)\n\n____________\n\n\n## 1. A single layer atmosphere\n____________\n\nWe will make our first attempt at quantifying the greenhouse effect in the simplest possible greenhouse model: a single layer of atmosphere that is able to absorb and emit longwave radiation.\n\n\n\n### Assumptions\n\n- Atmosphere is a single layer of air at temperature $T_a$\n- Atmosphere is **completely transparent to shortwave** solar radiation.\n- The **surface** absorbs shortwave radiation $(1-\\alpha) Q$\n- Atmosphere is **completely opaque to infrared** radiation\n- Both surface and atmosphere emit radiation as **blackbodies** ($\\sigma T_s^4, \\sigma T_a^4$)\n- Atmosphere radiates **equally up and down** ($\\sigma T_a^4$)\n- There are no other heat transfer mechanisms\n\nWe can now use the concept of energy balance to ask what the temperature need to be in order to balance the energy budgets at the surface and the atmosphere, i.e. the **radiative equilibrium temperatures**.\n\n\n### Energy balance at the surface\n\n\\begin{align}\n\\text{energy in} &= \\text{energy out} \\\\\n(1-\\alpha) Q + \\sigma T_a^4 &= \\sigma T_s^4 \\\\\n\\end{align}\n\nThe presence of the atmosphere above means there is an additional source term: downwelling infrared radiation from the atmosphere.\n\nWe call this the **back radiation**.\n\n### Energy balance for the atmosphere\n\n\\begin{align}\n\\text{energy in} &= \\text{energy out} \\\\\n\\sigma T_s^4 &= A\\uparrow + A\\downarrow = 2 \\sigma T_a^4 \\\\\n\\end{align}\n\nwhich means that \n$$ T_s = 2^\\frac{1}{4} T_a \\approx 1.2 T_a $$\n\nSo we have just determined that, in order to have a purely **radiative equilibrium**, we must have $T_s > T_a$. \n\n*The surface must be warmer than the atmosphere.*\n\n### Solve for the radiative equilibrium surface temperature\n\nNow plug this into the surface equation to find\n\n$$ \\frac{1}{2} \\sigma T_s^4 = (1-\\alpha) Q $$\n\nand use the definition of the emission temperature $T_e$ to write\n\n$$ (1-\\alpha) Q = \\sigma T_e^4 $$\n\n*In fact, in this model, $T_e$ is identical to the atmospheric temperature $T_a$, since all the OLR originates from this layer.*\n\nSolve for the surface temperature:\n$$ T_s = 2^\\frac{1}{4} T_e $$\n\nPutting in observed numbers, $T_e = 255$ K gives a surface temperature of \n$$T_s = 303 ~\\text{K}$$\n\nThis model is one small step closer to reality: surface is warmer than atmosphere, emissions to space generated in the atmosphere, atmosphere heated from below and helping to keep surface warm.\n\nBUT our model now overpredicts the surface temperature by about 15\u00baC (or K).\n\nIdeas about why?\n\nBasically we just need to read our **list of assumptions** above and realize that none of them are very good approximations:\n\n- Atmosphere absorbs some solar radiation.\n- Atmosphere is NOT a perfect absorber of longwave radiation\n- Absorption and emission varies strongly with wavelength *(atmosphere does not behave like a blackbody)*.\n- Emissions are not determined by a single temperature $T_a$ but by the detailed *vertical profile* of air temperture.\n- Energy is redistributed in the vertical by a variety of dynamical transport mechanisms (e.g. convection and boundary layer turbulence).\n\n\n\n____________\n\n\n## 2. Introducing the two-layer grey gas model\n____________\n\nLet's generalize the above model just a little bit to build a slighly more realistic model of longwave radiative transfer.\n\nWe will address two shortcomings of our single-layer model:\n1. No vertical structure\n2. 100% longwave opacity\n\nRelaxing these two assumptions gives us what turns out to be a very useful prototype model for **understanding how the greenhouse effect works**.\n\n### Assumptions\n\n- The atmosphere is **transparent to shortwave radiation** (still)\n- Divide the atmosphere up into **two layers of equal mass** (the dividing line is thus at 500 hPa pressure level)\n- Each layer **absorbs only a fraction $\\epsilon$ ** of whatever longwave radiation is incident upon it.\n- We will call the fraction $\\epsilon$ the **absorptivity** of the layer.\n- Assume $\\epsilon$ is the same in each layer\n\nThis is called the **grey gas** model, where grey here means the emission and absorption have no spectral dependence.\n\nWe can think of this model informally as a \"leaky greenhouse\".\n\nNote that the assumption that $\\epsilon$ is the same in each layer is appropriate if the absorption is actually carried out by a gas that is **well-mixed** in the atmosphere.\n\nOut of our two most important absorbers:\n\n- CO$_2$ is well mixed\n- H$_2$O is not (mostly confined to lower troposphere due to strong temperature dependence of the saturation vapor pressure).\n\nBut we will ignore this aspect of reality for now.\n\nIn order to build our model, we need to introduce one additional piece of physics known as **Kirchoff's Law**:\n\n$$ \\text{absorptivity} = \\text{emissivity} $$\n\nSo if a layer of atmosphere at temperature $T$ absorbs a fraction $\\epsilon$ of incident longwave radiation, it must emit\n\n$$ \\epsilon ~\\sigma ~T^4 $$\n\nboth up and down.\n\n### A sketch of the radiative fluxes in the 2-layer atmosphere\n\n\n\n- Surface temperature is $T_s$\n- Atm. temperatures are $T_0, T_1$ where $T_0$ is closest to the surface.\n- absorptivity of atm layers is $\\epsilon$\n- Surface emission is $\\sigma T_s^4$\n- Atm emission is $\\epsilon \\sigma T_0^4, \\epsilon \\sigma T_1^4$ (up and down)\n- Absorptivity = emissivity for atmospheric layers\n- a fraction $(1-\\epsilon)$ of the longwave beam is **transmitted** through each layer\n\nLet's think about the upwelling beam of longwave radiation, which we denote $U$.\n\nWe start at the surface. The upward flux from the surface to layer 0 is \n\n$$U_0 = \\sigma T_s^4$$\n\nFollowing this beam upward, we can write the upward flux from layer 0 to layer 1 as the sum of the transmitted part of what is incident from below, and the new emissions from layer 0:\n\n$$ U_1 = (1-\\epsilon) \\sigma T_s^4 + \\epsilon \\sigma T_0^4 $$\n\nContinuing to follow the same beam, the upwelling flux above layer 1 is\n\n$$ U_2 = (1-\\epsilon) U_1 + \\epsilon \\sigma T_1^4 $$\n\nSince there is no more atmosphere above layer 1, this upwelling beam is our OLR for this model:\n\n$$ OLR = U_2 = (1-\\epsilon) U_1 + \\epsilon \\sigma T_1^4 $$\n\nwhich works out to\n\n$$ OLR= (1-\\epsilon)^2 \\sigma T_s^4 + \\epsilon(1-\\epsilon)\\sigma T_0^4 + \\epsilon \\sigma T_1^4 $$\n\nHere the three terms represent **contributions to the total OLR that originate from each of the three levels**\n\n*What happens to this expression if $\\epsilon=1$? What does this represent physically?*\n\n*What about $\\epsilon=0$? *\n\nBy allowing the atmosphere to partially absorb emissions from other levels, we now see that the OLR includes emissions from every level - and therefore affected by temperature at every level!\n\n____________\n\n\n## 3. Tuning the grey gas model to observations\n____________\n\nIn building our new model we have introduced exactly one parameter, the absorptivity $\\epsilon$. We need to choose a value for $\\epsilon$.\n\nWe will tune our model so that it **reproduces the observed global mean OLR** given **observed global mean temperatures**.\n\nTo get appropriate temperatures for $T_s, T_0, T_1$, let's revisit the [global, annual mean lapse rate plot from NCEP Reanalysis data](L06_Radiation.ipynb) from the previous lecture.\n\n### Temperatures\n\nFirst, we set \n$$T_s = 288 \\text{ K} $$\n\nFrom the lapse rate plot, an average temperature for the layer between 1000 and 500 hPa is \n\n$$ T_0 = 275 \\text{ K}$$\n\nDefining an average temperature for the layer between 500 and 0 hPa is more ambiguous because of the lapse rate reversal at the tropopause. We will choose\n\n$$ T_1 = 230 \\text{ K}$$\n\nFrom the graph, this is approximately the observed global mean temperature at 275 hPa or about 10 km.\n\n### OLR\n\nFrom the [observed global energy budget](Lecture01 -- Planetary energy budget.ipynb) we set \n\n$$ OLR = 238.5 \\text{ W m}^{-2} $$\n\n### Solving for $\\epsilon$\n\nWe wrote down the expression for OLR as a function of temperatures and absorptivity in our model above. \n\nAll we need to do is plug the observed values into the above expression for OLR, and solve for $\\epsilon$.\n\nIt is a quadratic equation. We could work out the exact solution using the quadratic formula.\n\nBut let's do it graphically, using Python!\n\n### Exercise\n\n- Write a Python function that implements the OLR formula for the leaky greenhouse model\n- The function should accept four input parameters:\n - The three temperatures $T_s, T_0, T_1$\n - The emissivity $\\epsilon$\n- Using this function, make a graph of OLR vs. $\\epsilon$ *for the observed temperature values*\n- For the graph, $\\epsilon$ should range between 0 and 1.\n\n\n```python\n%matplotlib inline\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nsigma = 5.67E-8\n\ndef OLR(e, Ts=288, T0=275, T1=230):\n return (1-e)*((1-e)*sigma*Ts**4 + e*sigma*T0**4) + e*sigma*T1**4\n```\n\n\n```python\nimport numpy as np\ne = np.linspace(0,1)\nplt.plot(e, OLR(e))\nplt.plot(e, 238.5 * np.ones_like(e))\nplt.xlabel('epsilon (grey absorptivity)')\nplt.ylabel('OLR (W/m2)')\nplt.grid()\n```\n\nNote if you solve the quadratic equation algebraically you will get two solutions: \n\n- $\\epsilon = 0.58$ \n- $\\epsilon = 3.93$\n\nWhy is the second solution not physically meaningful?\n\nFrom the graphical solution, we end up choosing $\\epsilon = 0.58$. \n\nThis is the absorptivity that guarantees that **our model reproduces the observed OLR given the observed temperatures**.\n\n____________\n\n\n## 4. Level of emission\n____________\n\nNow that we have tuned up our model, we can see exactly how strongly each level contributes to the OLR. \n\nThe three components of the OLR are\n\n\\begin{align*}\nOLR_s &= (1-\\epsilon)^2 \\sigma T_s^4 \\\\\nOLR_0 &= \\epsilon(1-\\epsilon)\\sigma T_0^4 \\\\\nOLR_1 &= \\epsilon \\sigma T_1^4 \n\\end{align*}\n\nwhich of course add up to the total OLR we wrote down above.\n\n**Write some simple Python code to calculate each term in the OLR using the observed temperatures and the tuned value $\\epsilon = 0.58$. Fill out the list below using your calculated numbers.** \n\n\n```python\n\n```\n\n**Contributions to the OLR originating from each level, in W/m2:**\n\n- Surface: 68.8\n- Level 0: 79.0\n- Level 1: 92.0\n\n\n```python\n68.8 + 79. + 92.\n```\n\n\n\n\n 239.8\n\n\n\nNotice that the largest single contribution is coming from the top layer. \n\n*This is in spite of the fact that the emissions from this layer are weak, because it is so cold.*\n\n### Changing the level of emission by adding absorbers\n\nAdding some extra greenhouse absorbers will mean that a greater fraction of incident longwave radiation is absorbed in each layer.\n\nThus **$\\epsilon$ must increase** as we add greenhouse gases.\n\nSuppose we have $\\epsilon$ initially, and the absorptivity increases to $\\epsilon_2 = \\epsilon + \\Delta \\epsilon$.\n\nSuppose further that this increase happens **abruptly** so that there is no time for the temperatures to respond to this change. **We hold the temperatures fixed** in the column and ask how the radiative fluxes change.\n\n**Do you expect the OLR to increase or decrease?**\n\nLet's use our two-layer leaky greenhouse model to investigate the answer.\n\nThe components of the OLR before the perturbation are\n\n\\begin{align*}\nOLR_s &= (1-\\epsilon)^2 \\sigma T_s^4 \\\\\nOLR_0 &= \\epsilon(1-\\epsilon)\\sigma T_0^4 \\\\\nOLR_1 &= \\epsilon \\sigma T_1^4 \n\\end{align*}\n\n\n\nand after the perturbation we have\n\n\\begin{align*}\nOLR_s &= (1-\\epsilon - \\Delta \\epsilon)^2 \\sigma T_s^4 \\\\\nOLR_0 &= (\\epsilon + \\Delta \\epsilon)(1-\\epsilon - \\Delta \\epsilon)\\sigma T_0^4 \\\\\nOLR_1 &= (\\epsilon + \\Delta \\epsilon) \\sigma T_1^4 \n\\end{align*}\n\nLet's subtract off the original components to get the contributions to the **change in OLR** from each layer:\n\n\\begin{align*}\n\\Delta OLR_s &= \\left[(1-\\epsilon - \\Delta \\epsilon)^2 - (1-\\epsilon)^2\\right]\\sigma T_s^4 \\\\\n\\Delta OLR_0 &= \\left[(\\epsilon + \\Delta \\epsilon)(1-\\epsilon - \\Delta \\epsilon) - \\epsilon(1-\\epsilon) \\right] \\sigma T_0^4 \\\\\n\\Delta OLR_1 &= \\left[(\\epsilon + \\Delta \\epsilon) - \\epsilon \\right] \\sigma T_1^4 \n\\end{align*}\n\nNow expand this out, but to make things easier to deal with, neglect term in $\\Delta \\epsilon^2$ (very small - we will be considering changes of less than 10% in $\\epsilon$):\n\n\\begin{align*}\n\\Delta OLR_s &\\approx (\\Delta \\epsilon) \\left[ -2(1-\\epsilon) \\right] \\sigma T_s^4 \\\\\n\\Delta OLR_0 &\\approx (\\Delta \\epsilon) (1 - 2 \\epsilon) \\sigma T_0^4 \\\\\n\\Delta OLR_1 &\\approx (\\Delta \\epsilon) \\sigma T_1^4 \n\\end{align*}\n\nNow look at the **sign** of each term. Recall that $0 < \\epsilon < 1$. **Which terms in the OLR go up and which go down?**\n\n**THIS IS VERY IMPORTANT, SO STOP AND THINK ABOUT IT.**\n\nThe contribution from the **surface** must **decrease**, while the contribution from the **top layer** must **increase**.\n\n**When we add absorbers, the average level of emission goes up!**\n\n____________\n\n\n## 5. Radiative forcing in the 2-layer grey gas model\n____________\n\nWe now define a very important quantity:\n\n### \"Radiative forcing\" is the change in radiative flux at TOA after adding absorbers\n\nIn this model, only the longwave flux can change, so we define the radiative forcing as\n\n$$ R = - \\Delta OLR $$\n\n(with the minus sign so that $R$ is positive when the climate system is gaining extra energy).\n\nWe just worked out that whenever we add some extra absorbers, the emissions to space (on average) will originate from higher levels in the atmosphere. \n\nWhat does this mean for OLR? Will it increase or decrease?\n\nTo get the answer, we just have to sum up the three contributions we wrote above:\n\n\\begin{align*}\nR &= -\\Delta OLR_s - \\Delta OLR_0 - \\Delta OLR_1 \\\\\n &= -\\Delta \\epsilon \\left[ -2(1-\\epsilon) \\sigma T_s^4 + (1 - 2 \\epsilon) \\sigma T_0^4 + \\sigma T_1^4 \\right]\n\\end{align*}\n\nIs this a positive or negative number? The key point is this:\n\n**It depends on the temperatures, i.e. on the lapse rate.**\n\n### Greenhouse effect for an isothermal atmosphere\n\nStop and think about this question:\n\nIf the **surface and atmosphere are all at the same temperature**, does the OLR go up or down when $\\epsilon$ increases (i.e. we add more absorbers)?\n\nUnderstanding this question is key to understanding how the greenhouse effect works.\n\n#### Let's solve the isothermal case\n\nWe will just set $T_s = T_0 = T_1$ in the above expression for the radiative forcing.\n\nWhat do you get?\n\n\n```python\n\n```\n\n#### The answer is $R=0$ \n\nFor an isothermal atmosphere, there is **no change** in OLR when we add extra greenhouse absorbers. Hence, no radiative forcing and no greenhouse effect.\n\nWhy?\n\nThe level of emission still must go up. But since the temperature at the upper level is the **same** as everywhere else, the emissions are exactly the same.\n\n### The radiative forcing (change in OLR) depends on the lapse rate!\n\nFor a more realistic example of radiative forcing due to an increase in greenhouse absorbers, we our observed temperatures and the tuned value for $\\epsilon$. \n\nWe'll express the answer in W m$^{-2}$ for a 2% increase in $\\epsilon$:\n\n$$ \\Delta \\epsilon = 0.02 \\times 0.58 $$\n\n\n```python\nepsilon = 0.58\ndelta_epsilon = 0.02 * epsilon\ndelta_epsilon\n```\n\n\n\n\n 0.0116\n\n\n\nCalculate the three components of the radiative forcing:\n\n\n```python\nsigma = 5.67E-8\nTs = 288.\nT0 = 275.\nT1 = 230.\n```\n\n\n```python\n# Component originating from the surface\nRs = -delta_epsilon * (-2*(1-epsilon)*sigma * Ts**4)\nRs\n```\n\n\n\n\n 3.800933621091533\n\n\n\n\n```python\n# Component originating from level 0\nR0 = -delta_epsilon * (1-2*epsilon) * sigma * T0**4\nR0\n```\n\n\n\n\n 0.6018549074999996\n\n\n\n\n```python\n# Component originating from level 1\nR1 = -delta_epsilon * sigma * T1**4\nR1\n```\n\n\n\n\n -1.8405702252\n\n\n\nSo just add them up to get the total radiative forcing:\n\n\n```python\nR = Rs + R0 + R1 \nR\n```\n\n\n\n\n 2.5622183033915324\n\n\n\nSo in our example, **the OLR decreases by 2.6 W m$^{-2}$**, or equivalently, the **radiative forcing is +2.6 W m$^{-2}$.**\n\nWhat we have just calculated is this:\n\n*Given the observed lapse rates, a small increase in absorbers will cause a small decrease in OLR.*\n\nThe greenhouse effect thus gets stronger, and energy will begin to accumulate in the system -- which will eventually cause temperatures to increase as the system adjusts to a new equilibrium.\n\n____________\n\n\n## 6. Summary\n____________\n\n## Key physical lessons\n\n- Putting a **layer of longwave absorbers** above the surface keeps the **surface substantially warmer**, because of the **backradiation** from the atmosphere (greenhouse effect).\n- The **grey gas** model assumes that each layer absorbs and emits a fraction $\\epsilon$ of its blackbody value, independent of wavelength.\n\n- With **incomplete absorption** ($\\epsilon < 1$), there are contributions to the OLR from every level and the surface (there is no single **level of emission**)\n- Adding more absorbers means that **contributions to the OLR** from **upper levels** go **up**, while contributions from the surface go **down**.\n- This upward shift in the weighting of different levels is what we mean when we say the **level of emission goes up**.\n\n- The **radiative forcing** caused by an increase in absorbers **depends on the lapse rate**.\n- For an **isothermal atmosphere** the radiative forcing is zero and there is **no greenhouse effect**\n- The radiative forcing is positive for our atmosphere **because tropospheric temperatures tends to decrease with height**.\n\n\n```python\n\n```\n", "meta": {"hexsha": "7461edca75c77ee55b5fdebc233ef6f7fc6964ad", "size": 49267, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notes/L07_ElementaryGreenhouse.ipynb", "max_stars_repo_name": "brian-rose/env-415-2018", "max_stars_repo_head_hexsha": "a3023ed667391ba2da8ef50a31d06e6470c35171", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-11-19T04:14:33.000Z", "max_stars_repo_stars_event_max_datetime": "2020-11-19T04:14:33.000Z", "max_issues_repo_path": "notes/L07_ElementaryGreenhouse.ipynb", "max_issues_repo_name": "brian-rose/env-415", "max_issues_repo_head_hexsha": "a3023ed667391ba2da8ef50a31d06e6470c35171", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notes/L07_ElementaryGreenhouse.ipynb", "max_forks_repo_name": "brian-rose/env-415", "max_forks_repo_head_hexsha": "a3023ed667391ba2da8ef50a31d06e6470c35171", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 40.4490968801, "max_line_length": 16692, "alphanum_fraction": 0.6973836442, "converted": true, "num_tokens": 4726, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.6825737214979746, "lm_q1q2_score": 0.4397311284168303}} {"text": "```python\nimport pandas as pd\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport seaborn as sns\nimport iirsBenchmark.regressors as regressors\n\nimport Auxiliary\nimport re\n\nfrom sympy import sympify\nfrom IPython.display import display\nfrom CriticalDiagrams import draw_cd_diagram\nfrom ExpressionParser import count_expression_nodes\n\n\n# Setting everything up\nAuxiliary.create_global_file_path_variables(results_path='../../results')\nAuxiliary.set_mpl_sns_params(abnt=False)\n\n%matplotlib inline\n\nanalyzed_path = f'../post hoc analysis files/' # Where to save the plots\n```\n\n## Post processing reported expressions\n\n\n```python\ndef post_processing_regression_models(info_to_extract, index=False, digest=True):\n \n data = None\n \n # Only methods that returns an analytical expression\n for regressor in ['ITEA_regressor', 'Linear_regressor', 'Lasso_regressor', 'Operon_regressor']:\n df_regression = pd.read_csv(f'{Auxiliary.regression_path}{regressor}.csv')\n\n regressor_no_sufix = regressor.replace('_regressor', '')\n\n cols = [col_name for col_name in df_regression.columns\n if col_name.startswith(info_to_extract)]\n\n def simplify_and_count_nodes(s):\n try:\n return count_expression_nodes(\n str(\n sympify(\n s.replace(\"^\", \"**\").replace('id', '')\n )\n ).replace(\"**\", \"^\")\n )\n except Exception as e:\n #print(e)\n return np.nan\n \n df_regression[f'expr_size_{regressor_no_sufix}'] = \\\n df_regression['text_representation'].apply(simplify_and_count_nodes)\n \n df_regression[f'text_representation_{regressor_no_sufix}'] = \\\n df_regression['text_representation']\n \n df_regression = df_regression.drop('text_representation', axis=1)\n \n if digest == False:\n data = df_regression if data is None else pd.merge(\n data, df_regression, on=\"dataset\", how='outer')\n \n continue\n\n new_df = df_regression[['dataset']].reset_index(drop=True).join(\n pd.DataFrame(\n df_regression.loc[:, cols+[f'expr_size_{regressor_no_sufix}']].values,\n columns=cols+[f'expr_size_{regressor_no_sufix}']\n )\n ).groupby(['dataset']).mean().reset_index(level=[0])\n\n new_df.columns = [\n col_name + f'_{regressor_no_sufix}' if col_name.startswith(info_to_extract) else col_name\n for col_name in new_df.columns]\n \n data = new_df if data is None else pd.merge(\n data, new_df, on=\"dataset\", how='outer')\n\n if index:\n data.set_index('dataset', inplace=True)\n \n return data\n```\n\n\n```python\ndf_expressions = results_regression_analysis_models('rmse_test', index=True, digest=False)\n\ndisplay(df_expressions)\n\ntop1 = pd.read_csv(\n '../../datasets/FeynmanEquations.csv')[['Filename', '# variables']].nlargest(1, '# variables')\n\nfor regressor in ['ITEA', 'Operon', 'Lasso', 'Linear']:\n display(df_expressions.nsmallest(1, f'expr_size_{regressor}')[[f'text_representation_{regressor}', f'expr_size_{regressor}']].values[0])\n```\n\n ITEA_regressor\n Linear_regressor\n Lasso_regressor\n Operon_regressor\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    rep_xrmse_train_xrmse_test_xr2_train_xr2_test_xtot_time_xexpr_size_ITEAtext_representation_ITEArep_yrmse_train_y...expr_size_Lassotext_representation_Lassorep_yrmse_train_yrmse_test_yr2_train_yr2_test_ytot_time_yexpr_size_Operontext_representation_Operon
    dataset
    I.12.1232.842589e-073.272662e-071.01.0404.790747670.184*log(x_0^-3 * x_1^-2) + 0.0*cos(x_0^-1 * ...01.327760...92.96*x_0 + 2.987*x_1 + -8.893197.387918e-078.117035e-071.0000001.0000000.1751145.0(0.00000 + (0.23037 * ((1.44886 * x_1) * (2.99...
    I.12.1232.842589e-073.272662e-071.01.0404.790747670.184*log(x_0^-3 * x_1^-2) + 0.0*cos(x_0^-1 * ...01.327760...92.96*x_0 + 2.987*x_1 + -8.893256.921345e-078.643262e-071.0000001.0000000.3023775.0(0.00000 + ((-0.45102) * ((0.92112 * x_2) * ((...
    I.12.1232.842589e-073.272662e-071.01.0404.790747670.184*log(x_0^-3 * x_1^-2) + 0.0*cos(x_0^-1 * ...01.327760...92.96*x_0 + 2.987*x_1 + -8.893146.183685e-076.952388e-071.0000001.0000000.3047055.0(0.00000 + (1.76753 * ((1.60208 * x_2) * (0.35...
    I.12.1232.842589e-073.272662e-071.01.0404.790747670.184*log(x_0^-3 * x_1^-2) + 0.0*cos(x_0^-1 * ...01.327760...92.96*x_0 + 2.987*x_1 + -8.89309.744334e-071.156478e-061.0000001.0000000.1827455.0(0.00000 + (0.70875 * ((3.03057 * x_1) * (0.46...
    I.12.1232.842589e-073.272662e-071.01.0404.790747670.184*log(x_0^-3 * x_1^-2) + 0.0*cos(x_0^-1 * ...01.327760...92.96*x_0 + 2.987*x_1 + -8.893185.708457e-078.220077e-071.0000001.0000000.2358735.0(0.00000 + (0.27914 * (((2.68410 * x_1) * (0.7...
    ..................................................................
    I.43.43144.907660e-055.860011e-051.01.05062.816287160178.171*sqrt(x_0^-4 * x_1^2 * x_2^-2 * x_3^2) ...00.852137...17-0.738*x_0 + 0.54*x_1 + -0.649*x_2 + 0.531*x_3...109.372554e-049.790063e-041.0000001.0000001.99905941.0(0.00017 + ((-27.56030) * (((((0.59361 * x_4) ...
    I.43.43144.907660e-055.860011e-051.01.05062.816287160178.171*sqrt(x_0^-4 * x_1^2 * x_2^-2 * x_3^2) ...00.852137...17-0.738*x_0 + 0.54*x_1 + -0.649*x_2 + 0.531*x_3...134.804711e-035.772241e-030.9999910.9999881.87071891.0(0.00201 + ((-0.07452) * ((((((-0.03067) * x_1...
    I.43.43144.907660e-055.860011e-051.01.05062.816287160178.171*sqrt(x_0^-4 * x_1^2 * x_2^-2 * x_3^2) ...00.852137...17-0.738*x_0 + 0.54*x_1 + -0.649*x_2 + 0.531*x_3...66.390276e-037.025082e-030.9999840.9999822.00166348.0(0.00080 + (4134.54883 * ((((((exp(1.97547) * ...
    I.43.43144.907660e-055.860011e-051.01.05062.816287160178.171*sqrt(x_0^-4 * x_1^2 * x_2^-2 * x_3^2) ...00.852137...17-0.738*x_0 + 0.54*x_1 + -0.649*x_2 + 0.531*x_3...44.640332e-034.769003e-030.9999910.9999921.71746958.0((-0.02785) + (0.01793 * ((((((-2.89637) * x_2...
    I.43.43144.907660e-055.860011e-051.01.05062.816287160178.171*sqrt(x_0^-4 * x_1^2 * x_2^-2 * x_3^2) ...00.852137...17-0.738*x_0 + 0.54*x_1 + -0.649*x_2 + 0.531*x_3...199.878633e-031.072610e-020.9999610.9999591.00573649.0((-0.00092) + ((-0.11420) * (((((1.23495 * (0....
    \n

    89100 rows \u00d7 32 columns

    \n
    \n\n\n\n array(['-0.0*sqrt(x_0^2 * x_1^2) + 0.0*log(x_0^-2 * x_1^4) + -0.0*log(x_0 * x_1^3) + 1.0*id(x_0 * x_1) + 0.0*id(x_0 * x_1^-2) + -0.0*expn(x_0^4) + -0.0*id(x_1^-3) + 0.0',\n 5], dtype=object)\n\n\n\n array(['((-2.97555) + ((-0.00000) * ((((exp((1.55768 * x_3)) ^ 2) / (((((-1.00114) * x_3) ^ 2) / ((-0.19546) * x_2) / ((0.04506 / (((-1.00114) * x_3) ^ 2)) ^ 2)) * exp(((-0.19546) * x_3)))) - (((-0.33103) * x_1) + ((-0.19546) * x_2))) / (((((0.04506 / (((-1.00114) * x_3) ^ 2)) ^ 2) * exp(((-0.19546) * x_3))) * exp(exp(((-0.19546) * x_2)))) * exp((1.55768 * x_3)) * 0.04506) / ((((0.04506 / (((-1.00114) * x_3) ^ 2)) ^ 2) * (2.48929 / ((-1.00114) * x_3))) * exp((1.55768 * x_3))))))',\n 1.0], dtype=object)\n\n\n\n array(['2.138', 1], dtype=object)\n\n\n\n array(['2.961*x_0 + 2.988*x_1 + -8.898', 9], dtype=object)\n\n\n\n```python\nregression_analysis_results = results_regression_analysis_models('rmse_test', index=True)\n\nfig, ax = plt.subplots(1, 1, figsize=(6, 2.0))\n\ncols = [col_name for col_name in regression_analysis_results.columns if col_name.startswith('expr_size_')]\n\nres_exprSize = regression_analysis_results.loc[:, cols]\n\nmeds = res_exprSize.median()\nmeds.sort_values(ascending=(True), inplace=True)\nres_exprSize = res_exprSize[meds.index]\n\nflierprops = dict(marker='o', markerfacecolor='black', markersize=2, markeredgecolor='black')\n\nsns.boxplot(data=res_exprSize, orient=\"h\",medianprops={'color': 'k'},\n showfliers=True, ax=ax, flierprops=flierprops) #, x=x, y=y, order=order)\n\nfor box, color in zip(ax.artists, sns.color_palette(\"Blues\", len(ax.artists))):\n box.set_color(color)\n\nax.set_yticklabels(\n [s.replace(f'expr_size_', '')\n for s in meds.index],\n rotation = 0, fontsize = 12, ha='right')\n\nfor spine in ['right', 'top', 'bottom']:\n ax.spines[spine].set_visible(False)\n\nax.grid()\n\nplt.tight_layout()\nplt.savefig(f'{analyzed_path}expression_sizes.pdf', bbox_inches='tight')\nplt.show()\n```\n\n\n```python\nfig, axs = plt.subplots(1, 1, figsize=(6, 1.5))\n \nres_exprSize.columns = [col.replace('expr_size_', '') for col in res_exprSize.columns]\n\nmelted_res = pd.melt(res_exprSize.reset_index(), id_vars=['dataset'])\nmelted_res.columns = ['dataset_name', 'classifier_name', 'accuracy']\n\n\nmelted_res['accuracy'] = np.max(melted_res['accuracy']) - melted_res['accuracy']\n \ndraw_cd_diagram(\n df_perf=melted_res,\n labels=False, ax=axs, width=8, textspace=1.0, reverse=False)\n\nplt.tight_layout()\nplt.savefig(f'{analyzed_path}expression_sizes_criticaldiagram.pdf', bbox_inches='tight')\nplt.show()\n```\n", "meta": {"hexsha": "3c5663249e85c4af5586015e7371c238973cbc22", "size": 39035, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "post hoc analysis/scripts and notebooks/Expression sizes.ipynb", "max_stars_repo_name": "gAldeia/iirsBenchmark", "max_stars_repo_head_hexsha": "2211b4755405eb32178a09f1a01143d53dc6516d", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "post hoc analysis/scripts and notebooks/Expression sizes.ipynb", "max_issues_repo_name": "gAldeia/iirsBenchmark", "max_issues_repo_head_hexsha": "2211b4755405eb32178a09f1a01143d53dc6516d", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "post hoc analysis/scripts and notebooks/Expression sizes.ipynb", "max_forks_repo_name": "gAldeia/iirsBenchmark", "max_forks_repo_head_hexsha": "2211b4755405eb32178a09f1a01143d53dc6516d", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 52.0466666667, "max_line_length": 6564, "alphanum_fraction": 0.5329576022, "converted": true, "num_tokens": 5402, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259583, "lm_q2_score": 0.727975460709318, "lm_q1q2_score": 0.4396475548936777}} {"text": "
    \n

    Universidad Nacional de C\u00f3rdoba - Facultad de Matem\u00e1tica, Astronom\u00eda, F\u00edsica y Computaci\u00f3n

    \n

    Diplomatura en Ciencia de Datos, Aprendizaje Autom\u00e1tico y sus Aplicaciones

    \n
    \n\n

    Regresi\u00f3n Linear - Ejemplo

    \n

    An\u00e1lisis y Visualizaci\u00f3n de Datos - 2019

    \n\nEn este ejemplo veremos c\u00f3mo implementar una regresi\u00f3n log\u00edstica para predecir una variable num\u00e9rica. Volveremos a utilizar el dataset [Human Freedom Index 2018](https://www.cato.org/human-freedom-index-new) de el instituto Cato.\n\nUsaremos la misma versi\u00f3n del conjunto de datos que en el pr\u00e1ctico.\n\nEn esta notebook vamos a tratar de estimar una funci\u00f3n lineal que modele el cambio a trav\u00e9s del tiempo de la libertad personal y la econ\u00f3mica.\n\n\n```python\n%matplotlib inline\n\nimport numpy\nimport pandas\nimport matplotlib.pyplot as plt\nimport seaborn\n```\n\n\n```python\nseaborn.set_context(context='talk', font_scale=1.2)\nseaborn.set_style(\"darkgrid\")\n```\n\n\n```python\nBLUE = '#35A7FF'\nRED = '#FF5964'\nGREEN = '#6BF178'\nYELLOW = '#FFE74C'\n```\n\n\n```python\ndataset = pandas.read_csv(\n 'https://object.cato.org/sites/cato.org/files/human-freedom-index-files/human-freedom-index-2019.csv')\ndataset.shape\n```\n\n\n\n\n (1620, 120)\n\n\n\n\n```python\nscore_cols = [col for col in dataset.columns if 'pf_identity' in col] + [\n 'pf_score', # Personal Freedom (score)\n 'pf_rank', # Personal Freedom (rank)\n 'ef_score', # Economic Freedom (score)\n 'ef_rank', # Economic Freedom (rank)\n 'hf_score', # Human Freedom (score)\n 'hf_rank', # Human Freedom (rank)\n]\n\nimportant_cols = ['year', 'ISO_code', 'countries', 'region'] + score_cols\n```\n\n\n```python\ndataset = dataset[important_cols].replace('-', numpy.nan)\nfor score_col in score_cols:\n dataset[score_col] = pandas.to_numeric(dataset[score_col])\ndataset\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    yearISO_codecountriesregionpf_identity_legalpf_identity_sex_malepf_identity_sex_femalepf_identity_sexpf_identity_divorcepf_identitypf_scorepf_rankef_scoreef_rankhf_scorehf_rank
    02017ALBAlbaniaEastern Europe0.010.010.010.07.55.88.0146.07.6730.07.8438.0
    12017DZAAlgeriaMiddle East & North AfricaNaN0.00.00.00.00.05.20146.04.77159.04.99155.0
    22017AGOAngolaSub-Saharan Africa10.00.00.00.05.05.05.98121.04.83158.05.40151.0
    32017ARGArgentinaLatin America & the Caribbean10.010.010.010.010.010.08.0441.05.67147.06.8677.0
    42017ARMArmeniaCaucasus & Central Asia7.010.010.010.07.58.27.1572.07.7027.07.4254.0
    ...................................................
    16152008AUSAustraliaOceaniaNaN10.010.010.010.010.09.297.08.186.08.734.0
    16162008DNKDenmarkWestern EuropeNaN10.010.010.010.010.09.493.07.989.08.734.0
    16172008CHESwitzerlandWestern EuropeNaN10.010.010.010.010.09.316.08.354.08.833.0
    16182008NZLNew ZealandOceaniaNaN10.010.010.010.010.09.424.08.463.08.942.0
    16192008HKGHong KongEast AsiaNaN10.010.010.010.010.09.1312.09.111.09.121.0
    \n

    1620 rows \u00d7 16 columns

    \n
    \n\n\n\nEn el pr\u00e1ctico hab\u00edamos trabajado sobre las variables `ef_score` y `pf_score`, que hacen referencia a los \u00edndices de libertad personal y libertad econ\u00f3mica de cada p\u00e1is. Adem\u00e1s, sabemos que el dataset incluye una medici\u00f3n del \u00edndice anual por pa\u00eds desde 2008 hasta 2016, aunque hay datos faltantes de algunos indicadores.\n\nLa motivaci\u00f3n de este an\u00e1lisis comienza con este gr\u00e1fico, que muestra una tendencia decreciente de la libertad personal y una tendencia ascendiente de la libertad econ\u00f3mica. La libertad humana o `pf_score` es el promedio de ambos indicadores\n\n\n```python\nplt.figure(figsize=(10,6))\nseaborn.lineplot(data=dataset,\n x='year', y='ef_score',\n label='Economic Freedom')\nseaborn.lineplot(data=dataset,\n x='year', y='pf_score',\n label='Personal Freedom')\n\nseaborn.lineplot(data=dataset,\n x='year', y='hf_score',\n label='Human Freedom')\nplt.legend()\nplt.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.)\nplt.ylabel('Index score')\nseaborn.despine()\n```\n\nEste fen\u00f3meno podr\u00eda estar dado por varios factores:\n * Hay pocos pa\u00edses en los que la libertad personal est\u00e1 decreciendo, pero su libertad econ\u00f3mica se mantiene constante.\n * Los pa\u00edses para los cuales sube la libertad econ\u00f3mica decrecen en libertad personal.\n * **\u00bfOtras?**\n\nVeamos qu\u00e9 sucede en Argentina. Si graficamos ambas variables, vemos que \"van bajando\". Formalmente, esto significa que hay la recta que las modela tiene una pendiente negativa.\n\n**\u00bfY esto, es grave?**\n\n\n```python\nplt.figure(figsize=(15, 8))\nseaborn.regplot(data=dataset[(dataset.ISO_code == 'ARG')],\n x='year', y='ef_score', label='Economic Freedom')\nseaborn.regplot(data=dataset[(dataset.ISO_code == 'ARG')],\n x='year', y='pf_score', label='Personal Freedom')\nseaborn.regplot(data=dataset[(dataset.ISO_code == 'ARG')],\n x='year', y='hf_score', label='Human Freedom')\nplt.legend()\nplt.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.)\nseaborn.despine()\nplt.xlim(2008, 2016)\n```\n\nPodemos graficar varios pa\u00edses, pero es dif\u00edcil comparar visualmente entre tantas variables, qu\u00e9 pa\u00edses \"decrecen\" m\u00e1s r\u00e1pido que otros.\n\n\n```python\ncountries = ['Argentina', 'Brazil', 'Mexico', 'Bolivia',\n 'Uruguay', 'Peru', 'Colombia', 'Venezuela']\ng = seaborn.FacetGrid(dataset, col=\"countries\",\n margin_titles=True, aspect=2, col_wrap=2,\n col_order=countries)\ng.map(seaborn.lineplot, \"year\", \"ef_score\", color=BLUE)\ng.map(seaborn.lineplot, \"year\", \"pf_score\", color=RED)\ng.set(xlim=(2008, 2016))\n\nprint('Puntajes de libertad para Am\u00e9rica Latina');\n```\n\nPara poder comparar la situaci\u00f3n de Argentina con otros pa\u00edses, podemos comparar la pendiente de la recta de la regresi\u00f3n lineal. A partir del gr\u00e1fico anterior pudimos ver que la mayor\u00eda de los pa\u00edses tiene tendencias similares y que se pueden estimar con una recta sin perder generalidad. Esto es posible tambi\u00e9n, en cierta medida, porque tenemos pocos puntos para estimar.\n\n## Regresi\u00f3n lineal\n\nQueremos ver cu\u00e1l es el coeficiente que relaciona ambas variables.\n\n\nPrimero: **\u00bfCu\u00e1l es la variable dependiente? \u00bfCu\u00e1l la independiente?**\n\nUna vez que hemos determinado eso, lo que queremos encontrar es la funci\u00f3n de la siguiente forma:\n\n$$ef = a * year + b$$\n\nReescribiremos esto como una funci\u00f3n $e$ (por economic), cuyo par\u00e1metro es el valor $y$ (por year):\n\n$$e(y) = a * y + b$$\n\nVamos a describir los ejemplos como pares $(x_y, x_e)$, donde $x_y$ denota el `year` y $x_e$ denota `ef_score`. \n\nPara encontrar la recta $e$ que mejor describe los datos, queremos minimizar el error cuadr\u00e1tico medio, definido como:\n\n$$mse = \\frac{1}{|X|} \\sum_{x \\in X} (e(x_y) - x_e)^2 $$\n\nRecordemos que para minimizar una funci\u00f3n, una buena opci\u00f3n es comenzar por buscar los puntos estacionarios, donde la derivada se anula. Por suerte, la funci\u00f3n $mse$ es convexa, y por lo tanto tiene todos sus puntos estacionarios son minimizadores. El minimizador es el valor de los par\u00e1metros $a$ y $b$ que minimizan la funci\u00f3n. Ahora, en hemos cambiado nuestras \"variables\", lo que buscamos es encontrar la funci\u00f3n adecuada, por lo tanto lo que cambia son los valores de los par\u00e1metros que definen la funci\u00f3n. \n\nPrimero, notemos que:\n\n$$\\frac{\\partial}{\\partial a}e(y) = x_p$$\n\n$$\\frac{\\partial}{\\partial b}e(y) = 1$$\n\nCon eso, calculamos las derivadas parciales para cada par\u00e1metro de la funci\u00f3n $mse$.\n\n$$\\frac{\\partial}{\\partial a}mse = \\frac{2}{|X|} \\sum_{x \\in X} (e(x_p) - x_e) \\frac{\\partial}{\\partial a} (e(x_p) - x_e) = \\frac{2}{|X|} \\sum_{x \\in X} (e(x_p) - x_e) e_p $$\n\n$$\\frac{\\partial}{\\partial b}mse = \\frac{2}{|X|} \\sum_{x \\in X} \\frac{\\partial}{\\partial b} e(x_p) - x_e = \\frac{2}{|X|} \\sum_{x \\in X} e(x_p) - x_e $$\n\n\nA pesar del formuler\u00edo, es bastante simple. S\u00f3lo reemplazamos $mse$ por su definici\u00f3n, y luego aplicamos un par de reglas como \"la derivada de la suma es la suma de las derivadas\", la regla de la cadena, o la definici\u00f3n de la derivada de la funci\u00f3n cuadr\u00e1tica.\n\nUna vez que tenemos esos valores, tenemos que igualarlos a cero para encontrar los puntos estacionarios.\n\n\\begin{align}\n \\frac{\\partial}{\\partial a}mse &= \\frac{2}{|X|} \\sum_{x \\in X} (e(x_y) - x_e) x_y = 0 \\\\\n &\\Rightarrow a = \\frac{\\bar{x_y} \\bar{x_e} - \\overline{x_yx_e}}{(\\bar{x_y})^2 - \\overline{x_y^2}} \n\\end{align}\n\n\\begin{align}\n \\frac{\\partial}{\\partial b}mse &= \\frac{2}{|X|} \\sum_{x \\in X} e(x_y) - x_e = 0 \\\\\n &\\Rightarrow b = \\bar{x_e} - a \\bar{x_y}\n\\end{align}\n\nDonde $\\bar{x}$ es la media del valor para todos los ejemplos. Vamos a confiar en estas f\u00f3rmulas, pero una demostraci\u00f3n de las mismas est\u00e1 en:\n\nhttps://medium.freecodecamp.org/machine-learning-mean-squared-error-regression-line-c7dde9a26b93\n\n\n```python\ndef estimate_params(X_y, X_e):\n \"\"\"Caculates the value of a using all the examples.\"\"\"\n num = numpy.mean(X_y)*numpy.mean(X_e) - numpy.mean(numpy.multiply(X_y, X_e))\n denom = numpy.mean(X_y)**2 - numpy.mean(numpy.multiply(X_y, X_y))\n a = num / denom\n b = numpy.mean(X_e) - a * numpy.mean(X_y)\n return a, b\n```\n\n\n```python\n# Asumimos que todos los registros que tienen pf_score tienen el a\u00f1o.\na, b = estimate_params(\n dataset[(dataset.ISO_code == 'ARG') &\n (dataset.pf_score.notnull())].year.dropna(),\n dataset[dataset.ISO_code == 'ARG'].pf_score)\na, b\n```\n\n\n\n\n (-0.02048484848483261, 49.32575757572563)\n\n\n\n\n```python\ndef base_linear_regression(x_y, a):\n return a * x_y\n```\n\n\n```python\ndef regplot2(data, x, y, reg_func, **reg_func_args):\n \"\"\"Plots the x, y columns from data and builds a\n line with the regression reg_func.\"\"\"\n plt.figure(figsize=(8, 6))\n seaborn.scatterplot(data=data, x=x, y=y, color=BLUE)\n minimum = data[x].min()\n maximum = data[x].max()\n plt.plot([minimum, maximum],\n [reg_func(minimum, **reg_func_args),\n reg_func(maximum, **reg_func_args)],\n color=GREEN)\n seaborn.despine()\n plt.show()\n```\n\n\n```python\nregplot2(dataset[dataset.ISO_code == 'ARG'],\n x='year', y='pf_score', reg_func=base_linear_regression,\n a=a)\n```\n\nVemos que la recta va en el sentido correcto, pero est\u00e1 demasiado abajo. Esto ocurre porque no hemos usado el t\u00e9rmino de bias.\n\nRedefinamos entonces la regresi\u00f3n log\u00edstica\n\n\n```python\ndef linear_regression(x_y, a, b):\n return a * x_y + b\n```\n\n\n```python\nregplot2(dataset[dataset.ISO_code == 'ARG'],\n x='year', y='pf_score', reg_func=linear_regression,\n a=a, b=b)\n```\n\n## Continuamos el an\u00e1lisis\n\nPerfecto! Ahora podemos calcular las pendientes y los biases para todos los a\u00f1os, para regresiones que estimen el `pf_score`.\n\n\n```python\ndef build_regressions(data, x_var='year', y_var='pf_score'):\n records = []\n for code in data.ISO_code.unique():\n record = [code, data[data.ISO_code == code].region.values[0],\n data[data.ISO_code == code].countries.values[0]]\n y_data = data[data.ISO_code == code][y_var].dropna()\n # Comprobamos que hay datos en el intervalo\n if len(y_data) <= 1:\n continue\n x_data = data[(data.ISO_code == code) &\n (data[y_var].notnull())][x_var].dropna()\n # Estimamos los par\u00e1metros\n a, b = estimate_params(x_data, y_data)\n # Calculamos el error cuadr\u00e1tico medio de la regresi\u00f3n lineal estimada\n predictions = numpy.apply_along_axis(\n lambda x: linear_regression(x, a, b), 0, x_data)\n mse = numpy.mean(numpy.power(predictions - y_data, 2))\n record.extend([a, b, mse])\n # Agregamos el registro\n records.append(record)\n return pandas.DataFrame.from_records(\n records, columns=['ISO_code', 'region', 'country',\n 'slope', 'bias', 'mse']\n )\n```\n\n\n```python\npf_regressions = build_regressions(dataset).set_index('ISO_code')\npf_regressions[:10]\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    regioncountryslopebiasmse
    ISO_code
    ALBEastern EuropeAlbania-0.03690982.0845450.044086
    DZAMiddle East & North AfricaAlgeria-0.00612117.5879390.009140
    AGOSub-Saharan AfricaAngola0.087636-170.6781820.039059
    ARGLatin America & the CaribbeanArgentina-0.02048549.3257580.003178
    ARMCaucasus & Central AsiaArmenia-0.01666740.7566670.008773
    AUSOceaniaAustralia-0.01157632.5132120.002516
    AUTWestern EuropeAustria0.037394-66.0483030.007005
    AZECaucasus & Central AsiaAzerbaijan-0.105758218.9581210.060136
    BHSLatin America & the CaribbeanBahamas-0.03290974.1705450.009934
    BHRMiddle East & North AfricaBahrain-0.089515186.3882420.048762
    \n
    \n\n\n\n\n```python\nplt.figure(figsize=(10,6))\nseaborn.histplot(pf_regressions.slope, color=RED, label='Global')\nseaborn.histplot(\n pf_regressions[pf_regressions.region == 'Latin America & the Caribbean'].slope,\n color=BLUE, label='Latam y Caribe')\nplt.xlabel('Slope of linear regression that fits the pf_score of each country')\nplt.legend()\nseaborn.despine()\n```\n\n\n```python\ndef plot_regressions(regressions):\n plt.figure(figsize=(10,6))\n colors = seaborn.color_palette(\"cubehelix\", len(regressions))\n for color, (year, row) in zip(colors, regressions.iterrows()):\n minimum, maximum = 2008, 2016\n plt.plot([minimum, maximum],\n [linear_regression(minimum, row['slope'], row['bias']),\n linear_regression(maximum, row['slope'], row['bias'])],\n color=color, label=str(year), linestyle='--')\n plt.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.)\n seaborn.despine()\n```\n\n\n```python\nplot_regressions(pf_regressions.loc[['ARG', 'BRA', 'VEN', 'CAN']])\nplt.xlabel('year')\nplt.ylabel('pf_score')\nplt.ylim(4, 10)\n```\n\n### Libertad Econ\u00f3mica\n\n\n```python\nef_regressions = build_regressions(dataset, y_var='ef_score').set_index('ISO_code')\nef_regressions[:10]\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    regioncountryslopebiasmse
    ISO_code
    ALBEastern EuropeAlbania0.047333-87.8013330.003397
    DZAMiddle East & North AfricaAlgeria-0.04260690.8056970.009553
    AGOSub-Saharan AfricaAngola0.015394-25.7863030.084069
    ARGLatin America & the CaribbeanArgentina-0.083273173.0183640.132188
    ARMCaucasus & Central AsiaArmenia0.013333-19.1653330.003609
    AUSOceaniaAustralia-0.01090930.0265450.002294
    AUTWestern EuropeAustria-0.00848524.8197580.000690
    AZECaucasus & Central AsiaAzerbaijan0.042242-78.7878790.007204
    BHSLatin America & the CaribbeanBahamas-0.02666761.0386670.003569
    BHRMiddle East & North AfricaBahrain-0.01036428.0978180.009243
    \n
    \n\n\n\n\n```python\nplot_regressions(ef_regressions.loc[['ARG', 'BRA', 'VEN', 'CAN']])\nplt.xlabel('year')\nplt.ylabel('ef_score')\n```\n\n\n```python\nplt.figure(figsize=(10,6))\nseaborn.histplot(ef_regressions.slope, color=RED, label='Global')\nseaborn.histplot(\n ef_regressions[ef_regressions.region == 'Latin America & the Caribbean'].slope,\n color=BLUE, label='Latam y Caribe')\nplt.xlabel('Slope of linear regression that fits the ef_score of each country')\nplt.legend()\nseaborn.despine()\n```\n\n## An\u00e1lisis conjunto\n\n**\u00bfCu\u00e1les es el 10% de pa\u00edses en los que la libertad humana disminuye m\u00e1s r\u00e1pidamente?**\n\n\n```python\nquantil = pf_regressions.slope.quantile(0.1)\npf_regressions[pf_regressions.slope < quantil].country\n```\n\n\n\n\n ISO_code\n BTN Bhutan\n BRA Brazil\n BRN Brunei Darussalam\n BDI Burundi\n EGY Egypt\n GMB Gambia, The\n MUS Mauritius\n NPL Nepal\n NER Niger\n RWA Rwanda\n SYR Syria\n TJK Tajikistan\n THA Thailand\n TUR Turkey\n UKR Ukraine\n VEN Venezuela\n YEM Yemen, Rep.\n Name: country, dtype: object\n\n\n\n**\u00bfCu\u00e1les es el 10% de pa\u00edses en los que la libertad econ\u00f3mica disminuye m\u00e1s r\u00e1pidamente?**\n\n\n```python\nquantil = ef_regressions.slope.quantile(0.1)\nef_regressions[ef_regressions.slope < quantil].country\n```\n\n\n\n\n ISO_code\n ARG Argentina\n BRA Brazil\n BRN Brunei Darussalam\n EGY Egypt\n FJI Fiji\n GHA Ghana\n IRQ Iraq\n KWT Kuwait\n LBR Liberia\n LBY Libya\n PNG Pap. New Guinea\n SLE Sierra Leone\n SDN Sudan\n SYR Syria\n TUN Tunisia\n VEN Venezuela\n ZMB Zambia\n Name: country, dtype: object\n\n\n\n**\u00bfCu\u00e1les son los paises en los que la libertad econ\u00f3mica aumenta pero la libertad personal disminuye (r\u00e1pidamente)?**\n\n\n```python\nall_countries = dataset.ISO_code.unique()\ncodes = []\nfor code in all_countries:\n if (code in ef_regressions.index and code in pf_regressions.index and\n ef_regressions.loc[code].slope > 0.02 and\n pf_regressions.loc[code].slope < -0.02):\n codes.append(code)\nef_regressions.loc[codes].country\n```\n\n\n\n\n ISO_code\n ALB Albania\n AZE Azerbaijan\n BGR Bulgaria\n BDI Burundi\n KHM Cambodia\n CPV Cape Verde\n CHN China\n GMB Gambia, The\n GTM Guatemala\n GIN Guinea\n ISL Iceland\n IDN Indonesia\n KAZ Kazakhstan\n LAO Laos\n MYS Malaysia\n MLT Malta\n MEX Mexico\n MAR Morocco\n MMR Myanmar\n NPL Nepal\n NER Niger\n PRY Paraguay\n PHL Philippines\n RUS Russia\n RWA Rwanda\n ESP Spain\n TZA Tanzania\n VNM Vietnam\n Name: country, dtype: object\n\n\n\n# Errores\n\nCalculamos el mse pero nunca lo usamos. Veamos c\u00f3mo son los pa\u00edses para los que la regresi\u00f3n linear no produce una buena aproximaci\u00f3n\n\n\n```python\npf_regressions.mse.sort_values()[-10:]\n```\n\n\n\n\n ISO_code\n TLS 0.106220\n SYC 0.107328\n TUR 0.126490\n CAF 0.142989\n VEN 0.145224\n COD 0.165123\n LBY 0.182368\n GNB 0.194737\n YEM 0.195064\n BDI 0.206774\n Name: mse, dtype: float64\n\n\n\n\n```python\nplt.figure(figsize=(10,6))\ncountries = ['BDI', 'YEM', 'GNB', 'LBY']\nseaborn.lineplot(\n data=dataset[dataset.ISO_code.isin(countries)], x='year', y='hf_score',\n hue='countries')\nplt.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.)\nseaborn.despine()\n```\n\nClaramente se ve que estas funciones no pod\u00edan ser estimadas satisfactoriamente con una recta, pero a\u00fan as\u00ed, la tendencia general (descendiente o ascendiente) habr\u00eda sido aproximada\n", "meta": {"hexsha": "870674b62206e3a1daef536209f4bd4eb729526e", "size": 573764, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "analisisvis/notebooks/09_regresion_lineal.ipynb", "max_stars_repo_name": "ssulca/dipdata", "max_stars_repo_head_hexsha": "1872b21cb9adc11017a70fe9a44b5edc87f10f7e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-11-20T22:59:31.000Z", "max_stars_repo_stars_event_max_datetime": "2020-11-20T22:59:31.000Z", "max_issues_repo_path": "analisisvis/notebooks/09_regresion_lineal.ipynb", "max_issues_repo_name": "ser0090/dipdata", "max_issues_repo_head_hexsha": "1872b21cb9adc11017a70fe9a44b5edc87f10f7e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "analisisvis/notebooks/09_regresion_lineal.ipynb", "max_forks_repo_name": "ser0090/dipdata", "max_forks_repo_head_hexsha": "1872b21cb9adc11017a70fe9a44b5edc87f10f7e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-07-20T14:02:21.000Z", "max_forks_repo_forks_event_max_datetime": "2021-07-20T14:02:21.000Z", "avg_line_length": 309.473570658, "max_line_length": 99936, "alphanum_fraction": 0.913476621, "converted": true, "num_tokens": 8463, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784220301065, "lm_q2_score": 0.7662936484231889, "lm_q1q2_score": 0.4396061310391083}} {"text": "# + \n# **First Notebook: Virtual machine test and assignment submission**\n#### This notebook will test that the virtual machine (VM) is functioning properly and will show you how to submit an assignment to the autograder. To move through the notebook just run each of the cells. You will not need to solve any problems to complete this lab. You can run a cell by pressing \"shift-enter\", which will compute the current cell and advance to the next cell, or by clicking in a cell and pressing \"control-enter\", which will compute the current cell and remain in that cell. At the end of the notebook you will export / download the notebook and submit it to the autograder.\n#### ** This notebook covers: **\n#### *Part 1:* Test Spark functionality\n#### *Part 2:* Check class testing library\n#### *Part 3:* Check plotting\n#### *Part 4:* Check MathJax formulas\n#### *Part 5:* Export / download and submit\n\n### ** Part 1: Test Spark functionality **\n\n#### ** (1a) Parallelize, filter, and reduce **\n\n\n```python\n# Check that Spark is working\nlargeRange = sc.parallelize(xrange(100000))\nreduceTest = largeRange.reduce(lambda a, b: a + b)\nfilterReduceTest = largeRange.filter(lambda x: x % 7 == 0).sum()\n\nprint reduceTest\nprint filterReduceTest\n\n# If the Spark jobs don't work properly these will raise an AssertionError\nassert reduceTest == 4999950000\nassert filterReduceTest == 714264285\n```\n\n 4999950000\n 714264285\n\n\n#### ** (1b) Loading a text file **\n\n\n```python\n# Check loading data with sc.textFile\nimport os.path\nbaseDir = os.path.join('data')\ninputPath = os.path.join('cs100', 'lab1', 'shakespeare.txt')\nfileName = os.path.join(baseDir, inputPath)\n\nrawData = sc.textFile(fileName)\nshakespeareCount = rawData.count()\n\nprint shakespeareCount\n\n# If the text file didn't load properly an AssertionError will be raised\nassert shakespeareCount == 122395\n```\n\n 122395\n\n\n### ** Part 2: Check class testing library **\n\n#### ** (2a) Compare with hash **\n\n\n```python\n# TEST Compare with hash (2a)\n# Check our testing library/package\n# This should print '1 test passed.' on two lines\nfrom test_helper import Test\n\ntwelve = 12\nTest.assertEquals(twelve, 12, 'twelve should equal 12')\nTest.assertEqualsHashed(twelve, '7b52009b64fd0a2a49e6d8a939753077792b0554',\n 'twelve, once hashed, should equal the hashed value of 12')\n```\n\n 1 test passed.\n 1 test passed.\n\n\n#### ** (2b) Compare lists **\n\n\n```python\n# TEST Compare lists (2b)\n# This should print '1 test passed.'\nunsortedList = [(5, 'b'), (5, 'a'), (4, 'c'), (3, 'a')]\nTest.assertEquals(sorted(unsortedList), [(3, 'a'), (4, 'c'), (5, 'a'), (5, 'b')],\n 'unsortedList does not sort properly')\n```\n\n 1 test passed.\n\n\n### ** Part 3: Check plotting **\n\n#### ** (3a) Our first plot **\n#### After executing the code cell below, you should see a plot with 50 blue circles. The circles should start at the bottom left and end at the top right.\n\n\n```python\n# Check matplotlib plotting\nimport matplotlib.pyplot as plt\nimport matplotlib.cm as cm\nfrom math import log\n\n# function for generating plot layout\ndef preparePlot(xticks, yticks, figsize=(10.5, 6), hideLabels=False, gridColor='#999999', gridWidth=1.0):\n plt.close()\n fig, ax = plt.subplots(figsize=figsize, facecolor='white', edgecolor='white')\n ax.axes.tick_params(labelcolor='#999999', labelsize='10')\n for axis, ticks in [(ax.get_xaxis(), xticks), (ax.get_yaxis(), yticks)]:\n axis.set_ticks_position('none')\n axis.set_ticks(ticks)\n axis.label.set_color('#999999')\n if hideLabels: axis.set_ticklabels([])\n plt.grid(color=gridColor, linewidth=gridWidth, linestyle='-')\n map(lambda position: ax.spines[position].set_visible(False), ['bottom', 'top', 'left', 'right'])\n return fig, ax\n\n# generate layout and plot data\nx = range(1, 50)\ny = [log(x1 ** 2) for x1 in x]\nfig, ax = preparePlot(range(5, 60, 10), range(0, 12, 1))\nplt.scatter(x, y, s=14**2, c='#d6ebf2', edgecolors='#8cbfd0', alpha=0.75)\nax.set_xlabel(r'$range(1, 50)$'), ax.set_ylabel(r'$\\log_e(x^2)$')\npass\n```\n\n### ** Part 4: Check MathJax Formulas **\n\n#### ** (4a) Gradient descent formula **\n#### You should see a formula on the line below this one: $$ \\scriptsize \\mathbf{w}_{i+1} = \\mathbf{w}_i - \\alpha_i \\sum_j (\\mathbf{w}_i^\\top\\mathbf{x}_j - y_j) \\mathbf{x}_j \\,.$$\n \n#### This formula is included inline with the text and is $ \\scriptsize (\\mathbf{w}^\\top \\mathbf{x} - y) \\mathbf{x} $.\n\n#### ** (4b) Log loss formula **\n#### This formula shows log loss for single point. Log loss is defined as: $$ \\begin{align} \\scriptsize \\ell_{log}(p, y) = \\begin{cases} -\\log (p) & \\text{if } y = 1 \\\\\\ -\\log(1-p) & \\text{if } y = 0 \\end{cases} \\end{align} $$\n\n### ** Part 5: Export / download and submit **\n\n#### ** (5a) Time to submit **\n\n#### You have completed the lab. To submit the lab for grading you will need to download it from your IPython Notebook environment. You can do this by clicking on \"File\", then hovering your mouse over \"Download as\", and then clicking on \"Python (.py)\". This will export your IPython Notebook as a .py file to your computer.\n#### To upload this file to the course autograder, go to the edX website and find the page for submitting this assignment. Click \"Choose file\", then navigate to and click on the downloaded .py file. Now click the \"Open\" button and then the \"Check\" button. Your submission will be graded shortly and will be available on the page where you submitted. Note that when submission volumes are high, it may take as long as an hour to receive results.\n", "meta": {"hexsha": "5b9b31a703caa6889b2dafe37aebfefb8d935299", "size": 62117, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Week 0 - Course Software Setup/lab0_student.ipynb", "max_stars_repo_name": "rh01/ml-scalable", "max_stars_repo_head_hexsha": "e6e22cca7cb187bb4d131f66cda16f9665cfb068", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 117, "max_stars_repo_stars_event_min_datetime": "2015-09-14T05:07:48.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-26T17:03:22.000Z", "max_issues_repo_path": "Week 1 - Data Science Background and Course Software Setup/lab0_student.ipynb", "max_issues_repo_name": "bhimmetoglu/BerkeleyX-CS100.1x-Big-Data-with-Apache-Spark", "max_issues_repo_head_hexsha": "67d88ae76ded1a296f9bb69e0607409bd3adb0fc", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Week 1 - Data Science Background and Course Software Setup/lab0_student.ipynb", "max_forks_repo_name": "bhimmetoglu/BerkeleyX-CS100.1x-Big-Data-with-Apache-Spark", "max_forks_repo_head_hexsha": "67d88ae76ded1a296f9bb69e0607409bd3adb0fc", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 129, "max_forks_repo_forks_event_min_datetime": "2015-09-02T18:42:29.000Z", "max_forks_repo_forks_event_max_datetime": "2021-02-18T08:33:03.000Z", "avg_line_length": 204.3322368421, "max_line_length": 52768, "alphanum_fraction": 0.8965822561, "converted": true, "num_tokens": 1544, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5195213219520929, "lm_q2_score": 0.8459424373085146, "lm_q1q2_score": 0.43948513332589495}} {"text": "# Naive Bayes Spam Filter From Scratch\n\nAlthough referred to as \"Idiot Bayes\" by some, Naive Bayes models perform surprisingly well despite strong independence assumptions in the model. In this project we will build a spam classifier from scratch for SMS messages using Multinomial Naive Bayes which is known to suit situations where data can be turned into counts, such as word counts in text.\n\nYou can find more information about the dataset by clicking [here](https://archive.ics.uci.edu/ml/datasets/sms+spam+collection).\n\n## 1. Reading in the data\n\n\n```python\nimport pandas as pd\nimport numpy as np\nimport re\n```\n\n\n```python\ncollection = pd.read_csv(\"SMSSpamCollection\", sep=\"\\t\", header=None, names=[\"Label\", \"SMS\"])\ncollection.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    LabelSMS
    0hamGo until jurong point, crazy.. Available only ...
    1hamOk lar... Joking wif u oni...
    2spamFree entry in 2 a wkly comp to win FA Cup fina...
    3hamU dun say so early hor... U c already then say...
    4hamNah I don't think he goes to usf, he lives aro...
    \n
    \n\n\n\n\n```python\ncollection[\"Label\"].value_counts(dropna=False)\n```\n\n\n\n\n ham 4825\n spam 747\n Name: Label, dtype: int64\n\n\n\nAs we can see, non-spam messages are labelled as \"ham\" and there's a lot more non-spam mails compared to spam mails.\n\n## 2. Preprocessing\n\nAs a first step, we will randomize the entire dataset, then split it into the training and test sets. Training set will consist of 80% of the dataset, leaving 20% for testing purposes.\n\n\n```python\n#Specify random state to make sure the results are reproducable:\nsampled_set = collection.sample(frac=1, random_state=1)\n\n#Split into 80% - 20% sets:\ntrain_set = sampled_set[:4459].copy().reset_index(drop=True)\ntest_set = sampled_set[4459:].copy().reset_index(drop=True)\n\ntrain_set[\"Label\"].value_counts(normalize=True)\n```\n\n\n\n\n ham 0.865441\n spam 0.134559\n Name: Label, dtype: float64\n\n\n\n\n```python\ntest_set[\"Label\"].value_counts(normalize=True)\n```\n\n\n\n\n ham 0.867925\n spam 0.132075\n Name: Label, dtype: float64\n\n\n\nWhen we first read the dataset, we have checked value counts for the labels. We have found that 87% of the messages were labelled \"ham\" whereas 13% were labelled spam. We can see that our test and training sets have similar proportions for each value.\n\nWe will now work on the SMS column, starting with removing punctuation and making all letters lowercase.\n\n\n```python\ntrain_set[\"SMS\"] = train_set[\"SMS\"].str.replace(r\"\\W\", \" \").str.lower()\ntrain_set.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    LabelSMS
    0hamyep by the pretty sculpture
    1hamyes princess are you going to make me moan
    2hamwelp apparently he retired
    3hamhavent
    4hami forgot 2 ask \u00fc all smth there s a card on ...
    \n
    \n\n\n\nWe will now create a vocabulary for the training set. We will use sets for this to prevent duplicates.\n\n\n```python\ntrain_set[\"SMS\"] = train_set[\"SMS\"].str.split()\n\nvocab = []\nfor message in train_set[\"SMS\"]:\n for word in message:\n vocab.append(word)\n \nvocab = set(vocab) #removing duplicates\nvocab = list(vocab)\nprint(vocab[:5])\n```\n\n ['ibm', 'monkey', 'sam', 'perform', 'still']\n\n\nWe will now create a dictionary containing the words as keys and word counts per SMS. Then, we will form a DataFrame from it.\n\n\n```python\nword_counts_per_sms = {word: [0] * len(train_set[\"SMS\"]) for word in vocab}\n\nfor index, message in enumerate(train_set[\"SMS\"]):\n for word in message:\n word_counts_per_sms[word][index] += 1\n \nword_counts = pd.DataFrame(word_counts_per_sms)\ntraining_set = pd.concat([train_set, word_counts], axis=1)\ntraining_set.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    LabelSMSibmmonkeysamperformstillsecretaryin2figure...rgenthelpalivefneibhpubjogduo08000930705unconscious
    0ham[yep, by, the, pretty, sculpture]00000000...0000000000
    1ham[yes, princess, are, you, going, to, make, me,...00000000...0000000000
    2ham[welp, apparently, he, retired]00000000...0000000000
    3ham[havent]00000000...0000000000
    4ham[i, forgot, 2, ask, \u00fc, all, smth, there, s, a,...00000000...0000000000
    \n

    5 rows \u00d7 7787 columns

    \n
    \n\n\n\n## 3. Creating our spam filter\n\nWe will be using the probabilities of the equations below for classification:\n\n\\begin{equation}\nP(spam | w_1,w_2, ..., w_n) \\propto P(spam) \\cdot \\prod_{i=1}^{n}P(w_i|spam) \\\\\nP(ham | w_1,w_2, ..., w_n) \\propto P(ham) \\cdot \\prod_{i=1}^{n}P(w_i|ham)\n\\end{equation}\n\nFor each P(w|Spam) and P(w|Ham) in the formulas above, we will be using these equations with Laplace smoothing:\n\n\\begin{equation}\nP(w_i|spam) = \\frac{N_{w_i|spam} + \\alpha}{N_{spam} + \\alpha \\cdot N_{vocabulary}} \\\\\nP(w_i|ham) = \\frac{N_{w_i|ham} + \\alpha}{N_{ham} + \\alpha \\cdot N_{vocabulary}}\n\\end{equation}\n\n\n```python\nn_spam = sum([len(row[1]) for row in training_set if row[0] == \"spam\"])\nn_ham = sum([len(row[1]) for row in training_set if row[0] == \"ham\"])\nn_vocab = len(vocab)\n\np_spam = training_set[\"Label\"].value_counts()[\"spam\"] / training_set[\"Label\"].shape[0]\np_ham = training_set[\"Label\"].value_counts()[\"ham\"] / training_set[\"Label\"].shape[0]\n\nprint(p_spam, p_ham)\n```\n\n 0.13455931823278763 0.8654406817672123\n\n\nNow that we have calculated the constants, we will move on to calculating the parameters.\n\n\n```python\nspam_df = training_set[training_set[\"Label\"] == \"spam\"].copy()\nham_df = training_set[training_set[\"Label\"] == \"ham\"].copy()\n\nalpha = 1 #Since we use Laplace smoothing\n\nspam_params = {word: (sum(count) + alpha) / (n_spam + (alpha * n_vocab)) for word, count in spam_df.iloc[:, 2:].iteritems()}\nham_params = {word: (sum(count) + alpha) / (n_ham + (alpha * n_vocab)) for word, count in ham_df.iloc[:, 2:].iteritems()}\n```\n\nWe have now calculated our parameters, too. We can now begin writing our function.\n\n## 4. Classification process\n\n\n```python\ndef classify(message, verbose=True):\n\n message = re.sub('\\W', ' ', message).lower().split()\n \n try:\n p_spam_given_message = p_spam * np.prod([spam_params[word] for word in message])\n p_ham_given_message = p_ham * np.prod([ham_params[word] for word in message])\n except:\n p_spam_given_message = p_spam\n p_ham_given_message = p_ham\n \n if verbose == True:\n print(\"P(Spam|message):\", p_spam_given_message)\n print(\"P(Ham|message):\", p_ham_given_message)\n \n if p_ham_given_message > p_spam_given_message:\n print(\"Label: Non-spam\")\n elif p_ham_given_message < p_spam_given_message:\n print(\"Label: Spam\")\n else:\n print(\"This message carries equal probabilities of being non-spam vs. spam. Please consult a human for accurate classification.\")\n else:\n if p_ham_given_message > p_spam_given_message:\n return \"ham\"\n elif p_ham_given_message < p_spam_given_message:\n return \"spam\"\n else:\n return \"needs human classification\"\n\nclassify(\"success\")\n```\n\n P(Spam|message): 1.728443394127009e-05\n P(Ham|message): 0.00022233543526453754\n Label: Non-spam\n\n\nOur function seems to work well!\n\n## 5. Measuring Accuracy\n\nIn this step, we will be measuring our function's accuracy on the test set using the formula below:\n\n\\begin{equation}\n\\text{Accuracy} = \\frac{\\text{number of correctly classified messages}}{\\text{total number of classified messages}}\n\\end{equation}\n\n\n```python\ncorrectly_classified = 0\ntotal_classified = test_set.shape[0]\n\ntest_set[\"prediction\"] = test_set[\"SMS\"].apply(classify, verbose=False)\n\nfor row in test_set.iterrows():\n row = row[1]\n if row[\"Label\"] == row[\"prediction\"]:\n correctly_classified += 1\n \ntest_set[\"prediction\"].value_counts()\n```\n\n\n\n\n ham 1062\n spam 51\n Name: prediction, dtype: int64\n\n\n\n\n```python\naccuracy = correctly_classified / total_classified\nprint(\"Predictions made by our function are {pred}% accurate.\\nCorrect: {correct}\\nIncorrect: {inc}\\nTotal: {tot}\".format(pred=round(accuracy * 100, 2), correct=correctly_classified, inc=total_classified - correctly_classified, tot=total_classified))\n```\n\n Predictions made by our function are 91.37% accurate.\n Correct: 1017\n Incorrect: 96\n Total: 1113\n\n\n## 6. Conclusion\n\nIn this project, we have built a spam classifier from scratch. As we can see in the final step's output, our spam filter has an accuracy of ~91.37% with 1017 correctly classified messages out of 1113.\n", "meta": {"hexsha": "aaaeeaf4b57f99ee96fbc099c837155c3b3351c6", "size": 23382, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Spam Filter.ipynb", "max_stars_repo_name": "dilarakarabey/spam-filter-from-scratch", "max_stars_repo_head_hexsha": "693064f600ef8b624d35fe0437e53653e7f7d168", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Spam Filter.ipynb", "max_issues_repo_name": "dilarakarabey/spam-filter-from-scratch", "max_issues_repo_head_hexsha": "693064f600ef8b624d35fe0437e53653e7f7d168", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Spam Filter.ipynb", "max_forks_repo_name": "dilarakarabey/spam-filter-from-scratch", "max_forks_repo_head_hexsha": "693064f600ef8b624d35fe0437e53653e7f7d168", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.8063241107, "max_line_length": 365, "alphanum_fraction": 0.4263963733, "converted": true, "num_tokens": 3924, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.6859494421679929, "lm_q1q2_score": 0.43944380726024335}} {"text": "# \u039c\u03ad\u03c4\u03c1\u03b7\u03c3\u03b7 \u03c4\u03b7\u03c2 \u03c1\u03bf\u03ae\u03c2 \u03c4\u03b7\u03c2 \u03ba\u03b1\u03c4\u03b1\u03bd\u03bf\u03bc\u03ae\u03c2 \u03bd\u03b5\u03c4\u03c1\u03bf\u03bd\u03af\u03c9\u03bd \u03c3\u03b5 \u03c5\u03c0\u03bf\u03ba\u03c1\u03af\u03c3\u03b9\u03bc\u03bf \u03c0\u03c5\u03c1\u03b7\u03bd\u03b9\u03ba\u03cc \u03b1\u03bd\u03c4\u03b9\u03b4\u03c1\u03b1\u03c3\u03c4\u03ae\u03c1\u03b1\n\n## \u0398\u03b5\u03c9\u03c1\u03b7\u03c4\u03b9\u03ba\u03ae \u03b5\u03b9\u03c3\u03b1\u03b3\u03c9\u03b3\u03ae\n\n### \u0391\u03bd\u03c4\u03b9\u03b4\u03c1\u03b1\u03c3\u03c4\u03ae\u03c1\u03b1\u03c2\n\u03a3\u03c4\u03bf \u03c0\u03b5\u03af\u03c1\u03b1\u03bc\u03b1 \u03b1\u03c5\u03c4\u03cc \u03bc\u03b5\u03bb\u03b5\u03c4\u03ac\u03c4\u03b1\u03b9 \u03b7 \u03ba\u03b1\u03c4\u03b1\u03bd\u03bf\u03bc\u03ae \u03c4\u03c9\u03bd \u03bd\u03b5\u03c4\u03c1\u03bf\u03bd\u03af\u03c9\u03bd \u03c3\u03b5 \u03c5\u03c0\u03bf\u03ba\u03c1\u03af\u03c3\u03b9\u03bc\u03bf \u03c0\u03c5\u03c1\u03b7\u03bd\u03b9\u03ba\u03cc \u03b1\u03bd\u03c4\u03b9\u03b4\u03c1\u03b1\u03c3\u03c4\u03ae\u03c1\u03b1 ($k_{eff} \\approx 0.8$). \u039f \u03b1\u03bd\u03c4\u03b9\u03b4\u03c1\u03b1\u03c3\u03c4\u03ae\u03c1\u03b1\u03c2 \u03b1\u03c0\u03bf\u03c4\u03b5\u03bb\u03b5\u03af\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c1\u03ac\u03b2\u03b4\u03bf\u03c5\u03c2 \u03c6\u03c5\u03c3\u03b9\u03ba\u03bf\u03cd \u03bf\u03c5\u03c1\u03b1\u03bd\u03af\u03bf\u03c5 $U_3O_8(UO_2 . 2UO_3)$, \u03b5\u03c0\u03b9\u03b2\u03c1\u03b1\u03b4\u03c5\u03bd\u03c4\u03ae (\u03ba\u03b1\u03b9 \u03b1\u03bd\u03b1\u03ba\u03bb\u03b1\u03c3\u03c4\u03ae) \u03b5\u03bb\u03b1\u03c6\u03c1\u03cd \u03cd\u03b4\u03c9\u03c1 $H_2O$. \u03a4\u03bf \u03bf\u03c5\u03c1\u03ac\u03bd\u03b9\u03bf \u03b2\u03c1\u03af\u03c3\u03ba\u03b5\u03c4\u03b1\u03b9 \u03c3\u03b5 \u03c3\u03c9\u03bb\u03ae\u03bd\u03b5\u03c2 \u03b1\u03bb\u03bf\u03c5\u03bc\u03b9\u03bd\u03af\u03bf\u03c5 \u03b3\u03b9\u03b1 \u03c4\u03b7\u03bd \u03b1\u03c0\u03bf\u03c6\u03c5\u03b3\u03ae \u03b4\u03b9\u03b1\u03c1\u03c1\u03bf\u03ae\u03c2 \u03c1\u03b1\u03b4\u03b9\u03b5\u03bd\u03b5\u03c1\u03b3\u03bf\u03cd \u03c5\u03bb\u03b9\u03ba\u03bf\u03cd. \n\n\u03a3\u03c4\u03bf \u03ba\u03ad\u03bd\u03c4\u03c1\u03bf \u03c4\u03bf\u03c5 \u03b1\u03bd\u03c4\u03b9\u03b4\u03c1\u03b1\u03c3\u03c4\u03ae\u03c1\u03b1 \u03c4\u03bf\u03c0\u03bf\u03b8\u03b5\u03c4\u03b5\u03af\u03c4\u03b1\u03b9 \u03c0\u03b7\u03b3\u03ae \u03bd\u03b5\u03c4\u03c1\u03bf\u03bd\u03af\u03c9\u03bd 5Ci $\\ ^{241}AmBe$. \n\n\u03a4\u03b1 \u03bd\u03b5\u03c4\u03c1\u03cc\u03bd\u03b9\u03b1 \u03c4\u03b7\u03c2 \u03c0\u03b7\u03b3\u03ae\u03c2 \u03b8\u03b5\u03c1\u03bc\u03bf\u03c0\u03bf\u03b9\u03bf\u03cd\u03bd\u03c4\u03b1\u03b9 \u03ba\u03b1\u03b8\u03ce\u03c2 \u03c3\u03c5\u03b3\u03ba\u03c1\u03bf\u03cd\u03bd\u03c4\u03b1\u03b9 \u03bc\u03b5 \u03c4\u03bf\u03c5\u03c2 \u03c0\u03c5\u03c1\u03ae\u03bd\u03b5\u03c2 \u03c5\u03b4\u03c1\u03bf\u03b3\u03cc\u03bd\u03bf\u03c5 \u03c3\u03c4\u03bf \u03bd\u03b5\u03c1\u03cc, \u03ba\u03b1\u03b9 \u03b5\u03bd \u03c3\u03c5\u03bd\u03b5\u03c7\u03b5\u03af\u03b1 \u03bc\u03ad\u03c1\u03bf\u03c2 \u03c4\u03bf\u03c5\u03c2 \u03b1\u03bd\u03c4\u03b9\u03b4\u03c1\u03ac \u03bc\u03b5 \u03c4\u03bf $\\ ^{235}U$ \u03c0\u03c1\u03bf\u03ba\u03b1\u03bb\u03ce\u03bd\u03c4\u03b1\u03c2 \u03c3\u03c7\u03ac\u03c3\u03b7 \u03ba\u03b1\u03b9 \u03b1\u03c0\u03b5\u03bb\u03b5\u03c5\u03b8\u03ad\u03c1\u03c9\u03c3\u03b7 \u03b5\u03c0\u03b9\u03c0\u03bb\u03ad\u03bf\u03bd \u03bd\u03b5\u03c4\u03c1\u03bf\u03bd\u03af\u03c9\u03bd. \u03a4\u03bf \u03bd\u03b5\u03c1\u03cc \u03b4\u03c1\u03b1 \u03ba\u03b1\u03b9 \u03c9\u03c2 \u03b1\u03bd\u03b1\u03ba\u03bb\u03b1\u03c3\u03c4\u03ae\u03c2, \u03ce\u03c3\u03c4\u03b5 \u03bc\u03ad\u03c1\u03bf\u03c2 \u03c4\u03c9\u03bd \u03bd\u03b5\u03c4\u03c1\u03bf\u03bd\u03af\u03c9\u03bd \u03c0\u03bf\u03c5 \u03ba\u03b1\u03bd\u03bf\u03bd\u03b9\u03ba\u03ac \u03b8\u03b1 \u03b4\u03b9\u03ad\u03c6\u03b5\u03c5\u03b3\u03b1\u03bd \u03c4\u03bf\u03c5 \u03b1\u03bd\u03c4\u03b9\u03b4\u03c1\u03b1\u03c3\u03c4\u03ae\u03c1\u03b1 \u03bd\u03b1 \u03b1\u03bd\u03b1\u03ba\u03bb\u03ce\u03bd\u03c4\u03b1\u03b9 \u03c0\u03af\u03c3\u03c9. \n\n### \u0394\u03b9\u03ac\u03c7\u03c5\u03c3\u03b7 \u03bd\u03b5\u03c4\u03c1\u03bf\u03bd\u03af\u03c9\u03bd\n\u0398\u03b5\u03c9\u03c1\u03ce\u03bd\u03c4\u03b1\u03c2 \u03cc\u03c4\u03b9 \u03c4\u03b1 \u03bd\u03b5\u03c4\u03c1\u03cc\u03bd\u03b9\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bc\u03bf\u03bd\u03bf\u03b5\u03bd\u03b5\u03c1\u03b3\u03b5\u03b9\u03b1\u03ba\u03ac \u03bc\u03c0\u03bf\u03c1\u03bf\u03cd\u03bc\u03b5 \u03bd\u03b1 \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03b9\u03ae\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b7\u03bd \u03bc\u03ad\u03b8\u03bf\u03b4\u03bf \u03b4\u03b9\u03ac\u03c7\u03c5\u03c3\u03b7\u03c2 \u03bc\u03b9\u03b1\u03c2 \u03bf\u03bc\u03ac\u03b4\u03b1\u03c2. \u03a3\u03cd\u03bc\u03c6\u03c9\u03bd\u03b1 \u03bc\u03b5 \u03c4\u03b7 \u03bc\u03ad\u03b8\u03bf\u03b4\u03bf \u03b1\u03c5\u03c4\u03ae, \u03cc\u03c4\u03b1\u03bd \u03bf \u03b1\u03bd\u03c4\u03b9\u03b4\u03c1\u03b1\u03c3\u03c4\u03ae\u03c1\u03b1\u03c2 \u03bb\u03b5\u03b9\u03c4\u03bf\u03c5\u03c1\u03b3\u03b5\u03af \u03c3\u03b5 \u03c3\u03c4\u03b1\u03b8\u03b5\u03c1\u03ad\u03c2 \u03c3\u03c5\u03bd\u03b8\u03ae\u03ba\u03b5\u03c2, \u03b3\u03b9\u03b1 \u03c4\u03b1 \u03bd\u03b5\u03c4\u03c1\u03cc\u03bd\u03b9\u03b1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9: \n\n$$ created - escaped - absorbed = 0 $$\n\n\u03ae \n\n$$ D\\nabla^2\\Phi - \\Sigma_a\\Phi + S = 0$$\n\n\u03cc\u03c0\u03bf\u03c5 \u03a6 \u03b7 \u03c1\u03bf\u03ae \u03c4\u03c9\u03bd \u03b8\u03b5\u03c1\u03bc\u03b9\u03ba\u03ce\u03bd \u03bd\u03b5\u03c4\u03c1\u03bf\u03bd\u03af\u03c9\u03bd, D \u03bf \u03c3\u03c5\u03bd\u03c4\u03b5\u03bb\u03b5\u03c3\u03c4\u03ae\u03c2 \u03b4\u03b9\u03ac\u03c7\u03c5\u03c3\u03b7\u03c2, $\\Sigma_a$ \u03b7 \u03bc\u03b1\u03ba\u03c1\u03bf\u03c3\u03ba\u03bf\u03c0\u03b9\u03ba\u03ae \u03b5\u03bd\u03b5\u03c1\u03b3\u03cc\u03c2 \u03b4\u03b9\u03b1\u03c4\u03bf\u03bc\u03ae \u03b1\u03c0\u03bf\u03c1\u03c1\u03cc\u03c6\u03b7\u03c3\u03b7\u03c2, \u03ba\u03b1\u03b9 S \u03bf \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 n \u03b1\u03bd\u03ac \u03bc\u03bf\u03bd\u03ac\u03b4\u03b1 \u03cc\u03b3\u03ba\u03bf\u03c5 \u03b1\u03bd\u03ac \u03bc\u03bf\u03bd\u03ac\u03b4\u03b1 \u03c7\u03c1\u03cc\u03bd\u03bf\u03c5.\n\n\u0395\u03c0\u03b9\u03bb\u03cd\u03bf\u03bd\u03c4\u03b1\u03c2 \u03c4\u03b7\u03bd \u03c0\u03b1\u03c1\u03b1\u03c0\u03ac\u03bd\u03c9 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 \u03ba\u03b1\u03c4\u03b1\u03bb\u03ae\u03b3\u03bf\u03c5\u03bc\u03b5 \u03c3\u03c4\u03b7\u03bd: \n\n$$ \\Phi(r,z)=A\\cdot J_0\\cdot (2.405/R_{ex}r)\\cdot e^{-\\gamma z} $$\n\n\u03ae \n\n$$ln\\Phi(r,z) = ln[A\\cdot J_0\\cdot (2.405/R_{ex}r)] -\\gamma z \\qquad (1)$$\n\n\n\u0393\u03b9\u03b1 \u03b4\u03b5\u03b4\u03bf\u03bc\u03ad\u03bd\u03bf $r=r_0$ \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03c4\u03bf\u03c5 $ln\\Phi$ \u03c9\u03c2 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03bf \u03cd\u03c8\u03bf\u03c2 $z$ \u03b5\u03af\u03bd\u03b1\u03b9 \u03b5\u03c5\u03b8\u03b5\u03af\u03b1 \u03bc\u03b5 \u03ba\u03bb\u03af\u03c3\u03b7 $\\gamma$. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03c4\u03bf \u03cd\u03c8\u03bf\u03c2 \u03b1\u03c0\u03cc \u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf $r_0$ \u03c3\u03c4\u03bf \u03bf\u03c0\u03bf\u03af\u03bf \u03b7 \u03c1\u03bf\u03ae \u03bc\u03b5\u03b9\u03ce\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf $1/e$ \u03c4\u03b7\u03c2 \u03b1\u03c1\u03c7\u03b9\u03ba\u03ae\u03c2, \u03b5\u03af\u03bd\u03b1\u03b9 $l=1/\\gamma$ \u03ba\u03b1\u03b9 \u03bf\u03bd\u03bf\u03bc\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 _\u03bc\u03ae\u03ba\u03bf\u03c2 \u03c7\u03b1\u03bb\u03ac\u03c1\u03c9\u03c3\u03b7\u03c2_. \n\n### \u0399\u03b4\u03b9\u03b1\u03b9\u03c4\u03b5\u03c1\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2 \u03c0\u03b5\u03b9\u03c1\u03ac\u03bc\u03b1\u03c4\u03bf\u03c2\n\n\u039b\u03cc\u03b3\u03c9 \u03c3\u03c5\u03bd\u03c4\u03ae\u03c1\u03b7\u03c3\u03b7\u03c2 \u03c4\u03c9\u03bd \u03c1\u03ac\u03b2\u03b4\u03c9\u03bd \u03ba\u03b1\u03c5\u03c3\u03af\u03bc\u03bf\u03c5 \u03bf \u03b1\u03bd\u03c4\u03b9\u03b4\u03c1\u03b1\u03c3\u03c4\u03ae\u03c1\u03b1\u03c2 \u03c0\u03b5\u03c1\u03b9\u03b5\u03af\u03c7\u03b5 \u03bc\u03cc\u03bd\u03bf \u03c4\u03b7\u03bd \u03c0\u03b7\u03b3\u03ae $\\ ^{241}AmBe$ \u03ba\u03b1\u03b9 \u03c4\u03bf \u03bd\u03b5\u03c1\u03cc, \u03c7\u03c9\u03c1\u03af\u03c2 \u03ba\u03b1\u03bc\u03af\u03b1 \u03c1\u03ac\u03b2\u03b4\u03bf \u03ba\u03b1\u03c5\u03c3\u03af\u03bc\u03bf\u03c5. \u0395\u03c0\u03af\u03c3\u03b7\u03c2, \u03b8\u03b5\u03c9\u03c1\u03ae\u03b8\u03b7\u03ba\u03b5 \u03cc\u03c4\u03b9 \u03b7 \u03b5\u03bd\u03b5\u03c1\u03b3\u03bf\u03c0\u03bf\u03af\u03b7\u03c3\u03b7 \u03c4\u03bf\u03c5 $In$ \u03bf\u03c6\u03b5\u03af\u03bb\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c0\u03bf\u03ba\u03bb\u03b5\u03b9\u03c3\u03c4\u03b9\u03ba\u03ac \u03c3\u03b5 \u03b8\u03b5\u03c1\u03bc\u03b9\u03ba\u03ac \u03bd\u03b5\u03c4\u03c1\u03cc\u03bd\u03b9\u03b1.\n\n---\n\n## \u03a0\u03b5\u03af\u03c1\u03b1\u03bc\u03b1\n\n\u03a6\u03cd\u03bb\u03bb\u03b1 $In$ \u03c4\u03bf\u03c0\u03bf\u03b8\u03b5\u03c4\u03ae\u03b8\u03b7\u03ba\u03b1\u03bd \u03c3\u03c4\u03bf\u03bd \u03b1\u03bd\u03c4\u03b9\u03b4\u03c1\u03b1\u03c3\u03c4\u03ae\u03c1\u03b1 \u03c3\u03b5 \u03c3\u03c5\u03b3\u03ba\u03b5\u03ba\u03c1\u03b9\u03bc\u03ad\u03bd\u03b7 \u03b8\u03ad\u03c3\u03b7, \u03b1\u03bb\u03bb\u03ac \u03c3\u03b5 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03b5\u03c4\u03b9\u03ba\u03cc \u03cd\u03c8\u03bf\u03c2 \u03bf \u03ba\u03b1\u03b8\u03ad\u03bd\u03b1\u03c2. \u03a3\u03c4\u03bf \u03c6\u03c5\u03c3\u03b9\u03ba\u03cc $In$ \u03c5\u03c0\u03ac\u03c1\u03c7\u03b5\u03b9 95.7(1)% $\\ ^{115}In$. \u039c\u03b5\u03c4\u03ac \u03b1\u03c0\u03cc \u03b1\u03ba\u03c4\u03b9\u03bd\u03bf\u03b2\u03cc\u03bb\u03b7\u03c3\u03b7 4-5 \u03c9\u03c1\u03ce\u03bd (\u03c0\u03b5\u03c1\u03af\u03c0\u03bf\u03c5 5 \u03b7\u03bc\u03b9\u03b6\u03c9\u03ad\u03c2 \u03c4\u03bf\u03c5 $\\ ^{116}In$ \u03ce\u03c3\u03c4\u03b5 \u03b7 \u03b5\u03bd\u03b5\u03c1\u03b3\u03cc\u03c4\u03b7\u03c4\u03b1 \u03bd\u03b1 \u03c6\u03c4\u03ac\u03c3\u03b5\u03b9 \u03c0\u03b5\u03c1\u03af\u03c0\u03bf\u03c5 \u03c4\u03bf 99(1)% \u03c4\u03b7\u03c2 \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03b7\u03c2), \u03c4\u03b1 \u03c6\u03cd\u03bb\u03bb\u03b1 \u03b1\u03c6\u03b1\u03b9\u03c1\u03ad\u03b8\u03b7\u03ba\u03b1\u03bd. \n\n\n\u039f \u03c1\u03c5\u03b8\u03bc\u03cc\u03c2 \u03c5\u03c0\u03bf\u03b2\u03ac\u03b8\u03c1\u03bf\u03c5 \u03c7\u03c9\u03c1\u03af\u03c2 $In$ \u03ae\u03c4\u03b1\u03bd $8\\ cpm$ \u03ba\u03b1\u03b9 \u03bc\u03b5 **\u03bc\u03b7** \u03b5\u03bd\u03b5\u03c1\u03b3\u03bf\u03c0\u03bf\u03b9\u03b7\u03bc\u03ad\u03bd\u03bf $In$ \u03ae\u03c4\u03b1\u03bd $9\\ cpm$, \u03ba\u03b1\u03b9 \u03b1\u03c6\u03b1\u03b9\u03c1\u03b5\u03af\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03b1\u03bb\u03b7\u03b8\u03ae \u03c1\u03c5\u03b8\u03bc\u03cc. \n\n\u0397 \u03bf\u03bc\u03ac\u03b4\u03b1 \u03bc\u03b1\u03c2 \u03bc\u03ad\u03c4\u03c1\u03b7\u03c3\u03b5 \u03c4\u03bf\u03c5\u03c2 \u03c8\u03b5\u03c5\u03b4\u03b5\u03af\u03c2 \u03c1\u03c5\u03b8\u03bc\u03bf\u03cd\u03c2 R' \u03b1\u03c0\u03cc \u03c4\u03b9\u03c2 \u03c0\u03b7\u03b3\u03ad\u03c2 3,6,9. \u039f\u03b9 R' \u03bc\u03b5\u03c4\u03b1\u03c4\u03c1\u03ac\u03c0\u03b7\u03ba\u03b1\u03bd \u03c3\u03b5 \u03b1\u03bb\u03b7\u03b8\u03b5\u03af\u03c2 \u03bc\u03ad\u03c3\u03c9 \u03c4\u03b7\u03c2 \u03c3\u03c7\u03ad\u03c3\u03b7\u03c2: \n\n$$ R = \\frac{R'}{1-\\tau R'} $$\n\n\u03cc\u03c0\u03bf\u03c5 $\\tau=50\\ \\mu s$ o \u03bd\u03b5\u03ba\u03c1\u03cc\u03c2 \u03c7\u03c1\u03cc\u03bd\u03bf\u03c2 \u03c4\u03bf\u03c5 \u03b1\u03bd\u03b9\u03c7\u03bd\u03b5\u03c5\u03c4\u03ae Geiger. \n\n\n\u0397 \u03b1\u03c0\u03cc\u03b4\u03bf\u03c3\u03b7 \u03c4\u03bf\u03c5 Geiger \u03c3\u03c4\u03b9\u03c2 \u03c3\u03c5\u03b3\u03ba\u03b5\u03ba\u03c1\u03b9\u03bc\u03ad\u03bd\u03b5\u03c2 \u03b5\u03bd\u03ad\u03c1\u03b3\u03b5\u03b9\u03b5\u03c2 \u03c4\u03c9\u03bd $\\beta^-$ \u03b5\u03af\u03bd\u03b1\u03b9 $\\varepsilon=100\\%$ \u03ba\u03b1\u03b9 \u03bf \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03bd\u03c4\u03b1\u03c2 \u03b3\u03b5\u03c9\u03bc\u03b5\u03c4\u03c1\u03af\u03b1\u03c2 $g = 0.4$. H \u03b5\u03bd\u03b5\u03c1\u03b3\u03cc\u03c4\u03b7\u03c4\u03b1 $A'$ \u03b4\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03c3\u03c7\u03ad\u03c3\u03b7: \n\n$$ A' = \\frac{R}{\\varepsilon\\cdot g} $$\n\n\n\n\u0397 \u03b5\u03bd\u03b5\u03c1\u03b3\u03cc\u03c4\u03b7\u03c4\u03b1 \u03b1\u03bd\u03ac gr $A$ \u03b4\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03c3\u03c7\u03ad\u03c3\u03b7: \n\n$$A = \\frac{A'}{m}$$\n\n\u03cc\u03c0\u03bf\u03c5 $m$ \u03b7 \u03bc\u03ac\u03b6\u03b1 \u03c4\u03bf\u03c5 $In$ \u03c3\u03b5 gr.\n\n\u0398\u03b5\u03c9\u03c1\u03bf\u03cd\u03bc\u03b5 \u03cc\u03c4\u03b9 \u03bf\u03b9 \u03b1\u03bd\u03c4\u03af\u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03c2 \u03b5\u03bd\u03b5\u03c1\u03b3\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2 \u03c0\u03bf\u03c5 \u03c5\u03c0\u03bf\u03bb\u03bf\u03b3\u03af\u03c3\u03b1\u03bc\u03b5 \u03b1\u03bd\u03c4\u03b9\u03c3\u03c4\u03bf\u03b9\u03c7\u03bf\u03cd\u03bd \u03c3\u03c4\u03bf \u03bc\u03ad\u03c3\u03bf \u03c4\u03b7\u03c2 \u03b4\u03b9\u03ac\u03c1\u03ba\u03b5\u03b9\u03b1\u03c2 \u03bc\u03ad\u03c4\u03c1\u03b7\u03c3\u03b7\u03c2. \u03a0\u03c7. \u03b1\u03bd $t_{exit}=18\\ min$ o \u03c7\u03c1\u03cc\u03bd\u03bf\u03c2 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03ad\u03be\u03bf\u03b4\u03bf \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03b1\u03bd\u03c4\u03b9\u03b4\u03c1\u03b1\u03c3\u03c4\u03ae\u03c1\u03b1 \u03ba\u03b1\u03b9 $t_0=60\\ s$ \u03b7 \u03b4\u03b9\u03ac\u03c1\u03ba\u03b5\u03b9\u03b1 \u03bc\u03ad\u03c4\u03c1\u03b7\u03c3\u03b7\u03c2, \u03c4\u03cc\u03c4\u03b5 \u03b7 \u03b5\u03bd\u03b5\u03c1\u03b3\u03cc\u03c4\u03b7\u03c4\u03b1 \u03b1\u03bd\u03c4\u03b9\u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03af \u03c3\u03c4\u03b7\u03bd \u03c7\u03c1\u03bf\u03bd\u03b9\u03ba\u03ae \u03c3\u03c4\u03b9\u03b3\u03bc\u03ae $t=18.5\\ min$. \n\nH \u03c1\u03bf\u03ae \u03c4\u03c9\u03bd \u03b8\u03b5\u03c1\u03bc\u03b9\u03ba\u03ce\u03bd \u03bd\u03b5\u03c4\u03c1\u03bf\u03bd\u03af\u03c9\u03bd \u03b4\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03c3\u03c7\u03ad\u03c3\u03b7: \n\n$$ \\Phi_\\theta = \\frac {A_\\theta} {\\sigma \\cdot M (1-e^{-\\lambda t_{irr} }) }$$\n\n\u03cc\u03c0\u03bf\u03c5: \n - $t_{irr}$ \u03bf \u03c7\u03c1\u03cc\u03bd\u03bf\u03c2 \u03b1\u03ba\u03c4\u03b9\u03bd\u03bf\u03b2\u03cc\u03bb\u03b7\u03c3\u03b7\u03c2 \u03c3\u03c4\u03bf\u03bd \u03b1\u03bd\u03c4\u03b9\u03b4\u03c1\u03b1\u03c3\u03c4\u03ae\u03c1\u03b1\n - $\\sigma = 162\\ barn$ \n - $M = N_A\\cdot m/A$, \u03bc\u03b5 $A=115$\n\n\n```python\nimport numpy\nimport matplotlib as mpl\nimport matplotlib.pyplot as plt\nimport pandas as pd\nimport scipy.optimize\nimport scipy.constants\nimport sympy\nsympy.init_printing() \n\n# Used for latex-labels in plots .\n# (if you don't have the dependencies from http://stackoverflow.com/a/37218925/4230591 installed, \n# it will raise an error during plotting)\nfrom matplotlib import rc\nrc('text', usetex=True)\n```\n\n\n```python\n# ====================================================================================================\n# CONSTANTS AND MISC DATA\n#\n# One-letter variables should not be used to store data, \n# since they'll be needed for `sympy.symbols()` to display functions.\n# ====================================================================================================\n\n\nt_count = 60 # secs\nt_count_in_mins = t_count / 60.\nt_exit_reactor = 18 * 60.\nt_end = 66 * 60.\nt_between_measurements = 2 * 60.\n\nR_background = 8 /60.\nR_background_In = 9 /60.\nm_In = 1 /1000 # 1gr\nindium_115_ratio = .957\n\nM_constant = scipy.constants.N_A * m_In * indium_115_ratio / 115\n\nirradiation_start = '3:00'\nirradiation_t = 4.5 * 3600 # 4-5h\ng_times_e = 0.4\n\u03c4_constant=50 * 10**-6\n\nhalf_life = 54.29 * 60. #sec\ndecay_constant = numpy.log(2) / half_life\n\n\u03c3_In = 162 * 10e-28\n\n\n# =========================================================\n# IRRELEVANT DATA (not used in calculations)\n# =========================================================\n#\n# ~40\u03bcSv/h near reactor\n# AmBe ~10e+6 n/s\n# k_eff = ~.8\n# v_geiger = 500V\n```\n\n\n```python\n# =========================================================\n# R \n# =========================================================\n\nR_\u03b1\u03bb\u03b7\u03b8\u03ae\u03c2, \u03c3_R_\u03b1\u03bb\u03b7\u03b8\u03ae\u03c2, R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2, \u03c4, \u03c3_R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2, t = sympy.symbols(\"R, \\sigma_R, R' tau, \\sigma_R', t\")\n_R_\u03b1\u03bb\u03b7\u03b8\u03ae\u03c2_f = lambda R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2, \u03c4: R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2/(1-R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2*\u03c4)\nR_\u03b1\u03bb\u03b7\u03b8\u03ae\u03c2_f = sympy.lambdify((R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2, \u03c4), _R_\u03b1\u03bb\u03b7\u03b8\u03ae\u03c2_f(R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2, \u03c4), (sympy, numpy))\n\nprint('\\n\u039f \u03b1\u03bb\u03b7\u03b8\u03ae\u03c2 \u03c1\u03c5\u03b8\u03bc\u03cc\u03c2 R \u03b4\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03c3\u03c7\u03ad\u03c3\u03b7:')\nsympy.Eq(R_\u03b1\u03bb\u03b7\u03b8\u03ae\u03c2, R_\u03b1\u03bb\u03b7\u03b8\u03ae\u03c2_f(R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2, \u03c4))\n```\n\n\n```python\n# =========================================================\n# R' error\n# =========================================================\n_\u03c3_R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2_f = lambda R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2, t: sympy.sqrt(R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2) / t\n# (Currently `lambdify` can't handle both arrays *and* sympy functions,)\n# (therefor one function is used for pretty printing and a different one for calculations.)\n\u03c3_R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2_f = sympy.lambdify((R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2, t), _\u03c3_R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2_f(R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2, t), modules=['numpy'])\n\nprint('\\n\u03a4\u03bf \u03c3\u03c6\u03ac\u03bb\u03bc\u03b1 \u03c3\u03c4\u03bf\u03bd \u03c8\u03b5\u03c5\u03b4\u03ae \u03c1\u03c5\u03b8\u03bc\u03cc \u03b4\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03c3\u03c7\u03ad\u03c3\u03b7:')\nsympy.Eq(\u03c3_R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2, _\u03c3_R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2_f(R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2, t))\n```\n\n\n```python\n# =========================================================\n# R error \n# =========================================================\n\n_err_R_\u03b1\u03bb\u03b7\u03b8\u03ae\u03c2_sq = sympy.diff(R_\u03b1\u03bb\u03b7\u03b8\u03ae\u03c2_f(R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2, \u03c4), R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2)**2 * \u03c3_R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2**2\n_err_R_\u03b1\u03bb\u03b7\u03b8\u03ae\u03c2 = sympy.sqrt(_err_R_\u03b1\u03bb\u03b7\u03b8\u03ae\u03c2_sq)\nerr_R_\u03b1\u03bb\u03b7\u03b8\u03ae\u03c2_f = sympy.lambdify((R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2, \u03c3_R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2, \u03c4), _err_R_\u03b1\u03bb\u03b7\u03b8\u03ae\u03c2, modules=['numpy'])\n\nprint('\\n\u03a4\u03bf \u03c3\u03c6\u03ac\u03bb\u03bc\u03b1 \u03c3\u03c4\u03bf\u03bd \u03b1\u03bb\u03b7\u03b8\u03ae \u03c1\u03c5\u03b8\u03bc\u03cc \u03b4\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03c3\u03c7\u03ad\u03c3\u03b7:')\nsympy.Eq(\u03c3_R_\u03b1\u03bb\u03b7\u03b8\u03ae\u03c2, _err_R_\u03b1\u03bb\u03b7\u03b8\u03ae\u03c2)\n```\n\n\n```python\n# ====================================================================================================\n# DATA (sources 3, 6, 9)\n# ====================================================================================================\n\n\ndf3 = pd.read_csv('source3', names=['t (min)', \"R'(cpm)\"])\ndf6 = pd.read_csv('source6', names=['t (min)', \"R'(cpm)\"])\ndf9 = pd.read_csv('source9', names=['t (min)', \"R'(cpm)\"])\n\ndfs_3_6_9 = {3: df3, 6: df6, 9: df9}\n\n\nfor source_n, _df in dfs_3_6_9.items():\n # the measured rate is considered to have occured exactly in the middle of the measurement's duration\n _df['t (min)'] += t_count_in_mins / 2\n _df['t (s)'] = _df['t (min)'] * 60\n\n _df[\"R'(cps)\"] = _df[\"R'(cpm)\"] / 60 \n _df[\"R(cps)\"] = R_\u03b1\u03bb\u03b7\u03b8\u03ae\u03c2_f(_df[\"R'(cps)\"], \u03c4_constant) - R_background_In\n \n # errors\n _df[\"error \u03c3R'(cps)\"] = \u03c3_R_\u03c8\u03b5\u03c5\u03b4\u03ae\u03c2_f(_df[\"R'(cpm)\"], 60)\n _df[\"error \u03c3R(cps)\"] = err_R_\u03b1\u03bb\u03b7\u03b8\u03ae\u03c2_f(_df[\"R(cps)\"], _df[\"error \u03c3R'(cps)\"], \u03c4_constant)\n _df[\"relative \u03c3R %\"] = _df[\"error \u03c3R(cps)\"] / _df[\"R(cps)\"] * 100\n \n \n print('\\n' + '='*70)\n print('\u03a0\u03b7\u03b3\u03ae {}:\\n'.format(source_n))\n print(_df)\n\n source_label = r'source: {}'.format(source_n)\n plt.plot(_df['t (s)'], _df[\"R(cps)\"], marker='.', label=source_label)\n plt.ylim(ymin=0, ymax=160)\n\n\nplt.xlabel('t(s)')\nplt.ylabel('R(cps)')\nplt.legend(loc='upper right')\nplt.title('FIGURE 1:\\nRates vs time of sources 3,6 and 9. \\n(Errorbars were omitted since they are too small.)')\nplt.grid('on')\nplt.show()\n```\n\n### \u03a5\u03c0\u03bf\u03bb\u03bf\u03b3\u03b9\u03c3\u03bc\u03cc\u03c2 \u03c1\u03c5\u03b8\u03bc\u03bf\u03cd R \u03ba\u03b1\u03c4\u03ac \u03c4\u03b7\u03bd \u03ad\u03be\u03bf\u03b4\u03bf \u03c4\u03bf\u03c5 In \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03b1\u03bd\u03c4\u03b9\u03b4\u03c1\u03b1\u03c3\u03c4\u03ae\u03c1\u03b1\n\n[\u0398\u03b5\u03c9\u03c1\u03ce\u03bd\u03c4\u03b1\u03c2 \u03cc\u03c4\u03b9 \u03c4\u03bf \u03c3\u03c6\u03ac\u03bb\u03bc\u03b1 \u03c3\u03c4\u03bf\u03c5\u03c2 \u03c1\u03c5\u03b8\u03bc\u03bf\u03cd\u03c2](http://www.mathworks.com/help/stats/examples/pitfalls-in-fitting-nonlinear-models-by-transforming-to-linearity.html) \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03b1\u03c3\u03c4\u03b9\u03ba\u03cc \u03ba\u03b1\u03b9 \u03cc\u03c7\u03b9 \u03c0\u03c1\u03bf\u03c3\u03b8\u03b5\u03c4\u03b9\u03ba\u03cc, \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03b9\u03bf\u03cd\u03bc\u03b5 \u03bc\u03ad\u03b8\u03bf\u03b4\u03bf \u03b5\u03bb\u03b1\u03c7\u03af\u03c3\u03c4\u03c9\u03bd \u03c4\u03b5\u03c4\u03c1\u03b1\u03b3\u03ce\u03bd\u03c9\u03bd, \u03b1\u03c6\u03bf\u03cd \u03bc\u03b5\u03c4\u03b1\u03c4\u03c1\u03ad\u03c8\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b7\u03bd \u03b5\u03ba\u03b8\u03b5\u03c4\u03b9\u03ba\u03ae \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03c3\u03b5 \u03b3\u03c1\u03b1\u03bc\u03bc\u03b9\u03ba\u03ae.\n\n\n```python\n# ====================================================================================================\n# Linear fit\n# ====================================================================================================\n\n\n_f = lambda x, a, b: a+x*b\nlin_popt, lin_pcov = scipy.optimize.curve_fit(_f, df3['t (s)'], scipy.log(df3['R(cps)']), \n sigma=_df[\"error \u03c3R(cps)\"], absolute_sigma=True)\nlin_perr = numpy.sqrt(numpy.diag(lin_pcov))\n\nprint('a = {:.6}, b = {:.6}'.format(*lin_popt))\nprint('Ro = {:.4}'.format(scipy.e**lin_popt[0]))\nprint('\u03bb/b = {:.1%}'.format(-decay_constant/lin_popt[1]))\nprint('\u03c3\u03c6\u03ac\u03bb\u03bc\u03b1 \u03c3\u03c4\u03bf Ro: {:.4}'.format(scipy.e**lin_perr[0]))\n# (using python variables in markdown is not supported yet, therefor it has to be done manually)\n```\n\n a = 5.26188, b = -0.00021825\n Ro = 192.8\n \u03bb/b = 97.5%\n \u03c3\u03c6\u03ac\u03bb\u03bc\u03b1 \u03c3\u03c4\u03bf Ro: 2.819\n\n\n\u039f \u03c1\u03c5\u03b8\u03bc\u03cc\u03c2 \u03ba\u03b1\u03c4\u03ac \u03c4\u03bf \u03c4\u03ad\u03bb\u03bf\u03c2 \u03c4\u03b7\u03c2 \u03b1\u03ba\u03c4\u03b9\u03bd\u03bf\u03b2\u03cc\u03bb\u03b7\u03c3\u03b7\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 $R_0 = 193(3) cps$.\n\n\n```python\n# ====================================================================================================\n# Non linear fit (not used in further calculations; displayed just for comparison)\n# ====================================================================================================\n\n\ndef exp_f(x, a,b):\n return a * scipy.e**(b*x)\n\nexp_popt, _ = scipy.optimize.curve_fit(exp_f, df3['t (s)'], df3['R(cps)'],sigma=_df[\"error \u03c3R(cps)\"], p0=(300, -decay_constant))\nprint('a = {:.6}, b = {:.6}'.format(*exp_popt))\nprint('\u03bb/b = {:.6}'.format(-decay_constant/exp_popt[1]))\n```\n\n a = 194.294, b = -0.000221309\n \u03bb/b = 0.961513\n\n\n## \u039a\u03b1\u03c4\u03b1\u03bd\u03bf\u03bc\u03ae \u03c1\u03bf\u03ae\u03c2 \u03bd\u03b5\u03c4\u03c1\u03bf\u03bd\u03af\u03c9\u03bd \u03c3\u03c5\u03bd\u03b1\u03c1\u03c4\u03ae\u03c3\u03b5\u03b9 \u03c4\u03bf\u03c5 \u03cd\u03c8\u03bf\u03c5\u03c2 z\n\n\u0391\u03c1\u03c7\u03b9\u03ba\u03ac \u03c5\u03c0\u03bf\u03bb\u03bf\u03b3\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b7\u03bd \u03b5\u03bd\u03b5\u03c1\u03b3\u03cc\u03c4\u03b7\u03c4\u03b1 \u03c4\u03b7\u03c2 \u03b5\u03ba\u03ac\u03c3\u03c4\u03bf\u03c4\u03b5 \u03c0\u03b7\u03b3\u03ae\u03c2 \u03c3\u03c4\u03bf \u03c4\u03ad\u03bb\u03bf\u03c2 \u03c4\u03b7\u03c2 \u03b1\u03ba\u03c4\u03b9\u03bd\u03bf\u03b2\u03cc\u03bb\u03b7\u03c3\u03b7\u03c2, \u03bb\u03b1\u03bc\u03b2\u03ac\u03bd\u03bf\u03bd\u03c4\u03b1\u03c2 \u03c5\u03c0\u03cc\u03c8\u03b9\u03bd \u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03bd\u03c4\u03b1 \u03b3\u03b5\u03c9\u03bc\u03b5\u03c4\u03c1\u03af\u03b1\u03c2 $g=0.4$ \u03ba\u03b1\u03b9 \u03c4\u03b7\u03bd \u03b1\u03c0\u03cc\u03b4\u03bf\u03c3\u03b7 $\\varepsilon=1$ \u03c4\u03bf\u03c5 Geiger \u03c3\u03c4\u03b9\u03c2 \u03c3\u03c5\u03b3\u03ba\u03b5\u03ba\u03c1\u03b9\u03bc\u03ad\u03bd\u03b5\u03c2 \u03b5\u03bd\u03ad\u03c1\u03b3\u03b5\u03b9\u03b5\u03c2 \u03c4\u03c9\u03bd $\\beta^-$.\n\n\n```python\n# ============================================================================================\n# DATA (all sources' R at the end of irradiation)\n# ============================================================================================\n\n\ndf_all = pd.read_csv('all_sources', names=['z (cm)', \"R(cps)\"])\ndf_all['A(Bq)'] = df_all[\"R(cps)\"] / g_times_e\n```\n\n\u03a3\u03c4\u03b7\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03b3\u03c1\u03b1\u03c6\u03b9\u03ba\u03ae \u03c0\u03b1\u03c1\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7 \u03c0\u03b1\u03c1\u03bf\u03c5\u03c3\u03b9\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 \u03b7 \u03c1\u03bf\u03ae \u03c4\u03c9\u03bd \u03bd\u03b5\u03c4\u03c1\u03bf\u03bd\u03af\u03c9\u03bd \u03c3\u03c5\u03bd\u03b1\u03c1\u03c4\u03ae\u03c3\u03b5\u03b9 \u03c4\u03bf\u03c5 \u03cd\u03c8\u03bf\u03c5\u03c2. \u03a4\u03bf $z=0$ \u03b1\u03bd\u03c4\u03b9\u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03af \u03c3\u03c4\u03bf \u03cd\u03c8\u03bf\u03c2 \u03c4\u03b7\u03c2 \u03c0\u03b7\u03b3\u03ae\u03c2 $\\ ^{241}AmBe$.\n\n\u03a0\u03b1\u03c1\u03b1\u03c4\u03b7\u03c1\u03b5\u03af\u03c4\u03b1\u03b9 \u03cc\u03c4\u03b9 \u03b7 \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03b7 \u03c1\u03bf\u03ae \u03b2\u03c1\u03af\u03c3\u03ba\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf \u03cd\u03c8\u03bf\u03c2 \u03c4\u03b7\u03c2 \u03c0\u03b7\u03b3\u03ae\u03c2 $\\ ^{241}AmBe$. \u0397 \u03c1\u03bf\u03ae \u03b4\u03b5\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bc\u03bc\u03b5\u03c4\u03c1\u03b9\u03ba\u03ae, \u03bb\u03cc\u03b3\u03c9 \u03c4\u03b7\u03c2 \u03ba\u03b1\u03c4\u03b1\u03c3\u03ba\u03b5\u03c5\u03ae\u03c2 \u03c4\u03bf\u03c5 \u03b1\u03bd\u03c4\u03b9\u03b4\u03c1\u03b1\u03c3\u03c4\u03ae\u03c1\u03b1. \u0397 \u03b5\u03c0\u03ac\u03bd\u03c9 \u03b5\u03c0\u03b9\u03c6\u03ac\u03bd\u03b5\u03b9\u03b1 \u03c4\u03bf\u03c5 \u03b1\u03bd\u03c4\u03b9\u03b4\u03c1\u03b1\u03c3\u03c4\u03ae\u03c1\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b1\u03bd\u03bf\u03b9\u03c7\u03c4\u03ae (\u03b1\u03ad\u03c1\u03b1\u03c2), \u03b5\u03bd\u03ce \u03c3\u03c4\u03b7\u03bd \u03ba\u03ac\u03c4\u03c9 \u03b2\u03c1\u03af\u03c3\u03ba\u03b5\u03c4\u03b1\u03b9 \u03bf \u03bc\u03b5\u03c4\u03b1\u03bb\u03bb\u03b9\u03ba\u03cc\u03c2 \u03c4\u03bf\u03c5 \u03c0\u03ac\u03c4\u03bf\u03c2 \u03ba\u03b1\u03b8\u03ce\u03c2 \u03ba\u03b1\u03b9 \u03c4\u03bf \u03c4\u03c3\u03b9\u03bc\u03ad\u03bd\u03c4\u03bf. \u03a9\u03c2 \u03b1\u03c0\u03bf\u03c4\u03ad\u03bb\u03b5\u03c3\u03bc\u03b1 \u03b1\u03c5\u03c4\u03bf\u03cd, \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03b5\u03c0\u03ac\u03bd\u03c9 \u03b5\u03c0\u03b9\u03c6\u03ac\u03bd\u03b5\u03b9\u03b1 \u03c4\u03b1 \u03bd\u03b5\u03c4\u03c1\u03cc\u03bd\u03b9\u03b1 \u03b4\u03b9\u03b1\u03c6\u03b5\u03cd\u03b3\u03bf\u03c5\u03bd \u03b5\u03bb\u03b5\u03cd\u03b8\u03b5\u03c1\u03b1 \u03c4\u03bf\u03c5 \u03b1\u03bd\u03c4\u03b9\u03b4\u03c1\u03b1\u03c3\u03c4\u03ae\u03c1\u03b1 \u03bc\u03cc\u03bb\u03b9\u03c2 \u03b5\u03be\u03ad\u03bb\u03b8\u03bf\u03c5\u03bd \u03c4\u03bf\u03c5 \u03bd\u03b5\u03c1\u03bf\u03cd, \u03b5\u03bd\u03ce \u03c3\u03c4\u03b7\u03bd \u03ba\u03ac\u03c4\u03c9 \u03b5\u03c0\u03b9\u03c6\u03ac\u03bd\u03b5\u03b9\u03b1 \u03bc\u03ad\u03c1\u03bf\u03c2 \u03c4\u03c9\u03bd \u03bd\u03b5\u03c4\u03c1\u03bf\u03bd\u03af\u03c9\u03bd \u03bf\u03c0\u03b9\u03c3\u03b8\u03bf\u03c3\u03ba\u03b5\u03b4\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03b1 \u03c5\u03bb\u03b9\u03ba\u03ac \u03c0\u03bf\u03c5 \u03c0\u03c1\u03bf\u03b1\u03bd\u03b1\u03c6\u03ad\u03c1\u03b8\u03b7\u03ba\u03b1\u03bd. \n\n\n```python\n# ============================================================================================\n# Neutron flux formula\n# ============================================================================================\n\n\nA_\u03b8, t , \u03bb, \u03c3, t_irr, M, \u03a6 = sympy.symbols('A_th, t, \\lambda, \\sigma, t_irr, M, \\Phi')\n_flux_f = lambda A_\u03b8, \u03bb, t_irr, M, \u03c3: A_\u03b8 / (\u03c3 * M * (1-sympy.E**(-\u03bb * t_irr)))\nflux_f = sympy.lambdify((A_\u03b8, \u03bb, t_irr, M, \u03c3), _flux_f(A_\u03b8, \u03bb, t_irr, M, \u03c3))\n\nsympy.Eq(\u03a6, _flux_f(A_\u03b8, \u03bb, t_irr, M, \u03c3))\n```\n\n\n```python\n# ============================================================================================\n# FLUX CALCULATION\n# ============================================================================================\n\n\n# flux unit: n per cm^3 per sec\ndf_all['\u03a6'] = flux_f(df_all[\"R(cps)\"], decay_constant, irradiation_t, M_constant, \u03c3_In)/ (100**3)\ndf_all['\u03a6'] = df_all['\u03a6'].astype(scipy.float64, copy=False)\nprint(df_all)\n\n# --------------------------------------------------------------------\nplt.plot(df_all['\u03a6'],df_all['z (cm)'])\n\nplt.xlabel('flux$(n/cm^3/s)$')\nplt.ylabel('height (cm)')\nplt.title('FIGURE 2: \\nThermal neutron flux')\nplt.grid('on')\n\nplt.show()\n```\n\n## \u039c\u03ae\u03ba\u03bf\u03c2 \u03c7\u03b1\u03bb\u03ac\u03c1\u03c9\u03c3\u03b7\u03c2\n\n\n\n\n```python\n# ====================================================================================================\n# Least squares\n# ====================================================================================================\n\n\n_f = lambda x, a, b: a+x*b\n# Converting flux to n/m^3/s first, and z to meters.\n# (No errors were given for \u03a6)\nlin_popt, _ = scipy.optimize.curve_fit(_f, df_all['z (cm)']/100, scipy.log(df_all['\u03a6']*(100**3)))\nprint('a = {:.6}, b = {:.6}'.format(*lin_popt))\nrelaxation_length = -1/lin_popt[1]\nprint('\u03bc\u03ae\u03ba\u03bf\u03c2 \u03c7\u03b1\u03bb\u03ac\u03c1\u03c9\u03c3\u03b7\u03c2: {:.5} cm'.format(relaxation_length * 100))\n```\n\n a = 18.2358, b = -8.33343\n \u03bc\u03ae\u03ba\u03bf\u03c2 \u03c7\u03b1\u03bb\u03ac\u03c1\u03c9\u03c3\u03b7\u03c2: 12.0 cm\n\n\n\u03a4\u03bf \u03bc\u03ae\u03ba\u03bf\u03c2 \u03c7\u03b1\u03bb\u03ac\u03c1\u03c9\u03c3\u03b7\u03c2 \u03c0\u03c1\u03bf\u03ba\u03cd\u03c0\u03c4\u03b5\u03b9 12.0cm. \n\n(\u039b\u03cc\u03b3\u03c9 \u03c4\u03bf\u03c5 \u03cc\u03c4\u03b9 \u03b4\u03b5\u03bd \u03bc\u03b1\u03c2 \u03b4\u03cc\u03b8\u03b7\u03ba\u03b1\u03bd \u03c4\u03b1 \u03c3\u03c6\u03ac\u03bb\u03bc\u03b1\u03c4\u03b1 \u03b1\u03c0\u03cc \u03c4\u03b9\u03c2 \u03c5\u03c0\u03cc\u03bb\u03bf\u03b9\u03c0\u03b5\u03c2 \u03c0\u03b7\u03b3\u03ad\u03c2, \u03b4\u03b5\u03bd \u03c4\u03b1 \u03bb\u03b1\u03bc\u03b2\u03ac\u03bd\u03bf\u03c5\u03bc\u03b5 \u03c5\u03c0\u03cc\u03c8\u03b9\u03bd \u03c3\u03c4o\u03c5\u03c2 \u03c5\u03c0\u03bf\u03bb\u03bf\u03b3\u03b9\u03c3\u03bc\u03bf\u03cd\u03c2 \u03ba\u03b1\u03b9 \u03b4\u03b5\u03bd \u03bc\u03c0\u03bf\u03c1\u03bf\u03cd\u03bc\u03b5 \u03bd\u03b1 \u03c5\u03c0\u03bf\u03bb\u03bf\u03b3\u03af\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf \u03c3\u03c6\u03ac\u03bb\u03bc\u03b1 \u03c3\u03c4\u03bf \u03bc\u03ae\u03ba\u03bf\u03c2 \u03c7\u03b1\u03bb\u03ac\u03c1\u03c9\u03c3\u03b7\u03c2.)\n\n## \u03a0\u03b9\u03b8\u03b1\u03bd\u03ad\u03c2 \u03c0\u03b7\u03b3\u03ad\u03c2 \u03c3\u03c6\u03b1\u03bb\u03bc\u03ac\u03c4\u03c9\u03bd\n\n### \u039f\u03c0\u03b9\u03c3\u03b8\u03bf\u03c3\u03ba\u03ad\u03b4\u03b1\u03c3\u03b7 \u03c3\u03c9\u03bc\u03b1\u03c4\u03b9\u03b4\u03af\u03c9\u03bd $\\beta^-$ \u03b1\u03c0\u03cc \u03c4\u03bf \u03c5\u03bb\u03b9\u03ba\u03cc \u03ba\u03ac\u03c4\u03c9 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03c0\u03b7\u03b3\u03ae. \n\n\u0397 \u03bf\u03c0\u03b9\u03c3\u03b8\u03bf\u03c3\u03ba\u03ad\u03b4\u03b1\u03c3\u03b7 \u03c4\u03c9\u03bd $\\beta^-$ \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03b2\u03ac\u03c3\u03b7 \u03c4\u03bf\u03c0\u03bf\u03b8\u03ad\u03c4\u03b7\u03c3\u03b7\u03c2, \u03c4\u03bf\u03bd \u03c0\u03ac\u03b3\u03ba\u03bf \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03c5\u03bb\u03b9\u03ba\u03ac \u03b3\u03cd\u03c1\u03c9 \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03b1\u03bd\u03b9\u03c7\u03bd\u03b5\u03c5\u03c4\u03ae, \u03c3\u03c5\u03bc\u03c0\u03b5\u03c1\u03b9\u03bb\u03b1\u03bc\u03b2\u03b1\u03bd\u03bf\u03bc\u03ad\u03bd\u03bf\u03c5 \u03ba\u03b1\u03b9 \u03c4\u03b7\u03c2 \u03b8\u03ad\u03c3\u03b7\u03c2 \u03c4\u03b7\u03c2 \u03b5\u03c1\u03b5\u03c5\u03bd\u03ae\u03c4\u03c1\u03b9\u03b1\u03c2 \u03ba\u03b1\u03c4\u03ac \u03c4\u03b9\u03c2 \u03bc\u03b5\u03c4\u03c1\u03ae\u03c3\u03b5\u03b9\u03c2, \u03c0\u03b9\u03b8\u03b1\u03bd\u03cc\u03bd \u03c0\u03c1\u03bf\u03ba\u03ac\u03bb\u03b5\u03c3\u03b5 \u03c5\u03c0\u03b5\u03c1\u03b5\u03ba\u03c4\u03af\u03bc\u03b7\u03c3\u03b7 \u03c4\u03c9\u03bd \u03c1\u03c5\u03b8\u03bc\u03ce\u03bd.\n\n### \u0395\u03c0\u03b9\u03bb\u03bf\u03b3\u03ae \u03c7\u03c1\u03cc\u03bd\u03bf\u03c5 \u03bc\u03ad\u03c4\u03c1\u03b7\u03c3\u03b7\u03c2 \u03c3\u03c4\u03bf \u03bc\u03ad\u03c3\u03bf \u03c4\u03b7\u03c2 \u03b4\u03b9\u03ac\u03c1\u03ba\u03b5\u03b9\u03b1\u03c2 \u03bc\u03ad\u03c4\u03c1\u03b7\u03c3\u03b7\u03c2\n\n\u039f \u03b5\u03ba\u03ac\u03c3\u03c4\u03bf\u03c4\u03b5 \u03c1\u03c5\u03b8\u03bc\u03cc\u03c2 \u03b1\u03c0\u03bf\u03b4\u03cc\u03b8\u03b7\u03ba\u03b5 \u03c3\u03c4\u03bf \u03bc\u03ad\u03c3\u03bf \u03c4\u03b7\u03c2 \u03b4\u03b9\u03ac\u03c1\u03ba\u03b5\u03b9\u03b1\u03c2 \u03bc\u03ad\u03c4\u03c1\u03b7\u03c3\u03b7\u03c2 $t_0$ (\u03b4\u03b7\u03bb. $t_0 = t_1 + \\frac{t_2-t_1}{2}$). \u03a9\u03c3\u03c4\u03cc\u03c3\u03bf, \u03bb\u03cc\u03b3\u03c9 \u03c4\u03b7\u03c2 \u03b5\u03ba\u03b8\u03b5\u03c4\u03b9\u03ba\u03ae\u03c2 \u03c0\u03c4\u03ce\u03c3\u03b7\u03c2 \u03c4\u03b7\u03c2 \u03b5\u03bd\u03b5\u03c1\u03b3\u03cc\u03c4\u03b7\u03c4\u03b1\u03c2, \u03b7 \u03c5\u03c0\u03bf\u03bb\u03bf\u03b3\u03b9\u03b6\u03cc\u03bc\u03b5\u03bd\u03b7 \u03b5\u03bd\u03b5\u03c1\u03b3\u03cc\u03c4\u03b7\u03c4\u03b1 \u03c3\u03c4\u03b7\u03bd \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc\u03c4\u03b7\u03c4\u03b1 \u03b1\u03bd\u03c4\u03b9\u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03af \u03c3\u03b5 \u03c7\u03c1\u03cc\u03bd\u03bf \u03c0\u03c1\u03b9\u03bd \u03c4\u03bf\u03c5 \u03bc\u03ad\u03c3\u03bf\u03c5. \n\n\n\n### \u03a5\u03c0\u03b5\u03c1\u03b5\u03ba\u03c4\u03af\u03bc\u03b7\u03c3\u03b7 \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03bd\u03c4\u03b1 \u03b3\u03b5\u03c9\u03bc\u03b5\u03c4\u03c1\u03af\u03b1\u03c2 \u03bb\u03cc\u03b3\u03c9 \u03b5\u03b3\u03b3\u03cd\u03c4\u03b7\u03c4\u03b1\u03c2 \u03c0\u03b7\u03b3\u03ae\u03c2-\u03b1\u03bd\u03b9\u03c7\u03bd\u03b5\u03c5\u03c4\u03ae\n\n\u03a3\u03b5 \u03b5\u03c0\u03b1\u03c1\u03ba\u03ce\u03c2 \u03bc\u03b5\u03b3\u03ac\u03bb\u03b5\u03c2 \u03b1\u03c0\u03bf\u03c3\u03c4\u03ac\u03c3\u03b5\u03b9\u03c2 \u03bf \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03bd\u03c4\u03b1\u03c2 \u03b3\u03b5\u03c9\u03bc\u03b5\u03c4\u03c1\u03af\u03b1\u03c2 \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03b1\u03b3\u03bd\u03bf\u03b5\u03af \u03c4\u03b9\u03c2 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ac\u03c3\u03b5\u03b9\u03c2 \u03c4\u03bf\u03c5 \u03b1\u03bd\u03b9\u03c7\u03bd\u03b5\u03c5\u03c4\u03ae, \u03ba\u03b1\u03b9 \u03bd\u03b1 \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03b9\u03b5\u03af\u03c4\u03b1\u03b9 \u03c9\u03c2 \u03ad\u03c7\u03b5\u03b9 \u03b3\u03b9\u03b1 \u03c4\u03bf\u03bd \u03c5\u03c0\u03bf\u03bb\u03bf\u03b3\u03b9\u03c3\u03bc\u03cc \u03c4\u03c9\u03bd \u03c3\u03c9\u03bc\u03b1\u03c4\u03b9\u03b4\u03af\u03c9\u03bd \u03c0\u03bf\u03c5 \u03b4\u03b9\u03ad\u03c1\u03c7\u03bf\u03bd\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03bc\u03ad\u03c3\u03b1 \u03c4\u03bf\u03c5. \n\n\u0393\u03b9\u03b1 \u03c0\u03bf\u03bb\u03cd \u03ba\u03bf\u03bd\u03c4\u03b9\u03bd\u03ad\u03c2 \u03b1\u03c0\u03bf\u03c3\u03c4\u03ac\u03c3\u03b5\u03b9\u03c2 \u03cc\u03bc\u03c9\u03c2, \u03c3\u03b5 \u03bc\u03b5\u03b3\u03ac\u03bb\u03b5\u03c2 \u03b3\u03c9\u03bd\u03af\u03b5\u03c2 \u03c0\u03c1\u03cc\u03c3\u03c0\u03c4\u03c9\u03c3\u03b7\u03c2 (\u03b4\u03ad\u03c3\u03bc\u03b7 2 \u03c3\u03c4\u03b7\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03b5\u03b9\u03ba\u03cc\u03bd\u03b1) \u03c4\u03b1 \u03c3\u03c9\u03bc\u03b1\u03c4\u03af\u03b4\u03b9\u03b1 \u03b4\u03b9\u03ad\u03c1\u03c7\u03bf\u03bd\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03bb\u03b9\u03b3\u03cc\u03c4\u03b5\u03c1\u03bf \u03c5\u03bb\u03b9\u03ba\u03cc \u03c4\u03bf\u03c5 \u03b1\u03bd\u03b9\u03c7\u03bd\u03b5\u03c5\u03c4\u03ae, \u03c3\u03b5 \u03b1\u03bd\u03c4\u03af\u03b8\u03b5\u03c3\u03b7 \u03bc\u03b5 \u03c3\u03c9\u03bc\u03b1\u03c4\u03af\u03b4\u03b9\u03b1 \u03c0\u03bf\u03c5 \u03c0\u03c1\u03bf\u03c3\u03c0\u03af\u03c0\u03c4\u03bf\u03c5\u03bd \u03ba\u03ac\u03b8\u03b5\u03c4\u03b1 \u03c3\u03c4\u03b7\u03bd \u03b5\u03c0\u03b9\u03c6\u03ac\u03bd\u03b5\u03b9\u03ac \u03c4\u03bf\u03c5 (\u03b4\u03ad\u03c3\u03bc\u03b7 1). \n\n\n\n\u03a9\u03c2 \u03b1\u03c0\u03bf\u03c4\u03ad\u03bb\u03b5\u03c3\u03bc\u03b1, \u03ba\u03ac\u03c0\u03bf\u03b9\u03b1 \u03c3\u03c9\u03bc\u03b1\u03c4\u03af\u03b4\u03b9\u03b1 \u03c3\u03b5 \u03bc\u03b5\u03b3\u03ac\u03bb\u03b5\u03c2 \u03b3\u03c9\u03bd\u03af\u03b5\u03c2 \u03b4\u03b5\u03bd \u03c0\u03c1\u03bf\u03ba\u03b1\u03bb\u03bf\u03cd\u03bd \u03b5\u03c0\u03b1\u03c1\u03ba\u03ae \u03b9\u03bf\u03bd\u03b9\u03c3\u03bc\u03cc \u03ce\u03c3\u03c4\u03b5 \u03bd\u03b1 \u03ba\u03b1\u03c4\u03b1\u03c7\u03c9\u03c1\u03b7\u03b8\u03bf\u03cd\u03bd \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03b1\u03bd\u03b9\u03c7\u03bd\u03b5\u03c5\u03c4\u03ae. \u03a4\u03bf \u03c6\u03b1\u03b9\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf \u03b1\u03c5\u03c4\u03cc \u03af\u03c3\u03c9\u03c2 \u03bd\u03b1 \u03bc\u03b7\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03cc\u03c3\u03bf \u03c3\u03b7\u03bc\u03b1\u03bd\u03c4\u03b9\u03ba\u03cc \u03b3\u03b9\u03b1 \u03ac\u03bc\u03b5\u03c3\u03b1 \u03b9\u03bf\u03bd\u03af\u03b6\u03bf\u03c5\u03c3\u03b5\u03c2 \u03b1\u03ba\u03c4\u03b9\u03bd\u03bf\u03b2\u03bf\u03bb\u03af\u03b5\u03c2, \u03cc\u03c0\u03c9\u03c2 \u03c4\u03b1 $\\beta^-$ \u03c4\u03bf\u03c5 \u03c0\u03b5\u03b9\u03c1\u03ac\u03bc\u03b1\u03c4\u03bf\u03c2, \u03b1\u03bb\u03bb\u03ac \u03c9\u03c3\u03c4\u03cc\u03c3\u03bf \u03cc\u03c7\u03b9 \u03b1\u03bc\u03b5\u03bb\u03b7\u03c4\u03ad\u03bf.\n\n### \u03a4\u03b1\u03c5\u03c4\u03cc\u03c7\u03c1\u03bf\u03bd\u03b7 \u03bc\u03b5\u03c4\u03b1\u03ba\u03af\u03bd\u03b7\u03c3\u03b7/\u03bc\u03ad\u03c4\u03c1\u03b7\u03c3\u03b7 \u03ac\u03bb\u03bb\u03c9\u03bd \u03c0\u03b7\u03b3\u03ce\u03bd \u03c3\u03b5 \u03b4\u03b9\u03c0\u03bb\u03b1\u03bd\u03bf\u03cd\u03c2 Geiger\n\n\u03a4\u03b1\u03c5\u03c4\u03cc\u03c7\u03c1\u03bf\u03bd\u03b1 \u03bc\u03b5 \u03c4\u03b7\u03bd \u03bc\u03ad\u03c4\u03c1\u03b7\u03c3\u03ae \u03bc\u03b1\u03c2 \u03b4\u03b9\u03b5\u03be\u03ac\u03b3\u03bf\u03bd\u03c4\u03b1\u03bd \u03ba\u03b1\u03b9 \u03ac\u03bb\u03bb\u03b5\u03c2 \u03bc\u03b5\u03c4\u03c1\u03ae\u03c3\u03b5\u03b9\u03c2 \u03c3\u03b5 \u03b1\u03c0\u03cc\u03c3\u03c4\u03b1\u03c3\u03b7 \u03bc\u03b9\u03ba\u03c1\u03cc\u03c4\u03b5\u03c1\u03b7 \u03c4\u03c9\u03bd 2 \u03bc\u03ad\u03c4\u03c1\u03c9\u03bd, \u03c0\u03b9\u03b8\u03b1\u03bd\u03cc\u03bd \u03b1\u03c5\u03be\u03ac\u03bd\u03bf\u03bd\u03c4\u03b1\u03c2 \u03c3\u03b7\u03bc\u03b1\u03bd\u03c4\u03b9\u03ba\u03ac \u03c4\u03bf \u03c5\u03c0\u03cc\u03b2\u03b1\u03b8\u03c1\u03bf. \u0395\u03bd\u03b4\u03ad\u03c7\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c5\u03c4\u03cc \u03c4\u03bf \u03c3\u03c6\u03ac\u03bb\u03bc\u03b1 \u03bd\u03b1 \u03b5\u03c0\u03b7\u03c1\u03ad\u03b1\u03c3\u03b5 \u03b9\u03b4\u03b9\u03b1\u03af\u03c4\u03b5\u03c1\u03b1 \u03c4\u03b9\u03c2 \u03bc\u03b5\u03c4\u03c1\u03ae\u03c3\u03b5\u03b9\u03c2 \u03c0\u03b7\u03b3\u03ce\u03bd \u03bc\u03b5 \u03bc\u03b9\u03ba\u03c1\u03ad\u03c2 \u03b5\u03bd\u03b5\u03c1\u03b3\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2 (\u03c0\u03c7. \u03c0\u03b7\u03b3\u03ae 9).\n\n\n\n\n```python\nplt.plot(df9['t (s)'], df9[\"R(cps)\"], marker='.', label='Source 9')\nplt.errorbar(df9['t (s)'], df9[\"R(cps)\"], df9[\"error \u03c3R(cps)\"], label=None)\nplt.ylim(ymin=0)\nplt.xlabel('t(s)')\nplt.ylabel('R(cps)')\nplt.legend(loc='upper right')\nplt.title('FIGURE 3: \\nRate of source 9.')\nplt.grid('on')\n\nplt.show()\n```\n\n### \u039c\u03b7 \u03b8\u03b5\u03c1\u03bc\u03b9\u03ba\u03ac \u03bd\u03b5\u03c4\u03c1\u03cc\u03bd\u03b9\u03b1 \n\n\u039b\u03cc\u03b3\u03c9 \u03c4\u03b7\u03c2 \u03bc\u03b7 \u03c7\u03c1\u03ae\u03c3\u03b7\u03c2 $Cd$, \u03c4\u03b1\u03c5\u03c4\u03cc\u03c7\u03c1\u03bf\u03bd\u03b1 \u03ba\u03b1\u03b9 \u03bc\u03b7 \u03b8\u03b5\u03c1\u03bc\u03b9\u03ba\u03ac \u03bd\u03b5\u03c4\u03c1\u03cc\u03bd\u03b9\u03b1 \u03b1\u03c0\u03bf\u03c1\u03c1\u03bf\u03c6\u03cc\u03bd\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03bf $\\ ^{115}In$. \u03a9\u03c2 \u03b1\u03c0\u03bf\u03c4\u03ad\u03bb\u03b5\u03c3\u03bc\u03b1 \u03b1\u03c5\u03be\u03ac\u03bd\u03b5\u03b9 \u03bf \u03bc\u03b5\u03c4\u03c1\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03c2 \u03c1\u03c5\u03b8\u03bc\u03cc\u03c2, \u03bb\u03cc\u03b3\u03c9 \u03c4\u03bf\u03c5 \u03cc\u03c4\u03b9 \u03c0\u03bf\u03bb\u03bb\u03ac \u03b1\u03c0\u03cc \u03b1\u03c5\u03c4\u03ac \u03c4\u03b1 \u03bd\u03b5\u03c4\u03c1\u03cc\u03bd\u03b9\u03b1 \u03b1\u03c5\u03c4\u03ac \u03b8\u03b1 \u03b4\u03b9\u03ad\u03c6\u03b5\u03c5\u03b3\u03b1\u03bd \u03ae \u03b8\u03b1 \u03b1\u03c0\u03bf\u03c1\u03c1\u03bf\u03c6\u03cc\u03bd\u03c4\u03b1\u03bd \u03c0\u03c1\u03b9\u03bd \u03b8\u03b5\u03c1\u03bc\u03bf\u03c0\u03bf\u03b9\u03b7\u03b8\u03bf\u03cd\u03bd, \u03b5\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03b4\u03b5\u03bd \u03b8\u03b1 \u03ae\u03c4\u03b1\u03bd \u03b4\u03b9\u03b1\u03b8\u03ad\u03c3\u03b9\u03bc\u03b1 \u03c9\u03c2 \u03b8\u03b5\u03c1\u03bc\u03b9\u03ba\u03ac. \n\n## \u0391\u03bd\u03b1\u03c6\u03bf\u03c1\u03ad\u03c2\n\n\u0393\u03b9\u03b1 \u03c4\u03b7\u03bd \u03c3\u03c5\u03b3\u03b3\u03c1\u03b1\u03c6\u03ae \u03c4\u03b7\u03c2 \u03b5\u03c1\u03b3\u03b1\u03c3\u03af\u03b1\u03c2 \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03b9\u03ae\u03b8\u03b7\u03ba\u03b1\u03bd \u03bf\u03b9 \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03b1\u03bd\u03b1\u03c6\u03bf\u03c1\u03ad\u03c2:\n\n1. \u0397 \u03a0\u03a5\u03a1\u0397\u039d\u0399\u039a\u0397 \u03a6\u03a5\u03a3\u0399\u039a\u0397 \u03a3\u03a4\u039f \u0395\u03a1\u0393\u0391\u03a3\u03a4\u0397\u03a1\u0399\u039f - \u03a6\u039f\u0399\u03a4\u0397\u03a4\u0399\u039a\u0395\u03a3 \u0391\u03a3\u039a\u0397\u03a3\u0395\u0399\u03a3, X. \u0395\u039b\u0395\u03a5\u0398\u0395\u03a1\u0399\u0391\u0394\u0397\u03a3, \u039c. \u0396\u0391\u039c\u0391\u039d\u0397, \u0391. \u039b\u0399\u039f\u039b\u0399\u039f\u03a3, \u039c. \u039c\u0391\u039d\u03a9\u039b\u039f\u03a0\u039f\u03a5\u039b\u039f\u03a5, X. \u03a0\u0395\u03a4\u03a1\u0399\u0394\u039f\u03a5, \u0397. \u03a3\u0391\u0392\u0392\u0399\u0394\u0397\u03a3, \u0395\u03ba\u03b4\u03cc\u03c3\u03b5\u03b9\u03c2: COPY CITY\n\n2. https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html \n\n\n", "meta": {"hexsha": "398ecb7c9b6ce575cba238cf89902f735e385fe1", "size": 133978, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "neutron_flux.ipynb", "max_stars_repo_name": "FermiParadox/subcrit_reactor_n_flow", "max_stars_repo_head_hexsha": "ebc30cc64f511d118f10a9e477c0c09c5d142a32", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-12-01T20:44:19.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-01T20:44:19.000Z", "max_issues_repo_path": "neutron_flux.ipynb", "max_issues_repo_name": "FermiParadox/subcritical_nuclear_reactor_neutron_flow", "max_issues_repo_head_hexsha": "ebc30cc64f511d118f10a9e477c0c09c5d142a32", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "neutron_flux.ipynb", "max_forks_repo_name": "FermiParadox/subcritical_nuclear_reactor_neutron_flow", "max_forks_repo_head_hexsha": "ebc30cc64f511d118f10a9e477c0c09c5d142a32", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 158.9300118624, "max_line_length": 39356, "alphanum_fraction": 0.8479227933, "converted": true, "num_tokens": 8847, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8152324803738429, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.43939662788110306}} {"text": "###### Content under Creative Commons Attribution license CC-BY 4.0, code under MIT license (c)2014 L.A. Barba, C. Cooper, G.F. Forsyth, A. Krishnan.\n\n# Phugoid Motion\n\nWelcome to [**\"Practical Numerical Methods with Python!\"**](http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about) This course is a collaborative, online, open education project, where we aim to give a foundation in scientific computing. The focus is on numerical solution of problems modeled by ordinary and partial differential equations.\n\nThis IPython Notebook introduces the problem we'll be studying in the **first module** of the course: the _phugoid model of glider flight_. We'll start with some background, explaining the physics, and working out the mathematical model. \n\nFirst, we'll look at an idealized motion where there is no drag, resulting in a simple harmonic motion. We can plot some interesting trajectories that will pique your imagination. In the next notebook, you'll learn to numerically integrate the differential equation using Euler's method. But hang on ... first things first. \n\nThe term \"phugoid\" is used in aeronautics to refer to a motion pattern where an aircraft oscillates up and down \u2014nose-up and climb, then nose-down and descend\u2014 around an equilibrium trajectory. The aircraft oscillates in altitude, speed and pitch, with only small (neglected) variations in the angle of attack, as it repeatedly exchanges kinetic and potential energy.\n\nA low-amplitude phugoid motion can be just a nuisance, as the aircraft does not exceed the stall angle of attack and nothing bad happens. But the mode can also be unstable leading to a stall or even a loop!\n\nLook at this video showing a Cessna single-engine airplane in phugoid motion:\n\n\n```python\nfrom IPython.display import YouTubeVideo\nYouTubeVideo('ysdU4mnRYdM')\n```\n\n\n\n\n\n\n\n\n\n\nThat doesn't look too good! What's happening? \n\nIt can get a lot worse when an aircraft enters one of these modes that is unstable. For example, one of [NASA's Helios Solar Powered Aircraft](http://www.nasa.gov/centers/dryden/history/pastprojects/Helios/) prototype broke up in mid air due to extreme phugoid oscillations!\n\nHelios was a proof-of-concept solar electric-powered flying wing that broke the world altitude record for a non-rocket-powered aircraft in August 2001. But in June 26, 2003, it broke something else. The aircraft entered phugoid motion after encountering turbulence near the Hawaiian Island of Kauai. The high speed in the oscillatory movement exceeded the design limits, and it ended up wrecked in the Pacific Ocean. Luckily, the Helios was remotely operated, and nobody got hurt.\n\n## The physics of phugoids\n\nThe phugoid oscillation has the aircraft pitching up and down, as it decelerates and accelerates. The trajectory might look like a sinusoid, as in the figure below. The assumption is that the forward velocity of the aircraft, $v$, varies in such a way that the angle of attack remains (nearly) constant, which means that we can assume a constant lift coefficient.\n\n\n#### Figure 1. Trajectory of an aircraft in phugoid motion.\n\nIn the descending portion of the trajectory, the aircraft's velocity increases as it proceeds from a peak to the minimum height\u2014gaining kinetic energy at the expense of potential energy. The contrary happens in the upward segment, as its velocity decreases there.\n\nWe measure the pitch angle (between the aircraft's longitudinal axis and the horizontal) as positive when the aircraft's nose is pointing up. In the portion of the trajectory below the center-line, where it curves upwards, the pitch angle $\\theta$ is increasing: $\\dot{\\theta}>0$. And where the trajectory curves down, the pitch angle is decreasing: $\\dot{\\theta}<0$, as shown in the figure.\n\nLet's remind ourselves of the forces affecting an aircraft in a downward glide. Look at the figure below: we show the flight path, the forces on the glider (no thrust), and the _glide angle_ or flight path angle, $\\gamma$, between the flight path and the horizontal.\n\n\n#### Figure 2. Forces on a glider.\n\nThe force of lift, $L$ \u2014created by the airflow around the wings\u2014 is perpendicular to the trajectory, and the force of drag, $D$, is parallel to the trajectory. Both forces are expressed in terms of coefficients of lift and drag, $C_L$ and $C_D$, respectively, that depend on the wing design and _angle of attack_\u2014the angle between the wing chord and the flight path.\n\nIf you are not familiar with airplane aerodynamics, you might be getting confused with some terms here ... and all those angles! But be patient and look things up, if you need to. We're giving you a quick summary here.\n\nLift and drag are proportional to a surface area, $S$, and the dynamic pressure: $1/2 \\rho v^2$, where $\\rho$ is the density of air, and $v$ the forward velocity of the aircraft. The equations for lift and drag are:\n\n$$\\begin{eqnarray}\nL &=& C_L S \\times \\frac{1}{2} \\rho v^2 \\\\\nD &=& C_D S \\times \\frac{1}{2} \\rho v^2\n\\end{eqnarray}$$\n\nIf the glider were in equilibrium, the forces would balance each other. We can equate the forces in the directions perpendicular and parallel to the trajectory, as follows:\n\n$$\\begin{equation}\nL = W \\cos \\gamma \\quad \\text{and} \\quad D = W \\sin \\gamma\n\\end{equation}$$\n\nwhere $W$ repesents the weight of the glider.\n\nIn the figure, we've drawn the angle $\\gamma$ as the _glide angle_, formed between the direction of motion and the horizontal. We are not bothered with the _sign_ of the angle, because we draw a free-body diagram and take the direction of the forces into account in writing our balance equations. But later on, we will need to be careful with the sign of the angles. It can cause you a real headache to keep this straight, so be patient!\n\nIt looks like we've set this up to do a little bit of mathematics. Are you ready?\n\nBut before, a short glimpse of the history.\n\n## Lanchester's Aerodonetics\n\n\"Phugoid theory\" was first described by the British engineer Frederick W. Lanchester in _\"Aerodonetics\"_ (1909). This book is so old that it is now in the public domain, so you can actually download [from Google Books](http://books.google.com/books?id=6hxDAAAAIAAJ&dq=%22phugoid%20theory%20deals%20with%20the%20longitudinal%20stability%22&pg=PA37#v=onepage&q=%22phugoid%20theory%20deals%20with%20the%20longitudinal%20stability%22&f=false) a PDF file of a scan, or read it online. \n\nLanchester defines phugoid theory as the study of longitudinal stability of a flying machine (aerodone). He first considered the simplification where drag and moment of inertia are neglected. Then he included these effects, obtaining an equation of stability. In addition to describing many experiments by himself and others, Lanchester also reports on _\"numerical work ... done by the aid of an ordinary 25-cm slide rule.\"_ Go figure!\n\n### Ideal case of zero drag\n\nIn this section, we follow the derivation given by Milne-Thompson (1966), which we find a little bit easier than that of the original in \"Aerodonetics\"!\n\nAn aircraft flying in a steady, straight horizontal flight has a lift equal to its weight. The velocity in this condition is sometimes called _trim velocity_ (\"trim\" is what pilots do to set the controls to just stay in a steady flight). Let's use $v_t$ for the trim velocity, and from $L=W$ deduce that:\n\n$$\\begin{equation}\nW = C_L S \\times\\frac{1}{2} \\rho v_t^2\n\\end{equation}$$\n\nThe weight $W$ is constant for the aircraft, but the lift at any other flight condition depends on the flight speed, $v$. We can use the expression for the weight in terms of $v_t$ to obtain the ratio $L/W$ at any other flight velocity, as follows:\n\n$$\\begin{equation}\n\\frac{L}{W}= \\frac{v^2}{v_t^2}\n\\end{equation}$$\n\nImagine that the aircraft experienced a little upset, a wind gust, and it finds itself off the \"trim\" level, in a curved path with an instantaneous angle $\\theta$. In the sketch below, we exaggerate the curved trajectory of flight to help you visualize what we'll do next. The angle $\\theta$ (using the same name as Milne-Thompson) is between the _trajectory_ and the horizontal, positive up.\n\n\n#### Figure 3. Curved trajectory of the aircraft going up.\n\nA balance of forces now has to take into account that our reference frame is moving with the aircraft, in a rotating frame: we have a centrifugal force. The balance in the direction of lift is thus:\n\n$$\\begin{equation}\nL- W \\cos \\theta = \\frac{W}{g} \\frac{v^2}{R}\n\\end{equation}$$\n\nwhere $R$ is the radius of curvature of the trajectory, and $g$ the acceleration of gravity. Recall that the centrifugal acceleration is $v^2/R$. Rearrange this by dividing the equation by the weight, and use the expression we found for $L/W$, above. The following equation results:\n\n$$\\begin{equation}\n\\frac{v^2}{v_t^2}-\\cos \\theta = \\frac{v^2}{g R}\n\\end{equation}$$\n\nRecall that we simplified the problem assuming that there is no friction, which means that the total energy is constant (the lift does no work). If $z$ represents the depth below a reference horizontal line, the energy per unit mass is (kinetic plus potential energy):\n\n$$\\begin{equation}\n\\frac{1}{2}v^2-g z = \\text{constant}\n\\end{equation}$$\n\nTo get rid of that pesky constant, we can choose the reference horizontal line at the level that makes the constant energy equal to zero, so $v^2 = 2 g z$. That helps us re-write the phugoid equation in terms of $z$ as follows:\n\n$$\\begin{equation}\n\\frac{z}{z_t}-\\cos \\theta = \\frac{2z}{R}\n\\end{equation}$$\n\nLet $ds$ represent a small arc-length of the trajectory. We can write \n\n$$\\begin{equation}\n\\frac{1}{R} = \\frac{d\\theta}{ds} \\quad \\text{and}\\quad \\sin\\theta = -\\frac{dz}{ds}\n\\end{equation}$$\n\nEmploying the chain rule of calculus,\n\n$$\\begin{equation}\n\\frac{1}{R} = \\frac{d\\theta}{ds} = \\frac{dz}{ds}\\frac{d\\theta}{dz} = -\\sin \\theta\\frac{d\\theta}{dz}\n\\end{equation}$$\n\nMultiply the phugoid equation by $\\frac{1}{2\\sqrt{z}}$ to get:\n\n$$\\begin{equation}\n\\frac{\\sqrt{z}}{2z_t} - \\frac{\\cos\\theta}{2\\sqrt{z}} = \\frac{\\sqrt{z}}{R}\n\\end{equation}$$\n\nSubstituting for $1/R$ on the right hand side and bringing the cosine term over to the right, we get:\n\n$$\\begin{equation}\n\\frac{\\sqrt{z}}{2z_t} = \\frac{\\cos \\theta}{2 \\sqrt{z}} - \\sqrt{z} \\sin \\theta \\frac{d\\theta}{dz}\n\\end{equation}$$\n\nThe right-hand-side is an exact derivative! We can rewrite it as:\n\n$$\\begin{equation}\n\\frac{d}{dz} \\left(\\sqrt{z}\\cos\\theta \\right) = \\frac{\\sqrt{z}}{2z_t}\n\\end{equation}$$\n\nIntegrating this equation, we add an arbitrary constant, chosen as $C\\sqrt{z_t}$ which (after dividing through by $\\sqrt{z}$) gives:\n\n$$\\begin{equation}\n\\cos \\theta = \\frac{1}{3}\\frac{z}{z_t} + C\\sqrt{\\frac{z_t}{z}}\n\\end{equation}$$\n\nTaking the derivative of both sides of equation (15) and applying the relations from equation (10) yields:\n\n$$\\begin{equation}\n\\frac{z_t}{R} = \\frac{1}{3} - \\frac{C}{2}\\sqrt{\\frac{z_t^3}{z^3}}\n\\end{equation}$$\n\nMake sure you have followed the derivation, and perhaps write it out on paper!\n\n## Phugoid Curves\n\nEquation (15) is non-linear, which usually means we are hard-pressed to write a clean expression for the variable of interest, $z$. In fact, Lanchester himself said that he was unable to _\"reduce this expression to a form suitable for co-ordinate plotting.\"_ If the great polymath couldn't do it, we can't either!\n\nBut Lanchester _was_ able to plot a suitable approximation of the phugoid flight path using what he called the \"trammel\" method. If you're interested in seeing how he did it, his explanation begins on page [48 of Aerodonetics](http://books.google.com/books?id=6hxDAAAAIAAJ&pg=PA49&lpg=PA48&dq=aerodonetics+the+use+of+the+trammel&source=bl&ots=lB6EVKYQuT&sig=aVE2kiDWZoWftaWczMIrcYftMOs&hl=en&sa=X&ei=gTD_U82fGYjzgwT3moGwCQ&ved=0CCAQ6AEwAA#v=onepage&q=aerodonetics%20the%20use%20of%20the%20trammel&f=false). It's a trip.\n\nLanchester used Equations (15) and (16) to solve for the constant $C$ and the radius of curvature $R$ and then iteratively plotted small arcs of the phugoid path. By hand.\n\nWe wrote a neat little code that duplicates the manual trammel method, but it might be a bit much for you to absorb in the first lesson. If you want to look it over, you are more than welcome to. If you are just starting with Python, skip it for the moment and we'll return to it at the end of this module. \n\n### Plotting the flight path\n\nAs we mentioned, we wrote a Python code to reproduce programmatically what Lanchester did graphically. Here's a neat feature of IPython Notebooks: you can run external programs with the magical keyword ... wait for it ... `run`. And the jargon of IPython _is_ to call this \"magic.\" In fact, there are a bunch of [magic functions](http://ipython.org/ipython-doc/dev/interactive/tutorial.html) that you will learn about. They will make you a happy camper.\n\nLet's do it:\n\n\n```python\n%run phugoid.py\n%matplotlib inline\n```\n\nThis code cell loaded our simulated-trammel code, `phugoid.py`. The code defined a function for you in the background, called `plot_flight_path`, taking three inputs: $z_t$, $z$ and $\\theta$. \n\nLook again at Equation (15), where we take the positive square root. There are several possibilities, depending on the value that the constant $C$ takes. \n\n* There are no physical solutions for $C>2/3$, because it would result in $\\cos\\theta>1$. \n\n* If $C=2/3$, then the solution is a horizontal straight line, because $\\cos\\theta=1$, $\\theta=0$ and $R=\\infty$.\n\n* Any value of $C$ for which $0 < C < \\frac{2}{3}$ will produce \"trochoidal\"-like paths. What does this look like? Let's use our custom function `plot_flight_path` to find out!\n\n\n```python\n#zt = 64, z = 16, theta=0\nplot_flight_path(64, 16, 0)\n```\n\nCool! Note that the plot title tells us what the calculated value of $C$ was for our input conditions. We have a value of $C$ between $0$ and $\\frac{2}{3}$ and our path is trochoidal, like we announced it would be.\n\n* For negative values of $C$, the resultant flight path consists of a series of loops. Let's try it out!\n\n\n```python\nplot_flight_path(64,16,numpy.pi)\n```\n\nYou can play around with the input values and see what kind of behavior results. Just note that any value of $C > \\frac{2}{3}$ will result in $\\cos \\theta > 1$, which doesn't exist. Python will probably throw a few errors if you hit that condition, but just try again!\n\n* The last case is $C = 0$. Take another look at Equation (16) and plug in $C = 0$, what should happen? It looks like it will just reduce to \n\n$$R = 3z_t$$\n\nIt's a constant radius of curvature! In fact, this solution is a series of semi-circles, with a cusp between them. One way to force $C = 0$ that we can figure out from Equation (15), is to make:\n\n\n$$z = 3z_t\\ \\ \\ ,\\ \\ \\ \\theta = 0$$\n\n\n```python\nplot_flight_path(16,48,0.)\n```\n\nThat looks an awful lot like a quarter circle. And what's the radius of the arc? It's $$r = 48 = 3z_t.$$\n\nWe can also get a semi-circle out of our simulated trammel by changing to another configuration where $C$ is (near) zero. Here's one example:\n\n\n```python\nplot_flight_path(64,16,-numpy.pi/2)\n```\n\nThat is so nice. We have reproduced the trajectories that Lanchester found more than a hundred years ago, painstakingly drawing them by hand with a contraption called a \"trammel.\" It must have taken him days!\n\nHere is how the different phugoid curves are drawn in von K\u00e1rm\u00e1n's book, _Aerodynamics_ (1957). He never says _how_ he drew them, but we're guessing by hand, also. We did pretty good!\n\n\n\n#### Figure 4. Phugoid curves in von K\u00e1rm\u00e1n (1957).\n\nIn the next notebook of this series, we'll look at the differential equation that arises when you consider small perturbations on the horizontal phugoid, and we'll learn to numerically integrate that to get the flight paths.\n\n## References\n\n1. Lanchester, F. W. _Aerodonetics_, D. van Nostrand Company: New York, 1909. On the public domain. [Get it from Google Books](http://books.google.com/books?id=6hxDAAAAIAAJ&pg=PP1#v=onepage&q&f=false).\n\n2. Milne-Thompson, L. M. _Theoretical Aerodynamics_, Dover 2012 reprint of the revised 1966 edition. [Read on Google Books](http://books.google.com/books?id=EMfCAgAAQBAJ&lpg=PP1&pg=PP1#v=onepage&q&f=false) (see section 18.5)\n\n3. Sinha, N. K. and Ananthkrishnan, N. _Elementary Flight Dynamics with an introduction to Bifurcation and Continuation Methods_, CRC Press, 2013. [Read on Google Books](http://books.google.com/books?id=yXL6AQAAQBAJ&lpg=PP1&pg=PP1#v=onepage&q&f=false) (see chapter 5)\n\n4. von K\u00e1rm\u00e1n, T. _Aerodynamics_, Dover 2004 reprint of the 1957 2nd edition. (see pages 149\u2013151)\n\n---\n\n###### The cell below loads the style of this notebook. \n\n\n```python\n# Execute this cell to load the notebook's style sheet, then ignore it\nfrom IPython.core.display import HTML\ncss_file = '../../styles/numericalmoocstyle.css'\nHTML(open(css_file, \"r\").read())\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n", "meta": {"hexsha": "8fff4cf0d98baf45cc3f0834f00958878f1f8087", "size": 137257, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lessons/01_phugoid/01_01_Phugoid_Theory.ipynb", "max_stars_repo_name": "rolando-contribute/numerical-mooc", "max_stars_repo_head_hexsha": "5f2115666006bf6e6367320fff46ddc1e0e32044", "max_stars_repo_licenses": ["CC-BY-3.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2017-05-01T20:26:02.000Z", "max_stars_repo_stars_event_max_datetime": "2017-05-01T20:26:02.000Z", "max_issues_repo_path": "lessons/01_phugoid/01_01_Phugoid_Theory.ipynb", "max_issues_repo_name": "rolando-contribute/numerical-mooc", "max_issues_repo_head_hexsha": "5f2115666006bf6e6367320fff46ddc1e0e32044", "max_issues_repo_licenses": ["CC-BY-3.0"], "max_issues_count": 25, "max_issues_repo_issues_event_min_datetime": "2017-04-14T14:47:50.000Z", "max_issues_repo_issues_event_max_datetime": "2018-10-17T16:04:48.000Z", "max_forks_repo_path": "lessons/01_phugoid/01_01_Phugoid_Theory.ipynb", "max_forks_repo_name": "rolando-contribute/numerical-mooc", "max_forks_repo_head_hexsha": "5f2115666006bf6e6367320fff46ddc1e0e32044", "max_forks_repo_licenses": ["CC-BY-3.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 192.7766853933, "max_line_length": 35624, "alphanum_fraction": 0.8749499115, "converted": true, "num_tokens": 5498, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.8152324803738429, "lm_q1q2_score": 0.43939662788110306}} {"text": "```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nplt.style.use('classic')\n%matplotlib inline\n```\n\n# Class 5: Managing Data with Pandas \n\nPandas is a Python library for managing datasets. Documentation and examples are available on the website for Pandas: http://pandas.pydata.org/. \n\nIn this Notebook, we'll make use of a dataset containing long-run averages of inflation, money growth, and real GDP. The dataset is available here: https://www.briancjenkins.com/data/csv/qtyTheoryData.csv (Python code to generate the dataset: https://github.com/letsgoexploring/economic-data). Recall that the quantity theory of money implies the following linear relationship between the long-run rate of money growth, the long-run rate of inflation, and the long-run rate of real GDP growth in a country:\n\n\\begin{align}\n\\text{inflation} & = \\text{money growth} - \\text{real GDP growth},\n\\end{align}\n\nGenerally, we treat real GDP growth and money supply growth as exogenous so this is a theory about the determination of inflation.\n\n### Import Pandas\n\n\n```python\nimport pandas as pd\n```\n\n### Import data from a csv file\n\nPandas has a function called `read_csv()` for reading data from a csv file into a Pandas `DataFrame` object.\n\n\n```python\n# Import quantity theory data into a Pandas DataFrame called 'df' with country names as the index.\n\n# Directly from internet\ndf = pd.read_csv('https://www.briancjenkins.com/data/csv/qtyTheoryData.csv')\n\n# From current working directory\n# df = pd.read_csv('qtyTheoryData.csv')\n```\n\n\n```python\n# Print the first 5 rows\nprint(df.head())\n```\n\n country iso code observations inflation money growth \\\n 0 Albania ALB 21 0.05186 0.12230 \n 1 Algeria DZA 51 0.10688 0.16024 \n 2 Angola AGO 20 0.82783 1.00940 \n 3 Antigua and Barbuda ATG 38 0.03943 0.09630 \n 4 Argentina ARG 54 0.73168 0.77245 \n \n gdp growth \n 0 0.04594 \n 1 0.03818 \n 2 0.08146 \n 3 0.03575 \n 4 0.02422 \n\n\n\n```python\n# Print the last 10 rows\nprint(df.tail(10))\n```\n\n country iso code observations inflation money growth \\\n 168 Ukraine UKR 23 0.58513 0.55392 \n 169 United Arab Emirates ARE 40 0.03566 0.13659 \n 170 United States USA 54 0.03377 0.05644 \n 171 Uruguay URY 54 0.36846 0.40479 \n 172 Vanuatu VUT 36 0.03509 0.08102 \n 173 Venezuela, RB VEN 53 0.20298 0.26996 \n 174 West Bank and Gaza PSE 17 0.03632 0.11749 \n 175 Yemen, Rep. YEM 24 0.14113 0.13238 \n 176 Zambia ZMB 27 0.21917 0.23465 \n 177 Zimbabwe ZWE 32 -0.00124 -0.12919 \n \n gdp growth \n 168 -0.01023 \n 169 0.04688 \n 170 0.03081 \n 171 0.02251 \n 172 0.02889 \n 173 0.02740 \n 174 0.03186 \n 175 0.03838 \n 176 0.01163 \n 177 0.00813 \n\n\n\n```python\n# Print the type of variable 'df'\nprint(type(df))\n```\n\n \n\n\n### Properties of `DataFrame` objects\n\nLike entries in a spreadsheet file, elements in a `DataFrame` object have row (or *index*) and column coordinates. Column names are always strings. Index elements can be integers, strings, or dates.\n\n\n```python\n# Print the columns of df\nprint(df.columns)\n```\n\n Index(['country', 'iso code', 'observations', 'inflation', 'money growth',\n 'gdp growth'],\n dtype='object')\n\n\n\n```python\n# Create a new variable called money equal to the 'money growth' column and print\nmoney = df['money growth']\nprint(money)\n```\n\n 0 0.12230\n 1 0.16024\n 2 1.00940\n 3 0.09630\n 4 0.77245\n 5 0.46411\n 6 0.08128\n 7 0.10181\n 8 0.07491\n 9 0.50149\n 10 0.06965\n 11 0.08263\n 12 0.13523\n 13 0.10833\n 14 0.61733\n 15 0.06306\n 16 0.11148\n 17 0.10734\n 18 0.18748\n 19 0.42180\n 20 0.19611\n 21 0.15783\n 22 0.89717\n 23 0.02883\n 24 0.34614\n 25 0.10509\n 26 0.13430\n 27 0.09847\n 28 0.21306\n 29 0.09689\n ... \n 148 0.12114\n 149 0.14659\n 150 0.09518\n 151 0.09639\n 152 0.29333\n 153 0.29243\n 154 0.14436\n 155 0.10587\n 156 0.06059\n 157 0.15539\n 158 0.30609\n 159 0.20918\n 160 0.09645\n 161 0.18530\n 162 0.09973\n 163 0.09616\n 164 0.11841\n 165 0.10958\n 166 0.37305\n 167 0.36831\n 168 0.55392\n 169 0.13659\n 170 0.05644\n 171 0.40479\n 172 0.08102\n 173 0.26996\n 174 0.11749\n 175 0.13238\n 176 0.23465\n 177 -0.12919\n Name: money growth, Length: 178, dtype: float64\n\n\n\n```python\n# Print the type of the variable money\nprint(type(money))\n```\n\n \n\n\nA Pandas `Series` stores one column of data. Like a `DataFrame`, a `Series` object has an index. Note that `money` has the same index as `df`. Instead of having a column, the `Series` has a `name` attribute.\n\n\n```python\n# Print the name of the 'money' variable\nprint(money.name)\n```\n\n money growth\n\n\nSelect multiple columns of a `DataFrame` by puting the desired column names in a set a of square brackets (i.e., in a `list`).\n\n\n```python\n# Print the first 5 rows of just the inflation, money growth, and gdp growth columns\nprint(df[['inflation','money growth','gdp growth']].head())\n```\n\n inflation money growth gdp growth\n 0 0.05186 0.12230 0.04594\n 1 0.10688 0.16024 0.03818\n 2 0.82783 1.00940 0.08146\n 3 0.03943 0.09630 0.03575\n 4 0.73168 0.77245 0.02422\n\n\nAs mentioned, the set of row coordinates is the index. Unless specified otherwise, Pandas automatically assigns an integer index starting at 0 to rows of the `DataFrame`.\n\n\n```python\n# Print the index of 'df'\nprint(df.index)\n```\n\n RangeIndex(start=0, stop=178, step=1)\n\n\nNote that in the index of the `df` is the numbers 0 through 177. We could have specified a different index when we imported the data using `read_csv()`. For example, suppose we want to the country names to be the index of `df`. Since country names are in the first column of the data file, we can pass the argument `index_col=0` to `read_csv()`\n\n\n```python\n# Import quantity theory data into a Pandas DataFrame called 'df' with country names as the index.\ndf = pd.read_csv('https://www.briancjenkins.com/data/csv/qtyTheoryData.csv',index_col=0)\n\n# Print first 5 rows of df\nprint(df.head())\n```\n\n iso code observations inflation money growth \\\n country \n Albania ALB 21 0.05186 0.12230 \n Algeria DZA 51 0.10688 0.16024 \n Angola AGO 20 0.82783 1.00940 \n Antigua and Barbuda ATG 38 0.03943 0.09630 \n Argentina ARG 54 0.73168 0.77245 \n \n gdp growth \n country \n Albania 0.04594 \n Algeria 0.03818 \n Angola 0.08146 \n Antigua and Barbuda 0.03575 \n Argentina 0.02422 \n\n\nUse the `loc` attribute to select rows of the `DataFrame` by index *values*.\n\n\n```python\n# Create a new variable called 'df_usa' equal to the 'United States' row and print\ndf_usa = df.loc['United States']\nprint(df_usa)\n```\n\n iso code USA\n observations 54\n inflation 0.03377\n money growth 0.05644\n gdp growth 0.03081\n Name: United States, dtype: object\n\n\nUse `iloc` attribute to select row based on integer location (starting from 0).\n\n\n```python\n# Create a new variable called 'df_third' equal to the third row in the DataFrame and print\ndf_first = df.iloc[2]\nprint(df_first)\n```\n\n iso code AGO\n observations 20\n inflation 0.82783\n money growth 1.0094\n gdp growth 0.08146\n Name: Angola, dtype: object\n\n\nThere are several ways to return a single element of a Pandas `DataFrame`. For example, here are three that we want to return the value of inflation for the United States from the DataFrame `df`:\n\n1. `df.loc['United States','inflation']`\n2. `df.loc['United States']['inflation']`\n3. `df['inflation'].loc['United States']`\n\nThe first method points directly to the element in the `df` while the second and third methods return *copies* of the element. That means that you can modify the value of inflation for the United States by running:\n\n df.loc['United States','inflation'] = new_value\n \nBut running either:\n\n df.loc['United States']['inflation'] = new_value\n \nor:\n\n df['inflation'].loc['United States'] = new_value\n\nwill return an error.\n\n\n```python\n# Print the inflation rate of the United States (By index and column together)\nprint('Long-run average inflation in US: ',df.loc['United States','inflation'])\n```\n\n Long-run average inflation in US: 0.03377\n\n\n\n```python\n# Print the inflation rate of the United States (first by index, then by column)\nprint('Long-run average inflation in US: ',df.loc['United States']['inflation'])\n```\n\n Long-run average inflation in US: 0.03377\n\n\n\n```python\n# Print the inflation rate of the United States (first by column, then by index)\nprint('Long-run average inflation in US: ',df['inflation'].loc['United States'])\n```\n\n Long-run average inflation in US: 0.03377\n\n\nNew columns are easily created as functions of existing columns.\n\n\n```python\n# Create a new column called 'difference' equal to the money growth column minus \n# the inflation column and print the modified DataFrame\ndf['difference'] = df['money growth'] - df['inflation']\nprint(df['difference'])\n```\n\n country\n Albania 0.07044\n Algeria 0.05336\n Angola 0.18157\n Antigua and Barbuda 0.05687\n Argentina 0.04077\n Armenia 0.02154\n Aruba 0.04959\n Australia 0.05106\n Austria 0.05845\n Azerbaijan 0.07387\n Bahamas, The 0.02115\n Bahrain 0.05318\n Bangladesh 0.05657\n Barbados 0.04723\n Belarus 0.02929\n Belgium 0.04654\n Belize 0.08926\n Benin 0.05905\n Bhutan 0.12045\n Bolivia 0.08569\n Bosnia and Herzegovina 0.15565\n Botswana 0.06462\n Brazil 0.03037\n Brunei Darussalam -0.02624\n Bulgaria 0.06415\n Burkina Faso 0.06647\n Burundi 0.04095\n Cabo Verde 0.06715\n Cambodia 0.15604\n Cameroon 0.04300\n ... \n Sri Lanka 0.02800\n St. Kitts and Nevis 0.11041\n St. Lucia 0.05407\n St. Vincent and the Grenadines 0.04987\n Sudan 0.04654\n Suriname 0.03204\n Swaziland 0.05194\n Sweden 0.03408\n Switzerland 0.04355\n Syrian Arab Republic 0.06709\n Tajikistan 0.13993\n Tanzania 0.06030\n Thailand 0.05509\n Timor-Leste 0.09552\n Togo 0.05267\n Tonga 0.04220\n Trinidad and Tobago 0.04894\n Tunisia 0.05190\n Turkey 0.06002\n Uganda 0.09962\n Ukraine -0.03121\n United Arab Emirates 0.10093\n United States 0.02267\n Uruguay 0.03633\n Vanuatu 0.04593\n Venezuela, RB 0.06698\n West Bank and Gaza 0.08117\n Yemen, Rep. -0.00875\n Zambia 0.01548\n Zimbabwe -0.12795\n Name: difference, Length: 178, dtype: float64\n\n\n\n```python\n# Print the average difference between money growth and inflation\nprint(df.difference.mean())\n```\n\n 0.06082084269662919\n\n\n\n```python\n# Remove the following columns from the DataFrame: 'iso code','observations','difference'\ndf = df.drop(['observations','difference'],axis=1)\n\n# Print the modified DataFrame\nprint(df)\n```\n\n iso code inflation money growth gdp growth\n country \n Albania ALB 0.05186 0.12230 0.04594\n Algeria DZA 0.10688 0.16024 0.03818\n Angola AGO 0.82783 1.00940 0.08146\n Antigua and Barbuda ATG 0.03943 0.09630 0.03575\n Argentina ARG 0.73168 0.77245 0.02422\n Armenia ARM 0.44257 0.46411 0.05471\n Aruba ABW 0.03169 0.08128 0.00825\n Australia AUS 0.05075 0.10181 0.03461\n Austria AUT 0.01646 0.07491 0.01446\n Azerbaijan AZE 0.42762 0.50149 0.05468\n Bahamas, The BHS 0.04850 0.06965 0.01829\n Bahrain BHR 0.02945 0.08263 0.03984\n Bangladesh BGD 0.07866 0.13523 0.04486\n Barbados BRB 0.06110 0.10833 0.02450\n Belarus BLR 0.58804 0.61733 0.04919\n Belgium BEL 0.01652 0.06306 0.01456\n Belize BLZ 0.02222 0.11148 0.04995\n Benin BEN 0.04829 0.10734 0.03684\n Bhutan BTN 0.06703 0.18748 0.07094\n Bolivia BOL 0.33611 0.42180 0.03016\n Bosnia and Herzegovina BIH 0.04046 0.19611 0.04219\n Botswana BWA 0.09321 0.15783 0.06992\n Brazil BRA 0.86680 0.89717 0.04129\n Brunei Darussalam BRN 0.05507 0.02883 0.01031\n Bulgaria BGR 0.28199 0.34614 0.01943\n Burkina Faso BFA 0.03862 0.10509 0.04196\n Burundi BDI 0.09335 0.13430 0.02690\n Cabo Verde CPV 0.03132 0.09847 0.06541\n Cambodia KHM 0.05702 0.21306 0.04995\n Cameroon CMR 0.05389 0.09689 0.03480\n ... ... ... ... ...\n Sri Lanka LKA 0.09314 0.12114 0.04756\n St. Kitts and Nevis KNA 0.03618 0.14659 0.03959\n St. Lucia LCA 0.04111 0.09518 0.03606\n St. Vincent and the Grenadines VCT 0.04652 0.09639 0.03776\n Sudan SDN 0.24679 0.29333 0.03785\n Suriname SUR 0.26039 0.29243 0.01445\n Swaziland SWZ 0.09242 0.14436 0.04881\n Sweden SWE 0.07179 0.10587 0.02789\n Switzerland CHE 0.01704 0.06059 0.01700\n Syrian Arab Republic SYR 0.08830 0.15539 0.05270\n Tajikistan TJK 0.16616 0.30609 0.07200\n Tanzania TZA 0.14888 0.20918 0.04978\n Thailand THA 0.04136 0.09645 0.05997\n Timor-Leste TLS 0.08978 0.18530 0.07596\n Togo TGO 0.04706 0.09973 0.03432\n Tonga TON 0.05396 0.09616 0.01657\n Trinidad and Tobago TTO 0.06947 0.11841 0.02876\n Tunisia TUN 0.05768 0.10958 0.04585\n Turkey TUR 0.31303 0.37305 0.04647\n Uganda UGA 0.26869 0.36831 0.05673\n Ukraine UKR 0.58513 0.55392 -0.01023\n United Arab Emirates ARE 0.03566 0.13659 0.04688\n United States USA 0.03377 0.05644 0.03081\n Uruguay URY 0.36846 0.40479 0.02251\n Vanuatu VUT 0.03509 0.08102 0.02889\n Venezuela, RB VEN 0.20298 0.26996 0.02740\n West Bank and Gaza PSE 0.03632 0.11749 0.03186\n Yemen, Rep. YEM 0.14113 0.13238 0.03838\n Zambia ZMB 0.21917 0.23465 0.01163\n Zimbabwe ZWE -0.00124 -0.12919 0.00813\n \n [178 rows x 4 columns]\n\n\n### Methods\n\nA Pandas `DataFrame` has a bunch of useful methods defined for it. `describe()` returns some summary statistics.\n\n\n```python\n# Print the summary statistics for 'df'\nprint(df.describe())\n```\n\n inflation money growth gdp growth\n count 178.000000 178.000000 178.000000\n mean 0.123952 0.184773 0.036382\n std 0.165354 0.163258 0.021763\n min -0.001240 -0.129190 -0.023680\n 25% 0.041170 0.099833 0.023447\n 50% 0.068250 0.137390 0.036875\n 75% 0.125903 0.198862 0.046857\n max 1.220970 1.171190 0.159540\n\n\nThe `corr()` method returns a `DataFrame` containing the correlation coefficients of the specified `DataFrame`.\n\n\n```python\n# Create a variable called 'correlations' containg the correlation coefficients for columns in 'df'\ncorrelations = df.corr()\n\n# Print the correlation coefficients\nprint(correlations)\n```\n\n inflation money growth gdp growth\n inflation 1.000000 0.972160 -0.043356\n money growth 0.972160 1.000000 0.084328\n gdp growth -0.043356 0.084328 1.000000\n\n\n\n```python\n# Print the correlation coefficient for inflation and money growth\nprint('corr of inflation and money growth: ',round(correlations.loc['inflation','money growth'],4))\n\n# Print the correlation coefficient for inflation and real GDP growth\nprint('corr of inflation and gdp growth: ',round(correlations.loc['inflation','gdp growth'],4))\n\n# Print the correlation coefficient for money growth and real GDP growth\nprint('corr of money growth and gdp growth:',round(correlations.loc['money growth','gdp growth'],4))\n```\n\n corr of inflation and money growth: 0.9722\n corr of inflation and gdp growth: -0.0434\n corr of money growth and gdp growth: 0.0843\n\n\n`sort_values()` returns a copy of the original `DataFrame` sorted along the given column. The optional argument `ascending` is set to `True` by default, but can be changed to `False` if you want to print the lowest first.\n\n\n```python\n# Print rows for the countries with the 10 lowest inflation rates\nprint(df.sort_values('inflation').head(10))\n```\n\n iso code inflation money growth gdp growth\n country \n Zimbabwe ZWE -0.00124 -0.12919 0.00813\n Djibouti DJI 0.00033 0.07611 0.01616\n Germany DEU 0.01049 0.07344 0.01155\n Ireland IRL 0.01375 0.12348 0.03326\n Hong Kong SAR, China HKG 0.01391 0.09846 0.03618\n France FRA 0.01461 0.06584 0.01233\n Kosovo KSV 0.01540 0.08659 0.03359\n Greece GRC 0.01611 0.01423 -0.00452\n Austria AUT 0.01646 0.07491 0.01446\n Finland FIN 0.01647 0.06528 0.01387\n\n\n\n```python\n# Print rows for the countries with the 10 highest inflation rates\nprint(df.sort_values('inflation',ascending=False).head(10))\n```\n\n iso code inflation money growth gdp growth\n country \n Congo, Dem. Rep. COD 1.22097 1.17119 -0.00232\n Brazil BRA 0.86680 0.89717 0.04129\n Angola AGO 0.82783 1.00940 0.08146\n Argentina ARG 0.73168 0.77245 0.02422\n Nicaragua NIC 0.61301 0.66175 0.02475\n Belarus BLR 0.58804 0.61733 0.04919\n Ukraine UKR 0.58513 0.55392 -0.01023\n Peru PER 0.47231 0.53655 0.03444\n Armenia ARM 0.44257 0.46411 0.05471\n Azerbaijan AZE 0.42762 0.50149 0.05468\n\n\nNote that `sort_values` and `sort_index` return *copies* of the original `DataFrame`. If, in the previous example, we had wanted to actually modify `df`, we would have need to explicitly overwrite it:\n\n df = df.sort_index(ascending=False)\n\n\n```python\n# Print first 10 rows with the index sorted in descending alphabetical order\nprint(df.sort_index(ascending=False).head(10))\n```\n\n iso code inflation money growth gdp growth\n country \n Zimbabwe ZWE -0.00124 -0.12919 0.00813\n Zambia ZMB 0.21917 0.23465 0.01163\n Yemen, Rep. YEM 0.14113 0.13238 0.03838\n West Bank and Gaza PSE 0.03632 0.11749 0.03186\n Venezuela, RB VEN 0.20298 0.26996 0.02740\n Vanuatu VUT 0.03509 0.08102 0.02889\n Uruguay URY 0.36846 0.40479 0.02251\n United States USA 0.03377 0.05644 0.03081\n United Arab Emirates ARE 0.03566 0.13659 0.04688\n Ukraine UKR 0.58513 0.55392 -0.01023\n\n\n### Quick plotting example\n\nConstruct a graph that visually confirms the quantity theory of money by making a scatter plot with average money growth on the horizontal axis and average inflation on the vertical axis. Set the marker size `s` to 50 and opacity (`alpha`) 0.25. Add a 45 degree line, axis labels, and a title. Lower and upper limits for the horizontal and vertical axes should be -0.2 and 1.2.\n\n\n```python\n# Create data for 45 degree line\nx45 = [-0.2,1.2]\ny45 = [-0.2,1.2]\n\n# Create figure and axis\nfig = plt.figure()\nax = fig.add_subplot(1,1,1)\n\n\n# Plot 45 degree line and create legend in lower right corner\nax.plot(x45,y45,'-r',label = '$45^{\\circ}$')\nax.legend(loc='lower right')\n\n# Scatter plot of data inflation against money growth\nax.scatter(df['money growth'],df['inflation'],s=50,alpha = 0.25)\nax.set_xlim([-0.2,1.2])\nax.set_ylim([-0.2,1.2])\nax.set_xlabel('money growth')\nax.set_ylabel('inflation')\nax.set_title('Average inflation against average money growth \\nfor '+str(len(df.index))+' countries.')\nax.grid()\n```\n\n### Exporting a `DataFrame` to csv\n\nUse the DataFrame method `to_csv()` to export DataFrame to a csv file.\n\n\n```python\n# Export the DataFrame 'df' to a csv file called 'modified_data.csv'.\ndf.to_csv('modified_data.csv')\n```\n", "meta": {"hexsha": "9bf54f0a2d7f683cac9674e09cea45440f659b0a", "size": 62063, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Lecture Notebooks/Econ126_Class_05.ipynb", "max_stars_repo_name": "pmezap/computational-macroeconomics", "max_stars_repo_head_hexsha": "b703f46176bb16e712badf752784f8a7b996cdb1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 30, "max_stars_repo_stars_event_min_datetime": "2020-02-29T06:09:03.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-25T13:14:13.000Z", "max_issues_repo_path": "Lecture Notebooks/Econ126_Class_05.ipynb", "max_issues_repo_name": "letsgoexploring/computational-macroeconomics", "max_issues_repo_head_hexsha": "b703f46176bb16e712badf752784f8a7b996cdb1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Lecture Notebooks/Econ126_Class_05.ipynb", "max_forks_repo_name": "letsgoexploring/computational-macroeconomics", "max_forks_repo_head_hexsha": "b703f46176bb16e712badf752784f8a7b996cdb1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 18, "max_forks_repo_forks_event_min_datetime": "2019-09-24T07:48:49.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-22T21:36:30.000Z", "avg_line_length": 60.0803484995, "max_line_length": 26156, "alphanum_fraction": 0.671672333, "converted": true, "num_tokens": 7181, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891451980404, "lm_q2_score": 0.7461390043208003, "lm_q1q2_score": 0.439393160453393}} {"text": "# N-Gram Language Models\n\n#### Preliminaries: Install KenLM\n\n[KenLM](https://kheafield.com/code/kenlm/) is used in the latter part of this lab.\n\nTo install it, run the following commands **in the jupyter terminal** (see screenshot):\n\n sudo apt-get install build-essential libboost-all-dev cmake zlib1g-dev libbz2-dev liblzma-dev\n \n cd /home/jupyter\n wget -O - https://kheafield.com/code/kenlm.tar.gz |tar xz\n mkdir kenlm/build\n cd kenlm/build\n cmake ..\n make -j 4\n \n cd /home/jupyter/kenlm\n python setup.py install\n \n\n\n\n\n\n```python\n!pip install jsonlines\n```\n\n Collecting jsonlines\n Downloading https://files.pythonhosted.org/packages/4f/9a/ab96291470e305504aa4b7a2e0ec132e930da89eb3ca7a82fbe03167c131/jsonlines-1.2.0-py2.py3-none-any.whl\n Requirement already satisfied: six in /home/aims/anaconda3/envs/aims/lib/python3.7/site-packages (from jsonlines) (1.12.0)\n Installing collected packages: jsonlines\n Successfully installed jsonlines-1.2.0\n\n\n\n```python\n%pylab inline\nimport os, sys, glob, json, math\nimport numpy as np\nimport pandas as pd\nfrom tqdm import tqdm\nfrom pprint import pprint\nfrom collections import defaultdict\npd.set_option('display.max_colwidth', -1)\n```\n\n Populating the interactive namespace from numpy and matplotlib\n\n\n### N-gram Language Modeling\n\nIn **language modeling** we want to model the probability of variable length sequences, $$\\large p(w_1,\\ldots,w_T)=\\prod_{t=1}^T p(w_t|w_{`, and **end token**, ``:\n\n\n```python\ndata = [[''] + d + [''] for d in data_raw]\ndata\n```\n\n\n\n\n [['', 'A', 'A', 'B', 'B', ''],\n ['', 'A', 'A', 'B', ''],\n ['', 'A', 'A', 'B', 'C', ''],\n ['', 'A', 'A', 'A', ''],\n ['', 'A', 'A', 'A', 'A', '']]\n\n\n\nNow let's estimate a bigram model:\n\n\\begin{align}\np(w_t|w_{t-1}) &= \\frac{\\text{count}(w_{t-1}w_{t})}{\\sum_{w_{t'}}\\text{count}(w_{t-1}w_{t'})}\\\\\n &= \\texttt{count[prefix][wt] / totals[prefix]}\n\\end{align} \n\nwhere $\\texttt{prefix}$ is $w_{t-1}$ in this case.\n\n\n```python\ncount = defaultdict(lambda: defaultdict(float))\ntotal = defaultdict(float)\n\nn = 2\nfor sequence in data:\n for i in range(len(sequence)-n+1): # for each ngram\n ngram = tuple(sequence[i:i+n])\n prefix, word = ngram[:-1], ngram[-1]\n count[prefix][word] += 1 # count(w_{t-n+1}...w_t)\n total[prefix] += 1 # count(w_{t-n+1}...w_{t-1})\n```\n\nLet's see if the counts and totals make sense:\n\n- How many times did (A, B) occur? What about (B, B)?\n- How many times did (A) occur? What about (C)?\n\n\n```python\nprint(\"Counts:\")\npprint(dict(count))\nprint(\"\\nTotals:\")\npprint(dict(total))\n```\n\n Counts:\n {('',): defaultdict(, {'A': 5.0}),\n ('A',): defaultdict(, {'A': 8.0, 'B': 3.0, '': 2.0}),\n ('B',): defaultdict(, {'B': 1.0, '': 2.0, 'C': 1.0}),\n ('C',): defaultdict(, {'': 1.0})}\n \n Totals:\n {('',): 5.0, ('A',): 13.0, ('B',): 4.0, ('C',): 1.0}\n\n\n#### Conditional probability queries\n\nWe can now query a conditional probability:\n\n\\begin{align}\n\\texttt{p(word|prefix)} =&\\ \\texttt{count[prefix][word] / totals[prefix]}\n\\end{align}\n\n\n```python\nqueries = [('', 'A'),\n ('B', 'C')]\n\nfor query in queries:\n prefix, word = query[:-1], query[-1]\n p = count[prefix][word] / total[prefix] # We'll discuss the case when `total[prefix] = 0` below.\n print(\"p( %s | %s) = \\t%.5f\" % (word, ', '.join(prefix), p))\n```\n\n p( A | ) = \t1.00000\n p( C | B) = \t0.25000\n\n\n**Exercise**: Look at the training set and convince yourself that these conditional probabilities are correct according to the count-based estimation procedure.\n\n#### Sequence Probability\n\nWe can compute the probability of a sequence using the conditional probabilities along with the chain rule of probability:\n\n\\begin{align}\np(w_1,\\ldots,w_T)&\\approx\\prod_{t=1}^T p(w_t|w_{t-1})\n\\end{align}\n\n(Here $w_0$ is `` and $w_T$ is ``)\n\n\n```python\nsequence = ['', 'A', 'A', 'B', '']\n\ndef sequence_p(sequence, log=False):\n total_p = 1\n\n for i in range(len(sequence)-n+1):\n ngram = tuple(sequence[i:i+n])\n prefix = ngram[:-1]\n word = ngram[-1]\n p = count[prefix][word] / max(total[prefix], 1)\n if log:\n print(\"p(%s | %s) =\\t%.3f\" % (word, ', '.join(prefix), p))\n\n total_p *= p\n return total_p\n \n\nprint(\"\\nProduct: p(%s) = %.3f\" % (''.join(sequence[1:-1]), sequence_p(sequence, log=True)))\n```\n\n p(A | ) =\t1.000\n p(A | A) =\t0.615\n p(B | A) =\t0.231\n p( | B) =\t0.500\n \n Product: p(AAB) = 0.071\n\n\n### Real Example: Dialogue Utterances\n\nNow lets use the same ideas on a more realistic text corpus.\n\nWe will use utterances from a dialogue dataset called **Persona-Chat**. This dataset is relatively small and centers on a single domain, but it is simple and interpretable for our purposes here.\n\nWe'll also use an off-the-shelf ngram modeling package called `KenLM`.\n\n#### Loading the dataset\n\n\n```python\nimport kenlm\n```\n\n\n```python\ntrain = []\nvalid = []\nimport jsonlines\n\nfor ds, name in [(train, 'train'), (valid, 'valid')]:\n for line in jsonlines.Reader(open('data/personachat/personachat_all_sentences_%s.jsonl' % name, 'r')):\n ds.append(line['tokens'])\n \nvocab = list(set([t for ts in train for t in ts])) \nprint(\"Number of train examples: %d\" % (len(train)))\nprint(\"Number of valid examples: %d\" % (len(valid)))\nprint(\"Vocab size: %d\" % (len(vocab)))\n\nprint(\"\\nExamples:\")\npprint(train[:3])\n```\n\n### KenLM\n\nKenLM estimates n-gram language models using **modified Kneser-Ney smoothing**, and has a fast and memory-efficient implementation. \n- While we won't go into details here, **smoothing** is a technique used to account for ngrams that do not occur in the training corpus. \n- Normally, these ngrams would receive zero-probability mass. Smoothing ensures every ngram receives some probability.\n\n\n\nPlease see page 48 of the [lecture note](https://github.com/nyu-dl/NLP_DL_Lecture_Note/blob/master/lecture_note.pdf) for an overview of Kneser-Ney smoothing, and [[Chen & Goodman 1998]](https://dash.harvard.edu/bitstream/handle/1/25104739/tr-10-98.pdf?sequence=1) for further details.\n\n\n```python\nKENLM_DIR='/home/jupyter/kenlm'\n```\n\n#### Tokenize data\n\n\n```python\nif not os.path.exists('data/tokenized'):\n os.makedirs('data/tokenized')\nwith open('data/tokenized/pchat_train', 'w') as f:\n for line in tqdm(train):\n f.write(' '.join(line))\n f.write('\\n')\n```\n\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 133176/133176 [00:00<00:00, 795531.18it/s]\n\n\n#### Build kenlm n-gram models\n\nThis uses the `kenlm` commands `lmplz` to estimate the language model, then `build_binary` to convert it to an efficient format. We load the resulting model using the `kenlm` python wrapper.\n\nWe do this for n-gram models of order `2,3,4`:\n\n\n```python\nimport kenlm\n\ndata_prefix = 'pchat'\ndataset = 'pchat_train'\nif not os.path.exists('models'):\n os.makedirs('models')\n\nmodels = {}\nfor n in [2,3,4]:\n model_temp = 'models/%s_%d.arpa' % (data_prefix, n)\n model_name = 'models/%s_%d.klm' % (data_prefix, n)\n ! cat ./data/tokenized/$dataset | $KENLM_DIR/build/bin/lmplz -o $n > $model_temp\n ! $KENLM_DIR/build/bin/build_binary $model_temp $model_name\n models[n] = kenlm.LanguageModel(model_name)\n```\n\n === 1/5 Counting and sorting n-grams ===\n File stdin isn't normal. Using slower read() instead of mmap(). No progress bar.\n Unigram tokens 1602042 types 19156\n === 2/5 Calculating and sorting adjusted counts ===\n Chain sizes: 1:229872 2:10928379904\n Statistics:\n 1 19156 D1=0.588857 D2=1.0348 D3+=1.3388\n 2 231267 D1=0.708637 D2=1.06132 D3+=1.37629\n Memory estimate for binary LM:\n type kB\n probing 4551 assuming -p 1.5\n probing 4626 assuming -r models -p 1.5\n trie 1747 without quantization\n trie 1099 assuming -q 8 -b 8 quantization \n trie 1747 assuming -a 22 array pointer compression\n trie 1099 assuming -a 22 -q 8 -b 8 array pointer compression and quantization\n === 3/5 Calculating and sorting initial probabilities ===\n Chain sizes: 1:229872 2:3700272\n ----5---10---15---20---25---30---35---40---45---50---55---60---65---70---75---80---85---90---95--100\n ####################################################################################################\n === 4/5 Calculating and writing order-interpolated probabilities ===\n Chain sizes: 1:229872 2:3700272\n ----5---10---15---20---25---30---35---40---45---50---55---60---65---70---75---80---85---90---95--100\n ####################################################################################################\n === 5/5 Writing ARPA model ===\n ----5---10---15---20---25---30---35---40---45---50---55---60---65---70---75---80---85---90---95--100\n ****************************************************************************************************\n Name:lmplz\tVmPeak:10874216 kB\tVmRSS:10720 kB\tRSSMax:2938548 kB\tuser:0.848\tsys:1.288\tCPU:2.13913\treal:2.13823\n Reading models/pchat_2.arpa\n ----5---10---15---20---25---30---35---40---45---50---55---60---65---70---75---80---85---90---95--100\n ****************************************************************************************************\n SUCCESS\n === 1/5 Counting and sorting n-grams ===\n File stdin isn't normal. Using slower read() instead of mmap(). No progress bar.\n Unigram tokens 1602042 types 19156\n === 2/5 Calculating and sorting adjusted counts ===\n Chain sizes: 1:229872 2:3801175808 3:7127204352\n Statistics:\n 1 19156 D1=0.588857 D2=1.0348 D3+=1.3388\n 2 231267 D1=0.722265 D2=1.09097 D3+=1.44737\n 3 617624 D1=0.798924 D2=1.0703 D3+=1.33013\n Memory estimate for binary LM:\n type kB\n probing 16763 assuming -p 1.5\n probing 18193 assuming -r models -p 1.5\n trie 6683 without quantization\n trie 3625 assuming -q 8 -b 8 quantization \n trie 6354 assuming -a 22 array pointer compression\n trie 3296 assuming -a 22 -q 8 -b 8 array pointer compression and quantization\n === 3/5 Calculating and sorting initial probabilities ===\n Chain sizes: 1:229872 2:3700272 3:12352480\n ----5---10---15---20---25---30---35---40---45---50---55---60---65---70---75---80---85---90---95--100\n ####################################################################################################\n === 4/5 Calculating and writing order-interpolated probabilities ===\n Chain sizes: 1:229872 2:3700272 3:12352480\n ----5---10---15---20---25---30---35---40---45---50---55---60---65---70---75---80---85---90---95--100\n ####################################################################################################\n === 5/5 Writing ARPA model ===\n ----5---10---15---20---25---30---35---40---45---50---55---60---65---70---75---80---85---90---95--100\n ****************************************************************************************************\n Name:lmplz\tVmPeak:10866020 kB\tVmRSS:22604 kB\tRSSMax:2500264 kB\tuser:1.408\tsys:1.228\tCPU:2.63778\treal:2.52355\n Reading models/pchat_3.arpa\n ----5---10---15---20---25---30---35---40---45---50---55---60---65---70---75---80---85---90---95--100\n ****************************************************************************************************\n SUCCESS\n === 1/5 Counting and sorting n-grams ===\n File stdin isn't normal. Using slower read() instead of mmap(). No progress bar.\n Unigram tokens 1602042 types 19156\n === 2/5 Calculating and sorting adjusted counts ===\n Chain sizes: 1:229872 2:1860149760 3:3487780608 4:5580449280\n Statistics:\n 1 19156 D1=0.588857 D2=1.0348 D3+=1.3388\n 2 231267 D1=0.722265 D2=1.09097 D3+=1.44737\n 3 617624 D1=0.821472 D2=1.14168 D3+=1.40217\n 4 932153 D1=0.863174 D2=1.12888 D3+=1.29333\n Memory estimate for binary LM:\n type kB\n probing 36767 assuming -p 1.5\n probing 41816 assuming -r models -p 1.5\n trie 15838 without quantization\n trie 8356 assuming -q 8 -b 8 quantization \n trie 14567 assuming -a 22 array pointer compression\n trie 7085 assuming -a 22 -q 8 -b 8 array pointer compression and quantization\n === 3/5 Calculating and sorting initial probabilities ===\n Chain sizes: 1:229872 2:3700272 3:12352480 4:22371672\n ----5---10---15---20---25---30---35---40---45---50---55---60---65---70---75---80---85---90---95--100\n ####################################################################################################\n === 4/5 Calculating and writing order-interpolated probabilities ===\n Chain sizes: 1:229872 2:3700272 3:12352480 4:22371672\n ----5---10---15---20---25---30---35---40---45---50---55---60---65---70---75---80---85---90---95--100\n ####################################################################################################\n === 5/5 Writing ARPA model ===\n ----5---10---15---20---25---30---35---40---45---50---55---60---65---70---75---80---85---90---95--100\n ****************************************************************************************************\n Name:lmplz\tVmPeak:10865048 kB\tVmRSS:44524 kB\tRSSMax:2182940 kB\tuser:2.32\tsys:1.384\tCPU:3.7062\treal:3.30219\n Reading models/pchat_4.arpa\n ----5---10---15---20---25---30---35---40---45---50---55---60---65---70---75---80---85---90---95--100\n ****************************************************************************************************\n SUCCESS\n\n\n## Evaluation\n\n### Perplexity\n\nIntuitively, a good model should assign high probabilities to sequences from the 'true' distribution that it is modeling.\n\nA common way of quantifying this is with **perplexity**, a metric inversely-proportional to the probability that the model assigns to a set of sequences, e.g. a 'test set':\n\n\\begin{align}\n\\large \\text{ppl}(p, D) &\\large\\ = 2^{-\\frac{1}{N_{total}}\\log_2 p(D)}\n\\end{align}\n\n\nwhere $D=\\{(w_1,\\ldots,w_{N_i})_i\\}_{i=1}^M$ is a dataset of $M$ sequences with total length $N_{\\text{total}}=\\sum_{i}N_i$.\n\nPerplexity is defined on $[1,\\infty)$, with 1 being a perfect model (assigning probability 1 to $D$), and a 'worse' model as perplexity increases.\n\nIntuitively, _perplexity measures the average rank of the true next-token, when tokens are ordered by the model's conditional probabilities_.\n\n\n\n\n#### Evaluate Perplexity\n\n`kenlm` outputs log probabilities in **log base 10**. For the standard definition of perplexity we need **log base 2**. See the code below:\n\n\n```python\ndef perplexity_kenlm(model, sequences):\n n_total = 0\n logp_total = 0\n for sequence in sequences:\n text = ' '.join(sequence)\n logp_total += model.score(text)\n n_total += len(sequence) + 1 # add 1 for     \n \n # Convert log10 to log2\n log2p_total = logp_total / np.log10(2)\n ppl = 2 ** (- (1.0 / n_total) * log2p_total)\n return ppl\n```\n\n\n```python\nprint(\"=== Train ===\")\ndf = pd.DataFrame([(n, perplexity_kenlm(models[n], train)) for n in [2, 3, 4]], columns=['n', 'ppl'])\ndf.style.hide_index()\n```\n\n === Train ===\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n
    n ppl
    239.9248
    315.462
    48.91961
    \n\n\n\n\n```python\nprint(\"=== Valid ===\")\ndf = pd.DataFrame([(n, perplexity_kenlm(models[n], valid)) for n in [2, 3, 4]], columns=['n', 'ppl'])\ndf.style.hide_index()\n```\n\n === Valid ===\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n
    n ppl
    259.188
    344.9771
    443.2188
    \n\n\n\n#### Sequence probabilities:\n\\begin{align}\np(w_1,\\ldots,w_T)&\\approx\\prod_{t=1}^T p(w_t|w_{t-2},w_{t-1})\\\\\n&=\\sum_{t=1}^T \\log p(w_t|w_{t-2},w_{t-1}).\n\\end{align}\n\nwhere we use log probabilities in practice to avoid a product of many small numbers.\n\n\n```python\nsentences = [\n 'i like my pet dog .',\n 'i like my pet zebra .',\n 'i like my pet lion .',\n 'i live in the united states .',\n 'i live in the united states of america .'\n]\n\nfor sentence in sentences:\n print(sentence)\n for n in [2, 3, 4]:\n log10p = models[n].score(sentence)\n log2p = log10p / np.log10(2)\n print(\"n: %d\\t logp: %.3f\" % (n, log2p))\n print()\n```\n\n i like my pet dog .\n n: 2\t logp: -29.216\n n: 3\t logp: -28.843\n n: 4\t logp: -29.219\n \n i like my pet zebra .\n n: 2\t logp: -32.303\n n: 3\t logp: -31.157\n n: 4\t logp: -32.101\n \n i like my pet lion .\n n: 2\t logp: -38.352\n n: 3\t logp: -41.017\n n: 4\t logp: -42.320\n \n i live in the united states .\n n: 2\t logp: -22.599\n n: 3\t logp: -20.095\n n: 4\t logp: -17.854\n \n i live in the united states of america .\n n: 2\t logp: -40.958\n n: 3\t logp: -26.627\n n: 4\t logp: -24.311\n \n\n", "meta": {"hexsha": "d6a779a736aa5c2967b1a621daeb1dd990dc1d75", "size": 33891, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Part 02/001_LM/ngram_lm_lab/ngram_lms.ipynb", "max_stars_repo_name": "bandiang2/AMMI-NLP", "max_stars_repo_head_hexsha": "8d4c8f35f720d25ecf56d030618a7c5ec3585743", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Part 02/001_LM/ngram_lm_lab/ngram_lms.ipynb", "max_issues_repo_name": "bandiang2/AMMI-NLP", "max_issues_repo_head_hexsha": "8d4c8f35f720d25ecf56d030618a7c5ec3585743", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Part 02/001_LM/ngram_lm_lab/ngram_lms.ipynb", "max_forks_repo_name": "bandiang2/AMMI-NLP", "max_forks_repo_head_hexsha": "8d4c8f35f720d25ecf56d030618a7c5ec3585743", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 38.7325714286, "max_line_length": 1588, "alphanum_fraction": 0.5029063763, "converted": true, "num_tokens": 6954, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891451980403, "lm_q2_score": 0.7461389873857264, "lm_q1q2_score": 0.43939315048051175}} {"text": "# Further information\n\n## Why do we use `@` for matrix multiplication and not `*`?\n\nWith `sympy` it is in fact possible to use the `*` operator for matrix\nmultiplication:\n\n\n```python\nimport sympy as sym\n\nmatrix = sym.Matrix([[sym.S(1) / 5, 1], [1, 1]])\nother_matrix = sym.Matrix([[sym.S(4) / 5, 0], [0, 0]])\nmatrix * other_matrix\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}\\frac{4}{25} & 0\\\\\\frac{4}{5} & 0\\end{matrix}\\right]$\n\n\n\nHowever there are other libraries that can be used for linear algebra and in\nthose libraries the `*` does not do matrix multiplication, it does element wise\nmultiplication instead. So for clarity it is preferred to use `@` throughout.\n\n## I have read that `numpy` is a library for linear algebra?\n\n`numpy` is one of the most popular and important libraries in the Python\necosystem. It is in fact the best library to use when doing linear algebra as it\nis computationally efficient, **however** it cannot handle symbolic variables\nwhich is why we are seeing how to use `Sympy` here. {ref}`numpy` gives an\nintroduction to `numpy`.\n", "meta": {"hexsha": "c33e61bdde63e18b3a4fab16fa68f5ae39891504", "size": 2252, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "book/tools-for-mathematics/04-matrices/why/.main.md.bcp.ipynb", "max_stars_repo_name": "11michalis11/pfm", "max_stars_repo_head_hexsha": "c91b1eda70d7cde3fbe065db4667f84853947850", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 8, "max_stars_repo_stars_event_min_datetime": "2020-09-24T21:02:41.000Z", "max_stars_repo_stars_event_max_datetime": "2020-10-14T08:37:21.000Z", "max_issues_repo_path": "book/tools-for-mathematics/04-matrices/why/.main.md.bcp.ipynb", "max_issues_repo_name": "11michalis11/pfm", "max_issues_repo_head_hexsha": "c91b1eda70d7cde3fbe065db4667f84853947850", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 87, "max_issues_repo_issues_event_min_datetime": "2020-09-21T15:54:23.000Z", "max_issues_repo_issues_event_max_datetime": "2021-12-19T23:26:15.000Z", "max_forks_repo_path": "book/tools-for-mathematics/04-matrices/why/.main.md.bcp.ipynb", "max_forks_repo_name": "11michalis11/pfm", "max_forks_repo_head_hexsha": "c91b1eda70d7cde3fbe065db4667f84853947850", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2020-10-02T09:21:27.000Z", "max_forks_repo_forks_event_max_datetime": "2021-07-08T14:46:27.000Z", "avg_line_length": 25.8850574713, "max_line_length": 106, "alphanum_fraction": 0.5497335702, "converted": true, "num_tokens": 289, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.7461389930307512, "lm_q1q2_score": 0.43939314303786453}} {"text": "```python\n%matplotlib inline\n```\n\n\nHooks for autograd saved tensors\n=======================\n\n\nPyTorch typically computes gradients using backpropagation. However,\ncertain operations require intermediary results to be saved in order to\nperform backpropagation. This tutorial walks through how these tensors\nare saved/retrieved and how you can define hooks to control the\npacking/unpacking process.\n\nThis tutorial assumes you are familiar with how backpropagation works in\ntheory. If not, read this first:\nhttps://colab.research.google.com/drive/1aWNdmYt7RcHMbUk-Xz2Cv5-cGFSWPXe0#scrollTo=AHcEJ6nXUb7W\n\n\n\n\nSaved tensors\n-------------\n\n\n\n\nTraining a model usually consumes more memory than running it for\ninference. Broadly speaking, one can say that it is because \u201cPyTorch\nneeds to save the computation graph, which is needed to call\n``backward``\u201d, hence the additional memory usage. One goal of this\ntutorial is to finetune this understanding.\n\nIn fact, the graph in itself sometimes does not consume much more memory\nas it never copies any tensors. However, the graph can keep *references*\nto tensors that would otherwise have gone out of scope: those are\nreferred to as **saved tensors**.\n\n\n\n\nWhy does training a model (typically) requires more memory than evaluating it?\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n\n\n\nWe start with a simple example: :math: `y = a \\mapsto \\cdot b` , for which\nwe know the gradients of $y$ with respect to $a$ and\n$b$:\n\n\\begin{align}\\frac{\\partial y}{\\partial a} = b\\end{align}\n\n\\begin{align}\\frac{\\partial y}{\\partial b} = a\\end{align}\n\n\n\n\n\n```python\nimport torch\n\na = torch.randn(5, requires_grad=True)\nb = torch.ones(5, requires_grad=True)\ny = a * b\n```\n\nUsing a torchviz, we can visualize the computation graph\n\n .. figure:: https://user-images.githubusercontent.com/8019486/130124513-72e016a3-c36f-42b9-88e2-53baf3e016c5.png\n :width: 300\n :align: center\n\n\n\nIn this example, PyTorch saves intermediary values $a$ and\n$b$ in order to compute the gradient during the backward.\n\n .. figure:: https://user-images.githubusercontent.com/8019486/130124538-3da50977-6f0b-46d0-8909-5456ade9b598.png\n :width: 300\n :align: center\n\n\n\nThose intermediary values (in orange above) can be accessed (for\ndebugging purposes) by looking for attributes of the ``grad_fn`` of\n``y`` which start with the prefix ``_saved``:\n\n\n\n\n\n```python\nprint(y.grad_fn._saved_self)\nprint(y.grad_fn._saved_other)\n```\n\nAs the computation graph grows in depth, it will store more *saved\ntensors*. Meanwhile, those tensors would have gone out of scope if not\nfor the graph.\n\n\n\n\n\n```python\ndef f(x):\n return x * x\n\nx = torch.randn(5, requires_grad=True)\ny = f(f(f(x)))\n```\n\n.. figure:: https://user-images.githubusercontent.com/8019486/130124570-f1074098-1bb3-459e-bf5a-03bf6f65b403.png\n :width: 500\n :align: center\n\n\n\nIn the example above, executing without grad would only have kept ``x``\nand ``y`` in the scope, But the graph additionnally stores ``f(x)`` and\n``f(f(x)``. Hence, running a forward pass during training will be more\ncostly in memory usage than during evaluation (more precisely, when\nautograd is not required).\n\n\n\n\nThe concept of packing / unpacking\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n\n\n\nGoing back to the first example: ``y.grad_fn._saved_self`` and\n``y.grad_fn._saved_other`` point to the original tensor object,\nrespectively ``a`` and ``b``.\n\n\n\n\n\n```python\na = torch.randn(5, requires_grad=True)\nb = torch.ones(5, requires_grad=True)\ny = a * b\n\nprint(y.grad_fn._saved_self is a) # True\nprint(y.grad_fn._saved_other is b) # True\n```\n\nHowever, that may not always be the case.\n\n\n\n\n\n```python\na = torch.randn(5, requires_grad=True)\ny = torch.exp(a)\nprint(y.grad_fn._saved_result.equal(y)) # True\nprint(y.grad_fn._saved_result is y) # False\n```\n\nUnder the hood, PyTorch has **packed** and **unpacked** the tensor\n``y`` to prevent reference cycles.\n\nAs a rule of thumb, you should *not* rely on the fact that accessing\nthe tensor saved for backward will yield the same tensor object as the\noriginal tensor. They will however share the same *storage*.\n\n\n\n\nSaved tensors hooks\n-------------------\n\n\n\n\nPyTorch provides an API to control how saved tensors should be packed /\nunpacked.\n\n\n\n\n\n```python\ndef pack_hook(x):\n print(\"Packing\", x)\n return x\n\ndef unpack_hook(x):\n print(\"Unpacking\", x)\n return x\na = torch.ones(5, requires_grad=True)\nb = torch.ones(5, requires_grad=True) * 2\n\nwith torch.autograd.graph.saved_tensors_hooks(pack_hook, unpack_hook):\n y = a * b\n\ny.sum().backward()\n```\n\nThe ``pack_hook`` function will be called everytime an operation saves\na tensor for backward.\nThe output of ``pack_hook`` is then stored in the computation graph\ninstead of the original tensor.\nThe ``unpack_hook`` uses that return value to compute a new tensor,\nwhich is the one actually used during the backward pass.\nIn general, you want ``unpack_hook(pack_hook(t))`` to be equal to\n``t``.\n\n\n\n\n\n```python\nx = torch.randn(5, requires_grad=True)\nwith torch.autograd.graph.saved_tensors_hooks(lambda x: x * 4, lambda x: x / 4):\n y = torch.pow(x, 2)\ny.sum().backward()\nassert(x.grad.equal(2 * x))\n```\n\nOne thing to note is that the output of ``pack_hook`` can be *any Python\nobject*, as long as ``unpack_hook`` can derive a tensor with the correct\nvalue from it.\n\n\n\n\nSome unconventional examples\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n\n\n\nFirst, some silly examples to illustrate what is possible but you\nprobably don\u2019t ever want to do it.\n\n\n\n\n**Returning and int**\n\n\n\n\n```python\n# Returning the index of a Python list\n# Relatively harmless but with debatable usefulness\n\nstorage = []\n\ndef pack(x):\n storage.append(x)\n return len(storage) - 1\n\ndef unpack(x):\n return storage[x]\n\nx = torch.randn(5, requires_grad=True)\nwith torch.autograd.graph.saved_tensors_hooks(pack, unpack):\n y = x * x\ny.sum().backward()\n\nassert(x.grad.equal(2 * x))\n```\n\n**Returning a tuple**\n\n\n\n\n```python\n# Returning some tensor and a function how to unpack it\n# Quite unlikely to be useful in its current form\n\ndef pack(x):\n delta = torch.randn(*x.size())\n return x - delta, lambda x: x + delta\n\ndef unpack(packed):\n x, f = packed\n return f(x)\n\n\nx = torch.randn(5, requires_grad=True)\nwith torch.autograd.graph.saved_tensors_hooks(pack, unpack):\n y = x * x\ny.sum().backward()\n\nassert(torch.allclose(x.grad, 2 * x))\n```\n\n**Returning a str**\n\n\n\n\n```python\n# Returning the __repr__ of the tensor\n# Probably never do this\n\nx = torch.randn(5, requires_grad=True)\nwith torch.autograd.graph.saved_tensors_hooks(lambda x: repr(x), lambda x: eval(\"torch.\" + x)):\n y = x * x\ny.sum().backward()\nassert(torch.all(x.grad - 2 * x <= 1e-4))\n```\n\nAlthough those examples will not be useful in practice, they\nillustrate that the output of ``pack_hook`` can really be any Python\nobject as long as it contains enough information to retrieve the\ncontent of the original tensor.\nIn the next sections, we focus on more useful applications.\n\n\n\n\nSaving tensors to CPU\n~~~~~~~~~~~~~~~~~~~~~\n\n\n\n\nVery often, the tensors involved in the computation graph live on GPU.\nKeeping a reference to those tensors in the graph is what causes most\nmodels to run out of GPU memory during training while they would have\ndone fine during evaluation.\n\nHooks provide a very simple way to implement that.\n\n\n\n\n\n```python\ndef pack_hook(x):\n return (x.device, x.cpu())\n\ndef unpack_hook(packed):\n device, tensor = packed\n return tensor.to(device)\n\nx = torch.randn(5, requires_grad=True)\nwith torch.autograd.graph.saved_tensors_hooks(pack, unpack):\n y = x * x\ny.sum().backward()\n\ntorch.allclose(x.grad, (2 * x))\n```\n\nIn fact, PyTorch provides an API to conveniently use those hooks (as\nwell as the ability to use pinned memory).\n\n\n\n\n\n```python\nimport torch.nn as nn\n\nclass Model(nn.Module):\n def __init__(self):\n super().__init__()\n self.w = nn.Parameter(torch.randn(5))\n\n def forward(self, x):\n with torch.autograd.graph.save_on_cpu(pin_memory=True):\n # some computation\n return self.w * x\n\nx = torch.randn(5)\nmodel = Model()\nloss = model(x).sum()\nloss.backward()\n```\n\nIn practice, on a A100 GPU, for a resnet-152 with batch size 256, this\ncorresponds to a GPU memory usage reduction from 48GB to 5GB, at the\ncost of a 6x slowdown.\n\nOf course, you can modulate the tradeoff by only saving to CPU certain\nparts of the network.\n\nFor instance, you could define a special ``nn.Module`` that wraps any\nmodule and saves its tensors to CPU.\n\n\n\n\n\n```python\nclass SaveToCpu(nn.Module):\n def __init__(self, module):\n super().__init__()\n self.module = module\n\n def forward(self, *args, **kwargs):\n with torch.autograd.graph.save_on_cpu(pin_memory=True):\n return self.module(*args, **kwargs)\n\nmodel = nn.Sequential(\n nn.Linear(10, 100),\n SaveToCpu(nn.Linear(100, 100)),\n nn.Linear(100, 10),\n)\n\nx = torch.randn(10)\nloss = model(x).sum()\nloss.backward()\n```\n\nSaving tensors to disk\n~~~~~~~~~~~~~~~~~~~~~~\n\n\n\n\nSimilarly, you may want to save those tensors to disk. Again, this is\nachievable with those hooks.\n\n\n\n\nA naive version would look like this.\n\n\n\n\n\n```python\n# Naive version - HINT: Don't do this\n\nimport uuid\ntmp_dir = \"temp\"\n\ndef pack_hook(tensor):\n name = os.path.join(tmp_dir, str(uuid.uuid4()))\n torch.save(tensor, name)\n return name\n\ndef unpack_hook(name):\n return torch.load(name)\n```\n\nThe reason the above code is bad is that we are leaking files on the\ndisk and they are never cleared. Fixing this is not as trivial as it\nseems.\n\n\n\n\n\n```python\n# Incorrect version - HINT: Don't do this\n\nimport uuid\nimport os\nimport tempfile\ntmp_dir_obj = tempfile.TemporaryDirectory()\ntmp_dir = tmp_dir_obj.name\n\ndef pack_hook(tensor):\n name = os.path.join(tmp_dir, str(uuid.uuid4()))\n torch.save(tensor, name)\n return name\n\ndef unpack_hook(name):\n tensor = torch.load(name)\n os.remove(name)\n return tensor\n```\n\nThe reason the above code doesn\u2019t work is that ``unpack_hook`` can be\ncalled multiple times. If we delete the file during unpacking the first\ntime, it will not be available when the saved tensor is accessed a\nsecond time, which will raise an error.\n\n\n\n\n\n```python\nx = torch.ones(5, requires_grad=True)\nwith torch.autograd.graph.saved_tensors_hooks(pack_hook, unpack_hook):\n y = x.pow(2)\nprint(y.grad_fn._saved_self)\ntry:\n print(y.grad_fn._saved_self)\n print(\"Double access succeeded!\")\nexcept:\n print(\"Double access failed!\")\n```\n\nTo fix this, we can write a version of those hooks that takes advantage\nof the fact that PyTorch automatically releases (deletes) the saved data\nwhen it is no longer needed.\n\n\n\n\n\n```python\nclass SelfDeletingTempFile():\n def __init__(self):\n self.name = os.path.join(tmp_dir, str(uuid.uuid4()))\n\n def __del__(self):\n os.remove(self.name)\n\ndef pack_hook(tensor):\n temp_file = SelfDeletingTempFile()\n torch.save(tensor, temp_file.name)\n return temp_file\n\ndef unpack_hook(temp_file):\n return torch.load(temp_file.name)\n```\n\nWhen we call ``backward``, the output of ``pack_hook`` will be deleted,\nwhich causes the file to be removed, so we\u2019re no longer leaking the\nfiles.\n\nThis can then be used in your model, in the following way:\n\n\n\n\n\n```python\n# Only save on disk tensors that have size >= 1000\nSAVE_ON_DISK_THRESHOLD = 1000\n\ndef pack_hook(x):\n if x.numel() < SAVE_ON_DISK_THRESHOLD:\n return x\n temp_file = SelfDeletingTempFile()\n torch.save(tensor, temp_file.name)\n return temp_file\n\ndef unpack_hook(tensor_or_sctf):\n if isinstance(tensor_or_sctf, torch.Tensor):\n return tensor_or_sctf\n return torch.load(tensor_or_sctf.name)\n\nclass SaveToDisk(nn.Module):\n def __init__(self, module):\n super().__init__()\n self.module = module\n\n def forward(self, *args, **kwargs):\n with torch.autograd.graph.saved_tensors_hooks(pack_hook, unpack_hook):\n return self.module(*args, **kwargs)\n\nnet = nn.DataParallel(SaveToDisk(Model()))\n```\n\nIn this last example, we also demonstrate how to filter which tensors\nshould be saved (here, those whose number of elements is greater than\n1000) and how to combine this feature with ``nn.DataParallel``.\n\n\n\n\nIf you\u2019ve made it this far, congratulations! You now know how to use\nsaved tensor hooks and how they can be useful in a few scenarios to\ntradeoff memory for compute.\n\n\n\n", "meta": {"hexsha": "71291477279f43c71b5e7cb046ea308a80cedfec", "size": 20486, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/_downloads/d16174e25728fee27ada4c1dd3e37e0a/autograd_saved_tensors_hooks_tutorial.ipynb", "max_stars_repo_name": "spongebob03/PyTorch-tutorials-kr", "max_stars_repo_head_hexsha": "efe11ebede0d3384aacd1bdad5881ea8794223c8", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 44, "max_stars_repo_stars_event_min_datetime": "2021-12-07T14:51:04.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-30T10:34:17.000Z", "max_issues_repo_path": "docs/_downloads/d16174e25728fee27ada4c1dd3e37e0a/autograd_saved_tensors_hooks_tutorial.ipynb", "max_issues_repo_name": "spongebob03/PyTorch-tutorials-kr", "max_issues_repo_head_hexsha": "efe11ebede0d3384aacd1bdad5881ea8794223c8", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 128, "max_issues_repo_issues_event_min_datetime": "2021-12-02T18:11:58.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-27T05:16:39.000Z", "max_forks_repo_path": "docs/_downloads/d16174e25728fee27ada4c1dd3e37e0a/autograd_saved_tensors_hooks_tutorial.ipynb", "max_forks_repo_name": "spongebob03/PyTorch-tutorials-kr", "max_forks_repo_head_hexsha": "efe11ebede0d3384aacd1bdad5881ea8794223c8", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 14, "max_forks_repo_forks_event_min_datetime": "2021-12-02T18:56:13.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-18T07:18:23.000Z", "avg_line_length": 40.4063116371, "max_line_length": 778, "alphanum_fraction": 0.5740505711, "converted": true, "num_tokens": 3102, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891163376236, "lm_q2_score": 0.7461389930307512, "lm_q1q2_score": 0.43939313227092336}} {"text": "```python\nfrom IPython.core.display import HTML, Image\ncss_file = 'style.css'\nHTML(open(css_file, 'r').read())\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n```python\nfrom sympy import init_printing, Matrix, symbols, sqrt, Rational\nfrom warnings import filterwarnings\n```\n\n\n```python\ninit_printing(use_latex = 'mathjax')\nfilterwarnings('ignore')\n```\n\n+ This chapter starts with explanations using sympy\n+ A proper method using numpy and sympy is described in the example problem at the end\n\n# Singular value decomposition (SVD)\n\n## Derivation\n\n* This is the final form of matrix factorization\n* The factors are an orthogonal matrix A, a diagonal matrix Σ, and an orthogonal matrix V\n$$ {A}={U}{\\Sigma}{V}^{T} $$\n\n+ In case the matrix A is symmetric positive definite, the decomposition is akin to the following\n$$ {A}={Q}{\\Lambda}{Q}^{T} $$\n\n* Consider a vector *v*1 in ℝn row space, transformed into a vector *u*1 in ℝm column space by the matrix A\n$$ {u}_{1}={A}{v}_{1} $$\n\n* What we are looking for is an orthogonal basis in ℝn row space, transformed into an orthogonal basis in ℝm column space\n$$ {u}_{1}={A}{v}_{1} \\\\ { v }_{ 1 }\\bot { v }_{ 2 };{ u }_{ 1 }\\bot { u }_{ 2 }$$\n\n* It's easy to calculate an orthogonal basis in the row space using Gram-Schmidt\n* Now, though, we need something special in A that would ensure that the basis *u*i in in ℝm column space is also orthogonal (and at the same time make it orthonormal, so that *v*i ends up as σi*u*i)\n* The two nullspaces are not required\n* So, we are looking for the following\n$$ A\\begin{bmatrix} \\vdots & \\vdots & \\vdots & \\vdots \\\\ { v }_{ 1 } & { v }_{ 2 } & \\cdots & { v }_{ r } \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\end{bmatrix}=\\begin{bmatrix} \\vdots & \\vdots & \\vdots & \\vdots \\\\ { u }_{ 1 } & u_{ 2 } & \\cdots & { u }_{ r } \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\end{bmatrix}\\begin{bmatrix} { \\sigma }_{ 1 } & \\quad & \\quad & \\quad & \\quad \\\\ \\quad & { \\sigma }_{ 2 } & \\quad & \\quad & \\quad \\\\ \\quad & \\quad & \\quad \\ddots & \\quad & \\quad \\\\ \\quad & \\quad & \\quad & { \\sigma }_{ r }\\quad & \\quad \\\\ \\quad & \\quad & \\quad & \\quad & \\left( 0 \\right) \\end{bmatrix} \\\\ {A}{V}={U}{\\Sigma}$$\n\n+ In the case that we are not changing spaces V and U would be the same matrix Q (and then Q-1)\n\n### Example problem explaining the derivation\n\n+ Look at the next matrix A that is square and invertible (i.e. rank 2)\n\n\n```python\nA = Matrix([[4, 4], [-3, 3]])\nA\n```\n\n\n\n\n$$\\left[\\begin{matrix}4 & 4\\\\-3 & 3\\end{matrix}\\right]$$\n\n\n\n* We are looking for *v*1 and *v*2 in the ℝ2 rowspace and *u*1 and *u*2 in the ℝ2 columnspace, as well as the scaling factors *σ*1>0 and *σ*2>0\n\n+ Just to be complete, we extend V until *v*n with zero columns and U with zero columns until *u*m, as well as zeros for Σ to include the nullspaces\n\n+ Now A is not symmetric so that their eigenvectors are not orthogonal (Q), so we can't go that route\n\n+ From above we have the following and because V is square and orthogonal we have\n$$ {A}={U}{\\Sigma}{V}^{-1} \\\\ {A}={U}{\\Sigma}{V}^{T} $$\n\n+ Multiplying both sides by AT we will have a left-hand side that is square and definte (semi)definte\n$$ A=U\\Sigma { V }^{ T }\\\\ { A }^{ T }A=V{ \\Sigma }^{ T }{ U }^{ T }U\\Sigma { V }^{ T }\\\\ \\because \\quad { U }^{ T }U=I\\\\ \\because \\quad { \\Sigma }^{ T }\\Sigma =\\dots { \\sigma }_{ i }^{ 2 }\\dots \\\\ { A }^{ T }A=V{ \\Sigma }^{ T }\\Sigma { V }^{ T } $$\n\n+ Because ATA is now definite (semi)positive, we have a perfect situation akin to being able to use QΛQT\n+ The eigenvalues are the squares of the σi values\n+ To get U we use AAT and use its eigenvalues and eigenvectors\n\n* All of this is easy to accomplish with the mpmath submodule of sympy\n\n\n```python\nfrom sympy.mpmath import svd\n```\n\n\n```python\nU, S, V = svd(A)\n```\n\n\n```python\nU # The numbers round to zero!!! Please see it as zero\n```\n\n\n\n\n$$\\left[\\begin{matrix}1.0 & 3.33066907387547 \\cdot 10^{-16}\\\\1.11022302462516 \\cdot 10^{-16} & -1.0\\end{matrix}\\right]$$\n\n\n\n\n```python\nS # Not the final Sigma matrix\n```\n\n\n\n\n matrix(\n [['5.65685424949238'],\n ['4.24264068711928']])\n\n\n\n\n```python\nV\n```\n\n\n\n\n matrix(\n [['0.707106781186547', '0.707106781186548'],\n ['0.707106781186548', '-0.707106781186547']])\n\n\n\n+ There are square roots, so the values are given instead of symbols\n\n* Now let's do it step-by-step\n\n\n```python\nA.transpose() * A\n```\n\n\n\n\n$$\\left[\\begin{matrix}25 & 7\\\\7 & 25\\end{matrix}\\right]$$\n\n\n\n\n```python\n(A.transpose() * A).eigenvals()\n```\n\n\n\n\n$$\\begin{Bmatrix}18 : 1, & 32 : 1\\end{Bmatrix}$$\n\n\n\n\n```python\n(A.transpose() * A).eigenvects()\n```\n\n\n\n\n$$\\begin{bmatrix}\\begin{pmatrix}18, & 1, & \\begin{bmatrix}\\left[\\begin{matrix}-1\\\\1\\end{matrix}\\right]\\end{bmatrix}\\end{pmatrix}, & \\begin{pmatrix}32, & 1, & \\begin{bmatrix}\\left[\\begin{matrix}1\\\\1\\end{matrix}\\right]\\end{bmatrix}\\end{pmatrix}\\end{bmatrix}$$\n\n\n\n* These are not normalized, though\n* Also remember to take the square roots of the eigenvalues\n* ... and to add zeros to incorporate the correct size for *m* and *n*\n* ... and to take the transpose\n\n* Now let's tackle U\n\n\n```python\nA * A.transpose()\n```\n\n\n\n\n$$\\left[\\begin{matrix}32 & 0\\\\0 & 18\\end{matrix}\\right]$$\n\n\n\n\n```python\n(A * A.transpose()).eigenvals()\n```\n\n\n\n\n$$\\begin{Bmatrix}18 : 1, & 32 : 1\\end{Bmatrix}$$\n\n\n\n* The eigenvalues are always the same\n\n\n```python\n(A * A.transpose()).eigenvects()\n```\n\n\n\n\n$$\\begin{bmatrix}\\begin{pmatrix}18, & 1, & \\begin{bmatrix}\\left[\\begin{matrix}0\\\\1\\end{matrix}\\right]\\end{bmatrix}\\end{pmatrix}, & \\begin{pmatrix}32, & 1, & \\begin{bmatrix}\\left[\\begin{matrix}1\\\\0\\end{matrix}\\right]\\end{bmatrix}\\end{pmatrix}\\end{bmatrix}$$\n\n\n\n* Also remember to normalize (see example problem below)\n\n* We now have U, Σ and VT (although Σ must still be constructed; see below)\n\n### Example problem to explain the derivation for dependent rows, columns\n\n+ Let's consider this rank=1, 2×2 singular matrix\n+ The rowspace is just a line (the second row is a constant multiple of the first)\n+ The nullspace of this row picture is a line perpendicular to this\n+ The columnspace is also on a line, with the nullspace of AT being a line perpendicular to this\n\n\n```python\nA = Matrix([[4, 3], [8, 6]])\nA\n```\n\n\n\n\n$$\\left[\\begin{matrix}4 & 3\\\\8 & 6\\end{matrix}\\right]$$\n\n\n\n+ Let's use *svd*() first\n\n\n```python\nU, S, V = svd(A, full_matrices = True, compute_uv = True)\n```\n\n* This is likely to be different to the value you calculate for U\n* We are talking unit basis vectors, though, which can be in a different direction depending on your choice\n\n\n```python\nU\n```\n\n\n\n\n$$\\left[\\begin{matrix}-0.447213595499958 & -0.894427190999916\\\\-0.894427190999916 & 0.447213595499958\\end{matrix}\\right]$$\n\n\n\n\n```python\nS\n```\n\n\n\n\n matrix(\n [['11.1803398874989'],\n ['0.0']])\n\n\n\n+ Note that the size of our Σ matrix is wrong\n+ It has to be 2×2 and we have to create it from this info\n+ Since A has rank = 1 and all off-diagonal entries must be zero, we will only have a value in the first row, first column position\n+ Below I show you how to correct this\n\n\n```python\nV\n```\n\n\n\n\n matrix(\n [['-0.8', '-0.6'],\n ['-0.6', '0.8']])\n\n\n\n\n```python\nA.transpose() * A # Which will be symmetric positive definite and of rank = 1\n```\n\n\n\n\n$$\\left[\\begin{matrix}80 & 60\\\\60 & 45\\end{matrix}\\right]$$\n\n\n\n\n```python\n(A.transpose() * A).eigenvals() # One eigenvalue will be zero and the other must then be the trace\n```\n\n\n\n\n$$\\begin{Bmatrix}0 : 1, & 125 : 1\\end{Bmatrix}$$\n\n\n\n\n```python\n(A.transpose() * A).eigenvects()\n```\n\n\n\n\n$$\\begin{bmatrix}\\begin{pmatrix}0, & 1, & \\begin{bmatrix}\\left[\\begin{matrix}- \\frac{3}{4}\\\\1\\end{matrix}\\right]\\end{bmatrix}\\end{pmatrix}, & \\begin{pmatrix}125, & 1, & \\begin{bmatrix}\\left[\\begin{matrix}\\frac{4}{3}\\\\1\\end{matrix}\\right]\\end{bmatrix}\\end{pmatrix}\\end{bmatrix}$$\n\n\n\n\n```python\nA * A.transpose()\n```\n\n\n\n\n$$\\left[\\begin{matrix}25 & 50\\\\50 & 100\\end{matrix}\\right]$$\n\n\n\n\n```python\n(A * A.transpose()).eigenvals()\n```\n\n\n\n\n$$\\begin{Bmatrix}0 : 1, & 125 : 1\\end{Bmatrix}$$\n\n\n\n\n```python\n(A * A.transpose()).eigenvects()\n```\n\n\n\n\n$$\\begin{bmatrix}\\begin{pmatrix}0, & 1, & \\begin{bmatrix}\\left[\\begin{matrix}-2\\\\1\\end{matrix}\\right]\\end{bmatrix}\\end{pmatrix}, & \\begin{pmatrix}125, & 1, & \\begin{bmatrix}\\left[\\begin{matrix}\\frac{1}{2}\\\\1\\end{matrix}\\right]\\end{bmatrix}\\end{pmatrix}\\end{bmatrix}$$\n\n\n\n+ Inserted below is the three resultant matrices from our calculations above (normalized, etc)\n\n\n```python\nMatrix([[1 / sqrt(5), 2 / sqrt(5)], [2 / sqrt(5), 1 / sqrt(5)]]), Matrix([[sqrt(125), 0], [0, 0]]), Matrix([[0.8, 0.6], [0.6, -0.8]])\n```\n\n\n\n\n$$\\begin{pmatrix}\\left[\\begin{matrix}\\frac{\\sqrt{5}}{5} & \\frac{2 \\sqrt{5}}{5}\\\\\\frac{2 \\sqrt{5}}{5} & \\frac{\\sqrt{5}}{5}\\end{matrix}\\right], & \\left[\\begin{matrix}5 \\sqrt{5} & 0\\\\0 & 0\\end{matrix}\\right], & \\left[\\begin{matrix}0.8 & 0.6\\\\0.6 & -0.8\\end{matrix}\\right]\\end{pmatrix}$$\n\n\n\n+ Now let me show you how to correct the *svd*() solutions\n\n\n```python\nU\n```\n\n\n\n\n$$\\left[\\begin{matrix}-0.447213595499958 & -0.894427190999916\\\\-0.894427190999916 & 0.447213595499958\\end{matrix}\\right]$$\n\n\n\n\n```python\nS\n```\n\n\n\n\n matrix(\n [['11.1803398874989'],\n ['0.0']])\n\n\n\n\n```python\nS = Matrix([[11.1803398874989, 0], [0, 0]])\nS # Composed by hand (proper method further below)\n```\n\n\n\n\n$$\\left[\\begin{matrix}11.1803398874989 & 0\\\\0 & 0\\end{matrix}\\right]$$\n\n\n\n\n```python\nV\n```\n\n\n\n\n matrix(\n [['-0.8', '-0.6'],\n ['-0.6', '0.8']])\n\n\n\n\n```python\nV = Matrix([[-0.8, -0.6], [-0.6, 0.8]])\nV # Remember that this is actually V transpose\n```\n\n\n\n\n$$\\left[\\begin{matrix}-0.8 & -0.6\\\\-0.6 & 0.8\\end{matrix}\\right]$$\n\n\n\n+ Let's calculate AΣVT\n\n\n```python\nU * S * V\n```\n\n\n\n\n$$\\left[\\begin{matrix}3.99999999999998 & 2.99999999999999\\\\7.99999999999996 & 5.99999999999997\\end{matrix}\\right]$$\n\n\n\n+ Compensating for rounding, this is the original matrix A\n\n## Summary\n\n+ The orthonormal basis for the rowspace is\n$$ {v}_{1},{v}_{2},\\dots,{v}_{r} $$\n+ The orthonormal basis for the columnspace is\n$$ {u}_{1},{u}_{2},\\dots,{u}_{r} $$\n+ The orthonormal basis for the nullspace is\n$$ {v}_{r+1},{v}_{r+2},\\dots,{v}_{n} $$\n+ The orthonormal basis for the nullspace of AT\n$$ {u}_{r+1},{u}_{r+2},\\dots,{u}_{m} $$\n\n## Example problem\n\n### Example problem 1\n\n* Find the singular value decomposition of the matrix\n$$ \\begin{bmatrix}5&5\\\\-1&7\\end{bmatrix} $$\n\n#### Solution\n\n+ First off, I'll show you how to make proper use of numpy and scipy (as opposed to sympy) to solve singular value decomposition problems\n\n\n```python\nfrom numpy import matrix, transpose # Importing the matrix object and the \n# transpose object from numerical python (numpy)\nfrom numpy.linalg import svd, det # Importing the svd and determinant\n# methods from the linalg submodule \nfrom scipy.linalg import diagsvd\n```\n\n\n```python\ntype(transpose) # Type tells us what 'something' is (sometimes)\n```\n\n\n\n\n function\n\n\n\n\n```python\nC = matrix([[5, 5], [-1, 7]]) # Using the numpy matrix object\nC\n```\n\n\n\n\n matrix([[ 5, 5],\n [-1, 7]])\n\n\n\n* We can see from the determinant that the rows and columns are independent\n\n\n```python\ndet(C) # Notice the difference in syntax\n```\n\n\n\n\n$$40.0$$\n\n\n\n+ Let's calculate U by looking at ATA\n\n\n```python\ntranspose(C) *C # Notice the difference in synmtax\n```\n\n\n\n\n matrix([[26, 18],\n [18, 74]])\n\n\n\n+ This is symmetric, positive definite\n+ One eigenvalue will be 0 and the other, the trace (since they (the eigenvalues) must sum to the trace)\n+ Remember that the eigenvalues are the squares of the *σ*i values\n\n+ Now let's put numpy and sympy to good use\n\n\n```python\nU, S, VT = svd(C) # I use the computer variable VT to remind us that\n# this is the transpose of V\n```\n\n+ S will only indicate the eigenvalues and must be converted to the correct sized matrix\n\n\n```python\nM, N = C.shape # Shape returns a tuple (two values), indicating\n# row and column size\nM, N\n```\n\n\n\n\n$$\\begin{pmatrix}2, & 2\\end{pmatrix}$$\n\n\n\n\n```python\nSig = diagsvd(S, M, N) # Creating a m times n matrix from S\nSig\n```\n\n\n\n\n array([[ 8.94427191, 0. ],\n [ 0. , 4.47213595]])\n\n\n\n\n```python\nVT\n```\n\n\n\n\n matrix([[ 0.31622777, 0.9486833 ],\n [ 0.9486833 , -0.31622777]])\n\n\n\n+ Let's check if it worked!\n\n\n```python\nU * Sig * VT\n```\n\n\n\n\n matrix([[ 5., 5.],\n [-1., 7.]])\n\n\n\nNow, let's use good old sympy\n\n\n```python\nC = Matrix([[5, 5], [-1, 7]])\nC\n```\n\n\n\n\n$$\\left[\\begin{matrix}5 & 5\\\\-1 & 7\\end{matrix}\\right]$$\n\n\n\n+ We need to work with a positive (semi)definite matrix\n\n\n```python\nCTC = C.transpose() * C # Using the computer variable CTC to remind that\n# it is C transpose times C\nCTC, CTC.det()\n```\n\n\n\n\n$$\\begin{pmatrix}\\left[\\begin{matrix}26 & 18\\\\18 & 74\\end{matrix}\\right], & 1600\\end{pmatrix}$$\n\n\n\n+ Let's look at the eigenvalues\n\n\n```python\nCTC.eigenvals()\n```\n\n\n\n\n$$\\begin{Bmatrix}20 : 1, & 80 : 1\\end{Bmatrix}$$\n\n\n\n+ Σ will contain along its main diagonal the square root of these eigenvalues\n\n\n```python\nSig = Matrix([[sqrt(20), 0], [0, sqrt(80)]])\nSig\n```\n\n\n\n\n$$\\left[\\begin{matrix}2 \\sqrt{5} & 0\\\\0 & 4 \\sqrt{5}\\end{matrix}\\right]$$\n\n\n\n+ For V we require the eigenvectors of CTC\n+ We need to remember to normalize each vector (dividing each component by the length (norm) of that vector\n\n\n```python\nCTC.eigenvects()\n```\n\n\n\n\n$$\\begin{bmatrix}\\begin{pmatrix}20, & 1, & \\begin{bmatrix}\\left[\\begin{matrix}-3\\\\1\\end{matrix}\\right]\\end{bmatrix}\\end{pmatrix}, & \\begin{pmatrix}80, & 1, & \\begin{bmatrix}\\left[\\begin{matrix}\\frac{1}{3}\\\\1\\end{matrix}\\right]\\end{bmatrix}\\end{pmatrix}\\end{bmatrix}$$\n\n\n\n+ Let's normalize each *v*i by calculating the length (norm) of each\n\n\n```python\nv1 = Matrix([-3, 1])\nv1.norm()\n```\n\n\n\n\n$$\\sqrt{10}$$\n\n\n\n\n```python\nv2 = Matrix([Rational(1, 3), 1])\nv2.norm()\n```\n\n\n\n\n$$\\frac{\\sqrt{10}}{3}$$\n\n\n\n+ We'll get each element of V by dividing by these norms\n\n\n```python\n-3 / v1.norm(), 1 / v1.norm()\n```\n\n\n\n\n$$\\begin{pmatrix}- \\frac{3 \\sqrt{10}}{10}, & \\frac{\\sqrt{10}}{10}\\end{pmatrix}$$\n\n\n\n\n```python\nRational(1, 3) / v2.norm(), 1 / v2.norm()\n```\n\n\n\n\n$$\\begin{pmatrix}\\frac{\\sqrt{10}}{10}, & \\frac{3 \\sqrt{10}}{10}\\end{pmatrix}$$\n\n\n\n\n```python\nV = Matrix([[-3 / sqrt(10), 1 / sqrt(10)], [1 / sqrt(10), 3 / sqrt(10)]])\n# Just remember to put the elements of V in the correct place\nV\n```\n\n\n\n\n$$\\left[\\begin{matrix}- \\frac{3 \\sqrt{10}}{10} & \\frac{\\sqrt{10}}{10}\\\\\\frac{\\sqrt{10}}{10} & \\frac{3 \\sqrt{10}}{10}\\end{matrix}\\right]$$\n\n\n\n+ Remember that it is equal to the transpose of V\n\n\n```python\nV == V.transpose()\n```\n\n\n\n\n True\n\n\n\n+ Now for U using CCT\n\n\n```python\nCCT = C * C.transpose() # Using the computer variable CCT\nCCT\n```\n\n\n\n\n$$\\left[\\begin{matrix}50 & 30\\\\30 & 50\\end{matrix}\\right]$$\n\n\n\n+ The eigenvalues will be the same\n\n\n```python\nCCT.eigenvals()\n```\n\n\n\n\n$$\\begin{Bmatrix}20 : 1, & 80 : 1\\end{Bmatrix}$$\n\n\n\n\n```python\nCCT.eigenvects()\n```\n\n\n\n\n$$\\begin{bmatrix}\\begin{pmatrix}20, & 1, & \\begin{bmatrix}\\left[\\begin{matrix}-1\\\\1\\end{matrix}\\right]\\end{bmatrix}\\end{pmatrix}, & \\begin{pmatrix}80, & 1, & \\begin{bmatrix}\\left[\\begin{matrix}1\\\\1\\end{matrix}\\right]\\end{bmatrix}\\end{pmatrix}\\end{bmatrix}$$\n\n\n\n\n```python\nu1 = Matrix([-1, 1])\nu2 = Matrix([1, 1])\n```\n\n\n```python\n-1 / u1.norm(), 1 / u1.norm()\n```\n\n\n\n\n$$\\begin{pmatrix}- \\frac{\\sqrt{2}}{2}, & \\frac{\\sqrt{2}}{2}\\end{pmatrix}$$\n\n\n\n\n```python\n1 / u2.norm(), 1 / u2.norm()\n```\n\n\n\n\n$$\\begin{pmatrix}\\frac{\\sqrt{2}}{2}, & \\frac{\\sqrt{2}}{2}\\end{pmatrix}$$\n\n\n\n\n```python\nU = Matrix([[-sqrt(2) / 2, sqrt(2) / 2], [sqrt(2) / 2, sqrt(2) / 2]])\n# Just remember to put the elements of U in the correct place\nU\n```\n\n\n\n\n$$\\left[\\begin{matrix}- \\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$$\n\n\n\n+ Let's see if it worked!\n\n\n```python\nU * Sig * V\n```\n\n\n\n\n$$\\left[\\begin{matrix}5 & 5\\\\-1 & 7\\end{matrix}\\right]$$\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "283845d2bd93d8c636f2684eaaa3e1e07a1ce273", "size": 46492, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Mathematics/Linear Algebra/0.0 MIT-18.06 - Jupyter/Lecture_29_Singular_value_decomposition.ipynb", "max_stars_repo_name": "okara83/Becoming-a-Data-Scientist", "max_stars_repo_head_hexsha": "f09a15f7f239b96b77a2f080c403b2f3e95c9650", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Mathematics/Linear Algebra/0.0 MIT-18.06 - Jupyter/Lecture_29_Singular_value_decomposition.ipynb", "max_issues_repo_name": "okara83/Becoming-a-Data-Scientist", "max_issues_repo_head_hexsha": "f09a15f7f239b96b77a2f080c403b2f3e95c9650", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Mathematics/Linear Algebra/0.0 MIT-18.06 - Jupyter/Lecture_29_Singular_value_decomposition.ipynb", "max_forks_repo_name": "okara83/Becoming-a-Data-Scientist", "max_forks_repo_head_hexsha": "f09a15f7f239b96b77a2f080c403b2f3e95c9650", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2022-02-09T15:41:33.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-11T07:47:40.000Z", "avg_line_length": 23.5164390491, "max_line_length": 737, "alphanum_fraction": 0.4527660673, "converted": true, "num_tokens": 6253, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858117, "lm_q2_score": 0.7606506526772884, "lm_q1q2_score": 0.43927222644669667}} {"text": "# 13 Euler-Maclaurin\u306e\u548c\u516c\u5f0f\n\n\u9ed2\u6728\u7384\n\n2018-07-04\n\n* Copyright 2018 Gen Kuroki\n* License: MIT https://opensource.org/licenses/MIT\n* Repository: https://github.com/genkuroki/Calculus\n\n\u3053\u306e\u30d5\u30a1\u30a4\u30eb\u306f\u6b21\u306e\u5834\u6240\u3067\u304d\u308c\u3044\u306b\u95b2\u89a7\u3067\u304d\u308b:\n\n* http://nbviewer.jupyter.org/github/genkuroki/Calculus/blob/master/13%20Euler-Maclaurin%20summation%20formula.ipynb\n\n* https://genkuroki.github.io/documents/Calculus/13%20Euler-Maclaurin%20summation%20formula.pdf\n\n\u3053\u306e\u30d5\u30a1\u30a4\u30eb\u306f Julia Box \u3067\u5229\u7528\u3067\u304d\u308b.\n\n\u81ea\u5206\u306e\u30d1\u30bd\u30b3\u30f3\u306bJulia\u8a00\u8a9e\u3092\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u3057\u305f\u3044\u5834\u5408\u306b\u306f\n\n* Windows\u3078\u306eJulia\u8a00\u8a9e\u306e\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\n\n\u3092\u53c2\u7167\u305b\u3088.\n\n\u8ad6\u7406\u7684\u306b\u5b8c\u74a7\u306a\u8aac\u660e\u3092\u3059\u308b\u3064\u3082\u308a\u306f\u306a\u3044. \u7d30\u90e8\u306e\u3044\u3044\u52a0\u6e1b\u306a\u90e8\u5206\u306f\u81ea\u5206\u3067\u8a02\u6b63\u30fb\u4fee\u6b63\u305b\u3088.\n\n$\n\\newcommand\\eps{\\varepsilon}\n\\newcommand\\ds{\\displaystyle}\n\\newcommand\\Z{{\\mathbb Z}}\n\\newcommand\\R{{\\mathbb R}}\n\\newcommand\\C{{\\mathbb C}}\n\\newcommand\\QED{\\text{\u25a1}}\n\\newcommand\\root{\\sqrt}\n\\newcommand\\bra{\\langle}\n\\newcommand\\ket{\\rangle}\n\\newcommand\\d{\\partial}\n\\newcommand\\sech{\\operatorname{sech}}\n\\newcommand\\cosec{\\operatorname{cosec}}\n\\newcommand\\sign{\\operatorname{sign}}\n\\newcommand\\real{\\operatorname{Re}}\n\\newcommand\\imag{\\operatorname{Im}}\n$\n\n

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    \n\n\n```julia\nusing Plots\ngr(); ENV[\"PLOTS_TEST\"] = \"true\"\n#clibrary(:colorcet)\nclibrary(:misc)\n\nfunction pngplot(P...; kwargs...)\n sleep(0.1)\n pngfile = tempname() * \".png\"\n savefig(plot(P...; kwargs...), pngfile)\n showimg(\"image/png\", pngfile)\nend\npngplot(; kwargs...) = pngplot(plot!(; kwargs...))\n\nshowimg(mime, fn) = open(fn) do f\n base64 = base64encode(f)\n display(\"text/html\", \"\"\"\"\"\")\nend\n\nusing SymPy\n#sympy[:init_printing](order=\"lex\") # default\n#sympy[:init_printing](order=\"rev-lex\")\n\nusing SpecialFunctions\nusing QuadGK\n```\n\n## Bernoulli\u591a\u9805\u5f0f\n\n### Bernoulli\u591a\u9805\u5f0f\u306e\u5b9a\u7fa9\n\n**\u5b9a\u7fa9(Bernoulli\u591a\u9805\u5f0f):** Bernoulli\u591a\u9805\u5f0f** $B_n(x)$ ($n=0,1,2,\\ldots$)\u3092\n\n$$\n\\frac{ze^{zx}}{e^z-1} = \\sum_{n=0}^\\infty \\frac{B_n(x)}{n!}z^n\n$$\n\n\u306b\u3088\u3063\u3066\u5b9a\u7fa9\u3059\u308b. $\\QED$\n\n\n\n### Bernoulli\u591a\u9805\u5f0f\u306e\u57fa\u672c\u6027\u8cea\n\n**\u4e00\u822c\u5316Bernoulli\u591a\u9805\u5f0f\u306e\u57fa\u672c\u6027\u8cea:** Bernoulli\u591a\u9805\u5f0f $B_n(x)$ \u306f\u4ee5\u4e0b\u306e\u6027\u8cea\u3092\u6e80\u305f\u3057\u3066\u3044\u308b:\n\n(1) $B_0(x)=1$.\n\n(2) $\\ds\\int_0^1 B_n(x)\\,dx = \\delta_{n,0}$.\n\n(3) $\\ds B_n(x+h) = \\sum_{k=0}^n\\binom{n}{k}B_{n-k}(x)h^k = \n\\sum_{k=0}^n \\binom{n}{k} B_k(x) h^{n-k}$.\n\n(4) $B_n'(x)=nB_{n-1}(x)$.\n\n(5) $\\ds B_n(x+1)=B_n(x)+nx^{n-1}$.\n\n(6) $B_n(1-x)=(-1)^n B_n(x)$.\n\n(7) $B_n(1)=B_n(0)+\\delta_{n,1}$ \u3068\u306a\u308b.\n\n(8) $B_n(0)=1$, $\\ds B_n(0)=-\\frac{1}{2}$ \u3068\u306a, $n$ \u304c3\u4ee5\u4e0a\u306e\u5947\u6570\u306a\u3089\u3070 $B_n(0)=0$ \u3068\u306a\u308b.\n\n**\u8a3c\u660e:** (1) $e^{zx}=1+O(z)$, $\\ds\\frac{e^z-1}{z}=1+O(z)$ \u3088\u308a, $\\ds\\frac{ze^{zx}}{e^z-1}=1+O(z)$ \u306a\u306e\u3067 $B_0(x) = 1$.\n\n(2)\u3092\u793a\u305d\u3046.\n\n$$\n\\begin{aligned}\n&\n\\int_0^1 \\frac{ze^{zx}}{e^z-1}\\,dx = \\frac{z}{e^z-1}\\int_0^1 e^{zx}\\,dx = \n\\frac{z}{e^z-1}\\frac{e^z-1}{z} = 1, \n\\\\ &\n\\int_0^1\\frac{ze^{zx}}{e^z-1}\\,dx = \\sum_{n=0}^\\infty\\frac{z^n}{n!}\\int_0^1 B_n(x)\\,dx\n\\end{aligned}\n$$\n\n\u306a\u306e\u3067, \u3053\u308c\u3089\u3092\u6bd4\u8f03\u3057\u3066 $\\ds\\int_0^1 B_n(x)\\,dx = \\delta_{n,0}$.\n\n(3) \u4e8c\u9805\u5b9a\u7406\u3088\u308a,\n\n$$\n\\int_0^1 (x+y)^n\\,dy = \n\\sum_{k=0}^n \\binom{n}{k} x^{n-k} \\int_0^1 y^k\\,dy.\n$$\n\n\u3086\u3048\u306b, $x$ \u306e\u51fd\u6570\u3092 $x$ \u306e\u51fd\u6570\u306b\u79fb\u3059\u7dda\u5f62\u5199\u50cf(\u524d\u65b9\u79fb\u52d5\u5e73\u5747)\n\n$$\nf(x)\\mapsto \\int_0^1 f(x+y)\\,dy\n$$\n\n\u306f\u591a\u9805\u5f0f\u3092\u591a\u9805\u5f0f\u306b\u79fb\u3057, \u6700\u9ad8\u6b21\u306e\u4fc2\u6570\u304c1\u306e\u591a\u9805\u5f0f\u3092\u6700\u9ad8\u6b21\u306e\u4fc2\u6570\u304c1\u306e\u540c\u6b21\u306e\u591a\u9805\u5f0f\u306b\u79fb\u3059. \u3053\u308c\u3088\u308a, \u7dda\u5f62\u5199\u50cf $\\ds f(x)\\mapsto \\int_0^1 f(x+y)\\,dy$ \u306f\u591a\u9805\u5f0f\u3069\u3046\u3057\u306e\u4e00\u5bfe\u4e00\u5bfe\u5fdc\u3092\u4e0e\u3048\u308b\u7dda\u5f62\u5199\u50cf\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u305d\u3057\u3066,\n\n$$\n\\begin{aligned}\n&\n\\int_0^1\\frac{ze^{z(x+y)}}{e^z-1}\\,dx = \n\\sum_{n=0}^\\infty\\frac{\\int_0^1 B_n(x+y)\\,dy}{n!}z^n, \n\\\\ &\n\\int_0^1\\frac{ze^{z(x+y)}}{e^z-1}\\,dx = \n\\frac{ze^{zx}}{e^z-1}\\int_0^1 e^{zy}\\,dy =\n\\frac{ze^{zx}}{e^z-1}\\frac{e^z-1}{z} =\ne^{zx} =\n\\sum_{n=0}^\\infty \\frac{x^n}{n!}z^n\n\\end{aligned}\n$$\n\n\u306a\u306e\u3067, \u3053\u308c\u3089\u3092\u6bd4\u8f03\u3057\u3066,\n\n$$\n\\int_0^1 B_n(x+y)\\,dy = x^n\n$$\n\n\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u3086\u3048\u306b, \n\n$$\n\\int_0^1 B_n(x+h+y)\\,dy = (x+h)^n = \\sum_{k=0}^n \\binom{n}{k}x^{n-k}h^k =\n\\int_0^1 \\sum_{k=0}^n \\binom{n}{k}B_{n-k}(x+y)h^k \\,dy\n$$\n\n\u3088\u308a\n\n$$\nB_n(x+h) = \\sum_{k=0}^n \\binom{n}{k}B_{n-k}(x)h^k.\n$$\n\n(4) \u3059\u3050\u4e0a\u306e\u7b49\u5f0f\u306e\u53f3\u8fba\u306e $h$ \u306e\u4fc2\u6570\u3092\u898b\u308b\u3053\u3068\u306b\u3088\u3063\u3066,\n\n$$\nB_n'(x) = n B_{n-1}(x).\n$$\n\n(5) Bernoulli\u591a\u9805\u5f0f\u306e\u6bcd\u51fd\u6570\u306e $x$ \u306b $x+1$ \u3092\u4ee3\u5165\u3059\u308b\u3068,\n\n$$\n\\frac{ze^{z(x+1)}}{e^z-1} = \\frac{ze^z e^{zx}}{e^z-1} =\n\\frac{z(1+(e^z-1))e^{zx}}{e^z-1} = \\frac{ze^{zx}}{e^z-1} + ze{zx}\n$$\n\n\u306a\u306e\u3067\u4e21\u8fba\u3092 $z$ \u306b\u3064\u3044\u3066\u5c55\u958b\u3057\u3066\u6bd4\u8f03\u3059\u308c\u3070(5)\u304c\u5f97\u3089\u308c\u308b.\n\n(6) Bernoulli\u591a\u9805\u5f0f\u306e\u6bcd\u51fd\u6570\u306e $x$ \u306b $1-x$ \u3092\u4ee3\u5165\u3059\u308b\u3068,\n\n$$\n\\frac{ze^{z(1-x)}}{e^z-1} = \\frac{ze^z e^{-zx}}{e^z-1} =\n\\frac{ze^{-zx}}{1-e^{-z}} = \\frac{-ze^{-zx}}{e^{-z}-1}\n$$\n\n\u3068Bernoulli\u591a\u9805\u5f0f\u306e\u6bcd\u51fd\u6570\u306e $z$ \u306b $-z$ \u3092\u4ee3\u5165\u3057\u305f\u3082\u306e\u306b\u306a\u308b\u306e\u3067, \u4e21\u8fba\u3092 $z$ \u306b\u3064\u3044\u3066\u5c55\u958b\u3057\u3066\u6bd4\u8f03\u3059\u308c\u3070(5)\u304c\u5f97\u3089\u308c\u308b.\n\n(7) \u4e0a\u306e(2)\u3068(4)\u3088\u308a, $n$ \u304c2\u4ee5\u4e0a\u306e\u3068\u304d,\n\n$$\nB_n(1)-B_n(0) = \\int_0^1 B_n'(x)\\,dx = n\\int_0^1 B_{n-1}(x)\\,dx = n\\delta_{n-1,0} = \\delta_{n,1}\n$$\n\n\u3086\u3048\u306b $n$ \u304c2\u4ee5\u4e0a\u306e\u3068\u304d $B_n(1)=B_n(0)+\\delta_{n,1}$.\n\n(8) \u6b21\u306e\u51fd\u6570\u304c $z$ \u306e\u5076\u51fd\u6570\u3067 $z\\to 0$ \u3067 $1$ \u306b\u306a\u308b\u3053\u3068\u304b\u3089, (6)\u304c\u5f97\u3089\u308c\u308b:\n\n$$\n\\frac{z}{e^z-1} + \\frac{z}{2} = \\frac{z}{2}\\frac{e^{z/2}+e^{-z/2}}{e^{z/2}-e^{-z/2}}.\n\\qquad \\QED\n$$\n\n**\u6ce8\u610f:** $B_n=B_n(0)$ \u306f**Bernoulli\u6570**\u3068\u547c\u3070\u308c\u3066\u3044\u308b. (3)\u3067 $(x,h)$ \u3092 $(0,x)$ \u3067\u7f6e\u304d\u63db\u3048\u308b\u3068, Bernoulli\u591a\u9805\u5f0f\u304cBernoulli\u6570\u3067\u8868\u308f\u3055\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u308b:\n\n$$\nB_n(x) = \\sum_{k=0}^n \\binom{n}{k}B_k x^{n-k}.\n$$\n\n\u4e0a\u306e\u5b9a\u7406\u306e\u6761\u4ef6(1),(2),(4)\u306b\u3088\u3063\u3066Bernoulli\u591a\u9805\u5f0f $B_n(x)$ \u304c $n$ \u306b\u3064\u3044\u3066\u5e30\u7d0d\u7684\u306b\u4e00\u610f\u7684\u306b\u6c7a\u307e\u308b. $\\QED$\n\n**\u4f8b:** \n$$\nB_0 = 1, \\quad B_1 = -\\frac{1}{2}, \\quad\nB_2 = \\frac{1}{6}, \\quad B_3=0, \\quad B_4 = -\\frac{1}{30}\n$$\n\n\u306a\u306e\u3067\n\n$$\n\\begin{aligned}\n&\nB_0(x)=1, \\quad \nB_1(x)=x-\\frac{1}{2}, \\quad\nB_2(x)=x^2-x+\\frac{1}{6}, \n\\\\ &\nB_3(x)=x^3-\\frac{3}{2}x^2+\\frac{1}{2}x, \\quad\nB_4(x)=x^4-2x^3+x^2-\\frac{1}{30}.\n\\qquad\\QED\n\\end{aligned}\n$$\n\n\n```julia\nBernoulliPolynomial(n,x) = sympy[:bernoulli](n,x)\nx = symbols(\"x\", real=true)\n[BernoulliPolynomial(n,x) for n in 0:10]\n```\n\n\n\n\n\\begin{bmatrix}1\\\\x - \\frac{1}{2}\\\\x^{2} - x + \\frac{1}{6}\\\\x^{3} - \\frac{3 x^{2}}{2} + \\frac{x}{2}\\\\x^{4} - 2 x^{3} + x^{2} - \\frac{1}{30}\\\\x^{5} - \\frac{5 x^{4}}{2} + \\frac{5 x^{3}}{3} - \\frac{x}{6}\\\\x^{6} - 3 x^{5} + \\frac{5 x^{4}}{2} - \\frac{x^{2}}{2} + \\frac{1}{42}\\\\x^{7} - \\frac{7 x^{6}}{2} + \\frac{7 x^{5}}{2} - \\frac{7 x^{3}}{6} + \\frac{x}{6}\\\\x^{8} - 4 x^{7} + \\frac{14 x^{6}}{3} - \\frac{7 x^{4}}{3} + \\frac{2 x^{2}}{3} - \\frac{1}{30}\\\\x^{9} - \\frac{9 x^{8}}{2} + 6 x^{7} - \\frac{21 x^{5}}{5} + 2 x^{3} - \\frac{3 x}{10}\\\\x^{10} - 5 x^{9} + \\frac{15 x^{8}}{2} - 7 x^{6} + 5 x^{4} - \\frac{3 x^{2}}{2} + \\frac{5}{66}\\end{bmatrix}\n\n\n\n\n```julia\n# (2)\n\nBernoulliPolynomial(n,x) = sympy[:bernoulli](n,x)\nx = symbols(\"x\", real=true)\n[integrate(BernoulliPolynomial(n,x), (x,0,1)) for n = 0:10]'\n```\n\n\n\n\n 1\u00d711 RowVector{Any,ConjArray{Any,1,Array{SymPy.Sym,1}}}:\n 1 0 0 0 0 0 0 0 0 0 0\n\n\n\n\n```julia\n# (3)\n\nBernoulliNumber(n) = sympy[:bernoulli](n)\nBernoulliPolynomial(n,x) = sympy[:bernoulli](n,x)\nBinomCoeff(n,k) = sympy[:binomial_coefficients_list](n)[k+1]\nx, h = symbols(\"x h\", real=true)\n[BernoulliPolynomial(n,x) == sum(k->BinomCoeff(n,k)*BernoulliNumber(k)*x^(n-k), 0:n) for n in 0:10]'\n```\n\n\n\n\n 1\u00d711 RowVector{Bool,Array{Bool,1}}:\n true true true true true true true true true true true\n\n\n\n\n```julia\n# (4)\n\nBernoulliPolynomial(n,x) = sympy[:bernoulli](n,x)\nx = symbols(\"x\", real=true)\n[diff(BernoulliPolynomial(n,x), x) == n*BernoulliPolynomial(n-1,x) for n = 1:10]'\n```\n\n\n\n\n 1\u00d710 RowVector{Bool,Array{Bool,1}}:\n true true true true true true true true true true\n\n\n\n\n```julia\n# (5)\n\nBernoulliPolynomial(n,x) = sympy[:bernoulli](n,x)\nx = symbols(\"x\", real=true)\n[simplify(BernoulliPolynomial(n,x+1) - BernoulliPolynomial(n,x)) for n in 0:10]\n```\n\n\n\n\n\\begin{bmatrix}0\\\\1\\\\2 x\\\\3 x^{2}\\\\4 x^{3}\\\\5 x^{4}\\\\6 x^{5}\\\\7 x^{6}\\\\8 x^{7}\\\\9 x^{8}\\\\10 x^{9}\\end{bmatrix}\n\n\n\n\n```julia\n# (6)\n\nBernoulliPolynomial(n,x) = sympy[:bernoulli](n,x)\nx = symbols(\"x\", real=true)\n[expand(BernoulliPolynomial(n,1-x)) == (-1)^n*BernoulliPolynomial(n,x) for n in 0:10]'\n```\n\n\n\n\n 1\u00d711 RowVector{Bool,Array{Bool,1}}:\n true true true true true true true true true true true\n\n\n\n\n```julia\n# (7)\n\nBernoulliPolynomial(n,x) = sympy[:bernoulli](n,x)\nx = symbols(\"x\", real=true)\n[expand(BernoulliPolynomial(n,1)) - BernoulliPolynomial(n,0) for n in 0:10]'\n```\n\n\n\n\n 1\u00d711 RowVector{Any,ConjArray{Any,1,Array{SymPy.Sym,1}}}:\n 0 1 0 0 0 0 0 0 0 0 0\n\n\n\n\n```julia\n# (8)\n\nBernoulliNumber(n) = sympy[:bernoulli](n)\n[(n, BernoulliNumber(n)) for n in 0:10]\n```\n\n\n\n\n 11-element Array{Tuple{Int64,SymPy.Sym},1}:\n (0, 1) \n (1, -1/2) \n (2, 1/6) \n (3, 0) \n (4, -1/30)\n (5, 0) \n (6, 1/42) \n (7, 0) \n (8, -1/30)\n (9, 0) \n (10, 5/66)\n\n\n\n### \u3079\u304d\u4e57\u548c\n\n$m$ \u306f\u6b63\u306e\u6574\u6570\u3067\u3042\u308b\u3059\u308b. Bernoulli\u591a\u9805\u5f0f\u306b\u3064\u3044\u3066, \n\n$$\nB_{m+1}(x+1)-B_{m+1}(x) = (m+1)x^m, \n\\quad\\text{i.e.}\\quad\nx^m = \\frac{B_{m+1}(x+1)-B_{m+1}(x)}{m+1}\n$$\n\n\u304c\u6210\u7acb\u3057\u3066\u3044\u308b\u306e\u3067, \u3053\u308c\u3092 $x=0,1,\\ldots,n$ \u306b\u3064\u3044\u3066\u8db3\u3057\u4e0a\u3052\u308b\u3068,\n\n$$\n\\sum_{j=1}^n j^m = \\frac{B_{m+1}(n+1)-B_{m+1}}{m+1}.\n\\qquad \\QED\n$$\n\n\n```julia\nPowerSum(m, n) = sum(j->j^m, 1:n)\nBernoulliNumber(n) = sympy[:bernoulli](n)\nBernoulliPolynomial(n,x) = sympy[:bernoulli](n,x)\nPowerSumFormula(m, n) = (BernoulliPolynomial(m+1,n+1)-BernoulliNumber(m+1))/(m+1)\n[(m, PowerSum(m,10), PowerSumFormula(m, 10)) for m in 1:10]\n```\n\n\n\n\n 10-element Array{Tuple{Int64,Int64,SymPy.Sym},1}:\n (1, 55, 55) \n (2, 385, 385) \n (3, 3025, 3025) \n (4, 25333, 25333) \n (5, 220825, 220825) \n (6, 1978405, 1978405) \n (7, 18080425, 18080425) \n (8, 167731333, 167731333) \n (9, 1574304985, 1574304985) \n (10, 14914341925, 14914341925)\n\n\n\n### Bernoulli\u6570\u306e\u8a08\u7b97\u6cd5\n\nBernoulli\u6570 $B_n$ \u306f\n\n$$\\displaystyle\n\\frac{z}{e^z-1}=\\sum_{n=1}^\\infty B_n\\frac{z^n}{n!}\n$$\n\n\u3067\u5b9a\u7fa9\u3055\u308c\u308b. \u3057\u304b\u3057, \u3053\u306e\u5c55\u958b\u3092\u76f4\u63a5\u8a08\u7b97\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066 Bernoulli \u6570\u3092\u6c42\u3081\u308b\u306e\u306f\u52b9\u7387\u304c\u60aa\u3044.\n\n\u307e\u305a, \u5de6\u8fba\u306e $z\\to 0$ \u306e\u6975\u9650\u3092\u53d6\u308b\u3053\u3068\u306b\u3088\u3063\u3066 $B_0=1$ \u3067\u3042\u308b\u3053\u3068\u306f\u3059\u3050\u306b\u308f\u304b\u308b.\n\n\u6b21\u306b, $n$ \u304c $3$ \u4ee5\u4e0a\u306e\u5947\u6570\u306e\u3068\u304d $B_n=0$ \u3068\u306a\u308b\u3053\u3068\u3092(\u518d\u3073)\u793a\u305d\u3046. \n\n$$\\displaystyle\n\\frac{z}{e^z-1} + \\frac z2\n=\\frac z2\\frac{e^z+1}{e^z-1} \n=\\frac z2\\frac{e^{z/2}+e^{-z/2}}{e^{z/2}-e^{-z/2}}\n$$\n\n\u3088\u308a, \u5de6\u8fba\u306f\u5076\u51fd\u6570\u306b\u306a\u308b\u306e\u3067, \u305d\u306e\u5c55\u958b\u306e\u5947\u6570\u6b21\u306e\u9805\u306f\u6d88\u3048\u308b. \u3053\u306e\u3053\u3068\u304b\u3089, $B_1=-1/2$ \u3067\u304b\u3064, $0=B_3=B_5=B_7=\\cdots$ \u3067\u3042\u308b\u3053\u3068\u3082\u308f\u304b\u308b.\n\n$$\\displaystyle\n\\frac{ze^z}{e^z-1}\n=\\sum_{j,k=0}^\\infty \\frac{z^j}{j!}\\frac{B_k z^k}{k!}\n=\\sum_{n=0}^\\infty\\left(\\sum_{k=0}^n \\binom{n}{k} B_k\\right)\\frac{z^n}{n!}\n$$\n\n\u3067\u304b\u3064\n\n$$\\displaystyle\n\\frac{ze^z}{e^z-1}\n=\\frac{z}{e^z-1}+z\n=\\sum_{n=0}^\\infty(B_n+\\delta_{n1})\\frac{z^n}{n!}\n$$\n\n\u306a\u306e\u3067, \u3053\u308c\u3089\u3092\u6bd4\u8f03\u3059\u308b\u3068\n\n$$\\displaystyle\n\\sum_{k=0}^{n-1} \\binom{n}{k} B_k = \\delta_{n1}.\n$$\n\n\u3086\u3048\u306b, $n$ \u3092 $n+1$ \u3067\u7f6e\u304d\u63db\u3048, $n\\geqq 1$ \u3068\u3057, $B_n$ \u3092\u4ed6\u3067\u8868\u308f\u3059\u5f0f\u306b\u66f8\u304d\u76f4\u3059\u3068\n\n$$\\displaystyle\nB_n = -\\frac{1}{n+1}\\sum_{k=0}^{n-1}\\binom{n+1}{k}B_k\n\\qquad (n\\geqq 1).\n$$\n\n\u3053\u308c\u3092\u4f7f\u3048\u3070\u5e30\u7d0d\u7684\u306b $B_n$ \u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b. $B_0=1$, $B_1=-1/2$, $0=B_3=B_5=B_7=\\cdots$ \u3067\u3042\u308b\u3053\u3068\u3092\u4f7f\u3046\u3068, \n\n$$\\displaystyle\nB_{2m} = -\\frac{1}{2m+1}\\left(\n1 -\\frac{2m+1}{2}\n+\\sum_{k=1}^{m-1}\\binom{2m+1}{2k}B_{2k}\n\\right).\n$$\n\n**\u554f\u984c:** \u4e0a\u306e\u65b9\u3067\u306fSymPy\u306b\u304a\u3051\u308bBernoulli\u6570\u306e\u51fd\u6570\u3092\u5229\u7528\u3057\u305f. Bernoulli\u6570\u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u81ea\u5206\u3067\u66f8\u3051. $\\QED$\n\n**\u89e3\u7b54\u4f8b:** \u6b21\u306e\u30bb\u30eb\u306e\u901a\u308a. $\\QED$\n\n\n```julia\n# binomial coefficient: binom(n,k) = n(n-1)\u30fb(n-k+1)/k!\n#\nmydiv(a, b) = a / b\nmydiv(a::Integer, b::Integer) = a \u00f7 b\nfunction binom(n, k)\n k < 0 && return zero(n)\n k == 0 && return one(n)\n b = one(n)\n for j in 1:k\n b = mydiv(b*(n-k+j), j)\n end\n b\nend\n \n@show binom(Rational(big\"100\")/3, 30)\n\n# Bernoulli numbers: B(n) = Bernoulli[n+1] = B_n\n#\nstruct Bernoulli{T}\n B::Array{T,1}\nend\nfunction Bernoulli(; maxn=200)\n B = zeros(Rational{BigInt},maxn+1)\n B[1] = 1 # B_0\n B[2] = -1//2 # B_1\n for n in big\"2\":2:maxn+1\n B[n+1] = -(1//(n+1))*sum(j->binom(n+1,j)*B[j+1], 0:n-1)\n # B_n = -(1/(n+1)) \u03a3_{j=0}^{n-1} binom(n+1,j)*B_j\n end\n Bernoulli(B)\nend\n(B::Bernoulli)(n) = B.B[n+1]\n\nmaxn = 200\n@time B = Bernoulli(maxn=maxn) # B_n \u3092 B_{maxn} \u307e\u3067\u8a08\u7b97\nBB(n) = float(B(n)) # B(n) = B_n \u3067\u3042\u308b. BB(n)\u306f\u305d\u306e\u6d6e\u52d5\u5c0f\u6570\u70b9\u7248\n\n# SymPy\u306eBernoulli\u6570\u3068\u6bd4\u8f03\u3057\u3066\u6b63\u3057\u304f\u8a08\u7b97\u3067\u304d\u3066\u3044\u308b\u304b\u3069\u3046\u304b\u3092\u78ba\u8a8d\n#\nBernoulliNumber(n) = sympy[:bernoulli](n)\n@show B_eq_B = [B(n) == BernoulliNumber(n) for n in 0:maxn]\nprintln()\n@show all(B_eq_B)\n\nmaxnprint = 30\nprintln()\nfor n in [0; 1; 2:2:maxnprint]\n println(\"B($n) = \", B(n))\nend\nprintln()\nfor n in [0; 1; 2:2:maxnprint]\n println(\"BB($n) = \", BB(n))\nend\n```\n\n binom(Rational(@big_str(\"100\")) / 3, 30) = 11240781188817808072725280//984770902183611232881\n 1.215169 seconds (16.07 M allocations: 363.244 MiB, 19.05% gc time)\n B_eq_B = [B(n) == BernoulliNumber(n) for n = 0:maxn] = Bool[true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true]\n \n all(B_eq_B) = true\n \n B(0) = 1//1\n B(1) = -1//2\n B(2) = 1//6\n B(4) = -1//30\n B(6) = 1//42\n B(8) = -1//30\n B(10) = 5//66\n B(12) = -691//2730\n B(14) = 7//6\n B(16) = -3617//510\n B(18) = 43867//798\n B(20) = -174611//330\n B(22) = 854513//138\n B(24) = -236364091//2730\n B(26) = 8553103//6\n B(28) = -23749461029//870\n B(30) = 8615841276005//14322\n \n BB(0) = 1.000000000000000000000000000000000000000000000000000000000000000000000000000000\n BB(1) = -5.000000000000000000000000000000000000000000000000000000000000000000000000000000e-01\n BB(2) = 1.666666666666666666666666666666666666666666666666666666666666666666666666666674e-01\n BB(4) = -3.333333333333333333333333333333333333333333333333333333333333333333333333333359e-02\n BB(6) = 2.380952380952380952380952380952380952380952380952380952380952380952380952380947e-02\n BB(8) = -3.333333333333333333333333333333333333333333333333333333333333333333333333333359e-02\n BB(10) = 7.57575757575757575757575757575757575757575757575757575757575757575757575757578e-02\n BB(12) = -2.531135531135531135531135531135531135531135531135531135531135531135531135531131e-01\n BB(14) = 1.166666666666666666666666666666666666666666666666666666666666666666666666666661\n BB(16) = -7.092156862745098039215686274509803921568627450980392156862745098039215686274513\n BB(18) = 5.497117794486215538847117794486215538847117794486215538847117794486215538847111e+01\n BB(20) = -5.291242424242424242424242424242424242424242424242424242424242424242424242424247e+02\n BB(22) = 6.192123188405797101449275362318840579710144927536231884057971014492753623188388e+03\n BB(24) = -8.658025311355311355311355311355311355311355311355311355311355311355311355311313e+04\n BB(26) = 1.425517166666666666666666666666666666666666666666666666666666666666666666666661e+06\n BB(28) = -2.729823106781609195402298850574712643678160919540229885057471264367816091954032e+07\n BB(30) = 6.015808739006423683843038681748359167714006423683843038681748359167714006423638e+08\n\n\n### \u5468\u671f\u7684Bernoulli\u591a\u9805\u5f0f\u306eFourier\u7d1a\u6570\u5c55\u958b\n\n$\\widetilde{B}_k(x) = B_k(x-\\lfloor x\\rfloor)$ \u3092**\u5468\u671f\u7684Bernoulli\u591a\u9805\u5f0f**\u3068\u547c\u3076\u3053\u3068\u306b\u3059\u308b. \u5468\u671f\u7684Bernoulli\u591a\u9805\u5f0f\u306f $\\widetilde{B}_k(x+1)=\\widetilde{B}_k(x)$ \u3092\u6e80\u305f\u3057\u3066\u3044\u308b. \n\n\u5468\u671f\u7684Bernoulli\u591a\u9805\u5f0f\u306e\u6bcd\u51fd\u6570 $\\ds\\frac{z e^{z(x-\\lfloor x\\rfloor)}}{e^z-1}$ \u306e $x$ \u306e\u51fd\u6570\u3068\u3057\u3066\u306e Fourier\u4fc2\u6570 $a_n(z)$ \u306f\u6b21\u306e\u3088\u3046\u306b\u6c42\u307e\u308b:\n\n$$\n\\frac{e^z-1}{z}a_n(z) = \\int_0^1 e^{zx}e^{-2\\pi inx}\\,dx =\n\\left[\\frac{e^{(z-2\\pi in)x}}{z-2\\pi in}\\right]_{x=0}^{x=1} = \n\\frac{e^z-1}{z-2\\pi in},\n\\qquad\na_n(z) = \\frac{z}{z-2\\pi in}.\n$$\n\n\u3086\u3048\u306b $a_0(z)=1$ \u3067\u3042\u308a, $n\\ne 0$ \u306e\u3068\u304d\n\n$$\na_n(z) = -\\sum_{k=1}^\\infty \\frac{z^k}{(2\\pi in)^k}\n$$\n\n\u3053\u308c\u3088\u308a, $\\widetilde{B}_k(x)$ \u306eFourier\u4fc2\u6570 $a_{k,n}$ \u306f, $a_{0,n}=\\delta_{n,0}$, $a_{k,0}=\\delta_{k,0}$ \u3092\u6e80\u305f\u3057, $k\\ne 0$, $n\\geqq 1$ \u306e\u3068\u304d\n\n$$\na_{k,n} = -\\frac{k!}{(2\\pi in)^k}\n$$\n\n\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u3057\u305f\u304c\u3063\u3066, Fourier\u7d1a\u6570\u8ad6\u3088\u308a, $k=1$ \u306e\u3068\u304d\u306f\u6574\u6570\u3067\u306f\u306a\u3044\u5b9f\u6570 $x$ \u306b\u3064\u3044\u3066, $k\\geqq 2$ \u306e\u5834\u5408\u306b\u306f\u3059\u3079\u3066\u306e\u5b9f\u6570 $x$ \u306b\u3064\u3044\u3066\u6b21\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308b:\n\n$$\n\\widetilde{B}_k(x) = B_k(x-\\lfloor x\\rfloor) =\n-k!\\sum_{n\\ne 0} \\frac{e^{2\\pi inx}}{(2\\pi in)^k}.\n$$\n\n\u3059\u306a\u308f\u3061, $k=1,2,3,\\ldots$ \u306b\u3064\u3044\u3066\n\n$$\n\\widetilde{B}_{2k-1}(x) = \n(-1)^k 2(2k-1)!\\sum_{n=1}^\\infty\\frac{\\sin(2\\pi nx)}{(2\\pi n)^{2k-1}}, \n\\qquad\n\\widetilde{B}_{2k}(x) = \n(-1)^{k-1} 2(2k)!\\sum_{n=1}^\\infty \\frac{\\cos(2\\pi nx)}{(2\\pi n)^{2k}}. \n$$\n\n\u3053\u306e\u3053\u3068\u304b\u3089, $k$ \u304c\u5927\u304d\u3044\u3068\u304d(\u5b9f\u969b\u306b\u306f $k=5,6$ \u7a0b\u5ea6\u3067\u3059\u3067\u306b), \u5468\u671f\u7684Bernoulli\u591a\u9805\u5f0f\u306f $n=1$ \u306e\u9805\u3060\u3051\u3067\n\n$$\n\\widetilde{B}_{2k-1}(x) \\approx\n(-1)^k 2(2k-1)!\\frac{\\sin(2\\pi x)}{(2\\pi)^{2k-1}}, \n\\qquad\n\\widetilde{B}_{2k}(x) \\approx\n(-1)^{k-1} 2(2k)!\\frac{\\cos(2\\pi x)}{(2\\pi)^{2k}}\n$$\n\n\u3068\u8fd1\u4f3c\u3067\u304d\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u9069\u5f53\u306b\u30b9\u30b1\u30fc\u30eb\u3059\u308c\u3070\u5468\u671f\u7684Bernoulli\u591a\u9805\u5f0f\u306f $k\\to\\infty$ \u3067\u4e09\u89d2\u51fd\u6570\u306b\u53ce\u675f\u3059\u308b.\n\n\n```julia\nBBB = Bernoulli(Float64.(B.B)) # Float64 Bernoulli numbers\nBP(k,x) = sum(j->binom(k,j)*BBB(k-j)*x^j, 0:k) # Float64 Bernoulli polynomial\nPBP(k,x) = BP(k, x - floor(x)) # periodic Bernoulli polynomial\n\n# partial sum of Fourier series of periodic Bernoulli polynomial\nfunction PSFS(k, N, x)\n k == 0 && return zero(x)\n if isodd(k)\n return (-1)^((k+1)\u00f72)*2*factorial(k)*sum(n->sin(2\u03c0*n*x)/(2\u03c0*n)^k, 1:N)\n else\n return (-1)^(k\u00f72-1)*2*factorial(k)*sum(n->cos(2\u03c0*n*x)/(2\u03c0*n)^k, 1:N)\n end\nend\n\nPP = []\nx = -1.0:0.001:0.999\nfor (k,N) in [(1,20), (2,10), (3,3), (4,2), (5,1), (6,1)]\n y = PBP.(k,x)\n z = PSFS.(k, N, x)\n ymin = 1.2*minimum(y)\n ymax = 2.7*maximum(y)\n P = plot(legend=:topleft, size=(400, 250), ylim=(ymin, ymax))\n plot!(x, y, label=\"B_$k(x-[x])\")\n plot!(x, z, label=\"partial sum of Fourier series (N=$N)\")\n push!(PP, P)\nend\n\nplot(PP[1:2]..., size=(750, 280))\n```\n\n\n\n\n \n\n \n\n\n\n\n```julia\nplot(PP[3:4]..., size=(750, 280))\n```\n\n\n\n\n \n\n \n\n\n\n\n```julia\nplot(PP[5:6]..., size=(750, 280))\n```\n\n\n\n\n \n\n \n\n\n\n## Euler-Maclaurin\u306e\u548c\u516c\u5f0f\n\n### Euler-Maclaurin\u306e\u548c\u516c\u5f0f\u306e\u5c0e\u51fa\n\nBernoulli\u591a\u9805\u5f0f $B_n(x)$ \u3068Bernoulli\u6570 $B_n$ \u306b\u3064\u3044\u3066\n\n$$\n\\begin{aligned}\n&\nB_0(x) = 1, \\quad \\frac{d}{dx}\\frac{B_n(x)}{n!} = \\frac{B_{n-1}(x)}{(n-1)!}, \n\\\\ &\nB_1(0)=-\\frac{1}{2}, \\quad B_1(1)=\\frac{1}{2},\n\\\\ &\nB_n(1)=B_n(0)=B_n \\quad (n=0,2,3,4,5,\\ldots) \n\\\\ &\nB_{2j+1} = 0 \\quad (j=1,2,3,\\ldots)\n\\end{aligned}\n$$\n\n\u304c\u6210\u7acb\u3057\u3066\u3044\u308b. \u4ee5\u4e0b\u3067\u306f\u3057\u3070\u3089\u304f\u306e\u3042\u3044\u3060\u3053\u308c\u3089\u306e\u6761\u4ef6\u3057\u304b\u4f7f\u308f\u306a\u3044.\n\n\u90e8\u5206\u7a4d\u5206\u3092\u7e70\u308a\u8fd4\u3059\u3053\u3068\u306b\u3088\u3063\u3066,\n\n$$\n\\begin{aligned}\n\\int_0^1 f(x)\\,dx &= \\int_0^1 B_0(x)f(x)\\,dx \n\\\\ &=\n[B_1(x)f(x)]_0^1 - \\int_0^1 B_1(x)f'(x)\\,dx \n\\\\ &=\n[B_1(x)f(x)]_0^1 - \\frac{1}{2}[B_2(x)f'(x)]_0^1 + \\int_0^1 \\frac{B_2(x)}{2}f''(x)\\,dx \n\\\\ &=\n[B_1(x)f(x)]_0^1 - \\frac{1}{2}[B_2(x)f'(x)]_0^1 + \\frac{1}{3!}[B_3(x)f''(x)]_0^1 - \\int_0^1 \\frac{B_3(x)}{3!}f'''(x)\\,dx\n\\\\ &=\n\\cdots\\cdots\\cdots\\cdots\\cdots\n\\\\ &=\n\\sum_{k=1}^n \\frac{(-1)^{k-1}}{k!}\\left[B_k(x)f^{(k-1)}(x)\\right]_0^1 + \n(-1)^n\\int_0^1 \\frac{B_n(x)}{n!}f^{(n)}(x)\\,dx\n\\\\ &=\n\\frac{f(0)+f(1)}{2} + \\sum_{k=2}^n(-1)^{k-1}\\frac{B_k}{k!} (f^{(k-1)}(1)-f^{(k-1)}(0)) + \n(-1)^n\\int_0^1 \\frac{B_n(x)}{n!}f^{(n)}(x)\\,dx.\n\\end{aligned}\n$$\n\n\u5b9f\u6570 $x$ \u306b\u5bfe\u3057\u3066, $x$ \u4ee5\u4e0b\u306e\u6700\u5927\u306e\u6574\u6570\u3092 $\\lfloor x\\rfloor$ \u3068\u66f8\u304f. \u3053\u306e\u3068\u304d, $x-\\lfloor x\\rfloor$ \u306f $x$ \u306e\u300c\u5c0f\u6570\u90e8\u5206\u300d\u306b\u306a\u308b. \u3053\u306e\u3088\u3046\u306b\u8a18\u53f7\u3092\u6e96\u5099\u3057\u3066\u304a\u304f\u3068, \u6574\u6570 $j$ \u306b\u5bfe\u3057\u3066, \n\n$$\n\\begin{aligned}\n\\int_j^{j+1} f(x)\\,dx &= \\int_0^1 f(x+j)\\,dx\n\\\\ &=\n\\frac{f(j)+f(j+1)}{2} + \\sum_{k=2}^n (-1)^{k-1} \\frac{B_k}{k!} (f^{(k-1)}(j+1)-f^{(k-1)}(j)) + \n(-1)^n\\int_0^1 \\frac{B_n(x)}{n!}f^{(n)}(x+j)\\,dx\n\\\\ &=\n\\frac{f(j)+f(j+1)}{2} + \\sum_{k=2}^n (-1)^{k-1}\\frac{B_k}{k!} (f^{(k-1)}(j+1)-f^{(k-1)}(j)) + \n(-1)^n\\int_j^{j+1} \\frac{B_n(x-\\lfloor x\\rfloor)}{n!}f^{(n)}(x)\\,dx.\n\\end{aligned}\n$$\n\n$af(j), a+1:b-1)\n - sum(k -> (\n BernoulliNumber(k)/factorial(Sym(k))\n * (diff(f(x), x, k-1)(x=>b) - diff(f(x), x, k-1)(x=>a))\n ), 2:n)\n )\nend\n\nfunction EulerMaclaurinRemainder(f, a, b, n)\n x = symbols(\"x\", real=true)\n g = diff(f(x), x, n)\n (-1)^(n-1) * sum(k -> (\n integrate(BernoulliPolynomial(n,x)*g(x=>x+k), (x,0,1))\n ), a:b-1)/factorial(Sym(n))\nend\n\nx = symbols(\"x\", real=true)\n\n[integrate(x^m, (x, 0, 10)) for m in 7:15] |> display\n\n[\n EulerMaclaurinIntegral(x->x^m, 0, 10, 5) - EulerMaclaurinRemainder(x->x^m, 0, 10, 5)\n for m in 7:15\n] |> display\n```\n\n\n\\begin{bmatrix}12500000\\\\\\frac{1000000000}{9}\\\\1000000000\\\\\\frac{100000000000}{11}\\\\\\frac{250000000000}{3}\\\\\\frac{10000000000000}{13}\\\\\\frac{50000000000000}{7}\\\\\\frac{200000000000000}{3}\\\\625000000000000\\end{bmatrix}\n\n\n\n\\begin{bmatrix}12500000\\\\\\frac{1000000000}{9}\\\\1000000000\\\\\\frac{100000000000}{11}\\\\\\frac{250000000000}{3}\\\\\\frac{10000000000000}{13}\\\\\\frac{50000000000000}{7}\\\\\\frac{200000000000000}{3}\\\\625000000000000\\end{bmatrix}\n\n\n**Euler-Maclaurin\u306e\u548c\u516c\u5f0f\u306e\u89e3\u91c82:** Euler-Maclaurin\u306e\u548c\u516c\u5f0f\u306f\u6b21\u306e\u3088\u3046\u306b\u66f8\u304d\u76f4\u3055\u308c\u308b:\n\n$$\n\\begin{aligned}\n&\n\\sum_{j=a}^b f(j) = \n\\int_a^b f(x)\\,dx + \\frac{f(a)+f(b)}{2} + \n\\sum_{1\\leqq i\\leqq n/2} \\frac{B_{2i}}{(2i)!} (f^{(2i-1)}(b)-f^{(2i-1)}(a)) + R_n,\n\\\\ &\nR_n = (-1)^{n-1}\\int_a^b \\frac{B_n(x-\\lfloor x\\rfloor)}{n!}f^{(n)}(x)\\,dx\n\\end{aligned}\n$$\n\n\u3053\u308c\u306f $n$ \u304c3\u4ee5\u4e0a\u306e\u5947\u6570\u306e\u3068\u304d $B_n=0$ \u3068\u306a\u308b\u3053\u3068\u3092\u4f7f\u3046\u3068\u6b21\u306e\u3088\u3046\u306b\u66f8\u304d\u76f4\u3055\u308c\u308b:\n\n$$\n\\begin{aligned}\n&\n\\sum_{j=a}^b f(j) = \n\\int_a^b f(x)\\,dx + \\frac{f(a)+f(b)}{2} + \n\\sum_{k=2}^n \\frac{B_k}{k!} (f^{(k-1)}(b)-f^{(k-1)}(a)) + R_n,\n\\\\ &\nR_n = (-1)^{n-1}\\int_a^b \\frac{B_n(x-\\lfloor x\\rfloor)}{n!}f^{(n)}(x)\\,dx\n\\end{aligned}\n$$\n\n\u3053\u306e\u7b49\u5f0f\u306f\u51fd\u6570 $f$ \u306e\u6574\u6570\u306b\u304a\u3051\u308b\u5024\u306e\u548c $\\ds\\sum_{j=a}^b f(j)$ \u3092\u7a4d\u5206 $\\ds\\int_a^b f(x)\\,dx$ \u3067\u8fd1\u4f3c\u3057\u305f\u3068\u304d\u306e\u8aa4\u5dee\u304c\n\n$$\n\\frac{f(a)+f(b)}{2} + \n\\sum_{1\\leqq i\\leqq n/2} \\frac{B_{2i}}{(2i)!} (f^{(2i-1)}(b)-f^{(2i-1)}(a)) + R_n\n$$\n\n\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u3066\u3044\u308b. \u4f8b\u3048\u3070, $n=1$ \u306e\u5834\u5408\u306b\u306f, $\\ds B_1(x)=x-\\frac{1}{2}$ \u306a\u306e\u3067,\n\n$$\n\\sum_{j=a}^b f(j) = \n\\int_a^b f(x)\\,dx + \\frac{f(a)+f(b)}{2} + \n\\int_a^b\\left(x-\\lfloor x\\rfloor-\\frac{1}{2}\\right)f'(x)\\,dx.\n$$\n\n$n=2$ \u306e\u5834\u5408\u306b\u306f $\\ds B_2(x)=x^2-x+\\frac{1}{6}$, $\\ds B_2=\\frac{1}{6}$ \u3067\u3042\u308a,\n\n$$\n\\sum_{j=a}^b f(j) = \n\\int_a^b f(x)\\,dx + \\frac{f(a)+f(b)}{2} +\n\\frac{f'(b)-f'(a)}{12} -\n\\int_a^b\\frac{B_2(x-\\lfloor x\\rfloor)}{2}f''(x)\\,dx.\n$$\n\n\u3068\u306a\u308b. $\\QED$\n\n\n```julia\n# \u3059\u3050\u4e0a\u306e\u516c\u5f0f\u3092\u691c\u8a3c\n\nPowerSum(m, n) = sum(j->j^m, 1:n)\nBernoulliNumber(n) = sympy[:bernoulli](n)\nBernoulliPolynomial(n,x) = sympy[:bernoulli](n,x)\n\nfunction EulerMaclaurinSum(f, a, b, n)\n x = symbols(\"x\", real=true)\n (\n integrate(f(x), (x, a, b))\n + (f(a)+f(b))/Sym(2)\n + sum(k -> (\n BernoulliNumber(k)/factorial(Sym(k))\n * (diff(f(x), x, k-1)(x=>b) - diff(f(x), x, k-1)(x=>a))\n ), 2:n)\n )\nend\n\nfunction EulerMaclaurinRemainder(f, a, b, n)\n x = symbols(\"x\", real=true)\n g = diff(f(x), x, n)\n (-1)^(n-1) * sum(k -> (\n integrate(BernoulliPolynomial(n,x)*g(x=>x+k), (x,0,1))\n ), a:b-1)/factorial(Sym(n))\nend\n\n[PowerSum(m, 10) for m in 1:10] |> display\n\n[EulerMaclaurinSum(x->x^m, 1, 10, m+1) for m in 1:10] |> display\n\n[\n EulerMaclaurinSum(x->x^m, 1, 10, m-1) + EulerMaclaurinRemainder(x->x^m, 1, 10, m-1)\n for m in 3:10\n] |> display\n\n[\n EulerMaclaurinSum(x->x^m, 1, 10, m-2) + EulerMaclaurinRemainder(x->x^m, 1, 10, m-2)\n for m in 4:10\n] |> display\n\n[\n EulerMaclaurinSum(x->x^m, 1, 10, m-3) + EulerMaclaurinRemainder(x->x^m, 1, 10, m-3)\n for m in 5:10\n] |> display\n```\n\n\n 10-element Array{Int64,1}:\n 55\n 385\n 3025\n 25333\n 220825\n 1978405\n 18080425\n 167731333\n 1574304985\n 14914341925\n\n\n\n\\begin{bmatrix}55\\\\385\\\\3025\\\\25333\\\\220825\\\\1978405\\\\18080425\\\\167731333\\\\1574304985\\\\14914341925\\end{bmatrix}\n\n\n\n\\begin{bmatrix}3025\\\\25333\\\\220825\\\\1978405\\\\18080425\\\\167731333\\\\1574304985\\\\14914341925\\end{bmatrix}\n\n\n\n\\begin{bmatrix}25333\\\\220825\\\\1978405\\\\18080425\\\\167731333\\\\1574304985\\\\14914341925\\end{bmatrix}\n\n\n\n\\begin{bmatrix}220825\\\\1978405\\\\18080425\\\\167731333\\\\1574304985\\\\14914341925\\end{bmatrix}\n\n\n### Euler-Maclaurin\u306e\u548c\u516c\u5f0f\u306e\u5f62\u5f0f\u7684\u5c0e\u51fa\n\n\u51fd\u6570 $f(x)$ \u306b\u5bfe\u3057\u3066, \u3042\u308b\u51fd\u6570 $F(x)$ \u3067\n\n$$\nF(x+1) - F(x) = f(x+h)\n$$\n\n\u3068\u3044\u3046\u6761\u4ef6\u3092\u6e80\u305f\u3059\u3082\u306e\u3092\u6c42\u3081\u308b\u554f\u984c\u3092\u8003\u3048\u308b. \u305d\u306e\u3068\u304d, $\\ds D=\\frac{\\d}{\\d x}$ \u3068\u304a\u304f\u3068, \u5f62\u5f0f\u7684\u306b\u305d\u306e\u6761\u4ef6\u306f\n\n$$\n(e^D-1)F(x) = e^{hD}f(x) = De^{hD}\\int f(x)\\,dx\n$$\n\n\u3068\u66f8\u304d\u76f4\u3055\u308c\u308b. \u3053\u308c\u3088\u308a, \u5f62\u5f0f\u7684\u306b\u306f\n\n$$\nF(x) = \\frac{De^{hD}}{e^D-1}\\int f(x)\\,dx =\n\\sum_{k=0}^\\infty \\frac{B_k(h)}{k!}D^k \\int f(x)\\,dx =\n\\int f(x)\\,dx + \\sum_{k=1}^\\infty \\frac{B_k(h)}{k!}f^{(k-1)}(x).\n$$\n\n\u3053\u308c\u3088\u308a, \u6574\u6570 $an$ \u306e\u3068\u304d, \n\n$$\n\\begin{aligned}\n\\log n! &= \\log N! + \\log n - \\sum_{j=n}^N \\log j\n\\\\ &= \\log N! + \\log n -\\left(\n\\int_n^N \\log x\\,dx + \\frac{\\log n+\\log N}{2} +\n\\sum_{k=2}^{K-1}\\frac{B_k}{k(k-1)} \\left(\\frac{1}{N^{k-1}} - \\frac{1}{n^{k-1}}\\right) + \nR_{K,N}\n\\right)\n\\\\ &=\n\\log N! - \\left(N\\log N - N + \\frac{1}{2}\\log N\\right) - \n\\sum_{k=2}^{K-1} \\frac{B_k}{k(k-1)} \\frac{1}{N^{k-1}}\n\\\\ &\\,+\nn\\log n - n +\\frac{1}{2}\\log n +\n\\sum_{k=2}^{K-1}\\frac{B_k}{k(k-1)} \\frac{1}{n^{k-1}} + R_{K,N},\n\\\\ \nR_{K,N} &= (-1)^{K-1}\\int_n^N \\frac{\\tilde{B}_K(x)}{K}\\frac{(-1)^{K-1}}{x^K}\\,dx\n\\end{aligned}\n$$\n\n\u305f\u3060\u3057, $\\tilde{B}_n(x)=B_n(\\lfloor x\\rfloor)$ \u3068\u304a\u3044\u305f. \n\n\u3053\u3053\u3067\u306f, $N\\to\\infty$ \u306e\u3068\u304d\n\n$$\n\\log N! - \\left(N\\log N - N + \\frac{1}{2}\\log N\\right) \\to \\sqrt{2\\pi}\n$$\n\n\u3068\u306a\u308b\u3053\u3068\u306f\u65e2\u77e5\u3067\u3042\u308b\u3082\u306e\u3068\u3059\u308b. \u4f8b\u3048\u3070, \u30ce\u30fc\u30c8\u300c10 Gauss\u7a4d\u5206, \u30ac\u30f3\u30de\u51fd\u6570, \u30d9\u30fc\u30bf\u51fd\u6570\u300d\u300c12 Fourier\u89e3\u6790\u300d\u306eStirling\u306e\u8fd1\u4f3c\u516c\u5f0f\u306e\u7bc0\u3092\u53c2\u7167\u3057\u3066\u6b32\u3057\u3044. \u4ee5\u4e0b\u3067\u306f\u305d\u308c\u3089\u306e\u30ce\u30fc\u30c8\u3088\u308a\u3082\u7cbe\u5bc6\u306a\u7d50\u679c\u3092\u5f97\u308b.\n\n\u3053\u306e\u3068\u304d, \u4e0a\u306e\u7d50\u679c\u3067 $N\\to\\infty$ \u3068\u3059\u308b\u3068,\n\n$$\n\\begin{aligned}\n&\n\\log n! =\nn\\log n - n +\\frac{1}{2}\\log n + \\log\\sqrt{2\\pi} +\n\\sum_{k=2}^{K-1}\\frac{B_k}{k(k-1)} \\frac{1}{n^{k-1}} + R_K,\n\\\\ & \nR_K = (-1)^{K-1}\\int_n^\\infty \\frac{\\tilde{B}_K(x)}{K}\\frac{(-1)^{K-1}}{x^K}\\,dx = \nO\\left(\\frac{1}{n^{K-1}}\\right).\n\\end{aligned}\n$$\n\n$K=2L+1$ \u3068\u304a\u304f\u3053\u3068\u306b\u3088\u3063\u3066\u6b21\u304c\u5f97\u3089\u308c\u308b: \u6b63\u306e\u6574\u6570 $L$ \u306b\u5bfe\u3057\u3066,\n\n$$\n\\log n! =\nn\\log n - n + \\frac{1}{2}\\log n + \\log\\sqrt{2\\pi} +\n\\sum_{l=1}^L \\frac{B_{2l}}{(2l)(2l-1)}\\frac{1}{n^{2l-1}} + O\\left(\\frac{1}{n^{2L}}\\right).\n$$\n\n\u3053\u308c\u304c\u6c42\u3081\u3066\u3044\u305f\u7d50\u679c\u3067\u3042\u308b.\n\n\u4f8b\u3048\u3070, $L=2$ \u306e\u3068\u304d, $\\ds B_2=\\frac{1}{6}$, $\\ds B_4=-\\frac{1}{30}$ \u306a\u306e\u3067,\n\n$$\n\\log n! =\nn\\log n - n + \\frac{1}{2}\\log n + \\log\\sqrt{2\\pi} +\n\\frac{1}{12n} - \\frac{1}{360n^3} + O\\left(\\frac{1}{n^4}\\right).\n$$\n\n\u3053\u308c\u3088\u308a, \n\n$$\nn! = n^n e^{-n}\\sqrt{2\\pi n}\n\\left(1+\\frac{1}{12n} + \\frac{1}{288n^2} - \\frac{139}{51840n^3} + O\\left(\\frac{1}{n^4}\\right)\\right).\n$$\n\n\n```julia\nx = symbols(\"x\")\nseries(exp(x/12-x^3/360), x, n=4)\n```\n\n\n\n\n$$1 + \\frac{x}{12} + \\frac{x^{2}}{288} - \\frac{139 x^{3}}{51840} + \\mathcal{O}\\left(x^{4}\\right)$$\n\n\n\n### Poisson\u306e\u548c\u516c\u5f0f\u3068Euler-Maclaurin\u306e\u548c\u516c\u5f0f\u306e\u95a2\u4fc2\n\nPoisson\u306e\u548c\u516c\u5f0f\u3068\u306f, \u6025\u6e1b\u5c11\u51fd\u6570 $f(x)$ \u306b\u5bfe\u3057\u3066,\n\n$$\n\\sum_{m\\in\\Z} f(m) = \\sum_{n\\in\\Z} \\hat{f}(n), \\qquad\n\\hat{f}(p) = \\int_\\R f(x)e^{2\\pi i px}\\,dx\n$$\n\n\u304c\u6210\u7acb\u3059\u308b\u3068\u3044\u3046\u7d50\u679c\u3067\u3042\u3063\u305f. \u3053\u308c\u306e\u53f3\u8fba\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u5f62\u3067\u304d\u308b:\n\n$$\n\\begin{aligned}\n\\sum_{n\\in\\Z} \\hat{f}(n) &=\n\\sum_{n\\in\\Z} \\int_\\R f(x)e^{2\\pi i nx}\\,dx =\n\\int_\\R f(x)\\,dx + 2\\sum_{n=1}^\\infty\\int_\\R f(x)\\cos(2\\pi nx)\\,dx\n\\\\ &=\n\\int_\\R f(x)\\,dx - \\sum_{n=1}^\\infty\\int_\\R f'(x)\\frac{\\sin(2\\pi nx)}{\\pi n}\\,dx\n\\\\ &=\n\\int_\\R f(x)\\,dx + \\int_\\R \\left(-\\sum_{n=1}^\\infty\\frac{\\sin(2\\pi nx)}{\\pi n}\\right)f'(x)\\,dx\n\\end{aligned}\n$$\n\n2\u3064\u76ee\u306e\u7b49\u53f7\u3067\u306f $e^{2\\pi inx}+e^{-2\\pi inx}=2\\cos(2\\pi nx)$ \u3092\u7528\u3044, 3\u3064\u76ee\u306e\u7b49\u53f7\u3067\u306f\u90e8\u5206\u7a4d\u5206\u3092\u5b9f\u884c\u3057, 4\u3064\u76ee\u306e\u7b49\u53f7\u3067\u306f\u7121\u9650\u548c\u3068\u7a4d\u5206\u306e\u9806\u5e8f\u3092\u4ea4\u63db\u3057\u305f. \u305d\u308c\u3089\u306e\u64cd\u4f5c\u306f $f(x)$ \u304c\u6025\u6e1b\u5c11\u51fd\u6570\u3067\u3042\u308c\u3070\u5bb9\u6613\u306b\u6b63\u5f53\u5316\u3055\u308c\u308b. \n\n\u4e00\u65b9, Euler-Maclaurin\u306e\u548c\u516c\u5f0f\u306e\n\n$$\nB_1(x-\\lfloor x\\rfloor) = x - \\lfloor x\\rfloor - \\frac{1}{2}\n$$\n\n\u3092\u4f7f\u3046\u5834\u5408\u304b\u3089, \n\n$$\n\\sum_{m\\in\\Z} f(m) =\n\\int_\\R f(x)\\,dx + \\int_\\R \\left(x - \\lfloor x\\rfloor - \\frac{1}{2}\\right) f'(x)\\,dx\n$$\n\n\u304c\u5c0e\u304b\u308c\u308b. \u3053\u308c\u306f\u90e8\u5206\u7a4d\u5206\u306b\u3088\u3063\u3066\u5f97\u3089\u308c\u308b\u6b21\u306e\u516c\u5f0f\u304b\u3089\u305f\u3060\u3061\u306b\u5c0e\u304b\u308c\u308b\u6613\u3057\u3044\u516c\u5f0f\u3067\u3042\u308b\u3053\u3068\u306b\u3082\u6ce8\u610f\u305b\u3088:\n\n$$\n\\begin{aligned}\n\\int_n^{n+1} \\left(x - n - \\frac{1}{2}\\right) f'(x)\\,dx &=\n\\left[\\left(x - n - \\frac{1}{2}\\right)f(x)\\right]_n^{n+1} - \\int_n^{n+1}f(x)\\,dx - n\n\\\\ &=\n\\frac{f(n+1)-f(n)}{2} - \\int_n^{n+1}f(x)\\,dx.\n\\end{aligned}\n$$\n\n\u4ee5\u4e0a\u306e2\u3064\u306e\u7d50\u679c\u3092\u6bd4\u8f03\u3059\u308b\u3068, Poisson\u306e\u548c\u516c\u5f0f\u3068Euler-Maclaurin\u306e\u548c\u516c\u5f0f\u306e $B_1(x-\\lfloor x\\rfloor)$ \u3092\u4f7f\u3063\u305f\u5834\u5408\u306f, \n\n$$\nx - \\lfloor x\\rfloor - \\frac{1}{2} =\n-\\sum_{n=1}^\\infty\\frac{\\sin(2\\pi nx)}{\\pi n}\n\\tag{$*$}\n$$\n\n\u3068\u3044\u3046\u516c\u5f0f\u3067\u7d50\u3073\u4ed8\u3044\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u3053\u306e\u516c\u5f0f\u3092\u8a8d\u3081\u308c\u3070, Euler-Maclaurin\u306e\u548c\u516c\u5f0f\u306e $B_1(x-\\lfloor x\\rfloor)$ \u3092\u4f7f\u3063\u305f\u5834\u5408\u304b\u3089Poisson\u306e\u548c\u516c\u5f0f\u304c\u5c0e\u304b\u308c\u308b. \n\n\u516c\u5f0f($*$)\u306e\u5de6\u8fba\u306f\u3044\u308f\u3086\u308b**\u306e\u3053\u304e\u308a\u6ce2**\u3067\u3042\u308a, \u53f3\u8fba\u306f\u305d\u306eFourier\u7d1a\u6570\u3067\u3042\u308b. \u516c\u5f0f($*$)\u306fFourier\u7d1a\u6570\u8ad6\u306b\u304a\u3051\u308b\u975e\u5e38\u306b\u6709\u540d\u306a\u516c\u5f0f\u3067\u3042\u308a, \u672c\u8cea\u7684\u306b\u305d\u308c\u3068\u540c\u3058\u516c\u5f0f\u306fFourier\u7d1a\u6570\u8ad6\u306b\u3064\u3044\u3066\u66f8\u304b\u308c\u305f\u6587\u732e\u306b\u306f\u4f8b\u3068\u3057\u3066\u5fc5\u305a\u8f09\u3063\u3066\u3044\u308b\u3068\u8a00\u3063\u3066\u3088\u3044\u304f\u3089\u3044\u3067\u3042\u308b. (Fourier\u7d1a\u6570\u8ad6\u3088\u308a, \u516c\u5f0f($*$)\u306f $x$ \u304c\u6574\u6570\u3067\u306a\u3044\u3068\u304d\u306b\u306f\u5b9f\u969b\u306b\u6210\u7acb\u3057\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b.)\n\n\u3053\u306e\u3088\u3046\u306b, \u306e\u3053\u304e\u308a\u6ce2\u306eFourier\u7d1a\u6570\u5c55\u958b\u3068\u3044\u3046\u975e\u5e38\u306b\u7279\u6b8a\u306a\u516c\u5f0f\u306fPoisson\u306e\u548c\u516c\u5f0f\u3068\u3044\u3046\u4e00\u822c\u7684\u306a\u516c\u5f0f\u3092\u5c0e\u304f\u3060\u3051\u306e\u529b\u3092\u6301\u3063\u3066\u3044\u308b\u306e\u3067\u3042\u308b. \n\n**\u307e\u3068\u3081:** \u306e\u3053\u304e\u308a\u6ce2\u306eFourier\u7d1a\u6570\u5c55\u958b\u306f\u90e8\u5206\u7a4d\u5206\u3092\u901a\u3057\u3066Poisson\u306e\u548c\u516c\u5f0f\u3068\u672c\u8cea\u7684\u306b\u540c\u5024\u3067\u3042\u308b! $\\QED$\n\n\u3053\u306e\u7bc0\u3067\u89e3\u8aac\u3057\u305f\u3053\u3068\u306f\u6b21\u306e\u6587\u732e\u3067\u6307\u6458\u3055\u308c\u3066\u3044\u308b:\n\n* Tim Jameson, An elementary derivation of the Poisson summation formula\n\n\n```julia\nB_1(x) = x - 1/2\nb(x) = B_1(x - floor(x))\nS(N,x) = -sum(n->sin(2\u03c0*n*x)/(\u03c0*n), 1:N)\nx = -2:0.001:1.999\nN = 10\nplot(size=(400,200), ylim=(-0.6,1.2), legend=:top)\nplot!(x, b.(x), label=\"B_1(x-[x]) = x - [x] -1/2\")\nplot!(x, S.(N,x), label=\"partial sum of Fourier series (N=$N)\")\n```\n\n\n\n\n \n\n \n\n\n\n**\u88dc\u8db3:** \u3053\u306e\u30ce\u30fc\u30c8\u306e\u4e0a\u306e\u65b9\u306e\u5468\u671f\u7684Bernoulli\u591a\u9805\u5f0f $B_k(x-\\lfloor x\\rfloor)$ \u306eFourier\u7d1a\u6570\u5c55\u958b\u306e\u7bc0\u3092\u898b\u308c\u3070\u308f\u304b\u308b\u3088\u3046\u306b, \n\n$$\n\\sum_{n=1}^\\infty \\frac{\\cos(2\\pi nx)}{n^k}, \\quad\n\\sum_{n=1}^\\infty \\frac{\\sin(2\\pi nx)}{n^k}\n$$\n\n\u306e\u578b\u306eFourier\u7d1a\u6570\u306e\u53ce\u675f\u5148\u306f\u5e73\u884c\u79fb\u52d5\u3068\u5b9a\u6570\u500d\u306e\u9055\u3044\u3092\u9664\u3044\u3066\u5468\u671f\u7684Bernoulli\u591a\u9805\u5f0f\u306b\u306a\u308b. $\\QED$\n\n### \u53f0\u5f62\u516c\u5f0f\u3068Poisson\u306e\u548c\u516c\u5f0f\u306e\u95a2\u4fc2\n\n\u7c21\u5358\u306e\u305f\u3081 $f(x)$ \u306f $\\R$ \u4e0a\u306e\u6025\u6e1b\u5c11\u51fd\u6570\u3067\u3042\u308b\u3068\u3057, $a,b\\in\\Z$ \u304b\u3064 $a1$ \u306e\u3068\u304d(\u3088\u308a\u4e00\u822c\u306b\u306f $\\real s>1$ \u306e\u3068\u304d),\n\n$$\n\\zeta(s) = \\sum_{n=1}^\\infty \\frac{1}{n^s}\n$$\n\n\u306f\u7d76\u5bfe\u53ce\u675f\u3057\u3066\u3044\u308b\u306e\u3067\u3042\u3063\u305f. \u3053\u308c\u306bEuler-Maclaurin\u306e\u548c\u516c\u5f0f\n\n$$\n\\begin{aligned}\n&\n\\sum_{j=a}^b f(j) = \n\\int_a^b f(x)\\,dx + \\frac{f(a)+f(b)}{2} + \n\\sum_{k=2}^n \\frac{B_k}{k!} (f^{(k-1)}(b)-f^{(k-1)}(a)) + R_n,\n\\\\ &\nR_n = (-1)^{n-1}\\int_a^b \\frac{B_n(x-\\lfloor x\\rfloor)}{n!}f^{(n)}(x)\\,dx\n\\end{aligned}\n$$\n\n\u3092\u9069\u7528\u3057\u3066\u307f\u3088\u3046.\n\n### \u89e3\u6790\u63a5\u7d9a\n\n$\\real s > 1$ \u3067\u3042\u308b\u3068\u3057, $f(x)=x^{-s}$ \u3068\u304a\u304f. \u3053\u306e\u3068\u304d, \n\n$$\n\\begin{aligned}\n&\n\\int_a^\\infty f(x)\\,dx = \\int_1^\\infty x^{-s}\\,dx = \n\\left[\\frac{x^{-s+1}}{-s+1}\\right]_1^\\infty = \\frac{a^{-(s-1)}}{s-1}, \\qquad\nf(b)=b^{-s}\\to 0 \\quad(b\\to\\infty).\n\\\\ &\n\\frac{B_k}{k!}f^{(k-1)}(x) = \n\\frac{B_k}{k}\\binom{-s}{k-1} x^{-s-k+1}, \\quad\n\\frac{B_n(x-\\lfloor x\\rfloor)}{n!}f^{(n)}(x) = \n\\binom{-s}{n}B_n(x-\\lfloor x\\rfloor)x^{-s-n}\n\\end{aligned}\n$$\n\n\u306a\u306e\u3067, 2\u4ee5\u4e0a\u306e\u6574\u6570 $n$ \u306b\u3064\u3044\u3066,\n\n$$\n\\begin{aligned}\n&\n\\zeta(s) = \\frac{1}{s-1} + \\frac{1}{2} - \n\\sum_{k=2}^n \\frac{B_k}{k}\\binom{-s}{k-1} + R_n,\n\\\\ &\nR_n = (-1)^{n-1}\\binom{-s}{n}\\int_1^\\infty B_n(x-\\lfloor x\\rfloor)x^{-s-n}\\,dx.\n\\end{aligned}\n$$\n\n\u7a4d\u5206 $R_n$ \u306f $\\real s+n>1$ \u306a\u3089\u3070\u7d76\u5bfe\u53ce\u675f\u3057\u3066\u3044\u308b. \u3086\u3048\u306b, \u8907\u7d20\u5e73\u9762\u5168\u4f53\u306b $\\zeta(s)$ \u3092\u81ea\u7136\u306b\u62e1\u5f35\u3059\u308b\u65b9\u6cd5(\u89e3\u6790\u63a5\u7d9a\u3059\u308b\u65b9\u6cd5)\u304c\u5f97\u3089\u308c\u305f.\n\n$\\ds \\sum_{k=1}^\\infty \\frac{1}{n^s}$ \u305d\u306e\u3082\u306e\u3067\u306f\u306a\u304f, $n=a$ \u304b\u3089\u59cb\u307e\u308b\u7121\u9650\u548c $\\ds \\sum_{k=a}^\\infty \\frac{1}{n^s}=\\zeta(s)-\\sum_{n=1}^{a-1}\\frac{1}{n^s}$ \u306bEuler-Maclaurin\u306e\u548c\u516c\u5f0f\u3092\u9069\u7528\u3059\u308b\u3068,\n\n$$\n\\begin{aligned}\n&\n\\zeta(s) = \\sum_{n=1}^{a-1} \\frac{1}{n^s} - \\frac{a^{1-s}}{1-s} + \n\\frac{1}{2a^s} - \\sum_{k=2}^n \\frac{B_k}{k a^{s+k-1}}\\binom{-s}{k-1} + R_{n,a},\n\\\\ &\nR_{n,a} = (-1)^{n-1}\\binom{-s}{n}\\int_a^\\infty B_n(x-\\lfloor x\\rfloor)x^{-s-n}\\,dx.\n\\end{aligned}\n$$\n\n\n```julia\n# \u4e0a\u306e\u516c\u5f0f\u306b\u304a\u3051\u308b \u03b6(s) - R_{n,a} \u306e\u51fd\u6570\u5316\n\n# \u03b6(s) - R_{n,a} = \u03a3_{m=1}^{a-1} m^{-s} - a^{1-s}/(1-s) + 1/(2a^s)\n# - \u03a3_{k=2}^n B_k/(k a^{s+k-1}) binom(-s,k-1) (k is even)\n#\nfunction ApproxZeta(a, n, s)\n ss = float(big(s))\n z = zero(ss)\n z += (a \u2264 1 ? zero(ss) : sum(m->m^(-ss), 1:a-1)) # \u03a3_{m=1}^{a-1} m^{-s}\n z += -a^(1-ss)/(1-ss) # -a^{1-s}/(1-s)\n n == 0 && return z\n z += 1/(2*a^ss) # 1/(2a^s)\n n == 1 && return z\n z -= sum(k -> BB(k)/(k*a^(ss+k-1))*binom(-ss,k-1), 2:2:n)\n # -\u03a3_{k=2}^n B_k/(k a^{s+k-1}) binom(-s,k-1) (k is even)\nend\n\nA = ApproxZeta(40, 80, big\"0.5\")\nZ = zeta(big\"0.5\")\n@show A\n@show Z;\n```\n\n A = -1.460354508809586812889499152515298012467229331012581490542886087825530529474572\n Z = -1.460354508809586812889499152515298012467229331012581490542886087825530529474503\n\n\n$\\real s > 0$ \u306e\u3068\u304d, \n\n$$\n\\frac{1}{2a^s} - \\sum_{k=2}^n \\frac{B_k}{k a^{s+k-1}}\\binom{-s}{k-1} + R_{n,a}\n$$\n\n\u306f $a\\to\\infty$ \u3067 $0$ \u306b\u53ce\u675f\u3059\u308b\u306e\u3067,\n\n$$\n\\zeta(s) = \\lim_{a\\to\\infty}\\left(\\sum_{n=1}^{a-1} \\frac{1}{n^s} - \\frac{a^{1-s}}{1-s}\\right)\n\\quad (\\real s > 0)\n$$\n\n\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u3053\u308c\u306f, Dirichlet\u7d1a\u6570\u306e\u90e8\u5206\u548c $\\ds\\sum_{n=1}^{a-1}\\frac{1}{n^s}$ \u304b\u3089\u88dc\u6b63\u9805\n\n$$\n\\frac{a^{1-s}}{1-s}\n$$\n\n\u3092\u5f15\u304d\u53bb\u3063\u3066\u304b\u3089, Dirichlet\u7d1a\u6570\u306e\u7dcf\u548c\u3092\u53d6\u308c\u3070, $0 < \\real s < 1$ \u3067\u3082\u53ce\u675f\u3057\u3066, $\\zeta(s)$ \u306e\u6b63\u78ba\u306a\u5024\u304c\u5f97\u3089\u308c\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u3066\u3044\u308b.\n\n\n```julia\n# \u4e0a\u306e\u7d50\u679c\u306e\u30d7\u30ed\u30c3\u30c8\n\nApproxZeta0(a, s) = sum(n->n^(-s), 1:a-1) - a^(1-s)/(1-s)\na = 100\ns = 0.05:0.01:0.95\n@time z = zeta.(s)\n@time w = ApproxZeta0.(a, s)\nplot(size=(400, 250), legend=:bottomleft, xlabel=\"s\")\nplot!(s, z, label=\"zeta(s)\", lw=2)\nplot!(s, w, label=\"Euler-Maclaurin sum for n=0, a=$a\", lw=2, ls=:dash)\n```\n\n 0.057854 seconds (29.42 k allocations: 1.510 MiB)\n 0.049310 seconds (19.07 k allocations: 1007.852 KiB)\n\n\n\n\n\n \n\n \n\n\n\n\n```julia\n# \u3055\u3089\u306b\u9805\u306e\u6570\u30921\u3064\u5897\u3084\u3057\u305f\u5834\u5408\u306e\u30d7\u30ed\u30c3\u30c8\n\n# \u03b6(s) - R_{1,a} = \u03a3_{n=1}^{a-1} n^{-s} - a^{1-s}/(1-s) + 1/(2a^s)\n#\nApproxZeta1(a, s) = sum(n->n^(-s), 1:a-1) - a^(1-s)/(1-s) + 1/(2*a^s)\n\ns = -0.95:0.01:0.5\na = 10^3\n@time z = zeta.(s)\n@time w = ApproxZeta1.(a,s)\nplot(size=(400, 250), legend=:bottomleft, xlabel=\"s\")\nplot!(s, z, label=\"zeta(s)\", lw=2)\nplot!(s, w, label=\"Euler-Maclaurin sum for n=1, a=$a\", lw=2, ls=:dash)\n```\n\n 0.000181 seconds (29 allocations: 2.406 KiB)\n 0.058011 seconds (19.04 k allocations: 1005.415 KiB)\n\n\n\n\n\n \n\n \n\n\n\n\n```julia\n# \u3055\u3089\u306b\u4e00\u822c\u306e\u5834\u5408\u306e\u30d7\u30ed\u30c3\u30c8\n#\n# Euler-Maclaurin\u306e\u548c\u516c\u5f0f\u3067 \u03b6(s) \u306e\u8ca0\u306e s \u3067\u306e\u5024\u3092\u3074\u3063\u305f\u308a\u8fd1\u4f3c\u3067\u304d\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b.\n\n[(-m, zeta(-m), Float64(ApproxZeta(2, 17, -m))) for m = 0:12] |> display\n\nn = 10\ns = -1.5:0.05:0.5\na = 10\n@time z = zeta.(s)\n@time w = ApproxZeta.(a, n, s)\nP1 = plot(size=(400, 250), legend=:bottomleft, xlabel=\"s\")\nplot!(s, z, label=\"zeta(s)\", lw=2)\nplot!(s, w, label=\"Euler-Maclaurin sum for a=$a, n=$n\", lw=2, ls=:dash)\n\nn = 17\ns = -16:0.05:-2.0\na = 2\n@time z = zeta.(s)\n@time w = ApproxZeta.(a, n, s)\nP2 = plot(size=(400, 250), legend=:topright, xlabel=\"s\")\nplot!(s, z, label=\"zeta(s)\", lw=2)\nplot!(s, w, label=\"Euler-Maclaurin sum for a=$a, n=$n\", lw=2, ls=:dash)\n```\n\n\n 13-element Array{Tuple{Int64,Float64,Float64},1}:\n (0, -0.5, -0.5) \n (-1, -0.0833333, -0.0833333) \n (-2, -0.0, -1.29543e-77) \n (-3, 0.00833333, 0.00833333) \n (-4, -0.0, -3.45447e-77) \n (-5, -0.00396825, -0.00396825)\n (-6, -0.0, 0.0) \n (-7, 0.00416667, 0.00416667) \n (-8, -0.0, 0.0) \n (-9, -0.00757576, -0.00757576)\n (-10, -0.0, -4.42172e-75) \n (-11, 0.0210928, 0.0210928) \n (-12, -0.0, 0.0) \n\n\n 0.000099 seconds (29 allocations: 1.625 KiB)\n 0.181011 seconds (122.74 k allocations: 5.447 MiB)\n 0.000319 seconds (29 allocations: 3.563 KiB)\n 0.071032 seconds (565.80 k allocations: 20.997 MiB, 18.17% gc time)\n\n\n\n\n\n \n\n \n\n\n\n\n```julia\ndisplay(P1)\n```\n\n\n \n\n \n\n\n\u4e0a\u3068\u4e0b\u306e\u30b0\u30e9\u30d5\u3092\u898b\u308c\u3070\u308f\u304b\u308b\u3088\u3046\u306b, Euler-Maclaurin\u306e\u548c\u516c\u5f0f\u306b\u3088\u3063\u3066\u8ca0\u306e\u5b9f\u6570\u3067\u306e $\\zeta$ \u51fd\u6570\u306e\u5024\u3092\u975e\u5e38\u306b\u3088\u304f\u8fd1\u4f3c\u3067\u304d\u3066\u3044\u308b. \u5b9f\u306f $\\zeta(s)$ \u3092\u5b9f\u90e8\u304c\u8ca0\u306e\u8907\u7d20\u6570\u307e\u3067\u62e1\u5f35\u3057\u3066\u3082\u3053\u306e\u8fd1\u4f3c\u306f\u3046\u307e\u304f\u884c\u3063\u3066\u3044\u308b.\n\n\n```julia\ndisplay(P2)\n```\n\n\n \n\n \n\n\n### \u03b6(2)\u306e\u8fd1\u4f3c\u8a08\u7b97\n\n$\\ds\\zeta(2)=\\sum_{n=1}^\\infty \\frac{1}{n^2}$ \u3092\u8a08\u7b97\u305b\u3088\u3068\u3044\u3046\u554f\u984c\u306f**Basel\u554f\u984c**\u3068\u547c\u3070\u308c\u3066\u3044\u308b\u3089\u3057\u3044. Basel\u554f\u984c\u306fEuler\u306b\u3088\u3063\u30661743\u5e74\u3053\u308d\u306b\u89e3\u304b\u308c\u305f\u3089\u3057\u3044. Euler\u304c\u3069\u306e\u3088\u3046\u306b\u8003\u3048\u305f\u304b\u306b\u3064\u3044\u3066\u306f\u6b21\u306e\u6587\u732e\u3092\u53c2\u7167\u305b\u3088.\n\n* \u6749\u672c\u654f\u592b, \u30d0\u30fc\u30bc\u30eb\u554f\u984c\u3068\u30aa\u30a4\u30e9\u30fc, 2007\u5e748\u670823\u65e5, \u6570\u7406\u89e3\u6790\u7814\u7a76\u6240\u8b1b\u7a76\u9332, \u7b2c1583\u5dfb, 2008\u5e74, pp.159-167\n\nEuler\u306f $\\zeta(2)$ \u306e\u8fd1\u4f3c\u5024\u3092\u81ea\u3089\u958b\u767a\u3057\u305fEuler-Maclaurin\u306e\u548c\u516c\u5f0f\u3092\u4f7f\u3063\u3066\u7cbe\u5bc6\u306b\u8a08\u7b97\u3057\u305f\u3089\u3057\u3044.\n\n\u8fd1\u4f3c\u5f0f\n\n$$\n\\zeta(s) \\approx\n\\sum_{n=1}^{a-1} \\frac{1}{n^s} - \\frac{a^{1-s}}{1-s} + \n\\frac{1}{2a^s} - \\sum_{k=2}^n \\frac{B_k}{k a^{s+k-1}}\\binom{-s}{k-1} \n$$\n\n\u3092\u7528\u3044\u3066, $\\zeta(2)$ \u3092\u8a08\u7b97\u3057\u3066\u307f\u3088\u3046. 3\u4ee5\u4e0a\u306e\u5947\u6570 $n$ \u306b\u3064\u3044\u3066 $B_n=0$ \u3068\u306a\u308b\u306e\u3067, $n=2m$ \u306e\u3068\u304d, \u53f3\u8fba\u306e\u9805\u6570\u306f $a+m+1$ \u306b\u306a\u308b.\n\n\u4f8b\u3048\u3070, $a=10$, $m=9$ \u3068\u3057, 20\u9805\u306e\u548c\u3092\u53d6\u308b\u3068,\n\n$$\n\\zeta(2) \\approx 1.64493\\;40668\\;4749\\cdots\n$$\n\n\u3068\u306a\u308a, \u6b63\u78ba\u306a\u5024 $\\ds\\frac{\\pi^2}{6}=1.64493\\;40668\\;4822\\cdots$ \u3068\u5c0f\u6570\u70b9\u4ee5\u4e0b\u7b2c11\u6841\u307e\u3067\u4e00\u81f4\u3057\u3066\u3044\u308b. \n\nEuler\u306f\u5f8c\u306b $\\ds\\zeta(2)=\\frac{\\pi^2}{6}$ \u3092\u5f97\u308b. Euler\u306f\u7af6\u4e89\u76f8\u624b\u306b\u8b70\u8ad6\u306b\u53b3\u5bc6\u6027\u306b\u6b20\u3051\u308b\u3068\u3057\u3066\u69d8\u3005\u306a\u6279\u5224\u3092\u53d7\u3051\u305f\u306e\u3060\u304c, \u4ee5\u4e0a\u306e\u3088\u3046\u306a\u6570\u5024\u8a08\u7b97\u306e\u7d50\u679c\u3092\u77e5\u3063\u3066\u3044\u305f\u306e\u3067, \u6b63\u89e3\u3092\u5f97\u305f\u3068\u3044\u3046\u78ba\u4fe1\u306f\u5fae\u5875\u3082\u63fa\u3089\u304c\u306a\u304b\u3063\u305f\u3060\u308d\u3046\u3068\u601d\u308f\u308c\u308b.\n\n**\u6ce8\u610f:** \u8ad6\u7406\u7684\u306b\u53b3\u5bc6\u306a\u8a3c\u660e\u306e\u65b9\u6cd5\u304c\u767a\u9054\u3057\u305f\u73fe\u4ee3\u306b\u304a\u3044\u3066\u3082, \u4eba\u9593\u306f\u5e38\u306b\u8a3c\u660e\u3092\u9593\u9055\u3046\u53ef\u80fd\u6027\u304c\u3042\u308b. \u4eba\u9593\u304c\u884c\u3063\u305f\u8a3c\u660e\u306f\u7d76\u5bfe\u7684\u306b\u306f\u4fe1\u7528\u3067\u304d\u306a\u3044. \u3060\u304b\u3089, \u305f\u3068\u3048\u8a3c\u660e\u304c\u5b8c\u6210\u3057\u305f\u3068\u601d\u3063\u3066\u3044\u305f\u3068\u3057\u3066\u3082, \u53ef\u80fd\u306a\u3089\u3070\u6570\u5024\u8a08\u7b97\u306b\u3088\u3063\u3066\u8ad6\u7406\u7684\u306b\u53b3\u5bc6\u306a\u8a3c\u660e\u4ee5\u5916\u306e\u8a3c\u62e0\u3092\u4f5c\u3063\u3066\u3044\u305f\u65b9\u304c\u5b89\u5168\u3060\u3068\u601d\u308f\u308c\u308b. $\\QED$\n\n**\u6ce8\u610f:** \u6570\u5b66\u306e\u30ce\u30fc\u30c8\u3092\u4f5c\u308a\u306a\u304c\u3089, \u6c17\u8efd\u306b\u6570\u5024\u7684\u8a3c\u62e0\u3082\u540c\u6642\u306b\u5f97\u308b\u305f\u3081\u306e\u9053\u5177\u3068\u3057\u3066, \u7b46\u8005\u304c\u3053\u306e\u30ce\u30fc\u30c8\u4f5c\u6210\u306e\u305f\u3081\u306b\u7528\u3044\u3066\u3044\u308bJulia\u8a00\u8a9e\u3068Jupyter\u3068Nbextensions\u306eLive Markdown Preview\u306f\u3053\u308c\u3092\u66f8\u3044\u3066\u3044\u308b\u6642\u70b9\u3067\u76f8\u5f53\u306b\u512a\u79c0\u306a\u9053\u5177\u3067\u3042\u308b\u3088\u3046\u306b\u601d\u308f\u308c\u308b. $\\QED$\n\n\n```julia\n# 20\u9805\u306e\u548c\n\nN = 20\n[(m, N-m-1, 2m, ApproxZeta(N-m-1, 2m, 2) - big(\u03c0)^2/6) for m in 2:N\u00f72-1] |> display\n\nm = 9\na = N-m-1\nZ = big(\u03c0)^2/6\nA = ApproxZeta(a, m, 2)\n@show a,m\n@show Z\n@show A;\n```\n\n\n 8-element Array{Tuple{Int64,Int64,Int64,BigFloat},1}:\n (2, 17, 4, -5.77451793863474833797788940478358503699585407578399357681001619323664145261758e-11) \n (3, 16, 6, 4.808127352395625095013460112150325878389866054958153137408430487774776509324829e-13) \n (4, 15, 8, -8.630887513943044224615236465465206970650911046136527708026292186723865816966492e-15) \n (5, 14, 10, 3.116217978527385328054235573023871173466586797186396436897662720414552852297451e-16) \n (6, 13, 12, -2.200847274100542514575619216657515798396053843860275532661691594239630209744406e-17)\n (7, 12, 14, 3.035248943857815147777677383711316694019656935103319432181355248871820406062584e-18) \n (8, 11, 16, -8.335321043122531064769674746337938450627967961329547742403411230422546148897753e-19)\n (9, 10, 18, 4.746601814392005312714027578027970306539540935051342164737224161514796063021067e-19) \n\n\n (a, m) = (10, 9)\n Z = 1.644934066848226436472415166646025189218949901206798437735558229370007470403185\n A = 1.644934066847493071302595112118921642731166540690350214159737969261778785588307\n\n\n### s = 1\u3067\u306e\u03b6(s)\u306e\u5b9a\u6570\u9805\u304cEuler\u5b9a\u6570\u306b\u306a\u308b\u3053\u3068\n\n$\\zeta(s)=\\ds\\sum_{n=1}^\\infty \\frac{1}{n^s}$ \u306bEuler-Maclaurin\u306e\u548c\u516c\u5f0f\u3092\u4f7f\u3063\u3066, 2\u4ee5\u4e0a\u306e $n$ \u306b\u3064\u3044\u3066\u6b21\u306e\u516c\u5f0f\u304c\u5f97\u3089\u308c\u308b\u306e\u3067\u3042\u3063\u305f:\n\n$$\n\\begin{aligned}\n&\n\\zeta(s) = \\frac{1}{s-1} + \\frac{1}{2} - \n\\sum_{k=2}^n \\frac{B_k}{k}\\binom{-s}{k-1} + R_n,\n\\\\ &\nR_n = (-1)^{n-1}\\binom{-s}{n}\\int_1^\\infty B_n(x-\\lfloor x\\rfloor)x^{-s-n}\\,dx.\n\\end{aligned}\n$$\n\n$n=1$ \u306e\u5834\u5408\u306b\u306f\n\n\\begin{aligned}\n\\sum_{j=a}^b f(j) &= \n\\int_a^b f(x)\\,dx + f(a) + \\int_a^b (x-\\lfloor x\\rfloor)f'(x)\\,dx\n\\\\ &=\n\\int_a^b f(x)\\,dx + f(a) + \\sum_{j=a}^{b-1}\\int_0^1 x f'(x+j)\\,dx\n\\end{aligned}\n\n\u3092 $f(x)=x^{-s}$, $f'(x)=-sx^{-s-1}$, $a=1$, $b=\\infty$ \u306e\u5834\u5408\u306b\u9069\u7528\u3057\u3066,\n\n$$\n\\zeta(s) = \n\\frac{1}{s-1} + 1 - s\\sum_{j=1}^\\infty\\int_0^1 \\frac{x}{(x+j)^{s+1}}\\,dx\n$$\n\n\u3092\u5f97\u308b. \u3057\u305f\u304c\u3063\u3066,\n\n$$\n\\lim_{s\\to 1}\\left(\\zeta(s)-\\frac{1}{s-1}\\right) =\n1 - \\sum_{j=1}^\\infty\\int_0^1 \\frac{x}{(x+j)^2}\\,dx.\n$$\n\n\u305d\u3057\u3066, $x=t-j$ \u3068\u7f6e\u63db\u3059\u308b\u3068, \n\n$$\n\\begin{align}\n-\\int_0^1\\frac{x}{(x+j)^2}\\,dx &= \n-\\int_j^{j+1}\\frac{-(t-j)}{t^2}\\,dt = \n-\\left[\\log t + \\frac{j}{t}\\right]_j^{j+1} \n\\\\ &=\n-\\log(j+1)+\\log j -\\frac{j}{j+1}+1 =\n\\frac{1}{j+1} + \\log j - \\log(j+1)\n\\end{align}\n$$\n\n\u306a\u306e\u3067, \u3053\u308c\u3092 $j=1$ \u304b\u3089 $j=N-1$ \u307e\u3067\u8db3\u3057\u4e0a\u3052\u308b\u3053\u3068\u306b\u3088\u3063\u3066,\n\n$$\n1 - \\sum_{j=1}^{n-1}\\int_0^1\\frac{x}{(x+j)^2}\\,dx =\n\\sum_{j=1}^N\\frac{1}{j} - \\log N.\n$$\n\n\u3053\u308c\u306e $N\\to\\infty$ \u3067\u306e\u6975\u9650\u306fEuler\u5b9a\u6570 $\\gamma=0.5772\\cdots$ \u306e\u5b9a\u7fa9\u3067\u3042\u3063\u305f. \u4ee5\u4e0a\u306b\u3088\u3063\u3066\u6b21\u304c\u793a\u3055\u308c\u305f:\n\n$$\n\\lim_{s\\to 1}\\left(\\zeta(s)-\\frac{1}{s-1}\\right) = \\gamma = 0.5772\\cdots.\n$$\n\n### \u8ca0\u306e\u6574\u6570\u306b\u304a\u3051\u308b\u30bc\u30fc\u30bf\u51fd\u6570\u306e\u7279\u6b8a\u5024\u306e\u8a08\u7b97\n\nEuler-Maclaurin\u306e\u548c\u516c\u5f0f: $3$ \u4ee5\u4e0a\u306e\u6574\u6570 $k$ \u306b\u3064\u3044\u3066 $B_k=0$ \u306a\u306e\u3067, \u4ee5\u4e0b\u306e\u516c\u5f0f\u3067 $k$ \u306f\u5076\u6570\u306e\u307f\u3092\u52d5\u304f\u3068\u3057\u3066\u3088\u3044:\n\n$$\n\\begin{aligned}\n&\n\\sum_{n=a}^b f(n) = \n\\int_a^b f(x)\\,dx + \\frac{f(a)+f(b)}{2} + \n\\sum_{k=2}^m \\frac{B_k}{k!}(f^{(k-1)}(b) - f^{(k-1)}(a)) + R_m,\n\\\\ &\nR_n = (-1)^{m-1}\\int_a^b \\frac{\\tilde{B}_m(x)}{m!} f^{(m)}(x)\\,dx.\n\\end{aligned}\n$$\n\n\u3053\u3053\u3067 $\\tilde{B}_m(x)=B_m(x-\\lfloor x\\rfloor)$ \u3068\u304a\u3044\u305f.\n\nEuler-Maclaurin\u306e\u548c\u516c\u5f0f\u3092 $f(x)=n^{-s}$, $a=1$, $b=\\infty$ \u306e\u5834\u5408\u306b\u9069\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066 $\\zeta(s)$ \u306f\u6b21\u306e\u5f62\u3067 $\\Re s > 1-m$ \u307e\u3067\u81ea\u7136\u306b\u5ef6\u9577(\u89e3\u6790\u63a5\u7d9a)\u3055\u308c\u308b\u306e\u3067\u3042\u3063\u305f:\n\n$$\n\\zeta(s) = \n\\frac{1}{s-1} + \\frac{1}{2} -\n\\frac{1}{1-s}\\sum_{k=2}^m \\binom{1-s}{k} B_k + \n(-1)^{m-1}\\int_a^b \\binom{-s}{m} \\tilde{B}_m(x) x^{-s-m}\\,dx.\n$$\n\n\u3053\u306e\u516c\u5f0f\u3068 $k\\geqq 2$ \u306e\u3068\u304d $\\ds\\binom{1}{k}=0$ \u3068\u306a\u308b\u3053\u3068\u3088\u308a, \n\n$$\n\\zeta(0) = \\frac{1}{0-1} + \\frac{1}{2} = -\\frac{1}{2}.\n$$\n\n$r$ \u306f\u6b63\u306e\u6574\u6570\u3067\u3042\u308b\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $m>r$ \u3068\u3059\u308b\u3068 $\\ds\\binom{r}{m}=0$ \u3068\u306a\u308b\u306e\u3067, $B_0=1$, $B_1=-1/2$ \u306a\u306e\u3067,\n\n$$\n\\begin{aligned}\n\\zeta(-r) &=\n-\\frac{1}{r+1} + \\frac{1}{2} -\n\\frac{1}{r+1}\\sum_{k=2}^{r+1} \\binom{m+1}{k} B_k\n\\\\ =&\n-\\frac{1}{r+1}\\sum_{k=0}^{r+1} \\binom{m+1}{k} B_k =\n-\\frac{B_{r+1}}{r+1}.\n\\end{aligned}\n$$\n\n\u6700\u5f8c\u306e\u7b49\u53f7\u3067, Bernoulli\u6570\u3092\u5e30\u7d0d\u7684\u306b\u8a08\u7b97\u3059\u308b\u305f\u3081\u306b\u4f7f\u3048\u308b\u516c\u5f0f $\\ds\\sum_{k=0}^r \\binom{r+1}{k}B_k=0$ \u3092\u7528\u3044\u305f. \u4f8b\u3048\u3070, $r=1$ \u306e\u3068\u304d $B_0+2B_1=1+2(-1/2)=0$ \u3068\u306a\u308a, $r=2$ \u306e\u3068\u304d, $B_0+3B_1+3B_2=1+3(-1/2)+3(1/6)=0$ \u3068\u306a\u308b.\n\n\u4ee5\u4e0a\u306b\u3088\u3063\u3066\u6b21\u304c\u8a3c\u660e\u3055\u308c\u305f:\n\n$$\n\\zeta(0)=-\\frac{1}{2}, \\quad\n\\zeta(-r) = -\\frac{B_{r+1}}{r+1} \\quad (r=1,2,3,\\ldots).\n$$\n\n\u3053\u308c\u3089\u306e\u516c\u5f0f\u306f $B_n(1)=B_n+\\delta_{n,1}$, $B_1=-1/2$ \u3092\u4f7f\u3046\u3068, \n\n$$\n\\zeta(-r) = -\\frac{B_{r+1}(1)}{r+1} \\quad (r=0,1,2,\\ldots)\n$$\n\n\u306e\u5f62\u306b\u307e\u3068\u3081\u3089\u308c\u308b.\n\n### \u767a\u6563\u7d1a\u6570\u306e\u6709\u9650\u90e8\u5206\u3068 \u03b6(-r) \u306e\u95a2\u4fc2\n\n\u524d\u7bc0\u306e\u7d50\u679c $\\ds\\zeta(-r)=-\\frac{B_{r+1}(1)}{r+1}$ ($r=0,1,2,\\ldots$) \u306f\n\n$$\n\\begin{aligned}\n&\n1+1+1+1+\\cdots = -\\frac{1}{2},\n\\\\ &\n1+2+3+4+\\cdots = -\\frac{1}{12}\n\\end{aligned}\n$$\n\n\u306e\u3088\u3046\u306a\u5370\u8c61\u7684\u306a\u5f62\u5f0f\u3067\u66f8\u304b\u308c\u308b\u3053\u3068\u3082\u3042\u308b. \u305f\u3060\u3057, \u305d\u306e\u5834\u5408\u306b\u306f\u5de6\u8fba\u304c\u901a\u5e38\u306e\u7121\u9650\u548c\u3067\u306f\u306a\u304f, \u30bc\u30fc\u30bf\u51fd\u6570 $\\zeta(s)$ \u306e\u89e3\u6790\u63a5\u7d9a\u306e\u610f\u5473\u3067\u3042\u308b\u3053\u3068\u3092\u4e86\u89e3\u3057\u3066\u304a\u304b\u306a\u3051\u308c\u3070\u3044\u3051\u306a\u3044. \n\n\u5b9f\u306f\u3055\u3089\u306b\u89e3\u6790\u63a5\u7d9a\u3068\u3057\u3066\u7406\u89e3\u3059\u308b\u3060\u3051\u3067\u306f\u306a\u304f, \u300c\u5de6\u8fba\u306e\u767a\u6563\u3059\u308b\u7121\u9650\u548c\u304b\u3089\u9069\u5207\u306b\u7121\u9650\u5927\u3092\u5f15\u304d\u53bb\u308c\u3070\u53f3\u8fba\u306b\u7b49\u3057\u304f\u306a\u308b\u300d\u3068\u3044\u3046\u3088\u3046\u306a\u30bf\u30a4\u30d7\u306e\u547d\u984c\u3092\u3046\u307e\u304f\u4f5c\u308b\u3053\u3068\u3082\u3067\u304d\u308b. \u4ee5\u4e0b\u3067\u306f\u305d\u306e\u3053\u3068\u3092\u89e3\u8aac\u3057\u3088\u3046.\n\n\u4ee5\u4e0b, $\\eta$ \u306f\u975e\u8ca0\u306e\u5b9f\u6570\u306b\u5024\u3092\u6301\u3064 $\\R$ \u4e0a\u306e**\u6025\u6e1b\u5c11\u51fd\u6570**\u3067\u3042\u308b\u3068\u4eee\u5b9a\u3059\u308b. ($\\R$ \u4e0a\u306e\u6025\u6e1b\u5c11\u51fd\u6570\u3068\u306f $\\R$ \u4e0a\u306e $C^\\infty$ \u51fd\u6570\u3067\u305d\u308c\u81ea\u8eab\u304a\u3088\u3073\u305d\u306e\u3059\u3079\u3066\u306e\u968e\u6570\u306e\u5c0e\u51fd\u6570\u306b\u4efb\u610f\u306e\u591a\u9805\u5f0f\u51fd\u6570\u3092\u304b\u3051\u305f\u3082\u306e\u304c $|x|\\to\\infty$ \u3067 $0$ \u306b\u53ce\u675f\u3059\u308b\u3082\u306e\u306e\u3053\u3068\u3067\u3042\u308b.) \u3055\u3089\u306b, \n\n$$\n\\eta(0)=1, \\quad \\eta'(0)=0\n$$\n\n\u3068\u4eee\u5b9a\u3059\u308b. \u4f8b\u3048\u3070 $\\eta(x)=e^{-x^2}$ \u306f\u305d\u306e\u3088\u3046\u306a\u51fd\u6570\u306e\u4f8b\u306b\u306a\u3063\u3066\u3044\u308b.\n\n\u3053\u306e\u3068\u304d, $\\eta(x)$ \u304c\u6025\u6e1b\u5c11\u51fd\u6570\u3067\u3042\u308b\u3053\u3068\u3088\u308a, $N>0$ \u306e\u3068\u304d, \u7d1a\u6570\n\n$$\n\\sum_{n=1}^\\infty n^r \\eta(n/N) = 1^r\\eta(1/N) + 2^r\\eta(2/N) + 3^r\\eta(3/N) + \\cdots\n$$\n\n\u306f\u5e38\u306b\u7d76\u5bfe\u53ce\u675f\u3059\u308b. $r$ \u304c\u975e\u8ca0\u306e\u6574\u6570\u306e\u3068\u304d, $N\\to\\infty$ \u3068\u3059\u308b\u3068, \u3053\u306e\u7d1a\u6570\u306f\u767a\u6563\u7d1a\u6570 $1^r+2^r+3^r+\\cdots$ \u306b\u306a\u3063\u3066\u3057\u307e\u3046. \u4ee5\u4e0b\u306e\u76ee\u6a19\u306f, Euler-Maclaurin\u306e\u548c\u516c\u5f0f\u3092\u4f7f\u3046\u3068, \u305d\u306e $N\\to\\infty$ \u3067\u306e\u767a\u6563\u90e8\u5206\u304c $CN^{r+1}$ ($C$ \u306f $\\eta$ \u3068 $r$ \u3067\u5177\u4f53\u7684\u306b\u6c7a\u307e\u308b\u5b9a\u6570) \u306e\u5f62\u306b\u307e\u3068\u307e\u308b\u3053\u3068\u3092\u793a\u3059\u3053\u3068\u3067\u3042\u308b. \u305d\u3057\u3066, \u6b8b\u3063\u305f\u6709\u9650\u90e8\u5206\u306f**\u5e38\u306b** $\\zeta(-r)$ \u306b\u53ce\u675f\u3059\u308b\u3053\u3068\u3082\u793a\u3055\u308c\u308b.\n\n$\\tilde{B}_n(x)=B_n(x-\\lfloor x\\rfloor)$ \u3068\u66f8\u304f\u3053\u3068\u306b\u3059\u308b.\n\n\u3053\u306e\u3068\u304d, $f(x)=\\eta(x/N)$ \u306bEuler-Maclaurin\u306e\u548c\u516c\u5f0f\u3092\u9069\u7528\u3059\u308b\u3068, $f(0)=1$, $f'(0)=f(\\infty)=f'(\\infty)=0$ \u3088\u308a, \n$$\n\\begin{aligned}\n1+\\sum_{n=1}^\\infty\\eta(x/N) &= \n\\sum_{n=0}^\\infty\\eta(x/N) \n\\\\ &= \n\\int_0^\\infty\\eta(x/N)\\,dx + \\frac{1}{2} +B_2(f'(\\infty)-f'(0)) - \n\\int_0^\\infty\\frac{\\tilde{B}_2(x)}{2!}\\frac{1}{N^2}\\eta''(x/N)\\,dx\n\\\\ &=\nN\\int_0^\\infty\\eta(y)\\,dy + \\frac{1}{2} -\n\\frac{1}{N}\\int_0^\\infty\\frac{\\tilde{B}_2(Ny)}{2!}\\eta''(y)\\,dy.\n\\end{aligned}\n$$\n\n\u3086\u3048\u306b, $\\zeta(0)=-1/2$ \u3092\u4f7f\u3046\u3068,\n\n$$\n\\sum_{n=1}^\\infty\\eta(x/N) - N\\int_0^\\infty\\eta(y)\\,dy =\n\\zeta(0) + O(1/N).\n$$\n\n\u3053\u308c\u306f $N\\to\\infty$ \u3067\u767a\u6563\u7d1a\u6570 $1+1+1+1+\\cdots$ \u306b\u306a\u308b\u7121\u9650\u548c $\\ds \\sum_{n=1}^\\infty\\eta(x/N)$ \u304b\u3089, \u305d\u306e\u767a\u6563\u90e8\u5206 $\\ds N\\int_0^\\infty\\eta(y)\\,dy$ \u3092\u5f15\u304d\u53bb\u3063\u3066, $N\\to\\infty$ \u306e\u6975\u9650\u3092\u53d6\u308b\u3068, $\\zeta(0)$ \u306b\u53ce\u675f\u3059\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u3066\u3044\u308b. \u3053\u308c\u304c\u6b32\u3057\u3044\u7d50\u679c\u306e1\u3064\u76ee\u3067\u3042\u308b.\n\n$r$ \u306f\u6b63\u306e\u6574\u6570\u3067\u3042\u308b\u3068\u3057, $f(x)=x^r\\eta(x/N)$ \u3068\u304a\u304f. \u305d\u306e\u3068\u304d, Leibnitz\u5247\n\n$$\n(\\varphi(x) \\psi(x))^{(m)} = \\sum_{i=0}^r \\binom{m}{i}\\varphi^{(i)}(x)\\psi^{(m-i)}(x)\n\\\\\n$$\n\n\u3092\u4f7f\u3046\u3068,\n\n$$\nf^{(r+2)}(x) = \\frac{1}{N^2}F(x/N), \\quad\nF(y) = \\binom{r+2}{0}y^r\\eta^{(r+2)}(y) + \\cdots + \\binom{r+2}{r}r!\\eta(y)\n$$\n\n\u305d\u306e $f(x)$ \u306bEuler-Maclaurin\u306e\u548c\u516c\u5f0f\u3092\u9069\u7528\u3059\u308b\u3068, $f^{(k)}(\\infty)=f^{(k)}(\\infty)=0$ \u304a\u3088\u3073,\n\n$$\nf(0) = f'(0) = \\cdots = f^{(r-1)}(0) = f^{(r+1)}(0) = 0, \\quad\nf^{(r)}(0) = r!\n$$\n\n\u3088\u308a, \n\n$$\n\\begin{aligned}\n\\sum_{n=1}^\\infty n^r\\eta(n/N) &=\n\\sum_{n=0}^\\infty f(n) =\n\\int_0^\\infty f(x)\\,dx - \\frac{B_{r+1}}{(r+1)!}r! - \\frac{B_{r+2}}{(r+2)!}0 + \n(-1)^{r+1}\\int_0^\\infty \\frac{\\tilde{B}_{r+2}(x)}{(r+2)!} f^{(r+2)}(x)\\,dx\n\\\\ &=\nN^{r+1}\\int_0^\\infty y^r\\eta(y)\\,dy - \\frac{B_{r+1}}{r+1} +\n(-1)^{r+1}\\frac{1}{N}\\int_0^\\infty \\frac{\\tilde{B}_{r+2}(Ny)}{(r+2)!} F(y)\\,dy\n\\\\ &=\nN^{r+1}\\int_0^\\infty y^r\\eta(y)\\,dy - \\frac{B_{r+1}}{r+1} + O(1/N).\n\\end{aligned}\n$$\n\n\u3086\u3048\u306b, $\\ds\\zeta(-r)=-\\frac{B_{r+1}}{r+1}$ \u3092\u4f7f\u3046\u3068,\n\n$$\n\\sum_{n=1}^\\infty n^r\\eta(n/N) - N^{r+1}\\int_0^\\infty y^r\\eta(y)\\,dy =\n\\zeta(-r) + O(1/N).\n$$\n\n\u3053\u308c\u306f $N\\to\\infty$ \u3067\u767a\u6563\u7d1a\u6570 $1^r+2^r+3^r+4^r+\\cdots$ \u306b\u306a\u308b\u7121\u9650\u548c $\\ds \\sum_{n=1}^\\infty n^r\\eta(x/N)$ \u304b\u3089, \u305d\u306e\u767a\u6563\u90e8\u5206 $\\ds N^{r+1}\\int_0^\\infty y^r\\eta(y)\\,dy$ \u3092\u5f15\u304d\u53bb\u3063\u3066, $N\\to\\infty$ \u306e\u6975\u9650\u3092\u53d6\u308b\u3068, $\\zeta(-r)$ \u306b\u53ce\u675f\u3059\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u3066\u3044\u308b. \u3053\u308c\u304c\u6b32\u3057\u3044\u7d50\u679c\u3067\u3042\u308b.\n\n**\u6ce8\u610f:** \u4ee5\u4e0a\u306e\u8a08\u7b97\u306e\u30dd\u30a4\u30f3\u30c8\u306f, \u975e\u8ca0\u306e\u6025\u6e1b\u5c11\u51fd\u6570 $\\eta(x)$ \u3067 $\\eta(0)=1$, $\\eta'(0)=0$ \u3092\u6e80\u305f\u3059\u3082\u306e\u3067\u767a\u6563\u7d1a\u6570\u3092\u6b63\u5247\u5316\u3057\u3066\u5f97\u3089\u308c\u308b\u7d1a\u6570\u306e\u5834\u5408\u306b\u306f, Euler-Maclaurin\u306e\u548c\u516c\u5f0f\u306e\u300c\u9014\u4e2d\u306e\u9805\u300d\u304c\u307b\u3068\u3093\u3069\u6d88\u3048\u3066\u3057\u307e\u3046\u3053\u3068\u3067\u3042\u308b. $C N^{r+1}$ \u578b\u306e\u767a\u6563\u9805\u3068\u5b9a\u6570\u9805\u3068 $O(1/N)$ \u306e\u90e8\u5206\u306e3\u3064\u306e\u9805\u3057\u304b\u751f\u304d\u6b8b\u3089\u306a\u3044. $\\QED$\n\n**\u6ce8\u610f:** \u4ee5\u4e0a\u306e\u7d50\u679c\u306b\u95a2\u3059\u308b\u3088\u308a\u9032\u3093\u3060\u89e3\u8aac\u306b\u3064\u3044\u3066\u306f\u6b21\u306e\u30ea\u30f3\u30af\u5148\u3092\u53c2\u7167\u305b\u3088:\n\n* Terence Tao, The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, Blog: What's new, 10 April, 2010.\n\n\u3053\u306e\u30d6\u30ed\u30b0\u8a18\u4e8b\u306f\u304b\u306a\u308a\u8aad\u307f\u6613\u3044. $\\QED$\n\n**\u554f\u984c:** \u4ee5\u4e0a\u306e\u7d50\u679c\u3092\u6570\u5024\u8a08\u7b97\u3067\u3082\u78ba\u8a8d\u3057\u3066\u307f\u3088. $\\QED$\n\n**\u30d2\u30f3\u30c8:** $\\eta(x)=e^{-x^2}$ \u306e\u5834\u5408\u3092\u8a66\u3057\u3066\u307f\u3088. \u305d\u306e\u3068\u304d,\n\n$$\n\\int_0^\\infty y^r\\eta(y)\\,dy = \n\\int_0^\\infty y^r e^{-y^2}\\,dy = \n\\frac{1}{2}\\Gamma\\left(\\frac{r+1}{2}\\right)\n$$\n\n\u3068\u306a\u3063\u3066\u3044\u308b. $\\QED$\n\n**\u89e3\u7b54\u4f8b:** \u6b21\u306e\u30ea\u30f3\u30af\u5148\u306e\u30ce\u30fc\u30c8\u3092\u898b\u3088.\n\n* \u9ed2\u6728\u7384, \u03b6(s) \u306e Re s \uff1c 1 \u3067\u306e\u69d8\u5b50 $\\QED$\n\n\n```julia\ny = symbols(\"y\", real=true)\nr = symbols(\"r\", positive=true)\nintegrate(y^r*exp(-y^2), (y, 0, oo))\n```\n\n\n\n\n$$\\frac{1}{2} \\Gamma{\\left(\\frac{r}{2} + \\frac{1}{2} \\right)}$$\n\n\n\n\n```julia\n\n```\n", "meta": {"hexsha": "a7a0c6e7a87803ecdeb67c1383acf6362fbd2c6e", "size": 791099, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Old_Ver_for_Julia_v0.6/13 Euler-Maclaurin summation formula.ipynb", "max_stars_repo_name": "genkuroki/Calculus", "max_stars_repo_head_hexsha": "424ef53bf493242ce48c58ba39e43b8e601eb403", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 20, "max_stars_repo_stars_event_min_datetime": "2018-06-22T13:24:20.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-17T00:04:57.000Z", "max_issues_repo_path": "Old_Ver_for_Julia_v0.6/13 Euler-Maclaurin summation formula.ipynb", "max_issues_repo_name": "genkuroki/Calculus", "max_issues_repo_head_hexsha": "424ef53bf493242ce48c58ba39e43b8e601eb403", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Old_Ver_for_Julia_v0.6/13 Euler-Maclaurin summation formula.ipynb", "max_forks_repo_name": "genkuroki/Calculus", "max_forks_repo_head_hexsha": "424ef53bf493242ce48c58ba39e43b8e601eb403", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 5, "max_forks_repo_forks_event_min_datetime": "2019-12-28T19:57:41.000Z", "max_forks_repo_forks_event_max_datetime": "2021-09-06T23:23:46.000Z", "avg_line_length": 102.7134510517, "max_line_length": 3580, "alphanum_fraction": 0.6439990444, "converted": true, "num_tokens": 30417, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.7606506418255927, "lm_q1q2_score": 0.4392722201798928}} {"text": "

    ✅ Put your name here

    \n\n#

    In-Class Assignment 23: Classical and Quantum Bits

    \n\nIn yesterday's pre-class assignment, we got introduced to bits and qubits. Today, we'll get more practice working with both units of information.\n\n##

    Itinerary

    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    AssignmentTopicDescription
    Pre Class 23Background for Quantum ComputingHow Computers Store Information
    In Class 23Classsical and Quantum BitsInformation in Quantum States
    Pre Class 24Software for Quantum ComputingHigh Level Software and the Circuit Model
    In Class 24Programming Quantum ComputersManipulating Quantum Bits to Perform Useful Computations
    \n\n##

    Learning Goals for Today's In-Class Assignment

    \n\nThe purpose of this notebook is to understand how bits are used in classical computers and how qubits are used in quantum computers. In particular, by the end of today's assignment, you should:\n\n1. Know the difference between a bit and a qubit. (And thus the difference between classical and quantum computing.)\n1. Understand how information is stored and retrieved from bits, and what operations can be done on a bit.\n1. Understand how information is stored and retrieved from qubits, and what operations can be done on a qubit.\n\n\n```python\n\"\"\"Imports for the assignment.\"\"\"\nimport numpy as np\nimport matplotlib.pyplot as plt\n```\n\n##

    Recap of the Pre-Class Assignment

    \n\nIn the pre-class assignment, we learned that all information is stored in bits on (classical) computers. The key difference in quantum computers is that they store information in quantum bits, or qubits.\n\nTo get a deeper understanding of qubits, we reviewed/learned three important topics:\n\n* A complex number has real and imaginary parts and a complex conjugate that allows us to compute it's modulus squared.\n\n* A probability distribution is a list of numbers (or vector, now that we know what that is) that add up to one. \n\n* A vector is a list of numbers (real, complex, etc.) that we can form linear combinations, or superpositions, with.\n\nQuestion: Is this a valid probability distribution?\n\n\n```python\n\"\"\"Exercise: is this a valid probability distribution?\"\"\"\ndistribution = np.array([0.6, 0.4])\n```\n\n✎ **Answer:** Erase the contents of this cell and put your answer here!\n\nQuestion: If you answered yes, what could this probability distribution represent? (Give an example.)\n\n✎ **Answer:** Erase the contents of this cell and put your answer here!\n\nQuestion: What's the complex conjugate and modulus squared of\n\n\\begin{equation}\n \\alpha = 2 + 4i?\n\\end{equation}\n\n✎ **Answer:** Write code in the following cell to answer the above question.\n\n\n```python\n\"\"\"Exercise: compute the complex conjugate and modulus squared of the given complex number.\"\"\"\n# complex number\nalpha = 2 + 4j\n\n# TODO: compute and print out the complex conjugate\n\n\n# TODO: compute and print out the modulus squared\n\n```\n\n#

    Working with Bits

    \n\nRemember that a bit can have the values of either 1 or 0 (True or False, on or off, yes or no.) All information on (classical) computers is stored in bits. Computers process information by operating on bits.\n\nQuestion: How can you encode one letter of the alphabet (a, b, c, ..., x, y, z) using only bits? How many bits do you need at minimum?\n\n✎ **Answer:** Erase the contents of this cell and put your answer here!\n\nQuestion: In our laptops, bits are represented by electrical signals. Think of other physical systems that we could use to represent bits of information. List as many as you can (at least three).\n\n✎ **Answer:** Erase the contents of this cell and put your answer here!\n\n##

    Writing a Bit Class

    \n\nNow that we know what binary digits are and how to use them to represent information (like letters in the alphabet), let's do a bit of coding (excuse the pun). Specifically, let's code up a `Bit` class to understand them better. A skeleton of the class is provided below, with some methods implemented, which you should not change.\n\nYou're recommended to make all edits to the class here (rather than copying and pasting the class several times below). Unfortunately, this requires some scrolling back and forth between directions and the code. You may wish to have another copy of the notebook open and use one for reading instructions and the other for writing code.\n\n\n```python\n\"\"\"Code cell with a bit class. Keep your class here and modify it as you work through the notebook. \nA skeleton of the class is provided for you.\"\"\"\n\nclass Bit:\n \"\"\"Binary digit class.\"\"\"\n \n\n def display_value(self):# <-- do not modify this method\n \"\"\"Displays the value of the Bit.\"\"\"\n print(\"The bit's value is \", self.value, \".\", sep=\"\")\n```\n\nWe know that every class needs an `__init__` method, which here will create our `Bit`. Let's agree by convention to always start our bit in the \"off\" state, represented by 0, unless a different initial value is provided. We'll do this by using keyword arguments as described below.\n\n✎ Do the following to your class:\n\n(1) Define an `__init__` method. This method should have a keyword argument (input to the function) called `initial_value` whose default value is 0. Create a class attribute called `value` and set it to be `initial_value`.\n\n\n```python\n\"\"\"Create a bit and display it's value.\"\"\"\nb = Bit()\nb.display_value()\n```\n\n✎ Do the following to your class:\n\n(2) Write a method called `measure` which returns the `value` of the bit.\n\nNow run the following code block.\n\n\n```python\n\"\"\"Create a bit, display its value, and print out its state.\"\"\"\nb = Bit()\nb.display_value()\nprint(\"The measured state of the bit is {}\".format(b.measure()))\n```\n\nI know what you're thinking! \"Well duh! The measured state of a bit is just going to be it's value...\" Hold that thought! We're going to see a big difference when we write a class for a quantum bit.\n\nFirst, let's talk about the operations we can perform on a bit.\n\n###

    Operations on a Bit

    \n\nThere's only one non-trivial operation that can be performed on a bit -- negating, or flipping, its value. We'll call this operation the `NOT` operation, which has the following effect:\n\n\\begin{align}\n \\text{NOT(0)} &= 1 \\\\\n \\text{NOT(1)} &= 0\n\\end{align}\n\n✎ Do the following to your class:\n\n(3) Define a method called `NOT` which negates the `value` of the bit. This method should NOT return a value (no pun intended).\n\nNow run the following code with your class.\n\n\n```python\n\"\"\"Perform operations on a bit.\"\"\"\n# create a bit\nb = Bit()\nb.display_value()\nprint(\"The measured state of the bit is {}.\\n\".format(b.measure()))\n\n# apply a NOT operation\nb.NOT()\nb.display_value()\nprint(\"The measured state of the bit is {}.\\n\".format(b.measure()))\n\n# apply another NOT operation\nb.NOT()\nb.display_value()\nprint(\"The measured state of the bit is {}.\\n\".format(b.measure()))\n```\n\nNote that applying two `NOT` gates in a row gets us back to the same state, as you might expect.\n\nWe can certainly do more operations with multiple bits of information. For example, if we have two bits, we can take the `AND` or the `OR` of them. The `AND` takes in two bits and returns one if both input bits are one and zero otherwise. The `OR` returns one if either of the input bits is one (including both).\n\nOperations on multiple bits are crucial for information processing (i.e., computation), but for simplicity we won't discuss them in more detail here. \n\nOne reason we do mention multiple bits is for copying information, another crucial component of (classical) computation.\n\n##

    Copying Bits

    \n\nWe can copy a single classical bit into as many bits as we want. How? Well, we just look at the bit, record its value, then prepare another bit with that value.\n\n✎ Do the following to your class:\n\n(4) Define a method called `copy` which copies the `value` of the `Bit` to a new `Bit` and returns the new `Bit`. Then execute the following cell.\n\n\n```python\n\"\"\"Copy a bit.\"\"\"\nb = Bit()\nnew_bit = b.copy()\n\nb.display_value()\nnew_bit.display_value()\n\nprint(b == new_bit)\n```\n\nNote that the bits have the same value, but they are not equivalent, since they are different objects. With bits, we are able to directly \"measure\" the bits value and write it into a new bit of information.\n\nWe highlighted this feature, as well as the others, to now contrast bits with qubits.\n\n#

    Working with Qubits

    \n\nWhereas classical computers represent information using bits, quantum computers represent information using qubits. Here's a short refresher on what a qubit is from the Pre-Class Assignment.\n\nA qubit is a vector of complex numbers\n\n\\begin{equation}\n |\\psi\\rangle = \\alpha |0\\rangle + \\beta |1\\rangle\n =\n \\left[ \\begin{matrix}\n \\alpha \\\\\n \\beta \\\\\n \\end{matrix} \\right] .\n\\end{equation}\n\nThese complex numbers determine the probability of measuring 0 or 1, as we'll see today, so we require that $|\\alpha|^2 + |\\beta|^2 = 1$.\n\nAs a reminder, the vector $|0\\rangle$, sometimes called the ground state, is\n\n\\begin{equation}\n |0\\rangle = \\left[ \\begin{matrix}\n 1 \\\\\n 0 \\\\\n \\end{matrix} \\right]\n\\end{equation}\n\nand the vector $|1\\rangle$, sometimes called the excited state, is \n\n\\begin{equation}\n |1\\rangle = \\left[ \\begin{matrix}\n 0 \\\\\n 1 \\\\\n \\end{matrix} \\right] .\n\\end{equation}\n\nThe Greek symbol $\\psi$ (psi, pronounced: \"sigh\") is commonly used to represent qubits.\n\nQuestion: Bits are made of classical physical systems like light switches or electricity. What kind of quantum systems do people make qubits out of? Search the web to find out and record at least three of your findings below. Cite your source(s).\n\n✎ **Answer:** Erase the contents of this cell and put your answer here!\n\n##

    Writing a Qubit Class

    \n\nLet's now get more practice with qubits by writing a `Qubit` class in the same way that we wrote a `Bit` class. This will allow you to see the similarites and differences between classical and quantum bits.\n\n\n```python\n\"\"\"Code cell with a qubit class. Keep your class here and modify it as you work through the notebook. \nA skeleton of the class is provided for you.\"\"\"\n\nclass Qubit:\n \"\"\"Quantum bit class.\"\"\"\n\n\n def display_wavefunction(self):# <-- do not modify this method!\n \"\"\"Prints the wavefunction of the Qubit.\"\"\"\n print(\"The Qubit's wavefunction is\", self.wavefunction[0], self.wavefunction[1], sep=\"\\n\")\n```\n\n✎ Do the following to your `Qubit` class:\n\n(1) Define an `__init__` method. In this method:\n\n(a) Create a class attribute called `zero` which represents the vector $|0\\rangle$ above. This attribute should be a Numpy array. Make sure the datatype (`dtype`) is complex, for example using `np.complex64`.\n\n(b) Create a class attribute called `one` which represents the vector $|1\\rangle$ above. This attribute should be a Numpy array. Make sure the datatype (`dtype`) is complex, for example using `np.complex64`.\n\nBy convention, let's agree to always initialize a `Qubit` in the ground state $|0\\rangle$.\n\n✎ Do the following to your `Qubit` class:\n\n(1) In the `__init__` method, create an attribute called `wavefunction` of the `Qubit` and set it to be equal to the $|0\\rangle$ state. (The term wavefunction is physics jargon. We can say \"a qubit is...\" or \"a qubit's wavefunction is...\" interchangeably.)\n\nNow run the following code.\n\n\n```python\n\"\"\"Initialize a qubit and display its wavefunction.\"\"\"\nq = Qubit()\nq.display_wavefunction()\nprint()\n```\n\nIn the above code, we initialize a qubit and then use the provided `display_wavefunction` method to print out it's wavefunction. If your `__init__` method is correct, you should see the qubit's wavefunction as the $|0\\rangle$ vector.\n\n###

    Measuring a Qubit

    \n\nNow let's write a method to measure our `Qubit`. The two measurement rules of a qubit \n\n\\begin{equation}\n |\\psi\\rangle = \\alpha |0\\rangle + \\beta |1\\rangle\n =\n \\left[ \\begin{matrix}\n \\alpha \\\\\n \\beta \\\\\n \\end{matrix} \\right] .\n\\end{equation}\n\nare listed below:\n\nThe First Measurement Rule\n\n(1) The probability that $|\\psi\\rangle$ is in the ground state $|0\\rangle$ is $|\\alpha|^2$. The probability that $|\\psi\\rangle$ is in the excited state $|1\\rangle$ is $|\\beta|^2 = 1 - |\\alpha|^2$.\n\nKey Concept: There are two possible measurement outcomes of a qubit. Thus, the measurement outcome of a qubit is a bit. This is why we used 0 and 1 as labels for the vectors all along! When we measure the ground state $|0\\rangle$, we call this outcome $0$. When we measure the excited state $|1\\rangle$, we call this outcome $1$.\n\nThe Second Measurement Rule\n\n(2) If we measure the ground state, the wavefunction becomes $|0\\rangle$. If we measure the excited state, the wavefunction becomes $|1\\rangle$.\n\n\n\n✎ Do the following to your `Qubit` class:\n\n(1) Write a method called `measure` which measures a `Qubit` according to the above rules. Specifically, your method should return a bit (0 if the ground state was measured, or 1 if the excited state was measured) and modify the `wavefunction` of the `Qubit` appropriately.\n\nHints:\n\n(i) Compute the probability the qubit is in the ground state ($|\\alpha|^2$).\n\n(ii) Generate a random number between 0 and 1.\n\n(iii) If the random number is less than $|\\alpha|^2$, set the `wavefunction` to be the `zero` vector, and return the number (bit) 0. Otherwise, set the `wavefunction` to be the `one` vector, and return the number (bit) 1.\n\nNow run the following code.\n\n\n```python\n\"\"\"Initialize a qubit and measure it.\"\"\"\nq = Qubit()\nq.display_wavefunction()\nprint(\"The bit we obtain from measuring the qubit is {}.\\n\".format(q.measure()))\nq.display_wavefunction()\n```\n\nYou should have seen the measurement result 0.\n\nQuestion: Run the cell above many times and note the measurement result (i.e., bit) after each time you run it. (The keyboard shortcut \"control + enter\" is useful here.) What measured state do you always get? Why?\n\n✎ **Answer:** Erase the contents of this cell and put your answer here!\n\n##

    Operations on a Qubit

    \n\nOn a classical bit, we could only do one operation, the `NOT` operation, because the bit only had two states. With our `Qubit`, we have an underlying `wavefunction` that helps determine what our measured state will be, as we have seen above. Operations on a `Qubit` act on its `wavefunction`. As such, there's a lot more operations we can do to it! (In fact, there's infinitely many operations we can do on a qubit.)\n\nOne example of an operation is called the `X` or `NOT` operation. Why is it called this? Well, it has the effect\n\n\\begin{align}\n \\text{NOT($|0\\rangle$)} &= |1\\rangle \\\\\n \\text{NOT($|1\\rangle$)} &= |0\\rangle\n\\end{align}\n\nQubit operations can be written as matrices that act on a qubit's wavefunction to implement the operation. You don't have to know how to do matrix-vector multiplication, just how to do it in Python. An example is shown below.\n\n\n```python\n\"\"\"Example of a matrix vector multiplication using numpy.\"\"\"\n# an example matrix\nmatrix = np.array([[1, 1], [1, 1]])\n\n# an example vector\nvector = np.array([1, 0])\n\n# the matrix-vector product\nprint(np.dot(matrix, vector))\n\n# another way to do the matrix-vector product\nprint(matrix @ vector)\n```\n\nIt can be shown that a matrix representation for `NOT` is\n\n\\begin{equation}\n \\text{NOT} = \\left[ \\begin{matrix}\n 0 & 1 \\\\\n 1 & 0 \\\\\n \\end{matrix} \\right] .\n\\end{equation}\n\n(If you know linear algebra, prove this to youreself. If not, just take our word for it.)\n\n✎ Do the following to your `Qubit` class:\n\n(1) Write a method called `NOT` which multiplies the `wavefunction` the `NOT` matrix given above. (This method should NOT return a value (still no pun intended), only modify the `wavefunction`.)\n\nNow run the following code.\n\n\n```python\n\"\"\"Perform a NOT operation on a qubit.\"\"\"\nq = Qubit()\nq.display_wavefunction()\nprint()\n\nq.NOT()\nq.display_wavefunction()\n```\n\nYou should have a `Qubit` whose wavefunction is $|1\\rangle$ after executing the above cell. Now let's measure such a qubit to see what we get.\n\n\n```python\n\"\"\"Measure a qubit with wavefunction |1>.\"\"\"\nq = Qubit()\nq.NOT()\nq.display_wavefunction()\nprint(\"The bit we obtain from measuring the qubit is {}.\\n\".format(q.measure()))\nq.display_wavefunction()\n```\n\nYou should have seen 1 as the measurement result.\n\nQuestion: Run the above cell many times and observe your measurement results after each run. What measured state do you always get? Why?\n\n✎ **Answer:** Erase the contents of this cell and put your answer here!\n\nAnother quantum operation is called the Hadamard operation or Hadamard gate. A matrix representation for the Hadamard gate is given by\n\n\\begin{equation}\n H = \\frac{1}{\\sqrt{2}}\\left[ \\begin{matrix}\n 1 & 1 \\\\\n 1 & -1 \\\\\n \\end{matrix} \\right]\n\\end{equation}\n\n✎ Do the following to your `Qubit` class:\n\n(1) Write a method called `H` that applies the Hadamard gate given above to the `Qubit`s `wavefunction`.\n\nNow run the following code.\n\n\n```python\n\"\"\"Performing a Hadamard transform on a qubit.\"\"\"\nq = Qubit()\nq.H()\nq.display_wavefunction()\n```\n\nYou should see that the qubit has equal amplitudes (components of the wavefunction). Now let's measure the `Qubit` to see what we get.\n\n\n```python\n\"\"\"Measuring a qubit after performing a Hadamard gate.\"\"\"\nq = Qubit()\nq.H()\nq.display_wavefunction()\nprint(\"The bit we obtain from measuring the qubit is {}.\\n\".format(q.measure()))\nq.display_wavefunction()\n```\n\nWhat measurement result did you get?\n\nQuestion: Run the above cell several times and observe your measurement results after each run. Write a sentence describing your observation.\n\n✎ **Answer:** Erase the contents of this cell and put your answer here!\n\nQuestion: Write code to create a qubit, perform the Hadamard gate, and measure the qubit 1000 times. Record each measurement outcome, then make a histogram of the probability of measuring 0 and the probability of measuring 1.\n\n\n```python\n\"\"\"Put your code here.\"\"\"\n\n```\n\nQuestion: Reflect on your results. What can you say about the probability of measuring 0 and the probability of measuring 1?\n\n✎ **Answer:** Erase the contents of this cell and put your answer here!\n\n##

    Copying Qubits

    \n\nRemember the question of how to copy a bit? Well, now let's ask this for qubits:\n\nQuestion: How do you copy a qubit?\n\nAttempted Answer 1: You just copy it's wavefunction?\n\nThis won't work! Remember the wavefunction is just a mathematical tool that we use to help us calculate probabilities of what state the system is in. If we have a particle, say an electron, there's no wavefunction that we can just look at and then copy over it's information. (Unlike a light switch, a classical system, which we could look at and see with no issues.\n\nAttempted Answer 2: Well what if we just measure it then copy the measurement result into a new qubit?\n\nThis also won't work! Remember the measurement rules above. When we measure a qubit, we inherently change its wavefunction. The wavefunction it changes to is not, in general, the same as it was before measurement.\n\nThere is a way to copy one qubit to another, which is known as quantum teleportation, and involves a total of three qubits to get the job done. You'll get a chance to look at this in an upcoming assignment!\n\n#

    Assignment Wrap-up

    \n\n##

    Installing Qiskit\n\nFor the next assignments, we'll be using the Quantum Information Science Kit, or Qiskit, which is a Python package for quantum computing. Try to install Qiskit v0.7.0 on your computer now by executing the following cell. We'll be walking around to troubleshoot problems.\n\nNote: Why version 0.7.0? These quantum software packages are new and tend to change a bit. We'll use this version to make sure all the code in future assignments works as anticipated.\n\n\n```python\n\"\"\"Attempt to install Qiskit using pip. Uncomment the following two lines and run the cell.\"\"\"\n# !pip install --upgrade pip\n# !pip install qiskit==0.7.0\n```\n\n##

    Instructions to Get an API Key to Use a Quantum Computer

    \n\nYou can use Qiskit to program an actual quantum computer. To do so, one needs to register for an API key. If this interests you, follow the instructions below. (These instructions are also in the next Pre Class Assignment if you're short on time.) This is optional, not required.\n\n1. Navigate to the IBM Quantum Experience website https://quantumexperience.ng.bluemix.net/qx.\n\n1. Click \"Sign In\" in the upper right hand corner of the page (blue box with white text).\n\n1. In the pop-up screen, select \"Sign Up\" or \"Sign up using Github.\" The first requires an email, the second requires you to log into GitHub and authorize access to your account (to get an email).\n\n1. Fill out the form, then click \"Sign up\" at the bottom.\n\nOnce you have created an account, you can sign in (follow the first two steps above). Then, click the user icon in the upper right hand corner of the page, then click \"My Account.\" On the new screen, click the \"Advanced\" tab. Here, you can see your API key and copy it to your clipboard. You'll need to enter this in your notebook to use the real quantum computer backends.\n\n##

    Survey

    \n\n\n```python\nfrom IPython.display import HTML\nHTML(\n\"\"\"\n\n\"\"\"\n)\n```\n\n##

    Congrats, You're Finished!

    \n\nNow, you just need to submit this assignment by uploading it to the course Desire2Learn web page for today's submission folder. (Don't forget to add your name in the first cell.)\n\n

    © Copyright 2019, Michigan State University Board of Trustees.

    \n", "meta": {"hexsha": "ddac2b1a7a733e53bd7cd61ab407074b1ad6ee4f", "size": 36849, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "quantum/Day-23_In-Class-ClassicalAndQuantumBits-STUDENT.ipynb", "max_stars_repo_name": "rmlarose/teaching-project", "max_stars_repo_head_hexsha": "eec3c4e5454d34ee0de8a7e58438fc016ef05373", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "quantum/Day-23_In-Class-ClassicalAndQuantumBits-STUDENT.ipynb", "max_issues_repo_name": "rmlarose/teaching-project", "max_issues_repo_head_hexsha": "eec3c4e5454d34ee0de8a7e58438fc016ef05373", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "quantum/Day-23_In-Class-ClassicalAndQuantumBits-STUDENT.ipynb", "max_forks_repo_name": "rmlarose/teaching-project", "max_forks_repo_head_hexsha": "eec3c4e5454d34ee0de8a7e58438fc016ef05373", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 33.5295723385, "max_line_length": 425, "alphanum_fraction": 0.5838964422, "converted": true, "num_tokens": 6144, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.577495350642608, "lm_q2_score": 0.7606506472514406, "lm_q1q2_score": 0.43927221225099744}} {"text": "# Deep Residual Learning for Image Recognition\n\n## Degradation Problem\n\nDeep convolutional neural networks\uac00 \ub098\uc628 \uc774\ud6c4\ub85c \ub9ce\uc740 \ubc1c\uc804\uc774 \uc788\uc5c8\uc2b5\ub2c8\ub2e4.
    \n\uae30\ubcf8\uc801\uc73c\ub85c deep networks\ub294 features\ub4e4\uc744 \uc2a4\uc2a4\ub85c low/mid/high level features\ub85c \ub098\ub204\uba70$ ^{[01]} $, \uadfc\ub798\uc758 \ubaa8\ub378\ub4e4\uc758 \uacbd\uc6b0 layers\uce35\uc744 \ub354 \uae4a\uc774 \uc788\uac8c \uc313\uc544\uc11c features\ub4e4\uc758 levels\uc744 \ub354\uc6b1 \uc138\ubd84\ud654\ud558\uace0\uc790\ud558\ub294 \uc2dc\ub3c4\uac00 \uc788\uc73c\uba70$ ^{[02]} $, ImageNet \ucc4c\ub9b0\uc9c0\uc5d0\uc11c 16\uac1c \ub610\ub294 30\uac1c\ub4f1\uc758 layers\ub97c \uc0ac\uc6a9\ud558\ub294 \ub9e4\uc6b0 \uae4a\uc740 \ubaa8\ub378\ub4e4\uc744 \uc0ac\uc6a9\ud558\uae30\ub3c4 \ud558\uc600\uc2b5\ub2c8\ub2e4.$ ^{[03]} $\n\n\ub2e8\uc21c\ud788 layers\ub97c \ub354 \ub9ce\uc774 \uc313\uc73c\uba74 \ub354 \uc88b\uc740 \uacb0\uacfc\ub97c \ub0bc \uac83\uc778\uac00? \ub77c\ub294 \uc9c8\ubb38\uc5d0\ub294.. \uc0ac\uc2e4 \ubb38\uc81c\uac00 \uc788\uc2b5\ub2c8\ub2e4.
    \n\uc774\ubbf8 \uc798 \uc54c\ub824\uc9c4 vanishing/exploding gradients$ ^{[04]} $ $ ^{[05]} $\uc758 \ubb38\uc81c\ub294 convergence\uc790\uccb4\ub97c \ubabb\ud558\uac8c \ub9cc\ub4ed\ub2c8\ub2e4.
    \n\uc774\ub7ec\ud55c \ubb38\uc81c\ub294 normalized initialization$ ^{[06]} $ $ ^{[04]} $ $ ^{[07]} $, \uadf8\ub9ac\uace0 intermediate normalization layers $ ^{[08]} $\uc5d0 \uc758\ud574\uc11c \ub2e4\uc18c \ud574\uacb0\uc774 \ub418\uc5b4 \uc218\uc2ed\uce35\uc758 layers\ub4e4\uc774 SGD\ub97c \ud1b5\ud574 convergence\ub420 \uc218 \uc788\ub3c4\ub85d \ub3c4\uc640\uc90d\ub2c8\ub2e4. \n\nDeeper networks\ub97c \uc0ac\uc6a9\ud560\ub54c **degradation problem**\uc774 \ubc1c\uacac\ub418\uc5c8\uc2b5\ub2c8\ub2e4. degradation problem\uc740 network\uc758 depth\uac00 \ucee4\uc9c8\uc218\ub85d accuracy\ub294 saturated (\ub9c8\uce58 \ubb54\uac00 \uac00\ub4dd\ucc28\uc11c \ud604\uc0c1\ud0dc\uc5d0\uc11c \ub354 \uc9c4\uc804\uc774 \uc5c6\uc5b4\uc838 \ubc84\ub9ac\ub294 \uc0c1\ud0dc)\uac00 \ub418\uace0 degradation\uc774 \uc9c4\ud589\ub429\ub2c8\ub2e4. \uc774\ub54c degradation\uc740 overfitting\uc5d0 \uc758\ud574\uc11c \uc0dd\uaca8\ub098\ub294 \uac83\uc774 \uc544\ub2c8\uba70, \ub354 \ub9ce\uc740 layers\ub97c \ub123\uc744\uc218\ub85d training error\uac00 \ub354 \ub192\uc544\uc9d1\ub2c8\ub2e4.$ ^{[09]} $ (\ub9cc\uc57d overfitting\uc774\uc5c8\ub2e4\uba74 training error\ub294 \ub9e4\uc6b0 \ub0ae\uc544\uc57c \ud569\ub2c8\ub2e4.)\n\n\n\n
    \nCIFAR-10 \ub370\uc774\ud130\uc5d0 \ub300\ud55c training error(\uc67c\ucabd) \uadf8\ub9ac\uace0 test error(\uc624\ub978\ucabd) \uadf8\ub798\ud504.
    \n 20-layers \uadf8\ub9ac\uace0 56-layers\ub97c \uc0ac\uc6a9\ud588\uc73c\uba70, \ub354 \uae4a\uc740 \ub124\ud2b8\uc6cc\ud06c\uc77c\uc218\ub85d training error\uac00 \ub192\uc73c\uba70, \ub530\ub77c\uc11c test error\ub610\ud55c \ub192\ub2e4.    \n
    \n\n\ud55c\uac00\uc9c0 \uc2e4\ud5d8\uc5d0\uc11c \uc774\ub97c \ub4b7\ubc1b\uce68\ud560 \uadfc\uac70\ub97c \ub0b4\ub193\uc2b5\ub2c8\ub2e4.
    \nshallow network\uc5d0\uc11c \ud559\uc2b5\ub41c \ubaa8\ub378\uc704\uc5d0 \ub2e4\uce35\uc758 layers\ub97c \ucd94\uac00\uc801\uc73c\ub85c \uc313\uc2b5\ub2c8\ub2e4. \uc774\ub860\uc801\uc73c\ub85c\ub294 deeper \ubaa8\ub378\uc774 shallower \ubaa8\ub378\uc5d0 \ucd94\uac00\ub41c \uac83\uc774\uae30 \ub54c\ubb38\uc5d0 \ub354 \ub0ae\uc740 training error\ub97c \ubcf4\uc5ec\uc57c \ud569\ub2c8\ub2e4. \ud558\uc9c0\ub9cc \ud559\uc2b5\ub41c shallower \ubaa8\ub378\uc5d0 layers\ub97c \ub354 \ucd94\uac00\uc2dc\ucf1c\ub3c4, \uadf8\ub0e5 shallow \ubaa8\ub378\ubcf4\ub2e4 \ub354 \ub192\uc740 training error\ub97c \ubcf4\uc5ec\uc90d\ub2c8\ub2e4.\n\n\n\n
    \n**Constructed Solution**
    Shallower model(\uc67c\ucabd) \uadf8\ub9ac\uace0 Deeper model(\uc624\ub978\ucabd).
    Shallower model\ub85c \ubd80\ud130 \ud559\uc2b5\ub41c \uc9c0\uc2dd\uc744 \ubcf5\uc0ac\ud558\uace0, Identity mapping\ub85c\uc11c layers\ub97c \ucd94\uac00\ud558\uc600\ub2e4.
    Deeper model\uc740 shallower model\uacfc \ube44\uad50\ud558\uc5ec \ub354 \ub0ae\uac70\ub098 \uac19\uc740 training error\ub97c \ubcf4\uc5ec\uc57c \ud558\uc9c0\ub9cc
    \uc2e4\uc81c\ub294 degradation\ud604\uc0c1\uc73c\ub85c \uc778\ud558\uc5ec layers\uac00 \uae4a\uc5b4\uc9c8\uc218\ub85d training error\ub294 \ub192\uc544\uc9c4\ub2e4    \n
    \n\n\uc704\uc758 \uadf8\ub9bc\uc5d0\uc11c \ubcf4\ub4ef\uc774, shallower model\ub85c \ubd80\ud130 \ud559\uc2b5\ub41c \uc9c0\uc2dd\uc744 \ubcf5\uc0ac\ud558\uace0, \ucd94\uac00\uc801\uc73c\ub85c identity mapping\uc73c\ub85c\uc11c layers\ub97c \ub354 \ucd94\uac00\ud558\uc600\uc2b5\ub2c8\ub2e4.
    \n\uc5ec\uae30\uc11c identity mapping\uc774\ub77c\ub294 \ub73b\uc740 $ f(x) = x $ \uc758 \uc758\ubbf8\ub85c \uae30\uc874 \ud559\uc2b5\ub41c layers\uc5d0\uc11c \ub098\uc628 output\uc744 \ucd94\uac00\ub41c layers\uc5d0\uc11c \ub3d9\uc77c\ud55c output\uc744 \uc0dd\uc131\ud558\ub294 \uac83\uc785\ub2c8\ub2e4. \ub530\ub77c\uc11c identity mapping\uc73c\ub85c\uc11c \ucd94\uac00\ub41c layers\ub4e4\uc740 \ucd5c\uc18c\ud55c shallower model\uc5d0\uc11c \ub098\uc628 \uc608\uce21\uce58\uc640 \ub3d9\uc77c\ud558\uac70\ub098 \ub610\ub294 \ub354 \uae4a\uac8c \ub4e4\uc5b4\uac14\uc73c\ub2c8 \ub354 \uc798 \ud559\uc2b5\uc774 \ub418\uc5b4\uc57c \ud569\ub2c8\ub2e4.\n\n\ud558\uc9c0\ub9cc \ud604\uc2e4\uc740.. layers\uac00 \uae4a\uc5b4\uc9c0\uba74 \uae4a\uc5b4\uc9c8\uc218\ub85d training error\uac00 \ub354 \ub192\uc544\uc9c0\uba70, \ub530\ub77c\uc11c test error\ub610\ud55c \ub3d9\uc77c\ud558\uac8c \ub192\uc544\uc9d1\ub2c8\ub2e4.
    \n\uc774\ub7ec\ud55c \ud604\uc0c1\uc744 degradation problem\uc774\ub77c\uace0 \ud558\uba70, Deep Residual Network$ ^{[10]} $\uac00 \ud574\uacb0\ud558\ub824\ub294 \ubd80\ubd84\uc785\ub2c8\ub2e4.\n\n## Residual \n\n\uba3c\uc800 residual\uc5d0 \ub300\ud574\uc11c \uc54c\uc544\uc57c \ud569\ub2c8\ub2e4.
    \n\uac04\ub2e8\ud558\uac8c \uc774\uc57c\uae30 \ud558\uba74 residual\uc774\ub780 \uad00\uce21\uce58(observed data)\uc640 \uc608\uce21\uac12(estimated value)\uc0ac\uc774\uc758 \ucc28\uc774\uc785\ub2c8\ub2e4.
    \nLinear least square (\ucd5c\uc18c\ud68c\uadc0\uc9c1\uc120) residual\uc758 \ud569\uc740 0\uc774 \ub429\ub2c8\ub2e4.
    \n\uc608\ub97c \ub4e4\uc5b4 \ub2e4\uc74c\uc758 \uc218\uc2dd\uc5d0\uc11c true\uac12\uc778 $ x $\ub97c \ucc3e\uace0\uc790 \ud569\ub2c8\ub2e4.\n\n$$ f(x) = b $$\n\n\uc774\ub54c $ x $\uc758 \uadfc\uc0ac\uce58(approximation)\uc778 $ x_0 $\uac00 \uc8fc\uc5b4\uc84c\uc744\ub54c, residual\uac12\uc740 \ub2e4\uc74c\uacfc \uac19\uc2b5\ub2c8\ub2e4.\n\n$$ b - f(x_0) $$\n\n\ubc18\uba74\uc5d0 error\ub294 true\uac12\uc5d0\uc11c \uadfc\uc0ac\uce58(approximation)\uc758 \ucc28\uc774\uc774\uba70 \ub2e4\uc74c\uacfc \uac19\uc2b5\ub2c8\ub2e4.
    \n(\ud558\ub098\uc758 \uc608\ub85c.. \uadfc\uc0ac\uac12 3.14 - $ \\pi $ \uac00 \ubc14\ub85c \uc624\ucc28\uc785\ub2c8\ub2e4.)\n\n$$ x - x_0 $$\n\n\uc880\ub354 \uc798 \uc124\uba85\ud55c [\uc601\uc0c1](https://www.youtube.com/watch?v=snG7sa5CcJQ)\uc744 \ucc38\uace0 \ud569\ub2c8\ub2e4.\n\n\n## ResNet Explained\n\nDegradation\uc5d0\uc11c \uc5b8\uae09\ud55c \ud604\uc0c1\uc744 \ubcf4\uba74, \uc9c1\uad00\uc801\uc73c\ub85c \ubcf4\uba74 deep neural network\uc5d0 \ub354 \ub9ce\uc740 layers\ub97c \ucd94\uac00\uc2dc\ud0b5\uc73c\ub85c\uc11c \uc131\ub2a5\uc744 \ud5a5\uc0c1\uc2dc\ud0ac\uc218 \uc788\uc744\uac70 \uac19\uc9c0\ub9cc \uadf8\uc640\ub294 \uc815 \ubc18\ub300\uc758 \uacb0\uacfc\uac00 \ub098\uc654\uc2b5\ub2c8\ub2e4. \uc774\uac83\uc774 \uc758\ubbf8\ud558\ub294 \ubc14\ub294 multiple nonlinear layers\uc548\uc5d0\uc11c identity mappings\uc744 \uc2dc\ud0a4\ub294\ub370(approximate) \uc5b4\ub824\uc6c0\uc774 \uc788\ub2e4\ub294 \uac83\uc785\ub2c8\ub2e4. \uc774\ub294 \ud754\ud788 \ub525\ub7ec\ub2dd\uc5d0\uc11c \ub098\ud0c0\ub098\ub294 vanishing/exploding gradients \uc774\uc288 \uadf8\ub9ac\uace0 curse of dimensionality problem\ub4f1\uc73c\ub85c \ub098\ud0c0\ub098\ub294 \ud604\uc0c1\uc73c\ub85c \uc0dd\uac01\uc774 \ub429\ub2c8\ub2e4. \n\nResNets\uc740 \uc774\ub7ec\ud55c \ubb38\uc81c\ub97c \ud574\uacb0\ud558\uae30 \uc704\ud558\uc5ec, residual learning\uc744 \ud1b5\ud574 \uac15\uc81c\ub85c Identity mapping (function)\uc744 \ud559\uc2b5\ud558\ub3c4\ub85d \ud558\uc600\uc2b5\ub2c8\ub2e4.
    \n\n* $ x $: \ud574\ub2f9 \ub808\uc774\uc5b4\ub4e4\uc758 input\n* $ H(x) $: (\uc804\uccb4\uac00 \uc544\ub2cc) \uc18c\uaddc\ubaa8\uc758 \ub2e4\uce35 \ub808\uc774\uc5b4(a few stacked layers)\uc758 output \n* $ id(x) $: Identity mapping(function)\uc740 \ub2e8\uc21c\ud788 $ id(x) = x $ \uc73c\ub85c\uc11c, $ x $\uac12\uc744 \ubc1b\uc73c\uba74 \ub3d9\uc77c\ud55c $ x $\ub97c \ub9ac\ud134\uc2dc\ud0b5\ub2c8\ub2e4\n* $ H(x) $ \uadf8\ub9ac\uace0 $ x $ \ub294 \ub3d9\uc77c\ud55c dimension\uc744 \uac16\uace0 \uc788\ub2e4\uace0 \uac00\uc815\n\n\uc77c\ubc18\uc801\uc778 Neural Network\ub294 $ H(x) $ \uc790\uccb4\ub97c \ud559\uc2b5\ub2c8\ub2e4. \n\n\n\n\nResNet\uc758 \uacbd\uc6b0\uc5d0\ub294 residual function\uc744 \ud559\uc2b5\ud558\ub3c4\ub85d \uac15\uc81c\ud569\ub2c8\ub2e4.\n\n$$ F(x) = H(x) - id(x) $$\n\n\uc6b0\ub9ac\ub294 \uc2e4\uc81c true\uac12\uc744 \uc54c\uace0\uc790 \ud558\ub294 \uac83\uc774\uae30 \ub54c\ubb38\uc5d0 \uc704\uc758 \uacf5\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc7ac\uc815\ub9bd(reformulation)\ud560\uc218 \uc788\uc2b5\ub2c8\ub2e4.\n\n$$ \\begin{align}\nH(x) &= F(x) + id(x) \\\\\n&= F(x) + x\n\\end{align} $$\n\n\uc989 \uc544\ub798\uc758 \uadf8\ub9bc\ucc98\ub7fc \uadf8\ub798\ud504\uac00 \uadf8\ub824\uc9d1\ub2c8\ub2e4.\n\n\n\n\uc774\ub860\uc801\uc73c\ub85c identity mappings\uc774 \ucd5c\uc801\ud654(optimal)\ub418\uc5c8\ub2e4\uba74, \ub2e4\uc911 \ub808\uc774\uc5b4\uc758 weights\uc5f0\uc0b0 $ F(x) $ \uc758 \uac12\uc744 0\uc73c\ub85c \ub9cc\ub4e4\uac83\uc785\ub2c8\ub2e4. $ F(x) $ \uac00 0\uc774 \ub41c\ud6c4 $ id(x) $ \ub97c \ub354\ud558\uae30 \ub54c\ubb38\uc5d0 \ud574\ub2f9 subnetwork\ub294 identity function\uc73c\ub85c\uc11c \uae30\ub2a5\uc744 \ud558\uac8c \ub429\ub2c8\ub2e4. \n\n\uc2e4\uc81c\ub85c\ub294 identity mappings (layers)\uac00 \ucd5c\uc801\ud654\ub418\uc5b4 0\uc73c\ub85c \uc218\ub834\ud558\ub294 \uac83\uc740 \uc77c\uc5b4\ub098\uae30 \ud798\ub4ec\ub2c8\ub2e4.
    \n\ub2e4\ub9cc reformulation \ub41c \uacf5\uc2dd\uc548\uc5d0 identity function\uc774 \uc874\uc7ac\ud558\uae30 \ub54c\ubb38\uc5d0 reference\uac00 \ub420 \uc218 \uc788\uace0, \ub530\ub77c\uc11c neural network\uac00 \ud559\uc2b5\ud558\ub294\ub370 \ub3c4\uc6c0\uc744 \uc904 \uc218 \uc788\uc2b5\ub2c8\ub2e4.\n\n## Shortcut Connection\n\n\uc704\uc5d0\uc11c \uc774\ubbf8 \uc5b8\uae09\ud55c \uadf8\ub798\ud504\uc5d0\uc11c \ub17c\ubb38\uc5d0\uc11c\ub294 building block\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uace0 \uc788\uc2b5\ub2c8\ub2e4.
    \n(\uacf5\uc2dd\uc744 \uac04\ub7b5\ud558\uac8c \ud558\uae30 \uc704\ud574\uc11c bias\uc5d0 \ub300\ud55c \ubd80\ubd84\uc740 \uc758\ub3c4\uc801\uc73c\ub85c \ub204\ub77d\ub418\uc5b4 \uc788\uc2b5\ub2c8\ub2e4. \ub2f9\uc5f0\ud788 \uc2e4\uc81c \uad6c\ud604\uc2dc\uc5d0\ub294 \ud544\uc694\ud569\ub2c8\ub2e4.)\n\n$$ y = F(x\\ |\\ W_i) + x $$\n\n$ F(x\\ |\\ W_i) $ \ub294 \ud559\uc2b5\ub418\uc57c\ud560 residual mapping\uc744 \ub098\ud0c0\ub0b4\uba70, $ x $ \uadf8\ub9ac\uace0 $ y $\ub294 \uac01\uac01 input \uadf8\ub9ac\uace0 output\uc744 \ub098\ud0c0\ub0c5\ub2c8\ub2e4.
    \n\uc704\uc758 \uacf5\uc2dd\uc740 \uc544\uc8fc \uac04\ub7b5\ud558\uac8c \ud45c\ud604\ud558\uae30 \uc704\ud574\uc11c \ub098\ud0c0\ub0b8\uac83\uc774\uace0 2\uac1c\uc758 \ub808\uc774\uc5b4\ub97c \uc0ac\uc6a9\ud558\ub294 \uacbd\uc6b0\uc5d0\ub294 $ F $ \ud568\uc218\uc5d0\ub300\ud55c \uc815\uc758\uac00 \ubc14\ub01d\ub2c8\ub2e4.\n\n$$ F = W_2 \\sigma \\left(W_1 x \\right) $$\n\n\uc5ec\uae30\uc11c $ \\sigma $\ub294 ReLU\ub97c \uac00\ub974\ud0b5\ub2c8\ub2e4. $ F + x $ \ub294 **shortcut connection**\uc744 \ub098\ud0c0\ub0b4\uba70 element-wise addition\uc744 \uc5f0\uc0b0\ud569\ub2c8\ub2e4.
    \n\ud574\ub2f9 addtion \uc774\ud6c4! \ub450\ubc88\uc9f8 nonlinearity\ub97c \uc801\uc6a9\ud569\ub2c8\ub2e4. (\uc989 ReLU\ub97c addition\uc774\ud6c4\uc5d0 \uc801\uc6a9\ud558\uba74 \ub428)\n\n$ F + x $ \ub97c \uc5f0\uc0b0\ud560\ub54c \uc911\uc694\ud55c\uc810\uc740 **dimension\uc774 \uc11c\ub85c \ub3d9\uc77c**\ud574\uc57c \ud569\ub2c8\ub2e4.\n\ub9cc\uc57d \uc11c\ub85c dimension\uc774 \ub2e4\ub974\ub2e4\uba74 (\uc608\ub97c \ub4e4\uc5b4\uc11c input/output\uc758 channels\uc774 \uc11c\ub85c \ub2e4\ub984) linear projection $ W_s $ \ub97c shorcut connection\uc5d0 \uc801\uc6a9\uc2dc\ucf1c\uc11c dimension\uc744 \uc11c\ub85c \ub3d9\uc77c\ud558\uac8c \ub9cc\ub4e4\uc5b4\uc904\uc218 \uc788\uc2b5\ub2c8\ub2e4.\n\n$$ y = F(x\\ |\\ W_i) + W_s x $$\n\nResidual function $ F $\ub294 \uc0ac\uc2e4 \uc0c1\ub2f9\ud788 \uc720\uc5f0\ud569\ub2c8\ub2e4.\uc989 $ F $\ub294 2\uac1c, 3\uac1c \ub610\ub294 3\uac1c \uc774\uc0c1\uc758 \ub2e4\uce35\uc744 \uc0ac\uc6a9\ud558\ub294 \uac83\uc774 \uac00\ub2a5\ud569\ub2c8\ub2e4.
    \n\ud558\uc9c0\ub9cc \ub9cc\uc57d $ F $\uc548\uc5d0 1\uac1c\uc758 \ub808\uc774\uc5b4\ub9cc \uac16\uace0 \uc788\ub2e4\uba74 linear layer\uc640 \ub3d9\uc77c\ud574\uc9c0\uac8c \ub429\ub2c8\ub2e4.\n\n$$ y = W_1x + x $$\n\n\ub530\ub77c\uc11c 1\uac1c\ub9cc \uac16\uace0 \uc788\ub294 $ F $\ub294 \uc0ac\uc2e4\uc0c1 \uc758\ubbf8\uac00 \uc5c6\uc2b5\ub2c8\ub2e4.
    \n\ub610\ud55c $ F $\ub294 fully-connected layer \ub610\ub294 convolution\uac19\uc740 \ub2e4\uc591\ud55c \ubc29\ubc95\uc73c\ub85c \ubaa8\ub378\ub9c1\uc744 \ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4.\n\n\n## Implementation\n\n\ub17c\ubb38\uc5d0\uc11c\ub294 ImageNet\uc5d0 \ub300\ud55c \uad6c\ud604\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ud558\uc600\uc2b5\ub2c8\ub2e4. \n\n#### Data Augmentation\n\n224 x 224 \uc2f8\uc774\uc988\ub85c random crop \ub610\ub294 horizontal flip\uc774 \ub418\uc5c8\uc73c\uba70, \ud53d\uc139\ub9c8\ub2e4 \ud3c9\uade0\uac12\uc73c\ub85c subtract \ub418\uc5c8\uc2b5\ub2c8\ub2e4.
    \nStandard color augmentation\uc744 \uc2e4\ud589\ud588\uc2b5\ub2c8\ub2e4.\n\n#### ResNet Layer\n\nConv -> Batch -> ReLU -> Conv -> Batch -> Addition -> RELU\n\n# References\n\n\uc778\uc6a9\ud55c \ubb38\uc11c\ub4e4..\n\n* [01] [Visualizing and understanding convolutional neural networks](https://www.cs.nyu.edu/~fergus/papers/zeilerECCV2014.pdf)\n* [02] [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://arxiv.org/abs/1409.1556)\n* [03] [Going Deeper with Convolutions](https://arxiv.org/pdf/1409.4842.pdf)\n* [04] [Understanding the difficulty of training\ndeep feedforward neural networks](http://www-prima.imag.fr/jlc/Courses/2016/PRML/XavierInitialisation.pdf)\n* [05] [Learning long-term dependencies\nwith gradient descent is difficult](http://www.iro.umontreal.ca/~lisa/pointeurs/ieeetrnn94.pdf)\n* [06] [Efficient backprop. In Neural Networks: Tricks of the Trade](http://yann.lecun.com/exdb/publis/pdf/lecun-98b.pdf)\n* [07] [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://arxiv.org/pdf/1312.6120.pdf)\n* [08] [Batch normalization: Accelerating deep\nnetwork training by reducing internal covariate shift](https://arxiv.org/abs/1502.03167)\n* [09] [Convolutional neural networks at constrained time cost](https://arxiv.org/abs/1412.1710)\n* [10] [Deep Residual Learning for Image Recognition](https://arxiv.org/pdf/1512.03385.pdf)\n\n\uae00 \uc4f0\uba74\uc11c \ucc38\uace0\ud55c \ubb38\uc11c\ub4e4..\n\n* [Deep Residual Learning for Image Recognition - \uc6d0\ub798 ResNet Paper](https://arxiv.org/pdf/1512.03385.pdf)\n* [Identity Mappings in Deep Residual Networks - \uac1c\uc120\ub41c Paper](https://arxiv.org/pdf/1603.05027.pdf)\n* [Residual neural networks are an exciting area of deep learning research](https://blog.init.ai/residual-neural-networks-are-an-exciting-area-of-deep-learning-research-acf14f4912e9)\n* [Deep Residual Networks ICML 2016 Tutorial](http://icml.cc/2016/tutorials/icml2016_tutorial_deep_residual_networks_kaiminghe.pdf)\n", "meta": {"hexsha": "3c44efb43ced59f3840b82a280fdd5e66e150676", "size": 10187, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "deep-residual-learning-for-image-recognition.ipynb", "max_stars_repo_name": "AndersonJo/residual-network", "max_stars_repo_head_hexsha": "290aa59141fae688cd702c04420614817993a59a", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2018-05-24T12:44:09.000Z", "max_stars_repo_stars_event_max_datetime": "2019-06-12T06:54:39.000Z", "max_issues_repo_path": "deep-residual-learning-for-image-recognition.ipynb", "max_issues_repo_name": "AndersonJo/residual-network", "max_issues_repo_head_hexsha": "290aa59141fae688cd702c04420614817993a59a", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "deep-residual-learning-for-image-recognition.ipynb", "max_forks_repo_name": "AndersonJo/residual-network", "max_forks_repo_head_hexsha": "290aa59141fae688cd702c04420614817993a59a", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 48.9759615385, "max_line_length": 358, "alphanum_fraction": 0.6129380583, "converted": true, "num_tokens": 3691, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.7310585903489891, "lm_q1q2_score": 0.4387728558646381}} {"text": "## Configurations for Colab\n\n\n```python\nimport sys\nIN_COLAB = \"google.colab\" in sys.modules\n\nif IN_COLAB:\n !apt install python-opengl\n !apt install ffmpeg\n !apt install xvfb\n !pip install pyvirtualdisplay\n !pip install gym[all]\n from pyvirtualdisplay import Display\n \n # Start virtual display\n dis = Display(visible=0, size=(400, 400))\n dis.start()\n```\n\n# 05. Noisy Networks for Exploration\n\n[M. Fortunato et al., \"Noisy Networks for Exploration.\" arXiv preprint arXiv:1706.10295, 2017.](https://arxiv.org/pdf/1706.10295.pdf)\n\n\nNoisyNet is an exploration method that learns perturbations of the network weights to drive exploration. The key insight is that a single change to the weight vector can induce a consistent, and potentially very complex, state-dependent change in policy over multiple time steps.\n\nFirstly, let's take a look into a linear layer of a neural network with $p$ inputs and $q$ outputs, represented by\n\n$$\ny = wx + b,\n$$\n\nwhere $x \\in \\mathbb{R}^p$ is the layer input, $w \\in \\mathbb{R}^{q \\times p}$, and $b \\in \\mathbb{R}$ the bias.\n\nThe corresponding noisy linear layer is defined as:\n\n$$\ny = (\\mu^w + \\sigma^w \\odot \\epsilon^w) x + \\mu^b + \\sigma^b \\odot \\epsilon^b,\n$$\n\nwhere $\\mu^w + \\sigma^w \\odot \\epsilon^w$ and $\\mu^b + \\sigma^b \\odot \\epsilon^b$ replace $w$ and $b$ in the first linear layer equation. The parameters $\\mu^w \\in \\mathbb{R}^{q \\times p}, \\mu^b \\in \\mathbb{R}^q, \\sigma^w \\in \\mathbb{R}^{q \\times p}$ and $\\sigma^b \\in \\mathbb{R}^q$ are learnable, whereas $\\epsilon^w \\in \\mathbb{R}^{q \\times p}$ and $\\epsilon^b \\in \\mathbb{R}^q$ are noise random variables which can be generated by one of the following two ways:\n\n1. **Independent Gaussian noise**: the noise applied to each weight and bias is independent, where each random noise entry is drawn from a unit Gaussian distribution. This means that for each noisy linear layer, there are $pq + q$ noise variables (for $p$ inputs to the layer and $q$ outputs).\n2. **Factorised Gaussian noise:** This is a more computationally efficient way. It produces 2 random Gaussian noise vectors ($p, q$) and makes $pq + q$ noise entries by outer product as follows:\n\n$$\n\\begin{align}\n\\epsilon_{i,j}^w &= f(\\epsilon_i) f(\\epsilon_j),\\\\\n\\epsilon_{j}^b &= f(\\epsilon_i),\\\\\n\\text{where } f(x) &= sgn(x) \\sqrt{|x|}.\n\\end{align}\n$$\n\nIn all experiements of the paper, the authors used Factorised Gaussian noise, so we will go for it as well.\n\n\n```python\nimport math\nimport os\nfrom typing import Dict, List, Tuple\n\nimport gym\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport torch.optim as optim\nfrom IPython.display import clear_output\n```\n\n## Replay buffer\n\nPlease see *01.dqn.ipynb* for detailed description.\n\n\n```python\nclass ReplayBuffer:\n \"\"\"A simple numpy replay buffer.\"\"\"\n\n def __init__(self, obs_dim: int, size: int, batch_size: int = 32):\n self.obs_buf = np.zeros([size, obs_dim], dtype=np.float32)\n self.next_obs_buf = np.zeros([size, obs_dim], dtype=np.float32)\n self.acts_buf = np.zeros([size], dtype=np.float32)\n self.rews_buf = np.zeros([size], dtype=np.float32)\n self.done_buf = np.zeros(size, dtype=np.float32)\n self.max_size, self.batch_size = size, batch_size\n self.ptr, self.size, = 0, 0\n\n def store(\n self,\n obs: np.ndarray,\n act: np.ndarray, \n rew: float, \n next_obs: np.ndarray, \n done: bool,\n ):\n self.obs_buf[self.ptr] = obs\n self.next_obs_buf[self.ptr] = next_obs\n self.acts_buf[self.ptr] = act\n self.rews_buf[self.ptr] = rew\n self.done_buf[self.ptr] = done\n self.ptr = (self.ptr + 1) % self.max_size\n self.size = min(self.size + 1, self.max_size)\n\n def sample_batch(self) -> Dict[str, np.ndarray]:\n idxs = np.random.choice(self.size, size=self.batch_size, replace=False)\n return dict(obs=self.obs_buf[idxs],\n next_obs=self.next_obs_buf[idxs],\n acts=self.acts_buf[idxs],\n rews=self.rews_buf[idxs],\n done=self.done_buf[idxs])\n\n def __len__(self) -> int:\n return self.size\n```\n\n## Noisy Layer\n\n**References:**\n- https://github.com/higgsfield/RL-Adventure/blob/master/5.noisy%20dqn.ipynb\n- https://github.com/Kaixhin/Rainbow/blob/master/model.py\n\n\n```python\nclass NoisyLinear(nn.Module):\n \"\"\"Noisy linear module for NoisyNet.\n \n Attributes:\n in_features (int): input size of linear module\n out_features (int): output size of linear module\n std_init (float): initial std value\n weight_mu (nn.Parameter): mean value weight parameter\n weight_sigma (nn.Parameter): std value weight parameter\n bias_mu (nn.Parameter): mean value bias parameter\n bias_sigma (nn.Parameter): std value bias parameter\n \n \"\"\"\n\n def __init__(self, in_features: int, out_features: int, std_init: float = 0.5):\n \"\"\"Initialization.\"\"\"\n super(NoisyLinear, self).__init__()\n \n self.in_features = in_features\n self.out_features = out_features\n self.std_init = std_init\n\n self.weight_mu = nn.Parameter(torch.Tensor(out_features, in_features))\n self.weight_sigma = nn.Parameter(\n torch.Tensor(out_features, in_features)\n )\n self.register_buffer(\n \"weight_epsilon\", torch.Tensor(out_features, in_features)\n )\n\n self.bias_mu = nn.Parameter(torch.Tensor(out_features))\n self.bias_sigma = nn.Parameter(torch.Tensor(out_features))\n self.register_buffer(\"bias_epsilon\", torch.Tensor(out_features))\n\n self.reset_parameters()\n self.reset_noise()\n\n def reset_parameters(self):\n \"\"\"Reset trainable network parameters (factorized gaussian noise).\"\"\"\n mu_range = 1 / math.sqrt(self.in_features)\n self.weight_mu.data.uniform_(-mu_range, mu_range)\n self.weight_sigma.data.fill_(\n self.std_init / math.sqrt(self.in_features)\n )\n self.bias_mu.data.uniform_(-mu_range, mu_range)\n self.bias_sigma.data.fill_(\n self.std_init / math.sqrt(self.out_features)\n )\n\n def reset_noise(self):\n \"\"\"Make new noise.\"\"\"\n epsilon_in = self.scale_noise(self.in_features)\n epsilon_out = self.scale_noise(self.out_features)\n\n # outer product\n self.weight_epsilon.copy_(epsilon_out.ger(epsilon_in))\n self.bias_epsilon.copy_(epsilon_out)\n\n def forward(self, x: torch.Tensor) -> torch.Tensor:\n \"\"\"Forward method implementation.\n \n We don't use separate statements on train / eval mode.\n It doesn't show remarkable difference of performance.\n \"\"\"\n return F.linear(\n x,\n self.weight_mu + self.weight_sigma * self.weight_epsilon,\n self.bias_mu + self.bias_sigma * self.bias_epsilon,\n )\n \n @staticmethod\n def scale_noise(size: int) -> torch.Tensor:\n \"\"\"Set scale to make noise (factorized gaussian noise).\"\"\"\n x = torch.FloatTensor(np.random.normal(loc=0.0, scale=1.0, size=size))\n\n return x.sign().mul(x.abs().sqrt())\n```\n\n## Noisy Network\n\nWe use NoisyLinear for the last two FC layers, and there is a method to reset noise at every step.\nThese are the only differences from the example of *01.dqn.ipynb*.\n\n\n```python\nclass Network(nn.Module):\n def __init__(self, in_dim: int, out_dim: int):\n \"\"\"Initialization.\"\"\"\n super(Network, self).__init__()\n\n self.feature = nn.Linear(in_dim, 128)\n self.noisy_layer1 = NoisyLinear(128, 128)\n self.noisy_layer2 = NoisyLinear(128, out_dim)\n\n def forward(self, x: torch.Tensor) -> torch.Tensor:\n \"\"\"Forward method implementation.\"\"\"\n feature = F.relu(self.feature(x))\n hidden = F.relu(self.noisy_layer1(feature))\n out = self.noisy_layer2(hidden)\n \n return out\n \n def reset_noise(self):\n \"\"\"Reset all noisy layers.\"\"\"\n self.noisy_layer1.reset_noise()\n self.noisy_layer2.reset_noise()\n```\n\n## DQN + NoisyNet Agent (w/o DuelingNet)\n\nHere is a summary of DQNAgent class.\n\n| Method | Note |\n| --- | --- |\n|select_action | select an action from the input state. |\n|step | take an action and return the response of the env. |\n|compute_dqn_loss | return dqn loss. |\n|update_model | update the model by gradient descent. |\n|target_hard_update| hard update from the local model to the target model.|\n|train | train the agent during num_frames. |\n|test | test the agent (1 episode). |\n|plot | plot the training progresses. |\n\nIn the paper, NoisyNet is used as a component of the Dueling Network Architecture, which includes Double-DQN and Prioritized Experience Replay. However, we don't implement them to simplify the tutorial. One thing to note is that NoisyNet is an alternertive to $\\epsilon$-greedy method, so all $\\epsilon$ related lines are removed. Please check all comments with *NoisyNet*.\n\n\n```python\nclass DQNAgent:\n \"\"\"DQN Agent interacting with environment.\n \n Attribute:\n env (gym.Env): openAI Gym environment\n memory (ReplayBuffer): replay memory to store transitions\n batch_size (int): batch size for sampling\n target_update (int): period for target model's hard update\n gamma (float): discount factor\n dqn (Network): model to train and select actions\n dqn_target (Network): target model to update\n optimizer (torch.optim): optimizer for training dqn\n transition (list): transition information including\n state, action, reward, next_state, done\n \"\"\"\n\n def __init__(\n self, \n env: gym.Env,\n memory_size: int,\n batch_size: int,\n target_update: int,\n gamma: float = 0.99,\n ):\n \"\"\"Initialization.\n \n Args:\n env (gym.Env): openAI Gym environment\n memory_size (int): length of memory\n batch_size (int): batch size for sampling\n target_update (int): period for target model's hard update\n gamma (float): discount factor\n \"\"\"\n # NoisyNet: All attributes related to epsilon are removed\n obs_dim = env.observation_space.shape[0]\n action_dim = env.action_space.n\n \n self.env = env\n self.memory = ReplayBuffer(obs_dim, memory_size, batch_size)\n self.batch_size = batch_size\n self.target_update = target_update\n self.gamma = gamma\n \n # device: cpu / gpu\n self.device = torch.device(\n \"cuda\" if torch.cuda.is_available() else \"cpu\"\n )\n print(self.device)\n\n # networks: dqn, dqn_target\n self.dqn = Network(obs_dim, action_dim).to(self.device)\n self.dqn_target = Network(obs_dim, action_dim).to(self.device)\n self.dqn_target.load_state_dict(self.dqn.state_dict())\n self.dqn_target.eval()\n \n # optimizer\n self.optimizer = optim.Adam(self.dqn.parameters())\n\n # transition to store in memory\n self.transition = list()\n \n # mode: train / test\n self.is_test = False\n\n def select_action(self, state: np.ndarray) -> np.ndarray:\n \"\"\"Select an action from the input state.\"\"\"\n # NoisyNet: no epsilon greedy action selection\n selected_action = self.dqn(\n torch.FloatTensor(state).to(self.device)\n ).argmax()\n selected_action = selected_action.detach().cpu().numpy()\n \n if not self.is_test:\n self.transition = [state, selected_action]\n \n return selected_action\n\n def step(self, action: np.ndarray) -> Tuple[np.ndarray, np.float64, bool]:\n \"\"\"Take an action and return the response of the env.\"\"\"\n next_state, reward, done, _ = self.env.step(action)\n\n if not self.is_test:\n self.transition += [reward, next_state, done]\n self.memory.store(*self.transition)\n \n return next_state, reward, done\n\n def update_model(self) -> torch.Tensor:\n \"\"\"Update the model by gradient descent.\"\"\"\n samples = self.memory.sample_batch()\n\n loss = self._compute_dqn_loss(samples)\n\n self.optimizer.zero_grad()\n loss.backward()\n self.optimizer.step()\n \n # NoisyNet: reset noise\n self.dqn.reset_noise()\n self.dqn_target.reset_noise()\n\n return loss.item()\n \n def train(self, num_frames: int, plotting_interval: int = 200):\n \"\"\"Train the agent.\"\"\"\n self.is_test = False\n \n state = self.env.reset()\n update_cnt = 0\n losses = []\n scores = []\n score = 0\n\n for frame_idx in range(1, num_frames + 1):\n action = self.select_action(state)\n next_state, reward, done = self.step(action)\n\n state = next_state\n score += reward\n \n # NoisyNet: removed decrease of epsilon\n\n # if episode ends\n if done:\n state = self.env.reset()\n scores.append(score)\n score = 0\n\n # if training is ready\n if len(self.memory) >= self.batch_size:\n loss = self.update_model()\n losses.append(loss)\n update_cnt += 1\n \n # if hard update is needed\n if update_cnt % self.target_update == 0:\n self._target_hard_update()\n\n # plotting\n if frame_idx % plotting_interval == 0:\n self._plot(frame_idx, scores, losses)\n \n self.env.close()\n \n def test(self) -> List[np.ndarray]:\n \"\"\"Test the agent.\"\"\"\n self.is_test = True\n \n state = self.env.reset()\n done = False\n score = 0\n \n frames = []\n while not done:\n frames.append(self.env.render(mode=\"rgb_array\"))\n action = self.select_action(state)\n next_state, reward, done = self.step(action)\n\n state = next_state\n score += reward\n \n print(\"score: \", score)\n self.env.close()\n \n return frames\n\n def _compute_dqn_loss(self, samples: Dict[str, np.ndarray]) -> torch.Tensor:\n \"\"\"Return dqn loss.\"\"\"\n device = self.device # for shortening the following lines\n state = torch.FloatTensor(samples[\"obs\"]).to(device)\n next_state = torch.FloatTensor(samples[\"next_obs\"]).to(device)\n action = torch.LongTensor(samples[\"acts\"].reshape(-1, 1)).to(device)\n reward = torch.FloatTensor(samples[\"rews\"].reshape(-1, 1)).to(device)\n done = torch.FloatTensor(samples[\"done\"].reshape(-1, 1)).to(device)\n \n # G_t = r + gamma * v(s_{t+1}) if state != Terminal\n # = r otherwise\n curr_q_value = self.dqn(state).gather(1, action)\n next_q_value = self.dqn_target(next_state).max(\n dim=1, keepdim=True\n )[0].detach()\n mask = 1 - done\n target = (reward + self.gamma * next_q_value * mask).to(self.device)\n\n # calculate dqn loss\n loss = F.smooth_l1_loss(curr_q_value, target)\n\n return loss\n\n def _target_hard_update(self):\n \"\"\"Hard update: target <- local.\"\"\"\n self.dqn_target.load_state_dict(self.dqn.state_dict())\n \n def _plot(\n self, \n frame_idx: int, \n scores: List[float], \n losses: List[float], \n ):\n \"\"\"Plot the training progresses.\"\"\"\n clear_output(True)\n plt.figure(figsize=(20, 5))\n plt.subplot(131)\n plt.title('frame %s. score: %s' % (frame_idx, np.mean(scores[-10:])))\n plt.plot(scores)\n plt.subplot(132)\n plt.title('loss')\n plt.plot(losses)\n plt.show()\n```\n\n## Environment\n\nYou can see the [code](https://github.com/openai/gym/blob/master/gym/envs/classic_control/cartpole.py) and [configurations](https://github.com/openai/gym/blob/master/gym/envs/__init__.py#L53) of CartPole-v0 from OpenAI's repository.\n\n\n```python\n# environment\nenv_id = \"CartPole-v0\"\nenv = gym.make(env_id)\nif IN_COLAB:\n env = gym.wrappers.Monitor(env, \"videos\", force=True)\n```\n\n## Set random seed\n\n\n```python\nseed = 777\n\ndef seed_torch(seed):\n torch.manual_seed(seed)\n if torch.backends.cudnn.enabled:\n torch.backends.cudnn.benchmark = False\n torch.backends.cudnn.deterministic = True\n\nnp.random.seed(seed)\nseed_torch(seed)\nenv.seed(seed)\n```\n\n\n\n\n [777]\n\n\n\n## Initialize\n\n\n```python\n# parameters\nnum_frames = 20000\nmemory_size = 2000\nbatch_size = 64\ntarget_update = 150\n\n# train\nagent = DQNAgent(env, memory_size, batch_size, target_update)\n```\n\n cpu\n\n\n## Train\n\n\n```python\nagent.train(num_frames)\n```\n\n## Test\n\nRun the trained agent (1 episode).\n\n\n```python\nframes = agent.test()\n```\n\n score: 200.0\n\n\n## Render\n\n\n```python\nif IN_COLAB: # for colab\n import base64\n import glob\n import io\n import os\n\n from IPython.display import HTML, display\n\n\n def ipython_show_video(path: str) -> None:\n \"\"\"Show a video at `path` within IPython Notebook.\"\"\"\n if not os.path.isfile(path):\n raise NameError(\"Cannot access: {}\".format(path))\n\n video = io.open(path, \"r+b\").read()\n encoded = base64.b64encode(video)\n\n display(HTML(\n data=\"\"\"\n \n \"\"\".format(encoded.decode(\"ascii\"))\n ))\n\n list_of_files = glob.glob(\"videos/*.mp4\")\n latest_file = max(list_of_files, key=os.path.getctime)\n print(latest_file)\n ipython_show_video(latest_file)\n \nelse: # for jupyter\n from matplotlib import animation\n from JSAnimation.IPython_display import display_animation\n from IPython.display import display\n\n\n def display_frames_as_gif(frames: List[np.ndarray]) -> None:\n \"\"\"Displays a list of frames as a gif, with controls.\"\"\"\n patch = plt.imshow(frames[0])\n plt.axis('off')\n\n def animate(i):\n patch.set_data(frames[i])\n\n anim = animation.FuncAnimation(\n plt.gcf(), animate, frames = len(frames), interval=50\n )\n display(display_animation(anim, default_mode='loop'))\n\n\n # display \n display_frames_as_gif(frames)\n```\n\n\n\n\n\n
    \n \n
    \n \n
    \n \n \n \n \n \n \n \n \n \n
    \n Once \n Loop \n Reflect \n
    \n
    \n\n\n\n\n\n", "meta": {"hexsha": "37efffad0bec0c9cb5ea0b369872a3b1af4b8738", "size": 477510, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "05.noisy_net.ipynb", "max_stars_repo_name": "UmbertoJr/rainbow-is-all-you-need", "max_stars_repo_head_hexsha": "812edac69ba1ce1b8926df01ff7ff88d3eab786c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-06-28T16:08:57.000Z", "max_stars_repo_stars_event_max_datetime": "2021-06-28T16:08:57.000Z", "max_issues_repo_path": "05.noisy_net.ipynb", "max_issues_repo_name": "serereuk/rainbow-is-all-you-need", "max_issues_repo_head_hexsha": "812edac69ba1ce1b8926df01ff7ff88d3eab786c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "05.noisy_net.ipynb", "max_forks_repo_name": "serereuk/rainbow-is-all-you-need", "max_forks_repo_head_hexsha": "812edac69ba1ce1b8926df01ff7ff88d3eab786c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-06-21T13:50:40.000Z", "max_forks_repo_forks_event_max_datetime": "2021-06-21T13:50:40.000Z", "avg_line_length": 417.03930131, "max_line_length": 57708, "alphanum_fraction": 0.9416493895, "converted": true, "num_tokens": 4782, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191460821871, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.4387271747454299}} {"text": "```python\nimport numpy as np\nfrom matplotlib import pyplot as plt\nimport real_space_electrostatic_sum\n```\n\n# Benchmarking\n\nUsing the shared library `real-space-electrostatic-sum.so`, together with the Python wrapper in `real_space_electrostatic_sum.py`, this notebook reproduces results from [Pickard, *Phys. Rev. Mat.* **2**, 013806 (2018)](https://doi.org/10.1103/PhysRevMaterials.2.013806).\n\n## Reproducing part of Fig. 1(b)\n\nFig. 1(b) in the paper demonstrates the convergence of the real-space method for a simple cubic lattice with unit spacing.\n\n\n```python\n# lattice vectors\na_1 = np.array([1.0, 0.0, 0.0])\na_2 = np.array([0.0, 1.0, 0.0])\na_3 = np.array([0.0, 0.0, 1.0])\n\n# ion locations and charge array\nloc = np.zeros([1,3])\nchg = np.ones(1)\n\n# loop over cutoff radii\nr_c = np.linspace(0.001,30,500)\nr_d = 1.5\nene = np.zeros(len(r_c))\nfor i, r in enumerate(r_c):\n ene[i] = real_space_electrostatic_sum.energy(\n a_1, a_2, a_3, loc.shape[0], loc[:,0], loc[:,1], loc[:,2], chg, r, r_d)\n \n# generate part of Fig. 1(b)\nplt.plot(r_c, ene, 'r')\nplt.title('Fig. 1(b)')\nplt.xlim([0,30]); plt.ylim([-1.5,-1.25])\nplt.xlabel('$R_c$'); plt.ylabel('$E_i$')\nplt.show()\n```\n\n## Reproducing data in Table I\n\nTable I in the paper contains ion-ion energies for four crystals obtained with the real-space method, as well as the Ewald method. The real-space method data are re-generated here, exhibiting near perfect agreement with Table I. The Ewald energies reported below were re-obtained with CASTEP using the exact lattice parameters employed here.\n\n### _Al_\n\n\n```python\n# lattice vectors\na_1 = np.array([5.41141973394663, 0.00000000000000, 0.00000000000000])\na_2 = np.array([2.70570986697332, 4.68642696013821, 0.00000000000000])\na_3 = np.array([2.70570986697332, 1.56214232004608, 4.41840571073226])\n\n# ion locations\nloc = np.zeros([1,3])\n\n# charge array\nchg = 3.0 * np.ones(loc.shape[0])\n\n# length scale\nh_max = 4.42\n\n# reference energy\newald = -2.69595457432924945\nprint('Ewald: energy = {0:12.9f}'.format(-2.69595457432924945))\n\n# real-space-method energies\nr_d_hat = 2.0\nene = real_space_electrostatic_sum.energy(\n a_1, a_2, a_3, loc.shape[0], loc[:,0], loc[:,1], loc[:,2], chg, 3.0*r_d_hat**2*h_max, r_d_hat*h_max)\nprint('R\\u0302d = {0:3.1f}: energy = {1:12.9f}'.format(r_d_hat, ene))\nr_d_hat = 1.5\nene = real_space_electrostatic_sum.energy(\n a_1, a_2, a_3, loc.shape[0], loc[:,0], loc[:,1], loc[:,2], chg, 3.0*r_d_hat**2*h_max, r_d_hat*h_max)\nprint('R\\u0302d = {0:3.1f}: energy = {1:12.9f}'.format(r_d_hat, ene))\nr_d_hat = 1.0\nene = real_space_electrostatic_sum.energy(\n a_1, a_2, a_3, loc.shape[0], loc[:,0], loc[:,1], loc[:,2], chg, 3.0*r_d_hat**2*h_max, r_d_hat*h_max)\nprint('R\\u0302d = {0:3.1f}: energy = {1:12.9f}'.format(r_d_hat, ene))\n```\n\n Ewald: energy = -2.695954574\n R\u0302d = 2.0: energy = -2.695954574\n R\u0302d = 1.5: energy = -2.695954574\n R\u0302d = 1.0: energy = -2.696027958\n\n\n### _Si_\n\n\n```python\n# lattice vectors\na_1 = np.array([7.25654832321381, 0.00000000000000, 0.00000000000000])\na_2 = np.array([3.62827416160690, 6.28435519169252, 0.00000000000000])\na_3 = np.array([3.62827416160690, 2.09478506389751, 5.92494689524090])\n\n# ion locations\nloc = np.array([[0.0, 0.0, 0.0],\n [0.25, 0.25, 0.25]])\nloc = (np.vstack((a_1, a_2, a_3)).T).dot(loc.T).T # convert to cartesian\n\n# charge array\nchg = 4.0 * np.ones(loc.shape[0])\n\n# length scale\nh_max = 5.92\n\n# reference energy\newald = -8.39857465282205418\nprint('Ewald: energy = {0:12.9f}, per ion = {1:12.9f}'.format(ewald, ewald/loc.shape[0]))\n\n# real-space-method energies\nr_d_hat = 2.0\nene = real_space_electrostatic_sum.energy(\n a_1, a_2, a_3, loc.shape[0], loc[:,0], loc[:,1], loc[:,2], chg, 3.0*r_d_hat**2*h_max, r_d_hat*h_max)\nprint('R\\u0302d = {0:3.1f}: energy = {1:12.9f}, per ion = {2:12.9f}'.format(r_d_hat, ene, ene/loc.shape[0]))\nr_d_hat = 1.5\nene = real_space_electrostatic_sum.energy(\n a_1, a_2, a_3, loc.shape[0], loc[:,0], loc[:,1], loc[:,2], chg, 3.0*r_d_hat**2*h_max, r_d_hat*h_max)\nprint('R\\u0302d = {0:3.1f}: energy = {1:12.9f}, per ion = {2:12.9f}'.format(r_d_hat, ene, ene/loc.shape[0]))\nr_d_hat = 1.0\nene = real_space_electrostatic_sum.energy(\n a_1, a_2, a_3, loc.shape[0], loc[:,0], loc[:,1], loc[:,2], chg, 3.0*r_d_hat**2*h_max, r_d_hat*h_max)\nprint('R\\u0302d = {0:3.1f}: energy = {1:12.9f}, per ion = {2:12.9f}'.format(r_d_hat, ene, ene/loc.shape[0]))\n```\n\n Ewald: energy = -8.398574653, per ion = -4.199287326\n R\u0302d = 2.0: energy = -8.398574653, per ion = -4.199287326\n R\u0302d = 1.5: energy = -8.398574651, per ion = -4.199287326\n R\u0302d = 1.0: energy = -8.398589712, per ion = -4.199294856\n\n\n### _SiO2_\n\n\n```python\n# lattice vectors\na_1 = np.array([ 9.28422445623683, 0.00000000000000, 0.00000000000000])\na_2 = np.array([-4.64211222811842, 8.04037423353787, 0.00000000000000])\na_3 = np.array([ 0.00000000000000, 0.00000000000000, 10.2139697101486])\n\n# ion locations\nloc = np.array([[0.41500, 0.27200, 0.21300],\n [0.72800, 0.14300, 0.54633],\n [0.85700, 0.58500, 0.87967],\n [0.27200, 0.41500, 0.78700],\n [0.14300, 0.72800, 0.45367],\n [0.58500, 0.85700, 0.12033],\n [0.46500, 0.00000, 0.33333],\n [0.00000, 0.46500, 0.66667],\n [0.53500, 0.53500, 0.00000]])\nloc = (np.vstack((a_1, a_2, a_3)).T).dot(loc.T).T # convert to cartesian\n\n# charge array\nchg = 6.0 * np.ones(loc.shape[0]) # most are O\nchg[6:] = 4.0 # three are Si\n\n# length scale\nh_max = 10.21\n\n# reference energy\newald = -69.48809871723248932\nprint('Ewald: energy = {0:12.9f}, per ion = {1:12.9f}'.format(ewald, ewald/loc.shape[0]))\n\n# real-space-method energies\nr_d_hat = 2.0\nene = real_space_electrostatic_sum.energy(\n a_1, a_2, a_3, loc.shape[0], loc[:,0], loc[:,1], loc[:,2], chg, 3.0*r_d_hat**2*h_max, r_d_hat*h_max)\nprint('R\\u0302d = {0:3.1f}: energy = {1:12.9f}, per ion = {2:12.9f}'.format(r_d_hat, ene, ene/loc.shape[0]))\nr_d_hat = 1.5\nene = real_space_electrostatic_sum.energy(\n a_1, a_2, a_3, loc.shape[0], loc[:,0], loc[:,1], loc[:,2], chg, 3.0*r_d_hat**2*h_max, r_d_hat*h_max)\nprint('R\\u0302d = {0:3.1f}: energy = {1:12.9f}, per ion = {2:12.9f}'.format(r_d_hat, ene, ene/loc.shape[0]))\nr_d_hat = 1.0\nene = real_space_electrostatic_sum.energy(\n a_1, a_2, a_3, loc.shape[0], loc[:,0], loc[:,1], loc[:,2], chg, 3.0*r_d_hat**2*h_max, r_d_hat*h_max)\nprint('R\\u0302d = {0:3.1f}: energy = {1:12.9f}, per ion = {2:12.9f}'.format(r_d_hat, ene, ene/loc.shape[0]))\n```\n\n Ewald: energy = -69.488098717, per ion = -7.720899857\n R\u0302d = 2.0: energy = -69.488098717, per ion = -7.720899857\n R\u0302d = 1.5: energy = -69.488098713, per ion = -7.720899857\n R\u0302d = 1.0: energy = -69.487434045, per ion = -7.720826005\n\n\n### _Al2SiO5_\n\n\n```python\n# lattice vectors\na_1 = np.array([14.7289033699982, 0.00000000000000, 0.00000000000000])\na_2 = np.array([0.00000000000000, 14.9260018049230, 0.00000000000000])\na_3 = np.array([0.00000000000000, 0.00000000000000, 10.5049875335275])\n\n# ion locations\nloc = np.array([[0.23030, 0.13430, 0.23900],\n [0.76970, 0.86570, 0.23900],\n [0.26970, 0.63430, 0.26100],\n [0.73030, 0.36570, 0.26100],\n [0.76970, 0.86570, 0.76100],\n [0.23030, 0.13430, 0.76100],\n [0.73030, 0.36570, 0.73900],\n [0.26970, 0.63430, 0.73900],\n [0.00000, 0.00000, 0.24220],\n [0.50000, 0.50000, 0.25780],\n [0.00000, 0.00000, 0.75780],\n [0.50000, 0.50000, 0.74220],\n [0.37080, 0.13870, 0.50000],\n [0.42320, 0.36270, 0.50000],\n [0.62920, 0.86130, 0.50000],\n [0.57680, 0.63730, 0.50000],\n [0.12920, 0.63870, 0.00000],\n [0.07680, 0.86270, 0.00000],\n [0.87080, 0.36130, 0.00000],\n [0.92320, 0.13730, 0.00000],\n [0.24620, 0.25290, 0.00000],\n [0.42400, 0.36290, 0.00000],\n [0.10380, 0.40130, 0.00000],\n [0.75380, 0.74710, 0.00000],\n [0.57600, 0.63710, 0.00000],\n [0.89620, 0.59870, 0.00000],\n [0.25380, 0.75290, 0.50000],\n [0.07600, 0.86290, 0.50000],\n [0.39620, 0.90130, 0.50000],\n [0.74620, 0.24710, 0.50000],\n [0.92400, 0.13710, 0.50000],\n [0.60380, 0.09870, 0.50000]])\nloc = (np.vstack((a_1, a_2, a_3)).T).dot(loc.T).T # convert to cartesian\n\n# charge array\nchg = 6.0 * np.ones(loc.shape[0]) # most are O\nchg[8:13] = 3.0 # eight are Al\nchg[14] = 3.0\nchg[16] = 3.0\nchg[18] = 3.0\nchg[20] = 4.0 # four are Si\nchg[23] = 4.0\nchg[26] = 4.0\nchg[29] = 4.0\n\n# length scale\nh_max = 14.93\n\n# reference energy\newald = -244.05500850908111943\nprint('Ewald: energy = {0:14.9f}, per ion = {1:12.9f}'.format(ewald, ewald/loc.shape[0]))\n\n# real-space-method energies\nr_d_hat = 2.0\nene = real_space_electrostatic_sum.energy(\n a_1, a_2, a_3, loc.shape[0], loc[:,0], loc[:,1], loc[:,2], chg, 3.0*r_d_hat**2*h_max, r_d_hat*h_max)\nprint('R\\u0302d = {0:3.1f}: energy = {1:14.9f}, per ion = {2:12.9f}'.format(r_d_hat, ene, ene/loc.shape[0]))\nr_d_hat = 1.5\nene = real_space_electrostatic_sum.energy(\n a_1, a_2, a_3, loc.shape[0], loc[:,0], loc[:,1], loc[:,2], chg, 3.0*r_d_hat**2*h_max, r_d_hat*h_max)\nprint('R\\u0302d = {0:3.1f}: energy = {1:14.9f}, per ion = {2:12.9f}'.format(r_d_hat, ene, ene/loc.shape[0]))\nr_d_hat = 1.0\nene = real_space_electrostatic_sum.energy(\n a_1, a_2, a_3, loc.shape[0], loc[:,0], loc[:,1], loc[:,2], chg, 3.0*r_d_hat**2*h_max, r_d_hat*h_max)\nprint('R\\u0302d = {0:3.1f}: energy = {1:14.9f}, per ion = {2:12.9f}'.format(r_d_hat, ene, ene/loc.shape[0]))\n```\n\n Ewald: energy = -244.055008509, per ion = -7.626719016\n R\u0302d = 2.0: energy = -244.055008514, per ion = -7.626719016\n R\u0302d = 1.5: energy = -244.055008508, per ion = -7.626719016\n R\u0302d = 1.0: energy = -244.054921810, per ion = -7.626716307\n\n\n## Madelung energy for NaCl\n\nSee the discussion around Eq. (1) in [Mamode, _J. Math. Chem._ __55__, 734 (2017)](https://doi.org/10.1007/s10910-016-0705-9).\n\nWith $M_{\\mathrm{NaCl}}$ as the Madelung energy and $E_{\\mathrm{NaCl}}$ as the energy of a two-atom primitive cell having $z_{1,2}=\\pm 1$, the following identities hold\n\n\\begin{equation}\n\\begin{split}\nM_{\\mathrm{NaCl}} \n&= E_{\\mathrm{NaCl}} \\\\\n&= \\sum_{i\\in\\{1,2\\}} \\sum_{j\\ne i}^\\infty \\frac{z_i z_j}{2r_{ij}} \\\\\n&= \\sum_{i\\in\\{1,2\\}}\n \\left[ \\sum_{\\substack{j\\ne i \\\\z_iz_j>0}}^\\infty \\frac{z_i z_j}{2r_{ij}} -\n \\sum_{\\substack{j\\ne i \\\\z_iz_j<0}}^\\infty \\frac{|z_i z_j|}{2r_{ij}} \\right] \\\\\n&= \\sum_{i\\in\\{1,2\\}}\n \\left[ 2 \\sum_{\\substack{j\\ne i \\\\z_iz_j>0}}^\\infty \\frac{z_i z_j}{2r_{ij}} -\n \\sum_{j\\ne i}^\\infty \\frac{|z_i z_j|}{2r_{ij}} \\right] \\\\\n&= 4 E_{\\mathrm{FCC}} - \n \\sum_{i\\in\\{1,2\\}} \\sum_{j\\ne i}^\\infty \\frac{|z_i z_j|}{2r_{ij}}\n\\end{split}\n\\end{equation}\n\nand the final result should be $M_{\\mathrm{NaCl}} = \u22121.747 564 594 633 \\cdots$.\n\n\n```python\n# lattice vectors\na_1 = np.array([1.0, 1.0, 0.0])\na_2 = np.array([0.0, 1.0, 1.0])\na_3 = np.array([1.0, 0.0, 1.0])\n\n# length scale and cutoff\nh_max = np.sqrt(4.0/3.0)\nr_d_hat = 3.0\n\n# compute FCC energy\nloc = np.zeros([1,3])\nchg = np.ones(loc.shape[0])\nE_FCC = real_space_electrostatic_sum.energy(\n a_1, a_2, a_3, loc.shape[0], loc[:,0], loc[:,1], loc[:,2], chg, 3.0*r_d_hat**2*h_max, r_d_hat*h_max)\n\n# compute second term\nloc = np.zeros([2,3])\nloc[1,:] = np.array([1.0, 1.0, 1.0])\nchg = np.ones(loc.shape[0])\nE_2 = real_space_electrostatic_sum.energy(\n a_1, a_2, a_3, loc.shape[0], loc[:,0], loc[:,1], loc[:,2], chg, 3.0*r_d_hat**2*h_max, r_d_hat*h_max)\n\n# print result\nprint('M = {0:15.12f}'.format(4*E_FCC - E_2))\n```\n\n M = -1.747564594633\n\n", "meta": {"hexsha": "100694ea3fed00cc2f261a3af0a89fd2a2e7f16c", "size": 25366, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "python/benchmarking.ipynb", "max_stars_repo_name": "zhubonan/real-space-electrostatic-sum", "max_stars_repo_head_hexsha": "63390ff5dd7ba911ca86dc02c5f9a38f00e34830", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2020-07-08T17:39:15.000Z", "max_stars_repo_stars_event_max_datetime": "2021-02-07T04:05:48.000Z", "max_issues_repo_path": "python/benchmarking.ipynb", "max_issues_repo_name": "zhubonan/real-space-electrostatic-sum", "max_issues_repo_head_hexsha": "63390ff5dd7ba911ca86dc02c5f9a38f00e34830", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "python/benchmarking.ipynb", "max_forks_repo_name": "zhubonan/real-space-electrostatic-sum", "max_forks_repo_head_hexsha": "63390ff5dd7ba911ca86dc02c5f9a38f00e34830", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2020-01-17T04:41:22.000Z", "max_forks_repo_forks_event_max_datetime": "2020-07-16T19:51:19.000Z", "avg_line_length": 55.5054704595, "max_line_length": 8332, "alphanum_fraction": 0.6559173697, "converted": true, "num_tokens": 5049, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191460821871, "lm_q2_score": 0.61878043374385, "lm_q1q2_score": 0.4387271747454298}} {"text": "```python\n# This cell is added by sphinx-gallery\n!pip install mrsimulator --quiet\n\n\n%matplotlib inline\n\nimport mrsimulator\nprint(f'You are using mrsimulator v{mrsimulator.__version__}')\n```\n\n\n# Arbitrary spin transition (single-quantum)\n\n\u00b2\u2077Al (I=5/2) quadrupolar spectrum simulation.\n\n\nThe mrsimulator built-in one-dimensional methods, BlochDecaySpectrum and\nBlochDecayCTSpectrum, are designed to simulate spectrum from all single\nquantum transitions or central transition selective transition, respectively. In this\nexample, we show how you can simulate any arbitrary transition using the generic\nMethod1D method.\n\n\n\n\n```python\nimport matplotlib.pyplot as plt\n\nfrom mrsimulator import Simulator, SpinSystem, Site\nfrom mrsimulator.methods import Method1D\nfrom mrsimulator.method.event import SpectralEvent\nfrom mrsimulator.spin_system.tensors import SymmetricTensor\n```\n\nCreate a single-site arbitrary spin system.\n\n\n\n\n```python\nsite = Site(\n name=\"27Al\",\n isotope=\"27Al\",\n isotropic_chemical_shift=35.7, # in ppm\n quadrupolar=SymmetricTensor(Cq=5.959e6, eta=0.32), # Cq is in Hz\n)\nspin_system = SpinSystem(sites=[site])\n```\n\n## Selecting spin transitions for simulation\n\nThe arguments of the Method1D object are the same as the arguments of the\nBlochDecaySpectrum method; however, unlike a BlochDecaySpectrum method, the\n`spectral_dim_api` object in Method1D contains additional argument---`events`.\n\nThe `event_api` object is a collection of attributes, which are local to the\nevent. It is here where we define a `transition_query` to select one or more\ntransitions for simulating the spectrum. The two attributes of the `transition_query`\nare `P` and `D`, which are given as,\n\n\\begin{align}P &= m_f - m_i \\\\\n D &= m_f^2 - m_i^2,\\end{align}\n\nwhere $m_f$ and $m_i$ are the spin quantum numbers for the final and\ninitial energy states. Based on the query, the method selects all transitions from\nthe spin system that satisfy the query selection criterion. For example, to simulate\na spectrum for the satellite transition, $|-1/2\\rangle\\rightarrow|-3/2\\rangle$,\nset the value of\n\n\\begin{align}P &= \\left(-\\frac{3}{2}\\right) - \\left(-\\frac{1}{2}\\right) = -1 \\\\\n D &= \\frac{9}{4} - \\frac{1}{4} = 2.\\end{align}\n\nFor illustrative purposes, let's look at the infinite speed spectrum from this\nsatellite transition.\n\n\n\n\n```python\nmethod = Method1D(\n name=\"Inner Satellite Spectrum\",\n channels=[\"27Al\"],\n magnetic_flux_density=21.14, # in T\n rotor_frequency=1e9, # in Hz\n spectral_dimensions=[\n dict(\n count=1024,\n spectral_width=1e4, # in Hz\n reference_offset=1e4, # in Hz\n events=[\n SpectralEvent(\n # Selecting the inner satellite transitions\n transition_query=[dict(P=[-1], D=[2])],\n )\n ],\n )\n ],\n)\n\n# A graphical representation of the method object.\nplt.figure(figsize=(5, 3))\nmethod.plot()\nplt.show()\n```\n\nCreate the Simulator object and add the method and the spin system object.\n\n\n\n\n```python\nsim = Simulator()\nsim.spin_systems += [spin_system] # add the spin system\nsim.methods += [method] # add the method\n```\n\nSimulate the spectrum.\n\n\n\n\n```python\nsim.run()\n\n# The plot of the simulation before signal processing.\nplt.figure(figsize=(4.25, 3.0))\nax = plt.subplot(projection=\"csdm\")\nax.plot(sim.methods[0].simulation.real, color=\"black\", linewidth=1)\nax.invert_xaxis()\nplt.tight_layout()\nplt.show()\n```\n\n## Selecting both inner and outer-satellite transitions\nYou may use the same transition query selection criterion to select multiple\ntransitions. Consider the following transitions with respective P and D values.\n\n- $|-1/2\\rangle\\rightarrow|-3/2\\rangle$ ($P=-1, D=2$)\n- $|-3/2\\rangle\\rightarrow|-5/2\\rangle$ ($P=-1, D=4$)\n\n\n\n\n```python\nmethod2 = Method1D(\n name=\"Satellite Spectrum\",\n channels=[\"27Al\"],\n magnetic_flux_density=21.14, # in T\n rotor_frequency=1e9, # in Hz\n spectral_dimensions=[\n {\n \"count\": 1024,\n \"spectral_width\": 1e4, # in Hz\n \"reference_offset\": 1e4, # in Hz\n \"events\": [\n {\n \"transition_query\": [\n {\"P\": [-1], \"D\": [2]}, # <-- select inter satellite transitions\n {\"P\": [-1], \"D\": [4]}, # <-- select outer satellite transitions\n ]\n }\n ],\n }\n ],\n)\n\n# A graphical representation of the method object.\nplt.figure(figsize=(5, 3))\nmethod2.plot()\nplt.show()\n```\n\nUpdate the method object in the Simulator object.\n\n\n\n\n```python\nsim.methods[0] = method2 # add the method\n```\n\nSimulate the spectrum.\n\n\n\n\n```python\nsim.run()\n\n# The plot of the simulation before signal processing.\nplt.figure(figsize=(4.25, 3.0))\nax = plt.subplot(projection=\"csdm\")\nax.plot(sim.methods[0].simulation.real, color=\"black\", linewidth=1)\nax.invert_xaxis()\nplt.tight_layout()\nplt.show()\n```\n", "meta": {"hexsha": "485613df90b3c3e06ba8fe4108aa621ee5bd6341", "size": 8101, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/notebooks/examples/1D_simulation(crystalline)/plot_3_satellite_transition_sim.ipynb", "max_stars_repo_name": "mgiammar/mrsimulator", "max_stars_repo_head_hexsha": "5ec5551a922d110874ae0c73a6e185a9f674fbe5", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/notebooks/examples/1D_simulation(crystalline)/plot_3_satellite_transition_sim.ipynb", "max_issues_repo_name": "mgiammar/mrsimulator", "max_issues_repo_head_hexsha": "5ec5551a922d110874ae0c73a6e185a9f674fbe5", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 15, "max_issues_repo_issues_event_min_datetime": "2021-05-28T16:51:57.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-04T17:13:25.000Z", "max_forks_repo_path": "docs/notebooks/examples/1D_simulation(crystalline)/plot_3_satellite_transition_sim.ipynb", "max_forks_repo_name": "mgiammar/mrsimulator", "max_forks_repo_head_hexsha": "5ec5551a922d110874ae0c73a6e185a9f674fbe5", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 43.320855615, "max_line_length": 1245, "alphanum_fraction": 0.5623997037, "converted": true, "num_tokens": 1284, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.6926419831347361, "lm_q1q2_score": 0.43872654117171533}} {"text": "```\nimport sympy\nfrom phasor.utilities.ipynb.displays import *\nfrom phasor.utilities.ipynb.ipy_sympy import *\nimport scipy.linalg\n\n\nimport numpy.testing as np_test\nimport declarative as decl\nfrom declarative.bunch import (\n DeepBunch,\n)\n\n#import numpy as np\n\nfrom phasor import system\nfrom phasor import readouts\nfrom phasor import optics\nfrom phasor.optics.nonlinear_crystal import NonlinearCrystal\nfrom phasor.utilities.print import pprint\n\nfrom phasor.optics.models.KTP_test_stand import KTPTestStand\n```\n\n Populating the interactive namespace from numpy and matplotlib\n Sympy version: 1.0\n\n\n\n```\ndb = DeepBunch()\ndb.test.ktp.length.val = 20#np.linspace(0, 20, 100)\ndb.test.ktp.N_ode = 100\ndb.test.include_AC = False\nsys = system.BGSystem(\n ctree = db,\n)\nfrom phasor.optics.models.KTP_test_stand import KTPTestStand\nsys.own.test = KTPTestStand()\ndb = sys.ctree_shadow()\nprint(sys.test.ktp.ctree_as_yaml())\nsys.test.DC_R.DC_readout\n```\n\n N_ode: 100\n length: {units: millimeter, val: 20}\n nlg: 0.1\n \n Number of states: 4\n Number of states: 4\n PERTURB: 1\n COUPLING_SIZE: 108\n PRE-PURGING\n FRAC REMOVED: 0.06896551724137931 8\n PERTURB: 2\n COUPLING_SIZE: 124\n PRE-PURGING\n FRAC REMOVED: 0.03125 4\n PERTURB: 3\n COUPLING_SIZE: 124\n PRE-PURGING\n FRAC REMOVED: 0.03125 4\n PRE-PURGING\n FRAC REMOVED: 0.1447811447811448 86\n\n\n\n\n\n array(0.06923723470180326)\n\n\n\n\n```\nprint(sys.test.DC_R.DC_readout)\nprint(sys.test.DC_G.DC_readout)\n```\n\n 0.06923723470180326\n 0.9295115580241431\n\n\n\n```\ndb = DeepBunch()\ndb.test.ktp.length.val = np.linspace(0, 25, 100)\ndb.test.ktp.N_ode = 200\ndb.test.include_AC = False\nsys = system.BGSystem(\n ctree = db,\n)\nfrom phasor.optics.models.KTP_test_stand import KTPTestStand\nsys.own.test = KTPTestStand()\ndb = sys.ctree_shadow()\nprint(sys.test.ktp.ctree_as_yaml())\nprint(sys.test.DC_R.DC_readout)\n```\n\n N_ode: 200\n length:\n units: millimeter\n val: !!python/object/apply:numpy.core.multiarray._reconstruct\n args:\n - !!python/name:numpy.ndarray ''\n - !!python/tuple [0]\n - !!binary |\n Yg==\n state: !!python/tuple\n - 1\n - !!python/tuple [100]\n - !!python/object/apply:numpy.dtype\n args: [f8, 0, 1]\n state: !!python/tuple [3, <, null, null, null, -1, -1, 0]\n - false\n - !!binary |\n AAAAAAAAAAB/pUCtXynQP3+lQK1fKeA/Pvjggw8+6D9/pUCtXynwP9/OkJi3M/Q/Pvjggw8++D+e\n ITFvZ0j8P3+lQK1fKQBAL7rooosuAkDfzpCYtzMEQI/jOI7jOAZAPvjggw8+CEDuDIl5O0MKQJ4h\n MW9nSAxATjbZZJNNDkB/pUCtXykQQNevFKj1KxFAL7rooosuEkCHxLydITETQN/OkJi3MxRAN9lk\n k002FUCP4ziO4zgWQOftDIl5OxdAPvjggw8+GECWArV+pUAZQO4MiXk7QxpARhdddNFFG0CeITFv\n Z0gcQPYrBWr9Sh1ATjbZZJNNHkCmQK1fKVAfQH+lQK1fKSBAq6qqqqqqIEDXrxSo9SshQAO1fqVA\n rSFAL7rooosuIkBbv1Kg1q8iQIfEvJ0hMSNAs8kmm2yyI0DfzpCYtzMkQAvU+pUCtSRAN9lkk002\n JUBj3s6QmLclQI/jOI7jOCZAu+iiiy66JkDn7QyJeTsnQBPzdobEvCdAPvjggw8+KEBq/UqBWr8o\n QJYCtX6lQClAwgcffPDBKUDuDIl5O0MqQBoS83aGxCpARhdddNFFK0ByHMdxHMcrQJ4hMW9nSCxA\n yiabbLLJLED2KwVq/UotQCIxb2dIzC1ATjbZZJNNLkB6O0Ni3s4uQKZArV8pUC9A0kUXXXTRL0B/\n pUCtXykwQBWo9SsFajBAq6qqqqqqMEBBrV8pUOswQNevFKj1KzFAbbLJJptsMUADtX6lQK0xQJm3\n MyTm7TFAL7rooosuMkDFvJ0hMW8yQFu/UqDWrzJA8cEHH3zwMkCHxLydITEzQB3HcRzHcTNAs8km\n m2yyM0BJzNsZEvMzQN/OkJi3MzRAddFFF110NEAL1PqVArU0QKHWrxSo9TRAN9lkk002NUDN2xkS\n 83Y1QGPezpCYtzVA+eCDDz74NUCP4ziO4zg2QCXm7QyJeTZAu+iiiy66NkBR61cK1Po2QOftDIl5\n OzdAffDBBx98N0AT83aGxLw3QKn1KwVq/TdAPvjggw8+OEDU+pUCtX44QGr9SoFavzhAAAAAAAAA\n OUA=\n nlg: 0.1\n \n\n\n /usr/lib64/python3.5/site-packages/yaml/representer.py:135: FutureWarning: comparison to `None` will result in an elementwise object comparison in the future.\n if data in [None, ()]:\n\n\n Number of states: 4\n Number of states: 4\n PERTURB: 1\n COUPLING_SIZE: 108\n PRE-PURGING\n FRAC REMOVED: 0.06896551724137931 8\n PERTURB: 2\n COUPLING_SIZE: 124\n PRE-PURGING\n FRAC REMOVED: 0.03125 4\n PERTURB: 3\n COUPLING_SIZE: 124\n PRE-PURGING\n FRAC REMOVED: 0.03125 4\n PRE-PURGING\n FRAC REMOVED: 0.1447811447811448 86\n [ 1. 0.9993594 0.99744086 0.9942542 0.98981564 0.98414763\n 0.97727865 0.96924287 0.9600798 0.94983388 0.93855407 0.9262933\n 0.91310803 0.89905766 0.88420404 0.86861091 0.85234333 0.83546722\n 0.8180488 0.80015414 0.78184866 0.76319678 0.74426145 0.72510388\n 0.70578316 0.68635607 0.66687678 0.64739671 0.62796434 0.60862517\n 0.58942157 0.57039279 0.55157489 0.53300084 0.51470049 0.49670066\n 0.47902521 0.46169517 0.4447288 0.42814177 0.41194722 0.39615598\n 0.38077665 0.36581575 0.35127789 0.33716592 0.32348101 0.31022287\n 0.29738982 0.28497896 0.27298628 0.26140677 0.25023454 0.23946296\n 0.22908468 0.21909182 0.20947599 0.20022837 0.19133985 0.18280102\n 0.17460227 0.16673387 0.15918598 0.15194869 0.14501212 0.13836641\n 0.13200177 0.12590849 0.12007699 0.11449785 0.10916178 0.10405971\n 0.09918273 0.09452214 0.09006947 0.08581646 0.08175508 0.07787753\n 0.07417626 0.07064392 0.06727343 0.06405792 0.06099078 0.05806561\n 0.05527625 0.05261675 0.05008139 0.04766466 0.04536128 0.04316616\n 0.0410744 0.03908131 0.03718239 0.03537332 0.03364995 0.03200831\n 0.0304446 0.02895518 0.02753655 0.02618537]\n\n\n\n```\naxB = mplfigB(Nrows=1)\ntest = sys.test\naxB.ax0.plot(test.ktp.length_mm.val, test.DC_R.DC_readout, color = 'red')\naxB.ax0.plot(test.ktp.length_mm.val, test.DC_G.DC_readout, color = 'green')\naxB.ax0.plot(test.ktp.length_mm.val, test.DC_R.DC_readout + test.DC_G.DC_readout, color = 'black')\naxB.ax0.plot(test.ktp.length_mm.val, 1 * np.tanh(.100 * test.ktp.length_mm.val)**2, ls = '--', color = 'blue')\n#axB.ax0.set_ylim(0, 1.1)\n```\n\n\n```\ndb = DeepBunch()\ndb.test.ktp.length.val = np.linspace(0, 2.5, 100)\ndb.test.ktp.nlg = 1\ndb.test.ktp.N_ode = 100\ndb.test.include_AC = True\nsys = system.BGSystem(\n ctree = db,\n)\nsys.own.test = KTPTestStand()\ndb = sys.ctree_shadow()\n#print(sys.test.ktp.ctree_as_yaml())\n```\n\n\n```\naxB = mplfigB(Nrows=3)\ntest = sys.test\nX_NLG = test.ktp.length_mm.val\naxB.ax0.plot(\n X_NLG,\n test.DC_R.DC_readout,\n color = 'red',\n label = '1064 power',\n)\naxB.ax0.plot(\n X_NLG,\n test.DC_G.DC_readout,\n color = 'green',\n label = '532 power',\n)\naxB.ax0.plot(\n X_NLG,\n test.DC_R.DC_readout + test.DC_G.DC_readout,\n color = 'black',\n label = 'total power [W]',\n)\naxB.ax0.plot(\n X_NLG,\n 1 * np.tanh(1 * test.ktp.length_mm.val)**2,\n ls = '--',\n color = 'purple',\n label = r'tanh$^2$(1 * NLG)',\n)\naxB.ax0.set_ylim(0, 1.1)\naxB.ax0.set_ylabel('Readout Power [W]')\naxB.ax0.legend(\n fontsize = 8,\n loc = 'center left'\n)\naxB.ax1.plot(\n X_NLG,\n test.AC_R.AC_CSD_IQ[0, 0],\n color = 'red',\n label = 'amplitude quadrature',\n)\naxB.ax1.plot(\n X_NLG,\n test.AC_R.AC_CSD_IQ[1, 1],\n color = 'orange',\n label = 'phase quadrature',\n)\naxB.ax1.plot(\n X_NLG,\n test.AC_R.AC_CSD_ellipse.max,\n color = 'blue',\n label = 'ellipse max',\n ls = '--'\n)\naxB.ax1.plot(\n X_NLG,\n test.AC_R.AC_CSD_ellipse.min,\n color = 'purple',\n label = 'ellipse min',\n ls = '--',\n)\naxB.ax1.plot(\n X_NLG,\n test.AC_R.AC_CSD_ellipse.min**.25 * test.AC_R.AC_CSD_ellipse.max**.25,\n color = 'black',\n ls = '--',\n label = 'geometric mean',\n)\naxB.ax1.set_ylabel('1064 PSD/ShotN\\n[quanta/Hz]')\naxB.ax1.set_yscale('log')\naxB.ax1.legend(\n fontsize = 8,\n loc = 'center left'\n)\naxB.ax2.set_ylabel('532 PSD/ShotN\\n[quanta/Hz]')\naxB.ax2.plot(\n X_NLG,\n test.AC_G.AC_CSD_IQ[0, 0],\n color = 'green',\n label = 'amplitude quadrature',\n)\naxB.ax2.plot(\n X_NLG,\n test.AC_G.AC_CSD_IQ[1, 1],\n color = 'blue',\n label = 'phase quadrature',\n)\naxB.ax2.plot(\n X_NLG,\n test.AC_G.AC_CSD_ellipse.max,\n color = 'cyan',\n label = 'ellipse max',\n ls = '--',\n)\naxB.ax2.plot(\n X_NLG,\n test.AC_G.AC_CSD_ellipse.min,\n color = 'purple',\n label = 'ellipse min',\n ls = '--',\n)\naxB.ax2.plot(\n X_NLG,\n test.AC_G.AC_CSD_ellipse.min**.5 * test.AC_G.AC_CSD_ellipse.max**.5,\n color = 'black',\n label = 'geometric mean',\n ls = '--',\n)\naxB.ax2.set_xlabel('Crystal Total Nonlinear Gain [rtW / W]')\naxB.ax2.legend(\n fontsize = 8,\n loc = 'center left'\n)\n\n```\n\n\n```\naxB = mplfigB()\nfor src, CSD in sys.test.AC_R.noise.CSD_by_source.items():\n if not np.all(CSD['ps_In', 'ps_In'] == 0):\n axB.ax0.plot(\n sys.test.ktp.length_mm.val, \n np.ones_like(sys.test.ktp.length_mm.val) * CSD['ps_In', 'ps_In'].real,\n label = str(src),\n )\naxB.ax0.legend(\n fontsize = 8,\n)\n \n```\n\n\n```\ndb = DeepBunch()\ndb.test.ktp.length.val = 1\ndb.test.ktp.nlg = 1\ndb.test.ktp.N_ode = 100\ndb.test.include_AC = True\nsys = system.BGSystem(\n ctree = db,\n)\nsys.own.test = KTPTestStand()\ndb = sys.ctree_shadow()\n\nLSTARR = sys.test.full_noise_matrix()\n#print(sys.test.ktp.ctree_as_yaml())\n```\n\n Number of states: 20\n Number of states: 20\n PERTURB: 1\n COUPLING_SIZE: 636\n PRE-PURGING\n FRAC REMOVED: 0.28054298642533937 248\n PERTURB: 2\n COUPLING_SIZE: 700\n PRE-PURGING\n FRAC REMOVED: 0.2584745762711864 244\n PERTURB: 3\n COUPLING_SIZE: 716\n PRE-PURGING\n FRAC REMOVED: 0.24152542372881355 228\n PRE-PURGING\n FRAC REMOVED: 0.4016045304388863 1702\n RI RQ GI GQ\n -- -------- -------- ------- --------\n RI 0.309651 0 0 0\n RQ 0 0.947028 0 0.243494\n GI 0 0 1.58019 0\n GQ 0 0.243494 0 2.59602\n\n\n\n```\nprint(sys.AC_R.AC_CSD_IQ[:,:].real)\nprint(sys.AC_G.AC_CSD_IQ[:,:].real)\nprint(sys.AC_RGI.AC_CSD_IQ[:,:].real)\nprint(sys.AC_R.AC_CSD_IQ[:,:].imag)\nprint(sys.AC_G.AC_CSD_IQ[:,:].imag)\nprint(sys.AC_RGI.AC_CSD_IQ[:,:].imag)\n#print(sys.AC_G.AC_CSD_IQ)\n```\n\n\n```\nsys = system.BGSystem()\nsys.own.PSLR = optics.Laser(\n F = sys.system.F_carrier_1064,\n power_W = 1.,\n)\nsys.own.PSLG = optics.Laser(\n F = sys.system.F_carrier_1064,\n power_W = 1.,\n multiple = 2,\n)\nsys.own.dither = optics.AM()\n\nsys.own.ktp = NonlinearCrystal(\n nlg = .1,\n length_mm = np.linspace(0, 10, 100),\n N_ode = 100,\n)\n\nsys.own.mDC2 = optics.HarmonicMirror(\n mirror_H1 = optics.Mirror(\n T_hr = 1,\n ),\n mirror_H2 = optics.Mirror(\n T_hr = 0,\n ),\n AOI_deg = 45,\n)\nsys.own.PD_R = optics.MagicPD()\nsys.own.PD_G = optics.MagicPD()\nsys.own.hPD_R = optics.HiddenVariableHomodynePD(\n source_port = sys.PSLR.po_Fr.o,\n)\nsys.own.hPD_G = optics.HiddenVariableHomodynePD(\n source_port = sys.PSLG.po_Fr.o,\n)\n\nsys.system.bond_sequence(\n sys.PSLR.po_Fr,\n sys.dither.po_Fr,\n sys.ktp.po_Fr,\n sys.mDC2.po_FrA,\n sys.PD_R.po_Fr,\n sys.hPD_R.po_Fr,\n)\nsys.system.bond_sequence(\n sys.mDC2.po_FrB,\n sys.PD_G.po_Fr,\n sys.hPD_G.po_Fr,\n)\nsys.own.DC_R = readouts.DCReadout(\n port = sys.PD_R.Wpd.o,\n)\nsys.own.DC_G = readouts.DCReadout(\n port = sys.PD_G.Wpd.o,\n)\nsys.own.AC_G = readouts.HomodyneACReadout(\n portNI = sys.hPD_G.rtWpdI.o,\n portNQ = sys.hPD_G.rtWpdQ.o,\n portD = sys.dither.Drv.i,\n)\nsys.own.AC_R = readouts.HomodyneACReadout(\n portNI = sys.hPD_R.rtWpdI.o,\n portNQ = sys.hPD_R.rtWpdQ.o,\n portD = sys.dither.Drv.i,\n)\nsys.own.AC_RGI = readouts.HomodyneACReadout(\n portNI = sys.hPD_R.rtWpdI.o,\n portNQ = sys.hPD_G.rtWpdI.o,\n portD = sys.dither.Drv.i,\n)\nsys.own.AC_N = readouts.NoiseReadout(\n port_map = dict(\n RI = sys.hPD_R.rtWpdI.o,\n RQ = sys.hPD_R.rtWpdQ.o,\n GI = sys.hPD_G.rtWpdI.o,\n GQ = sys.hPD_G.rtWpdQ.o,\n \n)\n#print(\"A\")\n#pprint(sys.ctree.test.PSL)\n#print(\"sys.DC_R.DC_readout\", sys.DC_R.DC_readout, 2)\n#print(\"sys.DC_G.DC_readout\", sys.DC_G.DC_readout, 1)\n\n```\n\n\n```\n1.802e-19 - 2*2.36e-20\n```\n\n\n```\nsys.AC_RGI.AC_CSD_IQ[0,0]\n```\n\n\n```\nsys.AC_RGI.AC_CSD_ellipse\n```\n\n\n```\nsys = system.BGSystem()\nsys.own.PSLG = optics.Laser(\n F = sys.system.F_carrier_1064,\n power_W = 1.,\n multiple = 2,\n)\nsys.own.PSLR = optics.Laser(\n F = sys.system.F_carrier_1064,\n power_W = 0.001,\n multiple = 1,\n)\nsys.own.dither = optics.AM()\n\nsys.own.ktp = NonlinearCrystal(\n nlg = .1,\n length_mm = 10, #np.linspace(0, 20, 2),\n N_ode = 20,\n)\n\nsys.own.mDC1 = optics.HarmonicMirror(\n mirror_H1 = optics.Mirror(\n T_hr = 0,\n ),\n mirror_H2 = optics.Mirror(\n T_hr = 1,\n ),\n AOI_deg = 45,\n)\nsys.own.mDC2 = optics.HarmonicMirror(\n mirror_H1 = optics.Mirror(\n T_hr = 1,\n ),\n mirror_H2 = optics.Mirror(\n T_hr = 0,\n ),\n AOI_deg = 45,\n)\nsys.own.PD_R = optics.MagicPD()\nsys.own.PD_G = optics.MagicPD()\nsys.own.hPD_R = optics.HiddenVariableHomodynePD(\n source_port = sys.PSLR.po_Fr.o,\n)\nsys.own.hPD_G = optics.HiddenVariableHomodynePD()\n\nsys.system.bond_sequence(\n sys.PSLG.po_Fr,\n sys.mDC1.po_FrA,\n sys.dither.po_Fr,\n sys.ktp.po_Fr,\n sys.mDC2.po_FrA,\n sys.PD_R.po_Fr,\n sys.hPD_R.po_Fr,\n)\n#sys.system.bond_sequence(\n# sys.PSLR.po_Fr,\n# sys.mDC1.po_BkB,\n#)\nsys.system.bond_sequence(\n sys.mDC2.po_FrB,\n sys.PD_G.po_Fr,\n sys.hPD_G.po_Fr,\n)\nsys.own.DC_R = readouts.DCReadout(\n port = sys.PD_R.Wpd.o,\n)\nsys.own.DC_G = readouts.DCReadout(\n port = sys.PD_G.Wpd.o,\n)\nsys.own.AC_G = readouts.HomodyneACReadout(\n portNI = sys.hPD_G.rtWpdI.o,\n portNQ = sys.hPD_G.rtWpdQ.o,\n portD = sys.dither.Drv.i,\n)\nsys.own.AC_R = readouts.HomodyneACReadout(\n portNI = sys.hPD_R.rtWpdI.o,\n portNQ = sys.hPD_R.rtWpdQ.o,\n portD = sys.dither.Drv.i,\n)\nsys.own.AC_RGI = readouts.HomodyneACReadout(\n portNI = sys.hPD_R.rtWpdI.o,\n portNQ = sys.hPD_G.rtWpdI.o,\n portD = sys.dither.Drv.i,\n)\nsys.own.AC_N = readouts.NoiseReadout(\n port_map = dict(\n RI = sys.hPD_R.rtWpdI.o,\n RQ = sys.hPD_R.rtWpdQ.o,\n GI = sys.hPD_G.rtWpdI.o,\n GQ = sys.hPD_G.rtWpdQ.o,\n )\n)\n#print(\"A\")\n#pprint(sys.ctree.test.PSL)\n#print(\"sys.DC_R.DC_readout\", sys.DC_R.DC_readout, 2)\n#print(\"sys.DC_G.DC_readout\", sys.DC_G.DC_readout, 1)\n\n```\n\n\n```\nprint(sys.AC_R.AC_CSD_IQ[:,:])\nprint(sys.AC_G.AC_CSD_IQ[:,:])\nprint(sys.AC_RGI.AC_CSD_IQ[:,:])\nprint((sys.AC_R.AC_CSD_ellipse.min * sys.AC_R.AC_CSD_ellipse.max)**.5)\nprint(sys.AC_R.AC_CSD_ellipse.min / (sys.AC_R.AC_CSD_ellipse.min * sys.AC_R.AC_CSD_ellipse.max)**.5)\nsys.AC_R.AC_CSD_ellipse\n```\n\n\n```\naxB = mplfigB(Nrows=1)\naxB.ax0.plot(sys.ktp.length_mm.val, sys.DC_R.DC_readout, color = 'red')\naxB.ax0.plot(sys.ktp.length_mm.val, sys.DC_G.DC_readout, color = 'green')\naxB.ax0.plot(sys.ktp.length_mm.val, sys.DC_R.DC_readout + sys.DC_G.DC_readout, color = 'black')\naxB.ax0.set_ylim(0, 1.1)\n```\n\n\n```\nsys = system.BGSystem()\nsys.own.PSLG = optics.Laser(\n F = sys.system.F_carrier_1064,\n power_W = 1.,\n multiple = 2,\n)\nsys.own.PSLR = optics.Laser(\n F = sys.system.F_carrier_1064,\n power_W = 0.001,\n multiple = 1,\n)\nsys.own.PD_R = optics.MagicPD()\nsys.own.PD_G = optics.MagicPD()\nsys.own.dither = optics.AM()\n\nsys.own.hPD_R = optics.HiddenVariableHomodynePD(\n source_port = sys.PSLR.po_Fr.o,\n)\nsys.own.hPD_G = optics.HiddenVariableHomodynePD()\n\nsys.system.bond_sequence(\n sys.PSLG.po_Fr,\n sys.PD_G.po_Fr,\n sys.hPD_G.po_Fr,\n)\nsys.system.bond_sequence(\n sys.PSLR.po_Fr,\n sys.PD_R.po_Fr,\n sys.hPD_R.po_Fr,\n)\nsys.own.DC_R = readouts.DCReadout(\n port = sys.PD_R.Wpd.o,\n)\nsys.own.DC_G = readouts.DCReadout(\n port = sys.PD_G.Wpd.o,\n)\nsys.own.AC_G = readouts.HomodyneACReadout(\n portNI = sys.hPD_G.rtWpdI.o,\n portNQ = sys.hPD_G.rtWpdQ.o,\n portD = sys.dither.Drv.i,\n)\nsys.own.AC_R = readouts.HomodyneACReadout(\n portNI = sys.hPD_R.rtWpdI.o,\n portNQ = sys.hPD_R.rtWpdQ.o,\n portD = sys.dither.Drv.i,\n)\nsys.own.AC_RGI = readouts.HomodyneACReadout(\n portNI = sys.hPD_R.rtWpdI.o,\n portNQ = sys.hPD_G.rtWpdI.o,\n portD = sys.dither.Drv.i,\n)\nsys.own.AC_N = readouts.NoiseReadout(\n port_map = dict(\n RI = sys.hPD_R.rtWpdI.o,\n RQ = sys.hPD_R.rtWpdQ.o,\n GI = sys.hPD_G.rtWpdI.o,\n GQ = sys.hPD_G.rtWpdQ.o,\n )\n)\n#print(\"A\")\n#pprint(sys.ctree.test.PSL)\n#print(\"sys.DC_R.DC_readout\", sys.DC_R.DC_readout, 2)\n#print(\"sys.DC_G.DC_readout\", sys.DC_G.DC_readout, 1)\n\n```\n\n\n```\nprint(sys.AC_R.AC_CSD_IQ[:,:])\nprint(sys.AC_G.AC_CSD_IQ[:,:])\nprint(sys.AC_RGI.AC_CSD_IQ[:,:])\nprint((sys.AC_R.AC_CSD_ellipse.min * sys.AC_R.AC_CSD_ellipse.max)**.5)\nprint(sys.AC_R.AC_CSD_ellipse.min / (sys.AC_R.AC_CSD_ellipse.min * sys.AC_R.AC_CSD_ellipse.max)**.5)\nsys.AC_R.AC_CSD_ellipse\n```\n\n\n```\nfrom phasor.optics.models.KTP_test_stand import KTPTestStand\nsys = system.BGSystem()\nsys.own.test = KTPTestStand()\nLSTARR = sys.test.full_noise_matrix()\n```\n\n\n```\n\n```\n\n\n```\n\n```\n", "meta": {"hexsha": "dd76c0dca0cf00d9dcb6e6fd40b12190eeaf289d", "size": 302020, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "phasor/nonlinear_crystal/show_NLC.ipynb", "max_stars_repo_name": "mccullerlp/phasor-doc", "max_stars_repo_head_hexsha": "d4255d015023c51b762340e51c15dde609715212", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": 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YES\n2. YES", "lm_q1_score": 0.819893353516963, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.4387236451083298}} {"text": "# Causality and supervised machine learning\n*Andreas Bjerre-Nielsen*\n\n## Introduction\n\nWhat is the objective of empirical policy research? \n\n1. *causation*: what is the effect of a particular variable on an outcome? \n2. *prediction*: find some function that provides a good prediction of $y$ as a function of $x$\n\n## Intution\n\n$$ y = \\alpha + \\beta x + \\varepsilon $$\n\n- *causation*: interested in $\\hat{\\beta}$ \n\n- *prediction*: interested in $\\hat{y}$ \n\n\n## Preparation\n\n\n```python\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport pandas as pd\nimport seaborn as sns\nimport statsmodels.formula.api as smf # module for various regression models\n\nsns.set()\n\n%matplotlib inline\n```\n\n# Causal Inference\n\n## Introduction\n\nMost statistical methods in social science theory is focused on estimating **causal effects**\n\nCausal effect: how does the factor A affect B?\n\nExamples of causal questions:\n\n- what is the effect of being assigned to a shared group on communication?\n- what is the effect of sterotypical names on job interview success? \n\n## Intuition\n\nVariable of interest (often called *treatment*): $D_i$\n\nOutcome of interest: $Y_i$\n\n**Potential outcome framework**\n$$\nY_i = \\left\\{\n\\begin{array}{rl}\nY_{1i} & \\text{if } D_i = 1,\\\\\nY_{0i} & \\text{if } D_i = 0\n\\end{array} \\right.\n$$\n\nThe observed outcome $Y_i$ can be written in terms of potential outcomes as\n$$ Y_i = Y_{0i} + (Y_{1i}-Y_{0i})D_i$$\n\n$Y_{1i}-Y_{0i}$ is the *causal* effect of $D_i$ on $Y_i$. \n\nBut we never observe the same individual $i$ in both states. This is the **fundamental problem of causal inference**. \n\n## Selection Bias I\n\nWe need some way of estimating the state we do not observe (the ***counterfactual***)\n\nUsually, our sample contains individuals from both states - treated and untreated.\n\nSo why not do a naive comparison of averages by treatment status? i.e. $E[Y_i|D_i = 1] - E[Y_i|D_i = 0]$\n\n## Selection Bias II\nWe can rewrite into:\n\\begin{align}\n\\nonumber E[Y_i|D_i = 1] - E[Y_i|D_i = 0] = &E[Y_{1i}|D_i = 1] - E[Y_{0i}|D_i = 1] + \\\\\n \\nonumber &E[Y_{0i}|D_i = 1] - E[Y_{0i}|D_i = 0] \n\\end{align}\n\n\nThe decomposition:\n\n - $E[Y_{1i}|D_i = 1] - E[Y_{0i}|D_i = 1] = E[Y_{1i} - Y_{0i}|D_i = 1]$: the average *causal* effect of $D_i$ on $Y$. \n\n- $E[Y_{0i}|D_i = 1] - E[Y_{0i}|D_i = 0]$: difference in average $Y_{0i}$ between the two groups. Likely to be different from 0 when individuals are allowed to self-select into treatment. Often referred to as ***selection bias***. \n\n## Random assignment solves the problem\n\nRandom assignment of $D_i$ solves the problem because random assignment makes $D_i$ independent of potential outcomes\n\nThat means that $E[Y_{0i}|D_i = 1] = E[Y_{0i}|D_i = 0]$ and thus that the selection bias term is zero\n\nIntuition: with random assignment, non-treated individuals can be used as counterfactuals for treated (*what would have happened to individual $i$ had he not received the treatment*?)\n\nThis allows us to overcome the fundamental problem of causal inference\n\n\n## Randomization\n\nHolland and Rubin (1986)\n\n> no causation without manipulation\n\n\nAs mentioned, we need to worry when individuals are allowed to self-select\n\nThis means that a lot of thought has to go into the *randomization phase*\n\nRandomization into treatment groups has to be manipulated by someone \n\nBut what about effect of *immutable characteristics* such as race, gender, etc.?\n\n\n## Quasi Experiments\n\n*Quasi-experiments*: randomization happens by \"accident\"\n\n- Differences in Differences\n- Regression Discontinuity Design\n- Instrumental variables\n\n\n\n## Randomized Controlled Trials\n\n*Randomized controlled trials (RCT)*: randomization done by researcher\n\n- Survey experiments\n- Field experiments\n\nNote: difficult to say one is strictly better than the other. Randomization can be impractical and/or unethical. \n\n\n\n\n## External & internal validity\n\n*Internal validity*: Refers to the validity of causal conclusions\n\n*External validity*: Refers to the extent to which the conclusions of a particular study can be generalized beyond a particular setting\n\nRCTs - external and internal validity\n- Kosuke Imai (2016): There is tradeoff.\n- Cyrus Samii (2016): No such tradeoff. \n\n\n## Observational study\n\nIn many cases, social scientists are unable to randomize treatment assignment for ethical or logistic reasons\n\n*Observational study*: No random manipulation of treatment\n\nStrategy: Statistical control (control variables, fixed effects, matching, etc)\n\nRisks: selection & confounding bias and endogeneity. \n\n\n## Case: Racial Discrimination in the Labor Market\n\nDoes racial discrimination exist in the labor market?\n\n*Experiment*: In response to newspaper ads, researchers send out resumes of fictitious job candidates, varying only the names of the job applicants while leaving all other information in the resumes unchanges\n\nNames were randomized between stereotypically black- and white-sounding names (Lakisha, Jamal, Emily, Greg, etc.)\n\n## Case: Racial Discrimination in the Labor Market (2)\n\n\n```python\ngh_raw = \"https://raw.githubusercontent.com/\"\nuser = \"kosukeimai/\"\nrepo = 'qss/'\nbranch = \"master/\"\nfilepath = \"CAUSALITY/resume.csv\"\nurl = gh_raw + user + repo + branch + filepath\n\ndf = pd.read_csv(url,dtype={'race':'category','sex':'category'})\n```\n\n## Case: Racial Discrimination in the Labor Market (3)\n\nCan we use a boxplot? No boxplot not good for binary data.\n\n\n```python\nf, ax = plt.subplots(1,2,figsize=(12,4))\nsns.barplot(x='race', y='call', data=df, ax=ax[0])\nsns.barplot(x='sex', hue='race', y='call', data=df, ax=ax[1])\n```\n\n## Case: Racial Discrimination in the Labor Market (4)\n\n\n```python\nmodel = smf.ols(formula='call~race*sex', data=df)\nresults = model.fit()\nresults.summary()\n```\n\n\n\n\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
    OLS Regression Results
    Dep. Variable: call R-squared: 0.004
    Model: OLS Adj. R-squared: 0.003
    Method: Least Squares F-statistic: 5.973
    Date: Mon, 14 Aug 2017 Prob (F-statistic): 0.000464
    Time: 09:40:34 Log-Likelihood: -561.75
    No. Observations: 4870 AIC: 1131.
    Df Residuals: 4866 BIC: 1157.
    Df Model: 3
    Covariance Type: nonrobust
    \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
    coef std err t P>|t| [0.025 0.975]
    Intercept 0.0663 0.006 10.595 0.000 0.054 0.079
    race[T.white] 0.0326 0.009 3.677 0.000 0.015 0.050
    sex[T.male] -0.0080 0.013 -0.606 0.544 -0.034 0.018
    race[T.white]:sex[T.male] -0.0022 0.018 -0.121 0.904 -0.038 0.034
    \n\n\n \n\n\n \n\n\n \n\n\n \n\n
    Omnibus: 2968.362 Durbin-Watson: 1.441
    Prob(Omnibus): 0.000 Jarque-Bera (JB): 18913.265
    Skew: 3.067 Prob(JB): 0.00
    Kurtosis: 10.455 Cond. No. 6.64
    \n\n\n\n## Case: Racial Discrimination in the Labor Market (5)\n\n\n```python\nsumm_pd = results.summary2() # get summary as pandas tables\nparams = summ_pd.tables[1].iloc[1:] # pick coefficient tables\n\nm = len(params)\nerr_args = {'fmt':'.', 'capsize':5, 'markeredgewidth':2}\nplt.errorbar(y=range(m), \n x=params['Coef.'], \n xerr=params['Std.Err.'],\n **err_args)\nplt.yticks(range(m),params.index)\nplt.plot((0, 0), (0, m-1), '--', color='0.6')\n```\n\n# Machine learning\n\n## Topics in machine learning\n\n*Supervised Learning*: Models designed to infer a relationship between input and **labeled** training data. These models are used for **prediction**. Examples: OLS, logistic reg.\n\n*Unsupervised Learning*: Models designed to infer a relationship from **unlabeled** training data. This may involve clustering, dimensionality reduction and more.\n\n# Supervised machine learning\n\n## Prediction\n\nMany policy problems are not about causality but rather about prediction\n\nSometimes called *prediction policy problems*\n\n- How many people will sign up for Obamacare?\n- Who will win the U.S general election in November?\n- Who should the Department of Economics hire in the future?\n\n## Who predicts?\n\n* Local governments -> crime, childcare usage, pension payments etc.\n* FB/GOOG/AMZN/NTFL/etc. > estimating 'preferences' to improve customer experience / sales\n * ads, status updates, music, movies, news\n* Insurance companies -> what your risk of death is\n* Stock traders > to trade \n* Robots -> understanding their environment (e.g. self-driving cars)\n* You? -> will *Social Data Science* be a fun/rewarding/interesting course to follow?\n\n\n## Why predict? Glory!\n\n
    \n\n\n\n\n\n## The Netflix Contest\n\n\nCompetition started in October 2006. Training data is ratings for 18K movies by 400K Netflix customers, each rating between 1 and 5\n\nTraining data is very sparse - about 98% missing\n\nObjective is to predict the rating for a set of 1 million customer-movie pairs that are missing in the training data\n\nWinner: Averaged 800 models \n\n## Why predict? Riches!\n\n
    \n\n[http://www.heritagehealthprize.com/c/hhp](http://www.heritagehealthprize.com/c/hhp)\n\n\n## Prediction problem types\n\nPredicition problems are often described in terms of the data types they are predicting. \n- The first is **regression** which uses numeric (continuous) variables.\n- The second is **classification** which uses categorical (discrete) variables.\n\nExamples?\n\n## Case: Predicting gender from body measures\n\nWhat type of prediction problem is this? Classification.\n\n- Classification!\n\n## Case: Predicting gender from body measures (2)\n\n\n```python\nprint(body.head())\n```\n\n Gender Height Weight Male\n 0 Male 189.048364 109.819678 1.0\n 1 Male 176.081674 73.688955 1.0\n 2 Male 189.721870 96.584348 1.0\n 3 Male 183.631305 99.899282 1.0\n 4 Male 178.897397 93.682809 1.0\n\n\n\n```python\ngh_raw = \"https://raw.githubusercontent.com/\"\nuser = \"johnmyleswhite/\"\nrepo = 'ML_for_Hackers/'\nbranch = \"master/\"\nfilepath = \"02-Exploration/data/01_heights_weights_genders.csv\"\nurl = gh_raw + user + repo + branch + filepath\n\nbody = pd.read_csv(url)\nbody['Male'] = (body.Gender=='Male').astype(float) # Binary variable\nbody.Height = body.Height*2.56 # convert inches to cm\nbody.Weight = body.Weight*0.454 # convert pounds to kg\n```\n\n## Case: Predicting gender from body measures (3)\n*Do we already know any machine learning models?*\n\n\n```python\nfrom sklearn.linear_model import LogisticRegression\n\nX = body[['Weight', 'Height']] # features\ny = body.Male # labels\n\n# make classifier and fit model\nclf = LogisticRegression().fit(X, y)\n\n# parameter estimates\npd.Series(clf.coef_[0], index=['Weight', 'Height'])\n```\n\n\n\n\n Weight 0.433517\n Height -0.186886\n dtype: float64\n\n\n\n## Case: Predicting gender from body measures (4)\n\nLogit estimates\n\n$$ P(Y_i = 1 |X_i = x_i) = \\frac{1}{1 + e^{-x_i\\beta}}$$\n\nThis probability is .5 when $x_i\\beta = 0$. Thus we can classify predicted gender based on height and weight with the following rule:\n\n$$\n\\hat{y} = \\left\\{\n\\begin{array}{rl}\n1 & \\text{if } x_i\\beta \\geq 0,\\\\\n0 & \\text{otherwise}\n\\end{array} \\right.\n$$\n\nClassifier threshold:\n$$ H = \\frac{-\\alpha - \\beta_W W}{\\beta_H} $$\n\n## Case: Predicting gender from body measures (5)\n\n\n```python\nxx, yy = np.mgrid[25:125:.1, 130:210:.08]\ngrid = np.c_[xx.ravel(), yy.ravel()]\nprobs = clf.predict_proba(grid)[:, 1].reshape(xx.shape)\n\nf, ax = plt.subplots(figsize=(13, 7))\ncontour = ax.contourf(xx, yy, probs, 25, cmap=\"RdBu\", \n vmin=0, vmax=1)\nax_c = f.colorbar(contour)\nax_c.set_label(\"$P(y = 1)$\")\nax_c.set_ticks([0, .25, .5, .75, 1])\n\nplt.scatter(X.iloc[:,0], X.iloc[:,1], c=y, s=10,\n cmap=\"RdBu\", vmin=-.2, vmax=1.2,\n edgecolor=\"white\", linewidth=.5, alpha=.3)\n\nax.set(aspect=\"equal\",\n xlim=(25, 125), ylim=(130, 210),\n xlabel=\"Weight\", ylabel=\"Height\")\n```\n\n## Case: Predicting gender from body measures (6)\n\n**Decision boundary**\n\n\n```python\nf\n```\n\n## Case: Predicting gender from body measures (7)\nAnother model: nearest neighbor\n\n\n```python\nfrom sklearn.ensemble import RandomForestClassifier\nfrom sklearn.neighbors import KNeighborsClassifier\n\nX = body[['Weight', 'Height']]\ny = body.Male\n\n# make model on data - fifty nearest neighbors\nclf = KNeighborsClassifier(n_neighbors=50).fit(X, y)\n```\n\n## Case: Predicting gender from body measures (8)\n\nComputing nearest neighbor decision boundary\n\n\n```python\nxx, yy = np.mgrid[25:125:.1, 130:210:.08]\ngrid = np.c_[xx.ravel(), yy.ravel()]\nprobs = clf.predict_proba(grid)[:, 1].reshape(xx.shape)\n\nf, ax = plt.subplots(figsize=(13, 7))\ncontour = ax.contourf(xx, yy, probs, 25, cmap=\"RdBu\", \n vmin=0, vmax=1)\nax_c = f.colorbar(contour)\nax_c.set_label(\"$P(y = 1)$\")\nax_c.set_ticks([0, .25, .5, .75, 1])\n\nplt.scatter(X.iloc[:,0], X.iloc[:,1], c=y, s=10,\n cmap=\"RdBu\", vmin=-.2, vmax=1.2,\n edgecolor=\"white\", linewidth=.5, alpha=.1)\n\nax.set(aspect=\"equal\",\n xlim=(25, 125), ylim=(130, 210),\n xlabel=\"Weight\", ylabel=\"Height\")\n```\n\n## Case: Predicting gender from body measures (9)\n\nDecision boundary: ***non-linear***\n\n\n```python\nf\n```\n\n# Bias and variance\n\nAre OLS and logistic regression good for prediction?\n\n## The bias-variance tradeoff\n\nOLS is designed to minimize *in sample error*: the error rate you get on the same data set you used to build your predictor.\n\n$$ \\text{arg min}_{\\beta} \\sum_{i = 1}^{n} (y_i - \\hat{y}_i)^2 = \\text{arg min}_{\\beta} (V(\\hat{f}(x_0)) + \\sigma^2) $$\n\nBut for prediction we are interested in minimizing *out of sample error*: the error rate you get on a new data set\n\n\n\n## Prediction\n\nToo see this, consider a prediction at a new point (out-of-sample), $x_0$. Our prediction for $y_0$ is then $\\hat{f}(x_0)$ and the mean squared error (MSE ) can be decomposed as \n\n$$ \\mathbb{E}[(y_0 - \\hat{f}(x_0))^2] = \n\\underset{Bias(\\hat{f}(x))}{\\underbrace{\\mathbb{E}[\\hat{f}(x_0)-f(x_0)]}}^2 + \n\\underset{Var(\\hat{f}(x))}{\\underbrace{\\mathbb{E}[\\hat{f}(x_0)^2]-\\mathbb{E}[\\hat{f}(x_0)]^2}} +\n\\sigma^2$$\n\nBy ensuring zero bias within sample, OLS picks a solution which not be optimal for prediction \n- in many cases we can lower variance while increasing bias a little. \n\n## Bias and variance\n\nWhat do we mean by the *variance* and *bias* of an estimator?\n\n- *Bias*: Comes from using erroneous model assumptions, e.g. fitting non-linear fct. $f$ with linear fct. $\\hat{f}$. \n - Can lead to missing relevant patterns in data, i.e. *underfitting*.\n\n- *Variance*: Refers to model complexity. If the model is too complex then small changes to the data will cause the solution to change a lot. \n - Can lead to finding spurious patterns in data, i.e. *overfitting*.\n\nMachine learning techniques were developed specifically to maximize prediction performance by providing an empirical way to make this bias-variance trade off\n\n## Bias and variance (2)\n\n*So why do we care about bias?*\n\n- By not modelling bias: allows *inference*, i.e. testing hypotheses! (model parameters converge to true parameters) ~ interested in $\\hat{\\beta}$\n \n- By modelling bias: allows better predictive models as they trade off bias and variance. Interested in $\\hat{y}$\n\n# Out-of-sample measures\n\n## Key concepts\n\n- Training data: where we estimate our model\n- Test data: where we evaluate the model's accuracy\n\n## Error\n\nStatistical learning models are designed to minimize *out of sample error*: the error rate you get on a new data set\n\nKey ideas\n\n- Out of sample error is what you care about\n- In sample error $<$ out of sample error\n- The reason is overfitting (matching your algorithm to the data you have)\n\n## Error measures (continuous variables)\n\n**Mean absolute error (MAE)**:\n\n$$MAE = \\frac{1}{n} \\sum_{i=1}^{n} |\\hat{y}_i - y_i|$$\n\n**Mean squared error (MSE)**:\n\n$$MSE = \\frac{1}{n} \\sum_{i=1}^{n} (\\hat{y}_i - y_i)^2$$\n\n**Root mean squared error (RMSE)**:\n\n$$\\sqrt{MSE}$$\n\n**Question:** what is the difference?\n\n\n\n```python\nfrom sklearn.metrics import mean_absolute_error as mae, mean_squared_error as mse\n```\n\n## Case: Longevity\n\n\n```python\ngh_raw = \"https://raw.githubusercontent.com/\"\nuser = \"johnmyleswhite/\"\nrepo = \"ML_for_Hackers/\"\nbranch = \"master/\"\nfilepath = \"05-Regression/data/longevity.csv\"\nurl = gh_raw + user + repo + branch + filepath\n\nlongevity = pd.read_csv(url)\n\nprint(longevity.head())\n```\n\n Smokes AgeAtDeath\n 0 1 75\n 1 1 72\n 2 1 66\n 3 1 74\n 4 1 69\n\n\n## Case: Longevity (2)\n\n\n```python\ny_true = longevity.AgeAtDeath\n\ndef rmse_year(y): \n '''\n Takes as input: year (y)\n Outputs: root mean squared of guessing y \n '''\n y_pred = [y]*len(longevity)\n return np.sqrt(mse(y_true, y_pred))\n\n\n# normal for loop\nrmse_years = [] \n\nfor y in range(60,91):\n rmse_years.append((y, rmse_year(y)))\n \n# list comprehenseion:\nrmse_years = [(y, rmse_year(y)) for y in range(60,91)]\n \nrmse_data = pd.DataFrame(data = rmse_years,\n columns = ['age', 'rmse'])\n\n# print ('Best guess:', rmse_data.set_index('year').rmse.idxmin())\n#rmse_data.plot.scatter('age', 'rmse')\n```\n\n## Case: Longevity (3)\n\n\n```python\nf,ax = plt.subplots(figsize=(12,6))\nprint ('Best guess:', rmse_data.set_index('age').rmse.idxmin())\nrmse_data.plot.scatter('age', 'rmse', ax=ax)\n```\n\n# Cross validation (CV)\n\n## Test and training data \n\nAccuracy on the training set (resubstitution accuracy) does not capture bias and therefore is optimistic. A better estimate comes from an independent set (test set accuracy). Strategy:\n\n- Use share of observations for training\n- Use other of observations for testing model (out-of-sample)\n\nSo we estimate the test data accuracy with the model calibrated on training data.\n\n## Data split\n\nWhy not just divide data randomly the data into a test and training set?\n\nTwo drawbacks\n\n1. RMSE is very sensitive to which observations are used for test and training. \n2. RMSE is artificially large as not all observations are used for training model (model becomes more accurate for more observations)\n\nOne very useful refinement of the test-training data approach is ***cross-validation*** use multiple splits.\n\n## K-fold Cross Validation\n\n1. Divide the data into $k$ roughly equal subsets and label them $s = 1, ..., k$. \n2. Fit your model using the $k-1$ subsets other than subset $s$ \n3. Predict for subset $s$ and calculate RMSE\n4. Stop if $s = k$, otherwise increment $s$ by $1$ and continue\n\nThe $k$ fold CV estimate is computed by averaging the mean squared errors ($\\text{MSE}_1, ..., \\text{MSE}_k$)\n\n$$\\text{CV}_k = \\frac{1}{k}\\sum_{i = 1}^{k} \\text{MSE}_i$$\n\nCommon choices for $k$ are 3, 5 and 10. \n\nCV can (and should) be used both to decide hyperparameters and to report goodness-of-fit measures. \n\n## K-fold Cross Validation (2)\n\n\n\n\n## Fitting polynomial\nPolyonomial: $f(x) = 2+8*x^4$\n\nTry models of higher and higher polynomials. Iteration n: $y = \\sum_{k=0}^{n}(\\beta_k\\cdot x^k)+\\varepsilon$.\n\n\n```python\ndef my_polynomial(x):\n 'Polyonomial: f(x) = 2+8*x^4'\n return 2+8*x**4\n\nnp.random.seed(1234)\n\nx_range = pd.Series(np.arange(0,1,.02))\n\ny_s = []\nfor x in x_range:\n y = my_polynomial(x) \n y_s.append(y)\n \nfct_values = pd.Series(y_s) # true values\nerrors = pd.Series(np.random.normal(size=(50))) # measurement error\n\ny = fct_values + errors # observed outcome data\n```\n\n## Fitting polynomial (2)\n\n*Compute within sample error for OLS*\n\n\n\n```python\nfrom sklearn.linear_model import LinearRegression\n\nX_s = {}\nrmse_in_sample = []\n\nfor i in range(1,10): # for each order 1-9\n X_s['order '+str(i)]= x_range**i # add polynomial of order i \n X = pd.concat(X_s,axis=1) # concatename polynomials of order 1,..,i\n\n model = LinearRegression().fit(X,y) # fit linear regression model \n y_pred = model.predict(X) # make predictions within-sample\n \n rmse_in_sample.append(np.sqrt(mse(y, y_pred)))\n \nprint(X.iloc[:,:5].head(2))\n```\n\n order 1 order 2 order 3 order 4 order 5\n 0 0.00 0.0000 0.000000 0.000000e+00 0.000000e+00\n 1 0.02 0.0004 0.000008 1.600000e-07 3.200000e-09\n\n\n## Fitting polynomial (3)\n\n*Compute out-of-sample errors for OLS*\n\n\n```python\nfrom sklearn.model_selection import KFold\n\nkf = KFold(n_splits=10, random_state=123)\n\nX_s = {}\nrmse_CV = []\ncoef_CV = []\n\nfor i in range(1,10):\n X_s['order '+str(i)] = x_range**i\n X = pd.concat(X_s, axis=1)\n \n rmse_fold = []\n coef_fold = []\n \n # cross validation loop\n for train_index, test_index in kf.split(X):\n X_train, X_test = X.iloc[train_index], X.iloc[test_index]\n y_train, y_test = y.iloc[train_index], y.iloc[test_index] \n \n model = LinearRegression().fit(X_train,y_train) \n y_pred = model.predict(X_test)\n \n rmse = np.sqrt(mse(y_test, y_pred))\n rmse_fold.append(rmse)\n \n coef_fold.append(pd.Series(model.coef_, index=range(1,i+1)))\n \n rmse_CV.append(np.mean(rmse_fold))\n coef_CV.append(pd.concat(coef_fold,1).T.mean().rename(i)) \n```\n\n## Fitting polynomial (4)\n\nHow does in-sample and out-of-sample estimates of RMSE diverse?\n\n\n```python\nf,ax = plt.subplots(figsize=(12,6))\nrmse_df = pd.DataFrame(data = [rmse_in_sample, rmse_CV],\n index = ['OLS in-sample', 'OLS out-of-sample (CV)'],\n columns = range(1,10)).T\n\nrmse_df.columns.name = 'RMSE'\nrmse_df.index.name = 'polynomial order'\nrmse_df.plot(ax=ax)\nax.set_yscale('log')\n```\n\n## Fitting polynomial (5)\n\nWhy does in-sample and out-of-sample estimates of RMSE diverse? Well, the coefficient of OLS are increasingly mis-estimated for higher and higher number of variables.\n\n\n```python\npd.concat(coef_CV,1).T\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
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    16.234355NaNNaNNaNNaNNaNNaNNaNNaN
    2-6.14666712.590554NaNNaNNaNNaNNaNNaNNaN
    3-0.781354-1.1048079.249710NaNNaNNaNNaNNaNNaN
    4-6.39443322.107885-25.07807416.501495NaNNaNNaNNaNNaN
    57.678056-73.795322227.908655-266.336210112.593443NaNNaNNaNNaN
    6-24.034620197.092328-785.5779261574.624054-1482.858265528.364240NaNNaNNaN
    7-26.234597139.009441-235.967168-318.9848101647.278049-1974.405448777.934915NaNNaN
    8-2.480596-149.0251751496.517741-6011.77218212283.632851-13177.7163976927.394820-1354.677227NaN
    9-174.3111122248.717158-16219.51743969637.464867-182832.881188294545.334322-282663.341134147727.564074-32259.887872
    \n
    \n\n\n\n## Regularization\n\n*Why do we regularize?*\n\n- To avoid overfitting and thus have better predictions\n\n*How do we regularize?*\n\nWe make models which are less complex by reducing the number and/or size of the coefficients.\n\n\n## Regularization (2)\n\n*What does regularization look like?*\n\nWe add a penalty term our optimization procedure:\n \n$$ \\text{arg min}_\\beta \\, \\underset{\\text{MSE}}{\\underbrace{E[(y_0 - \\hat{f}(x_0))^2]}} + \\underset{\\text{penalty}}{\\underbrace{\\lambda \\cdot R(\\beta)}}$$\n\nIntroduction of penalties implies that increased model complexity has to be met with high increases precision of estimates.\n\n## Regularization (3)\n\n*What are some used penalty functions?*\n\nThe two most common penalty functions are L1 and L2 regularization.\n\n- L1 regularization (***Lasso***): $R(\\beta)=\\sum_{j=1}^{p}|\\beta_j|$ \n - Makes coefficients sparse, i.e. selects variables by removing some (if $\\lambda$ is high)\n \n \n- L2 regularization (***Ridge***): $R(\\beta)=\\sum_{j=1}^{p}\\beta_j^2$\n - Reduce coefficient size\n - Fast due to analytical solution\n \n*To note:* The *Elastic Net* uses a combination of L1 and L2 regularization.\n\n## Regularization (4)\n\n*How the Lasso (L1 reg.) deviates from OLS*\n\n\n\n## Regularization (5)\n\n*How the Ridge regression (L2 reg.) deviates from OLS*\n\n\n\n# Models for supervised machine learning\n\n## Model overview\n\n*Parametric models*\n- regression models \n - unbiased: OLS \n - biased: ***Lasso*** (L1), ***Ridge*** (L2) - (regulariation)\n- classifier\n - unbiased: logistic\n \n*Other models*\n- **random forest**\n- **nearest neighbor**\n- neural networks\n\n## Fitting polynomial (6)\n\nTraining the model with Lasso\n\n\n```python\nfrom sklearn.model_selection import KFold\nfrom sklearn.linear_model import Lasso\n\nkf = KFold(n_splits=10, random_state=123)\n\nX_s = {}\nrmse_CV_L = []\n\nfor i in range(1,10):\n X_s['order '+str(i)] = x_range**i\n X = pd.concat(X_s, axis=1)\n \n rmse_fold = []\n \n # cross validation loop\n for train_index, test_index in kf.split(X):\n X_train, X_test = X.iloc[train_index], X.iloc[test_index]\n y_train, y_test = y.iloc[train_index], y.iloc[test_index] \n \n # use Lasso with lambda = .05\n y_pred = Lasso(alpha=0.05)\n .fit(X_train, y_train)\n .predict(X_test)\n \n rmse = np.sqrt(mse(y_test, y_pred))\n rmse_fold.append(rmse)\n \n rmse_CV_L.append(np.mean(rmse_fold))\n```\n\n## Fitting polynomial (7)\n\nLasso vs OLS performance\n\n\n```python\nf,ax = plt.subplots(figsize=(12,6))\nrmse_df['Lasso out-of-sample (CV)'] = rmse_CV_L\nrmse_df.plot(ax=ax)\nax.set_yscale('log')\n```\n\n## Fitting polynomial (8)\n\nTuning The Lasso\n\n\n```python\nX = pd.concat(X_s, axis=1)\n\nrmse_CV_lambda = []\ncoef_CV_lambda = []\n\nfor l in np.arange(.0001,.01,.0001): # loop over different values of lambda\n \n rmse_fold = []\n \n for train_index, test_index in kf.split(X):\n X_train, X_test = X.iloc[train_index], X.iloc[test_index]\n y_train, y_test = y.iloc[train_index], y.iloc[test_index] \n\n model = Lasso(alpha=l).fit(X_train,y_train)\n y_pred = model.predict(X_test)\n\n rmse = np.sqrt(mse(y_test, y_pred))\n rmse_fold.append(rmse)\n \n coef_CV_lambda.append(model.coef_) \n rmse_CV_lambda.append([l, np.mean(rmse_fold)])\n```\n\n## Fitting polynomial (9)\n\nWe search the $\\lambda$ grid : 0.0001-0.01 to find the optimal $\\lambda$ parameter\n\n\n```python\nrmse_lambda_df = pd.DataFrame(rmse_CV_lambda,columns=['lambda', 'RMSE']).set_index('lambda')\n\nrmse_lambda_df.plot()\n```\n\n## Fitting polynomial (10)\n\nModel for optimal lambda\n\n\n```python\nlambda_opt = rmse_lambda_df.idxmin()\nlambda_opt\n```\n\n\n\n\n RMSE 0.0022\n dtype: float64\n\n\n\nParameters for optimal lambda\n\n\n```python\ncoefs = pd.DataFrame(coef_CV_lambda,columns=range(1,10),index=np.arange(.0001,.01,.0001))\nprint(coefs.loc[lambda_opt])\n```\n\n 1 2 3 4 5 6 7 8 9\n 0.0022 -0.077694 0.0 6.793418 0.0 0.0 -0.0 -0.0 -0.0 -0.0\n\n\n## Summary causality\n\n*Selection bias*: Issue for observational studies where treatment is correlated with baseline outcome.\n\n*Randomization*: Enactment of treatments that are assigned randomly and thus do not suff selection biases. \n\n## Summary supervised learning\n\n*Prediction/postdiction*: Application of model to estimate response/output associated with input. \n\n*Bias-variance tradeoff*: There exist models beside OLS which can improve better at out-of-sample predictions, however, they have biased parameter estimates.\n\n*Prediction types*: \n- when the response is numeric (continuous) the problem is called *regression*\n- when response is categorical (discrete) the problem is called *classification*\n\n*MAE/RMSE*: measures of prediction accuracy for regression problems\n\n*Cross validation*: Split data in test and training data. Train model on training data, test it on test data.\n\n*Regularization*: A technique used to model bias in an attempt to solve overfitting problems\n", "meta": {"hexsha": "f461eb05536185101c525a8aaa19780808142396", "size": 590247, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "slides/causal_supervised.ipynb", "max_stars_repo_name": "0933klatm/SDS", "max_stars_repo_head_hexsha": "3218a7075ee0befa8d5f423a4e0dab5b14a4cd98", "max_stars_repo_licenses": ["MIT", "Unlicense"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "slides/causal_supervised.ipynb", "max_issues_repo_name": "0933klatm/SDS", "max_issues_repo_head_hexsha": "3218a7075ee0befa8d5f423a4e0dab5b14a4cd98", "max_issues_repo_licenses": ["MIT", "Unlicense"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "slides/causal_supervised.ipynb", "max_forks_repo_name": "0933klatm/SDS", "max_forks_repo_head_hexsha": "3218a7075ee0befa8d5f423a4e0dab5b14a4cd98", "max_forks_repo_licenses": ["MIT", "Unlicense"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 238.6765062677, "max_line_length": 131076, "alphanum_fraction": 0.9034319531, "converted": true, "num_tokens": 9264, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665855647394, "lm_q2_score": 0.7401743620390163, "lm_q1q2_score": 0.43867661187222307}} {"text": "```python\nfrom IPython.display import Latex\nfrom IPython.display import Markdown as md\n```\n\n\n```latex\n%%latex\n\nSome words\n\\begin{align}\\label{eq:1}\nx & = y\\\\\nz & = \\beta\n\\end{align}\n\nSome more words\n\n$$\\frac{1}{2}$$\n```\n\n\n\nSome words\n\\begin{align}\\label{eq:1}\nx & = y\\\\\nz & = \\beta\n\\end{align}\n\nSome more words\n\n$$\\frac{1}{2}$$\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "1a981f498ddc4775e3bfb4ed5f572403d66c5cf6", "size": 1602, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Untitled.ipynb", "max_stars_repo_name": "bcreiner/IDD", "max_stars_repo_head_hexsha": "3353c6ad2b57e1abc3612c66b680dd5c2bc53b54", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Untitled.ipynb", "max_issues_repo_name": "bcreiner/IDD", "max_issues_repo_head_hexsha": "3353c6ad2b57e1abc3612c66b680dd5c2bc53b54", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Untitled.ipynb", "max_forks_repo_name": "bcreiner/IDD", "max_forks_repo_head_hexsha": "3353c6ad2b57e1abc3612c66b680dd5c2bc53b54", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 18.4137931034, "max_line_length": 48, "alphanum_fraction": 0.4531835206, "converted": true, "num_tokens": 125, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328917, "lm_q2_score": 0.7217432003123989, "lm_q1q2_score": 0.438576760907168}} {"text": "Copyright (c) 2015, 2016\n[Sebastian Raschka](http://sebastianraschka.com/)\n[Li-Yi Wei](http://www.liyiwei.org/)\n\nhttps://github.com/1iyiwei/pyml\n
    \n[MIT License](https://github.com/1iyiwei/pyml/blob/master/LICENSE.txt)\n\n# Introduction\n* What is machine learning\n* Why machine learning\n* Types of machine learning\n* Machine learning pipeline\n* Relations to other fields\n* Historical perspective\n\n# What is machine learning?\nThe construction and study of algorithms/programs that can learn from data .\n\n\n\n\n# Traditional programming\n\nSteps: formulate problem $\\rightarrow$ design algorithm $\\rightarrow$ write program $\\rightarrow$ test\n\nThe program remains invariant with different input data, unless the programmer manually changes it.\n\n\n# Example: minimum finding\nProblem\n* Given a sequence of numbers, find the smallest one\n\nAlgorithm\n* Record the currently minimum, initialized to $\\infty$\n* Loop through the numbers one by one\n * If this number $<$ minimum, minimum $\\leftarrow$ this number\n \nAnalysis\n* The time complexity of the above algorithm is $O(N)$ where $N$ is the sequence size.\n\n\n```python\n# a simple python program to find minimum numbers\n# note that it remains the same regardless of the input data\n\nimport math\n\n# function\ndef findmin(numbers):\n answer = math.inf\n for value in numbers:\n if value < answer:\n answer = value\n return answer\n\n# main\ntest = [3.14, 2.2, 8, -9.2, 100000, 0]\n\nprint(findmin(test))\n\n```\n\n -9.2\n\n\n# Example: sorting\n\nProblem\n* Given a sequence of numbers, order them from small to large\n\nAlgorithm\n* Pick one number (randomly) as anchor\n* Go through each other number\n * If this number $\\leq$ anchor, append it to the left sequence\n * Otherwise, append it to the right sequence\n* Apply the same method recursively to the left and right sequences\n* Concatenate left sequence, anchor, right sequence\n\n\n```python\n# code is left as an exercise\n```\n\n# Other examples\nThink about the programs you have written in the past; how many of them can learn from data?\n\n# Machine learning\n\nThe program can learn from data and change structure/behavior\n\nThe programmer still writes (part of) the program, but it is not fixed\n* Models with parameters that can change with input data, like brains\n* Programmer selects and initializes the model; the model parameters change with data (training via optimization)\n* The trained model deals with future situations\n\n[](http://www.wired.com/2016/03/final-game-alphago-lee-sedol-big-deal-humanity/)\n\n# Why learning from data\n\nSome algorithms/programs are hard/impossible to design/code manually/explicitly\n\nThe algorithms/programs might need to deal with unforseeable situations (generalization)\n\n\n\n\n\n# Example: handwriting digit recognition\n\n\n\nProblem\n* Input: a digit represented by a $28 \\times 28$ image (MNIST)\n* Output: one of the digits in [0 .. 9]\n\nTraditional programming?\n* Give it a try :-)\n\nMachine learning\n* Collect data - pairs of images and labels\n* Select and initialize a model; train the model (parameters) with the data\n* The model, if properly trained, can recognize handwritings not in the original dataset for training\n\nSometimes it is much easier to say what (example data) instead of how (algorithm)\n* [Soon We Won\u2019t Program Computers. We\u2019ll Train Them Like Dogs](http://www.wired.com/2016/05/the-end-of-code/)\n\n# Other applications\n\n* Self-driving cars\n* Language translation\n* Speech analysis & synthesis\n* Spam filtering\n* Recommendation systems\n* Fraud detection\n* Market prediction\n\n# Types of machine learning\n\n\n\n# Supervised learning\n\nGiven examples of inputs and corresponding desired outputs, predict outputs for future inputs\n* classification, regression, time series prediction\n\n\n\n# Classification vs. regression\n\n\n\n\n\n## Classification: discrete output\nClass labels: spam or not spam, positive or negative for disease, types of animals, etc.\n\n\n## Classifying flowers based on various features\n\n\n## Regression: continous output\nfunction fitting\n\n\n\n# Unsupervised learning\n\nGiven only inputs (without desired output examples), automatically discover representations, features, structure, etc. \n* clustering, outlier detection, density estimation, dimensionality reduction\n\n# Clustering\n\nPut data samples into different groups\n\n\n\n# Dimensionality reduction\n\nProject data from a higher dimensional space into a lower dimensional space\n* compression, visualization\n\n\n\n# Reinforcement learning\n\nGiven sequences of actions of an agent and feedbacks from an environment, learn to select action sequences in a way that maximises the expected reward\n\n\n\n* playing games\n * state: game board configuration\n * reward: expected winning chance\n \n* self driving cars\n * state: position, direction, speed, sensors, etc.\n * reward: not hitting anything, remaining time for destination, fuel consumption, etc.\n\n# Summary of types\n\nTypes of learning\n\n* Supervised learning\n * Given sample inputs and outputs, learn how to predict outputs for future inputs\n\n* Unsupervised learning\n * Given only sample inputs, learn to transform or organize them in some way\n\n* Reinforcement learning\n * No direct inputs or outputs, learn best behavior from environment feedbacks (states, rewards)\n\n\n \nTypes of data\n* Discrete/continuous $\\times$ input/output\n * Applies to different types of learning above\n\n* Discrete output\n * Classification for supervised learning\n * Clustering for unsupervised learning\n \n* Continuous output\n * Regression for supervised learning\n * Dimensionality reduction for unsupervised learning\n\n# Representation\n\n## Model\nWe can represent the model as a function $f$, with a set of parameters $\\Theta$.\n
    \nGiven a set of inputs $\\mathbf{X}$, the model computes outcomes $\\mathbf{Y}$.\n$$\\mathbf{Y} = f(\\mathbf{X}, \\Theta)$$\n\nFor example, in digit recognition, $\\mathbf{X}$ and $\\mathbf{Y}$ are the digit images and digit labels (0 to 9), respectively.\n\nThe parameters $\\Theta$ consist of those optimized automatically and those manually picked by humans.\nThe latter are called hyper-parameters.\n\n## Loss\nEvery machine learning task as a goal, which can be formalized as a loss function:\n$$L(\\mathbf{X}, \\mathbf{T}, \\mathbf{Y})$$\n, where $\\mathbf{T}$ is some form of target or auxiliary information, such as:\n* labels for supervised classification\n* number of clusters for unsupervised clustering\n* environment for reinforcement learning\n\n## Regularization\nIn addition to the objective, we often care about the simplicity of the model, for better efficiency and generalization (avoiding over-fitting).\nThe complexity of the model can be measured by another penalty function:\n$$P(\\Theta)$$\nSome common penalty functions include number and/or magnitude of parameters.\n\n## Objective\nWe can sum up both the loss and regularization terms as the total objective:\n$$\\Phi(\\mathbf{X}, \\mathbf{T}, \\Theta) = L\\left(\\mathbf{X}, \\mathbf{T}, \\mathbf{Y}=f(\\mathbf{X}, \\Theta)\\right) + P(\\Theta)$$\n\nDuring training, the goal is to optimize the parameters $\\Theta$ with respect to the given training data $\\mathbf{X}$ and $\\mathbf{T}$:\n$$argmin_\\Theta \\; \\Phi(\\mathbf{X}, \\mathbf{T}, \\Theta)$$\nAnd hope the trained model with generalize well to future data. \n\n## Example: curve fitting\n\n\n\nGiven a set of data points $\\left(\\mathbf{X}, \\mathbf{Y}\\right)$, fit a model curve to describe their relationship.\n\nThis is actually a regression problem, but we have all seen this in prior math/coding classes to serve as a good example for machine learning.\n\nRecall $\\mathbf{Y} = f(\\mathbf{X}, \\Theta)$ is our model.\n\nFor 2D linear curve fitting, the model is a straight line:\n$y = w_1 x + w_0$, so the parameters $\\Theta = \\{w_0, w_1\\}$.\n\nThe loss function is $L\\left(\\mathbf{X}, \\mathbf{T}, \\mathbf{Y}\\right) = \\sum_i \\left( T^{(i)} - Y^{(i)}\\right)^2 = \\sum_i \\left( T^{(i)} - w_1 X^{(i)} - w_0 \\right)^2$.\n
    \n($\\mathbf{X}$ is a matrix/tensor, and each data sample is a row. We denote the ith sample/row as $\\mathbf{X}^{(i)}$.)\n\nFor this simple example we don't care about regularization, thus $P(\\Theta) = 0$.\n\nThe goal is to optimize $\\Theta = \\{w_0, w_1 \\}$ with given $\\left(\\mathbf{X}, \\mathbf{Y}\\right)$ to minimize $L$.\nFor simple cases like this, we can directly optimize via calculus:\n$$\n\\begin{align}\n\\frac{\\partial L}{\\partial w_0} & = 0 \\\\\n\\frac{\\partial L}{\\partial w_1} & = 0\n\\end{align}\n$$\n\nThe math and coding will be left as an exercise.\n\n# Steps for building a machine learning system\n\nWe will talk about individual steps for the rest of this course.\n\n\n\n# Machine learning and other related fields\n\nData mining\n* discovering (i.e. mining) useful information from large data sets\n\nPattern recognition\n* originated from engineering, more on hand-crafted algorithms\n* machine learning originated from computer science\n\n\nArtificial intelligence\n* machine learning is a subset\n* traditional AI can be rule-based, deterministic\n* machine learnign tends to be data-driven, probabilistic\n * statistics\n\nCognitive science\n* reverse engineers the brain\n* machine learning focuses on the forward process\n * understanding biology can help, but not the goal\n\n# Deep learning\n\nDeep learning $\\subset$ neural networks $\\subset$ machine learning\n\nAlgorithms existed since the 80's, but lacked sufficient computing power\n\nMoore law enabled simple algorithms to process deep architecture and large data\n\nGeoffrey Hinton, the godfather of \u2018deep learning\u2019\u2014which helped Google\u2019s AlphaGo beat a grandmaster\u2014on the past, present and future of AI\n\n# A DARPA Perspective on Artificial Intelligence \n\nhttps://youtu.be/-O01G3tSYpU\n\n\n\n```python\nfrom IPython.display import YouTubeVideo\nYouTubeVideo(\"-O01G3tSYpU\")\n```\n\n\n\n\n\n\n\n\n\n\n# Reading\n* PML - Chapter 1\n* IML - Chapter 1\n\n# Coding\n* Install anaconda, git\n* Review/learn python, ipynb\n* See [README.md](https://github.com/1iyiwei/pyml/blob/master/code/ch01/README.md) for quick instructions\n\n# Assignment\n* [ex01](http://nbviewer.jupyter.org/github/1iyiwei/pyml/blob/master/code/ch01/ex01.ipynb)\n\n
    \n

    Fin

    \n
    \n", "meta": {"hexsha": "948bd649265cbbafc9d4c2404913cb30b7210706", "size": 61600, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "code/ch01/ch01.ipynb", "max_stars_repo_name": "1iyiwei/pyml", "max_stars_repo_head_hexsha": "9bc0fa94abd8dcb5de92689c981fbd9de2ed1940", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 27, "max_stars_repo_stars_event_min_datetime": "2016-12-29T05:58:14.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-17T10:27:32.000Z", "max_issues_repo_path": "code/ch01/ch01.ipynb", "max_issues_repo_name": "1iyiwei/pyml", "max_issues_repo_head_hexsha": "9bc0fa94abd8dcb5de92689c981fbd9de2ed1940", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "code/ch01/ch01.ipynb", "max_forks_repo_name": "1iyiwei/pyml", "max_forks_repo_head_hexsha": "9bc0fa94abd8dcb5de92689c981fbd9de2ed1940", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 34, "max_forks_repo_forks_event_min_datetime": "2016-09-02T04:59:40.000Z", "max_forks_repo_forks_event_max_datetime": "2020-10-05T02:11:37.000Z", "avg_line_length": 80.522875817, "max_line_length": 41451, "alphanum_fraction": 0.8285876623, "converted": true, "num_tokens": 2486, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.7799929002541068, "lm_q1q2_score": 0.4384936793806191}} {"text": "

    \n \n\n

    \n\n## Data Analytics \n\n### Bootstrap Confidence Intervals in Python \n\n#### Michael Pyrcz, Associate Professor, University of Texas at Austin \n\n##### [Twitter](https://twitter.com/geostatsguy) | [GitHub](https://github.com/GeostatsGuy) | [Website](http://michaelpyrcz.com) | [GoogleScholar](https://scholar.google.com/citations?user=QVZ20eQAAAAJ&hl=en&oi=ao) | [Book](https://www.amazon.com/Geostatistical-Reservoir-Modeling-Michael-Pyrcz/dp/0199731446) | [YouTube](https://www.youtube.com/channel/UCLqEr-xV-ceHdXXXrTId5ig) | [LinkedIn](https://www.linkedin.com/in/michael-pyrcz-61a648a1)\n\n\n### Bootstrap Confidence Intervals in Python \n\nHere's a simple workflow, demonstration of bootstrap for modeling workflows. This should help you get started with this important data analytics method to evaluate and integrate uncertainty in any sample statistics or model. \n\n#### Reporting Uncertainty and Significance\n\nWith confidence intervals and hypothesis testing we have the opportunity to report uncertainty and to report significance in our statistics. These methods are based on standard methods with their associated limitations and assumptions. For additional learning content to support learning about confidence intervals see [Confidence Intervals Lecture](https://youtu.be/oaXCcTWcU04) for more details and the [Confidence Intervals Interactive Demonstration](https://github.com/GeostatsGuy/PythonNumericalDemos/blob/master/Interactive_Confidence_Interval.ipynb) to play with and observe confidence intervals in real-time!\n\nAlso, for more information about statistical bootstrap see the [Bootstrap Lecture](https://youtu.be/oaXCcTWcU04) and [Bootstrap Workflow](https://git.io/fhgUW) for a general, empirical approach to assess uncertainty in statistics and models and [Bootstrap Interactive Demonstration](https://github.com/GeostatsGuy/PythonNumericalDemos/blob/master/Interactive_Bootstrap_Simple.ipynb).\n\nThis is a tutorial / demonstration of **Bootstrap Confidence Intervals in Python** for data analytics and data science workflows. In Python, the SciPy package, specifically the [SciPy Stats Functions](https://docs.scipy.org/doc/scipy/reference/stats.html) provides excellent statistics tools. \n\n#### Bootstrap\n\nThe uncertainty in an estimated population parameter from a sample, represented as a range, lower and upper bound, based on a specified probability interval known as the **confidence level**.\n\n* one source of uncertainty is the paucity of data.\n* do 200 or even less sample data provide a precise (and accurate estimate) of the mean? standard deviation? skew? 13th percentile / P13? 3rd central moment? experimental variogram? mutual information? Shannon entropy? etc.\n\nWould it be useful to know the uncertainty due to limited sampling?\n* what is the impact of uncertainty in the mean porosity e.g. 20%+/-2%?\n\n**Bootstrap** is a method to assess the uncertainty in a sample statistic by repeated random sampling with replacement.\n\nAssumptions\n* sufficient, representative sampling, identical, idependent samples\n\nLimitations\n1. assumes the samples are representative \n2. assumes stationarity\n3. only accounts for uncertainty due to too few samples, e.g. no uncertainty due to changes away from data\n4. does not account for boundary of area of interest \n5. assumes the samples are independent\n6. does not account for other local information sources\n\nThe Bootstrap Approach (Efron, 1982)\n\nStatistical resampling procedure to calculate uncertainty in a calculated statistic from the data itself.\n* Does this work? Prove it to yourself, for uncertainty in the mean solution is standard error: \n\n\\begin{equation}\n\\sigma^2_\\overline{x} = \\frac{\\sigma^2_s}{n}\n\\end{equation}\n\nExtremely powerful - could calculate uncertainty in any statistic! e.g. P13, skew etc.\n* Would not be possible access general uncertainty in any statistic without bootstrap.\n* Advanced forms account for spatial information and sampling strategy (game theory and Journel\u2019s spatial bootstrap (1993).\n\nSteps: \n\n1. assemble a sample set, must be representative, reasonable to assume independence between samples\n\n2. optional: build a cumulative distribution function (CDF)\n * may account for declustering weights, tail extrapolation\n * could use analogous data to support\n\n3. For $\\ell = 1, \\ldots, L$ realizations, do the following:\n\n * For $i = \\alpha, \\ldots, n$ data, do the following:\n\n * Draw a random sample with replacement from the sample set or Monte Carlo simulate from the CDF (if available). \n\n6. Calculate a realization of the sammary statistic of interest from the $n$ samples, e.g. $m^\\ell$, $\\sigma^2_{\\ell}$. Return to 3 for another realization.\n\n7. Compile and summarize the $L$ realizations of the statistic of interest.\n\n* calculate and display the histogram or CDF to see the full distribution\n\n* report the percentiles representing the lower and upper confidence intervals, e.g. for a confidence level of 95%, the 2.5 and 97.5 percentiles.\n\nThis is a very powerful method. Let's demonstrate a bunch of measures.\n\n* when available, I compare to the distributions from the analytic expressions.\n\nLet's demonstrate for a variety of parameters / statistics:\n\n* mean / arithmetic average\n\n* proportion\n\n* interquartile range\n\n* coefficient of variation\n\n* correlation coefficient\n\n#### Objective \n\nThe objective is to remove the hurdles of subsurface modeling workflow construction by providing building blocks and sufficient examples. This is not a coding class per se, but we need the ability to 'script' workflows working with numerical methods. \n\n#### Getting Started\n\nHere's the steps to get setup in Python with the GeostatsPy package:\n\n1. Install Anaconda 3 on your machine (https://www.anaconda.com/download/). \n2. From Anaconda Navigator (within Anaconda3 group), go to the environment tab, click on base (root) green arrow and open a terminal. \n3. In the terminal type: pip install geostatspy. \n4. Open Jupyter and in the top block get started by copy and pasting the code block below from this Jupyter Notebook to start using the geostatspy functionality. \n\nYou will need to copy the data file to your working directory. They are available here:\n\n* Tabular data - sample_data_biased.csv at https://git.io/fh0CW\n\n#### Load the required libraries\n\nThe following code loads the required libraries.\n\n\n```python\nimport numpy as np # ndarrys for gridded data\nimport pandas as pd # DataFrames for tabular data\nimport os # set working directory, run executables\nimport matplotlib.pyplot as plt # plotting\nimport matplotlib # plotting\nfrom matplotlib.ticker import (MultipleLocator, AutoMinorLocator) # control of axes ticks\nplt.rc('axes', axisbelow=True) # set axes and grids in the background for all plots\nimport seaborn as sns # plotting\nfrom scipy import stats # summary statistics\nimport math # trig etc.\nimport scipy.signal as signal # kernel for moving window calculation\nimport random # random sampling\nfrom scipy.stats import gaussian_kde # for PDF calculation\nfrom scipy.stats import t # Student's t distribution for analytical solution\n```\n\n#### Set the working directory\n\nI always like to do this so I don't lose files and to simplify subsequent read and writes (avoid including the full address each time). \n\n\n```python\n#os.chdir(\"c:/PGE383\") # set the working directory\n```\n\n#### Set the Random Number Seed\n\nFor repeatability, set the random number seed.\n\n* this ensures that the same random samples are drawn each run and everyone in the class will have the same results.\n\n* change this seed number for a new set of random realizations\n\n\n```python\nseed = 73073\nnp.random.seed(seed = seed)\n```\n\n#### Loading Tabular Data\n\nHere's the command to load our comma delimited data file in to a Pandas' DataFrame object. \n\n\n```python\n#df = pd.read_csv('sample_data_biased.csv') # load our data table\ndf = pd.read_csv('https://raw.githubusercontent.com/GeostatsGuy/GeoDataSets/master/sample_data_biased.csv') # load the data from Dr. Pyrcz's GitHub repository\n```\n\nLet's drop some samples so that we increase the variations in bootstrap samples for our demonstration below.\n\n\n```python\ndf = df.sample(frac = 0.2) # extract around 50 random samples to reduce the size of the dataset \nprint('Using ' + str(len(df)) + ' number of samples')\n```\n\n Using 58 number of samples\n\n\nVisualizing the DataFrame would be useful and we already learned about these methods in this [Working with Tabular Data Demo](https://git.io/fNgRW). \n\nWe can preview the DataFrame by printing a slice or by utilizing the 'head' DataFrame member function (with a nice and clean format, see below). With the slice we could look at any subset of the data table and with the head command, add parameter 'n=13' to see the first 13 rows of the dataset. \n\n\n```python\ndf.head() # display first 4 samples in the table as a preview\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    XYFaciesPorosityPerm
    150206900.1044843.003225
    1740060010.163054317.883581
    6864052910.12054710.801778
    23449028910.14452516.251107
    21447089910.2119441302.937858
    \n
    \n\n\n\n#### Summary Statistics for Tabular Data\n\nThe table includes X and Y coordinates (meters), Facies 1 and 0 (1 is sandstone and 0 interbedded sand and mudstone), Porosity (fraction), and permeability as Perm (mDarcy). \n\nThere are a lot of efficient methods to calculate summary statistics from tabular data in DataFrames. The describe command provides count, mean, minimum, maximum, and quartiles all in a nice data table. We use transpose just to flip the table so that features are on the rows and the statistics are on the columns.\n\n\n```python\ndf.describe().transpose()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    countmeanstdmin25%50%75%max
    X58.0418.275862231.45715310.000000272.500000400.000000560.000000990.000000
    Y58.0493.224138287.39361439.000000279.000000499.500000676.500000989.000000
    Facies58.00.8103450.3954520.0000001.0000001.0000001.0000001.000000
    Porosity58.00.1365010.0404730.0743490.1056620.1261920.1470670.223661
    Perm58.0270.961923590.8051850.0932524.11240015.10565179.3144342372.383732
    \n
    \n\n\n\n#### Visualizing Tabular Data with Location Maps \n\nIt is natural to set the x and y coordinate and feature ranges manually. e.g. do you want your color bar to go from 0.05887 to 0.24230 exactly? Also, let's pick a color map for display. I heard that plasma is known to be friendly to the color blind as the color and intensity vary together (hope I got that right, it was an interesting Twitter conversation started by Matt Hall from Agile if I recall correctly). We will assume a study area of 0 to 1,000m in x and y and omit any data outside this area.\n\n\n```python\nxmin = 0.0; xmax = 1000.0 # range of x values\nymin = 0.0; ymax = 1000.0 # range of y values\npormin = 0.05; pormax = 0.25; # range of porosity values\npermmin = 0.01; permmax = 10000\nnx = 100; ny = 100; csize = 10.0\ncmap = plt.cm.plasma # color map\ncumul_prob = np.linspace(0.0,1.0,100) # list of cumulative probabilities\n```\n\nLet's try out location maps, histograms and scatter plots. \n\n\n```python\nplt.subplot(231)\nplt.scatter(df['X'],df['Y'],s = 40,c = df['Porosity'],cmap = plt.cm.inferno,linewidths = 0.3,edgecolor = 'black',alpha = 0.8,vmin = pormin,vmax = pormax)\nplt.colorbar(); plt.xlabel('X (m)'); plt.ylabel('Y (m)'); plt.title('Porosity Location Map')\n\nplt.subplot(234)\nplt.hist(df['Porosity'],color = 'darkorange',alpha = 0.7,edgecolor='black',bins = np.linspace(pormin,pormax,int(len(df)/3)))\nplt.xlabel('Porosity (fraction)'); plt.ylabel('Frequency'); plt.title('Porosity Histogram')\nplt.grid(True, which='major',linewidth = 1.0); plt.grid(True, which='minor',linewidth = 0.2) # add y grids\nplt.gca().tick_params(which='major',length=7); plt.gca().tick_params(which='minor', length=4)\nplt.gca().xaxis.set_minor_locator(AutoMinorLocator()); plt.gca().yaxis.set_minor_locator(AutoMinorLocator()) # turn on minor ticks\n\nplt.subplot(232)\nplt.scatter(df['X'],df['Y'],s = 40,c = df['Perm'],cmap = plt.cm.inferno,linewidths = 0.3,edgecolor = 'black',alpha = 0.8,vmin = permmin,vmax = permmax,norm=matplotlib.colors.LogNorm())\nplt.colorbar(); plt.xlabel('X (m)'); plt.ylabel('Y (m)'); plt.title('Permeability Location Map')\n\nplt.subplot(235)\nplt.hist(df['Perm'],color = 'darkorange',alpha = 0.7,edgecolor='black',bins=np.logspace(np.log10(permmin),np.log10(permmax),int(len(df)/3)))\n#sns.kdeplot(x=df['Perm'],color = 'black',alpha = 0.2,levels = 1,log_scale = True,bw_adjust = 1)\nplt.xlabel('Permeability (mD)'); plt.ylabel('Frequency'); plt.title('Permeability Histogram'); plt.xscale('log')\nplt.grid(True, which='major',linewidth = 1.0); plt.grid(True, which='minor',linewidth = 0.2) # add y grids\nplt.gca().tick_params(which='major',length=7); plt.gca().tick_params(which='minor', length=4)\n\nplt.subplot(233)\nplt.scatter(df['Porosity'],df['Perm'],s = 40,color = 'darkorange',alpha = 0.7,edgecolor='black')\n#plt.contour(df['Porosity'],df['Perm'] Z, colors='black');\nplt.ylabel('Permeability (mD)'); plt.xlabel('Porosity (fraction)'); plt.title('Permeability-Porosity Scatter Plot')\nplt.yscale('log')\nsns.kdeplot(x=df['Porosity'],y=df['Perm'],color = 'black',alpha = 0.2,levels = 4)\n\nplt.subplots_adjust(left=0.0, bottom=0.0, right=3.0, top=2.2, wspace=0.2, hspace=0.2)\nplt.show()\n```\n\n### Bootstrap Method in Python \n\nIf you are new to bootstrap and Python, here's the most simple code possible for bootstrap.\n\n* specify the number of bootstrap realizations, $L$\n* declare a list to store the bootstrap realizations of the statistic of interest\n* loop over L bootstrap realizations\n * n MCS, random samples with replacement for a new realization of the data\n * calculate the realization of the statistic from the realization of the data\n* summarize the resulting uncertainty model, histogram, summary statistics etc.\n\n#### Bootstrap of the Sample Mean, Arithmetic Average\n\nLet's demonstrate bootstrap for uncertainty in the arithmetic average with a simple workflow.\n\n```python\npor_avg_real = [] # declare an empty list to store the bootstrap realizations of the statistic \nfor k in range(0,L): # loop over the L bootstrap realizations\n samples = random.choices(df['Porosity'].values, k=len(df)) # n Monte Carlo simulations\n por_avg_real.append(np.average(samples)) # calculate the statistic of interest from the new bootstrap dataset\n```\n\nWe will compare the bootstrap uncertainty in the sample arithmetic average distribution with the analytical expression:\n\n\\begin{equation}\n\\overline{x} \\pm t_{\\frac{\\alpha}{2},n-1}\\sqrt{\\frac{s^2}{n}}\n\\end{equation}\n\nThe remaining code in this block is just a super cool set of plots with the results.\n\n\n```python\n# Bootstrap method for uncertainty in the sample mean\nL = 1000 # set the number of bootstrap realizations \npor_avg_real = [] # declare an empty list to store the bootstrap realizations of the statistic \nfor k in range(0,L): # loop over the L bootstrap realizations\n samples = random.choices(df['Porosity'].values, k=len(df)) # n Monte Carlo simulations\n por_avg_real.append(np.average(samples)) # calculate the statistic of interest from the new bootstrap dataset\n\n# Analytical solution for the uncertainty in the sample mean, Student's t distribution with the correct mean and standard deviation\nanalytical = t.pdf(np.linspace(pormin,pormax,100), len(df)-1,loc=np.average(df['Porosity']),scale=math.sqrt(np.var(df['Porosity'].values)/len(df)))\n\n# Plot of the original data and bootstap and analytical results \nfig = plt.subplot(131) \nplt.hist(df['Porosity'],color = 'darkorange',alpha = 0.8,edgecolor='black',linewidth=2,bins = np.linspace(pormin,pormax,int(len(df)/3)))\n#plt.plot([np.average(df['Porosity']),np.average(df['Porosity'])],[0,100])\nplt.axvline(x=np.average(df['Porosity']),linestyle=\"--\",c='black')\nplt.text(np.average(df['Porosity'])+0.005, 8.8, r'Average = ' + str(round(np.average(df['Porosity']),3)), fontsize=12)\nplt.text(np.average(df['Porosity'])+0.005, 8.3, r'St.Dev. = ' + str(round(np.std(df['Porosity']),3)), fontsize=12)\nplt.text(np.average(df['Porosity'])+0.005, 7.8, r'P10 = ' + str(round(np.percentile(df['Porosity'],10),3)), fontsize=12)\nplt.text(np.average(df['Porosity'])+0.005, 7.3, r'P90 = ' + str(round(np.percentile(df['Porosity'],90),3)), fontsize=12)\nplt.xlabel('Porosity (fraction)'); plt.ylabel('Frequency'); plt.title('Original Porosity Histogram')\n\nplt.subplot(132)\nplt.hist(por_avg_real,color = 'darkorange',alpha = 0.8,edgecolor = 'black',bins=np.linspace(pormin,pormax,100),label = 'bootstrap',density = True) # plot the distribution, could also calculate any summary statistics\nplt.plot(np.linspace(pormin,pormax,100),analytical,color = 'black',linewidth=3,label = 'analytical',alpha=0.8)\nplt.axvline(x=np.average(por_avg_real),linestyle=\"--\",c='black')\nplt.fill_between(np.linspace(pormin,pormax,100), 0, analytical, where = np.linspace(pormin,pormax,100) <= np.percentile(por_avg_real,10), facecolor='red', interpolate=True, alpha = 0.5)\nplt.fill_between(np.linspace(pormin,pormax,100), 0, analytical, where = np.linspace(pormin,pormax,100) >= np.percentile(por_avg_real,90), facecolor='red', interpolate=True, alpha = 0.5)\nplt.text(np.average(por_avg_real)+0.009, L*0.07, r'Average = ' + str(round(np.average(por_avg_real),3)), fontsize=12)\nplt.text(np.average(por_avg_real)+0.009, L*0.066, r'St.Dev. = ' + str(round(np.std(por_avg_real),3)), fontsize=12)\nplt.text(np.average(por_avg_real)+0.009, L*0.062, r'P90 = ' + str(round(np.percentile(por_avg_real,90),3)), fontsize=12)\nplt.text(np.average(por_avg_real)+0.009, L*0.058, r'P10 = ' + str(round(np.percentile(por_avg_real,10),3)), fontsize=12)\nplt.xlabel('Bootstrap Realizations and Analytical Uncertainty Distribution for Mean'); plt.ylabel('Frequency'); plt.title('Mean Bootstrap Realizations')\nplt.legend(loc = 'upper left')\n\nfig = plt.subplot(133) \nplt.hist(df['Porosity'],color = 'darkorange',alpha = 0.8,edgecolor='black',linewidth=2,bins = np.linspace(pormin,pormax,int(len(df)/3)))\n#plt.plot([np.average(df['Porosity']),np.average(df['Porosity'])],[0,100])\nplt.axvline(x=np.average(df['Porosity']),c='black')\nplt.axvline(x=np.percentile(por_avg_real,90),linestyle=\"--\",c='black')\nplt.axvline(x=np.percentile(por_avg_real,10),linestyle=\"--\",c='black')\nplt.text(np.percentile(por_avg_real,90)+0.009,8.8, r'Average = ' + str(round(np.average(por_avg_real),3)), fontsize=12)\nplt.text(np.percentile(por_avg_real,90)+0.009,8.3, r'Average P90 = ' + str(round(np.percentile(por_avg_real,90),3)), fontsize=12)\nplt.text(np.percentile(por_avg_real,90)+0.009,7.8, r'Average P10 = ' + str(round(np.percentile(por_avg_real,10),3)), fontsize=12)\nplt.xlabel('Porosity (fraction)'); plt.ylabel('Frequency'); plt.title('Porosity Histogram with Uncertainty in Average')\n\nplt.subplots_adjust(left=0.0, bottom=0.0, right=2.0, top=1.2, wspace=0.2, hspace=0.2)\nplt.show()\n```\n\n#### Bootstrap of the Proportion\n\nLet's demonstrate another case for which we have the analytical solution. \n\nWe will compare the bootstrap results with the analytical expression:\n\n\\begin{equation}\n\\hat{p} \\pm t_{\\frac{\\alpha}{2},n-1}\\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}\n\\end{equation}\n\n\n```python\nL = 100000 # set the number of bootstrap realizations \nsand_prop_real = [] # declare an empty list to store the bootstrap realizations of the statistic \nshale_prop_real = []\nfor k in range(0,L): # loop over the L bootstrap realizations\n samples = random.choices(df['Facies'].values, k=len(df)) # n Monte Carlo simulations\n sand_prop_real.append(samples.count(1)/len(df)) # calculate the statistic of interest from the new bootstrap dataset\n shale_prop_real.append(samples.count(0)/len(df)) # calculate the statistic of interest from the new bootstrap dataset \nsand_prop = np.sum(df['Facies'] == 1)/len(df); shale_prop = np.sum(df['Facies'] == 0)/len(df)\n\nanalytical_shale = t.pdf(np.linspace(0.0,1.0,100), len(df)-1,loc=shale_prop,scale=math.sqrt(shale_prop*(1.0-shale_prop)/len(df)))\n\nfig = plt.subplot(121) \nbarlist = plt.bar(x=['Shale','Sand'],height = [1-sand_prop,sand_prop],color = 'gold',alpha = 0.8,edgecolor='black',linewidth=2)\nbarlist[0].set_color('brown'); barlist[0].set_edgecolor('black')\nplt.text(-.3, shale_prop+0.02, r'Prop. Shale = ' + str(round(shale_prop,2)), fontsize=12)\nplt.text(0.7, sand_prop+0.02, r'Prop. Sand = ' + str(round(sand_prop,2)), fontsize=12)\nplt.xlabel('Rock Type / Facies'); plt.ylabel('Proportion'); plt.title('Sample Histogram')\nplt.ylim([0,1]); plt.yticks(np.arange(0, 1.1, 0.1))\n\nplt.subplot(122) \nplt.hist(sand_prop_real,color = 'gold',alpha = 0.8,edgecolor = 'black',linewidth=2,bins=np.linspace(0.0,1.0,40),label = 'Sand',density = True) # plot the distribution, could also calculate any summary statistics\nanalytical = t.pdf(np.linspace(0.0,1.0,100), len(df)-1,loc=sand_prop,scale=math.sqrt(sand_prop*(1.0-sand_prop)/len(df)))\nplt.plot(np.linspace(0.0,1.0,100),analytical,color = 'black',label = 'Analytical',alpha=0.8)\nplt.axvline(x=sand_prop,linestyle=\"--\",c='black')\nplt.fill_between(np.linspace(0.0,1.0,100), 0, analytical, where = np.linspace(0.0,1.0,100) <= np.percentile(sand_prop_real,10), facecolor='red', interpolate=True, alpha = 0.8,label='Outside')\nplt.fill_between(np.linspace(0.0,1.0,100), 0, analytical, where = np.linspace(0.0,1.0,100) >= np.percentile(sand_prop_real,90), facecolor='red', interpolate=True, alpha = 0.8)\nplt.text(np.average(sand_prop_real)+0.04, 7.0, r'Prop. Exp = ' + str(round(sand_prop,2)), fontsize=8)\nplt.text(np.average(sand_prop_real)+0.04, 6.4, r'Prop. P90 = ' + str(round(np.percentile(sand_prop_real,90),2)), fontsize=8)\nplt.text(np.average(sand_prop_real)+0.04, 5.8, r'Prop. P10 = ' + str(round(np.percentile(sand_prop_real,10),2)), fontsize=8)\n\nplt.hist(shale_prop_real,color = 'brown',alpha = 0.8,edgecolor = 'black',linewidth=2,bins=np.linspace(0.0,1.0,40),label = 'Shale',density = True) # plot the distribution, could also calculate any summary statistics\nplt.plot(np.linspace(0.0,1.0,100),analytical_shale,color = 'black',alpha=0.8)\nplt.axvline(x=shale_prop,linestyle=\"--\",c='black')\nplt.fill_between(np.linspace(0.0,1.0,100), 0, analytical_shale, where = np.linspace(0.0,1.0,100) <= np.percentile(shale_prop_real,10), facecolor='blue', interpolate=True, alpha = 0.8,label='Outside')\nplt.fill_between(np.linspace(0.0,1.0,100), 0, analytical_shale, where = np.linspace(0.0,1.0,100) >= np.percentile(shale_prop_real,90), facecolor='blue', interpolate=True, alpha = 0.8)\nplt.text(np.average(shale_prop_real)+0.04, 7.0, r'Prop. Exp = ' + str(round(shale_prop,2)), fontsize=8)\nplt.text(np.average(shale_prop_real)+0.04, 6.4, r'Prop. P90 = ' + str(round(np.percentile(shale_prop_real,90),2)), fontsize=8)\nplt.text(np.average(shale_prop_real)+0.04, 5.8, r'Prop. P10 = ' + str(round(np.percentile(shale_prop_real,10),2)), fontsize=8)\n\nplt.xlabel('Rock Type / Facies Proportion (fraction)'); plt.ylabel('Frequency'); plt.title('Proportion Bootstrap Realizations and Analytical Uncertainty Distributions')\nplt.legend(loc = 'upper center')\n\nplt.subplots_adjust(left=0.0, bottom=0.0, right=2.0, top=1.2, wspace=0.2, hspace=0.2)\nplt.show()\n```\n\n#### Bootstrap of the Interquartile Range\n\nTo prove that we can bootstrap any statistic let's select a more complicated measure of dispersion, the interquartile range.\n\n\n```python\nL = 1000 # set the number of bootstrap realizations \niqr_real = [] # declare an empty list to store the bootstrap realizations of the statistic \nfor k in range(0,L): # loop over the L bootstrap realizations\n samples = random.choices(df['Porosity'].values, k=len(df)) # n Monte Carlo simulations\n iqr_real.append(np.percentile(samples,75) - np.percentile(samples,25)) # calculate the statistic of interest from the new bootstrap dataset\n\niqr = np.percentile(df['Porosity'],75) - np.percentile(df['Porosity'],25)\n\nplt.subplot(111)\nplt.hist(iqr_real,color = 'darkorange',alpha = 0.8,edgecolor = 'black',linewidth=2,bins=np.linspace(0.0,0.125,100),label = 'bootstrap',density = True) # plot the distribution, could also calculate any summary statistics\nplt.axvline(x=iqr,c='black')\nplt.axvline(x=np.percentile(iqr_real,10),linestyle=\"--\",c='black')\nplt.axvline(x=np.percentile(iqr_real,90),linestyle=\"--\",c='black')\nsns.kdeplot(x=iqr_real,color = 'black',alpha = 0.7,linewidth = 2,levels = 1,bw_adjust = 1)\nplt.text(np.percentile(iqr_real,90)+0.009, L*0.04, r'Average = ' + str(round(np.average(iqr_real),3)), fontsize=12)\nplt.text(np.percentile(iqr_real,90)+0.009, L*0.036, r'St.Dev. = ' + str(round(np.std(iqr_real),3)), fontsize=12)\nplt.text(np.percentile(iqr_real,90)+0.009, L*0.032, r'P90 = ' + str(round(np.percentile(iqr_real,90),3)), fontsize=12)\nplt.text(np.percentile(iqr_real,90)+0.009, L*0.028, r'P10 = ' + str(round(np.percentile(iqr_real,10),3)), fontsize=12)\nplt.xlabel('Boostrap Realizations of Interquartile Range'); plt.ylabel('Frequency'); plt.title('Bootstrap Uncertainty Distribution of Interquartile Range')\nplt.gca().grid(True, which='major',axis='both',linewidth = 1.0); plt.gca().grid(True, which='minor',axis='x',linewidth = 0.2) # add y grids\nplt.gca().tick_params(which='major',length=7); plt.gca().tick_params(which='minor', length=4)\nplt.gca().xaxis.set_minor_locator(AutoMinorLocator()); plt.gca().yaxis.set_minor_locator(AutoMinorLocator()) # turn on minor ticks\n\nplt.subplots_adjust(left=0.0, bottom=0.0, right=2.0, top=1.2, wspace=0.2, hspace=0.2)\nplt.show()\n```\n\n#### Bootstrap of the Coefficient of Variation\n\nHere's another statistic that requires multiple measures from the each bootstrap realization of the data.\n\n* this reinforces that we bootstrap dataset realizations and then calculate the statistic on this dataset realization\n\nFor the coefficient of variation we will:\n \n* calculate a bootstrap realization of the dataset with $n$ samples with replacement\n* calculate the mean and standard deviation from this bootstrapped realization of the dataset\n* calculate a boostrap realization of the coefficient of variation as the standard deviation divided by the mean\n\nRepeat this $L$ times and then evaluate the resulting distribution.\n\n\n```python\nL = 1000 # set the number of bootstrap realizations \ncv_real = [] # declare an empty list to store the bootstrap realizations of the statistic \n\nfor k in range(0,L): # loop over the L bootstrap realizations\n samples = random.choices(df['Porosity'].values, k=len(df)) # n Monte Carlo simulations\n cv_real.append(np.std(samples)/np.average(samples)) # calculate the statistic of interest from the new bootstrap dataset\n\ncv = np.std(df['Porosity'])/np.average(df['Porosity'])\n\nplt.subplot(111)\nplt.hist(cv_real,color = 'darkorange',alpha = 0.8,edgecolor = 'black',linewidth=2,bins=np.linspace(0.1,0.4,100),label = 'bootstrap',density = True) # plot the distribution, could also calculate any summary statistics\nplt.axvline(x=cv,c='black')\nplt.axvline(x=np.percentile(cv_real,10),linestyle=\"--\",c='black')\nplt.axvline(x=np.percentile(cv_real,90),linestyle=\"--\",c='black')\nsns.kdeplot(x=cv_real,color = 'black',alpha = 0.8,linewidth=2,levels = 1,bw_adjust = 1)\nplt.text(np.percentile(cv_real,90)+0.009, 16, r'Average = ' + str(round(np.average(cv_real),3)), fontsize=12)\nplt.text(np.percentile(cv_real,90)+0.009, 15, r'St.Dev. = ' + str(round(np.std(cv_real),3)), fontsize=12)\nplt.text(np.percentile(cv_real,90)+0.009, 14, r'P90 = ' + str(round(np.percentile(cv_real,90),3)), fontsize=12)\nplt.text(np.percentile(cv_real,90)+0.009, 13, r'P10 = ' + str(round(np.percentile(cv_real,10),3)), fontsize=12)\nplt.xlabel('Boostrap Realizations of Coefficient of Variation'); plt.ylabel('Frequency'); plt.title('Bootstrap Uncertainty Distribution of Coefficient of Variation')\nplt.gca().grid(True, which='major',axis='both',linewidth = 1.0); plt.gca().grid(True, which='minor',axis='x',linewidth = 0.2) # add y grids\nplt.gca().tick_params(which='major',length=7); plt.gca().tick_params(which='minor', length=4)\nplt.gca().xaxis.set_minor_locator(AutoMinorLocator()); plt.gca().yaxis.set_minor_locator(AutoMinorLocator()) # turn on minor ticks\n\nplt.subplots_adjust(left=0.0, bottom=0.0, right=2.0, top=1.2, wspace=0.2, hspace=0.2)\nplt.show()\n```\n\n#### Bootstrap of the Correlation Coefficient\n\nHere's a statistic that requires us to work with multiple, paired features at once. \n* this reinforces that we boostrap for a new realization of the dataset, a set of samples with all their features.\n\nFor the correlation coefficient we will:\n \n* calculate a bootstrap realization of the dataset with $n$ samples with replacement as a new DataFrame with all features\n* calculate the the correlation coefficient between 2 paired features\n\nRepeat this $L$ times and then evaluate the resulting distribution.\n\n\n```python\nL = 1000 # set the number of bootstrap realizations \ncorr_real = [] # declare an empty list to store the bootstrap realizations of the statistic \n\nfor k in range(0,L): # loop over the L bootstrap realizations\n samples = df.sample(n=len(df),replace=True,random_state = seed + k) # n random samples with replacement as a new DataFrame \n corr_real.append(samples[['Porosity','Perm']].corr()['Porosity'][1]) # calculate the statistic of interest from the new bootstrap dataset\n\ncorr = df[['Porosity','Perm']].corr()['Porosity'][1]\n\nplt.subplot(111)\nplt.hist(corr_real,color = 'darkorange',alpha = 0.8,edgecolor = 'black',linewidth=2,bins=np.linspace(0.5,1.0,100),label = 'bootstrap',density = True) # plot the distribution, could also calculate any summary statistics\nplt.axvline(x=corr,c='black')\nplt.axvline(x=np.percentile(corr_real,10),linestyle=\"--\",c='black')\nplt.axvline(x=np.percentile(corr_real,90),linestyle=\"--\",c='black')\nsns.kdeplot(x=corr_real,color = 'black',linewidth=2,alpha = 0.8,levels = 1,bw_adjust = 1)\n# plt.text(np.percentile(corr_real,90)+0.009, 16, r'Average = ' + str(round(np.average(corr_real),3)), fontsize=12)\n# plt.text(np.percentile(corr_real,90)+0.009, 15, r'St.Dev. = ' + str(round(np.std(corr_real),3)), fontsize=12)\n# plt.text(np.percentile(corr_real,90)+0.009, 14, r'P90 = ' + str(round(np.percentile(corr_real,90),3)), fontsize=12)\n# plt.text(np.percentile(corr_real,90)+0.009, 13, r'P10 = ' + str(round(np.percentile(corr_real,10),3)), fontsize=12)\nplt.xlabel('Boostrap Realizations of Correlation Coefficient'); plt.ylabel('Frequency'); plt.title('Bootstrap Uncertainty Distribution of Correlation Coefficient')\nplt.gca().grid(True, which='major',axis='both',linewidth = 1.0); plt.gca().grid(True, which='minor',axis='x',linewidth = 0.2) # add y grids\nplt.gca().tick_params(which='major',length=7); plt.gca().tick_params(which='minor', length=4)\nplt.gca().xaxis.set_minor_locator(AutoMinorLocator()); plt.gca().yaxis.set_minor_locator(AutoMinorLocator()) # turn on minor ticks\n\n\nplt.subplots_adjust(left=0.0, bottom=0.0, right=2.0, top=1.2, wspace=0.2, hspace=0.2)\nplt.show()\n```\n\n#### Comments\n\nThis was a basic demonstration of bootstrap. \n\n* you can use bootstrap to calculate uncertainty in any statistic!\n\n* you are calculating a set of realizations of the statistic, representing uncertainty due to small sample size\n\n* note the assumptions of bootstrap, including stationarity and representativity\n\n* remember, get a dataset realization by bootstrap and then calculate the realization of the statistic from the dataset realization\n\n* if your statistic has multiple inputs (e.g. P25 and P75), calculate each from the same bootstrap realization of the dataset.\n\nI have other demonstrations on the basics of working with DataFrames, ndarrays, univariate statistics, plotting data, declustering, data transformations, trend modeling and many other workflows available at https://github.com/GeostatsGuy/PythonNumericalDemos and https://github.com/GeostatsGuy/GeostatsPy. \n \nI hope this was helpful,\n\n*Michael*\n\n#### The Author:\n\n### Michael Pyrcz, Associate Professor, University of Texas at Austin \n*Novel Data Analytics, Geostatistics and Machine Learning Subsurface Solutions*\n\nWith over 17 years of experience in subsurface consulting, research and development, Michael has returned to academia driven by his passion for teaching and enthusiasm for enhancing engineers' and geoscientists' impact in subsurface resource development. \n\nFor more about Michael check out these links:\n\n#### [Twitter](https://twitter.com/geostatsguy) | [GitHub](https://github.com/GeostatsGuy) | [Website](http://michaelpyrcz.com) | [GoogleScholar](https://scholar.google.com/citations?user=QVZ20eQAAAAJ&hl=en&oi=ao) | [Book](https://www.amazon.com/Geostatistical-Reservoir-Modeling-Michael-Pyrcz/dp/0199731446) | [YouTube](https://www.youtube.com/channel/UCLqEr-xV-ceHdXXXrTId5ig) | [LinkedIn](https://www.linkedin.com/in/michael-pyrcz-61a648a1)\n\n#### Want to Work Together?\n\nI hope this content is helpful to those that want to learn more about subsurface modeling, data analytics and machine learning. Students and working professionals are welcome to participate.\n\n* Want to invite me to visit your company for training, mentoring, project review, workflow design and / or consulting? I'd be happy to drop by and work with you! \n\n* Interested in partnering, supporting my graduate student research or my Subsurface Data Analytics and Machine Learning consortium (co-PIs including Profs. Foster, Torres-Verdin and van Oort)? My research combines data analytics, stochastic modeling and machine learning theory with practice to develop novel methods and workflows to add value. We are solving challenging subsurface problems!\n\n* I can be reached at mpyrcz@austin.utexas.edu.\n\nI'm always happy to discuss,\n\n*Michael*\n\nMichael Pyrcz, Ph.D., P.Eng. Associate Professor The Hildebrand Department of Petroleum and Geosystems Engineering, Bureau of Economic Geology, The Jackson School of Geosciences, The University of Texas at Austin\n\n#### More Resources Available at: [Twitter](https://twitter.com/geostatsguy) | [GitHub](https://github.com/GeostatsGuy) | [Website](http://michaelpyrcz.com) | [GoogleScholar](https://scholar.google.com/citations?user=QVZ20eQAAAAJ&hl=en&oi=ao) | [Book](https://www.amazon.com/Geostatistical-Reservoir-Modeling-Michael-Pyrcz/dp/0199731446) | [YouTube](https://www.youtube.com/channel/UCLqEr-xV-ceHdXXXrTId5ig) | [LinkedIn](https://www.linkedin.com/in/michael-pyrcz-61a648a1)\n\n", "meta": {"hexsha": "9bc200211b076479bbac06fc982c82252f1af3d6", "size": 412844, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "PythonDataBasics_BootstrapConfidence.ipynb", "max_stars_repo_name": "caf3676/PythonNumericalDemos", "max_stars_repo_head_hexsha": "206a3d876f79e137af88b85ba98aff171e8d8e06", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "PythonDataBasics_BootstrapConfidence.ipynb", "max_issues_repo_name": "caf3676/PythonNumericalDemos", "max_issues_repo_head_hexsha": "206a3d876f79e137af88b85ba98aff171e8d8e06", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "PythonDataBasics_BootstrapConfidence.ipynb", "max_forks_repo_name": "caf3676/PythonNumericalDemos", "max_forks_repo_head_hexsha": "206a3d876f79e137af88b85ba98aff171e8d8e06", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-03-14T03:28:32.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-14T03:28:32.000Z", "avg_line_length": 409.5674603175, "max_line_length": 115468, "alphanum_fraction": 0.9215611708, "converted": true, "num_tokens": 10228, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.7853085859124002, "lm_q1q2_score": 0.43845898239823355}} {"text": "```python\nfrom IPython.core.display import HTML\nHTML(\"\")\n```\n\n\n\n\n\n\n\n\n# Lecture 5: Example of using available software: scipy.optimize\n\n# Optimization software\n* When we want to optimize something, we do not of course need to start everything from scratch. It is good to know how algorithms work, but if the development of new algorithms is not the main point, then one can just use packages and libraries that have been premade. \n* First, we have a look at some of the available software and, then, we have a closer look at scipy.optimize\n\n\n\n## Have you used any optimization software before? Please share your experiences.\n\n## Wolfram Alpha\n* Free web version of Mathematica\n* http://www.wolframalpha.com/\n* Can perform either symbolic or numerical calculations\n* Includes also some basic optimization\n\n## Rosenbrock function\nA non-convex function\n$$\nf(x) = (1-x_1)^2 +100(x_2-x_1^2)^2\n$$\nthat has a global minimum in $x^*=(1,1)^T$ where $f(x^*)=0$. The minimum is located in a narrow, banana-shaped valley.\n\nThe coefficient of the second term can be adjusted but it does not affect the position of the global minimum. The Rosenbrock function is used to test optimization algorithms.\n\n\n\n```python\ndef f_rosenb(x):\n return (1.0 -x[0])**2 + 100*(x[1] - x[0]**2)**2\n```\n\n\n\n\n\n## Matlab - Optimization toolbox\n* Interactive environment for numerical computing\n* Subroutines for unconstrained optimization:\n * fminbnd: find minimum of single-variable function on fixed interval\n * fminsearch: find minimum of unconstrained multivariable function using derivative-free method\n * fminunc: find minimum of unconstrained multivariable function using gradient-based method\n* https://se.mathworks.com/products/optimization.html?s_tid=srchtitle_optimization_1 \n* Matlab codes for the subroutines can be found in the directory where Matlab is installed:\n ..\\MATLAB\\R2013a\\toolbox\\optim\\optim\\\n* You can also use Octave (https://www.gnu.org/software/octave/) which is an open source software having compatibility with many Matlab scripts\n\n\n# Optimization with scipy.optimize\nIn Python, there are multiple packages for optimization. At this lecture, we are going to take a look at *scipy.optimize* package.\n\n## Starting up\n\nWhen we want to study a package in Python, we can import it..\n\n\n```python\nfrom scipy.optimize import minimize\n```\n\nIf we want to see the documentation, we can write the name of the package and two question marks and hit enter:\n\n\n```python\nminimize??\n```\n\nor\n\nhttps://docs.scipy.org/doc/scipy/tutorial/optimize.html\n\n## Optimization of multiple variables\n\nLet us define again our friendly objective function:\n\n\n```python\ndef f_simple(x):\n return (x[0] - 10.0)**2 + (x[1] + 5.0)**2+x[0]**2\n```\n\n### Method: `Nelder-Mead'\n\n* Uses a simplex which is a shape consisting $n+1$ vertices in n dimensional space. \n\n* E.g., a triangle in 2D, a tetrahedron in 3D, 5-cell pentachoron (https://en.wikipedia.org/wiki/5-cell). etc.\n\n* Begin with $n+1$ arbitrarily selected points in nD (3 in 2D) to form the simplex.\n* Follow the following steps in each iteration:\n * Sort the selected points ($f1\n Method :ref:`Nelder-Mead ` uses the\n Simplex algorithm [1]_, [2]_. This algorithm has been successful\n in many applications but other algorithms using the first and/or\n second derivatives information might be preferred for their better\n performances and robustness in general.\n...\n References\n ----------\n .. [1] Nelder, J A, and R Mead. 1965. A Simplex Method for Function\n Minimization. The Computer Journal 7: 308-13.\n .. [2] Wright M H. 1996. Direct search methods: Once scorned, now\n respectable, in Numerical Analysis 1995: Proceedings of the 1995\n Dundee Biennial Conference in Numerical Analysis (Eds. D F\n Griffiths and G A Watson). Addison Wesley Longman, Harlow, UK.\n 191-208.\n\n\n\n```python\nres = minimize(f_simple,[0,0],method='Nelder-Mead', \n options={'disp': True})\nprint(res.x)\nres = minimize(f_simple,[0,0],method='Powell', \n options={'disp': True})\nprint(res.x)\n```\n\n Optimization terminated successfully.\n Current function value: 50.000000\n Iterations: 99\n Function evaluations: 189\n [ 5.00003542 -4.99997315]\n Optimization terminated successfully.\n Current function value: 50.000000\n Iterations: 2\n Function evaluations: 99\n [ 5. -5.00000004]\n\n\n\n```python\nprint(type(res))\nprint(res)\nprint(res.message)\n```\n\n \n direc: array([[1., 0.],\n [0., 1.]])\n fun: 50.0\n message: 'Optimization terminated successfully.'\n nfev: 99\n nit: 2\n status: 0\n success: True\n x: array([ 5. , -5.00000004])\n Optimization terminated successfully.\n\n\n\n```python\nres = minimize(f_rosenb,[-2.0,-10],method='Nelder-Mead', \n options={'disp': True})\n```\n\n Optimization terminated successfully.\n Current function value: 0.000000\n Iterations: 155\n Function evaluations: 288\n\n\n\n```python\nprint(res)\n```\n\n final_simplex: (array([[1.00000755, 1.00001315],\n [1.00001942, 1.00004104],\n [0.99996779, 0.99993545]]), array([4.36964903e-10, 8.57224979e-10, 1.03919729e-09]))\n fun: 4.369649026757306e-10\n message: 'Optimization terminated successfully.'\n nfev: 288\n nit: 155\n status: 0\n success: True\n x: array([1.00000755, 1.00001315])\n\n\n### Method: Conjugate Gradient (`CG`)\n* Idea is to improve convergence properties of steepest descent\n* A search direction is a combination of the current search direction and a previous search direction\n* Steps in directions that are **orthogonal**\n* Memory consumption \ud835\udc42(\ud835\udc5b)\n* Best suited for large scale problems\n\n\n\nThe documentation has the following to say:\n
    \n    Method :ref:`CG ` uses a nonlinear conjugate\n    gradient algorithm by Polak and Ribiere, a variant of the\n    Fletcher-Reeves method described in [5]_ pp.  120-122. Only the\n    first derivatives are used.\n...\n   References\n    ----------\n...\n    .. [5] Nocedal, J, and S J Wright. 2006. Numerical Optimization.\n       Springer New York.\n
    \nThe Conjugate gradient method needs the gradient. The documentation has the following to say\n
    \n    jac : bool or callable, optional\n        Jacobian (gradient) of objective function. Only for CG, BFGS,\n        Newton-CG, L-BFGS-B, TNC, SLSQP, dogleg, trust-ncg.\n        If `jac` is a Boolean and is True, `fun` is assumed to return the\n        gradient along with the objective function. If False, the\n        gradient will be estimated numerically.\n        `jac` can also be a callable returning the gradient of the\n        objective. In this case, it must accept the same arguments as `fun`.\n
    \n\n### Estimating the gradient numerically:\n\n\n```python\nres = minimize(f_simple, [0,0], method='CG', #Conjugate gradient method\n options={'disp': True})\nprint(res.x)\n```\n\n Optimization terminated successfully.\n Current function value: 50.000000\n Iterations: 3\n Function evaluations: 21\n Gradient evaluations: 7\n [ 4.99999999 -5.00000006]\n\n\n### Giving the gradient with ad\n\n\n```python\nimport ad\nres = minimize(f_simple, [0,0], method='CG', #Conjugate gradient method\n options={'disp': True}, jac=ad.gh(f_simple)[0])\nprint(res.x)\n```\n\n Optimization terminated successfully.\n Current function value: 50.000000\n Iterations: 3\n Function evaluations: 7\n Gradient evaluations: 7\n [ 5. -5.]\n\n\n### Newton-Conjugate-Gradient algorithm (method=`Newton-CG`) \n\nNewton-CG method uses a Newton-CG algorithm [5] pp. 168 (also known as the truncated Newton method). It uses a CG method to the compute the search direction. See also *TNC* method for a box-constrained minimization with a similar algorithm.\n\n References\n ----------\n .. [5] Nocedal, J, and S J Wright. 2006. Numerical Optimization.\n Springer New York.\n\n\nThe Newton-CG algorithm needs the Jacobian and the Hessian. The documentation has the following to say:\n
    \n    hess, hessp : callable, optional\n        Hessian (matrix of second-order derivatives) of objective function or\n        Hessian of objective function times an arbitrary vector p.  Only for\n        Newton-CG, dogleg, trust-ncg.\n        Only one of `hessp` or `hess` needs to be given.  If `hess` is\n        provided, then `hessp` will be ignored.  If neither `hess` nor\n        `hessp` is provided, then the Hessian product will be approximated\n        using finite differences on `jac`. `hessp` must compute the Hessian\n        times an arbitrary vector.\n
    \n\n### Trying without reading the documentation\n\n\n```python\nimport numpy as np\nres = minimize(f_simple, [0,0], method='Newton-CG', #Newton-CG method\n options={'disp': True})\nprint(res.x)\n```\n\n### Giving the gradient\n\n\n```python\nimport ad\nres = minimize(f_simple, [0,0], method='Newton-CG', #Newton-CG method\n options={'disp': True},jac=ad.gh(f_simple)[0])\nprint(res.x)\n```\n\n Optimization terminated successfully.\n Current function value: 50.000000\n Iterations: 7\n Function evaluations: 7\n Gradient evaluations: 15\n Hessian evaluations: 0\n [ 4.99999999 -4.99999999]\n\n\n### Giving also the hessian\n\n\n```python\nimport ad\nres = minimize(f_simple, [0,0], method='Newton-CG', #Newton-CG method\n options={'disp': True},jac=ad.gh(f_simple)[0],\n hess=ad.gh(f_simple)[1])\nprint(res.x)\n```\n\n Optimization terminated successfully.\n Current function value: 50.000000\n Iterations: 7\n Function evaluations: 7\n Gradient evaluations: 7\n Hessian evaluations: 7\n [ 5. -5.]\n\n\n## Another example\n$$\n\\begin{align}\n\\min \\quad & (x_1-2)^4+(x_1 - 2x_2)^2\\\\\n\\text{s.t.}\\quad &x_1,x_2\\in\\mathbb R\n\\end{align} \n$$\nOptimal solution clearly is $x^*=(2,1)^T$\n\n\n```python\ndef f_simple2(x):\n return (x[0] - 2.0)**4 + (x[0] - 2.0*x[1])**2\n```\n\n\n```python\nimport numpy as np\nx0 = np.array([-96,-2000])\nres = minimize(f_simple2,x0,method='Nelder-Mead', \n options={'disp': True})\nprint(res.x)\nres = minimize(f_simple2,x0,method='Powell', \n options={'disp': True})\nprint(res.x)\nres = minimize(f_simple2, x0, method='CG', #Conjugate gradient method\n options={'disp': True}, jac=ad.gh(f_simple2)[0])\nprint(res.x)\nres = minimize(f_simple2, x0, method='Newton-CG', #Newton-CG method\n options={'disp': True},jac=ad.gh(f_simple2)[0],\n hess=ad.gh(f_simple2)[1])\nprint(res.x)\n```\n\n Optimization terminated successfully.\n Current function value: 0.000000\n Iterations: 79\n Function evaluations: 148\n [2.0000004 1.00000021]\n Optimization terminated successfully.\n Current function value: 0.000000\n Iterations: 29\n Function evaluations: 784\n [1.99999649 0.99999824]\n Optimization terminated successfully.\n Current function value: 0.000000\n Iterations: 14\n Function evaluations: 32\n Gradient evaluations: 32\n [2.00005457 1.00002728]\n Optimization terminated successfully.\n Current function value: 0.000000\n Iterations: 28\n Function evaluations: 30\n Gradient evaluations: 30\n Hessian evaluations: 28\n [1.99649493 0.99824742]\n\n\n\n```python\n# Add here optimization of the Rosenbrock function with gradient based algorithms\n```\n\n\n```python\nx0 = np.array([-2.0,-10])\nres = minimize(f_rosenb, x0, method='CG', #Conjugate gradient method\n options={'disp': True}, jac=ad.gh(f_rosenb)[0])\nprint(res.x)\nres = minimize(f_rosenb, x0, method='Newton-CG', #Newton-CG method\n options={'disp': True},jac=ad.gh(f_rosenb)[0],\n hess=ad.gh(f_rosenb)[1])\nprint(res.x)\n```\n\n Optimization terminated successfully.\n Current function value: 0.000000\n Iterations: 30\n Function evaluations: 71\n Gradient evaluations: 71\n [0.99999053 0.99998103]\n Optimization terminated successfully.\n Current function value: 0.000000\n Iterations: 31\n Function evaluations: 36\n Gradient evaluations: 36\n Hessian evaluations: 31\n [0.99999884 0.99999768]\n\n\n## Line search\n\n\n```python\ndef f_singlevar(x):\n return 2+(1-x)**2\n```\n\n\n```python\nfrom scipy.optimize import minimize_scalar\nminimize_scalar??\n```\n\n### Method: `Golden`\n\nThe documentation has the following to say:\n
    \n    Method :ref:`Golden ` uses the\n    golden section search technique. It uses analog of the bisection\n    method to decrease the bracketed interval. It is usually\n    preferable to use the *Brent* method.\n
    \n\n\n```python\nminimize_scalar(f_singlevar,method='golden',tol=0.00001)\n```\n\n\n\n\n fun: 2.0\n nfev: 30\n nit: 25\n success: True\n x: 1.0\n\n\n\n### Method: `Brent`\n\nThe documentation has the following to say about the Brent method:\n\n Method *Brent* uses Brent's algorithm to find a local minimum.\n The algorithm uses inverse parabolic interpolation when possible to\n speed up convergence of the golden section method.\n\n\n```python\nminimize_scalar(f_singlevar,method='brent')\n```\n\n\n\n\n fun: 2.0\n nfev: 8\n nit: 4\n success: True\n x: 0.99999998519\n\n\n", "meta": {"hexsha": "e62d136582ad9ca4d9724dee8ff8f45a18adc465", "size": 50874, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Lecture 5, scipy.optimize.ipynb", "max_stars_repo_name": "bshavazipour/TIES483-2022", "max_stars_repo_head_hexsha": "93dfabbfe1e953e5c5f83c44412963505ecf575a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Lecture 5, scipy.optimize.ipynb", "max_issues_repo_name": "bshavazipour/TIES483-2022", "max_issues_repo_head_hexsha": "93dfabbfe1e953e5c5f83c44412963505ecf575a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Lecture 5, scipy.optimize.ipynb", "max_forks_repo_name": "bshavazipour/TIES483-2022", "max_forks_repo_head_hexsha": "93dfabbfe1e953e5c5f83c44412963505ecf575a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-02-03T09:40:02.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-03T09:40:02.000Z", "avg_line_length": 27.6639477977, "max_line_length": 1320, "alphanum_fraction": 0.526673743, "converted": true, "num_tokens": 3693, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.7905303137346446, "lm_q1q2_score": 0.43832571062176084}} {"text": "# Paralelismo en Julia\n\n## 1. Introducci\u00f3n\n\nJulia proporciona distintos mecanismos para paralelizar c\u00f3digo. Algunas de las estrategias y desaf\u00edos para escribir algoritmos paralelos son los siguientes: \n\n* Estrategias de paralelismo\n * SIMD\n * Multi-hilo\n * Tareas\n * Multiproceso\n * Memoria compartida\n * Memoria distribuida\n * Programaci\u00f3n de GPU\n\n* Desaf\u00edos de la computaci\u00f3n paralela\n * Orden de ejecuci\u00f3n\n * ejecuci\u00f3n de fuera de orden de posibilidad\n * acceso y mutaci\u00f3n simult\u00e1neos\n * Acceso y movimiento de datos\n * C\u00f3digo de acceso y movimiento\n * Adaptaci\u00f3n adecuada de la estrategia de paralelismo a las capacidades de su m\u00e1quina\n * Hacer coincidir adecuadamente la estrategia de paralelismo con el problema en cuesti\u00f3n\n\n## \u00bfQu\u00e9 es lo que est\u00e1 sucediendo con nuestras computadoras?\n\n\n\n\n## Lo dif\u00edcil de la computaci\u00f3n paralela\n * No pensamos en paralelo\n * Aprendemos a escribir y razonar sobre programas en serie.\n * El deseo de paralelismo a menudo surge _despu\u00e9s_ de haber escrito su algoritmo (\u00a1y lo encontr\u00f3 demasiado lento!)\n\n## Resumen:\n * Las arquitecturas computacionales actuales nos empujan hacia la programaci\u00f3n paralela para un rendimiento m\u00e1ximo, \u00a1incluso si no estamos en un cl\u00faster!\n * Pero es dif\u00edcil dise\u00f1ar buenos algoritmos paralelos\n * Es dif\u00edcil de expresar y razonar sobre esos algoritmos.\n\n## 2. SIMD: El paralelismo que puede (a veces) suceder autom\u00e1ticamente\n\nSIMD: Instrucci\u00f3n \u00fanica, datos m\u00faltiples (Single Instruction Multiple Data)\n\n**Nota:** Tambi\u00e9n llamado confusamente vectorizaci\u00f3n\n\n### Arquitectura\n\nEn lugar de calcular cuatro sumas secuencialmente:\n\n\\begin{align}\nx_1 + y_1 &\\rightarrow z_1 \\\\\nx_2 + y_2 &\\rightarrow z_2 \\\\\nx_3 + y_3 &\\rightarrow z_3 \\\\\nx_4 + y_4 &\\rightarrow z_4\n\\end{align}\n\nProcesadores modernos tienen unidades de procesamiento vectorial que pueden hacer lo anterior a la vez:\n\n$$\n\\left(\\begin{array}{cc}\nx_1 \\\\\nx_2 \\\\\nx_3 \\\\\nx_4\n\\end{array}\\right)\n+\n\\left(\\begin{array}{cc}\ny_1 \\\\\ny_2 \\\\\ny_3 \\\\\ny_4\n\\end{array}\\right)\n\\rightarrow\n\\left(\\begin{array}{cc}\nz_1 \\\\\nz_2 \\\\\nz_3 \\\\\nz_4\n\\end{array}\\right)\n$$\n\n### \u00bfC\u00f3mo se logra?\n\n\n```julia\nusing BenchmarkTools\n```\n\n\n```julia\nA = rand(100_000)\nfunction simplesum(A)\n result = zero(eltype(A))\n for i in eachindex(A)\n @inbounds result += A[i]\n end\n return result\nend\n\nsimplesum(A)\n```\n\n\n\n\n 50154.62310354247\n\n\n\nComo muchos lenguajes de programaci\u00f3n modernos, Julia utiliza la verificaci\u00f3n de l\u00edmites ([_boundchecking_](https://docs.julialang.org/en/v1/devdocs/boundscheck/)) para garantizar la seguridad del programa al acceder a arreglos.\nEn bucles internos u otras situaciones cr\u00edticas de rendimiento, es posible que se desee omitir estas comprobaciones de l\u00edmites para mejorar el rendimiento en tiempo de ejecuci\u00f3n.\n\nEn consecuencia, Julia incluye la macro `@inbounds(...)` para decirle al compilador que omita dichas comprobaciones de l\u00edmites dentro del bloque dado.\n\n\n```julia\n@btime simplesum($A)\n```\n\n 143.232 \u03bcs (0 allocations: 0 bytes)\n\n\n\n\n\n 50154.62310354247\n\n\n\n\u00bfese tiempo es bueno?\n\n\n```julia\n@btime sum($A)\n```\n\n 17.611 \u03bcs (0 allocations: 0 bytes)\n\n\n\n\n\n 50154.62310354205\n\n\n\nDise\u00f1amos una funci\u00f3n m\u00e1s lenta que la suma predise\u00f1ada `sum()`, \u00a1y tambi\u00e9n estamos obteniendo una respuesta diferente! Veamos qu\u00e9 sucede con un flotante de 32 bits en lugar de uno de 64 bits. Cada elemento tiene la mitad del n\u00famero de bits, por lo que tambi\u00e9n permite duplicar la longitud (para que el n\u00famero total de bits procesados permanezca constante).\n\n\n```julia\nA32 = rand(Float32, length(A)*2)\n@btime simplesum($A32)\n@btime sum($A32)\n```\n\n 286.429 \u03bcs (0 allocations: 0 bytes)\n 19.587 \u03bcs (0 allocations: 0 bytes)\n\n\n\n\n\n 99875.3f0\n\n\n\n\u00a1Eso es aun peor! \u00bfQue est\u00e1 pasando aqui? \nEstamos viendo m\u00faltiples diferencias en el desempe\u00f1o: \u00bfquiz\u00e1s la suma incorporada de Julia est\u00e1 usando alg\u00fan paralelismo?\n\nIntentemos usar SIMD nosotros mismos:\n\n\n```julia\nfunction simdsum(A)\n result = zero(eltype(A))\n @simd for i in eachindex(A)\n @inbounds result += A[i]\n end\n return result\nend\n@btime simdsum($A)\n@btime simdsum($A32)\n```\n\n 16.293 \u03bcs (0 allocations: 0 bytes)\n 16.354 \u03bcs (0 allocations: 0 bytes)\n\n\n\n\n\n 99875.305f0\n\n\n\n\u00bfQu\u00e9 hizo y por qu\u00e9 no siempre usamos (usa Julia pues) `@simd` para cada bucle **for** autom\u00e1ticamente?\n\nVeamos los resultados:\n\n\n```julia\nsimplesum(A), simdsum(A), sum(A)\n```\n\n\n\n\n (50008.443227500014, 50008.44322750009, 50008.443227500095)\n\n\n\n\n```julia\nsimplesum(A32), simdsum(A32), sum(A32)\n```\n\n\n\n\n (99940.69f0, 99940.2f0, 99940.19f0)\n\n\n\n\u00bfPor qu\u00e9 no son iguales?\n\nSin `@simd`, Julia est\u00e1 haciendo _exactamente_ lo que le dijimos que hiciera: est\u00e1 tomando cada elemento de nuestro arreglo y lo agrega a una gran pila secuencialmente. Nuestra respuesta es m\u00e1s peque\u00f1a de lo que la \"suma\" incorporada de Julia cree que es: eso es porque, como la pila se hace m\u00e1s grande, comenzamos a perder las partes inferiores de cada elemento que estamos sumando, \u00a1y esas peque\u00f1as p\u00e9rdidas comienzan a acumularse!\n\nLa macro `@simd` le dice a Julia que puede reorganizar las adiciones de punto flotante -\nincluso si cambiara la respuesta. Dependiendo de su CPU, esto puede llevar a 2x o 4x\no incluso un paralelismo 8x. B\u00e1sicamente, Julia est\u00e1 calculando sumas independientes para\nlos \u00edndices pares y los \u00edndices impares simult\u00e1neamente:\n\n\\begin{align}\nodds &\\leftarrow 0 \\\\\nevens &\\leftarrow 0 \\\\\n\\text{loop}&\\ \\text{odd}\\ i: \\\\\n &\\left(\\begin{array}{cc}\nodds \\\\\nevens\n\\end{array}\\right)\n\\leftarrow\n\\left(\\begin{array}{cc}\nodds \\\\\nevens\n\\end{array}\\right)\n+\n\\left(\\begin{array}{cc}\nx_{i} \\\\\nx_{i+1}\n\\end{array}\\right) \\\\\ntotal &\\leftarrow evens + odds\n\\end{align}\n\n\nEn muchos casos, Julia puede y sabe que un bucle for puede ser vectorizado (SIMD-ed) y aprovechar\u00e1 esto por defecto.\n\n\n```julia\nB = rand(1:10, 100_000)\n@btime simplesum($B)\n@btime sum($B)\nB32 = rand(Int32(1):Int32(10), length(B)*2)\n@btime simplesum($B32)\n@btime simdsum($B32)\n```\n\n 16.291 \u03bcs (0 allocations: 0 bytes)\n 17.251 \u03bcs (0 allocations: 0 bytes)\n 16.338 \u03bcs (0 allocations: 0 bytes)\n 16.333 \u03bcs (0 allocations: 0 bytes)\n\n\n\n\n\n 1101303\n\n\n\n\u00bfC\u00f3mo inspeccionamos que se est\u00e1 vectorizando?\n\n\n```julia\n@code_llvm simdsum(A32)\n```\n\n \n ; @ In[25]:2 within `simdsum'\n define float @julia_simdsum_18131(%jl_value_t addrspace(10)* nonnull align 16 dereferenceable(40)) {\n top:\n ; @ In[25]:3 within `simdsum'\n ; \u250c @ simdloop.jl:69 within `macro expansion'\n ; \u2502\u250c @ abstractarray.jl:212 within `eachindex'\n ; \u2502\u2502\u250c @ abstractarray.jl:95 within `axes1'\n ; \u2502\u2502\u2502\u250c @ abstractarray.jl:75 within `axes'\n ; \u2502\u2502\u2502\u2502\u250c @ array.jl:155 within `size'\n %1 = addrspacecast %jl_value_t addrspace(10)* %0 to %jl_value_t addrspace(11)*\n %2 = bitcast %jl_value_t addrspace(11)* %1 to %jl_value_t addrspace(10)* addrspace(11)*\n %3 = getelementptr inbounds %jl_value_t addrspace(10)*, %jl_value_t addrspace(10)* addrspace(11)* %2, i64 3\n %4 = bitcast %jl_value_t addrspace(10)* addrspace(11)* %3 to i64 addrspace(11)*\n %5 = load i64, i64 addrspace(11)* %4, align 8\n ; \u2502\u2502\u2502\u2502\u2514\n ; \u2502\u2502\u2502\u2502\u250c @ tuple.jl:157 within `map'\n ; \u2502\u2502\u2502\u2502\u2502\u250c @ range.jl:320 within `OneTo' @ range.jl:311\n ; \u2502\u2502\u2502\u2502\u2502\u2502\u250c @ promotion.jl:409 within `max'\n %6 = icmp sgt i64 %5, 0\n %7 = select i1 %6, i64 %5, i64 0\n ; \u2502\u2514\u2514\u2514\u2514\u2514\u2514\n ; \u2502 @ simdloop.jl:72 within `macro expansion'\n br i1 %6, label %L13.lr.ph, label %L30\n \n L13.lr.ph: ; preds = %top\n %8 = bitcast %jl_value_t addrspace(11)* %1 to float addrspace(13)* addrspace(11)*\n %9 = load float addrspace(13)*, float addrspace(13)* addrspace(11)* %8, align 8\n ; \u2502 @ simdloop.jl:75 within `macro expansion'\n %min.iters.check = icmp ult i64 %7, 32\n br i1 %min.iters.check, label %scalar.ph, label %vector.ph\n \n vector.ph: ; preds = %L13.lr.ph\n %n.vec = and i64 %7, 9223372036854775776\n br label %vector.body\n \n vector.body: ; preds = %vector.body, %vector.ph\n ; \u2502 @ simdloop.jl:78 within `macro expansion'\n ; \u2502\u250c @ int.jl:53 within `+'\n %index = phi i64 [ 0, %vector.ph ], [ %index.next, %vector.body ]\n %vec.phi = phi <8 x float> [ zeroinitializer, %vector.ph ], [ %18, %vector.body ]\n %vec.phi10 = phi <8 x float> [ zeroinitializer, %vector.ph ], [ %19, %vector.body ]\n %vec.phi11 = phi <8 x float> [ zeroinitializer, %vector.ph ], [ %20, %vector.body ]\n %vec.phi12 = phi <8 x float> [ zeroinitializer, %vector.ph ], [ %21, %vector.body ]\n ; \u2502\u2514\n ; \u2502 @ simdloop.jl:77 within `macro expansion' @ In[25]:4\n ; \u2502\u250c @ array.jl:788 within `getindex'\n %10 = getelementptr inbounds float, float addrspace(13)* %9, i64 %index\n %11 = bitcast float addrspace(13)* %10 to <8 x float> addrspace(13)*\n %wide.load = load <8 x float>, <8 x float> addrspace(13)* %11, align 4\n %12 = getelementptr inbounds float, float addrspace(13)* %10, i64 8\n %13 = bitcast float addrspace(13)* %12 to <8 x float> addrspace(13)*\n %wide.load13 = load <8 x float>, <8 x float> addrspace(13)* %13, align 4\n %14 = getelementptr inbounds float, float addrspace(13)* %10, i64 16\n %15 = bitcast float addrspace(13)* %14 to <8 x float> addrspace(13)*\n %wide.load14 = load <8 x float>, <8 x float> addrspace(13)* %15, align 4\n %16 = getelementptr inbounds float, float addrspace(13)* %10, i64 24\n %17 = bitcast float addrspace(13)* %16 to <8 x float> addrspace(13)*\n %wide.load15 = load <8 x float>, <8 x float> addrspace(13)* %17, align 4\n ; \u2502\u2514\n ; \u2502\u250c @ float.jl:400 within `+'\n %18 = fadd fast <8 x float> %vec.phi, %wide.load\n %19 = fadd fast <8 x float> %vec.phi10, %wide.load13\n %20 = fadd fast <8 x float> %vec.phi11, %wide.load14\n %21 = fadd fast <8 x float> %vec.phi12, %wide.load15\n ; \u2502\u2514\n ; \u2502 @ simdloop.jl:78 within `macro expansion'\n ; \u2502\u250c @ int.jl:53 within `+'\n %index.next = add i64 %index, 32\n %22 = icmp eq i64 %index.next, %n.vec\n br i1 %22, label %middle.block, label %vector.body\n \n middle.block: ; preds = %vector.body\n ; \u2502\u2514\n ; \u2502 @ simdloop.jl:77 within `macro expansion' @ In[25]:4\n ; \u2502\u250c @ float.jl:400 within `+'\n %bin.rdx = fadd fast <8 x float> %19, %18\n %bin.rdx16 = fadd fast <8 x float> %20, %bin.rdx\n %bin.rdx17 = fadd fast <8 x float> %21, %bin.rdx16\n %rdx.shuf = shufflevector <8 x float> %bin.rdx17, <8 x float> undef, <8 x i32> \n %bin.rdx18 = fadd fast <8 x float> %bin.rdx17, %rdx.shuf\n %rdx.shuf19 = shufflevector <8 x float> %bin.rdx18, <8 x float> undef, <8 x i32> \n %bin.rdx20 = fadd fast <8 x float> %bin.rdx18, %rdx.shuf19\n %rdx.shuf21 = shufflevector <8 x float> %bin.rdx20, <8 x float> undef, <8 x i32> \n %bin.rdx22 = fadd fast <8 x float> %bin.rdx20, %rdx.shuf21\n %23 = extractelement <8 x float> %bin.rdx22, i32 0\n %cmp.n = icmp eq i64 %7, %n.vec\n ; \u2502\u2514\n ; \u2502 @ simdloop.jl:75 within `macro expansion'\n br i1 %cmp.n, label %L30, label %scalar.ph\n \n scalar.ph: ; preds = %middle.block, %L13.lr.ph\n %bc.resume.val = phi i64 [ %n.vec, %middle.block ], [ 0, %L13.lr.ph ]\n %bc.merge.rdx = phi float [ %23, %middle.block ], [ 0.000000e+00, %L13.lr.ph ]\n br label %L13\n \n L13: ; preds = %scalar.ph, %L13\n %value_phi16 = phi i64 [ %bc.resume.val, %scalar.ph ], [ %27, %L13 ]\n %value_phi5 = phi float [ %bc.merge.rdx, %scalar.ph ], [ %26, %L13 ]\n ; \u2502 @ simdloop.jl:77 within `macro expansion' @ In[25]:4\n ; \u2502\u250c @ array.jl:788 within `getindex'\n %24 = getelementptr inbounds float, float addrspace(13)* %9, i64 %value_phi16\n %25 = load float, float addrspace(13)* %24, align 4\n ; \u2502\u2514\n ; \u2502\u250c @ float.jl:400 within `+'\n %26 = fadd fast float %value_phi5, %25\n ; \u2502\u2514\n ; \u2502 @ simdloop.jl:78 within `macro expansion'\n ; \u2502\u250c @ int.jl:53 within `+'\n %27 = add nuw nsw i64 %value_phi16, 1\n ; \u2502\u2514\n ; \u2502 @ simdloop.jl:75 within `macro expansion'\n ; \u2502\u250c @ int.jl:49 within `<'\n %28 = icmp ult i64 %27, %7\n ; \u2502\u2514\n br i1 %28, label %L13, label %L30\n \n L30: ; preds = %L13, %middle.block, %top\n %value_phi2 = phi float [ 0.000000e+00, %top ], [ %26, %L13 ], [ %23, %middle.block ]\n ; \u2514\n ; @ In[25]:6 within `simdsum'\n ret float %value_phi2\n }\n\n\nEntonces, \u00bfcu\u00e1les son los desaf\u00edos?:\n\n- El mayor obst\u00e1culo es que tienes que convencer a Julia y LLVM de que puede usar instrucciones SIMD para tu algoritmo dado. Eso no siempre es posible.\n- Hay muchas limitaciones de lo que se puede y no se puede vectorizar\n- Es necesario pensar en las consecuencias de reordenar su algoritmo\n\n## Resumen\n\nSIMD:\n- Explota el paralelismo integrado en un procesador\n- Ideal para bucles internos peque\u00f1os (y estrechos)\n- A menudo ocurre autom\u00e1ticamente si tienes cuidado\n - Siga las [mejores pr\u00e1cticas de rendimiento](https://docs.julialang.org/en/v1/manual/performance-tips/)\n - Usa `@inbounds` para cualquier acceso a un arreglo\n - Evita ramas o llamadas a funciones\n- Dependiendo del procesador y los tipos involucrados, puede producir ganancias de 2-16x con una sobrecarga extraordinariamente peque\u00f1a\n\n\n\n```julia\n\n```\n", "meta": {"hexsha": "b40bf4de980391f03c19923c96d5be109f14fdb6", "size": 20874, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Curso Julia/1 Introduccion Julia/S8-OperacionesVectorizadas.ipynb", "max_stars_repo_name": "antonyayal/ProyectoFinalCursoJuliaAgo21", "max_stars_repo_head_hexsha": "79d199175ae9fbadb8bf90e685f34bc808efdff5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Curso Julia/1 Introduccion Julia/S8-OperacionesVectorizadas.ipynb", "max_issues_repo_name": "antonyayal/ProyectoFinalCursoJuliaAgo21", "max_issues_repo_head_hexsha": "79d199175ae9fbadb8bf90e685f34bc808efdff5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Curso Julia/1 Introduccion Julia/S8-OperacionesVectorizadas.ipynb", "max_forks_repo_name": "antonyayal/ProyectoFinalCursoJuliaAgo21", "max_forks_repo_head_hexsha": "79d199175ae9fbadb8bf90e685f34bc808efdff5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.820754717, "max_line_length": 444, "alphanum_fraction": 0.5324326914, "converted": true, "num_tokens": 4581, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.7905303087996143, "lm_q1q2_score": 0.4383257078854323}} {"text": "```python\nfrom jupyter_to_medium._latex import is_latex_cell, replicate_alignment\nfrom nbformat import read\n```\n\n\n```python\nnb = read('/Users/jamisonm/dev/jupyter_to_medium/tests/notebooks/Test Latex.ipynb', as_version=4)\n```\n\n\n```python\nmd = [x for x in nb['cells'] if x['cell_type'] == 'markdown']\n```\n\n\n```python\nlts = [x for x in md if is_latex_cell(x)]\n```\n\n\n```python\nl = lts[1]['source']\n```\n\n\n```python\ndef format_multiline_latex(latex: list, align_offset: int = 4) -> str:\n\n # we have a multi-liner - we want to:\n # - remove the new lines, we will replace later\n lt = [x.replace(\"\\n\", \"\") for x in latex]\n # - remove latex new line \\\\ at end of lines\n lt = [x[:-2] if x[-2:] == r\"\\\\\" else x for x in lt]\n # - remove '$$' at top and bottom of list\n lt = [x for x in lt if x != \"$$\"]\n # - remove '$$' even if other latex on that line\n if \"$$\" in lt[0]:\n lt[0] = lt[0][2:]\n if \"$$\" in lt[-1]:\n lt[-1] = lt[-1][:-2]\n # - check if latex contains \\begin and \\end statements\n has_beg_end = [\"\\\\begin\" in x or \"\\\\end\" in x for x in lt]\n # - if so then we need to handle as mpl can't render\n if max(has_beg_end):\n # check if '{align}' is in the brackets\n has_align = [\"align\" in x for x in lt]\n # if aligns are present\n if max(has_align):\n # remove \\begin{align } and \\end{align}\n lt = [x for x in lt if x != \"\\\\begin{align}\"]\n lt = [x for x in lt if x != \"\\\\end{align}\"]\n else:\n # remove \\begin{align } and \\end{align}\n lt = [x for x in lt if \"\\\\begin\" in x]\n lt = [x for x in lt if \"\\\\end\" in x]\n \n return lt\n```\n\n\n```python\nll = format_multiline_latex(l.split('\\n'))\n```\n\n\n```python\nll\n```\n\n\n\n\n ['r^l_T &= ln(\\\\frac{S_T}{S_0}) ',\n ' &= ln(\\\\frac{S_T}{S_{T-1}}) + ln(\\\\frac{S_{T-1}}{S_{T-2}}) + \\\\ldots + ln(\\\\frac{S_1}{S_{0}}) ',\n ' &= r^l_{T-1} + r^l_{T-2} + \\\\ldots + r^l_0']\n\n\n", "meta": {"hexsha": "1ec96f405a2940baa99fcc546b9eedef27fed5ba", "size": 3948, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tests/notebooks/Latex Support.ipynb", "max_stars_repo_name": "kefirbandi/jupyter_to_medium", "max_stars_repo_head_hexsha": "a1b7496d9d9f9a16890633a6650f5ebf9b52b41b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 17, "max_stars_repo_stars_event_min_datetime": "2020-05-10T05:28:46.000Z", "max_stars_repo_stars_event_max_datetime": "2020-05-23T14:20:01.000Z", "max_issues_repo_path": "tests/notebooks/Latex Support.ipynb", "max_issues_repo_name": "kefirbandi/jupyter_to_medium", "max_issues_repo_head_hexsha": "a1b7496d9d9f9a16890633a6650f5ebf9b52b41b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tests/notebooks/Latex Support.ipynb", "max_forks_repo_name": "kefirbandi/jupyter_to_medium", "max_forks_repo_head_hexsha": "a1b7496d9d9f9a16890633a6650f5ebf9b52b41b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2020-05-12T04:48:43.000Z", "max_forks_repo_forks_event_max_datetime": "2020-05-24T19:50:20.000Z", "avg_line_length": 26.32, "max_line_length": 124, "alphanum_fraction": 0.4627659574, "converted": true, "num_tokens": 653, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.7490872075132153, "lm_q1q2_score": 0.4382917897148576}} {"text": "###### Content under Creative Commons Attribution license CC-BY 4.0, code under MIT license (c)2014 L.A. Barba, C. Cooper, G.F. Forsyth, A. Krishnan.\n\n# Phugoid Motion\n\nWelcome to [**\"Practical Numerical Methods with Python!\"**](http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about) This course is a collaborative, online, open education project, where we aim to give a foundation in scientific computing. The focus is on numerical solution of problems modeled by ordinary and partial differential equations.\n\nThis IPython Notebook introduces the problem we'll be studying in the **first module** of the course: the _phugoid model of glider flight_. We'll start with some background, explaining the physics, and working out the mathematical model. \n\nFirst, we'll look at an idealized motion where there is no drag, resulting in a simple harmonic motion. We can plot some interesting trajectories that will pique your imagination. In the next notebook, you'll learn to numerically integrate the differential equation using Euler's method. But hang on ... first things first. \n\nThe term \"phugoid\" is used in aeronautics to refer to a motion pattern where an aircraft oscillates up and down \u2014nose-up and climb, then nose-down and descend\u2014 around an equilibrium trajectory. The aircraft oscillates in altitude, speed and pitch, with only small (neglected) variations in the angle of attack, as it repeatedly exchanges kinetic and potential energy.\n\nA low-amplitude phugoid motion can be just a nuisance, as the aircraft does not exceed the stall angle of attack and nothing bad happens. But the mode can also be unstable leading to a stall or even a loop!\n\nLook at this video showing a Cessna single-engine airplane in phugoid motion:\n\n\n```python\nfrom IPython.display import YouTubeVideo\nYouTubeVideo('ysdU4mnRYdM')\n```\n\n\n\n\n\n\n\n\n\n\nThat doesn't look too good! What's happening? \n\nIt can get a lot worse when an aircraft enters one of these modes that is unstable. For example, one of [NASA's Helios Solar Powered Aircraft](http://www.nasa.gov/centers/dryden/history/pastprojects/Helios/) prototype broke up in mid air due to extreme phugoid oscillations!\n\nHelios was a proof-of-concept solar electric-powered flying wing that broke the world altitude record for a non-rocket-powered aircraft in August 2001. But in June 26, 2003, it broke something else. The aircraft entered phugoid motion after encountering turbulence near the Hawaiian Island of Kauai. The high speed in the oscillatory movement exceeded the design limits, and it ended up wrecked in the Pacific Ocean. Luckily, the Helios was remotely operated, and nobody got hurt.\n\n## The physics of phugoids\n\nThe phugoid oscillation has the aircraft pitching up and down, as it decelerates and accelerates. The trajectory might look like a sinusoid, as in the figure below. The assumption is that the forward velocity of the aircraft, $v$, varies in such a way that the angle of attack remains (nearly) constant, which means that we can assume a constant lift coefficient.\n\n\n####Figure 1. Trajectory of an aircraft in phugoid motion.\n\nIn the descending portion of the trajectory, the aircraft's velocity increases as it proceeds from a peak to the minimum height\u2014gaining kinetic energy at the expense of potential energy. The contrary happens in the upward segment, as its velocity decreases there.\n\nWe measure the pitch angle (between the aircraft's longitudinal axis and the horizontal) as positive when the aircraft's nose is pointing up. In the portion of the trajectory below the center-line, where it curves upwards, the pitch angle $\\theta$ is increasing: $\\dot{\\theta}>0$. And where the trajectory curves down, the pitch angle is decreasing: $\\dot{\\theta}<0$, as shown in the figure.\n\nLet's remind ourselves of the forces affecting an aircraft in a downward glide. Look at the figure below: we show the flight path, the forces on the glider (no thrust), and the _glide angle_ or flight path angle, $\\gamma$, between the flight path and the horizontal.\n\n\n####Figure 2. Forces on a glider.\n\nThe force of lift, $L$ \u2014created by the airflow around the wings\u2014 is perpendicular to the trajectory, and the force of drag, $D$, is parallel to the trajectory. Both forces are expressed in terms of coefficients of lift and drag, $C_L$ and $C_D$, respectively, that depend on the wing design and _angle of attack_\u2014the angle between the wing chord and the flight path.\n\nIf you are not familiar with airplane aerodynamics, you might be getting confused with some terms here ... and all those angles! But be patient and look things up, if you need to. We're giving you a quick summary here.\n\nLift and drag are proportional to a surface area, $S$, and the dynamic pressure: $1/2 \\rho v^2$, where $\\rho$ is the density of air, and $v$ the forward velocity of the aircraft. The equations for lift and drag are:\n\n$$\\begin{eqnarray}\nL &=& C_L S \\times \\frac{1}{2} \\rho v^2 \\\\\nD &=& C_D S \\times \\frac{1}{2} \\rho v^2\n\\end{eqnarray}$$\n\nIf the glider were in equilibrium, the forces would balance each other. We can equate the forces in the directions perpendicular and parallel to the trajectory, as follows:\n\n$$\\begin{equation}\nL = W \\cos \\gamma \\quad \\text{and} \\quad D = W \\sin \\gamma\n\\end{equation}$$\n\nwhere $W$ repesents the weight of the glider.\n\nIn the figure, we've drawn the angle $\\gamma$ as the _glide angle_, formed between the direction of motion and the horizontal. We are not bothered with the _sign_ of the angle, because we draw a free-body diagram and take the direction of the forces into account in writing our balance equations. But later on, we will need to be careful with the sign of the angles. It can cause you a real headache to keep this straight, so be patient!\n\nIt looks like we've set this up to do a little bit of mathematics. Are you ready?\n\nBut before, a short glimpse of the history.\n\n## Lanchester's Aerodonetics\n\n\"Phugoid theory\" was first described by the British engineer Frederick W. Lanchester in _\"Aerodonetics\"_ (1909). This book is so old that it is now in the public domain, so you can actually download [from Google Books](http://books.google.com/books?id=6hxDAAAAIAAJ&dq=%22phugoid%20theory%20deals%20with%20the%20longitudinal%20stability%22&pg=PA37#v=onepage&q=%22phugoid%20theory%20deals%20with%20the%20longitudinal%20stability%22&f=false) a PDF file of a scan, or read it online. \n\nLanchester defines phugoid theory as the study of longitudinal stability of a flying machine (aerodone). He first considered the simplification where drag and moment of inertia are neglected. Then he included these effects, obtaining an equation of stability. In addition to describing many experiments by himself and others, Lanchester also reports on _\"numerical work ... done by the aid of an ordinary 25-cm slide rule.\"_ Go figure!\n\n### Ideal case of zero drag\n\nIn this section, we follow the derivation given by Milne-Thompson (1966), which we find a little bit easier than that of the original in \"Aerodonetics\"!\n\nAn aircraft flying in a steady, straight horizontal flight has a lift equal to its weight. The velocity in this condition is sometimes called _trim velocity_ (\"trim\" is what pilots do to set the controls to just stay in a steady flight). Let's use $v_t$ for the trim velocity, and from $L=W$ deduce that:\n\n$$\\begin{equation}\nW = C_L S \\times\\frac{1}{2} \\rho v_t^2\n\\end{equation}$$\n\nThe weight $W$ is constant for the aircraft, but the lift at any other flight condition depends on the flight speed, $v$. We can use the expression for the weight in terms of $v_t$ to obtain the ratio $L/W$ at any other flight velocity, as follows:\n\n$$\\begin{equation}\n\\frac{L}{W}= \\frac{v^2}{v_t^2}\n\\end{equation}$$\n\nImagine that the aircraft experienced a little upset, a wind gust, and it finds itself off the \"trim\" level, in a curved path with an instantaneous angle $\\theta$. In the sketch below, we exaggerate the curved trajectory of flight to help you visualize what we'll do next. The angle $\\theta$ (using the same name as Milne-Thompson) is between the _trajectory_ and the horizontal, positive up.\n\n\n####Figure 3. Curved trajectory of the aircraft going up.\n\nA balance of forces now has to take into account that our reference frame is moving with the aircraft, in a rotating frame: we have a centrifugal force. The balance in the direction of lift is thus:\n\n$$\\begin{equation}\nL- W \\cos \\theta = \\frac{W}{g} \\frac{v^2}{R}\n\\end{equation}$$\n\nwhere $R$ is the radius of curvature of the trajectory, and $g$ the acceleration of gravity. Recall that the centrifugal acceleration is $v^2/R$. Rearrange this by dividing the equation by the weight, and use the expression we found for $L/W$, above. The following equation results:\n\n$$\\begin{equation}\n\\frac{v^2}{v_t^2}-\\cos \\theta = \\frac{v^2}{g R}\n\\end{equation}$$\n\nRecall that we simplified the problem assuming that there is no friction, which means that the total energy is constant (the lift does no work). If $z$ represents the depth below a reference horizontal line, the energy per unit mass is (kinetic plus potential energy):\n\n$$\\begin{equation}\n\\frac{1}{2}v^2-g z = \\text{constant}\n\\end{equation}$$\n\nTo get rid of that pesky constant, we can choose the reference horizontal line at the level that makes the constant energy equal to zero, so $v^2 = 2 g z$. That helps us re-write the phugoid equation in terms of $z$ as follows:\n\n$$\\begin{equation}\n\\frac{z}{z_t}-\\cos \\theta = \\frac{2z}{R}\n\\end{equation}$$\n\nLet $ds$ represent a small arc-length of the trajectory. We can write \n\n$$\\begin{equation}\n\\frac{1}{R} = \\frac{d\\theta}{ds} \\quad \\text{and}\\quad \\sin\\theta = -\\frac{dz}{ds}\n\\end{equation}$$\n\nEmploying the chain rule of calculus,\n\n$$\\begin{equation}\n\\frac{1}{R} = \\frac{d\\theta}{ds} = \\frac{dz}{ds}\\frac{d\\theta}{dz} = -\\sin \\theta\\frac{d\\theta}{dz}\n\\end{equation}$$\n\nMultiply the phugoid equation by $\\frac{1}{2\\sqrt{z}}$ to get:\n\n$$\\begin{equation}\n\\frac{\\sqrt{z}}{2z_t} - \\frac{\\cos\\theta}{2\\sqrt{z}} = \\frac{\\sqrt{z}}{R}\n\\end{equation}$$\n\nSubstituting for $1/R$ on the right hand side and bringing the cosine term over to the right, we get:\n\n$$\\begin{equation}\n\\frac{\\sqrt{z}}{2z_t} = \\frac{\\cos \\theta}{2 \\sqrt{z}} - \\sqrt{z} \\sin \\theta \\frac{d\\theta}{dz}\n\\end{equation}$$\n\nThe right-hand-side is an exact derivative! We can rewrite it as:\n\n$$\\begin{equation}\n\\frac{d}{dz} \\left(\\sqrt{z}\\cos\\theta \\right) = \\frac{\\sqrt{z}}{2z_t}\n\\end{equation}$$\n\nIntegrating this equation, we add an arbitrary constant, chosen as $C\\sqrt{z_t}$ which (after dividing through by $\\sqrt{z}$) gives:\n\n$$\\begin{equation}\n\\cos \\theta = \\frac{1}{3}\\frac{z}{z_t} + C\\sqrt{\\frac{z_t}{z}}\n\\end{equation}$$\n\nTaking the derivative of both sides of equation (15) and applying the relations from equation (10) yields:\n\n$$\\begin{equation}\n\\frac{z_t}{R} = \\frac{1}{3} - \\frac{C}{2}\\sqrt{\\frac{z_t^3}{z^3}}\n\\end{equation}$$\n\nMake sure you have followed the derivation, and perhaps write it out on paper!\n\n## Phugoid Curves\n\nEquation (15) is non-linear, which usually means we are hard-pressed to write a clean expression for the variable of interest, $z$. In fact, Lanchester himself said that he was unable to _\"reduce this expression to a form suitable for co-ordinate plotting.\"_ If the great polymath couldn't do it, we can't either!\n\nBut Lanchester _was_ able to plot a suitable approximation of the phugoid flight path using what he called the \"trammel\" method. If you're interested in seeing how he did it, his explanation begins on page [48 of Aerodonetics](http://books.google.com/books?id=6hxDAAAAIAAJ&pg=PA49&lpg=PA48&dq=aerodonetics+the+use+of+the+trammel&source=bl&ots=lB6EVKYQuT&sig=aVE2kiDWZoWftaWczMIrcYftMOs&hl=en&sa=X&ei=gTD_U82fGYjzgwT3moGwCQ&ved=0CCAQ6AEwAA#v=onepage&q=aerodonetics%20the%20use%20of%20the%20trammel&f=false). It's a trip.\n\nLanchester used Equations (15) and (16) to solve for the constant $C$ and the radius of curvature $R$ and then iteratively plotted small arcs of the phugoid path. By hand.\n\nWe wrote a neat little code that duplicates the manual trammel method, but it might be a bit much for you to absorb in the first lesson. If you want to look it over, you are more than welcome to. If you are just starting with Python, skip it for the moment and we'll return to it at the end of this module. \n\n###Plotting the flight path\n\nAs we mentioned, we wrote a Python code to reproduce programmatically what Lanchester did graphically. Here's a neat feature of IPython Notebooks: you can run external programs with the magical keyword ... wait for it ... `run`. And the jargon of IPython _is_ to call this \"magic.\" In fact, there are a bunch of [magic functions](http://ipython.org/ipython-doc/dev/interactive/tutorial.html) that you will learn about. They will make you a happy camper.\n\nLet's do it:\n\n\n```python\n%run phugoid.py\n%matplotlib inline\n```\n\nThis code cell loaded our simulated-trammel code, `phugoid.py`. The code defined a function for you in the background, called `plot_flight_path`, taking three inputs: $z_t$, $z$ and $\\theta$. \n\nLook again at Equation (15), where we take the positive square root. There are several possibilities, depending on the value that the constant $C$ takes. \n\n* There are no physical solutions for $C>2/3$, because it would result in $\\cos\\theta>1$. \n\n* If $C=2/3$, then the solution is a horizontal straight line, because $\\cos\\theta=1$, $\\theta=0$ and $R=\\infty$.\n\n* Any value of $C$ for which $0 < C < \\frac{2}{3}$ will produce \"trochoidal\"-like paths. What does this look like? Let's use our custom function `plot_flight_path` to find out!\n\n\n```python\n#zt = 64, z = 16, theta=0\nplot_flight_path(64, 16, 0)\n```\n\nCool! Note that the plot title tells us what the calculated value of $C$ was for our input conditions. We have a value of $C$ between $0$ and $\\frac{2}{3}$ and our path is trochoidal, like we announced it would be.\n\n* For negative values of $C$, the resultant flight path consists of a series of loops. Let's try it out!\n\n\n```python\nplot_flight_path(75,10,numpy.pi)\n```\n\nYou can play around with the input values and see what kind of behavior results. Just note that any value of $C > \\frac{2}{3}$ will result in $\\cos \\theta > 1$, which doesn't exist. Python will probably throw a few errors if you hit that condition, but just try again!\n\n* The last case is $C = 0$. Take another look at Equation (16) and plug in $C = 0$, what should happen? It looks like it will just reduce to \n\n$$R = 3z_t$$\n\nIt's a constant radius of curvature! In fact, this solution is a series of semi-circles, with a cusp between them. One way to force $C = 0$ that we can figure out from Equation (15), is to make:\n\n\n$$z = 3z_t\\ \\ \\ ,\\ \\ \\ \\theta = 0$$\n\n\n```python\nplot_flight_path(16,48,0.)\n```\n\nThat looks an awful lot like a quarter circle. And what's the radius of the arc? It's $$r = 48 = 3z_t.$$\n\nWe can also get a semi-circle out of our simulated trammel by changing to another configuration where $C$ is (near) zero. Here's one example:\n\n\n```python\nplot_flight_path(64,16,-numpy.pi/2)\n```\n\nThat is so nice. We have reproduced the trajectories that Lanchester found more than a hundred years ago, painstakingly drawing them by hand with a contraption called a \"trammel.\" It must have taken him days!\n\nHere is how the different phugoid curves are drawn in von K\u00e1rm\u00e1n's book, _Aerodynamics_ (1957). He never says _how_ he drew them, but we're guessing by hand, also. We did pretty good!\n\n\n\n####Figure 4. Phugoid curves in von K\u00e1rm\u00e1n (1957).\n\nIn the next notebook of this series, we'll look at the differential equation that arises when you consider small perturbations on the horizontal phugoid, and we'll learn to numerically integrate that to get the flight paths.\n\n## References\n\n1. Lanchester, F. W. _Aerodonetics_, D. van Nostrand Company: New York, 1909. On the public domain. [Get it from Google Books](http://books.google.com/books?id=6hxDAAAAIAAJ&pg=PP1#v=onepage&q&f=false).\n\n2. Milne-Thompson, L. M. _Theoretical Aerodynamics_, Dover 2012 reprint of the revised 1966 edition. [Read on Google Books](http://books.google.com/books?id=EMfCAgAAQBAJ&lpg=PP1&pg=PP1#v=onepage&q&f=false) (see section 18.5)\n\n3. Sinha, N. K. and Ananthkrishnan, N. _Elementary Flight Dynamics with an introduction to Bifurcation and Continuation Methods_, CRC Press, 2013. [Read on Google Books](http://books.google.com/books?id=yXL6AQAAQBAJ&lpg=PP1&pg=PP1#v=onepage&q&f=false) (see chapter 5)\n\n4. von K\u00e1rm\u00e1n, T. _Aerodynamics_, Dover 2004 reprint of the 1957 2nd edition. (see pages 149\u2013151)\n\n## About this course\n\nThis course is a collaborative project in open education. Three professors across the world are teaching connected courses, developing and reviewing course materials, and interacting with the community of learners that follow the course online. They are:\n\n* Lorena A. Barba, the George Washington University, United States\n* Carlos Jerez, Pontificia Universidad Cat\u00f3lica de Chile\n* Ian Hawke, Southampton University, United Kingdom\n\n---\n\n######The cell below loads the style of this notebook. \n\n\n```python\n# Execute this cell to load the notebook's style sheet, then ignore it\nfrom IPython.core.display import HTML\ncss_file = '../../styles/numericalmoocstyle.css'\nHTML(open(css_file, \"r\").read())\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n", "meta": {"hexsha": "bd30a262c2fdc544daeba70a9d10d31ef339fefd", "size": 134536, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lessons/01_phugoid/01_01_Phugoid_Theory.ipynb", "max_stars_repo_name": "rbds/Numerical_Methods_working_folder", "max_stars_repo_head_hexsha": "d929ed7506054e7aa7ba059623c37ecf7d6ae993", "max_stars_repo_licenses": ["CC-BY-3.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "lessons/01_phugoid/01_01_Phugoid_Theory.ipynb", "max_issues_repo_name": "rbds/Numerical_Methods_working_folder", "max_issues_repo_head_hexsha": "d929ed7506054e7aa7ba059623c37ecf7d6ae993", "max_issues_repo_licenses": ["CC-BY-3.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "lessons/01_phugoid/01_01_Phugoid_Theory.ipynb", "max_forks_repo_name": "rbds/Numerical_Methods_working_folder", "max_forks_repo_head_hexsha": "d929ed7506054e7aa7ba059623c37ecf7d6ae993", "max_forks_repo_licenses": ["CC-BY-3.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 184.295890411, "max_line_length": 32254, "alphanum_fraction": 0.872539692, "converted": true, "num_tokens": 5587, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.8311430415844384, "lm_q1q2_score": 0.43827545911942506}} {"text": "# **Deutsch-Jozsa with Cirq**\n\n\n```python\npip install cirq\n```\n\n Collecting cirq\n Downloading cirq-0.13.1-py3-none-any.whl (7.7 kB)\n Collecting cirq-google==0.13.1\n Downloading cirq_google-0.13.1-py3-none-any.whl (437 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 437 kB 52 kB/s s eta 0:00:01\n \u001b[?25hCollecting cirq-aqt==0.13.1\n Downloading cirq_aqt-0.13.1-py3-none-any.whl (18 kB)\n Collecting cirq-pasqal==0.13.1\n Downloading cirq_pasqal-0.13.1-py3-none-any.whl (29 kB)\n Collecting cirq-web==0.13.1\n Downloading cirq_web-0.13.1-py3-none-any.whl (328 kB)\n \u001b[K 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rfc3986, pyasn1, idna, h11, rsa, pyasn1-modules, httpcore, certifi, cachetools, rfc3339, retrying, python-rapidjson, pyjwt, pydantic, py, msgpack, iso8601, httpx, googleapis-common-protos, google-auth, attrs, rpcq, retry, qcs-api-client, lark, grpcio, google-api-core, duet, pyquil, cirq-core, cirq-web, cirq-rigetti, cirq-pasqal, cirq-ionq, cirq-google, cirq-aqt, cirq\n Attempting uninstall: idna\n Found existing installation: idna 3.3\n Uninstalling idna-3.3:\n Successfully uninstalled idna-3.3\n Attempting uninstall: certifi\n Found existing installation: certifi 2021.10.8\n Uninstalling certifi-2021.10.8:\n Successfully uninstalled certifi-2021.10.8\n Attempting uninstall: pyjwt\n Found existing installation: PyJWT 2.3.0\n Uninstalling PyJWT-2.3.0:\n Successfully uninstalled PyJWT-2.3.0\n Attempting uninstall: msgpack\n Found existing installation: msgpack 1.0.3\n Uninstalling msgpack-1.0.3:\n Successfully uninstalled msgpack-1.0.3\n Attempting uninstall: attrs\n Found existing installation: attrs 21.4.0\n Uninstalling attrs-21.4.0:\n Successfully uninstalled attrs-21.4.0\n Successfully installed attrs-21.4.0 cachetools-4.2.4 certifi-2021.5.30 cirq-0.13.1 cirq-aqt-0.13.1 cirq-core-0.13.1 cirq-google-0.13.1 cirq-ionq-0.13.1 cirq-pasqal-0.13.1 cirq-rigetti-0.13.1 cirq-web-0.13.1 duet-0.2.3 google-api-core-1.31.5 google-auth-1.35.0 googleapis-common-protos-1.54.0 grpcio-1.43.0 h11-0.9.0 httpcore-0.11.1 httpx-0.15.5 idna-3.3 iso8601-0.1.16 lark-0.11.3 msgpack-0.6.2 py-1.11.0 pyasn1-0.4.8 pyasn1-modules-0.2.8 pydantic-1.8.2 pyjwt-1.7.1 pyquil-3.0.1 python-rapidjson-1.5 qcs-api-client-0.8.0 retry-0.9.2 retrying-1.3.3 rfc3339-6.2 rfc3986-1.5.0 rpcq-3.9.2 rsa-4.8\n Note: you may need to restart the kernel to use updated packages.\n\n\n\n```python\nimport cirq\nimport numpy as np\nimport matplotlib.pyplot as plt\n```\n\n :219: RuntimeWarning: scipy._lib.messagestream.MessageStream size changed, may indicate binary incompatibility. Expected 56 from C header, got 64 from PyObject\n\n\n\n```python\nprint(cirq.google.Bristlecone)\n```\n\n (0, 5)\u2500\u2500\u2500\u2500(0, 6)\n \u2502 \u2502\n \u2502 \u2502\n (1, 4)\u2500\u2500\u2500(1, 5)\u2500\u2500\u2500\u2500(1, 6)\u2500\u2500\u2500\u2500(1, 7)\n \u2502 \u2502 \u2502 \u2502\n \u2502 \u2502 \u2502 \u2502\n (2, 3)\u2500\u2500\u2500(2, 4)\u2500\u2500\u2500(2, 5)\u2500\u2500\u2500\u2500(2, 6)\u2500\u2500\u2500\u2500(2, 7)\u2500\u2500\u2500(2, 8)\n \u2502 \u2502 \u2502 \u2502 \u2502 \u2502\n \u2502 \u2502 \u2502 \u2502 \u2502 \u2502\n (3, 2)\u2500\u2500\u2500(3, 3)\u2500\u2500\u2500(3, 4)\u2500\u2500\u2500(3, 5)\u2500\u2500\u2500\u2500(3, 6)\u2500\u2500\u2500\u2500(3, 7)\u2500\u2500\u2500(3, 8)\u2500\u2500\u2500(3, 9)\n \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502\n \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502\n (4, 1)\u2500\u2500\u2500(4, 2)\u2500\u2500\u2500(4, 3)\u2500\u2500\u2500(4, 4)\u2500\u2500\u2500(4, 5)\u2500\u2500\u2500\u2500(4, 6)\u2500\u2500\u2500\u2500(4, 7)\u2500\u2500\u2500(4, 8)\u2500\u2500\u2500(4, 9)\u2500\u2500\u2500(4, 10)\n \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502\n \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502\n (5, 0)\u2500\u2500\u2500(5, 1)\u2500\u2500\u2500(5, 2)\u2500\u2500\u2500(5, 3)\u2500\u2500\u2500(5, 4)\u2500\u2500\u2500(5, 5)\u2500\u2500\u2500\u2500(5, 6)\u2500\u2500\u2500\u2500(5, 7)\u2500\u2500\u2500(5, 8)\u2500\u2500\u2500(5, 9)\u2500\u2500\u2500(5, 10)\u2500\u2500\u2500(5, 11)\n \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502\n \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502\n (6, 1)\u2500\u2500\u2500(6, 2)\u2500\u2500\u2500(6, 3)\u2500\u2500\u2500(6, 4)\u2500\u2500\u2500(6, 5)\u2500\u2500\u2500\u2500(6, 6)\u2500\u2500\u2500\u2500(6, 7)\u2500\u2500\u2500(6, 8)\u2500\u2500\u2500(6, 9)\u2500\u2500\u2500(6, 10)\n \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502\n \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502 \u2502\n (7, 2)\u2500\u2500\u2500(7, 3)\u2500\u2500\u2500(7, 4)\u2500\u2500\u2500(7, 5)\u2500\u2500\u2500\u2500(7, 6)\u2500\u2500\u2500\u2500(7, 7)\u2500\u2500\u2500(7, 8)\u2500\u2500\u2500(7, 9)\n \u2502 \u2502 \u2502 \u2502 \u2502 \u2502\n \u2502 \u2502 \u2502 \u2502 \u2502 \u2502\n (8, 3)\u2500\u2500\u2500(8, 4)\u2500\u2500\u2500(8, 5)\u2500\u2500\u2500\u2500(8, 6)\u2500\u2500\u2500\u2500(8, 7)\u2500\u2500\u2500(8, 8)\n \u2502 \u2502 \u2502 \u2502\n \u2502 \u2502 \u2502 \u2502\n (9, 4)\u2500\u2500\u2500(9, 5)\u2500\u2500\u2500\u2500(9, 6)\u2500\u2500\u2500\u2500(9, 7)\n \u2502 \u2502\n \u2502 \u2502\n (10, 5)\u2500\u2500\u2500(10, 6)\n\n\n /tmp/ipykernel_361/1631590671.py:1: DeprecationWarning: cirq.google was used but is deprecated.\n it will be removed in cirq v0.14.\n Use cirq_google instead.\n \n print(cirq.google.Bristlecone)\n\n\nThe following code gives the operations for these functions where we take two input qubits and compute the function in the third qubit.\n\n\n```python\nimport warnings\nwarnings.filterwarnings('ignore')\n\n\"\"\"Operations to query all possible functions on two bits.\nTwo of these functions are constant, and six of these functions are balanced.\n\"\"\"\n# Define three qubits to use.\nq0, q1, q2 = cirq.LineQubit.range(3)\n\n# Define the operations to query each of the two constant functions.\nconstant = (\n [], \n [cirq.X(q2)]\n)\n\n# Define the operations to query each of the six balanced functions.\nbalanced = (\n [cirq.CNOT(q0, q2)], \n [cirq.CNOT(q1, q2)], \n [cirq.CNOT(q0, q2), cirq.CNOT(q1, q2)],\n [cirq.CNOT(q0, q2), cirq.X(q2)], \n [cirq.CNOT(q1, q2), cirq.X(q2)], \n [cirq.CNOT(q0, q2), cirq.CNOT(q1, q2), cirq.X(q2)]\n)\n```\n\n\n```python\n# Define a function\n\ndef deJo_circuit(oracle):\n # Phase kickback trick.\n yield cirq.X(q2), cirq.H(q2)\n\n # Get an equal superposition over input bits.\n yield cirq.H(q0), cirq.H(q1)\n\n # Query the function.\n yield oracle\n\n # Use interference to get result, put last qubit into |1>.\n yield cirq.H(q0), cirq.H(q1), cirq.H(q2)\n\n # Use a final OR gate to put result in final qubit.\n yield cirq.X(q0), cirq.X(q1), cirq.CCX(q0, q1, q2)\n yield cirq.measure(q2)\n```\n\n\n```python\n\"\"\"Verify \"\"\"\nsimulator = cirq.Simulator()\n\nprint(\"\\nResult on constant functions:\")\nfor oracle in constant:\n result = simulator.run(cirq.Circuit(deJo_circuit(oracle)), repetitions=10)\n print(result)\n\nprint(\"\\nResult on balanced functions:\")\nfor oracle in balanced:\n result = simulator.run(cirq.Circuit(deJo_circuit(oracle)), repetitions=10)\n print(result)\n```\n\n \n Result on constant functions:\n 2=0000000000\n 2=0000000000\n \n Result on balanced functions:\n 2=1111111111\n 2=1111111111\n 2=1111111111\n 2=1111111111\n 2=1111111111\n 2=1111111111\n\n", "meta": {"hexsha": "44b0d64256958d79cd7cd46b815fbf32271df475", "size": 22980, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Project Files/ML Files/Deutsch-Jozsa with Cirq.ipynb", "max_stars_repo_name": "thirasit/Quantum-Chemistry-Learning-Project", "max_stars_repo_head_hexsha": "fb83f950ca916a9ab68bb6f845f06112afa90d91", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Project Files/ML Files/Deutsch-Jozsa with Cirq.ipynb", "max_issues_repo_name": "thirasit/Quantum-Chemistry-Learning-Project", "max_issues_repo_head_hexsha": "fb83f950ca916a9ab68bb6f845f06112afa90d91", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Project Files/ML Files/Deutsch-Jozsa with Cirq.ipynb", "max_forks_repo_name": "thirasit/Quantum-Chemistry-Learning-Project", "max_forks_repo_head_hexsha": "fb83f950ca916a9ab68bb6f845f06112afa90d91", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 55.9124087591, "max_line_length": 603, "alphanum_fraction": 0.529329852, "converted": true, "num_tokens": 6844, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.7690802476562641, "lm_q1q2_score": 0.4382624182771787}} {"text": "###### Content under Creative Commons Attribution license CC-BY 4.0, code under MIT license (c)2014 L.A. Barba, C. Cooper, G.F. Forsyth, A. Krishnan.\n\n# Phugoid Motion\n\nWelcome to [**\"Practical Numerical Methods with Python!\"**](http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about) This course is a collaborative, online, open education project, where we aim to give a foundation in scientific computing. The focus is on numerical solution of problems modeled by ordinary and partial differential equations.\n\nThis IPython Notebook introduces the problem we'll be studying in the **first module** of the course: the _phugoid model of glider flight_. We'll start with some background, explaining the physics, and working out the mathematical model. \n\nFirst, we'll look at an idealized motion where there is no drag, resulting in a simple harmonic motion. We can plot some interesting trajectories that will pique your imagination. In the next notebook, you'll learn to numerically integrate the differential equation using Euler's method. But hang on ... first things first. \n\nThe term \"phugoid\" is used in aeronautics to refer to a motion pattern where an aircraft oscillates up and down \u2014nose-up and climb, then nose-down and descend\u2014 around an equilibrium trajectory. The aircraft oscillates in altitude, speed and pitch, with only small (neglected) variations in the angle of attack, as it repeatedly exchanges kinetic and potential energy.\n\nA low-amplitude phugoid motion can be just a nuisance, as the aircraft does not exceed the stall angle of attack and nothing bad happens. But the mode can also be unstable leading to a stall or even a loop!\n\nLook at this video showing a Cessna single-engine airplane in phugoid motion:\n\n\n```python\nfrom IPython.display import YouTubeVideo\nYouTubeVideo('ysdU4mnRYdM')\n```\n\n\n\n\n\n\n\n\n\n\nThat doesn't look too good! What's happening? \n\nIt can get a lot worse when an aircraft enters one of these modes that is unstable. For example, one of [NASA's Helios Solar Powered Aircraft](http://www.nasa.gov/centers/dryden/history/pastprojects/Helios/) prototype broke up in mid air due to extreme phugoid oscillations!\n\nHelios was a proof-of-concept solar electric-powered flying wing that broke the world altitude record for a non-rocket-powered aircraft in August 2001. But in June 26, 2003, it broke something else. The aircraft entered phugoid motion after encountering turbulence near the Hawaiian Island of Kauai. The high speed in the oscillatory movement exceeded the design limits, and it ended up wrecked in the Pacific Ocean. Luckily, the Helios was remotely operated, and nobody got hurt.\n\n## The physics of phugoids\n\nThe phugoid oscillation has the aircraft pitching up and down, as it decelerates and accelerates. The trajectory might look like a sinusoid, as in the figure below. The assumption is that the forward velocity of the aircraft, $v$, varies in such a way that the angle of attack remains (nearly) constant, which means that we can assume a constant lift coefficient.\n\n\n####Figure 1. Trajectory of an aircraft in phugoid motion.\n\nIn the descending portion of the trajectory, the aircraft's velocity increases as it proceeds from a peak to the minimum height\u2014gaining kinetic energy at the expense of potential energy. The contrary happens in the upward segment, as its velocity decreases there.\n\nWe measure the pitch angle (between the aircraft's longitudinal axis and the horizontal) as positive when the aircraft's nose is pointing up. In the portion of the trajectory below the center-line, where it curves upwards, the pitch angle $\\theta$ is increasing: $\\dot{\\theta}>0$. And where the trajectory curves down, the pitch angle is decreasing: $\\dot{\\theta}<0$, as shown in the figure.\n\nLet's remind ourselves of the forces affecting an aircraft in a downward glide. Look at the figure below: we show the flight path, the forces on the glider (no thrust), and the _glide angle_ or flight path angle, $\\gamma$, between the flight path and the horizontal.\n\n\n####Figure 2. Forces on a glider.\n\nThe force of lift, $L$ \u2014created by the airflow around the wings\u2014 is perpendicular to the trajectory, and the force of drag, $D$, is parallel to the trajectory. Both forces are expressed in terms of coefficients of lift and drag, $C_L$ and $C_D$, respectively, that depend on the wing design and _angle of attack_\u2014the angle between the wing chord and the flight path.\n\nIf you are not familiar with airplane aerodynamics, you might be getting confused with some terms here ... and all those angles! But be patient and look things up, if you need to. We're giving you a quick summary here.\n\nLift and drag are proportional to a surface area, $S$, and the dynamic pressure: $1/2 \\rho v^2$, where $\\rho$ is the density of air, and $v$ the forward velocity of the aircraft. The equations for lift and drag are:\n\n$$\\begin{eqnarray}\nL &=& C_L S \\times \\frac{1}{2} \\rho v^2 \\\\\nD &=& C_D S \\times \\frac{1}{2} \\rho v^2\n\\end{eqnarray}$$\n\nIf the glider were in equilibrium, the forces would balance each other. We can equate the forces in the directions perpendicular and parallel to the trajectory, as follows:\n\n$$\\begin{equation}\nL = W \\cos \\gamma \\quad \\text{and} \\quad D = W \\sin \\gamma\n\\end{equation}$$\n\nwhere $W$ repesents the weight of the glider.\n\nIn the figure, we've drawn the angle $\\gamma$ as the _glide angle_, formed between the direction of motion and the horizontal. We are not bothered with the _sign_ of the angle, because we draw a free-body diagram and take the direction of the forces into account in writing our balance equations. But later on, we will need to be careful with the sign of the angles. It can cause you a real headache to keep this straight, so be patient!\n\nIt looks like we've set this up to do a little bit of mathematics. Are you ready?\n\nBut before, a short glimpse of the history.\n\n## Lanchester's Aerodonetics\n\n\"Phugoid theory\" was first described by the British engineer Frederick W. Lanchester in _\"Aerodonetics\"_ (1909). This book is so old that it is now in the public domain, so you can actually download [from Google Books](http://books.google.com/books?id=6hxDAAAAIAAJ&dq=%22phugoid%20theory%20deals%20with%20the%20longitudinal%20stability%22&pg=PA37#v=onepage&q=%22phugoid%20theory%20deals%20with%20the%20longitudinal%20stability%22&f=false) a PDF file of a scan, or read it online. \n\nLanchester defines phugoid theory as the study of longitudinal stability of a flying machine (aerodone). He first considered the simplification where drag and moment of inertia are neglected. Then he included these effects, obtaining an equation of stability. In addition to describing many experiments by himself and others, Lanchester also reports on _\"numerical work ... done by the aid of an ordinary 25-cm slide rule.\"_ Go figure!\n\n### Ideal case of zero drag\n\nIn this section, we follow the derivation given by Milne-Thompson (1966), which we find a little bit easier than that of the original in \"Aerodonetics\"!\n\nAn aircraft flying in a steady, straight horizontal flight has a lift equal to its weight. The velocity in this condition is sometimes called _trim velocity_ (\"trim\" is what pilots do to set the controls to just stay in a steady flight). Let's use $v_t$ for the trim velocity, and from $L=W$ deduce that:\n\n$$\\begin{equation}\nW = C_L S \\times\\frac{1}{2} \\rho v_t^2\n\\end{equation}$$\n\nThe weight $W$ is constant for the aircraft, but the lift at any other flight condition depends on the flight speed, $v$. We can use the expression for the weight in terms of $v_t$ to obtain the ratio $L/W$ at any other flight velocity, as follows:\n\n$$\\begin{equation}\n\\frac{L}{W}= \\frac{v^2}{v_t^2}\n\\end{equation}$$\n\nImagine that the aircraft experienced a little upset, a wind gust, and it finds itself off the \"trim\" level, in a curved path with an instantaneous angle $\\theta$. In the sketch below, we exaggerate the curved trajectory of flight to help you visualize what we'll do next. The angle $\\theta$ (using the same name as Milne-Thompson) is between the _trajectory_ and the horizontal, positive up.\n\n\n####Figure 3. Curved trajectory of the aircraft going up.\n\nA balance of forces now has to take into account that our reference frame is moving with the aircraft, in a rotating frame: we have a centrifugal force. The balance in the direction of lift is thus:\n\n$$\\begin{equation}\nL- W \\cos \\theta = \\frac{W}{g} \\frac{v^2}{R}\n\\end{equation}$$\n\nwhere $R$ is the radius of curvature of the trajectory, and $g$ the acceleration of gravity. Recall that the centrifugal acceleration is $v^2/R$. Rearrange this by dividing the equation by the weight, and use the expression we found for $L/W$, above. The following equation results:\n\n$$\\begin{equation}\n\\frac{v^2}{v_t^2}-\\cos \\theta = \\frac{v^2}{g R}\n\\end{equation}$$\n\nRecall that we simplified the problem assuming that there is no friction, which means that the total energy is constant (the lift does no work). If $z$ represents the depth below a reference horizontal line, the energy per unit mass is (kinetic plus potential energy):\n\n$$\\begin{equation}\n\\frac{1}{2}v^2-g z = \\text{constant}\n\\end{equation}$$\n\nTo get rid of that pesky constant, we can choose the reference horizontal line at the level that makes the constant energy equal to zero, so $v^2 = 2 g z$. That helps us re-write the phugoid equation in terms of $z$ as follows:\n\n$$\\begin{equation}\n\\frac{z}{z_t}-\\cos \\theta = \\frac{2z}{R}\n\\end{equation}$$\n\nLet $ds$ represent a small arc-length of the trajectory. We can write \n\n$$\\begin{equation}\n\\frac{1}{R} = \\frac{d\\theta}{ds} \\quad \\text{and}\\quad \\sin\\theta = -\\frac{dz}{ds}\n\\end{equation}$$\n\nEmploying the chain rule of calculus,\n\n$$\\begin{equation}\n\\frac{1}{R} = \\frac{d\\theta}{ds} = \\frac{dz}{ds}\\frac{d\\theta}{dz} = -\\sin \\theta\\frac{d\\theta}{dz}\n\\end{equation}$$\n\nMultiply the phugoid equation by $\\frac{1}{2\\sqrt{z}}$ to get:\n\n$$\\begin{equation}\n\\frac{\\sqrt{z}}{2z_t} - \\frac{\\cos\\theta}{2\\sqrt{z}} = \\frac{\\sqrt{z}}{R}\n\\end{equation}$$\n\nSubstituting for $1/R$ on the right hand side and bringing the cosine term over to the right, we get:\n\n$$\\begin{equation}\n\\frac{\\sqrt{z}}{2z_t} = \\frac{\\cos \\theta}{2 \\sqrt{z}} - \\sqrt{z} \\sin \\theta \\frac{d\\theta}{dz}\n\\end{equation}$$\n\nThe right-hand-side is an exact derivative! We can rewrite it as:\n\n$$\\begin{equation}\n\\frac{d}{dz} \\left(\\sqrt{z}\\cos\\theta \\right) = \\frac{\\sqrt{z}}{2z_t}\n\\end{equation}$$\n\nIntegrating this equation, we add an arbitrary constant, chosen as $C\\sqrt{z_t}$ which (after dividing through by $\\sqrt{z}$) gives:\n\n$$\\begin{equation}\n\\cos \\theta = \\frac{1}{3}\\frac{z}{z_t} + C\\sqrt{\\frac{z_t}{z}}\n\\end{equation}$$\n\nTaking the derivative of both sides of equation (15) and applying the relations from equation (10) yields:\n\n$$\\begin{equation}\n\\frac{z_t}{R} = \\frac{1}{3} - \\frac{C}{2}\\sqrt{\\frac{z_t^3}{z^3}}\n\\end{equation}$$\n\nMake sure you have followed the derivation, and perhaps write it out on paper!\n\n## Phugoid Curves\n\nEquation (15) is non-linear, which usually means we are hard-pressed to write a clean expression for the variable of interest, $z$. In fact, Lanchester himself said that he was unable to _\"reduce this expression to a form suitable for co-ordinate plotting.\"_ If the great polymath couldn't do it, we can't either!\n\nBut Lanchester _was_ able to plot a suitable approximation of the phugoid flight path using what he called the \"trammel\" method. If you're interested in seeing how he did it, his explanation begins on page [48 of Aerodonetics](http://books.google.com/books?id=6hxDAAAAIAAJ&pg=PA49&lpg=PA48&dq=aerodonetics+the+use+of+the+trammel&source=bl&ots=lB6EVKYQuT&sig=aVE2kiDWZoWftaWczMIrcYftMOs&hl=en&sa=X&ei=gTD_U82fGYjzgwT3moGwCQ&ved=0CCAQ6AEwAA#v=onepage&q=aerodonetics%20the%20use%20of%20the%20trammel&f=false). It's a trip.\n\nLanchester used Equations (15) and (16) to solve for the constant $C$ and the radius of curvature $R$ and then iteratively plotted small arcs of the phugoid path. By hand.\n\nWe wrote a neat little code that duplicates the manual trammel method, but it might be a bit much for you to absorb in the first lesson. If you want to look it over, you are more than welcome to. If you are just starting with Python, skip it for the moment and we'll return to it at the end of this module. \n\n###Plotting the flight path\n\nAs we mentioned, we wrote a Python code to reproduce programmatically what Lanchester did graphically. Here's a neat feature of IPython Notebooks: you can run external programs with the magical keyword ... wait for it ... `run`. And the jargon of IPython _is_ to call this \"magic.\" In fact, there are a bunch of [magic functions](http://ipython.org/ipython-doc/dev/interactive/tutorial.html) that you will learn about. They will make you a happy camper.\n\nLet's do it:\n\n\n```python\n%run phugoid.py\n%matplotlib inline\n```\n\nThis code cell loaded our simulated-trammel code, `phugoid.py`. The code defined a function for you in the background, called `plot_flight_path`, taking three inputs: $z_t$, $z$ and $\\theta$. \n\nLook again at Equation (15), where we take the positive square root. There are several possibilities, depending on the value that the constant $C$ takes. \n\n* There are no physical solutions for $C>2/3$, because it would result in $\\cos\\theta>1$. \n\n* If $C=2/3$, then the solution is a horizontal straight line, because $\\cos\\theta=1$, $\\theta=0$ and $R=\\infty$.\n\n* Any value of $C$ for which $0 < C < \\frac{2}{3}$ will produce \"trochoidal\"-like paths. What does this look like? Let's use our custom function `plot_flight_path` to find out!\n\n\n```python\n#zt = 64, z = 16, theta=0\nplot_flight_path(64, 16, 0)\n```\n\nCool! Note that the plot title tells us what the calculated value of $C$ was for our input conditions. We have a value of $C$ between $0$ and $\\frac{2}{3}$ and our path is trochoidal, like we announced it would be.\n\n* For negative values of $C$, the resultant flight path consists of a series of loops. Let's try it out!\n\n\n```python\nplot_flight_path(64,16,numpy.pi)\n```\n\nYou can play around with the input values and see what kind of behavior results. Just note that any value of $C > \\frac{2}{3}$ will result in $\\cos \\theta > 1$, which doesn't exist. Python will probably throw a few errors if you hit that condition, but just try again!\n\n* The last case is $C = 0$. Take another look at Equation (16) and plug in $C = 0$, what should happen? It looks like it will just reduce to \n\n$$R = 3z_t$$\n\nIt's a constant radius of curvature! In fact, this solution is a series of semi-circles, with a cusp between them. One way to force $C = 0$ that we can figure out from Equation (15), is to make:\n\n\n$$z = 3z_t\\ \\ \\ ,\\ \\ \\ \\theta = 0$$\n\n\n```python\nplot_flight_path(16,48,0.)\n```\n\nThat looks an awful lot like a quarter circle. And what's the radius of the arc? It's $$r = 48 = 3z_t.$$\n\nWe can also get a semi-circle out of our simulated trammel by changing to another configuration where $C$ is (near) zero. Here's one example:\n\n\n```python\nplot_flight_path(64,16,-numpy.pi/2)\n```\n\nThat is so nice. We have reproduced the trajectories that Lanchester found more than a hundred years ago, painstakingly drawing them by hand with a contraption called a \"trammel.\" It must have taken him days!\n\nHere is how the different phugoid curves are drawn in von K\u00e1rm\u00e1n's book, _Aerodynamics_ (1957). He never says _how_ he drew them, but we're guessing by hand, also. We did pretty good!\n\n\n\n####Figure 4. Phugoid curves in von K\u00e1rm\u00e1n (1957).\n\nIn the next notebook of this series, we'll look at the differential equation that arises when you consider small perturbations on the horizontal phugoid, and we'll learn to numerically integrate that to get the flight paths.\n\n## References\n\n1. Lanchester, F. W. _Aerodonetics_, D. van Nostrand Company: New York, 1909. On the public domain. [Get it from Google Books](http://books.google.com/books?id=6hxDAAAAIAAJ&pg=PP1#v=onepage&q&f=false).\n\n2. Milne-Thompson, L. M. _Theoretical Aerodynamics_, Dover 2012 reprint of the revised 1966 edition. [Read on Google Books](http://books.google.com/books?id=EMfCAgAAQBAJ&lpg=PP1&pg=PP1#v=onepage&q&f=false) (see section 18.5)\n\n3. Sinha, N. K. and Ananthkrishnan, N. _Elementary Flight Dynamics with an introduction to Bifurcation and Continuation Methods_, CRC Press, 2013. [Read on Google Books](http://books.google.com/books?id=yXL6AQAAQBAJ&lpg=PP1&pg=PP1#v=onepage&q&f=false) (see chapter 5)\n\n4. von K\u00e1rm\u00e1n, T. _Aerodynamics_, Dover 2004 reprint of the 1957 2nd edition. (see pages 149\u2013151)\n\n## About this course\n\nThis course is a collaborative project in open education. Three professors across the world are teaching connected courses, developing and reviewing course materials, and interacting with the community of learners that follow the course online. They are:\n\n* Lorena A. Barba, the George Washington University, United States\n* Carlos Jerez, Pontificia Universidad Cat\u00f3lica de Chile\n* Ian Hawke, Southampton University, United Kingdom\n\n---\n\n######The cell below loads the style of this notebook. \n\n\n```python\n# Execute this cell to load the notebook's style sheet, then ignore it\nfrom IPython.core.display import HTML\ncss_file = '../../styles/numericalmoocstyle.css'\nHTML(open(css_file, \"r\").read())\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n", "meta": {"hexsha": "37b5652a86a2e32eb1f74ed475baaf278c188ede", "size": 140276, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lessons/01_phugoid/01_01_Phugoid_Theory.ipynb", "max_stars_repo_name": "nbatista0630/batista--numerical-mooc", "max_stars_repo_head_hexsha": "fa3ae1bc698bbe66914300f424bde5a4a8625d51", "max_stars_repo_licenses": ["CC-BY-3.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "lessons/01_phugoid/01_01_Phugoid_Theory.ipynb", "max_issues_repo_name": "nbatista0630/batista--numerical-mooc", "max_issues_repo_head_hexsha": "fa3ae1bc698bbe66914300f424bde5a4a8625d51", "max_issues_repo_licenses": ["CC-BY-3.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2015-10-05T23:25:30.000Z", "max_issues_repo_issues_event_max_datetime": "2015-10-14T00:59:36.000Z", "max_forks_repo_path": "lessons/01_phugoid/01_01_Phugoid_Theory.ipynb", "max_forks_repo_name": "nbatista0630/batista--numerical-mooc", "max_forks_repo_head_hexsha": "fa3ae1bc698bbe66914300f424bde5a4a8625d51", "max_forks_repo_licenses": ["CC-BY-3.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 71.169964485, "max_line_length": 532, "alphanum_fraction": 0.7843964755, "converted": true, "num_tokens": 5587, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5156199157230156, "lm_q2_score": 0.8499711737573762, "lm_q1q2_score": 0.43826206497977094}} {"text": "# \u0412\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0432 \u0447\u0438\u0441\u043b\u0435\u043d\u043d\u044b\u0435 \u043c\u0435\u0442\u043e\u0434\u044b \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438 (\u042e. \u0415. \u041d\u0435\u0441\u0442\u0435\u0440\u043e\u0432 \u0412\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0432 \u0432\u044b\u043f\u0443\u043a\u043b\u0443\u044e \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u044e, \u0433\u043b. 1 $\\S$ 1.1)\n\n 1. \u041e\u0431\u0437\u043e\u0440 \u043c\u0430\u0442\u0435\u0440\u0438\u0430\u043b\u0430 \u0432\u0435\u0441\u0435\u043d\u043d\u0435\u0433\u043e \u0441\u0435\u043c\u0435\u0441\u0442\u0440\u0430\n 2. \u041f\u043e\u0441\u0442\u0430\u043d\u043e\u0432\u043a\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\n 3. \u041e\u0431\u0449\u0430\u044f \u0441\u0445\u0435\u043c\u0430 \u0440\u0435\u0448\u0435\u043d\u0438\u044f\n 4. \u0421\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435 \u043c\u0435\u0442\u043e\u0434\u043e\u0432 \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438\n 5. \u041c\u0435\u0442\u043e\u0434\u044b \u043e\u0434\u043d\u043e\u043c\u0435\u0440\u043d\u043e\u0439 \u043c\u0438\u043d\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438\n\n\n## \u041e\u0431\u0437\u043e\u0440 \u043c\u0430\u0442\u0435\u0440\u0438\u0430\u043b\u0430 \u0432\u0435\u0441\u0435\u043d\u043d\u0435\u0433\u043e \u0441\u0435\u043c\u0435\u0441\u0442\u0440\u0430\n\n\u0422\u0430\u043a\u0436\u0435 \u043d\u0430 [\u0441\u0442\u0440\u0430\u043d\u0438\u0446\u0435 \u043a\u0443\u0440\u0441\u0430](https://github.com/amkatrutsa/MIPT-Opt#spring-term).\n\n1. \u041c\u0435\u0442\u043e\u0434\u044b \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u0437\u0430\u0434\u0430\u0447 **\u0431\u0435\u0437\u0443\u0441\u043b\u043e\u0432\u043d\u043e\u0439** \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438\n - \u041e\u0434\u043d\u043e\u043c\u0435\u0440\u043d\u0430\u044f \u043c\u0438\u043d\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u044f (**\u0443\u0436\u0435 \u0441\u0435\u0433\u043e\u0434\u043d\u044f!**)\n - \u0413\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043d\u044b\u0439 \u0441\u043f\u0443\u0441\u043a\n - \u041c\u0435\u0442\u043e\u0434 \u041d\u044c\u044e\u0442\u043e\u043d\u0430\n - \u041a\u0432\u0430\u0437\u0438\u043d\u044c\u044e\u0442\u043e\u043d\u043e\u0432\u0441\u043a\u0438\u0435 \u043c\u0435\u0442\u043e\u0434\u044b\n - \u041c\u0435\u0442\u043e\u0434 \u0441\u043e\u043f\u0440\u044f\u0436\u0451\u043d\u043d\u044b\u0445 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043e\u0432 \n - \u041e\u043f\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u043e:\n - \u0420\u0435\u0448\u0435\u043d\u0438\u0435 \u0437\u0430\u0434\u0430\u0447\u0438 \u043d\u0430\u0438\u043c\u0435\u043d\u044c\u0448\u0438\u0445 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043e\u0432\n - \u041e\u043f\u0442\u0438\u043c\u0430\u043b\u044c\u043d\u044b\u0435 \u043c\u0435\u0442\u043e\u0434\u044b \u0438 \u043d\u0438\u0436\u043d\u0438\u0435 \u043e\u0446\u0435\u043d\u043a\u0438\n2. \u041c\u0435\u0442\u043e\u0434\u044b \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u0437\u0430\u0434\u0430\u0447 **\u0443\u0441\u043b\u043e\u0432\u043d\u043e\u0439** \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438\n - \u041c\u0435\u0442\u043e\u0434\u044b \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u0438 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u0430 \u0438 \u0443\u0441\u043b\u043e\u0432\u043d\u043e\u0433\u043e \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u0430\n - \u041c\u0435\u0442\u043e\u0434\u044b \u0448\u0442\u0440\u0430\u0444\u043d\u044b\u0445 \u0438 \u0431\u0430\u0440\u044c\u0435\u0440\u043d\u044b\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439\n - \u041c\u0435\u0442\u043e\u0434 \u043c\u043e\u0434\u0438\u0444\u0438\u0446\u0438\u0440\u043e\u0432\u0430\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u041b\u0430\u0433\u0440\u0430\u043d\u0436\u0430\n\n## \u041e\u0440\u0433\u0430\u043d\u0438\u0437\u0430\u0446\u0438\u044f \u0440\u0430\u0431\u043e\u0442\u044b \u0432 \u0441\u0435\u043c\u0435\u0441\u0442\u0440\u0435\n\n1. \u0421\u0435\u043c\u0438\u043d\u0430\u0440 \u0438 \u043b\u0435\u043a\u0446\u0438\u044f \u0440\u0430\u0437 \u0432 \u043d\u0435\u0434\u0435\u043b\u044e\n2. \u0414\u0432\u0430 \u0437\u0430\u0434\u0430\u043d\u0438\u044f \u0432 \u0442\u0435\u0447\u0435\u043d\u0438\u0435 \u0441\u0435\u043c\u0435\u0441\u0442\u0440\u0430\n3. Midterm \u0432 \u0441\u0435\u0440\u0435\u0434\u0438\u043d\u0435 \u0441\u0435\u043c\u0435\u0441\u0442\u0440\u0430\n4. \u0418\u0442\u043e\u0433\u043e\u0432\u0430\u044f \u043a\u043e\u043d\u0442\u0440\u043e\u043b\u044c\u043d\u0430\u044f \u0432 \u043a\u043e\u043d\u0446\u0435 \u0441\u0435\u043c\u0435\u0441\u0442\u0440\u0430\n5. \u042d\u043a\u0437\u0430\u043c\u0435\u043d \u0432 \u043a\u043e\u043d\u0446\u0435 \u0441\u0435\u043c\u0435\u0441\u0442\u0440\u0430 (\u0441\u0445\u0435\u043c\u0430 \u0432\u044b\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u043e\u0446\u0435\u043d\u043a\u0438 \u0430\u043d\u0430\u043b\u043e\u0433\u0438\u0447\u043d\u0430 \u043e\u0441\u0435\u043d\u043d\u0435\u043c\u0443 \u0441\u0435\u043c\u0435\u0441\u0442\u0440\u0443)\n6. \u041c\u0438\u043d\u0438\u043a\u043e\u043d\u0442\u0440\u043e\u043b\u044c\u043d\u044b\u0435 \u0432 \u043d\u0430\u0447\u0430\u043b\u0435 \u043a\u0430\u0436\u0434\u043e\u0433\u043e \u0441\u0435\u043c\u0438\u043d\u0430\u0440\u0430\n7. \u0414\u043e\u043c\u0430\u0448\u043d\u0435\u0435 \u0437\u0430\u0434\u0430\u043d\u0438\u0435 \u043f\u043e\u0447\u0442\u0438 \u043a\u0430\u0436\u0434\u0443\u044e \u043d\u0435\u0434\u0435\u043b\u044e: $\\TeX$ \u0438\u043b\u0438 Jupyter Notebook\n\n## \u041f\u043e\u0441\u0442\u0430\u043d\u043e\u0432\u043a\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\n\n\\begin{equation}\n\\begin{split}\n& \\min_{x \\in S} f_0(x)\\\\\n\\text{s.t. } & f_j(x) = 0, \\; j = 1,\\ldots,m\\\\\n& g_k(x) \\leq 0, \\; k = 1,\\ldots,p\n\\end{split}\n\\end{equation}\n\u0433\u0434\u0435 $S \\subseteq \\mathbb{R}^n$, $f_j: S \\rightarrow \\mathbb{R}, \\; j = 0,\\ldots,m$, $g_k: S \\rightarrow \\mathbb{R}, \\; k=1,\\ldots,p$\n\n\u0412\u0441\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043a\u0430\u043a \u043c\u0438\u043d\u0438\u043c\u0443\u043c \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u044b. \n\n\u0412\u0430\u0436\u043d\u044b\u0439 \u0444\u0430\u043a\u0442: \u0437\u0430\u0434\u0430\u0447\u0438 **\u043d\u0435\u043b\u0438\u043d\u0435\u0439\u043d\u043e\u0439** \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438 \n\n\u0432 \u0438\u0445 \u0441\u0430\u043c\u043e\u0439 \u043e\u0431\u0449\u0435\u0439 \u0444\u043e\u0440\u043c\u0435 \u044f\u0432\u043b\u044f\u044e\u0442\u0441\u044f **\u0447\u0438\u0441\u043b\u0435\u043d\u043d\u043e \u043d\u0435\u0440\u0430\u0437\u0440\u0435\u0448\u0438\u043c\u044b\u043c\u0438**!\n\n## \u0410\u043d\u0430\u043b\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u044b\n- \u041d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e\u0435 \u0443\u0441\u043b\u043e\u0432\u0438\u0435 \u043f\u0435\u0440\u0432\u043e\u0433\u043e \u043f\u043e\u0440\u044f\u0434\u043a\u0430: \n\n\u0435\u0441\u043b\u0438 $x^*$ \u0442\u043e\u0447\u043a\u0430 \u043b\u043e\u043a\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043c\u0438\u043d\u0438\u043c\u0443\u043c\u0430 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 $f(x)$, \u0442\u043e\u0433\u0434\u0430 \n$$\nf'(x^*) = 0\n$$\n- \u041d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e\u0435 \u0443\u0441\u043b\u043e\u0432\u0438\u0435 \u0432\u0442\u043e\u0440\u043e\u0433\u043e \u043f\u043e\u0440\u044f\u0434\u043a\u0430 \n\n\u0435\u0441\u043b\u0438 $x^*$ \u0442\u043e\u0447\u043a\u0430 \u043b\u043e\u043a\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043c\u0438\u043d\u0438\u043c\u0443\u043c\u0430 \u0434\u0432\u0430\u0436\u0434\u044b \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 $f(x)$, \u0442\u043e\u0433\u0434\u0430 \n\n$$\nf'(x^*) = 0 \\quad \\text{\u0438} \\quad f''(x^*) \\succeq 0\n$$\n- \u0414\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e\u0435 \u0443\u0441\u043b\u043e\u0432\u0438\u0435: \n\n\u043f\u0443\u0441\u0442\u044c $f(x)$ \u0434\u0432\u0430\u0436\u0434\u044b \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f, \u0438 \u043f\u0443\u0441\u0442\u044c \u0442\u043e\u0447\u043a\u0430 $x^*$ \u0443\u0434\u043e\u0432\u043b\u0435\u0442\u0432\u043e\u0440\u044f\u0435\u0442 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u043c\n\n$$\nf'(x^*) = 0 \\quad f''(x^*) \\succ 0,\n$$\n\u0442\u043e\u0433\u0434\u0430 $x^*$ \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0442\u043e\u0447\u043a\u043e\u0439 \u0441\u0442\u0440\u043e\u0433\u043e \u043b\u043e\u043a\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043c\u0438\u043d\u0438\u043c\u0443\u043c\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 $f(x)$.\n\n**\u0417\u0430\u043c\u0435\u0447\u0430\u043d\u0438\u0435**: \u0443\u0431\u0435\u0434\u0438\u0442\u0435\u0441\u044c, \u0447\u0442\u043e \u0412\u044b \u043f\u043e\u043d\u0438\u043c\u0430\u0435\u0442\u0435, \u043a\u0430\u043a \u0434\u043e\u043a\u0430\u0437\u044b\u0432\u0430\u0442\u044c \u044d\u0442\u0438\n\n\u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u044b!\n\n## \u041e\u0441\u043e\u0431\u0435\u043d\u043d\u043e\u0441\u0442\u0438 \u0447\u0438\u0441\u043b\u0435\u043d\u043d\u043e\u0433\u043e \u0440\u0435\u0448\u0435\u043d\u0438\u044f\n\n1. \u0422\u043e\u0447\u043d\u043e \u0440\u0435\u0448\u0438\u0442\u044c \u0437\u0430\u0434\u0430\u0447\u0443 \u043f\u0440\u0438\u043d\u0446\u0438\u043f\u0438\u0430\u043b\u044c\u043d\u043e \u043d\u0435\u0432\u043e\u0437\u043c\u043e\u0436\u043d\u043e \u0438\u0437-\u0437\u0430 \u043f\u043e\u0433\u0440\u0435\u0448\u043d\u043e\u0441\u0442\u0438 \u043c\u0430\u0448\u0438\u043d\u043d\u043e\u0439 \u0430\u0440\u0438\u0444\u043c\u0435\u0442\u0438\u043a\u0438\n2. \u041d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e \u0437\u0430\u0434\u0430\u0442\u044c \u043a\u0440\u0438\u0442\u0435\u0440\u0438\u0439 \u043e\u0431\u043d\u0430\u0440\u0443\u0436\u0435\u043d\u0438\u044f \u0440\u0435\u0448\u0435\u043d\u0438\u044f\n3. \u041d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u0442\u044c, \u043a\u0430\u043a\u0443\u044e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u044e \u043e \u0437\u0430\u0434\u0430\u0447\u0435 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c\n\n## \u041e\u0431\u0449\u0430\u044f \u0438\u0442\u0435\u0440\u0430\u0442\u0438\u0432\u043d\u0430\u044f \u0441\u0445\u0435\u043c\u0430\n\n\u0414\u0430\u043d\u043e: \u043d\u0430\u0447\u0430\u043b\u044c\u043d\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435 $x$, \u0442\u0440\u0435\u0431\u0443\u0435\u043c\u0430\u044f \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u044c $\\varepsilon$.\n\n```python\ndef GeneralScheme(x, epsilon):\n \n while StopCriterion(x) > epsilon:\n \n OracleResponse = RequestOracle(x)\n \n UpdateInformation(I, x, OracleResponse)\n \n x = NextPoint(I, x)\n \n return x\n```\n\n### \u0412\u043e\u043f\u0440\u043e\u0441\u044b\n1. \u041a\u0430\u043a\u0438\u0435 \u043a\u0440\u0438\u0442\u0435\u0440\u0438\u0438 \u043e\u0441\u0442\u0430\u043d\u043e\u0432\u043a\u0438 \u043c\u043e\u0433\u0443\u0442 \u0431\u044b\u0442\u044c?\n2. \u0427\u0442\u043e \u0442\u0430\u043a\u043e\u0435 \u043e\u0440\u0430\u043a\u0443\u043b \u0438 \u0437\u0430\u0447\u0435\u043c \u043e\u043d \u043d\u0443\u0436\u0435\u043d?\n3. \u0427\u0442\u043e \u0442\u0430\u043a\u043e\u0435 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u0430\u044f \u043c\u043e\u0434\u0435\u043b\u044c?\n4. \u041a\u0430\u043a \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u0435\u0442\u0441\u044f \u043d\u043e\u0432\u0430\u044f \u0442\u043e\u0447\u043a\u0430?\n\n#### \u041a\u0440\u0438\u0442\u0435\u0440\u0438\u0438 \u043e\u0441\u0442\u0430\u043d\u043e\u0432\u043a\u0438\n1. \u0421\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u044c \u043f\u043e \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442\u0443: \n$$\n\\| x_k - x^* \\|_2 < \\varepsilon\n$$ \n2. \u0421\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u044c \u043f\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u0438: \n$$\n\\| f_k - f^* \\|_2 < \\varepsilon\n$$ \n3. \u0412\u044b\u043f\u043e\u043b\u043d\u0435\u043d\u0438\u0435 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e\u0433\u043e \u0443\u0441\u043b\u043e\u0432\u0438\u044f \n$$\n\\| f'(x_k) \\|_2 < \\varepsilon\n$$\n\n\u041d\u043e \u0432\u0435\u0434\u044c $x^*$ \u043d\u0435\u0438\u0437\u0432\u0435\u0441\u0442\u043d\u0430!\n\n\u0422\u043e\u0433\u0434\u0430\n\n\\begin{align*}\n& \\|x_{k+1} - x_k \\| = \\|x_{k+1} - x_k + x^* - x^* \\| \\leq \\\\\n& \\|x_{k+1} - x^* \\| + \\| x_k - x^* \\| \\leq 2\\varepsilon\n\\end{align*}\n\n\u0410\u043d\u0430\u043b\u043e\u0433\u0438\u0447\u043d\u043e \u0434\u043b\u044f \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u0438 \u043f\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \n\n\u043e\u0434\u043d\u0430\u043a\u043e \u0438\u043d\u043e\u0433\u0434\u0430 \u043c\u043e\u0436\u043d\u043e \u043e\u0446\u0435\u043d\u0438\u0442\u044c $f^*$! \n\n**\u0417\u0430\u043c\u0435\u0447\u0430\u043d\u0438\u0435**: \u043b\u0443\u0447\u0448\u0435 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u044b\u0435 \u0438\u0437\u043c\u0435\u043d\u0435\u043d\u0438\u044f \n\n\u044d\u0442\u0438\u0445 \u0432\u0435\u043b\u0438\u0447\u0438\u043d! \n\n\u041d\u0430\u043f\u0440\u0438\u043c\u0435\u0440 $\\dfrac{\\|x_{k+1} - x_k \\|_2}{\\| x_k \\|_2}$\n\n\n#### \u0427\u0442\u043e \u0442\u0430\u043a\u043e\u0435 \u043e\u0440\u0430\u043a\u0443\u043b?\n**\u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435**: \u043e\u0440\u0430\u043a\u0443\u043b\u043e\u043c \u043d\u0430\u0437\u044b\u0432\u0430\u044e\u0442 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u0430\u0431\u0441\u0442\u0440\u0430\u043a\u0442\u043d\u043e\u0435 \n\n\u0443\u0441\u0442\u0440\u043e\u0439\u0441\u0442\u0432\u043e, \u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u043e\u0442\u0432\u0435\u0447\u0430\u0435\u0442 \u043d\u0430 \u043f\u043e\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0435 \u0432\u043e\u043f\u0440\u043e\u0441\u044b \n\n\u043c\u0435\u0442\u043e\u0434\u0430\n\n\u0410\u043d\u0430\u043b\u043e\u0433\u0438\u044f \u0438\u0437 \u041e\u041e\u041f: \n\n- \u043e\u0440\u0430\u043a\u0443\u043b - \u044d\u0442\u043e \u0432\u0438\u0440\u0442\u0443\u0430\u043b\u044c\u043d\u044b\u0439 \u043c\u0435\u0442\u043e\u0434 \u0431\u0430\u0437\u043e\u0432\u043e\u0433\u043e \u043a\u043b\u0430\u0441\u0441\u0430\n- \u043a\u0430\u0436\u0434\u0430\u044f \u0437\u0430\u0434\u0430\u0447\u0430 - \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0439 \u043a\u043b\u0430\u0441\u0441\n- \u043e\u0440\u0430\u043a\u0443\u043b \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u044f\u0435\u0442\u0441\u044f \u0434\u043b\u044f \u043a\u0430\u0436\u0434\u043e\u0439 \u0437\u0430\u0434\u0430\u0447\u0438 \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u043e \u0441\u043e\u0433\u043b\u0430\u0441\u043d\u043e \u043e\u0431\u0449\u0435\u043c\u0443 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044e \u0432 \u0431\u0430\u0437\u043e\u0432\u043e\u043c \u043a\u043b\u0430\u0441\u0441\u0435\n\n**\u041a\u043e\u043d\u0446\u0435\u043f\u0446\u0438\u044f \u0447\u0451\u0440\u043d\u043e\u0433\u043e \u044f\u0449\u0438\u043a\u0430**\n1. \u0415\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0439 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u0435\u0439, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c\u043e\u0439 \u0432 \u0445\u043e\u0434\u0435 \u0440\u0430\u0431\u043e\u0442\u044b \u0438\u0442\u0435\u0440\u0430\u0442\u0438\u0432\u043d\u043e\u0433\u043e \u043c\u0435\u0442\u043e\u0434\u0430, \u044f\u0432\u043b\u044f\u044e\u0442\u0441\u044f \u043e\u0442\u0432\u0435\u0442\u044b \u043e\u0440\u0430\u043a\u0443\u043b\u0430\n2. \u041e\u0442\u0432\u0435\u0442\u044b \u043e\u0440\u0430\u043a\u0443\u043b\u0430 \u044f\u0432\u043b\u044f\u044e\u0442\u0441\u044f *\u043b\u043e\u043a\u0430\u043b\u044c\u043d\u044b\u043c\u0438*\n\n#### \u0418\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u044f \u043e \u0437\u0430\u0434\u0430\u0447\u0435\n1. \u041a\u0430\u0436\u0434\u044b\u0439 \u043e\u0442\u0432\u0435\u0442 \u043e\u0440\u0430\u043a\u0443\u043b\u0430 \u0434\u0430\u0451\u0442 **\u043b\u043e\u043a\u0430\u043b\u044c\u043d\u0443\u044e** \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u044e \u043e \u043f\u043e\u0432\u0435\u0434\u0435\u043d\u0438\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0432 \u0442\u043e\u0447\u043a\u0435\n2. \u0410\u0433\u0440\u0435\u0433\u0438\u0440\u0443\u044f \u0432\u0441\u0435 \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u043d\u044b\u0435 \u043e\u0442\u0432\u0435\u0442\u044b \u043e\u0440\u0430\u043a\u0443\u043b\u0430, \u043e\u0431\u043d\u043e\u0432\u043b\u044f\u0435\u043c \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u044e \u043e **\u0433\u043b\u043e\u0431\u0430\u043b\u044c\u043d\u043e\u043c** \u0432\u0438\u0434\u0435 \u0446\u0435\u043b\u0435\u0432\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438:\n - \u043a\u0440\u0438\u0432\u0438\u0437\u043d\u0430\n - \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u0443\u0431\u044b\u0432\u0430\u043d\u0438\u044f\n - etc\n\n#### \u0412\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0435 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0439 \u0442\u043e\u0447\u043a\u0438\n\n$$\nx_{k+1} = x_{k} + \\alpha_k h_k\n$$\n\n- **\u041b\u0438\u043d\u0435\u0439\u043d\u044b\u0439 \u043f\u043e\u0438\u0441\u043a**: \u0444\u0438\u043a\u0441\u0438\u0440\u0443\u0435\u0442\u0441\u044f \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435 $h_k$ \u0438 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u0438\u0442\u0441\u044f \u043f\u043e\u0438\u0441\u043a \u043f\u043e \u044d\u0442\u043e\u043c\u0443 \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u044e \"\u043e\u043f\u0442\u0438\u043c\u0430\u043b\u044c\u043d\u043e\u0433\u043e\" \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f $\\alpha_k$\n- **\u041c\u0435\u0442\u043e\u0434 \u0434\u043e\u0432\u0435\u0440\u0438\u0442\u0435\u043b\u044c\u043d\u044b\u0445 \u043e\u0431\u043b\u0430\u0441\u0442\u0435\u0439**: \u0444\u0438\u043a\u0441\u0438\u0440\u0443\u0435\u0442\u0441\u044f \u0434\u043e\u043f\u0443\u0441\u0442\u0438\u043c\u044b\u0439 \u0440\u0430\u0437\u043c\u0435\u0440 *\u043e\u0431\u043b\u0430\u0441\u0442\u0438* \u043f\u043e \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u043e\u0439 \u043d\u043e\u0440\u043c\u0435 $\\| \\cdot \\| \\leq \\alpha$ \u0438 *\u043c\u043e\u0434\u0435\u043b\u044c* \u0446\u0435\u043b\u0435\u0432\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u0445\u043e\u0440\u043e\u0448\u043e \u0435\u0451 \u0430\u043f\u043f\u0440\u043e\u043a\u0441\u0438\u043c\u0438\u0440\u0443\u0435\u0442 \u0432 \u0432\u044b\u0431\u0440\u0430\u043d\u043d\u043e\u0439 \u043e\u0431\u043b\u0430\u0441\u0442\u0438. \n \n \u0414\u0430\u043b\u0435\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u0438\u0442\u0441\u044f \u043f\u043e\u0438\u0441\u043a \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u044f $h_k$, \u043c\u0438\u043d\u0438\u043c\u0438\u0437\u0438\u0440\u0443\u044e\u0449\u0435\u0433\u043e \u043c\u043e\u0434\u0435\u043b\u044c \u0446\u0435\u043b\u0435\u0432\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0438 \u043d\u0435 \u0432\u044b\u0432\u043e\u0434\u044f\u0449\u0435\u0433\u043e \u0442\u043e\u0447\u043a\u0443 $x_k + h_k$ \u0437\u0430 \u043f\u0440\u0435\u0434\u0435\u043b\u044b \u0434\u043e\u0432\u0435\u0440\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043e\u0431\u043b\u0430\u0441\u0442\u0438\n\n#### \u0412\u043e\u043f\u0440\u043e\u0441\u044b\n1. \u041a\u0430\u043a \u0432\u044b\u0431\u0440\u0430\u0442\u044c $\\alpha_k$?\n2. \u041a\u0430\u043a \u0432\u044b\u0431\u0440\u0430\u0442\u044c $h_k$?\n3. \u041a\u0430\u043a \u0432\u044b\u0431\u0440\u0430\u0442\u044c \u043c\u043e\u0434\u0435\u043b\u044c?\n4. \u041a\u0430\u043a \u0432\u044b\u0431\u0440\u0430\u0442\u044c \u043e\u0431\u043b\u0430\u0441\u0442\u044c?\n5. \u041a\u0430\u043a \u0432\u044b\u0431\u0440\u0430\u0442\u044c \u0440\u0430\u0437\u043c\u0435\u0440 \u043e\u0431\u043b\u0430\u0441\u0442\u0438? \n\n\n \u0412 \u043a\u0443\u0440\u0441\u0435 \u0440\u0430\u0441\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u0435\u0442\u0441\u044f \u0442\u043e\u043b\u044c\u043a\u043e \u043b\u0438\u043d\u0435\u0439\u043d\u044b\u0439 \u043f\u043e\u0438\u0441\u043a! \n \n\u041e\u0434\u043d\u0430\u043a\u043e \u043d\u0435\u0441\u043a\u043e\u043b\u044c\u043a\u043e \u0440\u0430\u0437 \u043a\u043e\u043f\u0446\u0435\u043f\u0446\u0438\u044f \u043c\u0435\u0442\u043e\u0434\u0430 \u0434\u043e\u0432\u0435\u0440\u0438\u0442\u0435\u043b\u044c\u043d\u044b\u0445 \u043e\u0431\u043b\u0430\u0441\u0442\u0435\u0439 \n\n\u0431\u0443\u0434\u0435\u0442 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0430.\n\n## \u041a\u0430\u043a \u0441\u0440\u0430\u0432\u043d\u0438\u0432\u0430\u0442\u044c \u043c\u0435\u0442\u043e\u0434\u044b \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438?\n\u0414\u043b\u044f \u0437\u0430\u0434\u0430\u043d\u043d\u043e\u0433\u043e \u043a\u043b\u0430\u0441\u0441\u0430 \u0437\u0430\u0434\u0430\u0447 \u0441\u0440\u0430\u0432\u043d\u0438\u0432\u0430\u044e\u0442 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0435 \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u044b:\n1. \u0421\u043b\u043e\u0436\u043d\u043e\u0441\u0442\u044c\n - \u0430\u043d\u0430\u043b\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u0430\u044f: \u0447\u0438\u0441\u043b\u043e \u043e\u0431\u0440\u0430\u0449\u0435\u043d\u0438\u0439 \u043a \u043e\u0440\u0430\u043a\u0443\u043b\u0443 \u0434\u043b\u044f \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u0437\u0430\u0434\u0430\u0447\u0438 \u0441 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u044c\u044e $\\varepsilon$\n - \u0430\u0440\u0438\u0444\u043c\u0435\u0442\u0438\u0447\u0435\u0441\u043a\u0430\u044f: \u043e\u0431\u0449\u0435\u0435 \u0447\u0438\u0441\u043b\u043e \u0432\u0441\u0435\u0445 \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0439, \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u044b\u0445 \u0434\u043b\u044f \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u0437\u0430\u0434\u0430\u0447\u0438 \u0441 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u044c\u044e $\\varepsilon$\n2. \u0421\u043a\u043e\u0440\u043e\u0441\u0442\u044c \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u0438\n3. \u042d\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442\u044b\n\n### \u0421\u043a\u043e\u0440\u043e\u0441\u0442\u0438 \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u0438 \n1. \u0421\u0443\u0431\u043b\u0438\u043d\u0435\u0439\u043d\u0430\u044f\n$$\n\\| x_{k+1} - x^* \\|_2 \\leq C k^{\\alpha},\n$$\n\u0433\u0434\u0435 $\\alpha < 0$ \u0438 $ 0 < C < \\infty$\n2. \u041b\u0438\u043d\u0435\u0439\u043d\u0430\u044f (\u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u043f\u0440\u043e\u0433\u0440\u0435\u0441\u0441\u0438\u044f)\n$$\n\\| x_{k+1} - x^* \\|_2 \\leq Cq^k, \n$$\n\u0433\u0434\u0435 $q \\in (0, 1)$ \u0438 $ 0 < C < \\infty$\n\n3. \u0421\u0432\u0435\u0440\u0445\u043b\u0438\u043d\u0435\u0439\u043d\u0430\u044f \n$$\n\\| x_{k+1} - x^* \\|_2 \\leq Cq^{k^p}, \n$$\n\u0433\u0434\u0435 $q \\in (0, 1)$, $ 0 < C < \\infty$ \u0438 $p > 1$\n4. \u041a\u0432\u0430\u0434\u0440\u0430\u0442\u0438\u0447\u043d\u0430\u044f\n$$\n\\| x_{k+1} - x^* \\|_2 \\leq C\\| x_k - x^* \\|^2_2, \\qquad \\text{\u0438\u043b\u0438} \\qquad \\| x_{k+1} - x^* \\|_2 \\leq C q^{2^k}\n$$\n\u0433\u0434\u0435 $q \\in (0, 1)$ \u0438 $ 0 < C < \\infty$\n\n### \u041e\u043f\u0442\u0438\u043c\u0430\u043b\u044c\u043d\u044b\u0435 \u043c\u0435\u0442\u043e\u0434\u044b: can we do better?\n- \u0414\u043e\u043a\u0430\u0437\u044b\u0432\u0430\u044e\u0442 \u043d\u0438\u0436\u043d\u0438\u0435 \u043e\u0446\u0435\u043d\u043a\u0438 \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u0435\u0439 \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u0438 \u0434\u043b\u044f \u043a\u043b\u0430\u0441\u0441\u0430 \u0437\u0430\u0434\u0430\u0447 \u0438 \u043c\u0435\u0442\u043e\u0434\u043e\u0432 \u0444\u0438\u043a\u0441\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u043e\u0433\u043e \u043f\u043e\u0440\u044f\u0434\u043a\u0430\n- \u041f\u0440\u0435\u0434\u043b\u0430\u0433\u0430\u044e\u0442 \u043c\u0435\u0442\u043e\u0434\u044b, \u043d\u0430 \u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u044d\u0442\u0438 \u043d\u0438\u0436\u043d\u0438\u0435 \u043e\u0446\u0435\u043d\u043a\u0438 \u0434\u043e\u0441\u0442\u0438\u0433\u0430\u044e\u0442\u0441\u044f $\\Rightarrow$ \u0434\u043e\u043a\u0430\u0437\u0430\u043d\u0430 \u043e\u043f\u0442\u0438\u043c\u0430\u043b\u044c\u043d\u043e\u0441\u0442\u044c \n- \u041d\u0438\u0436\u0435 \u043f\u0440\u043e \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u0442\u0435\u043e\u0440\u0435\u043c \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u0438\n\n\u041e\u043f\u0442\u0438\u043c\u0430\u043b\u044c\u043d\u044b\u043c \u043c\u0435\u0442\u043e\u0434\u0430\u043c \u0438 \u043d\u0438\u0436\u043d\u0438\u043c \u043e\u0446\u0435\u043d\u043a\u0430\u043c \u0431\u0443\u0434\u0435\u0442, \u0432\u043e\u0437\u043c\u043e\u0436\u043d\u043e, \n\n\u043f\u043e\u0441\u0432\u044f\u0449\u0451\u043d \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u044b\u0439 \u0441\u0435\u043c\u0438\u043d\u0430\u0440 \u0438\u043b\u0438 \u0447\u0430\u0441\u0442\u044c \u0434\u043e\u043c\u0430\u0448\u043d\u0435\u0433\u043e \u0437\u0430\u0434\u0430\u043d\u0438\u044f.\n\n\n```python\n%matplotlib inline\nimport matplotlib.pyplot as plt\n\nUSE_COLAB = False\nif not USE_COLAB:\n plt.rc(\"text\", usetex=True)\n\nimport numpy as np\nC = 10\nalpha = -0.5\nq = 0.9\nnum_iter = 7\nsublinear = np.array([C * k**alpha for k in range(1, num_iter + 1)])\nlinear = np.array([C * q**k for k in range(1, num_iter + 1)])\nsuperlinear = np.array([C * q**(k**2) for k in range(1, num_iter + 1)])\nquadratic = np.array([C * q**(2**k) for k in range(1, num_iter + 1)])\nplt.figure(figsize=(12,8))\nplt.semilogy(np.arange(1, num_iter+1), sublinear, \n label=r\"Sublinear, $\\alpha = -0.5$\")\nplt.semilogy(np.arange(1, num_iter+1), superlinear, \n label=r\"Superlinear, $q = 0.5, p=2$\")\nplt.semilogy(np.arange(1, num_iter+1), linear, \n label=r\"Linear, $q = 0.5$\")\nplt.semilogy(np.arange(1, num_iter+1), quadratic, \n label=r\"Quadratic, $q = 0.5$\")\nplt.xlabel(\"Number of iterations, $k$\", fontsize=28)\nplt.ylabel(\"Error rate upper bound\", fontsize=28)\nplt.legend(loc=\"best\", fontsize=26)\nplt.xticks(fontsize = 28)\n_ = plt.yticks(fontsize = 28)\n```\n\n### \u0417\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u0442\u0435\u043e\u0440\u0435\u043c \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u0438 (\u0411.\u0422. \u041f\u043e\u043b\u044f\u043a \u0412\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0432 \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u044e, \u0433\u043b. 1, $\\S$ 6)\n1. \u0427\u0442\u043e \u0434\u0430\u044e\u0442 \u0442\u0435\u043e\u0440\u0435\u043c\u044b \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u0438\n - \u043a\u043b\u0430\u0441\u0441 \u0437\u0430\u0434\u0430\u0447, \u0434\u043b\u044f \u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u043c\u043e\u0436\u043d\u043e \u0440\u0430\u0441\u0441\u0447\u0438\u0442\u044b\u0432\u0430\u0442\u044c \u043d\u0430 \u043f\u0440\u0438\u043c\u0435\u043d\u0438\u043c\u043e\u0441\u0442\u044c \u043c\u0435\u0442\u043e\u0434\u0430 (\u0432\u0430\u0436\u043d\u043e \u043d\u0435 \u0437\u0430\u0432\u044b\u0448\u0430\u0442\u044c \u0443\u0441\u043b\u043e\u0432\u0438\u044f!)\n - \u0432\u044b\u043f\u0443\u043a\u043b\u043e\u0441\u0442\u044c\n - \u0433\u043b\u0430\u0434\u043a\u043e\u0441\u0442\u044c\n - \u043a\u0430\u0447\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0435 \u043f\u043e\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u043c\u0435\u0442\u043e\u0434\u0430\n - \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u043e \u043b\u0438 \u043d\u0430\u0447\u0430\u043b\u044c\u043d\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435\n - \u043f\u043e \u043a\u0430\u043a\u043e\u043c\u0443 \u0444\u0443\u043d\u043a\u0446\u0438\u043e\u043d\u0430\u043b\u0443 \u0435\u0441\u0442\u044c \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u044c\n - \u043e\u0446\u0435\u043d\u043a\u0443 \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u0438 \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u0438\n - \u0442\u0435\u043e\u0440\u0435\u0442\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u043e\u0446\u0435\u043d\u043a\u0430 \u043f\u043e\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u043c\u0435\u0442\u043e\u0434\u0430 \u0431\u0435\u0437 \u043f\u0440\u043e\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u044d\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442\u043e\u0432\n - \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0444\u0430\u043a\u0442\u043e\u0440\u043e\u0432, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u0432\u043b\u0438\u044f\u044e\u0442 \u043d\u0430 \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u044c (\u043e\u0431\u0443\u0441\u043b\u043e\u0432\u043b\u0435\u043d\u043d\u043e\u0441\u0442\u044c, \u0440\u0430\u0437\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u044c, etc)\n - \u0438\u043d\u043e\u0433\u0434\u0430 \u0437\u0430\u0440\u0430\u043d\u0435\u0435 \u043c\u043e\u0436\u043d\u043e \u0432\u044b\u0431\u0440\u0430\u0442\u044c \u0447\u0438\u0441\u043b\u043e \u0438\u0442\u0435\u0440\u0430\u0446\u0438\u0439 \u0434\u043b\u044f \u0434\u043e\u0441\u0442\u0438\u0436\u0435\u043d\u0438\u044f \u0437\u0430\u0434\u0430\u043d\u043d\u043e\u0439 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u0438 \n\n2. \u0427\u0442\u043e **\u041d\u0415** \u0434\u0430\u044e\u0442 \u0442\u0435\u043e\u0440\u0435\u043c\u044b \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u0438\n - \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u044c \u043c\u0435\u0442\u043e\u0434\u0430 **\u043d\u0438\u0447\u0435\u0433\u043e \u043d\u0435 \u0433\u043e\u0432\u043e\u0440\u0438\u0442** \u043e \u0446\u0435\u043b\u0435\u0441\u043e\u043e\u0431\u0440\u0430\u0437\u043d\u043e\u0441\u0442\u0438 \u0435\u0433\u043e \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u044f\n - \u043e\u0446\u0435\u043d\u043a\u0438 \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u0438 \u0437\u0430\u0432\u0438\u0441\u044f\u0442 \u043e\u0442 \u043d\u0435\u0438\u0437\u0432\u0435\u0441\u0442\u043d\u044b\u0445 \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442 - \u043d\u0435\u043a\u043e\u043d\u0441\u0442\u0440\u0443\u043a\u0442\u0438\u0432\u043d\u044b\u0439 \u0445\u0430\u0440\u0430\u043a\u0442\u0435\u0440\n - \u0443\u0447\u0451\u0442 \u043e\u0448\u0438\u0431\u043e\u043a \u043e\u043a\u0440\u0443\u0433\u043b\u0435\u043d\u0438\u044f \u0438 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u0438 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u0432\u0441\u043f\u043e\u043c\u043e\u0433\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0445 \u0437\u0430\u0434\u0430\u0447\n \n**\u041c\u043e\u0440\u0430\u043b\u044c**: \u043d\u0443\u0436\u043d\u043e \u043f\u0440\u043e\u044f\u0432\u043b\u044f\u0442\u044c \u0440\u0430\u0437\u0443\u043c\u043d\u0443\u044e \u043e\u0441\u0442\u043e\u0440\u043e\u0436\u043d\u043e\u0441\u0442\u044c \n\n\u0438 \u0437\u0434\u0440\u0430\u0432\u044b\u0439 \u0441\u043c\u044b\u0441\u043b!\n\n## \u041a\u043b\u0430\u0441\u0441\u0438\u0444\u0438\u043a\u0430\u0446\u0438\u044f \u0437\u0430\u0434\u0430\u0447\n1. \u0411\u0435\u0437\u0443\u0441\u043b\u043e\u0432\u043d\u0430\u044f \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u044f\n - \u0446\u0435\u043b\u0435\u0432\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043b\u0438\u043f\u0448\u0438\u0446\u0435\u0432\u0430\n - \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442 \u0446\u0435\u043b\u0435\u0432\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043b\u0438\u043f\u0448\u0438\u0446\u0435\u0432\n2. \u0423\u0441\u043b\u043e\u0432\u043d\u0430\u044f \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u044f\n - \u043c\u043d\u043e\u0433\u043e\u0433\u0440\u0430\u043d\u043d\u0438\u043a\n - \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e \u043f\u0440\u043e\u0441\u0442\u043e\u0439 \u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u044b\n - \u043e\u0431\u0449\u0435\u0433\u043e \u0432\u0438\u0434\u0430\n\n## \u041a\u043b\u0430\u0441\u0441\u0438\u0444\u0438\u043a\u0430\u0446\u0438\u044f \u043c\u0435\u0442\u043e\u0434\u043e\u0432\n1. \u041c\u0435\u0442\u043e\u0434\u044b \u043d\u0443\u043b\u0435\u0432\u043e\u0433\u043e \u043f\u043e\u0440\u044f\u0434\u043a\u0430: \u043e\u0440\u0430\u043a\u0443\u043b \u0432\u043e\u0437\u0432\u0440\u0430\u0449\u0430\u0435\u0442 \u0442\u043e\u043b\u044c\u043a\u043e \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 $f(x)$\n\n2. \u041c\u0435\u0442\u043e\u0434\u044b \u043f\u0435\u0440\u0432\u043e\u0433\u043e \u043f\u043e\u0440\u044f\u0434\u043a\u0430: \u043e\u0440\u0430\u043a\u0443\u043b \u0432\u043e\u0437\u0432\u0440\u0430\u0449\u0430\u0435\u0442 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 $f(x)$ \u0438 \u0435\u0451 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442 $f'(x)$\n\n3. \u041c\u0435\u0442\u043e\u0434\u044b \u0432\u0442\u043e\u0440\u043e\u0433\u043e \u043f\u043e\u0440\u044f\u0434\u043a\u0430: \u043e\u0440\u0430\u043a\u0443\u043b \u0432\u043e\u0437\u0432\u0440\u0430\u0449\u0430\u0435\u0442 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 $f(x)$, \u0435\u0451 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442 $f'(x)$ \u0438 \u0433\u0435\u0441\u0441\u0438\u0430\u043d $f''(x)$.\n\n**\u0412\u043e\u043f\u0440\u043e\u0441**: \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u044e\u0442 \u043b\u0438 \u043c\u0435\u0442\u043e\u0434\u044b \u0431\u043e\u043b\u0435\u0435 \u0432\u044b\u0441\u043e\u043a\u043e\u0433\u043e \u043f\u043e\u0440\u044f\u0434\u043a\u0430?\n\n1. \u041e\u0434\u043d\u043e\u0448\u0430\u0433\u043e\u0432\u044b\u0435 \u043c\u0435\u0442\u043e\u0434\u044b \n$$\nx_{k+1} = \\Phi(x_k)\n$$\n2. \u041c\u043d\u043e\u0433\u043e\u0448\u0430\u0433\u043e\u0432\u044b\u0435 \u043c\u0435\u0442\u043e\u0434\u044b\n$$\nx_{k+1} = \\Phi(x_k, x_{k-1}, ...)\n$$\n\n## \u041e\u0434\u043d\u043e\u043c\u0435\u0440\u043d\u0430\u044f \u043c\u0438\u043d\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u044f\n**\u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435**. \u0424\u0443\u043d\u043a\u0446\u0438\u044f $f(x)$ \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u0443\u043d\u0438\u043c\u043e\u0434\u0430\u043b\u044c\u043d\u043e\u0439 \u043d\u0430 $[a, b]$, \u0435\u0441\u043b\u0438 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u0442\u0430\u043a\u0430\u044f \u0442\u043e\u0447\u043a\u0430 $x^* \\in [a, b]$, \u0447\u0442\u043e \n- $f(x_1) > f(x_2)$ \u0434\u043b\u044f \u043b\u044e\u0431\u044b\u0445 $a \\leq x_1 < x_2 < x^*$, \n\n\u0438 \n- $f(x_1) < f(x_2)$ \u0434\u043b\u044f \u043b\u044e\u0431\u044b\u0445 $x^* < x_1 < x_2 \\leq b$.\n\n**\u0412\u043e\u043f\u0440\u043e\u0441**: \u043a\u0430\u043a\u0430\u044f \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f \u0443\u043d\u0438\u043c\u043e\u0434\u0430\u043b\u044c\u043d\u044b\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439?\n\n### \u041c\u0435\u0442\u043e\u0434 \u0434\u0438\u0445\u043e\u0442\u043e\u043c\u0438\u0438\n\n\u0418\u0434\u0435\u044f \u0438\u0437 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0438 \u043f\u0435\u0440\u0432\u043e\u0433\u043e \u0441\u0435\u043c\u0435\u0441\u0442\u0440\u0430: \n\n\u0434\u0435\u043b\u0438\u043c \u043e\u0442\u0440\u0435\u0437\u043e\u043a $[a,b]$ \u043d\u0430 \u0434\u0432\u0435 \u0440\u0430\u0432\u043d\u044b\u0435 \u0447\u0430\u0441\u0442\u0438 \n\n\u043f\u043e\u043a\u0430 \u043d\u0435 \u043d\u0430\u0439\u0434\u0451\u043c \u043c\u0438\u043d\u0438\u043c\u0443\u043c \u0443\u043d\u0438\u043c\u043e\u0434\u0430\u043b\u044c\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438.\n\n- $N$ - \u0447\u0438\u0441\u043b\u043e \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 $f$\n- $K = \\frac{N - 1}{2}$ - \u0447\u0438\u0441\u043b\u043e \u0438\u0442\u0435\u0440\u0430\u0446\u0438\u0439\n\n\u0422\u043e\u0433\u0434\u0430\n$$\n|x_{K+1} - x^*| \\leq \\frac{b_{K+1} - a_{K+1}}{2} = \\left( \\frac{1}{2} \\right)^{\\frac{N-1}{2}} (b - a) \\approx 0.5^{K} (b - a) \n$$\n\n\n```python\ndef binary_search(f, a, b, epsilon, callback=None):\n c = (a + b) / 2.0\n while abs(b - a) > epsilon:\n# Check left subsegment\n y = (a + c) / 2.0\n if f(y) <= f(c):\n b = c\n c = y\n else:\n# Check right subsegment\n z = (b + c) / 2.0\n if f(c) <= f(z):\n a = y\n b = z\n else:\n a = c\n c = z\n if callback is not None:\n callback(a, b)\n return c\n```\n\n\n```python\ndef my_callback(a, b, left_bound, right_bound, approximation):\n left_bound.append(a)\n right_bound.append(b)\n approximation.append((a + b) / 2.0)\n```\n\n\n```python\nimport numpy as np\n\nleft_boud_bs = []\nright_bound_bs = []\napproximation_bs = []\n\ncallback_bs = lambda a, b: my_callback(a, b, \n left_boud_bs, right_bound_bs, approximation_bs)\n\n# Target unimodal function on given segment\nf = lambda x: (x - 2) * x * (x + 2)**2 # np.power(x+2, 2)\n# f = lambda x: -np.sin(x)\nx_true = -2\n# x_true = np.pi / 2.0\na = -3\nb = -1.5\nepsilon = 1e-8\nx_opt = binary_search(f, a, b, epsilon, callback_bs)\nprint(np.abs(x_opt - x_true))\nplt.figure(figsize=(10,6))\nplt.plot(np.linspace(a,b), f(np.linspace(a,b)))\nplt.title(\"Objective function\", fontsize=28)\nplt.xticks(fontsize = 28)\n_ = plt.yticks(fontsize = 28)\n```\n\n### \u041c\u0435\u0442\u043e\u0434 \u0437\u043e\u043b\u043e\u0442\u043e\u0433\u043e \u0441\u0435\u0447\u0435\u043d\u0438\u044f\n\u0418\u0434\u0435\u044f: \n\n\u0434\u0435\u043b\u0438\u0442\u044c \u043e\u0442\u0440\u0435\u0437\u043e\u043a $[a,b]$ \u043d\u0435 \u043d\u0430 \u0434\u0432\u0435 \u0440\u0430\u0432\u043d\u044b\u0435 \u043d\u0430\u0441\u0442\u0438, \n\n\u0430 \u0432 \u043f\u0440\u043e\u043f\u043e\u0440\u0446\u0438\u0438 \"\u0437\u043e\u043b\u043e\u0442\u043e\u0433\u043e \u0441\u0435\u0447\u0435\u043d\u0438\u044f\".\n\n\u041e\u0446\u0435\u043d\u0438\u043c \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u044c \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u0438 \u0430\u043d\u0430\u043b\u043e\u0433\u0438\u0447\u043d\u043e \u043c\u0435\u0442\u043e\u0434\u0443 \u0434\u0438\u0445\u043e\u0442\u043e\u043c\u0438\u0438:\n\n$$\n|x_{K+1} - x^*| \\leq b_{K+1} - a_{K+1} = \\left( \\frac{1}{\\tau} \\right)^{N-1} (b - a) \\approx 0.618^K(b-a),\n$$\n\u0433\u0434\u0435 $\\tau = \\frac{\\sqrt{5} + 1}{2}$.\n\n- \u041a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u043f\u0440\u043e\u0433\u0440\u0435\u0441\u0441\u0438\u0438 **\u0431\u043e\u043b\u044c\u0448\u0435**, \u0447\u0435\u043c \u0443 \u043c\u0435\u0442\u043e\u0434\u0430 \u0434\u0438\u0445\u043e\u0442\u043e\u043c\u0438\u0438\n- \u041a\u043e\u043b\u0438\u0447\u0435\u0441\u0442\u0432\u043e \u0432\u044b\u0437\u043e\u0432\u043e\u0432 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 **\u043c\u0435\u043d\u044c\u0448\u0435**, \u0447\u0435\u043c \u0443 \u043c\u0435\u0442\u043e\u0434\u0430 \u0434\u0438\u0445\u043e\u0442\u043e\u043c\u0438\u0438\n\n\n```python\ndef golden_search(f, a, b, tol=1e-5, callback=None):\n tau = (np.sqrt(5) + 1) / 2.0\n y = a + (b - a) / tau**2\n z = a + (b - a) / tau\n while b - a > tol:\n if f(y) <= f(z):\n b = z\n z = y\n y = a + (b - a) / tau**2\n else:\n a = y\n y = z\n z = a + (b - a) / tau\n if callback is not None:\n callback(a, b)\n return (a + b) / 2.0\n```\n\n\n```python\nleft_boud_gs = []\nright_bound_gs = []\napproximation_gs = []\n\ncb_gs = lambda a, b: my_callback(a, b, left_boud_gs, right_bound_gs, approximation_gs)\nx_gs = golden_search(f, a, b, epsilon, cb_gs)\n\nprint(f(x_opt))\nprint(f(x_gs))\nprint(np.abs(x_opt - x_true))\n```\n\n 6.93889390875399e-18\n 9.549014390504221e-18\n 9.313225746154785e-10\n\n\n### \u0421\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435 \u043c\u0435\u0442\u043e\u0434\u043e\u0432 \u043e\u0434\u043d\u043e\u043c\u0435\u0440\u043d\u043e\u0439 \u043c\u0438\u043d\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438\n\n\n```python\nplt.figure(figsize=(10,6))\nplt.semilogy(np.arange(1, len(approximation_bs) + 1), np.abs(x_true - np.array(approximation_bs, dtype=np.float64)), label=\"Binary search\")\nplt.semilogy(np.arange(1, len(approximation_gs) + 1), np.abs(x_true - np.array(approximation_gs, dtype=np.float64)), label=\"Golden search\")\nplt.xlabel(r\"Number of iterations, $k$\", fontsize=26)\nplt.ylabel(\"Error rate upper bound\", fontsize=26)\nplt.legend(loc=\"best\", fontsize=26)\nplt.xticks(fontsize = 26)\n_ = plt.yticks(fontsize = 26)\n```\n\n\n```python\n%timeit binary_search(f, a, b, epsilon)\n%timeit golden_search(f, a, b, epsilon)\n```\n\n 22.6 \u00b5s \u00b1 1.13 \u00b5s per loop (mean \u00b1 std. dev. of 7 runs, 10000 loops each)\n 84.1 \u00b5s \u00b1 6.64 \u00b5s per loop (mean \u00b1 std. dev. of 7 runs, 10000 loops each)\n\n\n## \u041f\u0440\u0438\u043c\u0435\u0440 \u0438\u043d\u043e\u0433\u043e \u043f\u043e\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u043c\u0435\u0442\u043e\u0434\u043e\u0432\n\n$$\nf(x) = \\sin(\\sin(\\sin(\\sqrt{x}))), \\; x \\in [2, 60]\n$$\n\n\n```python\nf = lambda x: np.sin(np.sin(np.sin(np.sqrt(x))))\nx_true = (3 * np.pi / 2)**2\na = 2\nb = 60\nepsilon = 1e-8\nplt.plot(np.linspace(a,b), f(np.linspace(a,b)))\nplt.xticks(fontsize = 28)\n_ = plt.yticks(fontsize = 28)\n```\n\n## \u0421\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435 \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u0438 \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u0438 \u0438 \u0432\u0440\u0435\u043c\u0435\u043d\u0438 \u0440\u0430\u0431\u043e\u0442\u044b \u043c\u0435\u0442\u043e\u0434\u043e\u0432\n\n### \u041c\u0435\u0442\u043e\u0434 \u0434\u0438\u0445\u043e\u0442\u043e\u043c\u0438\u0438\n\n\n```python\nleft_boud_bs = []\nright_bound_bs = []\napproximation_bs = []\n\ncallback_bs = lambda a, b: my_callback(a, b, \n left_boud_bs, right_bound_bs, approximation_bs)\n\nx_opt = binary_search(f, a, b, epsilon, callback_bs)\nprint(np.abs(x_opt - x_true))\n```\n\n 2.1968899233115735e-07\n\n\n### \u041c\u0435\u0442\u043e\u0434 \u0437\u043e\u043b\u043e\u0442\u043e\u0433\u043e \u0441\u0435\u0447\u0435\u043d\u0438\u044f\n\n\n```python\nleft_boud_gs = []\nright_bound_gs = []\napproximation_gs = []\n\ncb_gs = lambda a, b: my_callback(a, b, left_boud_gs, right_bound_gs, approximation_gs)\nx_gs = golden_search(f, a, b, epsilon, cb_gs)\n\nprint(np.abs(x_opt - x_true))\n```\n\n 2.1968899233115735e-07\n\n\n### \u0421\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u044c\n\n\n```python\nplt.figure(figsize=(8,6))\nplt.semilogy(np.abs(x_true - np.array(approximation_bs, dtype=np.float64)), label=\"Binary\")\nplt.semilogy(np.abs(x_true - np.array(approximation_gs, dtype=np.float64)), label=\"Golden\")\nplt.legend(fontsize=28)\nplt.xticks(fontsize=28)\n_ = plt.yticks(fontsize=28)\nplt.xlabel(r\"Number of iterations, $k$\", fontsize=26)\nplt.ylabel(\"Error rate upper bound\", fontsize=26)\n```\n\n### \u0412\u0440\u0435\u043c\u044f \u0440\u0430\u0431\u043e\u0442\u044b\n\n\n```python\n%timeit binary_search(f, a, b, epsilon)\n%timeit golden_search(f, a, b, epsilon)\n```\n\n 455 \u00b5s \u00b1 35.8 \u00b5s per loop (mean \u00b1 std. dev. of 7 runs, 1000 loops each)\n 416 \u00b5s \u00b1 26.2 \u00b5s per loop (mean \u00b1 std. dev. of 7 runs, 1000 loops each)\n\n\n## \u0420\u0435\u0437\u044e\u043c\u0435\n1. \u0412\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0432 \u0447\u0438\u0441\u043b\u0435\u043d\u043d\u044b\u0435 \u043c\u0435\u0442\u043e\u0434\u044b \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438\n2. \u041e\u0431\u0449\u0430\u044f \u0441\u0445\u0435\u043c\u0430 \u0440\u0430\u0431\u043e\u0442\u044b \u043c\u0435\u0442\u043e\u0434\u0430\n3. \u0421\u043f\u043e\u0441\u043e\u0431\u044b \u0441\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f \u043c\u0435\u0442\u043e\u0434\u043e\u0432 \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438\n4. \u0417\u043e\u043e\u043f\u0430\u0440\u043a \u0437\u0430\u0434\u0430\u0447 \u0438 \u043c\u0435\u0442\u043e\u0434\u043e\u0432\n5. \u041e\u0434\u043d\u043e\u043c\u0435\u0440\u043d\u0430\u044f \u043c\u0438\u043d\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u044f\n", "meta": {"hexsha": "f2376985ebc1965210f2688e0deba1965a26bd22", "size": 206541, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Spring2017-2019/12-NumMethods/Seminar12.ipynb", "max_stars_repo_name": "Jhomanik/MIPT-Opt", "max_stars_repo_head_hexsha": "c1629b93b7608081f2237278afd92ee426760a84", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 132, "max_stars_repo_stars_event_min_datetime": "2016-09-05T09:24:55.000Z", "max_stars_repo_stars_event_max_datetime": 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YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.8128673178375735, "lm_q1q2_score": 0.43812184495985473}} {"text": "```python\nimport os\nimport numpy as np\nimport matplotlib.pyplot as plt\nplt.rcParams['mathtext.fontset'] = 'stix'\n```\n\nLoad 200ns Aib9 trajectory\n\n\n```python\ninfile = '../../DATA/Train/AIB9/sum_phi_200ns.npy'\ninput_x = np.load(infile)\n\nbins=np.arange(-15., 17, 1)\nnum_bins=len(bins)\nidx_200ns=np.digitize(input_x, bins)\n\ndi=1\nN_mean=np.sum(np.abs(idx_200ns[:-di]-idx_200ns[di:])==1)\nN_mean/=len(idx_200ns)\n\nN0=len(np.where(idx_200ns<=15)[0])\nN1=len(np.where(idx_200ns>=16)[0])\nkappa_in = N0/N1\n\nprint('kappa:', kappa_in)\nprint('Nearest neighbor:', N_mean)\n```\n\n kappa: 1.148993438587424\n Nearest neighbor: 0.39906640023339995\n\n\nCheck 100ns Aib9 trajectory\n\n\n```python\nidx_100ns = idx_200ns[:2000000]\n\ndi=1\nN_mean=np.sum(np.abs(idx_100ns[:-di]-idx_100ns[di:])==1)\nN_mean/=len(idx_100ns)\n\nN0=len(np.where(idx_100ns<=15)[0])\nN1=len(np.where(idx_100ns>=16)[0])\nkappa_in = N0/N1\n\nprint('kappa:', kappa_in)\nprint('Nearest neighbor:', N_mean)\n```\n\n kappa: 1.0115786466903496\n Nearest neighbor: 0.397483\n\n\n# Calculate Nearest neighbor $\\langle N\\rangle$ sampled from the first training\n\nIn the first training, we let 800 independent LSTMs predict 800 trajectories of 100$ns$. Since we are using LSTM as a generative model, we can also train just one LSTM and use it to generate 800 predictions, starting from either the same initial condition or different initial conditions.\n\nData location: `./Output/`\n\n\n```python\nN_mean_list=[]\noutput_dir='./Output'\nfor i in range(800):\n\n pred_dir=os.path.join(output_dir, '{}/prediction.npy'.format(i))\n prediction=np.load(pred_dir)\n\n di=1\n N_mean=np.sum(np.abs(prediction[:-di]-prediction[di:])==1)\n N_mean/=len(prediction)\n N_mean_list.append(N_mean)\n \nN_mean_arr=np.array(N_mean_list)\n```\n\nPlot distribution\n\n\n```python\nhist = np.histogram( N_mean_arr, bins=50 )\nprob = hist[0].T\nmids = 0.5*(hist[1][1:]+hist[1][:-1])\n\nfig, ax = plt.subplots(figsize=(5,4))\n\nax.set_title('Distribution', size=20)\nax.plot(mids, prob)\n\nax.tick_params(axis='both', which='both', direction='in', labelsize=14)\nax.set_xlabel('$\\langle N\\\\rangle$', size=16)\nax.set_ylabel('Counts', size=16)\n\nplt.show()\n```\n\n# Determine $\\Delta\\lambda$\n\nFollowing the reference, we want to solve the following equation for $\\Delta\\lambda$\n\n\\begin{align}\n \\bar{s}^{(j)}_2&=\\sum_{\\Gamma}P^{(2)}_{\\Gamma}s^{(j)}_{\\Gamma} \\nonumber \\\\\n &=\\frac{\\sum_{k\\in\\Omega} s^{(j)}_k e^{-\\Delta\\lambda_j s^{(j)}_k} }{\\sum_{k\\in\\Omega} e^{-\\Delta\\lambda_j s^{(j)}_k}} \\\\\n &=f(\\Delta\\lambda)\n \\label{eq:lambda_solver}\n\\end{align}\n\n\nTo determine the $\\Delta\\lambda$ value, we can calculate the above equation and plot it versus $\\Delta\\lambda$, and find $\\Delta\\lambda=\\Delta\\lambda_{\\ast}$ which gives \n\\begin{align}\n\\bar{s}^{(j)}_2=f(\\Delta\\lambda_{\\ast})=s^{\\rm target}\n\\end{align}\n\nWe would also like to predict the variance of $\\bar{s}^{(j)}_2$, which can be obtained through derivative of $f$.\n\\begin{align}\nf'(\\lambda)=-\\sigma^2_{\\bar{s}^{(j)}_2}\n\\end{align}\n\n\n### $s = \\langle N\\rangle$\n\n\n```python\ndef f(lm):\n return np.sum(N_mean_arr*np.exp(-lm*N_mean_arr))/np.sum(np.exp(-lm*N_mean_arr))\n\ndef df(lm):\n return f(lm)**2-np.sum(N_mean_arr*N_mean_arr*np.exp(-lm*N_mean_arr))/np.sum(np.exp(-lm*N_mean_arr))\n\nlm_arr = np.linspace(0,1000)\nf_arr = [f(lm_i) for lm_i in lm_arr]\n\n\nfig, ax=plt.subplots(figsize=(5,3))\n\nax.plot(lm_arr, f_arr, label='$f$')\n\nax.tick_params(axis='both', which='both', direction='in', labelsize=14)\nax.set_xlabel('$\\lambda$', size=16)\nax.set_ylabel('$f(\\lambda)$', size=16)\n\nax.legend(fontsize=16)\n\nplt.show()\n\nlm=62\nprint( 'f({:.1f}) = {:.6f}'.format(lm, f(lm)) )\nprint( 'Standard error stderr[f(0)]={:.6f}'.format(np.std(N_mean_arr)/np.sqrt(len(N_mean_arr))) )\n\nprint( 'df({:.1f}) = {:.6f}'.format(lm, df(lm)) )\nprint( 'Expected standard error for new N_mean = {:.6f}'.format( np.sqrt(-df(lm))/np.sqrt(10) ) )\n```\n\nLet's see if select 10 predictions to build the subset is enough.\n\n\n```python\nlm_ast=62\np=np.exp(-lm_ast*(N_mean_arr))\np/=np.sum(p)\n\nsubset_mean_arr = []\nsubset_stdv_arr = []\nfor i in range(200):\n idx = np.random.choice(len(N_mean_arr), 10, p=p)\n selected = N_mean_arr[idx]\n \n mean=np.mean(selected)\n stdv=np.std(selected)/np.sqrt(len(selected))\n \n subset_mean_arr.append(mean)\n subset_stdv_arr.append(stdv)\n\nfig, ax = plt.subplots(figsize=(12,5), nrows=1, ncols=2)\n\nax[0].plot(subset_mean_arr)\nax[0].plot(np.arange(len(subset_mean_arr)), [0.38]*len(subset_mean_arr))\n\nax[1].plot(subset_stdv_arr)\nax[1].plot(np.arange(len(subset_stdv_arr)), [0.004]*len(subset_stdv_arr))\n\nax[0].tick_params(axis='both', which='both', direction='in', labelsize=16)\nax[0].set_xlabel('indices', size=16)\nax[0].set_ylabel('$\\langle N\\\\rangle$', size=16)\nax[0].set_ylim(0.3,0.5)\n\nax[1].tick_params(axis='both', which='both', direction='in', labelsize=16)\nax[1].set_xlabel('indices', size=16)\nax[1].set_ylabel('$\\sigma_{N}$', size=16)\nax[1].set_ylim(0.0,0.01)\n\n\nplt.show()\n```\n\nSo we will constrain our $\\langle N\\rangle$ to 0.38 with standard error 0.0041. Although we have shown above that subset size=10 is sufficient, there could be variance in mean constraint. Therefore, we will also constrain our sampling until we reach a reasonable value of $\\langle N\\rangle$ of the subset.\n\n\n```python\nlm_ast=62\np=np.exp(-lm_ast*(N_mean_arr))\np/=np.sum(p)\n\nmean=np.inf\nstdv=np.inf\nwhile abs(mean-0.380)>0.001 or abs(stdv-0.0041)>0.0001:\n idx = np.random.choice(len(N_mean_arr), 10, p=p)\n selected = N_mean_arr[idx]\n \n mean=np.mean(selected)\n stdv=np.std(selected)/np.sqrt(len(selected))\n \nprint( 'mean of selected sample = {:.3f}'.format(np.mean(selected)) )\nprint( 'Standard error stderr[selected sample] = {:.3f}'.format(np.std(selected)/np.sqrt(len(selected))) )\n\nfor ki in kappa_arr[idx]:\n print('{:.3f}'.format(ki))\n```\n\n mean of selected sample = 0.380\n Standard error stderr[selected sample] = 0.004\n 0.488\n 0.270\n 1.270\n 0.078\n 0.099\n 7.103\n 0.495\n 5.780\n 3.174\n 3.875\n\n\n# Concatenate subset as a new training set\n\nConcatenate the subset to a single trajectory, this concatenated trajectory is then used later to re-train a new LSTM.\n\n\n```python\nconc=[]\noutput_dir='./Output'\n\nfor i in idx:\n pred_dir=os.path.join(output_dir, '{}/prediction.npy'.format(i))\n prediction=np.load(pred_dir)\n conc.extend(prediction)\n \nconc = np.array(conc)\n```\n\nWe can also check what the $\\langle N\\rangle$ as well as $\\kappa$ value of concatenated trajectory is.\n\n\n```python\nN0=len(np.where(conc<=15)[0])\nN1=len(np.where(conc>=16)[0])\nkappa_conc = N0/N1\n\ndi=1\nN_mean_conc=np.sum(np.abs(conc[:-di]-conc[di:])==1)\nN_mean_conc/=len(conc)\n\n\nprint('kappa:', kappa_conc)\nprint('Nearest neighbor:', N_mean_conc)\n```\n\n kappa: 0.9525129329575397\n Nearest neighbor: 0.37985445\n\n", "meta": {"hexsha": "18fa2c2b6f7a6c39e8e582d25bddbb3aae550b98", "size": 106729, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "path_sampling_NN0_38.ipynb", "max_stars_repo_name": "tiwarylab/ps-LSTM", "max_stars_repo_head_hexsha": "2b9a7b825a2236abf279cd0e5f8b522e2c780dfa", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2022-03-02T12:56:22.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-02T21:13:25.000Z", "max_issues_repo_path": "path_sampling_NN0_38.ipynb", "max_issues_repo_name": "tiwarylab/ps-LSTM", "max_issues_repo_head_hexsha": "2b9a7b825a2236abf279cd0e5f8b522e2c780dfa", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "path_sampling_NN0_38.ipynb", "max_forks_repo_name": "tiwarylab/ps-LSTM", "max_forks_repo_head_hexsha": "2b9a7b825a2236abf279cd0e5f8b522e2c780dfa", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 222.3520833333, "max_line_length": 59780, "alphanum_fraction": 0.9146717387, "converted": true, "num_tokens": 2167, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175139669997, "lm_q2_score": 0.6654105653819836, "lm_q1q2_score": 0.4381179702261813}} {"text": "#
    Quantum Advantage For Hubbard Model On a Near-Term QuantumComputer
    \nGround state calculations using classical methods require the storage of three vectors of the size of the number of states. The number of states scales exponentially with the size of the system, which limits the number of sites in the lattice to about $20\\times 20$ on currently available classical hardware. For low temperatures, convergence problems appear that lead to an exponential growth of computational effort with decreasing temperature due to the so-called fermion sign problem. \n\nFor the Hubbard-type problems we use quantum simulation to map the system Hamiltonian on qubits system (quantum computer) and run quantum algorithms on them to demonstrate trend of advantage that quantum computers promise. The advantage is expected to be exponential as a quantum computer is able to store exponential amount of information on polynomial number of qubits.\n\nbe exponential as a quantum computer is able to store exponentialamount of information on polynomial number of qubits.\n\n### Hubbard model\nThe Hubbard model [1] is an approximate model used, especially in solid-state physics, to describe the transition between conducting and insulating systems. The Hubbard model is the simplest model of interacting particles in a lattice, with only two terms in the Hamiltonian: a kinetic term allowing for tunneling (\"hopping\") of particles between sites of the lattice and a potential term consisting of an on-site interaction. The Hubbard model is also used as a model for high-temperature superconductivity as well as to describe the behavior of ultracold atoms trapped in optical lattices. For electrons in a solid, the Hubbard model can be considered as an improvement on the tight-binding model, which includes only the hopping term. For strong interactions, it can give qualitatively different behavior from the tight-binding model, and correctly predicts the existence of so-called Mott insulators, which are prevented from becoming conducting by the strong repulsion between the particles.\n\nThe fact that the Hubbard model has not been solved analytically in arbitrary dimensions has led to intense research into numerical methods for these strongly correlated electron systems. One major goal of this research is to determine the low-temperature phase diagram of this model, particularly in two-dimensions. \n\n\nIt is known that quantum many-body problems require exponential time, in the size of the problem,\nto solve on a classical computer without approximations. As Feynman envisioned, one could use a quantum computer to simulate quantum systems to overcome such difficulties [2,3].\nWith quantum computers one can prepare initial states of the system under study as well as have\ncontrol over the Hamiltonians that governs the time evolutions of those states.\nRecent advances in quantum-computing hardware brought us quite close to Feynman\u2019s vision.\nDeveloping optimized simulation algorithms is of highest importance as current near-term quantum\nhardwares have limitations in scaling, and errors in state preparation, qubit-qubit couplings, gate\nsequences, and readout. Significant progress has been made in this direction [4-7].\nSeveral quantum algorithms exist to simulate correlated fermions on linear and 2D qubit arrays with\nnearest-neighbor couplings which are typical for superconducting transmon qubits.\nWe will use the Fermi-Hubbard model [1] to demonstrate these algorithms. The Hubbard model\napproximates the long-range Coulomb interaction of electrons in a crystal with a local on-site\ninteraction. This locality reduces the resources required for simulating the model and makes it a\nprime candidate for the early applications of quantum simulations [8]. \nFor a lattice with only one type of site and edges from each site only to itself and its neighbors, the Hamiltonian for the spinful model has the form\n\n\\begin{eqnarray}\n H &=& - \\sum_{a < b} t_{a, b}^{(\\mathrm{onsite})}\n \\sum_{i} \\sum_{\\sigma}\n (a^\\dagger_{i, a, \\sigma} a_{i, b, \\sigma} +\n a^\\dagger_{i, b, \\sigma} a_{i, a, \\sigma})\\nonumber\n \\\\&-& \\sum_{a} t_{a, a}^{(\\mathrm{nghbr})}\n \\sum_{\\{i, j\\}} \\sum_{\\sigma}\n (a^\\dagger_{i, a, \\sigma} a_{j, a, \\sigma} +\n a^\\dagger_{j, a, \\sigma} a_{i, a, \\sigma})\\nonumber\n \\\\ &-& \\sum_{a < b} t_{a, b}^{(\\mathrm{nghbr})}\n \\sum_{(i, j)} \\sum_{\\sigma}\n (a^\\dagger_{i, a, \\sigma} a_{j, b, \\sigma} +\n a^\\dagger_{j, b, \\sigma} a_{i, a, \\sigma})\\nonumber\n + \\sum_{a < b} U_{a, b}^{(\\mathrm{onsite}, +)}\n \\sum_{i} \\sum_{\\sigma}\n n_{i, a, \\sigma} n_{i, b, \\sigma}\\nonumber\n \\\\&+& \\sum_{a} U_{a, a}^{(\\mathrm{nghbr}, +)}\n \\sum_{\\{i, j\\}} \\sum_{\\sigma}\n n_{i, a, \\sigma} n_{j, a, \\sigma}\n + \\sum_{a < b} U_{a, b}^{(\\mathrm{nghbr}, +)}\n \\sum_{(i, j)} \\sum_{\\sigma}\n n_{i, a, \\sigma} n_{j, b, \\sigma}\\nonumber\n \\\\&+& \\sum_{a \\leq b} U_{a, b}^{(\\mathrm{onsite}, -)}\n \\sum_{i} \\sum_{\\sigma}\n n_{i, a, \\sigma} n_{i, b, -\\sigma}\\nonumber\n + \n \\sum_{a} U_{a, a}^{(\\mathrm{nghbr}, -)}\n \\sum_{\\{ i, j \\}} \\sum_{\\sigma}\n n_{i, a, \\sigma} n_{j, a, -\\sigma}\n \\\\&+& \\sum_{a < b} U_{a, b}^{(\\mathrm{nghbr}, -)}\n \\sum_{( i, j )} \\sum_{\\sigma}\n n_{i, a, \\sigma} n_{j, b, -\\sigma}\\nonumber\n \\\\&-& \\sum_{a} \\mu_a\n \\sum_i \\sum_{\\sigma} n_{i, a, \\sigma}\n - h \\sum_{i} \\sum_{a}\n \\left(n_{i, a, \\uparrow} - n_{i, a, \\downarrow}\\right) \\qquad (1)\n \\end{eqnarray}\n\nThe tunneling parameter corresponds to the terms\n$$\nt \\sum_{(i, j) \\in E^{(\\mathrm{edge type})}} \n \\sum_{\\sigma}\n \\left(a_{i, a, \\sigma}^{\\dagger} a_{j, b, \\sigma} \n + a_{j, b, \\sigma}^{\\dagger} a_{i, a, \\sigma}\\right)\n$$\nOne can also construct the Hamiltonian for the spinless model, which \u0570as the form\n\n\n\\begin{eqnarray}\n H = &-& \\sum_{a < b} t_{a, b}^{(\\mathrm{onsite})}\n \\sum_{i}\n (a^\\dagger_{i, a} a_{i, b} +\n a^\\dagger_{i, b} a\n a_{i, a}) -\n \\sum_{a} t_{a, a}^{(\\mathrm{nghbr})}\n \\sum_{\\{i, j\\}}\n (a^\\dagger_{i, a} a_{j, a} +\n a^\\dagger_{j, a} a_{i, a})\n \\\\&-& \n \\sum_{a < b} t_{a, b}^{(\\mathrm{nghbr})}\n \\sum_{(i, j)}\n (a^\\dagger_{i, a} a_{j, b} +\n a^\\dagger_{j, b} a_{i, a})\n +\n \\sum_{a < b} U_{a, b}^{(\\mathrm{onsite})}\n \\sum_{i}\n n_{i, a} n_{i, b}\n \\\\\n &+& \\sum_{a} U_{a, a}^{(\\mathrm{nghbr})}\n \\sum_{\\{i, j\\}}\n n_{i, a} n_{j, a}\n + \\sum_{a < b} U_{a, b}^{(\\mathrm{nghbr})}\n \\sum_{(i, j)}\n n_{i, a} n_{j, b}\\nonumber\n - \\sum_{a} \\mu_a\n \\sum_i n_{i, a}\n\\end{eqnarray}\n\nThe tunneling parameter corresponds to the terms\n$$\n-t \\sum_{(i, j) \\in E^{(\\mathrm{edge type})}} \n \\left(a_{i, a}^{\\dagger} a_{j, b} \n + a_{j, b}^{\\dagger} a_{i, a}\\right)$$\n\n### Fermi-Hubbard Model\n\nThe Fermi-Hubbard Hamiltonian for the spinful model has the form \n\n\\begin{eqnarray}\n H = &-& t \\sum_{\\langle i,j \\rangle} \\sum_{\\sigma}\n (a^\\dagger_{i, \\sigma} a_{j, \\sigma} +\n a^\\dagger_{j, \\sigma} a_{i, \\sigma})\n + U \\sum_{i} a^\\dagger_{i, \\uparrow} a_{i, \\uparrow}\n a^\\dagger_{i, \\downarrow} a_{i, \\downarrow}\\nonumber\n \\\\\n &-& \\mu \\sum_i \\sum_{\\sigma} a^\\dagger_{i, \\sigma} a_{i, \\sigma}\n - h \\sum_i (a^\\dagger_{i, \\uparrow} a_{i, \\uparrow} -\n a^\\dagger_{i, \\downarrow} a_{i, \\downarrow})\n\\end{eqnarray}\n\nThe single-band Fermi-Hubbard model, which is a radical simplification of the full electronic structure Hamiltonian, is described by the following Hamiltonian\n\n\\begin{eqnarray}\\label{1}\n H_{FH} = &-&\\sum_{\\langle j,k\\rangle,\\sigma}\n t_{j,k}(a_{j,\\sigma}^\\dagger a_{k,\\sigma}+ \\mathrm{h.c.}) + U \\sum_j n_{j,\\uparrow}n_{j,\\downarrow}\\nonumber\\\\&+&\n \\sum_{j,\\sigma}(\\varepsilon_j-\\mu)n_j-\n \\sum_j h_j(n_{j,\\uparrow} -n_{j,\\downarrow})\n\\end{eqnarray}\nwhere $a_{j,\\sigma}^\\dagger (a_{j,\\sigma})$\nis the creation (annihilation) operator for the $j$th site with spin $\\sigma$\nand $n_{j,\\sigma} = a_{j,\\sigma}^\\dagger a_{j,\\sigma}$ is the fermion occupation number operator and $n_j = \\sum_\\sigma n_{j,\\sigma}$. From all the\nelectronic bands, this model keeps only a single orbital close to the Fermi energy and reduces the\nlong-range Coulomb repulsion to just the local repulsion $U$ within this orbital. The first term on the right side of Eq. (1) describes fermions hopping between sites, the second term describes the on-site interactions, and the remaining two terms describe a local potential and a magnetic field. One can consider a translationally invariant model and then drop the indices in $t_{j,k}$\nand $\\varepsilon_j$ since all sites are\nequivalent.\n\n### Bose-Hubbard model\nThe Hamiltonian for the Bose-Hubbard model has the form \n\n$$H = - t \\sum_{\\langle i, j \\rangle} (b_i^\\dagger b_j + b_j^\\dagger b_i)\n + V \\sum_{\\langle i, j \\rangle} b_i^\\dagger b_i b_j^\\dagger b_j\n + \\frac{U}{2} \\sum_i b_i^\\dagger b_i (b_i^\\dagger b_i - 1)\n - \\mu \\sum_i b_i^\\dagger b_i.\n$$\n\n### Complexity\n\nThe computational complexity of the Hubbard model is reduced compared to the full electronic\nstructure problem, since only a single orbital is used per unit cell instead of dozens needed for the\nfull problem, and the number of terms is linear in $N$ instead of scaling with the fourth power $N^4$.\nOn a lattice of $20\\times 20$ unit cells this means a reduction of the total number of terms from about $10^{12}$ to $10^{3}$, which makes such a simulation feasible on a quantum computer.\n\nThe one-dimensional (1D) Hubbard model was solved by \\cite{Lieb}, however, it remains an open question for a full theoretical analysis of the 2D Hubbard model which requires going beyond the validity of\nmean-field and perturbation theory arguments.\n%The phase diagram of the 2D Fermi-Hubbard model is depicted in the picture below. \n%The critical $T_c$ temperature depends on the value of hole doping, AF stands for antiferromagnet. \nTo characterize the phases one can compute key quantities such as the quasiparticle energy gap and phase stiffness (superfluid density) in a superconducting phase. To answer these questions, one needs to determine the ground-state wave function for a range of interaction parameters and densities and then measure\nground-state correlation functions.\n\nAt half-filling, which is the case of one electron per unit cell, the Hubbard model gives a simple account of Mott insulating behavior. For $U=0$, the ground state is metallic. There is a single band\nand it is half filled, the Fermi surface is $|k_x \\pm k_y| = \\pi/a$, where $a$ is the lattice constant. However,\nfor $U>0$, the ground state is insulating. The physics is different in the $U \\ll t$\nand $U \\gg t$ limits.\nFor $U\\ll t$, the system lowers its energy by developing antiferromagnetic order, as a result of which the unit cell doubles in size. If there are two electrons per unit cell the system is effectively a band\ninsulator. If the Hubbard model captures the physics of the cuprate superconductors, then\nsuperconducting order with $d_{x_2-y_2}$\npairing symmetry must occur for a density of $1-x$ electrons per unit cell, with in the range $0.05 < x < 0.25$. If the Hubbard model is not superconducting in\nthis doping range, then superconductivity in the cuprates must be due to effects that are missing in\nthe Hubbard model, such as longer-ranged interactions, longer-ranged hopping, interlayer coupling,\nand phonons.\n\n\n\nExact numerical diagonalization of Hubbard model is limited to about 20 sites (40 logical qubits)\n[10, 11], which is too small for a finite-size scaling analysis. Classical approximate methods such as\nquantum Monte Carlo simulations or many-body theory expansions have been used to simulate\nsystems with hundreds of sites, which allows for extrapolation to the thermodynamic limit. Monte\nCarlo methods suffer from sign problems which prevent them from simulating systems at very low\ntemperatures [13]. Density matrix renormalization group (DMRG) [14] applied to the 2D Hubbard\nmodel requires mapping to an effective 1D problem. Other numerical methods to determine the\nphase diagrams of the Hubbard model include the dynamical cluster approximation [15, 16] and the\ndensity matrix embedding theory [10, 17]. These methods can asymptotically approach the exact\nsolution with increasingly larger clusters which requires an exponential amount of computing resources on a classical computer. \n\nOn a quantum computer, simulating the Fermi-Hubbard model, one maps the fermionic operators to qubit operators. In the second quantization picture, a particular spin orbital being unoccupied(occupied) is represented by the qubit state $|0\\rangle (|1\\rangle)$. To account for the parities of qubits\ncorresponding to other spin orbitals, e.g., by using the Jordan-Wigner transformation (JWT) [18, 27], fermionic anticommutation relations must be satisfied.\nThe last two terms of Eq. (1) can be implemented using single qubit operators, the on-site\ninteraction term can be implemented with two-qubit interactions. Because of the nonlocal parity\noperators in the JWT, the hopping terms cannot be implemented in more than one spatial\ndimension. It is of practical importance to reduce the depth of the quantum circuits for these terms\nwith only local qubit interactions [7]. Several quantum algorithms to simulate fermionic systems on\nnear-term devices with nearest-neighbor qubit-qubit couplings has been reported recently [20]. This\nopens a possibility to simulate many body physics, including the Fermi-Hubbard model, in the noisy\nintermediate-scale quantum era [21].\n\n\n### Mapping to code\n\nLet's build the code structure and show the approach, where we describe the transition between conducting and insulating systems. \nWe now give the solving approach of a restricted case of Fermi-Hubbard model.\n\n\nBy exploiting the open source OpenFermion library [22] one can prepare an arbitrary Slater determinant and map fermions to qubits. First, we describe the interface between OpenFermion and Rigetti\u2019s quantum simulation environment called Forest. The interface provides a method of transforming data generated in OpenFermion to a similar representation\nin pyQuil [24]. For this example we use OpenFermion to build a four-site single-band periodic boundary Hubbard model and apply first-order Trotter time-evolution to a starting state of two localized electrons of opposite spin.\nThe Forest-OpenFermion plugin provides the routines to inter-convert between the OpenFermion `QubitOperator` data structure and the synonymous data structure in pyQuil called a `PauliSum`.\n\nThe `FermionOperator` in OpenFermion can be used to translate the mathematical expression of the Hamiltonian directly to executable code. While we show how this model can be built in one line using the OpenFermion hamiltonians module, here we take the opportunity to demonstrate the ease of creating such models for study that could be\neasily modified as desired. Given the Hamiltonian of the Hubbard system \n\n\\begin{equation}\n H_{FH} = - \\sum_{\\langle j,k\\rangle,\\sigma}\n t_{j,k}(a_{j,\\sigma}^\\dagger a_{k,\\sigma}+ \\mathrm{h.c.}) + 4 \\sum_j n_{j,\\uparrow}n_{j,\\downarrow},\n\\end{equation} \n\nwhere we first consider the problem without local potential and magnetic field. We also used $U=4$ for numerical procedures. The example code to build this Hamiltonian is as follows\n\n\n```python\nfrom openfermion.transforms import jordan_wigner \nfrom openfermion.ops import FermionOperator\nfrom openfermion.ops import hermitian_conjugated\n\nhubbard_hamiltonian = FermionOperator()\nspatial_orbitals = 4\nfor i in range(spatial_orbitals):\n electron_hop_alpha = FermionOperator(((2*i, 1),\n (2*((i+1) % spatialorbitals), 0)))\n electron_hop_beta = FermionOperator(((2*i+1, 1),\n ((2*((i+1) % spatial_orbitals) + 1 ),0)))\n hubbard_hamiltonian +=-1*(electron_hop_alpha +\n hermitian_conjugated(electron_hop_alpha))\n hubbard_hamiltonian += -1*(electron_hop_beta +\n hermitian_conjugated(electron_hop_beta))\n hubbard_hamiltonian += FermionOperator (((2*i, 1), \n (2*i, 0), (2*i+1, 1), (2*i+1, 0)), 4.0)\n```\n\nHere, we have implicitly used even indexes [0, 2, 4, 6] as $\\alpha$ spin-orbitals and odd indexes [1, 3, 5, 7] as $\\beta$\nspin-orbitals. \n\nThe same model can be built using the OpenFermion Hubbard model builder routine in the hamiltonians module with a single function call:\n\n```python\nfrom openfermion.hamiltoniansi mport fermi_hubbard\n\nx_dim = 4\ny_dim = 1\nperiodic = True\nchemical_potential = 0\ntunneling = 1.0\ncoulomb = 4.0\n\nof_hubbard_hamiltonian = fermi_hubbard(x_dim, y_dim, tunneling,\n coulomb, chemical_potential=None, spinless=False)\n```\nThe above variables are self-explanatory.\n\nUsing the Jordan-Wigner transform functionality of OpenFermion, the Hubbard Hamiltonian can be transformed to a sum of QubitOperators which are then transformed to pyQuil PauliSum objects using routines in the Forest-OpenFermion plugin imported earlier.\n\n```python\nhubbard_term_generator = jordan_wigner(hubbard_hamiltonian)\npyquil_hubbard_generator = qubitop_to_pyquilpauli(hubbard_term_generator)\n```\n\nNow, with the data successfully transformed to a pyQuil representation, one can use them to construct quantum circuits on rigetti's simulator or hardware and make experiments.\n\nOften, one would like to obtain a sparse matrix representation of an operator which can be analyzed numerically. There is code in both openfermion.transforms and openfermion.utils which facilitates this. The function get\\_sparse\\_operator converts either a FermionOperator, a QubitOperator or other more advanced classes such as InteractionOperator to a scipy.sparse.csc matrix. There are numerous functions in openfermion.utils which one can call on the sparse operators such as \"get\\_gap\", \"get\\_hartree\\_fock\\_state\", \"get\\_ground\\_state\", etc. We show this off by computing the ground state energy of the Hubbard model. To do that, we use code from the openfermion.hamiltonians module which constructs lattice models of fermions such as Hubbard models.\n\n```python\nfrom openfermion.hamiltonians import fermi_hubbard\nfrom openfermion.transforms import get_sparse_operator \nfrom openfermion.transforms import jordan_wigner\nfrom openfermion.utils import get_ground_state\n\n# Set model.\nx_dim = 2\ny_dim = 2\ntunneling = 2.\ncoulomb = 1.\nmagnetic_field = 0.5\nchemical_potential = 0.25\nperiodic = 1\nspinless = 1\n\n# Get fermion operator.\nhubbard_model = fermi_hubbard(\n x_dimension, y_dimension, tunneling, coulomb, \n chemical_potential, magnetic_field, periodic, spinless)\n\n# Get qubit operator under Jordan-Wigner.\njw_hamiltonian = jordan_wigner(hubbard_model)\njw_hamiltonian.compress()\n\n# Get scipy.sparse.csc representation.\nsparse_operator = get_sparse_operator(hubbard_model)\n\nprint('\\nEnergy of the model is {}'.format(\n get_ground_state(sparse_operator)[0]))\n```\n\n\n### Measuring an observable\n\nTo measure observables or correlation functions on a quantum computer we use Paili group expansion of the related operator. \n\\begin{equation}\n O = \\sum_{i=1}^n \\alpha_i P_i,\n\\end{equation}\nwhere $n$ is the basis dimension needed for expansion and $P_i \\in \\{I, \\sigma_x, \\sigma_y, \\sigma_z\\}^{\\otimes N}$ is a tensor product of single-qubit Pauli operators on $N$ qubits.\nThis is needed as quantum circuits are natural for sampling in Pauli basis.\nThen the expectation value of the observable $O$ is \n\\begin{equation}\n \\langle O\\rangle = \\sum_{i=1}^n \\alpha_i \\langle P_i\\rangle,\n\\end{equation}\n\nFor constructing Slater determinants to approximate the ground state, we use the variational quantum eigensolver algorithm (VQE), where the ansatz state is prepared via a parametrized quantum circuit and optimized for minimum energy. The advantage of VQE over classical simulation methods is that it can prepare trial states that are not amenable to efficient classical numerics. We have described the VQE algorithm in [[26]](https://github.com/gate42qc/Tutorials/blob/master/VQE/VQEforhydrogenmoleculeenergy.ipynb).\n\n\nSo let's prepare the necessary packages and start modeling the problem\n\n\n```python\nfrom openfermion.hamiltonians import fermi_hubbard\nfrom openfermion.transforms import get_fermion_operator, jordan_wigner\nfrom openfermion.utils import get_ground_state\nfrom openfermion.ops import QubitOperator \nfrom forestopenfermion import pyquilpauli_to_qubitop \nfrom forestopenfermion import qubitop_to_pyquilpauli\n\nimport numpy as np\nimport random\nimport matplotlib.pyplot as plt\n%matplotlib inline\n\nfrom pyquil.quil import Program\nfrom pyquil.gates import *\nfrom pyquil.paulis import exponentiate\nfrom pyquil.quil import Program\nimport pyquil.api as api\n# we also import a function from module random_circuit we created\nfrom random_circuit import get_ansatz_circuit_generator\n```\n\n\n```python\n# Setting the model\nx_dimension = 3\ny_dimension = 2\ntunneling = 2.\ncoulomb = 1.0\nmagnetic_field = 0.5\nchemical_potential = 0.25\nperiodic = 1\nspinless = 0\n\n# create range of parameters for scan to model different material specifics\n# change to desired values\ntunnelings = np.arange(0.5, 1.5, 0.02)\ncoulombs = np.arange(0, 1.5, 0.05)\nmagnetic_fields = np.arange(0.5, 20, 2)\nchemical_potentials = np.arange(-500., 500, 50)\n\n# Get fermion operators\nhubbard_tunnels = [fermi_hubbard(x_dimension, y_dimension, tunneling, coulomb, \n chemical_potential, magnetic_field, periodic, spinless)\n for tunneling in tunnelings]\n\nhubbard_coulombs = [fermi_hubbard(x_dimension, y_dimension, tunneling, coul, \n chemical_potential, magnetic_field, periodic, spinless)\n for coul in coulombs]\n\nhubbard_magn = [fermi_hubbard(x_dimension, y_dimension, tunneling, coulomb, \n chemical_potential, magnetic_field, periodic, spinless)\n for magn in magnetic_fields]\n\nhubbard_chem = [fermi_hubbard(x_dimension, y_dimension, tunneling, coulomb, \n chemical_potential, magnetic_field, periodic, spinless)\n for chem in chemical_potentials]\n```\n\n\n```python\ncoulombs\n```\n\n\n\n\n array([0. , 0.05, 0.1 , 0.15, 0.2 , 0.25, 0.3 , 0.35, 0.4 , 0.45, 0.5 ,\n 0.55, 0.6 , 0.65, 0.7 , 0.75, 0.8 , 0.85, 0.9 , 0.95, 1. , 1.05,\n 1.1 , 1.15, 1.2 , 1.25, 1.3 , 1.35, 1.4 , 1.45])\n\n\n\n\n```python\n# Get qubit operator Hubbard hamiltonian under Jordan-Wigner transformation.\n\njw_ham_tunnels = [jordan_wigner(hubbard_model) for hubbard_model in hubbard_tunnels]\njw_ham_coulombs = [jordan_wigner(hubbard_model) for hubbard_model in hubbard_coulombs]\njw_ham_magn = [jordan_wigner(hubbard_model) for hubbard_model in hubbard_magn]\njw_ham_chem = [jordan_wigner(hubbard_model) for hubbard_model in hubbard_chem]\n\n#pyquil_hubbard_generator = qubitop_to_pyquilpauli(jw_hamiltonian)\njw_pyquil_hubbard_tunnels = [qubitop_to_pyquilpauli(jw_hamiltonian) for jw_hamiltonian in jw_ham_tunnels]\njw_pyquil_hubbard_coulombs = [qubitop_to_pyquilpauli(jw_hamiltonian) for jw_hamiltonian in jw_ham_coulombs]\njw_pyquil_hubbard_magn = [qubitop_to_pyquilpauli(jw_hamiltonian) for jw_hamiltonian in jw_ham_magn]\njw_pyquil_hubbard_chem = [qubitop_to_pyquilpauli(jw_hamiltonian) for jw_hamiltonian in jw_ham_chem]\n```\n\n### Ground state preparation using VQE\nFor usage of VQE see [our tutorial](https://github.com/gate42qc/Tutorials/blob/master/VQE/VQE_for_hydrogen_molecule_energy.ipynb)\n\n\n```python\n# depth of the ansatz circuit\nd = 1 \n# number of qubits are defined by lattice dimensions\nqubits = list(range(2* x_dimension * y_dimension))\nnumber_of_qubits = len(qubits)\n```\n\n\n```python\n\nvariational_state_evolve = get_ansatz_circuit_generator(qubits, d)\n\n```\n\n\n```python\nthetas = np.random.rand(number_of_qubits*(d+1))*np.pi\n\n```\n\n\n```python\n# Let's see the first Hubbard hamiltonian represented as a Pauli expansion\nprint(jw_pyquil_hubbard_tunnels[1])\n```\n\n (-0.26+0j)*Y0*Z1*Y2 + (-0.26+0j)*X0*Z1*X2 + (-0.26+0j)*Y1*Z2*Y3 + (-0.26+0j)*X1*Z2*X3 + (-0.26+0j)*Y0*Z1*Z2*Z3*Z4*Z5*Y6 + (-0.26+0j)*X0*Z1*Z2*Z3*Z4*Z5*X6 + (-0.26+0j)*Y1*Z2*Z3*Z4*Z5*Z6*Y7 + (-0.26+0j)*X1*Z2*Z3*Z4*Z5*Z6*X7 + (-0.375+0j)*Z1 + (0.125+0j)*Z0 + (0.25+0j)*Z0*Z1 + (-0.26+0j)*Y2*Z3*Y4 + (-0.26+0j)*X2*Z3*X4 + (-0.26+0j)*Y3*Z4*Y5 + (-0.26+0j)*X3*Z4*X5 + (-0.26+0j)*Y2*Z3*Z4*Z5*Z6*Z7*Y8 + (-0.26+0j)*X2*Z3*Z4*Z5*Z6*Z7*X8 + (-0.26+0j)*Y3*Z4*Z5*Z6*Z7*Z8*Y9 + (-0.26+0j)*X3*Z4*Z5*Z6*Z7*Z8*X9 + (-0.375+0j)*Z3 + (0.125+0j)*Z2 + (0.25+0j)*Z2*Z3 + (-0.26+0j)*X0*Z1*Z2*Z3*X4 + (-0.26+0j)*Y0*Z1*Z2*Z3*Y4 + (-0.26+0j)*X1*Z2*Z3*Z4*X5 + (-0.26+0j)*Y1*Z2*Z3*Z4*Y5 + (-0.26+0j)*Y4*Z5*Z6*Z7*Z8*Z9*Y10 + (-0.26+0j)*X4*Z5*Z6*Z7*Z8*Z9*X10 + (-0.26+0j)*Y5*Z6*Z7*Z8*Z9*Z10*Y11 + (-0.26+0j)*X5*Z6*Z7*Z8*Z9*Z10*X11 + (-0.375+0j)*Z5 + (0.125+0j)*Z4 + (0.25+0j)*Z4*Z5 + (-0.26+0j)*Y6*Z7*Y8 + (-0.26+0j)*X6*Z7*X8 + (-0.26+0j)*Y7*Z8*Y9 + (-0.26+0j)*X7*Z8*X9 + (-0.375+0j)*Z7 + (0.125+0j)*Z6 + (0.25+0j)*Z6*Z7 + (-0.26+0j)*Y8*Z9*Y10 + (-0.26+0j)*X8*Z9*X10 + (-0.26+0j)*Y9*Z10*Y11 + (-0.26+0j)*X9*Z10*X11 + (-0.375+0j)*Z9 + (0.125+0j)*Z8 + (0.25+0j)*Z8*Z9 + (-0.26+0j)*X6*Z7*Z8*Z9*X10 + (-0.26+0j)*Y6*Z7*Z8*Z9*Y10 + (-0.26+0j)*X7*Z8*Z9*Z10*X11 + (-0.26+0j)*Y7*Z8*Z9*Z10*Y11 + (-0.375+0j)*Z11 + (0.125+0j)*Z10 + (0.25+0j)*Z10*Z11\n\n\n\n```python\nfrom pyquil.api import get_qc\nqc = get_qc(\"12q-qvm\")\n```\n\n\n```python\n#from grove.pyvqe.vqe import VQE\nfrom vqe import VQE\nfrom scipy.optimize import minimize\n```\n\n\n```python\ninitial_parameters = thetas\n\nimport time\ninst = VQE(minimizer=minimize, minimizer_kwargs={'method': 'COBYLA'}) # 'BFGS'\n```\n\n\n```python\nstart = time.time()\nprint(\"-------------------starting optimization---------------------\")\n\n# # the VQE result at each interatomic distance is stored results_wf_sim\ncoulombs_results_wf_sim = [inst.vqe_run(variational_state_evolve, \n ham, \n initial_params=initial_parameters,\n qc=qc, samples=None)\n for ham in jw_pyquil_hubbard_coulombs]\n\n\nend = time.time()\nprint(\"optimization done - elapsed time is:\")\nprint((end - start)/60, 'minutes')\n```\n\n -------------------starting optimization---------------------\n optimization done - elapsed time is:\n 33.0844568490982 minutes\n\n\n\n```python\nenergies_coul = [res.fun for res in coulombs_results_wf_sim]\nenergies_coul\n```\n\n\n\n\n [-1.1426890813730481,\n -1.3330018265168322,\n -1.5126914585368993,\n -1.6720591321126526,\n -1.7176992812524543,\n -1.9007224635754165,\n -2.0448595511300165,\n -2.1627945731765847,\n -2.327116903337025,\n -2.412578791711057,\n -2.7363764168681803,\n -2.904921603669198,\n -3.026266834452335,\n -3.1297902726221527,\n -3.2178460715050567,\n -3.3124344873688756,\n -3.3869322998856144,\n -3.4598857315806173,\n -3.510606693961045,\n -3.5602624282936617,\n -3.5968501116056193,\n -3.6272681189294693,\n -3.9906241303561383,\n -3.9996540559853995,\n -3.649124520467428,\n -3.634780693407558,\n -3.610896092727703,\n -3.576449924238313,\n -3.595830848595052,\n -3.478665025105412]\n\n\n\n\n```python\nplt.scatter(coulombs, energies_coul, label='Fermi_Hubbard_coulomb')\nplt.ylabel('Energy (in units of T and J)')\nplt.savefig('images/FH_energy_no_noise.png', format='png', dpi=1000)\nplt.show()\n```\n\nThe above simulation is done algebraically. We can do same using sampling approach.\n\n\n```python\nstart = time.time()\nprint(\"-------------------starting optimization---------------------\")\n\n# the VQE result at each interatomic distance is stored results_wf_sim\nresults_sampl = [inst.vqe_run(variational_state_evolve, \n ham, \n initial_params=initial_parameters,\n qc=qc, samples=100)\n for ham in jw_pyquil_hubbard_tunnels]\n\n\nend = time.time()\nprint(\"------------optimization done - elapsed time is:--------------\")\nprint((end - start)/60, 'minutes')\n```\n\n\n```python\nenergies_sampl = [res.fun for res in results_sampl]\nprint(energies_sampl)\n```\n\n\n```python\nplt.scatter(coulombs, energies_sampl, label='Fermi_Hubbard')\n#plt.title('Sampling without noise')\nplt.xlabel('tunnelings')\nplt.ylabel('Energy (in units of T and J)')\nplt.savefig('images/FH_energy_samp_no_noise.png', format='png', dpi=1000)\nplt.show()\n```\n\n### References\n[1] J. Hubbard, \u201dElectron Correlations in Narrow Energy Bands\u201d. Proceed-ings of the Royal Society of London.276, 238 (1963) \n[2] R. P. Feynman, \u201cSimulating physics with computers,\u201d InternationalJournal of Theoretical Physics21, 467 (1982). \n[3] I. Georgescu, et. al., \u201cQuantum simulation,\u201d Reviews of Modern Physics86, 153 (2014). \n[4] M. B. Hastings, et. al., \u201cImproving Quantum Algorithms for QuantumChemistry,\u201d Quantum Info. Comput.15, 1 (2015) \n[5] A. Kandala, et. al., \u201cHardware-efficient variational quantum eigensolverfor small molecules and quantum magnets,\u201d Nature549, 242 (2017). \n[6] M. Reiher, et. al., \u201cElucidating reaction mechanisms on quantum com-puters,\u201d Proceedings of the National Academy of Sciences114, 7555(2017). \n[7] R. Babbush, et. al., \u201cLow Depth Quantum Simulation of ElectronicStructure,\u201d arXiv:1706.00023 (2017) \n[8] D. Wecker, et. al., \u201cSolving strongly correlated electron models on aquantum computer,\u201d Physical Review A92, 062318 (2015) \n[9] E. H. Lieb and F. Y. Wu, \u201cAbsence of Mott transition in an exactsolution of the short-range, one-band model in one dimension,\u201d Phys.Rev. Lett.20, 1445 (1968). \n[10] B.-X. Zheng and G. K.-L. Chan, \u201cGround-state phase diagram of thesquare lattice Hubbard model from density matrix embedding theory,\u201dPhysical Review B93, 035126 (2016). \n[11] C. J. Jia, et. al., \u201cFidelity study of the superconducting phase diagramin the two-dimensional single-band Hubbard model,\u201d Phys. Rev. B84,125113 (2011). \n[12] S. Yamada, et. al., \u201c16.447 tflops and 159-billion-dimensional exact-diagonalization for trapped fermion-hubbard model on the earth sim-ulator,\u201d in Supercomputing, 2005. Proceedings of the ACM/IEEE SC2005 Conference (2005) pp. 44. \n[13] M. Troyer and U.-J. Wiese, \u201cComputational Complexity and Funda-mental Limitations to Fermionic Quantum Monte Carlo Simulations,\u201dPhysical Review Letters94, 170201 (2005). \n[14] S. R. White, \u201cDensity-matrix algorithms for quantum renormalizationgroups,\u201d Physical Review B48, 10345 (1993). \n[15] M. H. Hettler, et. al, \u201cDynamical cluster approximation: Nonlocal dy-namics of correlated electron systems,\u201d Physical Review B61, 12739(2000). \n[16] G. Kotliar, et. al., \u201cElectronic structure calculations with dynamicalmean-field theory,\u201d Reviews of Modern Physics78, 865 (2006). \n[17] G. Knizia and G. K.-L. Chan, \u201cDensity Matrix Embedding: A SimpleAlternative to Dynamical Mean-Field Theory,\u201d Physical Review Letters109, 186404 (2012). \n[18] G. Ortiz, et. al., \u201cQuantum algorithms for fermionic simulations,\u201d Phys-ical Review A64, 022319 (2001). \n[19] J. D. Whitfield, et. al., \u201cSimulation of electronic structure Hamiltoniansusing quantum computers,\u201d Molecular Physics109, 735 (2011). \n[20] Z. Jiang, et. al., \u201cQuantum Algorithms to Simulate Many-Body Physicsof Correlated Fermions\u201d, Physical Review Applied9, 044036 (2018). \n[21] J. Preskill, \u201cQuantum computing in the NISQ era and beyond,\u201darXiv:1801.00862 (2018). \n[22] J. R. McClean, et. al., \u201dOpenFermion: The electronic structure packagefor quantum computers,\u201d arXiv:1710.07629 (2017). \n[23] S. B. Bravyi and A. Y. Kitaev, \u201cFermionic Quantum Computation,\u201dAnnals of Physics298, 210 (2002). \n[24] https://github.com/rigetti/pyquil \n[25] R. S. Smith, M. J. Curtis, and W. J. Zeng, \u201dA practical quantum in-struction set architecture,\u201d arXiv:1608.03355 (2016). \n[26] https://github.com/gate42qc/Tutorials/blob/master/VQE/VQEforhydrogenmoleculeenergy.ipynb \n[27] J. D. Whitfield, et. al., \u201cLocal spin operators for fermion simulations,\u201dPhys. Rev. A94, 030301 (2016). \n[28] I. D. Kivlichan, et. al., \u201cQuantum simulation of electronic structure withlinear depth and connectivity,\u201d arXiv:1711.04789 (2017). \n[29] Arute et. al., \u201dQuantum supremacy using a programmable supercon-ducting processor\u201d, Nature574, 505 (2019). \n[30] V. Bach, E. H. Lieb, and J. P. Solovej, \u201cGeneralized Hartree-Fock theoryand the Hubbard model,\u201d Journal of Statistical Physics76, 3 (1994).\n", "meta": {"hexsha": "445f26a8c60d522b81793cae1d54812e975229cd", "size": 51597, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Fermi_Hubbard/Fermi-Hubbard.ipynb", "max_stars_repo_name": "gate42qc/Tutorials", "max_stars_repo_head_hexsha": "06444c70a56d7ac72ea417fcc5470d8be96573e1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2019-09-14T02:54:37.000Z", "max_stars_repo_stars_event_max_datetime": "2020-04-25T19:12:00.000Z", "max_issues_repo_path": "Fermi_Hubbard/Fermi-Hubbard.ipynb", "max_issues_repo_name": "gate42qc/Tutorials", "max_issues_repo_head_hexsha": "06444c70a56d7ac72ea417fcc5470d8be96573e1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2019-09-13T10:52:56.000Z", "max_issues_repo_issues_event_max_datetime": "2019-09-13T10:52:56.000Z", "max_forks_repo_path": "Fermi_Hubbard/Fermi-Hubbard.ipynb", "max_forks_repo_name": "gate42qc/Tutorials", "max_forks_repo_head_hexsha": "06444c70a56d7ac72ea417fcc5470d8be96573e1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2019-09-13T07:16:52.000Z", "max_forks_repo_forks_event_max_datetime": "2020-02-11T23:34:31.000Z", "avg_line_length": 64.0956521739, "max_line_length": 9656, "alphanum_fraction": 0.6755819137, "converted": true, "num_tokens": 9948, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.665410572017153, "lm_q1q2_score": 0.43811796567485367}} {"text": "```python\nimport pandas as pd\nimport numpy as np\nfrom scipy.stats import norm\nimport scipy\n```\n\n\n```python\nEuroOptionClean = pd.read_csv(r'C:\\Users\\HP\\Desktop\\Fintech\\final\\final_project\\EuropeanOptionCleanData.csv')\n\nEuroOptionClean=EuroOptionClean.drop(columns='Unnamed: 0')\nmyData=EuroOptionClean.copy()\n```\n\n\n```python\n#myData=myData.drop([245,269,357,648,779,831,834])\n\n#myData.set_index(pd.Index(index))\n```\n\n\n```python\nmyData\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    currentDateExpDateStrikePriceTickerTypeLastIVChgStockPriceTImpliedVolatility (caluclated)Difference
    02/17/20173/17/201722.5ARRCall0.080.13956-0.0121.7099990.0766600.1374280.002132
    12/17/20173/17/201720.0ARRPut0.200.32783NaN21.7099990.0766600.336280-0.008450
    22/17/20173/17/2017180.0MSGPut6.450.18380-0.25175.3000030.0766600.187619-0.003819
    32/17/20175/19/2017185.0MSGCall3.100.18440NaN175.3000030.2491440.190170-0.005770
    42/17/20175/19/2017170.0MSGPut4.200.18729NaN175.3000030.2491440.191963-0.004673
    .......................................
    22662/17/20175/19/201710.0KPTIPut1.350.81455NaN10.6000000.2491440.8047350.009815
    22672/17/20178/18/201722.5KELYACall1.050.19477NaN21.9900000.4982890.202215-0.007445
    22682/17/20173/17/20175.0KERXCall0.450.75406-0.325.0600000.0766600.754426-0.000366
    22692/17/20173/17/20175.5KERXCall0.280.80470-0.025.0600000.0766600.8023530.002347
    22702/17/20173/17/20175.0KERXPut0.400.78303NaN5.0600000.0766600.7760350.006995
    \n

    2271 rows \u00d7 12 columns

    \n
    \n\n\n\n\n```python\nCall = myData[myData['Type']=='Call']\nPut = myData[myData['Type']=='Put']\nType=myData['Type'].values\nS = myData['StockPrice'].values\nK = myData['StrikePrice'].values\nT = myData['T'].values\nP=myData['Last'].values\nVol =myData['IV'].values\n```\n\nFunction definition\n\n\n```python\ndef OptionValue(S, K, T, r , Type ,sigma):\n \n d1 = (np.log(S /K) + (r + 0.5 * sigma**2) * T )/(sigma * np.sqrt(T))\n \n d2 = (np.log(S /K) + (r - 0.5 * sigma**2) * T )/(sigma * np.sqrt(T))\n\n if Type == 'Call':\n p = (S * norm.cdf(d1, 0, 1) - K * np.exp(-r * T) * norm.cdf(d2, 0, 1)) \n elif Type == 'Put':\n p = (K*np.exp(-r*T)*norm.cdf(-d2, 0.0, 1.0) - S*norm.cdf(-d1, 0.0, 1.0))\n return p\n```\n\n\n```python\n\ndef vega(S, K, T, sigma, r = 0.03):\n \n d1 = (np.log(S/K) + (r + 0.5*sigma**2)*T)/(sigma*np.sqrt(T))\n vega = (S * norm.pdf(d1, 0, 1) * np.sqrt(T)) \n return vega\ndef vomma(S, K, T, sigma, r = 0.03):\n d1 = (np.log(S/K) + (r + 0.5*sigma**2)*T)/(sigma*np.sqrt(T))\n d2=d1-sigma*np.sqrt(T)\n vomma=vega(S, K, T, sigma, r = 0.03)*d1*d2/sigma\n return vomma\n\ndef Bisection(S,K,T,l,r,rf,price,Type,tol=0.000000000001):\n count=1\n while r-l>tol:\n count=count+1\n mid=float((l+r)/2);\n if OptionValue(S,K,T,rf,Type,mid)>price:\n r=mid\n else:\n l=mid\n \n return l,count\n\n\n \ndef imp_vol_using_Newton(S, K, T, r, Price,Type,e,x0):\n \n count=1\n def newtons_method(S, K, T, r, Price,Type,x0, e):\n global count\n \n count=1\n delta = OptionValue (S,K,T,r,Type,x0) - (Price)\n while delta > e:\n count=count+1\n #print(count)\n x0 = (x0 - (OptionValue (S,K,T,r,Type,x0) - Price)/vega (S,K,T,x0,0.03))\n delta = abs(OptionValue (S,K,T,r,Type,x0) - Price)\n return x0,count\n sig ,count= newtons_method(S, K, T, r, Price,Type,x0 , e) \n return sig,count\n\nfrom scipy import optimize\n\n\ndef implied_vol_using_blent(S, K, T, r, Price,Type):\n def blent(x0):\n p1=OptionValue (S,K,T,r,Type,x0)-Price\n return p1\n\n root=optimize.brentq(blent,0.0000001,0.9999999)\n return root\n\ndef imp_vol_using_Halley(S, K, T, r, Price,Type,e,x0):\n count=1\n def Halley_method(S, K, T, r, Price,Type,x0, e):\n global count\n count=1\n delta = OptionValue (S,K,T,r,Type,x0) - (Price)\n while delta > e:\n count=count+1\n v=vega(S, K, T, x0, r = 0.03)\n vv=vomma(S, K, T, x0, r = 0.03)\n x0 = x0 - 2*delta*v/(2*v*v-vv*delta)\n delta = abs(OptionValue (S,K,T,r,Type,x0) - Price)\n return x0,count\n sig,count = Halley_method(S, K, T, r, Price,Type,x0 , e) \n return sig,count\n\ndef Muller(S, K, T, x0, x1, x2, Price, r = 0.03, Type = 'Call'):\n f0 = OptionValue(S, K, T, r, Type,x0)-Price\n f1 = OptionValue(S, K, T, r, Type,x0)-Price\n f2 = OptionValue(S, K, T, r, Type,x0)-Price \n c = f2\n b = ((x0-x2)**2 * (f1-f2)-(x1-x2)**2 * (f0-f2))/((x0-x2)*(x1-x2)*(x0-x1))\n a = ((x1-x2)*(f0-f2)-(x0-x2)*(f1-f2))/((x0-x2)*(x1-x2)*(x0-x1))\n if ((b-np.sqrt(b**2-4*a*c))>(b+np.sqrt(b**2-4*a*c))):\n x3 = x2-2*c/(b-np.sqrt(b**2-4*a*c))\n return x3\n \n else:\n x3 = x2-2*c/(b+np.sqrt(b**2-4*a*c))\n return x3\n\ndef MullerBisection(S, K, T, Xsmall, Xbig, Price, eps, r = 0.03, Type = 'Call'):\n \n count = 1\n while Xbig-Xsmall>eps:\n count = count + 1\n Xmid = float((Xsmall+Xbig)/2);\n XmiddleNew = Muller(S, K, T, Xsmall, Xbig, Xmid, Price, r, Type)\n if OptionValue(S, K, T, r, Type ,Xmid ) > Price:\n Xbig = Xmid\n if (Xsmall < XmiddleNew < Xbig):\n \n Xmiddle = XmiddleNew\n else:\n Xmiddle = (Xsmall+Xbig)/2.0 \n else:\n Xsmall = Xmid\n if (Xsmall < XmiddleNew < Xbig):\n Xmiddle = XmiddleNew \n else:\n Xmiddle = (Xsmall+Xbig)/2.0\n return Xsmall,count\n \n\n```\n\n\n```python\nMullerBisection(S[245], K[245], T[245], 0.000001, 0.99999, P[245], 0.00000000001, 0.03, Type [245])\n\n```\n\n D:\\Anaconda\\lib\\site-packages\\ipykernel_launcher.py:82: RuntimeWarning: divide by zero encountered in double_scalars\n\n\n\n\n\n (1e-06, 38)\n\n\n\nApply all methods to the whole dataset and get the est sigma\n\n\n\n```python\nsig_Bisection=[]\nsig_Brent=[]\nsig_MullerSection=[]\nsig_NewTon=[]\nsig_Halley=[]\nfor i in range(len(myData)):\n sig_Bisection.append(Bisection(S[i],K[i],T[i],0.00001,0.99999,0.03,P[i],Type[i],0.000000000001))\n sig_NewTon.append(imp_vol_using_Newton(S[i], K[i], T[i], 0.03, P[i],Type[i],0.000000000001,1))\n sig_MullerSection.append(MullerBisection(S[i], K[i], T[i], 0.00000001, 0.999999, P[i], 0.000000000001, 0.03, Type[i]))\n sig_Halley.append(imp_vol_using_Halley(S[i], K[i], T[i], 0.03, P[i],Type[i],0.000000000001,1))\n \n try:\n \n sig_Brent.append(implied_vol_using_blent(S[i], K[i], T[i], 0.03, P[i],Type[i]))\n except:\n sig_Brent.append(-1)\n \n```\n\n D:\\Anaconda\\lib\\site-packages\\ipykernel_launcher.py:82: RuntimeWarning: divide by zero encountered in double_scalars\n D:\\Anaconda\\lib\\site-packages\\ipykernel_launcher.py:37: RuntimeWarning: divide by zero encountered in double_scalars\n D:\\Anaconda\\lib\\site-packages\\ipykernel_launcher.py:3: RuntimeWarning: invalid value encountered in double_scalars\n This is separate from the ipykernel package so we can avoid doing imports until\n D:\\Anaconda\\lib\\site-packages\\ipykernel_launcher.py:5: RuntimeWarning: invalid value encountered in double_scalars\n \"\"\"\n D:\\Anaconda\\lib\\site-packages\\ipykernel_launcher.py:64: RuntimeWarning: invalid value encountered in double_scalars\n\n\n\n```python\nsig_new_Newton=[]\nsig_new_Halley=[]\nfor i in range(len(myData)):\n if(sig_Brent[i]==1):\n sig_new_Newton.append(-1)\n sig_new_Halley.append(-1)\n else:\n sig_new_Newton.append(imp_vol_using_Newton(S[i], K[i], T[i], 0.03, P[i],Type[i],0.000000000001,sig_Brent[i]))\n sig_new_Halley.append(imp_vol_using_Halley(S[i], K[i], T[i], 0.03, P[i],Type[i],0.000000000001,sig_Brent[i])) \n```\n\n\n```python\npd.DataFrame(sig_Bisection).iloc[:,1]\n```\n\n\n\n\n 0 41\n 1 41\n 2 41\n 3 41\n 4 41\n ..\n 2266 41\n 2267 41\n 2268 41\n 2269 41\n 2270 41\n Name: 1, Length: 2271, dtype: int64\n\n\n\n\n```python\nsig_NewTon\n```\n\n\n\n\n [(0.13273864438271227, 6),\n (0.3412234560295547, 6),\n (0.20113328358443297, 5),\n (0.1797851122456537, 5),\n (0.20442399137175243, 5),\n (0.20652540267958533, 5),\n (0.22107497489372754, 5),\n (0.3316289597174576, 4),\n (0.3215890948474279, 5),\n (0.5514395513529742, 5),\n (0.4606638701372737, 4),\n (0.4532345852915086, 4),\n (0.4449835616801641, 4),\n (0.2355353971268414, 5),\n (0.3316289597174576, 4),\n (0.3215890948474279, 5),\n (0.3316289597174576, 4),\n (0.3215890948474279, 5),\n (0.5514395513529742, 5),\n (0.4606638701372737, 4),\n (0.4532345852915086, 4),\n (0.4449835616801641, 4),\n (0.6017863102582938, 4),\n (0.6742365487870382, 4),\n (0.4246596723003124, 5),\n (0.31828564457230507, 5),\n (0.6029929311546932, 4),\n (0.9581086375875975, 4),\n (0.44057645739700824, 4),\n (0.8216310487724615, 4),\n (0.9041651252028005, 4),\n (0.9288577954310914, 4),\n (0.15060246349144768, 5),\n (0.1586446005432378, 4),\n (0.16928744173294497, 5),\n (0.17647905739984351, 6),\n (0.20692487299687026, 7),\n (0.24903682983918538, 7),\n (0.17865749390339722, 5),\n (0.19547151668108761, 5),\n (0.18582829607520032, 6),\n (0.2853126513769948, 6),\n (0.19546576336325255, 4),\n (0.1831114752884834, 6),\n (0.17998051581469798, 5),\n (0.1901834202108988, 4),\n (0.1897293780123207, 4),\n (0.19228375286183041, 5),\n (0.1767662909430255, 5),\n (0.17135685543607823, 4),\n (0.2179674072236606, 6),\n (0.21389214491811404, 6),\n (0.21378822110449158, 5),\n (0.24233330823408755, 6),\n (0.22180984959174724, 6),\n (0.22266010408057116, 5),\n (0.24485995898244595, 6),\n (0.22680937788934327, 6),\n (0.22804616438650888, 5),\n (0.21328453045535367, 4),\n (0.2213842038894756, 4),\n (0.2335796243977106, 5),\n (0.25660376768794513, 6),\n (0.23258525372755273, 6),\n (0.2483465360183011, 5),\n (0.22000206176674408, 5),\n (0.21642380384396329, 4),\n (0.2237851736132857, 4),\n (0.17784406715237666, 6),\n (0.22339132972525752, 5),\n (0.22737243017601477, 5),\n (0.23259708929873366, 4),\n (0.29547398516116835, 5),\n (0.31615992428371803, 5),\n (0.3166729006167617, 5),\n (0.23174551905474977, 5),\n (0.2324253247550381, 5),\n (0.30705093596560745, 5),\n (0.2663443442134263, 6),\n (0.24446300892513878, 5),\n (0.18281773037220098, 5),\n (0.22644632263548597, 5),\n (0.19997224068994013, 5),\n (0.192170348762103, 5),\n (0.24401278519169223, 5),\n (0.25833419889603376, 5),\n (0.21169726115688792, 5),\n (0.2776251092923188, 5),\n (0.2849917171354414, 5),\n (0.3687437384055337, 5),\n (0.25388573134537806, 5),\n (0.7299690413624434, 4),\n (0.24389826755588737, 6),\n (0.29609200574817196, 5),\n (0.26083429238899364, 5),\n (0.3183383220678213, 5),\n (0.4178425825739514, 6),\n (0.4352632259722094, 4),\n (0.45854428126132535, 4),\n (0.4770249019230556, 4),\n (0.483620810153457, 5),\n (0.4865132431394197, 5),\n (0.502558640634601, 5),\n (0.4996177050636277, 4),\n (0.5331784392665264, 4),\n (0.21424944774983745, 4),\n (0.3439927227707353, 5),\n (0.2345684476359573, 5),\n (0.20148241332690695, 6),\n (0.20321611065278108, 8),\n (0.3779942631363656, 6),\n (0.31046786928930087, 7),\n (0.195540765098365, 5),\n (0.2277594504299288, 7),\n (0.2028849118097202, 6),\n (0.227419263720426, 5),\n (0.20246190474516437, 6),\n (0.2176263036748814, 5),\n (0.26363262654827757, 7),\n (0.22450508251721124, 6),\n (0.2159174806435591, 5),\n (0.2190152864233003, 4),\n (0.23945751299964507, 5),\n (0.22138923212512326, 6),\n (0.23623334658564535, 5),\n (0.2265664290170863, 5),\n (0.29345467663703617, 5),\n (0.48845618847622685, 4),\n (0.47652096103173697, 5),\n (0.5469846116032664, 5),\n (0.48727675277377197, 4),\n (0.2765949416782976, 4),\n (0.32831514489977454, 6),\n (0.28751870419695447, 5),\n (0.29702286942403083, 6),\n (0.31012942940086174, 5),\n (0.2821484544148872, 5),\n (0.30214878065500633, 4),\n (0.478316718888235, 5),\n (0.37269074415002623, 5),\n (0.34380877586726055, 5),\n (0.31605002414589456, 4),\n (0.3370919900985545, 5),\n (0.41106495217167954, 4),\n (0.3448090311578161, 5),\n (0.2251932306549486, 6),\n (0.19035826625057772, 7),\n (0.20077720016404837, 6),\n (0.2600367856314781, 5),\n (0.33619316663579824, 5),\n (0.3242087615955621, 4),\n (0.4349683103238019, 5),\n (0.40972365741836053, 4),\n (0.5056576178043761, 4),\n (0.44267603696818286, 5),\n (0.35342982100513354, 4),\n (0.2794868878633494, 5),\n (0.43543401875495447, 5),\n (0.4545175450049643, 5),\n (0.5972889473226425, 4),\n (0.6165632894330058, 5),\n (0.6340838756459856, 5),\n (0.09387990115371533, 7),\n (0.1287771605919215, 9),\n (0.25406105684209995, 8),\n (0.22329946286619456, 9),\n (0.2479925228000151, 7),\n (0.16180352913308937, 8),\n (0.16403895745080613, 8),\n (0.17701845126157287, 7),\n (0.12595480547561344, 7),\n (0.11761249042579279, 6),\n (0.1158869685085099, 5),\n (0.11192424751779062, 5),\n (0.11829960949525752, 4),\n (0.09278253482101904, 6),\n (0.10148774343053861, 5),\n (0.10974609719520033, 5),\n (0.09781466492269572, 4),\n 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6),\n (0.20143472755371575, 8),\n (0.1870987043036772, 8),\n (0.1839721198464224, 8),\n (0.1732361108958545, 7),\n (0.15935836719760538, 7),\n (0.15097836402896825, 6),\n (0.1413274112189823, 4),\n (0.14147801462432374, 5),\n (0.1417896077171351, 4),\n (0.15571208472121836, 5),\n (0.14320016760528723, 5),\n (0.18533376248868938, 7),\n (0.16519649747282034, 6),\n (0.15411081567230403, 6),\n (0.13358240366892257, 5),\n (0.1366914223890436, 5),\n (0.13313766042600866, 5),\n (0.1335365564314213, 5),\n (0.1356934145323032, 6),\n (0.1409672743163309, 7),\n (0.17864427281802892, 6),\n (0.18972010843740808, 6),\n (0.17391899256300206, 6),\n (0.17960521783537056, 5),\n (0.16723511848427852, 5),\n (0.1774126073795841, 5),\n (0.16648792116997188, 5),\n (0.18806941571970354, 4),\n (0.15960783051355756, 5),\n (0.17360454533675795, 5),\n (-inf, 6),\n (0.15127219778026768, 5),\n (0.14626315385285768, 5),\n (0.14977281921145089, 5),\n (0.14809871113445222, 5),\n (0.14851015556758668, 6),\n (0.15069597708532612, 6),\n (0.1491427851929602, 6),\n (0.20761994210386664, 6),\n (0.2091410463620896, 5),\n (0.19010096714105154, 5),\n (0.19352154214046707, 5),\n (0.18812122257045052, 5),\n (0.2001483480629838, 5),\n (0.2324141186822195, 5),\n (0.2134654392060555, 5),\n (0.15510027554880157, 5),\n (0.15835627422875928, 5),\n (0.14869204092970761, 6),\n (0.21844968688999905, 5),\n (0.20374930585354264, 5),\n (0.26560202659395676, 5),\n (0.25332451287987207, 5),\n (0.23268410072691026, 6),\n (-inf, 7),\n (0.06425095014701922, 8),\n (0.09071882432698337, 6),\n (0.08565813852003702, 4),\n (0.10318138903567221, 8),\n (0.12773013308607467, 8),\n (0.1827877659538591, 8),\n (0.2877257715457169, 8),\n (0.13869395985515834, 8),\n (0.13676932646893256, 7),\n (0.11672213715126249, 6),\n (0.11243146737978531, 5),\n (0.10642919901344822, 6),\n (0.11415163928794794, 6),\n (0.11560599777725049, 7),\n (0.1206629820197471, 7),\n (0.14850826535112704, 9),\n (0.18001635126011162, 8),\n (0.10158462883069913, 6),\n (0.10814574924964172, 6),\n (0.1095734949141835, 5),\n (0.09201581093425899, 5),\n (0.1092284334694289, 4),\n (0.11172705609434835, 4),\n (0.11164618679406099, 5),\n (0.1159815208069952, 4),\n (0.11721629818478653, 6),\n (0.11787591195215144, 7),\n (0.11638737003873474, 5),\n (0.11647760929480407, 5),\n (0.11752310502412483, 5),\n (0.1288834945339188, 8),\n (0.13002985304203174, 8),\n (0.12027266898105717, 6),\n (0.10306280822320828, 6),\n (0.10255143467886546, 5),\n (0.11924255489747138, 4),\n (0.1213196355471951, 5),\n (0.12188479627949646, 5),\n (0.11081487964249354, 6),\n (0.13103339120436566, 7),\n (0.16452230681211635, 7),\n (0.15661115169965817, 8),\n (0.21993495525952309, 7),\n (0.13900996820097758, 6),\n (0.13276977365805992, 5),\n (0.12693588371791692, 5),\n (0.12617309800427817, 4),\n (0.1266402695226423, 4),\n (0.13876960251487344, 4),\n (0.20063506300845219, 7),\n (0.19843230908001527, 7),\n (0.1927432568760887, 7),\n (0.18201922641246113, 7),\n (0.17675922653282977, 6),\n (0.15595567310519115, 6),\n (0.1393201339015975, 6),\n (0.19304696355513787, 8),\n (0.18135269036862797, 7),\n (0.1788765271990486, 7),\n (0.16263643805243191, 7),\n (0.15636101459361515, 6),\n (0.1497955685915651, 6),\n (0.1426065327213369, 5),\n (0.14116567738820965, 5),\n (0.16678918702695703, 4),\n (0.14255553629436774, 6),\n (0.21069163086038967, 7),\n (0.16637974996523297, 6),\n (0.15211733055028043, 6),\n (0.17049911839589837, 5),\n (0.16719629170885175, 5),\n (0.1407045836004415, 5),\n (0.13682369619730728, 5),\n (0.13538287824966708, 5),\n (0.13744560406050943, 6),\n (0.1363865825273716, 6),\n (0.134738944209213, 7),\n (0.14121335896821413, 7),\n (0.19326657106902195, 6),\n (0.2007073099782655, 6),\n (0.20364355944414275, 6),\n (0.1806367756311675, 6),\n (0.18074022536776044, 5),\n (0.16678492929929245, 5),\n (0.16385563524220584, 4),\n (0.1906335659928311, 5),\n (0.17020241352778812, 5),\n (inf, 13),\n (0.14502435574693734, 5),\n (0.15201453161386808, 5),\n (0.15366726519664642, 5),\n (0.1522132114144064, 5),\n (0.1539275160607906, 5),\n (0.15374494267494795, 5),\n (0.20949620349030676, 6),\n (0.20572365361964584, 6),\n (0.19630046954706828, 5),\n (0.19792931817440326, 5),\n (0.1866320978429273, 5),\n (0.22311353290583186, 4),\n (0.23662640412460184, 4),\n (0.154401913371829, 5),\n (0.1542056130510585, 5),\n (0.15608366140033983, 5),\n (0.1514800276808683, 6),\n (0.15672376502633134, 6),\n (0.14813329572285117, 6),\n (0.15748945441543893, 6),\n (0.22686760820915025, 5),\n (0.24425008318212213, 5),\n (0.20321376174345673, 5),\n (0.2431279014720266, 5),\n (0.21252227144545044, 5),\n (0.2307853612853407, 5),\n (0.23888511827261324, 5),\n (0.23591686011746912, 5),\n (0.2562477752914985, 5),\n (0.22142941165275748, 5),\n (0.2613180677200388, 6),\n (0.25847045531443763, 5),\n (0.2576133120360073, 5),\n (0.3683268054489068, 4),\n (0.11621191903804633, 5),\n (0.12298212297940735, 4),\n (0.12130598371803313, 5),\n 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(0.32812192580476274, 4),\n (0.36536565682456795, 5),\n (0.45457667467884555, 5),\n (0.5654277334545597, 4),\n (0.4858498189866753, 6),\n (0.5949864313956996, 4),\n (0.6484220832915248, 5),\n (0.5614594897934861, 4),\n (0.5857996171079172, 4),\n (0.652782450127564, 4),\n (0.9164782217726231, 4),\n (0.24855412783801908, 4),\n (0.2553325198969807, 4),\n (0.2614697061874378, 5),\n (0.2769204535039051, 5),\n (0.49583774140254194, 4),\n (0.47187084311281535, 5),\n (0.33511706131823765, 5),\n (0.39726491420055365, 5),\n (0.14805965025845524, 6),\n (0.21683398288987227, 5),\n (0.14384718472568325, 6),\n (0.2527717478371747, 5),\n (0.19010412317344963, 4),\n (0.24912059353019933, 4),\n (0.22242070345802867, 5),\n (0.30755161660629443, 5),\n (0.2831419367706306, 5),\n (inf, 6),\n (0.15827564610340644, 5),\n (0.167796417956539, 5),\n (0.18205356634294062, 5),\n (0.17935626082343087, 6),\n (0.1855180324963683, 7),\n (0.18890297228791952, 7),\n (0.27460265995408767, 8),\n (0.22043550945821377, 7),\n (0.21451871179914656, 7),\n (0.21659496410136092, 5),\n (0.1774345411593533, 6),\n (0.19538897470237676, 6),\n (0.17403303074976725, 6),\n (0.18637888523924087, 4),\n (0.18679087344501222, 4),\n (0.19161556035308897, 5),\n (0.202097069275007, 5),\n (0.200958142924022, 5),\n (0.2067122310067438, 6),\n (0.17701125988524885, 7),\n (0.24018016774422207, 6),\n (0.22489626697129014, 5),\n (0.21275319505053591, 4),\n (0.22952503085324366, 5),\n (0.23245096948386185, 5),\n (0.22548330588690998, 5),\n (0.25442226551156877, 5),\n (0.24832881621238995, 5),\n (0.2320787102922262, 4),\n (0.2565439831754603, 4),\n (0.21560035499860178, 5),\n (0.21470361240551608, 6),\n ...]\n\n\n\nLocate the invalid data\n\n\n```python\nx=[]\nfor i in range(len(sig_Brent)):\n if sig_Brent[i]==-1:\n x.append(i)\nx\n```\n\n\n\n\n [245,\n 269,\n 357,\n 648,\n 779,\n 833,\n 834,\n 932,\n 967,\n 1079,\n 1440,\n 1615,\n 1893,\n 1896,\n 1902,\n 1981,\n 2196]\n\n\n\n\n```python\nmyData.iloc[x,:]\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    currentDateExpDateStrikePriceTickerTypeLastIVChgStockPriceTImpliedVolatility (caluclated)Difference
    2452/17/20176/16/2017775.0GOOGCall58.800.11782NaN828.0700070.3258040.118389-0.000569
    2692/17/20172/24/2017825.0GOOGLCall21.800.101594.00846.5499880.0191650.103802-0.002212
    3572/17/20176/16/2017785.0GOOGLCall66.330.11885NaN846.5499880.3258040.1186370.000213
    6482/17/20171/18/201975.0ADICall9.200.08506NaN82.4800031.9164960.087129-0.002069
    7792/17/20171/18/201992.5ADPCall10.850.11406NaN99.6800001.9164960.1132600.000800
    8332/17/20179/15/20176.0BCRXPut2.121.31231NaN6.5600000.5749491.317493-0.005183
    8342/17/20179/15/20172.0BDSIPut0.620.98398NaN1.8800000.5749490.9797520.004228
    9322/17/20173/17/201710.0CALACall0.901.186920.309.2000000.0766601.192867-0.005947
    9672/17/20176/16/201755.0CDWCall5.400.13518NaN59.9900020.3258040.141453-0.006273
    10792/17/20173/3/201733.0CSCOCall0.770.08481-0.0833.7400020.0383300.085637-0.000827
    14402/17/20173/17/201727.0ESPRCall2.001.00901-0.4225.0000000.0766601.0090050.000005
    16152/17/20178/18/201730.0BUSECall1.400.08406NaN31.0599990.4982890.0802720.003788
    18932/17/20175/19/201716.0INSMPut4.001.21677NaN15.4300000.2491441.2071630.009607
    18962/17/20175/19/201710.0INSYPut2.111.27191-0.1611.1000000.2491441.274413-0.002503
    19022/17/20172/24/201735.5INTCCall0.990.102530.0236.4800000.0191650.106462-0.003932
    19812/17/20174/21/2017110.0INTUCall10.160.13788NaN119.8600010.1724850.140908-0.003028
    21962/17/20173/24/2017280.0IBBCall15.040.12684NaN294.3500060.0958250.128524-0.001684
    \n
    \n\n\n\nUse nsolve from sympy to get a more accurate implied Volatility\n\n\n```python\nfrom sympy import nsolve,Symbol\nimport sympy\nvol=Symbol('sigma')\n# European call option\n\n\n#d1=(log(s/k)+(r-d+sigma*sigma/2)*tao)/(sigma*math.sqrt(tao))\n#d2=d1-sigma*math.sqrt(tao)\n\ndef normcdf(x):\n return (1+sympy.erf(x/sympy.sqrt(2)))/2\n\n\n\ndef Euro(s,k,sigma,tao,r,d,Type):\n if Type=='Call':\n d1=(sympy.log(s/k)+(r-d+sigma*sigma/2)*tao)/(sigma*sympy.sqrt(tao))\n d2=d1-sigma*sympy.sqrt(tao)\n call=s*sympy.exp(-d*tao)*normcdf(d1)-k*sympy.exp(-r*tao)*normcdf(d2)\n \n return call\n else:\n d1=(sympy.log(s/k)+(r-d+sigma*sigma/2)*tao)/(sigma*sympy.sqrt(tao))\n d2=d1-sigma*sympy.sqrt(tao)\n put=k*sympy.exp(-r*tao)*normcdf(-d2)-s*sympy.exp(-d*tao)*normcdf(-d1)\n return put\n\n\nImVol=[]\ntag=[]\nfor i in range(len(myData)):\n try:\n ImVol.append(nsolve(Euro(S[i],K[i],vol,T[i],0.03,0,Type[i])-P[i],vol,1))\n except:\n ImVol.append(str(i)+'--1')\n```\n\nCreate a df and drop the invalid rows\n\n\n```python\n#est vol value\nsig_Bisection_v=pd.DataFrame(sig_Bisection).iloc[:,0]\nsig_Brent_v=pd.DataFrame(sig_Brent).iloc[:,0]\nsig_MullerSection_v=pd.DataFrame(sig_MullerSection).iloc[:,0]\nsig_NewTon_v=pd.DataFrame(sig_NewTon).iloc[:,0]\nsig_Halley_v=pd.DataFrame(sig_Halley).iloc[:,0]\nsig_new_Newton_v=pd.DataFrame(sig_new_Newton).iloc[:,0]\nsig_new_Halley_v=pd.DataFrame(sig_new_Halley).iloc[:,0]\nImVol_v=pd.DataFrame(ImVol).iloc[:,0]\n\n#steps \nsig_Bisection_s=pd.DataFrame(sig_Bisection).iloc[:,1]\n#sig_Brent_s=pd.DataFrame(sig_Brent).iloc[:,0]\nsig_MullerSection_s=pd.DataFrame(sig_MullerSection).iloc[:,1]\nsig_NewTon_s=pd.DataFrame(sig_NewTon).iloc[:,1]\nsig_Halley_s=pd.DataFrame(sig_Halley).iloc[:,1]\nsig_new_Newton_s=pd.DataFrame(sig_new_Newton).iloc[:,1]\nsig_new_Halley_s=pd.DataFrame(sig_new_Halley).iloc[:,1]\n#ImVol_s=pd.DataFrame(ImVol).iloc[:,1]\n```\n\n\n```python\ndf_step=pd.DataFrame(list(zip(sig_Bisection_s,sig_MullerSection_s,sig_NewTon_s,sig_Halley_s,sig_new_Newton_s,sig_new_Halley_s)),columns=['Bisection','MullerSection','NewTon','Halley','new_Newton','new_Halley'])\ndf_step=df_step.drop(x)\n\nidx=pd.Series(list(range(2254)))\ndf_step=df_step.set_index([idx])\ndf_step=df_step.drop(1130)\nidx=pd.Series(list(range(2253)))\ndf_step=df_step.set_index([idx])\ndf_step\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    BisectionMullerSectionNewTonHalleynew_Newtonnew_Halley
    041416511
    141416511
    241415411
    341415511
    441415411
    .....................
    224841414311
    224941414411
    225041414311
    225141414411
    225241414311
    \n

    2253 rows \u00d7 6 columns

    \n
    \n\n\n\n\n```python\ndf=pd.DataFrame(list(zip(sig_Bisection_v,sig_Brent_v,sig_MullerSection_v,sig_NewTon_v,sig_Halley_v,sig_new_Newton_v,sig_new_Halley_v,ImVol_v)),columns=['Bisection','Brent','MullerSection','NewTon','Halley','new_Newton','new_Halley','ImVol'])\ndf=df.drop(x)\n\nidx=pd.Series(list(range(2254)))\ndf=df.set_index([idx])\ndf=df.drop(1130)\nidx=pd.Series(list(range(2253)))\ndf=df.set_index([idx])\ndf\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    BisectionBrentMullerSectionNewTonHalleynew_Newtonnew_HalleyImVol
    00.1327390.1327390.1327390.1327390.1327390.1327390.1327390.132738644382711
    10.3412230.3412230.3412230.3412230.3412230.3412230.3412230.341223456029555
    20.2011330.2011330.2011330.2011330.2011330.2011330.2011330.201133283584433
    30.1797850.1797850.1797850.1797850.1797850.1797850.1797850.179785112245654
    40.2044240.2044240.2044240.2044240.2044240.2044240.2044240.204423991371752
    ...........................
    22480.8209990.8209990.8209990.8209990.8209990.8209990.8209990.820999374001134
    22490.1828240.1828240.1828240.1828240.1828240.1828240.1828240.182824210572376
    22500.7460200.7460200.7460200.7460200.7460200.7460200.7460200.746020173737849
    22510.7962140.7962140.7962140.7962140.7962140.7962140.7962140.796213529078633
    22520.7850430.7850430.7850430.7850430.7850430.7850430.7850430.785042616447992
    \n

    2253 rows \u00d7 8 columns

    \n
    \n\n\n\nEvaluation\n\n\n```python\ndef mse(df):\n \n M=[]\n for j in range(7): #del last col\n sum=0\n for i in range(len(df)): \n sum=sum+(df.iloc[i,j]-df.iloc[i,-1])**2\n \n mean=sum/len(df)\n #print(mean)\n M.append(mean)\n return M\n\nimport math\ndef Efficiency(mse,DF_Step):\n meanStep=DF_Step.mean().tolist()\n del mse[1] #delete blent's column,\n \n M=[]\n for i in range(len(mse)):\n M.append(1/((1+mse[i])*math.log2(1+meanStep[i])))\n return M\n```\n\n\n```python\nMse_ans=mse(df)\n# 7 values\nMse_ans\n\n```\n\n\n\n\n [2.71138015583870e-25,\n 1.32853314961418e-26,\n 2.75159178888677e-25,\n 7.29410522846084e-27,\n 1.10227816034781e-27,\n 7.58407284211001e-27,\n 7.58407284211001e-27]\n\n\n\n\n```python\nMse1=Mse_ans.copy()\n#6 values \neffi=Efficiency(Mse1,df_step)\n\neffi\n```\n\n\n\n\n [0.185449023415369,\n 0.185449023415369,\n 0.373349971502113,\n 0.406742462765455,\n 0.953196442899999,\n 0.953196442899999]\n\n\n\nVisualization\n\n\n```python\nMse=Mse_ans.copy()\n\nMse\n```\n\n\n\n\n [2.71138015583870e-25,\n 1.32853314961418e-26,\n 2.75159178888677e-25,\n 7.29410522846084e-27,\n 1.10227816034781e-27,\n 7.58407284211001e-27,\n 7.58407284211001e-27]\n\n\n\n\n```python\nnames=list(df.columns)\nnames\n```\n\n\n\n\n ['Bisection',\n 'Brent',\n 'MullerSection',\n 'NewTon',\n 'Halley',\n 'new_Newton',\n 'new_Halley',\n 'ImVol']\n\n\n\n\n```python\nimport matplotlib.pyplot as plt\nnames=list(df.columns) \ndel names[-1]\ndel names[1]\n\nsteps=df_step.mean().tolist() \ndel Mse[1] #del Brent\n\n\n\nvalues=Mse\nplt.figure(figsize=(9, 9))\nplt.suptitle('MSE Comparation')\n\nplt.bar(names,values)\n\n```\n\n\n```python\nplt.figure(figsize=(9, 9))\nplt.suptitle('Step Comparation')\nplt.bar(names,steps,color='g')\n```\n\n\n```python\n\nplt.figure(figsize=(9, 9))\nplt.suptitle('Efficiency Comparation')\nplt.bar(names,effi,color='r')\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "c55617a442ec0fdee426d9f2fc9c1d73059b41eb", "size": 130551, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "P2/fintech_final.ipynb", "max_stars_repo_name": "Vincenqwu/Financial-Programming-Project", "max_stars_repo_head_hexsha": "23efc429c4ba313882e98d015356f7bffef7e0d0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "P2/fintech_final.ipynb", "max_issues_repo_name": "Vincenqwu/Financial-Programming-Project", "max_issues_repo_head_hexsha": "23efc429c4ba313882e98d015356f7bffef7e0d0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "P2/fintech_final.ipynb", "max_forks_repo_name": "Vincenqwu/Financial-Programming-Project", "max_forks_repo_head_hexsha": "23efc429c4ba313882e98d015356f7bffef7e0d0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 47.421358518, "max_line_length": 12520, "alphanum_fraction": 0.5842927285, "converted": true, "num_tokens": 24662, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.658417500561683, "lm_q1q2_score": 0.4381179569374302}} {"text": "\n\n# Natural Language Processing Tutorial\n\nWelcome to the second lab session of the [Mediterranean Machine Learning summer school](https://www.m2lschool.org/home)!\n\n\n[](https://colab.research.google.com/github/m2lschool/tutorials2021/blob/main/2_nlp/NLP_tutorial.ipynb)\n\n\nIn this tutorial, you will learn the fundamental components and main tasks in the Natural Language Processing (NLP) domain.\n\nThe tutorial is structured as follows:\n\n* Imports and downloads (shared across sections)\n* Part I: *Inner mechanisms of a vanilla Recurrent Neural Network (RNN)*\n* Part II: *Character-level language modeling with Long-Short Term Memory (LSTM) networks*\n* Part III: *Non-negative Matrix Factorization for topic modeling*\n* Part IV: *From traditional (bag-of-words) to pretrained representations (BERT) for text classification*\n\n\nAll sections can be executed independently after running the initial import and download cells.\n\nThe sections marked as \\[EXERCISE\\] contain cells with missing code that you should complete.\n\n\nCredits:\n**[Federico Bianchi](https://federicobianchi.io)**, **[Debora Nozza](https://dnozza.github.io/)** and **[Francesco Visin](https://scholar.google.it/citations?user=kaAnZw0AAAAJ)**\n\n### All Imports and Downloads\n\nHere we are going to install and import everything we are going to need for this tutorial. \n\n**Note**: *You can double-click the title of the collapsed cells (as the ones below) to expand them and read their content.*\n\n\n```python\n# @title Downloads and libraries installation\n%%capture\n!pip install git+https://github.com/deepmind/dm-haiku\n!pip install optax\n!pip install tensorflow-datasets\n!pip install emoji\n!pip install sentence-transformers\n```\n\n\n```python\n# @title Imports\nimport emoji\nfrom functools import partial\nimport re\nimport time\nfrom tqdm import tqdm\n\nimport matplotlib.pyplot as plt\nimport numpy as np\nfrom sklearn.datasets import fetch_20newsgroups\nfrom sklearn.feature_extraction.text import TfidfVectorizer\nfrom sentence_transformers import SentenceTransformer\n\nimport haiku as hk\nimport jax\nimport jax.numpy as jnp\nfrom jax.nn.initializers import glorot_normal, normal, glorot_uniform, zeros\nfrom jax import grad, jit, vmap, value_and_grad\nfrom jax import random\nfrom typing import Iterator, Mapping\nfrom jax import lax\nfrom jax.scipy.special import logsumexp\nfrom jax import ops\nimport optax\nimport tensorflow.compat.v2 as tf\nimport tensorflow_datasets as tfds\n\nfrom nltk.tokenize import RegexpTokenizer\n```\n\n# Part I. Vanilla RNN\n\n**Goal**: Train a vanilla RNN to predict the next value of a temporal series given its previous values.\n\nSpecifically, we aim to predict\n\n$$ \nsin (x +t \\epsilon) \n$$ \n\nfrom\n\n$$ \nsin (x), sin (x + \\epsilon), ..., sin (x + (t-1) \\epsilon) \n$$\n\nIn particular, we want the network to generate multiple predictions conditioned only on a few initial values. To do so, we will predict the next value of the function in a loop, conditioning on the value at the previous time-step (be it the initial, given, values or the ones predicted by the network at each previous step).\n\nTo learn the prediction model, we will use **teacher forcing**. This means that *during training* we will condition the model on values *coming from the ground truth* (i.e., the real sequence), rather than the output produced by the model at $t-1$. This makes training easier, because errors do not compound (a bad prediction at time $t$ does not influence the next prediction at time $t+1$).\n\nAt *inference time* (i.e., when we want to generate data from the model) we do not have access to the true sequence, so we must condition on the values *predicted by the model in previous time-steps*. This often leads to some compounding error, which generally makes it harder and harder to generate long sequences accurately.\n\nTo alleviate this problem we will also experiment with **warm starting**, which amounts to feeding the model with values coming from the ground truth (as done when teacher forcing) *only for a few initial steps*, and then feed it with its previous predictions instead. The rationale for this is to make training easier initially, to help the model learn the true function, and then feed it with its predictions to become robust to imperfect data.\n\n## Training data\n\nLet's store the sine wave over the interval $[0, 2\\pi]$ as our training data. This will be our **ground truth**.\n\nNote that differently from what is usually done, we don't have two separate training and test datasets here. Instead, we will train on random subsamples of the curve and then verify the ability of the network to generate the full curve when conditioned on a few initial values:\n\n* Predict a few future values from a random starting point $x_s$;\n* Generate the full trajectory, conditioned on a few initial points.\n\n\n```python\nt = np.arange(0, 2*np.pi, 0.1).reshape(-1, 1)\nsin_t = np.sin(t)\n\nplt.scatter(t, sin_t)\nplt.xlabel('$t$')\nplt.ylabel('$sin(t)$')\nplt.show()\n```\n\n\n```python\n# @title Hyperparameters\nWARM_START = 10 #@param {type:'integer'}\nTEACHER_FORCING = False #@param {type:'boolean'}\nSTEP_SIZE = 0.0001 #@param {type:'number'}\nUNROLL_LENGTH = 30 #@param {type:'integer'}\nHIDDEN_DIMENSION = 20 #@param {type:'integer'}\nNUM_ITERATIONS = 10000 #@param {type:'integer'}\nREPORTING_INTERVAL = 2000 #@param {type:'integer'}\n\n\n```\n\nAs you have seen in the introductory lab, to use pseudo-random functions in JAX we need to instantiate a random number generator, and pass it explicitely to all the operations that work with random numbers (e.g. model initialization, dropout, etc, ...).\n\n\n```python\nrnd_key = random.PRNGKey(1)\n```\n\n## RNN cell \\[EXERCISE\\]\n\nImplement a basic RNN cell using *jax.numpy (jnp)* functions\n\n$$ h_t = f( Wx_t + Vh_{t-1} + b) $$\n \nWhere,\n\n* $x_t$ input at time $t$,\n* $h_t$ hidden state at time $t$,\n* $W$ input-to-hidden mapping (trainable),\n* $V$ hidden-to-hidden mapping (trainable),\n* $b$ bias (trainable),\n* $f$ non-linearity chosen (we use *tanh*)\n\nStart by implementing a function to return the initial parameters:\n\n\n```python\ndef initialize_parameters(rnd_key):\n \"\"\"\n Initialize and return the Vanilla RNN parameters.\n\n Args:\n rnd_key: random key\n\n Returns:\n A dict of weights with keys `V`, `W`, `bias`, `decoder`, `decoder_bias`.\n \"\"\"\n\n input_dimension = 1 # the input is a scalar\n output_dimension = 1 # the output is a scalar\n\n ######################\n # YOUR CODE HERE #\n ######################\n # Initialize the Vanilla RNN parameters\n # using a glorot uniform distribution for W, \n # V, decoder and zero values for bias and \n # decoder_bias. Pay attention to the correct \n # shapes of all parameters.\n\n keys = random.split(rnd_key, 5)\n params = {\n\n }\n\n assert params is not None, 'Params should be initialized'\n\n return params\n```\n\nand then implement the RNN core itself:\n\n\n```python\ndef RNN_cell(params, x_t, current_state):\n \"\"\"\n This function will be called in a loop, when RNN core is connected to\n inputs and previous states.\n\n Args:\n params: neural network paramters\n x_t: jax DeviceArray containing the current input `x_{t}`\n prev_state: jax DeviceArray containing the previous state `h_{t-1}`\n\n Returns:\n A pair of RNN embedding and state.\n \"\"\"\n\n ######################\n # YOUR CODE HERE #\n ######################\n # Compute the output of the RNN Cell by implementing the equation above.\n output = None\n\n assert output is not None, 'The RNN cell must compute the output'\n\n # The RNN cell returns pairs of (o_t, h_t), respectively the output and the \n # state at time $t$. For vanilla RNN these are the same.\n return output, output\n\ndef get_initial_rnn_state(rnd_key):\n return zeros(rnd_key, (HIDDEN_DIMENSION,))\n```\n\nNow let's check that the output of the RNN is as expected.\n\n\n```python\nrnd_key, k1, k2 = random.split(rnd_key, 3)\nparams = initialize_parameters(k1)\nsingle_input = sin_t[0]\ninitial_state = get_initial_rnn_state(k2)\n\n# RNN_cell returns two vectors, the output and the state. When the input is a \n# single element (i.e., not a sequence) each should have shape HIDDEN_DIMENSION.\nassert len(RNN_cell(params, single_input, initial_state)[0]) == HIDDEN_DIMENSION\nassert len(RNN_cell(params, single_input, initial_state)[1]) == HIDDEN_DIMENSION\n```\n\n## Next element prediction [EXERCISE]\n\nThe `predict` method defined below supports both *teacher forcing* and *warmup*. For educational purposes, we unrolled the RNN loop manually so it is easier to understand and debug the code. You need to implement the missing bits, applying the `RNN_cell` and returning a sequence of scalar predictions.\n\nThe `sin(x)` input sequence is composed of scalars, which is what the `RNN_cell` expects so you don't need to preprocess them. The predictions of the `RNN_cell` have shape `(HIDDEN_DIMENSION,),` so you need to apply the $\\textbf{decoder}$ to recover the sequence of scalars we want to predict.\n\nWhen you are done debugging, uncomment the **`@jit` decorator** to ensure the code is optimized and runs faster.\n\n\n```python\ndef encoder_decoder(params, encoded_input, current_state):\n rnn_embedding, new_state = RNN_cell(params, encoded_input, current_state)\n return jnp.dot(rnn_embedding, params['decoder']) + (\n params['decoder_bias'], new_state)\n\n#@jit # Uncomment for full speed!\ndef predict(params, initial_rnn_state, inputs, training=True):\n \"\"\"Predict the next value of a sequence.\n\n Here we compute *one-step predictions*, i.e., we predict `sin_{t+1}` from \n `sin_{t}`. Note however that depending on the values of the TEACHER_FORCING \n and WARM_START parameters, sometimes we feed the model with the actual\n `sin_{t}` coming from the ground truth (i.e., `inputs`) and some other times\n with the last prediction of the model (which might or might not be \n accurate). This means that in some cases not all the values of `inputs` will\n be used for training.\n\n This code a simplified implementation of `hk.static_unroll` for educational \n purposes, do not use it in actual code. Statically unrolling the network has\n some caveats, e.g., it won't behave as expected if the length of the input\n sequence is variable. It is usually preferable to use `hk.dynamic_unroll`.\n\n Args:\n params: the parameters of the RNN\n initial_rnn_state: the initial state of the RNN\n inputs: the data passed in input to the RNN\n training: sets teacher forcing to True or False\n \n Returns:\n The next value predictions.\n \"\"\"\n if training:\n teacher_forcing = TEACHER_FORCING\n else:\n teacher_forcing = False\n\n predictions = []\n current_state = initial_rnn_state\n\n # Unroll the rnn loop by hand.\n for t in range(len(inputs)):\n\n # When teacher forcing, the input is always the actual previous value \n # (from the ground truth). Otherwise, we will feed the model with the \n # actual previous values during the warm start period only, and then \n # input the previous predictions instead.\n if teacher_forcing or t <= WARM_START:\n input_ = inputs[t]\n else:\n input_ = prediction\n\n ######################\n # YOUR CODE HERE #\n ######################\n # Connect the RNN: we apply our cell to (input, state) pairs, and we \n # expect it to return (output, next_state) pairs; note how the current\n # state is updated with the next_state.\n rnn_embedding, next_state = None, None\n\n assert rnn_embedding is not None, 'The RNN must return an output'\n\n current_state = next_state\n\n ######################\n # YOUR CODE HERE #\n ######################\n # Create a linear mapping from RNN output to the target (scalar)\n # using the decoder and decoder bias parameters\n prediction = None\n\n assert prediction is not None, ('Connect the rnn embeddings and the '\n 'decoder to get prediction')\n\n predictions.append(prediction)\n\n return jnp.stack(predictions)\n```\n\n### Shape check\n\nLet's check that our RNN predictions returns the desired shape\n\n\n```python\nrnd_key, k1, k2 = random.split(rnd_key, 3)\nparams = initialize_parameters(k1)\ninitial_rnn_state = get_initial_rnn_state(k2)\npredictions = predict(params, initial_rnn_state, sin_t[:UNROLL_LENGTH])\nassert len(predictions) == UNROLL_LENGTH\n```\n\n## Loss and weights update [EXERCISE]\n\nWe update the parameters of the network using the **ADAM** optimizer. In this section you need to complete the `loss_fn` that must compute the loss as the mean squared error of the prediction and the targets. As before, once you are done debugging uncomment the `@jit` decorator to ensure the code runs at full speed.\n\nThe `optimize` function uses this `loss_fn` function to compute the loss and the gradient of the loss w.r.t. the parameters of the network, and takes one step of gradient descent to reduce such loss. This is effectively one step of training.\n\n\n```python\n# @jit\ndef loss_fn(params, rnn_state, inputs, targets):\n \"\"\"\n Loss function is a simple mean squared error. \n\n Args:\n params: the parameters of the neural network\n rnn_state: the state of the rnn\n inputs: a sequence of inputs\n targets: a sequence of expected outputs\n\n Returns:\n The mean squared error.\n \"\"\"\n\n ######################\n # YOUR CODE HERE #\n ######################\n # Obtain the prediction given the current state and input data\n # and compute the mean squared error with the targets.\n predictions = None\n mse = None\n\n assert predictions is not None, 'Get the RNN predictions'\n assert mse is not None, ('Compute the mean squared error between the '\n 'predictions and the target')\n assert predictions.shape == targets.shape, (\n f'The predictions should have the same shape as the input and target, '\n f'but are respectively {predictions.shape}, {inputs.shape}, '\n f'{targets.shape}')\n\n return mse\n\n@jit\ndef optimize(params, opt_state, rnn_state, inputs, targets):\n \"\"\"\n Updates the parameters and returns the loss values.\n\n Args:\n params: the parameters of the neural network\n opt_state: the state of the optimizer\n rnn_state: the state of the rnn\n inputs: a sequence of inputs\n targets: a sequence of expected outputs\n \n Returns:\n The loss, the new optimizer state and the updated network parameters.\n \"\"\"\n loss_value, grads = jax.value_and_grad(loss_fn)(params, rnn_state, inputs,\n targets)\n updates, opt_state = optimizer(grads, opt_state)\n params = optax.apply_updates(params, updates)\n return loss_value, opt_state, params\n```\n\n## Training [EXERCISE]\n\nIn this section we will finally use all the functions defined above to train the network.\n\nThe following code prints an example of input and output data for an unroll length of 4. Note how the target at time $t$ is the input at time $t+1$. This is because we train the network to predict the next value in the sequence.\n\n\n```python\ntraining_input = jnp.array(sin_t[:4])\ntraining_target = jnp.array(sin_t[1:5])\n\nprint('Example of training sample')\nprint('input-> ', training_input.ravel())\nprint('target-> ', training_target.ravel())\n```\n\nAt this point, all the functions and parameters have been defined. We can create the training loop, run it and visualize the output plot predictions\n\n\n```python\n######################\n# YOUR CODE HERE #\n######################\n# Initialize parameters and get the initial state of the RNN\nparams = None\ninitial_rnn_state = None\n\nassert params is not None, 'Initialize params of the RNN.'\nassert initial_rnn_state is not None, 'Get the initial state of the RNN.'\n\n\nopt_init, optimizer = optax.adam(STEP_SIZE)\nopt_state = opt_init(params)\n\nlosses = []\nfor i in range(NUM_ITERATIONS+1):\n # Training.\n start = np.random.choice(range(len(sin_t) - UNROLL_LENGTH))\n training_input = jnp.array(sin_t[start: start+UNROLL_LENGTH])\n training_target = jnp.array(sin_t[start+1: start+UNROLL_LENGTH+1])\n # Note: `optimize` calls prediction, computes the loss and updates the \n # parameters of the network and the state of the optimizer.\n loss_value, opt_state, params = optimize(params, opt_state, \n initial_rnn_state, training_input, \n training_target)\n losses.append(loss_value.tolist())\n\n # Full sequence generation and plotting.\n if i % REPORTING_INTERVAL == 0:\n\n # Generate the full sequence (from the first to the last element).\n y_gen = predict(params, initial_rnn_state, sin_t[:-1], training=False)\n sampling_loss = loss_fn(params, initial_rnn_state, y_gen, sin_t[1:])\n plt.figure(figsize=(10, 5))\n\n plt.title(f'Training Loss {loss_value.tolist():.3f}, sampling loss '\n f'{sampling_loss.tolist():.3f}, iteration {i}')\n\n plt.plot(t[1:].ravel(), sin_t[1:].ravel(), c='blue',\n label='Ground truth',\n linestyle=':', lw=6)\n\n if TEACHER_FORCING:\n plt.plot(t[start: start+UNROLL_LENGTH].ravel(),\n training_input.ravel(),\n alpha=0.7, lw=5, c='green', label='Training input')\n else:\n plt.plot(t[start: start+WARM_START].ravel(),\n training_input[:WARM_START].ravel(),\n alpha=0.7, lw=5, c='green', label='Training input')\n plt.plot(t[start+1: start+UNROLL_LENGTH+1].ravel(),\n training_target.ravel(),\n alpha=0.7, lw=2, c='black', label='Training target')\n\n plt.plot(t[start: start+UNROLL_LENGTH].ravel(),\n predict(params, initial_rnn_state, training_input),\n alpha=0.7, lw=3, c='gold', label='Network training prediction')\n\n plt.plot(t[1:].ravel(),\n y_gen,\n alpha=0.7, c='r', label='Network full generation')\n\n plt.legend()\n plt.show()\n\nplt.figure()\nplt.title('Training Loss')\n_ = plt.plot(losses)\n```\n\nNow try changing the hyperparameters and see how they affect the training (e.g., how accurately does the network learn to model the curve, how fast does it converge, etc, ...)\n\n## **What is worth trying/understanding here?**\n- Difference between teacher forcing and learning on own samples:\n - What are the pros and cons of teacher forcing?\n - Why is the model struggling to learn in one of the setups?\n - What is it that we actually care about when we model sequences? How can this be negatively affected by teacher forcing?\n- Effect of warm starting:\n - How does warm starting affect our training when teacher forcing is disabled? Why?\n - How does warm starting affect our sampling when teacher forcing is enabled? Why?\n- What happens if the structure of interest is much longer than the unroll length?\n\n\n# Part II: Character-level language modeling\n\n**Goal**: Train a character level LSTM on text data - specifically Shakespeare's sonnets. \n\n**What is an LSTM**: An LSTM is an advanced variant of the RNN. As opposed to the vanilla RNN the output and the state are separate, which allows for much more flexibility in deciding what to store. The LSTM is also characterized by a more advanced mechanism to determine which parts of the input to store into the memory and what to potentially forget. *For an in-depth analysis of the differences between various kinds of RNNs, we recommend you to read [this excellent guide](https://colah.github.io/posts/2015-08-Understanding-LSTMs/) by Chris Olah.*\n\n**Haiku**: In order to develop our network we will use [Haiku](https://github.com/deepmind/dm-haiku), which provides many common building blocks, including LSTMs! \n\n**This tutorial**: Throughout this section you will build the network, train an LSTM on a dataset of Shakespeare's sonnets and finally you will be able to see how the quality of the generated text improves over time as the network trains.\n\n\n```python\n# @title Hyper-parameters\nBATCH_SIZE = 32 #@param {type:'integer'}\nSEQUENCE_LENGTH = 128 #@param {type:'integer'}\nHIDDEN_SIZE = 256 #@param {type:'integer'}\nSAMPLE_LENGTH = 128 #@param {type:'integer'}\nLEARNING_RATE = 1e-3 #@param {type:'number'}\nTRAINING_STEPS = 6500 #@param {type:'integer'}\nSAMPLING_INTERVAL = 200 #@param {type:'integer'}\nSEED = 43 #@param {type:'integer'}\nNUM_CHARS = 128 #@param {type:'integer'}\n```\n\n\n```python\n#@title Dataset loader with encoding and decoding functions { form-width: \"170px\" }\nBatch = Mapping[str, np.ndarray]\n\ndef load(split: tfds.Split, *, batch_size: int,sequence_length: int):\n '''Creates the Tiny Shakespeare dataset as a character modelling task.'''\n\n def preprocess_fn(x: Mapping[str, tf.Tensor]) -> Mapping[str, tf.Tensor]:\n x = x['text']\n x = tf.strings.unicode_split(x, 'UTF-8')\n x = tf.squeeze(tf.io.decode_raw(x, tf.uint8), axis=-1)\n x = tf.cast(x, tf.int32)\n return {'input': x[:-1], 'target': x[1:]}\n\n ds = tfds.load(name='tiny_shakespeare', split=split)\n ds = ds.map(preprocess_fn)\n ds = ds.unbatch()\n ds = ds.batch(sequence_length, drop_remainder=True)\n ds = ds.shuffle(100)\n ds = ds.repeat()\n ds = ds.batch(batch_size)\n ds = ds.map(lambda b: tf.nest.map_structure(tf.transpose, b)) # Time major.\n return tfds.as_numpy(ds)\n\ndef decode(x: np.ndarray) -> str:\n return ''.join([chr(x) for x in x])\n\ndef encode(x: str) -> np.ndarray:\n return np.array([ord(s) for s in x])\n```\n\n## Build the network with Haiku [EXERCISE]\n\nIn this section you will need to use Haiku to build the network. To do so, you will make use of the `hk.DeepRNN` ([doc](https://dm-haiku.readthedocs.io/en/latest/api.html#haiku.DeepRNN)) module to wrap together several layers into a single RNN core.\n\nImplement the following architecture:\n\n- Convert the input to a one-hot representation with `NUM_CHARS` elements (hint: use a lambda function `lambda x: jax.nn.one_hot(...)`) [(doc)](https://github.com/google/jax/blob/6c8fc1b031275c85b02cb819c6caa5afa002fa1d/jax/_src/nn/functions.py#L261)\n- LSTM of `HIDDEN_SIZE` size ([doc](https://dm-haiku.readthedocs.io/en/latest/api.html?highlight=lstm#lstm))\n- ReLU (non-linearity) ([doc](https://jax.readthedocs.io/en/latest/_autosummary/jax.nn.relu.html))\n- LSTM of `HIDDEN_SIZE` size ([doc](https://dm-haiku.readthedocs.io/en/latest/api.html?highlight=lstm#lstm))\n- An two-layer MLP composed of a layer of `HIDDEN_SIZE` size with ReLU activation, followed by a linear (no activation) layer mapping to the final problem size (number of characters: `NUM_CHARS`) ([docstring](https://github.com/deepmind/dm-haiku/blob/300e6a40be31e35940f0725ae7ed3457b737a5a3/haiku/_src/nets/mlp.py#L38))\n\n\n```python\ndef make_network() -> hk.RNNCore:\n \"\"\"Define the RNN core.\"\"\"\n ######################\n # YOUR CODE HERE #\n ######################\n # Design the LSTM with Haiku\n # following the architecture\n # defined in the text above.\n \n rnn_core = hk.DeepRNN([\n # add code\n ])\n\n assert rnn_core is not None, 'LSTM model should be initialized'\n\n return rnn_core\n```\n\n## Loss and Update [EXERCISE]\nAs before, we update the parameters of the network using the **ADAM** optimizer. \n\nIn this section you need to complete the `loss_fn`, which should compute the loss as the categorical cross-entropy of the prediction and the targets. The `optimize` function (which you don't need to modify) uses this `loss_fn` function to compute the loss and the gradient of the loss w.r.t. the parameters of the network, and takes one step of gradient descent to reduce such loss. This is effectively one step of training.\n\nAs before, once you are done debugging uncomment the `@jit` decorator **to ensure the code runs at full speed**.\n\n### Categorical Cross-entropy Recap\nThe categorial cross-entropy loss is computed as\n\n$$\\mathcal{L}_{\\text{xent}} = - \\mathbf{y} \\cdot \\log(\\mathbf{\\hat{y}})$$\n\nwhere $\\mathbf{y}$ is the true class probability and $\\mathbf{\\hat{y}}$ is the predicted class probability.\n\n**Example**\n\nLet's take as an example a classification problem with 3 classes. Assume our **target vector** for the current sample to be $[0, 1, 0]$ (i.e., the second class is the correct one, according to the ground truth). If our network's **predicted class probability** is $[0.2, 0.7, 0.1]$, the categorical cross-entropy loss for this prediction will be\n\n$$\n\\begin{align}\n \\mathcal{L}_{\\text{xent}} =& - \\bigg( 0 \\cdot \\log(0.2) + 1 \\cdot \\log(0.7) + 0 \\cdot \\log(0.1) \\bigg)\\\\\n =& - \\log(0.7) \\\\ \n =& \\; 0.35\n\\end{align}\n$$\n\nIn this case, the network was fairly confident that the input belonged to the second class, which was correct, so the error is not very large.\n\nLet's see what would have happened if the network was mistakenly confident that the true class was another one instead. Suppose the predicted vector was $[0.8, 0.05, 0.15]$, i.e., that the network was assigning a very low probability to the true class. The loss in this case would have been $-\\log(0.05) = 3$, which is much higher than before.\n\n\n```python\ndef loss_fn(batch):\n \"\"\"\n Compute the categorical cross-entropy loss.\n\n Unroll the network over a sequence of batched inputs & targets, and\n compute the categorical cross-entropy loss.\n\n Args:\n logits: the logits from the network\n batch: a batch of input and target data\n\n Returns:\n The categorical cross-entropy loss.\n \"\"\"\n sequence_length, batch_size = batch['input'].shape\n\n ######################\n # YOUR CODE HERE #\n ######################\n # create the network and unroll it (dynamic unrolling)\n # Obtain the prediction by computing the log_softmax on logits\n # Convert the log probabilities to one hot labels\n # and compute the categorical cross-entropy loss\n loss = None\n\n assert loss is not None, 'Loss should be computed'\n\n return loss\n\n\n# @jit\ndef optimize(params, opt_state, batch):\n \"\"\"\n Applies an update to the parameters.\n\n Args:\n params: the parameters of the neural network\n opt_state: the state of the optimizer\n batch: batch of input and target data\n\n Returns:\n The new optimizer state and the updated network parameters.\n \"\"\"\n ######################\n # YOUR CODE HERE #\n ######################\n # Similar to the RNN update function\n # - initialize the Adam optimizer\n # - apply grad function to loss\n # - compute params updates\n # - apply updates on params\n new_params = None\n new_opt_state = None\n\n # Note that since JAX is stateless (the state is passed explicitly to each \n # operation) here we create another optimizer function for simplicity, \n # instead of passing the one we create when we first initialize the state\n # of the optimizer.\n\n assert new_opt_state is not None, 'Compute the updated optimizer\\'s state.'\n assert new_params is not None, 'Compute the updated parameters.'\n\n return new_params, new_opt_state\n```\n\n## Sampling function\n\nThis function draws samples from the model, given an initial context.\n\n\n```python\ndef sample_fn(rng_key: jnp.ndarray, context: jnp.ndarray, sample_length: int):\n \"\"\"\n Draws samples from the model, given an initial context.\n\n Note: this function is impure; we will hk.transform() it (and jit it) in the\n training loop.\n\n Args:\n rng_key: random key\n context: initial context given to the model\n sample_length: length of the sample\n\n Returns:\n the generated tokens\n \"\"\"\n assert context.ndim == 1 # Sequence only, no batch\n\n def body_fn(t, values):\n tokens, state, rng_key = values\n token = tokens[t]\n next_logits, next_state = rnn_core(token, state)\n rng_key, k1 = jax.random.split(rng_key)\n next_token = jax.random.categorical(k1, next_logits, axis=-1)\n new_tokens = ops.index_update(tokens, ops.index[t + 1], next_token)\n return new_tokens, next_state, rng_key\n\n # Unroll over context (initial prompt).\n rnn_core = make_network()\n initial_state = rnn_core.initial_state(None) # no batch here!\n logits, state = hk.dynamic_unroll(rnn_core, context, initial_state)\n\n # Sample the first continuation token and initialize the output array.\n rng_key, k1 = jax.random.split(rng_key)\n first_token = jax.random.categorical(k1, logits[-1])\n tokens = np.zeros(sample_length, dtype=np.int32)\n tokens = ops.index_update(tokens, ops.index[0], first_token)\n\n # Sample the other tokens in a loop.\n initial_values = tokens, state, rng_key\n new_tokens, _, _ = lax.fori_loop(0, sample_length, body_fn, initial_values)\n\n return new_tokens\n```\n\n## Train and generate language\nFinally, this training loop puts everything together. The network is trained for a fixed number of `TRAINING_STEPS` and is evaluated at fixed intervals. During the evaluation, given a *context* sentence provided as input, the network tries to generate some text to complete that prompt.\n\nAs you can see, for the first few iterations the model generates nonsensical sentences, but after some training you can observe that the model is actually learning to reproduce text that resembles Shakespeare's sonnets.\n\n**Important:** make sure you uncommented the `@jit` decorators in the previous\ncode to enjoy accelerated training. Training should take less than 7 minutes.\n\n\n\n```python\n%%time\n# Note that since we use Haiku, we use their PRNG instead of the one from JAX.\nrng = hk.PRNGSequence(SEED)\n\ntrain_data = load(tfds.Split.TRAIN, batch_size=BATCH_SIZE,\n sequence_length=SEQUENCE_LENGTH)\ntrain_data = iter(train_data)\n\n# Out network doesn't have a nondifferentiable state (the inner states of the \n# LSTMs is differentiable), so we don't need to use `hk.transform_with_state` \n# and can use the typical `hk.transform` instead. Further, our network does not\n# have stochastic operations, so we can wrap it in a `without_apply_rng` call.\n# Note that here we transform `loss_fn` as a way to initialize the network\n# parameters a few lines below.\ntransformed_loss = hk.without_apply_rng(hk.transform(loss_fn))\n_, sample = hk.without_apply_rng(hk.transform(sample_fn))\nsample = jax.jit(sample, static_argnums=[3]) # sample_length is static\n\n# Setup optimizer.\nparams = transformed_loss.init(next(rng), next(train_data))\n# Similar to what we do for the loss, here we create the optimizer just to get\n# the initial optimizer state.\noptimizer = optax.adam(LEARNING_RATE)\nopt_state = optimizer.init(params)\n\nBOLD = '\\033[1m'\nEND = '\\033[0m'\nSEP = '\\n------------------------------\\n'\nfor step in range(TRAINING_STEPS):\n train_batch = next(train_data) # includes input and target\n \n # Run one step of training and update the parameters.\n params, opt_state = optimize(params, opt_state, train_batch)\n\n if step % SAMPLING_INTERVAL == 0:\n # Use the input text of the batch as context.\n context = train_batch['input'][:, 0] # drop the batch\n rng_key = next(rng)\n # Sample generated text given context.\n samples = sample(params, rng_key, context, SAMPLE_LENGTH)\n # Decode context and samples to actual sentences.\n prompt = decode(context)\n continuation = decode(samples)\n\n print(f'{BOLD}>>> Prompt (from the dataset):{SEP}{END}\\n{prompt}\\n')\n print(f'{BOLD}Continuation (from the network):{END}\\n{continuation}\\n')\n```\n\n## **What is worth trying/understanding here?**\n- After how many iterations the model is able to learn a sentence which makes sense?\n- Why the first generated sentences do not make sense and what pattern can you see?\n- How is the sample length parameters affecting the results?\n\nAnswers:\n\n- same words repeated\n\n# Part III: Topic modeling with Non-negative Matrix Factorization\n\n**Goal**: Extract topics from a set of documents exploiting *matrix factorization*.\n\n**What is topic modeling**: *Topic modeling* is a Natural Language Processing task whose aim is to discover the abstract *topics* that occur in a set of documents. Intuitively the idea is that, depending on the topic discussed in a document or a set of documents, specific words will appear more or less frequently. For instance, 'dog' and 'bone' will appear more often in documents about dogs, while 'cat' and 'meow' will appear in documents about cats. Common words, like 'the' and 'is', will tend to appear approximately uniformely in all kinds of documents so are not very informative to tell the documents apart. We can exploit this to cluster the documents in groups, according to their topic.\n\n**Credits:** this code is an adaptation of [this fast.ai tutorial](https://nbviewer.jupyter.org/github/fastai/numerical-linear-algebra/blob/master/nbs/2.%20Topic%20Modeling%20with%20NMF%20and%20SVD.ipynb).\n\n## Load dataset\n\nIn this section, we will use the well-known benchmark dataset **20newsgroups** to model the main topics discussed in a group of conversations. Newsgroups are discussion groups on Usenet, which was popular in the 80s and 90s before the web really took off. This dataset includes 18,000 newsgroups posts on 20 different topics.\n\n\n```python\ncategories = ['rec.motorcycles', 'sci.med', 'comp.graphics', 'sci.space']\nremove = ('headers', 'footers', 'quotes')\nnewsgroups_train = fetch_20newsgroups(subset='train', categories=categories, remove=remove)\n```\n\n\n```python\n# Print three random documents.\nprint('\\n\\n-----\\n'.join(newsgroups_train.data[:3]))\n```\n\n\n```python\n# Print the category of the previous documents.\nnp.array(newsgroups_train.target_names)[newsgroups_train.target[:3]]\n```\n\n## Document discrete representations\n\nIn order to represent the documents as a **term-document matrix**, we need to transform unstructured data (text) to structured data.\n\nThis can be performed by extracting count and/or frequencies of words in each documents.\n\n- The **Bag of Words (BoW)** model is the simplest way to represent text in numerical form. As the name suggests (nomen omen!), given a vocabulary of words we can represent any sentence as a *bag of words vector* (i.e., a vector of numbers) using the *counts*, i.e. we count how many times each word in the dictionary appears in the documents.\n\n- The **TF-IDF** (Term Frequency - Inverse Document Frequency) representation model is similar, but accounts for both the frequencies of words and their importance. For example, if a words appears in all the documents, it is not very important, as it does not allow us to differentiate documents.\n\nWe will use **TF-IDF** representations, which we will extract using the [sklearn](https://scikit-learn.org/) library. \n\n\n\n```python\n# Initialize the TF-IDF representation model.\ntf_idf = TfidfVectorizer(stop_words='english', min_df=10)\n# Apply TF-IDF representation to the documents.\nX_train = tf_idf.fit_transform(newsgroups_train['data'])\nX_train = X_train.todense()\nX_train = jnp.array(X_train)\n```\n\n\n```python\nX_train.shape\n```\n\n\n```python\n# Each column of the representation matrix corresponds to a word.\nvocab = np.array(tf_idf.get_feature_names())\nprint(vocab)\n```\n\n## Non-Negative Matrix Factorization [EXERCISE]\n\n**Non-Negative Matrix Factorization** is a statistical method to reduce the dimension of the input corpora. It uses the [factor analysis](https://en.wikipedia.org/wiki/Factor_analysis) method to provide comparatively less weight to the words with least coherence.\n\nTo reason on a practical example, assume we have an input matrix $V$ of shape $m \\times n$. This method factorizes $V$ into two matrices $W$ and $H$, such that the dimension of $W$ is $m \\times k$ and that of $H$ is $k \\times n$. In our case, $V$ represent the *term-document matrix* (the frequency of the terms that occur in the collection of documents), each row of the matrix $H$ is a word embedding and each column of the matrix $W$ represent the weight of each word in each of the sentences (i.e., the semantic relation of words with each sentence). \n\n\n\nThe assumption is that all the entries of $W$ and $H$ are positive, since all the entries of $V$ are positive.\n\n\n\n```python\ndef initialize_parameters(rnd_key, X_train, k):\n \"\"\"\n This function will initialize and returns the Vanilla RNN parameters.\n \n Args:\n rnd_key: random key\n X_train: term-document matrix\n k: hidden dimension\n\n Returns:\n A list of parameters [W, V, bias, decoder, decoder_bias].\n \"\"\"\n ######################\n # YOUR CODE HERE #\n ######################\n # Initialize the NNMF parameters (W and H) using normal distribution\n # (mean zero and unit standard deviation) and correct shapes\n m, n = X_train.shape\n k1, k2 = random.split(rnd_key)\n params = None\n\n assert params is not None, 'Params should be initialized'\n\n return params\n \n```\n\n## Frobenius Norm\n\nA way to perform NMF is by using the Frobenius norm, which is defined by the square root of the sum of the absolute squares of its elements. \n\n$$ ||A||_F = \\sqrt{\\sum_{i=1}^m \\sum_{j=1}^n | a_{i,j}|^2}$$\n\nWe want to minimize the Frobenious norm of\n\n$$ V - WH $$\n\nhowever, we also want to penalize the matrix $W$ and $H$ when they are negative. Thus, we will add a specific penalty function.\n\n### Penalty [EXERCISE]\n\n\n```python\ndef compute_penalty(matrix):\n \"\"\"\n Return the average of the square of the negative values in the matrix.\n\n Args:\n matrix: the matrix on which we want to compute the penality\n\n Returns:\n The penalty\n\n \"\"\"\n # Equivalent to `(matrix[matrix < 0]**2).mean()`\n return jnp.power(jnp.clip(matrix, a_max=0.), 2).mean()\n```\n\nLet's see this function in action: the \"more negative\" the matrix is, the higher the penalty will be\n\n\n```python\nA = jnp.array([[0, 1, 4], [5, 6, 7]]) # positive\nB = jnp.array([[0, 1, -4], [5, 6, 7]]) # some negative elments\nC = jnp.array([[0, 1, 4], [-5, -6, -7]]) # more negative elements\n\nprint(\"A\", compute_penalty(A))\nprint(\"B\", compute_penalty(B))\nprint(\"C\", compute_penalty(C))\n```\n\nWe can now use this into our general loss function\n\n\n```python\ndef get_penalty():\n \"\"\"\n We combine the penality for both matrices\n \"\"\"\n return (compute_penalty(params['W']).mean() + \n compute_penalty(params['H']).mean())\n\ndef loss_fn(params, V):\n \"\"\"\n Compute the Frobenius norm factorization loss\n \n The Frobenius norm of the matrix V and its matrix factorization (W*H),\n plus the penalty scaled by the regularization parameter.\n \n Args:\n params: params to optimize (W and H)\n V: matrix\n Returns:\n The Frobenius norm factorization loss.\n \"\"\"\n ######################\n # YOUR CODE HERE #\n ######################\n # Compute the loss and sum the regularization.\n # Combine the frobenious norm with the penality (weight the penality by the \n # regularization parameter)\n loss = None\n regularization_param=1e6\n\n assert loss is not None, 'Loss should be computed'\n\n return loss\n```\n\n## Parameters update function\nAs in the previous sections, the `optimize` function (which you don't need to modify) uses the `loss_fn` function to compute the loss and the gradient of the loss w.r.t. the parameters of the network, and takes one step of gradient descent to reduce such loss. This is effectively one step of training.\n\n\n```python\n@jit\ndef optimize(params, opt_state, x):\n loss_value, grads = jax.value_and_grad(loss_fn)(params, x)\n updates, opt_state = opt_update(grads, opt_state)\n params = optax.apply_updates(params, updates)\n return loss_value, opt_state, params\n```\n\n## Train\n\n\n```python\nrnd_key = random.PRNGKey(1)\nrnd_key, k1, k2 = random.split(rnd_key, 3)\nparams = initialize_parameters(k1, X_train, 5)\nopt_init, opt_update = optax.adam(0.01)\nopt_state = opt_init(params)\nfor _ in range(0, 200):\n loss_value, opt_state, params = optimize(params, opt_state, X_train)\n```\n\n## Visualize output topics\n\n\n```python\n# The number of most-common words that we want to visualize.\nnum_top_words = 8 #@param {type:\"integer\"}\n\ndef show_topics(H_matrix):\n \"\"\"\n Shows the topic coming from the H matrix. \n To be used after we have trained our NMF model.\n\n Args:\n H_matrix: the matrix that contains the topics\n\n Returns: \n A list of lists with the topic words for each topic\n \"\"\"\n top_words = lambda t: [vocab[i] for i in np.argsort(t)[:-num_top_words-1:-1]]\n topic_words = ([top_words(t) for t in H_matrix])\n return [' '.join(t) for t in topic_words]\n\nshow_topics(params['H'])\n```\n\n# Part IV: Text classification with neural networks\n\n**Goal**: Perform text classification exploiting traditional and advanced representation models and assess their impact on classification performance.\n\n**What is text classification**: *Text classification* is the most basic task of Natural Language Processing. The goal is to classify textual data as belonging to one (yes/no) or more categories.\n\nIn this section, we take the specific task of **Sentiment Analysis** applied to reviews. Sentiment analysis, also called *opinion mining*, is the field of study that analyzes people\u2019s opinions, sentiments, evaluations, appraisals, attitudes, and emotions towards entities such as products, services, organizations, individuals, issues, events, topics, and their attributes. Commonly, Sentiment Analysis is performed as a classification problem where a text should be classified as being *positive* or *negative*.\n\nTo address this task, we will investigate various representation models with a simple neural network. The goal of this section is to show the impact of the different representations on the classification accuracy.\n\nThe code has been adapted from [[link](https://github.com/google/jax/blob/master/docs/notebooks/neural_network_with_tfds_data.ipynb)]\n\n\n\n\n\n## Load dataset\n\nHere we load the *IMDB reviews* dataset for binary sentiment classification. The training set is composed of 25,000 highly movie reviews.\n\n\n```python\n# Load and preprocess the data.\ntrain_data, test_data = tfds.load(name='imdb_reviews', split=['train', 'test'], \n batch_size=-1, as_supervised=True)\ntrain_examples, train_labels = tfds.as_numpy(train_data)\ntest_examples, test_labels = tfds.as_numpy(test_data)\ntrain_examples = list(map(lambda x : x.decode('utf-8') , train_examples))\ntest_examples = list(map(lambda x : x.decode('utf-8') , test_examples))\n\ndef batch(X, y, batch_size=32):\n \"\"\"\n This function will allow to select batches of training data.\n\n Args:\n X: train raw data\n y: train labels\n batch_size: size of the batch\n \"\"\"\n total_len = len(X)\n for ndx in range(0, total_len, batch_size):\n yield (X[ndx:min(ndx + batch_size, total_len)], \n y[ndx:min(ndx + batch_size, total_len)])\n```\n\n## Initialize Variables\n\n\n```python\nPARAM_SCALE = 0.1 #@param {type:'number'}\nSTEP_SIZE = 0.01 #@param {type:'number'}\nNUM_EPOCHS = 10 #@param {type:'integer'}\nBATCH_SIZE = 128 #@param {type:'integer'}\nN_TARGETS = 2 #@param {type:'integer'}\n```\n\n## Traditional Representation: TF-IDF [EXERCISE]\n\nSimilar to what we've previously seen, we use TF-IDF representation for the discrete representation of textual data.\n\n\n```python\ndef get_discrete_data(train_examples,test_examples):\n\n ######################\n # YOUR CODE HERE #\n ######################\n # Transform test and train data with TF-IDF (using Sklearn TfidfVectorizer,\n # from sklearn) with min document frequency set to 80\n # transform arrays to dense arrays \n X_train = None\n X_test = None\n \n # we make these matrices dense\n X_train = X_train.todense()\n X_test = X_test.todense()\n\n assert X_train is not None, 'Data should be transformed with TF-IDF vectorizer'\n\n return X_train, X_test\n```\n\n## Initialize parameters [EXERCISE]\n\n\n```python\ndef random_layer_params(m, n, key, scale=1e-2):\n w_key, b_key = random.split(key)\n return (scale * random.normal(w_key, (n, m)),\n scale * random.normal(b_key, (n,)))\n\ndef init_network_params(sizes, rnd_key):\n \"\"\"\n Initialize neural network parameters. \n\n Args:\n sizes: layers' size\n key: random key\n\n Returns:\n The initialized parameters.\n \"\"\"\n keys = random.split(rnd_key, len(sizes))\n return [random_layer_params(m, n, k) \n for m, n, k in zip(sizes[:-1], sizes[1:], keys)]\n```\n\n\n```python\ndef predict(params, sentence):\n \"\"\"\n Predict class given a sentence. \n\n Args:\n params: the parameters of the neural network\n sentence: input data\n\n Returns:\n The predictions of the network\n \"\"\"\n activations = sentence\n\n ######################\n # YOUR CODE HERE #\n ######################\n # For each layer in the neural network\n # compute the basic MLP (with weights and bias)\n # using relu as activation function.\n # Pay attention to the last layer\n\n output_last_layer = None\n\n assert output_last_layer is not None, 'Design MLP architecture'\n\n return jax.nn.log_softmax(output_last_layer)\n\n# We use vmap to make the prediction batched.\nbatched_predict = vmap(predict, in_axes=(None, 0))\n```\n\n\n```python\n def accuracy(params, x, y):\n \"\"\"\n Compute accuracy measure.\n\n Args:\n x: input data\n y: target data\n\n Returns:\n The classification accuracy\n \"\"\"\n\n ######################\n # YOUR CODE HERE #\n ######################\n # Compute the accuracy.\n #\n # Reminder: the accuracy is \n # the average number of correctly \n # predicted classes.\n #\n # Hint: Use batched_predict to get the\n # classes ids predicted by the network,\n # and compare them with the target\n # classes from the ground truth.\n \n accuracy = None\n\n assert accuracy is not None, 'Accuracy should be computed'\n\n return accuracy\n```\n\n## Loss and update\n\n\n```python\ndef loss(params, x, y):\n \"\"\"\n Loss function. \n\n Args:\n params: the parameters of the neural network\n x: input data\n y: target data (class)\n\n Returns:\n The computed loss\n \"\"\"\n preds = batched_predict(params, x)\n return -jnp.mean(preds * y)\n\n@jit\ndef update(params, x, y):\n \"\"\"\n The optimization function. \n\n Args:\n params: the parameters of the neural network\n x: input data\n y: target data (class)\n \n Returns:\n The updated parameters\n \"\"\"\n loss_value, grads = jax.value_and_grad(loss)(params, x, y)\n return [(w - STEP_SIZE * dw, b - STEP_SIZE * db)\n for (w, b), (dw, db) in zip(params, grads)], loss_value\n```\n\n## Train\n\n\n```python\ndef train_loop(train_examples, test_examples, train_labels, test_labels, \n layer_sizes=None):\n # Visualize learning progress with a bar.\n pbar = tqdm(total=NUM_EPOCHS, position=0, leave=True)\n\n X_train, X_test = get_discrete_data(train_examples, test_examples)\n if layer_sizes is None:\n layer_sizes = [X_train.shape[1], 256, 2]\n params = init_network_params(layer_sizes, random.PRNGKey(0))\n\n # Transform target data to one-hot representation.\n y = jax.nn.one_hot(train_labels, N_TARGETS)\n y_test = jax.nn.one_hot(test_labels, N_TARGETS)\n\n for epoch in range(NUM_EPOCHS):\n # Update parameters for each batch.\n for x_t, y_t in batch(X_train, y):\n params, loss_value = update(params, x_t, y_t)\n pbar.update(1)\n \n # Compute accuracy on training and test.\n train_acc = accuracy(params, X_train, y)\n test_acc = accuracy(params, X_test, y_test)\n\n pbar.set_description('Loss value is {0:.2f}, training accuracy is '\n '{1:.5}, test accuracy is {2:.5}'.format(loss_value,\n train_acc,\n test_acc))\n```\n\n\n```python\ntrain_loop(train_examples, test_examples, train_labels, test_labels)\n```\n\n## Preprocessing + TF-IDF\n\nText preprocessing is traditionally an important step for NLP tasks. It transforms text into a more digestible form so that machine learning algorithms can perform better. For example:\n- it removes 'not meaningful words' such as a,an,the,for, etc. (so-called *stopwords*)\n- lowercase everything\n- remove symbols\n- remove or process hashtags, mentions or emoticons (particularly important for social media text)\n\nWe expect that 'cleaning' the text will help us on obtaining better performance.\n\n\n```python\n# Preprocessing functions\ndef split_hashtag(tag):\n pattern = re.compile(r'[A-Z][a-z]+|\\d+|[A-Z]+(?![a-z])')\n return pattern.findall(tag)\n\n\ndef quick_preprocessing(sentence, language, process_urls=False, \n process_mentions=False, process_emoticon=False,\n split_hashtags=False):\n \"\"\"\n A quick preprocessing function that removes unwanted tokens from the text\n\n Args:\n sentence: the sentence to be preprocessed\n language: the language the sentence is in\n process_urls: whether to process URLs\n process_mentions: whether to process mentions\n process_emoticon: whether to process emoticons\n split_hashtags: whether to split hashtags\n\n Returns:\n The modified sentence given the preprocessing configuration\n\n \"\"\"\n url_re = (r'http[s]?://(?:[a-z]|[0-9]|[$-_@.&+]|[!*\\(\\),]|(?:%[0-9a-f]'\n r'[0-9a-f]))+')\n mention_re = r'(?:@[\\w_]+)'\n emoticons_re = r'(?::|;|=)(?:-)?(?:\\)|\\(|D|P)'\n\n if process_urls:\n sentence = re.sub(url_re, 'TAGurl', sentence, flags=re.MULTILINE)\n if process_mentions:\n sentence = re.sub(mention_re, 'TAGmention', sentence,\n flags=re.MULTILINE)\n if process_emoticon:\n sentence = re.sub(emoticons_re, 'TAGemoticon', sentence,\n flags=re.MULTILINE)\n \n if split_hashtags:\n hashtag_list = list({tag.strip('#') for tag in sentence.split() \n if tag.startswith('#')})\n hashtag_split_list = [' '.join(split_hashtag(h)) for h in hashtag_list]\n for i in range(len(hashtag_list)):\n sentence = sentence.replace(hashtag_list[i],\n ' ' + hashtag_split_list[i] + ' ')\n\n sentence = sentence.lower()\n sentence = ' '.join(\n re.sub('(@[A-Za-z0-9]+)|([^0-9A-Za-z \\t])|(\\w+:\\/\\/\\S+)',' ',\n sentence).split())\n tokenizer = RegexpTokenizer(r'\\w+')\n words = tokenizer.tokenize(sentence) \n good_words = [w for w in words]\n return ' '.join(good_words)\n\n\ndef complete_preprocessing(sentence, language):\n return quick_preprocessing(sentence, language, True, True, True, True)\n```\n\n\n```python\n# Apply preprocessing to train and test samples.\npreprocessed_training = []\npreprocessed_testing = []\n\npbar = tqdm(total=len(train_examples), position=0, leave=True)\nfor tr_example, ts_example in zip(train_examples, test_examples):\n preprocessed_training.append(complete_preprocessing(tr_example, 'english'))\n preprocessed_testing.append(complete_preprocessing(ts_example, 'english'))\n pbar.update(1)\n\npbar.close()\n```\n\nWe use the same code as before for performing the training\n\n\n```python\ntrain_loop(preprocessed_training, preprocessed_testing, train_labels, \n test_labels)\n```\n\n## Entering BERT Embeddings [EXERCISE]\n\nRecently, pretrained sentence embeddings, lead by Bidirectional Encoder Representations from Transformers ([BERT](https://www.aclweb.org/anthology/N19-1423/)), have become the standard in any NLP application. This means that sentences can be transformed into a dense and meaningful representation exploiting these large pretrained language models.\n\nIn this section, we use the [Sentence-Transformers](https://github.com/UKPLab/sentence-transformers) library that permits to compute dense vector representations for sentences and paragraphs using transformer networks like BERT / RoBERTa / XLM-RoBERTa etc. \n\n**NOTE**: this might generate an OOM errror because these embeddings are generated with pytroch (that is fighting with Jax for the GPU). One thing you can do is: restart the notebook, generate the embeddings, and then train this new network with jax. You can look at jax memory allocation settings [here](https://jax.readthedocs.io/en/latest/gpu_memory_allocation.html).\n\n%%capture\nmodel_bert = SentenceTransformer('distilbert-base-nli-mean-tokens') \n\n\n```python\n%%capture\nmodel_bert = SentenceTransformer('distilbert-base-nli-mean-tokens') \n```\n\n\n```python\n# Transform the training and testing data into sentence embeddings.\n# This might take a few minutes.\nembeddings_train = jnp.array(model_bert.encode(train_examples, show_progress_bar=True))\nembeddings_test = jnp.array(model_bert.encode(test_examples, show_progress_bar=True))\n```\n\n\n```python\ndef bert_train_loop(X_train, X_test, train_labels, test_labels, \n layer_sizes=None):\n # Visualize learning progress with a bar.\n pbar = tqdm(total=NUM_EPOCHS, position=0, leave=True)\n\n if layer_sizes is None:\n layer_sizes = [X_train.shape[1], 256, 2]\n params = init_network_params(layer_sizes, random.PRNGKey(0))\n\n # Transform target data to one-hot representation.\n y = jax.nn.one_hot(train_labels, N_TARGETS)\n y_test = jax.nn.one_hot(test_labels, N_TARGETS)\n\n for epoch in range(NUM_EPOCHS):\n # Update parameters for each batch.\n for x_t, y_t in batch(X_train, y):\n params, loss_value = update(params, x_t, y_t)\n pbar.update(1)\n \n # Compute accuracy on training and test.\n train_acc = accuracy(params, X_train, y)\n test_acc = accuracy(params, X_test, y_test)\n\n pbar.set_description('Loss value is {0:.2f}, training accuracy is '\n '{1:.5}, test accuracy is {2:.5}'.format(loss_value,\n train_acc,\n test_acc))\n return params\n```\n\n\n```python\nbert_network = bert_train_loop(embeddings_train, embeddings_test, train_labels, test_labels)\n```\n\n## Let's play with our new network :)\n\nWe are going to test our sentiment model on two positive documents and 1 negative one. Let's see if the model is able to predict the classes as expected.\nYou can add custom sentences and see the results!\n\n\n```python\nexamples = jnp.array(model_bert.encode([\"wow, they nice movie totally recommend it\", \n \"worst movie ever, I will never go to the cinema again\", \n \"I liked it a lot, best movie ever\"]))\n```\n\n\n```python\njnp.argmax(jnp.exp(batched_predict(bert_network, examples)), axis=1)\n```\n", "meta": {"hexsha": "517aa70939366753787930167e6981176ccd86e5", "size": 86328, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "To_be_Solved_NLP_Tutorial_Jax_Haiku_Language_Modeling_Classification.ipynb", "max_stars_repo_name": "linker81/tutorials2021", "max_stars_repo_head_hexsha": "f3479e7510545e4ef94aa3795683c8db7039f786", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "To_be_Solved_NLP_Tutorial_Jax_Haiku_Language_Modeling_Classification.ipynb", "max_issues_repo_name": "linker81/tutorials2021", "max_issues_repo_head_hexsha": "f3479e7510545e4ef94aa3795683c8db7039f786", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "To_be_Solved_NLP_Tutorial_Jax_Haiku_Language_Modeling_Classification.ipynb", "max_forks_repo_name": "linker81/tutorials2021", "max_forks_repo_head_hexsha": "f3479e7510545e4ef94aa3795683c8db7039f786", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 40.3590462833, "max_line_length": 712, "alphanum_fraction": 0.5199703457, "converted": true, "num_tokens": 13124, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.712232184238947, "lm_q1q2_score": 0.4380853668330579}} {"text": "```python\n# %load /Users/facaiyan/Study/book_notes/preconfig.py\n%matplotlib inline\n\nimport matplotlib.pyplot as plt\nimport seaborn as sns\nsns.set(color_codes=True)\nplt.rcParams['axes.grid'] = False\n\nimport numpy as np\n\n#from IPython.display import SVG\ndef show_image(filename, figsize=None, res_dir=True):\n if figsize:\n plt.figure(figsize=figsize)\n\n if res_dir:\n filename = './res/{}'.format(filename)\n\n plt.imshow(plt.imread(filename))\n```\n\nChapter 6 Deep Feedforward Networks\n===================================\n\nfeedforward: no feedback connections.\n\ninput layer -> hidden layers -> output layer\n\n\\begin{align}\n y &= f(x; \\theta) \\approx f^*(x) \\\\\n &= W^T \\phi(x) + b\n\\end{align}\n\nhow to choose the mapping $\\phi$?\n\n+ use a very generic $\\phi$, such as RBF kernel $\\implies$ generation remains poor. \n+ manually engineer $\\phi$ $\\implies$ classical feature engineer.\n+ learn $\\phi$ by deep learning. $y = f(x; \\theta, w) = \\phi(x; \\theta)^T w$.\n - advantage: only needs to find the right general function family *VS* find the right function.\n\n### 6.1 Example: Learning XOR\n\n$h = g(W^T x + c)$, where an affine transformation followed by an activation function $g$.\n\n\n```python\nshow_image(\"fig6_2.png\", figsize=(5, 8))\n```\n\ndefault activation function is rectified linear unit (ReLU): $g(z) = max\\{0, z\\}$\n+ advantage:\n 1. piecewise linear function: very close to linear.\n 2. easy to optimize with gradient-base methods.\n\n\n```python\nrelu = lambda x: np.maximum(0, x)\n\nx = np.linspace(-2, 2, 1000)\ny = relu(x)\n\nplt.ylim([-1, 3])\nplt.grid(True)\nplt.plot(x, y)\n```\n\n### 6.2 Gradient-Based Learning\n\nimportant:\n\n+ initialize all weights to small random values.\n+ initialize biases to zero or small positive values (push result to right area of ReLU).\n\n\n#### 6.2.1 Cost Functions\n\nIn most cases, \n\n+ our parametric model defines a distribution $p(y | x; \\theta)$,\n+ simply use the priciple of maximum likelihood.\n\n$\\implies$ [cross-entropy](https://en.wikipedia.org/wiki/Cross_entropy) as the cost function.\n\n\n##### 6.2.1.1 Learning Conditional Distributions with Maximum Likelihood\n\nmaximum likelihdood in neural networks => cost function is simply the negative log-likelihood == cross-entropy.\n\n\\begin{equation}\n J(\\theta) = - \\mathbb{E}_{x, y \\sim \\hat{p}_{data}} \\log p_{model}(y | x)\n\\end{equation}\n\nadvantage:\n\n+ Specifying a model p(y | x) automatically determines a cost function log p(y | x). => removes the burden of designing cost functions for each model.\n+ Undo the exp of some output units => avoid staturate problem (flat area, very small gradient).\n\nunusual property of the cross-entropy cose: does not have a minimum value (negative infinity). => regularization.\n\n\n##### 6.2.1.2 Learning Conditional Statistics\n\ncost L2: learn mean of y when x is given.\n\ncost L1: learn median of y when x is given.\n\n#### 6.2.2 Output Units\n\n##### 6.2.2.1 Linear Units for Gaussian Output Distributions\n\nLinear output layers are often used to produce the mean of a conditional Gaussian distribution.\n\n\u8fd9\u91cc\u610f\u601d\u662f\u8bf4\uff0c\u7ed9\u5b9a$x$\uff0c\u5b83\u5bf9\u5e94\u7684\u6837\u672c\u96c6$y$\u5e94\u662f\u9ad8\u65af\u5206\u5e03\u3002\u800c\u7528\u7ebf\u6027\u6a21\u578b\u6765\u5b66\u4e60\uff0c\u9884\u6d4b\u7684\u6b63\u662f\u6837\u672c\u96c6\u5747\u503c$f(x) = \\bar{y}$\u3002\u53ef\u89c1\uff0c\u8fd9\u79cd\u60c5\u51b5\u5e38\u89c1\u4e8e\u56de\u5f52\u95ee\u9898\u3002\n\n\n##### 6.2.2.2 Sigmoid Units for Bernoulli Output Distributions\n\nbinary classification\n\n\\begin{align}\n P(y) &= \\delta((2y - 1) z) \\quad \\text{where } z = w^T h + b \\\\\n J(\\theta) &= - \\log P(y | x) \\quad \\text{undo exp} \\\\\n &= \\zeta ((1 - 2y) z)\n\\end{align}\n\nmaximum likelihood is almost always the preferred approach to training sigmoid\noutput units.\n\n\n##### 6.2.2.3 Softmax Units for Multinoulli Output Distributions\n\nmultiple classification\n\n\\begin{equation}\n \\operatorname{softmax}(z)_i = \\frac{\\exp(z_i)}{\\sum_j \\exp(z_j)}\n\\end{equation}\n\n\\begin{align}\n \\log \\operatorname{softmax}(z)_i &= z_i - log \\sum_j \\exp(z_j) \\\\\n & \\approx z_i - \\max_j (z_j)\n\\end{align}\n\n+ Overall, unregularized maximum likelihood will drive the softmax to predict the fraction of counts of counts of each outcome observed in the training set.\n\n+ The argument $z$ can be produced in two different ways:\n 1. one vs rest: N estimators\n 2. choose one class as \"pivot\" class: N - 1 estimators\n \n+ softmax provides a \"softened\" version of the argmax. \n\n\n##### 6.2.2.4 Other Output Types\n\nIn general, think neural network as representing a function $f(x; \\thetha) = w$, which provides the parameters for a distribution over $y$. Our loss function can then be inperpreted as $- \\log p(y; w(x))$.\n\n### 6.3 Hidden Units\n\n+ Rectified linear units are an excellent default choice of hidden unit.\n+ In practice, gradient descent still performs well enough for functions which are not actually differentiable.\n\n#### 6.3.1 Rectified Linear Units and Their Generalizations\n\n#### 6.3.2 Logistic Sigmoid and Hyperbolic Tangent\n\nThe widespread saturation of sigmoidal units => use as hidden units is now discouraged.\n\n##### 6.3.3 Other Hidden Units\n\n### 6.4 Architecture Design\n\n+ layers: group of units\n+ chain structure\n 1. the depth of the network\n 2. the width of each layer\n\n#### 6.4.1 Universal Approximation Properties and Depth\n\nuniversal approximation theorem: large MLP will be able to **represent** any function. However, we are not guaranteed that the training algorithm will be able to **learn** that function.\n\nEmpirically, greater depth does seem to result in better generalization.\n\n#### 6.4.2 Other Architectural Considerations\n\n### 6.5 Back-Propagation and Other Differentiation Algorithms\n\nchain rule:\n\n\\begin{equation}\n \\frac{dz}{dx} = \\frac{dz}{dy} \\frac{dy}{dx}\n\\end{equation}\n\n### 6.6 Historical Notes\n\n\n```python\n\n```\n", "meta": {"hexsha": "f51914bc80b7d5b11db50543e3b68a95ab0fc92f", "size": 67546, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "deep_learning/Deep_Feedforward_Networks/note.ipynb", "max_stars_repo_name": "ningchi/book_notes", "max_stars_repo_head_hexsha": "c6f8001f7d5f873896c4b3a8b1409b21ef33c328", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2017-12-31T12:10:42.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-10T15:49:34.000Z", "max_issues_repo_path": "deep_learning/Deep_Feedforward_Networks/note.ipynb", "max_issues_repo_name": "ningchi/book_notes", "max_issues_repo_head_hexsha": "c6f8001f7d5f873896c4b3a8b1409b21ef33c328", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2017-12-05T13:04:14.000Z", "max_issues_repo_issues_event_max_datetime": "2017-12-07T16:24:50.000Z", "max_forks_repo_path": "deep_learning/Deep_Feedforward_Networks/note.ipynb", "max_forks_repo_name": "ningchi/book_notes", "max_forks_repo_head_hexsha": "c6f8001f7d5f873896c4b3a8b1409b21ef33c328", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2017-06-27T07:19:28.000Z", "max_forks_repo_forks_event_max_datetime": "2017-11-19T08:57:35.000Z", "avg_line_length": 202.2335329341, "max_line_length": 45912, "alphanum_fraction": 0.8963373109, "converted": true, "num_tokens": 1550, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.712232184238947, "lm_q1q2_score": 0.4380853668330579}} {"text": "# Defining Custom Display Logic for Your Own Objects\n\n## Overview\n\nIn Python, objects can declare their textual representation using the `__repr__` method. IPython expands on this idea and allows objects to declare other, richer representations including:\n\n* HTML\n* JSON\n* PNG\n* JPEG\n* SVG\n* LaTeX\n\nThis Notebook shows how you can add custom display logic to your own classes, so that they can be displayed using these rich representations. There are two ways of accomplishing this:\n\n1. Implementing special display methods such as `_repr_html_`.\n2. Registering a display function for a particular type.\n\nIn this Notebook we show how both approaches work.\n\nBefore we get started, we will import the various display functions for displaying the different formats we will create.\n\n\n```\nfrom IPython.display import display\nfrom IPython.display import (\n display_html, display_jpeg, display_png,\n display_javascript, display_svg, display_latex\n)\n```\n\n## Implementing special display methods\n\nThe main idea of the first approach is that you have to implement special display methods, one for each representation you want to use. Here is a list of the names of the special methods and the values they must return:\n\n* `_repr_html_`: return raw HTML as a string\n* `_repr_json_`: return raw JSON as a string\n* `_repr_jpeg_`: return raw JPEG data\n* `_repr_png_`: return raw PNG data\n* `_repr_svg_`: return raw SVG data as a string\n* `_repr_latex_`: return LaTeX commands in a string surrounded by \"$\".\n\n### Model Citizen: pandas\n\nA prominent example of a package that has IPython-aware rich representations of its objects is [pandas](http://pandas.pydata.org/).\n\nA pandas DataFrame has a rich HTML table representation,\nusing `_repr_html_`.\n\n\n\n```\nimport io\nimport pandas\n```\n\n\n```\n%%writefile data.csv\nDate,Open,High,Low,Close,Volume,Adj Close\n2012-06-01,569.16,590.00,548.50,584.00,14077000,581.50\n2012-05-01,584.90,596.76,522.18,577.73,18827900,575.26\n2012-04-02,601.83,644.00,555.00,583.98,28759100,581.48\n2012-03-01,548.17,621.45,516.22,599.55,26486000,596.99\n2012-02-01,458.41,547.61,453.98,542.44,22001000,540.12\n2012-01-03,409.40,458.24,409.00,456.48,12949100,454.53\n\n```\n\n Writing data.csv\n\n\n\n```\ndf = pandas.read_csv(\"data.csv\")\npandas.set_option('display.notebook_repr_html', False)\ndf\n```\n\n\n\n\n Date Open High Low Close Volume Adj Close\n 0 2012-06-01 569.16 590.00 548.50 584.00 14077000 581.50\n 1 2012-05-01 584.90 596.76 522.18 577.73 18827900 575.26\n 2 2012-04-02 601.83 644.00 555.00 583.98 28759100 581.48\n 3 2012-03-01 548.17 621.45 516.22 599.55 26486000 596.99\n 4 2012-02-01 458.41 547.61 453.98 542.44 22001000 540.12\n 5 2012-01-03 409.40 458.24 409.00 456.48 12949100 454.53\n\n\n\nrich HTML can be activated via `pandas.set_option`.\n\n\n```\npandas.set_option('display.notebook_repr_html', True)\ndf\n```\n\n\n\n\n
    \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    DateOpenHighLowCloseVolumeAdj Close
    0 2012-06-01 569.16 590.00 548.50 584.00 14077000 581.50
    1 2012-05-01 584.90 596.76 522.18 577.73 18827900 575.26
    2 2012-04-02 601.83 644.00 555.00 583.98 28759100 581.48
    3 2012-03-01 548.17 621.45 516.22 599.55 26486000 596.99
    4 2012-02-01 458.41 547.61 453.98 542.44 22001000 540.12
    5 2012-01-03 409.40 458.24 409.00 456.48 12949100 454.53
    \n
    \n\n\n\n\n```\nlines = df._repr_html_().splitlines()\nprint \"\\n\".join(lines[:20])\n```\n\n
    \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n### Exercise\n\nWrite a simple `Circle` Python class. Don't even worry about properties such as radius, position, colors, etc. To help you out use the following representations (remember to wrap them in Python strings):\n\nFor HTML:\n\n ○\n\nFor SVG:\n\n \n \n \n\nFor LaTeX (wrap with `$` and use a raw Python string):\n\n \\bigcirc\n\nAfter you write the class, create an instance and then use `display_html`, `display_svg` and `display_latex` to display those representations.\n\nTips : you can slightly tweek the representation to know from which `_repr_*_` method it came from. \nFor example in my solution the svg representation is blue, and the HTML one show \"`HTML`\" between brackets.\n\n### Solution\n\nHere is my simple `MyCircle` class:\n\n\n```\n%load soln/mycircle.py\n```\n\nNow create an instance and use the display methods:\n\n\n```\nc = MyCircle()\n```\n\n\n```\ndisplay_html(c)\n```\n\n\n○ (html)\n\n\n\n```\ndisplay_svg(c)\n```\n\n\n \n\n \n\n\n\n```\ndisplay_latex(c)\n```\n\n\n$\\bigcirc \\LaTeX$\n\n\n\n```\ndisplay_javascript(c)\n```\n\n\n\n## Adding IPython display support to existing objects\n\nWhen you are directly writing your own classes, you can adapt them for display in IPython by following the above example. But in practice, we often need to work with existing code we can't modify. We now illustrate how to add these kinds of extended display capabilities to existing objects. To continue with our example above, we will add a PNG representation to our `Circle` class using Matplotlib.\n\n### Model citizen: sympy\n\n[SymPy](http://sympy.org) is another model citizen that defines rich representations of its object.\nUnlike pandas above, sympy registers display formatters via IPython's display formatter API, rather than declaring `_repr_mime_` methods.\n\n\n```\nfrom sympy import Rational, pi, exp, I, symbols\nx, y, z = symbols(\"x y z\")\n```\n\n\n```\nr = Rational(3,2)*pi + exp(I*x) / (x**2 + y)\nr\n```\n\n\n\n\n 3*pi/2 + exp(I*x)/(x**2 + y)\n\n\n\nSymPy provides an `init_printing` function that sets up advanced $\\LaTeX$\nrepresentations of its objects.\n\n\n```\nfrom sympy.interactive.printing import init_printing\ninit_printing()\nr\n```\n\nTo add a display method to an existing class, we must use IPython's display formatter API. Here we show all of the available formatters:\n\n\n```\nip = get_ipython()\nfor mime, formatter in ip.display_formatter.formatters.items():\n print '%24s : %s' % (mime, formatter.__class__.__name__)\n\n```\n\n text/html : HTMLFormatter\n image/jpeg : JPEGFormatter\n image/svg+xml : SVGFormatter\n image/png : PNGFormatter\n application/javascript : JavascriptFormatter\n text/latex : LatexFormatter\n application/json : JSONFormatter\n text/plain : PlainTextFormatter\n\n\nLet's grab the PNG formatter:\n\n\n```\npng_f = ip.display_formatter.formatters['image/png']\n```\n\nWe will use the `for_type` method to register our display function.\n\n\n```\npng_f.for_type?\n```\n\nAs the docstring describes, we need to define a function the takes the object as a parameter and returns the raw PNG data.\n\n\n```\n%matplotlib inline\nimport matplotlib.pyplot as plt\n```\n\n\n```\nclass AnotherCircle(object):\n def __init__(self, radius=1, center=(0,0), color='r'):\n self.radius = radius\n self.center = center\n self.color = color\n \n def __repr__(self):\n return \"<%s Circle with r=%s at %s>\" % (\n self.color,\n self.radius,\n self.center,\n )\n \nc = AnotherCircle()\nc\n```\n\n\n\n\n \n\n\n\n\n```\nfrom IPython.core.pylabtools import print_figure\n\ndef png_circle(circle):\n \"\"\"Render AnotherCircle to png data using matplotlib\"\"\"\n fig, ax = plt.subplots()\n patch = plt.Circle(circle.center,\n radius=circle.radius,\n fc=circle.color,\n )\n ax.add_patch(patch)\n plt.axis('scaled')\n data = print_figure(fig, 'png')\n # We MUST close the figure, otherwise IPython's display machinery\n # will pick it up and send it as output, resulting in a double display\n plt.close(fig)\n return data\n```\n\n\n```\nc = AnotherCircle()\nprint repr(png_circle(c)[:10])\n```\n\n '\\x89PNG\\r\\n\\x1a\\n\\x00\\x00'\n\n\nNow we register the display function for the type:\n\n\n```\npng_f.for_type(AnotherCircle, png_circle)\n```\n\nNow all `Circle` instances have PNG representations!\n\n\n```\nc2 = AnotherCircle(radius=2, center=(1,0), color='g')\nc2\n```\n\n\n```\ndisplay_png(c2)\n```\n\n## return the object\n\n\n```\n# for demonstration purpose, I do the same with a circle that has no _repr_javascript method\nclass MyNoJSCircle(MyCircle):\n \n def _repr_javascript_(self):\n return\n\ncNoJS = MyNoJSCircle()\n```\n\nOf course you can now still return the object, and this will use compute all the representations, store them in the notebook and show you the appropriate one.\n\n\n```\ncNoJS\n```\n\nOr just use `display(object)` if you are in a middle of a loop\n\n\n```\nfor i in range(3):\n display(cNoJS)\n```\n\nAdvantage of using `display()` versus `display_*()` is that all representation will be stored in the notebook document and notebook file, they are then availlable for other frontends or post-processing tool like `nbconvert`.\n\nLet's compare `display()` vs `display_html()` for our circle in the Notebook Web-app and we'll see later the difference in nbconvert.\n\n\n```\nprint \"I should see a nice html circle in web-app, but\"\nprint \"nothing if the format I'm viewing the notebook in\"\nprint \"does not support html\"\ndisplay_html(cNoJS)\n```\n\n\n```\nprint \"Whatever the format I will see a representation\"\nprint \"of my circle\"\ndisplay(cNoJS)\n```\n\n\n```\nprint \"Same if I return the object\"\ncNoJS\n```\n\n\n```\nprint \"But not if I print it\"\nprint cNoJS\n```\n", "meta": {"hexsha": "8b4bda3e6b2195b89c52fa48722620573ee0a05d", "size": 63920, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/IPython Kernel/Old Custom Display Logic.ipynb", "max_stars_repo_name": "kaishuocheng/jupyter", "max_stars_repo_head_hexsha": "96ae75723eb62d30cb02768295422898aace79ef", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 748, "max_stars_repo_stars_event_min_datetime": "2015-01-05T05:48:49.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-27T01:05:42.000Z", "max_issues_repo_path": "examples/IPython Kernel/Old Custom Display Logic.ipynb", "max_issues_repo_name": "kaishuocheng/jupyter", "max_issues_repo_head_hexsha": "96ae75723eb62d30cb02768295422898aace79ef", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 32, "max_issues_repo_issues_event_min_datetime": "2015-04-02T22:25:41.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-18T05:31:46.000Z", "max_forks_repo_path": "examples/IPython Kernel/Old Custom Display Logic.ipynb", "max_forks_repo_name": "kaishuocheng/jupyter", "max_forks_repo_head_hexsha": "96ae75723eb62d30cb02768295422898aace79ef", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 816, "max_forks_repo_forks_event_min_datetime": "2015-01-04T04:19:15.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-17T20:57:19.000Z", "avg_line_length": 66.7920585162, "max_line_length": 19610, "alphanum_fraction": 0.7894399249, "converted": true, "num_tokens": 3284, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.7122321781307374, "lm_q1q2_score": 0.4380853630759723}} {"text": "# Fake News Detection: an application of classic NLP techniques\n**Universidade de Bras\u00edlia**
    \nFaculdade de Tecnologia
    \nPrograma de P\u00f3s-gradua\u00e7\u00e3o em Engenharia El\u00e9trica (PPGEE)\n\n## Author: Stefano M P C Souza (stefanomozart@ieee.org)
    Advisor: Daniel G Silva
    Advisor: Anderson C A Nascimento\n\n\n## 1. Experiment design\n\nWe want to study the impact of various NLP preprocessing techniques in the task of text classification for fake news detection. We are going to use the pipeline from [[1](#bot)] for model traing, tuning (hyper-parameter search) and comparison. The following ML algorithms are used:\n1. Naive Bayes:\n2. Decision Trees:\n2. K-Nearest Neighbour:\n3. Logistic Regression:\n3. Suport-Vector Machines:\n4. Random Forest:\n5. XGBoost:\n\nAll models are trained and tested on a binary (*fake*/real) classification task. The *pipeline*, written by the author, extends the `sklearn.pipeline.Pipeline` class, from scikit-learn, and consists of the following steps:\n1. **Training and tuning**: uses a random search algorithm to select the best hyper-parameters for each ML model;\n2. **Selection**: for each dataset, selects the models with best performance, on the selected metric, for the validation set. The selected model is trained one more time with the concatanation of the training and the valiudation set;\n5. **Test**: the models selected on the previous step, and trained on training+validation sets are used to classify texts in the test set. The final score, on the selected metric, is record so we can compare .\n\n## 2. Natural Language Processing\n\n### 2.1. Selected techniques\n\n1. **Tokenization**: the text, a sequence of caracters, is transformed in a ordered collection of tokens (words, punctiation marks, emojis, etc);\n2. **Stopword removal (SwR)**: removing words that do not add information, in the statistical learning sense, to any specific class in the sample. Most algorithms rely on experts dictionaries or on statistical measures such as *Mutual Information*;\n3. **Stemming**: Stemming is the reduction of variant forms of a word, eliminating inflectional morphemes such as verbal tense or plural suffixes, in order to provide a common representation, the root or stem. The intuition is to perform a dimensionality reduction on the dataset, removing rare morphological word variants, and reduce the risk of bias on word statistics measured on the documents;\n4. **Lemmatization:** Lemmatization consists on the reduction of each token to a linguistically valid root or lemma. The goal, from the statistical perspective, is exactly the same as in stemming: reduce variance in term frequency. It is sometimes compared to the normalization of the word sample, and aims to provide more accurate transformations than stemming, from the linguistic perspective;\n5. **Bag-of-Words (BoW)**: The BoW algorithm used in most NLP libraries is based on the *Vector Space Model* (VSM) and associates the tokens with with the corresponding term frequency: the number of occurrences of that token in that document. This algorithm produces an unordered set that does not retain any information on word order or proximity in the document ;\n6. **Term Frequency/Inverse Document Frequency (TF-IDF)**: Similar to the Vector Space Model Bag-of-Words, the TF-IDF (sometimes expressed as TF*IDF) document representation will associate each token in a document with a normalized or smoothed term frequency, weighted by the inverse of the frequency at which the term occurs in $D$, the corpus, or in the list of documents under processing. That is, $f_{t_i, d_j}$, the number of occurrences of token $t_i$ in document $d_j$, is replaced by $\\mathrm{tf\\cdot{idf}}$, where:\n \n\\begin{equation}\n\\begin{split}\n \\mathrm{tf}(t_i,d_j) &=1 + \\log \\frac{f_{t_i,d_j}}{\\sum_{t\\in d_j}{f_{t,d_j}}} \\\\\n \\mathrm{idf}(t_i, D) &= 1 + \\log \\frac{|D|+1}{|\\{d \\in D : t_i \\in d\\}|+1}\n\\end{split}\n\\end{equation}\n\n### 2.2. Datasets\n\nWe selected 2 datasets in English and 2 in Portuguese. Each pair has a dataset with full-length news\narticles and a dataset comprised of short statements, or sentences. The purpose of experimenting\nwith different languages and text sizes was to observe how these variables may impact preprocessing\nand training cost, and, ultimately, model performance.\n\nThe selected datasets are:\n - **Liar Dataset (liar):** curated by the UC Santa Barbara NLP Group, contains 12791 claims\n by North-American politicians and celebrities, classified as `true`, `mostly-true`, `half-true`, \n `barely-true`, `false` and `pants-on-fire` [[2](#liar)];\n\n - **Source Based Fake News Classification (sbnc):** 2020 full-length news manually labeled\n as `Real` or `Fake` [[3](#sbnc)];\n \n - **FactCk.br:** 1313 claims by Brazilian politicians, manually annotated by fact checking agencies\\footnote{\\url{https://piaui.folha.uol.com.br/lupa}, \\url{https://www.aosfatos.org} and \\url{https://apublica.org}} as `true`, `false`, `imprecise` and `others` [[4](#factckbr)];\n\n - **Fake.br:** 7200 full-length news articles, with text and metadata, manually flagged as `real` or `fake` news [[5](#fakebr)].\n\nThe classification experiments were preceded by a dataset preparation so that each dataset would have the same structure: \n1. **label**: (boolean) indicating if that text was labeled as *fake news*;\n2. **text**: (string) a concatenation of title (when available) and news body. \n\n## 3. Processing\n#### Daset preparation\n\n\n```python\n# importando bibliotecas de prop\u00f3sito geral, utilizada na manipula\u00e7\u00e3o dos datasets\nimport pandas as pd\nimport numpy as np\nimport joblib\n\nimport os, sys, inspect, time\nsys.path.insert(0, os.path.dirname(os.path.abspath(inspect.getfile(inspect.currentframe()))))\n```\n\n\n```python\n# Os datasets ser\u00e3o armezenados em um dicion\u00e1rio a fim de facilitar \n# a itera\u00e7\u00e3o de cada experimento sobre todos os datasets\ndatasets = [\n # Dataset 1: Liar \n {'name': 'liar', 'lang': 'en', 'df': pd.read_csv('datasets/liar/liar.csv')},\n \n # Dataset 2: Source Based FK Detection\n {'name': 'sbnc', 'lang': 'en', 'df': pd.read_csv('datasets/sbnc/sbnc.csv')},\n\n # Dataset 3: Fake.br\n {'name': 'fake.br', 'lang': 'pt', 'df': pd.read_csv('datasets/fake.br/fake.br.csv')},\n\n # Dataset 4: FactCk.br\n {'name': 'factck.br', 'lang': 'pt', 'df': pd.read_csv(\"datasets/factck.br/factck.br.csv\")}\n]\n\nexperiments = {\n \"E01\": {'preprocessing_time': {}, 'name': 'bow'},\n \"E02\": {'preprocessing_time': {}, 'name': 'bow.swr'},\n \"E03\": {'preprocessing_time': {}, 'name': 'bow.stem'},\n \"E04\": {'preprocessing_time': {}, 'name': 'bow.lemm'},\n \"E05\": {'preprocessing_time': {}, 'name': 'bow.lemm.swr'},\n \"E06\": {'preprocessing_time': {}, 'name': 'tfidf'},\n \"E07\": {'preprocessing_time': {}, 'name': 'tfidf.swr'},\n \"E08\": {'preprocessing_time': {}, 'name': 'tfidf.stem'},\n \"E09\": {'preprocessing_time': {}, 'name': 'tfidf.lemm'},\n \"E10\": {'preprocessing_time': {}, 'name': 'tfidf.lemm.swr'},\n}\n```\n\n#### Data split\n\n\n```python\nfrom sklearn.model_selection import train_test_split\n\nfor d in datasets:\n train_valid, test = train_test_split(d['df'], stratify=d['df'].label, test_size=0.2, random_state=42)\n train, valid = train_test_split(train_valid, stratify=train_valid.label, test_size=0.2, random_state=42)\n \n train_valid.to_csv(f\"datasets/{d['name']}/train.valid.csv\", index=False)\n train.to_csv(f\"datasets/{d['name']}/train.csv\", index=False)\n valid.to_csv(f\"datasets/{d['name']}/valid.csv\", index=False)\n test.to_csv(f\"datasets/{d['name']}/test.csv\", index=False)\n \n d['train.valid'] = train_valid\n d['train'] = train\n d['valid'] = valid\n d['test'] = test\n```\n\n### 3.1. Bag of Words (BoW)\n\n\n```python\nfrom sklearn.feature_extraction.text import CountVectorizer\nfrom scipy.sparse import save_npz\n\nfor d in datasets:\n t = time.process_time()\n \n cv = CountVectorizer() \n train = cv.fit_transform(d['train'].text)\n save_npz(f\"datasets/{d['name']}/train.bow.npz\", train)\n valid = cv.transform(d['valid'].text)\n save_npz(f\"datasets/{d['name']}/valid.bow.npz\", valid)\n \n cv = CountVectorizer()\n train = cv.fit_transform(d['train.valid'].text)\n save_npz(f\"datasets/{d['name']}/train.valid.bow.npz\", train)\n test = cv.transform(d['test'].text)\n save_npz(f\"datasets/{d['name']}/test.bow.npz\", test)\n \n experiments[\"E01\"]['preprocessing_time'][d['name']] = time.process_time() - t\n\n```\n\n### 3.2. BoW and Stopword Removal (BoW + SwR):\n\n\n```python\nimport nltk\n\nswr = {\n 'en': nltk.corpus.stopwords.words(\"english\"), \n 'pt': nltk.corpus.stopwords.words(\"portuguese\")\n}\n\nfor d in datasets:\n t = time.process_time()\n \n cv = CountVectorizer(stop_words=swr[d['lang']])\n train = cv.fit_transform(d['train'].text)\n save_npz(f\"datasets/{d['name']}/train.bow.swr.npz\", train)\n valid = cv.transform(d['valid'].text)\n save_npz(f\"datasets/{d['name']}/valid.bow.swr.npz\", valid)\n \n cv = CountVectorizer(stop_words=swr[d['lang']])\n train = cv.fit_transform(d['train.valid'].text)\n save_npz(f\"datasets/{d['name']}/train.valid.bow.swr.npz\", train)\n test = cv.transform(d['test'].text)\n save_npz(f\"datasets/{d['name']}/test.bow.swr.npz\", test)\n \n experiments[\"E02\"]['preprocessing_time'][d['name']] = time.process_time() - t\n```\n\n### 3.3. BoW and Stemming\n\n\n```python\ncv_analyzer = CountVectorizer().build_analyzer()\n\nsnowball = {\n 'en': nltk.stem.SnowballStemmer('english'),\n 'pt': nltk.stem.SnowballStemmer('portuguese')\n}\n\ndef en_stemmer(doc):\n return (snowball['en'].stem(w) for w in cv_analyzer(doc))\n\ndef pt_stemmer(doc):\n return (snowball['pt'].stem(w) for w in cv_analyzer(doc))\n\ncv_stemmer = {\n 'en': en_stemmer,\n 'pt': pt_stemmer\n}\n```\n\n\n```python\nfor d in datasets:\n t = time.process_time()\n \n cv = CountVectorizer(analyzer=cv_stemmer[d['lang']])\n train = cv.fit_transform(d['train'].text)\n save_npz(f\"datasets/{d['name']}/train.bow.stem.npz\", train)\n valid = cv.transform(d['valid'].text)\n save_npz(f\"datasets/{d['name']}/valid.bow.stem.npz\", valid)\n \n train = cv.fit_transform(d['train.valid'].text)\n save_npz(f\"datasets/{d['name']}/train.valid.bow.stem.npz\", train)\n test = cv.transform(d['test'].text)\n save_npz(f\"datasets/{d['name']}/test.bow.stem.npz\", test)\n \n experiments[\"E03\"]['preprocessing_time'][d['name']] = time.process_time() - t\n```\n\n### 3.4. BoW and Lemmatization\n\n\n```python\nimport stanza\nstanza_pt = stanza.Pipeline(lang='pt', processors='tokenize,mwt,pos,lemma')\n\nwordnet = nltk.stem.WordNetLemmatizer()\n\ndef en_lemma(doc):\n return [wordnet.lemmatize(token) for token in nltk.word_tokenize(doc)]\n \ndef pt_lemma(doc):\n d = stanza_pt(doc).sentences\n return [w.lemma for s in d for w in s.words]\n\nlemmatizer = {\n 'en': en_lemma,\n 'pt': pt_lemma\n}\n\n```\n\n 2021-06-08 13:57:05 INFO: Loading these models for language: pt (Portuguese):\n =======================\n | Processor | Package |\n -----------------------\n | tokenize | bosque |\n | mwt | bosque |\n | pos | bosque |\n | lemma | bosque |\n =======================\n \n 2021-06-08 13:57:05 INFO: Use device: gpu\n 2021-06-08 13:57:05 INFO: Loading: tokenize\n 2021-06-08 13:57:09 INFO: Loading: mwt\n 2021-06-08 13:57:09 INFO: Loading: pos\n 2021-06-08 13:57:11 INFO: Loading: lemma\n 2021-06-08 13:57:11 INFO: Done loading processors!\n\n\n\n```python\nfor d in datasets:\n t = time.process_time()\n \n cv = CountVectorizer(tokenizer=lemmatizer[d['lang']])\n train = cv.fit_transform(d['train'].text)\n save_npz(f\"datasets/{d['name']}/train.bow.lemm.npz\", train)\n valid = cv.transform(d['valid'].text)\n save_npz(f\"datasets/{d['name']}/valid.bow.lemm.npz\", valid)\n \n cv = CountVectorizer(tokenizer=lemmatizer[d['lang']])\n train = cv.fit_transform(d['train.valid'].text)\n save_npz(f\"datasets/{d['name']}/train.valid.bow.lemm.npz\", train)\n test = cv.transform(d['test'].text)\n save_npz(f\"datasets/{d['name']}/test.bow.lemm.npz\", test)\n \n experiments[\"E04\"]['preprocessing_time'][d['name']] = time.process_time() - t\n```\n\n### 3.5. BoW, Lemmatization and SwR\n\n\n```python\nfor d in datasets:\n t = time.process_time()\n \n cv = CountVectorizer(tokenizer=lemmatizer[d['lang']], stop_words=swr[d['lang']]) \n train = cv.fit_transform(d['train'].text)\n save_npz(f\"datasets/{d['name']}/train.bow.lemm.swr.npz\", train)\n valid = cv.transform(d['valid'].text)\n save_npz(f\"datasets/{d['name']}/valid.bow.lemm.swr.npz\", valid)\n \n cv = CountVectorizer(tokenizer=lemmatizer[d['lang']], stop_words=swr[d['lang']]) \n train = cv.fit_transform(d['train.valid'].text)\n save_npz(f\"datasets/{d['name']}/train.valid.bow.lemm.swr.npz\", train)\n test = cv.transform(d['test'].text)\n save_npz(f\"datasets/{d['name']}/test.bow.lemm.swr.npz\", test)\n \n experiments[\"E05\"]['preprocessing_time'][d['name']] = time.process_time() - t\n```\n\n /home/dev/.local/lib/python3.8/site-packages/sklearn/feature_extraction/text.py:388: UserWarning: Your stop_words may be inconsistent with your preprocessing. Tokenizing the stop words generated tokens [\"'d\", \"'ll\", \"'re\", \"'s\", \"'ve\", 'could', 'doe', 'ha', 'might', 'must', \"n't\", 'need', 'sha', 'wa', 'wo', 'would'] not in stop_words.\n warnings.warn('Your stop_words may be inconsistent with '\n /home/dev/.local/lib/python3.8/site-packages/sklearn/feature_extraction/text.py:388: UserWarning: Your stop_words may be inconsistent with your preprocessing. Tokenizing the stop words generated tokens ['estar', 'estivar', 'f\u00f4r', 'haver', 'ir', 'm', 'ser', 'ter', 'v\u00f3s'] not in stop_words.\n warnings.warn('Your stop_words may be inconsistent with '\n\n\n### 3.6. Term-Frequency/Inverse Document Frequency (TF-IDF)\n\n\n```python\nfrom sklearn.feature_extraction.text import TfidfVectorizer\n\nfor d in datasets:\n t = time.process_time()\n \n tv = TfidfVectorizer() \n train = tv.fit_transform(d['train'].text)\n save_npz(f\"datasets/{d['name']}/train.tfidf.npz\", train) \n valid = tv.transform(d['valid'].text)\n save_npz(f\"datasets/{d['name']}/valid.tfidf.npz\", valid)\n \n tv = TfidfVectorizer()\n train = tv.fit_transform(d['train.valid'].text)\n save_npz(f\"datasets/{d['name']}/train.valid.tfidf.npz\", train)\n test = tv.transform(d['test'].text)\n save_npz(f\"datasets/{d['name']}/test.tfidf.npz\", test) \n \n experiments[\"E06\"]['preprocessing_time'][d['name']] = time.process_time() - t\n \n```\n\n### 3.7. TF-IDF and SwR\n\n\n```python\nfrom sklearn.feature_extraction.text import TfidfVectorizer\n\nfor d in datasets:\n t = time.process_time()\n \n tv = TfidfVectorizer(stop_words=swr[d['lang']])\n train = tv.fit_transform(d['train'].text)\n save_npz(f\"datasets/{d['name']}/train.tfidf.swr.npz\", train)\n valid = tv.transform(d['valid'].text)\n save_npz(f\"datasets/{d['name']}/valid.tfidf.swr.npz\", valid)\n \n tv = TfidfVectorizer(stop_words=swr[d['lang']])\n train = tv.fit_transform(d['train.valid'].text)\n save_npz(f\"datasets/{d['name']}/train.valid.tfidf.swr.npz\", train)\n test = tv.transform(d['test'].text)\n save_npz(f\"datasets/{d['name']}/test.tfidf.swr.npz\", test)\n \n experiments[\"E07\"]['preprocessing_time'][d['name']] = time.process_time() - t\n\n```\n\n### 3.8. TF-IDF and Stemming\n\n\n```python\n#norm_count_vec = TfidfVectorizer(use_idf=False, norm='l2')\ntf_analyzer = TfidfVectorizer().build_analyzer()\n\nsnowball = {\n 'en': nltk.stem.SnowballStemmer('english'),\n 'pt': nltk.stem.SnowballStemmer('portuguese')\n}\n\ndef en_stemmer(doc):\n return (snowball['en'].stem(w) for w in tf_analyzer(doc))\n\ndef pt_stemmer(doc):\n return (snowball['pt'].stem(w) for w in tf_analyzer(doc))\n\ntf_stemmer = {\n 'en': en_stemmer,\n 'pt': pt_stemmer\n}\n```\n\n\n```python\nfrom sklearn.feature_extraction.text import TfidfVectorizer\n\nfor d in datasets:\n t = time.process_time()\n \n tv = TfidfVectorizer(tokenizer=tf_stemmer[d['lang']])\n train = tv.fit_transform(d['train'].text)\n save_npz(f\"datasets/{d['name']}/train.tfidf.stem.npz\", train)\n valid = tv.transform(d['valid'].text)\n save_npz(f\"datasets/{d['name']}/valid.tfidf.stem.npz\", valid)\n \n tv = TfidfVectorizer(tokenizer=tf_stemmer[d['lang']])\n train = tv.fit_transform(d['train.valid'].text)\n save_npz(f\"datasets/{d['name']}/train.valid.tfidf.stem.npz\", train)\n test = tv.transform(d['test'].text)\n save_npz(f\"datasets/{d['name']}/test.tfidf.stem.npz\", test)\n \n experiments[\"E08\"]['preprocessing_time'][d['name']] = time.process_time() - t\n\n```\n\n### 3.9. TF-IDF and Lemmatization\n\n\n```python\nfrom sklearn.feature_extraction.text import TfidfVectorizer\n\nfor d in datasets:\n t = time.process_time()\n \n tv = TfidfVectorizer(tokenizer=lemmatizer[d['lang']])\n train = tv.fit_transform(d['train'].text)\n save_npz(f\"datasets/{d['name']}/train.tfidf.lemm.npz\", train) \n valid = tv.transform(d['valid'].text)\n save_npz(f\"datasets/{d['name']}/valid.tfidf.lemm.npz\", valid)\n \n tv = TfidfVectorizer(tokenizer=lemmatizer[d['lang']])\n train = tv.fit_transform(d['train.valid'].text)\n save_npz(f\"datasets/{d['name']}/train.valid.tfidf.lemm.npz\", train) \n test = tv.transform(d['test'].text)\n save_npz(f\"datasets/{d['name']}/test.tfidf.lemm.npz\", test)\n \n experiments[\"E09\"]['preprocessing_time'][d['name']] = time.process_time() - t\n\n```\n\n### 3.10. TF-IDF, Lemmatization and SwR\n\n\n```python\nfrom sklearn.feature_extraction.text import TfidfVectorizer\n\nfor d in datasets:\n t = time.process_time()\n \n tv = TfidfVectorizer(tokenizer=lemmatizer[d['lang']], stop_words=swr[d['lang']])\n train = tv.fit_transform(d['train'].text)\n save_npz(f\"datasets/{d['name']}/train.tfidf.lemm.swr.npz\", train)\n valid = tv.transform(d['valid'].text)\n save_npz(f\"datasets/{d['name']}/valid.tfidf.lemm.swr.npz\", valid)\n \n tv = TfidfVectorizer(tokenizer=lemmatizer[d['lang']], stop_words=swr[d['lang']])\n train = tv.fit_transform(d['train.valid'].text)\n save_npz(f\"datasets/{d['name']}/train.valid.tfidf.lemm.swr.npz\", train)\n test = tv.transform(d['test'].text)\n save_npz(f\"datasets/{d['name']}/test.tfidf.lemm.swr.npz\", test)\n \n experiments[\"E10\"]['preprocessing_time'][d['name']] = time.process_time() - t\n\n```\n\n### 4. Saving the pre-processed datasets\n\n\n```python\nimport joblib\n\njoblib.dump(datasets, 'datasets.pyd')\njoblib.dump(experiments, 'experiments.pyd')\n```\n\n\n\n\n ['experiments.pyd']\n\n\n\n## References\n\n[1]: Souza, S.M.P. et al. *Tuning machine learning models to detect bots on Twitter*. 2020 Workshop on Communication Networks and Power Systems (WCNPS). Brasilia, 2020.\n\n\n[2] Wlliam Yang Wang, \"Liar, Liar Pants on Fire\": A New Benchmark Dataset for Fake News Detection, to appear in Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (ACL 2017), short paper, Vancouver, BC, Canada, July 30-August 4, ACL.\n\n\n[3]. A. Bharadwaj, B. Ashar, P. Barbhaya, R. Bhatia, Z. Shaikh, Source based fake news classification using machine learning (Aug 2020).URL https://kaggle.com/ruchi798/source-based-news-classification\n\n\n[4]. J. a. Moreno, G. Bressan, Factck.br: A new dataset to study fake news,in: Proceedings of the 25th Brazillian Symposium on Multimedia andthe Web, WebMedia \u201919, Association for Computing Machinery, NewYork, NY, USA, 2019, p. 525\u2013527. doi:10.1145/3323503.3361698.\n\n\n[5]. Monteiro R.A., Santos R.L.S., Pardo T.A.S., de Almeida T.A., Ruiz E.E.S., Vale O.A. (2018) Contributions to the Study of Fake News in Portuguese: New Corpus and Automatic Detection Results. In: Villavicencio A. et al. (eds) Computational Processing of the Portuguese Language. PROPOR 2018. Lecture Notes in Computer Science, vol 11122. Springer, Cham.\n\n", "meta": {"hexsha": "d3bb3d18a8bdcc14bd3ace42fb53fa5bf014539c", "size": 29470, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "experiments/classic_nlp.ipynb", "max_stars_repo_name": "stefanomozart/ppml_fake_news", "max_stars_repo_head_hexsha": "3a0780196ec5b4e3a7a18041e5daa2be40637192", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "experiments/classic_nlp.ipynb", "max_issues_repo_name": "stefanomozart/ppml_fake_news", "max_issues_repo_head_hexsha": "3a0780196ec5b4e3a7a18041e5daa2be40637192", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "experiments/classic_nlp.ipynb", "max_forks_repo_name": "stefanomozart/ppml_fake_news", "max_forks_repo_head_hexsha": "3a0780196ec5b4e3a7a18041e5daa2be40637192", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 38.0749354005, "max_line_length": 534, "alphanum_fraction": 0.5719375636, "converted": true, "num_tokens": 5694, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421276, "lm_q2_score": 0.6688802669716107, "lm_q1q2_score": 0.438046161199999}} {"text": "```python\nimport pandas as pd\nimport numpy as np\nfrom matplotlib import pyplot as plt\nfrom sympy import *\nimport seaborn as sns\n```\n\n#### Get Data & Inspect Columns (Gapminder Dataset)\n\n\n```python\nlife_expectancy= pd.read_excel(r'C:\\Users\\denni\\Desktop\\data_vault_2021\\visual_data_analysis\\gapminder_lifeexpectancy.xlsx', index_col=0)\npopulation = pd.read_excel(r'C:\\Users\\denni\\Desktop\\data_vault_2021\\visual_data_analysis\\gapminder_population.xlsx', index_col=0)\nfertility = pd.read_csv(r'C:\\Users\\denni\\Desktop\\data_vault_2021\\visual_data_analysis\\gapminder_total_fertility.csv', index_col=0)\n```\n\n\n```python\nlife_expectancy.index.rename('life_expectancy', inplace=True)\nlife_expectancy\n```\n\n\n\n\n
    \n\n
    DateOpenHighLowCloseVolumeAdj Close
    0 2012-06-01 569.16 590.00
    \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    1800180118021803180418051806180718081809...2007200820092010201120122013201420152016
    life_expectancy
    AbkhaziaNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    Afghanistan28.2128.2028.1928.1828.1728.1628.1528.1428.1328.12...52.452.853.353.654.054.454.854.953.852.72
    Akrotiri and DhekeliaNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    Albania35.4035.4035.4035.4035.4035.4035.4035.4035.4035.40...76.676.877.077.277.477.577.777.978.078.10
    Algeria28.8228.8228.8228.8228.8228.8228.8228.8228.8228.82...75.375.575.776.076.176.276.376.376.476.50
    ..................................................................
    YugoslaviaNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    Zambia32.6032.6032.6032.6032.6032.6032.6032.6032.6032.60...49.051.152.353.153.754.755.656.356.757.10
    Zimbabwe33.7033.7033.7033.7033.7033.7033.7033.7033.7033.70...46.447.348.049.151.654.255.757.059.361.69
    \u00c5landNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    South Sudan26.6726.6726.6726.6726.6726.6726.6726.6726.6726.67...55.555.655.856.055.956.056.056.156.156.10
    \n

    260 rows \u00d7 217 columns

    \n
    \n\n\n\n\n```python\npopulation.index.rename('total_population', inplace=True)\npopulation\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    1800181018201830184018501860187018801890...2006200720082009201020112012201320142015
    total_population
    AbkhaziaNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    Afghanistan3280000.03280000.03323519.03448982.03625022.03810047.03973968.04169690.04419695.04710171.0...25183615.025877544.026528741.027207291.027962207.028809167.029726803.030682500.031627506.032526562.0
    Akrotiri and DhekeliaNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...15700.015700.015700.0NaNNaNNaNNaNNaNNaNNaN
    Albania410445.0423591.0438671.0457234.0478227.0506889.0552800.0610036.0672544.0741688.0...3050741.03010849.02968026.02929886.02901883.02886010.02880667.02883281.02889676.02896679.0
    Algeria2503218.02595056.02713079.02880355.03082721.03299305.03536468.03811028.04143163.04525691.0...33749328.034261971.034811059.035401790.036036159.036717132.037439427.038186135.038934334.039666519.0
    ..................................................................
    Northern MarianasNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    South Georgia and the South Sandwich IslandsNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    US Minor Outlying IslandsNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    Virgin IslandsNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    West BankNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    \n

    275 rows \u00d7 81 columns

    \n
    \n\n\n\n\n```python\nfertility.index.rename('total_fertility_rate', inplace=True)\nfertility\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    1800180118021803180418051806180718081809...2006200720082009201020112012201320142015
    total_fertility_rate
    AbkhaziaNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    Afghanistan7.007.007.007.007.007.007.007.007.007.00...6.706.466.205.935.665.405.144.904.684.47
    Akrotiri and DhekeliaNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    Albania4.604.604.604.604.604.604.604.604.604.60...1.851.801.761.741.741.751.761.771.781.78
    Algeria6.996.996.996.996.996.996.996.996.996.99...2.582.662.732.782.822.832.822.802.762.71
    ..................................................................
    YugoslaviaNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    Zambia6.716.716.716.716.716.716.716.716.716.71...5.935.915.885.855.815.775.735.695.645.59
    Zimbabwe6.756.756.756.756.756.756.756.756.756.75...3.943.903.853.793.723.643.563.493.413.35
    \u00c5landNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    \u00c5landNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    \n

    260 rows \u00d7 216 columns

    \n
    \n\n\n\n\n```python\nprint(life_expectancy.isnull().sum())\nprint(population.isnull().sum())\nprint(fertility.isnull().sum())\n```\n\n 1800 59\n 1801 59\n 1802 59\n 1803 59\n 1804 59\n ..\n 2012 52\n 2013 52\n 2014 52\n 2015 52\n 2016 52\n Length: 217, dtype: int64\n 1800 46\n 1810 46\n 1820 46\n 1830 46\n 1840 46\n ..\n 2011 42\n 2012 42\n 2013 42\n 2014 44\n 2015 44\n Length: 81, dtype: int64\n 1800 59\n 1801 59\n 1802 59\n 1803 59\n 1804 59\n ..\n 2011 59\n 2012 59\n 2013 59\n 2014 61\n 2015 61\n Length: 216, dtype: int64\n\n\n\n```python\nlife_without_na=life_expectancy.dropna()\npopulation_without_na=population.dropna()\nfertility_without_na=fertility.dropna()\n```\n\n\n```python\nlife_without_na.isna().sum()\n```\n\n\n\n\n 1800 0\n 1801 0\n 1802 0\n 1803 0\n 1804 0\n ..\n 2012 0\n 2013 0\n 2014 0\n 2015 0\n 2016 0\n Length: 217, dtype: int64\n\n\n\n\n```python\nprint(life_without_na.shape)\nprint(population_without_na.shape)\nprint(fertility_without_na.shape)\n```\n\n (201, 217)\n (229, 81)\n (199, 216)\n\n\n#### Inspect variables (descriptive statistics & exploratory plots)\n\n\n```python\nlife_expectancy.describe()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    1800180118021803180418051806180718081809...2007200820092010201120122013201420152016
    count201.000000201.000000201.000000201.000000201.000000201.000000201.000000201.000000201.000000201.000000...208.000000208.000000208.000000208.000000208.000000208.000000208.000000208.000000208.00000208.000000
    mean31.48602031.44890531.46348331.37741331.44631831.56253731.61597031.57313431.37676631.310448...70.13971270.44716370.76774070.96990471.32437571.66307771.91610672.08812572.3210172.556635
    std3.7635853.7557393.8782043.9019663.8771563.9479574.0392613.9173394.0172283.972970...8.9532558.8004528.6103418.8988598.3765408.2174668.1208528.0749907.902027.738535
    min23.39000023.39000023.39000019.60000023.39000023.39000023.39000023.39000012.48000013.430000...43.30000044.50000045.50000032.20000046.70000046.10000045.60000045.40000047.1000048.860000
    25%29.00000028.95000028.90000028.90000028.95000029.00000029.00000029.00000028.95000028.820000...64.82500064.87500065.22500065.47500065.60000066.07500066.47500066.77500067.0500067.175000
    50%31.80000031.70000031.60000031.50000031.60000031.70000031.80000031.80000031.60000031.500000...72.75000073.00000073.35000073.70000073.75000074.05000074.15000074.30000074.4000074.500000
    75%33.90000033.90000033.90000033.80000033.87000033.90000034.00000034.00000033.87000033.800000...76.92500077.15000077.42500077.65000077.82500078.12500078.30000078.40000078.5000078.650000
    max42.85000040.30000044.37000044.84000042.83000044.27000045.82000043.56000043.55000041.740000...84.50000084.60000084.60000084.70000084.70000084.70000084.80000084.80000084.8000084.800000
    \n

    8 rows \u00d7 217 columns

    \n
    \n\n\n\n\n```python\npopulation.describe()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    1800181018201830184018501860187018801890...2006200720082009201020112012201320142015
    count2.290000e+022.290000e+022.290000e+022.290000e+022.290000e+022.290000e+022.290000e+022.290000e+022.290000e+022.290000e+02...2.560000e+022.560000e+022.560000e+022.380000e+022.380000e+022.330000e+022.330000e+022.330000e+022.310000e+022.310000e+02
    mean4.137832e+064.330692e+064.561516e+064.822455e+065.066513e+065.250450e+065.398263e+065.609144e+065.945787e+066.376547e+06...2.813600e+072.846652e+072.880011e+072.876698e+072.911667e+073.010052e+073.046134e+073.082280e+073.135238e+073.171457e+07
    std2.422799e+072.601555e+072.789334e+072.941603e+073.018315e+072.995424e+072.898546e+072.846866e+072.904625e+073.022539e+07...1.149097e+081.160131e+081.171146e+081.212191e+081.223602e+081.247404e+081.258755e+081.270003e+081.286516e+081.297556e+08
    min1.000000e+021.000000e+021.000000e+021.000000e+021.000000e+021.090000e+021.390000e+021.910000e+022.590000e+023.520000e+02...5.000000e+015.000000e+015.000000e+015.000000e+015.000000e+017.990000e+027.990000e+027.990000e+027.990000e+028.000000e+02
    25%4.390800e+044.390800e+044.467300e+044.689400e+044.764300e+045.340500e+046.446900e+046.542200e+046.817000e+046.922200e+04...2.868232e+052.928408e+052.989525e+053.174302e+053.243505e+053.994430e+054.055120e+054.114990e+054.175585e+054.209290e+05
    50%4.010000e+054.133260e+054.235270e+054.535070e+054.782270e+055.443020e+055.959280e+056.599300e+057.139570e+057.432100e+05...4.521912e+064.561442e+064.636136e+064.523432e+064.581304e+064.953945e+065.018367e+065.240088e+065.307188e+065.373502e+06
    75%1.894389e+061.904962e+061.983383e+062.101048e+062.186704e+062.236808e+062.399360e+062.596105e+062.822978e+063.088808e+06...1.711520e+071.764428e+071.819443e+071.676626e+071.691918e+072.011166e+071.994495e+071.932259e+071.894310e+071.900687e+07
    max3.216750e+083.505430e+083.800553e+084.023735e+084.112134e+084.027113e+083.800475e+083.636612e+083.655442e+083.771353e+08...1.312601e+091.319625e+091.326691e+091.333807e+091.340969e+091.348174e+091.355387e+091.362514e+091.369436e+091.376049e+09
    \n

    8 rows \u00d7 81 columns

    \n
    \n\n\n\n\n```python\nfertility.describe()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    1800180118021803180418051806180718081809...2006200720082009201020112012201320142015
    count201.000000201.000000201.000000201.000000201.000000201.000000201.000000201.000000201.000000201.000000...201.000000201.000000201.000000201.000000201.000000201.000000201.000000201.000000199.000000199.000000
    mean6.0840806.0812446.0843286.0833836.0836826.0834336.0793536.0781596.0706476.059701...2.9994032.9716422.9438812.9157712.8849252.8534332.8232842.7941792.7751762.744523
    std0.7774650.7808570.7754310.7748060.7749990.7719390.7769310.7779620.7916150.817953...1.5911731.5613091.5317751.5024061.4752641.4477391.4183261.3898481.3621671.332827
    min4.0400004.0400003.9100004.0500003.9400004.0600004.0700004.0500004.0000003.210000...0.8700000.9100000.9400000.9700000.9000001.0300001.0600001.0700001.1100001.130000
    25%5.6200005.6200005.6200005.6200005.6400005.6200005.6200005.6200005.6200005.620000...1.8200001.8300001.8300001.8400001.8400001.8200001.8000001.8000001.7950001.785000
    50%6.1500006.1500006.1500006.1500006.1500006.1500006.1500006.1500006.1300006.130000...2.4500002.4400002.4300002.4100002.3800002.3500002.3300002.2900002.2600002.230000
    75%6.6900006.6900006.6900006.6900006.6900006.6900006.6900006.6900006.6900006.690000...3.9100003.8300003.7600003.7400003.7200003.6400003.5600003.4900003.5650003.495000
    max8.1000008.1000008.1000008.1000008.1000008.1000008.1000008.1000008.1000008.100000...7.6000007.5900007.5900007.5900007.5800007.5800007.5700007.5600007.5400007.510000
    \n

    8 rows \u00d7 216 columns

    \n
    \n\n\n\n\n```python\nlife_exp_tr=life_without_na.transpose()\nsns.heatmap(life_exp_tr)\nlife_exp_tr\n```\n\n\n```python\nlife_expectancy.loc[life_expectancy[1800]==life_expectancy[1800].min(),:]\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    1800180118021803180418051806180718081809...2007200820092010201120122013201420152016
    life_expectancy
    Yemen23.3923.3923.3923.3923.3923.3923.3923.3923.3923.39...65.265.766.266.666.666.767.167.166.064.92
    \n

    1 rows \u00d7 217 columns

    \n
    \n\n\n\n\n```python\nlife_expectancy.loc[life_expectancy[1800]==life_expectancy[1800].max(),:]\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    1800180118021803180418051806180718081809...2007200820092010201120122013201420152016
    life_expectancy
    Iceland42.8533.8827.6219.624.7630.8545.8243.5643.5540.42...82.182.482.582.882.983.183.283.383.383.3
    \n

    1 rows \u00d7 217 columns

    \n
    \n\n\n\n\n```python\npopulation\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    1800181018201830184018501860187018801890...2006200720082009201020112012201320142015
    total_population
    AbkhaziaNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    Afghanistan3280000.03280000.03323519.03448982.03625022.03810047.03973968.04169690.04419695.04710171.0...25183615.025877544.026528741.027207291.027962207.028809167.029726803.030682500.031627506.032526562.0
    Akrotiri and DhekeliaNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...15700.015700.015700.0NaNNaNNaNNaNNaNNaNNaN
    Albania410445.0423591.0438671.0457234.0478227.0506889.0552800.0610036.0672544.0741688.0...3050741.03010849.02968026.02929886.02901883.02886010.02880667.02883281.02889676.02896679.0
    Algeria2503218.02595056.02713079.02880355.03082721.03299305.03536468.03811028.04143163.04525691.0...33749328.034261971.034811059.035401790.036036159.036717132.037439427.038186135.038934334.039666519.0
    ..................................................................
    Northern MarianasNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    South Georgia and the South Sandwich IslandsNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    US Minor Outlying IslandsNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    Virgin IslandsNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    West BankNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
    \n

    275 rows \u00d7 81 columns

    \n
    \n\n\n\n\n```python\nprint(population.loc[population[1800] > 100000000].count())\nprint(population.loc[population[2015] > 100000000].count())\n```\n\n 1800 2\n 1810 2\n 1820 2\n 1830 2\n 1840 2\n ..\n 2011 2\n 2012 2\n 2013 2\n 2014 2\n 2015 2\n Length: 81, dtype: int64\n 1800 12\n 1810 12\n 1820 12\n 1830 12\n 1840 12\n ..\n 2011 12\n 2012 12\n 2013 12\n 2014 12\n 2015 12\n Length: 81, dtype: int64\n\n\n#### Merge the different Datasources & Data wrangling\n\n\n```python\nfert = fertility\nlife= life_expectancy\npop= population\n```\n\n\n```python\nfert.shape\n```\n\n\n\n\n (260, 216)\n\n\n\n\n```python\nlife.shape\n\n```\n\n\n\n\n (260, 217)\n\n\n\n\n```python\npop.shape\n```\n\n\n\n\n (275, 81)\n\n\n\n\n```python\nfert.columns\n```\n\n\n\n\n Index(['1800', '1801', '1802', '1803', '1804', '1805', '1806', '1807', '1808',\n '1809',\n ...\n '2006', '2007', '2008', '2009', '2010', '2011', '2012', '2013', '2014',\n '2015'],\n dtype='object', length=216)\n\n\n\n\n```python\nncol=[int(x) for x in fert.columns]\nfert.set_axis(axis=1, labels=ncol, inplace=True)\n```\n\n\n```python\nlife.columns\n```\n\n\n\n\n Int64Index([1800, 1801, 1802, 1803, 1804, 1805, 1806, 1807, 1808, 1809,\n ...\n 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016],\n dtype='int64', length=217)\n\n\n\n\n```python\nfert.columns\n```\n\n\n\n\n Int64Index([1800, 1801, 1802, 1803, 1804, 1805, 1806, 1807, 1808, 1809,\n ...\n 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015],\n dtype='int64', length=216)\n\n\n\n\n```python\npop.columns\n```\n\n\n\n\n Int64Index([1800, 1810, 1820, 1830, 1840, 1850, 1860, 1870, 1880, 1890, 1900,\n 1910, 1920, 1930, 1940, 1950, 1951, 1952, 1953, 1954, 1955, 1956,\n 1957, 1958, 1959, 1960, 1961, 1962, 1963, 1964, 1965, 1966, 1967,\n 1968, 1969, 1970, 1971, 1972, 1973, 1974, 1975, 1976, 1977, 1978,\n 1979, 1980, 1981, 1982, 1983, 1984, 1985, 1986, 1987, 1988, 1989,\n 1990, 1991, 1992, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,\n 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011,\n 2012, 2013, 2014, 2015],\n dtype='int64')\n\n\n\n\n```python\nsfert=fert.stack()\nslife=life.stack()\nspop=pop.stack()\n```\n\n\n```python\nd={'fertility': sfert, 'lifeexp': slife, 'population':spop}\ndf2=pd.DataFrame(data=d)\n```\n\n\n```python\ndf3=df2.stack()\n```\n\n\n```python\ndf4=df3.unstack((0,2))\n```\n\n\n```python\ndf4.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    AfghanistanAkrotiri and DhekeliaAlbaniaAlgeria...YemenYugoslaviaZambiaZimbabwe\u00c5land
    fertilitylifeexppopulationpopulationfertilitylifeexppopulationfertilitylifeexppopulation...populationpopulationfertilitylifeexppopulationfertilitylifeexppopulationpopulationlifeexp
    18007.028.213280000.0NaN4.635.4410445.06.9928.822503218.0...2593000.0NaN6.7132.6747000.06.7533.71085814.0NaNNaN
    18017.028.20NaNNaN4.635.4NaN6.9928.82NaN...NaNNaN6.7132.6NaN6.7533.7NaNNaNNaN
    18027.028.19NaNNaN4.635.4NaN6.9928.82NaN...NaNNaN6.7132.6NaN6.7533.7NaNNaNNaN
    18037.028.18NaNNaN4.635.4NaN6.9928.82NaN...NaNNaN6.7132.6NaN6.7533.7NaNNaNNaN
    18047.028.17NaNNaN4.635.4NaN6.9928.82NaN...NaNNaN6.7132.6NaN6.7533.7NaNNaNNaN
    \n

    5 rows \u00d7 667 columns

    \n
    \n\n\n\n\n```python\ndf6=df3.unstack(1)\ndf6s=df6\ndf6=df6[1880]\ndf6=df6.unstack(1)\ndf6.plot.scatter('fertility', 'lifeexp', s=0.1)\n# immer df6.head um die Daten zu sehen!\n```\n\n\n```python\ndf6\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    fertilitylifeexppopulation
    Afghanistan7.0027.394419695.0
    Akrotiri and DhekeliaNaNNaNNaN
    Albania4.6035.40672544.0
    Algeria6.9928.824143163.0
    American SamoaNaNNaN6582.0
    ............
    Yemen6.8823.392960773.0
    YugoslaviaNaNNaNNaN
    Zambia6.7132.60744157.0
    Zimbabwe6.7533.701661683.0
    \u00c5landNaNNaNNaN
    \n

    256 rows \u00d7 3 columns

    \n
    \n\n\n\n\n```python\ndf6.sort_values(by=['population', 'lifeexp'], ascending=False)\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    fertilitylifeexppopulation
    China5.5032.00365544192.0
    India5.9525.44223020377.0
    Russia6.8030.2053996807.0
    United States4.7539.4151256498.0
    Germany5.0638.9143577358.0
    ............
    USSRNaNNaNNaN
    United Korea (former)NaNNaNNaN
    West GermanyNaNNaNNaN
    YugoslaviaNaNNaNNaN
    \u00c5landNaNNaNNaN
    \n

    256 rows \u00d7 3 columns

    \n
    \n\n\n\n\n```python\ndef country_rating(fertility,population):\n if (fertility>=3) and (population>=100000000):\n return 1\n else:\n return 0\n```\n\n\n```python\ndf6['dynamic'] = df6.apply(lambda x: country_rating(x['fertility'],x['population']),axis=1)\n```\n\n\n```python\ndf6.loc[df6.loc[:,\"dynamic\"]==1]\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    fertilitylifeexppopulationdynamic
    China5.5032.00365544192.01
    India5.9525.44223020377.01
    \n
    \n\n\n\n#### Visualize findings (relevant facts & explanatory plots & key findings)\n\n\n```python\nsns.set_context(\"talk\", font_scale=1.1)\nplt.figure(figsize=(10,6))\nsns.scatterplot(x=\"fertility\", \n y=\"lifeexp\",\n size=\"population\", \n data=df6)\nplt.xlabel(\"fertility\")\nplt.ylabel(\"life_expectancy\")\nplt.tight_layout()\n#plt.savefig(\"gapminder1880.png\",format='png',dpi=150)\n```\n\n\n```python\nimport matplotlib.pyplot as plt\nlife_expectancy[1800].hist(bins=15)\nlife_expectancy[2015].hist(bins=15, alpha=0.5)\n#plt.savefig('histo.png')\n```\n\n\n```python\nsns.distplot(life_expectancy[2015] , kde=True, color=\"red\", bins=30)\nsns.distplot(life_expectancy[1800] , kde=True, color=\"blue\", bins=30)\n```\n\n\n```python\nlife_expectancy_stack=life_expectancy.stack()\nlife_expectancy_stack['Yemen'].hist(bins=15)\nlife_expectancy_stack['Iceland'].hist(bins=15)\n#plt.savefig('histo1.png')\n#Life_exp1990stack['Ethiopia','Andorra']\n```\n\n\n```python\nlife_expectancy[2015].hist(facecolor='green',alpha=0.75,histtype='bar',bins=5)\nplt.axis([50.0, 85.0,0.0, 80.0])\nplt.title('Life expectancy 2015')\nplt.xlabel('Years')\nplt.ylabel('countries')\n\n```\n\n\n```python\nplt.plot(life_expectancy.loc['Yemen'])\nplt.plot(life_expectancy.loc['Iceland'])\n```\n\n\n```python\n# Reproduce a Plot\nsubset2 = life_expectancy[[1880, 1900, 2000, 2015]]\n#stichprobe lander mit subset\nsubset2.plot(kind='hist')\n```\n\n\n```python\n#Create a Boxplot\n#pd.DataFrame.boxplot()\n#Slicing!\n#x und y label schreiben!\nlife_without_na.boxplot(column=[1880,2015],rot=90)\nplt.figure(figsize=(100,16))\n```\n\n\n```python\ndf4[['Germany', 'France', 'Sweden']].plot()\n```\n\n\n```python\ndf5=df3.unstack(2)\ndf5.plot.scatter('fertility', 'lifeexp', s=0.1)\n```\n\n\n```python\nplt.figure(figsize=(20,20))\ncmap = plt.get_cmap('tab20', lut = len(df6)).colors\nsns.lmplot(x=\"fertility\", y=\"lifeexp\",data=df6,height=10)\nplt.legend(loc=0)\nplt.axis((0,10,0,90))\n```\n\n#### Create an animated scatterplot\n\n\n```python\ndf6\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    fertilitylifeexppopulation
    Afghanistan7.6726.857752118.0
    Akrotiri and DhekeliaNaNNaN10661.0
    Albania5.8054.481263171.0
    Algeria7.6542.778872247.0
    American SamoaNaNNaN18937.0
    ............
    Yemen7.2723.514402320.0
    YugoslaviaNaNNaN16285527.0
    Zambia6.7142.852316950.0
    Zimbabwe6.7548.462746854.0
    \u00c5landNaNNaN21649.0
    \n

    256 rows \u00d7 3 columns

    \n
    \n\n\n\n\n```python\ncmap = plt.get_cmap('tab20', lut=len(df6)). colors\ndf6.plot.scatter('fertility', 'lifeexp', s=0.1, c=cmap)\nplt.legend(loc=0)\nplt.axis((0,10,0,90))\n#plt.savefig(\"lifeexp_1960.png\")\n```\n\n\n```python\nfor i in range(1950,2016):\n df6=df6s[i]\n df6=df6.unstack(1)\n\n cmap = plt.get_cmap('tab20', lut=len(df6)). colors\n df6.plot.scatter('fertility', 'lifeexp', s=3, c=cmap)\n plt.legend(loc=0)\n plt.axis((0,10,0,90))\n filename=str(\"lifeexp_\"+str(i)+ \".png\")\n plt.savefig(filename)\n \n save as lifeexp_[i].png \n```\n\n\n```python\nimport imageio\nimages = []\n\nfor i in range(1950, 2016):\n filename = 'lifeexp_{}.png'.format(i)\n images.append(imageio.imread(filename))\n\nimageio.mimsave('output.gif', images, fps=20)\n```\n", "meta": {"hexsha": "7ab1fb1077b5dd313d73c83e1d72bea7377c503c", "size": 575164, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Visual_Data_Analysis.ipynb", "max_stars_repo_name": "Pijanes/Portfolio", "max_stars_repo_head_hexsha": "86bc97f5b0ee0c541eefeef9bd81c51c0505cc5e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Visual_Data_Analysis.ipynb", "max_issues_repo_name": "Pijanes/Portfolio", "max_issues_repo_head_hexsha": "86bc97f5b0ee0c541eefeef9bd81c51c0505cc5e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Visual_Data_Analysis.ipynb", "max_forks_repo_name": "Pijanes/Portfolio", "max_forks_repo_head_hexsha": "86bc97f5b0ee0c541eefeef9bd81c51c0505cc5e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 114.6430137532, "max_line_length": 89268, "alphanum_fraction": 0.7760517, "converted": true, "num_tokens": 26099, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.668880247169804, "lm_q2_score": 0.6548947425132315, "lm_q1q2_score": 0.43804615724245544}} {"text": "# Deep Cox Mixtures with Heterogenous Effects (CMHE) Demo\n
    \n\nAuthor: ***Mononito Goswami*** <mgoswami@cs.cmu.edu>\n\n
    \n\n\n
    \n\n# Contents\n\n\n\n### 1. [Introduction](#introduction) \n\n\n### 2. [Synthetic Data](#syndata) \n####               1.1 [Generative Process for the Synthetic Dataset.](#gensyndata)\n####               1.2 [Loading and Visualizing the Dataset.](#vissyndata)\n####               1.2 [Split Dataset into Train and Test.](#splitdata)\n\n \n### 3. [Counterfactual Phenotyping](#phenotyping)\n\n####               3.1 [Phenotyping with CMHE](#phenocmhe)\n\n####               3.1 [Comparison with Clustering](#clustering)\n\n\n\n### 4. [Factual Regression](#regression)\n\n####               3.1 [Factual Regression with CMHE](#regcmhe)\n\n\n####               3.1 [Comparison with a Deep Cox Proportional Hazards Model](#deepcph)\n\n
    \n\n\n\n\n## 1. Introduction\n\nFigure A: Schematic Description of CMHE:The set of features (confounders) $\\mathbf{x}$ are passed through an encoder to obtain deep non-linear representations. These representations then describe the latent phenogroups $\\mathbf{P}(Z|X=\\mathbf{x})$ and $\\mathbf{P}(\\mathbf{\\phi}|X=\\mathbf{x})$ that determine the base survival rate and the treatment effect respectively. Finally, the individual level hazard (survival) curve under an intervention $A=\\mathbf{a}$ is described by marginalizing over $Z$ and $\\mathbf{\\phi}$ as $\\mathbf{S}(t|X=x, A=a) = \\mathbf{E}_{(Z,\\mathbf{\\phi)}\\sim \\mathbf{P}(\\cdot|X)}\\big[ \\mathbf{S}(t|A=\\mathbf{a}, X, Z, \\mathbf{\\phi})\\big]$. \n\n\n\n\n\n\n\n

    Cox Mixture with Heterogenous Effects (CMHE) is a flexible approach to recover counterfactual phenotypes of individuals that demonstrate heterogneous effects to an intervention in terms of censored Time-to-Event outcomes. CMHE is not restricted by the strong Cox Proportional Hazards assumption or any parametric assumption on the time to event distributions. CMHE achieves this by describing each individual as belonging to two different latent groups, \n$Z$ that mediate the base survival rate and $\\phi$ the effect of the treatment. CMHE can also be employed to model individual level counterfactuals or for standard factual survival regression.\n\n**Figure B (Right)**: CMHE in Plate Notation. $\\mathbf{x}$ confounds treatment assignment $A$ and outcome $T$ (Model parameters and censoring distribution have been abstracted out).\n\n \n \n*For full details on Cox Mixtures with Heterogenous Effects, please refer to our preprint*:\n\n[Counterfactual Phenotyping with Censored Time-to-Events, arXiv preprint, C. Nagpal, M. Goswami, K. Dufendach, A. Dubrawski](https://arxiv.org/abs/2202.11089)\n\n\n\n\n\n## 2. Synthetic Data Example\n\n\n\n```python\nimport pandas as pd\nimport torch\nfrom tqdm import tqdm \nimport sys\nsys.path.append('../')\n\nfrom auton_survival.datasets import load_dataset\nfrom cmhe_demo_utils import * \n```\n\n\n### 2.1. Generative Process for the Synthetic Data\n\n1. Features $x_1$, $x_2$ and the base survival phenotypes $Z$ are sampled from $\\texttt{scikit-learn's make_blobs(...)}$ function which generates isotropic Gaussian blobs:\n$$[x_1, x_2], Z \\sim \\texttt{sklearn.datasets.make_blobs(K = 3)}$$\n2. Features $x_3$ and $x_4$ are sampled uniformly, whereas the underlying treatment effect phenotypes $\\phi$ are defined according to an $L_1$-ball:\n$$ [x_1, x_2] \\sim \\texttt{Uniform}(-2, 2) $$\n$$ \\phi \\triangleq \\mathbb{1}\\{|x_3| + |x_3| > 2\\} $$\n3. We then sample treat assignments from a Bernoulli distribution:\n$$ A \\sim \\texttt{Bernoulli}(\\frac{1}{2}) $$\n4. Next, the time-to-event $T$ conditioned on the confounders $x$, latent $Z$ and latent effect group $\\phi$ are generated from a Gompertz distribution:\n$$ T^{*}| (Z=k, {\\phi}=m, A={a}) \\sim \\nonumber \\texttt{Gompertz}\\big({\\beta}_{k}^{\\top}{x} +({-a}^m)\\big) $$\n5. Finally, the observed time $T$ is obtained after censoring some of the events and censoring time is chosen uniformly at random upto $T^*$:\n$$\\delta \\sim \\texttt{Bernoulli}(\\frac{3}{4}), \\quad C \\sim \\texttt{Uniform}(0, {T}^{*})$$\n$$ T = \\begin{cases} T^*, & \\text{if } \\delta = 1 \\\\ C, & \\text{if } \\delta = 0 \\end{cases} $$\n\n\n```python\n# Load the synthetic dataset\noutcomes, features, interventions = load_dataset(dataset='SYNTHETIC')\n\n# Let's take a look at take the dataset\nfeatures.head(5)\n```\n\n\n\n\n

    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    X1X2X3X4X5X6X7X8
    00.1487451.8924840.1952540.8607570.6965230.4836970.3395510.374794
    11.139439-0.9433300.4110540.1795330.4286860.6830570.6009480.070483
    2-0.9612370.782706-0.3053810.5835760.1574780.0705560.0345900.776005
    30.4665080.694348-0.2496511.5670920.8509590.4161780.9688410.863598
    4-0.249002-0.5520911.854651-0.4662340.8603850.3671840.9543470.748930
    \n
    \n\n\n\n\n### 2.2. Visualizing the Synthetic Data\n\n\n```python\nplot_synthetic_data(outcomes, features, interventions)\n```\n\n\n### 2.3 Split Dataset into Train and Test folds\n\n\n```python\n# Hyper-parameters\nrandom_seed = 0\ntest_size = 0.25\n\n# Split the synthetic data into training and testing data\nimport numpy as np\n\nnp.random.seed(random_seed)\nn = features.shape[0] \n\ntest_idx = np.zeros(n).astype('bool')\ntest_idx[np.random.randint(n, size=int(n*test_size))] = True \n\nfeatures_tr = features.iloc[~test_idx] \noutcomes_tr = outcomes.iloc[~test_idx]\ninterventions_tr = interventions[~test_idx]\nprint(f'Number of training data points: {len(features_tr)}')\n\nfeatures_te = features.iloc[test_idx] \noutcomes_te = outcomes.iloc[test_idx]\ninterventions_te = interventions[test_idx]\nprint(f'Number of test data points: {len(features_te)}')\n\nx_tr = features_tr.values.astype('float32')\nt_tr = outcomes_tr['time'].values.astype('float32')\ne_tr = outcomes_tr['event'].values.astype('float32')\na_tr = interventions_tr.values.astype('float32')\n\nx_te = features_te.values.astype('float32')\nt_te = outcomes_te['time'].values.astype('float32')\ne_te = outcomes_te['event'].values.astype('float32')\na_te = interventions_te.values.astype('float32')\n\nprint('Training Data Statistics:')\nprint(f'Shape of covariates: {x_tr.shape} | times: {t_tr.shape} | events: {e_tr.shape} | interventions: {a_tr.shape}')\n```\n\n Number of training data points: 3899\n Number of test data points: 1101\n Training Data Statistics:\n Shape of covariates: (3899, 8) | times: (3899,) | events: (3899,) | interventions: (3899,)\n\n\n\n```python\ndef find_max_treatment_effect_phenotype(g, zeta_probs, factual_outcomes):\n \"\"\"\n Find the group with the maximum treatement effect phenotype\n \"\"\"\n mean_differential_survival = np.zeros(zeta_probs.shape[1]) # Area under treatment phenotype group\n outcomes_train, interventions_train = factual_outcomes \n\n # Assign each individual to their treatment phenotype group\n for gr in range(g): # For each treatment phenotype group\n # Probability of belonging the the g^th treatment phenotype\n zeta_probs_g = zeta_probs[:, gr] \n # Consider only those individuals who are in the top 75 percentiles in this phenotype\n z_mask = zeta_probs_g>np.quantile(zeta_probs_g, 0.75) \n\n mean_differential_survival[gr] = find_mean_differential_survival(\n outcomes_train.loc[z_mask], interventions_train.loc[z_mask]) \n\n return np.nanargmax(mean_differential_survival)\n```\n\n\n## 3. Counterfactual Phenotyping\n\n\n### 3.1 Counterfactual Phenotyping with CMHE\n\n\n```python\n# Hyper-parameters to train model\nk = 1 # number of underlying base survival phenotypes\ng = 2 # number of underlying treatment effect phenotypes.\nlayers = [50, 50] # number of neurons in each hidden layer.\n\nrandom_seed = 3\niters = 100 # number of training epochs\nlearning_rate = 0.001\nbatch_size = 128 \nvsize = 0.15 # size of the validation split\npatience = 3\noptimizer = \"Adam\"\n```\n\n\n```python\nfrom auton_survival.models.cmhe import DeepCoxMixturesHeterogenousEffects\n\ntorch.manual_seed(random_seed)\nnp.random.seed(random_seed)\n\n# Instantiate the CMHE model\nmodel = DeepCoxMixturesHeterogenousEffects(k=k, g=g, layers=layers)\n\nmodel = model.fit(x_tr, t_tr, e_tr, a_tr, vsize=vsize, val_data=None, iters=iters, \n learning_rate=learning_rate, batch_size=batch_size, \n optimizer=optimizer, random_state=random_seed, \n patience=patience)\n```\n\n 88%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2589 | 88/100 [00:41<00:05, 2.12it/s]\n\n\n\n```python\nprint(f'Treatment Effect for the {g} groups: {model.torch_model[0].omega.detach()}')\n\nzeta_probs_train = model.predict_latent_phi(x_tr)\nzeta_train = np.argmax(zeta_probs_train, axis=1)\nprint(f'Distribution of individuals in each treatement phenotype in the training data: \\\n{np.unique(zeta_train, return_counts=True)[1]}')\n\nmax_treat_idx_CMHE = find_max_treatment_effect_phenotype(\n g=2, zeta_probs=zeta_probs_train, factual_outcomes=(outcomes_tr, interventions_tr))\nprint(f'\\nGroup {max_treat_idx_CMHE} has the maximum restricted mean survival time on the training data!')\n```\n\n Treatment Effect for the 2 groups: tensor([-0.5053, 0.3926])\n Distribution of individuals in each treatement phenotype in the training data: [2006 1893]\n \n Group 1 has the maximum restricted mean survival time on the training data!\n\n\n /Users/chiragn/anaconda3/lib/python3.8/site-packages/lifelines/fitters/__init__.py:200: ApproximationWarning: Approximating using linear interpolation`.\n \n warnings.warn(\"Approximating using linear interpolation`.\\n\", exceptions.ApproximationWarning)\n\n\n### Evaluate CMHE on Test Data\n\n\n```python\n# Now for each individual in the test data, let's find the probability that \n# they belong to the max treatment effect group\n\nzeta_probs_test_CMHE = model.predict_latent_phi(x_te)\nzeta_test = np.argmax(zeta_probs_test_CMHE, axis=1)\nprint(f'Distribution of individuals in each treatement phenotype in the test data: \\\n{np.unique(zeta_test, return_counts=True)[1]}')\n\n# Now let us evaluate our performance\nplot_phenotypes_roc(outcomes_te, zeta_probs_test_CMHE[:, max_treat_idx_CMHE])\n```\n\n\n### 3.3 Comparison with the Clustering phenotyper\n\nWe compare the ability of CMHE against dimensionality reduction followed by clustering for counterfactual phenotyping. Specifically, we first perform dimensionality reduction of the input confounders, $\\mathbf{x}$, followed by clustering. Due to a small number of confounders in the synthetic data, in the following experiment, we directly perform clustering using a Gaussian Mixture Model (GMM) with 2 components and diagonal covariance matrices.\n\n\n```python\nfrom phenotyping import ClusteringPhenotyper\nfrom sklearn.metrics import auc\n\nclustering_method = 'gmm'\ndim_red_method = None # We would not perform dimensionality reduction for the synthetic dataset\nn_components = None \nn_clusters = 2 # Number of underlying treatment effect phenotypes\n\n# Running the phenotyper\nphenotyper = ClusteringPhenotyper(clustering_method=clustering_method, \n dim_red_method=dim_red_method, \n n_components=n_components, \n n_clusters=n_clusters)\n```\n\n\n```python\nzeta_probs_train = phenotyper.fit_phenotype(features_tr.values)\nzeta_train = np.argmax(zeta_probs_train, axis=1)\nprint(f'Distribution of individuals in each treatement phenotype in the training data: \\\n{np.unique(zeta_train, return_counts=True)[1]}')\n\nmax_treat_idx_CP = find_max_treatment_effect_phenotype(\n g=2, zeta_probs=zeta_probs_train, factual_outcomes=(outcomes_tr, interventions_tr))\nprint(f'\\nGroup {max_treat_idx_CP} has the maximum restricted mean survival time on the training data!')\n```\n\n No Dimensionaity reduction specified...\n Proceeding to learn clusters with the raw features...\n Fitting the following Clustering Model:\n GaussianMixture(covariance_type='diag', n_components=3)\n Distribution of individuals in each treatement phenotype in the training data: [1438 1312 1149]\n \n Group 2 has the maximum restricted mean survival time on the training data!\n\n\n /Users/chiragn/anaconda3/lib/python3.8/site-packages/lifelines/fitters/__init__.py:200: ApproximationWarning: Approximating using linear interpolation`.\n \n warnings.warn(\"Approximating using linear interpolation`.\\n\", exceptions.ApproximationWarning)\n\n\n### Evaluate Clustering Phenotyper on Test Data\n\n\n```python\n# Now for each individual in the test data, let's find the probability that \n# they belong to the max treatment effect group\n\n# Use the phenotyper trained on training data to phenotype on testing data\nzeta_probs_test_CP = phenotyper.phenotype(x_te)\nzeta_test_CP = np.argmax(zeta_probs_test_CP, axis=1)\nprint(f'Distribution of individuals in each treatement phenotype in the test data: \\\n{np.unique(zeta_test_CP, return_counts=True)[1]}')\n\n# Now let us evaluate our performance\nplot_phenotypes_roc(outcomes_te, zeta_probs_test_CP[:, max_treat_idx_CP])\n```\n\n\n## 4. CMHE for Factual Regression\n\nFor completeness, we further evaluate the performance of CMHE in estimating factual risk over multiple time horizons using the standard survival analysis metrics, including: \n\n1. $\\textbf{Brier Score} \\ (\\textrm{BS})$: Defined as the Mean Squared Error (MSE) around the probabilistic prediction at a certain time horizon.\n\\begin{align}\n\\text{BS}(t) = \\mathop{\\mathbf{E}}_{x\\sim\\mathcal{D}}\\big[ ||\\mathbf{1}\\{ T > t \\} - \\widehat{\\mathbf{P}}(T>t|X)\\big)||_{_\\textbf{2}}^\\textbf{2} \\big]\n\\end{align}\n2. $ \\textbf{Time Dependent Concordance Index} \\ (C^{\\text{td}}$): A rank order statistic that computes model performance in ranking patients based on their estimated risk at a specfic time horizon.\n\\begin{align}\nC^{td }(t) = \\mathbf{P}\\big( \\hat{F}(t| \\mathbf{x}_i) > \\hat{F}(t| \\mathbf{x}_j) | \\delta_i=1, T_i\n\n### 4.1 Factual Regression Performance of CMHE\n\n\n```python\nhorizons = [1, 3, 5]\n\n# Now let us predict survival using CMHE\npredictions_test_CMHE = model.predict_survival(x_te, a_te, t=horizons)\n\nCI1, CI3, CI5, IBS = factual_evaluate((x_tr, t_tr, e_tr, a_tr), (x_te, t_te, e_te, a_te), \n horizons, predictions_test_CMHE)\nprint(f'Concordance Index (1 Year): {np.around(CI1, 4)} (3 Year) {np.around(CI3, 4)}: (5 Year): {np.around(CI5, 4)}')\nprint(f'Integrated Brier Score: {np.around(IBS, 4)}')\n```\n\n Concordance Index (1 Year): 0.6894 (3 Year) 0.6978: (5 Year): 0.6987\n Integrated Brier Score: 0.1524\n\n\n\n```python\nx_te\n```\n\n\n\n\n array([[ 0.1487453 , 1.8924836 , 0.19525401, ..., 0.4836966 ,\n 0.33955073, 0.37479353],\n [ 2.2802675 , -0.73033774, -1.7158557 , ..., 0.47515342,\n 0.8169348 , 0.59493804],\n [ 1.4902165 , -0.91186345, 0.7905248 , ..., 0.08489922,\n 0.37587634, 0.62569857],\n ...,\n [ 2.0787652 , -2.0501418 , -0.36273366, ..., 0.63403505,\n 0.97710544, 0.81890947],\n [-1.5852283 , -0.79666543, 0.9420089 , ..., 0.6343607 ,\n 0.5449456 , 0.01124444],\n [ 0.25414062, 1.5835027 , -0.41139466, ..., 0.369376 ,\n 0.05497523, 0.5677395 ]], dtype=float32)\n\n\n\n\n### 4.2 Comparison with Deep Cox-Proportional Hazards Model\n\n\n```python\nfrom auton_survival.estimators import SurvivalModel\n\n# Now let us train a Deep Cox-proportional Hazard model with two linear layers and tanh activations\nrandom_seed = 0\nhyperparams = {'layers':[[50, 50]], \n 'lr':[1e-3],\n 'bs':[128], \n 'activation':['tanh']}\n```\n\n\n```python\ndcph_model = SurvivalModel(model='dcph', random_seed=0, hyperparams=hyperparams)\n\ninterventions_tr.name, interventions_te.name = 'treat', 'treat'\nfeatures_tr_dcph = pd.concat([features_tr, interventions_tr], axis=1)\nfeatures_te_dcph = pd.concat([features_te, interventions_te], axis=1)\n\n# Train the DCPH model\ndcph_model = dcph_model.fit(features_tr_dcph, outcomes_tr)\n```\n\n 0:\t[0s / 0s],\t\ttrain_loss: 3.4539,\tval_loss: 3.4712\n 1:\t[0s / 0s],\t\ttrain_loss: 3.3983,\tval_loss: 3.4378\n 2:\t[0s / 0s],\t\ttrain_loss: 3.3631,\tval_loss: 3.4173\n 3:\t[0s / 0s],\t\ttrain_loss: 3.3495,\tval_loss: 3.4090\n 4:\t[0s / 0s],\t\ttrain_loss: 3.3353,\tval_loss: 3.3997\n 5:\t[0s / 0s],\t\ttrain_loss: 3.3275,\tval_loss: 3.3949\n 6:\t[0s / 0s],\t\ttrain_loss: 3.3231,\tval_loss: 3.3874\n 7:\t[0s / 0s],\t\ttrain_loss: 3.3233,\tval_loss: 3.3898\n 8:\t[0s / 0s],\t\ttrain_loss: 3.3157,\tval_loss: 3.3892\n 9:\t[0s / 0s],\t\ttrain_loss: 3.3086,\tval_loss: 3.3821\n 10:\t[0s / 0s],\t\ttrain_loss: 3.2997,\tval_loss: 3.3888\n 11:\t[0s / 0s],\t\ttrain_loss: 3.2978,\tval_loss: 3.3864\n 12:\t[0s / 0s],\t\ttrain_loss: 3.2952,\tval_loss: 3.3854\n 13:\t[0s / 0s],\t\ttrain_loss: 3.3009,\tval_loss: 3.3910\n 14:\t[0s / 0s],\t\ttrain_loss: 3.2954,\tval_loss: 3.3866\n 15:\t[0s / 0s],\t\ttrain_loss: 3.2926,\tval_loss: 3.3890\n 16:\t[0s / 0s],\t\ttrain_loss: 3.2925,\tval_loss: 3.3853\n 17:\t[0s / 0s],\t\ttrain_loss: 3.2928,\tval_loss: 3.3838\n 18:\t[0s / 0s],\t\ttrain_loss: 3.2794,\tval_loss: 3.3780\n 19:\t[0s / 0s],\t\ttrain_loss: 3.2956,\tval_loss: 3.3802\n 20:\t[0s / 0s],\t\ttrain_loss: 3.2847,\tval_loss: 3.3787\n 21:\t[0s / 0s],\t\ttrain_loss: 3.2823,\tval_loss: 3.3785\n 22:\t[0s / 0s],\t\ttrain_loss: 3.2827,\tval_loss: 3.3784\n 23:\t[0s / 0s],\t\ttrain_loss: 3.2775,\tval_loss: 3.3852\n 24:\t[0s / 0s],\t\ttrain_loss: 3.2831,\tval_loss: 3.3793\n 25:\t[0s / 0s],\t\ttrain_loss: 3.2764,\tval_loss: 3.3689\n 26:\t[0s / 0s],\t\ttrain_loss: 3.2639,\tval_loss: 3.3785\n 27:\t[0s / 0s],\t\ttrain_loss: 3.2696,\tval_loss: 3.3704\n 28:\t[0s / 0s],\t\ttrain_loss: 3.2651,\tval_loss: 3.3758\n 29:\t[0s / 0s],\t\ttrain_loss: 3.2729,\tval_loss: 3.3691\n 30:\t[0s / 0s],\t\ttrain_loss: 3.2721,\tval_loss: 3.3706\n 31:\t[0s / 1s],\t\ttrain_loss: 3.2599,\tval_loss: 3.3760\n 32:\t[0s / 1s],\t\ttrain_loss: 3.2659,\tval_loss: 3.3685\n 33:\t[0s / 1s],\t\ttrain_loss: 3.2632,\tval_loss: 3.3655\n 34:\t[0s / 1s],\t\ttrain_loss: 3.2614,\tval_loss: 3.3643\n 35:\t[0s / 1s],\t\ttrain_loss: 3.2646,\tval_loss: 3.3647\n 36:\t[0s / 1s],\t\ttrain_loss: 3.2546,\tval_loss: 3.3661\n 37:\t[0s / 1s],\t\ttrain_loss: 3.2581,\tval_loss: 3.3605\n 38:\t[0s / 1s],\t\ttrain_loss: 3.2541,\tval_loss: 3.3634\n 39:\t[0s / 1s],\t\ttrain_loss: 3.2509,\tval_loss: 3.3617\n 40:\t[0s / 1s],\t\ttrain_loss: 3.2515,\tval_loss: 3.3631\n 41:\t[0s / 1s],\t\ttrain_loss: 3.2558,\tval_loss: 3.3641\n 42:\t[0s / 1s],\t\ttrain_loss: 3.2572,\tval_loss: 3.3597\n 43:\t[0s / 1s],\t\ttrain_loss: 3.2445,\tval_loss: 3.3635\n 44:\t[0s / 1s],\t\ttrain_loss: 3.2508,\tval_loss: 3.3605\n 45:\t[0s / 1s],\t\ttrain_loss: 3.2433,\tval_loss: 3.3623\n 46:\t[0s / 1s],\t\ttrain_loss: 3.2479,\tval_loss: 3.3560\n 47:\t[0s / 1s],\t\ttrain_loss: 3.2356,\tval_loss: 3.3593\n 48:\t[0s / 1s],\t\ttrain_loss: 3.2412,\tval_loss: 3.3614\n 49:\t[0s / 1s],\t\ttrain_loss: 3.2410,\tval_loss: 3.3589\n\n\n### Evaluate DCPH on Test Data\n\n\n```python\n# Find suvival scores in the test data\npredictions_test_DCPH = dcph_model.predict_survival(features_te_dcph, horizons)\n\nCI1, CI3, CI5, IBS = factual_evaluate((x_tr, t_tr, e_tr, a_tr), (x_te, t_te, e_te, a_te), \n horizons, predictions_test_DCPH)\nprint(f'Concordance Index (1 Year): {np.around(CI1, 4)} (3 Year) {np.around(CI3, 4)}: (5 Year): {np.around(CI5, 4)}')\nprint(f'Integrated Brier Score: {np.around(IBS, 4)}')\n```\n\n Concordance Index (1 Year): 0.6894 (3 Year) 0.6925: (5 Year): 0.6942\n Integrated Brier Score: 0.1535\n\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "ef3bd6db6163dffa30fb79c062be69318f3f372d", "size": 627015, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/Demo of CMHE on Synthetic Data.ipynb", "max_stars_repo_name": "vedant-sanil/auton-survival", "max_stars_repo_head_hexsha": "3e6c5ffed1dfe30d6edcfc57a952fce23455e585", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "examples/Demo of CMHE on Synthetic Data.ipynb", "max_issues_repo_name": "vedant-sanil/auton-survival", "max_issues_repo_head_hexsha": "3e6c5ffed1dfe30d6edcfc57a952fce23455e585", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "examples/Demo of CMHE on Synthetic Data.ipynb", "max_forks_repo_name": "vedant-sanil/auton-survival", "max_forks_repo_head_hexsha": "3e6c5ffed1dfe30d6edcfc57a952fce23455e585", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 645.7415036045, "max_line_length": 555628, "alphanum_fraction": 0.9421736322, "converted": true, "num_tokens": 7485, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.668880247169804, "lm_q2_score": 0.6548947357776795, "lm_q1q2_score": 0.4380461527371778}} {"text": "\n\n# Tutorial 3: Deep linear neural networks\n**Week 1, Day 2: Linear Deep Learning**\n\n**By Neuromatch Academy**\n\n__Content creators:__ Saeed Salehi, Spiros Chavlis, Andrew Saxe\n\n__Content reviewers:__ Polina Turishcheva, Antoine De Comite\n\n__Content editors:__ Anoop Kulkarni\n\n__Production editors:__ Khalid Almubarak, Spiros Chavlis\n\n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n

    \n\n---\n# Tutorial Objectives\n\n* Deep linear neural networks\n* Learning dynamics and singular value decomposition\n* Representational Similarity Analysis\n* Illusory correlations & ethics\n\n\n```python\n# @title Tutorial slides\n\n# @markdown These are the slides for the videos in this tutorial\nfrom IPython.display import IFrame\nIFrame(src=f\"https://mfr.ca-1.osf.io/render?url=https://osf.io/bncr8/?direct%26mode=render%26action=download%26mode=render\", width=854, height=480)\n```\n\n---\n# Setup\n\nThis a GPU-Free tutorial!\n\n\n```python\n# @title Install dependencies\n!pip install git+https://github.com/NeuromatchAcademy/evaltools --quiet\n\nfrom evaltools.airtable import AirtableForm\n```\n\n\n```python\n# Imports\nimport math\nimport torch\nimport matplotlib\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nimport torch.nn as nn\nimport torch.optim as optim\n```\n\n\n```python\n# @title Figure settings\n\nfrom matplotlib import gridspec\nfrom ipywidgets import interact, IntSlider, FloatSlider, fixed\nfrom ipywidgets import FloatLogSlider, Layout, VBox\nfrom ipywidgets import interactive_output\nfrom mpl_toolkits.axes_grid1 import make_axes_locatable\n\nimport warnings\nwarnings.filterwarnings(\"ignore\")\n\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/content-creation/main/nma.mplstyle\")\n```\n\n\n```python\n# @title Plotting functions\n\ndef plot_x_y_hier_data(im1, im2, subplot_ratio=[1, 2]):\n fig = plt.figure(figsize=(12, 5))\n gs = gridspec.GridSpec(1, 2, width_ratios=subplot_ratio)\n ax0 = plt.subplot(gs[0])\n ax1 = plt.subplot(gs[1])\n ax0.imshow(im1, cmap=\"cool\")\n ax1.imshow(im2, cmap=\"cool\")\n # plt.suptitle(\"The whole dataset as imshow plot\", y=1.02)\n ax0.set_title(\"Labels of all samples\")\n ax1.set_title(\"Features of all samples\")\n ax0.set_axis_off()\n ax1.set_axis_off()\n plt.tight_layout()\n plt.show()\n\n\ndef plot_x_y_hier_one(im1, im2, subplot_ratio=[1, 2]):\n fig = plt.figure(figsize=(12, 1))\n gs = gridspec.GridSpec(1, 2, width_ratios=subplot_ratio)\n ax0 = plt.subplot(gs[0])\n ax1 = plt.subplot(gs[1])\n ax0.imshow(im1, cmap=\"cool\")\n ax1.imshow(im2, cmap=\"cool\")\n ax0.set_title(\"Labels of a single sample\")\n ax1.set_title(\"Features of a single sample\")\n ax0.set_axis_off()\n ax1.set_axis_off()\n plt.tight_layout()\n plt.show()\n\n\ndef plot_tree_data(label_list = None, feature_array = None, new_feature = None):\n cmap = matplotlib.colors.ListedColormap(['cyan', 'magenta'])\n n_features = 10\n n_labels = 8\n im1 = np.eye(n_labels)\n if feature_array is None:\n im2 = np.array([[1, 1, 1, 1, 1, 1, 1, 1],\n [0, 0, 0, 0, 0, 0, 0, 0],\n [0, 0, 0, 0, 1, 1, 1, 1],\n [1, 1, 1, 1, 0, 0, 0, 0],\n [0, 0, 0, 0, 0, 0, 1, 1],\n [0, 0, 1, 1, 0, 0, 0, 0],\n [1, 1, 0, 0, 0, 0, 0, 0],\n [0, 0, 0, 0, 1, 1, 0, 0],\n [0, 1, 1, 1, 0, 0, 0, 0],\n [0, 0, 0, 0, 1, 1, 0, 1]]).T\n im2[im2 == 0] = -1\n feature_list = ['can_grow',\n 'is_mammal',\n 'has_leaves',\n 'can_move',\n 'has_trunk',\n 'can_fly',\n 'can_swim',\n 'has_stem',\n 'is_warmblooded',\n 'can_flower']\n else:\n im2 = feature_array\n if label_list is None:\n label_list = ['Goldfish', 'Tuna', 'Robin', 'Canary',\n 'Rose', 'Daisy', 'Pine', 'Oak']\n fig = plt.figure(figsize=(12, 7))\n gs = gridspec.GridSpec(1, 2, width_ratios=[1, 1.35])\n ax1 = plt.subplot(gs[0])\n ax2 = plt.subplot(gs[1])\n ax1.imshow(im1, cmap=cmap)\n if feature_array is None:\n implt = ax2.imshow(im2, cmap=cmap, vmin=-1.0, vmax=1.0)\n else:\n implt = ax2.imshow(im2[:, -n_features:], cmap=cmap, vmin=-1.0, vmax=1.0)\n divider = make_axes_locatable(ax2)\n cax = divider.append_axes(\"right\", size=\"5%\", pad=0.1)\n cbar = plt.colorbar(implt, cax=cax, ticks=[-0.5, 0.5])\n cbar.ax.set_yticklabels(['no', 'yes'])\n ax1.set_title(\"Labels\")\n ax1.set_yticks(ticks=np.arange(n_labels))\n ax1.set_yticklabels(labels=label_list)\n ax1.set_xticks(ticks=np.arange(n_labels))\n ax1.set_xticklabels(labels=label_list, rotation='vertical')\n ax2.set_title(\"{} random Features\".format(n_features))\n ax2.set_yticks(ticks=np.arange(n_labels))\n ax2.set_yticklabels(labels=label_list)\n if feature_array is None:\n ax2.set_xticks(ticks=np.arange(n_features))\n ax2.set_xticklabels(labels=feature_list, rotation='vertical')\n else:\n ax2.set_xticks(ticks=[n_features-1])\n ax2.set_xticklabels(labels=[new_feature], rotation='vertical')\n plt.tight_layout()\n plt.show()\n\n\ndef plot_loss(loss_array, title=\"Training loss (Mean Squared Error)\", c=\"r\"):\n plt.figure(figsize=(10, 5))\n plt.plot(loss_array, color=c)\n plt.xlabel(\"Epoch\")\n plt.ylabel(\"MSE\")\n plt.title(title)\n plt.show()\n\n\ndef plot_loss_sv(loss_array, sv_array):\n n_sing_values = sv_array.shape[1]\n sv_array = sv_array / np.max(sv_array)\n cmap = plt.cm.get_cmap(\"Set1\", n_sing_values)\n\n _, (plot1, plot2) = plt.subplots(2, 1, sharex=True, figsize=(10, 10))\n plot1.set_title(\"Training loss (Mean Squared Error)\")\n plot1.plot(loss_array, color='r')\n\n plot2.set_title(\"Evolution of singular values (modes)\")\n for i in range(n_sing_values):\n plot2.plot(sv_array[:, i], c=cmap(i))\n plot2.set_xlabel(\"Epoch\")\n plt.show()\n\n\ndef plot_loss_sv_twin(loss_array, sv_array):\n n_sing_values = sv_array.shape[1]\n sv_array = sv_array / np.max(sv_array)\n cmap = plt.cm.get_cmap(\"winter\", n_sing_values)\n\n fig = plt.figure(figsize=(10, 5))\n ax1 = plt.gca()\n ax1.set_title(\"Learning Dynamics\")\n ax1.set_xlabel(\"Epoch\")\n ax1.set_ylabel(\"Mean Squared Error\", c='r')\n ax1.tick_params(axis='y', labelcolor='r')\n ax1.plot(loss_array, color='r')\n\n ax2 = ax1.twinx()\n ax2.set_ylabel(\"Singular values (modes)\", c='b')\n ax2.tick_params(axis='y', labelcolor='b')\n for i in range(n_sing_values):\n ax2.plot(sv_array[:, i], c=cmap(i))\n\n fig.tight_layout()\n plt.show()\n\n\ndef plot_ills_sv_twin(ill_array, sv_array, ill_label):\n n_sing_values = sv_array.shape[1]\n sv_array = sv_array / np.max(sv_array)\n cmap = plt.cm.get_cmap(\"winter\", n_sing_values)\n\n fig = plt.figure(figsize=(10, 5))\n ax1 = plt.gca()\n ax1.set_title(\"Network training and the Illusory Correlations\")\n ax1.set_xlabel(\"Epoch\")\n ax1.set_ylabel(ill_label, c='r')\n ax1.tick_params(axis='y', labelcolor='r')\n ax1.plot(ill_array, color='r', linewidth=3)\n ax1.set_ylim(-1.05, 1.05)\n # ax1.set_yticks([-1, 0, 1])\n # ax1.set_yticklabels(['False', 'Not sure', 'True'])\n\n ax2 = ax1.twinx()\n ax2.set_ylabel(\"Singular values (modes)\", c='b')\n ax2.tick_params(axis='y', labelcolor='b')\n for i in range(n_sing_values):\n ax2.plot(sv_array[:, i], c=cmap(i))\n\n fig.tight_layout()\n plt.show()\n\n\ndef plot_loss_sv_rsm(loss_array, sv_array, rsm_array, i_ep):\n n_ep = loss_array.shape[0]\n rsm_array = rsm_array / np.max(rsm_array)\n sv_array = sv_array / np.max(sv_array)\n\n n_sing_values = sv_array.shape[1]\n cmap = plt.cm.get_cmap(\"winter\", n_sing_values)\n\n fig = plt.figure(figsize=(14, 5))\n gs = gridspec.GridSpec(1, 2, width_ratios=[5, 3])\n\n ax0 = plt.subplot(gs[1])\n ax0.yaxis.tick_right()\n implot = ax0.imshow(rsm_array[i_ep], cmap=\"Purples\", vmin=0.0, vmax=1.0)\n divider = make_axes_locatable(ax0)\n cax = divider.append_axes(\"right\", size=\"5%\", pad=0.9)\n cbar = plt.colorbar(implot, cax=cax, ticks=[])\n cbar.ax.set_ylabel('Similarity', fontsize=12)\n ax0.set_title(\"RSM at epoch {}\".format(i_ep), fontsize=16)\n # ax0.set_axis_off()\n ax0.set_yticks(ticks=np.arange(n_sing_values))\n ax0.set_yticklabels(labels=item_names)\n # ax0.set_xticks([])\n ax0.set_xticks(ticks=np.arange(n_sing_values))\n ax0.set_xticklabels(labels=item_names, rotation='vertical')\n\n ax1 = plt.subplot(gs[0])\n ax1.set_title(\"Learning Dynamics\", fontsize=16)\n ax1.set_xlabel(\"Epoch\")\n ax1.set_ylabel(\"Mean Squared Error\", c='r')\n ax1.tick_params(axis='y', labelcolor='r', direction=\"in\")\n ax1.plot(np.arange(n_ep), loss_array, color='r')\n ax1.axvspan(i_ep-2, i_ep+2, alpha=0.2, color='m')\n\n ax2 = ax1.twinx()\n ax2.set_ylabel(\"Singular values\", c='b')\n ax2.tick_params(axis='y', labelcolor='b', direction=\"in\")\n for i in range(n_sing_values):\n ax2.plot(np.arange(n_ep), sv_array[:, i], c=cmap(i))\n ax1.set_xlim(-1, n_ep+1)\n ax2.set_xlim(-1, n_ep+1)\n\n plt.show()\n```\n\n\n```python\n#@title Helper functions\n\natform = AirtableForm('appn7VdPRseSoMXEG',\\\n 'W1D2_T3','https://portal.neuromatchacademy.org/api/redirect/to/f60119ed-1c22-4dae-9e18-b6a767f477e1')\n\ndef build_tree(n_levels, n_branches, probability, to_np_array=True):\n \"\"\"Builds a tree\n \"\"\"\n assert 0.0 <= probability <= 1.0\n\n tree = {}\n\n tree[\"level\"] = [0]\n for i in range(1, n_levels+1):\n tree[\"level\"].extend([i]*(n_branches**i))\n\n tree[\"pflip\"] = [probability]*len(tree[\"level\"])\n\n tree[\"parent\"] = [None]\n k = len(tree[\"level\"])-1\n for j in range(k//n_branches):\n tree[\"parent\"].extend([j]*n_branches)\n\n if to_np_array:\n tree[\"level\"] = np.array(tree[\"level\"])\n tree[\"pflip\"] = np.array(tree[\"pflip\"])\n tree[\"parent\"] = np.array(tree[\"parent\"])\n\n return tree\n\n\ndef sample_from_tree(tree, n):\n \"\"\" Generates n samples from a tree\n \"\"\"\n items = [i for i, v in enumerate(tree[\"level\"]) if v == max(tree[\"level\"])]\n n_items = len(items)\n x = np.zeros(shape=(n, n_items))\n rand_temp = np.random.rand(n, len(tree[\"pflip\"]))\n flip_temp = np.repeat(tree[\"pflip\"].reshape(1, -1), n, 0)\n samp = (rand_temp > flip_temp) * 2 - 1\n\n for i in range(n_items):\n j = items[i]\n prop = samp[:, j]\n while tree[\"parent\"][j] is not None:\n j = tree[\"parent\"][j]\n prop = prop * samp[:, j]\n x[:, i] = prop.T\n return x\n\n\ndef generate_hsd():\n # building the tree\n n_branches = 2 # 2 branches at each node\n probability = .15 # flipping probability\n n_levels = 3 # number of levels (depth of tree)\n tree = build_tree(n_levels, n_branches, probability, to_np_array=True)\n tree[\"pflip\"][0] = 0.5\n n_samples = 10000 # Sample this many features\n\n tree_labels = np.eye(n_branches**n_levels)\n tree_features = sample_from_tree(tree, n_samples).T\n return tree_labels, tree_features\n\n\ndef linear_regression(X, Y):\n \"\"\"Analytical Linear regression\n\n \"\"\"\n assert isinstance(X, np.ndarray)\n assert isinstance(Y, np.ndarray)\n M, Dx = X.shape\n N, Dy = Y.shape\n assert Dx == Dy\n W = Y @ X.T @ np.linalg.inv(X @ X.T)\n return W\n\n\ndef add_feature(existing_features, new_feature):\n assert isinstance(existing_features, np.ndarray)\n assert isinstance(new_feature, list)\n new_feature = np.array([new_feature]).T\n # return np.hstack((tree_features, new_feature*2-1))\n return np.hstack((tree_features, new_feature))\n\n\ndef net_svd(model, in_dim):\n \"\"\"Performs a Singular Value Decomposition on a given model weights\n\n Args:\n model (torch.nn.Module): neural network model\n in_dim (int): the input dimension of the model\n\n Returns:\n U, \u03a3, V (Tensors): Orthogonal, diagonal, and orthogonal matrices\n \"\"\"\n W_tot = torch.eye(in_dim)\n for weight in model.parameters():\n W_tot = weight.detach() @ W_tot\n U, SIGMA, V = torch.svd(W_tot)\n return U, SIGMA, V\n\n\ndef net_rsm(h):\n \"\"\"Calculates the Representational Similarity Matrix\n\n Arg:\n h (torch.Tensor): activity of a hidden layer\n\n Returns:\n (torch.Tensor): Representational Similarity Matrix\n \"\"\"\n rsm = h @ h.T\n return rsm\n\n\ndef initializer_(model, gamma=1e-12):\n \"\"\"(in-place) Re-initialization of weights\n\n Args:\n model (torch.nn.Module): PyTorch neural net model\n gamma (float): initialization scale\n \"\"\"\n for weight in model.parameters():\n n_out, n_in = weight.shape\n sigma = gamma / math.sqrt(n_in + n_out)\n nn.init.normal_(weight, mean=0.0, std=sigma)\n\n\ndef test_initializer_ex(seed):\n torch.manual_seed(seed)\n model = LNNet(5000, 5000, 1)\n try:\n ex_initializer_(model, gamma=1)\n std = torch.std(next(iter(model.parameters())).detach()).item()\n if -1e-5 <= (std - 0.01) <= 1e-5:\n print(\"Well done! Seems to be correct!\")\n else:\n print(\"Please double check your implementation!\")\n except:\n print(\"Faulty Implementation!\")\n\n\ndef test_net_svd_ex(seed):\n torch.manual_seed(seed)\n model = LNNet(8, 30, 100)\n try:\n U_ex, \u03a3_ex, V_ex = ex_net_svd(model, 8)\n U, \u03a3, V = net_svd(model, 8)\n if (torch.all(torch.isclose(U_ex.detach(), U.detach(), atol=1e-6)) and\n torch.all(torch.isclose(\u03a3_ex.detach(), \u03a3.detach(), atol=1e-6)) and\n torch.all(torch.isclose(V_ex.detach(), V.detach(), atol=1e-6))):\n print(\"Well done! Seems to be correct!\")\n else:\n print(\"Please double check your implementation!\")\n except:\n print(\"Faulty Implementation!\")\n\n\ndef test_net_rsm_ex(seed):\n torch.manual_seed(seed)\n x = torch.rand(7, 17)\n try:\n y_ex = ex_net_rsm(x)\n y = x @ x.T\n if (torch.all(torch.isclose(y_ex, y, atol=1e-6))):\n print(\"Well done! Seems to be correct!\")\n else:\n print(\"Please double check your implementation!\")\n except:\n print(\"Faulty Implementation!\")\n```\n\n\n```python\n#@title Set random seed\n\n#@markdown Executing `set_seed(seed=seed)` you are setting the seed\n\n# for DL its critical to set the random seed so that students can have a\n# baseline to compare their results to expected results.\n# Read more here: https://pytorch.org/docs/stable/notes/randomness.html\n\n# Call `set_seed` function in the exercises to ensure reproducibility.\nimport random\nimport torch\n\ndef set_seed(seed=None, seed_torch=True):\n if seed is None:\n seed = np.random.choice(2 ** 32)\n random.seed(seed)\n np.random.seed(seed)\n if seed_torch:\n torch.manual_seed(seed)\n torch.cuda.manual_seed_all(seed)\n torch.cuda.manual_seed(seed)\n torch.backends.cudnn.benchmark = False\n torch.backends.cudnn.deterministic = True\n\n print(f'Random seed {seed} has been set.')\n\n\n# In case that `DataLoader` is used\ndef seed_worker(worker_id):\n worker_seed = torch.initial_seed() % 2**32\n np.random.seed(worker_seed)\n random.seed(worker_seed)\n```\n\n\n```python\n#@title Set device (GPU or CPU). Execute `set_device()`\n# especially if torch modules used.\n\n# inform the user if the notebook uses GPU or CPU.\n\ndef set_device():\n device = \"cuda\" if torch.cuda.is_available() else \"cpu\"\n if device != \"cuda\":\n print(\"GPU is not enabled in this notebook. \\n\"\n \"If you want to enable it, in the menu under `Runtime` -> \\n\"\n \"`Hardware accelerator.` and select `GPU` from the dropdown menu\")\n else:\n print(\"GPU is enabled in this notebook. \\n\"\n \"If you want to disable it, in the menu under `Runtime` -> \\n\"\n \"`Hardware accelerator.` and select `None` from the dropdown menu\")\n\n return device\n```\n\n\n```python\nSEED = 2021\nset_seed(seed=SEED)\nDEVICE = set_device()\n```\n\n Random seed 2021 has been set.\n GPU is enabled in this notebook. \n If you want to disable it, in the menu under `Runtime` -> \n `Hardware accelerator.` and select `None` from the dropdown menu\n\n\nThis colab notebook is GPU free!\n\n---\n# Section 0: Prelude\n*Time estimate: ~10 mins*\n\n\n\n**A note on the exercises**: Most of the exercises are marked Optional (Bonus) and should only read through them if you are in a tight timeline. Therefore we would not rely on the implementation of the exercises. If necessary, you can look at the *Helper Functions* cell above to find the functions and classes used in this tutorial.\n\nThroughout this tutorial, we will use a linear neural net with a single hidden layer. We have also excluded `bias` from the layers. Please note that the forward loop returns the hidden activation, besides the network output (prediction). we will need it in section 3.\n\n\n```python\nclass LNNet(nn.Module):\n \"\"\"A Linear Neural Net with one hidden layer\n \"\"\"\n\n def __init__(self, in_dim, hid_dim, out_dim):\n \"\"\"\n Args:\n in_dim (int): input dimension\n out_dim (int): ouput dimension\n hid_dim (int): hidden dimension\n \"\"\"\n super().__init__()\n self.in_hid = nn.Linear(in_dim, hid_dim, bias=False)\n self.hid_out = nn.Linear(hid_dim, out_dim, bias=False)\n\n def forward(self, x):\n \"\"\"\n Args:\n x (torch.Tensor): input tensor\n \"\"\"\n hid = self.in_hid(x) # hidden activity\n out = self.hid_out(hid) # output (prediction)\n return out, hid\n```\n\nOther than `net_svd` and `net_rsm` functions, the training loop should be mostly familiar to you. We will define these functions in the coming sections.\n\n**important**: Please note that the two functions are part of inner training loop and are therefore executed and recorded at every iteration.\n\n\n```python\ndef train(model, inputs, targets, n_epochs, lr, illusory_i=0):\n \"\"\"Training function\n\n Args:\n model (torch nn.Module): the neural network\n inputs (torch.Tensor): features (input) with shape `[batch_size, input_dim]`\n targets (torch.Tensor): targets (labels) with shape `[batch_size, output_dim]`\n n_epochs (int): number of training epochs (iterations)\n lr (float): learning rate\n illusory_i (int): index of illusory feature\n\n Returns:\n np.ndarray: record (evolution) of training loss\n np.ndarray: record (evolution) of singular values (dynamic modes)\n np.ndarray: record (evolution) of representational similarity matrices\n np.ndarray: record of network prediction for the last feature\n \"\"\"\n in_dim = inputs.size(1)\n\n losses = np.zeros(n_epochs) # loss records\n modes = np.zeros((n_epochs, in_dim)) # singular values (modes) records\n rs_mats = [] # representational similarity matrices\n illusions = np.zeros(n_epochs) # prediction for the given feature\n\n optimizer = optim.SGD(model.parameters(), lr=lr)\n criterion = nn.MSELoss()\n\n for i in range(n_epochs):\n optimizer.zero_grad()\n predictions, hiddens = model(inputs)\n loss = criterion(predictions, targets)\n loss.backward()\n optimizer.step()\n\n # Section 2 Singular value decomposition\n U, \u03a3, V = net_svd(model, in_dim)\n\n # Section 3 calculating representational similarity matrix\n RSM = net_rsm(hiddens.detach())\n\n # Section 4 network prediction of illusory_i inputs for the last feature\n pred_ij = predictions.detach()[illusory_i, -1]\n\n # logging (recordings)\n losses[i] = loss.item()\n modes[i] = \u03a3.detach().numpy()\n rs_mats.append(RSM.numpy())\n illusions[i] = pred_ij.numpy()\n\n return losses, modes, np.array(rs_mats), illusions\n```\n\nWe also need take over the initialization of the weights. In PyTorch, [`nn.init`](https://pytorch.org/docs/stable/nn.init.html) provides us with the functions to initialize tensors from a given distribution.\n\n## Coding Exercise 0: Re-initialization (Optional)\n\nComplete the function `ex_initializer_`, such that the weights are sampled from the following distribution:\n\n\\begin{equation}\n\\mathcal{N}\\left(\\mu=0, ~~\\sigma=\\gamma \\sqrt{\\dfrac{1}{n_{in} + n_{out}}} \\right)\n\\end{equation}\n\nwhere $\\gamma$ is the initialization scale, $n_{in}$ and $n_{out}$ are respectively input and output dimensions of the layer. the Underscore (\"_\") in `ex_initializer_` and other functions, denotes \"[in-place](https://discuss.pytorch.org/t/what-is-in-place-operation/16244/2)\" operation.\n\n**important note**: since we did not include bias in the layers, the `model.parameters()` would only return the weights in each layer.\n\n\n```python\n#add event to airtable\natform.add_event('Coding Exercise 0: Re-initialization')\n\ndef ex_initializer_(model, gamma=1e-12):\n \"\"\"(in-place) Re-initialization of weights\n\n Args:\n model (torch.nn.Module): PyTorch neural net model\n gamma (float): initialization scale\n \"\"\"\n for weight in model.parameters():\n n_out, n_in = weight.shape\n #################################################\n ## Define the standard deviation (sigma) for the normal distribution\n # as given in the equation above\n # Complete the function and remove or comment the line below\n #raise NotImplementedError(\"Function `ex_initializer_`\")\n #################################################\n sigma = gamma * ( 1 / np.sqrt(n_out + n_in) )\n nn.init.normal_(weight, mean=0.0, std=sigma)\n\n## uncomment and run\ntest_initializer_ex(SEED)\n```\n\n Well done! Seems to be correct!\n\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content-dl/tree/main//tutorials/W1D2_LinearDeepLearning/solutions/W1D2_Tutorial3_Solution_e471aef3.py)\n\n\n\n---\n# Section 1: Deep Linear Neural Nets\n\n*Time estimate: ~20 mins*\n\n\n```python\n# @title Video 1: Intro to Representation Learning\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1iM4y1T7eJ\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"DqMSU4Bikt0\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 1: Intro to Representation Learning')\n\ndisplay(out)\n```\n\n Video available at https://www.bilibili.com/video/BV1iM4y1T7eJ\n\n\n\n\n\n\n\n\n Video available at https://youtube.com/watch?v=DqMSU4Bikt0\n\n\n\n\n\n\n\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nSo far, depth just seems to slow down the learning. And we know that a single nonlinear hidden layer (given enough number of neurons and infinite training samples) has the potential to approximate any function. So it seems fair to ask: **What is depth good for**?\n\nOne reason can be that shallow nonlinear neural networks hardly meet their true potential in practice. In the contrast, deep neural nets are often surprisingly powerful in learning complex functions without sacrificing generalization. A core intuition behind deep learning is that deep nets derive their power through learning internal representations. How does this work? To address representation learning, we have to go beyond the 1D chain.\n\nFor this and the next couple of exercises, we use syntactically generated hierarchically structured data through a *branching diffusion process* (see [this reference](https://www.pnas.org/content/pnas/suppl/2019/05/16/1820226116.DCSupplemental/pnas.1820226116.sapp.pdf) for more details).\n\n
    \n\n
    hierarchically structured data (a tree)
    \n\nThe inputs to the network are labels (i.e. names), while the outputs are the features (i.e. attributes). For example, for the label \"Goldfish\", the network has to learn all the (artificially created) features, such as \"*can swim*\", \"*is cold-blooded*\", \"*has fins*\", and more. Given that we are training on hierarchically structured data, network could also learn the tree structure, that Goldfish and Tuna have rather similar features, and Robin has more in common with Tuna, compared to Rose.\n\n\n```python\n# @markdown #### Run to generate and visualize training samples from tree\n\ntree_labels, tree_features = generate_hsd()\n\n# convert (cast) data from np.ndarray to torch.Tensor\nlabel_tensor = torch.tensor(tree_labels).float()\nfeature_tensor = torch.tensor(tree_features).float()\n\nitem_names = ['Goldfish', 'Tuna', 'Robin', 'Canary',\n 'Rose', 'Daisy', 'Pine', 'Oak']\nplot_tree_data()\n\n# dimensions\nprint(\"---------------------------------------------------------------\")\nprint(\"Input Dimension: {}\".format(tree_labels.shape[1]))\nprint(\"Output Dimension: {}\".format(tree_features.shape[1]))\nprint(\"Number of samples: {}\".format(tree_features.shape[0]))\n```\n\nTo continue this tutorial, it is vital to understand the premise of our training data and what the task is. Therefore, please take your time to discuss them with your pod.\n\n
    \n\n
    The neural network used for this tutorial
    \n\n## Interactive Demo 1: Training the deep LNN\n\nTraining a neural net on our data is straight forward. But before executing the next cell, remember the training loss curve from previous tutorial.\n\n\n```python\n# @markdown #### Make sure you execute this cell to train the network and plot\n\nlr = 100.0 # learning rate\ngamma = 1e-12 # initialization scale\nn_epochs = 250 # number of epochs\ndim_input = 8 # input dimension = `label_tensor.size(1)`\ndim_hidden = 30 # hidden neurons\ndim_output = 10000 # output dimension = `feature_tensor.size(1)`\n\n# model instantiation\ndlnn_model = LNNet(dim_input, dim_hidden, dim_output)\n\n# weights re-initialization\ninitializer_(dlnn_model, gamma)\n\n# training\nlosses, *_ = train(dlnn_model,\n label_tensor,\n feature_tensor,\n n_epochs=n_epochs,\n lr=lr)\n\n# plotting\nplot_loss(losses)\n```\n\n**Think!**\n\nWhy haven't we seen these \"bumps\" in training before? And should we look for them in the future? What do these bumps mean?\n\nRecall from previous tutorial, that we are always interested in learning rate ($\\eta$) and initialization ($\\gamma$) that would give us the fastest but yet stable (reliable) convergence. Try finding the optimal $\\eta$ and $\\gamma$ using the following widgets. More specifically, try large $\\gamma$ and see if we can recover the bumps by tuning the $\\eta$.\n\n\n```python\n# @markdown #### Make sure you execute this cell to enable the widget!\n\ndef loss_lr_init(lr, gamma):\n \"\"\"Trains and plots the loss evolution given lr and initialization\n Args:\n lr (float): learning rate\n gamma (float): initialization scale\n \"\"\"\n n_epochs = 250 # number of epochs\n dim_input = 8 # input dimension = `label_tensor.size(1)`\n dim_hidden = 30 # hidden neurons\n dim_output = 10000 # output dimension = `feature_tensor.size(1)`\n\n # model instantiation\n dlnn_model = LNNet(dim_input, dim_hidden, dim_output)\n\n # weights re-initialization\n initializer_(dlnn_model, gamma)\n\n losses, *_ = train(dlnn_model,\n label_tensor,\n feature_tensor,\n n_epochs=n_epochs,\n lr=lr)\n\n plot_loss(losses)\n\n_ = interact(loss_lr_init,\n lr = FloatSlider(min=1.0, max=200.0,\n step=1.0, value=100.0,\n continuous_update=False,\n readout_format='.1f',\n description='eta'),\n epochs = fixed(250),\n gamma = FloatLogSlider(min=-15, max=1,\n step=1, value=1e-12, base=10,\n continuous_update=False,\n description='gamma')\n )\n```\n\n---\n# Section 2: Singular Value Decomposition (SVD)\n\n*Time estimate: ~20 mins*\n\n\n```python\n# @title Video 2: SVD\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1bw411R7DJ\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"18oNWRziskM\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 2: SVD')\n\ndisplay(out)\n```\n\n Video available at https://www.bilibili.com/video/BV1bw411R7DJ\n\n\n\n\n\n\n\n\n Video available at https://youtube.com/watch?v=18oNWRziskM\n\n\n\n\n\n\n\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nIn this section, we intend to study the learning (training) dynamics we just saw. First, we should know that a linear neural network is performing sequential matrix multiplications, which can be simplified to:\n\n\\begin{align}\n\\mathbf{y} &= \\mathbf{W}_{L}~\\mathbf{W}_{L-1}~\\dots~\\mathbf{W}_{1} ~ \\mathbf{x} \\\\\n &= (\\prod_{i=1}^{L}{\\mathbf{W}_{i}}) ~ \\mathbf{x} \\\\\n &= \\mathbf{W}_{tot} ~ \\mathbf{x}\n\\end{align}\n\nwhere $L$ denotes the number of layers in our network.\n\n[Saxe et al. (2013)](https://arxiv.org/abs/1312.6120) showed that to analyze and to understanding the nonlinear learning dynamics of a deep LNN, we can use [Singular Value Decomposition (SVD)](https://en.wikipedia.org/wiki/Singular_value_decomposition) to decompose the $\\mathbf{W}_{tot}$ into orthogonal vectors, where orthogonality of the vectors would ensure their \"individuality (independence)\". This means we can break a deep wide LNN into multiple deep narrow LNN, so their activity is untangled from each other.\n\n
    \n\n__A Quick intro to SVD__\n\nAny real-valued matix $A$ (yes, ANY) can be decomposed (factorized) to 3 matrices:\n\n\\begin{equation}\n\\mathbf{A} = \\mathbf{U} \\mathbf{\u03a3} \\mathbf{V}^{\\top}\n\\end{equation}\n\nwhere $U$ is an orthogonal matrix, $\\Sigma$ is a diagonal matrix, and $V$ is again an orthogonal matrix. The diagonal elements of $\\Sigma$ are called **singular values**.\n\nThe main difference between SVD and EigenValue Decomposition (EVD), is that EVD requires $A$ to be squared and does not guarantee the eigenvectors to be orthogonal. \n\nWe strongly recommend the [Singular Value Decomposition (the SVD)](https://www.youtube.com/watch?v=mBcLRGuAFUk) by the amazing [Gilbert Strang](http://www-math.mit.edu/~gs/) if you would like to learn more.\n\n\n## Coding Exercise 2: SVD (Optional)\n\nThe goal is to perform the SVD on $\\mathbf{W}_{tot}$ in every epoch, and record the singular values (modes) during the training.\n\nComplete the function `ex_net_svd`, by first calculating the $\\mathbf{W}_{tot} = \\prod_{i=1}^{L}{\\mathbf{W}_{i}}$ and finally performing SVD on the $\\mathbf{W}_{tot}$. Please use the PyTorch [`torch.svd`](https://pytorch.org/docs/stable/generated/torch.svd.html) instead of NumPy [`np.linalg.svd`](https://numpy.org/doc/stable/reference/generated/numpy.linalg.svd.html).\n\n\n```python\ndef ex_net_svd(model, in_dim):\n \"\"\"Performs a Singular Value Decomposition on a given model weights\n\n Args:\n model (torch.nn.Module): neural network model\n in_dim (int): the input dimension of the model\n\n Returns:\n U, \u03a3, V (Tensors): Orthogonal, diagonal, and orthogonal matrices\n \"\"\"\n W_tot = torch.eye(in_dim)\n for weight in model.parameters():\n #################################################\n ## Calculate the W_tot by multiplication of all weights\n # and then perform SVD on the W_tot using pytorch's `torch.svd`\n # Remember that weights need to be `.detach()` from the graph\n # Complete the function and remove or comment the line below\n #raise NotImplementedError(\"Function `ex_net_svd`\")\n #################################################\n W_tot = weight @ W_tot\n U, \u03a3, V = torch.svd(W_tot) \n return U, \u03a3, V\n\n#add event to airtable\natform.add_event('Coding Exercise 2: SVD')\n\n## Uncomment and run\ntest_net_svd_ex(SEED)\n```\n\n Well done! Seems to be correct!\n\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content-dl/tree/main//tutorials/W1D2_LinearDeepLearning/solutions/W1D2_Tutorial3_Solution_51b2e553.py)\n\n\n\n\n```python\n# @markdown #### Make sure you execute this cell to train the network and plot\n\nlr = 100.0 # learning rate\ngamma = 1e-12 # initialization scale\nn_epochs = 250 # number of epochs\ndim_input = 8 # input dimension = `label_tensor.size(1)`\ndim_hidden = 30 # hidden neurons\ndim_output = 10000 # output dimension = `feature_tensor.size(1)`\n\n# model instantiation\ndlnn_model = LNNet(dim_input, dim_hidden, dim_output)\n\n# weights re-initialization\ninitializer_(dlnn_model, gamma)\n\n# training\nlosses, modes, *_ = train(dlnn_model,\n label_tensor,\n feature_tensor,\n n_epochs=n_epochs,\n lr=lr)\n\nplot_loss_sv_twin(losses, modes)\n```\n\n**Think!**\n\nIn EigenValue decomposition, the amount of variance explained by eigenvectors is proportional to the corresponding eigenvalues. What about the SVD? We see that the gradient descent guides the network to first learn the features that carry more information (have higher singular value)!\n\n\n```python\n# @title Video 3: SVD - Discussion\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1t54y1J7Tb\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"JEbRPPG2kUI\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 3: SVD - Discussion')\n\ndisplay(out)\n```\n\n Video available at https://www.bilibili.com/video/BV1t54y1J7Tb\n\n\n\n\n\n\n\n\n Video available at https://youtube.com/watch?v=JEbRPPG2kUI\n\n\n\n\n\n\n\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n---\n# Section 3: Representational Similarity Analysis (RSA)\n\n*Time estimate: ~20 mins*\n\n\n```python\n# @title Video 4: RSA\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV19f4y157zD\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"YOs1yffysX8\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 4: RSA')\n\ndisplay(out)\n```\n\n Video available at https://www.bilibili.com/video/BV19f4y157zD\n\n\n\n\n\n\n\n\n Video available at https://youtube.com/watch?v=YOs1yffysX8\n\n\n\n\n\n\n\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nThe previous section ended with an interesting remark. SVD helped to break our deep \"wide\" linear neural net into 8 deep \"narrow\" linear neural nets.\n\nThe first narrow net (highest singular value) converges fastest, while the last four narrow nets, converge almost simultaneously and have the smallest singular values. Can it be that the narrow net with larger mode is learning the difference between \"living things\" and \"objects\", while another narrow net with smaller mode is learning the difference between Fish and Birds? how could we check this hypothesis?\n\nRepresentational Similarity Analysis (RSA) is an approach that could help us understand the internal representation of our network. The main idea is that the activity of hidden units (neurons) in the network must be similar when the network is presented with similar input. For our dataset (hierarchically structured data), we expect the activity of neurons in the hidden layer to be more similar for Tuna and Canary, and less similar for Tuna and Oak.\n\nFor similarity measure, we can use the good old dot (scalar) product, which is also called cosine similarity. For calculating the dot product between multiple vectors (which would be our case), we can simply use matrix multiplication. Therefore the Representational Similarity Matrix for multiple-input (batch) activity could be calculated as follow:\n\n\\begin{equation}\nRSM = \\mathbf{H} \\mathbf{H}^{\\top}\n\\end{equation}\n\nwhere $\\mathbf{H} = \\mathbf{X} \\mathbf{W_1}$ is the activity of hidden neurons for a given batch $\\mathbf{X}$.\n\n\n## Coding Exercise 3: RSA (Optional)\n\nThe task is simple. We would need to measure the similarity between hidden layer activities $~\\mathbf{h} = \\mathbf{x} ~\\mathbf{W_1}$) for every input $\\mathbf{x}$.\n\nIf we perform RSA in every iteration, we could also see the evolution of representation learning.\n\n\n```python\ndef ex_net_rsm(h):\n \"\"\"Calculates the Representational Similarity Matrix\n\n Arg:\n h (torch.Tensor): activity of a hidden layer\n\n Returns:\n (torch.Tensor): Representational Similarity Matrix\n \"\"\"\n #################################################\n ## Calculate the Representational Similarity Matrix\n # Complete the function and remove or comment the line below\n #raise NotImplementedError(\"Function `ex_net_rsm`\")\n #################################################\n rsm = h @ h.T\n return rsm\n\n#add event to airtable\natform.add_event(' Coding Exercise 3: RSA')\n\n## Uncomment and run\ntest_net_rsm_ex(SEED)\n```\n\n Well done! Seems to be correct!\n\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content-dl/tree/main//tutorials/W1D2_LinearDeepLearning/solutions/W1D2_Tutorial3_Solution_394c7d85.py)\n\n\n\nNow we can train the model while recording the losses, modes, and RSMs at every iteration. First, use the epoch slider to explore the evolution of RSM without changing default lr ($\\eta$) and initialization ($\\gamma$). Then, as we did before, set $\\eta$ and $\\gamma$ to larger values to see whether you can retrieve the sequential structured learning of representations.\n\n\n```python\n#@markdown #### Make sure you execute this cell to enable widgets\n\ndef loss_svd_rsm_lr_gamma(lr, gamma, i_ep):\n \"\"\"\n Args:\n lr (float): learning rate\n gamma (float): initialization scale\n i_ep (int): which epoch to show\n\n \"\"\"\n n_epochs = 250 # number of epochs\n dim_input = 8 # input dimension = `label_tensor.size(1)`\n dim_hidden = 30 # hidden neurons\n dim_output = 10000 # output dimension = `feature_tensor.size(1)`\n\n # model instantiation\n dlnn_model = LNNet(dim_input, dim_hidden, dim_output)\n\n # weights re-initialization\n initializer_(dlnn_model, gamma)\n\n # training\n losses, modes, rsms, _ = train(dlnn_model,\n label_tensor,\n feature_tensor,\n n_epochs=n_epochs,\n lr=lr)\n plot_loss_sv_rsm(losses, modes, rsms, i_ep)\n\ni_ep_slider = IntSlider(min=10, max=241, step=1, value=61,\n continuous_update=False,\n description='Epoch',\n layout=Layout(width='630px'))\n\nlr_slider = FloatSlider(min=20.0, max=200.0, step=1.0, value=100.0,\n continuous_update=False,\n readout_format='.1f',\n description='eta')\n\ngamma_slider = FloatLogSlider(min=-15, max=1, step=1,\n value=1e-12, base=10,\n continuous_update=False,\n description='gamma')\n\nwidgets_ui = VBox([lr_slider, gamma_slider, i_ep_slider])\n\nwidgets_out = interactive_output(loss_svd_rsm_lr_gamma,\n {'lr': lr_slider,\n 'gamma': gamma_slider,\n 'i_ep': i_ep_slider})\n\ndisplay(widgets_ui, widgets_out)\n```\n\nLet's take a moment to analyze this more. A deep neural net is learning the representations, rather than a naive mapping (look-up table). This is thought to be the reason for deep neural nets supreme generalization and transfer learning ability. Unsurprisingly, neural nets with no hidden layer are incapable of representation learning, even with extremely small initialization.\n\n\n```python\n# @title Video 5: RSA - Discussion\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV18y4y1j7Xr\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"vprldATyq1o\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 5: RSA - Discussion')\ndisplay(out)\n```\n\n Video available at https://www.bilibili.com/video/BV18y4y1j7Xr\n\n\n\n\n\n\n\n\n Video available at https://youtube.com/watch?v=vprldATyq1o\n\n\n\n\n\n\n\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n---\n# Section 4: Illusory Correlations\n\n*Time estimate: ~20-30 mins*\n\n\n```python\n# @title Video 6: Illusory Correlations\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1vv411E7Sq\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"RxsAvyIoqEo\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 6: Illusory Correlations')\n\ndisplay(out)\n```\n\n Video available at https://www.bilibili.com/video/BV1vv411E7Sq\n\n\n\n\n\n\n\n\n Video available at https://youtube.com/watch?v=RxsAvyIoqEo\n\n\n\n\n\n\n\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nLet's recall the training loss curves. There was often a long plateau (where the weights are stuck at a saddle point), followed by a sudden drop. For very deep complex neural nets, such plateaus can take hours of training, and we are often tempted to stop the training, because we believe it is \"as good as it gets\"! Another side effect of \"immature interruption\" of training is the network finding (learning) illusory correlations.\n\nTo better understand this, let's do the next demonstration and exercise.\n\n## Demonstration: Illusory Correlations\n\nOur original dataset has 4 animals: Canary, Robin, Goldfish, and Tuna. These animals all have bones. Therefore if we include a \"has bone\" feature, the network would learn it at the second level (i.e. second bump, second mode convergence), when it learns the animal-plants distinction.\n\nWhat if the dataset has Shark instead of Goldfish. Sharks don't have bones (their skeletons are made of cartilaginous, which is much lighter than true bone and more flexible). Then we will have a feature which is *True* (i.e. +1) for Tuna, Robin, and Canary, but *False* (i.e. -1) for all the plants and the shark! Let's see what the network does.\n\nFirst, we add the new feature to the targets. We then start training our LNN and in every epoch, record the network prediction for \"sharks having bones\".\n\n
    \n\n\n```python\n# sampling new data from the tree\ntree_labels, tree_features = generate_hsd()\n\n# replacing Goldfish with Shark\nitem_names = ['Shark', 'Tuna', 'Robin', 'Canary',\n 'Rose', 'Daisy', 'Pine', 'Oak']\n\n# index of label to record\nillusion_idx = 0 # Shark is the first element\n\n# the new feature (has bones) vector\nnew_feature = [-1, 1, 1, 1, -1, -1, -1, -1]\nits_label = 'has_bones'\n\n# adding feature has_bones to the feature array\ntree_features = add_feature(tree_features, new_feature)\n\n# plotting\nplot_tree_data(item_names, tree_features, its_label)\n```\n\nYou can see the new feature shown in the last column of the plot above.\n\nNow we can train the network on the new data, and record the network prediction (output) for Shark (indexed 0) label and \"has bone\" feature (last feature, indexed -1), during the training.\n\nHere is the snippet from the training loop that keeps track of network prediction for `illusory_i`th label and last (`-1`) feature:\n\n```python\npred_ij = predictions.detach()[illusory_i, -1]\n```\n\n\n```python\n#@markdown #### Make sure you execute this cell to train the network and plot\n\n# convert (cast) data from np.ndarray to torch.Tensor\nlabel_tensor = torch.tensor(tree_labels).float()\nfeature_tensor = torch.tensor(tree_features).float()\n\nlr = 100.0 # learning rate\ngamma = 1e-12 # initialization scale\nn_epochs = 250 # number of epochs\ndim_input = 8 # input dimension = `label_tensor.size(1)`\ndim_hidden = 30 # hidden neurons\ndim_output = feature_tensor.size(1)\n\n# model instantiation\ndlnn_model = LNNet(dim_input, dim_hidden, dim_output)\n\n# weights re-initialization\ninitializer_(dlnn_model, gamma)\n\n# training\n_, modes, _, ill_predictions = train(dlnn_model,\n label_tensor,\n feature_tensor,\n n_epochs=n_epochs,\n lr=lr,\n illusory_i=illusion_idx)\n\n# a label for the plot\nill_label = f\"Prediction for {item_names[illusion_idx]} {its_label}\"\n\n# plotting\nplot_ills_sv_twin(ill_predictions, modes, ill_label)\n```\n\nIt seems that the network starts by learning an \"illusory correlation\" that sharks have bones, and in later epochs, as it learns deeper representations, it can see (learn) beyond the illusory correlation. This is important to remember that we never presented the network with any data saying that sharks have bones.\n\n## Exercise 4: Illusory Correlations\n\nThis exercise is just for you to explore the idea of illusory correlations. Think of medical, natural, or possibly social illusory correlations which can test the learning power of deep linear neural nets.\n\n**important notes**: the generated data is independent of tree labels, therefore the names are just for convenience.\n\nHere is our example for **Non-human Living things do not speak**. The lines marked by `{edit}` are for you to change in your example.\n\n\n\n```python\n# sampling new data from the tree\ntree_labels, tree_features = generate_hsd()\n\n# {edit} replacing Canary with Parrot\nitem_names = ['Goldfish', 'Tuna', 'Robin', 'Parrot',\n 'Rose', 'Daisy', 'Pine', 'Oak']\n\n# {edit} index of label to record\nillusion_idx = 3 # Parrot is the fourth element\n\n# {edit} the new feature (cannot speak) vector\nnew_feature = [1, 1, 1, -1, 1, 1, 1, 1]\nits_label = 'cannot_speak'\n\n# adding feature has_bones to the feature array\ntree_features = add_feature(tree_features, new_feature)\n\n# plotting\nplot_tree_data(item_names, tree_features, its_label)\n```\n\n\n```python\n# @markdown #### Make sure you execute this cell to train the network and plot\n\n# convert (cast) data from np.ndarray to torch.Tensor\nlabel_tensor = torch.tensor(tree_labels).float()\nfeature_tensor = torch.tensor(tree_features).float()\n\nlr = 100.0 # learning rate\ngamma = 1e-12 # initialization scale\nn_epochs = 250 # number of epochs\ndim_input = 8 # input dimension = `label_tensor.size(1)`\ndim_hidden = 30 # hidden neurons\ndim_output = feature_tensor.size(1)\n\n# model instantiation\ndlnn_model = LNNet(dim_input, dim_hidden, dim_output)\n\n# weights re-initialization\ninitializer_(dlnn_model, gamma)\n\n# training\n_, modes, _, ill_predictions = train(dlnn_model,\n label_tensor,\n feature_tensor,\n n_epochs=n_epochs,\n lr=lr,\n illusory_i=illusion_idx)\n\n# a label for the plot\nill_label = f\"Prediction for {item_names[illusion_idx]} {its_label}\"\n\n# plotting\nplot_ills_sv_twin(ill_predictions, modes, ill_label)\n```\n\n\n```python\n# @title Video 7: Illusory Correlations - Discussion\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1vv411E7rg\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"6VLHKQjQJmI\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 7: Illusory Correlations')\n\ndisplay(out)\n```\n\n Video available at https://www.bilibili.com/video/BV1vv411E7rg\n\n\n\n\n\n\n\n\n Video available at https://youtube.com/watch?v=6VLHKQjQJmI\n\n\n\n\n\n\n\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n---\n# Summary\n\nThe second day of the course has ended. So, in the third tutorial of the linear deep learning day we have learned more advanced topics. In the beginning we implemented a deep linear neural network and then we studied its learning dynamics using the linear algebra tool called singular value decomposition. Then, we learned about the representational similarity analysis and the illusory correlation.\n\n\n```python\n# @title Video 8: Outro\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1AL411n7ns\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"N2szOIsKyXE\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 8: Outro')\n\ndisplay(out)\n```\n\n Video available at https://www.bilibili.com/video/BV1AL411n7ns\n\n\n\n\n\n\n\n\n Video available at https://youtube.com/watch?v=N2szOIsKyXE\n\n\n\n\n\n\n\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n\n```python\n# @title Airtable Submission Link\nfrom IPython import display as IPydisplay\nIPydisplay.HTML(\n f\"\"\"\n
    \n \n \n
    \"\"\" )\n```\n\n\n\n\n\n
    \n \n \n
    \n\n\n\n---\n# Bonus\n\n*Time estimate: ~20-30 mins*\n\n\n```python\n# @title Video 9: Linear Regression\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1Pf4y1L71L\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"uULOAbhYaaE\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n\n Video available at https://www.bilibili.com/video/BV1Pf4y1L71L\n\n\n\n\n\n\n\n\n Video available at https://youtube.com/watch?v=uULOAbhYaaE\n\n\n\n\n\n\n\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n## Section 5.1: Linear Regression\n\nGenerally, *regression* refers to a set of methods for modeling the mapping (relationship) between one (or more) independent variable(s) (i.e., features) and one (or more) dependent variable(s) (i.e., labels). For example, if we want to examine the relative impacts of calendar date, GPS coordinates, and time of the say (the independent variables) on air temperature (the dependent variable). On the other hand, regression can be used for predictive analysis. Thus the independent variables are also called predictors. When the model contains more than one predictor, then the method is called *multiple regression*, and if it contains more than one dependent variable called *multivariate regression*. Regression problems pop up whenever we want to predict a numerical (usually continuous) value.\n\nThe independent variables are collected in vector $\\mathbf{x} \\in \\mathbb{R}^M$, where $M$ denotes the number of independent variables, while the dependent variables are collected in vector $\\mathbf{y} \\in \\mathbb{R}^N$, where $N$ denotes the number of dependent variables. And the mapping between them is represented by the weight matrix $\\mathbf{W} \\in \\mathbb{R}^{N \\times M}$ and a bias vector $\\mathbf{b} \\in \\mathbb{R}^{N}$ (generalizing to affine mappings).\n\nThe multivariate regression model can be written as:\n\n\\begin{equation}\n\\mathbf{y} = \\mathbf{W} ~ \\mathbf{x} + \\mathbf{b}\n\\end{equation}\n\nor it can be written in matrix format as:\n\n\\begin{equation}\n\\begin{bmatrix} y_{1} \\\\ y_{2} \\\\ \\vdots \\\\ y_{N} \\\\ \\end{bmatrix} = \\begin{bmatrix} w_{1,1} & w_{1,2} & \\dots & w_{1,M} \\\\ w_{2,1} & w_{2,2} & \\dots & w_{2,M} \\\\ \\vdots & \\ddots & \\ddots & \\vdots \\\\ w_{N,1} & w_{N,2} & \\dots & w_{N,M} \\end{bmatrix} \\begin{bmatrix} x_{1} \\\\ x_{2} \\\\ \\vdots \\\\ x_{M} \\\\ \\end{bmatrix} + \\begin{bmatrix} b_{1} \\\\ b_{2} \\\\ \\vdots \\\\b_{N} \\\\ \\end{bmatrix}\n\\end{equation}\n\n\n## Section 5.2: Vectorized regression\n\nLinear regression can be simply extended to multi-samples ($D$) input-output mapping, which we can collect in a matrix $\\mathbf{X} \\in \\mathbb{R}^{M \\times D}$, sometimes called the design matrix. The sample dimension also shows up in the output matrix $\\mathbf{Y} \\in \\mathbb{R}^{N \\times D}$. Thus, linear regression takes the following form:\n\n\\begin{equation}\n\\mathbf{Y} = \\mathbf{W} ~ \\mathbf{X} + \\mathbf{b}\n\\end{equation}\n\nwhere matrix $\\mathbf{W} \\in \\mathbb{R}^{N \\times M}$ and the vector $\\mathbf{b} \\in \\mathbb{R}^{N}$ (broadcasted over sample dimension) are the desired parameters to find.\n\n## Section 5.3: Analytical Linear Regression\n\nLinear regression is a relatively simple optimization problem. Unlike most other models that we will see in this course, linear regression for mean squared loss can be solved analytically.\n\nFor $D$ samples (batch size), $\\mathbf{X} \\in \\mathbb{R}^{M \\times D}$, and $\\mathbf{Y} \\in \\mathbb{R}^{N \\times D}$, the goal of linear regression is to find $\\mathbf{W} \\in \\mathbb{R}^{N \\times M}$ such that:\n\n\\begin{equation}\n\\mathbf{Y} = \\mathbf{W} ~ \\mathbf{X}\n\\end{equation}\n\nGiven the Squared Error loss function, we have:\n\n\\begin{equation}\nLoss(\\mathbf{W}) = ||\\mathbf{Y} - \\mathbf{W} ~ \\mathbf{X}||^2\n\\end{equation}\n\nSo, using matrix notation, the optimization problem is given by:\n\n\\begin{align}\n\\mathbf{W^{*}} &= \\underset{\\mathbf{W}}{\\mathrm{argmin}} \\left( Loss (\\mathbf{W}) \\right) \\\\\n &= \\underset{\\mathbf{W}}{\\mathrm{argmin}} \\left( ||\\mathbf{Y} - \\mathbf{W} ~ \\mathbf{X}||^2 \\right) \\\\\n&= \\underset{\\mathbf{W}}{\\mathrm{argmin}} \\left( \\left( \\mathbf{Y} - \\mathbf{W} ~ \\mathbf{X}\\right)^{\\top} \\left( \\mathbf{Y} - \\mathbf{W} ~ \\mathbf{X}\\right) \\right)\n\\end{align}\n\nTo solve the minimization problem, we can simply set the derivative of the loss with respect to $\\mathbf{W}$ to zero.\n\n\\begin{equation}\n\\dfrac{\\partial Loss}{\\partial \\mathbf{W}} = 0\n\\end{equation}\n\nAssuming that $\\mathbf{X}\\mathbf{X}^{\\top}$ is full-rank, and thus it is invertible we can write:\n\n\\begin{equation}\n\\mathbf{W}^{\\mathbf{*}} = \\mathbf{Y} \\mathbf{X}^{\\top} \\left( \\mathbf{X} \\mathbf{X}^{\\top} \\right) ^{-1}\n\\end{equation}\n\n\n\n### Coding Exercise 5.3.1: Analytical solution to LR\n\nComplete the function `linear_regression` for finding the analytical solution to linear regression.\n\n\n\n```python\ndef linear_regression(X, Y):\n \"\"\"Analytical Linear regression\n\n Args:\n X (np.ndarray): design matrix\n Y (np.ndarray): target ouputs\n\n return:\n np.ndarray: estimated weights (mapping)\n \"\"\"\n assert isinstance(X, np.ndarray)\n assert isinstance(Y, np.ndarray)\n M, Dx = X.shape\n N, Dy = Y.shape\n assert Dx == Dy\n #################################################\n ## Complete the linear_regression_exercise function\n # Complete the function and remove or comment the line below\n raise NotImplementedError(\"Linear Regression `linear_regression`\")\n #################################################\n W = ...\n\n return W\n\n\nW_true = np.random.randint(low=0, high=10, size=(3, 3)).astype(float)\n\nX_train = np.random.rand(3, 37) # 37 samples\nnoise = np.random.normal(scale=0.01, size=(3, 37))\nY_train = W_true @ X_train + noise\n\n## Uncomment and run\n# W_estimate = linear_regression(X_train, Y_train)\n# print(f\"True weights:\\n {W_true}\")\n# print(f\"\\nEstimated weights:\\n {np.round(W_estimate, 1)}\")\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content-dl/tree/main//tutorials/W1D2_LinearDeepLearning/solutions/W1D2_Tutorial3_Solution_55ea556e.py)\n\n\n\n## Demonstration: Linear Regression vs. DLNN\n\nA linear neural network with NO hidden layer is very similar to linear regression in its core. We also know that no matter how many hidden layers a linear network has, it can be compressed to linear regression (no hidden layers).\n\nIn this demonstration, we use the hierarchically structured data to:\n\n* analytically find the mapping between features and labels\n* train a zero-depth LNN to find the mapping \n* compare them to the $W_{tot}$ from the already trained deep LNN\n\n\n```python\n# sampling new data from the tree\ntree_labels, tree_features = generate_hsd()\n\n# convert (cast) data from np.ndarray to torch.Tensor\nlabel_tensor = torch.tensor(tree_labels).float()\nfeature_tensor = torch.tensor(tree_features).float()\n```\n\n\n```python\n# calculating the W_tot for deep network (already trained model)\n\nlr = 100.0 # learning rate\ngamma = 1e-12 # initialization scale\nn_epochs = 250 # number of epochs\ndim_input = 8 # input dimension = `label_tensor.size(1)`\ndim_hidden = 30 # hidden neurons\ndim_output = 10000 # output dimension = `feature_tensor.size(1)`\n\n# model instantiation\ndlnn_model = LNNet(dim_input, dim_hidden, dim_output)\n\n# weights re-initialization\ninitializer_(dlnn_model, gamma)\n\n# training\nlosses, modes, rsms, ills = train(dlnn_model,\n label_tensor,\n feature_tensor,\n n_epochs=n_epochs,\n lr=lr)\n\ndeep_W_tot = torch.eye(dim_input)\nfor weight in dlnn_model.parameters():\n deep_W_tot = weight @ deep_W_tot\ndeep_W_tot = deep_W_tot.detach().numpy()\n```\n\n\n```python\n# analytically estimation of weights\n# our data is batch first dimension, so we need to transpose our data\nanalytical_weights = linear_regression(tree_labels.T, tree_features.T)\n```\n\n\n```python\nclass LRNet(nn.Module):\n \"\"\"A Linear Neural Net with ZERO hidden layer (LR net)\n \"\"\"\n\n def __init__(self, in_dim, out_dim):\n \"\"\"\n Args:\n in_dim (int): input dimension\n hid_dim (int): hidden dimension\n \"\"\"\n super().__init__()\n self.in_out = nn.Linear(in_dim, out_dim, bias=False)\n\n def forward(self, x):\n \"\"\"\n Args:\n x (torch.Tensor): input tensor\n \"\"\"\n out = self.in_out(x) # output (prediction)\n return out\n```\n\n\n```python\nlr = 1000.0 # learning rate\ngamma = 1e-12 # initialization scale\nn_epochs = 250 # number of epochs\ndim_input = 8 # input dimension = `label_tensor.size(1)`\ndim_output = 10000 # output dimension = `feature_tensor.size(1)`\n\n# model instantiation\nLR_model = LRNet(dim_input, dim_output)\noptimizer = optim.SGD(LR_model.parameters(), lr=lr)\ncriterion = nn.MSELoss()\n\nlosses = np.zeros(n_epochs) # loss records\nfor i in range(n_epochs): # training loop\n optimizer.zero_grad()\n predictions = LR_model(label_tensor)\n loss = criterion(predictions, feature_tensor)\n loss.backward()\n optimizer.step()\n losses[i] = loss.item()\n\n# trained weights from zero_depth_model\nLR_model_weights = next(iter(LR_model.parameters())).detach().numpy()\n\nplot_loss(losses, \"Training loss for zero depth LNN\", c=\"r\")\n```\n\n\n```python\nprint(\"The final weights from all methods are approximately equal?! \"\n\"{}!\".format(\n (np.allclose(analytical_weights, LR_model_weights, atol=1e-02) and \\\n np.allclose(analytical_weights, deep_W_tot, atol=1e-02))\n )\n)\n```\n\nAs you may have guessed, they all arrive at the same results but through very different paths.\n\n\n```python\n# @title Video 10: Linear Regression - Discussion\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV18v411E7Wg\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"gG15_J0i05Y\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n", "meta": {"hexsha": "1106833cad680c0eab686eeb3f8a8c8fcfe9cdfc", "size": 954224, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W1D2_LinearDeepLearning/student/ED_W1D2_Tutorial3.ipynb", "max_stars_repo_name": "eduardojdiniz/course-content-dl", "max_stars_repo_head_hexsha": "8d66641683651bce7b0179b6d890aef5a048a8b9", "max_stars_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tutorials/W1D2_LinearDeepLearning/student/ED_W1D2_Tutorial3.ipynb", "max_issues_repo_name": "eduardojdiniz/course-content-dl", "max_issues_repo_head_hexsha": "8d66641683651bce7b0179b6d890aef5a048a8b9", "max_issues_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/W1D2_LinearDeepLearning/student/ED_W1D2_Tutorial3.ipynb", "max_forks_repo_name": "eduardojdiniz/course-content-dl", "max_forks_repo_head_hexsha": "8d66641683651bce7b0179b6d890aef5a048a8b9", "max_forks_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 291.9902080783, "max_line_length": 120136, "alphanum_fraction": 0.9162439846, "converted": true, "num_tokens": 18298, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660543, "lm_q2_score": 0.6723317057447908, "lm_q1q2_score": 0.4379265272060957}} {"text": "   \n\n# Tutorial 1: Variational Autoencoders (VAEs)\n\n**Week 2, Day 5: Generative Models**\n\n**By Neuromatch Academy**\n\n__Content creators:__ Saeed Salehi, Spiros Chavlis, Vikash Gilja\n\n__Content reviewers:__ Diptodip Deb, Kelson Shilling-Scrivo\n\n__Content editor:__ Charles J Edelson, Spiros Chavlis \n\n__Production editors:__ Saeed Salehi, Spiros Chavlis \n\n*Inspired from UPenn course*:\n__Instructor:__ Konrad Kording, __Original Content creators:__ Richard Lange, Arash Ash\n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n

    \n\n---\n# Tutorial Objectives\nIn the first tutorial of the *Generative Models* day, we are going to\n\n- Think about unsupervised learning / Generative Models and get a bird's eye view of why it is useful\n- Build intuition about latent variables\n- See the connection between AutoEncoders and PCA\n- Start thinking about neural networks as generative models by contrasting AutoEncoders and Variational AutoEncoders\n\n\n```python\n# @title Tutorial slides\n\n# @markdown These are the slides for the videos in this tutorial\n\n# @markdown If you want to locally download the slides, click [here](https://osf.io/rd7ng/download)\nfrom IPython.display import IFrame\nIFrame(src=f\"https://mfr.ca-1.osf.io/render?url=https://osf.io/rd7ng/?direct%26mode=render%26action=download%26mode=render\", width=854, height=480)\n```\n\n---\n# Setup\n\n\n```python\n# @title Install dependencies\n!pip install torchvision --quiet\n\n# @markdown Install *Huggingface BigGAN* library\n\n# @markdown Please ignore `Errors`/`Warnings` during installation.\n!pip install pytorch-pretrained-biggan --quiet\n!pip install Pillow libsixel-python --quiet\n\n!pip install git+https://github.com/NeuromatchAcademy/evaltools --quiet\nfrom evaltools.airtable import AirtableForm\n\n# generate airtable form\natform = AirtableForm('appn7VdPRseSoMXEG','W2D5_T1','https://portal.neuromatchacademy.org/api/redirect/to/9c55f6cb-cdf9-4429-ac1c-ec44fe64c303')\n```\n\n\n```python\n# Imports\nimport torch\nimport random\n\nimport numpy as np\nimport matplotlib.pylab as plt\nfrom sklearn.decomposition import PCA\n\nimport torch.nn as nn\nimport torch.nn.functional as F\nfrom torch.utils.data import DataLoader\n\nimport torchvision\nfrom torchvision import datasets, transforms\n\nfrom pytorch_pretrained_biggan import BigGAN\nfrom pytorch_pretrained_biggan import one_hot_from_names\nfrom pytorch_pretrained_biggan import truncated_noise_sample\n\nfrom tqdm.notebook import tqdm, trange\n```\n\n\n```python\n# @title Figure settings\nimport ipywidgets as widgets # interactive display\nfrom ipywidgets import interact, IntSlider, FloatSlider, interact_manual, fixed\nfrom ipywidgets import FloatLogSlider, HBox, Layout, VBox, interactive, Label\nfrom ipywidgets import interactive_output, Dropdown\n\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/content-creation/main/nma.mplstyle\")\n```\n\n\n```python\n# @title Helper functions\n\n#@title Helper functions\n\ndef image_moments(image_batches, n_batches=None):\n \"\"\"\n Compute mean an covariance of all pixels from batches of images\n \"\"\"\n m1, m2 = torch.zeros((), device=DEVICE), torch.zeros((), device=DEVICE)\n n = 0\n for im in tqdm(image_batches, total=n_batches, leave=False,\n desc='Computing pixel mean and covariance...'):\n im = im.to(DEVICE)\n b = im.size()[0]\n im = im.view(b, -1)\n m1 = m1 + im.sum(dim=0)\n m2 = m2 + (im.view(b,-1,1) * im.view(b,1,-1)).sum(dim=0)\n n += b\n m1, m2 = m1/n, m2/n\n cov = m2 - m1.view(-1,1)*m1.view(1,-1)\n return m1.cpu(), cov.cpu()\n\n\ndef interpolate(A, B, num_interps):\n if A.shape != B.shape:\n raise ValueError('A and B must have the same shape to interpolate.')\n alphas = np.linspace(0, 1, num_interps)\n return np.array([(1-a)*A + a*B for a in alphas])\n\n\ndef kl_q_p(zs, phi):\n \"\"\"Given [b,n,k] samples of z drawn from q, compute estimate of KL(q||p).\n phi must be size [b,k+1]\n\n This uses mu_p = 0 and sigma_p = 1, which simplifies the log(p(zs)) term to\n just -1/2*(zs**2)\n \"\"\"\n b, n, k = zs.size()\n mu_q, log_sig_q = phi[:,:-1], phi[:,-1]\n log_p = -0.5*(zs**2)\n log_q = -0.5*(zs - mu_q.view(b,1,k))**2 / log_sig_q.exp().view(b,1,1)**2 - log_sig_q.view(b,1,-1)\n # Size of log_q and log_p is [b,n,k]. Sum along [k] but mean along [b,n]\n return (log_q - log_p).sum(dim=2).mean(dim=(0,1))\n\n\ndef log_p_x(x, mu_xs, sig_x):\n \"\"\"Given [batch, ...] input x and [batch, n, ...] reconstructions, compute\n pixel-wise log Gaussian probability\n\n Sum over pixel dimensions, but mean over batch and samples.\n \"\"\"\n b, n = mu_xs.size()[:2]\n # Flatten out pixels and add a singleton dimension [1] so that x will be\n # implicitly expanded when combined with mu_xs\n x = x.reshape(b, 1, -1)\n _, _, p = x.size()\n squared_error = (x - mu_xs.view(b, n, -1))**2 / (2*sig_x**2)\n\n # Size of squared_error is [b,n,p]. log prob is by definition sum over [p].\n # Expected value requires mean over [n]. Handling different size batches\n # requires mean over [b].\n return -(squared_error + torch.log(sig_x)).sum(dim=2).mean(dim=(0,1))\n\n\ndef pca_encoder_decoder(mu, cov, k):\n \"\"\"\n Compute encoder and decoder matrices for PCA dimensionality reduction\n \"\"\"\n mu = mu.view(1,-1)\n u, s, v = torch.svd_lowrank(cov, q=k)\n W_encode = v / torch.sqrt(s)\n W_decode = u * torch.sqrt(s)\n\n def pca_encode(x):\n # Encoder: subtract mean image and project onto top K eigenvectors of\n # the data covariance\n return (x.view(-1,mu.numel()) - mu) @ W_encode\n\n def pca_decode(h):\n # Decoder: un-project then add back in the mean\n return (h @ W_decode.T) + mu\n\n return pca_encode, pca_decode\n\n\ndef cout(x, layer):\n \"\"\"Unnecessarily complicated but complete way to\n calculate the output depth, height and width size for a Conv2D layer\n\n Args:\n x (tuple): input size (depth, height, width)\n layer (nn.Conv2d): the Conv2D layer\n\n returns:\n (int): output shape as given in [Ref]\n\n Ref:\n https://pytorch.org/docs/stable/generated/torch.nn.Conv2d.html\n \"\"\"\n assert isinstance(layer, nn.Conv2d)\n p = layer.padding if isinstance(layer.padding, tuple) else (layer.padding,)\n k = layer.kernel_size if isinstance(layer.kernel_size, tuple) else (layer.kernel_size,)\n d = layer.dilation if isinstance(layer.dilation, tuple) else (layer.dilation,)\n s = layer.stride if isinstance(layer.stride, tuple) else (layer.stride,)\n in_depth, in_height, in_width = x\n out_depth = layer.out_channels\n out_height = 1 + (in_height + 2 * p[0] - (k[0] - 1) * d[0] - 1) // s[0]\n out_width = 1 + (in_width + 2 * p[-1] - (k[-1] - 1) * d[-1] - 1) // s[-1]\n return (out_depth, out_height, out_width)\n```\n\n\n```python\n# @title Plotting functions\n\ndef plot_gen_samples_ppca(therm1, therm2, therm_data_sim):\n plt.plot(therm1, therm2, '.', c='c', label='training data')\n plt.plot(therm_data_sim[0], therm_data_sim[1], '.', c='m', label='\"generated\" data')\n plt.axis('equal')\n plt.xlabel('Thermometer 1 ($^\\circ$C)')\n plt.ylabel('Thermometer 2 ($^\\circ$C)')\n plt.legend()\n plt.show()\n\n\ndef plot_linear_ae(lin_losses):\n plt.figure()\n plt.plot(lin_losses)\n plt.ylim([0, 2*torch.as_tensor(lin_losses).median()])\n plt.xlabel('Training batch')\n plt.ylabel('MSE Loss')\n plt.show()\n\n\ndef plot_conv_ae(lin_losses, conv_losses):\n plt.figure()\n plt.plot(lin_losses)\n plt.plot(conv_losses)\n plt.legend(['Lin AE', 'Conv AE'])\n plt.xlabel('Training batch')\n plt.ylabel('MSE Loss')\n plt.ylim([0,\n 2*max(torch.as_tensor(conv_losses).median(),\n torch.as_tensor(lin_losses).median())])\n plt.show()\n\n\ndef plot_images(images, h=3, w=3, plt_title=''):\n plt.figure(figsize=(h*2, w*2))\n plt.suptitle(plt_title, y=1.03)\n for i in range(h*w):\n plt.subplot(h, w, i + 1)\n plot_torch_image(images[i])\n plt.axis('off')\n plt.show()\n\n\ndef plot_phi(phi, num=4):\n plt.figure(figsize=(12, 3))\n for i in range(num):\n plt.subplot(1, num, i + 1)\n plt.scatter(zs[i, :, 0], zs[i, :, 1], marker='.')\n th = torch.linspace(0, 6.28318, 100)\n x, y = torch.cos(th), torch.sin(th)\n # Draw 2-sigma contours\n plt.plot(\n 2*x*phi[i, 2].exp().item() + phi[i, 0].item(),\n 2*y*phi[i, 2].exp().item() + phi[i, 1].item()\n )\n plt.xlim(-5, 5)\n plt.ylim(-5, 5)\n plt.grid()\n plt.axis('equal')\n plt.suptitle('If rsample() is correct, then most but not all points should lie in the circles')\n plt.show()\n\n\ndef plot_torch_image(image, ax=None):\n ax = ax if ax is not None else plt.gca()\n c, h, w = image.size()\n if c==1:\n cm = 'gray'\n else:\n cm = None\n\n # Torch images have shape (channels, height, width) but matplotlib expects\n # (height, width, channels) or just (height,width) when grayscale\n im_plt = torch.clip(image.detach().cpu().permute(1,2,0).squeeze(), 0.0, 1.0)\n ax.imshow(im_plt, cmap=cm)\n ax.set_xticks([])\n ax.set_yticks([])\n ax.spines['right'].set_visible(False)\n ax.spines['top'].set_visible(False)\n```\n\n\n```python\n# @title Set random seed\n\n# @markdown Executing `set_seed(seed=seed)` you are setting the seed\n\n# for DL its critical to set the random seed so that students can have a\n# baseline to compare their results to expected results.\n# Read more here: https://pytorch.org/docs/stable/notes/randomness.html\n\n# Call `set_seed` function in the exercises to ensure reproducibility.\nimport random\nimport torch\n\ndef set_seed(seed=None, seed_torch=True):\n if seed is None:\n seed = np.random.choice(2 ** 32)\n random.seed(seed)\n np.random.seed(seed)\n if seed_torch:\n torch.manual_seed(seed)\n torch.cuda.manual_seed_all(seed)\n torch.cuda.manual_seed(seed)\n torch.backends.cudnn.benchmark = False\n torch.backends.cudnn.deterministic = True\n\n print(f'Random seed {seed} has been set.')\n\n\n# In case that `DataLoader` is used\ndef seed_worker(worker_id):\n worker_seed = torch.initial_seed() % 2**32\n np.random.seed(worker_seed)\n random.seed(worker_seed)\n```\n\n\n```python\n# @title Set device (GPU or CPU). Execute `set_device()`\n# especially if torch modules used.\n\n# inform the user if the notebook uses GPU or CPU.\n\ndef set_device():\n device = \"cuda\" if torch.cuda.is_available() else \"cpu\"\n if device != \"cuda\":\n print(\"WARNING: For this notebook to perform best, \"\n \"if possible, in the menu under `Runtime` -> \"\n \"`Change runtime type.` select `GPU` \")\n else:\n print(\"GPU is enabled in this notebook.\")\n\n return device\n```\n\n\n```python\nSEED = 2021\nset_seed(seed=SEED)\nDEVICE = set_device()\n```\n\n\n```python\n# @title Download `wordnet` dataset\n\n# import nltk\n# nltk.download('wordnet')\n\nimport requests, zipfile\n\nfname = 'wordnet.zip'\nurl = 'https://osf.io/ekjxy/download'\nr = requests.get(url, allow_redirects=True)\n\nwith open('wordnet.zip', 'wb') as fd:\n fd.write(r.content)\n\nwith zipfile.ZipFile(fname, 'r') as zip_ref:\n zip_ref.extractall('/root/nltk_data/corpora')\n```\n\n---\n# Section 1: Generative models\n\n*Time estimate: ~15mins*\n\n**Please** run the cell after the video to download BigGAN (a generative model) and a few standard image datasets while the video plays.\n\n\n```python\n# @title Video 1: Generative Modeling\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1Vy4y1j7cN\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"5EEx0sdyR_U\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 1: Generative Modeling')\n\n\ndisplay(out)\n```\n\n\n```python\n# @markdown Download BigGAN (a generative model) and a few standard image datasets\n\n## Initially was downloaded directly\n# biggan_model = BigGAN.from_pretrained('biggan-deep-256')\n\nurl = \"https://osf.io/3yvhw/download\"\nfname = \"biggan_deep_256\"\nr = requests.get(url, allow_redirects=True)\nwith open(fname, 'wb') as fd:\n fd.write(r.content)\n\nbiggan_model = torch.load(fname)\n```\n\n## Section 1.1: Generating Images from BigGAN\n\nTo demonstrate the power of generative models, we are giving you a sneak peek of a fully trained generative model called BigGAN. You\u2019ll see it again (with more background under your belt) later today. For now, let\u2019s just focus on BigGAN as a generative model. Specifically, BigGAN is a class conditional generative model for 128 x 128 images. The classes are based on categorical labels that describe the images and images are generated based upon a vector (z from the video lecture) and the probability that the image comes from a specific discrete category.\n\nFor now, don\u2019t worry about the specifics of the model other than the fact that it generates images based on the vector and the category label.\n\nTo explore the space of generated images, we\u2019ve provided you with a widget that allows you to select a category label and to alter the value of the vector. The vector is a 128-D, which may seem high dimensional, but is much lower-dimensional than a 128 x 128 image! To simplify usability the widget limits the magnitude of the vector and constrains all entries to be equal (so you are only exploring a subset of the possible images that can be generated).\n\n### Interactive Demo 1.1: BigGAN Generator\n\n\n```python\n# @markdown BigGAN Image Generator (the updates may take a few seconds)\n\n# category = 'German shepherd' # @param ['tench', 'magpie', 'jellyfish', 'German shepherd', 'bee', 'acoustic guitar', 'coffee mug', 'minibus', 'monitor']\n# z_magnitude = -16 # @param {type:\"slider\", min:-50, max:50, step:1}\n\ndef sample_from_biggan(category, z_magnitude):\n unit_vector = np.ones((1, 128))/np.sqrt(128)\n z = z_magnitude * unit_vector\n y = one_hot_from_names(category, batch_size=1)\n\n z = torch.from_numpy(z)\n z = z.float()\n y = torch.from_numpy(y)\n\n # Move to GPU\n z = z.to(device=set_device())\n y = y.to(device=set_device())\n biggan_model.to(device=set_device())\n\n with torch.no_grad():\n output = biggan_model(z, y, 1)\n\n # Back to CPU\n output = output.to('cpu')\n\n # The output layer of BigGAN has a tanh layer, resulting the range of [-1, 1] for the output image\n # Therefore, we normalize the images properly to [0, 1] range.\n # Clipping is only in case of numerical instability problems\n\n output = torch.clip(((output.detach().clone() + 1) / 2.0), 0, 1)\n\n plt.imshow(output.squeeze().moveaxis(0,-1))\n plt.axis('off')\n\nz_slider = IntSlider(min=-15, max=15, step=1, value=0,\n continuous_update=False,\n description='Z Magnitude',\n layout=Layout(width='440px'))\n\ncategory_dropdown = Dropdown(\n options=['tench', 'magpie', 'jellyfish', 'German shepherd', 'bee',\n 'acoustic guitar', 'coffee mug', 'minibus', 'monitor'],\n value=\"German shepherd\",\n description=\"Category: \")\n\nwidgets_ui = VBox([category_dropdown, z_slider])\n\nwidgets_out = interactive_output(sample_from_biggan,\n {\n 'z_magnitude': z_slider,\n 'category': category_dropdown\n }\n )\n\ndisplay(widgets_ui, widgets_out)\n```\n\n### Think! 1.1: Generated images\n\nAs you alter the magnitude of the vector, what do you note about the relationship between the different generated images? What do you note about the relationship between the image and the category label as you increase the magnitude of the vector?\n\n\n```python\n# @title Student Response\nfrom ipywidgets import widgets\n\n\ntext=widgets.Textarea(\n value='Type your answer here and click on `Submit!`',\n placeholder='Type something',\n description='',\n disabled=False\n)\n\nbutton = widgets.Button(description=\"Submit!\")\n\ndisplay(text,button)\n\ndef on_button_clicked(b):\n atform.add_answer('q1' , text.value)\n print(\"Submission successful!\")\n\n\nbutton.on_click(on_button_clicked)\n```\n\n\n```python\n# to_remove explanation\n\n\"\"\"\nThis is called the truncation trick, i.e., truncating a z vector by resampling\nthe values with magnitude above a chosen threshold lead to improvement in\nindividual sample quality at the cost of a reduction in overall sample variety.\n\"\"\";\n```\n\n## Section 1.2: Interpolating Images with BigGAN\nThis next widget allows you to interpolate between two generated images. It does this by linearly interpolating between the probability of each category you select and linearly interpolating between the latent vector values.\n\n### Interactive Demo 1.2: BigGAN Interpolation\n\n\n```python\n# @markdown BigGAN Interpolation Widget (the updates may take a few seconds)\n\ndef interpolate_biggan(category_A, z_magnitude_A, category_B, z_magnitude_B):\n num_interps = 16\n\n # category_A = 'jellyfish' #@param ['tench', 'magpie', 'jellyfish', 'German shepherd', 'bee', 'acoustic guitar', 'coffee mug', 'minibus', 'monitor']\n # z_magnitude_A = 0 #@param {type:\"slider\", min:-10, max:10, step:1}\n\n # category_B = 'German shepherd' #@param ['tench', 'magpie', 'jellyfish', 'German shepherd', 'bee', 'acoustic guitar', 'coffee mug', 'minibus', 'monitor']\n # z_magnitude_B = 0 #@param {type:\"slider\", min:-10, max:10, step:1}\n\n\n def interpolate_and_shape(A, B, num_interps):\n interps = interpolate(A, B, num_interps)\n return (interps.transpose(1, 0, *range(2, len(interps.shape)))\n .reshape(num_interps, *interps.shape[2:]))\n\n unit_vector = np.ones((1, 128))/np.sqrt(128)\n z_A = z_magnitude_A * unit_vector\n z_B = z_magnitude_B * unit_vector\n y_A = one_hot_from_names(category_A, batch_size=1)\n y_B = one_hot_from_names(category_B, batch_size=1)\n\n z_interp = interpolate_and_shape(z_A, z_B, num_interps)\n y_interp = interpolate_and_shape(y_A, y_B, num_interps)\n\n # Convert to tensor\n z_interp = torch.from_numpy(z_interp)\n z_interp = z_interp.float()\n y_interp = torch.from_numpy(y_interp)\n\n # Move to GPU\n z_interp = z_interp.to(DEVICE)\n y_interp = y_interp.to(DEVICE)\n biggan_model.to(DEVICE)\n\n with torch.no_grad():\n output = biggan_model(z_interp, y_interp, 1)\n\n # Back to CPU\n output = output.to('cpu')\n\n # The output layer of BigGAN has a tanh layer, resulting the range of [-1, 1] for the output image\n # Therefore, we normalize the images properly to [0, 1] range.\n # Clipping is only in case of numerical instability problems\n\n output = torch.clip(((output.detach().clone() + 1) / 2.0), 0, 1)\n output = output\n\n # Make grid and show generated samples\n output_grid = torchvision.utils.make_grid(output,\n nrow=min(4, output.shape[0]),\n padding=5)\n plt.axis('off');\n plt.imshow(output_grid.permute(1, 2, 0))\n plt.show()\n\n\nz_A_slider = IntSlider(min=-10, max=10, step=1, value=0,\n continuous_update=False, description='Z Magnitude A',\n layout=Layout(width='440px'), style={'description_width': 'initial'})\n\nz_B_slider = IntSlider(min=-10, max=10, step=1, value=0,\n continuous_update=False, description='Z Magntude B',\n layout=Layout(width='440px'), style={'description_width': 'initial'})\n\ncategory_A_dropdown = Dropdown(\n options=['tench', 'magpie', 'jellyfish', 'German shepherd', 'bee',\n 'acoustic guitar', 'coffee mug', 'minibus', 'monitor'],\n value=\"German shepherd\", description=\"Category A: \")\n\ncategory_B_dropdown = Dropdown(\n options=['tench', 'magpie', 'jellyfish', 'German shepherd', 'bee',\n 'acoustic guitar', 'coffee mug', 'minibus', 'monitor'],\n value=\"jellyfish\", description=\"Category B: \")\n\n\n\nwidgets_ui = VBox([HBox([category_A_dropdown, z_A_slider]),\n HBox([category_B_dropdown, z_B_slider])])\n\nwidgets_out = interactive_output(interpolate_biggan,\n {'category_A': category_A_dropdown,\n 'z_magnitude_A': z_A_slider,\n 'category_B': category_B_dropdown,\n 'z_magnitude_B': z_B_slider})\n\ndisplay(widgets_ui, widgets_out)\n```\n\n### Think! 1.2: Interpolating samples from the same category\n\nTry interpolating between samples from the same category, samples from similar categories, and samples from very different categories. Do you notice any trends? What does this suggest about the representations of images in the latent space?\n\n\n```python\n# @title Student Response\nfrom ipywidgets import widgets\n\n\ntext=widgets.Textarea(\n value='Type your answer here and click on `Submit!`',\n placeholder='Type something',\n description='',\n disabled=False\n)\n\nbutton = widgets.Button(description=\"Submit!\")\n\ndisplay(text,button)\n\ndef on_button_clicked(b):\n atform.add_answer('q2' , text.value)\n print(\"Submission successful!\")\n\n\nbutton.on_click(on_button_clicked)\n```\n\n\n```python\n# to_remove explanation\n\n\"\"\"\nOur model convincingly interpolates between disparate samples and the nearest\nneighbors for its samples are visually distinct, suggesting that our model\ndoes not simply memorize training data.\n\"\"\";\n```\n\n---\n# Section 2: Latent Variable Models\n\n*Time estimate: ~15mins* excluding the Bonus\n\n\n```python\n# @title Video 2: Latent Variable Models\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1Db4y167Ys\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"_e0nKUeBDFo\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 2: Latent Variable Models')\n\ndisplay(out)\n```\n\nIn the video, the concept of a latent variable model was introduced. We saw how PCA (principal component analysis) can be extended into a generative model with latent variables called pPCA. For pPCA the latent variables (z in the video) are the projections onto the principal component axes. \n\nThe dimensionality of the principal components is typically set to be substantially lower-dimensional than the original data. Thus, the latent variables (the projection onto the principal component axes) are a lower-dimensional representation of the original data (dimensionality reduction!). With pPCA we can estimate the original distribution of the high dimensional data. This allows us to generate data with a distribution that \u201clooks\u201d more like the original data than if we were to only use PCA to generate data from the latent variables. Let\u2019s see how that might look with a simple example. \n\n## Coding Exercise 2: pPCA (Bonus)\n\nAssume we have two noisy thermometers measuring the temperature of the same room. They both make noisy measurements. The room tends to be around 25°C (that's 77°F), but can vary around that temperature. If we take lots of readings from the two thermometers over time and plot the paired readings, we might see something like the plot generated below:\n\n\n```python\n# @markdown Generate example datapoints from the two thermometers\n\ndef generate_data(n_samples, mean_of_temps, cov_of_temps, seed):\n \"\"\"\n Generate random data, normally distributed\n\n Args:\n n_samples : int\n The number of samples to be generated\n mean_of_temps : numpy.ndarray\n 1D array with the mean of temparatures, Kx1\n cov_of_temps : numpy.ndarray\n 2D array with the covariance, , KxK\n seed : int\n Set random seed for the psudo random generator\n Returns:\n therm1 : numpy.ndarray\n therm2 : numpy.ndarray\n \"\"\"\n\n np.random.seed(seed)\n therm1, therm2 = np.random.multivariate_normal(mean_of_temps,\n cov_of_temps,\n n_samples).T\n return therm1, therm2\n\n\nn_samples = 2000\nmean_of_temps = np.array([25, 25])\ncov_of_temps = np.array([[10, 5], [5, 10]])\ntherm1, therm2 = generate_data(n_samples, mean_of_temps, cov_of_temps, seed=SEED)\n\nplt.plot(therm1, therm2, '.')\nplt.axis('equal')\nplt.xlabel('Thermometer 1 ($^\\circ$C)')\nplt.ylabel('Thermometer 2 ($^\\circ$C)')\nplt.show()\n```\n\nLet\u2019s model these data with a single principal component. Given that the thermometers are measuring the same actual temperature, the principal component axes will be the identity line. The direction of this axes can be indicated by the unit vector $[1 ~~ 1]~/~\\sqrt2$. We could estimate this axes by applying PCA. We can plot this axes, it tells us something about the data, but we can\u2019t generate from it:\n\n\n```python\n# @markdown Add first PC axes to the plot\n\nplt.plot(therm1, therm2, '.')\nplt.axis('equal')\nplt.xlabel('Thermometer 1 ($^\\circ$C)')\nplt.ylabel('Thermometer 2 ($^\\circ$C)')\nplt.plot([plt.axis()[0], plt.axis()[1]],\n [plt.axis()[0], plt.axis()[1]])\nplt.show()\n```\n\n**Step 1:** Calculate the parameters of the pPCA model\n\nThis part is completed already, so you don't need to make any edits:\n\n\n```python\n# Project Data onto the principal component axes.\n# We could have \"learned\" this from the data by applying PCA,\n# but we \"know\" the value from the problem definition.\npc_axes = np.array([1.0, 1.0]) / np.sqrt(2.0)\n\n# thermometers data\ntherm_data = np.array([therm1, therm2])\n\n# Zero center the data\ntherm_data_mean = np.mean(therm_data, 1)\ntherm_data_center = np.outer(therm_data_mean, np.ones(therm_data.shape[1]))\ntherm_data_zero_centered = therm_data - therm_data_center\n\n# Calculate the variance of the projection on the PC axes\npc_projection = np.matmul(pc_axes, therm_data_zero_centered);\npc_axes_variance = np.var(pc_projection)\n\n# Calculate the residual variance (variance not accounted for by projection on the PC axes)\nsensor_noise_std = np.mean(np.linalg.norm(therm_data_zero_centered - np.outer(pc_axes, pc_projection), axis=0, ord=2))\nsensor_noise_var = sensor_noise_std **2\n```\n\n**Step 2**: \"Generate\" from the pPCA model of the thermometer data.\n\nComplete the code so it properly generates from the pPCA model. At present we aren't accounting for the \"sensor noise\" the sensor noise is the variance that the PC axes isn't accounting for. Thus, you'll note that the current output sits on the PC axes and doesn't look like the original data distribution!!\n\nHere is the equation for sampling from the pPCA model:\n\n\\begin{equation}\nx = \\mu + W z + \\epsilon, \\,\\text{where}\\,~~ \\epsilon \\sim \\mathcal{N}(0,~\\sigma^2 \\mathbf{I})\n\\end{equation}\n\n\n```python\ndef gen_from_pPCA(noise_var, data_mean, pc_axes, pc_variance):\n \"\"\"\n Args:\n noise_var (np.ndarray): sensor noise variance\n data_mean (np.ndarray): thermometer data mean\n pc_axes (np.ndarray): principal component axes\n pc_variance (np.ndarray): the variance of the projection on the PC axes\n \"\"\"\n # We are matching this value to the thermometer data so the visualizations look similar\n n_samples = 1000\n\n # Randomly sample from z (latent space value)\n z = np.random.normal(0.0, np.sqrt(pc_variance), n_samples)\n\n # sensor noise covariance matrix (\u2211)\n epsilon_cov = [[noise_var, 0.0], [0.0, noise_var]]\n\n # data mean reshaped for the generation\n sim_mean = np.outer(data_mean, np.ones(n_samples))\n\n ####################################################################\n # Fill in all missing code below (...),\n # then remove or comment the line below to test your class\n raise NotImplementedError(\"Please complete the `gen_from_pPCA` function\")\n ####################################################################\n # draw `n_samples` from `np.random.multivariate_normal`\n rand_eps = ...\n rand_eps = rand_eps.T\n\n # generate (simulate, draw) `n_samples` from pPCA model\n therm_data_sim = ...\n\n return therm_data_sim\n\n\n# add event to airtable\natform.add_event('Coding Exercise 2: pPCA')\n\n## Uncomment to test your code\n# therm_data_sim = gen_from_pPCA(sensor_noise_var, therm_data_mean, pc_axes, pc_axes_variance)\n# plot_gen_samples_ppca(therm1, therm2, therm_data_sim)\n```\n\n\n```python\n# to_remove solution\ndef gen_from_pPCA(noise_var, data_mean, pc_axes, pc_variance):\n \"\"\"\n Args:\n noise_var (np.ndarray): sensor noise variance\n data_mean (np.ndarray): thermometer data mean\n pc_axes (np.ndarray): principal component axes\n pc_variance (np.ndarray): the variance of the projection on the PC axes\n \"\"\"\n # We are matching this value to the thermometer data so the visualizations look similar\n n_samples = 1000\n\n # Randomly sample from z (latent space value)\n z = np.random.normal(0.0, np.sqrt(pc_variance), n_samples)\n\n # sensor noise covariance matrix (\u2211)\n epsilon_cov = [[noise_var, 0.0], [0.0, noise_var]]\n\n # data mean reshaped for the generation\n sim_mean = np.outer(data_mean, np.ones(n_samples))\n\n # draw `n_samples` from `np.random.multivariate_normal`\n rand_eps = np.random.multivariate_normal([0.0, 0.0], epsilon_cov, n_samples)\n rand_eps = rand_eps.T\n\n # generate (simulate, draw) `n_samples` from pPCA model\n therm_data_sim = sim_mean + np.outer(pc_axes, z) + rand_eps\n\n return therm_data_sim\n\n\n# add event to airtable\natform.add_event('Coding Exercise 2: pPCA')\n\n## Uncomment to test your code\ntherm_data_sim = gen_from_pPCA(sensor_noise_var, therm_data_mean, pc_axes, pc_axes_variance)\nwith plt.xkcd():\n plot_gen_samples_ppca(therm1, therm2, therm_data_sim)\n```\n\n---\n# Section 3: Autoencoders\n\n*Time estimate: ~30mins*\n\n**Please** run the cell after the video to download MNIST and CIFAR10 image datasets while the video plays.\n\n\n```python\n# @title Video 3: Autoenconders\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV16b4y167Z2\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"MlyIL1PmDCA\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 3: Autoenconders')\n\ndisplay(out)\n```\n\n\n```python\n# @markdown Download MNIST and CIFAR10 datasets\nimport tarfile, requests, os\n\nfname = 'MNIST.tar.gz'\nname = 'mnist'\nurl = 'https://osf.io/y2fj6/download'\n\nif not os.path.exists(name):\n print('\\nDownloading MNIST dataset...')\n r = requests.get(url, allow_redirects=True)\n with open(fname, 'wb') as fh:\n fh.write(r.content)\n print('\\nDownloading MNIST completed..\\n')\n\nif not os.path.exists(name):\n with tarfile.open(fname) as tar:\n tar.extractall(name)\n os.remove(fname)\nelse:\n print('MNIST dataset has been dowloaded.\\n')\n\n\nfname = 'cifar-10-python.tar.gz'\nname = 'cifar10'\nurl = 'https://osf.io/jbpme/download'\n\nif not os.path.exists(name):\n print('\\nDownloading CIFAR10 dataset...')\n r = requests.get(url, allow_redirects=True)\n with open(fname, 'wb') as fh:\n fh.write(r.content)\n print('\\nDownloading CIFAR10 completed.')\n\nif not os.path.exists(name):\n with tarfile.open(fname) as tar:\n tar.extractall(name)\n os.remove(fname)\nelse:\n print('CIFAR10 dataset has been dowloaded.')\n```\n\n\n```python\n# @markdown Load MNIST and CIFAR10 image datasets\n# See https://pytorch.org/docs/stable/torchvision/datasets.html\n\n# MNIST\nmnist = datasets.MNIST('./mnist/',\n train=True,\n transform=transforms.ToTensor(),\n download=False)\nmnist_val = datasets.MNIST('./mnist/',\n train=False,\n transform=transforms.ToTensor(),\n download=False)\n\n# CIFAR 10\ncifar10 = datasets.CIFAR10('./cifar10/',\n train=True,\n transform=transforms.ToTensor(),\n download=False)\ncifar10_val = datasets.CIFAR10('./cifar10/',\n train=False,\n transform=transforms.ToTensor(),\n download=False)\n```\n\n### Select a dataset\n\nWe've built today's tutorial to be flexible. It should work more-or-less out of the box with both MNIST and CIFAR (and other image datasets). MNIST is in many ways simpler, and the results will likely look better and run a bit faster if using MNIST. But we are leaving it up to you to pick which one you want to experiment with!\n\nWe encourage pods to coordinate so that some members use MNIST and others use CIFAR10. Keep in mind that the CIFAR dataset may require more learning epochs (longer training required).\n\n\n```python\ndef get_data(name='mnist'):\n if name == 'mnist':\n my_dataset_name = \"MNIST\"\n my_dataset = mnist\n my_valset = mnist_val\n my_dataset_shape = (1, 28, 28)\n my_dataset_size = 28 * 28\n elif name == 'cifar10':\n my_dataset_name = \"CIFAR10\"\n my_dataset = cifar10\n my_valset = cifar10_val\n my_dataset_shape = (3, 32, 32)\n my_dataset_size = 3 * 32 * 32\n\n return my_dataset, my_dataset_name, my_dataset_shape, my_dataset_size, my_valset\n\n\ntrain_set, dataset_name, data_shape, data_size, valid_set = get_data(name='mnist')\n```\n\n## Section 3.1: Conceptual introduction to AutoEncoders\n\nNow we'll create our first autoencoder. It will reduce images down to $K$ dimensions. The architecture will be quite simple: the input will be linearly mapped to a single hidden (or latent) layer $\\mathbf{h}$ with $K$ units, which will then be linearly mapped back to an output that is the same size as the input:\n\n\\begin{equation}\n\\mathbf{x} \\longrightarrow \\mathbf{h} \\longrightarrow \\mathbf{x'}\n\\end{equation}\n\nThe loss function we'll use will simply be mean squared error (MSE) quantifying how well the reconstruction ($\\mathbf{x'}$) matches the original image ($\\mathbf{x}$):\n\n\\begin{equation}\n\\text{MSE Loss} = \\sum_{i=1}^{N} ||\\mathbf{x}_i - \\mathbf{x'}_i||^2_2\n\\end{equation}\n\nIf all goes well, then the AutoEncoder will learn, **end to end**, a good \"encoding\" or \"compression\" of inputs to a latent representation ($\\mathbf{x \\longrightarrow h}$) as well as a good \"decoding\" of that latent representation to a reconstruction of the original input ($\\mathbf{h \\longrightarrow x'}$).\n\nThe first choice to make is the dimensionality of $\\mathbf{h}$. We'll see more on this below, but For MNIST, 5 to 20 is plenty. For CIFAR, we need more like 50 to 100 dimensions.\n\nCoordinate with your pod to try a variety of values for $K$ in each dataset so you can compare results.\n\n### Coding Exercise 3.1: Linear AutoEncoder Architecture\n\nComplete the missing parts of the `LinearAutoEncoder` class.\n\nThe `LinearAutoEncoder` as two stages: an `encoder` which linearly maps from inputs of size `x_dim = my_dataset_dim` to a hidden layer of size `h_dim = K` (with no nonlinearity), and a `decoder` which maps back from `K` up to the number of pixels in each image.\n\n\n```python\n# @markdown #### Run to define the `train_autoencoder` function.\n# @markdown Feel free to inspect the training function if the time allows.\n\n# @markdown `train_autoencoder(autoencoder, dataset, device, epochs=20, batch_size=250, seed=0)`\n\n\ndef train_autoencoder(autoencoder, dataset, device, epochs=20, batch_size=250,\n seed=0):\n autoencoder.to(device)\n optim = torch.optim.Adam(autoencoder.parameters(),\n lr=1e-3,\n weight_decay=1e-5)\n loss_fn = nn.MSELoss()\n g_seed = torch.Generator()\n g_seed.manual_seed(seed)\n loader = DataLoader(dataset,\n batch_size=batch_size,\n shuffle=True,\n pin_memory=True,\n num_workers=2,\n worker_init_fn=seed_worker,\n generator=g_seed)\n\n mse_loss = torch.zeros(epochs * len(dataset) // batch_size, device=device)\n i = 0\n for epoch in trange(epochs, desc='Epoch'):\n for im_batch, _ in loader:\n im_batch = im_batch.to(device)\n optim.zero_grad()\n reconstruction = autoencoder(im_batch)\n # write the loss calculation\n loss = loss_fn(reconstruction.view(batch_size, -1),\n target=im_batch.view(batch_size, -1))\n loss.backward()\n optim.step()\n\n mse_loss[i] = loss.detach()\n i += 1\n # After training completes, make sure the model is on CPU so we can easily\n # do more visualizations and demos.\n autoencoder.to('cpu')\n return mse_loss.cpu()\n```\n\n\n```python\nclass LinearAutoEncoder(nn.Module):\n def __init__(self, x_dim, h_dim):\n \"\"\"A Linear AutoEncoder\n\n Args:\n x_dim (int): input dimension\n h_dim (int): hidden dimension, bottleneck dimension, K\n \"\"\"\n super().__init__()\n ####################################################################\n # Fill in all missing code below (...),\n # then remove or comment the line below to test your class\n raise NotImplementedError(\"Please complete the LinearAutoEncoder class!\")\n ####################################################################\n # encoder layer (a linear mapping from x_dim to K)\n self.enc_lin = ...\n # decoder layer (a linear mapping from K to x_dim)\n self.dec_lin = ...\n\n def encode(self, x):\n ####################################################################\n # Fill in all missing code below (...),\n raise NotImplementedError(\"Please complete the `encode` function!\")\n ####################################################################\n h = ...\n return h\n\n def decode(self, h):\n ####################################################################\n # Fill in all missing code below (...),\n raise NotImplementedError(\"Please complete the `decode` function!\")\n ####################################################################\n x_prime = ...\n return x_prime\n\n def forward(self, x):\n flat_x = x.view(x.size(0), -1)\n h = self.encode(flat_x)\n return self.decode(h).view(x.size())\n\n\n# add event to airtable\natform.add_event('Coding Exercise 3.1: Linear AutoEncoder Architecture')\n\n# Pick your own K\nK = 20\nset_seed(seed=SEED)\n## Uncomment to test your code\n# lin_ae = LinearAutoEncoder(data_size, K)\n# lin_losses = train_autoencoder(lin_ae, train_set, device=DEVICE, seed=SEED)\n# plot_linear_ae(lin_losses)\n```\n\n\n```python\n# to_remove solution\nclass LinearAutoEncoder(nn.Module):\n def __init__(self, x_dim, h_dim):\n \"\"\"A Linear AutoEncoder\n\n Args:\n x_dim (int): input dimension\n h_dim (int): hidden dimension, bottleneck dimension, K\n \"\"\"\n super().__init__()\n # encoder layer (a linear mapping from x_dim to K)\n self.enc_lin = nn.Linear(x_dim, h_dim)\n # decoder layer (a linear mapping from K to x_dim)\n self.dec_lin = nn.Linear(h_dim, x_dim)\n\n def encode(self, x):\n h = self.enc_lin(x)\n return h\n\n def decode(self, h):\n x_prime = self.dec_lin(h)\n return x_prime\n\n def forward(self, x):\n flat_x = x.view(x.size(0), -1)\n h = self.encode(flat_x)\n return self.decode(h).view(x.size())\n\n\n# add event to airtable\natform.add_event('Coding Exercise 3.1: Linear AutoEncoder Architecture')\n\n# Pick your own K\nK = 20\nset_seed(seed=SEED)\n## Uncomment to test your code\nlin_ae = LinearAutoEncoder(data_size, K)\nlin_losses = train_autoencoder(lin_ae, train_set, device=DEVICE, seed=SEED)\nwith plt.xkcd():\n plot_linear_ae(lin_losses)\n```\n\n### Comparison to PCA\n\nOne way to think about AutoEncoders is as a form of dimensionality-reduction. The dimensionality of $\\mathbf{h}$ is much smaller than the dimensionality of $\\mathbf{x}$.\n\nAnother common technique for dimensionality reduction is to project data onto the top $K$ **principal components** (Principal Component Analysis or PCA). For comparison, let's also apply PCA for dimensionality reduction.\n\n\n```python\n# PCA requires finding the top K eigenvectors of the data covariance. Start by\n# finding the mean and covariance of the pixels in our dataset\ng_seed = torch.Generator()\ng_seed.manual_seed(SEED)\n\nloader = DataLoader(train_set,\n batch_size=32,\n pin_memory=True,\n num_workers=2,\n worker_init_fn=seed_worker,\n generator=g_seed)\n\nmu, cov = image_moments((im for im, _ in loader),\n n_batches=len(train_set) // 32)\n\npca_encode, pca_decode = pca_encoder_decoder(mu, cov, K)\n```\n\nLet's visualize some of the reconstructions ($\\mathbf{x'}$) side-by-side with the input images ($\\mathbf{x}$).\n\n\n```python\n# @markdown Visualize the reconstructions $\\mathbf{x}'$, run this code a few times to see different examples.\n\nn_plot = 7\nplt.figure(figsize=(10, 4.5))\nfor i in range(n_plot):\n idx = torch.randint(len(train_set), size=())\n image, _ = train_set[idx]\n # Get reconstructed image from autoencoder\n with torch.no_grad():\n reconstruction = lin_ae(image.unsqueeze(0)).reshape(image.size())\n\n # Get reconstruction from PCA dimensionality reduction\n h_pca = pca_encode(image)\n recon_pca = pca_decode(h_pca).reshape(image.size())\n\n plt.subplot(3, n_plot, i + 1)\n plot_torch_image(image)\n if i == 0:\n plt.ylabel('Original\\nImage')\n\n plt.subplot(3, n_plot, i + 1 + n_plot)\n plot_torch_image(reconstruction)\n if i == 0:\n plt.ylabel(f'Lin AE\\n(K={K})')\n\n plt.subplot(3, n_plot, i + 1 + 2*n_plot)\n plot_torch_image(recon_pca)\n if i == 0:\n plt.ylabel(f'PCA\\n(K={K})')\nplt.show()\n```\n\n### Think! 3.1: PCA vs. Linear autoenconder\n\nCompare the PCA-based reconstructions to those from the linear autoencoder. Is one better than the other? Are they equally good? Equally bad? How does the choice of $K$ impact reconstruction quality?\n\n\n```python\n# @title Student Response\nfrom ipywidgets import widgets\n\n\ntext=widgets.Textarea(\n value='Type your answer here and click on `Submit!`',\n placeholder='Type something',\n description='',\n disabled=False\n)\n\nbutton = widgets.Button(description=\"Submit!\")\n\ndisplay(text,button)\n\ndef on_button_clicked(b):\n atform.add_answer('q3' , text.value)\n print(\"Submission successful!\")\n\n\nbutton.on_click(on_button_clicked)\n```\n\n\n```python\n# to_remove explanation\n\n\"\"\"\nSingle layered Autoencoder with linear activation function is similar to PCA.\nHowever, the Autoencoder can better capture the curvature in the data and hence,\nattempts to encode the most important features since they are capable of\nreconstruction anyway but are also prone to overfitting due to a higher number of parameters.\nPCA retains projections onto planes with maximum variance (and minimum error) and loses data.\nThe Autoencoder performs poorly if there are no underlying relationships between the features.\n\"\"\";\n```\n\n## Section 3.2: Building a nonlinear convolutional autoencoder\n\n**Nonlinear:** We'd like to apply autoencoders to learn a more flexible nonlinear mapping between the latent space and the images. Such a mapping can provide a more \"expressive\" model that better describes the image data than a linear mapping. This can be achieved by adding nonlinear activation functions to our encoder and decoder!\n\n**Convolutional:** As you saw on the day dedicated to RNNs and CNNs, parameter sharing is often a good idea for images! It's quite common to use convolutional layers in autoencoders to share parameters across locations in the image.\n\n**Side Note:** The `nn.Linear` layer (used in the linear autoencoder above) has a \"bias\" term, which is a learnable offset parameter separate for each output unit. Just like PCA \"centers\" the data by subtracting off the mean image (`mu`) before encoding and adds the average back in during decoding, a bias term in the decoder can effectively account for the first moment (mean) of the data (i.e. the average of all images in the training set). Convolution layers do have bias parameters, but the bias is applied per filter rather than per pixel location. If we're generating grayscale images (like those in MNIST), then `Conv2d` will learn only one bias across the entire image.\n\nFor some conceptual continuity with both PCA and the `nn.Linear` layers above, the next block defines a custom `BiasLayer` for adding a learnable per-pixel offset. This custom layer will be used twice: as the first stage of the encoder and as the final stage of the decoder. Ideally, this means that the rest of the neural net can focus on fitting more interesting fine-grained structure.\n\n\n```python\nclass BiasLayer(nn.Module):\n def __init__(self, shape):\n super(BiasLayer, self).__init__()\n init_bias = torch.zeros(shape)\n self.bias = nn.Parameter(init_bias, requires_grad=True)\n\n def forward(self, x):\n return x + self.bias\n```\n\nWith that out of the way, we will next define a **nonlinear** and **convolutional** autoencoder. Here's a quick tour of the architecture:\n\n1. The **encoder** once again maps from images to $\\mathbf{h}\\in\\mathbb{R}^K$. This will use a `BiasLayer` followed by two convolutional layers (`nn.Conv2D`), followed by flattening and linearly projecting down to $K$ dimensions. The convolutional layers will have `ReLU` nonlinearities on their outputs. \n1. The **decoder** inverts this process, taking in vectors of length $K$ and outputting images. Roughly speaking, its architecture is a \"mirror image\" of the encoder: the first decoder layer is linear, followed by two **deconvolution** layers ([`ConvTranspose2d`](https://pytorch.org/docs/stable/generated/torch.nn.ConvTranspose2d.html)). The `ConvTranspose2d`layers will have `ReLU` nonlinearities on their _inputs_. This \"mirror image\" between the encoder and decoder is a useful and near-ubiquitous convention. The idea is that the decoder can then learn to approximately invert the encoder, but it is not a strict requirement (and it does not guarantee the decoder will be an exact inverse of the encoder!).\n\nBelow is a schematic of the architecture for MNIST. Notice that the width and height dimensions of the image planes reduce after each `nn.Conv2d` and increase after each `nn.ConvTranspose2d`. With CIFAR10, the architecture is the same but the exact sizes will differ.\n\n\n\n[`torch.nn.ConvTranspose2d`](https://pytorch.org/docs/stable/generated/torch.nn.ConvTranspose2d.html) module can be seen as the gradient of `Conv2d` with respect to its input. It is also known as a fractionally-strided convolution or a deconvolution (although it is not an actual deconvolution operation). The following code demonstrates this change in sizes:\n\n\n```python\ndummy_image = torch.rand(data_shape).unsqueeze(0)\nin_channels = data_shape[0]\nout_channels = 7\n\ndummy_conv = nn.Conv2d(in_channels=in_channels,\n out_channels=out_channels,\n kernel_size=5)\n\ndummy_deconv = nn.ConvTranspose2d(in_channels=out_channels,\n out_channels=in_channels,\n kernel_size=5)\n\nprint(f'Size of image is {dummy_image.shape}')\nprint(f'Size of Conv2D(image) {dummy_conv(dummy_image).shape}')\nprint(f'Size of ConvTranspose2D(Conv2D(image)) {dummy_deconv(dummy_conv(dummy_image)).shape}')\n```\n\n### Coding Exercise 3.2: Fill in code for the `ConvAutoEncoder` module\n\nComplete the `ConvAutoEncoder` class. We use the helper function `cout(torch.Tensor, nn.Conv2D)` to calculate the output shape of a [`nn.Conv2D`](https://pytorch.org/docs/stable/generated/torch.nn.Conv2d.html) layer given a tensor with shape (channels, height, width).\n\n\n```python\nclass ConvAutoEncoder(nn.Module):\n def __init__(self, x_dim, h_dim, n_filters=32, filter_size=5):\n \"\"\"A Convolutional AutoEncoder\n\n Args:\n x_dim (tuple): input dimensions (channels, height, widths)\n h_dim (int): hidden dimension, bottleneck dimension, K\n n_filters (int): number of filters (number of output channels)\n filter_size (int): kernel size\n \"\"\"\n super().__init__()\n channels, height, widths = x_dim\n\n # encoder input bias layer\n self.enc_bias = BiasLayer(x_dim)\n\n # first encoder conv2d layer\n self.enc_conv_1 = nn.Conv2d(channels, n_filters, filter_size)\n\n # output shape of the first encoder conv2d layer given x_dim input\n conv_1_shape = cout(x_dim, self.enc_conv_1)\n\n # second encoder conv2d layer\n self.enc_conv_2 = nn.Conv2d(n_filters, n_filters, filter_size)\n\n # output shape of the second encoder conv2d layer given conv_1_shape input\n conv_2_shape = cout(conv_1_shape, self.enc_conv_2)\n\n # The bottleneck is a dense layer, therefore we need a flattenning layer\n self.enc_flatten = nn.Flatten()\n\n # conv output shape is (depth, height, width), so the flatten size is:\n flat_after_conv = conv_2_shape[0] * conv_2_shape[1] * conv_2_shape[2]\n\n # encoder Linear layer\n self.enc_lin = nn.Linear(flat_after_conv, h_dim)\n\n ####################################################################\n # Fill in all missing code below (...),\n # then remove or comment the line below to test your class\n # Remember that decoder is \"undo\"-ing what the encoder has done!\n raise NotImplementedError(\"Please complete the `ConvAutoEncoder` class!\")\n ####################################################################\n # decoder Linear layer\n self.dec_lin = ...\n\n # unflatten data to (depth, height, width) shape\n self.dec_unflatten = nn.Unflatten(dim=-1, unflattened_size=conv_2_shape)\n\n # first \"deconvolution\" layer\n self.dec_deconv_1 = nn.ConvTranspose2d(n_filters, n_filters, filter_size)\n\n # second \"deconvolution\" layer\n self.dec_deconv_2 = ...\n\n # decoder output bias layer\n self.dec_bias = BiasLayer(x_dim)\n\n def encode(self, x):\n s = self.enc_bias(x)\n s = F.relu(self.enc_conv_1(s))\n s = F.relu(self.enc_conv_2(s))\n s = self.enc_flatten(s)\n h = self.enc_lin(s)\n return h\n\n def decode(self, h):\n s = F.relu(self.dec_lin(h))\n s = self.dec_unflatten(s)\n s = F.relu(self.dec_deconv_1(s))\n s = self.dec_deconv_2(s)\n x_prime = self.dec_bias(s)\n return x_prime\n\n def forward(self, x):\n return self.decode(self.encode(x))\n\n\n# add event to airtable\natform.add_event('Coding Exercise 3.2: Fill in code for the ConvAutoEncoder module')\n\nK = 20\nset_seed(seed=SEED)\n## Uncomment to test your solution\n# trained_conv_AE = ConvAutoEncoder(data_shape, K)\n# assert trained_conv_AE.encode(train_set[0][0].unsqueeze(0)).numel() == K, \"Encoder output size should be K!\"\n# conv_losses = train_autoencoder(trained_conv_AE, train_set, device=DEVICE, seed=SEED)\n# plot_conv_ae(lin_losses, conv_losses)\n```\n\n\n```python\n# to_remove solution\nclass ConvAutoEncoder(nn.Module):\n def __init__(self, x_dim, h_dim, n_filters=32, filter_size=5):\n \"\"\"A Convolutional AutoEncoder\n\n Args:\n x_dim (tuple): input dimensions (channels, height, widths)\n h_dim (int): hidden dimension, bottleneck dimension, K\n n_filters (int): number of filters (number of output channels)\n filter_size (int): kernel size\n \"\"\"\n super().__init__()\n channels, height, widths = x_dim\n\n # encoder input bias layer\n self.enc_bias = BiasLayer(x_dim)\n\n # first encoder conv2d layer\n self.enc_conv_1 = nn.Conv2d(channels, n_filters, filter_size)\n\n # output shape of the first encoder conv2d layer given x_dim input\n conv_1_shape = cout(x_dim, self.enc_conv_1)\n\n # second encoder conv2d layer\n self.enc_conv_2 = nn.Conv2d(n_filters, n_filters, filter_size)\n\n # output shape of the second encoder conv2d layer given conv_1_shape input\n conv_2_shape = cout(conv_1_shape, self.enc_conv_2)\n\n # The bottleneck is a dense layer, therefore we need a flattenning layer\n self.enc_flatten = nn.Flatten()\n\n # conv output shape is (depth, height, width), so the flatten size is:\n flat_after_conv = conv_2_shape[0] * conv_2_shape[1] * conv_2_shape[2]\n\n # encoder Linear layer\n self.enc_lin = nn.Linear(flat_after_conv, h_dim)\n\n # decoder Linear layer\n self.dec_lin = nn.Linear(h_dim, flat_after_conv)\n\n # unflatten data to (depth, height, width) shape\n self.dec_unflatten = nn.Unflatten(dim=-1, unflattened_size=conv_2_shape)\n\n # first \"deconvolution\" layer\n self.dec_deconv_1 = nn.ConvTranspose2d(n_filters, n_filters, filter_size)\n\n # second \"deconvolution\" layer\n self.dec_deconv_2 = nn.ConvTranspose2d(n_filters, channels, filter_size)\n\n # decoder output bias layer\n self.dec_bias = BiasLayer(x_dim)\n\n def encode(self, x):\n s = self.enc_bias(x)\n s = F.relu(self.enc_conv_1(s))\n s = F.relu(self.enc_conv_2(s))\n s = self.enc_flatten(s)\n h = self.enc_lin(s)\n return h\n\n def decode(self, h):\n s = F.relu(self.dec_lin(h))\n s = self.dec_unflatten(s)\n s = F.relu(self.dec_deconv_1(s))\n s = self.dec_deconv_2(s)\n x_prime = self.dec_bias(s)\n return x_prime\n\n def forward(self, x):\n return self.decode(self.encode(x))\n\n\n# add event to airtable\natform.add_event('Coding Exercise 3.2: Fill in code for the ConvAutoEncoder module')\n\nK = 20\nset_seed(seed=SEED)\n## Uncomment to test your solution\ntrained_conv_AE = ConvAutoEncoder(data_shape, K)\nassert trained_conv_AE.encode(train_set[0][0].unsqueeze(0)).numel() == K, \"Encoder output size should be K!\"\nconv_losses = train_autoencoder(trained_conv_AE, train_set, device=DEVICE, seed=SEED)\nwith plt.xkcd():\n plot_conv_ae(lin_losses, conv_losses)\n```\n\nYou should see that the `ConvAutoEncoder` achieved lower MSE loss than the linear one. If not, you may need to retrain it (or run another few training epochs from where it left off). We make fewer guarantees on this working with CIFAR10, but it should definitely work with MNIST.\n\nNow let's visually compare the reconstructed images from the linear and nonlinear autoencoders. Keep in mind that both have the same dimensionality for $\\mathbf{h}$!\n\n\n```python\n# @markdown Visualize the linear and nonlinear AE outputs\nn_plot = 7\nplt.figure(figsize=(10, 4.5))\nfor i in range(n_plot):\n idx = torch.randint(len(train_set), size=())\n image, _ = train_set[idx]\n with torch.no_grad():\n # Get reconstructed image from linear autoencoder\n lin_recon = lin_ae(image.unsqueeze(0))[0]\n\n # Get reconstruction from deep (nonlinear) autoencoder\n nonlin_recon = trained_conv_AE(image.unsqueeze(0))[0]\n\n plt.subplot(3, n_plot, i+1)\n plot_torch_image(image)\n if i == 0:\n plt.ylabel('Original\\nImage')\n\n plt.subplot(3, n_plot, i + 1 + n_plot)\n plot_torch_image(lin_recon)\n if i == 0:\n plt.ylabel(f'Lin AE\\n(K={K})')\n\n plt.subplot(3, n_plot, i + 1 + 2*n_plot)\n plot_torch_image(nonlin_recon)\n if i == 0:\n plt.ylabel(f'NonLin AE\\n(K={K})')\nplt.show()\n```\n\n---\n# Section 4: Variational Auto-Encoders (VAEs)\n\n*Time estimate: ~25mins*\n\n**Please** run the cell after the video to train a VAE for MNIST while watching it.\n\n\n```python\n# @title Video 4: Variational Autoencoders\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV17v411E7ye\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"srWb_Gp6OGA\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 4: Variational Autoencoders')\n\ndisplay(out)\n```\n\n\n```python\n# @markdown Train a VAE for MNIST while watching the video. (Note: this VAE has a 2D latent space. If you are feeling ambitious, edit the code and modify the latent space dimensionality and see what happens.)\nK_VAE = 2\n\n\nclass ConvVAE(nn.Module):\n def __init__(self, K, num_filters=32, filter_size=5):\n super(ConvVAE, self).__init__()\n\n # With padding=0, the number of pixels cut off from each image dimension\n # is filter_size // 2. Double it to get the amount of pixels lost in\n # width and height per Conv2D layer, or added back in per\n # ConvTranspose2D layer.\n filter_reduction = 2 * (filter_size // 2)\n\n # After passing input through two Conv2d layers, the shape will be\n # 'shape_after_conv'. This is also the shape that will go into the first\n # deconvolution layer in the decoder\n self.shape_after_conv = (num_filters,\n data_shape[1]-2*filter_reduction,\n data_shape[2]-2*filter_reduction)\n flat_size_after_conv = self.shape_after_conv[0] \\\n * self.shape_after_conv[1] \\\n * self.shape_after_conv[2]\n\n # Define the recognition model (encoder or q) part\n self.q_bias = BiasLayer(data_shape)\n self.q_conv_1 = nn.Conv2d(data_shape[0], num_filters, 5)\n self.q_conv_2 = nn.Conv2d(num_filters, num_filters, 5)\n self.q_flatten = nn.Flatten()\n self.q_fc_phi = nn.Linear(flat_size_after_conv, K+1)\n\n # Define the generative model (decoder or p) part\n self.p_fc_upsample = nn.Linear(K, flat_size_after_conv)\n self.p_unflatten = nn.Unflatten(-1, self.shape_after_conv)\n self.p_deconv_1 = nn.ConvTranspose2d(num_filters, num_filters, 5)\n self.p_deconv_2 = nn.ConvTranspose2d(num_filters, data_shape[0], 5)\n self.p_bias = BiasLayer(data_shape)\n\n # Define a special extra parameter to learn scalar sig_x for all pixels\n self.log_sig_x = nn.Parameter(torch.zeros(()))\n\n def infer(self, x):\n \"\"\"Map (batch of) x to (batch of) phi which can then be passed to\n rsample to get z\n \"\"\"\n s = self.q_bias(x)\n s = F.relu(self.q_conv_1(s))\n s = F.relu(self.q_conv_2(s))\n flat_s = s.view(s.size()[0], -1)\n phi = self.q_fc_phi(flat_s)\n return phi\n\n def generate(self, zs):\n \"\"\"Map [b,n,k] sized samples of z to [b,n,p] sized images\n \"\"\"\n # Note that for the purposes of passing through the generator, we need\n # to reshape zs to be size [b*n,k]\n b, n, k = zs.size()\n s = zs.view(b*n, -1)\n s = F.relu(self.p_fc_upsample(s)).view((b*n,) + self.shape_after_conv)\n s = F.relu(self.p_deconv_1(s))\n s = self.p_deconv_2(s)\n s = self.p_bias(s)\n mu_xs = s.view(b, n, -1)\n return mu_xs\n\n def decode(self, zs):\n # Included for compatability with conv-AE code\n return self.generate(zs.unsqueeze(0))\n\n def forward(self, x):\n # VAE.forward() is not used for training, but we'll treat it like a\n # classic autoencoder by taking a single sample of z ~ q\n phi = self.infer(x)\n zs = rsample(phi, 1)\n return self.generate(zs).view(x.size())\n\n def elbo(self, x, n=1):\n \"\"\"Run input end to end through the VAE and compute the ELBO using n\n samples of z\n \"\"\"\n phi = self.infer(x)\n zs = rsample(phi, n)\n mu_xs = self.generate(zs)\n return log_p_x(x, mu_xs, self.log_sig_x.exp()) - kl_q_p(zs, phi)\n\n\ndef expected_z(phi):\n return phi[:, :-1]\n\n\ndef rsample(phi, n_samples):\n \"\"\"Sample z ~ q(z;phi)\n Ouput z is size [b,n_samples,K] given phi with shape [b,K+1]. The first K\n entries of each row of phi are the mean of q, and phi[:,-1] is the log\n standard deviation\n \"\"\"\n b, kplus1 = phi.size()\n k = kplus1-1\n mu, sig = phi[:, :-1], phi[:,-1].exp()\n eps = torch.randn(b, n_samples, k, device=phi.device)\n return eps*sig.view(b,1,1) + mu.view(b,1,k)\n\n\ndef train_vae(vae, dataset, epochs=10, n_samples=1000):\n opt = torch.optim.Adam(vae.parameters(), lr=1e-3, weight_decay=0)\n elbo_vals = []\n vae.to(DEVICE)\n vae.train()\n loader = DataLoader(dataset, batch_size=250, shuffle=True, pin_memory=True)\n for epoch in trange(epochs, desc='Epochs'):\n for im, _ in tqdm(loader, total=len(dataset) // 250, desc='Batches', leave=False):\n im = im.to(DEVICE)\n opt.zero_grad()\n loss = -vae.elbo(im)\n loss.backward()\n opt.step()\n\n elbo_vals.append(-loss.item())\n vae.to('cpu')\n vae.eval()\n return elbo_vals\n\n\ntrained_conv_VarAE = ConvVAE(K=K_VAE)\nelbo_vals = train_vae(trained_conv_VarAE, train_set, n_samples=10000)\n\nprint(f'Learned sigma_x is {torch.exp(trained_conv_VarAE.log_sig_x)}')\n\n# Uncomment below if you'd like to see the the training\n# curve of the evaluated ELBO loss function\n# ELBO is the loss function used to train VAEs (see lecture!)\nplt.figure()\nplt.plot(elbo_vals)\nplt.xlabel('Batch #')\nplt.ylabel('ELBO')\nplt.show()\n```\n\n## Section 4.1: Components of a VAE\n\n*Recognition models and density networks*\n\n\nVariational AutoEncoders (VAEs) are a lot like the classic AutoEncoders (AEs), but where we explicitly think about probability distributions. In the language of VAEs, the __encoder__ is replaced with a __recognition model__, and the __decoder__ is replaced with a __density network__.\n\nWhere in a classic autoencoder the encoder maps from images to a single hidden vector,\n\n\\begin{equation}\n\\mathbf{x} \\overset{\\text{AE}}{\\longrightarrow} \\mathbf{h} \\, ,\n\\end{equation}\n\nin a VAE we would say that a recognition model maps from inputs to entire __distributions__ over hidden vectors,\n\n\\begin{equation}\n\\mathbf{x} \\overset{\\text{VAE}}{\\longrightarrow} q_{\\mathbf{w_e}}(\\mathbf{z}) \\, ,\n\\end{equation}\n\nwhich we will then sample from. Here $\\mathbf{w_e}$ refers to the weights of the recognition model, which parametarize our distribution generating network. We'll say more in a moment about what kind of distribution $q_{\\mathbf{w_e}}(\\mathbf{z})$ is.\nPart of what makes VAEs work is that the loss function will require good reconstructions of the input not just for a single $\\mathbf{z}$, but _on average_ from samples of $\\mathbf{z} \\sim q_{\\mathbf{w_e}}(\\mathbf{z})$.\n\nIn the classic autoencoder, we had a decoder which maps from hidden vectors to reconstructions of the input:\n\n\\begin{equation}\n\\mathbf{h} \\overset{\\text{AE}}{\\longrightarrow} \\mathbf{x'} \\, .\n\\end{equation}\n\nIn a density network, reconstructions are expressed in terms of a distribution:\n\n\\begin{equation}\n\\mathbf{z} \\overset{\\text{VAE}}{\\longrightarrow} p_{\\mathbf{w_d}}(\\mathbf{x}|\\mathbf{z})\n\\end{equation}\n\nwhere, as above, $p_{\\mathbf{w_d}}(\\mathbf{x}|\\mathbf{z})$ is defined by mapping $\\mathbf{z}$ through a density network then treating the resulting $f(\\mathbf{z};\\mathbf{w_d})$ as the mean of a (Gaussian) distribution over $\\mathbf{x}$. Similarly, our reconstruction distribution is parametarized by the weights of the density network.\n\n## Section 4.2: Generating novel images from the decoder\n\nIf we isolate the decoder part of the AutoEncoder, what we have is a neural network that takes as input a vector of size $K$ and produces as output an image that looks something like our training data. Recall that in our earlier notation, we had an input $\\mathbf{x}$ that was mapped to a low-dimensional hidden representation $\\mathbf{h}$ which was then decoded into a reconstruction of the input, $\\mathbf{x'}$:\n\n\\begin{equation}\n\\mathbf{x} \\overset{\\text{encode}}{\\longrightarrow} \\mathbf{h} \\overset{\\text{decode}}{\\longrightarrow} \\mathbf{x'}\\, .\n\\end{equation}\n\nPartly as a matter of convention, and partly to distinguish where we are going next from the previous section, we're going to introduce a new variable, $\\mathbf{z} \\in \\mathbb{R}^K$, which will take the place of $\\mathbf{h}$. The key difference is that while $\\mathbf{h}$ is produced by the encoder for a particular $\\mathbf{x}$, $\\mathbf{z}$ will be drawn out of thin air from a prior of our choosing:\n\n\\begin{equation}\n\\mathbf{z} \\sim p(\\mathbf{z})\\\\ \\mathbf{z} \\overset{\\text{decode}}{\\longrightarrow} \\mathbf{x}\\, .\n\\end{equation}\n\n(Note that it is also common convention to drop the \"prime\" on $\\mathbf{x}$ when it is no longer being thought of as a \"reconstruction\").\n\n### Coding Exercise 4.2: Generating images\n\n\n```python\ndef generate_images(autoencoder, K, n_images=1):\n \"\"\"Generate n_images 'new' images from the decoder part of the given\n autoencoder.\n\n returns (n_images, channels, height, width) tensor of images\n \"\"\"\n # Concatenate tuples to get (n_images, channels, height, width)\n output_shape = (n_images,) + data_shape\n with torch.no_grad():\n ####################################################################\n # Fill in all missing code below (...),\n # then remove or comment the line below to test your function\n raise NotImplementedError(\"Please complete the `generate_images` function!\")\n ####################################################################\n # sample z from a unit gaussian, pass through autoencoder.decode()\n z = ...\n x = ...\n\n return x.reshape(output_shape)\n\n\n# add event to airtable\natform.add_event('Coding Exercise 4.2: Generating images')\n\nset_seed(seed=SEED)\n## Uncomment to test your solution\n# images = generate_images(trained_conv_AE, K, n_images=9)\n# plot_images(images, plt_title='Images Generated from the Conv-AE')\n# images = generate_images(trained_conv_VarAE, K_VAE, n_images=9)\n# plot_images(images, plt_title='Images Generated from a Conv-Variational-AE')\n```\n\n\n```python\n# to_remove solution\ndef generate_images(autoencoder, K, n_images=1):\n \"\"\"Generate n_images 'new' images from the decoder part of the given\n autoencoder.\n\n returns (n_images, channels, height, width) tensor of images\n \"\"\"\n # Concatenate tuples to get (n_images, channels, height, width)\n output_shape = (n_images,) + data_shape\n with torch.no_grad():\n # sample z from a unit gaussian, pass through autoencoder.decode()\n z = torch.randn(n_images, K)\n x = autoencoder.decode(z)\n\n return x.reshape(output_shape)\n\n\n# add event to airtable\natform.add_event('Coding Exercise 4.2: Generating images')\n\nset_seed(seed=SEED)\n## Uncomment to test your solution\nimages = generate_images(trained_conv_AE, K, n_images=9)\nplot_images(images, plt_title='Images Generated from the Conv-AE')\nimages = generate_images(trained_conv_VarAE, K_VAE, n_images=9)\nplot_images(images, plt_title='Images Generated from a Conv-Variational-AE')\n```\n\n### Think! 4.2: AutoEncoders vs. Variational AutoEncoders\n\nCompare the images generated by the AutoEncoder to the images generated by the Variational AutoEncoder. You can run the code a few times to see a variety of examples.\n\nDoes one set look more like the training set (handwritten digits) than the other? What is driving this difference?\n\n\n```python\n# to_remove explanation\n\n\"\"\"\nAn Autoencoder accepts input, compresses it, and recreates it. On the other hand,\nVAEs assume that the source data has some underlying distribution and attempts\nto find the distribution parameters. So, VAEs are similar to GANs\n(but note that GANs work differently, as we will see in the next tutorials).\n\"\"\";\n```\n\n---\n# Section 5: State of the art VAEs and Wrap-up (Bonus)\n\n\n```python\n# @title Video 5: State-Of-The-Art VAEs\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1hg411M7KY\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"PXBl3KwRfh4\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 5: State-Of-The-Art VAEs')\n\ndisplay(out)\n```\n\n---\n# Summary\nThrough this tutorial, we have learned\n- What a generative model is and why we are interested in them.\n- How latent variable models relate to generative models with the example of pPCA.\n- What a basic AutoEncoder is and how they relate to other latent variable models.\n- The basics of Variational AutoEncoders and how they function as generative models.\n- An introduction to the broad applications of VAEs.\n\nIn the next two tutorials we will cover GANs and how to train them.\n\n\n```python\n# @title Airtable Submission Link\nfrom IPython import display as IPydisplay\nIPydisplay.HTML(\n f\"\"\"\n
    \n \n \n
    \"\"\" )\n```\n", "meta": {"hexsha": "b922b6d2717bddaa1244cbf7107536e0438cd55f", "size": 98257, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W2D5_GenerativeModels/W2D5_Tutorial1.ipynb", "max_stars_repo_name": "justynaekert/course-content-dl", "max_stars_repo_head_hexsha": "aa64d9feb1ae92ad4b7afaf13b13616b3a020c20", "max_stars_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-09-04T01:57:41.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-04T01:57:41.000Z", "max_issues_repo_path": "tutorials/W2D5_GenerativeModels/W2D5_Tutorial1.ipynb", "max_issues_repo_name": "justynaekert/course-content-dl", "max_issues_repo_head_hexsha": "aa64d9feb1ae92ad4b7afaf13b13616b3a020c20", "max_issues_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/W2D5_GenerativeModels/W2D5_Tutorial1.ipynb", "max_forks_repo_name": "justynaekert/course-content-dl", "max_forks_repo_head_hexsha": "aa64d9feb1ae92ad4b7afaf13b13616b3a020c20", "max_forks_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 37.3175085454, "max_line_length": 724, "alphanum_fraction": 0.5700560774, "converted": true, "num_tokens": 18366, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.6992544147913993, "lm_q1q2_score": 0.43782009513452363}} {"text": "# Day 10\n\n\n```python\n! cat README.md\n```\n\n\n```python\nwith open('input.txt', 'rt') as f:\n adapters = sorted(list(map(int, f.read().splitlines())))\n```\n\n\n```python\ndiff = [adapters[0]] + [adapters[i+1] - x if i < len(adapters)-1 else 3 for i, x in enumerate(adapters)]\ndiff.count(1) * diff.count(3)\n```\n--- Part Two ---\nTo completely determine whether you have enough adapters, you'll need to figure out how many different ways they can be arranged. Every arrangement needs to connect the charging outlet to your device. The previous rules about when adapters can successfully connect still apply.\n\nThe first example above (the one that starts with 16, 10, 15) supports the following arrangements:\n\n(0), 1, 4, 5, 6, 7, 10, 11, 12, 15, 16, 19, (22)\n(0), 1, 4, 5, 6, 7, 10, 12, 15, 16, 19, (22)\n(0), 1, 4, 5, 7, 10, 11, 12, 15, 16, 19, (22)\n(0), 1, 4, 5, 7, 10, 12, 15, 16, 19, (22)\n(0), 1, 4, 6, 7, 10, 11, 12, 15, 16, 19, (22)\n(0), 1, 4, 6, 7, 10, 12, 15, 16, 19, (22)\n(0), 1, 4, 7, 10, 11, 12, 15, 16, 19, (22)\n(0), 1, 4, 7, 10, 12, 15, 16, 19, (22)\n(The charging outlet and your device's built-in adapter are shown in parentheses.) Given the adapters from the first example, the total number of arrangements that connect the charging outlet to your device is 8.\n\nThe second example above (the one that starts with 28, 33, 18) has many arrangements. Here are a few:\n\n(0), 1, 2, 3, 4, 7, 8, 9, 10, 11, 14, 17, 18, 19, 20, 23, 24, 25, 28, 31,\n32, 33, 34, 35, 38, 39, 42, 45, 46, 47, 48, 49, (52)\n\n(0), 1, 2, 3, 4, 7, 8, 9, 10, 11, 14, 17, 18, 19, 20, 23, 24, 25, 28, 31,\n32, 33, 34, 35, 38, 39, 42, 45, 46, 47, 49, (52)\n\n(0), 1, 2, 3, 4, 7, 8, 9, 10, 11, 14, 17, 18, 19, 20, 23, 24, 25, 28, 31,\n32, 33, 34, 35, 38, 39, 42, 45, 46, 48, 49, (52)\n\n(0), 1, 2, 3, 4, 7, 8, 9, 10, 11, 14, 17, 18, 19, 20, 23, 24, 25, 28, 31,\n32, 33, 34, 35, 38, 39, 42, 45, 46, 49, (52)\n\n(0), 1, 2, 3, 4, 7, 8, 9, 10, 11, 14, 17, 18, 19, 20, 23, 24, 25, 28, 31,\n32, 33, 34, 35, 38, 39, 42, 45, 47, 48, 49, (52)\n\n(0), 3, 4, 7, 10, 11, 14, 17, 20, 23, 25, 28, 31, 34, 35, 38, 39, 42, 45,\n46, 48, 49, (52)\n\n(0), 3, 4, 7, 10, 11, 14, 17, 20, 23, 25, 28, 31, 34, 35, 38, 39, 42, 45,\n46, 49, (52)\n\n(0), 3, 4, 7, 10, 11, 14, 17, 20, 23, 25, 28, 31, 34, 35, 38, 39, 42, 45,\n47, 48, 49, (52)\n\n(0), 3, 4, 7, 10, 11, 14, 17, 20, 23, 25, 28, 31, 34, 35, 38, 39, 42, 45,\n47, 49, (52)\n\n(0), 3, 4, 7, 10, 11, 14, 17, 20, 23, 25, 28, 31, 34, 35, 38, 39, 42, 45,\n48, 49, (52)\nIn total, this set of adapters can connect the charging outlet to your device in 19208 distinct arrangements.\n\nYou glance back down at your bag and try to remember why you brought so many adapters; there must be more than a trillion valid ways to arrange them! Surely, there must be an efficient way to count the arrangements.\n\nWhat is the total number of distinct ways you can arrange the adapters to connect the charging outlet to your device?\n\n```python\n# divide and conquer with limited brute force!\n\nimport math\nfrom functools import reduce\n\n\ndef arrangements_diff_one(n):\n \n queue, completed = [0], 0\n while len(queue) > 0:\n head = queue.pop(0)\n children = [head + i for i in [1,2,3] if head + i < n] \n if len(children) == 0:\n completed += 1\n else:\n for y in children:\n queue.append(y)\n return completed\n\n\ndef divide_and_conquer(adapters):\n \n # diffs\n diff = [adapters[0]] + [adapters[i+1] - x if i < len(adapters)-1 else 3 for i, x in enumerate(adapters)]\n \n # chunks\n tractsizes, acc = [], 0\n for i, x in enumerate(diff):\n if diff[i] == 1:\n acc += 1\n else:\n if acc > 1:\n tractsizes.append(acc+1)\n acc = 0\n \n # arrangements\n acc = 1\n for n in tractsizes:\n acc *= arrangements_diff_one(n)\n return acc\n```\n\n\n```python\n# test\nadapters_1 = sorted([16,10,15,5,1,11,7,19,6,12,4])\nadapters_2 = sorted([28,33,18,42,31,14,46,20,48,47,24,23,49,45,19,38,39,11,1,32,25,35,8,17,7,9,4,2,34,10,3])\n\nassert(divide_and_conquer(adapters_1) == 8)\nassert(divide_and_conquer(adapters_2) == 19208)\n```\n\n\n```python\ndivide_and_conquer(adapters)\n```\n\n\n```python\n# extra elegant solution\n# formula-type of approach assuming diffs are {1,3} alone\n# it uses generating-functions: see tutorial in slack\n\nfrom functools import reduce\nfrom sympy import var\n\nx = var('x')\ng = x / (1 - x - x**2 - x**3) # gf how many times a piece is used\n\ndef diff_one_tract_lengths(adapters):\n diff = [adapters[0]] + [adapters[i+1] - x if i < len(adapters)-1 else 3 for i, x in enumerate(adapters)]\n return [len(x) + 1 for x in ''.join(list(map(str, diff))).split('3') if len(x) > 0]\n\ndef arrangements_diff_one_formula(n):\n return g.series(x, 0, n+1).coeff(x ** n)\n \nreduce(lambda x,y: x*y, list(map(arrangements_diff_one_formula, diff_one_tract_lengths(adapters))), 1)\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "cc8c5d889bcf6ce485f6f022c76c86d1a4b0958a", "size": 7413, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "2020/ferran/10/10.ipynb", "max_stars_repo_name": "bbglab/adventofcode", "max_stars_repo_head_hexsha": "65b6d8331d10f229b59232882d60024b08d69294", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "2020/ferran/10/10.ipynb", "max_issues_repo_name": "bbglab/adventofcode", "max_issues_repo_head_hexsha": "65b6d8331d10f229b59232882d60024b08d69294", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "2020/ferran/10/10.ipynb", "max_forks_repo_name": "bbglab/adventofcode", "max_forks_repo_head_hexsha": "65b6d8331d10f229b59232882d60024b08d69294", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2016-12-02T09:20:42.000Z", "max_forks_repo_forks_event_max_datetime": "2021-12-01T13:31:07.000Z", "avg_line_length": 32.9466666667, "max_line_length": 286, "alphanum_fraction": 0.5081613382, "converted": true, "num_tokens": 2045, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259583, "lm_q2_score": 0.724870282120402, "lm_q1q2_score": 0.4377722386944271}} {"text": "# Facebook Friend Recommender\n\nFirst we will load our dataset from Kaggle and perform exploratory data analysis on our given data set such as number of followers and followees of each person. Then we will generate some datapoints which were not present in our given data-set, since we have only class label 1 data. Then we will do some feature engineering on dataset like finding shortest path, kartz centrality, jaccard distances, page rank, preferential attachements etc. After performing exploratory data analysis and feature engineering, we will split whole dataset into train and test and perform random forest and xgboost taking f1-score as our metric. At the end we will plot confusion matrix and pretty-table for both algorithm and finf best hyperparameters.\n\n## Setup\n\n\n```python\nimport math\nimport random\nimport pickle\nimport os\nimport csv\nimport pandas as pd\nimport datetime\nimport time\nimport numpy as np\n\nimport matplotlib\nimport matplotlib.pylab as plt\nimport seaborn as sns\nfrom matplotlib import rcParams\nfrom sklearn.cluster import MiniBatchKMeans, KMeans\nfrom tqdm.notebook import tqdm\nfrom sklearn.model_selection import train_test_split\n\nimport xgboost as xgb\nimport networkx as nx\nimport pdb\nimport pickle\n\nimport warnings\nwarnings.filterwarnings(\"ignore\")\n```\n\n## Load dataset\n\n\n\n\n```python\n!pip install -q -U kaggle\n!pip install --upgrade --force-reinstall --no-deps kaggle\n!mkdir ~/.kaggle\n!cp /content/drive/MyDrive/kaggle.json ~/.kaggle/\n!chmod 600 ~/.kaggle/kaggle.json\n!kaggle competitions download -c FacebookRecruiting\n```\n\n Collecting kaggle\n Downloading kaggle-1.5.12.tar.gz (58 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 58 kB 4.5 MB/s \n \u001b[?25hBuilding wheels for collected packages: kaggle\n Building wheel for kaggle (setup.py) ... \u001b[?25l\u001b[?25hdone\n Created wheel for kaggle: filename=kaggle-1.5.12-py3-none-any.whl size=73052 sha256=940b16c3e71a3ef30f3c537442dadc0e4ad95f67fd31f490151951ac308ec731\n Stored in directory: /root/.cache/pip/wheels/62/d6/58/5853130f941e75b2177d281eb7e44b4a98ed46dd155f556dc5\n Successfully built kaggle\n Installing collected packages: kaggle\n Attempting uninstall: kaggle\n Found existing installation: kaggle 1.5.12\n Uninstalling kaggle-1.5.12:\n Successfully uninstalled kaggle-1.5.12\n Successfully installed kaggle-1.5.12\n Downloading FacebookRecruiting.zip to /content\n 84% 161M/191M [00:04<00:01, 26.3MB/s]\n 100% 191M/191M [00:04<00:00, 42.8MB/s]\n\n\n\n```python\n!unzip FacebookRecruiting.zip\n```\n\n Archive: FacebookRecruiting.zip\n inflating: bfs_benchmark.csv \n inflating: random_benchmark.csv \n inflating: test.csv \n inflating: train.7z \n inflating: train.csv \n inflating: train.gz \n inflating: train.zip \n\n\n## Reading graph\n\n\n```python\ntraincsv = pd.read_csv('train.csv')\ntraincsv.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    source_nodedestination_node
    01690569
    11315892
    21189226
    32834328
    421615927
    \n
    \n\n\n\n\n```python\ntraincsv.describe()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    source_nodedestination_node
    count9.437519e+069.437519e+06
    mean9.306740e+059.312252e+05
    std5.383368e+055.380682e+05
    min1.000000e+001.000000e+00
    25%4.638685e+054.647640e+05
    50%9.303910e+059.316830e+05
    75%1.397245e+061.397560e+06
    max1.862220e+061.862220e+06
    \n
    \n\n\n\n\n```python\nprint(traincsv[traincsv.isna().any(1)])\nprint(traincsv.info())\nprint(\"Number of diplicate entries: \",sum(traincsv.duplicated()))\ntraincsv.to_csv('train_woheader.csv',header=False,index=False)\nprint(\"saved the graph into file\")\n```\n\n Empty DataFrame\n Columns: [source_node, destination_node]\n Index: []\n \n RangeIndex: 9437519 entries, 0 to 9437518\n Data columns (total 2 columns):\n # Column Dtype\n --- ------ -----\n 0 source_node int64\n 1 destination_node int64\n dtypes: int64(2)\n memory usage: 144.0 MB\n None\n Number of diplicate entries: 0\n saved the graph into file\n\n\n\n```python\ng = nx.read_edgelist('train_woheader.csv',delimiter=',',create_using=nx.DiGraph(),nodetype=int)\nprint(nx.info(g))\n```\n\n Name: \n Type: DiGraph\n Number of nodes: 1862220\n Number of edges: 9437519\n Average in degree: 5.0679\n Average out degree: 5.0679\n\n\n\n```python\ntraincsv.head(20).to_csv('train_woheader_sample.csv',header=False,index=False)\n \nsubgraph=nx.read_edgelist('train_woheader_sample.csv',delimiter=',',create_using=nx.DiGraph(),nodetype=int)\n# https://stackoverflow.com/questions/9402255/drawing-a-huge-graph-with-networkx-and-matplotlib\n\npos=nx.spring_layout(subgraph)\nnx.draw(subgraph,pos,node_color='#A0CBE2',edge_color='#00bb5e',width=1,edge_cmap=plt.cm.Blues,with_labels=True)\nplt.savefig(\"graph_sample.pdf\")\nprint(nx.info(subgraph))\n```\n\n## Exploratory data analysis\n\n\n```python\n# No of Unique persons \nprint(\"The number of unique persons\",len(g.nodes()))\n```\n\n The number of unique persons 1862220\n\n\n\n```python\n# No of followers of each person\nindegree_dist = list(dict(g.in_degree()).values())\nindegree_dist.sort()\nplt.figure(figsize=(10,6))\nplt.plot(indegree_dist)\nplt.xlabel('Index No')\nplt.ylabel('No Of Followers')\nplt.show()\n```\n\n\n```python\nindegree_dist = list(dict(g.in_degree()).values())\nindegree_dist.sort()\nplt.figure(figsize=(10,6))\nplt.plot(indegree_dist[0:1500000])\nplt.xlabel('Index No')\nplt.ylabel('No Of Followers')\nplt.show()\n```\n\n\n```python\n# No Of people each person is following\noutdegree_dist = list(dict(g.out_degree()).values())\noutdegree_dist.sort()\nplt.figure(figsize=(10,6))\nplt.plot(outdegree_dist)\nplt.xlabel('Index No')\nplt.ylabel('No Of people each person is following')\nplt.show()\n```\n\n\n```python\nprint('No of persons who are not following anyone are {} ({:.2%})'.format(sum(np.array(outdegree_dist)==0),\n sum(np.array(outdegree_dist)==0)/len(outdegree_dist)))\n```\n\n No of persons who are not following anyone are 274512 (14.74%)\n\n\n\n```python\nprint('No of persons having zero followers are {} ({:.2%})'.format(sum(np.array(indegree_dist)==0),\n sum(np.array(indegree_dist)==0)/len(indegree_dist)))\n```\n\n No of persons having zero followers are 188043 (10.10%)\n\n\n\n```python\ncount=0\nfor i in g.nodes():\n if len(list(g.predecessors(i)))==0 :\n if len(list(g.successors(i)))==0:\n count+=1\nprint('No of persons those are not following anyone and also not having any followers are',count)\n```\n\n No of persons those are not following anyone and also not having any followers are 0\n\n\n## Negative sampling\n\nGenerating some edges which are not present in graph for supervised learning. In other words, we are generating bad links from graph which are not in graph and whose shortest path is greater than 2.\n\n\n```python\nr = csv.reader(open('train_woheader.csv','r'))\nedges = dict()\nfor edge in r:\n edges[(edge[0], edge[1])] = 1\nmissing_edges = set([])\n\nwith tqdm(total=9437519) as pbar:\n while (len(missing_edges)<9437519):\n a=random.randint(1, 1862220) \n b=random.randint(1, 1862220)\n tmp = edges.get((a,b),-1)\n if tmp == -1 and a!=b:\n try:\n if nx.shortest_path_length(g,source=a,target=b) > 2: \n missing_edges.add((a,b))\n else:\n continue \n except: \n missing_edges.add((a,b)) \n else:\n continue\n pbar.update(1)\npickle.dump(missing_edges,open('missing_edges_final.p','wb'))\n```\n\n\n HBox(children=(FloatProgress(value=0.0, max=9437519.0), HTML(value='')))\n\n\n \n\n\n\n```python\nlist(missing_edges)[:10]\n```\n\n\n\n\n [(885577, 1583706),\n (1487373, 176918),\n (1796282, 916021),\n (167023, 569005),\n (1204330, 1443051),\n (823309, 780941),\n (731061, 1684320),\n (283674, 455265),\n (412300, 691150),\n (586754, 854524)]\n\n\n\n## Train/test split\n\n> Tip: We will split positive links and negative links seperatly because we need only positive training data for creating graph and for feature generation.\n\n\n```python\n#reading total data df\ndf_pos = pd.read_csv('train.csv')\ndf_neg = pd.DataFrame(list(missing_edges), columns=['source_node', 'destination_node'])\n\nprint(\"Number of nodes in the graph with edges\", df_pos.shape[0])\nprint(\"Number of nodes in the graph without edges\", df_neg.shape[0])\n\n#Trian test split \n#Spiltted data into 80-20\nX_train_pos, X_test_pos, y_train_pos, y_test_pos = train_test_split(df_pos,np.ones(len(df_pos)),test_size=0.2, random_state=9)\nX_train_neg, X_test_neg, y_train_neg, y_test_neg = train_test_split(df_neg,np.zeros(len(df_neg)),test_size=0.2, random_state=9)\n\nprint('='*60)\nprint(\"Number of nodes in the train data graph with edges\", X_train_pos.shape[0],\"=\",y_train_pos.shape[0])\nprint(\"Number of nodes in the train data graph without edges\", X_train_neg.shape[0],\"=\", y_train_neg.shape[0])\nprint('='*60)\nprint(\"Number of nodes in the test data graph with edges\", X_test_pos.shape[0],\"=\",y_test_pos.shape[0])\nprint(\"Number of nodes in the test data graph without edges\", X_test_neg.shape[0],\"=\",y_test_neg.shape[0])\n\n#removing header and saving\nX_train_pos.to_csv('train_pos_after_eda.csv',header=False, index=False)\nX_test_pos.to_csv('test_pos_after_eda.csv',header=False, index=False)\nX_train_neg.to_csv('train_neg_after_eda.csv',header=False, index=False)\nX_test_neg.to_csv('test_neg_after_eda.csv',header=False, index=False)\n```\n\n Number of nodes in the graph with edges 9437519\n Number of nodes in the graph without edges 9437519\n ============================================================\n Number of nodes in the train data graph with edges 7550015 = 7550015\n Number of nodes in the train data graph without edges 7550015 = 7550015\n ============================================================\n Number of nodes in the test data graph with edges 1887504 = 1887504\n Number of nodes in the test data graph without edges 1887504 = 1887504\n\n\n\n```python\ntrain_graph=nx.read_edgelist('train_pos_after_eda.csv',delimiter=',',create_using=nx.DiGraph(),nodetype=int)\ntest_graph=nx.read_edgelist('test_pos_after_eda.csv',delimiter=',',create_using=nx.DiGraph(),nodetype=int)\nprint(nx.info(train_graph))\nprint(nx.info(test_graph))\n\n# finding the unique nodes in both train and test graphs\ntrain_nodes_pos = set(train_graph.nodes())\ntest_nodes_pos = set(test_graph.nodes())\n\ntrY_teY = len(train_nodes_pos.intersection(test_nodes_pos))\ntrY_teN = len(train_nodes_pos - test_nodes_pos)\nteY_trN = len(test_nodes_pos - train_nodes_pos)\n\nprint('no of people common in train and test -- ',trY_teY)\nprint('no of people present in train but not present in test -- ',trY_teN)\nprint('no of people present in test but not present in train -- ',teY_trN)\nprint(' % of people not there in Train but exist in Test in total Test data are {} %'.format(teY_trN/len(test_nodes_pos)*100))\n```\n\n Name: \n Type: DiGraph\n Number of nodes: 1780722\n Number of edges: 7550015\n Average in degree: 4.2399\n Average out degree: 4.2399\n Name: \n Type: DiGraph\n Number of nodes: 1144623\n Number of edges: 1887504\n Average in degree: 1.6490\n Average out degree: 1.6490\n no of people common in train and test -- 1063125\n no of people present in train but not present in test -- 717597\n no of people present in test but not present in train -- 81498\n % of people not there in Train but exist in Test in total Test data are 7.1200735962845405 %\n\n\n\n```python\nX_train_pos = pd.read_csv('train_pos_after_eda.csv', names=['source_node', 'destination_node'])\nX_test_pos = pd.read_csv('test_pos_after_eda.csv', names=['source_node', 'destination_node'])\nX_train_neg = pd.read_csv('train_neg_after_eda.csv', names=['source_node', 'destination_node'])\nX_test_neg = pd.read_csv('test_neg_after_eda.csv', names=['source_node', 'destination_node'])\n\nprint('='*60)\nprint(\"Number of nodes in the train data graph with edges\", X_train_pos.shape[0])\nprint(\"Number of nodes in the train data graph without edges\", X_train_neg.shape[0])\nprint('='*60)\nprint(\"Number of nodes in the test data graph with edges\", X_test_pos.shape[0])\nprint(\"Number of nodes in the test data graph without edges\", X_test_neg.shape[0])\n\nX_train = X_train_pos.append(X_train_neg,ignore_index=True)\ny_train = np.concatenate((y_train_pos,y_train_neg))\nX_test = X_test_pos.append(X_test_neg,ignore_index=True)\ny_test = np.concatenate((y_test_pos,y_test_neg)) \n\nX_train.to_csv('train_after_eda.csv',header=False,index=False)\nX_test.to_csv('test_after_eda.csv',header=False,index=False)\npd.DataFrame(y_train.astype(int)).to_csv('train_y.csv',header=False,index=False)\npd.DataFrame(y_test.astype(int)).to_csv('test_y.csv',header=False,index=False)\n```\n\n\n```python\nprint(\"Data points in train data\",X_train.shape)\nprint(\"Data points in test data\",X_test.shape)\nprint(\"Shape of traget variable in train\",y_train.shape)\nprint(\"Shape of traget variable in test\", y_test.shape)\n```\n\n Data points in train data (15100030, 2)\n Data points in test data (3775008, 2)\n Shape of traget variable in train (15100030,)\n Shape of traget variable in test (3775008,)\n\n\n## Feature engineering\n\n### Similarity measures\n\n#### Jaccard distance\n\n\\begin{equation}\nj = \\frac{|X\\cap Y|}{|X \\cup Y|} \n\\end{equation}\n\n\n```python\ndef jaccard_for_followees(a,b):\n try:\n if len(set(train_graph.successors(a))) == 0 | len(set(train_graph.successors(b))) == 0:\n return 0\n sim = (len(set(train_graph.successors(a)).intersection(set(train_graph.successors(b)))))/\\\n (len(set(train_graph.successors(a)).union(set(train_graph.successors(b)))))\n except:\n return 0\n return sim\n```\n\n\n```python\ndef jaccard_for_followers(a,b):\n try:\n if len(set(train_graph.predecessors(a))) == 0 | len(set(g.predecessors(b))) == 0:\n return 0\n sim = (len(set(train_graph.predecessors(a)).intersection(set(train_graph.predecessors(b)))))/\\\n (len(set(train_graph.predecessors(a)).union(set(train_graph.predecessors(b)))))\n return sim\n except:\n return 0\n```\n\n#### Cosine distance\n\n\\begin{equation}\nCosineDistance = \\frac{|X\\cap Y|}{|X|\\cdot|Y|} \n\\end{equation}\n\n\n```python\ndef cosine_for_followees(a,b):\n try:\n if len(set(train_graph.successors(a))) == 0 | len(set(train_graph.successors(b))) == 0:\n return 0\n sim = (len(set(train_graph.successors(a)).intersection(set(train_graph.successors(b)))))/\\\n (math.sqrt(len(set(train_graph.successors(a)))*len((set(train_graph.successors(b))))))\n return sim\n except:\n return 0\n```\n\n\n```python\ndef cosine_for_followers(a,b):\n try:\n \n if len(set(train_graph.predecessors(a))) == 0 | len(set(train_graph.predecessors(b))) == 0:\n return 0\n sim = (len(set(train_graph.predecessors(a)).intersection(set(train_graph.predecessors(b)))))/\\\n (math.sqrt(len(set(train_graph.predecessors(a))))*(len(set(train_graph.predecessors(b)))))\n return sim\n except:\n return 0\n```\n\n### Ranking measures\n\n#### Pagerank\n\n\n```python\ntrain_graph=nx.read_edgelist('train_pos_after_eda.csv',delimiter=',',create_using=nx.DiGraph(),nodetype=int)\npr = nx.pagerank(train_graph, alpha=0.85)\npickle.dump(pr,open('page_rank.p','wb'))\n```\n\n\n```python\nprint('min',pr[min(pr, key=pr.get)])\nprint('max',pr[max(pr, key=pr.get)])\n#for imputing to nodes which are not there in Train data\nprint('mean_pr',float(sum(pr.values())) / len(pr))\n```\n\n min 1.6556497245737814e-07\n max 2.7098251341935827e-05\n mean_pr 5.615699699389075e-07\n\n\n### Other graph features\n\n#### Shortest path\n\nGetting Shortest path between two nodes, and if any 2 given nodes have a direct path i.e directly connected then we are removing that edge and calculating path.\n\n\n```python\ndef compute_shortest_path_length(a,b):\n p=-1\n try:\n if train_graph.has_edge(a,b):\n train_graph.remove_edge(a,b)\n p= nx.shortest_path_length(train_graph,source=a,target=b)\n train_graph.add_edge(a,b)\n else:\n p= nx.shortest_path_length(train_graph,source=a,target=b)\n return p\n except:\n return -1\n```\n\n\n```python\n# unit test 1\ncompute_shortest_path_length(77697, 826021)\n```\n\n\n\n\n 10\n\n\n\n\n```python\n# unit test 2\ncompute_shortest_path_length(669354, 1635354)\n```\n\n\n\n\n -1\n\n\n\n#### Same community\n\n\n```python\nwcc = list(nx.weakly_connected_components(train_graph))\n```\n\n\n```python\ndef belongs_to_same_wcc(a,b):\n index = []\n if train_graph.has_edge(b,a):\n return 1\n if train_graph.has_edge(a,b):\n for i in wcc:\n if a in i:\n index= i\n break\n if (b in index):\n train_graph.remove_edge(a,b)\n if compute_shortest_path_length(a,b)==-1:\n train_graph.add_edge(a,b)\n return 0\n else:\n train_graph.add_edge(a,b)\n return 1\n else:\n return 0\n else:\n for i in wcc:\n if a in i:\n index= i\n break\n if(b in index):\n return 1\n else:\n return 0\n```\n\n#### Admaic/Adar index\n\nAdamic/Adar measures is defined as inverted sum of degrees of common neighbours for given two vertices: $A(x,y)=\\sum_{u \\in N(x) \\cap N(y)}\\frac{1}{log(|N(u)|)}$\n\n\n```python\ndef calc_adar_in(a,b):\n sum=0\n try:\n n=list(set(train_graph.successors(a)).intersection(set(train_graph.successors(b))))\n if len(n)!=0:\n for i in n:\n sum=sum+(1/np.log10(len(list(train_graph.predecessors(i)))))\n return sum\n else:\n return 0\n except:\n return 0\n```\n\n### Is person following back?\n\n\n```python\ndef follows_back(a,b):\n if train_graph.has_edge(b,a):\n return 1\n else:\n return 0\n```\n\n#### Katz centrality\n\nKatz centrality computes the centrality for a node based on the centrality of its neighbors. It is a generalization of the eigenvector centrality. The Katz centrality for node i is: $x_i = \\alpha \\sum_{j} A_{ij} x_j + \\beta$\n\n\n```python\nkatz = nx.katz.katz_centrality(train_graph,alpha=0.005,beta=1)\npickle.dump(katz,open('katz.p','wb'))\n```\n\n\n```python\nprint('min',katz[min(katz, key=katz.get)])\nprint('max',katz[max(katz, key=katz.get)])\nprint('mean',float(sum(katz.values())) / len(katz))\n```\n\n min 0.0007313532484065916\n max 0.003394554981699122\n mean 0.0007483800935562018\n\n\n## Checkpointing\n\n\n```python\n# !mkdir fbfndrec\n# %cd fbfndrec\n\n# !mv ../train.csv .\n# !mv ../test.csv .\n\n# !mv ../train_pos_after_eda.csv .\n# !mv ../test_pos_after_eda.csv .\n# !mv ../train_neg_after_eda.csv .\n# !mv ../test_neg_after_eda.csv .\n\n# !mv ../train_after_eda.csv .\n# !mv ../test_after_eda.csv .\n# !mv ../train_y.csv .\n# !mv ../test_y.csv .\n\n# !mv ../page_rank.p .\n# !mv ../katz.p .\n\n# !zip fbfndrec.zip ./*\n\n# !mv fbfndrec.zip /content/drive/MyDrive/TempData\n```\n", "meta": {"hexsha": "840811e00dbc9aede207ab7f0d78562a63bcdfad", "size": 238777, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "_notebooks/2022-01-28-fbfriend-recs.ipynb", "max_stars_repo_name": "recohut/notebook", "max_stars_repo_head_hexsha": "610670666a1c3d8ef430d42f712ff72ecdbd8f86", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "_notebooks/2022-01-28-fbfriend-recs.ipynb", "max_issues_repo_name": "recohut/notebook", "max_issues_repo_head_hexsha": "610670666a1c3d8ef430d42f712ff72ecdbd8f86", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2022-01-12T05:40:57.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-12T05:40:57.000Z", "max_forks_repo_path": "_notebooks/2022-01-28-fbfriend-recs.ipynb", "max_forks_repo_name": "recohut/notebook", "max_forks_repo_head_hexsha": "610670666a1c3d8ef430d42f712ff72ecdbd8f86", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 119388.5, "max_line_length": 238776, "alphanum_fraction": 0.9116037139, "converted": true, "num_tokens": 5842, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.4377591042952579}} {"text": "# Optimised symbolic finite difference computation with Devito\n\nWelcome to the interactive hands-on tutorial for [Devito](http://www.devitoproject.org). Devito is a domain-specific language (DSL) and code generation framework for the design of highly optimised finite difference kernels, and was primarily designed for use in seismic inversion methods. Devito utilises SymPy to allow the definition of matrix-free finite difference operators from high-level symbolic equations and generates optimised and automatically tuned code specific to a given target architecture.\n\nThis hands-on tutorial is intended to give you an initial flavour of the Devito framework and the power of symbolic computation. We will demonstrate how quickly explicit finite difference operators can be created from only a few lines of Python code, and how we can use them to implement complex imaging algorithms in very little time (literally!). \n\n## Installation and setup\n\nIf you're seeing this on the projector, navigate to the following url to access the Azure notebooks project:\n\n### TODO Add URL\n\nOnce here, clone the project to create your personal copy. Next, click on TODO to start the notebook container. \n\nNow you should see a set of Jupyter notebooks, inlcuding this one called ```00_index.ipynb```. Open your copy of this notebook and proceed further. \n\n\n```python\n!pip install -r requirements.txt\n```\n\n Collecting git+https://github.com/inducer/cgen (from -r requirements.txt (line 1))\n Cloning https://github.com/inducer/cgen to /tmp/pip-req-build-5ydpe78j\n Requirement already satisfied (use --upgrade to upgrade): cgen==2018.1 from git+https://github.com/inducer/cgen in /venv/lib/python3.6/site-packages (from -r requirements.txt (line 1))\n Collecting git+https://github.com/inducer/codepy (from -r requirements.txt (line 2))\n Cloning https://github.com/inducer/codepy to /tmp/pip-req-build-mpn21s66\n Requirement already satisfied (use --upgrade to upgrade): codepy==2017.2.2 from git+https://github.com/inducer/codepy in /venv/lib/python3.6/site-packages (from -r requirements.txt (line 2))\n Collecting git+https://github.com/opesci/devito (from -r requirements.txt (line 3))\n Cloning https://github.com/opesci/devito to /tmp/pip-req-build-axo_sfyj\n Requirement already satisfied: pytools>=2015.1.2 in /venv/lib/python3.6/site-packages (from cgen==2018.1->-r requirements.txt (line 1)) (2019.1)\n Requirement already satisfied: numpy>=1.6 in /venv/lib/python3.6/site-packages (from cgen==2018.1->-r requirements.txt (line 1)) (1.16.2)\n Requirement already satisfied: appdirs>=1.4.0 in /venv/lib/python3.6/site-packages (from codepy==2017.2.2->-r requirements.txt (line 2)) (1.4.3)\n Requirement already satisfied: six in /venv/lib/python3.6/site-packages (from codepy==2017.2.2->-r requirements.txt (line 2)) (1.12.0)\n Requirement already satisfied: sympy>=1.4 in /venv/lib/python3.6/site-packages (from devito==3.4+873.gc57eec36->-r requirements.txt (line 3)) (1.4)\n Requirement already satisfied: scipy in /venv/lib/python3.6/site-packages (from devito==3.4+873.gc57eec36->-r requirements.txt (line 3)) (1.2.1)\n Requirement already satisfied: pytest-runner in /venv/lib/python3.6/site-packages (from devito==3.4+873.gc57eec36->-r requirements.txt (line 3)) (4.4)\n 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devito==3.4+873.gc57eec36->-r requirements.txt (line 3)) (0.15.0)\n Requirement already satisfied: click in /venv/lib/python3.6/site-packages (from devito==3.4+873.gc57eec36->-r requirements.txt (line 3)) (7.0)\n Requirement already satisfied: codecov in /venv/lib/python3.6/site-packages (from devito==3.4+873.gc57eec36->-r requirements.txt (line 3)) (2.0.15)\n Requirement already satisfied: pytest-cov in /venv/lib/python3.6/site-packages (from devito==3.4+873.gc57eec36->-r requirements.txt (line 3)) (2.6.1)\n Requirement already satisfied: multidict in /venv/lib/python3.6/site-packages (from devito==3.4+873.gc57eec36->-r requirements.txt (line 3)) (4.5.2)\n Requirement already satisfied: frozendict in /venv/lib/python3.6/site-packages (from devito==3.4+873.gc57eec36->-r requirements.txt (line 3)) (1.2)\n Requirement already satisfied: anytree>=2.4.3 in /venv/lib/python3.6/site-packages (from devito==3.4+873.gc57eec36->-r requirements.txt (line 3)) (2.6.0)\n Requirement already 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requirements.txt (line 3)) (0.14.11)\n Requirement already satisfied: ptyprocess>=0.5 in /venv/lib/python3.6/site-packages (from pexpect; sys_platform != \"win32\"->ipython>=5.0.0->ipykernel->nbval->devito==3.4+873.gc57eec36->-r requirements.txt (line 3)) (0.6.0)\n Requirement already satisfied: wcwidth in /venv/lib/python3.6/site-packages (from prompt-toolkit<2.1.0,>=2.0.0->ipython>=5.0.0->ipykernel->nbval->devito==3.4+873.gc57eec36->-r requirements.txt (line 3)) (0.1.7)\n Installing collected packages: devito\n Running setup.py install for devito ... \u001b[?25ldone\n \u001b[?25hSuccessfully installed devito-3.4+873.gc57eec36\n\n\nOnce the above setup has finished, we should run a quick sanity check that we have everything that we need. The following cell should simply complete without any errors.\n\n\n```python\nfrom devito import *\nfrom examples.seismic import Model\n```\n\nFor one section of this tutorial we will have a quick look at the stencil compiler YASK. To verify YASK works smoothly you can run:\n\n\n```python\nimport yask\n```\n\n## Instructors and helpers\n\nThis tutorial will be given by:\n\n* **Navjot Kukreja** \n* **Lucas Cavalcante**\nTODO\n\n## Learning objectives\n\n* How to use [SymPy](http://www.sympy.org) and [Devito](http://www.devitoproject.org) to create simple finite difference expressions from governing equations\n* Creating Devito operators to perform highly optimized stencil computations from the symbolic kernel definitions\n* Create basic seismic modelling operator to model wave propagation for a seismic survey\n* Implement a functional FWI algorithm usign high-level components from [Devito](http://www.devitoproject.org) and `scipy.optimize`\n* Gain an overview of the various performance optimization techniques used in Devito operators\n\nTODO\n\n## Outline\n\nTODO\n\n* [Session 1: **Introduction to Devito**](01_introduction.ipynb)\n * Functions and derivatives **[5min]**\n * Exercise: A linear convection operator **[10min]**\n * Second derivatives and high-order stencils **[5min]**\n * Exercise 2: Making a wave! **[10min]**\n\n* Session 2: **Seismic Imaging**\n * [Full Waveform Inversion (FWI) with Devito](02a_fwi.ipynb)\n * [Integration with Scipy.optimize](02b_scipy_optimize.ipynb)\n * [Distributed processing with Dask](02c_dask.ipynb)\n * [Advanced imaging with Skimage](02d_skimage_tv.ipynb)\n \n\n* Session 3: [**Performance Optimization and Analysis**](03_performance.ipynb)\n * Introduction to performance optimization in Devito **[2min]**\n * Setup for shared-memory parallelism **[5min]**\n * Devito Symbolic Engine (DSE) **[5min]**\n * Devito Loop Engine (DLE) **[5min]**\n * Exercise 4: performance analysis of a TTI forward operator **[8min]**\n * A sneak peek at the YASK backend **[5min]**\n\n### Bonus Material and further reading\n\nTODO\n\n* [Opesci project webpage](http://www.opesci.org/)\n * [Devito documentation](http://www.opesci.org/devito/)\n* More detailed [introductory tutorials](http://www.opesci.org/devito/tutorials.html), covering the following topics:\n * Introduction to Devito with CFD\n * Introdcution to seismic imaging\n\n### References\n\n* M. Lange, N. Kukreja, F. Luporini, M. Louboutin, C. Yount, J. H\u00fcckelheim and G. Gorman. Optimised finite difference computation from symbolic equations. Accepted for publication in Proceedings of the 15th Python in Science Conference, 2017. [[doi:10.25080/shinma-7f4c6e7-00d](http://conference.scipy.org/proceedings/scipy2017/michael_lange.html)] [[arxiv](http://arxiv.org/abs/1707.03776)]\n\n* M. Louboutin, M. Lange, N. Kukreja, F. Herrmann, and G. Gorman. _Performance\nprediction of finite-difference solvers for different computer architectures_. Accepted\nfor publication in Computers & Geosciences, 2016, [doi:10.1016/j.cageo.2017.04.014](http://www.sciencedirect.com/science/article/pii/S0098300416304034)\n\n\nThis notebook is part of the tutorial \"Optimised Symbolic Finite Difference Computation with Devito\" presented at University of Sao Paulo April 2019.\n\n\n```python\n\n```\n", "meta": {"hexsha": "11a2392f0a39c34c03aec116b47c0da47a9c3c90", "size": 23208, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "00_index.ipynb", "max_stars_repo_name": "navjotk/devitoworkshop", "max_stars_repo_head_hexsha": "ebb5dcd40ba32caf2be520bfc420251c32ad2079", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "00_index.ipynb", "max_issues_repo_name": "navjotk/devitoworkshop", "max_issues_repo_head_hexsha": "ebb5dcd40ba32caf2be520bfc420251c32ad2079", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "00_index.ipynb", "max_forks_repo_name": "navjotk/devitoworkshop", "max_forks_repo_head_hexsha": "ebb5dcd40ba32caf2be520bfc420251c32ad2079", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 73.4430379747, "max_line_length": 515, "alphanum_fraction": 0.6719665633, "converted": true, "num_tokens": 6188, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953797290153, "lm_q2_score": 0.7577943767446202, "lm_q1q2_score": 0.43762275135464695}} {"text": "```python\n#mapping some polygons - \nimport pandas as pd\nimport geopandas as gpd\nfrom shapely.geometry.polygon import Polygon\nimport cartopy.crs as ccrs\nimport cartopy.feature as cfeature\nimport matplotlib.pyplot as plt\nfrom matplotlib import cm\nimport folium\nimport numpy as np\n%matplotlib inline\n```\n\n\n```python\n#dataframes for csv files - Buffers\ndfbuffer = pd.read_csv('bufferlatlongV2.csv', delimiter = ',').astype(float)\ndfhurdat = pd.read_csv('hurdatlatlong.csv', delimiter = ',')\ndfcoast = pd.read_csv('coastlatlongV2.csv', delimiter = ',').astype(float)\ndfbasin = pd.read_csv('jamesbasinlatlongV2.csv', delimiter = ',').astype(float)\n\n\n```\n\n\n```python\n#create the polygons\nbuffer_geom =Polygon(zip(dfbuffer['Lon'],dfbuffer['Lat']))\nbasin_geom = Polygon(zip(dfbasin['Lon'],dfbasin['Lat']))\ncrs = {'init' : 'epsg:4326'}\nbufferpoly = gpd.GeoDataFrame(index = [0], crs = crs, geometry = [buffer_geom])\nbufferpoly.to_file(filename = 'buffer.geojson', driver = 'GeoJSON')\nbasinpoly = gpd.GeoDataFrame(index = [0], crs = crs, geometry = [basin_geom])\nbasinpoly.to_file(filename = 'basin.geojson', driver = 'GeoJSON')\n```\n\n /Users/williampc/opt/anaconda3/envs/geop/lib/python3.9/site-packages/pyproj/crs/crs.py:53: FutureWarning: '+init=:' syntax is deprecated. ':' is the preferred initialization method. When making the change, be mindful of axis order changes: https://pyproj4.github.io/pyproj/stable/gotchas.html#axis-order-changes-in-proj-6\n return _prepare_from_string(\" \".join(pjargs))\n\n\n\n```python\nimport pandas as pd\nfrom datetime import datetime\ndef lat_lon_to_float (v):\n \"\"\"Convert strings from NHC to float locations\"\"\"\n if (v[-1] == 'S') or (v[-1] == 'W'):\n multiplier = -1\n else:\n multiplier = 1\n return float(v[:-1])*multiplier\n```\n\n\n```python\nhurdata = []\nwith open ('hurdat2.txt', 'r') as f:\n for line in f.readlines():\n if line.startswith('AL'):\n storm_id = line.split(',')\n storm_number = storm_id[0].strip()\n storm_name = storm_id[1].strip()\n else:\n location_line = line.split(',')\n dt = datetime.strptime(location_line[0] + location_line[1],\"%Y%m%d %H%M\")\n storm_status = location_line[3].strip()\n storm_lat = lat_lon_to_float(location_line[4].strip())\n storm_lon = lat_lon_to_float(location_line[5].strip())\n max_speed = float(location_line[6].strip())\n hurdata.append([storm_number,storm_name,storm_status,storm_lat,storm_lon,dt,max_speed])\n```\n\n\n```python\ndf = pd.DataFrame(hurdata, columns = ['Storm Number','Storm Name', 'Storm Status', 'Lat', 'Lon','Time', 'Max Speed'])\ndf.head()\nlen(df)\n\n```\n\n\n\n\n 51817\n\n\n\n\n```python\nfrom sympy import Point, Polygon\nimport pandas as pd\nimport geopandas as gpd\nfrom shapely.geometry import *\n# # #changing to a GeoDataFrame to create geometry series\n# hurdatgdf = gpd.GeoDataFrame(df, geometry=gpd.points_from_xy(df.Lon,df.Lat))\n# # hurdatgdf.head()\n# # len(hurdatgdf)\n# T = pd.merge(dfbuffer, dfhurdat, how='inner', on=['Lat', 'Lon'])\n# print(T)\n# len(T)\n# T\ndfinbuffer = pd.merge(dfhurdat,dfbuffer,on=['Lat','Lon'])\npd.set_option(\"display.max_rows\", None, \"display.max_columns\", None)\nprint(dfinbuffer)\n```\n\n Unnamed: 0 Storm Number Storm Name Storm Status Lat Lon \\\n 0 2874 AL021876 UNNAMED HU 36.0 -77.3 \n 1 3620 AL021879 UNNAMED HU 37.3 -75.4 \n 2 3620 AL021879 UNNAMED HU 37.3 -75.4 \n 3 4906 AL081885 UNNAMED EX 39.0 -78.0 \n 4 4906 AL081885 UNNAMED EX 39.0 -78.0 \n 5 4906 AL081885 UNNAMED EX 39.0 -78.0 \n 6 4906 AL081885 UNNAMED EX 39.0 -78.0 \n 7 4906 AL081885 UNNAMED EX 39.0 -78.0 \n 8 4906 AL081885 UNNAMED EX 39.0 -78.0 \n 9 4947 AL021886 UNNAMED TD 38.4 -76.9 \n 10 4947 AL021886 UNNAMED TD 38.4 -76.9 \n 11 4947 AL021886 UNNAMED TD 38.4 -76.9 \n 12 4947 AL021886 UNNAMED TD 38.4 -76.9 \n 13 4947 AL021886 UNNAMED TD 38.4 -76.9 \n 14 4947 AL021886 UNNAMED TD 38.4 -76.9 \n 15 8951 AL031899 UNNAMED HU 36.4 -75.3 \n 16 8951 AL031899 UNNAMED HU 36.4 -75.3 \n 17 8951 AL031899 UNNAMED HU 36.4 -75.3 \n 18 8951 AL031899 UNNAMED HU 36.4 -75.3 \n 19 9621 AL031901 UNNAMED HU 36.1 -75.6 \n 20 9621 AL031901 UNNAMED HU 36.1 -75.6 \n 21 9621 AL031901 UNNAMED HU 36.1 -75.6 \n 22 9621 AL031901 UNNAMED HU 36.1 -75.6 \n 23 9621 AL031901 UNNAMED HU 36.1 -75.6 \n 24 9621 AL031901 UNNAMED HU 36.1 -75.6 \n 25 9621 AL031901 UNNAMED HU 36.1 -75.6 \n 26 9622 AL031901 UNNAMED HU 36.0 -75.8 \n 27 9622 AL031901 UNNAMED HU 36.0 -75.8 \n 28 9622 AL031901 UNNAMED HU 36.0 -75.8 \n 29 45637 AL072007 GABRIELLE TS 36.0 -75.8 \n 30 45637 AL072007 GABRIELLE TS 36.0 -75.8 \n 31 45637 AL072007 GABRIELLE TS 36.0 -75.8 \n 32 9901 AL101901 UNNAMED EX 37.8 -81.4 \n 33 9901 AL101901 UNNAMED EX 37.8 -81.4 \n 34 20141 AL071943 UNNAMED TS 37.7 -75.7 \n 35 20141 AL071943 UNNAMED TS 37.7 -75.7 \n 36 20141 AL071943 UNNAMED TS 37.7 -75.7 \n 37 20141 AL071943 UNNAMED TS 37.7 -75.7 \n 38 20141 AL071943 UNNAMED TS 37.7 -75.7 \n 39 20141 AL071943 UNNAMED TS 37.7 -75.7 \n 40 22903 AL021952 ABLE TS 36.6 -79.9 \n 41 22903 AL021952 ABLE TS 36.6 -79.9 \n 42 22903 AL021952 ABLE TS 36.6 -79.9 \n 43 22903 AL021952 ABLE TS 36.6 -79.9 \n 44 22903 AL021952 ABLE TS 36.6 -79.9 \n 45 22903 AL021952 ABLE TS 36.6 -79.9 \n 46 27462 AL131964 UNNAMED TD 36.2 -75.5 \n 47 27462 AL131964 UNNAMED TD 36.2 -75.5 \n 48 27462 AL131964 UNNAMED TD 36.2 -75.5 \n 49 27462 AL131964 UNNAMED TD 36.2 -75.5 \n 50 27462 AL131964 UNNAMED TD 36.2 -75.5 \n 51 28542 AL141967 DORIA TS 37.3 -75.3 \n 52 28542 AL141967 DORIA TS 37.3 -75.3 \n 53 29875 AL011970 ALMA EX 37.0 -75.5 \n 54 32268 AL131975 ELOISE EX 37.5 -81.5 \n 55 32268 AL131975 ELOISE EX 37.5 -81.5 \n 56 38896 AL031994 BERYL TD 37.5 -81.5 \n 57 38896 AL031994 BERYL TD 37.5 -81.5 \n 58 40093 AL061996 FRAN TD 39.2 -79.9 \n 59 40093 AL061996 FRAN TD 39.2 -79.9 \n 60 42103 AL012001 ALLISON SD 35.9 -76.8 \n 61 42103 AL012001 ALLISON SD 35.9 -76.8 \n \n Time \n 0 1876-09-17 18:00:00 \n 1 1879-08-18 18:00:00 \n 2 1879-08-18 18:00:00 \n 3 1885-10-13 18:00:00 \n 4 1885-10-13 18:00:00 \n 5 1885-10-13 18:00:00 \n 6 1885-10-13 18:00:00 \n 7 1885-10-13 18:00:00 \n 8 1885-10-13 18:00:00 \n 9 1886-06-23 06:00:00 \n 10 1886-06-23 06:00:00 \n 11 1886-06-23 06:00:00 \n 12 1886-06-23 06:00:00 \n 13 1886-06-23 06:00:00 \n 14 1886-06-23 06:00:00 \n 15 1899-08-19 00:00:00 \n 16 1899-08-19 00:00:00 \n 17 1899-08-19 00:00:00 \n 18 1899-08-19 00:00:00 \n 19 1901-07-11 06:00:00 \n 20 1901-07-11 06:00:00 \n 21 1901-07-11 06:00:00 \n 22 1901-07-11 06:00:00 \n 23 1901-07-11 06:00:00 \n 24 1901-07-11 06:00:00 \n 25 1901-07-11 06:00:00 \n 26 1901-07-11 07:00:00 \n 27 1901-07-11 07:00:00 \n 28 1901-07-11 07:00:00 \n 29 2007-09-10 00:00:00 \n 30 2007-09-10 00:00:00 \n 31 2007-09-10 00:00:00 \n 32 1901-09-29 00:00:00 \n 33 1901-09-29 00:00:00 \n 34 1943-10-01 00:00:00 \n 35 1943-10-01 00:00:00 \n 36 1943-10-01 00:00:00 \n 37 1943-10-01 00:00:00 \n 38 1943-10-01 00:00:00 \n 39 1943-10-01 00:00:00 \n 40 1952-09-01 00:00:00 \n 41 1952-09-01 00:00:00 \n 42 1952-09-01 00:00:00 \n 43 1952-09-01 00:00:00 \n 44 1952-09-01 00:00:00 \n 45 1952-09-01 00:00:00 \n 46 1964-07-24 06:00:00 \n 47 1964-07-24 06:00:00 \n 48 1964-07-24 06:00:00 \n 49 1964-07-24 06:00:00 \n 50 1964-07-24 06:00:00 \n 51 1967-09-16 18:00:00 \n 52 1967-09-16 18:00:00 \n 53 1970-05-27 06:00:00 \n 54 1975-09-24 18:00:00 \n 55 1975-09-24 18:00:00 \n 56 1994-08-17 18:00:00 \n 57 1994-08-17 18:00:00 \n 58 1996-09-07 00:00:00 \n 59 1996-09-07 00:00:00 \n 60 2001-06-15 18:00:00 \n 61 2001-06-15 18:00:00 \n\n\n\n```python\n# total map with all storms and buffers\nimport matplotlib.pyplot as plt\nimport cartopy.crs as ccrs\nimport cartopy.feature as cfeature\n```\n\n\n```python\nplot_crs = ccrs.LambertConformal(central_longitude =-100., central_latitude = 45)\ndata_crs = ccrs.PlateCarree()\n```\n\n\n```python\nimport matplotlib.patches as mpatches\nimport matplotlib.pyplot as plt\nimport shapely.geometry as sgeom\nimport cartopy.crs as ccrs\nimport cartopy.io.shapereader as shpreader\n\n\ndef basincoords():\n \"\"\"\n Return a list of latitudes and a list of longitudes (lons, lats)\n for James River Basin \n\n \"\"\"\n basinlon = dfbasin['Lon']\n basinlat = dfbasin['Lat']\n\n return basinlon, basinlat\n\ndef buffercoords():\n \"\"\"\n Return a list of latitudes and a list of longitudes (lons, lats)\n for James River Basin \n\n \"\"\"\n bufferlon = dfbuffer['Lon']\n bufferlat = dfbuffer['Lat']\n \n\n return bufferlon, bufferlat\nbufferlon, bufferlat = buffercoords()\nbasinlon, basinlat = basincoords()\n\n#ax.set_title('James River Basin and Buffer - Virginia, USA')\n\n# turn the lons and lats into a shapely LineString\nbuffer = sgeom.LineString(zip(bufferlon, bufferlat))\nbasin = sgeom.LineString(zip(basinlon, basinlat))\n\n\nfig = plt.figure(figsize = (7,7))\nax = plt.subplot(1,1,1,projection = plot_crs)\n\nax.set_extent([-85,-70,32,40],data_crs)\nax.coastlines('50m', edgecolor = 'k', linewidth = 0.75)\nax.add_feature(cfeature.STATES, linewidth = 0.5)\nax.add_feature(cfeature.RIVERS, linewidth = 0.85)\nax.add_feature(cfeature.OCEAN)\nax.add_geometries([buffer], ccrs.PlateCarree(),facecolor='#C8A2C8', alpha=0.5)\nax.add_geometries([basin], ccrs.PlateCarree(),facecolor='rgb', edgecolor='k')\n\n\n\n#for storm_number in vahurc['Storm Number'].unique():\n #data = df[jameshurc['Storm Number'] == storm_number]\n #print(vahurc)\nax.plot(dfinbuffer['Lon'], dfinbuffer['Lat'])#,dfbuffer['Lon'],dfbuffer['Lat'],dfbasin['Lat'],dfbasin['Lon'], transform = data_crs)\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "f43431b2fc810e6c5b62f31ea16137d29b5fc1a2", "size": 85343, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "buffer_plus_storms.ipynb", "max_stars_repo_name": "williampc8985/VT-JamesRiver", "max_stars_repo_head_hexsha": "6bacd10f4fd6158db74973ddc1abd89b650efc9f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "buffer_plus_storms.ipynb", "max_issues_repo_name": "williampc8985/VT-JamesRiver", "max_issues_repo_head_hexsha": "6bacd10f4fd6158db74973ddc1abd89b650efc9f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, 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YES\n2. YES", "lm_q1_score": 0.7577943712746407, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.4376227371750013}} {"text": "\n# Week 9 March 1-5: Resampling Techniques, Bootstrap and Blocking\n\n \n**Morten Hjorth-Jensen Email morten.hjorth-jensen@fys.uio.no**, Department of Physics and Center fo Computing in Science Education, University of Oslo, Oslo, Norway and Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, East Lansing, Michigan, USA\n\nDate: **Mar 19, 2021**\n\nCopyright 1999-2021, Morten Hjorth-Jensen Email morten.hjorth-jensen@fys.uio.no. Released under CC Attribution-NonCommercial 4.0 license\n\n\n\n\n## Overview of week 9, March 1-5\n**Topics.**\n\n* Top down approach first, what we need to code\n\n* Resampling Techniques and statistics: Bootstrap and Blocking\n\n\n\n**Teaching Material, videos and written material.**\n\n* Overview video on the [Bootstrap method](https://www.youtube.com/watch?v=O_Fj4q8lgmc&ab_channel=MarinStatsLectures-RProgramming%26Statistics)\n\n* These Lecture notes\n\n* [Marius Johnson's Master thesis on the Blocking Method](https://www.duo.uio.no/bitstream/handle/10852/68360/PhysRevE.98.043304.pdf?sequence=2&isAllowed=y)\n\n\n\n\n## The top-down approach, part 1\n\nLast week we discusse dhow to implement a gradient descent method like the simplest possible gradient descent with a simple learning rate as parameter to tune. We repeat the codes here.\n\n\n```python\n%matplotlib inline\n\n# 2-electron VMC code for 2dim quantum dot with importance sampling\n# Using gaussian rng for new positions and Metropolis- Hastings \n# Added energy minimization\n# Common imports\nfrom math import exp, sqrt\nfrom random import random, seed, normalvariate\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom mpl_toolkits.mplot3d import Axes3D\nfrom matplotlib import cm\nfrom matplotlib.ticker import LinearLocator, FormatStrFormatter\nimport sys\n\n\n\n# Trial wave function for the 2-electron quantum dot in two dims\ndef WaveFunction(r,alpha,beta):\n r1 = r[0,0]**2 + r[0,1]**2\n r2 = r[1,0]**2 + r[1,1]**2\n r12 = sqrt((r[0,0]-r[1,0])**2 + (r[0,1]-r[1,1])**2)\n deno = r12/(1+beta*r12)\n return exp(-0.5*alpha*(r1+r2)+deno)\n\n# Local energy for the 2-electron quantum dot in two dims, using analytical local energy\ndef LocalEnergy(r,alpha,beta):\n \n r1 = (r[0,0]**2 + r[0,1]**2)\n r2 = (r[1,0]**2 + r[1,1]**2)\n r12 = sqrt((r[0,0]-r[1,0])**2 + (r[0,1]-r[1,1])**2)\n deno = 1.0/(1+beta*r12)\n deno2 = deno*deno\n return 0.5*(1-alpha*alpha)*(r1 + r2) +2.0*alpha + 1.0/r12+deno2*(alpha*r12-deno2+2*beta*deno-1.0/r12)\n\n# Derivate of wave function ansatz as function of variational parameters\ndef DerivativeWFansatz(r,alpha,beta):\n \n WfDer = np.zeros((2), np.double)\n r1 = (r[0,0]**2 + r[0,1]**2)\n r2 = (r[1,0]**2 + r[1,1]**2)\n r12 = sqrt((r[0,0]-r[1,0])**2 + (r[0,1]-r[1,1])**2)\n deno = 1.0/(1+beta*r12)\n deno2 = deno*deno\n WfDer[0] = -0.5*(r1+r2)\n WfDer[1] = -r12*r12*deno2\n return WfDer\n\n# Setting up the quantum force for the two-electron quantum dot, recall that it is a vector\ndef QuantumForce(r,alpha,beta):\n\n qforce = np.zeros((NumberParticles,Dimension), np.double)\n r12 = sqrt((r[0,0]-r[1,0])**2 + (r[0,1]-r[1,1])**2)\n deno = 1.0/(1+beta*r12)\n qforce[0,:] = -2*r[0,:]*alpha*(r[0,:]-r[1,:])*deno*deno/r12\n qforce[1,:] = -2*r[1,:]*alpha*(r[1,:]-r[0,:])*deno*deno/r12\n return qforce\n \n\n# Computing the derivative of the energy and the energy \ndef EnergyMinimization(alpha, beta):\n\n NumberMCcycles= 10000\n # Parameters in the Fokker-Planck simulation of the quantum force\n D = 0.5\n TimeStep = 0.05\n # positions\n PositionOld = np.zeros((NumberParticles,Dimension), np.double)\n PositionNew = np.zeros((NumberParticles,Dimension), np.double)\n # Quantum force\n QuantumForceOld = np.zeros((NumberParticles,Dimension), np.double)\n QuantumForceNew = np.zeros((NumberParticles,Dimension), np.double)\n\n # seed for rng generator \n seed()\n energy = 0.0\n DeltaE = 0.0\n EnergyDer = np.zeros((2), np.double)\n DeltaPsi = np.zeros((2), np.double)\n DerivativePsiE = np.zeros((2), np.double)\n #Initial position\n for i in range(NumberParticles):\n for j in range(Dimension):\n PositionOld[i,j] = normalvariate(0.0,1.0)*sqrt(TimeStep)\n wfold = WaveFunction(PositionOld,alpha,beta)\n QuantumForceOld = QuantumForce(PositionOld,alpha, beta)\n\n #Loop over MC MCcycles\n for MCcycle in range(NumberMCcycles):\n #Trial position moving one particle at the time\n for i in range(NumberParticles):\n for j in range(Dimension):\n PositionNew[i,j] = PositionOld[i,j]+normalvariate(0.0,1.0)*sqrt(TimeStep)+\\\n QuantumForceOld[i,j]*TimeStep*D\n wfnew = WaveFunction(PositionNew,alpha,beta)\n QuantumForceNew = QuantumForce(PositionNew,alpha, beta)\n GreensFunction = 0.0\n for j in range(Dimension):\n GreensFunction += 0.5*(QuantumForceOld[i,j]+QuantumForceNew[i,j])*\\\n\t (D*TimeStep*0.5*(QuantumForceOld[i,j]-QuantumForceNew[i,j])-\\\n PositionNew[i,j]+PositionOld[i,j])\n \n GreensFunction = exp(GreensFunction)\n ProbabilityRatio = GreensFunction*wfnew**2/wfold**2\n #Metropolis-Hastings test to see whether we accept the move\n if random() <= ProbabilityRatio:\n for j in range(Dimension):\n PositionOld[i,j] = PositionNew[i,j]\n QuantumForceOld[i,j] = QuantumForceNew[i,j]\n wfold = wfnew\n DeltaE = LocalEnergy(PositionOld,alpha,beta)\n DerPsi = DerivativeWFansatz(PositionOld,alpha,beta)\n DeltaPsi += DerPsi\n energy += DeltaE\n DerivativePsiE += DerPsi*DeltaE\n \n # We calculate mean values\n energy /= NumberMCcycles\n DerivativePsiE /= NumberMCcycles\n DeltaPsi /= NumberMCcycles\n EnergyDer = 2*(DerivativePsiE-DeltaPsi*energy)\n return energy, EnergyDer\n\n\n#Here starts the main program with variable declarations\nNumberParticles = 2\nDimension = 2\n# guess for variational parameters\nalpha = 0.95\nbeta = 0.3\n# Set up iteration using stochastic gradient method\nEnergy = 0\nEDerivative = np.zeros((2), np.double)\n# Learning rate eta, max iterations, need to change to adaptive learning rate\neta = 0.01\nMaxIterations = 50\niter = 0\n\nEnergies = np.zeros(MaxIterations)\nEnergyDerivatives1 = np.zeros(MaxIterations)\nEnergyDerivatives2 = np.zeros(MaxIterations)\nAlphaValues = np.zeros(MaxIterations)\nBetaValues = np.zeros(MaxIterations)\n\nwhile iter < MaxIterations:\n Energy, EDerivative = EnergyMinimization(alpha,beta)\n alphagradient = EDerivative[0]\n betagradient = EDerivative[1]\n alpha -= eta*alphagradient\n beta -= eta*betagradient \n Energies[iter] = Energy\n EnergyDerivatives1[iter] = EDerivative[0] \n EnergyDerivatives2[iter] = EDerivative[1] \n AlphaValues[iter] = alpha\n BetaValues[iter] = beta\n iter += 1\n\n#nice printout with Pandas\nimport pandas as pd\nfrom pandas import DataFrame\npd.set_option('max_columns', 6)\ndata ={'Alpha':AlphaValues,'Beta':BetaValues,'Energy':Energies,'Alpha Derivative':EnergyDerivatives1,'Beta Derivative':EnergyDerivatives2}\n\nframe = pd.DataFrame(data)\nprint(frame)\n```\n\n Alpha Beta Energy Alpha Derivative Beta Derivative\n 0 0.952471 0.302037 3.016387 -0.247135 -0.203671\n 1 0.954795 0.303940 3.029544 -0.232340 -0.190346\n 2 0.956858 0.305676 3.015198 -0.206349 -0.173568\n 3 0.958977 0.307486 3.006236 -0.211873 -0.181037\n 4 0.960851 0.309150 3.012114 -0.187402 -0.166353\n 5 0.962882 0.310941 3.007021 -0.203075 -0.179168\n 6 0.964631 0.312468 2.999393 -0.174974 -0.152620\n 7 0.966714 0.314139 3.009335 -0.208262 -0.167151\n 8 0.968463 0.315547 3.020319 -0.174855 -0.140742\n 9 0.970040 0.316911 3.014918 -0.157758 -0.136428\n 10 0.971736 0.318407 3.001852 -0.169535 -0.149638\n 11 0.973292 0.319758 3.004794 -0.155687 -0.135037\n 12 0.974575 0.320990 2.997130 -0.128237 -0.123228\n 13 0.976202 0.322293 2.999610 -0.162677 -0.130268\n 14 0.977609 0.323469 3.013079 -0.140761 -0.117682\n 15 0.978797 0.324532 3.008125 -0.118754 -0.106286\n 16 0.980063 0.325636 3.000419 -0.126608 -0.110420\n 17 0.981101 0.326593 3.009269 -0.103811 -0.095629\n 18 0.982087 0.327532 3.004387 -0.098621 -0.093912\n 19 0.983044 0.328518 2.999726 -0.095676 -0.098639\n 20 0.984256 0.329586 2.995441 -0.121211 -0.106785\n 21 0.985339 0.330563 2.998627 -0.108343 -0.097689\n 22 0.986245 0.331438 3.001941 -0.090580 -0.087516\n 23 0.987186 0.332305 2.998803 -0.094048 -0.086666\n 24 0.988028 0.333181 2.994558 -0.084252 -0.087604\n 25 0.988790 0.333976 3.003508 -0.076148 -0.079508\n 26 0.989604 0.334765 2.998644 -0.081400 -0.078893\n 27 0.990332 0.335491 3.003694 -0.072851 -0.072578\n 28 0.991020 0.336210 3.001020 -0.068826 -0.071905\n 29 0.991715 0.336880 3.004798 -0.069469 -0.067018\n 30 0.992301 0.337509 3.007172 -0.058630 -0.062906\n 31 0.992932 0.338159 3.003623 -0.063095 -0.065009\n 32 0.993519 0.338815 3.003110 -0.058667 -0.065637\n 33 0.994145 0.339486 2.998508 -0.062584 -0.067014\n 34 0.994722 0.340136 2.998125 -0.057742 -0.065075\n 35 0.995276 0.340754 3.000603 -0.055382 -0.061730\n 36 0.995745 0.341244 3.010393 -0.046852 -0.049045\n 37 0.996204 0.341782 3.002136 -0.045980 -0.053780\n 38 0.996676 0.342352 2.997916 -0.047181 -0.056985\n 39 0.997130 0.342935 2.998770 -0.045353 -0.058314\n 40 0.997575 0.343455 3.000545 -0.044469 -0.052040\n 41 0.997916 0.343902 3.005916 -0.034128 -0.044679\n 42 0.998283 0.344348 3.006643 -0.036676 -0.044615\n 43 0.998697 0.344871 3.000212 -0.041480 -0.052282\n 44 0.998969 0.345272 3.003977 -0.027170 -0.040150\n 45 0.999345 0.345747 3.000847 -0.037629 -0.047456\n 46 0.999667 0.346191 3.003118 -0.032147 -0.044381\n 47 1.000018 0.346676 2.997754 -0.035130 -0.048558\n 48 1.000334 0.347132 2.998584 -0.031554 -0.045563\n 49 1.000564 0.347502 3.003228 -0.023081 -0.036966\n\n\n## What have we done?\nThe exact energy is $3.0$ for an oscillator frequency $\\omega =1$\n(with $\\hbar =1$). We note however that with this learning rate and\nnumber of iterations, the energies and the derivatives are not yet\nconverged.\n\nWe can improve upon this by using the algorithms provided by the **optimize** package in Python.\nOne of these algorithms is Broyden\u2013Fletcher\u2013Goldfarb\u2013Shanno (BFGS) algorithm. \n\nThe optimization problem is to minimize $f(\\mathbf {x} )$ where\n$\\mathbf {x}$ is a vector in $R^{n}$, and $f$ is a differentiable\nscalar function. There are no constraints on the values that $\\mathbf{x}$ can take.\n\nThe algorithm begins at an initial estimate for the optimal value\n$\\mathbf {x}_{0}$ and proceeds iteratively to get a better estimate at\neach stage.\n\nThe search direction $p_k$ at stage $k$ is given by the solution of the analogue of the Newton equation\n\n$$\nB_{k}\\mathbf {p} _{k}=-\\nabla f(\\mathbf {x}_{k}),\n$$\n\nwhere $B_{k}$ is an approximation to the Hessian matrix, which is\nupdated iteratively at each stage, and $\\nabla f(\\mathbf {x} _{k})$\nis the gradient of the function\nevaluated at $x_k$. \nA line search in the direction $p_k$ is then used to\nfind the next point $x_{k+1}$ by minimising\n\n$$\nf(\\mathbf {x}_{k}+\\alpha \\mathbf {p}_{k}),\n$$\n\nover the scalar $\\alpha > 0$.\n\n\n\n## Code part 2\nThe modified code here uses the BFGS algorithm but performs now a\nproduction run and writes to file all average values of the\nenergy.\n\n\n```python\n# 2-electron VMC code for 2dim quantum dot with importance sampling\n# Using gaussian rng for new positions and Metropolis- Hastings \n# Added energy minimization\nfrom math import exp, sqrt\nfrom random import random, seed, normalvariate\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom mpl_toolkits.mplot3d import Axes3D\nfrom matplotlib import cm\nfrom matplotlib.ticker import LinearLocator, FormatStrFormatter\nfrom scipy.optimize import minimize\nimport sys\nimport os\n\n# Where to save data files\nPROJECT_ROOT_DIR = \"Results\"\nDATA_ID = \"Results/EnergyMin\"\n\nif not os.path.exists(PROJECT_ROOT_DIR):\n os.mkdir(PROJECT_ROOT_DIR)\n\nif not os.path.exists(DATA_ID):\n os.makedirs(DATA_ID)\n\ndef data_path(dat_id):\n return os.path.join(DATA_ID, dat_id)\n\noutfile = open(data_path(\"Energies.dat\"),'w')\n\n\n# Trial wave function for the 2-electron quantum dot in two dims\ndef WaveFunction(r,alpha,beta):\n r1 = r[0,0]**2 + r[0,1]**2\n r2 = r[1,0]**2 + r[1,1]**2\n r12 = sqrt((r[0,0]-r[1,0])**2 + (r[0,1]-r[1,1])**2)\n deno = r12/(1+beta*r12)\n return exp(-0.5*alpha*(r1+r2)+deno)\n\n# Local energy for the 2-electron quantum dot in two dims, using analytical local energy\ndef LocalEnergy(r,alpha,beta):\n \n r1 = (r[0,0]**2 + r[0,1]**2)\n r2 = (r[1,0]**2 + r[1,1]**2)\n r12 = sqrt((r[0,0]-r[1,0])**2 + (r[0,1]-r[1,1])**2)\n deno = 1.0/(1+beta*r12)\n deno2 = deno*deno\n return 0.5*(1-alpha*alpha)*(r1 + r2) +2.0*alpha + 1.0/r12+deno2*(alpha*r12-deno2+2*beta*deno-1.0/r12)\n\n# Derivate of wave function ansatz as function of variational parameters\ndef DerivativeWFansatz(r,alpha,beta):\n \n WfDer = np.zeros((2), np.double)\n r1 = (r[0,0]**2 + r[0,1]**2)\n r2 = (r[1,0]**2 + r[1,1]**2)\n r12 = sqrt((r[0,0]-r[1,0])**2 + (r[0,1]-r[1,1])**2)\n deno = 1.0/(1+beta*r12)\n deno2 = deno*deno\n WfDer[0] = -0.5*(r1+r2)\n WfDer[1] = -r12*r12*deno2\n return WfDer\n\n# Setting up the quantum force for the two-electron quantum dot, recall that it is a vector\ndef QuantumForce(r,alpha,beta):\n\n qforce = np.zeros((NumberParticles,Dimension), np.double)\n r12 = sqrt((r[0,0]-r[1,0])**2 + (r[0,1]-r[1,1])**2)\n deno = 1.0/(1+beta*r12)\n qforce[0,:] = -2*r[0,:]*alpha*(r[0,:]-r[1,:])*deno*deno/r12\n qforce[1,:] = -2*r[1,:]*alpha*(r[1,:]-r[0,:])*deno*deno/r12\n return qforce\n \n\n# Computing the derivative of the energy and the energy \ndef EnergyDerivative(x0):\n\n \n # Parameters in the Fokker-Planck simulation of the quantum force\n D = 0.5\n TimeStep = 0.05\n # positions\n PositionOld = np.zeros((NumberParticles,Dimension), np.double)\n PositionNew = np.zeros((NumberParticles,Dimension), np.double)\n # Quantum force\n QuantumForceOld = np.zeros((NumberParticles,Dimension), np.double)\n QuantumForceNew = np.zeros((NumberParticles,Dimension), np.double)\n\n energy = 0.0\n DeltaE = 0.0\n alpha = x0[0]\n beta = x0[1]\n EnergyDer = 0.0\n DeltaPsi = 0.0\n DerivativePsiE = 0.0 \n #Initial position\n for i in range(NumberParticles):\n for j in range(Dimension):\n PositionOld[i,j] = normalvariate(0.0,1.0)*sqrt(TimeStep)\n wfold = WaveFunction(PositionOld,alpha,beta)\n QuantumForceOld = QuantumForce(PositionOld,alpha, beta)\n\n #Loop over MC MCcycles\n for MCcycle in range(NumberMCcycles):\n #Trial position moving one particle at the time\n for i in range(NumberParticles):\n for j in range(Dimension):\n PositionNew[i,j] = PositionOld[i,j]+normalvariate(0.0,1.0)*sqrt(TimeStep)+\\\n QuantumForceOld[i,j]*TimeStep*D\n wfnew = WaveFunction(PositionNew,alpha,beta)\n QuantumForceNew = QuantumForce(PositionNew,alpha, beta)\n GreensFunction = 0.0\n for j in range(Dimension):\n GreensFunction += 0.5*(QuantumForceOld[i,j]+QuantumForceNew[i,j])*\\\n\t (D*TimeStep*0.5*(QuantumForceOld[i,j]-QuantumForceNew[i,j])-\\\n PositionNew[i,j]+PositionOld[i,j])\n \n GreensFunction = exp(GreensFunction)\n ProbabilityRatio = GreensFunction*wfnew**2/wfold**2\n #Metropolis-Hastings test to see whether we accept the move\n if random() <= ProbabilityRatio:\n for j in range(Dimension):\n PositionOld[i,j] = PositionNew[i,j]\n QuantumForceOld[i,j] = QuantumForceNew[i,j]\n wfold = wfnew\n DeltaE = LocalEnergy(PositionOld,alpha,beta)\n DerPsi = DerivativeWFansatz(PositionOld,alpha,beta)\n DeltaPsi += DerPsi\n energy += DeltaE\n DerivativePsiE += DerPsi*DeltaE\n \n # We calculate mean values\n energy /= NumberMCcycles\n DerivativePsiE /= NumberMCcycles\n DeltaPsi /= NumberMCcycles\n EnergyDer = 2*(DerivativePsiE-DeltaPsi*energy)\n return EnergyDer\n\n\n# Computing the expectation value of the local energy \ndef Energy(x0):\n # Parameters in the Fokker-Planck simulation of the quantum force\n D = 0.5\n TimeStep = 0.05\n # positions\n PositionOld = np.zeros((NumberParticles,Dimension), np.double)\n PositionNew = np.zeros((NumberParticles,Dimension), np.double)\n # Quantum force\n QuantumForceOld = np.zeros((NumberParticles,Dimension), np.double)\n QuantumForceNew = np.zeros((NumberParticles,Dimension), np.double)\n\n energy = 0.0\n DeltaE = 0.0\n alpha = x0[0]\n beta = x0[1]\n #Initial position\n for i in range(NumberParticles):\n for j in range(Dimension):\n PositionOld[i,j] = normalvariate(0.0,1.0)*sqrt(TimeStep)\n wfold = WaveFunction(PositionOld,alpha,beta)\n QuantumForceOld = QuantumForce(PositionOld,alpha, beta)\n\n #Loop over MC MCcycles\n for MCcycle in range(NumberMCcycles):\n #Trial position moving one particle at the time\n for i in range(NumberParticles):\n for j in range(Dimension):\n PositionNew[i,j] = PositionOld[i,j]+normalvariate(0.0,1.0)*sqrt(TimeStep)+\\\n QuantumForceOld[i,j]*TimeStep*D\n wfnew = WaveFunction(PositionNew,alpha,beta)\n QuantumForceNew = QuantumForce(PositionNew,alpha, beta)\n GreensFunction = 0.0\n for j in range(Dimension):\n GreensFunction += 0.5*(QuantumForceOld[i,j]+QuantumForceNew[i,j])*\\\n\t (D*TimeStep*0.5*(QuantumForceOld[i,j]-QuantumForceNew[i,j])-\\\n PositionNew[i,j]+PositionOld[i,j])\n \n GreensFunction = exp(GreensFunction)\n ProbabilityRatio = GreensFunction*wfnew**2/wfold**2\n #Metropolis-Hastings test to see whether we accept the move\n if random() <= ProbabilityRatio:\n for j in range(Dimension):\n PositionOld[i,j] = PositionNew[i,j]\n QuantumForceOld[i,j] = QuantumForceNew[i,j]\n wfold = wfnew\n DeltaE = LocalEnergy(PositionOld,alpha,beta)\n energy += DeltaE\n if Printout: \n outfile.write('%f\\n' %(DeltaE))# ('%f\\n' %(energy/(MCcycle+1.0))) \n # We calculate mean values\n energy /= NumberMCcycles\n return energy\n\n#Here starts the main program with variable declarations\nNumberParticles = 2\nDimension = 2\n# seed for rng generator \nseed()\n# Monte Carlo cycles for parameter optimization\nPrintout = False\nNumberMCcycles= 10000\n# guess for variational parameters\nx0 = np.array([0.9,0.2])\n# Using Broydens method to find optimal parameters\nres = minimize(Energy, x0, method='BFGS', jac=EnergyDerivative, options={'gtol': 1e-4,'disp': True})\nx0 = res.x\n# Compute the energy again with the optimal parameters and increased number of Monte Cycles\nNumberMCcycles= 2**19\nPrintout = True\nFinalEnergy = Energy(x0)\nEResult = np.array([FinalEnergy,FinalEnergy])\noutfile.close()\n#nice printout with Pandas\nimport pandas as pd\nfrom pandas import DataFrame\ndata ={'Optimal Parameters':x0, 'Final Energy':EResult}\nframe = pd.DataFrame(data)\nprint(frame)\n```\n\n Warning: Desired error not necessarily achieved due to precision loss.\n Current function value: 2.998422\n Iterations: 3\n Function evaluations: 22\n Gradient evaluations: 10\n Optimal Parameters Final Energy\n 0 0.962938 3.000951\n 1 0.457425 3.000951\n\n\nNote that the **minimize** function returns the final values for the\nvariable $\\alpha=x0[0]$ and $\\beta=x0[1]$ in the array $x$.\n\nWhen we have found the minimum, we use these optimal parameters to perform a production run of energies.\nThe output is in turn written to file and is used, together with resampling methods like the **blocking method**,\nto obtain the best possible estimate for the standard deviation. The optimal minimum is, even with our guess, rather close to the exact value of $3.0$ a.u.\n\nThe [sampling\nfunctions](https://github.com/CompPhysics/ComputationalPhysics2/tree/gh-pages/doc/Programs/Resampling)\ncan be used to perform both a blocking analysis, or a standard\nbootstrap and jackknife analysis.\n\n## How do we proceed?\n\nThere are several paths which can be chosen. One is to extend the\nbrute force gradient descent method with an adapative stochastic\ngradient. There are several examples of this. A recent approach based\non [the Langevin equations](https://arxiv.org/pdf/1805.09416.pdf)\nseems like a promising approach for general and possibly non-convex\noptimization problems.\n\nHere we would like to point out that our next step is now to use the\noptimal values for our variational parameters and use these as inputs\nto a production run. Here we would output values of the energy and\nperform for example a blocking analysis of the results in order to get\na best possible estimate of the standard deviation.\n\n\n## Resampling analysis\n\nThe next step is then to use the above data sets and perform a\nresampling analysis, either using say the Bootstrap method or the\nBlocking method. Since the data will be correlated, we would recommend\nto use the non-iid Bootstrap code here. The theoretical background for these resampling methods is found in the [statistical analysis lecture notes](http://compphysics.github.io/ComputationalPhysics2/doc/pub/statanalysis/html/statanalysis.html)\n\nHere we have tailored the codes to the output file from the previous example. We present first the bootstrap resampling with non-iid stochastic event.\n\n\n```python\n# Common imports\nimport os\n\n# Where to save the figures and data files\nDATA_ID = \"Results/EnergyMin\"\n\ndef data_path(dat_id):\n return os.path.join(DATA_ID, dat_id)\n\ninfile = open(data_path(\"Energies.dat\"),'r')\n\nfrom numpy import std, mean, concatenate, arange, loadtxt, zeros, ceil\nfrom numpy.random import randint\nfrom time import time\n\n\ndef tsboot(data,statistic,R,l):\n t = zeros(R); n = len(data); k = int(ceil(float(n)/l));\n inds = arange(n); t0 = time()\n \n # time series bootstrap\n for i in range(R):\n # construct bootstrap sample from\n # k chunks of data. The chunksize is l\n _data = concatenate([data[j:j+l] for j in randint(0,n-l,k)])[0:n];\n t[i] = statistic(_data)\n\n # analysis\n print (\"Runtime: %g sec\" % (time()-t0)); print (\"Bootstrap Statistics :\")\n print (\"original bias std. error\")\n print (\"%8g %14g %15g\" % (statistic(data), \\\n mean(t) - statistic(data), \\\n std(t) ))\n return t\n# Read in data\nX = loadtxt(infile)\n# statistic to be estimated. Takes two args.\n# arg1: the data\ndef stat(data):\n return mean(data)\nt = tsboot(X, stat, 2**12, 2**10)\n```\n\n Runtime: 2.91977 sec\n Bootstrap Statistics :\n original bias std. error\n 3.00095 -3.49515e-06 0.000311354\n\n\nThe blocking code, based on the article of [Marius Jonsson](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.98.043304) is given here\n\n\n```python\n# Common imports\nimport os\n\n# Where to save the figures and data files\nDATA_ID = \"Results/EnergyMin\"\n\ndef data_path(dat_id):\n return os.path.join(DATA_ID, dat_id)\n\ninfile = open(data_path(\"Energies.dat\"),'r')\n\nfrom numpy import log2, zeros, mean, var, sum, loadtxt, arange, array, cumsum, dot, transpose, diagonal, sqrt\nfrom numpy.linalg import inv\n\ndef block(x):\n # preliminaries\n n = len(x)\n d = int(log2(n))\n s, gamma = zeros(d), zeros(d)\n mu = mean(x)\n\n # estimate the auto-covariance and variances \n # for each blocking transformation\n for i in arange(0,d):\n n = len(x)\n # estimate autocovariance of x\n gamma[i] = (n)**(-1)*sum( (x[0:(n-1)]-mu)*(x[1:n]-mu) )\n # estimate variance of x\n s[i] = var(x)\n # perform blocking transformation\n x = 0.5*(x[0::2] + x[1::2])\n \n # generate the test observator M_k from the theorem\n M = (cumsum( ((gamma/s)**2*2**arange(1,d+1)[::-1])[::-1] ) )[::-1]\n\n # we need a list of magic numbers\n q =array([6.634897,9.210340, 11.344867, 13.276704, 15.086272, 16.811894, 18.475307, 20.090235, 21.665994, 23.209251, 24.724970, 26.216967, 27.688250, 29.141238, 30.577914, 31.999927, 33.408664, 34.805306, 36.190869, 37.566235, 38.932173, 40.289360, 41.638398, 42.979820, 44.314105, 45.641683, 46.962942, 48.278236, 49.587884, 50.892181])\n\n # use magic to determine when we should have stopped blocking\n for k in arange(0,d):\n if(M[k] < q[k]):\n break\n if (k >= d-1):\n print(\"Warning: Use more data\")\n return mu, s[k]/2**(d-k)\n\n\nx = loadtxt(infile)\n(mean, var) = block(x) \nstd = sqrt(var)\nimport pandas as pd\nfrom pandas import DataFrame\ndata ={'Mean':[mean], 'STDev':[std]}\nframe = pd.DataFrame(data,index=['Values'])\nprint(frame)\n```\n\n Mean STDev\n Values 3.000951 0.000314\n\n\n## Why resampling methods ?\n**Statistical analysis.**\n\n * Our simulations can be treated as *computer experiments*. This is particularly the case for Monte Carlo methods\n\n * The results can be analysed with the same statistical tools as we would use analysing experimental data.\n\n * As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.\n\n \n\n## Statistical analysis\n * As in other experiments, many numerical experiments have two classes of errors:\n\n * Statistical errors\n\n * Systematical errors\n\n\n * Statistical errors can be estimated using standard tools from statistics\n\n * Systematical errors are method specific and must be treated differently from case to case.\n\n \n\n## Statistics\nThe *probability distribution function (PDF)* is a function\n$p(x)$ on the domain which, in the discrete case, gives us the\nprobability or relative frequency with which these values of $X$ occur:\n\n$$\np(x) = \\mathrm{prob}(X=x)\n$$\n\nIn the continuous case, the PDF does not directly depict the\nactual probability. Instead we define the probability for the\nstochastic variable to assume any value on an infinitesimal interval\naround $x$ to be $p(x)dx$. The continuous function $p(x)$ then gives us\nthe *density* of the probability rather than the probability\nitself. The probability for a stochastic variable to assume any value\non a non-infinitesimal interval $[a,\\,b]$ is then just the integral:\n\n$$\n\\mathrm{prob}(a\\leq X\\leq b) = \\int_a^b p(x)dx\n$$\n\nQualitatively speaking, a stochastic variable represents the values of\nnumbers chosen as if by chance from some specified PDF so that the\nselection of a large set of these numbers reproduces this PDF.\n\n\n\n\n## Statistics, moments\nA particularly useful class of special expectation values are the\n*moments*. The $n$-th moment of the PDF $p$ is defined as\nfollows:\n\n$$\n\\langle x^n\\rangle \\equiv \\int\\! x^n p(x)\\,dx\n$$\n\nThe zero-th moment $\\langle 1\\rangle$ is just the normalization condition of\n$p$. The first moment, $\\langle x\\rangle$, is called the *mean* of $p$\nand often denoted by the letter $\\mu$:\n\n$$\n\\langle x\\rangle = \\mu \\equiv \\int\\! x p(x)\\,dx\n$$\n\n## Statistics, central moments\nA special version of the moments is the set of *central moments*,\nthe n-th central moment defined as:\n\n$$\n\\langle (x-\\langle x \\rangle )^n\\rangle \\equiv \\int\\! (x-\\langle x\\rangle)^n p(x)\\,dx\n$$\n\nThe zero-th and first central moments are both trivial, equal $1$ and\n$0$, respectively. But the second central moment, known as the\n*variance* of $p$, is of particular interest. For the stochastic\nvariable $X$, the variance is denoted as $\\sigma^2_X$ or $\\mathrm{var}(X)$:\n\n\n
    \n\n$$\n\\begin{equation}\n\\sigma^2_X\\ \\ =\\ \\ \\mathrm{var}(X) = \\langle (x-\\langle x\\rangle)^2\\rangle =\n\\int\\! (x-\\langle x\\rangle)^2 p(x)\\,dx\n\\label{_auto1} \\tag{1}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n = \\int\\! \\left(x^2 - 2 x \\langle x\\rangle^{2} +\n \\langle x\\rangle^2\\right)p(x)\\,dx\n\\label{_auto2} \\tag{2}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n = \\langle x^2\\rangle - 2 \\langle x\\rangle\\langle x\\rangle + \\langle x\\rangle^2\n\\label{_auto3} \\tag{3}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n = \\langle x^2\\rangle - \\langle x\\rangle^2\n\\label{_auto4} \\tag{4}\n\\end{equation}\n$$\n\nThe square root of the variance, $\\sigma =\\sqrt{\\langle (x-\\langle x\\rangle)^2\\rangle}$ is called the *standard deviation* of $p$. It is clearly just the RMS (root-mean-square)\nvalue of the deviation of the PDF from its mean value, interpreted\nqualitatively as the *spread* of $p$ around its mean.\n\n\n\n## Statistics, covariance\nAnother important quantity is the so called covariance, a variant of\nthe above defined variance. Consider again the set $\\{X_i\\}$ of $n$\nstochastic variables (not necessarily uncorrelated) with the\nmultivariate PDF $P(x_1,\\dots,x_n)$. The *covariance* of two\nof the stochastic variables, $X_i$ and $X_j$, is defined as follows:\n\n$$\n\\mathrm{cov}(X_i,\\,X_j) \\equiv \\langle (x_i-\\langle x_i\\rangle)(x_j-\\langle x_j\\rangle)\\rangle\n\\nonumber\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n=\n\\int\\!\\cdots\\!\\int\\!(x_i-\\langle x_i \\rangle)(x_j-\\langle x_j \\rangle)\\,\nP(x_1,\\dots,x_n)\\,dx_1\\dots dx_n\n\\label{eq:def_covariance} \\tag{5}\n\\end{equation}\n$$\n\nwith\n\n$$\n\\langle x_i\\rangle =\n\\int\\!\\cdots\\!\\int\\!x_i\\,P(x_1,\\dots,x_n)\\,dx_1\\dots dx_n\n$$\n\n## Statistics, more covariance\nIf we consider the above covariance as a matrix $C_{ij}=\\mathrm{cov}(X_i,\\,X_j)$, then the diagonal elements are just the familiar\nvariances, $C_{ii} = \\mathrm{cov}(X_i,\\,X_i) = \\mathrm{var}(X_i)$. It turns out that\nall the off-diagonal elements are zero if the stochastic variables are\nuncorrelated. This is easy to show, keeping in mind the linearity of\nthe expectation value. Consider the stochastic variables $X_i$ and\n$X_j$, ($i\\neq j$):\n\n\n
    \n\n$$\n\\begin{equation}\n\\mathrm{cov}(X_i,\\,X_j) = \\langle(x_i-\\langle x_i\\rangle)(x_j-\\langle x_j\\rangle)\\rangle\n\\label{_auto5} \\tag{6}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n=\\langle x_i x_j - x_i\\langle x_j\\rangle - \\langle x_i\\rangle x_j + \\langle x_i\\rangle\\langle x_j\\rangle\\rangle \n\\label{_auto6} \\tag{7}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n=\\langle x_i x_j\\rangle - \\langle x_i\\langle x_j\\rangle\\rangle - \\langle \\langle x_i\\rangle x_j\\rangle +\n\\langle \\langle x_i\\rangle\\langle x_j\\rangle\\rangle\n\\label{_auto7} \\tag{8}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n=\\langle x_i x_j\\rangle - \\langle x_i\\rangle\\langle x_j\\rangle - \\langle x_i\\rangle\\langle x_j\\rangle +\n\\langle x_i\\rangle\\langle x_j\\rangle\n\\label{_auto8} \\tag{9}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n=\\langle x_i x_j\\rangle - \\langle x_i\\rangle\\langle x_j\\rangle\n\\label{_auto9} \\tag{10}\n\\end{equation}\n$$\n\n## Statistics, independent variables\nIf $X_i$ and $X_j$ are independent, we get \n$\\langle x_i x_j\\rangle =\\langle x_i\\rangle\\langle x_j\\rangle$, resulting in $\\mathrm{cov}(X_i, X_j) = 0\\ \\ (i\\neq j)$.\n\nAlso useful for us is the covariance of linear combinations of\nstochastic variables. Let $\\{X_i\\}$ and $\\{Y_i\\}$ be two sets of\nstochastic variables. Let also $\\{a_i\\}$ and $\\{b_i\\}$ be two sets of\nscalars. Consider the linear combination:\n\n$$\nU = \\sum_i a_i X_i \\qquad V = \\sum_j b_j Y_j\n$$\n\nBy the linearity of the expectation value\n\n$$\n\\mathrm{cov}(U, V) = \\sum_{i,j}a_i b_j \\mathrm{cov}(X_i, Y_j)\n$$\n\n## Statistics, more variance\nNow, since the variance is just $\\mathrm{var}(X_i) = \\mathrm{cov}(X_i, X_i)$, we get\nthe variance of the linear combination $U = \\sum_i a_i X_i$:\n\n\n
    \n\n$$\n\\begin{equation}\n\\mathrm{var}(U) = \\sum_{i,j}a_i a_j \\mathrm{cov}(X_i, X_j)\n\\label{eq:variance_linear_combination} \\tag{11}\n\\end{equation}\n$$\n\nAnd in the special case when the stochastic variables are\nuncorrelated, the off-diagonal elements of the covariance are as we\nknow zero, resulting in:\n\n2\n2\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\mathrm{var}(\\sum_i a_i X_i) = \\sum_i a_i^2 \\mathrm{var}(X_i)\n$$\n\nwhich will become very useful in our study of the error in the mean\nvalue of a set of measurements.\n\n\n\n## Statistics and stochastic processes\nA *stochastic process* is a process that produces sequentially a\nchain of values:\n\n$$\n\\{x_1, x_2,\\dots\\,x_k,\\dots\\}.\n$$\n\nWe will call these\nvalues our *measurements* and the entire set as our measured\n*sample*. The action of measuring all the elements of a sample\nwe will call a stochastic *experiment* since, operationally,\nthey are often associated with results of empirical observation of\nsome physical or mathematical phenomena; precisely an experiment. We\nassume that these values are distributed according to some \nPDF $p_X^{\\phantom X}(x)$, where $X$ is just the formal symbol for the\nstochastic variable whose PDF is $p_X^{\\phantom X}(x)$. Instead of\ntrying to determine the full distribution $p$ we are often only\ninterested in finding the few lowest moments, like the mean\n$\\mu_X^{\\phantom X}$ and the variance $\\sigma_X^{\\phantom X}$.\n\n\n\n\n\n## Statistics and sample variables\nIn practical situations a sample is always of finite size. Let that\nsize be $n$. The expectation value of a sample, the *sample mean*, is then defined as follows:\n\n$$\n\\bar{x}_n \\equiv \\frac{1}{n}\\sum_{k=1}^n x_k\n$$\n\nThe *sample variance* is:\n\n$$\n\\mathrm{var}(x) \\equiv \\frac{1}{n}\\sum_{k=1}^n (x_k - \\bar{x}_n)^2\n$$\n\nits square root being the *standard deviation of the sample*. The\n*sample covariance* is:\n\n$$\n\\mathrm{cov}(x)\\equiv\\frac{1}{n}\\sum_{kl}(x_k - \\bar{x}_n)(x_l - \\bar{x}_n)\n$$\n\n## Statistics, sample variance and covariance\nNote that the sample variance is the sample covariance without the\ncross terms. In a similar manner as the covariance in Eq. ([5](#eq:def_covariance)) is a measure of the correlation between\ntwo stochastic variables, the above defined sample covariance is a\nmeasure of the sequential correlation between succeeding measurements\nof a sample.\n\nThese quantities, being known experimental values, differ\nsignificantly from and must not be confused with the similarly named\nquantities for stochastic variables, mean $\\mu_X$, variance $\\mathrm{var}(X)$\nand covariance $\\mathrm{cov}(X,Y)$.\n\n\n\n## Statistics, law of large numbers\nThe law of large numbers\nstates that as the size of our sample grows to infinity, the sample\nmean approaches the true mean $\\mu_X^{\\phantom X}$ of the chosen PDF:\n\n$$\n\\lim_{n\\to\\infty}\\bar{x}_n = \\mu_X^{\\phantom X}\n$$\n\nThe sample mean $\\bar{x}_n$ works therefore as an estimate of the true\nmean $\\mu_X^{\\phantom X}$.\n\nWhat we need to find out is how good an approximation $\\bar{x}_n$ is to\n$\\mu_X^{\\phantom X}$. In any stochastic measurement, an estimated\nmean is of no use to us without a measure of its error. A quantity\nthat tells us how well we can reproduce it in another experiment. We\nare therefore interested in the PDF of the sample mean itself. Its\nstandard deviation will be a measure of the spread of sample means,\nand we will simply call it the *error* of the sample mean, or\njust sample error, and denote it by $\\mathrm{err}_X^{\\phantom X}$. In\npractice, we will only be able to produce an *estimate* of the\nsample error since the exact value would require the knowledge of the\ntrue PDFs behind, which we usually do not have.\n\n\n\n\n## Statistics, more on sample error\nLet us first take a look at what happens to the sample error as the\nsize of the sample grows. In a sample, each of the measurements $x_i$\ncan be associated with its own stochastic variable $X_i$. The\nstochastic variable $\\overline X_n$ for the sample mean $\\bar{x}_n$ is\nthen just a linear combination, already familiar to us:\n\n$$\n\\overline X_n = \\frac{1}{n}\\sum_{i=1}^n X_i\n$$\n\nAll the coefficients are just equal $1/n$. The PDF of $\\overline X_n$,\ndenoted by $p_{\\overline X_n}(x)$ is the desired PDF of the sample\nmeans.\n\n\n\n## Statistics\nThe probability density of obtaining a sample mean $\\bar x_n$\nis the product of probabilities of obtaining arbitrary values $x_1,\nx_2,\\dots,x_n$ with the constraint that the mean of the set $\\{x_i\\}$\nis $\\bar x_n$:\n\n$$\np_{\\overline X_n}(x) = \\int p_X^{\\phantom X}(x_1)\\cdots\n\\int p_X^{\\phantom X}(x_n)\\ \n\\delta\\!\\left(x - \\frac{x_1+x_2+\\dots+x_n}{n}\\right)dx_n \\cdots dx_1\n$$\n\nAnd in particular we are interested in its variance $\\mathrm{var}(\\overline X_n)$.\n\n\n\n\n\n## Statistics, central limit theorem\nIt is generally not possible to express $p_{\\overline X_n}(x)$ in a\nclosed form given an arbitrary PDF $p_X^{\\phantom X}$ and a number\n$n$. But for the limit $n\\to\\infty$ it is possible to make an\napproximation. The very important result is called *the central limit theorem*. It tells us that as $n$ goes to infinity,\n$p_{\\overline X_n}(x)$ approaches a Gaussian distribution whose mean\nand variance equal the true mean and variance, $\\mu_{X}^{\\phantom X}$\nand $\\sigma_{X}^{2}$, respectively:\n\n\n
    \n\n$$\n\\begin{equation}\n\\lim_{n\\to\\infty} p_{\\overline X_n}(x) =\n\\left(\\frac{n}{2\\pi\\mathrm{var}(X)}\\right)^{1/2}\ne^{-\\frac{n(x-\\bar x_n)^2}{2\\mathrm{var}(X)}}\n\\label{eq:central_limit_gaussian} \\tag{12}\n\\end{equation}\n$$\n\n## Statistics, more technicalities\nThe desired variance\n$\\mathrm{var}(\\overline X_n)$, i.e. the sample error squared\n$\\mathrm{err}_X^2$, is given by:\n\n\n
    \n\n$$\n\\begin{equation}\n\\mathrm{err}_X^2 = \\mathrm{var}(\\overline X_n) = \\frac{1}{n^2}\n\\sum_{ij} \\mathrm{cov}(X_i, X_j)\n\\label{eq:error_exact} \\tag{13}\n\\end{equation}\n$$\n\nWe see now that in order to calculate the exact error of the sample\nwith the above expression, we would need the true means\n$\\mu_{X_i}^{\\phantom X}$ of the stochastic variables $X_i$. To\ncalculate these requires that we know the true multivariate PDF of all\nthe $X_i$. But this PDF is unknown to us, we have only got the measurements of\none sample. The best we can do is to let the sample itself be an\nestimate of the PDF of each of the $X_i$, estimating all properties of\n$X_i$ through the measurements of the sample.\n\n\n\n\n## Statistics\nOur estimate of $\\mu_{X_i}^{\\phantom X}$ is then the sample mean $\\bar x$\nitself, in accordance with the the central limit theorem:\n\n$$\n\\mu_{X_i}^{\\phantom X} = \\langle x_i\\rangle \\approx \\frac{1}{n}\\sum_{k=1}^n x_k = \\bar x\n$$\n\nUsing $\\bar x$ in place of $\\mu_{X_i}^{\\phantom X}$ we can give an\n*estimate* of the covariance in Eq. ([13](#eq:error_exact))\n\n$$\n\\mathrm{cov}(X_i, X_j) = \\langle (x_i-\\langle x_i\\rangle)(x_j-\\langle x_j\\rangle)\\rangle\n\\approx\\langle (x_i - \\bar x)(x_j - \\bar{x})\\rangle,\n$$\n\nresulting in\n\n$$\n\\frac{1}{n} \\sum_{l}^n \\left(\\frac{1}{n}\\sum_{k}^n (x_k -\\bar x_n)(x_l - \\bar x_n)\\right)=\\frac{1}{n}\\frac{1}{n} \\sum_{kl} (x_k -\\bar x_n)(x_l - \\bar x_n)=\\frac{1}{n}\\mathrm{cov}(x)\n$$\n\n## Statistics and sample variance\nBy the same procedure we can use the sample variance as an\nestimate of the variance of any of the stochastic variables $X_i$\n\n$$\n\\mathrm{var}(X_i)=\\langle x_i - \\langle x_i\\rangle\\rangle \\approx \\langle x_i - \\bar x_n\\rangle\\nonumber,\n$$\n\nwhich is approximated as\n\n\n
    \n\n$$\n\\begin{equation}\n\\mathrm{var}(X_i)\\approx \\frac{1}{n}\\sum_{k=1}^n (x_k - \\bar x_n)=\\mathrm{var}(x)\n\\label{eq:var_estimate_i_think} \\tag{14}\n\\end{equation}\n$$\n\nNow we can calculate an estimate of the error\n$\\mathrm{err}_X^{\\phantom X}$ of the sample mean $\\bar x_n$:\n\n$$\n\\mathrm{err}_X^2\n=\\frac{1}{n^2}\\sum_{ij} \\mathrm{cov}(X_i, X_j) \\nonumber\n$$\n\n$$\n\\approx\\frac{1}{n^2}\\sum_{ij}\\frac{1}{n}\\mathrm{cov}(x) =\\frac{1}{n^2}n^2\\frac{1}{n}\\mathrm{cov}(x)\\nonumber\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n=\\frac{1}{n}\\mathrm{cov}(x)\n\\label{eq:error_estimate} \\tag{15}\n\\end{equation}\n$$\n\nwhich is nothing but the sample covariance divided by the number of\nmeasurements in the sample.\n\n\n\n## Statistics, uncorrelated results\n\nIn the special case that the measurements of the sample are\nuncorrelated (equivalently the stochastic variables $X_i$ are\nuncorrelated) we have that the off-diagonal elements of the covariance\nare zero. This gives the following estimate of the sample error:\n\n$$\n\\mathrm{err}_X^2=\\frac{1}{n^2}\\sum_{ij} \\mathrm{cov}(X_i, X_j) =\n\\frac{1}{n^2} \\sum_i \\mathrm{var}(X_i),\n$$\n\nresulting in\n\n\n
    \n\n$$\n\\begin{equation}\n\\mathrm{err}_X^2\\approx \\frac{1}{n^2} \\sum_i \\mathrm{var}(x)= \\frac{1}{n}\\mathrm{var}(x)\n\\label{eq:error_estimate_uncorrel} \\tag{16}\n\\end{equation}\n$$\n\nwhere in the second step we have used Eq. ([14](#eq:var_estimate_i_think)).\nThe error of the sample is then just its standard deviation divided by\nthe square root of the number of measurements the sample contains.\nThis is a very useful formula which is easy to compute. It acts as a\nfirst approximation to the error, but in numerical experiments, we\ncannot overlook the always present correlations.\n\n\n\n## Statistics, computations\nFor computational purposes one usually splits up the estimate of\n$\\mathrm{err}_X^2$, given by Eq. ([15](#eq:error_estimate)), into two\nparts\n\n$$\n\\mathrm{err}_X^2 = \\frac{1}{n}\\mathrm{var}(x) + \\frac{1}{n}(\\mathrm{cov}(x)-\\mathrm{var}(x)),\n$$\n\nwhich equals\n\n\n
    \n\n$$\n\\begin{equation}\n\\frac{1}{n^2}\\sum_{k=1}^n (x_k - \\bar x_n)^2 +\\frac{2}{n^2}\\sum_{k**This module has been validated against the LALSuite SEOBNRv3/SEOBNRv3_opt code that was reviewed and approved for LIGO parameter estimation by the LIGO Scientific Collaboration.**\n\n\n\n# Table of Contents\n$$\\label{toc}$$\n\nThis module is organized as follows, matching the \"steps\" listed in [BCD2006](https://arxiv.org/abs/gr-qc/0508067):\n\n0. [Introduction:](#intro) The Physical System of Interest\n1. [Step 1:](#step1) Initial Coordinate Choice\n1. [Step 2:](#step2) Compute ${\\bf r}$, ${\\bf p}_{r}$, ${\\bf p}_{\\theta}$, and ${\\bf p}_{\\phi}$\n1. [Step 3:](#step3) Rotate $\\hat{\\bf L} \\to {\\bf e}_{z}$\n1. [Step 4:](#step4) Compute $\\dot{\\bf r}$\n1. [Step 5:](#step5) Invert the rotation of Step 3\n1. [Output](#latex_pdf_output): Output this module to $\\LaTeX$-formatted PDF\n\n\n\n# Introduction: The Physical System of Interest \\[Back to [top](#toc)\\]\n$$\\label{intro}$$\n\nConsider two compact objects (e.g. black holes or neutron stars) with masses $m_{1}$, $m_{2}$ (in solar masses) and spin angular momenta ${\\bf S}_{1}$, ${\\bf S}_{2}$ in a binary system. The spinning effective one-body (\"SEOB\") Hamiltonian $H_{\\rm real}$ (see [BB2010](https://arxiv.org/abs/0912.3517) Equation (5.69)) describes the dynamics of this system. We seek initial conditions for nonadiabatic evolutions of such a system, and follow [BCD2006](https://arxiv.org/abs/gr-qc/0508067) Section IV A.\n\nPlease note that throughout this notebook we adpot the following conventions:\n1. $c = G = 1$ where $c$ is the speed of light in a vacuum and $G$ is Newton's gravitational constant,\n1. hatted vectors (e.g. $\\hat{\\bf L}_{N}$) usually denote scaled or unit vectors, and\n1. the initial inclination angle $\\iota$ is chosen to be zero.\n\nChoose a spatial coordinate basis $\\left\\{ {\\bf e}_{0}, {\\bf e}_{1}, {\\bf e}_{2} \\right\\}$ so that the initial separation vector ${\\bf r}$ between the compact objects lies along the ${\\bf e}_{0}$-axis. We start with the following parameters:\n1. binary parameters $m_{1}$, $m_{2}$, ${\\bf S}_{1}$, and ${\\bf S}_{2}$,\n1. initial frequency $f$, and\n1. SEOB model parameters (constants dependent on $m_{1}$, $m_{2}$).\n\nOur goal is to produce initial dynamical variables\n1. ${\\bf x} = \\left( x, y, z \\right)$, and\n1. ${\\bf p} = \\left( p_{x}, p_{y}, p_{z} \\right)$.\n\nWe include below the physical parameters necessary to compute the initial conditions. Solving for the initial conditions requires computing $\\frac{ \\partial H_{\\rm real} }{ \\partial p_{i} }$ for $i \\in \\left\\{ r, \\theta, \\phi \\right\\}$, which we defer to another module.\n\nPlease note that in [BCD2006](https://arxiv.org/abs/gr-qc/0508067) the initial conditions are solved for given an initial separation; here we use a given initial frequency instead. The difference is in our approach to solving Equation (4.8). Our approach also differs from that found in LALSuite's XLALSimIMRSpinEOBInitialConditionsPrec() function (file: LALSimIMRSpinEOBInitialConditionsPrec.c) because we choose our intial coordinate system so that the inclination angle $\\iota$ is zero (this simplifies their step one and step five).\n\nBesides the physical parameters, we also need the [Euler\u2013Mascheroni constant](https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant) $\\gamma$ and the [geomtrized](https://en.wikipedia.org/wiki/Geometrized_unit_system) solar mass $\\mathcal{M}_{\\odot}$, both hard-coded in LALSuite with the significant digits shown below. (The following links are directed to the appropriate LALSuite documentation: [$\\gamma$](https://lscsoft.docs.ligo.org/lalsuite/lal/group___l_a_l_constants__h.html#gac6af32574ff8acaeeafe8bf422281e98) and [$\\mathcal{M}_{\\odot}$](https://lscsoft.docs.ligo.org/lalsuite/lal/group___l_a_l_constants__h.html#gab83f8c705dda3fd0bb2d5f2470bb9cdd).)\n\n\n```python\nimport sympy as sp\nm1,m2,S10,S11,S12,S20,S21,S22,f = sp.symbols(\"m1 m2 S10 S11 S12 S20 S21 S22 f\",real=True)\ndHdpx,dHdpy,dHdpz = sp.symbols(\"dHdpx dHdpy dHdpz\",real=True)\n\ngamma = 0.577215664901532860606512090082402431\nMsol = 4.925491025543575903411922162094833998e-6\n```\n\n\n\n# Step 1: Initial Coordinate Choice \\[Back to [top](#toc)\\]\n$$\\label{step1}$$\n\n\n\n## Mass terms \\[Back to [top](#toc)\\]\n$$\\label{massterms}$$\n\nWe begin by defining the following useful quantities:\n\n\\begin{align*}\n M &= m_{1} + m_{2} \\\\\n \\eta &= \\frac{ m_{1} m_{2} }{ M^{2} }\n\\end{align*}\n\n\n```python\nM = m1 + m2\nMsq = M*M\neta = m1*m2/Msq\n```\n\n\n\n## Spin terms \\[Back to [top](#toc)\\]\n$$\\label{spinterms}$$\n\nCan we find a source that says how the basis for the orientation of ${\\bf S}_{1}$, ${\\bf S}_{2}$ is chosen? For now I assume it's in the $\\hat{\\bf r}, \\hat{\\bf v}, \\hat{\\bf L}_{N}$-frame, which is suggestd by the fact that LALSuite rotates them using the same rotation matrix as rotates $\\hat{\\bf r}$, $\\hat{\\bf v}$, and $\\hat{\\bf L}_{N}$ if $\\iota \\not= 0$. Since we assumed $G = c = 1$, we make the spin angular momenta dimensionless via:\n\n\\begin{align*}\n \\hat{\\bf S}_{1} &= \\frac{ 1 }{ M^{2} } {\\bf S}_{1} \\\\\n \\hat{\\bf S}_{2} &= \\frac{ 1 }{ M^{2} } {\\bf S}_{2}\n\\end{align*}\n\n\n```python\nS1hat0 = S10/Msq\nS1hat1 = S11/Msq\nS1hat2 = S12/Msq\nS2hat0 = S20/Msq\nS2hat1 = S21/Msq\nS2hat2 = S22/Msq\n```\n\n\n\n## Normalized Position $\\hat{\\bf r}$ \\[Back to [top](#toc)\\]\n$$\\label{rhat}$$\n\nWe assumed that the initial separation vector ${\\bf r}$ lies along the ${\\bf e}_{0}$-axis, so the normalized initial separation vector $\\hat{\\bf r}$ is given by\n\n\\begin{align*}\n \\hat{\\bf r}^{0} &= 1 \\\\\n \\hat{\\bf r}^{1} &= 0 \\\\\n \\hat{\\bf r}^{2} &= 0.\n\\end{align*}\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Lines 803--805.\n\n\n```python\nrhat0 = 1.\nrhat1 = 0.\nrhat2 = 0.\n```\n\n\n\n## Normalized Orbital Angular Momenutm $\\hat{\\bf L}_{N}$ \\[Back to [top](#toc)\\]\n$$\\label{ln}$$\n\nSince we assume that the initial separation vector ${\\bf r}$ between $m_{1}$ and $m_{2}$ lies along the ${\\bf e}_{0}$-axis, the initial orbital plane coincides with the ${\\bf e}_{0},{\\bf e}_{1}$-plane. Thus the normalized inital orbital angular momentum vector $\\hat{\\bf L}_{N}$ is given by\n\n\\begin{align*}\n \\hat{\\bf L}_{N}^{0} &= 0 \\\\\n \\hat{\\bf L}_{N}^{1} &= 0 \\\\\n \\hat{\\bf L}_{N}^{2} &= 1.\n\\end{align*}\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Lines 788--790.\n\n\n```python\nLNhat0 = 0.\nLNhat1 = 0.\nLNhat2 = 1.\n```\n\n\n\n## Normalized Velocity $\\hat{\\bf v}$ \\[Back to [top](#toc)\\]\n$$\\label{vhat}$$\n\nGiven normalized orbital angular momentum ($\\hat{\\bf L}_{N}$) and normalized position ($\\hat{\\bf r}$), the normalized velocity vector ($\\hat{\\bf v}$) is given by\n\n\\begin{equation*}\n \\hat{\\bf v} = \\frac{ \\hat{\\bf L}_{N} \\times \\hat{\\bf r} }{ \\left\\lvert \\hat{\\bf L}_{N} \\times \\hat{\\bf r} \\right\\rvert }.\n\\end{equation*}\n\nGiven $\\hat{\\bf L}_{N} = \\begin{bmatrix} 0 \\\\ 0 \\\\ 1 \\end{bmatrix}$ and $\\hat{\\bf r} = \\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \\end{bmatrix}$ it is clear that $\\hat{\\bf v} = \\begin{bmatrix} 0 \\\\ 1 \\\\ 0 \\end{bmatrix}$.\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Lines 812--814.\n\n\n```python\nvhat0 = 0.\nvhat1 = 1.\nvhat2 = 0.\n```\n\n## Note \\[Back to [top](#toc)\\]\n\n\nSince we began assuming $\\iota = 0$, we do not need to rotate $\\hat{\\bf r}$, $\\hat{\\bf v}$, $\\hat{\\bf L}_{N}$, ${\\bf S}_{1}$, ${\\bf S}_{2}$, $\\hat{\\bf S}_{1}$, or $\\hat{\\bf S}_{2}$ as is done at LALSimIMRSpinEOBInitialConditionsPrec.c Lines 840.\n\n\n\n# Step 2: Compute ${\\bf r}$ and ${\\bf p}$ in spherical coordinates \\[Back to [top](#toc)\\]\n$$\\label{step2}$$\n\n\n\n## $\\omega$ \\[Back to [top](#toc)\\]\n$$\\label{omega}$$\n\nIs there a paper reference for this formula? It's not quite Newtonian ($\\omega = 2 \\pi f)$.\n \n\\begin{equation*}\n \\omega = M \\mathcal{M}_{\\odot} \\pi f.\n\\end{equation*}\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Line 893.\n\n\n```python\nomega = M*Msol*sp.pi*f\n```\n\n\n\n## Initial Velocity $v$ \\[Back to [top](#toc)\\]\n$$\\label{velocity}$$\n\nIs there a paper reference for this formula?\n\n\\begin{equation*}\n v = \\sqrt[3]{ \\omega }.\n\\end{equation*}\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Line 894.\n\n\n```python\nv = sp.cbrt(omega)\n```\n\n\n\n## Root-finding...\n\n$$\\label{roots}$$\n\n**Note: LALSuite scales the initial guesses given to the root-finding routine; see LALSimIMRSpinEOBInitialConditionsPrec.c Line 899.**\n\nWe will write components of the momentum vector ${\\bf p}$ in spherical coordinates with components ${\\bf p}^{r}$, ${\\bf p}^{\\theta}$, and ${\\bf p}^{\\phi}$.\n\nThese are initial guesses for the root-finding routine:\n\n\\begin{align*}\n {\\bf r}^{r} &= \\frac{ 1 }{ v^{2} } \\\\\n {\\bf p}^{\\phi} &= v \\\\\n {\\bf p}^{\\theta} &= 0.2.\n\\end{align*}\n\nThe root finder seeks to solve\n\n\\begin{equation*}\n \\begin{bmatrix} \\frac{ \\partial H }{ \\partial {\\bf r}^{r} } \\\\ \\frac{ \\partial H }{ \\partial {\\bf p}^{\\theta} } \\\\ \\frac{ \\partial H }{ \\partial {\\bf p}^{\\phi} } - \\omega \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\end{bmatrix}.\n\\end{equation*}\n\nAs the Hamiltonian is given in Cartesian coordinates, this requires computing $\\frac{ \\partial H }{ \\partial {\\bf r}^{0} }$, $\\frac{ \\partial H }{ \\partial {\\bf p}^{1} }$, and $\\frac{ \\partial H }{ \\partial {\\bf p}^{2} }$ and then converting to spherical coordinates via\n\n\\begin{align*}\n \\frac{\\partial H}{\\partial {\\bf r}^{r}} &= \\frac{\\partial H}{\\partial {\\bf r}^{0}} - \\frac{\\frac{\\partial H}{\\partial {\\bf p}^{1}}{\\bf p}^{\\phi}}{\\left({\\bf r}^{r}\\right)^{2}} + \\frac{\\frac{\\partial H}{\\partial {\\bf p}^{2}}{\\bf p}^{\\theta}}{\\left({\\bf r}^{r}\\right)^{2}} \\\\\n \\frac{\\partial H}{\\partial {\\bf p}^{\\theta}} &= -\\frac{\\frac{\\partial H}{\\partial {\\bf p}^{2}}}{{\\bf r}^{r}} \\\\\n \\frac{\\partial H}{\\partial {\\bf p}^{\\phi}} &= \\frac{\\frac{\\partial H}{\\partial {\\bf p}^{1}}}{{\\bf r}^{r}}.\n\\end{align*}\n\nIn the end, we should have a cartesian postition vector ${\\bf q}$ and momentum vector ${\\bf p}$. It seems that because we stared with $\\iota = 0$ that we can use $\\hat{\\bf r}$ instead of $\\hat{\\bf q}$?\n\n\n```python\nq0 = 1.\nq1 = 1.\nq2 = 1.\np0 = 1.\np1 = 1.\np2 = 1.\n```\n\n\n\n# Step 3: Rotate $\\hat{\\bf L} \\to {\\bf e}_{z}$ \\[Back to [top](#toc)\\]\n$$\\label{step3}$$\n\n## Note \\[Back to [top](#toc)\\]\n\n\nAt this point, see LALSimIMRSpinEOBInitialConditionsPrec.c normalizes the Cartesian separation and momentum vectors constructed in [Step 2](#step2). We already have a normalized separation vector $\\hat{\\bf r}$, so we skip that step.\n\n\n\n## Normalize ${\\bf p}$ \\[Back to [top](#toc)\\]\n$$\\label{normp}$$\n\nNext we normalize the new position vector ${\\bf p}$ that we found in [Step 2](#step2):\n\n\\begin{equation*}\n \\hat{\\bf p} = \\frac{ {\\bf p} }{ \\left\\lvert {\\bf p} \\right\\rvert}.\n\\end{equation*}\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Line 1101.\n\n\n```python\npnorminv = 1/sp.sqrt(p0*p0 + p1*p1 + p2*p2)\nphat0 = p0*pnorminv\nphat1 = p1*pnorminv\nphat2 = p2*pnorminv\n```\n\n\n\n## $\\hat{\\bf L}$ \\[Back to [top](#toc)\\]\n$$\\label{lhat}$$\n\nWe compute the normalized relativistic angular momentum vector $\\hat{\\bf L}$ (find source for this formula?):\n\n\\begin{equation*}\n \\hat{\\bf L} = \\frac{ \\hat{\\bf r} \\times \\hat{\\bf p} }{ \\left\\lvert \\hat{\\bf r} \\times \\hat{\\bf p} \\right\\rvert }.\n\\end{equation*}\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Lines 1098--1100.\n\n\n```python\nLhat0 = rhat1*phat2 - rhat2*phat1\nLhat1 = rhat2*phat0 - rhat0*phat2\nLhat2 = rhat0*phat1 - rhat1*phat0\nLhatnorminv = 1./sp.sqrt(Lhat0*Lhat0 + Lhat1*Lhat1 + Lhat2*Lhat2)\nLhat0 /= Lhatnorminv\nLhat1 /= Lhatnorminv\nLhat2 /= Lhatnorminv\n```\n\n\n\n## Rotation matrix \\[Back to [top](#toc)\\]\n$$\\label{rotationmatrix}$$\n\nThe rotation matrix from the $\\left\\{ \\hat{\\bf r}, {\\bf v}, \\hat{\\bf L}_{N} \\right\\}$ frame to the $\\left\\{ \\hat{\\bf r}, {\\bf p}, \\hat{\\bf L} \\right\\}$ frame is given by\n\n\\begin{equation*}\n \\begin{bmatrix} \\hat{\\bf r}^{0} & \\hat{\\bf r}^{1} & \\hat{\\bf r}^{2} \\\\\n \\hat{\\bf p}^{0} & \\hat{\\bf p}^{1} & \\hat{\\bf p}^{2} \\\\\n \\hat{\\bf L}^{0} & \\hat{\\bf L}^{1} & \\hat{\\bf L}^{2}\\end{bmatrix}.\n\\end{equation*}\n\nThe matrix to invert the rotation is simply the transpose:\n\n\\begin{equation*}\n \\begin{bmatrix} \\hat{\\bf r}^{0} & \\hat{\\bf p}^{0} & \\hat{\\bf L}^{0} \\\\\n \\hat{\\bf r}^{1} & \\hat{\\bf p}^{1} & \\hat{\\bf L}^{1} \\\\\n \\hat{\\bf r}^{2} & \\hat{\\bf p}^{2} & \\hat{\\bf L}^{2}\\end{bmatrix}.\n\\end{equation*}\n\nTo see that this is indeed the correct matrix inverse, note that by construction $\\hat{\\bf q}$, $\\hat{\\bf p}$, and $\\hat{\\bf L}$ are all unit vectors orthogonal to one another. See LALSimIMRSpinEOBInitialConditionsPrec.c Line 1107.\n\n\n```python\nrotate00 = rhat0\nrotate01 = rhat1\nrotate02 = rhat2\nrotate10 = phat0\nrotate11 = phat1\nrotate12 = phat2\nrotate20 = Lhat0\nrotate21 = Lhat1\nrotate22 = Lhat2\n```\n\n\n\n## Rotate $\\hat{\\bf r}$ \\[Back to [top](#toc)\\]\n$$\\label{rotatesrhat}$$\n\nWe now rotate $\\hat{\\bf r}$. We'll use primes to denote the rotated vector.\n\n\\begin{equation*}\n \\hat{\\bf r}^{\\prime} = \\begin{bmatrix} \\hat{\\bf r}^{0} & \\hat{\\bf r}^{1} & \\hat{\\bf r}^{2} \\\\\n \\hat{\\bf p}^{0} & \\hat{\\bf p}^{1} & \\hat{\\bf p}^{2} \\\\\n \\hat{\\bf L}^{0} & \\hat{\\bf L}^{1} & \\hat{\\bf L}^{2}\\end{bmatrix}\n \\begin{bmatrix} \\hat{\\bf r}^{0} \\\\ \\hat{\\bf r}^{1} \\\\ \\hat{\\bf r}^{2} \\end{bmatrix}\n\\end{equation*}\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Line 1112.\n\n\n```python\nrhatprm0 = rhat0*rhat0 + rhat1*rhat1 + rhat2*rhat2\nrhatprm1 = phat0*rhat0 + phat1*rhat1 + phat2*rhat2\nrhatprm2 = Lhat0*rhat0 + Lhat1*rhat1 + Lhat2*rhat2\n```\n\n\n\n## Rotate $\\hat{\\bf v}$ \\[Back to [top](#toc)\\]\n$$\\label{rotatevhat}$$\n\nWe rotate $\\hat{\\bf v}$. We'll use primes to denote the rotated vector.\n\n\\begin{equation*}\n \\hat{\\bf v}^{\\prime} = \\begin{bmatrix} \\hat{\\bf r}^{0} & \\hat{\\bf r}^{1} & \\hat{\\bf r}^{2} \\\\\n \\hat{\\bf p}^{0} & \\hat{\\bf p}^{1} & \\hat{\\bf p}^{2} \\\\\n \\hat{\\bf L}^{0} & \\hat{\\bf L}^{1} & \\hat{\\bf L}^{2}\\end{bmatrix}\n \\begin{bmatrix} \\hat{\\bf v}^{0} \\\\ \\hat{\\bf v}^{1} \\\\ \\hat{\\bf v}^{2} \\end{bmatrix}\n\\end{equation*}\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Line 1113.\n\n\n```python\nvhatprm0 = rhat0*vhat0 + rhat1*vhat1 + rhat2*vhat2\nvhatprm1 = phat0*vhat0 + phat1*vhat1 + phat2*vhat2\nvhatprm2 = Lhat0*vhat0 + Lhat1*vhat1 + Lhat2*vhat2\n```\n\n\n\n## Rotate $\\hat{\\bf L}_{N}$ \\[Back to [top](#toc)\\]\n$$\\label{rotatelnhat}$$\n\nWe rotate $\\hat{\\bf L}_{N}$. We'll use primes to denote the rotated vector.\n\n\\begin{equation*}\n \\hat{\\bf L}_{N}^{\\prime} = \\begin{bmatrix} \\hat{\\bf r}^{0} & \\hat{\\bf r}^{1} & \\hat{\\bf r}^{2} \\\\\n \\hat{\\bf p}^{0} & \\hat{\\bf p}^{1} & \\hat{\\bf p}^{2} \\\\\n \\hat{\\bf L}^{0} & \\hat{\\bf L}^{1} & \\hat{\\bf L}^{2}\\end{bmatrix}\n \\begin{bmatrix} \\hat{\\bf L}_{N}^{0} \\\\ \\hat{\\bf L}_{N}^{1} \\\\ \\hat{\\bf L}_{N}^{2} \\end{bmatrix}\n\\end{equation*}\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Line 1114.\n\n\n```python\nLNhatprm0 = rhat0*LNhat0 + rhat1*LNhat1 + rhat2*LNhat2\nLNhatprm1 = phat0*LNhat0 + phat1*LNhat1 + phat2*LNhat2\nLNhatprm2 = Lhat0*LNhat0 + Lhat1*LNhat1 + Lhat2*LNhat2\n```\n\n\n\n## Rotate ${\\bf S}_{1}$ \\[Back to [top](#toc)\\]\n$$\\label{rotates1}$$\n\nWe rotate ${\\bf S}_{1}$. We'll use primes to denote the rotated vector.\n\n\\begin{equation*}\n {\\bf S}_{1}^{\\prime} = \\begin{bmatrix} \\hat{\\bf r}^{0} & \\hat{\\bf r}^{1} & \\hat{\\bf r}^{2} \\\\\n \\hat{\\bf p}^{0} & \\hat{\\bf p}^{1} & \\hat{\\bf p}^{2} \\\\\n \\hat{\\bf L}^{0} & \\hat{\\bf L}^{1} & \\hat{\\bf L}^{2}\\end{bmatrix}\n \\begin{bmatrix} {\\bf S}_{1}^{0} \\\\ {\\bf S}_{1}^{1} \\\\ {\\bf S}_{1}^{2} \\end{bmatrix}\n\\end{equation*}\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Line 1115.\n\n\n```python\nS1prm0 = rhat0*S10 + rhat1*S11 + rhat2*S12\nS1prm1 = phat0*S10 + phat1*S11 + phat2*S12\nS1prm2 = Lhat0*S10 + Lhat1*S11 + Lhat2*S12\n```\n\n\n\n## Rotate ${\\bf S}_{2}$ \\[Back to [top](#toc)\\]\n$$\\label{rotates2}$$\n\nWe rotate ${\\bf S}_{2}$. We'll use primes to denote the rotated vector.\n\n\\begin{equation*}\n {\\bf S}_{2}^{\\prime} = \\begin{bmatrix} \\hat{\\bf r}^{0} & \\hat{\\bf r}^{1} & \\hat{\\bf r}^{2} \\\\\n \\hat{\\bf p}^{0} & \\hat{\\bf p}^{1} & \\hat{\\bf p}^{2} \\\\\n \\hat{\\bf L}^{0} & \\hat{\\bf L}^{1} & \\hat{\\bf L}^{2}\\end{bmatrix}\n \\begin{bmatrix} {\\bf S}_{2}^{0} \\\\ {\\bf S}_{2}^{1} \\\\ {\\bf S}_{2}^{z} \\end{bmatrix}\n\\end{equation*}\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Line 1116.\n\n\n```python\nS2prm0 = rhat0*S20 + rhat1*S21 + rhat2*S22\nS2prm1 = phat0*S20 + phat1*S21 + phat2*S22\nS2prm2 = Lhat0*S20 + Lhat1*S21 + Lhat2*S22\n```\n\n\n\n## Rotate $\\hat{\\bf S}_{1}$ \\[Back to [top](#toc)\\]\n$$\\label{rotates1hat}$$\n\nWe rotate $\\hat{\\bf S}_{1}$. We'll use primes to denote the rotated vector.\n\n\\begin{equation*}\n \\hat{\\bf S}_{1}^{\\prime} = \\begin{bmatrix} \\hat{\\bf r}^{0} & \\hat{\\bf r}^{1} & \\hat{\\bf r}^{2} \\\\\n \\hat{\\bf p}^{0} & \\hat{\\bf p}^{1} & \\hat{\\bf p}^{2} \\\\\n \\hat{\\bf L}^{0} & \\hat{\\bf L}^{1} & \\hat{\\bf L}^{2}\\end{bmatrix}\n \\begin{bmatrix} \\hat{\\bf S}_{1}^{0} \\\\ \\hat{\\bf S}_{1}^{1} \\\\ \\hat{\\bf S}_{1}^{1} \\end{bmatrix}\n\\end{equation*}\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Line 1117.\n\n\n```python\nS1hatprm0 = rhat0*S1hat0 + rhat1*S1hat1 + rhat2*S1hat2\nS1hatprm1 = phat0*S1hat0 + phat1*S1hat1 + phat2*S1hat2\nS1hatprm2 = Lhat0*S1hat0 + Lhat1*S1hat1 + Lhat2*S1hat2\n```\n\n\n\n## Rotate $\\hat{\\bf S}_{2}$ \\[Back to [top](#toc)\\]\n$$\\label{rotates2hat\\hat}$$\n\nWe rotate $\\hat{\\bf S}_{2}$. We'll use primes to denote the rotated vector.\n\n\\begin{equation*}\n \\hat{\\bf S}_{2}^{\\prime} = \\begin{bmatrix} \\hat{\\bf r}^{0} & \\hat{\\bf r}^{1} & \\hat{\\bf r}^{2} \\\\\n \\hat{\\bf p}^{0} & \\hat{\\bf p}^{1} & \\hat{\\bf p}^{2} \\\\\n \\hat{\\bf L}^{0} & \\hat{\\bf L}^{1} & \\hat{\\bf L}^{2}\\end{bmatrix}\n \\begin{bmatrix} \\hat{\\bf S}_{2}^{0} \\\\ \\hat{\\bf S}_{2}^{1} \\\\ \\hat{\\bf S}_{2}^{2} \\end{bmatrix}\n\\end{equation*}\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Line 1118.\n\n\n```python\nS2hatprm0 = rhat0*S2hat0 + rhat1*S2hat1 + rhat2*S2hat2\nS2hatprm1 = phat0*S2hat0 + phat1*S2hat1 + phat2*S2hat2\nS2hatprm2 = Lhat0*S2hat0 + Lhat1*S2hat1 + Lhat2*S2hat2\n```\n\n\n\n## Rotate ${\\bf r}$ \\[Back to [top](#toc)\\]\n$$\\label{rotater}$$\n\nWe rotate ${\\bf p}$. We'll use primes to denote the rotated vector.\n\n\\begin{equation*}\n {\\bf r}^{\\prime} = \\begin{bmatrix} \\hat{\\bf r}^{0} & \\hat{\\bf r}^{1} & \\hat{\\bf r}^{2} \\\\\n \\hat{\\bf p}^{0} & \\hat{\\bf p}^{1} & \\hat{\\bf p}^{2} \\\\\n \\hat{\\bf L}^{0} & \\hat{\\bf L}^{1} & \\hat{\\bf L}^{2}\\end{bmatrix}\n \\begin{bmatrix} {\\bf r}^{0} \\\\ {\\bf r}^{1} \\\\ {\\bf r}^{2} \\end{bmatrix}\n\\end{equation*}\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Line 1119.\n\n\n```python\nrprm0 = rhat0*p0 + rhat1*p1 + rhat2*p2\nrprm1 = phat0*p0 + phat1*p1 + phat2*p2\nrprm2 = Lhat0*p0 + Lhat1*p1 + Lhat2*p2\n```\n\n\n\n## Rotate ${\\bf p}$ \\[Back to [top](#toc)\\]\n$$\\label{rotatep}$$\n\nWe rotate ${\\bf p}$. We'll use primes to denote the rotated vector.\n\n\\begin{equation*}\n {\\bf p}^{\\prime} = \\begin{bmatrix} \\hat{\\bf r}^{0} & \\hat{\\bf r}^{1} & \\hat{\\bf r}^{2} \\\\\n \\hat{\\bf p}^{0} & \\hat{\\bf p}^{1} & \\hat{\\bf p}^{2} \\\\\n \\hat{\\bf L}^{0} & \\hat{\\bf L}^{1} & \\hat{\\bf L}^{2}\\end{bmatrix}\n \\begin{bmatrix} {\\bf p}^{0} \\\\ {\\bf p}^{y} \\\\ {\\bf p}^{2} \\end{bmatrix}\n\\end{equation*}\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Line 1120.\n\n\n```python\npprm0 = rhat0*p0 + rhat1*p1 + rhat2*p2\npprm1 = phat0*p0 + phat1*p1 + phat2*p2\npprm2 = Lhat0*p0 + Lhat1*p1 + Lhat2*p2\n```\n\n\n\n# Step 4: Compute $\\dot{\\bf r}$ \\[Back to [top](#toc)\\]\n$$\\label{step4}$$\n\n\n\n## Convert from Cartesian to Spherical \\[Back to [top](#toc)\\]\n$$\\label{carttosph}$$\n\nWe convert position and momentum into spherical coordinates. In the special case where $\\theta = \\frac{ \\pi }{ 2 }$ and $\\phi = 0$, the spherical position vector ${\\bf r} = \\left( {\\bf r}^{r}, {\\bf r}^{\\theta}, {\\bf r}^{\\phi} \\right)$ is given by\n\n\\begin{align*}\n {\\bf r}^{r} &= {\\bf r}^{0} \\\\\n {\\bf r}^{\\theta} &= \\frac{ \\pi }{ 2 } \\\\\n {\\bf r}^{\\phi} &= 0\n\\end{align*}\n\nand the spherical momentum vector ${\\bf p} = \\left( {\\bf p}^{r}, {\\bf p}^{\\theta}, {\\bf p}^{\\phi} \\right)$ is given by\n\n\\begin{align*}\n {\\bf p}^{r} &= {\\bf p}^{0} \\\\\n {\\bf p}^{\\theta} &= - {\\bf r}^{0}{\\bf p}^{2} \\\\\n {\\bf p}^{\\phi} &= {\\bf r}^{0}{\\bf p}^{1} \\\\\n\\end{align*}\n\nWe call a Cartesian to spherical routine at LALSimIMRSpinEOBInitialConditionsPrec.c Line 1139, and the function itself is defined on Lines 243--285.\n\n\n```python\nrr = rprm0\nrtheta = sp.pi/2.\nrphi = 0.\npr = pprm0\nptheta = -rprm0*pprm2\npphi = rprm0*pprm1\n```\n\n\n\n## Second derivatives of $H_{\\rm real}$ \\[Back to [top](#toc)\\]\n$$\\label{seconderiv}$$\n\nWe need to compute $\\frac{ \\partial H }{ \\partial {\\bf p}^{\\phi} }$, $\\frac{ \\partial^{2} H_{\\rm real} }{ \\partial r^{2} }$, and $\\frac{ \\partial^{2} H_{\\rm real} }{ \\partial r \\partial {\\bf p}^{\\phi} }$ (in another module).\n\nNote: be sure that, following this, we use normalized spins.\n\n\n```python\ndHdpphi = 0.\nd2Hdr2 = 0.\nd2Hdrdpphi = 1.\n```\n\n\n\n## $\\frac{ \\partial E }{ \\partial r }$ \\[Back to [top](#toc)\\]\n$$\\label{dEdr}$$\n\nI don't know what this is or means (see [BCD2006](https://arxiv.org/abs/gr-qc/0508067) Equations (4.14) and (3.7)), but we compute $\\frac{ \\partial E }{ \\partial r }$ in spherical coordinates (when ${\\bf r}$ is directed along the ${\\bf e}_{0}$ axis) via\n\n\\begin{equation*}\n \\frac{ \\partial E }{ \\partial r } = -\\frac{ \\frac{ \\partial H }{ \\partial {\\bf p}^{\\phi} } \\frac{ \\partial^{2} H }{ \\left(\\partial {\\bf r}^{r} \\right)^{2} } }{ \\frac{ \\partial^{2} H }{ \\partial H \\partial {\\bf p}^{\\phi} } }.\n\\end{equation*}\n\n\n\n\n```python\ndEdr = -dHdpphi*d2Hdr2/d2Hdrdpphi\n```\n\n\n\n## ${\\bf S}_{\\rm Kerr}$ \\[Back to [top](#toc)\\]\n$$\\label{skerr}$$\n\nFrom [BB2010](https://arxiv.org/abs/0912.3517) Equations (5.2), (5.63), and (5.67) we have\n\n\\begin{equation*}\n {\\bf S}_{\\rm Kerr} = {\\bf S}_{1} + {\\bf S}_{2}.\n\\end{equation*}\n\nTaking the square of [BB2010](https://arxiv.org/abs/0912.3517) Equation (4.9) (be careful with the factor or $M$...),\n\n\\begin{equation*}\n a^{2} = \\frac{ {\\bf S}_{Kerr} \\cdot {\\bf S}_{Kerr} }{ M^{2} }\n\\end{equation*}\n\nso that\n\n\\begin{equation*}\n a = \\sqrt{ a^{2} }.\n\\end{equation*}\n\n\n```python\nSKerr0 = S10 + S20\nSKerr1 = S11 + S21\nSKerr2 = S12 + S22\nasq = (SKerr0*SKerr0 + SKerr1*SKerr1 + SKerr2*SKerr2)/(M*M)\na = sp.sqrt(asq)\n```\n\n\n\n## $\\boldsymbol{\\sigma}^{*}$ \\[Back to [top](#toc)\\]\n$$\\label{sigmastar}$$\n\nFrom [BB2010](https://arxiv.org/abs/0912.3517) Equation (5.3),\n\n\\begin{equation*}\n \\boldsymbol{\\sigma}^{*} = \\frac{ m_{2} }{ m_{1} } {\\bf S}_{1} + \\frac{ m_{1} }{ m_{2} }{\\bf S}_{2}.\n\\end{equation*}\n\n\n```python\nsigmastar0 = S10*m2/m1 + S20*m1/m2\nsigmastar1 = S11*m2/m1 + S21*m1/m2\nsigmastar2 = S12*m2/m1 + S22*m1/m2\n```\n\n\n\n## $H_{\\rm real}$ \\[Back to [top](#toc)\\]\n$$\\label{Hreal}$$\n\nWe now compute $H_{\\rm real}$ (LALSimIMRSpinEOBInitialConditionsPrec.c Line 1217) (another module).\n\n\n```python\nHreal = 0\n```\n\n\n\n## Polar data \\[Back to [top](#toc)\\]\n$$\\label{polardata}$$\n\nAt LALSimIMRSpinEOBInitialConditionsPrec.c Lines 1234--1238, we set the following polar data ${\\bf P}$ (need to find reference for this?):\n\n\\begin{align*}\n {\\bf P}^{0} &= {\\bf r}^{r} \\\\\n {\\bf P}^{1} &= 0 \\\\\n {\\bf P}^{2} &= {\\bf p}^{r} \\\\\n {\\bf P}^{3} &= {\\bf p}^{\\phi}\n\\end{align*}\n\n\n```python\npolar0 = rr\npolar1 = 0.\npolar2 = pr\npolar3 = pphi\n```\n\n\n\n## Compute ${\\bf L}$ \\[Back to [top](#toc)\\]\n$$\\label{l}$$\n\nWe compute ${\\bf L} = {\\bf r}^{\\prime}\\times{\\bf p}^{\\prime}$ and normalize as follows.\n\n\\begin{equation*}\n {\\bf L} = \\frac{ {\\bf r}^{\\prime} \\times {\\bf p}^{\\prime} }{ \\left\\lvert {\\bf r}^{\\prime} \\times {\\bf p}^{\\prime} \\right\\rvert }\n\\end{equation*}\n\nSee LALSimIMRSpinEOBFactorizedFluxPrec_v3opt.c lines 121--129.\n\n\n```python\nL0 = rprm1*pprm2 - rprm2*pprm1\nL1 = rprm2*pprm0 - rprm0*pprm2\nL2 = rprm0*pprm1 - rprm1*pprm0\nLmaginv = 1./sp.sqrt(rprmcrosspprm0*rprmcrosspprm0\n + rprmcrosspprm1*rprmcrosspprm1\n + rprmcrosspprm2*rprmcrosspprm2)\nL0 /= Lmaginv\nL1 /= Lmaginv\nL2 /= Lmaginv\n```\n\n\n\n## ${\\bf S}_{1}\\cdot{\\bf L}$ \\[Back to [top](#toc)\\]\n$$\\label{s1dotl}$$\n\nWe compute ${\\bf r}^{\\prime}\\times{\\bf p}^{\\prime}$ and normalize as follows.\n\nSee LALSimIMRSpinEOBFactorizedFluxPrec_v3opt.c lines 121--129.\n\n\n\n# Step 5: Invert the rotation of Step 3 \\[Back to [top](#toc)\\]\n$$\\label{step5}$$\n\n\n\n## Inverse Rotation Matrix \\[Back to [top](#toc)\\]\n$$\\label{invrotationmatrix}$$\n\nThe matrix to invert the rotation applied in [Step 3](#step3) is:\n\n\\begin{equation*}\n \\begin{bmatrix} \\hat{\\bf r}^{0} & \\hat{\\bf p}^{0} & \\hat{\\bf L}^{0} \\\\\n \\hat{\\bf r}^{1} & \\hat{\\bf p}^{1} & \\hat{\\bf L}^{1} \\\\\n \\hat{\\bf r}^{2} & \\hat{\\bf p}^{2} & \\hat{\\bf L}^{2}\\end{bmatrix}.\n\\end{equation*}\n\nTo see that this is indeed the correct matrix inverse, note that by construction $\\hat{\\bf q}$, $\\hat{\\bf p}$, and $\\hat{\\bf L}$ are all unit vectors orthogonal to one another. See LALSimIMRSpinEOBInitialConditionsPrec.c Line 1107.\n\n\n```python\ninvert00 = rhat0\ninvert01 = phat0\ninvert02 = Lhat0\ninvert10 = rhat1\ninvert11 = phat1\ninvert12 = Lhat1\ninvert20 = rhat2\ninvert21 = phat2\ninvert22 = Lhat2\n```\n\n\n\n## Rotate $\\hat{\\bf r}^{\\prime}$ \\[Back to [top](#toc)\\]\n$$\\label{invrotaterhat}$$\n\nWe rotate $\\hat{\\bf r}^{\\prime}$ and call the new separation vector ${\\bf r}$.\n\n\\begin{equation*}\n \\hat{\\bf r} = \\begin{bmatrix} \\hat{\\bf r}^{0} & \\hat{\\bf p}^{0} & \\hat{\\bf L}^{0} \\\\\n \\hat{\\bf r}^{1} & \\hat{\\bf p}^{1} & \\hat{\\bf L}^{1} \\\\\n \\hat{\\bf r}^{2} & \\hat{\\bf p}^{2} & \\hat{\\bf L}^{2} \\end{bmatrix}\n \\begin{bmatrix} \\hat{\\bf r}^{\\prime 0} \\\\ \\hat{\\bf r}^{\\prime 1} \\\\ \\hat{\\bf r}^{\\prime 2} \\end{bmatrix}\n\\end{equation*}\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Line 1315.\n\n\n```python\nrhat0 = rhat0*rhatprm0 + phat0*rhatprm1 + Lhat0*rhatprm2\nrhat1 = rhat1*rhatprm0 + phat1*rhatprm1 + Lhat1*rhatprm2\nrhat0 = rhat2*rhatprm0 + phat2*rhatprm1 + Lhat2*rhatprm2\n```\n\n\n\n## Rotate $\\hat{\\bf v}^{\\prime}$ \\[Back to [top](#toc)\\]\n$$\\label{invrotatevhat}$$\n\nWe rotate $\\hat{\\bf v}^{\\prime}$ and call the new separation vector ${\\bf v}$.\n\n\\begin{equation*}\n \\hat{\\bf v} = \\begin{bmatrix} \\hat{\\bf r}^{0} & \\hat{\\bf p}^{0} & \\hat{\\bf L}^{0} \\\\\n \\hat{\\bf r}^{1} & \\hat{\\bf p}^{1} & \\hat{\\bf L}^{1} \\\\\n \\hat{\\bf r}^{2} & \\hat{\\bf p}^{2} & \\hat{\\bf L}^{2} \\end{bmatrix}\n \\begin{bmatrix} \\hat{\\bf v}^{\\prime 0} \\\\ \\hat{\\bf v}^{\\prime 1} \\\\ \\hat{\\bf v}^{\\prime 2} \\end{bmatrix}\n\\end{equation*}\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Line 1316.\n\n\n```python\nvhat0 = rhat0*vhatprm0 + phat0*vhatprm1 + Lhat0*vhatprm2\nvhat1 = rhat1*vhatprm0 + phat1*vhatprm1 + Lhat1*vhatprm2\nvhat2 = rhat2*vhatprm0 + phat2*vhatprm1 + Lhat2*vhatprm2\n```\n\n\n\n## Rotate $\\hat{\\bf L}_{N}^{\\prime}$ \\[Back to [top](#toc)\\]\n$$\\label{invrotatelnhat}$$\n\nWe rotate $\\hat{\\bf L}_{N}^{\\prime}$ and call the new separation vector ${\\bf L}_{N}$.\n\n\\begin{equation*}\n \\hat{\\bf L}_{N} = \\begin{bmatrix} \\hat{\\bf r}^{0} & \\hat{\\bf p}^{0} & \\hat{\\bf L}^{0} \\\\\n \\hat{\\bf r}^{1} & \\hat{\\bf p}^{1} & \\hat{\\bf L}^{1} \\\\\n \\hat{\\bf r}^{2} & \\hat{\\bf p}^{2} & \\hat{\\bf L}^{2} \\end{bmatrix}\n \\begin{bmatrix} \\hat{\\bf L}_{N}^{\\prime 0} \\\\ \\hat{\\bf L}_{N}^{\\prime 1} \\\\ \\hat{\\bf L}_{N}^{\\prime 2} \\end{bmatrix}\n\\end{equation*}\n\nSee LALSimIMRSpinEOBInitialConditionsPrec.c Line 1317.\n\n\n```python\nLNhat0 = rhat0*LNhatprm0 + phat0*LNhatprm1 + Lhat0*LNhatprm2\nLNhat1 = rhat1*LNhatprm0 + phat1*LNhatprm1 + Lhat1*LNhatprm2\nLNhat2 = rhat2*LNhatprm0 + phat2*LNhatprm1 + Lhat2*LNhatprm2\n```\n\n\n\n# Tortoise Conversion Matrix \\[Back to [top](#toc)\\]\n$$\\label{tortoise_matrix}$$\n\nWe're now back to LALSpinPrecHcapRvecDerivative_v3opt.c, Lines 92--96.\n\nFrom [Pan, Buonanno, Buchman, et. al. (2010)](https://arxiv.org/abs/0912.3466v2) Equation (A3) the matrix for the coordinate conversion to tortoise coordinates is\n\n\\begin{align*}\n \\begin{pmatrix} 1 + \\frac{ x^{2} }{ r^{2} } \\left( \\xi - 1 \\right) & \\frac{ x y }{ r^{2} } \\left( \\xi - 1 \\right) & \\frac{ x z }{ r^{2} } \\left( \\xi - 1 \\right) \\\\\n \\frac{ x y }{ r^{2} } \\left( \\xi - 1 \\right) & 1 + \\frac{ y^{2} }{ r^{2} } \\left( \\xi - 1 \\right) & \\frac{ y z }{ r^{2} } \\left( \\xi - 1 \\right) \\\\\n \\frac{ x z }{ r^{2} } \\left( \\xi - 1 \\right) & \\frac{ y z }{ r^{2} } \\left( \\xi - 1 \\right) & 1 + \\frac{ z^{2} }{ r^{2} } \\left( \\xi - 1 \\right) \\end{pmatrix}\n\\end{align*}\n\n\n```python\nximinus1 = xi - 1\ntoTort = sp.Array([[1 + x*x*ximinus1/(r*r), x*y*ximinus1/(r*r), x*z*ximinus1/(r*r)],\n [x*y*ximinus1/(r*r), 1 + y*y*ximinus1/(r*r), y*z*ximinus1/(r*r)],\n [x*z*ximinus1/(r*r), y*z*ximinus1/(r*r), 1 + z*z*ximinus1/(r*r)]])\n```\n\n\n\n# Output: Output this module to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n$$\\label{latex_pdf_output}$$\n\n\n```python\n!jupyter nbconvert --to latex --template latex_nrpy_style.tplx Tutorial-SEOBNR_Initial_Conditions.ipynb\n!pdflatex -interaction=batchmode Tutorial-SEOBNR_Initial_Conditions.tex\n!pdflatex -interaction=batchmode Tutorial-SEOBNR_Initial_Conditions.tex\n!pdflatex -interaction=batchmode Tutorial-SEOBNR_Initial_Conditions.tex\n!rm -f Tut*.out Tut*.aux Tut*.log\n```\n", "meta": {"hexsha": "a28d68120b134b09afd45b2cd30a30f82e489137", "size": 47444, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Tutorial-SEOBNR_Initial_Conditions.ipynb", "max_stars_repo_name": "Lituchy/nrpyunittesting", "max_stars_repo_head_hexsha": "c611891a2f6f07afd762121e59d2c5ab857d925b", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Tutorial-SEOBNR_Initial_Conditions.ipynb", "max_issues_repo_name": "Lituchy/nrpyunittesting", "max_issues_repo_head_hexsha": "c611891a2f6f07afd762121e59d2c5ab857d925b", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Tutorial-SEOBNR_Initial_Conditions.ipynb", "max_forks_repo_name": "Lituchy/nrpyunittesting", "max_forks_repo_head_hexsha": "c611891a2f6f07afd762121e59d2c5ab857d925b", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.5516178737, "max_line_length": 1470, "alphanum_fraction": 0.5040257988, "converted": true, "num_tokens": 11958, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8056321889812553, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.43734813513981646}} {"text": " Trusted Notebook\" width=\"500 px\" align=\"left\">\n\n## _*Relaxation and Decoherence*_ \n\nThe latest version of this notebook is available on https://github.com/qiskit/qiskit-tutorial.\n\n***\n### Contributors\nMartin Sandberg, Hanhee Paik, Antonio C\u00f3rcoles, Doug McClure, Jay Gambetta, and Yael Ben-Haim\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nimport qiskit\nfrom qiskit.providers.aer.noise.errors.standard_errors import amplitude_damping_error\nfrom qiskit.providers.aer.noise.errors.standard_errors import phase_damping_error\nfrom qiskit.providers.aer.noise import NoiseModel\n\nfrom qiskit.ignis.characterization.coherence import T1Fitter, T2StarFitter, T2Fitter\nfrom qiskit.ignis.characterization.coherence import t1_circuits, t2_circuits, t2star_circuits\n```\n\n## Introduction\n\nIn an ideal world, quantum systems would be well-isolated from their environment, which would prevent unwanted dynamics of the quantum information we encode in them. For example, suppose we prepared a qubit in the $|1>$ state, but through interaction with the environment, the state is flipped to $|0>$. That flip could affect the outcome of a quantum algorithm that's being run using that qubit, meaning the answers we get out of the quantum device would change. For this reason, we seek to isolate quantum computers from the surrounding environment.\n\nHowever, perfect isolation is not possible: after all, we have to be able to control the quantum computer, which means coupling it to external systems to manipulate quantum information. This tradeoff is sometimes referred to as the \"Tao of quantum computing\". Because our controls introduce coupling between qubits and the environment, we expect some unwanted interactions can occur.\n\nThese unwanted interactions introduce _noise_ into the qubits, which affects their behavior. The rate of these interactions sets characteristic timescales over which information encoded in qubits can be reliably stored and manipulated. (If the interaction has a rate $\\Gamma$, the characteristic timescale is $\\sim 1/\\Gamma$.) In this tutorial, we discuss two timescales that arise from energy relaxation and decoherence -- usually referred to as $T_{1}$ and $T_{2}$, respectively -- and show how they can be measured.\n\n# Measuring $T_1$ time\n\n**Theory**\n\nThe $T_{1}$ time is the characteristic timescale over which the state of a qubit damps toward the $|0>$ state. Given an arbitrary initial single-qubit state $\\rho(0)$, represented by a $2\\times 2$ matrix as\n$$\\rho(0) = \\begin{pmatrix}\\rho_{00} & \\rho_{01} \\\\ \\rho_{01}^{\\star} & \\rho_{11}\\end{pmatrix},$$\nunder amplitude damping noise, the state of the changes as\n$$\\rho(t) = \\begin{pmatrix}\\rho_{00} + (1-e^{-\\Gamma_{1}t})\\rho_{11} & e^{-\\Gamma_{1}t/2}\\rho_{01} \\\\ e^{-\\Gamma_{1}t/2}\\rho_{01}^{\\star} & e^{-\\Gamma_{1} t}\\rho_{11}\\end{pmatrix} \\underset{t\\rightarrow \\infty}{\\longrightarrow} |0><0|.$$\n\nNotice that amplitude damping noise also removes any coherences between $|0>$ and $|1>$ of the state (the off-diagonal elements.) The rate at which this _decoherence_ occurs is half that of $\\Gamma_{1}$.\n\nThe time evolution of the state under amplitude damping noise can be derived as the continuous-time limit of an amplitude damping channel\n$$\\mathcal{E}[\\rho] = M_{0} \\rho M_{0}^{\\dagger} + M_{1}\\rho M_{1}^{\\dagger},$$\nwhere\n$$M_{0} = \\begin{pmatrix} 1 & 0 \\\\0& \\sqrt{1-p}\\end{pmatrix}~,~M_{1} = \\begin{pmatrix} 0 & \\sqrt{p} \\\\ 0 & 0 \\end{pmatrix},$$\nand the probability of decay $p$ is $\\Gamma_{1}\\Delta t$.\n\nThe decay rate $\\Gamma_{1}$ sets a natural time scale for the decay process; namely, $\\Gamma^{-1}$. This number is often called the $T_{1}$ time. Notice the off-diagonal elements also decay, with characteristic decay rate $\\Gamma /2$.\n\nNotice that the probability of the qubit remaining in the $|1>$ state is given by\n\n$$P_{1}(t) = \\mathrm{Tr}\\left[ |1><1| \\rho(t)\\right] = e^{-\\Gamma_{1} t}\\rho_{11}.$$\n\nIf the qubit was prepared in the $|1>$ state, then $P_{1}(t) =e^{-\\Gamma_{1} t}$.\n\nA simple way of estimating the $T_{1}$ time is to collect statistics about the decay curve for $P_{1}(t)$ when the qubit is initialized to $|1>$. This can be done by choosing a variety of times $t_{1}, t_{2}, \\cdots t_{N}$, and then running the following experiment many times:\n* Prepare the qubit in $|1>$.\n* Wait a delay time $t_{j}$.\n* Measure the qubit in the $|0>, |1>$ basis.\n\nAn estimate of $P_{1}(t_{j})$ is the number of times the qubit was observed to be in $|1>$, divided by the total number of times the experiment was repeated. Given several estimated values of $P_{1}$ for a variety of $(t_{j})$, we can fit the resulting decay curve is fit to an exponential and extract an estimate of $\\Gamma_{1}$, and hence, the $T_{1}$ time.\n\nThe IBM Q Experience does not currently support delays of arbitrary length, so for now, we just append a series of identity operations after the initial excitation pulse. Each identity operation has the same duration of a single-qubit gate and is followed by a -shorter- buffer time. These parameters are backend-dependent.\n\n**Code**\n\nThe code blocks below walk through constructing the requisite experiments to estimate the $T_{1}$ time of a qubit, sending those experiments to a simulator, and then fitting the data the simulator sends back.\n\n\n```python\n# 12 numbers ranging from 10 to 1000, logarithmically spaced\n# extra point at 1500\nnum_of_gates = np.append((np.logspace(1, 3, 12)).astype(int), np.array([1500]))\ngate_time = 0.1\n\n# Select the qubits whose T1 are to be measured\nqubits = [0]\n\n# Generate experiments\ncircs, xdata = t1_circuits(num_of_gates, gate_time, qubits)\n```\n\nOne of the features of the fitters are that we can split the circuits into multiple jobs and then give the results to the fitter as a list. Demonstrated below.\n\n\n```python\n# Set the simulator with amplitude damping noise\nt1 = 25.0\ngamma = 1 - np.exp(-gate_time/t1)\nerror = amplitude_damping_error(gamma)\nnoise_model = NoiseModel()\nnoise_model.add_quantum_error(error, 'id', [0])\n\n# Run the simulator\nbackend = qiskit.Aer.get_backend('qasm_simulator')\nshots = 200\nbackend_result = qiskit.execute(circs, backend,\n shots=shots, noise_model=noise_model).result()\n```\n\n\n```python\n%matplotlib inline\n# Fit the data to an exponential\n# The correct answers are a=1, and c=0, and t1=25/15 for qubit 0/2\n# The user does not know the correct answer exactly,\n# so starts the fit from a different but close location\n\ninitial_t1 = t1*1.2\ninitial_a = 1.0\ninitial_c = 0.0\n\nfit = T1Fitter(backend_result, xdata, qubits,\n fit_p0=[initial_a, initial_t1, initial_c],\n fit_bounds=([0, 0, -1], [2, initial_t1*2, 1]))\n\nfit.plot(0)\n```\n\n# Measuring $T_2^*$ time\n\n**Theory**\n\nAmplitude damping noise affects the off-diagonal elements of the density matrix in addition to the on-diagonal elements. However, there are other noise processes that only affect the off-diagonal elements, while keeping the on-diagonal elements the same. These kinds of noise processes cause _decoherence_.\n\nAs a simple example of decoherence, consider the pure superposition state\n$$| \\psi(\\theta) > = \\frac{1}{\\sqrt{2}}\\left(|0> + e^{i\\theta} |1>\\right).$$\nExpressed as a density matrix, this state is\n$$\\rho(\\theta) = | \\psi(\\theta) >< \\psi(\\theta) | = \\frac{1}{2}\\begin{pmatrix}1 & e^{-i\\theta} \\\\ e^{i\\theta} & 1\\end{pmatrix}.$$\n\nThis state has _coherence_ between $|0>$ and $|1>$, which manifests itself in the non-zero off-diagonal terms. If the state had _decohered_, those off-diagonal terms would be zero:\n$$\\rho_{\\mathrm{decohered}} = \\frac{1}{2}\\begin{pmatrix}1 & 0 \\\\ 0 & 1\\end{pmatrix}.$$\nWhen the state has decohered, it can be written as a classical _mixture_:\n$$\\rho_{\\mathrm{decohered}} = \\frac{1}{2}\\left(|0><0| + |1><1| \\right).$$\n\nOne mechanism by which decoherence happens is _dephasing_. Under dephasing noise, the state of the qubit evolves as\n$$\\rho(t) = \\begin{pmatrix}\\rho_{00} & e^{-\\Gamma_{2}t}\\rho_{01} \\\\ e^{-\\Gamma_{2}t}\\rho_{01}^{\\star} & \\rho_{11}\\end{pmatrix} \\underset{t\\rightarrow \\infty}{\\longrightarrow} \\begin{pmatrix}\\rho_{00} & 0\\\\ 0& \\rho_{11}\\end{pmatrix}.$$\n\nThe time evolution of $\\rho$ under dephasing noise can be derived as the continuous-time limit of the following noise channel:\n$$\\mathcal{E}[\\rho] = M_{0}\\rho M_{0}^{\\dagger} + M_{1} \\rho M_{1}^{\\dagger} + M_{2}\\rho M_{2}^{\\dagger},$$\nwhere\n$$M_{0} =\\sqrt{1-p}I~,~M_{1} = \\sqrt{p}\\begin{pmatrix}1 &0 \\\\ 0 & 0 \\end{pmatrix}~,~M_{2} = \\sqrt{p}\\begin{pmatrix}0 & 0 \\\\ 0 & 1\\end{pmatrix}.$$\n\n\nThe rate of decay in the coherences can be measured by the following experiment:\n\n* Prepare the qubit in the $|+>$ state, which can be done by initializing the qubit to $|0>$ and applying a Hadamard gate, $H$.\n* Wait a delay time $t_{j}$.\n* Measure the qubit in the $| \\pm >$ basis, which can be done by applying a Hadamard and then measuring in the computational basis.\n\nIf decoherence processes are present, then after a delay time $t_{j}$, the state of the qubit is\n\n$$\\rho(t_{j}) = \\frac{1}{2}\\begin{pmatrix}1 & e^{-\\Gamma_{2}t_{j}} \\\\ e^{-\\Gamma_{2}t_{j}} & 1\\end{pmatrix}.$$\n\nMeasuring in the $| \\pm >$ basis, the probability of observing the outcome $|+>$ is given by\n\n$$P_{+}(t_{j}) = \\mathrm{Tr}\\left(|+><+| \\rho(t_{j})\\right) = \\frac{1}{2}\\left(1 + e^{-\\Gamma_{2}t_{j}}\\right).$$\n\nAgain, by estimating $P_{+}(t_{j})$ for a variety of $t_{j}$, we can then fit a decay curve to extract an estimate of $\\Gamma_{2}$.\n\nIn the actual experiment, we change the phase of the pulse before the measurement in order to create oscillations in the observed dynamics of $P_{+}(t_{j})$. If we just did two Hadamard gates separated by a delay, we would observe a decay of characteristic time $T^*_2$, but with a strong dependence on any deviation of the calibrated qubit frequency from the actual one. By implementing the qubit pulses with different phases, we shift the frequency dependence into the oscillating feature of the dynamics, and can fit the decaying envelope for a more faithful measure of the coherence time.\n\n
    \n\nThere is one subtle point of note. In the discussion of $T_{1}$ time above, we saw that amplitude damping noise also causes the off-diagonal elements to decay (at a rate $\\Gamma_{1}/2$). Suppose a qubit was affected by only amplitude damping noise, and dephasing noise was absent. In this scenaro, _the rate of decay of coherences can be non-zero, even in the absense of dephasing noise_.\n\nFor this reason, it's important to recognize there are many noise processes contributing to the rate at which coherences decay. In this tutorial, we assume that the total rate of decoherence, $\\Gamma$, can be decomposed into a sum of independent rates:\n\n$$\\Gamma = \\Gamma_{T_{1}} + \\Gamma_{2} + \\Gamma_{\\mathrm{other}}. $$\n\nPhenomenologically, the rate $\\Gamma_{\\mathrm{other}}$ quantifies the rate of decoherence due to other noise processes in addition to pure amplitude damping and pure dephasing about the $Z$-axis. Note that because general noise can cause dephasing about the $Z$-axis -- in addition to doing other things to the qubit -- echo sequences are typically used to help mitigate the effects of those kind(s) of noise on $T_{2}^{\\star}$. (Echo sequences are discussed below, in the sections $T_{2}$ echo and CPMG measurement.)\n\nIf decoherence at a rate $\\Gamma$ is taking place, then the state of the qubit changes as\n$$\\begin{equation}\n\\rho(t) = \\begin{pmatrix}\\rho_{00} & e^{-\\Gamma t}\\rho_{01} \\\\ e^{-\\Gamma t}\\rho_{01}^{\\star} & 1-\\rho_{00}\\end{pmatrix}.\n\\end{equation}$$\n\nThe timescale associated with this decay rate is called $T_{2}$, and is given by $T_{2} = 1/\\Gamma$.\n\n\n$T_{2}$ relates to the other timescales introduced previously as\n$$T_{2} = \\left(\\frac{2}{T_{1}} + \\frac{1}{T_{2}} + \\frac{1}{T_{\\mathrm{other}}}\\right)^{-1} = T_{2}^{\\star}\\left( 1 + \\frac{2T_{2}^{\\star}}{T_{1}} + \\frac{T^{\\star}_{2}}{T_{\\mathrm{other}}}\\right)^{-1} \\leq T_{2}^{\\star},$$\nwhere we've defined $T_{\\mathrm{other}} = 1 /\\Gamma_{\\mathrm{other}}$.\n\n
    \n\n\n```python\n# 50 points linearly spaced in two regions (fine and coarse)\n# 30 from 10->150, 20 from 160->450\nnum_of_gates = np.append((np.linspace(10, 150, 30)).astype(int), (np.linspace(160,450,20)).astype(int))\ngate_time = 0.1\n\n# Select the qubits whose T2* are to be measured\nqubits = [0]\n\n# Generate experiments\ncircs, xdata, osc_freq = t2star_circuits(num_of_gates, gate_time, qubits, nosc=5)\n```\n\n\n```python\nbackend = qiskit.Aer.get_backend('qasm_simulator')\n\n# Set the simulator with phase damping noise\nt2 = 10\np = 1 - np.exp(-2*gate_time/t2)\nerror = phase_damping_error(p)\nnoise_model = NoiseModel()\nnoise_model.add_quantum_error(error, 'id', [0])\n\n# Run the simulator\nshots = 300\nbackend_result = qiskit.execute(circs, backend,\n shots=shots, noise_model=noise_model).result()\n```\n\n\n```python\n%matplotlib inline\n# Fit the data to an oscillator\n# The correct answers are a=0.5, f=osc_freq, phi=0, c=0.5, and t2=10/5 for qubit 0/2\n# The user does not know the correct answer exactly,\n# so starts the fit from a different but close location\n\ninitial_t2 = t2*1.1\ninitial_a = 0.5\ninitial_c = 0.5\ninitial_f = osc_freq \ninitial_phi = -np.pi/20\n\nfit = T2StarFitter(backend_result, xdata, qubits,\n fit_p0=[initial_a, initial_t2, initial_f, initial_phi, initial_c],\n fit_bounds=([-0.5, 0, 0, -np.pi, -0.5],\n [1.5, 2*t2, 2*osc_freq, np.pi, 1.5]))\n\nfit.plot(0)\n```\n\n## Measuring T2 Time using a single echo pulse\n\nWe have referred to the previous experiment's characteristic time as \ud835\udc47\u22172 and not \ud835\udc472\n\nby analogy to nuclear magnetic resonance (NMR). Indeed, one can isolate different frequency components to the decoherence process by devising increasingly elaborated pulse sequences. To illustrate the analogy with NMR, one can think about an ensemble of nuclear spins precessing in an external DC magnetic field. Due to field inhomogeneities, each spin might precess with a slightly different Larmor frequency. This certainly will affect the observed coherence time of the ensemble. However, it is possible to echo away this low-frequency decoherence process by applying a pi-pulse to the system halfway through the delay. The effect of this pi-pulse is to reverse the direction of the precession of each individual spin due to field inhomogeneities. Thus, the spins that had precessed more now start precessing in the opposite direction faster than the spins that had precessed less, and after an equal delay, all the spins in the system recover the initial coherence, except for other, higher-frequency, decoherence mechanisms.\n\nHere, we are measuring only a single qubit rather than an ensemble of spins. Consequently coherence measurements require averaging an ensemble of measurements in order to eliminate projection noise, and run-to-run fluctuations in the qubit's frequency which will similarly manifest themselves as decoherence if they are not canceled out. By running this \ud835\udc472\necho sequence, we can therefore remove low-frequency components of the decoherence.\n\n\n```python\n# 50 points linearly spaced to 300\nnum_of_gates = (np.linspace(10, 300, 50)).astype(int)\ngate_time = 0.1\n\n# Select the qubits whose T2 are to be measured\nqubits = [0]\n\n# Generate experiments\ncircs, xdata = t2_circuits(num_of_gates, gate_time, qubits)\n```\n\n\n```python\nbackend = qiskit.Aer.get_backend('qasm_simulator')\n\n# Set the simulator with phase damping noise\nt2 = 10\np = 1 - np.exp(-2*gate_time/t2)\nerror = phase_damping_error(p)\nnoise_model = NoiseModel()\nnoise_model.add_quantum_error(error, 'id', [0])\n\n# Run the simulator\nshots = 300\nbackend_result = qiskit.execute(circs, backend,\n shots=shots, noise_model=noise_model).result()\n```\n\n\n```python\n%matplotlib inline\n# Fit the data to an exponent\n# The correct answers are a=1, c=0, and t2=10/5 for qubit 0/2\n# The user does not know the correct answer exactly,\n# so starts the fit from a different but close location\n\ninitial_t2 = t2*1.1\ninitial_a = 0.5\ninitial_c = 0.5\n\nfit = T2Fitter(backend_result, xdata, qubits,\n fit_p0=[initial_a, initial_t2, initial_c],\n fit_bounds=([-0.5, 0, -0.5],\n [1.5, 2*t2, 1.5]))\n\nfit.plot(0)\n```\n\n## Measuring T2 Time by a CPMG sequence\n\nAs explained above, the echo sequence removes low-frequency decoherence mechanisms. This noise-filtering procedure can be extended with increased number of pi-pulses within the delay. In the following experiment, we implement an echo experiment with seven pi-pulses during the delay between the initial and final pulses. This kind of echo with several pi-pulses is referred to as a CPMG experiment, after Carr, Purcell, Meiboom, and Gill.\n\n\n```python\nnum_of_gates = (np.linspace(1, 30, 30)).astype(int)\ngate_time = 0.1\n\n# Select the qubits whose T2 are to be measured\nqubits = [0]\n\n# Echo parameters\nn_echos = 5\nalt_phase_echo = True\n\n# Generate experiments\ncircs, xdata = t2_circuits(num_of_gates, gate_time, qubits, n_echos, alt_phase_echo)\n```\n\n\n```python\nbackend = qiskit.Aer.get_backend('qasm_simulator')\n\n# Set the simulator with phase damping noise\nt2 = 10\np = 1 - np.exp(-2*gate_time/t2)\nerror = phase_damping_error(p)\nnoise_model = NoiseModel()\nnoise_model.add_quantum_error(error, 'id', [0])\n\n# Run the simulator\nshots = 300\nbackend_result = qiskit.execute(circs, backend,\n shots=shots, noise_model=noise_model).result()\n```\n\n\n```python\n%matplotlib inline\n# Fit the data to an exponent\n# The correct answers are a=1, c=0, and t2=10/5 for qubit 0/2\n# The user does not know the correct answer exactly,\n# so starts the fit from a different but close location\n\ninitial_t2 = t2*1.1\ninitial_a = 0.5\ninitial_c = 0.5\n\nfit = T2Fitter(backend_result, xdata, qubits,\n fit_p0=[initial_a, initial_t2, initial_c],\n fit_bounds=([-0.5, 0, -0.5],\n [1.5, 2*t2, 1.5]))\n\nfit.plot(0)\n```\n", "meta": {"hexsha": "847af4db3c0c4b9fe396d4d33f8e973ac362b6bd", "size": 136230, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "community/ignis/coherence-overview.ipynb", "max_stars_repo_name": "sebhofer/qiskit-tutorials", "max_stars_repo_head_hexsha": "1efb5977b00345373b4c4d9889c1823859a248c1", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-04-29T15:11:27.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-09T20:52:21.000Z", "max_issues_repo_path": "community/ignis/coherence-overview.ipynb", "max_issues_repo_name": "Nishant-codex/qiskit-tutorials", "max_issues_repo_head_hexsha": "55b46f2bc879f98b4483e4c4126ea66e2f1b3391", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-05-08T20:25:11.000Z", "max_issues_repo_issues_event_max_datetime": "2020-05-08T20:25:11.000Z", "max_forks_repo_path": "community/ignis/coherence-overview.ipynb", "max_forks_repo_name": "Nishant-codex/qiskit-tutorials", "max_forks_repo_head_hexsha": "55b46f2bc879f98b4483e4c4126ea66e2f1b3391", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2019-03-24T21:00:25.000Z", "max_forks_repo_forks_event_max_datetime": "2019-03-24T21:57:10.000Z", "avg_line_length": 224.0625, "max_line_length": 29892, "alphanum_fraction": 0.8984071056, "converted": true, "num_tokens": 5147, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.6825737279551494, "lm_q1q2_score": 0.4372812130662668}} {"text": "```python\nimport gk\n```\n\n\n```python\nimport sympy as sym\n```\n\n\n```python\ndir(gk)\n```\n\n\n\n\n ['AA',\n 'AbelianGroup',\n 'AbelianGroupMorphism',\n 'AbelianGroupWithValues',\n 'AbelianVariety',\n 'AdditiveAbelianGroup',\n 'AdditiveAbelianGroupWrapper',\n 'AdditiveAbelianGroupWrapperElement',\n 'AdditiveMagmas',\n 'AffineCryptosystem',\n 'AffineGroup',\n 'AffineHypersurface',\n 'AffineNilTemperleyLiebTypeA',\n 'AffinePermutationGroup',\n 'AffineSpace',\n 'AffineToricVariety',\n 'AffineWeylGroups',\n 'AlarmInterrupt',\n 'Algebra',\n 'AlgebraIdeals',\n 'AlgebraModules',\n 'AlgebraicField',\n 'AlgebraicNumber',\n 'AlgebraicReal',\n 'AlgebraicRealField',\n 'Algebras',\n 'AlgebrasWithBasis',\n 'AllCusps',\n 'AllExactCovers',\n 'Alphabet',\n 'AlphabeticStrings',\n 'AlternatingGroup',\n 'AlternatingSignMatrices',\n 'AlternatingSignMatrix',\n 'ArithmeticSubgroup_Permutation',\n 'Arrangements',\n 'ArtinGroup',\n 'AsymptoticRing',\n 'AtkinModularCorrespondenceDatabase',\n 'AtkinModularPolynomialDatabase',\n 'AugmentedLatticeDiagramFilling',\n 'Automaton',\n 'Axiom',\n 'BackslashOperator',\n 'BaxterPermutations',\n 'Bessel',\n 'BezoutianQuadraticForm',\n 'Bialgebras',\n 'BialgebrasWithBasis',\n 'Bimodules',\n 'BinaryQF',\n 'BinaryQF_reduced_representatives',\n 'BinaryRecurrenceSequence',\n 'BinaryStrings',\n 'BinaryTree',\n 'BinaryTrees',\n 'BipartiteGraph',\n 'Bitset',\n 'BlockDesign',\n 'BooleanPolynomialRing',\n 'BraidGroup',\n 'BranchingRule',\n 'BrandtModule',\n 'BrauerAlgebra',\n 'BruhatTitsQuotient',\n 'CBF',\n 'CC',\n 'CDF',\n 'CFiniteSequence',\n 'CFiniteSequences',\n 'CIF',\n 'CLF',\n 'CPRFanoToricVariety',\n 'CRT',\n 'CRT_basis',\n 'CRT_list',\n 'CRT_vectors',\n 'CachedFunction',\n 'CallableSymbolicExpressionRing',\n 'CartanMatrix',\n 'CartanType',\n 'CartesianProduct',\n 'Category',\n 'ChainComplex',\n 'ChainComplexMorphism',\n 'ChainComplexes',\n 'Chi',\n 'Ci',\n 'ClassFunction',\n 'ClassicalCrystals',\n 'ClassicalModularPolynomialDatabase',\n 'CliffordAlgebra',\n 'ClusterAlgebra',\n 'ClusterComplex',\n 'ClusterQuiver',\n 'ClusterSeed',\n 'Coalgebras',\n 'CoalgebrasWithBasis',\n 'Color',\n 'ColoredPermutations',\n 'Combinations',\n 'CombinatorialAlgebra',\n 'CombinatorialClass',\n 'CombinatorialFreeModule',\n 'CombinatorialObject',\n 'CombinatorialSpecies',\n 'CommutativeAdditiveGroups',\n 'CommutativeAdditiveMonoids',\n 'CommutativeAdditiveSemigroups',\n 'CommutativeAlgebraElement',\n 'CommutativeAlgebraIdeals',\n 'CommutativeAlgebras',\n 'CommutativeRing',\n 'CommutativeRingElement',\n 'CommutativeRingIdeals',\n 'CommutativeRings',\n 'CompleteDiscreteValuationFields',\n 'CompleteDiscreteValuationRings',\n 'ComplexBallField',\n 'ComplexDoubleElement',\n 'ComplexDoubleField',\n 'ComplexField',\n 'ComplexIntervalField',\n 'ComplexIntervalFieldElement',\n 'ComplexLazyField',\n 'ComplexNumber',\n 'Complexes',\n 'Composition',\n 'CompositionTableau',\n 'CompositionTableaux',\n 'Compositions',\n 'Cone',\n 'CongruenceSubgroup',\n 'Conic',\n 'ConjugacyClass',\n 'ConjugacyClassGAP',\n 'ConstantFunction',\n 'Constellation',\n 'Constellations',\n 'ContreTableaux',\n 'ConwayPolynomials',\n 'CooperativeGame',\n 'CoordinatePatch',\n 'Core',\n 'Cores',\n 'CoveringDesign',\n 'CoxeterGroup',\n 'CoxeterGroups',\n 'CoxeterMatrix',\n 'CoxeterType',\n 'CremonaDatabase',\n 'CremonaModularSymbols',\n 'Crystals',\n 'CubeGroup',\n 'CubicalComplex',\n 'Curve',\n 'Cusp',\n 'CuspFamily',\n 'CuspForms',\n 'Cusps',\n 'CyclicPermutationGroup',\n 'CyclicPermutations',\n 'CyclicPermutationsOfPartition',\n 'CyclicSievingCheck',\n 'CyclicSievingPolynomial',\n 'CyclotomicField',\n 'Cylindrical',\n 'DLXCPP',\n 'DLXMatrix',\n 'DOT_SAGE',\n 'DWT',\n 'DeBruijnSequences',\n 'DedekindDomain',\n 'DedekindDomainElement',\n 'DedekindEtaModularCorrespondenceDatabase',\n 'DedekindEtaModularPolynomialDatabase',\n 'DegreeSequences',\n 'DeltaComplex',\n 'Derangements',\n 'DescentAlgebra',\n 'DiCyclicGroup',\n 'DiGraph',\n 'DiagonalQuadraticForm',\n 'DifferentialForm',\n 'DifferentialForms',\n 'DifferentialWeylAlgebra',\n 'DihedralGroup',\n 'DirichletGroup',\n 'DiscreteProbabilitySpace',\n 'DiscreteRandomVariable',\n 'DiscreteValuationFields',\n 'DiscreteValuationRings',\n 'DiscreteValueGroup',\n 'DisjointSet',\n 'DisjointUnionEnumeratedSets',\n 'DivisionRings',\n 'Dokchitser',\n 'Domains',\n 'DyckWord',\n 'DyckWords',\n 'DynamicalSystem',\n 'DynamicalSystem_affine',\n 'DynamicalSystem_projective',\n 'DynkinDiagram',\n 'E',\n 'ECM',\n 'Ei',\n 'EisensteinForms',\n 'EisensteinIntegers',\n 'ElementWrapper',\n 'Elements',\n 'EllipticCurve',\n 'EllipticCurveIsogeny',\n 'EllipticCurve_from_c4c6',\n 'EllipticCurve_from_cubic',\n 'EllipticCurve_from_j',\n 'EllipticCurves_with_good_reduction_outside_S',\n 'EmptySetError',\n 'End',\n 'EnumeratedSets',\n 'EquationOrder',\n 'EtaGroup',\n 'EtaGroupElement',\n 'EtaProduct',\n 'EuclideanDomain',\n 'EuclideanDomainElement',\n 'EuclideanDomains',\n 'EuclideanGroup',\n 'EuclideanSpace',\n 'Euler_Phi',\n 'Expression',\n 'ExtendedAffineWeylGroup',\n 'ExteriorAlgebra',\n 'FFT',\n 'FaceFan',\n 'Factorization',\n 'Family',\n 'Fan',\n 'Fan2d',\n 'FanMorphism',\n 'FareySymbol',\n 'FastFourierTransform',\n 'Field',\n 'FieldElement',\n 'Fields',\n 'FilteredCombinatorialClass',\n 'FilteredVectorSpace',\n 'FiniteCoxeterGroups',\n 'FiniteCrystals',\n 'FiniteDimensionalAlgebra',\n 'FiniteDimensionalAlgebrasWithBasis',\n 'FiniteDimensionalBialgebrasWithBasis',\n 'FiniteDimensionalCoalgebrasWithBasis',\n 'FiniteDimensionalHopfAlgebrasWithBasis',\n 'FiniteDimensionalModulesWithBasis',\n 'FiniteEnumeratedSet',\n 'FiniteEnumeratedSets',\n 'FiniteField',\n 'FiniteFields',\n 'FiniteGroups',\n 'FiniteLatticePosets',\n 'FiniteMonoids',\n 'FinitePermutationGroups',\n 'FinitePosets',\n 'FiniteRankFreeModule',\n 'FiniteSemigroups',\n 'FiniteSetMaps',\n 'FiniteSets',\n 'FiniteStateMachine',\n 'FiniteWeylGroups',\n 'FiniteWords',\n 'FockSpace',\n 'ForgetfulFunctor',\n 'FormalSum',\n 'FormalSums',\n 'Frac',\n 'FractionField',\n 'FreeAbelianMonoid',\n 'FreeAlgebra',\n 'FreeAlgebraQuotient',\n 'FreeGroup',\n 'FreeModule',\n 'FreeModules',\n 'FreeMonoid',\n 'FreeQuadraticModule',\n 'FreeQuasisymmetricFunctions',\n 'FreeSymmetricFunctions',\n 'FriCAS',\n 'FrozenBitset',\n 'FullyPackedLoop',\n 'FullyPackedLoops',\n 'FunctionField',\n 'FunctionFields',\n 'FusionRing',\n 'G1list',\n 'GCD',\n 'GF',\n 'GHlist',\n 'GL',\n 'GO',\n 'GSets',\n 'GU',\n 'Gamma',\n 'Gamma0',\n 'Gamma0_NFCusps',\n 'Gamma1',\n 'GammaH',\n 'Gap',\n 'Gap3',\n 'GaussValuation',\n 'GaussianIntegers',\n 'GcdDomains',\n 'GelfandTsetlinPattern',\n 'GelfandTsetlinPatterns',\n 'GeneralDihedralGroup',\n 'GeneralDiscreteDistribution',\n 'GenericGraphQuery',\n 'Genus',\n 'Genus2reduction',\n 'Gfan',\n 'Giac',\n 'Gp',\n 'GradedAlgebras',\n 'GradedAlgebrasWithBasis',\n 'GradedBialgebras',\n 'GradedBialgebrasWithBasis',\n 'GradedCoalgebras',\n 'GradedCoalgebrasWithBasis',\n 'GradedCommutativeAlgebra',\n 'GradedHopfAlgebras',\n 'GradedHopfAlgebrasWithBasis',\n 'GradedModules',\n 'GradedModulesWithBasis',\n 'Graph',\n 'GraphDatabase',\n 'GraphPaths',\n 'GraphQuery',\n 'Graphics',\n 'GroupAlgebra',\n 'GroupAlgebras',\n 'GroupExp',\n 'GroupExpElement',\n 'GroupExp_Class',\n 'GroupSemidirectProduct',\n 'GroupSemidirectProductElement',\n 'Groupoid',\n 'Groups',\n 'GrowthDiagram',\n 'GrowthDiagramBinWord',\n 'GrowthDiagramBurge',\n 'GrowthDiagramDomino',\n 'GrowthDiagramRSK',\n 'GrowthDiagramSylvester',\n 'GrowthDiagramYoungFibonacci',\n 'HallAlgebra',\n 'Hasse_bounds',\n 'HeckeAlgebraSymmetricGroupT',\n 'HeckeModules',\n 'HeilbronnCremona',\n 'HeilbronnMerel',\n 'HexadecimalStrings',\n 'HighestWeightCrystals',\n 'HilbertClassPolynomialDatabase',\n 'HillCryptosystem',\n 'Hom',\n 'Homset',\n 'HomsetWithBase',\n 'HopfAlgebras',\n 'HopfAlgebrasWithBasis',\n 'HyperbolicPlane',\n 'HyperbolicPlane_quadratic_form',\n 'HyperellipticCurve',\n 'HyperellipticCurve_from_invariants',\n 'Hypergraph',\n 'HyperplaneArrangements',\n 'I',\n 'Ideal',\n 'Ideals',\n 'IdentityFunctor',\n 'IncidenceStructure',\n 'IncreasingTableau',\n 'IncreasingTableaux',\n 'IndexedSequence',\n 'InfiniteAbstractCombinatorialClass',\n 'InfiniteEnumeratedSets',\n 'InfinitePolynomialRing',\n 'InfiniteWords',\n 'Infinity',\n 'InfinityRing',\n 'InnerProductSpace',\n 'IntList',\n 'Integer',\n 'IntegerListsLex',\n 'IntegerMod',\n 'IntegerModRing',\n 'IntegerRange',\n 'IntegerRing',\n 'IntegerVectors',\n 'IntegerVectorsModPermutationGroup',\n 'Integers',\n 'IntegrableRepresentation',\n 'IntegralDomain',\n 'IntegralDomainElement',\n 'IntegralDomains',\n 'IntegralLattice',\n 'InteractiveLPProblem',\n 'InteractiveLPProblemStandardForm',\n 'IwahoriHeckeAlgebra',\n 'J0',\n 'J1',\n 'JH',\n 'Jacobian',\n 'JoinSemilattice',\n 'JonesDatabase',\n 'JordanAlgebra',\n 'Kash',\n 'KazhdanLusztigPolynomial',\n 'KleinFourGroup',\n 'KleshchevPartitions',\n 'Knot',\n 'Knots',\n 'KnutsonTaoPuzzleSolver',\n 'KodairaSymbol',\n 'KostkaFoulkesPolynomial',\n 'KoszulComplex',\n 'KummerSurface',\n 'LCM',\n 'LFSRCryptosystem',\n 'LFunctionZeroSum',\n 'LabelledBinaryTree',\n 'LabelledBinaryTrees',\n 'LabelledOrderedTree',\n 'LabelledOrderedTrees',\n 'LabelledRootedTree',\n 'LabelledRootedTrees',\n 'LatexExpr',\n 'LatinSquare',\n 'LatinSquare_generator',\n 'LatticeDiagram',\n 'LatticePolytope',\n 'LatticePoset',\n 'LatticePosets',\n 'LaurentPolynomialRing',\n 'LaurentSeries',\n 'LaurentSeriesRing',\n 'LazyLaurentSeriesRing',\n 'LazyPowerSeriesRing',\n 'LeftModules',\n 'Li',\n 'LiE',\n 'LieAlgebra',\n 'LieAlgebras',\n 'LinearCode',\n 'LinearCodeFromVectorSpace',\n 'Link',\n 'Lisp',\n 'LittlewoodRichardsonTableau',\n 'LittlewoodRichardsonTableaux',\n 'LocalComponent',\n 'LyndonWord',\n 'LyndonWords',\n 'MPComplexField',\n 'MSymbol',\n 'MacMahonOmega',\n 'Macaulay2',\n 'Magma',\n 'Magmas',\n 'Manifold',\n 'MapCombinatorialClass',\n 'Maple',\n 'Mat',\n 'MatchingGame',\n 'Mathematica',\n 'MathieuGroup',\n 'Matlab',\n 'Matrix',\n 'MatrixAlgebras',\n 'MatrixGroup',\n 'MatrixSpace',\n 'Matroid',\n 'Maxima',\n 'MeetSemilattice',\n 'Mestre_conic',\n 'Minimog',\n 'MixedIntegerLinearProgram',\n 'Mod',\n 'ModularAbelianVarieties',\n 'ModularForms',\n 'ModularFormsRing',\n 'ModularSymbols',\n 'ModularSymbols_clear_cache',\n 'Modules',\n 'ModulesWithBasis',\n 'Moebius',\n 'MonoidAlgebras',\n 'Monoids',\n 'MonotoneTriangles',\n 'Morphism',\n 'MultiFilteredVectorSpace',\n 'MultiSkewTableau',\n 'MultiSkewTableaux',\n 'Mupad',\n 'Mutability',\n 'Mwrank',\n 'N',\n 'NFCusp',\n 'NFCusps',\n 'NN',\n 'NaN',\n 'Necklaces',\n 'NefPartition',\n 'Newform',\n 'Newforms',\n 'NilCoxeterAlgebra',\n 'NonCommutativeSymmetricFunctions',\n 'NonDecreasingParkingFunction',\n 'NonDecreasingParkingFunctions',\n 'NonNegativeIntegerSemiring',\n 'NonNegativeIntegers',\n 'NonSymmetricMacdonaldPolynomials',\n 'NonattackingFillings',\n 'NormalFan',\n 'NormalFormGame',\n 'NumberField',\n 'NumberFieldElement',\n 'NumberFieldTower',\n 'NumberFields',\n 'O',\n 'Objects',\n 'OctalStrings',\n 'Octave',\n 'OneExactCover',\n 'OpenInterval',\n 'OrderedMonoids',\n 'OrderedMultisetPartitionIntoSets',\n 'OrderedMultisetPartitionsIntoSets',\n 'OrderedPartitions',\n 'OrderedSetPartition',\n 'OrderedSetPartitions',\n 'OrderedSets',\n 'OrderedTree',\n 'OrderedTrees',\n 'OverconvergentDistributions',\n 'OverconvergentModularForms',\n 'P1List',\n 'P1NFList',\n 'PGL',\n 'PGU',\n 'PSL',\n 'PSU',\n 'PSage',\n 'PSp',\n 'Parallelism',\n 'ParallelogramPolyomino',\n 'ParallelogramPolyominoes',\n 'ParametrizedSurface3D',\n 'Parent',\n 'ParentWithBase',\n 'ParentWithGens',\n 'Pari',\n 'PariError',\n 'PariGroup',\n 'PariRing',\n 'ParkingFunction',\n 'ParkingFunctions',\n 'PartiallyOrderedMonoids',\n 'PartiallyOrderedSets',\n 'Partition',\n 'PartitionAlgebra',\n 'PartitionTuple',\n 'PartitionTuples',\n 'Partitions',\n 'PartitionsGreatestEQ',\n 'PartitionsGreatestLE',\n 'PartitionsInBox',\n 'PerfectMatching',\n 'PerfectMatchings',\n 'PeriodicSolitonCellularAutomata',\n 'Permutation',\n 'PermutationGroup',\n 'PermutationGroupElement',\n 'PermutationGroupMap',\n 'PermutationGroupMorphism',\n 'PermutationGroupMorphism_id',\n 'PermutationGroupMorphism_im_gens',\n 'PermutationGroup_generic',\n 'PermutationGroup_subgroup',\n 'PermutationGroups',\n 'Permutations',\n 'PlanarAlgebra',\n 'PlanePartition',\n 'PlanePartitions',\n 'PointConfiguration',\n 'PointedSets',\n 'PollackStevensModularSymbols',\n 'PolyhedralSets',\n 'Polyhedron',\n 'Polynomial',\n 'PolynomialQuotientRing',\n 'PolynomialQuotientRingElement',\n 'PolynomialRing',\n 'Poset',\n 'Posets',\n 'PositiveIntegers',\n 'PowComputer',\n 'PowComputer_ext_maker',\n 'PowerSeries',\n 'PowerSeriesRing',\n 'PrimarySimilarityClassType',\n 'PrimarySimilarityClassTypes',\n 'Primes',\n 'PrimitiveGroup',\n 'PrimitiveGroups',\n 'PrincipalIdealDomain',\n 'PrincipalIdealDomainElement',\n 'PrincipalIdealDomains',\n 'ProductProjectiveSpaces',\n 'Profiler',\n 'ProjectiveHypersurface',\n 'ProjectiveSpace',\n 'PropagatingIdeal',\n 'QQ',\n 'QQbar',\n 'QSystem',\n 'Qp',\n 'QpCR',\n 'QpFP',\n 'QpLC',\n 'QpLF',\n 'Qq',\n 'QqCR',\n 'QqFP',\n 'QuadraticBernoulliNumber',\n 'QuadraticField',\n 'QuadraticForm',\n 'QuadraticSpace',\n 'QuantumGroup',\n 'QuarticCurve',\n 'QuasiSymmetricFunctions',\n 'QuaternionAlgebra',\n 'QuaternionGroup',\n 'QuaternionMatrixGroupGF3',\n 'QuiverMutationType',\n 'QuotientFields',\n 'QuotientRing',\n 'R',\n 'RBF',\n 'RDF',\n 'RIF',\n 'RLF',\n 'RR',\n 'RSK',\n 'RSK_inverse',\n 'Radix64Strings',\n 'Rational',\n 'RationalCherednikAlgebra',\n 'RationalField',\n 'Rationals',\n 'RealBallField',\n 'RealDistribution',\n 'RealDoubleElement',\n 'RealDoubleField',\n 'RealField',\n 'RealInterval',\n 'RealIntervalField',\n 'RealLazyField',\n 'RealLine',\n 'RealNumber',\n 'RealSet',\n 'Realizations',\n 'Reals',\n 'RecursivelyEnumeratedSet',\n 'ReflectionGroup',\n 'ReflexivePolytope',\n 'ReflexivePolytopes',\n 'RegularCrystals',\n 'ResidueField',\n 'RibbonGraph',\n 'RibbonShapedTableau',\n 'RibbonShapedTableaux',\n 'RibbonTableau',\n 'RibbonTableaux',\n 'Riemann_Map',\n 'RiggedConfigurations',\n 'RightAngledArtinGroup',\n 'RightModules',\n 'Ring',\n 'RingElement',\n 'RingIdeals',\n 'RingModules',\n 'Rings',\n 'Rngs',\n 'RootSystem',\n 'RootedTree',\n 'RootedTrees',\n 'RowStandardTableau',\n 'RowStandardTableauTuple',\n 'RowStandardTableauTuples',\n 'RowStandardTableaux',\n 'RubiksCube',\n 'SAGE_DB',\n 'SAGE_DOC_SRC',\n 'SAGE_ENV',\n 'SAGE_LOCAL',\n 'SAGE_ROOT',\n 'SAGE_SRC',\n 'SAGE_TMP',\n 'SAT',\n 'SL',\n 'SL2Z',\n 'SO',\n 'SQLDatabase',\n 'SQLQuery',\n 'SR',\n 'SU',\n 'Sage',\n 'SageObject',\n 'Sandpile',\n 'SandpileConfig',\n 'SandpileDivisor',\n 'Schemes',\n 'SchubertPolynomialRing',\n 'SchurAlgebra',\n 'SchurTensorModule',\n 'SemidefiniteProgram',\n 'SemidihedralGroup',\n 'Semigroups',\n 'SemimonomialTransformationGroup',\n 'Semirings',\n 'SemistandardMultiSkewTableaux',\n 'SemistandardSkewTableaux',\n 'SemistandardTableau',\n 'SemistandardTableaux',\n 'Sequence',\n 'Set',\n 'SetPartition',\n 'SetPartitions',\n 'SetPartitionsAk',\n 'SetPartitionsBk',\n 'SetPartitionsIk',\n 'SetPartitionsPRk',\n 'SetPartitionsPk',\n 'SetPartitionsRk',\n 'SetPartitionsSk',\n 'SetPartitionsTk',\n 'Sets',\n 'SetsWithGrading',\n 'SetsWithPartialMaps',\n 'Shi',\n 'ShiftCryptosystem',\n 'ShiftedPrimedTableau',\n 'ShiftedPrimedTableaux',\n 'ShrinkingGeneratorCryptosystem',\n 'ShuffleAlgebra',\n 'Si',\n 'Sigma',\n 'SignalError',\n 'SignedCompositions',\n 'SignedPermutations',\n 'SimilarityClassType',\n 'SimilarityClassTypes',\n 'Simplex',\n 'SimplicialComplex',\n 'SimplicialComplexMorphism',\n 'SimplicialComplexes',\n 'SineGordonYsystem',\n 'Singular',\n 'SixVertexModel',\n 'SkewPartition',\n 'SkewPartitions',\n 'SkewPolynomialRing',\n 'SkewTableau',\n 'SkewTableaux',\n 'SloaneEncyclopedia',\n 'SolitonCellularAutomata',\n 'Sp',\n 'Spec',\n 'Spherical',\n 'SphericalDistribution',\n 'SphericalElevation',\n 'Spline',\n 'SplitMetacyclicGroup',\n 'Sq',\n 'StandardBracketedLyndonWords',\n 'StandardRibbonShapedTableaux',\n 'StandardSkewTableaux',\n 'StandardTableau',\n 'StandardTableauTuple',\n 'StandardTableauTuples',\n 'StandardTableaux',\n 'SteenrodAlgebra',\n 'SteinWatkinsAllData',\n 'SteinWatkinsPrimeData',\n 'StrongTableau',\n 'StrongTableaux',\n 'Subsets',\n 'SubstitutionCryptosystem',\n 'SubwordComplex',\n 'Subwords',\n 'Sudoku',\n 'SuperPartition',\n 'SuperPartitions',\n 'SupersingularModule',\n 'SuzukiGroup',\n 'SymbolicData',\n 'SymbolicLogic',\n 'Symk',\n 'SymmetricFunctions',\n 'SymmetricFunctionsNonCommutingVariables',\n 'SymmetricFunctionsNonCommutingVariablesDual',\n 'SymmetricGroup',\n 'SymmetricGroupAlgebra',\n 'SymmetricGroupRepresentation',\n 'SymmetricGroupRepresentations',\n 'Tableau',\n 'TableauTuple',\n 'TableauTuples',\n 'Tableaux',\n 'Tachyon',\n 'TamariIntervalPoset',\n 'TamariIntervalPosets',\n 'TateAlgebra',\n 'TemperleyLiebAlgebra',\n 'TensorAlgebra',\n 'TermOrder',\n 'TernaryQF',\n 'TestSuite',\n 'TimeSeries',\n 'ToricIdeal',\n 'ToricLattice',\n 'ToricVariety',\n 'TorsionQuadraticForm',\n 'TotallyOrderedFiniteSet',\n 'Transducer',\n 'TransitiveGroup',\n 'TransitiveGroups',\n 'TranspositionCryptosystem',\n 'TropicalSemiring',\n 'TruncatedStaircases',\n 'Tuples',\n 'UnionCombinatorialClass',\n 'UniqueFactorizationDomains',\n 'UniqueRepresentation',\n 'UnitGroup',\n 'UniversalCyclotomicField',\n 'Unknown',\n 'UnorderedTuples',\n 'UnsignedInfinityRing',\n 'UnwrappingMorphism',\n 'VectorPartition',\n 'VectorPartitions',\n 'VectorSpace',\n 'VectorSpaces',\n 'VigenereCryptosystem',\n 'VoronoiDiagram',\n 'WaveletTransform',\n 'WeakReversePlanePartition',\n 'WeakReversePlanePartitions',\n 'WeakTableau',\n 'WeakTableaux',\n 'WehlerK3Surface',\n 'WeierstrassForm',\n 'WeightRing',\n 'WeightedIntegerVectors',\n 'WeylCharacterRing',\n 'WeylDim',\n 'WeylGroup',\n 'WeylGroupElement',\n 'WeylGroups',\n 'Word',\n 'WordMorphism',\n 'WordOptions',\n 'WordPaths',\n 'WordQuasiSymmetricFunctions',\n 'Words',\n 'YangBaxterGraph',\n 'Yangian',\n 'ZZ',\n 'Zmod',\n 'Zp',\n 'ZpCA',\n 'ZpCR',\n 'ZpFM',\n 'ZpFP',\n 'ZpLC',\n 'ZpLF',\n 'Zq',\n 'ZqCA',\n 'ZqCR',\n 'ZqFM',\n 'ZqFP',\n '__builtins__',\n '__cached__',\n '__doc__',\n '__file__',\n '__loader__',\n '__name__',\n '__package__',\n '__spec__',\n 'a0',\n 'a1',\n 'a2',\n 'a3',\n 'a4',\n 'a5',\n 'a6',\n 'abs_symbolic',\n 'absolute_igusa_invariants_kohel',\n 'absolute_igusa_invariants_wamelen',\n 'absolute_import',\n 'abstract_method',\n 'acos',\n 'acosh',\n 'acot',\n 'acoth',\n 'acsc',\n 'acsch',\n 'add',\n 'addgp',\n 'addition_names',\n 'additive_order',\n 'affinity_network',\n 'airy_ai',\n 'airy_ai_prime',\n 'airy_bi',\n 'airy_bi_prime',\n 'alarm',\n 'algdep',\n 'algebras',\n 'all_max_clique',\n 'animate',\n 'arc',\n 'arccos',\n 'arccosh',\n 'arccot',\n 'arccoth',\n 'arccsc',\n 'arccsch',\n 'arcsec',\n 'arcsech',\n 'arcsin',\n 'arcsinh',\n 'arctan',\n 'arctan2',\n 'arctanh',\n 'arg',\n 'arrow',\n 'arrow2d',\n 'arrow3d',\n 'ascii_art',\n 'asec',\n 'asech',\n 'asin',\n 'asinh',\n 'assume',\n 'assuming',\n 'assumptions',\n 'asym',\n 'asymptotic_expansions',\n 'atan',\n 'atan2',\n 'atanh',\n 'attach',\n 'attached_files',\n 'attrcall',\n 'automata',\n 'axiom',\n 'backtrack_all',\n 'balanced_sum',\n 'banner',\n 'bar_chart',\n 'base_field',\n 'base_ring',\n 'basis',\n 'bell_number',\n 'bell_polynomial',\n 'benchmark',\n 'berlekamp_massey',\n 'bernoulli',\n 'bernoulli_mod_p',\n 'bernoulli_mod_p_single',\n 'bernoulli_polynomial',\n 'bessel_I',\n 'bessel_J',\n 'bessel_K',\n 'bessel_Y',\n 'beta',\n 'betavariate',\n 'bezier3d',\n 'bezier_path',\n 'binomial',\n 'binomial_coefficients',\n 'block_diagonal_matrix',\n 'block_matrix',\n 'branching_rule',\n 'branching_rule_from_plethysm',\n 'browse_sage_doc',\n 'bsgs',\n 'build_alphabet',\n 'buzzard_tpslopes',\n 'cached_function',\n 'cached_in_parent_method',\n 'cached_method',\n 'cancel_alarm',\n 'canonical_coercion',\n 'cartesian_product',\n 'cartesian_product_iterator',\n 'cases',\n 'catalan',\n 'catalan_number',\n 'category',\n 'ceil',\n 'channels',\n 'characteristic_polynomial',\n 'charpoly',\n 'chebyshev_T',\n 'chebyshev_U',\n 'choice',\n 'circle',\n 'class_graph',\n 'clear_vars',\n 'clebsch_gordan',\n 'clebsch_invariants',\n 'clique_number',\n 'cm_j_invariants',\n 'cm_j_invariants_and_orders',\n 'cm_orders',\n 'codes',\n 'coerce',\n 'coercion_model',\n 'coercion_traceback',\n 'coincidence_discriminant',\n 'coincidence_index',\n 'colormaps',\n 'colors',\n 'column_matrix',\n 'companion_matrix',\n 'complex_cubic_spline',\n 'complex_plot',\n 'complex_root_of',\n 'compose',\n 'conjugate',\n 'constructions',\n 'continuant',\n 'continued_fraction',\n 'continued_fraction_list',\n 'contour_plot',\n 'convergents',\n 'convolution',\n 'conway_polynomial',\n 'copy',\n 'copying',\n 'copyright',\n 'cos',\n 'cos_integral',\n 'cosh',\n 'cosh_integral',\n 'cot',\n 'coth',\n 'cputime',\n 'cremona_curves',\n 'cremona_optimal_curves',\n 'crt',\n 'crt_basis',\n 'crystals',\n 'csc',\n 'csch',\n 'cube',\n 'cubical_complexes',\n 'cunningham_prime_factors',\n 'current_randstate',\n 'cyclotomic_polynomial',\n 'cyclotomic_value',\n 'cylindrical_plot3d',\n 'cython',\n 'cython_create_local_so',\n 'cython_lambda',\n 'db',\n 'db_save',\n 'debug',\n 'decomposition',\n 'dedekind_sum',\n 'deepcopy',\n 'default_mip_solver',\n 'default_sdp_solver',\n 'degree_lowest_rational_function',\n 'delta_complexes',\n 'delta_lseries',\n 'delta_qexp',\n 'denominator',\n 'density_plot',\n 'derivative',\n 'designs',\n 'designs_from_XML',\n 'designs_from_XML_url',\n 'desolve',\n 'desolve_laplace',\n 'desolve_mintides',\n 'desolve_odeint',\n 'desolve_rk4',\n 'desolve_system',\n 'desolve_system_rk4',\n 'desolve_tides_mpfr',\n 'desolvers',\n 'det',\n 'detach',\n 'developer',\n 'diagonal_matrix',\n 'dickman_rho',\n 'diff',\n 'differences',\n 'digraphs',\n 'dilog',\n 'dim',\n 'dimension',\n 'dimension_cusp_forms',\n 'dimension_eis',\n 'dimension_modular_forms',\n 'dimension_new_cusp_forms',\n 'dimension_supersingular_module',\n 'dirac_delta',\n 'direct_product_permgroups',\n 'disc',\n 'discrete_log',\n 'discrete_log_generic',\n 'discrete_log_lambda',\n 'discrete_log_rho',\n 'discriminant',\n 'disk',\n 'disk_cached_function',\n 'divisors',\n 'dodecahedron',\n 'dumps',\n 'e',\n 'ecm',\n 'edit',\n 'eisenstein_series_lseries',\n 'eisenstein_series_qexp',\n 'elementary_matrix',\n 'ellipse',\n 'ellipsis_iter',\n 'ellipsis_range',\n 'elliptic_curves',\n 'elliptic_e',\n 'elliptic_ec',\n 'elliptic_eu',\n 'elliptic_f',\n 'elliptic_j',\n 'elliptic_kc',\n 'elliptic_pi',\n 'email',\n 'end',\n 'enum_affine_finite_field',\n 'enum_affine_rational_field',\n 'enumerate_totallyreal_fields_all',\n 'enumerate_totallyreal_fields_prim',\n 'enumerate_totallyreal_fields_rel',\n 'eratosthenes',\n 'erf',\n 'erfc',\n 'erfi',\n 'erfinv',\n 'eta',\n 'eta_poly_relations',\n 'euler_gamma',\n 'euler_number',\n 'euler_phi',\n 'eulers_method',\n 'eulers_method_2x2',\n 'eulers_method_2x2_plot',\n 'exists',\n 'exists_conway_polynomial',\n 'exp',\n 'exp_integral_e',\n 'exp_integral_e1',\n 'exp_integral_ei',\n 'exp_polar',\n 'expand',\n 'experimental_packages',\n 'explain_pickle',\n 'expnums',\n 'exponential_integral_1',\n 'expovariate',\n 'extend_to_primitive',\n 'external_ray',\n 'factor',\n 'factorial',\n 'falling_factorial',\n 'false',\n 'fast_callable',\n 'fast_float',\n 'fcp',\n 'fibonacci',\n 'fibonacci_sequence',\n 'fibonacci_xrange',\n 'finance',\n 'find_a_ternary_qf_by_level_disc',\n 'find_all_ternary_qf_by_level_disc',\n 'find_fit',\n 'find_local_maximum',\n 'find_local_minimum',\n 'find_root',\n 'findstat',\n 'firing_graph',\n 'flatten',\n 'floor',\n 'forall',\n 'forget',\n 'fork',\n 'fortran',\n 'four_squares',\n 'four_ti_2',\n 'frac',\n 'free_module_element',\n 'frequency_distribution',\n 'fresnel_cos',\n 'fresnel_sin',\n 'fricas',\n 'frobby',\n 'func_persist',\n 'function',\n 'fundamental_discriminant',\n 'game_theory',\n 'gamma',\n 'gamma__exact',\n 'gamma_inc',\n 'gamma_inc_lower',\n 'gammavariate',\n 'gap',\n 'gap3',\n 'gap3_version',\n 'gap_reset_workspace',\n 'gaunt',\n 'gauss',\n 'gaussian_binomial',\n 'gcd',\n 'gegenbauer',\n 'gen',\n 'gen_laguerre',\n 'gen_legendre_P',\n 'gen_legendre_Q',\n 'gens',\n 'genus2reduction',\n 'get_coercion_model',\n 'get_display_manager',\n 'get_gcd',\n 'get_inverse_mod',\n 'get_memory_usage',\n 'get_remote_file',\n 'get_verbose',\n 'get_verbose_files',\n 'getattr_debug',\n 'getrandbits',\n 'gfan',\n 'giac',\n 'glaisher',\n 'gnuplot',\n 'golden_ratio',\n 'gp',\n 'gp_version',\n 'graph_classes',\n 'graph_coloring',\n 'graph_db_info',\n 'graph_editor',\n 'graphics_array',\n 'graphs',\n 'graphs_list',\n 'groups',\n 'hadamard_matrix',\n 'hadamard_matrix_www',\n 'half_integral_weight_modform_basis',\n 'hankel1',\n 'hankel2',\n 'harmonic_number',\n 'heaviside',\n 'hecke_operator',\n 'hecke_operator_on_basis',\n 'hecke_operator_on_qexp',\n 'hecke_series',\n 'heegner_point',\n 'heegner_points',\n 'help',\n 'hermite',\n 'hermite_constant',\n 'hilbert_class_polynomial',\n 'hilbert_conductor',\n 'hilbert_conductor_inverse',\n 'hilbert_symbol',\n 'histogram',\n 'hmm',\n 'hold',\n 'hom',\n 'html',\n 'hue',\n 'hurwitz_zeta',\n 'hyperbolic_arc',\n 'hyperbolic_polygon',\n 'hyperbolic_regular_polygon',\n 'hyperbolic_triangle',\n 'hypergeometric',\n 'hypergeometric_M',\n 'hypergeometric_U',\n 'hypergraphs',\n 'hyperplane_arrangements',\n 'i',\n 'icosahedron',\n 'ideal',\n 'identity_matrix',\n 'igusa_clebsch_invariants',\n 'imag',\n 'imag_part',\n 'image',\n 'imaginary',\n 'implicit_multiplication',\n 'implicit_plot',\n 'implicit_plot3d',\n 'import_statements',\n 'infinity',\n 'infix_operator',\n 'initial_seed',\n 'inotebook',\n 'install_scripts',\n 'installed_packages',\n 'integer_ceil',\n 'integer_floor',\n 'integral',\n 'integral_closure',\n 'integral_numerical',\n 'integrate',\n 'interact',\n 'interacts',\n 'interfaces',\n 'interval',\n 'invariant_theory',\n 'inverse_jacobi',\n 'inverse_jacobi_cd',\n 'inverse_jacobi_cn',\n 'inverse_jacobi_cs',\n 'inverse_jacobi_dc',\n 'inverse_jacobi_dn',\n 'inverse_jacobi_ds',\n 'inverse_jacobi_nc',\n 'inverse_jacobi_nd',\n 'inverse_jacobi_ns',\n 'inverse_jacobi_sc',\n 'inverse_jacobi_sd',\n 'inverse_jacobi_sn',\n 'inverse_laplace',\n 'inverse_mod',\n 'is_ProductProjectiveSpaces',\n 'is_ProjectiveSpace',\n 'is_commutative',\n 'is_even',\n 'is_field',\n 'is_fundamental_discriminant',\n 'is_integrally_closed',\n 'is_iterator',\n 'is_odd',\n 'is_pAdicField',\n 'is_pAdicRing',\n 'is_package_installed',\n 'is_power_of_two',\n 'is_prime',\n 'is_prime_power',\n 'is_pseudoprime',\n 'is_pseudoprime_power',\n 'is_real_place',\n 'is_square',\n 'is_squarefree',\n 'is_triangular_number',\n 'isogeny_codomain_from_kernel',\n 'isqrt',\n 'j_invariant_qexp',\n 'jacobi',\n 'jacobi_P',\n 'jacobi_am',\n 'jacobi_cd',\n 'jacobi_cn',\n 'jacobi_cs',\n 'jacobi_dc',\n 'jacobi_dn',\n 'jacobi_ds',\n 'jacobi_nc',\n 'jacobi_nd',\n 'jacobi_ns',\n 'jacobi_sc',\n 'jacobi_sd',\n 'jacobi_sn',\n 'jacobi_symbol',\n 'jacobian',\n 'jordan_block',\n 'julia_plot',\n 'kash',\n 'kash_version',\n 'kernel',\n 'khinchin',\n 'kronecker',\n 'kronecker_character',\n 'kronecker_character_upside_down',\n 'kronecker_delta',\n 'kronecker_symbol',\n 'krull_dimension',\n 'laguerre',\n 'lambert_w',\n 'laplace',\n 'latex',\n 'lattice_polytope',\n 'lazy_attribute',\n 'lazy_class_attribute',\n 'lazy_import',\n 'lcalc',\n 'lcm',\n 'least_quadratic_nonresidue',\n 'legendre_P',\n 'legendre_Q',\n 'legendre_phi',\n 'legendre_symbol',\n 'lfsr_autocorrelation',\n 'lfsr_connection_polynomial',\n 'lfsr_sequence',\n 'li',\n 'libgap',\n 'license',\n 'lie',\n 'lie_algebras',\n 'lift',\n 'lift_to_sl2z',\n 'lim',\n 'limit',\n 'line',\n 'line2d',\n 'line3d',\n 'linear_program',\n 'linear_relation',\n 'linear_transformation',\n 'lisp',\n 'list_plot',\n 'list_plot3d',\n 'list_plot_loglog',\n 'list_plot_semilogx',\n 'list_plot_semilogy',\n 'ln',\n 'load',\n 'load_attach_mode',\n 'load_attach_path',\n 'load_session',\n 'loads',\n 'local_print_mode',\n 'localvars',\n 'log',\n 'log2',\n 'log_b',\n 'log_gamma',\n 'log_integral',\n 'log_integral_offset',\n 'lognormvariate',\n 'logstr',\n 'lucas_number1',\n 'lucas_number2',\n 'macaulay2',\n 'magma',\n 'magma_free',\n 'mandelbrot_plot',\n 'manifolds',\n 'manual',\n 'map_threaded',\n 'maple',\n 'math',\n 'mathematica',\n 'mathml',\n 'matlab',\n 'matlab_version',\n 'matrix',\n 'matrix_plot',\n 'matroids',\n 'max_clique',\n 'max_symbolic',\n 'maxima',\n 'maxima_calculus',\n 'mean',\n 'median',\n 'merge_points',\n 'mertens',\n 'min_symbolic',\n 'minimal_polynomial',\n 'minimize',\n 'minimize_constrained',\n 'minpoly',\n 'mod',\n 'mode',\n 'moebius',\n 'monomials',\n 'monsky_washnitzer',\n 'moving_average',\n 'mq',\n 'mqrr_rational_reconstruction',\n 'mrange',\n 'mrange_iter',\n 'mul',\n 'multinomial',\n 'multinomial_coefficients',\n 'multiple',\n 'multiples',\n 'multiplication_names',\n 'multiplicative_order',\n 'mupad',\n 'mwrank',\n 'mwrank_EllipticCurve',\n 'mwrank_MordellWeil',\n 'mwrank_get_precision',\n 'mwrank_initprimes',\n 'mwrank_set_precision',\n 'n',\n 'nest',\n 'newton_method_sizes',\n 'next_prime',\n 'next_prime_power',\n 'next_probable_prime',\n 'ngens',\n 'norm',\n 'normalvariate',\n 'notebook',\n 'nth_prime',\n 'ntl',\n 'num_cusps_of_width',\n 'number_field_elements_from_algebraics',\n 'number_of_divisors',\n 'number_of_partitions',\n 'number_of_tuples',\n 'number_of_unordered_tuples',\n 'numbers_abc',\n 'numerator',\n 'numerical_approx',\n 'numerical_eigenforms',\n 'numerical_integral',\n 'objgen',\n 'objgens',\n 'octahedron',\n 'octave',\n 'odd_part',\n 'ode_solver',\n 'ode_system',\n 'oeis',\n 'ones_matrix',\n 'oo',\n 'operator',\n 'optional_packages',\n 'order',\n 'order_from_bounds',\n 'order_from_multiple',\n 'os',\n 'pAdicExtension',\n 'pAdicField',\n 'pAdicRing',\n 'pAdicWeightSpace',\n 'package_versions',\n 'pad_zeros',\n 'padic_printing',\n 'pager',\n 'parallel',\n 'parallel_firing_graph',\n 'parametric_plot',\n 'parametric_plot3d',\n 'parent',\n 'paretovariate',\n 'pari',\n 'pari_gen',\n 'partial_sieve_function',\n 'permutation_action',\n 'pi',\n 'pickle_function',\n 'piecewise',\n 'plot',\n 'plot3d',\n 'plot_loglog',\n 'plot_semilogx',\n 'plot_semilogy',\n 'plot_slope_field',\n 'plot_step_function',\n 'plot_vector_field',\n 'plot_vector_field3d',\n 'point',\n 'point2d',\n 'point3d',\n 'points',\n 'polar_plot',\n 'polygen',\n 'polygens',\n 'polygon',\n 'polygon2d',\n 'polygon3d',\n 'polygon_spline',\n 'polygons3d',\n 'polylog',\n 'polymake',\n 'polytopes',\n 'posets',\n 'povray',\n 'power',\n 'power_mod',\n 'powerset',\n 'preparse',\n 'preparser',\n 'pretty_print',\n 'pretty_print_default',\n 'previous_prime',\n 'previous_prime_power',\n 'prime_divisors',\n 'prime_factors',\n 'prime_pi',\n 'prime_powers',\n 'prime_range',\n 'prime_to_m_part',\n 'primes',\n 'primes_first_n',\n 'primitive_root',\n 'prod',\n 'product',\n 'proof',\n 'propcalc',\n 'psi',\n 'python_help',\n 'q_binomial',\n 'qepcad',\n 'qepcad_formula',\n 'qepcad_version',\n 'qexp_eta',\n 'qsieve',\n 'quadratic_L_function__exact',\n 'quadratic_L_function__numerical',\n 'quadratic_residues',\n 'quit_sage',\n 'quo',\n 'quotient',\n 'r',\n 'r_version',\n 'racah',\n 'radical',\n 'rainbow',\n 'randint',\n 'random',\n 'random_DAG',\n 'random_WehlerK3Surface',\n 'random_cone',\n 'random_matrix',\n 'random_prime',\n 'random_quadraticform',\n 'random_quadraticform_with_conditions',\n 'random_sublist',\n 'random_ternaryqf',\n 'random_ternaryqf_with_conditions',\n 'random_vector',\n 'randrange',\n 'rank',\n 'ranker',\n 'rational_reconstruction',\n 'raw_getattr',\n 'read_data',\n 'real',\n 'real_part',\n 'reciprocal_trig_functions',\n 'reduce',\n 'reference',\n 'region_plot',\n 'register_unpickle_override',\n 'regulator',\n 'repr_lincomb',\n 'reset',\n 'reset_load_attach_path',\n 'restore',\n 'revolution_plot3d',\n 'rising_factorial',\n 'robinson_schensted_knuth',\n 'robinson_schensted_knuth_inverse',\n 'round',\n 'run_doctests',\n 'running_total',\n 'runsnake',\n 'sage',\n 'sage0',\n 'sage0_version',\n 'sage_eval',\n 'sage_globals',\n 'sage_input',\n 'sage_wraps',\n 'sageobj',\n 'sample',\n 'sandpiles',\n 'save',\n 'save_session',\n 'scatter_plot',\n 'schonheim',\n 'scilab',\n 'search_def',\n 'search_doc',\n 'search_src',\n 'sec',\n 'sech',\n 'seed',\n 'self_orthogonal_binary_codes',\n 'seq',\n 'series_precision',\n 'set_default_variable_name',\n 'set_edit_template',\n 'set_gap_memory_pool_size',\n 'set_modsym_print_mode',\n 'set_random_seed',\n 'set_series_precision',\n 'set_verbose',\n 'set_verbose_files',\n 'sgn',\n 'sh',\n 'show',\n 'show_identifiers',\n 'shuffle',\n 'sidon_sets',\n 'sig_on_count',\n 'sigma',\n 'sign',\n 'simplicial_complexes',\n 'simplicial_sets',\n 'simplify',\n 'sin',\n 'sin_integral',\n 'singular',\n 'singular_version',\n 'sinh',\n 'sinh_integral',\n 'sleep',\n 'sloane',\n 'solve',\n 'solve_diophantine',\n 'solve_ineq',\n 'solve_mod',\n 'sort_complex_numbers_for_display',\n 'span',\n 'specialize',\n 'species',\n 'sphere',\n 'spherical_bessel_J',\n 'spherical_bessel_Y',\n 'spherical_hankel1',\n 'spherical_hankel2',\n 'spherical_harmonic',\n 'spherical_plot3d',\n 'spike_function',\n 'spline',\n 'sqrt',\n 'squarefree_divisors',\n 'squarefree_part',\n 'srange',\n 'standard_packages',\n 'stats',\n 'std',\n 'steenrod_algebra_basis',\n 'stieltjes',\n 'stirling_number1',\n 'stirling_number2',\n 'streamline_plot',\n 'strip_encoding',\n 'structure_description',\n 'struve_H',\n 'struve_L',\n 'sturm_bound',\n 'subfactorial',\n 'subsets',\n 'sudoku',\n 'sum',\n 'sum_of_k_squares',\n 'supersingular_D',\n 'supersingular_j',\n 'surfaces',\n 'sxrange',\n 'sym',\n 'symbolic_expression',\n 'symmetrica',\n 'sympow',\n 'sys',\n 'table',\n 'tachyon_rt',\n 'tan',\n 'tanh',\n 'taylor',\n 'tensor',\n 'tests',\n 'tetrahedron',\n 'text',\n 'text3d',\n 'theta2_qexp',\n 'theta_qexp',\n 'three_squares',\n 'timeit',\n 'tmp_dir',\n 'tmp_filename',\n 'toric_plotter',\n 'toric_varieties',\n 'trace',\n 'transducers',\n 'transpose',\n 'trial_division',\n 'triangle_sandpile',\n 'trivial_character',\n 'trivial_covering_design',\n 'true',\n 'ttest',\n 'tuples',\n 'tutorial',\n 'twinprime',\n 'two_squares',\n 'type_debug',\n 'ultraspherical',\n 'unicode_art',\n 'uniform',\n 'union',\n 'uniq',\n 'unit_step',\n 'units',\n 'unordered_tuples',\n 'unpickle_appends',\n 'unpickle_build',\n 'unpickle_extension',\n 'unpickle_function',\n 'unpickle_global',\n 'unpickle_instantiate',\n 'unpickle_newobj',\n 'unpickle_persistent',\n 'unset_verbose_files',\n 'unsigned_infinity',\n 'v',\n 'valuation',\n 'valuations',\n 'var',\n 'variance',\n 'varmap',\n 'varmapsr',\n 'vector',\n 'verbose',\n 'version',\n 'victor_miller_basis',\n 'view',\n 'vonmisesvariate',\n 'walltime',\n 'walsh_matrix',\n 'warnings',\n 'wave',\n 'wedge',\n 'weibullvariate',\n 'wigner_3j',\n 'wigner_6j',\n 'wigner_9j',\n 'wilmes_algorithm',\n 'word_problem',\n 'words',\n 'wronskian',\n 'x',\n 'x0',\n 'x1',\n 'xgcd',\n 'xinterval',\n 'xkcd',\n 'xlcm',\n 'xmrange',\n 'xmrange_iter',\n 'xsr',\n 'xsrange',\n 'xsym',\n 'y',\n 'y0',\n 'ysr',\n 'ysym',\n 'ysymraw',\n 'zero_matrix',\n 'zero_vector',\n 'zeta',\n 'zeta__exact',\n 'zeta_symmetric',\n 'zeta_zeros',\n 'zetaderiv']\n\n\n\n\n```python\ngk.ysymraw\n```\n\n\n\n\n [0, a0 + a1 + a2*x1 - x1 + (-a3*x1 + (a4 + a5 + a6*x1 - x1)**2.0)**0.5]\n\n\n\n\n```python\ngk.ysym[1]\n```\n\n\n\n\n$\\displaystyle a_{0} + a_{1} + a_{2} x_{1} - x_{1} + \\left(- a_{3} x_{1} + \\left(a_{4} + a_{5} + a_{6} x_{1} - x_{1}\\right)^{2.0}\\right)^{0.5}$\n\n\n\n\n```python\nvarmap = {\n# 'x0': 'S',\n# 'x1': 1,\n# #'a0': 'u',\n# 'a0': 'v',\n# 'a1': 'u',\n# 'a2': 'J',\n# 'a3': 'K',\n 'x1': 'R',\n 'a6': 'v',\n 'a4': \"v\",\n 'a5': -sym.var(\"u\")*sym.var(\"R\"),\n}\n```\n\n\n```python\ngk.ysym[1].subs(varmap)\n```\n\n\n\n\n$\\displaystyle R a_{2} - R + a_{0} + a_{1} + \\left(- R a_{3} + \\left(- R u + R v - R + v\\right)^{2.0}\\right)^{0.5}$\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "4ca1203ac51613d477c1fa7ef5d700517cf8cb44", "size": 58738, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Debugging Goldbeter Koshland.ipynb", "max_stars_repo_name": "twright/bondwb", "max_stars_repo_head_hexsha": "5557788f8cdf780fa2899ca29eb926ed5c3ab205", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2018-05-04T20:00:47.000Z", "max_stars_repo_stars_event_max_datetime": "2019-08-16T11:54:15.000Z", "max_issues_repo_path": "Debugging Goldbeter Koshland.ipynb", "max_issues_repo_name": "twright/bondwb", "max_issues_repo_head_hexsha": "5557788f8cdf780fa2899ca29eb926ed5c3ab205", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Debugging Goldbeter Koshland.ipynb", "max_forks_repo_name": "twright/bondwb", "max_forks_repo_head_hexsha": "5557788f8cdf780fa2899ca29eb926ed5c3ab205", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 28.8638820639, "max_line_length": 157, "alphanum_fraction": 0.4583063775, "converted": true, "num_tokens": 12028, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.640635854839898, "lm_q1q2_score": 0.4372812078365008}} {"text": "

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    Escuela de Ciencias B\u00e1sicas, Tecnolog\u00eda e Ingenier\u00eda

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    ECBTI

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    \u00bfUsted sabe por qu\u00e9 debe mantener la distancia social recomendada de personas con s\u00edntomas de la COVID-19? !Venga le cuento!

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    Equipo de Trabajo en Ciencia de Datos - UNAD

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    Expositores: Carlos Alvarez - Jeinny Peralta - Tatiana S\u00e1nchez

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    Abril 21 de 2020

    \n    \n\n\n\n# Modelo matem\u00e1tico y computacional para predecir el comportamiento de la Covid-19\n\n## Distanciamiento social como estrategia para ralentizar la propagaci\u00f3n\n\n***Nota:*** basado en los art\u00edculos publicados en las p\u00e1ginas web:\n\n- [Social Distancing to Slow the Coronavirus - Modeling the flattening of the COVID-19 peaks](https://towardsdatascience.com/social-distancing-to-slow-the-coronavirus-768292f04296 \"Towardsdatascience\"): By Christian Hubbs, in Towardsdatascience.com, Mar 12, 2020\n\n\n- [Why outbreaks like coronavirus spread exponentially, and how to \u201cflatten the curve\u201d](https://www.washingtonpost.com/graphics/2020/world/corona-simulator/): By Harry Stevens, in The Washington Post, Mar 14, 2020\n\n\n- [Infectious Disease Modelling: Understanding the models that are used to model Coronavirus](https://towardsdatascience.com/infectious-disease-modelling-part-i-understanding-sir-28d60e29fdfc): By Henri Froese, in Towardsdatascience.com, Apr 06,2020\n\n\n- [Infectious Disease Modelling: Beyond the Basic SIR Model](https://towardsdatascience.com/infectious-disease-modelling-part-i-understanding-sir-28d60e29fdfc): By Henri Froese, in Towardsdatascience.com, Apr 11,2020 \n\n# Contextualizaci\u00f3n\n\nComo es de \u00e1mplio conocimiento de todos hoy en d\u00eda, la Covid-19 ha crecido r\u00e1pidamente en todo el mundo. La gran mayor\u00eda de paises han tomado medidas dr\u00e1sticas frente a la movilidad de sus ciudadanos. Estas medidas se toman con la intenci\u00f3n de retrasar la propagaci\u00f3n (no detenerla) de la enfermedad. Este tipo de estrategia se denomina *Distanciamiento Social*.\n\nLa idea detr\u00e1s de esta pol\u00edtica de salud p\u00fablica es reducir el contacto *persona-a-persona* para que sea menos probable la propagaci\u00f3n de la enfermedad y la capacidad de los sistemas de salud local no se saturen, ayudando a garantizar la adecuada atenci\u00f3n a la poblaci\u00f3n enferma y disminyendo la cantidad de decesos. Los efectos de una pol\u00edtica en este sentido se pueden observar en la siguiente gr\u00e1fica.\n\n\n\n# Distanciamiento social - Definici\u00f3n\n\n\u201cConjunto de medidas no farmac\u00e9uticas de control de infecciones, con el objetivo de detener o desacelerar la propagaci\u00f3n de una\u00a0enfermedad contagiosa.\u201d \n\n

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    \n\n# Distanciamiento social - Definici\u00f3n\n\n***Lev\u00edtico, 13:46:*** \n\n\n

    \u201cTodo el tiempo que la llaga estuviere en \u00e9l, ser\u00e1 inmundo; estar\u00e1 impuro, y habitar\u00e1 solo; fuera del campamento ser\u00e1 su morada.\u201d

    \n    \n\n\n\n\nSe han establecido colonias de leprosos y\u00a0lazaretos\u00a0como medios para impedir la transmisi\u00f3n de la\u00a0lepra\u00a0y otras enfermedades infecciosas.\n\n***Lazareto - Ancona, Italia, 1700's***\n\n

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    \n\n***Agua de Dios, Colombia, 1870's***\n

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    \n\n# Modelos matem\u00e1ticos\n\nLos modelos matem\u00e1ticos son aproximaciones, bien sustentadas y justificadas, de la realidad y en este contexto intentan brindar elementos a la toma de decisiones de pol\u00edticas p\u00fablicas para intentan aplanar la curva de propagaci\u00f3n de la enfermedad.\n\n# Simulaci\u00f3n f\u00edsica de dispersi\u00f3n de got\u00edculas\n\nVeamos primer una serie de simulaciones computacionales de la dispersi\u00f3n de part\u00edculas producto de la tos humana empleando t\u00e9cnicas de la [CFD](https://en.wikipedia.org/wiki/Computational_fluid_dynamics \"CFD\") (Din\u00e1mica de Fluidos Computaconal) y empleando un software comercial\n\n\n```python\nfrom IPython.display import IFrame\n\nURL = 'https://www.ansys.com/about-ansys/covid-19-simulation-insights'\nIFrame(src = URL, width = 980, height = 500)\n```\n\n\n\n\n\n\n\n\n\n\n\n```python\nURL = 'https://www.washingtonpost.com/graphics/2020/world/corona-simulator-spanish/'\nIFrame(src = URL, width = 980, height = 600)\n```\n\n\n\n\n\n\n\n\n\n\n# Modelo S.I.R\n\nEs una representaci\u00f3n simplificada de la propagaci\u00f3n entre miembros de una poblaci\u00f3n de enfermedades contagiosas. \n\nEl modelo SIR es un modelo comportamental, es decir separa una poblaci\u00f3n en diferentes grupos de acuerdo a sus comportamientos:\n\n- **S**usceptible (que puede ser futuramente infectado, \"sano\")\n\n\n- **I**nfectado\n\n\n- **R**ecuperado (ha sido infectado y no volver\u00e1 a infectarse)\n\nPara definir las variables involucradas en este simple modelo veamos el siguiente ejemplo.\n\n## Ejemplo\n\nConsideremos una poblaci\u00f3n de $1000$ habitantes ($N=1000$), adem\u00e1s, consideremos que han pasado $7$ d\u00edas despu\u00e9s del brote de la epidemia. Para este instante, se sabe que $400$ de ellas est\u00e1n infectadas y $50$ se han recuperado, lo anterior puede ser denotado como:\n \n* $S(7) = 550$ (considerando que toda la poblaci\u00f3n es susceptible al inicio de la pandemia).\n* $I(7) = 400$.\n* $R(7) = 50$.\n\n\nPara esta enfermedad consideremos que la probabilidad de que una persona infectada contagie a una persona sana es del $20\\%$, y supongamos que el n\u00famero promedio de personas con las que una persona est\u00e1 en contacto por d\u00eda es de $5$. Entonces, por d\u00eda, un individuo infectado se encuentra con $5$ personas resultando infectadas el $20\\%$ de ellas, es decir, la persona enferma infectar\u00e1 a $1$ persona por d\u00eda. Esto es denotado con el par\u00e1metro $\\beta$, que es la cantidad esperada de personas que una persona infectada contagia por d\u00eda.\n\nSea $D$ la cantidad de d\u00edas que una persona puede transmitir la enfermedad, este dato es extremadamente importante. Si $D = 5$, una persona infectada camina durante cinco d\u00edas propagando la enfermedad e infecta a $1$ persona por d\u00eda ($\\beta = 1$). Entonces, esperamos que una persona infectada infecte a $5$ personas m\u00e1s. Este es el n\u00famero de reproducci\u00f3n b\u00e1sico, $R_0$, que representa el n\u00famero total de personas que infecta una persona infectada, es decir, $R_0=\\beta \\times D$.\n\nSi se piensa en $D$ como el n\u00famero de d\u00edas que una persona infectada tiene la enfermedad, entonces se puede pensar en otro par\u00e1metro $\\gamma$, como la tasa de recuperaci\u00f3n, o la proporci\u00f3n de infectados que se recuperan por d\u00eda. Por ejemplo, si se tienen actualmente $400$ personas infectadas y $D=5$, entonces, por d\u00eda $1/5$ de ellas se recuperar\u00e1n (aproximadamente $80$ personas), es decir, $\\gamma=1/5$. Siendo $\\gamma=1/D$, entonces $D=1/\\gamma$ y $R_0=\\beta \\times D$, se deduce que $R_0=\\beta / \\gamma$.\n\nResumiendo, las variables involucradas en el ejemplo anterior son:\n\n- $N$: Total de la poblaci\u00f3n\n\n\n- $S(t)$: N\u00famero de personas susceptibles de ser infectadas en un d\u00eda $t$\n\n\n- $I(t)$: N\u00famero de personas infectadas en un d\u00eda $t$\n\n\n- $R(t)$: N\u00famero de personas recuperadas en un d\u00eda $t$\n\n\n\n- $\\beta$: Cantidad esperada de personas que una persona infectada infecta por d\u00eda\n\n\n- $D$: N\u00famero de d\u00edas que una persona infectada tiene y puede transmitir la enfermedad\n\n\n- $\\gamma$: Proporci\u00f3n de infectados recuperados por d\u00eda ($\\gamma = 1/D$)\n\n\n\n- $R_0$: N\u00famero total de personas infectadas por una persona infectada ($R_0=\\beta / \\gamma$)\n\n\n\n\n# Derivaci\u00f3n de las ecuaciones\n\nDeseamos determinar el n\u00famero de *infectados*, *susceptibles* y *recuperados* para todos los d\u00edas, dados \u00fanicamente los par\u00e1metros $\\gamma$, $\\beta$ y $N$. Es dif\u00edcil obtener una f\u00f3rmula directa para $S(t)$, $I(t)$ y $R(t)$, sin embargo, es muy simple describir el cambio por d\u00eda de $S$, $I$ y $R$, es decir, como cambia el n\u00famero de *susceptibles* / *infectados* / *recuperados* dependiendo de los actuales valores.\n\nAhora estamos en el d\u00eda $t$ despu\u00e9s del brote de la enfermedad. A\u00fan as\u00ed, la cantidad esperada de personas que una persona infectada infecta por d\u00eda es $1$ ($\\beta = 1$) y el n\u00famero de d\u00edas que una persona infectada tiene y puede transmitir la enfermedad es $5$ ($\\gamma =1/5$ y $D=5$).\n\nEn el d\u00eda $t=7$, tenemos los datos dados anteriormente, \u00bfc\u00f3mo cambian $S(t)$, $I(t)$ y $R(t)$ al d\u00eda siguiente?\n\nSe tienen $400$ personas infectadas. Cada uno de ellos infecta a $\\beta = 1$ persona por d\u00eda. Sin embargo, solo $550/1000 = 55\\%$ de las personas todav\u00eda son susceptibles y pueden infectarse (eso es $S(t) / N$). Entonces, infectan a $400 \\times 1 \\times 55/1000 = 22$. Entonces, $22$ personas de los susceptibles se infectan, por lo que $S(t)$ cambia en menos $22$. Al conectar las variables, acabamos de derivar la primera f\u00f3rmula:\n\n$$\\text{cambio de } S(t) \\text{ al d\u00eda siguiente } = -\\beta \\cdot I(t) \\cdot S(t) / N$$\n\nLo anterior es la expresi\u00f3n de la deriva, $S'(t) = dS/dt$, y reescribiendo\n\n$$S'(t) = dS(t)/dt = -\\beta \\cdot I(t) \\cdot S(t) / N$$\n\nAhora, \u00bfc\u00f3mo cambia la cantidad de infectados? hay algunas personas nuevas infectadas. Exactamente la cantidad de personas que \"*salen*\" de $S(t)$ \"*llegan*\" a $I(t)$. Entonces, tenemos $22$ nuevos infectados y ya sabemos que la f\u00f3rmula ser\u00e1 similar a esta: $I'(t)= + \\beta \\cdot I(t) \\cdot S(t) / N$ (se gana la cantidad exacta que $S(t)$ pierde, por lo que simplementese cambia el signo). Solo falta una cosa: algunas personas se recuperan. Recuerde, tenemos $\\gamma$ para eso, es la proporci\u00f3n de infectados que se recuperan por d\u00eda, \u00a1eso es justo lo que necesitamos!\n\nTenemos $400$ infectados y $\\gamma = 1/5$, por lo que un quinto de los $400$ se recupera. Eso es $1/5 \\cdot 400 = 80$. Finalmente, obtenemos la f\u00f3rmula:\n\n$$I'(t) = dI(t)/dt = \\beta \\cdot I(t) \\cdot S(t) / N - \\gamma \\cdot I(t)$$\n\nEl primer t\u00e9rmino del lado derecho de la ecuaci\u00f3n es el reci\u00e9n infectado de los susceptibles. La segunda parte son las recuperaciones.\n\nFinalmente, llegamos a la \u00faltima f\u00f3rmula, el cambio en las recuperaciones. Los reci\u00e9n recuperados son exactamente los $80$ que acabamos de calcular; no hay personas saliendo del compartimento \"*recuperado*\". Se asume que una vez recuperados, permanecen inmunes:\n\n$$R'(t) = dR(t)/dt = \\gamma \\cdot I(t)$$\n\nYa contamos con las f\u00f3rmulas que se buscaban:\n\n$$\n\\begin{align}\n\\frac{dS}{dt} & = -\\beta \\cdot I \\cdot \\frac{S}{N} \\\\\n\\frac{dI}{dt} & = \\beta \\cdot I \\cdot \\frac{S}{N} - \\gamma \\cdot I \\\\\n\\frac{dR}{dt} & = \\gamma \\cdot I\n\\end{align}\n$$\n\nEste es un conjunto de Ecuaciones Diferenciales Ordinarias (ODEs).\n\nAhora podemos describir el cambio en el n\u00famero de personas *susceptibles*, *infectadas* y *recuperadas*. A partir de estas f\u00f3rmulas, afortunadamente, podemos calcular los n\u00fameros que realmente nos interesan: $S(t)$, $I(t)$ y $R(t)$, el n\u00famero de personas *susceptibles*, *infectadas* y *recuperadas* por cada d\u00eda $t$.\n\n# Codificando el modelo\n\nAhora vamos a codificar las ecuaciones. Se emplear\u00e1n los c\u00f3digos implementados en [Infectious Disease Modelling: Understanding the models that are used to model Coronavirus](https://towardsdatascience.com/infectious-disease-modelling-part-i-understanding-sir-28d60e29fdfc): By Henri Froese, in Towardsdatascience.com, Apr 06, 2020. El c\u00f3digo completo lo pueden descargar de [su sitio en GitHub](https://github.com/hf2000510/infectious_disease_modelling).\n\nInicialmente se definiran los par\u00e1metros $N$, $\\beta$, $D$, $\\gamma$, y las condiciones iniciales, en $t=0$, para $S(t)$, $I(t)$ y $R_0(t)$. \n\n\n```python\nimport numpy as np\nfrom scipy.integrate import odeint\nimport matplotlib.pyplot as plt\n```\n\nAhora se implementar\u00e1n las f\u00f3rmulas determinadas\n\n\n```python\ndef deriv(y, t, N, beta, gamma):\n S, I, R = y\n dSdt = -beta * S * I / N\n dIdt = beta * S * I / N - gamma * I\n dRdt = gamma * I\n return dSdt, dIdt, dRdt\n```\n\n\n```python\ndef plotsir(t, S, I, R):\n f, ax = plt.subplots(1,1,figsize=(10,4))\n ax.plot(t, S, 'b', alpha=0.7, linewidth=2, label='Susceptible')\n ax.plot(t, I, 'y', alpha=0.7, linewidth=2, label='Infected')\n ax.plot(t, R, 'g', alpha=0.7, linewidth=2, label='Recovered')\n \n ax.set_xlabel('Time (days)')\n \n ax.yaxis.set_tick_params(length=0)\n ax.xaxis.set_tick_params(length=0)\n ax.grid(b=True, which='major', c='w', lw=2, ls='-')\n legend = ax.legend()\n legend.get_frame().set_alpha(0.5)\n for spine in ('top', 'right', 'bottom', 'left'):\n ax.spines[spine].set_visible(False)\n plt.show();\n \n```\n\n\n```python\nN = 1000\nbeta = 1.0 # infected person infects 1 other person per day\nD = 5.0 # infections lasts four days\ngamma = 1.0 / D\n\nS0, I0, R0 = 999, 1, 0 # initial conditions: one infected, rest susceptible\n```\n\nAhora aqu\u00ed es donde ocurre la magia: obtenemos nuestros valores $S(t)$, $I(t)$ y $R(t)$ de la funci\u00f3n `odeint` de `Python` que toma las f\u00f3rmulas que definimos anteriormente, las condiciones iniciales y nuestras variables $N$, $\\beta$ y $\\gamma$, y calcula $S$, $I$ y $R$ para cada uno de los siguientes $50$ d\u00edas.\n\n\n```python\nt = np.linspace(0, 50, 50) # Grid of time points (in days)\ny0 = S0, I0, R0 # Initial conditions vector\n\n# Integrate the SIR equations over the time grid, t.\nret = odeint(deriv, y0, t, args=(N, beta, gamma))\nS, I, R = ret.T\n```\n\n\n```python\nplotsir(t, S, I, R)\n```\n\nComo puede ver, solo toma poco m\u00e1s de $10$ d\u00edas para que casi una poblaci\u00f3n completa de $1000$ personas se infecte. Por supuesto, la enfermedad modelada aqu\u00ed tiene un valor $R_0$ muy alto de $5.0$. Simplemente cambiando el n\u00famero de personas que una persona infectada infecta por d\u00eda $\\beta$ a $0.5$ resulta en un escenario completamente diferente:\n\n\n```python\nN = 1000\nbeta = 0.50 # infected person infects 1 other person per day\nD = 5.0 # infections lasts four days\ngamma = 1.0 / D\n\nS0, I0, R0 = 999, 1, 0 # initial conditions: one infected, rest susceptible\n```\n\n\n```python\nt = np.linspace(0, 50, 50) # Grid of time points (in days)\ny0 = S0, I0, R0 # Initial conditions vector\n\n# Integrate the SIR equations over the time grid, t.\nret = odeint(deriv, y0, t, args=(N, beta, gamma))\nS, I, R = ret.T\n```\n\n\n```python\nplotsir(t, S, I, R)\n```\n\nComo puede ver, estos sistemas de EDO son extremadamente sensibles a los par\u00e1metros iniciales. Esa es tambi\u00e9n la raz\u00f3n por la cual es tan dif\u00edcil modelar correctamente un brote emergente de una nueva enfermedad: simplemente no sabemos cu\u00e1les son los par\u00e1metros, e incluso los cambios leves producen resultados muy diferentes.\n\n# Conclusiones\n\nEl modelo S.I.R. modela simplificadamente la manera como se puede transmitir una enfermedad contagiosa en un grupo de personas, sin embargo, como en todo modelo, muchas son las inquietudes que surgen. Todo depender\u00e1 de los par\u00e1metros que se tienen en cuenta y la calidad de la informaci\u00f3n, datos, con que se cuente.\n\n\n```python\nURL = 'https://www.geogebra.org/classic/q2avpqqy'\nIFrame(src = URL, width = 980, height = 500)\n```\n\n\n\n\n\n\n\n\n\n\n\n```python\nURL = 'https://www.geogebra.org/classic/mxhj5ryw'\nIFrame(src = URL, width = 980, height = 500)\n```\n\n\n\n\n\n\n\n\n\n\n\n```python\nURL = 'https://www.geogebra.org/classic/yasmhuqf'\nIFrame(src = URL, width = 980, height = 500)\n```\n\n\n\n\n\n\n\n\n\n\n# *S.E.I.R*\n\n\nEn la literatura existen otros modelos m\u00e1s completos, y complejos, como el SEIR, que es una extensi\u00f3n del S.I.R. y que considera, adem\u00e1s de los par\u00e1metros vistos, otros como:\n\n- Un estado \"***Deceso***\" que contempla las personas que fallecen debido a la enfermedad.\n\n\n- Un estado \"***Expuesto***\" para individuos que han contra\u00eddo la enfermedad pero que a\u00fan no son infecciosos (este es conocido como el modelo ***S.E.I.R.***)\n\n\n- Valores de $R_0$ dependientes del tiempo que nos permitir\u00e1n modelar cuarentenas, bloqueos, ...\n\n\n- Tasas de mortalidad dependientes de los recursos y la edad que nos permitir\u00e1n modelar hospitales superpoblados, poblaciones con muchos j\u00f3venes, ...\n\n\nentre muchos otros posibles par\u00e1metros adicionales, que podr\u00edan complementar el modelo, pero lo har\u00edan mucho m\u00e1s complejo.\n\nAl derivar las ecuaciones, ya las consideramos intuitivamente como \"*instrucciones*\" que nos dicen lo que le sucede a la poblaci\u00f3n al d\u00eda siguiente (por ejemplo, cuando $10$ personas est\u00e1n infectadas y la recuperaci\u00f3n se produce a un ritmo de $\\gamma = 1/5$, entonces el n\u00famero de individuos recuperados al d\u00eda siguiente deber\u00eda aumentar en $1/5 \\times 10 = 2$). Ahora solidificamos esta comprensi\u00f3n de las ecuaciones como \"*direcciones*\" o \"*transiciones*\" de un compartimento $S$, $I$ o $R$ a otro; esto simplificar\u00e1 enormemente las cosas cuando introduzcamos m\u00e1s compartimentos m\u00e1s adelante y las ecuaciones se vuelvan confusas.\n\n## Definiciones\n\nLos *compartimientos* son cajas, que denotan \"*estados*\", como este:\n\n

    \n \n

    \n\nLas *transiciones* de un compartimento a otro se representan mediante flechas, con el siguiente etiquetado:\n\n

    \n \n

    \n\nLa *tasa* describe cu\u00e1nto tiempo dura la transici\u00f3n, la *poblaci\u00f3n* es el grupo de individuos a los que se aplica esta transici\u00f3n, y la *probabilidad* es la probabilidad de que la transici\u00f3n tenga lugar para un individuo.\n\n### Ejemplo\n\nSupongamos la transici\u00f3n de *Susceptibles* a *Infectados* en las ecuaciones $S.I.R.$, con $\\beta=2$, una poblaci\u00f3n total de $100$, $10$ infectados y $90$ susceptibles. La tasa es $1$, ya que las infecciones ocurren de inmediato; la poblaci\u00f3n a la que se aplica la transici\u00f3n es $2 \\times 10 = 20$ individuos, ya que los $10$ infectados infectan a $2$ personas; la probabilidad es del $90\\%$, ya que $90/100$ personas a\u00fan pueden estar infectadas. Corresponde a esta notaci\u00f3n intuitiva:\n\n

    \n \n

    \n\n\nDe forma m\u00e1s general, para todo el modelo (para $I \\rightarrow R$, la *tasa* es $\\gamma$ y la probabilidad es $1$ a medida que todos se recuperar\u00e1n)\n\n

    \n \n

    \n\nComo puede verse, las flechas que apuntan hacia un compartimento se agregan en la ecuaci\u00f3n; las flechas que apuntan lejos de un compartimento se restan. Es una forma gr\u00e1fica de presentar las ecuaciones determinadas arriba.\n\n$$\n\\begin{align}\n\\frac{dS}{dt} & = -\\beta \\cdot I \\cdot \\frac{S}{N} \\\\\n\\frac{dI}{dt} & = \\beta \\cdot I \\cdot \\frac{S}{N} - \\gamma \\cdot I \\\\\n\\frac{dR}{dt} & = \\gamma \\cdot I\n\\end{align}\n$$\n\nAunque ya tengamos un mejor entendimiento del modelo $S.I.R.$, e incluso se haya codificado en un lenguaje de programaci\u00f3n como `python`, los resultados obtenidos no representan muy bien la realidad, y por ahora no deja de ser un simple \"juguete\" interesante. \n\nVamos a tratar de \"mejorar\" este modelo b\u00e1sico incluyendo otros \"compartimientos\"\n\n# Introduciendo nuevos Compartimientos\n\n## Obteniendo el compartimiento de *Expuestos*\n\nMuchas enfermedades infecciosas tienen un per\u00edodo de incubaci\u00f3n antes de ser infecciosas durante el cual el hu\u00e9sped a\u00fan no puede transmitir la enfermedad. Llamaremos a tales individuos, y a todo el compartimento, *expuestos*.\n\nIntuitivamente, tendremos transiciones de la forma $S \\rightarrow E \\rightarrow I \\rightarrow R$: las personas susceptibles pueden contraer el virus y as\u00ed quedar expuestos, luego infectados y luego recuperados. La nueva transici\u00f3n $S \\rightarrow E$ tendr\u00e1 la misma flecha que la transici\u00f3n $S \\rightarrow I$ actual, ya que la probabilidad es la misma (todos los *susceptibles* pueden estar *expuestos*), la tasa es la misma (\"*exposici\u00f3n*\" ocurre inmediatamente) y la poblaci\u00f3n es igual (los individuos infecciosos pueden propagar la enfermedad y cada uno expone a $\\beta$ nuevos individuos por d\u00eda). Tampoco hay raz\u00f3n para que la transici\u00f3n de $I$ a $R$ cambie. La \u00fanica nueva transici\u00f3n es la de $E$ a $I$: la probabilidad es $1$ (todos los que est\u00e1n expuestos se infectan), la poblaci\u00f3n es $E$ (todos los expuestos se infectar\u00e1n) y la tasa obtiene una nueva variable, $\\delta$(delta). Llegamos a estas transiciones:\n\n

    \n \n

    \n\n\n\nDe estas transiciones, podemos derivar inmediatamente las nuevas ecuaciones:\n\n$$\n\\begin{align}\n\\frac{dS}{dt} & = -\\beta \\cdot I \\cdot \\frac{S}{N} \\\\\n\\frac{dE}{dt} & = \\beta \\cdot I \\cdot \\frac{S}{N} - \\delta \\cdot E\\\\\n\\frac{dI}{dt} & = \\delta \\cdot E - \\gamma \\cdot I \\\\\n\\frac{dR}{dt} & = \\gamma \\cdot I\n\\end{align}\n$$\n\n## Programando el compartimiento de *Expuestos*\n\nRetomando el c\u00f3digo realizado antes y cambiando algunas l\u00edneas para adicionarle los nuevos compartimientos. Como ejemplo se modelar\u00e1 una enfermedad altamente infecciosa, $R_0=5.0$ en una poblaci\u00f3n de $1$ mill\u00f3n, con un periodo de incubaci\u00f3n de $5$ d\u00edas y una recuperaci\u00f3n de $7$ d\u00edas.\n\n\n```python\ndef plotsir(t, S, E, I, R):\n f, ax = plt.subplots(1,1,figsize=(10,4))\n ax.plot(t, S, 'b', alpha=0.7, linewidth=2, label='Susceptible')\n ax.plot(t, E, 'y', alpha=0.7, linewidth=2, label='Expuesto')\n ax.plot(t, I, 'r', alpha=0.7, linewidth=2, label='Infectado')\n ax.plot(t, R, 'g', alpha=0.7, linewidth=2, label='Recuperado')\n \n ax.set_xlabel('Tiempo (dias)')\n ax.set_ylabel('Poblaci\u00f3n (habitantes)')\n \n ax.yaxis.set_tick_params(length=0)\n ax.xaxis.set_tick_params(length=0)\n ax.grid(b=True, which='major', c='w', lw=2, ls='-')\n legend = ax.legend()\n legend.get_frame().set_alpha(0.5)\n for spine in ('top', 'right', 'bottom', 'left'):\n ax.spines[spine].set_visible(False)\n plt.show();\n```\n\nFunci\u00f3n para determinar las derivadas de las funciones\n\n\n```python\ndef deriv(y, t, N, beta, gamma, delta):\n S, E, I, R = y\n dSdt = -beta * S * I / N\n dEdt = beta * S * I / N - delta * E\n dIdt = delta * E - gamma * I\n dRdt = gamma * I\n return dSdt, dEdt, dIdt, dRdt\n```\n\nDeterminaci\u00f3n de los valores iniciales\n\n\n```python\nN = 1000000\nD = 4.0 # infections lasts four days\ngamma = 1.0 / D\ndelta = 1.0 / 5.0 # incubation period of three days\nR_0 = 5.0\nbeta = R_0 * gamma # infected person infects 1 other person per day\n\nS0, E0, I0, R0 = N-1, 1, 0, 0 # initial conditions: one exposed\n```\n\nC\u00e1lculo de $S,E,I$ y $R$ para todo el tiempo de simulaci\u00f3n (100 d\u00edas)\n\n\n```python\nt = np.linspace(0, 100, 100) # Grid of time points (in days)\ny0 = S0, E0, I0, R0 # Initial conditions vector\n\n# Integrate the SIR equations over the time grid, t.\nret = odeint(deriv, y0, t, args=(N, beta, gamma, delta))\nS, E, I, R = ret.T\n```\n\n\n```python\nplotsir(t, S, E, I, R)\n```\n\nCada vez tiene mejor cara... pero a\u00fan falta lo m\u00e1s cruel de todo, representar los posibles decesos. \n\n## Obteniendo el compartimiento de \"*Decesos*\"\n\nRecordemos el modelo $S.E.I.R.$\n\n

    \n \n

    \n\n\u00bfCu\u00e1ndo puede morir una persona por la enfermedad? \u00a1S\u00f3lo mientras est\u00e9 infectado!, esto significa que hay qu\u00e9 agregar una transici\u00f3n $I \\rightarrow D$. \n\n\nPor supuesto, las personas no mueren de inmediato; Definimos una nueva variable $\\rho$(rho) para la tasa a la que las personas mueren (por ejemplo, cuando tarda $6$ d\u00edas en morir, $\\rho=1/6$). No hay raz\u00f3n para que la tasa de recuperaci\u00f3n, $\\gamma$, cambie. Entonces, nuestro nuevo modelo se ver\u00e1:\n\n

    \n \n

    \n\nLo \u00fanico que falta son las probabilidades de pasar de infectado a recuperado y de infectado a muerto. Esa ser\u00e1 una variable m\u00e1s (\u00a1la \u00faltima por ahora!), La tasa de mortalidad $\\alpha$. Por ejemplo, si $\\alpha= 5\\%$, $\\rho = 1$ y $\\gamma=1$ (por lo que las personas mueren o se recuperan en $1$ d\u00eda, eso es un ejemplo m\u00e1s f\u00e1cil) y $100$ personas est\u00e1n infectadas, entonces $5\\% \\times 100 = 5$ personas morir\u00e1n. Eso deja $95\\% \\times 100 = 95$ personas en recuperaci\u00f3n. En resumen, la probabilidad de $I \\rightarrow D$ es $\\alpha$ y, por lo tanto, la probabilidad de $I \\rightarrow R$ es $1-\\alpha$. Finalmente llegamos a este modelo:\n\n

    \n \n

    \n\nAdicionando esta nueva ecuaci\u00f3n al conjunto que ten\u00edamos de antes:\n\n$$\n\\begin{align}\n\\frac{dS}{dt} & = -\\beta \\cdot I \\cdot \\frac{S}{N} \\\\\n\\frac{dE}{dt} & = \\beta \\cdot I \\cdot \\frac{S}{N} - \\delta \\cdot E\\\\\n\\frac{dI}{dt} & = \\delta \\cdot E - (1-\\alpha) \\cdot \\gamma \\cdot I - \\alpha \\cdot \\rho \\cdot I\\\\\n\\frac{dR}{dt} & = (1-\\alpha) \\cdot \\gamma \\cdot I \\\\\n\\frac{dD}{dt} & = \\alpha \\cdot \\rho \\cdot I\n\\end{align}\n$$\n\n## Programando el compartimiento de *Decesos*\n\nModificando el c\u00f3digo que venimos trayendo para incluir los decesos. Solo es necesario realizar algunos cambios menores, como es incluir $\\alpha=20\\%$ y $\\rho=1/9$:\n\n\n```python\ndef deriv(y, t, N, beta, gamma, delta, alpha, rho):\n S, E, I, R, D = y\n dSdt = -beta * S * I / N\n dEdt = beta * S * I / N - delta * E\n dIdt = delta * E - (1 - alpha) * gamma * I - alpha * rho * I\n dRdt = (1 - alpha) * gamma * I\n dDdt = alpha * rho * I\n return dSdt, dEdt, dIdt, dRdt, dDdt\n```\n\n\n```python\nD = 4.0 # infections lasts four days\ngamma = 1.0 / D\ndelta = 1.0 / 5.0 # incubation period of five days\nR_0 = 5.0\nbeta = R_0 * gamma # R_0 = beta / gamma, so beta = R_0 * gamma\nalpha = 0.2 # 10% death rate\nrho = 1/9 # 9 days from infection until death\nS0, E0, I0, R0, D0 = N-1, 1, 0, 0, 0 # initial conditions: one exposed\n```\n\n\n```python\nt = np.linspace(0, 100, 100) # Grid of time points (in days)\ny0 = S0, E0, I0, R0, D0 # Initial conditions vector\n\n# Integrate the SIR equations over the time grid, t.\nret = odeint(deriv, y0, t, args=(N, beta, gamma, delta, alpha, rho))\nS, E, I, R, D = ret.T\n```\n\n\n```python\ndef plotseird(t, S, E, I, R, D):\n f, ax = plt.subplots(1,1,figsize=(10,4))\n ax.plot(t, S, 'b', alpha=0.7, linewidth=2, label='Susceptible')\n ax.plot(t, E, 'y', alpha=0.7, linewidth=2, label='Exposed')\n ax.plot(t, I, 'r', alpha=0.7, linewidth=2, label='Infected')\n ax.plot(t, R, 'g', alpha=0.7, linewidth=2, label='Recovered')\n ax.plot(t, D, 'k', alpha=0.7, linewidth=2, label='Dead')\n ax.plot(t, S+E+I+R+D, 'c--', alpha=0.7, linewidth=2, label='Total')\n\n ax.set_xlabel('Time (days)')\n\n ax.yaxis.set_tick_params(length=0)\n ax.xaxis.set_tick_params(length=0)\n ax.grid(b=True, which='major', c='w', lw=2, ls='-')\n legend = ax.legend()\n legend.get_frame().set_alpha(0.5)\n for spine in ('top', 'right', 'bottom', 'left'):\n ax.spines[spine].set_visible(False)\n plt.show();\n```\n\n\n```python\nplotseird(t, S, E, I, R, D)\n```\n\nSe adicion\u00f3 una variable \"*total*\" que suma $S, E, I, R$ y $D$ para cada paso de tiempo como un \"*control de sanidad*\": los compartimentos siempre tienen que sumar $N$; Esto puede dar una pista sobre si las ecuaciones son correctas.\n\nYa sabemos c\u00f3mo ir agregando nuevos compartimentos al modelo: piense qu\u00e9 transiciones deben agregarse y cambiarse; pensar en las probabilidades, poblaciones y tasas de estas nuevas transiciones; dibuja el diagrama; y finalmente escribe las ecuaciones. \u00a1La codificaci\u00f3n definitivamente no es la parte dif\u00edcil para estos modelos!\n\nPor ejemplo, es posible que desee agregar un compartimento \"*UCI*\" para las personas infectadas que necesitan ir a una *UCI* (lo haremos en el pr\u00f3ximo art\u00edculo). Piense desde qu\u00e9 compartimiento las personas pueden ir a la *UCI*, a d\u00f3nde pueden ir despu\u00e9s de la *UCI*, etc.\n\n# Variables dependientes del tiempo\n\nResumamos el listado de las variables empleadas hasta ahora:\n\n- $N$: poblaci\u00f3n total\n\n\n- $S(t)$: n\u00famero de personas susceptibles en el d\u00eda $t$\n\n\n- $E(t)$: n\u00famero de personas expuestas el d\u00eda $t$\n\n\n- $I(t)$: n\u00famero de personas infectadas el d\u00eda $t$\n\n\n- $R(t)$: n\u00famero de personas recuperadas el d\u00eda $t$\n\n\n- $D(t)$: n\u00famero de personas fallecidas el d\u00eda $t$\n\n\n- $\\beta$: cantidad esperada de personas que una persona infectada infecta por d\u00eda\n\n\n- $D$: n\u00famero de d\u00edas que una persona infectada tiene y puede transmitir la enfermedad\n\n\n- $\\gamma$: la proporci\u00f3n de infectados que se recuperan por d\u00eda ($\\gamma = 1/D$)\n\n\n- $R_0$: el n\u00famero total de personas que infecta una persona infectada ($R_0= \\beta / \\gamma$)\n\n\n- $\\delta$: duraci\u00f3n del per\u00edodo de incubaci\u00f3n\n\n\n- $\\alpha$: tasa de mortalidad\n\n\n- $\\rho$: tasa a la que muere la gente (= 1 / d\u00edas desde la infecci\u00f3n hasta la muerte)\n\nComo puede ver, solo los compartimentos cambian con el tiempo (no son constantes). \u00a1Por supuesto, esto es muy poco realista! Como ejemplo, \u00bfpor qu\u00e9 el valor $R_0$ deber\u00eda ser constante? Seguramente, los bloqueos a nivel nacional reducen la cantidad de personas que infecta una persona infectada, \u00a1de eso se trata! Naturalmente, para acercarnos a modelar desarrollos del mundo real, tenemos que hacer que nuestras variables cambien con el tiempo.\n\nEsto har\u00e1 parte de la continuaci\u00f3n de este trabajo.\n\n# Muchas gracias por su atenci\u00f3n!!!\n", "meta": {"hexsha": "a923526f8aaff836ad730129032d83ed7ab380af", "size": 177701, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Covid-19_SEIRD_UNAD.ipynb", "max_stars_repo_name": "UNADCdD/Covid-19-SEIRD", "max_stars_repo_head_hexsha": "397ead1141658bfd92ad9706420504a68a3186b7", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Covid-19_SEIRD_UNAD.ipynb", "max_issues_repo_name": "UNADCdD/Covid-19-SEIRD", "max_issues_repo_head_hexsha": "397ead1141658bfd92ad9706420504a68a3186b7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Covid-19_SEIRD_UNAD.ipynb", "max_forks_repo_name": "UNADCdD/Covid-19-SEIRD", "max_forks_repo_head_hexsha": "397ead1141658bfd92ad9706420504a68a3186b7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 121.629705681, "max_line_length": 33630, "alphanum_fraction": 0.8255496593, "converted": true, "num_tokens": 8668, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030761371502, "lm_q2_score": 0.7520125682019722, "lm_q1q2_score": 0.43714721918960486}} {"text": "```python\nfrom IPython.display import Image \nImage('../../../python_for_probability_statistics_and_machine_learning.jpg')\n```\n\n\n\n\n \n\n \n\n\n\n[Python for Probability, Statistics, and Machine Learning](https://www.springer.com/fr/book/9783319307152)\n\n\n```python\nfrom pprint import pprint\nimport textwrap\nimport sys, re\n```\n\n# Useful Inequalities\n\nIn practice, few quantities can be analytically calculated. Some knowledge\nof bounding inequalities helps find the ballpark for potential solutions. This\nsections discusses three key inequalities that are important for \nprobability, statistics, and machine learning.\n\n## Markov's Inequality\n\nLet $X$ be a non-negative random variable\nand suppose that $\\mathbb{E}(X) < \\infty$. Then,\nfor any $t>0$,\n\n$$\n\\mathbb{P}(X>t)\\leq \\frac{\\mathbb{E}(X)}{t}\n$$\n\n This is a foundational inequality that is\nused as a stepping stone to other inequalities. It is easy\nto prove. Because $X>0$, we have the following,\n\n$$\n\\begin{align*}\n\\mathbb{E}(X)&=\\int_0^\\infty x f_x(x)dx =\\underbrace{\\int_0^t x f_x(x)dx}_{\\text{omit this}}+\\int_t^\\infty x f_x(x)dx \\\\\\ \n &\\ge\\int_t^\\infty x f_x(x)dx \\ge t\\int_t^\\infty x f_x(x)dx = t \\mathbb{P}(X>t)\n\\end{align*}\n$$\n\n The step that establishes the inequality is the part where the\n$\\int_0^t x f_x(x)dx$ is omitted. For a particular $f_x(x)$ that my be\nconcentrated around the $[0,t]$ interval, this could be a lot to throw out.\nFor that reason, the Markov Inequality is considered a *loose* inequality,\nmeaning that there is a substantial gap between both sides of the inequality.\nFor example, as shown in [Figure](#fig:ProbabilityInequalities_001), the\n$\\chi^2$ distribution has a lot of its mass on the left, which would be omitted\nin the Markov Inequality. [Figure](#fig:ProbabilityInequalities_002) shows\nthe two curves established by the Markov Inequality. The gray shaded region is\nthe gap between the two terms and indicates that looseness of the bound\n(fatter shaded region) for this case.\n\n\n\n
    \n\n

    The $\\chi_1^2$ density has much of its weight on the left, which is excluded in the establishment of the Markov Inequality.

    \n\n\n\n\n\n\n\n
    \n\n

    The shaded area shows the region between the curves on either side of the Markov Inequality.

    \n\n\n\n\n\n## Chebyshev's Inequality\n\nChebyshev's Inequality drops out directly from the Markov Inequality. Let\n$\\mu=\\mathbb{E}(X)$ and $\\sigma^2=\\mathbb{V}(X)$. Then, we have\n\n$$\n\\mathbb{P}(\\vert X-\\mu\\vert \\ge t) \\le \\frac{\\sigma^2}{t^2}\n$$\n\n Note that if we normalize so that $Z=(X-\\mu)/\\sigma$, we\nhave $\\mathbb{P}(\\vert Z\\vert \\ge k) \\le 1/k^2$. In particular,\n$\\mathbb{P}(\\vert Z\\vert \\ge 2) \\le 1/4$. We can illustrate this\ninequality using Sympy statistics module,\n\n\n```python\nimport sympy\nimport sympy.stats as ss\nt=sympy.symbols('t',real=True)\nx=ss.ChiSquared('x',1)\n```\n\n To get the left side of the Chebyshev inequality, we\nhave to write this out as the following conditional probability,\n\n\n```python\nr = ss.P((x-1) > t,x>1)+ss.P(-(x-1) > t,x<1)\n```\n\n This is because of certain limitations in the statistics module at\nthis point in its development regarding the absolute value function. We could\ntake the above expression, which is a function of $t$ and attempt to compute\nthe integral, but that would take a very long time (the expression is very long\nand complicated, which is why we did not print it out above). This is because\nSympy is a pure-python module that does not utilize any C-level optimizations\nunder the hood. In this situation, it's better to use the built-in cumulative\ndensity function as in the following (after some rearrangement of the terms),\n\n\n```python\nw=(1-ss.cdf(x)(t+1))+ss.cdf(x)(1-t)\n```\n\n To plot this, we can evaluated at a variety of `t` values by using\nthe `.subs` substitution method, but it is more convenient to use the\n`lambdify` method to convert the expression to a function.\n\n\n```python\nfw=sympy.lambdify(t,w)\n```\n\n Then, we can evaluate this function using something like\n\n\n```python\nmap(fw,[0,1,2,3,4])\n```\n\n\n\n\n [1.0,\n 0.1572992070502851,\n 0.08326451666355028,\n 0.04550026389635853,\n 0.0253473186774682]\n\n\n\n to produce the following [Figure](#fig:ProbabilityInequalities_003). \n\n\n\n
    \n\n

    The shaded area shows the region between the curves on either side of the Chebyshev Inequality.

    \n\n\n\n\n\n**Programming Tip.**\n\nNote that we cannot use vectorized inputs for the `lambdify` function because\nit contains embedded functions that are only available in Sympy. Otherwise, we\ncould have used `lambdify(t,fw,numpy)` to specify the corresponding functions\nin Numpy to use for the expression.\n\n\n\n## Hoeffding's Inequality\n
    \n\nHoeffding's Inequality is similar, but less loose, than Markov's Inequality.\nLet $X_1,\\ldots,X_n$ be iid observations such that $\\mathbb{E}(X_i)=\\mu$ and\n$a\\le X_i \\le b$. Then, for any $\\epsilon>0$, we have\n\n$$\n\\mathbb{P}(\\vert \\overline{X}_n -\\mu\\vert \\ge \\epsilon) \\le 2 \\exp(-2 n\\epsilon^2/(b-a)^2)\n$$\n\n where $\\overline{X}_n = \\tfrac{1}{n}\\sum_i^n X_i$. Note that we\nfurther assume that the individual random variables are bounded.\n\n**Corollary.** If $X_1,\\ldots,X_n$ are independent with $\\mathbb{P}(a\\le X_i\\le b)=1$\nand all with $\\mathbb{E}(X_i)=\\mu$. Then, we have\n\n$$\n\\vert\\overline{X}_n-\\mu\\vert \\le \\sqrt{\\frac{c}{2 n}\\log \\frac{2}{\\delta}}\n$$\n\n where $c=(b-a)^2$. We will see this inequality again in the machine\nlearning chapter. [Figure](#fig:ProbabilityInequalities_004) shows the Markov\nand Hoeffding bounds for the case of ten identically and uniformly distributed\nrandom variables, $X_i \\sim \\mathcal{U}[0,1]$. The solid line shows\n$\\mathbb{P}(\\vert \\overline{X}_n - 1/2 \\vert > \\epsilon)$. Note that the\nHoeffding Inequality is tighter than the Markov Inequality and that both of\nthem merge when $\\epsilon$ gets big enough.\n\n\n\n
    \n\n

    This shows the Markov and Hoeffding bounds for the case of ten identically and uniformly distributed random variables.

    \n\n\n\n", "meta": {"hexsha": "b4532dc3cde8aad7384bbb2c8c2de30cdfcfaf46", "size": 126628, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "chapters/probability/notebooks/ProbabilityInequalities.ipynb", "max_stars_repo_name": "nsydn/Python-for-Probability-Statistics-and-Machine-Learning", "max_stars_repo_head_hexsha": "d3e0f8ea475525a694a975dbfd2bf80bc2967cc6", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 570, "max_stars_repo_stars_event_min_datetime": "2016-05-05T19:08:27.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T05:09:19.000Z", "max_issues_repo_path": "chapters/probability/notebooks/ProbabilityInequalities.ipynb", "max_issues_repo_name": "crlsmcl/https-github.com-unpingco-Python-for-Probability-Statistics-and-Machine-Learning", "max_issues_repo_head_hexsha": "6fd69459a28c0b76b37fad79b7e8e430d09a86a5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2016-05-12T22:18:58.000Z", "max_issues_repo_issues_event_max_datetime": "2019-11-06T14:37:06.000Z", "max_forks_repo_path": "chapters/probability/notebooks/ProbabilityInequalities.ipynb", "max_forks_repo_name": "crlsmcl/https-github.com-unpingco-Python-for-Probability-Statistics-and-Machine-Learning", "max_forks_repo_head_hexsha": "6fd69459a28c0b76b37fad79b7e8e430d09a86a5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 276, "max_forks_repo_forks_event_min_datetime": "2016-05-27T01:42:05.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-27T11:20:27.000Z", "avg_line_length": 338.577540107, "max_line_length": 114721, "alphanum_fraction": 0.9258931674, "converted": true, "num_tokens": 2098, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.7826624789529375, "lm_q1q2_score": 0.43698158945286686}} {"text": "# Hemoglobin Model Construction\n\nBased on Chapter 13 of Systems Biology: Simulation of Dynamic Network States\n\nTo construct a module of hemoglobin, first we import **MASSpy** and other essential packages. Constants used throughout the notebook are also defined.\n\n\n```python\nfrom os import path\n\nimport matplotlib.pyplot as plt\n\nfrom sympy import Equality, Symbol, solveset, sympify, pprint\n\nfrom cobra import DictList\n\nfrom mass import (\n MassConfiguration, MassMetabolite, MassModel,\n MassReaction, Simulation, UnitDefinition)\nfrom mass.example_data import create_example_model\nfrom mass.io import json, sbml \nfrom mass.util import strip_time, qcqa_model\n\nmass_config = MassConfiguration()\n\nmass_config.irreversible_Keq = float(\"inf\")\n```\n\n## Model Construction \n\nThe first step of creating a model of hemoglobin is to define the `MassModel`. \n\n\n```python\nhemoglobin = MassModel(\"Hemoglobin\")\n```\n\n Set parameter Username\n\n\n### Metabolites\n\nThe next step is to define all of the metabolites using the `MassMetabolite` object. Some considerations for this step include the following:\n\n1. It is important to use a clear and consistent format for identifiers and names when defining the `MassMetabolite` objects for various reasons, some of which include improvements to model clarity and utility, assurance of unique identifiers (required to add metabolites to the model), and consistency when collaborating and communicating with others. \n\n2. In order to ensure our model is physiologically accurate, it is important to provide the `formula` argument with a string representing the chemical formula for each metabolite, and the `charge` argument with an integer representing the metabolite's ionic charge (Note that neutrally charged metabolites are provided with 0). These attributes can always be set later if necessary using the `formula` and `charge` attribute set methods. To include the Hemoglobin macromolecule in the formula, brackets are used (e.g., [HB]).\n\n3. To indicate that the cytosol is the cellular compartment in which the reactions occur, the string \"c\" is provided to the `compartment` argument.\n\nThis model will be created using identifiers and names found in the [BiGG Database](http://bigg.ucsd.edu/).\n\nIn this model, there are 13 metabolites inside the cytosol compartment. Note that for metabolites without BiGG identifiers are given ones that are similar to BiGG style. \n\n\n```python\nhb_c = MassMetabolite(\n \"hb_c\", \n name=\"Hemoglobin\", \n formula=\"[HB]\",\n charge=0,\n compartment=\"c\",\n fixed=False)\n\nhb_1o2_c = MassMetabolite(\n \"hb_1o2_c\", \n name=\"Oxyhemoglobin (1)\", \n formula=\"[HB]-O2\",\n charge=0,\n compartment=\"c\",\n fixed=False)\n\nhb_2o2_c = MassMetabolite(\n \"hb_2o2_c\", \n name=\"Oxyhemoglobin (2)\", \n formula=\"[HB]-O4\",\n charge=0,\n compartment=\"c\",\n fixed=False)\n\nhb_3o2_c = MassMetabolite(\n \"hb_3o2_c\", \n name=\"Oxyhemoglobin (3)\", \n formula=\"[HB]-O6\",\n charge=0,\n compartment=\"c\",\n fixed=False)\n\nhb_4o2_c = MassMetabolite(\n \"hb_4o2_c\", \n name=\"Oxyhemoglobin (4)\", \n formula=\"[HB]-O8\",\n charge=0,\n compartment=\"c\",\n fixed=False)\n\ndhb_c = MassMetabolite(\n \"dhb_c\", \n name=\"Deoxyhemoglobin\", \n formula=\"[HB]-C3H3O10P2\",\n charge=-5,\n compartment=\"c\",\n fixed=False)\n\n_23dpg_c = MassMetabolite(\n \"_23dpg_c\", \n name=\"2,3-Disphospho-D-glycerate\", \n formula=\"C3H3O10P2\",\n charge=-5,\n compartment=\"c\",\n fixed=False)\n\n_13dpg_c = MassMetabolite(\n \"_13dpg_c\",\n name=\"3-Phospho-D-glyceroyl phosphate\",\n formula=\"C3H4O10P2\",\n charge=-4,\n compartment=\"c\",\n fixed=False)\n\n_3pg_c = MassMetabolite(\n \"_3pg_c\",\n name=\"3-Phospho-D-glycerate\",\n formula=\"C3H4O7P\",\n charge=-3,\n compartment=\"c\",\n fixed=False)\n\no2_c = MassMetabolite(\n \"o2_c\",\n name=\"Oxygen\",\n formula=\"O2\",\n charge=0,\n compartment=\"c\",\n fixed=False)\n\nh_c = MassMetabolite(\n \"h_c\",\n name=\"H+\",\n formula=\"H\",\n charge=1,\n compartment=\"c\",\n fixed=False)\n\npi_c = MassMetabolite(\n \"pi_c\",\n name=\"Phosphate\",\n formula=\"HPO4\",\n charge=-2,\n compartment=\"c\",\n fixed=False)\n\nh2o_c = MassMetabolite(\n \"h2o_c\",\n name=\"H2O\",\n formula=\"H2O\",\n charge=0,\n compartment=\"c\",\n fixed=False)\n```\n\n### Reactions\n\nOnce all of the `MassMetabolite` objects for each metabolite, the next step is to define all of the reactions that occur and their stoichiometry.\n\n1. As with the metabolites, it is also important to use a clear and consistent format for identifiers and names when defining when defining the `MassReaction` objects.\n\n2. To make this model useful for integration with other models, it is important to provide a string to the `subsystem` argument. By providing the subsystem, the reactions can be easily obtained even when integrated with a significantly larger model through the `subsystem` attribute.\n\n3. After the creation of each `MassReaction` object, the metabolites are added to the reaction using a dictionary where keys are the `MassMetabolite` objects and values are the stoichiometric coefficients (reactants have negative coefficients, products have positive ones). \n\nThis model will be created using identifiers and names found in the [BiGG Database](http://bigg.ucsd.edu/).\n\nIn this model, there are 7 reactions occuring inside the cytosol compartment.\n\n\n```python\nDPGase = MassReaction(\n \"DPGase\",\n name=\"Diphosphoglycerate phosphatase\",\n subsystem=hemoglobin.id, \n reversible=False)\nDPGase.add_metabolites({\n h2o_c: -1,\n _23dpg_c: -1,\n _3pg_c: 1,\n pi_c: 1})\n\nDPGM = MassReaction(\n \"DPGM\",\n name=\"Diphosphoglyceromutase\",\n subsystem=hemoglobin.id,\n reversible=True)\nDPGM.add_metabolites({\n _13dpg_c: -1,\n _23dpg_c: 1,\n h_c: 1})\n\nHBDPG = MassReaction(\n \"HBDPG\",\n name=\"Hemoglobin-23dpg binding\",\n subsystem=hemoglobin.id,\n reversible=True)\nHBDPG.add_metabolites({\n hb_c: -1,\n _23dpg_c: -1,\n dhb_c: 1})\n\nHBO1 = MassReaction(\n \"HBO1\",\n name=\"Oxygen Loading (1)\",\n subsystem=hemoglobin.id,\n reversible=True)\nHBO1.add_metabolites({\n hb_c: -1,\n o2_c: -1,\n hb_1o2_c: 1})\n\nHBO2 = MassReaction(\n \"HBO2\",\n name=\"Oxygen Loading (2)\",\n subsystem=hemoglobin.id,\n reversible=True)\nHBO2.add_metabolites({\n hb_1o2_c: -1,\n o2_c: -1,\n hb_2o2_c: 1})\n\nHBO3 = MassReaction(\n \"HBO3\",\n name=\"Oxygen Loading (3)\",\n subsystem=hemoglobin.id,\n reversible=True)\nHBO3.add_metabolites({\n hb_2o2_c: -1,\n o2_c: -1,\n hb_3o2_c: 1})\n\nHBO4 = MassReaction(\n \"HBO4\",\n name=\"Oxygen Loading (4)\",\n subsystem=hemoglobin.id,\n reversible=True)\nHBO4.add_metabolites({\n hb_3o2_c: -1,\n o2_c: -1,\n hb_4o2_c: 1})\n```\n\nAfter generating the reactions, all reactions are added to the model through the `MassModel.add_reactions` class method. Adding the `MassReaction` objects will also add their associated `MassMetabolite` objects if they have not already been added to the model. \n\n\n```python\nhemoglobin.add_reactions([\n DPGase, DPGM, HBDPG, HBO1, HBO2, HBO3, HBO4])\n\nfor reaction in hemoglobin.reactions:\n print(reaction)\n```\n\n DPGase: _23dpg_c + h2o_c --> _3pg_c + pi_c\n DPGM: _13dpg_c <=> _23dpg_c + h_c\n HBDPG: _23dpg_c + hb_c <=> dhb_c\n HBO1: hb_c + o2_c <=> hb_1o2_c\n HBO2: hb_1o2_c + o2_c <=> hb_2o2_c\n HBO3: hb_2o2_c + o2_c <=> hb_3o2_c\n HBO4: hb_3o2_c + o2_c <=> hb_4o2_c\n\n\n### Boundary reactions\n\nAfter generating the reactions, the next step is to add the boundary reactions and boundary conditions (the concentrations of the boundary 'metabolites' of the system). This can easily be done using the `MassModel.add_boundary` method. With the generation of the boundary reactions, the system becomes an open system, allowing for the flow of mass through the biochemical pathways of the model. Once added, the model will be able to return the boundary conditions as a dictionary through the `MassModel.boundary_conditions` attribute.\n\nAll boundary reactions are originally created with the metabolite as the reactant. However, there are times where it would be preferable to represent the metabolite as the product. For these situtations, the `MassReaction.reverse_stoichiometry` method can be used with its `inplace` argument to create a new `MassReaction` or simply reverse the stoichiometry for the current `MassReaction.` \n\nIn this model, there is 1 boundary reaction that must be defined.\n\n\n```python\nSK_o2_c = hemoglobin.add_boundary(\n metabolite=o2_c, boundary_type=\"sink\", subsystem=\"Pseudoreaction\",\n boundary_condition=0.0200788)\n\nprint(\"Boundary Reactions and Values\\n-----------------------------\")\nfor reaction in hemoglobin.boundary:\n boundary_met = reaction.boundary_metabolite\n bc_value = hemoglobin.boundary_conditions.get(boundary_met)\n print(\"{0}\\n{1}: {2}\\n\".format(\n reaction, boundary_met, bc_value))\n```\n\n Boundary Reactions and Values\n -----------------------------\n SK_o2_c: o2_c <=> \n o2_b: 0.0200788\n \n\n\n### Ordering of internal species and reactions\n\nSometimes, it is also desirable to reorder the metabolite and reaction objects inside the model to follow the physiology. To reorder the internal objects, one can use `cobra.DictList` containers and the `DictList.get_by_any` method with the list of object identifiers in the desirable order. To ensure all objects are still present and not forgotten in the model, a small QA check is also performed. \n\n\n```python\nnew_metabolite_order = [\n \"_23dpg_c\", \"hb_c\", \"hb_1o2_c\", \"hb_2o2_c\", \n \"hb_3o2_c\", \"hb_4o2_c\", \"dhb_c\", \"_13dpg_c\",\n \"_3pg_c\", \"o2_c\", \"pi_c\", \"h_c\", \"h2o_c\"]\n\nif len(hemoglobin.metabolites) == len(new_metabolite_order):\n hemoglobin.metabolites = DictList(\n hemoglobin.metabolites.get_by_any(new_metabolite_order))\n \nnew_reaction_order = [\n \"DPGM\", \"DPGase\", \"HBO1\", \"HBO2\", \n \"HBO3\", \"HBO4\", \"HBDPG\", \"SK_o2_c\"]\n\nif len(hemoglobin.reactions) == len(new_reaction_order):\n hemoglobin.reactions = DictList(\n hemoglobin.reactions.get_by_any(new_reaction_order))\n \nhemoglobin.update_S(array_type=\"DataFrame\", dtype=int)\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    DPGMDPGaseHBO1HBO2HBO3HBO4HBDPGSK_o2_c
    _23dpg_c1-10000-10
    hb_c00-1000-10
    hb_1o2_c001-10000
    hb_2o2_c0001-1000
    hb_3o2_c00001-100
    hb_4o2_c00000100
    dhb_c00000010
    _13dpg_c-10000000
    _3pg_c01000000
    o2_c00-1-1-1-10-1
    pi_c01000000
    h_c10000000
    h2o_c0-1000000
    \n
    \n\n\n\n### Computing the steady state concentrations. \n\nThe binding of the two ligands, oxygen and DPG23, to hemoglobin is a rapid process. Since hemoglobin is confined to the RBC, we can use equilibrium assumptions for the binding reactions. \n\n1. The binding of oxygen is at equilibrium for each form of oxygenated hemoglobin.\n2. The binding of DPG23 to hemoglobin is also at equilibrium \n3. The total mass of hemoglobin is a constant \n\nThese six equations have six unknowns (the six forms of Hb) and need to be solved simultaneously as a function of the oxygen and DPG23 concentrations. The equilibrium relationships can be combined with the $\\text{Hb}_{\\mathrm{tot}}$ mass balance, and this equation is solved for $\\text{Hb}_{\\mathrm{0}}$ for given oxygen and 23DPG concentrations. Then the steady state concentrations for all other forms of hemoglobin can be computed from the equilibrium relationships.\n\nTo do this, the **SymPy** package is utilized. The metabolites and equilibrium constants are defined as `sympy.Symbol` objects, and then the equilibrium expressions are converted into `sympy.Equality` objects for symbolic calculations.\n\n\n```python\nmetabolites = {metabolite.id: Symbol(metabolite.id) \n for metabolite in hemoglobin.metabolites}\n\nconcentration_equations = {}\n# Iterate through reactions assumed to be at equilibrium\nfor reaction in [HBO1, HBO2, HBO3, HBO4, HBDPG]:\n equilibrium_expression = Equality(\n Symbol(reaction.Keq_str),\n strip_time(reaction.get_mass_action_ratio()))\n # Find the hemoglobin form being made as a product (bound to most oxygen)\n hb_product = [\n Symbol(metabolite.id) for metabolite in reaction.products\n if metabolite.id not in [\"_23dpg_c\", \"hb_c\", \"o2_c\"]].pop()\n # Solve equation for the desired form hemoglobin \n equation = solveset(equilibrium_expression, hb_product)\n equation = next(iter(equation))\n # Update equilibrium expression dict with the equation\n # for the bound form of hemoglobin. These equations will\n # be dependent on hb_c, o2_c, and _23dpg_c.\n concentration_equations.update({\n hb_product: equation.subs(concentration_equations)})\n# Specify an equation for the total amount of hemoglobin\nHB_total_symbol = Symbol(\"HB-Total\")\nHB_total = Equality(\n HB_total_symbol,\n sympify(\"+\".join([\n metabolite.id for metabolite in hemoglobin.metabolites\n if \"hb\" in metabolite.id]), locals=metabolites))\nHB_total = HB_total.subs(concentration_equations)\npprint(HB_total)\n```\n\n 4 \n HB-Total = Keq_HBDPG\u22c5_23dpg_c\u22c5hb_c + Keq_HBO1\u22c5Keq_HBO2\u22c5Keq_HBO3\u22c5Keq_HBO4\u22c5hb_c\u22c5o_2_c + Keq_HBO1\u22c5Keq_HBO2\n \n 3 2 \n \u22c5Keq_HBO3\u22c5hb_c\u22c5o_2_c + Keq_HBO1\u22c5Keq_HBO2\u22c5hb_c\u22c5o_2_c + Keq_HBO1\u22c5hb_c\u22c5o_2_c + hb_c\n\n\nAt this point, the numerical values for the equilibrium constant and the total concetration of hemoglobin are specified. The total amount of hemoglobin is a constant, at circa 7.3 mM. These values are substituted into the current equations. \n\n\n```python\nnumerical_values = {HB_total_symbol: 7.3}\n\nDPGM.Keq = 2.3*1e6\nHBO1.Keq = 41.8352\nHBO2.Keq = 73.2115\nHBO3.Keq = 177.799 \nHBO4.Keq = 1289.92 \nHBDPG.Keq = 1/4\nSK_o2_c.Keq = 1\n\nnumerical_values.update({\n Symbol(reaction.Keq_str): reaction.Keq \n for reaction in hemoglobin.reactions})\n\nconcentration_equations.update({\n hb_form: equation.subs(numerical_values)\n for hb_form, equation in concentration_equations.items()})\nHB_total = HB_total.subs(numerical_values)\npprint(HB_total)\n```\n\n 4 3 \n 7.3 = 0.25\u22c5_23dpg_c\u22c5hb_c + 702446487.27335\u22c5hb_c\u22c5o_2_c + 544565.932207695\u22c5hb_c\u22c5o_2_c + 3062.8177448\u22c5hb_\n \n 2 \n c\u22c5o_2_c + 41.8352\u22c5hb_c\u22c5o_2_c + hb_c\n\n\nTo find the steady state, we have to specify the numerical values of the variables that characterize the network environment. The flux through the Rapoport-Luebering shunt is typically about 0.44 mM/hr\u00a0(Schrader 1993). The steady state concentration of 23DPG is typically about 3.1 mM\u00a0(Mehta 2005). The concentration of oxygen that we chose to solve for the steady state is 70 mmHg, that is mid way between 100 mmHg in the lung, and 40 mmHg in tissue. Using these numbers, the computed steady state concentrations are obtained, as: \n\n\n```python\n# Define known concentrations\nconcentrations = {\n metabolites[\"_23dpg_c\"]: 3.1, \n metabolites[\"o2_c\"]: 70*2.8684*1e-4}\n# Convert the solution into a numerical value\nhb_conc = next(iter(solveset(\n HB_total.subs(concentrations),\n Symbol(\"hb_c\"))))\nconcentrations.update({metabolites[\"hb_c\"]: hb_conc})\n# Solve for the rest of the hemoglobin concentrations\nfor hb_form, equation in concentration_equations.items():\n equation = equation.subs(concentrations)\n concentrations.update({hb_form: equation})\n```\n\nOnce the steady state concentrations have been determined, the hemoglobin module can be updated. The remaining concentrations are obtained from the glycolysis module. \n\n\n```python\nglycolysis = create_example_model(\"SB2_Glycolysis.json\")\n\nfor metabolite_symbol, value_symbol in concentrations.items():\n metabolite = hemoglobin.metabolites.get_by_id(str(metabolite_symbol))\n metabolite.ic = float(value_symbol)\n\nfor met in hemoglobin.metabolites:\n if met.ic is None:\n met.ic = glycolysis.metabolites.get_by_id(str(met)).ic\n\nfor metabolite, concentration in hemoglobin.initial_conditions.items():\n print(\"{0}: {1:.6f}\".format(metabolite, concentration))\n```\n\n _23dpg_c: 3.100000\n hb_c: 0.059625\n hb_1o2_c: 0.050085\n hb_2o2_c: 0.073625\n hb_3o2_c: 0.262842\n hb_4o2_c: 6.807613\n dhb_c: 0.046210\n _13dpg_c: 0.000243\n _3pg_c: 0.077300\n o2_c: 0.020079\n pi_c: 2.500000\n h_c: 0.000090\n h2o_c: 1.000000\n\n\nWith the steady state concentrations and steady state flux values, the PERCs can be calculated. For this module, the PERCs for the binding of hemoglobin to oxygen will be set manually to better reflect the physiology.\n\n__Note:__ Reactions at equilibrium have a steady state flux of 0. \n\n\n```python\nDPGM.v = 0.441\nDPGase.v = 0.441\nHBO1.v = 0\nHBO2.v = 0\nHBO3.v = 0\nHBO4.v = 0\nHBDPG.v = 0\nSK_o2_c.v = 0\n\nhemoglobin.calculate_PERCs(update_reactions=True)\n\nHBO1.kf = 506935\nHBO2.kf = 511077\nHBO3.kf = 509243\nHBO4.kf = 501595\nHBDPG.kf =519613\nSK_o2_c.kf = 509726\n```\n\n## QC/QA Model\n\nBefore simulating the model, it is important to ensure that the model is elementally balanced, and that the model can simulate. Therefore, the `qcqa_model` function from `mass.util.qcqa` is used to provide a report on the model quality and indicate whether simulation is possible and if not, what parameters and/or initial conditions are missing. \n\n\n```python\nqcqa_model(hemoglobin, parameters=True, concentrations=True, \n fluxes=True, superfluous=True, elemental=True)\n```\n\n \u2552\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2555\n \u2502 MODEL ID: Hemoglobin \u2502\n \u2502 SIMULATABLE: True \u2502\n \u2502 PARAMETERS NUMERICALY CONSISTENT: True \u2502\n \u255e\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2561\n \u2558\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255b\n\n\nFrom the results of the QC/QA test, it can be seen that the model can be simulated and is numerically consistent.\n\n## Steady State and Model Validation \n\nIn order to determine whether the module can be successfully integrated into a model, another model can be loaded, merged with the module, and simulated. To validate this module, it will be merged with a glycolysis model. \n\nTo find the steady state of the model and perform simulations, the model must first be loaded into a `Simulation`. In order to load a model into a `Simulation`, the model must be simulatable, meaning there are no missing numerical values that would prevent the integration of the ODEs that comprise the model. The `verbose` argument can be used while loading a model to produce a message indicating the successful loading of a model, or why a model could not load.\n\nOnce loaded into a `Simulation`, the `find_steady_state` method can be used with the `update_values` argument in order to update the initial conditions and fluxes of the model to a steady state (if necessary). The model can be simulated using the `simulate` method by passing the model to simulate, and a tuple containing the start time and the end time. The number of time points can also be included, but is optional.\n\nAfter a successful simulation, two `MassSolution` objects are returned. The first `MassSolution` contains the concentration results of the simulation, and the second contains the flux results of the simulation. \n\nTo visually validate the steady state of the model, concentration and flux solutions can be plotted using the `plot_time_profile` function from `mass.visualization`. Alternatively, the `MassSolution.view_time_profile` property can be used to quickly generate a time profile for the results.\n\n\n```python\nglyc_hb = glycolysis.merge(hemoglobin, inplace=False)\n\n# Setup simulation object, ensure model is at steady state\nsim = Simulation(glyc_hb, verbose=True)\nsim.find_steady_state(glyc_hb, strategy=\"simulate\", update_values=True)\n# Simulate from 0 to 1000 with 10001 points in the output\nconc_sol, flux_sol = sim.simulate(glyc_hb, time=(0, 1e3))\n# Quickly render and display time profiles\nconc_sol.view_time_profile()\n```\n\n### Storing information and references\n#### Compartment\nBecause the character \"c\" represents the cytosol compartment, it is recommended to define and set the compartment in the `MassModel.compartments` attribute.\n\n\n```python\nhemoglobin.compartments = {\"c\": \"Cytosol\"}\nprint(hemoglobin.compartments)\n```\n\n {'c': 'Cytosol'}\n\n\n#### Units\nAll of the units for the numerical values used in this model are \"Millimoles\" for amount and \"Liters\" for volume (giving a concentration unit of 'Millimolar'), and \"Hours\" for time. In order to ensure that future users understand the numerical values for model, it is important to define the `MassModel.units` attribute.\n\nThe `MassModel.units` is a `cobra.DictList` that contains only `UnitDefinition` objects from the `mass.core.unit` submodule. Each `UnitDefinition` is created from `Unit` objects representing the base units that comprise the `UnitDefinition`. These `Units` are stored in the `list_of_units` attribute. Pre-built units can be viewed using the `print_defined_unit_values` function from the `mass.core.unit` submodule. Alternatively, custom units can also be created using the `UnitDefinition.create_unit` method. For more information about units, please see the module docstring for `mass.core.unit` submodule.\n\n__Note:__ It is important to note that this attribute will NOT track units, but instead acts as a reference for the user and others so that they can perform necessary unit conversions.\n\n\n```python\n# Using pre-build units to define UnitDefinitions\nconcentration = UnitDefinition(\"mM\", name=\"Millimolar\",\n list_of_units=[\"millimole\", \"per_litre\"])\ntime = UnitDefinition(\"hr\", name=\"hour\", list_of_units=[\"hour\"])\n\n# Add units to model\nhemoglobin.add_units([concentration, time])\nprint(hemoglobin.units)\n```\n\n [, ]\n\n\n## Export\n\nAfter validation, the model is ready to be saved. The model can either be exported as a \".json\" file or as an \".sbml\" (\".xml\") file using their repsective submodules in `mass.io`.\n\nTo export the model, only the path to the directory and the model object itself need to be specified.\n\n### Export using SBML\n\n\n```python\nsbml.write_sbml_model(mass_model=hemoglobin, filename=\"SB2_\" + hemoglobin.id + \".xml\")\n```\n\n### Export using JSON\n\n\n```python\njson.save_json_model(mass_model=hemoglobin, filename=\"SB2_\" + hemoglobin.id + \".json\")\n```\n", "meta": {"hexsha": "04167c3028a4675a74f7c1ae52ad4ea5cb05c24b", "size": 80438, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/education/sb2/model_construction/sb2_hemoglobin.ipynb", "max_stars_repo_name": "z-haiman/MASSpy", "max_stars_repo_head_hexsha": "aeeed1e3f9d1058e9485247a86f85cb94eeecbc9", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/education/sb2/model_construction/sb2_hemoglobin.ipynb", "max_issues_repo_name": "z-haiman/MASSpy", "max_issues_repo_head_hexsha": "aeeed1e3f9d1058e9485247a86f85cb94eeecbc9", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/education/sb2/model_construction/sb2_hemoglobin.ipynb", "max_forks_repo_name": "z-haiman/MASSpy", "max_forks_repo_head_hexsha": "aeeed1e3f9d1058e9485247a86f85cb94eeecbc9", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 67.1996658312, "max_line_length": 37596, "alphanum_fraction": 0.7398990527, "converted": true, "num_tokens": 7322, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370308082623217, "lm_q2_score": 0.6859494614282922, "lm_q1q2_score": 0.43697093984076923}} {"text": "# SpectralDNS\n\n\n\n\n\n**Rayleigh-B\u00e9nard** convection computed with spectral accuracy using a [spectralDNS solver](https://github.com/spectralDNS/spectralDNS/blob/master/demo/RayleighBenard.py).\n\nThe [spectralDNS](https://github.com/spectralDNS) project revolves around\nimplementing high-performance flow solvers in [Python](https://www.python.org), which is a modern and very\nhigh-level programming language. The project is supported through several grants from [King\nAbdullahs University of Science and Technology](https://www.hpc.kaust.edu.sa),\ngranting access to some of the world's largest supercomputers. The work has\nbeen presented at several conferences and as invited talks\n\n * [11'th International Conference on Scientific Computing and Applications](http://tianyuan.xmu.edu.cn/activities/19-20/ICSCA2019/index.html)\n\n * [International Conference on Computational Science and Engineering](https://cseconf2017.simula.no)\n\n * [I3MS Seminar Series of the Institute for Modeling and Simulation, 6 Nov 2017 RWTH AAchen University](https://www.aices.rwth-aachen.de/en/media-and-seminars/events/mortensen-seminar)\n\n * [Predictive Complex Computational Fluid Dynamics, KAUST, May 2017](https://pccfd.kaust.edu.sa/speaker?si=4)\n\n * [MekIT'17 National Conference on computational Mechanics, Trondheim, May 2017](http://arxiv.org/abs/1708.03188)\n\n * [EuroScipy, Cambridge, August 2015](https://www.euroscipy.org/2015/schedule/presentation/6/)\n\nThe *spectralDNS* project on github contains several repositories, each representing a smaller part of the overall project. The most important are presented beolw.\n\n## spectralDNS\n\n\n\n**Strong scaling** of triply periodic Navier-Stokes solver on the Shaheen II supercomputer at KAUST.\n\nThe [spectralDNS](https://github.com/spectralDNS/spectralDNS) repository is home to several different pseudo-spectral Navier-Stokes and MagnetoHydroDynamics solvers. Most solvers are for triply periodic domains. The simplest possible Navier-Stokes solver is described by {% cite Mortensen2016 %}, who show that a highly efficient solver can be created using no more than 100 lines of code, using nothing more than standard\ntools like *Numpy* and *MPI for Python*. The DNS solver has been tested for a\ntransitional Taylor-Green vortex using a computational box of size $2048^3$. Accuracy is, well spectral, and in benchmark tests on the Shaheen II supercomputer at KAUST it has been found to scale well up to 64,000 cores.\nA state-of-the-art spectral channel flow solver that is making extensive use of *shenfun*, has been described by {% cite mortensen2017spectral %}. Turbulent flow at $Re_{\\tau}=2000$ is shown in the movie below.\n\n\n\nWith colleagues at the Extreme Computing Research Center (ECRC), King Abdullah University of Science and Technology (KAUST), we have been using [spectralDNS](https://github.com/spectralDNS) to investigate time integration of Fourier pseudospectral Direct Numerical Simulations {% cite ketcheson2020 %}. We investigate the use of higher\u2010order Runge\u2010Kutta pairs and automatic step size control based on local error estimation. We find that the fifth\u2010order accurate Runge\u2010Kutta pair of Bogacki and Shampine gives much greater accuracy at a significantly reduced computational cost.\n\n## Shenfun\n\nWith the [shenfun](https://github.com/spectralDNS/shenfun) Python module\n{% cite shenfun %} an\neffort is made towards automating the implementation of the spectral Galerkin\nmethod for simple tensor product domains, consisting of non-periodic and\nperiodic directions. The user interface to *shenfun* is intentionally made very\nsimilar to [FEniCS](https://fenicsproject.org). Partial Differential Equations\nare represented through weak variational forms and solved using efficient direct\nsolvers where available. MPI decomposition is achieved through the \n[mpi4py-fft](https://bitbucket.org/mpi4py/mpi4py-fft) module, and all developed solvers may,\nwith no additional effort, be run on supercomputers using thousands of\nprocessors.\n\nAn introduction to *shenfun* is given in {% cite shenfun %}, on [readthedocs](https://shenfun.readthedocs.io)\nand the recent paper {% cite mortensen_joss %}. Introduction to *mpi4py-fft*\nis given [here](https://mpi4py-fft.readthedocs.io/en/latest/) and\nin {% cite mpi4py-fft_joss jpdc_fft %}. Further documentation is found at \n\n[](https://shenfun.readthedocs.io/en/latest/)\n\nTry shenfun in the computational cell below. Modify to own liking and run interactively.\n\n\n```python\nfrom sympy import symbols, cos, sin, pi\nfrom shenfun import *\nimport matplotlib.pyplot as plt\n%matplotlib inline\n\nx, y = symbols(\"x,y\")\n\nC = Basis(20, 'Chebyshev')\nF = Basis(24, 'Fourier', dtype='d')\nT = TensorProductSpace(comm, (C, F))\nu = project(sin(2*x)*cos(pi*y), T)\nX = T.local_mesh(True)\nplt.contourf(X[0], X[1], u.backward())\nplt.colorbar()\nplt.show()\n\n```\n\n \n\n\n## mpi4py-fft \n\nmpi4py-fft is a Python package for computing Fast Fourier Transforms (FFTs).\nLarge arrays are distributed and communications are handled under the hood by\nMPI for Python (mpi4py). To distribute large arrays we are using a\n[new and completely generic algorithm](https://arxiv.org/abs/1804.09536)\nthat allows for any index set of a multidimensional array to be distributed. We\ncan distribute just one index (a slab decomposition), two index sets (pencil\ndecomposition) or even more for higher-dimensional arrays.\n\nmpi4py-fft comes with its own Python interface to the serial\n[FFTW](http://www.fftw.org) library. This interface can be used\nmuch like [pyfftw](https://hgomersall.github.io/pyFFTW), and even for\nreal-to-real transforms, like discrete cosine or sine transforms. Further documentation is found at\n\n[](https://mpi4py-fft.readthedocs.io/en/latest/)\n\nNumber of downloads from conda-forge:\n\n[](https://anaconda.org/conda-forge/mpi4py-fft)\n\n\n\n## References\n\n{% bibliography --cited %}\n", "meta": {"hexsha": "33b21b119636791a7490c84a9eab51be443f8590", "size": 91630, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "homepage/content/research/spectraldns.ipynb", "max_stars_repo_name": "mikaem/jekyll-homepage", "max_stars_repo_head_hexsha": "e74a173b23c8889253d5dd5240a41a75fff6e5e2", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "homepage/content/research/spectraldns.ipynb", "max_issues_repo_name": "mikaem/jekyll-homepage", "max_issues_repo_head_hexsha": "e74a173b23c8889253d5dd5240a41a75fff6e5e2", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "homepage/content/research/spectraldns.ipynb", "max_forks_repo_name": "mikaem/jekyll-homepage", "max_forks_repo_head_hexsha": "e74a173b23c8889253d5dd5240a41a75fff6e5e2", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 465.1269035533, "max_line_length": 69951, "alphanum_fraction": 0.7258430645, "converted": true, "num_tokens": 1472, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.6859494614282923, "lm_q1q2_score": 0.436970930387056}} {"text": "# 10 - Ensemble Methods - Continuation\n\n\nby [Alejandro Correa Bahnsen](albahnsen.com/)\n\nversion 0.2, May 2016\n\n## Part of the class [Machine Learning for Security Informatics](https://github.com/albahnsen/ML_SecurityInformatics)\n\n\nThis notebook is licensed under a [Creative Commons Attribution-ShareAlike 3.0 Unported License](http://creativecommons.org/licenses/by-sa/3.0/deed.en_US). Special thanks goes to [Kevin Markham](https://github.com/justmarkham)\n\nWhy are we learning about ensembling?\n\n- Very popular method for improving the predictive performance of machine learning models\n- Provides a foundation for understanding more sophisticated models\n\n# Part 1: Combination of classifiers - Majority Voting\n\n The most typical form of an ensemble is made by combining $T$ different base classifiers.\n Each base classifier $M(\\mathcal{S}_j)$ is trained by applying algorithm $M$ to a random subset \n $\\mathcal{S}_j$ of the training set $\\mathcal{S}$. \n For simplicity we define $M_j \\equiv M(\\mathcal{S}_j)$ for $j=1,\\dots,T$, and \n $\\mathcal{M}=\\{M_j\\}_{j=1}^{T}$ a set of base classifiers.\n Then, these models are combined using majority voting to create the ensemble $H$ as follows\n $$\n f_{mv}(\\mathcal{S},\\mathcal{M}) = max_{c \\in \\{0,1\\}} \\sum_{j=1}^T \n \\mathbf{1}_c(M_j(\\mathcal{S})).\n $$\n\n\n\n```python\n# read in and prepare the chrun data\n# Download the dataset\nimport pandas as pd\nimport numpy as np\n\ndata = pd.read_csv('../datasets/churn.csv')\n\n# Create X and y\n\n# Select only the numeric features\nX = data.iloc[:, [1,2,6,7,8,9,10]].astype(np.float)\n# Convert bools to floats\nX = X.join((data.iloc[:, [4,5]] == 'no').astype(np.float))\n\ny = (data.iloc[:, -1] == 'True.').astype(np.int)\n```\n\n\n```python\nX.head()\n```\n\n\n\n\n
    \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    Account LengthArea CodeVMail MessageDay MinsDay CallsDay ChargeEve MinsInt'l PlanVMail Plan
    0128.0415.025.0265.1110.045.07197.41.00.0
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    \n\n\n\n\n```python\ny.value_counts().to_frame('count').assign(percentage = lambda x: x/x.sum())\n```\n\n\n\n\n
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    \n\n\n\n\n```python\nfrom sklearn.cross_validation import train_test_split\nX_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.33, random_state=42)\n```\n\nCreate 100 decision trees\n\n\n```python\nn_estimators = 100\n# set a seed for reproducibility\nnp.random.seed(123)\n\nn_samples = X_train.shape[0]\n\n# create bootstrap samples (will be used to select rows from the DataFrame)\nsamples = [np.random.choice(a=n_samples, size=n_samples, replace=True) for _ in range(n_estimators)]\n```\n\n\n```python\nfrom sklearn.tree import DecisionTreeClassifier\n\nnp.random.seed(123) \nseeds = np.random.randint(1, 10000, size=n_estimators)\n\ntrees = {}\nfor i in range(n_estimators):\n trees[i] = DecisionTreeClassifier(max_features=\"sqrt\", max_depth=None, random_state=seeds[i])\n trees[i].fit(X_train.iloc[samples[i]], y_train.iloc[samples[i]])\n```\n\n\n```python\n# Predict \ny_pred_df = pd.DataFrame(index=X_test.index, columns=list(range(n_estimators)))\nfor i in range(n_estimators):\n y_pred_df.ix[:, i] = trees[i].predict(X_test)\n\ny_pred_df.head()\n```\n\n\n\n\n
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    \n\n\n\nPredict using majority voting\n\n\n```python\ny_pred_df.sum(axis=1)[:10]\n```\n\n\n\n\n 438 2\n 2674 5\n 1345 35\n 1957 17\n 2148 3\n 3106 4\n 1786 22\n 321 6\n 3082 10\n 2240 5\n dtype: int64\n\n\n\n\n```python\ny_pred = (y_pred_df.sum(axis=1) >= (n_estimators / 2)).astype(np.int)\n\nfrom sklearn import metrics\nmetrics.f1_score(y_pred, y_test)\n```\n\n\n\n\n 0.52459016393442637\n\n\n\n\n```python\nmetrics.accuracy_score(y_pred, y_test)\n```\n\n\n\n\n 0.89454545454545453\n\n\n\n### Using majority voting with sklearn\n\n\n```python\nfrom sklearn.ensemble import BaggingClassifier\nclf = BaggingClassifier(base_estimator=DecisionTreeClassifier(), n_estimators=100, bootstrap=True,\n random_state=42, n_jobs=-1, oob_score=True)\n```\n\n\n```python\nclf.fit(X_train, y_train)\ny_pred = clf.predict(X_test)\nmetrics.f1_score(y_pred, y_test), metrics.accuracy_score(y_pred, y_test)\n```\n\n\n\n\n (0.53600000000000003, 0.89454545454545453)\n\n\n\n# Part 2: Combination of classifiers - Weighted Voting\n\nThe majority voting approach gives the same weight to each classfier regardless of the performance of each one. Why not take into account the oob performance of each classifier\n\nFirst, in the traditional approach, a \nsimilar comparison of the votes of the base classifiers is made, but giving a weight $\\alpha_j$ \nto each classifier $M_j$ during the voting phase\n$$\n f_{wv}(\\mathcal{S},\\mathcal{M}, \\alpha)\n =\\max_{c \\in \\{0,1\\}} \\sum_{j=1}^T \\alpha_j \\mathbf{1}_c(M_j(\\mathcal{S})),\n$$\nwhere $\\alpha=\\{\\alpha_j\\}_{j=1}^T$.\nThe calculation of $\\alpha_j$ is related to the performance of each classifier $M_j$.\nIt is usually defined as the normalized misclassification error $\\epsilon$ of the base \nclassifier $M_j$ in the out of bag set $\\mathcal{S}_j^{oob}=\\mathcal{S}-\\mathcal{S}_j$\n\\begin{equation}\n \\alpha_j=\\frac{1-\\epsilon(M_j(\\mathcal{S}_j^{oob}))}{\\sum_{j_1=1}^T \n 1-\\epsilon(M_{j_1}(\\mathcal{S}_{j_1}^{oob}))}.\n\\end{equation}\n\nSelect each oob sample\n\n\n```python\nsamples_oob = []\n# show the \"out-of-bag\" observations for each sample\nfor sample in samples:\n samples_oob.append(sorted(set(range(n_samples)) - set(sample)))\n```\n\nEstimate the oob error of each classifier\n\n\n```python\nerrors = np.zeros(n_estimators)\n\nfor i in range(n_estimators):\n y_pred_ = trees[i].predict(X_train.iloc[samples_oob[i]])\n errors[i] = 1 - metrics.accuracy_score(y_train.iloc[samples_oob[i]], y_pred_)\n```\n\n\n```python\n%matplotlib inline\nimport matplotlib.pyplot as plt\nplt.style.use('fivethirtyeight')\n\nplt.scatter(range(n_estimators), errors)\nplt.xlim([0, n_estimators])\nplt.title('OOB error of each tree')\n```\n\nEstimate $\\alpha$\n\n\n```python\nalpha = (1 - errors) / (1 - errors).sum()\n```\n\n\n```python\nweighted_sum_1 = ((y_pred_df) * alpha).sum(axis=1)\n```\n\n\n```python\nweighted_sum_1.head(20)\n```\n\n\n\n\n 438 0.019993\n 2674 0.050009\n 1345 0.350236\n 1957 0.170230\n 2148 0.030047\n 3106 0.040100\n 1786 0.219819\n 321 0.059707\n 3082 0.100178\n 2240 0.050128\n 1910 0.180194\n 2124 0.190111\n 2351 0.049877\n 1736 0.950014\n 879 0.039378\n 785 0.219632\n 2684 0.010104\n 787 0.710568\n 170 0.220390\n 1720 0.020166\n dtype: float64\n\n\n\n\n```python\ny_pred = (weighted_sum_1 >= 0.5).astype(np.int)\n\nmetrics.f1_score(y_pred, y_test), metrics.accuracy_score(y_pred, y_test)\n```\n\n\n\n\n (0.52674897119341557, 0.8954545454545455)\n\n\n\n### Using Weighted voting with sklearn\n\n\n```python\nclf = BaggingClassifier(base_estimator=DecisionTreeClassifier(), n_estimators=100, bootstrap=True,\n random_state=42, n_jobs=-1, oob_score=True)\nclf.fit(X_train, y_train)\ny_pred = clf.predict(X_test)\nmetrics.f1_score(y_pred, y_test), metrics.accuracy_score(y_pred, y_test)\n```\n\n\n\n\n (0.53600000000000003, 0.89454545454545453)\n\n\n\n\n```python\nerrors = np.zeros(clf.n_estimators)\ny_pred_all_ = np.zeros((X_test.shape[0], clf.n_estimators))\n\nfor i in range(clf.n_estimators):\n oob_sample = ~clf.estimators_samples_[i]\n y_pred_ = clf.estimators_[i].predict(X_train.values[oob_sample])\n errors[i] = metrics.accuracy_score(y_pred_, y_train.values[oob_sample])\n y_pred_all_[:, i] = clf.estimators_[i].predict(X_test)\n \nalpha = (1 - errors) / (1 - errors).sum()\ny_pred = (np.sum(y_pred_all_ * alpha, axis=1) >= 0.5).astype(np.int)\n```\n\n\n```python\nmetrics.f1_score(y_pred, y_test), metrics.accuracy_score(y_pred, y_test)\n```\n\n\n\n\n (0.55335968379446643, 0.89727272727272722)\n\n\n\n# Part 3: Combination of classifiers - Stacking\n\nThe staking method consists in combining the different base classifiers by learning a \nsecond level algorithm on top of them. In this framework, once the base \nclassifiers are constructed using the training set $\\mathcal{S}$, a new set is constructed \nwhere the output of the base classifiers are now considered as the features while keeping the \nclass labels.\n\nEven though there is no restriction on which algorithm can be used as a second level learner, \nit is common to use a linear model, such as \n$$\n f_s(\\mathcal{S},\\mathcal{M},\\beta) =\n g \\left( \\sum_{j=1}^T \\beta_j M_j(\\mathcal{S}) \\right),\n$$\nwhere $\\beta=\\{\\beta_j\\}_{j=1}^T$, and $g(\\cdot)$ is the sign function \n$g(z)=sign(z)$ in the case of a linear regression or the sigmoid function, defined \nas $g(z)=1/(1+e^{-z})$, in the case of a logistic regression. \n\nLets first get a new training set consisting of the output of every classifier\n\n\n```python\nX_train_2 = pd.DataFrame(index=X_train.index, columns=list(range(n_estimators)))\n\nfor i in range(n_estimators):\n X_train_2[i] = trees[i].predict(X_train)\n```\n\n\n```python\nX_train_2.head()\n```\n\n\n\n\n
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    \n\n\n\n\n```python\nfrom sklearn.linear_model import LogisticRegressionCV\n```\n\n\n```python\nlr = LogisticRegressionCV()\nlr.fit(X_train_2, y_train)\n```\n\n\n\n\n LogisticRegressionCV(Cs=10, class_weight=None, cv=None, dual=False,\n fit_intercept=True, intercept_scaling=1.0, max_iter=100,\n multi_class='ovr', n_jobs=1, penalty='l2', random_state=None,\n refit=True, scoring=None, solver='lbfgs', tol=0.0001, verbose=0)\n\n\n\n\n```python\nlr.coef_\n```\n\n\n\n\n array([[ 0.10093102, 0.1042197 , 0.09431205, 0.09652843, 0.09709429,\n 0.09902616, 0.11100235, 0.09662288, 0.09340919, 0.09112994,\n 0.10012606, 0.09821902, 0.09383543, 0.09553507, 0.09147579,\n 0.09649564, 0.08965686, 0.09196857, 0.09684012, 0.09020758,\n 0.09839592, 0.09513808, 0.1044603 , 0.10028703, 0.09671603,\n 0.09725639, 0.10912207, 0.10590827, 0.10275491, 0.10275279,\n 0.10607316, 0.09803225, 0.10319411, 0.0926599 , 0.09702325,\n 0.09524124, 0.088848 , 0.09960894, 0.09053403, 0.09010282,\n 0.0990557 , 0.0987997 , 0.10538386, 0.09584352, 0.09633964,\n 0.09001206, 0.09181887, 0.08995095, 0.10130986, 0.10827168,\n 0.10064992, 0.09771002, 0.08922346, 0.10078438, 0.10173442,\n 0.1052274 , 0.09743252, 0.09597317, 0.08932798, 0.10033609,\n 0.10346122, 0.10145004, 0.09017084, 0.10348697, 0.09335995,\n 0.09795824, 0.10166729, 0.09306547, 0.09538575, 0.10997592,\n 0.09352845, 0.09860336, 0.1059772 , 0.09583408, 0.09823145,\n 0.09995048, 0.10224689, 0.10065135, 0.10208938, 0.11257989,\n 0.09956423, 0.11515946, 0.09798322, 0.10092449, 0.10150098,\n 0.10275192, 0.09180693, 0.0990442 , 0.10016612, 0.10145948,\n 0.09848122, 0.10322931, 0.09913907, 0.08925477, 0.09950337,\n 0.10277594, 0.09249331, 0.0954106 , 0.1053263 , 0.09849884]])\n\n\n\n\n```python\ny_pred = lr.predict(y_pred_df)\n```\n\n\n```python\nmetrics.f1_score(y_pred, y_test), metrics.accuracy_score(y_pred, y_test)\n```\n\n\n\n\n (0.53658536585365846, 0.89636363636363636)\n\n\n\n### Using sklearn\n\n\n```python\ny_pred_all_ = np.zeros((X_test.shape[0], clf.n_estimators))\nX_train_3 = np.zeros((X_train.shape[0], clf.n_estimators))\n\nfor i in range(clf.n_estimators):\n\n X_train_3[:, i] = clf.estimators_[i].predict(X_train)\n y_pred_all_[:, i] = clf.estimators_[i].predict(X_test)\n \nlr = LogisticRegressionCV()\nlr.fit(X_train_3, y_train)\n\ny_pred = lr.predict(y_pred_all_)\nmetrics.f1_score(y_pred, y_test), metrics.accuracy_score(y_pred, y_test)\n```\n\n\n\n\n (0.56250000000000011, 0.89818181818181819)\n\n\n\nvs using only one dt\n\n\n```python\ndt = DecisionTreeClassifier()\ndt.fit(X_train, y_train)\ny_pred = dt.predict(X_test)\nmetrics.f1_score(y_pred, y_test), metrics.accuracy_score(y_pred, y_test)\n```\n\n\n\n\n (0.44510385756676557, 0.82999999999999996)\n\n\n\n# Part 4: Boosting\n\nWhile boosting is not algorithmically constrained, most boosting algorithms consist of iteratively learning weak classifiers with respect to a distribution and adding them to a final strong classifier. When they are added, they are typically weighted in some way that is usually related to the weak learners' accuracy. After a weak learner is added, the data is reweighted: examples that are misclassified gain weight and examples that are classified correctly lose weight (some boosting algorithms actually decrease the weight of repeatedly misclassified examples, e.g., boost by majority and BrownBoost). Thus, future weak learners focus more on the examples that previous weak learners misclassified. (Wikipedia)\n\n\n```python\nfrom IPython.display import Image\nImage(url= \"http://vision.cs.chubu.ac.jp/wp/wp-content/uploads/2013/07/OurMethodv81.png\", width=900)\n```\n\n\n\n\n\n\n\n\n## Adaboost\n\nAdaBoost (adaptive boosting) is an ensemble learning algorithm that can be used for classification or regression. Although AdaBoost is more resistant to overfitting than many machine learning algorithms, it is often sensitive to noisy data and outliers.\n\nAdaBoost is called adaptive because it uses multiple iterations to generate a single composite strong learner. AdaBoost creates the strong learner (a classifier that is well-correlated to the true classifier) by iteratively adding weak learners (a classifier that is only slightly correlated to the true classifier). During each round of training, a new weak learner is added to the ensemble and a weighting vector is adjusted to focus on examples that were misclassified in previous rounds. The result is a classifier that has higher accuracy than the weak learners\u2019 classifiers.\n\nAlgorithm:\n\n* Initialize all weights ($w_i$) to 1 / n_samples\n* Train a classifier $h_t$ using weights\n* Estimate training error $e_t$\n* set $alpha_t = log\\left(\\frac{1-e_t}{e_t}\\right)$\n* Update weights \n$$w_i^{t+1} = w_i^{t}e^{\\left(\\alpha_t \\mathbf{I}\\left(y_i \\ne h_t(x_t)\\right)\\right)}$$\n* Repeat while $e_t<0.5$ and $t= 1).astype(np.int)\n```\n\n\n```python\nmetrics.f1_score(y_pred, y_test.values), metrics.accuracy_score(y_pred, y_test.values)\n```\n\n\n\n\n (0.51051051051051044, 0.85181818181818181)\n\n\n\n### Using sklearn\n\n\n```python\nfrom sklearn.ensemble import AdaBoostClassifier\n```\n\n\n```python\nclf = AdaBoostClassifier()\nclf\n```\n\n\n\n\n AdaBoostClassifier(algorithm='SAMME.R', base_estimator=None,\n learning_rate=1.0, n_estimators=50, random_state=None)\n\n\n\n\n```python\nclf.fit(X_train, y_train)\ny_pred = clf.predict(X_test)\nmetrics.f1_score(y_pred, y_test.values), metrics.accuracy_score(y_pred, y_test.values)\n```\n\n\n\n\n (0.29107981220657275, 0.86272727272727268)\n\n\n\n### Gradient Boosting\n\n\n```python\nfrom sklearn.ensemble import GradientBoostingClassifier\n\nclf = GradientBoostingClassifier()\nclf\n```\n\n\n\n\n GradientBoostingClassifier(init=None, learning_rate=0.1, loss='deviance',\n max_depth=3, max_features=None, max_leaf_nodes=None,\n min_samples_leaf=1, min_samples_split=2,\n min_weight_fraction_leaf=0.0, n_estimators=100,\n presort='auto', random_state=None, subsample=1.0, verbose=0,\n warm_start=False)\n\n\n\n\n```python\nclf.fit(X_train, y_train)\ny_pred = clf.predict(X_test)\nmetrics.f1_score(y_pred, y_test.values), metrics.accuracy_score(y_pred, y_test.values)\n```\n\n\n\n\n (0.52892561983471076, 0.89636363636363636)\n\n\n", "meta": {"hexsha": "8e1e8e6399fb83bb54b3e35d44d3c88a6d8afbd2", "size": 68281, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/10_EnsembleMethods_cont.ipynb", "max_stars_repo_name": "S7evenrg/ML_SecurityInformatics", "max_stars_repo_head_hexsha": "d5f156dc40dda5f7606a4b1a6150ee478bd75aa7", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 54, "max_stars_repo_stars_event_min_datetime": "2016-04-29T00:42:29.000Z", "max_stars_repo_stars_event_max_datetime": "2021-04-07T07:34:54.000Z", "max_issues_repo_path": "notebooks/10_EnsembleMethods_cont.ipynb", "max_issues_repo_name": "S7evenrg/ML_SecurityInformatics", "max_issues_repo_head_hexsha": "d5f156dc40dda5f7606a4b1a6150ee478bd75aa7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/10_EnsembleMethods_cont.ipynb", "max_forks_repo_name": "S7evenrg/ML_SecurityInformatics", "max_forks_repo_head_hexsha": "d5f156dc40dda5f7606a4b1a6150ee478bd75aa7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 34, "max_forks_repo_forks_event_min_datetime": "2016-05-03T11:57:16.000Z", "max_forks_repo_forks_event_max_datetime": "2021-05-17T12:49:53.000Z", "avg_line_length": 36.6511003757, "max_line_length": 18726, "alphanum_fraction": 0.5911307684, "converted": true, "num_tokens": 8603, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.7279754548076477, "lm_q1q2_score": 0.4369223938027036}} {"text": "```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport scipy \nfrom sklearn.model_selection import ParameterGrid\nfrom sklearn.manifold import Isomap\nimport time\nfrom tqdm import tqdm\n\nimport librosa\nfrom librosa import cqt\nfrom librosa.core import amplitude_to_db\nfrom librosa.display import specshow\n\nimport os\nimport glob\n```\n\n\n```python\ndata_dir = '/Users/sripathisridhar/Desktop/SOL'\n```\n\n\n```python\nfile_paths= sorted(glob.glob(os.path.join(data_dir, '**', '*.wav')))\n\nfile_names= []\nfor file_path in file_paths:\n file_names.append(os.path.basename(file_path))\n```\n\n\n```python\nhop_size= 512\nq= 24\n```\n\n\n```python\nimport h5py \n\nwith h5py.File(\"TinySOL.h5\", \"r\") as f:\n features_dict = {key:f[key][()] for key in f.keys()}\n```\n\n\n```python\ngrid = {\n 'Q': [24],\n 'k': [3],\n 'comp': ['log'],\n 'instr': ['Hp-ord'],\n 'dyn': ['all']\n}\n\nsettings = list(ParameterGrid(grid))\n\nfor setting in settings:\n \n if setting[\"instr\"] == 'all':\n setting['instr'] = ''\n \n if setting['dyn'] == 'all':\n setting['dyn'] = ''\n```\n\n\n```python\nbatch_str = []\nCQT_OCTAVES = 7\n\nfeatures_keys = list(features_dict.keys())\n\nfor setting in settings:\n \n q = setting['Q']\n # Batch process and store in a folder\n batch_str = [setting['instr'], setting['dyn']]\n\n batch_features = []\n for feature_key in features_keys:\n # Get features that match setting\n \n if all(x in feature_key for x in batch_str):\n batch_features.append(features_dict[feature_key])\n \n batch_features = np.stack(batch_features, axis=1)\n \n # Isomap parameters\n hop_size = 512\n compression = 'log'\n features = amplitude_to_db(batch_features)\n n_neighbors = setting['k']\n n_dimensions = 3\n n_octaves = 3 \n\n # Prune feature matrix\n bin_low = np.where((np.std(features, axis=1) / np.std(features)) > 0.1)[0][0] + q\n bin_high = bin_low + n_octaves*q \n X = features[bin_low:bin_high, :]\n\n # Z-score Standardization- improves contrast in correlation matrix\n mus = np.mean(X, axis=1)\n sigmas = np.std(X, axis=1)\n X_std = (X - mus[:, np.newaxis]) / (1e-6 + sigmas[:, np.newaxis]) # 1e-6 to avoid runtime division by zero\n\n # Pearson correlation matrix\n rho_std = np.dot(X_std, X_std.T) / X_std.shape[1]\n \n # Isomap embedding\n isomap = Isomap(n_components= n_dimensions, n_neighbors= n_neighbors)\n coords = isomap.fit_transform(rho_std)\n \n # Get note value\n freqs= librosa.cqt_frequencies(q*CQT_OCTAVES, fmin=librosa.note_to_hz('C1'), bins_per_octave=q) #librosa CQT default fmin is C1\n chroma_list= librosa.core.hz_to_note(freqs[bin_low:bin_high])\n \n notes = []\n reps = q//12\n for chroma in chroma_list:\n for i in range(reps):\n notes.append(chroma)\n```\n\n\n```python\ncurr_fig= plt.figure(figsize=(5.5, 2.75))\nax= curr_fig.add_subplot(121)\nax.axis('off')\n\nimport colorcet as cc\nsubsampled_color_ids = np.floor(np.linspace(0, 256, q, endpoint=False)).astype('int')\ncolor_list= [cc.cyclic_mygbm_30_95_c78[i] for i in subsampled_color_ids]\n\n# Plot embedding with color\nfor i in range(coords.shape[0]):\n plt.scatter(coords[i, 0], coords[i, 1], color= color_list[i%q], s=30.0)\n\nplt.plot(coords[:, 0], coords[:, 1], color='black', linewidth=0.2)\n\n# Plot Pearson correlation matrix\nrho_frequencies = freqs[bin_low:bin_high]\n\nfreq_ticklabels = ['A2', 'A3', 'A4']\nfreq_ticks = librosa.core.note_to_hz(freq_ticklabels)\n\ntick_bins = []\ntick_labels= []\nfor i,freq_tick in enumerate(freq_ticks):\n tick_bin = np.argmin(np.abs(rho_frequencies-freq_tick))\n tick_bins.append(tick_bin)\n tick_labels.append(freq_ticklabels[i])\n\nplt.figure(figsize=(2.5,2.5))\nplt.imshow(np.abs(rho_std), cmap='magma_r')\nplt.xticks(tick_bins)\nplt.gca().set_xticklabels(freq_ticklabels)\n# plt.xlabel('Log-frequency (octaves)')\nplt.yticks(tick_bins)\nplt.gca().set_yticklabels(freq_ticklabels)\n# plt.ylabel('Log-frequency (octaves)')\nplt.gca().invert_yaxis()\n\nplt.clim(0, 1)\n\n```\n\n### Circle projection\n\n\n```python\nimport circle_fit\nimport importlib\nimportlib.reload(circle_fit)\nfrom circle_fit import circle_fit\n\nA = np.transpose(coords[:,:-1])\nx, r, circle_residual = circle_fit(A, verbose=True)\n```\n\n\n```python\nimport matplotlib\nmatplotlib.rc('font', family='serif')\n\nfig, axes = plt.subplots()\nplt.scatter(A[0,:],A[1,:])\nplt.plot(x[0],x[1],'rx')\n\ncircle = plt.Circle(x, radius=r, fill=False, linestyle='-.')\n\naxes.set_aspect(1)\naxes.add_artist(circle)\n\n# axes.set_ylim([-5,6])\n# axes.set_xlim([-2,8])\n\nplt.title('Circle fit: TinySOL all instr', pad=10.0)\nplt.show()\n\nprint(np.sqrt(circle_residual)/72)\n```\n\n\n```python\nr\n```\n\n\n\n\n 123.68845666079883\n\n\n\n\n```python\ndef d_squared(a, b):\n # Takes two n-D tuples and returns euclidean distance between them\n \n # Cast to array for computation \n # Cast first to tuple in case a or b are Sympy Point objects\n p_a = np.array(tuple(a), dtype='float')\n p_b = np.array(tuple(b), dtype='float')\n \n return np.sum(np.square(p_a - p_b))\n```\n\n\n```python\nimport sympy\n\nfrom sympy.geometry import Circle, Point, Line\n\ncenter = Point(x, evaluate=False)\nc = Circle(center, r, evaluate=False)\n\nl = Line(Point(coords[0,:-1]), center, evaluate=False)\npoints = [tuple(p) for p in l.points]\n\nxy_prime = []\n\n# TODO: Optimize to a more pythonic manner\nfor x,y in coords[:,:2]:\n \n intersections = c.intersection(Line(Point(x,y), center, evaluate=False))\n \n if d_squared((x,y),intersections[0]) < d_squared((x,y), intersections[1]):\n xy_prime.append([float(p) for p in intersections[0]])\n else:\n xy_prime.append([float(p) for p in intersections[1]])\n \n```\n\n\n```python\nfig, axes = plt.subplots()\nplt.scatter(np.array(xy_prime)[:,0],np.array(xy_prime)[:,1], s=10, \n label='projected points')\nplt.scatter(A[0,:],A[1,:], s=0.5, label='isomap embedding points (2D)')\nplt.plot(center[0],center[1],'rx')\n\ncircle = plt.Circle([float(p) for p in center], radius=r, fill=False, \n linestyle='--', label='estimated circle fit')\n\naxes.set_aspect(1)\naxes.add_artist(circle)\n\nplt.title('Projected points on circle', pad=10.0)\nplt.legend(bbox_to_anchor=(1,1))\nplt.show()\n\n\n```\n\n### Line projection\n\n\n```python\nz = np.arange(len(coords[:,2]))\nz_fit = scipy.stats.linregress(z, coords[:,2])\nprint(z_fit.stderr)\n```\n\n 0.0018336634903286483\n\n\n\n```python\nplt.figure()\nplt.title('Line fit: TinySOL all instr')\nplt.scatter(np.arange(len(coords[:,2])), coords[:,2])\n\nplt.plot(z_fit.intercept + z_fit.slope*z, 'b')\n```\n\n\n```python\n# New line coordinates\nz_prime = [i * z_fit.slope + z_fit.intercept for i,_ in enumerate(coords[:,2])]\n```\n\n\n```python\ncoords_prime = np.append(np.array(xy_prime), np.expand_dims(np.array(z_prime), axis=1), axis=1)\ncoords_length = coords_prime.shape[0]\n```\n\n### Distance matrices \n\n\n```python\n# Projected helix self-distance matrix\n\nD_proj = np.zeros((coords_length, coords_length))\nfor i in range(coords_length):\n for j in range(i,coords_length):\n \n D_proj[i][j] = d_squared(coords_prime[i,:], coords_prime[j,:])\n```\n\n\n```python\n# Isomap embedding self-distance matrix\n\nD_isomap = np.zeros((coords_length, coords_length)) # Projected points same no. as isomap\nfor i in range(coords_length):\n for j in range(i, coords_length):\n \n D_isomap[i][j] = d_squared(coords[i,:], coords[j,:])\n```\n\n\n```python\n# Geodesic self-distance matrix\n\nD_geodesic = isomap.dist_matrix_\n\n# Convert to upper triangular sparse matrix\nfor i in range(coords_length):\n for j in range(i):\n D_geodesic[i,j] = 0\n```\n\n\n```python\n## Centering matrix\n\ndef centered(A, Q=24, J=3):\n # Returns centered distance matrix\n \n '''\n Inputs\n -----\n A - squared distance matrix\n Q - quality factor, 24 by default\n J - number of octaves, 3 by default\n \n Returns\n -----\n tau - MDS style diagonalized matrix of A\n '''\n \n coords_length = A.shape[0]\n H = np.zeros((coords_length, coords_length))\n\n const = 1/(Q*J)\n for i in range(coords_length):\n for j in range(coords_length):\n if j==i:\n H[i,j] = 1 - const\n else:\n H[i,j] = -const\n \n return -0.5 * np.matmul(np.matmul(H, A), H)\n```\n\n\n```python\ndef frobenius_distance(A, B):\n # Given two nxn matrices, return their 'Frobenius distance'\n \n return np.sqrt(np.sum(np.square(A - B)))\n```\n\n\n```python\nloss_isomap = frobenius_distance(centered(D_geodesic), centered(D_isomap))/coords_length\nloss_total = frobenius_distance(centered(D_geodesic), centered(D_proj))/coords_length\nloss_proj = frobenius_distance(centered(D_isomap), centered(D_proj))/coords_length\n```\n\n\n```python\nprint(f\"Isomap loss= {loss_isomap}\")\nprint(f\"Projection loss= {loss_proj}\")\nprint(f\"Total loss= {loss_total}\")\n```\n\n Isomap loss= 117.47799081825059\n Projection loss= 1.0433842670152964\n Total loss= 116.82039800129996\n\n\n\n```python\n(loss_total) - (loss_isomap + loss_proj) < 0\n```\n\n\n\n\n True\n\n\n\n\n```python\n## NOT STABLE- REWRITE\n\nhelicality_dict[setting['instr']] = [loss_isomap, loss_proj, loss_total]\n```\n\n\n```python\n\nhelicality_dict\n```\n\n\n\n\n {'TpC-ord': [14.780170171262531, 6.90431053444174, 13.687580004785238],\n 'Hp-ord': [117.47799081825059, 1.0433842670152964, 116.82039800129996],\n 'Fl-ord': [10.860110079953524, 7.292949241337281, 9.508129580612602]}\n\n\n\n\n```python\nimport json\n\nwith open(\"SOL_instr.json\", \"w\") as outfile:\n json.dump(helicality_dict, outfile)\n```\n\n\n```python\ninfile = open(\"SOL_instr.json\")\nhelicality_data = json.load(infile)\nprint(helicality_data)\n```\n\n {'TpC-ord': [14.780170171262531, 6.90431053444174, 13.687580004785238], 'Hp-ord': [117.47799081825059, 1.0433842670152964, 116.82039800129996], 'Fl-ord': [10.860110079953524, 7.292949241337281, 9.508129580612602]}\n\n", "meta": {"hexsha": "27680834484e1dee35418b03b3d3667200f1c4b3", "size": 83837, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "test-notebooks/helicalitySOL_instr.ipynb", "max_stars_repo_name": "sripathisridhar/sridhar2020ismir", "max_stars_repo_head_hexsha": "7e7b621fdf83a67784ab0b1fce37e483931094f8", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-01-14T10:00:28.000Z", "max_stars_repo_stars_event_max_datetime": "2021-01-14T10:00:28.000Z", "max_issues_repo_path": "test-notebooks/helicalitySOL_instr.ipynb", "max_issues_repo_name": "sripathisridhar/sridhar2020ismir", "max_issues_repo_head_hexsha": "7e7b621fdf83a67784ab0b1fce37e483931094f8", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-10-14T20:50:20.000Z", "max_issues_repo_issues_event_max_datetime": "2020-10-14T20:50:21.000Z", "max_forks_repo_path": "test-notebooks/helicalitySOL_instr.ipynb", "max_forks_repo_name": "sripathisridhar/sridhar2020ismir", "max_forks_repo_head_hexsha": "7e7b621fdf83a67784ab0b1fce37e483931094f8", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 115.3191196699, "max_line_length": 20120, "alphanum_fraction": 0.8683158987, "converted": true, "num_tokens": 2815, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754371026368, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.4369223831763621}} {"text": "\n\n\n\n
    \n\n\n
    \n\n### Latex Macros\n$\\newcommand{\\Re}[1]{{\\mathbb{R}^{{#1}}}}\n\\newcommand{\\Rez}{{\\mathbb{R}}}$\n\n\n```python\n%load_ext autoreload\n%autoreload 2\n#%tableofcontents\n```\n\n The autoreload extension is already loaded. To reload it, use:\n %reload_ext autoreload\n\n\n\n```python\nimport copy\n\nimport ipywidgets as widgets\nimport matplotlib.pyplot as plt\nimport networkx as nx\nimport numpy as np\nimport sympy\nimport torch\nfrom ipywidgets import fixed, interact, interact_manual, interactive\nfrom scipy.stats import ortho_group # compute unitary matrices\n\nimport spectral_function_library as spec_lib\n\n%matplotlib inline\n```\n\n# Convolution Neural Networks \nMaterial is taken from [this Blog](https://www.instapaper.com/read/1477946505)\n\nStarting from an RGB image: \n\n\n\nthe idea is pass this image through as series of steps in order to extract information. The filter is used for this task. \n\n\n\nafter image src.\n\n\n\n\n\n\n## Two important points about the convolutional layer: \n\n1. The filter is identical for each pixel. This reduces the number of parameters to calculate. \nThe constant filter helps satisfy the inductive bias of *translation invariance*. \n\n2. The convolution is local to the image pixel to which it is applied. Thus, the structure of the image is taken into account during the calculation. \n\nA typical CNN architecture: \n\n\n\njupyter nbextension enable --py widgetsnbextensionjupyter nbextension enable --py widgetsnbextensionjupyter nbextension enable --py widgetsnbextensionjupyter nbextension enable --py widgetsnbextension# Alternative view of CNN\n\n\n\n* An image can be considered to be a graph\n* The nodes $V$ are the centers of the pixels\n* If a filter has width 3, each nodes is connected to $8 * d$ adjacent nodes, where $d$ is the number of channels\n\n# Motivation\nConsider a set of nodes $x_i$, and associated attributes $y_i$. This can be graphed. Let us connect these nodes with edges $e_{ij} = (x_i, x_{i+1})$.\n\n\n```python\n@interact(N=(5, 40))\ndef plot1d(N):\n x = np.linspace(0, 10, N)\n plt.plot(x, 0 * x, \"-o\")\n plt.show()\n```\n\n\n interactive(children=(IntSlider(value=22, description='N', max=40, min=5), Output()), _dom_classes=('widget-in\u2026\n\n\nAdd an attribute to each of these nodes. I will add a random noise in $N(0,\\sigma)$ and $\\sigma=1.5$, which is fairly large. \n\nConsider the problem of computing *embeddings* of each node with the requirement that nearby nodes with similar attributes should have similar embeddings. \n\nWithout further constraints imposed on the problem (also called *inductive biases*, we will apply a local transformation to this function, and specifically an averaging operation. We will replace $y_i$ by the average of its neighbors :\n$$ y_i \\longrightarrow \\frac12 (y_{i-1} + y_{i+1})$$\nThe boundary points need special treatment. There are three main ehoices: \n1. Do not move the point\n2. Move the point in such a way as to satisfy some condition on the slope.\n3. Develop an algorithm that figures out the proper treatment\n\nWe will consider the first choice for simplicity. For future reference, we call the collection of points $V$, the collection of edges $E$. We denote the boundary nodes by $\\partial V$, and the boundary edges (edges attached to $\\partial V$ by $\\partial E$, which is a common notation in discrete and differential geometry. \n\n\n```python\n@interact(seed=(1, 100), eps=(0, 1.5), N=(5, 40))\ndef plot1d(seed, eps, N):\n np.random.seed(seed)\n x = np.linspace(0, 10, N)\n noise = eps * np.random.randn(N)\n y = np.sin((x / x[-1]) * 2 * np.pi * 2.5) + noise\n plt.plot(x, y, \"-o\")\n plt.show()\n```\n\n\n interactive(children=(IntSlider(value=50, description='seed', min=1), FloatSlider(value=0.75, description='eps\u2026\n\n\nMore generally, each point might have multiple attribute. Thus, the node $x_i$, would have $d$ attributes $y_0, \\cdots, y_{d-1}$. These attributes could be categorical or continuous, and the categorical attributes could be nominal (there is nor ordering, such as 'red', 'blue', 'orange') or ordinal (bad, poor, average, good, very good excellent). \n\n\n```python\ndSlider = widgets.IntSlider(min=1, max=5, value=3, description=\"Nb Attributes\")\nseedSlider = widgets.IntSlider(min=1, max=100, value=50, description=\"Seed\")\nepsSlider = widgets.FloatSlider(\n min=0.0, max=1.5, value=0.30, description=\"Noise $\\sigma$\"\n)\n\n\n@interact(seed=seedSlider, eps=epsSlider, N=(5, 40), d=dSlider, nb_blur_iter=(0, 5))\ndef plot1d(seed, eps, N, d, nb_blur_iter):\n np.random.seed(seed)\n eps = eps * np.array([1.0, 2.0, 0.5, 3.0, 4.0])\n x = np.linspace(0, 10, N)\n noise = np.random.randn(d, N)\n y = np.zeros([5, N])\n fcts = {}\n fcts[0] = np.sin((x / x[-1]) * 2 * np.pi * 2.5)\n fcts[1] = 1.5 * np.cos((x / x[-1]) * 2 * np.pi * 2.5) ** 2\n fcts[2] = x ** 2 / 10 * np.exp(3 - 0.5 * x)\n fcts[3] = np.cos((x / x[-1]) * 2 * np.pi * 4.5)\n fcts[4] = 1.5 * np.cos((x / x[-1]) * 2 * np.pi * 2.5)\n for i in range(0, 5):\n y[i] = fcts[i]\n\n for i in range(0, d):\n y[i] += eps[i] * noise[i]\n\n yy = copy.copy(y)\n\n for i in range(0, d):\n for n in range(0, nb_blur_iter):\n yy[i][0] = y[i][0]\n yy[i][N - 1] = y[i][N - 1]\n yy[i][1 : N - 2] = 0.5 * (y[i][0 : N - 3] + y[i][2 : N - 1])\n y = copy.copy(yy)\n\n for i in range(0, d):\n plt.plot(x, yy[i], \"-o\")\n plt.grid(True)\n plt.ylim(-2, 5)\n plt.show()\n```\n\n\n interactive(children=(IntSlider(value=50, description='Seed', min=1), FloatSlider(value=0.3, description='Nois\u2026\n\n\nSo far, I am describing vector-valued discrete functions of $x$, which is a 1-D representation of a graph $d$ attributes at each node $x_i$. More generally, nodes are points in *some* space, which can be 1-D, 2-D, higher-D, or more abstract, namely, a space of *points*. \n\nNow consider adding attributes $y_{Eij}$ to the edges. What kind of transformation functions should one consider? \n\nThis averaging function is an example of a local filter defined in physical space. This filter takes attributes at nodes and transforms them into a new set of number defined at these same nodes. More generally, in Graph Neural networks, we will consider operators that take attributes defined at nodes, edges, and the graph, and transform them into a new set of vectors defined on these same nodes, vectors and graphs. \n\nFilters can be defined either in physical space or in spectral space. We will illustrate the concept by considering the derivative operator on continuous and discrete grids.\n\n## First Derivative operator (also a filter) on 1D grid in physical space\nConsider points $x_i$, $i=0,\\cdots, N-1$ connected by edges $e_{i,i+1} = (x_i, x_{i+1})$. The central difference operator of the function $f_i = f(x_i)$ is defined by\n$$\nf'_i = \\frac{f_{i+1} - f_{i-1}}{x_{i+1} - x_{i-1}}\n$$ for $i=1,\\cdots,N-2$, with one-sided operators defined at the boundaries (which is one of many possibilities): \n\\begin{align}\nf'_0 &= \\frac{f_1 - f_0}{x_1-x_0} \\\\\nf'_{N-1} &= \\frac{f_{N-1} - f_{N-2}}{x_{N-1} - x_{N-2}}\n\\end{align}\nwhere $f'_i$ is the approximation of $f'(x)$ evaluated at $x=x_i$. Note that the derivative can be expressed as a vector \n$f' = (f'_0,\\cdots,f'_{N-1})$, and $f'_i$ is linear with respect to the values $f_j$. Therefore one can write the matrix \nexpression: \n$$ f' = D f $$\nwhere $D \\in \\Re{N\\times N}$ is an $N \\times N$ matrix. The matrix $D$ is a derivative filter. More specifically, it is a \n*global* filter since it updates the values at all nodes at once. To the contrary, a *local* filter is defined as the matrix that updates the derivative at a single point. Thus: \n$$\nf'_i = (\\begin{matrix}-\\alpha & 0 & \\alpha\\end{matrix})^T \n (\\begin{matrix} f_{i+1} & 0 & f_{i-1}) \\end{matrix}\n$$\nwhere a superscript $T$ denotes transpose, and $\\alpha = (x_{i+1} - x_{i-1})^{-1}$. Clearly, the local \nfilter is local to the point at which it applies. The new value only depends on the values of its immediate neighbors. \n\n***\n# Spectral Analysis of graphs\n## Continuous Fourier Transform (CFT)\nWhen working in the continuous domain $\\Rez$, a function $f(x)\\in\\Rez$ has a Fourier Transform $\\hat{f}(k)$ related by \n$$ \\hat{f}(k) = \\frac{1}{2\\pi} \\int_{-\\infty}^\\infty e^{\\iota k x} f(x) \\, dx $$\nConversely, one can apply a similar operation to recover $f(x)$ from its Fourier Transform: \n\n$$ f(x) = \\frac{1}{2\\pi} \\int_{-\\infty}^\\infty e^{-\\iota k x} \\hat{f}(k) \\, dk $$\n\nNotice the sign in the exponent: positive when transforming from physical to Fourier space, and negative when returning to physical space. The sign is a convention. Different authors might use the opposite sign. So always pay attention to the conventions in any paper you read. \n\n(you should all have learned about the Fourier transform previously). \n\nLet us compute the first derivative of $f(x)$: \n$$\\frac{d}{dx} f(x) = f'(x)$$\nThe conventional approach would be to calculate the derivative manually, or discretize the expression in physical space. However, the alternative is to compute the derivative by first transforming the expression to Fourier (also called spectral) space: \n\n\\begin{align}\n \\frac{d}{dx} f(x) &= \\frac{d}{dx} \\frac{1}{2\\pi} \\int_{-\\infty}^\\infty e^{-\\iota k x} \\hat{f}(k) d k \\\\ \n &= \\frac{1}{2\\pi} \\int_{-\\infty}^\\infty (-\\iota k) e^{-\\iota k x} \\hat{f}(k) dk \\\\\n &= \\cal{F}^{-1} [-\\iota k \\hat{f}(k)]\n\\end{align}\nwhere \n\\begin{align}\n\\cal{F}f(x) &= \\hat{f}(k) \\\\\n\\cal{F}^{-1} \\hat{f}(k) &= f(x) \\\\\n\\end{align}\n\nSo to given a function $f(x)$, one can compute the derivative with the following three steps: \n1. $f(x) \\longrightarrow \\hat{f}(k)$\n2. $\\hat{f}(k) \\longrightarrow (-\\iota k) \\hat{f}(k)$\n3. $(-\\iota k)\\hat{f}(k) \\longrightarrow \\cal{F}^{-1} \\left[(-\\iota k)\\hat{f}(k)\\right] = \\frac{d}{dx} f(x)$\n\nThus, the derivative operation is applied in Fourier space. A complex operation in physical space becomes a simple multiplication in Fourier space, *at the cost* of two Fourier Transforms. \n\n### Fourier Spectrum\n$\\hat{f}(k)$ is called the Fourier Spectrum and is generally a complex variable. \n$P(k) = |\\hat{f}(k)|^2$ is the power spectrum, and satisfies the property: \n$$\n\\int_{-\\infty}^\\infty P(k) dk = \\int_{-\\infty}^\\infty |\\hat{f}(k)|^2 dx = \\int_{-\\infty}^\\infty |f(x)|^2 dx\n$$\na rule that generalizes to and holds in $\\Re{n}$. \n\n### Filter\nThe coefficient $(-\\iota k)$ above is an example of a complex operator in Fourier space. This operator tranforms a function $\\hat{f}(k)$ into a \"filtered\" function $\\hat{g}(k)$: \n$$\n\\hat{g}(k) = (-\\iota k) \\hat{f}(k)\n$$\nand in this particular case, results in the Fourier transform of the $x$-derivative of $f(x)$. More generally, one can define an operator $\\hat{H}(k)$ acting on $\\hat{f}(k)$, which \"shapes\" the power spectrum, leading to filters with different characteristics: low-pass, band-pass, high-pass, custom. \n\nGiven a function $f(x)$, the resulting filtered function $f_H(x)$ can be defined similarly to the derivative: \n\n\\begin{align}\n f(x) & \\longrightarrow \\cal{F}(f(x)) = \\hat{f}(k) \\\\\n \\hat{f}(k) & \\longrightarrow \\hat{H}(k) \\hat{f}(k) \\\\\n \\hat{H}(k)\\hat{f}(k) & \\longrightarrow \\cal{F}^{-1} (\\hat{H}(k)\\hat{f}(k)) = f_H(x)\n\\end{align}\n\nWe will often omit the argument $x$ or $k$, letting the \"hat\" notation indicate whether or not we are in Fourier space. Thus, we can write\n$$\nf_H = \\cal{F}^{-1} [\\hat{H} \\; \\cal{F}(f) ]\n$$ or the equivalent form (requiring the definition of product of operators): \n\n\\begin{align}\nf_H &= (\\cal{F}^{-1} \\, \\hat{H} \\, \\cal{F}) \\; f \\\\\n &= H f\n\\end{align}\nwhich defines the filter $H(x)$ in physical space, acting on $f(x)$ to produce $f_H(x)$: \n$$\nf_H(x) = H(x) * f(x)\n$$\nwhere $*$ denotes the convolution operator: \n$$\nH(x) * f(x) = \\int_{-\\infty}^\\infty H(x-s) f(s) \\, ds\n$$\n\n## Formal proof of convolution theorem in continuous space\nWe start with the relation: \n$$ H = \\cal{F}^{-1} \\hat{H} \\cal{F} $$\nand express both sides of the equation in integral form: \n\\begin{align}\n\\int e^{-\\iota k x} \\left( \\hat{H}(k)\\hat{f}(k)\\right) \\, dk &=\n \\int e^{-\\iota k x}\\, dk \\left( \\int e^{\\iota k x''} H(x'')\\,dx'' \\int e^{\\iota k x'} f(x') \\, dx' \\right) \\\\\n &= \\int dk \\int e^{\\iota k (x'' + x' - x)} H(x'') f(x) \\, dx' \\, dx''\n\\end{align}\nNow make use of the following integral definition of the Dirac function: \n$$\n\\int e^{\\iota k x} \\, dk = 2\\pi \\delta(x)\n$$\nwhich leads to\n\\begin{align}\n\\int e^{-\\iota k x} \\left( \\hat{H}(k)\\hat{f}(k)\\right) \\, dk &=\n \\int dk \\int e^{\\iota k (x'' + x' - x)} H(x'') f(x') \\, dx' \\, dx'' \\\\\n&= 2\\pi \\int \\delta(x'' + x' - x) H(x'') f(x') \\, dx' \\, dx'' \\\\\n&= 2\\pi \\int H(x-x') f(x') \\, dx' \\\\\n&= C \\; H(x) * f(x) = L(x)\n\\end{align}\nwhere $C$ is a constant of proportionality. \nI was not careful with constants in front of the integrals when taking Fourier transforms and their \ninverses. \n\nWe thus find that \n$$\n\\cal{F}^{-1} \\left(\\hat{H}(k)\\hat{f}(k)\\right) = H * f\n$$\nCareful calculations show that the constant $C=1$. \n\nIntegrating $A(x)$ over $x$ leads to: \n$$\n\\int \\hat{H}(k) \\hat{f}(k) \\, dk = \\int H(x) f(x) \\, dx\n$$\noften referred to as [Plancherel's identity](https://en.wikipedia.org/wiki/Parseval%27s_identity).\n\nAll integrals are taken over the domain $[-\\infty, \\infty]$. \n\n---\n# Ideal Low-, Mid-, High-pass filters\n## Low-pass filter\n\n\\begin{align}\nH(k) &= 1, \\hspace{1in} k < k_0 \\\\\n&= 0, \\hspace{1in} k \\ge k_0 \n\\end{align}\n## Band-pass filter\n\n\\begin{align}\nH(k) &= 1, \\hspace{1in} k_0 < k < k_1, \\; k_0 < k_1 \\\\\n&= 0 \\hspace{1in} \\rm{otherwise}\n\\end{align}\n## High-pass filter\n\n\\begin{align}\nH(k) &= 1, \\hspace{1in} k > k_0 \\\\\n&= 0, \\hspace{1in} k \\le k_0 \n\\end{align}\n\n\n#### Notes: \n* np.fft uses the discrete Fourier Tranform since the grid is discrete (we skip over these details)\n* The $x$-domain is $[0,0.5]$. \n* $\\sin(2\\pi f_1 x)= 0$ at $x=0$ and $x=0.5$. The $x-derivative is $2\\pi f_1\\cos(f_1 2\\pi x)$, equal \nto $2\\pi f_1$ at $x=0$ and $2\\pi f_1 \\cos(\\pi f_1)$ at $x=0.5$, equal to 2\\pi f_1$ if $f_1$ is even. \nTherefore the function is periodic over the domain, since the $f_1$ slider ranges from -40 to 40 by increments of 10. \nOn the other hand, $\\cos(2\\pi f_3 x + 0.7)$ is not periodic over the $x$ domain (the phase is 0.7, which is not a multiple of $2\\pi$. The frequencies are obtained by \ndecomposing this function into a series of $\\sin$ and $\\cos$ at different frequencies with zero phase. \n\n\n```python\ngrid = widgets.GridspecLayout(3, 3)\n```\n\n\n```python\nfreq1Slider = widgets.IntSlider(min=0, max=60, value=30)\nfreq2Slider = widgets.IntSlider(min=30, max=120, value=70)\nfreq3Slider = widgets.IntSlider(min=90, max=200, value=110)\nampl1Slider = widgets.FloatSlider(min=-15, max=15, value=5)\nampl2Slider = widgets.FloatSlider(min=-15, max=15, value=10)\nampl3Slider = widgets.FloatSlider(min=-15, max=15, value=10)\nk0Slider = widgets.IntSlider(min=0, max=50, value=15)\nk1Slider = widgets.IntSlider(min=5, max=150, value=100, Description=\"k1\")\n```\n\n\n```python\n@interact_manual(\n freq1=freq1Slider, # (-20, 60, 10),\n freq2=freq2Slider, # (-90, 90, 10),\n freq3=freq3Slider, # (-300, 300, 15),\n ampl1=ampl1Slider, # 1,\n ampl2=ampl2Slider, # 0.5,\n ampl3=ampl3Slider, # 1,\n k0=k0Slider, # (0, 50, 5),\n k1=k1Slider, # (5, 150, 10),\n)\ndef plotSin2(freq1, freq2, freq3, ampl1, ampl2, ampl3, k0, k1):\n fig = plt.figure(figsize=(16, 7))\n x = np.linspace(0, 0.5, 500)\n k = np.linspace(0, 499, 500)\n\n # NOTE: These functions are NOT periodic over the domain.\n # Therefore, the spectrum is not exactly a collection of delta functions\n # I could be more precise, but that is not the point of this demonstration.\n\n s = (\n ampl1 * np.sin(freq1 * 2 * np.pi * x)\n + ampl2 * np.sin(freq2 * 2 * np.pi * x)\n + ampl3 * np.cos(freq3 * 2 * np.pi * x + 0.7)\n )\n nrows, ncols = 3, 2\n\n # ax1.clear() # to avoid flicker, does not work\n ax = fig.add_subplot(nrows, ncols, 1)\n # fig, axes = plt.subplots(nrows, ncols, figsize=(16, 5))\n ax.set_ylabel(\"Amplitude\")\n ax.set_xlabel(\"Time [s]\")\n ax.plot(x, s)\n\n fft = np.fft.fft(s)\n ifft = np.fft.ifft(s)\n # print(\"s: \", s[0:10])\n # print(\"ifft: \", ifft[0:11])\n # print(\"fft[0-10]: \", fft[0:11])\n # print(\"fft[:-10,:]: \", fft[-10:])\n power_spec = np.abs(fft) ** 2\n # power_spec[0] = 0 # REMOVE MEAN COMPONENT (simply equal to the mean of the function)\n ax2 = fig.add_subplot(nrows, ncols, 2)\n ax = ax2\n ax.plot(power_spec[0:250])\n ax.set_ylabel(\"Power Spectrum\")\n ax.set_xlabel(\"k\")\n\n heaviside = np.where((k > k0) & (k < k1), 1, 0)\n # Symmetrize this function with respect to $k=500/2$\n for i in range(1, 250): # 250 = 500/2\n heaviside[500 - i] = heaviside[i] # in Fourier space\n # print(heaviside)\n\n filtered_power_spectrum = power_spec * heaviside\n # print(list(zip(power_spec, heaviside, filtered_power_spectrum)))\n # print(\"power spec: \", power_spec[0:50])\n # print(\"filtered_spec: \", filtered_power_spectrum[0:50])\n filtered_function = np.fft.ifft(filtered_power_spectrum)\n\n ax = fig.add_subplot(nrows, ncols, 3)\n ax.plot(filtered_function)\n ax.set_ylabel(\"Filtered $f_H(x) = H(x) f(x)$\")\n ax.set_xlabel(\"x\")\n\n ax = fig.add_subplot(nrows, ncols, 4)\n ax.plot(filtered_power_spectrum[0:250])\n ax.set_xlabel(\"k\")\n ax.set_ylabel(\"Filtered Power Spectrum\")\n\n filter_phys = np.fft.ifft(heaviside)\n ax = fig.add_subplot(nrows, ncols, 5)\n ax.plot(filter_phys)\n ax.set_ylabel(\"Filter $H(x)$\")\n ax.set_xlabel(\"k\")\n\n ax = fig.add_subplot(nrows, ncols, 6)\n ax.plot(heaviside[0:250])\n ax.set_ylabel(\"Filter $\\hat{H}(k)$\")\n ax.set_xlabel(\"k\")\n\n plt.tight_layout()\n plt.show()\n sumf2 = np.sum(s ** 2)\n sump2 = np.sum(power_spec[0:250])\n sump3 = np.sum(power_spec)\n # print(sum2, sump2, sump2 / sumf2, sump3 / sumf2)\n # print(np.sum(power_spec[0:250]), np.sum(power_spec[0:500]), power_spec.shape)\n\n # The ratio sump2 / sumf2 = 250 (when there is no mean component)\n # The k=0 component has no complex conjugate. All other components have a complex conjugate.\n # These details are beyond the scope of this lecture.\n # = Number of points N / 2\n # sum f[i]^2 dx = sum f[i]^2 (0.5/N) = sum power_spectrum * normalizing constant\n # (one must be careful with this constant)\n\n\n# Alternative to @interact\n# interact(plotSin2, freq1=(-40,40,10), freq2=(-90,90,10), freq3=(-300,300,15), ampl1=1, ampl2=.5, ampl3=1)\n```\n\n\n interactive(children=(IntSlider(value=30, description='freq1', max=60), IntSlider(value=70, description='freq2\u2026\n\n\nThe strong oscilations in the Filter $H(x)$ are due to the discontinuity of the filter in Fourier space. \nA property of these 1-D filters is that localization in Fourier space (the filter is nonzero for very few $k$) leads \nto non-local filters $H(x)$ in physical space, and vice-versa. \n\nThe challenge is to construct filters local in both physical and Fourier space, which is the strength of wavelets (beyond the scope of these lectures). Note that the Fourier transform of a Gaussian is a Gaussian, and it is local in both spaces. (Demonstrate it for yourself as a homework exercise). \n\n\n### Discrete 1D domain\n* A set of nodes $x_i$, $i=0,1,\\cdots,N-1$, such that $x_i$ is connected to $x_{i+1}$. This graph is acyclic (there are no cycles. \n* If the first and last node are connected, we add the edge $(x_{N-1}, x_{0})$ and create a cyclic graph.\n* The adjacency matrix of the cyclic graph is as follows: \n$$\nA = \\left(\\begin{matrix}\n0 & 0 & 0 & \\cdots & 0 & 1 \\\\\n1 & 0 & 0 & \\cdots & 0 & 0 \\\\\n0 & 1 & 0 & \\cdots & 0 & 0 \\\\\n0 & 0 & 1 & \\cdots & 0 & 0 \\\\\n\\cdots\n\\end{matrix}\\right)\n$$\n* A signal $s$ on a graph is defined as the sequence of $N$ elements\n$$ x = (x_0, x_1, \\cdots, x_{N-1}) $$\nwhere each $x_i\\in\\Rez$.\n\n### 1-D Periodic Domain\n#### Fourier Filter\n### 1-D Non-periodic Domain\n## Fourier Transform, Discrete (DFT)\n### 1-D Periodic Domain \n### 1-D Non-periodic Domain\n## Graph Signal Processing, Discrete\n### 1-D cyclic graph\n### 2=D Discrete periodic \n### Adjoint $A$\n### Degree Matrix $D$\n### Laplacian $L$\n###\n\n\n```python\n# layout = ['circular','planar','random']\nseed_slider = widgets.IntSlider(min=100, max=120, step=2, value=110)\nN_slider = widgets.IntSlider(min=5, max=40, step=1, value=10)\n# matrix = ['Adjacency Matrix', 'Laplacian', 'D^-1 A', 'D^-1 L', 'D^-1/2 L D^-1/2']\n\n\n@interact(N=N_slider, seed=seed_slider)\ndef generate_graph_from_adjacency_matrix(N, seed):\n \"\"\"\n Arguments\n N: number of nodes\n \"\"\"\n np.random.seed(seed)\n ints = np.random.randint(0, 2, N * N).reshape(N, N)\n for i in range(N):\n ints[i,i] = 0\n\n # Symmetric array\n ints = ints + ints.transpose()\n ints = np.clip(ints, 0, 1) # the elements should be zero or 1\n\n # Different matrices\n A = ints\n D = np.sum(A, axis=0)\n D = np.diag(D)\n L = D - A\n invD = np.linalg.inv(D)\n invDA = A * invD\n invDL = invD * L\n invDLinvD = np.sqrt(invD) * L * np.sqrt(invD)\n\n matrix = [\"A\", \"D\", \"L\", \"invD\", \"invDA\", \"invDL\", \"invDinvD\"]\n matrices = [A, D, L, invD, invDA, invDL, invDLinvD]\n\n # Eigenvalues\n fig, axes = plt.subplots(3, 3, figsize=(10, 8))\n axes = axes.reshape(-1)\n fig.suptitle(\"Sorted Eigenvalues of various matrices\")\n for i, m in enumerate(matrices):\n ax = axes[i]\n eigs = np.linalg.eigvals(m)\n eigs = np.sort(eigs)[::-1]\n ax.set_title(matrix[i])\n ax.grid(True)\n ax.plot(eigs, \"-o\")\n for i in range(i + 1, axes.shape[-1]):\n axes[i].axis(\"off\")\n plt.tight_layout()\n plt.show()\n```\n\n\n interactive(children=(IntSlider(value=10, description='N', max=40, min=5), IntSlider(value=110, description='s\u2026\n\n\n### Notes\n* The eigenvalues (spectrum )of A and L are approximatley related (the plots look very similar) but not equal.\n* The spectra shape depend very little on the seed (A is filled with random numbers (0,1) and is symmetrized to make sure that the eigenvalues $\\lambda_i \\in \\Rez$.\n\n***\n## Same plot as above but allowing for different types of graph types. \n* Generate the graph, compute the adjacent matrix, and call the previous function\n\n\n\n\n```python\ndef generate_graph_from_adjacency_matrix_1(G, N, seed):\n \"\"\"\n Arguments\n N: number of nodes\n \"\"\"\n np.random.seed(seed)\n\n # Convert to np.ndArray\n A = nx.linalg.graphmatrix.adjacency_matrix(G).toarray()\n nx.linalg\n # print(\"Adj: \", A, \"\\n\", A.shape, \"\\n\", type(A))\n\n # Different matrices\n D = np.sum(A, axis=0)\n D = np.diag(D)\n L = D - A\n invD = np.linalg.inv(D)\n invDA = A * invD\n invDL = invD * L\n invDLinvD = np.sqrt(invD) * L * np.sqrt(invD)\n\n Ln = nx.normalized_laplacian_matrix(G)\n Ln = Ln.toarray() # from sparse array to ndarray\n\n matrix = [\"A\", \"D\", \"L\", \"invD\", \"invDA\", \"invDL\", \"invDinvD\", \"Ln\"]\n matrices = [A, D, L, invD, invDA, invDL, invDLinvD, Ln]\n\n # Eigenvalues\n fig, axes = plt.subplots(3, 3, figsize=(10, 8))\n axes = axes.reshape(-1)\n fig.suptitle(\"Eigenvalues of various matrices\")\n for i, m in enumerate(matrices):\n ax = axes[i]\n eigs = np.linalg.eigvals(m)\n eigs = np.sort(eigs)[::-1]\n ax.set_title(matrix[i])\n ax.grid(True)\n ax.plot(eigs, \"-o\")\n for i in range(i + 2, axes.shape[-1]):\n axes[i].axis(\"off\")\n plt.tight_layout()\n plt.show()\n```\n\n\n```python\nprob_slider = widgets.FloatSlider(min=0, max=1, step=0.1, value=0.5)\nnode_slider = widgets.IntSlider(min=3, max=30, step=1, value=10)\nnb_neigh_slider = widgets.IntSlider(min=1, max=10, step=1, value=4)\nnb_edges_per_node_slider = widgets.IntSlider(min=1, max=20, step=2, value=5)\nseed_slider = widgets.IntSlider(int=1, max=50, step=1, value=25)\ngraph_type = [\"connected_watts_strogatz\", \"powerlaw_cluster_graph\"]\n\n\n@interact(\n nb_nodes=node_slider,\n prob=prob_slider,\n nb_neigh=nb_neigh_slider,\n nb_edges_per_node=nb_edges_per_node_slider,\n seed=seed_slider,\n graph_type=graph_type,\n # directed=True,\n)\ndef drawGraph(nb_nodes, nb_neigh, prob, seed, nb_edges_per_node, graph_type):\n if graph_type == \"connected_watts_strogatz\":\n nb_edges_per_node_slider.style.handle_color = 'red'\n nb_neigh_slider.style.handle_color = 'black'\n nb_tries = 20\n edge_prob = prob\n G = nx.connected_watts_strogatz_graph(\n nb_nodes, nb_neigh, edge_prob, nb_tries, seed\n )\n elif graph_type == \"powerlaw_cluster_graph\":\n nb_neigh_slider.style.handle_color = 'red'\n nb_edges_per_node_slider.style.handle_color = 'black'\n add_tri_prob = prob\n if nb_edges_per_node >= nb_nodes:\n nb_edges_per_node = nb_nodes - 1\n G = nx.powerlaw_cluster_graph(nb_nodes, nb_edges_per_node, add_tri_prob, seed)\n\n generate_graph_from_adjacency_matrix_1(G, nb_nodes, seed)\n```\n\n\n interactive(children=(IntSlider(value=10, description='nb_nodes', max=30, min=3), IntSlider(value=4, descripti\u2026\n\n\n\n\n
    \nCreated with Jupyter, delivered by Fastly, rendered by Rackspace.\n
    \n\n# prob_slider = widgets.FloatSlider(\n min=0, max=1, step=0.1, value=0.5, description=\"Probability\"\n)\nnode_slider = widgets.IntSlider(min=3, max=20, step=1, value=7)\nnb_neigh_slider = widgets.IntSlider(min=1, max=10, step=1, value=4)\nnb_edges_per_node_slider = widgets.IntSlider(min=1, max=20, step=2, value=5)\nseed_slider = widgets.IntSlider(int=1, max=50, step=1, value=25)\ngraph_type = [\"connected_watts_strogatz\", \"powerlaw_cluster_graph\", \"circular_graph\"]\n\n# Also draw the eigenfunctions for the cyclic case where the nodes are arranged in a circular layout,\n# with labels in the nodes\n\n\n@interact_manual(\n nb_nodes=node_slider,\n prob=prob_slider,\n nb_neigh=nb_neigh_slider,\n nb_edges_per_node=nb_edges_per_node_slider,\n seed=seed_slider,\n graph_type=graph_type,\n)\ndef drawGraphEigenvalues(nb_nodes, nb_neigh, prob, seed, nb_edges_per_node, graph_type):\n if graph_type == \"connected_watts_strogatz\":\n nb_edges_per_node_slider.style.handle_color = \"red\"\n nb_neigh_slider.style.handle_color = \"black\"\n nb_tries = 20\n edge_prob = prob\n G = nx.connected_watts_strogatz_graph(\n nb_nodes, nb_neigh, edge_prob, nb_tries, seed\n )\n elif graph_type == \"powerlaw_cluster_graph\":\n nb_neigh_slider.style.handle_color = \"red\"\n nb_edges_per_node_slider.style.handle_color = \"black\"\n add_tri_prob = prob\n if nb_edges_per_node >= nb_nodes:\n nb_edges_per_node = nb_nodes - 1\n G = nx.powerlaw_cluster_graph(nb_nodes, nb_edges_per_node, add_tri_prob, seed)\n elif graph_type == \"circular_graph\":\n nb_neigh_slider.style.handle_color = \"red\"\n nb_edges_per_node_slider.style.handle_color = \"red\"\n nb_neigh_slider.style.handle_color = \"red\"\n prob_slider.style.handle_color = \"red\"\n seed_slider.style.handle_color = \"red\"\n\n G = nx.Graph()\n for n in range(nb_nodes):\n G.add_node(n)\n for n in range(nb_nodes):\n G.add_edge(n, n + 1)\n G.add_edge(nb_nodes - 1, 0)\n\n spec_lib.generate_eigenvectors_from_adjacency_matrix_1(G, nb_nodes, seed)\n\n\n```python\n# Test Eigenfunction, sorting, etc. by creating a matrix whose eigenvalues I know\nN_slider = widgets.IntSlider(min=3, max=10, step=1, value=5)\nseed_slider = widgets.IntSlider(min=100, max=200, step=1)\n\n\n@interact(N=N_slider, seed=seed_slider)\ndef test_eigen(N, seed):\n # generate eigenvalues\n np.random.seed(seed)\n # large variance for wider spread of spectrum\n eigens = (20.0 + 100.0 * np.random.randn(N)) / 20\n eigens = np.where(eigens < 0, -eigens, eigens)\n print(\"eigens= \", eigens)\n print(\"eigens[0]= \", eigens[0])\n print(\"eigens[1]= \\n\", eigens[1])\n # print(\"eigens= \\n\", eigens)\n eigens = np.diag(eigens)\n ee = np.linalg.eig(eigens)\n print(\"ee= \\n\", ee)\n print(\"ee[0]= \", ee[0], type(ee[0]))\n print(\"ee[1]= \\n\", ee[1])\n\n args = np.argsort(ee[0])\n print(\"args:\", args, type(args))\n ee0 = ee[0][args]\n ee1 = ee[1][:, args]\n print(\"sorted ee\")\n print(\"ee[0]= \", ee0)\n print(\"ee[1]= \\n\", ee1)\n recursivelyrecursively\n\n # create eigenvectors\n x = ortho_group.rvs(N)\n # Similarity transform (eigenvalues of A are invariant)\n A = x.T @ eigens @ x\n # A = x @ np.linalg.inv(x)\n # print(\"A= \\n\", A)\n # print(\"x.T= \\n\", x.T)\n # print(\"inv(x)= \\n\", np.linalg.inv(x))\n eigens = np.linalg.eig(A)\n args = np.argsort(eigens[0])\n print(\"===============================\")\n print(\"args: \\n\", args)\n eigs = eigens[0][args]\n print(\"unsorted eigs: \\n\", eigens[0])\n print(\"sorted eigs: \\n\", eigs)\n eigv = eigens[1][:, args]\n print(\"unsorted x:\\n \", x.T)\n print(\"unsorted eigv: \\n\", eigens[1])\n print(\"sorted x: \\n\", x.T[:, args])\n print(\"sorted eigv= \\n\", eigv)\n\n pass\n```\n\n\n interactive(children=(IntSlider(value=5, description='N', max=10, min=3), IntSlider(value=100, description='se\u2026\n\n\n# Exploration of eigenvalue and eigenfunctions for the 1-D cyclic and non-cyclic cases\nAs we have seen, a signal $s^1=(s_0, s_1, \\cdots, s_{N-1})\\in\\Re{N}$, is transformed into a signal $s^2\\in\\Re{N}$ by a filter $H$ according to \n$$ s^2 = H s^1$$ where $H$ is a matrix in $\\Re{N\\times N}$. Applying this filter recursively, one finds that \n\\begin{align}\ns^3 &= H s^2 \\\\\ns^4 &= H s^3 \\\\\ns^l &= H s^{l-1}\n\\end{align}\nIf this is done a large number of times, and if one assumes convergence of $s^l$ to a vector of finite norm, one finds in the limit: \n$$\ns^\\infty = H s^\\infty\n$$\nwhich states that $s^\\infty$ is an eigenvector of the filter $H$ with a unit eigenvalue $\\lambda=1$. \n\n## Cyclic case, directed graph\nThe adjoint matrix is\n$$\nA = \\left(\\begin{matrix}\n0 & 0 & 0 & \\cdots & 0 & 1 \\\\\n1 & 0 & 0 & \\cdots & 0 & 0 \\\\\n0 & 1 & 0 & \\cdots & 0 & 0 \\\\\n0 & 0 & 1 & \\cdots & 0 & 0 \\\\\n\\cdots\n\\end{matrix}\\right)\n$$\nRecall: $A_{i,j} = 1$ means an edge goes from node $j$ to node $i$. In this case, there is an edge from node $i+1$ to node $i$\nfor all nodes. There is also an edge from node $N-1$ to node $0$. This matrix is periodic. \n\nGiven a signal \n$$\ns = (s_0, s_1, \\cdots, s_{N-1})\n$$\nthe action of $A$ on $s$ simply shifts the value $s_i$ on node $i$ to node $i-1$: \n$$\ns^1 = A s = (s_{N-1}, s_0, s_1, \\cdots, s_{N-2})\n$$\n\nIn the next animation, we define a graph over a set of nodes, and a signal on this graph, and we apply the operator\n$A$ multiple times. \n\n\n```python\nj = -1\n\n\n@interact_manual(seed=(1, 100), eps=(0, 1.5), N=(5, 40))\ndef plot1d(seed, eps, N=15):\n global j\n np.random.seed(seed)\n # Define a NxN matrix\n A = np.zeros([N, N])\n for i in range(1, N):\n A[i, i - 1] = 1\n A[0, N - 1] = 1\n x = np.linspace(0, 10, N)\n\n # Signal s\n noise = eps * np.random.randn(N)\n s = np.sin((x / x[-1]) * 2 * np.pi * 2.5) + noise\n\n j += 1\n Aj = np.linalg.matrix_power(A, j)\n new_s = Aj @ s\n\n print(Aj)\n plt.plot(x, s, \"-o\", color=\"red\")\n plt.plot(x, new_s, \"-o\")\n plt.title(\"Press button to apply $A$\")\n plt.show()\n```\n\n\n interactive(children=(IntSlider(value=50, description='seed', min=1), FloatSlider(value=0.75, description='eps\u2026\n\n\nA is called the shift operator in 1-D signal processing. Application of $A$ to a time signal translates the signal by $\\Delta t$. The same is true with our graph. Of course, we are working with a special kind of graph. Let us now repeat this process with an undirected cyclic graph. Since node $i$ has a bidirectional connection to node $j$, each row of $A$ has two columns with a unit value. Thus, the adjacency matrix (now symmetric) becomes: \n$$\nA = \\left(\\begin{matrix}\n0 & 1 & 0 & \\cdots & 0 & 1 \\\\\n1 & 0 & 1 & \\cdots & 0 & 0 \\\\\n0 & 1 & 0 & \\cdots & 0 & 0 \\\\\n0 & 0 & 1 & \\cdots & 0 & 0 \\\\\n\\cdots \\\\\n0 & 0 & 0 & \\cdots & 0 & 1 \\\\\n1 & 0 & 0 & \\cdots & 1 & 0 \\\\\n\\end{matrix}\\right)\n$$\n\n\n\n```python\nj = -1\n\n\n@interact_manual(seed=(1, 100), eps=(0, 1.5), N=(5, 40))\ndef plot1d(seed, eps, N=15):\n global j\n np.random.seed(seed)\n # Define a NxN matrix\n A = np.zeros([N, N])\n for i in range(1, N):\n A[i, i - 1] = 1\n A[0, N - 1] = 1\n A = A + A.T\n\n x = np.linspace(0, 10, N)\n\n # Signal s\n noise = eps * np.random.randn(N)\n s = np.sin((x / x[-1]) * 2 * np.pi * 2.5) + noise\n\n j += 1\n Aj = np.linalg.matrix_power(A, j)\n new_s = Aj @ s\n\n print(Aj)\n plt.plot(x, s, \"-\", color=\"red\")\n plt.plot(x, new_s, \"-o\")\n plt.title(\"Press button to apply $A$\")\n plt.show()\n```\n\n\n interactive(children=(IntSlider(value=50, description='seed', min=1), FloatSlider(value=0.75, description='eps\u2026\n\n\nThe result: instability. The signal $A^n s$ goes to infinity as the number of iterations grows without bound (i.e., $n\\rightarrow\\infty$). Later, when working with neural networks, we want to avoid weights that converge towards infinity or zero. \n\nThis justifies the use of normalized adjacency matrices. The most common normalization is to premultiply $A$ by $D^{-1}$, where $D$ is the degree matrix. For our graph, all nodes have degree 2. Let us try again. We define a left normalization:\n$$\nA^* = D^{-1} A \n$$\nAnother popular normalization technique is the symmetric version of the preceding one: \n$$\nA^* = D^{-1/2} A D^{-1/2}\n$$\n\n\n```python\nj = -1\n\n\n@interact_manual(\n seed=(1, 100),\n eps=(0, 1.5),\n N=(5, 40),\n jincr=(1, 10),\n normalization=[\"left\", \"symmetric\"],\n)\ndef plot1d(seed, eps=0.1, N=15, normalization=\"left\", jincr=1):\n global j\n np.random.seed(seed)\n # Define a NxN matrix\n A = np.zeros([N, N])\n for i in range(1, N):\n A[i, i - 1] = 1\n A[0, N - 1] = 1\n A = A + A.T\n D = np.sum(A, axis=1) # works for all A\n Dinv = np.diag(1.0 / D)\n\n if normalization == \"left\":\n Dinv = np.diag(1.0 / D)\n A = Dinv @ A\n print(\"DinvSq @ A @ DinvSq= \", A)\n else:\n DinvSq = np.sqrt(Dinv)\n A = DinvSq @ A @ DinvSq\n\n x = np.linspace(0, 10, N)\n\n # Signal s\n noise = eps * np.random.randn(N)\n s = np.sin((x / x[-1]) * 2 * np.pi * 2.5) + noise\n print(\"mean(s) = \", np.mean(s))\n\n j += jincr\n Aj = np.linalg.matrix_power(A, j)\n new_s = Aj @ s\n print(\"mean(new_s) = \", np.mean(new_s))\n print(\"new_s= \", new_s)\n\n plt.plot(x, s, \"-\", color=\"red\")\n plt.plot(x, new_s, \"-o\")\n plt.title(\"Press button to apply $A$\")\n plt.show()\n```\n\n\n interactive(children=(IntSlider(value=50, description='seed', min=1), FloatSlider(value=0.1, description='eps'\u2026\n\n\nOne observes that after many repetitions of normalized (left or symmetric), $A$, the signal converges to a constant equal to the mean of the original signal: \n$$\n\\lim_{n\\rightarrow\\infty} s_{new} = \\text{mean}(s) = \\frac1N\\sum_0^{n-1} s_i\n$$\n\nFrom a theoretical point of view, if $s_{new}$ converges to a constant, it means that in the limit of $n\\rightarrow\\infty$, \n$$\n(A^*)^n s_{new} = (A^*)^{n-1} s_{new}\n$$\nwhich implies that \n$$ A^* s_{new} = s_{new} $$\nIn other words, $\\lambda=1$ is an eigenvalue of the normalized adjacency matrix (corresonding to a bidirectional cyclic graph), either \n$A^* = D^{-1} A$ or $A^* = D^{-1/2} A D^{-1/2}$. \n\nOne can easily show that if a single eigenvalue is greater than 1, $s_{new} \\rightarrow \\infty$. Since that does not happen, the maximum eigenvalue must be unity.\n\nWe check this out by computing the eigenvalues of the normalized matrix (which must be real since the matrix is symmetric). One also notices that since $A$ is symmetric, both normalizations produce the same results. \n\nExercise: Can you prove this? \n\n\n```python\n@interact_manual(N=(5, 40), normalization=[\"left\", \"symmetric\"])\ndef plot1d(N=15, normalization=\"left\"):\n # Define a NxN matrix\n A = np.zeros([N, N])\n # cyclic linear chain with two connections per node\n for i in range(1, N):\n A[i, i - 1] = 1\n A[0, N - 1] = 1\n A = A + A.T\n \n D = np.sum(A, axis=1) # works for all A\n Dinv = np.diag(1.0 / D)\n\n if normalization == \"left\":\n Dinv = np.diag(1.0 / D)\n A = Dinv @ A\n else:\n DinvSq = np.sqrt(Dinv)\n A = DinvSq @ A @ DinvSq\n\n print(\"A^*= \", A)\n evalue, evector = np.linalg.eig(A)\n print(\"\\nSorted eigenvalues: \", np.sort(evalue))\n print(f\"NOTE: the maximum eigenvalue = 1\")\n```\n\n\n interactive(children=(IntSlider(value=15, description='N', max=40, min=5), Dropdown(description='normalization\u2026\n\n\n---\n## Cyclic case, non-directed graph\nWe now repeat the last few experiments with a linear graph (i.e., a chain), but non-periodic: the boundary points are not considered as a single point. \n\n### Directed Graph\n $A_{i+1,i}=1$, for $i=0,\\cdots,N-2$. \n\n### Undirected Graph\n$A_{i+1,i}$ and $A_{i,i+1}=1$ for $i=0,\\cdots,N-2$. \n\nLet us apply the previous code to this case and see the effect of successive applications of $A$ on the signal. \n\nUndirected graphs lead to NaNs in the normalized matrices. \n\n\n```python\n@interact_manual(\n N=(5, 20),\n normalization=[\"none\", \"left\", \"symmetric\"],\n graph=[\"undirected\", \"directed\"],\n)\ndef plot1d(N=15, normalization=\"left\", graph=[\"undirected\"]):\n # Define a NxN matrix\n A = np.zeros([N, N])\n for i in range(1, N):\n A[i, i - 1] = 1\n\n if graph == \"undirected\":\n A = A + A.T\n\n D = np.sum(A, axis=1) # works for all A\n print(\"D= \", D)\n\n if normalization == \"left\":\n Dinv = np.diag(1.0 / D)\n An = Dinv @ A\n elif normalization == \"none\":\n An = A\n else:\n Dinv = np.diag(1.0 / D)\n DinvSq = np.sqrt(Dinv)\n An = DinvSq @ A @ DinvSq\n\n print(\"A = \", A)\n print(\"An= \", An)\n evalue, evector = np.linalg.eig(An)\n print(np.sort(evalue))\n```\n\n\n interactive(children=(IntSlider(value=15, description='N', max=20, min=5), Dropdown(description='normalization\u2026\n\n\nWhen the graph is directed, the first row of $A$ is zero, which leads to a zero eigenvalue, and the matrix is not invertible. \n\nWith no normalization, the maximum eigenvalue magnitude is greater than unity, which is not desirable for an iterative process. However, with both left and symmetric normalization, the eigenvalues are still greater than unity. \n\nThis leads to the idea of iterating with a matrix whose eigenvalues have better properties. This matrix is the Laplacian: \n$$\nL = D - A\n$$\nwhose rows sum to zero. One easily sees that this represents a first or second order approximation to the second derivative is the nodes are equally spaced. The Laplacian measure curvature. \n\nLet us compute the eigenvalues of $L$, and its normalized version: \n\\begin{align}\nL^* &= D^{-1} L \\\\\nL^* &= D^{-1/2} L D^{-1/2}\n\\end{align}\nwhere $D$ is still defined as the degree matrix of $A$. \n\n\n\n```python\nnp.diag([1, 2, 3])\n```\n\n\n\n\n array([[1, 0, 0],\n [0, 2, 0],\n [0, 0, 3]])\n\n\n\n\n```python\n@interact_manual(\n N=(5, 20),\n normalization=[\"none\", \"left\", \"symmetric\"],\n graph=[\"undirected\", \"directed\"],\n)\ndef plot1d(N=15, normalization=\"none\", graph=[\"undirected\"]):\n # Define a NxN matrix\n A = np.zeros([N, N])\n for i in range(1, N):\n A[i, i - 1] = 1\n\n if graph == \"undirected\":\n A = A + A.T\n\n diagD = np.sum(A, axis=1) # works for all A\n Dinv = np.diag(1 / diagD)\n D = np.diag(diagD)\n # print(\"D= \", D)\n # print(\"Dinv= \", Dinv)\n # print(\"diag(D) \", np.diag(D))\n\n # print(\"D= \", D)\n # print(\"A= \", A)\n L = D - A\n # print(\"L= \", L)\n\n # We will call L (normalized or not, the filter)\n H = L\n\n if normalization == \"left\":\n Hn = Dinv @ H # normalized\n elif normalization == \"none\":\n Hn = L\n else:\n DinvSq = np.sqrt(Dinv)\n Hn = DinvSq @ H @ DinvSq\n\n print(\"A= \", A)\n print(\"Dinv= \", Dinv)\n print(\"(Dinv@D)= \", (Dinv @ np.diag(D)))\n print(\"norm(Dinv@D-np.eye(N))= \", np.linalg.norm(Dinv @ np.diag(D) - np.eye(N)))\n print(\"L=H = \", L)\n print(\"Hn= \", Hn)\n evalue, evector = np.linalg.eig(Hn)\n print(\"Sorted eigenvalues: \", np.sort(evalue))\n```\n\n\n interactive(children=(IntSlider(value=15, description='N', max=20, min=5), Dropdown(description='normalization\u2026\n\n\nEverything works as expected for undirected graphs. The two normalizations (left and symmetric) produce real eigenvalues in the range $[0,2]$. The unormalized Laplacian has unbounded eigenvalues. $\\lambda=1$ is another eigenvalue, independent of the number of nodes, $N$. \n\nClearly, the iteration \n$$\ns^{n+1} = A^n s^n\n$$\ndiverges as $n\\rightarrow\\infty$. \n\nFrom linear algebra, any symmetric matrix $L$ can be expressed as\n$$\nL = U^{-1} \\Lambda U\n$$\nwhere the *columns* of $U^{-1}$ are the eigenvectors of $A$ and $\\Lambda$ is a diagonal matrix with the eigenvalues of $L$. This is easily seen by multiplying both sides by $U^{-1}$: \n$$\nL \\, U^{-1} = U^{-1} \\Lambda \n$$\nIn component notation: \n\\begin{align}\n\\sum_j L_{ij} U^{-1}_{jk} &= \\sum_j U^{-1}_{ij} \\Lambda_{jk} \\\\\n &= \\sum_j U^{-1}_{ij} \\delta_{jk} \\Lambda_{jk} \\\\\n &= U^{-1}_{ik} \\lambda_k\n\\end{align}\nwhere $\\Lambda\\in\\Re{N\\times N}$ is a diagonal matrix. If the eigenvectors are normalized, $U^{-1} = U^T$, which is a normal matrix (i.e., the eigenvectors have unit length, and are orthogonal). We made the implicit asumptions that all eigenvalues are different. Otherwise, one has to resort to the Jordan normal form, which is out of scope.\nThe LHS (left-hand side) of the last equation represents $L U^{-1}_k$, where $U^{-1}_k$ is the $k^{th}$ column of $U^{-1}$. Therefore, $U^{-1}_k$ is an eigenfunction of $L$ with eigenvalue $\\lambda_k$. Again, we assume all eigenvalues are different. If an eigenvalue $\\lambda_k$ has multiplicity $m_k$, the corresponding eigenvectors form a subspace $U^{-1}_k \\in \\Re{m_k\\times m_k}$. \n\n## Eigenvectors of various operators on the cyclic and non-cyclic chain\nWe write a program to plot the eigenvectors of $A$, normalized $A$ (left and symmetric), and $L$ (normalized or not). \n\nConsider a matrix that has a unit eigenvalue $\\lambda = 1$ with associated eigenvector $v$. Assume that $\\lambda=1$ is the largest eigenvector. Starting from a random signal $s$, we know that it can be expressed as a linear combination of the eigenvectors of $A$. Since the eigenvectors form a basis of $A$, this expansion is unique. Thus: \n$$\ns = \\sum_k a_k v_k\n$$\nwhere $v_k$, $k=0,1,\\cdots,N-1$ is the $k^{th}$ eigenvectors and $\\lambda_k$ is the $k^{th}$ eigenvalue. Apply $A$ to both sides: \n$$\nA s = \\sum_k a_k A_k v_k = \\sum_k a_k \\lambda_k v_k\n$$\nTherefore, applying $A$ multiple times to both sides: \n$$\nA^n s = \\sum_k a_k \\lambda^n_k v_k\n$$\nIf we assume that $\\lambda_{max}$ is the eigenvalue of maximum magnitude, we reexpress the equation above as\n\\begin{align}\nA^n s &= \\lambda_{max}^n \\sum_k a_k \\left(\\frac{\\lambda_k}{\\lambda_{max}}\\right)^n v_k \n\\end{align}\nAs $n\\rightarrow\\infty$, the term with the largest eigenvalue in magnitude will dominate the expression. Therefore, \n$$\nA^n s \\rightarrow a_{k^*} \\lambda_{k^*}^n v_{k^*}\n$$ \nfor very large $n$. Setting $\\lambda_{max}=1$, we find that\n$$\nA^n s \\rightarrow a_{k^*} v_{k^*}\n$$\nwhich is finite. This result holds for any matrix $A$. \n\nWe demonstrated this earlier in the case when $A$ is the shift operator of a linear undirected graph, that $v_{k^*} \\rightarrow \\text{mean}(s}$. In this case, the constant function is an eigenvector that corresponds to the unity eigenvalue. \n\n\n(NEED MORE DEVELOPMENT). I AM CONFUSED. WHAT INFORMATION AM I TRYING TO IMPART? \n\n\n\n```python\n# I would like which_eig to plot to have a maximum of N\nwhich_eig_slider = widgets.IntSlider(min=0, max=100, value=0)\n\n\n@interact(N=(5, 100), which_eig=(0, 100), graph=[\"undirected\", \"directed\"])\ndef plot1d(N=10, which_eig=which_eig_slider, graph=[\"undirected\"]):\n # poor programming but allows me to change the slider position\n # global which_eig_slider\n\n # Define a NxN matrix\n # if which_eig > N:\n # which_eig = N - 1 # count from 0\n # which_eig_slider.value = N - 1\n # print(which_eig_slider)\n # print(\"which_eig: \", which_eig)\n\n A = np.zeros([N, N])\n for i in range(1, N):\n A[i, i - 1] = 1\n\n if graph == \"undirected\":\n A = A + A.T\n\n diagD = np.sum(A, axis=1) # works for all A\n # The undirected version has an Inf in Dinv\n Dinv = np.diag(1 / diagD)\n D = np.diag(diagD)\n L = D - A\n # We will call L (normalized or not, the filter)\n H = L\n\n H_dict = {}\n eigval_dict = {}\n eigvec_dict = {}\n H_dict[\"none\"] = L\n\n # Next two matrices have NaNs in the undirected graph case\n H_dict[\"left\"] = Dinv @ H # normalized\n DinvSq = np.sqrt(Dinv)\n H_dict[\"symmetric\"] = DinvSq @ H @ DinvSq\n\n if graph == \"directed\":\n # Remove keys (works even when key is not in dict)\n H_dict.pop(\"left\", None)\n H_dict.pop(\"symmetric\", None)\n\n # Draw three columns: no normalization, left, and symmetric\n # Draw 5 eigenvectors for first five eigenvalues, sorted by magnitude\n # Below the eigenvectors, plot the first 10 eigenvalues, sorted by magnitude\n\n nrows = 3\n ncols = 3\n # rows and cols are used to access axes array elements\n row_eigf, row_eigv = 0, 1\n cols_dict = {\"none\": 0, \"left\": 1, \"symmetric\": 2}\n pos_none_eig = 2, 1\n pos_none_tot_var = 2, 0\n\n fig, axes = plt.subplots(nrows, ncols, figsize=(15, 6))\n\n for k, v in H_dict.items():\n eigval_dict[k], eigvec_dict[k] = np.linalg.eig(v)\n arg = np.argsort(eigval_dict[k])\n eigval_dict[k] = eigval_dict[k][arg]\n eigvec_dict[k] = eigvec_dict[k][:, arg]\n\n for k in H_dict.keys():\n ax = axes[row_eigf, cols_dict[k]]\n for i in range(0, 5):\n ax.plot(eigvec_dict[k][:, i], \"-o\", label=f\"$\\lambda_{i}$\")\n ax.set_xlabel(\"k\")\n ax.set_ylabel(\"v_k\")\n ax.legend(framealpha=0.5)\n\n ax = axes[row_eigv, cols_dict[k]]\n ax.plot(eigval_dict[k], \"-o\", color=\"black\")\n ax.set_ylim(0, 5)\n ax.set_xlabel(\"k\")\n ax.set_ylabel(\"$\\lambda_k$\")\n ax.grid(True)\n\n ax = axes[pos_none_eig] # [0], pos_none_eig[1]]\n ax.set_ylim(-0.2, 0.2)\n ax.grid(True)\n ax.set_title(\"Single Eigenvector, no normalization\")\n\n try:\n eigvec = eigvec_dict[\"none\"][:, which_eig]\n except:\n print(f\"which_eig must be < N! Reset value to ${N-1}$\")\n which_eig = N - 1\n eigvec = eigvec_dict[\"none\"][:, which_eig]\n\n # print(\"norm(eigvec): \", np.linalg.norm(eigvec, 2))\n # eig_string = \"$\\lambda_%s$\" % which_eig\n # print(\"eig_string: \", eig_string)\n ax.plot(eigvec, \"-o\", color=\"black\", label=f\"$\\lambda_{which_eig}$\")\n ax = axes[row_eigv, cols_dict[\"none\"]]\n ax.plot(which_eig, eigval_dict[\"none\"][which_eig], \"o\", ms=10, color=\"red\")\n ax.set_title(f\"Eigenvalues $\\lambda_k$\")\n\n ax = axes[pos_none_tot_var]\n\n def tot_var(L, v):\n \"\"\"\n Calculate the total variation: \\sum_i (s[i]-s[j])^2\n where s is a signal, which could be an eigenvector of $A$.\n The function is inefficient but will work on general graphs\n \"\"\"\n total_variat = 0\n for i in range(N):\n for j in range(N):\n if abs(A[i, j]) > 0.01:\n total_variat += (v[i] - v[j]) ** 2\n return total_variat\n\n # Calculate total variation for all eigenvalues, and for 'none' and 'symmetric' normaliz\n totvar = []\n for i in range(N):\n v = eigvec_dict[\"none\"][:, i]\n totvar.append(tot_var(L, v))\n\n ax.plot(totvar, \"-o\", color=\"black\")\n ax.plot(which_eig, totvar[which_eig], \"o\", ms=10, color=\"red\")\n ax.grid(True)\n ax.set_title(\"Total Variation, $L$, no normalization\")\n\n # Plot curve\n\n for k in H_dict.keys():\n ax = axes[0, cols_dict[k]]\n ax.set_title(k + \" normalization\")\n ax = axes[1, cols_dict[k]]\n ax.set_title(k + \" normalization\")\n\n plt.suptitle(\n \"Eigenvectors and eigenvalues for $L$ (left), $D^{-1}L$ (middle), $D^{-1/2}LD^{-1/2}$ (right)\",\n fontsize=16,\n )\n\n plt.tight_layout()\n # plt.show()\n```\n\n\n interactive(children=(IntSlider(value=10, description='N', min=5), IntSlider(value=50, description='which_eig'\u2026\n\n\n## Findings\n* The spectrum (eigenvalues) range is independent of $N$. \n* The eigenvector of the unnormalized Laplacian has a fixed range for most $N$. It always has unit $l_2$ norm. \n* The total variation $\\sum_{i,j} A_{i,j} (v_i-v_j)^2$ increases with the eigenvalue. Here, $v_j$ is the eigenvector $j$ that corresponds to eigenvalue $\\lambda_i$. \n\n## Code complexity\nThe plotting code above is getting complicated. It is therefore time to simplify the code by refactoring common operations. Different plots have different number of subplots, and each subplot draws one or more curves. They require an axis (`ax`) and dependent and independent variables, either one or a group. Therefore, smaller routines dedicated to drawing a single subplot would be useful. \nFurthermore, there is a need to create routines to create different kinds of matrices, alogn with their eigenvalues, and eigenvectors. Of course, the `Networkx` graph already does this, but doing it ourselves is good coding practice. \n\n# Code refactoring\n\n## Refactored version of previous function\n* The new functions are located in the file `spectral_function_library.py` in the main folder.\n* Pay attention to the first two lines of this notebook: \n * %load_ext autoreload\n * %autoreload 2\n \nThese two lines ensure that modules are automatically reloaded when changed on disk. \n\n\n```python\n# I would like which_eig to plot to have a maximum of N\nwhich_eig_slider = widgets.IntSlider(min=0, max=100, value=0)\n\n\n@interact(N=(5, 100), which_eig=(0, 100), graph=[\"undirected\", \"directed\"])\ndef plot1d(N=10, which_eig=which_eig_slider, graph=[\"undirected\"]):\n A = spec_lib.linear_acyclic_chain(N, graph)\n D = spec_lib.degree_matrix(A)\n H = L = D - A # H stands for filter\n\n norms = [\"none\", \"left\", \"symmetric\"]\n H_dict = {k: spec_lib.normalized_matrix(L, D, k) for k in norms}\n\n eigval_dict = {}\n eigvec_dict = {}\n\n if graph == \"directed\":\n # Remove keys (works even when key is not in dict)\n H_dict.pop(\"left\", None)\n H_dict.pop(\"symmetric\", None)\n\n # Draw three columns: no normalization, left, and symmetric\n # Draw 5 eigenvectors for first five eigenvalues, sorted by magnitude\n # Below the eigenvectors, plot the first 10 eigenvalues, sorted by magnitude\n\n for k, v in H_dict.items():\n eigval_dict[k], eigvec_dict[k] = np.linalg.eig(v)\n arg = np.argsort(eigval_dict[k])\n eigval_dict[k] = eigval_dict[k][arg]\n eigvec_dict[k] = eigvec_dict[k][:, arg]\n \n # Total variation (based on Laplaci\n totvar = [spec_lib.tot_var(A, eigvec_dict[\"none\"][:, i]) for i in range(N)]\n \n \"\"\"\n Six plots of eigenvalues and eigenvectors of L and two normalized versions, \n left and symmetric normalization by Dinv and sqrt(Dinv). \n Also plotted: \n 1) total variation of the signal a a function of eigenvalue\n 2) the k^{th} eigenvector of the Laplacian. The chosen eigenvector is controlled\n with a slider bar (which_eigen)\n \"\"\"\n spec_lib.plot_data1(H_dict, eigval_dict, eigvec_dict, totvar, which_eig)\n```\n\n\n interactive(children=(IntSlider(value=10, description='N', min=5), IntSlider(value=50, description='which_eig'\u2026\n\n\n# Example of a simple embedding calculation using a spectral approach. \n* We will not be concerned with efficiency\n* We will linearize any nonlinearities. \n\n---\n## Time to think about node embeddings and Neural networks\nThe simplest algorithm would be to iterate the following: \n$$\nH^{n+1}_{i,l} = \\sum_{j\\in\\cal{N}(v_j)\\cup v_j} (I_{i,j} +A_{i,j}) H^{n+1}_{j,k} W_{k,l} \n$H^{i,l}$ to feature $l$ on graph node $i$. Feature $l$ is also called the value of element $l$ of the node's embedding. The number of features on a node need not be equal to the number of embeddings. \n\nThe subscript refers to the iteration number. In practice, a nonlinear funciton is applied between iterations. Thus, \n$$\nH^{n+1} = \\sigma((I+A) H^{n} W) \n$$\nwhere $W$ is a weight matrix that will be determined by optimizing an appropriate cost function. \n\nLet us link together multiple iterations: \n\\begin{align}\nH^{1} &= \\sigma((I+A) H^0 W^0) \\\\\nH^{2} &= \\sigma((I+A) H^1 W^1) \\\\\n\\cdots &= \\cdots\n\\end{align}\nNote that $w^n$ could be independent of $n$, which reminds us of recursion, or have different values for each iteration, which reminds us of a multi-stage convolution network. The weight matrix $W^n \\in \\Re{d^{n}\\times d^{n+1}}$ where $d^n$ is the size of the embedding vector at iteration $n$. $H^0$ is usually chosen to be the existing feature matrix of the graph. \n\nNow let us remove the nonlinearity. This gives the linear algorithm :\n\\begin{align}\nH^{1} &= (I+A) H^0 W^0 \\\\\nH^{2} &= (I+A) H^1 W^1 \\\\\n &= (I+A)^2 W^0 W^1 \\\\\n\\cdots &= \\cdots\n\\end{align}\nSince $W^0$ and $W^1$ were computed by the algorithm being developed, their product can be replaced by a single matrix $W$. After $n$ iterations, we have: \n$$\nH^n = (I+A)^n H^0 W^0\n$$\nWe will actually replace $I+A$ by its symmetrized normalized form\n$$\n\\tilde{A} = (I+A)^{-1/2} (I+A) (I+A)^{-1/2}\n$$\nWe will use PyTorch to implement one layer of a GNN, namely\n$$\nH^{n+1} = \\sigma(\\tilde{A} H^{n} W) \n$$\nwhere we will assume an embedding in $\\Re{d}$, $W\\in\\Re{d^n\\times d^{n+1}}$. \n\n\n# def generate_graph_from_adjacency_matrix_1(G, N, seed):\n \"\"\"\n Arguments\n N: number of nodes\n \"\"\"\n np.random.seed(seed)\n\n # Convert to np.ndArray\n A = nx.linalg.graphmatrix.adjacency_matrix(G).toarray()\n nx.linalg\n # print(\"Adj: \", A, \"\\n\", A.shape, \"\\n\", type(A))\n\n # Different matrices\n D = np.sum(A, axis=0)\n D = np.diag(D)\n I = np.eye(N)\n Sqinv = np.sqrt(np.inv(I+A))\n An = Sqinv @ (A+I) @ Sqinv\n\n print(I)\n L = D - A\n Ln = nx.normalized_laplacian_matrix(G) # Is it symmetric?\n Ln = Ln.toarray() # from sparse array to ndarray\n\n # Create a signal on the graph. We will choose a sine wave. \n def sig(ivec, freq):\n return np.sin(2.*np.pi*freq*ivec / ivec[-1])\n \n ivec = np.asarray(list(range(D.shape[0])))\n s = sig(ivec, freq=2)\n \n ### WHAT NEXT?\n \n \n matrix = [\"A\", \"D\", \"L\", \"invD\", \"invDA\", \"invDL\", \"invDinvD\", \"Ln\"]\n matrices = [A, D, L, invD, invDA, invDL, invDLinvD, Ln]\n\n # Eigenvalues\n fig, axes = plt.subplots(3, 3, figsize=(10, 8))\n axes = axes.reshape(-1)\n fig.suptitle(\"Eigenvalues of various matrices\")\n for i, m in enumerate(matrices):\n ax = axes[i]\n eigs = np.linalg.eigvals(m)\n eigs = np.sort(eigs)[::-1]\n ax.set_title(matrix[i])\n ax.grid(True)\n ax.plot(eigs, \"-o\")\n for i in range(i + 2, axes.shape[-1]):\n axes[i].axis(\"off\")\n plt.tight_layout()\n plt.show()\n\n\n```python\nprob_slider = widgets.FloatSlider(min=0, max=1, step=0.1, value=0.5)\nnode_slider = widgets.IntSlider(min=3, max=30, step=1, value=10)\nnb_neigh_slider = widgets.IntSlider(min=1, max=10, step=1, value=4)\nnb_edges_per_node_slider = widgets.IntSlider(min=1, max=20, step=2, value=5)\nseed_slider = widgets.IntSlider(int=1, max=50, step=1, value=25)\ngraph_type = [\"connected_watts_strogatz\", \"powerlaw_cluster_graph\"]\n\n@interact(\n nb_nodes=node_slider,\n prob=prob_slider,\n nb_neigh=nb_neigh_slider,\n nb_edges_per_node=nb_edges_per_node_slider,\n seed=seed_slider,\n graph_type=graph_type,\n # directed=True,\n)\ndef drawGraph(nb_nodes, nb_neigh, prob, seed, nb_edges_per_node, graph_type):\n if graph_type == \"connected_watts_strogatz\":\n nb_edges_per_node_slider.style.handle_color = 'red'\n nb_neigh_slider.style.handle_color = 'black'\n nb_tries = 20\n edge_prob = prob\n G = nx.connected_watts_strogatz_graph(\n nb_nodes, nb_neigh, edge_prob, nb_tries, seed\n )\n elif graph_type == \"powerlaw_cluster_graph\":\n nb_neigh_slider.style.handle_color = 'red'\n nb_edges_per_node_slider.style.handle_color = 'black'\n add_tri_prob = prob\n if nb_edges_per_node >= nb_nodes:\n nb_edges_per_node = nb_nodes - 1\n G = nx.powerlaw_cluster_graph(nb_nodes, nb_edges_per_node, add_tri_prob, seed)\n\n generate_graph_from_adjacency_matrix_1(G, nb_nodes, seed)\n```\n\n\n interactive(children=(IntSlider(value=10, description='nb_nodes', max=30, min=3), IntSlider(value=4, descripti\u2026\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "ad1da27a564f889cdc614fe0849c52af8fdd7ddd", "size": 90742, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Spectral.ipynb", "max_stars_repo_name": "PhotoWolf/Advanced_Grad_Seminar_GNN", "max_stars_repo_head_hexsha": "3a577c1b953de276ac01ef96d49b37bad4088f75", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Spectral.ipynb", "max_issues_repo_name": "PhotoWolf/Advanced_Grad_Seminar_GNN", "max_issues_repo_head_hexsha": "3a577c1b953de276ac01ef96d49b37bad4088f75", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Spectral.ipynb", "max_forks_repo_name": "PhotoWolf/Advanced_Grad_Seminar_GNN", "max_forks_repo_head_hexsha": "3a577c1b953de276ac01ef96d49b37bad4088f75", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 37.5432354158, "max_line_length": 456, "alphanum_fraction": 0.5385268123, "converted": true, "num_tokens": 17852, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.7371581626286833, "lm_q1q2_score": 0.43688902187353096}} {"text": "Python for beginners\n=================\n\nPlease indicate your name below, since you will need to submit this notebook completed latest the day after the datalab.\n\nDon't forget to save your progress during the datalab to avoid any loss due to crashes.\n\n\n```python\nname=''\n```\n\n# Introduction\n\nPython is a general-purpose, high-level interpreted programming language. One of the main goals when designing the language was to make it easy to read and to write. Indeed, often writing a python code feels like writing pseudo-code (ie. using plain language to describe an algorithm). Since it is interpreted, the instructions are directly executed by the python interpreter, and users do not need to bother with compiling the code. Besides, python is freely usable and distributable. These make the code ideal for any introductory programming course.\n\nPython has two main versions, which coexisted in the recent years and have some differences: Python2 and Python3. In our opinion any novice user should only learn Python3 (you are welcome to search for debate articles), thus in this course we use Python3 interpreters and syntax.\n\nLearning a programming language at depth takes lot of time, espescially if it is your first programming language. It takes a lot of practice and dedication. This introduction is just a guidance to help you achieve your goal.\n\nThis datalab reviews\n- how to run your python code\n- how to create variables\n- the different data types\n- loops and conditions\n- definition of functions\n- how to read files\n\nAfter completing the tutorial you will be familiar with the basic python syntax and will be able to implement simple algorithms and mathematics on your own. Before running each of the code blocks, think what would you expect as an output, and compare the result with your expectation. If you get curious about some functionality, try to come up with small experiments to see how Python works.\n\nWe will try to cover most of the things what is needed for this course, however no one is expected to know every built-in functions and perks of python, so you will often search the web for answers. There are so many good tutorials ([like this one](https://www.w3schools.com/python/default.asp)) and forum entries, that I do not even try to list them. Besides, it is also an important skill for you to find them yourself. \n\nOf course the language in itself is just a tool, we as users also need to follow the Zen of Python if we would like to write great code (you can execute the cell below with the \"Run\" button, or with Shift+Enter).\n\n\n```python\nimport this\n```\n\nThere are several tools, or packages which build on python. These packages can be either installed through the package installer called *pip*, or an other convenient way is to install the [Anaconda](https://www.anaconda.com/products/individual) which readily installs several of the most important packages. During this course we will rely on some of them:\n\n- numpy: allows handling multi-dimensional arrays, numeric computing and linear algebra\n- scipy: provides tools for numerical analysis (integration, function fitting, numerical ODE solution) and more\n- matplotlib: allows the creation of figures and plots.\n\nWith the usage of these three packages the functionality of Python becomes comparable to Matlab. Indeed users experienced with Matlab will find even some of the syntax familiar. Other useful packages that will be used in this course are\n\n- Jupyter: allows the user to write interactive notebooks\n- Pandas: provides tools to manipulate data at a higher level through the application of the DataFrame object.\n\nBesides these packages there are several other packages written by the community. It often happens that the solution to your problem is already solved by someone, and you can just use an existing package.\n\n## How to run a python code\n\nIn this course we will be using Jupyter notebooks to run code, however it is useful if you know how to run python code without Jupyter as well.\n\nSince Python is an interpreted language it can be executed line by line, which allows to use it in an interactive way. To execute it you can use the python interpreter or a jupyter notebook interactivaly, or you can place your script in a file, and execute that. \n\n### Python interpreter\n\nAfter installing python on your computer you can type `python`in your terminal (Linux/Mac) or command prompt (Windows) to initiate the Python interpreter. You will see a command prompt starting with `>>>`. Then you can type commands.\n\n```\n >>> 299792458*299792458\n 89875517873681764\n >>> a=1.38\n >>> a*2\n 2.76\n```\n\nThis is convenient to use the language as a calculator or to test smaller code snippets. You can exit the interpreter by typing `exit()` or `quit()`.\n\n### Jupyter notebook\n\nA Jupyter notebook (such as this one) is an interactive terminal what you can open in a browser window, and allows you to mix documents (written in Markdown language and/or latex) with executable code and plots. It is very convenient if one wants to demonstrate something or provide some data analysis.\n\nYou can type `jupyter notebook` in your terminal to launch a notebook environment, and then write the code and the document. Also [Google's Colab](https://colab.research.google.com/) provides a way to create jupyter notebooks in the cloud. When using Colab, one does not need to install anything, since it offers the most often used packages readily.\n\n\n### Script from file\n\nIf one needs to write a longer script or program, it is often more convinient to save it into a file, and execute it. Let's consider you have written a script in *file.py*\n\n```python\n myNumber=4\n print('My square is: ',myNumber*myNumber)\n```\nThen one can execute this script with:\n\n```\n $ python file.py \n My square is: 16\n```\n\nFor even longer projects, which might be separated into several files it is often useful to create *module*s or *package*s, which we will touch upon briefly, however within this course will not need to worry about the creation of such.\n\n## How to use Jupyter notebooks\n\nThe usage of Jupyter notebooks is rather straightforward, since the layout of the application resembles a word editor. The content of the notebook is organized into Cells. There is a dropdown window from which you can select whether the Cell you are currently working on should contain Python *code* or documentation written in *Markdown*.\n\nUpon double clicking a Cell containing text you will see the source of that text. In this way you can explore the Markdown syntax. Besides that you can always rely on one of the [cheatsheets](https://github.com/adam-p/markdown-here/wiki/Markdown-Cheatsheet).\n\nIf you wanted to write LaTeX (eg. to render equations), you can just include the command within Markdown. Such as \n\n```latex\n \\begin{equation}\n E=mc^2\n \\end{equation}\n```\n\nrenders as\n\n\\begin{equation}\nE=mc^2\n\\end{equation}\n\nYou can Insert new Cells with the *+* button or from the *Insert* menu. Similarly you can delete, copy and paste cells with the corresponding button or by opening the *Edit* menu. You can *Run* (ie. execute) a single cell, or all of them at once with the *Cell* menu. \n\nSometimes you might encounter bugs or you overload the memory due to which your notebook environment does not respond anymore, in this case you can restart the kernel under the *Kernel* menu.\n\nFinally, if you would like to check the type of an object or variable, or you would like to reach the documentation of a method or function, you can type `?name of method/variable`.\n\n\n```python\na=3\n```\n\n\n```python\n?a\n```\n\n\n```python\n?print\n```\n\n# Introduction to python\n\nFrom now on we will focus on the syntax of the language. We will review how to declare variabes, how to to handle loops and conditions and how to write program functions.\n\n## Variables and arithmetics\n\nLet's calculate the number density of a nuclide with a known physical density\n\n$$N=\\frac{N_A\\rho}{A}$$\n\nwhere $N_A$ is Avogadro's number, $\\rho$ is the physical density and $A$ is the mass number of a given nuclide.\n\n\n```python\nN_A=6.022e23 #atom/mol\nrho=19.1 #density of uranium. g/cm3\nA=235 #mass number of U-235\n\nN=N_A*rho/A #/cm3\n\nmyString='My material is made of a nuclide with mass number %d and has a density %.1f g/cm3'%(A,rho)\nprint(myString)\nprint('The number density is %.2e'%N)\n```\n\nAnd now calculate the speed of a neutron travelling with a kinetic energy of 2 MeV.\n\n$$v=\\sqrt{\\frac{2E}{m}}$$\n\n\n```python\nimport math\n\nE=2 #MeV\nEJ=E*1.60217662E-13 #J\n\nmn = 1.67492749804E-27 #neutron mass in kg\n\nv = math.sqrt(2*EJ/mn)\n\nprint('The speed of the neutron is')\nprint(v)\nprint('m/s')\n\n```\n\nOr alternatively you could include escape characters (like newline `\\n`, or tab `\\t`) in the string:\n\n\n```python\nprint('The speed of the neutron is \\n \\t {} \\n m/s'.format(v))\n```\n\nAlthough we just did some basic calculations (essentially using the interpreter as a calculator), \nwe can already notice a lot of things.\n\n1. When declaring a variable we do not require to give a type, Python will figure that out\n\n\n```python\nprint(type(N_A))\nprint(type(rho))\nprint(type(A))\nprint(type(myString))\n```\n\nAnd Python even figures out that in case an integer is multiplied with a floating point number, the result should be a floating point number.\n\n\n```python\nprint(type(E))\nprint(type(EJ))\n```\n\nNevertheless, it is usually a good practice to define a variable as float if it is intended to be used as a float, and as an integer if it is intended to be used as an integer. For example, we should have written `E=2.0`.\n\n2. The `print()` function allows us to inspect the content of variables. It can handle strings and numbers. Without formatting the long format of the number is printed.\n3. If something is written between quotation marks `''` or double quotation marks `\"\"`, that is considered as a string variable, which can be formatted. Actually there are more ways of [string formatting](https://realpython.com/python-string-formatting/), here at most places we used one widespread method, by using spaceholders for the variable's content to be included: `%.nx`, where *.n* gives the number of floating points to be printed, and *x* clarifies the type of the variable (*d* for integers or decimals, *f* for floating point numbers, *e* for scientific notation, and *s* for string). This is similar as some other languages work, however not necessarily the most pythonic solution. You can see an example for using the more pythonic `.format()` method also.\n4. Mathematical operators look like as we would expect, +, -, \\*, / for addition, substraction, multiplication and division. There are three other operators %, \\*\\*, // for modulus, exponentiation and integer division (if we want to convert the result into integer). and we can have some shorthand operators, for example `x+=2` is the same as `x=x+2`. Here you can also notice that $x=x+2$ seems incorrect according to math, however in programming languages it is common: the left hand side will access the current value of the variable, then adds 2, and assigns the result to the variable. **Note**: the precedence of the operators is important (first exponentation, then division, multiplication, addition/substraction is performed, so one must use brackets `()` to change the precedence)\n\n\n```python\nc=299792458\nprint(c%2)\nprint(c%3)\nprint(c**2)\nprint(2//3,' vs ',2/3)\nprint(10//3,' vs ',10/3)\nprint(c+c/2) #1.5c\nprint((c+c)/2) #c\n```\n\n5. The hashtag symbol `#` marks that a comment follows: information not executed by the code, but guiding the reader of the code.\n6. With `import` we can load modules or packages. Importing has several alternatives, such as `from package import function`. Here we imported the math package, which contains basic mathetmatical functions (cosine, sine, square root, etc). We will not use it much later, because we will use numpy instead. We also see, that if we want to access a method or function of a package, we can use `.` as for `math.sqrt()`. If we just type `math.` and hit TAB, jupyter will show the available functions:\n\n\n```python\nmath.\n```\n\n\n```python\n#though be careful with imports like this, since several packages might have functions with the same name\nfrom math import sqrt\nprint(sqrt(2*EJ/mn))\n```\n\nAlso, in python since the type of a variable doesn't need to be defined, actually one can change the type dynamically. Although, as said before, it is a good practice to keep the same type for a variable throughout the code.\n\n\n```python\nx=42\nx='fortytwo'\n```\n\nWe can also modify the type with built in functions. This is often useful, for example when reading values as strings from a file, we will need to convert them to float if we want to make operations with them.\n\n\n```python\nx='42'\nprint(type(x),x)\nx=int(x)\nprint(type(x),x)\nx=float(x)\nprint(type(x),x)\n\n```\n\nBy the way, what happens if we do operations with strings:\n\n\n```python\na='42'\na*3\n```\n\n\n```python\na/3\n```\n\nThere is finally one last simple variable type: bool, which can take `True` or `False` value. As in other languages we can define comparison operators, which will return bool values.\n\n- `x==y`, `x!=y`\n- `x>=y`, `x>y`\n- `x<=y`, `x c\n```\n\nwith booleans we can define logic operators `and`, `or`.\n\n\n```python\nv < c and 2*v < c #True if both true\n```\n\n## Built-in data structures\n\nPython has several compound data structures. These are the list (ordered collection), tuple (immutable ordered collection), dictionary (unordered (key,value) mapping), set (unordered collection of unique values). Immutable means that it cannot be changed after being created. However here, we will only focus on lists and dictionaries.\n\n### Lists\n\nLists in other languages are called arrays or vectors. In Python they can store any data type, even a mix of data types. We can reach any element through its index (indexing starts at 0). A negative index means that we are going backwards (ie. `-1` refers to the last element of the list).\n\n\n```python\nemptyList=[] #we can create an empty list\nX=[1,2,3,4]\nprint(len(X)) #len() returns the number of elements\n\nprint(X[0])\n\nprint(X[-1]) #the last element, similarly -2, -3 etc would work\nprint(X[-2]) #the element before the last\n\nprint(X[1:3]) #from element with index 1 till element with index 3 (exlusive)\n\nX.append(5) #append value to the end\nprint(X)\n\nprint([0]*5) #what happens now?\n\nprint(X+[0]*5) #and now?\n```\n\nLists can contain any type, even other lists. So we can use them to create multidimensional arrays. But as we will see it later, we prefer numpy for that.\n\n\n```python\nA=[0,'one',2.0,[3,4,5]]\nA[3][1] #from element with index 3 take the element with index 1. ofc if we know that it is a list which has an element\n```\n\n\n```python\nA[2][1] #does't work\n```\n\n\n```python\nA[3][5] #does't work\n```\n\nStrings also behave like lists: they are lists of characters!\n\n\n```python\nmyStr = 'And this we will use often when reading data from a file'\n\nprint(myStr[4])\nprint(myStr[-4:])\nprint(myStr[:5])\nprint(myStr[3:20])\n```\n\n### Dictionaries\n\nDictionaries are mappings of keys to values, which provide a lot of flexibility. We will use them a lot in this course. You can define it as comma-separated `key: value` pairs within curly brackets. Similarly to lists, dictionaries can also be nested (ie. a value stores an other dictionary).\n\n\n```python\nmyDict={} #we can initialize an empty dict and later add keys: values\nZ={'H': 1, 'He': 2,'U': 92, 'Pu': 94} #or we can immediately initialize it with key: values.\nprint(Z['Pu'])\nprint(Z.keys())\nprint(Z.values())\nZ['C']=6 #creates a new item\nZ\n```\n\n### Identity and membership operations\n\nPython has operators to check whether an element is a member of a list or dictionary, and also to compare such compound objects:\n\n`is`, `is not`, `in`, `not in`. These operations will return booleans. \n\n\n```python\nprint('one' in A)\nprint('two' not in A)\nprint(2 in Z) # this looks for the keys in the dictionary!\nprint('He' in Z)\n```\n\n## Conditionals\n\nAs in other languages, you can define conditons, points in the code where an action follows a decision.\n\n**Note that indentation is a MUST in Python**, it is not only a question of aesthetics, but this is how you tell the interpreter when a block ends. Luckily most good python editors and jupyter will automatically indent after a `:`\n\n\n```python\nE=1.0 #MeV\n\nif E<=0.025e-6:\n print('Thermal neutron')\nelif E>0.025e-6 and E<=0.4e-6:\n print('Epihermal neutron')\nelif E>0.4e-6 and E<=0.5e-6:\n print('Cadmium neutron')\nelif E>0.5e-6 and E<=1.0e-6:\n print('EpiCadmium neutron')\nelif E>1.0e-6 and E<=10.0e-6:\n print('Slow neutron')\nelif E>10.0e-6 and E<=300e-6:\n print('Resonance neutron')\nelif E>300.0e-6 and E<=1.0:\n print('Intermediate neutron')\nelse:\n print('Fast neutron')\n```\n\n## Loops\n\nLoops allow to repeatedly execute code. In Python we can use `for` (to iterate through an array of numbers) and `while` loops (to execute the statement while some condition is met).\n\n\n```python\nfor i in [1,4,9,16]:\n print(i,math.sqrt(i))\n```\n\n\n```python\nfor j in range(5):\n print(j)\n```\n\n\n```python\n#or ofter you will want to iterate through the indices of a list\na=[1,4,9,16]\nfor j in range(len(a)):\n print(j,a[j])\n```\n\nNote, that here `range` is a built in function to produce a sequence of integers from start (inclusive)\nto stop (exclusive) by step. There are some other handy built in methods like `zip` allows you to iterate through several lists and `enumerate` can be used to iterate through both the elements of a list and the index of the elements.\n\n\n```python\nfor i,j in zip([1,4,9,16],[1,8,27,256]):\n print(i,j)\n```\n\n\n```python\nfor i,j in enumerate([1,4,9,16]):\n print(i,j)\n```\n\n\n```python\ni=1\nx=1\nwhile x<10:\n x=i**2\n print(i,x)\n i+=1\n```\n\n### List and Dictionary comprehension\n\nPythonic programming allows you to write certain code in a much shorter way. With that it is possible to write very powerful one-liner code. However one needs to compromise between having concise code and readable code.\n\n\n```python\na=[] #creates an empty list\n\nfor i in range(10):\n a.append(i)\n \nprint(a)\n\nb=[i for i in range(10)]\n\nprint(b)\n```\n\n## Functions and error handling\n\nWe often need to execute a certain sequence of statements repeatedly. In order to avoid copying code we can organize certain parts into functions (copying is both ugly and error-prone). This also allows our code to be reusable. Functions take parameters which need to be listed at the definition, and optionally they can `return` variables. These parameters have a local scope (they don't exist outside the function). If a variable used by the function is neither listed as an input parameter nor defined within the function, python will look for the variable in the global namespace. Sometimes such behaviour is intentional, but one needs to be careful, using the same variable names locally and globally can lead to bugs. (You can further read on *namespace* and *scope* [here](https://realpython.com/python-namespaces-scope/)). It is however possible to write a function without any parameters or returns. For example:\n\n\n```python\ndef myErrorMsg():\n print('Something is wrong')\n \nmyErrorMsg()\n```\n\nOne can also provide default values for parameters:\n\n\n```python\ndef myErrorMsg2(msg='Something is wrong'):\n print(msg)\n \nmyErrorMsg2()\nmyErrorMsg2('Not so wrong')\nmyErrorMsg2(msg='Not so wrong')#is the same, however in case of more parameters it might be necessary to tell which one you are refering to\n```\n\nLet's consider our neutron speed calculation. This looks like something we might want to repeat for various numbers, and we also need to make sure that we are trying to perform the operations on numbers, otherwise we will need to raise an [exception](https://docs.python.org/3/library/exceptions.html) to the user. Of course error handling is optional, and for small functions like this, it is an overkill (and anyhow `math.sqrt()` would perform error handling). But for more complicated functions it is a good practice to handle possible errors.\n\n\n```python\ndef speed(E,m):\n \"\"\" \n Function to calculate speed from energy and mass\n \n Parameters\n ----------\n E : float\n kinetic energy in J\n m : float\n mass in kg\n \n Returns\n -------\n v : float\n speed in m/s\n \n Note\n ----\n One could do certain checks here to make sure that the parameters \n indeed have the right type, nevertheless python will anyhow complain\n \"\"\"\n import math #we want to make sure that the math package is imported\n if (isinstance(E,int) or isinstance(E,float)) and (isinstance(m,int) or isinstance(m,float)):\n v = math.sqrt(2*E/m)\n else:\n raise TypeError('Energy and mass need to be float or int.')\n \n return v\n\n\nvneutron=speed(EJ,mn)\nprint(vneutron)\n```\n\n\n```python\nspeed('32','2')\n```\n\nHere you can see a different type of comment, encapsulated within triple quotation marks. This is called a *docstring*. This is not just simply a comment, but the information about the function, which is returned if `?speed` is typed. There are various styles which can be used to write docstrings (for example, here we have used the [numpy style](https://numpydoc.readthedocs.io/en/latest/format.html))\n\n\n```python\n?speed\n```\n\n## Reading and writing files\n\nAs a physicist or engineer you will find yourself often in a situation that you have to read data from other files, or that you have to write the results of your calculations into files. There are of course several good ways to write data (by structuring it according to some standard which facilitates loading the data later in memory; we will learn about this in the next datalab), but it often happens that a file contains content for which you need to write a parser to extract the relevant information. This is espescially true for the outputfiles generated by some old, legacy scientific softwares.\n\nTherefore, it is very important to read/write files. For this we have to open the file with `open(filename,'r')`, where `'r'` marks that we are opening the file for reading (we could have `'w'` for writing, when creating a new file, or `'a'` for appending when writing to the end of an existing file). We can see that this function will create an object of the `_io.TextIOWrapper` class (we will talk about classes a bit more in a following datalabs). This class has several methods (or functions) to read the content. \n\nLet's consider we have an output file '01-sample.txt' produced with some code, which calculated the radioactivity in a piece of iron after being irradiated with a neutron generator.\n\n\n```python\nmyFile = open(\"01-sample.txt\", \"r\") #check type of f\nprint(type(myFile))\nfilecontent=myFile.read() #notice that we call the function with .read(), showing that this is a method of the object.\nmyFile.close()\nfilecontent \n\n```\n\nNotice, that the file is read as a long string, with several `\\n` characters marking the beginning of a new line. This is not always what we want, espescially if our file resembles a table. \n\nwith `.readline()` we could read one single line, and with `.readlines()` we can read all lines in a list. However to get out the data we might be interested in we still need to locate the lines which contain it, and find the right information within the line string. We will need to `.strip()` the string (to remove the `\\n` characters which are unnecessary for us), and to `.split()` which as we will see splits the string into smaller pieces (the \"splitting rule\" can be input to the method, by default it will split at each white space).\n\n\n```python\nmyFile = open(\"01-sample.txt\", \"r\") #we have to reopen it, because we closed it\nfilecontentNew=myFile.readlines()\nmyFile.close()\nfilecontentNew \n```\n\n\n```python\nprint(filecontentNew[6])\nprint(filecontentNew[6].strip()) #removes the \\n character, and white space around lines\nprint(filecontentNew[6].strip().split()) #splits it into a list, split can handle different separators. default is whitespace\nprint(filecontentNew[6].strip().split()[3]) #We got our activity!\nprint(3*filecontentNew[6].strip().split()[3]) #ooh but it is a string!\nprint(float(filecontentNew[6].strip().split()[3])) #and now it is a float\n```\n\n\n```python\nmyFile = open(\"01-sample.txt\", \"r\")\nfor line in myFile.readlines(): #you can loop through the elements of the list\n print(line.strip().title()) #title makes capital letters small, except at the beginning of the line\nmyFile.close()\n```\n\n**Note**: there is a more pythonic \"modern\" way of opening files, however, I find that more confusing for beginners. This looks like the following:\n\n\n```python\nwith open('01-sample.txt', 'r') as myFile:\n content = myFile.readlines()\nprint(content)\n```\n\n# Exercises\n\n## 1\n\nWrite a function which can calculate the decay constant $\\lambda$ based on the half-life of an isotope. The function should return $\\lambda$ in $1/s$ units, however the user should be able to give the half-lifes with different units. Possible units can be `'s'` for seconds, `'m'` for minutes, `'h'` for hours,`'d'` for days,`'y'` for years. Raise an exception if other units are given. \n\n$$\\lambda=\\frac{\\ln(2)}{T_{1/2}}$$\n\n\n```python\ndef decayConst(hl,unit='h'):\n \"\"\"\n your docstring\n \"\"\"\n #your code\n return lam #notice lambda is a python keyword, so it cannot be a variable name\n```\n\n ## 2\n \nBelow we have a dictionary with the half-life (in days) of some isotopes, which are relevant when performing passive gamma spectroscopy measurements of spent nuclear fuel. Create an other dictionary (named `isotopeInfo`, see below), where the keys are the same but the values are other dictionaries, storing the half-life and the decay constant. Use your previously created function and loops. \n \n```python\n isotopeInfo={'Y-91': {'hl': some value, 'lambda': some value},\n ...}\n```\n\n\n```python\nisotopes={'Y-91': 58.5,\n 'Zr-95': 64,\n 'Nb-95': 35,\n 'Ru-103': 39.2,\n 'Ru-106': 372,\n 'Sb-125': 2.76*365,\n 'Cs-134': 2.065*365,\n 'Cs-137': 30.1*365,\n 'Eu-154': 8.6*365,\n 'Eu-155': 4.75*365, \n 'Ce-141': 32.5,\n 'Ce-144': 285}\n```\n\n## 3\n\nWrite a script which reads the content of the file '01-sample.txt', and organizes it as a dictionary with the keys being the nuclide names (in a 'Symbol-MassNumber' format), and the values being the activities as floats. Do not modify the file!\n\nThe solution should look like\n\n```python\n activity={'Mn-56': 2.8492E+05, \n 'Mn-57': 6.0933E+03,\n ...}\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "96a03fcbc70e3464c3a99305ca28edfaf55b66a5", "size": 56983, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Datalabs/Datalab01/1-BasicPython.ipynb", "max_stars_repo_name": "ezsolti/RFP", "max_stars_repo_head_hexsha": "5a410dd30ad61686b5d54d7778462e5e217be159", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 21, "max_stars_repo_stars_event_min_datetime": "2021-06-18T15:25:22.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-21T07:34:42.000Z", "max_issues_repo_path": "Datalabs/Datalab01/1-BasicPython.ipynb", "max_issues_repo_name": "ezsolti/RFP", "max_issues_repo_head_hexsha": "5a410dd30ad61686b5d54d7778462e5e217be159", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Datalabs/Datalab01/1-BasicPython.ipynb", "max_forks_repo_name": "ezsolti/RFP", "max_forks_repo_head_hexsha": "5a410dd30ad61686b5d54d7778462e5e217be159", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 5, "max_forks_repo_forks_event_min_datetime": "2021-06-19T00:28:02.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-09T18:58:27.000Z", "avg_line_length": 32.2302036199, "max_line_length": 928, "alphanum_fraction": 0.598529386, "converted": true, "num_tokens": 6540, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.78793120560257, "lm_q1q2_score": 0.436884581927287}} {"text": "## Configurations for Colab\n\n\n```python\nimport sys\nIN_COLAB = \"google.colab\" in sys.modules\n\nif IN_COLAB:\n !apt install python-opengl\n !apt install ffmpeg\n !apt install xvfb\n !pip install pyvirtualdisplay\n !pip install gym\n from pyvirtualdisplay import Display\n \n # Start virtual display\n dis = Display(visible=0, size=(400, 400))\n dis.start()\n```\n\n# 05. Noisy Networks for Exploration\n\n[M. Fortunato et al., \"Noisy Networks for Exploration.\" arXiv preprint arXiv:1706.10295, 2017.](https://arxiv.org/pdf/1706.10295.pdf)\n\n\nNoisyNet is an exploration method that learns perturbations of the network weights to drive exploration. The key insight is that a single change to the weight vector can induce a consistent, and potentially very complex, state-dependent change in policy over multiple time steps.\n\nFirstly, let's take a look into a linear layer of a neural network with $p$ inputs and $q$ outputs, represented by\n\n$$\ny = wx + b,\n$$\n\nwhere $x \\in \\mathbb{R}^p$ is the layer input, $w \\in \\mathbb{R}^{q \\times p}$, and $b \\in \\mathbb{R}$ the bias.\n\nThe corresponding noisy linear layer is defined as:\n\n$$\ny = (\\mu^w + \\sigma^w \\odot \\epsilon^w) x + \\mu^b + \\sigma^b \\odot \\epsilon^b,\n$$\n\nwhere $\\mu^w + \\sigma^w \\odot \\epsilon^w$ and $\\mu^b + \\sigma^b \\odot \\epsilon^b$ replace $w$ and $b$ in the first linear layer equation. The parameters $\\mu^w \\in \\mathbb{R}^{q \\times p}, \\mu^b \\in \\mathbb{R}^q, \\sigma^w \\in \\mathbb{R}^{q \\times p}$ and $\\sigma^b \\in \\mathbb{R}^q$ are learnable, whereas $\\epsilon^w \\in \\mathbb{R}^{q \\times p}$ and $\\epsilon^b \\in \\mathbb{R}^q$ are noise random variables which can be generated by one of the following two ways:\n\n1. **Independent Gaussian noise**: the noise applied to each weight and bias is independent, where each random noise entry is drawn from a unit Gaussian distribution. This means that for each noisy linear layer, there are $pq + q$ noise variables (for $p$ inputs to the layer and $q$ outputs).\n2. **Factorised Gaussian noise:** This is a more computationally efficient way. It produces 2 random Gaussian noise vectors ($p, q$) and makes $pq + q$ noise entries by outer product as follows:\n\n$$\n\\begin{align}\n\\epsilon_{i,j}^w &= f(\\epsilon_i) f(\\epsilon_j),\\\\\n\\epsilon_{j}^b &= f(\\epsilon_i),\\\\\n\\text{where } f(x) &= sgn(x) \\sqrt{|x|}.\n\\end{align}\n$$\n\nIn all experiements of the paper, the authors used Factorised Gaussian noise, so we will go for it as well.\n\n\n```python\nimport math\nimport os\nfrom typing import Dict, List, Tuple\n\nimport gym\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport torch.optim as optim\nfrom IPython.display import clear_output\n```\n\n## Replay buffer\n\nPlease see *01.dqn.ipynb* for detailed description.\n\n\n```python\nclass ReplayBuffer:\n \"\"\"A simple numpy replay buffer.\"\"\"\n\n def __init__(self, obs_dim: int, size: int, batch_size: int = 32):\n self.obs_buf = np.zeros([size, obs_dim], dtype=np.float32)\n self.next_obs_buf = np.zeros([size, obs_dim], dtype=np.float32)\n self.acts_buf = np.zeros([size], dtype=np.float32)\n self.rews_buf = np.zeros([size], dtype=np.float32)\n self.done_buf = np.zeros(size, dtype=np.float32)\n self.max_size, self.batch_size = size, batch_size\n self.ptr, self.size, = 0, 0\n\n def store(\n self,\n obs: np.ndarray,\n act: np.ndarray, \n rew: float, \n next_obs: np.ndarray, \n done: bool,\n ):\n self.obs_buf[self.ptr] = obs\n self.next_obs_buf[self.ptr] = next_obs\n self.acts_buf[self.ptr] = act\n self.rews_buf[self.ptr] = rew\n self.done_buf[self.ptr] = done\n self.ptr = (self.ptr + 1) % self.max_size\n self.size = min(self.size + 1, self.max_size)\n\n def sample_batch(self) -> Dict[str, np.ndarray]:\n idxs = np.random.choice(self.size, size=self.batch_size, replace=False)\n return dict(obs=self.obs_buf[idxs],\n next_obs=self.next_obs_buf[idxs],\n acts=self.acts_buf[idxs],\n rews=self.rews_buf[idxs],\n done=self.done_buf[idxs])\n\n def __len__(self) -> int:\n return self.size\n```\n\n## Noisy Layer\n\n**References:**\n- https://github.com/higgsfield/RL-Adventure/blob/master/5.noisy%20dqn.ipynb\n- https://github.com/Kaixhin/Rainbow/blob/master/model.py\n\n\n```python\nclass NoisyLinear(nn.Module):\n \"\"\"Noisy linear module for NoisyNet.\n \n Attributes:\n in_features (int): input size of linear module\n out_features (int): output size of linear module\n std_init (float): initial std value\n weight_mu (nn.Parameter): mean value weight parameter\n weight_sigma (nn.Parameter): std value weight parameter\n bias_mu (nn.Parameter): mean value bias parameter\n bias_sigma (nn.Parameter): std value bias parameter\n \n \"\"\"\n\n def __init__(self, in_features: int, out_features: int, std_init: float = 0.5):\n \"\"\"Initialization.\"\"\"\n super(NoisyLinear, self).__init__()\n \n self.in_features = in_features\n self.out_features = out_features\n self.std_init = std_init\n\n self.weight_mu = nn.Parameter(torch.Tensor(out_features, in_features))\n self.weight_sigma = nn.Parameter(\n torch.Tensor(out_features, in_features)\n )\n self.register_buffer(\n \"weight_epsilon\", torch.Tensor(out_features, in_features)\n )\n\n self.bias_mu = nn.Parameter(torch.Tensor(out_features))\n self.bias_sigma = nn.Parameter(torch.Tensor(out_features))\n self.register_buffer(\"bias_epsilon\", torch.Tensor(out_features))\n\n self.reset_parameters()\n self.reset_noise()\n\n def reset_parameters(self):\n \"\"\"Reset trainable network parameters (factorized gaussian noise).\"\"\"\n mu_range = 1 / math.sqrt(self.in_features)\n self.weight_mu.data.uniform_(-mu_range, mu_range)\n self.weight_sigma.data.fill_(\n self.std_init / math.sqrt(self.in_features)\n )\n self.bias_mu.data.uniform_(-mu_range, mu_range)\n self.bias_sigma.data.fill_(\n self.std_init / math.sqrt(self.out_features)\n )\n\n def reset_noise(self):\n \"\"\"Make new noise.\"\"\"\n epsilon_in = self.scale_noise(self.in_features)\n epsilon_out = self.scale_noise(self.out_features)\n\n # outer product\n self.weight_epsilon.copy_(epsilon_out.ger(epsilon_in))\n self.bias_epsilon.copy_(epsilon_out)\n\n def forward(self, x: torch.Tensor) -> torch.Tensor:\n \"\"\"Forward method implementation.\n \n We don't use separate statements on train / eval mode.\n It doesn't show remarkable difference of performance.\n \"\"\"\n return F.linear(\n x,\n self.weight_mu + self.weight_sigma * self.weight_epsilon,\n self.bias_mu + self.bias_sigma * self.bias_epsilon,\n )\n \n @staticmethod\n def scale_noise(size: int) -> torch.Tensor:\n \"\"\"Set scale to make noise (factorized gaussian noise).\"\"\"\n x = torch.randn(size)\n\n return x.sign().mul(x.abs().sqrt())\n```\n\n## Noisy Network\n\nWe use NoisyLinear for the last two FC layers, and there is a method to reset noise at every step.\nThese are the only differences from the example of *01.dqn.ipynb*.\n\n\n```python\nclass Network(nn.Module):\n def __init__(self, in_dim: int, out_dim: int):\n \"\"\"Initialization.\"\"\"\n super(Network, self).__init__()\n\n self.feature = nn.Linear(in_dim, 128)\n self.noisy_layer1 = NoisyLinear(128, 128)\n self.noisy_layer2 = NoisyLinear(128, out_dim)\n\n def forward(self, x: torch.Tensor) -> torch.Tensor:\n \"\"\"Forward method implementation.\"\"\"\n feature = F.relu(self.feature(x))\n hidden = F.relu(self.noisy_layer1(feature))\n out = self.noisy_layer2(hidden)\n \n return out\n \n def reset_noise(self):\n \"\"\"Reset all noisy layers.\"\"\"\n self.noisy_layer1.reset_noise()\n self.noisy_layer2.reset_noise()\n```\n\n## DQN + NoisyNet Agent (w/o DuelingNet)\n\nHere is a summary of DQNAgent class.\n\n| Method | Note |\n| --- | --- |\n|select_action | select an action from the input state. |\n|step | take an action and return the response of the env. |\n|compute_dqn_loss | return dqn loss. |\n|update_model | update the model by gradient descent. |\n|target_hard_update| hard update from the local model to the target model.|\n|train | train the agent during num_frames. |\n|test | test the agent (1 episode). |\n|plot | plot the training progresses. |\n\nIn the paper, NoisyNet is used as a component of the Dueling Network Architecture, which includes Double-DQN and Prioritized Experience Replay. However, we don't implement them to simplify the tutorial. One thing to note is that NoisyNet is an alternertive to $\\epsilon$-greedy method, so all $\\epsilon$ related lines are removed. Please check all comments with *NoisyNet*.\n\n\n```python\nclass DQNAgent:\n \"\"\"DQN Agent interacting with environment.\n \n Attribute:\n env (gym.Env): openAI Gym environment\n memory (ReplayBuffer): replay memory to store transitions\n batch_size (int): batch size for sampling\n target_update (int): period for target model's hard update\n gamma (float): discount factor\n dqn (Network): model to train and select actions\n dqn_target (Network): target model to update\n optimizer (torch.optim): optimizer for training dqn\n transition (list): transition information including\n state, action, reward, next_state, done\n \"\"\"\n\n def __init__(\n self, \n env: gym.Env,\n memory_size: int,\n batch_size: int,\n target_update: int,\n gamma: float = 0.99,\n ):\n \"\"\"Initialization.\n \n Args:\n env (gym.Env): openAI Gym environment\n memory_size (int): length of memory\n batch_size (int): batch size for sampling\n target_update (int): period for target model's hard update\n gamma (float): discount factor\n \"\"\"\n # NoisyNet: All attributes related to epsilon are removed\n obs_dim = env.observation_space.shape[0]\n action_dim = env.action_space.n\n \n self.env = env\n self.memory = ReplayBuffer(obs_dim, memory_size, batch_size)\n self.batch_size = batch_size\n self.target_update = target_update\n self.gamma = gamma\n \n # device: cpu / gpu\n self.device = torch.device(\n \"cuda\" if torch.cuda.is_available() else \"cpu\"\n )\n print(self.device)\n\n # networks: dqn, dqn_target\n self.dqn = Network(obs_dim, action_dim).to(self.device)\n self.dqn_target = Network(obs_dim, action_dim).to(self.device)\n self.dqn_target.load_state_dict(self.dqn.state_dict())\n self.dqn_target.eval()\n \n # optimizer\n self.optimizer = optim.Adam(self.dqn.parameters())\n\n # transition to store in memory\n self.transition = list()\n \n # mode: train / test\n self.is_test = False\n\n def select_action(self, state: np.ndarray) -> np.ndarray:\n \"\"\"Select an action from the input state.\"\"\"\n # NoisyNet: no epsilon greedy action selection\n selected_action = self.dqn(\n torch.FloatTensor(state).to(self.device)\n ).argmax()\n selected_action = selected_action.detach().cpu().numpy()\n \n if not self.is_test:\n self.transition = [state, selected_action]\n \n return selected_action\n\n def step(self, action: np.ndarray) -> Tuple[np.ndarray, np.float64, bool]:\n \"\"\"Take an action and return the response of the env.\"\"\"\n next_state, reward, done, _ = self.env.step(action)\n\n if not self.is_test:\n self.transition += [reward, next_state, done]\n self.memory.store(*self.transition)\n \n return next_state, reward, done\n\n def update_model(self) -> torch.Tensor:\n \"\"\"Update the model by gradient descent.\"\"\"\n samples = self.memory.sample_batch()\n\n loss = self._compute_dqn_loss(samples)\n\n self.optimizer.zero_grad()\n loss.backward()\n self.optimizer.step()\n \n # NoisyNet: reset noise\n self.dqn.reset_noise()\n self.dqn_target.reset_noise()\n\n return loss.item()\n \n def train(self, num_frames: int, plotting_interval: int = 200):\n \"\"\"Train the agent.\"\"\"\n self.is_test = False\n \n state = self.env.reset()\n update_cnt = 0\n losses = []\n scores = []\n score = 0\n\n for frame_idx in range(1, num_frames + 1):\n action = self.select_action(state)\n next_state, reward, done = self.step(action)\n\n state = next_state\n score += reward\n \n # NoisyNet: removed decrease of epsilon\n\n # if episode ends\n if done:\n state = self.env.reset()\n scores.append(score)\n score = 0\n\n # if training is ready\n if len(self.memory) >= self.batch_size:\n loss = self.update_model()\n losses.append(loss)\n update_cnt += 1\n \n # if hard update is needed\n if update_cnt % self.target_update == 0:\n self._target_hard_update()\n\n # plotting\n if frame_idx % plotting_interval == 0:\n self._plot(frame_idx, scores, losses)\n \n self.env.close()\n \n def test(self) -> List[np.ndarray]:\n \"\"\"Test the agent.\"\"\"\n self.is_test = True\n \n state = self.env.reset()\n done = False\n score = 0\n \n frames = []\n while not done:\n frames.append(self.env.render(mode=\"rgb_array\"))\n action = self.select_action(state)\n next_state, reward, done = self.step(action)\n\n state = next_state\n score += reward\n \n print(\"score: \", score)\n self.env.close()\n \n return frames\n\n def _compute_dqn_loss(self, samples: Dict[str, np.ndarray]) -> torch.Tensor:\n \"\"\"Return dqn loss.\"\"\"\n device = self.device # for shortening the following lines\n state = torch.FloatTensor(samples[\"obs\"]).to(device)\n next_state = torch.FloatTensor(samples[\"next_obs\"]).to(device)\n action = torch.LongTensor(samples[\"acts\"].reshape(-1, 1)).to(device)\n reward = torch.FloatTensor(samples[\"rews\"].reshape(-1, 1)).to(device)\n done = torch.FloatTensor(samples[\"done\"].reshape(-1, 1)).to(device)\n \n # G_t = r + gamma * v(s_{t+1}) if state != Terminal\n # = r otherwise\n curr_q_value = self.dqn(state).gather(1, action)\n next_q_value = self.dqn_target(next_state).max(\n dim=1, keepdim=True\n )[0].detach()\n mask = 1 - done\n target = (reward + self.gamma * next_q_value * mask).to(self.device)\n\n # calculate dqn loss\n loss = F.smooth_l1_loss(curr_q_value, target)\n\n return loss\n\n def _target_hard_update(self):\n \"\"\"Hard update: target <- local.\"\"\"\n self.dqn_target.load_state_dict(self.dqn.state_dict())\n \n def _plot(\n self, \n frame_idx: int, \n scores: List[float], \n losses: List[float], \n ):\n \"\"\"Plot the training progresses.\"\"\"\n clear_output(True)\n plt.figure(figsize=(20, 5))\n plt.subplot(131)\n plt.title('frame %s. score: %s' % (frame_idx, np.mean(scores[-10:])))\n plt.plot(scores)\n plt.subplot(132)\n plt.title('loss')\n plt.plot(losses)\n plt.show()\n```\n\n## Environment\n\nYou can see the [code](https://github.com/openai/gym/blob/master/gym/envs/classic_control/cartpole.py) and [configurations](https://github.com/openai/gym/blob/master/gym/envs/__init__.py#L53) of CartPole-v0 from OpenAI's repository.\n\n\n```python\n# environment\nenv_id = \"CartPole-v0\"\nenv = gym.make(env_id)\nif IN_COLAB:\n env = gym.wrappers.Monitor(env, \"videos\", force=True)\n```\n\n## Set random seed\n\n\n```python\nseed = 777\n\ndef seed_torch(seed):\n torch.manual_seed(seed)\n if torch.backends.cudnn.enabled:\n torch.backends.cudnn.benchmark = False\n torch.backends.cudnn.deterministic = True\n\nnp.random.seed(seed)\nseed_torch(seed)\nenv.seed(seed)\n```\n\n\n\n\n [777]\n\n\n\n## Initialize\n\n\n```python\n# parameters\nnum_frames = 20000\nmemory_size = 10000\nbatch_size = 128\ntarget_update = 150\n\n# train\nagent = DQNAgent(env, memory_size, batch_size, target_update)\n```\n\n cpu\n\n\n## Train\n\n\n```python\nagent.train(num_frames)\n```\n\n## Test\n\nRun the trained agent (1 episode).\n\n\n```python\nframes = agent.test()\n```\n\n score: 200.0\n\n\n## Render\n\n\n```python\nif IN_COLAB: # for colab\n import base64\n import glob\n import io\n import os\n\n from IPython.display import HTML, display\n\n\n def ipython_show_video(path: str) -> None:\n \"\"\"Show a video at `path` within IPython Notebook.\"\"\"\n if not os.path.isfile(path):\n raise NameError(\"Cannot access: {}\".format(path))\n\n video = io.open(path, \"r+b\").read()\n encoded = base64.b64encode(video)\n\n display(HTML(\n data=\"\"\"\n \n \"\"\".format(encoded.decode(\"ascii\"))\n ))\n\n list_of_files = glob.glob(\"videos/*.mp4\")\n latest_file = max(list_of_files, key=os.path.getctime)\n print(latest_file)\n ipython_show_video(latest_file)\n \nelse: # for jupyter\n from matplotlib import animation\n from JSAnimation.IPython_display import display_animation\n from IPython.display import display\n\n\n def display_frames_as_gif(frames: List[np.ndarray]) -> None:\n \"\"\"Displays a list of frames as a gif, with controls.\"\"\"\n patch = plt.imshow(frames[0])\n plt.axis('off')\n\n def animate(i):\n patch.set_data(frames[i])\n\n anim = animation.FuncAnimation(\n plt.gcf(), animate, frames = len(frames), interval=50\n )\n display(display_animation(anim, default_mode='loop'))\n\n\n # display \n display_frames_as_gif(frames)\n```\n\n\n\n\n\n
    \n \n
    \n \n
    \n \n \n \n \n \n \n \n \n \n
    \n Once \n Loop \n Reflect \n
    \n
    \n\n\n\n\n\n", "meta": {"hexsha": "412f48baee42a8248049b787676dd8ce897b23c1", "size": 458044, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "05.noisy_net.ipynb", "max_stars_repo_name": "Yueyuhou/rainbow-is-all-you-need", "max_stars_repo_head_hexsha": "0d87577a48f0c270d953ff895561ec71ad5c026d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1332, "max_stars_repo_stars_event_min_datetime": "2019-06-30T00:19:01.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T12:33:14.000Z", "max_issues_repo_path": "05.noisy_net.ipynb", "max_issues_repo_name": "Yueyuhou/rainbow-is-all-you-need", "max_issues_repo_head_hexsha": "0d87577a48f0c270d953ff895561ec71ad5c026d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 37, "max_issues_repo_issues_event_min_datetime": "2019-07-02T16:43:12.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-13T14:07:04.000Z", "max_forks_repo_path": "05.noisy_net.ipynb", "max_forks_repo_name": "Yueyuhou/rainbow-is-all-you-need", "max_forks_repo_head_hexsha": "0d87577a48f0c270d953ff895561ec71ad5c026d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 291, "max_forks_repo_forks_event_min_datetime": "2019-07-04T04:22:21.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-31T02:37:01.000Z", "avg_line_length": 400.0384279476, "max_line_length": 41720, "alphanum_fraction": 0.9406367074, "converted": true, "num_tokens": 4776, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.4367259656549011}} {"text": "\u0412\u043e\u0434\u043e\u043f\u044c\u044f\u043d \u0410.\u041e. \u0425\u0430\u0431\u0438\u0431\u0443\u043b\u043b\u0438\u043d \u0420.\u0410. 2019 - 2022 \u0433.\n\n## \u0421\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435:\n* [1.1. \u0414\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u043d\u0435\u0444\u0442\u0438 \u0433\u0430\u0437\u043e\u043c](#Pb)\n * [1.1.1. \u0414\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f \u0421\u0442\u0435\u043d\u0434\u0438\u043d\u0433\u0430](#Pb_Standing)\n * [1.1.2. \u0414\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f \u0412\u0430\u043b\u043a\u043e \u041c\u0430\u043a\u043a\u0435\u0439\u043d\u0430](#Pb_Valco)\n* [1.2. \u0413\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435](\u2116Rs)\n * [1.2.1. \u0413\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435, \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f \u0421\u0442\u0435\u043d\u0434\u0438\u043d\u0433\u0430](#Rs_Standing)\n * [1.2.2. \u0413\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435, \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f \u0412\u0435\u043b\u0430\u0440\u0434\u0435-\u041c\u0430\u043a\u043a\u0435\u0439\u043d\u0430](#Rs_McCain)\n * [1.2.3. \u041e\u0446\u0435\u043d\u043a\u0430 \u043e\u0431\u044a\u0435\u043c\u0430 \u0440\u0430\u0441\u0442\u0432\u043e\u0440\u0435\u043d\u043d\u043e\u0433\u043e \u0433\u0430\u0437\u0430 \u0432 \u0441\u0435\u043f\u0430\u0440\u0430\u0442\u043e\u0440\u0435, \u0434\u043b\u044f \u0443\u0442\u043e\u0447\u043d\u0435\u043d\u0438\u044f \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u044f \u043f\u043e \u0437\u0430\u043c\u0435\u0440\u0430\u043c \u043d\u0430 \u0441\u0435\u043f\u0430\u0440\u0430\u0442\u043e\u0440\u0435. \u041a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f \u041c\u0430\u043a\u043a\u0435\u0439\u043d\u0430.](#Rsb_McCain)\n* [1.3. \u041e\u0431\u044a\u0435\u043c\u043d\u044b\u0439 \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442 \u043d\u0435\u0444\u0442\u0438](#FVF)\n * [1.3.1. \u041e\u0431\u044a\u0435\u043c\u043d\u044b\u0439 \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442 \u043d\u0435\u0444\u0442\u0438 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u0432\u044b\u0448\u0435 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f](#FVF_above_Pb)\n\n\n# 1. PVT \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043f\u043b\u0430\u0441\u0442\u043e\u0432\u044b\u0445 \u0444\u043b\u044e\u0438\u0434\u043e\u0432\n\n\n## 1.1. \u0414\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u043d\u0435\u0444\u0442\u0438 \u0433\u0430\u0437\u043e\u043c\n\n\n### 1.1.1. \u0414\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f \u0421\u0442\u0435\u043d\u0434\u0438\u043d\u0433\u0430\n\n\n\u041a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f \u0421\u0442\u0435\u043d\u0434\u0438\u043d\u0433\u0430 (Standing, 1947) \u0434\u043b\u044f \u043e\u0446\u0435\u043d\u043a\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u043d\u0435\u0444\u0442\u0438 \u0433\u0430\u0437\u043e\u043c. \n\n$$ P_b = 0.5197 \\left( \\frac{R_{sb}}{\\gamma_g}\\right)^{0.83} 10 ^{y_g} \\tag{1.1.1.1} $$\n\n\u0433\u0434\u0435\n\n$P_b$ - \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, $\u041c\u041f\u0430$ \n\n$R_{sb}$ - \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, $\u043c^3/\u043c^3 $\n\n$\\gamma_g$ - \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u0433\u0430\u0437\u0430 (\u043f\u043e \u0432\u043e\u0437\u0434\u0443\u0445\u0443), \u0431\u0435\u0437\u0440\u0430\u0437\u043c\u0435\u0440\u043d\u0430\u044f \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u0430 \n\n$y_g$ - \u043c\u043e\u043b\u044c\u043d\u0430\u044f \u0434\u043e\u043b\u044f \u0433\u0430\u0437\u0430, $ y_g = 1.225 +0.00164 T - \\frac{ 1.769}{\\gamma_o}$\n\n$\\gamma_o$ - \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u043d\u0435\u0444\u0442\u0438 (\u043f\u043e \u0432\u043e\u0434\u0435), \u0431\u0435\u0437\u0440\u0430\u0437\u043c\u0435\u0440\u043d\u0430\u044f \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u0430 \n\n$ T $ - \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u0430, $ ^{\\circ}\\mathrm{K}$\n\n\u041a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u0438 Standing \u0431\u0430\u0437\u0438\u0440\u0443\u044e\u0442\u0441\u044f \u043d\u0430 105 \u044d\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442\u0430\u043b\u044c\u043d\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u044b\u0445 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f\u0445 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u043d\u0435\u0444\u0442\u044f\u043d\u044b\u0445 \u0441\u0438\u0441\u0442\u0435\u043c \u041a\u0430\u043b\u0438\u0444\u043e\u0440\u043d\u0438\u0438. \u0414\u0438\u0430\u043f\u0430\u0437\u043e\u043d\u044b \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0439 \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0445 \u0441\u0432\u043e\u0439\u0441\u0442\u0432, \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u043d\u044b\u0445 \u0434\u043b\u044f \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u043a\u0438 \u0434\u0430\u043d\u043d\u043e\u0439 \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u0438, \u043f\u0440\u0438\u0432\u0435\u0434\u0435\u043d\u044b \u0432 \u0442\u0430\u0431\u043b\u0438\u0446\u0435 \u043d\u0438\u0436\u0435. \n\n|

    \u041f\u0430\u0440\u0430\u043c\u0435\u0442\u0440 | \u0414\u0438\u0430\u043f\u0430\u0437\u043e\u043d |\n| :--- | :--- |\n|

    \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f,$P_b$ , $ \u041c\u041f\u0430 $ | 0.896\u202648.263 |\n|

    \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u0430, $^{\\circ}\\mathrm{K} $ | 310\u2026400 |\n|

    \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, $R_{sb}$ , $\u043c^3/\u043c^3 $ | 3.6\u2026254 |\n|

    \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u043d\u0435\u0444\u0442\u0438 \u043f\u043e \u0432\u043e\u0434\u0435, $\\gamma_o$ | 0.725\u20260.956 |\n|

    \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u0433\u0430\u0437\u0430, $\\gamma_g$ | 0.59\u20260.95 |\n\n\n \n \nref \"A Pressure-Volume-Temperature Correlation for Mixtures of California Oil and Gases\", M.B. Standing, Drill. & Prod. Prac., API, 1947.\n\n\n```python\nimport sys \nsys.path.append('..')\nimport neftpy.upvt as pvt\nimport neftpy.uconvert as uc\n```\n\n\n```python\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport scipy.constants as const\nfrom sympy import *\ninit_printing()\n```\n\n\n```python\n# \u041a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f \u0421\u0442\u0435\u043d\u0434\u0438\u043d\u0433\u0430 \u0434\u043b\u044f \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u0440\u0435\u0430\u043b\u0438\u0437\u043e\u0432\u0430\u043d\u0430 \n# \u0432 \u0432\u0438\u0434\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 unf_pb_Standing_MPaa \u0432 \u043c\u043e\u0434\u0443\u043b\u0435 neftpy.upvt\n\npvt.unf_pb_Standing_MPaa(rsb_m3m3=100, gamma_oil=0.86, gamma_gas=0.6, t_K=350)\n```\n\n\n\n\n array(20.1702107)\n\n\n\n\u0432 \u043f\u0440\u0438\u0432\u0435\u0434\u0435\u043d\u043e\u043c \u043a\u043e\u0434\u0435 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0430 \u043a\u043e\u0440\u0440\u0435\u043a\u0446\u0438\u044f \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0439 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u043f\u0440\u0438 \u043d\u0438\u0437\u043a\u0438\u0445 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f\u0445 \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u044f \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u0434\u043b\u044f \u043e\u0431\u0435\u0441\u043f\u0435\u0447\u0435\u043d\u0438\u044f \u0432\u044b\u0445\u043e\u0434\u0430 \u043d\u0430 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 $P_b = 1$ \u043f\u0440\u0438 $R{sb} = 0$\n\n\n\n\n\n```python\n# \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u044b \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u044f\u044e\u0449\u0438\u0435 \u0434\u0438\u0430\u043f\u0430\u0437\u043e\u043d\u044b \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0439 \u0434\u043b\u044f \u043f\u043e\u0441\u0442\u0440\u043e\u0435\u043d\u0438\u044f \u0433\u0440\u0430\u0444\u0438\u043a\u043e\u0432\nrsb_set=np.arange(1,300,10)\n\nt_set=np.arange(273,380,30)\nt_set_def=np.array([313])\n\ngg_set=np.arange(0.6,1,0.1)\ngg_set_def=np.array([0.8])\n\ngo_set=np.arange(0.8,1,0.05)\ngo_set_def=np.array([0.86])\n\n# \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0430\u0441\u0442\u0440\u043e\u0439\u043a\u0438 \u0433\u0440\u0430\u0444\u0438\u043a\u043e\u0432 \u043f\u043e \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044e \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u043e\u0442 \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u044f\ndef plot_pb_rsb(plt, func, \n tset, goset, ggset, \n plot_title, plot_xlab, plot_ylab):\n for t in tset:\n for gg in ggset:\n for go in goset: \n plt.plot(rsb_set, \n func(rsb_set,t_K = t,gamma_gas = gg,gamma_oil = go), \n label='t = %1.0f $ ^{\\circ}\\mathrm{K}$'%t +\n ' $\\gamma_g$ = %1.2f'%gg + \n ' $\\gamma_o$ = %1.2f'%go\n )\n plt.title(plot_title)\n plt.ylabel(plot_ylab, color = 'black')\n plt.xlabel(plot_xlab, color = 'black')\n plt.legend()\n```\n\n\n```python\n# \u043a\u043e\u0434 \u0434\u043b\u044f \u043f\u043e\u0441\u0442\u0440\u043e\u0435\u043d\u0438\u044f \u0433\u0440\u0430\u0444\u0438\u043a\u043e\u0432\nplt.figure(figsize=(15, 8))\n# \u0440\u0438\u0441\u0443\u0435\u043c \u043f\u0435\u0440\u0432\u044b\u0439 \u0433\u0440\u0430\u0444\u0438\u043a\nplt.subplot(221)\nplot_pb_rsb(plt, pvt.unf_pb_Standing_MPaa, \n t_set, go_set_def, gg_set_def,\n '\u0414\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u043e\u0442 \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u044f',\n '$R_{sb}, \u043c^3/\u043c^3$',\n '$P_b, MPa$')\nplt.grid()\n# \u0440\u0438\u0441\u0443\u0435\u043c \u0432\u0442\u043e\u0440\u043e\u0439 \u0433\u0440\u0430\u0444\u0438\u043a\nplt.subplot(222)\nplot_pb_rsb(plt, pvt.unf_pb_Standing_MPaa, \n t_set_def, go_set, gg_set_def,\n '\u0414\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u043e\u0442 \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u044f',\n '$R_{sb}, \u043c^3/\u043c^3$',\n '$P_b, MPa$')\nplt.grid()\n# \u0440\u0438\u0441\u0443\u0435\u043c \u0442\u0440\u0435\u0442\u0438\u0439 \u0433\u0440\u0430\u0444\u0438\u043a\nplt.subplot(223)\nplot_pb_rsb(plt, pvt.unf_pb_Standing_MPaa, \n t_set_def, go_set_def, gg_set,\n '\u0414\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u043e\u0442 \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u044f',\n '$R_{sb}, \u043c^3/\u043c^3$',\n '$P_b, MPa$')\n\nplt.subplots_adjust(top=0.92, bottom=0.08, left=0.10, right=0.95, hspace=0.25, wspace=0.35)\n# \u0440\u0438\u0441\u0443\u0435\u043c \u0432\u0441\u0435\nplt.grid()\nplt.show()\n```\n\n---\n\n### 1.1.2. \u0414\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f Valko McCain \n\n\n\u041a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f Valco McCain (2003) \u0434\u043b\u044f \u043e\u0446\u0435\u043d\u043a\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u043d\u0435\u0444\u0442\u0438 \u0433\u0430\u0437\u043e\u043c \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u0430\u043d\u0430 \u043d\u0430 \u043e\u0441\u043d\u043e\u0432\u0435 \u0431\u0430\u043d\u043a\u0430 \u0434\u0430\u043d\u043d\u044b\u0445 \u043d\u0435\u0444\u0442\u0435\u0439 \u0441\u043e \u0432\u0441\u0435\u0433\u043e \u043c\u0438\u0440\u0430. \u041d\u0430 \u0440\u0438\u0441\u0443\u043d\u043a\u0435 \u043f\u043e\u043a\u0430\u0437\u0430\u043d\u044b \u0438\u0441\u0442\u043e\u0447\u043d\u0438\u043a\u0438 \u0434\u0430\u043d\u043d\u044b\u0445, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u0431\u044b\u043b\u0438 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u044b \u0430\u0432\u0442\u043e\u0440\u0430\u043c\u0438 \u0434\u043b\u044f \u043d\u0430\u0441\u0442\u0440\u043e\u0439\u043a\u0438 \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u0438. \n\n\n\n\n\u0414\u0438\u0430\u043f\u0430\u0437\u043e\u043d\u044b \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0439 \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0445 \u0441\u0432\u043e\u0439\u0441\u0442\u0432 (1745 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0439), \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u043d\u044b\u0445 \u0434\u043b\u044f \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u043a\u0438 \u0434\u0430\u043d\u043d\u043e\u0439 \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u0438, \u043f\u0440\u0438\u0432\u0435\u0434\u0435\u043d\u044b \u0432 \u0442\u0430\u0431\u043b\u0438\u0446\u0435 \u043d\u0438\u0436\u0435. \n\n|

    \u041f\u0430\u0440\u0430\u043c\u0435\u0442\u0440 | \u041c\u0438\u043d|\u0421\u0440\u0435\u0434\u043d\u0435\u0435|\u041c\u0430\u043a\u0441|\n| :--- | :---: |:---:|:---:|\n|

    \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f,$P_b$ , $ \u041c\u041f\u0430 $ | 0.55 |15.0|45.5|\n|

    \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u0430, $^{\\circ}\\mathrm{\u0421} $ | 15 |85|172|\n|

    \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, $R_{sb}$ , $\u043c^3/\u043c^3 $ | 2 |104|395|\n|

    \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u043d\u0435\u0444\u0442\u0438 \u043f\u043e \u0432\u043e\u0434\u0435, $\\gamma_o$ | 0.724 |0.846|1.02|\n|

    \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u0433\u0430\u0437\u0430 \u043d\u0430 \u0441\u0435\u043f\u0430\u0440\u0430\u0442\u043e\u0440\u0435, $\\gamma_g$ | 0.555 |0.838|1.685|\n \n\u041f\u043e \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u0430\u043c \u0441\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0439 \u0441 \u0437\u0430\u043c\u0435\u0440\u0435\u043d\u043d\u044b\u043c\u0438 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f\u043c\u0438 \u0430\u0431\u0441\u043e\u043b\u044e\u0442\u043d\u0430\u044f \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u0441\u0440\u0435\u0434\u043d\u0435\u0439 \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043e\u0448\u0438\u0431\u043a\u0438 (AARE) \u0434\u043b\u044f \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u0438 \u0441\u043e\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u0442 \u043e\u043a\u043e\u043b\u043e 11%. \u0410\u0432\u0442\u043e\u0440\u044b \u043e\u0442\u043c\u0435\u0447\u0430\u044e\u0442, \u0447\u0442\u043e \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u043d\u0430\u044f \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u044c \u0441\u043e\u043e\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0443\u0435\u0442 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u0438 \u0437\u0430\u043c\u0435\u0440\u043e\u0432 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u043d\u044b\u0445 \u0434\u043b\u044f \u043f\u043e\u0441\u0442\u0440\u043e\u0435\u043d\u0438\u044f \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u0438 \u0438 \u0434\u043b\u044f \u043f\u043e\u0441\u0442\u0440\u043e\u0435\u043d\u0438\u044f \u0431\u043e\u043b\u0435\u0435 \u0442\u043e\u0447\u043d\u044b\u0445 \u0437\u0430\u0432\u0438\u0441\u0438\u043c\u043e\u0441\u0442\u0435\u0439 \u043f\u043e\u0442\u0440\u0435\u0431\u0443\u0435\u0442\u0441\u044f \u0441\u0431\u043e\u0440 \u043d\u043e\u0432\u044b\u0445 \u0434\u0430\u043d\u043d\u044b\u0445 \u0441 \u043f\u043e\u0432\u044b\u0448\u0435\u043d\u043d\u043e\u0439 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u044c\u044e.\n\n$$\nln P_b = 7.475 + 0.713 z + 0.0075 z^2 \\tag{1.1.2.1}\n$$\n\u0433\u0434\u0435 \n\n$$\nz = z_1 + z_2 + z_3 + z_4\n$$\n$$\nz_1 = -5.48 - 0.0375\\cdot ln R_{sb}+0.281\\cdot (ln R_{sb})^2 - 0.0206\\cdot (ln R_{sb})^3\n$$\n$$\nz_2 = 1.27 - 0.0449\\cdot API +4.36 \\cdot 10^{-4} API^2 -4.76 \\cdot 10^{-6} API^3\n$$\n$$\nz_3 = 4.51 - 10.84 \\cdot \\gamma_{gSP} +8.39\\cdot \\gamma_{gSP}^2 -2.34\\cdot \\gamma_{gSP}^3\n$$\n$$\nz_4 = -0.7835 + 6.23 \\cdot 10^{-3} \\cdot T_R - 1.22 \\cdot 10^{-5} \\cdot T_R^2+ 1.03 \\cdot 10^{-8} \\cdot T_R^3\n$$\n\n\u0433\u0434\u0435\n\n* $P_b$ - \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, $psia$\n* $R_{sb}$ - \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, ${scf}/{STB}$\n* $\\gamma_{gSP}$ - \u0443\u0434\u0435\u043b\u044c\u043d\u0430\u044f \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u0433\u0430\u0437\u0430, \u043e\u0442\u043e\u0431\u0440\u0430\u043d\u043d\u043e\u0433\u043e \u043d\u0430 \u0441\u0435\u043f\u0430\u0440\u0430\u0442\u043e\u0440\u0435, \u0431\u0435\u0437\u0440\u0430\u0437\u043c\u0435\u0440\u043d\u0430\u044f \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u0430\n* $T_R$ - \u043f\u043b\u0430\u0441\u0442\u043e\u0432\u0430\u044f \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u0430, $F$\n\n\nref Reservoir oil bubblepoint pressures revisited; solution gas-oil ratios and surface gas specific gravities. P.P.Valko, W.D.McCain Jr. Journal of petroleum science and engineering 37(2003) 153-169\n\n---\n#### \u041f\u0440\u0435\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u0435\u0434\u0438\u043d\u0438\u0446 \u0438\u0437\u043c\u0435\u0440\u0435\u043d\u0438\u044f \u0434\u043b\u044f \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u0438 \u0412\u0430\u043b\u043a\u043e \u041c\u0430\u043a\u043a\u0435\u0439\u043d\u0430\n\n\n```python\n# \u043e\u0431\u044a\u044f\u0432\u043b\u0435\u043d\u0438\u044f \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u044b\u0445 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u044b\u0445 \u0434\u043b\u044f \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u0435\u0434\u0438\u043d\u0438\u0446 \u0432 \u0432\u044b\u0440\u043e\u0436\u0435\u043d\u0438\u0438\nrsb_scfSTB, rsb_m3m3 = symbols('R_sb[scfSTB] R_sb[m3m3]')\nAPI, gamma_o = symbols('API gamma_o')\ngamma_gSP = symbols('gamma_gSP')\nT_RF,T_RK = symbols('T_R[F] T_R[K]')\nz,z1,z2,z3,z4 = symbols('z,z1,z2,z3,z4')\np_bpsia, p_bMPaa = symbols('p_b[psia],p_b[MPaa]')\n```\n\n\n```python\n# \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0430 \u0440\u0430\u0441\u0447\u0435\u0442\u0430 \u0432 \u0430\u043c\u0435\u0440\u0438\u043a\u0430\u043d\u0441\u043a\u0438\u0445 \u043f\u0440\u043e\u043c\u044b\u0441\u043b\u043e\u0432\u044b\u0445 \u0435\u0434\u0438\u043d\u0438\u0446\u0430\u0445\neq1 = Eq(z,z1+z2+z3+z4)\neq2 = Eq(z1, -5.48 - 0.03758 * ln(rsb_scfSTB)+ 0.281* ln(rsb_scfSTB)**2 - 0.0206* ln(rsb_scfSTB)**3)\neq3 = Eq(z2, 1.27 - 0.0449* API +4.36 * 10**-4 *API**2 -4.76 * 10**-6 *API**3)\neq4 = Eq(z3, 4.51- 10.84 *gamma_gSP +8.39*gamma_gSP**2 -2.34*gamma_gSP**3 )\neq5 = Eq(z4, -0.7835 + 6.23 * 10**-3 * T_RF - 1.22 * 10**-5 * T_RF**2+ 1.03 * 10**-8 * T_RF**3)\neq6 =Eq(ln(p_bpsia),(7.475 + 0.713 * z + 0.0075 * z**2))\n# \u043f\u043e\u043a\u0430\u0436\u0435\u043c \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u0432 \u043f\u0435\u0447\u0430\u0442\u043d\u043e\u043c \u0432\u0438\u0434\u0435\ndisplay(eq6)\ndisplay(eq1)\ndisplay(eq2)\ndisplay(eq3)\ndisplay(eq4)\ndisplay(eq5)\n```\n\n\n```python\n# \u041d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043f\u0435\u0440\u0435\u0432\u043e\u0434\u043d\u044b\u0435 \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u044b \nprint(uc.m3m3_2_scfstb(1))\nprint(uc.MPa_2_psi(1))\n```\n\n 5.614583333333334\n 145.03773773020922\n\n\n\n```python\n# \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u0434\u043b\u044f \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u0435\u0434\u0438\u043d\u0438\u0446 \u0438\u0437\u043c\u0435\u0440\u0435\u043d\u0438\u044f \u0438\u0437 \u0430\u043c\u0435\u0440\u0438\u043a\u0430\u043d\u0441\u043a\u0438\u0445 \u043f\u0440\u043e\u043c\u044b\u0441\u043b\u043e\u0432\u044b\u0445 \u0432 \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0435\nscfSTB_to_m3m3 = rsb_m3m3 * uc.m3m3_2_scfstb(1)\n#API_to_gamma_o = 141.5 / gamma_o - 131.5\nAPI_to_gamma_o = uc.gamma_oil_2_api(gamma_o)\n#F_to_K = T_RK * 9 / 5 - 459.67\nF_to_K = uc.K_2_F(T_RK)\npsi_to_MPa = p_bMPaa * uc.MPa_2_psi(1)\n# \u043f\u043e\u043a\u0430\u0436\u0435\u043c \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u0432 \u043f\u0435\u0447\u0430\u0442\u043d\u043e\u043c \u0432\u0438\u0434\u0435\ndisplay(Eq(rsb_scfSTB , scfSTB_to_m3m3))\ndisplay(Eq(API,API_to_gamma_o))\ndisplay(Eq(T_RF,F_to_K))\ndisplay(Eq(p_bpsia,psi_to_MPa))\n```\n\n\n```python\n# \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0430 \u0432 \u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0435\u0434\u0438\u043d\u0438\u0446\u044b \u0441 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u043c \u0441\u0438\u043c\u0432\u043e\u043b\u044c\u043d\u044b\u0445 \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0439\neq2_m=simplify(eq2.subs(rsb_scfSTB,scfSTB_to_m3m3))\neq3_m=simplify(eq3.subs(API,API_to_gamma_o))\neq5_m=simplify(eq5.subs(T_RF,F_to_K))\neq6_m=eq6.subs(p_bpsia, psi_to_MPa)\neq8=solve(eq6_m,p_bMPaa)\neq9=Eq(p_bMPaa, eq8[0])\n# \u0432\u044b\u0432\u043e\u0434 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u043e\u0432 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0439\ndisplay(eq9)\ndisplay(eq1)\ndisplay(eq2_m)\ndisplay(eq3_m)\ndisplay(eq4)\ndisplay(eq5_m)\n```\n\n\n```python\npvt.unf_pb_Valko_MPaa(rsb_m3m3 = 100, gamma_oil=0.86, gamma_gas=0.6, t_K=350)\n```\n\n\n\n\n array(23.29991091)\n\n\n\n\n```python\nplt.figure(figsize=(15,8))\nf = pvt.unf_pb_Valko_MPaa\n# \u0440\u0438\u0441\u0443\u0435\u043c \u043f\u0435\u0440\u0432\u044b\u0439 \u0433\u0440\u0430\u0444\u0438\u043a\nplt.subplot(221)\nplot_pb_rsb(plt, pvt.unf_pb_Valko_MPaa, \n t_set,go_set_def,gg_set_def,\n '\u0414\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u043e\u0442 \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u044f',\n '$R_{sb}, \u043c^3/\u043c^3$',\n '$P_b, MPa$')\nplt.grid()\n# \u0440\u0438\u0441\u0443\u0435\u043c \u0432\u0442\u043e\u0440\u043e\u0439 \u0433\u0440\u0430\u0444\u0438\u043a\nplt.subplot(222)\nplot_pb_rsb(plt, pvt.unf_pb_Valko_MPaa, \n t_set_def,go_set,gg_set_def,\n '\u0414\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u043e\u0442 \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u044f',\n '$R_{sb}, \u043c^3/\u043c^3$',\n '$P_b, MPa$')\nplt.grid()\n# \u0440\u0438\u0441\u0443\u0435\u043c \u0442\u0440\u0435\u0442\u0438\u0439 \u0433\u0440\u0430\u0444\u0438\u043a\nplt.subplot(223)\nplot_pb_rsb(plt, pvt.unf_pb_Valko_MPaa, \n t_set_def,go_set_def,gg_set,\n '\u0414\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u043e\u0442 \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u044f',\n '$R_{sb}, \u043c^3/\u043c^3$',\n '$P_b, MPa$')\n\nplt.subplots_adjust(top=0.92, bottom=0.08, left=0.10, right=0.95, hspace=0.25, wspace=0.35)\n# \u0440\u0438\u0441\u0443\u0435\u043c \u0432\u0441\u0435\nplt.grid()\nplt.show()\n```\n\n\u0432 \u043f\u0440\u0438\u0432\u0435\u0434\u0435\u043d\u043e\u043c \u043a\u043e\u0434\u0435 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0430 \u043a\u043e\u0440\u0440\u0435\u043a\u0446\u0438\u044f \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0439 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u043f\u0440\u0438 \u043d\u0438\u0437\u043a\u0438\u0445 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f\u0445 \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u044f \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u0434\u043b\u044f \u043e\u0431\u0435\u0441\u043f\u0435\u0447\u0435\u043d\u0438\u044f \u0432\u044b\u0445\u043e\u0434\u0430 \u043d\u0430 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 $P_b = 1$ \u043f\u0440\u0438 $R{sb} = 0$ \u0438 \u043f\u0440\u0438 \u0431\u043e\u043b\u044c\u0448\u0438\u0445 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f\u0445 \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u044f\n\n\n\n\u0441\u043b\u0435\u0434\u0443\u0435\u0442 \u043e\u0442\u043c\u0435\u0442\u0438\u0442\u044c, \u0447\u0442\u043e \u0432 \u043e\u0442\u043b\u0438\u0447\u0438\u0438 \u043e\u0442 \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u0439 \u0442\u0438\u043f\u0430 \u0421\u0442\u0435\u043d\u0434\u0438\u043d\u0433\u0430 \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f \u0412\u0430\u043b\u043a\u043e \u041c\u0430\u043a\u0435\u0439\u043d\u0430 \u0445\u043e\u0440\u043e\u0448\u043e \u043e\u043f\u0438\u0441\u044b\u0432\u0430\u0435\u0442 \u0438\u0441\u0445\u043e\u0434\u043d\u044b\u0439 \u043d\u0430\u0431\u043e\u0440 \u0434\u0430\u043d\u043d\u044b\u0445 \u0432 \u043f\u0440\u0435\u0434\u0435\u043b\u0430\u0445 \u043e\u0431\u043b\u0430\u0441\u0442\u0438 \u043f\u0440\u0438\u043c\u0435\u043d\u0438\u043c\u043e\u0441\u0442\u0438, \u043d\u043e \u0434\u0430\u0435\u0442 \u043d\u0435\u0444\u0438\u0437\u0438\u0447\u043d\u044b\u0435 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u044b \u0437\u0430 \u043f\u0440\u0435\u0434\u0435\u043b\u0430\u043c\u0438 \u0434\u0438\u0430\u043f\u0430\u0437\u043e\u043d\u0430 \u043f\u0440\u0438\u043c\u0435\u043d\u0438\u043c\u043e\u0441\u0442\u0438. \u041f\u0440\u0438\u0432\u0435\u0434\u0435\u043d\u043d\u0430\u044f \u0432 \u043a\u043e\u0434\u0435 \u043a\u043e\u0440\u0440\u0435\u043a\u0442\u0438\u0440\u043e\u0432\u043a\u0435 \u043c\u043e\u0436\u0435\u0442 \u0447\u0430\u0441\u0442\u0438\u0447\u043d\u043e \u0441\u0433\u043b\u0430\u0434\u0438\u0442\u044c \u044d\u043a\u0441\u043f\u0440\u0430\u043f\u043e\u043b\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u044b\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f, \u043d\u043e \u043b\u0443\u0447\u0448\u0435 \u043f\u0440\u0438 \u043f\u0440\u043e\u0432\u0435\u0434\u0435\u043d\u0438\u0438 \u0440\u0430\u0441\u0447\u0435\u0442\u043e\u0432 \u043a\u043e\u043d\u0442\u0440\u043e\u043b\u0438\u0440\u043e\u0432\u0430\u0442\u044c, \u0447\u0442\u043e\u0431\u044b \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u043b\u0430\u0441\u044c \u0432 \u043f\u0440\u0435\u0434\u0435\u043b\u0430\u0445 \u0434\u0438\u0430\u043f\u0430\u0437\u043e\u043d\u0430 \u043f\u0440\u0438\u043c\u0435\u043c\u043e\u0441\u0442\u0438. \n\n## 1.2. \u0413\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435\n\n\n\u0413\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 $R_s$ - \u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u0440\u0430\u0441\u0442\u0432\u043e\u0440\u0435\u043d\u043d\u043e\u0433\u043e \u0433\u0430\u0437\u0430 \u0432 \u043d\u0435\u0444\u0442\u0438 \u043f\u0440\u0438 \u0437\u0430\u0434\u0430\u043d\u043d\u044b\u0445 \u0442\u0435\u0440\u043c\u043e\u0431\u0430\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u0445 (solution gas ratio). \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u044f\u0435\u0442\u0441\u044f \u043a\u0430\u043a \u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435 \u043e\u0431\u044a\u0435\u043c\u0430 \u0433\u0430\u0437\u0430 \u0432\u044b\u0434\u0435\u043b\u0438\u0432\u0448\u0435\u0433\u043e\u0441\u044f \u0438\u0437 \u043d\u0435\u0444\u0442\u0438 \u043f\u0440\u0438 \u043f\u0440\u0438\u0432\u0435\u0434\u0435\u043d\u0438\u0438 \u0435\u0435 \u043a \u0441\u0442\u0430\u043d\u0434\u0430\u0440\u0442\u043d\u044b\u043c \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u043c \u043a \u043e\u0431\u044a\u0435\u043c\u0443 \u043e\u0441\u0442\u0430\u0432\u0448\u0435\u0439\u0441\u044f \u043d\u0435\u0444\u0442\u0438 \u0432 \u0441\u0442\u0430\u043d\u0434\u0430\u0440\u0442\u043d\u044b\u0445 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u0445 \n\n### 1.2.1 \u0413\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435, \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f \u0421\u0442\u0435\u043d\u0434\u0438\u043d\u0433\u0430\n\n\n\u0414\u043b\u044f \u0440\u0430\u0441\u0447\u0435\u0442\u0430 \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u044f \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u0442\u0441\u044f \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f, \u043e\u0431\u0440\u0430\u0442\u043d\u0430\u044f \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u0438 \u0421\u0442\u0435\u043d\u0434\u0438\u043d\u0433\u0430 (1.1.1.1) \u0434\u043b\u044f \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u043d\u0435\u0444\u0442\u0438 \u0433\u0430\u0437\u043e\u043c. \n\n$$ R_s = \\gamma_g \\left( \\frac{1.92 p}{\\ 10^{y_g}}\\right)^{1.204} \\tag{1.2.1.1} $$\n\n\u0433\u0434\u0435:\n\n$R_s$ - \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435, $\u043c^3/\u043c^3 $\n\n$P$ - \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0435, $\u041c\u041f\u0430$ \n\n$\\gamma_g$ - \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u0433\u0430\u0437\u0430, \u0431\u0435\u0437\u0440\u0430\u0437\u043c\u0435\u0440\u043d\u0430\u044f \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u0430 \n\n$y_g$ - \u043c\u043e\u043b\u044c\u043d\u0430\u044f \u0434\u043e\u043b\u044f \u0433\u0430\u0437\u0430, $ y_g = 1.225 +0.00164 T - \\dfrac{ 1.769}{\\gamma_o}$\n\n$\\gamma_o$ - \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u043d\u0435\u0444\u0442\u0438, \u0431\u0435\u0437\u0440\u0430\u0437\u043c\u0435\u0440\u043d\u0430\u044f \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u0430 \n\n$ T $ - \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u0430, $ ^{\\circ}\\mathrm{K}$\n\n\u0413\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043e\u0434\u043d\u0438\u043c \u0438\u0437 \u043a\u043b\u044e\u0447\u0435\u0432\u044b\u0445 \u0441\u0432\u043e\u0439\u0441\u0442\u0432 \u043d\u0435\u0444\u0442\u0438 \u043f\u0440\u0438 \u0440\u0430\u0441\u0447\u0451\u0442\u0430\u0445 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u0438 \u0441\u043a\u0432\u0430\u0436\u0438\u043d \u0438 \u0440\u0430\u0431\u043e\u0442\u044b \u0441\u043a\u0432\u0430\u0436\u0438\u043d\u043d\u043e\u0433\u043e \u043e\u0431\u043e\u0440\u0443\u0434\u043e\u0432\u0430\u043d\u0438\u044f. \u0414\u0438\u043d\u0430\u043c\u0438\u043a\u0430 \u0438\u0437\u043c\u0435\u043d\u0435\u043d\u0438\u044f \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u044f \u0432\u043e \u043c\u043d\u043e\u0433\u043e\u043c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u044f\u0435\u0442 \u043a\u043e\u043b\u0438\u0447\u0435\u0441\u0442\u0432\u043e \u0441\u0432\u043e\u0431\u043e\u0434\u043d\u043e\u0433\u043e \u0433\u0430\u0437\u0430 \u0432 \u043f\u043e\u0442\u043e\u043a\u0435 \u0438 \u0434\u043e\u043b\u0436\u043d\u0430 \u0443\u0447\u0438\u0442\u044b\u0432\u0430\u0442\u044c\u0441\u044f \u043f\u0440\u0438 \u043f\u0440\u043e\u0432\u0435\u0434\u0435\u043d\u0438\u0438 \u0440\u0430\u0441\u0447\u0451\u0442\u043e\u0432.\n\n\u0415\u0441\u043b\u0438 \u0438\u0437\u0432\u0435\u0441\u0442\u043d\u043e \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, \u0442\u043e \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f\u0445 \u043d\u0438\u0436\u0435 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u043c\u043e\u0436\u0435\u0442 \u0431\u044b\u0442\u044c \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u043e \u0438\u0437 \u043f\u0440\u043e\u043f\u043e\u0440\u0446\u0438\u0438:\n\n$$ R_s = \\ R_{sb}\\left( \\frac{P}{\\ P_b}\\right)^{1.204} \\tag{1.2.1.2} $$\n\n\u0433\u0434\u0435:\n\n$R_s$ - \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435, $\u043c^3/\u043c^3 $\n\n$P$ - \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0435, $\u041c\u041f\u0430$ \n\n$P_b$ - \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, $\u041c\u041f\u0430$\n\n$R_{sb}$ - \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, $\u043c^3/\u043c^3 $\n\n\u041a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u0438 Standing \u0431\u0430\u0437\u0438\u0440\u0443\u044e\u0442\u0441\u044f \u043d\u0430 105 \u044d\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442\u0430\u043b\u044c\u043d\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u044b\u0445 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f\u0445 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u043d\u0435\u0444\u0442\u044f\u043d\u044b\u0445 \u0441\u0438\u0441\u0442\u0435\u043c \u041a\u0430\u043b\u0438\u0444\u043e\u0440\u043d\u0438\u0438. \u0414\u0438\u0430\u043f\u0430\u0437\u043e\u043d\u044b \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0439 \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0445 \u0441\u0432\u043e\u0439\u0441\u0442\u0432, \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u043d\u044b\u0445 \u0434\u043b\u044f \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u043a\u0438 \u0434\u0430\u043d\u043d\u043e\u0439 \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u0438, \u043f\u0440\u0438\u0432\u0435\u0434\u0435\u043d\u044b \u0432 \u0442\u0430\u0431\u043b\u0438\u0446\u0435 \u043d\u0438\u0436\u0435. \n\n|

    \u041f\u0430\u0440\u0430\u043c\u0435\u0442\u0440 | \u0414\u0438\u0430\u043f\u0430\u0437\u043e\u043d |\n| :--- | :--- |\n|

    \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f,$P_b$ , $ \u041c\u041f\u0430 $ | 0.896\u202648.263 |\n|

    \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u0430, $^{\\circ}\\mathrm{K} $ | 310\u2026400 |\n|

    \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, $R_{sb}$ , $\u043c^3/\u043c^3 $ | 3.6\u2026254 |\n|

    \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u043d\u0435\u0444\u0442\u0438 \u043f\u043e \u0432\u043e\u0434\u0435, $\\gamma_o$ | 0.725\u20260.956 |\n|

    \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u0433\u0430\u0437\u0430, $\\gamma_g$ | 0.59\u20260.95 |\n\n\nref \"A Pressure-Volume-Temperature Correlation for Mixtures of California Oil and Gases\", M.B. Standing, Drill. & Prod. Prac., API, 1947.\n\n\n```python\npvt.unf_rs_Standing_m3m3(p_MPaa=3, pb_MPaa=10, rsb_m3m3=130, gamma_oil=0.86, gamma_gas=0.6, t_K=350)\n```\n\n\n\n\n array(30.50684834)\n\n\n\n\n```python\nnp.zeros_like(1.2)\n```\n\n\n\n\n array(0.)\n\n\n\n\n```python\n# \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u044b \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u044f\u044e\u0449\u0438\u0435 \u0434\u0438\u0430\u043f\u0430\u0437\u043e\u043d\u044b \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0439 \u0434\u043b\u044f \u043f\u043e\u0441\u0442\u0440\u043e\u0435\u043d\u0438\u044f \u0433\u0440\u0430\u0444\u0438\u043a\u043e\u0432\np_set=np.linspace(0.11,11,100)\nt_set=np.arange(294,400,30)\nt_set_def=np.array([313])\ngg_set=np.arange(0.6,1,0.1)\ngg_set_def=np.array([0.8])\ngo_set=np.arange(0.8,1,0.05)\ngo_set_def=np.array([0.86])\n# \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0434\u043b\u044f \u0430\u0432\u0442\u043e\u043c\u0430\u0442\u0438\u0437\u0430\u0446\u0438\u0438 \u043f\u043e\u0441\u0442\u0440\u043e\u0435\u043d\u0438\u044f \u0433\u0440\u0430\u0444\u0438\u043a\u043e\u0432 \u043f\u043e \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044e \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f\ndef plot_rs_p(plt, func, \n tset, goset, ggset,\n plot_title, plot_xlab, plot_ylab):\n for t in tset:\n for gg in ggset:\n for go in goset:\n plt.plot(p_set, func(p_set, t_K = t, gamma_gas = gg, gamma_oil = go), \n label='t = %1.0f $ ^{\\circ}\\mathrm{K}$'%t +\n ' $\\gamma_g$ = %1.2f'%gg + \n ' $\\gamma_o$ = %1.2f'%go\n )\n plt.title(plot_title)\n plt.ylabel(plot_ylab, color = 'black')\n plt.xlabel(plot_xlab, color = 'black')\n plt.legend()\n```\n\n\n```python\n# \u043a\u043e\u0434 \u0434\u043b\u044f \u043f\u043e\u0441\u0442\u0440\u043e\u0435\u043d\u0438\u044f \u0433\u0440\u0430\u0444\u0438\u043a\u043e\u0432\nplt.figure(figsize=(15,8))\nf = pvt.unf_rs_Standing_m3m3\n# \u0440\u0438\u0441\u0443\u0435\u043c \u043f\u0435\u0440\u0432\u044b\u0439 \u0433\u0440\u0430\u0444\u0438\u043a\nplt.subplot(221)\nplot_rs_p(plt, pvt.unf_rs_Standing_m3m3,\n t_set,go_set_def,gg_set_def,\n '\u0413\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043e\u0442 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f',\n '$P, MPa$',\n '$R_s, \u043c^3/\u043c^3$')\nplt.grid()\n# \u0440\u0438\u0441\u0443\u0435\u043c \u0432\u0442\u043e\u0440\u043e\u0439 \u0433\u0440\u0430\u0444\u0438\u043a\nplt.subplot(222)\nplot_rs_p(plt, pvt.unf_rs_Standing_m3m3,\n t_set_def,go_set,gg_set_def,\n '\u0413\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043e\u0442 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f',\n '$P, MPa$',\n '$R_s, \u043c^3/\u043c^3$')\nplt.grid()\n# \u0440\u0438\u0441\u0443\u0435\u043c \u0442\u0440\u0435\u0442\u0438\u0439 \u0433\u0440\u0430\u0444\u0438\u043a\nplt.subplot(223)\nplot_rs_p(plt, pvt.unf_rs_Standing_m3m3,\n t_set_def,go_set_def,gg_set,\n '\u0413\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043e\u0442 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f',\n '$P, MPa$',\n '$R_s, \u043c^3/\u043c^3$')\n\nplt.subplots_adjust(top=0.92, bottom=0.08, left=0.10, right=0.95, hspace=0.25, wspace=0.35)\n# \u0440\u0438\u0441\u0443\u0435\u043c \u0432\u0441\u0435\nplt.grid()\nplt.show()\n```\n\n### 1.2.2. \u0413\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435, \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f \u0412\u0435\u043b\u0430\u0440\u0434\u0435-\u041c\u0430\u043a\u043a\u0435\u0439\u043d\u0430\n\n\u041a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f \u0412\u0435\u043b\u0430\u0440\u0434\u0435-\u041c\u0430\u043a\u043a\u0435\u0439\u043d\u0430 (1999) \u0434\u043b\u044f \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u044f \u0431\u0430\u0437\u0438\u0440\u0443\u0435\u0442\u0441\u044f \u043d\u0430 718 \u043b\u0430\u0431\u043e\u0440\u0430\u0442\u043e\u0440\u043d\u044b\u0445 \u0438\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u043d\u0438\u044f\u0445 \u0440\u0430\u0437\u0433\u0430\u0437\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u0440\u0430\u0437\u043b\u0438\u0447\u043d\u044b\u0445 \u043d\u0435\u0444\u0442\u0435\u0439 \u0441\u043e \u0432\u0441\u0435\u0433\u043e \u043c\u0438\u0440\u0430.\n\n$$ R_s = R_{sb}R_{sr} \\tag{1.2.2.1} $$\n\n\u0433\u0434\u0435:\n\n$R_s$ - \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435, $\u043c^3/\u043c^3$\n\n$R_{sb}$ - \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, $\u043c^3/\u043c^3$\n\n$R_{sr}$ - \u043f\u0440\u0438\u0432\u0435\u0434\u0435\u043d\u043d\u043e\u0435 \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435\n\n$$R_{sr}=a_1p_r^{a_2} + (1-a_1)P_r^{a_3} \\tag{1.2.2.2}$$\n\n\u0433\u0434\u0435 $P_r$ - \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u043f\u0440\u0435\u0432\u044b\u0448\u0435\u043d\u0438\u044f \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f, $psig/psig$\n\n$$P_r=\\dfrac{(P-14,7)}{(P_b-14,7)} \\tag{1.2.2.3} $$ \n\n$P$ - \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0435, psia\n\n$P_b$ - \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, psia\n\n$$a_1=9.73 \\cdot 10^{-7}\\gamma_{gSP}^{1.672608}API^{0.929870}T^{0.247235}(P_b-14.7)^{1.056052} \\tag{1.2.2.4}$$\n\n$$a_2=0.022339 \\gamma_{gSP}^{-1.004750}API^{0.337711}T^{0.132795}(P_b-14.7)^{0.302065} \\tag{1.2.2.5}$$\n\n$$a_3=0.725167 \\gamma_{gSP}^{-1.485480}API^{-0.164741}T^{-0.091330}(P_b-14.7)^{0.047094} \\tag{1.2.2.6}$$\n\n\u0433\u0434\u0435 \u0432 \u0441\u0432\u043e\u044e \u043e\u0447\u0435\u0440\u0435\u0434\u044c\n\n$\\gamma_{gSP}$ - \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u0433\u0430\u0437\u0430 \u043d\u0430 \u0441\u0435\u043f\u0430\u0440\u0430\u0442\u043e\u0440\u0435\n\n$API$ - \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u043d\u0435\u0444\u0442\u0438 \u0432 \u0433\u0440\u0430\u0434\u0443\u0441\u0430\u0445 API \n\n$T$ - \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u0430, F\n\n\n\u0412 \u043f\u0440\u0438\u0432\u0435\u0434\u0435\u043d\u043d\u043e\u0439 \u0442\u0430\u0431\u043b\u0438\u0446\u0435 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u044b \u0434\u0438\u0430\u043f\u0430\u0437\u043e\u043d\u044b \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0439 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u043d\u044b\u0445 \u0434\u043b\u044f \u0441\u043e\u0437\u0434\u0430\u043d\u0438\u044f \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u0438:\n\n|

    \u041f\u0430\u0440\u0430\u043c\u0435\u0442\u0440 | \u041c\u0438\u043d|\u0421\u0440\u0435\u0434\u043d\u0435\u0435|\u041c\u0430\u043a\u0441|\n| :--- | :---: |:---:|:---:|\n|

    \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f,$P_b$ , $ \u041c\u041f\u0430 $ | 2.861 |15.706|53.434|\n|

    \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u0430, $^{\\circ}\\mathrm{\u0421} $ | 21 |86|160|\n|

    \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u0433\u0430\u0437\u0430 \u043d\u0430 \u0441\u0435\u043f\u0430\u0440\u0430\u0442\u043e\u0440\u0435, $\\gamma_g$ | 0.555 |0.793|1.472|\n|

    \u043e\u0431\u044a\u0435\u043c\u043d\u044b\u0439 \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442 \u043d\u0435\u0444\u0442\u0438 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, $B_{ob}$ , $ \u043c^3/\u043c^3 $ | 1.012 |1.358|2.042|\n\n>\"Correlation of Black Oil Properties at Pressures Below Bubblepoint Pressure\u2014A New Approach\",\n J. VELARDE, T.A. BLASINGAME Texas A&M University, W.D. MCCAIN, JR. S.A. Holditch & Associates, Inc 1999\n\n\n```python\nuc.MPa_2_psi(1)\n```\n\n\n```python\nuc.atm_2_bar(1)\n```\n\n\n```python\n14.6959 * 10 / 1.01325\n```\n\n\n```python\nA = np.array([9.73 * 10 ** (-7), 1.672608, 0.929870, 0.247235, 1.056052])\nB = np.array([0.022339, -1.004750, 0.337711, 0.132795, 0.302065])\nC = np.array([0.725167, -1.485480, -0.164741, -0.091330, 0.047094])\n\na1, a2, a3 = symbols('a1 a2 a3')\napi = symbols('API')\ngamma_gas = symbols('gamma_gas')\ngamma_o = symbols('gamma_o')\nt_F,t_K = symbols('T_[F] T_[K]')\npb_psia, p_bMPaa = symbols('p_b[psia],p_b[MPaa]')\n\neq1 = Eq(a1, A[0] * gamma_gas ** A[1] * api ** A[2] * t_F ** A[3] * (pb_psia - 14.7) ** A[4])\neq2 = Eq(a2, B[0] * gamma_gas ** B[1] * api ** B[2] * t_F ** B[3] * (pb_psia - 14.7) ** B[4])\neq3 = Eq(a3, C[0] * gamma_gas ** C[1] * api ** C[2] * t_F ** C[3] * (pb_psia - 14.7) ** C[4])\n\ndisplay(eq1)\ndisplay(eq2)\ndisplay(eq3)\n\n# \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u0434\u043b\u044f \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u0435\u0434\u0438\u043d\u0438\u0446 \u0438\u0437\u043c\u0435\u0440\u0435\u043d\u0438\u044f \u0438\u0437 \u0430\u043c\u0435\u0440\u0438\u043a\u0430\u043d\u0441\u043a\u0438\u0445 \u043f\u0440\u043e\u043c\u044b\u0441\u043b\u043e\u0432\u044b\u0445 \u0432 \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0435\napi_to_gamma_o = 141.5 / gamma_o - 131.5\nF_to_K = t_K * 9 / 5 - 459.67\npsi_to_MPa = p_bMPaa * 145.037737730209#14.6959 #* 10.1325\n# \u043f\u043e\u043a\u0430\u0436\u0435\u043c \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u0432 \u043f\u0435\u0447\u0430\u0442\u043d\u043e\u043c \u0432\u0438\u0434\u0435\ndisplay(Eq(api, api_to_gamma_o))\ndisplay(Eq(t_F, F_to_K))\ndisplay(Eq(pb_psia, psi_to_MPa))\n\neq1_m=simplify(eq1.subs(api, api_to_gamma_o)\n .subs(t_F, F_to_K)\n .subs(pb_psia, psi_to_MPa)\n )\ndisplay(eq1_m)\n\neq2_m=simplify(eq2.subs(api, api_to_gamma_o)\n .subs(t_F, F_to_K)\n .subs(pb_psia, psi_to_MPa)\n )\ndisplay(eq2_m)\n\neq3_m=simplify(eq3.subs(api, api_to_gamma_o)\n .subs(t_F, F_to_K)\n .subs(pb_psia, psi_to_MPa)\n )\ndisplay(eq3_m)\n```\n\n\n```python\n# \u043e\u0431\u044a\u044f\u0432\u043b\u0435\u043d\u0438\u044f \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u044b\u0445 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u044b\u0445 \u0434\u043b\u044f \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u0435\u0434\u0438\u043d\u0438\u0446 \u0432 \u0432\u044b\u0440\u043e\u0436\u0435\u043d\u0438\u0438\nb_o = symbols('b_o')\nrho_sto_lbft3, rho_or_lbft3 = symbols('rho_sto[lbft3] rho_or[lbft3]')\nrs_scfstb, gamma_g = symbols('r_s[scfstb] gamma_g')\nrs_m3m3 = symbols('r_s[m3m3]')\nrho_sto_kgm3, rho_or_kgm3 = symbols('rho_sto[kgm3] rho_or[kgm3]')\n\n# \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0430 \u0440\u0430\u0441\u0447\u0435\u0442\u0430 \u0432 \u0430\u043c\u0435\u0440\u0438\u043a\u0430\u043d\u0441\u043a\u0438\u0445 \u043f\u0440\u043e\u043c\u044b\u0441\u043b\u043e\u0432\u044b\u0445 \u0435\u0434\u0438\u043d\u0438\u0446\u0430\u0445\neq1 = Eq(b_o, (rho_sto_lbft3 + 0.01357 * rs_scfstb * gamma_g)/rho_or_lbft3)\n# \u043f\u043e\u043a\u0430\u0436\u0435\u043c \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u0432 \u043f\u0435\u0447\u0430\u0442\u043d\u043e\u043c \u0432\u0438\u0434\u0435\nprint('\u0438\u0441\u0445\u043e\u0434\u043d\u043e\u0435 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435')\ndisplay(eq1)\n\n# \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u0434\u043b\u044f \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u0435\u0434\u0438\u043d\u0438\u0446 \u0438\u0437\u043c\u0435\u0440\u0435\u043d\u0438\u044f \u0438\u0437 \u0430\u043c\u0435\u0440\u0438\u043a\u0430\u043d\u0441\u043a\u0438\u0445 \u043f\u0440\u043e\u043c\u044b\u0441\u043b\u043e\u0432\u044b\u0445 \u0432 \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0435\n# \u0434\u043b\u044f \u0440\u0430\u0431\u043e\u0442\u044b \u0441 \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442\u0430\u043c\u0438 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u043c \u043c\u043e\u0434\u0443\u043b\u044c neftpy.uconvert \nscfstb_to_m3m3 = rs_m3m3 * uc.m3m3_2_scfstb(1)\nsto_lbft3_to_kgm3 = rho_sto_kgm3 * uc.kgm3_2_lbft3(1)\nor_lbft3_to_kgm3 = rho_or_kgm3 * uc.kgm3_2_lbft3(1)\n\n# \u043f\u043e\u043a\u0430\u0436\u0435\u043c \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u0432 \u043f\u0435\u0447\u0430\u0442\u043d\u043e\u043c \u0432\u0438\u0434\u0435\nprint('\u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442\u044b \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f')\ndisplay(Eq(rs_scfstb , scfstb_to_m3m3))\ndisplay(Eq(rho_sto_lbft3 , sto_lbft3_to_kgm3))\n\n# \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0430 \u0432 \u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0435\u0434\u0438\u043d\u0438\u0446\u044b \u0441 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u043c \u0441\u0438\u043c\u0432\u043e\u043b\u044c\u043d\u044b\u0445 \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0439\neq1_m=simplify(eq1.subs(rs_scfstb, scfstb_to_m3m3)\n .subs(rho_sto_lbft3, sto_lbft3_to_kgm3)\n .subs(rho_or_lbft3, or_lbft3_to_kgm3)\n )\n# \u0432\u044b\u0432\u043e\u0434 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u043e\u0432 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0439\nprint('\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u043d\u043e\u0435 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435')\ndisplay(eq1_m)\n```\n\n\n```python\npvt._unf_rs_Velarde_m3m3_(1.1)\n```\n\n\n\n\n array(15.86745211)\n\n\n\n\n```python\npvt.unf_rs_Velarde_m3m3(1.1)\n```\n\n\n\n\n array(15.86745211)\n\n\n\n\n```python\n# \u043a\u043e\u0434 \u0434\u043b\u044f \u043f\u043e\u0441\u0442\u0440\u043e\u0435\u043d\u0438\u044f \u0433\u0440\u0430\u0444\u0438\u043a\u043e\u0432\nplt.figure(figsize=(15,8))\nf = pvt.unf_rs_Velarde_m3m3\n# \u0440\u0438\u0441\u0443\u0435\u043c \u043f\u0435\u0440\u0432\u044b\u0439 \u0433\u0440\u0430\u0444\u0438\u043a\nplt.subplot(221)\nplt.grid()\nplot_rs_p(plt, pvt.unf_rs_Velarde_m3m3,\n t_set,go_set_def,gg_set_def,\n '\u0413\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043e\u0442 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f',\n '$P, MPa$',\n '$R_s, \u043c^3/\u043c^3$')\n# \u0440\u0438\u0441\u0443\u0435\u043c \u0432\u0442\u043e\u0440\u043e\u0439 \u0433\u0440\u0430\u0444\u0438\u043a\nplt.subplot(222)\nplt.grid()\nplot_rs_p(plt, pvt.unf_rs_Velarde_m3m3,\n t_set_def,go_set,gg_set_def,\n '\u0413\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043e\u0442 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f',\n '$P, MPa$',\n '$R_s, \u043c^3/\u043c^3$')\n# \u0440\u0438\u0441\u0443\u0435\u043c \u0442\u0440\u0435\u0442\u0438\u0439 \u0433\u0440\u0430\u0444\u0438\u043a\nplt.subplot(223)\nplot_rs_p(plt, pvt.unf_rs_Velarde_m3m3,\n t_set_def,go_set_def,gg_set,\n '\u0413\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043e\u0442 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f',\n '$P, MPa$',\n '$R_s, \u043c^3/\u043c^3$')\n\nplt.subplots_adjust(top=0.92, bottom=0.08, left=0.10, right=0.95, hspace=0.25, wspace=0.35)\n# \u0440\u0438\u0441\u0443\u0435\u043c \u0432\u0441\u0435\nplt.grid()\nplt.show()\n```\n\n### 1.2.3. \u041e\u0446\u0435\u043d\u043a\u0430 \u043e\u0431\u044a\u0435\u043c\u0430 \u0440\u0430\u0441\u0442\u0432\u043e\u0440\u0435\u043d\u043d\u043e\u0433\u043e \u0433\u0430\u0437\u0430 \u0432 \u0441\u0435\u043f\u0430\u0440\u0430\u0442\u043e\u0440\u0435, \u0434\u043b\u044f \u0443\u0442\u043e\u0447\u043d\u0435\u043d\u0438\u044f \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u044f \u043f\u043e \u0437\u0430\u043c\u0435\u0440\u0430\u043c \u043d\u0430 \u0441\u0435\u043f\u0430\u0440\u0430\u0442\u043e\u0440\u0435. \u041a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f \u041c\u0430\u043a\u043a\u0435\u0439\u043d\u0430.\n\n\n\u0412\u043e \u043c\u043d\u043e\u0433\u0438\u0445 \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f\u0445 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u0442\u0441\u044f \u0432 \u043a\u0430\u0447\u0435\u0441\u0442\u0432\u0435 \u0438\u0441\u0445\u043e\u0434\u043d\u043e\u0433\u043e \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0430 - \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f. \u0417\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u044d\u0442\u043e\u0433\u043e \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0430 \u043c\u043e\u0436\u0435\u0442 \u0431\u044b\u0442\u044c \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u043e \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u043f\u043e\u043b\u0435\u0432\u044b\u0445 \u0434\u0430\u043d\u043d\u044b\u0445 \u043a\u0430\u043a \u0441\u0443\u043c\u043c\u0430 \u043e\u0442\u0434\u0435\u043b\u044f\u0435\u043c\u043e\u0433\u043e \u0433\u0430\u0437\u043e\u0432\u043e\u0433\u043e \u0444\u0430\u043a\u0442\u043e\u0440\u0430 \u0432 \u0441\u0435\u043f\u0430\u0440\u0430\u0442\u043e\u0440\u0435 \u0438 \u0440\u0435\u0437\u0435\u0440\u0432\u0443\u0430\u0440\u0435 \u0434\u043b\u044f \u0442\u043e\u0432\u0430\u0440\u043d\u043e\u0439 \u043d\u0435\u0444\u0442\u0438.\n\n\n$$ R_{sb} = R_{sp} + R_{st} \\tag{1.2.3.1} $$\n\n\u0433\u0434\u0435:\n\n$R_{sb}$ - \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, $\u043c^3/\u043c^3$\n\n$R_{sp}$ - \u0433\u0430\u0437\u043e\u0432\u044b\u0439 \u0444\u0430\u043a\u0442\u043e\u0440, \u043e\u0442\u0434\u0435\u043b\u044f\u0435\u043c\u044b\u0439 \u0432 \u0441\u0435\u043f\u0430\u0440\u0430\u0442\u043e\u0440\u0435, $\u043c^3/\u043c^3$\n\n$R_{st}$ - \u0433\u0430\u0437\u043e\u0432\u044b\u0439 \u0444\u0430\u043a\u0442\u043e\u0440 \u0432 \u0440\u0435\u0437\u0435\u0440\u0432\u0443\u0430\u0440\u0435 \u0434\u043b\u044f \u0442\u043e\u0432\u0430\u0440\u043d\u043e\u0439 \u043d\u0435\u0444\u0442\u0438, $\u043c^3/\u043c^3$\n\n\u0414\u0430\u043d\u043d\u043e\u0435 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435 \u0441\u043f\u0440\u0430\u0432\u0435\u0434\u043b\u0438\u0432\u043e \u0442\u043e\u043b\u044c\u043a\u043e \u0435\u0441\u043b\u0438 \u043f\u043b\u0430\u0441\u0442\u043e\u0432\u043e\u0435 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u0432\u044b\u0448\u0435 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f.\n\n---\n\n\n\u0420\u0430\u0441\u0445\u043e\u0434 \u0433\u0430\u0437\u0430 \u0438 \u0434\u0435\u0431\u0438\u0442 \u043d\u0435\u0444\u0442\u0438 \u0432 \u0441\u0435\u043f\u0430\u0440\u0430\u0442\u043e\u0440\u0435 \u043e\u0431\u044b\u0447\u043d\u043e \u0438\u0437\u043c\u0435\u0440\u044f\u044e\u0442\u0441\u044f, \u043a\u043e\u0433\u0434\u0430 \u043a\u0430\u043a \u0432 \u0440\u0435\u0437\u0435\u0440\u0432\u0443\u0430\u0440\u0435 \u0433\u0430\u0437 \u043e\u0431\u044b\u0447\u043d\u043e \u0432\u044b\u043f\u0443\u0441\u043a\u0430\u0435\u0442\u0441\u044f \u0438 \u043d\u0435 \u0437\u0430\u043c\u0435\u0440\u044f\u0435\u0442\u0441\u044f. \u041f\u043e\u044d\u0442\u043e\u043c\u0443 \u0434\u043b\u044f \u0431\u043e\u043b\u0435\u0435 \u0442\u043e\u0447\u043d\u043e\u0439 \u043e\u0446\u0435\u043d\u043a\u0438 \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u044f \u043f\u043b\u0430\u0441\u0442\u043e\u0432\u043e\u0439 \u043d\u0435\u0444\u0442\u0438 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e \u043e\u0446\u0435\u043d\u0438\u0442\u044c \u0433\u0430\u0437\u043e\u0432\u044b\u0439 \u0444\u0430\u043a\u0442\u043e\u0440 \u0432 \u0440\u0435\u0437\u0435\u0440\u0432\u0443\u0430\u0440\u0435.\n\u0422\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, \u0431\u044b\u043b \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u0430\u043d\u0430 \u0444\u043e\u0440\u043c\u0443\u043b\u0430 \u043d\u0430 \u043e\u0441\u043d\u043e\u0432\u0435 GRACE-\u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0430 \u043d\u0430 \u0431\u0430\u0437\u0435 898 \u0438\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u043d\u0438\u0439 \u043f\u043b\u0430\u0441\u0442\u043e\u0432\u043e\u0433\u043e \u0444\u043b\u044e\u0438\u0434\u0430. \u0412 \u0442\u0430\u0431\u043b\u0438\u0446\u0435 \u043f\u0440\u0438\u0432\u0435\u0434\u0435\u043d\u044b \u043d\u0430\u0431\u043e\u0440 \u0434\u0430\u043d\u043d\u044b\u0445 \u0434\u043b\u044f \u0438\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u043d\u0438\u044f.\n\n\n\n$$ ln R_{st} = 3.955 + 0.83z - 0.024z^2 + 0.075z^3 \\tag{1.2.3.2} $$\n\n\u0433\u0434\u0435 \n\n$$ z =\\sum_{n=1}^3 z_n $$\n\n$$ z_n = C0_n + C1_nV_n + C2_nV_n^2 $$\n\n|

    $n$ | $V$|$CO$|$C1$|$C2$|\n| :--- | :---: |:---:|:---:|:---:|\n|

    $1$ | $ln P_{sp} $ |$-8.005$| $2.7$|$-0.161$|\n|

    $2$ | $ln T_{sp}$ |$1.224$|$-0.5$|$0$|\n|

    $3$ | $API$ |$-1.587$|$0.0441$|$-2.29 \\cdot 10 ^{-5}$|\n \n$T_{sp}$ - \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u0430 \u043d\u0430 \u0441\u0435\u043f\u0430\u0440\u0430\u0442\u043e\u0440\u0435, \u00b0F\n \n$P_{sp}$ - \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430 \u0441\u0435\u043f\u0430\u0440\u0430\u0442\u043e\u0440\u0430, psia\n\n\u0412\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u0434\u043b\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u0433\u0430\u0437\u043e\u0432\u043e\u0433\u043e \u0444\u0430\u043a\u0442\u043e\u0440\u0430 \u0432 \u0440\u0435\u0437\u0435\u0440\u0432\u0443\u0430\u0440\u0435 \u0442\u0440\u0435\u0431\u0443\u0435\u0442 \u0437\u043d\u0430\u0442\u044c \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u0443 \u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u0432 \u0441\u0435\u043f\u0430\u0440\u0430\u0442\u043e\u0440\u0435, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043e\u0431\u044b\u0447\u043d\u043e \u043d\u0435 \u0432\u0441\u0435\u0433\u0434\u0430 \u0431\u044b\u0432\u0430\u044e\u0442 \u0438\u0437\u0432\u0435\u0441\u0442\u043d\u044b. \u041f\u043e\u044d\u0442\u043e\u043c\u0443 \u0432 \u044d\u0442\u043e\u043c \u0441\u043b\u0443\u0447\u0430\u0435 \u043c\u043e\u0436\u043d\u043e \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0435 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435.\n\n$$ R_{sb} = 1.1618 R_{sp} \\tag{1.2.3.3} $$\n \n> \"Reservoir oil bubblepoint pressures revisited; solution gas\u2013oil ratios and surface gas specific gravities\",\n J. VELARDE, W.D. MCCAIN, 2002\n\n\n```python\n# \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u044b \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u044f\u044e\u0449\u0438\u0435 \u0434\u0438\u0430\u043f\u0430\u0437\u043e\u043d\u044b \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0439 \u0434\u043b\u044f \u043f\u043e\u0441\u0442\u0440\u043e\u0435\u043d\u0438\u044f \u0433\u0440\u0430\u0444\u0438\u043a\u043e\u0432\np_set=np.arange(1,11,0.25)\nt_set=np.arange(294,400,30)\nt_set_def=np.array([313])\ngo_set=np.arange(0.8,1,0.05)\ngo_set_def=np.array([0.86])\nr_sp = 50\n# \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0434\u043b\u044f \u0430\u0432\u0442\u043e\u043c\u0430\u0442\u0438\u0437\u0430\u0446\u0438\u0438 \u043f\u043e\u0441\u0442\u0440\u043e\u0435\u043d\u0438\u044f \u0433\u0440\u0430\u0444\u0438\u043a\u043e\u0432 \u043f\u043e \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044e \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f\ndef plot_rsb_psep(plt,func,\n tset, goset,\n plot_title, plot_xlab, plot_ylab):\n for t in tset:\n for go in goset:\n plt.plot(p_set, func(r_sp,go,p_set,t), \n label='t = %1.0f $ ^{\\circ}\\mathrm{K}$'%t +\n ' $\\gamma_o$ = %1.2f'%go)\n plt.title(plot_title)\n plt.ylabel(plot_ylab, color = 'black')\n plt.xlabel(plot_xlab, color = 'black')\n plt.legend()\n```\n\n\n```python\n# \u043a\u043e\u0434 \u0434\u043b\u044f \u043f\u043e\u0441\u0442\u0440\u043e\u0435\u043d\u0438\u044f \u0433\u0440\u0430\u0444\u0438\u043a\u043e\u0432\nplt.figure(figsize=(15,8))\nf = pvt.unf_rsb_Mccain_m3m3\n# \u0440\u0438\u0441\u0443\u0435\u043c \u043f\u0435\u0440\u0432\u044b\u0439 \u0433\u0440\u0430\u0444\u0438\u043a\nplt.subplot(221)\nplt.grid()\nplot_rsb_psep(plt, f,\n t_set, go_set_def,\n '\u0413\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043e\u0442 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u0432 \u0441\u0435\u043f\u0430\u0440\u0430\u0442\u043e\u0440\u0435',\n '$P, MPa$',\n '$R_s, \u043c^3/\u043c^3$')\n# \u0440\u0438\u0441\u0443\u0435\u043c \u0432\u0442\u043e\u0440\u043e\u0439 \u0433\u0440\u0430\u0444\u0438\u043a\nplt.subplot(222)\nplt.grid()\nplot_rsb_psep(plt, f,\n t_set_def,go_set,\n '\u0413\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043e\u0442 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u0432 \u0441\u0435\u043f\u0430\u0440\u0430\u0442\u043e\u0440\u0435',\n '$P, MPa$',\n '$R_s, \u043c^3/\u043c^3$')\n\nplt.subplots_adjust(top=0.92, bottom=0.08, left=0.10, right=0.95, hspace=0.25, wspace=0.35)\n# \u0440\u0438\u0441\u0443\u0435\u043c \u0432\u0441\u0435\nplt.show()\n```\n\n## 1.3. \u041e\u0431\u044a\u0435\u043c\u043d\u044b\u0439 \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442 \u043d\u0435\u0444\u0442\u0438\n\n### 1.3.1. \u041a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u0432\u044b\u0448\u0435 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f\n\n\u041a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f \u0434\u043b\u044f \u043e\u0431\u044a\u0435\u043c\u043d\u043e\u0433\u043e \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442\u0430 \u043d\u0435\u0444\u0442\u0438 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u0432\u044b\u0448\u0435 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f \u0432 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u0438\u0441\u0442\u043e\u0447\u043d\u0438\u043a\u0430\u0445 \u0443\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f, \u0447\u0442\u043e \u043e\u043d\u0430 \u043f\u0440\u0438\u043d\u0430\u0434\u043b\u0435\u0436\u0438\u0442 \u0421\u0442\u0435\u043d\u0434\u0438\u043d\u0433\u0443, \u0432 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0445 Vasquez & Beggs. \u041d\u0430 \u0441\u0430\u043c\u043e\u043c \u0434\u0435\u043b\u0435 \u044d\u0442\u043e \u043d\u0435 \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f, \u0442\u0430\u043a \u043a\u0430\u043a \u043f\u0440\u0438\u0440\u043e\u0434\u0430 \u0435\u0435 \u043f\u0440\u043e\u0438\u0441\u0445\u043e\u0436\u0434\u0435\u043d\u0438\u044f \u043d\u0435 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u0447\u0435\u0441\u043a\u0430\u044f, \u0430 \u0432\u043f\u043e\u043b\u043d\u0435 \u0441\u0435\u0431\u0435 \u0444\u0438\u0437\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435.\n\n$$ B_o = B_{ob} \\cdot \\exp(c_o(p_b - p)) $$\n\n\u0433\u0434\u0435:\n\n$ B_o $ - \u043e\u0431\u044a\u0435\u043c\u043d\u044b\u0439 \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442 \u043d\u0435\u0444\u0442\u0438 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 p, \u043c3/\u043c3\n\n$ B_{ob} $ - \u043e\u0431\u044a\u0435\u043c\u043d\u044b\u0439 \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442 \u043d\u0435\u0444\u0442\u0438 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, \u043c3/\u043c3\n\n$ c_o $ - \u0441\u0436\u0438\u043c\u0430\u0435\u043c\u043e\u0441\u0442\u044c \u043d\u0435\u0444\u0442\u0438, 1/\u041c\u041f\u0430\n\n$ P $ - \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0435, \u041c\u041f\u0430\n\n$ P_b $ - \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f, \u041c\u041f\u0430\n\n\n### \u041a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u044f \u041c\u0430\u043a\u043a\u0435\u0439\u043d\u0430 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 \u043c\u0435\u043d\u044c\u0448\u0435 \u0438\u043b\u0438 \u0440\u0430\u0432\u043d\u043e\u043c \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u044e \u043d\u0430\u0441\u044b\u0449\u0435\u043d\u0438\u044f\n\n\u0423\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435 \u0432\u044b\u0432\u043e\u0434\u0438\u0442\u0441\u044f \u0438\u0437 \u043c\u0430\u0442\u0435\u0440\u0438\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0431\u0430\u043b\u0430\u043d\u0441\u0430 \u0438 \u043d\u0435 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043a\u043e\u0440\u0440\u0435\u043b\u044f\u0446\u0438\u0435\u0439.\n\n$$ b_o = \\left( \\frac{ \\rho_{STO} + 0.01357 R_s \\gamma_g}{\\rho_{or}}\\right) $$\n\n\u0433\u0434\u0435:\n\n$ b_o $ - \u043e\u0431\u044a\u0435\u043c\u043d\u044b\u0439 \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442 \u043d\u0435\u0444\u0442\u0438 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 $P$, \u043c3/\u043c3\n\n$ \\rho_{STO} $ - \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u0434\u0435\u0433\u0430\u0437\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u043e\u0439 \u043d\u0435\u0444\u0442\u0438, \u0444\u0443\u043d\u0442/\u0444\u04423 (\u043a\u0433/\u043c3)\n\n$ R_s $ - \u0433\u0430\u0437\u043e\u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u043f\u0440\u0438 \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0438 p, \u0444\u04423/\u0431\u0430\u0440\u0440\u0435\u043b\u044c (\u043c3/\u043c3)\n\n$ \\gamma_g $ - \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u0433\u0430\u0437\u0430 \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u043e \u0432\u043e\u0437\u0434\u0443\u0445\u0430\n\n$ \\rho_{or} $ - \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u043f\u043b\u0430\u0441\u0442\u043e\u0432\u043e\u0439 \u043d\u0435\u0444\u0442\u0438, \u0444\u0443\u043d\u0442/\u0444\u04423 (\u043a\u0433/\u043c3)\n\n#### \u0412\u043d\u0443\u0442\u0440\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0443\u0436\u0435 \u0440\u0435\u0430\u043b\u0438\u0437\u043e\u0432\u0430\u043d \u043f\u0435\u0440\u0435\u0432\u043e\u0434 \u0432\u0435\u043b\u0438\u0447\u0438\u043d, \u0435\u0434\u0438\u043d\u0438\u0446\u044b \u0438\u0437\u043c\u0435\u0440\u0435\u043d\u0438\u044f \u0432 \u0441\u043a\u043e\u0431\u043a\u0430\u0445 - \u0432\u0445\u043e\u0434\u043d\u044b\u0435 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u044b \u0432 \u0444\u0443\u043d\u043a\u0446\u0438\u044e\n\n\n```python\n# \u043e\u0431\u044a\u044f\u0432\u043b\u0435\u043d\u0438\u044f \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u044b\u0445 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u044b\u0445 \u0434\u043b\u044f \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u0435\u0434\u0438\u043d\u0438\u0446 \u0432 \u0432\u044b\u0440\u043e\u0436\u0435\u043d\u0438\u0438\nb_o = symbols('b_o')\nrho_sto_lbft3, rho_or_lbft3 = symbols('rho_sto[lbft3] rho_or[lbft3]')\nrs_scfstb, gamma_g = symbols('r_s[scfstb] gamma_g')\nrs_m3m3 = symbols('r_s[m3m3]')\nrho_sto_kgm3, rho_or_kgm3 = symbols('rho_sto[kgm3] rho_or[kgm3]')\n\n# \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0430 \u0440\u0430\u0441\u0447\u0435\u0442\u0430 \u0432 \u0430\u043c\u0435\u0440\u0438\u043a\u0430\u043d\u0441\u043a\u0438\u0445 \u043f\u0440\u043e\u043c\u044b\u0441\u043b\u043e\u0432\u044b\u0445 \u0435\u0434\u0438\u043d\u0438\u0446\u0430\u0445\neq1 = Eq(b_o, (rho_sto_lbft3 + 0.01357 * rs_scfstb * gamma_g)/rho_or_lbft3)\n# \u043f\u043e\u043a\u0430\u0436\u0435\u043c \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u0432 \u043f\u0435\u0447\u0430\u0442\u043d\u043e\u043c \u0432\u0438\u0434\u0435\nprint('\u0438\u0441\u0445\u043e\u0434\u043d\u043e\u0435 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435')\ndisplay(eq1)\n\n# \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u0434\u043b\u044f \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u0435\u0434\u0438\u043d\u0438\u0446 \u0438\u0437\u043c\u0435\u0440\u0435\u043d\u0438\u044f \u0438\u0437 \u0430\u043c\u0435\u0440\u0438\u043a\u0430\u043d\u0441\u043a\u0438\u0445 \u043f\u0440\u043e\u043c\u044b\u0441\u043b\u043e\u0432\u044b\u0445 \u0432 \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0435\n# \u0434\u043b\u044f \u0440\u0430\u0431\u043e\u0442\u044b \u0441 \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442\u0430\u043c\u0438 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u043c \u043c\u043e\u0434\u0443\u043b\u044c neftpy.uconvert \nscfstb_to_m3m3 = rs_m3m3 * uc.m3m3_2_scfstb(1)\nsto_lbft3_to_kgm3 = rho_sto_kgm3 * uc.kgm3_2_lbft3(1)\nor_lbft3_to_kgm3 = rho_or_kgm3 * uc.kgm3_2_lbft3(1)\n\n# \u043f\u043e\u043a\u0430\u0436\u0435\u043c \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u0432 \u043f\u0435\u0447\u0430\u0442\u043d\u043e\u043c \u0432\u0438\u0434\u0435\nprint('\u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442\u044b \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f')\ndisplay(Eq(rs_scfstb , scfstb_to_m3m3))\ndisplay(Eq(rho_sto_lbft3 , sto_lbft3_to_kgm3))\n\n# \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0430 \u0432 \u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0435\u0434\u0438\u043d\u0438\u0446\u044b \u0441 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u043c \u0441\u0438\u043c\u0432\u043e\u043b\u044c\u043d\u044b\u0445 \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0439\neq1_m = simplify(eq1.subs(rs_scfstb, scfstb_to_m3m3)\n .subs(rho_sto_lbft3, sto_lbft3_to_kgm3)\n .subs(rho_or_lbft3, or_lbft3_to_kgm3)\n )\n# \u0432\u044b\u0432\u043e\u0434 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u043e\u0432 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0439\nprint('\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u043d\u043e\u0435 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435')\ndisplay(eq1_m)\n```\n\n# \u0421\u043f\u0438\u0441\u043e\u043a \u043b\u0438\u0442\u0435\u0440\u0430\u0442\u0443\u0440\u044b\n\n1. \"A Pressure-Volume-Temperature Correlation for Mixtures of California Oil and Gases\", M.B. Standing, Drill. & Prod. Prac., API, 1947.\n2. \"Correlation of Black Oil Properties at Pressures Below Bubblepoint Pressure\u2014A New Approach\",\n J. VELARDE, T.A. BLASINGAME Texas A&M University, W.D. MCCAIN, JR. S.A. Holditch & Associates, Inc 1999\n3. \"Reservoir oil bubblepoint pressures revisited; solution gas\u2013oil ratios and surface gas specific gravities\",\n J. VELARDE, W.D. MCCAIN, 2002\n", "meta": {"hexsha": "038f5c72320f4aa67e3fed2b3c8bc13af7b787da", "size": 714414, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/01_01_pvt_pb.ipynb", "max_stars_repo_name": "unifloc/neftpy", "max_stars_repo_head_hexsha": "63232f00badcdfb56d0186dc9c50e2707ddce004", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-11-07T13:04:34.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-07T13:04:34.000Z", "max_issues_repo_path": "notebooks/01_01_pvt_pb.ipynb", "max_issues_repo_name": "unifloc/neftpy", "max_issues_repo_head_hexsha": "63232f00badcdfb56d0186dc9c50e2707ddce004", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/01_01_pvt_pb.ipynb", "max_forks_repo_name": "unifloc/neftpy", "max_forks_repo_head_hexsha": "63232f00badcdfb56d0186dc9c50e2707ddce004", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 402.2601351351, "max_line_length": 122676, "alphanum_fraction": 0.9270535012, "converted": true, "num_tokens": 11951, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.4367259656549011}} {"text": "```\n# default_exp resnetx\n```\n\n\n```\n#export\nfrom wong.imports import *\nfrom wong.core import *\nfrom wong.config import cfg, assert_cfg\n\nfrom torchvision.models.utils import load_state_dict_from_url\n\n```\n\n\n```\nfrom fastcore.all import * # test_eq\n```\n\n# ResNetX\n> a folded resnet\n\nThe key distinguishing feature of our proposed architecture is the use of concatenation-skip (addition(additive)-skip) connections like DenseNet (ResNet), but with selective long-range and short range skip connections rather than a dense connectivity.\n\nDespite various parameter-efficient depthwise-convolution-based designs, for GPU-based deployment ResNet architecture provide a comparable or better speed-accuracy trade-off.\n\nRef:\n\nXNect: Real-time Multi-person 3D Human Pose Estimation with a Single RGB Camera\n\n\nThe proposed networks reduces computations by 20% with equivalent or even superior accuary on the ImageNet dataset, and significantly outperforms state-of-the-art approaches in terms of AP_50 on the MS COCO object detection dataset. \n\nRef:\nCSPNet: A new backbone that can enhance learning capability of CNN\n\nis more accurate and more computationally efficient than the state of art ResNets networks.\n\nwhich achieve much better accuracy and efficiency than previous ConvNets.\n\nA residual network with multiple direct paths\n\nIn order to compare ResNetX with ResNet, we using ablation method. As ResNet is an special ResNetX when fold=1, we first express ResNet as ResNetX, then we change fold from 1, 2, 3, 4 to evaluate its performance. We first use transfer learning, we got pre-trained model of resnet152, then we fill the weights of ResNetX model with pretrained model, then fine tuning them, we got an better result ; Second method is training the model from scratch, we \n\nhttps://petewarden.com/2017/10/29/how-do-cnns-deal-with-position-differences/\n\nAs you go deeper into a network, the number of channels will typically increase, but the size of the image will shrink. This shrinking is done using pooling layers, traditionally with average pooling but more commonly using maximum pooling these days.\n\n\n```\n#export\ndef get_pred(l:int, d:int=1, start_id:int=None, end_id:int=None):\n \"get predecessor layer id.\"\n if start_id is None: start_id = d\n if end_id is None: end_id = l\n assert l >= 1 and start_id >= d and end_id > start_id\n if l < start_id or l > end_id or d == 1: # if the current layer index is less than the fold depth, or if fold depth == 1\n pred = l - 1\n else:\n remainder = (l-1-(start_id-d)) % (d-1)\n pred = l - 2 * (1+remainder)\n return pred\n```\n\nParameters:\n- l : current layer id.\n- start_id : index of the starting node\n- end_id : index of the ending node\n- d : fold depth.\n\nReturn:\n- The previous layer id that directly link to the current layer.\n\n\n\\begin{equation}\\label{eq:resnetx}\n i = \n \\left\\{\n \\begin{array}{ll}\n 1 & l < d \\lor d=1 ; \\\\\n 2 * (1 + (l-1) \\pmod{d-1}) & \\textrm{else} .\n \\end{array}\n \\right.\n\\end{equation}\n\n\n\n```\nget_pred(l=17, d=2, start_id=13)\n```\n\n\n\n\n 15\n\n\n\n\n```\nget_pred(l=50, d=5, start_id=8)\n```\n\n\n\n\n 44\n\n\n\n\n```\ntest_eq(get_pred(l=12, d=1, start_id=1), 11)\n\ntest_eq(get_pred(l=8, d=5, start_id=7), 4)\ntest_eq(get_pred(l=12, d=4, start_id=6), 10)\n\n```\n\n\n```\n#export\ndef layer_diff(cur:int, pred:int, num_nodes:tuple):\n \"layer difference between the current layer and the predecessor layer.\"\n assert cur > pred\n num_nodes = (1,) + num_nodes\n cumsum = 0 # start with 0\n for i, num in enumerate(num_nodes):\n if cumsum <= cur < cumsum + num:\n cur_layer = i\n if cur == cumsum:\n first = True\n else:\n first = False\n if cumsum <= pred < cumsum + num:\n pred_layer = i\n cumsum += num\n diff = cur_layer - pred_layer\n return diff, first\n```\n\n\n```\nnum_nodes = (3,4,6,3)\ncur, pred = 9,0\nlayer_diff(cur, pred, num_nodes)\n```\n\n\n\n\n (3, False)\n\n\n\nParameters:\n- Stem : the stemming stage, which accept original images, transform them, then input into the backbone network.\n- Unit : the operation at nodes.\n- Conn : the connections between nodes\n- fold : the fold depth\n- ni : number of input channels of the backbone network.\n- *num_stages : number of stages in the backbone network.*\n- num_nodes : number of nodes of every stage in the backbone network.\n- start_id : index of starting node of ResNetX\n- base : standard width of channels in the backbone network.\n- exp : expansion along with the increase of stages.\n- bottle_scale : bottleneck scale\n- first_downsample: dose down-sample at the start of the first stage.\n- deep_stem : using 7x7 or 3 3x3 conv in stemming stage.\n- c_in : number of input channels of the Start layer\n- c_out : number of classes in the output of the final classifier.\n- kwargs : arguments translate into `Unit`\n\n\n```\n#export\nclass ResNetX(nn.Module):\n \"A folded resnet.\"\n def __init__(self, Stem, Unit, Conn, Tail, fold:int, ni:int, num_nodes:tuple, start_id:int=None, end_id:int=None,\n base:int=64, exp:int=2, bottle_scale:int=1, first_downsample:bool=False,\n c_in:int=3, c_out:int=10, **kwargs):\n super(ResNetX, self).__init__()\n # fold depth should be less than the sum length of any two neighboring stages\n \n if start_id < fold: start_id = fold\n origin_ni = ni\n num_stages = len(num_nodes)\n nhs = [base * exp ** i for i in range(num_stages)] \n nos = [int(nh * bottle_scale) for nh in nhs]\n strides = [1 if i==0 and not first_downsample else 2 for i in range(num_stages)]\n# print('nhs=', nhs, 'nos=', nos, 'nus=', nus, 'strides=', strides)\n \n self.stem = Stem(c_in, no=ni) # , deep_stem\n \n units = []\n idmappings = []\n cur = 1\n for i, (nh, no, nu, stride) in enumerate(zip(nhs, nos, num_nodes, strides)):\n for j in range(nu):\n if j == 0: # the first node(layer) of each stage\n units += [Unit(ni, no, nh, stride=stride, **kwargs)]\n else:\n units += [Unit(no, no, nh, stride=1, **kwargs)]\n \n pred = get_pred(cur, fold, start_id, end_id) # \n diff, first = layer_diff(cur, pred, num_nodes)\n assert diff == 0 or diff == 1 or (diff == 2 and pred == 0), \\\n 'cur={}, pred={}, diff={} is not allowed.'.format(cur, pred, diff)\n# print('fold = {} , cur = {} , pred = {} ,diff = {}'.format(fold, cur, pred, diff))\n if diff == 0:\n idmappings += [Conn(no, no, stride=1)]\n elif diff == 1:\n# if first:\n idmappings += [Conn(ni, no, stride=stride)]\n# else:\n# idmappings += [Conn(no, no, stride=1)]\n elif diff == 2:\n idmappings += [Conn(origin_ni, no, stride=stride)]\n cur += 1\n ni = no\n self.units = nn.ModuleList(units)\n self.idmappings = nn.ModuleList(idmappings)\n \n self.classifier = Tail(nos[-1], c_out)\n self.fold, self.start_id, self.end_id = fold, start_id, end_id\n self.num_nodes = num_nodes\n init_cnn(self)\n \n def forward(self, x):\n results = {}\n results[0] = self.stem(x)\n cur = 0\n for i, (unit, idmapping) in enumerate(zip(self.units, self.idmappings)):\n cur += 1\n pred = get_pred(cur, self.fold, self.start_id, self.end_id)\n diff, first = layer_diff(cur, pred, self.num_nodes)\n# if diff == 0:\n results[cur % (2*self.fold-1)] = unit(results[(cur-1) % (2*self.fold-1)]) + idmapping(results[pred % (2*self.fold-1)])\n# else:\n# results[cur % (2*self.fold-1)] = unit(results[(cur-1) % (2*self.fold-1)]) + idmapping(results[(cur-1) % (2*self.fold-1)])\n x = results[cur % (2*self.fold-1)]\n\n x = self.classifier(x)\n return x\n \n def my_load_state_dict(self, state_dict, local_to_pretrained):\n error_msgs = []\n def load(module, prefix=''):\n local_name_params = itertools.chain(module._parameters.items(), module._buffers.items())\n local_state = {k: v.data for k, v in local_name_params if v is not None}\n\n new_prefix = local_to_pretrained.get(prefix, 'none')\n for name, param in local_state.items():\n key = new_prefix + name\n if key in state_dict:\n# print(key)\n input_param = state_dict[key]\n\n if input_param.shape != param.shape:\n # local shape should match the one in checkpoint\n error_msgs.append('size mismatch for {}: copying a param with shape {} from checkpoint, '\n 'the shape in current model is {}.'\n .format(key, input_param.shape, param.shape))\n continue\n\n try:\n param.copy_(input_param)\n except Exception:\n error_msgs.append('While copying the parameter named \"{}\", '\n 'whose dimensions in the model are {} and '\n 'whose dimensions in the checkpoint are {}.'\n .format(key, param.size(), input_param.size()))\n \n for name, child in module._modules.items():\n if child is not None:\n load(child, prefix + name + '.')\n load(self)\n load = None # break load->load reference cycle\n \n \n```\n\n\n```\n#export\ndef resnet_local_to_pretrained(num_nodes, fold, start_id, end_id):\n \"mapping from local state_dict to pretrained state_dict. the pretrained model is restricted to torchvision.models.resnet.\"\n local_to_pretrained = { # mapping from the names of local modules to the names of pretrained modules\n 'stem.0.': 'conv1.',\n 'stem.1.': 'bn1.',\n }\n\n cumsum = 0\n for i, num in enumerate(num_nodes):\n for j in range(num):\n key = 'units.' + str(cumsum + j) + '.'\n value = 'layer' + str(i+1) + '.' + str(j) + '.'\n downsample0 = 'layer' + str(i+1) + '.0.' + 'downsample.0.'\n downsample1 = 'layer' + str(i+1) + '.0.' + 'downsample.1.'\n\n pred = get_pred(cumsum + j + 1, fold, start_id, end_id) # \n diff = layer_diff(cumsum + j + 1, pred, num_nodes)\n if diff == 1:\n idmapping0 = 'idmappings.' + str(cumsum + j) + '.unit.0.'\n idmapping1 = 'idmappings.' + str(cumsum + j) + '.unit.1.'\n# print(idmapping0, downsample0)\n# print(idmapping1, downsample1)\n local_to_pretrained[idmapping0] = downsample0\n local_to_pretrained[idmapping1] = downsample1\n\n for a, b in zip(['1.','2.','4.','5.','7.','8.'], ['conv1.','bn1.','conv2.','bn2.','conv3.','bn3.']):\n# print (key + a, value + b)\n local_to_pretrained[key + a] = value + b\n\n cumsum += num\n \n return local_to_pretrained\n\n```\n\nThree priority levels to set configuration:\n- `default_cfg` the default configuration, which set all the option names and their default values\n- `cfg_file` the configuration file, which will override the default configuration\n- `cfg_list` the configuration list, which will override all the previous configurations.\n\n\n```\n#export\ndef resnetx(default_cfg:dict, cfg_file:str=None, cfg_list:list=None, pretrained:bool=False, **kwargs):\n \"wrapped resnetx\"\n assert default_cfg.__class__.__module__ == 'yacs.config' and default_cfg.__class__.__name__ == 'CfgNode' \n cfg = default_cfg\n if cfg_file is not None: cfg.merge_from_file(cfg_file)\n if cfg_list is not None: cfg.merge_from_list(cfg_list)\n assert_cfg(cfg)\n cfg.freeze()\n \n Stem = getattr(sys.modules[__name__], cfg.GRAPH.STEM)\n Unit = getattr(sys.modules[__name__], cfg.GRAPH.UNIT)\n Conn = getattr(sys.modules[__name__], cfg.GRAPH.CONN)\n Tail = getattr(sys.modules[__name__], cfg.GRAPH.TAIL)\n # start_id >= fold + 1, fold <= 6\n model = ResNetX(Stem=Stem, Unit=Unit, Conn=Conn, Tail=Tail, fold=cfg.GRAPH.FOLD, ni=cfg.GRAPH.NI, num_nodes=cfg.GRAPH.NUM_NODES, \n start_id=cfg.GRAPH.START_ID, end_id=cfg.GRAPH.END_ID, base=cfg.GRAPH.BASE, exp=cfg.GRAPH.EXP, bottle_scale=cfg.GRAPH.BOTTLE_SCALE,\n first_downsample=cfg.GRAPH.FIRST_DOWNSAMPLE, **kwargs)\n if pretrained:\n state_dict = load_state_dict_from_url(cfg.URL)\n local_to_pretrained = resnet_local_to_pretrained(cfg.GRAPH.NUM_NODES, cfg.GRAPH.FOLD,cfg.GRAPH.START_ID,cfg.GRAPH.END_ID)\n model.my_load_state_dict(state_dict, local_to_pretrained)\n for param in model.parameters(): # freeze all\n param.requires_grad = False\n return model\n```\n\n\n```\ncfg\n```\n\n\n\n\n CfgNode({'URL': 'https://download.pytorch.org/models/resnet152-b121ed2d.pth', 'GRAPH': CfgNode({'NUM_STAGES': 4, 'NUM_NODES': (3, 8, 36, 3), 'NUM_CHANNELS': (64, 128, 256, 512), 'STEM': 'resnet_stem', 'UNIT': 'mbconv', 'CONN': 'IdentityMapping', 'TAIL': 'Classifier', 'FOLD': 3, 'START_ID': 15, 'END_ID': 47, 'NI': 64, 'BASE': 64, 'EXP': 2, 'BOTTLE_SCALE': 0.5, 'FIRST_DOWNSAMPLE': False, 'DEEP_STEM': False})})\n\n\n\n\n```\nnum_nodes = (3, 8, 36, 3)\nnum_all_nodes = sum(num_nodes)\nfold = 3\nstart_id = num_nodes[0] + num_nodes[1] + fold + 1 \nend_id = num_nodes[0] + num_nodes[1] + num_nodes[0] + num_nodes[2] - 3 \ncfg_list = [\"GRAPH.STEM\", \"resnet_stem\",\n \"GRAPH.UNIT\", \"mbconv\", # resnet_bottleneck\n \"GRAPH.CONN\", \"IdentityMapping\",\n \"GRAPH.TAIL\", \"Classifier\",\n \"GRAPH.NUM_NODES\", num_nodes,\n \"GRAPH.FOLD\", fold,\n \"GRAPH.START_ID\", start_id,\n \"GRAPH.END_ID\", end_id,\n \"GRAPH.NI\", 64,\n \"GRAPH.BASE\", 64,\n \"GRAPH.EXP\", 2,\n \"GRAPH.BOTTLE_SCALE\", 0.5, # 4\n \"URL\", 'https://download.pytorch.org/models/resnet152-b121ed2d.pth',\n ]\nmodel = resnetx(cfg, cfg_list=cfg_list, pretrained=False, c_out=100, ks=5)\n```\n\n\n```\nnum_nodes = (24, 24, 24, 24)\nnum_all_nodes = sum(num_nodes)\nfold = 4\nstart_id = fold\nend_id = num_all_nodes\ncfg_list = [\"GRAPH.STEM\", \"conv_bn\",\n \"GRAPH.UNIT\", \"mbconv\", # resnet_bottleneck\n \"GRAPH.CONN\", \"IdentityMappingMaxPool\",\n \"GRAPH.TAIL\", \"Classifier\",\n \"GRAPH.NUM_NODES\", num_nodes,\n \"GRAPH.FOLD\", fold,\n \"GRAPH.START_ID\", start_id,\n \"GRAPH.END_ID\", end_id,\n \"GRAPH.NI\", 64,\n \"GRAPH.BASE\", 64,\n \"GRAPH.EXP\", 1,\n \"GRAPH.BOTTLE_SCALE\", 1., # 4\n \"URL\", 'https://download.pytorch.org/models/resnet152-b121ed2d.pth',\n ]\nmodel = resnetx(cfg, cfg_list=cfg_list, pretrained=False, c_out=100, ks=3)\nnum_params(model)\n```\n\n\n\n\n tensor(886948)\n\n\n\n**Tip** : Three methods to get `class` or `function` object from its string name:\n\n- `getattr(sys.modules[__name__], cfg.GRAPH.STEM)`\n- `globals()[cfg.GRAPH.STEM]`\n- `eval(cfg.GRAPH.STEM)`\n\n\n```\nx = torch.randn(2,3,64,64)\n```\n\n\n```\nwith torch.autograd.set_detect_anomaly(True):\n out = model(x)\n out.mean().backward()\n```\n\n\n```\n\"{:,}\".format(num_params(model))\n```\n\n\n\n\n '60,225,700'\n\n\n\n\n```\n\n```\n", "meta": {"hexsha": "ba4e746c02529d9d73aeb3e97c847a21c6a1eb88", "size": 22166, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "nbs/04_resnetx.ipynb", "max_stars_repo_name": "keepsimpler/wong", "max_stars_repo_head_hexsha": "9d2cd2db07ba5e05796855485706a92aee4d4a34", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "nbs/04_resnetx.ipynb", "max_issues_repo_name": "keepsimpler/wong", "max_issues_repo_head_hexsha": "9d2cd2db07ba5e05796855485706a92aee4d4a34", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2021-05-20T11:52:35.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-27T23:58:06.000Z", "max_forks_repo_path": "nbs/04_resnetx.ipynb", "max_forks_repo_name": "keepsimpler/wong", "max_forks_repo_head_hexsha": "9d2cd2db07ba5e05796855485706a92aee4d4a34", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 37.5059221658, "max_line_length": 460, "alphanum_fraction": 0.511368763, "converted": true, "num_tokens": 4123, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.7185943925708561, "lm_q1q2_score": 0.4366633464137477}} {"text": "# AM207 Final Project\n\n#### Team: ShiLe Wong, Xin Zeng (interested in research), Yujie Cai, Jiahui Tang\n\n#### Paper: Risk score learning for COVID-19 contact tracing apps\n\n# Notebook contents\n\n\u26d4 PLEASE DO NOT RERUN THE EXPERIMENT PART (II.4) SINCE IT TAKES EXTREMELY LONG TIME\n\nIf you need to return to the content page, kindly press `return to top` in each section.\n\n[I. Introduction](#I.-Introduction)\n\n - [Problem Statement](#Problem-Statement)\n \n - [Literature](#Literature)\n \n[II. Methods](#II.-Methods)\n\n - [1. High Level Technical Content](#1.-High-Level-Technical-Content)\n \n - [2. Detailed Technical Content](#2.-Detailed-Technical-Content)\n \n - [2.1 Synthetic data generator](#2.1-Synthetic-data-generator)\n \n - [2.2 Machine learning technique used](#2.2-Machine-learning-technique-used)\n \n - [3. Implementation](#3.-Implementation)\n \n - [A. Simulation of Toy Dataset](#A.-Simulation-of-Toy-Dataset)\n \n - [B. Bagging Simulator](#B.-Bagging-Simulator)\n \n - [C. Training the Model](#C.-Training-the-Model)\n \n - [Analytical Gradient](#Analytical-Gradient)\n \n - [Training Code](#Training-Code)\n \n - [4. Model and Experiments](#4.-Model-and-Experiments)\n \n - [1. Varying maximum bag size](#1.-Varying-maximum-bag-size)\n \n - [2. Varying censoring probability](#2.-Varying-censoring-probability)\n \n - [3. Varying Positive Bag Construction](#3.-Varying-Positive-Bag-Construction)\n \n - [4. Robustness to Model Mismatch](#4.-Robustness-to-Model-Mismatch)\n \n - [5. Reordering of data](#5.-Reordering-of-data)\n \n - [6. Examples of failure mode](#6.-Examples-of-failure-mode)\n \n[III. Discussion](#III.-Discussion)\n\n - [1. Evaluation](#1.-Evaluation)\n \n - [2. Future Work and Potential Modifications](#2.-Future-Work-and-Potential-Modifications)\n \n - [3. Broader Impact](#3.-Broader-Impact)\n \n[References](#References)\n\n\n# I. Introduction\n\n* Problem statement - what is the problem the paper aims to solve?\n* Context/scope - why is this problem important or interesting?\n* Existing work - what has been done in literature?\n* Contribution - what is gap in literature that the paper is trying to fill? What is the unique contribution\n\n\n### **Problem Statement**\n\nDuring the COVID-19 Pandemic, contact tracing came to the forefront of digital tools to help combat the spread of the virus. Part of the work in contact tracing is to accurately determine the risk of infection of an individual given their exposure to other individuals who have tested positive, so as to alert the individual should the risk be significant. This paper aims to solve the problem of accurately determining the probability of infection based on features of interpersonal interactions (e.g. duration and distance of interaction) and the contagiousness of the infected case.\n\nThis is particularly important since contact tracing is a key tool used by many places to contain the spread of the virus. When an individual is alerted of their risk of infection in a timely manner, they can be advised to isolate and thereby reduce the chances of them spreading the virus to other people should they be infected. Hence, accurately determining the risk of infection and using the information to advise individuals of relevant protocols can determine the efficacy of contact tracing as a tool, and more seriously, the extent of virus and pandemic progression.\n\n### **Literature**\n\n[Return to top](#Notebook-contents)\n\nThe paper references a few examples of work that has been done in contact tracing, and specifically in using data to train models for risk scoring. First, the paper centers on the risk score model for the Google/Apple Exposure Notification (GAEN) system [1] for contact tracing, which has predefined relationships of specified parameters that come from data which can be collected through mobile applications. The associated parameters are the main focus of the machine learning optimization problem covered in the paper, and that we hope to recreate.\n\nSecond, the paper highlights that many current risk score models or contact tracing apps assume a known model with fixed parameters. There is a publicly available resource put together by domain experts that suggest values for the parameters. [2] A couple of other papers cited also evaluate the effectiveness of contact tracing apps that assume a fixed, known model. [3,4] The authors find this landscape to present a gap in that these models are not dynamic or flexible enough to adjust to or reflect what the data might tell us. Having fixed parameters beforehand without accounting for what is happening in real life (as seen from data), even if they were determined through expert guidance, may still be biased or inaccurate. Hence, they suggest that their own model of training to fit data could perform better at estimating an individual's risk of infection, and compare their results to that of a Swiss model that uses fixed thresholds. [5]\n\nFinally, they acknowledge one paper that attempts to use data to train a model that optimizes for the risk parameters. [6] However, there were two gaps: (1) this earlier paper did not account for the infectiousness of the index case; and (2) this earlier paper performed supervised learning. For the first gap, by not accounting for the infectiousness, the model is potentially missing out on additional useful data that could inform the risk score, since the infectiousness of a patient you come across could greatly affect whether or not you would catch the virus from them. Moreover, the infectiousness is data that is captured by the GAEN app, so having a model that can utilize all the relevant data points collected by the app can maximize its efficacy and allow the model to be deployed directly. The second gap relates to the real-life applicability of the model. This current paper suggests that they have a better solution since their model takes into account imperfect information, for example, by not having complete infection labels for every single exposure, but an overall label for a bag of exposures that is prone to false negatives because of censoring. This is also potentially more realistic in a real-life situation where information collection is likely imperfect.\n\n# II. Methods\n\n## 1. High Level Technical Content \n\n[Return to top](#Notebook-contents)\n\nDue to privacy issues, the authors of the paper could not access real data collected by the covid tracing app. Therefore, the first contribution they made was to propose a mechanism to generate synthetic data, which we will provide detailed documentation of in II.2. Since the focus of our project is on understanding and testing their model for learning risk score parameters, we have decided to use the same synthetic data generator to replicate and test the experiments. \n\nSecond, with a dataset in hand, the authors performed several comparative experiments to prove that their algorithm is effective in estimating parameters of risk score models. They do this by doing a semi-supervised machine learning, where observations are clustered or \"bagged\" and their labels are aggregated (and in some cases, partially censored). The observation features are then fed into a machine learning model for the model to learn to predict risk scores that will maximize the log-likelihood of the corresponding bag labels. Producing risk score estimations involves optimizing for parameters such as attenuation thresholds and weights and contagiousness weights, which was performed using the `jax` package implementation of stochastic gradient descent.\n\n\n## 2. Detailed Technical Content\n\n[Return to top](#Notebook-contents)\n\n### 2.1 Synthetic data generator:\n\n\nAlthough data generation is not the main objective of this project, it is worth pointing out that the authors have done a tremendous job putting together a sensible physical model using scientific literature, and coming up with approximation tools to simulate the real life scenario, i.e. the transmission mechanism of COVID-19. We will elaborate on how the transmission is modelled in the physical world and analogous features collected on the mobile application, since this is what defines the generative model and is thus used to inform the eventual optimization.\n\nA bottom-up approach can be used to describe the transmission mechanism. That is, we first figure out the risk of infection from a single exposure, then we combine it with the social network effect to determine how aggregated exposures may have an effect on a person's overall infection outcome.\n\n#### **2.1.1 In the real world: Biological Model of Risk Scoring**\n\n\nSuppose a person is exposed to another person who has been infected. The hazard score of that event is defined to be a function of three arguments: duration of exposure $\\tau_n$, the distance of exposure $d_n$, and the time of sympton onset since the exposure $\\sigma_n$ (of the infected person), which is a proxy for the infectiousness of the index case. Namely, \n\n$$s_n = f(\\tau_n, d_n, \\sigma_n; \\phi) = \\tau_n \\times f_{dist}(d_n; \\phi) \\times f_{inf}(\\sigma_n; \\phi) \\quad \\quad (1)$$\n\nwhere $\\phi$ are parameters of the simulator. Notice the exact functional form of $f_{dist}$ and $f_{inf}$ follow fluid physics law as well as empirical conventions[1,2], which we choose not to write in details here given this is not our core implementation and innovation. \n\n\n#### **2.1.2 On a mobile application: Translating the biological model to a computational model**\n\n> ##### **From Hazard Score to Risk Score**\n\nSince all training features (essentially the model inputs) will be recorded by a mobile device, the data generator is modelled after the device used for recording. For this paper, the authors decide to follow the GAEN system as mentioned above in the Introduction. Specifically, the distance function $f_{dist}$ is approximated by bluetooth attentuation $f_{ble}$, and the infectiousness calculation function $f_{inf}$ is approximated by a quantized version of symptom onset time $f_{con}$. Rewriting equation (1) from above, we get the risk score model:\n\n$$r_n = f(\\tau_n, a_n, c_n; \\psi) = \\tau_n \\times f_{ble}(a_n; \\psi) \\times f_{con}(c_n; \\psi) \\quad \\quad (2)$$\n\nwhere $\\psi$ are parameters of the risk score model. The inputs to the model that are proxied by digital features are elaborated on below.\n\n> ##### **Distance as proxied by Bluetooth Attenuation**\n\nThe bluetooth attenuation function assigns different weights to different attenuation thresholds, and all the weights and thresholds are parameters to optimize for. To put it simply, the farther apart the devices, the less weight the attenuation function assigns. The actual math shows a relationship that maps bluetooth attenuation ($a_n$) to 4 categories defined by 3 attentuation threshold angles, $\\theta_b^\\text{ble}$ (distance), with each category having an output of one weight. \n\n\\begin{equation*}\nf_{\\text{ble}}(a_n; \\psi)=\\begin{cases}\n w_1^\\text{ble} \\quad &\\text{if} \\, a_n \\leq \\theta_1^\\text{ble} \\\\\n w_2^\\text{ble} \\quad &\\text{if} \\, \\theta_1^\\text{ble} < a_n \\leq \\theta_2^\\text{ble} \\\\\n w_3^\\text{ble} \\quad &\\text{if} \\, \\theta_2^\\text{ble} < a_n \\leq \\theta_3^\\text{ble} \\\\\n w_4^\\text{ble} \\quad &\\text{if} \\, \\theta_3^\\text{ble} < a_n \\\\\n \\end{cases}\n\\end{equation*}\n\nTwo modifications to this mapping function that is relevant for our work is (1) the assumption of monotocity of the weights; and (2) transforming this discretized categorization into a continuous function. These two modifications directly influence our mathematical work and interpretation.\n\n(1) Assumption of Monotonicity\n\nBiologically, it is reasonable to believe that the risk of infection only gets higher when the distance between an individual and infected case gets closer. Hence, to ensure this monotonic relationship is followed in the computational model, the authors opted to reparameterize the weights as follows:\n\n$$w^{\\text{ble}} = [w_1^{\\text{ble}}, w_2^{\\text{ble}} = w_1^{\\text{ble}} + \\Delta_2^{\\text{ble}}, w_3^{\\text{ble}} = w_2^{\\text{ble}} + \\Delta_3^{\\text{ble}}, w_4^{\\text{ble}} = w_3^{\\text{ble}} + \\Delta_4^{\\text{ble}}]$$\n \nHence, we will optimize over $(w^{\\text{ble}}_1, \\Delta_2^{\\text{ble}},\\Delta_3^{\\text{ble}},\\Delta_4^{\\text{ble}})$ instead of $(w_1^{\\text{ble}}, w_2^{\\text{ble}}, w_3^{\\text{ble}}, w_4^{\\text{ble}})$. We will have a function that maps the weights to the parameters we optimize, and vice versa.\n\n(2) Discrete to continuous mapping\n\nThe approach as represented by the function right now is known as hard binning. However, this means that the function is not continuous with respects to the attenuation thresholds. The motivation to ensure the function is continuous with respects to the thresholds is so that the function can be easily differentiated with respects to the thresholds and thus the gradient can be easily calculated too. Hence, the authors opted to replace this hard binning method with a soft thresholding approximation, by converting the discrete mapping to one that utilizes the sigmoid function, $\\sigma_\\tau$. In the equations below, $I \\left(\\theta^{\\text{ble}}_{b-1} < a_{nk} \\leq \\theta^{\\text{ble}}_{b}\\right)$ represents the output from the discrete function presented above.\n\n\\begin{aligned}\n\\tau_{nb} &= \\sum_k \\tau_{nk} I \\left(\\theta^{\\text{ble}}_{b-1} < a_{nk} \\leq \\theta^{\\text{ble}}_{b}\\right)\\\\\n& \\approx \\sum_k \\tau_{nk} \\sigma_\\tau (a_{nk} - \\theta^{\\text{ble}}_{b-1}) \\sigma_\\tau(\\theta^{\\text{ble}}_{b}-a_{nk})\\\\\n& \\approx \\sum_k \\tau_{nk} \\left[\\frac{1}{1+\\exp(-t(a_{nk} - \\theta^{\\text{ble}}_{b-1}))}\\right] \\left[\\frac{1}{1+\\exp(-t(\\theta^{\\text{ble}}_{b}-a_{nk}))}\\right]\\\\\n\\end{aligned}\n\n> ##### **Contagiousness of Index Case**\n\nThe days since symptom onset is calculated by health authorities and directly mapped to 3 contagiousness levels: none ($c_n = 1$), standard ($c_n = 2$), and high ($c_n = 3$). The eventual contagiousness level is then used in the risk scoring instead of having the calculated days be directly accessible by Google/Apple in view of privacy concerns. Again, zero weight is assigned to the none level, and two non-zero weights are assigned to the other two levels. \n\n\\begin{equation*}\nf_{\\text{con}}(c_n; \\psi)=\\begin{cases}\n 0 \\quad &\\text{if} \\, c_n = 1 \\\\\n w_2^{\\text{con}} \\quad &\\text{if} \\, c_n = 2 \\\\\n w_3^{\\text{con}} \\quad &\\text{if} \\, c_n = 3 \\\\\n \\end{cases}\n\\end{equation*}\n\nAgain, there is an assumption of monotonicity that is used to reparameterize the weights because there is a biological basis for believing that the risk of infection will only increase with increasing contagiousness.\n\n$$w^{\\text{con}} = [w_2^{\\text{con}}, w_2^{\\text{con}} + \\Delta^\\text{con}]$$\n\nWe will optimize over $(w_2^{\\text{con}}, \\Delta^\\text{con})$ and similarly have a function that maps between these parameters and the weights in the function.\n\n> ##### **Mapping Risk Score to Probability of Infection**\n\nFinally, to map the risk score into probability space, which is constrained to the bounds $[0,1]$, the authors use the standard exponential does response model [3]. Below is the form that uses the risk score model (instead of the hazard score model), which takes into account the digital features collected to determine the estimated probability of infection. \n\n$$q_n = \\mathbb{P}(y_n = 1|\\tilde x_n; \\psi) = 1 - e^{-\\mu r_n}$$\n\nHere $\\tilde x_n = (\\tau_n, a_n, c_n)$, $y_n = 1$ (infected) or $0$ (not infected).\n\n#### **2.1.3 Multiple exposures**\n\nAs an extension of the individual exposure modelling described above, the authors look at combining the risk from multiple exposures to reflect a single individual's experience, First, the authors recognize that different exposure events means different lengths of time spent in different distances to infected cases. This is represented by calculating the amount of time spent in each attenuation bucket (which is then transformed to the continuous sigmoid):\n\n$$\\underbrace{\\tau_{n b}}_{\\text{duration in bucket } b}= \\sum_{k=1}^{K_{n}} \\underbrace{\\tau_{n k}}_{\\text {duration}} \\times \\underbrace{\\mathbb{I}\\left(\\theta_{b-1}^{\\text {ble }}< a_{nk} \\leq \\theta_{b}^{\\text {ble}}\\right)}_{\\text{attenuation is in bucket } b}$$\n\nwhere $K_n$ is the number of exposures within the n'th exposure window. We then are able to compute the overall risk for the exposure:\n\n$$\\underbrace{r_{n}}_{\\text {risk score }}=\\underbrace{\\left[\\sum_{b=1}^{N_{B}} \\tau_{n b} w_{b}^{\\text {ble }}\\right]}_{\\text {weighted exposure minutes }} \\times \\underbrace{\\left[\\sum_{\\ell=1}^{N_{C}} \\mathbb{I}\\left(c_{n, \\ell}\\right) w_{\\ell}^{\\text {con }}\\right]}_{\\text {weighted contagiousness level }}$$\n\nIf we have multiple exposures for a user, we sum the risk scores to get $R_j = \\sum_{n \\in E_j} r_n$, which then translates into an overall probability of infection:\n\n$$Q_j = 1 - e^{-\\mu \\cdot R_j}$$\n\n#### **2.1.4 Bagging of exposure observations**\n\nThe individual exposure events are first randomly generated. Then, the exposure events are randomly grouped or bagged to represent the experience of an individual, and the probability of infection is calculated using the above aggregation. Each exposure event can affect multiple individuals, in a concept coined as the social network effect in the paper. However, this is not crucial to our calculations since it is taken care of in the bagging step of the model instead of in optimization. The probability of infection is then used to generate the ground truth label of whether a person is infected or not.\n\nHaving generated the data, the authors attempt to simulate the situation on the other side, which is in receiving the data and optimizing a model for the parameters. The same model is assumed for optimization, and the bagging of exposure events as described in the section above adds complexity in terms of noise and the potential for \"censoring\", where determinative exposure events (when an exposure significantly contributes to a person's risk but is not recorded) is not captured.\n\n\n### 2.2 Machine learning technique used\n\n + Multi-instance learning, Stochastic Gradient Descent\n\nThe paper is dealing with a multi-instance (MI) learning problem, which belongs to supervised learning. Where MI learning differs from the traditional scenario is in the nature of the learning examples. In MI learning each example is represented by a bag of feature vectors. Classification labels are only provided for entire bags, and the task is to learn a model that predicts the classification labels for unseen future bags. In our case, we have only one, binary label (a user gets infected or not) for a \"bag\" of exposure events. Potentially this type of problem gets more challenging to tackle as the bag getting larger or the feature in the bag is noisy, which correspond to more exposures and censoring recorded on the app. \n\n\nAs mentioned above, we have ten parameters of the infectious risk model to learn. To recap, two weights parameters $(\\Delta_{2:3}^{\\text{con}})$ come from the contageousness measure; three threshold angles $(\\theta_{1:3}^{\\text{ble}})$ and four weights $(w_1^{\\text{ble}} \\text{and} \\Delta_{2:4}^{\\text{ble}})$ from bluetooth attentuation; and $\\mu$, a constant in the exponential model. The objective is to maximize the log-likelihood (or minimizing the binary cross entropy) of th e following fuction, where $Q_j$ incorporates these parameters:\n\n$$\\mathcal{L}(\\psi) = -\\sum_{j=1}^J Y_j \\log Q_j + (1-Y_j)\\log (1-Q_j)$$\n\n$Y_j \\in \\{0,1\\}$ is the infection label for user $j$. \n\nThe exact techniques used by original authors are not complicated though, since relatively simple algorithm such as stochastic gradient descent (SGD) has proven the effectivness through their experiments. Gradient descent is an iterative algorithm, that starts from a random point on a function and travels down its slope in steps until it reaches the optimal point of function. Adding stochastic component can let the gradient descent to start at random point for each iteration. Some biggest advantages are to reduce the computations enormously and to avoid the algorithm stuck at undesired optima. However, we need to be careful on choosing a proper learning rate, as SGD is sensitive to it. \n\nIn the original experiment, the authors fit the model using 1000 iterations with batch size of 100. After the fitting, they choose AUC as metrics to discusses the performance with increasing problem complexity. \n\n\n## 3. Implementation\n\n[Return to top](#Notebook-contents)\n\nThere are 3 parts to the implementation: (A) Data Simulation and Generation, (B) Bagging Simulation, and (C) Learning Model Training. As mentioned, we have decided to use the original data simulation and generation code, as well as most of the bagging simulation, since the focus of our project is on testing the learning model. We have modified the original code to simplify processes where possible and to use simpler python functions as a way to test the robustness of the algorithm when reduced to less fancy functions.\n\n\n```python\n# it is advisory to install this version of jax package, and to run the notebook on intel-chip computer\n# !pip install jax==0.2.18\n```\n\n\n```python\n%matplotlib inline\nimport matplotlib.pyplot as plt\nimport time\nimport math\nimport itertools\nimport numpy as np\nfrom scipy.stats import nbinom\nfrom functools import partial\nimport scipy.stats\nfrom scipy.stats import bernoulli\nfrom dataclasses import dataclass, asdict\nimport sklearn.metrics as metrics\nfrom textwrap import wrap\nimport random\nfrom sklearn.model_selection import train_test_split\n\nimport jax\nimport jax.numpy as jnp\nimport jax.nn as jnn\n```\n\n### A. Simulation of Toy Dataset\n\nThe end goal of the simulation is to create a set of data points that have the features used to train the model (e.g. bluetooth attenuation, infectiousness, etc.), the labels for the data points (whether the exposure led to an infection) and the probability of infection, as calculated using the generative model and parameters. The code first includes a few helper functions for calculating different components of the risk score. Then we put it in a dictionary and calculate the probabilities and labels for subsequent use.\n\n\n```python\n### Original Data Generating Function\n\n@dataclass\nclass BleParams:\n slope: float = 0.21\n intercept: float = 3.92\n sigma: float = np.sqrt(0.33)\n tx: float = 0.0\n correction: float = 2.398\n name: str = 'briers-lognormal'\n\n@dataclass\nclass ModelParams:\n ble_params: BleParams = BleParams() # may want to change sigma\n distance_fun: str = 'quadratic' # quadratic or sigmoid\n distance_Dmin: float = 1.0\n distance_slope: float = 2.0\n distance_inflection: float = 2.0\n infectiousness_fun: str = 'skew-logistic' # gaussian or skew-logistic\n beta: float = 1e-1 # transmission rate (the simulation Colab uses 1e-3)\n\n@dataclass\nclass RiskConfig:\n ble_thresholds: jnp.array = jnp.array([])\n ble_weights: jnp.array = jnp.array([])\n inf_levels: jnp.array = jnp.array([])\n inf_weights: jnp.array = jnp.array([])\n name: str = ''\n beta: float = 3.1 * 1e-6 # Wilson table 1\n\n\ndef incubation_dist(t):\n mu = 1.621\n sig = 0.418\n rv = scipy.stats.lognorm(sig, scale=np.exp(mu))\n return rv.pdf(t)\n\n# Symptom days to infectiousness\ndef skew_logistic_scaled(x, alpha, mu, sigma):\n return scipy.stats.genlogistic.pdf(x, alpha, loc=mu, scale=sigma)\n\ndef ptost_conditional(ts, incubation):\n mu = -4\n sigma = 1.85\n alpha = 5.85\n tau = 5.42\n fpos = skew_logistic_scaled(ts, alpha, mu, sigma)\n fneg = skew_logistic_scaled(ts*tau/incubation, alpha, mu, sigma)\n ps = fpos\n neg = jnp.where(ts < 0)\n ps[neg] = fneg[neg]\n ps = ps/np.max(ps)\n return ps\n\ndef ptost_uncond(tost_times):\n incub_times = np.arange(1, 14, 1)\n incub_probs = incubation_dist(incub_times) \n tost_probs = np.zeros_like(tost_times, dtype=float)\n for k, incub in enumerate(incub_times):\n ps = ptost_conditional(tost_times, incub)\n tost_probs += incub_probs[k] * ps\n return tost_probs\n\ninfectiousness_curve_times = np.arange(-14, 14+1, 0.1)\ninfectiousness_curve_vals = ptost_uncond(infectiousness_curve_times)\n\ndef infectiousness_skew_logistic(delta):\n return np.interp(delta, infectiousness_curve_times, infectiousness_curve_vals)\n\n# Distance to dose\ndef transmission_vs_distance_quadratic(d, Dmin=1): \n m = np.power(Dmin,2)/np.power(d, 2)\n return np.minimum(1, m)\n\n# Generating input data grid\ndef uniform_input_data_grid(max_dur = 60, max_dist = 5, ngrid_dist = 20, ngrid_dur=20, min_dist=0.1, min_dur=5):\n distances = np.linspace(min_dist, max_dist, ngrid_dist) # meters\n durations = np.linspace(min_dur, max_dur, ngrid_dur) # minutes\n symptoms = np.arange(-10, 10+0.001, dtype=int) # must be int\n return distances, durations, symptoms\n\n# Generating input data and parameters\ndef make_input_data(sigma=0.1, nsamples=20, distances=None, durations=None, symptoms=None):\n if distances is None:\n distances, durations, symptoms = uniform_input_data_grid()\n ble_params = BleParams()\n attens = dist_to_atten(distances, ble_params)\n \n # Make 3d cross product of the three 1d inputs\n vals = itertools.product(distances, durations, symptoms)\n X = np.vstack([np.array(v) for v in vals])\n distance_grid = X[:,0]\n duration_grid = X[:,1]\n symptom_grid = np.array(X[:,2], dtype=int)\n n = len(distance_grid)\n print('Making grid of {} distances x {} durations x {} onsets = {} points'.format(\n len(distances), len(durations), len(symptoms), n))\n\n # noise-free atteniations \n atten_grid = dist_to_atten(distance_grid, ble_params) \n\n # noisy samples\n atten_grid_samples = []\n for n in range(nsamples):\n sample = dist_to_atten_sample(distance_grid, ble_params, sigma)\n atten_grid_samples.append(sample)\n\n \n # Make 2d matrix for surface plotting functions (for fixed duration)\n grid_2d_matrix = np.meshgrid(symptoms, attens)\n symptom_grid_2d_matrix, atten_grid_2d_matrix = grid_2d_matrix\n vals = [z for z in zip(*(x.flat for x in grid_2d_matrix))]\n X = np.vstack([np.array(v) for v in vals])\n symptom_grid_2d = np.array(X[:,0], dtype=int)\n atten_grid_2d = X[:,1]\n\n data = {\n 'distance_grid': distance_grid, 'atten_grid': atten_grid, \n 'duration_grid': duration_grid, 'symptom_grid': symptom_grid,\n 'symptom_grid_2d_matrix': symptom_grid_2d_matrix,\n 'atten_grid_2d_matrix': atten_grid_2d_matrix,\n 'symptom_grid_2d': symptom_grid_2d,\n 'atten_grid_2d': atten_grid_2d,\n 'atten_grid_samples': atten_grid_samples,\n 'noise_level': sigma, 'nsamples': nsamples}\n return data\n\n\n# Transmission model\ndef hazard_fun_batch(attenuations, durations, symptom_days, params, distances=None):\n \"\"\"\n Args:\n params = ModelParams() object.\n \"\"\"\n if distances is None:\n distances = atten_to_dist(attenuations, params.ble_params)\n if params.distance_fun == 'quadratic':\n fd = transmission_vs_distance_quadratic(distances, params.distance_Dmin)\n elif params.distance_fun == 'sigmoid':\n fd = transmission_vs_distance_sigmoid(distances, params.distance_slope, params.distance_inflection)\n elif params.distance_fun == 'spline':\n fd = transmission_vs_distance_spline(distances)\n if params.infectiousness_fun == 'gaussian':\n finf = infectiousness_gaussian(symptom_days) \n elif params.infectiousness_fun == 'skew-logistic':\n finf = infectiousness_skew_logistic(symptom_days)\n doses = durations * fd * finf \n return doses\n\n# calculate probability of infection, with option to do an approx with Taylor series\ndef prob_infection_batch(attenuations, durations, symptom_days, params, approx='taylor', exp_taylor_terms=np.inf, temp=1, distances=None):\n \"\"\"\n Args:\n params = ModelParams() object.\n \"\"\"\n doses = hazard_fun_batch(attenuations, durations, symptom_days, params, distances)\n prob_infect_exact = 1-np.exp(-params.beta * doses)\n if not (isinstance(temp, list) or isinstance(temp, np.ndarray)):\n temp = list(temp)\n\n if approx == 'none' or (approx == 'taylor' and exp_taylor_terms == np.inf) or (approx == 'tempered' and len(temp)==1 and temp[0] == 1):\n prob_infect_approx = prob_infect_exact\n elif approx == 'taylor' and exp_taylor_terms < np.inf: \n prob_infect_approx = []\n x = -params.beta * doses\n prob_infect = np.zeros(len(doses))\n for k in range(1,exp_taylor_terms+1):\n prob_infect -= x**k / math.factorial(k)\n prob_infect_approx.append(np.minimum(np.maximum(prob_infect, 0), 1))\n elif approx == 'tempered':\n prob_infect_approx = []\n x = -params.beta * doses\n for t in temp:\n prob_infect = 1 - np.power(np.maximum(1 + (1-t)*x,0), 1/(1-t))\n prob_infect_approx.append(prob_infect)\n else:\n raise ValueError(\"Unknnown approximation type: {}\".format(approx))\n return prob_infect_exact, prob_infect_approx\n\n\ndef prob_infection_grid(data, params, approx='none', exp_taylor_terms=np.inf, temp=1):\n \"\"\"\n Args:\n data = dictionary obtained from make_input_data().\n params = ModelParams() object.\n approx: Approximation for the exponential function for simulating model mismatch (none/taylor/tempered)\n exp_taylor_terms: Number of approximation terms to use for Taylor approximation\n temp: Array of temperature parameters for 'tempered' approximation \n \"\"\"\n ps = prob_infection_batch(data['atten_grid'], data['duration_grid'], data['symptom_grid'], params, approx, exp_taylor_terms, temp)\n return ps\n\ndef risk_score_grid(data, config):\n rs = risk_score_batch(data['atten_grid'], data['duration_grid'], data['symptom_grid'], config)\n return rs\n\ndef prob_risk_score_grid(data, config):\n qs = prob_risk_score_batch(data['atten_grid'], data['duration_grid'], data['symptom_grid'], config)\n return qs\n\ndef risk_score_grid_sample(data, config, ndx):\n rs = risk_score_batch(data['atten_grid_samples'][ndx], data['duration_grid'], data['symptom_grid'], config)\n return rs\n\ndef atten_to_rssi(atten, ble_params):\n return ble_params.tx - (atten + ble_params.correction)\n\ndef rssi_to_atten(rssi, ble_params):\n return ble_params.tx - (rssi + ble_params.correction)\n\ndef atten_to_dist(atten, ble_params):\n rssi = ble_params.tx - (atten + ble_params.correction)\n return np.exp((np.log(-rssi) - ble_params.intercept)/ble_params.slope)\n\ndef dist_to_atten(distance, ble_params):\n mu = ble_params.intercept + ble_params.slope * np.log(distance)\n rssi = -np.exp(mu)\n atten = ble_params.tx - (rssi + ble_params.correction)\n return atten\n\n# This will be used in the future to add noise\ndef dist_to_atten_sample(distances, ble_params, sigma):\n # if ble_params.sigma == 0:\n if sigma == 0:\n return dist_to_atten(distances, ble_params)\n N = len(distances)\n mus = ble_params.intercept + ble_params.slope * np.log(distances)\n # sigs = ble_params.sigma\n sigs = sigma\n rssi = -scipy.stats.lognorm(s=sigs, scale=np.exp(mus)).rvs()\n atten = ble_params.tx - (rssi + ble_params.correction)\n return atten\n\ndef make_infectiousness_params_v2():\n # Derived from Arizona by averaging some bins\n inf_pre = np.zeros((9), dtype=int)\n inf_post = np.zeros((5), dtype=int)\n inf_mid6 = np.array([1, 3, 4, 5, 6, 6, 6, 6, 5, 4, 3, 2, 2, 1, 1])\n inf_mid = np.ones_like(inf_mid6)\n ndx = (inf_mid6 >= 5)\n inf_mid[ndx] = 2\n #inf_mid = np.array([1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1])\n inf_levels = np.concatenate((inf_pre, inf_mid, inf_post))\n inf_weights = np.array([0, 10**1.6, 10**2])/100\n return inf_levels, inf_weights\n\ndef days_to_inf_levels(symptom_days):\n symptom_days = np.atleast_1d(symptom_days) # turn into array\n inf_levels, _ = make_infectiousness_params_v2()\n return inf_levels[symptom_days + 14]\n\n# initializing input data\ndistances, durations, symptoms = uniform_input_data_grid(max_dur = 60, max_dist = 5, ngrid_dist = 80, ngrid_dur=20, min_dist=0.1, min_dur=5)\ndata = make_input_data(distances=distances, durations=durations, symptoms=symptoms)\n\n# initializing parameters\nble_params = BleParams()\nrssi = atten_to_rssi(data['atten_grid'], ble_params)\nduration = data['duration_grid']\nsymptom_day = data['symptom_grid']\ninfectiousness = days_to_inf_levels(symptom_day)\n\nparams = ModelParams()\n\n# Create model inputs\nX = np.concatenate([rssi[:,None], duration[:,None], infectiousness[:,None]], axis=1)\napprox_type = 'taylor'\nexp_taylor_terms = 8\nexp_temperature = np.arange(0.5,1.,0.09)\n\n# calculating probability of infection\nprob_infect_exact, prob_infect_approx = prob_infection_grid(data, params, approx=approx_type, exp_taylor_terms=exp_taylor_terms, temp=exp_temperature)\nprob_infect_approx = np.array(prob_infect_approx)\n# manually inspect the approximation error (higher error for larger values -- away from 0)\nind = np.where(prob_infect_exact>0.6)\nprint('Approx:', prob_infect_approx[:,ind])\nprint('Exact:', prob_infect_exact[ind])\n\n\nprint (X.shape, prob_infect_exact.shape)\n\n\nsimdata = dict()\nsimdata['X'] = X\nsimdata['prob_infect_exact'] = prob_infect_exact\nsimdata['prob_infect_approx'] = prob_infect_approx\n\n```\n\n WARNING:absl:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)\n\n\n Making grid of 80 distances x 20 durations x 21 onsets = 33600 points\n Approx: [[[1. 1. 1. ... 0.97538178 0.92937435 0.92675691]]\n \n [[0.49996063 0.49998181 0.46092688 ... 0.49969697 0.49750601 0.49731772]]\n \n [[0.67110338 0.66968263 0.81008083 ... 0.65435507 0.63129513 0.62997964]]\n \n ...\n \n [[0.63518322 0.63414931 0.72087815 ... 0.62280329 0.60509304 0.60406036]]\n \n [[0.63539429 0.63435626 0.72199235 ... 0.62296994 0.60521186 0.60417686]]\n \n [[0.63536768 0.63433023 0.72181414 ... 0.62294962 0.60519806 0.60416337]]]\n Exact: [0.63537038 0.63433287 0.72183658 ... 0.62295162 0.60519936 0.60416464]\n (33600, 3) (33600,)\n\n\n\n```python\nX_epi = simdata['X'] \nprobabilities_true_epi = simdata['prob_infect_exact']\nprobabilities_true_epi_approx = simdata['prob_infect_approx'].T # N x num_approx_steps\nN = len(probabilities_true_epi)\nperm = np.random.permutation(N)\nX_epi = X_epi[perm, :]\nprobabilities_true_epi = probabilities_true_epi[perm]\nprobabilities_true_epi_approx = np.clip(probabilities_true_epi_approx[perm,:], 1e-5, 1-1e-5)\n \n# generating labels for the exposure events\ndef sample_labels(probabilities):\n Y = bernoulli.rvs(probabilities)\n N_pos = np.sum(Y)\n N_neg = np.sum(1-Y)\n print(\"total: {}, positives: {}, negatives: {}\".format(N, N_pos, N_neg))\n pos_neg_ratio= float(N_neg) / N_pos\n return Y, N_pos, N_neg, pos_neg_ratio\n\nY_epi, N_pos, N_neg, pos_neg_ratio = sample_labels(probabilities_true_epi)\n```\n\n total: 33600, positives: 5383, negatives: 28217\n\n\n\n```python\nsimdata\n```\n\n\n\n\n {'X': array([[-31.07666234, 5. , 0. ],\n [-31.07666234, 5. , 0. ],\n [-31.07666234, 5. , 0. ],\n ...,\n [-70.66723028, 60. , 1. ],\n [-70.66723028, 60. , 1. ],\n [-70.66723028, 60. , 0. ]]),\n 'prob_infect_exact': array([0.00123577, 0.00276287, 0.00595197, ..., 0.00616533, 0.00361119,\n 0.0021102 ]),\n 'prob_infect_approx': array([[0.00123653, 0.00276669, 0.00596975, ..., 0.00618441, 0.00361772,\n 0.00211243],\n [0.00123577, 0.00276287, 0.00595193, ..., 0.00616529, 0.00361118,\n 0.0021102 ],\n [0.00123577, 0.00276287, 0.00595197, ..., 0.00616533, 0.00361119,\n 0.0021102 ],\n ...,\n [0.00123577, 0.00276287, 0.00595197, ..., 0.00616533, 0.00361119,\n 0.0021102 ],\n [0.00123577, 0.00276287, 0.00595197, ..., 0.00616533, 0.00361119,\n 0.0021102 ],\n [0.00123577, 0.00276287, 0.00595197, ..., 0.00616533, 0.00361119,\n 0.0021102 ]])}\n\n\n\n### B. Bagging Simulator\n\n[Return to top](#Notebook-contents) \n\nNext, we need to bag the exposure events to tag them to an individual's aggregated risk of infection. The bagging simulator takes in parameters to determine the bag size, and various probabilities determining the makeup of the bag (e.g. number of positive and negative exposures, etc.).\n\n\n```python\nclass Bag_Simulator():\n def __init__(self, p_pos, r_pos, p_neg, r_neg, max_bag_size, censor_prob_pos, censor_prob_neg, max_pos_in_bag=3):\n self.p_pos = p_pos\n self.r_pos = r_pos\n self.p_neg = p_neg\n self.r_neg = r_neg\n self.max_bag_size = max_bag_size\n self.censor_prob_pos = censor_prob_pos\n self.censor_prob_neg = censor_prob_neg\n self.max_pos_in_bag = max_pos_in_bag\n\n @staticmethod\n def get_pos_neg_bag_sizes(bag_sizes, pos_bag_size_probs, neg_bag_size_probs, N_pos_Y, N_neg_Y,\n max_pos_in_bag=1, N_pos_bags=None):\n\n if max_pos_in_bag > max(bag_sizes):\n raise ValueError(\"max_pos_in_bag needs to be less than or equal to bag_sizes\")\n if N_pos_bags is None:\n raise ValueError(\"Please give an input for N_pos_bags\")\n # start from N_pos_bag, each time decrease by 10, until we get the required number of\n # negative and positive bag, and return the data simulation result\n for N_pos_bags_temp in range(N_pos_bags, 0, -10):\n pos_bag_size_samples = np.random.choice(bag_sizes, size=N_pos_bags_temp, replace=True, p=pos_bag_size_probs)\n # sample uniformly in order to prepare for the implementation -\n # \n pos_samples_per_bag = np.random.randint(low=1, high=max_pos_in_bag + 1, size=N_pos_bags_temp)\n N_pos_required = np.sum(pos_samples_per_bag)\n N_neg_bags = int(N_pos_bags_temp * N_neg_Y / N_pos_Y)\n neg_bag_size_samples = np.random.choice(bag_sizes, size=N_neg_bags, replace=True, p=neg_bag_size_probs)\n N_neg_required = np.sum(pos_bag_size_samples - pos_samples_per_bag) + np.sum(neg_bag_size_samples)\n if N_neg_required < N_neg_Y and N_pos_required < N_pos_Y:\n\n return pos_bag_size_samples, neg_bag_size_samples, pos_samples_per_bag, N_pos_bags_temp, \\\n int(N_pos_bags_temp * pos_neg_ratio)\n\n @staticmethod\n def get_neg_bino_norm(k, n, p):\n temp_pmf = nbinom.pmf(k, n, p)\n\n return temp_pmf / np.sum(temp_pmf)\n\n @staticmethod\n def train_test_split_func(mat, labels, N_pos_bags, N_neg_bags, train_ratio=0.8, order=True):\n N_bags = N_pos_bags + N_neg_bags\n N_pos_trn = int(N_pos_bags * train_ratio)\n N_neg_trn = int(N_neg_bags * train_ratio)\n if order:\n # if order = True, we use the same data splitting function as the original codebase\n # it will keep the first 80% as training data and keep the rest 20% as test data\n ind_trn = list(np.arange(N_pos_trn)) + list(np.arange(N_pos_bags, N_pos_bags + N_neg_trn))\n ind_tst = list(np.arange(N_pos_trn, N_pos_bags)) + list(np.arange(N_pos_bags + N_neg_trn, N_bags))\n np.random.shuffle(ind_trn)\n np.random.shuffle(ind_tst)\n assign_mat_trn, assign_mat_tst = mat[ind_trn], mat[ind_tst]\n bag_labels_trn, bag_labels_tst = labels[ind_trn], labels[ind_tst]\n else:\n # if order = False, we use our own data splitting function, which would not keep the order\n # of train and test data splitting\n assign_mat_trn_part_1, assign_mat_tst_part_1, bag_labels_trn_part_1, bag_labels_tst_part_1\\\n = train_test_split(mat[:N_pos_bags], labels[:N_pos_bags], train_size=train_ratio)\n assign_mat_trn_part_2, assign_mat_tst_part_2, bag_labels_trn_part_2, bag_labels_tst_part_2 \\\n = train_test_split(mat[N_pos_bags:], labels[N_pos_bags:], train_size=train_ratio)\n assign_mat_trn = np.concatenate((assign_mat_trn_part_1, assign_mat_trn_part_2), axis=0)\n assign_mat_tst = np.concatenate((assign_mat_tst_part_1, assign_mat_tst_part_2), axis=0)\n bag_labels_trn = np.concatenate((bag_labels_trn_part_1, bag_labels_trn_part_2), axis=0)\n bag_labels_tst = np.concatenate((bag_labels_tst_part_1, bag_labels_tst_part_2), axis=0)\n # deal with the censoring issues - a user may get exposed to events that are not recorded by their phone\n bag_labels_trn = bag_labels_trn[np.sum(assign_mat_trn, axis=1) > 0]\n bag_labels_tst = bag_labels_tst[np.sum(assign_mat_tst, axis=1) > 0]\n assign_mat_trn = assign_mat_trn[np.sum(assign_mat_trn, axis=1) > 0, :]\n assign_mat_tst = assign_mat_tst[np.sum(assign_mat_tst, axis=1) > 0, :]\n\n return assign_mat_trn, assign_mat_tst, bag_labels_trn, bag_labels_tst\n\n def simulate_bagged_data(self, X, Y, probabilities, order=True, visualize=True):\n # probabilities for bag sizes\n # next assigned a random bag of size k of these events to each user.\n # The value k is sampled from a truncated negative binomial distribution with parameters (p; r)\n # where the truncation parameter is the maximum bag size b.\n pos_bag_size_probs = self.get_neg_bino_norm(np.arange(self.max_bag_size), self.r_pos, 1-self.p_pos)\n neg_bag_size_probs = self.get_neg_bino_norm(np.arange(self.max_bag_size), self.r_neg, 1-self.p_neg)\n\n N_pos_Y = np.sum(Y)\n N_neg_Y = Y.shape[0] - N_pos_Y\n print(\"total: {}, positives: {}, negatives: {}\".format(N, N_pos_Y, N_neg_Y))\n bag_sizes = np.arange(1, self.max_bag_size + 1)\n pos_bag_size_samples, neg_bag_size_samples, pos_samples_per_bag, N_pos_bags, N_neg_bags = \\\n self.get_pos_neg_bag_sizes(bag_sizes, pos_bag_size_probs, neg_bag_size_probs, N_pos_Y, N_neg_Y, \\\n max_pos_in_bag=min(self.max_bag_size, self.max_pos_in_bag), N_pos_bags=710)\n N_pos_required = np.sum(pos_samples_per_bag)\n N_neg_required = np.sum(pos_bag_size_samples - pos_samples_per_bag) + np.sum(neg_bag_size_samples)\n\n # evaluate the correctness of N_neg_required and N_pos_required\n if N_neg_required > N_neg_Y:\n raise ValueError(\"the required number of negative bags should be less than the total number of \"\n \"negative bags\")\n if N_pos_required > N_pos_Y:\n raise ValueError(\"the required number of positive bags should be less than the total number of \"\n \"positive bags\")\n\n # deal with X and probabilities array, only keep the required number of samples\n valid_idx = list(np.where(np.array(Y == 1))[0][0:N_pos_required]) + \\\n list(np.where(np.array(Y == 0))[0][0:N_neg_required])\n np.random.shuffle(valid_idx)\n X_shuff, Y_shuff, probabilities_shuff = X[valid_idx, :], Y[valid_idx], probabilities[valid_idx]\n\n N_bags = N_pos_bags + N_neg_bags\n N_samples = np.sum(pos_bag_size_samples) + np.sum(neg_bag_size_samples)\n\n # get the statistics of empirical bag sizes to better evaluate the accuracy of our data simulation\n print('Empirical Bag sizes:')\n print('\\t Positive bags: mean size {:.3f}, median size {:d}'.format(np.mean(pos_bag_size_samples),\n int(np.median(pos_bag_size_samples))))\n print('\\t Negative bags: mean size {:.3f}, median size {:d}'.format(np.mean(neg_bag_size_samples),\n int(np.median(neg_bag_size_samples))))\n ind_pos_exposures = np.where(np.array(Y_shuff == 1))[0]\n ind_neg_exposures = np.where(np.array(Y_shuff == 0))[0]\n\n bag_labels = np.concatenate((np.ones(N_pos_bags), np.zeros(N_neg_bags)))\n assign_mat = np.zeros((N_bags, N_samples))\n curr_ind_pos, curr_ind_neg = 0, 0\n # We consider two scenarios for constructing positive bags: (i) each positive bag contains exactly one positive\n # exposure and rest are negative exposures, (ii) each positive bag contains N positive exposures,\n # where N is sampled uniformly from {1; 2; 3}\n # We keep the original format of constructing the matrix in order to better experiment our modeling\n for i in range(N_pos_bags):\n ind_pos = ind_pos_exposures[curr_ind_pos:curr_ind_pos + pos_samples_per_bag[i]]\n assign_mat[i, ind_pos] = 1 * bernoulli.rvs(1 - self.censor_prob_pos)\n curr_ind_pos += pos_samples_per_bag[i]\n if pos_bag_size_samples[i] > 1:\n ind_neg = ind_neg_exposures[curr_ind_neg:curr_ind_neg + pos_bag_size_samples[i] - pos_samples_per_bag[i]]\n assign_mat[i, ind_neg] = 1 * bernoulli.rvs(1 - self.censor_prob_neg)\n curr_ind_neg += pos_bag_size_samples[i] - pos_samples_per_bag[i]\n for i in range(N_neg_bags):\n ind1 = i + N_pos_bags\n ind2 = ind_neg_exposures[curr_ind_neg:curr_ind_neg + neg_bag_size_samples[i]]\n assign_mat[ind1, ind2] = 1 * bernoulli.rvs(1 - self.censor_prob_neg)\n curr_ind_neg += neg_bag_size_samples[i]\n\n assign_mat_trn, assign_mat_tst, bag_labels_trn, bag_labels_tst = \\\n self.train_test_split_func(assign_mat, bag_labels, N_pos_bags, N_neg_bags, 0.8, order)\n\n print('assign_mat size, X_shuff size:', assign_mat.shape, X_shuff.shape)\n print('assign_mat_trn size, assign_mat_tst size', assign_mat_trn.shape, assign_mat_tst.shape)\n print('Average positive samples per bag: {}'.format(np.mean(pos_samples_per_bag)))\n\n if visualize:\n self.visual_analysis(pos_bag_size_probs, neg_bag_size_probs)\n\n return X_shuff, probabilities_shuff, assign_mat_trn, assign_mat_tst, bag_labels_trn, bag_labels_tst\n\n def visual_analysis(self, pos_bag_size_probs, neg_bag_size_probs):\n fig, ax = plt.subplots(1, 2, figsize=(10, 5))\n ax[0].stem(np.arange(self.max_bag_size) + 1, pos_bag_size_probs, use_line_collection=True)\n ax[0].set_title('Probability of positive bag sizes')\n\n ax[1].stem(np.arange(self.max_bag_size) + 1, neg_bag_size_probs, use_line_collection=True)\n ax[1].set_title('Probability of negative bag sizes')\n\n plt.tight_layout()\n plt.show()\n```\n\n\n```python\n# Running a test to visualize the bagging simulator\nbag_sim = Bag_Simulator(p_pos=0.6, r_pos=2, p_neg=0.6, r_neg=2, max_bag_size=16, censor_prob_pos=0, censor_prob_neg=0)\nX, probabilities_true, assign_mat_trn, assign_mat_tst, bag_labels_trn, bag_labels_tst = \\\n bag_sim.simulate_bagged_data(X_epi, Y_epi, probabilities_true_epi)\n```\n\n### C. Training the Model\n\n\n[Return to top](#Notebook-contents)\n\n\nFinally, we recreate the learning model and train it to the simulated dataset. Again, we first define some helper functions such as loss function, conversion of parameters (for reparameterization), initialization of weights and unpacking them, etc. Next, we train the model using stochastic gradient descent code from class. Finally, we combine everything together with the bagging simulator above, such that the function will take in an instance of a bagging simulator class, and run the necessary steps for the inputs to be bagged and used in training.\n\n#### Analytical Gradient\n\nIntstead of directly applying the gradient descent, we have also re-derive the analytical gradient for each of the parameter that we need to train for. Although we don't have time to implement these derivations in the main body of training process, we expect that with the aid of analytical expression the algorithm will get more efficient. The followings are gradient expression with respect to our parameters:\n\n#### **Loss Function**\n\n\\begin{aligned}\nL(\\psi) &= -\\sum_j Y_j \\log (Q_j) + (1-Y_j) \\log(1-Q_j)\\\\\n\\frac{d L}{d Q} &= -\\sum_j \\frac{Y_j}{Q_j} + \\frac{1-Y_j}{1-Q_j}\\\\\n\\end{aligned}\n\nAs a reminder,\n\n\\begin{aligned}\nQ_j &= 1 - \\exp\\left[-\\mu R_j\\right] \\\\\n&= 1 - \\exp\\left[-\\mu \\sum_n f_\\text{risk} (\\bar{x_n}; \\psi)\\right]\\\\\nf_\\text{risk} &= \\left[\\sum_{b=1}^{N_B} \\tau_{nb} w_b^{\\text{ble}}\\right] \\; \\times \\; \\left[\\sum_{l=1}^{N_C} I(c_{n,l})w_l^{\\text{con}}\\right]\n\\end{aligned}\n\n\n### **Gradient wrt to scale factor, $\\mu$**\n\n\\begin{aligned}\n\\frac{d Q_j}{d \\mu} &= R_j \\exp\\left[-\\mu R_j\\right]\\\\\n&= \\sum_n f_\\text{risk} \\exp\\left[-\\mu \\sum_n f_\\text{risk} (\\bar{x_n}; \\psi)\\right]\n\\end{aligned}\n\n\n### **Gradient wrt to bluetooth attenuation weights, $w_{1:4}^\\text{ble}$**\n\nReminder of monotocity, where we are optimizing over $(w^{\\text{ble}}_1, \\Delta_2^{\\text{ble}},\\Delta_3^{\\text{ble}},\\Delta_4^{\\text{ble}})$.\n\n\n\\begin{aligned}\n\\frac{d Q_j}{d w^{\\text{ble}}_b} &= \\frac{d Q_j}{df} \\frac{df}{d w^{\\text{ble}}_b}\\\\\n&= \\left(\\mu \\exp\\left[-\\mu R_j\\right]\\right) \\sum_n \\left(\\tau_{nb} \\; \\times \\; \\left[\\sum_{l=1}^{N_C} I(c_{n,l})w_l^{\\text{con}}\\right]\\right)\\\\\n\\frac{d Q_j}{d w^{\\text{ble}}_1}\n&= \\left(\\mu \\exp\\left[-\\mu R_j\\right]\\right) \\sum_n \\left(\\left[\\sum_{b=1}^{N_B} \\tau_{nb} \\right] \\; \\times \\; \\left[\\sum_{l=1}^{N_C} I(c_{n,l})w_l^{\\text{con}}\\right]\\right)\\\\\n\\frac{d Q_j}{d \\Delta^{\\text{ble}}_1} &= \\left(\\mu \\exp\\left[-\\mu R_j\\right]\\right) \\sum_n \\left(\\left[\\sum_{b=2}^{N_B} \\tau_{nb} \\right] \\; \\times \\; \\left[\\sum_{l=1}^{N_C} I(c_{n,l})w_l^{\\text{con}}\\right]\\right)\\\\\n\\frac{d Q_j}{d \\Delta^{\\text{ble}}_2} &= \\left(\\mu \\exp\\left[-\\mu R_j\\right]\\right) \\sum_n \\left(\\left[\\sum_{b=3}^{N_B} \\tau_{nb} \\right] \\; \\times \\; \\left[\\sum_{l=1}^{N_C} I(c_{n,l})w_l^{\\text{con}}\\right]\\right)\\\\\n\\frac{d Q_j}{d \\Delta^{\\text{ble}}_3} &= \\left(\\mu \\exp\\left[-\\mu R_j\\right]\\right) \\sum_n \\left(\\left[\\sum_{b=4}^{N_B} \\tau_{nb} \\right] \\; \\times \\; \\left[\\sum_{l=1}^{N_C} I(c_{n,l})w_l^{\\text{con}}\\right]\\right)\\\\\n\\end{aligned}\n\n### **Gradient wrt to bluetooth attenuation thresholds, $\\theta_{1:3}^\\text{ble}$**\n\nReminder of soft thresholding, where we are replacing hard binning with sigmoid function:\n\n\\begin{aligned}\n\\tau_{nb} &= \\sum_k \\tau_{nk} I \\left(\\theta^{\\text{ble}}_{b-1} < a_{nk} \\leq \\theta^{\\text{ble}}_{b}\\right)\\\\\n& \\approx \\sum_k \\tau_{nk} \\sigma_\\tau (a_{nk} - \\theta^{\\text{ble}}_{b-1}) \\sigma_\\tau(\\theta^{\\text{ble}}_{b}-a_{nk})\\\\\n& \\approx \\sum_k \\tau_{nk} \\left[\\frac{1}{1+\\exp(-t(a_{nk} - \\theta^{\\text{ble}}_{b-1}))}\\right] \\left[\\frac{1}{1+\\exp(-t(\\theta^{\\text{ble}}_{b}-a_{nk}))}\\right]\\\\\n\\\\\n\\end{aligned}\n\n\\begin{aligned}\n\\frac{d Q_j}{d \\theta^{\\text{ble}}_b} &= \\frac{d Q_j}{df} \\frac{df}{d\\tau}\\frac{d\\tau}{d \\theta^{\\text{ble}}_b}\\\\\n&= \\left(\\mu \\exp\\left[-\\mu R_j\\right]\\right) \\sum_n \\left(\\sum_{b=1}^{N_b} w_b^\\text{ble} \\; \\times \\; \\left[\\sum_{l=1}^{N_C} I(c_{n,l})w_l^{\\text{con}}\\right]\\right) \\frac{d\\tau}{d \\theta^{\\text{ble}}_b}\\\\\n\\frac{d\\tau}{d \\theta^{\\text{ble}}_b}\n&= \\sum_k \\tau_{nk} \\left[\\frac{-t}{(1+\\exp(-t(a_{nk} - \\theta^{\\text{ble}}_{b})))^2}\\right] \\left[\\frac{1}{1+\\exp(-t(\\theta^{\\text{ble}}_{b+1}-a_{nk}))}\\right]\\\\\n&+ \\sum_k \\tau_{nk} \\left[\\frac{1}{1+\\exp(-t(a_{nk} - \\theta^{\\text{ble}}_{b-1}))}\\right] \\left[\\frac{t}{(1+\\exp(-t(\\theta^{\\text{ble}}_{b}-a_{nk})))^2}\\right]\n\\end{aligned}\n\n### **Gradient wrt to contagiousness weights, $w_{2:3}^\\text{con}$**\n\nReminder of monotocity, where we are optimizing over $(w^{\\text{con}}_2, \\Delta^{\\text{con}})$.\n\n\n\\begin{aligned}\n\\frac{d Q_j}{d w^{\\text{ble}}_l} &= \\frac{d Q_j}{df} \\frac{df}{d w^{\\text{con}}_l}\\\\\n&= \\left(\\mu \\exp\\left[-\\mu R_j\\right]\\right) \\sum_n \\left(\\sum_{b=1}^{N_B} \\tau_{nb} w_b^{\\text{ble}} \\; \\times \\; \\sum_{l=1}^{N_C} I(c_{n,l})\\right)\\\\\n\\frac{d Q_j}{d w^{\\text{con}}_2}\n&= \\left(\\mu \\exp\\left[-\\mu R_j\\right]\\right) \\sum_n \\left(\\sum_{b=1}^{N_B} \\tau_{nb} w_b^{\\text{ble}} \\; \\times \\; \\sum_{l=2}^{N_C} I(c_{n,l})\\right)\\\\\n\\frac{d Q_j}{d \\Delta^{\\text{con}}} &= \\left(\\mu \\exp\\left[-\\mu R_j\\right]\\right) \\sum_n \\left(\\sum_{b=1}^{N_B} \\tau_{nb} w_b^{\\text{ble}} \\; \\times \\; I(c_{n,3})\\right)\\\\\n\\end{aligned}\n\n#### Training Code\n\n[Return to top](#Notebook-contents)\n\n\n```python\n# basic configuration\nn_rssi_buckets = 4\nn_infect_levels = 3\nrssi_lowest_th = -120\nn_rssi_th = n_rssi_buckets-1 # number of thresholds for rssi buckets\nmodel_dim = 1 + n_rssi_buckets + n_rssi_th + n_infect_levels # total number of parameters\n\nn_trials = 1\nn_random_restarts_train = 5\n\n# helper functions\ndef sigmoid_func(x):\n\n return 1/(1 + np.exp(-x))\n\n# loss function\ndef cross_entropy_loss(y, q):\n return -np.mean(y*np.log(q) + (1-y)*np.log(1-q))\n```\n\n\n```python\n# initialize parameters\ndef get_init_parameters():\n beta = 0.1\n\n # rssi weights\n rssi_w_residual = np.random.rand(n_rssi_buckets) * 0.01\n\n # rssi thresholds\n rssi_th_residual = np.random.randint(low=10, high=40, size=n_rssi_th)\n\n # infectiousness weights\n infect_w_residual = np.random.rand(n_infect_levels) * 0.01\n\n return np.concatenate([[beta], rssi_w_residual, rssi_th_residual, infect_w_residual])\n\n\nget_beta = lambda weights: weights[0]\nget_rssi_w = lambda weights: weights[1:1+n_rssi_buckets]\nget_rssi_th = lambda weights: weights[1+n_rssi_buckets:1+n_rssi_buckets+n_rssi_th]\nget_infect_w = lambda weights: weights[1+n_rssi_buckets+n_rssi_th:1+n_rssi_buckets+n_rssi_th+n_infect_levels]\n\n# helper function to unpack weights\ndef unpack_weights(weights):\n return get_beta(weights), get_rssi_w(weights), get_rssi_th(weights), get_infect_w(weights)\n\n\ndef residual_to_scoring(weights):\n # rssi weights\n rssi_w_residual = get_rssi_w(weights)\n rssi_w = [np.sum(rssi_w_residual[:i+1]) for i in range(n_rssi_buckets)]\n\n # rssi thresholds\n rssi_th_residual = get_rssi_th(weights)\n rssi_th = [rssi_lowest_th+np.sum(rssi_th_residual[:i+1]) for i in range(n_rssi_th)]\n\n # infectiousness weights\n infect_w_residual = get_infect_w(weights)\n infect_w = [np.sum(infect_w_residual[:i+1]) for i in range(n_infect_levels)]\n\n return np.concatenate([[get_beta(weights)], rssi_w, rssi_th, infect_w])\n\n\ndef print_params(weights):\n print(\" beta\", get_beta(weights))\n print(\" rssi_w\", get_rssi_w(weights))\n print(\" rssi_th\", get_rssi_th(weights))\n print(\" infect_w\", get_infect_w(weights))\n```\n\n\n```python\n# continuous loss function, where attenuation is defined in terms of sigmoid functions\ndef loss_fn_ce(params, batch_x, batch_y, assign_mat, sigmoid_temp):\n\n f_rssi = batch_x[:, 0]\n beta, rssi_w, rssi_th, infect_w = unpack_weights(residual_to_scoring(params))\n infectiousness_bucket = np.eye(n_infect_levels)[np.array(batch_x[:, 2], dtype=int)]\n infectiousness_score = np.sum(np.multiply(np.array(infect_w), infectiousness_bucket), axis=1)\n\n rssi_score = rssi_w[0] * sigmoid_func(-sigmoid_temp * (f_rssi - rssi_th[0])) * \\\n sigmoid_func(10. * (f_rssi - rssi_lowest_th)) + rssi_w[-1] * \\\n sigmoid_func(sigmoid_temp * (f_rssi - rssi_th[n_rssi_buckets - 2]))\n rssi_score += np.dot(sigmoid_func(sigmoid_temp * (f_rssi[:, np.newaxis] - rssi_th[: n_rssi_buckets - 2])) \\\n * sigmoid_func(-sigmoid_temp * (f_rssi[:, np.newaxis] - rssi_th[1: n_rssi_buckets - 1])),\n rssi_w[1: n_rssi_buckets - 1])\n rssi_score *= batch_x[:, 1]\n\n score = beta * rssi_score * infectiousness_score\n bag_scores = np.dot(assign_mat, score)\n bag_probs = 1 - np.exp(-bag_scores)\n bag_probs = np.clip(bag_probs, 1e-5, 1 - 1e-5)\n loss = cross_entropy_loss(batch_y, bag_probs)\n\n return loss, bag_probs\n\n# discrete loss function, where attenuation is defined in terms of discrete bins\ndef loss_fn_stepbins_ce(params, batch_x, batch_y, assign_mat, return_scores=False):\n beta, rssi_w, rssi_th, infect_w = unpack_weights(residual_to_scoring(params))\n infectiousness_bucket = jax.nn.one_hot(batch_x[:, 2], num_classes=n_infect_levels)\n infectiousness_score = np.sum(np.multiply(np.array(infect_w), infectiousness_bucket), axis=1)\n rssi_bucket = np.eye(n_rssi_buckets)[np.array(np.digitize(batch_x[:, 0], rssi_th), dtype=int)]\n rssi_score = np.sum(np.multiply(np.array(rssi_w), rssi_bucket), axis=1) * batch_x[:, 1]\n\n score = beta * rssi_score * infectiousness_score\n if return_scores:\n return score\n bag_scores = np.dot(assign_mat, score)\n bag_probs = 1 - np.exp(-bag_scores)\n bag_probs = np.clip(bag_probs, 1e-5, 1 - 1e-5)\n loss = cross_entropy_loss(batch_y, bag_probs)\n\n return loss, bag_probs\n\n\n# numerical calculation of the gradient of the loss function\ndef grad_loss_fn_ce(params, batch_x, batch_y, assign_mat, sigmoid_temp, batch_loss):\n def grad(idx):\n delta = 1e-10\n params_d = np.copy(params)\n params_d[idx] += delta\n return (loss_fn_ce(params_d, batch_x, batch_y, assign_mat, sigmoid_temp)[0] - batch_loss)/delta\n return np.asarray([grad(i) for i in range(len(params))])\n```\n\n\n```python\ndef sigmoid_temp_fn(init, target, current_iter, num_iters):\n return np.interp(current_iter, [0, num_iters * 0.5, num_iters * 0.9, num_iters], [init, init, target, target])\n\n# implementation of stochastic gradient descent\ndef train(features, bag_labels, assign_mat, sigmoid_temp_init, sigmoid_temp_target, batch_size, num_iters, lr):\n num_samples = len(bag_labels)\n batch_size = np.minimum(batch_size, num_samples)\n\n model_params = get_init_parameters()\n loss_start, _ = loss_fn_ce(model_params, features, bag_labels, assign_mat, sigmoid_temp_init)\n loss_start_step, _ = loss_fn_stepbins_ce(model_params, features, bag_labels, assign_mat)\n print(\"Start\", \"sigmoid loss\", loss_start, \"step loss\", loss_start_step)\n\n for i in range(num_iters):\n l, r = i*batch_size%num_samples, (i+1)*batch_size%num_samples\n if l <= r:\n batch_y = bag_labels[l:r]\n batch_assign_mat = assign_mat[l:r]\n else:\n batch_y = np.concatenate([bag_labels[l:], bag_labels[:r]])\n batch_assign_mat = np.concatenate([assign_mat[l:], assign_mat[:r]], axis=0)\n\n sigmoid_temp = sigmoid_temp_fn(sigmoid_temp_init, sigmoid_temp_target, i, num_iters)\n\n loss, _ = loss_fn_ce(model_params, features, batch_y, batch_assign_mat, sigmoid_temp)\n grad = grad_loss_fn_ce(model_params, features, batch_y, batch_assign_mat, sigmoid_temp, loss)\n\n model_params -= lr * grad\n\n if i % 500 == 0:\n loss_step, _ = loss_fn_stepbins_ce(model_params, features, bag_labels, assign_mat)\n print(\"iter\", i, \"sigmoid loss\", loss, \"step loss\", loss_step, \"sigmoid temp\", sigmoid_temp)\n\n loss_final, _ = loss_fn_ce(model_params, features, bag_labels, assign_mat, sigmoid_temp_target)\n loss_final_step, probs_final = loss_fn_stepbins_ce(model_params, features, bag_labels, assign_mat)\n print(\"End\", \"sigmoid loss\", loss_final, \"step loss\", loss_final_step)\n print_params(model_params)\n\n return model_params, loss_final_step, probs_final\n\n# metric used for assessing method: AUC score\ndef auc(score_pred, labels):\n fpr, tpr, threshold = metrics.roc_curve(labels, score_pred)\n return metrics.auc(fpr, tpr)\n```\n\n\n```python\n# combining training, evaluation and bagging simulation\ndef train_and_eval_with_bag_config(bag_sim: Bag_Simulator, X_epi, probabilities_true_epi, n_trials=1, n_random_restarts=5, order=True):\n \"\"\"Train the risk score model on data generated according to the given bag configuration. \"\"\"\n auc_train_trials = []\n auc_test_trials = []\n for j in range(n_trials):\n # Y_epi, N_pos, N_neg, pos_neg_ratio = sample_labels(probabilities_true_epi)\n X, probabilities_true, assign_mat_trn, assign_mat_tst, bag_labels_trn, bag_labels_tst = bag_sim.simulate_bagged_data(X_epi, Y_epi, probabilities_true_epi, visualize=False, order=order)\n best_loss = np.inf\n best_model_params = None\n best_final_probs = None\n for i in range(n_random_restarts):\n print('----------- Trial {}/{}: Training run {}/{} ----------------'.format(j+1, n_trials, i+1, n_random_restarts))\n model_params, loss_st_step, final_probs = train(X, bag_labels_trn, assign_mat_trn,\n sigmoid_temp_init=0.1, sigmoid_temp_target=1,\n batch_size=200, num_iters=5000, lr=0.001)\n if loss_st_step < best_loss:\n best_loss = loss_st_step\n best_model_params = model_params\n best_final_probs = final_probs\n\n print(\"best loss\", best_loss)\n print(\"best scoring parameters\")\n print_params(residual_to_scoring(best_model_params))\n\n probs_bags_true_trn = 1 - np.exp(np.dot(assign_mat_trn, np.log(1-probabilities_true)))\n auc_train_trials.append({\n 'Learned': auc(best_final_probs, bag_labels_trn),\n 'True': auc(probs_bags_true_trn, bag_labels_trn),\n })\n\n scores_learned = scores_learned = loss_fn_stepbins_ce(best_model_params, X, None, None, return_scores=True)\n scores_learned_bags_tst = np.dot(assign_mat_tst, scores_learned)\n probs_learned_bags_tst = 1 - np.exp(-1*scores_learned_bags_tst)\n probs_bags_true_tst = 1 - np.exp(np.dot(assign_mat_tst, np.log(1-probabilities_true)))\n auc_test_trials.append({\n 'Learned': auc(probs_learned_bags_tst, bag_labels_tst),\n 'True': auc(probs_bags_true_tst, bag_labels_tst),\n })\n\n return auc_train_trials, auc_test_trials\n```\n\n## 4. Model and Experiments\n\n[Return to top](#Notebook-contents)\n\n\n\u26d4 PLEASE DO NOT RERUN THE EXPERIMENT PART SINCE IT TAKES EXTREMELY LONG TIME\n\n\nA few experiments were performed around tweaking different features of the data to check that the parameter learning model is behaving as we expect it to. These experiments were geared towards an outcome-based focus (such that we want to see the outputs and use the expected trend of the output as a proxy for the \"correctness\" of the model), rather than a specific mechanics correctness check. The authors also compared the outcomes to the performance of a ground truth model (\"True\", in subsequent plots) and that of the Swiss model with fixed parameters. We decided to focus our efforts on comparing it to the ground truth to elucidate its general performance trends, since the Swiss model is not our focus.\n\nAn overview of the experiments and a brief summary of the results are listed below. Specific elaboration of each experiment follows in the respective subsections for the experiments.\n\n1. **Maximum bag sizes were varied, while holding other parameters constant.** In particular, while the maximum bag size can go up to 32, the maximum number of positive exposures per bag remains fixed at 1.\n2. **Censoring probability is varied, while holding other parameters constant.** Censoring probability refers to the chances of an exposure not being recorded and thus not being a part of the data. We test this with different censor probabilities, for fixed maximum bag sizes of 16 and 32, with a maximum of 1 positive exposure per bag.\n3. **Varying positive bag construction, with varied maximum bag sizes.** The maximum number of positive exposures per bag can now go up to 3, while maximum bag sizes increases up to 32.\n4. **Robustness to model mismatch.** Since we are working with simulated data, there is a probability that the simulated data is not reflective of the real world, and that data in the real world is much less precise. Hence, the original generative model is approximated and the resulting dataset tested on to see the performance of the current learning model to potentially different generative models.\n\nIn addition, we ran one of our own experiments just as corroboration to make sure that the learning model is not training or fitting specifically to a specific order of the data, again as a check for the robustness of the model in general.\n\n5. **Reordering of data in training.** Training data is reordered so the model will learn parameters on different sequences of data points each time.\n\nFor all experiments, the metric used in the paper and which we are emulating is the AUC score. A higher AUC score reflects higher accuracy of the model in predicting the parameters that will generate a risk score which gives the maximum log-likelihood for the infection observation. Since we are using our own basic python code for the training, we do not have the efficiency gains of an ML package like `jax`. Hence, our code takes a very long while to run (about 20 minutes for 1 trial with 5 random starting points) and we were unable to run it multiple times to get replicates. Instead, we were only able to initialize the algorithm from 5 different starting points and plotted the results from the best output.\n\n### 1. Varying maximum bag size\n\n[Return to top](#Notebook-contents)\n\nThis experiment is iterated over increasing bag sizes to show the effect that increasing bag size has on the performance of the learning algorithm. Increasing bag size would increase uncertainty in assigning a risk score, since there is more noise with more observations, while the maximum number of positive exposures in the bag still remains fixed at 1. Hence, the signal of the one positive exposure (if it exists) gets drowned out as bag size increases, and it is harder to push the algorithm to an optimum which captures these signals well. \n\nWe expect the performance of the algorithm to decrease, i.e. the AUC score to fall, as bag size increases. This is reflected in the two plots below, where the AUC drops from a high of around 0.9 to plateau arounnd 0.7. We also note that the performance of the learning algorithm is worse than that of the true parameters, which is to be expected. Overall, the trends for this experiment definitely supports the claim being made.\n\n\n```python\nbag_size = [1, 2, 4, 8, 16, 32]\nn_trials = 1\nn_random_restarts_train = 5\nidx = 0\n\nauc_train_learned = np.zeros((len(bag_size),n_trials))\nauc_train_true = np.zeros((len(bag_size),n_trials))\nauc_test_learned = np.zeros((len(bag_size),n_trials))\nauc_test_true = np.zeros((len(bag_size),n_trials))\nfor bs in bag_size:\n bag_sim = Bag_Simulator(p_pos=0.6,r_pos=2,p_neg=0.6,r_neg=2,max_bag_size=bs,censor_prob_pos=0.,censor_prob_neg=0,max_pos_in_bag=1)\n auc_train_trials, auc_test_trials = train_and_eval_with_bag_config(bag_sim, X_epi,\n probabilities_true_epi, n_trials=n_trials,\n n_random_restarts=n_random_restarts_train)\n for i in range(n_trials):\n auc_train_learned[idx, i] = dict(auc_train_trials[i])['Learned']\n auc_train_true[idx, i] = dict(auc_train_trials[i])['True']\n auc_test_learned[idx, i] = dict(auc_test_trials[i])['Learned']\n auc_test_true[idx, i] = dict(auc_test_trials[i])['True']\n \n idx += 1\n```\n\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 1.000, median size 1\n \t Negative bags: mean size 1.000, median size 1\n assign_mat size, X_shuff size: (4431, 4431) (4431, 3)\n assign_mat_trn size, assign_mat_tst size (3544, 4431) (887, 4431)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.2196481992672172 step loss 1.222210239360569\n iter 0 sigmoid loss 0.9862754799858239 step loss 0.814513886000415 sigmoid temp 0.1\n iter 500 sigmoid loss 0.34829134088286573 step loss 0.33055410146157227 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3654958522934435 step loss 0.3173655889164178 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3163487501708123 step loss 0.30776392023344307 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.34303746484624625 step loss 0.29532438249387144 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3010745930260712 step loss 0.30725786676460737 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.25729091373058494 step loss 0.30843284917153 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2339990858005359 step loss 0.2852264088550915 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2946117784621882 step loss 0.2834389517223261 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2538802932606835 step loss 0.2821023762972173 sigmoid temp 1.0\n End sigmoid loss 0.27690590014658967 step loss 0.28207150875750864\n beta 0.31814946321585535\n rssi_w [-0.02672788 0.00308919 0.05818862 0.29809107]\n rssi_th [22.00970926 12.00974559 26.00950492]\n infect_w [0.01011637 0.06290278 0.2990673 ]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.3492833660134254 step loss 1.3501762423425883\n iter 0 sigmoid loss 1.0852725920246198 step loss 0.7766639402810296 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3449052873836508 step loss 0.3249453762118487 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3599655661093599 step loss 0.3105596722543181 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.30729405882520316 step loss 0.30051551777960184 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.33754079587338753 step loss 0.2902281956425082 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2979583363513365 step loss 0.5535757176672355 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.2560376819566245 step loss 0.317482220030642 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.23342053877684785 step loss 0.285559235787496 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2941459574845776 step loss 0.283339030795168 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.253804805694507 step loss 0.2821701700847701 sigmoid temp 1.0\n End sigmoid loss 0.27694375783764374 step loss 0.28211412077414877\n beta 0.3189380318706746\n rssi_w [-0.01563539 0.04994793 0.29846174 0.04497514]\n rssi_th [33.0100867 27.00981929 38.99981351]\n infect_w [0.01009629 0.06257141 0.29982455]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.314431752960136 step loss 1.3223808409892337\n iter 0 sigmoid loss 1.0759138775539359 step loss 0.7317077515354588 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3454184108267995 step loss 0.330822231178298 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3625657197554941 step loss 0.31805266467399923 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3099096490193773 step loss 0.30815852269488203 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3367237385775653 step loss 0.2946497053107379 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.29421561235440014 step loss 0.29308202272137457 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.25772315453353195 step loss 0.281640689359193 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.22826954898652813 step loss 0.2806432100951332 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2812712329182976 step loss 0.2807752707018698 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.24316066811962977 step loss 0.2795575323582204 sigmoid temp 1.0\n End sigmoid loss 0.27076288414010735 step loss 0.2797176575118505\n beta 0.34214272559275327\n rssi_w [-0.04834133 -0.03303423 0.11643114 0.30446053]\n rssi_th [13.00561404 33.0056144 16.00453658]\n infect_w [0.01037641 0.06674242 0.3237918 ]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.1360164106822468 step loss 1.1594077049401923\n iter 0 sigmoid loss 0.9226898994047119 step loss 0.881699584307315 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3447300594788919 step loss 0.33690844808999 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.359397899639995 step loss 0.3248854310213846 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.30207885745859847 step loss 0.31890956532086456 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3363709486149912 step loss 0.31702291494556284 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.29367699023145855 step loss 0.3737019028574219 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.2732222263239344 step loss 0.314808378874788 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.25300941727592285 step loss 0.3063724230940409 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2910865674946439 step loss 0.3029864787099586 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2509768667767587 step loss 0.3015879776365636 sigmoid temp 1.0\n End sigmoid loss 0.2875716933333297 step loss 0.30173233143928085\n beta 0.3819275465859583\n rssi_w [-0.00133491 0.05422526 0.35900911 0.07284581]\n rssi_th [29.98777505 38.98744486 23.99938031]\n infect_w [0.01104492 0.07244988 0.36374857]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.2403519668438656 step loss 1.2490852580666376\n iter 0 sigmoid loss 1.032214014006301 step loss 0.867480483443673 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3463001030266875 step loss 0.33955959339768405 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3629528043739739 step loss 0.33079954989487065 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3128960099281651 step loss 0.32652306274269177 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.34278448432940706 step loss 0.32245603872072054 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3048556793162368 step loss 0.3689122821855853 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.2944052874933162 step loss 0.3705178964001299 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2900462691737964 step loss 0.36841679192008675 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.33615207228610167 step loss 0.36798555368067787 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2768485580437252 step loss 0.3678325509947183 sigmoid temp 1.0\n End sigmoid loss 0.31629904388682406 step loss 0.36768966712358037\n beta 0.29173950815964955\n rssi_w [-0.02533958 0.01802206 0.20917493 0.18011978]\n rssi_th [27.00841907 25.00835387 30.99784968]\n infect_w [0.01045732 0.0630152 0.27083847]\n best loss 0.2797176575118505\n best scoring parameters\n beta 0.34214272559275327\n rssi_w [-0.04834133 -0.08137557 0.03505558 0.3395161 ]\n rssi_th [-106.99438596 -73.98877156 -57.98423498]\n infect_w [0.01037641 0.07711883 0.40091062]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 1.531, median size 2\n \t Negative bags: mean size 1.530, median size 2\n assign_mat size, X_shuff size: (4431, 6781) (6781, 3)\n assign_mat_trn size, assign_mat_tst size (3544, 6781) (887, 6781)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.2542579502446776 step loss 1.2487382612923148\n iter 0 sigmoid loss 1.0991440368194825 step loss 0.7495445106839727 sigmoid temp 0.1\n iter 500 sigmoid loss 0.33032400626120334 step loss 0.36628090369556265 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.41974103946148633 step loss 0.3547656606709759 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.37204511361807846 step loss 0.347653789696991 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3406001112522178 step loss 0.4036199915562351 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.30990460618709664 step loss 0.4444212303274524 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.30229686299501124 step loss 0.43658300450524196 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2832648501719038 step loss 0.4184632280591647 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.32453606002701746 step loss 0.4026785886173063 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.37495171800940574 step loss 0.3490513812784508 sigmoid temp 1.0\n End sigmoid loss 0.3371028905955947 step loss 0.3481930017405456\n beta 0.26619442821941275\n rssi_w [-0.00820623 0.01422462 0.21210928 0.13712829]\n rssi_th [18.01018765 36.01016367 32.99884028]\n infect_w [0.00944253 0.04811085 0.24852945]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.4386797658024109 step loss 1.437051178897278\n iter 0 sigmoid loss 1.26710050584824 step loss 0.540926449889315 sigmoid temp 0.1\n iter 500 sigmoid loss 0.33710987817393984 step loss 0.37760118046583463 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4228594436300864 step loss 0.3630179279868117 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3776594556156215 step loss 0.3559227381650932 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.33448178270814977 step loss 0.38326814155192285 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3066721859569305 step loss 0.4287893896730894 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.30562287351209627 step loss 0.4009645384006627 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.29224453283880003 step loss 0.39318677139244596 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.33107559809742104 step loss 0.383451019446852 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3996469968631663 step loss 0.37731657674175195 sigmoid temp 1.0\n End sigmoid loss 0.34887275574179116 step loss 0.3752514998105998\n beta 0.24507336526525803\n rssi_w [-0.07370648 -0.0252912 0.07195639 0.23113137]\n rssi_th [27.01000343 10.00993883 15.00963243]\n infect_w [0.01057806 0.04871464 0.25377082]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.2145943589558894 step loss 1.2292449096664293\n iter 0 sigmoid loss 1.0657432882995377 step loss 0.7813910710366821 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3276137433260783 step loss 0.358233910930718 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.41508424386466586 step loss 0.34236649022081955 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.36730344393186437 step loss 0.32975119174741874 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3300639430405276 step loss 0.49979604588623766 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3053861670318048 step loss 0.543407607064737 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.29371004814447277 step loss 0.5078765095440005 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2704012278262117 step loss 0.32778058524572434 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.31698399266707566 step loss 0.31781969888418493 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.31492519133804886 step loss 0.3168103721174622 sigmoid temp 1.0\n End sigmoid loss 0.31043643090370043 step loss 0.31669828216149226\n beta 0.2929171922210315\n rssi_w [-0.02779731 0.05170849 0.26305149 0.1024189 ]\n rssi_th [35.01225472 24.01199559 30.99939719]\n infect_w [0.00843376 0.04895851 0.27700931]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.1565252853743866 step loss 1.157828979867429\n iter 0 sigmoid loss 1.0141812626301707 step loss 0.7999139062573913 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3300350761205623 step loss 0.36078568683330753 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.41891618024698085 step loss 0.34673737999570814 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.37149819373131676 step loss 0.33654393882703854 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.33920007544697034 step loss 0.34077328018130315 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3091270281657758 step loss 0.48882039223925816 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.33685400055753106 step loss 0.3554555615783302 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3111267887904567 step loss 0.3487347387723489 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.330637352697524 step loss 0.34542802097490233 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3930632843702508 step loss 0.34140681203905016 sigmoid temp 1.0\n End sigmoid loss 0.3364206885214906 step loss 0.3368920906119168\n beta 0.14682351562035334\n rssi_w [0.08292439 0.12534931 0.39435069 0.07400231]\n rssi_th [29.00632536 28.00621409 36.99969133]\n infect_w [0.00671981 0.03269384 0.15098285]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.211002781684638 step loss 1.2453393225844651\n iter 0 sigmoid loss 1.0602174116097216 step loss 0.7758903826781265 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3234545002479548 step loss 0.35488784285481795 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4061939142812665 step loss 0.33750105316544055 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3574537604012478 step loss 0.33722094612732323 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.30536489883608153 step loss 0.6355887406828218 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2982555164212942 step loss 0.6406657923450457 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.29692938848060524 step loss 0.3673220005953579 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2740566414020418 step loss 0.31499985225024557 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.31079578982933553 step loss 0.30849609850343557 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.30445453980848597 step loss 0.307813102304284 sigmoid temp 1.0\n End sigmoid loss 0.29929800452213684 step loss 0.3078427541738534\n beta 0.32133188337658536\n rssi_w [-0.00046041 0.02589036 0.24405924 0.19126697]\n rssi_th [25.00418659 36.00412456 14.99856078]\n infect_w [0.00852617 0.04813495 0.30638078]\n best loss 0.3078427541738534\n best scoring parameters\n beta 0.32133188337658536\n rssi_w [-4.60413277e-04 2.54299438e-02 2.69489184e-01 4.60756159e-01]\n rssi_th [-94.99581341 -58.99168885 -43.99312806]\n infect_w [0.00852617 0.05666112 0.3630419 ]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 2.442, median size 2\n \t Negative bags: mean size 2.460, median size 2\n assign_mat size, X_shuff size: (4431, 10889) (10889, 3)\n assign_mat_trn size, assign_mat_tst size (3544, 10889) (887, 10889)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.0009368453708052 step loss 1.0248526300033787\n iter 0 sigmoid loss 0.8231362585072307 step loss 0.80889379428807 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3628610158793373 step loss 0.4012783201372545 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.36172196094055536 step loss 0.3941323763140489 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.34934776868486694 step loss 0.3914627814591779 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4367228915978661 step loss 0.3969166380659574 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3252530793532269 step loss 0.48469674641811367 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.39182889847809826 step loss 0.3922787187857123 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.38475436692661263 step loss 0.3852666409476967 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4190733923236498 step loss 0.3847234319673131 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.39157700924857425 step loss 0.38462227368192925 sigmoid temp 1.0\n End sigmoid loss 0.37826725600014494 step loss 0.38510683956852026\n beta 0.3302946622290136\n rssi_w [-0.04149985 0.08944592 0.29476089 0.06709602]\n rssi_th [37.99055327 36.98963673 18.99927916]\n infect_w [0.00883776 0.04646333 0.31384184]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.0169572395096111 step loss 1.0203426143383083\n iter 0 sigmoid loss 0.8412057684413875 step loss 0.767353303559227 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3678601794554108 step loss 0.40213354739378115 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3695608839609963 step loss 0.397648774933657 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.35900305527165016 step loss 0.39624251882817163 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4446735743051898 step loss 0.39415603956653644 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.33104566778180955 step loss 0.39244676338173845 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.397821579863171 step loss 0.3923947204503723 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3900990714252515 step loss 0.3919042920844357 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4202575141902929 step loss 0.3905120285741393 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.40110131135992083 step loss 0.3902550421131676 sigmoid temp 1.0\n End sigmoid loss 0.38577807715733203 step loss 0.3907950402788242\n beta 0.3335286758414767\n rssi_w [-0.07584819 -0.06677438 0.19058492 0.23899636]\n rssi_th [11.0005262 35.00052862 31.99398383]\n infect_w [0.01031902 0.05503392 0.3159286 ]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.0484433033742078 step loss 1.0587712699886058\n iter 0 sigmoid loss 0.8622807955403934 step loss 0.7670525044417578 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3629512560785062 step loss 0.38651290856272463 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.35881933266672017 step loss 0.3735329989261551 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.34054715434828803 step loss 0.36894619769280046 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.41951123149628833 step loss 0.541875603843097 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.32330832482106237 step loss 0.5689730730368658 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.357686665528317 step loss 0.48570311618842565 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.34634188376798747 step loss 0.4204324095957439 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3985344534770286 step loss 0.3637922260118371 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3648880718368695 step loss 0.36013264034437176 sigmoid temp 1.0\n End sigmoid loss 0.35149434911927163 step loss 0.3605828786598372\n beta 0.26986989814782303\n rssi_w [-0.01655406 0.02685145 0.15609169 0.203555 ]\n rssi_th [27.00710348 29.00703766 21.99790526]\n infect_w [0.00682411 0.03888786 0.25217181]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.021225537552424 step loss 1.0174427860005293\n iter 0 sigmoid loss 0.8422612533781575 step loss 0.7482279136250211 sigmoid temp 0.1\n iter 500 sigmoid loss 0.36889547851892823 step loss 0.3900729636002297 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.370134259979335 step loss 0.3813461153692668 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.35829194557512195 step loss 0.3733966558719989 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4378453946600935 step loss 0.4168825216240139 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3273189027184783 step loss 0.5102442485853002 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3752551399616602 step loss 0.4674887613737611 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.36226441503490264 step loss 0.4340540299336691 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.40403639990647877 step loss 0.3787452138671756 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3790056199808578 step loss 0.36950699873382176 sigmoid temp 1.0\n End sigmoid loss 0.3614832262563044 step loss 0.369813211204048\n beta 0.21193359183980787\n rssi_w [-0.02447912 -0.02369076 0.05872583 0.24376672]\n rssi_th [11.01113978 25.01114586 19.01097838]\n infect_w [0.00608109 0.0337888 0.20483724]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.1630920721450544 step loss 1.1778537357295034\n iter 0 sigmoid loss 0.9560840221778392 step loss 0.72463120453073 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3619910516719601 step loss 0.3850839928406265 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.35665167743095594 step loss 0.3699143093813687 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.33757430179609194 step loss 0.3637993720242214 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4171085368497141 step loss 0.5984291844121125 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.32255441301808885 step loss 0.6328150684560299 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.34344936461604286 step loss 0.5057578985171322 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.33197177895213076 step loss 0.35361564733886125 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.385408963562775 step loss 0.34864595236928425 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3612986607373753 step loss 0.34748513149915033 sigmoid temp 1.0\n End sigmoid loss 0.3412614695223663 step loss 0.3481679213500683\n beta 0.28151688568890376\n rssi_w [-0.0305788 0.04681709 0.19106086 0.1788425 ]\n rssi_th [35.00722378 24.00703123 18.99867989]\n infect_w [0.00705589 0.03596621 0.26577491]\n best loss 0.3481679213500683\n best scoring parameters\n beta 0.28151688568890376\n rssi_w [-0.0305788 0.01623829 0.20729915 0.38614165]\n rssi_th [-84.99277622 -60.985745 -41.9870651 ]\n infect_w [0.00705589 0.0430221 0.30879701]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 3.434, median size 3\n \t Negative bags: mean size 3.549, median size 3\n assign_mat size, X_shuff size: (4431, 15644) (15644, 3)\n assign_mat_trn size, assign_mat_tst size (3544, 15644) (887, 15644)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.0989571474625857 step loss 1.1018578231800826\n iter 0 sigmoid loss 1.1491678917969221 step loss 0.6052432478972876 sigmoid temp 0.1\n iter 500 sigmoid loss 0.43019727367072585 step loss 0.4303060231518011 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4221958864305085 step loss 0.42951982774716935 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4159549802963056 step loss 0.4298024230291397 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.47475275498545827 step loss 0.4299192160000223 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.428945102741489 step loss 0.4306723208746355 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.42646279131652903 step loss 0.42910824760835076 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5405900096016509 step loss 0.42926517851668833 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.34354036220633405 step loss 0.42916078951635594 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.44435030878977727 step loss 0.4290950126184315 sigmoid temp 1.0\n End sigmoid loss 0.42913390821474634 step loss 0.42913875156407705\n beta 0.18132076678712963\n rssi_w [-0.03284964 -0.01530286 0.03611927 0.14672396]\n rssi_th [22.00132156 10.00134886 10.001404 ]\n infect_w [0.01062208 0.02639868 0.16054654]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.2119670796864503 step loss 1.2229387054156347\n iter 0 sigmoid loss 1.2683595662648006 step loss 0.570528102632815 sigmoid temp 0.1\n iter 500 sigmoid loss 0.42813118261076055 step loss 0.417124536305877 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4038746277819699 step loss 0.39759907954873974 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3840140172937482 step loss 0.7650994889189427 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4199193013336431 step loss 0.8689976152125455 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.38364709348881654 step loss 0.8811888709028907 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3574572382830311 step loss 0.38289884093419563 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4669730734935383 step loss 0.36973486761395324 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.28145735030036656 step loss 0.3700513715525977 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3912889287350083 step loss 0.37003353814577594 sigmoid temp 1.0\n End sigmoid loss 0.36189237967312826 step loss 0.37063758401831925\n beta 0.2873728333601221\n rssi_w [-0.01242542 0.03102715 0.26516283 0.09476956]\n rssi_th [30.00057151 35.00046219 23.9995277 ]\n infect_w [0.00624657 0.03341343 0.28058031]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.089235446815925 step loss 1.0867601230048185\n iter 0 sigmoid loss 1.1384517746100855 step loss 0.6357239189450119 sigmoid temp 0.1\n iter 500 sigmoid loss 0.42497498357951646 step loss 0.4243888699108853 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4117079770049823 step loss 0.4174013021488541 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.39683855143774827 step loss 0.410110564388828 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4326121988220846 step loss 0.4144896440304075 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3801074999034559 step loss 0.4315787615785423 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3490388213558331 step loss 0.4007558547017735 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.47992287676022516 step loss 0.4001026047967735 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3001034681846128 step loss 0.40180583662212876 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.4206628851940893 step loss 0.40039472266095844 sigmoid temp 1.0\n End sigmoid loss 0.38871202477845457 step loss 0.4014085668291584\n beta 0.3192839032577609\n rssi_w [-0.05132325 -0.03842799 0.11761297 0.2893943 ]\n rssi_th [16.98970557 26.98970409 27.98812128]\n infect_w [0.00991987 0.03800155 0.30651046]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.2374652044087817 step loss 1.2697797102512385\n iter 0 sigmoid loss 1.2976138857638984 step loss 0.5763021466150031 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4241994071074896 step loss 0.4116796711420008 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4079279671152121 step loss 0.39619283423932644 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.39126940321822007 step loss 0.4218825181625615 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4249241941451551 step loss 0.7093116996353145 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.38229500069968725 step loss 0.7345878140197787 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.38007825460196515 step loss 0.4127735321746032 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.48574342987301306 step loss 0.3791740754076931 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2860272335881121 step loss 0.3782040757862411 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3918948375370654 step loss 0.377177215459471 sigmoid temp 1.0\n End sigmoid loss 0.3688068899537123 step loss 0.37754337545691624\n beta 0.2456382403564049\n rssi_w [-0.00466312 0.02261362 0.24071925 0.07923244]\n rssi_th [26.00967569 35.00961444 31.99964521]\n infect_w [0.00606669 0.03059335 0.23279802]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 0.9990187365202167 step loss 1.019347885136546\n iter 0 sigmoid loss 1.0410919190990617 step loss 0.7029442166259489 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4219953137915449 step loss 0.411482232579493 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.40595504998153326 step loss 0.39661792945829033 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3882560140898776 step loss 0.5540792308147697 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.42270296800306795 step loss 0.7158349577557306 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.38143377942134804 step loss 0.737033081734651 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3774064329720085 step loss 0.4055128882817777 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4850261034436116 step loss 0.3783232867447902 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2852441221257129 step loss 0.3776859913850543 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3926951459609441 step loss 0.37677072302007253 sigmoid temp 1.0\n End sigmoid loss 0.3684148357709782 step loss 0.3771969653459169\n beta 0.2556671974656254\n rssi_w [-0.01794022 0.03459107 0.22175838 0.11948838]\n rssi_th [35.00910756 26.00896472 24.99929547]\n infect_w [0.00630109 0.03116419 0.23967559]\n best loss 0.37063758401831925\n best scoring parameters\n beta 0.2873728333601221\n rssi_w [-0.01242542 0.01860172 0.28376456 0.37853411]\n rssi_th [-89.99942849 -54.9989663 -30.9994386 ]\n infect_w [0.00624657 0.03966 0.3202403 ]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 4.111, median size 4\n \t Negative bags: mean size 3.988, median size 3\n assign_mat size, X_shuff size: (4431, 17757) (17757, 3)\n assign_mat_trn size, assign_mat_tst size (3544, 17757) (887, 17757)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 0.9748990167748784 step loss 0.9859036567571016\n iter 0 sigmoid loss 0.9204370479055964 step loss 0.7205776203992035 sigmoid temp 0.1\n iter 500 sigmoid loss 0.48068134312406 step loss 0.41566616313443766 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.42690976696065575 step loss 0.4074721516275898 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.358682439677635 step loss 0.406635593633988 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.40626159439451826 step loss 1.1298781195394467 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.346180246001649 step loss 1.2188714588124245 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3371343632484743 step loss 0.40778335353739853 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.32875233721474884 step loss 0.3965907613235664 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3659329837963686 step loss 0.39277077285224915 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.40592914292714816 step loss 0.3912566532701605 sigmoid temp 1.0\n End sigmoid loss 0.3792666618154413 step loss 0.3938106767867642\n beta 0.31213998890878825\n rssi_w [-0.04467326 0.06984278 0.29299468 0.04111122]\n rssi_th [37.98794502 32.98738379 29.99973622]\n infect_w [0.00488498 0.03825571 0.30008489]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.152542444634709 step loss 1.1522866583609814\n iter 0 sigmoid loss 1.0858167025861456 step loss 0.6014132656539012 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4832140267495075 step loss 0.41256665937311987 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.43022222063782567 step loss 0.40195501631098035 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.35979231327915884 step loss 0.3912787577576726 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.40630284633328256 step loss 0.873480293500598 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.351230999589402 step loss 0.9469848240254967 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.33939729376038286 step loss 0.3893535992202328 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.31228573817122457 step loss 0.3785803080589754 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3389561677099671 step loss 0.37275639823519263 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.389688210338941 step loss 0.3713697156095296 sigmoid temp 1.0\n End sigmoid loss 0.3603329659509825 step loss 0.3770192666736725\n beta 0.27512365585448106\n rssi_w [-0.03679403 -0.01900935 0.07385382 0.29175778]\n rssi_th [18.99482058 19.99482397 27.99438988]\n infect_w [0.00583846 0.03403145 0.29139881]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.1597757367794224 step loss 1.1580803535819795\n iter 0 sigmoid loss 1.1011447432798462 step loss 0.5617375015403211 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4823674233072873 step loss 0.41294184464269407 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4272972456620168 step loss 0.39914260661616086 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3573633064707212 step loss 0.3903061870890647 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.39992844613333306 step loss 0.6591298642428416 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.36723951868724625 step loss 0.4674436792622308 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.34261832979773277 step loss 0.3885454792259474 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3192256841696225 step loss 0.3778430668941695 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3382443046100865 step loss 0.37235303248433815 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3895818842608566 step loss 0.3711448202500897 sigmoid temp 1.0\n End sigmoid loss 0.36007935471147745 step loss 0.37597411229998257\n beta 0.2336049182398824\n rssi_w [-0.04801234 -0.03228204 0.10058249 0.30689587]\n rssi_th [22.99510837 19.99512424 23.99424339]\n infect_w [0.00617988 0.03857789 0.31589681]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 0.9054840584248571 step loss 0.9128406414394724\n iter 0 sigmoid loss 0.8610691703889679 step loss 0.6880609068382071 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4832408382458469 step loss 0.4171116281463462 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4310355092018325 step loss 0.41039150607042313 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.36111102000859474 step loss 0.40375199495803615 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4106125035354161 step loss 0.4298056050266513 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.44379451848360346 step loss 0.43797578360279893 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4021829426619099 step loss 0.431245105312617 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.372676774577981 step loss 0.4246252405338701 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.40398728081359897 step loss 0.421373533969354 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.43679432779395777 step loss 0.4158077196241885 sigmoid temp 1.0\n End sigmoid loss 0.40470323843500783 step loss 0.4087460914588197\n beta 0.07536484219740058\n rssi_w [-0.02253801 -0.0067258 0.08045713 0.26329795]\n rssi_th [19.99400157 20.99403192 30.99339984]\n infect_w [0.10592904 0.08796898 0.3552571 ]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.1067272503033452 step loss 1.1115718601801887\n iter 0 sigmoid loss 1.0502086881486052 step loss 0.6732712002007268 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4823979767143507 step loss 0.41582542594842986 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.42901097496177143 step loss 0.4067246366256657 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.36016451263982235 step loss 0.40384476046103407 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.41060781824615517 step loss 1.0947507774682055 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.34674874653780835 step loss 1.1815050986917504 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.336159067952802 step loss 0.4067184808833454 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3230778763091036 step loss 0.3926995721898426 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.35841104268777735 step loss 0.3881869705657731 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.4008175564446487 step loss 0.38650891528584413 sigmoid temp 1.0\n End sigmoid loss 0.3736397474248406 step loss 0.389879186423813\n beta 0.30955920915222135\n rssi_w [-0.02489753 0.0481256 0.29801111 0.02520082]\n rssi_th [33.98849526 35.98819671 38.99988173]\n infect_w [0.00510078 0.03584486 0.29729084]\n best loss 0.37597411229998257\n best scoring parameters\n beta 0.2336049182398824\n rssi_w [-0.04801234 -0.08029438 0.0202881 0.32718397]\n rssi_th [-97.00489163 -77.00976739 -53.015524 ]\n infect_w [0.00617988 0.04475777 0.36065458]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 4.037, median size 3\n \t Negative bags: mean size 3.992, median size 3\n assign_mat size, X_shuff size: (4431, 17719) (17719, 3)\n assign_mat_trn size, assign_mat_tst size (3544, 17719) (887, 17719)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 0.9584355952189165 step loss 0.9595069357448618\n iter 0 sigmoid loss 0.9155362039793868 step loss 0.6752425610384393 sigmoid temp 0.1\n iter 500 sigmoid loss 0.44330682317406406 step loss 0.4239566373152859 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4413084124380372 step loss 0.41351743661258245 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3404559227328157 step loss 0.4058181748698215 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3682367589200335 step loss 0.9318070198402737 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3749802592091828 step loss 1.0089297398063108 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.45270445429834794 step loss 0.3963004660913007 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.35029233394839937 step loss 0.38915704120736727 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3475395871499697 step loss 0.3920814797293475 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.30526468054531264 step loss 0.3859755813327383 sigmoid temp 1.0\n End sigmoid loss 0.3706779935799182 step loss 0.38651332769284164\n beta 0.2949379518917382\n rssi_w [-0.02083399 -0.00368505 0.04657364 0.28543612]\n rssi_th [19.99435535 9.99436729 36.99420367]\n infect_w [0.00637619 0.02707732 0.28362898]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.0138017550285168 step loss 1.0256546806081344\n iter 0 sigmoid loss 0.9864079503365781 step loss 0.6784153598565799 sigmoid temp 0.1\n iter 500 sigmoid loss 0.43979824897093067 step loss 0.43347518476337255 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4368671576406108 step loss 0.4278280189570286 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3407430177530455 step loss 0.4269559566314634 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3792049194763557 step loss 0.44043125939459465 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3722549596178888 step loss 0.5202180820738351 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4940794675369483 step loss 0.42425933809560185 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.38633359349117724 step loss 0.42221971894193017 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.35854694940279286 step loss 0.4224683558308123 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.30470171812167707 step loss 0.422024863177062 sigmoid temp 1.0\n End sigmoid loss 0.41770753059157606 step loss 0.4219408704439557\n beta 0.3040799584879238\n rssi_w [-0.05819764 0.09544139 0.26171356 0.07078446]\n rssi_th [38.99352128 38.99245134 14.99919075]\n infect_w [0.00929784 0.03873178 0.28991958]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 0.9746253547817743 step loss 0.9735765249305646\n iter 0 sigmoid loss 0.931087828181728 step loss 0.6573153593484814 sigmoid temp 0.1\n iter 500 sigmoid loss 0.44385676703578186 step loss 0.42392621701631783 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4430958483340802 step loss 0.4163069058981526 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.34517216449511573 step loss 0.408610355698138 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3771845938640909 step loss 0.5193985479359687 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3797174405647921 step loss 0.5532277305555318 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.45946059306428977 step loss 0.5259614331752199 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3749086625005987 step loss 0.46090629116178283 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3353888036484151 step loss 0.4537383749257938 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.29288040128234494 step loss 0.41393621278548076 sigmoid temp 1.0\n End sigmoid loss 0.401287047060194 step loss 0.4124267397071571\n beta 0.19649131493816296\n rssi_w [-0.03739503 -0.02244763 0.06617088 0.1878629 ]\n rssi_th [16.00859271 23.00858779 16.00836327]\n infect_w [0.00614487 0.02404731 0.18514966]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 0.9877497029337007 step loss 0.9974353777040189\n iter 0 sigmoid loss 0.9505657346040545 step loss 0.658009446086451 sigmoid temp 0.1\n iter 500 sigmoid loss 0.44070108425446475 step loss 0.42043000755785187 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4335668877127297 step loss 0.40675983463846016 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3174385451688586 step loss 0.3875504087594457 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3505832120232563 step loss 0.38091119395678297 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.37586555920865244 step loss 0.3811394567968361 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4423536893168996 step loss 0.37536973167824284 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3431746867583359 step loss 0.3751973391236418 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3255857978257046 step loss 0.37794756566718213 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.29627796233145637 step loss 0.375840165759274 sigmoid temp 1.0\n End sigmoid loss 0.3650240756447838 step loss 0.3756677560178296\n beta 0.28135943247177264\n rssi_w [-0.07310736 -0.00284085 0.09900217 0.2554516 ]\n rssi_th [38.00308203 10.00298624 17.00126918]\n infect_w [0.00582939 0.02755165 0.27153014]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.0069936172229015 step loss 1.0087008191829636\n iter 0 sigmoid loss 0.979822516634024 step loss 0.6084432120245528 sigmoid temp 0.1\n iter 500 sigmoid loss 0.44211491371467326 step loss 0.42120138054691286 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4373179583337275 step loss 0.40639168664940567 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3298535064865496 step loss 0.39725737670134775 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.35756767995416483 step loss 0.7917337828834067 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.37557014755077717 step loss 0.8455885135647134 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.44920927619332957 step loss 0.38686193788397766 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.34458356758771136 step loss 0.3788719972030017 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3303508989710538 step loss 0.3825570222984718 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.29961822864174914 step loss 0.37699547246921594 sigmoid temp 1.0\n End sigmoid loss 0.3669119694932571 step loss 0.3768125801558805\n beta 0.2759777129572981\n rssi_w [-0.03090733 -0.02230944 0.0743536 0.26903605]\n rssi_th [16.00100298 23.00100458 26.00062674]\n infect_w [0.00573934 0.02682251 0.26771483]\n best loss 0.3756677560178296\n best scoring parameters\n beta 0.28135943247177264\n rssi_w [-0.07310736 -0.0759482 0.02305397 0.27850557]\n rssi_th [-81.99691797 -71.99393173 -54.99266255]\n infect_w [0.00582939 0.03338104 0.30491117]\n\n\n\n```python\n# the above two cells are deleted because of plotting redundancy\n\nfig, axs = plt.subplots(1, 2, figsize=(15,5))\naxs[0].plot(bag_size, np.mean(auc_train_learned, axis=1), '+-', label='train, learned')\naxs[0].plot(bag_size, np.mean(auc_train_true, axis=1), '+-', label='train, true')\naxs[0].set_xlabel(\"Maximum Bag Size\")\naxs[0].set_ylabel(\"AUC\")\naxs[0].set_title(\"Train: No Censoring\")\naxs[0].legend()\n\naxs[1].plot(bag_size, np.mean(auc_test_learned, axis=1), '+-', label='test, learned')\naxs[1].plot(bag_size, np.mean(auc_test_true, axis=1), '+-', label='test, true')\naxs[1].set_xlabel(\"Maximum Bag Size\")\naxs[1].set_ylabel(\"AUC\")\naxs[1].set_title(\"Test: No Censoring\")\naxs[1].legend()\n```\n\n### 2. Varying censoring probability\n\n[Return to top](#Notebook-contents)\n\nThis experiment is iterated over censoring probabilities to show the effect that increasing censoring probability has on the performance of the learning algorithm. Increasing censoring probability would reduce accuracy in assigning a risk score, since there is less signal when some positive exposures are censored or not taken into account in learning the model. With less data, the algorithm will definitely be biased against calculating a higher risk score. \n\nWe expect the performance of the algorithm to decrease, i.e. the AUC score to fall, as censoring probability increases. This is reflected in the below too sets of experiments. When we set the maximum bag size to be 16, two plots shows that the AUC drops from a high of around 0.8 to less than 0.6. We even try a differnt maximum bag size (32) to cross check, and again the AUC drops from 0.8 - 0.55. Overall, the trends for this experiment supports the claim being made, although it definitely begs the question of what is the point of making this claim and experiment, knowing that this is a fundamental design flaw that may not be helpful to pandemic efforts (e.g. being more cautious and compensating for potentially censored exposures, instead of just recognizing and accepting that the prediction and risk scoring will drop). A discussion on this is expounded on in part III.1, Evaluation.\n\n\n```python\n# max bag size 16\ncensor_prob = [0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7]\nn_trials = 1\nn_random_restarts_train = 5\n\nidx = 0\n\nauc_train_learned = np.zeros((len(censor_prob),n_trials))\nauc_train_true = np.zeros((len(censor_prob),n_trials))\nauc_test_learned = np.zeros((len(censor_prob),n_trials))\nauc_test_true = np.zeros((len(censor_prob),n_trials))\nfor prob in censor_prob:\n bag_sim = Bag_Simulator(p_pos=0.6,r_pos=2,p_neg=0.6,r_neg=2,max_bag_size=16,censor_prob_pos=prob,censor_prob_neg=0,max_pos_in_bag=1)\n auc_train_trials, auc_test_trials = train_and_eval_with_bag_config(bag_sim, X_epi,\n probabilities_true_epi, n_trials=n_trials,\n n_random_restarts=n_random_restarts_train)\n for i in range(n_trials):\n auc_train_learned[idx, i] = dict(auc_train_trials[i])['Learned']\n auc_train_true[idx, i] = dict(auc_train_trials[i])['True']\n auc_test_learned[idx, i] = dict(auc_test_trials[i])['Learned']\n auc_test_true[idx, i] = dict(auc_test_trials[i])['True']\n \n idx += 1\n```\n\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 4.000, median size 3\n \t Negative bags: mean size 3.971, median size 3\n assign_mat size, X_shuff size: (4431, 17617) (17617, 3)\n assign_mat_trn size, assign_mat_tst size (3532, 17617) (884, 17617)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 0.9485228239751493 step loss 0.9524340213955766\n iter 0 sigmoid loss 1.0343423184328464 step loss 0.6536440470489701 sigmoid temp 0.1\n iter 500 sigmoid loss 0.45214364711373617 step loss 0.42511188606857947 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3571010128038134 step loss 0.4195549973570463 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3435219386823791 step loss 0.41440946804973894 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.35325216987076385 step loss 0.5087466194759144 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3700036360287546 step loss 1.145408216296316 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.35588216299828945 step loss 0.41873685690255147 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.37595680910828255 step loss 0.40436777788332756 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.28164639033773736 step loss 0.40285287761041194 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.38987522206750796 step loss 0.40215533607564347 sigmoid temp 1.0\n End sigmoid loss 0.39344512772269913 step loss 0.40321511359522894\n beta 0.2947369887567163\n rssi_w [-0.02575168 -0.01621143 0.07142658 0.2794411 ]\n rssi_th [13.98907232 21.98907775 34.98866313]\n infect_w [0.00724824 0.03873066 0.28251154]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.14943818398294 step loss 1.1508897243176852\n iter 0 sigmoid loss 1.2557359585516663 step loss 0.5557833168387539 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4508196587804173 step loss 0.4275497969088081 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.35547297680540507 step loss 0.42168086961587464 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3395045998501148 step loss 0.42230080035011863 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.35170631784315 step loss 0.47126115663349716 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3880691986365569 step loss 0.503058190888692 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.41941342697912776 step loss 0.42516980241927976 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.476709972279983 step loss 0.4248801562847659 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3220774160970542 step loss 0.4245700188280031 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.4499855058706702 step loss 0.4250450174758641 sigmoid temp 1.0\n End sigmoid loss 0.4241541966654932 step loss 0.4246325811225136\n beta 0.02832020345723263\n rssi_w [0.24586174 0.27157564 0.35787701 0.17132981]\n rssi_th [33.00311548 18.00314873 29.99882307]\n infect_w [0.00710947 0.03132744 0.13540038]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.1304133199294242 step loss 1.1377520168495279\n iter 0 sigmoid loss 1.2363686285478745 step loss 0.5831857728900853 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4496449356136363 step loss 0.4260296490993249 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.35410033815079983 step loss 0.42073701360069365 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3371250959310653 step loss 0.4228881254935475 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3496888316439012 step loss 0.46747892965393 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.37915072213385625 step loss 0.48816113861820476 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3604408903221612 step loss 0.47979283347073026 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.42064154651723357 step loss 0.46680568432696795 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.31073096103622805 step loss 0.45254805595411923 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.41764578651059964 step loss 0.4478320891827461 sigmoid temp 1.0\n End sigmoid loss 0.41308277781443875 step loss 0.44774814315593897\n beta 0.23317912939125873\n rssi_w [-0.06249766 0.01661313 0.12618638 0.19569143]\n rssi_th [36.00253939 15.00238247 27.99636375]\n infect_w [0.01196022 0.04126821 0.22465949]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.015521709248962 step loss 1.0236195851683034\n iter 0 sigmoid loss 1.1090438251954853 step loss 0.6510692594580176 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4472961125835201 step loss 0.4147531359000531 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3507394110809725 step loss 0.40375726370060594 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.33003924076800023 step loss 0.5170095533094967 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3469403707402416 step loss 0.6085852197998672 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.37593652643270414 step loss 0.5641841946140157 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.34329560749563526 step loss 0.5135708478286853 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.39130642785188924 step loss 0.40101083372908636 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2854403753106812 step loss 0.3938515238159006 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.38026866786094776 step loss 0.39093751699549295 sigmoid temp 1.0\n End sigmoid loss 0.3849005996629174 step loss 0.39162909583420014\n beta 0.22502367030542414\n rssi_w [-0.03992202 0.05117245 0.16654142 0.13518303]\n rssi_th [39.00866455 19.00843891 23.99886942]\n infect_w [0.0068536 0.03650233 0.21240824]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.082907238827845 step loss 1.09193641848689\n iter 0 sigmoid loss 1.1831446035407525 step loss 0.5870042710141307 sigmoid temp 0.1\n iter 500 sigmoid loss 0.44897088966852394 step loss 0.41376247431167207 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3508920728870821 step loss 0.39930739698736 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3272176638255874 step loss 0.43324625213717066 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.37183166583388627 step loss 0.4705377784714426 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.35940255824816547 step loss 0.4723046575423035 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3445822496398017 step loss 0.4176612468243527 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3856794766657225 step loss 0.4097568209336681 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.28569697442994135 step loss 0.40228548046264695 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.35286496327226813 step loss 0.39312629071048066 sigmoid temp 1.0\n End sigmoid loss 0.3761087762489566 step loss 0.3926933560458921\n beta 0.2140592703618992\n rssi_w [-0.03871878 0.02148719 0.04755239 0.23628181]\n rssi_th [37.00740956 14.00732435 10.00771133]\n infect_w [0.00617474 0.03486513 0.20947108]\n best loss 0.39162909583420014\n best scoring parameters\n beta 0.22502367030542414\n rssi_w [-0.03992202 0.01125043 0.17779185 0.31297488]\n rssi_th [-80.99133545 -61.98289654 -37.98402712]\n infect_w [0.0068536 0.04335592 0.25576416]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 4.044, median size 3\n \t Negative bags: mean size 3.999, median size 3\n assign_mat size, X_shuff size: (4431, 17753) (17753, 3)\n assign_mat_trn size, assign_mat_tst size (3533, 17753) (887, 17753)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.0599834949409723 step loss 1.1062477182182773\n iter 0 sigmoid loss 1.4051344173269846 step loss 0.6107159758085565 sigmoid temp 0.1\n iter 500 sigmoid loss 0.43534891794879216 step loss 0.42904225919784367 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.41032156211156606 step loss 0.4211831537444761 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4855695857297 step loss 0.4364954725314619 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.34451759182698827 step loss 0.993579087498757 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.40491072670632616 step loss 1.0520217417097737 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.31771335902707915 step loss 0.41456826248461537 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.46659016913515244 step loss 0.3990308561022598 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.30898678638579113 step loss 0.4008058130243649 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.389049494861305 step loss 0.3980956753969632 sigmoid temp 1.0\n End sigmoid loss 0.3900882935916594 step loss 0.39802476815877347\n beta 0.2919483028078872\n rssi_w [-0.01151736 0.04029357 0.26601983 0.08592018]\n rssi_th [31.99408322 35.99393058 18.99959966]\n infect_w [0.01145329 0.03194882 0.27620047]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.0670571800086088 step loss 1.0623264794717502\n iter 0 sigmoid loss 1.4116791129770043 step loss 0.5488023859967738 sigmoid temp 0.1\n iter 500 sigmoid loss 0.44038700731743147 step loss 0.43057351112953757 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.41605959155227734 step loss 0.4245035909108968 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.49748894954635325 step loss 0.42108460273633647 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.35562365338324325 step loss 0.4936779908743291 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4096309855039331 step loss 0.5187795456942368 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3693601646905407 step loss 0.42457615238991764 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5121919810437153 step loss 0.45714890179585477 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.35674651166030247 step loss 0.43244054963694495 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.38383056655954023 step loss 0.4219241782839748 sigmoid temp 1.0\n End sigmoid loss 0.4173506410105693 step loss 0.4212092914177328\n beta 0.15634688993172433\n rssi_w [-0.05108632 -0.00804121 0.07442572 0.20719607]\n rssi_th [32.00888283 12.00885515 10.008674 ]\n infect_w [0.01165317 0.02915182 0.15113625]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 0.9900218081249702 step loss 0.9878817371270485\n iter 0 sigmoid loss 1.3060513544764973 step loss 0.5906340084967605 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4411355857669449 step loss 0.4326544955707184 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.41700377693824026 step loss 0.4278583581849244 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.5009678172858747 step loss 0.4254598334995291 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3603349956682841 step loss 0.485112322340791 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.41031594507744296 step loss 0.5201888679308462 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3714413946606516 step loss 0.504698254118722 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5183634630291492 step loss 0.48239529001687664 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3623522425378898 step loss 0.4249026489338522 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3897396113131552 step loss 0.4295295724610477 sigmoid temp 1.0\n End sigmoid loss 0.42103952877360656 step loss 0.42771110755550656\n beta 0.183620429469633\n rssi_w [-0.06142202 -0.01029964 0.07686148 0.15757274]\n rssi_th [32.00763326 11.00756941 10.00738694]\n infect_w [0.01266182 0.03220065 0.15987909]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 0.9107695609239481 step loss 0.9211004573134386\n iter 0 sigmoid loss 1.2006807383621378 step loss 0.631883229726673 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4382706947719744 step loss 0.4266256245333012 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.41350903398636646 step loss 0.4161222224171302 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4913200594844489 step loss 0.4068220773064525 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.348667910751725 step loss 0.6983159009674679 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4075734886155 step loss 0.7348744096513725 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3240645617208411 step loss 0.4093419681719954 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.46504018102797884 step loss 0.3949912411679229 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3143934442026155 step loss 0.3970401889845131 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.37780717461402935 step loss 0.39258860990515343 sigmoid temp 1.0\n End sigmoid loss 0.3850075575529071 step loss 0.39274639616054025\n beta 0.2621638751059251\n rssi_w [-0.02587634 -0.01999205 0.07087278 0.24118029]\n rssi_th [14.00258364 29.0025847 21.00226237]\n infect_w [0.01046306 0.0310202 0.24515122]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 0.9636217882332689 step loss 0.96514180036121\n iter 0 sigmoid loss 1.2725944228160733 step loss 0.6066144225576261 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4377295245514257 step loss 0.42521745816455425 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4133443446584828 step loss 0.41433022918426404 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.48894942205642683 step loss 0.40212468340248025 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3474029450603537 step loss 0.5327690768355758 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.408254859672487 step loss 0.5449602094631594 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3435723855367614 step loss 0.40368418346344065 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.48264767449873447 step loss 0.397441853844039 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3246068385453376 step loss 0.3983566264532936 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3635080556042601 step loss 0.3966021563632778 sigmoid temp 1.0\n End sigmoid loss 0.39388078359715795 step loss 0.3966434229814081\n beta 0.2259173942891323\n rssi_w [-0.0416623 -0.00861539 0.07337219 0.20025934]\n rssi_th [32.00927737 14.00924981 14.00915292]\n infect_w [0.01037495 0.03249099 0.20572465]\n best loss 0.39274639616054025\n best scoring parameters\n beta 0.2621638751059251\n rssi_w [-0.02587634 -0.04586839 0.02500438 0.26618468]\n rssi_th [-105.99741636 -76.99483166 -55.99256929]\n infect_w [0.01046306 0.04148326 0.28663448]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 3.970, median size 3\n \t Negative bags: mean size 3.968, median size 3\n assign_mat size, X_shuff size: (4431, 17584) (17584, 3)\n assign_mat_trn size, assign_mat_tst size (3523, 17584) (884, 17584)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.0091192812980763 step loss 1.0171784881714627\n iter 0 sigmoid loss 0.9615358359978212 step loss 0.7050878943806502 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4570312594234545 step loss 0.4448243319748371 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4613930570784137 step loss 0.4368654583817928 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4410105007238011 step loss 0.4346942147296875 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4449085487468312 step loss 0.6071722021043133 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.43372214227493067 step loss 1.0066692139456777 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3922170578826427 step loss 0.4257798638694882 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.36243982700599 step loss 0.42283419323365734 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3550462737092562 step loss 0.4216491061736858 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.38809020990084536 step loss 0.4190891636895279 sigmoid temp 1.0\n End sigmoid loss 0.4103341706022644 step loss 0.4197662291159915\n beta 0.2685724704270086\n rssi_w [-0.00712551 0.03621298 0.25114947 0.02801793]\n rssi_th [28.99620539 37.99608298 37.99989344]\n infect_w [0.01563272 0.0236943 0.25348238]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.049152173833648 step loss 1.0498266529794034\n iter 0 sigmoid loss 1.004021473765049 step loss 0.6420516187673501 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4570624570482974 step loss 0.451084093190618 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4633643529247766 step loss 0.4463916269720146 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.44109040562950513 step loss 0.4450100749690276 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.44598335308238113 step loss 0.47419185480498144 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4359588816634663 step loss 0.49109751194013423 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.41428474077082583 step loss 0.48647493188989605 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.40143255203548506 step loss 0.4792421447447725 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3712569898075199 step loss 0.4735901721702228 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.4287315544860122 step loss 0.46107723614891816 sigmoid temp 1.0\n End sigmoid loss 0.43725261080247857 step loss 0.46020117120608584\n beta 0.2260951504322705\n rssi_w [-0.03317102 0.02232079 0.08911527 0.19004575]\n rssi_th [35.00058746 16.00051223 27.99716342]\n infect_w [0.02437053 0.02479763 0.20749323]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 0.9986099356031158 step loss 1.0105788815711287\n iter 0 sigmoid loss 0.9534682455165338 step loss 0.676352305853482 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4549225267159791 step loss 0.4436914660345952 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.46102273543632394 step loss 0.4346812051356985 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.44278493282760034 step loss 0.4239584508694613 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4437346875873551 step loss 0.41826053726787266 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4351591404914782 step loss 0.42319932619469575 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.40076666317377985 step loss 0.4120463836205764 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.36330989453288587 step loss 0.41310077012530094 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3532656456386959 step loss 0.41354408135854304 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.38694567799424745 step loss 0.41204247925037135 sigmoid temp 1.0\n End sigmoid loss 0.408252422621521 step loss 0.41271684545185744\n beta 0.25942930198242775\n rssi_w [-0.05664674 0.00276783 0.08105622 0.22761729]\n rssi_th [36.00112004 11.00105374 18.00050734]\n infect_w [0.01505614 0.02384547 0.24395094]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.0397123963081616 step loss 1.0359454659286025\n iter 0 sigmoid loss 0.9952559380828041 step loss 0.645200347451241 sigmoid temp 0.1\n iter 500 sigmoid loss 0.46050170917813094 step loss 0.45444529388153526 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.47547094223423664 step loss 0.4528024525770459 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4448117384159485 step loss 0.45284885542238945 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4568898844689771 step loss 0.4534924482863655 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4811908392399674 step loss 0.45335260815131234 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.40299227278618427 step loss 0.4528990900158876 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4051636484221848 step loss 0.4528376358194307 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4050051151936872 step loss 0.453037931096741 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.43388693329971945 step loss 0.45282372976035773 sigmoid temp 1.0\n End sigmoid loss 0.4528009903738666 step loss 0.45280110597814105\n beta 0.14962466659819143\n rssi_w [0.00534867 0.01094172 0.03378874 0.1094588 ]\n rssi_th [15.00051979 10.00054239 14.00061888]\n infect_w [0.01993644 0.0151893 0.12671971]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.0664850666778414 step loss 1.0632348498290887\n iter 0 sigmoid loss 1.0151918477058215 step loss 0.6280593718103179 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4592750590553426 step loss 0.4543886229006539 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4742010140539216 step loss 0.4528903401506405 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4441011094783359 step loss 0.45316611031049475 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.45545634286189135 step loss 0.45464987005892565 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4762683976784608 step loss 0.45511956665112174 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.40274466104395484 step loss 0.45308026171178434 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.40502184491797194 step loss 0.45280731182105705 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.40491108464324244 step loss 0.4529946054090768 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.43373918548998297 step loss 0.45282601196206806 sigmoid temp 1.0\n End sigmoid loss 0.45277353147753197 step loss 0.45280007107418485\n beta 0.16622680848167307\n rssi_w [-0.01948132 -0.00832676 0.02009129 0.13530891]\n rssi_th [13.00131 15.00132324 16.00137398]\n infect_w [0.02242738 0.01710177 0.14286346]\n best loss 0.41271684545185744\n best scoring parameters\n beta 0.25942930198242775\n rssi_w [-0.05664674 -0.05387891 0.02717731 0.2547946 ]\n rssi_th [-83.99887996 -72.99782621 -54.99731888]\n infect_w [0.01505614 0.03890162 0.28285256]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 4.096, median size 4\n \t Negative bags: mean size 3.958, median size 3\n assign_mat size, X_shuff size: (4431, 17634) (17634, 3)\n assign_mat_trn size, assign_mat_tst size (3512, 17634) (879, 17634)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 0.902388605664256 step loss 0.9070246140432743\n iter 0 sigmoid loss 0.7241558607078243 step loss 0.718813812121556 sigmoid temp 0.1\n iter 500 sigmoid loss 0.49055629142088364 step loss 0.43204114920788733 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.41648183874465444 step loss 0.42618520805447424 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.43187041943915444 step loss 0.42350764609846864 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.33673175695719637 step loss 0.42902118640871656 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.43525049939971017 step loss 0.6125652034974723 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.39476749309960224 step loss 0.46659848067406245 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3504156770294445 step loss 0.4250678395624062 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3776823278487054 step loss 0.42255149335775216 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.5592986058864331 step loss 0.4211812926602127 sigmoid temp 1.0\n End sigmoid loss 0.4168989310742883 step loss 0.4211099955583812\n beta 0.19395527800215576\n rssi_w [0.00090942 0.03254 0.16168495 0.04005769]\n rssi_th [28.00577212 30.00571613 37.99989463]\n infect_w [0.02256417 0.01922161 0.16899231]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 0.8337433031905223 step loss 0.8352257331258391\n iter 0 sigmoid loss 0.6701696433728042 step loss 0.693789519440687 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4919169740664433 step loss 0.43604900258489937 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4170034012719984 step loss 0.4316392173988014 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.43160840237472187 step loss 0.42894434014740085 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3374151833971695 step loss 0.42685411957975505 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4345907151155172 step loss 0.4339665130572028 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.40664024487599304 step loss 0.4246139812662994 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.35779331884029064 step loss 0.42233932318225764 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3874802029718667 step loss 0.4229940978608172 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.5521448523976116 step loss 0.4219227480600603 sigmoid temp 1.0\n End sigmoid loss 0.4138927125426052 step loss 0.42231905250485285\n beta 0.2512865405173258\n rssi_w [-0.02314682 -0.01733824 0.0781462 0.22050568]\n rssi_th [10.99317894 30.99318184 26.99266361]\n infect_w [0.02357733 0.02688842 0.23132508]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 0.9437675731687688 step loss 0.9429834558221684\n iter 0 sigmoid loss 0.7578141972637173 step loss 0.6774755106290956 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4947849831945143 step loss 0.43691233001235974 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4182067950726893 step loss 0.4342226026056398 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4369429124230817 step loss 0.4333025823907796 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3428457514371722 step loss 0.4331499984515151 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.44204814389802644 step loss 0.43845937932836776 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4092696319561845 step loss 0.47438180934585955 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3437268120898675 step loss 0.47091061712031207 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3949515985875924 step loss 0.46203252996048644 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.5639393112901012 step loss 0.44000627145287435 sigmoid temp 1.0\n End sigmoid loss 0.4292607761268049 step loss 0.4376260251472908\n beta 0.1718736048455071\n rssi_w [-0.00449395 -0.00559749 0.01534216 0.14319203]\n rssi_th [11.00451968 12.00453178 29.00453209]\n infect_w [0.02466868 0.02067534 0.14287634]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 0.9799640407576043 step loss 0.9697776644365664\n iter 0 sigmoid loss 0.7876691247090409 step loss 0.6890607696721578 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4926341669459019 step loss 0.4396946857162193 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.41750314596419796 step loss 0.4372063355999314 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4340642676788278 step loss 0.43657272471780023 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.34080081894701636 step loss 0.4361515295940306 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4383829580693521 step loss 0.43569686698537613 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4272689670660425 step loss 0.43546193210359696 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3490875785363643 step loss 0.43483447986768653 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4203700976630776 step loss 0.4346885223986256 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.5583264551182314 step loss 0.43481660565114405 sigmoid temp 1.0\n End sigmoid loss 0.43350841571196264 step loss 0.4346455675254362\n beta 0.22289415707300578\n rssi_w [-0.03407153 -0.03412527 0.12764422 0.15161307]\n rssi_th [12.00009793 34.00010181 31.99781381]\n infect_w [0.03559379 0.02441238 0.19825241]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 0.929281172290601 step loss 0.9347226574695162\n iter 0 sigmoid loss 0.7492220474968144 step loss 0.7047325825809073 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4919797591122175 step loss 0.44009635331160546 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4172991690448767 step loss 0.4380714082056613 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.43385145764772015 step loss 0.43800635935500876 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3403227244479004 step loss 0.4387340891381862 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.43817508947658185 step loss 0.44021273654145143 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4193381290487607 step loss 0.440456268868668 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.34356428627545094 step loss 0.43825070479047995 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4083786121460907 step loss 0.4375553125700773 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.556668310210864 step loss 0.4375357264836558 sigmoid temp 1.0\n End sigmoid loss 0.43191327123962625 step loss 0.43727848027506444\n beta 0.2219142470310075\n rssi_w [-0.04144339 -0.03500091 0.14544069 0.13048948]\n rssi_th [11.00214576 38.00214534 31.99850534]\n infect_w [0.03555179 0.02664454 0.19672933]\n best loss 0.4211099955583812\n best scoring parameters\n beta 0.19395527800215576\n rssi_w [0.00090942 0.03344942 0.19513437 0.23519206]\n rssi_th [-91.99422788 -61.98851175 -23.98861712]\n infect_w [0.02256417 0.04178578 0.21077809]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 3.787, median size 3\n \t Negative bags: mean size 3.973, median size 3\n assign_mat size, X_shuff size: (4431, 17471) (17471, 3)\n assign_mat_trn size, assign_mat_tst size (3501, 17471) (879, 17471)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 0.9921783371561332 step loss 1.0173855798188418\n iter 0 sigmoid loss 0.9269219750114297 step loss 0.6837216725243115 sigmoid temp 0.1\n iter 500 sigmoid loss 0.32729082454972414 step loss 0.447413749704823 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.469697960861379 step loss 0.44175749928384195 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4329845844923987 step loss 0.4458342986412993 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.39460034159754187 step loss 0.5710946511257343 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4014219596999531 step loss 0.6729878219088087 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.47186534136545505 step loss 0.44868155300328794 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5589760550743925 step loss 0.4394366931665223 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.36744989880866813 step loss 0.43676535935872834 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.4774994539596357 step loss 0.43605662296477465 sigmoid temp 1.0\n End sigmoid loss 0.43233149377289454 step loss 0.43604196601607687\n beta 0.202078935859673\n rssi_w [0.00243876 0.0302765 0.09492518 0.1512279 ]\n rssi_th [31.00132295 26.00128486 15.99817122]\n infect_w [0.01928202 0.02754398 0.17879142]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.0270578280813023 step loss 1.0274578093088536\n iter 0 sigmoid loss 0.9571048151387903 step loss 0.6678341318614818 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3292984928623415 step loss 0.45009908827099887 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4734489246221591 step loss 0.447002506486519 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.43875549023993315 step loss 0.4460268773126735 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3971715961938319 step loss 0.44797486675768605 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4010728695519595 step loss 0.4873292246314941 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.46807274406255034 step loss 0.4883904470346781 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5765911302125063 step loss 0.47915065145357877 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.38568898716039995 step loss 0.4543885777305441 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.49693555174565757 step loss 0.447442200677125 sigmoid temp 1.0\n End sigmoid loss 0.44327288054591835 step loss 0.4467226480928144\n beta 0.18739755030795494\n rssi_w [-0.00300147 0.01735094 0.09389586 0.13374406]\n rssi_th [26.00158731 26.00157192 26.9986949 ]\n infect_w [0.02459057 0.02637592 0.16096391]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.1034456701829443 step loss 1.1117209417511473\n iter 0 sigmoid loss 1.0288873465198909 step loss 0.6032901547997278 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3290518010780096 step loss 0.445942669284933 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4733335666242821 step loss 0.4403782418276905 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4390823592786521 step loss 0.43797440790915143 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3977876713684324 step loss 0.43573631484728753 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.40110402464842354 step loss 0.4569443287891918 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.46750094578318674 step loss 0.43617701123858843 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5536453193468777 step loss 0.432975847976988 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3547087568689131 step loss 0.43184731342689153 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.4684117314478516 step loss 0.4315249023137672 sigmoid temp 1.0\n End sigmoid loss 0.42869919936636985 step loss 0.43169628551925854\n beta 0.1999487820285694\n rssi_w [-0.00169242 0.04222482 0.17627796 0.0261435 ]\n rssi_th [35.00407228 26.00394061 38.99993296]\n infect_w [0.01834921 0.02520295 0.17672264]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 0.898936323363491 step loss 0.9020127542934837\n iter 0 sigmoid loss 0.8383129717419874 step loss 0.6667525995413653 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3290484027898134 step loss 0.4469550072496422 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.47240470915674854 step loss 0.44137771320775715 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.43607405796312987 step loss 0.43701994760852786 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3953448952261091 step loss 0.43239025782956153 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4000471098803534 step loss 0.43328609087354586 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4660450139639387 step loss 0.4295922830538131 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5557301585050974 step loss 0.42887979643848245 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.35871067641682686 step loss 0.42892401302489136 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.46097907350054973 step loss 0.4290972989995316 sigmoid temp 1.0\n End sigmoid loss 0.4252629625814993 step loss 0.42919212006514657\n beta 0.22124534320978598\n rssi_w [-0.03638763 0.00101764 0.07510408 0.18647587]\n rssi_th [33.00156364 15.00153722 16.00124738]\n infect_w [0.01790256 0.02809339 0.19991486]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 0.8911573584412767 step loss 0.894239624125361\n iter 0 sigmoid loss 0.8312106161767413 step loss 0.689086468409968 sigmoid temp 0.1\n iter 500 sigmoid loss 0.33089490178206377 step loss 0.45184317934002316 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.47667432835912565 step loss 0.4490469656299259 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.44214666790780127 step loss 0.44852108393065737 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3986710137758685 step loss 0.4473048206825218 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.40317716055479047 step loss 0.4534008103795817 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4617861002945912 step loss 0.4452476835167671 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5875028729049019 step loss 0.4449817558279512 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3923250891050699 step loss 0.444744534754929 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.4868462528393252 step loss 0.44475480103905546 sigmoid temp 1.0\n End sigmoid loss 0.4413596786285821 step loss 0.4450004882973234\n beta 0.2202726240823182\n rssi_w [ 0.00022603 -0.0007246 0.05456782 0.19549985]\n rssi_th [ 9.99509767 24.99510684 37.99487477]\n infect_w [0.02286324 0.03503979 0.19672156]\n best loss 0.42919212006514657\n best scoring parameters\n beta 0.22124534320978598\n rssi_w [-0.03638763 -0.03536999 0.03973409 0.22620996]\n rssi_th [-86.99843636 -71.99689914 -55.99565176]\n infect_w [0.01790256 0.04599595 0.24591082]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 3.725, median size 3\n \t Negative bags: mean size 4.014, median size 3\n assign_mat size, X_shuff size: (4431, 17580) (17580, 3)\n assign_mat_trn size, assign_mat_tst size (3492, 17580) (873, 17580)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 0.9950644156091666 step loss 1.012646853090109\n iter 0 sigmoid loss 1.0735260363205448 step loss 0.6541289556518118 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3654219048225363 step loss 0.4596528135411609 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4831919534176488 step loss 0.4562648007985465 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.5363940931969442 step loss 0.4654407499868636 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4614557256265476 step loss 0.5922264458049383 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.46978525909034263 step loss 0.6420901484225616 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.32778899081258855 step loss 0.4711623330887816 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.44734698737486767 step loss 0.4519738967776688 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4759911488787735 step loss 0.4506514079080209 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.5029878105587683 step loss 0.4504755988233535 sigmoid temp 1.0\n End sigmoid loss 0.4481748235490813 step loss 0.4502061613989258\n beta 0.18634199244461627\n rssi_w [0.0071482 0.02470378 0.07567104 0.14387545]\n rssi_th [28.0015476 28.00152858 15.99865068]\n infect_w [0.0348379 0.02996212 0.15582672]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.0645818463258092 step loss 1.079394937692549\n iter 0 sigmoid loss 1.1456306067517632 step loss 0.6467044298579331 sigmoid temp 0.1\n iter 500 sigmoid loss 0.36573104537245504 step loss 0.46339362456023425 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.48342297591269767 step loss 0.46091972590621433 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.5368512927908423 step loss 0.46406811628620115 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.46097566759494285 step loss 0.47454843661941737 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4686033910039963 step loss 0.5139910418876927 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3330454303347901 step loss 0.4609062729581079 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.46523466002313635 step loss 0.4544839117035226 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.46503960542536843 step loss 0.4542186067445012 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.5217953975470853 step loss 0.4543675508268044 sigmoid temp 1.0\n End sigmoid loss 0.4500246961435221 step loss 0.45409405115101803\n beta 0.20770542735460965\n rssi_w [-0.00035298 0.04724827 0.17478361 0.05836451]\n rssi_th [33.99575933 34.99561849 16.99974228]\n infect_w [0.03498191 0.03907104 0.17853957]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 0.964833756816844 step loss 0.9620848336973198\n iter 0 sigmoid loss 1.0389925443581312 step loss 0.6348913103309652 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3669916036654247 step loss 0.46098677758397366 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4873985337965553 step loss 0.45795517719108964 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.5433998271313825 step loss 0.45646563346429314 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4753091680130083 step loss 0.4641168766616103 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.47172451946311583 step loss 0.5602415493076168 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.33463243915451957 step loss 0.536221644175283 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4463432473869466 step loss 0.45896087908162747 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4886280479348005 step loss 0.4565956065011295 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.5037251765639997 step loss 0.45609249818149655 sigmoid temp 1.0\n End sigmoid loss 0.4551672275198782 step loss 0.4558250747732186\n beta 0.15606698216809037\n rssi_w [-0.00550537 0.00963967 0.03718686 0.12374464]\n rssi_th [20.00477552 11.00478126 24.00475025]\n infect_w [0.03379308 0.02530598 0.11902364]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 0.9507303685143156 step loss 0.9747558112119313\n iter 0 sigmoid loss 1.026581021931174 step loss 0.674544726588081 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3648780182082942 step loss 0.46045185612268985 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.48210204537383355 step loss 0.45772105104247895 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.5350158613243763 step loss 0.4695010027997829 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4590246038717109 step loss 0.7226671638128339 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4690405818001785 step loss 0.7662299406062077 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.32584986133533733 step loss 0.4560760489842857 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.44972455299618724 step loss 0.44809053435853197 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4750980115120734 step loss 0.4476500208937995 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.513859067710874 step loss 0.44788982228389684 sigmoid temp 1.0\n End sigmoid loss 0.4444806046209674 step loss 0.4475494362985874\n beta 0.1949460476906694\n rssi_w [0.0076947 0.03256419 0.14218536 0.09548 ]\n rssi_th [32.0011637 30.00111571 14.9996392 ]\n infect_w [0.03118139 0.02945028 0.16688242]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.0526877450143948 step loss 1.0761272446630075\n iter 0 sigmoid loss 1.1316283220036987 step loss 0.6047454335499178 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3665577523534743 step loss 0.4601971191731279 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.48625430097737116 step loss 0.4558234594918321 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.5410294528965572 step loss 0.4544763872328382 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4676688215297252 step loss 0.46199424781405046 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4690886414704405 step loss 0.6575337008894859 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3250775610652015 step loss 0.4578395107000471 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4534267208546979 step loss 0.4475654985446088 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.47008423879156574 step loss 0.44731628381722427 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.5183449980710413 step loss 0.44751794466521644 sigmoid temp 1.0\n End sigmoid loss 0.444027607194745 step loss 0.447189863219822\n beta 0.1979141678847384\n rssi_w [0.00247538 0.00873832 0.03193863 0.17947988]\n rssi_th [10.00035868 20.00036399 34.00031369]\n infect_w [0.03017734 0.03241795 0.16946648]\n best loss 0.447189863219822\n best scoring parameters\n beta 0.1979141678847384\n rssi_w [0.00247538 0.0112137 0.04315233 0.22263222]\n rssi_th [-109.99964132 -89.99927734 -55.99896365]\n infect_w [0.03017734 0.06259529 0.23206177]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 3.854, median size 3\n \t Negative bags: mean size 4.031, median size 3\n assign_mat size, X_shuff size: (4431, 17737) (17737, 3)\n assign_mat_trn size, assign_mat_tst size (3503, 17737) (876, 17737)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 0.9305103147358927 step loss 0.9337027008608452\n iter 0 sigmoid loss 1.1188505394404797 step loss 0.6298369441454157 sigmoid temp 0.1\n iter 500 sigmoid loss 0.43863333933520016 step loss 0.47023422658365005 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.45090296633245464 step loss 0.4699071920731872 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3938175327282301 step loss 0.4693750792702582 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4444856692249602 step loss 0.47099154963092166 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.47139684255913944 step loss 0.4710584894227841 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4552011256543315 step loss 0.46973897459544167 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.467974674974956 step loss 0.4692967718525514 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.40048522538732895 step loss 0.46916989001668385 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.48509155925616626 step loss 0.4692366805153945 sigmoid temp 1.0\n End sigmoid loss 0.468095651497619 step loss 0.46917969915945995\n beta 0.15081064013978676\n rssi_w [-0.01404111 -0.01539842 0.02173585 0.11412656]\n rssi_th [13.00217513 23.00218222 12.00218891]\n infect_w [0.05146164 0.0151069 0.10626142]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 0.9783905657696419 step loss 0.977363479856561\n iter 0 sigmoid loss 1.1753600261246653 step loss 0.6360027815905505 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4375717239916106 step loss 0.47051027939427215 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.45068360055836104 step loss 0.4697509790936517 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3908917901068723 step loss 0.46902762935051556 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.44288873302202836 step loss 0.47005469090172525 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.46593852145069986 step loss 0.469842192942888 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4569536789398934 step loss 0.47006460393181004 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4639990495158676 step loss 0.4693212786918911 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3953482664646545 step loss 0.46898087654295745 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.47502680449679474 step loss 0.4691659821413699 sigmoid temp 1.0\n End sigmoid loss 0.46578494050342284 step loss 0.4690494088627622\n beta 0.17446013750023184\n rssi_w [-0.02464478 -0.01658293 0.11966385 0.08070362]\n rssi_th [18.00227707 31.00227531 30.99934227]\n infect_w [0.05941076 0.01877467 0.13452496]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 0.973755053916082 step loss 0.9697100564272692\n iter 0 sigmoid loss 1.1734622936010588 step loss 0.6274709039436911 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4382853746132894 step loss 0.4693539294281288 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4506428833739702 step loss 0.4675708101697039 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.39162106067303426 step loss 0.46694486703852744 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.44329084071618213 step loss 0.46605542783943565 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.46632209737949554 step loss 0.4657341367644723 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.45311289692532825 step loss 0.4650103655950317 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.46237436668484483 step loss 0.46484699087895587 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.39247852019912743 step loss 0.4647739045959348 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.47628314946056904 step loss 0.46468493961525265 sigmoid temp 1.0\n End sigmoid loss 0.46383277003317186 step loss 0.4646811464213035\n beta 0.17541367585961468\n rssi_w [-0.00826179 0.00939047 0.06866879 0.13081521]\n rssi_th [26.99857092 14.99857918 30.99819578]\n infect_w [0.05266577 0.01980319 0.13869218]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 0.9812416158866549 step loss 0.9858779557326423\n iter 0 sigmoid loss 1.178415803829379 step loss 0.6339396208161198 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4380748705507992 step loss 0.46891699459767494 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.45067128146570345 step loss 0.46678025973091053 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3911179671991769 step loss 0.4660798401491497 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4429603185695652 step loss 0.46487939841975884 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4651819899287035 step loss 0.4645094198594816 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4529272249231859 step loss 0.4635139138581381 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4611112640966595 step loss 0.46330016671475965 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.388531184593112 step loss 0.4631881808520622 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.4756053942593418 step loss 0.4631021735782821 sigmoid temp 1.0\n End sigmoid loss 0.46234005953145413 step loss 0.4630884433632045\n beta 0.17890276154665088\n rssi_w [-0.01030565 0.00993532 0.06763728 0.13726195]\n rssi_th [29.99898801 12.99899493 27.99858342]\n infect_w [0.04984021 0.01976952 0.1425016 ]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.093545430903593 step loss 1.0976030172074902\n iter 0 sigmoid loss 1.3135583069155263 step loss 0.6228266520493595 sigmoid temp 0.1\n iter 500 sigmoid loss 0.43812502173781936 step loss 0.4673543615012976 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.45097498236679223 step loss 0.46447293426605707 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3894926527534832 step loss 0.46391891708384536 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.44203659787084193 step loss 0.4627456615376147 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4631908877663934 step loss 0.46506362321575057 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4538346569748539 step loss 0.4623905159363272 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.46368895122225795 step loss 0.461520624692847 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3779325483052229 step loss 0.461169210732971 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.4797961508712353 step loss 0.46105185748543853 sigmoid temp 1.0\n End sigmoid loss 0.45963792481654603 step loss 0.46102484376426656\n beta 0.17375233201933896\n rssi_w [0.00905162 0.05287537 0.13475946 0.01689901]\n rssi_th [39.00077323 24.00061347 35.99995513]\n infect_w [0.04123245 0.01777155 0.14197923]\n best loss 0.46102484376426656\n best scoring parameters\n beta 0.17375233201933896\n rssi_w [0.00905162 0.06192699 0.19668645 0.21358546]\n rssi_th [-80.99922677 -56.99861331 -20.99865817]\n infect_w [0.04123245 0.059004 0.20098323]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 4.028, median size 3\n \t Negative bags: mean size 4.039, median size 3\n assign_mat size, X_shuff size: (4431, 17890) (17890, 3)\n assign_mat_trn size, assign_mat_tst size (3489, 17890) (864, 17890)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 0.8777273930922844 step loss 0.880192874593367\n iter 0 sigmoid loss 0.9566767614042072 step loss 0.6564652261532506 sigmoid temp 0.1\n iter 500 sigmoid loss 0.5253813903295778 step loss 0.4570790845520184 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.42830688622974494 step loss 0.4556820117083965 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4627952730710912 step loss 0.45516953869446297 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4921212728842522 step loss 0.4549555499055795 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.38845123482287675 step loss 0.45526154352456377 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.47355223133692315 step loss 0.4552503392758387 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.46096018927304555 step loss 0.45500551152998175 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4291620033327601 step loss 0.45491197087525537 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.4751036484476478 step loss 0.4548666263673317 sigmoid temp 1.0\n End sigmoid loss 0.45348202575275753 step loss 0.4548140698241487\n beta 0.14728609837023612\n rssi_w [0.00731248 0.01810238 0.0530715 0.09736733]\n rssi_th [29.00160072 18.00160384 13.00162932]\n infect_w [0.05039158 0.00865997 0.10098917]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 0.9529191524880816 step loss 0.9592558814167672\n iter 0 sigmoid loss 1.0367800971915972 step loss 0.6382908660567891 sigmoid temp 0.1\n iter 500 sigmoid loss 0.5260639471350342 step loss 0.45673070217867073 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4286933601643906 step loss 0.45508898605515363 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.46268530003331854 step loss 0.4542705530605495 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.49242573810027246 step loss 0.4535455751053015 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.38868950798012974 step loss 0.45321414385774184 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4759505023594929 step loss 0.45143743867939545 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4565122786878314 step loss 0.45099637794702585 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.42225567957504917 step loss 0.4507425498054632 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.47760837673619183 step loss 0.4506812338575852 sigmoid temp 1.0\n End sigmoid loss 0.44995066171262427 step loss 0.4506283274243954\n beta 0.16989530232370811\n rssi_w [0.01213265 0.01469823 0.03623819 0.13540469]\n rssi_th [25.99925863 9.99927661 32.99920824]\n infect_w [0.05879413 0.01367295 0.12841263]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 0.9049522700138144 step loss 0.9126308526028939\n iter 0 sigmoid loss 0.9863220184474112 step loss 0.6531845656771288 sigmoid temp 0.1\n iter 500 sigmoid loss 0.524837339351982 step loss 0.45660983833138274 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4280235500797482 step loss 0.4548942275120709 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4625420063155596 step loss 0.45396978014320727 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.49097321038528163 step loss 0.45316658381515146 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.38810167528407663 step loss 0.45262876026750487 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4749003647574081 step loss 0.45197322864016104 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4567370024919809 step loss 0.45144081073404374 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.421250777817563 step loss 0.45110047305052137 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.47673217008999413 step loss 0.4508074968262059 sigmoid temp 1.0\n End sigmoid loss 0.44977086674076955 step loss 0.45060527981062687\n beta 0.16713812783498697\n rssi_w [0.00790698 0.02221856 0.03989189 0.12961505]\n rssi_th [35.99967047 13.99967091 17.99975307]\n infect_w [0.05505863 0.01467598 0.12580068]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 0.999330405175377 step loss 1.0103048439599616\n iter 0 sigmoid loss 1.0820979818885035 step loss 0.6413962074520848 sigmoid temp 0.1\n iter 500 sigmoid loss 0.524513500686999 step loss 0.4579521561396509 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.42785895621513703 step loss 0.4568450077228733 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4624966162278007 step loss 0.45655298119616594 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4912322851541331 step loss 0.45662311618705215 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.388285933316554 step loss 0.4571297561345019 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.47379866132620696 step loss 0.45765723492224636 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4627808913125241 step loss 0.4574904586278754 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.42227637328079565 step loss 0.4568201786168088 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.4783643152108902 step loss 0.4562981791413302 sigmoid temp 1.0\n End sigmoid loss 0.4549348747016768 step loss 0.455953430606858\n beta 0.1524875393148605\n rssi_w [0.02116223 0.0258964 0.05469208 0.10194352]\n rssi_th [27.9998728 23.99987226 25.99928315]\n infect_w [0.05902218 0.01140717 0.10343262]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.059071450994615 step loss 1.073481603910481\n iter 0 sigmoid loss 1.1476513227041918 step loss 0.6410551683683031 sigmoid temp 0.1\n iter 500 sigmoid loss 0.5243366746589527 step loss 0.45723358561389016 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.42776387780379205 step loss 0.4557162965215706 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.46235875062669973 step loss 0.45524237515519034 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4903099513965333 step loss 0.4550117166766253 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.38808982109180024 step loss 0.4553635137804799 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4758340432116268 step loss 0.4522144687323552 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4567365346854237 step loss 0.4518595848521039 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.42254443066768965 step loss 0.4516615571090528 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.47843848422204543 step loss 0.45163137669979786 sigmoid temp 1.0\n End sigmoid loss 0.45045089954136036 step loss 0.4515903059313341\n beta 0.1714659920131311\n rssi_w [0.02799005 0.03368496 0.13269333 0.03104774]\n rssi_th [32.99901361 36.99896036 22.999895 ]\n infect_w [0.06189439 0.01348182 0.12946966]\n best loss 0.45060527981062687\n best scoring parameters\n beta 0.16713812783498697\n rssi_w [0.00790698 0.03012555 0.07001744 0.19963249]\n rssi_th [-84.00032953 -70.00065861 -52.00090554]\n infect_w [0.05505863 0.06973462 0.19553529]\n\n\n\n```python\nfig, axs = plt.subplots(1, 2, figsize=(15,5))\naxs[0].plot(censor_prob, np.mean(auc_train_learned, axis=1), '+-', label='train, learned')\naxs[0].plot(censor_prob, np.mean(auc_train_true, axis=1), '+-', label='train, true')\naxs[0].set_xlabel(\"Censor Probability\")\naxs[0].set_ylabel(\"AUC\")\naxs[0].set_title(\"Train: Max Bag Size = 16\")\naxs[0].legend()\n\naxs[1].plot(censor_prob, np.mean(auc_test_learned, axis=1), '+-', label='test, learned')\naxs[1].plot(censor_prob, np.mean(auc_test_true, axis=1), '+-', label='test, true')\naxs[1].set_xlabel(\"Censor Probability\")\naxs[1].set_ylabel(\"AUC\")\naxs[1].set_title(\"Test: Max Bag Size = 16\")\naxs[1].legend()\n```\n\n\n```python\n# max bag size 32\ncensor_prob = [0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7]\nn_trials = 1\nn_random_restarts_train = 5\n\nidx = 0\n\nauc_train_learned = np.zeros((len(censor_prob),n_trials))\nauc_train_true = np.zeros((len(censor_prob),n_trials))\nauc_test_learned = np.zeros((len(censor_prob),n_trials))\nauc_test_true = np.zeros((len(censor_prob),n_trials))\nfor prob in censor_prob:\n bag_sim = Bag_Simulator(p_pos=0.6,r_pos=2,p_neg=0.6,r_neg=2,max_bag_size=32,censor_prob_pos=prob,censor_prob_neg=0,max_pos_in_bag=1)\n auc_train_trials, auc_test_trials = train_and_eval_with_bag_config(bag_sim, X_epi,\n probabilities_true_epi, n_trials=n_trials,\n n_random_restarts=n_random_restarts_train)\n for i in range(n_trials):\n auc_train_learned[idx, i] = dict(auc_train_trials[i])['Learned']\n auc_train_true[idx, i] = dict(auc_train_trials[i])['True']\n auc_test_learned[idx, i] = dict(auc_test_trials[i])['Learned']\n auc_test_true[idx, i] = dict(auc_test_trials[i])['True']\n \n idx += 1\n```\n\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 4.287, median size 3\n \t Negative bags: mean size 4.009, median size 3\n assign_mat size, X_shuff size: (4431, 17962) (17962, 3)\n assign_mat_trn size, assign_mat_tst size (3539, 17962) (886, 17962)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 0.9118934098450943 step loss 0.909669808186479\n iter 0 sigmoid loss 0.6813132590804648 step loss 0.7224957709916452 sigmoid temp 0.1\n iter 500 sigmoid loss 0.42864037965183344 step loss 0.4251824955255295 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.35591760806873546 step loss 0.4196330665695337 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.49162052755086877 step loss 0.41328101346893126 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.42485674607065205 step loss 0.4216879154718225 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3722794338677258 step loss 1.1317223003385162 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.31748749229604406 step loss 0.4150162074509731 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.40417024974072224 step loss 0.4011409212391632 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.41871511007971535 step loss 0.3999600919866094 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.31842775698571024 step loss 0.4040927848602059 sigmoid temp 1.0\n End sigmoid loss 0.390089913120393 step loss 0.4014798039846761\n beta 0.2981239439174942\n rssi_w [-0.02881316 -0.00117054 0.05825849 0.28388062]\n rssi_th [19.98995763 12.98996206 37.98967797]\n infect_w [0.00900965 0.03121122 0.28614133]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.1497703253310148 step loss 1.174294552387767\n iter 0 sigmoid loss 0.8655076600989596 step loss 0.6337094162928667 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4258412556429518 step loss 0.41937980813968145 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.35119335200132423 step loss 0.4104788041451449 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.47470519842184195 step loss 0.3976514349940032 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.41071392049124883 step loss 0.3946224286390281 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3715305949935344 step loss 0.41600823111799035 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3058377415822132 step loss 0.38326474323395154 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.38204560604350424 step loss 0.38245973918063153 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.38997595062561835 step loss 0.382128668378131 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.28773397777850157 step loss 0.38647014271639407 sigmoid temp 1.0\n End sigmoid loss 0.3733260645380961 step loss 0.3838017198514066\n beta 0.27868130308684724\n rssi_w [-0.04905825 -0.02417877 0.09577571 0.25675278]\n rssi_th [26.00029499 21.00028112 18.99892468]\n infect_w [0.00779567 0.02798425 0.26789786]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 0.9227242049054967 step loss 0.9237348673029135\n iter 0 sigmoid loss 0.6907250483588215 step loss 0.7222826912634355 sigmoid temp 0.1\n iter 500 sigmoid loss 0.42556791298080526 step loss 0.42020876054859263 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3501606227479393 step loss 0.41103617708766876 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.46939825489370735 step loss 0.4033234127370597 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.40845050881944883 step loss 0.4050296808996737 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3710327462645864 step loss 0.40918198022448926 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.307846735081686 step loss 0.3974592885837757 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3805091993979822 step loss 0.39207751990783124 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.398161963829178 step loss 0.388963195594134 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2936414400328864 step loss 0.3926500146697445 sigmoid temp 1.0\n End sigmoid loss 0.37297687685611813 step loss 0.3900426466031585\n beta 0.2726467363516448\n rssi_w [-0.0403939 0.01818501 0.0535601 0.26725361]\n rssi_th [36.99786807 12.99778893 16.99793344]\n infect_w [0.00776285 0.02819645 0.26256143]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 0.9982740812919559 step loss 1.0097173266836759\n iter 0 sigmoid loss 0.7455446582472015 step loss 0.7447278628276126 sigmoid temp 0.1\n iter 500 sigmoid loss 0.42423375382082157 step loss 0.41445698728247304 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.35026711331265764 step loss 0.40305672088780253 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.47507111087436116 step loss 0.39609442585240884 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.41234194766211824 step loss 0.7178653503526092 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3714231616779135 step loss 0.7900546154341493 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3109240951592708 step loss 0.43056306123469784 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.39237632510502185 step loss 0.3840170891833283 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.390416816555756 step loss 0.3810514488853801 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2749145523572135 step loss 0.38279029122552805 sigmoid temp 1.0\n End sigmoid loss 0.3762288993019626 step loss 0.3814475744201258\n beta 0.24707054019069685\n rssi_w [-0.01005645 0.03055713 0.22518549 0.06520085]\n rssi_th [31.00740738 31.00730698 30.99973133]\n infect_w [0.00719336 0.02494017 0.23522369]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.0830921233415653 step loss 1.1022109753817362\n iter 0 sigmoid loss 0.8102095900865767 step loss 0.7062607766314764 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4221856679340618 step loss 0.4175881577785034 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.34780115001891226 step loss 0.4067512777955055 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.46587906604801593 step loss 0.40998638398813847 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4061901041829323 step loss 0.8094763266137135 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3686361850450399 step loss 0.8835916594964472 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.40385603234024287 step loss 0.4349540637002899 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.44284444834901265 step loss 0.42818054842871384 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.45304129869464077 step loss 0.416762289031831 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3183842326999284 step loss 0.39954799089426213 sigmoid temp 1.0\n End sigmoid loss 0.3833593340910248 step loss 0.3867271934148009\n beta 0.07789904671030567\n rssi_w [0.01751365 0.07240997 0.3029526 0.09553433]\n rssi_th [35.99924476 27.99901623 20.99960752]\n infect_w [0.01013795 0.05371529 0.31047018]\n best loss 0.3814475744201258\n best scoring parameters\n beta 0.24707054019069685\n rssi_w [-0.01005645 0.02050068 0.24568617 0.31088702]\n rssi_th [-88.99259262 -57.98528564 -26.98555431]\n infect_w [0.00719336 0.03213352 0.26735722]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 4.018, median size 3\n \t Negative bags: mean size 4.099, median size 3\n assign_mat size, X_shuff size: (4431, 18104) (18104, 3)\n assign_mat_trn size, assign_mat_tst size (3537, 18104) (884, 18104)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.0105839180604048 step loss 1.0158629689557725\n iter 0 sigmoid loss 1.0200794858969855 step loss 0.686096997575165 sigmoid temp 0.1\n iter 500 sigmoid loss 0.436428643180973 step loss 0.42803830911001073 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.36950109971365025 step loss 0.4196116580563786 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3531968859765589 step loss 0.42159450032117274 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.45075324023855967 step loss 0.5990451949748531 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.444062340073537 step loss 0.641680364711601 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3475543826639715 step loss 0.533935468938191 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.37213341013094436 step loss 0.42603830184011793 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.43749801369477326 step loss 0.41519316920420074 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.32586533889966113 step loss 0.412635354219877 sigmoid temp 1.0\n End sigmoid loss 0.4089960427030807 step loss 0.41288195658665433\n beta 0.20245363613048398\n rssi_w [-0.02043003 0.03999343 0.16491407 0.08063926]\n rssi_th [33.0099112 24.00979046 34.99962448]\n infect_w [0.0074555 0.03742398 0.18014548]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.0509808439587844 step loss 1.0624912637332433\n iter 0 sigmoid loss 1.0620104545632159 step loss 0.6919799742447202 sigmoid temp 0.1\n iter 500 sigmoid loss 0.43708465341641306 step loss 0.44007726684970744 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3692436964218237 step loss 0.4366518657123465 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3540883287474905 step loss 0.4367854173780225 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4543005508205465 step loss 0.4433758145119965 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4503091204660633 step loss 0.4751853183848075 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.35885244080816286 step loss 0.4369952495199496 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.41264339710329523 step loss 0.4343200979286101 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4532139339427711 step loss 0.43283217822309594 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3422971378020802 step loss 0.43258681601429255 sigmoid temp 1.0\n End sigmoid loss 0.4294807781401373 step loss 0.43263333376568935\n beta 0.27043038202935177\n rssi_w [-0.0472705 0.08987954 0.22815365 0.06080994]\n rssi_th [38.9953336 38.99443632 16.9993819 ]\n infect_w [0.00971815 0.05254384 0.24949428]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 0.9911838567962058 step loss 0.9888170679475695\n iter 0 sigmoid loss 0.9964051923229036 step loss 0.6649731362317823 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4387127328371072 step loss 0.44170168730173043 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.37433287880530125 step loss 0.43952711161187896 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.35844025135195534 step loss 0.4385291837382651 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.46356807822971674 step loss 0.44021591818924943 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4498654286041603 step loss 0.4493531816408822 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3433725454672219 step loss 0.43819605748556684 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.41147545201028884 step loss 0.4364171986696831 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4625232390408203 step loss 0.4372105247372046 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3407851223559065 step loss 0.43605673688317437 sigmoid temp 1.0\n End sigmoid loss 0.43193153707596843 step loss 0.4368200147315756\n beta 0.28911491206363704\n rssi_w [-0.08828627 -0.06492814 0.18730441 0.17966138]\n rssi_th [21.00431314 26.00430221 34.99687293]\n infect_w [0.01197882 0.07054941 0.26559154]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.0722462776188868 step loss 1.087536373108656\n iter 0 sigmoid loss 1.082061308842106 step loss 0.6621553469796533 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4355679145737295 step loss 0.4350671479458526 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.36439230689888374 step loss 0.4293991147644412 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.34853039148584597 step loss 0.44100326860375316 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.441526433159095 step loss 1.1108620649785417 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.44549578029016146 step loss 1.2282347550509387 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.36079720073445676 step loss 0.42781821548921306 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.39451546655024616 step loss 0.4206543866588788 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4044165893378816 step loss 0.41812609708219667 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3096964394498815 step loss 0.41776163669813265 sigmoid temp 1.0\n End sigmoid loss 0.4053726118538696 step loss 0.41889227336738716\n beta 0.2890584461206046\n rssi_w [-0.03189624 0.0607351 0.26187921 0.06148749]\n rssi_th [36.98876955 33.98838765 21.9995927 ]\n infect_w [0.00893046 0.04231932 0.27286585]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 0.9931518853613719 step loss 0.9984355428424393\n iter 0 sigmoid loss 1.003789914087546 step loss 0.6830414039559528 sigmoid temp 0.1\n iter 500 sigmoid loss 0.43660947285282203 step loss 0.44018274816943437 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3688116799525528 step loss 0.4371553712052097 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3532287219350809 step loss 0.4382531709641948 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.452835449165738 step loss 0.4484568655498046 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.44984590138214714 step loss 0.49653446687666275 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.35821580836445704 step loss 0.4362251359115219 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.41273506099956025 step loss 0.4339643322189932 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.45353856630674955 step loss 0.4326363226370431 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.34239929789477636 step loss 0.43244315918307813 sigmoid temp 1.0\n End sigmoid loss 0.42947615189931454 step loss 0.4325132500921343\n beta 0.26857067741682383\n rssi_w [-0.04111931 0.08432171 0.22492491 0.07511917]\n rssi_th [38.99535554 38.99454458 11.99921869]\n infect_w [0.00968925 0.052482 0.24755704]\n best loss 0.41288195658665433\n best scoring parameters\n beta 0.20245363613048398\n rssi_w [-0.02043003 0.0195634 0.18447747 0.26511673]\n rssi_th [-86.9900888 -62.98029834 -27.98067386]\n infect_w [0.0074555 0.04487948 0.22502496]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 3.777, median size 3\n \t Negative bags: mean size 3.947, median size 3\n assign_mat size, X_shuff size: (4431, 17367) (17367, 3)\n assign_mat_trn size, assign_mat_tst size (3519, 17367) (881, 17367)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 0.975409330070861 step loss 0.9747734083481637\n iter 0 sigmoid loss 0.9947749792931379 step loss 0.6665896444573158 sigmoid temp 0.1\n iter 500 sigmoid loss 0.43156300927072844 step loss 0.44032029015651436 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.432681985277631 step loss 0.4351593424051562 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.48877041786288594 step loss 0.43269791395636126 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.46553317119450993 step loss 0.4700573735734595 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4098037254745317 step loss 0.5625555819629008 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3785314531256958 step loss 0.5292608753073194 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3602177079424901 step loss 0.47152276650698405 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.494661873887446 step loss 0.43790191944513346 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3887555243351633 step loss 0.4336439153846939 sigmoid temp 1.0\n End sigmoid loss 0.43026163333088885 step loss 0.4338004908622908\n beta 0.18551154673241968\n rssi_w [-0.00847205 0.02551247 0.13901872 0.07837722]\n rssi_th [28.00571798 26.00567502 34.99963387]\n infect_w [0.0189463 0.0229701 0.15949123]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 0.9365859613768692 step loss 0.9371835642768587\n iter 0 sigmoid loss 0.9519630975786008 step loss 0.6567671551083899 sigmoid temp 0.1\n iter 500 sigmoid loss 0.432197248107825 step loss 0.44079616662713506 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4347742928447135 step loss 0.43593253105243807 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.49239768791794625 step loss 0.4322804923202989 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.47043137061785745 step loss 0.43341420945433495 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.41111035556085523 step loss 0.5608960941455017 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3773053004792821 step loss 0.5301070707546228 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.35637525043582213 step loss 0.4469311496239352 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4966412670118065 step loss 0.4343129298379118 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3851130043216623 step loss 0.4314830869182055 sigmoid temp 1.0\n End sigmoid loss 0.4278895794449158 step loss 0.4318875045904978\n beta 0.18231107462286503\n rssi_w [-0.00606771 -0.0056085 0.03354293 0.15533621]\n rssi_th [11.00629479 18.00630289 26.0062607 ]\n infect_w [0.01750993 0.02275888 0.15568493]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 0.9837152628256155 step loss 0.9753088325935021\n iter 0 sigmoid loss 1.0021716967428251 step loss 0.6642001436522473 sigmoid temp 0.1\n iter 500 sigmoid loss 0.43177755313560257 step loss 0.4473882058097413 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4331932073040805 step loss 0.44552923350384005 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4896612168219148 step loss 0.44569422495029043 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.46658607305153543 step loss 0.4528133367905907 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4103173869600594 step loss 0.4761702541524535 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3811596614786715 step loss 0.4748197970874019 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.36592677182679767 step loss 0.47085293561892033 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.49588209912217535 step loss 0.4670960108397933 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.398794506696815 step loss 0.4668827851236352 sigmoid temp 1.0\n End sigmoid loss 0.4383300190202453 step loss 0.46584355405777456\n beta 0.21342259131907895\n rssi_w [-0.07090414 0.0141532 0.14261377 0.11291345]\n rssi_th [39.00201394 11.0018502 34.99927517]\n infect_w [0.02797307 0.02363476 0.18882392]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 0.9838212638467987 step loss 1.0053717094814325\n iter 0 sigmoid loss 1.001179400233784 step loss 0.6711324186475526 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4290967555538488 step loss 0.4359482981047094 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4279080977279142 step loss 0.4291051886917749 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.48001957032664555 step loss 0.5648794928110087 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.44984754121455145 step loss 0.7231473173777293 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.40669745015746445 step loss 0.7702475844889615 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.37067477197242643 step loss 0.4471504660523068 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.329119128213188 step loss 0.41859328667754353 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.5001296728779288 step loss 0.414932375732261 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.36372627213842057 step loss 0.41346849671287533 sigmoid temp 1.0\n End sigmoid loss 0.40759242732336964 step loss 0.4138035111793725\n beta 0.22980290120534477\n rssi_w [-0.00462991 0.02796374 0.0709572 0.19786224]\n rssi_th [32.00109709 26.00104867 9.99853786]\n infect_w [0.01489422 0.02707087 0.20936405]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.019815334213676 step loss 1.0151553138911897\n iter 0 sigmoid loss 1.0392836592530141 step loss 0.6554907089003936 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4320717185147129 step loss 0.4475771943711337 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.43288218485636276 step loss 0.44520999157557745 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.48901280870051733 step loss 0.4451941603708432 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4645925935434852 step loss 0.45242060677200746 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4091297975455231 step loss 0.4750023222792399 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.38105775721433277 step loss 0.47453857556351203 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.36389787274636604 step loss 0.4706415473335185 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.49558830710507523 step loss 0.4669394011813872 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.39928163008528217 step loss 0.46670531391782466 sigmoid temp 1.0\n End sigmoid loss 0.4379322204754461 step loss 0.4656914629743128\n beta 0.21724181014916014\n rssi_w [-0.07006987 0.01593608 0.13636348 0.12788232]\n rssi_th [39.00122381 11.00105945 32.99864619]\n infect_w [0.0278955 0.02380691 0.19310472]\n best loss 0.4138035111793725\n best scoring parameters\n beta 0.22980290120534477\n rssi_w [-0.00462991 0.02333382 0.09429103 0.29215327]\n rssi_th [-87.99890291 -61.99785423 -51.99931637]\n infect_w [0.01489422 0.04196509 0.25132914]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 4.042, median size 3\n \t Negative bags: mean size 3.967, median size 3\n assign_mat size, X_shuff size: (4431, 17632) (17632, 3)\n assign_mat_trn size, assign_mat_tst size (3513, 17632) (883, 17632)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.0394749757544528 step loss 1.04937901463078\n iter 0 sigmoid loss 1.1149425729781945 step loss 0.6603080591808738 sigmoid temp 0.1\n iter 500 sigmoid loss 0.450918372682334 step loss 0.4361282774183669 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.42328549538595195 step loss 0.42969104576917655 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.47424223109354086 step loss 0.4249257712257901 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4304975514512563 step loss 0.4486850978440188 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.45915491586915613 step loss 0.6487664598410355 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.40073780081057064 step loss 0.4296323232120459 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4249094862148429 step loss 0.4204359167060696 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.47356624038210965 step loss 0.41954594869861744 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.4638139970914715 step loss 0.41898770496485516 sigmoid temp 1.0\n End sigmoid loss 0.41657905349704866 step loss 0.4190370887657087\n beta 0.20896930802140048\n rssi_w [0.00543773 0.02632381 0.17959965 0.04736566]\n rssi_th [26.00705993 34.00702138 33.9998527 ]\n infect_w [0.01631651 0.02651848 0.18627774]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 0.8856128878508515 step loss 0.8944196747374434\n iter 0 sigmoid loss 0.9566773344609482 step loss 0.6677822226829427 sigmoid temp 0.1\n iter 500 sigmoid loss 0.45021513218700465 step loss 0.4368597081644636 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.422885438607405 step loss 0.43096000610022134 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4731204745386524 step loss 0.4270496561837842 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.42900536014490315 step loss 0.45944477002123213 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.45836236391293583 step loss 0.492531239820557 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4029102322491031 step loss 0.4457225632801077 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4252347879890415 step loss 0.43119864973596406 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.47404769821446135 step loss 0.4152101617377285 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.4484554089665083 step loss 0.4136497944162795 sigmoid temp 1.0\n End sigmoid loss 0.41132540582375293 step loss 0.4141210932190844\n beta 0.21983051884101315\n rssi_w [ 0.00362804 0.0271268 -0.00037571 0.19871599]\n rssi_th [29.00405231 22.00403507 11.00480535]\n infect_w [0.01625747 0.02808354 0.19794873]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 0.9326424440023182 step loss 0.9398708208267889\n iter 0 sigmoid loss 1.0219158037987843 step loss 0.6760467881427428 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4515854028353184 step loss 0.4426532597708663 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4234818164225321 step loss 0.4393237251295167 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.47596014398624253 step loss 0.4384028746243495 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.43212322227072947 step loss 0.4528955068239608 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4607318479522253 step loss 0.5367253921583635 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.40661824117733397 step loss 0.5274892754478812 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4449335531642361 step loss 0.4893162679311381 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.48384585834582094 step loss 0.44358637384446464 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.486935627162822 step loss 0.439345110688053 sigmoid temp 1.0\n End sigmoid loss 0.43537555230918973 step loss 0.4386412444494034\n beta 0.19275089807303517\n rssi_w [-0.00147577 0.02558054 0.08989928 0.14075895]\n rssi_th [30.00212776 23.00210033 26.99846332]\n infect_w [0.02181818 0.02998092 0.16539211]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 0.9842572198957232 step loss 0.9890190438358423\n iter 0 sigmoid loss 1.070055167707073 step loss 0.6472942074830003 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4508460235926078 step loss 0.43820944593818517 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.42305099146228675 step loss 0.43185433959169994 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4732355152131003 step loss 0.4249003193908451 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4290477037330709 step loss 0.4187520596593889 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4584385605550256 step loss 0.41715914436361073 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.39872922863398086 step loss 0.4139509115596956 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4207050854361289 step loss 0.4137276526300545 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4729500743148563 step loss 0.4139690844304706 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.4429121692830191 step loss 0.4138665249757335 sigmoid temp 1.0\n End sigmoid loss 0.4095065604693693 step loss 0.41422775182367133\n beta 0.23103801987934816\n rssi_w [-0.02846403 0.00295602 0.05646949 0.20439562]\n rssi_th [35.00303933 13.00302307 15.00356182]\n infect_w [0.01590755 0.02791553 0.21030352]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 0.9313221708448703 step loss 0.949567022327887\n iter 0 sigmoid loss 1.0100984004721116 step loss 0.699364429662078 sigmoid temp 0.1\n iter 500 sigmoid loss 0.45020106396049137 step loss 0.4387158893890138 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4223598489186554 step loss 0.43359943417627067 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.47232494356086263 step loss 0.4301579223658677 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.42869662198001746 step loss 0.4545631837670273 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.45898956986640543 step loss 0.8709494595576058 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.39417172958864766 step loss 0.4239317591613941 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.42272327881070554 step loss 0.418822531322704 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.46590205384581856 step loss 0.418142479203301 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.442300083096001 step loss 0.4169653648582701 sigmoid temp 1.0\n End sigmoid loss 0.40978285621959804 step loss 0.41772675728652225\n beta 0.25341725057016473\n rssi_w [0.00216955 0.027851 0.2284861 0.05936821]\n rssi_th [27.99701162 37.99693915 23.99978396]\n infect_w [0.01614029 0.02850586 0.23415236]\n best loss 0.4141210932190844\n best scoring parameters\n beta 0.21983051884101315\n rssi_w [0.00362804 0.03075484 0.03037912 0.22909511]\n rssi_th [-90.99594769 -68.99191262 -57.98710727]\n infect_w [0.01625747 0.04434101 0.24228974]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 3.939, median size 3\n \t Negative bags: mean size 4.015, median size 3\n assign_mat size, X_shuff size: (4431, 17735) (17735, 3)\n assign_mat_trn size, assign_mat_tst size (3512, 17735) (877, 17735)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.087839254180437 step loss 1.0932722472974823\n iter 0 sigmoid loss 1.1580944565529783 step loss 0.6042819237385558 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3883770423088354 step loss 0.46206789398650605 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.5598798704688486 step loss 0.4588964477639659 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.43083801627709406 step loss 0.457866080495336 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.45593633232047337 step loss 0.4732778790464496 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.36669955481210076 step loss 0.4935591779545636 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3986793340759807 step loss 0.5000972359686572 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.41548718908569 step loss 0.4889929219894398 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.48563417894335825 step loss 0.4773407205231478 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.39261680405593696 step loss 0.4693484297042276 sigmoid temp 1.0\n End sigmoid loss 0.4522314210847185 step loss 0.46903112289832455\n beta 0.2022447088105949\n rssi_w [-0.02450282 0.02090326 0.07854919 0.16199313]\n rssi_th [34.99996329 15.99991352 28.9981647 ]\n infect_w [0.04027041 0.02481356 0.17741309]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 0.9769801545813155 step loss 0.9895869765171298\n iter 0 sigmoid loss 1.0414140169367325 step loss 0.6595855356211184 sigmoid temp 0.1\n iter 500 sigmoid loss 0.38710949783753035 step loss 0.45745335448537217 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.5607508857273084 step loss 0.45174979548768024 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4303093445104374 step loss 0.4482192323582017 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.45546137547121046 step loss 0.4501377978922481 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.36525160006765583 step loss 0.4749784391651915 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3872355513784562 step loss 0.4469832239240161 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4079235026817222 step loss 0.44564362227235493 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4855835885423269 step loss 0.44532519602709114 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.39529941781284317 step loss 0.44511394878810945 sigmoid temp 1.0\n End sigmoid loss 0.44260718421550255 step loss 0.44536095378854046\n beta 0.19333186501425673\n rssi_w [-0.01951442 -0.00453176 0.06498826 0.15840143]\n rssi_th [18.00463323 29.00462999 13.00463065]\n infect_w [0.0268492 0.01805363 0.16773854]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 0.9723191551487554 step loss 0.9868577068563075\n iter 0 sigmoid loss 1.0427272222422101 step loss 0.6843398815722586 sigmoid temp 0.1\n iter 500 sigmoid loss 0.38708424129706104 step loss 0.4579593209106654 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.5574917410898389 step loss 0.45210931352939426 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.43008506739714236 step loss 0.45699448865240566 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.45312912243249814 step loss 0.6434117157704216 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.36417514871955325 step loss 0.7567894783514513 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3858572613345401 step loss 0.4580473383766367 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.40952115503006253 step loss 0.44766484366093245 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4841621298853792 step loss 0.44585627763020985 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.39175513460170985 step loss 0.4451755282787314 sigmoid temp 1.0\n End sigmoid loss 0.44168459963243145 step loss 0.445334862952551\n beta 0.20282383666737616\n rssi_w [-0.00041208 0.03474548 0.117681 0.13389502]\n rssi_th [35.00277811 24.00271563 18.99906765]\n infect_w [0.02813538 0.01996431 0.17805803]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.3048400825421331 step loss 1.3313937166071235\n iter 0 sigmoid loss 1.3891621198346586 step loss 0.5352856558008654 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3872755940882426 step loss 0.45983023413021845 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.5580070072768042 step loss 0.4544503786871219 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.43102430414713994 step loss 0.45283753872370197 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4552990771752873 step loss 0.45464885045985487 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3669374861704011 step loss 0.48008000169158693 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3940395298771413 step loss 0.44585835676635827 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4039399058866395 step loss 0.4436712959593652 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4690872972798757 step loss 0.4433380233948163 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.38515572203357806 step loss 0.4432191164640763 sigmoid temp 1.0\n End sigmoid loss 0.4385595261527159 step loss 0.44384421094332654\n beta 0.23770571682947264\n rssi_w [-0.01477425 0.05253822 0.21173482 0.05693385]\n rssi_th [38.99611499 29.99583421 22.99974188]\n infect_w [0.03089407 0.02192226 0.2185698 ]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.1251017938771906 step loss 1.153248956285027\n iter 0 sigmoid loss 1.1827454649428086 step loss 0.6365536877881028 sigmoid temp 0.1\n iter 500 sigmoid loss 0.38767818118237946 step loss 0.4622200294211987 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.557755558522556 step loss 0.4582216430522008 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4308674040525117 step loss 0.45874610951171907 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4552375072002184 step loss 0.46421757835856936 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3672370398162992 step loss 0.49110771892266186 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.40246994269044684 step loss 0.4546094406474768 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.41621633451466883 step loss 0.4527603371725568 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.47350435903842486 step loss 0.4524828729245317 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3855785439914275 step loss 0.4523254252897514 sigmoid temp 1.0\n End sigmoid loss 0.44740894657739183 step loss 0.45295310314625864\n beta 0.2346472988661418\n rssi_w [-0.00837042 0.05131483 0.20556818 0.04912882]\n rssi_th [36.99563745 35.99537857 19.99970056]\n infect_w [0.03750086 0.02206022 0.21273202]\n best loss 0.44384421094332654\n best scoring parameters\n beta 0.23770571682947264\n rssi_w [-0.01477425 0.03776397 0.24949879 0.30643264]\n rssi_th [-81.00388501 -51.00805079 -28.00830892]\n infect_w [0.03089407 0.05281632 0.27138612]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 3.932, median size 3\n \t Negative bags: mean size 3.998, median size 3\n assign_mat size, X_shuff size: (4431, 17668) (17668, 3)\n assign_mat_trn size, assign_mat_tst size (3508, 17668) (872, 17668)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.050193211866523 step loss 1.0504721120240479\n iter 0 sigmoid loss 1.0908038676429923 step loss 0.6486547223238058 sigmoid temp 0.1\n iter 500 sigmoid loss 0.5034013090626045 step loss 0.4661657807572751 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.44914205157930764 step loss 0.4644501171456039 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.500902996841499 step loss 0.4640788510887457 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.44600130400909344 step loss 0.4698435042107657 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.517910589010403 step loss 0.5053340675700123 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.45413855404736986 step loss 0.5061476693904973 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5217557880555941 step loss 0.49768506457548 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.45008406554664077 step loss 0.46878389180573427 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.4684652942332036 step loss 0.4652570808824822 sigmoid temp 1.0\n End sigmoid loss 0.4602250509745371 step loss 0.46484924200515065\n beta 0.17107647304049892\n rssi_w [0.00147151 0.0171863 0.07599566 0.12505067]\n rssi_th [20.99973078 30.99972341 24.9985636 ]\n infect_w [0.04403552 0.02900909 0.13537564]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.2714348419789183 step loss 1.2781397097213405\n iter 0 sigmoid loss 1.3118618766502763 step loss 0.5229664789235658 sigmoid temp 0.1\n iter 500 sigmoid loss 0.5037496468317292 step loss 0.4684802471035948 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4482844251856782 step loss 0.46645439465193156 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.5002562779000669 step loss 0.46634781541732284 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.44473280644218494 step loss 0.47492613760498936 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.5191254114529005 step loss 0.47869154503565897 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4528244284834673 step loss 0.4778567175183733 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5217858486617992 step loss 0.4777762885414834 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4483311031411097 step loss 0.47665125656930607 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.47037748208785807 step loss 0.4767025883442265 sigmoid temp 1.0\n End sigmoid loss 0.4605190993798862 step loss 0.47634472010706275\n beta 0.1779951442320126\n rssi_w [-0.02227368 0.00961429 0.08297067 0.13461191]\n rssi_th [31.99876555 17.99874318 25.99782724]\n infect_w [0.0498416 0.03454115 0.15594837]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.4211125767465704 step loss 1.5096454360189335\n iter 0 sigmoid loss 1.4670603670159226 step loss 0.47421417040987024 sigmoid temp 0.1\n iter 500 sigmoid loss 0.5020515374474315 step loss 0.46432705501988975 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.45155169554735314 step loss 0.46429181742813425 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.5034332445937347 step loss 0.46306317901639144 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4517188227215982 step loss 0.46316789761902405 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.5165759995892931 step loss 0.4617988483669019 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.46213884963781354 step loss 0.46079046831098375 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5200335289759411 step loss 0.45928688369943077 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.45764754327681073 step loss 0.4576817769677528 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.46979927036151264 step loss 0.45592213499286605 sigmoid temp 1.0\n End sigmoid loss 0.4534181698523756 step loss 0.4543422159568081\n beta 0.12391024600534309\n rssi_w [0.06007068 0.0633297 0.2042788 0.03013188]\n rssi_th [27.99918662 37.99908902 27.999888 ]\n infect_w [0.03059275 0.02420756 0.09308692]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 0.9179277978930521 step loss 0.9211401127183539\n iter 0 sigmoid loss 0.9566137027413437 step loss 0.6785582489859799 sigmoid temp 0.1\n iter 500 sigmoid loss 0.5041566250954793 step loss 0.4668169494979409 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.44926402279345823 step loss 0.4651849442108793 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.5023524018451344 step loss 0.463681264668372 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4465302471160753 step loss 0.4646579061113853 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.5174123004701656 step loss 0.46599390560501763 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.45511254841026344 step loss 0.46264115860991323 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.526069320190685 step loss 0.46117359311344436 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4554983282816255 step loss 0.461312875756086 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.47464890303288404 step loss 0.4609603664765773 sigmoid temp 1.0\n End sigmoid loss 0.45840283626888306 step loss 0.46136527621673\n beta 0.19175056325805287\n rssi_w [-0.0003365 0.00584606 0.04666665 0.16771546]\n rssi_th [14.99655143 18.99656117 38.99640875]\n infect_w [0.04669049 0.03656233 0.1569127 ]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 0.953806040000221 step loss 0.9426649895129358\n iter 0 sigmoid loss 0.9931579255699364 step loss 0.6790931312128008 sigmoid temp 0.1\n iter 500 sigmoid loss 0.504107324037143 step loss 0.4693083638369083 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4498994930000185 step loss 0.468031228269876 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.5022529216598269 step loss 0.46848715624815146 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.44824177435478973 step loss 0.46809139709617914 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.5164704360998982 step loss 0.4699851543463971 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4582616427711632 step loss 0.4676402122765692 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5190849432831768 step loss 0.4677589469512025 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.45706927822379007 step loss 0.46731743311345936 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.46795109173886457 step loss 0.467358459699436 sigmoid temp 1.0\n End sigmoid loss 0.466141536086455 step loss 0.4673510778241987\n beta 0.18010156038462669\n rssi_w [-0.03095758 -0.01866058 0.11327139 0.1051141 ]\n rssi_th [26.00092401 21.00092798 34.99891061]\n infect_w [0.05830536 0.0340719 0.13844239]\n best loss 0.4543422159568081\n best scoring parameters\n beta 0.12391024600534309\n rssi_w [0.06007068 0.12340038 0.32767918 0.35781106]\n rssi_th [-92.00081338 -54.00172436 -26.00183636]\n infect_w [0.03059275 0.05480031 0.14788723]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 3.948, median size 3\n \t Negative bags: mean size 4.049, median size 3\n assign_mat size, X_shuff size: (4431, 17870) (17870, 3)\n assign_mat_trn size, assign_mat_tst size (3492, 17870) (875, 17870)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.122070744003032 step loss 1.1487630514560048\n iter 0 sigmoid loss 0.9670005357697355 step loss 0.6429736442012415 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4391366171580114 step loss 0.46400554142392275 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.42650789733560585 step loss 0.4626351132184985 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.47483163960839486 step loss 0.4649766873201727 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.48830498355050234 step loss 0.48461982969866285 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4006045710123388 step loss 0.610632968313629 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.39711732970059105 step loss 0.4633227125098513 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.562446188791133 step loss 0.4593151383215866 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.5266268773024599 step loss 0.45873157665882647 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.5645035440298435 step loss 0.4585271900171728 sigmoid temp 1.0\n End sigmoid loss 0.45703464710548436 step loss 0.4585006455335067\n beta 0.15908882494409796\n rssi_w [0.01513308 0.02794613 0.0730871 0.10456733]\n rssi_th [24.00113117 33.00112008 15.99910039]\n infect_w [0.0469209 0.03296887 0.11531267]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 0.9463269639812599 step loss 0.9539284222238102\n iter 0 sigmoid loss 0.8147533683674911 step loss 0.6746724053739839 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4392354276234232 step loss 0.46334593407064434 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.42747404306380504 step loss 0.46044028635678425 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4753787147090007 step loss 0.45824530217504644 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4883832089302419 step loss 0.45772565104118335 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.40178147626283645 step loss 0.4592724282528812 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.39660898944250556 step loss 0.45653347294572716 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5584773028676874 step loss 0.4563856511093013 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.524778438713505 step loss 0.456265717082645 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.5713573258225625 step loss 0.45620324803166007 sigmoid temp 1.0\n End sigmoid loss 0.45489873425246025 step loss 0.4562521957652737\n beta 0.162156224611698\n rssi_w [0.00391552 0.01332672 0.03535265 0.13037823]\n rssi_th [22.00057042 27.00056914 13.00103899]\n infect_w [0.04476564 0.03086553 0.11980603]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 0.9793296397888168 step loss 0.9866112925292869\n iter 0 sigmoid loss 0.8424879393116559 step loss 0.6688258576327764 sigmoid temp 0.1\n iter 500 sigmoid loss 0.43915883956429974 step loss 0.46698611093137926 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.42887832268090054 step loss 0.4660347363258706 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.47696116554576634 step loss 0.4654188240496694 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.49054323129625016 step loss 0.46520515945476454 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.40349286348234353 step loss 0.46564783782801605 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.39569417428362497 step loss 0.46552978067398454 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5685232407106143 step loss 0.4653655833051503 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.5409519847002876 step loss 0.4652105582593811 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.570612334345103 step loss 0.46507439782872756 sigmoid temp 1.0\n End sigmoid loss 0.46375106978552116 step loss 0.4651105668407215\n beta 0.16493052984168913\n rssi_w [-0.01016592 0.00281782 0.06761892 0.12286244]\n rssi_th [23.99819541 18.99820181 31.9978325 ]\n infect_w [0.06043972 0.04256316 0.11347867]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 0.8562330938515436 step loss 0.8555642570171704\n iter 0 sigmoid loss 0.7350063187641596 step loss 0.6831825046723977 sigmoid temp 0.1\n iter 500 sigmoid loss 0.43917261765707666 step loss 0.46774883766509645 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.42998780835643124 step loss 0.467991331298429 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.47769275150007545 step loss 0.4693814164995946 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.49117576979741123 step loss 0.4724126527400592 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.406411897846652 step loss 0.483174907252442 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.400786543277129 step loss 0.5101028315610763 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5740277900525801 step loss 0.499725028852719 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.5439905251491914 step loss 0.4745730665790884 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.5675739093544455 step loss 0.4712566443413256 sigmoid temp 1.0\n End sigmoid loss 0.4661934687834033 step loss 0.47009653628417264\n beta 0.13629956557560377\n rssi_w [-0.00888067 0.00384041 0.03459493 0.09561554]\n rssi_th [22.00134184 15.0013481 14.00133626]\n infect_w [0.05090801 0.02453106 0.08016311]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.1256068527230048 step loss 1.1262261282526855\n iter 0 sigmoid loss 0.9583826006474058 step loss 0.6094090846335091 sigmoid temp 0.1\n iter 500 sigmoid loss 0.43918600608344405 step loss 0.4643001242251231 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4275347085923646 step loss 0.4617292876795894 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.47569557303992566 step loss 0.45959928527088245 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.48876122203006017 step loss 0.45866498312181964 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4016517153378447 step loss 0.45928890346331636 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.39552829204950546 step loss 0.4573767231492989 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5682952912361178 step loss 0.4571635289111572 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.5319881967201165 step loss 0.45703978017372293 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.5640337192748328 step loss 0.4569387701570394 sigmoid temp 1.0\n End sigmoid loss 0.45628284177713574 step loss 0.45704768481908664\n beta 0.17374890642619648\n rssi_w [-0.00913231 -0.00420919 0.06215439 0.13768595]\n rssi_th [11.99872852 33.99872932 21.99848824]\n infect_w [0.05115039 0.04256908 0.13333738]\n best loss 0.4562521957652737\n best scoring parameters\n beta 0.162156224611698\n rssi_w [0.00391552 0.01724224 0.05259489 0.18297312]\n rssi_th [-97.99942958 -70.99886043 -57.99782144]\n infect_w [0.04476564 0.07563117 0.1954372 ]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 4.046, median size 3\n \t Negative bags: mean size 3.951, median size 3\n assign_mat size, X_shuff size: (4431, 17576) (17576, 3)\n assign_mat_trn size, assign_mat_tst size (3481, 17576) (878, 17576)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 0.868340760639224 step loss 0.8668661369703308\n iter 0 sigmoid loss 0.8612517119287328 step loss 0.6683878902035966 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4761673106096875 step loss 0.4608032744657371 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4262378736789351 step loss 0.4597934258526387 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.5085652774944142 step loss 0.45921051533791063 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4977037633250163 step loss 0.4586980481405367 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.5250271974262489 step loss 0.45831163061109365 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4345540268469537 step loss 0.45802535551316476 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4217222315784516 step loss 0.4579799389705816 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4932562526846435 step loss 0.45791290986815913 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.5054018519103042 step loss 0.4579194838652503 sigmoid temp 1.0\n End sigmoid loss 0.45674383587181566 step loss 0.4579974604116454\n beta 0.15848293904708868\n rssi_w [-0.00099646 0.01315242 0.05480578 0.11448192]\n rssi_th [30.99882895 14.99883482 24.9985695 ]\n infect_w [0.06258591 0.02474049 0.10678335]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.1270915751440902 step loss 1.1534528452426431\n iter 0 sigmoid loss 1.1266719449635263 step loss 0.5608836741123522 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4748054519891128 step loss 0.45993879240826313 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4266749951694694 step loss 0.4595759873933723 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.5078105587994715 step loss 0.4596379889766263 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.49744158738737143 step loss 0.4595297765101663 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.5248988424864947 step loss 0.46023428207536743 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.43567478779977975 step loss 0.4593431766490877 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.42617149132665494 step loss 0.45890050757651973 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4933529353459262 step loss 0.4583525596948272 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.5053208352563517 step loss 0.45813847409517683 sigmoid temp 1.0\n End sigmoid loss 0.45719931371167766 step loss 0.4580772092994998\n beta 0.13982105811761728\n rssi_w [0.02903556 0.04261583 0.06656545 0.08229681]\n rssi_th [34.00100779 22.00098932 18.99969886]\n infect_w [0.05203326 0.0155668 0.08700549]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.0384253749584487 step loss 1.0519137404921997\n iter 0 sigmoid loss 1.0369354999231997 step loss 0.6382203078284431 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4764514552664811 step loss 0.4595229694395131 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4258230342656073 step loss 0.45863533634359666 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.5076071789234551 step loss 0.4583963360261181 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.49791441988260265 step loss 0.4581512226582662 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.5253673311923457 step loss 0.4590757181578046 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4311654138225806 step loss 0.45585852120197784 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.408846499297153 step loss 0.45553006497757853 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4825783998740495 step loss 0.4551789061853551 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.5097922288676492 step loss 0.4551453368775139 sigmoid temp 1.0\n End sigmoid loss 0.4541349982600823 step loss 0.4552088700108696\n beta 0.16193951684885236\n rssi_w [0.02154052 0.03992957 0.1199036 0.02995197]\n rssi_th [30.99941021 35.99936545 25.99991558]\n infect_w [0.05743841 0.02339488 0.11805686]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.137670220625792 step loss 1.1551368911651974\n iter 0 sigmoid loss 1.1394049577713776 step loss 0.5897520640925555 sigmoid temp 0.1\n iter 500 sigmoid loss 0.47601413721844466 step loss 0.46013993496217304 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4268799745796349 step loss 0.4597083266064767 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.5071184934195999 step loss 0.45989210744918607 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4976863090625189 step loss 0.4601689572431519 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.5252190665335681 step loss 0.4616603438839094 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.43571786894740455 step loss 0.4603042673026627 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4251350726317086 step loss 0.4592311582477833 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.49243001301379463 step loss 0.45858156546825707 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.5077919123690401 step loss 0.458346305869169 sigmoid temp 1.0\n End sigmoid loss 0.4569501399256659 step loss 0.45827191517483307\n beta 0.1440730307804671\n rssi_w [0.0243433 0.0421321 0.06866924 0.07232369]\n rssi_th [37.00130932 20.0012856 20.99981308]\n infect_w [0.05522088 0.01606109 0.09416195]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 0.9060860548357565 step loss 0.8990630428028576\n iter 0 sigmoid loss 0.8971221576132505 step loss 0.6439398407988968 sigmoid temp 0.1\n iter 500 sigmoid loss 0.47718606782448303 step loss 0.4621716378101358 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4263760221554493 step loss 0.46170825664728915 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.5103875158561412 step loss 0.46190004101532045 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.49771970623734774 step loss 0.4630122843320353 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.5245215069128899 step loss 0.46445163608943507 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.44288875945336764 step loss 0.46270072966759307 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4381440293964037 step loss 0.462557877268341 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.5038733826667224 step loss 0.46263668479873443 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.5058299093613952 step loss 0.4625065212695499 sigmoid temp 1.0\n End sigmoid loss 0.46071251364931287 step loss 0.46241789166147035\n beta 0.13357306543883\n rssi_w [0.00064022 0.00855895 0.02395354 0.09057627]\n rssi_th [26.00070971 10.00072354 14.00073763]\n infect_w [0.05727278 0.01315329 0.07348374]\n best loss 0.4552088700108696\n best scoring parameters\n beta 0.16193951684885236\n rssi_w [0.02154052 0.0614701 0.1813737 0.21132567]\n rssi_th [-89.00058979 -53.00122434 -27.00130877]\n infect_w [0.05743841 0.08083329 0.19889015]\n\n\n\n```python\nfig, axs = plt.subplots(1, 2, figsize=(15,5))\naxs[0].plot(censor_prob, np.mean(auc_train_learned, axis=1), '+-', label='train, learned')\naxs[0].plot(censor_prob, np.mean(auc_train_true, axis=1), '+-', label='train, true')\naxs[0].set_xlabel(\"Censor Probability\")\naxs[0].set_ylabel(\"AUC\")\naxs[0].set_title(\"Train: Max Bag Size = 32\")\naxs[0].legend()\n\naxs[1].plot(censor_prob, np.mean(auc_test_learned, axis=1), '+-', label='test, learned')\naxs[1].plot(censor_prob, np.mean(auc_test_true, axis=1), '+-', label='test, true')\naxs[1].set_xlabel(\"Censor Probability\")\naxs[1].set_ylabel(\"AUC\")\naxs[1].set_title(\"Test: Max Bag Size = 32\")\naxs[1].legend()\n```\n\n### 3. Varying Positive Bag Construction\n\n[Return to top](#Notebook-contents)\n\nThis experiment first increases the maximum number of positive exposures per bag from 1 to 3, then iterates over increasing bag sizes. We want to compare the results here to that of Experiment 1., to show the effect that increasing maximum number of positive exposures has on the performance of the learning algorithm. Increasing maximum number of positive exposures would increase the signal received by the algorithm. Hence, the algorithm should perform better as long as the gain in signal (compared to experiment 1) is not outweighed by the gain in noise from increasing bag size. \n\nWe expect the performance of the algorithm to increase, i.e. the AUC score to increase, compared to experiment 1. However, this will only happen for maximum bag sizes up till around 3 or 4, and will decrease after. This is reflected in the two plots below, where the AUC increases slightly (from around 0.8 to 0.9) up until a maximum bag size of 4, then drops. This trend supports the claim being made, since having up to 3 positive exposures in a bag is significant when the number of positive exposures make up a substantial percentage of the bag size. In this case, a maximum of 3 positive exposures in a bag of 4 exposures would mean a signal strength of 75%. However as the bag gets larger but the positive exposures remain at a fixed maximum of 3, it faces the same problem as Experiment 1 where the noise of the additional exposures drown out the signal. \n\n\n```python\n# effect of varying bag size, given up to 3 positive exposures per bag\nbag_size = [1, 2, 4, 8, 16, 32]\nn_trials = 1\nn_random_restarts_train = 5\n\nidx = 0\n\nauc_train_learned = np.zeros((len(bag_size),n_trials))\nauc_train_true = np.zeros((len(bag_size),n_trials))\nauc_test_learned = np.zeros((len(bag_size),n_trials))\nauc_test_true = np.zeros((len(bag_size),n_trials))\nfor bs in bag_size:\n bag_sim = Bag_Simulator(p_pos=0.6,r_pos=2,p_neg=0.6,r_neg=2,max_bag_size=bs,censor_prob_pos=0.,censor_prob_neg=0,max_pos_in_bag=3)\n auc_train_trials, auc_test_trials = train_and_eval_with_bag_config(bag_sim, X_epi,\n probabilities_true_epi, n_trials=n_trials,\n n_random_restarts=n_random_restarts_train)\n for i in range(n_trials):\n auc_train_learned[idx, i] = dict(auc_train_trials[i])['Learned']\n auc_train_true[idx, i] = dict(auc_train_trials[i])['True']\n auc_test_learned[idx, i] = dict(auc_test_trials[i])['Learned']\n auc_test_true[idx, i] = dict(auc_test_trials[i])['True']\n \n idx += 1\n```\n\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 1.000, median size 1\n \t Negative bags: mean size 1.000, median size 1\n assign_mat size, X_shuff size: (4431, 4431) (4431, 3)\n assign_mat_trn size, assign_mat_tst size (3544, 4431) (887, 4431)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.435833557100843 step loss 1.4919998719849097\n iter 0 sigmoid loss 1.4457935865163787 step loss 0.6713896398738584 sigmoid temp 0.1\n iter 500 sigmoid loss 0.39454125185947403 step loss 0.33957002420014953 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3095893621262033 step loss 0.33017755074426913 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.33892465124191545 step loss 0.3257831404756726 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.26839281863461933 step loss 0.3239480777134672 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.28425681386804286 step loss 0.32995353237192987 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.25639420126707274 step loss 0.32061880256656455 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2892982448335 step loss 0.3165769226657959 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.21932085816532698 step loss 0.3147733684101123 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2629467147084974 step loss 0.31369451680656774 sigmoid temp 1.0\n End sigmoid loss 0.30482443933883296 step loss 0.3136665358227709\n beta 0.36454351959026565\n rssi_w [-0.00891451 0.08174279 0.33709637 0.08386947]\n rssi_th [36.9872118 34.98642146 17.99918278]\n infect_w [0.0102498 0.07409035 0.34600911]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.1792263453195573 step loss 1.1826108468348329\n iter 0 sigmoid loss 1.1990823699835427 step loss 0.8106054425825353 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3974026165081266 step loss 0.32538955075162307 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3096188217105617 step loss 0.31214543410696954 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3422317515591384 step loss 0.3033524503903451 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.26825366710442416 step loss 0.29548802482457254 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.28294446855487193 step loss 0.5030384986190611 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.22992604978931375 step loss 0.5015503347567078 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.24954707464115258 step loss 0.29951641071840024 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.22563704164832693 step loss 0.2928560175857487 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2649010382904389 step loss 0.2907834609730259 sigmoid temp 1.0\n End sigmoid loss 0.28367750290483273 step loss 0.29082108181776367\n beta 0.30473032824597224\n rssi_w [-1.46438908e-04 3.00162578e-02 2.69548916e-01 1.01503879e-01]\n rssi_th [27.01117852 31.01107618 29.99942152]\n infect_w [0.008287 0.06144131 0.28524676]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.1191399356421223 step loss 1.117764397265231\n iter 0 sigmoid loss 1.1367670741115896 step loss 0.8009906519988977 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3986518293753795 step loss 0.3388143991981894 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.30981340508745703 step loss 0.32953356025963493 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.342443260434693 step loss 0.3248614336007339 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.26776852413466234 step loss 0.31962083119983353 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.27856308835321664 step loss 0.316338204553544 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.24318170645122394 step loss 0.31393276372833945 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.277330900011492 step loss 0.31301754386668723 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.21250065032442744 step loss 0.31172182295338186 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.25444173212357796 step loss 0.3099931958763543 sigmoid temp 1.0\n End sigmoid loss 0.296660898888017 step loss 0.3099177658023278\n beta 0.38678136837489985\n rssi_w [-0.06358413 -0.05965949 0.18421374 0.31571538]\n rssi_th [ 9.9931627 36.99316644 23.98724104]\n infect_w [0.01057328 0.07909798 0.3676716 ]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.1297204879905975 step loss 1.1434506076765911\n iter 0 sigmoid loss 1.1484704952597613 step loss 0.8513777113552274 sigmoid temp 0.1\n iter 500 sigmoid loss 0.39385518470340813 step loss 0.3435475617052379 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3099017476281969 step loss 0.33473553740951906 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.33870253977027104 step loss 0.3314618130836289 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.2685811786062221 step loss 0.33088375195390374 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.28482503290620953 step loss 0.3368402483927518 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.26164328957725336 step loss 0.3293335173396089 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2995110392603005 step loss 0.32641051492866774 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.22931538191358253 step loss 0.32518955757664536 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2797674015726871 step loss 0.3245272774881173 sigmoid temp 1.0\n End sigmoid loss 0.3157458890404614 step loss 0.32454450314208916\n beta 0.35603490633185897\n rssi_w [-0.01570107 0.09986237 0.31855147 0.07725113]\n rssi_th [38.98969191 34.98860717 16.99908833]\n infect_w [0.01059267 0.07457185 0.33635803]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.1317548255216041 step loss 1.163480103409567\n iter 0 sigmoid loss 1.144695683373002 step loss 0.8538863053731299 sigmoid temp 0.1\n iter 500 sigmoid loss 0.39182661301601995 step loss 0.34242551969720547 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3089554289357865 step loss 0.33390019365089874 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3337516121299441 step loss 0.3309409216861679 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.2650709687756752 step loss 0.33437485581814014 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2759487549603042 step loss 0.36522202078372157 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.2569143065094508 step loss 0.3307036174404193 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.29361876308116797 step loss 0.3250149106759687 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.22243437662067908 step loss 0.3231074958143326 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.27036288106123685 step loss 0.3221558237212638 sigmoid temp 1.0\n End sigmoid loss 0.311077933292284 step loss 0.32216296608877387\n beta 0.36454288648558586\n rssi_w [-0.00448539 0.08228686 0.32755887 0.09916323]\n rssi_th [34.98756897 37.98693841 11.9988121 ]\n infect_w [0.01025086 0.0722514 0.34586982]\n best loss 0.29082108181776367\n best scoring parameters\n beta 0.30473032824597224\n rssi_w [-1.46438908e-04 2.98698189e-02 2.99418735e-01 4.00922614e-01]\n rssi_th [-92.98882148 -61.9777453 -31.97832378]\n infect_w [0.008287 0.06972831 0.35497506]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 1.554, median size 2\n \t Negative bags: mean size 1.560, median size 2\n assign_mat size, X_shuff size: (4431, 6908) (6908, 3)\n assign_mat_trn size, assign_mat_tst size (3544, 6908) (790, 6908)\n Average positive samples per bag: 1.4943661971830986\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.1837801759657274 step loss 1.2112377357946957\n iter 0 sigmoid loss 0.9477913519377271 step loss 0.7209081263101211 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3349688385253218 step loss 0.3232795960383862 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.33193688632245694 step loss 0.3123857349011004 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3263300072175103 step loss 0.30728753602285613 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3521643168437044 step loss 0.3044053616695704 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2929035110604238 step loss 0.34364100922277707 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.22718399095888608 step loss 0.3422480529431119 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3151558089963482 step loss 0.33936665296542945 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3020099866160886 step loss 0.3390493804195359 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2542830269043291 step loss 0.32905569172890275 sigmoid temp 1.0\n End sigmoid loss 0.2941304094978999 step loss 0.3288035508403608\n beta 0.29751522417621457\n rssi_w [-0.05757349 0.04433764 0.15860202 0.22673117]\n rssi_th [37.00570723 15.00545638 27.99653896]\n infect_w [0.00603402 0.04106273 0.28536993]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.0601334787383894 step loss 1.0654949194468064\n iter 0 sigmoid loss 0.8473871708757903 step loss 0.7719838610891679 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3358095408821296 step loss 0.31887620898804514 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3350853338556621 step loss 0.3064395364302617 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3291085029425044 step loss 0.2967381278475104 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3530951516511754 step loss 0.2887718105380951 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2933878997243796 step loss 0.7735290052018553 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.2063604325587362 step loss 0.28465096296298875 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.30693890477765573 step loss 0.27273308100803395 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.26097021568243856 step loss 0.27066882084426147 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2407030837320341 step loss 0.269901603201037 sigmoid temp 1.0\n End sigmoid loss 0.2561414628813297 step loss 0.2710686046457263\n beta 0.3634742521379074\n rssi_w [-0.01154868 -0.00428187 0.04535941 0.35006989]\n rssi_th [10.99261993 18.99264053 36.99242331]\n infect_w [0.00471397 0.04650676 0.35207539]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.1639796715373854 step loss 1.1811425386861225\n iter 0 sigmoid loss 0.9284115028915347 step loss 0.7294162285758815 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3335159830450552 step loss 0.3176963318497122 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3305163587440608 step loss 0.3043830246996297 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.32202154272876926 step loss 0.2931326156538014 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3443084953176339 step loss 0.2792229339554346 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2852249221041937 step loss 0.2860006081020005 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.199785821826446 step loss 0.26685282524331494 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2901449573196514 step loss 0.26719614341088943 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.25875819584754806 step loss 0.2672318451239024 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2333486454015771 step loss 0.2669578206589667 sigmoid temp 1.0\n End sigmoid loss 0.25200365815369447 step loss 0.26809192783551933\n beta 0.36607689856961156\n rssi_w [-0.06398688 -0.04166101 0.13257696 0.32276791]\n rssi_th [19.99852829 26.9985198 18.99590321]\n infect_w [0.00487024 0.04710775 0.35427045]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.2262461833797738 step loss 1.2520149662114275\n iter 0 sigmoid loss 0.9766126877462472 step loss 0.7122862492365746 sigmoid temp 0.1\n iter 500 sigmoid loss 0.33091606989769007 step loss 0.31243579527376303 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.32810681453381546 step loss 0.2989146787279141 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3192870760210069 step loss 0.28919696235115716 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.34272054687740366 step loss 0.42157176043497707 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.28416236270566086 step loss 0.4513712909451253 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.2051172326431882 step loss 0.4030877990070281 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.28913486226025037 step loss 0.36262301574766986 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2742578929935363 step loss 0.28407620779237336 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.22815258909756497 step loss 0.27994586458291176 sigmoid temp 1.0\n End sigmoid loss 0.27074257032024857 step loss 0.27995832094764345\n beta 0.3056887607460034\n rssi_w [-0.00532949 0.01914278 0.15780747 0.24799286]\n rssi_th [23.00639063 33.0063528 18.99640253]\n infect_w [0.0047028 0.04246963 0.29287029]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.021817981225012 step loss 1.0260073352275603\n iter 0 sigmoid loss 0.8173883088173843 step loss 0.7768269362778379 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3364914217610597 step loss 0.3296259681014695 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3361932172935505 step loss 0.3214757685543663 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3319806000143506 step loss 0.3186399166540513 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3594927558308194 step loss 0.3151109943646052 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3043574597304923 step loss 0.3144751384316902 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.23721909738585878 step loss 0.311844137700054 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3723133697155322 step loss 0.3082690947166901 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.30838675307275787 step loss 0.30692484717495894 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.27360888055421456 step loss 0.30658058234453833 sigmoid temp 1.0\n End sigmoid loss 0.2990636209262062 step loss 0.30696139900386\n beta 0.36303294459184554\n rssi_w [-0.04639609 -0.02993565 0.13173107 0.32121327]\n rssi_th [18.99071552 22.99073304 32.98891586]\n infect_w [0.00479624 0.05639082 0.35024903]\n best loss 0.26809192783551933\n best scoring parameters\n beta 0.36607689856961156\n rssi_w [-0.06398688 -0.10564789 0.02692907 0.34969698]\n rssi_th [-100.00147171 -73.00295191 -54.0070487 ]\n infect_w [0.00487024 0.05197799 0.40624844]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 2.414, median size 2\n \t Negative bags: mean size 2.426, median size 2\n assign_mat size, X_shuff size: (4431, 10742) (10742, 3)\n assign_mat_trn size, assign_mat_tst size (3544, 10742) (817, 10742)\n Average positive samples per bag: 2.0183098591549298\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.0483955193136627 step loss 1.0664725511442985\n iter 0 sigmoid loss 1.0458123143958244 step loss 0.6544109294534421 sigmoid temp 0.1\n iter 500 sigmoid loss 0.30327933245903926 step loss 0.314813838483179 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.2964877765415271 step loss 0.3108225514537911 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3178248920763817 step loss 0.3311268129989106 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.31292756466964344 step loss 0.33292217562247917 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3729603223219333 step loss 0.3879918284976411 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.26749964963095507 step loss 0.33596707377274204 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.21860730973363338 step loss 0.32194410773507515 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2604660968419151 step loss 0.3153186318609429 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.24858026958776364 step loss 0.3097385186582696 sigmoid temp 1.0\n End sigmoid loss 0.28119306946340394 step loss 0.3096328033667339\n beta 0.3058978226804342\n rssi_w [-0.01693843 -0.00571563 0.09103949 0.30506625]\n rssi_th [20.99563883 30.99562752 21.99290101]\n infect_w [-0.01157165 0.05782105 0.32857977]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.2178879164927356 step loss 1.206422856990227\n iter 0 sigmoid loss 1.21068714209863 step loss 0.5394622130786187 sigmoid temp 0.1\n iter 500 sigmoid loss 0.30674051948516295 step loss 0.3292190675603544 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.30020624930255047 step loss 0.3278972554982895 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.33253447719016066 step loss 0.3281798985735014 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3372285072959565 step loss 0.3283385456158762 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.27537759883751456 step loss 0.33254825030481727 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3098528363204261 step loss 0.3287167638352251 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.24777028023015824 step loss 0.3274240229948868 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2824104372556783 step loss 0.34094871888692 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.30168266503123276 step loss 0.3315158794176655 sigmoid temp 1.0\n End sigmoid loss 0.30991339316690386 step loss 0.32304373093927113\n beta 0.24706479922743652\n rssi_w [-0.09225507 -0.03835916 0.20779519 0.19616545]\n rssi_th [29.00383935 20.00377375 35.99687701]\n infect_w [-0.02313488 0.09997147 0.32300484]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.0268411340218473 step loss 1.0266487973957241\n iter 0 sigmoid loss 1.0223135992904222 step loss 0.6321602044295461 sigmoid temp 0.1\n iter 500 sigmoid loss 0.31230767514244673 step loss 0.31419583184840894 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.2939397419853703 step loss 0.2976057751234797 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3186886246721539 step loss 0.2805673564093317 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.31279275482712465 step loss 0.472573501767167 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.24889500846355056 step loss 0.5279823734863572 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.24573026265541792 step loss 0.28736376473271874 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.17095897572158655 step loss 0.2606262767320319 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.24094577156780503 step loss 0.2513552911294643 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.22306664110468652 step loss 0.2500201737192396 sigmoid temp 1.0\n End sigmoid loss 0.24085525018082787 step loss 0.2515791994867952\n beta 0.295672622549381\n rssi_w [-0.03164335 -0.02234683 0.06816968 0.30315742]\n rssi_th [12.00361296 27.00361929 25.00323664]\n infect_w [0.00306017 0.03513377 0.30808236]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 0.9390940890024152 step loss 0.9371965487314721\n iter 0 sigmoid loss 0.9343324391241532 step loss 0.6674767816547984 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3463482275755533 step loss 0.35843742586010635 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.30055663619528333 step loss 0.32850452856223694 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.30926887605252223 step loss 0.30852103166693134 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3026805140001392 step loss 0.30056177770798975 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2473837046743271 step loss 0.3185193435828881 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.2701627910954156 step loss 0.2939600376447682 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.21620013954411882 step loss 0.2862455865630266 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2684452149468559 step loss 0.2849267478331681 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2722652693491858 step loss 0.2821448956725052 sigmoid temp 1.0\n End sigmoid loss 0.2721432080114936 step loss 0.28275430960526104\n beta 0.32629780854765966\n rssi_w [-0.06765316 -0.02421641 0.1230735 0.35122977]\n rssi_th [26.98816212 15.98812446 28.98670742]\n infect_w [-0.00773397 0.05503144 0.38716006]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.009873767145957 step loss 1.0154980658978043\n iter 0 sigmoid loss 1.001346140742862 step loss 0.643558623756567 sigmoid temp 0.1\n iter 500 sigmoid loss 0.30219716488713844 step loss 0.3088504127602089 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.28792231793587797 step loss 0.29411022272910275 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.30755810374247067 step loss 0.27681133169618743 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3173552592422565 step loss 0.2916625780194401 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.33520682179030503 step loss 0.31351861265221614 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3114154007777099 step loss 0.29349192477470654 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.21028255381481412 step loss 0.28101508466886665 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2725427736444487 step loss 0.26386361178176526 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.22414357511092994 step loss 0.2555271479151027 sigmoid temp 1.0\n End sigmoid loss 0.30694891027731397 step loss 0.3083261280744437\n beta 0.03425330135909943\n rssi_w [0.06265861 0.0799978 0.12309871 0.37879034]\n rssi_th [21.99968989 27.99969156 14.00127127]\n infect_w [0.0685701 0.0758079 0.25987471]\n best loss 0.2515791994867952\n best scoring parameters\n beta 0.295672622549381\n rssi_w [-0.03164335 -0.05399018 0.0141795 0.31733692]\n rssi_th [-107.99638704 -80.99276775 -55.98953112]\n infect_w [0.00306017 0.03819394 0.3462763 ]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 3.434, median size 3\n \t Negative bags: mean size 3.483, median size 3\n assign_mat size, X_shuff size: (4431, 15399) (15399, 3)\n assign_mat_trn size, assign_mat_tst size (3544, 15399) (852, 15399)\n Average positive samples per bag: 2.0056338028169014\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.0764540839697798 step loss 1.0680860619494743\n iter 0 sigmoid loss 1.0571995477007587 step loss 0.5923029620621589 sigmoid temp 0.1\n iter 500 sigmoid loss 0.35349329524167694 step loss 0.3428925034902044 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3282976187412433 step loss 0.33319961108215057 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.2875475354758166 step loss 0.33293641250709827 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.27049013474742095 step loss 0.34437576566110933 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2856665415763507 step loss 0.34715589405350583 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.25219977081734934 step loss 0.34070224888127904 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3075139183737931 step loss 0.3326408952534338 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.34907251021461183 step loss 0.32289887930971967 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3354762252465908 step loss 0.31863772601471907 sigmoid temp 1.0\n End sigmoid loss 0.2982569591401199 step loss 0.31058687584263783\n beta 0.16773482136290807\n rssi_w [-0.03025482 0.02445459 0.08361863 0.3236848 ]\n rssi_th [31.99791559 18.99782999 19.99422803]\n infect_w [-0.00421518 0.05111055 0.35084305]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 0.9837427463527523 step loss 0.9853270351952809\n iter 0 sigmoid loss 0.9657276046484056 step loss 0.6374495920030804 sigmoid temp 0.1\n iter 500 sigmoid loss 0.36043785387132254 step loss 0.34884306367090734 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.34108294889277724 step loss 0.33786101785254563 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.31556670800770914 step loss 0.3250590433665159 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.270607229444227 step loss 0.8377032127425301 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3011086423980984 step loss 0.8309427974479386 sigmoid temp 0.1\n\n\n :19: RuntimeWarning: overflow encountered in exp\n bag_probs = 1 - np.exp(-bag_scores)\n\n\n iter 3000 sigmoid loss 0.361063284980753 step loss 0.3780265261237605 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.34023816510789084 step loss 0.34353019935326723 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.37793763709219325 step loss 0.32696753404057216 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3295722714835959 step loss 0.31347956858420006 sigmoid temp 1.0\n End sigmoid loss 0.29408189928379236 step loss 0.30341406381645697\n beta 0.1484755444118967\n rssi_w [-0.01813235 0.00640908 0.0569794 0.41795193]\n rssi_th [20.99382379 13.9938754 33.99351701]\n infect_w [-0.00643011 0.06611393 0.3235244 ]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.0169659204554367 step loss 1.0119708920559902\n iter 0 sigmoid loss 0.9965926446947005 step loss 0.611458029663613 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3546095576688197 step loss 0.3367901473824934 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.326642512513709 step loss 0.3229570700660556 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.27688344697871226 step loss 0.31763805316651095 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.25173923305856977 step loss 0.40874838635468697 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.37310314317143 step loss 0.42301222589244375 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.32332039586700345 step loss 0.39959244176702746 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.34804359614898234 step loss 0.35846935182046524 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3704301056595952 step loss 0.34568069579463845 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.31154266932138724 step loss 0.33259063746122214 sigmoid temp 1.0\n End sigmoid loss 0.2982127213464786 step loss 0.30633768993297\n beta 0.07736834789208158\n rssi_w [-0.02128961 -0.00127299 0.03391992 0.31068733]\n rssi_th [21.0046793 29.00467287 11.00732811]\n infect_w [0.06833397 0.03146502 0.45051903]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 0.8546811667772868 step loss 0.8548432792437243\n iter 0 sigmoid loss 0.8359833828888756 step loss 0.6537020282676334 sigmoid temp 0.1\n iter 500 sigmoid loss 0.40197094134946665 step loss 0.3803343009296961 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.33097630155924274 step loss 0.34460344288853473 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.2951258644571462 step loss 0.33704474602620077 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.2614404242516807 step loss 0.35376099087239266 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3056894909194974 step loss 0.4295898218880118 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.2823663444008604 step loss 0.3355202948496509 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.29809695373233 step loss 0.32701793984980787 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.38337026045441064 step loss 0.3235559202587417 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3226554971590123 step loss 0.32344237014282246 sigmoid temp 1.0\n End sigmoid loss 0.31722687366499713 step loss 0.3240929079117391\n beta 0.30009501972007013\n rssi_w [-0.06461549 -0.04439714 0.14008108 0.33291061]\n rssi_th [18.99103118 23.99103123 31.98874044]\n infect_w [0.00259953 0.03525011 0.37475352]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 0.9244275800577914 step loss 0.9322311183768813\n iter 0 sigmoid loss 0.9046230489237681 step loss 0.6887417478054525 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3540892576095851 step loss 0.3528404388882456 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.331726770278583 step loss 0.34909699551566103 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.30601224684148687 step loss 0.3505912417700841 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.2659980496724373 step loss 1.1367359919725215 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.30134408950073616 step loss 1.1685634029316383 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.2691861153941832 step loss 0.3480156886758861 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2784945991477422 step loss 0.3278522008633694 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.37157799266368174 step loss 0.32378965584476394 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.31792484603835364 step loss 0.3237929265458603 sigmoid temp 1.0\n End sigmoid loss 0.3074184051366011 step loss 0.32492785615681796\n beta 0.3460132558779193\n rssi_w [-0.04102027 0.06576481 0.33833382 0.08359598]\n rssi_th [35.98657379 36.98597718 18.9992383 ]\n infect_w [0.00306472 0.03050086 0.35469057]\n best loss 0.30341406381645697\n best scoring parameters\n beta 0.1484755444118967\n rssi_w [-0.01813235 -0.01172326 0.04525614 0.46320807]\n rssi_th [-99.00617621 -85.01230081 -51.01878381]\n infect_w [-0.00643011 0.05968383 0.38320823]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 3.931, median size 3\n \t Negative bags: mean size 3.930, median size 3\n assign_mat size, X_shuff size: (4431, 17416) (17416, 3)\n assign_mat_trn size, assign_mat_tst size (3544, 17416) (860, 17416)\n Average positive samples per bag: 1.9985915492957746\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.074308041622844 step loss 1.0921744240159226\n iter 0 sigmoid loss 0.8619189928794129 step loss 0.6518237903780099 sigmoid temp 0.1\n iter 500 sigmoid loss 0.26507007930097654 step loss 0.3485297353464759 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4446410466742245 step loss 0.7253296453474026 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4017298907253675 step loss 0.3675385912175915 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.2990026053676507 step loss 0.7399962098287965 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.367577239586316 step loss 0.36139885138598016 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.32620531303128203 step loss 0.32421556809137225 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.36144595221694326 step loss 0.3373443413782715 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3493079336246524 step loss 0.31924415633707787 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2074003097736241 step loss 0.2902538630602254 sigmoid temp 1.0\n End sigmoid loss 0.2724944469840246 step loss 0.28339924249703924\n beta 0.13189956687688068\n rssi_w [-0.00397016 0.02179501 0.35264055 0.12778646]\n rssi_th [28.99852922 35.99834323 19.99927207]\n infect_w [4.15408403e-04 4.62507655e-02 4.27374179e-01]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.1103325337201766 step loss 1.1133034319288388\n iter 0 sigmoid loss 0.8924771371095206 step loss 0.6400954788247705 sigmoid temp 0.1\n iter 500 sigmoid loss 0.26624823027622563 step loss 0.3447295402164742 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.43550623345766565 step loss 0.3436227707596619 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.36062109468326087 step loss 0.5540757471136449 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.2882741619257891 step loss 0.5828633049353199 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2935349141016008 step loss 0.5657585950665054 sigmoid temp 0.1\n iter 3000 sigmoid loss 2.3025930930340457 step loss 1.845195097136538 sigmoid temp 0.325\n iter 3500 sigmoid loss 2.129899361060242 step loss 1.845195097136538 sigmoid temp 0.55\n iter 4000 sigmoid loss 2.245028515709444 step loss 1.845195097136538 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.9210432372436185 step loss 1.845195097136538 sigmoid temp 1.0\n End sigmoid loss 1.845195097136538 step loss 1.845195097136538\n beta -0.00539213834981353\n rssi_w [0.07355895 0.13715468 0.26202025 0.09229795]\n rssi_th [34.00018558 26.99999996 29.99967401]\n infect_w [0.78599135 0.28040119 0.359496 ]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 0.9139636998553873 step loss 0.9196519283333209\n iter 0 sigmoid loss 0.7322910048120181 step loss 0.6752741017282132 sigmoid temp 0.1\n iter 500 sigmoid loss 0.33898886978636883 step loss 0.39964662994535144 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4005466970333886 step loss 0.4163286079163426 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.30548559455380114 step loss 0.40200219758226224 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.28871977080782335 step loss 0.41519961757655566 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.29267673011679984 step loss 0.41364174910859974 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.2968253711135796 step loss 0.4000035851690708 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3745890384429692 step loss 0.3561649864528792 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.38725731952714026 step loss 0.36662921300409795 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2500555710165875 step loss 0.36434578255789307 sigmoid temp 1.0\n End sigmoid loss 0.3388179772676223 step loss 0.3623049301904934\n beta 0.14613440990341997\n rssi_w [-0.10771422 -0.00399769 0.23925365 0.21491513]\n rssi_th [35.00733858 15.00708283 36.99725229]\n infect_w [-0.02483374 0.06426976 0.3276071 ]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.020502828122302 step loss 1.0352542578316004\n iter 0 sigmoid loss 0.8159785647510587 step loss 0.6886808369313763 sigmoid temp 0.1\n iter 500 sigmoid loss 0.2664697308527969 step loss 0.3406726903657647 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4181667215903628 step loss 0.5629326450987093 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3533212567049334 step loss 0.5344117634128035 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.30055805952540743 step loss 0.5780934627905712 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.37640688093125974 step loss 0.5600865389669549 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3420271277728741 step loss 0.426859316745658 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3380468728916101 step loss 0.33433941994464506 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.34053239520319406 step loss 0.3009536333940574 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.1969585954083506 step loss 0.2857748861956967 sigmoid temp 1.0\n End sigmoid loss 0.2742300279904177 step loss 0.2848308848231574\n beta 0.13760233263649937\n rssi_w [-0.01767127 0.03267217 0.2938986 0.07462121]\n rssi_th [34.00408502 29.00391847 29.99971742]\n infect_w [0.00529859 0.02420644 0.4476416 ]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.1543778445741473 step loss 1.163442077431679\n iter 0 sigmoid loss 0.9286415499628997 step loss 0.5845306563581355 sigmoid temp 0.1\n iter 500 sigmoid loss 0.26845824188101114 step loss 0.3582968875396573 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4284872456262534 step loss 0.36066583169919425 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.32027423714905423 step loss 0.7078032422219086 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3011595255506872 step loss 0.7247499019694966 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.28784294982672615 step loss 0.9059959322767932 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.26739782660480726 step loss 0.3163899687525215 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.29161851414725865 step loss 0.30686736682965626 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.31407569874828395 step loss 0.2993400721677094 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2076803107072706 step loss 0.3104873231794138 sigmoid temp 1.0\n End sigmoid loss 0.28354476097922127 step loss 0.3021113292328123\n beta 0.3193823764318203\n rssi_w [-0.0400224 -0.03125329 0.08747432 0.31810784]\n rssi_th [10.99042496 28.99042374 28.98979827]\n infect_w [0.00128175 0.02095975 0.3506989 ]\n best loss 0.28339924249703924\n best scoring parameters\n beta 0.13189956687688068\n rssi_w [-0.00397016 0.01782485 0.37046539 0.49825185]\n rssi_th [-91.00147078 -55.00312755 -35.00385548]\n infect_w [4.15408403e-04 4.66661739e-02 4.74040353e-01]\n total: 33600, positives: 5383, negatives: 28217\n Empirical Bag sizes:\n \t Positive bags: mean size 3.961, median size 3\n \t Negative bags: mean size 3.927, median size 3\n assign_mat size, X_shuff size: (4431, 17425) (17425, 3)\n assign_mat_trn size, assign_mat_tst size (3544, 17425) (856, 17425)\n Average positive samples per bag: 1.9464788732394367\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.3387237447419145 step loss 1.3447208390311294\n iter 0 sigmoid loss 1.4562781881829232 step loss 0.45767753117627785 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3481410442884669 step loss 0.36325707517502465 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.41914099900034957 step loss 0.3527017760912043 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3148681431432876 step loss 0.3473734658047806 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3123269871352352 step loss 0.8738140103561342 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.33996594301944816 step loss 0.9109644721564347 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3135136527935177 step loss 0.3344141973991924 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.34145795792662825 step loss 0.32457008597150916 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2348321126410569 step loss 0.3243639474936104 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2781944057533604 step loss 0.32525861347697055 sigmoid temp 1.0\n End sigmoid loss 0.3035439403433146 step loss 0.3189210492412512\n beta 0.31314490894486224\n rssi_w [-0.02473453 0.04985327 0.32375727 0.02382609]\n rssi_th [32.98770738 36.98733673 37.99985511]\n infect_w [-0.00811771 0.04365196 0.32575459]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 0.9786152153913475 step loss 0.9794428293547269\n iter 0 sigmoid loss 1.0669045608887566 step loss 0.5983972407645848 sigmoid temp 0.1\n iter 500 sigmoid loss 0.33318124606905813 step loss 0.3854177964486685 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3684217173033565 step loss 0.8204765893973324 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.27987095202144674 step loss 0.8822933899652167 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.29966499117265044 step loss 0.9240478917716214 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.32954169800155186 step loss 0.876664527690675 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.31359021857634845 step loss 0.3360958672345581 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.34050164612326583 step loss 0.3229324062558429 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.23454024657409522 step loss 0.32472663501047544 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.27863828994626744 step loss 0.3253870886276926 sigmoid temp 1.0\n End sigmoid loss 0.30395973462434234 step loss 0.31874734258432524\n beta 0.3013817790302852\n rssi_w [-0.02964597 -0.00723343 0.06357961 0.32498458]\n rssi_th [18.98797203 15.9879617 34.98770693]\n infect_w [-0.00862474 0.04493073 0.33733376]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 0.9544463217078807 step loss 0.9792354665788661\n iter 0 sigmoid loss 1.0404354182282909 step loss 0.64696648714194 sigmoid temp 0.1\n iter 500 sigmoid loss 0.34070062150823466 step loss 0.34770865527568123 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.40874321735420743 step loss 0.3453046412755628 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.27839820458634984 step loss 0.634447588869537 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.29600607370348214 step loss 0.7034770872398431 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.32839375928576586 step loss 0.6565211522569324 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.38445010795026796 step loss 0.4867650433511203 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4489346076791588 step loss 0.3692106762775337 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3252474315462943 step loss 0.3658628518973308 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3267543354740927 step loss 0.36295468380163853 sigmoid temp 1.0\n End sigmoid loss 0.35188881946630074 step loss 0.35800987868200324\n beta 0.034852975558285926\n rssi_w [-0.00366661 0.01793733 0.2103066 0.12058636]\n rssi_th [28.00225065 35.00217277 14.99958845]\n infect_w [0.3215021 0.19159309 0.38067324]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 0.9424909059652559 step loss 0.9580765746650205\n iter 0 sigmoid loss 1.02361018444839 step loss 0.6490522457922852 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3401700509676124 step loss 0.34721779199225933 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4087034668862255 step loss 0.3466362643498007 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.2800384019966229 step loss 0.5680180771727921 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.29683056268258956 step loss 0.6485383104878711 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3289864097379754 step loss 0.5942496496551237 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3314354943827539 step loss 0.30033253227540013 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3350205396755239 step loss 0.30194101077247143 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.23215721334821493 step loss 0.302532503440396 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3069407341937681 step loss 0.3014733324683184 sigmoid temp 1.0\n End sigmoid loss 0.2932807054399969 step loss 0.29980743486513833\n beta 0.2470613719689595\n rssi_w [-0.00613804 0.02256573 0.19495839 0.17723609]\n rssi_th [28.0084664 33.00839793 16.99841104]\n infect_w [-0.00620999 0.03736829 0.2656186 ]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 0.978260629044742 step loss 0.9807382829668692\n iter 0 sigmoid loss 1.064646446431703 step loss 0.5715855998477232 sigmoid temp 0.1\n iter 500 sigmoid loss 0.34838659944976974 step loss 0.35236041432211956 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.47517605188874584 step loss 0.3970444533548932 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3777939919011915 step loss 0.3701118754657488 sigmoid temp 0.1\n iter 2000 sigmoid loss 1.0856864543907967 step loss 1.0721274595945356 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.6982000927774817 step loss 0.7055016039169326 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3904644110731229 step loss 0.3961375999768722 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.4446908706051314 step loss 0.3536366337772732 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2869161457081573 step loss 0.34859547664356133 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3248866114726301 step loss 0.3455494051152873 sigmoid temp 1.0\n End sigmoid loss 0.3413835437974398 step loss 0.34130006350781694\n beta 0.05182453499387163\n rssi_w [0.14070578 0.14209401 0.1354966 0.33130185]\n rssi_th [12.99880172 13.99895686 37.99879138]\n infect_w [-0.0441648 0.08184584 0.18030195]\n best loss 0.29980743486513833\n best scoring parameters\n beta 0.2470613719689595\n rssi_w [-0.00613804 0.01642769 0.21138608 0.38862218]\n rssi_th [-91.9915336 -58.98313567 -41.98472463]\n infect_w [-0.00620999 0.03115829 0.29677689]\n\n\n\n```python\nfig, axs = plt.subplots(1, 2, figsize=(15,5))\naxs[0].plot(bag_size, np.mean(auc_train_learned, axis=1), '+-', label='train, learned')\naxs[0].plot(bag_size, np.mean(auc_train_true, axis=1), '+-', label='train, true')\naxs[0].set_xlabel(\"Maximum Bag Size\")\naxs[0].set_ylabel(\"AUC\")\naxs[0].set_title(\"Train: No Censoring\")\naxs[0].legend()\n\naxs[1].plot(bag_size, np.mean(auc_test_learned, axis=1), '+-', label='test, learned')\naxs[1].plot(bag_size, np.mean(auc_test_true, axis=1), '+-', label='test, true')\naxs[1].set_xlabel(\"Maximum Bag Size\")\naxs[1].set_ylabel(\"AUC\")\naxs[1].set_title(\"Test: No Censoring\")\naxs[1].legend()\n```\n\n### 4. Robustness to Model Mismatch\n\n[Return to top](#Notebook-contents)\n\nRecognizing that the simulator may not be completely accurate or reflective of real data, slightly modifying the simulator by using a Taylor series approximation and then testing the performance of the model to see its robustness to slightly different generating models.\n\nThis experiment is iterated over decreasing number of Taylor approximation terms to show the effect that increasing model mismatch has on the performance of the learning algorithm. Decreasing the number of terms used in the Taylor approximation of the generative function that directly calculates the probability of infection would increase model mismatch in that the approximation becomes less accurate to the original function. Hence, the probability of infection calculated should be less accurate as well.\n\nWe expect the performance of the algorithm to decrease, i.e. the AUC score to fall, as model mismatch increases, or as the number of Taylor approximation terms decreases. \n\nThis generic trend has been illustrated through the below two graphs. Reading from left to right following the x-axis (number of taylor terms increases), we do see a increase of AUC for the training set and testing sets. We can see they matche our expectation. However, for both sets the accuracy for using Inf terms (10 terms) is not clear that the infinity approximation seems to be less accurate than using less term. This phenomenon suggests two possibilities. First, our simulator may not be a very accruate model of how COVID is tranmitted, so that mismatch phenomenon is reflected not so completely. Second, our version of simulator is in general a robust one but with some noise. \n\n\n```python\n# initializing input data\ndistances, durations, symptoms = uniform_input_data_grid(max_dur = 60, max_dist = 5, ngrid_dist = 80, ngrid_dur=20, min_dist=0.1, min_dur=5)\ndata = make_input_data(distances=distances, durations=durations, symptoms=symptoms)\n\n# initializing parameters\nble_params = BleParams()\nrssi = atten_to_rssi(data['atten_grid'], ble_params)\nduration = data['duration_grid']\nsymptom_day = data['symptom_grid']\ninfectiousness = days_to_inf_levels(symptom_day)\n\nparams = ModelParams()\n\n# Create model inputs\nX = np.concatenate([rssi[:,None], duration[:,None], infectiousness[:,None]], axis=1)\napprox_type = 'taylor'\nexp_taylor_terms = 8\nexp_temperature = np.arange(0.5,1.,0.09)\n\n# calculating probability of infection\nprob_infect_exact, prob_infect_approx = prob_infection_grid(data, params, approx=approx_type, exp_taylor_terms=exp_taylor_terms, temp=exp_temperature)\nprob_infect_approx = np.array(prob_infect_approx)\n# manually inspect the approximation error (higher error for larger values -- away from 0)\nind = np.where(prob_infect_exact>0.6)\nprint('Approx:', prob_infect_approx[:,ind])\nprint('Exact:', prob_infect_exact[ind])\n\n\nprint (X.shape, prob_infect_exact.shape)\n\n\nsimdata = dict()\nsimdata['X'] = X\nsimdata['prob_infect_exact'] = prob_infect_exact\nsimdata['prob_infect_approx'] = prob_infect_approx\n\nX_epi = simdata['X'] \nprobabilities_true_epi = simdata['prob_infect_exact']\nprobabilities_true_epi_approx = simdata['prob_infect_approx'].T # N x num_approx_steps\nN = len(probabilities_true_epi)\nperm = np.random.permutation(N)\nX_epi = X_epi[perm, :]\nprobabilities_true_epi = probabilities_true_epi[perm]\nprobabilities_true_epi_approx = np.clip(probabilities_true_epi_approx[perm,:], 1e-5, 1-1e-5)\n \n# generating labels for the exposure events\ndef sample_labels(probabilities):\n Y = bernoulli.rvs(probabilities)\n N_pos = np.sum(Y)\n N_neg = np.sum(1-Y)\n print(\"total: {}, positives: {}, negatives: {}\".format(N, N_pos, N_neg))\n pos_neg_ratio= float(N_neg) / N_pos\n return Y, N_pos, N_neg, pos_neg_ratio\n\nY_epi, N_pos, N_neg, pos_neg_ratio = sample_labels(probabilities_true_epi)\n```\n\n Making grid of 80 distances x 20 durations x 21 onsets = 33600 points\n Approx: [[[1. 1. 1. ... 0.97538178 0.92937435 0.92675691]]\n \n [[0.49996063 0.49998181 0.46092688 ... 0.49969697 0.49750601 0.49731772]]\n \n [[0.67110338 0.66968263 0.81008083 ... 0.65435507 0.63129513 0.62997964]]\n \n ...\n \n [[0.63518322 0.63414931 0.72087815 ... 0.62280329 0.60509304 0.60406036]]\n \n [[0.63539429 0.63435626 0.72199235 ... 0.62296994 0.60521186 0.60417686]]\n \n [[0.63536768 0.63433023 0.72181414 ... 0.62294962 0.60519806 0.60416337]]]\n Exact: [0.63537038 0.63433287 0.72183658 ... 0.62295162 0.60519936 0.60416464]\n (33600, 3) (33600,)\n total: 33600, positives: 5383, negatives: 28217\n\n\n\n```python\n# parameters of training\nn_trials = 1\nn_random_restarts_train = 5\n\nidx = 0\n\n# taylor series approximation parameters\napprox_type = 'taylor'\nexp_taylor_terms = [np.inf, 8, 6, 4, 2]\nexp_temperature = np.arange(0.5,1.,0.09)\n\nauc_train_learned = np.zeros((len(exp_taylor_terms),n_trials))\nauc_train_true = np.zeros((len(exp_taylor_terms),n_trials))\nauc_test_learned = np.zeros((len(exp_taylor_terms),n_trials))\nauc_test_true = np.zeros((len(exp_taylor_terms),n_trials))\n\nfor num_taylor_term in exp_taylor_terms:\n # generate probability of infection using taylor approximation\n prob_infect_exact, prob_infect_approx = prob_infection_grid(data, params, approx=approx_type, exp_taylor_terms=num_taylor_term, temp=exp_temperature)\n prob_infect_approx = np.array(prob_infect_approx)\n\n # store data\n simdata = dict()\n simdata['X'] = X\n simdata['prob_infect_exact'] = prob_infect_exact\n simdata['prob_infect_approx'] = prob_infect_approx\n\n # shuffle data points\n X_epi = simdata['X'] \n probabilities_true_epi = simdata['prob_infect_exact']\n probabilities_true_epi_approx = simdata['prob_infect_approx'].T # N x num_approx_steps\n N = len(probabilities_true_epi)\n perm = np.random.permutation(N)\n X_epi = X_epi[perm, :]\n probabilities_true_epi = probabilities_true_epi[perm]\n probabilities_true_epi_approx = np.clip(probabilities_true_epi_approx[perm], 1e-5, 1-1e-5)\n \n # sample labels (exposure outcome) based on probability of infection\n Y_epi, N_pos, N_neg, pos_neg_ratio = sample_labels(probabilities_true_epi)\n\n # initialize bag simulator and run the training\n bag_sim = Bag_Simulator(p_pos=0.6,r_pos=2,p_neg=0.6,r_neg=2,max_bag_size=4,censor_prob_pos=0.,censor_prob_neg=0,max_pos_in_bag=3)\n auc_train_trials, auc_test_trials = train_and_eval_with_bag_config(bag_sim, X_epi,\n probabilities_true_epi, n_trials=n_trials,\n n_random_restarts=n_random_restarts_train)\n for i in range(n_trials):\n auc_train_learned[idx, i] = dict(auc_train_trials[i])['Learned']\n auc_train_true[idx, i] = dict(auc_train_trials[i])['True']\n auc_test_learned[idx, i] = dict(auc_test_trials[i])['Learned']\n auc_test_true[idx, i] = dict(auc_test_trials[i])['True']\n \n idx += 1\n```\n\n total: 33600, positives: 5403, negatives: 28197\n total: 33600, positives: 5403, negatives: 28197\n Empirical Bag sizes:\n \t Positive bags: mean size 2.455, median size 2\n \t Negative bags: mean size 2.431, median size 2\n assign_mat size, X_shuff size: (4415, 10750) (10750, 3)\n assign_mat_trn size, assign_mat_tst size (3532, 10750) (812, 10750)\n Average positive samples per bag: 2.0126760563380284\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 0.9761889113483901 step loss 0.9897511676748867\n iter 0 sigmoid loss 1.021417163339004 step loss 0.6673122182516648 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3314837361232751 step loss 0.31720750425336 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3065881446743266 step loss 0.3054449968220978 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.2324311130960724 step loss 0.3040847644574086 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.24234261433300083 step loss 0.4116515374821078 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2565302668174569 step loss 0.41345551129455405 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.2888643753801915 step loss 0.370467790100824 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.22727234067173363 step loss 0.3347709573328102 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.27341499163941485 step loss 0.2992725134475831 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3411246082128925 step loss 0.28775174730629194 sigmoid temp 1.0\n End sigmoid loss 0.277143018015852 step loss 0.2860360440538287\n beta 0.30307344167358463\n rssi_w [-0.00676378 0.01519916 0.07773313 0.29091915]\n rssi_th [24.001531 30.0015014 19.99473793]\n infect_w [-0.00799261 0.04951506 0.29049198]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 0.968023617363326 step loss 0.9728609858012375\n iter 0 sigmoid loss 1.0019860014155144 step loss 0.6730036255218729 sigmoid temp 0.1\n iter 500 sigmoid loss 0.33419449577896804 step loss 0.3181235045856165 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.31082430858523574 step loss 0.3082808225301001 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.2364051771117516 step loss 0.2955240472786161 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.24442268875306944 step loss 0.2889273655688886 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.24461151569188908 step loss 0.30457098414098405 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.28938259001506034 step loss 0.2733694808482109 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2102717666009019 step loss 0.27291754327663986 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.26802155851665466 step loss 0.2755259246007242 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.33567823229071964 step loss 0.27335155099550135 sigmoid temp 1.0\n End sigmoid loss 0.26304489213556775 step loss 0.2711930245305742\n beta 0.33133395060659365\n rssi_w [-0.06376326 -0.00828931 0.10052504 0.30415331]\n rssi_th [33.99891328 9.99887129 22.99795373]\n infect_w [-0.0052266 0.0459258 0.31978485]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 0.9944521363498497 step loss 0.9940919170459281\n iter 0 sigmoid loss 1.0384083235417358 step loss 0.6439706823245083 sigmoid temp 0.1\n iter 500 sigmoid loss 0.333426529884174 step loss 0.322875030243759 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.30902981955595704 step loss 0.3125074993805202 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.23420521370856184 step loss 0.3012790412153068 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.24356843531358588 step loss 0.28625936799066476 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.24583937628537794 step loss 0.2808984544305078 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.2857821862178712 step loss 0.2787694550609737 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.21190799338221397 step loss 0.27668228800121086 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.26873331200295086 step loss 0.27711588809784676 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3394647727301139 step loss 0.2763873465400654 sigmoid temp 1.0\n End sigmoid loss 0.2657202410380499 step loss 0.2763207169382355\n beta 0.3280076819016763\n rssi_w [-0.07689807 -0.00524139 0.1178018 0.31517038]\n rssi_th [36.99547059 10.99535589 20.99440692]\n infect_w [-0.00657197 0.04959745 0.31825546]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.2678925249196522 step loss 1.2849863547911433\n iter 0 sigmoid loss 1.3133883282475813 step loss 0.5132591518544045 sigmoid temp 0.1\n iter 500 sigmoid loss 0.33494757096069266 step loss 0.33178308448144983 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.31588530176053253 step loss 0.3279774439603084 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.24731791774147183 step loss 0.32728789901944394 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.2770053358352593 step loss 0.3268131596752183 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2804388225749945 step loss 0.33889945843864927 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.31972820714529376 step loss 0.3545033754665234 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2601006252869082 step loss 0.35035293433906095 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2987460512085311 step loss 0.34560161647766807 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.34932014788409305 step loss 0.3394198475696733 sigmoid temp 1.0\n End sigmoid loss 0.3133551508624793 step loss 0.3389175187900252\n beta 0.2687508768832835\n rssi_w [-0.03529322 -0.01387122 0.16055963 0.20727079]\n rssi_th [20.00218555 30.00217516 32.99673548]\n infect_w [-0.00533397 0.04133009 0.25431648]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.0021816309505907 step loss 1.003686388446959\n iter 0 sigmoid loss 1.0416508713322334 step loss 0.6862948741134087 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3332857461399182 step loss 0.3125781916197977 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3098418632510754 step loss 0.2991039513907258 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.23889016229828333 step loss 0.2880895023750064 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.25282917898125207 step loss 0.4359781078795731 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2509626681015402 step loss 0.5215287578032193 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3007228572258948 step loss 0.4114490334351089 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.21733861623692013 step loss 0.28792875652661587 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2625391389284848 step loss 0.2811229370730432 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.31146487878656487 step loss 0.2796410210849009 sigmoid temp 1.0\n End sigmoid loss 0.27094713895255557 step loss 0.2788822506106673\n beta 0.26602883623484386\n rssi_w [-0.00745859 0.02697996 0.25108037 0.05160191]\n rssi_th [28.00911343 32.00901453 38.99980941]\n infect_w [-0.00158539 0.03502961 0.2535229 ]\n best loss 0.2711930245305742\n best scoring parameters\n beta 0.33133395060659365\n rssi_w [-0.06376326 -0.07205256 0.02847248 0.33262579]\n rssi_th [-86.00108672 -76.00221543 -53.0042617 ]\n infect_w [-0.0052266 0.0406992 0.36048405]\n total: 33600, positives: 5424, negatives: 28176\n total: 33600, positives: 5424, negatives: 28176\n Empirical Bag sizes:\n \t Positive bags: mean size 2.423, median size 2\n \t Negative bags: mean size 2.445, median size 2\n assign_mat size, X_shuff size: (4398, 10739) (10739, 3)\n assign_mat_trn size, assign_mat_tst size (3518, 10739) (814, 10739)\n Average positive samples per bag: 1.976056338028169\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.0117763336022245 step loss 1.0069395557468184\n iter 0 sigmoid loss 0.8378884027795246 step loss 0.6708688470144607 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3368251838635717 step loss 0.41389610596273607 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3769180261606765 step loss 0.38656084659537177 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.38724303773576396 step loss 0.3750461597189268 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.39456613046966654 step loss 0.36778899923802166 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.34331204753007016 step loss 0.35566794951202285 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.37623683322062407 step loss 0.37753953648110766 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3246350333497148 step loss 0.34692035295846047 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.31200120179889007 step loss 0.3219838790594384 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.29435962872346266 step loss 0.3100061046808495 sigmoid temp 1.0\n End sigmoid loss 0.2847850168481446 step loss 0.29039190893193384\n beta 0.08023396979851541\n rssi_w [-0.01629494 -0.00301911 0.02996466 0.28172842]\n rssi_th [15.01033151 15.01037937 28.01033439]\n infect_w [0.03016635 0.10810942 0.41291908]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 0.9413684972978628 step loss 0.9413802128677877\n iter 0 sigmoid loss 0.7760000274832266 step loss 0.6944538456113053 sigmoid temp 0.1\n iter 500 sigmoid loss 0.360834658239298 step loss 0.3192338694126401 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3185238389103641 step loss 0.30709457505287957 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.28578073056880654 step loss 0.2848294022250643 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.2795304098014802 step loss 0.44957906487704946 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3462518974083888 step loss 0.3632899113097922 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3875866454946115 step loss 0.3498637453110776 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.28848690559138807 step loss 0.3353688580834498 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3013910658129923 step loss 0.3250415467353563 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.29929572451848363 step loss 0.3160620315463743 sigmoid temp 1.0\n End sigmoid loss 0.299865179916498 step loss 0.3067889802694238\n beta 0.04842205475187311\n rssi_w [-0.02531797 -0.01565455 0.05511326 0.35444505]\n rssi_th [15.00838154 22.00841588 23.0082105 ]\n infect_w [0.17417672 0.05982531 0.44173847]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.0849154183558944 step loss 1.0916862803444172\n iter 0 sigmoid loss 0.8872060700720827 step loss 0.6907112289596767 sigmoid temp 0.1\n iter 500 sigmoid loss 0.33469898108789153 step loss 0.33146786988992843 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.34743972960285563 step loss 0.33400224023608577 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.2809789448992591 step loss 0.2918311166841746 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.2726317369474811 step loss 0.3314347481340991 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3383864402563415 step loss 0.2926073506850617 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.29716423146324933 step loss 0.27055393461311317 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.23893931271230662 step loss 0.2688463944368926 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.24033934201229257 step loss 0.26358041453612013 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.24263334148932844 step loss 0.2631347961806768 sigmoid temp 1.0\n End sigmoid loss 0.2537617687363822 step loss 0.2627025116961782\n beta 0.3371289367641971\n rssi_w [-0.040126 -0.03296138 0.0959144 0.34913089]\n rssi_th [14.9927602 27.99276705 24.99180444]\n infect_w [0.00380624 0.03530965 0.34326312]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 0.9157290981238239 step loss 0.9228633898108545\n iter 0 sigmoid loss 0.7575297472053942 step loss 0.7152919746397449 sigmoid temp 0.1\n iter 500 sigmoid loss 0.35961934795340006 step loss 0.3208463532639252 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.2998315856349841 step loss 0.3082193754729461 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.2755034381134897 step loss 0.2889302792889755 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3302262540013634 step loss 0.3411623436801547 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2480004507474863 step loss 0.3141264680829214 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3040772272513412 step loss 0.2660123679225523 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2308845594677694 step loss 0.2579944498190688 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3525520554856123 step loss 0.3234347422412836 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.22816617770281725 step loss 0.2582776618850651 sigmoid temp 1.0\n End sigmoid loss 0.2518396800430423 step loss 0.2568939412738871\n beta 0.26095541288674295\n rssi_w [-0.06082025 -0.00910146 0.0928565 0.32482694]\n rssi_th [33.00618059 15.00612352 15.00611034]\n infect_w [0.00416903 0.03170053 0.27041203]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.1001888447798418 step loss 1.1127444288838548\n iter 0 sigmoid loss 0.906393036539964 step loss 0.6914389309506154 sigmoid temp 0.1\n iter 500 sigmoid loss 0.36205242434491663 step loss 0.33162999168572915 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3213998012702977 step loss 0.33121916061746826 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.29198747245155227 step loss 0.3192756752947955 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.29394620420100537 step loss 0.3407265647064768 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.25784585010464817 step loss 0.39982661886745574 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4330871047907751 step loss 0.44125337677557896 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3530396377931195 step loss 0.37438579795410154 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3858833679954858 step loss 0.3676571197269924 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.37151633191186745 step loss 0.3580590006084795 sigmoid temp 1.0\n End sigmoid loss 0.34516110389964894 step loss 0.3456239705596689\n beta 0.025185407978478458\n rssi_w [0.07754928 0.10957033 0.34185954 0.12549475]\n rssi_th [21.00738585 31.00735547 37.99925797]\n infect_w [ 0.12554983 -0.01694648 0.42556761]\n best loss 0.2568939412738871\n best scoring parameters\n beta 0.26095541288674295\n rssi_w [-0.06082025 -0.06992171 0.02293479 0.34776173]\n rssi_th [-86.99381941 -71.98769589 -56.98158555]\n infect_w [0.00416903 0.03586956 0.30628159]\n total: 33600, positives: 5476, negatives: 28124\n total: 33600, positives: 5476, negatives: 28124\n Empirical Bag sizes:\n \t Positive bags: mean size 2.444, median size 2\n \t Negative bags: mean size 2.403, median size 2\n assign_mat size, X_shuff size: (4356, 10498) (10498, 3)\n assign_mat_trn size, assign_mat_tst size (3484, 10498) (805, 10498)\n Average positive samples per bag: 1.9985915492957746\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.050700000937514 step loss 1.0723877280350322\n iter 0 sigmoid loss 1.1071175616223914 step loss 0.6711965237567618 sigmoid temp 0.1\n iter 500 sigmoid loss 0.43292503230846596 step loss 0.3246052990000388 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.2819879585152071 step loss 0.30853333476192896 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.31931761842551337 step loss 0.3941470826176214 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.31516532796343644 step loss 0.46906301056796923 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.38297519848429074 step loss 0.3840555453742579 sigmoid temp 0.1\n iter 3000 sigmoid loss 2.0723347837356414 step loss 1.8769721079933666 sigmoid temp 0.325\n iter 3500 sigmoid loss 1.554253587814231 step loss 1.8769721079933666 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.7845118971126357 step loss 1.8769721079933666 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 2.0723347837356414 step loss 1.8769721079933666 sigmoid temp 1.0\n End sigmoid loss 1.8769721079933666 step loss 1.8769721079933666\n beta -0.016475104006141827\n rssi_w [0.27656435 0.28514252 0.28777727 0.18103096]\n rssi_th [18.99926397 37.99922805 19.99945653]\n infect_w [0.66316829 0.50897839 0.21102604]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.0442526544167743 step loss 1.0503217830063107\n iter 0 sigmoid loss 1.1019051284865942 step loss 0.647633348613962 sigmoid temp 0.1\n iter 500 sigmoid loss 0.401343003586174 step loss 0.32537218980675503 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.28532678823428415 step loss 0.307952017425985 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.2760122183108975 step loss 0.4003428641982124 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.2841734466750489 step loss 0.4281370271624417 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.27573362407926766 step loss 0.3917739705524372 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.31819746347630734 step loss 0.3741094015001002 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.26420561746554555 step loss 0.33854965358335554 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3908630070758818 step loss 0.43167301741075464 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3366091313470006 step loss 0.29782514793951603 sigmoid temp 1.0\n End sigmoid loss 0.28899084562041794 step loss 0.29483027977583626\n beta 0.209520112423489\n rssi_w [-0.00697141 0.01390192 0.12561576 0.27034064]\n rssi_th [25.00654784 29.00651161 21.99549058]\n infect_w [0.00590323 0.03780186 0.3228653 ]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.0280442107310195 step loss 1.027821355378588\n iter 0 sigmoid loss 1.086722516859532 step loss 0.6408307967415539 sigmoid temp 0.1\n iter 500 sigmoid loss 0.40645069620904406 step loss 0.33775781913485664 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.30281622381367584 step loss 0.324872342092692 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.31669410942781384 step loss 0.3188704638686423 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.292316601531116 step loss 0.3654068035556563 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3500026588068616 step loss 5.423202209409318 sigmoid temp 0.1\n iter 3000 sigmoid loss 9.440600681288322 step loss 9.635963357030672 sigmoid temp 0.325\n iter 3500 sigmoid loss 9.958681877209935 step loss 9.635963357030672 sigmoid temp 0.55\n iter 4000 sigmoid loss 9.728423567911442 step loss 9.635963357030672 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 9.440600681288322 step loss 9.635963357030672 sigmoid temp 1.0\n End sigmoid loss 9.635963357030672 step loss 9.635963357030672\n beta 0.15241782341514065\n rssi_w [11.38831537 11.15463686 9.3498625 6.26207465]\n rssi_th [20.93561675 20.93706243 10.95421027]\n infect_w [1.16309737 0.31034784 0.26400701]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 0.9299611304686863 step loss 0.9388259102450195\n iter 0 sigmoid loss 0.9835188730115777 step loss 0.673397489057002 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4305610507039188 step loss 0.31813719189807527 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.28458566522726086 step loss 0.29950320718198026 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.31836648098289116 step loss 0.3473382807445598 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.5131856091263282 step loss 0.3561350332588275 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2888142112690752 step loss 0.3455049474178999 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.32146520965113157 step loss 0.31521138626629097 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2461729084251161 step loss 0.2883847846687525 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.282378135117418 step loss 0.2664535353397077 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3079096690862176 step loss 0.25603379796681586 sigmoid temp 1.0\n End sigmoid loss 0.24999508072537502 step loss 0.25622885412740715\n beta 0.26205820685399883\n rssi_w [-0.01048998 0.02100763 0.01090489 0.34379186]\n rssi_th [31.00800991 22.00796399 10.00617811]\n infect_w [0.00698751 0.02754638 0.26711649]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.1490020579454652 step loss 1.1836940977310941\n iter 0 sigmoid loss 1.207669691524286 step loss 0.6502496503305356 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4033087114580763 step loss 0.34174653270349653 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.2952713960601649 step loss 0.3304013412871835 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.31038383241224526 step loss 0.3269793162920554 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.2985578203428798 step loss 0.3313195293458792 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2612842539546426 step loss 0.4422703900669067 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.29233201013374543 step loss 0.330149188738576 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.25863148473779174 step loss 0.3183658604498668 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2990754464358117 step loss 0.31598758512110586 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.36941847939679 step loss 0.31476244973852785 sigmoid temp 1.0\n End sigmoid loss 0.30672246957309696 step loss 0.31471993724270686\n beta 0.3733514797848002\n rssi_w [-0.05910714 0.09629025 0.34586091 0.02998736]\n rssi_th [38.98748061 36.98628123 30.99969989]\n infect_w [0.00481378 0.03880205 0.36468845]\n best loss 0.25622885412740715\n best scoring parameters\n beta 0.26205820685399883\n rssi_w [-0.01048998 0.01051766 0.02142254 0.3652144 ]\n rssi_th [-88.99199009 -66.9840261 -56.97784799]\n infect_w [0.00698751 0.03453389 0.30165038]\n total: 33600, positives: 5306, negatives: 28294\n total: 33600, positives: 5306, negatives: 28294\n Empirical Bag sizes:\n \t Positive bags: mean size 2.386, median size 2\n \t Negative bags: mean size 2.420, median size 2\n assign_mat size, X_shuff size: (4496, 10856) (10856, 3)\n assign_mat_trn size, assign_mat_tst size (3596, 10856) (819, 10856)\n Average positive samples per bag: 2.0211267605633805\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 0.9856274873057295 step loss 0.9799558500979308\n iter 0 sigmoid loss 0.9682141608623652 step loss 0.6407077235801452 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3138466771710027 step loss 0.3335773062484404 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3587820237725902 step loss 0.3327384441805366 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.2799087525103408 step loss 0.3334165477158871 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.362688340581696 step loss 0.33640447250538597 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3039664099378745 step loss 0.36043142845852716 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3906976024585985 step loss 0.3654248607770023 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3844056309556697 step loss 0.3681576779759489 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.28805357498371115 step loss 0.36617920464128195 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.31005291915413635 step loss 0.3584751442071529 sigmoid temp 1.0\n End sigmoid loss 0.3246888608239906 step loss 0.35744582872938957\n beta 0.25709955780707217\n rssi_w [-0.07610314 -0.06083233 0.02919978 0.22351129]\n rssi_th [11.00994355 26.00994433 13.00972668]\n infect_w [-0.03124629 0.09417402 0.22495596]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 0.9829364712501389 step loss 0.98618475218312\n iter 0 sigmoid loss 0.9662277209094887 step loss 0.6600962774133198 sigmoid temp 0.1\n iter 500 sigmoid loss 0.294194812605973 step loss 0.32092538387292796 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.33961494459625585 step loss 0.2997224982076951 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.2528868064926071 step loss 0.2872706243514301 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3185889303067503 step loss 0.4712468215508598 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.29065161402500966 step loss 0.4805581735732966 sigmoid temp 0.1\n iter 3000 sigmoid loss 2.0723347837356414 step loss 1.8185127764890128 sigmoid temp 0.325\n iter 3500 sigmoid loss 1.8420764744372367 step loss 1.8185127764890128 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.439124433165029 step loss 1.8185127764890128 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 1.8996410517618378 step loss 1.8185127764890128 sigmoid temp 1.0\n End sigmoid loss 1.8185127764890128 step loss 1.8185127764890128\n beta -0.005400460026494636\n rssi_w [0.16806696 0.17362854 0.28914621 0.2071164 ]\n rssi_th [10.9998679 31.99986855 20.99946768]\n infect_w [0.41367786 0.21035569 0.27560763]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.0124933124392304 step loss 1.0075727421875813\n iter 0 sigmoid loss 0.9964176686503966 step loss 0.6400888507595542 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3103329003081068 step loss 0.31635495291732163 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3437536465964445 step loss 0.3080610825829011 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.2644132398566424 step loss 0.3870929268827619 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3315991686247593 step loss 0.3909501448255509 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3216697568217667 step loss 0.28588942709175796 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.326243343051572 step loss 0.34770304052909734 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.34072908004203994 step loss 0.32679764613498086 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2780862334392855 step loss 0.33468625264282315 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.32055760016884877 step loss 0.32741954611150187 sigmoid temp 1.0\n End sigmoid loss 0.32717552212168066 step loss 0.32705576593331515\n beta 0.012731011575406376\n rssi_w [0.78012639 0.79438412 0.87307363 0.39969228]\n rssi_th [17.99373204 26.99373631 15.99400732]\n infect_w [-0.01638142 0.05370691 0.16654075]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.042065751830827 step loss 1.0389162206987155\n iter 0 sigmoid loss 1.0246137928883052 step loss 0.6211005420099378 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3124142733881703 step loss 0.316827986625363 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3494799453971373 step loss 0.3094626189193042 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.26952435941887537 step loss 0.29834074814418965 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3365601036329474 step loss 0.4364273215589634 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.297895661128944 step loss 0.46096510727340156 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.29763943093813083 step loss 0.27937365442941425 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3166030577767268 step loss 0.27497476695186146 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.20904497888318738 step loss 0.27295966439280767 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2696691992383976 step loss 0.26679099215124763 sigmoid temp 1.0\n End sigmoid loss 0.2614414143350133 step loss 0.2670003280288604\n beta 0.25172498285531686\n rssi_w [-0.02143688 -0.00547803 0.0415944 0.27449505]\n rssi_th [19.00737525 10.00739364 32.00726635]\n infect_w [0.00449546 0.03511774 0.2705482 ]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 0.9499185074485678 step loss 0.9581813036171953\n iter 0 sigmoid loss 0.9339534369822425 step loss 0.6777168556619347 sigmoid temp 0.1\n iter 500 sigmoid loss 0.31571482323281 step loss 0.3301980860112648 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3501386846036756 step loss 0.3273568987853008 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.27333509707911274 step loss 0.32137190893397194 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.35043203630703135 step loss 0.3189594610745801 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2964463773065993 step loss 0.33420235305300644 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.31171578312938125 step loss 0.3177740056257855 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3081400130181046 step loss 0.31678844769216075 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.30598571490326715 step loss 0.3393066545390774 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2908917434975597 step loss 0.30499014020940796 sigmoid temp 1.0\n End sigmoid loss 0.29699592431707067 step loss 0.3057341102288989\n beta 0.31913771353077036\n rssi_w [-0.05692075 -0.02750026 0.12061046 0.28065534]\n rssi_th [20.99003246 19.99002813 32.98860046]\n infect_w [0.00407876 0.05471382 0.35025388]\n best loss 0.2670003280288604\n best scoring parameters\n beta 0.25172498285531686\n rssi_w [-0.02143688 -0.02691492 0.01467949 0.28917453]\n rssi_th [-100.99262475 -90.98523111 -58.97796476]\n infect_w [0.00449546 0.03961321 0.3101614 ]\n total: 33600, positives: 5510, negatives: 28090\n total: 33600, positives: 5510, negatives: 28090\n Empirical Bag sizes:\n \t Positive bags: mean size 2.383, median size 2\n \t Negative bags: mean size 2.414, median size 2\n assign_mat size, X_shuff size: (4329, 10429) (10429, 3)\n assign_mat_trn size, assign_mat_tst size (3463, 10429) (789, 10429)\n Average positive samples per bag: 1.9873239436619718\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 0.914559364292362 step loss 0.9091070973149527\n iter 0 sigmoid loss 0.6850023965866596 step loss 0.7190299283274696 sigmoid temp 0.1\n iter 500 sigmoid loss 0.36479061559759335 step loss 0.3712520841967357 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3449048437238417 step loss 0.3291356322202904 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.35349849189350957 step loss 0.3248451688220629 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4377822961184613 step loss 0.3216665881138618 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.24492095451646712 step loss 0.3792241977634512 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.6417051195828881 step loss 0.5991868337975524 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.30798276440867883 step loss 0.33293130081862454 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.28881082040451916 step loss 0.3334281096804071 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3068530629304855 step loss 0.3370259883995444 sigmoid temp 1.0\n End sigmoid loss 0.3159388875485679 step loss 0.3411207548587151\n beta 0.16738226254743294\n rssi_w [-0.03851589 -0.02039665 0.02329896 0.29512235]\n rssi_th [12.01273669 15.0127391 25.0126832 ]\n infect_w [0.00407227 0.02596316 0.20988878]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.030716042412783 step loss 1.035925898694786\n iter 0 sigmoid loss 0.7712868179511585 step loss 0.726381537577717 sigmoid temp 0.1\n iter 500 sigmoid loss 0.32528117019690833 step loss 0.34107476985408036 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.34133760287908177 step loss 0.33323405455534155 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.345129351517697 step loss 0.3292734190977211 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.42465129428966336 step loss 0.3251519246434918 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.23903897317542916 step loss 0.3376909178380343 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.27925366168525717 step loss 0.32589984700730656 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2762964176702473 step loss 0.3206407531473924 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.26100166659643276 step loss 0.31816899904342233 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.33302699075694114 step loss 0.31751599183779666 sigmoid temp 1.0\n End sigmoid loss 0.3080449790704457 step loss 0.317803072845173\n beta 0.3552943184630507\n rssi_w [-0.06737212 -0.0377545 0.14518859 0.31105866]\n rssi_th [24.9901867 17.99018529 31.98783453]\n infect_w [0.00420138 0.03954365 0.35156071]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.0951244122356836 step loss 1.1081920843845683\n iter 0 sigmoid loss 0.8166650632136219 step loss 0.7355853102254045 sigmoid temp 0.1\n iter 500 sigmoid loss 0.32418434374717703 step loss 0.32228506226247355 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3330378704731438 step loss 0.3033614402396351 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.325804793173278 step loss 0.28971548884964077 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3991380844121138 step loss 0.5507885516671671 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.22730013965960474 step loss 0.58603255241523 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4027272561509919 step loss 0.33919747116272164 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.31052814070000295 step loss 0.3380097357835549 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2922991940387058 step loss 0.33740914441950354 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.36262971186071014 step loss 0.3380808291700783 sigmoid temp 1.0\n End sigmoid loss 0.336763237504049 step loss 0.33648558349701463\n beta 0.025690725710580842\n rssi_w [0.63087279 0.64899758 0.49475422 0.10721602]\n rssi_th [25.99609975 35.99597462 27.99930408]\n infect_w [-0.02863309 0.0688601 0.18998051]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.1926071529654718 step loss 1.1982968367787146\n iter 0 sigmoid loss 0.8876284227004276 step loss 0.6650696211720296 sigmoid temp 0.1\n iter 500 sigmoid loss 0.32354361474313026 step loss 0.3261414057305246 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3365137584796406 step loss 0.31061992798608995 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.32957847173960403 step loss 0.2926619307057414 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.39726849559981003 step loss 0.5202248668725079 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2229864788795248 step loss 0.5946870570492035 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.33247612536952936 step loss 0.3030209125891851 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.23208702161638065 step loss 0.2821031192524758 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2227167751714479 step loss 0.2757008941060939 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.28283699565496795 step loss 0.2733812515856558 sigmoid temp 1.0\n End sigmoid loss 0.25953788538864353 step loss 0.2753305669220631\n beta 0.29792164768217544\n rssi_w [-0.04110388 -0.02637844 0.08425594 0.32156406]\n rssi_th [21.00020233 21.00021167 21.99968127]\n infect_w [0.00377954 0.0294282 0.29818869]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 0.9575317925295885 step loss 0.9529882541240144\n iter 0 sigmoid loss 0.7126572249482631 step loss 0.7202464905665807 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3273336272570387 step loss 0.34275693519703276 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.34734137935636483 step loss 0.34129994672236386 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.35779205047080465 step loss 0.3422998560445396 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4435500827620793 step loss 0.3468313689835928 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2506914868850697 step loss 0.3628676246018822 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.311011813653992 step loss 0.35217594302695915 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.32568904584998437 step loss 0.3463881643858418 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2999899123744154 step loss 0.3426121388213957 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3250710050167126 step loss 0.34309170517320714 sigmoid temp 1.0\n End sigmoid loss 0.3280195082024813 step loss 0.3415167353341498\n beta 0.28633403542356317\n rssi_w [-0.09732455 -0.08028568 0.02757418 0.24728275]\n rssi_th [15.00839724 23.00839426 11.00806361]\n infect_w [0.00529761 0.03335168 0.27845284]\n best loss 0.2753305669220631\n best scoring parameters\n beta 0.29792164768217544\n rssi_w [-0.04110388 -0.06748233 0.01677362 0.33833768]\n rssi_th [-98.99979767 -77.99958601 -55.99990474]\n infect_w [0.00377954 0.03320773 0.33139643]\n\n\n\n```python\ntaylor_terms = [10, 8, 6, 4, 2]\n\nfig, axs = plt.subplots(1, 2, figsize=(15,5))\naxs[0].plot(taylor_terms, np.mean(auc_train_learned, axis=1), '+-', label='train, learned')\naxs[0].plot(taylor_terms, np.mean(auc_train_true, axis=1), '+-', label='train, true')\naxs[0].set_xlabel(\"Number of Taylor Approx Terms\")\naxs[0].set_ylabel(\"AUC\")\naxs[0].set_title(\"Train: No Censoring\")\naxs[0].set_xticks([10, 8, 6, 4, 2])\naxs[0].set_xticklabels(['Inf', '8', '6', '4', '2'])\naxs[0].legend()\n\naxs[1].plot(taylor_terms, np.mean(auc_test_learned, axis=1), '+-', label='test, learned')\naxs[1].plot(taylor_terms, np.mean(auc_test_true, axis=1), '+-', label='test, true')\naxs[1].set_xlabel(\"Number of Taylor Approx Terms\")\naxs[1].set_ylabel(\"AUC\")\naxs[1].set_title(\"Test: No Censoring\")\naxs[1].set_xticks([10, 8, 6, 4, 2])\naxs[1].set_xticklabels(['Inf', '8', '6', '4', '2'])\n#axs[1].set_xticks(['Inf', 8, 6, 4, 2])\naxs[1].legend()\n```\n\n### 5. Reordering of data\n\n\n[Return to top](#Notebook-contents)\n\nTesting the robustness of the optimization routine to a reordering of the data points in the bagging simulation.\n\nThis experiment is iterated over increasing bag sizes, but the difference between this and experiment 1 is the shuffling of training data. We would expect the same trends as Experiment 1, and it acts more like a check to make sure that the learning model previously was not learning to a very specific sequence of the data and being biased. This is confirmed by two graphs below, where we can see the decreasing AUC following the increase of bag size. \n\n\nMore discussion on this can be found in III.2 Future Work and Potential Modifications, under optimization of SGD.\n\n\n```python\n# effect of reordering data points\nbag_size = [1, 2, 4, 8, 16, 32]\nn_trials = 1\nn_random_restarts_train = 5\n\n\nidx = 0\n\nauc_train_learned = np.zeros((len(bag_size),n_trials))\nauc_train_true = np.zeros((len(bag_size),n_trials))\nauc_test_learned = np.zeros((len(bag_size),n_trials))\nauc_test_true = np.zeros((len(bag_size),n_trials))\nfor bs in bag_size:\n bag_sim = Bag_Simulator(p_pos=0.6,r_pos=2,p_neg=0.6,r_neg=2,max_bag_size=bs,censor_prob_pos=0.,censor_prob_neg=0,max_pos_in_bag=1)\n auc_train_trials, auc_test_trials = train_and_eval_with_bag_config(bag_sim, X_epi,\n probabilities_true_epi, n_trials=n_trials,\n n_random_restarts=n_random_restarts_train,order=False)\n for i in range(n_trials):\n auc_train_learned[idx, i] = dict(auc_train_trials[i])['Learned']\n auc_train_true[idx, i] = dict(auc_train_trials[i])['True']\n auc_test_learned[idx, i] = dict(auc_test_trials[i])['Learned']\n auc_test_true[idx, i] = dict(auc_test_trials[i])['True']\n \n idx += 1\n```\n\n total: 33600, positives: 5510, negatives: 28090\n Empirical Bag sizes:\n \t Positive bags: mean size 1.000, median size 1\n \t Negative bags: mean size 1.000, median size 1\n assign_mat size, X_shuff size: (4329, 4329) (4329, 3)\n assign_mat_trn size, assign_mat_tst size (3463, 4329) (866, 4329)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.2200283407728685 step loss 1.2257949275655553\n iter 0 sigmoid loss 7.437925088221127 step loss 0.3781165192511059 sigmoid temp 0.1\n iter 500 sigmoid loss 0.13565879062049044 step loss 0.3403220812006198 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.12980633434680391 step loss 0.3493532206783222 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.11756154614373092 step loss 0.3355314300042378 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.15763043394481963 step loss 0.3303848238483888 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.15141073263349017 step loss 0.32850620685633036 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.15310144867096384 step loss 0.3316717097813872 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5488375933827389 step loss 0.3335557576191651 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.1832488143492679 step loss 0.32570398439821235 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.15688955796286805 step loss 0.33923525007776595 sigmoid temp 1.0\n End sigmoid loss 0.2796809976995267 step loss 0.2888149890934166\n beta 0.20610362977114816\n rssi_w [-0.00625257 0.01440334 0.07741165 0.32711023]\n rssi_th [19.00433372 33.00432231 10.00463654]\n infect_w [0.01095765 0.08261399 0.31108047]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.187653006200411 step loss 1.185981082804295\n iter 0 sigmoid loss 7.23677397155167 step loss 0.4032773586123913 sigmoid temp 0.1\n iter 500 sigmoid loss 0.13127321835355438 step loss 0.3503931164071771 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.14039694176189152 step loss 0.3392882549437097 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.13085767183433986 step loss 0.3324028931185966 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.1479991022642534 step loss 0.3319405336090666 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.14228501915834177 step loss 0.3310834296880523 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.15167023573118898 step loss 0.33533224040927584 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5772666690633678 step loss 0.33641974655095774 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.3174278032773215 step loss 0.3283139935015345 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.11642461866792911 step loss 0.32816102787902374 sigmoid temp 1.0\n End sigmoid loss 0.31222306629165697 step loss 0.3279148078630854\n beta 0.2634509662340466\n rssi_w [-0.04058278 -0.03244996 0.19453353 0.29935388]\n rssi_th [13.99820566 35.99820547 22.99244015]\n infect_w [0.00976668 0.08358958 0.31789537]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.2186466610692883 step loss 1.2136226664199206\n iter 0 sigmoid loss 7.426016870892952 step loss 0.383282987866722 sigmoid temp 0.1\n iter 500 sigmoid loss 0.13725768052210113 step loss 0.34490241092863017 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.13974890050734948 step loss 0.3206979099346905 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.12708760857211782 step loss 0.3039755876309025 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.14532817467251036 step loss 0.29978108008325205 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.15147548724336812 step loss 0.3159555108690549 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.16088648121713944 step loss 0.2939009866519037 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5421093409389842 step loss 0.29639571167587764 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.1687981456740553 step loss 0.2912369320397565 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.08634742874395535 step loss 0.2985765060197714 sigmoid temp 1.0\n End sigmoid loss 0.28743839026259366 step loss 0.29291809962000903\n beta 0.23225000095040682\n rssi_w [-0.05374608 -0.03261321 0.11852488 0.31084829]\n rssi_th [16.01035263 29.01034867 14.00972893]\n infect_w [0.00934486 0.07836315 0.31068516]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.1647577933003832 step loss 1.1753679680412414\n iter 0 sigmoid loss 7.096151377311994 step loss 0.4131697207886723 sigmoid temp 0.1\n iter 500 sigmoid loss 0.1316490790517923 step loss 0.3513190026454029 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.14089396029414517 step loss 0.3414178263660852 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.13308366930729265 step loss 0.336070213864582 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.15211849798533358 step loss 0.331907536438449 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.14022018584905463 step loss 0.33562144394177273 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.15347037820147105 step loss 0.3397330232532275 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.575227606622392 step loss 0.3444048118702118 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.3786838771619745 step loss 0.33296339251747575 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.10362854754961504 step loss 0.3339916968033601 sigmoid temp 1.0\n End sigmoid loss 0.3182217784048862 step loss 0.33211950193108114\n beta 0.2647550748606896\n rssi_w [-0.07892412 -0.00086087 0.21155644 0.2745972 ]\n rssi_th [35.00178431 15.00165248 24.99447006]\n infect_w [0.0099931 0.08278793 0.31404121]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.2214960561005235 step loss 1.2365344510250085\n iter 0 sigmoid loss 7.450523334888487 step loss 0.3781633948964664 sigmoid temp 0.1\n iter 500 sigmoid loss 0.1367412654293796 step loss 0.33994617633016927 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.13075185547902415 step loss 0.31447523496164587 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.11779435747072049 step loss 0.3968234568180847 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.1556889295723245 step loss 0.3897232232824614 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.1527833944540057 step loss 0.38696735212836436 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.6503643711312154 step loss 0.5883087875513432 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5614569276286376 step loss 0.3061276639359147 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.1765473992613467 step loss 0.2868906830962544 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.09369361482751451 step loss 0.289139120189195 sigmoid temp 1.0\n End sigmoid loss 0.27741837163815253 step loss 0.28472352617500724\n beta 0.1977112001032676\n rssi_w [-0.00839605 0.02810527 0.12311457 0.30931161]\n rssi_th [27.00463747 28.0045912 10.00153689]\n infect_w [0.01053864 0.09296143 0.36112924]\n best loss 0.28472352617500724\n best scoring parameters\n beta 0.1977112001032676\n rssi_w [-0.00839605 0.01970922 0.14282379 0.4521354 ]\n rssi_th [-92.99536253 -64.99077133 -54.98923445]\n infect_w [0.01053864 0.10350008 0.46462931]\n total: 33600, positives: 5510, negatives: 28090\n Empirical Bag sizes:\n \t Positive bags: mean size 1.551, median size 2\n \t Negative bags: mean size 1.542, median size 2\n assign_mat size, X_shuff size: (4329, 6683) (6683, 3)\n assign_mat_trn size, assign_mat_tst size (3463, 6683) (866, 6683)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.0815387297397805 step loss 1.091542172964426\n iter 0 sigmoid loss 6.631021374131824 step loss 0.4208986051654761 sigmoid temp 0.1\n iter 500 sigmoid loss 0.12022072428383755 step loss 0.37008517042190153 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.12624030563455102 step loss 0.35081647635693897 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.13325796343572321 step loss 0.4687466287244836 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.14700192963770298 step loss 0.47869314701454907 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.1960937825518546 step loss 0.47371785982954395 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.17319054864833042 step loss 0.3654403572949135 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5669620557041499 step loss 0.3379619090188579 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.4162690121617356 step loss 0.33056933399222826 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.09461132365252571 step loss 0.3387567309100637 sigmoid temp 1.0\n End sigmoid loss 0.3218828624437763 step loss 0.33170778241367754\n beta 0.1771504102270448\n rssi_w [-0.00140747 0.02011427 0.20230656 0.25473526]\n rssi_th [22.00489773 34.00487007 17.99651483]\n infect_w [0.0100112 0.06349208 0.31908927]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.102654276088049 step loss 1.104151060205447\n iter 0 sigmoid loss 6.78115741327719 step loss 0.4170991766068726 sigmoid temp 0.1\n iter 500 sigmoid loss 0.1231701906409905 step loss 0.3788378432461293 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.12997458136883897 step loss 0.36335639811675 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.14176886896335167 step loss 0.352392209629956 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.1484857388510517 step loss 0.3472376022782935 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.19322398247445466 step loss 0.35544639503900416 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.1986605311180117 step loss 0.3334681954166262 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.542676897333064 step loss 0.343733458488395 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.5321427376832872 step loss 0.32915745232378363 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.0757049895687735 step loss 0.34267096788356616 sigmoid temp 1.0\n End sigmoid loss 0.32294523863290703 step loss 0.3314322563234409\n beta 0.23393175755938012\n rssi_w [-0.01468774 0.00262154 0.06894324 0.39081965]\n rssi_th [14.98947235 17.98950006 35.98911853]\n infect_w [0.00964724 0.05892993 0.32698059]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.1122002293684359 step loss 1.1238060591398036\n iter 0 sigmoid loss 6.826740257969773 step loss 0.41714821828379006 sigmoid temp 0.1\n iter 500 sigmoid loss 0.12349190538643207 step loss 0.3762454720975128 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.13124726379717433 step loss 0.36407116311170934 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.1418816785194037 step loss 0.35535553589010344 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.14588684316961234 step loss 0.38175534995474775 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.1920186472724076 step loss 0.39212014583125865 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.20840575667741873 step loss 0.38407257102131986 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5793158478500376 step loss 0.39127544077240467 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.5244664077406542 step loss 0.35558156386784895 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.10566924324664198 step loss 0.3899346437747572 sigmoid temp 1.0\n End sigmoid loss 0.3424496208532756 step loss 0.3708216717381274\n beta 0.17508987778792606\n rssi_w [-0.01737704 0.01239242 0.19208603 0.25850809]\n rssi_th [25.00392251 27.0038956 24.99636256]\n infect_w [0.01014915 0.06059425 0.29385709]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.03226391012608 step loss 1.0363902006419898\n iter 0 sigmoid loss 6.331416726942313 step loss 0.4333569737718021 sigmoid temp 0.1\n iter 500 sigmoid loss 0.12258099971479136 step loss 0.37346098400546307 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.12948957599296906 step loss 0.3570488071671687 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.13928551743668968 step loss 0.34328578315214925 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.14547749936613086 step loss 0.34004437382303637 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.21186681514792297 step loss 0.33236146781286713 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.17012581088502465 step loss 0.3241231745920087 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.539953494298237 step loss 0.33226935817708214 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.4627264568194618 step loss 0.3210606571388934 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.07158360202716256 step loss 0.3359382779059328 sigmoid temp 1.0\n End sigmoid loss 0.3118067440231877 step loss 0.3240426154377944\n beta 0.21889059597924854\n rssi_w [-0.02841116 -0.01683682 0.09036948 0.37302917]\n rssi_th [13.99807141 25.99808497 24.99749394]\n infect_w [0.00931241 0.0563715 0.32713936]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.1430836117465435 step loss 1.1547010670553903\n iter 0 sigmoid loss 7.022288367140208 step loss 0.4109904410733168 sigmoid temp 0.1\n iter 500 sigmoid loss 0.12311626911806352 step loss 0.374892981868913 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.1297957647204809 step loss 0.3590699872593152 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.13940153723414608 step loss 0.34563125181341386 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.14445977458723988 step loss 0.33404184880546717 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.19075409973112037 step loss 0.3270383267217231 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.1832427950473994 step loss 0.3264126270301285 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5361472034251533 step loss 0.3352463411393033 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.494600391960139 step loss 0.3211834458901376 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.07122057835017419 step loss 0.3364612309104921 sigmoid temp 1.0\n End sigmoid loss 0.3151485206826657 step loss 0.3239716406839413\n beta 0.23310899840690558\n rssi_w [-0.05275615 -0.02889128 0.13080762 0.372498 ]\n rssi_th [16.99610808 27.99610909 21.99427465]\n infect_w [0.00946331 0.0579735 0.33028183]\n best loss 0.3239716406839413\n best scoring parameters\n beta 0.23310899840690558\n rssi_w [-0.05275615 -0.08164743 0.04916019 0.42165819]\n rssi_th [-103.00389192 -75.00778283 -53.01350818]\n infect_w [0.00946331 0.06743682 0.39771865]\n total: 33600, positives: 5510, negatives: 28090\n Empirical Bag sizes:\n \t Positive bags: mean size 2.415, median size 2\n \t Negative bags: mean size 2.414, median size 2\n assign_mat size, X_shuff size: (4329, 10451) (10451, 3)\n assign_mat_trn size, assign_mat_tst size (3463, 10451) (866, 10451)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.053705570186742 step loss 1.063177002850041\n iter 0 sigmoid loss 6.457958942867378 step loss 0.43494164175266936 sigmoid temp 0.1\n iter 500 sigmoid loss 0.1052512653155061 step loss 0.4049795872938209 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.13729552781125115 step loss 0.3905812967906323 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.14858454367340446 step loss 0.3791825439199998 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.1705019023612972 step loss 0.36958294815990594 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.18346907482288388 step loss 0.3638620583275751 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.2294095585731209 step loss 0.36145680020141974 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.6052969488029238 step loss 0.3772082210354107 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.704354199931724 step loss 0.35629297592315184 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.06769340409147469 step loss 0.37247188224537564 sigmoid temp 1.0\n End sigmoid loss 0.35022485737088627 step loss 0.35872971662725833\n beta 0.16818359543636632\n rssi_w [-0.0217611 -0.01176912 0.09029338 0.37345858]\n rssi_th [10.99009885 26.99011401 30.98945498]\n infect_w [0.00961328 0.04960597 0.31470109]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.1522173221272205 step loss 1.159812242752954\n iter 0 sigmoid loss 7.050833279996359 step loss 0.4675979654811026 sigmoid temp 0.1\n iter 500 sigmoid loss 0.11796345627286767 step loss 0.40914757921474243 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.13922431100286542 step loss 0.3915949126748629 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.14912610700165047 step loss 0.3844306660350423 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.17171346213672883 step loss 0.4066601417310028 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.18108716227771438 step loss 0.43398166071091643 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.24019000626626522 step loss 0.44256681340265197 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.6680054096412081 step loss 0.4314620719541576 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.6603400034995806 step loss 0.41147310466422055 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.11245254334407456 step loss 0.41772398717129133 sigmoid temp 1.0\n End sigmoid loss 0.3798510567585645 step loss 0.4090799689780174\n beta 0.11477311661890469\n rssi_w [-0.02792297 0.00383463 0.23981065 0.13318216]\n rssi_th [22.0088452 30.00881967 35.99897639]\n infect_w [0.01247098 0.05057298 0.28247723]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.1147611771318993 step loss 1.1297721113912131\n iter 0 sigmoid loss 6.821063003990839 step loss 0.45636408640060205 sigmoid temp 0.1\n iter 500 sigmoid loss 0.11076888560794645 step loss 0.40993654920150696 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.1368534662202971 step loss 0.39916566305188816 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.14638575816314914 step loss 0.39464656352550787 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.16855088635795043 step loss 0.39353721949737125 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.18441817881046998 step loss 0.40033207041983254 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.23489595227256885 step loss 0.4135334064245925 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.6628736949191318 step loss 0.42562206941008585 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.7892378051079532 step loss 0.39393937570918175 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.09102796400082257 step loss 0.39808651268490647 sigmoid temp 1.0\n End sigmoid loss 0.3802015544206742 step loss 0.39108184676363444\n beta 0.166866380210584\n rssi_w [-0.08163297 -0.06640081 0.2459255 0.2642401 ]\n rssi_th [12.00689424 37.00689314 27.99450928]\n infect_w [0.01220533 0.05270049 0.30375292]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.0321304595484966 step loss 1.0685276575825207\n iter 0 sigmoid loss 6.3138065809062525 step loss 0.4236786614568939 sigmoid temp 0.1\n iter 500 sigmoid loss 0.10773165012460705 step loss 0.40073600564240824 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.1322643619161537 step loss 0.38399135786986366 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.13641654365518893 step loss 0.6521141740238319 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.16510297080112551 step loss 0.7404199119257422 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.20415910047841748 step loss 0.6919693545513369 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.20264389321840198 step loss 0.3455845853841482 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.5795823038893281 step loss 0.3543706336030514 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.5407849239336588 step loss 0.3424476821283038 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.08154156013944641 step loss 0.35712833929803384 sigmoid temp 1.0\n End sigmoid loss 0.333913177363113 step loss 0.34569519566777007\n beta 0.1549323143916869\n rssi_w [0.00408048 0.02858292 0.28380265 0.18495712]\n rssi_th [25.99934943 35.99927492 11.99832912]\n infect_w [0.00964887 0.04464384 0.32383084]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.1207373014341917 step loss 1.128642341921689\n iter 0 sigmoid loss 6.858635372002917 step loss 0.4737856140168961 sigmoid temp 0.1\n iter 500 sigmoid loss 0.1166860134798031 step loss 0.41341071682124475 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.13505894581692893 step loss 0.3879721289861054 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.14192391074241725 step loss 0.3753317772802448 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.16791142359780842 step loss 0.36782302456470595 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.18671282687995192 step loss 0.3606128446287236 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.22890201120157216 step loss 0.36123675091219404 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.6022678791333866 step loss 0.377242459402326 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.7007338947227912 step loss 0.3561180663513722 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.06775587555775699 step loss 0.3722365196207162 sigmoid temp 1.0\n End sigmoid loss 0.3499993900423281 step loss 0.35862146010544993\n beta 0.16662135454416074\n rssi_w [-0.04016042 -0.01761184 0.1149367 0.38200887]\n rssi_th [22.99037791 17.99039994 27.9894151 ]\n infect_w [0.0096112 0.04939867 0.31517772]\n best loss 0.34569519566777007\n best scoring parameters\n beta 0.1549323143916869\n rssi_w [0.00408048 0.0326634 0.31646605 0.50142317]\n rssi_th [-94.00065057 -58.00137564 -46.00304652]\n infect_w [0.00964887 0.05429271 0.37812355]\n total: 33600, positives: 5510, negatives: 28090\n Empirical Bag sizes:\n \t Positive bags: mean size 3.417, median size 3\n \t Negative bags: mean size 3.479, median size 3\n assign_mat size, X_shuff size: (4329, 15018) (15018, 3)\n assign_mat_trn size, assign_mat_tst size (3463, 15018) (866, 15018)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.164021126924725 step loss 1.1739256520186758\n iter 0 sigmoid loss 7.127573798655419 step loss 0.6338654598158889 sigmoid temp 0.1\n iter 500 sigmoid loss 0.11540265879746801 step loss 0.43796685242844086 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.1447836993212887 step loss 0.4215727750071143 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.1547527149002575 step loss 0.41027740273474744 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.15824797422568862 step loss 0.4424844703171529 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.23234243519770106 step loss 0.46744954623081636 sigmoid temp 0.1\n iter 3000 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.325\n iter 3500 sigmoid loss 5.4110802685625075 step loss 1.888354205673647 sigmoid temp 0.55\n iter 4000 sigmoid loss 11.512925464970227 step loss 1.888354205673647 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 1.0\n End sigmoid loss 1.888354205673647 step loss 1.888354205673647\n beta -0.007476832919561888\n rssi_w [0.20842493 0.23653634 0.33238415 0.09566679]\n rssi_th [23.99945807 29.99942584 36.99968422]\n infect_w [0.07419803 0.09326014 0.22126786]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.1218073117927598 step loss 1.1209626608635979\n iter 0 sigmoid loss 6.86967328396954 step loss 0.7377151892029055 sigmoid temp 0.1\n iter 500 sigmoid loss 0.101350771405699 step loss 0.45655238789902175 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.14039947502666558 step loss 0.4403171318896573 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.1600722655572786 step loss 0.43486622616026605 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.16679906117220497 step loss 0.4344580005334438 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.24225653609378528 step loss 0.438308646506774 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.24056182924176636 step loss 0.4439873156763693 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.7945908387123688 step loss 0.457551100223333 sigmoid temp 0.55\n iter 4000 sigmoid loss 2.1659694804084615 step loss 0.4341830927352767 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.1117362213374855 step loss 0.4470468071969657 sigmoid temp 1.0\n End sigmoid loss 0.43731340118035145 step loss 0.43731344565298685\n beta 0.08266689088533\n rssi_w [0.01120219 0.014841 0.03104297 0.10816683]\n rssi_th [11.00062897 11.00066366 16.0008078 ]\n infect_w [0.01171971 0.05159829 0.22834407]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.0051416290783688 step loss 1.0049762239395061\n iter 0 sigmoid loss 6.152564907338749 step loss 0.4878570993439965 sigmoid temp 0.1\n iter 500 sigmoid loss 0.11192465555203544 step loss 0.43678619014399345 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.13490492429844764 step loss 0.41334825937508507 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.14360708181963666 step loss 0.39433581949533597 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.1626454118588886 step loss 0.3856371259314096 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2241067146674925 step loss 0.3881315636811367 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3875860919915975 step loss 0.4891066314654366 sigmoid temp 0.325\n iter 3500 sigmoid loss 5.4110802685625075 step loss 1.888354205673647 sigmoid temp 0.55\n iter 4000 sigmoid loss 11.512925464970227 step loss 1.888354205673647 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 1.0\n End sigmoid loss 1.888354205673647 step loss 1.888354205673647\n beta 3.854231831611978\n rssi_w [0.03898314 0.11924252 0.21172367 0.25933113]\n rssi_th [36.00123681 13.0011738 13.00137246]\n infect_w [-0.15446207 -0.04588106 0.19135725]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.0156343766376088 step loss 1.0157415786427302\n iter 0 sigmoid loss 6.21106125529065 step loss 0.4859054213430398 sigmoid temp 0.1\n iter 500 sigmoid loss 0.113698877633251 step loss 0.43975992474233044 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.137735760093946 step loss 0.42276106841005795 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.14553928392405577 step loss 0.4096883462317898 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.1603300045055186 step loss 0.40284929561354255 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.23506008715940863 step loss 0.4058910479758703 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.2520211278937669 step loss 0.4413631871209033 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.7130446512466945 step loss 0.43146512615272226 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.9621165339583422 step loss 0.4023719644033792 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.09830064513632376 step loss 0.40972063430022265 sigmoid temp 1.0\n End sigmoid loss 0.390137005806932 step loss 0.3986132616348588\n beta 0.12242582786138315\n rssi_w [-0.05770491 -0.0549296 0.18288553 0.3356233 ]\n rssi_th [12.99844323 35.9984498 21.99200472]\n infect_w [0.01019742 0.05365196 0.3116923 ]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.295637732281315 step loss 1.2946348438927167\n iter 0 sigmoid loss 7.981763621626328 step loss 1.8156209029856776 sigmoid temp 0.1\n iter 500 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.1\n iter 1000 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.1\n iter 1500 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.1\n iter 2000 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.1\n iter 2500 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.1\n iter 3000 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.325\n iter 3500 sigmoid loss 5.4110802685625075 step loss 1.888354205673647 sigmoid temp 0.55\n iter 4000 sigmoid loss 11.512925464970227 step loss 1.888354205673647 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 1.0\n End sigmoid loss 1.888354205673647 step loss 1.888354205673647\n beta -0.012438766509265203\n rssi_w [0.05955461 0.05888137 0.06438315 0.03392044]\n rssi_th [13.00002489 11.00002723 39.0000268 ]\n infect_w [0.4311061 0.26423096 0.07645602]\n best loss 0.3986132616348588\n best scoring parameters\n beta 0.12242582786138315\n rssi_w [-0.05770491 -0.11263451 0.07025102 0.40587432]\n rssi_th [-107.00155677 -71.00310696 -49.01110224]\n infect_w [0.01019742 0.06384938 0.37554168]\n total: 33600, positives: 5510, negatives: 28090\n Empirical Bag sizes:\n \t Positive bags: mean size 3.930, median size 3\n \t Negative bags: mean size 3.899, median size 3\n assign_mat size, X_shuff size: (4329, 16900) (16900, 3)\n assign_mat_trn size, assign_mat_tst size (3463, 16900) (866, 16900)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.191777071502548 step loss 1.2032794846638821\n iter 0 sigmoid loss 7.31237126254042 step loss 0.9047164958485454 sigmoid temp 0.1\n iter 500 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.1\n iter 1000 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.1\n iter 1500 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.1\n iter 2000 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.1\n iter 2500 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.1\n iter 3000 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.325\n iter 3500 sigmoid loss 5.4110802685625075 step loss 1.888354205673647 sigmoid temp 0.55\n iter 4000 sigmoid loss 11.512925464970227 step loss 1.888354205673647 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 1.0\n End sigmoid loss 1.888354205673647 step loss 1.888354205673647\n beta -0.003721064827200221\n rssi_w [0.18039048 0.17101061 0.1356006 0.04739228]\n rssi_th [33.00013938 18.00014615 26.00004293]\n infect_w [0.05783596 0.03033331 0.05701104]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.0403572920423185 step loss 1.0438179841220039\n iter 0 sigmoid loss 6.378760460648703 step loss 0.5464244957322628 sigmoid temp 0.1\n iter 500 sigmoid loss 0.10270788363073141 step loss 0.4517292973496229 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.13295988518962623 step loss 0.4408528160976033 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.14519283636646405 step loss 0.4346829555735615 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.1637853620788405 step loss 0.4306796745931068 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.21338299033910574 step loss 0.4306082280917841 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.28221213127929184 step loss 0.43918183003804057 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.7201638509340893 step loss 0.4537544556298907 sigmoid temp 0.55\n iter 4000 sigmoid loss 2.127182608071928 step loss 0.4289046892330713 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.11304817429579903 step loss 0.4418766706805477 sigmoid temp 1.0\n End sigmoid loss 0.43096742108453473 step loss 0.43246904545676546\n beta 0.09287520216495473\n rssi_w [-0.02847775 -0.00633366 0.14212326 0.26930627]\n rssi_th [19.99721371 21.99723596 36.99562863]\n infect_w [0.0117103 0.06518763 0.23658242]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.107450235319392 step loss 1.1186480106757377\n iter 0 sigmoid loss 6.791028917227407 step loss 0.6444695920408058 sigmoid temp 0.1\n iter 500 sigmoid loss 0.108283480992493 step loss 0.44986472233270897 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.1326399080912921 step loss 0.4396917225902593 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.14306690823519946 step loss 0.43669193442616167 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.1624274352011393 step loss 0.44133114096370996 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.211975819044941 step loss 0.4587017015738552 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.27496416314716343 step loss 0.46434679146888397 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.7131239518644469 step loss 0.47769306355674257 sigmoid temp 0.55\n iter 4000 sigmoid loss 2.113147247867269 step loss 0.43817150342426603 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.11275414343433042 step loss 0.4431118638638064 sigmoid temp 1.0\n End sigmoid loss 0.4295825590484723 step loss 0.43603093308792634\n beta 0.09961272784121392\n rssi_w [-0.08407973 -0.0556397 0.25369392 0.20153168]\n rssi_th [24.00872691 25.00872452 34.99716243]\n infect_w [0.01185002 0.0679517 0.24316877]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.0114586895928706 step loss 1.013411702008754\n iter 0 sigmoid loss 6.2007695415077 step loss 0.5170427611252294 sigmoid temp 0.1\n iter 500 sigmoid loss 0.10260860017214529 step loss 0.4424322583611398 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.12683262411535043 step loss 0.42253249169755314 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.13633376119958324 step loss 0.41187437686694456 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.16067200512307056 step loss 0.42647709293227815 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2090241423766638 step loss 0.44152922942766565 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.25770875104750535 step loss 0.3880295185937732 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.6495941917746005 step loss 0.4095078345497042 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.9498525054706488 step loss 0.3840016938580607 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.10342300877529037 step loss 0.4035864628182036 sigmoid temp 1.0\n End sigmoid loss 0.38171495944033906 step loss 0.3875661696787256\n beta 0.09934252134514701\n rssi_w [-0.01820138 -0.00172222 0.05710198 0.37194003]\n rssi_th [14.99997395 16.9999949 31.9997962 ]\n infect_w [0.0101385 0.05239699 0.28148766]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.0818744158036557 step loss 1.0784272013508163\n iter 0 sigmoid loss 6.637407803253684 step loss 0.5865157792280238 sigmoid temp 0.1\n iter 500 sigmoid loss 0.10695061989092451 step loss 0.44546700607240614 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.12907723340553778 step loss 0.4318019642272524 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.13791713784064769 step loss 0.42561908890733907 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.16087578167368025 step loss 0.4337746281326564 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.20984413193998655 step loss 0.4526791759030183 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.24763150641347145 step loss 0.43486795037324655 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.7131791053756638 step loss 0.44540832898113536 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.9955212953265964 step loss 0.4251568280273782 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.12281231563921245 step loss 0.44183163933961483 sigmoid temp 1.0\n End sigmoid loss 0.42038136219357364 step loss 0.4234471025523617\n beta 0.06432331769713465\n rssi_w [-0.01494058 0.02605825 0.20803604 0.2441002 ]\n rssi_th [26.0023819 25.0023565 27.99719324]\n infect_w [0.01618152 0.05375587 0.23206551]\n best loss 0.3875661696787256\n best scoring parameters\n beta 0.09934252134514701\n rssi_w [-0.01820138 -0.0199236 0.03717838 0.4091184 ]\n rssi_th [-105.00002605 -88.00003115 -56.00023494]\n infect_w [0.0101385 0.06253549 0.34402315]\n total: 33600, positives: 5510, negatives: 28090\n Empirical Bag sizes:\n \t Positive bags: mean size 3.845, median size 3\n \t Negative bags: mean size 4.032, median size 3\n assign_mat size, X_shuff size: (4329, 17321) (17321, 3)\n assign_mat_trn size, assign_mat_tst size (3463, 17321) (866, 17321)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.0580479987438371 step loss 1.0620600328450887\n iter 0 sigmoid loss 6.3882477690610555 step loss 0.5955776146276555 sigmoid temp 0.1\n iter 500 sigmoid loss 0.10098631391410401 step loss 0.4539695398560876 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.14084775438578742 step loss 0.425134120073966 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.13531202626375768 step loss 0.4075293958074853 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.17476631138161067 step loss 0.39855198444644374 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.20279853030090286 step loss 0.39905918375343635 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.22263443424284574 step loss 0.41516859608945467 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.6889018432843236 step loss 0.4327690608072723 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.9276338275759628 step loss 0.39707176561955776 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.08674491111617876 step loss 0.4151265905246291 sigmoid temp 1.0\n End sigmoid loss 0.3882652447434063 step loss 0.39789662552064126\n beta 0.11011825553898227\n rssi_w [-0.08757291 -0.03548074 0.17159975 0.37234285]\n rssi_th [31.99867138 15.99865913 19.99376357]\n infect_w [0.00802257 0.05318134 0.29495052]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.0296810494099158 step loss 1.0348304679499734\n iter 0 sigmoid loss 6.210073205206389 step loss 0.5243384118694412 sigmoid temp 0.1\n iter 500 sigmoid loss 0.10382402381447736 step loss 0.4438346017240038 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.13982582392519383 step loss 0.4364088081290756 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.13485922016598356 step loss 0.47003859425454786 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.17497068655452813 step loss 0.47041181880136235 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.20029877913182362 step loss 0.4607855141521135 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.2318718342306692 step loss 0.43001248472554987 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.6848364755088311 step loss 0.41757436925419905 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.8468070498780318 step loss 0.4008325942469534 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.09254963511065568 step loss 0.42518250007687014 sigmoid temp 1.0\n End sigmoid loss 0.38406147030609905 step loss 0.4039466357414682\n beta 0.09643303008611817\n rssi_w [-0.02188888 0.01399088 0.10405083 0.33952002]\n rssi_th [24.00095163 29.00092657 14.99708551]\n infect_w [0.00910659 0.05545171 0.28167103]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 0.9748760796835905 step loss 0.9827458499435262\n iter 0 sigmoid loss 5.870143356136887 step loss 0.49425481555851597 sigmoid temp 0.1\n iter 500 sigmoid loss 0.10434969581940613 step loss 0.45121872492943144 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.14395923344635142 step loss 0.43451923260832653 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.13541503531947657 step loss 0.42167351271235304 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.17484542380496174 step loss 0.41618235783235025 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.20618008467183288 step loss 0.42132573156916414 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.23182686523794974 step loss 0.44250139649972803 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.7008383489014542 step loss 0.4803683379057979 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.982900125901864 step loss 0.41684198525864724 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.09189806965085945 step loss 0.42573889294106926 sigmoid temp 1.0\n End sigmoid loss 0.40485450503207115 step loss 0.41438436025588954\n beta 0.11031200957957168\n rssi_w [-0.07140993 -0.05607187 0.19893517 0.3449718 ]\n rssi_th [15.99846453 32.99846658 22.99118566]\n infect_w [0.0086931 0.05723526 0.28413378]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.265710826846673 step loss 1.283553266874576\n iter 0 sigmoid loss 7.654549736372495 step loss 1.080420630741337 sigmoid temp 0.1\n iter 500 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.1\n iter 1000 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.1\n iter 1500 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.1\n iter 2000 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.1\n iter 2500 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.1\n iter 3000 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 0.325\n iter 3500 sigmoid loss 5.4110802685625075 step loss 1.888354205673647 sigmoid temp 0.55\n iter 4000 sigmoid loss 11.512925464970227 step loss 1.888354205673647 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 1.0000050000287821e-05 step loss 1.888354205673647 sigmoid temp 1.0\n End sigmoid loss 1.888354205673647 step loss 1.888354205673647\n beta -0.0018059424262585055\n rssi_w [0.07238158 0.06592042 0.03560238 0.01609686]\n rssi_th [36.00006136 28.00005124 21.00002046]\n infect_w [0.23672299 0.18271265 0.08036453]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 0.98091570736716 step loss 0.9842510926503855\n iter 0 sigmoid loss 5.905366200100873 step loss 0.49997594693830166 sigmoid temp 0.1\n iter 500 sigmoid loss 0.10408831239662106 step loss 0.45084133386652364 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.14611772224721808 step loss 0.4317680049269966 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.13683802872506226 step loss 0.41750885499357554 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.1743689888169363 step loss 0.40702879603267533 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2064813620716287 step loss 0.397817663410085 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.21712041834722792 step loss 0.40262683164040314 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.6981024161250547 step loss 0.4234940527162012 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.9389320658791565 step loss 0.3947969798878027 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.08977456255523972 step loss 0.41761568631159013 sigmoid temp 1.0\n End sigmoid loss 0.39059263325953814 step loss 0.3994417856858591\n beta 0.10664200541788951\n rssi_w [-0.0385116 -0.01404745 0.10043055 0.39078713]\n rssi_th [22.99330113 17.99333782 26.99247926]\n infect_w [0.00805373 0.05262691 0.28860174]\n best loss 0.39789662552064126\n best scoring parameters\n beta 0.11011825553898227\n rssi_w [-0.08757291 -0.12305364 0.04854611 0.42088896]\n rssi_th [-88.00132862 -72.00266949 -52.00890592]\n infect_w [0.00802257 0.06120391 0.35615443]\n\n\n\n```python\nfig, axs = plt.subplots(1, 2, figsize=(15,5))\naxs[0].plot(bag_size, np.mean(auc_train_learned, axis=1), '+-', label='train, learned')\naxs[0].plot(bag_size, np.mean(auc_train_true, axis=1), '+-', label='train, true')\naxs[0].set_xlabel(\"Maximum Bag Size\")\naxs[0].set_ylabel(\"AUC\")\naxs[0].set_title(\"Train: No Censoring\")\naxs[0].legend()\n\naxs[1].plot(bag_size, np.mean(auc_test_learned, axis=1), '+-', label='test, learned')\naxs[1].plot(bag_size, np.mean(auc_test_true, axis=1), '+-', label='test, true')\naxs[1].set_xlabel(\"Maximum Bag Size\")\naxs[1].set_ylabel(\"AUC\")\naxs[1].set_title(\"Test: No Censoring\")\naxs[1].legend()\n```\n\n### 6. Examples of failure mode\n\n[Return to top](#Notebook-contents)\n\nHere we test out two scenarios of \"failure\". First is using a large learning rate in the traning process. The learning rate problem is briefly mentioned in II.2.2 when we introduce SGD alogirthm that SGD is sensitive to the learning rate. We show that if the learning rate is large (0.1 vs 0.001 original), the algorithm could not converge to a descent accuracy. Compared to original implementation, the accuracy is only 0.5 for training and 0.17 for testing, where the original hits 0.87 for both traning and testing, with truth being 0.9. The large learning rate is thus considered an example of failure becuase it results in tremedous drop in accuracy and it isnot better than randomlly guessing half of the population is infected. \n\n\n\n```python\n# failed experiment- learning rate edge \ndef train_and_eval_with_bag_config_fail1(bag_sim: Bag_Simulator, X_epi, probabilities_true_epi, n_trials=1, n_random_restarts=5, order=True):\n \"\"\"Train the risk score model on data generated according to the given bag configuration. \"\"\"\n auc_train_trials = []\n auc_test_trials = []\n for j in range(n_trials):\n # Y_epi, N_pos, N_neg, pos_neg_ratio = sample_labels(probabilities_true_epi)\n X, probabilities_true, assign_mat_trn, assign_mat_tst, bag_labels_trn, bag_labels_tst = bag_sim.simulate_bagged_data(X_epi, Y_epi, probabilities_true_epi, visualize=False, order=order)\n best_loss = np.inf\n best_model_params = None\n best_final_probs = None\n for i in range(n_random_restarts):\n # here we replace learning rate by 0.1\n print('----------- Trial {}/{}: Training run {}/{} ----------------'.format(j+1, n_trials, i+1, n_random_restarts))\n model_params, loss_st_step, final_probs = train(X, bag_labels_trn, assign_mat_trn,\n sigmoid_temp_init=0.1, sigmoid_temp_target=1,\n batch_size=200, num_iters=5000, lr=0.1)\n if loss_st_step < best_loss:\n best_loss = loss_st_step\n best_model_params = model_params\n best_final_probs = final_probs\n\n print(\"best loss\", best_loss)\n print(\"best scoring parameters\")\n print_params(residual_to_scoring(best_model_params))\n\n probs_bags_true_trn = 1 - np.exp(np.dot(assign_mat_trn, np.log(1-probabilities_true)))\n auc_train_trials.append({\n 'Learned': auc(best_final_probs, bag_labels_trn),\n 'True': auc(probs_bags_true_trn, bag_labels_trn),\n })\n\n scores_learned = scores_learned = loss_fn_stepbins_ce(best_model_params, X, None, None, return_scores=True)\n scores_learned_bags_tst = np.dot(assign_mat_tst, scores_learned)\n probs_learned_bags_tst = 1 - np.exp(-1*scores_learned_bags_tst)\n probs_bags_true_tst = 1 - np.exp(np.dot(assign_mat_tst, np.log(1-probabilities_true)))\n auc_test_trials.append({\n 'Learned': auc(probs_learned_bags_tst, bag_labels_tst),\n 'True': auc(probs_bags_true_tst, bag_labels_tst),\n })\n\n return auc_train_trials, auc_test_trials\n```\n\n\n```python\n# original\nn_trials = 1\nn_random_restarts_train = 5\nauc_train_trials, auc_test_trials = train_and_eval_with_bag_config(Bag_Simulator(p_pos=0.7, r_pos=2, p_neg=0.7, r_neg=2, max_bag_size=1,\n censor_prob_pos=0, censor_prob_neg=0), X_epi, probabilities_true_epi,\n n_trials=n_trials, n_random_restarts=n_random_restarts_train)\n```\n\n total: 33600, positives: 5510, negatives: 28090\n Empirical Bag sizes:\n \t Positive bags: mean size 1.000, median size 1\n \t Negative bags: mean size 1.000, median size 1\n assign_mat size, X_shuff size: (4329, 4329) (4329, 3)\n assign_mat_trn size, assign_mat_tst size (3463, 4329) (866, 4329)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.1219768761036102 step loss 1.1134305430934133\n iter 0 sigmoid loss 0.9663027166935403 step loss 0.837428210184174 sigmoid temp 0.1\n iter 500 sigmoid loss 0.36271120524836975 step loss 0.3381407657863293 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.36149079470830925 step loss 0.3275225543097697 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3758836523867663 step loss 0.31989607206949394 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3699628598627152 step loss 0.3107080338066055 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.30020568475549064 step loss 0.4402199868299871 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.30898606758406016 step loss 0.4403082210510724 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.23457354990714596 step loss 0.3240916607391936 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.28006727480966737 step loss 0.31263913187145137 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3460729894233397 step loss 0.3077482800674307 sigmoid temp 1.0\n End sigmoid loss 0.2983593624725792 step loss 0.307999892932363\n beta 0.2841002950087045\n rssi_w [-0.01118165 0.00219091 0.02885734 0.26976416]\n rssi_th [14.01189321 10.0119251 32.01188942]\n infect_w [0.01068561 0.06612105 0.26178645]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.1457434532965907 step loss 1.1491762523270734\n iter 0 sigmoid loss 0.9902394561477001 step loss 0.8487481950915559 sigmoid temp 0.1\n iter 500 sigmoid loss 0.36180377961950355 step loss 0.340671065651749 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.35957862172517835 step loss 0.32983856362839264 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3720004789111704 step loss 0.32216890203338233 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.36085029049721595 step loss 0.314755913536715 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2891252498941304 step loss 0.34557096692299266 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.29112235898332783 step loss 0.3160033305059157 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2638199614556918 step loss 0.3053500017175401 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2796526439865578 step loss 0.3028299446152391 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3343538784399405 step loss 0.3014388400608691 sigmoid temp 1.0\n End sigmoid loss 0.29101660087516673 step loss 0.30134850584165435\n beta 0.36188739982169194\n rssi_w [-0.01955325 -0.00111057 0.07568212 0.34122789]\n rssi_th [16.9902897 17.99031881 32.98982456]\n infect_w [0.01152648 0.08235205 0.34068951]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.2020121659779306 step loss 1.2010239727032244\n iter 0 sigmoid loss 1.036905700820247 step loss 0.8143162087154622 sigmoid temp 0.1\n iter 500 sigmoid loss 0.36210924011377926 step loss 0.3367130158210591 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3591718675515695 step loss 0.3238694607180305 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3714104558996327 step loss 0.3138481536287908 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3588939342904793 step loss 0.30365594350010133 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2861934782124144 step loss 0.7864866695189553 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.2842112350200353 step loss 0.3169160178920426 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2257839986385018 step loss 0.2973027849812792 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.27447369839729996 step loss 0.29382376549418915 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.33888263356816517 step loss 0.29213685251488386 sigmoid temp 1.0\n End sigmoid loss 0.280771949612669 step loss 0.29230298359724893\n beta 0.3451541150638725\n rssi_w [-0.01504015 -0.00227843 0.06091929 0.32659457]\n rssi_th [ 9.99768374 22.99769922 30.99740691]\n infect_w [0.01085689 0.07305679 0.32561762]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.1399673984433492 step loss 1.1446680057496375\n iter 0 sigmoid loss 0.9831912052486962 step loss 0.8394458189424857 sigmoid temp 0.1\n iter 500 sigmoid loss 0.35998843337571296 step loss 0.3368872131987415 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3546987657773586 step loss 0.32399037660499697 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.36513790651788824 step loss 0.31271755682816343 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.34184305558628286 step loss 0.29808727913382205 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2692442831098379 step loss 0.30201994734881726 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.2880803291450651 step loss 0.2902010907477397 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.21727002293537295 step loss 0.28970412024695635 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.26733486823414665 step loss 0.28913330754522154 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.33862514731809507 step loss 0.28915179176908673 sigmoid temp 1.0\n End sigmoid loss 0.27705788092489003 step loss 0.2892166535870432\n beta 0.3250073971588051\n rssi_w [-0.05463631 0.00118022 0.10047149 0.30548269]\n rssi_th [33.00310092 16.00303798 13.00321009]\n infect_w [0.01062478 0.0709289 0.3043646 ]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.1626413073218032 step loss 1.1921772548396574\n iter 0 sigmoid loss 1.0003442518942394 step loss 0.8947840177353745 sigmoid temp 0.1\n iter 500 sigmoid loss 0.36092287211641577 step loss 0.34760736404978787 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3549934667427918 step loss 0.3376806412691283 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.36625373990167553 step loss 0.33317912434326447 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.35670347973880573 step loss 0.33202686881685917 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.28792205612352056 step loss 0.3524469092536834 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.31282368970016683 step loss 0.3336261396993105 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2817749232613088 step loss 0.32795175923211173 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2838234533596232 step loss 0.3258442208140686 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3552570275075493 step loss 0.32461680818996236 sigmoid temp 1.0\n End sigmoid loss 0.3145478356617899 step loss 0.32447164045091903\n beta 0.36140357933801537\n rssi_w [-0.02323071 0.09388365 0.33166136 0.05052908]\n rssi_th [37.9868323 33.98588074 26.99952635]\n infect_w [0.01204858 0.08662959 0.33868893]\n best loss 0.2892166535870432\n best scoring parameters\n beta 0.3250073971588051\n rssi_w [-0.05463631 -0.05345609 0.0470154 0.35249809]\n rssi_th [-86.99689908 -70.9938611 -57.99065101]\n infect_w [0.01062478 0.08155368 0.38591828]\n\n\n\n```python\n## Failure due to large learning rate \n# lr = 0.1\nauc_train_trials_fail1, auc_test_trials_fail1 = train_and_eval_with_bag_config_fail1(Bag_Simulator(p_pos=0.7, r_pos=2, p_neg=0.7, r_neg=2, max_bag_size=1,\n censor_prob_pos=0, censor_prob_neg=0), X_epi, probabilities_true_epi,\n n_trials=n_trials, n_random_restarts=n_random_restarts_train)\n```\n\n total: 33600, positives: 5510, negatives: 28090\n Empirical Bag sizes:\n \t Positive bags: mean size 1.000, median size 1\n \t Negative bags: mean size 1.000, median size 1\n assign_mat size, X_shuff size: (4329, 4329) (4329, 3)\n assign_mat_trn size, assign_mat_tst size (3463, 4329) (866, 4329)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.1040380849114508 step loss 1.1100025411852166\n iter 0 sigmoid loss 0.7472514014654863 step loss 6.940842522733289 sigmoid temp 0.1\n iter 500 sigmoid loss 1.5542535878142312 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 1000 sigmoid loss 2.0147702064110398 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 1500 sigmoid loss 1.7269473197880345 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 2000 sigmoid loss 1.8420764744372367 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 2500 sigmoid loss 2.3025930930340457 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 3000 sigmoid loss 2.1874639383848433 step loss 1.8883542056736473 sigmoid temp 0.325\n iter 3500 sigmoid loss 1.8996410517618378 step loss 1.8883542056736473 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.439124433165029 step loss 1.8883542056736473 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 1.6118181651388321 step loss 1.8883542056736473 sigmoid temp 1.0\n End sigmoid loss 1.8883542056736473 step loss 1.8883542056736473\n beta -1.2151600919414363\n rssi_w [0.28376745 0.28725701 0.26474128 0.12409393]\n rssi_th [19.00227773 31.0023046 25.00145728]\n infect_w [0.34213929 0.57401231 0.47431262]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.3234328330105503 step loss 1.322511194138947\n iter 0 sigmoid loss 0.8758795055776042 step loss 9.29335749463293 sigmoid temp 0.1\n iter 500 sigmoid loss 1.5542535878142312 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 1000 sigmoid loss 2.0147702064110398 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 1500 sigmoid loss 1.7269473197880345 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 2000 sigmoid loss 1.8420764744372367 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 2500 sigmoid loss 2.3025930930340457 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 3000 sigmoid loss 2.1874639383848433 step loss 1.8883542056736473 sigmoid temp 0.325\n iter 3500 sigmoid loss 1.8996410517618378 step loss 1.8883542056736473 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.439124433165029 step loss 1.8883542056736473 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 1.6118181651388321 step loss 1.8883542056736473 sigmoid temp 1.0\n End sigmoid loss 1.8883542056736473 step loss 1.8883542056736473\n beta -0.08411561164555223\n rssi_w [0.89211075 0.861778 0.72944081 0.27047382]\n rssi_th [31.00032396 19.00035183 26.00016742]\n infect_w [1.4491814 1.47404627 0.75252978]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.274822916393252 step loss 1.277161912970972\n iter 0 sigmoid loss 0.869610925271254 step loss 9.122991174765934 sigmoid temp 0.1\n iter 500 sigmoid loss 1.5542535878142312 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 1000 sigmoid loss 2.0147702064110398 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 1500 sigmoid loss 1.7269473197880345 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 2000 sigmoid loss 1.8420764744372367 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 2500 sigmoid loss 2.3025930930340457 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 3000 sigmoid loss 2.1874639383848433 step loss 1.8883542056736473 sigmoid temp 0.325\n iter 3500 sigmoid loss 1.8996410517618378 step loss 1.8883542056736473 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.439124433165029 step loss 1.8883542056736473 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 1.6118181651388321 step loss 1.8883542056736473 sigmoid temp 1.0\n End sigmoid loss 1.8883542056736473 step loss 1.8883542056736473\n beta -0.1941047405613517\n rssi_w [0.5955551 0.59215407 0.56512745 0.29579467]\n rssi_th [18.00038447 15.00040068 35.00034628]\n infect_w [1.5123345 1.54451291 1.25238962]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.1777957735059572 step loss 1.173851115715025\n iter 0 sigmoid loss 0.7901091955609181 step loss 8.519828197340118 sigmoid temp 0.1\n iter 500 sigmoid loss 1.5542535878142312 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 1000 sigmoid loss 2.0147702064110398 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 1500 sigmoid loss 1.7269473197880345 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 2000 sigmoid loss 1.8420764744372367 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 2500 sigmoid loss 2.3025930930340457 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 3000 sigmoid loss 2.1874639383848433 step loss 1.8883542056736473 sigmoid temp 0.325\n iter 3500 sigmoid loss 1.8996410517618378 step loss 1.8883542056736473 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.439124433165029 step loss 1.8883542056736473 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 1.6118181651388321 step loss 1.8883542056736473 sigmoid temp 1.0\n End sigmoid loss 1.8883542056736473 step loss 1.8883542056736473\n beta -0.7011228347371002\n rssi_w [0.30010918 0.29708958 0.2979463 0.25745628]\n rssi_th [10.00080402 11.00083999 30.00092892]\n infect_w [1.08725984 1.16024169 0.77224845]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.0889428309813716 step loss 1.0860534926777339\n iter 0 sigmoid loss 0.7383080390097849 step loss 7.006774104063675 sigmoid temp 0.1\n iter 500 sigmoid loss 1.5542535878142312 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 1000 sigmoid loss 2.0147702064110398 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 1500 sigmoid loss 1.7269473197880345 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 2000 sigmoid loss 1.8420764744372367 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 2500 sigmoid loss 2.3025930930340457 step loss 1.8883542056736473 sigmoid temp 0.1\n iter 3000 sigmoid loss 2.1874639383848433 step loss 1.8883542056736473 sigmoid temp 0.325\n iter 3500 sigmoid loss 1.8996410517618378 step loss 1.8883542056736473 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.439124433165029 step loss 1.8883542056736473 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 1.6118181651388321 step loss 1.8883542056736473 sigmoid temp 1.0\n End sigmoid loss 1.8883542056736473 step loss 1.8883542056736473\n beta -1.011445935286781\n rssi_w [0.32110423 0.32314634 0.32081075 0.23235056]\n rssi_th [10.00105593 17.00111999 34.00119171]\n infect_w [0.33296816 0.52918572 0.43347082]\n best loss 1.8883542056736473\n best scoring parameters\n beta -1.2151600919414363\n rssi_w [0.28376745 0.57102446 0.83576575 0.95985968]\n rssi_th [-100.99772227 -69.99541767 -44.99396039]\n infect_w [0.34213929 0.9161516 1.39046422]\n\n\n\n```python\nprint(auc_train_trials, auc_test_trials)\nprint(auc_train_trials_fail1, auc_test_trials_fail1)\n```\n\n [{'Learned': 0.8815031379989784, 'True': 0.9200731591622273}] [{'Learned': 0.8578709828028948, 'True': 0.898470936113921}]\n [{'Learned': 0.5, 'True': 0.9113959230338856}] [{'Learned': 0.17880417866313908, 'True': 0.9314790677768267}]\n\n\nThe second senario we want to test is using a very large bag size (128), compared to the largest bag size of 32 that we used in our main experiment. To save time, we do not incorporate random restart in this part but we compare the large bag size implementation with the original one. Notice that the train accuracy has dropoed to 0.5, and test accuracy has dropped to 0.34 The scenario is considered another example of failure because it is not better than randomlly guessing half of the population is infected or not.\n\n\n```python\n# failed experiment- very large bag size\n\n# original with only one restarts\nauc_train_trials2, auc_test_trials2 = train_and_eval_with_bag_config(Bag_Simulator(p_pos=0.7, r_pos=2, p_neg=0.7, r_neg=2, max_bag_size=1,\n censor_prob_pos=0, censor_prob_neg=0), X_epi, probabilities_true_epi,\n n_trials=n_trials, n_random_restarts=1)\n```\n\n total: 33600, positives: 5510, negatives: 28090\n Empirical Bag sizes:\n \t Positive bags: mean size 1.000, median size 1\n \t Negative bags: mean size 1.000, median size 1\n assign_mat size, X_shuff size: (4329, 4329) (4329, 3)\n assign_mat_trn size, assign_mat_tst size (3463, 4329) (866, 4329)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/1 ----------------\n Start sigmoid loss 1.2482413242950465 step loss 1.257868839025658\n iter 0 sigmoid loss 1.3089002228609719 step loss 0.8105117319211484 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3601045326008543 step loss 0.3504521219930372 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.32678743194839205 step loss 0.3402750577323596 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.2814995703769667 step loss 0.33519728759973993 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.28184545917561143 step loss 0.3321994978236391 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.28785784219343813 step loss 0.3420697972974473 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.32177173272940907 step loss 0.3327175800784395 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.30848847742173413 step loss 0.32835832822378885 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.34054734348224164 step loss 0.32629551911146465 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.2714758048137344 step loss 0.32536195273038765 sigmoid temp 1.0\n End sigmoid loss 0.3159437624260285 step loss 0.3252708364415301\n beta 0.3589744596548371\n rssi_w [-0.02972732 0.09826311 0.32823586 0.0456899 ]\n rssi_th [38.98712193 32.98602841 28.99960241]\n infect_w [0.013443 0.08319367 0.33627621]\n best loss 0.3252708364415301\n best scoring parameters\n beta 0.3589744596548371\n rssi_w [-0.02972732 0.06853579 0.39677166 0.44246156]\n rssi_th [-81.01287807 -48.02684966 -19.02724725]\n infect_w [0.013443 0.09663667 0.43291288]\n\n\n\n```python\n# failure due to large bag size\n# bag size = 128\nauc_train_trials_fail2, auc_test_trials_fail2 = train_and_eval_with_bag_config(Bag_Simulator(p_pos=0.7, r_pos=2, p_neg=0.7, r_neg=2, max_bag_size=128,\n censor_prob_pos=0, censor_prob_neg=0), X_epi, probabilities_true_epi,\n n_trials=n_trials, n_random_restarts=1)\n\n```\n\n total: 33600, positives: 5510, negatives: 28090\n Empirical Bag sizes:\n \t Positive bags: mean size 5.424, median size 5\n \t Negative bags: mean size 5.676, median size 5\n assign_mat size, X_shuff size: (4329, 24393) (24393, 3)\n assign_mat_trn size, assign_mat_tst size (3463, 24393) (853, 24393)\n Average positive samples per bag: 2.0225352112676056\n ----------- Trial 1/1: Training run 1/1 ----------------\n Start sigmoid loss 1.1043613021850538 step loss 1.106081606826197\n iter 0 sigmoid loss 0.9274505669663057 step loss 0.5786967530300093 sigmoid temp 0.1\n iter 500 sigmoid loss 0.40193995281904654 step loss 0.36923183783669455 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3345641027245263 step loss 0.35555054529122104 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3523339335787491 step loss 0.65126397413235 sigmoid temp 0.1\n iter 2000 sigmoid loss 2.01477020641104 step loss 1.888354205673647 sigmoid temp 0.1\n iter 2500 sigmoid loss 1.49668901048963 step loss 1.888354205673647 sigmoid temp 0.1\n iter 3000 sigmoid loss 1.6693827424634333 step loss 1.888354205673647 sigmoid temp 0.325\n iter 3500 sigmoid loss 2.53285140233245 step loss 1.888354205673647 sigmoid temp 0.55\n iter 4000 sigmoid loss 1.8996410517618378 step loss 1.888354205673647 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 1.7845118971126357 step loss 1.888354205673647 sigmoid temp 1.0\n End sigmoid loss 1.888354205673647 step loss 1.888354205673647\n beta -0.07472037844301627\n rssi_w [1.00751246 1.00079449 0.95560032 0.45536793]\n rssi_th [ 9.99592727 18.99598599 36.99606376]\n infect_w [0.1331177 0.08801679 0.22843648]\n best loss 1.888354205673647\n best scoring parameters\n beta -0.07472037844301627\n rssi_w [1.00751246 2.00830695 2.96390727 3.4192752 ]\n rssi_th [-110.00407273 -91.00808674 -54.01202298]\n infect_w [0.1331177 0.22113449 0.44957097]\n\n\n\n```python\nprint(auc_train_trials2, auc_test_trials2)\nprint(auc_train_trials_fail2, auc_test_trials_fail2)\n```\n\n [{'Learned': 0.8425077233695785, 'True': 0.9157514169646549}] [{'Learned': 0.8392002567893548, 'True': 0.916810948564314}]\n [{'Learned': 0.5, 'True': 0.8811789997324186}] [{'Learned': 0.3443671876547612, 'True': 0.8843277668825895}]\n\n\n# III. Discussion\n\n## 1. Evaluation\n\nOverall, this seems like a pretty generic workflow similar to what we are used to in class\u2013define a generative model, get data, use data to predict optimal parameter values for the model and check the accuracy with a predefined loss function. We do not see any major standout issues. However, we do see room to question some of their design choices.\n\n### 1.1 Choice of Loss Function\nFor example, we find that the author's choice of loss function does not reflect real world considerations fully. In the context of the pandemic, we would prefer a model that is more liberal in its estimates of risk because we would ideally want to err on the side of caution in the short run\u2014we would rather isolate individuals who may not be infected, than to let individuals who are infected roam freely. In this sense, we would prefer a model with a higher false positive rate than a lower false negative rate. The loss function that is being minimized over right now does not seem to take this into account by weighing equally the risk scores for those infected and those not infected. From a practical, policy standpoint, we might want to adjust this to be more liberal in predictions. Alternatively, recognizing this feature of the model can also lead to a practical implementation where the threshold for alerting users of their close contact status is lower to compensate for the predictions.\n\n### 1.2 Robustness to Model Mismatch\nThis discussion continues from Experiment 4 in the previous section. In the paper, the authors tested the robustness of their model to a potential model mismatch, by using a Taylor series approximation to the generative function for calculating the probability of infection. Poorer approximations (with fewer terms) would therefore mean a greater mismatch. However, this still assumes that the underlying model is fixed and is correct, while the approximations just appear to mimic noise in the real world. We find that the authors have not sufficiently proven that the model is truly robust to model mismatches where the generative model they have assumed is almost fundamentally flawed in design to begin with. This may potentially disadvantage their model in favor of an expert-driven model, since it is less flexible to change the whole structure of the learning model and re-learn parameters, compared to an expert changing guidance on thresholds immediately in response to the pandemic.\n\n### 1.3 Comparison with Expert Guided Fixed Thresholds\nAnother consideration we had was the author's claim that their model performs better than fixed models with expert-guided thresholds. While they did perform a comparison with the Swiss model, we are hard pressed to accept this comparison as sufficient evidence that their claim holds. For one, the data they used to test both models originated from their data generator with specific parameters. They have not provided us with enough evidence to show that the data generated is reflective of the real world (apart from the fact that the equations are based on real world physics, there is no explicit justification for the parameters used in generating data) and less that it is reflective of the situation in Switzerland, which the Swiss model was presumably being tested in. Second, since we (and the authors) did not have access to the data from the GAEN app, we can only take the author's word that the data generated from their simulation was similar enough to that collected by GAEN, that the model can be easily ported over for integration and use with GAEN.\n\n### 1.4 Accuracy and Uncertainty of Model\nNonetheless, it seems promising that we are seeing the right trends in the other experiments they conducted to test their own algorithm. For example, increasing bag size and censoring probability increases the difficulty of the optimization since there is potentially more noise or less signal, and we see a corresponding decrease in accuracy. Increasing signal through increasing positive bag constructions increases accuracy to a certain point where the benefit of the increased signal is trumped by the drawbacks of noise. These experiments give us confidence that the model is being uncertain in the right conditions as we expect. \n\nHowever, further work would need to be done to properly quantify the level of uncertainty and decide if it is practical. For example, we see that the inaccuracy drops as bag size increases, but the main experiments only tested until a bag size of 32. It is unclear if this is a realistic bag size to work with for real data, especially at the height of the pandemic when a single person may have many exposure events that it presents a separate problem just deciding which exposure events go in one bag and how to aggregate risk scores across different bags meaningfully. Moreover, if an ideal bag size is larger than 32, like in the demonstration of failure mode No.2, then we know for a fact that the accuracy will drop which begs the question of whether any potential gains in accuracy over an expert determined model is significant enough to justify an ML model. If anything, the experiments definitely highlighted that a lot of the parameters (e.g. maximum bag size, maximum number of positive exposures, censoring probability) can and probably should be tuned according to what we know of the real world constraints. For example, we would want to adjust the maximum bag size to make sure that there is substantive signal from positive exposures, given the average number of positive exposures a person may receive daily, but also make sure the bag size is still realistic and efficient to run. Censoring probability in real life would also come with knowledge of take-up rates of the contact tracing app and how well people use them. Depending on that, we would have a gauge of where the model could fail and compensate or guard against it.\n\nLast but no the least, through the failure mode No.1, we acknowledge that every algorithm has its advantages and drawbacks and some uncareful choice of parameters can result in decreasing model accuracy (increasing model uncertainty). For example, the SGD algorithm is sensitive to the learning rate, if the rate is set to be extremely low or high, the gradient search will stuck in a place or jump frantically. Other ML algorithms have issues, too, such as too easy to overfit. To conclude, the story warns us that we have to be aware of the inherited problems associated with specific alogorithm to choose in the problem. Some algorithm needs to be carefully tuned before experiments. As an extension, we actually optimize the algorithm in III.2 Future Work and Potential Modifications. \n\n\n## 2. Future Work and Potential Modifications\n\n### 2.1 Possible Modification 1: Optimization for SGD\n\nBased on the common tricks mentioned in LeCun et al.'s Backprop chapter [11], we could consider the following methods for improving the performance of the stochastic gradient descent.\n\n1. **Shuffling and Curriculum Learning.** Generally, we want to avoid providing the training examples in a meaningful, consistent order to our model as this may bias the optimization algorithm. Consequently, it is often a good idea to shuffle the training data after every epoch.\n\n2. **Batch Normalization.** To facilitate learning, we typically normalize the initial values of our parameters by initializing them with zero mean and unit variance. As training progresses and we update parameters to different extents, we lose this normalization, which slows down training and amplifies changes as the network becomes deeper.\n\n According to Ioffe and Szegedy's paper on Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift [12], batch normalization reestablishes these normalizations for every mini-batch and changes are back-propagated through the operation as well. By making normalization part of the model architecture, we are able to use higher learning rates and pay less attention to the initialization parameters. Batch normalization additionally acts as a regularizer, reducing (and sometimes even eliminating) the need for Dropout.\n\n3. **Early Stopping.** According to Geoff Hinton: \"Early stopping is beautiful free lunch\" (NIPS 2015 Tutorial slides, slide 63). We should thus always monitor error on a validation set during training and stop (with some patience) if our validation error does not improve enough.\n\nWe also recognize that even though these tricks have their advantages, we cannot blindly assume that their benefits hold true for all machine learning models. There is definitely room to understand our model better and see if there is even a need to be concerned about certain things, like batch normalizations, if the model was actually simple enough that the learning and update of parameters need not be complicated further.\n\n### 2.2 Possible Modification 2: Other Parameter Learning Methods\n\nHere, we explore specific learning methods that can enhance SGD for more efficient learning.\n\n1. **Momentum.** SGD has trouble navigating ravines, i.e. areas where the surface curves much more steeply in one dimension than in another, which are common around local optima. In these scenarios, SGD oscillates across the slopes of the ravine while only making hesitant progress along the bottom towards the local optimum. Momentum is a method that helps accelerate SGD in the relevant direction and dampens oscillations. It does this by adding a fraction $\\gamma$ of the update vector of the past time step to the current update vector: \n\\begin{align} \n\\begin{split} \nv_t &= \\gamma v_{t-1} + \\eta \\nabla_\\theta J( \\theta) \\\\ \n\\theta &= \\theta - v_t \n\\end{split} \n\\end{align}\n\n This could potentially help speed up the efficiency of the SGD algorithm, though careful tuning would be needed to avoid too big a momentum such that the SGD falls into a local optimum too quickly and the algorithm terminates prematurely.\n\n2. **Adagrad.** Adagrad is an algorithm for gradient-based optimization. It adapts the learning rate to the parameters, performing smaller updates (i.e. low learning rates) for parameters associated with frequently occurring features, and larger updates (i.e. high learning rates) for parameters associated with infrequent features. For this reason, it is well-suited to dealing with sparse data. Dean et al. found in their paper, Large Scale Distributed Deep Networks [13], that Adagrad greatly improved the robustness of SGD and used it for training large-scale neural nets at Google, which\u2014among other things\u2014learned to recognize cats in YouTube videos.\n\n In its update rule, Adagrad modifies the general learning rate $\\eta$ at each time step $t$ for every parameter $\\theta_i$ based on the past gradients that have been computed for $\\theta_i$:\n$$\\theta_{t+1, i} = \\theta_{t, i} - \\dfrac{\\eta}{\\sqrt{G_{t, ii} + \\epsilon}} \\cdot g_{t, i}$$\n\\\n$G_{t, ii}$ here is a diagonal matrix where each diagonal element $i$, $i$ is the sum of the squares of the gradients w.r.t. $\\theta_i$ up to time step $t$, while $\\epsilon$ is a smoothing term that avoids division by zero (usually on the order of 1e\u22128).\n\n While this method may have its benefits, for our particular model, it may not be completely relevant since we are not exactly working with sparse data (and also assume that our data is complete, save for the censoring). Nonetheless, in a real world setting, as the authors of our paper have mentioned, it is likely that the contact tracing app may miss picking up on a lot of exposure events. Hence, the problem may be reformulated to be one of determining risk score with sparse data, and this method may come in handy then.\n\n3. **Adadelta.** Adadelta is an extension of Adagrad that seeks to reduce its aggressive, monotonically decreasing learning rate. Instead of accumulating all past squared gradients, Adadelta restricts the window of accumulated past gradients to some fixed size $w$. With Adadelta, we do not even need to set a default learning rate, as it has been eliminated from the update rule.\n\n4. **Adam.** Adaptive Moment Estimation (Adam) is another method that computes adaptive learning rates for each parameter. In addition to storing an exponentially decaying average of past squared gradients $v_t$ like Adadelta, Adam also keeps an exponentially decaying average of past gradients $m_t$, similar to momentum. Whereas momentum can be seen as a ball running down a slope, Adam behaves like a heavy ball with friction, which thus prefers flat minima in the error surface. We compute the decaying averages of past and past squared gradients $m_t$ and $v_t$ respectively as follows:\n\n$$\n\\begin{align} \n\\begin{split} \nm_t &= \\beta_1 m_{t-1} + (1 - \\beta_1) g_t \\\\ \nv_t &= \\beta_2 v_{t-1} + (1 - \\beta_2) g_t^2 \n\\end{split} \n\\end{align}\n$$\n\nUltimately, it is a balance we need to strike between overcomplicating the model and optimizing methods for greater efficiency.\n\n\n### 2.3 Implementation of Optimization\n\nIn this part, we implement several proposed optimization methods. \n\n\n```python\n# Implementation of Optimization 1 - Shuffling and Curriculum Learning.\n\ndef train_and_eval_with_bag_config_mod_v1(bag_sim: Bag_Simulator, X_epi, probabilities_true_epi, n_trials=1, n_random_restarts=5):\n '''\n first modification - Shuffling and Curriculum Learning.\n '''\n auc_train_trials = []\n auc_test_trials = []\n for j in range(n_trials):\n # Since there is alrealy randomness inside the bag_sim.simulate_bagged_data function, we\n # don't implement shuffling here for each trial\n X, probabilities_true, assign_mat_trn, assign_mat_tst, bag_labels_trn, bag_labels_tst = \\\n bag_sim.simulate_bagged_data(X_epi, Y_epi, probabilities_true_epi, visualize=False)\n best_loss = np.inf\n best_model_params = None\n best_final_probs = None\n for i in range(n_random_restarts):\n print('----------- Trial {}/{}: Training run {}/{} ----------------'.format(j+1, n_trials, i+1, n_random_restarts))\n # Modification here, shuffling here before each iteration\n # Since our three input variables for training input are dependent of each other\n # need to wisely shuffle the data\n idx1 = np.arange(assign_mat_trn.shape[0])\n np.random.shuffle(idx1)\n bag_labels_trn = bag_labels_trn[idx1]\n assign_mat_trn = assign_mat_trn[idx1, :]\n\n model_params, loss_st_step, final_probs = train(X, bag_labels_trn, assign_mat_trn,\n sigmoid_temp_init=0.1, sigmoid_temp_target=1,\n batch_size=200, num_iters=5000, lr=0.001)\n if loss_st_step < best_loss:\n best_loss = loss_st_step\n best_model_params = model_params\n best_final_probs = final_probs\n\n print(\"best loss\", best_loss)\n print(\"best scoring parameters\")\n print_params(residual_to_scoring(best_model_params))\n\n probs_bags_true_trn = 1 - np.exp(np.dot(assign_mat_trn, np.log(1-probabilities_true)))\n auc_train_trials.append({\n 'Learned': auc(best_final_probs, bag_labels_trn),\n 'True': auc(probs_bags_true_trn, bag_labels_trn),\n })\n\n scores_learned = loss_fn_stepbins_ce(best_model_params, X, None, None, return_scores=True)\n scores_learned_bags_tst = np.dot(assign_mat_tst, scores_learned)\n probs_learned_bags_tst = 1 - np.exp(-1*scores_learned_bags_tst)\n probs_bags_true_tst = 1 - np.exp(np.dot(assign_mat_tst, np.log(1-probabilities_true)))\n auc_test_trials.append({\n 'Learned': auc(probs_learned_bags_tst, bag_labels_tst),\n 'True': auc(probs_bags_true_tst, bag_labels_tst),\n })\n\n return auc_train_trials, auc_test_trials\n```\n\n\n```python\n## Original Implementation for Comparison\nauc_train_trials, auc_test_trials = train_and_eval_with_bag_config(Bag_Simulator(p_pos=0.7, r_pos=2, p_neg=0.7, r_neg=2, max_bag_size=1,\n censor_prob_pos=0, censor_prob_neg=0), X_epi, probabilities_true_epi,\n n_trials=n_trials, n_random_restarts=n_random_restarts_train)\n```\n\n total: 33600, positives: 5468, negatives: 28132\n Empirical Bag sizes:\n \t Positive bags: mean size 1.000, median size 1\n \t Negative bags: mean size 1.000, median size 1\n assign_mat size, X_shuff size: (4362, 4362) (4362, 3)\n assign_mat_trn size, assign_mat_tst size (3489, 4362) (873, 4362)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.175454359499679 step loss 1.1832132363026815\n iter 0 sigmoid loss 1.117649362414514 step loss 0.8203591934372848 sigmoid temp 0.1\n iter 500 sigmoid loss 0.35707710172401136 step loss 0.35665064280738146 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.30134682717362193 step loss 0.3477959750554159 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.320967402585495 step loss 0.341930252024401 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3601606781223854 step loss 0.33620355160992454 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3108253803566541 step loss 0.33951994942820707 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.28368279314657735 step loss 0.3331322280170496 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.314742110941194 step loss 0.33052413621735105 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.35939014189468965 step loss 0.3285581887256153 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.307463261052493 step loss 0.3271980722345064 sigmoid temp 1.0\n End sigmoid loss 0.3202877715184831 step loss 0.3271534472272605\n beta 0.3674340512931472\n rssi_w [-0.07016319 -0.00715545 0.14010975 0.31979209]\n rssi_th [31.99115258 10.99110705 28.98919164]\n infect_w [0.01315177 0.09385931 0.34281078]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.0771338394573606 step loss 1.0800334238163982\n iter 0 sigmoid loss 1.0200397895368374 step loss 0.8348038708842997 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3565983926027269 step loss 0.3479807873602735 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3017770675288302 step loss 0.33726327930362854 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.32205472714417366 step loss 0.32825676561026407 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.35812455437549146 step loss 0.3163929406285835 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.30078649114517314 step loss 0.4756833347562294 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.29517980847367464 step loss 0.3158733075551057 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2980187283820313 step loss 0.3176714310632307 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.32633096066560213 step loss 0.31526712682861296 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.314697135741725 step loss 0.3138260052644261 sigmoid temp 1.0\n End sigmoid loss 0.3071843972833988 step loss 0.31400121918974294\n beta 0.2893414872111698\n rssi_w [-0.05872754 -0.01510624 0.09957395 0.25147242]\n rssi_th [29.013529 17.0134983 11.01355341]\n infect_w [0.01139964 0.07363368 0.26459577]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.2658655464677038 step loss 1.2648647336092953\n iter 0 sigmoid loss 1.2005092034804057 step loss 0.7671637316484229 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3587234401166512 step loss 0.36318192563046303 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3084965281169524 step loss 0.3587846285745685 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.33365306558763136 step loss 0.3585461112881062 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.371849249413504 step loss 0.357881401684053 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.33610950155419106 step loss 0.3582193351114027 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.31548939534678383 step loss 0.36399485480579796 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3237759781793778 step loss 0.3636256410313402 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3677576877697471 step loss 0.3637510573117591 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.34098157136074947 step loss 0.36377856631507416 sigmoid temp 1.0\n End sigmoid loss 0.3458668904556869 step loss 0.3631699033681599\n beta 0.2869283741684491\n rssi_w [-0.0643455 -0.03937188 0.01172672 0.26042512]\n rssi_th [21.01031292 11.01035528 18.01034316]\n infect_w [0.01277739 0.08036489 0.26069394]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.2491117478102614 step loss 1.2466017246337888\n iter 0 sigmoid loss 1.184584318792444 step loss 0.7927295298619816 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3581990012990465 step loss 0.3645337493112961 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.30545623206629335 step loss 0.3594676458345833 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3295871888669818 step loss 0.35813917691487246 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.36932011049499297 step loss 0.35696019385549377 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.33271993195063154 step loss 0.3559995648781122 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3255832476667148 step loss 0.35519173695117756 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.34447349539030125 step loss 0.35455609826491874 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.39078485255185486 step loss 0.35422522159283587 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.34867665841306467 step loss 0.3540481928463866 sigmoid temp 1.0\n End sigmoid loss 0.3514194889026039 step loss 0.3539206206837834\n beta 0.30841154746431554\n rssi_w [-0.05587615 0.00691536 0.16507394 0.23582111]\n rssi_th [30.99863468 12.99861645 37.99584031]\n infect_w [0.01422222 0.08892061 0.28151565]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.0898781162622277 step loss 1.0885482448899388\n iter 0 sigmoid loss 1.0333147144612085 step loss 0.8341307296189877 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3565955658199472 step loss 0.34559490782027014 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3014121175502259 step loss 0.33297535313989085 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3215791386731428 step loss 0.3219910995186105 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.359036105438005 step loss 0.3093666977728371 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.30517396146357795 step loss 0.5853857857179227 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.27697318350476435 step loss 0.32823649557537626 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.2852165098662003 step loss 0.31077106128836196 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.3070650714220685 step loss 0.30553376024950113 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.31609073052363934 step loss 0.3032528576032665 sigmoid temp 1.0\n End sigmoid loss 0.29446381922897863 step loss 0.3038440922688811\n beta 0.3131636109292678\n rssi_w [-0.0339953 -0.0240882 0.08959544 0.28331788]\n rssi_th [13.00798561 28.00799278 19.00756412]\n infect_w [0.01091791 0.0735068 0.29041621]\n best loss 0.3038440922688811\n best scoring parameters\n beta 0.3131636109292678\n rssi_w [-0.0339953 -0.0580835 0.03151194 0.31482982]\n rssi_th [-106.99201439 -78.9840216 -59.97645748]\n infect_w [0.01091791 0.08442472 0.37484093]\n\n\n\n```python\n## Implementation for Optimization\nauc_train_trials_mod1, auc_test_trials_mod1 = train_and_eval_with_bag_config_mod_v1(Bag_Simulator(p_pos=0.7, r_pos=2, p_neg=0.7, r_neg=2, max_bag_size=1,\n censor_prob_pos=0, censor_prob_neg=0), X_epi, probabilities_true_epi,\n n_trials=n_trials, n_random_restarts=n_random_restarts_train)\n```\n\n total: 33600, positives: 5468, negatives: 28132\n Empirical Bag sizes:\n \t Positive bags: mean size 1.000, median size 1\n \t Negative bags: mean size 1.000, median size 1\n assign_mat size, X_shuff size: (4362, 4362) (4362, 3)\n assign_mat_trn size, assign_mat_tst size (3489, 4362) (873, 4362)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.1401424218642982 step loss 1.1436560013052968\n iter 0 sigmoid loss 1.3462001999257598 step loss 0.7672636596642491 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3431212433022248 step loss 0.36417474510526776 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.33504211084725966 step loss 0.36078466496277367 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.34733871864549487 step loss 0.36087523781564873 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.31419139476353386 step loss 0.3608011422532609 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.309317508597252 step loss 0.3612132033152587 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.327192316094813 step loss 0.36042844923895306 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3423636813162942 step loss 0.36033438327150497 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.35162458931183965 step loss 0.36033896587576525 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3557900969785965 step loss 0.36034482315468946 sigmoid temp 1.0\n End sigmoid loss 0.36023190285979095 step loss 0.36033316958899464\n beta 0.220563615202407\n rssi_w [-0.0063548 0.01677183 0.05805691 0.18891131]\n rssi_th [22.00230957 10.00237039 13.00250901]\n infect_w [0.01045464 0.05911262 0.19295159]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.236647737918061 step loss 1.2465425193048665\n iter 0 sigmoid loss 1.1536670363636998 step loss 0.8154070671807274 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4602245433499331 step loss 0.35758811583294886 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3191523933281198 step loss 0.35025708933914 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.29336166486401233 step loss 0.3457561530116548 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4458199516811221 step loss 0.3409417831217647 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.29865361630729026 step loss 0.3379677414587886 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.25533930902344104 step loss 0.33586579630965396 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.45194367471393737 step loss 0.3352630417980278 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.35785893021515025 step loss 0.33436850390959016 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.277396723816177 step loss 0.3339110348509307 sigmoid temp 1.0\n End sigmoid loss 0.3233467556374652 step loss 0.3337275972575485\n beta 0.3663136913771957\n rssi_w [-0.06384534 -0.05667536 0.18342005 0.29154415]\n rssi_th [12.99664157 33.99664624 25.99059173]\n infect_w [0.01511239 0.10140753 0.33903767]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.1308300541177252 step loss 1.125942609666351\n iter 0 sigmoid loss 0.8777181184952721 step loss 0.8504996054781335 sigmoid temp 0.1\n iter 500 sigmoid loss 0.30895914326320795 step loss 0.3463762477876531 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.34878746974823116 step loss 0.33588911093376983 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.32853164737299084 step loss 0.3274769000651727 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.32923107953929753 step loss 0.31739036468188997 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.22810447808858847 step loss 0.4742797673096794 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.31538002947587784 step loss 0.4738937423059239 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.27214589953959456 step loss 0.3197845896602318 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2334844273737094 step loss 0.31478297859036886 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3478845236790072 step loss 0.31316407982935235 sigmoid temp 1.0\n End sigmoid loss 0.30677726632444346 step loss 0.3128142361174993\n beta 0.2831805127028296\n rssi_w [-0.00559402 0.00310695 0.03106268 0.26663911]\n rssi_th [15.01309386 10.0131303 32.01308958]\n infect_w [0.01202266 0.06888319 0.25878685]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.1603819678263663 step loss 1.1608008536832246\n iter 0 sigmoid loss 1.093479832574556 step loss 0.8138793917262754 sigmoid temp 0.1\n iter 500 sigmoid loss 0.382255474248154 step loss 0.35030401488695684 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3900769964429307 step loss 0.34174994807147796 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.32566082577075733 step loss 0.33584345197604465 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.29702984542444344 step loss 0.32898255475952765 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.30598455330526536 step loss 0.3279480882684482 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.27407759329779025 step loss 0.44600418283862 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.249496871951325 step loss 0.44482881978410244 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.2726846599319101 step loss 0.32926202146846145 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.30229395271218457 step loss 0.3265693588043742 sigmoid temp 1.0\n End sigmoid loss 0.3191916094531024 step loss 0.32577036956249406\n beta 0.26831623900204205\n rssi_w [-0.00338974 0.00707756 0.02132021 0.25065161]\n rssi_th [10.01414259 12.01416872 33.01416666]\n infect_w [0.0124663 0.06827428 0.24245894]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.2593495159768548 step loss 1.259782486120562\n iter 0 sigmoid loss 1.3756625129626878 step loss 0.7627413587588253 sigmoid temp 0.1\n iter 500 sigmoid loss 0.42494978611750867 step loss 0.36498363532509065 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3231099660663862 step loss 0.3605020882182618 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.39396145834816776 step loss 0.3599727421033583 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4012764525315393 step loss 0.35938847314509625 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2847547649145002 step loss 0.35943776540407124 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.3766327337415516 step loss 0.3586734458044755 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3811660763568456 step loss 0.35784678700403966 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.29049872288161893 step loss 0.3576404112663247 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.3930488448769449 step loss 0.35749610175527313 sigmoid temp 1.0\n End sigmoid loss 0.350913458149417 step loss 0.357345054774594\n beta 0.32195640186906116\n rssi_w [-0.06749454 -0.05558537 0.22927979 0.18732512]\n rssi_th [13.00790314 35.00790573 35.9971166 ]\n infect_w [0.01709291 0.09846075 0.29220741]\n best loss 0.3128142361174993\n best scoring parameters\n beta 0.2831805127028296\n rssi_w [-0.00559402 -0.00248707 0.02857561 0.29521471]\n rssi_th [-104.98690614 -94.97377583 -62.96068625]\n infect_w [0.01202266 0.08090585 0.3396927 ]\n\n\n\n```python\nprint(auc_train_trials, auc_test_trials)\nprint(auc_train_trials_mod1, auc_test_trials_mod1)\n```\n\n [{'Learned': 0.8707116027214296, 'True': 0.905479866532299}] [{'Learned': 0.8590778597714879, 'True': 0.8955896803529798}]\n [{'Learned': 0.5027758557507317, 'True': 0.9036879613869455}] [{'Learned': 0.8552725381013854, 'True': 0.9029546636866341}]\n\n\nAnalysis of optimization 1 - Shuffling and Curriculum Learning.\n\nThe Area Under the Curve (AUC) is the measure of the ability of a classifier to distinguish between classes and is used as a summary of the ROC curve. The higher the AUC, the better the performance of the model at distinguishing between the positive and negative classes.\n\nAs we could observe from the result, although the AUC in training data is far less than the true AUC, the optimization method achieves similar result in test data. We don't make significant improvement after implementing shuffling.\n\n\n```python\n# Implementation of Optimization 2 - Batch Normalization.\n\ndef get_init_parameters_mod_v2():\n beta = 0.1\n\n # rssi weights\n rssi_w_residual = np.random.rand(n_rssi_buckets) * 0.01\n # modification: normalises input by subtracting the mean and dividing it by the standard deviation\n rssi_w_residual = (rssi_w_residual - np.mean(rssi_w_residual)) / np.std(rssi_w_residual)\n\n # rssi thresholds\n rssi_th_residual = np.random.randint(low=10, high=40, size=n_rssi_th)\n\n # infectiousness weights\n infect_w_residual = np.random.rand(n_infect_levels) * 0.01\n # modification: normalises input by subtracting the mean and dividing it by the standard deviation\n infect_w_residual = (infect_w_residual - np.mean(infect_w_residual)) / np.std(infect_w_residual)\n\n return np.concatenate([[beta], rssi_w_residual, rssi_th_residual, infect_w_residual])\n\n\ndef train_mod_v2(features, bag_labels, assign_mat, sigmoid_temp_init, sigmoid_temp_target, batch_size, num_iters, lr):\n num_samples = len(bag_labels)\n batch_size = np.minimum(batch_size, num_samples)\n\n model_params = get_init_parameters_mod_v2()\n loss_start, _ = loss_fn_ce(model_params, features, bag_labels, assign_mat, sigmoid_temp_init)\n loss_start_step, _ = loss_fn_stepbins_ce(model_params, features, bag_labels, assign_mat)\n print(\"Start\", \"sigmoid loss\", loss_start, \"step loss\", loss_start_step)\n\n for i in range(num_iters):\n l, r = i*batch_size%num_samples, (i+1)*batch_size%num_samples\n if l <= r:\n batch_y = bag_labels[l:r]\n batch_assign_mat = assign_mat[l:r]\n else:\n batch_y = np.concatenate([bag_labels[l:], bag_labels[:r]])\n batch_assign_mat = np.concatenate([assign_mat[l:], assign_mat[:r]], axis=0)\n\n sigmoid_temp = sigmoid_temp_fn(sigmoid_temp_init, sigmoid_temp_target, i, num_iters)\n\n loss, _ = loss_fn_ce(model_params, features, batch_y, batch_assign_mat, sigmoid_temp)\n grad = grad_loss_fn_ce(model_params, features, batch_y, batch_assign_mat, sigmoid_temp, loss)\n\n model_params -= lr * grad\n\n if i % 500 == 0:\n loss_step, _ = loss_fn_stepbins_ce(model_params, features, bag_labels, assign_mat)\n print(\"iter\", i, \"sigmoid loss\", loss, \"step loss\", loss_step, \"sigmoid temp\", sigmoid_temp)\n\n loss_final, _ = loss_fn_ce(model_params, features, bag_labels, assign_mat, sigmoid_temp_target)\n loss_final_step, probs_final = loss_fn_stepbins_ce(model_params, features, bag_labels, assign_mat)\n print(\"End\", \"sigmoid loss\", loss_final, \"step loss\", loss_final_step)\n print_params(model_params)\n\n return model_params, loss_final_step, probs_final\n\n\ndef train_and_eval_with_bag_config_mod_v2(bag_sim: Bag_Simulator, X_epi, probabilities_true_epi, n_trials=1, n_random_restarts=5):\n '''\n second modification - Batch Normalization.\n '''\n auc_train_trials = []\n auc_test_trials = []\n for j in range(n_trials):\n X, probabilities_true, assign_mat_trn, assign_mat_tst, bag_labels_trn, bag_labels_tst = \\\n bag_sim.simulate_bagged_data(X_epi, Y_epi, probabilities_true_epi, visualize=False)\n best_loss = np.inf\n best_model_params = None\n best_final_probs = None\n for i in range(n_random_restarts):\n print('----------- Trial {}/{}: Training run {}/{} ----------------'.format(j+1, n_trials, i+1, n_random_restarts))\n model_params, loss_st_step, final_probs = train_mod_v2(X, bag_labels_trn, assign_mat_trn,\n sigmoid_temp_init=0.1, sigmoid_temp_target=1,\n batch_size=200, num_iters=5000, lr=0.001)\n if loss_st_step < best_loss:\n best_loss = loss_st_step\n best_model_params = model_params\n best_final_probs = final_probs\n\n print(\"best loss\", best_loss)\n print(\"best scoring parameters\")\n print_params(residual_to_scoring(best_model_params))\n\n probs_bags_true_trn = 1 - np.exp(np.dot(assign_mat_trn, np.log(1-probabilities_true)))\n auc_train_trials.append({\n 'Learned': auc(best_final_probs, bag_labels_trn),\n 'True': auc(probs_bags_true_trn, bag_labels_trn),\n })\n\n scores_learned = loss_fn_stepbins_ce(best_model_params, X, None, None, return_scores=True)\n scores_learned_bags_tst = np.dot(assign_mat_tst, scores_learned)\n probs_learned_bags_tst = 1 - np.exp(-1*scores_learned_bags_tst)\n probs_bags_true_tst = 1 - np.exp(np.dot(assign_mat_tst, np.log(1-probabilities_true)))\n auc_test_trials.append({\n 'Learned': auc(probs_learned_bags_tst, bag_labels_tst),\n 'True': auc(probs_bags_true_tst, bag_labels_tst),\n })\n\n return auc_train_trials, auc_test_trials\n```\n\n\n```python\nauc_train_trials_mod2, auc_test_trials_mod2 = train_and_eval_with_bag_config_mod_v2(Bag_Simulator(p_pos=0.7, r_pos=2, p_neg=0.7, r_neg=2, max_bag_size=1,\n censor_prob_pos=0, censor_prob_neg=0), X_epi, probabilities_true_epi,\n n_trials=n_trials, n_random_restarts=n_random_restarts_train)\n```\n\n total: 33600, positives: 5468, negatives: 28132\n Empirical Bag sizes:\n \t Positive bags: mean size 1.000, median size 1\n \t Negative bags: mean size 1.000, median size 1\n assign_mat size, X_shuff size: (4362, 4362) (4362, 3)\n assign_mat_trn size, assign_mat_tst size (3489, 4362) (873, 4362)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 2.0576858184833804 step loss 2.224771594043257\n iter 0 sigmoid loss 2.610820801456132 step loss 2.0280994003958766 sigmoid temp 0.1\n iter 500 sigmoid loss 0.585181008814583 step loss 0.6693544360189062 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.47957867281029437 step loss 0.6258575181357663 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.6236749701357784 step loss 0.6205437282138629 sigmoid temp 0.1\n iter 2000 sigmoid loss 1.074574248161077 step loss 1.4310497748259916 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.5070326492003061 step loss 0.5888809538616244 sigmoid temp 0.1\n iter 3000 sigmoid loss 2.5328514023324504 step loss 1.8742822798077217 sigmoid temp 0.325\n iter 3500 sigmoid loss 2.129899361060242 step loss 1.8742822798077217 sigmoid temp 0.55\n iter 4000 sigmoid loss 2.0147702064110398 step loss 1.8742822798077217 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 2.3601576703586473 step loss 1.8742822798077217 sigmoid temp 1.0\n End sigmoid loss 1.8742822798077217 step loss 1.8742822798077217\n beta -0.5623678566741731\n rssi_w [-0.42769027 0.63430937 1.61618956 -0.79318106]\n rssi_th [19.02500314 20.02450286 27.01693086]\n infect_w [ 0.90693189 -0.90111566 0.7014179 ]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 2.4829643871711484 step loss 2.5368917380905316\n iter 0 sigmoid loss 3.1327683089723894 step loss 2.2527256285289865 sigmoid temp 0.1\n iter 500 sigmoid loss 0.5555271725318686 step loss 1.1485436655340089 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.4453909705115127 step loss 0.7452029626223868 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.5917840638121035 step loss 0.6340292273068193 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.6364060756860055 step loss 0.7421289857951209 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.4650025450865789 step loss 0.568683977217117 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.6340334296328577 step loss 0.5560497975408962 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.46603434198709975 step loss 0.5216466258236958 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.5405592992519042 step loss 0.4975847035456155 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.5522115015920683 step loss 0.49427274502654617 sigmoid temp 1.0\n End sigmoid loss 0.46855340073767543 step loss 0.47300561258150925\n beta -0.013385554834088959\n rssi_w [-1.2221151 -0.72692072 1.25054845 -0.43165211]\n rssi_th [35.01440891 34.01357142 20.9965786 ]\n infect_w [-0.99936992 1.32445744 0.12416327]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 2.0106128557687963 step loss 1.962986581975038\n iter 0 sigmoid loss 2.548620417037077 step loss 1.6856846906943277 sigmoid temp 0.1\n iter 500 sigmoid loss 0.4893322655108115 step loss 0.45725725896987224 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3771756651596943 step loss 0.444471115144588 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4966903431780011 step loss 0.43492468423272296 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.4012244993761624 step loss 0.42862686608649586 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.39972187300883105 step loss 0.428982673156327 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.5143167090043073 step loss 0.5506648720189085 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3673409718445901 step loss 0.4197440189623897 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.4496141298414909 step loss 0.40789374211165996 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.4676322102100709 step loss 0.4039461139572588 sigmoid temp 1.0\n End sigmoid loss 0.396224980631607 step loss 0.40216821363707095\n beta 0.021998757312913148\n rssi_w [-1.54368523 0.56501408 1.03599983 0.46107105]\n rssi_th [18.00728505 17.00778995 21.00894507]\n infect_w [-0.55322778 1.6341344 -0.08281342]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.8742822798077217 step loss 1.8742822798077217\n iter 0 sigmoid loss 2.360157670358647 step loss 1.8742822798077217 sigmoid temp 0.1\n iter 500 sigmoid loss 1.8996410517618374 step loss 1.8742822798077217 sigmoid temp 0.1\n iter 1000 sigmoid loss 1.4391244331650288 step loss 1.8742822798077217 sigmoid temp 0.1\n iter 1500 sigmoid loss 2.2450285157094445 step loss 1.8742822798077217 sigmoid temp 0.1\n iter 2000 sigmoid loss 1.8996410517618378 step loss 1.8742822798077217 sigmoid temp 0.1\n iter 2500 sigmoid loss 1.8420764744372367 step loss 1.8742822798077217 sigmoid temp 0.1\n iter 3000 sigmoid loss 2.5328514023324504 step loss 1.8742822798077217 sigmoid temp 0.325\n iter 3500 sigmoid loss 2.129899361060242 step loss 1.8742822798077217 sigmoid temp 0.55\n iter 4000 sigmoid loss 2.0147702064110398 step loss 1.8742822798077217 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 2.3601576703586473 step loss 1.8742822798077217 sigmoid temp 1.0\n End sigmoid loss 1.8742822798077217 step loss 1.8742822798077217\n beta 0.1\n rssi_w [ 0.5720815 -1.64496397 0.09413032 0.97875215]\n rssi_th [10. 23. 27.]\n infect_w [ 0.69943048 0.7147554 -1.41418588]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.9785554399737868 step loss 1.9464938025210503\n iter 0 sigmoid loss 2.510710457132315 step loss 1.70607455926438 sigmoid temp 0.1\n iter 500 sigmoid loss 0.44077658315824664 step loss 0.4155803336641432 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3347406346811367 step loss 0.40990011574159985 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.4440817550010436 step loss 0.40274066985249457 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3619856077368651 step loss 0.3970030112478793 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.3382904849891028 step loss 0.39239590117132583 sigmoid temp 0.1\n iter 3000 sigmoid loss 0.4620923613688032 step loss 0.37223409219556036 sigmoid temp 0.325\n iter 3500 sigmoid loss 0.3208670158362542 step loss 0.368690018014641 sigmoid temp 0.55\n iter 4000 sigmoid loss 0.37165814713420375 step loss 0.3626144523012853 sigmoid temp 0.7749999999999999\n iter 4500 sigmoid loss 0.4245134174766849 step loss 0.3611950105922687 sigmoid temp 1.0\n End sigmoid loss 0.3547046180405894 step loss 0.36091263274430396\n beta 0.18390467681751446\n rssi_w [ 1.05992852 -1.57824209 0.55747086 0.32115715]\n rssi_th [19.01044504 23.0100067 14.01074522]\n infect_w [-1.16731969 1.25861033 0.27849697]\n best loss 0.36091263274430396\n best scoring parameters\n beta 0.18390467681751446\n rssi_w [ 1.05992852 -0.51831357 0.03915729 0.36031445]\n rssi_th [-100.98955496 -77.97954826 -63.96880304]\n infect_w [-1.16731969 0.09129064 0.36978761]\n\n\n\n```python\nprint(auc_train_trials, auc_test_trials)\nprint(auc_train_trials_mod2, auc_test_trials_mod2)\n```\n\n [{'Learned': 0.8708758456249307, 'True': 0.9058276395793452}] [{'Learned': 0.8575027456118378, 'True': 0.8938700603071232}]\n [{'Learned': 0.8492964376467638, 'True': 0.898958067129239}] [{'Learned': 0.8668089246835321, 'True': 0.9220149900772625}]\n\n\nAnalysis of optimization 2 - Batch Normalization: Based on the training performance of our optimized method, we found that the loss doesn't decrease well as expected for some random trials. Also, the best loss is greater than that without batch normalization. This shows that batch normalization might not be useful in stochatic gradient descent method. \n\n\n```python\n# # Implementation of Optimization 3 - Early Stopping\n# previously we only have training & testing data, need further splitting for validation data\ndef check_validation_loss(validation_loss_history, num_early_stop):\n if num_early_stop and len(validation_loss_history) > num_early_stop * 2:\n recent_best = min(validation_loss_history[-num_early_stop:])\n previous_best = min(validation_loss_history[:-num_early_stop])\n if recent_best > previous_best:\n return True\n\n return False\n\n\ndef train_mod_v3(features, bag_labels, assign_mat, sigmoid_temp_init, sigmoid_temp_target, batch_size, num_iters, lr,\n bag_labels_val, assign_mat_val, patient_early_stopping):\n num_samples = len(bag_labels)\n batch_size = np.minimum(batch_size, num_samples)\n\n model_params = get_init_parameters()\n loss_start, _ = loss_fn_ce(model_params, features, bag_labels, assign_mat, sigmoid_temp_init)\n loss_start_step, _ = loss_fn_stepbins_ce(model_params, features, bag_labels, assign_mat)\n print(\"Start\", \"sigmoid loss\", loss_start, \"step loss\", loss_start_step)\n\n validation_loss_history = []\n\n for i in range(num_iters):\n l, r = i*batch_size%num_samples, (i+1)*batch_size%num_samples\n if l <= r:\n batch_y = bag_labels[l:r]\n batch_assign_mat = assign_mat[l:r]\n else:\n batch_y = np.concatenate([bag_labels[l:], bag_labels[:r]])\n batch_assign_mat = np.concatenate([assign_mat[l:], assign_mat[:r]], axis=0)\n\n sigmoid_temp = sigmoid_temp_fn(sigmoid_temp_init, sigmoid_temp_target, i, num_iters)\n\n loss, _ = loss_fn_ce(model_params, features, batch_y, batch_assign_mat, sigmoid_temp)\n grad = grad_loss_fn_ce(model_params, features, batch_y, batch_assign_mat, sigmoid_temp, loss)\n\n model_params -= lr * grad\n\n # add early stopping\n loss_val, _ = loss_fn_ce(model_params, features, bag_labels_val, assign_mat_val, sigmoid_temp)\n validation_loss_history.append(loss_val)\n\n if check_validation_loss(validation_loss_history, patient_early_stopping):\n print(\"Early Stopping in Iteration, \", i)\n break\n\n if i % 500 == 0:\n loss_step, _ = loss_fn_stepbins_ce(model_params, features, bag_labels, assign_mat)\n print(\"iter\", i, \"sigmoid loss\", loss, \"step loss\", loss_step, \"sigmoid temp\", sigmoid_temp)\n\n loss_final, _ = loss_fn_ce(model_params, features, bag_labels, assign_mat, sigmoid_temp_target)\n loss_final_step, probs_final = loss_fn_stepbins_ce(model_params, features, bag_labels, assign_mat)\n print(\"End\", \"sigmoid loss\", loss_final, \"step loss\", loss_final_step)\n print_params(model_params)\n\n return model_params, loss_final_step, probs_final\n\n\ndef train_and_eval_with_bag_config_mod_v3(bag_sim: Bag_Simulator, X_epi, probabilities_true_epi, n_trials=1, n_random_restarts=5, patient_early_stopping=30):\n '''\n third modification - Early Stopping.\n '''\n auc_train_trials = []\n auc_test_trials = []\n for j in range(n_trials):\n X, probabilities_true, assign_mat_trn, assign_mat_tst, bag_labels_trn, bag_labels_tst = \\\n bag_sim.simulate_bagged_data(X_epi, Y_epi, probabilities_true_epi, visualize=False)\n ## Split the validation data for early stopping evaluation\n assign_mat_trn, assign_mat_val, bag_labels_trn, bag_labels_val = \\\n train_test_split(assign_mat_trn, bag_labels_trn, train_size=0.8)\n best_loss = np.inf\n best_model_params = None\n best_final_probs = None\n for i in range(n_random_restarts):\n print('----------- Trial {}/{}: Training run {}/{} ----------------'.format(j+1, n_trials, i+1, n_random_restarts))\n model_params, loss_st_step, final_probs = train_mod_v3(X, bag_labels_trn, assign_mat_trn,\n sigmoid_temp_init=0.1, sigmoid_temp_target=1,\n batch_size=200, num_iters=5000, lr=0.001, assign_mat_val=assign_mat_val,\n bag_labels_val=bag_labels_val, patient_early_stopping=patient_early_stopping)\n if loss_st_step < best_loss:\n best_loss = loss_st_step\n best_model_params = model_params\n best_final_probs = final_probs\n\n print(\"best loss\", best_loss)\n print(\"best scoring parameters\")\n print_params(residual_to_scoring(best_model_params))\n\n probs_bags_true_trn = 1 - np.exp(np.dot(assign_mat_trn, np.log(1-probabilities_true)))\n auc_train_trials.append({\n 'Learned': auc(best_final_probs, bag_labels_trn),\n 'True': auc(probs_bags_true_trn, bag_labels_trn),\n })\n\n scores_learned = loss_fn_stepbins_ce(best_model_params, X, None, None, return_scores=True)\n scores_learned_bags_tst = np.dot(assign_mat_tst, scores_learned)\n probs_learned_bags_tst = 1 - np.exp(-1*scores_learned_bags_tst)\n probs_bags_true_tst = 1 - np.exp(np.dot(assign_mat_tst, np.log(1-probabilities_true)))\n auc_test_trials.append({\n 'Learned': auc(probs_learned_bags_tst, bag_labels_tst),\n 'True': auc(probs_bags_true_tst, bag_labels_tst),\n })\n\n return auc_train_trials, auc_test_trials\n```\n\n\n```python\nauc_train_trials_mod3, auc_test_trials_mod3 = train_and_eval_with_bag_config_mod_v3(Bag_Simulator(p_pos=0.7, r_pos=2, p_neg=0.7, r_neg=2, max_bag_size=1,\n censor_prob_pos=0, censor_prob_neg=0), X_epi, probabilities_true_epi,\n n_trials=n_trials, n_random_restarts=n_random_restarts_train, patient_early_stopping=30)\n```\n\n total: 33600, positives: 5468, negatives: 28132\n Empirical Bag sizes:\n \t Positive bags: mean size 1.000, median size 1\n \t Negative bags: mean size 1.000, median size 1\n assign_mat size, X_shuff size: (4362, 4362) (4362, 3)\n assign_mat_trn size, assign_mat_tst size (3489, 4362) (873, 4362)\n Average positive samples per bag: 1.0\n ----------- Trial 1/1: Training run 1/5 ----------------\n Start sigmoid loss 1.0720328100187655 step loss 1.0779339026915968\n iter 0 sigmoid loss 1.2420871342086806 step loss 0.8217326385528234 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3405446915232291 step loss 0.35552467003539606 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3747062675032058 step loss 0.3434560007590296 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.36744567088141017 step loss 0.33300717598266677 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3123300817613699 step loss 0.32083193866737864 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2623574702836697 step loss 0.31499328352848865 sigmoid temp 0.1\n Early Stopping in Iteration, 2826\n End sigmoid loss 0.30281256362246667 step loss 0.31375909865081475\n beta 0.31954446927097135\n rssi_w [-0.06863074 0.00311495 0.14251487 0.26133037]\n rssi_th [37.00149179 12.00138963 18.00026373]\n infect_w [0.01568274 0.10734184 0.28510776]\n ----------- Trial 1/1: Training run 2/5 ----------------\n Start sigmoid loss 1.3629694683181406 step loss 1.4387220184582845\n iter 0 sigmoid loss 1.573063902066644 step loss 0.7160343995122965 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3360930213589836 step loss 0.3692667593372829 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3716686374809008 step loss 0.36199502195967426 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.36436528149483477 step loss 0.35987625189523215 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3275896944202663 step loss 0.3613010902057594 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.27467378919278035 step loss 0.3720045867645901 sigmoid temp 0.1\n Early Stopping in Iteration, 2588\n End sigmoid loss 0.35919049440447465 step loss 0.36820270226217\n beta 0.2814950784106983\n rssi_w [-0.01581254 0.09027835 0.2336265 0.10522025]\n rssi_th [38.99962747 36.99874658 10.99880576]\n infect_w [0.02330598 0.09264656 0.24716957]\n ----------- Trial 1/1: Training run 3/5 ----------------\n Start sigmoid loss 1.2062706330640343 step loss 1.2020663634238067\n iter 0 sigmoid loss 1.3924612040712918 step loss 0.8041389397904805 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3418397643821111 step loss 0.3492138537432713 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.377264334743824 step loss 0.3354951305851122 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.3680721180686166 step loss 0.3257878977794353 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3284298549916748 step loss 0.3229556406232951 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.27238727098296783 step loss 0.5441717024991753 sigmoid temp 0.1\n Early Stopping in Iteration, 2784\n End sigmoid loss 0.47255945538278143 step loss 0.5451207765267403\n beta 0.30743119999444557\n rssi_w [-0.03667396 0.01352073 0.28143558 0.06825052]\n rssi_th [28.00181649 30.00168424 36.99972077]\n infect_w [0.01700541 0.10693578 0.27226173]\n ----------- Trial 1/1: Training run 4/5 ----------------\n Start sigmoid loss 1.1987813934708635 step loss 1.2008472384560758\n iter 0 sigmoid loss 1.392568152273663 step loss 0.782357827034807 sigmoid temp 0.1\n iter 500 sigmoid loss 0.3434168717819195 step loss 0.36953015725709654 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.38147955258237914 step loss 0.363540179863 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.36873111136408726 step loss 0.3613075130407875 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.336574490685103 step loss 0.3594748611263109 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2816990688285593 step loss 0.35826733388141324 sigmoid temp 0.1\n Early Stopping in Iteration, 2727\n End sigmoid loss 0.351964261905139 step loss 0.35831071075348847\n beta 0.28977674154374566\n rssi_w [-0.0366212 -0.01887151 0.16452663 0.21554626]\n rssi_th [19.00075415 28.00075908 29.99902396]\n infect_w [0.02473776 0.10142327 0.2531886 ]\n ----------- Trial 1/1: Training run 5/5 ----------------\n Start sigmoid loss 1.212299884281231 step loss 1.214254490360228\n iter 0 sigmoid loss 1.415410057940195 step loss 0.7787554910983174 sigmoid temp 0.1\n iter 500 sigmoid loss 0.34313478430573136 step loss 0.3564871172527192 sigmoid temp 0.1\n iter 1000 sigmoid loss 0.3796552443286441 step loss 0.3444288498825448 sigmoid temp 0.1\n iter 1500 sigmoid loss 0.36829851998973395 step loss 0.3350212464435135 sigmoid temp 0.1\n iter 2000 sigmoid loss 0.3272691370052284 step loss 0.32519908914318557 sigmoid temp 0.1\n iter 2500 sigmoid loss 0.2673572972598275 step loss 0.34010995163915964 sigmoid temp 0.1\n Early Stopping in Iteration, 2826\n End sigmoid loss 0.30730715959937455 step loss 0.32137454983371894\n beta 0.31875908529089825\n rssi_w [-0.04539322 -0.01999173 0.10096825 0.28298262]\n rssi_th [22.00064079 20.00065298 24.9998323 ]\n infect_w [0.02195661 0.11924217 0.27954828]\n best loss 0.31375909865081475\n best scoring parameters\n beta 0.31954446927097135\n rssi_w [-0.06863074 -0.06551579 0.07699908 0.33832945]\n rssi_th [-82.99850821 -70.99711858 -52.99685485]\n infect_w [0.01568274 0.12302458 0.40813234]\n\n\n\n```python\nprint(auc_train_trials, auc_test_trials)\nprint(auc_train_trials_mod3, auc_test_trials_mod3)\n```\n\n [{'Learned': 0.8707116027214296, 'True': 0.905479866532299}] [{'Learned': 0.8590778597714879, 'True': 0.8955896803529798}]\n [{'Learned': 0.854169753863936, 'True': 0.9041179815657276}] [{'Learned': 0.8653783164100884, 'True': 0.9093755418970734}]\n\n\nAnalysis of optimization 3 - Early Stopping. \n\nWith early stopping, we can partially solve the problem of slow running, without sacrificing too much accuracy. Also, compared the AUC and loss between test dataset, we achieve better result and avoid overfitting.\n\n## 3. Broader Impact\n\n[Return to top](#Notebook-contents)\n\n+ **The Tracing App**\n\nThe world has been fighting COVID-19 for two years. From a perspective, we have witnessed how technological advances could aid in preventing the spread of the virus. The covid tracing app is an innovative example that benefits lots of people. Each terminal user who agrees to share information with the platform can get timely feedback from the app about the risk of infection. For larger parties like medical providers who need to monitor close contacts or relevant public agencies that need to enact policies, the existence of a tracing app saves time from endless searching.\n\n\nIt is necessary to obtain great accuracy for risk score modeling since it will provide more helpful information and saves time. The inacurrate contact tracing, on the contrary, will falsely occupy the medical resources greatly. As we mentioned before, one of the main contributions from original authors is to build off the model from real-world, bio-medically sound laws. With more realistic constraints considered, the original authors' parameters are more helpful in improving the tracing content. As for us, we manage to replicate their approach for optimizing the risk score model parameters and experiment with more optimization methods on top of their data management and fundamental SGD algorithm. We hope that we have made positive impacts throughout the process and contributed to smoother future iterations of this application.\n\n+ **The Inspiration for ML Community**\n\nThe successful application for using machine learning technologies on covid-19 risk score modeling could give confidence to the ML community and encourage engineers/scientists to explore more. Essentially, we provide several possible modifications and enhancements that can be applied to the original learning algorithm. It shows that others are incentivized to improve the performance based on a good starting point (the original work done by paper authors). Combining with the background of the study, we incentivize more ML engineers to contribute to real-world problems.\n\nFurthermore, the work could have broadened the community\u2019s perspective on what ML can achieve. Now it is only an example for making an accurate covid tracing app. Still, in the future, the procedure can be used for other infectious diseases or be prepared for future public health events. Other fields are possible, too, since we have shown the capability of ML to model some real-life scenarios with actual physical law captured. All in all, we can propose a bright future for ML applications, but there are caveats that we have to be careful about as well. \n\n\n+ **Privacy Issue in Data Usage**\n\nThe trade-off between data privacy and prediction accuracy has bothered data scientists for a long time. In the original work and our replicated version, we use simulated data for exposure events and interactions between human users. However, companies in the real world will finally adopt the tracing app and similar applications, and they will feed in real data. It could eventually become a privacy issue, and such an issue has already shown its lousy influence in some countries. In China, the government strictly controls the covid cases, and the media will keep publishing the case information over the internet (Weibo, Weixin official account, etc.). Several internet violence events have happened: a young lady from Chengdu has been reported as PCR test positive, and then not only her itineraries, but also her name, ID number, family background were extensively searched for and reported on the internet. The impact is detrimental for her and further warns us of the problem of using real data in public health scenarios. \n \nIt seems that maybe the higher-level administrators must come up with a legislative solution to improve the situation. Meanwhile, for our end, as engineers and data scientists, we recommend adding privacy filters onto the dataset manually. We could incorporate methods for the de-identification process, such as suppressing, adding synthetic records, or applying a differential privacy algorithm. \n \nHowever, when manipulating de-identification on the dataset, we potentially result in some level of inaccuracy. One aspect is misrepresentation associated with a biased and distorted dataset. Like in crime analysis, misrepresentation and underrepresentation cause a big problem of prioritizing some races. The other aspect is information loss. In this specific scenario of the covid tracing app, using too much de-identification processing harms the integrity of information, which is one of the bedrocks that we want to base on improving the algorithm. Moreover, we are alternatively creating more censorship, potentially increasing the learning complexity for machines. \n \nAlthough we have a good intention of protecting privacy while using public data by some means, we shall think of the consequences. Our goal of contributing to the world needs to be reassessed with the abovementioned ethical issues in a delicate balance. In conclusion, albeit no more action we could take during this step of the project, we urge researchers to be aware of the tradeoff of data privacy and data accuracy and incorporate methods to mitigate the problem.\n\n\n# References\n\n[Return to top](#Notebook-contents)\n\n[1] Google-Apple. Exposure notifications: Using technology to help public health authorities fight covid-19, 2020. URL https://www.google.com/covid19/exposurenotifications/.\n\n[2] LFPH. Configuring Exposure Notification Risk Scores for COVID-19 (Linux Foundation Public Health). URL https://github.com/lfph/gaen-risk-scoring/blob/main/risk-scoring.md.\n\n[3] Luca Ferretti, Chris Wymant, Michelle Kendall, Lele Zhao, Anel Nurtay, Lucie Abeler-D \u0308orner, Michael Parker, David Bonsall, and Christophe Fraser. Quantifying SARS-CoV-2 transmission suggests epidemic control with digital contact tracing. Science, 368(6491), May 2020b. URL http://dx.doi.org/10.1126/science.abb6936.\n\n[4] Matthew Abueg, Robert Hinch, Neo Wu, Luyang Liu, William J M Probert, Austin Wu, Paul Eastham, Yusef Shafi, Matt Rosencrantz, Michael Dikovsky, Zhao Cheng, Anel Nur- tay, Lucie Abeler-D \u0308orner, David G Bonsall, Michael V McConnell, Shawn O\u2019Banion, and Christophe Fraser. Modeling the combined effect of digital exposure notification and non- pharmaceutical interventions on the COVID-19 epidemic in washington state. NPJ Digital Medicine, (medrxiv;2020.08.29.20184135v1), 2021. URL https://www.medrxiv.org/content/ 10.1101/2020.08.29.20184135v1.abstract.\n\n[5] Marcel Salathe, Christian Althaus, Nanina Anderegg, Daniele Antonioli, Tala Ballouz, Edouard Bugnon, Srdjan C\u02c7apkun, Dennis Jackson, Sang-Il Kim, Jim Larus, Nicola Low, Wouter Lueks, Dominik Menges, C \u0301edric Moullet, Mathias Payer, Julien Riou, Theresa Stadler, Carmela Troncoso, Effy Vayena, and Viktor von Wyl. Early evidence of effectiveness of digital contact tracing for SARS-CoV-2 in switzerland. Swiss Med. Wkly, 150:w20457, December 2020. URL http: //dx.doi.org/10.4414/smw.2020.20457.\n\n[6] Felix Sattler, Jackie Ma, Patrick Wagner, David Neumann, Markus Wenzel, Ralf Sch \u0308afer, Wojciech Samek, Klaus-Robert Mu \u0308ller, and Thomas Wiegand. Risk estimation of SARS-CoV-2 transmission from bluetooth low energy measurements. npj Digital Medicine, 3(1):129, October 2020. URL https://doi.org/10.1038/s41746-020-00340-0.\n\n[7] Mark Briers, Marcos Charalambides, and Chris Holmes. Risk scoring calculation for the current NHSx contact tracing app. May 2020. URL http://arxiv.org/abs/2005.11057.\n\n[8] Luca Ferretti, Alice Ledda, Chris Wymant, Lele Zhao, Virginia Ledda, Lucie Abeler-Dorner, Michelle Kendall, Anel Nurtay, Hao-Yuan Cheng, Ta-Chou Ng, Hsien-Ho Lin, Rob Hinch, Joanna Masel, A Marm Kilpatrick, and Christophe Fraser. The timing of COVID-19 trans- mission. September 2020a. URL https://www.medrxiv.org/content/10.1101/2020.09.04. 20188516v1.abstract.\n\n[9] Timo Smieszek. A mechanistic model of infection: why duration and intensity of contacts should be included in models of disease spread. Theor. Biol. Med. Model., 6:25, November 2009. URL http://dx.doi.org/10.1186/1742-4682-6-25.\n\n[10] Charles N Haas. Quantitative Microbial Risk Assessment (2nd edn). Wiley, July 2014. URL https://openlibrary.org/books/OL27557844M/Quantitative_Microbial_Risk_Assessment.\n\n[11] Yann A. LeCun, L\u00e9on Bottou, Genevieve B. Orr and Klaus-Robert M\u00fclle. Efficient BackProp. Neural Networks: Tricks of the Trade, 1524: 9\u201350, 1998. \n\n[12] Sergey Ioffe and Christian Szegedy. Batch Normalization : Accelerating Deep Network Training by Reducing Internal Covariate Shift. March 2015. URL https://arxiv.org/abs/1502.03167\n\n[13] Jeffrey Dean, Greg S. Corrado, Rajat Monga, Kai Chen, Matthieu Devin, Quoc V. Le, Mark Z. Mao, Marc\u2019Aurelio Ranzato, Andrew Senior, Paul Tucker, Ke Yang and Andrew Y. Ng. Large Scale Distributed Deep Networks. Neural Information Processing Systems, 1\u201311, 2012.\n\n[Return to top](#Notebook-contents)\n\n\n```python\n\n```\n", "meta": {"hexsha": "994a1e68e2503e83d0694bad5ad098f38309d5bc", "size": 838789, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "risk_score_learning/Project_Implementation_Final.ipynb", "max_stars_repo_name": "TangJiahui/AM207-Stochastic-Optimization", "max_stars_repo_head_hexsha": "4288efeb7c017d8d7fa1432ef0b4fcb2f1d42f1d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "risk_score_learning/Project_Implementation_Final.ipynb", "max_issues_repo_name": "TangJiahui/AM207-Stochastic-Optimization", "max_issues_repo_head_hexsha": "4288efeb7c017d8d7fa1432ef0b4fcb2f1d42f1d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "risk_score_learning/Project_Implementation_Final.ipynb", "max_forks_repo_name": "TangJiahui/AM207-Stochastic-Optimization", "max_forks_repo_head_hexsha": "4288efeb7c017d8d7fa1432ef0b4fcb2f1d42f1d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 108.3286839726, "max_line_length": 49972, "alphanum_fraction": 0.7828941486, "converted": true, "num_tokens": 156889, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.6893056231680122, "lm_q1q2_score": 0.43661325652546856}} {"text": "\n\n## Setup\n\n\n```python\nimport torch\nprint(torch.__version__)\n```\n\n 1.3.0+cu100\n\n\n\n```python\nimport numpy as np\n\nimport torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nfrom torch.autograd import Variable\nfrom torch.optim import Adam\nfrom torchvision import datasets, transforms\n\nfrom time import time\n\nUSE_CUDA = True\n```\n\n## CIFAR-10 data loader/generator.\n\nCode below setups image generators from folder './data'\n\nNormalization values for CIFAR10 are taken from pytorch website (usual normalization values for the task).\n\n\n```python\nclass Cifar10:\n def __init__(self, batch_size):\n dataset_transform = transforms.Compose([\n transforms.ToTensor(),\n transforms.Normalize((0.4914, 0.4822, 0.4465), (0.247, 0.243, 0.261))\n ])\n\n train_dataset = datasets.CIFAR10('./data', train=True, download=True, transform=dataset_transform)\n test_dataset = datasets.CIFAR10('./data', train=False, download=True, transform=dataset_transform)\n \n self.train_loader = torch.utils.data.DataLoader(train_dataset, batch_size=batch_size, shuffle=True)\n self.test_loader = torch.utils.data.DataLoader(test_dataset, batch_size=batch_size, shuffle=True) \n```\n\n## Network\n\nRecall the architecture of CapsNet. This tutorial walks you through building process of it. Note that actual values of parameters such as \"number of capsules\", \"number of filters in the first layer\" etc. are not taken from MNIST implementation in the original paper, but instead from CIFAR10 implementation.\n\n### Pre-capsule layer\n\nThis is a usual convolution layer that extracts basic features from images.\n\n\n```python\nclass ConvLayer(nn.Module):\n def __init__(self, in_channels=3, out_channels=256, kernel_size=9):\n super(ConvLayer, self).__init__()\n\n self.conv = nn.Conv2d(in_channels=in_channels,\n out_channels=out_channels,\n kernel_size=kernel_size,\n stride=1\n )\n\n def forward(self, x):\n return F.relu(self.conv(x))\n```\n\n### First capsule layer (PrimaryCaps)\n\nThis is the second layer of the network and the first one which contains capsules (recall that capsules are just groups of neurons).\n\nThe squash operation is the following one:\n\n\\begin{align}\nv_j & = \\frac{(\\|s_j\\|^2)}{(1 + \\|s_j\\|^2)} \\frac{s_j}{\\|s_j\\|}\\\\\n\\end{align}\n\nIt takes a vector s_j as input, normalizes it to unit norm and then adds some non-linearity so that large vectors become close to 1 and small vectors close to zero. Recall that it is needed to enforce the property of v_j's norm being the probability (or certainty) that object is detected by the capsule v_j.\n\n\n```python\nclass PrimaryCaps(nn.Module):\n def __init__(self, num_capsules=8, in_channels=256, out_channels=64, kernel_size=9):\n super(PrimaryCaps, self).__init__()\n\n self.capsules = nn.ModuleList([\n nn.Conv2d(in_channels=in_channels, out_channels=out_channels, kernel_size=kernel_size, stride=2, padding=0) \n for _ in range(num_capsules)])\n \n def forward(self, x):\n u = [capsule(x) for capsule in self.capsules]\n u = torch.stack(u, dim=1)\n u = u.view(x.size(0), 64 * 8 * 8, -1)\n return self.squash(u)\n \n def squash(self, input_tensor):\n squared_norm = (input_tensor ** 2).sum(-1, keepdim=True)\n output_tensor = squared_norm * input_tensor / ((1. + squared_norm) * torch.sqrt(squared_norm))\n return output_tensor\n```\n\n### Second capsule layer (DigitCaps)\n\nThis is the final layer of the network and the one that contains digit-capsules (or in case of CIFAR10 - class-capsules) which predict the class on the image.\n\nBelow you may see the dynamic routing algorithm from the original paper under the forward section of the layer.\n\n\n\n\n```python\nclass DigitCaps(nn.Module):\n def __init__(self, num_capsules=10, num_routes=64 * 8 * 8, in_channels=8, out_channels=16):\n super(DigitCaps, self).__init__()\n\n self.in_channels = in_channels\n self.num_routes = num_routes\n self.num_capsules = num_capsules\n\n self.W = nn.Parameter(torch.randn(1, num_routes, num_capsules, out_channels, in_channels))\n\n def forward(self, x):\n batch_size = x.size(0)\n x = torch.stack([x] * self.num_capsules, dim=2).unsqueeze(4)\n\n W = torch.cat([self.W] * batch_size, dim=0)\n u_hat = torch.matmul(W, x)\n\n b_ij = Variable(torch.zeros(1, self.num_routes, self.num_capsules, 1))\n if USE_CUDA:\n b_ij = b_ij.cuda()\n\n num_iterations = 3\n for iteration in range(num_iterations):\n c_ij = F.softmax(b_ij)\n c_ij = torch.cat([c_ij] * batch_size, dim=0).unsqueeze(4)\n\n s_j = (c_ij * u_hat).sum(dim=1, keepdim=True)\n v_j = self.squash(s_j)\n \n if iteration < num_iterations - 1:\n a_ij = torch.matmul(u_hat.transpose(3, 4), torch.cat([v_j] * self.num_routes, dim=1))\n b_ij = b_ij + a_ij.squeeze(4).mean(dim=0, keepdim=True)\n\n return v_j.squeeze(1)\n \n def squash(self, input_tensor):\n squared_norm = (input_tensor ** 2).sum(-1, keepdim=True)\n output_tensor = squared_norm * input_tensor / ((1. + squared_norm) * torch.sqrt(squared_norm))\n return output_tensor\n```\n\n### Reconstruction part of network (decoder)\n\nThis is the second task for the network, namely, to reconstruct the image from the final class-capsules. \n\nThis is a useful technique of regularization to prevent overfitting and also to enforce the property of capsules representing the 'instantiation parameters' of the object. In other words, final capsule should contain information about the class it predicts and that information (implicitly) may be: rotation angle, distortion, illumination etc.\n\nThe reconstruction is done by a simple decoder (stack of fully-connected layers). Below is the picture for MNIST.\n\n\n\n\n```python\nclass Decoder(nn.Module):\n def __init__(self):\n super(Decoder, self).__init__()\n \n self.reconstraction_layers = nn.Sequential(\n nn.Linear(16 * 10, 512),\n nn.ReLU(inplace=True),\n nn.Linear(512, 1024),\n nn.ReLU(inplace=True),\n nn.Linear(1024, 1024*3),\n nn.Sigmoid()\n )\n \n def forward(self, x, data):\n classes = torch.sqrt((x ** 2).sum(2))\n classes = F.softmax(classes)\n \n _, max_length_indices = classes.max(dim=1)\n masked = Variable(torch.sparse.torch.eye(10))\n if USE_CUDA:\n masked = masked.cuda()\n masked = masked.index_select(dim=0, index=Variable(max_length_indices.squeeze(1).data))\n \n reconstructions = self.reconstraction_layers((x * masked[:, :, None, None]).view(x.size(0), -1))\n reconstructions = reconstructions.view(-1, 3, 32, 32)\n \n return reconstructions, masked\n```\n\n### Full network (CapsNet)\n\nThis is a final forward pass for the whole network. The only new part here is the custom loss from the original paper.\n\n\n\n\n```python\nclass CapsNet(nn.Module):\n def __init__(self):\n super(CapsNet, self).__init__()\n self.conv_layer = ConvLayer()\n self.primary_capsules = PrimaryCaps()\n self.digit_capsules = DigitCaps()\n self.decoder = Decoder()\n \n self.mse_loss = nn.MSELoss()\n \n def forward(self, data):\n output = self.digit_capsules(self.primary_capsules(self.conv_layer(data)))\n reconstructions, masked = self.decoder(output, data)\n return output, reconstructions, masked\n \n def loss(self, data, x, target, reconstructions):\n return self.margin_loss(x, target) + self.reconstruction_loss(data, reconstructions)\n \n def margin_loss(self, x, labels, size_average=True):\n batch_size = x.size(0)\n\n v_c = torch.sqrt((x**2).sum(dim=2, keepdim=True))\n\n left = (F.relu(0.9 - v_c)**2).view(batch_size, -1)\n right = (F.relu(v_c - 0.1)**2).view(batch_size, -1)\n\n loss = labels * left + 0.5 * (1.0 - labels) * right\n loss = loss.sum(dim=1).mean()\n\n return loss\n \n def reconstruction_loss(self, data, reconstructions):\n loss = self.mse_loss(reconstructions.view(reconstructions.size(0), -1), data.view(reconstructions.size(0), -1))\n return loss * 0.0005\n```\n\nHere the model is compiled with Adam optimizer with basic parameters.\n\n\n```python\ncapsule_net = CapsNet()\nif USE_CUDA:\n capsule_net = capsule_net.cuda()\noptimizer = Adam(capsule_net.parameters())\n```\n\n## Training\n\nNote that one epoch takes a lot of time even on GPU, so don't rush and plan everything ahead and try to justify your ideas prior to implementing them.\n\n\n```python\nbatch_size = 8\n# dataset = Mnist(batch_size)\ndataset = Cifar10(batch_size)\n\nn_epochs = 5\n\n\nfor epoch in range(n_epochs):\n ep_start = time()\n capsule_net.train()\n train_loss = 0\n train_accuracy = 0\n for batch_id, (data, target) in enumerate(dataset.train_loader):\n st = time()\n\n target = torch.sparse.torch.eye(10).index_select(dim=0, index=target)\n data, target = Variable(data), Variable(target)\n\n if USE_CUDA:\n data, target = data.cuda(), target.cuda()\n\n optimizer.zero_grad()\n output, reconstructions, masked = capsule_net(data)\n loss = capsule_net.loss(data, output, target, reconstructions)\n loss.backward()\n optimizer.step()\n\n train_loss = train_loss+loss.data\n \n tr_accuracy = sum(np.argmax(masked.data.cpu().numpy(), 1) == \n np.argmax(target.data.cpu().numpy(), 1)) / float(batch_size)\n train_accuracy += tr_accuracy\n if batch_id % 10 == 0 or batch_id == 99:\n print (\"train accuracy [batch {}]:\".format(batch_id), tr_accuracy)\n print (\"Train loss:{}\\n\".format(loss.data))\n en = time()\n# print 'Sec per batch', round(en-st, 2)\n ep_end = time()\n print ('Total train loss', train_loss / len(dataset.train_loader))\n print ('Total train accuracy', train_accuracy / len(dataset.train_loader))\n print ('Total time for training an epoch: {}'.format(int(ep_end - ep_start)) ) \n capsule_net.eval()\n test_loss = 0\n test_accuracy = 0\n for batch_id, (data, target) in enumerate(dataset.test_loader):\n\n target = torch.sparse.torch.eye(10).index_select(dim=0, index=target)\n data, target = Variable(data), Variable(target)\n\n if USE_CUDA:\n data, target = data.cuda(), target.cuda()\n\n output, reconstructions, masked = capsule_net(data)\n loss = capsule_net.loss(data, output, target, reconstructions)\n\n test_loss += loss.data\n ts_accuracy = sum(np.argmax(masked.data.cpu().numpy(), 1) == \n np.argmax(target.data.cpu().numpy(), 1)) / float(batch_size)\n test_accuracy += ts_accuracy\n if batch_id % 25 == 0 or batch_id == 99:\n print (\"test accuracy [batch {}]:\".format(batch_id), ts_accuracy)\n \n print ('Total test loss', test_loss / len(dataset.test_loader))\n print ('Total test accuracy', test_accuracy / len(dataset.test_loader))\n```\n\n Files already downloaded and verified\n Files already downloaded and verified\n\n\n /usr/local/lib/python3.6/dist-packages/ipykernel_launcher.py:24: UserWarning: Implicit dimension choice for softmax has been deprecated. Change the call to include dim=X as an argument.\n /usr/local/lib/python3.6/dist-packages/ipykernel_launcher.py:16: UserWarning: Implicit dimension choice for softmax has been deprecated. Change the call to include dim=X as an argument.\n app.launch_new_instance()\n\n\n train accuracy [batch 0]: 0.75\n Train loss:0.44040122628211975\n \n train accuracy [batch 10]: 0.75\n Train loss:0.256026953458786\n \n train accuracy [batch 20]: 0.25\n Train loss:0.3877621591091156\n \n train accuracy [batch 30]: 0.5\n Train loss:0.3283441364765167\n \n train accuracy [batch 40]: 0.625\n Train loss:0.221501424908638\n \n train accuracy [batch 50]: 0.625\n Train loss:0.2758632302284241\n \n train accuracy [batch 60]: 0.5\n Train loss:0.3989644944667816\n \n train accuracy [batch 70]: 0.5\n Train loss:0.5103580951690674\n \n train accuracy [batch 80]: 0.875\n Train loss:0.2092967927455902\n \n train accuracy [batch 90]: 0.25\n Train loss:0.3581005930900574\n \n train accuracy [batch 99]: 0.625\n Train loss:0.2962380349636078\n \n train accuracy [batch 100]: 0.375\n Train loss:0.45387619733810425\n \n train accuracy [batch 110]: 0.5\n Train loss:0.3117087781429291\n \n train accuracy [batch 120]: 0.5\n Train loss:0.24584147334098816\n \n train accuracy [batch 130]: 0.625\n Train loss:0.4087778627872467\n \n train accuracy [batch 140]: 0.5\n Train loss:0.3685236871242523\n \n train accuracy [batch 150]: 0.5\n Train loss:0.3082018494606018\n \n train accuracy [batch 160]: 0.75\n Train loss:0.27456679940223694\n \n train accuracy [batch 170]: 0.375\n Train loss:0.35793745517730713\n \n train accuracy [batch 180]: 0.375\n Train loss:0.456869900226593\n \n train accuracy [batch 190]: 0.625\n Train loss:0.28419938683509827\n \n train accuracy [batch 200]: 0.625\n Train loss:0.2701646089553833\n \n train accuracy [batch 210]: 0.875\n Train loss:0.24805375933647156\n \n train accuracy [batch 220]: 0.75\n Train loss:0.22128085792064667\n \n train accuracy [batch 230]: 0.625\n Train loss:0.25438639521598816\n \n train accuracy [batch 240]: 0.625\n Train loss:0.2738930881023407\n \n train accuracy [batch 250]: 0.625\n Train loss:0.21997350454330444\n \n train accuracy [batch 260]: 0.5\n Train loss:0.28495484590530396\n \n train accuracy [batch 270]: 0.75\n Train loss:0.2025119960308075\n \n train accuracy [batch 280]: 0.5\n Train loss:0.35066333413124084\n \n train accuracy [batch 290]: 0.375\n Train loss:0.42032527923583984\n \n train accuracy [batch 300]: 0.625\n Train loss:0.30019789934158325\n \n train accuracy [batch 310]: 0.5\n Train loss:0.28026774525642395\n \n train accuracy [batch 320]: 0.25\n Train loss:0.39849853515625\n \n train accuracy [batch 330]: 0.375\n Train loss:0.3418615758419037\n \n train accuracy [batch 340]: 0.625\n Train loss:0.4191482961177826\n \n train accuracy [batch 350]: 0.375\n Train loss:0.43522709608078003\n \n train accuracy [batch 360]: 0.25\n Train loss:0.3068813681602478\n \n train accuracy [batch 370]: 0.875\n Train loss:0.20144468545913696\n \n train accuracy [batch 380]: 0.75\n Train loss:0.3681528568267822\n \n train accuracy [batch 390]: 0.75\n Train loss:0.2626267373561859\n \n train accuracy [batch 400]: 0.625\n Train loss:0.2844865322113037\n \n train accuracy [batch 410]: 0.375\n Train loss:0.27677732706069946\n \n train accuracy [batch 420]: 0.625\n Train loss:0.27561578154563904\n \n train accuracy [batch 430]: 0.375\n Train loss:0.41790035367012024\n \n train accuracy [batch 440]: 0.625\n Train loss:0.29906609654426575\n \n train accuracy [batch 450]: 0.75\n Train loss:0.3039526343345642\n \n train accuracy [batch 460]: 0.75\n Train loss:0.3176349997520447\n \n train accuracy [batch 470]: 0.5\n Train loss:0.27933916449546814\n \n train accuracy [batch 480]: 0.375\n Train loss:0.34871789813041687\n \n train accuracy [batch 490]: 0.625\n Train loss:0.3364467918872833\n \n train accuracy [batch 500]: 0.375\n Train loss:0.3233523368835449\n \n train accuracy [batch 510]: 0.875\n Train loss:0.21484608948230743\n \n train accuracy [batch 520]: 0.5\n Train loss:0.3602026104927063\n \n train accuracy [batch 530]: 0.375\n Train loss:0.3080466389656067\n \n train accuracy [batch 540]: 0.875\n Train loss:0.18648548424243927\n \n train accuracy [batch 550]: 0.5\n Train loss:0.3541390597820282\n \n train accuracy [batch 560]: 0.375\n Train loss:0.3873644173145294\n \n train accuracy [batch 570]: 0.75\n Train loss:0.3770657777786255\n \n train accuracy [batch 580]: 0.625\n Train loss:0.3363107144832611\n \n train accuracy [batch 590]: 0.375\n Train loss:0.30930468440055847\n \n train accuracy [batch 600]: 0.75\n Train loss:0.23976346850395203\n \n train accuracy [batch 610]: 0.5\n Train loss:0.32667776942253113\n \n train accuracy [batch 620]: 0.75\n Train loss:0.26391318440437317\n \n train accuracy [batch 630]: 0.5\n Train loss:0.3432902693748474\n \n train accuracy [batch 640]: 0.375\n Train loss:0.3769707679748535\n \n train accuracy [batch 650]: 0.625\n Train loss:0.3419474959373474\n \n train accuracy [batch 660]: 0.75\n Train loss:0.2903408110141754\n \n train accuracy [batch 670]: 0.5\n Train loss:0.3938432037830353\n \n train accuracy [batch 680]: 0.75\n Train loss:0.22902695834636688\n \n train accuracy [batch 690]: 0.875\n Train loss:0.2416389435529709\n \n train accuracy [batch 700]: 0.375\n Train loss:0.392365038394928\n \n train accuracy [batch 710]: 0.75\n Train loss:0.2579198181629181\n \n train accuracy [batch 720]: 0.875\n Train loss:0.19818130135536194\n \n train accuracy [batch 730]: 0.625\n Train loss:0.2502315044403076\n \n train accuracy [batch 740]: 0.625\n Train loss:0.4022776186466217\n \n train accuracy [batch 750]: 0.5\n Train loss:0.40264248847961426\n \n train accuracy [batch 760]: 0.5\n Train loss:0.39997991919517517\n \n train accuracy [batch 770]: 0.75\n Train loss:0.24252311885356903\n \n train accuracy [batch 780]: 0.25\n Train loss:0.3961404860019684\n \n train accuracy [batch 790]: 0.625\n Train loss:0.35996294021606445\n \n train accuracy [batch 800]: 0.5\n Train loss:0.33650514483451843\n \n train accuracy [batch 810]: 0.75\n Train loss:0.19573794305324554\n \n train accuracy [batch 820]: 0.375\n Train loss:0.3242988586425781\n \n train accuracy [batch 830]: 0.625\n Train loss:0.28145354986190796\n \n train accuracy [batch 840]: 0.5\n Train loss:0.34811508655548096\n \n train accuracy [batch 850]: 0.875\n Train loss:0.24334490299224854\n \n train accuracy [batch 860]: 0.75\n Train loss:0.27299195528030396\n \n train accuracy [batch 870]: 0.625\n Train loss:0.29947999119758606\n \n train accuracy [batch 880]: 0.625\n Train loss:0.2732141613960266\n \n train accuracy [batch 890]: 0.5\n Train loss:0.2902957499027252\n \n train accuracy [batch 900]: 0.25\n Train loss:0.4077756702899933\n \n train accuracy [batch 910]: 0.625\n Train loss:0.30798065662384033\n \n train accuracy [batch 920]: 0.625\n Train loss:0.3299465477466583\n \n train accuracy [batch 930]: 0.75\n Train loss:0.2978147566318512\n \n train accuracy [batch 940]: 0.875\n Train loss:0.23485146462917328\n \n train accuracy [batch 950]: 0.5\n Train loss:0.27012646198272705\n \n train accuracy [batch 960]: 0.75\n Train loss:0.21245306730270386\n \n train accuracy [batch 970]: 0.875\n Train loss:0.1920769214630127\n \n train accuracy [batch 980]: 0.5\n Train loss:0.36695992946624756\n \n train accuracy [batch 990]: 0.75\n Train loss:0.25608330965042114\n \n train accuracy [batch 1000]: 0.5\n Train loss:0.31384149193763733\n \n train accuracy [batch 1010]: 0.375\n Train loss:0.28369322419166565\n \n train accuracy [batch 1020]: 0.75\n Train loss:0.2694675326347351\n \n train accuracy [batch 1030]: 0.5\n Train loss:0.35276177525520325\n \n train accuracy [batch 1040]: 0.625\n Train loss:0.41912832856178284\n \n train accuracy [batch 1050]: 0.125\n Train loss:0.5802091360092163\n \n train accuracy [batch 1060]: 0.5\n Train loss:0.30287423729896545\n \n train accuracy [batch 1070]: 0.375\n Train loss:0.3688276410102844\n \n train accuracy [batch 1080]: 0.625\n Train loss:0.2742982506752014\n \n train accuracy [batch 1090]: 0.375\n Train loss:0.46933409571647644\n \n train accuracy [batch 1100]: 0.625\n Train loss:0.35423707962036133\n \n train accuracy [batch 1110]: 0.75\n Train loss:0.2606743276119232\n \n train accuracy [batch 1120]: 0.75\n Train loss:0.25097745656967163\n \n train accuracy [batch 1130]: 0.625\n Train loss:0.3753395676612854\n \n train accuracy [batch 1140]: 0.625\n Train loss:0.33310821652412415\n \n train accuracy [batch 1150]: 0.375\n Train loss:0.38652393221855164\n \n train accuracy [batch 1160]: 0.625\n Train loss:0.2716743052005768\n \n train accuracy [batch 1170]: 0.75\n Train loss:0.3554810881614685\n \n train accuracy [batch 1180]: 0.75\n Train loss:0.2680845856666565\n \n train accuracy [batch 1190]: 0.375\n Train loss:0.36345821619033813\n \n train accuracy [batch 1200]: 0.375\n Train loss:0.29094114899635315\n \n train accuracy [batch 1210]: 0.375\n Train loss:0.36037948727607727\n \n train accuracy [batch 1220]: 0.75\n Train loss:0.2269488126039505\n \n train accuracy [batch 1230]: 0.625\n Train loss:0.3188307583332062\n \n train accuracy [batch 1240]: 0.5\n Train loss:0.3136765956878662\n \n train accuracy [batch 1250]: 0.5\n Train loss:0.17316696047782898\n \n train accuracy [batch 1260]: 0.375\n Train loss:0.4376659393310547\n \n train accuracy [batch 1270]: 0.5\n Train loss:0.40528810024261475\n \n train accuracy [batch 1280]: 0.625\n Train loss:0.3518957793712616\n \n train accuracy [batch 1290]: 0.5\n Train loss:0.35087117552757263\n \n train accuracy [batch 1300]: 0.375\n Train loss:0.37797075510025024\n \n train accuracy [batch 1310]: 0.375\n Train loss:0.3883957862854004\n \n train accuracy [batch 1320]: 0.625\n Train loss:0.34037408232688904\n \n train accuracy [batch 1330]: 0.25\n Train loss:0.38131818175315857\n \n train accuracy [batch 1340]: 0.25\n Train loss:0.37814074754714966\n \n train accuracy [batch 1350]: 0.375\n Train loss:0.36812624335289\n \n train accuracy [batch 1360]: 0.75\n Train loss:0.2254808098077774\n \n train accuracy [batch 1370]: 0.375\n Train loss:0.3604638874530792\n \n train accuracy [batch 1380]: 0.5\n Train loss:0.3031318187713623\n \n train accuracy [batch 1390]: 0.875\n Train loss:0.199925497174263\n \n train accuracy [batch 1400]: 0.875\n Train loss:0.19246995449066162\n \n train accuracy [batch 1410]: 0.75\n Train loss:0.21276119351387024\n \n train accuracy [batch 1420]: 0.75\n Train loss:0.28103107213974\n \n train accuracy [batch 1430]: 0.625\n Train loss:0.2369115799665451\n \n train accuracy [batch 1440]: 0.75\n Train loss:0.22428740561008453\n \n train accuracy [batch 1450]: 0.375\n Train loss:0.3785833716392517\n \n train accuracy [batch 1460]: 0.625\n Train loss:0.3029880225658417\n \n train accuracy [batch 1470]: 0.5\n Train loss:0.3803355097770691\n \n train accuracy [batch 1480]: 0.375\n Train loss:0.37404772639274597\n \n train accuracy [batch 1490]: 0.5\n Train loss:0.38900116086006165\n \n train accuracy [batch 1500]: 0.5\n Train loss:0.26412731409072876\n \n train accuracy [batch 1510]: 0.625\n Train loss:0.28007644414901733\n \n train accuracy 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Train loss:0.30144554376602173\n \n Total train loss tensor(0.3061, device='cuda:0')\n Total train accuracy 0.5801\n Total time for training an epoch: 1924\n test accuracy [batch 0]: 0.625\n test accuracy [batch 25]: 0.375\n test accuracy [batch 50]: 0.625\n test accuracy [batch 75]: 0.75\n test accuracy [batch 99]: 0.5\n test accuracy [batch 100]: 0.625\n test accuracy [batch 125]: 0.625\n test accuracy [batch 150]: 0.5\n test accuracy [batch 175]: 0.375\n test accuracy [batch 200]: 0.75\n test accuracy [batch 225]: 0.5\n test accuracy [batch 250]: 0.625\n test accuracy [batch 275]: 0.625\n test accuracy [batch 300]: 0.625\n test accuracy [batch 325]: 0.625\n test accuracy [batch 350]: 0.625\n test accuracy [batch 375]: 0.5\n test accuracy [batch 400]: 0.375\n test accuracy [batch 425]: 0.5\n test accuracy [batch 450]: 0.375\n test accuracy [batch 475]: 0.625\n test accuracy [batch 500]: 0.75\n test accuracy [batch 525]: 0.875\n test accuracy [batch 550]: 0.75\n test accuracy [batch 575]: 0.75\n test accuracy [batch 600]: 0.5\n test accuracy [batch 625]: 0.625\n test accuracy [batch 650]: 0.625\n test accuracy [batch 675]: 0.375\n test accuracy [batch 700]: 0.875\n test accuracy [batch 725]: 0.5\n test accuracy [batch 750]: 0.875\n test accuracy [batch 775]: 0.625\n test accuracy [batch 800]: 0.625\n test accuracy [batch 825]: 0.375\n test accuracy [batch 850]: 0.75\n test accuracy [batch 875]: 0.75\n test accuracy [batch 900]: 0.625\n test accuracy [batch 925]: 0.75\n test accuracy [batch 950]: 0.5\n test accuracy [batch 975]: 0.375\n test accuracy [batch 1000]: 0.25\n test accuracy [batch 1025]: 0.375\n test accuracy [batch 1050]: 0.625\n test accuracy [batch 1075]: 0.5\n test accuracy [batch 1100]: 0.625\n test accuracy [batch 1125]: 0.625\n test accuracy [batch 1150]: 0.625\n test accuracy [batch 1175]: 0.5\n test accuracy [batch 1200]: 0.625\n test accuracy [batch 1225]: 0.5\n Total test loss tensor(0.2964, device='cuda:0')\n Total test accuracy 0.5906\n 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0.625\n Train loss:0.346942275762558\n \n train accuracy [batch 6120]: 0.375\n Train loss:0.30234137177467346\n \n train accuracy [batch 6130]: 0.625\n Train loss:0.21421201527118683\n \n train accuracy [batch 6140]: 0.625\n Train loss:0.1932411789894104\n \n train accuracy [batch 6150]: 0.875\n Train loss:0.19806978106498718\n \n train accuracy [batch 6160]: 0.625\n Train loss:0.2829047739505768\n \n train accuracy [batch 6170]: 0.5\n Train loss:0.28501737117767334\n \n train accuracy [batch 6180]: 0.75\n Train loss:0.1826426386833191\n \n train accuracy [batch 6190]: 0.5\n Train loss:0.2669743001461029\n \n train accuracy [batch 6200]: 0.25\n Train loss:0.4461513161659241\n \n train accuracy [batch 6210]: 0.875\n Train loss:0.23501677811145782\n \n train accuracy [batch 6220]: 0.75\n Train loss:0.20516276359558105\n \n train accuracy [batch 6230]: 0.75\n Train loss:0.20258349180221558\n \n train accuracy [batch 6240]: 0.75\n Train loss:0.22129452228546143\n \n Total train loss tensor(0.2771, device='cuda:0')\n Total train accuracy 0.6224\n Total time for training an epoch: 1923\n test accuracy [batch 0]: 0.75\n test accuracy [batch 25]: 0.375\n test accuracy [batch 50]: 0.5\n test accuracy [batch 75]: 0.875\n test accuracy [batch 99]: 0.5\n test accuracy [batch 100]: 0.875\n test accuracy [batch 125]: 0.75\n test accuracy [batch 150]: 0.75\n test accuracy [batch 175]: 0.25\n test accuracy [batch 200]: 0.75\n test accuracy [batch 225]: 0.875\n test accuracy [batch 250]: 0.75\n test accuracy [batch 275]: 0.875\n test accuracy [batch 300]: 0.5\n test accuracy [batch 325]: 0.5\n test accuracy [batch 350]: 0.875\n test accuracy [batch 375]: 0.625\n test accuracy [batch 400]: 0.875\n test accuracy [batch 425]: 0.75\n test accuracy [batch 450]: 0.375\n test accuracy [batch 475]: 0.5\n test accuracy [batch 500]: 0.5\n test accuracy [batch 525]: 0.625\n test accuracy [batch 550]: 0.625\n test accuracy [batch 575]: 0.625\n test accuracy [batch 600]: 0.5\n test accuracy [batch 625]: 0.625\n test accuracy [batch 650]: 0.75\n test accuracy [batch 675]: 0.75\n test accuracy [batch 700]: 0.5\n test accuracy [batch 725]: 0.5\n test accuracy [batch 750]: 0.375\n test accuracy [batch 775]: 0.875\n test accuracy [batch 800]: 0.75\n test accuracy [batch 825]: 0.625\n test accuracy [batch 850]: 0.375\n test accuracy [batch 875]: 0.625\n test accuracy [batch 900]: 1.0\n test accuracy [batch 925]: 0.75\n test accuracy [batch 950]: 0.625\n test accuracy [batch 975]: 0.5\n test accuracy [batch 1000]: 0.625\n test accuracy [batch 1025]: 0.875\n test accuracy [batch 1050]: 0.75\n test accuracy [batch 1075]: 0.5\n test accuracy [batch 1100]: 0.375\n test accuracy [batch 1125]: 0.625\n test accuracy [batch 1150]: 0.625\n test accuracy [batch 1175]: 0.875\n test accuracy [batch 1200]: 0.75\n test accuracy [batch 1225]: 0.625\n Total test loss tensor(0.2710, device='cuda:0')\n Total test accuracy 0.6265\n train accuracy [batch 0]: 0.625\n Train loss:0.3005981147289276\n \n train accuracy [batch 10]: 0.75\n Train loss:0.20637445151805878\n \n train accuracy [batch 20]: 0.5\n Train loss:0.35502853989601135\n \n train accuracy [batch 30]: 0.75\n Train loss:0.23378358781337738\n \n train accuracy [batch 40]: 0.75\n Train loss:0.22386406362056732\n \n train accuracy [batch 50]: 0.875\n Train loss:0.16856589913368225\n \n train accuracy [batch 60]: 0.75\n Train loss:0.33903855085372925\n \n train accuracy [batch 70]: 0.625\n Train loss:0.2664778232574463\n \n train accuracy [batch 80]: 0.625\n Train loss:0.1525537222623825\n \n train accuracy [batch 90]: 0.625\n Train loss:0.3325689136981964\n \n train accuracy [batch 99]: 0.5\n Train loss:0.2661738097667694\n \n train accuracy [batch 100]: 0.5\n Train loss:0.2856035530567169\n \n train accuracy [batch 110]: 0.5\n Train loss:0.3319566249847412\n \n train accuracy [batch 120]: 0.5\n Train loss:0.3282714784145355\n \n train accuracy [batch 130]: 0.75\n Train loss:0.20729497075080872\n \n train accuracy [batch 140]: 0.5\n Train loss:0.35374733805656433\n \n train accuracy [batch 150]: 0.375\n Train loss:0.35978609323501587\n \n train accuracy [batch 160]: 0.75\n Train loss:0.21894784271717072\n \n train accuracy [batch 170]: 0.875\n Train loss:0.18460573256015778\n \n train accuracy [batch 180]: 0.5\n Train loss:0.3361012637615204\n \n train accuracy [batch 190]: 0.625\n Train loss:0.279468834400177\n \n train accuracy [batch 200]: 0.5\n Train loss:0.4201899766921997\n \n train accuracy [batch 210]: 0.875\n Train loss:0.12063197791576385\n \n train accuracy [batch 220]: 1.0\n Train loss:0.23107749223709106\n \n train accuracy [batch 230]: 0.625\n Train loss:0.3311820924282074\n \n train accuracy [batch 240]: 0.875\n Train loss:0.23892205953598022\n \n train accuracy [batch 250]: 1.0\n Train loss:0.11629598587751389\n \n train accuracy [batch 260]: 0.875\n Train loss:0.17000387609004974\n \n train accuracy [batch 270]: 0.625\n Train loss:0.28185462951660156\n \n train accuracy [batch 280]: 0.875\n Train loss:0.21023783087730408\n \n train accuracy [batch 290]: 0.375\n Train loss:0.4303801953792572\n \n train accuracy [batch 300]: 0.625\n Train loss:0.2677255868911743\n \n train accuracy [batch 310]: 0.5\n Train loss:0.3666248619556427\n \n train accuracy [batch 320]: 0.625\n Train loss:0.21638591587543488\n \n train accuracy [batch 330]: 0.5\n Train loss:0.2794554829597473\n \n train accuracy [batch 340]: 0.875\n Train loss:0.2524922788143158\n \n train accuracy [batch 350]: 0.5\n Train loss:0.35941460728645325\n \n train accuracy [batch 360]: 0.5\n Train loss:0.28441187739372253\n \n train accuracy [batch 370]: 0.75\n Train loss:0.2860966622829437\n \n train accuracy [batch 380]: 0.75\n Train loss:0.18357889354228973\n \n train accuracy [batch 390]: 0.75\n Train loss:0.2797137498855591\n \n train accuracy [batch 400]: 0.875\n Train loss:0.21824193000793457\n \n train accuracy [batch 410]: 0.625\n Train loss:0.22499339282512665\n \n train accuracy [batch 420]: 0.625\n Train loss:0.2459859549999237\n \n train accuracy [batch 430]: 0.5\n Train loss:0.33415019512176514\n \n train accuracy [batch 440]: 0.75\n Train loss:0.25296545028686523\n \n train accuracy [batch 450]: 0.75\n Train loss:0.22271980345249176\n \n train accuracy [batch 460]: 0.75\n Train loss:0.18293055891990662\n \n train accuracy [batch 470]: 0.625\n Train loss:0.2730482220649719\n \n train accuracy [batch 480]: 0.75\n Train loss:0.25706785917282104\n \n train accuracy [batch 490]: 0.375\n Train loss:0.31653136014938354\n \n train accuracy [batch 500]: 0.625\n Train loss:0.2370682954788208\n \n train accuracy [batch 510]: 0.625\n Train loss:0.233169287443161\n \n train accuracy [batch 520]: 0.625\n Train loss:0.2064540535211563\n \n train accuracy [batch 530]: 0.5\n Train loss:0.2457168847322464\n \n train accuracy [batch 540]: 0.875\n Train loss:0.23437722027301788\n \n train accuracy [batch 550]: 0.25\n Train 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loss:0.17595402896404266\n \n train accuracy [batch 700]: 0.75\n Train loss:0.18778693675994873\n \n train accuracy [batch 710]: 0.5\n Train loss:0.4212041199207306\n \n train accuracy [batch 720]: 1.0\n Train loss:0.08350937813520432\n \n train accuracy [batch 730]: 0.625\n Train loss:0.293171763420105\n \n train accuracy [batch 740]: 0.625\n Train loss:0.2106294482946396\n \n train accuracy [batch 750]: 0.625\n Train loss:0.3049262762069702\n \n train accuracy [batch 760]: 0.75\n Train loss:0.1501106470823288\n \n train accuracy [batch 770]: 0.75\n Train loss:0.2677222490310669\n \n train accuracy [batch 780]: 0.75\n Train loss:0.2825045585632324\n \n train accuracy [batch 790]: 0.375\n Train loss:0.452310174703598\n \n train accuracy [batch 800]: 0.5\n Train loss:0.37337252497673035\n \n train accuracy [batch 810]: 0.5\n Train loss:0.35063257813453674\n \n train accuracy [batch 820]: 0.5\n Train loss:0.39701586961746216\n \n train accuracy [batch 830]: 0.75\n Train 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Train loss:0.22325441241264343\n \n train accuracy [batch 5140]: 0.75\n Train loss:0.21965761482715607\n \n train accuracy [batch 5150]: 0.625\n Train loss:0.34656333923339844\n \n train accuracy [batch 5160]: 0.875\n Train loss:0.23684890568256378\n \n train accuracy [batch 5170]: 0.5\n Train loss:0.3483738899230957\n \n train accuracy [batch 5180]: 0.625\n Train loss:0.2874879837036133\n \n train accuracy [batch 5190]: 0.875\n Train loss:0.10905739665031433\n \n train accuracy [batch 5200]: 0.75\n Train loss:0.24058939516544342\n \n train accuracy [batch 5210]: 0.25\n Train loss:0.3438631296157837\n \n train accuracy [batch 5220]: 0.625\n Train loss:0.2687462568283081\n \n train accuracy [batch 5230]: 0.875\n Train loss:0.219723641872406\n \n train accuracy [batch 5240]: 0.75\n Train loss:0.19102095067501068\n \n train accuracy [batch 5250]: 0.75\n Train loss:0.22995297610759735\n \n train accuracy [batch 5260]: 0.75\n Train loss:0.17057062685489655\n \n train accuracy [batch 5270]: 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5410]: 0.875\n Train loss:0.19173382222652435\n \n train accuracy [batch 5420]: 0.875\n Train loss:0.2084059864282608\n \n train accuracy [batch 5430]: 0.875\n Train loss:0.16327621042728424\n \n train accuracy [batch 5440]: 0.75\n Train loss:0.2196284979581833\n \n train accuracy [batch 5450]: 0.625\n Train loss:0.23250775039196014\n \n train accuracy [batch 5460]: 0.625\n Train loss:0.22014006972312927\n \n train accuracy [batch 5470]: 0.625\n Train loss:0.33843347430229187\n \n train accuracy [batch 5480]: 0.375\n Train loss:0.4169428050518036\n \n train accuracy [batch 5490]: 0.75\n Train loss:0.24862976372241974\n \n train accuracy [batch 5500]: 0.625\n Train loss:0.27877673506736755\n \n train accuracy [batch 5510]: 0.625\n Train loss:0.2788218855857849\n \n train accuracy [batch 5520]: 0.625\n Train loss:0.3683173358440399\n \n train accuracy [batch 5530]: 0.75\n Train loss:0.2461317628622055\n \n train accuracy [batch 5540]: 0.875\n Train loss:0.12661212682724\n \n train accuracy [batch 5550]: 0.625\n Train loss:0.29230520129203796\n \n train accuracy [batch 5560]: 0.375\n Train loss:0.5075621604919434\n \n train accuracy [batch 5570]: 0.375\n Train loss:0.3223779499530792\n \n train accuracy [batch 5580]: 1.0\n Train loss:0.20726576447486877\n \n train accuracy [batch 5590]: 0.75\n Train loss:0.29706865549087524\n \n train accuracy [batch 5600]: 0.75\n Train loss:0.1799827665090561\n \n train accuracy [batch 5610]: 0.75\n Train loss:0.27623793482780457\n \n train accuracy [batch 5620]: 0.375\n Train loss:0.35575559735298157\n \n train accuracy [batch 5630]: 0.75\n Train loss:0.11526127904653549\n \n train accuracy [batch 5640]: 0.625\n Train loss:0.3524117171764374\n \n train accuracy [batch 5650]: 0.75\n Train loss:0.17760053277015686\n \n train accuracy [batch 5660]: 0.875\n Train loss:0.2302509993314743\n \n train accuracy [batch 5670]: 0.625\n Train loss:0.3611142635345459\n \n train accuracy [batch 5680]: 0.875\n Train loss:0.09404448419809341\n \n train accuracy [batch 5690]: 0.625\n Train loss:0.23340293765068054\n \n train accuracy [batch 5700]: 1.0\n Train loss:0.15751366317272186\n \n train accuracy [batch 5710]: 0.75\n Train loss:0.24035745859146118\n \n train accuracy [batch 5720]: 0.625\n Train loss:0.3132080137729645\n \n train accuracy [batch 5730]: 0.625\n Train loss:0.2487853467464447\n \n train accuracy [batch 5740]: 0.75\n Train loss:0.25853079557418823\n \n train accuracy [batch 5750]: 1.0\n Train loss:0.11772987991571426\n \n train accuracy [batch 5760]: 0.875\n Train loss:0.1493472456932068\n \n train accuracy [batch 5770]: 0.75\n Train loss:0.22357672452926636\n \n train accuracy [batch 5780]: 0.75\n Train loss:0.20110644400119781\n \n train accuracy [batch 5790]: 0.75\n Train loss:0.23017370700836182\n \n train accuracy [batch 5800]: 1.0\n Train loss:0.15060952305793762\n \n train accuracy [batch 5810]: 0.75\n Train loss:0.25380152463912964\n \n train accuracy [batch 5820]: 0.625\n Train 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Train loss:0.13912256062030792\n \n train accuracy [batch 5970]: 0.875\n Train loss:0.14650161564350128\n \n train accuracy [batch 5980]: 0.75\n Train loss:0.213309183716774\n \n train accuracy [batch 5990]: 0.875\n Train loss:0.21924848854541779\n \n train accuracy [batch 6000]: 0.5\n Train loss:0.28526735305786133\n \n train accuracy [batch 6010]: 0.625\n Train loss:0.38406670093536377\n \n train accuracy [batch 6020]: 0.5\n Train loss:0.32226628065109253\n \n train accuracy [batch 6030]: 0.5\n Train loss:0.2613353133201599\n \n train accuracy [batch 6040]: 0.625\n Train loss:0.20578421652317047\n \n train accuracy [batch 6050]: 0.625\n Train loss:0.2646269202232361\n \n train accuracy [batch 6060]: 0.75\n Train loss:0.2303260713815689\n \n train accuracy [batch 6070]: 0.75\n Train loss:0.17271631956100464\n \n train accuracy [batch 6080]: 0.875\n Train loss:0.15978410840034485\n \n train accuracy [batch 6090]: 0.625\n Train loss:0.2796916961669922\n \n train accuracy [batch 6100]: 0.25\n Train loss:0.4931305944919586\n \n train accuracy [batch 6110]: 0.625\n Train loss:0.2700378894805908\n \n train accuracy [batch 6120]: 0.875\n Train loss:0.2472865730524063\n \n train accuracy [batch 6130]: 0.5\n Train loss:0.36011308431625366\n \n train accuracy [batch 6140]: 0.625\n Train loss:0.36410805583000183\n \n train accuracy [batch 6150]: 0.625\n Train loss:0.3245472311973572\n \n train accuracy [batch 6160]: 0.75\n Train loss:0.22623655200004578\n \n train accuracy [batch 6170]: 1.0\n Train loss:0.1308673620223999\n \n train accuracy [batch 6180]: 0.75\n Train loss:0.2410961538553238\n \n train accuracy [batch 6190]: 0.625\n Train loss:0.18547740578651428\n \n train accuracy [batch 6200]: 0.75\n Train loss:0.22186173498630524\n \n train accuracy [batch 6210]: 0.5\n Train loss:0.3835524916648865\n \n train accuracy [batch 6220]: 0.5\n Train loss:0.26176708936691284\n \n train accuracy [batch 6230]: 0.75\n Train loss:0.21424700319766998\n \n train accuracy [batch 6240]: 0.375\n Train loss:0.3731403648853302\n \n Total train loss tensor(0.2577, device='cuda:0')\n Total train accuracy 0.653\n Total time for training an epoch: 1926\n test accuracy [batch 0]: 0.875\n test accuracy [batch 25]: 0.875\n test accuracy [batch 50]: 0.625\n test accuracy [batch 75]: 0.75\n test accuracy [batch 99]: 0.375\n test accuracy [batch 100]: 0.625\n test accuracy [batch 125]: 0.375\n test accuracy [batch 150]: 0.875\n test accuracy [batch 175]: 0.5\n test accuracy [batch 200]: 0.625\n test accuracy [batch 225]: 0.75\n test accuracy [batch 250]: 0.5\n test accuracy [batch 275]: 0.75\n test accuracy [batch 300]: 0.625\n test accuracy [batch 325]: 0.625\n test accuracy [batch 350]: 0.625\n test accuracy [batch 375]: 0.375\n test accuracy [batch 400]: 0.375\n test accuracy [batch 425]: 0.5\n test accuracy [batch 450]: 0.625\n test accuracy [batch 475]: 0.375\n test accuracy [batch 500]: 0.375\n test accuracy [batch 525]: 0.875\n test accuracy [batch 550]: 0.625\n test accuracy [batch 575]: 0.75\n test accuracy [batch 600]: 0.75\n test accuracy [batch 625]: 0.375\n test accuracy [batch 650]: 1.0\n test accuracy [batch 675]: 0.5\n test accuracy [batch 700]: 0.375\n test accuracy [batch 725]: 0.75\n test accuracy [batch 750]: 0.625\n test accuracy [batch 775]: 1.0\n test accuracy [batch 800]: 0.75\n test accuracy [batch 825]: 0.5\n test accuracy [batch 850]: 0.5\n test accuracy [batch 875]: 0.875\n test accuracy [batch 900]: 0.625\n test accuracy [batch 925]: 0.75\n test accuracy [batch 950]: 0.75\n test accuracy [batch 975]: 0.875\n test accuracy [batch 1000]: 0.5\n test accuracy [batch 1025]: 0.375\n test accuracy [batch 1050]: 0.5\n test accuracy [batch 1075]: 0.75\n test accuracy [batch 1100]: 0.875\n test accuracy [batch 1125]: 0.5\n test accuracy [batch 1150]: 0.25\n test accuracy [batch 1175]: 0.75\n test accuracy [batch 1200]: 0.75\n test accuracy [batch 1225]: 0.625\n Total test loss tensor(0.2645, device='cuda:0')\n Total test accuracy 0.6384\n train accuracy [batch 0]: 0.625\n Train loss:0.3546221852302551\n \n train accuracy [batch 10]: 0.625\n Train loss:0.28719407320022583\n \n train accuracy [batch 20]: 0.625\n Train loss:0.2495260238647461\n \n train accuracy [batch 30]: 0.75\n Train loss:0.2010570615530014\n \n train accuracy [batch 40]: 0.625\n Train loss:0.16678905487060547\n \n train accuracy [batch 50]: 0.875\n Train loss:0.20412378013134003\n \n train accuracy [batch 60]: 0.625\n Train loss:0.2676551640033722\n \n train accuracy [batch 70]: 0.375\n Train loss:0.3213568329811096\n \n train accuracy [batch 80]: 0.5\n Train loss:0.2960965931415558\n \n train accuracy [batch 90]: 0.625\n Train loss:0.26752394437789917\n \n train accuracy [batch 99]: 0.5\n Train loss:0.2967164218425751\n \n train accuracy [batch 100]: 0.625\n Train loss:0.3532872498035431\n \n train accuracy [batch 110]: 0.5\n Train loss:0.23537392914295197\n \n train accuracy [batch 120]: 0.625\n Train loss:0.23355765640735626\n \n train accuracy [batch 130]: 0.875\n Train loss:0.18960945308208466\n \n train accuracy [batch 140]: 0.625\n Train loss:0.2907230854034424\n \n train accuracy [batch 150]: 0.625\n Train loss:0.3114067316055298\n \n train accuracy [batch 160]: 0.625\n Train loss:0.2879076302051544\n \n train accuracy [batch 170]: 0.75\n Train loss:0.20882320404052734\n \n train accuracy [batch 180]: 0.625\n Train loss:0.2858717143535614\n \n train accuracy [batch 190]: 0.625\n Train loss:0.24521897733211517\n \n train accuracy [batch 200]: 0.875\n Train loss:0.23874910175800323\n \n train accuracy [batch 210]: 0.75\n Train loss:0.26370173692703247\n \n train accuracy [batch 220]: 1.0\n Train loss:0.09792857617139816\n \n train accuracy [batch 230]: 0.75\n Train loss:0.1968042403459549\n \n train accuracy [batch 240]: 0.625\n Train loss:0.28378456830978394\n \n train accuracy [batch 250]: 0.5\n Train loss:0.33257049322128296\n \n train accuracy [batch 260]: 0.75\n Train loss:0.23279695212841034\n \n train accuracy [batch 270]: 0.5\n Train loss:0.2828330397605896\n \n train accuracy [batch 280]: 0.625\n Train loss:0.24596761167049408\n \n train accuracy [batch 290]: 0.625\n Train loss:0.21631589531898499\n \n train accuracy [batch 300]: 0.625\n Train loss:0.3333890438079834\n \n train accuracy [batch 310]: 0.625\n Train loss:0.20025154948234558\n \n train accuracy [batch 320]: 0.625\n Train loss:0.26392343640327454\n \n train accuracy [batch 330]: 0.75\n Train loss:0.18936434388160706\n \n train accuracy [batch 340]: 0.625\n Train loss:0.2686009705066681\n \n train accuracy [batch 350]: 0.375\n Train loss:0.3425871431827545\n \n train accuracy [batch 360]: 0.875\n Train loss:0.11567188054323196\n \n train accuracy [batch 370]: 0.625\n Train loss:0.28372281789779663\n \n train accuracy [batch 380]: 0.375\n Train loss:0.3210271894931793\n \n train accuracy [batch 390]: 0.75\n Train loss:0.254921019077301\n \n train accuracy [batch 400]: 0.5\n Train loss:0.18595345318317413\n \n train accuracy [batch 410]: 0.5\n Train loss:0.28018659353256226\n \n train accuracy [batch 420]: 0.625\n Train loss:0.19634145498275757\n \n train accuracy [batch 430]: 0.5\n Train loss:0.2577197551727295\n \n train accuracy [batch 440]: 0.375\n Train loss:0.27107569575309753\n \n train accuracy [batch 450]: 0.875\n Train loss:0.15134331583976746\n \n train accuracy [batch 460]: 0.875\n Train loss:0.13931432366371155\n \n train accuracy [batch 470]: 0.875\n Train loss:0.13641878962516785\n \n train accuracy [batch 480]: 0.75\n Train loss:0.21931461989879608\n \n train accuracy [batch 490]: 0.625\n Train loss:0.3042968809604645\n \n train accuracy [batch 500]: 0.875\n Train loss:0.17744643986225128\n \n train accuracy [batch 510]: 1.0\n Train loss:0.21428492665290833\n \n train accuracy [batch 520]: 0.875\n Train loss:0.1825569123029709\n \n train accuracy [batch 530]: 0.625\n Train loss:0.17262002825737\n \n train accuracy [batch 540]: 0.75\n Train loss:0.2243228554725647\n \n train 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loss:0.322172075510025\n \n train accuracy [batch 6230]: 0.75\n Train loss:0.25454846024513245\n \n train accuracy [batch 6240]: 0.875\n Train loss:0.18068768084049225\n \n Total train loss tensor(0.2419, device='cuda:0')\n Total train accuracy 0.6761\n Total time for training an epoch: 1924\n test accuracy [batch 0]: 0.625\n test accuracy [batch 25]: 0.625\n test accuracy [batch 50]: 0.625\n test accuracy [batch 75]: 0.375\n test accuracy [batch 99]: 0.75\n test accuracy [batch 100]: 0.75\n test accuracy [batch 125]: 0.75\n test accuracy [batch 150]: 0.875\n test accuracy [batch 175]: 0.625\n test accuracy [batch 200]: 0.875\n test accuracy [batch 225]: 0.375\n test accuracy [batch 250]: 0.5\n test accuracy [batch 275]: 0.625\n test accuracy [batch 300]: 0.75\n test accuracy [batch 325]: 0.625\n test accuracy [batch 350]: 0.75\n test accuracy [batch 375]: 0.375\n test accuracy [batch 400]: 0.75\n test accuracy [batch 425]: 0.375\n test accuracy [batch 450]: 0.75\n test accuracy [batch 475]: 0.25\n test accuracy [batch 500]: 0.75\n test accuracy [batch 525]: 0.875\n test accuracy [batch 550]: 0.625\n test accuracy [batch 575]: 0.5\n test accuracy [batch 600]: 0.625\n test accuracy [batch 625]: 0.75\n test accuracy [batch 650]: 0.625\n test accuracy [batch 675]: 0.625\n test accuracy [batch 700]: 0.375\n test accuracy [batch 725]: 0.75\n test accuracy [batch 750]: 0.375\n test accuracy [batch 775]: 0.625\n test accuracy [batch 800]: 0.375\n test accuracy [batch 825]: 0.875\n test accuracy [batch 850]: 0.875\n test accuracy [batch 875]: 0.625\n test accuracy [batch 900]: 0.75\n test accuracy [batch 925]: 0.5\n test accuracy [batch 950]: 0.75\n test accuracy [batch 975]: 0.625\n test accuracy [batch 1000]: 0.5\n test accuracy [batch 1025]: 0.5\n test accuracy [batch 1050]: 0.625\n test accuracy [batch 1075]: 0.625\n test accuracy [batch 1100]: 0.75\n test accuracy [batch 1125]: 0.625\n test accuracy [batch 1150]: 1.0\n test accuracy [batch 1175]: 0.875\n test accuracy [batch 1200]: 0.75\n test accuracy [batch 1225]: 0.75\n Total test loss tensor(0.2522, device='cuda:0')\n Total test accuracy 0.6577\n train accuracy [batch 0]: 0.625\n Train loss:0.2715083956718445\n \n train accuracy [batch 10]: 0.75\n Train loss:0.21335981786251068\n \n train accuracy [batch 20]: 0.625\n Train loss:0.28606128692626953\n \n train accuracy [batch 30]: 0.875\n Train loss:0.20880332589149475\n \n train accuracy [batch 40]: 1.0\n Train loss:0.11379777640104294\n \n train accuracy [batch 50]: 0.875\n Train loss:0.19915175437927246\n \n train accuracy [batch 60]: 0.875\n Train loss:0.18792009353637695\n \n train accuracy [batch 70]: 0.75\n Train loss:0.18493203818798065\n \n train accuracy [batch 80]: 0.5\n Train loss:0.2936200797557831\n \n train accuracy [batch 90]: 0.75\n Train loss:0.1539417952299118\n \n train accuracy [batch 99]: 0.625\n Train loss:0.23578068614006042\n \n train accuracy [batch 100]: 0.625\n Train loss:0.3232159912586212\n \n train accuracy [batch 110]: 0.75\n Train loss:0.23732510209083557\n \n train accuracy [batch 120]: 0.75\n Train loss:0.19611471891403198\n \n train accuracy [batch 130]: 0.875\n Train loss:0.11603718996047974\n \n train accuracy [batch 140]: 0.625\n Train loss:0.15396635234355927\n \n train accuracy [batch 150]: 0.75\n Train loss:0.16583478450775146\n \n train accuracy [batch 160]: 0.75\n Train loss:0.2252095490694046\n \n train accuracy [batch 170]: 1.0\n Train loss:0.12730935215950012\n \n train accuracy [batch 180]: 0.75\n Train loss:0.24965083599090576\n \n train accuracy [batch 190]: 0.875\n Train loss:0.1255081444978714\n \n train accuracy [batch 200]: 0.75\n Train loss:0.23591633141040802\n \n train accuracy [batch 210]: 0.75\n Train loss:0.18210482597351074\n \n train accuracy [batch 220]: 0.875\n Train loss:0.1961209923028946\n \n train accuracy [batch 230]: 0.625\n Train loss:0.2213779091835022\n \n train accuracy [batch 240]: 0.625\n Train loss:0.21701647341251373\n \n train accuracy [batch 250]: 0.875\n Train loss:0.21938712894916534\n \n train accuracy [batch 260]: 0.625\n Train loss:0.33233174681663513\n \n train accuracy [batch 270]: 0.375\n Train loss:0.4112550616264343\n \n train accuracy [batch 280]: 0.25\n Train loss:0.3917049765586853\n \n train accuracy [batch 290]: 0.5\n Train loss:0.319137305021286\n \n train accuracy [batch 300]: 0.625\n Train loss:0.3292703926563263\n \n train accuracy [batch 310]: 0.5\n Train loss:0.25502803921699524\n \n train accuracy [batch 320]: 0.625\n Train loss:0.22084853053092957\n \n train accuracy [batch 330]: 0.875\n Train loss:0.1377730667591095\n \n train accuracy [batch 340]: 0.875\n Train loss:0.17081837356090546\n \n train accuracy [batch 350]: 0.875\n Train loss:0.12906523048877716\n \n train accuracy [batch 360]: 0.625\n Train loss:0.3099055886268616\n \n train accuracy [batch 370]: 0.5\n Train loss:0.24414370954036713\n \n train accuracy [batch 380]: 1.0\n Train loss:0.12359140068292618\n \n train accuracy [batch 390]: 0.875\n Train loss:0.23222941160202026\n \n train accuracy [batch 400]: 0.5\n Train loss:0.27194589376449585\n \n train accuracy [batch 410]: 0.375\n Train loss:0.3619565963745117\n \n train accuracy [batch 420]: 0.625\n Train loss:0.23759602010250092\n \n train accuracy [batch 430]: 0.5\n Train loss:0.4003154933452606\n \n train accuracy [batch 440]: 0.875\n Train loss:0.23253314197063446\n \n train accuracy [batch 450]: 0.875\n Train loss:0.21972310543060303\n \n train accuracy [batch 460]: 0.75\n Train loss:0.2747349143028259\n \n train accuracy [batch 470]: 0.375\n Train loss:0.3330291509628296\n \n train accuracy [batch 480]: 0.75\n Train loss:0.1795983463525772\n \n train accuracy [batch 490]: 0.75\n Train loss:0.18681764602661133\n \n train accuracy [batch 500]: 0.75\n Train loss:0.21338224411010742\n \n train accuracy [batch 510]: 0.625\n Train loss:0.22907480597496033\n \n train accuracy [batch 520]: 0.5\n Train loss:0.3175472915172577\n \n train accuracy [batch 530]: 0.875\n Train 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loss:0.22692930698394775\n \n train accuracy [batch 4970]: 0.625\n Train loss:0.3183352053165436\n \n train accuracy [batch 4980]: 0.875\n Train loss:0.12595482170581818\n \n train accuracy [batch 4990]: 0.375\n Train loss:0.33668041229248047\n \n train accuracy [batch 5000]: 0.875\n Train loss:0.09505360573530197\n \n train accuracy [batch 5010]: 0.75\n Train loss:0.2570771276950836\n \n train accuracy [batch 5020]: 0.625\n Train loss:0.3427368700504303\n \n train accuracy [batch 5030]: 0.625\n Train loss:0.2466358095407486\n \n train accuracy [batch 5040]: 0.75\n Train loss:0.2089557945728302\n \n train accuracy [batch 5050]: 0.75\n Train loss:0.20964358747005463\n \n train accuracy [batch 5060]: 0.5\n Train loss:0.22853337228298187\n \n train accuracy [batch 5070]: 0.75\n Train loss:0.2408680021762848\n \n train accuracy [batch 5080]: 0.875\n Train loss:0.1558910608291626\n \n train accuracy [batch 5090]: 0.75\n Train loss:0.20512604713439941\n \n train accuracy [batch 5100]: 0.5\n Train loss:0.29321345686912537\n \n train accuracy [batch 5110]: 1.0\n Train loss:0.09881318360567093\n \n train accuracy [batch 5120]: 0.625\n Train loss:0.22015494108200073\n \n train accuracy [batch 5130]: 0.75\n Train loss:0.2037847638130188\n \n train accuracy [batch 5140]: 0.875\n Train loss:0.10143602639436722\n \n train accuracy [batch 5150]: 0.625\n Train loss:0.20159511268138885\n \n train accuracy [batch 5160]: 0.75\n Train loss:0.19360314309597015\n \n train accuracy [batch 5170]: 0.75\n Train loss:0.22437292337417603\n \n train accuracy [batch 5180]: 0.625\n Train loss:0.26927822828292847\n \n train accuracy [batch 5190]: 0.625\n Train loss:0.2736431956291199\n \n train accuracy [batch 5200]: 0.375\n Train loss:0.40672567486763\n \n train accuracy [batch 5210]: 0.75\n Train loss:0.29113179445266724\n \n train accuracy [batch 5220]: 0.875\n Train loss:0.12431342899799347\n \n train accuracy [batch 5230]: 0.875\n Train loss:0.1559259444475174\n \n train accuracy [batch 5240]: 0.5\n Train loss:0.12298212200403214\n \n train accuracy [batch 5250]: 0.625\n Train loss:0.33093875646591187\n \n train accuracy [batch 5260]: 0.75\n Train loss:0.146297886967659\n \n train accuracy [batch 5270]: 0.625\n Train loss:0.2894054353237152\n \n train accuracy [batch 5280]: 0.625\n Train loss:0.2725050449371338\n \n train accuracy [batch 5290]: 0.75\n Train loss:0.1852465569972992\n \n train accuracy [batch 5300]: 0.75\n Train loss:0.3438527286052704\n \n train accuracy [batch 5310]: 0.75\n Train loss:0.21213388442993164\n \n train accuracy [batch 5320]: 0.75\n Train loss:0.1642754077911377\n \n train accuracy [batch 5330]: 1.0\n Train loss:0.14563892781734467\n \n train accuracy [batch 5340]: 0.875\n Train loss:0.133686363697052\n \n train accuracy [batch 5350]: 0.625\n Train loss:0.21912823617458344\n \n train accuracy [batch 5360]: 0.75\n Train loss:0.14399003982543945\n \n train accuracy [batch 5370]: 1.0\n Train loss:0.11438806354999542\n \n train accuracy [batch 5380]: 0.5\n Train loss:0.3619755804538727\n \n train accuracy [batch 5390]: 0.875\n Train loss:0.12036847323179245\n \n train accuracy [batch 5400]: 0.5\n Train loss:0.29926273226737976\n \n train accuracy [batch 5410]: 0.875\n Train loss:0.16055433452129364\n \n train accuracy [batch 5420]: 0.625\n Train loss:0.319177508354187\n \n train accuracy [batch 5430]: 0.75\n Train loss:0.11768341809511185\n \n train accuracy [batch 5440]: 0.5\n Train loss:0.3003620207309723\n \n train accuracy [batch 5450]: 0.625\n Train loss:0.20779038965702057\n \n train accuracy [batch 5460]: 0.75\n Train loss:0.23391634225845337\n \n train accuracy [batch 5470]: 0.5\n Train loss:0.1544954627752304\n \n train accuracy [batch 5480]: 0.75\n Train loss:0.1864781528711319\n \n train accuracy [batch 5490]: 0.625\n Train loss:0.2022000253200531\n \n train accuracy [batch 5500]: 0.625\n Train loss:0.20388546586036682\n \n train accuracy [batch 5510]: 0.75\n Train loss:0.20768077671527863\n \n train accuracy [batch 5520]: 0.875\n Train loss:0.1185218095779419\n \n train accuracy [batch 5530]: 0.625\n Train loss:0.2842881977558136\n \n train accuracy [batch 5540]: 0.5\n Train loss:0.26945406198501587\n \n train accuracy [batch 5550]: 0.75\n Train loss:0.28778618574142456\n \n train accuracy [batch 5560]: 0.625\n Train loss:0.21580395102500916\n \n train accuracy [batch 5570]: 0.75\n Train loss:0.21567320823669434\n \n train accuracy [batch 5580]: 0.5\n Train loss:0.2917247712612152\n \n train accuracy [batch 5590]: 0.875\n Train loss:0.17989042401313782\n \n train accuracy [batch 5600]: 0.875\n Train loss:0.18704749643802643\n \n train accuracy [batch 5610]: 0.75\n Train loss:0.15175576508045197\n \n train accuracy [batch 5620]: 0.75\n Train loss:0.19318777322769165\n \n train accuracy [batch 5630]: 0.625\n Train loss:0.23376743495464325\n \n train accuracy [batch 5640]: 0.625\n Train loss:0.16147667169570923\n \n train accuracy [batch 5650]: 0.875\n Train loss:0.16571716964244843\n \n train accuracy [batch 5660]: 0.625\n Train loss:0.27856844663619995\n \n train accuracy [batch 5670]: 0.375\n Train loss:0.3565319776535034\n \n train accuracy [batch 5680]: 0.5\n Train loss:0.24198587238788605\n \n train accuracy [batch 5690]: 0.75\n Train loss:0.3071681559085846\n \n train accuracy [batch 5700]: 0.625\n Train loss:0.26649364829063416\n \n train accuracy [batch 5710]: 0.75\n Train loss:0.2388439029455185\n \n train accuracy [batch 5720]: 0.75\n Train loss:0.25441044569015503\n \n train accuracy [batch 5730]: 0.5\n Train loss:0.5091224908828735\n \n train accuracy [batch 5740]: 0.625\n Train loss:0.2352447509765625\n \n train accuracy [batch 5750]: 0.625\n Train loss:0.3418582081794739\n \n train accuracy [batch 5760]: 0.5\n Train loss:0.261846125125885\n \n train accuracy [batch 5770]: 0.5\n Train loss:0.23078352212905884\n \n train accuracy [batch 5780]: 0.625\n Train loss:0.26396071910858154\n \n train accuracy [batch 5790]: 0.875\n Train loss:0.2117684781551361\n \n train accuracy [batch 5800]: 0.625\n Train loss:0.24975432455539703\n \n train accuracy [batch 5810]: 0.75\n Train loss:0.17721940577030182\n \n train accuracy [batch 5820]: 0.75\n Train loss:0.2639155387878418\n \n train accuracy [batch 5830]: 0.5\n Train loss:0.27949264645576477\n \n train accuracy [batch 5840]: 0.625\n Train loss:0.22457215189933777\n \n train accuracy [batch 5850]: 0.625\n Train loss:0.259468138217926\n \n train accuracy [batch 5860]: 0.75\n Train loss:0.1991502195596695\n \n train accuracy [batch 5870]: 0.625\n Train loss:0.19163347780704498\n \n train accuracy [batch 5880]: 0.375\n Train loss:0.4133496582508087\n \n train accuracy [batch 5890]: 0.875\n Train loss:0.18311095237731934\n \n train accuracy [batch 5900]: 0.75\n Train loss:0.244059756398201\n \n train accuracy [batch 5910]: 0.625\n Train loss:0.21618492901325226\n \n train accuracy [batch 5920]: 0.75\n Train loss:0.18840216100215912\n \n train accuracy [batch 5930]: 0.875\n Train loss:0.19808417558670044\n \n train accuracy [batch 5940]: 0.625\n Train loss:0.23389877378940582\n \n train accuracy [batch 5950]: 0.75\n Train loss:0.1332731693983078\n \n train accuracy [batch 5960]: 0.5\n Train loss:0.3378540277481079\n \n train accuracy [batch 5970]: 0.5\n Train loss:0.3078005015850067\n \n train accuracy [batch 5980]: 0.5\n Train loss:0.27882829308509827\n \n train accuracy [batch 5990]: 0.5\n Train loss:0.27486905455589294\n \n train accuracy [batch 6000]: 0.625\n Train loss:0.19619759917259216\n \n train accuracy [batch 6010]: 0.875\n Train loss:0.1722298413515091\n \n train accuracy [batch 6020]: 0.875\n Train loss:0.12124376744031906\n \n train accuracy [batch 6030]: 0.5\n Train loss:0.3026106059551239\n \n train accuracy [batch 6040]: 0.5\n Train loss:0.36057257652282715\n \n train accuracy [batch 6050]: 0.75\n Train loss:0.24072059988975525\n \n train accuracy [batch 6060]: 0.625\n Train loss:0.23582738637924194\n \n train accuracy [batch 6070]: 0.875\n Train loss:0.18452346324920654\n \n train accuracy [batch 6080]: 0.875\n Train loss:0.18011625111103058\n \n train accuracy [batch 6090]: 0.5\n Train loss:0.30030810832977295\n \n train accuracy [batch 6100]: 1.0\n Train loss:0.10892714560031891\n \n train accuracy [batch 6110]: 0.625\n Train loss:0.27430784702301025\n \n train accuracy [batch 6120]: 0.75\n Train loss:0.1481596976518631\n \n train accuracy [batch 6130]: 0.5\n Train loss:0.40821921825408936\n \n train accuracy [batch 6140]: 0.5\n Train loss:0.3034548759460449\n \n train accuracy [batch 6150]: 0.625\n Train loss:0.27740678191185\n \n train accuracy [batch 6160]: 0.75\n Train loss:0.22054055333137512\n \n train accuracy [batch 6170]: 0.875\n Train loss:0.16000376641750336\n \n train accuracy [batch 6180]: 0.75\n Train loss:0.1460401564836502\n \n train accuracy [batch 6190]: 1.0\n Train loss:0.10990487039089203\n \n train accuracy [batch 6200]: 0.75\n Train loss:0.1698562055826187\n \n train accuracy [batch 6210]: 0.5\n Train loss:0.25032222270965576\n \n train accuracy [batch 6220]: 0.5\n Train loss:0.4649367928504944\n \n train accuracy [batch 6230]: 1.0\n Train loss:0.03791125863790512\n \n train accuracy [batch 6240]: 0.75\n Train loss:0.22967521846294403\n \n Total train loss tensor(0.2288, device='cuda:0')\n Total train accuracy 0.69634\n Total time for training an epoch: 1934\n test accuracy [batch 0]: 0.75\n test accuracy [batch 25]: 0.75\n test accuracy [batch 50]: 0.625\n test accuracy [batch 75]: 0.5\n test accuracy [batch 99]: 0.5\n test accuracy [batch 100]: 0.625\n test accuracy [batch 125]: 0.75\n test accuracy [batch 150]: 0.625\n test accuracy [batch 175]: 0.625\n test accuracy [batch 200]: 0.625\n test accuracy [batch 225]: 0.875\n test accuracy [batch 250]: 0.5\n test accuracy [batch 275]: 0.875\n test accuracy [batch 300]: 0.75\n test accuracy [batch 325]: 1.0\n test accuracy [batch 350]: 0.75\n test accuracy [batch 375]: 1.0\n test accuracy [batch 400]: 0.75\n test accuracy [batch 425]: 0.625\n test accuracy [batch 450]: 0.625\n test accuracy [batch 475]: 0.875\n test accuracy [batch 500]: 0.625\n test accuracy [batch 525]: 0.5\n test accuracy [batch 550]: 0.625\n test accuracy [batch 575]: 0.75\n test accuracy [batch 600]: 0.5\n test accuracy [batch 625]: 0.625\n test accuracy [batch 650]: 0.875\n test accuracy [batch 675]: 0.75\n test accuracy [batch 700]: 0.75\n test accuracy [batch 725]: 0.75\n test accuracy [batch 750]: 1.0\n test accuracy [batch 775]: 0.5\n test accuracy [batch 800]: 0.625\n test accuracy [batch 825]: 0.625\n test accuracy [batch 850]: 0.75\n test accuracy [batch 875]: 0.875\n test accuracy [batch 900]: 0.5\n test accuracy [batch 925]: 0.75\n test accuracy [batch 950]: 0.75\n test accuracy [batch 975]: 0.625\n test accuracy [batch 1000]: 0.5\n test accuracy [batch 1025]: 0.5\n test accuracy [batch 1050]: 0.75\n test accuracy [batch 1075]: 0.75\n test accuracy [batch 1100]: 0.5\n test accuracy [batch 1125]: 0.625\n test accuracy [batch 1150]: 0.875\n test accuracy [batch 1175]: 1.0\n test accuracy [batch 1200]: 0.625\n test accuracy [batch 1225]: 0.875\n Total test loss tensor(0.2454, device='cuda:0')\n Total test accuracy 0.6653\n\n\n## Reconstructions\n\nHere you can view the reconstructions obtained by your CapsNet. Nothing special here, just fun to visualize them. Actually, for mnist the reconstructions are great, however for CIFAR10 they are rubbish (see the original paper for clues on that).\n\nBe careful when running reconstructions after `keyboard_interrupt`, because this may result in wrong input-target values.\n\n\n```python\nimport matplotlib\nimport matplotlib.pyplot as plt\n\ndef plot_images_separately(images):\n \"Plot the six MNIST images separately.\"\n fig = plt.figure()\n for j in range(1, 7):\n ax = fig.add_subplot(1, 6, j)\n ax.matshow(images[j-1], cmap = matplotlib.cm.binary)\n plt.xticks(np.array([]))\n plt.yticks(np.array([]))\n plt.show()\n```\n\n\n```python\nplot_images_separately(data[:6,0].data.cpu().numpy())\n```\n\n\n```python\nplot_images_separately(reconstructions[:6,0].data.cpu().numpy())\n```\n\n### To-Do\n\n- Stack more convolutional layers before capsule layers.\n- Increase the size of the capsule layers (more capsules, larger capsules etc.). Note that it may take a lot of time.\n- Play with number of routing iterations in forward pass.\n- Play with kernel size of convolutions in the first layer (don't forget to change parameters of subsequent layers).\n- Play with kernel size of capsules in the second layer (again, pay attention to the parameters of subsequent computations).\n\n- Try different variants of original implementation's loss function (change m+, m-, lambda, get rid of square etc.).\n- Try different loss functions (make it pure Hinge or pure MSE, maybe even cross-entropy!).\n- Try different implementation of capsules (not usual convolution operation, but maybe fully connected groups of neurons).\n- Try different non-linearities for capsules (changing ^2 to ^4 doesn't count!).\n- Try different weights for reconstruction loss.\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "05f21eb76c235e2b9e320bad86bd6e7cb12ab7fd", "size": 428555, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Caps_CIFAR.ipynb", "max_stars_repo_name": "QuickLearner171998/CapsNet", "max_stars_repo_head_hexsha": "239183fef76819513c91e08ce8d2d6c816b10450", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2019-10-07T08:05:01.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-07T10:34:24.000Z", "max_issues_repo_path": "Caps_CIFAR.ipynb", "max_issues_repo_name": "QuickLearner171998/CapsNet", "max_issues_repo_head_hexsha": "239183fef76819513c91e08ce8d2d6c816b10450", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Caps_CIFAR.ipynb", "max_forks_repo_name": "QuickLearner171998/CapsNet", "max_forks_repo_head_hexsha": "239183fef76819513c91e08ce8d2d6c816b10450", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 41.1123369148, "max_line_length": 16786, "alphanum_fraction": 0.4969747174, "converted": true, "num_tokens": 90814, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056295505783, "lm_q2_score": 0.6334102567576901, "lm_q1q2_score": 0.436613255798153}} {"text": "##### Run this cell to set your notebook up on Google Colab\n\n\n```python\n!apt-get install -y xvfb python-opengl ffmpeg > /dev/null 2>&1\n\n!git clone https://github.com/yfletberliac/rlss2019-hands-on.git > /dev/null 2>&1\n!pip install -q torch==1.1.0 torchvision pyvirtualdisplay piglet > /dev/null 2>&1\n```\n\n# Deep Q Networks\n------------\nYou can find the original paper [here](https://www.cs.toronto.edu/~vmnih/docs/dqn.pdf).\n\n## Preliminaries: Q Learning\n\n#### Q-Value\n\n**Q-Value** is a measure of the overall expected reward assuming the agent is in state $s$ and performs action $a$, and then continues playing until the end of the episode following some policy $\\pi$. It is defined mathematically as:\n\n\\begin{equation}\nQ^{\\pi}\\left(s_{t}, a_{t}\\right)=E\\left[R_{t+1}+\\gamma R_{t+2}+\\gamma^{2} R_{t+3}+\\ldots | s_{t}, a_{t}\\right]\n\\end{equation}\n\nwhere $R_{t+1}$ is the immediate reward received after performing action $a_{t}$ in state $s_{t}$ and $\\gamma$ is the discount factor and controls the importance of the future rewards versus the immediate ones: the lower the discount factor is, the less important future rewards are.\n\n#### Bellman Optimality Equation\n\nFormally, the Bellman equation defines the relationships between a given state (or, in our case, a **state-action pair**) and its successors. While many forms exist, one of the most common is the **Bellman Optimality Equation** for the optimal **Q-Value**, which is given by:\n\n\\begin{equation}\nQ^{*}(s, a)=\\sum_{s^{\\prime}, r} p\\left(s^{\\prime}, r | s, a\\right)\\left[r+\\gamma \\max _{a^{\\prime}} Q^{*}\\left(s^{\\prime}, a^{\\prime}\\right)\\right]\n\\end{equation}\n\nOf course, when no uncertainty exists (transition probabilities are either 0 or 1), we have:\n\n\\begin{equation}\nQ^{*}(s, a)=r(s, a)+\\gamma \\max _{a^{\\prime}} Q^{*}\\left(s^{\\prime}, a^{\\prime}\\right)\n\\end{equation}\n\n#### Q-Value Iteration\n\nWe define the corresponding Bellman backup operator:\n\\begin{equation}\n[\\mathcal{T} Q]\\left(s, a\\right)=r(s, a)+\\gamma \\max _{a^{\\prime}} Q\\left(s^{\\prime}, a^{\\prime}\\right)\n\\end{equation}\n\n$Q$ is a fixed point of $\\mathcal{T}$:\n\\begin{equation}\n\\mathcal{T} Q^{*}=Q^{*}\n\\end{equation}\n\nIf we apply the Bellman operator $\\mathcal{T}$ repeatedly to any initial $Q$, the series converges to $Q^{*}$:\n\\begin{equation}\nQ, \\mathcal{T} Q, \\mathcal{T}^{2} Q, \\cdots \\rightarrow Q^{*}\n\\end{equation}\n\n# Imports\n\n\n```python\nimport sys\nsys.path.insert(0, './rlss2019-hands-on/utils')\n# If using the Docker image, replace by:\n# sys.path.insert(0, '../utils')\n\nimport gym, random, os.path, math, glob, csv, base64\nfrom pathlib import Path\nfrom timeit import default_timer as timer\nfrom datetime import timedelta\nimport numpy as np\n\nimport torch\nimport torch.optim as optim\nimport torch.nn as nn\nimport torch.nn.functional as F\n\nimport matplotlib\n%matplotlib inline\n\nfrom qfettes_plot import plot_all_data\nfrom qfettes_wrappers import *\nfrom openai_wrappers import make_atari, wrap_deepmind\nfrom gym.wrappers import Monitor\n\nfrom pyvirtualdisplay import Display\nfrom IPython import display as ipythondisplay\nfrom IPython.display import clear_output\n```\n\n------------\n\n# Deep Q learning\n\nUsually in Deep RL, the **Q-Value** is defined as $Q(s,a;\\theta)$ where $\\theta$ represents the parameters of the function approximation used.\n\n\n\nFor *MuJoCo* or *Roboschool* environments, we usually use a simple 2- or 3-layer MLP whereas when using **raw pixels for observations** such as in *Atari 2600* games, we usually use a 1-, 2- or 3-layer CNN.\n\nIn our case, since we want to train DQN on *CartPole*, we will use a 3-layer perceptron for our function approximation. \n\n## Network declaration\n\nIn this section, we build $Q(s,a;\\theta)$ function approximation. Since the input is composed of 4 scalars, namely:\n

    [position of cart, velocity of cart, angle of pole, rotation rate of pole]
    \nwe build a FCN -> ReLU -> FCN -> ReLU -> FCN neural network. As an exercice, change the architecture of the network:\n\n1. Change the 1st fully-connected layer from 8 hidden neurons to 16\n2. Create `self.fc2` in `__init__` with 16 neurons\n3. Create `self.fc3` with `self.num_actions` as the output size\n4. Add it to the network in `forward` with no activation function\n\n\n```python\nclass DQN(nn.Module):\n def __init__(self, input_shape, num_actions):\n super().__init__()\n \n self.input_shape = input_shape\n self.num_actions = num_actions\n\n self.fc1 = nn.Linear(self.input_shape[0], 8)\n self.fc2 = ...\n self.fc3 = ...\n \n def forward(self, x):\n x = F.relu(self.fc2(F.relu(self.fc1(x))))\n x = ...\n\n return x\n```\n\n## Safety checks\n\n#### Network architecture\n\nAs a *safety check*, inspect the resulting network in the next cell. For instance, the total number of trainable parameters should change with the architecture. Check the correctness of `in_features` and `out_features`.\n\n\n```python\nenv_id = 'CartPole-v0'\nenv = gym.make(env_id)\nnetwork = DQN(env.observation_space.shape, env.action_space.n)\n\nprint(\"Observation space:\\n\", env.observation_space.shape, \"\\n\")\nprint(\"Network architecture:\\n\", network, \"\\n\")\n\nmodel_parameters = filter(lambda p: p.requires_grad, network.parameters())\nprint(\"Total number of trainable parameters:\\n\", sum([np.prod(p.size()) for p in model_parameters]))\n```\n\n Observation space:\n (4,) \n \n Network architecture:\n DQN(\n (fc1): Linear(in_features=4, out_features=8, bias=True)\n (fc2): Linear(in_features=8, out_features=8, bias=True)\n (fc3): Linear(in_features=8, out_features=2, bias=True)\n ) \n \n Total number of trainable parameters:\n 130\n\n\n#### Run a Policy with Random Actions\n\nWhat the working environment looks like? It's always useful to know the details about the environment you train your policy on. For instance, its dynamics, the size of action and observation space, etc. Below we display three different random policies on `CartPole-v0`.\n\n\n```python\ndisplay = Display(visible=0, size=(1400, 900))\ndisplay.start()\n\ndef show_video():\n html = []\n for mp4 in Path(\"videos\").glob(\"*.mp4\"):\n video_b64 = base64.b64encode(mp4.read_bytes())\n html.append(''''''.format(mp4, video_b64.decode('ascii')))\n ipythondisplay.display(ipythondisplay.HTML(data=\"
    \".join(html)))\n \nenv = Monitor(env, './videos', force=True, video_callable=lambda episode: True)\n\nfor episode in range(2):\n done = False\n obs = env.reset()\n while not done:\n action = env.action_space.sample()\n obs, reward, done, info = env.step(action)\nenv.close()\nshow_video()\n```\n\n xdpyinfo was not found, X start can not be checked! Please install xdpyinfo!\n\n\n\n
    \n\n\nWe can see the episode ending prematurely because the pole drops.\n\n-----\n**Question**:\n\nIt is also important to identify some of the characteristics of the problem. `CartPole-v0` can be described as a **fully-observable**, **deterministic**, **continuous state space**, with a **discrete action space** and **frequent rewards**. Take some time to understand each of these terms :-) Try to find the opposite term for each of them, e.g. deterministic <> stochastic.\n\n## Experience Replay Memory\n\nAs usual RL tasks have no pre-generated training sets which they can learn from, in off-policy learning, our agent must keep records of all the state-transitions it encountered so it can **learn from them later**. The memory-buffer used to store this is often referred to as the **Experience Replay Memory**. There are several types and architectures of these memory buffers\u200a\u2014\u200abut some very common ones are:\n- the *cyclic memory buffers*: they make sure the agent keeps training over its new behavior rather than things that might no longer be relevant\n- the *reservoir-sampling-based memory buffers*: they guarantee each state-transition recorded has an even probability to be inserted to the buffer\n\nWe use a combination of both.\n\nIn `push`:\n1. Append the transition to memory\n2. Create the if statement which deletes an old transition from the memory\n\n\n```python\nclass ExperienceReplayMemory:\n def __init__(self, capacity):\n self.capacity = capacity\n self.memory = []\n\n def push(self, transition):\n # Append the transition below\n ...\n \n # Now, we need an `if` statement in order to keep the capacity to its limit. Write it below.\n # Hint: `del something` will delete something if something is an array\n if ...:\n \n raise NotImplementedError\n\n def sample(self, batch_size):\n return random.sample(self.memory, batch_size)\n\n def __len__(self):\n return len(self.memory)\n```\n\n------------\n\nNow we have:\n- the **DQN** network,\n- the **ExperienceReplayMemory**.\n\nLet's build the **Agent** class !\n\n## Agent declaration\n\nIn the cell below:\n1. Create `self.target_model` in `declare_networks`\n2. Complete the epsilon-greedy algorithm in `get_action`\n\n\n```python\nclass Agent(object):\n def __init__(self, config, env, log_dir='/tmp/gym'):\n self.log_dir = log_dir\n self.rewards = []\n self.action_log_frequency = config.ACTION_SELECTION_COUNT_FREQUENCY\n self.action_selections = [0 for _ in range(env.action_space.n)]\n \n # Define the DQN networks\n def declare_networks(self):\n self.model = DQN(self.num_feats, self.num_actions)\n # Create `self.target_model` with the same network architecture\n self.target_model = ...\n raise NotImplementedError\n\n # Define the Replay Memory\n def declare_memory(self):\n self.memory = ExperienceReplayMemory(self.experience_replay_size)\n \n # Append the new transition to the Replay Memory\n def append_to_replay(self, s, a, r, s_):\n self.memory.push((s, a, r, s_))\n \n # Sample transitions from the Replay Memory\n def sample_minibatch(self):\n transitions = self.memory.sample(self.batch_size)\n batch_state, batch_action, batch_reward, batch_next_state = zip(*transitions)\n\n shape = (-1,)+self.num_feats\n\n batch_state = torch.tensor(batch_state, device=self.device, dtype=torch.float).view(shape)\n batch_action = torch.tensor(batch_action, device=self.device, dtype=torch.long).squeeze().view(-1, 1)\n batch_reward = torch.tensor(batch_reward, device=self.device, dtype=torch.float).squeeze().view(-1, 1)\n \n non_final_mask = torch.tensor(tuple(map(lambda s: s is not None, batch_next_state)), device=self.device, dtype=torch.uint8)\n # Sometimes all next states are false\n try:\n non_final_next_states = torch.tensor([s for s in batch_next_state if s is not None], device=self.device, dtype=torch.float).view(shape)\n empty_next_state_values = False\n except:\n non_final_next_states = None\n empty_next_state_values = True\n\n return batch_state, batch_action, batch_reward, non_final_next_states, non_final_mask, empty_next_state_values\n \n # Sample action\n def get_action(self, s, eps=0.1):\n with torch.no_grad():\n # Epsilon-greedy\n if np.random.random() >= eps:\n X = torch.tensor([s], device=self.device, dtype=torch.float)\n a = self.model(X).max(1)[1].view(1, 1)\n return a.item()\n else:\n ...\n```\n\n-----\n**Question**:\n\nRemember we define the objective function as\n\\begin{equation}\nJ=\\left(r+\\gamma \\max _{a^{\\prime}} Q\\left(s^{\\prime}, a^{\\prime}, \\mathbf{\\theta}^{-}\\right)-Q(s, a, \\mathbf{\\theta})\\right)^{2},\n\\end{equation}\nwhere $\\theta^{-}$ are the target parameters.\n\nWhy do we need a target network in the first place ?\n\n## Learning\n\nIn the cell below, and from the above objective fonction:\n1. Write the value `expected_q_values`\n2. Write `diff`\n3. The `update` function needs some work\n\n\n```python\nclass Learning(Agent):\n def __init__(self, env=None, config=None, log_dir='/tmp/gym'):\n super().__init__(config=config, env=env, log_dir=log_dir)\n \n # Compute loss from the Bellman Optimality Equation\n def compute_loss(self, batch_vars):\n batch_state, batch_action, batch_reward, non_final_next_states, non_final_mask, empty_next_state_values = batch_vars\n\n # Estimate\n current_q_values = self.model(batch_state).gather(1, batch_action)\n \n # Target\n with torch.no_grad():\n max_next_q_values = torch.zeros(self.batch_size, device=self.device, dtype=torch.float).unsqueeze(dim=1)\n if not empty_next_state_values:\n max_next_action = self.get_max_next_state_action(non_final_next_states)\n max_next_q_values[non_final_mask] = self.target_model(non_final_next_states).gather(1, max_next_action)\n # From the equation above, write the value `expected_q_values`.\n expected_q_values = ...\n \n # From the equation above, write the value `diff`.\n diff = ...\n loss = self.MSE(diff)\n loss = loss.mean()\n \n raise NotImplementedError\n return loss\n\n # Update both networks (the agent and the target)\n def update(self, s, a, r, s_, sample_idx=0):\n self.append_to_replay(s, a, r, s_)\n \n # When not to update ?\n # There is a concise way to write to skip the update, fill in the 2 blanks in the `if` statement below.\n # Hint: the sample count should be < the learn_start hyperparameter and respect the update_freq.\n if ... or ...:\n raise NotImplementedError\n return None\n\n batch_vars = self.sample_minibatch()\n loss = self.compute_loss(batch_vars)\n\n # Optimize the model\n self.optimizer.zero_grad()\n loss.backward()\n for param in self.model.parameters():\n param.grad.data.clamp_(-1, 1)\n self.optimizer.step()\n\n self.update_target_model()\n self.save_td(loss.item(), sample_idx)\n\n def update_target_model(self):\n # Copy weights from model to target_model following `target_net_update_freq`.\n self.update_count+=1\n if self.update_count % self.target_net_update_freq == 0:\n self.target_model.load_state_dict(self.model.state_dict())\n```\n\n## Model declaration\n\n\n```python\nclass Model(Learning):\n def __init__(self, env=None, config=None, log_dir='/tmp/gym'):\n super().__init__(config=config, env=env, log_dir=log_dir)\n self.device = config.device\n\n # Hyperparameters\n self.gamma = config.GAMMA\n self.target_net_update_freq = config.TARGET_NET_UPDATE_FREQ\n self.experience_replay_size = config.EXP_REPLAY_SIZE\n self.batch_size = config.BATCH_SIZE\n self.learn_start = config.LEARN_START\n self.update_freq = config.UPDATE_FREQ\n\n # Environment specific parameters\n self.num_feats = env.observation_space.shape\n self.num_actions = env.action_space.n\n self.env = env\n\n self.declare_networks()\n self.declare_memory()\n self.target_model.load_state_dict(self.model.state_dict())\n self.optimizer = optim.Adam(self.model.parameters(), lr=config.LR)\n \n # Move to correct device\n self.model = self.model.to(self.device)\n self.target_model.to(self.device)\n \n self.model.train()\n self.target_model.train()\n \n self.update_count = 0\n \n def save_td(self, td, tstep):\n with open(os.path.join(self.log_dir, 'td.csv'), 'a') as f:\n writer = csv.writer(f)\n writer.writerow((tstep, td))\n\n def get_max_next_state_action(self, next_states):\n return self.target_model(next_states).max(dim=1)[1].view(-1, 1)\n \n def MSE(self, x):\n return 0.5 * x.pow(2)\n\n def save_reward(self, reward):\n self.rewards.append(reward)\n\n def save_action(self, action, tstep):\n self.action_selections[int(action)] += 1.0/self.action_log_frequency\n if (tstep+1) % self.action_log_frequency == 0:\n with open(os.path.join(self.log_dir, 'action_log.csv'), 'a') as f:\n writer = csv.writer(f)\n writer.writerow(list([tstep]+self.action_selections))\n self.action_selections = [0 for _ in range(len(self.action_selections))]\n \n def save_w(self):\n if not os.path.exists(\"../saved_agents\"):\n os.makedirs(\"../saved_agents\")\n torch.save(self.model.state_dict(), '../saved_agents/model.dump')\n torch.save(self.optimizer.state_dict(), '../saved_agents/optim.dump')\n```\n\n## Hyperparameters\n\n\n```python\nclass Config(object):\n def __init__(self):\n self.device = torch.device(\"cuda\" if torch.cuda.is_available() else \"cpu\")\n \n # Main agent variables\n self.GAMMA=0.99\n self.LR=1e-3\n \n # Epsilon variables\n self.epsilon_start = 1.0\n self.epsilon_final = 0.01\n self.epsilon_decay = 10000\n self.epsilon_by_sample = lambda sample_idx: config.epsilon_final + (config.epsilon_start - config.epsilon_final) * math.exp(-1. * sample_idx / config.epsilon_decay)\n\n # Memory\n self.TARGET_NET_UPDATE_FREQ = 1000\n self.EXP_REPLAY_SIZE = 10000\n self.BATCH_SIZE = 64\n\n # Learning control variables\n self.LEARN_START = 1000\n self.MAX_SAMPLES = 50000\n self.UPDATE_FREQ = 1\n\n # Data logging parameters\n self.ACTION_SELECTION_COUNT_FREQUENCY = 1000\n \n \nconfig = Config()\n```\n\n## Training\n\n\n```python\nimport gym\nfrom openai_monitor import Monitor\nfrom IPython import display\nimport matplotlib\nimport matplotlib.pyplot as plt\n%matplotlib inline\n\nstart=timer()\n\nlog_dir = \"/tmp/gym/\"\ntry:\n os.makedirs(log_dir)\nexcept OSError:\n files = glob.glob(os.path.join(log_dir, '*.monitor.csv')) \\\n + glob.glob(os.path.join(log_dir, '*td.csv')) \\\n + glob.glob(os.path.join(log_dir, '*action_log.csv'))\n for f in files:\n os.remove(f)\n\nenv_id = 'CartPole-v0'\nenv = gym.make(env_id)\nenv = Monitor(env, os.path.join(log_dir, env_id))\n \nmodel = Model(env=env, config=config, log_dir=log_dir)\n\nepisode_reward = 0\n\nobservation = env.reset()\nfor sample_idx in range(1, config.MAX_SAMPLES + 1):\n \n epsilon = config.epsilon_by_sample(sample_idx)\n\n action = model.get_action(observation, epsilon)\n # Log action selection\n model.save_action(action, sample_idx)\n\n prev_observation=observation\n observation, reward, done, _ = env.step(action)\n observation = None if done else observation\n\n model.update(prev_observation, action, reward, observation, sample_idx)\n episode_reward += reward\n\n if done:\n observation = env.reset()\n model.save_reward(episode_reward)\n episode_reward = 0\n if sample_idx % 1000 == 0:\n try:\n clear_output(True)\n plot_all_data(log_dir, env_id, 'DQN', config.MAX_SAMPLES, bin_size=(10, 100, 100, 1), smooth=1, time=timedelta(seconds=int(timer()-start)), ipynb=True)\n except IOError:\n pass\n\nmodel.save_w()\nenv.close()\n```\n\nBy observing the plots, does the learning appear to be stable?\n\nIf your answer is *yes*, then start a second run, and a third, with the same hyperparameters. ;-)\n\nYou have just faced reproducibility concerns, which is quite a serious problem in deep RL and which can be dealt with by e.g. running your experiments on a sufficient number of seeds (~ 6-8 min.)\n\n## Visualize the agent\n\n\n```python\nfrom gym.wrappers import Monitor\n\n# Loading the agent\nfname_model = \"../saved_agents/model.dump\"\nfname_optim = \"../saved_agents/optim.dump\"\nlog_dir = \"/tmp/gym/\"\n\nmodel = Model(env=env, config=config, log_dir=log_dir)\n\nif os.path.isfile(fname_model):\n model.model.load_state_dict(torch.load(fname_model))\n model.target_model.load_state_dict(model.model.state_dict())\n\nif os.path.isfile(fname_optim):\n model.optimizer.load_state_dict(torch.load(fname_optim))\n\nenv_id = 'CartPole-v0'\nenv = gym.make(env_id)\nenv = Monitor(env, './videos', force=True, video_callable=lambda episode: True)\n\nfor episode in range(3):\n done = False\n obs = env.reset()\n while not done:\n action = model.get_action(obs)\n obs, _, done, _ = env.step(action)\n\nenv.close()\nshow_video()\n```\n\nYou can experiment with modifying the hypermarameters (learning rate, batch size, experience replay size, etc.) to see if you can make its performance improve !\n\n-------------\n", "meta": {"hexsha": "bd5463243762d1877dbe1fa82e9c8d0ba210a834", "size": 53394, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Reinforcement Learning Summer School 2019 (Lille, France)/practical_drl_2/practical_drl_2.ipynb", "max_stars_repo_name": "xuedong/rlss2019", "max_stars_repo_head_hexsha": "d7468c2fcf269d8afd6fb0f44993aa9797867944", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Reinforcement Learning Summer School 2019 (Lille, France)/practical_drl_2/practical_drl_2.ipynb", "max_issues_repo_name": "xuedong/rlss2019", "max_issues_repo_head_hexsha": "d7468c2fcf269d8afd6fb0f44993aa9797867944", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Reinforcement Learning Summer School 2019 (Lille, France)/practical_drl_2/practical_drl_2.ipynb", "max_forks_repo_name": "xuedong/rlss2019", "max_forks_repo_head_hexsha": "d7468c2fcf269d8afd6fb0f44993aa9797867944", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 49.3475046211, "max_line_length": 9885, "alphanum_fraction": 0.6232348204, "converted": true, "num_tokens": 5182, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102498375401, "lm_q2_score": 0.689305616785446, "lm_q1q2_score": 0.43661324294248904}} {"text": "# Kamodo-CLI\n\nThe kamodo command-line interface should be installed automatically with pip\n\n pip install kamodo\n\n!!! warning\n The command line app is a work in progress. Expect changes!\n\n!!! note\n You can run these examples in a notebook with `!kamodo` \n\nRun the command line interface with\n\n kamodo .=value\n \n### instantiation\nTo create a simple function ready for evaluation:\n\n```console\nkamodo model.params.f[cm]=x**2\n lhs rhs symbol units\nf(x) f x**2 f(x) cm\n\\begin{equation}f{\\left(x \\right)} [cm] = x^{2}\\end{equation}\n```\n \n \n\nKamodo will print a description of the loaded model when `verbose` output is on (the default).\n\n### evaluation\n\nTo evaluate a model, provide arguments for each function.\n\n```console\nkamodo model.params.f[cm]=x**2 model.evaluate.f.x=[-3,-1,1,3] verbose=false\nf(x=[-3, -1, 1, 3]) cm = \n\n[9 1 1 9]\n\n```\n\n### visualization\n\nTo visualize model output, provide plot parameters similar to evaluation\n\n```console\nkamodo model.params.f[cm]=x**2 model.plot.f.x.min=-2 model.plot.f.x.max=2 model.plot.f.x.n=25 verbose=False\n```\n\nAn interactive plot will open with your figure\n\n{! notebooks/outputs/2019-12-20/13-28-47/temp-plot.html !}\n\n### Kamodofied models\n\nTo work with a kamodofied model, specify the `model.class`.\n\n```console\nkamodo model.class=kamodo.readers.tiegcm.TIEGCM_Kamodo model.params.filename=$PWD/s001.nc model.plot.EFLUX.lon=[0]\n```\n\n\n\n### Configuration\n\nKamodo can use configuration files so that arguments do not have to be passed manually.\nTo create your own configuration, create a config.yaml file in your project's directory:\n```yaml\n{! notebooks/config.yaml !}\n```\nRunning kamodo from a directory containing the above `config.yaml` will produce the same plot as before:\n\n\n```python\n! kamodo\n```\n\n lhs rhs symbol units\r\n f(x) f x**2 f(x) cm\r\n \\begin{equation}f{\\left(x \\right)} [cm] = x^{2}\\end{equation}\r\n\n\n### Configuration Priority\n\nKamodo is built on hydra, which prioritizes configuration using the following rules:\n\n1. If there are two configurations that define the same value, the second one would win.\n2. If two configurations are contributing to the same dictionary the result would be the combined dictionary.\n\n### Help\n```console\nkamodo --help\nA low-coding command line interface for Kamodo\n\nThis application allows users to work with kamodo-compatible models and data\ndirectly from the command line. \n\nCustom models, data, and expressions may be composed by editing config files\nwithout needing to write python.\n\n== Configuration groups ==\nCompose your configuration from those groups (group=option)\n\n== Config ==\nOverride anything in the config (foo.bar=value)\nmodel:\n class: kamodo.Kamodo\n evaluate: {}\n fig_layout: {}\n params: {}\n plot: {}\n plot_conf:\n animation_opts: null\n auto_open: true\n auto_play: true\n config: null\n filename: temp-plot.html\n image: null\n image_filename: plot_image\n include_mathjax: cdn\n include_plotlyjs: true\n link_text: Export to plot.ly\n output_type: file\n show_link: false\n validate: true\nverbose: true\n\nPowered by Hydra (https://hydra.cc)\nUse --hydra-help to view Hydra specific help\n```\n\n### tab completion\n\nKamodo supports tab completion for bash. To set up bash tab completion, run the following:\n\n eval \"$(kamodo -sc install=bash)\"\n", "meta": {"hexsha": "6047053f10761df2f0f40f5faba7e8af8cf8b0d7", "size": 5967, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/notebooks/CommandLineInterface.ipynb", "max_stars_repo_name": "dfm/kamodo-core", "max_stars_repo_head_hexsha": "aa7f678f0123ddffa2d0450361e9789728e9d7b8", "max_stars_repo_licenses": ["NASA-1.3"], "max_stars_count": 28, "max_stars_repo_stars_event_min_datetime": "2019-08-30T16:18:53.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-13T16:33:51.000Z", "max_issues_repo_path": "docs/notebooks/CommandLineInterface.ipynb", "max_issues_repo_name": "dfm/kamodo-core", "max_issues_repo_head_hexsha": "aa7f678f0123ddffa2d0450361e9789728e9d7b8", "max_issues_repo_licenses": ["NASA-1.3"], "max_issues_count": 60, "max_issues_repo_issues_event_min_datetime": "2021-09-28T17:53:32.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-31T18:39:21.000Z", "max_forks_repo_path": "docs/notebooks/CommandLineInterface.ipynb", "max_forks_repo_name": "dfm/kamodo-core", "max_forks_repo_head_hexsha": "aa7f678f0123ddffa2d0450361e9789728e9d7b8", "max_forks_repo_licenses": ["NASA-1.3"], "max_forks_count": 20, "max_forks_repo_forks_event_min_datetime": "2019-08-16T21:22:40.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-07T15:39:18.000Z", "avg_line_length": 25.6094420601, "max_line_length": 124, "alphanum_fraction": 0.5403050109, "converted": true, "num_tokens": 881, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011686727232, "lm_q2_score": 0.7461389817407016, "lm_q1q2_score": 0.4365667902087602}} {"text": "```python\n%load_ext autoreload\n%autoreload 2\n```\n\n\n```python\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nfrom sklearn.linear_model import LinearRegression, Lasso, LogisticRegression\nfrom sklearn.ensemble import GradientBoostingRegressor, GradientBoostingClassifier\nfrom sklearn.ensemble import RandomForestRegressor, RandomForestClassifier\nfrom sklearn.preprocessing import PolynomialFeatures, StandardScaler\nfrom sklearn.pipeline import Pipeline\nimport scipy.special\n```\n\n# NLSYM DATA\n\n\n```python\n# Preprocess data\ndf = pd.read_csv(\"data/card.csv\")\ndata_filter = df['educ'].values >= 6\nT = df['educ'].values[data_filter]\nZ = df['nearc4'].values[data_filter]\ny = df['lwage'].values[data_filter]\n\n# Impute missing values with mean, add dummy columns\n# I excluded the columns 'weights' as we don't know what it is\nX_df = df[['exper', 'expersq']].copy()\nX_df['fatheduc'] = df['fatheduc'].fillna(value=df['fatheduc'].mean())\nX_df['fatheduc_nan'] = df['fatheduc'].isnull()*1\nX_df['motheduc'] = df['motheduc'].fillna(value=df['motheduc'].mean())\nX_df['motheduc_nan'] = df['motheduc'].isnull()*1\nX_df[['momdad14', 'sinmom14', 'reg661', 'reg662',\n 'reg663', 'reg664', 'reg665', 'reg666', 'reg667', 'reg668', 'reg669', 'south66']] = df[['momdad14', 'sinmom14', \n 'reg661', 'reg662','reg663', 'reg664', 'reg665', 'reg666', 'reg667', 'reg668', 'reg669', 'south66']]\nX_df[['black', 'smsa', 'south', 'smsa66']] = df[['black', 'smsa', 'south', 'smsa66']]\ncolumns_to_scale = ['fatheduc', 'motheduc', 'exper', 'expersq']\nscaler = StandardScaler()\nX_raw = X_df.values[data_filter]\nX_df[columns_to_scale] = scaler.fit_transform(X_df[columns_to_scale])\nX = X_df.values[data_filter]\n```\n\n C:\\ProgramData\\Anaconda3\\lib\\site-packages\\sklearn\\preprocessing\\data.py:645: DataConversionWarning: Data with input dtype int64, float64 were all converted to float64 by StandardScaler.\n return self.partial_fit(X, y)\n C:\\ProgramData\\Anaconda3\\lib\\site-packages\\sklearn\\base.py:464: DataConversionWarning: Data with input dtype int64, float64 were all converted to float64 by StandardScaler.\n return self.fit(X, **fit_params).transform(X)\n\n\n\n```python\nX_df.columns\n```\n\n\n\n\n Index(['exper', 'expersq', 'fatheduc', 'fatheduc_nan', 'motheduc',\n 'motheduc_nan', 'momdad14', 'sinmom14', 'reg661', 'reg662', 'reg663',\n 'reg664', 'reg665', 'reg666', 'reg667', 'reg668', 'reg669', 'south66',\n 'black', 'smsa', 'south', 'smsa66'],\n dtype='object')\n\n\n\n# 2. Semi-Synthetic Data with Binary Instrument and Continuous Treatment\n\nData generating process uning real covariates $X$ and instrument $Z$, but synthetic $y$ and $T$ based on the \"intent to treat\" instrument setting with non-compliance. The instrument corresponds to a fully randomized recommendation of treatment. Then each sample complies with the recommendation to some degree. This probability also depends on an unobserved confounder that has a direct effect on the outcome. Moreover, compliance also depends on the observed feature $X$.\n\n\\begin{align}\nX \\tag{ real features}\\\\\nZ \\tag{real instrument}\\\\\n\\nu \\sim \\; & \\text{U}[0, 1] \\tag{unobserved confounder}\\\\\nC = \\; & c\\cdot X[i], \\; c \\;(const)\\sim \\text{U}[.2, .3] \\tag{compliance degree}\\\\\nT = \\; & C\\cdot Z + g(X) + \\nu \\tag{treatment}\\\\\ny \\sim \\; & \\text{Normal}(\\mu=\\theta(X) \\cdot (T + \\nu) + f(X),\\; \\sigma=.1) \\tag{outcome}\n\\end{align}\n\nMoreover:\n\\begin{align}\n\\theta(X) = \\; & \\alpha + \\beta \\cdot X[i] \\tag{CATE}\\\\\nf(X) = \\; & X[i] \\tag{Nuissance function}\\\\\ng(X) = \\; & X[i] \\tag{Nuissance function}\n\\end{align}\n\n\n\n```python\ndef dgp_bin_Z_cont_T(X, Z, hetero_col, true_fn, random_seed=None):\n np.random.seed(random_seed)\n n, d = X.shape\n nu = np.random.uniform(-1, 1, size=(n,))\n c = np.random.uniform(0.2, 0.3)\n C = c * X[:, hetero_col] # Compliers when recomended\n T = C * Z + X[:, hetero_col] + nu # Treatment with compliance\n y = true_fn(X) * (T + nu) + 0.05*X[:, hetero_col] + np.random.normal(0, .1, size=(n,))\n return y, T\n\nhetero_col = 4 # Mother's education\nhetero_col_2 = 7\ntrue_fn = lambda X: 0.1 + 0.05*X[:, hetero_col] - 0.1*X[:, hetero_col_2]\n\nnp.random.seed(1237)\ny, T = dgp_bin_Z_cont_T(X_raw, Z, hetero_col, true_fn)\n\nplt.figure(figsize=(10, 2))\nplt.subplot(1, 2, 1)\nplt.hist(T[Z==0])\nplt.title(\"T[Z=0]: Total: {}, Mean(T): {:.2f}\".format(T[Z==0].shape[0], np.mean(T[Z==0])))\nplt.subplot(1, 2, 2)\nplt.hist(T[Z==1])\nplt.title(\"T[Z=1]: Total: {}, Mean(T): {:.2f}\".format(T[Z==1].shape[0], np.mean(T[Z==1])))\nplt.show()\n```\n\n# ANALYSIS\n\n### Defining some hyperparameters\n\n\n```python\nrandom_seed = 12345 # random seed for each experiment\nN_SPLITS = 10 # number of splits for cross-fitting\nCOV_CLIP = 20/X.shape[0] # covariance clipping in driv\nprint(COV_CLIP)\n```\n\n 0.006686726847208292\n\n\n### Defining some generic non-parametric regressors and classifiers\n\n\n```python\nfrom utilities import RegWrapper\nfrom sklearn.model_selection import GridSearchCV\nfrom sklearn.linear_model import LassoCV, LogisticRegressionCV\nfrom xgboost import XGBClassifier, XGBRegressor\nfrom xgb_utilities import XGBWrapper\n\n# XGB forest models for Regression and Classification\nmodel = lambda: XGBWrapper(XGBRegressor(gamma=0.001, n_estimators=100, min_child_weight=20, n_jobs=10),\n early_stopping_rounds=5, eval_metric='rmse', binary=False)\n\nmodel_clf = lambda: RegWrapper(XGBWrapper(XGBClassifier(gamma=0.001, n_estimators=100, min_child_weight=20, n_jobs=10),\n early_stopping_rounds=5, eval_metric='logloss', binary=True))\n```\n\n### Some utility functions\n\n\n```python\ndef nuisance_diagnostic(cate, nuisance_model, property_name, property_fn, \n index_names=None, statistic=np.std, threshold=None):\n std = statistic([property_fn(ns) for ns in cate.fitted_nuisances[nuisance_model]], axis=0)\n if hasattr(std, '__len__'):\n if threshold is None:\n coefs = np.argmax(std).flatten()\n else:\n coefs = np.argwhere(std >= threshold).flatten()\n if index_names is None:\n index_names = np.arange(std.shape[0])\n for high_var in coefs:\n plt.title(\"{}: {}[{}] Across Folds\".format(nuisance_model, property_name, index_names[high_var]))\n plt.plot([property_fn(ns)[high_var] for ns in cate.fitted_nuisances[nuisance_model]])\n plt.xlabel('fold')\n plt.ylabel('property')\n plt.show()\n else:\n plt.title(\"{}: {} Across Folds\".format(nuisance_model, property_name)) \n plt.plot([property_fn(ns) for ns in cate.fitted_nuisances[nuisance_model]])\n plt.xlabel('fold')\n plt.ylabel('property')\n plt.show()\n \n```\n\n# ATE via DMLATEIV\n\n\n```python\nfrom dml_ate_iv import DMLATEIV\n\nnp.random.seed(random_seed)\n\n# We need to specify models to be used for each of these residualizations\nmodel_Y_X = lambda: model() # model for E[Y | X]\nmodel_T_X = lambda: model() # model for E[T | X]. We use a regressor since T is continuous\nmodel_Z_X = lambda: model_clf() # model for E[Z | X]. We use a classifier since Z is binary\n\ndmlate = DMLATEIV(model_Y_X(), model_T_X(), model_Z_X(),\n n_splits=N_SPLITS, # n_splits determines the number of splits to be used for cross-fitting.\n binary_instrument=True, # a flag whether to stratify cross-fitting by instrument\n binary_treatment=False # a flag whether to stratify cross-fitting by treatment\n )\n```\n\n C:\\ProgramData\\Anaconda3\\lib\\site-packages\\numba\\errors.py:105: UserWarning: Insufficiently recent colorama version found. Numba requires colorama >= 0.3.9\n warnings.warn(msg)\n\n\n\n```python\n# We fit DMLATEIV with these models\ndmlate.fit(y, T, X, Z)\n```\n\n\n\n\n \n\n\n\n\n```python\n# We call effect() to get the ATE\nta_effect = dmlate.effect()\n```\n\n\n```python\n# Comparison with true ATE\nprint(\"ATE Estimate: {:.3f}\".format(ta_effect))\nprint(\"True ATE: {:.3f}\".format(np.mean(true_fn(X_raw))))\n# CATE MSE\nprint(\"CATE MSE: {:.2f}\".format(np.mean((true_fn(X_raw) - ta_effect)**2)))\n```\n\n ATE Estimate: 0.648\n True ATE: 0.609\n CATE MSE: 0.03\n\n\n\n```python\n# We can call normal_effect_interval to get confidence intervals based\n# based on the asympotic normal approximation\nta_effect = dmlate.normal_effect_interval(lower=2.5, upper=97.5)\n# Comparison with true ATE\nprint(\"ATE Estimate Interval: ({:.3f}, {:.3f})\".format(ta_effect[0], ta_effect[1]))\nprint(\"True ATE: {:.3f}\".format(np.mean(true_fn(X_raw))))\n```\n\n ATE Estimate Interval: (0.601, 0.696)\n True ATE: 0.609\n\n\n\n```python\ndef get_dmlateiv_coverage(true_effect, iteration):\n y, T = dgp_bin_Z_cont_T(X_raw, Z, hetero_col, true_fn, random_seed=iteration)\n dmlate = DMLATEIV(model_Y_X(), model_T_X(), model_Z_X(),\n n_splits=N_SPLITS, # n_splits determines the number of splits to be used for cross-fitting.\n binary_instrument=True, # a flag whether to stratify cross-fitting by instrument\n binary_treatment=False # a flag whether to stratify cross-fitting by treatment\n )\n dmlate.fit(y, T, X, Z)\n left, right = dmlate.normal_effect_interval(lower=2.5, upper=97.5)\n if true_effect >= left and true_effect <= right:\n return 1\n return 0\n\nfrom joblib import Parallel, delayed\nn_experiments=100\ntrue_ate = np.mean(true_fn(X_raw))\nif True:\n contains_truth = np.array(Parallel(n_jobs=-1, verbose=3)(\n delayed(get_dmlateiv_coverage)(true_ate, it) for it in range(n_experiments)))\n print(\"Coverage: {}\".format(contains_truth.mean()))\n```\n\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 12 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 8 tasks | elapsed: 30.7s\n\n\n Coverage: 0.72\n\n\n [Parallel(n_jobs=-1)]: Done 100 out of 100 | elapsed: 2.0min finished\n\n\n# ATE and CATE via DMLIV\n\n\n```python\nfrom dml_iv import DMLIV\nfrom utilities import SelectiveLasso, SeparateModel\nfrom sklearn.linear_model import LassoCV, LogisticRegressionCV\nfrom econml.utilities import hstack\n\nnp.random.seed(random_seed)\n\n# For DMLIV we also need a model for E[T | X, Z]. To allow for heterogeneity in the compliance, i.e.\n# T = beta(X)*Z + gamma(X)\n# we train a separate model for Z=1 and Z=0. The model for Z=1 learns the\n# quantity beta(X) + gamma(X) and the model for Z=0 learns gamma(X).\nmodel_T_XZ = lambda: SeparateModel(model(), model())\n\n# We now specify the features to be used for heterogeneity. We will fit a CATE model of the form\n# theta(X) = \n# for some set of features phi(X). The featurizer needs to support fit_transform, that takes\n# X and returns phi(X). We need to include a bias if we also want a constant term.\ndmliv_featurizer = lambda: PolynomialFeatures(degree=1, include_bias=True)\n\n# Then we need to specify a model to be used for fitting the parameters theta in the linear form.\n# This model will minimize the square loss:\n# (Y - E[Y|X] - * (E[T|X,Z] - E[T|X]))**2\n#dmliv_model_effect = lambda: LinearRegression(fit_intercept=False)\n\n\n# Potentially with some regularization on theta. Here we use an ell_1 penalty on theta\n# If we also have a prior that there is no effect heterogeneity we can use a selective lasso\n# that does not penalize the constant term in the CATE model\ndmliv_model_effect = lambda: SelectiveLasso(np.arange(1, X.shape[1]+1), LassoCV(cv=5, fit_intercept=False))\n\n\n# We initialize DMLIV with all these models and call fit\ncate = DMLIV(model_Y_X(), model_T_X(), model_T_XZ(), \n dmliv_model_effect(), dmliv_featurizer(),\n n_splits=N_SPLITS, # number of splits to use for cross-fitting\n binary_instrument=True, # a flag whether to stratify cross-fitting by instrument\n binary_treatment=False # a flag whether to stratify cross-fitting by treatment\n )\n```\n\n\n```python\ncate.fit(y, T, X, Z)\n```\n\n\n\n\n \n\n\n\n\n```python\n# To get the CATE at every X we call effect(X)\ndml_effect = cate.effect(X)\nplt.hist(dml_effect, label='est')\nplt.hist(true_fn(X_raw), alpha=.2, label='true')\nplt.legend()\nplt.show()\n```\n\n\n```python\n# To get the parameter theta we call coef_. The first entry is the intercept of the CATE model\nprint(cate.coef_)\n```\n\n [ 0.60084499 0. 0. 0.00162356 -0. 0.11428089\n 0. -0. -0. 0. -0. 0.\n 0. 0. -0. -0. 0. -0.\n -0. -0. -0.00789543 -0. -0. ]\n\n\n\n```python\n# We can average the CATE to get an ATE\nprint(\"ATE Estimate: {:.3f}\".format(np.mean(dml_effect)))\nprint(\"True ATE: {:.3f}\".format(np.mean(true_fn(X_raw))))\n```\n\n ATE Estimate: 0.596\n True ATE: 0.609\n\n\n\n```python\n# We can also see how it compares to the true CATE at each target point and calculate MSE\nplt.title(\"DMLIV CATE as Function of {}. MSE={:.3f}\".format(X_df.columns[4], np.mean((dml_effect-true_fn(X_raw))**2)))\nplt.scatter(X[:, 4], dml_effect, label='est')\nplt.scatter(X[:, 4], true_fn(X_raw), label='true', alpha=.2)\nplt.legend()\nplt.show()\n```\n\n# ATE and Projected CATE via DRIV\n\n\n```python\nfrom dml_iv import DMLIV\nfrom dr_iv import DRIV, ProjectedDRIV\nfrom utilities import SubsetWrapper, StatsModelLinearRegression, ConstantModel\nfrom sklearn.dummy import DummyRegressor\n\nnp.random.seed(random_seed)\n\n# For DRIV we need a model for predicting E[T*Z | X]. We use a classifier\nmodel_TZ_X = lambda: model()\n\n# We also need a model for the final regression that will fit the function theta(X)\n# If we want to fit an ATE, we simply fit a constant functin theta(X) = theta\n# We can do this with a pipeline where the preprocessing step only creates a bias column\n# and the regression step fits a linear regression with no intercept.\n# To get normal confidence intervals easily we can use a statsmodels linear regression\n# wrapped in an sklearn interface\nconst_driv_model_effect = lambda: ConstantModel()\n\n# As in OrthoDMLIV we need a perliminary estimator of the CATE.\n# We use a DMLIV estimator with no cross-fitting (n_splits=1)\ndmliv_prel_model_effect = DMLIV(model_Y_X(), model_T_X(), model_T_XZ(),\n dmliv_model_effect(), dmliv_featurizer(),\n n_splits=1, binary_instrument=True, binary_treatment=False)\n\nconst_dr_cate = DRIV(model_Y_X(), model_T_X(), model_Z_X(), # same as in DMLATEIV\n dmliv_prel_model_effect, # preliminary model for CATE, must support fit(y, T, X, Z) and effect(X)\n model_TZ_X(), # model for E[T * Z | X]\n const_driv_model_effect(), # model for final stage of fitting theta(X)\n cov_clip=COV_CLIP, # covariance clipping to avoid large values in final regression from weak instruments\n n_splits=N_SPLITS, # number of splits to use for cross-fitting\n binary_instrument=True, # a flag whether to stratify cross-fitting by instrument\n binary_treatment=False, # a flag whether to stratify cross-fitting by treatment\n opt_reweighted=False # whether to optimally re-weight samples. Valid only for flexible final model\n )\n```\n\n\n```python\nconst_dr_cate.fit(y, T, X, Z, store_final=True)\n```\n\n\n\n\n \n\n\n\n\n```python\n# To get the statsmodel summary we look at the effect_model, which is the pipeline, we then look\n# at the reg step of the pipeline which is the statsmodel wrapper and then we look\n# at the model attribute of the statsmodel wrapper and print the summary()\nconst_dr_cate.effect_model.summary()\n```\n\n C:\\ProgramData\\Anaconda3\\lib\\site-packages\\statsmodels\\regression\\linear_model.py:1554: RuntimeWarning: invalid value encountered in double_scalars\n return self.ess/self.df_model\n C:\\ProgramData\\Anaconda3\\lib\\site-packages\\scipy\\stats\\_distn_infrastructure.py:877: RuntimeWarning: invalid value encountered in greater\n return (self.a < x) & (x < self.b)\n C:\\ProgramData\\Anaconda3\\lib\\site-packages\\scipy\\stats\\_distn_infrastructure.py:877: RuntimeWarning: invalid value encountered in less\n return (self.a < x) & (x < self.b)\n C:\\ProgramData\\Anaconda3\\lib\\site-packages\\scipy\\stats\\_distn_infrastructure.py:1831: RuntimeWarning: invalid value encountered in less_equal\n cond2 = cond0 & (x <= self.a)\n\n\n\n\n\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
    OLS Regression Results
    Dep. Variable: y R-squared: 0.000
    Model: OLS Adj. R-squared: 0.000
    Method: Least Squares F-statistic: nan
    Date: Sat, 01 Jun 2019 Prob (F-statistic): nan
    Time: 16:39:54 Log-Likelihood: -6509.5
    No. Observations: 2991 AIC: 1.302e+04
    Df Residuals: 2990 BIC: 1.303e+04
    Df Model: 0
    Covariance Type: nonrobust
    \n\n\n \n\n\n \n\n
    coef std err t P>|t| [0.025 0.975]
    const 0.6501 0.039 16.668 0.000 0.574 0.727
    \n\n\n \n\n\n \n\n\n \n\n\n \n\n
    Omnibus: 2934.155 Durbin-Watson: 1.968
    Prob(Omnibus): 0.000 Jarque-Bera (JB): 1860178.825
    Skew: 3.818 Prob(JB): 0.00
    Kurtosis: 124.934 Cond. No. 1.00


    Warnings:
    [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n\n\n\n\n```python\ndef get_driv_coverage(true_effect, iteration):\n y, T = dgp_bin_Z_cont_T(X_raw, Z, hetero_col, true_fn, random_seed=iteration)\n dmliv_prel_model_effect = DMLIV(model_Y_X(), model_T_X(), model_T_XZ(),\n dmliv_model_effect(), dmliv_featurizer(),\n n_splits=1, binary_instrument=True, binary_treatment=True)\n const_dr_cate = DRIV(model_Y_X(), model_T_X(), model_Z_X(), # same as in DMLATEIV\n dmliv_prel_model_effect, # preliminary model for CATE, must support fit(y, T, X, Z) and effect(X)\n model_TZ_X(), # model for E[T * Z | X]\n const_driv_model_effect(), # model for final stage of fitting theta(X)\n cov_clip=COV_CLIP, # covariance clipping to avoid large values in final regression from weak instruments\n n_splits=N_SPLITS, # number of splits to use for cross-fitting\n binary_instrument=True, # a flag whether to stratify cross-fitting by instrument\n binary_treatment=False # a flag whether to stratify cross-fitting by treatment\n )\n const_dr_cate.fit(y, T, X, Z, store_final=True)\n left, right = const_dr_cate.effect_model.est.model.conf_int(alpha=0.05)[0]\n if true_effect >= left and true_effect <= right:\n return 1\n return 0\n```\n\n\n```python\nfrom joblib import Parallel, delayed\nn_experiments=100\ntrue_ate = np.mean(true_fn(X_raw))\nif True:\n contains_truth = np.array(Parallel(n_jobs=-1, verbose=3)(\n delayed(get_driv_coverage)(true_ate, it) for it in range(n_experiments)))\n print(\"Coverage: {}\".format(contains_truth.mean()))\n```\n\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 12 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 8 tasks | elapsed: 31.2s\n\n\n Coverage: 0.93\n\n\n [Parallel(n_jobs=-1)]: Done 100 out of 100 | elapsed: 4.3min finished\n\n\n## Projecting CATE to a pre-chosen subset of variables in final model\n\n\n```python\nfrom dml_iv import DMLIV\nfrom dr_iv import DRIV, ProjectedDRIV\nfrom utilities import SubsetWrapper\n\nnp.random.seed(random_seed)\n\n# We could also fit a projection on a subset of the features by using the\n# subset wrapper from our utilities.\n\n# Example: including everything for expository purposes, but any array-like of indices would work\nsubset_names = set(['motheduc', 'sinmom14'])\n# list of indices of features X to use in the final model\nfeature_inds = np.argwhere([(x in subset_names) for x in X_df.columns.values]).flatten() #[0] #np.arange(X.shape[1]) \nprint(feature_inds)\n# Because we are projecting to a low dimensional model space, we can\n# do valid inference and we can use statsmodel linear regression to get all\n# the hypothesis testing capability\nproj_driv_model_effect = lambda: SubsetWrapper(StatsModelLinearRegression(),\n feature_inds # list of indices of features X to use in the final model\n )\n```\n\n [4 7]\n\n\n\n```python\nX_df.columns\n```\n\n\n\n\n Index(['exper', 'expersq', 'fatheduc', 'fatheduc_nan', 'motheduc',\n 'motheduc_nan', 'momdad14', 'sinmom14', 'reg661', 'reg662', 'reg663',\n 'reg664', 'reg665', 'reg666', 'reg667', 'reg668', 'reg669', 'south66',\n 'black', 'smsa', 'south', 'smsa66'],\n dtype='object')\n\n\n\n\n```python\nproj_dr_cate = const_dr_cate.refit_final(proj_driv_model_effect())\n```\n\n\n```python\n# To get the CATE at every X we call effect(X[:, feature_inds])\nproj_dr_effect = proj_dr_cate.effect(X[:, feature_inds])\n```\n\n\n```python\n# To get the statsmodel summary we look at the effect_model, which is\n# an instance of SubsetWrapper, we look at the model of the SubsetWrapper which is \n# and instance of the pipeline, we then look at the reg step of the pipeline which is the statsmodel wrapper and\n# call summary() of the wrapper (most prob there is a better API for this, but we can go with this for now :)\nproj_dr_cate.effect_model.summary(alpha=.05, xname=['const']+list(X_df.columns[feature_inds]))\n```\n\n\n\n\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
    OLS Regression Results
    Dep. Variable: y R-squared: 0.002
    Model: OLS Adj. R-squared: 0.001
    Method: Least Squares F-statistic: 2.934
    Date: Sat, 01 Jun 2019 Prob (F-statistic): 0.0533
    Time: 16:44:16 Log-Likelihood: -6506.5
    No. Observations: 2991 AIC: 1.302e+04
    Df Residuals: 2988 BIC: 1.304e+04
    Df Model: 2
    Covariance Type: nonrobust
    \n\n\n \n\n\n \n\n\n \n\n\n \n\n
    coef std err t P>|t| [0.025 0.975]
    const 0.6352 0.041 15.433 0.000 0.554 0.716
    motheduc 0.0915 0.040 2.306 0.021 0.014 0.169
    sinmom14 0.1403 0.131 1.070 0.285 -0.117 0.397
    \n\n\n \n\n\n \n\n\n \n\n\n \n\n
    Omnibus: 2951.753 Durbin-Watson: 1.972
    Prob(Omnibus): 0.000 Jarque-Bera (JB): 1886680.784
    Skew: 3.859 Prob(JB): 0.00
    Kurtosis: 125.798 Cond. No. 3.41


    Warnings:
    [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n\n\n\n\n```python\n# We can access the coefficient by looking at the coefficient attribute of the final step of the pipeline\nprint(\"Estimated Params: {}, {}\".format(proj_dr_cate.intercept_, proj_dr_cate.coef_))\n# True coefficients of projection\nprint(\"True Params: {}\".format(\n LinearRegression(fit_intercept=False).fit(PolynomialFeatures(degree=1,\n include_bias=True).fit_transform(X[:, feature_inds]),\n true_fn(X_raw)).coef_))\n```\n\n Estimated Params: 0.6351902202433224, [0.091538 0.14032943]\n True Params: [ 0.61740685 0.14934234 -0.1 ]\n\n\n\n```python\n# We can also evaluate coverage and create prediction intervals using statsmodels attributes\nfrom statsmodels.sandbox.regression.predstd import wls_prediction_std\nres = proj_dr_cate.effect_model.model\npredictions = res.get_prediction(PolynomialFeatures(degree=1, include_bias=True).fit_transform(X[:, feature_inds]))\nframe = predictions.summary_frame(alpha=0.05)\npred = frame['mean']\niv_l = frame['mean_ci_lower']\niv_u = frame['mean_ci_upper']\n\n# This is the true CATE functions\ntheta_true = true_fn(X_raw)\n# This is the true projection of the CATE function on the subspace of linear functions of the\n# subset of the features used in the projection\ntrue_proj = LinearRegression().fit(X[:, feature_inds], theta_true).predict(X[:, feature_inds])\n\n# Are we covering the true projection\ncovered = (true_proj <= iv_u) & (true_proj >= iv_l)\nprint(\"Coverage of True Projection: {:.2f}\".format(np.mean(covered)))\n\nfig, ax = plt.subplots(figsize=(8,6))\n\norder = np.argsort(X[:, feature_inds[0]])\nax.plot(X[order, feature_inds[0]], iv_u[order], 'r--')\nax.plot(X[order, feature_inds[0]], iv_l[order], 'r--')\nax.plot(X[order, feature_inds[0]], pred[order], 'g--.', label=\"pred\")\nax.plot(X[order, feature_inds[0]], theta_true[order], 'b-', label=\"True\", alpha=.3)\nax.plot(X[order, feature_inds[0]], true_proj[order], 'b-', label=\"TrueProj\", alpha=.3)\nax.legend(loc='best')\nplt.show()\n```\n\n# CATE via Re-Weighted DRIV: DRIV-RW\n\n## Lasso CATE\n\n\n```python\nfrom dml_iv import DMLIV\nfrom dr_iv import DRIV, ProjectedDRIV\nfrom utilities import SubsetWrapper, StatsModelLinearRegression, ConstantModel, WeightWrapper\nfrom sklearn.dummy import DummyRegressor\n\nnp.random.seed(random_seed)\n\n# For DRIV we need a model for predicting E[T*Z | X]. We use a classifier\nmodel_TZ_X = lambda: model()\n\n# We also need a model for the final regression that will fit the function theta(X)\n# This model now needs to accept sample weights at fit time.\nconst_driv_model_effect = lambda: WeightWrapper(Pipeline([('bias', PolynomialFeatures(degree=1, include_bias=True)),\n ('reg', SelectiveLasso(np.arange(1, X.shape[1]+1),\n LassoCV(cv=5, fit_intercept=False)))]))\n\n# As in OrthoDMLIV we need a perliminary estimator of the CATE.\n# We use a DMLIV estimator with no cross-fitting (n_splits=1)\ndmliv_prel_model_effect = DMLIV(model_Y_X(), model_T_X(), model_T_XZ(),\n dmliv_model_effect(), dmliv_featurizer(),\n n_splits=1, binary_instrument=True, binary_treatment=False)\n\nconst_dr_cate = DRIV(model_Y_X(), model_T_X(), model_Z_X(), # same as in DMLATEIV\n dmliv_prel_model_effect, # preliminary model for CATE, must support fit(y, T, X, Z) and effect(X)\n model_TZ_X(), # model for E[T * Z | X]\n const_driv_model_effect(), # model for final stage of fitting theta(X)\n cov_clip=COV_CLIP, # covariance clipping to avoid large values in final regression from weak instruments\n n_splits=N_SPLITS, # number of splits to use for cross-fitting\n binary_instrument=True, # a flag whether to stratify cross-fitting by instrument\n binary_treatment=False, # a flag whether to stratify cross-fitting by treatment\n opt_reweighted=True # whether to optimally re-weight samples. Valid only for flexible final model\n )\n```\n\n\n```python\nconst_dr_cate.fit(y, T, X, Z, store_final=True)\n```\n\n\n\n\n \n\n\n\n\n```python\n# We can average the CATE to get an ATE\ndr_effect = const_dr_cate.effect(X)\nprint(\"ATE Estimate: {:.3f}\".format(np.mean(dr_effect)))\nprint(\"True ATE: {:.3f}\".format(np.mean(true_fn(X_raw))))\n```\n\n ATE Estimate: 0.599\n True ATE: 0.609\n\n\n\n```python\n# We can also see how it compares to the true CATE at each target point and calculate MSE\nplt.title(\"DMLIV CATE as Function of {}: MSE={:.3f}\".format(X_df.columns[4], np.mean((dr_effect-true_fn(X_raw))**2)))\nplt.scatter(X[:, 4], dr_effect, label='est')\nplt.scatter(X[:, 4], true_fn(X_raw), label='true', alpha=.2)\nplt.legend()\nplt.show()\n```\n\n## Random Forest CATE\n\n\n```python\nfrom dml_iv import DMLIV\nfrom dr_iv import DRIV, ProjectedDRIV\nfrom utilities import SubsetWrapper\nfrom sklearn.ensemble import RandomForestRegressor, GradientBoostingRegressor\n \nnp.random.seed(random_seed)\n\nrf_driv_model_effect = lambda: RandomForestRegressor(n_estimators=5000, max_depth=3, min_impurity_decrease=0.00001,\n min_samples_leaf=100, bootstrap=True)\n```\n\n\n```python\nrf_dr_cate = const_dr_cate.refit_final(rf_driv_model_effect())\n```\n\n\n```python\nrf_dr_effect = rf_dr_cate.effect(X)\n```\n\n\n```python\nprint(\"ATE Estimate: {:.2f}\".format(np.mean(rf_dr_effect)))\nprint(\"True ATE: {:.2f}\".format(np.mean(true_fn(X_raw))))\n```\n\n ATE Estimate: 0.60\n True ATE: 0.61\n\n\n\n```python\nplt.title(\"DRIV CATE: MSE {:.2}\".format(np.mean((true_fn(X_raw) - rf_dr_effect)**2)))\nplt.scatter(X[:, 4], rf_dr_effect, label='est')\nplt.scatter(X[:, 4], true_fn(X_raw), label='true', alpha=.2)\nplt.legend()\nplt.show()\n```\n\n\n```python\nimport shap\nimport pandas as pd\n\nXdf = pd.DataFrame(X, columns=X_df.columns)\n# explain the model's predictions using SHAP values\nexplainer = shap.TreeExplainer(rf_dr_cate.effect_model)\nshap_values = explainer.shap_values(Xdf)\n\n# visualize the first prediction's explanation (use matplotlib=True to avoid Javascript)\nshap.force_plot(explainer.expected_value, shap_values[0,:], Xdf.iloc[0,:], matplotlib=True)\n```\n\n\n```python\nshap.summary_plot(shap_values, Xdf)\n```\n\n\n```python\nshap.summary_plot(shap_values, Xdf, plot_type='bar')\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "59fe505c6d63e7bcb7df429e6976c5addf382094", "size": 255095, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "prototypes/dml_iv/NLSYM_Semi_Synthetic_GBM.ipynb", "max_stars_repo_name": "khwilson/EconML", "max_stars_repo_head_hexsha": "80264390106a8c57e2286177a2c4f8a47b51a32e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1846, "max_stars_repo_stars_event_min_datetime": "2019-05-06T21:14:19.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T11:52:21.000Z", "max_issues_repo_path": "prototypes/dml_iv/NLSYM_Semi_Synthetic_GBM.ipynb", "max_issues_repo_name": "khwilson/EconML", "max_issues_repo_head_hexsha": "80264390106a8c57e2286177a2c4f8a47b51a32e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 393, "max_issues_repo_issues_event_min_datetime": "2019-05-08T00:55:32.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-31T14:26:16.000Z", "max_forks_repo_path": "prototypes/dml_iv/NLSYM_Semi_Synthetic_GBM.ipynb", "max_forks_repo_name": "khwilson/EconML", "max_forks_repo_head_hexsha": "80264390106a8c57e2286177a2c4f8a47b51a32e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 414, "max_forks_repo_forks_event_min_datetime": "2019-05-14T03:51:08.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-31T09:32:17.000Z", "avg_line_length": 167.0563195809, "max_line_length": 49500, "alphanum_fraction": 0.8622238774, "converted": true, "num_tokens": 9124, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746213017459, "lm_q2_score": 0.6926419894793246, "lm_q1q2_score": 0.43620834662202956}} {"text": "\\title{Floating Point (Q) format and Floating Point Rounding in myHDL}\n\\author{Steven K Armour}\n\\maketitle\n\n# Referances\nhttps://timetoexplore.net/blog/fixed-point-numbers-in-verilog\n\n\n```python\nfrom myhdl import *\nfrom myhdlpeek import Peeker\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\n%matplotlib inline\n\nfrom sympy import *\ninit_printing()\n\nfrom bitstring import BitArray\n\n#https://github.com/jrjohansson/version_information\n%load_ext version_information\n%version_information myhdl, myhdlpeek, numpy, pandas, matplotlib, sympy, bitstring\n```\n\n\n\n\n
    SoftwareVersion
    Python3.6.5 64bit [GCC 7.2.0]
    IPython6.4.0
    OSLinux 4.15.0 30 generic x86_64 with debian buster sid
    myhdl0.10
    myhdlpeek0.0.7
    numpy1.14.3
    pandas0.23.0
    matplotlib2.2.2
    sympy1.1.1
    bitstring3.1.5
    Sat Aug 25 17:54:05 2018 MDT
    \n\n\n\n\n```python\n#helper functions to read in the .v and .vhd generated files into python\ndef VerilogTextReader(loc, printresult=True):\n with open(f'{loc}.v', 'r') as vText:\n VerilogText=vText.read()\n if printresult:\n print(f'***Verilog modual from {loc}.v***\\n\\n', VerilogText)\n return VerilogText\n\ndef VHDLTextReader(loc, printresult=True):\n with open(f'{loc}.vhd', 'r') as vText:\n VerilogText=vText.read()\n if printresult:\n print(f'***VHDL modual from {loc}.vhd***\\n\\n', VerilogText)\n return VerilogText\n```\n\n\n```python\n#4 bit int, 4 bit float\nQ=(4,4)\nQlen=Q[0]+Q[1]\nQscale=2**(Q[1]); Qscale\n```\n\n# Postive Addition\n\n\n```python\na=3.6250; b=4.0625\nc=a+b; c\n```\n\n\n```python\naQ=int(a*Qscale); bQ=int(b*Qscale)\nf'aQ:{aQ}; bA:{bQ}'\n```\n\n\n\n\n 'aQ:58; bA:65'\n\n\n\n\n```python\naQBV=intbv(aQ)[Qlen:]; bQBV=intbv(bQ)[Qlen:]\nf'aQBV: {bin(aQBV, Qlen)}; bQBV: {bin(bQBV, Qlen)}'\n```\n\n\n\n\n 'aQBV: 00111010; bQBV: 01000001'\n\n\n\n\n```python\ncQ=aQBV+bQBV; cQ\n```\n\n\n```python\nc==cQ/Qscale\n```\n\n\n\n\n True\n\n\n\n\n```python\nclass AddPosTVGen():\n \"\"\"\n Class to generate postive random numbers to be Qed for testing \n \"\"\"\n def __init__(self, Q, N):\n \"\"\"\n Take in arguments and create output holds\n Args:\n Q(tuple): Q notation tuple where Q[0] is int bit len and Q[1] is\n dec bit len\n N(int): number of values to generate\n \"\"\"\n self.Q=Q; self.N=N\n self.Qlen=self.Q[0]+self.Q[1]; self.Qmax=2**self.Qlen\n self.Qscale=2**self.Q[1]\n \n self.aTV=np.zeros(0); self.aTVQ=np.zeros(0)\n self.bTV=np.zeros(0); self.bTVQ=np.zeros(0)\n self.cK=np.zeros(0); self.cKQ=np.zeros(0)\n\n def Genrator(self):\n \"\"\"\n Random Number genrator in floating point and supsequent Qed version\n \"\"\"\n self.V1=np.array((1/np.random.ranf())).round(decimals=self.Q[1])\n \n #needed to force np.random to generate a differint random num\n np.random.seed(np.random.randint(self.Qmax))\n \n self.V2=np.array((1/np.random.ranf())).round(decimals=self.Q[1])\n \n self.V1Q=(self.V1*self.Qscale).astype(int)\n self.V2Q=(self.V2*self.Qscale).astype(int)\n \n def GenratorCheckAndAdd(self):\n \"\"\"\n Cheacks if the sum of the two randome numbers generated are going to break the Qmax\n if they do dont append to retrun holds\n \"\"\"\n self.V1pV2=(self.V1+self.V2).round(decimals=self.Q[1])\n self.V1pV2Q=(self.V1pV2*self.Qscale).astype(int)\n if (self.V1Q+self.V2Q)\n\n\n\n\n\n```python\nPAOData=Peeker.to_dataframe()\n#load in the source floating values\nPAOData['aTV']=aTV; PAOData['bTV']=bTV\n#get the predicted floating Point Sum\nPAOData['aTV+bTV']=aTV+bTV\n#get the predicted fixed point sum\nPAOData['aQ+bQ']=aTVQ+bTVQ\n#reorder\nPAOData=PAOData[['a', 'aTV', 'b', 'bTV', 'aTV+bTV', 'aQ+bQ', 'c']]\n#load the sourced Qed sum\nPAOData['cKTVQ']=cKTVQ\n#de Q the testbench gen sum \nPAOData['cdQ']=PAOData['c']/Qscale\n#load the sourced floting sum\nPAOData['cKTV']=cKTV\nPAOData\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    aaTVbbTVaTV+bTVaQ+bQccKTVQcdQcKTV
    0332.0671332.06714.13426666764.1254.7942
    1251.5798251.57983.15965050503.1253.1322
    2422.6264422.62645.25288484975.2506.0666
    3201.2822201.28222.56444040492.5003.1142
    4181.1643181.16432.328636362282.25014.2690
    5201.2822201.28222.56444040492.5003.1142
    6181.1643181.16432.328636362282.25014.2690
    7201.2822201.28222.56444040492.5003.1142
    8181.1643181.16432.328636362282.25014.2690
    9201.2822201.28222.56444040492.5003.1142
    10181.1643181.16432.328636362282.25014.2690
    11201.2822201.28222.56444040492.5003.1142
    12181.1643181.16432.328636362282.25014.2690
    13201.2822201.28222.56444040492.5003.1142
    14181.1643181.16432.328636362282.25014.2690
    15201.2822201.28222.56444040492.5003.1142
    16181.1643181.16432.328636362282.25014.2690
    17201.2822201.28222.56444040492.5003.1142
    18181.1643181.16432.328636362282.25014.2690
    19201.2822201.28222.56444040492.5003.1142
    20181.1643181.16432.328636362282.25014.2690
    21201.2822201.28222.56444040492.5003.1142
    22181.1643181.16432.328636362282.25014.2690
    23201.2822201.28222.56444040492.5003.1142
    24181.1643181.16432.328636362282.25014.2690
    25201.2822201.28222.56444040492.5003.1142
    26181.1643181.16432.328636362282.25014.2690
    27201.2822201.28222.56444040492.5003.1142
    28181.1643181.16432.328636362282.25014.2690
    29201.2822201.28222.56444040492.5003.1142
    .................................
    71201.2822201.28222.56444040492.5003.1142
    72181.1643181.16432.328636362282.25014.2690
    73201.2822201.28222.56444040492.5003.1142
    74181.1643181.16432.328636362282.25014.2690
    75201.2822201.28222.56444040492.5003.1142
    76181.1643181.16432.328636362282.25014.2690
    77201.2822201.28222.56444040492.5003.1142
    78181.1643181.16432.328636362282.25014.2690
    79201.2822201.28222.56444040492.5003.1142
    80181.1643181.16432.328636362282.25014.2690
    81201.2822201.28222.56444040492.5003.1142
    82181.1643181.16432.328636362282.25014.2690
    83201.2822201.28222.56444040492.5003.1142
    84181.1643181.16432.328636362282.25014.2690
    85201.2822201.28222.56444040492.5003.1142
    86181.1643181.16432.328636362282.25014.2690
    87201.2822201.28222.56444040492.5003.1142
    88181.1643181.16432.328636362282.25014.2690
    89201.2822201.28222.56444040492.5003.1142
    90181.1643181.16432.328636362282.25014.2690
    91201.2822201.28222.56444040492.5003.1142
    92181.1643181.16432.328636362282.25014.2690
    93201.2822201.28222.56444040492.5003.1142
    94181.1643181.16432.328636362282.25014.2690
    95201.2822201.28222.56444040492.5003.1142
    96181.1643181.16432.328636362282.25014.2690
    97201.2822201.28222.56444040492.5003.1142
    98181.1643181.16432.328636362282.25014.2690
    99201.2822201.28222.56444040492.5003.1142
    100181.1643181.16432.328636362282.25014.2690
    \n

    101 rows \u00d7 10 columns

    \n
    \n\n\n\n\n```python\n#dataframe of error measures\nPAODataErr=pd.DataFrame()\nPAODataErr['aQ+bQ_c']=np.abs(PAOData['aQ+bQ']-PAOData['c'])\nPAODataErr['c_cKTVQ']=np.abs(PAOData['c']-PAOData['cKTVQ'])\nPAODataErr['cdQ_cKTV']=np.abs(PAOData['cdQ']-PAOData['cKTV'])\nPAODataErr['c_cKTVQ__cdQ_cKTV']=np.abs((PAODataErr['c_cKTVQ']/ Qscale)- PAODataErr['cdQ_cKTV'])\nPAODataErr\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    aQ+bQ_cc_cKTVQcdQ_cKTVc_cKTVQ__cdQ_cKTV
    00100.66920.0442
    1000.00720.0072
    20130.81660.0041
    3090.61420.0517
    4019212.01900.0190
    5090.61420.0517
    6019212.01900.0190
    7090.61420.0517
    8019212.01900.0190
    9090.61420.0517
    10019212.01900.0190
    11090.61420.0517
    12019212.01900.0190
    13090.61420.0517
    14019212.01900.0190
    15090.61420.0517
    16019212.01900.0190
    17090.61420.0517
    18019212.01900.0190
    19090.61420.0517
    20019212.01900.0190
    21090.61420.0517
    22019212.01900.0190
    23090.61420.0517
    24019212.01900.0190
    25090.61420.0517
    26019212.01900.0190
    27090.61420.0517
    28019212.01900.0190
    29090.61420.0517
    ...............
    71090.61420.0517
    72019212.01900.0190
    73090.61420.0517
    74019212.01900.0190
    75090.61420.0517
    76019212.01900.0190
    77090.61420.0517
    78019212.01900.0190
    79090.61420.0517
    80019212.01900.0190
    81090.61420.0517
    82019212.01900.0190
    83090.61420.0517
    84019212.01900.0190
    85090.61420.0517
    86019212.01900.0190
    87090.61420.0517
    88019212.01900.0190
    89090.61420.0517
    90019212.01900.0190
    91090.61420.0517
    92019212.01900.0190
    93090.61420.0517
    94019212.01900.0190
    95090.61420.0517
    96019212.01900.0190
    97090.61420.0517
    98019212.01900.0190
    99090.61420.0517
    100019212.01900.0190
    \n

    101 rows \u00d7 4 columns

    \n
    \n\n\n\n\n```python\nPAODataErr.describe()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    aQ+bQ_cc_cKTVQcdQ_cKTVc_cKTVQ__cdQ_cKTV
    count101.0101.000000101.000000101.000000
    mean0.097.7425746.1437600.034850
    std0.091.9597365.7320470.016739
    min0.00.0000000.0072000.004100
    25%0.09.0000000.6142000.019000
    50%0.010.0000000.6692000.019000
    75%0.0192.00000012.0190000.051700
    max0.0192.00000012.0190000.051700
    \n
    \n\n\n\n\n```python\nDUT.convert()\nVerilogTextReader('AdderBehaverial');\n```\n\n ***Verilog modual from AdderBehaverial.v***\n \n // File: AdderBehaverial.v\n // Generated by MyHDL 0.10\n // Date: Sat Aug 25 17:54:15 2018\n \n \n `timescale 1ns/10ps\n \n module AdderBehaverial (\n a,\n b,\n c\n );\n \n \n input [7:0] a;\n input [7:0] b;\n output [7:0] c;\n wire [7:0] c;\n \n \n \n \n \n assign c = (a + b);\n \n endmodule\n \n\n\n# Negative Values\n\n\n```python\na=3.6250; aQ=int(a*Qscale);a, aQ\n```\n\n\n```python\nb=-1.5; bMagQ=int(abs(b)*Qscale); bMagQ\n```\n\n\n```python\nbMagQBV=bin(bMagQ, Qlen); bMagQBV\n```\n\n\n\n\n '00011000'\n\n\n\n\n```python\nbQBVComp=\"\".join([str(int(not(int(i)))) for i in bMagQBV]); bQBVComp\n```\n\n\n\n\n '11100111'\n\n\n\n\n```python\nbQComp=int(bQBVComp, 2); bQComp\n```\n\n\n```python\nbQ2Comp=bQComp+1; bQ2Comp\n```\n\n\n```python\nbQBV2Comp=bin(bQ2Comp, 2); bQBV2Comp\n```\n\n\n\n\n '11101000'\n\n\n\n\n```python\n(BitArray(bin=bQBV2Comp).int)/ Qscale\n```\n\n\n```python\naQBV=intbv(aQ)[Qlen:].signed()\naQBV, bin(aQBV, Qlen), aQBV.min, aQBV.max\n```\n\n\n\n\n (intbv(58), '00111010', -128, 128)\n\n\n\n\n```python\nbQBV=intbv(int(b*Qscale))[Qlen:].signed()\nbQBV, bin(bQBV, Qlen)\n```\n\n\n\n\n (intbv(-24), '11101000')\n\n\n\n\n```python\nbQBV2Comp==bin(bQBV, Qlen)\n```\n\n\n\n\n True\n\n\n\n\n```python\na+b\n```\n\n\n```python\nc=aQBV+bQBV; c, c/Qscale\n```\n\n\n```python\nclass AddPosNegTVGen():\n \"\"\"\n Class to generate postive random numbers to be Qed for testing \n \"\"\"\n def __init__(self, Q, N):\n \"\"\"\n Take in arguments and create output holds\n Args:\n Q(tuple): Q notation tuple where Q[0] is int bit len and Q[1] is\n dec bit len\n N(int): number of values to generate\n \"\"\"\n self.Q=Q; self.N=N\n self.Qlen=self.Q[0]+self.Q[1]\n self.Qmin=-(2**(Qlen-1)); self.Qmax=2**(self.Qlen-1) -1\n self.Qscale=2**self.Q[1]\n \n self.aTV=np.zeros(0); self.aTVQ=np.zeros(0)\n self.bTV=np.zeros(0); self.bTVQ=np.zeros(0)\n self.cK=np.zeros(0); self.cKQ=np.zeros(0)\n\n def Genrator(self):\n \"\"\"\n Random Number genrator in floating point and supsequent Qed version\n \"\"\"\n self.V1=np.array((1/np.random.ranf())).round(decimals=self.Q[1])\n \n #needed to force np.random to generate a differint random num\n np.random.seed(np.random.randint(self.Qmax))\n \n self.V2=np.array((1/np.random.ranf())).round(decimals=self.Q[1])\n \n #needed to force np.random to generate a differint random num\n np.random.seed(np.random.randint(self.Qmax))\n \n self.Sign=np.random.randint(2)\n if self.Sign==1:\n self.V2=-self.V2\n self.V1Q=(self.V1*self.Qscale).astype(int)\n self.V2Q=(self.V2*self.Qscale).astype(int)\n \n def GenratorCheckAndAdd(self):\n \"\"\"\n Cheacks if the sum of the two randome numbers generated are going to break the Qmax\n if they do dont append to retrun holds\n \"\"\"\n self.V1pV2=(self.V1+self.V2).round(decimals=self.Q[1])\n self.V1pV2Q=(self.V1pV2*self.Qscale).astype(int)\n \n check=self.V1Q+self.V2Q\n if self.V1Q>self.Qmin and self.V1Qself.Qmin and self.V2Qself.Qmin and check\n\n\n\n\n\n```python\nAOData=Peeker.to_dataframe()\n#load in the source floating values\nAOData['aTV']=aTV; AOData['bTV']=bTV\n#get the predicted floating Point Sum\nAOData['aTV+bTV']=aTV+bTV\n#get the predicted fixed point sum\nAOData['aQ+bQ']=aTVQ+bTVQ\n#reorder\nAOData=AOData[['a', 'aTV', 'b', 'bTV', 'aTV+bTV', 'aQ+bQ', 'c']]\n#load the sourced Qed sum\nAOData['cKTVQ']=cKTVQ\n#de Q the testbench gen sum \nAOData['cdQ']=AOData['c']/Qscale\n#load the sourced floting sum\nAOData['cKTV']=cKTV\nAOData\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    aaTVbbTVaTV+bTVaQ+bQccKTVQcdQcKTV
    0201.2822201.83203.11424040492.5003.1142
    1382.377638-1.55600.82167676134.7500.8216
    2623.8753622.66556.54081241241047.7506.5408
    3221.436622-1.17810.2585444442.7500.2585
    4352.2410351.97354.21457070674.3754.2145
    5493.080649-1.80521.27549898206.1251.2754
    6231.439723-1.14930.2904464642.8750.2904
    7191.220619-3.4958-2.27523838-362.375-2.2752
    8483.0613483.37556.436896961026.0006.4368
    9523.2753521.29654.5718104104736.5004.5718
    10231.490023-2.3142-0.82424646-132.875-0.8242
    11372.323137-1.36560.95757474154.6250.9575
    12191.2145191.28582.50033838402.3752.5003
    13382.377638-1.55600.82167676134.7500.8216
    14623.8753622.66556.54081241241047.7506.5408
    15221.436622-1.17810.2585444442.7500.2585
    16352.2410351.97354.21457070674.3754.2145
    17493.080649-1.80521.27549898206.1251.2754
    18231.439723-1.14930.2904464642.8750.2904
    19191.220619-3.4958-2.27523838-362.375-2.2752
    20483.0613483.37556.436896961026.0006.4368
    21523.2753521.29654.5718104104736.5004.5718
    22231.490023-2.3142-0.82424646-132.875-0.8242
    23372.323137-1.36560.95757474154.6250.9575
    24191.2145191.28582.50033838402.3752.5003
    25382.377638-1.55600.82167676134.7500.8216
    26623.8753622.66556.54081241241047.7506.5408
    27221.436622-1.17810.2585444442.7500.2585
    28352.2410351.97354.21457070674.3754.2145
    29493.080649-1.80521.27549898206.1251.2754
    .................................
    71372.323137-1.36560.95757474154.6250.9575
    72191.2145191.28582.50033838402.3752.5003
    73382.377638-1.55600.82167676134.7500.8216
    74623.8753622.66556.54081241241047.7506.5408
    75221.436622-1.17810.2585444442.7500.2585
    76352.2410351.97354.21457070674.3754.2145
    77493.080649-1.80521.27549898206.1251.2754
    78231.439723-1.14930.2904464642.8750.2904
    79191.220619-3.4958-2.27523838-362.375-2.2752
    80483.0613483.37556.436896961026.0006.4368
    81523.2753521.29654.5718104104736.5004.5718
    82231.490023-2.3142-0.82424646-132.875-0.8242
    83372.323137-1.36560.95757474154.6250.9575
    84191.2145191.28582.50033838402.3752.5003
    85382.377638-1.55600.82167676134.7500.8216
    86623.8753622.66556.54081241241047.7506.5408
    87221.436622-1.17810.2585444442.7500.2585
    88352.2410351.97354.21457070674.3754.2145
    89493.080649-1.80521.27549898206.1251.2754
    90231.439723-1.14930.2904464642.8750.2904
    91191.220619-3.4958-2.27523838-362.375-2.2752
    92483.0613483.37556.436896961026.0006.4368
    93523.2753521.29654.5718104104736.5004.5718
    94231.490023-2.3142-0.82424646-132.875-0.8242
    95372.323137-1.36560.95757474154.6250.9575
    96191.2145191.28582.50033838402.3752.5003
    97382.377638-1.55600.82167676134.7500.8216
    98623.8753622.66556.54081241241047.7506.5408
    99221.436622-1.17810.2585444442.7500.2585
    100352.2410351.97354.21457070674.3754.2145
    \n

    101 rows \u00d7 10 columns

    \n
    \n\n\n\n\n```python\n#dataframe of error measures\nAODataErr=pd.DataFrame()\nAODataErr['aQ+bQ_c']=np.abs(AOData['aQ+bQ']-AOData['c'])\nAODataErr['c_cKTVQ']=np.abs(AOData['c']-AOData['cKTVQ'])\nAODataErr['cdQ_cKTV']=np.abs(AOData['cdQ']-AOData['cKTV'])\nAODataErr['c_cKTVQ__cdQ_cKTV']=np.abs((AODataErr['c_cKTVQ']/ Qscale)- AODataErr['cdQ_cKTV'])\nAODataErr\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    aQ+bQ_cc_cKTVQcdQ_cKTVc_cKTVQ__cdQ_cKTV
    0090.61420.0517
    10633.92840.0091
    20201.20920.0408
    30402.49150.0085
    4030.16050.0270
    50784.84960.0254
    60422.58460.0404
    70744.65020.0252
    8060.43680.0618
    90311.92820.0093
    100593.69920.0117
    110593.66750.0200
    12020.12530.0003
    130633.92840.0091
    140201.20920.0408
    150402.49150.0085
    16030.16050.0270
    170784.84960.0254
    180422.58460.0404
    190744.65020.0252
    20060.43680.0618
    210311.92820.0093
    220593.69920.0117
    230593.66750.0200
    24020.12530.0003
    250633.92840.0091
    260201.20920.0408
    270402.49150.0085
    28030.16050.0270
    290784.84960.0254
    ...............
    710593.66750.0200
    72020.12530.0003
    730633.92840.0091
    740201.20920.0408
    750402.49150.0085
    76030.16050.0270
    770784.84960.0254
    780422.58460.0404
    790744.65020.0252
    80060.43680.0618
    810311.92820.0093
    820593.69920.0117
    830593.66750.0200
    84020.12530.0003
    850633.92840.0091
    860201.20920.0408
    870402.49150.0085
    88030.16050.0270
    890784.84960.0254
    900422.58460.0404
    910744.65020.0252
    92060.43680.0618
    930311.92820.0093
    940593.69920.0117
    950593.66750.0200
    96020.12530.0003
    970633.92840.0091
    980201.20920.0408
    990402.49150.0085
    100030.16050.0270
    \n

    101 rows \u00d7 4 columns

    \n
    \n\n\n\n\n```python\nAODataErr.describe()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    aQ+bQ_cc_cKTVQcdQ_cKTVc_cKTVQ__cdQ_cKTV
    count101.0101.000000101.000000101.000000
    mean0.039.1188122.4381370.023496
    std0.026.3857111.6467840.016939
    min0.02.0000000.1253000.000300
    25%0.09.0000000.6142000.009100
    50%0.040.0000002.4915000.025200
    75%0.059.0000003.6992000.040400
    max0.078.0000004.8496000.061800
    \n
    \n\n\n\n\n```python\nDUT.convert()\nVerilogTextReader('AdderBehaverial');\n```\n\n ***Verilog modual from AdderBehaverial.v***\n \n // File: AdderBehaverial.v\n // Generated by MyHDL 0.10\n // Date: Sat Aug 25 17:54:24 2018\n \n \n `timescale 1ns/10ps\n \n module AdderBehaverial (\n a,\n b,\n c\n );\n \n \n input signed [7:0] a;\n input signed [7:0] b;\n output signed [7:0] c;\n wire signed [7:0] c;\n \n \n \n \n \n assign c = (a + b);\n \n endmodule\n \n\n\n# Multiblication\n\n\n```python\n#Q4.4 *Q4.4 -> Q8.8\nQ2=(Q[0]*2, Q[1]*2)\nQ2len=Q2[0]+Q2[1]\nQ2scale=2**(Q2[1]); Q2scale\n```\n\n\n```python\na=3.2500; aQ=int(a*Qscale)\nb=-2.065; bQ=int(b*Qscale)\naQ, bQ\nbin(aQ, Qlen), bin(bQ, Qlen)\n```\n\n\n\n\n ('00110100', '11011111')\n\n\n\n\n```python\nab=a*b; ab\nabQ=int(ab*Qscale); abQ\nabdQ=abQ/ Qscale; abdQ, ab\n```\n\n\n```python\naQBV=intbv(aQ)[Qlen:].signed(); bQBV=intbv(bQ)[Qlen:].signed()\nf'aQBV: {bin(aQBV, Qlen)}; bQBV: {bin(bQBV, Qlen)}'\n```\n\n\n\n\n 'aQBV: 00110100; bQBV: 11011111'\n\n\n\n\n```python\nabQ=aQBV*bQBV; abQ\n```\n\n\n```python\nabdQ=abQ/ Qscale; abdQ, ab\n```\n\n\n```python\nabdQ=abQ/ Q2scale; abdQ,ab\n```\n\n\n```python\nclass MultPosNegTVGen():\n \"\"\"\n Class to generate postive random numbers to be Qed for testing \n \"\"\"\n def __init__(self, Q, N):\n \"\"\"\n Take in arguments and create output holds\n Args:\n Q(tuple): Q notation tuple where Q[0] is int bit len and Q[1] is\n dec bit len\n N(int): number of values to generate\n \"\"\"\n self.Q=Q; self.N=N\n self.Qlen=self.Q[0]+self.Q[1]\n self.Qmin=-(2**(self.Qlen-1)); self.Qmax=2**(self.Qlen-1) -1\n self.Qscale=2**self.Q[1]\n \n #Q4.4 *Q4.4 -> Q8.8\n self.Q2=(self.Q[0]*2, self.Q[1]*2)\n self.Q2len=self.Q2[0]+self.Q2[1]\n self.Q2min=-(2**(self.Q2len-1)); self.Q2max=2**(self.Q2len-1) -1\n self.Q2scale=2**(Q2[1])\n \n \n \n self.aTV=np.zeros(0); self.aTVQ=np.zeros(0)\n self.bTV=np.zeros(0); self.bTVQ=np.zeros(0)\n self.cK=np.zeros(0); self.cKQ=np.zeros(0)\n\n def Genrator(self):\n \"\"\"\n Random Number genrator in floating point and supsequent Qed version\n \"\"\"\n self.V1=np.array((1/np.random.ranf())).round(decimals=self.Q[1])\n \n #needed to force np.random to generate a differint random num\n np.random.seed(np.random.randint(self.Qmax))\n \n self.V2=np.array((1/np.random.ranf())).round(decimals=self.Q[1])\n \n #needed to force np.random to generate a differint random num\n np.random.seed(np.random.randint(self.Qmax))\n \n self.Sign=np.random.randint(2)\n if self.Sign==1:\n self.V2=-self.V2\n self.V1Q=(self.V1*self.Qscale).astype(int)\n self.V2Q=(self.V2*self.Qscale).astype(int)\n \n def GenratorCheckAndMul(self):\n \"\"\"\n Cheacks if the sum of the two randome numbers generated are going to break the Qmax\n if they do dont append to retrun holds\n \"\"\"\n self.V1tV2=(self.V1*self.V2).round(decimals=self.Q2[1])\n self.V1tV2Q=(self.V1tV2*self.Q2scale).astype(int)\n check=self.V1Q*self.V2Q\n if self.V1Q>self.Qmin and self.V1Qself.Qmin and self.V2Qself.Q2min and check\n\n\n\n\n\n```python\nMultiData=Peeker.to_dataframe()\n#load in the source floating values\nMultiData['aTV']=aTV; MultiData['bTV']=bTV\n#get the predicted floating Point Sum\nMultiData['aTV*bTV']=aTV*bTV\n#get the predicted fixed point sum\nMultiData['aQ*bQ']=aTVQ*bTVQ\n#reorder\nMultiData=MultiData[['a', 'aTV', 'b', 'bTV', 'aTV*bTV', 'aQ*bQ', 'c']]\n#load the sourced Qed sum\nMultiData['cKTVQ']=cKTVQ\n#de Q the testbench gen sum \nMultiData['cdQ']=MultiData['c']/Q2scale\n#load the sourced floting sum\nMultiData['cKTV']=cKTV\nMultiData\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    aaTVbbTVaTV*bTVaQ*bQccKTVQcdQcKTV
    0493.0806-28-1.8052-5.561099-1372-1372-1423-5.359375-5.561099
    1231.4397-18-1.1493-1.654647-414-414-423-1.617188-1.654647
    2191.2206-55-3.4958-4.266973-1045-1045-1092-4.082031-4.266973
    3483.0613543.375510.33341825922592264510.12500010.333418
    4523.2753201.29654.2464261040104010874.0625004.246426
    5231.4900-37-2.3142-3.448158-851-851-882-3.324219-3.448158
    6372.3231-21-1.3656-3.172425-777-777-812-3.035156-3.172425
    7191.2145201.28581.5616043803803991.4843751.561604
    8382.3776-24-1.5560-3.699546-912-912-947-3.562500-3.699546
    9623.8753422.665510.32961226042604264410.17187510.329612
    10221.4366-18-1.1781-1.692458-396-396-433-1.546875-1.692458
    11352.2410311.97354.4226141085108511324.2382814.422613
    12493.0806-28-1.8052-5.561099-1372-1372-1423-5.359375-5.561099
    13231.4397-18-1.1493-1.654647-414-414-423-1.617188-1.654647
    14191.2206-55-3.4958-4.266973-1045-1045-1092-4.082031-4.266973
    15483.0613543.375510.33341825922592264510.12500010.333418
    16523.2753201.29654.2464261040104010874.0625004.246426
    17231.4900-37-2.3142-3.448158-851-851-882-3.324219-3.448158
    18372.3231-21-1.3656-3.172425-777-777-812-3.035156-3.172425
    19191.2145201.28581.5616043803803991.4843751.561604
    20382.3776-24-1.5560-3.699546-912-912-947-3.562500-3.699546
    21623.8753422.665510.32961226042604264410.17187510.329612
    22221.4366-18-1.1781-1.692458-396-396-433-1.546875-1.692458
    23352.2410311.97354.4226141085108511324.2382814.422613
    24493.0806-28-1.8052-5.561099-1372-1372-1423-5.359375-5.561099
    25231.4397-18-1.1493-1.654647-414-414-423-1.617188-1.654647
    26191.2206-55-3.4958-4.266973-1045-1045-1092-4.082031-4.266973
    27483.0613543.375510.33341825922592264510.12500010.333418
    28523.2753201.29654.2464261040104010874.0625004.246426
    29231.4900-37-2.3142-3.448158-851-851-882-3.324219-3.448158
    .................................
    71352.2410311.97354.4226141085108511324.2382814.422613
    72493.0806-28-1.8052-5.561099-1372-1372-1423-5.359375-5.561099
    73231.4397-18-1.1493-1.654647-414-414-423-1.617188-1.654647
    74191.2206-55-3.4958-4.266973-1045-1045-1092-4.082031-4.266973
    75483.0613543.375510.33341825922592264510.12500010.333418
    76523.2753201.29654.2464261040104010874.0625004.246426
    77231.4900-37-2.3142-3.448158-851-851-882-3.324219-3.448158
    78372.3231-21-1.3656-3.172425-777-777-812-3.035156-3.172425
    79191.2145201.28581.5616043803803991.4843751.561604
    80382.3776-24-1.5560-3.699546-912-912-947-3.562500-3.699546
    81623.8753422.665510.32961226042604264410.17187510.329612
    82221.4366-18-1.1781-1.692458-396-396-433-1.546875-1.692458
    83352.2410311.97354.4226141085108511324.2382814.422613
    84493.0806-28-1.8052-5.561099-1372-1372-1423-5.359375-5.561099
    85231.4397-18-1.1493-1.654647-414-414-423-1.617188-1.654647
    86191.2206-55-3.4958-4.266973-1045-1045-1092-4.082031-4.266973
    87483.0613543.375510.33341825922592264510.12500010.333418
    88523.2753201.29654.2464261040104010874.0625004.246426
    89231.4900-37-2.3142-3.448158-851-851-882-3.324219-3.448158
    90372.3231-21-1.3656-3.172425-777-777-812-3.035156-3.172425
    91191.2145201.28581.5616043803803991.4843751.561604
    92382.3776-24-1.5560-3.699546-912-912-947-3.562500-3.699546
    93623.8753422.665510.32961226042604264410.17187510.329612
    94221.4366-18-1.1781-1.692458-396-396-433-1.546875-1.692458
    95352.2410311.97354.4226141085108511324.2382814.422613
    96493.0806-28-1.8052-5.561099-1372-1372-1423-5.359375-5.561099
    97231.4397-18-1.1493-1.654647-414-414-423-1.617188-1.654647
    98191.2206-55-3.4958-4.266973-1045-1045-1092-4.082031-4.266973
    99483.0613543.375510.33341825922592264510.12500010.333418
    100523.2753201.29654.2464261040104010874.0625004.246426
    \n

    101 rows \u00d7 10 columns

    \n
    \n\n\n\n\n```python\n#dataframe of error measures\nMultiDataErr=pd.DataFrame()\nMultiDataErr['aQ*bQ_c']=np.abs(MultiData['aQ*bQ']-MultiData['c'])\nMultiDataErr['c_cKTVQ']=np.abs(MultiData['c']-MultiData['cKTVQ'])\nMultiDataErr['cdQ_cKTV']=np.abs(MultiData['cdQ']-MultiData['cKTV'])\nMultiDataErr['c_cKTVQ__cdQ_cKTV']=np.abs((MultiDataErr['c_cKTVQ']/ Q2scale)- MultiDataErr['cdQ_cKTV'])\nMultiDataErr\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    aQ*bQ_cc_cKTVQcdQ_cKTVc_cKTVQ__cdQ_cKTV
    00510.2017240.002505
    1090.0374600.002303
    20470.1849420.001348
    30530.2084180.001387
    40470.1839260.000333
    50310.1239390.002845
    60350.1372690.000550
    70190.0772290.003010
    80350.1370460.000327
    90400.1577370.001487
    100370.1455830.001052
    110470.1843320.000738
    120510.2017240.002505
    13090.0374600.002303
    140470.1849420.001348
    150530.2084180.001387
    160470.1839260.000333
    170310.1239390.002845
    180350.1372690.000550
    190190.0772290.003010
    200350.1370460.000327
    210400.1577370.001487
    220370.1455830.001052
    230470.1843320.000738
    240510.2017240.002505
    25090.0374600.002303
    260470.1849420.001348
    270530.2084180.001387
    280470.1839260.000333
    290310.1239390.002845
    ...............
    710470.1843320.000738
    720510.2017240.002505
    73090.0374600.002303
    740470.1849420.001348
    750530.2084180.001387
    760470.1839260.000333
    770310.1239390.002845
    780350.1372690.000550
    790190.0772290.003010
    800350.1370460.000327
    810400.1577370.001487
    820370.1455830.001052
    830470.1843320.000738
    840510.2017240.002505
    85090.0374600.002303
    860470.1849420.001348
    870530.2084180.001387
    880470.1839260.000333
    890310.1239390.002845
    900350.1372690.000550
    910190.0772290.003010
    920350.1370460.000327
    930400.1577370.001487
    940370.1455830.001052
    950470.1843320.000738
    960510.2017240.002505
    97090.0374600.002303
    980470.1849420.001348
    990530.2084180.001387
    1000470.1839260.000333
    \n

    101 rows \u00d7 4 columns

    \n
    \n\n\n\n\n```python\nMultiDataErr.describe()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    aQ*bQ_cc_cKTVQcdQ_cKTVc_cKTVQ__cdQ_cKTV
    count101.0101.000000101.000000101.000000
    mean0.037.7722770.1490430.001495
    std0.012.9150150.0500540.000920
    min0.09.0000000.0374600.000327
    25%0.035.0000000.1370460.000738
    50%0.040.0000000.1577370.001387
    75%0.047.0000000.1849420.002303
    max0.053.0000000.2084180.003010
    \n
    \n\n\n\n\n```python\nDUT.convert()\nVerilogTextReader('MultiBehaverial');\n```\n\n ***Verilog modual from MultiBehaverial.v***\n \n // File: MultiBehaverial.v\n // Generated by MyHDL 0.10\n // Date: Sat Aug 25 17:54:33 2018\n \n \n `timescale 1ns/10ps\n \n module MultiBehaverial (\n a,\n b,\n c\n );\n \n \n input signed [7:0] a;\n input signed [7:0] b;\n output signed [15:0] c;\n wire signed [15:0] c;\n \n \n \n \n \n assign c = (a * b);\n \n endmodule\n \n\n\n# Trunction (Unsighned)\n\n\n```python\n#Q4.4 *Q4.4 -> Q8.8\nQ2=(Q[0]*2, Q[1]*2)\nQ2len=Q2[0]+Q2[1]\nQ2scale=2**(Q2[1]); Q2scale\n```\n\n\n```python\na=3.2500; aQ=int(a*Qscale)\nb=2.0625; bQ=int(b*Qscale)\naQ, bQ\n#bin(aQ, Qlen), bin(bQ, Qlen)\n```\n\n\n```python\nab=a*b; ab\nabQ=int(ab*Qscale); abQ\nabdQ=abQ/ Qscale; abdQ, ab\n```\n\n\n```python\naQBV=intbv(aQ)[Qlen:]; bQBV=intbv(bQ)[Qlen:]\nf'aQBV: {bin(aQBV, Qlen)}; bQBV: {bin(bQBV, Qlen)}'\n```\n\n\n\n\n 'aQBV: 00110100; bQBV: 00100001'\n\n\n\n\n```python\nabQ=aQBV*bQBV; abQ\n```\n\n\n```python\nabQBV=intbv(abQ)[Q2len:].signed(); abQBV, bin(abQBV), len(bin(abQBV))\n```\n\n\n\n\n (intbv(1716), '11010110100', 11)\n\n\n\n\n```python\nfor j in range(Q2[1]):\n Trunc=abQBV[Q2len:j]\n TruncDQ=Trunc/(2**(Q2[1]-j))\n print(bin(Trunc), TruncDQ, np.abs(ab-TruncDQ))\n```\n\n 11010110100 6.703125 0.0\n 1101011010 6.703125 0.0\n 110101101 6.703125 0.0\n 11010110 6.6875 0.015625\n 1101011 6.6875 0.015625\n 110101 6.625 0.078125\n 11010 6.5 0.203125\n 1101 6.5 0.203125\n\n\n# Trunction (sighned)\n\n\n```python\na=3.2500; aQ=int(a*Qscale)\nb=-2.0625; bQ=int(b*Qscale)\naQ, bQ\n#bin(aQ, Qlen), bin(bQ, Qlen)\n```\n\n\n```python\nab=a*b; ab\nabQ=int(ab*Qscale); abQ\nabdQ=abQ/ Qscale; abdQ, ab\n```\n\n\n```python\naQBV=intbv(aQ)[Qlen:].signed(); bQBV=intbv(bQ)[Qlen:].signed()\nf'aQBV: {bin(aQBV, Qlen)}; bQBV: {bin(bQBV, Qlen)}'\n```\n\n\n\n\n 'aQBV: 00110100; bQBV: 11011111'\n\n\n\n\n```python\nabQ=aQBV*bQBV; abQ\n```\n\n\n```python\nabQBV=intbv(abQ)[Q2len:].signed(); abQBV, bin(abQBV), len(bin(abQBV))\n```\n\n\n\n\n (intbv(-1716), '100101001100', 12)\n\n\n\n\n```python\nfor j in range(Q2[1]):\n Trunc=abQBV[Q2len:j].signed()\n TruncDQ=Trunc/(2**(Q2[1]-j))\n print(bin(Trunc), TruncDQ, np.abs(ab-TruncDQ))\n```\n\n 100101001100 -6.703125 0.0\n 10010100110 -6.703125 0.0\n 1001010011 -6.703125 0.0\n 100101001 -6.71875 0.015625\n 10010100 -6.75 0.046875\n 1001010 -6.75 0.046875\n 100101 -6.75 0.046875\n 10010 -7.0 0.296875\n\n\n\n```python\nfor j in range(Q2[1]):\n Trunc=(abQBV>>j).signed()\n TruncDQ=Trunc/(2**(Q2[1]-j))\n print(bin(Trunc), TruncDQ, np.abs(ab-TruncDQ))\n```\n\n 100101001100 -6.703125 0.0\n 10010100110 -6.703125 0.0\n 1001010011 -6.703125 0.0\n 100101001 -6.71875 0.015625\n 10010100 -6.75 0.046875\n 1001010 -6.75 0.046875\n 100101 -6.75 0.046875\n 10010 -7.0 0.296875\n\n\n# Round Half Up\n\n\n```python\na=3.2500; aQ=int(a*Qscale)\nb=-2.0625; bQ=int(b*Qscale)\naQ, bQ\nbin(aQ, Qlen), bin(bQ, Qlen)\n```\n\n\n\n\n ('00110100', '11011111')\n\n\n\n\n```python\nab=a*b; ab\nabQ=int(ab*Qscale); abQ\nabdQ=abQ/ Qscale; abdQ, ab\n```\n\n\n```python\naQBV=intbv(aQ)[Qlen:].signed(); bQBV=intbv(bQ)[Qlen:].signed()\nf'aQBV: {bin(aQBV, Qlen)}; bQBV: {bin(bQBV, Qlen)}'\n```\n\n\n\n\n 'aQBV: 00110100; bQBV: 11011111'\n\n\n\n\n```python\nabQ=aQBV*bQBV; abQ\n```\n\n\n```python\nabQBV=intbv(abQ)[Q2len:].signed(); abQBV, bin(abQBV), len(bin(abQBV))\n```\n\n\n\n\n (intbv(-1716), '100101001100', 12)\n\n\n\n\n```python\nab, floor(ab+.5), -ceiling(-ab-.5), ceiling(floor(2*ab)/2)\n```\n\n\n```python\nRound=abQBV[Q2len-1:0].signed()\nRoundDQ=Round/(2**(Q2[1]))\nprint(bin(Round), RoundDQ, np.abs(ab-RoundDQ))\n```\n\n 100101001100 -6.703125 0.0\n\n\n`{ {(OWID){1'b0}}, 1'b1, {(IWID-OWID-1){1'b0}} }`, .5\n\n `i_data[(IWID-1):0]+ { {(OWID){1'b0}}, 1'b1, {(IWID-OWID-1){1'b0}} }`, x+.5\n \n `w_halfup[(IWID-1):(IWID-OWID)]`, floor(x+.5)\n\n\n```python\nconcat(intbv(0)[8:], True, intbv(0)[16-8-1:])\n```\n\n\n\n\n intbv(128)\n\n\n\n\n```python\nPointFive=intbv(int(.5*Q2scale))[16:]; PointFive, bin(PointFive, 16)\n```\n\n\n\n\n (intbv(128), '0000000010000000')\n\n\n\n\n```python\nabQBVP5=intbv(abQBV+PointFive)[16:].signed()\nabQBVP5\n```\n\n\n\n\n intbv(-1588)\n\n\n\n\n```python\nabQBVP5=abQBVP5[Q2len-1:Q2len-Qlen].signed(); abQBVP5\n```\n\n\n\n\n intbv(-7)\n\n\n\n\n```python\nabQBVP5, floor(ab+.5)\n```\n\n\n\n\n (intbv(-7), -7)\n\n\n\n# Round towards zero\n\n\n```python\n\n```\n\n# Round Away from Zero\n\n# Round Half to Even\n\n\n```python\n\n```\n", "meta": {"hexsha": "07e0ef706e90a4bd086a4611bfa0f366675c2a44", "size": 236550, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "myHDL_DigitalSignalandSystems/FloatingNumAndRounding/QRound.ipynb", "max_stars_repo_name": "PyLCARS/PythonUberHDL", "max_stars_repo_head_hexsha": "f7ae2293d6efaca7986d62540798cdf061383d06", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 31, "max_stars_repo_stars_event_min_datetime": "2017-10-09T12:15:14.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-28T09:05:21.000Z", "max_issues_repo_path": "myHDL_DigitalSignalandSystems/FloatingNumAndRounding/QRound.ipynb", "max_issues_repo_name": "cfelton/PythonUberHDL", "max_issues_repo_head_hexsha": "f7ae2293d6efaca7986d62540798cdf061383d06", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "myHDL_DigitalSignalandSystems/FloatingNumAndRounding/QRound.ipynb", "max_forks_repo_name": "cfelton/PythonUberHDL", "max_forks_repo_head_hexsha": "f7ae2293d6efaca7986d62540798cdf061383d06", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 12, "max_forks_repo_forks_event_min_datetime": "2018-02-09T15:36:20.000Z", "max_forks_repo_forks_event_max_datetime": "2021-04-20T21:39:12.000Z", "avg_line_length": 34.0115025162, "max_line_length": 3141, "alphanum_fraction": 0.3743267808, "converted": true, "num_tokens": 41199, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419831347362, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.43620833300049283}} {"text": "# Realization of Recursive Filters\n\n*This jupyter notebook is part of a [collection of notebooks](../index.ipynb) on various topics of Digital Signal Processing. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*\n\n## Quantization of Filter Coefficients\n\nThe finite numerical resolution of digital number representations has impact on the properties of filters, as already discussed for [non-recursive filters](../nonrecursive_filters/quantization_effects.ipynb#Quantization-Effects). The quantization of coefficients, state variables, algebraic operations and signals plays an important role in the design of recursive filters. Compared to non-recursive filters, the impact of quantization is often more prominent due to the feedback. Severe degradations from the desired characteristics and instability are potential consequences of a finite word length in practical implementations.\n\nA recursive filter of order $N \\geq 2$ can be [decomposed into second-order sections (SOS)](../recursive_filters/cascaded_structures.ipynb). Due to the grouping of poles/zeros to filter coefficients with a limited amplitude range, a realization by cascaded SOS is favorable in practice. We therefore limit our investigation of quantization effects to SOS. The transfer function of a SOS is given as\n\n\\begin{equation}\nH(z) = \\frac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}}\n\\end{equation}\n\nThis can be [split into a non-recursive part and a recursive part](../recursive_filters/introduction.ipynb#Recursive-Filters). The quantization effects of non-recursive filters have already been discussed. We therefore focus here on the recursive part given by the transfer function\n\n\\begin{equation}\nH(z) = \\frac{1}{1 + a_1 z^{-1} + a_2 z^{-2}}\n\\end{equation}\n\nThis section investigates the consequences of quantization in recursive filters. As for non-recursive filters, we first take a look at the quantization of filter coefficients. The structure used for the realization of the filter has impact on the quantization effects. We begin with the direct form followed by the coupled form, as example for an alternative structure.\n\n### Direct Form\n\nAbove transfer function of the recursive part of a SOS can be rewritten in terms of its complex conjugate poles $z_{\\infty}$ and $z_{\\infty}^*$ as\n\n\\begin{equation}\nH(z) = \\frac{1}{(z-z_{\\infty}) (z-z_{\\infty}^*)} = \\frac{z^{-2}}{ 1 \\underbrace{- 2 r \\cos(\\varphi)}_{a_1} \\; z^{-1} + \\underbrace{r^2}_{a_2} \\; z^{-2} }\n\\end{equation}\n\nwhere $r = |z_{\\infty}|$ and $\\varphi = \\arg \\{z_{\\infty}\\}$ denote the absolute value and phase of the pole $z_{\\infty}$, respectively. Let's assume a [linear uniform quantization](../quantization/linear_uniform_quantization_error.ipynb#Quantization-Error-of-a-Linear-Uniform-Quantizer) of the coefficients $a_1$ and $a_2$ with quantization step $Q$. Discarding clipping, the following relations for the locations of the poles can be found\n\n\\begin{align}\nr_n &= \\sqrt{n \\cdot Q} \\\\\n\\varphi_{nm} &= \\arccos \\left( \\sqrt{\\frac{m^2 Q}{4 n}} \\right)\n\\end{align}\nfor $n \\in \\mathbb{N}_0$ and $m \\in \\mathbb{Z}$. Quantization of the filter coefficients $a_1$ and $a_2$ into a finite number of amplitude values leads to a finite number of pole locations. In the $z$-plane the possible pole locations are given by the intersections of\n\n* circles whose radii $r_n$ are given by $r_n = \\sqrt{n \\cdot Q}$ with\n* equidistant vertical lines which intersect the horizontal axis at $\\frac{1}{2} m \\cdot Q$.\n\nThe finite number of pole locations may lead to deviations from a desired filter characteristic since a desired pole location is moved to the next possible pole location. The filter may even get unstable, when poles are moved outside the unit circle. For illustration, the resulting pole locations for a SOS realized in direct form are computed and plotted.\n\n\n```python\n%matplotlib inline\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom matplotlib.patches import Circle\nimport scipy.signal as sig\nimport itertools\n\n\ndef compute_pole_locations(Q):\n a1 = np.arange(-2, 2+Q, Q)\n a2 = np.arange(0, 1+Q, Q)\n \n p = np.asarray([np.roots([1, n, m]) for (n,m) in itertools.product(a1, a2)])\n p = p[np.imag(p)!=0]\n\n return p\n\n\ndef plot_pole_locations(p, Q):\n ax = plt.gca()\n for n in np.arange(np.ceil(2/Q)+1):\n circle = Circle((0,0), radius=np.sqrt(n*Q), fill=False, color='black', ls='solid', alpha=0.05)\n ax.add_patch(circle)\n ax.axvline(.5*n*Q, color='0.95')\n ax.axvline(-.5*n*Q, color='0.95')\n\n unit_circle = Circle((0,0), radius=1, fill=False, color='red', ls='solid')\n ax.add_patch(unit_circle) \n\n plt.plot(np.real(p), np.imag(p), 'b.', ms = 4)\n plt.xlabel(r'Re{$z$}')\n plt.ylabel(r'Im{$z$}')\n plt.axis([-1.1, 1.1, -1.1, 1.1])\n\n# compute and plot pole locations\nfor w in [5,6]:\n Q = 2/(2**(w-1)) # quantization stepsize\n plt.figure(figsize=(5, 5))\n p = compute_pole_locations(Q)\n plot_pole_locations(p, Q)\n plt.title(r'Direct form coefficient quantization to $w=%d$ bits'%w)\n```\n\n**Exercise**\n\n* What consequences does the distribution of pole locations on the desired characteristics of a filter have for e.g. low/high frequencies?\n\nSolution: Quantization of the original filter coefficients leads to a limited number of possible pole and zero locations. These locations are not uniformly distributed over the $z$-plane, as can be observed from above illustrations. The density of potential locations is especially low for low frequencies and close to the Nyquist frequency. The properties of a designed filter having poles and/or zeros at low/high frequencies will potentially deviate more when quantizing its coefficients, as a consequence.\n\n### Coupled Form\n\nBesides the quantization step $Q$, the pole distribution depends also on the topology of the filter. In order to gain a different distribution of pole locations after quantization, one has to derive structures where the coefficients of the multipliers are given by other values than the direct form coefficients $a_1$ and $a_2$. \n\nOne of these alternative structures is the coupled form (also known as Gold & Rader structure)\n\n\n\nwhere $\\Re\\{z_\\infty\\} = r \\cdot \\cos \\varphi$ and $\\Im\\{z_\\infty\\} = r \\cdot \\sin \\varphi$ denote the real- and imaginary part of the complex pole $z_\\infty$, respectively. Analysis of the structure reveals its difference equation as\n\n\\begin{align}\nw[k] &= x[k] + \\Re\\{z_\\infty\\} \\, w[k-1] - \\Im\\{z_\\infty\\} \\, y[k-1] \\\\\ny[k] &= \\Im\\{z_\\infty\\} \\, w[k-1] + \\Re\\{z_\\infty\\} \\, y[k-1]\n\\end{align}\n\nand its transfer function as\n\n\\begin{equation}\nH(z) = \\frac{\\Im\\{z_\\infty\\} \\; z^{-1}}{ 1 - 2 \\Re\\{z_\\infty\\} \\; z^{-1} + (\\Re\\{z_\\infty\\}^2 + \\Im\\{z_\\infty\\}^2) \\; z^{-2} }\n\\end{equation}\n\nNote that the numerator of the transfer function differs from the recursive only SOS given above. However, this can be considered in the design of the transfer function of a general SOS.\n\nThe real- and imaginary part of the pole $z_\\infty$ occur directly as coefficients for the multipliers in the coupled form. Quantization of these coefficients results therefore in a Cartesian grid of possible pole locations in the $z$-plane. This is illustrated in the following.\n\n\n```python\ndef compute_pole_locations(w):\n Q = 1/(2**(w-1)) # quantization stepsize\n a1 = np.arange(-1, 1+Q, Q)\n a2 = np.arange(-1, 1+Q, Q)\n \n p = np.asarray([n+1j*m for (n,m) in itertools.product(a1, a2) if n**2+m**2 <= 1])\n\n return p\n\ndef plot_pole_locations(p):\n ax = plt.gca()\n \n unit_circle = Circle((0,0), radius=1, fill=False, color='red', ls='solid')\n ax.add_patch(unit_circle) \n\n plt.plot(np.real(p), np.imag(p), 'b.', ms = 4)\n plt.xlabel(r'Re{$z$}')\n plt.ylabel(r'Im{$z$}')\n plt.axis([-1.1, 1.1, -1.1, 1.1])\n\n \n# compute and plot pole locations\nfor w in [5,6]:\n plt.figure(figsize=(5, 5))\n p = compute_pole_locations(w)\n plot_pole_locations(p)\n plt.title(r'Coupled form coefficient quantization to $w=%d$ bits'%w)\n```\n\n**Excercise**\n\n* What is the benefit of this representation in comparison to the direct from discussed in the previous section?\n\nSolution: A befit of the coupled form is a uniform distribution of potential pole and zero locations in the $z$-plane. This holds especially for low frequencies and close to the Nyquist frequency.\n\n### Example - Influence of coefficient quantization\n\nThe following example illustrates the effects of coefficient quantization for a recursive [Butterworth filter](https://en.wikipedia.org/wiki/Butterworth_filter) realized in cascaded SOSs in transposed direct form II.\n\n\n```python\nw = 16 # wordlength of filter coefficients\nN = 7 # order of filter\n\n\ndef uniform_midtread_quantizer(x, w, xmin=1):\n # quantization step\n Q = xmin/(2**(w-1))\n # limiter\n x = np.copy(x)\n idx = np.where(x <= -xmin)\n x[idx] = -1\n idx = np.where(x > xmin - Q)\n x[idx] = 1 - Q\n # linear uniform quantization\n xQ = Q * np.floor(x/Q + 1/2)\n \n return xQ\n\n\ndef zplane(z, p, title='Poles and Zeros'):\n \"Plots zero and pole locations in the complex z-plane\"\n ax = plt.gca()\n \n ax.plot(np.real(z), np.imag(z), 'bo', fillstyle='none', ms = 10)\n ax.plot(np.real(p), np.imag(p), 'rx', fillstyle='none', ms = 10)\n unit_circle = Circle((0,0), radius=1, fill=False,\n color='black', ls='solid', alpha=0.9)\n ax.add_patch(unit_circle)\n ax.axvline(0, color='0.7')\n ax.axhline(0, color='0.7')\n \n plt.title(title)\n plt.xlabel(r'Re{$z$}')\n plt.ylabel(r'Im{$z$}')\n plt.axis('equal')\n plt.xlim((-2, 2))\n plt.ylim((-2, 2))\n plt.grid()\n\n \n# coefficients of recursive filter\nb, a = sig.butter(N, 0.2, 'low')\n# decomposition into SOS\nsos = sig.tf2sos(b, a, pairing='nearest')\nsos = sos/np.amax(np.abs(sos))\n# quantization of SOS coefficients\nsosq = uniform_midtread_quantizer(sos, w, xmin=1)\n# compute overall transfer function of (quantized) filter\nH = np.ones(512)\nHq = np.ones(512)\nfor n in range(sos.shape[0]):\n Om, Hn = sig.freqz(sos[n, 0:3], sos[n, 3:6])\n H = H * Hn\n Om, Hn = sig.freqz(sosq[n, 0:3], sosq[n, 3:6])\n Hq = Hq * Hn\n\n\n# plot magnitude responses\nplt.figure(figsize=(10, 3))\nplt.plot(Om, 20 * np.log10(abs(H)), label='continuous')\nplt.plot(Om, 20 * np.log10(abs(Hq)), label='quantized')\nplt.title('Magnitude response')\nplt.xlabel(r'$\\Omega$')\nplt.ylabel(r'$|H(e^{j \\Omega})|$ in dB')\nplt.legend(loc=3)\nplt.grid()\n# plot phase responses\nplt.figure(figsize=(10, 3))\nplt.plot(Om, np.unwrap(np.angle(H)), label='continuous')\nplt.plot(Om, np.unwrap(np.angle(Hq)), label='quantized')\nplt.title('Phase')\nplt.xlabel(r'$\\Omega$')\nplt.ylabel(r'$\\varphi (\\Omega)$ in rad')\nplt.legend(loc=3)\nplt.grid()\n```\n\n**Exercise**\n\n* Decrease the word length `w` of the filter. What happens? At what word length does the filter become unstable?\n* Increase the order `N` of the filter for a fixed word length `w`. What happens?\n\nSolution: The deviations from the continuous (desired) realization of the filter increase with decreasing word length. The filter with order `N=5` becomes unstable for `w < 10`. Increasing the order `N` of the filter for a fixed word length results also in instabilities. Consequently, for a high order filter also a higher word length is required.\n\n**Copyright**\n\nThis notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Digital Signal Processing - Lecture notes featuring computational examples, 2016-2018*.\n", "meta": {"hexsha": "25b7ca2a0c65dc3feeab1071dc7b97782c2ba2bd", "size": 237429, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "recursive_filters/quantization_of_coefficients.ipynb", "max_stars_repo_name": "hustcxl/digital-signal-processing-lecture", "max_stars_repo_head_hexsha": "1d6d9af39ed8cc2fc768a9af523cfa97ec4123f8", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-11-04T03:40:49.000Z", "max_stars_repo_stars_event_max_datetime": "2020-11-04T03:40:49.000Z", "max_issues_repo_path": "recursive_filters/quantization_of_coefficients.ipynb", "max_issues_repo_name": "cphysics/signal", "max_issues_repo_head_hexsha": "2e47bb4f0cf368418ee9a1108f0cea24a5dc812d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "recursive_filters/quantization_of_coefficients.ipynb", "max_forks_repo_name": "cphysics/signal", "max_forks_repo_head_hexsha": "2e47bb4f0cf368418ee9a1108f0cea24a5dc812d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 576.2839805825, "max_line_length": 71680, "alphanum_fraction": 0.9385753215, "converted": true, "num_tokens": 3299, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160666, "lm_q2_score": 0.7090191399336402, "lm_q1q2_score": 0.43610906230162866}} {"text": "# Summas de Riemann: punto intermedio.\r\n## Definici\u00f3n de suma de Riemann.\r\nLas sumas de Riemman son una aproximacion para obtener el area bajo una curva entre dos puntos especificos. Esta aproximacion se realiza usando poligonos regulares que cubren el area objetivo, de los cuales se puede hallar su area con mayor facilidad.\r\n\r\n##Suma de punto medio.\r\nSon una aproximaci\u00f3n de Riemann que toman la evaluacion de cada punto de los intervalos en donde este punto es la mitad del ancho de la figura geometrica. \r\nSe observa que esta aproximacion, respecto a la derecha y a la izquierda, es mas precisa debido a que intenta distribuir las areas no cubiertas con las areas sobrantes de la figura semejante proxima. \r\nSe comprende mejor al observar la siguiente grafica:\r\n\r\n\r\n\r\nGrafica tomada de https://es.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-2/v/midpoint-sums\r\n\r\n###Referencias bibliogr\u00e1ficas.\r\n- Khan Academy (s.f). Sumas de punto medio. tomado de https://es.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-2/v/midpoint-sums\r\n\r\n###Bibliograf\u00eda.\r\n\r\n- Khan Academy (s.f). Introducci\u00f3n a la aproximaci\u00f3n de Riemann. tomado de \r\nhttps://es.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-2/v/simple-riemann-approximation-using-rectangles\r\n\r\n- Khan Academy (s.f). Sumas de punto medio. tomado de https://es.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-2/v/midpoint-sums\n\n\n```\n\r\n# ej. B :: punto medio :: funcion = x**3\r\n## Ejecutar :: imprime todo el cuadro de una vez.\r\n\r\n\r\nfrom sympy import integrate, init_printing\r\nfrom sympy.abc import x\r\n#init_printing(use_latex=\"mathjax\")\r\n\r\n# valor real: se halla con sympy: con la funcion integrate\r\ninteg_v_exacto = integrate( x**3, (x, lim_inf, lim_sup) )\r\n\r\n# en una funcion obtiene las sumas de Riemman para todas las particiones 3, 10, 25, 100 que estan en la tupla\r\n\r\ndef todas_las_sumas_Rnn( lim_inf, lim_sup, n_particiones ) :\r\n global particiones_ , evaluacion_Particion\r\n particiones_ , evaluacion_Particion = {}, {}\r\n sumasRiemann = []\r\n for n_particion in n_particiones :\r\n append = []\r\n append.append(n_particion)\r\n delta = (lim_sup - lim_inf) / n_particion\r\n ### |--- I N T E R M E D I A !! ---|\r\n valores_x_delta = [ ((lim_inf + delta / 2) + (i * delta)) for i in range( 0, n_particion, 1) ] \r\n particiones_[n_particion] = valores_x_delta\r\n eval_func_parti = [ ( valor **3 ) for valor in valores_x_delta ]\r\n evaluacion_Particion[n_particion] = eval_func_parti\r\n suma = 0\r\n for altura in eval_func_parti : suma += delta * altura \r\n append.append(suma)\r\n error = ( (integ_v_exacto - suma) / integ_v_exacto ) * 100 \r\n append.append(error)\r\n sumasRiemann.append(append)\r\n return sumasRiemann\r\n\r\n\r\nlim_sup = 2\r\nlim_inf = 0\r\nn_particiones = ( 3, 10, 25, 100 )\r\n\r\n\r\nsumasRiemann = todas_las_sumas_Rnn( lim_inf, lim_sup, n_particiones )\r\n\r\n\r\nfrom tabulate import tabulate\r\nprint(tabulate( sumasRiemann , headers=['Partici\u00f3n', 'Suma', 'Error'], tablefmt= 'francy_grid'))\r\nprint('')\r\n\r\n\n```\n\n Partici\u00f3n Suma Error\n ----------- ------- -------\n 3 3.77778 5.55556\n 10 3.98 0.5\n 25 3.9968 0.08\n 100 3.9998 0.005\n \n\n\n\n```\n\r\nimport matplotlib.pyplot as plot\r\nimport numpy as np\r\n\r\ndef F_particionInter( n_part ):\r\n ret = particiones_[ n_part ]\r\n #print(ret)\r\n return ret \r\n\r\ndef F_evaluacionParticion( n_part ):\r\n print('n_part: ', n_part)\r\n ret = evaluacion_Particion[ n_part ]\r\n print('ret',ret)\r\n return ret\r\n \r\n\r\nx1 = np.arange(0, 10, 0.01)\r\ny1 = x1**3\r\n\r\nprint('particione_ : ', particiones_)\r\nprint('evaluacion_Particion: ', evaluacion_Particion)\r\n\r\nfor n_part in n_particiones:\r\n x1 = np.arange(0, n_part, 0.01)\r\n y1 = x1**3\r\n x = F_particionInter( n_part )\r\n y = F_evaluacionParticion(n_part )\r\n print('x: ', x)\r\n print('y: ', y)\r\n #y=[0.03703703703703703, 1.0, 4.629629629629628]\r\n plot.plot( x, y , label = ' particiones ' )\r\n for i in range( 0, len(x) ):\r\n the_x = x[i]\r\n print('the x: ', the_x)\r\n the_y = [ ]\r\n since = y[i]\r\n init = y[i]\r\n delt = init/9\r\n for i in range(since,10,-delt ):\r\n the_y.append(init)\r\n init -= delt \r\n x_s = [ the_x for k in range(10) ]\r\n print('x_s: ', len(x_s))\r\n print('the_y: ' , len(the_y))\r\n plot.plot(x_s, the_y, label=\"pruebas\" )\r\n plot.plot(x1, y1, label=' funcion objetivo ... ')\r\n\r\n plot.xlabel(' intervalos - delta ')\r\n plot.ylabel(' evaluacion de la funcion ')\r\n plot.title(' sima Riemann: intermedias ' )\r\n plot.legend()\r\n plot.show()\n```\n", "meta": {"hexsha": "e11a36b902ed43c9ef5ad2507fd144156bd6bc3c", "size": 33842, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Riemman_sums_aprox.ipynb", "max_stars_repo_name": "henrymorenoespitia/numerical_methods_and_analysis", "max_stars_repo_head_hexsha": "494b5503cef01dc0d6b745edb454bb77e03481e5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Riemman_sums_aprox.ipynb", "max_issues_repo_name": "henrymorenoespitia/numerical_methods_and_analysis", "max_issues_repo_head_hexsha": "494b5503cef01dc0d6b745edb454bb77e03481e5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Riemman_sums_aprox.ipynb", "max_forks_repo_name": "henrymorenoespitia/numerical_methods_and_analysis", "max_forks_repo_head_hexsha": "494b5503cef01dc0d6b745edb454bb77e03481e5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 16921.0, "max_line_length": 33841, "alphanum_fraction": 0.869068022, "converted": true, "num_tokens": 1438, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.8031737892899222, "lm_q1q2_score": 0.4360135602120291}} {"text": "\n\n\n# Standalone Fishbone-Moncrief C Code\n\nWe start with the NRPy+ expressions generated in the [Tutorial-FishboneMoncriefID](Tutorial-FishboneMoncriefID.ipynb), and output them to the C file \"FishboneMoncriefID/FMstandalone.h\".\n\nFurther, $\\Gamma = \\alpha u^0$ is given by (as shown [here](Tutorial-u0_smallb_Poynting-Cartesian.ipynb)):\n$$\n\\Gamma = \\alpha u^0 = \\sqrt{\\frac{1}{1 - \\gamma_{ij}v^i_{(n)}v^j_{(n)}}}.\n$$\n\n\n```python\nimport sympy as sp # SymPy: The Python computer algebra package upon which NRPy+ depends\nfrom outputC import lhrh # NRPy+: Core C code output module\nimport indexedexp as ixp # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support\nimport finite_difference as fin # NRPy+: Finite difference C code generation module\nimport FishboneMoncriefID.FishboneMoncriefID as fmid\n\n# Step 1: Set up the Fishbone-Moncrief initial data. This sets all the ID gridfunctions.\nfmid.FishboneMoncriefID(\"Spherical\")\n\ngammaDD = ixp.zerorank2()\n\nDIM = 3\nfor i in range(DIM):\n for j in range(DIM):\n if i<=j:\n gammaDD[i][j] = fmid.IDgammaDD[i][j]\n else:\n gammaDD[i][j] = fmid.IDgammaDD[j][i]\n\n# gamma_{ij} v^i_{(n)} v^j_{(n)}\nGammacontraction = sp.sympify(0)\nfor i in range(DIM):\n for j in range(DIM):\n Gammacontraction += gammaDD[i][j] * fmid.IDValencia3velocityU[i] * fmid.IDValencia3velocityU[j]\n\nGammafactor = sp.sqrt(1 / (1 - Gammacontraction))\n\n# -={ F-M quantities: Generate C code from expressions and output to file }=-\nFishboneMoncrief_to_print = [\\\n lhrh(lhs=\"alpha\",rhs=fmid.IDalpha),\\\n lhrh(lhs=\"betaU0\",rhs=fmid.IDbetaU[0]),\\\n lhrh(lhs=\"betaU1\",rhs=fmid.IDbetaU[1]),\\\n lhrh(lhs=\"betaU2\",rhs=fmid.IDbetaU[2]),\\\n lhrh(lhs=\"Gammafactor\",rhs=Gammafactor),\\\n lhrh(lhs=\"Gamma_times_ValenciavU0\",rhs=Gammafactor*fmid.IDValencia3velocityU[0]),\\\n lhrh(lhs=\"Gamma_times_ValenciavU1\",rhs=Gammafactor*fmid.IDValencia3velocityU[1]),\\\n lhrh(lhs=\"Gamma_times_ValenciavU2\",rhs=Gammafactor*fmid.IDValencia3velocityU[2]),\\\n lhrh(lhs=\"uKS4U1\",rhs=fmid.uKS4U[1]),\\\n lhrh(lhs=\"uKS4U2\",rhs=fmid.uKS4U[2]),\\\n lhrh(lhs=\"uKS4U3\",rhs=fmid.uKS4U[3]),\\\n lhrh(lhs=\"uBL4U1\",rhs=fmid.uBL4U[1]),\\\n lhrh(lhs=\"uBL4U2\",rhs=fmid.uBL4U[2]),\\\n lhrh(lhs=\"uBL4U3\",rhs=fmid.uBL4U[3])\n ]\n# print(fmid.uKS4U[3])\nfin.FD_outputC(\"FishboneMoncriefID/FM_standalone.h\",FishboneMoncrief_to_print,params=\"outCverbose=False,CSE_enable=False\")\n```\n\n Wrote to file \"FishboneMoncriefID/FM_standalone.h\"\n\n\n\n```python\n%%writefile FishboneMoncriefID/FM_standalone.c\n\n#include \"stdio.h\"\n#include \"stdlib.h\"\n#include \"math.h\"\n\nconst double a = 0.9375;\nconst double M = 1.0;\nconst double r_at_max_density = 12.0;\nconst double r_in = 6.0;\n\nint main(int argc, const char *argv[]) {\n\n // Step 0a: Read command-line input, error out if nonconformant\n double xx0,xx1,xx2;\n/*\n if(argc != 4) {\n printf(\"Error: Expected three command-line arguments: ./FM_standalone r theta phi\\n\");\n exit(1);\n }\n xx0 = strtod(argv[1],NULL);\n xx1 = strtod(argv[2],NULL);\n xx2 = strtod(argv[3],NULL);\n*/\n\n// printf(\"# Output: r,th,ph, alpha, betaU0, betaU1, betaU2, Gamma, Gamma*vValenciaU0, Gamma*vValenciaU1, Gamma*vValenciaU2\\n\");\n for(double xx0=1.6;xx0<50.0;xx0+=0.2) {\n xx1 = 1.56463634120e0; //M_PI/2.0;\n xx2 = 0.0;\n double alpha,betaU0,betaU1,betaU2,Gammafactor,Gamma_times_ValenciavU0,Gamma_times_ValenciavU1,Gamma_times_ValenciavU2;\n double uKS4U1,uKS4U2,uKS4U3,uBL4U1,uBL4U2,uBL4U3;\n#include \"FM_standalone.h\"\n if(xx0 < r_in) {\n Gammafactor = 1.0;\n Gamma_times_ValenciavU0 = Gamma_times_ValenciavU1 = Gamma_times_ValenciavU2 = 0.0;\n uKS4U1 = uKS4U2 = uKS4U3 = 0.0;\n uBL4U1 = uBL4U2 = uBL4U3 = 0.0;\n }\n printf(\"%e %e %e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e\\n\",\n xx0,xx1,xx2,\n alpha,betaU0,betaU1,betaU2,\n Gammafactor,\n Gamma_times_ValenciavU0, // util1(1) in FMtorus.f90; util(1,i,j,k) near the write statement\n Gamma_times_ValenciavU1, // util1(3) in FMtorus.f90.\n Gamma_times_ValenciavU2, // util1(2) in FMtorus.f90.\n uKS4U1,uKS4U2,uKS4U3,\n uBL4U1,uBL4U2,uBL4U3);\n }\n return 0;\n}\n```\n\n Writing FishboneMoncriefID/FM_standalone.c\n\n\n\n```python\n!gcc -O2 FishboneMoncriefID/FM_standalone.c -o FM_standalone -lm\n```\n\n\n```python\n!./FM_standalone > out.txt\n```\n\n\n```python\n%matplotlib inline\n\nimport matplotlib.pyplot as plt\nimport mpmath as mp\nimport csv\n\n# Download torus_cuts.csv:\nURL = \"http://astro.phys.wvu.edu/zetienne/torus_cuts.csv\"\noutfile = \"torus_cuts.csv\"\ntry:\n with open(outfile,\"w\") as file:\n file.write(urllib.request.urlopen(URL).read().decode(\"utf-8\"))\nexcept:\n try:\n with open(outfile,\"w\") as file:\n file.write(urllib.urlopen(URL).read().decode(\"utf-8\"))\n except:\n # If all else fails, hope wget does the job\n !wget -O $outfile $URL\n\ndef file_reader(filename,list_of_cols,delim=\" \"):\n with open(filename) as file:\n reader = csv.reader(file, delimiter=delim)\n data = list(zip(*reader))\n# print(data)\n # data is a tuple of strings. Tuples are immutable, and we need to perform math on\n # the data, so here we convert tuple to lists of floats:\n# data_output = [[sp.sympify(0) for i in range(len(list_of_cols))] for j in range(len(data[0]))]\n data_output = [[sp.sympify(0) for i in range(len(data[0]))] for j in range(len(list_of_cols))]\n for i in range(len(data[0])):\n for j in range(len(list_of_cols)):\n# print(i,j,data[list_of_cols[j]][i])\n data_output[j][i] = float(data[list_of_cols[j]][i])\n return data_output\n\nNRPy_data_output = file_reader('out.txt', [0,7,8,9,10])\nstd_data_output = file_reader('torus_cuts.csv',[0,4,1,3,2])\n\n\nylabels = ['Lorentz Gamma_{KS}=G','G*v^r_{KS,Val.}','G*v^{\\\\theta}_{KS,Val.}','G*v^{\\phi}_{KS,Val.}']\n\nfor i in range(len(ylabels)):\n # https://matplotlib.org/gallery/text_labels_and_annotations/legend.html#sphx-glr-gallery-text-labels-and-annotations-legend-py\n fig, ax = plt.subplots()\n plt.title(\"NRPy's FM solve with FMtorus.f90: \"+ylabels[i])\n plt.xlabel(\"r/M\")\n plt.ylabel(ylabels[i])\n ax.plot(NRPy_data_output[0], NRPy_data_output[i+1], 'k--', label='NRPyFMSolve')\n ax.plot(std_data_output[0], std_data_output[i+1], 'k-', label='FMtorus.f90')\n legend = ax.legend(loc='upper right', shadow=True, fontsize='x-large')\n legend.get_frame().set_facecolor('C1')\n plt.show()\n```\n", "meta": {"hexsha": "4849bfc09039e27d0f69624667130f19a977004b", "size": 111027, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Tutorial-Start_to_Finish-FishboneMoncriefID_standalone.ipynb", "max_stars_repo_name": "fedelopezar/nrpytutorial", "max_stars_repo_head_hexsha": "753acd954be4a2f99639c9f9fd5e623689fc7493", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": 66, "max_stars_repo_stars_event_min_datetime": "2018-06-26T22:18:09.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-09T21:12:33.000Z", "max_issues_repo_path": "Tutorial-Start_to_Finish-FishboneMoncriefID_standalone.ipynb", "max_issues_repo_name": "fedelopezar/nrpytutorial", "max_issues_repo_head_hexsha": "753acd954be4a2f99639c9f9fd5e623689fc7493", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": 14, "max_issues_repo_issues_event_min_datetime": "2020-02-13T16:09:29.000Z", "max_issues_repo_issues_event_max_datetime": "2021-11-12T14:59:59.000Z", "max_forks_repo_path": "Tutorial-Start_to_Finish-FishboneMoncriefID_standalone.ipynb", "max_forks_repo_name": "fedelopezar/nrpytutorial", "max_forks_repo_head_hexsha": "753acd954be4a2f99639c9f9fd5e623689fc7493", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": 30, "max_forks_repo_forks_event_min_datetime": "2019-01-09T09:57:51.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-08T18:45:08.000Z", "avg_line_length": 311.0, "max_line_length": 27300, "alphanum_fraction": 0.9157772434, "converted": true, "num_tokens": 2227, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432182679956, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.43588370528786513}} {"text": "# Building Better Models for Inference:\n\n# How to construct practical models for existing tools\n\nIn this notebook, we will walk through fitting an observed optical light curve from a tidal disruption event (TDE), the destruction and accretion of a star by a supermassive black hole, using two different approaches.\n\nAs mentioned in the lecture, there are different kinds of models one can apply to a set of data. A code I have written, MOSFiT, is an attempt to provide a framework for building models that can be used within other optimizers/samplers. While MOSFiT can run independently with its own built-in samplers, in the notebook below we will simple be using it as a \"black box\" function for use in external optimization routines.\n\nOur first approach will be using the `tde` model in MOSFiT. This model uses both interpolation tables and integrations, making an analytical derivative not available. Our second approach will be to construct a simple analytical function to fit the same data. We will then be comparing performance, both in terms of the quality of the resulting solution, but also the speed by which the solution was computed, and in how we relate our solution to what transpired in this event.\n\n* * *\n\nBy J Guillochon (Harvard)\n\n*We will be mostly using the mosfit package and scipy routines. Both are available via conda.*\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport mosfit\nimport time\n\n# Disable \"retina\" line below if your monitor doesn't support it.\n%matplotlib inline\n%config InlineBackend.figure_format = 'retina'\n```\n\n## Problem 1) Fitting data with a blackbox model\n\nIn this first cell, we load the data of a particularly well-sampled tidal disruption event from the Pan-STARRS survey, PS1-10jh. This even is notable because it was caught on the rise, peak, and decay, with solid cadence.\n\nThe datafile can be aquired from https://northwestern.app.box.com/s/ekwpbf8ufe1ivogpxq9yyex302zx0t96.\n\n\n```python\n# Load the data from the Open Supernova Catalog.\n# Note: if loading the data doesn't work, I have a local copy.\nmy_printer = mosfit.printer.Printer(quiet=True) # just to avoid spamming from MOSFiT routines.\nmy_fetcher = mosfit.fetcher.Fetcher()\n\nfetched = my_fetcher.fetch('PS1-10jh.json')[0]\n\nmy_model = mosfit.model.Model(model='tde', printer=my_printer)\nfetched_data = my_fetcher.load_data(fetched)\nmy_model.load_data(\n fetched_data, event_name=fetched['name'],\n exclude_bands=['u', 'r', 'i', 'z', 'F225W', 'NUV'], # ignore all bands but g when computing ln_likelihood.\n smooth_times=100, # for plotting smooth fits later\n user_fixed_parameters=['covariance']) # don't use GP objective function.\n\n# Generate 100 random parameter realizations.\nx = np.random.rand(100, my_model.get_num_free_parameters())\n\n# Compute time per function call.\nstart_time = time.time()\nln_likes = [my_model.ln_likelihood(xx) for xx in x]\nstop_time = time.time()\n\nprint('{}s per function call.'.format((stop_time - start_time)/100.0))\n```\n\n**Problem 1a**\n\nFirst, let's visualize the data we have downloaded. MOSFiT loads data in a format conforming to the OAC schema specification, which is a JSON dictionary where the top level of the structure is each event's name. The code snippet below will load a JSON dictionary for the event in question, plot the full time series of photometric data (with error bars) within the `photometry` key below.\n\n*Hint: The photometry is a mixture of different data types, and not every entry has the same set of keys. Optical/UV/IR photometry will always have a `band` key. Ignore upper limits (indicated with the `upperlimit` attribute). Use the `.get()` function liberally, and make sure everything is a `float`!*\n\n\n```python\ntimes = []\nmags = []\nerrs = []\nfor x in fetched_data[fetched['name']]['photometry']:\n # complete\n\nplt.errorbar(times, mags, yerr=errs, fmt='o')\nplt.gca().invert_yaxis()\nplt.show()\n```\n\n**Problem 1b**\n\nWe know what the data looks like, and we've loaded a model that can be used to fit the data which computes a likelihood. Let's minimize the parameters of this model using various `scipy.optimize` routines. Note that since we are trying to **maximize** the likelihood, we have constructed a wrapper function around `ln_likelihood`, `my_func`, to reverse its sign, and to handle bad function evaluations.\n\nMost `optimize` routines in `scipy` require a derivative. Since we don't have this available, `scipy` must construct an approximate one, unless the method doesn't require a gradient to be computed (like `differential_evolution`). For this first sub-problem, optimize `my_func` using `differential_evolution`.\n\n*Hints: Each variable is bounded to the range (0, 1), but problems can arise if an optimizer attempts to compute values outside or right at the boundaries. Therefore, it is recommended to use a bounded optimizer in `scipy`, where the bounds do **not include** 0 or 1.*\n\n\n```python\nimport scipy\n\ndef my_func(x):\n try:\n fx = -float(my_model.ln_likelihood(x))\n except:\n fx = np.inf\n return fx\n\neps = 0.00001\nbounds = # complete\nresults = scipy.optimize.differential_evolution( # complete\nbest_x = results.x\n\nprint('All done! Best score: `{}`.'.format(-results.fun))\n```\n\nThis might take a while; try to limit the execution time of the above to ~5 minutes by playing with the `maxiter` and similar options of the `scipy` optimizers.\n\nOnce the above has finished evaluating, compare the score you got to your neighbors. Is there a significant difference between your scores? Let's plot your result against the data.\n\nModel output is provided in the `output` object below, the format is a dictionary of arrays of the same length. The times of observation are in the `times` array, and magnitudes are in the `model_observations` array.\n\n*Hint: `times` is given relative to the time of the first detection, so add `min(times)` to your time to overplot onto the data.*\n\n\n```python\noutput = my_model.run_stack(best_x, root='output')\n\nfor ti, t in enumerate(output['times']):\n # complete\n\nplt.errorbar( # complete\nplt.plot( # complete\nplt.gca().invert_yaxis()\nplt.show()\n```\n\n**Problem 1c**\n\nTry optimizing the same function using **another** minimization routine in scipy that can take a derivative as an input (examples: `L-BFGS-B`, `SLSQP`, `basinhopping`, etc.).\n\n\n```python\nresults = scipy.optimize.basinhopping( #complete\nbest_x = results.x\n\nprint('All done! Best score: `{}`.'.format(-results.fun))\n```\n\nNow, plot the results of the above minimization alongside your original `differential_evolution` solution.\n\n\n```python\n# complete\n\nfor ti, t in enumerate(output['times']):\n # complete\n\n# complete\n```\n\nAfter this process, **some** of you **might** have gotten a good solution with a runtime of a few minutes. In practice, guaranteed convergence to the best solution can take a very long time. Whats more, we only attempted to find the **best** solution available, usually we are interested in posterior distributions that (usually) include the best solution. These take even longer to compute (tens of thousands of function evaluations for a problem of this size).\n\n## Problem 2\n\nNow, we'll construct our own simpler model that is **analytically differentiable**. We'll partly motivate the shape of this function based upon our knowledge of how tidal disruption events are expected to behave theoretically, but there will be limitations.\n\nFirst, let's define a function that loosely mimics a tidal disruption event's temporal evolution. Tidal disruption events rise exponentially, then decay as a power-law. Canonically, the decay rate is -5/3, and the rise is very unconstrained, being mediated by complicated dynamics and accretion physics that have yet to be determined. So, we use the following agnostic form,\n\n$$L(t) = L_0 \\left(1-e^{-\\frac{t}{t_0}}\\right)^{\\alpha } \\left(\\frac{t}{t_0}\\right)^{-\\beta }.$$\n\nTidal disruption observations are usually reported in magnitudes, thus the expression we'll actually compare against observations is\n\n$$m(t) = m_0 - 2.5 \\log_{10}\\left[\\left(1-e^{-\\frac{t}{t_0}}\\right)^{\\alpha } \\left(\\frac{t}{t_0}\\right)^{-\\beta }\\right].$$\n\nTo calculate the likelihood, we want to subtract the above from the observations. We'll make the gross assumption that the color of a tidal disruption is constant in time (which turns out to not be a terrible assumption) and thus $L_{\\rm g}(t) \\propto L(t)$.\n\nOur likelihood function will be defined as the product of the squares of differences between our model and observation,\n\n$$p = \\prod_i \\frac{1}{\\sqrt{2\\pi (\\sigma_i^2 + \\sigma^2)}} \\left[\\frac{\\left(m_{{\\rm g}, i} - \\bar{m}_{{\\rm g}, i}\\right)^2}{2\\left(\\sigma_i^2 + \\sigma^2\\right)}\\right],$$\n\nand thus our log likelihood is the sum of these squared differences, plus a separate sum for the variances,\n\n$$\\log p = -\\frac{1}{2} \\left\\{\\sum_i \\left[\\frac{\\left(m_{{\\rm g}, i} - \\bar{m}_{{\\rm g}, i}\\right)^2}{\\sigma_i^2 + \\sigma^2}\\right] + \\log 2\\pi \\left(\\sigma_i^2 + \\sigma^2\\right)\\right\\}.$$\n\n**Problem 2a**\n\nWrite the above expression as a python function:\n\n\n```python\ndef analytic_f(x, tt, mm, vv):\n m0, alpha, beta, tau, t0, sigma = tuple(x)\n t = np.array(tt)\n m = np.array(mm)\n v = np.array(vv)\n \n # complete\n```\n\n** Problem 2a **\n\nCompute the derivative for $\\log p$ (above expression) with respect to $m_0$ (Mathematica might be helpful here). Below are the derivatives for the other five free parameters $\\alpha$, $\\beta$, $\\tau$, $t_0$, and $\\sigma$:\n\n$$\n\\begin{align}\n\\frac{\\partial\\log p}{\\partial \\alpha} &= \\sum_i -\\frac{5 \\log \\left(1-e^{-\\frac{t+t_0}{\\tau }}\\right) \\left\\{\\log (100) (\\bar{m}-m_0)+5 \\log \\left[\\left(1-e^{-\\frac{t+t_0}{\\tau }}\\right)^{\\alpha } \\left(\\frac{t+t_0}{\\tau }\\right)^{-\\beta }\\right]\\right\\}}{4 \\log ^2(10) \\left(\\sigma_i^2+\\sigma^2\\right)}\\\\\n\\frac{\\partial\\log p}{\\partial \\beta} &= \\sum_i \\frac{5 \\log \\left(\\frac{t+t_0}{\\tau }\\right) \\left\\{\\log (100) (\\bar{m}-m_0)+5 \\log \\left[\\left(1-e^{-\\frac{t+t_0}{\\tau }}\\right)^{\\alpha } \\left(\\frac{t+t_0}{\\tau }\\right)^{-\\beta }\\right]\\right\\}}{4 \\log ^2(10) \\left(\\sigma_i^2+\\sigma^2\\right)}\\\\\n\\frac{\\partial\\log p}{\\partial \\tau} &= \\sum_i \\frac{5 \\left(\\alpha (t+t_0)-\\beta \\tau \\left(e^{\\frac{t+t_0}{\\tau }}-1\\right)\\right) \\left(\\log (100) (\\bar{m}-m_0)+5 \\log \\left(\\left(1-e^{-\\frac{t+t_0}{\\tau }}\\right)^{\\alpha } \\left(\\frac{t+t_0}{\\tau }\\right)^{-\\beta }\\right)\\right)}{4 \\tau ^2 \\log ^2(10) \\left(\\sigma_i^2 + \\sigma^2\\right) \\left(e^{\\frac{t+t_0}{\\tau }}-1\\right)}\\\\\n\\frac{\\partial\\log p}{\\partial t_0} &= \\sum_i \\frac{5 \\left(\\alpha (t+t_0)-\\beta \\tau \\left(e^{\\frac{t+t_0}{\\tau }}-1\\right)\\right) \\left(\\log (100) (m_0-\\bar{m})-5 \\log \\left(\\left(1-e^{-\\frac{t+t_0}{\\tau }}\\right)^{\\alpha } \\left(\\frac{t+t_0}{\\tau }\\right)^{-\\beta }\\right)\\right)}{4 \\tau \\log ^2(10) \\left(\\sigma_i^2+\\sigma^2\\right) (t+t_0) \\left(e^{\\frac{t+t_0}{\\tau }}-1\\right)}\\\\\n\\frac{\\partial\\log p}{\\partial \\sigma} &= \\sum_i \\frac{\\sigma_i}{4 \\log ^2(10) \\left(\\sigma_i^2+\\sigma^2\\right)^2} \\left\\{5 \\log \\left[\\left(1-e^{-\\frac{t+t_0}{\\tau }}\\right)^{\\alpha } \\left(\\frac{t+t_0}{\\tau }\\right)^{-\\beta }\\right]\\right.\\\\&\\times\\left.\\left(4 \\log (10) (\\bar{m}-m_0)+5 \\log \\left[\\left(1-e^{-\\frac{t+t_0}{\\tau }}\\right)^{\\alpha } \\left(\\frac{t+t_0}{\\tau }\\right)^{-\\beta }\\right]\\right)+4 \\log ^2(10) \\left((m_0-\\bar{m})^2-\\sigma_i^2-\\sigma^2\\right)\\right\\}\n\\end{align}\n$$\n\n**Problem 2b**\n\nWe now need to write each of these derivatives as python functions. These functions should accept a single vector argument `x` with length equal to the number of free parameters, plus a vector $t$ (the times of the observation) vector $m$ (the magnitudes of each observation), and finally errors $v$ (the measurement error of each observation). Again, 5 of the 6 parameters have already been written for you (you must provide the 6th).\n\n\n```python\ndef dlogp_dalpha(x, tt, mm, vv):\n m0, alpha, beta, tau, t0, sigma = tuple(x)\n t = np.array(tt)\n m = np.array(mm)\n v = np.array(vv)\n \n derivs = np.sum(-5.0 * np.log(1.0 - np.exp(-(t + t0) / tau)) * (np.log(100.0) * (m - m0) + 5.0 * lf(\n alpha, beta, tau, t0, t)) / (4.0 * np.log(10.0) ** 2 * (v ** 2 + sigma ** 2)))\n \n return derivs\n\ndef dlogp_dbeta(x, tt, mm, vv):\n m0, alpha, beta, tau, t0, sigma = tuple(x)\n t = np.array(tt)\n m = np.array(mm)\n v = np.array(vv)\n \n derivs = np.sum(5.0 * np.log((t + t0) / tau) * (np.log(100.0) * (m - m0) + 5.0 * lf(\n alpha, beta, tau, t0, t)) / (4.0 * np.log(10.0) ** 2 * (v ** 2 + sigma ** 2)))\n \n return derivs\n\ndef dlogp_dtau(x, tt, mm, vv):\n m0, alpha, beta, tau, t0, sigma = tuple(x)\n t = np.array(tt)\n m = np.array(mm)\n v = np.array(vv)\n \n derivs = np.sum(5.0 * (alpha * (t + t0) - beta * tau * (np.exp((t + t0)/tau) - 1.0)) * (\n np.log(100.0) * (m - m0) + 5.0 * lf(\n alpha, beta, tau, t0, t)) / (4.0 * tau ** 2 * np.log(10.0) ** 2 * (v ** 2 + sigma ** 2) * (\n np.exp((t + t0)/tau) - 1.0)))\n \n return derivs\n\ndef dlogp_dt0(x, tt, mm, vv):\n m0, alpha, beta, tau, t0, sigma = tuple(x)\n t = np.array(tt)\n m = np.array(mm)\n v = np.array(vv)\n \n derivs = np.sum(-5.0 * (alpha * (t + t0) - beta * tau * (np.exp((t + t0)/tau) - 1.0)) * (\n np.log(100.0) * (m - m0) + 5.0 * lf(\n alpha, beta, tau, t0, t)) / (4.0 * tau * (t + t0) * np.log(10.0) ** 2 * (v ** 2 + sigma ** 2) * (\n np.exp((t + t0)/tau) - 1.0)))\n \n return derivs\n\ndef dlogp_dsigma(x, tt, mm, vv):\n m0, alpha, beta, tau, t0, sigma = tuple(x)\n t = np.array(tt)\n m = np.array(mm)\n v = np.array(vv)\n \n derivs = np.sum(sigma/(4.0 * np.log(10.0) ** 2 * (v**2 + sigma**2)**2) * (5.0 * lf(\n alpha, beta, tau, t0, t) * (4.0 * np.log(10.0) * (m - m0) + 5.0 * lf(\n alpha, beta, tau, t0, t)) + 4.0 * np.log(10.0) ** 2 * ((m0 - m) ** 2 - v ** 2 - sigma ** 2)))\n \n return derivs\n```\n\n**Problem 2c**\n\nMake sure the derivatives for all the above function are consistent with the finite differences of the objective function. How large is the error for an `eps = 1e-8` (the default distance used when no Jacobian is provided)? Make a histogram for each derivative of these errors for 100 random parameter combinations drawn from the bounds (in other words, six plots with 100 samples each).\n\n*Hint: you will likely have to remove some nans.*\n\n\n```python\n# Set up bounds/test parameters.\nabounds = [\n [0.0, 30.0],\n [0.1, 50.0],\n [0.1, 10.0],\n [0.1, 200.0],\n [0.1, 200.0],\n [0.001, 1.0]\n]\n\ntest_times = [1.0, 20.0]\ntest_mags = [23.0, 19.0]\ntest_errs = [0.1, 0.2]\n\n# Draw a random parameter combo to test with.\nn = 100\ndm0_diff = np.zeros(n)\nfor p in range(n):\n test_x = [abounds[i][0] + x * (abounds[i][1] - abounds[i][0]) for i, x in enumerate(np.random.rand(6))]\n\n # Check that every derivative expression is close to finite difference.\n teps = 1e-10\n\n xp = list(test_x)\n xp[0] += teps\n exactd = dlogp_dm0(test_x, test_times, test_mags, test_errs)\n dm0_diff[p] = (exactd - (\n analytic_f(test_x, test_times, test_mags, test_errs) - analytic_f(\n xp, test_times, test_mags, test_errs)) / teps) / exactd\n\n # complete for rest of parameters\n\nplt.subplot(321)\nplt.hist(dm0_diff[~np.isnan(dm0_diff)]);\n# complete for rest of parameters\n```\n\nWhich derivatives seem to have the least accurate finite differences? Why?\n\n## Problem 3\n\nNow we have an analytical function with analytical derivatives that should be accurate to near-machine precision. \n\n** Problem 3a **\n\nFirst, let's optimize our function using `differential_evolution`, as we did above with the MOSFiT output, without using the derivatives we have constructed (as `differential_evolution` does not use them).\n\n\n```python\nimport scipy\n\ntimes0 = np.array(times) - min(times)\nresults = scipy.optimize.differential_evolution(# complete\nbest_x = results.x\n\nprint('All done! Best score: `{}`, took `{}` function evaluations.'.format(results.fun, results.nfev))\n```\n\nNow plot the result:\n\n\n```python\n# complete\n```\n\nHow good is the approximation?\n\n** Problem 3b **\n\nLet's minimize using the `basinhopping` algorithm now, again not using our derivatives.\n\n\n```python\ntimes0 = np.array(times) - min(times)\nresults = scipy.optimize.basinhopping( # complete\nbest_x = results.x\n\nprint('All done! Best score: `{}`, took `{}` function evaluations.'.format(results.fun, results.nfev))\n```\n\nThis algorithm, which depends on finite differencing, seems to have taken more function evaluations than `differential_evolution`. Let's give it some help: construct a jacobian using the derivative functions defined above.\n\n*Hint: mind the sign of the Jacobian since we are **minimizing** the function.*\n\n\n```python\ndef jac(x, tt, mm, vv):\n m0, alpha, beta, tau, t0, sigma = tuple(x)\n t = np.array(tt)\n m = np.array(mm)\n v = np.array(vv)\n \n # complete\n \n return jac\n\nresults = scipy.optimize.basinhopping( # complete\nbest_x = results.x\n\nprint('All done! Best score: `{}`, took `{}` function evaluations.'.format(results.fun, results.nfev))\n\n# plot the resulting fit\n```\n\nIf all went well, the jacobian version of the optimization should have taken ~8x fewer function evaluations. But is it faster?\n\n**Problem 3c**\n\nCompute how many times the Jacobian was called, and estimate how expensive the Jacobian is to compute relative to the objective function. How does this compare to the run that only used finite differencing?\n\n\n```python\nglobal jcount\njcount = 0\n\n# complete\n```\n\nCan you think of a reason why using the Jacobian version may be preferable, even if it is slower?\n\n## Challenge Problem(s)\n\n**Select one (or more) of the following:**\n\n- Fit a different event using either MOSFiT or the analytical formula. Any supernova can be loaded by name from the internet via the `fetch` method of the `Fetcher` class. Examples: SN1987A, SN2011fe, PTF09ge. If you are using the analytical model, exclude all but one of the bands in the dataset.\n- Optimize the Jacobian function to reuse common functions that are shared between each derivative component (example: $1 - e^{((t + t_0)/\\tau)}$ appears frequently in the expressions, it only needs to be computed once).\n- Sample the posterior in a Monte Carlo framework (using priors of your choice). Samplers like `emcee` are versatile and work even when derivatives aren't available, but we **do** have derivatives, so more powerful methods like Hamiltonian MCMC are available to us. A simple HMC for our purposes is available via `pip install pyhmc`, see the README for instructions on how to construct the input function: https://github.com/rmcgibbo/pyhmc. Plot the resulting samples using the `corner` package.\n\n\n```python\nfrom pyhmc import hmc\n# complete\n```\n\n## Concluding questions\n\n- What did we learn from fitting the analytic model about the **physics** of the disruption?\n- Does the analytic function have utility for generating simulated data?\n- Where else might the analytic function have a use?\n", "meta": {"hexsha": "becb13d241db391a943c33eb74e5984ef6357f5b", "size": 26667, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Session6/Day1/BuildingBetterModels.ipynb", "max_stars_repo_name": "hsnee/LSSTC-DSFP-Sessions", "max_stars_repo_head_hexsha": "5d90992179c80efbd63e9ecc95fe0fef7a0d83c1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-08-10T06:07:17.000Z", "max_stars_repo_stars_event_max_datetime": "2020-08-10T06:07:17.000Z", "max_issues_repo_path": "Session6/Day1/BuildingBetterModels.ipynb", "max_issues_repo_name": "hsnee/LSSTC-DSFP-Sessions", "max_issues_repo_head_hexsha": "5d90992179c80efbd63e9ecc95fe0fef7a0d83c1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2021-09-28T22:44:49.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-08T20:36:34.000Z", "max_forks_repo_path": "Session6/Day1/BuildingBetterModels.ipynb", "max_forks_repo_name": "hsnee/LSSTC-DSFP-Sessions", "max_forks_repo_head_hexsha": "5d90992179c80efbd63e9ecc95fe0fef7a0d83c1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-06-19T10:18:44.000Z", "max_forks_repo_forks_event_max_datetime": "2020-06-19T10:18:44.000Z", "avg_line_length": 39.1585903084, "max_line_length": 545, "alphanum_fraction": 0.5797802527, "converted": true, "num_tokens": 5535, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432182679956, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.43588370528786513}} {"text": "# Import Libraries\n\n\n```python\nfrom __future__ import print_function\nimport neptune\nimport os, sys, math\nimport matplotlib.pyplot as plt\nroot = os.path.abspath(os.getcwd())\nlibs = root + '/libs'\ndata = root + '/data'\nsys.path.append(libs)\nsys.path.append(data)\nfrom nn_utils import *\nfrom utilsData import *\nfrom utilsData import read_dataset\nfrom utilsData import transform_labels\nimport keras\nfrom keras.utils import plot_model\nfrom keras.models import Sequential, load_model\nfrom keras.models import Model\nfrom keras.datasets import mnist\nfrom keras.layers import *\nfrom keras.utils import to_categorical\nimport pandas as pd\nimport json\nimport numpy as np\nimport ctypes\nimport sklearn \n%matplotlib inline\nimport plotly.graph_objects as go\nimport time\n```\n\n# Define Constants and Variables\n\n\n```python\n# Define Path Constants\ndataRoot = \"data\"\ndataset_name = 'Cloth'\nrootWithSlash = dataRoot + '/' \ndataTest = dataset_name + \"_TEST.txt\"\ndataTrain = dataset_name + \"_TRAIN.txt\"\n\n# Skript Constants\nINTERPOLATION = 2\nNUMBER_OF_SAMPLES_CNN = 250\nADM_LOWER_LIMIT_INT_8 = 127\nADM_UPPER_LIMIT_INT_8 = ADM_LOWER_LIMIT_INT_8\nMAX_NUMBER_OF_DATA_POINTS = 2927\nbudget = 100\nalpha = 0.5\nwindow = 10\n\npickleObjectNameW = '/lowstandardhighvolumetipswater.pkl'\npickleObjectNameG = '/lowstandardhighvolumetipsglycerin.pkl'\n```\n\n# Load Acquired Aspiration Data\n\n\n```python\n# Load data from pickle Objects\nsql_DF_Water = pd.read_pickle(dataRoot+pickleObjectNameW)\nsql_DF_Glycerol = pd.read_pickle(dataRoot+pickleObjectNameG)\nsql_DF = pd.concat([sql_DF_Glycerol, sql_DF_Water])\n\n# Select just pressure datas \nsql_DFFilteredAsp = sql_DF.loc[sql_DF.loc[:,'J_Read Pressure Data Active'] == 1,:]\n# Select just class ids and pressure datas \nsql_DFFilteredAspColumns = sql_DFFilteredAsp.iloc[:,-MAX_NUMBER_OF_DATA_POINTS:]\n# Add Class id to dataframe\nsql_DFFilteredAspColumns['ClassId'] = sql_DFFilteredAsp.loc[:,'ClassId']\n# Reset index\nsql_DFFilteredAspColumns = sql_DFFilteredAspColumns.reset_index()\nsql_DFFilteredAspColumns = sql_DFFilteredAspColumns.drop(['index'], axis=1)\n```\n\n# Different Aspiration Curves Raw Data\n\n\n```python\nplt.close()\nfig = go.Figure()\n\ntitle = 'Different Aspiration Curves Raw Data'\n# Add traces\n\nfor i in range(0, sql_DFFilteredAspColumns.shape[0]):\n # Normal Aspiration\n if int(sql_DFFilteredAspColumns.loc[i,'ClassId']) == 0:\n fig.add_trace(go.Scatter(x=np.arange(len(np.array(sql_DFFilteredAspColumns.iloc[i,:]))), y=np.array(sql_DFFilteredAspColumns.iloc[i,:]),name='Asp. Normal', \n line=dict(color='rgb(0,0,0)', width=1)))\n # Air Aspiration\n elif int(sql_DFFilteredAspColumns.loc[i,'ClassId']) == 1:\n fig.add_trace(go.Scatter(x=np.arange(len(np.array(sql_DFFilteredAspColumns.iloc[i,:]))), y=np.array(sql_DFFilteredAspColumns.iloc[i,:]),name='Asp. Air', \n line=dict(color='rgb(255,0,0)', width=1))) \n # Cloth Aspiration\n elif int(sql_DFFilteredAspColumns.loc[i,'ClassId']) == 2:\n fig.add_trace(go.Scatter(x=np.arange(len(np.array(sql_DFFilteredAspColumns.iloc[i,:]))), y=np.array(sql_DFFilteredAspColumns.iloc[i,:]),name='Asp. Cloth', \n line=dict(color='rgb(0,0,255)', width=1))) \n \n \nannotations = [] \n \nannotations.append(dict(xref='paper', yref='paper', x=0.0, y=1.05,\n xanchor='left', yanchor='bottom',\n text=title,\n font=dict(family='Arial',\n size=30,\n color='rgb(37,37,37)'),\n showarrow=False)) \n\nfig.update_layout(annotations=annotations,\n yaxis_title='Pressure [pa]',\n xaxis_title='Number of Samples [N] ')\n\nfig.show()\n```\n\n# Scale Time Data to 250 Points\n\nTwo different pressure data cases are possible\n - Aspiration pressure curve contains less than 250 points -> upsampling case\n - Aspiration pressure curve contains more than 250 points -> Downsampling case\n \nThe idea for this problem is to transform a digital signal to a nearly continuous signal by interpolation and resample it such that the whole curve in time contains 250 samples.\n\n# Linear Interpolation of Curve\n\n\n```python\nsql_DFFilteredAspColumnsCopy = sql_DFFilteredAspColumns.copy()\nsql_DFFilteredAspColumnsCopy = sql_DFFilteredAspColumnsCopy.drop(['ClassId'], axis= 1)\nnewDataframe = sql_DFFilteredAspColumnsCopy\nfor interpol in range(0,INTERPOLATION):\n newDataframe = pd.DataFrame()\n for i in range(0, sql_DFFilteredAspColumns.shape[0]):\n sql_DFFilteredAspColumnsWithoutNan = np.array(sql_DFFilteredAspColumnsCopy.iloc[i,:].dropna())\n dataNumberingArray = np.arange(len(sql_DFFilteredAspColumnsWithoutNan)) + 1\n zeroPaddedData = np.insert(sql_DFFilteredAspColumnsWithoutNan, dataNumberingArray, 0)\n x = 0\n while x < (len(zeroPaddedData) - 2):\n zeroPaddedData[x + 1] = ((zeroPaddedData[x] - zeroPaddedData[x + 2]) / 2.0) + zeroPaddedData[x + 2]\n x = x + 2\n \n interpolatedData = pd.DataFrame(zeroPaddedData)\n interpolatedDataT = interpolatedData.T\n \n newDataframe = pd.concat([newDataframe, interpolatedDataT])\n \n newDataframe = newDataframe.reset_index()\n newDataframe = newDataframe.drop(['index'], axis=1)\n \n sql_DFFilteredAspColumnsCopy = newDataframe \n\n```\n\n# Resample Curves Such That Wohle Curve Is Sampled Equally\n\n\n```python\ntotalResampledData = pd.DataFrame()\nfor i in range(0, sql_DFFilteredAspColumns.shape[0]):\n sampleLengthInterpolatedCurve = len(sql_DFFilteredAspColumnsCopy.iloc[i, :].dropna())\n numberOfOffsetSamples = sampleLengthInterpolatedCurve/NUMBER_OF_SAMPLES_CNN\n sampleIteration = 0\n newDataframeWithoutNan = sql_DFFilteredAspColumnsCopy.iloc[i, :].dropna()\n resampledData = []\n for k in range(0, NUMBER_OF_SAMPLES_CNN):\n resampledData.append(newDataframeWithoutNan[int(sampleIteration)])\n sampleIteration +=numberOfOffsetSamples\n \n resampledDataPd = pd.DataFrame(np.array(resampledData))\n totalResampledData = pd.concat([totalResampledData, resampledDataPd.T])\n \ntotalResampledData = totalResampledData.reset_index()\ntotalResampledData = totalResampledData.drop(['index'], axis=1)\nsql_DFFilteredAspColumnsResIndex = sql_DFFilteredAspColumns.reset_index()\nsql_DFFilteredAspColumnsResIndex = sql_DFFilteredAspColumnsResIndex.drop(['index'], axis=1)\ntotalResampledData['ClassId'] = list(sql_DFFilteredAspColumnsResIndex.loc[:,'ClassId'])\n# Make sure ClassId stands at first position\ntotalResampledDataColumns = totalResampledData.columns.tolist()\ntotalResampledDataColumns = totalResampledDataColumns[-1:] + totalResampledDataColumns[:-1]\ntotalResampledData = totalResampledData[totalResampledDataColumns]\n```\n\n# Save to CSV \n\n\n```python\ntotalResampledData.to_csv(rootWithSlash+dataTest, index=False, header=False)\ntotalResampledData.to_csv(rootWithSlash+dataTrain, index=False, header=False)\n```\n\n# Convert Test/Train Cap Level Json File to Dataframe and Numpy Array\n\n\n```python\n# Load text file\ndatasets_dict = read_dataset(dataRoot, dataset_name) \nx_train = datasets_dict[dataset_name][0]\ny_train = datasets_dict[dataset_name][1]\nx_test = datasets_dict[dataset_name][2]\ny_test = datasets_dict[dataset_name][3]\ny_trainClass = datasets_dict[dataset_name][1]\ny_testClass = datasets_dict[dataset_name][3]\n\nnb_classes = len(np.unique(np.concatenate((y_train,y_test),axis =0)))\n# make the min to zero of labels\ny_train,y_test = transform_labels(y_train,y_test)\n\n# save orignal y because later we will use binary\ny_true = y_test.astype(np.int64) \n# transform the labels from integers to one hot vectors\nenc = sklearn.preprocessing.OneHotEncoder()\nenc.fit(np.concatenate((y_train,y_test),axis =0).reshape(-1,1))\ny_train = enc.transform(y_train.reshape(-1,1)).toarray()\ny_test = enc.transform(y_test.reshape(-1,1)).toarray()\n\nif len(x_train.shape) == 2: # if univariate \n # add a dimension to make it multivariate with one dimension \n x_train = x_train.reshape((x_train.shape[0],x_train.shape[1],1))\n x_test = x_test.reshape((x_test.shape[0],x_test.shape[1],1))\n\ninput_shape = x_train.shape[1:]\n```\n\n# Scale Aspiration Data\n\n\\begin{align}\n\\Delta_s = max(S(t)) - min(S(t)) \\\\\nS_s(t) = \\frac{\\Delta_{s128}}{\\Delta_s} \\Bigg(S(t) - \\bigg|\\frac{|max(S(t))| - |min(S(t))|}{2}\\bigg|\\Bigg) \\hspace{1cm} |max(S(t))| > |min(S(t))|\\\\\nS_s(t) = \\frac{\\Delta_{s128}}{\\Delta_s} \\Bigg(S(t) + \\bigg|\\frac{|max(S(t))| - |min(S(t))|}{2}\\bigg|\\Bigg) \\hspace{1cm} |min(S(t))| > |max(S(t))|\n\\end{align}\n\nwhere $S_s(t)$ defines the scaled pressure data, $S(t)$ defines the unscaled pressure data, $\\Delta_{s128}$ the max range of a int_8 datatype (255) and $\\Delta_s$ the max range of the unscaled pressure data $S(t)$\n\n\n```python\nplt.close()\nfig = go.Figure()\n\ntitle = 'Different Aspiration Curves Test Data'\n# Add traces\n\nfor i in range(0,x_test.shape[0]):\n # Normal Aspiration\n if int(y_testClass[i]) == 0:\n fig.add_trace(go.Scatter(x=np.arange(len(x_test[i])), y=x_test[i,:].ravel(),name='Asp. Normal', \n line=dict(color='rgb(0,0,0)', width=1)))\n # Air Aspiration\n elif int(y_testClass[i]) == 1:\n fig.add_trace(go.Scatter(x=np.arange(len(x_test[i])), y=x_test[i,:].ravel(),name='Asp. Air', \n line=dict(color='rgb(255,0,0)', width=1))) \n # Cloth Aspiration\n elif int(y_testClass[i]) == 2:\n fig.add_trace(go.Scatter(x=np.arange(len(x_test[i])), y=x_test[i,:].ravel(),name='Asp. Cloth', \n line=dict(color='rgb(0,0,255)', width=1))) \n \n \nannotations = [] \n \nannotations.append(dict(xref='paper', yref='paper', x=0.0, y=1.05,\n xanchor='left', yanchor='bottom',\n text=title,\n font=dict(family='Arial',\n size=30,\n color='rgb(37,37,37)'),\n showarrow=False)) \n\nfig.update_layout(annotations=annotations,\n yaxis_title='Pressure [pa]',\n xaxis_title='Number of Samples [N] ')\n\nfig.show()\n```\n\n# Display Train X (Time) and Y (Normal-, Cloth-, Air-Aspiration)\n\n\n```python\nfig = go.Figure()\ntitle = 'Different Aspiration Curves Train Data'\nscaledTrain = x_train\n# Add traces\nfor i in range(0,x_train.shape[0]):\n # Scale data\n temp = []\n data = x_train[i]\n delta_s = max(data) - min(data)\n if(abs(max(data)) > abs(min(data))):\n tempScaled = 253.0/delta_s * np.array(data) \n scaledTrain[i,:] = tempScaled + (ADM_UPPER_LIMIT_INT_8 - max(tempScaled))\n elif(abs(min(data)) > abs(max(data))): \n tempScaled = 253.0/delta_s * np.array(data) \n scaledTrain[i,:] = tempScaled + (-ADM_LOWER_LIMIT_INT_8 - min(tempScaled))\n \n # Normal Aspiration\n if int(y_trainClass[i]) == 0:\n fig.add_trace(go.Scatter(x=np.arange(len(x_train[i])), y=scaledTrain[i,:].ravel(),name='Asp. Normal', \n line=dict(color='rgb(0,0,0)', width=1)))\n # Air Aspiration Aspiration\n elif int(y_trainClass[i]) == 1:\n fig.add_trace(go.Scatter(x=np.arange(len(x_train[i])), y=scaledTrain[i,:].ravel(),name='Asp. Air', \n line=dict(color='rgb(255,0,0)', width=1))) \n # Cloth Aspiration\n elif int(y_trainClass[i]) == 2 :\n fig.add_trace(go.Scatter(x=np.arange(len(x_train[i])), y=scaledTrain[i,:].ravel(),name='Asp. Cloth', \n line=dict(color='rgb(0,0,255)', width=1))) \n \nannotations = [] \n \nannotations.append(dict(xref='paper', yref='paper', x=0.0, y=1.0,\n xanchor='left', yanchor='bottom',\n text=title,\n font=dict(family='Arial',\n size=30,\n color='rgb(37,37,37)'),\n showarrow=False)) \n\nfig.update_layout(annotations=annotations,\n yaxis_title='Pressure [pa]',\n xaxis_title='Number of Samples [N] ')\nfig.show()\n\n```\n\n# Display Test X (Time) and Y (Normal-, Cloth-, Air-Aspiration)\n\n\n```python\nplt.close()\nfig = go.Figure()\n\ntitle = 'Different Aspiration Curves Test Data'\nscaledTest = x_test\n# Add traces\nfor i in range(0,x_test.shape[0]):\n # Scale data\n temp = []\n data = x_test[i]\n delta_s = max(data) - min(data)\n if(abs(max(data)) > abs(min(data))):\n tempScaled = 253.0/delta_s * np.array(data) \n scaledTest[i,:] = tempScaled + (ADM_UPPER_LIMIT_INT_8 - max(tempScaled))\n elif(abs(min(data)) > abs(max(data))): \n tempScaled = 253.0/delta_s * np.array(data) \n scaledTest[i,:] = tempScaled + (-ADM_LOWER_LIMIT_INT_8 - min(tempScaled))\n\nfor i in range(0,x_test.shape[0]):\n # Normal Aspiration\n if int(y_testClass[i]) == 0:\n fig.add_trace(go.Scatter(x=np.arange(len(x_test[i])), y=scaledTest[i,:].ravel(),name='Asp. Normal', \n line=dict(color='rgb(0,0,0)', width=1)))\n # Air Aspiration\n elif int(y_testClass[i]) == 1:\n fig.add_trace(go.Scatter(x=np.arange(len(x_test[i])), y=scaledTest[i,:].ravel(),name='Asp. Air', \n line=dict(color='rgb(255,0,0)', width=1))) \n # Cloth Aspiration\n elif int(y_testClass[i]) == 2:\n fig.add_trace(go.Scatter(x=np.arange(len(x_test[i])), y=scaledTest[i,:].ravel(),name='Asp. Cloth', \n line=dict(color='rgb(0,0,255)', width=1))) \n \n \nannotations = [] \n \nannotations.append(dict(xref='paper', yref='paper', x=0.0, y=1.05,\n xanchor='left', yanchor='bottom',\n text=title,\n font=dict(family='Arial',\n size=30,\n color='rgb(37,37,37)'),\n showarrow=False)) \n\nfig.update_layout(annotations=annotations,\n yaxis_title='Pressure [pa]',\n xaxis_title='Number of Samples [N] ')\n\nfig.show()\n```\n\n# Define Layers of CNN Algorithm\n\n\n```python\n#https://adventuresinmachinelearning.com/keras-tutorial-cnn-11-lines/\ndef KModel(x_train, y_train, x_test, y_test):\n # Shuffle\n permutation = np.random.permutation(y_train.shape[0])\n x_train = x_train[permutation, :, :]\n y_train = y_train[permutation]\n \n inputs = Input(shape=x_train.shape[1:])\n # Layer Block 1\n x = Conv1D(32, kernel_size=(9), strides=(2), padding='same')(inputs)\n x = BatchNormalization()(x)\n x = Dropout(0.2)(x)\n x = ReLU()(x)\n x = MaxPool1D(2, strides=2)(x)\n\n # Layer Block 2\n x1 = Conv1D(32, kernel_size=(5), strides=(1), padding=\"same\")(x)\n x1 = BatchNormalization()(x1)\n x1 = Dropout(0.2)(x1)\n x1 = ReLU()(x1)\n x1 = MaxPool1D(2, strides=2)(x1)\n\n # Layer Block 3\n x2 = Conv1D(32, kernel_size=(3), strides=(1), padding=\"same\")(x)\n x2 = BatchNormalization()(x2)\n x2 = Dropout(0.2)(x2)\n x2 = ReLU()(x2)\n x2 = MaxPool1D(2, strides=2)(x2)\n\n # Layer Block 4\n x3 = MaxPool1D(2, strides=2)(x)\n x3 = Dropout(0.2)(x3)\n\n # concate all inception layers\n x = concatenate([ x1, x2,x3], axis=-1)\n\n # conclusion\n x = Conv1D(48, kernel_size=(3), strides=(1), padding=\"same\")(x)\n x = ReLU()(x)\n x = MaxPool1D(2, strides=2)(x)\n x = Dropout(0.2)(x)\n\n # our netowrk is not that deep, so a hidden fully connected layer is introduce\n x = Flatten()(x)\n x = Dense(64)(x)\n x = Dropout(0.2)(x)\n x = ReLU()(x)\n x = Dense(3)(x)\n y = Softmax()(x)\n \n model = Model(inputs=inputs, outputs=y)\n model.compile(loss=\"categorical_crossentropy\", optimizer=\"Adam\", metrics=['accuracy'])\n model.fit(x_train, y_train, batch_size=16, epochs=100, shuffle=True, verbose=2, validation_data=(x_test, y_test))\n return model\n```\n\n# Train CNN Algorithm\n\n\n```python\nif os.path.exists('.shift_list'):\n os.remove(\".shift_list\")\nmodel = KModel(scaledTrain, y_train, scaledTest, y_test)\nmodel.save(dataRoot+'model.h5')\n```\n\n# Generate C Code for NN Library and Transfer Weights File SVN\n\n\n```python\ngenerate_model(model, x_test, name='NN_weights.h')\n```\n", "meta": {"hexsha": "f9edb5447016b2fc5127c410bfcb9812d7a06a1b", "size": 24165, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "phenomenDetection.ipynb", "max_stars_repo_name": "achenbachsven/learningSkript", "max_stars_repo_head_hexsha": "7af067cbf0c8d7eed010806923f8af2e38977be2", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "phenomenDetection.ipynb", "max_issues_repo_name": "achenbachsven/learningSkript", "max_issues_repo_head_hexsha": "7af067cbf0c8d7eed010806923f8af2e38977be2", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2020-03-24T15:59:06.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-10T01:53:37.000Z", "max_forks_repo_path": "phenomenDetection.ipynb", "max_forks_repo_name": "achenbachsven/learningSkript", "max_forks_repo_head_hexsha": "7af067cbf0c8d7eed010806923f8af2e38977be2", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 38.1151419558, "max_line_length": 221, "alphanum_fraction": 0.5131388372, "converted": true, "num_tokens": 4531, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743735019595, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.43588064342819355}} {"text": "```python\n# setup and imports\n\n# the usual suspects\n%matplotlib inline\n%config InlineBackend.figure_format = 'png'\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\n\nfrom matplotlib.colors import LogNorm\nimport seaborn as sns\n\nfrom ipywidgets import interact\n\n# used for generating graphs\nfrom graphviz import Digraph\n\n# data mining algorithms\nfrom mlxtend.frequent_patterns import apriori\nfrom mlxtend.frequent_patterns import association_rules\nfrom sklearn_extra.cluster import KMedoids\n\n# config\nplt.rcParams[\"figure.figsize\"] = (10,10)\n```\n\n\n```python\n# load the data\n# note: this if randomly generated fake data, not real enrolment data\ndf = pd.read_csv('sample.csv')\ndf.drop('UNIQUEID', axis=1, inplace=True)\n```\n\n# Course Co-occurence Analysis\n\nLearning Analytics, Visual Analytics @ UBC\n\nCraig Thompson, CTLT\n\nAcademic programs often prescribe some number of official pathways. However, students may also choose to take combinations of courses other than those we intend.\n\nWe'd like to reveal those patterns.\n\n\n\n\n[Desire path](https://flic.kr/p/5kDxUt) by [wetwebwork](https://www.flickr.com/photos/wetwebwork/][/url]), on Flickr [(CC BY 2.0)](https://creativecommons.org/licenses/by/2.0/)\n\n# Data we have\n\n- For every student who earns a degree, we have a record of all the courses they took and counted towards their degree requirements.\n\n- We've limited the dataset to courses from a single department.\n\n- Our dataset is two dimensional **binary indicator matrix**:\n - Across the horizontal axis: all the courses offered by the department\n - Down the vertical axis: IDs for each student who earned a degree\n - For each student/course pair we indicate whether the student took the course\n - This is fake data\n\n\n```python\ndf.head(20)\n```\n\n# Data we don't have\n\n- We don't have data for non-majors or anyone who did not complete their degree.\n\n- We don't have performance data, so we don't know how well any student did in any particular course.\n\n- We don't have any temporal information, so we don't know:\n - Which courses students took in sequence\n - Whether they took a pair of courses in back to back terms or with gaps\n - Which courses they took in concurrently\n\n# About the analysis\n\n- We are doing an exploratory analysis. This is not experimental.\n\n- We will try to answer questions about *what* has happened, but we are unable to address *why*.\n\n- We'd like to say \"students are discovering their own pathways through our planned degrees\". The reality is that students may be taking these groupings of courses because they:\n - Fit nicely in their timetable\n - Are offered by instructors they like\n - Have a reputation of being easy or fun courses\n - Have free or inexpensive textbooks\n - Are the only courses left at registration time\n\n# Analysis via clustering\n\nCommon questions:\n- What does an \"average\" student look like, in terms of the courses they study?\n- If there were $N$ prototypical students, what would they look like?\n\nAnswer:\n- We can formulate a *mathematically* average student, but there is no *pedagogically meaningful* average student.\n- This sort of analysis is messy and hard to interpret.\n- We'll do it anyway just to see!\n\n\n```python\nim = plt.imshow(df.head(100), cmap=plt.cm.binary)\n```\n\n\n```python\n# most common courses\ndf.sum().sort_values(ascending=False).head(10)\n```\n\n\n```python\n# how many courses does each student take?\ndf.sum(axis=1).value_counts().sort_index().plot(kind='bar');\n```\n\n\n```python\n# helper functions for clustering students in course-space\n\ndef cluster(n, df):\n kmedoids = KMedoids(n_clusters=n, metric='manhattan', random_state=0).fit(df)\n nearest_medoid = kmedoids.predict(df)\n distances = kmedoids.transform(df)\n nearest_distance = distances[[np.arange(distances.shape[0])],nearest_medoid].T\n return (kmedoids, nearest_medoid, distances, nearest_distance)\n\ndef describe_clusters(kmedoids, nearest_medoid, distances, nearest_distance):\n plt.figure(figsize=(10, 10))\n\n for i in range(kmedoids.cluster_centers_.shape[0]):\n print(\"cluster\", i+1, \"centroid:\", list(df.columns[kmedoids.cluster_centers_[i,:] == 1]))\n print(\"number of students in this cluster:\", (nearest_medoid == i).sum())\n cluster_member_distances = nearest_distance[nearest_medoid == i]\n if cluster_member_distances.size > 0:\n print(\"minimum distance to centroid:\", cluster_member_distances.min())\n print(\"maximum distance to centroid:\", cluster_member_distances.max())\n print(\"mean distance to centroid:\", cluster_member_distances.mean())\n print(\"median distance to centroid:\", np.median(cluster_member_distances))\n print()\n plt.plot(sorted(cluster_member_distances))\n\n\n```\n\n\n```python\ndescribe_clusters(*cluster(4, df))\n```\n\n# Lessons (re-)learned\n\n- Course enrolment datasets are big, and hard to construct a clear mental picture of.\n- We're working with (only) 50 students and 22 courses.\n- Within academic programs, there aren't usually clear, strong, non-prescribed patterns at the level of whole-enrolment histories\n\nSo, let's try something different...\n\n# History\n\nWhat other domains work with similarly shaped data? Consumer purchases!\n\n- Each individual shopper collects a bunch of items\n- When a customer checks out, a sales invoice is generated listing all the items that were purchased together\n\nFrom all the sales invoices, we may wish to look for patterns in consumer behaviour:\n- Are there items that are **frequently** purchased together?\n- Are some items good **predictors** of other items being purchased?\n\nWhy would someone care?\n- If consumers buy hot dogs whenever they buy hotdog buns, then grocery stores can attempt to manipulate custormers into buying hotdogs by putting hotdog buns on sale. **Profit!**\n\n# Content warning\n\nThe following slides contain math.\n\n# Set theory\n\n- a *set* is an unordered collection of distinct objects.\n\n- For this analysis, each student's course enrolment history is being treated as a set. Sets are often written like this: $\\{a,b,c\\}$\n\n- All the student enrolment histories are jointly represented as a collection of sets.\n - They are not a set-of-sets, because sets have distinct elements, and two students are able to have exactly the same course enrolment history.\n - So, this collection of sets is called a *multiset* or a *bag*, to denote that it may contain duplicate elements.\n\n\n\n# Frequent itemsets\n\nGiven a multiset (such as a stack of grocery store receipts, or a table of student-course enrolments), how do we find the frequently occurring subsets (or itemsets)?\n\nExample: Given $[\\{a\\},\\{a,b\\},\\{a,c\\},\\{a,b,c,d\\}]$\n\nWe can see that:\n- $\\{a\\}$ occurs in all 4 sets\n- $\\{a,b\\}$ and $\\{a,c\\}$ each occur in 2 sets\n\n\n\n\\begin{align}\n&\\mathrm{Apriori}(T,\\epsilon)\\\\\n&\\quad L_1 \\gets \\{\\textrm{large 1 item sets}\\}\\\\\n&\\quad k \\gets 2\\\\\n&\\quad \\textbf{while}\\ L_{k-1} \\neq \\emptyset\\\\\n&\\quad \\quad C_k \\gets \\{c = a \\cup \\{b\\} \\mid a \\in L_{k-1} \\land b \\notin a, \\{s \\subseteq c \\mid |s| = k - 1 \\} \\subseteq L_{k-1} \\}\\\\\n&\\quad \\quad \\textbf{for}\\ \\textrm{transactions}\\ t \\in T\\\\\n&\\quad \\quad \\quad D_t \\gets \\{c \\in C_k \\mid c \\subseteq t \\}\\\\\n&\\quad \\quad \\quad \\textbf{for}\\ \\textrm{candidates}\\ c \\in D_t\\\\\n&\\quad \\quad \\quad \\quad count[c] \\gets count[c] + 1\\\\\n&\\quad \\quad L_k \\gets \\{c \\in C_k \\mid count[c] \\geq \\epsilon \\}\\\\\n&\\quad \\quad k \\gets k + 1\\\\\n&\\quad \\textbf{return} \\bigcup_k L_k\n\\end{align}\n\n\nRakesh Agrawal and Ramakrishnan Srikant. 1994. Fast Algorithms for Mining Association Rules in Large Databases. In Proceedings of the 20th International Conference on Very Large Data Bases (VLDB \u201994). Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 487\u2013499. ( > 26k citations!)\n\nLet $X$ be an itemset and $T$ be the set of transactions/records in the database\n\n\n\n\\begin{align}\n&\\textrm{Support}(X) = \\frac{\\mid \\{t \\in T \\mid X \\subseteq t \\}\\mid }{\\mid T \\mid}\n\\end{align}\n\n\n\n*Support* indicates how frequently a given itemset appears in the transactions of the database.\n- A support of 1 indicates the itemset appears in every transaction.\n- A support of 0.5 indicates the itemset appears in half of the transactions.\n\n\n```python\ncourse_frequency = apriori(df, min_support=np.nextafter(0,1), max_len=1, use_colnames=True)\ncourse_frequency['course'] = course_frequency['itemsets'].apply(lambda x: set(x).pop())\ncourse_frequency['course_number'] = course_frequency['course'].apply(lambda x: x[4:])\ncourse_frequency[['support', 'course']].sort_values(by='support',ascending=False)\n```\n\n\n```python\ncf = course_frequency[['support', 'course']].set_index('course').sort_values(by='support',ascending=False)\n\ndef f(limit):\n cf.head(limit).plot(kind='bar')\n\ni = interact(f, limit=(1,20))\n```\n\n\n```python\nfrequent_itemsets = apriori(df, min_support=np.nextafter(0,1), max_len=2, use_colnames=True)\nfrequent_itemsets.sort_values(by='support',ascending=False)\n```\n\n# Association rules\n\n\\begin{equation*}\nX \\Rightarrow Y, \\textrm{where}\\ X,Y \\subseteq I\n\\end{equation*}\n\nX is called the *antecedent* and Y is the *consequent*.\n\n$I$ is the set of all items (e.g. courses).\n\nexample: $\\textrm{Math110} \\Rightarrow \\textrm{Math210}$ would be read as \"if Math110, then Math210\".\n\nNow we have a notation for a relationship between two itemsets (in this case, the two itemsets each contain a single item), but we need to describe the *qualities* of that relationship...\n\nRakesh Agrawal, Tomasz Imieli\u0144ski, and Arun Swami. 1993. Mining association rules between sets of items in large databases. In Proceedings of the 1993 ACM SIGMOD international conference on Management of data (SIGMOD \u201993). Association for Computing Machinery, New York, NY, USA, 207\u2013216. DOI:https://doi.org/10.1145/170035.170072 (> 22k citations!)\n\n# Metrics for quantifying association rules: Support\n\n- *Antecedent Support*: indicates how frequently the antecedent item set appears in the database.\n\n$$\\textrm{Antecedent Support}(X \\Rightarrow Y) = \\frac{\\mid \\{t \\in T \\mid X \\subseteq t \\}\\mid }{\\mid T \\mid}$$\n\n- *Consequent Support*: indicates how frequently the consequent item set appears in the database.\n\n$$\\textrm{Consequent Support}(X \\Rightarrow Y) = \\frac{\\mid \\{t \\in T \\mid Y \\subseteq t \\}\\mid }{\\mid T \\mid}$$\n\n- *(Rule) Support*: indicates how frequently the all them items of the antecedent and consequent jointly appear in the database.\n\n$$\\textrm{Support}(X \\Rightarrow Y) = \\frac{\\mid \\{t \\in T \\mid X \\cup Y \\subseteq t \\}\\mid }{\\mid T \\mid}$$\n\n## Metrics for quantifying association rules: Confidence\n\n$$\\textrm{Confidence}(X \\Rightarrow Y) = \\frac{ \\textrm{Support}(X \\Rightarrow Y) }{ \\textrm{Support}(X) }$$\n\n*Confidence*: the ratio of rule support to antecedent support. \n - Or, given that the antecedent has been observed, how likely are we to also observe the consequent?\n\nIf a rule has high confidence, and the antecedent is observed, then we can be fairly confident that the consequent will be observed as well.\n\n## Metrics for quantifying association rules: Lift\n\n$$\\textrm{Lift}(X \\Rightarrow Y) = \\frac{ \\textrm{Confidence}(X \\Rightarrow Y) }{ \\textrm{Support}(Y) }$$\n\n*Lift*: ratio of confidence to consequent support. Lift is a measure of how much more often the antecedent and the consequent occur together than would be expected if they were statistically independent. When the antecedent of a rule with high lift is observed, we can be more confident that the consequent will also be observed.\n\nConfidence and lift are both descriptors of the \"power\" of a rule.\n\n\n```python\nrules = association_rules(frequent_itemsets, metric=\"support\", min_threshold=np.nextafter(0,1))\n\nrules['antecedent_course'] = rules['antecedents'].apply(lambda x: set(x).pop())\nrules['consequent_course'] = rules['consequents'].apply(lambda x: set(x).pop())\n\nrules['antecedent_course_number'] = rules['antecedent_course'].apply(lambda x: int(x[4:]))\nrules['consequent_course_number'] = rules['consequent_course'].apply(lambda x: int(x[4:]))\n\nrules['antecedent_year_level'] = rules['antecedent_course_number'].apply(lambda x: x//100 )\nrules['consequent_year_level'] = rules['consequent_course_number'].apply(lambda x: x//100)\n```\n\n\n```python\nrules\n```\n\n\n```python\npairwise_rules = rules[(rules['antecedent_year_level']==3) & (rules['consequent_year_level']==3)]\npairwise_support = pairwise_rules.pivot(index='antecedent_course',columns='consequent_course',values='support').fillna(0)\nax = sns.heatmap(pairwise_support, xticklabels=True, yticklabels=True, cmap='BuPu')\n```\n\n\n```python\npairwise_rules = rules[(rules['antecedent_year_level']==3) & (rules['consequent_year_level']==3)]\npairwise_confidence = pairwise_rules.pivot(index='antecedent_course',columns='consequent_course',values='confidence').fillna(0)\nax = sns.heatmap(pairwise_confidence, xticklabels=True, yticklabels=True, cmap='BuPu')\n```\n\n\n```python\npairwise_rules = rules[(rules['antecedent_year_level']==3) & (rules['consequent_year_level']==3)]\npairwise_lift = pairwise_rules.pivot(index='antecedent_course',columns='consequent_course',values='lift').fillna(0.1)\n#pairwise_lift = pairwise_lift.applymap(lambda x: x if x >=1 else 0.01)\nax = sns.heatmap(pairwise_lift, xticklabels=True, yticklabels=True, cmap='BuPu', norm=LogNorm())\n```\n\n\n```python\n# exploring 'significant' rules\nsig_rules = rules[\n (rules['support'] > 0.01)\n #& (rules['antecedent support'] > 0.01)\n #& (rules['consequent support'] > 0.01)\n & (rules['antecedent_year_level'] <= rules['consequent_year_level'])\n & (rules['confidence'] > 0.5)\n & (rules['lift'] > 1.5)\n ].sort_values(by='lift',ascending=False)\nsig_rules\n```\n\n\n```python\ndef plot_rules(sig_rules):\n antecedents = sig_rules[['antecedent_course','antecedent_course_number']]\n antecedents.columns = ['course','course_number']\n\n consequents = sig_rules[['consequent_course','consequent_course_number']]\n consequents.columns = ['course','course_number']\n\n figure_courses = pd.concat([antecedents, consequents]).drop_duplicates()\n\n dot = Digraph()\n for course in figure_courses.itertuples():\n dot.node(str(course.course_number),course.course)\n\n for association in sig_rules.itertuples():\n dot.edge(str(association.antecedent_course_number), str(association.consequent_course_number), label=f\"{association.lift:.2f}\")\n\n dot.graph_attr['overlap'] = 'False'\n dot.engine = 'neato'\n return dot\n```\n\n\n```python\ndot = plot_rules(sig_rules)\ndot\n```\n\n# What next?\n\n- This analysis was (nearly) totally naive to the official structure of the courses/programs offered. That means we 'discovered' several results that are obvious and uninteresting to anyone with domain expertise.\n - For example, we may have 'discovered' that if you take a senior course, then you are very likely to also take the prerequisites.\n - We should filter the result set to remove any association rules that reflect formal prerequisite relationships.\n\n- We've looked at *what* courses students take, but we have not investigated ordering, or the effects of ordering. An analysis of sequencing would be useful for planning connections across courses.\n\n# Thank You\n\nQuestions?\n\nContact: craig.thompson@ubc.ca\n", "meta": {"hexsha": "64852fa46911b326be4d073270d41517e92753da", "size": 22056, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "analysis.ipynb", "max_stars_repo_name": "craigdsthompson/course-cooccurrence", "max_stars_repo_head_hexsha": "44708ba318ad031ab5a294c4d0a6c8ebb5b7a252", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "analysis.ipynb", "max_issues_repo_name": "craigdsthompson/course-cooccurrence", "max_issues_repo_head_hexsha": "44708ba318ad031ab5a294c4d0a6c8ebb5b7a252", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "analysis.ipynb", "max_forks_repo_name": "craigdsthompson/course-cooccurrence", "max_forks_repo_head_hexsha": "44708ba318ad031ab5a294c4d0a6c8ebb5b7a252", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.4561983471, "max_line_length": 354, "alphanum_fraction": 0.6019677185, "converted": true, "num_tokens": 3915, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499941, "lm_q2_score": 0.658417500561683, "lm_q1q2_score": 0.4358216054817543}} {"text": "```javascript\n%%javascript\n$('#appmode-leave').hide();\n$('#copy-binder-link').hide();\n$('#visit-repo-link').hide();\n```\n\n\n```python\nimport ipywidgets as ipw\nimport json\nimport random\nimport time\nimport pandas as pd\nimport os\nimport webbrowser\nimport math\nfrom IPython.display import display, Markdown, Math, Latex, clear_output\n\n# set kinetic parameters\nwith open(\"rate_parameters.json\") as infile:\n jsdata = json.load(infile)\n\nparams = jsdata[\"equi4\"]\n```\n\nThe hydrolysis Sucrose into Glucose and Fructose is catalysed by the enzyme Invertase.\n\\begin{equation}\nSucrose + Invertase + \\mathrm{H_2O} \\to Glucose + Fructose\n\\end{equation}\nThere are however several substances that can inhibit the efficacy of the catalyst\n\nImagine performing a series of experiments using different initial concentration of Sucrose where you measure the rate of formation of Glucose with. The results of your experiments are affected by the presence of a contaminating substance that interferes with the catalytic reaction. Although you can somewhat control the concentration of the contaminant, you cannot completely eliminate it.\n\n1. Determine whether the contaminating substance inhibits the catalytic reaction and the type of the inhibition mechanism, *e.g.* Competitive, Uncompetitive, Non-competitive or Mixed.\n\n2. Determine the maximum rate achieved by the reaction, $V_{max}$ and the Michaelis constant, $K_M$, in the case you could completely eliminate the contamininat.\n\n### Tips:\n- Note that every time you restart the experiment the type of the inhibition mechanism may change.\n\n### Instructions:\n\n- Use the slide bar below to select temperature at which you perform the virtual experiment, \n- Click `Perform measurement` to run the virtual experiment and obtain the result of the experiment,\n- Click `Download CSV` to export the complete data set for all the experiments as a CSV file.\n\n\n```python\n# define path to results.csv file\nrespath = os.path.join(os.getcwd(), \"..\", \"results.csv\")\n\n# delete existing result file and setup rng\nif os.path.exists(respath):\n os.remove(respath)\n\nclass system:\n def __init__(self, vol=0, conc=0, press=0):\n self.vol = vol\n self.conc = conc\n self.press = press\n self.inhibition = 0\n self.seed = 0\n self.Vm = 0\n self.Km = 0\n self.Ki = 0\n self.Kip= 0\n\nclass data:\n def __init__(self, start=-1, error=0, label='none', units='pure', value=0,\n minval=-1, maxval=3, text='none'):\n self.start = start\n self.minval = minval\n self.maxval = maxval\n self.error = error\n self.label = label\n self.units = units\n self.value = value\n self.text = text\n \n# Experiment setup (+ hidden paramters)\nsystem = system()\ndef initialiseExperiment():\n global n\n global system \n global columns_list\n global scatter\n \n scatter = 0.01\n \n n = []\n columns_list = []\n\n n.append(len(args)) # number of input adjustable parameters \n n.append(len(result)) # number of results for the experiment\n\n for i in range(0, n[0]):\n columns_list.append(f\"{args[i].label} [{args[i].units}]\")\n for i in range(0, n[1]):\n columns_list.append(f\"{result[i].label} [{result[i].units}]\")\n\n # Random number seed\n t = int( time.time() * 1000.0 )\n system.seed = ((t & 0xff000000) >> 24) + ((t & 0x00ff0000) >> 8) +((t & 0x0000ff00) << 8) +((t & 0x000000ff) << 24)\n random.seed(system.seed)\n\n # Random inhibition type\n rnd = random.random()\n system.inhibition = int(5 * rnd)\n if (system.inhibition > 4):\n system.inhibition = 4\n \n system.Vm = params[\"Vm\"] * (1 + random.random()/2)\n system.Km = params[\"Km\"] * (1 + random.random()/2)\n system.Ki = system.Km * random.random()\n system.Kip= system.Km * random.random()\n\n\n```\n\n\n```python\n# Adjustable input parameters\ndef initialiseVariables():\n global logScale\n logScale = True\n global args\n args = []\n args.append(\n data(\n label = \"[S]\",\n minval = -3,\n maxval = 1,\n start = 0.001,\n units = \"mol/L\",\n value = 0.\n )\n )\n\n args.append(\n data(\n label = \"[I]\",\n minval = -3,\n maxval = 0,\n start = 0.001,\n units = \"mol/L\",\n value = 0.\n )\n )\n\n\n# Results\ndef initialiseResults():\n global result\n result = []\n result.append(\n data(\n label = \"Reaction Rate\",\n start = 0.,\n error = random.random() / 10.,\n units = \"mol/L\u00b7min\"\n )\n )\n\ndef measure():\n concS = float(args[0].text.value)\n concI = float(args[0].text.value)\n \n Vm = system.Vm\n Km = system.Km\n Ki = system.Ki\n Kip= system.Kip\n \n # no inhibition\n a = 1\n ap = 1\n \n # competitive\n if (system.inhibition == 1):\n a = 1 + concI / Ki\n ap = 1\n adp = 1\n\n # non-competitive\n elif (system.inhibition == 4):\n a = 1\n ap = 1 + concI / Ki\n adp = 1\n \n # un-competitive\n elif (system.inhibition == 2):\n a = 1\n ap = 1\n adp = 1. / (1 + concI / Kip)\n\n # mixed\n elif (system.inhibition == 3):\n a = 1 + concI / Ki\n ap = 1\n adp = 1. / (1 + concI / Kip)\n\n res = (ap * adp) * Vm * concS / ((a * adp)*Km + concS)\n return res\n\ninitialiseVariables()\n\n```\n\n\n```python\nout_P = ipw.Output()\nout_L = ipw.Output()\nout_X = ipw.Output()\n\nwith out_L:\n display(Markdown(\"[Download CSV](../results.csv)\"))\n \ndef calc(btn):\n out_P.clear_output()\n \n # Measurement result\n result[0].value = measure()\n \n # Random error\n result[0].error = result[0].value * scatter * (0.5 - random.random()) * 2\n \n # Output result\n out_R[0].value = f\"{result[0].value + result[0].error:.3e}\"\n\n # Read previous lines\n res = pd.read_csv(respath) \n \n var_list = []\n for i in range(0, n[0]):\n var_list.append(args[i].text.value)\n for i in range(0, n[1]):\n var_list.append(result[i].value + result[i].error)\n \n # Append result\n res.loc[len(res)] = var_list\n res.to_csv(respath, index=False)\n with out_P:\n display(res.tail(50))\n\ndef reset(btn):\n if os.path.exists(respath):\n os.remove(respath)\n\n initialiseResults()\n initialiseExperiment()\n \n res = pd.DataFrame(columns=columns_list)\n res.to_csv(respath, index=False)\n with out_P:\n out_P.clear_output()\n display(res.tail(1))\n\n with out_X:\n out_X.clear_output()\n \nbtn_reset = ipw.Button(description=\"Restart Laboratory\", layout=ipw.Layout(width=\"150px\"))\nbtn_reset.on_click(reset)\n\nbtn_calc = ipw.Button(description=\"Perform measurement\", layout=ipw.Layout(width=\"150px\"))\nbtn_calc.on_click(calc)\n# ---\n\nrows = []\nreset(btn_reset)\n\nargs[0].text = ipw.Text(str(args[0].start))\nrows.append(ipw.HBox([ipw.Label('Initial concentration of ' + args[0].label + ' : '),args[0].text]))\n\nargs[1].text = ipw.Text(str(args[1].start))\nrows.append(ipw.HBox([ipw.Label('Initial concentration of ' + args[1].label + ' : '),args[1].text]))\n\nout_R = []\nfor i in range(0, n[1]):\n out_R.append(ipw.Label(value=\"\"))\n rows.append(ipw.HBox([ipw.Label(value=f\"Measured {result[i].label} [{result[i].units}]:\",\n layout=ipw.Layout(width=\"250px\")),\n out_R[i]]))\n\nrows.append(ipw.HBox([btn_reset, btn_calc, out_L]))\n\ndef calc2(btn):\n random.seed(system.seed)\n rnd = random.random()\n iType = int(4 * rnd) + 1\n \n with out_X:\n out_X.clear_output()\n if (iType == 1):\n display(Markdown(r'Competitive inhibition'))\n elif (iType == 2):\n display(Markdown(r'Un-Competitive inhibition'))\n elif (iType == 3):\n display(Markdown(r'Mixed inhibition'))\n elif (iType == 4):\n display(Markdown(r'Non-Competitive inhibition'))\n else:\n display(Markdown(r'No inhibition'))\n\n display(Markdown(r'$K_M$ = 'rf'{system.Km:7.5}'))\n display(Markdown(r'$V_{max}$ = 'rf'{system.Ki:7.5}'))\n if (iType == 1) or (iType == 3) or (iType == 4):\n display(Markdown(r'$K_i$ = 'rf'{system.Ki:7.5}'))\n if (iType == 2) or (iType == 3) or (iType == 4):\n display(Markdown(r'$K_i^\\prime$ = 'rf'{system.Kip:7.5}'))\n\ndisplay(out_X)\n\nbtn_calc2 = ipw.Button(description=\"Check Inhibition Type\", layout=ipw.Layout(width=\"150px\"))\nbtn_calc2.on_click(calc2)\nrows.append(ipw.HBox([btn_calc2]))\n\nrows.append(ipw.HBox([out_P]))\nipw.VBox(rows)\n\n\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "dc1c4e47a6f7b3190f9ce79b88e1c8af23824a07", "size": 12673, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "CEK_problems/equilibrium_04.ipynb", "max_stars_repo_name": "blake-armstrong/TeachingNotebook", "max_stars_repo_head_hexsha": "30cdca5bffd552eaecc0368c3e92744c4d6d368c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "CEK_problems/equilibrium_04.ipynb", "max_issues_repo_name": "blake-armstrong/TeachingNotebook", "max_issues_repo_head_hexsha": "30cdca5bffd552eaecc0368c3e92744c4d6d368c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "CEK_problems/equilibrium_04.ipynb", "max_forks_repo_name": "blake-armstrong/TeachingNotebook", "max_forks_repo_head_hexsha": "30cdca5bffd552eaecc0368c3e92744c4d6d368c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-02-23T11:36:12.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-23T11:36:12.000Z", "avg_line_length": 32.8316062176, "max_line_length": 400, "alphanum_fraction": 0.4832320682, "converted": true, "num_tokens": 2386, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.6584174938590246, "lm_q1q2_score": 0.4358216010451115}} {"text": "\n# The Coupled Cluster Method\n\n \n**Thomas Papenbrock**, The University of Tennessee, Knoxville, tpapenbr@utk.edu\n\nDate: **Aug 4, 2018**\n\nCopyright 2018, Thomas Papenbrock. Released under CC Attribution-NonCommercial 4.0 license\n\n\n\n\n\n\n\n\n\n\n# The Coupled-Cluster Method\n\n\n# Introduction\n\nThe coupled-cluster method is an efficient tool to compute atomic\nnuclei with an effort that grows polynomial with system size. While\nthis might still be expensive, it is now possible to compute nuclei\nwith mass numbers about $A\\approx 100$ with this method. Recall that\nfull configuration interaction (FCI) such as the no-core shell model\nexhibits an exponential cost and is therefore limited to light nuclei.\n\n\n\n
    \n\n

    Realistic computations of atomic nuclei with interactions from chiral EFT. The slow increase prior to 2015 is based on quantum Monte Carlo and the no-core shell model. These methods are exponentially expensive (in mass number $A$) and meet with exponentially increasing computer power (Moore's law), thus leading to a progress that is linear in time. Methods such as coupled clusters and in-medium SRG carry a polynomial cost in mass number are transforming the field.

    \n\n\n\n\n\n\n# The normal-ordered Hamiltonian\n\nWe start from the reference state\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{HFref} \\tag{1}\n\\vert\\Phi_0\\rangle = \\prod_{i=1}^A a^\\dagger_i \\vert 0\\rangle \n\\end{equation}\n$$\n\nfor the description of a nucleus with mass number $A$. Usually, this\nreference is the Hartree-Fock state, but that is not necessary. In the\nshell-model picture, it could also be a product state where the lowest\n$A$ harmonic oscillator states are occupied. Here and in what\nfollows, the indices $i,j,k,\\ldots$ run over hole states,\ni.e. orbitals occupied in the reference state ([1](#HFref)), while\n$a,b,c,\\ldots$ run over particle states, i.e. unoccupied\norbitals. Indices $p,q,r,s$ can identify any orbital. Let $n_u$ be\nthe number of unoccupied states, and $A$ is of course the number of\noccupied states. We consider the Hamiltonian\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{Ham} \\tag{2} H =\n\\sum_{pq} \\varepsilon^p_q a^\\dagger_p a_q +\n\\frac{1}{4}\\sum_{pqrs}\\langle pq\\vert V\\vert rs\\rangle\na^\\dagger_pa^\\dagger_q a_sa_r\n\\end{equation}\n$$\n\nThe reference state ([1](#HFref)) is a non-trivial vacuum of our theory. \nWe normal order this Hamiltonian with respect to the nontrivial vacuum\nstate given by the Hartree-Fock reference and obtain the\nnormal-ordered Hamiltonian\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{HN} \\tag{3}\nH_N = \\sum_{pq} f_{pq} \\left\\{a^\\dagger_p a_q\\right\\} + \\frac{1}{4}\\sum_{pqrs}\\langle pq\\vert V\\vert rs\\rangle \\left\\{a^\\dagger_pa^\\dagger_q a_sa_r\\right\\}.\n\\end{equation}\n$$\n\nHere,\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{Fock} \\tag{4}\nf^p_q = \\varepsilon^p_q + \\sum_i \\langle pi\\vert V\\vert qi\\rangle\n\\end{equation}\n$$\n\nis the Fock matrix. We note that the Fock matrix is diagonal in the\nHartree-Fock basis. The brackets $\\{\\cdots\\}$ in Eq. ([3](#HN)) denote\nnormal ordering, i.e. all operators that annihilate the nontrivial\nvaccum ([1](#HFref)) are to the right of those operators that create\nwith respect to that vaccum. Normal ordering implies that $\\langle\n\\Phi_0\\vert H_N\\vert \\Phi_0\\rangle = 0$.\n\n\n\n\n\n\n## Exercise 1: Practice in normal ordering\n
    \n\nNormal order the expression $\\sum\\limits_{pq}\\varepsilon_q^p a^\\dagger_p a_q$.\n\n\n\n**Hint.**\n\n\n
    \n\n$$\n\\begin{equation}\n\\sum_{pq}\\varepsilon_q^p a^\\dagger_p a_q\n=\\sum_{ab}\\varepsilon_b^a a^\\dagger_a a_b\n+\\sum_{ai}\\varepsilon_i^a a^\\dagger_a a_i\n+\\sum_{ai}\\varepsilon_a^i a^\\dagger_i a_a \n+\\sum_{ij}\\varepsilon_j^i a^\\dagger_i a_j\n\\label{_auto1} \\tag{5}\n\\end{equation}\n$$\n\n\n\n\n\n**Answer.**\nWe have to move all operators that annihilate the reference state to the right of those that create on the reference state. Thus,\n\n\n
    \n\n$$\n\\begin{equation}\n\\sum_{pq}\\varepsilon_q^p a^\\dagger_p a_q\n=\\sum_{ab}\\varepsilon_b^a a^\\dagger_a a_b\n+\\sum_{ai}\\varepsilon_i^a a^\\dagger_a a_i\n+\\sum_{ai}\\varepsilon_a^i a^\\dagger_i a_a\n+\\sum_{ij}\\varepsilon_j^i a^\\dagger_i a_j\n\\label{_auto2} \\tag{6}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n=\\sum_{ab}\\varepsilon_b^a a^\\dagger_a a_b\n+\\sum_{ai}\\varepsilon_i^a a^\\dagger_a a_i\n+\\sum_{ai}\\varepsilon_a^i a^\\dagger_i a_a\n+\\sum_{ij}\\varepsilon_j^i \\left(-a_ja^\\dagger_i +\\delta_i^j\\right)\n\\label{_auto3} \\tag{7}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n=\\sum_{ab}\\varepsilon_b^a a^\\dagger_a a_b\n+\\sum_{ai}\\varepsilon_i^a a^\\dagger_a a_i\n+\\sum_{ai}\\varepsilon_a^i a^\\dagger_i a_a\n-\\sum_{ij}\\varepsilon_j^i a_ja^\\dagger_i +\\sum_i \\varepsilon_i^i\n\\label{_auto4} \\tag{8}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n=\\sum_{pq}\\varepsilon_q^p \\left\\{a^\\dagger_p a_q\\right\\} +\\sum_i \\varepsilon_i^i\n\\label{_auto5} \\tag{9}\n\\end{equation}\n$$\n\n\n\n\n\n\n===== =====\n\n\n\n\nWe note that $H = E_{HF} + H_N$, where\n\n\n
    \n\n$$\n\\begin{equation}\nE_{HF} \\equiv \\langle\\Phi_0\\vert H\\vert \\Phi_0\\rangle = \\sum_{i} \\varepsilon^i_i +\\frac{1}{2}\\sum_{ij}\\langle ij\\vert V\\vert ij\\rangle\n\\label{_auto6} \\tag{10}\n\\end{equation}\n$$\n\nis the Hartree-Fock energy.\nThe coupled-cluster method is a very efficient tool to compute nuclei\nwhen a \"good\" reference state is available. Let us assume that the\nreference state results from a Hartree-Fock calculation.\n\n\n\n\n\n\n\n\n\n\n## Exercise 2: What does \"good\" mean?\n
    \n\nHow do you know whether a Hartree-Fock state is a \"good\" reference?\nWhich results of the Hartree-Fock computation will inform you?\n\n\n\n**Answer.**\nOnce the Hartree-Fock equations are solved, the Fock matrix\n([4](#Fock)) becomes diagonal, and its diagonal elements can be viewed\nas single-particle energies. Hopefully, there is a clear gap in the\nsingle-particle spectrum at the Fermi surface, i.e. after $A$ orbitals\nare filled.\n\n\n\n\n\n\n===== =====\n\n\n\nIf symmetry-restricted Hartree-Fock is used, one is limited to compute\nnuclei with closed subshells for neutrons and for protons. On a first\nview, this might seem as a severe limitation. But is it? \n\n\n\n\n\n\n\n## Exercise 3: How many nuclei are accessible with the coupled cluster method based on spherical mean fields?\n
    \n\nIf one limits oneself to nuclei with mass number up to\nmass number $A=60$, how many nuclei can potentially be described with\nthe coupled-cluster method? Which of these nuclei are potentially\ninteresting? Why?\n\n\n\n**Answer.**\nNuclear shell closures are at $N,Z=2,8,20,28,50,82,126$, and subshell\nclosures at $N,Z=2,6,8,14,16,20,28,32,34,40,50,\\ldots$. \n\nIn the physics of nuclei, the evolution of nuclear structure as\nneutrons are added (or removed) from an isotope is a key\n\n\n\ninterest. Examples are the rare isotopes of helium (He-8,10)\noxygen (O-22,24,28), calcium (Ca-52,54,60), nickel (Ni-78) and tin\n(Sn-100,132). The coupled-cluster method has the\npotential to address questions regarding these nuclei, and in a\nseveral cases was used to make predictions before experimental data\nwas available. In addition, the method can be used to compute\nneighbors of nuclei with closed subshells.\n\n\n\n\n\n\n===== =====\n\n\n\n\n# The similarity transformed Hamiltonian\n\n\nThere are several ways to view and understand the coupled-cluster\nmethod. A first simple view of coupled-cluster theory is that the\nmethod induces correlations into the reference state by expressing a\ncorrelated state as\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{psi} \\tag{11}\n\\vert\\Psi\\rangle = e^T \\vert\\Phi_0\\rangle ,\n\\end{equation}\n$$\n\nHere, $T$ is an operator that induces correlations. We can now demand\nthat the correlate state ([11](#psi)) becomes and eigenstate of the\nHamiltonian $H_N$, i.e. $H_N\\vert \\Psi\\rangle = E\\vert \\Psi\\rangle$. This view,\nwhile correct, is not the most productive one. Instead, we\nleft-multiply the Schroedinger equation with $e^{-T}$ and find\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{Schroedinger} \\tag{12}\n\\overline{H_N}\\vert \\Phi_0\\rangle = E_c \\vert \\Phi_0\\rangle . \n\\end{equation}\n$$\n\nHere, $E_c$ is the correlation energy, and the total energy is\n$E=E_c+E_{HF}$. The similarity-transformed Hamiltonian is defined as\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{Hsim} \\tag{13}\n\\overline{H_N} \\equiv e^{-T} H_N e^T .\n\\end{equation}\n$$\n\nA more productive view on coupled-cluster theory thus emerges: This\nmethod seeks a similarity transformation such that the uncorrelated\nreference state ([1](#HFref)) becomes an exact eigenstate of the\nsimilarity-transformed Hamiltonian ([13](#Hsim)).\n\n\n\n\n\n## Exercise 4: What $T$ leads to Hermitian $\\overline{H_N}$ ?\n
    \n\nWhat are the conditions on $T$ such that $\\overline{H_N}$ is Hermitian?\n\n\n\n**Answer.**\nFor a Hermitian $\\overline{H_N}$, we need a unitary $e^T$, i.e. an\nanti-Hermitian $T$ with $T = -T^\\dagger$\n\n\n\n\n\n\n===== =====\n\n\n\nAs we will see below, coupld-cluster theory employs a non-Hermitian Hamiltonian.\n\n\n\n\n\n## Exercise 5: Understanding (non-unitary) similarity transformations\n
    \n\nShow that $\\overline{H_N}$ has the same eigenvalues as $H_N$ for\narbitrary $T$. What is the spectral decomposition of a non-Hermitian\n$\\overline{H_N}$ ?\n\n\n\n**Answer.**\nLet $H_N\\vert E\\rangle = E\\vert E\\rangle$. Thus\n\n$$\n\\begin{align*}\nH_N e^{T} e^{-T} \\vert E\\rangle &= E\\vert E\\rangle , \\\\\n\\left(e^{-T} H_N e^T\\right) e^{-T} \\vert E\\rangle &= Ee^{-T} \\vert E\\rangle , \\\\\n\\overline{H_N} e^{-T} \\vert E\\rangle &= E e^{-T}\\vert E\\rangle .\n\\end{align*}\n$$\n\nThus, if $\\vert E\\rangle$ is an eigenstate of $H_N$ with eigenvalue $E$,\nthen $e^{-T}\\vert E\\rangle$ is eigenstate of $\\overline{H_N}$ with the same\neigenvalue.\n\nA non-Hermitian $\\overline{H_N}$ has eigenvalues $E_\\alpha$\ncorresponding to left $\\langle L_\\alpha\\vert $ and right $\\vert R_\\alpha\n\\rangle$ eigenstates. Thus\n\n\n
    \n\n$$\n\\begin{equation}\n\\overline{H_N} = \\sum_\\alpha \\vert R_\\alpha\\rangle E_\\alpha \\langle L_\\alpha \\vert \n\\label{_auto7} \\tag{14}\n\\end{equation}\n$$\n\nwith bi-orthonormal $\\langle L_\\alpha\\vert R_\\beta\\rangle = \\delta_\\alpha^\\beta$.\n\n\n\n\n\n\n===== =====\n\n\n\n\nTo make progress, we have to specify the cluster operator $T$. In\ncoupled cluster theory, this operator is\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{Top} \\tag{15}\nT \\equiv \\sum_{ia} t_i^a a^\\dagger_a a_i + \\frac{1}{4}\\sum_{ijab}t_{ij}^{ab}\na^\\dagger_aa^\\dagger_ba_ja_i + \\cdots\n+ \\frac{1}{(A!)^2}\\sum_{i_1\\ldots i_A a_1 \\ldots a_A}\nt_{i_1\\ldots i_A}^{a_1\\ldots a_A} a^\\dagger_{a_1}\\cdots a^\\dagger_{a_A} a_{i_A}\\cdots a_{i_1} .\n\\end{equation}\n$$\n\nThus, the operator ([15](#Top)) induces particle-hole (p-h)\nexcitations with respect to the reference. In general, $T$ generates\nup to $Ap-Ah$ excitations, and the unknown parameters are the cluster amplitides\n$t_i^a$, $t_{ij}^{ab}$, ..., $t_{i_1,\\ldots,i_A}^{a_1,\\ldots,a_A}$.\n\n\n\n\n\n## Exercise 6: How many unknowns?\n
    \n\nShow that the number of unknowns is as large as the FCI dimension of\nthe problem, using the numbers $A$ and $n_u$.\n\n\n\n**Answer.**\nWe have to sum up all $np-nh$ excitations, and there are\n$\\binom{n_u}{n}$ particle states and $\\binom{A}{A-n}$ hole states for\neach $n$. Thus, we have for the total number\n\n\n
    \n\n$$\n\\begin{equation}\n\\sum_{n=0}^A \\binom{n_u}{n} \\binom{A}{A-n}= \\binom{A+n_u}{A} .\n\\label{_auto8} \\tag{16}\n\\end{equation}\n$$\n\nThe right hand side are obviously all ways to distribute $A$ fermions over $n_0+A$ orbitals.\n\n\n\n\n\n\n===== =====\n\n\n\n\nThus, the coupled-cluster method with the full cluster operator\n([15](#Top)) is exponentially expensive, just as FCI. To make progress,\nwe need to make an approximation by truncating the operator. Here, we\nwill use the CCSD (coupled clusters singles doubles) approximation,\nwhere\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{Tccsd} \\tag{17}\nT \\equiv \\sum_{ia} t_i^a a^\\dagger_a a_i + \\frac{1}{4}\\sum_{ijab}t_{ij}^{ab}\na^\\dagger_aa^\\dagger_ba_ja_i .\n\\end{equation}\n$$\n\nWe need to determine the unknown cluster amplitudes that enter in CCSD. Let\n\n\n
    \n\n$$\n\\begin{equation}\n\\vert\\Phi_i^a\\rangle = a^\\dagger_a a_i \\vert \\Phi_0\\rangle , \n\\label{_auto9} \\tag{18}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n\\vert\\Phi_{ij}^{ab}\\rangle = a^\\dagger_a a^\\dagger_b a_j a_i \\vert \\Phi_0\\rangle\n\\label{_auto10} \\tag{19}\n\\end{equation}\n$$\n\nbe 1p-1h and 2p-2h excitations of the reference. Computing matrix\nelements of the Schroedinger Equation ([12](#Schroedinger)) yields\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{ccsd} \\tag{20}\n\\langle \\Phi_0\\vert \\overline{H_N}\\vert \\Phi_0\\rangle = E_c , \n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n\\langle \\Phi_i^a\\vert \\overline{H_N}\\vert \\Phi_0\\rangle = 0 , \n\\label{_auto11} \\tag{21}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n\\langle \\Phi_{ij}^{ab}\\vert \\overline{H_N}\\vert \\Phi_0\\rangle = 0 .\n\\label{_auto12} \\tag{22}\n\\end{equation}\n$$\n\nThe first equation states that the coupled-cluster correlation energy\nis an expectation value of the similarity-transformed Hamiltonian. The\nsecond and third equations state that the similarity-transformed\nHamiltonian exhibits no 1p-1h and no 2p-2h excitations. These\nequations have to be solved to find the unknown amplitudes $t_i^a$ and\n$t_{ij}^{ab}$. Then one can use these amplitudes and compute the\ncorrelation energy from the first line of Eq. ([20](#ccsd)).\n\nWe note that in the CCSD approximation the reference state is not an\nexact eigenstates. Rather, it is decoupled from simple states but\n$\\overline{H}$ still connects this state to 3p-3h, and 4p-4h states\netc.\n\nAt this point, it is important to recall that we assumed starting from\na \"good\" reference state. In such a case, we might reasonably expect\nthat the inclusion of 1p-1h and 2p-2h excitations could result in an\naccurate approximation. Indeed, empirically one finds that CCSD\naccounts for about 90% of the corelation energy, i.e. of the\ndifference between the exact energy and the Hartree-Fock energy. The\ninclusion of triples (3p-3h excitations) typically yields 99% of the\ncorrelation energy.\n\nWe see that the coupled-cluster method in its CCSD approximation\nyields a similarity-transformed Hamiltonian that is of a two-body\nstructure with respect to a non-trivial vacuum. When viewed in this\nlight, the coupled-cluster method \"transforms\" an $A$-body problem\n(in CCSD) into a two-body problem, albeit with respect to a nontrivial\nvacuum.\n\n\n\n\n\n\n## Exercise 7: Why is CCD not exact?\n
    \n\nAbove we argued that a similarity transformation preserves all eigenvalues. Nevertheless, the CCD correlation energy is not the exact correlation energy. Explain!\n\n\n\n**Answer.**\nThe CCD approximation does not make $\\vert\\Phi_0\\rangle$ an exact\neigenstate of $\\overline{H_N}$; it is only an eigenstate when the\nsimilarity-transformed Hamiltonian is truncated to at most 2p-2h\nstates. The full $\\overline{H_N}$, with $T=T_2$, would involve\nsix-body terms (do you understand this?), and this full Hamiltonian\nwould reproduce the exact correlation energy. Thus CCD is a similarity\ntransformation plus a truncation, which decouples the ground state only\nfrom 2p-2h states.\n\n\n\n\n\n\n===== =====\n\n\n\n# Computing the similarity-transformed Hamiltonian\n\nThe solution of the CCSD equations, i.e. the second and third line of\nEq. ([20](#ccsd)), and the computation of the correlation energy\nrequires us to compute matrix elements of the similarity-transformed\nHamiltonian ([13](#Hsim)). This can be done with the\nBaker-Campbell-Hausdorff expansion\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{BCH} \\tag{23}\n\\overline{H_N} = e^{-T} H_N e^T \n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n=H_N + \\left[ H_N, T\\right]+ \\frac{1}{2!}\\left[ \\left[ H_N, T\\right], T\\right]\n+ \\frac{1}{3!}\\left[\\left[ \\left[ H_N, T\\right], T\\right], T\\right] +\\ldots .\n\\label{_auto13} \\tag{24}\n\\end{equation}\n$$\n\nWe now come to a key element of coupled-cluster theory: the cluster\noperator ([15](#Top)) consists of sums of terms that consist of particle\ncreation and hole annihilation operators (but no particle annihilation\nor hole creation operators). Thus, all terms that enter $T$ commute\nwith each other. This means that the commutators in the\nBaker-Campbell-Hausdorff expansion ([23](#BCH)) can only be non-zero\nbecause each $T$ must connect to $H_N$ (but no $T$ with another\n$T$). Thus, the expansion is finite.\n\n\n\n\n\n## Exercise 8: When does CCSD truncate?\n
    \n\nIn CCSD and for two-body Hamiltonians, how many nested\ncommutators yield nonzero results? Where does the\nBaker-Campbell-Hausdorff expansion terminate? What is the (many-body) rank of the resulting $\\overline{H_N}$?\n\n\n\n**Answer.**\nCCSD truncates for two-body operators at four-fold nested commutators,\nbecause each of the four annihilation and creation operators in\n$\\overline{H_N}$ can be knocked out with one term of $T$.\n\n\n\n\n\n\n===== =====\n\n\n\n\nWe see that the (disadvantage of having a) non-Hermitian Hamiltonian\n$\\overline{H_N}$ leads to the advantage that the\nBaker-Campbell-Hausdorff expansion is finite, thus leading to the\npossibility to compute $\\overline{H_N}$ exactly. In contrast, the\nIMSRG deals with a Hermitian Hamiltonian throughout, and the infinite\nBaker-Campbell-Hausdorff expansion is truncated at a high order when\nterms become very small.\n\nWe write the similarity-transformed Hamiltonian as\n\n\n
    \n\n$$\n\\begin{equation}\n\\overline{H_N}=\\sum_{pq} \\overline{H}^p_q a^\\dagger_q a_p + {1\\over 4} \\sum_{pqrs} \\overline{H}^{pq}_{rs} a^\\dagger_p a^\\dagger_q a_s a_r + \\ldots\n\\label{_auto14} \\tag{25}\n\\end{equation}\n$$\n\nwith\n\n\n
    \n\n$$\n\\begin{equation}\n\\overline{H}^p_q \\equiv \\langle p\\vert \\overline{H_N}\\vert q\\rangle , \n\\label{_auto15} \\tag{26}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n\\overline{H}^{pq}_{rs} \\equiv \\langle pq\\vert \\overline{H_N}\\vert rs\\rangle .\n\\label{_auto16} \\tag{27}\n\\end{equation}\n$$\n\nThus, the CCSD Eqs. ([20](#ccsd)) for the amplitudes can be written as\n$\\overline{H}_i^a = 0$ and $\\overline{H}_{ij}^{ab}=0$.\n\n\n\n\n\n## Exercise 9: Compute the matrix element $\\overline{H}_{ab}^{ij}\\equiv \\langle ij\\vert \\overline{H_N}\\vert ab\\rangle$\n
    \n\n\n\n**Answer.**\nThis is a simple task. This matrix element is part of the operator\n$\\overline{H}_{ab}^{ij}a^\\dagger_ia^\\dagger_ja_ba_a$, i.e. particles\nare annihilated and holes are created. Thus, no contraction of the\nHamiltonian $H$ with any cluster operator $T$ (remember that $T$\nannihilates holes and creates particles) can happen, and we simply\nhave $\\overline{H}_{ab}^{ij} = \\langle ij\\vert V\\vert ab\\rangle$.\n\n\n\n\n\n\n===== =====\n\n\n\n\nWe need to work out the similarity-transformed Hamiltonian of\nEq. ([23](#BCH)). To do this, we write $T=T_1 +T_2$ and $H_N= F +V$,\nwhere $T_1$ and $F$ are one-body operators, and $T_2$ and $V$ are\ntwo-body operators.\n\n## Example: The contribution of $[F, T_2]$ to $\\overline{H_N}$\n
    \n\nThe commutator $[F, T_2]$ consists of two-body and one-body terms. Let\nus compute first the two-body term, as it results from a single\ncontraction (i.e. a single application of $[a_p, a^\\dagger_q] =\n\\delta_p^q$). We denote this as $[F, T_2]_{2b}$ and find\n\n$$\n\\begin{align*}\n[F, T_2]_{2b} &= \\frac{1}{4}\\sum_{pq}\\sum_{rsuv} f_p^q t_{ij}^{ab}\\left[a^\\dagger_q a_p, a^\\dagger_a a^\\dagger_b a_j a_i \\right]_{2b} \\\\\n&= \\frac{1}{4}\\sum_{pq}\\sum_{abij} f_p^q t_{ij}^{ab}\\delta_p^a a^\\dagger_q a^\\dagger_b a_j a_i \\\\\n&- \\frac{1}{4}\\sum_{pq}\\sum_{abij} f_p^q t_{ij}^{ab}\\delta_p^b a^\\dagger_q a^\\dagger_a a_j a_i \\\\\n&- \\frac{1}{4}\\sum_{pq}\\sum_{abij} f_p^q t_{ij}^{ab}\\delta_q^j a^\\dagger_a a^\\dagger_b a_p a_i \\\\\n&+ \\frac{1}{4}\\sum_{pq}\\sum_{abij} f_p^q t_{ij}^{ab}\\delta_q^i a^\\dagger_a a^\\dagger_b a_p a_j \\\\\n&= \\frac{1}{4}\\sum_{qbij}\\left(\\sum_{a} f_a^q t_{ij}^{ab}\\right)a^\\dagger_q a^\\dagger_b a_j a_i \\\\\n&- \\frac{1}{4}\\sum_{qaij}\\left(\\sum_{b} f_b^q t_{ij}^{ab}\\right)a^\\dagger_q a^\\dagger_a a_j a_i \\\\\n&- \\frac{1}{4}\\sum_{pabi}\\left(\\sum_{j} f_p^j t_{ij}^{ab}\\right)a^\\dagger_a a^\\dagger_b a_p a_i \\\\\n&+ \\frac{1}{4}\\sum_{pabj}\\left(\\sum_{i} f_p^i t_{ij}^{ab}\\right)a^\\dagger_a a^\\dagger_b a_p a_j \\\\\n&= \\frac{1}{2}\\sum_{qbij}\\left(\\sum_{a} f_a^q t_{ij}^{ab}\\right)a^\\dagger_q a^\\dagger_b a_j a_i \\\\\n&- \\frac{1}{2}\\sum_{pabi}\\left(\\sum_{j} f_p^j t_{ij}^{ab}\\right)a^\\dagger_a a^\\dagger_b a_p a_i .\n\\end{align*}\n$$\n\nHere we exploited the antisymmetry $t_{ij}^{ab} = -t_{ji}^{ab} =\n-t_{ij}^{ba} = t_{ji}^{ba}$ in the last step. Using $a^\\dagger_q a^\\dagger_b a_j a_i = -a^\\dagger_b a^\\dagger_q a_j a_i $ and $a^\\dagger_a a^\\dagger_b a_p a_i = a^\\dagger_a a^\\dagger_b a_i a_p$, we can make the expression \nmanifest antisymmetric, i.e.\n\n$$\n\\begin{align*}\n[F, T_2]_{2b}\n&= \\frac{1}{4}\\sum_{qbij}\\left[\\sum_{a} \\left(f_a^q t_{ij}^{ab}-f_a^b t_{ij}^{qa}\\right)\\right]a^\\dagger_q a^\\dagger_b a_j a_i \\\\\n&- \\frac{1}{4}\\sum_{pabi}\\left[\\sum_{j} \\left(f_p^j t_{ij}^{ab}-f_i^j t_{pj}^{ab}\\right)\\right]a^\\dagger_a a^\\dagger_b a_p a_i .\n\\end{align*}\n$$\n\nThus, the contribution of $[F, T_2]_{2b}$ to the matrix element $\\overline{H}_{ij}^{ab}$ is\n\n$$\n\\begin{align*}\n\\overline{H}_{ij}^{ab} \\leftarrow \\sum_{c} \\left(f_c^a t_{ij}^{cb}-f_c^b t_{ij}^{ac}\\right) - \\sum_{k} \\left(f_j^k t_{ik}^{ab}-f_i^k t_{jk}^{ab}\\right)\n\\end{align*}\n$$\n\nHere we used an arrow to indicate that this is just one contribution\nto this matrix element. We see that the derivation straight forward,\nbut somewhat tedious. As no one likes to commute too much (neither in\nthis example nor when going to and from work), and so we need a better\napproach. This is where diagramms come in handy.\n\n===== =====\n\n\n\n\n### Diagrams\n\nThe pictures in this Subsection are taken from Crawford and Schaefer.\n\nBy convention, hole lines (labels $i, j, k,\\ldots$) are pointing down. \n\n\n
    \n\n

    This is a hole line.

    \n\n\n\n\n\nBy convention, particle lines (labels $a, b, c,\\ldots$) are pointing up. \n\n\n
    \n\n

    This is a particle line.

    \n\n\n\n\n\nLet us look at the one-body operator of the normal-ordered Hamiltonian, i.e. Fock matrix. Its diagrams are as follows.\n\n\n\n
    \n\n

    The diagrams corresponding to $f_a^b$. The dashed line with the 'X' denotes the interaction $F$ between the incoming and outgoing lines. The labels $a$ and $b$ are not denoted, but you should label the outgoing and incoming lines accordingly.

    \n\n\n\n\n\n\n\n
    \n\n

    The diagrams corresponding to $f_i^j$. The dashed line with the 'X' denotes the interaction $F$ between the incoming and outgoing lines.

    \n\n\n\n\n\n\n\n
    \n\n

    The diagrams corresponding to $f_a^i$. The dashed line with the 'X' denotes the interaction $F$ between the incoming and outgoing lines.

    \n\n\n\n\n\n\n\n
    \n\n

    The diagrams corresponding to $f_i^a$. The dashed line with the 'X' denotes the interaction $F$ between the incoming and outgoing lines.

    \n\n\n\n\n\nWe now turn to the two-body interaction. It is denoted as a horizontal\ndashed line with incoming and outgoing lines attached to it. We start\nby noting that the following diagrams of the interaction are all\nrelated by permutation symmetry.\n\n\n\n
    \n\n

    The diagrams corresponding to $\\langle ai\\vert V\\vert jb \\rangle = - \\langle ai\\vert V\\vert bj \\rangle = -\\langle ia\\vert V\\vert jb \\rangle = \\langle ia\\vert V\\vert bj\\rangle$.

    \n\n\n\n\n\n\n\n\n\n## Exercise 10: Assign the correct matrix element $\\langle pq\\vert V\\vert rs\\rangle$ to each of the following diagrams of the interaction\n
    \n\nRemember: $\\langle\\rm{left-out, right-out}\\vert V\\vert \\rm{left-in, right-in}\\rangle$.\n\n\n**a)**\n\n\n\n

    \n\n\n\n\n\n\n\n**Answer.**\n$\\langle ab\\vert V\\vert cd\\rangle + \\langle ij\\vert V\\vert kl\\rangle + \\langle ia\\vert V\\vert bj\\rangle$\n\n\n\n**b)**\n\n\n\n

    \n\n\n\n\n\n\n\n**Answer.**\n$\\langle ai\\vert V\\vert bc\\rangle + \\langle ij\\vert V\\vert ka\\rangle + \\langle ab\\vert V\\vert ci\\rangle$\n\n\n\n**c)**\n\n\n\n

    \n\n\n\n\n\n\n\n**Answer.**\n$\\langle ia\\vert V\\vert jk\\rangle + \\langle ab\\vert V\\vert ij\\rangle + \\langle ij\\vert V\\vert ab\\rangle$\n\n\n\n\n\n\n===== =====\n\n\n\n\nFinally, we have the following diagrams for the $T_1$ and $T_2$ amplitudes.\n\n\n
    \n\n

    The horizontal full line is the cluster amplitude with incoming hole lines and outgoing particle lines as indicated.

    \n\n\n\n\n\nWe are now in the position to construct the diagrams of the\nsimilarity-transformed Hamiltonian, keeping in mind that these\ndiagrams correspond to matrix elements of $\\overline{H_N}$. The rules\nare as follows.\n\n1. Write down all *topologically different* diagrams corresponding to the desired matrix element. Topologically different diagrams differ in the number and type of lines (particle or hole) that connect the Fock matrix $F$ or the interaction $V$ to the cluster amplitudes $T$, but not whether these connections are left or right (as those are related by antisymmetry). As an example, all diagrams in Fig. [fig-symmetry](#fig-symmetry) are topologically identical, because they consist of incoming particle and hole lines and of outgoing particle and hole lines. \n\n2. Write down the matrix elements that enter the diagram, and sum over all internal lines. \n\n3. The overall sign is $(-1)$ to the power of [(number of hole lines) - (number of loops)].\n\n4. Symmetry factor: For each pair of equivalent lines (i.e. lines that connect the same two operators) multiply with a factor $1/2$. For $n$ identical vertices, multiply the algebraic expression by the symmery factor $1/n!$ to account properly for the number of ways the diagram can be constructed. \n\n5. Antisymmetrize the outgoing and incoming lines as necessary.\n\nPlease note that this really works. You could derive these rules for\nyourself from the commutations and factors that enter the\nBaker-Campbell-Hausdorff expansion. The sign comes obviously from the\narrangement of creation and annilhilation operators, while the\nsymmetry factor stems from all the different ways, one can contract\nthe cluster operator with the normal-ordered Hamiltonian.\n\n\n## Example: CCSD correlation energy\n
    \nThe CCSD correlation energy, $E_c= \\langle\n\\Phi_0\\vert \\overline{H_N}\\vert \\Phi_0\\rangle$, is the first of the CCSD\nequations ([20](#ccsd)). It is a vacuum expectation value and thus\nconsists of all diagrams with no external legs. There are three such diagrams:\n\n\n\n
    \n\n

    Three diagrams enter for the CCSD correlation energy, i.e. all diagrams that leave no external legs.

    \n\n\n\n\n\nThe correponding algebraic expression is $E_c=\\sum_{ia}f^i_a t_i^a +{1\\over 4}\\sum_{ijab} \\langle ij\\vert V\\vert ab\\rangle t_{ij}^{ab} + {1\\over 2} \\sum_{ijab} \\langle ij\\vert V\\vert ab\\rangle t_i^a t_j^b$.\n\nThe first algebraic expression is clear. We have one hole line and one\nloop, giving it a positive sign. There are no equivalent lines or\nvertices, giving it no symmetry factor. The second diagram has two\nloops and two hole lines, again leading to a positive sign. We have a\npair of equivalent hole lines and a pair of equivalent particle lines,\neach giving a symmetry factor of $1/2$. The third diagram has two\nloops and two hole lines, again leading to a positive sign. We have\ntwo indentical vertices (each connecting to a $T_1$ in the same way)\nand thus a symmetry factor $1/2$.\n\n===== =====\n\n\n\n\n# CCD Approximation\n\nIn what follows, we will consider the coupled cluster doubles (CCD)\napproximation. This approximation is valid in cases where the system\ncannot exhibit any particle-hole excitations (such as nuclear matter\nwhen formulated on a momentum-space grid) or for the pairing model (as\nthe pairing interactions only excites pairs of particles). In this\ncase $t_i^a=0$ for all $i, a$, and $\\overline{H}_i^a=0$. The CCD\napproximation is also of some sort of leading order approximation in\nthe Hartree-Fock basis (as the Hartree-Fock Hamiltonian exhibits no\nparticle-hole excitations).\n\n\n\n\n\n\n## Exercise 11: Derive the CCD equations!\n
    \n\nLet us consider the matrix element $\\overline{H}_{ij}^{ab}$. Clearly,\nit consists of all diagrams (i.e. all combinations of $T_2$, and a\nsingle $F$ or $V$ that have two incoming hole lines and two outgoing\nparticle lines. Write down all these diagrams.\n\n\n\n**Hint.**\nStart systematically! Consider all combinations of $F$ and $V$ diagrams with 0, 1, and 2 cluster amplitudes $T_2$.\n\n\n\n\n\n**Answer.**\n\n\n
    \n\n

    The diagrams for the $T_2$ equation, i.e. the matrix elements of $\\overline{H}_{ij}^{ab}$. Taken from Baardsen et al (2013).

    \n\n\n\n\n\nThe corresponding algebraic expression is\n\n$$\n\\begin{align*}\n\\overline{H}_{ij}^{ab} &= \\langle ab\\vert V\\vert ij\\rangle + P(ab)\\sum_c f_c^bt_{ij}^{ac} - P(ij)\\sum_k f_j^k t_{ik}^{ab} \\\\\n&+ {1\\over 2} \\sum_{cd} \\langle ab\\vert V\\vert cd\\rangle t_{ij}^{cd}+ {1\\over 2} \\sum_{kl} \\langle kl\\vert V\\vert ij\\rangle t_{kl}^{ab} + P(ab)P(ij)\\sum_{kc} \\langle kb\\vert V\\vert cj \\rangle t_{ik}^{ac} \\\\\n&+ {1\\over 2} P(ij)P(ab)\\sum_{kcld} \\langle kl\\vert V\\vert cd\\rangle t_{ik}^{ac}t_{lj}^{db} \n+ {1\\over 2} P(ij)\\sum_{kcld} \\langle kl\\vert V\\vert cd\\rangle t_{ik}^{cd}t_{lj}^{ab}\\\\\n&+ {1\\over 2} P(ab)\\sum_{kcld} \\langle kl\\vert V\\vert cd\\rangle t_{kl}^{ac}t_{ij}^{db}\n+ {1\\over 4} \\sum_{kcld} \\langle kl\\vert V\\vert cd\\rangle t_{ij}^{cd}t_{kl}^{ab} . \n\\end{align*}\n$$\n\n\n\n\n\n\n===== =====\n\n\n\n\nLet us now turn to the computational cost of a CCD computation.\n\n\n\n\n\n## Exercise 12: Computational scaling of CCD\n
    \n\nFor each of the diagrams in Fig. [fig-ccd](#fig-ccd) write down the\ncomputational cost in terms of the number of occupied $A$ and the\nnumber of unoccupied $n_u$ orbitals.\n\n\n\n**Answer.**\nThe cost is $A^2 n_u^2$, $A^2 n_u^3$, $A^3 n_u^2$,\n$A^2 n_u^4$, $A^4 n_u^2$, $A^3 n_u^3$,\n$A^4 n_u^4$, $A^4 n_u^4$,\n$A^4 n_u^4$, and $A^4 n_u^4$ for the respective diagrams.\n\n\n\n\n\n\n===== =====\n\n\n\nNote that $n_u\\gg A$ in general. In textbooks, one reads that CCD (and\nCCSD) cost only $A^2n_u^4$. Our most expensive diagrams, however are\n$A^4n_u^4$. What is going on?\n\nTo understand this puzzle, let us consider the last diagram of\nFig. [fig-ccd](#fig-ccd). We break up the computation into two steps,\ncomputing first the intermediate\n\n\n
    \n\n$$\n\\begin{equation}\n\\chi_{ij}^{kl}\\equiv {1\\over 2} \\sum_{cd} \\langle kl\\vert V\\vert cd\\rangle t_{ij}^{cd}\n\\label{_auto17} \\tag{28}\n\\end{equation}\n$$\n\nat a cost of $A^4n_u^2$, and then\n\n\n
    \n\n$$\n\\begin{equation}\n{1\\over 2} \\sum_{kl} \\chi_{ij}^{kl} t_{kl}^{ab} \n\\label{_auto18} \\tag{29}\n\\end{equation}\n$$\n\nat a cost of $A^4n_u^2$. This is affordable. The price to pay is the\nstorage of the intermediate $\\chi_{ij}^{kl}$, i.e. we traded\nmemory for computational cycles. This trick is known as\n\"factorization.\" \n\n\n\n\n\n\n## Exercise 13: Factorize the remaining diagrams of the CCD equation\n
    \n\nDiagrams 7, 8, and 9 of Fig. [fig-ccd](#fig-ccd) also need to be factorized.\n\n\n\n**Answer.**\nFor diagram number 7, we compute\n\n\n
    \n\n$$\n\\begin{equation}\n\\chi_{id}^{al}\\equiv\\sum_{kc} \\langle kl\\vert V\\vert cd\\rangle t_{ik}^{ac}\n\\label{_auto19} \\tag{30}\n\\end{equation}\n$$\n\nat a cost of $A^3 n_u^3$ and then compute\n\n\n
    \n\n$$\n\\begin{equation}\n{1\\over 2} P(ij)P(ab) \\sum_{ld} \\chi_{id}^{al} t_{lj}^{db} \n\\label{_auto20} \\tag{31}\n\\end{equation}\n$$\n\nat the cost of $A^3 n_u^3$.\n\nFor diagram number 8, we compute\n\n\n
    \n\n$$\n\\begin{equation}\n\\chi_{i}^{l}\\equiv -{1\\over 2} \\sum_{kcd} \\langle kl\\vert V\\vert cd\\rangle t_{ik}^{cd}\n\\label{_auto21} \\tag{32}\n\\end{equation}\n$$\n\nat a cost of $A^3 n_u^2$, and then compute\n\n\n
    \n\n$$\n\\begin{equation}\n-P(ij) \\sum_l \\chi_i^l t_{lj}^{ab}\n\\label{_auto22} \\tag{33}\n\\end{equation}\n$$\n\nat the cost of $A^3 n_u^2$.\n\nFor diagram number 9, we compute\n\n\n
    \n\n$$\n\\begin{equation}\n\\chi_d^a\\equiv{1\\over 2} \\sum_{kcl} \\langle kl\\vert V\\vert cd\\rangle t_{kl}^{ac}\n\\label{_auto23} \\tag{34}\n\\end{equation}\n$$\n\nat a cost of $A^2 n_u^3$ and then compute\n\n\n
    \n\n$$\n\\begin{equation}\nP(ab)\\sum_d \\chi_d^a t_{ij}^{db}\n\\label{_auto24} \\tag{35}\n\\end{equation}\n$$\n\nat the cost of $A^3 n_u^3$.\n\n\n\n\n\n\n===== =====\n\n\n\n\nWe are now ready, to derive the full CCSD equations, i.e. the matrix\nelements of $\\overline{H}_i^a$ and $\\overline{H}_{ij}^{ab}$. \n\n\n\n\n\n\n## Project 14: (Optional) Derive the CCSD equations!\n
    \n\n\n**a)**\nLet us consider the matrix element $\\overline{H}_i^a$ first. Clearly, it consists of all diagrams (i.e. all combinations of $T_1$, $T_2$, and a single $F$ or $V$ that have an incoming hole line and an outgoing particle line. Write down all these diagrams.\n\n\n\n**Answer.**\n\n\n
    \n\n

    The diagrams for the $T_1$ equation, i.e. the matrix elements of $\\overline{H}_i^a$. Taken from Crawford and Schaefer. Here $\\langle pq\\vert \\vert rs\\rangle \\equiv \\langle pq\\vert V\\vert rs\\rangle$ and $f_{pq}\\equiv f^p_q$.

    \n\n\n\n\n\n\n\n**b)**\nLet us now consider the matrix element $\\overline{H}_{ij}^{ab}$. Clearly, it consists of all diagrams (i.e. all combinations of $T_1$, $T_2$, and a single $F$ or $V$ that have two incoming hole lines and two outgoing particle lines. Write down all these diagrams and corresponding algebraic expressions.\n\n\n\n**Answer.**\n\n\n
    \n\n

    The diagrams for the $T_2$ equation, i.e. the matrix elements of $\\overline{H}_{ij}^{ab}$. Taken from Crawford and Schaefer. Here $\\langle pq\\vert \\vert rs\\rangle \\equiv \\langle pq\\vert V\\vert rs\\rangle$, $f_{pq}\\equiv f^p_q$, and $P(ab) = 1 - (a\\leftrightarrow b)$ antisymmetrizes.

    \n\n\n\n\n\n\n\n\n\n\n===== =====\n\n\n\nWe can now turn to the solution of the coupled-cluster equations.\n\n\n\n## Solving the CCD equations\n\nThe CCD equations, depicted in Fig. [fig-ccd](#fig-ccd), are nonlinear in the\ncluster amplitudes. How do we solve $\\overline{H}_{ij}^{ab}=0$? We\nsubtract $(f_a^a +f_b^b -f_i^i -f_j^j)t_{ij}^{ab}$ from both sides of\n$\\overline{H}_{ij}^{ab}=0$ (because this term is contained in\n$\\overline{H}_{ij}^{ab}$) and find\n\n$$\n\\begin{align*}\n(f_i^i +f_j^j -f_a^a -f_b^b)t_{ij}^{ab} &= (f_i^i +f_j^j -f_a^a -f_b^b)t_{ij}^{ab} +\\overline{H}_{ij}^{ab}\n\\end{align*}\n$$\n\nDividing by $(f_i^i +f_j^j -f_a^a -f_b^b)$ yields\n\n\n
    \n\n$$\n\\begin{equation}\nt_{ij}^{ab} = t_{ij}^{ab} + \\frac{\\overline{H}_{ij}^{ab}}{f_i^i +f_j^j -f_a^a -f_b^b}\n\\label{iter} \\tag{36}\n\\end{equation}\n$$\n\nThis equation is of the type $t=f(t)$, and we solve it by iteration,\ni.e. we start with a guess $t_0$ and iterate $t_{n+1}=f(t_n)$, and\nhope that this will converge to a solution. We take the perturbative result\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{pert} \\tag{37}\n\\left(t_{ij}^{ab}\\right)_0 = \\frac{\\langle ab\\vert V\\vert ij\\rangle}{f_i^i +f_j^j -f_a^a -f_b^b}\n\\end{equation}\n$$\n\nas a starting point, compute $\\overline{H}_{ij}^{ab}$, and find a new\n$t_{ij}^{ab}$ from the right-hand side of Eq. ([36](#iter)). We repeat\nthis process until the amplitudes (or the CCD energy) converge.\n\n\n\n# CCD for the pairing Hamiltonian\n\n\nYou learned about the pairing Hamiltonian earlier in this\nschool. Convince yourself that this Hamiltonian does not induce any\n1p-1h excitations. Let us solve the CCD equations for this\nproblem. This consists of the following steps\n\n1. Write a function that compute the potential, i.e. it returns a four-indexed array (or tensor). We need $\\langle ab\\vert V\\vert cd\\rangle$, $\\langle ij\\vert V\\vert kl\\rangle$, and $\\langle ab\\vert V\\vert ij\\rangle$. Why is there no $\\langle ab\\vert V\\vert id\\rangle$ or $\\langle ai\\vert V\\vert jb\\rangle$ ?\n\n2. Write a function that computes the Fock matrix, i.e. a two-indexed array. We only need $f_a^b$ and $f_i^j$. Why? \n\n3. Initialize the cluster amplitudes according to Eq. ([37](#pert)), and solve Eq. ([36](#iter)) by iteration. The cluster amplitudes $T_1$ and $T_2$ are two- and four-indexed arrays, respectively.\n\nPlease note that the contraction of tensors (i.e. the summation over\ncommon indices in products of tensors) is very user friendly and\nelegant in python when `numpy.einsum` is used.\n\n\n\n\n\n## Project 15: Solve the CCD equations for the pairing problem\n
    \n\nThe Hamiltonian is\n\n\n
    \n\n$$\n\\begin{equation}\nH = \\delta \\sum_{p=1}^\\Omega (p-1)\\left(a^\\dagger_{p+}a_{p+} + a^\\dagger_{p-}a_{p-}\\right)\n-{g \\over 2} \\sum_{p, q=1}^\\Omega a^\\dagger_{p+}a^\\dagger_{p-} a_{q-} a_{q+} .\n\\label{_auto25} \\tag{38}\n\\end{equation}\n$$\n\nCheck your results and reproduce Fig 8.5 and Table 8.12 from Lecture Notes in Physics 936.\n\n\n\n**Answer.**\n[Click for IPython notebook for FCI and CCD solutions](https://github.com/NuclearTalent/ManyBody2018/tree/master/doc/Programs/Python/PairingModel)\n\n\n```\n## Coupled clusters in CCD approximation\n## Implemented for the pairing model of Lecture Notes in Physics 936, Chapter 8.\n## Thomas Papenbrock, June 2018\n\nimport numpy as np\n\n\ndef init_pairing_v(g,pnum,hnum):\n \"\"\"\n returns potential matrices of the pairing model in three relevant channels\n \n param g: strength of the pairing interaction, as in Eq. (8.42)\n param pnum: number of particle states\n param hnum: number of hole states\n \n return v_pppp, v_pphh, v_hhhh: np.array(pnum,pnum,pnum,pnum), \n np.array(pnum,pnum,hnum,hnum), \n np.array(hnum,hnum,hnum,hnum), \n The interaction as a 4-indexed tensor in three channels.\n \"\"\"\n v_pppp=np.zeros((pnum,pnum,pnum,pnum))\n v_pphh=np.zeros((pnum,pnum,hnum,hnum))\n v_hhhh=np.zeros((hnum,hnum,hnum,hnum))\n \n gval=-0.5*g\n for a in range(0,pnum,2):\n for b in range(0,pnum,2):\n v_pppp[a,a+1,b,b+1]=gval\n v_pppp[a+1,a,b,b+1]=-gval\n v_pppp[a,a+1,b+1,b]=-gval\n v_pppp[a+1,a,b+1,b]=gval\n \n for a in range(0,pnum,2):\n for i in range(0,hnum,2):\n v_pphh[a,a+1,i,i+1]=gval\n v_pphh[a+1,a,i,i+1]=-gval\n v_pphh[a,a+1,i+1,i]=-gval\n v_pphh[a+1,a,i+1,i]=gval\n \n for j in range(0,hnum,2):\n for i in range(0,hnum,2):\n v_hhhh[j,j+1,i,i+1]=gval\n v_hhhh[j+1,j,i,i+1]=-gval\n v_hhhh[j,j+1,i+1,i]=-gval\n v_hhhh[j+1,j,i+1,i]=gval\n \n return v_pppp, v_pphh, v_hhhh\n \n \ndef init_pairing_fock(delta,g,pnum,hnum):\n \"\"\"\n initializes the Fock matrix of the pairing model\n \n param delta: Single-particle spacing, as in Eq. (8.41)\n param g: pairing strength, as in eq. (8.42)\n param pnum: number of particle states\n param hnum: number of hole states\n \n return f_pp, f_hh: The Fock matrix in two channels as numpy arrays np.array(pnum,pnum), np.array(hnum,hnum). \n \"\"\"\n# the Fock matrix for the pairing model. No f_ph needed, because we are in Hartree-Fock basis \n deltaval=0.5*delta\n gval=-0.5*g\n f_pp = np.zeros((pnum,pnum))\n f_hh = np.zeros((hnum,hnum))\n\n for i in range(0,hnum,2):\n f_hh[i ,i ] = deltaval*i+gval\n f_hh[i+1,i+1] = deltaval*i+gval\n \n for a in range(0,pnum,2):\n f_pp[a ,a ] = deltaval*(hnum+a)\n f_pp[a+1,a+1] = deltaval*(hnum+a)\n \n return f_pp, f_hh\n\n\ndef init_t2(v_pphh,f_pp,f_hh):\n \"\"\"\n Initializes t2 amlitudes as in MBPT2, see first equation on page 345\n \n param v_pphh: pairing tensor in pphh channel\n param f_pp: Fock matrix in pp channel\n param f_hh: Fock matrix in hh channel\n \n return t2: numpy array in pphh format, 4-indices tensor\n \"\"\"\n pnum = len(f_pp)\n hnum = len(f_hh)\n t2_new = np.zeros((pnum,pnum,hnum,hnum))\n for i in range(hnum):\n for j in range(hnum):\n for a in range(pnum):\n for b in range(pnum):\n t2_new[a,b,i,j] = v_pphh[a,b,i,j] / (f_hh[i,i]+f_hh[j,j]-f_pp[a,a]-f_pp[b,b])\n return t2_new\n\n\n# CCD equations. Note that the \"->abij\" assignment is redundant, because indices are ordered alphabetically.\n# Nevertheless, we retain it for transparency.\ndef ccd_iter(v_pppp,v_pphh,v_hhhh,f_pp,f_hh,t2):\n \"\"\"\n Performs one iteration of the CCD equations (8.34), using also intermediates for the nonliniar terms\n \n param v_pppp: pppp-channel pairing tensor, numpy array\n param v_pphh: pphh-channel pairing tensor, numpy array\n param v_hhhh: hhhh-channel pairing tensor, numpy array\n param f_pp: Fock matrix in pp channel\n param f_hh: Fock matrix in hh channel\n param t2: Initial t2 amplitude, tensor in form of pphh channel\n \n return t2_new: new t2 amplitude, tensor in form of pphh channel\n \"\"\"\n pnum = len(f_pp)\n hnum = len(f_hh)\n Hbar_pphh = ( v_pphh \n + np.einsum('bc,acij->abij',f_pp,t2) \n - np.einsum('ac,bcij->abij',f_pp,t2) \n - np.einsum('abik,kj->abij',t2,f_hh)\n + np.einsum('abjk,ki->abij',t2,f_hh)\n + 0.5*np.einsum('abcd,cdij->abij',v_pppp,t2) \n + 0.5*np.einsum('abkl,klij->abij',t2,v_hhhh)\n )\n\n # hh intermediate, see (8.47)\n chi_hh = 0.5* np.einsum('cdkl,cdjl->kj',v_pphh,t2)\n\n Hbar_pphh = Hbar_pphh - ( np.einsum('abik,kj->abij',t2,chi_hh) \n - np.einsum('abik,kj->abji',t2,chi_hh) )\n\n # pp intermediate, see (8.46)\n chi_pp = -0.5* np.einsum('cdkl,bdkl->cb',v_pphh,t2)\n\n Hbar_pphh = Hbar_pphh + ( np.einsum('acij,cb->abij',t2,chi_pp) \n - np.einsum('acij,cb->baij',t2,chi_pp) )\n\n # hhhh intermediate, see (8.48)\n chi_hhhh = 0.5 * np.einsum('cdkl,cdij->klij',v_pphh,t2)\n\n Hbar_pphh = Hbar_pphh + 0.5 * np.einsum('abkl,klij->abij',t2,chi_hhhh)\n\n # phph intermediate, see (8.49)\n chi_phph= + 0.5 * np.einsum('cdkl,dblj->bkcj',v_pphh,t2)\n\n\n Hbar_pphh = Hbar_pphh + ( np.einsum('bkcj,acik->abij',chi_phph,t2)\n - np.einsum('bkcj,acik->baij',chi_phph,t2)\n - np.einsum('bkcj,acik->abji',chi_phph,t2)\n + np.einsum('bkcj,acik->baji',chi_phph,t2) )\n \n t2_new=np.zeros((pnum,pnum,hnum,hnum))\n for i in range(hnum):\n for j in range(hnum):\n for a in range(pnum):\n for b in range(pnum):\n t2_new[a,b,i,j] = ( t2[a,b,i,j] \n + Hbar_pphh[a,b,i,j] / (f_hh[i,i]+f_hh[j,j]-f_pp[a,a]-f_pp[b,b]) )\n\n return t2_new\n\n\ndef ccd_energy(v_pphh,t2):\n \"\"\"\n Computes CCD energy. Call as \n energy = ccd_energy(v_pphh,t2)\n \n param v_pphh: pphh-channel pairing tensor, numpy array\n param t2: t2 amplitude, tensor in form of pphh channel\n \n return energy: CCD correlation energy\n \"\"\"\n erg = 0.25*np.einsum('abij,abij',v_pphh,t2)\n return erg\n\n###############################\n######## Main Program\n\n# set parameters as for model\npnum = 20 # number of particle states\nhnum = 10 # number of hole states\ndelta = 1.0\n\ng = 0.5\n\nprint(\"parameters\")\nprint(\"delta =\", delta, \", g =\", g)\n\n\n# Initialize pairing matrix elements and Fock matrix\nv_pppp, v_pphh, v_hhhh = init_pairing_v(g,pnum,hnum)\nf_pp, f_hh = init_pairing_fock(delta,g,pnum,hnum)\n\n# Initialize T2 amplitudes from MBPT2\nt2 = init_t2(v_pphh,f_pp,f_hh)\nerg = ccd_energy(v_pphh,t2)\n\n# Exact MBPT2 for comparison, see last equation on page 365 \nexact_mbpt2 = -0.25*g**2*(1.0/(2.0+g) + 2.0/(4.0+g) + 1.0/(6.0+g))\nprint(\"MBPT2 energy =\", erg, \", compared to exact:\", exact_mbpt2)\n \n \n# iterate CCD equations niter times\nniter=200\nmix=0.3\nerg_old=0.0\neps=1.e-8\nfor iter in range(niter):\n t2_new = ccd_iter(v_pppp,v_pphh,v_hhhh,f_pp,f_hh,t2)\n erg = ccd_energy(v_pphh,t2_new)\n myeps = abs(erg-erg_old)/abs(erg)\n if myeps < eps: break\n erg_old=erg\n print(\"iter=\", iter, \"erg=\", erg, \"myeps=\", myeps)\n t2 = mix*t2_new + (1.0-mix)*t2 \n \nprint(\"Energy = \", erg)\n```\n\n\n\n\n\n\n# Nucleonic Matter\n\nWe want to compute nucleonic matter using coupled cluster or IMSRG\nmethods, and start with considering the relevant symmetries.\n\n\n\n\n\n## Exercise 16: Which symmetries are relevant for nuclear matter?\n
    \n\n\n**a)**\nEnumerate continuous and discrete symmetries of nuclear matter.\n\n\n\n**Answer.**\nThe symmetries are the same as for nuclei. Continuous symmetries:\ntranslational and rotational invariance. Discrete symmetries: Parity\nand time reversal invariance.\n\n\n\n**b)**\nWhat basis should we use to implement these symmetries? Why do we have to make a choice between the two continuous symmetries? Which basis is most convenient and why?\n\n\n\n**Answer.**\nAngular momentum and momentum do not commute. Thus, there is no basis\nthat respects both symmetries simultaneously. If we choose the\nspherical basis, we are computing a spherical blob of nuclear matter\nand have to contend with surface effects, i.e. with finite size\neffects. We also need a partial-wave decomposition of the nuclear\ninteraction. This approach has been done followed in\n[\"Coupled-cluster studies of infinite nuclear matter, \" G. Baardsen,\nA. Ekstr\u00f6m, G. Hagen, M. Hjorth-Jensen, arXiv:1306.5681, Phys. Rev. C\n88, 054312 (2013)]. If we choose a basis of discrete momentum states,\ntranslational invariance can be respected. This also facilitates the\nimplementetation of modern nuclear interactions (which are often\nformulated in momentum space in effective field theories). However, we\nhave to think about the finite size effects imposed by periodic\nboundary conditions (or generalized Bloch waves). This approch was\nimplemented in [\"Coupled-cluster calculations of nucleonic matter,\"\nG. Hagen, T. Papenbrock, A. Ekstr\u00f6m, K. A. Wendt, G. Baardsen,\nS. Gandolfi, M. Hjorth-Jensen, C. J. Horowitz, arXiv:1311.2925,\nPhys. Rev. C 89, 014319 (2014)].\n\n\n\n\n\n\n===== =====\n\n\n\n\n# Basis states\n\nIn what follows, we employ a basis made from discrete momentum states,\ni.e. those states $\\vert k_x, k_y, k_z\\rangle$ in a cubic box of size $L$ that\nexhibit periodic boundary conditions, i.e. $\\psi_k(x+L) =\\psi_k(x)$.\n\n\n\n\n\n## Exercise 17: Determine the basis states\n
    \n\nWhat are the discrete values of momenta admissable in $(k_x, k_y, k_z)$?\n\n\n\n**Answer.**\nIn 1D position space $\\psi_k(x) \\propto e^{i k x}$ with $k =\n{2\\pi n\\over L}$ and $n=0, \\pm 1, \\pm 2, \\ldots$ fulfill $\\psi_k(x+L)\n= \\psi_k(x)$.\n\n\n\n\n\n\n===== =====\n\n\n\n\nThus, we use a cubic lattice in momentum space. Note that the\nmomentum states $e^{i k x}$ are not invariant under time reversal\n(i.e. under $k\\to -k$), and also do not exhibit good parity ($x\\to\n-x$). The former implies that the Hamiltonian matrix will in general\nbe complex Hermitian and that the cluster amplitudes will in general\nbe complex.\n\n\n\n\n\n\n\n\n## Exercise 18: How large should the basis be?\n
    \n\nWhat values should be chosen for the box size $L$ and for the maximum number $n_{\\rm max}$ , i.e.\nfor the discrete momenta $k = {2\\pi n\\over L}$ and $n=0, \\pm 1,\\\n \\pm 2, \\ldots, \\pm n_{\\rm max}$?\n\n\n\n**Answer.**\nUsually $n_{\\rm max}$ is fixed by computational cost, because we have\n$(2n_{\\rm max}+1)^3$ basis states. We have used $n_{\\rm max}=4$ or up\nto $n_{\\rm max}=6$ in actual calculations to get converged results.\n\nThe maximum momentum must fulfill $k_{n_{\\rm max}}> \\Lambda$, where\n$\\Lambda$ is the momemtum cutoff of the interaction. This then fixes\n$L$ for a given $n_{\\rm max}$.\n\n\n\n\n\n\n===== =====\n\n\n\n\nCoupled cluster and IMSRG start from a Hartree-Fock reference state,\nand we need to think about this next. What are the magic numbers of a\ncubic lattice for neutron matter?\n\n\n\n\n\n## Exercise 19: Determine the lowest few magic numbers for a cubic lattice.\n
    \n\n\n\n**Answer.**\nAs the spin-degeneracy is $g_s=2$, we have the magic numbers $g_s N$\nwith $N=1, 7, 19, 27, 33, 57, 66, \\ldots$ for neutrons.\n\n\n\n\n\n\n===== =====\n\n\n\nGiven $n_{\\rm max}$ and $L$ for the basis parameters, we can choose a\nmagic neutron number $N$. Clearly, the density of the system is then\n$\\rho=N/L^3$. This summarizes the requirements for the basis. We\nchoose $n_{\\rm max}$ as large as possible, i.e. as large as\ncomputationally feasible. Then $L$ and $N$ are constrained by the UV\ncutoff and density of the system.\n\n\n## Finite size effects\nWe could also have considered the case of a more generalized boundary\ncondition, i.e. $\\psi_k(x+L) =e^{i\\theta}\\psi_k(x)$. Admissable\nmomenta that fulfill such a boundary condition are $k_n(\\theta) = {2\\pi n\n+\\theta \\over L}$. Avering over the \"twist\" angle $\\theta$ removes\nfinite size effects, because the discrete momenta are really drawn\nform a continuum. In three dimensions, there are three possible twist\nangles, and averaging over twist angles implies summing over many\nresults corresponding to different angles. Thus, the removal of\nfinite-size effects significantly increases the numerical expense.\nAn example\nis shown in [Figure](#fig-finite), where we compute the kinetic\nenergy per particle\n\n\n
    \n\n$$\n\\begin{equation}\nT_N(\\theta_x,\\theta_y,\\theta_z)=g_s\\sum_{n_x, n_y, n_z \\in N} {\\hbar^2 \\left( k_{n_x}^2(\\theta_x) +k_{n_y}^2(\\theta_y) +k_{n_z}^2(\\theta_z)\\right)\\over 2m}\n\\label{_auto26} \\tag{39}\n\\end{equation}\n$$\n\nand compare with the infinite result $T_{\\rm inf} = {3\\over 10} \n{\\hbar^2 k_F^2\\over m} N$ valid for the free Fermi gas. We clearly see\nstrong shell effects (blue dashed line) and that averaging over the twist angles\n(red full line) very much reduces the shell oscillations. We also note that\nthe neutron number 66 is quite attractive as it exhibits smaller\nfinite size effects than other of the accessible magic numbers.\n\n\n\n\n
    \n\n

    Relative finite-size corrections for the kinetic energy in pure neutron matter at the Fermi momentum $k_F = 1.6795 {\\rm fm}^{-1}$ versus the neutron number A. TABC10 are twist-averaged boundary conditions with 10 Gauss-Legendre points in each spatial direction.

    \n\n\n\n\n\n# Channel structure of Hamiltonian and cluster amplitudes\n\nGood quantum numbers for the nuclear interaction (i.e. operators that\ncommute with the Hamiltonian and with each other) are total momentum,\nand the number of neutrons and protons, and - for simple interactions\n%s\n : \n also the spin (this is really spin, not orbital anular momentum or\n\ntotal angular momentum, as the latter two do not commute with\nmomentum). Thus the Hamiltonian (and the cluster amplitudes) will\nconsist of blocks, one for each set of quantum numbers. We call the\nset of quantum numbers that label each such block as a \"channel.\"\nAs the interaction is block diagonal, a numerically efficient\nimplementation of nuclear matter has to take advantage of this channel\nstructure. In fact, neutron matter cannot be computed in a numerically\nconverged way (i.e. for large enough $n_{\\rm max}$) if one does not\nexploit the channel structure.\n\nThe Hamiltonian is of the structure\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{HQ} \\tag{40}\nH = \\sum_{\\vec{k}, \\sigma} \\varepsilon_{\\vec{k}, \\sigma}^{\\vec{k}, \\sigma} a^\\dagger_{\\vec{k}, \\sigma}a_{\\vec{k}, \\sigma}\n+ \\sum_{\\vec{Q},\\vec{p},\\vec{k},\\sigma_s} V_{\\sigma_1\\sigma_2}^{\\sigma_3\\sigma_4}(\\vec{p},\\vec{k}) a^\\dagger_{\\vec{Q/2}+\\vec{p}, \\sigma_3}a^\\dagger_{\\vec{Q/2}-\\vec{p}, \\sigma_4} a_{\\vec{Q/2}-\\vec{k}, \\sigma_2}a_{\\vec{Q/2}+\\vec{k}, \\sigma_1}\n\\end{equation}\n$$\n\nwith $\\varepsilon_{\\vec{k}, \\sigma}^{\\vec{k}, \\sigma} = {k^2\\over\n2m}$. In Eq. ([40](#HQ)) we expressed the single-particle momenta in\nterms of center-of-mass momentum $\\vec{Q}$ and relative momenta\n$(\\vec{k},\\vec{p})$, i.e. the incoming momenta $(\\vec{k}_1,\n\\vec{k}_2)$ and outgoing momenta $(\\vec{k}_3, \\vec{k}_3)$ are\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{CoM} \\tag{41}\n\\vec{k}_1 = \\vec{Q}/2 +\\vec{k} ,\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n\\vec{k}_2 = \\vec{Q}/2 -\\vec{k} ,\n\\label{_auto27} \\tag{42}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n\\vec{k}_3 = \\vec{Q}/2 +\\vec{p} ,\n\\label{_auto28} \\tag{43}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n\\vec{k}_4 = \\vec{Q}/2 -\\vec{p} .\n\\label{_auto29} \\tag{44}\n\\end{equation}\n$$\n\nThe conservation of momentum is obvious in the two-body interaction as\nboth the annihilation operators and the creation operators depend on\nthe same center-of-mass momentum $\\vec{Q}$. We note that the two-body\ninteraction $V$ depends only on the relative momenta\n$(\\vec{k},\\vec{p})$ but not on the center-of-mass momentum. We also\nnote that a local interaction (i.e. an interaction that is\nmultiplicative in position space) depends only on the momentum\ntransfer $\\vec{k}-\\vec{p}$. The spin projections $\\pm 1/2$ are denoted\nas $\\sigma$.\n\nThus, the $T_2$ operator is\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{t2Q} \\tag{45}\nT_2 = {1\\over 4} \\sum_{\\vec{Q},\\vec{p},\\vec{k},\\sigma_s} t_{\\sigma_1\\sigma_2}^{\\sigma_3\\sigma_4}(Q; \\vec{p},\\vec{k}) a^\\dagger_{\\vec{Q/2}+\\vec{p}, \\sigma_3}a^\\dagger_{\\vec{Q/2}-\\vec{p}, \\sigma_4} a_{\\vec{Q/2}-\\vec{k}, \\sigma_2}a_{\\vec{Q/2}+\\vec{k}, \\sigma_1} . \n\\end{equation}\n$$\n\nWe note that the amplitude $t_{\\sigma_1\\sigma_2}^{\\sigma_3\\sigma_4}(Q;\n\\vec{p},\\vec{k})$ depends on the center-of-mass momentum $\\vec{Q}$, in\ncontrast to the potential matrix element\n$V_{\\sigma_1\\sigma_2}^{\\sigma_3\\sigma_4}(\\vec{p},\\vec{k})$.\n\nIn the expressions ([40](#HQ)) and ([45](#t2Q)) we supressed that\n$\\sigma_1+\\sigma_2 = \\sigma_3+\\sigma_4$. So, a channel is defined by\n$\\vec{Q}$ and total spin projection $\\sigma_1+\\sigma_2$.\n\nBecause of this channel structure, the simple solution we implemented\nfor the pairing problem cannot be really re-used when computing\nneutron matter. Let us take a look at the Minnesota potential\n\n\n
    \n\n$$\n\\begin{equation}\nV(r) = \\left( V_R(r) + {1\\over 2}(1+P_{12}^\\sigma)V_T(r) + {1\\over 2}(1-P_{12}^\\sigma)V_S(r)\\right) {1\\over 2}(1-P_{12}^\\sigma P_{12}^\\tau). \n\\label{_auto30} \\tag{46}\n\\end{equation}\n$$\n\nHere,\n\n$$\nP^\\sigma_{12}= {1\\over 2}(1+\\vec{\\sigma}_1\\cdot\\vec{\\sigma}_2) , \\nonumber\n$$\n\n\n
    \n\n$$\n\\begin{equation} \nP^\\tau_{12} = {1\\over 2}(1+\\vec{\\tau}_1\\cdot\\vec{\\tau}_2) \n\\label{_auto31} \\tag{47}\n\\end{equation}\n$$\n\nare spin and isospin exchange operators, respectively, and\n$\\vec{\\sigma}$ and $\\vec{\\tau}$ are vectors of Pauli matrices in spin\nand isospin space, respectively. Thus,\n\n\n
    \n\n$$\n\\begin{equation}\n{1\\over 2}(1-P_{12}^\\sigma P_{12}^\\tau) = \\vert\nS_{12}=0, T_{12}=1\\rangle\\langle S_{12}=0,T_{12}=1\\vert + \\vert\nS_{12}=1, T_{12}=0\\rangle\\langle S_{12}=1,T_{12}=0\\vert\n\\label{_auto32} \\tag{48}\n\\end{equation}\n$$\n\nprojects onto two-particle spin-isospin states as indicated, while\n\n\n
    \n\n$$\n\\begin{equation}\n{1\\over 2}(1-P_{12}^\\sigma) = \\vert\nS_{12}=0\\rangle\\langle S_{12}=0\\vert , \n\\label{_auto33} \\tag{49}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} {1\\over 2}(1+P_{12}^\\sigma)\n = \\vert S_{12}=1\\rangle\\langle S_{12}=1\\vert\n\\label{_auto34} \\tag{50}\n\\end{equation}\n$$\n\nproject onto spin singlet and spin triplet combinations,\nrespectively. For neutron matter, two-neutron states have isospin\n$T_{12}=1$, and the Minnesota potential has no triplet term $V_T$.\nFor the spin-exchange operator (and spins $s_1, s_2=\\pm 1/2$), we have\n$P_{12}^\\sigma\\vert s_1s_2\\rangle= \\vert s_2s_1\\rangle$. For neutron\nmatter, $P_{12}^\\tau=1$, because the states are symmetric under exchange of\nisospin. Thus, the Minnesota potential simplifies significantly for\nneutron matter as $V_T$ does not contribute.\n\nWe note that the spin operator has matrix elements\n\n\n
    \n\n$$\n\\begin{equation}\n\\langle s_1' s_2'\\vert {1\\over 2}(1-P_{12}^{\\sigma})\\vert s_1 s_2\\rangle\n= {1\\over 2} \\left(\n \\delta_{s_1}^{s_1'}\\delta_{s_2}^{s_2'}\n-\\delta_{s_1}^{s_2'}\\delta_{s_2}^{s_1'}\\right) . \n\\label{_auto35} \\tag{51}\n\\end{equation}\n$$\n\nThe radial functions are\n\n\n
    \n\n$$\n\\begin{equation}\nV_R(r) =V_R e^{-\\kappa_R r^2} , \n\\label{_auto36} \\tag{52}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \nV_S(r) =V_S e^{-\\kappa_S r^2} , \n\\label{_auto37} \\tag{53}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \nV_T(r) =V_T e^{-\\kappa_T r^2} , \n\\label{_auto38} \\tag{54}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n\\label{_auto39} \\tag{55}\n\\end{equation}\n$$\n\nand the parameters are as follows\n\n\n\n\n\n\n\n\n\n\n
    $\\alpha$ $V_\\alpha$ $\\kappa_\\alpha$
    $R$ +200 MeV 1.487 fm $^{-2}$
    $S$ -91.85 MeV 0.465 fm $^{-2}$
    $T$ -178 MeV 0.639 fm $^{-2}$
    \nNote that $\\kappa_\\alpha^{1/2}$ sets the momentum scale of the\nMinnesota potential. We see that we deal with a short-ranged repulsive\ncore (the $V_R$ term) and longer ranged attractive terms in the\nsinglet (the term $V_S$) and triplet (the term $V_T$) channels.\n\n\nA Fourier transform (in the finite cube of length $L$) yields the momentum-space form of the potential\n\n\n
    \n\n$$\n\\begin{equation}\n\\langle k_p k_q \\vert V_\\alpha\\vert k_r k_s \\rangle = {V_\\alpha\\over L^3} \\left({\\pi\\over\\kappa_\\alpha}\\right)^{3/2}\ne^{- {q^2 \\over 4\\kappa_\\alpha}} \\delta_{k_p+k_q}^{k_r+k_s} . \n\\label{_auto40} \\tag{56}\n\\end{equation}\n$$\n\nHere, $q\\equiv {1\\over 2}(k_p-k_q-k_r+k_s)$ is the momentum transfer,\nand the momentum conservation $k_p+k_q=k_r+k_s$ is explicit.\n\nAs we are dealing only with neutrons, the potential matrix elements\n(including spin) are for $\\alpha = R, S$\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{ME_of_V} \\tag{57}\n\\langle\tk_p s_p k_q s_q\\vert V_\\alpha\\vert k_r s_r k_s s_s\\rangle = \\langle\tk_p k_q \\vert V_\\alpha\\vert k_r k_s \\rangle\n{1\\over 2}\\left(\\delta_{s_p}^{s_r}\\delta_{s_q}^{s_s} - \\delta_{s_p}^{s_s}\\delta_{s_q}^{s_r}\\right) , \n\\end{equation}\n$$\n\nand it is understood that there is no contribution from $V_T$. Please note that the matrix elements ([57](#ME_of_V)) are not yet antisymmetric under exchange, but $\\langle k_p s_p k_q s_q\\vert V_\\alpha\\vert k_r s_r k_s s_s\\rangle -\n\\langle k_p s_p k_q s_q\\vert V_\\alpha\\vert k_s s_s k_r s_r\\rangle$ is. \n\n\n\n## Example: Channel structure and its usage\n
    \n\nWe have single-particle states with momentum and spin, namely\n\n\n
    \n\n$$\n\\begin{equation}\n\\vert r\\rangle \\equiv \\vert k_r s_r\\rangle .\n\\label{_auto41} \\tag{58}\n\\end{equation}\n$$\n\nNaively, two-body states are then\n\n\n
    \n\n$$\n\\begin{equation}\n\\vert r s \\rangle \\equiv \\vert k_r s_r k_s s_s\\rangle ,\n\\label{_auto42} \\tag{59}\n\\end{equation}\n$$\n\nbut using the center-of-mass transformation ([41](#CoM)) we can rewrite\n\n\n
    \n\n$$\n\\begin{equation}\n\\vert r s \\rangle \\equiv \\vert P_{rs} k_{rs} s_r s_s\\rangle ,\n\\label{_auto43} \\tag{60}\n\\end{equation}\n$$\n\nwhere $P_{rs} = k_r +k_s$ is the total momentum and\n$k_{rs}=(k_r-k_s)/2$ is the relative momentum. This representation of\ntwo-body states is well adapated to our problem, because the\ninteraction and the $T_2$ amplitudes preserve the total momentum.\nThus, we store the cluster amplitudes $t_{ij}^{ab}$ as matrices\n\n\n
    \n\n$$\n\\begin{equation}\nt_{ij}^{ab} \\to \\left[t(P_{ij})\\right]_{\\vert k_{ij} s_i s_j\\rangle}^{\\vert k_{ab} s_a s_b\\rangle}\n\\equiv t_{\\vert P_{ij} k_{ij} s_i s_j\\rangle}^{\\vert P_{ij} k_{ab} s_a s_b\\rangle} , \n\\label{_auto44} \\tag{61}\n\\end{equation}\n$$\n\nand the conservation of total momentum is explicit.\n\nLikewise, the pppp, pphh, and hhhh parts of the interaction can be written in this form, namely\n\n\n
    \n\n$$\n\\begin{equation}\nV_{cd}^{ab} \\to \\left[V(P_{ab})\\right]_{\\vert k_{cd} s_c s_d\\rangle}^{\\vert k_{ab} s_a s_b\\rangle}\n\\equiv V_{\\vert P_{ab} k_{cd} s_c s_d\\rangle}^{\\vert P_{ab} k_{ab} s_a s_b\\rangle} , \n\\label{_auto45} \\tag{62}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \nV_{ij}^{ab} \\to \\left[V(P_{ij})\\right]_{\\vert k_{ij} s_i s_j\\rangle}^{\\vert k_{ab} s_a s_b\\rangle}\n\\equiv V_{\\vert P_{ij} k_{ij} s_i s_j\\rangle}^{\\vert P_{ij} k_{ab} s_a s_b\\rangle} , \n\\label{_auto46} \\tag{63}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \nV_{ij}^{kl} \\to \\left[V(P_{ij})\\right]_{\\vert k_{ij} s_i s_j\\rangle}^{\\vert k_{kl} s_k s_l\\rangle}\n\\equiv V_{\\vert P_{ij} k_{ij} s_i s_j\\rangle}^{\\vert P_{ij} k_{kl} s_k s_l\\rangle} , \n\\label{_auto47} \\tag{64}\n\\end{equation}\n$$\n\nand we also have\n\n\n
    \n\n$$\n\\begin{equation}\n\\overline{H}_{ij}^{ab} \\to \\left[\\overline{H}(P_{ij})\\right]_{\\vert k_{ij} s_i s_j\\rangle}^{\\vert k_{ab} s_a s_b\\rangle}\n\\equiv \\overline{H}_{\\vert P_{ij} k_{ij} s_i s_j\\rangle}^{\\vert P_{ij} k_{ab} s_a s_b\\rangle} , \n\\label{_auto48} \\tag{65}\n\\end{equation}\n$$\n\nUsing these objects, diagrams (1), (4), and (5) of [Figure](#fig-ccd) can be done for each block of momentum $P_{ij}$ as a\ncopy and matrix-matrix multiplications, respectively\n\n\n
    \n\n$$\n\\begin{equation}\n\\left[\\overline{H}(P_{ij})\\right]_{\\vert k_{ij} s_i s_j\\rangle}^{\\vert k_{ab} s_a s_b\\rangle} =\n\\left[V(P_{ij})\\right]_{\\vert k_{ij} s_i s_j\\rangle}^{\\vert k_{ab} s_a s_b\\rangle} \n\\label{_auto49} \\tag{66}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n+ {1\\over 2} \\sum_{\\vert k_{kl} s_k s_l\\rangle}\n\\left[t(P_{ij})\\right]_{\\vert k_{kl} s_k s_l\\rangle}^{\\vert k_{ab} s_a s_b\\rangle}\n\\left[V(P_{ij})\\right]_{\\vert k_{ij} s_i s_j\\rangle}^{\\vert k_{kl} s_k s_l\\rangle} \n\\label{_auto50} \\tag{67}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n+ {1\\over 2} \\sum_{\\vert k_{cd} s_c s_d\\rangle}\n\\left[V(P_{ij})\\right]_{\\vert k_{cd} s_c s_d\\rangle}^{\\vert k_{ab} s_a s_b\\rangle}\n\\left[t(P_{ij})\\right]_{\\vert k_{ij} s_i s_j\\rangle}^{\\vert k_{cd} s_c s_d\\rangle} .\n\\label{_auto51} \\tag{68}\n\\end{equation}\n$$\n\nSimilarly, the CCD correlation energy results from\n\n\n
    \n\n$$\n\\begin{equation}\nE_c = {1\\over 4} \\sum_{P_{ij}} \\sum_{\\vert k_{ij} s_i s_j\\rangle}\\sum_{\\vert k_{ab} s_a s_b\\rangle }\n\\left[t(P_{ij})\\right]_{\\vert k_{ij} s_i s_j\\rangle}^{\\vert k_{ab} s_a s_b\\rangle}\n\\left[V(P_{ij})\\right]_{\\vert k_{ij} s_i s_j\\rangle}^{\\vert k_{ab} s_a s_b\\rangle}\n\\label{_auto52} \\tag{69}\n\\end{equation}\n$$\n\nThese efficiences cannot be used for the sixth diagram of [Figure](#fig-ccd). One could either change to a ph formulation, noting that\n$k_a-k_i = k_k -k_c$ is also a preserved quantity in $t_{ik}^{ac}$ and\nthat $k_k-k_c = k_j-k_b$ is preserved in $V_{cj}^{kb}$. Thus\n$\\sum_{kc} t_{ik}^{ac}V_{cj}^{kb}$ has a conserved quantity $k_k-k_c$\nin the loop, and we can again use matrix-matrix multiplications for\nthis diagram. This requires us to store the $T_2$ amplitude in a phhp\nand in the usual pphh formulation. Alternatively, we could simply code\nthis diagram with the loops over single-particle states. If this seems\ntoo tedious, one can also limit CCD to the first five diagrams in\n[Figure](#fig-ccd) (these are the pp and hh ladders), which gives a\ngood description for neutron matter, see the comparison between this\nand full CCD in [\"Coupled-cluster calculations of nucleonic matter,\"\nG. Hagen, T. Papenbrock, A. Ekstr\u00f6m, K. A. Wendt, G. Baardsen,\nS. Gandolfi, M. Hjorth-Jensen, C. J. Horowitz, arXiv:1311.2925,\nPhys. Rev. C 89, 014319 (2014)].\n\n===== =====\n\n \n\n\nThe steps towards the solution of the CCD equations for neutron matter are as follows\n\n1. For a given density and UV cutoff, set up the lattice, i.e. determine the single-particle basis.\n\n2. Determine the channels allowed by the (Minnesota) interaction, i.e. sets of two-body states that are connected by the interaction.\n\n3. Exploit this channel structure when computing the diagrams.\n\n4. Solve the coupled-cluster equations. Here we start first with the pp and hh ladders, i.e. using only the first five diagrams of [Figure](#fig-ccd).\n\n\n\n\n\n## Exercise 20: Write a CCD code for neutron matter, focusing first on ladder approximation, i.e. including the first five diagrams in [Figure](#fig-ccd).\n
    \n\n\n\n**Answer.**\n[Click for IPython notebook](https://github.com/NuclearTalent/ManyBody2018/tree/master/doc/Programs/Python/NeutronMatter)\n\n\n```\nimport numpy as np\n\n##############################################################\n# CCD Program for neutron matter with the Minnesota potential.\n#\n# Thomas Papenbrock, July/August 2018\n#\n# License: Free Software following Version 3 of GNU General Public License, \n# see https://www.gnu.org/licenses/gpl.html\n#######\n\n\n##########################\n# Class for neutron matter \n#######\n\nclass MomSpaceBasis:\n \"\"\"\n momentum-space basis class\n The constructor has the form\n \n MomSpaceBasis(Nmax,kmax)\n \n param Nmax: Number of lattice points in positive kx-direction\n param kmax: Highest lattice momentum (in 1/fm)\n \n return: MomSpaceBasis as a single-partcle basis. \n attributes of MomSpaceBasis are\n \n dk : lattice spacing in 1/fm\n Lbox : linear dimension (in fm) of cubic box\n nvec : lattice vectors (integers)\n kvec : lattice momentum vectors (floats, in 1/fm)\n ngrid : total number of lattice points \n \"\"\"\n def __init__(self,Nmax,kmax,ordered=True):\n \"\"\"\n the constructor\n \n Generates a cubic lattice in momentum space\n param Nmax: Number of lattice points in positive kx-direction\n param kmax: Highest lattice momentum (in 1/fm)\n param ordered: Optional parameter, True by default, will order lattice points by kinetic energy\n \n return MomSpaceBasis\n \"\"\"\n self.Nmax = Nmax\n self.dim = 0\n self.ngrid = 0\n self._kvec=[]\n self._nvec=[]\n\n dk = kmax / Nmax\n self.dk = dk\n self.Lbox = 2.0*np.pi/dk\n \n nx=[]\n nvec=[]\n for i in range(-Nmax,Nmax+1):\n self.dim=self.dim+1\n nx.append(i)\n \n #print('nx=',nx)\n \n for i in nx:\n for j in nx:\n for k in nx:\n nvec.append(np.array([i,j,k], dtype=int))\n \n #print('nvec=',nvec)\n self.ngrid=len(nvec)\n \n if ordered:\n #print(\"ordered\")\n norm=np.zeros(self.ngrid,dtype=int)\n for i, vec in enumerate(nvec):\n npvec=np.array(vec,dtype=int)\n norm[i]=np.dot(npvec,npvec)\n # print(i, vec, norm[i])\n \n index=np.argsort(norm)\n #print(index)\n self._nvec=[]\n for i, ind in enumerate(index):\n #print(i, ind, nvec[ind])\n self._nvec.append(nvec[ind])\n \n else: \n self._nvec=nvec # a list\n \n self._kvec = np.array(self._nvec)*dk # a numpy array\n\n \n def kvec(self,indx=-1):\n \"\"\"\n MomSpaceBasis.kvec(i) returns ith momentum vector\n MomSpaceBasis.kvec() returns all momentum vectors\n \n param indx: index of k-vector to be returned, optional\n return 3-vector (if index non-negative), or all vectors if no index specified\n \"\"\"\n if indx == -1:\n return self._kvec\n else:\n return self._kvec[indx]\n \n def nvec(self,indx=-1):\n \"\"\"\n MomSpaceBasis.nvec(i) returns ith lattice vector\n MomSpaceBasis.nvec() returns all lattice vectors\n \n param indx: index of lattice vector to be returned, optional\n return 3-vector (if index non-negative), or all lattice vectors if no index specified\n \"\"\"\n if indx == -1:\n return self._nvec\n else:\n return self._nvec[indx]\n \n def dens(self,num):\n \"\"\"\n returns density of system if num particles are present\n param num: int, number of particles\n return dens: float\n \"\"\"\n return num/(self.Lbox)**3\n \n def update(self,dk):\n \"\"\"\n Uses dk as new lattice spacing and rescales existing lattice\n param dk: in 1/fm lattice spacing in momentum space\n \"\"\"\n self.Lbox=2.0*np.pi/dk\n self._kvec = np.array(self._nvec)*dk\n \n def __len__(self):\n \"\"\"\n overloading of the 'len' function\n \"\"\"\n return self.ngrid\n \n \n############\n# useful functions\n\ndef magic_numbers(basis):\n \"\"\"\n param basis: MomSpaceBasis object\n return magic: array of magic numbers\n \"\"\"\n nvecs = basis.nvec()\n vec=np.array(nvecs[0],dtype=int)\n norm = np.dot(vec,vec)\n magic=[]\n for i in range(1,len(nvecs)):\n vec=np.array(nvecs[i],dtype=int)\n norm2 = np.dot(vec,vec)\n if norm2 > norm: \n magic.append(2*i)\n norm=norm2\n return magic\n\n\ndef get_dk(rho,Num):\n \"\"\"\n param rho: desired density\n param Num: magic number of particles\n return dk: grid spacing in momentum space (in 1/fm)\n \"\"\"\n Lbox = (Num/rho)**(1.0/3.0)\n dk = 2.0*np.pi/Lbox\n return dk\n\ndef spbasis_from_MomSpaceBasis(lattice_vecs,st_degen):\n \"\"\"\n converts a lattice to a single particle basis for spin-isospin degeneracy st_degen\n param lattice_vecs: list of lattice vectors for 1st particle\n param st_degen: spin-isospin degeneracy\n return: basis as a list of momenta\n \"\"\"\n if st_degen != 2: # for now only neutron matter\n print(\"Unexpected parameter st_degen\")\n return lattice_vecs\n \n basis=[]\n for vec in lattice_vecs:\n for st in range(st_degen):\n basis.append(np.array(vec,dtype=int))\n \n return basis\n\n\n\n#########################################################\n# Functions for comparisons with infinite free Fermi gas\n \ndef kF_from_density(rho,st_degen=2):\n \"\"\"\n Computes Fermi momentum for given density and spin/isospin degeneracy.\n \n param rho: density in inverse fm cubed\n param st_degen: spin-isospin degeneracy; default 2\n return: Fermi momentum in inverse fm\n \"\"\"\n res = (6.0*(np.pi)**2*rho/st_degen)**(1.0/3.0)\n return res\n\ndef EnergyDensity_FermiGas(kF,st_degen=2):\n \"\"\"\n Computes energy density of free Fermi gas at Fermi momentum and spin/isospin degeneracy\n param kF: Fermi momentum in inverse fm\n param st_degen: spin-isospin degeneracy; default 2\n return: Energy density in MeV/fm**3\n \"\"\"\n pvec = np.array([kF,0.0,0.0])\n erg = (st_degen*kF**3/(10.0*np.pi**2)) * Tkin(pvec)\n return erg\n\n\n########################################################################################\n# Functions for CCD of neutron matter\n# Implementation uses only pp and hh ladders\n# \n########################################################################################\n\n\nfrom numba import jit \n# compile a few functions to gain speed; should probably done in Fortran or C++, \n# and called from Python\n\n@jit(nopython=True)\ndef minnesota_nn(p_out,s1_out,s2_out,p_in,s1_in,s2_in,Lbox):\n \"\"\"\n The Minnesota potential between two neutrons, not yet anti-symmetrized \n param p_out: relative out momentum\n param p_in : relative in momentum\n param s1_out, s2_out: spin projections of out particles 1 and 2\n param s1_in, s2_in : spin projections of in particles 1 and 2\n Lbox : size of momentum box\n return: value of potential in MeV; not anti-symmetrized!\n \"\"\"\n # parameters. VT is not active between two neutrons (no triplet)\n VR = 200.0\n VS = -91.85 # sign typo in Lecture Notes Physics 936, Chap. 8\n kappaR = 1.487\n kappaS = 0.465\n \n qvec=p_out-p_in\n q2=np.dot(qvec,qvec)\n \n s1_i =spin2spinor(s1_in)\n s2_i =spin2spinor(s2_in)\n s1_o =spin2spinor(s1_out)\n s2_o =spin2spinor(s2_out)\n \n spin_part = 0.5 * ( np.dot(s1_i,s1_o)*np.dot(s2_i,s2_o)\n -np.dot(s1_i,s2_o)*np.dot(s2_i,s1_o) )\n \n \n pot = spin_part * ( VR*np.exp(-0.25*q2/kappaR) / (Lbox*np.sqrt(kappaR))**3 \n + VS*np.exp(-0.25*q2/kappaS) / (Lbox*np.sqrt(kappaS))**3 )\n \n pot = pot*(np.sqrt(np.pi))**3 \n\n \n return pot\n\n@jit\ndef spin_of_index(i):\n \"\"\"\n Even indices of the lattive have spin up, odds have spin down\n param i: index of sp_basis\n return: spin as +/- 1\n \"\"\"\n spin = 1-2*np.remainder(i,2)\n return spin\n\n@jit\ndef spin2spinor(s):\n \"\"\"\n Makes a two-component spinor of an integer s\n param s: spin = +/- 1\n return: two-component numpy array [1,0] for up and [0,1] for down\n \"\"\"\n up =np.array([1.0,0.0])\n down=np.array([0.0,1.0])\n if s == 1:\n return up\n else:\n return down\n\n@jit\ndef Tkin(pvec):\n \"\"\"\n Kinetic energy for a momentum vector\n param pvec: 3-component numpy array in inverse fm\n return: kinetic energy of that momentum in MeV\n \"\"\"\n nucleon_mass = 938.92\n hbarc = 197.33\n# More precise numbers for neutron mass and hbar follow.\n# For N=14, this yields E_HF = 10.3337 MeV per nucleon in HF. Benchmarked with Ragnar Stroberg.\n# nucleon_mass = 939.56563\n# hbarc = 197.3269718\n p2 = np.dot(pvec,pvec)\n res = 0.5*hbarc**2*p2/nucleon_mass\n return res\n \n@jit\ndef compute_total_Tkin(Nocc,sp_basis,dk):\n \"\"\"\n Computes total kinetic energy of reference state\n param Nocc, sp_basis, dk: particle number, integer s.p. lattice, delta k \n return: total kinetic energy\n \"\"\"\n erg=0.0\n for i in range(Nocc):\n mom_vec = sp_basis[i]\n vec=np.array(mom_vec)*dk\n erg=erg+Tkin(vec)\n \n return erg\n\n\n\n@jit\ndef Fock(pvec,s,sp_basis,Nocc,dk,Lbox):\n \"\"\"\n Fock matrix of momentum pvec in hh space\n param pvec: 3-component numpy array in inverse fm\n param s: spin as +/- 1 of state\n param_sp_basis, Nocc, dk, Lbox : parameters of s.p. basis and system\n \n return: Fock matrix element = kinetic energy of that momentum in MeV\n \"\"\"\n res = Tkin(pvec)\n \n dum=0.0\n for i in range(Nocc):\n vec=sp_basis[i]*dk\n si=spin_of_index(i)\n p_in = 0.5*(vec-pvec)\n p_out= p_in\n dum = dum + ( minnesota_nn(p_out,s,si, p_in, s,si,Lbox) \n -minnesota_nn(p_out,s,si,-p_in,si, s,Lbox) ) #antisymmetrized Minnesota\n \n res = res+dum\n return res\n\ndef compute_E_HF_simple(Nocc,sp_basis,dk):\n \"\"\"\n Computes HF energy of reference state\n param Nocc, sp_basis, dk: particle number, integer s.p. lattice, delta k \n return: total HF energy\n \"\"\"\n erg=compute_total_Tkin(Nocc,sp_basis,dk)\n\n pot=0.0\n for i in range(Nocc):\n momi=sp_basis[i]*dk\n si = spin_of_index(i)\n for j in range(Nocc):\n momj=sp_basis[j]*dk\n sj = spin_of_index(j)\n p_rel = 0.5*(momi-momj)\n pot = pot + 0.5* ( minnesota_nn(p_rel,si,sj, p_rel,si,sj,Lbox)\n - minnesota_nn(p_rel,si,sj,-p_rel,sj,si,Lbox) )\n \n erg = erg+pot\n return erg\n\n\ndef get_channels(sp_basis,start1,end1,start2,end2,identical,other_channels=None):\n \"\"\"\n Returns channels for coupled cluster based on Minnesota potential\n param sp_Basis: A single-particle basis\n param start1: index to start for particle 1\n param end1: index to end for particle 1\n param start2: index to start for particle 2\n param end2: index to end for particle 2\n param identical: True for hh or pp, False for hp\n param other_channels: list of other channels to compare with\n return: channels, p_rel, t2amp. channels is a list of p12, where p12 is a momentum vector; \n p_rel is a nested list with relative momenta and spins for each channel\n \"\"\"\n channel=[]\n p_rel=[]\n for i, mom_vecs1 in enumerate(sp_basis[start1:end1]):\n #vec1=np.array(mom_vecs1,dtype=int)\n vec1=mom_vecs1\n spin1=spin_of_index(i)\n \n for j, mom_vecs2 in enumerate(sp_basis[start2:end2]):\n if identical and i==j: continue #Fortran cycle\n #vec2=np.array(mom_vecs2,dtype=int)\n vec2=mom_vecs2\n spin2=spin_of_index(j)\n \n p12 = vec1+vec2\n prel= vec1-vec2\n spins=np.array([spin1,spin2],dtype=int)\n ps=[prel,spins]\n\n new=True\n needed=True\n if other_channels is not None: #check whether we need this channel\n needed=False\n for chan_o in other_channels:\n if (chan_o==p12).all(): \n needed=True\n break\n if needed: #check whether this channel exists already\n for ipos, chan in enumerate(channel):\n if (chan==p12).all(): \n new=False\n break\n \n if needed and new: \n channel.append(p12)\n p_rel.append([ps])\n \n if needed and not new:\n p_rel[ipos].append(ps)\n \n return channel, p_rel \n \n\n \ndef setup_T2_amplitudes(sp_basis,NN,st_degen):\n \"\"\"\n returns the t2 amplitudes and t2 channels\n param sp_basis: a sp_basis\n param NN: neutron number\n param st_degen: 2 for the moment, spin-isospin degeneracy\n return: hh_channels, pp_channels, p_rel_hh, p_rel_pp, t2amp\n these are the hh and pp channels of T2, lists of the relative momenta, \n and t2amps as a list of numpy arrays set to zero \n \"\"\"\n num_states = len(sp_basis)\n \n hh_channels, p_rel_hh = get_channels(sp_basis,0,NN,0,NN,True)\n print('hh channels=', len(hh_channels))\n\n pp_channels, p_rel_pp = get_channels(sp_basis,NN,num_states,NN,num_states,True,hh_channels)\n print('pp channels=', len(pp_channels))\n \n if len(pp_channels) != len(hh_channels): print('pp and hh channels do not match')\n \n\n ordered_pp_channel=[]\n ordered_p_rel_pp=[]\n for i, chanhh in enumerate(hh_channels):\n for j, chanpp in enumerate(pp_channels):\n if (chanpp==chanhh).all():\n ordered_pp_channel.append(chanpp)\n ordered_p_rel_pp.append(p_rel_pp[j]) \n break\n \n pp_channels = ordered_pp_channel\n p_rel_pp = ordered_p_rel_pp\n \n # set t2 amplitudes to zero in each channel\n t2amp = fill_pot(Lbox, dk, pp_channels, hh_channels, p_rel_pp, p_rel_hh, True)\n \n return hh_channels, pp_channels, p_rel_hh, p_rel_pp, t2amp\n\ndef fill_pot(Lbox, dk, channels_out, channels_in, p_rel_out, p_rel_in, T2amp=False):\n \"\"\"\n Fills lists of matrices such as Vhhhh, Vpphh, Vpppp, t2_pphh\n param Lbox: Lbox\n param dk: dk\n param channels_out, channels_in: the channels we have\n param p_rel_out, p_rel_in: the list of [prel, [s1,s2]]\n param T2amp=False: Set to True if t2_pphh needs to be computed\n return: The object of desire as a list of numpy matrices. \n Contain matrix elements for potentials, zeros if T2amp=True is requested. \n \"\"\"\n Vpot=[]\n for i, chan_in in enumerate(channels_in):\n dim_in = len(p_rel_in[i])\n dim_out= len(p_rel_out[i])\n Vpot_chan=np.zeros((dim_out,dim_in))\n if not T2amp: \n for ii, ps_i in enumerate(p_rel_in[i]):\n [pii, [s1, s2]] = ps_i\n pii = pii*dk*0.5\n for jj, ps_j in enumerate(p_rel_out[i]):\n if dim_in == dim_out and jj > ii: continue\n [pjj, [ss1, ss2]] = ps_j\n pjj = pjj*dk*0.5\n Vpot_chan[jj,ii] = ( minnesota_nn( pjj,ss1,ss2, pii,s1,s2,Lbox)\n -minnesota_nn(-pjj,ss2,ss1, pii,s1,s2,Lbox) )\n if dim_in == dim_out : Vpot_chan[ii,jj] = Vpot_chan[jj,ii]\n \n Vpot.append(Vpot_chan)\n return Vpot\n\n\ndef init_V(Lbox, dk, hhchannels, ppchannels, p_relhh, p_relpp,zeros=False):\n \"\"\"\n Sets up Vhhhh, Vpphh, and Vpppp. \n \n return: Vhhhh, Vpphh, Vpppp as a lists of numpy arrays\n \"\"\"\n Vhhhh = fill_pot(Lbox, dk, hhchannels, hhchannels, p_relhh, p_relhh,zeros)\n Vpphh = fill_pot(Lbox, dk, ppchannels, hhchannels, p_relpp, p_relhh,zeros)\n Vpppp = fill_pot(Lbox, dk, ppchannels, ppchannels, p_relpp, p_relpp,zeros)\n \n return Vhhhh, Vpphh, Vpppp \n \n@jit\ndef make_diagram(obj1,obj2,fac):\n \"\"\"\n Makes diagrams for pp or hh ladders as matrix-matrix multiplications\n \"\"\"\n hbar_pphh=[]\n dim1=len(obj1)\n for chan in range(dim1):\n mat1 = obj1[chan]\n mat2 = obj2[chan]\n hbar_pphh.append( fac*np.matmul(mat1,mat2) )\n \n return hbar_pphh\n\n\ndef make_diagrams2_3(t2_pphh,fabij):\n hbar_pphh=[]\n for i, t2_mat in enumerate(t2_pphh):\n f_mat = fabij[i]\n hbar_mat = t2_mat*f_mat\n hbar_pphh.append(hbar_mat)\n return hbar_pphh\n\n\ndef compute_hbar(v_pppp,v_pphh,v_hhhh,t2_pphh,fabij):\n diagram1 = v_pphh.copy()\n diagram23 = make_diagrams2_3(t2_pphh, fabij)\n diagram4 = make_diagram(v_pppp,t2_pphh,0.5)\n diagram5 = make_diagram(t2_pphh,v_hhhh,0.5)\n \n hbar_pphh=[]\n for i in range(len(t2_pphh)):\n mat = ( diagram1[i]\n + diagram23[i]\n + diagram4[i]\n + diagram5[i] )\n hbar_pphh.append(mat)\n \n return hbar_pphh\n \ndef get_energy_denominator(hh_channels,p_rel_pp,p_rel_hh,sp_basis,Nocc,dk,Lbox):\n res=[]\n fabij=[]\n for i, Ptot in enumerate(hh_channels):\n dimhh=len(p_rel_hh[i])\n dimpp=len(p_rel_pp[i])\n res_mat = np.zeros((dimpp,dimhh))\n f_mat = np.zeros((dimpp,dimhh))\n for ii, psh_rel in enumerate(p_rel_hh[i]):\n [pij, [si, sj]] = psh_rel \n p_i = (Ptot+pij)//2\n p_i = p_i + np.array([Nmax,Nmax,Nmax],dtype=int)\n p_j = (Ptot-pij)//2\n p_j = p_j + np.array([Nmax,Nmax,Nmax],dtype=int)\n ssi = (1-si)//2\n ssj = (1-sj)//2\n fii = fock_mtx4[p_i[0],p_i[1],p_i[2],ssi]\n fjj = fock_mtx4[p_j[0],p_j[1],p_j[2],ssj]\n for jj, psp_rel in enumerate(p_rel_pp[i]):\n [pab, [sa, sb]] = psp_rel\n p_a = (Ptot+pab)//2\n p_a = p_a + np.array([Nmax,Nmax,Nmax],dtype=int)\n p_b = (Ptot-pab)//2\n p_b = p_b + np.array([Nmax,Nmax,Nmax],dtype=int)\n ssa = (1-sa)//2\n ssb = (1-sb)//2\n faa = fock_mtx4[p_a[0],p_a[1],p_a[2],ssa]\n fbb = fock_mtx4[p_b[0],p_b[1],p_b[2],ssb]\n \n res_mat[jj,ii] = 1.0 / (fii + fjj - faa - fbb) \n f_mat[jj,ii] = faa + fbb - fii - fjj\n res.append(res_mat)\n fabij.append(f_mat)\n return res, fabij\n\n\ndef get_t2_from_mbpt(Vpphh,denom):\n \"\"\"\n param Vpphh: Vpphh\n param denom: energy denominator in pphh format\n return t2: quotient of both, element for element \n \"\"\"\n res = []\n for i, vv in enumerate(Vpphh):\n dd = denom[i]\n res_mat = vv*dd #how simple in python; element by element multiply\n res.append(res_mat)\n return res\n\n\ndef compute_E_CCD(Vpphh,T2pphh):\n erg=0.0\n# erg2=0.0\n for i, t2mat in enumerate(T2pphh):\n vmat = Vpphh[i]\n erg = erg + 0.25*np.sum(vmat*t2mat)\n return erg\n\n\ndef compute_Fock_4(sp_basis,Nocc,dk,Lbox,Nmax):\n fock_mtx4=np.zeros(shape=(2*Nmax+1, 2*Nmax+1, 2*Nmax+1, 2))\n for i, vec in enumerate(sp_basis):\n pvec=vec*dk\n spin=spin_of_index(i)\n si = (1 - spin)//2\n px=vec[0]+Nmax\n py=vec[1]+Nmax\n pz=vec[2]+Nmax\n fock_mtx4[px,py,pz,si] = Fock(pvec,spin,sp_basis,Nocc,dk,Lbox)\n return fock_mtx4\n\n#####################################\n########### Main Program starts here \n\nfrom timeit import default_timer as timer\n# for timing purposes\n\nprogstart=timer()\n\nNmax=1\nkmax=1.0\nmbase = MomSpaceBasis(Nmax,kmax)\nlattice=mbase.nvec()\n\n## set particle number\nNN=14\nst_degen=2 # spin up and down\nprint(\"chosen N =\", NN)\nprint(\"magic numbers\", magic_numbers(mbase))\n\n## set density\nrho=0.08\n\ndk = get_dk(rho,NN)\n\nmbase.update(dk)\nLbox = mbase.Lbox\n\n\n## get single particle basis\n\nsp_basis = spbasis_from_MomSpaceBasis(lattice,st_degen)\nnum_states = len(sp_basis)\n\nprint('number of s.p. states:', num_states)\n\n# print out a few facts of the reference state\ntotal_Tkin = compute_total_Tkin(NN,sp_basis,dk)\nprint('total Tkin per particle =', total_Tkin/NN )\n\nk_fermi = kF_from_density(rho)\n\nprint(\"Fermi momentum =\", k_fermi)\n\nE_gas = EnergyDensity_FermiGas(k_fermi)\n\nprint(\"Energy per neutron of infinite free Fermi gas\", E_gas/rho)\n\nE_HF = compute_E_HF_simple(NN,sp_basis,dk)\nE_HF = E_HF/NN\nprint(\"HF energy per neutron =\", E_HF)\n\n## now we start our business ...\n## get all channels and two-body states within those channels; set T2 to zero\nhh_channels, pp_channels, p_rel_hh, p_rel_pp, t2_pphh = setup_T2_amplitudes(sp_basis,NN,st_degen)\n\n# get some insight in how big this all is\ncount=0\nfor i, channel in enumerate(p_rel_hh):\n dim=len(p_rel_hh[i])\n count=count+dim\n\nprint('hh number of total combinations', count)\n\ncount=0\nfor i, channel in enumerate(p_rel_pp):\n dim=len(p_rel_pp[i])\n count=count+dim\n \nprint('pp number of total combinations', count)\n\n\nprint(\"get v_hhhh, v_pphh, v_pppp\")\nstart = timer()\nv_hhhh, v_pphh, v_pppp = init_V(Lbox, dk, hh_channels, pp_channels, p_rel_hh, p_rel_pp)\nend = timer()\nprint(\"what a hog!\", end-start, 'seconds')\n\n\nprint(\"compute energy denominator\")\nstart = timer()\nfock_mtx4 = compute_Fock_4(sp_basis,NN,dk,Lbox,Nmax)\ndenom_pphh, f_abij = get_energy_denominator(pp_channels,p_rel_pp,p_rel_hh,sp_basis,NN,dk,Lbox)\nend = timer()\nprint(\"that's faster\", end-start, 'seconds')\n\nprint(\"Initialize T2 from MBPT2\")\nt2_pphh = get_t2_from_mbpt(v_pphh,denom_pphh)\n\nerg = compute_E_CCD(v_pphh,t2_pphh)\nprint('MBPT2 correlation energy per neutron =', erg/NN)\n\n\nprint(\"start CCD iterations ...\")\n\nniter=200\nmix=0.99\nerg_old=0.0\neps=1.e-8\nfor iter in range(niter):\n \n start = timer()\n hbar_pphh = compute_hbar(v_pppp,v_pphh,v_hhhh,t2_pphh,f_abij)\n end = timer()\n print(\"time of making Hbar:\", end-start, 'seconds')\n \n t2_new = get_t2_from_mbpt(hbar_pphh,denom_pphh)\n \n for i in range(len(t2_new)):\n t2_new[i] = t2_pphh[i] + t2_new[i]\n \n erg = compute_E_CCD(v_pphh,t2_new)\n \n myeps = abs(erg-erg_old)/abs(erg)\n if myeps < eps: break\n erg_old=erg\n\n print(\"iter=\", iter, \"Correlation energy per neutron=\", erg/NN, \", epsilon=\", myeps)\n \n for i in range(len(t2_pphh)):\n t2_pphh[i] = mix*t2_new[i] + (1.0-mix)*t2_pphh[i]\n \nprint(\"Correlation energy per neutron= \", erg/NN)\n\nprogend=timer()\nprint('total time in seconds', progend-progstart)\n```\n\n\n\n\n\n\n===== =====\n\n\n\n\n\n\n# Benchmarks with the Minnesota potential\n\nFor the benchmarks, let us use a nucleon mass $m=938.92$ MeV, $\\hbar =\n197.33$ MeV fm, and $c=1$. For $N=14$ neutrons at a density $\\rho=0.08\n{\\rm fm}^{-3}$ one finds $T_{\\rm kin}(\\vert\\phi_0\\rangle)/N =\n22.427553$ MeV, and $E_{HF}/N = 10.3498$ MeV. In model spaces with\n$N_{\\rm max}=1$ and $N_{\\rm max}=2$, and using only the first five diagrams\nof [Figure](#fig-ccd) for the CCD calculation, yields the correlation\nenergies per particle of $E_{c}/N =-0.2118$ MeV and $E_{c}/N =-0.6923$\nMeV, respectively.\n\n# From Structure to Reactions\n\nNuclear coupled-cluster theory has also been used to describe aspects\nof nuclear reactions, namely photo reactions and computations of\noptical potentials. In what follows, we want to discuss these approaches.\n\n# Electroweak reactions\n\nLet us assume we probe a nucleus with an electroweak probe (e.g. via\nphoton or $Z$-boson exchange). The corresponding operator\n$\\hat{\\Theta}$ introduces transitions in the nucleus. For photo\nreactions, the relevant operator is the dipole operator\n\n\n
    \n\n$$\n\\begin{equation}\n\\hat{\\Theta} = \\sum_{1=1}^A q_i (\\vec{r}_i - \\vec{R}_{CoM}) .\n\\label{_auto53} \\tag{70}\n\\end{equation}\n$$\n\nHere $q_i$ is the charge of nucleon $i$, and $\\vec{R}_{CoM}$ is the\nposition of the center of mass. The structure function or response\nfunction describing the reaction is\n\n\n
    \n\n$$\n\\begin{equation}\nS(\\omega) \\equiv \\sum_f \\langle\\psi_0|\\hat{\\Theta}^\\dagger|\\psi_f\\rangle\\langle \\psi_f |\\hat{\\Theta}|\\psi_0\\rangle \\delta(E_f-E_0-\\omega).\n\\label{_auto54} \\tag{71}\n\\end{equation}\n$$\n\nHere, the sum is over all final states. The structure function is\ndifficult to compute because the sum is over (infinitely many)\ncontinuum states, and we seek a simpler formulation. The key idea is\nthat the Lorentz integral transform (LIT) of the structure function\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{LIT} \\tag{72}\nL(\\omega_0,\\Gamma) \\equiv {\\Gamma\\over\\pi} \\int d\\omega \\frac{S(\\omega)}{(\\omega-\\omega_0)^2+\\Gamma^2} \n\\end{equation}\n$$\n\n$$\n= {\\Gamma\\over\\pi}\\langle\\psi_0|\\hat{\\Theta}^\\dagger {1\\over H-E_0-\\omega_0+i\\Gamma} {1\\over H-E_0-\\omega_0-i\\Gamma}\\hat{\\Theta}|\\psi_0\\rangle \\nonumber.\n$$\n\nWe note that the LIT $L(\\omega_0,\\Gamma)$ of the structure function is\na ground-state expectation value and thus much easier to compute than\nthe structure function itself. We also note that the LIT is not\ninvertible (mathematically speaking), but making some assumptions\nabout the structure function, and imposing a finite resolution\n$\\Gamma$ alleviates this problem in practical computation.\n\nWe next rewrite the LIT for coupled cluster using the shorthand\n$z\\equiv E_0+\\omega_0+i\\Gamma$ as\n\n\n
    \n\n$$\n\\begin{equation}\nL(\\omega_0,\\Gamma) \\equiv {\\Gamma\\over\\pi} \\langle\\tilde{\\psi}_L(z^*) \\vert \\tilde{\\psi}_R(z)\\rangle , \n\\label{_auto55} \\tag{73}\n\\end{equation}\n$$\n\nwith $\\vert\\tilde{\\psi}_R(z)\\rangle$ and $\\langle\\tilde{\\psi}_L(z^*)\\vert$ fulfilling\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{keyLR} \\tag{74}\n\\left(\\overline{H}-z\\right) \\vert\\tilde{\\psi}_R(z)\\rangle = \\overline{\\Theta} \\vert\\phi_0\\rangle , \n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n\\langle\\tilde{\\psi}_L(z^*)\\vert \\left(\\overline{H}-z^*\\right) = \\langle \\phi_L\\vert .\n\\label{_auto56} \\tag{75}\n\\end{equation}\n$$\n\nHere, $\\langle \\phi_L\\vert$ is the left ground state, i.e. the left\neigenstate of the similarity-transformed Hamiltonian. Note that in the\ncoupled-cluster formulation we have distinguished between left and\nright states (as these are not adjoints of each other), and replaced\nall operators by their similarity transformations. Making the ansatz\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{ansatzR} \\tag{76}\n\\vert\\tilde{\\psi}_R(z)\\rangle = \\hat{R}\\vert\\phi_0\\rangle \n\\end{equation}\n$$\n\n$$\n= \\left( r_0(z) + \\sum_{ia} r_i^a(z) a_a^\\dagger a_i + {1\\over 4} \\sum_{ijab} r_{ij}^{ab}(z) a_a^\\dagger a^\\dagger_b a_j a_i +\\cdots\\right)\\vert\\phi_0\\rangle \\nonumber\n$$\n\nfor the right state, and\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{ansatzL} \\tag{77}\n\\langle\\tilde{\\psi}_L(z^*)\\vert= \\langle\\phi_L\\vert \\hat{L} \n\\end{equation}\n$$\n\n$$\n= \\langle \\phi_L\\vert \\left( l_0(z) + \\sum_{ia} l^i_a(z) a_i^\\dagger a_a + {1\\over 4} \\sum_{ijab} l^{ij}_{ab}(z) a_i^\\dagger a^\\dagger_j a_b a_a +\\cdots\\right) \\nonumber\n$$\n\nfor the left state, make the Eq. ([74](#keyLR)) linear systems of\nequations that can be solved for the parameters $r_0, r_i^a,\nr_{ij}^{ab}$ and $l_0, l_a^i, l_{ab}^{ij}$ for each value of $z$. The LIT then becomes\n\n\n
    \n\n$$\n\\begin{equation}\nL(z) = l_0(z) r_0(z) + \\sum_{ia} l_a^i(z) r_i^a(z) +{1\\over 4} l_{ab}^{ij}(z) r_{ij}^{ab}(z) \n\\label{_auto57} \\tag{78}\n\\end{equation}\n$$\n\nin the CCSD approximation. For details, please see [\"Giant and pigmy dipole resonances in 4He, 16,22O, and 40Ca from chiral nucleon-nucleon interactions,\" S. Bacca, N. Barnea, G. Hagen, M. Miorelli, G. Orlandini, and T. Papenbrock,\nPhys. Rev. C 90, 064619 (2014)].\n\n\n# Computing optical potentials from microscopic input\n\nThe single-particle Green's function\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{Green} \\tag{79}\nG(\\alpha, \\beta, E) \\equiv \\langle\\psi_0\\vert a_\\alpha {1\\over E-(H-E_0)+i\\eta}a^\\dagger_\\beta\\vert\\psi_0\\rangle\n+ \\langle\\psi_0\\vert a_\\beta^\\dagger {1\\over E-(H-E_0)-i\\eta}a_\\alpha\\vert\\psi_0\\rangle \n\\end{equation}\n$$\n\ndescribes the propagation of particles and holes in the nucleus with a\nground state $\\vert\\psi_0\\rangle$ and energy $E_0$. The Green's function fulfills the Dyson equation\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{Dyson} \\tag{80}\nG = G^{(0)} +G^{(0)} \\Sigma^* G , \n\\end{equation}\n$$\n\nwhere $\\Sigma^*$ is the self energy and $G^{(0)}$ is the Hartree-Fock Green's function\n\n\n
    \n\n$$\n\\begin{equation}\n\\label{HFGreen} \\tag{81}\nG^{(0)}(\\alpha, \\beta, E) \\equiv \\langle\\phi_0\\vert a_\\alpha {1\\over E-(H_{HF}-E_{HF})+i\\eta}a^\\dagger_\\beta\\vert\\phi_0\\rangle\n+ \\langle\\phi_0\\vert a_\\beta^\\dagger {1\\over E-(H_{HF}-E_{HF})-i\\eta}a_\\alpha\\vert\\phi_0\\rangle \n\\end{equation}\n$$\n\n$$\n= \\delta_\\alpha^\\beta \\left({\\Theta(\\alpha-F) \\over E-\\varepsilon_\\alpha +i\\eta} + {\\Theta(F-\\alpha) \\over E-\\varepsilon_\\alpha -i\\eta} \\right) \\nonumber.\n$$\n\nHere, $\\Theta()$ denotes the unit step function and $F$ labels the\nindex of the Fermi surface. The key point is now that the optical\npotential $\\Sigma'$ (which describes the reaction of a single nucleon with the\nnucleus) is related to the self energy and the Hartree-Fock potential\n$U_{HF}$ by [F. Capuzzi and C. Mahaux, \"Projection operator approach to the\nself-energy,\" Ann. Phys. (NY) 245, 147 (1996)]\n\n\n
    \n\n$$\n\\begin{equation}\n\\Sigma' \\equiv \\Sigma^* +U_{HF} .\n\\label{_auto58} \\tag{82}\n\\end{equation}\n$$\n\nThe idea is thus as follows. Starting from a Hartree-Fock calculations\nenables us to compute the Hartree-Fock potential $U_{HF}$ and the\nHartree-Fock Green's function ([81](#HFGreen)). Computing the Green's\nfunction ([79](#Green)) in coupled clusters, and inverting the Dyson\nequation ([80](#Dyson)) using\n\n\n
    \n\n$$\n\\begin{equation}\n\\Sigma^* = \\left(G^{(0)}\\right)^{-1} - G^{-1}\n\\label{_auto59} \\tag{83}\n\\end{equation}\n$$\n\nthus allows us to compute the optical potential. We note that the\nGreen's function ([79](#Green)) resembles in its structure the LIT\n([72](#LIT)), and we will indeed use a similar approach to compute this\nobject with the coupled-cluster method. Indeed, we compute\n\n\n
    \n\n$$\n\\begin{equation}\n\\left(E-(\\overline{H}-E_0)+ i\\eta\\right) \\vert\\tilde{\\psi}_R\\rangle = \\overline{a^\\dagger_\\beta} \\vert\\phi_0\\rangle , \n\\label{_auto60} \\tag{84}\n\\end{equation}\n$$\n\n\n
    \n\n$$\n\\begin{equation} \n\\langle\\tilde{\\psi}_L\\vert \\left(E-(E_0-\\overline{H})-i\\eta\\right) = \\langle \\phi_L\\vert \\overline{a_\\beta^\\dagger} , \n\\label{_auto61} \\tag{85}\n\\end{equation}\n$$\n\nby making the ansatz ([76](#ansatzR)) and ([77](#ansatzL)) for the right\nand left states, respectively, solve the resulting linear systems, and\nthen compute $G(\\alpha,\\beta,E) = \\langle\\phi_L\\vert\n\\overline{a_\\alpha} \\vert\\tilde{\\psi}_R\\rangle + \\langle\n\\tilde{\\psi}_L\\vert \\overline{a_\\alpha}\\vert\\phi_0\\rangle$.\n\nWe note that a high quality optical potential can only be obtained if\nthe structure of the nucleus is computed with a good accuracy, i.e. in\ngood agreement to data on binding and separation energies, and charge\nand matter radii. For details, please see [\"Optical potential from\nfirst principles,\" J. Rotureau, P. Danielewicz, G. Hagen, F. Nunes,\nand T. Papenbrock, Phys. Rev. C 95, 024315 (2017); arXiv:1611.04554].\nWe also note that this procedure must be repeated for any nucleus of\ninterest, because the optical potential depends on the nucleus under\nconsideration. Once the optical potential $\\Sigma'$ is computed, the\nsingle-particle Schroedinger equation\n\n\n
    \n\n$$\n\\begin{equation}\n\\left( -{\\hbar^2\\Delta\\over 2m} +\\Sigma'\\right)\\xi = E\\xi \n\\label{_auto62} \\tag{86}\n\\end{equation}\n$$\n\ndescribes the interaction of a single nucleon (and wave function\n$\\xi$) with the nucleus.\n\nFinally, we note that this approach does not depend on solving the\nnucleus $A$ with the coupled-cluster method. Alternatives, such as the\nIMSRG or Self-Consistent Green's Function methods could also be used.\n", "meta": {"hexsha": "712d7e363f015da7de25d307a8317ef9501c497f", "size": 153506, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "doc/pub/CCM/ipynb/CCM.ipynb", "max_stars_repo_name": "NuclearTalent/ManyBody2018", "max_stars_repo_head_hexsha": "2339ed834777fa10f6156344f17494b9a7c0bf91", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2018-07-17T01:09:17.000Z", "max_stars_repo_stars_event_max_datetime": "2021-04-08T02:34:02.000Z", "max_issues_repo_path": "doc/pub/CCM/ipynb/CCM.ipynb", "max_issues_repo_name": "NuclearTalent/ManyBody2018", "max_issues_repo_head_hexsha": "2339ed834777fa10f6156344f17494b9a7c0bf91", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/pub/CCM/ipynb/CCM.ipynb", "max_forks_repo_name": "NuclearTalent/ManyBody2018", "max_forks_repo_head_hexsha": "2339ed834777fa10f6156344f17494b9a7c0bf91", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 8, "max_forks_repo_forks_event_min_datetime": "2018-07-16T06:31:54.000Z", "max_forks_repo_forks_event_max_datetime": "2021-05-01T07:53:38.000Z", "avg_line_length": 35.6990697674, "max_line_length": 577, "alphanum_fraction": 0.5270738603, "converted": true, "num_tokens": 33462, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417487156366, "lm_q2_score": 0.6619228691808012, "lm_q1q2_score": 0.43582159221735517}} {"text": "```python\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nfrom collections import defaultdict\nfrom scipy.optimize import minimize\n\nimport networkx as nx\nfrom networkx.generators.random_graphs import erdos_renyi_graph\n\nfrom IPython.display import Image\n```\n\n\n```python\nfrom qiskit import QuantumCircuit, execute, Aer\nfrom qiskit.tools.visualization import circuit_drawer, plot_histogram\n```\n\n\n```python\nfrom quantuminspire.credentials import get_authentication\nfrom quantuminspire.api import QuantumInspireAPI\nfrom quantuminspire.qiskit import QI\n\nQI_URL = 'https://api.quantum-inspire.com/'\n```\n\nIn this notebook you will apply what you have just learned about cqasm and Quantum Inspire. We will consider a simple quantum algorithm, the quantum approximate optimization algorithm (QAOA), for which you will code the circuit in cqasm and send some jobs to real quantum hardware on the Quantum Inspire platform.\n\n## 1. Recap: QAOA and MAXCUT\n\n### Introduction to the Quantum Approximate Optimization Algorithm\n\n$$\\newcommand{\\ket}[1]{\\left|{#1}\\right\\rangle}$$\n$$\\newcommand{\\bra}[1]{\\left\\langle{#1}\\right|}$$\n$$\\newcommand{\\braket}[2]{\\left\\langle{#1}\\middle|{#2}\\right\\rangle}$$\n\nConsider some combinatorial optimization problem with objective function $C:x\\rightarrow \\mathbb{R}$ acting on $n$-bit strings $x\\in \\{0,1\\}^n$, domain $\\mathcal{D} \\subseteq \\{0,1\\}^n$, and objective\n\n\\begin{align}\n \\max_{x \\in \\mathcal{D}} C(x).\n\\end{align}\n\nIn maximization, an approximate optimization algorithm aims to find a string $x'$ that achieves a desired approximation ratio $\\alpha$, i.e.\n\n\\begin{equation}\n \\frac{C(x')}{C^*}\\geq \\alpha,\n\\end{equation}\n\nwhere $C^* = \\max_{x \\in \\mathcal{D}} C(x)$.\nIn QAOA, such combinatorial optimization problems are encoded into a cost Hamiltonian $H_C$, a mixing Hamiltonian $H_M$ and some initial quantum state $\\ket{\\psi_0}$. The cost Hamiltonian is diagonal in the computational basis by design, and represents $C$ if its eigenvalues satisfy\n\n\\begin{align}\n H_C \\ket{x} = C(x) \\ket{x} \\text{ for all } x \\in \\{0,1\\}^n.\n\\end{align}\n\nThe mixing Hamiltonian $H_M$ depends on $\\mathcal{D}$ and its structure, and is in the unconstrained case (i.e. when $\\mathcal{D}=\\{0,1\\}^n$) usually taken to be the transverse field Hamiltonian $H_M = \\sum_{j} X_j$. Constraints (i.e. when $\\mathcal{D}\\subset \\{0,1\\}^n$) can be incorporated directly into the mixing Hamiltonian or are added as a penalty function in the cost Hamiltonian. The initial quantum state $\\ket{\\psi_0}$ is usually taken as the uniform superposition over all possible states in the domain. $\\text{QAOA}_p$, parametrized in $\\gamma=(\\gamma_0,\\gamma_1,\\dots,\\gamma_{p-1}),\\beta=(\\beta_0,\\beta_1,\\dots,\\beta_{p-1})$, refers to a level-$p$ QAOA circuit that applies $p$ steps of alternating time evolutions of the cost and mixing Hamiltonians on the initial state. At step $k$, the unitaries of the time evolutions are given by\n\n\\begin{align}\n U_C(\\gamma_k) = e^{-i \\gamma_k H_C }, \\label{eq:UC} \\\\\n U_M(\\beta_k) = e^{-i \\beta_k H_M }. \\label{eq:UM}\n\\end{align}\n\nSo the final state $\\ket{\\gamma,\\beta}$ of $\\text{QAOA}_p$ is given by \n\n\\begin{align}\n \\ket{\\gamma,\\beta} = \\prod_{k=0}^{p-1} U_M(\\beta_k) U_C(\\gamma_k) \\ket{\\psi_0}.\n\\end{align}\n\nThe expectation value $ F_p(\\gamma,\\beta)$ of the cost Hamiltonian for state $\\ket{\\gamma,\\beta}$ is given by\n\n\\begin{align}\n F_p(\\gamma,\\beta) = \n \\bra{\\gamma,\\beta}H_C\\ket{\\gamma,\\beta},\n \\label{eq:Fp}\n\\end{align}\n\nand can be statistically estimated by taking samples of $\\ket{\\gamma,\\beta}$. The achieved approximation ratio (in expectation) of $\\text{QAOA}_p$ is then\n\n\\begin{equation}\n \\alpha = \\frac{F_p(\\gamma,\\beta)}{C^*}.\n\\end{equation}\n\nThe parameter combinations of $\\gamma,\\beta$ are usually found through a classical optimization procedure that uses $F_p(\\gamma,\\beta)$ as a black-box function to be maximized.\n\n### Example application: MAXCUT\n\nMaxCut is an NP-hard optimisation problem that looks for an optimal 'cut' for a graph $G(V,E)$, in the sense that the cut generates a subset of nodes $S \\subset V$ that shares the largest amount of edges with its complement $ V\\setminus S$. In slightly modified form (omitting the constant), it has the following objective function\n\n\\begin{align}\n\\max_{s} \\frac{1}{2} \\sum_{\n\\langle i,j \\rangle \\in E} 1-s_i s_j,\n\\end{align}\n\nwhere the $s_i\\in\\{-1,1\\}$ are the variables and $i,j$ are the edge indices. This function can be easily converted into an Ising cost Hamiltonian, which takes the form\n\n\\begin{align}\nH_C = \\frac{1}{2}\\sum_{\\langle i,j\\rangle \\in E} I-Z_i Z_j.\n\\end{align}\n\nWe use the standard mixing Hamiltonian that sums over all nodes:\n\n\\begin{align}\nH_M = \\sum_{v \\in V} X_v.\n\\end{align}\n\nAs the initial state $\\ket{\\Psi_0}$ we take the uniform superposition, given by\n\n\\begin{align}\n\\ket{\\psi_0} = \\frac{1}{\\sqrt{2^{|V|}}}\\sum_{x=0}^{2^{|V|}-1} \\ket{x} \n\\end{align}\n\n\nThe goal of this workshop is to guide you through an implemented code that simulates a small quantum computer running the QAOA algorithm applied to the MAXCUT problem. We will use qiskit as well as cqasm as SDK's. For the sake of run time, you will always run the classical optimization part using the qiskit simulator: it would take too long for our purposes to do the actual function evualtions in the classical optimization step on the hardware.\n\n## 2. Some useful functions and intializations\n\nWe first define some useful functions to be used later throughout the code.\n\n\n```python\n# Just some function to draw graphs\ndef draw_cc_graph(G,node_color='b',fig_size=4):\n plt.figure(figsize=(fig_size,fig_size))\n nx.draw(G, G.pos, \n node_color= node_color,\n with_labels=True,\n node_size=1000,font_size=14)\n plt.show()\n```\n\n\n```python\n# Define the objective function\ndef maxcut_obj(x,G):\n cut = 0\n for i, j in G.edges():\n if x[i] != x[j]:\n # the edge is cut, negative value in agreement with the optimizer (which is a minimizer)\n cut -= 1\n return cut\n\n# Brute force method\ndef brute_force(G):\n n = len(G.nodes)\n costs = np.zeros(0)\n costs=[]\n for i in range(2**n):\n calc_costs = -1*maxcut_obj(bin(i)[2:].zfill(n),G)\n costs.append(calc_costs)\n max_costs_bf = max(costs)\n index_max = costs.index(max(costs))\n max_sol_bf = bin(index_max)[2:].zfill(n)\n return max_costs_bf, max_sol_bf,costs\n\n```\n\n\n```python\n# Generating the distribution resulting from random guessing the solution\ndef random_guessing_dist(G):\n dictio= dict()\n n = len(G.nodes())\n for i in range(2**n):\n key = bin(i)[2:].zfill(n)\n dictio[key] = maxcut_obj(bin(i)[2:].zfill(n),G)\n RG_energies_dist = defaultdict(int)\n for x in dictio:\n RG_energies_dist[maxcut_obj(x,G)] += 1\n return RG_energies_dist\n\n# Visualize multiple distributions\ndef plot_E_distributions(E_dists,p,labels):\n plt.figure()\n x_min = 1000\n x_max = - 1000\n width = 0.25/len(E_dists)\n for index,E_dist in enumerate(E_dists):\n pos = width*index-width*len(E_dists)/4 \n label = labels[index]\n X_list,Y_list = zip(*E_dist.items())\n X = -np.asarray(X_list)\n Y = np.asarray(Y_list)\n plt.bar(X + pos, Y/np.sum(Y), color = 'C'+str(index), width = width,label= label+', $p=$'+str(p))\n if np.min(X)x_max:\n x_max = np.max(X)\n plt.xticks(np.arange(x_min,x_max+1))\n plt.legend()\n plt.xlabel('Objective function value')\n plt.ylabel('Probability')\n plt.show()\n\n\n# Determinet the expected objective function value from the random guessing distribution\ndef energy_random_guessing(RG_energies_dist):\n energy_random_guessing = 0\n total_count = 0\n for energy in RG_energies_dist.keys():\n count = RG_energies_dist[energy]\n energy_random_guessing += energy*count\n total_count += count\n energy_random_guessing = energy_random_guessing/total_count\n return energy_random_guessing\n```\n\n### Test instances\n\n\n```python\nw2 = np.matrix([\n [0, 1],\n [1, 0]])\nG2 = nx.from_numpy_matrix(w2)\npositions = nx.circular_layout(G2)\nG2.pos=positions\nprint('G2:')\ndraw_cc_graph(G2)\n\n\nw3 = np.matrix([\n [0, 1, 1],\n [1, 0, 1],\n [1, 1, 0]])\nG3 = nx.from_numpy_matrix(w3)\npositions = nx.circular_layout(G3)\nG3.pos=positions\nprint('G3:')\ndraw_cc_graph(G3)\n```\n\n## 3. Circuit generators\n\nWe provide you with an example written in qiskit. You have to write the one for cqasm yourself.\n\n### Qiskit generators\n\n\n```python\nclass Qiskit(object): \n # Cost operator:\n def get_cost_operator_circuit(G, gamma):\n N = G.number_of_nodes()\n qc = QuantumCircuit(N,N)\n for i, j in G.edges():\n qc.cx(i,j)\n qc.rz(2*gamma, j)\n qc.cx(i,j)\n return qc\n\n # Mixing operator\n def get_mixer_operator_circuit(G, beta):\n N = G.number_of_nodes()\n qc = QuantumCircuit(N,N)\n for n in G.nodes():\n qc.rx(2*beta, n)\n return qc\n \n # Build the circuit:\n def get_qaoa_circuit(G, beta, gamma):\n assert(len(beta) == len(gamma))\n p = len(beta) # number of unitary operations\n N = G.number_of_nodes()\n qc = QuantumCircuit(N,N)\n # first step: apply Hadamards to obtain uniform superposition\n qc.h(range(N))\n # second step: apply p alternating operators\n for i in range(p):\n qc.compose(Qiskit.get_cost_operator_circuit(G,gamma[i]),inplace=True)\n qc.compose(Qiskit.get_mixer_operator_circuit(G,beta[i]),inplace=True)\n # final step: measure the result\n qc.barrier(range(N))\n qc.measure(range(N), range(N))\n return qc\n\n\n```\n\n\n```python\n# Show the circuit for the G3 (triangle) graph\np = 1\nbeta = np.random.rand(p)*2*np.pi\ngamma = np.random.rand(p)*2*np.pi\nqc = Qiskit.get_qaoa_circuit(G3,beta, gamma)\nqc.draw(output='mpl')\n\n```\n\n### cqasm generators\n\nNow it is up to you to apply what we have learned about cqasm to write the script for the cost and mixing operators:\n\n\n```python\nclass Cqasm(object):\n \n ### We give them this part\n def get_qasm_header(N_qubits):\n \"\"\"\n Create cQASM header for `N_qubits` qubits and prepare all in |0>-state.\n \"\"\"\n header = f\"\"\"\nversion 1.0\nqubits {N_qubits}\nprep_z q[0:{N_qubits-1}]\n\"\"\"\n return header\n \n def get_cost_operator(graph, gamma, p=1):\n \"\"\"\n Create cost operator for given angle `gamma`.\n \"\"\"\n layer_list = graph.number_of_edges()*[None]\n for n, (i,j) in enumerate(graph.edges()):\n layer_list[n] = '\\n'.join([f\"CNOT q[{i}], q[{j}]\", \n f\"Rz q[{j}], {2*gamma}\", \n f\"CNOT q[{i}], q[{j}]\"])\n\n return f\".U_gamma_{p}\\n\" + '\\n'.join(layer_list) + '\\n'\n\n def get_mixing_operator(graph, beta, p=1):\n \"\"\"\n Create mixing operator for given angle `beta`. \n Use parallel application of single qubit gates.\n \"\"\"\n U_beta = \"{\" + ' | '.join([f\"Rx q[{i}], {2*beta}\" for i in graph.nodes()]) + \"}\"\n return f\".U_beta_{p}\\n\" + U_beta + '\\n'\n\n def get_qaoa_circuit(graph, beta, gamma):\n \"\"\"\n Create full QAOA circuit for given `graph` and angles `beta` and `gamma`.\n \"\"\"\n assert len(beta) == len(gamma)\n p = len(beta) # number of layers\n N_qubits = graph.number_of_nodes()\n circuit_str = Cqasm.get_qasm_header(5) #N_qubits)\n\n # first step: apply Hadamards to obtain uniform superposition\n circuit_str += \"{\" + ' | '.join([f\"H q[{i}]\" for i in graph.nodes()]) + \"}\\n\\n\"\n # second step: apply p alternating operators\n circuit_str += '\\n'.join([Cqasm.get_cost_operator(graph, gamma[i], i+1) \n + Cqasm.get_mixing_operator(graph, beta[i], i+1) for i in range(p)])\n # final step: measure the result\n circuit_str += \"\\n\"\n circuit_str += \"measure_all\"\n\n return circuit_str\n```\n\n## 4. Hybrid-quantum classical optimization\n\nSince QAOA is usually adopted as a hybrid quantum-classical algorithm, we need to construct an outer loop which optimizes the estimated $\\bra{\\gamma,\\beta}H\\ket{\\gamma,\\beta}$.\n\n\n```python\n# Black-box function that describes the energy output of the QAOA quantum circuit\ndef get_black_box_objective(G, p, SDK = 'qiskit', backend = None, shots=2**10):\n if SDK == 'cqasm':\n if not backend:\n backend = 'QX single-node simulator'\n backend_type = qi.get_backend_type_by_name(backend)\n def f(theta):\n # first half is betas, second half is gammas\n beta = theta[:p]\n gamma = theta[p:]\n qc = Cqasm.get_qaoa_circuit(G, beta, gamma)\n result = qi.execute_qasm(qc, backend_type=backend_type, number_of_shots=shots)\n counts = result['histogram'] \n # return the energy\n return compute_maxcut_energy(counts, G)\n\n if SDK == 'qiskit':\n if not backend:\n backend = 'qasm_simulator'\n backend = Aer.get_backend(backend)\n def f(theta):\n # first half is betas, second half is gammas\n beta = theta[:p]\n gamma = theta[p:]\n qc = Qiskit.get_qaoa_circuit(G,beta, gamma)\n counts = execute(qc, backend,shots=shots).result().get_counts()\n # return the energy\n return compute_maxcut_energy(counts, G)\n else:\n return 'error: SDK not found'\n return f\n\n# Estimate the expectation value based on the circuit output\ndef compute_maxcut_energy(counts, G):\n energy = 0\n total_counts = 0\n for meas, meas_count in counts.items():\n obj_for_meas = maxcut_obj(meas, G)\n energy += obj_for_meas * meas_count\n total_counts += meas_count\n return energy / total_counts\n```\n\n## 5. A simple instance on the quantum inspire platform: 2-qubit case\n\nLet us first consider the most simple MAXCUT instance. We have just two nodes, and an optimal cut with objective value 1 would be to place both nodes in its own set.\n\n\n```python\nG=G2\nmax_costs_bf, max_sol_bf,costs = brute_force(G)\nprint(\"brute force method best cut: \",max_costs_bf)\nprint(\"best string brute force method:\",max_sol_bf)\n\ncolors = ['red' if x == '0' else 'b' for x in max_sol_bf]\ndraw_cc_graph(G,node_color = colors)\n```\n\nUsing qiskit, the circuit would look the following:\n\n\n```python\n# Test and show circuit for some beta,gamma\np = 1\nbeta = np.random.rand(p)*np.pi\ngamma = np.random.rand(p)*2*np.pi\nqc = Qiskit.get_qaoa_circuit(G,beta, gamma)\nqc.draw(output='mpl')\n```\n\nNow let's run our hybrid-quantum algorithm simulation using qiskit:\n\n\n```python\n# Parameters that can be changed:\np = 1\nlb = np.zeros(2*p)\nub = np.hstack([np.full(p, np.pi), np.full(p, 2*np.pi)])\ninit_point = np.random.uniform(lb, ub, 2*p)\nshots = 2**10\noptimiser = 'COBYLA'\nmax_iter = 100\n\n# Training of the parameters beta and gamma\nobj = get_black_box_objective(G,p,SDK='qiskit',shots=shots)\n# Lower and upper bounds: beta \\in {0, pi}, gamma \\in {0, 2*pi}\nbounds = [lb,ub]\n\n# Maximum number of iterations: 100\nres = minimize(obj, init_point, method=optimiser, bounds = bounds, options={'maxiter':max_iter, 'disp': True})\nprint(res)\n```\n\n /home/redwombat/miniconda3/envs/qi-py38/lib/python3.8/site-packages/scipy/optimize/_minimize.py:544: RuntimeWarning: Method COBYLA cannot handle bounds.\n warn('Method %s cannot handle bounds.' % method,\n\n\n fun: -1.0\n maxcv: 0.0\n message: 'Optimization terminated successfully.'\n nfev: 26\n status: 1\n success: True\n x: array([-0.39933054, 0.79140574])\n\n\n\n```python\n#Determine the approximation ratio:\nprint('Approximation ratio is',-res['fun']/max_costs_bf)\n```\n\n Approximation ratio is 1.0\n\n\n\n```python\n# Extract the optimal values for beta and gamma and run a new circuit with these parameters\noptimal_theta = res['x']\nqc = Qiskit.get_qaoa_circuit(G, optimal_theta[:p], optimal_theta[p:])\ncounts = execute(qc,backend = Aer.get_backend('qasm_simulator'),shots=shots).result().get_counts()\n```\n\n\n```python\nplt.bar(counts.keys(), counts.values())\nplt.xlabel('String')\nplt.ylabel('Count')\nplt.show()\n```\n\n\n```python\nRG_dist = random_guessing_dist(G)\n```\n\n\n```python\n# Measurement distribution \nE_dist = defaultdict(int)\nfor k, v in counts.items():\n E_dist[maxcut_obj(k,G)] += v\n\nplot_E_distributions([E_dist,RG_dist],p,['Qiskit','random guessing'])\n\nE_random_guessing = energy_random_guessing(RG_dist)\nprint('Energy from random guessing is', E_random_guessing)\n```\n\n\n```python\nX_list,Y_list = zip(*E_dist.items())\nX = -np.asarray(X_list)\nY = np.asarray(Y_list)\nprint('Probability of measuring the optimal solution is',Y[np.argmax(X)]/shots)\n```\n\n Probability of measuring the optimal solution is 1.0\n\n\nNow that we have obtained some good values for $\\beta$ and $\\gamma$ through classical simulation, let's see what Starmon-5 would give us.\n\nThe figure below shows the topology of Starmon-5. Since q0 is not connected to q1, we have to relabel the nodes. Networkx as such an option, by using 'nx.relabel_nodes(G,{1:2}' we can relabel node 1 as node 2. Since q0 is connected to q2, this does allow us to run our cqasm code on Starmon-5. For qiskit, this step is irrelevant as we have all-to-all connectivity in the simulation.\n\n\n```python\nImage(filename='Starmon5.png')\n```\n\n\n```python\nqc_Cqasm = Cqasm.get_qaoa_circuit(nx.relabel_nodes(G, {1: 2}), optimal_theta[:p], optimal_theta[p:])\nprint(qc_Cqasm)\n```\n\n \n version 1.0\n qubits 5\n prep_z q[0:4]\n {H q[0] | H q[2]}\n \n .U_gamma_1\n CNOT q[0], q[2]\n Rz q[2], 1.5828114789648722\n CNOT q[0], q[2]\n .U_beta_1\n {Rx q[0], -0.798661078152829 | Rx q[2], -0.798661078152829}\n \n measure_all\n\n\nNow we run the Cqasm-circuit on the Starmon-5 Hardware.\n\n\n```python\nauthentication = get_authentication()\nQI.set_authentication(authentication, QI_URL)\n```\n\n\n```python\nqiapi = QuantumInspireAPI(QI_URL, authentication)\nresult = qiapi.execute_qasm(qc_Cqasm, backend_type=qiapi.get_backend_type('Starmon-5'), number_of_shots=2**10)\ncounts_QI = result['histogram']\n```\n\nInspecting 'counts_QI', we see that it returns the integer corresponding to the bit string result of the measurement \n\n\n```python\ncounts_QI\n```\n\n\n\n\n OrderedDict([('0', 0.056640625),\n ('1', 0.4208984375),\n ('2', 0.005859375),\n ('3', 0.021484375),\n ('4', 0.3994140625),\n ('5', 0.0283203125),\n ('6', 0.0234375),\n ('7', 0.001953125),\n ('8', 0.001953125),\n ('9', 0.0068359375),\n ('12', 0.015625),\n ('13', 0.001953125),\n ('17', 0.0009765625),\n ('20', 0.0078125),\n ('22', 0.001953125),\n ('28', 0.0048828125)])\n\n\n\nNote that we measure more than just the two relevant qubits, since we had the 'measure all' command in the the cqasm code. The distribution over the strings looks the following:\n\n\n```python\ncounts_bin = {}\nfor k,v in counts_QI.items():\n counts_bin[f'{int(k):05b}'] = v\nprint(counts_bin)\nplt.bar(counts_bin.keys(), counts_bin.values())\nplt.xlabel('State')\nplt.ylabel('Measurement probability')\nplt.xticks(rotation='vertical')\nplt.show()\n```\n\nLet's create another counts dictionary with only the relevant qubits, which are q0 and q2:\n\n\n```python\ncounts_bin_red = defaultdict(float)\nfor string in counts_bin:\n q0 = string[-1]\n q1 = string[-3]\n counts_bin_red[(q0+q1)]+=counts_bin[string]\n```\n\n\n```python\ncounts_bin_red\n```\n\n\n\n\n defaultdict(float,\n {'00': 0.064453125,\n '10': 0.4501953125,\n '01': 0.453125,\n '11': 0.0322265625})\n\n\n\nWe now plot all distributions (qiskit, Starmon-5, and random guessing) in a single plot.\n\n\n```python\n#Determine the approximation ratio:\nprint('Approximation ratio on the hardware is',-compute_maxcut_energy(counts_bin_red,G)/max_costs_bf)\n\n# Random guessing distribution\nRG_dist = random_guessing_dist(G)\n\n# Measurement distribution \nE_dist_S5 = defaultdict(int)\nfor k, v in counts_bin_red.items():\n E_dist_S5[maxcut_obj(k,G)] += v\n \nplot_E_distributions([E_dist,E_dist_S5,RG_dist],p,['Qiskit','Starmon-5','random guessing'])\n\n\nX_list,Y_list = zip(*E_dist_S5.items())\nX = -np.asarray(X_list)\nY = np.asarray(Y_list)\nprint('Probability of measuring the optimal solution is',Y[np.argmax(X)])\n\n\nE_random_guessing = energy_random_guessing(RG_dist)\nprint('Expected approximation ratio random guessing is', -E_random_guessing/max_costs_bf)\n```\n\n## 6. Compilation issues: the triangle graph\n\n\nFor the graph with just two nodes we already had some minor compilation issues, but this was easily fixed by relabeling the nodes. We will now consider an example for which relabeling is simply not good enough to get it mapped to the Starmon-5 toplogy.\n\n\n```python\nG=G3\nmax_costs_bf, max_sol_bf,costs = brute_force(G)\nprint(\"brute force method best cut: \",max_costs_bf)\nprint(\"best string brute force method:\",max_sol_bf)\n\ncolors = ['red' if x == '0' else 'b' for x in max_sol_bf]\ndraw_cc_graph(G,node_color = colors)\n```\n\nDue to the topology of Starmon-5 this graph cannot be executed without any SWAPS. Therefore, we ask you to write a new circuit generator that uses SWAPS in order to make the algorithm work with the Starmon-5 topology. Let's also swap back to the original graph configuration, so that we can in the end measure only the qubits that correspond to a node in the graph (this is already written for you)\n\n\n```python\ndef QAOA_triangle_circuit_cqasm(graph, beta, gamma):\n circuit_str = Cqasm.get_qasm_header(5)\n circuit_str += \"{\" + ' | '.join([f\"H q[{i}]\" for i in graph.nodes()]) + \"}\\n\\n\"\n \n def get_triangle_cost_operator(graph, gamma, p):\n layer_list = graph.number_of_edges() * [None]\n for n, edge in enumerate(graph.edges()):\n if 0 in edge and 1 in edge:\n layer_list[n] = '\\n'.join([f\"SWAP q[{edge[0]}], q[2]\",\n f\"CNOT q[2], q[{edge[1]}]\", \n f\"Rz q[{edge[1]}], {2*gamma}\", \n f\"CNOT q[2], q[{edge[1]}]\",\n f\"SWAP q[{edge[0]}], q[2]\" ])\n else:\n layer_list[n] = '\\n'.join([f\"CNOT q[{edge[0]}], q[{edge[1]}]\", \n f\"Rz q[{edge[1]}], {2*gamma}\", \n f\"CNOT q[{edge[0]}], q[{edge[1]}]\"])\n\n return f\".U_gamma_{p}\\n\" + '\\n'.join(layer_list) + '\\n'\n \n circuit_str += '\\n'.join([get_triangle_cost_operator(graph, gamma[i], i+1) \n + Cqasm.get_mixing_operator(graph, beta[i], i+1) for i in range(p)]) \n circuit_str += \"\\n\"\n circuit_str += \"{\" + ' | '.join([f\"measure q[{i}]\" for i in graph.nodes()]) + \"}\\n\" \n return circuit_str \n```\n\nWe now run the same procedure as before to obtain good parameter values\n\n\n```python\n# Parameters that can be changed:\np = 1\nlb = np.zeros(2*p)\nub = np.hstack([np.full(p, np.pi), np.full(p, 2*np.pi)])\ninit_point = np.random.uniform(lb, ub, 2*p)\nshots = 2**10\noptimiser = 'COBYLA'\nmax_iter = 100\n\n# Training of the parameters beta and gamma\nobj = get_black_box_objective(G,p,SDK='qiskit',shots=shots)\n# Lower and upper bounds: beta \\in {0, pi}, gamma \\in {0, 2*pi}\nbounds = [lb,ub]\n\n# Maximum number of iterations: 100\nres = minimize(obj, init_point, method=optimiser, bounds = bounds,options={'maxiter':max_iter, 'disp': True})\nprint(res)\n```\n\n /home/redwombat/miniconda3/envs/qi-py38/lib/python3.8/site-packages/scipy/optimize/_minimize.py:544: RuntimeWarning: Method COBYLA cannot handle bounds.\n warn('Method %s cannot handle bounds.' % method,\n\n\n fun: -2.0\n maxcv: 0.0\n message: 'Optimization terminated successfully.'\n nfev: 32\n status: 1\n success: True\n x: array([1.89052483, 4.39449308])\n\n\n\n```python\n#Determine the approximation ratio:\nprint('Approximation ratio is',-res['fun']/max_costs_bf)\n\n# Extract the optimal values for beta and gamma and run a new circuit with these parameters\noptimal_theta = res['x']\nqc = Qiskit.get_qaoa_circuit(G, optimal_theta[:p], optimal_theta[p:])\ncounts = execute(qc,backend = Aer.get_backend('qasm_simulator'),shots=shots).result().get_counts()\n\n# Random guessing distribution\nRG_dist = random_guessing_dist(G)\n\n# Measurement distribution \nE_dist = defaultdict(int)\nfor k, v in counts.items():\n E_dist[maxcut_obj(k,G)] += v\n\nX_list,Y_list = zip(*E_dist.items())\nX = -np.asarray(X_list)\nY = np.asarray(Y_list)\nprint('Probability of measuring the optimal solution is',Y[np.argmax(X)]/shots)\n\nE_random_guessing = energy_random_guessing(RG_dist)\nprint('Expected approximation ratio random guessing is', -E_random_guessing/max_costs_bf)\n```\n\n Approximation ratio is 1.0\n Probability of measuring the optimal solution is 1.0\n Expected approximation ratio random guessing is 0.75\n\n\n\n```python\nplt.bar(counts.keys(), counts.values())\nplt.xlabel('String')\nplt.ylabel('Count')\nplt.show()\n```\n\nLet's run it on Starmon-5 again!\n\n\n```python\n# Extract the optimal values for beta and gamma and run a new circuit with these parameters\noptimal_theta = res['x']\nqasm_circuit = QAOA_triangle_circuit_cqasm(G, optimal_theta[:p], optimal_theta[p:])\nqiapi = QuantumInspireAPI(QI_URL, authentication)\nresult = qiapi.execute_qasm(qasm_circuit, backend_type=qiapi.get_backend_type('Starmon-5'), number_of_shots=shots)\ncounts = result['histogram']\n\nprint(qasm_circuit)\nprint(result)\n```\n\n \n version 1.0\n qubits 5\n prep_z q[0:4]\n {H q[0] | H q[1] | H q[2]}\n \n .U_gamma_1\n SWAP q[0], q[2]\n CNOT q[2], q[1]\n Rz q[1], 8.788986167232244\n CNOT q[2], q[1]\n SWAP q[0], q[2]\n CNOT q[0], q[2]\n Rz q[2], 8.788986167232244\n CNOT q[0], q[2]\n CNOT q[1], q[2]\n Rz q[2], 8.788986167232244\n CNOT q[1], q[2]\n .U_beta_1\n {Rx q[0], 3.781049654543756 | Rx q[1], 3.781049654543756 | Rx q[2], 3.781049654543756}\n \n {measure q[0] | measure q[1] | measure q[2]}\n \n {'id': 7069985, 'url': 'https://api.quantum-inspire.com/results/7069985/', 'job': 'https://api.quantum-inspire.com/jobs/7078006/', 'created_at': '2021-11-26T12:22:32.339692Z', 'number_of_qubits': 5, 'execution_time_in_seconds': 0.155648, 'raw_text': '', 'raw_data_url': 'https://api.quantum-inspire.com/results/7069985/raw-data/c34440ffc174351f859a320acde97bf1f5596d01f3bbd9acacd93a0771764fe5/', 'histogram': OrderedDict([('0', 0.0390625), ('1', 0.1611328125), ('2', 0.08203125), ('3', 0.240234375), ('4', 0.1025390625), ('5', 0.15625), ('6', 0.15625), ('7', 0.0625)]), 'histogram_url': 'https://api.quantum-inspire.com/results/7069985/histogram/c34440ffc174351f859a320acde97bf1f5596d01f3bbd9acacd93a0771764fe5/', 'measurement_mask': 7, 'quantum_states_url': 'https://api.quantum-inspire.com/results/7069985/quantum-states/c34440ffc174351f859a320acde97bf1f5596d01f3bbd9acacd93a0771764fe5/', 'measurement_register_url': 'https://api.quantum-inspire.com/results/7069985/measurement-register/c34440ffc174351f859a320acde97bf1f5596d01f3bbd9acacd93a0771764fe5/', 'calibration': 'https://api.quantum-inspire.com/calibration/109027/'}\n\n\n\n```python\ncounts\n```\n\n\n\n\n OrderedDict([('0', 0.0390625),\n ('1', 0.1611328125),\n ('2', 0.08203125),\n ('3', 0.240234375),\n ('4', 0.1025390625),\n ('5', 0.15625),\n ('6', 0.15625),\n ('7', 0.0625)])\n\n\n\n\n```python\ncounts_bin = {}\nfor k,v in counts.items():\n counts_bin[f'{int(k):03b}'] = v\nprint(counts_bin)\nplt.bar(counts_bin.keys(), counts_bin.values())\nplt.xlabel('String')\nplt.ylabel('Probability')\nplt.show()\n```\n\n\n```python\n#Determine the approximation ratio:\nprint('Approximation ratio on the hardware is',-compute_maxcut_energy(counts_bin,G)/max_costs_bf)\n\n# Random guessing distribution\nRG_dist = random_guessing_dist(G)\n\n# Measurement distribution \nE_dist_S5 = defaultdict(int)\nfor k, v in counts_bin.items():\n E_dist_S5[maxcut_obj(k,G)] += v\n \nplot_E_distributions([E_dist,E_dist_S5,RG_dist],p,['Qiskit','Starmon-5','random guessing'])\n\n\nX_list,Y_list = zip(*E_dist_S5.items())\nX = -np.asarray(X_list)\nY = np.asarray(Y_list)\nprint('Probability of measuring the optimal solution is',Y[np.argmax(X)])\n\n\nE_random_guessing = energy_random_guessing(RG_dist)\nprint('Expected approximation ratio random guessing is', -E_random_guessing/max_costs_bf)\n```\n\n## 7. More advanced questions\n\nSome questions you could look at:\n\n- What is the performance on other graph instances?\n- How scalable is this hardware for larger problem sizes?\n- How much can the circuit be optimized for certain graph instances?\n- Are the errors perfectly random or is there some correlation?\n- Are there tricks to find good parameters? \n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "d0f16a961a8224ccb91fdd8b54c53ba31c743f3d", "size": 253708, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/notebooks/eqtc_workshop/EQTC_workshop_QAOA_tutorial_solution.ipynb", "max_stars_repo_name": "QuTech-Delft/quantum-inspire-examples", "max_stars_repo_head_hexsha": "04e9bb879bf0b8a12cede0d67f7384028c40788f", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-09-13T11:05:46.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-27T12:28:21.000Z", "max_issues_repo_path": "docs/notebooks/eqtc_workshop/EQTC_workshop_QAOA_tutorial_solution.ipynb", "max_issues_repo_name": "QuTech-Delft/quantum-inspire-examples", "max_issues_repo_head_hexsha": "04e9bb879bf0b8a12cede0d67f7384028c40788f", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2021-06-21T12:29:39.000Z", "max_issues_repo_issues_event_max_datetime": "2021-11-28T10:12:09.000Z", "max_forks_repo_path": "docs/notebooks/eqtc_workshop/EQTC_workshop_QAOA_tutorial_solution.ipynb", "max_forks_repo_name": "QuTech-Delft/quantum-inspire-examples", "max_forks_repo_head_hexsha": "04e9bb879bf0b8a12cede0d67f7384028c40788f", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2021-06-21T12:09:52.000Z", "max_forks_repo_forks_event_max_datetime": "2021-07-26T10:54:56.000Z", "avg_line_length": 163.5770470664, "max_line_length": 90600, "alphanum_fraction": 0.8847769877, "converted": true, "num_tokens": 8270, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.6654105653819836, "lm_q1q2_score": 0.43577387191760303}} {"text": "```python\nimport pandapower as pp\n```\n\n\n```python\nfrom sympy import symbols\n```\n\n\n```python\nnet = pp.create_empty_network() \nb1 = pp.create_bus(net, vn_kv=13.2)\nb2 = pp.create_bus(net, vn_kv=13.2)\nb3 = pp.create_bus(net, vn_kv=13.2)\n\npp.create_line(net, from_bus=b1, to_bus=b2, length_km=0.8, std_type=\"NAYY 4x50 SE\")\npp.create_line(net, from_bus=b2, to_bus=b3, length_km=1.2, std_type=\"NAYY 4x50 SE\")\n\npp.create_ext_grid(net, bus=b1)\n\npp.create_load(net, bus=b3, p_mw=0.350)\n```\n\n\n\n\n 0\n\n\n\n\n```python\n pp.runpp(net)\n```\n\n\n```python\n\n```\n\n\n```python\nprint(net.res_bus.vm_pu)\nprint(net.res_line)\n```\n\n 0 1.000000\n 1 0.998973\n 2 0.997425\n Name: vm_pu, dtype: float64\n p_from_mw q_from_mvar p_to_mw q_to_mvar pl_mw ql_mvar \\\n 0 0.350909 -0.022814 -0.350545 1.367420e-02 0.000364 -0.009140 \n 1 0.350545 -0.013674 -0.350000 -9.562987e-10 0.000545 -0.013674 \n \n i_from_ka i_to_ka i_ka vm_from_pu va_from_degree vm_to_pu \\\n 0 0.015381 0.015360 0.015381 1.000000 0.000000 0.998973 \n 1 0.015360 0.015348 0.015360 0.998973 -0.010749 0.997425 \n \n va_to_degree loading_percent \n 0 -0.010749 10.831460 \n 1 -0.023998 10.816756 \n\n\n\n```python\npp.create_sgen(net, b2, p_mw=0.05, q_mvar=0.025, max_p_mw=0.2, max_q_mvar=0.2)\n```\n\n\n\n\n 0\n\n\n\n\n```python\npp.runpp(net)\n```\n\n\n```python\nprint(net.res_bus.vm_pu)\nprint(net.res_line)\n```\n\n 0 1.000000\n 1 0.999130\n 2 0.997582\n Name: vm_pu, dtype: float64\n p_from_mw q_from_mvar p_to_mw q_to_mvar pl_mw ql_mvar \\\n 0 0.300817 -0.047832 -0.300544 3.867856e-02 0.000272 -0.009153 \n 1 0.350544 -0.013679 -0.350000 -7.725773e-10 0.000544 -0.013679 \n \n i_from_ka i_to_ka i_ka vm_from_pu va_from_degree vm_to_pu \\\n 0 0.013323 0.013265 0.013323 1.00000 0.000000 0.999130 \n 1 0.015357 0.015346 0.015357 0.99913 -0.013882 0.997582 \n \n va_to_degree loading_percent \n 0 -0.013882 9.382118 \n 1 -0.027127 10.815054 \n\n\n\n```python\nprint(net)\n```\n\n This pandapower network includes the following parameter tables:\n - bus (3 elements)\n - load (1 element)\n - sgen (1 element)\n - ext_grid (1 element)\n - line (2 elements)\n and the following results tables:\n - res_bus (3 elements)\n - res_line (2 elements)\n - res_ext_grid (1 element)\n - res_load (1 element)\n - res_sgen (1 element)\n\n\n\n```python\ndef net_add_optfw(net):\n net.ext_grid['initial_cost'] = 0.0\n net.ext_grid['operation_cost'] = 0.0\n net.ext_grid['ap_mw'] = 100.0\n net.ext_grid['op_mw'] = 0.0\n net.ext_grid['constraints'] = None\n\n net.load['initial_cost'] = 0.0\n net.load['operation_cost'] = 0.0\n net.load['ap_mw'] = 100.0\n net.load['op_mw'] = 0.0\n net.load['constraints'] = None\n\n net.sgen['initial_cost'] = 0.0\n net.sgen['operation_cost'] = 0.0\n net.sgen['ap_mw'] = 100.0\n net.sgen['op_mw'] = 0.0\n net.sgen['constraints'] = None \n```\n\n\n```python\nnet_add_optfw(net)\n```\n\n\n```python\nnet.ext_grid\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    namebusvm_puva_degreein_serviceinitial_costoperation_costap_mwop_mwconstraints
    0None01.00.0True0.00.0100.00.0None
    \n
    \n\n\n\nVariables que define el modelo:\n\nt para el tiempo, (en horas)\n\n\n```python\n\n```\n\n\n```python\nnet.ext_grid\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    namebusvm_puva_degreein_service
    0None01.00.0True
    \n
    \n\n\n\n\n```python\nnet.ext_grid.__class__\n```\n\n\n```python\nnet.load.__class__\n```\n\n\n```python\nnet.load.columns\n```\n\n\n```python\nnet.line\n```\n\n\n```python\nnet.load\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    namebusp_mwq_mvarconst_z_percentconst_i_percentsn_mvascalingin_servicetype
    0None20.350.00.00.0NaN1.0Truewye
    \n
    \n\n\n\n\n```python\nnet.load['bus'][0]\n```\n\n\n\n\n 2\n\n\n\n\n```python\nnet.bus.loc[2]['vn_kv']+5\n```\n\n\n\n\n 18.2\n\n\n\n\n```python\nnet.load['ap_mw']=net.load['p_mw']\n```\n\n\n```python\nnet.load\n```\n\n\n```python\nnet.load['name'][0]='Carga'\n```\n\n\n```python\nnet.load\n```\n\n\n```python\nnet.ext_grid\n```\n\n\n```python\nnet.switch\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "07ce353bcef4309d7036b1fc3feb34df899f614a", "size": 14820, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Notebooks/Prueba SimPy.ipynb", "max_stars_repo_name": "ExoticAnt/MMGO", "max_stars_repo_head_hexsha": "cd4237958d3ea56344468cf2e1d62a894a5b11c7", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-09-30T18:08:16.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-30T18:08:16.000Z", "max_issues_repo_path": "Notebooks/Prueba SimPy.ipynb", "max_issues_repo_name": "ExoticAnt/Modular-Microgrids-Optimization", "max_issues_repo_head_hexsha": "cd4237958d3ea56344468cf2e1d62a894a5b11c7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 27, "max_issues_repo_issues_event_min_datetime": "2021-06-13T20:10:41.000Z", "max_issues_repo_issues_event_max_datetime": "2021-07-30T16:00:48.000Z", "max_forks_repo_path": "Notebooks/Prueba SimPy.ipynb", "max_forks_repo_name": "ExoticAnt/Modular-Microgrids-Optimization", "max_forks_repo_head_hexsha": "cd4237958d3ea56344468cf2e1d62a894a5b11c7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 23.3385826772, "max_line_length": 94, "alphanum_fraction": 0.4350877193, "converted": true, "num_tokens": 2257, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548782017745, "lm_q2_score": 0.6688802603710085, "lm_q1q2_score": 0.4356784205255294}} {"text": "## Lecture 5 - Convolutional neural networks\n\n> A deep dive into convolutional neural networks for jet tagging\n\n## Learning objectives\n\n* Know how to implement and train a convolutional neural network in PyTorch\n* Learn how to debug unstable training runs with activation statistics\n* Understand what the 1-cycle training policy is\n* Understand what batch normalization is and how to incorporate it into a CNN\n\n## References\n\n* Chapter 13 of [_Deep Learning for Coders with fastai & PyTorch_](https://github.com/fastai/fastbook) by Jeremy Howard and Sylvain Gugger.\n* Chapter 11 of [_Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow_](https://www.amazon.com/Hands-Machine-Learning-Scikit-Learn-TensorFlow/dp/1492032646/ref=sr_1_1?crid=29Z1GSCOWEPDV&keywords=hands+on+machine+learning+with+scikit-learn+and+tensorflow+2&qid=1653288575&sprefix=hands+on+ma%2Caps%2C160&sr=8-1) by Aur\u00e9lien Geron\n\n## Setup\n\n\n```python\n# Uncomment and run this cell if using Colab, Kaggle etc\n# %pip install fastai==2.6.0 datasets\n```\n\n## Imports\n\n\n```python\nfrom datasets import load_dataset\nfrom fastai.callback.hook import *\nfrom fastai.vision.all import *\nfrom sklearn.metrics import accuracy_score\nfrom torch.utils.data import TensorDataset\nfrom torchvision.transforms import ToTensor\n```\n\n\n```python\nimport datasets\n\n# Suppress logs to keep things tidy\ndatasets.logging.set_verbosity_error()\n```\n\n## Loading the data\n\nLast lecture we fine-tuned a pretrained CNN on the top tagging dataset, where each jet was represented as a 2D image of \"energy pixels\". Today, we'll take a look at training our own CNN from scratch, and explore some of the techniques that fastai utilizes to stabilise the training of these neural nets. To get started, let's download the same dataset of jet images from last lecture:\n\n\n```python\nimages_ds = load_dataset(\"dl4phys/top_tagging_images\")\n\n# Peek at one example\nplt.imshow(images_ds[\"train\"][0][\"image\"]);\n```\n\nAs we saw last lecture, we need to convert these PIL images into PyTorch tensors, so let's reuse the same helper function to create a training and validation set to experiment with:\n\n\n```python\ndef get_dataset(dataset, num_examples=None):\n if num_examples is not None:\n dataset = dataset.shuffle(seed=42).select(range(num_examples))\n\n x = torch.cat([ToTensor()(img) for img in dataset[\"image\"]]).unsqueeze(1)\n y = torch.cat([torch.tensor(l).unsqueeze(0) for l in dataset[\"label\"]])\n\n return TensorDataset(x, y)\n```\n\nWith this function, we can now generate a sample of jets as follows:\n\n\n```python\n# Lower the training size if Colab RAM explodes \ud83d\udca3\ntrain_ds = get_dataset(images_ds[\"train\"], num_examples=350_000)\nvalid_ds = get_dataset(images_ds[\"validation\"], num_examples=35_000)\n```\n\nSince we'll be experimenting a bit with the batch size in this lecture, we'll also implement a helper function to return the dataloaders associated with these datasets:\n\n\n```python\ndef get_dls(bs=128):\n train_dl = DataLoader(train_ds, bs=bs, shuffle=True)\n valid_dl = DataLoader(valid_ds, bs=bs)\n return DataLoaders(train_dl, valid_dl)\n```\n\nNow that we have this function, we can get the dataloaders and grab a batch of data to test with:\n\n\n```python\ndls = get_dls()\n\nxb, yb = first(dls.train)\nxb.shape, yb.shape\n```\n\n\n\n\n (torch.Size([128, 1, 40, 40]), torch.Size([128]))\n\n\n\n## Creating a CNN\n\nRecall in lecture 3, that we created a neural network for $N$-subjettiness features. We used the `nn.Sequential()` class to create a network that involved stacking fully-connected layers and ReLU activation functions:\n\n\n```python\nmodel = nn.Sequential(\n nn.Linear(20, 200),\n nn.ReLU(),\n nn.Linear(200, 2),\n)\n\nmodel\n```\n\n\n\n\n Sequential(\n (0): Linear(in_features=20, out_features=200, bias=True)\n (1): ReLU()\n (2): Linear(in_features=200, out_features=2, bias=True)\n )\n\n\n\nIn this lecture, we'll take a similar approach, but this time using _convolutional layers_ instead of linear ones. In PyTorch, convolutional layers are created by using the `nn.Conv2d()` module. Let's try replacing the `nn.Linear()` layers in our previous architecture with `nn.Conv2d()` ones instead:\n\n\n```python\nbroken_cnn = nn.Sequential(\n nn.Conv2d(1, 30, kernel_size=3, padding=1),\n nn.ReLU(),\n nn.Conv2d(30, 1, kernel_size=3, padding=1),\n)\n```\n\nNote that unlike linear layers, convolutional layers don't require us to specify the number of features in the input. That's because convolutions are applied across each pixel automatically, so the weights only depend on the number of channels and the kernel size. \n\nLet's now see what happens if we pass a minibatch of data through this model:\n\n\n```python\n# Feed a minibatch to the model\noutputs = broken_cnn(xb)\noutputs.shape\n```\n\n\n\n\n torch.Size([128, 1, 40, 40])\n\n\n\nHmm, this output isn't quite what we want for classification: we need a tensor of shape `(batch_size, num_classes)` but instead have a $40\\times 40$ map of activations. The standard way to handle this is to apply a sequence of _stride-2 convolutions_, which decrease the size of the outputs so that the final layer size is 1. To see why this is the case, recall that for an image of height $n_H$ and width $n_W$, the dimension of the output activation map is given by:\n\n$$\\left( \\left\\lfloor\\frac{n_H + 2p - k}{s} + 1 \\right\\rfloor , \\left\\lfloor\\frac{n_W + 2p - k}{s} + 1 \\right\\rfloor \\right) \\,,$$\n\nwhere $p$ is the padding, $k$ the kernel of size $k\\times k$, and $s$ the stride. For fixed $p,f$ and $s>1$, we can see that the dimension of the activation map is shrunk with each convolutional layer.\n\nWith this in mind, let's create a stack of stride-2 convolutional layers with $3\\times 3$ kernels. We'll intersperse each convolutional layer with a ReLU activation, so let's write a helper function that returns the two together:\n\n\n```python\ndef conv(ni, nf, ks=3, act=True):\n layer = nn.Conv2d(ni, nf, stride=2, kernel_size=ks, padding=ks // 2)\n if act:\n layer = nn.Sequential(layer, nn.ReLU())\n return layer\n```\n\nAs discussed in the fastai book, it's good practice to increase the number of output features `nf` with each convolutional layer. That's because, stride-2 convolutions decrease the size of the activation map, so we increase the number of output features to avoid compressing the capacity of our model. We also set the kernel size `ks` to the typical value of $3\\times 3$, and set the padding to $ks//2$ to ensure the output activation map is the same shape as the input image.\n\nWe can now build a CNN by stacking the `conv()` function until we reach a $1\\times 1$ activation map:\n\n\n```python\nsimple_cnn = nn.Sequential(\n conv(1, 4), # 20x20\n conv(4, 8), # 10x10\n conv(8, 16), # 5x5\n conv(16, 32), # 3x3\n conv(32, 16), # 2x2\n conv(16, 2, act=False), # 1x1\n Flatten(),\n)\n\nlearn = Learner(\n dls, simple_cnn, metrics=[accuracy, RocAucBinary()], loss_func=F.cross_entropy\n)\nlearn.summary()\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n Sequential (Input shape: 128 x 1 x 40 x 40)\n ============================================================================\n Layer (type) Output Shape Param # Trainable \n ============================================================================\n 128 x 4 x 20 x 20 \n Conv2d 40 True \n ReLU \n ____________________________________________________________________________\n 128 x 8 x 10 x 10 \n Conv2d 296 True \n ReLU \n ____________________________________________________________________________\n 128 x 16 x 5 x 5 \n Conv2d 1168 True \n ReLU \n ____________________________________________________________________________\n 128 x 32 x 3 x 3 \n Conv2d 4640 True \n ReLU \n ____________________________________________________________________________\n 128 x 16 x 2 x 2 \n Conv2d 4624 True \n ReLU \n ____________________________________________________________________________\n 128 x 2 x 1 x 1 \n Conv2d 290 True \n ____________________________________________________________________________\n 128 x 2 \n Flatten \n ____________________________________________________________________________\n \n Total params: 11,058\n Total trainable params: 11,058\n Total non-trainable params: 0\n \n Optimizer used: \n Loss function: \n \n Callbacks:\n - TrainEvalCallback\n - Recorder\n - ProgressCallback\n\n\n\nSince the final convolutional layer will be a tensor of shape $128\\times2\\times1\\times1$, we've stripped off those final $1\\times 1$ axes via the `Flatten()` module. We can now verify that the output of this model is in a form suitable for classification:\n\n\n```python\nsimple_cnn(xb).shape\n```\n\n\n\n\n torch.Size([128, 2])\n\n\n\nOkay, this looks good, so let's find a good learning rate and train for a few epochs:\n\n\n```python\nlearn.lr_find()\n```\n\n\n```python\nlearn.fit(3, 3e-3)\n```\n\n\n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    epochtrain_lossvalid_lossaccuracyroc_auc_scoretime
    00.2519220.2579910.8922860.95785300:12
    10.2481500.2434930.8991140.96300300:12
    20.2312960.2372530.9005430.96425800:12
    \n\n\nThis model is much worse than the pretrained ResNet34 model that we fine-tuned in the previous lecture. We can also see that we've started overfitting quite strongly after the first epoch (because the training and validation losses are diverging). Let's now see if we can make the model more accurate and train it even faster by using some of the techniques that the factory learners (like `vision_learner()`) provide under the hood.\n\n## Debugging training with activation statistics\n\nOne way to train a more accurate CNN is simply double the number of filters in each stride-2 layer. However, increasing the number of filters also requires us to increase the kernel size. For example, if our first layer increases the number of filters from 4 to 8, then a $3 \\times 3$ kernel will not force the CNN to learn any useful features. To handle that, we'll increase the kernel size to $5\\times 5$ and adjust the CNN architecture as follows:\n\n\n```python\ndef simple_cnn():\n return nn.Sequential(\n conv(1, 8, ks=5), # 20x20\n conv(8, 16), # 10x10\n conv(16, 32), # 5x5\n conv(32, 64), # 3x3\n conv(64, 128), # 2x2\n conv(128, 2, act=False), # 1x1\n Flatten(),\n ).to(\"cuda\")\n```\n\nSince we'll be training this network in several ways, let's wrap the creation of the `Learner` and training in a `fit()` function:\n\n\n```python\ndef fit(epochs=1, lr=0.06):\n learn = Learner(\n dls,\n simple_cnn(),\n metrics=[accuracy, RocAucBinary()],\n loss_func=F.cross_entropy,\n cbs=ActivationStats(with_hist=True),\n )\n learn.fit(epochs, lr)\n return learn\n```\n\nTo give a sense for what can go wrong when training CNNs, we've used a much larger learning rate than before. We've also use the `ActivationStats()` callback, which records the statistics of the activations in each layer - this will be handy for debugging. Let's see how well this model performs after 1 epoch:\n\n\n```python\nlearn = fit()\n```\n\n /home/lewis/miniconda3/envs/dl4phys/lib/python3.9/site-packages/fastai/callback/core.py:67: UserWarning: You are shadowing an attribute (modules) that exists in the learner. Use `self.learn.modules` to avoid this\n warn(f\"You are shadowing an attribute ({name}) that exists in the learner. Use `self.learn.{name}` to avoid this\")\n\n\n\n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    epochtrain_lossvalid_lossaccuracyroc_auc_scoretime
    00.6946010.6939600.5003430.50000000:19
    \n\n\nOkay, it seems the large learning rate has given us a model than doesn't perform better than random chance. To diagnose the problem, we can use the `plot_layer_stats()` method that the `ActivationStats` callback has tracked. For example, here are the mean and standard deviation of the activations in the first layer:\n\n\n```python\nlearn.activation_stats.plot_layer_stats(0)\n```\n\nHere we can see that the mean and standard deviation have a somewhat dramatic spike / drop in the early stages of training. A bigger problem is that the fraction of activations that are near zero is almost 1 after a few iterations. This is a problem because activations that are zero in one layer, will propagate zeros to the next layer, and so on. As a result, these parts of the network are effectively turned off, which makes the learning process suboptimal. \n\nLet's now have a look at the last convolutional layer in the CNN:\n\n\n```python\nlearn.activation_stats.plot_layer_stats(-2)\n```\n\nHere we see a similar pattern, with the fraction of near zero activations reaching almost 1 in fewer iterations. Let's see if we can imprve the situation by increasing the batch size.\n\n### Increasing the batch size\n\nIf you're GPU can handle it, increasing the batch size is one technique that can sometimes stabilise training. This is because a larger batch size implies more accurate gradients, so SGD can potentially train more efficiently. Let's try increasing our batch size by a factor of 4, training the model and investigating the activation statistics:\n\n\n```python\ndls = get_dls(512)\nlearn = fit()\nlearn.activation_stats.plot_layer_stats(-2)\n```\n\nOkay, increasing the batch size hasn't helped much. Clearly the problem is that our learning rate is too large, which is causing training to diverge. One way to handle this is to use a _dynamic_ learning rate that is adjusted from low to high, and back to low values during training itself! Let's take a look at 1-cycle training and finally understand what's going one when we call `Learner.fit_one_cycle()`.\n\n### 1-cycle training\n\nThe basic idea behind 1-cycle training is to split training into two phases:\n\n* **Warmup:** grow the learning rate from some minimum to maximum value. By starting with a small learning rate, we can avoid hopping out of a local minimum in the loss and diverging.\n* **Annealing:** decrease the learning rate from the maximum value to the minimum. By gradually lowering the learning rate, we avoid skipping a good local minimum towards the end of training.\n\nThis technique is one of the main aspects that allows fastai to train fast and accurate models. As we've done in previous lectures, we can use the `fit_one_cycle()` method, so let's adjust our `fit()` function and train again:\n\n\n```python\ndef fit(epochs=1, lr=0.06):\n learn = Learner(\n dls,\n simple_cnn(),\n metrics=[accuracy, RocAucBinary()],\n loss_func=F.cross_entropy,\n cbs=ActivationStats(with_hist=True),\n )\n learn.fit_one_cycle(epochs, lr)\n return learn\n\nlearn = fit()\n```\n\n /home/lewis/miniconda3/envs/dl4phys/lib/python3.9/site-packages/fastai/callback/core.py:67: UserWarning: You are shadowing an attribute (modules) that exists in the learner. Use `self.learn.modules` to avoid this\n warn(f\"You are shadowing an attribute ({name}) that exists in the learner. Use `self.learn.{name}` to avoid this\")\n\n\n\n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    epochtrain_lossvalid_lossaccuracyroc_auc_scoretime
    00.2550700.2575230.8920290.95734400:12
    \n\n\nNice, we've finally got a model that performs better than random! All fastai `Learner`s come with a `Recorder()` callback, which tracks various aspect of training. Here we can use it to inspect the evolution of the learning rate:\n\n\n```python\nlearn.recorder.plot_sched()\n```\n\nHere we can see the learning rate increases linearly until the maximum value, before being annealed with a cosine function. The second plot refers to a hyperparameter called _momentum_, which takes values between $[0,1]$ and is often denoted by $\\beta$. This hyperparameter belongs to a technique called _momentum optimization_, which extends SGD to speed up training. Recall that in SGD, we update our weights and biases according to the following update rule:\n\n$$ \\theta \\to \\theta' = \\theta - \\eta \\nabla_\\theta L(\\theta) .$$\n\nOne problem with this rule, is that if the gradients are small in some region, then the subsequent updates will also be small and training can be slow. To deal with this, momentum optimization uses an _exponentially weighted average_ to modify the update rule:\n\n\\begin{align}\n\\mathbf{m}'&= \\beta\\mathbf{m} + (1-\\beta)\\nabla_\\theta L(\\theta) \\\\\n\\theta' &= \\theta - \\eta\\mathbf{m}'\n\\end{align}\n\nIntuitively, the momentum vector $\\mathbf{m}$ is storing a running average of past gradient values, and this enables the update step to pick directions based on these averages. The result is that SGD with momentum allows us to traverse the loss landscape much faster (especially through plateaus). \n\nLet's now have a look at the activation statistics in the final `conv()` layer:\n\n\n```python\nlearn.activation_stats.plot_layer_stats(-2)\n```\n\nOkay, we've getting somewhat better, but we still have quite a few activations getting close to zero towards the end of training. Let's look at one last technique that can help us deal with this.\n\n### Batch normalization\n\nA very effective technique to deal with vanishing activations in training is to use _batch normalization_ (or batchnorm for short). This techniuqe tries to maintain a good distribution of activations during training by applying a normalization operation just before or after each hidden layer. The way this works is to zero-center and normalize the activations in each layer by introducing two new parameters $\\gamma$ and $\\beta$ (not the same $\\beta$ from momentum!). The parameters are used to learn the optimal scale of the inputs across each layer, and the batchnorm operation can be summarised in the following equations:\n\n\\begin{align}\n\\mathbf{\\mu}_B &= \\frac{1}{m_B} \\sum_i \\mathbf{x}^{(i)} \\\\\n\\sigma_B^2 &= \\frac{1}{m_B} \\sum_i \\left(\\mathbf{x}^{(i)} - \\mathbf{\\mu}_B\\right)^2 \\\\\n\\hat{\\mathbf{x}}^{(i)} &= \\frac{\\mathbf{x}^{(i)} - \\mathbf{\\mu}_B}{\\sqrt{\\mathbf{\\sigma}_B^2 + \\epsilon}} \\\\\n\\mathbf{z}^{(i)} &= \\mathbf{\\gamma} \\odot \\hat{\\mathbf{x}}^{(i)} + \\mathbf{\\beta}\n\\end{align}\n\nThese statistics make it easier to train models, since we don't have to enforce a global normalization on the data (as we did with the `MinMaxScaler()` in previous lectures).\n\nImplementing batchnorm in PyTorch is rather simple: we just add a `nn.BatchNorm2d()` layer after each convolution:\n\n\n```python\ndef conv(ni, nf, ks=3, act=True):\n layers = [nn.Conv2d(ni, nf, stride=2, kernel_size=ks, padding=ks // 2)]\n layers.append(nn.BatchNorm2d(nf))\n if act:\n layers.append(nn.ReLU())\n return nn.Sequential(*layers)\n```\n\nLet's now train the model again with 1-cycle policy training:\n\n\n```python\nlearn = fit()\n```\n\n /home/lewis/miniconda3/envs/dl4phys/lib/python3.9/site-packages/fastai/callback/core.py:67: UserWarning: You are shadowing an attribute (modules) that exists in the learner. Use `self.learn.modules` to avoid this\n warn(f\"You are shadowing an attribute ({name}) that exists in the learner. Use `self.learn.{name}` to avoid this\")\n\n\n\n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    epochtrain_lossvalid_lossaccuracyroc_auc_scoretime
    00.2165970.2144710.9097140.97049000:22
    \n\n\nNice! This is pretty great result after just one epoch of training, and gets is quite close to the results quoted for CNNs in the top taggin review paper. Let's also inspect the activation statistics:\n\n\n```python\nlearn.activation_stats.plot_layer_stats(-2)\n```\n\nThis is looking much better: the activations evolve smoothly and we've managed to prevent half of the activations from vanishing. Since batchnorm claims to work with large learning rates, let's ramp this up to 0.1 and see what we get:\n\n\n```python\nlearn = fit(10, lr=0.1)\n```\n\n /home/lewis/miniconda3/envs/dl4phys/lib/python3.9/site-packages/fastai/callback/core.py:67: UserWarning: You are shadowing an attribute (modules) that exists in the learner. Use `self.learn.modules` to avoid this\n warn(f\"You are shadowing an attribute ({name}) that exists in the learner. Use `self.learn.{name}` to avoid this\")\n\n\n\n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    epochtrain_lossvalid_lossaccuracyroc_auc_scoretime
    00.2416210.3245240.8795710.95173600:22
    10.2320911.6578440.6559140.91325500:21
    20.2226610.8197990.6120860.90605200:21
    30.2203390.2760020.8854290.95714500:21
    40.2179050.4103750.8132570.92018700:21
    50.2090550.4336100.8029710.92020400:21
    60.2047820.3634770.8329430.94595100:21
    70.1985090.2067480.9134290.97291100:21
    80.1916560.2025640.9160000.97443400:21
    90.1821510.1962920.9188860.97523700:21
    \n\n\nGreat, this has given us a small boost and only took a few minutes to train a CNN form scratch!\n\n## Exercises\n\n* Implement the same CNN architecture from the review. Can you get close to their results?\n* Read the [1-cycle policy training paper](https://arxiv.org/abs/1708.07120)\n\n\n```python\n\n```\n", "meta": {"hexsha": "4295342774ee32c93186e3b8617df696477ebcdc", "size": 220982, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lecture05.ipynb", "max_stars_repo_name": "lewtun/dl4phys", "max_stars_repo_head_hexsha": "5cc0b4373d501602854a46a2528486b1bb3606d3", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "lecture05.ipynb", "max_issues_repo_name": "lewtun/dl4phys", "max_issues_repo_head_hexsha": "5cc0b4373d501602854a46a2528486b1bb3606d3", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "lecture05.ipynb", "max_forks_repo_name": "lewtun/dl4phys", "max_forks_repo_head_hexsha": "5cc0b4373d501602854a46a2528486b1bb3606d3", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 139.5969677827, "max_line_length": 37840, "alphanum_fraction": 0.8664642369, "converted": true, "num_tokens": 7037, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660542, "lm_q2_score": 0.6688802603710086, "lm_q1q2_score": 0.43567841147175335}} {"text": "## Example 1: `Hello world`\n\n\n```python\n# Print the character string `Hello world!`\nprint('Hello world!')\n```\n\n Hello world!\n\n\n## Example 2: Make a plot of the US unemployment rate\n\nDownload unemployment rate data from FRED (https://fred.stlouisfed.org/series/UNRATE/) and make a well-labeled plot.\n\n\n```python\n# Import the pandas library as pd\nimport pandas as pd\n\n# Import the plotting library matplotlib.pyplot as plt\nimport matplotlib.pyplot as plt\n\n# Download the unemployment rate data (PROVIDED)\nunemployment = pd.read_csv('https://fred.stlouisfed.org/data/UNRATE.txt',skiprows=24,sep='\\s+',index_col=0,parse_dates = True)\n\n# Print first 5 rows of unemployment variable\nunemployment.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    VALUE
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    1948-01-013.4
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    1948-03-014.0
    1948-04-013.9
    1948-05-013.5
    \n
    \n\n\n\n\n```python\n# Create a well-labeled plot of the US unemployment rate\nunemployment.plot(legend=False,lw=3,alpha=0.75)\nplt.ylabel('Percent')\nplt.xlabel('Date')\nplt.title('US Unemployment Rate')\n\n# Save the figure to the current working directory at 120 dots per inch resolution\nplt.savefig('unemployment.png',dpi=120)\n```\n\n## Example 2: Date and time\n\nUse the `datetime` module to print the current date and time\n\n\n```python\n# Import the datetime module\nimport datetime\n\n# Create a variable called `today` that stores the current date and time\ntoday = datetime.datetime.today()\n\n# Print the value of `today`\nprint(today)\n```\n\n 2021-01-05 12:02:08.861515\n\n\n\n```python\n# Print today's date formatted as: month day, year\nprint(today.strftime('%B %d, %Y'))\n```\n\n January 05, 2021\n\n\n\n```python\n# Print the time component of `today` as: HH:MM:SS\nprint(today.strftime('%I:%M:%S'))\n```\n\n 12:02:08\n\n\n## Example 3: Approximate a geometric series\n\n\nConsider the infinite geometric series:\n\n\\begin{align}\n\\sum_{k=0}^{\\infty} r^k & = r + r + r^2 + r^3 + \\cdots\n\\end{align}\n\nIf $|r|<1$, then the series converges to:\n\n\\begin{align}\n\\frac{1}{1+r}\n\\end{align}\n\nWe can verify this fact numerically by computing the *truncated* series, i.e., compute the series to a finite number of terms. We will do this for $r=0.2$. The idea is to create a variable $s$ that is initially equal to zero. Then we let $k$ increment from 0, 1, up to $N$ and each time add $r^k$ to $s$.\n\n\n```python\n# Set the value of r\nr = 0.2\n\n# Create a variable `N` that stores the desired number of terms to compute\nN = 25\n\n# Initialize a variable that will store the value of the summation. Set equal to tzero\ns = 0\n\n# Iterate over values of k fron 0 to N. Add the new term to the sum and print the current value of the sum\nfor k in range(N+1):\n s+=r**k\n print(s)\n```\n\n 1.0\n 1.2\n 1.24\n 1.248\n 1.2496\n 1.2499200000000001\n 1.2499840000000002\n 1.2499968000000001\n 1.2499993600000001\n 1.249999872\n 1.2499999744\n 1.24999999488\n 1.249999998976\n 1.2499999997952\n 1.24999999995904\n 1.249999999991808\n 1.2499999999983618\n 1.2499999999996725\n 1.2499999999999347\n 1.2499999999999871\n 1.2499999999999976\n 1.2499999999999996\n 1.25\n 1.25\n 1.25\n 1.25\n\n\n## Example 4: Draw random values from a normal probability distribution\n\nThe `numpy` library has a bunch of powerful tools for working with numbers. Here we'll use `numpy` to draw a random sample from the normal probability distribution. Specifically, 100 values from $\\mathcal{N}(2,0.2^2)$.\n\n\n```python\n# Import the numpy numerical tools library as np \nimport numpy as np\n\n# Set the seed or state of the numpy random number generator\nnp.random.seed(126)\n\n# Create a variable called 'x' that contains 100 draws from the normal(2,0.2^2) didistribution\nx = np.random.normal(loc=2,scale=0.2,size=100)\n\nplt.plot(x)\n```\n\n\n```python\n# Print the mean of the values in the variable 'x'\nprint(np.mean(x))\n```\n\n 1.983109892630882\n\n\n\n```python\n# Print the standard deviation of the values in the variable 'x'\nprint(np.std(x))\n```\n\n 0.21843109555638324\n\n", "meta": {"hexsha": "dcbe496cc00fc16fc4e6693bbe734a58d094c4fb", "size": 69786, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Lecture Notebooks/Econ126_Class_01.ipynb", "max_stars_repo_name": "letsgoexploring/econ126", "max_stars_repo_head_hexsha": "05f50d2392dd1c7c38b14950cb8d7eff7ff775ee", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-12-12T16:28:44.000Z", "max_stars_repo_stars_event_max_datetime": "2021-02-24T12:11:04.000Z", "max_issues_repo_path": "Lecture Notebooks/Econ126_Class_01.ipynb", "max_issues_repo_name": "letsgoexploring/econ126", "max_issues_repo_head_hexsha": "05f50d2392dd1c7c38b14950cb8d7eff7ff775ee", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2019-04-29T08:50:41.000Z", "max_issues_repo_issues_event_max_datetime": "2019-04-29T08:51:05.000Z", "max_forks_repo_path": "Lecture Notebooks/Econ126_Class_01.ipynb", "max_forks_repo_name": "letsgoexploring/econ126", "max_forks_repo_head_hexsha": "05f50d2392dd1c7c38b14950cb8d7eff7ff775ee", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 19, "max_forks_repo_forks_event_min_datetime": "2019-03-08T18:49:19.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-07T23:27:16.000Z", "avg_line_length": 171.0441176471, "max_line_length": 35652, "alphanum_fraction": 0.9028315135, "converted": true, "num_tokens": 1473, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5078118642792044, "lm_q2_score": 0.8577680995361899, "lm_q1q2_score": 0.4355848177447028}} {"text": "Probabilistic Programming and Bayesian Methods for Hackers \n========\n\nWelcome to *Bayesian Methods for Hackers*. The full Github repository is available at [github/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers](https://github.com/CamDavidsonPilon/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers). The other chapters can be found on the project's [homepage](https://camdavidsonpilon.github.io/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers/). We hope you enjoy the book, and we encourage any contributions!\n\n#### Looking for a printed version of Bayesian Methods for Hackers?\n\n_Bayesian Methods for Hackers_ is now a published book by Addison-Wesley, available on [Amazon](http://www.amazon.com/Bayesian-Methods-Hackers-Probabilistic-Addison-Wesley/dp/0133902838)! \n\n\n\nChapter 1\n======\n***\n\nThe Philosophy of Bayesian Inference\n------\n \n> You are a skilled programmer, but bugs still slip into your code. After a particularly difficult implementation of an algorithm, you decide to test your code on a trivial example. It passes. You test the code on a harder problem. It passes once again. And it passes the next, *even more difficult*, test too! You are starting to believe that there may be no bugs in this code...\n\nIf you think this way, then congratulations, you already are thinking Bayesian! Bayesian inference is simply updating your beliefs after considering new evidence. A Bayesian can rarely be certain about a result, but he or she can be very confident. Just like in the example above, we can never be 100% sure that our code is bug-free unless we test it on every possible problem; something rarely possible in practice. Instead, we can test it on a large number of problems, and if it succeeds we can feel more *confident* about our code, but still not certain. Bayesian inference works identically: we update our beliefs about an outcome; rarely can we be absolutely sure unless we rule out all other alternatives. \n\n\n### The Bayesian state of mind\n\n\nBayesian inference differs from more traditional statistical inference by preserving *uncertainty*. At first, this sounds like a bad statistical technique. Isn't statistics all about deriving *certainty* from randomness? To reconcile this, we need to start thinking like Bayesians. \n\nThe Bayesian world-view interprets probability as measure of *believability in an event*, that is, how confident we are in an event occurring. In fact, we will see in a moment that this is the natural interpretation of probability. \n\nFor this to be clearer, we consider an alternative interpretation of probability: *Frequentist*, known as the more *classical* version of statistics, assumes that probability is the long-run frequency of events (hence the bestowed title). For example, the *probability of plane accidents* under a frequentist philosophy is interpreted as the *long-term frequency of plane accidents*. This makes logical sense for many probabilities of events, but becomes more difficult to understand when events have no long-term frequency of occurrences. Consider: we often assign probabilities to outcomes of presidential elections, but the election itself only happens once! Frequentists get around this by invoking alternative realities and saying across all these realities, the frequency of occurrences defines the probability. \n\nBayesians, on the other hand, have a more intuitive approach. Bayesians interpret a probability as measure of *belief*, or confidence, of an event occurring. Simply, a probability is a summary of an opinion. An individual who assigns a belief of 0 to an event has no confidence that the event will occur; conversely, assigning a belief of 1 implies that the individual is absolutely certain of an event occurring. Beliefs between 0 and 1 allow for weightings of other outcomes. This definition agrees with the probability of a plane accident example, for having observed the frequency of plane accidents, an individual's belief should be equal to that frequency, excluding any outside information. Similarly, under this definition of probability being equal to beliefs, it is meaningful to speak about probabilities (beliefs) of presidential election outcomes: how confident are you candidate *A* will win?\n\nNotice in the paragraph above, I assigned the belief (probability) measure to an *individual*, not to Nature. This is very interesting, as this definition leaves room for conflicting beliefs between individuals. Again, this is appropriate for what naturally occurs: different individuals have different beliefs of events occurring, because they possess different *information* about the world. The existence of different beliefs does not imply that anyone is wrong. Consider the following examples demonstrating the relationship between individual beliefs and probabilities:\n\n- I flip a coin, and we both guess the result. We would both agree, assuming the coin is fair, that the probability of Heads is 1/2. Assume, then, that I peek at the coin. Now I know for certain what the result is: I assign probability 1.0 to either Heads or Tails (whichever it is). Now what is *your* belief that the coin is Heads? My knowledge of the outcome has not changed the coin's results. Thus we assign different probabilities to the result. \n\n- Your code either has a bug in it or not, but we do not know for certain which is true, though we have a belief about the presence or absence of a bug. \n\n- A medical patient is exhibiting symptoms $x$, $y$ and $z$. There are a number of diseases that could be causing all of them, but only a single disease is present. A doctor has beliefs about which disease, but a second doctor may have slightly different beliefs. \n\n\nThis philosophy of treating beliefs as probability is natural to humans. We employ it constantly as we interact with the world and only see partial truths, but gather evidence to form beliefs. Alternatively, you have to be *trained* to think like a frequentist. \n\nTo align ourselves with traditional probability notation, we denote our belief about event $A$ as $P(A)$. We call this quantity the *prior probability*.\n\nJohn Maynard Keynes, a great economist and thinker, said \"When the facts change, I change my mind. What do you do, sir?\" This quote reflects the way a Bayesian updates his or her beliefs after seeing evidence. Even — especially — if the evidence is counter to what was initially believed, the evidence cannot be ignored. We denote our updated belief as $P(A |X )$, interpreted as the probability of $A$ given the evidence $X$. We call the updated belief the *posterior probability* so as to contrast it with the prior probability. For example, consider the posterior probabilities (read: posterior beliefs) of the above examples, after observing some evidence $X$:\n\n1\\. $P(A): \\;\\;$ the coin has a 50 percent chance of being Heads. $P(A | X):\\;\\;$ You look at the coin, observe a Heads has landed, denote this information $X$, and trivially assign probability 1.0 to Heads and 0.0 to Tails.\n\n2\\. $P(A): \\;\\;$ This big, complex code likely has a bug in it. $P(A | X): \\;\\;$ The code passed all $X$ tests; there still might be a bug, but its presence is less likely now.\n\n3\\. $P(A):\\;\\;$ The patient could have any number of diseases. $P(A | X):\\;\\;$ Performing a blood test generated evidence $X$, ruling out some of the possible diseases from consideration.\n\n\nIt's clear that in each example we did not completely discard the prior belief after seeing new evidence $X$, but we *re-weighted the prior* to incorporate the new evidence (i.e. we put more weight, or confidence, on some beliefs versus others). \n\nBy introducing prior uncertainty about events, we are already admitting that any guess we make is potentially very wrong. After observing data, evidence, or other information, we update our beliefs, and our guess becomes *less wrong*. This is the alternative side of the prediction coin, where typically we try to be *more right*. \n\n\n\n### Bayesian Inference in Practice\n\n If frequentist and Bayesian inference were programming functions, with inputs being statistical problems, then the two would be different in what they return to the user. The frequentist inference function would return a number, representing an estimate (typically a summary statistic like the sample average etc.), whereas the Bayesian function would return *probabilities*.\n\nFor example, in our debugging problem above, calling the frequentist function with the argument \"My code passed all $X$ tests; is my code bug-free?\" would return a *YES*. On the other hand, asking our Bayesian function \"Often my code has bugs. My code passed all $X$ tests; is my code bug-free?\" would return something very different: probabilities of *YES* and *NO*. The function might return:\n\n\n> *YES*, with probability 0.8; *NO*, with probability 0.2\n\n\n\nThis is very different from the answer the frequentist function returned. Notice that the Bayesian function accepted an additional argument: *\"Often my code has bugs\"*. This parameter is the *prior*. By including the prior parameter, we are telling the Bayesian function to include our belief about the situation. Technically this parameter in the Bayesian function is optional, but we will see excluding it has its own consequences. \n\n\n#### Incorporating evidence\n\nAs we acquire more and more instances of evidence, our prior belief is *washed out* by the new evidence. This is to be expected. For example, if your prior belief is something ridiculous, like \"I expect the sun to explode today\", and each day you are proved wrong, you would hope that any inference would correct you, or at least align your beliefs better. Bayesian inference will correct this belief.\n\n\nDenote $N$ as the number of instances of evidence we possess. As we gather an *infinite* amount of evidence, say as $N \\rightarrow \\infty$, our Bayesian results (often) align with frequentist results. Hence for large $N$, statistical inference is more or less objective. On the other hand, for small $N$, inference is much more *unstable*: frequentist estimates have more variance and larger confidence intervals. This is where Bayesian analysis excels. By introducing a prior, and returning probabilities (instead of a scalar estimate), we *preserve the uncertainty* that reflects the instability of statistical inference of a small $N$ dataset. \n\nOne may think that for large $N$, one can be indifferent between the two techniques since they offer similar inference, and might lean towards the computationally-simpler, frequentist methods. An individual in this position should consider the following quote by Andrew Gelman (2005)[1], before making such a decision:\n\n> Sample sizes are never large. If $N$ is too small to get a sufficiently-precise estimate, you need to get more data (or make more assumptions). But once $N$ is \"large enough,\" you can start subdividing the data to learn more (for example, in a public opinion poll, once you have a good estimate for the entire country, you can estimate among men and women, northerners and southerners, different age groups, etc.). $N$ is never enough because if it were \"enough\" you'd already be on to the next problem for which you need more data.\n\n### Are frequentist methods incorrect then? \n\n**No.**\n\nFrequentist methods are still useful or state-of-the-art in many areas. Tools such as least squares linear regression, LASSO regression, and expectation-maximization algorithms are all powerful and fast. Bayesian methods complement these techniques by solving problems that these approaches cannot, or by illuminating the underlying system with more flexible modeling.\n\n\n#### A note on *Big Data*\nParadoxically, big data's predictive analytic problems are actually solved by relatively simple algorithms [2][4]. Thus we can argue that big data's prediction difficulty does not lie in the algorithm used, but instead on the computational difficulties of storage and execution on big data. (One should also consider Gelman's quote from above and ask \"Do I really have big data?\" )\n\nThe much more difficult analytic problems involve *medium data* and, especially troublesome, *really small data*. Using a similar argument as Gelman's above, if big data problems are *big enough* to be readily solved, then we should be more interested in the *not-quite-big enough* datasets. \n\n\n### Our Bayesian framework\n\nWe are interested in beliefs, which can be interpreted as probabilities by thinking Bayesian. We have a *prior* belief in event $A$, beliefs formed by previous information, e.g., our prior belief about bugs being in our code before performing tests.\n\nSecondly, we observe our evidence. To continue our buggy-code example: if our code passes $X$ tests, we want to update our belief to incorporate this. We call this new belief the *posterior* probability. Updating our belief is done via the following equation, known as Bayes' Theorem, after its discoverer Thomas Bayes:\n\n\\begin{align}\n P( A | X ) = & \\frac{ P(X | A) P(A) } {P(X) } \\\\\\\\[5pt]\n& \\propto P(X | A) P(A)\\;\\; (\\propto \\text{is proportional to } )\n\\end{align}\n\nThe above formula is not unique to Bayesian inference: it is a mathematical fact with uses outside Bayesian inference. Bayesian inference merely uses it to connect prior probabilities $P(A)$ with an updated posterior probabilities $P(A | X )$.\n\n##### Example: Mandatory coin-flip example\n\nEvery statistics text must contain a coin-flipping example, I'll use it here to get it out of the way. Suppose, naively, that you are unsure about the probability of heads in a coin flip (spoiler alert: it's 50%). You believe there is some true underlying ratio, call it $p$, but have no prior opinion on what $p$ might be. \n\nWe begin to flip a coin, and record the observations: either $H$ or $T$. This is our observed data. An interesting question to ask is how our inference changes as we observe more and more data? More specifically, what do our posterior probabilities look like when we have little data, versus when we have lots of data. \n\nBelow we plot a sequence of updating posterior probabilities as we observe increasing amounts of data (coin flips).\n\n\n```python\n\"\"\"\nThe book uses a custom matplotlibrc file, which provides the unique styles for\nmatplotlib plots. If executing this book, and you wish to use the book's\nstyling, provided are two options:\n 1. Overwrite your own matplotlibrc file with the rc-file provided in the\n book's styles/ dir. See http://matplotlib.org/users/customizing.html\n 2. Also in the styles is bmh_matplotlibrc.json file. This can be used to\n update the styles in only this notebook. Try running the following code:\n\n import json, matplotlib\n s = json.load( open(\"../styles/bmh_matplotlibrc.json\") )\n matplotlib.rcParams.update(s)\n\n\"\"\"\n\n# The code below can be passed over, as it is currently not important, plus it\n# uses advanced topics we have not covered yet. LOOK AT PICTURE, MICHAEL!\n%matplotlib inline\nfrom IPython.core.pylabtools import figsize\nimport numpy as np\nfrom matplotlib import pyplot as plt\nfigsize(11, 9)\n\nimport scipy.stats as stats\n\ndist = stats.beta\nn_trials = [0, 1, 2, 3, 4, 5, 8, 15, 50, 500]\ndata = stats.bernoulli.rvs(0.5, size=n_trials[-1])\nx = np.linspace(0, 1, 100)\n\n# For the already prepared, I'm using Binomial's conj. prior.\nfor k, N in enumerate(n_trials):\n sx = plt.subplot(len(n_trials) / 2, 2, k + 1)\n plt.xlabel(\"$p$, probability of heads\") \\\n if k in [0, len(n_trials) - 1] else None\n plt.setp(sx.get_yticklabels(), visible=False)\n heads = data[:N].sum()\n y = dist.pdf(x, 1 + heads, 1 + N - heads)\n plt.plot(x, y, label=\"observe %d tosses,\\n %d heads\" % (N, heads))\n plt.fill_between(x, 0, y, color=\"#348ABD\", alpha=0.4)\n plt.vlines(0.5, 0, 4, color=\"k\", linestyles=\"--\", lw=1)\n\n leg = plt.legend()\n leg.get_frame().set_alpha(0.4)\n plt.autoscale(tight=True)\n\n\nplt.suptitle(\"Bayesian updating of posterior probabilities\",\n y=1.02,\n fontsize=14)\n\nplt.tight_layout()\n```\n\nThe posterior probabilities are represented by the curves, and our uncertainty is proportional to the width of the curve. As the plot above shows, as we start to observe data our posterior probabilities start to shift and move around. Eventually, as we observe more and more data (coin-flips), our probabilities will tighten closer and closer around the true value of $p=0.5$ (marked by a dashed line). \n\nNotice that the plots are not always *peaked* at 0.5. There is no reason it should be: recall we assumed we did not have a prior opinion of what $p$ is. In fact, if we observe quite extreme data, say 8 flips and only 1 observed heads, our distribution would look very biased *away* from lumping around 0.5 (with no prior opinion, how confident would you feel betting on a fair coin after observing 8 tails and 1 head). As more data accumulates, we would see more and more probability being assigned at $p=0.5$, though never all of it.\n\nThe next example is a simple demonstration of the mathematics of Bayesian inference. \n\n##### Example: Bug, or just sweet, unintended feature?\n\n\nLet $A$ denote the event that our code has **no bugs** in it. Let $X$ denote the event that the code passes all debugging tests. For now, we will leave the prior probability of no bugs as a variable, i.e. $P(A) = p$. \n\nWe are interested in $P(A|X)$, i.e. the probability of no bugs, given our debugging tests $X$ pass. To use the formula above, we need to compute some quantities.\n\nWhat is $P(X | A)$, i.e., the probability that the code passes $X$ tests *given* there are no bugs? Well, it is equal to 1, for code with no bugs will pass all tests. \n\n$P(X)$ is a little bit trickier: The event $X$ can be divided into two possibilities, event $X$ occurring even though our code *indeed has* bugs (denoted $\\sim A\\;$, spoken *not $A$*), or event $X$ without bugs ($A$). $P(X)$ can be represented as:\n\n\\begin{align}\nP(X ) & = P(X \\text{ and } A) + P(X \\text{ and } \\sim A) \\\\\\\\[5pt]\n & = P(X|A)P(A) + P(X | \\sim A)P(\\sim A)\\\\\\\\[5pt]\n& = P(X|A)p + P(X | \\sim A)(1-p)\n\\end{align}\n\nWe have already computed $P(X|A)$ above. On the other hand, $P(X | \\sim A)$ is subjective: our code can pass tests but still have a bug in it, though the probability there is a bug present is reduced. Note this is dependent on the number of tests performed, the degree of complication in the tests, etc. Let's be conservative and assign $P(X|\\sim A) = 0.5$. Then\n\n\\begin{align}\nP(A | X) & = \\frac{1\\cdot p}{ 1\\cdot p +0.5 (1-p) } \\\\\\\\\n& = \\frac{ 2 p}{1+p}\n\\end{align}\nThis is the posterior probability. What does it look like as a function of our prior, $p \\in [0,1]$? \n\n\n```python\nfigsize(12.5, 4)\np = np.linspace(0, 1, 50)\nplt.plot(p, 2 * p / (1 + p), color=\"#348ABD\", lw=3)\n# plt.fill_between(p, 2*p/(1+p), alpha=.5, facecolor=[\"#A60628\"])\nplt.scatter(0.2, 2 * (0.2) / 1.2, s=140, c=\"#348ABD\")\nplt.xlim(0, 1)\nplt.ylim(0, 1)\nplt.xlabel(\"Prior, $P(A) = p$\")\nplt.ylabel(\"Posterior, $P(A|X)$, with $P(A) = p$\")\nplt.title(\"Is my code bug-free?\")\n```\n\nWe can see the biggest gains if we observe the $X$ tests passed when the prior probability, $p$, is low. Let's settle on a specific value for the prior. I'm a strong programmer (I think), so I'm going to give myself a realistic prior of 0.20, that is, there is a 20% chance that I write code bug-free. To be more realistic, this prior should be a function of how complicated and large the code is, but let's pin it at 0.20. Then my updated belief that my code is bug-free is 0.33. \n\nRecall that the prior is a probability: $p$ is the prior probability that there *are no bugs*, so $1-p$ is the prior probability that there *are bugs*.\n\nSimilarly, our posterior is also a probability, with $P(A | X)$ the probability there is no bug *given we saw all tests pass*, hence $1-P(A|X)$ is the probability there is a bug *given all tests passed*. What does our posterior probability look like? Below is a chart of both the prior and the posterior probabilities. \n\n\n\n```python\nfigsize(12.5, 4)\ncolours = [\"#348ABD\", \"#A60628\"]\n\nprior = [0.20, 0.80]\nposterior = [1. / 3, 2. / 3]\nplt.bar([0, .7], prior, alpha=0.70, width=0.25,\n color=colours[0], label=\"prior distribution\",\n lw=\"3\", edgecolor=colours[0])\n\nplt.bar([0 + 0.25, .7 + 0.25], posterior, alpha=0.7,\n width=0.25, color=colours[1],\n label=\"posterior distribution\",\n lw=\"3\", edgecolor=colours[1])\n\nplt.ylim(0,1)\nplt.xticks([0.20, .95], [\"Bugs Absent\", \"Bugs Present\"])\nplt.title(\"Prior and Posterior probability of bugs present\")\nplt.ylabel(\"Probability\")\nplt.legend(loc=\"upper left\");\n```\n\nNotice that after we observed $X$ occur, the probability of bugs being absent increased. By increasing the number of tests, we can approach confidence (probability 1) that there are no bugs present.\n\nThis was a very simple example of Bayesian inference and Bayes rule. Unfortunately, the mathematics necessary to perform more complicated Bayesian inference only becomes more difficult, except for artificially constructed cases. We will later see that this type of mathematical analysis is actually unnecessary. First we must broaden our modeling tools. The next section deals with *probability distributions*. If you are already familiar, feel free to skip (or at least skim), but for the less familiar the next section is essential.\n\n_______\n\n## Probability Distributions\n\n\n**Let's quickly recall what a probability distribution is:** Let $Z$ be some random variable. Then associated with $Z$ is a *probability distribution function* that assigns probabilities to the different outcomes $Z$ can take. Graphically, a probability distribution is a curve where the probability of an outcome is proportional to the height of the curve. You can see examples in the first figure of this chapter. \n\nWe can divide random variables into three classifications:\n\n- **$Z$ is discrete**: Discrete random variables may only assume values on a specified list. Things like populations, movie ratings, and number of votes are all discrete random variables. Discrete random variables become more clear when we contrast them with...\n\n- **$Z$ is continuous**: Continuous random variable can take on arbitrarily exact values. For example, temperature, speed, time, color are all modeled as continuous variables because you can progressively make the values more and more precise.\n\n- **$Z$ is mixed**: Mixed random variables assign probabilities to both discrete and continuous random variables, i.e. it is a combination of the above two categories. \n\n#### Expected Value\nExpected value (EV) is one of the most important concepts in probability. The EV for a given probability distribution can be described as \"the mean value in the long run for many repeated samples from that distribution.\" To borrow a metaphor from physics, a distribution's EV acts like its \"center of mass.\" Imagine repeating the same experiment many times over, and taking the average over each outcome. The more you repeat the experiment, the closer this average will become to the distributions EV. (side note: as the number of repeated experiments goes to infinity, the difference between the average outcome and the EV becomes arbitrarily small.)\n\n### Discrete Case\nIf $Z$ is discrete, then its distribution is called a *probability mass function*, which measures the probability $Z$ takes on the value $k$, denoted $P(Z=k)$. Note that the probability mass function completely describes the random variable $Z$, that is, if we know the mass function, we know how $Z$ should behave. There are popular probability mass functions that consistently appear: we will introduce them as needed, but let's introduce the first very useful probability mass function. We say $Z$ is *Poisson*-distributed if:\n\n$$P(Z = k) =\\frac{ \\lambda^k e^{-\\lambda} }{k!}, \\; \\; k=0,1,2, \\dots, \\; \\; \\lambda \\in \\mathbb{R}_{>0} $$\n\n$\\lambda$ is called a parameter of the distribution, and it controls the distribution's shape. For the Poisson distribution, $\\lambda$ can be any positive number. By increasing $\\lambda$, we add more probability to larger values, and conversely by decreasing $\\lambda$ we add more probability to smaller values. One can describe $\\lambda$ as the *intensity* of the Poisson distribution. \n\nUnlike $\\lambda$, which can be any positive number, the value $k$ in the above formula must be a non-negative integer, i.e., $k$ must take on values 0,1,2, and so on. This is very important, because if you wanted to model a population you could not make sense of populations with 4.25 or 5.612 members. \n\nIf a random variable $Z$ has a Poisson mass distribution, we denote this by writing\n\n$$Z \\sim \\text{Poi}(\\lambda) $$\n\nOne useful property of the Poisson distribution is that its expected value is equal to its parameter, i.e.:\n\n$$E\\large[ \\;Z\\; | \\; \\lambda \\;\\large] = \\lambda $$\n\nWe will use this property often, so it's useful to remember. Below, we plot the probability mass distribution for different $\\lambda$ values. The first thing to notice is that by increasing $\\lambda$, we add more probability of larger values occurring. Second, notice that although the graph ends at 15, the distributions do not. They assign positive probability to every non-negative integer.\n\n\n```python\nfigsize(12.5, 4)\n\nimport scipy.stats as stats\na = np.arange(16)\npoi = stats.poisson\nlambda_ = [1.5, 4.25]\ncolours = [\"#348ABD\", \"#A60628\"]\n\nplt.bar(a, poi.pmf(a, lambda_[0]), color=colours[0],\n label=\"$\\lambda = %.1f$\" % lambda_[0], alpha=0.60,\n edgecolor=colours[0], lw=\"3\")\n\nplt.bar(a, poi.pmf(a, lambda_[1]), color=colours[1],\n label=\"$\\lambda = %.1f$\" % lambda_[1], alpha=0.60,\n edgecolor=colours[1], lw=\"3\")\n\nplt.xticks(a + 0.4, a)\nplt.legend()\nplt.ylabel(\"probability of $k$\")\nplt.xlabel(\"$k$\")\nplt.title(\"Probability mass function of a Poisson random variable; differing \\\n$\\lambda$ values\")\n```\n\n### Continuous Case\nInstead of a probability mass function, a continuous random variable has a *probability density function*. This might seem like unnecessary nomenclature, but the density function and the mass function are very different creatures. An example of continuous random variable is a random variable with *exponential density*. The density function for an exponential random variable looks like this:\n\n$$f_Z(z | \\lambda) = \\lambda e^{-\\lambda z }, \\;\\; z\\ge 0$$\n\nLike a Poisson random variable, an exponential random variable can take on only non-negative values. But unlike a Poisson variable, the exponential can take on *any* non-negative values, including non-integral values such as 4.25 or 5.612401. This property makes it a poor choice for count data, which must be an integer, but a great choice for time data, temperature data (measured in Kelvins, of course), or any other precise *and positive* variable. The graph below shows two probability density functions with different $\\lambda$ values. \n\nWhen a random variable $Z$ has an exponential distribution with parameter $\\lambda$, we say *$Z$ is exponential* and write\n\n$$Z \\sim \\text{Exp}(\\lambda)$$\n\nGiven a specific $\\lambda$, the expected value of an exponential random variable is equal to the inverse of $\\lambda$, that is:\n\n$$E[\\; Z \\;|\\; \\lambda \\;] = \\frac{1}{\\lambda}$$\n\n\n```python\na = np.linspace(0, 4, 100)\nexpo = stats.expon\nlambda_ = [0.5, 1]\n\nfor l, c in zip(lambda_, colours):\n plt.plot(a, expo.pdf(a, scale=1. / l), lw=3,\n color=c, label=\"$\\lambda = %.1f$\" % l)\n plt.fill_between(a, expo.pdf(a, scale=1. / l), color=c, alpha=.33)\n\nplt.legend()\nplt.ylabel(\"PDF at $z$\")\nplt.xlabel(\"$z$\")\nplt.ylim(0, 1.2)\nplt.title(\"Probability density function of an Exponential random variable;\\\n differing $\\lambda$\");\n```\n\n\n### But what is $\\lambda \\;$?\n\n\n**This question is what motivates statistics**. In the real world, $\\lambda$ is hidden from us. We see only $Z$, and must go backwards to try and determine $\\lambda$. The problem is difficult because there is no one-to-one mapping from $Z$ to $\\lambda$. Many different methods have been created to solve the problem of estimating $\\lambda$, but since $\\lambda$ is never actually observed, no one can say for certain which method is best! \n\nBayesian inference is concerned with *beliefs* about what $\\lambda$ might be. Rather than try to guess $\\lambda$ exactly, we can only talk about what $\\lambda$ is likely to be by assigning a probability distribution to $\\lambda$.\n\nThis might seem odd at first. After all, $\\lambda$ is fixed; it is not (necessarily) random! How can we assign probabilities to values of a non-random variable? Ah, we have fallen for our old, frequentist way of thinking. Recall that under Bayesian philosophy, we *can* assign probabilities if we interpret them as beliefs. And it is entirely acceptable to have *beliefs* about the parameter $\\lambda$. \n\n\n\n##### Example: Inferring behaviour from text-message data\n\nLet's try to model a more interesting example, one that concerns the rate at which a user sends and receives text messages:\n\n> You are given a series of daily text-message counts from a user of your system. The data, plotted over time, appears in the chart below. You are curious to know if the user's text-messaging habits have changed over time, either gradually or suddenly. How can you model this? (This is in fact my own text-message data. Judge my popularity as you wish.)\n\n\n\n```python\nfigsize(12.5, 3.5)\ncount_data = np.loadtxt(\"data/txtdata.csv\")\nn_count_data = len(count_data)\nplt.bar(np.arange(n_count_data), count_data, color=\"#348ABD\")\nplt.xlabel(\"Time (days)\")\nplt.ylabel(\"count of text-msgs received\")\nplt.title(\"Did the user's texting habits change over time?\")\nplt.xlim(0, n_count_data);\n```\n\nBefore we start modeling, see what you can figure out just by looking at the chart above. Would you say there was a change in behaviour during this time period? \n\nHow can we start to model this? Well, as we have conveniently already seen, a Poisson random variable is a very appropriate model for this type of *count* data. Denoting day $i$'s text-message count by $C_i$, \n\n$$ C_i \\sim \\text{Poisson}(\\lambda) $$\n\nWe are not sure what the value of the $\\lambda$ parameter really is, however. Looking at the chart above, it appears that the rate might become higher late in the observation period, which is equivalent to saying that $\\lambda$ increases at some point during the observations. (Recall that a higher value of $\\lambda$ assigns more probability to larger outcomes. That is, there is a higher probability of many text messages having been sent on a given day.)\n\nHow can we represent this observation mathematically? Let's assume that on some day during the observation period (call it $\\tau$), the parameter $\\lambda$ suddenly jumps to a higher value. So we really have two $\\lambda$ parameters: one for the period before $\\tau$, and one for the rest of the observation period. In the literature, a sudden transition like this would be called a *switchpoint*:\n\n$$\n\\lambda = \n\\begin{cases}\n\\lambda_1 & \\text{if } t \\lt \\tau \\cr\n\\lambda_2 & \\text{if } t \\ge \\tau\n\\end{cases}\n$$\n\n\nIf, in reality, no sudden change occurred and indeed $\\lambda_1 = \\lambda_2$, then the $\\lambda$s posterior distributions should look about equal.\n\nWe are interested in inferring the unknown $\\lambda$s. To use Bayesian inference, we need to assign prior probabilities to the different possible values of $\\lambda$. What would be good prior probability distributions for $\\lambda_1$ and $\\lambda_2$? Recall that $\\lambda$ can be any positive number. As we saw earlier, the *exponential* distribution provides a continuous density function for positive numbers, so it might be a good choice for modeling $\\lambda_i$. But recall that the exponential distribution takes a parameter of its own, so we'll need to include that parameter in our model. Let's call that parameter $\\alpha$.\n\n\\begin{align}\n&\\lambda_1 \\sim \\text{Exp}( \\alpha ) \\\\\\\n&\\lambda_2 \\sim \\text{Exp}( \\alpha )\n\\end{align}\n\n$\\alpha$ is called a *hyper-parameter* or *parent variable*. In literal terms, it is a parameter that influences other parameters. Our initial guess at $\\alpha$ does not influence the model too strongly, so we have some flexibility in our choice. A good rule of thumb is to set the exponential parameter equal to the inverse of the average of the count data. Since we're modeling $\\lambda$ using an exponential distribution, we can use the expected value identity shown earlier to get:\n\n$$\\frac{1}{N}\\sum_{i=0}^N \\;C_i \\approx E[\\; \\lambda \\; |\\; \\alpha ] = \\frac{1}{\\alpha}$$ \n\nAn alternative, and something I encourage the reader to try, would be to have two priors: one for each $\\lambda_i$. Creating two exponential distributions with different $\\alpha$ values reflects our prior belief that the rate changed at some point during the observations.\n\nWhat about $\\tau$? Because of the noisiness of the data, it's difficult to pick out a priori when $\\tau$ might have occurred. Instead, we can assign a *uniform prior belief* to every possible day. This is equivalent to saying\n\n\\begin{align}\n& \\tau \\sim \\text{DiscreteUniform(1,70) }\\\\\\\\\n& \\Rightarrow P( \\tau = k ) = \\frac{1}{70}\n\\end{align}\n\nSo after all this, what does our overall prior distribution for the unknown variables look like? Frankly, *it doesn't matter*. What we should understand is that it's an ugly, complicated mess involving symbols only a mathematician could love. And things will only get uglier the more complicated our models become. Regardless, all we really care about is the posterior distribution.\n\nWe next turn to PyMC, a Python library for performing Bayesian analysis that is undaunted by the mathematical monster we have created. \n\n\nIntroducing our first hammer: PyMC\n-----\n\nPyMC is a Python library for programming Bayesian analysis [3]. It is a fast, well-maintained library. The only unfortunate part is that its documentation is lacking in certain areas, especially those that bridge the gap between beginner and hacker. One of this book's main goals is to solve that problem, and also to demonstrate why PyMC is so cool.\n\nWe will model the problem above using PyMC. This type of programming is called *probabilistic programming*, an unfortunate misnomer that invokes ideas of randomly-generated code and has likely confused and frightened users away from this field. The code is not random; it is probabilistic in the sense that we create probability models using programming variables as the model's components. Model components are first-class primitives within the PyMC framework. \n\nB. Cronin [5] has a very motivating description of probabilistic programming:\n\n> Another way of thinking about this: unlike a traditional program, which only runs in the forward directions, a probabilistic program is run in both the forward and backward direction. It runs forward to compute the consequences of the assumptions it contains about the world (i.e., the model space it represents), but it also runs backward from the data to constrain the possible explanations. In practice, many probabilistic programming systems will cleverly interleave these forward and backward operations to efficiently home in on the best explanations.\n\nBecause of the confusion engendered by the term *probabilistic programming*, I'll refrain from using it. Instead, I'll simply say *programming*, since that's what it really is. \n\nPyMC code is easy to read. The only novel thing should be the syntax, and I will interrupt the code to explain individual sections. Simply remember that we are representing the model's components ($\\tau, \\lambda_1, \\lambda_2$ ) as variables:\n\n\n```python\nimport pymc as pm\n\nalpha = 1.0 / count_data.mean() # Recall count_data is the\n # variable that holds our txt counts\nlambda_1 = pm.Exponential(\"lambda_1\", alpha)\nlambda_2 = pm.Exponential(\"lambda_2\", alpha)\n\ntau = pm.DiscreteUniform(\"tau\", lower=0, upper=n_count_data)\n```\n\nIn the code above, we create the PyMC variables corresponding to $\\lambda_1$ and $\\lambda_2$. We assign them to PyMC's *stochastic variables*, so-called because they are treated by the back end as random number generators. We can demonstrate this fact by calling their built-in `random()` methods.\n\n\n```python\nprint(\"Random output:\", tau.random(), tau.random(), tau.random())\n```\n\n Random output: 64 5 12\n\n\n\n```python\n@pm.deterministic\ndef lambda_(tau=tau, lambda_1=lambda_1, lambda_2=lambda_2):\n out = np.zeros(n_count_data)\n out[:tau] = lambda_1 # lambda before tau is lambda1\n out[tau:] = lambda_2 # lambda after (and including) tau is lambda2\n return out\n```\n\nThis code creates a new function `lambda_`, but really we can think of it as a random variable: the random variable $\\lambda$ from above. Note that because `lambda_1`, `lambda_2` and `tau` are random, `lambda_` will be random. We are **not** fixing any variables yet.\n\n`@pm.deterministic` is a decorator that tells PyMC this is a deterministic function. That is, if the arguments were deterministic (which they are not), the output would be deterministic as well. Deterministic functions will be covered in Chapter 2. \n\n\n```python\nobservation = pm.Poisson(\"obs\", lambda_, value=count_data, observed=True)\n\nmodel = pm.Model([observation, lambda_1, lambda_2, tau])\n```\n\nThe variable `observation` combines our data, `count_data`, with our proposed data-generation scheme, given by the variable `lambda_`, through the `value` keyword. We also set `observed = True` to tell PyMC that this should stay fixed in our analysis. Finally, PyMC wants us to collect all the variables of interest and create a `Model` instance out of them. This makes our life easier when we retrieve the results.\n\nThe code below will be explained in Chapter 3, but I show it here so you can see where our results come from. One can think of it as a *learning* step. The machinery being employed is called *Markov Chain Monte Carlo* (MCMC), which I also delay explaining until Chapter 3. This technique returns thousands of random variables from the posterior distributions of $\\lambda_1, \\lambda_2$ and $\\tau$. We can plot a histogram of the random variables to see what the posterior distributions look like. Below, we collect the samples (called *traces* in the MCMC literature) into histograms.\n\n\n```python\n# Mysterious code to be explained in Chapter 3.\nmcmc = pm.MCMC(model)\nmcmc.sample(40000, 10000, 1)\n```\n\n [-----------------100%-----------------] 40000 of 40000 complete in 6.5 sec\n\n\n```python\nlambda_1_samples = mcmc.trace('lambda_1')[:]\nlambda_2_samples = mcmc.trace('lambda_2')[:]\ntau_samples = mcmc.trace('tau')[:]\n```\n\n\n```python\nfigsize(12.5, 10)\n# histogram of the samples:\n\nax = plt.subplot(311)\nax.set_autoscaley_on(False)\n\nplt.hist(lambda_1_samples, histtype='stepfilled', bins=30, alpha=0.85,\n label=\"posterior of $\\lambda_1$\", color=\"#A60628\", normed=True)\nplt.legend(loc=\"upper left\")\nplt.title(r\"\"\"Posterior distributions of the variables\n $\\lambda_1,\\;\\lambda_2,\\;\\tau$\"\"\")\nplt.xlim([15, 30])\nplt.xlabel(\"$\\lambda_1$ value\")\n\nax = plt.subplot(312)\nax.set_autoscaley_on(False)\nplt.hist(lambda_2_samples, histtype='stepfilled', bins=30, alpha=0.85,\n label=\"posterior of $\\lambda_2$\", color=\"#7A68A6\", normed=True)\nplt.legend(loc=\"upper left\")\nplt.xlim([15, 30])\nplt.xlabel(\"$\\lambda_2$ value\")\n\nplt.subplot(313)\nw = 1.0 / tau_samples.shape[0] * np.ones_like(tau_samples)\nplt.hist(tau_samples, bins=n_count_data, alpha=1,\n label=r\"posterior of $\\tau$\",\n color=\"#467821\", weights=w, rwidth=2.)\nplt.xticks(np.arange(n_count_data))\n\nplt.legend(loc=\"upper left\")\nplt.ylim([0, .75])\nplt.xlim([35, len(count_data) - 20])\nplt.xlabel(r\"$\\tau$ (in days)\")\nplt.ylabel(\"probability\");\n```\n\n### Interpretation\n\nRecall that Bayesian methodology returns a *distribution*. Hence we now have distributions to describe the unknown $\\lambda$s and $\\tau$. What have we gained? Immediately, we can see the uncertainty in our estimates: the wider the distribution, the less certain our posterior belief should be. We can also see what the plausible values for the parameters are: $\\lambda_1$ is around 18 and $\\lambda_2$ is around 23. The posterior distributions of the two $\\lambda$s are clearly distinct, indicating that it is indeed likely that there was a change in the user's text-message behaviour.\n\nWhat other observations can you make? If you look at the original data again, do these results seem reasonable? \n\nNotice also that the posterior distributions for the $\\lambda$s do not look like exponential distributions, even though our priors for these variables were exponential. In fact, the posterior distributions are not really of any form that we recognize from the original model. But that's OK! This is one of the benefits of taking a computational point of view. If we had instead done this analysis using mathematical approaches, we would have been stuck with an analytically intractable (and messy) distribution. Our use of a computational approach makes us indifferent to mathematical tractability.\n\nOur analysis also returned a distribution for $\\tau$. Its posterior distribution looks a little different from the other two because it is a discrete random variable, so it doesn't assign probabilities to intervals. We can see that near day 45, there was a 50% chance that the user's behaviour changed. Had no change occurred, or had the change been gradual over time, the posterior distribution of $\\tau$ would have been more spread out, reflecting that many days were plausible candidates for $\\tau$. By contrast, in the actual results we see that only three or four days make any sense as potential transition points. \n\n### Why would I want samples from the posterior, anyways?\n\n\nWe will deal with this question for the remainder of the book, and it is an understatement to say that it will lead us to some amazing results. For now, let's end this chapter with one more example.\n\nWe'll use the posterior samples to answer the following question: what is the expected number of texts at day $t, \\; 0 \\le t \\le 70$ ? Recall that the expected value of a Poisson variable is equal to its parameter $\\lambda$. Therefore, the question is equivalent to *what is the expected value of $\\lambda$ at time $t$*?\n\nIn the code below, let $i$ index samples from the posterior distributions. Given a day $t$, we average over all possible $\\lambda_i$ for that day $t$, using $\\lambda_i = \\lambda_{1,i}$ if $t \\lt \\tau_i$ (that is, if the behaviour change has not yet occurred), else we use $\\lambda_i = \\lambda_{2,i}$. \n\n\n```python\nfigsize(12.5, 5)\n# tau_samples, lambda_1_samples, lambda_2_samples contain\n# N samples from the corresponding posterior distribution\nN = tau_samples.shape[0]\nexpected_texts_per_day = np.zeros(n_count_data)\nfor day in range(0, n_count_data):\n # ix is a bool index of all tau samples corresponding to\n # the switchpoint occurring prior to value of 'day'\n ix = day < tau_samples\n # Each posterior sample corresponds to a value for tau.\n # for each day, that value of tau indicates whether we're \"before\"\n # (in the lambda1 \"regime\") or\n # \"after\" (in the lambda2 \"regime\") the switchpoint.\n # by taking the posterior sample of lambda1/2 accordingly, we can average\n # over all samples to get an expected value for lambda on that day.\n # As explained, the \"message count\" random variable is Poisson distributed,\n # and therefore lambda (the poisson parameter) is the expected value of\n # \"message count\".\n expected_texts_per_day[day] = (lambda_1_samples[ix].sum()\n + lambda_2_samples[~ix].sum()) / N\n\n\nplt.plot(range(n_count_data), expected_texts_per_day, lw=4, color=\"#E24A33\",\n label=\"expected number of text-messages received\")\nplt.xlim(0, n_count_data)\nplt.xlabel(\"Day\")\nplt.ylabel(\"Expected # text-messages\")\nplt.title(\"Expected number of text-messages received\")\nplt.ylim(0, 60)\nplt.bar(np.arange(len(count_data)), count_data, color=\"#348ABD\", alpha=0.65,\n label=\"observed texts per day\")\n\nplt.legend(loc=\"upper left\");\n```\n\nOur analysis shows strong support for believing the user's behavior did change ($\\lambda_1$ would have been close in value to $\\lambda_2$ had this not been true), and that the change was sudden rather than gradual (as demonstrated by $\\tau$'s strongly peaked posterior distribution). We can speculate what might have caused this: a cheaper text-message rate, a recent weather-to-text subscription, or perhaps a new relationship. (In fact, the 45th day corresponds to Christmas, and I moved away to Toronto the next month, leaving a girlfriend behind.)\n\n\n##### Exercises\n\n1\\. Using `lambda_1_samples` and `lambda_2_samples`, what is the mean of the posterior distributions of $\\lambda_1$ and $\\lambda_2$?\n\n\n```python\n# type your code here.\n```\n\n2\\. What is the expected percentage increase in text-message rates? `hint:` compute the mean of `lambda_1_samples/lambda_2_samples`. Note that this quantity is very different from `lambda_1_samples.mean()/lambda_2_samples.mean()`.\n\n\n```python\n# type your code here.\n```\n\n3\\. What is the mean of $\\lambda_1$ **given** that we know $\\tau$ is less than 45. That is, suppose we have been given new information that the change in behaviour occurred prior to day 45. What is the expected value of $\\lambda_1$ now? (You do not need to redo the PyMC part. Just consider all instances where `tau_samples < 45`.)\n\n\n```python\n# type your code here.\n```\n\n### References\n\n\n- [1] Gelman, Andrew. N.p.. Web. 22 Jan 2013. [N is never large enough](http://andrewgelman.com/2005/07/31/n_is_never_larg/).\n- [2] Norvig, Peter. 2009. [The Unreasonable Effectiveness of Data](http://static.googleusercontent.com/media/research.google.com/en//pubs/archive/35179.pdf).\n- [3] Patil, A., D. Huard and C.J. Fonnesbeck. 2010. \nPyMC: Bayesian Stochastic Modelling in Python. Journal of Statistical \nSoftware, 35(4), pp. 1-81. \n- [4] Jimmy Lin and Alek Kolcz. Large-Scale Machine Learning at Twitter. Proceedings of the 2012 ACM SIGMOD International Conference on Management of Data (SIGMOD 2012), pages 793-804, May 2012, Scottsdale, Arizona.\n- [5] Cronin, Beau. \"Why Probabilistic Programming Matters.\" 24 Mar 2013. Google, Online Posting to Google . Web. 24 Mar. 2013. .\n\n\n```python\nfrom IPython.core.display import HTML\n\n\ndef css_styling():\n styles = open(\"../styles/custom.css\", \"r\").read()\n return HTML(styles)\ncss_styling()\n```\n\n\n\n\n\n\n\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "1bd11b8c28abf2d22e0cdac7600d5694958af337", "size": 342815, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "1.3.BayesianInference/exercises/srcs/2/lab2.ipynb", "max_stars_repo_name": "mihaighidoveanu/machine-learning-examples", "max_stars_repo_head_hexsha": "e5a7ab71e52ae2809115eb7d7c943b46ebf394f3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "1.3.BayesianInference/exercises/srcs/2/lab2.ipynb", "max_issues_repo_name": "mihaighidoveanu/machine-learning-examples", "max_issues_repo_head_hexsha": "e5a7ab71e52ae2809115eb7d7c943b46ebf394f3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "1.3.BayesianInference/exercises/srcs/2/lab2.ipynb", "max_forks_repo_name": "mihaighidoveanu/machine-learning-examples", "max_forks_repo_head_hexsha": "e5a7ab71e52ae2809115eb7d7c943b46ebf394f3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-05-02T13:12:21.000Z", "max_forks_repo_forks_event_max_datetime": "2021-05-02T13:12:21.000Z", "avg_line_length": 312.2176684882, "max_line_length": 89232, "alphanum_fraction": 0.8998701924, "converted": true, "num_tokens": 11683, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.7799929053683038, "lm_q1q2_score": 0.43549109445720124}} {"text": "\n\n# Xarray in 45 minutes\n\nIn this lesson, we discuss cover the basics of Xarray data structures. By the\nend of the lesson, we will be able to:\n\n- Understand the basic data structures in Xarray\n- Inspect `DataArray` and `Dataset` objects.\n- Read and write netCDF files using Xarray.\n- Understand that there are many packages that build on top of xarray\n\n\n## A practical example\n\n\n\n```python\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport xarray as xr\n\n%matplotlib inline\n```\n\n\n```python\n# load tutorial dataset\nds = xr.tutorial.load_dataset(\"air_temperature\")\n```\n\n## What's in a dataset? many DataArrays\n\n\n\n```python\n# dataset repr\nds\n```\n\nDatasets are dict-like containers of DataArrays i.e. they are a mapping of\nvariable name to DataArray.\n\n\n\n```python\n# pull out \"air\" dataarray with dictionary syntax\nds[\"air\"]\n```\n\nYou can save some typing by using the \"attribute\" or \"dot\" notation. This won't\nwork for variable names that clash with a built-in method name (like `mean` for\nexample).\n\n\n\n```python\n# pull out dataarray using dot notation\nds.air ## same as ds[\"air\"]\n```\n\n## What's in a DataArray? data + (a lot of) metadata\n\n### Named dimensions `.dims`\n\n\n\n```python\nds.air.dims\n```\n\n### Coordinate variables or \"tick labels\" (`.coords`)\n\n`.coords` is a simple\n[data container](https://xarray.pydata.org/en/stable/data-structures.html#coordinates)\nfor coordinate variables.\n\n\n\n```python\nds.air.coords\n```\n\nCoordinates objects support similar indexing notation\n\n\n\n```python\n# extracting coordinate variables\nds.air.lon\n```\n\n\n```python\n# extracting coorindate variables from .coords\nds.coords[\"lon\"]\n```\n\nIt is useful to think of the values in these coordinate variables as axis\n\"labels\" such as \"tick labels\" in a figure. These are coordinate locations on a\ngrid at which you have data.\n\n\n### Arbitrary attributes (`.attrs`)\n\n`.attrs` is a dictionary that can contain arbitrary python objects. Your only\nlimitation is that some attributes may not be writeable to a netCDF file\n\n\n\n```python\nds.air.attrs\n```\n\n\n```python\n# assign your own attribute\nds.air.attrs[\"who_is_awesome\"] = \"xarray\"\nds.air.attrs\n```\n\n### Underlying data (`.data`)\n\nXarray structures wrap underlying simpler data structures. In this case, the\nunderlying data is a numpy array which you may be familiar with.\n\nThis part of xarray is quite extensible allowing for GPU arrays, sparse arrays,\narrays with units etc. See the demo at the end.\n\n\n\n```python\nds.air.data\n```\n\n\n```python\n# what is the type of the underlying data\ntype(ds.air.data)\n```\n\nA numpy array!\n\n\n\n\n### Review\n\nXarray provides two main data structures\n\n- DataArrays that wrap underlying data containers (e.g. numpy arrays) and\n contain associated metadata\n- Datasets that are dict-like containers of DataArrays\n\nFor more see\n\n- https://xarray.pydata.org/en/stable/data-structures.html#dataset\n- https://xarray.pydata.org/en/stable/data-structures.html#dataarray\n\n\n---\n\n## Why xarray? Use metadata for fun and ~profit~ papers!\n\n### Analysis without xarray: `X(`\n\n\n\n```python\n# plot the first timestep\nlat = ds.air.lat.data # numpy array\nlon = ds.air.lon.data # numpy array\ntemp = ds.air.data # numpy array\nplt.figure()\nplt.pcolormesh(lon, lat, temp[0, :, :])\n```\n\n\n```python\ntemp.mean(axis=1) ## what did I just do? I can't tell by looking at this line.\n```\n\n### Analysis with xarray `=)`\n\nHow readable is this code?\n\n\n\n```python\nds.air.isel(time=1).plot(x=\"lon\")\n```\n\nUse dimension names instead of axis numbers\n\n\n\n```python\nds.air.mean(\"time\")\n```\n\n---\n\n## Extracting data or \"indexing\" : `.sel`, `.isel`\n\nXarray supports\n\n- label-based indexing using `.sel`\n- position-based indexing using `.isel`\n\nFor more see https://xarray.pydata.org/en/stable/indexing.html\n\n\n### Label-based indexing\n\nXarray inherits its label-based indexing rules from pandas; this means great\nsupport for dates and times!\n\n\n\n```python\n# pull out data for all of 2013-May\nds.sel(time=\"2013-05\")\n```\n\n\n```python\n# demonstrate slicing\nds.sel(time=slice(\"2013-05\", \"2013-07\"))\n```\n\n\n```python\n# demonstrate \"nearest\" indexing\nds.sel(lon=240.2, method=\"nearest\")\n```\n\n\n```python\n# \"nearest indexing at multiple points\"\nds.sel(lon=[240.125, 234], lat=[40.3, 50.3], method=\"nearest\")\n```\n\n### Position-based indexing\n\nThis is similar to your usual numpy `array[0, 2, 3]` but with the power of named\ndimensions!\n\n\n\n```python\n# pull out time index 0 and lat index 0\nds.air.isel(time=0, lat=0) # much better than ds.air[0, 0, :]\n```\n\n\n```python\n# demonstrate slicing\nds.air.isel(lat=slice(10))\n```\n\n---\n\n## Concepts for computation\n\n\n### Broadcasting: expanding data\n\nLet's try to calculate grid cell area associated with the air temperature data.\nWe may want this to make a proper area-weighted domain-average for example\n\nA very approximate formula is\n\n\\begin{equation} \u0394lat \\times \u0394lon \\times \\cos(\\text{latitude}) \\end{equation}\n\nassuming that $\u0394lon$ = 111km and $\u0394lat$ = 111km\n\n\n\n```python\ndlon = np.cos(ds.air.lat * np.pi / 180) * 111e3\ndlon\n```\n\n\n```python\ndlat = 111e3 * xr.ones_like(ds.air.lon)\ndlat\n```\n\n\n```python\ncell_area = dlon * dlat\ncell_area\n```\n\nThe result has two dimensions because xarray realizes that dimensions `lon` and\n`lat` are different so it automatically \"broadcasts\" to get a 2D result. See the\nlast row in this image from _Jake VanderPlas Python Data Science Handbook_\n\n\n\nBecause xarray knows about dimension names we avoid having to create unnecessary\nsize-1 dimensions using `np.newaxis` or `.reshape`. For more, see\nhttps://xarray.pydata.org/en/stable/computation.html#broadcasting-by-dimension-name\n\n\n---\n\n### Alignment: putting data on the same grid\n\nWhen doing arithmetic operations xarray automatically \"aligns\" i.e. puts the\ndata on the same grid. In this case `cell_area` and `ds.air` are at the same\nlat, lon points so things are multiplied as you would expect\n\n\n\n```python\n(cell_area * ds.air.isel(time=1))\n```\n\nNow lets make `cell_area` unaligned i.e. change the coordinate labels\n\n\n\n```python\n# make a copy of cell_area\n# then add 1e-5 to lat\ncell_area_bad = cell_area.copy(deep=True)\ncell_area_bad[\"lat\"] = cell_area.lat + 1e-5\ncell_area_bad\n```\n\n\n```python\ncell_area_bad * ds.air.isel(time=1)\n```\n\n**Tip:** If you notice extra NaNs or missing points after xarray computation, it\nmeans that your xarray coordinates were not aligned _exactly_.\n\nFor more, see\nhttps://xarray.pydata.org/en/stable/computation.html#automatic-alignment\n\n\n---\n\n## High level computation: `groupby`, `resample`, `rolling`, `coarsen`, `weighted`\n\nXarray has some very useful high level objects that let you do common\ncomputations:\n\n1. `groupby` :\n [Bin data in to groups and reduce](https://xarray.pydata.org/en/stable/groupby.html)\n1. `resample` :\n [Groupby specialized for time axes. Either downsample or upsample your data.](https://xarray.pydata.org/en/stable/time-series.html#resampling-and-grouped-operations)\n1. `rolling` :\n [Operate on rolling windows of your data e.g. running mean](https://xarray.pydata.org/en/stable/computation.html#rolling-window-operations)\n1. `coarsen` :\n [Downsample your data](https://xarray.pydata.org/en/stable/computation.html#coarsen-large-arrays)\n1. `weighted` :\n [Weight your data before reducing](https://xarray.pydata.org/en/stable/computation.html#weighted-array-reductions)\n\n\n### groupby\n\n\n\n```python\n# seasonal groups\nds.groupby(\"time.season\")\n```\n\n\n```python\n# make a seasonal mean\nseasonal_mean = ds.groupby(\"time.season\").mean()\nseasonal_mean\n```\n\nThe seasons are out of order (they are alphabetically sorted). This is a common\nannoyance. The solution is to use `.reindex`\n\n\n\n```python\nseasonal_mean = seasonal_mean.reindex(season=[\"DJF\", \"MAM\", \"JJA\", \"SON\"])\nseasonal_mean\n```\n\n### resample\n\n\n\n```python\n# resample to monthly frequency\nds.resample(time=\"M\").mean()\n```\n\n### weighted\n\n\n\n```python\n# weight by cell_area and take mean over (time, lon)\nds.weighted(cell_area).mean([\"lon\", \"time\"]).air.plot()\n```\n\n---\n\n## Visualization: `.plot`\n\nFor more see https://xarray.pydata.org/en/stable/plotting.html and\nhttps://xarray.pydata.org/en/stable/examples/visualization_gallery.html\n\nWe have seen very simple plots earlier. Xarray has some support for visualizing\n3D and 4D datasets by presenting multiple facets (or panels or subplots) showing\nvariations across rows and/or columns.\n\n\n\n```python\n# facet the seasonal_mean\nseasonal_mean.air.plot(col=\"season\")\n```\n\n\n```python\n# contours\nseasonal_mean.air.plot.contour(col=\"season\", levels=20, add_colorbar=True)\n```\n\n\n```python\n# line plots too? wut\nseasonal_mean.air.mean(\"lon\").plot.line(hue=\"season\", y=\"lat\")\n```\n\n---\n\n## Reading and writing to disk\n\nXarray supports many disk formats. Below is a small example using netCDF. For\nmore see https://xarray.pydata.org/en/stable/io.html\n\n\n\n```python\n# write ds to netCDF\nds.to_netcdf(\"my-example-dataset.nc\")\n```\n\n\n```python\n# read from disk\nfromdisk = xr.open_dataset(\"my-example-dataset.nc\")\nfromdisk\n```\n\n\n```python\n# check that the two are identical\nds.identical(fromdisk)\n```\n\n**Tip:** A common use case to read datasets that are a collection of many netCDF\nfiles. See\nhttps://xarray.pydata.org/en/stable/io.html#reading-multi-file-datasets for how\nto handle that\n\n\n---\n\n## More information\n\n1. A description of common terms used in the xarray documentation:\n https://xarray.pydata.org/en/stable/terminology.html\n1. For information on how to create a DataArray from an existing numpy array:\n https://xarray.pydata.org/en/stable/data-structures.html#creating-a-dataarray\n1. Answers to common questions on \"how to do X\" are here:\n https://xarray.pydata.org/en/stable/howdoi.html\n1. Our more extensive Scipy 2020 tutorial material:\n https://xarray-contrib.github.io/xarray-tutorial/\n1. Ryan Abernathey has a book on data analysis with a chapter on Xarray:\n https://earth-env-data-science.github.io/lectures/xarray/xarray_intro.html\n\n\n---\n\n## The scientific python / pangeo ecosystem: demo\n\nXarray ties in to the larger scientific python ecosystem and in turn many\npackages build on top of xarray. A long list of such packages is here:\nhttps://xarray.pydata.org/en/stable/related-projects.html.\n\nNow we will demonstrate some cool features.\n\n\n### Pandas: tabular data structures\n\nYou can easily convert between xarray and pandas structures:\nhttps://pandas.pydata.org/\n\nThis allows you to conveniently use the extensive pandas ecosystem of packages\n(like seaborn) for your work.\n\nSee https://xarray.pydata.org/en/stable/pandas.html\n\n\n\n```python\n# convert to pandas dataframe\ndf = ds.isel(time=slice(10)).to_dataframe()\ndf\n```\n\n\n```python\n# convert dataframe to xarray\ndf.to_xarray()\n```\n\n### xarray can wrap other array types, not just numpy\n\n\n\n**dask** : parallel arrays https://xarray.pydata.org/en/stable/dask.html &\nhttps://docs.dask.org/en/latest/array.html\n\n\n\n**pydata/sparse** : sparse arrays http://sparse.pydata.org\n\n\n\n**cupy** : GPU arrays http://cupy.chainer.org\n\n\n\n**pint** : unit-aware computations https://pint.readthedocs.org &\nhttps://github.com/xarray-contrib/pint-xarray\n\n\n### Xarray + dask\n\nDask cuts up NumPy arrays into blocks and parallelizes your analysis code across\nthese blocks\n\n\n\n\n\n```python\n# make dask cluster; this is for demo purposes\nimport dask\nimport distributed\n\ncluster = distributed.LocalCluster()\n```\n\n\n```python\nclient = distributed.Client(cluster)\nclient\n```\n\n\n```python\n# demonstrate dask dataset\ndasky = xr.tutorial.open_dataset(\n \"air_temperature\",\n chunks={\"time\": 10}, # 10 time steps in each block\n)\n\ndasky.air\n```\n\nAll computations with dask-backed xarray objects are lazy, allowing you to build\nup a complicated chain of analysis steps quickly\n\n\n\n```python\n# demonstrate lazy mean\ndasky.air.mean(\"lat\")\n```\n\nTo get concrete values, call `.compute` or `.load`\n\n\n\n```python\n# \"compute\" the mean\ndasky.air.mean(\"lat\").compute()\n```\n\n### holoviews: javascript interactive plots\n\nthe `hvplot` package is a nice easy way to access\n[holoviews](http://holoviews.org/) functionality. It attaches itself to all\nxarray objects under the `.hvplot` namespace. So instead of using `.plot` use\n`.hvplot`\n\n\n\n```python\nimport hvplot.xarray\n\nds.air.hvplot(groupby=\"time\", clim=(270, 300))\n```\n\nTry the slider!\n\n\n### cf_xarray : use even more metadata for even more fun and ~profit~ papers\n\n[cf_xarray](https://cf-xarray.readthedocs.io/) is a new project that tries to\nlet you make use of other CF attributes that xarray ignores. It attaches itself\nto all xarray objects under the `.cf` namespace.\n\nWhere xarray allows you to specify dimension names for analysis, `cf_xarray`\nlets you specify logical names like `\"latitude\"` or `\"longitude\"` instead as\nlong as the appropriate CF attributes are set.\n\n\n\n```python\nimport cf_xarray\n```\n\n\n```python\n# describe cf attributes in dataset\nds.air.cf.describe()\n```\n\nThe following `mean` operation will work with any dataset that has appropriate\nattributes set that allow detection of the \"latitude\" variable (e.g.\n`units: \"degress_north\"` or `standard_name: \"latitude\"`)\n\n\n\n```python\n# demonstrate equivalent of .mean(\"lat\")\nds.air.cf.mean(\"latitude\")\n```\n\n\n```python\n# demonstrate indexing\nds.air.cf.sel(longitude=242.5, method=\"nearest\")\n```\n\n### Other cool packages\n\n- [xgcm](https://xgcm.readthedocs.io/) : grid-aware operations with xarray\n objects\n- [xrft](https://xrft.readthedocs.io/) : fourier transforms with xarray\n- [xclim](https://xclim.readthedocs.io/) : calculating climate indices with\n xarray objects\n- [intake-xarray](https://intake-xarray.readthedocs.io/) : forget about file\n paths\n- [rioxarray](https://corteva.github.io/rioxarray/stable/index.html) : raster\n files and xarray\n- [xesmf](https://xesmf.readthedocs.io/) : regrid using ESMF\n- [MetPy](https://unidata.github.io/MetPy/latest/index.html) : tools for working\n with weather data\n\nMore here: https://xarray.pydata.org/en/stable/related-projects.html\n\n", "meta": {"hexsha": "88deb832ff6d6ab071700dd424dbe266b5082d11", "size": 31363, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "oceanhackweek-2020/xarray-oceanhackweek20.ipynb", "max_stars_repo_name": "dcherian/xarray-tutorial", "max_stars_repo_head_hexsha": "c133a80c2d911ef841ee6197f88ec0a0d87fbd94", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 78, "max_stars_repo_stars_event_min_datetime": "2020-04-29T20:36:42.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-21T08:54:23.000Z", "max_issues_repo_path": "oceanhackweek-2020/xarray-oceanhackweek20.ipynb", "max_issues_repo_name": "dcherian/xarray-tutorial", "max_issues_repo_head_hexsha": "c133a80c2d911ef841ee6197f88ec0a0d87fbd94", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 39, "max_issues_repo_issues_event_min_datetime": "2020-04-29T18:19:34.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-07T17:51:00.000Z", "max_forks_repo_path": "oceanhackweek-2020/xarray-oceanhackweek20.ipynb", "max_forks_repo_name": "dcherian/xarray-tutorial", "max_forks_repo_head_hexsha": "c133a80c2d911ef841ee6197f88ec0a0d87fbd94", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 51, "max_forks_repo_forks_event_min_datetime": "2020-05-19T18:55:33.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-18T18:09:28.000Z", "avg_line_length": 24.8912698413, "max_line_length": 177, "alphanum_fraction": 0.5598316488, "converted": true, "num_tokens": 3601, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.7799929002541068, "lm_q1q2_score": 0.435491091601807}} {"text": "```python\nimport sys\nfrom __future__ import division\n```\n\n\n```python\nimport numpy as np\nfrom phasor.utilities.ipynb.displays import *\n#from YALL.utilities.tabulate import tabulate\n\nimport declarative\n\nfrom declarative.bunch import (\n DeepBunch\n)\n\nimport phasor.math.dispatched as dmath \n#import phasor.math.dispatch_sympy\n```\n\n Populating the interactive namespace from numpy and matplotlib\n\n\n\n```python\nimport phasor.utilities.version as version\nprint(version.foundations_version())\n\nfrom phasor.utilities.np import logspaced\nfrom phasor.utilities.ipynb.sympy import *\n\nfrom phasor import optics\nfrom phasor import base \nfrom phasor import signals \nfrom phasor import system\nfrom phasor import readouts \nfrom phasor import mechanical\n```\n\n b'2017-09-22 09:45:51 -0400 (a76310c5d4476904171a3f1b18117db454719432)'\n Sympy version: 1.0\n\n\n\n```python\nd = sympy.var('d')\nN = sympy.var('N')\nl = sympy.var('l')\nA = sympy.var('A')\ns = sympy.var('s')\nk = sympy.var('k')\nepsilon = sympy.var('epsilon')\nepsilon_0 = sympy.var('epsilon_0')\n\nd_l1 = sympy.var('l_delta_1')\nF1 = sympy.var('F_1')\nd_l2 = sympy.var('l_delta_2')\nF2 = sympy.var('F_2')\n\nC = sympy.var('C')\nV1 = sympy.var('V_1')\nI1 = sympy.var('I_1')\nV2 = sympy.var('V_2')\nI2 = sympy.var('I_2')\n\nZ_e = sympy.var('Z_e', real = True)\nZ_m = sympy.var('Z_m', real = True)\nk_e = sympy.sqrt(1/sympy.re(Z_e))\nk_m = sympy.sqrt(1/sympy.im(Z_m))\nk_e = sympy.var('k_e', real = True)\nk_m = sympy.var('k_m', real = True)\nk_e = sympy.sympify(1)\nk_m = sympy.sympify(1)\n\nexpr1 = d * F1 - I1/s + C * V1\ndisplay(expr1)\nexpr2 = -d_l1 * k + F1 + k * d * V1\ndisplay(expr2)\n\nrel = sympy.Matrix([\n [C, -1/s, -C, +1/s, d, 0, -d, 0],\n [d*k, 0, -d*k, 0, 1, -k, -1, +k],\n [0,1,0,1,0,0,0,0],\n [0,0,0,0,0,1,0,1],\n])\nvar = sympy.Matrix([V1, I1, V2, I2, F1, d_l1, F2, d_l2])\nrel * var\n```\n\n\n```python\ntrans = sympy.Matrix([\n [k_e/2, k_e/2 * Z_e , 0, 0, 0, 0, 0, 0], \n [k_e/2, -k_e/2 * Z_e.conjugate(), 0, 0, 0, 0, 0, 0], \n [0, 0, k_e/2, k_e/2 * Z_e , 0, 0, 0, 0], \n [0, 0, k_e/2, -k_e/2 * Z_e.conjugate(), 0, 0, 0, 0], \n [0, 0, 0, 0, k_m/2, k_m/2 * Z_m , 0, 0], \n [0, 0, 0, 0, k_m/2, -k_m/2 * Z_m.conjugate(), 0, 0], \n [0, 0, 0, 0, 0, 0, k_m/2, k_m/2 * Z_m ], \n [0, 0, 0, 0, 0, 0, k_m/2, -k_m/2 * Z_m.conjugate()], \n])\ntrans * var\n```\n\n\n```python\nrel_ab = rel * trans**-1\nrel_a = rel_ab[:,::2]\nrel_b = rel_ab[:,1::2]\n\n```\n\n\n```python\nrel_b\n```\n\n\n```python\nrel_atob = -rel_b**-1 * rel_a\nrel_atob.simplify()\nrel_atob\n```\n\n\n```python\nfrom sympy.utilities.lambdify import lambdastr\n```\n\n\n```python\ne = rel_atob[1,1]\ndef pyval(expr):\n sym = list(expr.free_symbols)\n return lambdastr(sym, expr).split(':')[1].strip()\n```\n\n\n```python\npyval(rel_atob[0,0])\n```\n\n\n\n\n '((Z_m + k)/(C*Z_e*Z_m*s + C*Z_e*k*s - Z_e*Z_m*d**2*k*s + Z_m + k))'\n\n\n\n\n```python\npyval(rel_atob[1,1])\n```\n\n\n\n\n '((Z_m + k)/(C*Z_e*Z_m*s + C*Z_e*k*s - Z_e*Z_m*d**2*k*s + Z_m + k))'\n\n\n\n\n```python\npyval(rel_atob[0,1])\n```\n\n\n\n\n '(Z_e*s*(-C*Z_m - C*k + Z_m*d**2*k)/(-C*Z_e*Z_m*s - C*Z_e*k*s + Z_e*Z_m*d**2*k*s - Z_m - k))'\n\n\n\n\n```python\npyval(rel_atob[1,0])\n```\n\n\n\n\n '(Z_e*s*(-C*Z_m*k - C + d**2*k)/(-C*Z_e*Z_m*k*s - C*Z_e*s + Z_e*d**2*k*s - Z_m*k - 1))'\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "72da2bbe2d3ef83bd0666599098bc21ff4ffca3d", "size": 49249, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "phasor/mechanical/PZT/PZT_series.ipynb", "max_stars_repo_name": "mccullerlp/phasor-doc", "max_stars_repo_head_hexsha": "d4255d015023c51b762340e51c15dde609715212", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "phasor/mechanical/PZT/PZT_series.ipynb", "max_issues_repo_name": "mccullerlp/phasor-doc", "max_issues_repo_head_hexsha": "d4255d015023c51b762340e51c15dde609715212", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "phasor/mechanical/PZT/PZT_series.ipynb", "max_forks_repo_name": "mccullerlp/phasor-doc", "max_forks_repo_head_hexsha": "d4255d015023c51b762340e51c15dde609715212", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 93.2746212121, "max_line_length": 16408, "alphanum_fraction": 0.6929074702, "converted": true, "num_tokens": 1432, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929002541067, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.4354910916018069}} {"text": "```python\nimport sys\nfrom importlib import reload\nimport sympy\nfrom itertools import product\nfrom collections import defaultdict\n```\n\n\n```python\nsys.path.insert(0, '../')\n```\n\n\n```python\nimport multivector as mv\nimport examples as ex\nimport embedding as emb\n```\n\n```\nt*(x0,x1,x2,y0,y1,y2) = (t^2*x0, t*x1, t*x2, t-2*y0, t^-1*y1, t^-1*y2)\n\ninvariants:\nx0*y0\nx1*y1\nx2*y2\nx1*y2\nx2*y1\nx0*y1^2\nx0*y1*y2\nx0*y2^2\ny0*x1^2\ny0*x1*x2\ny0*x2^2\n```\n\n\n```python\nq = sympy.symbols(tuple('q%d' %x for x in range(5)))\na = sympy.symbols(tuple('a%d' %x for x in range(3)))\nb = sympy.symbols(tuple('b%d' %x for x in range(3)))\n#z = sympy.symbols('z0 z1')\n\nx = sympy.symbols('x0 x1 x2')\ny = sympy.symbols('y0 y1 y2')\nstdbrac = mv.Mv(x+y, {(x[0],y[0]):1, (x[1],y[1]):1, (x[2],y[2]):1}).brac\n```\n\n\n```python\nrmap = {\n q[0]: x[0]*y[0], \n q[1]: x[1]*y[1],\n q[2]: x[2]*y[2],\n q[3]: x[1]*y[2],\n q[4]: x[2]*y[1],\n a[0]: x[0]*(y[1]**2),\n a[1]: x[0]*(y[1]*y[2]),\n a[2]: x[0]*(y[2]**2),\n b[0]: y[0]*(x[1]**2),\n b[1]: y[0]*(x[1]*x[2]),\n b[2]: y[0]*(x[2]**2)\n}\n\nembcoords, invariants = list(zip(*rmap.items()))\n```\n\n\n```python\ngbK = sympy.polys.groebner([ai-xi for (ai,xi) in rmap.items()])\nK = [g for g in gbK if not (set(g.as_poly().gens) - set(embcoords))]\n```\n\n\n```python\n{k:(Pi1.brac(q[0], k)/k).simplify() for k in K}\n```\n\n\n\n\n {-a0*b2 + q0*q4**2: 0,\n b0*b2 - b1**2: 2,\n a0*b2*q3 - a1*b1*q4: 0,\n -a2*b2 + q0*q2**2: 0,\n -b1*q4 + b2*q1: 1,\n -a1*b0 + q0*q1*q3: 0,\n -a0*b1 + q0*q1*q4: 0,\n -a1*q3 + a2*q1: -1,\n -a0*b0 + q0*q1**2: 0,\n a0*b1*q3 - a1*b0*q4: 0,\n -a1*b2 + q0*q2*q4: 0,\n b0*q2 - b1*q3: 1,\n -a1*b1 + q0*q3*q4: 0,\n -a2*b1 + q0*q2*q3: 0,\n a0*a2 - a1**2: -2,\n a1*q2 - a2*q4: -1,\n -b0*q4 + b1*q1: 1,\n -a2*b0 + q0*q3**2: 0,\n q1*q2 - q3*q4: 0,\n a1*b2*q3 - a2*b1*q4: 0,\n a0*q2 - a1*q4: -1,\n b1*q2 - b2*q3: 1,\n a1*b1*q3 - a2*b0*q4: 0,\n -a0*q3 + a1*q1: -1}\n\n\n\n\n```python\n{k:(Pi1.brac((q[1] - q[2])/2, k)/k).simplify() for k in K}\n```\n\n\n\n\n {-a0*b2 + q0*q4**2: 2,\n b0*b2 - b1**2: 0,\n a0*b2*q3 - a1*b1*q4: 1,\n -a2*b2 + q0*q2**2: 0,\n -b1*q4 + b2*q1: 1,\n -a1*b0 + q0*q1*q3: -1,\n -a0*b1 + q0*q1*q4: 1,\n -a1*q3 + a2*q1: -1,\n -a0*b0 + q0*q1**2: 0,\n a0*b1*q3 - a1*b0*q4: 0,\n -a1*b2 + q0*q2*q4: 1,\n b0*q2 - b1*q3: -1,\n -a1*b1 + q0*q3*q4: 0,\n -a2*b1 + q0*q2*q3: -1,\n a0*a2 - a1**2: 0,\n a1*q2 - a2*q4: 0,\n -b0*q4 + b1*q1: 0,\n -a2*b0 + q0*q3**2: -2,\n q1*q2 - q3*q4: 0,\n a1*b2*q3 - a2*b1*q4: 0,\n a0*q2 - a1*q4: 1,\n b1*q2 - b2*q3: 0,\n a1*b1*q3 - a2*b0*q4: -1,\n -a0*q3 + a1*q1: 0}\n\n\n\n\n```python\nKm = [\n q[1]*q[2] - q[3]*q[4],\n -1*a[2]*b[0] + q[0]*q[3]**2,\n -1*a[2]*b[1] + q[0]*q[2]*q[3],\n -1*a[1]*b[0] + q[0]*q[1]*q[3],\n -1*a[2]*b[2] + q[0]*q[2]**2,\n -1*a[1]*b[1] + q[0]*q[3]*q[4],\n -1*a[0]*b[0] + q[0]*q[1]**2,\n -1*a[0]*b[1] + q[0]*q[1]*q[4],\n -1*a[1]*b[2] + q[0]*q[2]*q[4],\n -1*a[0]*b[2] + q[0]*q[4]**2,\n -1*a[1]*q[3] + a[2]*q[1],\n -1*a[0]*q[3] + a[1]*q[1],\n a[1]*q[2] - a[2]*q[4],\n a[0]*q[2] - a[1]*q[4],\n b[0]*q[2] - b[1]*q[3],\n -b[0]*q[4] + b[1]*q[1],\n b[1]*q[2] - b[2]*q[3],\n -b[1]*q[4] + b[2]*q[1],\n a[1]*b[1]*q[3] - a[2]*b[0]*q[4],\n a[0]*b[1]*q[3] - a[1]*b[0]*q[4],\n a[1]*b[2]*q[3] - a[2]*b[1]*q[4],\n a[0]*b[2]*q[3] - a[1]*b[1]*q[4],\n a[0]*a[2] - a[1]**2,\n b[0]*b[2] - b[1]**2\n]\n```\n\n\n```python\nKcoords = sympy.symbols(\n ['C'] + \n ['MQ%d' %x for x in range(9)] + \n ['AQ%d' %x for x in range(4)] + \n ['BQ%d' %x for x in range(4)] +\n ['MM%d' %x for x in range(4)] + \n ['AA%d' %x for x in range(1)] + \n ['BB%d' %x for x in range(1)]\n)\n```\n\n\n```python\nKHom = dict(zip(Kcoords, Km))\nkerCoords, kerPolys = list(zip(*KHom.items()))\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\nP = {}\nfor k1, k2 in product(rmap.keys(), rmap.keys()):\n if (k2, k1) not in P.keys():\n P[(k1, k2)] = stdbrac(rmap[k1],rmap[k2])\n```\n\n\n```python\nlift = emb.lift(rmap, embcoords, x+y)\n```\n\n\n```python\nPi = mv.Mv(embcoords, P).mapCoeffs(lambda x:lift(x))\n```\n\n\n```python\nPi.mapCoeffs(lambda x:x.simplify())\n```\n\n\n\n\n -a0*dq0*da0 + 2*a0*dq1*da0 + a0*dq4*da1 - a1*dq0*da1 + a1*dq1*da1 + a1*dq2*da1 + 2*a1*dq3*da0 + 2*a1*dq4*da2 - a2*dq0*da2 + 2*a2*dq2*da2 + a2*dq3*da1 - b0*db0*dq0 - 2*b0*dq1*db0 - b0*dq3*db1 + 2*b1*db0*dq4 + b1*dq0*db1 - b1*dq1*db1 - b1*dq2*db1 - 2*b1*dq3*db2 + b2*dq0*db2 - 2*b2*dq2*db2 - b2*dq4*db1 + q1*(4*q0 - q1)*db0*da0 + q2*(-4*q0 + q2)*da2*db2 - q3**2*db0*da2 + q3*(-2*q0 + q2)*da2*db1 + q3*(2*q0 - q1)*db0*da1 - q3*dq1*dq3 + q3*dq2*dq3 - q4**2*db2*da0 + q4*(2*q0 - q1)*db1*da0 + q4*(2*q0 - q2)*db2*da1 + q4*dq1*dq4 - q4*dq2*dq4 + (-q1 + q2)*dq3*dq4 + (q0*q1 + q2*(q0 - q1))*db1*da1\n\n\n\n\n```python\nmv.sBr(Pi,Pi).sort().mapCoeffs(lambda x:x.simplify()/2) == 0\n```\n\n\n\n\n False\n\n\n\n\n```python\nJac = mv.sBr(Pi,Pi).sort().mapCoeffs(lambda x:x.simplify()/2)\n```\n\n\n```python\n\n```\n\n\n```python\nJac.mapCoeffs(liftK)\n```\n\n\n\n\n 3*AQ0*da1*da2*db1 + 4*AQ0*da2*db0*da0 + 4*AQ1*da1*db0*da0 - 4*AQ2*db2*da1*da2 - 3*AQ3*da1*db1*da0 + 4*AQ3*db2*da2*da0 - 3*BQ0*da1*db0*db1 - 4*BQ0*db2*da2*db0 + 4*BQ1*db0*db1*da0 - 4*BQ2*db2*da2*db1 + 3*BQ3*db2*da1*db1 - 4*BQ3*db2*db0*da0 - C*da1*dq4*db0 + C*da2*dq4*db1 - C*db2*da1*dq3 + C*dq3*db1*da0 + (2*AQ1 - 2*AQ2)*da2*db1*da0 + (2*BQ1 - 2*BQ2)*db2*da1*db0\n\n\n\n\n```python\nPi_degs = Pi.deg()\n```\n\n\n```python\nreload(emb)\nliftK = emb.lift(KHom, kerCoords, embcoords)\n```\n\n\n```python\nfor c in sorted(embcoords, key=lambda x:str(x)):\n print(c,'\\t',-1*mv.sBr(Pi, mv.Mv(embcoords, {():c})).sort().mapCoeffs(lambda x:x.simplify()))\n```\n\n a0 \t a0*dq0 - 2*a0*dq1 - 2*a1*dq3 - q1*(4*q0 - q1)*db0 + q4**2*db2 - q4*(2*q0 - q1)*db1\n a1 \t -a0*dq4 + a1*dq0 - a1*dq1 - a1*dq2 - a2*dq3 - q3*(2*q0 - q1)*db0 - q4*(2*q0 - q2)*db2 + (-q0*q1 - q2*(q0 - q1))*db1\n a2 \t -2*a1*dq4 + a2*dq0 - 2*a2*dq2 - q2*(4*q0 - q2)*db2 + q3**2*db0 - q3*(2*q0 - q2)*db1\n b0 \t -b0*dq0 + 2*b0*dq1 + 2*b1*dq4 - q1*(-4*q0 + q1)*da0 - q3**2*da2 - q3*(-2*q0 + q1)*da1\n b1 \t b0*dq3 - b1*dq0 + b1*dq1 + b1*dq2 + b2*dq4 - q3*(-2*q0 + q2)*da2 - q4*(-2*q0 + q1)*da0 + (q0*q1 + q2*(q0 - q1))*da1\n b2 \t 2*b1*dq3 - b2*dq0 + 2*b2*dq2 - q2*(-4*q0 + q2)*da2 - q4**2*da0 - q4*(-2*q0 + q2)*da1\n q0 \t -a0*da0 - a1*da1 - a2*da2 + b0*db0 + b1*db1 + b2*db2\n q1 \t 2*a0*da0 + a1*da1 - 2*b0*db0 - b1*db1 - q3*dq3 + q4*dq4\n q2 \t a1*da1 + 2*a2*da2 - b1*db1 - 2*b2*db2 + q3*dq3 - q4*dq4\n q3 \t 2*a1*da0 + a2*da1 - b0*db1 - 2*b1*db2 + q3*dq1 - q3*dq2 + (-q1 + q2)*dq4\n q4 \t a0*da1 + 2*a1*da2 - 2*b1*db0 - b2*db1 - q4*dq1 + q4*dq2 + (q1 - q2)*dq3\n\n\n\n```python\nfor qk in K:\n print(qk,'\\t',-1*mv.sBr(Pi, mv.Mv(embcoords, {():qk})).sort().mapCoeffs(liftK), end='\\n\\n')\n```\n\n -a0*b0 + q0*q1**2 \t -2*MQ2*dq3 + 2*MQ6*dq4 + (-AQ0*q1 - AQ1*q3)*da2 + (2*AQ1*q0 - AQ1*q1)*da1 + (-2*BQ1*q0 + BQ1*q1)*db1 + (BQ1*q4 + BQ3*q1)*db2\n \n -a1*b0 + q0*q1*q3 \t AQ0*q0*da1 - AQ0*q3*da2 - MQ0*dq3 + MQ2*dq1 - MQ2*dq2 + (-2*AQ1*q0 + AQ1*q1)*da0 + (BQ0*q0 + BQ1*q3 - C*b0)*db1 + (-BQ0*q4 - 2*BQ1*q0 + BQ3*q3)*db2 + (C*q0 + 2*MQ4 - MQ5)*dq4\n \n -a0*b1 + q0*q1*q4 \t -BQ3*q0*db1 + BQ3*q4*db2 - MQ6*dq1 + MQ6*dq2 + MQ8*dq4 + (2*BQ1*q0 - BQ1*q1)*db0 + (-AQ0*q4 + 2*AQ1*q0 + AQ3*q3)*da2 + (-AQ1*q4 - AQ3*q0 + C*a0)*da1 + (-C*q0 - 2*MQ4 + MQ5)*dq3\n \n -a2*b2 + q0*q2**2 \t 2*MQ1*dq3 - 2*MQ7*dq4 + (2*AQ2*q0 - AQ2*q2)*da1 + (-AQ2*q4 - AQ3*q2)*da0 + (BQ0*q2 + BQ2*q3)*db0 + (-2*BQ2*q0 + BQ2*q2)*db1\n \n -a2*b1 + q0*q2*q3 \t -BQ0*q0*db1 + BQ0*q3*db0 + MQ0*dq3 + MQ1*dq1 - MQ1*dq2 + (2*BQ2*q0 - BQ2*q2)*db2 + (-AQ0*q0 - AQ2*q3 + C*a2)*da1 + (AQ0*q4 + 2*AQ2*q0 - AQ3*q3)*da0 + (-C*q0 + MQ3 - 2*MQ4)*dq4\n \n -a1*b2 + q0*q2*q4 \t AQ3*q0*da1 - AQ3*q4*da0 - MQ7*dq1 + MQ7*dq2 - MQ8*dq4 + (-2*AQ2*q0 + AQ2*q2)*da2 + (BQ0*q4 - 2*BQ2*q0 - BQ3*q3)*db0 + (BQ2*q4 + BQ3*q0 - C*b2)*db1 + (C*q0 - MQ3 + 2*MQ4)*dq3\n \n -a2*b0 + q0*q3**2 \t AQ0*q3*da1 - BQ0*q3*db1 + 2*MQ0*dq1 - 2*MQ0*dq2 + (2*MQ1 - 2*MQ2)*dq4 + (-4*AQ0*q0 + AQ0*q1 + AQ1*q3)*da0 + (4*BQ0*q0 - BQ0*q2 - BQ2*q3)*db2\n \n -a1*b1 + q0*q3*q4 \t AQ1*q4*da0 + AQ2*q3*da2 - BQ1*q3*db0 - BQ2*q4*db2 + (-MQ1 + MQ2)*dq3 + (-MQ6 + MQ7)*dq4 + (-AQ1*q0 - AQ2*q0 + C*a1)*da1 + (BQ1*q0 + BQ2*q0 - C*b1)*db1\n \n -a0*b2 + q0*q4**2 \t AQ3*q4*da1 - BQ3*q4*db1 - 2*MQ8*dq1 + 2*MQ8*dq2 + (2*MQ6 - 2*MQ7)*dq3 + (AQ2*q4 - 4*AQ3*q0 + AQ3*q2)*da2 + (-BQ1*q4 + 4*BQ3*q0 - BQ3*q1)*db0\n \n q1*q2 - q3*q4 \t 2*AQ0*da2 + 2*AQ3*da0 - 2*BQ0*db0 - 2*BQ3*db2 + (AQ1 + AQ2)*da1 + (-BQ1 - BQ2)*db1\n \n -a0*q3 + a1*q1 \t -AA0*da1 - AQ0*dq3 + AQ1*dq0 - AQ1*dq1 - AQ1*dq2 - AQ3*dq4 + 2*MQ2*db0 + (C*q4 - 2*MQ6)*db2 + (-C*q0 + C*q1 + MQ4 - MQ5)*db1\n \n -a1*q3 + a2*q1 \t 2*AA0*da0 + AQ0*dq0 - 2*AQ0*dq2 + 2*MQ0*db0 + (-AQ1 - AQ2)*dq4 + (MQ1 - MQ2)*db1 + (-4*C*q0 + C*q2 - 2*MQ4)*db2\n \n -b0*q4 + b1*q1 \t BB0*db1 + BQ0*dq3 - BQ1*dq0 + BQ1*dq1 + BQ1*dq2 + BQ3*dq4 - 2*MQ6*da0 + (-C*q3 + 2*MQ2)*da2 + (C*q0 - C*q1 - MQ4 + MQ5)*da1\n \n -b1*q4 + b2*q1 \t -2*BB0*db0 - BQ3*dq0 + 2*BQ3*dq2 - 2*MQ8*da0 + (BQ1 + BQ2)*dq3 + (MQ6 - MQ7)*da1 + (4*C*q0 - C*q2 + 2*MQ4)*da2\n \n a0*q2 - a1*q4 \t 2*AA0*da2 + AQ3*dq0 - 2*AQ3*dq1 + 2*MQ8*db2 + (-AQ1 - AQ2)*dq3 + (MQ6 - MQ7)*db1 + (-4*C*q0 + C*q1 - 2*MQ4)*db0\n \n a1*q2 - a2*q4 \t -AA0*da1 - AQ0*dq3 + AQ2*dq0 - AQ2*dq1 - AQ2*dq2 - AQ3*dq4 + 2*MQ7*db2 + (C*q3 - 2*MQ1)*db0 + (-C*q0 + C*q2 - MQ3 + MQ4)*db1\n \n b0*q2 - b1*q3 \t -2*BB0*db2 - BQ0*dq0 + 2*BQ0*dq1 - 2*MQ0*da2 + (BQ1 + BQ2)*dq4 + (MQ1 - MQ2)*da1 + (4*C*q0 - C*q1 + 2*MQ4)*da0\n \n b1*q2 - b2*q3 \t BB0*db1 + BQ0*dq3 - BQ2*dq0 + BQ2*dq1 + BQ2*dq2 + BQ3*dq4 - 2*MQ1*da2 + (-C*q4 + 2*MQ7)*da0 + (C*q0 - C*q2 + MQ3 - MQ4)*da1\n \n a0*b1*q3 - a1*b0*q4 \t (2*AA0*b0 - AQ3*q3**2 + 2*MQ0*a0)*da2 + (-AQ1*b0 + BQ0*a1 - MM0)*dq3 + (2*BB0*a0 - BQ0*q4**2 + 2*MQ8*b0)*db2 + (-BQ1*a0 + BQ2*a0 + 2*MM3)*dq4 + (AQ1*q1*q4 + 2*MQ4*a0 - 4*MQ6*a1)*da0 + (BQ1*q1*q3 - 4*MQ2*b1 + 2*MQ4*b0)*db0 + (2*AA0*b1 + AQ1*q3*q4 - C*a0*q3 + MQ1*a0 + MQ2*a0 - 2*MQ4*a1)*da1 + (BB0*a1 + BQ0*q0*q4 + BQ1*q3*q4 - C*b0*q4 - MQ4*b1 + MQ6*b0)*db1\n \n a0*b2*q3 - a1*b1*q4 \t -MM3*dq1 + MM3*dq2 + (AQ1*q4**2 - 2*MQ8*a1)*da0 + (AQ3*b2 - BQ3*a0)*dq4 + (-BQ2*q4**2 + 2*MQ8*b1)*db2 + (4*AA0*b1 - AQ3*q2*q3 + 4*MQ1*a0 - 2*MQ4*a1)*da2 + (-4*BB0*a1 + BQ3*q1*q3 - 4*MQ2*b2 + 2*MQ4*b1)*db0 + (-BQ1*a1 + BQ2*a1 + 2*MM1 - MM2)*dq3 + (AA0*b2 - AQ3*q3*q4 + C*a1*q4 + 2*MQ4*a0 - MQ6*a1 - MQ7*a1)*da1 + (-BB0*a0 + BQ3*q3*q4 - C*b1*q4 - 2*MQ4*b2 + MQ6*b1 + MQ7*b1)*db1\n \n a1*b1*q3 - a2*b0*q4 \t MM0*dq1 - MM0*dq2 + (-AQ0*b0 + BQ0*a2)*dq3 + (-AQ2*q3**2 + 2*MQ0*a1)*da2 + (BQ1*q3**2 - 2*MQ0*b1)*db0 + (4*BQ0*q0*q4 - BQ0*q2*q4 + 2*MQ4*b1)*db2 + (-4*AA0*b1 + AQ0*q1*q4 + 2*MQ4*a1 - 4*MQ6*a2)*da0 + (-BQ1*a1 + BQ2*a1 - MM1 + 2*MM2)*dq4 + (-AA0*b0 + AQ0*q3*q4 - C*a1*q3 + MQ1*a1 + MQ2*a1 - 2*MQ4*a2)*da1 + (BB0*a2 - BQ0*q3*q4 + C*b1*q3 - MQ1*b1 - MQ2*b1 + 2*MQ4*b0)*db1\n \n a1*b2*q3 - a2*b1*q4 \t (-2*AA0*b2 + AQ0*q4**2 - 2*MQ8*a2)*da0 + (-AQ0*b1 + BQ2*a2 + MM0)*dq3 + (AQ2*b2 - BQ3*a1 - MM3)*dq4 + (-2*BB0*a2 + BQ3*q3**2 - 2*MQ0*b2)*db0 + (-AQ2*q2*q3 + 4*MQ1*a1 - 2*MQ4*a2)*da2 + (-BQ2*q2*q4 - 2*MQ4*b2 + 4*MQ7*b1)*db2 + (-2*AA0*b1 - AQ2*q3*q4 + C*a2*q4 + 2*MQ4*a1 - MQ6*a2 - MQ7*a2)*da1 + (-2*BB0*a1 - BQ2*q3*q4 + C*b2*q3 - MQ1*b2 - MQ2*b2 + 2*MQ4*b1)*db1\n \n a0*a2 - a1**2 \t 2*AA0*dq0 - 2*AA0*dq1 - 2*AA0*dq2 + (-4*AQ0*q0 + AQ0*q1 - AQ1*q3)*db0 + (-AQ2*q4 - 4*AQ3*q0 + AQ3*q2)*db2 + (AQ0*q4 + 2*AQ1*q0 + 2*AQ2*q0 + AQ3*q3 - 2*C*a1)*db1\n \n b0*b2 - b1**2 \t -2*BB0*dq0 + 2*BB0*dq1 + 2*BB0*dq2 + (4*BQ0*q0 - BQ0*q2 + BQ2*q3)*da2 + (BQ1*q4 + 4*BQ3*q0 - BQ3*q1)*da0 + (-BQ0*q4 - 2*BQ1*q0 - 2*BQ2*q0 - BQ3*q3 + 2*C*b1)*da1\n \n\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "eb422b20352298d62a9e543fcab3659d8e2632b2", "size": 19082, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/poisson-embedding-211-action.ipynb", "max_stars_repo_name": "afraenkel/schouten-calculus", "max_stars_repo_head_hexsha": "8b7df46a7240570d05a1688618eea6af9891ed31", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/poisson-embedding-211-action.ipynb", "max_issues_repo_name": "afraenkel/schouten-calculus", "max_issues_repo_head_hexsha": "8b7df46a7240570d05a1688618eea6af9891ed31", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/poisson-embedding-211-action.ipynb", "max_forks_repo_name": "afraenkel/schouten-calculus", "max_forks_repo_head_hexsha": "8b7df46a7240570d05a1688618eea6af9891ed31", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.1930379747, "max_line_length": 599, "alphanum_fraction": 0.4543548894, "converted": true, "num_tokens": 6798, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872131147276, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4354467121457265}} {"text": "# Surfinpy\n\n#### Tutorial 3 - Pressure\n\nChemical potential can be converted to pressure values using\n\n\\begin{align}\nP & = \\frac{\\mu_O}{k_B T} ,\n\\end{align}\n\nwhere P is the pressure, $\\mu$ is the chemical potential of oxygen, $k_B$ is the Boltzmnann constant and T is the temperature. \n\n\n```python\nimport matplotlib.pyplot as plt\nfrom surfinpy import mu_vs_mu\nfrom surfinpy import utils as ut\nfrom surfinpy import data\n```\n\n\n```python\nOxygen_exp = ut.fit_nist(\"O2.txt\")[298]\nWater_exp = ut.fit_nist(\"H2O.txt\")[298]\nOxygen_corrected = (-9.08 + -0.86 + Oxygen_exp) / 2 \nWater_corrected = -14.84 + 0.55 + Water_exp\n```\n\n\n```python\nbulk = data.ReferenceDataSet(cation = 1, anion = 2, energy = -780.0, funits = 4)\n\npure = data.DataSet(cation = 24, x = 48, y = 0, area = 60.0, energy = -575.0, label = \"0.00 $TiO_2$\", nspecies = 1)\nH2O = data.DataSet(cation = 24, x = 48, y = 2, area = 60.0, energy = -612.0, label = \"0.16 $TiO_2$\", nspecies = 1)\nH2O_2 = data.DataSet(cation = 24, x = 48, y = 4, area = 60.0, energy = -640.0, label = \"0.32 $TiO_2$\", nspecies = 1)\nH2O_3 = data.DataSet(cation = 24, x = 48, y = 8, area = 60.0, energy = -676.0, label = \"0.64 $TiO_2$\", nspecies = 1)\nVo = data.DataSet(cation = 24, x = 46, y = 0, area = 60.0, energy = -558.0, label = \"0.00 $TiO_1.9$\", nspecies = 1)\nH2O_Vo_1 = data.DataSet(cation = 24, x = 46, y = 2, area = 60.0, energy = -594.0, label = \"0.00 $TiO_1.9$\", nspecies = 1)\nH2O_Vo_2 = data.DataSet(cation = 24, x = 46, y = 4, area = 60.0, energy = -624.0, label = \"0.16 $TiO_1.9$\", nspecies = 1)\nH2O_Vo_3 = data.DataSet(cation = 24, x = 46, y = 6, area = 60.0, energy = -640.0, label = \"0.32 $TiO_1.9$\", nspecies = 1)\nH2O_Vo_4 = data.DataSet(cation = 24, x = 46, y = 8, area = 60.0, energy = -670.0, label = \"0.64 $TiO_1.9$\", nspecies = 1)\n\ndata = [pure, Vo, H2O, H2O_Vo_1, H2O_2, H2O_Vo_2, H2O_3, H2O_Vo_3, H2O_Vo_4]\n\n```\n\n\n```python\ndeltaX = {'Range': [ -12, -6], 'Label': 'O'}\ndeltaY = {'Range': [ -19, -12], 'Label': 'H_2O'}\n```\n\n\n```python\nsystem = mu_vs_mu.calculate(data, bulk, deltaX, deltaY, x_energy=Oxygen_corrected, y_energy=Water_corrected)\n```\n\nAs before we can generate a basic plot of oxygen chemical potential vs water chemical potential at 298 K\n\n\n```python\nax = system.plot_phase(temperature=298, cbar_title=\"$H_2O$ $/$ $nm^2$\", figsize=(6, 5))\nplt.savefig(\"../../../docs/source/Figures/Surfaces_4.png\", dpi=600)\nplt.show()\n```\n\nWe can also generate the same plot but with the $\\mu$ values converted to pressure.\n\n\n```python\nsystem.plot_pressure(temperature=298, cbar_title=\"$H_2O$ $/$ $nm^2$\", figsize=(6, 5))\nplt.savefig(\"../../../docs/source/Figures/Surfaces_5.png\", dpi=600)\nplt.show()\n```\n\nFinally, we can also combine these two plots into one\n\n\n```python\nsystem.plot_mu_p(temperature=298, cbar_title=\"$H_2O$ $/$ $nm^2$\", figsize=(6, 5))\nplt.savefig(\"../../../docs/source/Figures/Surfaces_6.png\", dpi=600)\nplt.show()\n```\n", "meta": {"hexsha": "f66b1bdbfaf20ccee4e70872cf2708f1407acac1", "size": 74017, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/Notebooks/Surfaces/Tutorial_3.ipynb", "max_stars_repo_name": "jstse/SurfinPy", "max_stars_repo_head_hexsha": "ff3a79f9415c170885e109ab881368271f3dcc19", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 30, "max_stars_repo_stars_event_min_datetime": "2019-01-28T17:47:24.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-22T03:26:00.000Z", "max_issues_repo_path": "examples/Notebooks/Surfaces/Tutorial_3.ipynb", "max_issues_repo_name": "jstse/SurfinPy", "max_issues_repo_head_hexsha": "ff3a79f9415c170885e109ab881368271f3dcc19", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 14, "max_issues_repo_issues_event_min_datetime": "2018-09-03T15:49:06.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-08T22:09:51.000Z", "max_forks_repo_path": "examples/Notebooks/Surfaces/Tutorial_3.ipynb", "max_forks_repo_name": "jstse/SurfinPy", "max_forks_repo_head_hexsha": "ff3a79f9415c170885e109ab881368271f3dcc19", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 19, "max_forks_repo_forks_event_min_datetime": "2019-02-11T09:11:29.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-11T08:47:24.000Z", "avg_line_length": 333.4099099099, "max_line_length": 33088, "alphanum_fraction": 0.9353526892, "converted": true, "num_tokens": 1116, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.43544591824361073}} {"text": "**1. Create the below pattern using nested for loop in Python**\n$$\n\\begin{align}\n&*\\\\\n&**\\\\\n&***\\\\\n&****\\\\\n&*****\\\\\n&****\\\\\n&***\\\\\n&**\\\\\n&*\n\\end{align}\n$$\n\n\n```python\nnum = 5\nfor star in range(1,10): #we take range upto 10 because we have to print 9 lines\n if star <= 5: #Stars of less than and equal to 5 have to print in increasing order of numbers \n print('*'*star)\n if star>5: #Stars less than 5 have to print in decreasing order of numbers\n print('*'*(num-1))\n num -= 1\n```\n\n *\n **\n ***\n ****\n *****\n ****\n ***\n **\n *\n\n\n**2. Write a Python program to reverse a word after accepting the input from the user.**\n

    \n Input word: ineuron
    \n Output word: norueni\n

    \n\n\n```python\nuser_input = input(\"Enter the Word: \")\n#Let say ineuron has 7 alphabet so, the loop is starting from 6 to -1 at step of -1 \n#so loops runs 7 times from 6 to -1(excluded) and print in reverse order\nfor char in range(len(user_input)-1,-1,-1): \n print(user_input[char], end='')\n```\n\n Enter the Word: ineuron\n norueni\n", "meta": {"hexsha": "fadc123a42cf29de70f64076d18f3dcaf2a8ba7d", "size": 2431, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Python Assignment/Python Assignment 2.ipynb", "max_stars_repo_name": "Abhishek20182/Machine-Learning-And-Deep-Learning-Masters", "max_stars_repo_head_hexsha": "7457a4de5d7b89d49a20f33958296ce35f728d3f", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Python Assignment/Python Assignment 2.ipynb", "max_issues_repo_name": "Abhishek20182/Machine-Learning-And-Deep-Learning-Masters", "max_issues_repo_head_hexsha": "7457a4de5d7b89d49a20f33958296ce35f728d3f", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Python Assignment/Python Assignment 2.ipynb", "max_forks_repo_name": "Abhishek20182/Machine-Learning-And-Deep-Learning-Masters", "max_forks_repo_head_hexsha": "7457a4de5d7b89d49a20f33958296ce35f728d3f", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 22.1, "max_line_length": 107, "alphanum_fraction": 0.4652406417, "converted": true, "num_tokens": 341, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.7690802370707281, "lm_q1q2_score": 0.435313669235044}} {"text": "```python\nimport numpy as np\nimport math\nfrom scipy import optimize\nimport numba as nb\nimport scipy.stats as stats\n\nimport ps3_functions as func\n\n# For plots:\nimport matplotlib as mpl\nimport matplotlib.pyplot as plt\n%matplotlib inline\nplt.style.use('seaborn-whitegrid')\nmpl.style.use('seaborn')\nprop_cycle = plt.rcParams[\"axes.prop_cycle\"]\ncolors = prop_cycle.by_key()[\"color\"]\nimport ipywidgets as widgets\nfrom ipywidgets import interact, interact_manual\n```\n\nThe formulas from Q1 are still valid. Now note that:\n\n* $c_1^*>c_1^M$ and $c_2^*c_1^M$ and $c_2^*1$. What does this imply for optimal $c_0$? And for the credit constraint?\n\n\n\n.\n\n.\n\n.\n\nYou should end up with the demand for capital:\n$$\\begin{align}\nk_1 &= \\dfrac{q_0k_0-b_0}{q_0-\\theta} \\\\ \n &= k_0+\\dfrac{k_0\\theta -b_0}{q_0-\\theta}.\n\\end{align}$$\n\n### Q3: Discuss expression for capital demand:\n\nNote that $q_0$ is both the price for new capital $k_1$ purchased at $t=0$ and the value of existing capital $k_0$.\n\nComment on capital demand and how $q_0$ affects it. You may think of two channels from above as a $\\color{red}{\\text{price}}$ effect and a $\\color{blue}{\\text{wealth}}$ effect. \n\n\n## Q4: The financier problem and the market clearing condition for capital.\n\n**Financier problem:**\n\nWrite the introduction text as a mathematical problem. \n\nSimilar setup to $E$, except that\n\n* No borrowing (endowments are large enough to finance their optimal level $k_t^F$).\n* Concave production function, not linear (and both in $t=0,1$).\n* This yields a finite optimal level of $k_t^F$.\n\nSolve for the optimal $k_1^F$.\n\n\n**Market clearing:**\n\nWith a fixed supply of capital the market clearing reads:\n\n$$\\begin{align}\n \\bar{k} = k_1^F+k_1^E.\n\\end{align}$$\n\nFor convenience we will define the *residual supply curve* that entrepreneurs face, by substitution of $(k_1^F)^*$ into the market clearing condition:\n\n$$\\begin{align}\n k_1^E = \\min\\left(\\bar{k}, \\mbox{ }\\bar{k}-(1+\\theta-q_0)\\right).\n\\end{align}$$\n\n## Q5: Show and depict graphically that when $b_0<\\theta k_0$ there is a unique equilibrium.\n\n\n## Q6: Show that when $b_0>\\theta k_0$ there can be multiple equilibria: A 'bad' and a 'good' one. Explain.\n\nIn both Q5 and Q6 it is important to keep track of different relevant ranges of the price $q_0$: \n\n* When $b_0<\\theta k_0$ is demand increasing? In which region of $q_0?$ (remember to take assumption (A1) into account).\n* Is supply increasing? What is the minimum/maximum level supply attains (on the relevant range of $q_0$)?\n\nIn the figure below, you can quickly investigate how the different parameter values affect the demand and supply of capital.\n\n\n```python\na, amin, amax = 1, 0.5, 1.5 # gives baseline value a, minimum and maximum in sliders below.\nb_0, bmin, bmax =0.6, 0.1, 1 # gives baseline value b, minimum and maximum in sliders below.\nk_0, kmin, kmax = 0.9, 0.1, 1 # gives baseline value k_0, minimum and maximum in sliders below.\ntheta, thetamin, thetamax = 0.5, 0, 1 # gives baseline value theta, minimum and maximum in sliders below.\nkbar=0.9 # Fixed level of supply.\nq_0 = np.linspace(0, 2, 1000) # range of q to consider.\nkplot_max = 1 # Maximum \nfunc.interactive_capdemand(q_0,a,a,amin,amax,b_0,b_0,bmin,bmax,k_0,k_0,kmin,kmax,theta,theta,thetamin,thetamax,kbar,kplot_max)\n```\n\n\n interactive(children=(FloatSlider(value=0.5, description='$\\\\theta$', max=1.0, step=0.05), FloatSlider(value=1\u2026\n\n\n## Q7: How can an appropriate reduction in $b_0$ eliminate the bad equilibrium?\n\nVerify that it is the case in the figure above.\n\nWhat is the intuition behind this?\n\n## Q8: Suppose there were multiple equilibria and the government stands ready to buy assets at the good equilibrium price. How many assets would it have to buy to implement this policy?\n\n## Q9: Compare costs of Q7 and Q8 as policies for working in the financial crisis. Q7 interpreted as injecting capital into the banking system and Q8 as a plan to buy toxic securities with governnment money. Discuss.\n\n# PS3, part 2: Impatient households and risky investments \n\nThe setup is a Diamond-Dybvig (1983) type of model. Specifically:\n\n* Three periods $t=0,1,2$.\n* Two storage technologies \n * One period tech. with return $1$.\n * Two period tech. with return $R>1$. Can be prematurely liquidated $L<1$.\n* Continuum of agents with unit endowment. \n* A fraction $\\pi$ will be impatient, $1-\\pi$ patient. However, there is uncertainty (at time $0$) about what type the agent'll be. The expected utility is then given by:\n $$\\begin{align}\n u = \\pi\\sqrt{c_1}+(1-\\pi)\\rho \\sqrt{c_2}.\n \\end{align}$$\n At time $t=1$ when the type is known the utility is:\n $$\\begin{align}\n \\text{If impatient}: && u_1^{ip} &= \\sqrt{c_1} \\\\ \n \\text{If patient}: && u_1^{p} &= \\rho\\sqrt{c_2}\n \\end{align}$$\n where $\\rho$ reflects standard discounting of the future.\n \nIf you are not familiar with the DD model, you may consult the brief recap in \"DD model.pdf\".\n\n## Q1: Characterize optimal allocation and market allocation.\n\nWe define the market allocation as in Diamond-Dybvig (1983):\n\n* At time $t=0$ households choose long run investment $I$ (and short run investment $(1-I)$). \n* At time $t=1$ they recognize their type.\n* A bond market is included as follows: \n * Households that turns out to be patient: Use short run investment $1-I$ to purchase bonds at $t=1$, receiving $(1-I)/p$ at $t=2$.\n * Impatient consumers receive $p$ units of income at time $t=1$, for each unit of income they repay at $t=2$. \n * Equilibrium price $p=1/R$.\n \nWrite up the $c_1$ and $c_2$ constraints. Show that the market allocation (M) is given by:\n\n$$\\begin{align}\n c_1^M = 1, && c_2^M = R.\n\\end{align}$$\n\n### Social Planner solution (optimal):\n\n\n\nA social planner allocates $I$ of the initial endowment to long run investments, and $1-I$ to short run investments. As the planner does not operate under uncertanity, $I$ is naturally chosen such that there is no premature liquidation of long run assets. This means that short run investments are allocated to impatient households (share $\\pi$), and return on long run investments are allocated to patient households. Show that this setup implies the planner solution:\n$$\\begin{align}\n c_1^* = \\dfrac{1}{\\pi+(1-\\pi)\\rho^2R}, && c_2^* = \\dfrac{(\\rho R)^2}{\\pi+(1-\\pi)\\rho^2R}.\n\\end{align}$$\n\nComment on the result. Compare to the market outcome.\n\n## Q2, part 1: Is it possible to introduce financial intermediaries, and does this implement the optimal allocation?\n\n(*State-contingent contract: The outcome of the contract depends on the realization of some state (e.g. which type of agent the consumer is at $t=1$)).*\n\nYes. If the financial intermediaries offer **state-contingent contracts** the market is said to be **complete**, in which case the uncertainty no longer entails inefficiency. Try to outline how such a contract might look like. In particular, include the fact that the contract must be **incentive compatible** and obey the **participation constraint** for the financial intermediary.\n\n## Q2, part 2: Is it possible to have bank runs, and why are they inefficient?\n\n\n* Assume that a patient household expects all other households to withdraw early. What happens to this household's incentives to withdraw early/keep patient?\n* Of the patient households withdraw early, the bank has to liquidate long run investments prematurely. What is the maximum amount of income they can amass in $t=1$? Denote this $BR$.\n* Show that a bank run is an equilibrium if $BR \n\n\n\n




    \n# IWI131\n## Programaci\u00f3n de Computadores\n\n### Sebasti\u00e1n Flores\n\nhttp://progra.usm.cl/ \n\nhttps://www.github.com/usantamaria/iwi131\n\n\n## \u00bfQu\u00e9 contenido aprenderemos?\n\n* Listas\n* Tuplas\n\n## \u00bfPorqu\u00e9 aprenderemos ese contenido?\n\n* Listas\n* Tuplas\n\nPorque utilizar colecciones de elementos resulta natural y permite simplificar operaciones.\n\n## Motivaci\u00f3n\n\nCalcule el promedio y la desviaci\u00f3n est\u00e1ndar de $N$ datos, $x_1$, ..., $x_n$, ingresados por el usuario:\n\n$$ \\begin{align}\nmean &= \\frac{1}{n} \\sum_{i=1} x_i \\\\\n(std)^2 &= \\frac{1}{n} \\sum_{i=1} (x_i- mean)^2\n\\end{align}\n$$\n\n\n```python\nN = int(raw_input(\"Ingrese numero de datos:\"))\n# Calculo del promedio\nprint \"Ingrese numeros (para calcular promedio)\"\nj = 1\nmean = 0.0\nwhile j<=N:\n x_i = float(raw_input(\"Ingrese numero: \"))\n mean += x_i\n j += 1\nmean = mean / N\n# Calculo de desviacion estandar\nprint \"Ingrese los MISMOS numeros (para calcular desviacion estandar)\"\nj = 1\nvar = 0.0\nwhile j<=N:\n x_i = float(raw_input(\"Ingrese numero: \"))\n var += (x_i-mean)**2\n j += 1\nstd = (var/N)**0.5\n# Imprimir resultados\nprint \"promedio\", mean\nprint \"desviacion estandar\", std\n```\n\n## An\u00e1lisis del problema anterior\nEl problema anterior resultaba engorroso para el usuario porque necesitabamos realizar 2 ciclos while en los datos: \n\n* La primera vez para calcular el promedio. \n* La segunda vez para calcular la desviaci\u00f3n est\u00e1ndar.\n\nProceso ser\u00eda m\u00e1s f\u00e1cil si primero almacen\u00e1ramos los datos, y luego realiz\u00e1ramos los c\u00e1lculos necesarios sin molestar al usuario. \n\n## Tuplas\n * Colecciones ***inmutables*** de datos.\n * Tienen un n\u00famero reducido de operaciones.\n * Se utilizan cuando el n\u00famero de elementos y su contenido no var\u00eda: por ejemplo, coordenadas de un punto\n ***(x,y,z)*** o n\u00famero complejo ***(Real,Imaginario)***.\n\n## Listas\n * Colecciones ***mutables*** de datos.\n * Tienen un n\u00famero amplio de operaciones.\n * Se utilizan cuando el n\u00famero de elementos es variable.\n\n\n\n\n## Tuplas\nColecciones heterog\u00e9neas e ***inmutables*** de datos.\n \nSe definen con par\u00e9ntesis redondos y como colecci\u00f3n pueden tener datos de distintos tipos:\n\n\n```python\nposicion_alfil = (7, 6)\n\nalumno = ('Fulano', 'De Tal', '201199001-1')\n\ncarta = (5, 'corazones')\n\nfecha = (2011, 4, 12)\n\ntriangulo = ((5, 1), (2, 4), (-2, 0))\n\npersonaje = ('Arturo Prat', (1848, 4.0, 3), (1879, 5, 21))\n```\n\n\n```python\nprint personaje[0][0:5]\n```\n\n#### Tuplas\n## Desempaquetado de Tuplas\n\nSi una tupla tiene $n$ datos, es posible \"desempaquetar\" en variables de manera directa:\n\n\n```python\npunto = (6.6, -2.4, 3.7)\nx, y, z = punto\nprint x\n```\n\n\n```python\npersonaje = (\"Bernardo O'Higgins\", (1778, 8, 20), (1842, 10, 24))\nnombre, nacimiento, _ = personaje\nan, mn, dn = nacimiento\nad, md, dd = defuncion\nprint ad - an\nprint personaje\n```\n\n#### Tuplas\n## \u00bfC\u00f3mo reconocer una tupla?\n\nUtilice **type**\n\n\n```python\na = (1,2)\nb = (-1,-2)\nc1 = a + b\nprint c1\nc2 = (a[0] + b[0], a[1] + b[1])\nprint c2\nprint type(a[1])\n```\n\n#### Tuplas\n## \u00bfDonde hemos utilizado ya tuplas?\n\nEn las funciones de m\u00faltiples retornos:\n\n\n```python\ndef rectangulo(a,b):\n peri = 2*(a+b)\n area = a*b\n return peri, area\n\ns = rectangulo(1,2)\nprint type(s)\nprint s\n\np, a = rectangulo(2,3) # Esto de clase anterior era una desempaquetado!\nprint type(p)\nprint p\nprint type(s)\nprint s\n```\n\n#### Tuplas\n## \u00bfQu\u00e9 m\u00e9todos puedo aplicar a tuplas?\n* **len**: cuantos elementos hay en la tupla\n* **sum**: sacar elementos de la tupla\n\n\n```python\ns_mixto = (1, 1.0, True, \"1\")\nprint len(s_mixto)\n#print sum(s_mixto)\nprint sum(s_mixto[:3])\n\ns = (1, 1.0, 2., 2, 3.)\nprint len(s)\nprint s[1:-1]\nprint sum(s)\nprint sum(s[1:-1])\n```\n\n#### Listas\n## Otros m\u00e9todos\n\nTestear pertenencia en tupla\n\n dato in tupla\n \nregresar\u00e1 siempre un booleano: True si dato est\u00e1 en la tupla, False si no est\u00e1 en la tupla.\n\n\n```python\nmi_tupla = (\"sebastian\", \"carlos\", \"jose\")\n\nprint \"sebastian\" in mi_tupla\n\nprint \"seba\" in mi_tupla\n\nprint \"carlos\" in mi_tupla\n\nprint \"pepe\" in mi_tupla\n```\n\n#### Tuplas\n## Acceder valores vs Cambiar valores\n* Los valores de una tupla pueden accederse mediante \u00edndices: **[i], [i:], [:j], [i:j]**.\n\n mi_tupla[mi_indice]\n\n* Los valores de una tupla no pueden cambiarse. La tupla es completamente inmutable.\n\n\n```python\n# Posicion original\npunto = (0, 0, 0)\nprint punto\n\n# Acceso a valores\nv = (1,2,3)\ndt = 2.0\nprint punto[0] + v[0]*dt\nprint punto[1] + v[1]*dt\nprint punto[2] + v[2]*dt\n\n# Actualizaci\u00f3n de valores\n#punto[0] = 1\n\ns = (1, 1.0, 2., 2, 3.)\nprint s[1:-1]\nprint len(s)\nprint sum(s)\nprint sum(s[1:-1])\n```\n\n#### Tuplas\n## Ejemplo: Fibonacci con tuplas\nImprima los primeros $n$ numeros de fibonacci\n\n\n```python\nn = int(raw_input(\"Ingrese n: \"))\nj = 1\na, b = 0, 1\nwhile j<=n:\n print b, \n a, b = b, a+b # Mira Ma, sin variables auxiliares\n j+= 1\n```\n\n## Listas\nColecciones heterog\u00e9neas y ***mutables*** de datos.\n\nSe definen con par\u00e9ntesis cuadrados: [ ]\n\n\n```python\nmi_lista_1 = [1, 1., \"1\", True]\nmi_lista_2 = [1, 2, 4, 10, 55]\n\nprint mi_lista_1\nprint mi_lista_2\n\n# Convertir de tupla a lista\nmi_tupla = tuple(mi_lista_1)\nprint mi_tupla, type(mi_tupla)\nmi_nueva_lista = list(mi_tupla)\nprint mi_nueva_lista, type(mi_nueva_lista)\n\n```\n\n#### Listas\n## Creando listas\nLas listas se dicen mutables porque se pueden crear din\u00e1micamente utilizando el m\u00e9todo ***append***.\n\nEl m\u00e9todo append agrega un dato **al final** de la lista.\n\n\n```python\nvalores = []\nprint valores\n\nvalores.append(5)\nprint valores\n\nvalores.append(1)\nprint valores\n\nvalores.append(6)\nprint valores\n\nvalores.append(-4)\nprint valores\n```\n\n#### Listas\n## Creando listas con datos del usuario\n\n\n\n```python\n# Terminamos de pedir datos cuando se ingresa \"fin\".\nmi_lista = [] # Lista vacia\nwhile True:\n mi_dato = raw_input(\"Ingrese dato: \")\n if mi_dato==\"fin\":\n break\n else:\n if mi_dato in (\"True\", \"False\"):\n mi_lista.append(bool(mi_dato))\n else:\n mi_lista.append(mi_dato)\n \n# Imprimamos la lista\nprint mi_lista\n\nprint bool(\"True\")\nprint bool(\"False\")\nprint bool(\"\")\n```\n\n#### Listas\n## Accesando listas\n\nLos elementos de la lista se accesan utilizando **[i:j]** de la misma forma que strings y tuplas.\n\n\n```python\na = [0, 10, 20, 30]\nprint a[0]\nprint a[-1]\nprint a\n\na.append(2)\na.append(-3)\nprint a[0]\nprint a[-1]\nprint a\n```\n\n#### Listas\n## Otros m\u00e9todos\n\n* len(lista): regresa la cantidad de objetos de la lista\n* sum(lista): regresa la suma de los elementos de la lista (si se puede).\n* lista.pop(): regresa el ultimo elemento de la lista y lo saca de la lista.\n* lista.sort(): ordena los elementos de la lista de menor a mayor.\n* lista.reverse(): ordena los elementos de la lista en el orden reverso al original.\n\n\n```python\nx = [1,3,5,7,9,0,2,4,6,8]\nprint len(x)\n\nprint sum(x)\n\nx.reverse() # Revierte orden de la lista (y no regresa nada)\nprint x\n\nx.sort() # Ordena la lista (y no regresa nada)\nprint x\n\nx.reverse() # Revierte orden de la lista (y no regresa nada)\nprint x\n\nxi = x.pop()\nprint xi\nprint x\n```\n\n#### Listas\n## Otros m\u00e9todos\n\nTestear pertenencia en lista\n\n dato in lista\n \nregresar\u00e1 siempre un booleano: True si dato est\u00e1 en la lista, False si no est\u00e1 en la lista.\n\n\n```python\nmi_lista = [\"sebastian\", \"carlos\", \"jose\"]\n\nprint \"sebastian\" in mi_lista\n\nprint \"seba\" in mi_lista\n\nprint \"carlos\" in mi_lista\n\nprint \"pepe\" in mi_lista\n```\n\n#### Listas\n## Otros m\u00e9todos: range\n\nEl m\u00e9todo range es pr\u00e1ctico para **generar listas de n\u00fameros enteros**:\n\n* range(m) regresa la lista [0,1,2,..,m-1] \n* range(n,m) regresa la lista [n,n+1,n+2,...,m-1]\n* range(n,m,k) regresa la lista [n,n+k, n+2 k,...,x] donde x es el mayor numero n+j k < m\n\n\n```python\nprint range(5)\nprint range(10)\n\nprint range(2, 5)\nprint range(2, 10)\n\nprint range(2, 5, 3)\nprint range(2, 11, 3)\n```\n\n#### Listas\n## Ejemplo\nCalcula la suma de los rec\u00edprocos de los primeros n numeros naturales:\n\n$$ \\frac{1}{1}+\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}+..\\frac{1}{n}$$\n\n\n```python\nn = 10\nl = []\nj = 1\nwhile j<=n:\n l.append(1./j)\n j+= 1\nprint sum(l)\n```\n\n#### Listas\n## Copiando listas: CUIDADO\n\nCuando las listas se asignan, en realidad hacen referencia a una lista com\u00fan. Esto se hace para ahorrar memoria RAM y est\u00e1 implementado as\u00ed en python.\n\n\n```python\na = [5, 1, 4]\nb = list(a) # a\nb.append(10)\n# Modifiquemos b\nb[0] = 1000\nprint b\nprint a\n# la lista a tambien se vio modificada\n```\n\nPara evitar este comportamiento, debemos utilizar\n\n b = list(a)\n \n\n\n#### Listas\n## Aplicaci\u00f3n al ejemplo\nCalcule el promedio y la desviaci\u00f3n est\u00e1ndar de $N$ datos, $x_1$, ..., $x_n$, ingresados por el usuario:\n\n$$ \\begin{align}\nmean &= \\frac{1}{n} \\sum_{i=1} x_i \\\\\n(std)^2 &= \\frac{1}{n} \\sum_{i=1} (x_i- mean)^2\n\\end{align}\n$$\n\n\n```python\n###################################################################\n# Cambiar para utilizar listas\n###################################################################\nN = int(raw_input(\"Ingrese numero de datos:\"))\n# Calculo del promedio\nprint \"Ingrese numeros (para calcular promedio)\"\nj = 1\nmean = 0.0\nwhile j<=N:\n x_i = float(raw_input(\"Ingrese numero: \"))\n mean += x_i\n j += 1\nmean = mean / N\n# Calculo de desviacion estandar\nprint \"Ingrese los MISMOS numeros (para calcular desviacion estandar)\"\nj = 1\nvar = 0.0\nwhile j<=N:\n x_i = float(raw_input(\"Ingrese numero: \"))\n var += (x_i-mean)**2\n j += 1\nstd = (var/N)**0.5\n# Imprimir resultados\nprint \"promedio\", mean\nprint \"desviacion estandar\", std\n```\n\n#### Listas\n## Soluci\u00f3n v1\nCalcule el promedio y la desviaci\u00f3n est\u00e1ndar de $N$ datos, $x_1$, ..., $x_n$, ingresados por el usuario.\n\n\n```python\nN = int(raw_input(\"Ingrese numero de datos:\"))\nlista_datos = []\n# Crear la lista\nj = 1\nwhile j<=N:\n x_i = float(raw_input(\"Ingrese numero: \"))\n lista_datos.append(x_i)\n j += 1\n# Calcular el promedio\ni = 0\nmean = 0.0\nwhile i>> pelicula_por_pais(cartelera, 'FRANCIA')\n [('El muelle', 1962), ('La dama de honor', 2004), ('Melo', 1986)]\n\n\n```python\n# Solucion estudiantes\n```\n\n#### Ejercicio Tipo Certamen: C2 1S 2014\n## Soluci\u00f3n pregunta 1\n\nDesarrolle la funci\u00f3n pelicula_por_pais(cartelera, pais) que recibe la lista de la cartelera y el nombre de un pa\u00eds, y que retorne la lista con las pel\u00edculas realizadas en dicho pa\u00eds. Cada elemento de esta lista resultante es una tupla con el nombre de la pel\u00edcula y el a\u00f1o de filmaci\u00f3n.\n\n >>> pelicula_por_pais(cartelera, 'FRANCIA')\n [('El muelle', 1962), ('La dama de honor', 2004), ('Melo', 1986)]\n\n\n```python\n# Solucion\n\ndef pelicula_por_pais(cartelera, pais):\n n = len(cartelera)\n j = 0\n peliculas_pais = []\n while j\n\n# Tutorial 3: \"Why\" models\n**Week 1, Day 1: Model Types**\n\n**By Neuromatch Academy**\n\n__Content creators:__ Matt Laporte, Byron Galbraith, Konrad Kording\n\n__Content reviewers:__ Dalin Guo, Aishwarya Balwani, Madineh Sarvestani, Maryam Vaziri-Pashkam, Michael Waskom, Ella Batty\n\nWe would like to acknowledge [Steinmetz _et al._ (2019)](https://www.nature.com/articles/s41586-019-1787-x) for sharing their data, a subset of which is used here.\n\n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n

    \n\n___\n# Tutorial Objectives\n\n*Estimated timing of tutorial: 45 minutes*\n\nThis is tutorial 3 of a 3-part series on different flavors of models used to understand neural data. In parts 1 and 2 we explored mechanisms that would produce the data. In this tutorial we will explore models and techniques that can potentially explain *why* the spiking data we have observed is produced the way it is.\n\nTo understand why different spiking behaviors may be beneficial, we will learn about the concept of entropy. Specifically, we will:\n\n- Write code to compute formula for entropy, a measure of information\n- Compute the entropy of a number of toy distributions\n- Compute the entropy of spiking activity from the Steinmetz dataset\n\n\n```python\n# @title Tutorial slides\n\n# @markdown These are the slides for the videos in all tutorials today\nfrom IPython.display import IFrame\nIFrame(src=f\"https://mfr.ca-1.osf.io/render?url=https://osf.io/6dxwe/?direct%26mode=render%26action=download%26mode=render\", width=854, height=480)\n```\n\n\n```python\n# @title Video 1: \u201cWhy\u201d models\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = 'https://player.bilibili.com/player.html?bvid={0}&page={1}'.format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=\"BV16t4y1Q7DR\", width=854, height=480, fs=1)\n print('Video available at https://www.bilibili.com/video/{0}'.format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=\"OOIDEr1e5Gg\", width=854, height=480, fs=1, rel=0)\n print('Video available at https://youtube.com/watch?v=' + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n# Setup\n\n\n```python\n# Imports\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy import stats\n```\n\n\n```python\n#@title Figure Settings\nimport ipywidgets as widgets #interactive display\n\n%matplotlib inline\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/course-content/master/nma.mplstyle\")\n```\n\n\n```python\n#@title Plotting Functions\n\ndef plot_pmf(pmf,isi_range):\n \"\"\"Plot the probability mass function.\"\"\"\n ymax = max(0.2, 1.05 * np.max(pmf))\n pmf_ = np.insert(pmf, 0, pmf[0])\n plt.plot(bins, pmf_, drawstyle=\"steps\")\n plt.fill_between(bins, pmf_, step=\"pre\", alpha=0.4)\n plt.title(f\"Neuron {neuron_idx}\")\n plt.xlabel(\"Inter-spike interval (s)\")\n plt.ylabel(\"Probability mass\")\n plt.xlim(isi_range);\n plt.ylim([0, ymax])\n```\n\n\n```python\n#@title Download Data\nimport io\nimport requests\nr = requests.get('https://osf.io/sy5xt/download')\nif r.status_code != 200:\n print('Could not download data')\nelse:\n steinmetz_spikes = np.load(io.BytesIO(r.content), allow_pickle=True)['spike_times']\n```\n\n---\n# Section 1: Optimization and Information\n\n*Remember that the notation section is located after the Summary for quick reference!*\n\nNeurons can only fire so often in a fixed period of time, as the act of emitting a spike consumes energy that is depleted and must eventually be replenished. To communicate effectively for downstream computation, the neuron would need to make good use of its limited spiking capability. This becomes an optimization problem: \n\nWhat is the optimal way for a neuron to fire in order to maximize its ability to communicate information?\n\nIn order to explore this question, we first need to have a quantifiable measure for information. Shannon introduced the concept of entropy to do just that, and defined it as\n\n\\begin{align}\n H_b(X) = -\\sum_{x\\in X} p(x) \\log_b p(x)\n\\end{align}\n\nwhere $H$ is entropy measured in units of base $b$ and $p(x)$ is the probability of observing the event $x$ from the set of all possible events in $X$. See the Bonus Section 1 for a more detailed look at how this equation was derived.\n\nThe most common base of measuring entropy is $b=2$, so we often talk about *bits* of information, though other bases are used as well (e.g. when $b=e$ we call the units *nats*).\n\nFirst, let's explore how entropy changes between some simple discrete probability distributions. In the rest of this tutorial we will refer to these as probability mass functions (PMF), where $p(x_i)$ equals the $i^{th}$ value in an array, and mass refers to how much of the distribution is contained at that value.\n\nFor our first PMF, we will choose one where all of the probability mass is located in the middle of the distribution.\n\n\n```python\nn_bins = 50 # number of points supporting the distribution\nx_range = (0, 1) # will be subdivided evenly into bins corresponding to points\n\nbins = np.linspace(*x_range, n_bins + 1) # bin edges\n\npmf = np.zeros(n_bins)\npmf[len(pmf) // 2] = 1.0 # middle point has all the mass\n\n# Since we already have a PMF, rather than un-binned samples, `plt.hist` is not\n# suitable. Instead, we directly plot the PMF as a step function to visualize\n# the histogram:\npmf_ = np.insert(pmf, 0, pmf[0]) # this is necessary to align plot steps with bin edges\nplt.plot(bins, pmf_, drawstyle=\"steps\")\n# `fill_between` provides area shading\nplt.fill_between(bins, pmf_, step=\"pre\", alpha=0.4)\nplt.xlabel(\"x\")\nplt.ylabel(\"p(x)\")\nplt.xlim(x_range)\nplt.ylim(0, 1);\n```\n\nIf we were to draw a sample from this distribution, we know exactly what we would get every time. Distributions where all the mass is concentrated on a single event are known as *deterministic*.\n\nHow much entropy is contained in a deterministic distribution? We will compute this in the next exercise.\n\n## Coding Exercise 1: Computing Entropy\n\nYour first exercise is to implement a method that computes the entropy of a discrete probability distribution, given its mass function. Remember that we are interested in entropy in units of _bits_, so be sure to use the correct log function. \n\nRecall that $\\log(0)$ is undefined. When evaluated at $0$, NumPy log functions (such as `np.log2`) return `np.nan` (\"Not a Number\"). By convention, these undefined terms\u2014 which correspond to points in the distribution with zero mass\u2014are excluded from the sum that computes the entropy.\n\n\n```python\ndef entropy(pmf):\n \"\"\"Given a discrete distribution, return the Shannon entropy in bits.\n\n This is a measure of information in the distribution. For a totally\n deterministic distribution, where samples are always found in the same bin,\n then samples from the distribution give no more information and the entropy\n is 0.\n\n For now this assumes `pmf` arrives as a well-formed distribution (that is,\n `np.sum(pmf)==1` and `not np.any(pmf < 0)`)\n\n Args:\n pmf (np.ndarray): The probability mass function for a discrete distribution\n represented as an array of probabilities.\n Returns:\n h (number): The entropy of the distribution in `pmf`.\n\n \"\"\"\n ############################################################################\n # Exercise for students: compute the entropy of the provided PMF\n # 1. Exclude the points in the distribution with no mass (where `pmf==0`).\n # Hint: this is equivalent to including only the points with `pmf>0`.\n # 2. Implement the equation for Shannon entropy (in bits).\n # When ready to test, comment or remove the next line\n ############################################################################\n\n # reduce to non-zero entries to avoid an error from log2(0)\n pmf = pmf[pmf>0]\n\n # implement the equation for Shannon entropy (in bits)\n h = -np.sum(pmf*np.log2(pmf))\n\n # return the absolute value (avoids getting a -0 result)\n return np.abs(h)\n\n# Call entropy function and print result\nprint(f\"{entropy(pmf):.2f} bits\")\n```\n\n 0.00 bits\n\n\n\n```python\n# to_remove solution\ndef entropy(pmf):\n \"\"\"Given a discrete distribution, return the Shannon entropy in bits.\n\n This is a measure of information in the distribution. For a totally\n deterministic distribution, where samples are always found in the same bin,\n then samples from the distribution give no more information and the entropy\n is 0.\n\n For now this assumes `pmf` arrives as a well-formed distribution (that is,\n `np.sum(pmf)==1` and `not np.any(pmf < 0)`)\n\n Args:\n pmf (np.ndarray): The probability mass function for a discrete distribution\n represented as an array of probabilities.\n Returns:\n h (number): The entropy of the distribution in `pmf`.\n \"\"\"\n # reduce to non-zero entries to avoid an error from log2(0)\n pmf = pmf[pmf > 0]\n\n # implement the equation for Shannon entropy (in bits)\n h = -np.sum(pmf * np.log2(pmf))\n\n # return the absolute value (avoids getting a -0 result)\n return np.abs(h)\n\n# Call entropy function and print result\nprint(f\"{entropy(pmf):.2f} bits\")\n```\n\n 0.00 bits\n\n\nWe expect zero surprise from a deterministic distribution. If we had done this calculation by hand, it would simply be $-1\\log_2 1 = -0=0$.\n\nNote that changing the location of the peak (i.e. the point and bin on which all the mass rests) doesn't alter the entropy. The entropy is about how predictable a sample is with respect to a distribution. A single peak is deterministic regardless of which point it sits on - the following plot shows a PMF that would also have zero entropy.\n\n\n```python\n# @markdown Execute this cell to visualize another PMF with zero entropy\npmf = np.zeros(n_bins)\npmf[2] = 1.0 # arbitrary point has all the mass\n\npmf_ = np.insert(pmf, 0, pmf[0])\nplt.plot(bins, pmf_, drawstyle=\"steps\")\nplt.fill_between(bins, pmf_, step=\"pre\", alpha=0.4)\nplt.xlabel(\"x\")\nplt.ylabel(\"p(x)\")\nplt.xlim(x_range)\nplt.ylim(0, 1);\n```\n\nWhat about a distribution with mass split equally between two points?\n\n\n```python\n# @markdown Execute this cell to visualize a PMF with split mass\n\npmf = np.zeros(n_bins)\npmf[len(pmf) // 3] = 0.5\npmf[2 * len(pmf) // 3] = 0.5\n\npmf_ = np.insert(pmf, 0, pmf[0])\nplt.plot(bins, pmf_, drawstyle=\"steps\")\nplt.fill_between(bins, pmf_, step=\"pre\", alpha=0.4)\nplt.xlabel(\"x\")\nplt.ylabel(\"p(x)\")\nplt.xlim(x_range)\nplt.ylim(0, 1);\n```\n\nHere, the entropy calculation is: $-(0.5 \\log_2 0.5 + 0.5\\log_2 0.5)=1$\n\nThere is 1 bit of entropy. This means that before we take a random sample, there is 1 bit of uncertainty about which point in the distribution the sample will fall on: it will either be the first peak or the second one. \n\nLikewise, if we make one of the peaks taller (i.e. its point holds more of the probability mass) and the other one shorter, the entropy will decrease because of the increased certainty that the sample will fall on one point and not the other: : $-(0.2 \\log_2 0.2 + 0.8\\log_2 0.8)\\approx 0.72$\n\n\n\nTry changing the definition of the number and weighting of peaks, and see how the entropy varies.\n\nIf we split the probability mass among even more points, the entropy continues to increase. Let's derive the general form for $N$ points of equal mass, where $p_i=p=1/N$:\n\n\\begin{align}\n -\\sum_i p_i \\log_b p_i&= -\\sum_i^N \\frac{1}{N} \\log_b \\frac{1}{N}\\\\\n &= -\\log_b \\frac{1}{N} \\\\\n &= \\log_b N\n\\end{align}\n\n\nIf we have $N$ discrete points, the _uniform distribution_ (where all points have equal mass) is the distribution with the highest entropy: $\\log_b N$. This upper bound on entropy is useful when considering binning strategies, as any estimate of entropy over $N$ discrete points (or bins) must be in the interval $[0, \\log_b N]$.\n\n\n\n```python\n# @markdown Execute this cell to visualize a PMF of uniform distribution\n\npmf = np.ones(n_bins) / n_bins # [1/N] * N\n\npmf_ = np.insert(pmf, 0, pmf[0])\nplt.plot(bins, pmf_, drawstyle=\"steps\")\nplt.fill_between(bins, pmf_, step=\"pre\", alpha=0.4)\nplt.xlabel(\"x\")\nplt.ylabel(\"p(x)\")\nplt.xlim(x_range);\nplt.ylim(0, 1);\n```\n\nHere, there are 50 points and the entropy of the uniform distribution is $\\log_2 50\\approx 5.64$. If we construct _any_ discrete distribution $X$ over 50 points (or bins) and calculate an entropy of $H_2(X)>\\log_2 50$, something must be wrong with our implementation of the discrete entropy computation.\n\n---\n# Section 2: Information, neurons, and spikes\n\n*Estimated timing to here from start of tutorial: 20 min*\n\n\n```python\n# @title Video 2: Entropy of different distributions\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = 'https://player.bilibili.com/player.html?bvid={0}&page={1}'.format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=\"BV1df4y1976g\", width=854, height=480, fs=1)\n print('Video available at https://www.bilibili.com/video/{0}'.format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=\"o6nyrx3KH20\", width=854, height=480, fs=1, rel=0)\n print('Video available at https://youtube.com/watch?v=' + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nRecall the discussion of spike times and inter-spike intervals (ISIs) from Tutorial 1. What does the information content (or distributional entropy) of these measures say about our theory of nervous systems? \n\nWe'll consider three hypothetical neurons that all have the same mean ISI, but with different distributions:\n\n1. Deterministic\n2. Uniform\n3. Exponential\n\nFixing the mean of the ISI distribution is equivalent to fixing its inverse: the neuron's mean firing rate. If a neuron has a fixed energy budget and each of its spikes has the same energy cost, then by fixing the mean firing rate, we are normalizing for energy expenditure. This provides a basis for comparing the entropy of different ISI distributions. In other words: if our neuron has a fixed budget, what ISI distribution should it express (all else being equal) to maximize the information content of its outputs?\n\nLet's construct our three distributions and see how their entropies differ.\n\n\n```python\nn_bins = 50\nmean_isi = 0.025\nisi_range = (0, 0.25)\n\nbins = np.linspace(*isi_range, n_bins + 1)\nmean_idx = np.searchsorted(bins, mean_isi)\n\n# 1. all mass concentrated on the ISI mean\npmf_single = np.zeros(n_bins)\npmf_single[mean_idx] = 1.0\n\n# 2. mass uniformly distributed about the ISI mean\npmf_uniform = np.zeros(n_bins)\npmf_uniform[0:2*mean_idx] = 1 / (2 * mean_idx)\n\n# 3. mass exponentially distributed about the ISI mean\npmf_exp = stats.expon.pdf(bins[1:], scale=mean_isi)\npmf_exp /= np.sum(pmf_exp)\n```\n\n\n```python\n#@title\n#@markdown Run this cell to plot the three PMFs\nfig, axes = plt.subplots(ncols=3, figsize=(18, 5))\n\ndists = [# (subplot title, pmf, ylim)\n (\"(1) Deterministic\", pmf_single, (0, 1.05)),\n (\"(1) Uniform\", pmf_uniform, (0, 1.05)),\n (\"(1) Exponential\", pmf_exp, (0, 1.05))]\n\nfor ax, (label, pmf_, ylim) in zip(axes, dists):\n pmf_ = np.insert(pmf_, 0, pmf_[0])\n ax.plot(bins, pmf_, drawstyle=\"steps\")\n ax.fill_between(bins, pmf_, step=\"pre\", alpha=0.4)\n ax.set_title(label)\n ax.set_xlabel(\"Inter-spike interval (s)\")\n ax.set_ylabel(\"Probability mass\")\n ax.set_xlim(isi_range);\n ax.set_ylim(ylim);\n```\n\n\n```python\nprint(\n f\"Deterministic: {entropy(pmf_single):.2f} bits\",\n f\"Uniform: {entropy(pmf_uniform):.2f} bits\",\n f\"Exponential: {entropy(pmf_exp):.2f} bits\",\n sep=\"\\n\",\n)\n```\n\n Deterministic: 0.00 bits\n Uniform: 3.32 bits\n Exponential: 3.77 bits\n\n\n---\n# Section 3: Calculate entropy of ISI distributions from data\n\n*Estimated timing to here from start of tutorial: 25 min*\n\n## Section 3.1: Computing probabilities from histogram\n\n\n```python\n# @title Video 3: Probabilities from histogram\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = 'https://player.bilibili.com/player.html?bvid={0}&page={1}'.format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=\"BV1Jk4y1B7cz\", width=854, height=480, fs=1)\n print('Video available at https://www.bilibili.com/video/{0}'.format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=\"e2U_-07O9jo\", width=854, height=480, fs=1, rel=0)\n print('Video available at https://youtube.com/watch?v=' + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nIn the previous example we created the PMFs by hand to illustrate idealized scenarios. How would we compute them from data recorded from actual neurons?\n\nOne way is to convert the ISI histograms we've previously computed into discrete probability distributions using the following equation:\n\n\\begin{align}\np_i = \\frac{n_i}{\\sum\\nolimits_{i}n_i}\n\\end{align}\n\nwhere $p_i$ is the probability of an ISI falling within a particular interval $i$ and $n_i$ is the count of how many ISIs were observed in that interval.\n\n### Coding Exercise 3.1: Probabilty Mass Function\n\nYour second exercise is to implement a method that will produce a probability mass function from an array of ISI bin counts.\n\nTo verify your solution, we will compute the probability distribution of ISIs from real neural data taken from the Steinmetz dataset.\n\n\n```python\ndef pmf_from_counts(counts):\n \"\"\"Given counts, normalize by the total to estimate probabilities.\"\"\"\n ###########################################################################\n # Exercise: Compute the PMF. Remove the next line to test your function\n ###########################################################################\n\n pmf = counts/np.sum(counts)\n\n return pmf\n\n\n# Get neuron index\nneuron_idx = 283\n\n# Get counts of ISIs from Steinmetz data\nisi = np.diff(steinmetz_spikes[neuron_idx])\nbins = np.linspace(*isi_range, n_bins + 1)\ncounts, _ = np.histogram(isi, bins)\n\n# Compute pmf\npmf = pmf_from_counts(counts)\n\n# Visualize\nplot_pmf(pmf,isi_range)\n```\n\n\n```python\n# to_remove solution\ndef pmf_from_counts(counts):\n \"\"\"Given counts, normalize by the total to estimate probabilities.\"\"\"\n pmf = counts / np.sum(counts)\n return pmf\n\n\n# Get neuron index\nneuron_idx = 283\n\n# Get counts of ISIs from Steinmetz data\nisi = np.diff(steinmetz_spikes[neuron_idx])\nbins = np.linspace(*isi_range, n_bins + 1)\ncounts, _ = np.histogram(isi, bins)\n\n# Compute pmf\npmf = pmf_from_counts(counts)\n\n# Visualize\nwith plt.xkcd():\n plot_pmf(pmf,isi_range)\n```\n\n## Section 3.2: Calculating entropy from pmf\n\n\n```python\n# @title Video 4: Calculating entropy from pmf\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = 'https://player.bilibili.com/player.html?bvid={0}&page={1}'.format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=\"BV1vA411e7Cd\", width=854, height=480, fs=1)\n print('Video available at https://www.bilibili.com/video/{0}'.format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=\"Xjy-jj-6Oz0\", width=854, height=480, fs=1, rel=0)\n print('Video available at https://youtube.com/watch?v=' + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nNow that we have the probability distribution for the actual neuron spiking activity, we can calculate its entropy.\n\n\n```python\nprint(f\"Entropy for Neuron {neuron_idx}: {entropy(pmf):.2f} bits\")\n```\n\n Entropy for Neuron 283: 3.36 bits\n\n\n### Interactive Demo 3.2: Entropy of neurons\n\nWe can combine the above distribution plot and entropy calculation with an interactive widget to explore how the different neurons in the dataset vary in spiking activity and relative information. Note that the mean firing rate across neurons is not fixed, so some neurons with a uniform ISI distribution may have higher entropy than neurons with a more exponential distribution.\n\n\n\n\n\n```python\n#@title\n#@markdown **Run the cell** to enable the sliders.\n\ndef _pmf_from_counts(counts):\n \"\"\"Given counts, normalize by the total to estimate probabilities.\"\"\"\n pmf = counts / np.sum(counts)\n return pmf\n\ndef _entropy(pmf):\n \"\"\"Given a discrete distribution, return the Shannon entropy in bits.\"\"\"\n # remove non-zero entries to avoid an error from log2(0)\n pmf = pmf[pmf > 0]\n h = -np.sum(pmf * np.log2(pmf))\n # absolute value applied to avoid getting a -0 result\n return np.abs(h)\n\n@widgets.interact(neuron=widgets.IntSlider(0, min=0, max=(len(steinmetz_spikes)-1)))\ndef steinmetz_pmf(neuron):\n \"\"\" Given a neuron from the Steinmetz data, compute its PMF and entropy \"\"\"\n isi = np.diff(steinmetz_spikes[neuron])\n bins = np.linspace(*isi_range, n_bins + 1)\n counts, _ = np.histogram(isi, bins)\n pmf = _pmf_from_counts(counts)\n\n plot_pmf(pmf,isi_range)\n plt.title(f\"Neuron {neuron}: H = {_entropy(pmf):.2f} bits\")\n```\n\n\n interactive(children=(IntSlider(value=0, description='neuron', max=733), Output()), _dom_classes=('widget-inte\u2026\n\n\n---\n# Section 4: Reflecting on why models\n\n*Estimated timing to here from start of tutorial: 35 min*\n\n## Think! 3: Reflecting on how models\n\nPlease discuss the following questions for around 10 minutes with your group:\n\n- Have you seen why models before?\n- Have you ever done one?\n- Why are why models useful?\n- When are they possible? Does your field have why models?\n- What do we learn from constructing them?\n\n---\n# Summary\n\n*Estimated timing of tutorial: 45 minutes*\n\n\n```python\n# @title Video 5: Summary of model types\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = 'https://player.bilibili.com/player.html?bvid={0}&page={1}'.format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=\"BV1F5411e7ww\", width=854, height=480, fs=1)\n print('Video available at https://www.bilibili.com/video/{0}'.format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=\"X4K2RR5qBK8\", width=854, height=480, fs=1, rel=0)\n print('Video available at https://youtube.com/watch?v=' + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nCongratulations! You've finished your first NMA tutorial. In this 3 part tutorial series, we used different types of models to understand the spiking behavior of neurons recorded in the Steinmetz data set. \n\n - We used \"what\" models to discover that the ISI distribution of real neurons is closest to an exponential distribution\n - We used \"how\" models to discover that balanced excitatory and inbhitiory inputs, coupled with a leaky membrane, can give rise to neuronal spiking with exhibiting such an exponential ISI distribution\n - We used \"why\" models to discover that exponential ISI distributions contain the most information when the mean spiking is constrained\n\n\n\n---\n# Notation\n\n\\begin{align}\nH(X) &\\quad \\text{entropy of random variable X}\\\\\nb &\\quad \\text{base, e.g. b=2 or b=e}\\\\\nx &\\quad \\text{event x}\\\\\np(x) &\\quad \\text{probability of observing event x}\\\\\n\\text{ISI} &\\quad \\text{interspike interval}\\\\\nn_i &\\quad \\text{count of observed ISIs in interval i}\\\\\np_i &\\quad \\text{probability of of an ISI falling within a particular interval i}\n\\end{align}\n\n---\n# Bonus\n\n---\n## Bonus Section 1: The foundations for Entropy\n\nIn his foundational [1948 paper](https://en.wikipedia.org/wiki/A_Mathematical_Theory_of_Communication) on information theory, Claude Shannon began with three criteria for a function $H$ defining the entropy of a discrete distribution of probability masses $p_i\\in p(X)$ over the points $x_i\\in X$:\n1. $H$ should be continuous in the $p_i$. \n - That is, $H$ should change smoothly in response to smooth changes to the mass $p_i$ on each point $x_i$.\n2. If all the points have equal shares of the probability mass, $p_i=1/N$, $H$ should be a non-decreasing function of $N$. \n - That is, if $X_N$ is the support with $N$ discrete points and $p(x\\in X_N)$ assigns constant mass to each point, then $H(X_1) < H(X_2) < H(X_3) < \\dots$\n3. $H$ should be preserved by (invariant to) the equivalent (de)composition of distributions.\n - For example (from Shannon's paper) if we have a discrete distribution over three points with masses $(\\frac{1}{2},\\frac{1}{3},\\frac{1}{6})$, then their entropy can be represented in terms of a direct choice between the three and calculated $H(\\frac{1}{2},\\frac{1}{3},\\frac{1}{6})$. However, it could also be represented in terms of a series of two choices: \n 1. either we sample the point with mass $1/2$ or not (_not_ is the other $1/2$, whose subdivisions are not given in the first choice), \n 2. if (with probability $1/2$) we _don't_ sample the first point, we sample one of the two remaining points, masses $1/3$ and $1/6$.\n \n Thus in this case we require that $H(\\frac{1}{2},\\frac{1}{3},\\frac{1}{6})=H(\\frac{1}{2},\\frac{1}{2}) + \\frac{1}{2}H(\\frac{1}{3}, \\frac{1}{6})$\n\nThere is a unique function (up to a linear scaling factor) which satisfies these 3 requirements: \n\n\\begin{align}\n H_b(X) = -\\sum_{x\\in X} p(x) \\log_b p(x)\n\\end{align}\n\nWhere the base of the logarithm $b>1$ controls the units of entropy. The two most common cases are $b=2$ for units of _bits_, and $b=e$ for _nats_.\n\nWe can view this function as the expectation of the self-information over a distribution:\n\n\\begin{align}\nH_b(X) = E_{x\\in X} \\left[I_b(x)\\right]\\\\\nI_b(x) =-\\log_b p(x)\n\\end{align}\n\nSelf-information is just the negative logarithm of probability, and is a measure of how surprising an event sampled from the distribution would be. Events with $p(x)=1$ are certain to occur, and their self-information is zero (as is the entropy of the distribution they compose) meaning they are totally unsurprising. The smaller the probability of an event, the higher its self-information, and the more surprising the event would be to observe. \n\n", "meta": {"hexsha": "36f7fa49b3b68eb58da4ae2524825de7f3bb0677", "size": 599005, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W1D1_ModelTypes/W1D1_Tutorial3.ipynb", "max_stars_repo_name": "luisarai/NMA2021", "max_stars_repo_head_hexsha": "d6cd66bf32d929f3030d0d66c2c92de55bd2d886", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tutorials/W1D1_ModelTypes/W1D1_Tutorial3.ipynb", "max_issues_repo_name": "luisarai/NMA2021", "max_issues_repo_head_hexsha": "d6cd66bf32d929f3030d0d66c2c92de55bd2d886", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/W1D1_ModelTypes/W1D1_Tutorial3.ipynb", "max_forks_repo_name": "luisarai/NMA2021", "max_forks_repo_head_hexsha": "d6cd66bf32d929f3030d0d66c2c92de55bd2d886", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 189.1395642564, "max_line_length": 105418, "alphanum_fraction": 0.8762380948, "converted": true, "num_tokens": 7539, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.7634837689358857, "lm_q1q2_score": 0.4350732500397881}} {"text": "# Lagrangian mechanics in generalized coordinates\n\n> Marcos Duarte \n> [Laboratory of Biomechanics and Motor Control](http://pesquisa.ufabc.edu.br/bmclab) \n> Federal University of ABC, Brazil\n\nThe Lagrangian mechanics can be formulated completely independent of the Newtonian mechanics and Cartesian coordinates; Lagrange developed this new formalism based on the [principle of least action](http://nbviewer.jupyter.org/github/BMClab/bmc/blob/master/notebooks/principle_of_least_action.ipynb). \nIn this notebook, we will take a less noble path, we will deduce the Lagrange's equation from Newtonian mechanics.\n\n

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    \n
    \n\n## Review on Newton's laws of motion\n\nThe [Newton's laws of motion](https://en.wikipedia.org/wiki/Newton's_laws_of_motion) laid the foundation for classical mechanics. They describe the relationship between the motion of a body and the possible forces acting upon it.\n\nConsider the motion of a particle in three-dimensional space, its position in time can be represented by the following vector:\n

    \n\n\\begin{equation}\n\\vec{r}(t) = x(t)\\hat{i} + y(t)\\hat{j} + z(t)\\hat{k}\n\\label{}\n\\end{equation}\n\n \nAnd given its position, the particle's velocity and acceleration are:\n

    \n\n\\begin{equation} \\begin{array}{l}\n\\vec{v}(t) = \\dfrac{\\mathrm d \\vec{r}(t)}{\\mathrm d t} = \\dfrac{d x(t)}{\\mathrm d t}\\hat{i} + \\dfrac{d y(t)}{\\mathrm d t}\\hat{j} + \\dfrac{d z(t)}{\\mathrm d t}\\hat{k} \\\\\n\\vec{a}(t) = \\dfrac{\\mathrm d \\vec{v}(t)}{\\mathrm d t} = \\dfrac{\\mathrm d^2 \\vec{r}(t)}{\\mathrm d t^2} = \\dfrac{d^2 x(t)}{\\mathrm d t^2}\\hat{i} + \\dfrac{d^2 y(t)}{\\mathrm d t^2}\\hat{j} + \\dfrac{d^2 z(t)}{\\mathrm d t^2}\\hat{k} \n\\label{}\n\\end{array} \\end{equation}\n\n\nThe particle's linear momentum is defined as:\n

    \n\n\\begin{equation}\n\\vec{p}(t) = m\\vec{v}(t)\n\\label{}\n\\end{equation}\n\n\nwhere $m$ and $\\vec{v}$ are the mass and velocity of the body.\n\nNewton's second law relates the resultant force applied on the particle to the rate of variation of its linear momentum, and if the mass is constant:\n

    \n\n\\begin{equation} \\begin{array}{l}\n\\vec{F}(t) = \\dfrac{\\mathrm d \\vec{p}(t)}{\\mathrm d t} = \\dfrac{\\mathrm d (m\\vec{v}(t))}{\\mathrm d t} \\\\\n\\vec{F}(t) = m\\vec{a}(t) \n\\label{}\n\\end{array} \\end{equation}\n\n\nFrom Newton's second law, if the position of the particle at any time is known, one can determine the resultant force acting on it. If the position is not known, but the resultant force is, the position of the particle can determined solving the following second order ordinary differential equation:\n

    \n\n\\begin{equation}\n\\frac{\\mathrm d^2 \\vec{r}(t)}{\\mathrm d t^2} = \\frac{\\vec{F}(t)}{m}\n\\label{}\n\\end{equation}\n\n \nThe differential equation above is referred as the equation of motion (EOM) of the particle. For example, a system of $N$ particles will require $3N$ EOMs to describe their motion. \nThe EOM has the general solution:\n

    \n\n\\begin{equation}\n\\vec{r}(t) = \\int\\!\\bigg(\\int\\frac{\\vec{F}(t)}{m} \\, \\mathrm{d}t\\bigg) \\, \\mathrm{d}t\n\\label{}\n\\end{equation}\n\n \nwhich requires the determination of two constants, the initial position and velocity.\n\n### Mechanical energy\n\nA related physical quantity is the mechanical energy, which is the sum of kinetic and potential energies. \nThe kinetic energy, $T$ of a particle is given by:\n

    \n\n\\begin{equation}\nT = \\frac{1}{2}m v^2\n\\label{}\n\\end{equation}\n\n \nWhich can be expressed in terms of its linear momentum as:\n

    \n\n\\begin{equation}\nT = \\frac{1}{2m} p^2\n\\label{}\n\\end{equation}\n\n \nAnd for a given coordinate of the particle's motion, its linear momentum can be obtained from its kinetic energy by:\n

    \n\n\\begin{equation}\n\\vec{p} = \\frac{\\partial T}{\\partial \\vec{v}}\n\\label{eq11}\n\\end{equation}\n\n \nThe potential energy $V$ is the stored energy of a particle and its formulation is dependent on the force acting on the particle. For a conservative force dependent solely on the particle position, such as due to the gravitational field near the Earth surface or due to a linear spring, the force can be expressed in terms of the gradient of the potential energy:\n

    \n\n\\begin{equation}\n\\vec{F} = -\\nabla V(\\vec{r}) = -\\frac{\\partial V}{\\partial x}\\hat{i} - \\frac{\\partial V}{\\partial y}\\hat{j} - \\frac{\\partial V}{\\partial z}\\hat{k}\n\\label{eq12}\n\\end{equation}\n\n\n## Lagrange's equation in Cartesian Coordinates\n\nFor simplicity, let's first deduce the Lagrange's equation for a particle in Cartesian Coordinates and from Newton's second law.\n\nBecause we want to deduce the laws of motion based on the mechanical energy of the particle, one can see that the time derivative of the expression for the linear momentum as a function of the kinetic energy, cf. Eq. (\\ref{eq11}), is equal to the force acting on the particle and we can substitute the force in Newton's second law by this term:\n\n\\begin{equation} \n\\frac{\\mathrm d }{\\mathrm d t}\\bigg(\\frac{\\partial T}{\\partial \\dot x}\\bigg) = m\\ddot x \n\\label{eq13}\n\\end{equation}\n\nWe saw that a conservative force can also be expressed in terms of the potential energy of the particle, cf. Eq. (\\ref{eq12}); substituting the right side of the equation above by this expression, we have:\n\n\\begin{equation} \n\\frac{\\mathrm d }{\\mathrm d t}\\bigg(\\frac{\\partial T}{\\partial \\dot x}\\bigg) = -\\frac{\\partial V}{\\partial x} \n\\label{eq14}\n\\end{equation}\n\nUsing the fact that:\n\n\\begin{equation} \n\\frac{\\partial T}{\\partial x} = 0 \\quad and \\quad \\frac{\\partial V}{\\partial \\dot x} = 0 \n\\label{eq15}\n\\end{equation}\n\nWe can write:\n\n\\begin{equation} \n\\frac{\\mathrm d }{\\mathrm d t}\\bigg(\\frac{\\partial (T-V)}{\\partial \\dot x}\\bigg) - \\frac{\\partial (T-V)}{\\partial x} = 0 \n\\label{eq16}\n\\end{equation}\n\nDefining the Lagrange or Lagrangian function, $\\mathcal{L}$, as the difference between the kinetic and potential energy in the system:\n\n\\begin{equation} \n\\mathcal{L} = T - V \n\\label{eq17}\n\\end{equation}\n\nWe have the Lagrange's equation in Cartesian Coordinates for a conservative force acting on a particle:\n\n\\begin{equation} \n\\frac{\\mathrm d }{\\mathrm d t}\\bigg(\\frac{\\partial \\mathcal{L}}{\\partial \\dot x}\\bigg) - \\frac{\\partial \\mathcal{L}}{\\partial x} = 0 \n\\label{eq18}\n\\end{equation}\n\nOnce all derivatives of the Lagrangian function are calculated, this equation will be the equation of motion for the particle. If there are $N$ independent particles in a three-dimensional space, there will be $3N$ equations of motion for the system. \nThe set of equations above for a system are known as Euler\u2013Lagrange's equations, or Lagrange's equations of the second kind.\n\n## Generalized coordinates\n\nThe direct application of Newton's laws to mechanical systems results in a set of equations of motion in terms of Cartesian coordinates of each of the particles that make up the system. In many cases, this is not the most convenient coordinate system to solve the problem or describe the movement of the system. For example, for a serial chain of rigid links, such as a member of the human body or from a robot manipulator, it may be simpler to describe the positions of each link by the angles between links. \n\nCoordinate systems such as angles of a chain of links are referred as [generalized coordinates](https://en.wikipedia.org/wiki/Generalized_coordinates). Generalized coordinates uniquely specify the positions of the particles in a system. Although there may be several generalized coordinates to describe a system, usually a judicious choice of generalized coordinates provides the minimum number of independent coordinates that define the configuration of a system (which is the number of degrees of freedom of the system), turning the problem simpler to solve. In this case, when the number of generalized coordinates equals the number of degrees of freedom, the system is referred as a holonomic system. In a non-holonomic system, the number of generalized coordinates necessary do describe the system depends on the path taken by the system. \n\nBeing a little more technical, according to [Wikipedia](https://en.wikipedia.org/wiki/Configuration_space_(physics)): \n\"In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the vector space defined by these coordinates is called the configuration space of the physical system. It is often the case that these parameters satisfy mathematical constraints, such that the set of actual configurations of the system is a manifold in the space of generalized coordinates. This manifold is called the configuration manifold of the system.\"\n\nIn problems where it is desired to use generalized coordinates, one can write Newton's equations of motion in terms of Cartesian coordinates and then transform them into generalized coordinates. However, it would be desirable and convenient to have a general method that would directly establish the equations of motion in terms of a set of convenient generalized coordinates. In addition, general methods for writing, and perhaps solving, the equations of motion in terms of any coordinate system would also be desirable. The [Lagrangian mechanics](https://en.wikipedia.org/wiki/Lagrangian_mechanics) is such a method.\n\nWhen describing a system of particles using any set of generalized coordinates, $q_1,\\dotsc,q_{3N}$, these are related to, for example, the Cartesian coordinates by:\n \n\\begin{equation} \\begin{array}{rcl}\nq_i =q_i (x_1,\\dotsc,x_{3N} ) \\quad i=1,\\dotsc,3N \\\\\nx_i =x_i (q_1,\\dotsc,q_{3N} ) \\quad i=1,\\dotsc,3N \n\\label{qx}\n\\end{array} \\end{equation}\n\nThe Cartesian components of velocity as a function of generalized coordinates are:\n\n\\begin{equation}\n\\dot{x}_i =\\frac{\\mathrm d x_i (q_1, q_2,\\dotsc,q_{3N} \n)}{\\mathrm d t}=\\sum\\limits_{j=1}^{3N} {\\frac{\\partial x_i }{\\partial q_j }} \n\\frac{\\mathrm d q_j }{\\mathrm d t}\n\\label{eq_xdotqdot}\n\\end{equation}\n\nwhere for simplicity we omitted the explicit mention of the temporal dependence of each coordinate.\n\nThat is, any Cartesian component of the particle velocity as a function of generalized coordinates is a function of all the components of position and velocity in the generalized coordinates:\n\n\\begin{equation} \n\\dot{x}_i = \\dot{x}_i (q_1,\\dotsc,q_{3N} ,\\dot{q}_1,\\dotsc,\\dot{q}_{3N} ) \\quad i=1,\\dotsc,3N \n\\label{eq27}\n\\end{equation}\n\n## Lagrange's equation\n\nIn analogy to Newtonian mechanics, one can think that the equations of motion can be obtained by equating the generalized force, $F_i$, to the temporal rate of change of each generalized momentum, $p_i$:\n\n\\begin{equation} \nF_i =\\frac{\\partial p_i }{\\partial t} \n\\label{eq28}\n\\end{equation}\n\nIn the formula above, let's substitute the quantity $p_i$ by its definition in terms of the kinetic energy:\n\n\\begin{equation} \n\\frac{\\partial p_i }{\\partial t} =\\frac{\\partial }{\\partial t}\\left( {\\frac{\\partial T}{\\partial \n\\dot{q}_i }} \\right)=\\frac{\\partial }{\\partial t}\\left( \n{\\sum\\limits_{j=1}^{3N} {m_j \\dot{x}_j \\frac{\\partial \\dot{x}_j \n}{\\partial \\dot{q}_i }} } \\right)\n\\label{eq29}\n\\end{equation}\n\nwhere we used:\n\n\\begin{equation} \n\\frac{\\partial T}{\\partial \\dot{q}_i }=\\sum\\limits_{j=1}^{3N} \n{\\frac{\\partial T}{\\partial \\dot{x}_j }\\frac{\\partial \\dot{x}_j \n}{\\partial \\dot{q}_i }} \n\\label{eq30}\n\\end{equation}\n\nUsing the [product rule](https://en.wikipedia.org/wiki/Product_rule), the derivative of the product in Eq. (\\ref{eq29}) is:\n\n\\begin{equation} \n\\frac{\\partial p_i }{\\partial t}=\\sum\\limits_{j=1}^{3N} {m_j \n\\ddot{x}_j \\frac{\\partial \\dot{x}_j }{\\partial \\dot{q}_i }} \n+\\sum\\limits_{j=1}^{3N} {m_j \\dot{x}_j \\frac{\\mathrm d }{\\mathrm d t}\\left( \n{\\frac{\\partial \\dot{x}_j }{\\partial \\dot{q}_i }} \\right)} \n\\label{eq31}\n\\end{equation}\n\nBut:\n\n\\begin{equation} \n\\frac{\\partial \\dot{x}_i }{\\partial \\dot{q}_j }=\\frac{\\partial x_i \n}{\\partial q_j } \\quad because \\quad \\frac{\\partial \n\\dot{x}_i }{\\partial \\dot{q}_j }=\\frac{\\partial x_i }{\\partial \nt}\\frac{\\partial t}{\\partial q_j }=\\frac{\\partial x_i }{\\partial q_j} \n\\label{eq32}\n\\end{equation}\n\nThen:\n\n\\begin{equation} \n\\frac{\\partial p_i }{\\partial t}=\\sum\\limits_{j=1}^{3N} {m_j \n\\ddot{x}_j \\frac{\\partial x_j }{\\partial q_i }} \n+\\sum\\limits_{j=1}^{3N} {m_j \\dot{x}_j \\frac{\\mathrm d }{\\mathrm d t}\\left( \n{\\frac{\\partial x_j }{\\partial q_i }} \\right)} \n\\label{eq33}\n\\end{equation}\n\nThe first term on the right side of the equation above is proportional to $m_j \n\\ddot{x}_j$ and we will define as the generalized force, $Q_i$. But, different from Newtonian mechanics, the temporal variation of the generalized momentum is equal to the generalized force plus another term, which will investigate now. The last part of this second term can be derived as:\n\n\\begin{equation}\n\\frac{\\mathrm d }{\\mathrm d t}\\left( {\\frac{\\partial x_j }{\\partial q_i }} \\right) = \n\\sum\\limits_{k=1}^{3N} {\\frac{\\mathrm d }{\\mathrm d q_k }\\left( {\\frac{\\partial \nx_j }{\\partial q_i }} \\right)\\frac{\\mathrm d q_k }{\\mathrm d t}} =\\sum\\limits_{k=1}^{3N} \n{\\frac{\\partial^2 x_j }{\\partial q_k \\partial q_i }\\dot{q}_k }\n\\label{eq34}\n\\end{equation}\n\nwhere we used the [chain rule](https://en.wikipedia.org/wiki/Chain_rule) for the differentiation:\n\\begin{equation}\n\\frac{\\mathrm d }{\\mathrm d t}\\Big( {f\\big({g(t)}\\big)}\\Big) = \\frac{\\partial f}{\\partial g}\\frac{\\partial g}{\\partial t}\n\\label{eq35}\n\\end{equation}\n\nBut if we look at Eq. (\\ref{eq_xdotqdot}) we see that the term at the right side of the Eq. (\\ref{eq34}) can be obtained by:\n\n\\begin{equation}\n\\frac{\\partial \\dot{x}_j }{\\partial q_i } = \\frac{\\partial }{\\partial q_i }\\left(\\sum\\limits_{k=1}^{3N} \\frac{\\partial \nx_j }{\\partial q_i }\\dot{q}_k \\right) = \\sum\\limits_{k=1}^{3N} \n{\\frac{\\partial^2 x_j }{\\partial q_k \\partial q_i }\\dot{q}_k }\n\\label{eq36}\n\\end{equation}\n\nComparing the Eq. (\\ref{eq34}) and Eq. (\\ref{eq36}) we have:\n\n\\begin{equation}\n\\frac{\\mathrm d }{\\mathrm d t}\\left( {\\frac{\\partial x_j }{\\partial q_i }} \\right) = \n\\frac{\\mathrm d }{\\mathrm d q_i}\\left( {\\frac{\\partial x_j }{\\partial t }} \\right)\n\\label{eq_dotdxdq}\n\\end{equation}\n\nOn the other hand, it is possible to relate the term $\\partial \\dot{x}_j / \\partial q_i$ to the derivative of kinetic energy with respect to the coordinate $q_i$:\n\n\\begin{equation} \n\\frac{\\partial T}{\\partial q_i }=\\frac{\\partial }{\\partial q_i }\\left( \n{\\sum\\limits_{j=1}^{3N} {\\frac{1}{2}m_j \\dot{x}_j^2} } \n\\right)=\\sum\\limits_{j=1}^{3N} {m_j \\dot{x}_j } \\frac{\\partial \n\\dot{x}_j }{\\partial q_i } \n\\label{eq38}\n\\end{equation}\n\nwhere once again we used the chain rule for the differentiation.\n\nUsing Eq. (\\ref{eq_dotdxdq}), Eq. (\\ref{eq38}) becomes\n\n\\begin{equation} \n\\frac{\\partial T}{\\partial q_i }=\\sum\\limits_{j=1}^{3N} {m_j \n\\dot{x}_j } \\frac{\\mathrm d }{\\mathrm d t}\\left( {\\frac{\\partial x_j }{\\partial q_i }} \n\\right) \n\\label{eq39}\n\\end{equation}\n\nReturning to Eq. (\\ref{eq33}), it can be rewritten as:\n\n\\begin{equation} \n\\frac{\\mathrm d }{\\mathrm d t}\\left( {\\frac{\\partial T}{\\partial \\dot{q}_i }} \\right) = Q_i + \\frac{\\partial T}{\\partial q_i } \n\\label{eq40}\n\\end{equation}\n\nand\n\n\\begin{equation} \n\\frac{\\mathrm d }{\\mathrm d t}\\left( {\\frac{\\partial T}{\\partial \\dot{q}_i }} \\right) - \\frac{\\partial T}{\\partial q_i } = Q_i \n\\label{eq41}\n\\end{equation}\n\nNow let's look at $Q_i$, the generalized force. It can be decomposed into two terms: \n\nThe first term, composed of the conservative forces, i.e. forces that can be written as potential gradients:\n\n\\begin{equation} \nQ_C =-\\frac{\\partial V}{\\partial q_i } \\quad , \\quad V=V\\left( {q_1,\\dotsc,q_{3N} } \\right) \n\\label{eq42}\n\\end{equation}\n\nAn example of conservative force is the gravitational force.\n\nAnd the second term, encompassing all non-conservative forces, $Q_{NC}$. \n\nThen:\n\n\\begin{equation} Q_i =-\\frac{\\partial V}{\\partial q_i }+Q_{NCi} \\quad , \\quad V=V\\left( {q_1,\\dotsc,q_{3N} } \\right) \\end{equation}\n\nThe Eq. (\\ref{eq41}) becomes\n\n\\begin{equation} \n\\frac{\\mathrm d }{\\mathrm d t}\\left( {\\frac{\\partial T}{\\partial \\dot{q}_i }} \n\\right)-\\frac{\\partial T}{\\partial q_i }=-\\frac{\\partial V}{\\partial q_i} + Q_{NCi}\n\\label{eq43}\n\\end{equation}\n\nRearranging, we have:\n\n\\begin{equation} \\frac{\\mathrm d }{\\mathrm d t}\\left( {\\frac{\\partial \\left( {T-V} \\right)}{\\partial \n\\dot{q}_i }} \\right)-\\frac{\\partial \\left( {T-V} \\right)}{\\partial q_i} = Q_{NCi} \n\\label{eq44}\n\\end{equation}\n\nThis is possible because:\n\n\\begin{equation} \n\\frac{\\partial V}{\\partial \\dot{q}_i} = 0 \n\\label{eq45}\n\\end{equation}\n\nDefining:\n\n\\begin{equation} \n\\mathcal{L} \\equiv \\mathcal{L}(q_1,\\dotsc,q_{3N} ,\\dot{q}_1,\\dotsc,\\dot{q}_{3N} ) = T - V \n\\label{eq46}\n\\end{equation}\n\nas the Lagrange or Lagrangian function, we have the Lagrange's equation:\n\n\\begin{equation} \n\\frac{\\mathrm d }{\\mathrm d t}\\left( {\\frac{\\partial \\mathcal{L}}{\\partial \\dot{q}_i }} \n\\right)-\\frac{\\partial \\mathcal{L}}{\\partial q_i } = Q_{NCi} \\quad i=1,\\dotsc,3N \n\\label{eq47}\n\\end{equation}\n\nOnce all derivatives of the Lagrangian function are calculated, this equation will be the equation of motion for each particle. If there are $N$ independent particles in a three-dimensional space, there will be $3N$ equations for the system.\n\nThe set of equations above for a system are known as Euler\u2013Lagrange equations, or Lagrange's equations of the second kind.\n\n### Constraints\n \nAn important class of problems in mechanics, in which the Lagrangian equations are particularly useful, are composed of constrained systems. A constraint is a restriction on the freedom of movement of a particle or a system of particles (a constraint decreases the number of degrees of freedom of a system). A rigid body, or the movement of a pendulum, are examples of constrained systems. It can be shown, in a similar way, that the Lagrange equation, deduced here for a system of free particles, is also valid for a system of particles under the action of constraints. \nThe Euler-Lagrange equation, for a system of $3N$ particles and with $k$ constraints, is then defined as:\n\n\\begin{equation} \n\\frac{\\mathrm d }{\\mathrm d t}\\left( {\\frac{\\partial \\mathcal{L}}{\\partial \\dot{q}_i}} \\right)-\\frac{\\partial \\mathcal{L}}{\\partial q_i } = Q_{NCi} \\quad i=1,\\dotsc,3N-k \n\\label{eq48}\n\\end{equation}\n\n## Further reading\n\n- [The Principle of Least Action in ](https://www.feynmanlectures.caltech.edu/II_19.html) \n- Vandiver JK (MIT OpenCourseWare) [An Introduction to Lagrangian Mechanics](https://ocw.mit.edu/courses/mechanical-engineering/2-003sc-engineering-dynamics-fall-2011/lagrange-equations/MIT2_003SCF11_Lagrange.pdf)\n\n## Video lectures on the internet \n\n- iLectureOnline: [Lectures in Lagrangian Mechanics](http://www.ilectureonline.com/lectures/subject/PHYSICS/34/245) \n- MIT OpenCourseWare: [Introduction to Lagrange With Examples](https://youtu.be/zhk9xLjrmi4)\n\n## References\n\n- Goldstein H (1980) [Classical Mechanics](https://books.google.com.br/books?id=tJCuQgAACAAJ), 3rd ed., Addison-Wesley. \n- Marion JB (1970) [Classical Dynamics of particles and systems](https://books.google.com.br/books?id=Ss43BQAAQBAJ), 2nd ed., Academic Press. \n- Synge JL (1949) [Principles of Mechanics](https://books.google.com.br/books?id=qsYfENCRG5QC), 2nd ed., McGraw-hill. \n- Taylor J (2005) [Classical Mechanics](https://archive.org/details/JohnTaylorClassicalMechanics). University Science Books. \n", "meta": {"hexsha": "47c55947c3968fed4235bc6807893c90982e6d39", "size": 28036, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/lagrangian_mechanics_generalized.ipynb", "max_stars_repo_name": "e-moncao-lima/BMC", "max_stars_repo_head_hexsha": "98c3abbf89e630d64b695b535b0be4ddc8b2724b", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 293, "max_stars_repo_stars_event_min_datetime": "2015-01-17T12:36:30.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-13T13:13:12.000Z", "max_issues_repo_path": "notebooks/lagrangian_mechanics_generalized.ipynb", "max_issues_repo_name": "e-moncao-lima/BMC", "max_issues_repo_head_hexsha": "98c3abbf89e630d64b695b535b0be4ddc8b2724b", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 11, "max_issues_repo_issues_event_min_datetime": "2018-06-21T21:40:40.000Z", "max_issues_repo_issues_event_max_datetime": "2018-08-09T19:55:26.000Z", "max_forks_repo_path": "notebooks/lagrangian_mechanics_generalized.ipynb", "max_forks_repo_name": "e-moncao-lima/BMC", "max_forks_repo_head_hexsha": "98c3abbf89e630d64b695b535b0be4ddc8b2724b", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 162, "max_forks_repo_forks_event_min_datetime": "2015-01-16T22:54:31.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-14T21:14:43.000Z", "avg_line_length": 50.0642857143, "max_line_length": 1782, "alphanum_fraction": 0.6044014838, "converted": true, "num_tokens": 6329, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.7248702702332475, "lm_q1q2_score": 0.4350586981677942}} {"text": "```python\n%matplotlib inline\n```\n\n\nWriting mathematical expressions\n================================\n\nAn introduction to writing mathematical expressions in Matplotlib.\n\nYou can use a subset TeX markup in any matplotlib text string by placing it\ninside a pair of dollar signs ($).\n\nNote that you do not need to have TeX installed, since Matplotlib ships\nits own TeX expression parser, layout engine, and fonts. The layout engine\nis a fairly direct adaptation of the layout algorithms in Donald Knuth's\nTeX, so the quality is quite good (matplotlib also provides a ``usetex``\noption for those who do want to call out to TeX to generate their text (see\n:doc:`/tutorials/text/usetex`).\n\nAny text element can use math text. You should use raw strings (precede the\nquotes with an ``'r'``), and surround the math text with dollar signs ($), as\nin TeX. Regular text and mathtext can be interleaved within the same string.\nMathtext can use DejaVu Sans (default), DejaVu Serif, the Computer Modern fonts\n(from (La)TeX), `STIX `_ fonts (with are designed\nto blend well with Times), or a Unicode font that you provide. The mathtext\nfont can be selected with the customization variable ``mathtext.fontset`` (see\n:doc:`/tutorials/introductory/customizing`)\n\nHere is a simple example::\n\n # plain text\n plt.title('alpha > beta')\n\nproduces \"alpha > beta\".\n\nWhereas this::\n\n # math text\n plt.title(r'$\\alpha > \\beta$')\n\nproduces \":mathmpl:`\\alpha > \\beta`\".\n\n

    Note

    Mathtext should be placed between a pair of dollar signs ($). To make it\n easy to display monetary values, e.g., \"$100.00\", if a single dollar sign\n is present in the entire string, it will be displayed verbatim as a dollar\n sign. This is a small change from regular TeX, where the dollar sign in\n non-math text would have to be escaped ('\\\\\\$').

    \n\n

    Note

    While the syntax inside the pair of dollar signs ($) aims to be TeX-like,\n the text outside does not. In particular, characters such as::\n\n # $ % & ~ _ ^ \\ { } \\( \\) \\[ \\]\n\n have special meaning outside of math mode in TeX. Therefore, these\n characters will behave differently depending on the rcParam ``text.usetex``\n flag. See the :doc:`usetex tutorial ` for more\n information.

    \n\nSubscripts and superscripts\n---------------------------\n\nTo make subscripts and superscripts, use the ``'_'`` and ``'^'`` symbols::\n\n r'$\\alpha_i > \\beta_i$'\n\n\\begin{align}\\alpha_i > \\beta_i\\end{align}\n\nSome symbols automatically put their sub/superscripts under and over the\noperator. For example, to write the sum of :mathmpl:`x_i` from :mathmpl:`0` to\n:mathmpl:`\\infty`, you could do::\n\n r'$\\sum_{i=0}^\\infty x_i$'\n\n\\begin{align}\\sum_{i=0}^\\infty x_i\\end{align}\n\nFractions, binomials, and stacked numbers\n-----------------------------------------\n\nFractions, binomials, and stacked numbers can be created with the\n``\\frac{}{}``, ``\\binom{}{}`` and ``\\genfrac{}{}{}{}{}{}`` commands,\nrespectively::\n\n r'$\\frac{3}{4} \\binom{3}{4} \\genfrac{}{}{0}{}{3}{4}$'\n\nproduces\n\n\\begin{align}\\frac{3}{4} \\binom{3}{4} \\stackrel{}{}{0}{}{3}{4}\\end{align}\n\nFractions can be arbitrarily nested::\n\n r'$\\frac{5 - \\frac{1}{x}}{4}$'\n\nproduces\n\n\\begin{align}\\frac{5 - \\frac{1}{x}}{4}\\end{align}\n\nNote that special care needs to be taken to place parentheses and brackets\naround fractions. Doing things the obvious way produces brackets that are too\nsmall::\n\n r'$(\\frac{5 - \\frac{1}{x}}{4})$'\n\n.. math ::\n\n (\\frac{5 - \\frac{1}{x}}{4})\n\nThe solution is to precede the bracket with ``\\left`` and ``\\right`` to inform\nthe parser that those brackets encompass the entire object.::\n\n r'$\\left(\\frac{5 - \\frac{1}{x}}{4}\\right)$'\n\n.. math ::\n\n \\left(\\frac{5 - \\frac{1}{x}}{4}\\right)\n\nRadicals\n--------\n\nRadicals can be produced with the ``\\sqrt[]{}`` command. For example::\n\n r'$\\sqrt{2}$'\n\n.. math ::\n\n \\sqrt{2}\n\nAny base can (optionally) be provided inside square brackets. Note that the\nbase must be a simple expression, and can not contain layout commands such as\nfractions or sub/superscripts::\n\n r'$\\sqrt[3]{x}$'\n\n.. math ::\n\n \\sqrt[3]{x}\n\n\nFonts\n-----\n\nThe default font is *italics* for mathematical symbols.\n\n

    Note

    This default can be changed using the ``mathtext.default`` rcParam. This is\n useful, for example, to use the same font as regular non-math text for math\n text, by setting it to ``regular``.

    \n\nTo change fonts, e.g., to write \"sin\" in a Roman font, enclose the text in a\nfont command::\n\n r'$s(t) = \\mathcal{A}\\mathrm{sin}(2 \\omega t)$'\n\n\\begin{align}s(t) = \\mathcal{A}\\mathrm{sin}(2 \\omega t)\\end{align}\n\nMore conveniently, many commonly used function names that are typeset in\na Roman font have shortcuts. So the expression above could be written as\nfollows::\n\n r'$s(t) = \\mathcal{A}\\sin(2 \\omega t)$'\n\n\\begin{align}s(t) = \\mathcal{A}\\sin(2 \\omega t)\\end{align}\n\nHere \"s\" and \"t\" are variable in italics font (default), \"sin\" is in Roman\nfont, and the amplitude \"A\" is in calligraphy font. Note in the example above\nthe calligraphy ``A`` is squished into the ``sin``. You can use a spacing\ncommand to add a little whitespace between them::\n\n r's(t) = \\mathcal{A}\\/\\sin(2 \\omega t)'\n\n.. Here we cheat a bit: for HTML math rendering, Sphinx relies on MathJax which\n doesn't actually support the italic correction (\\/); instead, use a thin\n space (\\,) which is supported.\n\n\\begin{align}s(t) = \\mathcal{A}\\,\\sin(2 \\omega t)\\end{align}\n\nThe choices available with all fonts are:\n\n ========================= ================================\n Command Result\n ========================= ================================\n ``\\mathrm{Roman}`` :mathmpl:`\\mathrm{Roman}`\n ``\\mathit{Italic}`` :mathmpl:`\\mathit{Italic}`\n ``\\mathtt{Typewriter}`` :mathmpl:`\\mathtt{Typewriter}`\n ``\\mathcal{CALLIGRAPHY}`` :mathmpl:`\\mathcal{CALLIGRAPHY}`\n ========================= ================================\n\n.. role:: math-stix(mathmpl)\n :fontset: stix\n\nWhen using the `STIX `_ fonts, you also have the\nchoice of:\n\n ================================ =========================================\n Command Result\n ================================ =========================================\n ``\\mathbb{blackboard}`` :math-stix:`\\mathbb{blackboard}`\n ``\\mathrm{\\mathbb{blackboard}}`` :math-stix:`\\mathrm{\\mathbb{blackboard}}`\n ``\\mathfrak{Fraktur}`` :math-stix:`\\mathfrak{Fraktur}`\n ``\\mathsf{sansserif}`` :math-stix:`\\mathsf{sansserif}`\n ``\\mathrm{\\mathsf{sansserif}}`` :math-stix:`\\mathrm{\\mathsf{sansserif}}`\n ================================ =========================================\n\nThere are also three global \"font sets\" to choose from, which are\nselected using the ``mathtext.fontset`` parameter in `matplotlibrc\n`.\n\n``cm``: **Computer Modern (TeX)**\n\n\n\n\n``stix``: **STIX** (designed to blend well with Times)\n\n\n\n\n``stixsans``: **STIX sans-serif**\n\n\n\n\nAdditionally, you can use ``\\mathdefault{...}`` or its alias\n``\\mathregular{...}`` to use the font used for regular text outside of\nmathtext. There are a number of limitations to this approach, most notably\nthat far fewer symbols will be available, but it can be useful to make math\nexpressions blend well with other text in the plot.\n\nCustom fonts\n~~~~~~~~~~~~\n\nmathtext also provides a way to use custom fonts for math. This method is\nfairly tricky to use, and should be considered an experimental feature for\npatient users only. By setting the rcParam ``mathtext.fontset`` to ``custom``,\nyou can then set the following parameters, which control which font file to use\nfor a particular set of math characters.\n\n ============================== =================================\n Parameter Corresponds to\n ============================== =================================\n ``mathtext.it`` ``\\mathit{}`` or default italic\n ``mathtext.rm`` ``\\mathrm{}`` Roman (upright)\n ``mathtext.tt`` ``\\mathtt{}`` Typewriter (monospace)\n ``mathtext.bf`` ``\\mathbf{}`` bold italic\n ``mathtext.cal`` ``\\mathcal{}`` calligraphic\n ``mathtext.sf`` ``\\mathsf{}`` sans-serif\n ============================== =================================\n\nEach parameter should be set to a fontconfig font descriptor (as defined in the\nyet-to-be-written font chapter).\n\n.. TODO: Link to font chapter\n\nThe fonts used should have a Unicode mapping in order to find any\nnon-Latin characters, such as Greek. If you want to use a math symbol\nthat is not contained in your custom fonts, you can set the rcParam\n``mathtext.fallback_to_cm`` to ``True`` which will cause the mathtext system\nto use characters from the default Computer Modern fonts whenever a particular\ncharacter can not be found in the custom font.\n\nNote that the math glyphs specified in Unicode have evolved over time, and many\nfonts may not have glyphs in the correct place for mathtext.\n\nAccents\n-------\n\nAn accent command may precede any symbol to add an accent above it. There are\nlong and short forms for some of them.\n\n ============================== =================================\n Command Result\n ============================== =================================\n ``\\acute a`` or ``\\'a`` :mathmpl:`\\acute a`\n ``\\bar a`` :mathmpl:`\\bar a`\n ``\\breve a`` :mathmpl:`\\breve a`\n ``\\ddot a`` or ``\\''a`` :mathmpl:`\\ddot a`\n ``\\dot a`` or ``\\.a`` :mathmpl:`\\dot a`\n ``\\grave a`` or ``\\`a`` :mathmpl:`\\grave a`\n ``\\hat a`` or ``\\^a`` :mathmpl:`\\hat a`\n ``\\tilde a`` or ``\\~a`` :mathmpl:`\\tilde a`\n ``\\vec a`` :mathmpl:`\\vec a`\n ``\\overline{abc}`` :mathmpl:`\\overline{abc}`\n ============================== =================================\n\nIn addition, there are two special accents that automatically adjust to the\nwidth of the symbols below:\n\n ============================== =================================\n Command Result\n ============================== =================================\n ``\\widehat{xyz}`` :mathmpl:`\\widehat{xyz}`\n ``\\widetilde{xyz}`` :mathmpl:`\\widetilde{xyz}`\n ============================== =================================\n\nCare should be taken when putting accents on lower-case i's and j's. Note that\nin the following ``\\imath`` is used to avoid the extra dot over the i::\n\n r\"$\\hat i\\ \\ \\hat \\imath$\"\n\n\\begin{align}\\hat i\\ \\ \\hat \\imath\\end{align}\n\nSymbols\n-------\n\nYou can also use a large number of the TeX symbols, as in ``\\infty``,\n``\\leftarrow``, ``\\sum``, ``\\int``.\n\n.. math_symbol_table::\n\nIf a particular symbol does not have a name (as is true of many of the more\nobscure symbols in the STIX fonts), Unicode characters can also be used::\n\n ur'$\\u23ce$'\n\nExample\n-------\n\nHere is an example illustrating many of these features in context.\n\n.. figure:: ../../gallery/pyplots/images/sphx_glr_pyplot_mathtext_001.png\n :target: ../../gallery/pyplots/pyplot_mathtext.html\n :align: center\n :scale: 50\n\n Pyplot Mathtext\n\n", "meta": {"hexsha": "ca1696e324e69788b5239d9f1f09067f9d5a9ac4", "size": 12864, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Writing-mathematical-expressions.ipynb", "max_stars_repo_name": "swshadle/conda", "max_stars_repo_head_hexsha": "0f6aead8fef362816cf3ecc1ceaf21faad9315e0", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-10-14T13:43:03.000Z", "max_stars_repo_stars_event_max_datetime": "2019-10-14T13:43:03.000Z", "max_issues_repo_path": "Writing-mathematical-expressions.ipynb", "max_issues_repo_name": "swshadle/conda", "max_issues_repo_head_hexsha": "0f6aead8fef362816cf3ecc1ceaf21faad9315e0", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Writing-mathematical-expressions.ipynb", "max_forks_repo_name": "swshadle/conda", "max_forks_repo_head_hexsha": "0f6aead8fef362816cf3ecc1ceaf21faad9315e0", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 292.3636363636, "max_line_length": 12072, "alphanum_fraction": 0.5851212687, "converted": true, "num_tokens": 3029, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011686727232, "lm_q2_score": 0.7431680029241322, "lm_q1q2_score": 0.4348284670310835}} {"text": "# **CS224W - Colab 4**\n\nIn Colab 2 we constructed GNN models by using PyTorch Geometric's built in GCN layer, `GCNConv`. In Colab 3 we implemented the **GraphSAGE** ([Hamilton et al. (2017)](https://arxiv.org/abs/1706.02216)) layer. In this colab you'll use what you've learned and implement a more powerful layer: **GAT** ([Veli\u010dkovi\u0107 et al. (2018)](https://arxiv.org/abs/1710.10903)). Then we will run our models on the CORA dataset, which is a standard citation network benchmark dataset.\n\n**Note**: Make sure to **sequentially run all the cells in each section** so that the intermediate variables / packages will carry over to the next cell\n\nHave fun and good luck on Colab 4 :)\n\n# Device\nWe recommend using a GPU for this Colab.\n\nPlease click `Runtime` and then `Change runtime type`. Then set the `hardware accelerator` to **GPU**.\n\n## Installation\n\n\n```python\n# Install torch geometric\nimport os\nif 'IS_GRADESCOPE_ENV' not in os.environ:\n !pip install torch-scatter -f https://data.pyg.org/whl/torch-1.10.0+cu113.html\n !pip install torch-sparse -f https://data.pyg.org/whl/torch-1.10.0+cu113.html\n !pip install torch-geometric\n !pip install -q git+https://github.com/snap-stanford/deepsnap.git\n```\n\n\n```python\nimport torch_geometric\ntorch_geometric.__version__\n```\n\n# 1) GNN Layers\n\n## Implementing Layer Modules\n\nIn Colab 2, we implemented a GCN model for node and graph classification tasks. However, for that notebook we took advantage of PyG's built in GCN module. For Colabs 3 and 4, we provide a build upon a general Graph Neural Network Stack, into which we will be able to plugin our own module implementations: GraphSAGE and GAT.\n\nWe will then use our layer implemenations to complete node classification on the CORA dataset, a standard citation network benchmark. In this dataset, nodes correspond to documents and edges correspond to undirected citations. Each node or document in the graph is assigned a class label and features based on the documents binarized bag-of-words representation. Specifically, the Cora graph has 2708 nodes, 5429 edges, 7 prediction classes, and 1433 features per node. \n\n## GNN Stack Module\n\nBelow is the implementation of a general GNN stack, where we can plugin any GNN layer, such as **GraphSage**, **GAT**, etc. This module is provided for you. Your implementations of the **GraphSage** and **GAT** layers will function as components in the GNNStack Module.\n\n\n```python\nimport torch\nimport torch_scatter\nimport torch.nn as nn\nimport torch.nn.functional as F\n\nimport torch_geometric.nn as pyg_nn\nimport torch_geometric.utils as pyg_utils\n\nfrom torch import Tensor\nfrom typing import Union, Tuple, Optional\nfrom torch_geometric.typing import (OptPairTensor, Adj, Size, NoneType,\n OptTensor)\n\nfrom torch.nn import Parameter, Linear\nfrom torch_sparse import SparseTensor, set_diag\nfrom torch_geometric.nn.conv import MessagePassing\nfrom torch_geometric.utils import remove_self_loops, add_self_loops, softmax, degree\n\nclass GNNStack(torch.nn.Module):\n def __init__(self, input_dim, hidden_dim, output_dim, args, emb=False):\n super(GNNStack, self).__init__()\n conv_model = self.build_conv_model(args.model_type)\n self.convs = nn.ModuleList()\n self.convs.append(conv_model(input_dim, hidden_dim))\n assert (args.num_layers >= 1), 'Number of layers is not >=1'\n for l in range(args.num_layers-1):\n self.convs.append(conv_model(args.heads * hidden_dim, hidden_dim))\n\n # post-message-passing\n self.post_mp = nn.Sequential(\n nn.Linear(args.heads * hidden_dim, hidden_dim), nn.Dropout(args.dropout), \n nn.Linear(hidden_dim, output_dim))\n\n self.dropout = args.dropout\n self.num_layers = args.num_layers\n\n self.emb = emb\n\n def build_conv_model(self, model_type):\n if model_type == 'GraphSage':\n return GraphSage\n elif model_type == 'GAT':\n # When applying GAT with num heads > 1, you need to modify the \n # input and output dimension of the conv layers (self.convs),\n # to ensure that the input dim of the next layer is num heads\n # multiplied by the output dim of the previous layer.\n # HINT: In case you want to play with multiheads, you need to change the for-loop that builds up self.convs to be\n # self.convs.append(conv_model(hidden_dim * num_heads, hidden_dim)), \n # and also the first nn.Linear(hidden_dim * num_heads, hidden_dim) in post-message-passing.\n return GAT\n\n def forward(self, data):\n x, edge_index, batch = data.x, data.edge_index, data.batch\n \n for i in range(self.num_layers):\n x = self.convs[i](x, edge_index)\n x = F.relu(x)\n x = F.dropout(x, p=self.dropout,training=self.training)\n\n x = self.post_mp(x)\n\n if self.emb == True:\n return x\n\n return F.log_softmax(x, dim=1)\n\n def loss(self, pred, label):\n return F.nll_loss(pred, label)\n```\n\n## Creating Our Own Message Passing Layer\n\nNow let's start implementing our own message passing layers! Working through this part will help us become acutely familiar with the behind the scenes work of implementing Pytorch Message Passing Layers, allowing us to build our own GNN models. To do so, we will work with and implement 3 critcal functions needed to define a PyG Message Passing Layer: `forward`, `message`, and `aggregate`.\n\nBefore diving head first into the coding details, let us quickly review the key components of the message passing process. To do so, we will focus on a single round of messsage passing with respect to a single central node $x$. Before message passing, $x$ is associated with a feature vector $x^{l-1}$, and the goal of message passing is to update this feature vector as $x^l$. To do so, we implement the following steps: 1) each neighboring node $v$ passes its current message $v^{l-1}$ across the edge $(x, v)$ - 2) for the node $x$, we aggregate all of the messages of the neighboring nodes (for example through a sum or mean) - and 3) we transform the aggregated information by for example applying linear and non-linear transformations. Altogether, the message passing process is applied such that every node $u$ in our graph updates its embedding by acting as the central node $x$ in step 1-3 described above. \n\nNow, we extending this process to that of a single message passing layer, the job of a message passing layer is to update the current feature representation or embedding of each node in a graph by propagating and transforming information within the graph. Overall, the general paradigm of a message passing layers is: 1) pre-processing -> 2) **message passing** / propagation -> 3) post-processing. \n\nThe `forward` fuction that we will implement for our message passing layer captures this execution logic. Namely, the `forward` function handles the pre and post-processing of node features / embeddings, as well as initiates message passing by calling the `propagate` function. \n\n\nThe `propagate` function encapsulates the message passing process! It does so by calling three important functions: 1) `message`, 2) `aggregate`, and 3) `update`. Our implementation will vary slightly from this, as we will not explicitly implement `update`, but instead place the logic for updating node embeddings after message passing and within the `forward` function. To be more specific, after information is propagated (message passing), we can further transform the node embeddings outputed by `propagate`. Therefore, the output of `forward` is exactly the node embeddings after one GNN layer.\n\nLastly, before starting to implement our own layer, let us dig a bit deeper into each of the functions described above:\n\n1. \n\n```\ndef propagate(edge_index, x=(x_i, x_j), extra=(extra_i, extra_j), size=size):\n```\nCalling `propagate` initiates the message passing process. Looking at the function parameters, we highlight a couple of key parameters. \n\n - `edge_index` is passed to the forward function and captures the edge structure of the graph.\n - `x=(x_i, x_j)` represents the node features that will be used in message passing. In order to explain why we pass the tuple `(x_i, x_j)`, we first look at how our edges are represented. For every edge $(i, j) \\in \\mathcal{E}$, we can differentiate $i$ as the source or central node ($x_{central}$) and j as the neighboring node ($x_{neighbor}$). \n \n Taking the example of message passing above, for a central node $u$ we will aggregate and transform all of the messages associated with the nodes $v$ s.t. $(u, v) \\in \\mathcal{E}$ (i.e. $v \\in \\mathcal{N}_{u}$). Thus we see, the subscripts `_i` and `_j` allow us to specifcally differenciate features associated with central nodes (i.e. nodes recieving message information) and neighboring nodes (i.e. nodes passing messages). \n\n This is definitely a somewhat confusing concept; however, one key thing to remember / wrap your head around is that depending on the perspective, a node $x$ acts as a central node or a neighboring node. In fact, in undirected graphs we store both edge directions (i.e. $(i, j)$ and $(j, i)$). From the central node perspective, `x_i`, x is collecting neighboring information to update its embedding. From a neighboring node perspective, `x_j`, x is passing its message information along the edge connecting it to a different central node.\n\n - `extra=(extra_i, extra_j)` represents additional information that we can associate with each node beyond its current feature embedding. In fact, we can include as many additional parameters of the form `param=(param_i, param_j)` as we would like. Again, we highlight that indexing with `_i` and `_j` allows us to differentiate central and neighboring nodes. \n\n The output of the `propagate` function is a matrix of node embeddings after the message passing process and has shape $[N, d]$.\n\n2. \n```\ndef message(x_j, ...):\n```\nThe `message` function is called by propagate and constructs the messages from\nneighboring nodes $j$ to central nodes $i$ for each edge $(i, j)$ in *edge_index*. This function can take any argument that was initially passed to `propagate`. Furthermore, we can again differentiate central nodes and neighboring nodes by appending `_i` or `_j` to the variable name, .e.g. `x_i` and `x_j`. Looking more specifically at the variables, we have:\n\n - `x_j` represents a matrix of feature embeddings for all neighboring nodes passing their messages along their respective edge (i.e. all nodes $j$ for edges $(i, j) \\in \\mathcal{E}$). Thus, its shape is $[|\\mathcal{E}|, d]$!\n - In implementing GAT we will see how to access additional variables passed to propagate\n\n Critically, we see that the output of the `message` function is a matrix of neighboring node embeddings ready to be aggregated, having shape $[|\\mathcal{E}|, d]$.\n\n3. \n```\ndef aggregate(self, inputs, index, dim_size = None):\n```\nLastly, the `aggregate` function is used to aggregate the messages from neighboring nodes. Looking at the parameters we highlight:\n\n - `inputs` represents a matrix of the messages passed from neighboring nodes (i.e. the output of the `message` function).\n - `index` has the same shape as `inputs` and tells us the central node that corresponding to each of the rows / messages $j$ in the `inputs` matrix. Thus, `index` tells us which rows / messages to aggregate for each central node.\n\n The output of `aggregate` is of shape $[N, d]$.\n\n\nFor additional resources refer to the PyG documentation for implementing custom message passing layers: https://pytorch-geometric.readthedocs.io/en/latest/notes/create_gnn.html\n\n## GAT Implementation\n\nAttention mechanisms have become the state-of-the-art in many sequence-based tasks such as machine translation and learning sentence representations. One of the major benefits of attention-based mechanisms is their ability to focus on the most relevant parts of the input to make decisions. In this problem, we will see how attention mechanisms can be used to perform node classification over graph-structured data through the usage of Graph Attention Networks (GATs) ([Veli\u010dkovi\u0107 et al. (2018)](https://arxiv.org/abs/1710.10903)).\n\nThe building block of the Graph Attention Network is the graph attention layer, which is a variant of the aggregation function. Let $N$ be the number of nodes and $F$ be the dimension of the feature vector for each node. The input to each graph attentional layer is a set of node features: $\\mathbf{h} = \\{\\overrightarrow{h_1}, \\overrightarrow{h_2}, \\dots, \\overrightarrow{h_N}$\\}, $\\overrightarrow{h_i} \\in R^F$. The output of each graph attentional layer is a new set of node features, which may have a new dimension $F'$: $\\mathbf{h'} = \\{\\overrightarrow{h_1'}, \\overrightarrow{h_2'}, \\dots, \\overrightarrow{h_N'}\\}$, with $\\overrightarrow{h_i'} \\in \\mathbb{R}^{F'}$.\n\nWe will now describe how this transformation is performed for each graph attention layer. First, a shared linear transformation parametrized by the weight matrix $\\mathbf{W} \\in \\mathbb{R}^{F' \\times F}$ is applied to every node. \n\nNext, we perform self-attention on the nodes. We use a shared attention function $a$:\n\\begin{equation} \na : \\mathbb{R}^{F'} \\times \\mathbb{R}^{F'} \\rightarrow \\mathbb{R}.\n\\end{equation}\n\nthat computes the attention coefficients capturing the importance of node $j$'s features to node $i$:\n\\begin{equation}\ne_{ij} = a(\\mathbf{W_l}\\overrightarrow{h_i}, \\mathbf{W_r} \\overrightarrow{h_j})\n\\end{equation}\n\nThe most general formulation of self-attention allows every node to attend to all other nodes which drops all structural information. However, to utilize graph structure in the attention mechanisms, we use **masked attention**. In masked attention, we only compute attention coefficients $e_{ij}$ for nodes $j \\in \\mathcal{N}_i$ where $\\mathcal{N}_i$ is some neighborhood of node $i$ in the graph.\n\nTo easily compare coefficients across different nodes, we normalize the coefficients across $j$ using a softmax function:\n\\begin{equation}\n\\alpha_{ij} = \\text{softmax}_j(e_{ij}) = \\frac{\\exp(e_{ij})}{\\sum_{k \\in \\mathcal{N}_i} \\exp(e_{ik})}\n\\end{equation}\n\nFor this problem, our attention mechanism $a$ will be a single-layer feedforward neural network parametrized by a weight vectors $\\overrightarrow{a_l} \\in \\mathbb{R}^{F'}$ and $\\overrightarrow{a_r} \\in \\mathbb{R}^{F'}$, followed by a LeakyReLU nonlinearity (with negative input slope 0.2). Let $\\cdot^T$ represent transposition and $||$ represent concatenation. The coefficients computed by our attention mechanism may be expressed as:\n\n\\begin{equation}\n\\alpha_{ij} = \\frac{\\exp\\Big(\\text{LeakyReLU}\\Big(\\overrightarrow{a_l}^T \\mathbf{W_l} \\overrightarrow{h_i} + \\overrightarrow{a_r}^T\\mathbf{W_r}\\overrightarrow{h_j}\\Big)\\Big)}{\\sum_{k\\in \\mathcal{N}_i} \\exp\\Big(\\text{LeakyReLU}\\Big(\\overrightarrow{a_l}^T \\mathbf{W_l} \\overrightarrow{h_i} + \\overrightarrow{a_r}^T\\mathbf{W_r}\\overrightarrow{h_k}\\Big)\\Big)}\n\\end{equation}\n\nFor the following questions, we denote `alpha_l` = $\\alpha_l = [...,\\overrightarrow{a_l}^T \\mathbf{W_l} \\overrightarrow{h_i},...] \\in \\mathcal{R}^n$ and `alpha_r` = $\\alpha_r = [..., \\overrightarrow{a_r}^T \\mathbf{W_r} \\overrightarrow{h_j}, ...] \\in \\mathcal{R}^n$.\n\n\nAt every layer of GAT, after the attention coefficients are computed for that layer, the aggregation function can be computed by a weighted sum of neighborhood messages, where weights are specified by $\\alpha_{ij}$.\n\nNow, we use the normalized attention coefficients to compute a linear combination of the features corresponding to them. These aggregated features will serve as the final output features for every node.\n\n\\begin{equation}\nh_i' = \\sum_{j \\in \\mathcal{N}_i} \\alpha_{ij} \\mathbf{W_r} \\overrightarrow{h_j}.\n\\end{equation}\n\nAt this point, we have covered a lot of information! Before reading further about multi-head attention, we encourage you to go again through the excersize of thinking about what components of the attention mechanism correspond with the different functions: 1) `forward`, 2) `message`, and 3 `aggregate`. \n\n- Hint 1: Our aggregation is very similar to that of GraphSage except now we are using sum aggregation\n- Hint 2: The terms we aggregate over again represent the individual message that each neighbor node j sends. Thus, we see that $\\alpha_{ij}$ is part of the message each node sends and is thus computed during the message step. This makes sense since an attention weight is associated with each edge in the graph.\n- Hint 3: Look at the terms in the definition of $\\alpha_{ij}$. What values do we want to pre-process and pass as parameters to the `propagate` function. The parameters of `message(..., x_j, alpha_j, alpha_i, ...)` should give a good hint. \n\n### Multi-Head Attention\nTo stabilize the learning process of self-attention, we use multi-head attention. To do this we use $K$ independent attention mechanisms, or ``heads'' compute output features as in the above equations. Then, we concatenate these output feature representations:\n\n\\begin{equation}\n \\overrightarrow{h_i}' = ||_{k=1}^K \\Big(\\sum_{j \\in \\mathcal{N}_i} \\alpha_{ij}^{(k)} \\mathbf{W_r}^{(k)} \\overrightarrow{h_j}\\Big)\n\\end{equation}\n\nwhere $||$ is concentation, $\\alpha_{ij}^{(k)}$ are the normalized attention coefficients computed by the $k$-th attention mechanism $(a^k)$, and $\\mathbf{W}^{(k)}$ is the corresponding input linear transformation's weight matrix. Note that for this setting, $\\mathbf{h'} \\in \\mathbb{R}^{KF'}$.\n\n\n```python\nclass GAT(MessagePassing):\n\n def __init__(self, in_channels, out_channels, heads = 2,\n negative_slope = 0.2, dropout = 0., **kwargs):\n super(GAT, self).__init__(node_dim=0, **kwargs)\n\n self.in_channels = in_channels\n self.out_channels = out_channels\n self.heads = heads\n self.negative_slope = negative_slope\n self.dropout = dropout\n\n self.lin_l = None\n self.lin_r = None\n self.att_l = None\n self.att_r = None\n\n ############################################################################\n # TODO: Your code here! \n # Define the layers needed for the message functions below.\n # self.lin_l is the linear transformation that you apply to embeddings \n # BEFORE message passing.\n # \n # Pay attention to dimensions of the linear layers, since we're using \n # multi-head attention.\n # Our implementation is ~1 lines, but don't worry if you deviate from this.\n\n ############################################################################\n\n self.lin_r = self.lin_l\n\n ############################################################################\n # TODO: Your code here! \n # Define the attention parameters \\overrightarrow{a_l/r}^T in the above intro.\n # You have to deal with multi-head scenarios.\n # Use nn.Parameter instead of nn.Linear\n # Our implementation is ~2 lines, but don't worry if you deviate from this.\n\n ############################################################################\n\n self.reset_parameters()\n\n def reset_parameters(self):\n nn.init.xavier_uniform_(self.lin_l.weight)\n nn.init.xavier_uniform_(self.lin_r.weight)\n nn.init.xavier_uniform_(self.att_l)\n nn.init.xavier_uniform_(self.att_r)\n\n def forward(self, x, edge_index, size = None):\n \n H, C = self.heads, self.out_channels\n\n ############################################################################\n # TODO: Your code here! \n # Implement message passing, as well as any pre- and post-processing (our update rule).\n # 1. First apply linear transformation to node embeddings, and split that \n # into multiple heads. We use the same representations for source and\n # target nodes, but apply different linear weights (W_l and W_r)\n # 2. Calculate alpha vectors for central nodes (alpha_l) and neighbor nodes (alpha_r).\n # 3. Call propagate function to conduct the message passing. \n # 3.1 Remember to pass alpha = (alpha_l, alpha_r) as a parameter.\n # 3.2 See there for more information: https://pytorch-geometric.readthedocs.io/en/latest/notes/create_gnn.html\n # 4. Transform the output back to the shape of [N, H * C].\n # Our implementation is ~5 lines, but don't worry if you deviate from this.\n\n\n ############################################################################\n\n return out\n\n\n def message(self, x_j, alpha_j, alpha_i, index, ptr, size_i):\n\n ############################################################################\n # TODO: Your code here! \n # Implement your message function. Putting the attention in message \n # instead of in update is a little tricky.\n # 1. Calculate the final attention weights using alpha_i and alpha_j,\n # and apply leaky Relu.\n # 2. Calculate softmax over the neighbor nodes for all the nodes. Use \n # torch_geometric.utils.softmax instead of the one in Pytorch.\n # 3. Apply dropout to attention weights (alpha).\n # 4. Multiply embeddings and attention weights. As a sanity check, the output\n # should be of shape [E, H, C].\n # 5. ptr (LongTensor, optional): If given, computes the softmax based on\n # sorted inputs in CSR representation. You can simply pass it to softmax.\n # Our implementation is ~4-5 lines, but don't worry if you deviate from this.\n\n\n ############################################################################\n\n return out\n\n\n def aggregate(self, inputs, index, dim_size = None):\n\n ############################################################################\n # TODO: Your code here! \n # Implement your aggregate function here.\n # See here as how to use torch_scatter.scatter: https://pytorch-scatter.readthedocs.io/en/latest/_modules/torch_scatter/scatter.html\n # Pay attention to \"reduce\" parameter is different from that in GraphSage.\n # Our implementation is ~1 lines, but don't worry if you deviate from this.\n\n ############################################################################\n \n return out\n```\n\n## Building Optimizers\n\nThis function has been implemented for you. **For grading purposes please use the default Adam optimizer**, but feel free to play with other types of optimizers on your own.\n\n\n```python\nimport torch.optim as optim\n\ndef build_optimizer(args, params):\n weight_decay = args.weight_decay\n filter_fn = filter(lambda p : p.requires_grad, params)\n if args.opt == 'adam':\n optimizer = optim.Adam(filter_fn, lr=args.lr, weight_decay=weight_decay)\n elif args.opt == 'sgd':\n optimizer = optim.SGD(filter_fn, lr=args.lr, momentum=0.95, weight_decay=weight_decay)\n elif args.opt == 'rmsprop':\n optimizer = optim.RMSprop(filter_fn, lr=args.lr, weight_decay=weight_decay)\n elif args.opt == 'adagrad':\n optimizer = optim.Adagrad(filter_fn, lr=args.lr, weight_decay=weight_decay)\n if args.opt_scheduler == 'none':\n return None, optimizer\n elif args.opt_scheduler == 'step':\n scheduler = optim.lr_scheduler.StepLR(optimizer, step_size=args.opt_decay_step, gamma=args.opt_decay_rate)\n elif args.opt_scheduler == 'cos':\n scheduler = optim.lr_scheduler.CosineAnnealingLR(optimizer, T_max=args.opt_restart)\n return scheduler, optimizer\n```\n\n## Training and Testing\n\nHere we provide you with the functions to train and test. **Please do not modify this part for grading purposes.**\n\n\n```python\nimport time\n\nimport networkx as nx\nimport numpy as np\nimport torch\nimport torch.optim as optim\nfrom tqdm import trange\nimport pandas as pd\nimport copy\n\nfrom torch_geometric.datasets import TUDataset\nfrom torch_geometric.datasets import Planetoid\nfrom torch_geometric.data import DataLoader\n\nimport torch_geometric.nn as pyg_nn\n\nimport matplotlib.pyplot as plt\n\n\ndef train(dataset, args):\n \n print(\"Node task. test set size:\", np.sum(dataset[0]['test_mask'].numpy()))\n print()\n test_loader = loader = DataLoader(dataset, batch_size=args.batch_size, shuffle=False)\n\n # build model\n model = GNNStack(dataset.num_node_features, args.hidden_dim, dataset.num_classes, \n args)\n scheduler, opt = build_optimizer(args, model.parameters())\n\n # train\n losses = []\n test_accs = []\n best_acc = 0\n best_model = None\n for epoch in trange(args.epochs, desc=\"Training\", unit=\"Epochs\"):\n total_loss = 0\n model.train()\n for batch in loader:\n opt.zero_grad()\n pred = model(batch)\n label = batch.y\n pred = pred[batch.train_mask]\n label = label[batch.train_mask]\n loss = model.loss(pred, label)\n loss.backward()\n opt.step()\n total_loss += loss.item() * batch.num_graphs\n total_loss /= len(loader.dataset)\n losses.append(total_loss)\n\n if epoch % 10 == 0:\n test_acc = test(test_loader, model)\n test_accs.append(test_acc)\n if test_acc > best_acc:\n best_acc = test_acc\n best_model = copy.deepcopy(model)\n else:\n test_accs.append(test_accs[-1])\n \n return test_accs, losses, best_model, best_acc, test_loader\n\ndef test(loader, test_model, is_validation=False, save_model_preds=False, model_type=None):\n test_model.eval()\n\n correct = 0\n # Note that Cora is only one graph!\n for data in loader:\n with torch.no_grad():\n # max(dim=1) returns values, indices tuple; only need indices\n pred = test_model(data).max(dim=1)[1]\n label = data.y\n\n mask = data.val_mask if is_validation else data.test_mask\n # node classification: only evaluate on nodes in test set\n pred = pred[mask]\n label = label[mask]\n\n if save_model_preds:\n print (\"Saving Model Predictions for Model Type\", model_type)\n\n data = {}\n data['pred'] = pred.view(-1).cpu().detach().numpy()\n data['label'] = label.view(-1).cpu().detach().numpy()\n\n df = pd.DataFrame(data=data)\n # Save locally as csv\n df.to_csv('CORA-Node-' + model_type + '.csv', sep=',', index=False)\n \n correct += pred.eq(label).sum().item()\n\n total = 0\n for data in loader.dataset:\n total += torch.sum(data.val_mask if is_validation else data.test_mask).item()\n\n return correct / total\n \nclass objectview(object):\n def __init__(self, d):\n self.__dict__ = d\n\n```\n\n## Let's Start the Training!\n\nWe will be working on the CORA dataset on node-level classification.\n\nThis part is implemented for you. **For grading purposes, please do not modify the default parameters.** However, feel free to play with different configurations just for fun!\n\n**Submit your best accuracy and loss on Gradescope.**\n\n\n```python\nif 'IS_GRADESCOPE_ENV' not in os.environ:\n for args in [\n {'model_type': 'GAT', 'dataset': 'cora', 'num_layers': 2, 'heads': 1, 'batch_size': 32, 'hidden_dim': 32, 'dropout': 0.5, 'epochs': 500, 'opt': 'adam', 'opt_scheduler': 'none', 'opt_restart': 0, 'weight_decay': 5e-3, 'lr': 0.01},\n ]:\n args = objectview(args)\n for model in ['GAT']:\n args.model_type = model\n\n # Match the dimension.\n if model == 'GAT':\n args.heads = 2\n else:\n args.heads = 1\n\n if args.dataset == 'cora':\n dataset = Planetoid(root='/tmp/cora', name='Cora')\n else:\n raise NotImplementedError(\"Unknown dataset\") \n test_accs, losses, best_model, best_acc, test_loader = train(dataset, args) \n\n print(\"Maximum test set accuracy: {0}\".format(max(test_accs)))\n print(\"Minimum loss: {0}\".format(min(losses)))\n\n # Run test for our best model to save the predictions!\n test(test_loader, best_model, is_validation=False, save_model_preds=True, model_type=model)\n print()\n\n plt.title(dataset.name)\n plt.plot(losses, label=\"training loss\" + \" - \" + args.model_type)\n plt.plot(test_accs, label=\"test accuracy\" + \" - \" + args.model_type)\n plt.legend()\n plt.show()\n\n```\n\n## Question 1: What is the maximum accuracy obtained on test set for GAT? (10 points)\n\n\nRunning the training cell above will also save your best GAT model predictions as *CORA-Node-GAT.csv*. \n\nWhen you sumbit your assignment, you will have to download this file and attatch it to your submission. As with the other colabs, please zip this file (DON'T CHANGE ITS NAME) and the .csv file that's generated!\n\n", "meta": {"hexsha": "fcccda6a8a989351bf5812488737bceed4fe1823", "size": 38841, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "labs/CS224W_Colab4.ipynb", "max_stars_repo_name": "kdha0727/CS224W", "max_stars_repo_head_hexsha": "095d61eb751b43d97ad8ca8b3a07266cc5006631", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "labs/CS224W_Colab4.ipynb", "max_issues_repo_name": "kdha0727/CS224W", "max_issues_repo_head_hexsha": "095d61eb751b43d97ad8ca8b3a07266cc5006631", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "labs/CS224W_Colab4.ipynb", "max_forks_repo_name": "kdha0727/CS224W", "max_forks_repo_head_hexsha": "095d61eb751b43d97ad8ca8b3a07266cc5006631", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 55.9668587896, "max_line_length": 929, "alphanum_fraction": 0.556963003, "converted": true, "num_tokens": 6871, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.585101154203231, "lm_q2_score": 0.743168019989179, "lm_q1q2_score": 0.4348284662625985}} {"text": "```python\n%pylab inline\n```\n\n Populating the interactive namespace from numpy and matplotlib\n\n\n# Precalculus\n\n## Week 1\n\n**Topics:** Function definition, vertical line test, piecewise defined functions, domain and range, symmetry.\n\n**Problems:** 3, 7, 8, 9, 31, 32, 37, 38, 44, 46, 47, 48, 63, 69, 70, 73, 74, 77, additional problem set G.\n\n### Function definition\n\n### Vertical line test\n\n### Piecewise defined functions\n\nA piecewise defined function is defined as:\n\n$$\\begin{align}f(x)=\\begin{cases}2x\\quad &, x > 1\\\\\n-x\\quad&, x \\leq 1\\end{cases}\\end{align}$$\n\nNow we can define the **absolute value** function $|\\ x \\ |$:\n\n$$\\begin{align}f(x) = |\\ x \\ |=\\begin{cases}x\\quad &, x > 0\\\\\n-x\\quad&, x < 0\\end{cases}\\end{align}$$\n\n### Domain and range\n\n### Symmetry\n\n## Week 2\n\n**Topics:** Polynomials, power functions, rational functions, trigonometric functions, exponential functions.\n\n**Problems:** 1, 2, 3, 6, 7, 10, 14, 17, 20, 23, 27, 32, additional problem set B.\n\n### Polynomials\n\n### Power functions\n\n### Rational functions\n\n### Trigonometric functions\n\n### Exponential functions\n\n## Week 3\n\n**Topics:** Transformations, composite functions, solving power equations.\n\n**Problems:** 9, 10, 11, 12, 13, 16, 31, 32, 33, 34, 36.\n\n### Transformations\n\n### Composite functions\n\n### Solving power equations\n\n## Week 4\n\n**Topics:** Exponential and logarithmic functions (Basisvaardigheden H10), solving power equations.\n\n**Problems:** 1, 3, 4, 6, 7, 10, 11, 12, 15.\n\n### Exponential functions\n\nExponential functions come in the form of $y=2^x$. Notice that the exponent is a variable, which defines this as an exponential function unlike $y=x^2$ which is a power function.\n\nGraphing these functions yields:\n\n\n```python\npoints = np.arange(-5,5.1,0.1)\nplot(points, [2**x for x in points], c='b', lw=2)\nplot(points, [(1/2)**x for x in points], c='r', lw=2)\ngrid()\nlegend(['2^x', '(1/2)^x']);\n```\n\nWe can transform $2^x$:\n\n* Moving up by adding $5$ gives $2^x+5$.\n* Moving to the left by adding $5$ to the exponent gives $2^{x+5}$.\n\nThe effect can be seen in the plot:\n\n\n```python\npoints = np.arange(-2, 2.1, 0.1)\nplot(points, [2**x for x in points], c='b', lw=2)\nplot(points, [2**x+5 for x in points], c='g', lw=2)\nplot(points, [2**(x+5) for x in points], c='r', lw=2)\ngrid()\nlegend(['2^x', '2^x+5', '2^(x+5)']);\n```\n\n**Y-intercept**\n\nTo find the y-intercept we know that it happens where $x=0$, filling in $x=0$ into the equations in the example above gives:\n\n* $2^0=1$\n* $2^0+5=6$\n* $2^{(0+5)}=32$\n\n\n```python\nprint(2**0)\nprint(2**0+5)\nprint(2**(0+5))\n```\n\n 1\n 6\n 32\n\n\n**Horizontal asymptote**\n\nTo find the horizontal asymptote is rather easy. For any exponential function which is not shifted up or down this will be at $y=0$. For the above given examples we get:\n\n* $2^x$ at $y=0$.\n* $2^x+5$ at $y=5$.\n* $2^{(x+5)}$ at $y=0$.\n\n### Logarithmic functions\n\nThe logarithms is the inverse of the exponential function. We can use logarithms to solve for $x$ in $2^x=32$ for example.\n\n**Definition**\n\nThe logarithm is defined as $\\log_a(b)=c \\iff a^c=b$, where $a$ is called the base, and $c$ is the exponent.\n\n\n```python\npoints = np.arange(0.01,3.1,0.01)\nplot(points, [2**x for x in points])\nplot(points, [math.log(x,2) for x in points])\ngrid()\n```\n\n**Rules for logarithms**\n\n* $g^{\\log(a)}=a$\n* $^g\\log(g^a)=a$\n* $^g\\log(a)+\\ ^g\\log(b)=\\ ^g\\log(ab)$\n* $^g\\log(a)-\\ ^g\\log(b)=\\ ^g\\log(\\frac{a}{b})$\n* $^g\\log(a^p)=p\\cdot \\ ^g\\log(a)$\n* $^g\\log(a) = \\frac{^b\\log(a)}{^b\\log(g)}$ where $b$ is any base.\n\n\n```python\n\n```\n", "meta": {"hexsha": "2d04432cf9685b7140ff3957edd2223981c8b85d", "size": 54992, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Applied Math/Y1S2/Precalculus.ipynb", "max_stars_repo_name": "darkeclipz/jupyter-notebooks", "max_stars_repo_head_hexsha": "5de784244ad9db12cfacbbec3053b11f10456d7e", "max_stars_repo_licenses": ["Unlicense"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-08-28T12:16:12.000Z", "max_stars_repo_stars_event_max_datetime": "2018-08-28T12:16:12.000Z", "max_issues_repo_path": "Applied Math/Y1S2/Precalculus.ipynb", "max_issues_repo_name": "darkeclipz/jupyter-notebooks", "max_issues_repo_head_hexsha": "5de784244ad9db12cfacbbec3053b11f10456d7e", "max_issues_repo_licenses": ["Unlicense"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Applied Math/Y1S2/Precalculus.ipynb", "max_forks_repo_name": "darkeclipz/jupyter-notebooks", "max_forks_repo_head_hexsha": "5de784244ad9db12cfacbbec3053b11f10456d7e", "max_forks_repo_licenses": ["Unlicense"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 140.6445012788, "max_line_length": 16514, "alphanum_fraction": 0.8827102124, "converted": true, "num_tokens": 1243, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.6825737473266735, "lm_q1q2_score": 0.4348204965509632}} {"text": "```python\nimport pandas as pd\nimport numpy as np\nimport matplotlib.pyplot as plt\n%matplotlib inline\n```\n\n## Regress\u00e3o Linear\n\nA Regress\u00e3o Linear \u00e9 usada entender a rela\u00e7\u00e3o entre vari\u00e1veis que ir\u00e3o explicar determinado comportamento. \n\nPode ser usada para:\n \n* Predi\u00e7\u00e3o de demandas para produtos\n* Ajudar a entender o efeitos de certos tratamentos em um paciente\n* Entender quais fatores influenciam em uma economia equilibrada\n\nExistem v\u00e1rios tipos de regress\u00e3o, tais como:\n\n* Regress\u00e3o Linear Simples\n* Regress\u00e3o Linear M\u00faltipla\n* Regress\u00e3o Log\u00edstica\n* Regress\u00e3o Polinomial\n\n## 1. Regress\u00e3o Linear Simples\n\nA regress\u00e3o linear simples ir\u00e1 mostrar a rela\u00e7\u00e3o entre duas vari\u00e1veis. Como a varia\u00e7\u00e3o na vari\u00e1vel independente podem influenciar no valor da dependente. \n\nPode ser entendida como uma equa\u00e7\u00e3o da reta, onde existe uma vari\u00e1vel Y (dependente) cujo valor est\u00e1 relacionado \u00e0 X (independente). O objetivo dessa abordagem \u00e9 chegar ao valor de Y a partir de X. \n\n\\begin{equation}\n\\label{eq:linear_regression}\ny_1 = \\beta_01 + \\beta_1x_1\n\\end{equation}\n\n* $y_n$ = Vari\u00e1vel dependente \n* $x_n$ = Vari\u00e1vel independente \n\nPor exemplo, a vari\u00e1vel dependente \u00e9 o resultado da predi\u00e7\u00e3o de demandas de produtos em uma loja. \n\n### 2. O que signfica ser linear?\n\nTodos os termos s\u00e3o uma constante ou um par\u00e2metro multiplicado por uma vari\u00e1vel independente.\n\n\\begin{equation}\n\\label{eq:linear_regression}\ny_1 = \\beta_01 + \\beta_1x_1\n\\end{equation}\n\n* $\\beta_0$ = Vari\u00e1vel que intercepta o eixo Y \n* $\\beta_1$ = Vari\u00e1vel que que indica a inclina\u00e7\u00e3o da reta \n\n \n\nDe maneira geral, o objetivo do algoritmo \u00e9 gerar uma reta que fa\u00e7a a divis\u00e3o entre os pontos, por exemplo, do gr\u00e1fico abaixo.\n\n\n```python\nnp.random.seed(42)\nx = np.linspace(-10,10, 20)\ny = x * .1 + 3 + np.random.normal(0,2,size=20)\n\nfig, (ax1, ax2) = plt.subplots(1, 2)\nfig.suptitle('Regress\u00e3o Linear Simples')\n\nax1.scatter(x, y)\nax2.scatter(x, y)\nax2.scatter(1, 1 * .1 + 3, c='C4', s=100)\nax2.plot(x, x * .1 + 3, c='C1')\n```\n\n### 3. Como saber o qu\u00e3o boa \u00e9 a linha gerada? \n\nA melhor linha \u00e9 aquela onde todos os pontos possuem a menor dist\u00e2ncia at\u00e9 a reta gerada. \n\n \n\nUma maneira de se obter o melhor fit, \u00e9 por meio do c\u00e1lculo das fun\u00e7\u00f5es de custo. As mais comuns s\u00e3o: \n\n* Erro absoluto m\u00e9dio\n* Erro quadr\u00e1tico m\u00e9dio\n\n\u00c9 necess\u00e1rio desenvolver um algoritmo para minimizar essas fun\u00e7\u00f5es.\n\n### 3.1 Gradiente Descendente\n\nO gradiente descendente \u00e9 como se fosse uma b\u00fassola apontando para a dire\u00e7\u00e3o do m\u00ednimo.\n\nA taxa de aprendizagem indica a acelera\u00e7\u00e3o.\n\n\n\n### 3.2 Erro absoluto m\u00e9dio\n\nConsidere que o eixo y representa o valor a ser predito, considerando o valor da vari\u00e1vel x. Na figura abaixo, o ponto a ser predito est\u00e1 localizado na coordenada A(x,y) e o ponto identificado pelo algoritmo, aquele que intercepta a reta, \u00e9 identificado por B(x, \u0177). Logo, o erro para essa predi\u00e7\u00e3o \u00e9 a dist\u00e2ncia vertical do ponto (x,y) a (x, \u0177), ou seja, a subtra\u00e7\u00e3o dos valores y-\u0177. \n\n \n\nDessa forma, o c\u00e1lculo do erro absoluto ser\u00e1 a m\u00e9dia de todos os pontos acima e abaixo da reta.\n\n\\begin{equation}\nError=\\sum_{i=1}^m |y-\u0177|\n\\end{equation}\n\nPor que usamos o valor absoluto? \n\nPorque queremos que os pontos positivos n\u00e3o sejam anulados pelos pontos negativos durante o somat\u00f3rio, por isso utilizados |y-\u0177|.\n\n* Se o ponto estiver acima da reta, a dist\u00e2ncia ser\u00e1 y-\u0177\n* Caso contr\u00e1rio, ser\u00e1 \u0177-y\n\nSistemas lineares muito complexos com o n\u00famero de observa\u00e7\u00f5es maior que a dimens\u00e3o n\u00e3o possuem resolu\u00e7\u00e3o, normalmente. \u00c9 o caso do contexto atual. \n\nConsiderando tamb\u00e9m que \n\n\n```python\nX = [(2, -2), (5, 6), (-4, -4), (-7, 1), (8, 14)]\n\nsum_abs = 0\nm = len(X)\nfor x, y in X:\n y_hat = 1.2 * x + 2\n sum_abs+= abs(y-y_hat)\n\nmae = sum_abs/m\nmae\n```\n\n\n\n\n 3.88\n\n\n\n### 3.3 Erro quadr\u00e1tico m\u00e9dio\n\nO erro quadr\u00e1tico m\u00e9dio segue o mesmo processo do erro absoluto m\u00e9dio , entretanto, ao inv\u00e9s de realizar a dist\u00e2ncia vertical, \u00e9 tra\u00e7ado um quadrado.\n\n\n\nA \u00e1rea do quadrado ser\u00e1 (y-\u0177)\u00b2, um valor que sempre ser\u00e1 positivo. O erro quadr\u00e1tico m\u00e9dio ser\u00e1 a m\u00e9dia entre as s\u00e9ries de quadrados. \u00c0 medida que o algoritmo vai diminuindo sua percentagem de erro, a \u00e1rea do quadrado tamb\u00e9m diminui. \n\n\\begin{equation}\nError=\\sum_{i=1}^m |y-\u0177|\u00b2\n\\end{equation}\n\n\n\n```python\nX = [(2, -2), (5, 6), (-4, -4), (-7, 1), (8, 14)]\n\nsum_abs = 0\nm = len(X)\nfor x, y in X:\n y_hat = 1.2 * x + 2\n sum_abs+= abs(y-y_hat)**2\n\nmse = sum_abs/(2*m)\nmse\n```\n\n\n\n\n 10.692000000000002\n\n\n\n \n\n## Refer\u00eancias\n\n[Regress\u00e3o Linear Simples](https://www.ime.usp.br/~fmachado/MAE229/AULA10.pdf)\n\n[Regress\u00e3o Linear Simples II](https://edisciplinas.usp.br/pluginfile.php/1479289/mod_resource/content/0/regr_lin.pdf)\n\n[An\u00e1lise de Regress\u00e3o](https://www.ime.unicamp.br/~nancy/Cursos/me104/regressao.pdf)\n\n\n", "meta": {"hexsha": "2bc546c130e3b5bbc6b924a7f4d06f4e7fe58e9c", "size": 20354, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Regressao_Linear/regressao_linear_simples.ipynb", "max_stars_repo_name": "julianyraiol/machine_learning_nanodegree", "max_stars_repo_head_hexsha": "32059c674058dda53e9a8670b548d51499e38c86", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Regressao_Linear/regressao_linear_simples.ipynb", "max_issues_repo_name": "julianyraiol/machine_learning_nanodegree", "max_issues_repo_head_hexsha": "32059c674058dda53e9a8670b548d51499e38c86", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Regressao_Linear/regressao_linear_simples.ipynb", "max_forks_repo_name": "julianyraiol/machine_learning_nanodegree", "max_forks_repo_head_hexsha": "32059c674058dda53e9a8670b548d51499e38c86", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2019-06-25T03:08:00.000Z", "max_forks_repo_forks_event_max_datetime": "2019-06-25T03:08:00.000Z", "avg_line_length": 60.2189349112, "max_line_length": 11316, "alphanum_fraction": 0.7843175789, "converted": true, "num_tokens": 1575, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.8244619242200081, "lm_q1q2_score": 0.4347523955385228}} {"text": "# TPSC approximation\nAuthor: [Niklas Witt](mailto:niklas.witt@physik.uni-hamburg.de)\n\n## Theory of TPSC\n\nThe Two-Particle Self-Consistent (TPSC) approximation is a non-perturbative semi-analytical method that was first introduced by Vilk and Tremblay {cite:p}`Vilk1997`. TPSC can be used to study magnetic fluctuations, while it also obeys the Mermin-Wagner theorem in two dimensions, i.e., a phase transtition at finite temperatures is prohibited. In addition, the TPSC method satisfies several conservation laws, sum rules and the Pauli principle (actually, it is constructed in a way to fulfill these, since they are used to determine model parameters self-consistently). TPSC is applicable in the weak to intermediate coupling regime, but it breaks down in the strong coupling regime and it cannot describe the Mott transition unlike other non-perturbative methods like Dynamical Mean-Field Theory (DMFT).\n\nFor a (pedagogical) review, please have a look at {cite:p}`Allen2004,Tremblay2012` for the single-orbital case implemented here and {cite:p}`Zantout2021` for the more complex multi-orbital theory.\n\n\n#### Set of TPSC equations\nWe review the set of equations that need to be solved in the TPSC approximation assuming a one-band Hubbard model with interaction $U$ (it is not so easy to extend TPSC to models with more parameters, since sum rules to determine the additional parameters self-consistently need to be found) in the paramagnetic phase (SU(2) symmetric), i.e., $n = \\langle n\\rangle = 2n_{\\sigma}$ for the electron filling $n$. TPSC is constructed in a way to fulfill certain sum rules and the Pauli principle in the form $\\langle n^2\\rangle = \\langle n\\rangle$. The control quantitites are spin and charge correlation function (susceptibilities) which are evaluated in a Random-Phase-Approximation (RPA) like fashion\n\n$$ \\chi_{\\mathrm{sp}}^{\\mathrm{RPA}}(i\\nu_m, \\boldsymbol{q}) = \\frac{\\chi_0(i\\nu_m, \\boldsymbol{q})}{1-U\\chi_0(i\\nu_m, \\boldsymbol{q})}\\;,\\quad \\chi_{\\mathrm{ch}}^{\\mathrm{RPA}}(i\\nu_m, \\boldsymbol{q}) = \\frac{\\chi_0(i\\nu_m, \\boldsymbol{q})}{1+U\\chi_0(i\\nu_m, \\boldsymbol{q})} $$\n\nwith the irreducible susceptibility (\"bubble diagram\")\n\n$$ \\chi_0(i\\nu_m, \\boldsymbol{q}) = - \\frac{T}{N_{\\boldsymbol{k}}} \\sum_{n,\\boldsymbol{k}} G(i\\omega_n + i\\nu_m, \\boldsymbol{k} + \\boldsymbol{q})G(i\\omega_n, \\boldsymbol{k})\\;,$$\n\nwhere $\\nu_m = 2n\\pi T$ [$\\omega_n=(2n+1)\\pi T$] and $\\boldsymbol{q}$ [$\\boldsymbol{k}$] are bosonic [fermionic] Matsubara frequencies and momentum at temperature $T$, $N_{\\boldsymbol{k}}$ denotes the number of $\\boldsymbol{k}$-points, and $G_0(i\\omega_n,\\boldsymbol{k}) = [i\\omega_n - (\\varepsilon_{\\boldsymbol{k}}-\\mu)]^{-1}$ is the bare (non-interacting) Green function with with single-particle dispersion $\\varepsilon_{\\boldsymbol{k}}$ and chemical potential $\\mu$. Please note that sometimes a factor of 2 is included in the definition of $\\chi_0$ leading to slightly different factors in all equations given here. The convolution sum to calculate $\\chi_0$ can be easily evaluated by Fourier transforming to imaginary-time and real space, resulting in a simple multiplication\n\n$$ \\chi_0(\\tau, \\boldsymbol{r}) = - G(\\tau, \\boldsymbol{r})G(-\\tau,-\\boldsymbol{r}) = G(\\tau, \\boldsymbol{r})G(\\beta-\\tau,\\boldsymbol{r})\\;.$$\n\nIn our practical implementation, we will perform this step using the `sparse-ir` package. A similar calculation is necessary to set the chemical potential $\\mu$ for fixed electron density $n$, as \n\n$$ n = 2n_{\\sigma} = 2 - \\frac{2}{N_{\\boldsymbol{k}}} \\sum_{\\boldsymbol{k}} G(\\tau=0^+, \\boldsymbol{k}) $$\n\nwith a factor 2 from spin degeneracy and $0^+ = \\lim_{\\eta\\to 0+} \\eta$ needs to be solved by using some root finding algorithm like bisection method or Brent's method. The Fourier transformation to $\\tau=0^+$ can be easily performed with the `sparse-ir` package.\n\nThe above RPA definition of spin and charge susceptibility violate the Pauli principle. In TPSC, we overcome this problem by introducing two effective, renormalized interactions (\"irreducible vertices\") $U_{\\mathrm{sp}}$ and $U_{\\mathrm{ch}}$ that enter spin and charge correlation functions as \n\n$$ \\chi_{\\mathrm{sp}}(i\\nu_m, \\boldsymbol{q}) = \\frac{\\chi_0(i\\nu_m, \\boldsymbol{q})}{1-U_{\\mathrm{sp}}\\chi_0(i\\nu_m, \\boldsymbol{q})}\\;,\\quad \\chi_{\\mathrm{ch}}(i\\nu_m, \\boldsymbol{q}) = \\frac{\\chi_0(i\\nu_m, \\boldsymbol{q})}{1+U_{\\mathrm{ch}}\\chi_0(i\\nu_m, \\boldsymbol{q})}\\,. $$\n\nThese two effetive interactions are determined by the two local sum rules \n\n$$\n\\begin{align}\n 2 \\frac{T}{N_{\\boldsymbol{k}}} \\sum_{m,\\boldsymbol{q}} \\chi_{\\mathrm{sp}} &= \\left\\langle (n_{\\uparrow} - n_{\\downarrow})^2\\right\\rangle = n - 2\\langle n_{\\uparrow}n_{\\downarrow}\\rangle\\;,\\\\\n 2 \\frac{T}{N_{\\boldsymbol{k}}} \\sum_{m,\\boldsymbol{q}} \\chi_{\\mathrm{ch}} &= \\left\\langle (n_{\\uparrow} + n_{\\downarrow})^2\\right\\rangle - \\left\\langle n_{\\uparrow} + n_{\\downarrow}\\right\\rangle^2 = n + 2\\langle n_{\\uparrow}n_{\\downarrow}\\rangle - n^2\\;.\n\\end{align}\n$$\n\nBoth sum rules can be exactly derived from the Pauli principle ($\\langle n^2\\rangle = \\langle n\\rangle$). In principle, we could now determine $U_{\\mathrm{sp}}$ and $U_{\\mathrm{ch}}$ from local-spin and local-charge sum rule if we knew the double occupancy $\\langle n_{\\uparrow}n_{\\downarrow}\\rangle$. TPSC makes the ansatz\n\n$$ U_{\\mathrm{sp}}\\langle n_{\\uparrow}\\rangle\\langle n_{\\downarrow}\\rangle = U_{\\mathrm{sp}}\\frac{n^2}{4} = U\\langle n_{\\uparrow}n_{\\downarrow}\\rangle\\;,$$ \n\nwhich reproduces Kanamori-Brueckner type screening. The four equations above form a set of self-consistent equations for either $U_{\\mathrm{sp}}$ or equivalently $\\langle n_{\\uparrow}n_{\\downarrow}\\rangle$. In practice, we treat $U_{\\mathrm{sp}}$ as the parameter to be determined self-consistently by inserting the ansatz in the local-spin sum rule. Effectively, we then need to find the root of the function\n\n$$ f(U_{\\mathrm{sp}}) = 2\\frac{T}{N_{\\boldsymbol{k}}} \\sum_{m,\\boldsymbol{q}}\\chi_{\\mathrm{sp}}(U_{\\mathrm{sp}}) - n + \\frac{U_{\\mathrm{sp}}}{2U}n^2\\;. $$\n\nAfterwards we can calculate the double occupancy $\\langle n_{\\uparrow}n_{\\downarrow}\\rangle = \\frac{U_{\\mathrm{sp}}}{4U} n^2$ and then perform a similar root finding for $U_{\\mathrm{ch}}$ from the function \n\n$$ g(U_{\\mathrm{ch}}) = 2 \\frac{T}{N_{\\boldsymbol{k}}} \\sum_{m,\\boldsymbol{q}} \\chi_{\\mathrm{ch}}(U_{\\mathrm{ch}}) - n - 2\\langle n_{\\uparrow}n_{\\downarrow}\\rangle^2 + n^2\\;. $$\n\nIn TPSC, a self-energy $\\Sigma$ can be derived that is calculated from the interaction {cite:p}`Moukouri2000`\n\n$$ V(i\\nu_m, \\boldsymbol{q}) = \\frac{U}{4} \\left(3 U_{\\mathrm{sp}} \\chi_{\\mathrm{sp}}(i\\nu_m, \\boldsymbol{q}) + U_{\\mathrm{ch}} \\chi_{\\mathrm{ch}}(i\\nu_m, \\boldsymbol{q})\\right) + U\\;. $$\n\nThe self-energy itself is given by a convolution in $(i\\omega_n, \\boldsymbol{k})$ space\n\n$$ \\Sigma(i\\omega_n, \\boldsymbol{k}) = \\frac{T}{N_{\\boldsymbol{k}}} \\sum_{m,\\boldsymbol{q}} V(i\\nu_m, \\boldsymbol{q}) G(i\\omega_n - i\\nu_m, \\boldsymbol{k} - \\boldsymbol{q}) $$\n\nwhich Fourier transformed to $(\\tau,\\boldsymbol{r})$ space takes the form\n\n$$ \\Sigma(\\tau, \\boldsymbol{r}) = V(\\tau, \\boldsymbol{r}) G(\\tau, \\boldsymbol{r})\\;. $$ \n\nThe interacting Green function is determined by the Dyson equation\n\n$$\n\\begin{align}\nG(i\\omega_n,\\boldsymbol{k}) &= [G_0^{-1}(i\\omega_n,\\boldsymbol{k}) - \\Sigma(i\\omega_n,\\boldsymbol{k})]^{-1} \\\\\n& = [i\\omega_n - (\\varepsilon_{\\boldsymbol{k}}-\\mu) - \\Sigma(i\\omega_n,\\boldsymbol{k})]^{-1}.\n\\end{align}\n$$\n\n\n\n#### Notes on practical implementation\nWhen implementing the TPSC, a few points need to be treated carefully which we adress in the following:\n\n* The constant Hartree term $V_{\\mathrm{H}} = U$ in the interaction $V$ and respective self-energy term $\\Sigma_H = U n_{\\sigma} = U\\frac{n}{2}$ can be absorbed into the definition of the chemical potential $\\mu$. Otherwise we would have to treat them separately.\n* An upper bound for the renormalized spin vertex $U_{\\mathrm{sp}}$ exists. Since the denominator spin susceptibility $\\chi_{\\mathrm{sp}}$ should not diverge, the upper bound is given by the RPA critical interaction value $U_{\\mathrm{crit}} = 1/\\mathrm{max}\\{\\chi^0\\}$. Mathematically, the function $f(U_{\\mathrm{sp}}) = 2\\sum \\chi_{\\mathrm{sp}}(U_{\\mathrm{sp}}) - n + \\frac{U_{\\mathrm{sp}}}{2U}n^2$, from which $U_{\\mathrm{sp}}$ is determined, turns unstable for $U_{\\mathrm{sp}} \\geq U_{\\mathrm{crit}}$ (try plotting $f(U_{\\mathrm{sp}})$!). At this point, TPSC is not applicable and, e.g., the temperature $T$ is too low or the (unrenormalized) interaction $U$ too large.\n* An internal accuracy check $\\frac{1}{2}\\mathrm{Tr}(\\Sigma G) = U \\langle n_{\\uparrow} n_{\\downarrow}\\rangle$ can be employed to test the validity of TPSC (not done here).\n\n## Code implementation\nWe are implementing TPSC for the simple case of a square lattice model with dispersion $\\varepsilon_{\\boldsymbol{k}} = -2t\\,[\\cos(k_x) + \\cos(k_y)]$ with nearest-neighbor hopping $t$ which sets the energy scale of our system (bandwidth $W = 8t$). First, we load all necessary basic modules that we are going to need in implementing TPSC and visualizing results:\n\n\n```\nimport numpy as np\nimport scipy as sc\nimport scipy.optimize\nfrom warnings import warn\nimport sparse_ir\n%matplotlib inline\nimport matplotlib.pyplot as plt\n```\n\n#### Parameter setting\n\n\n```\n### System parameters\nt = 1 # hopping amplitude\nW = 8*t # bandwidth\nwmax = 10 # set wmax >= W\n\nT = 0.1 # temperature\nbeta = 1/T # inverse temperature\nn = 0.85 # electron filling, here per spin per lattice site (n=1: half filling)\nU = 4 # Hubbard interaction\n\n### Numerical parameters\nnk1, nk2 = 24, 24 # number of k_points along one repiprocal crystal lattice direction k1 = kx, k2 = ky\nnk = nk1*nk2\nIR_tol = 1e-10 # accuary for l-cutoff of IR basis functions\n```\n\n#### Generating meshes\nWe need to generate a $\\boldsymbol{k}$-mesh as well as set up the IR basis functions on a sparse $\\tau$ and $i\\omega_n$ grid. Then we can calculate the dispersion on this mesh.\nIn addition, we set calculation routines to Fourier transform $k\\leftrightarrow r$ and $\\tau\\leftrightarrow i\\omega_n$ (via IR basis).\n\n\n```\n#### Initiate fermionic and bosonic IR basis objects\nIR_basis_set = sparse_ir.FiniteTempBasisSet(beta, wmax, eps=IR_tol)\n\nclass Mesh:\n \"\"\"\n Holding class for k-mesh and sparsely sampled imaginary time 'tau' / Matsubara frequency 'iw_n' grids.\n Additionally it defines the Fourier transform routines 'r <-> k' and 'tau <-> l <-> wn'.\n \"\"\"\n def __init__(self,IR_basis_set,nk1,nk2):\n self.IR_basis_set = IR_basis_set\n\n # generate k-mesh and dispersion\n self.nk1, self.nk2, self.nk = nk1, nk2, nk1*nk2\n self.k1, self.k2 = np.meshgrid(np.arange(self.nk1)/self.nk1, np.arange(self.nk2)/self.nk2)\n self.ek = -2*t*( np.cos(2*np.pi*self.k1) + np.cos(2*np.pi*self.k2) ).reshape(nk)\n\n # lowest Matsubara frequency index\n self.iw0_f = np.where(self.IR_basis_set.wn_f == 1)[0][0]\n self.iw0_b = np.where(self.IR_basis_set.wn_b == 0)[0][0]\n\n ### Generate a frequency-momentum grid for iw_n and ek (in preparation for calculating the Green function)\n # frequency mesh (for Green function)\n self.iwn_f = 1j * self.IR_basis_set.wn_f * np.pi * T\n self.iwn_f_ = np.tensordot(self.iwn_f, np.ones(nk), axes=0)\n\n # ek mesh\n self.ek_ = np.tensordot(np.ones(len(self.iwn_f)), self.ek, axes=0)\n\n def smpl_obj(self, statistics):\n \"\"\" Return sampling object for given statistic \"\"\"\n smpl_tau = {'F': self.IR_basis_set.smpl_tau_f, 'B': self.IR_basis_set.smpl_tau_b}[statistics]\n smpl_wn = {'F': self.IR_basis_set.smpl_wn_f, 'B': self.IR_basis_set.smpl_wn_b }[statistics]\n return smpl_tau, smpl_wn\n\n \n def tau_to_wn(self, statistics, obj_tau):\n \"\"\" Fourier transform from tau to iw_n via IR basis \"\"\"\n smpl_tau, smpl_wn = self.smpl_obj(statistics)\n\n obj_tau = obj_tau.reshape((smpl_tau.tau.size, self.nk1, self.nk2))\n obj_l = smpl_tau.fit(obj_tau, axis=0)\n obj_wn = smpl_wn.evaluate(obj_l, axis=0).reshape((smpl_wn.wn.size, self.nk))\n return obj_wn\n\n def wn_to_tau(self, statistics, obj_wn):\n \"\"\" Fourier transform from tau to iw_n via IR basis \"\"\"\n smpl_tau, smpl_wn = self.smpl_obj(statistics)\n\n obj_wn = obj_wn.reshape((smpl_wn.wn.size, self.nk1, self.nk2))\n obj_l = smpl_wn.fit(obj_wn, axis=0)\n obj_tau = smpl_tau.evaluate(obj_l, axis=0).reshape((smpl_tau.tau.size, self.nk))\n return obj_tau\n\n \n def k_to_r(self,obj_k):\n \"\"\" Fourier transform from k-space to real space \"\"\"\n obj_k = obj_k.reshape(-1, self.nk1, self.nk2)\n obj_r = np.fft.fftn(obj_k,axes=(1,2))\n obj_r = obj_r.reshape(-1, self.nk)\n return obj_r\n\n def r_to_k(self,obj_r):\n \"\"\" Fourier transform from real space to k-space \"\"\"\n obj_r = obj_r.reshape(-1, self.nk1, self.nk2)\n obj_k = np.fft.ifftn(obj_r,axes=(1,2))/self.nk\n obj_k = obj_k.reshape(-1, self.nk)\n return obj_k\n```\n\n#### TPSC solver\nWe wrap the calculation steps of TPSC (i.e. determining $U_{\\mathrm{sp}},U_{\\mathrm{ch}}$) in the `TPSCSolver` class. We use the `Mesh` class defined above to perform calculation steps.\n\n\n```\nclass TPSCSolver:\n def __init__(self, mesh, U, n, U_sfc_tol=1e-12, verbose=True):\n \"\"\"\n Solver class to calculate the TPSC method.\n After initializing the Solver by `solver = TPSCSolver(mesh, U, n, **kwargs)` it \n can be run by `solver.solve()`.\n \"\"\"\n ## set internal parameters for the solve \n self.U = U\n self.n = n\n self.mesh = mesh\n self.U_sfc_tol = U_sfc_tol\n self.verbose = verbose\n \n ## set initial Green function and irreducible susceptibility\n # NOT running the TPSCSolver.solve instance corresponds to staying on RPA level\n self.sigma = 0\n \n self.mu = 0\n self.mu_calc()\n \n self.gkio_calc(self.mu)\n self.grit_calc()\n self.ckio_calc()\n \n # determine critical U_crit = 1/max(chi0) as an upper bound to U_sp\n self.U_crit = 1/np.amax(self.ckio.real)\n \n \n #%%%%%%%%%%% Solving instance\n def solve(self):\n \"\"\"\n Determine spin and charge vertex self-consistently from sum rules and calculate self-energy.\n \"\"\"\n # determine spin vertex U_sp\n self.spin_vertex_calc()\n \n # set double occupancy from Kanamori-Bruckner screening\n self.docc_calc()\n \n # determine charge vertex U_ch\n self.charge_vertex_calc()\n \n # set spin and charge susceptibility\n self.chi_spin = self.RPA_term_calc( self.U_sp)\n self.chi_charge = self.RPA_term_calc(-self.U_ch)\n\n # calculate interaction, self-energy and interacting Green function\n self.V_calc()\n self.sigma_calc()\n self.mu_calc()\n self.gkio_calc(self.mu)\n \n #%%%%%%%%%%% Calculation steps for self.energy\n def gkio_calc(self, mu):\n \"\"\" Calculate Green function G(iw,k) \"\"\"\n self.gkio = (self.mesh.iwn_f_ - (self.mesh.ek_ - mu) - self.sigma)**(-1)\n\n def grit_calc(self):\n \"\"\" Calculate real space Green function G(tau,r) [for calculating chi0 and sigma] \"\"\"\n # Fourier transform\n grit = self.mesh.k_to_r(self.gkio)\n self.grit = self.mesh.wn_to_tau('F', grit)\n\n def ckio_calc(self):\n \"\"\" Calculate irreducible susciptibility chi0(iv,q) \"\"\"\n ckio = self.grit * self.grit[::-1, :]\n\n # Fourier transform\n ckio = self.mesh.r_to_k(ckio)\n self.ckio = self.mesh.tau_to_wn('B', ckio)\n\n def V_calc(self):\n \"\"\" Calculate interaction V(tau,r) from RPA-like spin and charge susceptibility \"\"\"\n V = self.U/4 * (3*self.U_sp*self.chi_spin + self.U_ch*self.chi_charge)\n # Constant Hartree Term V ~ U needs to be treated extra, since they cannot be modeled by the IR basis.\n # In the single-band case, the Hartree term can be absorbed into the chemical potential.\n\n # Fourier transform\n V = self.mesh.k_to_r(V)\n self.V = self.mesh.wn_to_tau('B', V)\n\n def sigma_calc(self):\n \"\"\" Calculate self-energy Sigma(iwn,k) \"\"\"\n sigma = self.V * self.grit\n \n # Fourier transform\n sigma = self.mesh.r_to_k(sigma)\n self.sigma = self.mesh.tau_to_wn('F', sigma)\n\n\n #%%%%%%%%%%% Determining spin and charge vertex\n def RPA_term_calc(self, U):\n \"\"\" Set RPA-like susceptibility \"\"\"\n chi_RPA = self.ckio / (1 - U*self.ckio)\n return chi_RPA \n \n def chi_qtrace_calc(self, U):\n \"\"\" Calculate (iv_m, q) trace of chi_RPA term \"\"\"\n # chi_qtrace = sum_(m,q) chi(iv_m,q)\n chi_RPA = self.RPA_term_calc(U)\n chi_trace = np.sum(chi_RPA, axis=1)/self.mesh.nk\n chi_trace_l = self.mesh.IR_basis_set.smpl_wn_b.fit(chi_trace)\n chi_trace = self.mesh.IR_basis_set.basis_b.u(0)@chi_trace_l\n return chi_trace.real\n \n def docc_calc(self):\n \"\"\" Calculate double occupancy from Kanamori-Bruckner type screening \"\"\"\n self.docc = 0.25 * self.U_sp/self.U * self.n**2\n \n def spin_vertex_calc(self):\n \"\"\" Determine self-consistently from sum rule \"\"\"\n # interval [U_a, U_b] for root finding\n U_a = 0\n U_b = np.floor(self.U_crit*100)/100\n \n chi_trace = self.chi_qtrace_calc\n sfc_eq = lambda U_sp : 2*chi_trace(U_sp) - self.n + 0.5*(U_sp/self.U)*self.n**2\n\n if sfc_eq(U_b) > 0: \n self.U_sp = sc.optimize.brentq(sfc_eq, U_a, U_b, rtol = self.U_sfc_tol)\n else:\n warn(\"System underwent phase transition, U^sp > U_crit = {}! U is too large or T too low for given doping.\".format(self.U_crit))\n \n def charge_vertex_calc(self):\n \"\"\" Determine self-consistently from sum rule \"\"\"\n # interval [U_a, U_b] for root finding\n U_a = 0\n U_b = 100\n \n chi_trace = self.chi_qtrace_calc\n sfc_eq = lambda U_ch : 2*chi_trace(-U_ch) - self.n + (1 - 2*self.docc)*self.n**2\n\n self.U_ch = sc.optimize.brentq(sfc_eq, U_a, U_b, rtol = self.U_sfc_tol)\n\n \n #%%%%%%%%%%% Setting chemical potential mu\n def calc_electron_density(self, mu):\n \"\"\" Calculate chemical potential mu from Green function \"\"\"\n self.gkio_calc(mu)\n gio = np.sum(self.gkio,axis=1)/self.mesh.nk\n g_l = self.mesh.IR_basis_set.smpl_wn_f.fit(gio)\n g_tau0 = self.mesh.IR_basis_set.basis_f.u(0)@g_l\n \n n = 1 + np.real(g_tau0)\n n = 2*n #for spin\n return n\n\n def mu_calc(self):\n \"\"\" Find chemical potential for a given filling n0 via brentq root finding algorithm \"\"\"\n n_calc = self.calc_electron_density\n n0 = self.n\n f = lambda mu : n_calc(mu) - n0\n\n self.mu = sc.optimize.brentq(f, np.amax(self.mesh.ek)*3, np.amin(self.mesh.ek)*3)\n```\n\n### Execute TPSC solver\n\n\n```\n# initialize calculation\nIR_basis_set = sparse_ir.FiniteTempBasisSet(beta, wmax, eps=IR_tol)\nmesh = Mesh(IR_basis_set, nk1, nk2)\nsolver = TPSCSolver(mesh, U, n)\n\n# perform TPSC calculations\nsolver.solve()\n```\n\n#### Visualize results\n\n\n```\n# plot 2D k-dependence of lowest Matsubara frequency of e.g. green function\nplt.pcolormesh(2*mesh.k1.reshape(nk1,nk2), 2*mesh.k2.reshape(nk1,nk2), np.real(solver.gkio[mesh.iw0_f].reshape(mesh.nk1,mesh.nk2)), shading='auto')\nax = plt.gca()\nax.set_xlabel('$k_x/\\pi$')\nax.set_xlim([0,2])\nax.set_ylabel('$k_y/\\pi$')\nax.set_ylim([0,2])\nax.set_aspect('equal')\nax.set_title('Re $G(k,i\\omega_0)$')\nplt.colorbar()\nplt.show()\n```\n\n\n```\n# plot 2D k-dependence of lowest Matsubara frequency of e.g. self-energy\nplt.pcolormesh(2*mesh.k1.reshape(nk1,nk2), 2*mesh.k2.reshape(nk1,nk2), np.imag(solver.sigma[mesh.iw0_f].reshape(mesh.nk1,mesh.nk2)), shading='auto')\nax = plt.gca()\nax.set_xlabel('$k_x/\\pi$')\nax.set_xlim([0,2])\nax.set_ylabel('$k_y/\\pi$')\nax.set_ylim([0,2])\nax.set_aspect('equal')\nax.set_title('Im $\\Sigma(k,i\\omega_0)$')\nplt.colorbar()\nplt.show()\n```\n\n\n```\n# plot 2D k-dependence of lowest Matsubara frequency of e.g. chi_spin\nplt.pcolormesh(2*mesh.k1.reshape(nk1,nk2), 2*mesh.k2.reshape(nk1,nk2), np.real(solver.chi_spin[mesh.iw0_b].reshape(mesh.nk1,mesh.nk2)), shading='auto')\nax = plt.gca()\nax.set_xlabel('$k_x/\\pi$')\nax.set_xlim([0,2])\nax.set_ylabel('$k_y/\\pi$')\nax.set_ylim([0,2])\nax.set_aspect('equal')\nax.set_title('$\\chi_{\\mathrm{sp}}(k,i\\nu_0)$')\nplt.colorbar()\nplt.show()\n```\n\n## Example: Interaction dependent renormalization\nAs a simple example demonstration of our `sparse-ir` TPSC code developed above, we will reproduce Fig. 2 of {cite:p}`Vilk1997`. It shows the $U$ dependence of renormalized/effective spin and charge interactions $U_{\\mathrm{sp}}$ and $U_{\\mathrm{ch}}$ (irreducible vertices) at half filling $n=1$ and $T>T_{\\mathrm{crit}}$ for all considered $U$ (i.e. $U_{\\mathrm{sp}}= W\n\n# numerical parameters\nnk1, nk2 = 24, 24 # k-mesh sufficiently dense!\nnk = nk1*nk2\nIR_tol = 1e-8 # accuary for l-cutoff of IR basis functions\n\n\n# initialize meshes\nIR_basis_set = sparse_ir.FiniteTempBasisSet(beta, wmax, eps=IR_tol)\nmesh = Mesh(IR_basis_set, nk1, nk2)\n\n# set initial self_energy - will be set to previous calculation step afterwards\nsigma_init = 0\n\n# empty arrays for results later\nU_sp_array = np.empty((len(U_array)))\nU_ch_array = np.empty((len(U_array)))\n\n\n#%%%%%%%%%%%%%%% Calculations for different U values\nprint(\"Start TPSC loop...\")\nfor U_it, U in enumerate(U_array):\n #print(\"Now: U = {:.1f}\".format(U))\n \n # TPSC solver\n solver = TPSCSolver(mesh, U, n, verbose=False)\n solver.solve()\n \n # save data for plotting\n U_sp_array[U_it] = solver.U_sp\n U_ch_array[U_it] = solver.U_ch\nprint(\"Finished. Plotting now.\")\n\n\n#%%%%%%%%%%%%%%%% Plot results\nplt.plot(U_array, U_ch_array, '-', label='$U_{\\mathrm{ch}}$')\nplt.plot(U_array, U_sp_array, '-', label='$U_{\\mathrm{sp}}$')\nax = plt.gca()\nax.set_xlabel('$U$', fontsize=12)\nax.set_xlim([0,5])\nax.set_ylabel('Interaction', fontsize=12)\nax.set_ylim([0,20])\nax.legend(frameon=False, fontsize=12)\n```\n\n\n```\n\n```\n", "meta": {"hexsha": "5854af4ab1e5d6d1752697c4822bfb78e791641d", "size": 29486, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "src/TPSC_py.ipynb", "max_stars_repo_name": "SpM-lab/sparse-ir-tutorial", "max_stars_repo_head_hexsha": "346f0f03254746b8fd000351b4a4296bd7890e20", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2022-03-26T06:47:35.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-28T10:21:49.000Z", "max_issues_repo_path": "src/TPSC_py.ipynb", "max_issues_repo_name": "SpM-lab/sparse-ir-tutorial", "max_issues_repo_head_hexsha": "346f0f03254746b8fd000351b4a4296bd7890e20", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/TPSC_py.ipynb", "max_forks_repo_name": "SpM-lab/sparse-ir-tutorial", "max_forks_repo_head_hexsha": "346f0f03254746b8fd000351b4a4296bd7890e20", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 48.8178807947, "max_line_length": 813, "alphanum_fraction": 0.5681340297, "converted": true, "num_tokens": 6792, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.6076631698328917, "lm_q1q2_score": 0.4347368130522987}} {"text": "+ This notebook is part of lecture 17 *Orthogonal matrices and Gram-Scmidt* in the OCW MIT course 18.06 by Prof Gilbert Strang [1]\n+ Created by me, Dr Juan H Klopper\n + Head of Acute Care Surgery\n + Groote Schuur Hospital\n + University Cape Town\n + Email me with your thoughts, comments, suggestions and corrections \n
    Linear Algebra OCW MIT18.06 IPython notebook [2] study notes by Dr Juan H Klopper is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.\n\n+ [1] OCW MIT 18.06\n+ [2] Fernando P\u00e9rez, Brian E. Granger, IPython: A System for Interactive Scientific Computing, Computing in Science and Engineering, vol. 9, no. 3, pp. 21-29, May/June 2007, doi:10.1109/MCSE.2007.53. URL: http://ipython.org\n\n\n```python\nfrom IPython.core.display import HTML, Image\ncss_file = 'style.css'\nHTML(open(css_file, 'r').read())\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n```python\nfrom sympy import init_printing, symbols, Matrix, sin, cos, sqrt, Rational, GramSchmidt\nfrom warnings import filterwarnings\n```\n\n\n```python\ninit_printing(use_latex = 'mathjax')\nfilterwarnings('ignore')\n```\n\n\n```python\ntheta = symbols('theta')\n```\n\n# Orthogonal basis\n# Orthogonal matrix\n# Gram-Schmidt\n\n## Orthogonal basis\n\n* Here we mean vectors *q*1,*q*2,...,*q*n\n\n* We actually mean orthonormal vectors (for orthogonal or perpendicular and of unit length / normalized)\n* Vectors that are orthogonal have a dot product equal to zero\n * If they are orthogonal\n$$ {q}_{i}^{T}{q}_{j}={0} $$\n * If they are not\n$$ {q}_{i}^{T}{q}_{j}\\neq{0} $$\n\n## Orthogonal matrix\n\n* We can now put these (column) basis vectors into a matrix Q\n\n* This brings about\n$$ {Q}^{T}{Q}={I} $$\n\n* In the case of the matrix Q being square the word *orthogonal matrix* is used\n* When it is square we can calculate the inverse making\n$$ {Q}^{T}={Q}^{-1} $$\n\n* Consider the following permutation matrix with orthonormal column vectors\n\n\n```python\nQ = Matrix([[0, 0, 1], [1, 0, 0], [0, 1, 0]])\nQ, Q.transpose()\n```\n\n\n\n\n$$\\begin{pmatrix}\\left[\\begin{matrix}0 & 0 & 1\\\\1 & 0 & 0\\\\0 & 1 & 0\\end{matrix}\\right], & \\left[\\begin{matrix}0 & 1 & 0\\\\0 & 0 & 1\\\\1 & 0 & 0\\end{matrix}\\right]\\end{pmatrix}$$\n\n\n\n* In this example the transpose also contains orthonormal column vectors\n* Multiplication gives the identity matrix\n\n\n```python\nQ.transpose() * Q\n```\n\n\n\n\n$$\\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$\n\n\n\n* Consider this example\n\n\n```python\nQ = Matrix([[cos(theta), -sin(theta)], [sin(theta), cos(theta)]])\nQ, Q.transpose()\n```\n\n\n\n\n$$\\begin{pmatrix}\\left[\\begin{matrix}\\cos{\\left (\\theta \\right )} & - \\sin{\\left (\\theta \\right )}\\\\\\sin{\\left (\\theta \\right )} & \\cos{\\left (\\theta \\right )}\\end{matrix}\\right], & \\left[\\begin{matrix}\\cos{\\left (\\theta \\right )} & \\sin{\\left (\\theta \\right )}\\\\- \\sin{\\left (\\theta \\right )} & \\cos{\\left (\\theta \\right )}\\end{matrix}\\right]\\end{pmatrix}$$\n\n\n\n* The two column vectors are orthogonal and the length of each column vector is 1\n* It is thus an orthogonal matrix\n\n\n```python\nQ.transpose() * Q\n```\n\n\n\n\n$$\\left[\\begin{matrix}\\sin^{2}{\\left (\\theta \\right )} + \\cos^{2}{\\left (\\theta \\right )} & 0\\\\0 & \\sin^{2}{\\left (\\theta \\right )} + \\cos^{2}{\\left (\\theta \\right )}\\end{matrix}\\right]$$\n\n\n\n* The example below certainly has orthogonal column vectors, but they are not of unit length\n$$ {Q}=\\begin{bmatrix} 1 & 1 \\\\ 1 & {-1} \\end{bmatrix} $$\n\n* Well, we can change them into unit vectors by dividing each component by the length of that vector\n$$ \\sqrt { { \\left( 1 \\right) }^{ 2 }+{ \\left( 1 \\right) }^{ 2 } } =\\sqrt { 2 } \\\\ \\sqrt { { \\left( 1 \\right) }^{ 2 }+{ \\left( -1 \\right) }^{ 2 } } =\\sqrt { 2 } $$\n$$ Q=\\frac { 1 }{ \\sqrt { 2 } } \\begin{bmatrix} 1 & 1 \\\\ 1 & -1 \\end{bmatrix} $$\n\n* As it stands QTQ is not the identity matrix\n\n\n```python\nQ = Matrix([[1, 1], [1, -1]])\nQ.transpose() * Q\n```\n\n\n\n\n$$\\left[\\begin{matrix}2 & 0\\\\0 & 2\\end{matrix}\\right]$$\n\n\n\n* Turning it into an orthogonal matrix\n\n\n```python\nQ = (1 / sqrt(2)) * Matrix([[1, 1], [1, -1]])\nQ \n```\n\n\n\n\n$$\\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$$\n\n\n\n\n```python\nQ.transpose() * Q\n```\n\n\n\n\n$$\\left[\\begin{matrix}1 & 0\\\\0 & 1\\end{matrix}\\right]$$\n\n\n\n* Consider this example with orthogonal (but not orthonormal) column vectors\n\n\n```python\nQ = Matrix([[1, 1, 1, 1], [1, -1, 1, -1], [1, 1, -1, -1], [1, -1, -1, 1]])\nQ\n```\n\n\n\n\n$$\\left[\\begin{matrix}1 & 1 & 1 & 1\\\\1 & -1 & 1 & -1\\\\1 & 1 & -1 & -1\\\\1 & -1 & -1 & 1\\end{matrix}\\right]$$\n\n\n\n* Again, as it stands QTQ is not the identity matrix\n\n\n```python\nQ.transpose() * Q\n```\n\n\n\n\n$$\\left[\\begin{matrix}4 & 0 & 0 & 0\\\\0 & 4 & 0 & 0\\\\0 & 0 & 4 & 0\\\\0 & 0 & 0 & 4\\end{matrix}\\right]$$\n\n\n\n* But turning it into an orthogonal matrix works\n\n\n```python\nQ = Rational(1, 2) * Matrix([[1, 1, 1, 1], [1, -1, 1, -1], [1, 1, -1, -1], [1, -1, -1, 1]])\n# Rational() creates a mathematical fraction instead of a decimal\nQ\n```\n\n\n\n\n$$\\left[\\begin{matrix}\\frac{1}{2} & \\frac{1}{2} & \\frac{1}{2} & \\frac{1}{2}\\\\\\frac{1}{2} & - \\frac{1}{2} & \\frac{1}{2} & - \\frac{1}{2}\\\\\\frac{1}{2} & \\frac{1}{2} & - \\frac{1}{2} & - \\frac{1}{2}\\\\\\frac{1}{2} & - \\frac{1}{2} & - \\frac{1}{2} & \\frac{1}{2}\\end{matrix}\\right]$$\n\n\n\n\n```python\nQ.transpose() * Q\n```\n\n\n\n\n$$\\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\0 & 0 & 1 & 0\\\\0 & 0 & 0 & 1\\end{matrix}\\right]$$\n\n\n\n* Consider this matrix Q with orthogonal column vectors, but that is not square\n\n\n```python\nQ = Rational(1, 3) * Matrix([[1, -2], [2, -1], [2, 2]])\nQ\n```\n\n\n\n\n$$\\left[\\begin{matrix}\\frac{1}{3} & - \\frac{2}{3}\\\\\\frac{2}{3} & - \\frac{1}{3}\\\\\\frac{2}{3} & \\frac{2}{3}\\end{matrix}\\right]$$\n\n\n\n* We now have a matrix with two column vectors that are normalized and orthogonal to each other and they form a basis for a plane (subspace) in ℝ3\n\n* There must be a third column matrix of unit length, orthogonal to the other two so we end up with an orthogonal matrix \n\n\n```python\nQ = Rational(1, 3) * Matrix([[1, -2, 2], [2, -1, -2], [2, 2, 1]])\nQ\n```\n\n\n\n\n$$\\left[\\begin{matrix}\\frac{1}{3} & - \\frac{2}{3} & \\frac{2}{3}\\\\\\frac{2}{3} & - \\frac{1}{3} & - \\frac{2}{3}\\\\\\frac{2}{3} & \\frac{2}{3} & \\frac{1}{3}\\end{matrix}\\right]$$\n\n\n\n\n```python\nQ.transpose() * Q\n```\n\n\n\n\n$$\\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$\n\n\n\n* Let's make use of these matrices with orthonormal columns (which we will always denote with a letter Q) and project them onto their columnspace\n* What would the projection matrix be?\n$$ Q\\underline{x}={b} \\\\ {P}={Q}{\\left({Q}^{T}{Q}\\right)}^{-1}{Q}^{T} $$\n\n* Remember, though that for matrices with orthonormal column vectors we have QTQ is the identity matrix and we have\n$$ {P}={Q}{Q}^{T} $$\n\n* If additionally, Q is square, then we have independent columns and the columnspace contain the whole space ℝn and the projection matrix is the identity matrix in *n*\n * Remember QT = Q-1 in these cases making it easy to see that we get the identity matrix\n * Remember also that the projection matrix is symmetric\n * Lastly the projection matrix has the property of squaring it leaves us in the same spot, so here we will have (QQT)2=QQT\n\n* All of this has the final consequence that\n$$ {Q}^{T}Q\\hat{x}={Q}^{T}\\underline{b} \\\\ \\hat{x}={Q}^{T}\\underline{b} \\\\ \\hat{x}_{i}={q}_{i}^{T}{b} $$\n\n## Gram-Schmidt\n\n* All of the above makes things quite easy, so we should try and create orthogonal matrices\n\n* Good, let's start with two independent vectors **a** and **b** and try and create two orthogonal vectors **A** and **B** and then create two orthonormal vectors\n$$ { q }_{ 1 }=\\frac { A }{ \\left\\| A \\right\\| } \\\\ { q }_{ 2 }=\\frac { B }{ \\left\\| B \\right\\| } $$\n\n* We can choose one of them as our initial vector, say **a** = **A**, so we have to get an orthogonal projection (to **a**) for **B**\n* This is what we previously called the error vector **e**\n$$ \\underline{e}=\\underline{b}-\\underline{p} $$\n* Remembering how to get **p** we have the following\n$$ B=\\underline { b } -\\frac { { A }^{ T }\\underline { b } }{ { A }^{ T }A } A $$\n\n* Let's do an example\n\n\n```python\na = Matrix([1, 1, 1])\nb = Matrix([1, 0, 2])\na, b\n```\n\n\n\n\n$$\\begin{pmatrix}\\left[\\begin{matrix}1\\\\1\\\\1\\end{matrix}\\right], & \\left[\\begin{matrix}1\\\\0\\\\2\\end{matrix}\\right]\\end{pmatrix}$$\n\n\n\n\n```python\nA = a\nA\n```\n\n\n\n\n$$\\left[\\begin{matrix}1\\\\1\\\\1\\end{matrix}\\right]$$\n\n\n\n\n```python\nA.transpose() * b\n```\n\n\n\n\n$$\\left[\\begin{matrix}3\\end{matrix}\\right]$$\n\n\n\n\n```python\nA.transpose() * A\n```\n\n\n\n\n$$\\left[\\begin{matrix}3\\end{matrix}\\right]$$\n\n\n\n\n```python\nB = b - A\nB\n```\n\n\n\n\n$$\\left[\\begin{matrix}0\\\\-1\\\\1\\end{matrix}\\right]$$\n\n\n\n* Checking that they are perpendicular\n\n\n```python\nA.transpose() * B\n```\n\n\n\n\n$$\\left[\\begin{matrix}0\\end{matrix}\\right]$$\n\n\n\n* Now we have to create Q by turning **A** and **B** into unit vectors and place them in the same matrix\n\n\n```python\nA.normalized() # Easy way no normalize a matrix\n```\n\n\n\n\n$$\\left[\\begin{matrix}\\frac{\\sqrt{3}}{3}\\\\\\frac{\\sqrt{3}}{3}\\\\\\frac{\\sqrt{3}}{3}\\end{matrix}\\right]$$\n\n\n\n\n```python\nB.normalized()\n```\n\n\n\n\n$$\\left[\\begin{matrix}0\\\\- \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$$\n\n\n\n\n```python\nQ = Matrix([[sqrt(3) / 3, 0], [sqrt(3) / 3, -sqrt(2) / 2], [sqrt(3) / 3, sqrt(2) / 2]])\nQ\n```\n\n\n\n\n$$\\left[\\begin{matrix}\\frac{\\sqrt{3}}{3} & 0\\\\\\frac{\\sqrt{3}}{3} & - \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{3}}{3} & \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$$\n\n\n\n* The columnspace of the original matrix (of two column vectors) and Q are the same\n\n* In python™ we can use the following code\n\n\n```python\n# The column matrices (independant orthogonal column vectors) are entered indivisually inside square bracket []\nA = [Matrix([1, 1, 1]), Matrix([1, 0, 2])]\nA\n```\n\n\n\n\n$$\\begin{bmatrix}\\left[\\begin{matrix}1\\\\1\\\\1\\end{matrix}\\right], & \\left[\\begin{matrix}1\\\\0\\\\2\\end{matrix}\\right]\\end{bmatrix}$$\n\n\n\n\n```python\nQ = GramSchmidt(A, True) # The True argument normalizes the columns\nQ\n```\n\n\n\n\n$$\\begin{bmatrix}\\left[\\begin{matrix}\\frac{\\sqrt{3}}{3}\\\\\\frac{\\sqrt{3}}{3}\\\\\\frac{\\sqrt{3}}{3}\\end{matrix}\\right], & \\left[\\begin{matrix}0\\\\- \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2}\\end{matrix}\\right]\\end{bmatrix}$$\n\n\n\n## Example problems\n\n### Example problem 1\n\n* Create an orthogonal matrix from the following matrix\n$$ \\begin{bmatrix} 1 & 2 & 4 \\\\ 0 & 0 & 5 \\\\ 0 & 3 & 6 \\end{bmatrix} $$\n\n#### Solution\n\n\n```python\nA = [Matrix([1, 0, 0]), Matrix([2, 0, 3]), Matrix([4, 5, 6])]\nA\n```\n\n\n\n\n$$\\begin{bmatrix}\\left[\\begin{matrix}1\\\\0\\\\0\\end{matrix}\\right], & \\left[\\begin{matrix}2\\\\0\\\\3\\end{matrix}\\right], & \\left[\\begin{matrix}4\\\\5\\\\6\\end{matrix}\\right]\\end{bmatrix}$$\n\n\n\n\n```python\nQ = GramSchmidt(A, True)\nQ\n```\n\n\n\n\n$$\\begin{bmatrix}\\left[\\begin{matrix}1\\\\0\\\\0\\end{matrix}\\right], & \\left[\\begin{matrix}0\\\\0\\\\1\\end{matrix}\\right], & \\left[\\begin{matrix}0\\\\1\\\\0\\end{matrix}\\right]\\end{bmatrix}$$\n\n\n\n* We can also consider QR-factorization\n\n\n```python\nfrom sympy.mpmath import matrix, qr\n```\n\n\n```python\nA = matrix([[1, 2, 4], [0, 0, 5], [0, 3, 6]])\nprint(A)\n```\n\n [1.0 2.0 4.0]\n [0.0 0.0 5.0]\n [0.0 3.0 6.0]\n\n\n\n```python\nQ, R = qr(A)\n```\n\n\n```python\nprint(Q)\n```\n\n [1.0 0.0 0.0]\n [0.0 0.0 -1.0]\n [0.0 -1.0 0.0]\n\n\n\n```python\nprint(R)\n```\n\n [1.0 2.0 4.0]\n [0.0 -3.0 -6.0]\n [0.0 0.0 -5.0]\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "1bc54b24a672273981577d6dfe49995414fc6b97", "size": 34833, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "forks/MIT_OCW_Linear_Algebra_18_06-master/II_04_Orthogonal_matrices_Gram_Schmidt.ipynb", "max_stars_repo_name": "solomonxie/jupyter-notebooks", "max_stars_repo_head_hexsha": "65999f179e037242138de72f512dda4bf00c7379", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-02-13T05:52:05.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-08T09:52:35.000Z", "max_issues_repo_path": "forks/MIT_OCW_Linear_Algebra_18_06-master/II_04_Orthogonal_matrices_Gram_Schmidt.ipynb", "max_issues_repo_name": "solomonxie/jupyter-notebooks", "max_issues_repo_head_hexsha": "65999f179e037242138de72f512dda4bf00c7379", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "forks/MIT_OCW_Linear_Algebra_18_06-master/II_04_Orthogonal_matrices_Gram_Schmidt.ipynb", "max_forks_repo_name": "solomonxie/jupyter-notebooks", "max_forks_repo_head_hexsha": "65999f179e037242138de72f512dda4bf00c7379", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.4957805907, "max_line_length": 708, "alphanum_fraction": 0.435937186, "converted": true, "num_tokens": 4937, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.7154239957834732, "lm_q1q2_score": 0.43473681305229855}} {"text": "```python\n%matplotlib inline\n\nimport os\nimport unicodedata\nimport pandas as pd\n\nimport numpy\nimport sympy\n\nimport matplotlib.pyplot as plt\nimport matplotlib.ticker as ticker\nimport matplotlib.animation as animation\n\nfrom scipy.interpolate import make_interp_spline, BSpline\nfrom scipy.ndimage.filters import gaussian_filter1d\n\nfrom IPython.display import HTML\nfrom IPython import display\n\ns = sympy.symbols\n```\n\n\n```python\ngraphs = []\nlog_path = \"/home/datta/lab/Rethinking-Binarized-Neural-Network-Optimization/hyperparameter_logs/acc/\"\nfiles = [\"pit6g2\",\"pit6g3\",\"pit6g4\"]\nfor file in os.listdir(log_path):\n df = pd.read_json(log_path+file)#+\".json\")\n df = df[[1, 2]]\n df[2] = gaussian_filter1d(df[2], sigma=2)\n \n df['label'] = file.split(\".json\")[0]\n graphs.append(df)\n\n\ndef best_acc(df):\n return df[2].max()\n\nsorted_graphs = sorted(graphs, key = best_acc, reverse=True)\n```\n\n\n```python\n%matplotlib inline\nfig, ax = plt.subplots(1, 1)\n_ = plt.rcParams['figure.figsize'] = (15, 5)\n\nlabels = []\nfor df in sorted_graphs:\n _ = ax.plot(df[1]/909, df[2])\n labels.append(df['label'][0])\n\n# print(df['label'][0])\n# _ = plt.ylim(0.8, 1)\n\n_ = plt.grid(True)\n_ = plt.xlabel(\"Number of Epochs\")\n_ = plt.ylabel(r'$\\pi$'+ \" - Flip Ratio\")\n\n\nlabels = [\n r'$\\gamma = 10^{-2}$',\n r'$\\gamma = 10^{-3}$',\n r'$\\gamma = 10^{-4}$'\n]\n#03 632 53\n# _ = ax.legend(labels, loc=4)\n\n```\n\n\n```python\n# fig.savefig((str(time)+\".png\")\n```\n\n\n\n\n 1573058620.5257473\n\n\n\n\n```python\ndf\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
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    \n
    \n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "82a435adccc6d9243d09160deb904045ffff540b", "size": 105781, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "report/notebooks/parse_cifar_logs.ipynb", "max_stars_repo_name": "roman-bachmann/Rethinking-Binarized-Neural-Network-Optimization", "max_stars_repo_head_hexsha": "2c1dab8b7028eef803d7437b79f12369100becf3", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 12, "max_stars_repo_stars_event_min_datetime": "2019-12-16T11:44:43.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-18T12:41:03.000Z", "max_issues_repo_path": "report/notebooks/parse_cifar_logs.ipynb", "max_issues_repo_name": "roman-bachmann/Rethinking-Binarized-Neural-Network-Optimization", "max_issues_repo_head_hexsha": "2c1dab8b7028eef803d7437b79f12369100becf3", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "report/notebooks/parse_cifar_logs.ipynb", "max_forks_repo_name": "roman-bachmann/Rethinking-Binarized-Neural-Network-Optimization", "max_forks_repo_head_hexsha": "2c1dab8b7028eef803d7437b79f12369100becf3", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2020-01-14T05:53:35.000Z", "max_forks_repo_forks_event_max_datetime": "2020-12-04T07:54:17.000Z", "avg_line_length": 304.8443804035, "max_line_length": 96992, "alphanum_fraction": 0.9179814901, "converted": true, "num_tokens": 1501, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300449389326, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.4347118459253992}} {"text": "# Algorithmic Fairness: Considering Different Definitions\n\nApproximate notebook time: 2 hours \n\n## Introduction\n\nDecision making within the United States criminal justice system relies heavily on risk assessment, which determines the potential risk that a released defendant will fail to appear in court or cause harm to the public. Judges use these assessments to decide if bail can be set or if a defendant should be detained before trial. While this is not new in the legal system, the use of risk scores determined by an algorithm are gaining prevalence and support. Proponents promote the use of risk scores to guide judges in their decision making, arguing that machine learning could lead to greater efficiency, accountability, and less biased decisions compared with human judgment ([Henry](https://theappeal.org/risk-assessment-explained/)). On the other hand, critical voices raise the concern that biases can creep into these algorithms at any point in the process, and that algorithms are often applied to the wrong situations ([Henry](https://theappeal.org/risk-assessment-explained/)). Further, they exacerbate the racism embedded deep within the criminal justice system by perpetuating inequalities found in historical data ([Henry](https://theappeal.org/risk-assessment-explained/)).\n\nIn the debate about the use of risk assessment algorithms, people have used data analysis to determine the extent to which these algorithms are fair to different groups of people. In this homework, **you will explore some of the many definitions and metrics (different ways of operationalizing data to quantify those definitions) of fairness that can be applied to the risk assessment tool COMPAS**. In doing so, you will understand and provide evidence for or against the presence of bias within the algorithm. You will examine the arguments and analyses made by the company that created COMPAS and the critics of this risk assessment tool to gain a deeper understanding of the technical and societal interpretations and implications of fairness. \n\n**NOTE**: When we discuss bias in this module, we define it most generally as prejudice or an inclination in favor of one person, thing, or group compared to another. In the context of machine learning, bias is a \u201cphenomenon that occurs when an algorithm produces results that are systemically prejudiced due to erroneous assumptions in the machine learning process\u201d ([Rouse](https://searchenterpriseai.techtarget.com/definition/machine-learning-bias-algorithm-bias-or-AI-bias#:~:text=Machine%20learning%20bias%2C%20also%20sometimes,in%20the%20machine%20learning%20process)).\n\n## Table of Contents:\n* [Part 0. COMPAS](#part-zero)\n* [Part 1. ProPublica's Perspective](#part-one)\n* [Part 2. Northpointe's Perspective](#part-two)\n* [Part 3. Yet Another Definition of Fairness](#part-three)\n* [Part 4. Conclusion](#part-four)\n\n## Setup\n\nLet's begin by importing the packages we need.\n\n\n```python\nimport numpy as np\nimport pandas as pd\nfrom sklearn.linear_model import LogisticRegression\nfrom sklearn.metrics import roc_curve, roc_auc_score\nimport matplotlib.pyplot as plt\nimport seaborn as sns\n!pip install aif360\n!pip install BlackBoxAuditing\nfrom aif360.algorithms.preprocessing import DisparateImpactRemover\nfrom aif360.datasets import BinaryLabelDataset\n```\n\n Requirement already satisfied: aif360 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (0.3.0)\n Requirement already satisfied: pandas>=0.24.0 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from aif360) (1.0.5)\n Requirement already satisfied: scikit-learn>=0.21 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from aif360) (0.23.1)\n Requirement already satisfied: numpy>=1.16 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from aif360) (1.18.5)\n Requirement already satisfied: scipy>=1.2.0 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from aif360) (1.5.0)\n Requirement already satisfied: matplotlib in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from aif360) (3.2.2)\n Requirement already satisfied: python-dateutil>=2.6.1 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from pandas>=0.24.0->aif360) (2.8.1)\n Requirement already satisfied: pytz>=2017.2 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from pandas>=0.24.0->aif360) (2020.1)\n Requirement already satisfied: threadpoolctl>=2.0.0 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from scikit-learn>=0.21->aif360) (2.1.0)\n Requirement already satisfied: joblib>=0.11 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from scikit-learn>=0.21->aif360) (0.16.0)\n Requirement already satisfied: cycler>=0.10 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from matplotlib->aif360) (0.10.0)\n Requirement already satisfied: kiwisolver>=1.0.1 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from matplotlib->aif360) (1.2.0)\n Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.1 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from matplotlib->aif360) (2.4.7)\n Requirement already satisfied: six>=1.5 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from python-dateutil>=2.6.1->pandas>=0.24.0->aif360) (1.15.0)\n Requirement already satisfied: BlackBoxAuditing in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (0.1.54)\n Requirement already satisfied: numpy in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from BlackBoxAuditing) (1.18.5)\n Requirement already satisfied: pandas in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from BlackBoxAuditing) (1.0.5)\n Requirement already satisfied: networkx in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from BlackBoxAuditing) (2.4)\n Requirement already satisfied: matplotlib in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from BlackBoxAuditing) (3.2.2)\n Requirement already satisfied: python-dateutil>=2.6.1 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from pandas->BlackBoxAuditing) (2.8.1)\n Requirement already satisfied: pytz>=2017.2 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from pandas->BlackBoxAuditing) (2020.1)\n Requirement already satisfied: decorator>=4.3.0 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from networkx->BlackBoxAuditing) (4.4.2)\n Requirement already satisfied: kiwisolver>=1.0.1 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from matplotlib->BlackBoxAuditing) (1.2.0)\n Requirement already satisfied: cycler>=0.10 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from matplotlib->BlackBoxAuditing) (0.10.0)\n Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.1 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from matplotlib->BlackBoxAuditing) (2.4.7)\n Requirement already satisfied: six>=1.5 in /Users/eva/opt/anaconda3/lib/python3.8/site-packages (from python-dateutil>=2.6.1->pandas->BlackBoxAuditing) (1.15.0)\n\n\n WARNING:root:No module named 'numba.decorators': LFR will be unavailable. To install, run:\n pip install 'aif360[LFR]'\n WARNING:root:No module named 'tensorflow': AdversarialDebiasing will be unavailable. To install, run:\n pip install 'aif360[AdversarialDebiasing]'\n\n\n# Part 0. COMPAS: Why it was created and how it exists in the court system \n\nCOMPAS (Correctional Offender Management Profiling for Alternative Sanctions) is a commercial tool produced by the for-profit company Northpointe (acquired by equivant) known as a recidivism risk assessment system. **Tools like COMPAS are used to predict the risk of future crimes for an individual who has entered the US criminal justice system by outputting a risk score from 1-10**. While COMPAS was initially intended to aid decisions made by probation officers on treatment and supervision of those who are incarcerated, Northpointe has since emphasized the scalability of the tool to \u201cfit the needs of many different decision points\u201d including pre-screening assessments, pretrial release decisions (whether or not to hold an arrested individual in jail until their trial), and post-trial next steps for the defendant ([Northpointe](http://www.northpointeinc.com/files/downloads/FAQ_Document.pdf)). These algorithms are believed by many to provide the ability to make the court system more just by removing or correcting for bias of criminal justice officials.\n\n### Question 0a\nExplain 3 parties that are impacted by the COMPAS tool. In what ways are they impacted? (Can you think of impacts beyond those in the courtroom for at least one of your examples?)\n\n*Student Written Answer Here*\n\n### Question 0b\nBased on your initial reading, what is one problem of the criminal justice system that the COMPAS tool could potentially alleviate? What is one potential problem that using the COMPAS algorithm could introduce? \n\n*Student Written Answer Here*\n\n## Dataset Setup\n\nWe will be using the data that was obtained and used by ProPublica in their own analysis of the COMPAS tool from Broward County public records of people who were scored between 2013 and 2014 ([ProPublica](https://www.propublica.org/article/how-we-analyzed-the-compas-recidivism-algorithm)). In order to replicate ProPublica's analysis, we remove any cases where the charge was not within 30 days of the score (ProPublica did this in order to match the COMPAS score with the correct criminal case). We are left with 6172 rows in the dataset.\n\n\n```python\ndata = pd.read_csv('compas-scores-two-years.csv')\ndata = data.query('days_b_screening_arrest <= 30 & days_b_screening_arrest >= -30')\ndata\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    idnamefirstlastcompas_screening_datesexdobageage_catrace...v_decile_scorev_score_textv_screening_datein_custodyout_custodypriors_count.1startendeventtwo_year_recid
    01miguel hernandezmiguelhernandez2013-08-14Male1947-04-1869Greater than 45Other...1Low2013-08-142014-07-072014-07-140032700
    13kevon dixonkevondixon2013-01-27Male1982-01-223425 - 45African-American...1Low2013-01-272013-01-262013-02-050915911
    24ed philoedphilo2013-04-14Male1991-05-1424Less than 25African-American...3Low2013-04-142013-06-162013-06-16406301
    57marsha milesmarshamiles2013-11-30Male1971-08-224425 - 45Other...1Low2013-11-302013-11-302013-12-010185300
    68edward riddleedwardriddle2014-02-19Male1974-07-234125 - 45Caucasian...2Low2014-02-192014-03-312014-04-181454011
    ..................................................................
    720910996steven butlerstevenbutler2013-11-23Male1992-07-1723Less than 25African-American...5Medium2013-11-232013-11-222013-11-240186000
    721010997malcolm simmonsmalcolmsimmons2014-02-01Male1993-03-2523Less than 25African-American...5Medium2014-02-012014-01-312014-02-020179000
    721110999winston gregorywinstongregory2014-01-14Male1958-10-0157Greater than 45Other...1Low2014-01-142014-01-132014-01-140080800
    721211000farrah jeanfarrahjean2014-03-09Female1982-11-173325 - 45African-American...2Low2014-03-092014-03-082014-03-093075400
    721311001florencia sanmartinflorenciasanmartin2014-06-30Female1992-12-1823Less than 25Hispanic...4Low2014-06-302015-03-152015-03-152025801
    \n

    6172 rows \u00d7 53 columns

    \n
    \n\n\n\nWe are also able to filter out any information that was not used by ProPublica and select fields for severity of charge, number of priors, demographics, age, sex, compas scores, and whether each person was accused of a crime within two years.\n\n\n```python\nselect_data = data[[\"age\", \"c_charge_degree\", \"race\", \"age_cat\", \"score_text\", \"sex\", \"priors_count\", \n \"days_b_screening_arrest\", \"decile_score\", \"is_recid\", \"two_year_recid\", \"c_jail_in\", \"c_jail_out\"]]\n```\n\n### Question 0c\nExplore the dataset. What is the granularity of this dataset? \n\n*Student Written Answer Here*\n\n***Sensitive features*** are attributes within a dataset that are given special consideration and treatment for potential legal, social, or ethical reasons. Often, these features are recognized and protected by antidiscrimination or privacy laws. One example of a sensitive feature is age. \n\n### Question 0d\nIdentify 2 sensitive features in the dataset that we have not already mentioned. \n\n*Student Written Answer Here*\n\n### Question 0e \nPick one of the sensitive features you have identified. Identify at least 2 features in the dataset that are proxies for that sensitive feature.\n\n*Student Written Answer Here*\n\n### Question 0f\nAs a data scientist, why is it important to give special consideration to these kinds of features? \n\n*Student Written Answer Here*\n\n# Part 1. ProPublica\u2019s Perspective \n\n### Who is ProPublica?\n\nProPublica is a nonprofit organization that \u201cproduces investigative journalism with moral force\u201d ([ProPublica](https://www.propublica.org/about/)). ProPublica was founded as a nonpartisan newsroom aiming to expose and question abuses of power, justice, and public trust, often by systems and institutions deeply ingrained in the US.\n\nIn 2016, ProPublica investigated the COMPAS algorithm to assess the accuracy of and potential racial bias within the tool, as it became more popular within the United States court system nationwide. In their analysis, ProPublica used data from defendants with risk scores from Broward County, FL from 2013 to 2014 to test for statistical differences in outcomes for Black and white defendants, which ultimately highlighted racial disparities that exist within the algorithm. ProPublica came to the conclusion that COMPAS utilizes data from a criminal justice system with a history of racial injustices, thus continuing to disproportionately target and arrest Black people in comparison to their white counterparts. While the COMPAS algorithm treats unequal groups alike, which may appear neutral, ProPublica\u2019s data analysis and reporting emphasized the bias against Black defendants and their communities that COMPAS produced from this line of thinking, a claim that Northpointe has disputed (as we will see later).\n\nLet's retrace ProPublica's statistical analysis in order to better understand ProPublica's argument and engage with the metric of fairness that it uses.\n\n## Question 1. Logistic Regression: What are the odds of getting a high risk score?\n\nProPublica\u2019s first attempt at understanding the disparity in risk scores from the COMPAS tool was through logistic regression to model the chance of getting a \u201chigher\u201d (i.e. more \"risky\") score. COMPAS labels scores 1-4 as low, 5-7 as medium, and 8-10 as high scores. For the purposes of their analysis, ProPublica labeled any score above a low score as high.\n\n### Question 1a (i)\nCreate a logistic regression model to predict the score of defendants based on their sex, age, race, previous arrests, seriousness of the crime, and future criminal behavior. \n\n\n```python\n# Create independent variable decile score: 1 for \"high\" score, 0 for \"low\" score.\ny = np.where(select_data.decile_score>4, 1, 0)\n\n\n# Collect the dependent variables: Binarize categorical variables and take numerical variables from select_data.\n# Numerical Variables\nX = pd.DataFrame(select_data[[\"priors_count\", \"two_year_recid\"]])\n\n#Binarize sex, age categories, race, and charge degree\nX[\"sex\"] = pd.get_dummies(select_data[\"sex\"])[\"Female\"]\nX[[\"Greater than 45\", \"Less than 25\"]] = pd.get_dummies(select_data[\"age_cat\"])[[\"Greater than 45\", \"Less than 25\"]]\nX[[\"African-American\", \"Asian\", \"Hispanic\", \"Native American\", \"Other\"]] = pd.get_dummies(select_data[\"race\"])[[\"African-American\", \"Asian\", \"Hispanic\", \"Native American\", \"Other\"]]\nX[\"misdemeanor_charge\"] = pd.get_dummies(select_data[\"c_charge_degree\"])[\"M\"]\n\n\n# Create the model \nmodel = LogisticRegression()\nmodel.fit(X, y)\n```\n\n\n\n\n LogisticRegression()\n\n\n\n### Question 1a (ii)\nPrint out the coefficients paired with the corresponding feature names.\n\n\n```python\n# Pair the coefficients with feature names\nfeatures = list(X.columns)\nprint(list(zip(features, model.coef_[0])))\n```\n\n [('priors_count', 0.26841574060888973), ('two_year_recid', 0.6831840383941473), ('sex', 0.21881440417074133), ('Greater than 45', -1.345344452434324), ('Less than 25', 1.3015017693515076), ('African-American', 0.47767854348036104), ('Asian', -0.20805671230713116), ('Hispanic', -0.42133030966487106), ('Native American', 0.8956289051113875), ('Other', -0.8059219576573431), ('misdemeanor_charge', -0.3098895148947778)]\n\n\n### Question 1b\nWhat features are most predictive? \n\n*Student Written Answer Here*\n\n### Question 1c\nAre Black defendants more likely to get a high risk score opposed to white defendants? If so, by how much? Show your calculations. \n\n\n```python\nintercept = model.intercept_[0]\ncontrol = np.exp(intercept)/(1 + np.exp(intercept))\nblack_coef = model.coef_[0][5]\nnp.exp(black_coef)/(1 - control + (control * np.exp(black_coef)))\n```\n\n\n\n\n 1.4530844528826197\n\n\n\n*Student Written Answer Here* \n\n## Question 2. FPR and FNR: Does COMPAS overpredict or underpredict across groups?\n\nIn order to answer this question and understand the ways in which bias is present in the risk scores, ProPublica used the False Positive Rate (FPR) and False Negative Rate (FNR) as their metrics to understand and quantify fairness. \n\n### Question 2a\nComplete the following functions to calculate the FPR and FNR. Afterwards, apply these functions to each racial subgroup: Black defendants and white defendants. Keep in mind that ProPublica defines a high score as anything above 4, and therefore a false positive would be a defendant with a high score who did not recidivate. \n\n\n```python\ndef fpr(race_feature, data):\n # Return the False Positive Rate of scores for the specified race_feature\n \n subgroup = data[data[\"race\"] == race_feature]\n did_not_recidivate = subgroup[subgroup[\"two_year_recid\"] == 0]\n\n fp = did_not_recidivate[did_not_recidivate[\"decile_score\"] > 4].shape[0]\n tn = did_not_recidivate[did_not_recidivate[\"decile_score\"] <= 4].shape[0]\n return fp / (fp + tn)\n\n\ndef fnr(race_feature, data):\n # Return the False Negative Rate of scores for the specified race_feature\n \n subgroup = data[data[\"race\"] == race_feature]\n recidivated = subgroup[subgroup[\"two_year_recid\"] == 1]\n \n fn = recidivated[recidivated[\"decile_score\"] <= 4].shape[0]\n tp = recidivated[recidivated[\"decile_score\"] > 4].shape[0]\n return fn / (fn + tp)\n\n```\n\n\n```python\n# Apply the metrics to the dataset\nprint(\"FPR for Black defendants:\", fpr(\"African-American\", select_data))\nprint(\"FPR for white defendants:\", fpr(\"Caucasian\", select_data))\nprint(\"FNR for Black defendants:\", fnr(\"African-American\", select_data))\nprint(\"FNR for white defendants:\", fnr(\"Caucasian\", select_data))\n```\n\n FPR for Black defendants: 0.4233817701453104\n FPR for white defendants: 0.22014051522248243\n FNR for Black defendants: 0.2847682119205298\n FNR for white defendants: 0.49635036496350365\n\n\n### Question 2b\nWhat can you conclude from these metrics about the overprediction of risk scores for Black and white defendants? By how much is the tool overpredicting? (Hint: Look at your calculations for the FPR.)\n\n\n```python\n# Calculation of ratio for FPR\nfpr(\"African-American\", select_data) / fpr(\"Caucasian\", select_data)\n```\n\n\n\n\n 1.9232342111919953\n\n\n\n*Student Written Answer Here*\n\n### Question 2c\nWhat can you conclude from these metrics about the underprediction of risk scores for Black and white defendants? By how much is the tool underpredicting? (Hint: Look at your calculations for the FNR.)\n\n\n```python\n# Calculation of ratio for FNR\nfnr(\"Caucasian\", select_data) / fnr(\"African-American\", select_data)\n```\n\n\n\n\n 1.7429977932439313\n\n\n\n*Student Written Answer Here*\n\n### Question 2d\nWhat is the importance of overprediction and underprediction in regard to ProPublica\u2019s analysis? How might these observations have real impacts on the defendants who receive scores?\n\n*Student Written Answer Here*\n\n## Question 3.\n\n### Question 3a (i)\nUtilizing your answers from 1b and 2b, what problems does ProPublica highlight in the COMPAS algorithm? \n\n*Student Written Answer Here*\n\n### Question 3a (ii)\nHow would you describe ProPublica\u2019s definition of fairness, after learning and utilizing the metrics they used?\n\n*Student Written Answer Here*\n\n### Question 3b \nWhy did ProPublica choose to investigate bias between races rather than a different sensitive feature? (Hint: think about how ProPublica\u2019s conclusions reflect the racial disparities in our current criminal justice system.)\n\n*Student Written Answer Here*\n\n### Question 3c\nWhat is ProPublica\u2019s agenda as an investigative journalism organization? How do we see this in their analysis and conclusions?\n\n*Student Written Answer Here*\n\nWe mentioned earlier that Northpointe disagreed with ProPublica's argument that the COMPAS algorithm is racially biased. Now that we\u2019ve analyzed ProPublica\u2019s perspective and seen the way in which they define and operationalize the concept of fairness, let\u2019s move on to Northpointe\u2019s.\n\n# Part 2. Northpointe's Perspective \n\n### Who is Northpointe? \n\nNorthpointe (merged with two other companies to create *equivant* in 2017) is a for-profit computer software company that aims to advance justice by informing and instilling confidence in decision makers at every stage of the criminal justice system ([equivant](https://www.equivant.com/)). In addition to operating and continuing to develop COMPAS, *equivant* has developed a variety of technologies for use in court case management, attorney case management, inmate classification, and risk/needs assessment strategies. \n\nIn the wake of criticism from ProPublica and other researchers alike, Northpointe produced a [detailed response](http://go.volarisgroup.com/rs/430-MBX-989/images/ProPublica_Commentary_Final_070616.pdf) to ProPublica\u2019s allegations, claiming that these critiques of their tool utilized the wrong type of classification statistics in their analysis and portrayed the tool incorrectly. The company provided their own analysis of the COMPAS algorithm by using different statistical methods and responding individually to each of ProPublica\u2019s claims of racial bias against Black defendants. \n\nUpon examining their tool\u2019s fairness through accuracy equity and predictive parity (which are metrics that were left out of ProPublica\u2019s analysis), as well as the fact that the model was not trained with a race feature, Northpointe concluded that their algorithm treats all citizens and specified groups equally, and therefore does not exhibit signs of bias or inequality for specified groups. Now, let\u2019s take a look at how Northpointe supported this argument.\n\n## Question 4. Accuracy Equity: Is each group being discriminated against equally?\n\nInstead of analyzing and comparing the model errors FNR and FPR, Northpointe utilized the complement of FNR, known as the TPR (or what is often referred to as *Sensitivity*), paired with the FPR to prove what they refer to as ***Accuracy Equity*** through the use of a *ROC Curve*. Accuracy equity, according to [Northpointe](http://go.volarisgroup.com/rs/430-MBX-989/images/ProPublica_Commentary_Final_070616.pdf), is exhibited in the model \u201cif it can discriminate recidivists and nonrecidivists equally well for two different groups such as blacks and whites.\u201d Recall that we use ROC curves and the *Area Under the Curve* to understand how much a model is capable of distinguishing between classes. \n\n### Question 4a\nUtilize the sklearn metrics package to calculate TPR and FPR, visualize the ROC curve for both white and Black defendants, and calculate the AUC for each curve.\n\n\n```python\n# Calculate FPR and FNR from metrics package - Black defendants\nblack_def = select_data[select_data[\"race\"] == \"African-American\"]\n# True binary values\ny1 = black_def['two_year_recid']\n# Predicted values from COMPAS tool\npred1 = black_def[\"decile_score\"].replace([1, 2, 3, 4], 0).replace([5, 6, 7, 8, 9, 10], 1)\nfpr_black, tpr_black, threshold = roc_curve(y1, pred1)\n\n# Calculate FPR and FNR from metrics package - White defendants\nwhite_def = select_data[select_data[\"race\"] == \"Caucasian\"]\n# True binary values\ny2 = white_def['two_year_recid']\n# Predicted values from COMPAS tool\npred2 = white_def[\"decile_score\"].replace([1, 2, 3, 4], 0).replace([5, 6, 7, 8, 9, 10], 1)\nfpr_white, tpr_white, threshold = roc_curve(y2, pred2)\n\n# Plot the ROC \nplt.subplots(1, figsize=(10,10))\nplt.title('ROC - Black Defendents')\nplt.plot(fpr_black, tpr_black)\nplt.plot(fpr_white, tpr_white)\nplt.plot([0, 1], ls=\"--\")\nplt.ylabel('Sensitivity')\nplt.xlabel('1 - Specificity')\nplt.show()\n\n```\n\n\n```python\n# Calculate the AUC\nprint(\"AUC for Black defendants:\", roc_auc_score(y1, pred1))\nprint(\"AUC for white defendants:\", roc_auc_score(y2, pred2))\n```\n\n AUC for Black defendants: 0.6459250089670798\n AUC for white defendants: 0.641754559907007\n\n\n### Question 4b (i)\nWhat do you notice from the ROC curve and the AUC calculation? List at least two general observations. \n\n*Student Written Answer Here*\n\n### Question 4b (ii)\nWhat could Northpointe take away from this visualization to prove their point? Is accuracy equity being represented here? (Hint: Is each racial group being discriminated against equally?)\n\n*Student Written Answer Here*\n\n## Question 5. Predictive Parity: Is the likelihood of recidivism equal across groups?\n\nIn addition to the metric outlined above, Northpointe also utilized positive predictive values to explore the likelihood of defendants to reoffend, and to therefore prove that ***Predictive Parity*** is achieved. Predictive parity, according to [Northpointe](http://go.volarisgroup.com/rs/430-MBX-989/images/ProPublica_Commentary_Final_070616.pdf), is exhibited in a model \u201cif the classifier obtains similar predictive values for two different groups such as blacks and whites, for example, the probability of recidivating, given a high risk score, is similar for blacks and whites.\u201d Let\u2019s explore how they analyzed this. \n\n### Question 5a \n\nComplete the following functions to calculate the positive predictive values and negative predictive values. Afterwards, apply these functions to the data of white defendants and the data of Black defendants. \n\n\n```python\ndef ppv(race_feature, data):\n # Return the Positive Predictive Value of scores for the specified race_feature\n \n subgroup = data[data[\"race\"] == race_feature]\n recidivated = subgroup[subgroup[\"two_year_recid\"] == 1]\n did_not_recidivate = subgroup[subgroup[\"two_year_recid\"] == 0]\n \n fp = did_not_recidivate[did_not_recidivate[\"decile_score\"] > 4].shape[0]\n tp = recidivated[recidivated[\"decile_score\"] > 4].shape[0]\n return tp / (tp + fp)\n\n\ndef npv(race_feature, data):\n # Return the Negative Predictive Value of scores for the specified race_feature\n \n subgroup = data[data[\"race\"] == race_feature]\n recidivated = subgroup[subgroup[\"two_year_recid\"] == 1]\n did_not_recidivate = subgroup[subgroup[\"two_year_recid\"] == 0]\n\n fn = recidivated[recidivated[\"decile_score\"] <= 4].shape[0]\n tn = did_not_recidivate[did_not_recidivate[\"decile_score\"] <= 4].shape[0]\n return tn / (tn + fn)\n\n```\n\n\n```python\n# Apply metrics to the dataset\nprint(\"PPV for Black defendants:\", ppv(\"African-American\", select_data))\nprint(\"NPV for Black defendants:\", npv(\"African-American\", select_data))\nprint(\"PPV for white defendants:\", ppv(\"Caucasian\", select_data))\nprint(\"NPV for white defendants:\", npv(\"Caucasian\", select_data))\n```\n\n PPV for Black defendants: 0.6495352651722253\n NPV for Black defendants: 0.6485884101040119\n PPV for white defendants: 0.5948275862068966\n NPV for white defendants: 0.7100213219616205\n\n\n### Question 5b\nUse the metrics you calculated above to fill in the table below.\n\n| | White | Black |\n|---:|:-------------|:-----------|\n| Labeled higher risk, but didn't reoffend | 41% | 35% |\n| Labeled lower risk, but did reoffend | 29% | 35% |\n\n\n```python\n# High risk but did not re-offend - white\n1 - ppv(\"Caucasian\", select_data)\n# Low risk but did re-offend - white\n1 - npv(\"Caucasian\", select_data)\n# High risk but did not re-offend - Black\n1 - ppv(\"African-American\", select_data)\n# Low risk but did re-offend - Black\n1 - npv(\"African-American\", select_data)\n```\n\n\n\n\n 0.3514115898959881\n\n\n\n### Question 5c (i)\nWhat do you notice about the positive predictive values for each group? List at least one general observation.\n\n*Student Written Answer Here*\n\n### Question 5c (ii)\nWhat could Northpointe conclude from these findings? Is predictive parity represented here? (Hint: Is the likelihood of recidivism relatively equal for each racial group?)\n\n*Student Written Answer Here*\n\n## Question 6.\n\n### Question 6a\nHow would you describe Northpointe\u2019s definition of fairness, after learning and utilizing the metrics they used? How is this different from your description of ProPublica\u2019s definition from Q3aii? \n\n*Student Written Answer Here*\n\n### Question 6b \n\nIf anything, what are ProPublica and Northpointe each not considering in their definitions? (Hint: Think about other goodness metrics in ML, as well as your knowledge of the historical context of policing data)\n\n*Student Written Answer Here*\n\nSo far, we\u2019ve investigated ProPublica\u2019s and Northpointe\u2019s definitions of fairness. In the world of machine learning there are [many more](https://www.google.com/url?q=https://fairmlbook.org/tutorial2.html&sa=D&ust=1606727018134000&usg=AOvVaw06zU_fm8h7xp71d8igA8KI), so in the next section we will take a look at a third definition.\n\n# Part 3. Yet Another Definition of Fairness \n\nIn this section, you will go through yet another metric and definition used to evaluate fairness in machine learning: **disparate impact**. Disparate impact is a legal doctrine that determines if there is unintended discrimination towards a protected class ([Society for Human Resource Management](https://www.shrm.org/resourcesandtools/tools-and-samples/hr-qa/pages/disparateimpactdisparatetreatment.aspx)). In machine learning, disparate impact is a metric to evaluate fairness in a model. It is a form of bias within an algorithm that reflects systemic discrimination when a model\u2019s outputs are dependent on a ***sensitive feautre*** (the protected class). This is often considered unintentional (like the legal doctrine) due to the fact that the sensitive feature is omitted from the model, though it is still correlated with the output through proxy variables ([Wang et al.](https://arxiv.org/pdf/1801.05398.pdf#:~:text=Abstract%E2%80%94In%20the%20context%20of,e.g.%2C%20race%20or%20gender)).\n\nNot only will you evaluate the fairness of the tool (as Northpointe and ProPublica did) by measuring the bias reflected in the outputs of the model, but you will remove it to actually change those outputs and therefore eliminate the dependencies between the risk scores and the race feature. In order to computationally remove the disparate impact that we quantify, we can use tools like aif360\u2019s [Disparate Impact Remover](https://aif360.readthedocs.io/en/latest/modules/generated/aif360.algorithms.preprocessing.DisparateImpactRemover.html). aif360 is a package created by IBM\u2019s AI Research team, which contains a variety of tools to \u201chelp you examine, report, and mitigate discrimination and bias in machine learning models throughout the AI application lifecycle\u201d ([AI Fairness 360](https://aif360.mybluemix.net/)).\n\n## Question 7. Disparate Impact: Quantification and Removal\n\nFirst, let\u2019s visualize the disparity that we would like to remove from the dataset. In order to do that we need to distinguish between a privileged group and an unprivileged group. In technical terms, the privileged group receives higher scores from a trained model, so therefore the Black defendants will be considered \"privileged\" and the white defendants will be considered \"unprivileged\" in this case. \n\n### Question 7a\n\nUse a histogram to plot the scores for Black defendants and the scores for white defendants. Visualize both histograms on the same plot.\n\n\n```python\nunpriv = select_data[select_data[\"race\"] == \"Caucasian\"]\npriv = select_data[select_data[\"race\"] == \"African-American\"]\nsns.distplot(priv[\"decile_score\"], hist=True, rug=False)\nsns.distplot(unpriv[\"decile_score\"], hist=True, rug=False)\n```\n\n### Question 7b \n\nWhat do you notice from the plot? (Hint: how do the distributions differ across racial groups with respect to mean, shape of distribution, etc.) \n\n*Student Written Answer Here*\n\nNow, we need to quantify the disparate impact we are seeing in the plot. In machine learning, we can understand disparate impact as the proportion of individuals that get positive outcomes (did they get a high score) for the two groups described above:\n\n\\begin{align}\n\\Pr(Y=1|D=Unprivileged) \\ / \\ \\Pr(Y=1|D=Privileged)\n\\end{align}\n\nIn this equation Y is 1 if the defendant received a high score and 0 if they received a low score. \n\n### Question 7c (i)\n\nCreate a function to calculate the proportion of individuals from a specified racial group that get positive outcomes.\n\n\n```python\ndef proportion(data, group):\n # Returns the proportion of individuals in data from the group who recidivate\n race_group = data[data[\"race\"] == group]\n positive_outcomes = race_group[race_group[\"decile_score\"] > 4]\n return len(positive_outcomes) / len(race_group)\n```\n\n### Question 7c (ii)\n\nUse this function to calculate the disparate impact, using the equation from above.\n\n\n```python\nprob_priv = proportion(select_data, \"African-American\")\nprob_unpriv = proportion(select_data, \"Caucasian\")\nprob_unpriv / prob_priv\n```\n\n\n\n\n 0.5745131730114521\n\n\n\nIf the proportion of unprivileged individuals receiving positive outcomes to privileged individuals receiving positive outcomes is less than 80%, there is a disparate impact violation. In order to stop a trained model from replicating these biases in its output, we can now use aif360\u2019s Disparate Impact Remover to remove the bias we just calculated.\n\n### Question 7d\n\nCreate a Disparate Impact Remover type object and use the function fit_transform on our data. In order to do this you will need to first create a BinaryLabelDataset from our dataset to use for fit_transform. Check out the documentation [here](https://aif360.readthedocs.io/en/latest/modules/generated/aif360.datasets.BinaryLabelDataset.html#aif360.datasets.BinaryLabelDataset) to see how to implement this.\n\n\n```python\n# Create new DataFrame with just the necessary columns - Only numeric values for race, decile score, and two year recid\n\n# The BinaryLabelDataset requires decile_scores to be continuous -->\n# Use this line of code noise = np.random.normal(0, 0.1, race_data.shape[0]) to add noise to your decile_score column\n\nrace_data = select_data[(select_data[\"race\"] == \"Caucasian\") | (select_data[\"race\"] == \"African-American\")]\nrace_col = pd.get_dummies(race_data, \"race\")[\"race_Caucasian\"]\nnoise = np.random.normal(0, 0.1, race_data.shape[0])\ndecile_col = race_data[\"decile_score\"] + noise\nrecid_col = race_data[\"two_year_recid\"]\nnew_df = pd.DataFrame({\"race\": race_col, \"decile_score\": decile_col, \"two_year_recid\": recid_col})\n\n# Create BinaryLabelDataset\nBLD = BinaryLabelDataset(favorable_label=1, # Positive Outcome\n unfavorable_label=0, # Negative Outcome\n df=new_df,\n label_names=[\"two_year_recid\"],\n protected_attribute_names=[\"race\"],\n unprivileged_protected_attributes=[1])\n```\n\n\n```python\nremover = DisparateImpactRemover(repair_level=1.0, sensitive_attribute=\"race\")\ntransformed_data = remover.fit_transform(BLD)\n```\n\n### Question 7e\n\nSimilar to part a, use a histogram to plot the scores on the modified dataset. Afterwards, use the proportion function created above to calculate the disparate impact of the transformed dataset.\n\n\n```python\n# Transform output from DIRemover into usable DataFrame\ntransformed_df = pd.DataFrame(np.hstack([transformed_data.features, transformed_data.labels]),\n columns=[\"race\",\"decile_score\",\"two_year_recid\"])\n\nunpriv_t = transformed_df[transformed_df[\"race\"] == 1]\npriv_t = transformed_df[transformed_df[\"race\"] == 0]\nsns.distplot(priv_t[\"decile_score\"], hist=True, rug=False)\nsns.distplot(unpriv_t[\"decile_score\"], hist=True, rug=False)\n```\n\n\n```python\n# Calculate disparate impact\nprob_priv_t = proportion(transformed_df, 0)\nprob_unpriv_t = proportion(transformed_df, 1)\nprob_unpriv_t / prob_priv_t\n```\n\n\n\n\n 0.9996653950920561\n\n\n\n### Question 7f\n\nWhat has changed from our original histogram? Please explain why this change has happened. \n\n*Student Written Answer Here*\n\n### Question 7g\n\nHow would you describe this third definition of fairness, after learning and utilizing these new metrics?\n\n*Student Written Answer Here*\n\n### Question 7h\nHow does this definition of fairness differ from ProPublica\u2019s and Northpointe\u2019s? \n\n*Student Written Answer Here*\n\n## Question 8. Considering Expertise Outside of Data Science\n\nJust now, you used your technical data science skills to computationally remove bias from the data set. By removing bias, we\u2019ve made the outputs of the algorithm statistically fair in regards to one definition of fairness. However, it is important to consider many types of knowledge and experiences beyond data science expertise when analyzing and creating an algorithm like COMPAS. As such, you will think through issues of expertise and fairness in the next set of questions.\n\n### Question 8a\n\nLook back to your answer from Q0a. Now that you\u2019ve gone through several definitions of fairness, how would you add to or revise your answer: Explain 3 parties that are impacted by the COMPAS tool. In what ways are they impacted?\n\n*Student Written Answer Here*\n\n### Question 8b\n\nWhat expertise and lived experiences are necessary to understand and critically think about the issues produced by COMPAS?\n\n*Student Written Answer Here*\n\n### Question 8c\n\nWhy is this third definition of fairness still inadequate as a measurement of justice in the court system? (Hint: look at the previous two questions and answers).\n\n*Student Written Answer Here*\n\n # Part 4. Conclusion \n\n## Question 9. Which Definition Is Fair? And Who Decides?\n\nWe\u2019ve now gone through three definitions of fairness, each one with a different idea of how to operationalize fairness and to judge whether or not an algorithm is fair. As a data scientist, you may encounter situations where you will need to make decisions that affect real-world outcomes and people! Let\u2019s try to do this for COMPAS. \n\n### Question 9a\nIf you had to decide between the three definitions of fairness above, which definition do you think would make \u201cfair\u201d decisions for everyone who goes through the court system? What values did you consider as you made this decision? If you cannot come to a decision, what challenges did you come across when considering this? \n\n*Student Written Answer Here*\n\n### Question 9b\nTake a step back and think about how different actors who created, utilize, and are affected by COMPAS would consider which definition is most fair. Name two relevant actors, and discuss what they would value in *their own* definitions of fairness. Of the three definitions you have explored, which would they decide is most fair from the perspective of that actor? If you don\u2019t think they\u2019d choose any of them, explain why. (Examples of actors, which you\u2019re welcome to use: judges, defendants, police, policy makers, community members) \n\n*Student Written Answer Here*\n\nChoosing one definition of fairness can be incredibly difficult when you need to consider all the actors at play. Throughout this module we have examined where and how the COMPAS algorithm is appropriate to use. It is also important to recognize the problems that are not solvable by an algorithm and think through how we can make the ecosystem that COMPAS is in (which includes but is not limited to the legal system, affected communities, the tech industry, etc.) more equitable.\n\n### Question 9c\nWhat issues that are relevant to the COMPAS ecosystem but outside of the algorithm itself need to be addressed to be able to create a more equitable system, with or without the algorithm? \n\n*Student Written Answer Here*\n\nYou\u2019ve now begun to think through the very complex systems in which the COMPAS algorithm functions. **Congratulations!** Through considering a few of the differing definitions of fairness connected to COMPAS, hopefully you can begin to understand some of the human contexts of creating algorithms that intentionally affect people and their decision-making. \n\n\n```python\n\n```\n", "meta": {"hexsha": "c0e53d9f911f4d24aeed64723a3426fdf07370a8", "size": 149203, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "COMPAS/.ipynb_checkpoints/COMPAS Project-checkpoint.ipynb", "max_stars_repo_name": "ds-modules/HCE-Materials", "max_stars_repo_head_hexsha": "a3bc3b02f2c1a6a0af6646b021f2e605d074e4a2", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-06-10T22:51:02.000Z", "max_stars_repo_stars_event_max_datetime": "2021-06-10T22:51:02.000Z", "max_issues_repo_path": "COMPAS/.ipynb_checkpoints/COMPAS Project-checkpoint.ipynb", "max_issues_repo_name": "ds-modules/HCE-Materials", "max_issues_repo_head_hexsha": "a3bc3b02f2c1a6a0af6646b021f2e605d074e4a2", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "COMPAS/.ipynb_checkpoints/COMPAS Project-checkpoint.ipynb", "max_forks_repo_name": "ds-modules/HCE-Materials", "max_forks_repo_head_hexsha": "a3bc3b02f2c1a6a0af6646b021f2e605d074e4a2", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 82.1150247661, "max_line_length": 48964, "alphanum_fraction": 0.7808824219, "converted": true, "num_tokens": 12108, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.6859494421679929, "lm_q1q2_score": 0.43448741703329574}} {"text": "

    Table of Contents

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    \n\n\n```python\nimport numpy as np\nimport a301.radiation\n```\n\n# Question 1\n\n - (10) A satellite orbiting at an altitude of 36000 km observes the\n surface in a thermal channel with a wavelength range of\n $8\\ \\mu m < \\lambda < 10\\ \\mu m$.\n \n - Assuming that the atmosphere has density scale height of\n $H_\\rho=10$ km and a surface air density of $\\rho_{air}=1$\n and that the absorber has mass absorption coefficient of\n $k_\\lambda = 3 \\times 10^{-2}\\ m^2/kg$ at $\\lambda=9\\ \\mu m$\n and a mixing ratio $6 \\times 10^{-3}$ kg/kg, find the vertical\n optical thickness $\\tau$ and transmittance of the atmosphere\n directly beneath the satellite\n \n - If the surface is black with a temperature of 300 K, and the\n atmosphere has an average temperature of 270 K, find the\n \n - radiance observed by the satellite in at 9 $\\mu m$\n \n - brightness temperature of the pixel in Kelvin for that\n radiance\n \n - Given a pixel size 2 $km^2$, what is the flux, in , reaching\n the satellite in this channel?\n\n\n\n## Question 1a solution\n\nAssuming that the atmosphere has density scale height of\n$H_\\rho=10$ km and a surface air density of $\\rho_{air}=1$\nand that the absorber has mass absorption coefficient of\n$k_\\lambda = 3 \\times 10^{-2}\\ m^2/kg$ at $\\lambda=9\\ \\mu m$\nand a mixing ratio $6 \\times 10^{-3}$ kg/kg, find the vertical\noptical thickness $\\tau$ and transmittance of the atmosphere\ndirectly beneath the satellite\n\n$$\\rho_{atm} = \\rho_0 \\exp \\left ( -z/H \\right )$$\n\n$$H=10\\ km$$\n\n$$\\tau = \\int_0^{3.6e6}k \\rho_0 \\exp (-z/H ) r_{mix} dz$$\n\n$$\\tau = -H k \\exp(-z^\\prime/H ) \\rho_0 r_{mix} \\big \\rvert_0^{3.6e6} =0 - (-Hk \\rho_0 r_{mix})=H k \\rho_0 r_{mix} $$\n\n$$t=\\exp(-\\tau)$$\n\n\n```python\nH=10000.\nk=3.e-2\nrho0=1.\nrmix=6.e-3\ntau = H*k*rho0*rmix\nt=np.exp(-tau)\nprint(f'optical thickness \u03c4={tau} and transmittance t={t:5.2f}')\n```\n\n optical thickness \u03c4=1.8 and transmittance t= 0.17\n\n\n## Question 1b solution\n- If the surface is black with a temperature of 300 K, and the\n atmosphere has an average temperature of 270 K, find the\n \n - radiance observed by the satellite in at 9 $\\mu m$\n \n - brightness temperature of the pixel in Kelvin for that\n radiance\n\n$$L_{atm}= B(300)*\\exp(-\\tau) + (1 - \\exp(-\\tau))*B(270)$$\n\n\n```python\nt=np.exp(-tau)\ne=1 - t\nL270=a301.radiation.calc_radiance(9.e-6,270)\nL300=a301.radiation.calc_radiance(9.e-6,300)\nLsat = t*L300 + e*L270\nprint(Lsat)\nTbright=a301.radiation.planck_invert(9.e-6,Lsat)\nprint(f'radiance is {Lsat*1.e-6:5.2f} W/m^2/microns/sr')\nprint(f'brightness temperature is {Tbright:5.2f} K')\n```\n\n 6153879.87160191\n radiance is 6.15 W/m^2/microns/sr\n brightness temperature is 275.85 K\n\n\n## Question 1c solution:\n- Given a pixel size 2 $km^2$, what is the flux, in , reaching\n the satellite in this channel?\n\n\n\n$\\Delta \\omega = A/R^2 = 2/36000^2. = 1.54 \\times 10^{-9}$ sr\n\n$E = L \\Delta \\omega \\,\\Delta \\lambda = 6.15\\ W\\,m^2\\,\\mu^{-1} m \\times 1.54 \\times 10^{-9} \\times 2$\n\n\n\n```python\nEout=6.15*1.54e-9*2\nprint(f'flux in channel is {Eout:5.2g} W/m^2')\n```\n\n flux in channel is 1.9e-08 W/m^2\n\n\n# Question 2\n\n## Question 2a solution\n\n- (3) A cone has a spreading angle of 35 degrees between its\n center and its side. What is its subtended solid angle?\n \n$$\\omega = \\int_0^{2\\pi} \\int_0^{35} \\sin \\theta d\\theta d\\phi = 2\\pi (-\\cos \\theta \\big \\rvert_0^{35}) = 2 \\pi (1 - \\cos(35))$$\n\n\n```python\nomega = 2*np.pi*(1 - np.cos(35*np.pi/180.))\nprint(f'solid angle = {omega:5.2f} sr')\n```\n\n solid angle = 1.14 sr\n\n\n \n## Question 2b solution \n \n- (3) Assuming that radiance is independent of the distance $d$\n between an instrument and a surface, show that the flux from the\n surface decreases as $1/d^2$\n \nGiven a narrow field of view of a pixel the radiance is:\n\n$$E \\approx L \\Delta \\omega$$\n\nwhere $\\Delta \\omega = A/d^2$ with A the area of the pixel. Since $L$ is constant, $E \\propto 1/d^2$\n\n# Question 3 \n\nIntegrate the Schwartzchild equation for constant temperature\n\n## Question 3 solution\n \n1. We know the emission from an infinitesimally thin layer:\n\n $$ dL_{emission} = B_{\\lambda} (T_{layer}) de_\\lambda = B_{\\lambda} (T_{layer}) d\\tau_\\lambda$$\n\n\n2. Add the gain from $dL_{emission}$ to the loss from $dL_{absorption}$ to get\n the **Schwartzchild equation** without scattering:\n\n $$ dL_{\\lambda,absorption} + dL_{\\lambda,emission} = -L_\\lambda\\, d\\tau_\\lambda + B_\\lambda (T_{layer})\\, d\\tau_\\lambda $$\n\n3. We can rewrite :eq:$schwart1$ as:\n \n $$ \\frac{dL_\\lambda}{d\\tau_\\lambda} = -L_\\lambda + B_\\lambda (T_{layer})$$\n\n4. In class I used change of variables to derived the following: if the temperature $T_{layer}$ (and hence $B_\\lambda(T_{layer})$) is constant with height and the radiance arriving at the base of the layer is $L_{\\lambda 0} = B_{\\lambda} T_{skin}$ for a black surface with $e_\\lambda = 1$, then the total radiance exiting the top of the layer is $L_{\\lambda}$ where:\n\n $$ \\int_{L_{\\lambda 0}}^{L_\\lambda} \\frac{dL^\\prime_\\lambda}{L^\\prime_\\lambda -\n B_\\lambda} = - \\int_{0}^{\\tau_{T}} d\\tau^\\prime $$\n\n Where the limits of integration run from just above the black surface (where the radiance from\n the surface is $L_{\\lambda 0}$) and $\\tau=0$ to the top of the layer, (where the radiance is $L_\\lambda$) and the optical thickness is $\\tau_{\\lambda T}$.\n\n To integrate this, make the change of variables:\n\n\n\n\n\\begin{align}\n U^\\prime &= L^\\prime_\\lambda - B_\\lambda \\\\\n dU^\\prime &= dL^\\prime_\\lambda\\\\\n \\frac{dL^\\prime_\\lambda}{L^\\prime_\\lambda -\n B_\\lambda} &= \\frac{dU^\\prime}{U^\\prime} = d\\ln U^\\prime\n\\end{align}\n\n \n\n where I have made use of the fact that $dB_\\lambda = 0$ since the temperature is constant.\n\n This means that we can now solve this by integrating a perfect differential:\n \n $$\n \\int_{U_0}^U d\\ln U^\\prime = \\ln \\left (\\frac{U}{U_0} \\right ) = \\ln \\left (\\frac{L_\\lambda - B_\\lambda}{L_{\\lambda 0} - B_\\lambda} \\right ) = - \\tau_{\\lambda T} $$\n\n Taking the $\\exp$ of both sides:\n\n $$ L_\\lambda - B_\\lambda = (L_{\\lambda 0} - B_\\lambda) \\exp (-\\tau_{\\lambda T}) $$\n\n \n or rearranging and recognizing that the transmittance is $\\hat{t_\\lambda} = \\exp(-\\tau_{\\lambda T} )$:\n\n $$ L_\\lambda = L_{\\lambda 0} \\exp( -\\tau_{\\lambda T} ) + B_\\lambda (T_{layer})(1- \\exp( -\\tau_{\\lambda T} )) $$\n\n \n $$ L_\\lambda = L_{\\lambda 0} \\hat{t}_{\\lambda} + B_\\lambda (T_{layer})(1- \\hat{t}_{\\lambda}) $$\n\n $$ L_\\lambda = L_{\\lambda 0} \\hat{t}_{\\lambda} + B_\\lambda (T_{layer})a_\\lambda $$\n\n5. so bringing in Kirchoff's law, the radiance exiting the top of the isothermal layer of thickness $\\Delta \\tau$ is: \n\n $$ L_\\lambda = L_{\\lambda 0} \\hat{t}_{\\lambda} + e_\\lambda B_\\lambda $$\n\n\n\n# Question 4\n\n- Pyresample (10)\n\nConsider the following code:\n\n from pyresample import SwathDefinition, kd_tree, geometry\n proj_params = get_proj_params(m5_file)\n swath_def = SwathDefinition(lons_5km, lats_5km)\n area_def_lr=swath_def.compute_optimal_bb_area(proj_dict=proj_params)\n area_def_lr.name=\"ir wv retrieval modis 5 km resolution (lr=low resolution)\"\n area_def_lr.area_id='modis_ir_wv'\n area_def_lr.job_id = area_def_lr.area_id\n fill_value=-9999.\n image_wv_ir = kd_tree.resample_nearest(swath_def, wv_ir_scaled.ravel(),\n area_def_lr, radius_of_influence=5000, \n nprocs=2,fill_value=fill_value)\n image_wv_ir[image_wv_ir < -9000]=np.nan\n print(f'\\ndump area definition:\\n{area_def_lr}\\n')\n print((f'\\nx and y pixel dimensions in meters:'\n f'\\n{area_def_lr.pixel_size_x}\\n{area_def_lr.pixel_size_y}\\n'))\n\nIn the context of this snippet, explain what the following objects\n(i.e. their type, what some of their attributes are, etc.) and how\nthey are used to map a satellite image:\n\n## Question 4 solution\n\n- proj\\_params\n\ndictionary holding parameters for a map projection that\ncan be used by pyproj to map lat/lon to x/y: datum, lat\\_0, lon\\_0\nname of projection etc.\n\n- swath\\_def\n\nobject of type pyresample.geometry.SwathDefinition that holds data and\nfunctions needed to convert modis pixel lat/lon values to x,y -- pass as input\nto kd_tree_resample_nearest\n\n- area\\_def\\_lr\n\nobject of type pyresample.geometry.AreaDefinition that holds x,y array information\nlike number of rows, number of columns and image extent in x and y.\n\n- wv\\_ir\\_scaled.ravel()\n\nwater vapor data scaled to units of cm in the column and converted to a 1-dimensional\nvector using the ravel method.\n\n- kd\\_tree.resample\\_nearest\n\nfunction that takes water vapor values and sorts them onto an x,y grid based on\ntheir lat/lon values from the swath\\_def object. This is the mapped image.\n\n\n```python\n!pwd\n```\n\n /Users/phil/repos/a301_web/notebooks\r\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "c0efd21376f3608d25aedf6dc1469328233eca74", "size": 16836, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/midterm_2018_sols.ipynb", "max_stars_repo_name": "Pearl-Ayem/ATSC_Notebook_Data", "max_stars_repo_head_hexsha": "c075d166c235ac4e68a4b77750e02b2a5e77abd0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/midterm_2018_sols.ipynb", "max_issues_repo_name": "Pearl-Ayem/ATSC_Notebook_Data", "max_issues_repo_head_hexsha": "c075d166c235ac4e68a4b77750e02b2a5e77abd0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/midterm_2018_sols.ipynb", "max_forks_repo_name": "Pearl-Ayem/ATSC_Notebook_Data", "max_forks_repo_head_hexsha": "c075d166c235ac4e68a4b77750e02b2a5e77abd0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.9210526316, "max_line_length": 2024, "alphanum_fraction": 0.5288073177, "converted": true, "num_tokens": 3511, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953797290153, "lm_q2_score": 0.752012562644147, "lm_q1q2_score": 0.4342837804251716}} {"text": "

    Advanced Information Analytics

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    A statistical perspective on learning

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    IST 718 \u2013 Big Data Analytics

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    Daniel E. Acuna

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    http://acuna.io

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    • Small/medium data
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    \n\n# Outline of this unit\n- Definition of learning\n- Terminology\n- Function learning\n- Statistical learning\n- Prediction: reduce vs irreducible error\n- Inference\n- Parametric models vs non-parametric models\n- Supervised vs unsupervised learning\n- Regression vs classification\n- Interpretability vs flexibility\n\n# What is learning?\n\n# Learning\n- From Merriam-Webster:\n
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    1. The act or experience of one that learns.
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    \nUsing experience to improve future performance:\n\n - Experience = Data\n - Future = Data not seen before\n - Performance = Error function or loss function\n\n\n# What is statistical learning?\nFrom ISLR:\n
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    Statistical learning refers to a vast set of tools for understanding data. These tools can be classified as supervised or unsupervised. Broadly speaking, supervised statistical learning involves building a statistical model for predicting, or estimating, an output based on one or more inputs. With unsupervised statistical learning, there are inputs but no supervising output; nevertheless we can learn relationships and structure from such data.

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    \n\n# Terminology\n- What we measure: **features** (*input variables, predictors, or independent variables*.) \n\n- What we want to predict or associate with what we measure: **output** (*label, response, dependent variable*.) \n\n- In the case of unsupervised learning, we do not have outputs. \n\n- We will focus on supervised learning first.\n\n# Terminology (2)\n- What we measure: **features** \n\n- What we want to predict or associate with what we measure: **output** \n\n- Identify the features and outputs in the following examples:\n - Predict the price of a stock in 6 months from now, on the basis of company performance measures and economic data.\n - Predict whether a patient, hospitalized due to a heart attack, will have a second heart attack. The prediction is to be based on demographic, diet and clinical measurements for that patient.\n - Identify the numbers in a handwritten ZIP code, from a digitized image.\n - Predict the longitude and latitude of a car based on GPS measurements, accelerometer, and gyroscope.\n - Predict the hash tags of a tweet based on its text.\n\n# Mathematical formalization of learning\n- We will say that **we want to learn a function $f$ about a phenomenon**. \n
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    • $x$: market capitalization, trading volume, stock price, and economic data.
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    • $X$: space of all possible market capitalization, volumes, stock prices, and economic data.
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    \n\n# Mathematical formalization of learning (4)\n- Predict whether a patient, hospitalized due to a heart attack, will have a second heart attack. The prediction is to be based on demographic, diet and clinical measurements for that patient. \n
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    \n\n# Mathematical formalization of learning (5)\n- Predict whether a patient, hospitalized due to a heart attack, will have a second heart attack. The prediction is to be based on demographic, diet and clinical measurements for that patient. \n
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    • $x$: Demographics, diet, clinical measure of a patient hospitalized due to a heart attack.
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    • $X$: Space of demographics, diet, and clinical data of patients hospitalized for heart attack.
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    \n\n# Example: Diabetes dataset\n
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    • Ten baseline variables including age, sex, body mass index, average blood pressure, and six blood serum measurements were obtained for each of $n$ = 442 diabetes patients, as well as the response of interest, a quantitative measure of disease progression one year after baseline.\n
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    \n\n# A statistical perspective on learning\n- We wish to learn something from the data $D$.\n\n\n- Statistical learning means that the data have been generated by some unknown *random process*:\n$$x,y \\thicksim p()$$ \n\n\n- This reads *\"$x,y$ are sampled according to the probability distribution $p(\\cdot)$.\"* \n\n\n- **We wish to estimate the *unknown $p$* from such data**.\n\n# Quick probability review\n
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    • **Probability of an event** $A$ is denoted by $\\;p(A)$
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    \n\n# How bad are we at probabilities?\n- You go to a friend's house and you know that she has two children.\n- Suddenly, a girl runs between the two of you, and your friend says \"she is my daughter.\"\n- What is the probability that the other child is **also** a girl?\n\n$\\qquad$**Hint**: Conditioning selects one subset of the sample space \n\n$$p(x \\mid y) = p(x,y)/p(y)$$\n\n\n# How bad are we at probabilities? (2)\n- You go to a friend's house and you know that she has two children.\n- Suddenly, a girl runs between the two of you, and your friend says \"she is my daughter.\"\n- What is the probability that the other child is **also** a girl? \n
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    \n\n# A statistical perspective on learning\n- Statistical learning refers to a set of approaches for estimating $\\;f$ assuming some noise added into the system, which cannot be predicted from $x$. \n
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    \n\n# Why we estimate $f$ ?\n- Broadly speaking, we want to estimate $f$ to make **predictions** or **inference**, or both.\n- **Prediction** for a new data point never seen before: $\\;Y = f(X)$\n- E.g., for the diabetes case, we might want to predict disease progression based on the BMI of a patient.\n
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    \n\n# Why we estimate $f$ ? (2)\n
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    \n \n\n# Why we estimate $f$ ? (3)\n- There is always **reducible error** and **irreducible error**. \n\n- Squared error: \n$$\\begin{align}\nE[(Y-\\hat{Y})^2] &= E[(f(X)+\\epsilon-\\hat{f}(X))^2] \\\\\n&= E[(f(X)-\\hat{f}(X))^2+2(f(X)-\\hat{f}(X))\\epsilon + \\epsilon^2] \\\\\n&= E[(f(X)-\\hat{f}(X))^2]+\\overbrace{E[2(f(X)-\\hat{f}(X))\\epsilon]}^0 + E[\\epsilon^2] \\\\\n&= E[(f(X)-\\hat{f}(X))^2] + E[\\epsilon^2] \\\\\n\\end{align}$$\n\n- $E[(f(X)-\\hat{f}(X))^2]$ is **reducible** variance.\n- $E[\\epsilon^2]$ is **irreducible** variance.\n\n\n\n\n# Why we estimate $f$ ? (4)\n- **Inference**: sometimes we want to understand $f$ (look inside) \n\n - Which predictors are associated with $Y$? e.g., do we need to include gender in the prediction of blood pressure? \n - Relationship between $Y$ and each $X$. e.g., blood pressure is higher for older people? \n \n - Is the relationship between $Y$ and $X$ appropriately captured by the model? Is the linear relationship enough?\n\n# How do we estimate $f$ ?\n- We estimate using **training data**. For notation, we will assume that we have $n$ training points. $x_{ij}$ is the value of variable $j$ for data point $i$, and $y_i$ is the independent variable for that data point. \n\n- **Parametric methods**\n 1. Define the form of $f$ (e.g., linear model.)\n 2. Use a procedure to fit or train the model. \n \n- **Nonparametric methods**\n - These methods don\u2019t make assumptions about $f$.\n - Informally, they try to get as close as possible to the training data but not too close.\n - In general, the more data, the better the fit, but the harder to intepret.\n\n# How do we estimate $f$ ? (2)\n- Sometimes, the probability distribution that generated the data is known or assumed to be known, except for a set of parameters describing it. \n\n$$x,y \\thicksim p(\\theta)$$ \n\n- We need to infer $\\theta$\n\n# Two views on statistical learning\n- **Bayesian statistics**: use prior knowledge about the phenomenon and then combine evidence with that prior knowledge.\n\nMathematically, use prior over unknown parameters ($p(\\theta)$) and the likelihood of the data given a parameter value ($p(x,y \\mid \\theta)$). Use Baye's Theorem to infer the data generating distribution.\n\n$$\\hat{\\theta} \\thicksim p(\\theta \\mid x,y)= \\frac{p(x,y \\mid \\theta)p(\\theta)}{p(x,y)}$$\n\n# Two views on statistical learning (2)\n- **Frequentist statistics**: \n - Defines an *estimator* for a parameter based on data. \n \n - It assumes a procedure where samples from the same data will be observed an infinite amount of time. \n \n - It uses the resampled estimated to build a distribution over estimations.\n\n- In this class, we will take a **frequentist view**: \n \n - We will use the Maximum Likelihood Estimation (MLE): \n\n$$ \\hat{\\theta} = \\arg\\max_\\theta p(y \\mid \\theta)$$\n\n# Example: Diabetes dataset\n- Let\u2019s look at the disease progression distribution of the sample. \n\n
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    \n\n# Example: Diabetes dataset (2)\n- We will assume that disease progression is distribuited according to a Gaussian distribution:\n\n$$\\text{disease progression} \\thicksim p(\\mu,\\sigma)$$ \n\n
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    \n\n# Gaussian distribution\n- Gaussian or Normal distribution: most common probability distribution. \n\n$$y \\thicksim N(\\mu, \\sigma)=\\frac{1}{\\sigma\\sqrt{2\\pi}}\\exp\\left[-\\frac{1}{2\\sigma^2}(x-\\mu)^2\\right]$$\n\n
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    \n\n# Diabetes dataset and assumptions\n- We will assume that disease progression ($d$) is distributed according to a Gaussian distribution and each subject is independent of each other.\n\n$$p(d_1,d_2,\\ldots,d_{442} \\mid \\mu,\\sigma)=p(d_1 \\mid \\mu,\\sigma)p(d_2 \\mid \\mu,\\sigma) \\cdots p(d_{442} \\mid \\mu,\\sigma)$$\n\n- MLE: \n\n$$\\hat{\\mu}=\\arg\\max_\\mu \\prod_{i=1}^{442} {\\frac{1}{\\sigma\\sqrt{2\\pi}}\\exp\\left[-\\frac{1}{2\\sigma^2}(d_i-\\mu)^2\\right]}$$\n\n# How do we perform MLE?\n- Let\u2019s work through the math.\n\n$$\\begin{align}\np(d_1,d_2,\\ldots,d_{442} \\mid \\mu,\\sigma) &= \\prod_{i=1}^{442} {\\frac{1}{\\sigma\\sqrt{2\\pi}}\\exp\\left[-\\frac{1}{2\\sigma^2}(d_i-\\mu)^2\\right]} \\\\\np(d_1,d_2,\\ldots,d_{442} \\mid \\mu,\\sigma) &= \\left(\\frac{1}{\\sigma\\sqrt{2\\pi}}\\right)^{442}\\exp\\left[-\\frac{1}{2\\sigma^2}\\sum_{i=1}^{442}(d_i-\\mu)^2\\right] \\\\\n\\end{align}$$\n \n \n- Because $p$ is always positive, then we have the following property: \n\n$$\\arg\u2061\\max\u2061 p\u2061(\\cdot) = \\arg\u2061\\max\u2061 \\log p\u2061(\\cdot)$$\n\n# How do we perform MLE? (2)\n- Let\u2019s work through the math.\n\n$$\\log p(d_1,d_2,\\ldots,d_{442} \\mid \\mu,\\sigma) = 442\\log\\left(\\frac{1}{\\sigma\\sqrt{2\\pi}}\\right)-\\frac{1}{2\\sigma^2}\\sum_{i=1}^{442}(d_i-\\mu)^2$$\n \n \n- At the end, we want to maximize a **likelihood** function based on two variables: $\\mu$ and $\\sigma$ \n\n$$\\arg\u2061\\max_{\\mu,\\sigma} l(\\mu,\\sigma)$$ \n\n- Let\u2019s focus on estimating $\\mu$ for now:\n - How would you use calculus to find the maximum of $l(\\mu,\\sigma)$ with respect to $\\mu$?\n\n\n# How do we perform MLE? (3)\n- Let\u2019s rewrite to make it clear: \n\n$$l(\\mu) = f_1(\\sigma) - f_2(\\sigma)\\sum_{i=1}^{442}(d_i-\\mu)^2$$\n\n\n# How do we perform MLE? (4)\n- One way of finding maximum/minimum is to look at where the slopes are zero. \n\n
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    \n\n# How do we perform MLE? (4)\n- Let\u2019s rewrite to make it clear: \n$$\\begin{align}\n\\frac{dl(\\mu)}{d\\mu} &= 0 \\\\\n\\frac{dl(\\mu)}{d\\mu}\\left(f_1(\\sigma) - f_2(\\sigma)\\sum (d_i-\\mu)^2\\right) &=0 \\\\\n- f_2(\\sigma)\\sum \\frac{d(d_i-\\mu)^2}{d\\mu} &= 0 \\\\\n\\sum (d_i-\\mu) &= 0 \\\\\n\\sum d_i - \\sum\\mu &= 0 \\\\\n\\sum d_i - n\\mu &= 0 \\\\\n\\frac{\\sum d_i}{n} &= \\mu \\\\\n\\end{align}$$\n\n- Simply the empirical average! (Can you do the same with the standard deviation?)\n\n# Example: Diabetes dataset\n- The **statistical estimation** of the distribution that generates *disease progression* is the mean of the data! \n\n- Also, **maximizing the likelihood under Gaussian** assumption is equivalent to finding the **minimum squared error** between my estimation and the data.\n- Usually, algorithms try to **minimize the negative loglikelihood (nLL)**, which is the same as **maximizing the loglikelihood**\n\n# Summary of disease progression estimation\n- We assume a statistical perspective to **learn** dp:\n - We assume that we have uncertainty in our process. \n \n- We define a **model** to make such prediction:\n - We define a Gaussian distribution around dp. \n \n- We use **experience**:\n - We use data to find an appropriate Gaussian distribution. \n \n- We define a **loss function** as performance:\n - We use the mean squared error to evaluate performance (any problems with this?)\n\n\n\n# A more interesting problem\n- The diabetes dataset is intended for understanding disease progression.\n- Again, the dataset description:\n
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    Ten baseline variables, age, sex, body mass index, average blood pressure, and six blood serum measurements were obtained for each of $n$ = 442 diabetes patients, as well as **the response of interest, a quantitative measure of disease progression one year after baseline**.\n

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    \n- Let\u2019s suppose we want to predict disease progression.\n - How can we use this dataset?\n\n# More interesting statistical learning\n- Suppose there is a relationship: \n$$Y=f(X)+\\epsilon$$ \n\n where $X$ is a set of independent variables and $Y$ is an output variable, and $\\epsilon$ is some noise (e.g., unobserved phenomena, variability of subjects) with mean zero.\n- Given that $\\epsilon$ is a random variable, then the relationship is stochastic: \n$$p(Y \\mid X)\\thicksim N(f(X),\\sigma_\\epsilon)$$\n\n# Parametric method: linear regression\n- Linear regression assumes a linear relationship between $X$ and $Y$ plus some noise: \n\n$$Y = X\\beta+\\epsilon$$ \n\n- The noise ($\\epsilon$) is assumed to be Gaussian. \n\n- Therefore, we find the parameters ($\\beta$) by minimizing the squared error (or equivalently, maximizing the likelihood.)\n\n# Example: predicting progression based on BMI\n- Predict disease progression based on the BMI of a patient. \n\n- **Prediction**: \n\n Model: $\\;Y = \\beta_0 + \\text{bmi}\\beta_1 + \\epsilon$\n
    \n
    \n
    \n\n# Example: predicting progression based on BMI (interpretation)\n- Model: $\\;y = \\beta_0 + \\text{bmi}\\beta_1 + \\epsilon$ \n\n- This is, for each observation ($y_i,\\text{bmi}_i$), we will define:\n\n - $\\mu_i = \\beta_0 + \\text{bmi}\\beta_1$ \n \n - $p(y_i \\mid \\text{bmi}_i) = \\prod_{i=1}^{n} {\\frac{1}{\\sigma\\sqrt{2\\pi}}\\exp\\left[-\\frac{1}{2\\sigma^2}(y_i-\\mu_i)^2\\right]}$ (\\*) \n\n\n- Same as before, we want to find best $\\beta_0$ and $\\beta_1$ so as to maximize **(\\*)** or minimize the squared error:\n$$l(\\beta_0,\\beta_1) = \\sum (y_i-\\mu_i)^2$$\n\n\n# Example: predicting progression based on BMI (interpretation) (2)\n- Again, we can take the derivatives and set them to zero: \n\n$$\\begin{align}\n\\frac{dl(\\beta_0,\\beta_1)}{d\\beta_0} &= \\frac{d\\sum(y_i-\\mu_i)^2}{d\\beta_0} \\\\\n&= (y_1-\\mu_1)^2 + \\cdots + (y_n-\\mu_n)^2 \\\\ \n&= (y_1-(\\beta_0+\\beta_1\\text{bmi}_1))^2 + \\cdots \\\\ \n\\end{align}$$\n\n# Example: predicting progression based on BMI (interpretation) (3)\n- In general, we can express the loss function as: \n$$l(\\beta) = (Y-X\\beta)^T(Y-X\\beta)$$ \n\n\n- And we can try to find the minimum of that function by:\n$$\\frac{dl(\\beta)}{d\\beta} = 0$$ \n\n\n- The solution to this is out of the scope of this class.\n\n$$\\hat{\\beta} = (X^TX)^{-1}X^Ty$$ \n\n\n\n# Supervised learning: Linear regression\n
    \n$$ \\text{dp} = b_0 + b_\\text{bmi} * \\text{bmi} + \\epsilon$$\n
    \n
    \n
    \n\n# Supervised learning: Linear regression (2)\n- Maximizing probability is equivalent to **minimizing squared errors of model**: \n$$ \\text{dp} = b_0 + b_\\text{bmi} * \\text{bmi} + \\epsilon$$\n
    \n
    \n
    \n\n# Supervised learning: Linear regression (3)\n- Maximizing probability is equivalent to **minimizing squared errors of model**: \n$$ \\text{dp} = b_0 + b_\\text{bmi} * \\text{bmi} + \\epsilon$$\n
    \n
    \n
    \n\n# Supervised learning: Linear regression (4)\n- Maximizing probability is equivalent to **minimizing squared errors of model**: \n$$ \\text{dp} = b_0 + b_\\text{bmi} * \\text{bmi} + \\epsilon$$\n
    \n
    \n
    \n\n# Supervised learning: Linear regression (5)\n- Linear regression finds the parameters that maximize the probability of observing the data points.\n$$ \\text{dp} = -117 + 10.23 * \\text{bmi} + \\epsilon$$\n
    \n
    \n\n# What can we do with our model?\n
    \n$$ \\text{dp} = -117 + 10.23 * \\text{bmi} + \\epsilon$$\n
    \n
    \n\n# Nonparametric method: Nearest neighbors\n
    \n
    \n
    \n
    \n

    For a new data point $\\hat{X}$ return the mean $Y$ of the closest $k$ points in the training data:

    \n
      \n
    • Very simple to implement.
      • \n
    • Needs lots of data to work well.
      • \n
    • Hard to interpret.
      • \n
    \n
    \n
    \n
    \n
    \n
    \n \n\n\n# Regression vs classification\n- When the variable we are trying to predict ($Y$) is quantitative, then we talk about **regression**. \n\n- When the variable is categorical (no easy comparison), then we talk about **classification**. \n\n- Examples of classification?\n\n# Classification: statistical formulation\n- In regression, we learn a *continuous probability distribution* such as Gaussian describing our outcome variable\n- In classification, we learn a *discrete probability distribution* for our outcome variable\n- This is because the variable $y$ takes on discrete values $C_1, C_2, \\dots, C_k$\n\n# Example: two-class logistic regression for classification\n\n- One of the simplest models for classification\n- It makes use of the Bernoulli probability distribution\n$$ p(y \\mid \\theta) = \\theta^y (1-\\theta)^{y-1}$$\nwhere $y \\in \\{0, 1 \\}$ and represent two classes $C_1$ and $C_2$\n- Can you give examples of two-class classification problems?\n\n# Example: logistic regression\n- More specifically, we map each set of features $x$ into a value between 0 and 1 using the following transformation:\n- We first do a linear map of $x$ into $z$:\n\n$$z = b_0 + b_1 x_1 + \\dots + b_m x_m = x^T b$$\n\n- And then, we perform a *non-linear transformation* of $z$\n\n$$\\sigma(z) = \\frac{1}{1 + e^{-z}}$$\nwhich is called the **sigmoid** transform. The sigmoid is constrained between 0 and 1 (plot)\n- We make $\\sigma(z)$ equal to the probability of $y = 1$ or $\\theta$ for a Bernoulli distribution.\n\n# Example: logistic regression (2)\n- Therefore, the likelihood of each observation is as follows\n\n$$p(y_i \\mid x_i, b) = \\theta_i^{y_i} (1 - \\theta_i)^{1 - y_i}$$\n\nwhere\n\n$$\\theta_i = \\sigma(x_i^T b)$$\n\n- Therefore, the likelihood of a set of observations is as follows\n\n$$p(y_1,\\dots,y_n, \\mid b, x_1, \\dots, x_n ) = \\prod_{i=1}^n p(y_i, \\mid x_i, b)$$\n\n# Example: logistic regression (3)\n\n- Maximizing the likelihood is similar to what we did for the Gaussian model\n- We will express the *negative log-likelihood* and *minimize it*:\n$$\\begin{align}\n\\text{nLL}(b) &= - \\log {\\prod_{i=1}^n p(y_i, \\mid x_i, b)}\\\\\n & = - \\sum_{i=1}^n \\log \\left( \\theta_i^{y_i} (1 - \\theta_i)^{1 - y_i} \\right)\\\\\n & = - \\sum_{i=1}^n \\left( y_i \\log \\theta_i + (1 - y_i) \\log (1 - \\theta_i) \\right)\n \\end{align}\n $$\n\n# Example: logistic regression (4)\n- Minimizing by finding the parameters that make the derivative 0 cannot be done in one step\n- *This does not have a closs solution for logistic regression:*\n$$ \\nabla \\text{nLL}(b) = 0 $$\n\n# Example: logistic regression (5)\n- We have to do **gradient descent** where we perform the following operations several times\n$$ b^{t+1} \\leftarrow b^{t} - \\lambda \\nabla \\text{nLL}(b)$$\n\n$$ \\nabla \\text{nLL}(b) = (\\frac{d \\text{nLL}(b)}{d b_0} \\; \\dots \\; \\frac{d \\text{nLL}(b)}{d b_m})$$\n\n# Example: logistic regression (6)\n\n$$ \n\\begin{align}\n\\frac{d \\text{nLL}(b)}{d b_j} &= \\frac{d}{d b_j} \\left[ - \\sum_{i=1}^n \\left( y_i \\log \\theta_i + (1 - y_i) \\log (1 - \\theta_i) \\right) \\right]\\\\\n&= - \\sum_{i=1}^n \\left( y_i \\frac{d}{d b_j} \\log \\theta_i + (1 - y_i) \\frac{d}{d b_j} \\log (1 - \\theta_i) \\right)\\\\\n&= - \\sum_{i=1}^n \\left( y_i - \\theta_i \\right) x_j \\\\\n\\end{align}\n$$\n- Derivation?\n\n# Example: logistic regression (7)\n\n$$ \\frac{d \\text{nLL}(b)}{d b_j} = - \\sum_{i=1}^n \\left( y_i - \\theta_i \\right) x_j $$\n\n- $y_i - \\theta_i$: how wrong our prediction is\n- $x_j$: the value of feature $j$\n\n# Logistic regression: What can we do with our model?\n\n- Interpretation is difficult:\n$$p(y = 1 | x, b) = \\sigma(b_0 + \\sum_{j=1}^{m} b_j x_j) $$\n- How can we interpret $b_j$?\n- For example, suppose we can predict whether a customer buys a product based on its price in dollars:\n$$p(\\text{customer buys} | \\text{price}) = \\sigma(0.2 - \\frac{1}{2} \\text{price}) $$\n- How to interpret the weight of price?\n\n# Logistic regression: What can we do with our model? (2)\n\n- One typical solution is to look at the probability changes around the 50% threshold ($z=0$)\n$$d = p(y = 1 | x=0 \\text{ but } x_j=1) - \\frac{1}{2} $$\n\n# Logistic regression: What can we do with our model? (3)\n- This expression $d$ however is still difficult, but we can do a first order approximation around $z=0$\n\n$$\n\\begin{align}\nd & \\approx \\frac{d}{d x_j} \\sigma(z)\\\\\n& = \\sigma(z) (1 - \\sigma(z)) b_j\\\\\n& = \\sigma(0) (1 - \\sigma(0)) b_j \\quad \\left[z=0\\right]\\\\\n& = \\frac{1}{4} b_j\\\\\n\\end{align}\n$$\n- Interpret: $p(\\text{customer buys} | \\text{price}) = \\sigma(0.2 - \\frac{1}{2} \\text{price}) $\n\n# Accuracy vs. interpretability tradeoff\nIn general:\n - Simple models are less accurate but more interpretable.\n - Complex models are more accurate but less interpretable. \n
    \n
    \n\n# Supervised vs unsupervised learning\n- **Supervised**:\n - Each $X$ is associated with a $Y$. \n \n- **Unsupervised**:\n - We have $X$ but no association. \n\n\n- Supervised learning is in general easier because we know how well we are building the association. \n\n- Unsupervised learning is harder because there is no clear evaluation method. \n\n- This course will mostly deal with **supervised learning**.\n\n# Unsupervised learning\n- There is no output.\n- It can be seen as learning a function that maps the input into an intermediate representation.\n- That intermediate representation makes the data easier to interpret.\n- There should be a map back from such intermediate representation and the original space.\n- The map from intermediate to feature space ($g$) should be as close as possible to the original input.\n
    \n\n# Examples of unsupervised learning\n- **Clustering**: cluster the diabetic patients into groups.\n- **Topic modeling**: describe the content of a set of documents (e.g., tweets) based on topics (soft clustering.)\n- **Dimensionality reduction**: describe a large set of features into a smaller set of features while retaining the variability of the data. \n
    \n
    \n\n# Take-home message\n1. Statistical learning acknowledges uncertainty and tries to learn a stochastic model of the data. \n\n2. We need to define a model of the data. \n\n3. We need to estimate the parameters of such model. \n\n4. We can use that model to predict or interpret the results. \n\n5. We can use supervised learning to learn a relationship between variables. \n\n6. If variables are not quantitative, we use classification models.\n", "meta": {"hexsha": "fe1198d4003904794cd9946245b1d6b276f3cd50", "size": 46538, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "slides/unit-05-1_statistical_learning.ipynb", "max_stars_repo_name": "daniel-acuna/ist718", "max_stars_repo_head_hexsha": "0a83f373aa00dc9cd1ff2e8da74d0255f04c9728", "max_stars_repo_licenses": ["BSD-4-Clause-UC"], "max_stars_count": 13, "max_stars_repo_stars_event_min_datetime": "2018-09-17T14:02:59.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-31T19:08:07.000Z", "max_issues_repo_path": "slides/unit-05-1_statistical_learning.ipynb", "max_issues_repo_name": "wozhouwozhou/ist718", "max_issues_repo_head_hexsha": "565e9767f6f35f77f9c14f2a94b2d75a0a6e2c02", "max_issues_repo_licenses": ["BSD-4-Clause-UC"], "max_issues_count": 5, "max_issues_repo_issues_event_min_datetime": "2020-03-24T15:51:10.000Z", "max_issues_repo_issues_event_max_datetime": "2021-12-13T19:48:14.000Z", "max_forks_repo_path": "slides/unit-05-1_statistical_learning.ipynb", "max_forks_repo_name": "wozhouwozhou/ist718", "max_forks_repo_head_hexsha": "565e9767f6f35f77f9c14f2a94b2d75a0a6e2c02", "max_forks_repo_licenses": ["BSD-4-Clause-UC"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2018-09-25T13:35:39.000Z", "max_forks_repo_forks_event_max_datetime": "2021-09-04T15:29:42.000Z", "avg_line_length": 31.08750835, "max_line_length": 467, "alphanum_fraction": 0.5301044308, "converted": true, "num_tokens": 7729, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746213017459, "lm_q2_score": 0.6893056167854461, "lm_q1q2_score": 0.4341071837722207}} {"text": "This notebook is used for training an LSTM model to predict the Remaining Useful Life (RUL) of turbofan engines based on simulated sensor data.\n\n\n```python\nimport glob\nimport os\nimport pandas as pd\nimport numpy as np\nimport matplotlib.pyplot as plt\n%matplotlib inline\n\n# Set for reproducability\nnp.random.seed(101) \nPYTHONHASHSEED = 0\n\nimport tensorflow as tf\nimport keras\nfrom keras.optimizers import Adam\nfrom keras.callbacks import TensorBoard, ModelCheckpoint, EarlyStopping, LearningRateScheduler\nfrom keras.models import Sequential, load_model\nfrom keras.layers import Dense, Dropout, LSTM, GRU, Masking, Activation, RepeatVector, TimeDistributed\n\nfrom sklearn.pipeline import Pipeline\nfrom sklearn import preprocessing\nfrom sklearn.metrics import confusion_matrix, recall_score, precision_score\n\nfrom data_generator import TSDataGenerator, split_data, create_generators\nfrom util import set_log_dir, rmse\nfrom util import LRDecay\n\nfrom tqdm import tqdm, tqdm_notebook\n\n```\n\n Using TensorFlow backend.\n\n\n\n```python\nDATA_DIR = os.path.abspath(\"./data/\")\nMODEL_DIR = os.path.abspath(\"./model/\")\n\npersist_run_stats = True # Enable for saving results to CouchDB\n```\n\n### Load Data\n\nThe data used for this project is the NASA C-MAPSS Turbofan Engine Degradation Data Set https://ti.arc.nasa.gov/c/6/. This data is model based simulated data from the Commercial Modular Aero-Propulsion System Simulation (C-MAPSS).\n\n\n```python\n!ls {DATA_DIR}\n```\n\n CMAPSSDATA.zip\t\t\t RUL_FD004.txt train.csv\r\n 'Damage Propagation Modeling.pdf' test_FD001.txt train_FD001.txt\r\n README.md\t\t\t test_FD002.txt train_FD002.txt\r\n readme.txt\t\t\t test_FD003.txt train_FD003.txt\r\n RUL_FD001.txt\t\t\t test_FD004.txt train_FD004.txt\r\n RUL_FD002.txt\t\t\t test_x.csv\r\n RUL_FD003.txt\t\t\t test_y.csv\r\n\n\nThe data set is a multivariate time series. Each entry (row) reflects an operational cycle of a specific engine identified by engine id and cycle time. There are multiple entries per engine to represent different reporting times. Other columns represents different features 3 operational settings and 21 sensors:\n\n
    \n    1)      engine id\n    2)      time, in cycles\n    3)      operational setting 1\n    4)      operational setting 2\n    5)      operational setting 3\n    6)      sensor measurement  1\n    7)      sensor measurement  2\n    ...\n    26)     sensor measurement  21\n
    \n\n\n```python\ncols = ['id', 'cycle' ]\n\n# Three operational setting columns\nsetting_cols = ['setting' + str(i) for i in range(1,4)]\ncols.extend(setting_cols)\n\n# Twenty one sensor columns\nsensor_cols = ['s' + str(i) for i in range(1,22)]\ncols.extend(sensor_cols)\n\nsort_cols = ['id','cycle']\n\n```\n\nThe CMAPSS data is divided into training, test, and RUL data files. Each of these is further partitioned in 4 subsets that represents a different operational condition. The number of engines in each vary.\n\n\n```python\n\nfn_id_map = {\n \"train_FD001\": 1000,\n \"train_FD002\": 2000,\n \"train_FD003\": 3000,\n \"train_FD004\": 4000,\n \"test_FD001\": 5000,\n \"test_FD002\": 6000,\n \"test_FD003\": 7000,\n \"test_FD004\": 8000, \n \"RUL_FD001\": 5000,\n \"RUL_FD002\": 6000,\n \"RUL_FD003\": 7000,\n \"RUL_FD004\": 8000, \n}\n\n\n# Filename is mapped to a condition. Map:\n# ONE (Sea Level) to 0\n# SIX to 1\nfn_condition_map = {\n \"train_FD001\": 1,\n \"train_FD002\": 2,\n \"train_FD003\": 1,\n \"train_FD004\": 2,\n \"test_FD001\": 1,\n \"test_FD002\": 2,\n \"test_FD003\": 1,\n \"test_FD004\": 2, \n}\n```\n\n \n\n\n```python\n\ndef load_data(paths, col_names, sort_cols):\n # read data \n df = pd.DataFrame()\n for p in paths:\n instance_df = pd.read_csv(p, sep=\" \", header=None)\n instance_df.drop(instance_df.columns[[26, 27]], axis=1, inplace=True)\n instance_df.columns = col_names\n instance_df['filename'] = os.path.splitext(os.path.basename(p))[0]\n \n df = pd.concat((df, instance_df), sort=False) \n\n df['condition'] = df['filename'].apply( lambda f: fn_condition_map[f])\n df['id'] = df['id'] + df['filename'].apply( lambda f: fn_id_map[f])\n df.drop(['filename'], axis=1, inplace=True)\n df = df.sort_values(sort_cols)\n return df\n```\n\nRead training, validation, and test data \n\n\n\n```python\npath = os.path.join(DATA_DIR, \"train_FD*.txt\")\nall_files = glob.glob(path)\n\ntrain_df = load_data(all_files, cols, sort_cols)\nprint(\"Train: \", train_df.shape)\n\n```\n\n Train: (160359, 27)\n\n\n\n```python\ntrain_df.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    idcyclesetting1setting2setting3s1s2s3s4s5...s13s14s15s16s17s18s19s20s21condition
    010011-0.0007-0.0004100.0518.67641.821589.701400.6014.62...2388.028138.628.41950.033922388100.039.0623.41901
    1100120.0019-0.0003100.0518.67642.151591.821403.1414.62...2388.078131.498.43180.033922388100.039.0023.42361
    210013-0.00430.0003100.0518.67642.351587.991404.2014.62...2388.038133.238.41780.033902388100.038.9523.34421
    3100140.00070.0000100.0518.67642.351582.791401.8714.62...2388.088133.838.36820.033922388100.038.8823.37391
    410015-0.0019-0.0002100.0518.67642.371582.851406.2214.62...2388.048133.808.42940.033932388100.038.9023.40441
    \n

    5 rows \u00d7 27 columns

    \n
    \n\n\n\n---\n## Data Preparation\n\nTwo step process. First step is to calculate the Remaining Useful Life (RUL) that will be used as the label. The calculation different when using training or test files. Second, the data will be transformed using a min/max scaler.\n\n### Calculate Training Data RUL\n\n\n```python\ndef calc_training_rul(df):\n # Data Labeling - generate column RUL\n rul = pd.DataFrame(df.groupby('id')['cycle'].max()).reset_index()\n rul.columns = ['id', 'max']\n df = df.merge(rul, on=['id'], how='left')\n df['RUL'] = df['max'] - df['cycle']\n df.drop('max', axis=1, inplace=True)\n return df\n```\n\n\n```python\ntrain_df = calc_training_rul(train_df)\ntrain_df.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    idcyclesetting1setting2setting3s1s2s3s4s5...s14s15s16s17s18s19s20s21conditionRUL
    010011-0.0007-0.0004100.0518.67641.821589.701400.6014.62...8138.628.41950.033922388100.039.0623.41901191
    1100120.0019-0.0003100.0518.67642.151591.821403.1414.62...8131.498.43180.033922388100.039.0023.42361190
    210013-0.00430.0003100.0518.67642.351587.991404.2014.62...8133.238.41780.033902388100.038.9523.34421189
    3100140.00070.0000100.0518.67642.351582.791401.8714.62...8133.838.36820.033922388100.038.8823.37391188
    410015-0.0019-0.0002100.0518.67642.371582.851406.2214.62...8133.808.42940.033932388100.038.9023.40441187
    \n

    5 rows \u00d7 28 columns

    \n
    \n\n\n\n\n### Data Transform\n\nAll transforms will be done using a pipeline. At this time only the min_max scalar is used.\n\n\n```python\npipeline = Pipeline(steps=[\n # The default activation function for LSTM tanh, so we'll use a range of [-1,1].\n ('min_max_scaler', preprocessing.MinMaxScaler(feature_range=(-1, 1)))\n])\n```\n\nTransform training data.\n\n\n```python\n# Set up the columns that will be scaled\ntrain_df['cycle_norm'] = train_df['cycle']\n\n# Transform all columns except id, cycle, and RUL\ncols_transform = train_df.columns.difference(['id','cycle', 'RUL'])\n\nxform_train_df = pd.DataFrame(pipeline.fit_transform(train_df[cols_transform]), \n columns=cols_transform, \n index=train_df.index)\njoin_df = train_df[train_df.columns.difference(cols_transform)].join(xform_train_df)\ntrain_df = join_df.reindex(columns = train_df.columns)\ntrain_df.head()\n```\n\n /home/saad/anaconda3/envs/tf/lib/python3.6/site-packages/sklearn/preprocessing/data.py:323: DataConversionWarning: Data with input dtype int64, float64 were all converted to float64 by MinMaxScaler.\n return self.partial_fit(X, y)\n\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    idcyclesetting1setting2setting3s1s2s3s4s5...s15s16s17s18s19s20s21conditionRULcycle_norm
    010011-0.999619-0.9995251.01.00.9399800.8545850.8042231.0...-0.8191441.00.8367351.01.00.9441640.940747-1.0191-1.00000
    110012-0.999495-0.9992881.01.00.9460000.8659150.8163841.0...-0.8106921.00.8367351.01.00.9401280.941260-1.0190-0.99631
    210013-0.999791-0.9978641.01.00.9496490.8454470.8214591.0...-0.8203121.00.7959181.01.00.9367640.932408-1.0189-0.99262
    310014-0.999553-0.9985761.01.00.9496490.8176570.8103041.0...-0.8543941.00.8367351.01.00.9320550.935719-1.0188-0.98893
    410015-0.999676-0.9990511.01.00.9500140.8179780.8311311.0...-0.8123411.00.8571431.01.00.9334010.939119-1.0187-0.98524
    \n

    5 rows \u00d7 29 columns

    \n
    \n\n\n\n---\n## Model\n\nIdentify the columns that will be used for features and labels.\n\n\n```python\n# Build the feature column list \nfeature_cols = ['cycle_norm', 'condition']\n\n# Three operational setting columns\nsetting_cols = ['setting' + str(i) for i in range(1,4)]\nfeature_cols.extend(setting_cols)\n\n# Twenty one sensor columns\nsensor_cols = ['s' + str(i) for i in range(1,22)]\nfeature_cols.extend(sensor_cols)\n\n# Build the label column list\nlabel_cols = ['RUL']\n```\n\n### LSTM Network\nThe model is an LSTM network. The first layer is an LSTM layer with 128 units followed by a set of Dense layers with 64, 32, and 1 units respectively. \n\n[Keras LSTM](https://keras.io/layers/recurrent/) layers expect an input in the shape of a numpy array of 3 dimensions (samples, time steps, features) where samples is the number of training sequences, time steps is the look back window or sequence length and features is the number of features of each sequence at each time step. \n\nThe LSTM layer is followed by three Dense layers with 64, 32 and finally 1 unit. All use a RELU activation function.\n\nAn Adam optimizer is used and Root Mean Squared Error (RMSE) is used for the loss function:\n\n\\begin{equation}\n RMSE = \\sqrt{ \\frac{1}{n} \\sum_{i=1}^{n} (\\hat{y}^i-RUL^i)^2 }\n\\end{equation}\n\nMean Squared Error (MSE) and Mean Absolute Error (MAE) are also tracked as metrics.\n\n\n\n```python\n\ndef create_model(batch_size, seq_length, num_features, num_labels):\n # build the network\n\n model = Sequential()\n\n model.add(Masking(mask_value=0., input_shape=(sequence_length, num_features)))\n\n model.add(LSTM(\n input_shape=(sequence_length, num_features),\n units=128,\n batch_input_shape=(batch_size, sequence_length, num_features),\n stateful=False,\n dropout=0.05,\n return_sequences=False))\n\n model.add(Dense(units=64, activation='relu'))\n model.add(Dense(units=32, activation='relu'))\n model.add(Dense(units=num_labels, activation='relu'))\n \n return model\n\n```\n\n\n```python\n# Size of the series time window.\nsequence_length = 25\n\n# Number of time series sequences that will be train on per batch.\nbatch_size = 512\n\nnum_features = len(feature_cols)\nnum_labels = len(label_cols)\n\n# Create the model\nmodel = create_model(batch_size, sequence_length, num_features, num_labels)\n\nopt = Adam(lr=1e-3, decay=0.0, amsgrad=True)\nmodel.compile(loss=rmse, optimizer=opt, metrics=['mse', 'mae'])\n```\n\n\n```python\nprint(model.summary())\n```\n\n _________________________________________________________________\n Layer (type) Output Shape Param # \n =================================================================\n masking_1 (Masking) (None, 25, 26) 0 \n _________________________________________________________________\n lstm_1 (LSTM) (None, 128) 79360 \n _________________________________________________________________\n dense_1 (Dense) (None, 64) 8256 \n _________________________________________________________________\n dense_2 (Dense) (None, 32) 2080 \n _________________________________________________________________\n dense_3 (Dense) (None, 1) 33 \n =================================================================\n Total params: 89,729\n Trainable params: 89,729\n Non-trainable params: 0\n _________________________________________________________________\n None\n\n\n---\n## Train\n\n\n```python\n# Setup log directory\nlog_dir, checkpoint_path = set_log_dir(MODEL_DIR, \"engine\")\n\nprint(\"Log dir: \", log_dir)\nprint(\"Checkpoint path: \", checkpoint_path)\n\n# Save the pipeline for later use\nfrom sklearn.externals import joblib \npipeline_path = os.path.join(log_dir, 'engine_pipeline.pkl') \njoblib.dump(pipeline, pipeline_path) \n```\n\n Log dir: /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916\n Checkpoint path: /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n\n\n\n\n\n ['/home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_pipeline.pkl']\n\n\n\n\n```python\ntraining_runs = 4\nnum_epochs = 200\ninitial_epoch = 0\nepochs_before_decay = 15\nlrate = 1e-3\npatience = 10\n \ntensorboard = TensorBoard(log_dir=log_dir,\n histogram_freq=0, write_graph=True, write_images=False)\n\ncheckpointer = ModelCheckpoint(checkpoint_path, verbose=1, save_best_only=True)\n\nepoch_loss_history = []\nepoch_val_history = []\nepoch_lr_history = []\n\nfor tr_run in range(training_runs):\n\n print(\"Training run: {}, epoch: {}\".format(tr_run, initial_epoch))\n\n t_df, v_df = split_data(train_df, randomize=True, train_pct=.8)\n train_data_generator, val_data_generator = create_generators(t_df, v_df, \n feature_cols, \n label_cols, \n batch_size=batch_size, \n sequence_length=sequence_length, \n randomize=True, \n loop=True,\n pad=False,\n verbose=True)\n \n # Callbacks\n lr_decay = LRDecay(initial_lrate=lrate, epochs_step=epochs_before_decay)\n lr_scheduler = LearningRateScheduler(lr_decay.step_decay, verbose=1)\n \n earlystopper = EarlyStopping(patience=patience, verbose=1)\n \n callbacks = [ tensorboard, checkpointer, lr_scheduler, earlystopper]\n\n # fit the network\n history = model.fit_generator(\n generator=train_data_generator.generate(), \n validation_data=val_data_generator.generate(), \n initial_epoch=initial_epoch,\n epochs=num_epochs, \n steps_per_epoch=train_data_generator.summary()['max_iterations'],\n validation_steps=val_data_generator.summary()['max_iterations'],\n shuffle=False,\n verbose=1,\n callbacks=callbacks )\n \n # pick up after the last epoch\n if len(history.epoch) > 0: \n initial_epoch = history.epoch[-1] + 1\n \n # TODO fix, sometimes Keras is returning an empty history dict.\n try:\n # Save loss/val metrics\n epoch_loss_history += history.history['loss']\n epoch_val_history += history.history['val_loss']\n epoch_lr_history += lr_decay.history_lr\n except:\n pass\n \n # reduce starting lr as we iterate into another training loop\n #lrate /= 10\n \n print(\"Loading previous best weights: \", checkpoint_path)\n model.load_weights(checkpoint_path)\n \n```\n\n Training run: 0, epoch: 0\n Engine split: Training=0.80, Validation=0.20\n Cycle split: Training=0.79, Validation=0.21\n Number of items: 567\n Undersized items: 0\n Data shape: (127034, 29)\n Max steps: 113426\n Max iterations: 221 @ 512\n Number of items: 142\n Undersized items: 0\n Data shape: (33325, 29)\n Max steps: 29917\n Max iterations: 58 @ 512\n Epoch 1/200\n \n Epoch 00001: LearningRateScheduler setting learning rate to 0.001.\n 221/221 [==============================] - 175s 790ms/step - loss: 82.7410 - mean_squared_error: 7828.9939 - mean_absolute_error: 68.3616 - val_loss: 80.9200 - val_mean_squared_error: 7356.1490 - val_mean_absolute_error: 65.7779\n \n Epoch 00001: val_loss improved from inf to 80.91996, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 2/200\n \n Epoch 00002: LearningRateScheduler setting learning rate to 0.001.\n 221/221 [==============================] - 169s 766ms/step - loss: 73.4294 - mean_squared_error: 6116.0378 - mean_absolute_error: 60.8045 - val_loss: 78.9407 - val_mean_squared_error: 6887.0101 - val_mean_absolute_error: 64.6546\n \n Epoch 00002: val_loss improved from 80.91996 to 78.94069, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 3/200\n \n Epoch 00003: LearningRateScheduler setting learning rate to 0.001.\n 221/221 [==============================] - 171s 774ms/step - loss: 73.8117 - mean_squared_error: 6109.2091 - mean_absolute_error: 60.7736 - val_loss: 78.8043 - val_mean_squared_error: 6922.2818 - val_mean_absolute_error: 64.7921\n \n Epoch 00003: val_loss improved from 78.94069 to 78.80430, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 4/200\n \n Epoch 00004: LearningRateScheduler setting learning rate to 0.001.\n 221/221 [==============================] - 170s 771ms/step - loss: 72.2080 - mean_squared_error: 5966.2634 - mean_absolute_error: 59.4079 - val_loss: 77.2404 - val_mean_squared_error: 6565.2801 - val_mean_absolute_error: 62.5842\n \n Epoch 00004: val_loss improved from 78.80430 to 77.24043, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 5/200\n \n Epoch 00005: LearningRateScheduler setting learning rate to 0.001.\n 221/221 [==============================] - 171s 775ms/step - loss: 63.7693 - mean_squared_error: 4922.2971 - mean_absolute_error: 51.8710 - val_loss: 65.1067 - val_mean_squared_error: 4783.9595 - val_mean_absolute_error: 54.9560\n \n Epoch 00005: val_loss improved from 77.24043 to 65.10669, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 6/200\n \n Epoch 00006: LearningRateScheduler setting learning rate to 0.001.\n 221/221 [==============================] - 169s 766ms/step - loss: 60.8147 - mean_squared_error: 4614.4337 - mean_absolute_error: 49.4630 - val_loss: 66.0941 - val_mean_squared_error: 5270.6374 - val_mean_absolute_error: 52.8529\n \n Epoch 00006: val_loss did not improve from 65.10669\n Epoch 7/200\n \n Epoch 00007: LearningRateScheduler setting learning rate to 0.001.\n 221/221 [==============================] - 171s 775ms/step - loss: 59.5111 - mean_squared_error: 4367.9204 - mean_absolute_error: 48.7288 - val_loss: 64.8260 - val_mean_squared_error: 5220.2637 - val_mean_absolute_error: 52.5697\n \n Epoch 00007: val_loss improved from 65.10669 to 64.82600, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 8/200\n \n Epoch 00008: LearningRateScheduler setting learning rate to 0.001.\n 221/221 [==============================] - 170s 771ms/step - loss: 57.6040 - mean_squared_error: 4126.1646 - mean_absolute_error: 47.0149 - val_loss: 63.2307 - val_mean_squared_error: 5111.1676 - val_mean_absolute_error: 50.5345\n \n Epoch 00008: val_loss improved from 64.82600 to 63.23067, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 9/200\n \n Epoch 00009: LearningRateScheduler setting learning rate to 0.001.\n 221/221 [==============================] - 172s 776ms/step - loss: 55.1983 - mean_squared_error: 3995.0672 - mean_absolute_error: 45.0184 - val_loss: 68.2289 - val_mean_squared_error: 6030.8233 - val_mean_absolute_error: 54.4334\n \n Epoch 00009: val_loss did not improve from 63.23067\n Epoch 10/200\n \n Epoch 00010: LearningRateScheduler setting learning rate to 0.001.\n 221/221 [==============================] - 171s 774ms/step - loss: 55.8896 - mean_squared_error: 4050.7812 - mean_absolute_error: 45.2195 - val_loss: 59.0314 - val_mean_squared_error: 4356.2175 - val_mean_absolute_error: 47.0952\n \n Epoch 00010: val_loss improved from 63.23067 to 59.03142, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 11/200\n \n Epoch 00011: LearningRateScheduler setting learning rate to 0.001.\n 221/221 [==============================] - 170s 769ms/step - loss: 54.0163 - mean_squared_error: 3799.6754 - mean_absolute_error: 43.9449 - val_loss: 57.3357 - val_mean_squared_error: 4196.8567 - val_mean_absolute_error: 46.5990\n \n Epoch 00011: val_loss improved from 59.03142 to 57.33572, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 12/200\n \n Epoch 00012: LearningRateScheduler setting learning rate to 0.001.\n 221/221 [==============================] - 171s 772ms/step - loss: 53.7663 - mean_squared_error: 3807.5294 - mean_absolute_error: 43.5652 - val_loss: 62.2213 - val_mean_squared_error: 5003.8425 - val_mean_absolute_error: 50.3692\n \n Epoch 00012: val_loss did not improve from 57.33572\n Epoch 13/200\n \n Epoch 00013: LearningRateScheduler setting learning rate to 0.001.\n 221/221 [==============================] - 171s 774ms/step - loss: 53.5564 - mean_squared_error: 3725.8787 - mean_absolute_error: 43.0080 - val_loss: 53.3707 - val_mean_squared_error: 3677.1034 - val_mean_absolute_error: 43.7705\n \n Epoch 00013: val_loss improved from 57.33572 to 53.37070, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 14/200\n \n Epoch 00014: LearningRateScheduler setting learning rate to 0.001.\n 221/221 [==============================] - 171s 775ms/step - loss: 52.9694 - mean_squared_error: 3700.7633 - mean_absolute_error: 42.8857 - val_loss: 56.1607 - val_mean_squared_error: 3910.4719 - val_mean_absolute_error: 46.1223\n \n Epoch 00014: val_loss did not improve from 53.37070\n Epoch 15/200\n \n Epoch 00015: LearningRateScheduler setting learning rate to 0.00049067.\n 221/221 [==============================] - 170s 768ms/step - loss: 51.8930 - mean_squared_error: 3504.6674 - mean_absolute_error: 41.8107 - val_loss: 55.2183 - val_mean_squared_error: 3749.5455 - val_mean_absolute_error: 43.7075\n \n Epoch 00015: val_loss did not improve from 53.37070\n Epoch 16/200\n \n Epoch 00016: LearningRateScheduler setting learning rate to 0.00049067.\n 221/221 [==============================] - 170s 771ms/step - loss: 51.0752 - mean_squared_error: 3397.4955 - mean_absolute_error: 41.1946 - val_loss: 56.0740 - val_mean_squared_error: 4087.7122 - val_mean_absolute_error: 44.4838\n \n Epoch 00016: val_loss did not improve from 53.37070\n Epoch 17/200\n \n Epoch 00017: LearningRateScheduler setting learning rate to 0.00049067.\n 221/221 [==============================] - 170s 771ms/step - loss: 50.8982 - mean_squared_error: 3579.0307 - mean_absolute_error: 41.2423 - val_loss: 55.5619 - val_mean_squared_error: 3833.9144 - val_mean_absolute_error: 44.1653\n \n Epoch 00017: val_loss did not improve from 53.37070\n Epoch 18/200\n \n Epoch 00018: LearningRateScheduler setting learning rate to 0.00049067.\n 221/221 [==============================] - 171s 776ms/step - loss: 50.2696 - mean_squared_error: 3351.6375 - mean_absolute_error: 40.5340 - val_loss: 55.7762 - val_mean_squared_error: 4070.8367 - val_mean_absolute_error: 43.5560\n \n Epoch 00018: val_loss did not improve from 53.37070\n Epoch 19/200\n \n Epoch 00019: LearningRateScheduler setting learning rate to 0.00049067.\n 221/221 [==============================] - 172s 780ms/step - loss: 50.0818 - mean_squared_error: 3443.1074 - mean_absolute_error: 40.6007 - val_loss: 50.9779 - val_mean_squared_error: 3093.6141 - val_mean_absolute_error: 42.6931\n \n Epoch 00019: val_loss improved from 53.37070 to 50.97791, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 20/200\n \n Epoch 00020: LearningRateScheduler setting learning rate to 0.00049067.\n 221/221 [==============================] - 171s 772ms/step - loss: 48.6803 - mean_squared_error: 3214.4156 - mean_absolute_error: 39.3914 - val_loss: 57.4121 - val_mean_squared_error: 4480.9729 - val_mean_absolute_error: 45.5967\n \n Epoch 00020: val_loss did not improve from 50.97791\n Epoch 21/200\n \n Epoch 00021: LearningRateScheduler setting learning rate to 0.00049067.\n 221/221 [==============================] - 172s 779ms/step - loss: 50.1434 - mean_squared_error: 3312.1235 - mean_absolute_error: 40.1076 - val_loss: 53.4099 - val_mean_squared_error: 3548.8934 - val_mean_absolute_error: 41.4218\n \n Epoch 00021: val_loss did not improve from 50.97791\n Epoch 22/200\n \n Epoch 00022: LearningRateScheduler setting learning rate to 0.00049067.\n 221/221 [==============================] - 169s 767ms/step - loss: 49.8854 - mean_squared_error: 3334.8621 - mean_absolute_error: 40.2642 - val_loss: 51.5347 - val_mean_squared_error: 3337.7774 - val_mean_absolute_error: 40.7340\n \n Epoch 00022: val_loss did not improve from 50.97791\n Epoch 23/200\n \n Epoch 00023: LearningRateScheduler setting learning rate to 0.00049067.\n 221/221 [==============================] - 171s 776ms/step - loss: 48.2216 - mean_squared_error: 3072.3540 - mean_absolute_error: 38.5349 - val_loss: 56.3896 - val_mean_squared_error: 4049.4748 - val_mean_absolute_error: 45.0704\n \n Epoch 00023: val_loss did not improve from 50.97791\n Epoch 24/200\n \n Epoch 00024: LearningRateScheduler setting learning rate to 0.00049067.\n 221/221 [==============================] - 170s 770ms/step - loss: 50.7495 - mean_squared_error: 3443.1058 - mean_absolute_error: 40.8825 - val_loss: 52.8592 - val_mean_squared_error: 3608.8424 - val_mean_absolute_error: 42.5599\n \n Epoch 00024: val_loss did not improve from 50.97791\n Epoch 25/200\n \n Epoch 00025: LearningRateScheduler setting learning rate to 0.00049067.\n 221/221 [==============================] - 171s 772ms/step - loss: 49.9125 - mean_squared_error: 3316.0165 - mean_absolute_error: 40.1574 - val_loss: 53.2067 - val_mean_squared_error: 3555.6456 - val_mean_absolute_error: 42.5659\n \n Epoch 00025: val_loss did not improve from 50.97791\n Epoch 26/200\n \n Epoch 00026: LearningRateScheduler setting learning rate to 0.00049067.\n 221/221 [==============================] - 172s 779ms/step - loss: 48.4906 - mean_squared_error: 2995.8786 - mean_absolute_error: 38.9174 - val_loss: 56.8230 - val_mean_squared_error: 4331.9377 - val_mean_absolute_error: 44.5568\n \n Epoch 00026: val_loss did not improve from 50.97791\n Epoch 27/200\n \n Epoch 00027: LearningRateScheduler setting learning rate to 0.00049067.\n 221/221 [==============================] - 170s 770ms/step - loss: 49.5727 - mean_squared_error: 3350.3886 - mean_absolute_error: 39.8464 - val_loss: 52.2079 - val_mean_squared_error: 3428.1991 - val_mean_absolute_error: 41.4118\n \n Epoch 00027: val_loss did not improve from 50.97791\n Epoch 28/200\n \n Epoch 00028: LearningRateScheduler setting learning rate to 0.00049067.\n 221/221 [==============================] - 170s 768ms/step - loss: 49.6553 - mean_squared_error: 3201.2799 - mean_absolute_error: 39.3550 - val_loss: 58.5301 - val_mean_squared_error: 4443.4937 - val_mean_absolute_error: 45.9268\n \n Epoch 00028: val_loss did not improve from 50.97791\n Epoch 29/200\n \n Epoch 00029: LearningRateScheduler setting learning rate to 0.00049067.\n 221/221 [==============================] - 172s 777ms/step - loss: 49.3769 - mean_squared_error: 3244.2433 - mean_absolute_error: 39.1454 - val_loss: 53.9256 - val_mean_squared_error: 3593.5663 - val_mean_absolute_error: 42.8261\n \n Epoch 00029: val_loss did not improve from 50.97791\n Epoch 00029: early stopping\n Loading previous best weights: /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Training run: 1, epoch: 29\n Engine split: Training=0.80, Validation=0.20\n Cycle split: Training=0.80, Validation=0.20\n Number of items: 567\n Undersized items: 0\n Data shape: (127943, 29)\n Max steps: 114335\n Max iterations: 223 @ 512\n Number of items: 142\n Undersized items: 0\n Data shape: (32416, 29)\n Max steps: 29008\n Max iterations: 56 @ 512\n Epoch 30/200\n \n Epoch 00030: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 177s 793ms/step - loss: 51.7171 - mean_squared_error: 3482.9756 - mean_absolute_error: 41.6893 - val_loss: 52.1176 - val_mean_squared_error: 3708.3443 - val_mean_absolute_error: 41.9418\n \n Epoch 00030: val_loss did not improve from 50.97791\n Epoch 31/200\n \n Epoch 00031: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 172s 769ms/step - loss: 50.5687 - mean_squared_error: 3349.7572 - mean_absolute_error: 41.0085 - val_loss: 51.6841 - val_mean_squared_error: 3671.9269 - val_mean_absolute_error: 41.9979\n \n Epoch 00031: val_loss did not improve from 50.97791\n Epoch 32/200\n \n Epoch 00032: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 171s 765ms/step - loss: 50.5706 - mean_squared_error: 3388.0263 - mean_absolute_error: 41.0224 - val_loss: 55.6557 - val_mean_squared_error: 3885.5620 - val_mean_absolute_error: 45.9316\n \n Epoch 00032: val_loss did not improve from 50.97791\n Epoch 33/200\n \n Epoch 00033: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 170s 762ms/step - loss: 51.3301 - mean_squared_error: 3331.6447 - mean_absolute_error: 41.1051 - val_loss: 55.5534 - val_mean_squared_error: 4194.7698 - val_mean_absolute_error: 43.3639\n \n Epoch 00033: val_loss did not improve from 50.97791\n Epoch 34/200\n \n Epoch 00034: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 171s 767ms/step - loss: 50.0764 - mean_squared_error: 3256.9120 - mean_absolute_error: 40.6946 - val_loss: 54.3125 - val_mean_squared_error: 4002.5222 - val_mean_absolute_error: 44.7414\n \n Epoch 00034: val_loss did not improve from 50.97791\n Epoch 35/200\n \n Epoch 00035: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 170s 764ms/step - loss: 51.1510 - mean_squared_error: 3428.1141 - mean_absolute_error: 41.2382 - val_loss: 54.7492 - val_mean_squared_error: 3398.5948 - val_mean_absolute_error: 45.9703\n \n Epoch 00035: val_loss did not improve from 50.97791\n Epoch 36/200\n \n Epoch 00036: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 171s 768ms/step - loss: 49.8745 - mean_squared_error: 3249.3838 - mean_absolute_error: 40.4786 - val_loss: 54.1422 - val_mean_squared_error: 4278.7387 - val_mean_absolute_error: 43.8949\n \n Epoch 00036: val_loss did not improve from 50.97791\n Epoch 37/200\n \n Epoch 00037: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 171s 768ms/step - loss: 50.5978 - mean_squared_error: 3335.5922 - mean_absolute_error: 40.5832 - val_loss: 49.2238 - val_mean_squared_error: 3480.7149 - val_mean_absolute_error: 40.3681\n \n Epoch 00037: val_loss improved from 50.97791 to 49.22382, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 38/200\n \n Epoch 00038: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 172s 773ms/step - loss: 50.8264 - mean_squared_error: 3386.0788 - mean_absolute_error: 41.1672 - val_loss: 50.8602 - val_mean_squared_error: 3378.7921 - val_mean_absolute_error: 41.1133\n \n Epoch 00038: val_loss did not improve from 49.22382\n Epoch 39/200\n \n Epoch 00039: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 171s 765ms/step - loss: 50.0873 - mean_squared_error: 3203.8677 - mean_absolute_error: 40.3460 - val_loss: 61.0342 - val_mean_squared_error: 4968.7473 - val_mean_absolute_error: 47.8055\n \n Epoch 00039: val_loss did not improve from 49.22382\n Epoch 40/200\n \n Epoch 00040: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 170s 762ms/step - loss: 49.9364 - mean_squared_error: 3278.7989 - mean_absolute_error: 39.7868 - val_loss: 55.8622 - val_mean_squared_error: 3802.4901 - val_mean_absolute_error: 47.0354\n \n Epoch 00040: val_loss did not improve from 49.22382\n Epoch 41/200\n \n Epoch 00041: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 171s 767ms/step - loss: 50.2608 - mean_squared_error: 3228.0728 - mean_absolute_error: 40.2343 - val_loss: 52.3863 - val_mean_squared_error: 3343.8958 - val_mean_absolute_error: 43.5816\n \n Epoch 00041: val_loss did not improve from 49.22382\n Epoch 42/200\n \n Epoch 00042: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 171s 767ms/step - loss: 49.5806 - mean_squared_error: 3164.7433 - mean_absolute_error: 39.8622 - val_loss: 50.8606 - val_mean_squared_error: 3576.2737 - val_mean_absolute_error: 39.4924\n \n Epoch 00042: val_loss did not improve from 49.22382\n Epoch 43/200\n \n Epoch 00043: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 171s 768ms/step - loss: 47.8170 - mean_squared_error: 2967.2399 - mean_absolute_error: 38.5124 - val_loss: 50.4866 - val_mean_squared_error: 3405.0406 - val_mean_absolute_error: 41.0014\n \n Epoch 00043: val_loss did not improve from 49.22382\n Epoch 44/200\n \n Epoch 00044: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 171s 768ms/step - loss: 49.1404 - mean_squared_error: 3126.0838 - mean_absolute_error: 39.2404 - val_loss: 47.6679 - val_mean_squared_error: 3218.6241 - val_mean_absolute_error: 37.6643\n \n Epoch 00044: val_loss improved from 49.22382 to 47.66794, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 45/200\n \n Epoch 00045: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 172s 770ms/step - loss: 47.1094 - mean_squared_error: 2797.4657 - mean_absolute_error: 37.5768 - val_loss: 47.7376 - val_mean_squared_error: 3155.3952 - val_mean_absolute_error: 37.6434\n \n Epoch 00045: val_loss did not improve from 47.66794\n Epoch 46/200\n \n Epoch 00046: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 171s 766ms/step - loss: 48.8475 - mean_squared_error: 3120.9277 - mean_absolute_error: 38.7644 - val_loss: 49.7684 - val_mean_squared_error: 3474.2000 - val_mean_absolute_error: 39.4820\n \n Epoch 00046: val_loss did not improve from 47.66794\n Epoch 47/200\n \n Epoch 00047: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 170s 764ms/step - loss: 47.7777 - mean_squared_error: 2967.1358 - mean_absolute_error: 37.7790 - val_loss: 45.1925 - val_mean_squared_error: 2603.0956 - val_mean_absolute_error: 37.1032\n \n Epoch 00047: val_loss improved from 47.66794 to 45.19248, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 48/200\n \n Epoch 00048: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 172s 772ms/step - loss: 45.9461 - mean_squared_error: 2719.8844 - mean_absolute_error: 37.0818 - val_loss: 53.6253 - val_mean_squared_error: 3618.6470 - val_mean_absolute_error: 42.1127\n \n Epoch 00048: val_loss did not improve from 45.19248\n Epoch 49/200\n \n Epoch 00049: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 171s 768ms/step - loss: 49.2976 - mean_squared_error: 3116.8867 - mean_absolute_error: 39.1072 - val_loss: 52.4682 - val_mean_squared_error: 4236.1338 - val_mean_absolute_error: 41.9199\n \n Epoch 00049: val_loss did not improve from 45.19248\n Epoch 50/200\n \n Epoch 00050: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 171s 769ms/step - loss: 47.4937 - mean_squared_error: 2900.1720 - mean_absolute_error: 37.8936 - val_loss: 52.8261 - val_mean_squared_error: 3696.0184 - val_mean_absolute_error: 42.3109\n \n Epoch 00050: val_loss did not improve from 45.19248\n Epoch 51/200\n \n Epoch 00051: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 171s 768ms/step - loss: 47.2459 - mean_squared_error: 2922.2539 - mean_absolute_error: 37.5051 - val_loss: 45.3990 - val_mean_squared_error: 2767.1730 - val_mean_absolute_error: 36.6157\n \n Epoch 00051: val_loss did not improve from 45.19248\n Epoch 52/200\n \n Epoch 00052: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 171s 767ms/step - loss: 46.3145 - mean_squared_error: 2835.0806 - mean_absolute_error: 37.1046 - val_loss: 47.6797 - val_mean_squared_error: 2799.4648 - val_mean_absolute_error: 38.1494\n \n Epoch 00052: val_loss did not improve from 45.19248\n Epoch 53/200\n \n Epoch 00053: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 170s 764ms/step - loss: 46.9924 - mean_squared_error: 2858.3458 - mean_absolute_error: 37.3979 - val_loss: 47.6297 - val_mean_squared_error: 3311.7591 - val_mean_absolute_error: 37.5152\n \n Epoch 00053: val_loss did not improve from 45.19248\n Epoch 54/200\n \n Epoch 00054: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 171s 765ms/step - loss: 46.2032 - mean_squared_error: 2846.1004 - mean_absolute_error: 37.0614 - val_loss: 49.7124 - val_mean_squared_error: 3203.1269 - val_mean_absolute_error: 40.6326\n \n Epoch 00054: val_loss did not improve from 45.19248\n Epoch 55/200\n \n Epoch 00055: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 170s 763ms/step - loss: 45.5385 - mean_squared_error: 2679.5357 - mean_absolute_error: 36.2483 - val_loss: 50.0502 - val_mean_squared_error: 3288.3536 - val_mean_absolute_error: 41.0041\n \n Epoch 00055: val_loss did not improve from 45.19248\n Epoch 56/200\n \n Epoch 00056: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 171s 765ms/step - loss: 46.5553 - mean_squared_error: 2838.6112 - mean_absolute_error: 36.8946 - val_loss: 52.4525 - val_mean_squared_error: 3627.3432 - val_mean_absolute_error: 41.8988\n \n Epoch 00056: val_loss did not improve from 45.19248\n Epoch 57/200\n \n Epoch 00057: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 171s 765ms/step - loss: 46.6657 - mean_squared_error: 2840.6583 - mean_absolute_error: 37.0303 - val_loss: 48.9920 - val_mean_squared_error: 3234.9856 - val_mean_absolute_error: 39.5501\n \n Epoch 00057: val_loss did not improve from 45.19248\n Epoch 00057: early stopping\n Loading previous best weights: /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Training run: 2, epoch: 57\n Engine split: Training=0.80, Validation=0.20\n Cycle split: Training=0.80, Validation=0.20\n Number of items: 567\n Undersized items: 0\n Data shape: (128185, 29)\n Max steps: 114577\n Max iterations: 223 @ 512\n Number of items: 142\n Undersized items: 0\n Data shape: (32174, 29)\n Max steps: 28766\n Max iterations: 56 @ 512\n Epoch 58/200\n \n Epoch 00058: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 177s 794ms/step - loss: 50.9558 - mean_squared_error: 3356.9982 - mean_absolute_error: 40.4163 - val_loss: 41.5364 - val_mean_squared_error: 2359.7258 - val_mean_absolute_error: 32.9559\n \n Epoch 00058: val_loss improved from 45.19248 to 41.53640, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 59/200\n \n Epoch 00059: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 171s 766ms/step - loss: 50.0820 - mean_squared_error: 3166.6616 - mean_absolute_error: 39.9118 - val_loss: 44.1802 - val_mean_squared_error: 2852.1871 - val_mean_absolute_error: 35.0597\n \n Epoch 00059: val_loss did not improve from 41.53640\n Epoch 60/200\n \n Epoch 00060: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 171s 767ms/step - loss: 49.7509 - mean_squared_error: 3230.8675 - mean_absolute_error: 40.1815 - val_loss: 45.7048 - val_mean_squared_error: 2877.0496 - val_mean_absolute_error: 35.4414\n \n Epoch 00060: val_loss did not improve from 41.53640\n Epoch 61/200\n \n Epoch 00061: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 170s 762ms/step - loss: 50.0737 - mean_squared_error: 3236.2220 - mean_absolute_error: 40.0612 - val_loss: 43.6050 - val_mean_squared_error: 2518.5998 - val_mean_absolute_error: 32.8914\n \n Epoch 00061: val_loss did not improve from 41.53640\n Epoch 62/200\n \n Epoch 00062: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 171s 768ms/step - loss: 50.7214 - mean_squared_error: 3337.8501 - mean_absolute_error: 40.7784 - val_loss: 44.0467 - val_mean_squared_error: 2568.8256 - val_mean_absolute_error: 33.9893\n \n Epoch 00062: val_loss did not improve from 41.53640\n Epoch 63/200\n \n Epoch 00063: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 170s 764ms/step - loss: 49.9818 - mean_squared_error: 3214.3525 - mean_absolute_error: 40.3672 - val_loss: 40.9317 - val_mean_squared_error: 2367.0996 - val_mean_absolute_error: 32.8321\n \n Epoch 00063: val_loss improved from 41.53640 to 40.93173, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 64/200\n \n Epoch 00064: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 171s 768ms/step - loss: 49.3156 - mean_squared_error: 3158.0953 - mean_absolute_error: 39.4025 - val_loss: 43.7492 - val_mean_squared_error: 2391.9892 - val_mean_absolute_error: 34.7741\n \n Epoch 00064: val_loss did not improve from 40.93173\n Epoch 65/200\n \n Epoch 00065: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 170s 763ms/step - loss: 48.5260 - mean_squared_error: 3025.9707 - mean_absolute_error: 38.6469 - val_loss: 43.4658 - val_mean_squared_error: 2527.0157 - val_mean_absolute_error: 35.3154\n \n Epoch 00065: val_loss did not improve from 40.93173\n Epoch 66/200\n \n Epoch 00066: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 170s 763ms/step - loss: 49.3226 - mean_squared_error: 3199.6746 - mean_absolute_error: 39.3560 - val_loss: 41.5372 - val_mean_squared_error: 2242.5722 - val_mean_absolute_error: 33.9836\n \n Epoch 00066: val_loss did not improve from 40.93173\n Epoch 67/200\n \n Epoch 00067: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 171s 765ms/step - loss: 49.7449 - mean_squared_error: 3092.4901 - mean_absolute_error: 39.4234 - val_loss: 42.7124 - val_mean_squared_error: 2351.3885 - val_mean_absolute_error: 34.6492\n \n Epoch 00067: val_loss did not improve from 40.93173\n Epoch 68/200\n \n Epoch 00068: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 171s 767ms/step - loss: 49.8736 - mean_squared_error: 3249.7648 - mean_absolute_error: 39.6734 - val_loss: 42.6367 - val_mean_squared_error: 2571.7652 - val_mean_absolute_error: 32.8977\n \n Epoch 00068: val_loss did not improve from 40.93173\n Epoch 69/200\n \n Epoch 00069: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 171s 767ms/step - loss: 49.2586 - mean_squared_error: 3117.7979 - mean_absolute_error: 39.0369 - val_loss: 43.3282 - val_mean_squared_error: 2331.9929 - val_mean_absolute_error: 34.2937\n \n Epoch 00069: val_loss did not improve from 40.93173\n Epoch 70/200\n \n Epoch 00070: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 171s 765ms/step - loss: 48.0065 - mean_squared_error: 3019.8103 - mean_absolute_error: 38.5582 - val_loss: 43.6003 - val_mean_squared_error: 2545.8150 - val_mean_absolute_error: 35.5245\n \n Epoch 00070: val_loss did not improve from 40.93173\n Epoch 71/200\n \n Epoch 00071: LearningRateScheduler setting learning rate to 0.001.\n 223/223 [==============================] - 170s 761ms/step - loss: 50.7352 - mean_squared_error: 3299.1254 - mean_absolute_error: 40.4949 - val_loss: 41.4907 - val_mean_squared_error: 2311.1255 - val_mean_absolute_error: 31.9480\n \n Epoch 00071: val_loss did not improve from 40.93173\n Epoch 72/200\n \n Epoch 00072: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 172s 771ms/step - loss: 47.3094 - mean_squared_error: 2998.8858 - mean_absolute_error: 37.8157 - val_loss: 43.2020 - val_mean_squared_error: 2578.0438 - val_mean_absolute_error: 34.7546\n \n Epoch 00072: val_loss did not improve from 40.93173\n Epoch 73/200\n \n Epoch 00073: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 170s 762ms/step - loss: 48.9405 - mean_squared_error: 3027.6780 - mean_absolute_error: 38.7400 - val_loss: 40.8030 - val_mean_squared_error: 1902.8859 - val_mean_absolute_error: 33.5064\n \n Epoch 00073: val_loss improved from 40.93173 to 40.80302, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 74/200\n \n Epoch 00074: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 172s 770ms/step - loss: 48.7538 - mean_squared_error: 3057.5313 - mean_absolute_error: 39.1746 - val_loss: 42.0624 - val_mean_squared_error: 2608.1424 - val_mean_absolute_error: 32.2830\n \n Epoch 00074: val_loss did not improve from 40.80302\n Epoch 75/200\n \n Epoch 00075: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 169s 760ms/step - loss: 47.2065 - mean_squared_error: 3005.6398 - mean_absolute_error: 37.7168 - val_loss: 42.0927 - val_mean_squared_error: 2423.9991 - val_mean_absolute_error: 32.8521\n \n Epoch 00075: val_loss did not improve from 40.80302\n Epoch 76/200\n \n Epoch 00076: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 171s 765ms/step - loss: 47.3881 - mean_squared_error: 2876.3997 - mean_absolute_error: 37.5880 - val_loss: 39.6096 - val_mean_squared_error: 2095.8231 - val_mean_absolute_error: 31.5156\n \n Epoch 00076: val_loss improved from 40.80302 to 39.60961, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 77/200\n \n Epoch 00077: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 171s 767ms/step - loss: 46.8346 - mean_squared_error: 2885.7237 - mean_absolute_error: 37.3162 - val_loss: 39.8949 - val_mean_squared_error: 2220.7993 - val_mean_absolute_error: 30.8580\n \n Epoch 00077: val_loss did not improve from 39.60961\n Epoch 78/200\n \n Epoch 00078: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 172s 772ms/step - loss: 47.9390 - mean_squared_error: 2976.8742 - mean_absolute_error: 38.2056 - val_loss: 41.2636 - val_mean_squared_error: 1927.9644 - val_mean_absolute_error: 33.5595\n \n Epoch 00078: val_loss did not improve from 39.60961\n Epoch 79/200\n \n Epoch 00079: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 171s 765ms/step - loss: 46.9223 - mean_squared_error: 2983.6417 - mean_absolute_error: 37.4561 - val_loss: 44.0659 - val_mean_squared_error: 2579.6663 - val_mean_absolute_error: 36.0412\n \n Epoch 00079: val_loss did not improve from 39.60961\n Epoch 80/200\n \n Epoch 00080: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 170s 764ms/step - loss: 46.0406 - mean_squared_error: 2890.6206 - mean_absolute_error: 36.6235 - val_loss: 41.7041 - val_mean_squared_error: 2278.0436 - val_mean_absolute_error: 32.6757\n \n Epoch 00080: val_loss did not improve from 39.60961\n Epoch 81/200\n \n Epoch 00081: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 170s 763ms/step - loss: 46.7944 - mean_squared_error: 2844.2725 - mean_absolute_error: 37.3749 - val_loss: 40.4825 - val_mean_squared_error: 2262.4665 - val_mean_absolute_error: 31.0324\n \n Epoch 00081: val_loss did not improve from 39.60961\n Epoch 82/200\n \n Epoch 00082: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 171s 767ms/step - loss: 47.2630 - mean_squared_error: 2964.6351 - mean_absolute_error: 37.3166 - val_loss: 40.1976 - val_mean_squared_error: 2184.6751 - val_mean_absolute_error: 31.4182\n \n Epoch 00082: val_loss did not improve from 39.60961\n Epoch 83/200\n \n Epoch 00083: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 171s 766ms/step - loss: 46.7734 - mean_squared_error: 2884.1777 - mean_absolute_error: 37.2751 - val_loss: 39.8123 - val_mean_squared_error: 2192.6067 - val_mean_absolute_error: 30.2782\n \n Epoch 00083: val_loss did not improve from 39.60961\n Epoch 84/200\n \n Epoch 00084: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 170s 763ms/step - loss: 46.7485 - mean_squared_error: 2845.9328 - mean_absolute_error: 37.0914 - val_loss: 38.9177 - val_mean_squared_error: 1934.4509 - val_mean_absolute_error: 29.5185\n \n Epoch 00084: val_loss improved from 39.60961 to 38.91767, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 85/200\n \n Epoch 00085: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 170s 762ms/step - loss: 46.6343 - mean_squared_error: 2847.3621 - mean_absolute_error: 37.2241 - val_loss: 41.5255 - val_mean_squared_error: 2340.1127 - val_mean_absolute_error: 31.5391\n \n Epoch 00085: val_loss did not improve from 38.91767\n Epoch 86/200\n \n Epoch 00086: LearningRateScheduler setting learning rate to 0.00049067.\n 223/223 [==============================] - 171s 768ms/step - loss: 47.4999 - mean_squared_error: 2929.3795 - mean_absolute_error: 37.6484 - val_loss: 38.0931 - val_mean_squared_error: 1919.7987 - val_mean_absolute_error: 30.8227\n \n Epoch 00086: val_loss improved from 38.91767 to 38.09311, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 87/200\n \n Epoch 00087: LearningRateScheduler setting learning rate to 0.00023585.\n 223/223 [==============================] - 171s 768ms/step - loss: 43.3678 - mean_squared_error: 2372.3566 - mean_absolute_error: 34.3873 - val_loss: 37.0847 - val_mean_squared_error: 1730.3208 - val_mean_absolute_error: 29.0763\n \n Epoch 00087: val_loss improved from 38.09311 to 37.08471, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 88/200\n \n Epoch 00088: LearningRateScheduler setting learning rate to 0.00023585.\n 223/223 [==============================] - 171s 768ms/step - loss: 46.9710 - mean_squared_error: 2997.7096 - mean_absolute_error: 37.1876 - val_loss: 41.0807 - val_mean_squared_error: 2068.8274 - val_mean_absolute_error: 32.7934\n \n Epoch 00088: val_loss did not improve from 37.08471\n Epoch 89/200\n \n Epoch 00089: LearningRateScheduler setting learning rate to 0.00023585.\n 223/223 [==============================] - 171s 766ms/step - loss: 46.3130 - mean_squared_error: 2794.5652 - mean_absolute_error: 36.7077 - val_loss: 39.0119 - val_mean_squared_error: 2020.9802 - val_mean_absolute_error: 30.6546\n \n Epoch 00089: val_loss did not improve from 37.08471\n Epoch 90/200\n \n Epoch 00090: LearningRateScheduler setting learning rate to 0.00023585.\n 223/223 [==============================] - 171s 769ms/step - loss: 44.4006 - mean_squared_error: 2530.9755 - mean_absolute_error: 35.1859 - val_loss: 41.7850 - val_mean_squared_error: 2246.9836 - val_mean_absolute_error: 32.8345\n \n Epoch 00090: val_loss did not improve from 37.08471\n Epoch 91/200\n \n Epoch 00091: LearningRateScheduler setting learning rate to 0.00023585.\n 223/223 [==============================] - 171s 766ms/step - loss: 46.5176 - mean_squared_error: 2895.8857 - mean_absolute_error: 37.0492 - val_loss: 37.7842 - val_mean_squared_error: 1901.9025 - val_mean_absolute_error: 29.9140\n \n Epoch 00091: val_loss did not improve from 37.08471\n Epoch 92/200\n \n Epoch 00092: LearningRateScheduler setting learning rate to 0.00023585.\n 223/223 [==============================] - 171s 765ms/step - loss: 44.6569 - mean_squared_error: 2575.6439 - mean_absolute_error: 35.2938 - val_loss: 38.5559 - val_mean_squared_error: 2053.1744 - val_mean_absolute_error: 29.7295\n \n Epoch 00092: val_loss did not improve from 37.08471\n Epoch 93/200\n \n Epoch 00093: LearningRateScheduler setting learning rate to 0.00023585.\n 223/223 [==============================] - 171s 768ms/step - loss: 44.6224 - mean_squared_error: 2606.4915 - mean_absolute_error: 35.3451 - val_loss: 39.6345 - val_mean_squared_error: 2025.8955 - val_mean_absolute_error: 30.6322\n \n Epoch 00093: val_loss did not improve from 37.08471\n Epoch 94/200\n \n Epoch 00094: LearningRateScheduler setting learning rate to 0.00023585.\n 223/223 [==============================] - 172s 770ms/step - loss: 46.6396 - mean_squared_error: 2872.1600 - mean_absolute_error: 36.5488 - val_loss: 36.6945 - val_mean_squared_error: 1589.7511 - val_mean_absolute_error: 29.2713\n \n Epoch 00094: val_loss improved from 37.08471 to 36.69451, saving model to /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Epoch 95/200\n \n Epoch 00095: LearningRateScheduler setting learning rate to 0.00023585.\n 223/223 [==============================] - 171s 769ms/step - loss: 44.2760 - mean_squared_error: 2471.6439 - mean_absolute_error: 34.7294 - val_loss: 43.2578 - val_mean_squared_error: 2383.8426 - val_mean_absolute_error: 34.2095\n \n Epoch 00095: val_loss did not improve from 36.69451\n Epoch 96/200\n \n Epoch 00096: LearningRateScheduler setting learning rate to 0.00023585.\n 223/223 [==============================] - 170s 763ms/step - loss: 45.9795 - mean_squared_error: 2859.2075 - mean_absolute_error: 36.3889 - val_loss: 37.0568 - val_mean_squared_error: 1893.5547 - val_mean_absolute_error: 28.5351\n \n Epoch 00096: val_loss did not improve from 36.69451\n Epoch 97/200\n \n Epoch 00097: LearningRateScheduler setting learning rate to 0.00023585.\n 223/223 [==============================] - 171s 765ms/step - loss: 44.5854 - mean_squared_error: 2504.5123 - mean_absolute_error: 35.2989 - val_loss: 41.6517 - val_mean_squared_error: 2239.0874 - val_mean_absolute_error: 32.8417\n \n Epoch 00097: val_loss did not improve from 36.69451\n Epoch 98/200\n \n Epoch 00098: LearningRateScheduler setting learning rate to 0.00023585.\n 223/223 [==============================] - 171s 767ms/step - loss: 45.4487 - mean_squared_error: 2835.7435 - mean_absolute_error: 35.4623 - val_loss: 37.8632 - val_mean_squared_error: 1678.1420 - val_mean_absolute_error: 30.3175\n \n Epoch 00098: val_loss did not improve from 36.69451\n Epoch 99/200\n \n Epoch 00099: LearningRateScheduler setting learning rate to 0.00023585.\n 223/223 [==============================] - 171s 766ms/step - loss: 44.9849 - mean_squared_error: 2661.8143 - mean_absolute_error: 35.6733 - val_loss: 38.2597 - val_mean_squared_error: 2132.6596 - val_mean_absolute_error: 29.5080\n \n Epoch 00099: val_loss did not improve from 36.69451\n Epoch 100/200\n \n Epoch 00100: LearningRateScheduler setting learning rate to 0.00023585.\n 223/223 [==============================] - 171s 766ms/step - loss: 42.9926 - mean_squared_error: 2430.2934 - mean_absolute_error: 33.9323 - val_loss: 42.0223 - val_mean_squared_error: 2094.8205 - val_mean_absolute_error: 32.9810\n \n Epoch 00100: val_loss did not improve from 36.69451\n Epoch 101/200\n \n Epoch 00101: LearningRateScheduler setting learning rate to 0.00023585.\n 223/223 [==============================] - 171s 765ms/step - loss: 45.4798 - mean_squared_error: 2632.2893 - mean_absolute_error: 35.8055 - val_loss: 40.2772 - val_mean_squared_error: 2199.4724 - val_mean_absolute_error: 31.0075\n \n Epoch 00101: val_loss did not improve from 36.69451\n Epoch 102/200\n \n Epoch 00102: LearningRateScheduler setting learning rate to 0.00011101.\n 223/223 [==============================] - 172s 770ms/step - loss: 45.2292 - mean_squared_error: 2715.6062 - mean_absolute_error: 35.9004 - val_loss: 38.0430 - val_mean_squared_error: 1828.4453 - val_mean_absolute_error: 29.9727\n \n Epoch 00102: val_loss did not improve from 36.69451\n Epoch 103/200\n \n Epoch 00103: LearningRateScheduler setting learning rate to 0.00011101.\n 223/223 [==============================] - 171s 768ms/step - loss: 43.8983 - mean_squared_error: 2576.5037 - mean_absolute_error: 34.5217 - val_loss: 41.5556 - val_mean_squared_error: 2175.1995 - val_mean_absolute_error: 34.1130\n \n Epoch 00103: val_loss did not improve from 36.69451\n Epoch 104/200\n \n Epoch 00104: LearningRateScheduler setting learning rate to 0.00011101.\n 223/223 [==============================] - 170s 761ms/step - loss: 45.0925 - mean_squared_error: 2701.5656 - mean_absolute_error: 35.6956 - val_loss: 38.9283 - val_mean_squared_error: 1909.2236 - val_mean_absolute_error: 31.1378\n \n Epoch 00104: val_loss did not improve from 36.69451\n Epoch 00104: early stopping\n Loading previous best weights: /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n Training run: 3, epoch: 104\n Engine split: Training=0.80, Validation=0.20\n Cycle split: Training=0.80, Validation=0.20\n Number of items: 567\n Undersized items: 0\n Data shape: (128641, 29)\n Max steps: 115033\n Max iterations: 224 @ 512\n Number of items: 142\n Undersized items: 0\n Data shape: (31718, 29)\n Max steps: 28310\n Max iterations: 55 @ 512\n Epoch 105/200\n \n Epoch 00105: LearningRateScheduler setting learning rate to 0.001.\n 224/224 [==============================] - 178s 794ms/step - loss: 46.7740 - mean_squared_error: 2851.3663 - mean_absolute_error: 37.2373 - val_loss: 44.5731 - val_mean_squared_error: 2323.3556 - val_mean_absolute_error: 35.4615\n \n Epoch 00105: val_loss did not improve from 36.69451\n Epoch 106/200\n \n Epoch 00106: LearningRateScheduler setting learning rate to 0.001.\n 224/224 [==============================] - 172s 768ms/step - loss: 47.4780 - mean_squared_error: 2930.4046 - mean_absolute_error: 37.3320 - val_loss: 43.8179 - val_mean_squared_error: 2336.8530 - val_mean_absolute_error: 35.4118\n \n Epoch 00106: val_loss did not improve from 36.69451\n Epoch 107/200\n \n Epoch 00107: LearningRateScheduler setting learning rate to 0.001.\n 224/224 [==============================] - 171s 762ms/step - loss: 47.0984 - mean_squared_error: 2859.4039 - mean_absolute_error: 37.0442 - val_loss: 44.5796 - val_mean_squared_error: 2577.8519 - val_mean_absolute_error: 35.8468\n \n Epoch 00107: val_loss did not improve from 36.69451\n Epoch 108/200\n \n Epoch 00108: LearningRateScheduler setting learning rate to 0.001.\n 224/224 [==============================] - 171s 764ms/step - loss: 44.6807 - mean_squared_error: 2594.8742 - mean_absolute_error: 35.1308 - val_loss: 54.5514 - val_mean_squared_error: 3375.7863 - val_mean_absolute_error: 46.3016\n \n Epoch 00108: val_loss did not improve from 36.69451\n Epoch 109/200\n \n Epoch 00109: LearningRateScheduler setting learning rate to 0.001.\n 224/224 [==============================] - 171s 765ms/step - loss: 46.4760 - mean_squared_error: 2856.0634 - mean_absolute_error: 36.8565 - val_loss: 41.3253 - val_mean_squared_error: 2198.2020 - val_mean_absolute_error: 33.2083\n \n Epoch 00109: val_loss did not improve from 36.69451\n Epoch 110/200\n \n Epoch 00110: LearningRateScheduler setting learning rate to 0.001.\n 224/224 [==============================] - 170s 758ms/step - loss: 46.9323 - mean_squared_error: 2789.2062 - mean_absolute_error: 37.1449 - val_loss: 44.8855 - val_mean_squared_error: 2639.1379 - val_mean_absolute_error: 34.8277\n \n Epoch 00110: val_loss did not improve from 36.69451\n Epoch 111/200\n \n Epoch 00111: LearningRateScheduler setting learning rate to 0.001.\n 224/224 [==============================] - 171s 765ms/step - loss: 46.3458 - mean_squared_error: 2796.1569 - mean_absolute_error: 36.7753 - val_loss: 47.6357 - val_mean_squared_error: 2819.2200 - val_mean_absolute_error: 38.5139\n \n Epoch 00111: val_loss did not improve from 36.69451\n Epoch 112/200\n \n Epoch 00112: LearningRateScheduler setting learning rate to 0.001.\n 224/224 [==============================] - 172s 766ms/step - loss: 45.0972 - mean_squared_error: 2645.3111 - mean_absolute_error: 35.8573 - val_loss: 46.1476 - val_mean_squared_error: 2621.0218 - val_mean_absolute_error: 37.1280\n \n Epoch 00112: val_loss did not improve from 36.69451\n Epoch 113/200\n \n Epoch 00113: LearningRateScheduler setting learning rate to 0.001.\n 224/224 [==============================] - 170s 760ms/step - loss: 47.7392 - mean_squared_error: 3005.7941 - mean_absolute_error: 37.8205 - val_loss: 49.8040 - val_mean_squared_error: 3192.4084 - val_mean_absolute_error: 39.1004\n \n Epoch 00113: val_loss did not improve from 36.69451\n Epoch 114/200\n \n Epoch 00114: LearningRateScheduler setting learning rate to 0.001.\n 224/224 [==============================] - 171s 764ms/step - loss: 45.8265 - mean_squared_error: 2752.4556 - mean_absolute_error: 36.1591 - val_loss: 44.5941 - val_mean_squared_error: 2347.1558 - val_mean_absolute_error: 34.7990\n \n Epoch 00114: val_loss did not improve from 36.69451\n Epoch 115/200\n \n Epoch 00115: LearningRateScheduler setting learning rate to 0.001.\n 224/224 [==============================] - 172s 766ms/step - loss: 45.6714 - mean_squared_error: 2724.3944 - mean_absolute_error: 36.0908 - val_loss: 50.0545 - val_mean_squared_error: 2817.2430 - val_mean_absolute_error: 41.4499\n \n Epoch 00115: val_loss did not improve from 36.69451\n Epoch 116/200\n \n Epoch 00116: LearningRateScheduler setting learning rate to 0.001.\n 224/224 [==============================] - 171s 764ms/step - loss: 45.6454 - mean_squared_error: 2738.7780 - mean_absolute_error: 35.8476 - val_loss: 41.7414 - val_mean_squared_error: 2360.4926 - val_mean_absolute_error: 32.0672\n \n Epoch 00116: val_loss did not improve from 36.69451\n Epoch 117/200\n \n Epoch 00117: LearningRateScheduler setting learning rate to 0.001.\n 224/224 [==============================] - 172s 767ms/step - loss: 44.9922 - mean_squared_error: 2684.6346 - mean_absolute_error: 35.9968 - val_loss: 46.6646 - val_mean_squared_error: 3039.9081 - val_mean_absolute_error: 36.0352\n \n Epoch 00117: val_loss did not improve from 36.69451\n Epoch 118/200\n \n Epoch 00118: LearningRateScheduler setting learning rate to 0.001.\n 224/224 [==============================] - 170s 758ms/step - loss: 46.3994 - mean_squared_error: 2916.3936 - mean_absolute_error: 36.6927 - val_loss: 49.6271 - val_mean_squared_error: 2829.7702 - val_mean_absolute_error: 40.6010\n \n Epoch 00118: val_loss did not improve from 36.69451\n Epoch 119/200\n \n Epoch 00119: LearningRateScheduler setting learning rate to 0.00049067.\n 224/224 [==============================] - 171s 765ms/step - loss: 43.3925 - mean_squared_error: 2533.2320 - mean_absolute_error: 34.6782 - val_loss: 43.4537 - val_mean_squared_error: 2380.0416 - val_mean_absolute_error: 33.9435\n \n Epoch 00119: val_loss did not improve from 36.69451\n Epoch 00119: early stopping\n Loading previous best weights: /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n\n\nPlot the training and validation loss across all runs overlayed with the learning rate used.\n\n\n```python\nfig, ax1 = plt.subplots(figsize=(10,6))\n\nepochs = np.arange(len(epoch_loss_history))\nax1.plot(epochs, epoch_loss_history, label='loss')\nax1.plot(epochs, epoch_val_history, label='val_loss')\nax1.set_xlabel(\"Epoch\")\nax1.set_ylabel(\"RMSE\")\nax1.set_ylim(0)\nax1.legend()\n\n# Add a rhs scale for the LR.\nax2 = ax1.twinx()\nlr_epochs = []\nlr_steps = []\n\nfor e1, l1, l2 in epoch_lr_history:\n lr_epochs.append(e1)\n lr_steps.append(l1)\n\n\nax2.plot(lr_epochs, lr_steps, 'r.', label='lr')\nax2.set_ylabel(\"Learning Rate\")\nax2.legend(loc='lower left')\n\nfig.tight_layout()\nplt.show()\n```\n\n---\n## Test\n\nLoading and pre-processing the test data is similar to how validation data was handled.\n\n\n#### Load Model\n\nReload the latest, best model from this run.\n\n\n```python\nprint(\"Loading model: \", checkpoint_path)\ncustom_objects={'rmse':rmse}\ninf_model = load_model(checkpoint_path, custom_objects=custom_objects)\n\ninf_model.summary()\n```\n\n Loading model: /home/saad/workspaces/predictive-maintenance-lstm/model/engine20190122T0916/engine_model.h5\n _________________________________________________________________\n Layer (type) Output Shape Param # \n =================================================================\n masking_1 (Masking) (None, 25, 26) 0 \n _________________________________________________________________\n lstm_1 (LSTM) (None, 128) 79360 \n _________________________________________________________________\n dense_1 (Dense) (None, 64) 8256 \n _________________________________________________________________\n dense_2 (Dense) (None, 32) 2080 \n _________________________________________________________________\n dense_3 (Dense) (None, 1) 33 \n =================================================================\n Total params: 89,729\n Trainable params: 89,729\n Non-trainable params: 0\n _________________________________________________________________\n\n\n#### Load prepare test data\n\nCalculating RUL for test data is a bit more involved. For evaluation we'll be using the data from the test_FD003.txt files and the corresponding RUL_FD003.txt for the actual RUL ground truth. This allows us to utilize more data for training. The key calculation difference is that the last cycle in the these files must equal the ground truth RUL. Therefore we compute the RUL for each entry by working backwards from the last cycle entry.\n\n\n```python\ndataset_name = 'FD003'\n\ntest_X_path = os.path.join(DATA_DIR, 'test_' + dataset_name + '.txt')\ntest_y_path = os.path.join(DATA_DIR, 'RUL_' + dataset_name + '.txt')\n```\n\n\n```python\n\ndef load_rul_data(paths, col_names):\n \n # Filename is used to determine the condition\n col_names.append('filename')\n\n # read data \n df = pd.DataFrame()\n for p in paths:\n instance_df = pd.read_csv(p, sep=\" \", header=None)\n instance_df.drop(instance_df.columns[[1]], axis=1, inplace=True)\n instance_df['filename'] = os.path.splitext(os.path.basename(p))[0]\n instance_df = instance_df.reset_index()\n instance_df.columns = col_names\n \n df = pd.concat((df, instance_df), sort=False) \n\n df['id'] = df['id'] + df['filename'].apply( lambda f: fn_id_map[f]) + 1\n df.drop(['filename'], axis=1, inplace=True)\n return df\n\n\n```\n\n\n```python\ndef calc_test_rul( feature_df, label_df):\n # If index is not reset there will be int/str type issues when attempting the merge. \n cycle_count_df = feature_df.groupby('id').count().reset_index()[['id','cycle']].rename(index=str, columns={\"cycle\":\"cycles\"}).reset_index(drop=True)\n print(cycle_count_df.shape)\n\n # Join cycle and RUL dataframes\n assert cycle_count_df.shape[0] == label_df.shape[0]\n tmp_df = cycle_count_df.merge(label_df, on=\"id\", how='left')\n\n # The RUL actual column contains the value for the last cycle.\n # Adding the cycles column will give us the RUL for the first cycle.\n tmp_df['RUL_actual'] = tmp_df['cycles'] + tmp_df['RUL_actual']\n tmp_df.drop('cycles', axis=1, inplace=True)\n\n # Join the two data frames\n feature_df = feature_df.merge(tmp_df, on='id', how='left')\n\n\n # Use the cycle to decrement the RUL until the ground truth is reached.\n feature_df['RUL'] = feature_df['RUL_actual'] - feature_df['cycle']\n feature_df.drop('RUL_actual', axis=1, inplace=True)\n \n return feature_df\n\n```\n\n\n```python\n# Read in the features\ntest_df = load_data([test_X_path], cols, sort_cols)\n\n# Read in the labels (RUL)\ntest_rul_df = load_rul_data([test_y_path], ['id', 'RUL_actual'])\n\n# Calculate the RUL and merge back to the test dataframe\ntest_df = calc_test_rul(test_df, test_rul_df)\ntest_df.head()\n\n```\n\n (100, 2)\n\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    idcyclesetting1setting2setting3s1s2s3s4s5...s14s15s16s17s18s19s20s21conditionRUL
    070011-0.0017-0.0004100.0518.67641.941581.931396.9314.62...8133.488.37600.033912388100.039.0723.44681276
    1700120.0006-0.0002100.0518.67642.021584.861398.9014.62...8137.448.40620.033912388100.039.0423.48071275
    2700130.0014-0.0003100.0518.67641.681581.781391.9214.62...8138.258.35530.033912388100.039.1023.42441274
    3700140.00270.0001100.0518.67642.201584.531395.3414.62...8137.078.37090.033922388100.038.9723.47821273
    470015-0.00010.0001100.0518.67642.461589.031395.8614.62...8134.208.41460.033912388100.039.0923.39501272
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    5 rows \u00d7 28 columns

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    \n\n\n\nTransform test data\n\n\n```python\n\ntest_df['cycle_norm'] = test_df['cycle']\n\nnorm_test_df = pd.DataFrame(pipeline.transform(test_df[cols_transform]), \n columns=cols_transform, \n index=test_df.index)\ntest_join_df = test_df[test_df.columns.difference(cols_transform)].join(norm_test_df)\ntest_df = test_join_df.reindex(columns = test_df.columns)\ntest_df = test_df.reset_index(drop=True)\n```\n\n\n```python\ntest_df.head()[feature_cols]\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    cycle_normconditionsetting1setting2setting3s1s2s3s4s5...s12s13s14s15s16s17s18s19s20s21
    0-1.00000-1.0-0.999667-0.9995251.01.00.9421690.8130610.7866511.0...0.9237500.9859470.284547-0.8490351.00.8163271.01.00.9448370.943846
    1-0.99631-1.0-0.999557-0.9990511.01.00.9436290.8287200.7960841.0...0.9235540.9863330.302228-0.8282831.00.8163271.01.00.9428190.947625
    2-0.99262-1.0-0.999519-0.9992881.01.00.9374260.8122600.7626641.0...0.9247760.9859470.305845-0.8632581.00.8163271.01.00.9468550.941349
    3-0.98893-1.0-0.999457-0.9983381.01.00.9469120.8269560.7790391.0...0.9264870.9860580.300576-0.8525391.00.8367351.01.00.9381100.947347
    4-0.98524-1.0-0.999591-0.9983381.01.00.9516560.8510050.7815281.0...0.9238480.9861130.287762-0.8225111.00.8163271.01.00.9461820.938071
    \n

    5 rows \u00d7 26 columns

    \n
    \n\n\n\n\n```python\ntest_df.head()[label_cols]\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    RUL
    0276
    1275
    2274
    3273
    4272
    \n
    \n\n\n\n#### Evaluate\n\nCollect metrics across then entire test sample.\n\n\n```python\ntest_data_generator = TSDataGenerator(test_df, \n feature_cols, \n label_cols,\n batch_size=batch_size,\n seq_length=sequence_length, \n randomize=False,\n loop=False)\ntest_data_generator.print_summary()\n\nX = []\ny = []\nfor p in tqdm_notebook(test_data_generator.generate(), total=test_data_generator.summary()['max_iterations']):\n X.append(p[0])\n y.append(p[1])\n\ntest_X = np.vstack(X)\ntest_y = np.vstack(y)\n\n```\n\n Number of items: 100\n Undersized items: 0\n Data shape: (16596, 29)\n Max steps: 14196\n Max iterations: 27 @ 512\n\n\n\n HBox(children=(IntProgress(value=0, max=27), HTML(value='')))\n\n\n \n\n\n\n```python\n# Evaluation metrics are RMSE, MSE, MAE\nscore = inf_model.evaluate(test_X, test_y, verbose=1, batch_size=batch_size)\nprint('Test score:\\n\\tRMSE: {}\\n\\tMSE: {}\\n\\tMAE: {}'.format(*score))\n```\n\n 13824/13824 [==============================] - 0s 28us/step\n Test score:\n \tRMSE: 60.54784216704192\n \tMSE: 4545.844335485388\n \tMAE: 47.62541029188368\n\n\nPredict on one batch from the test data. This is a fragment of the data.\n\n\n```python\ntest_data_generator = TSDataGenerator(test_df, feature_cols, label_cols, batch_size=batch_size, seq_length=sequence_length, loop=False)\n\ng = test_data_generator.generate()\ntest_X, test_y = next(g)\ny_pred_array = inf_model.predict_on_batch(test_X)\n```\n\n\n```python\ndef plot_prediction(rul_actual, rul_predicted): \n fig = plt.figure(figsize=(25,5))\n cycles = np.arange(len(rul_actual))\n plt.scatter(cycles, rul_predicted, marker='.', label=\"Predicted\")\n plt.plot(cycles, rul_actual, 'r', label=\"Actual\")\n plt.xlabel(\"Cycle\")\n plt.xlim(0)\n plt.ylabel(\"RUL\")\n plt.ylim(0)\n\n plt.legend()\n plt.show()\n```\n\n\n```python\nplot_prediction(test_y, y_pred_array)\n```\n\nThe above plot shows the first batch. As the sequence approaches the last time series entry, the prediction converged closeer to the actual RUL. Running a second batch gives us:\n\n\n```python\ntest_X, test_y = next(g)\ny_pred_array = inf_model.predict_on_batch(test_X)\nplot_prediction(test_y, y_pred_array)\n```\n\n---\n\n## Persist\n\nSave parameters and results (requires couchdb and client)\n\n\n```python\nif persist_run_stats:\n print(\"Persisting training run.\")\n import sys\n import os\n sys.path.append(os.path.abspath('../ml_utils'))\n import ml_utils\n\n username = os.getenv('COUCHDB_USER')\n password = os.getenv('COUCHDB_PASSWORD')\n \n db_helper = ml_utils.couchdb.CouchDBHelper(\"engine\", username, password)\n db_helper.connect()\n\n payload = {\n 'score': score,\n 'model': inf_model.to_json(),\n 'epoch_lr_history': epoch_lr_history,\n 'epoch_loss_history': epoch_loss_history,\n 'epoch_val_history': epoch_val_history\n }\n\n db_helper.save(os.path.basename(log_dir), payload)\n db_helper.disconnect()\nelse:\n print(\"Not persisting training run.\")\n\n```\n\n Persisting training run.\n\n\n\n```python\n# This is to prevent overwrites\npersist_run_stats = False\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "5abb71fa121543f5fce797827f6cb5cc84e411cb", "size": 267083, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "train.ipynb", "max_stars_repo_name": "sabderra/predictive-maintenance-lstm", "max_stars_repo_head_hexsha": "6580c16d1c63d8d69ff9e40d721b088b87921171", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 36, "max_stars_repo_stars_event_min_datetime": "2019-03-18T14:31:06.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-24T02:14:47.000Z", "max_issues_repo_path": "train.ipynb", "max_issues_repo_name": "ainio1995/predictive-maintenance-lstm", "max_issues_repo_head_hexsha": "6580c16d1c63d8d69ff9e40d721b088b87921171", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-04-09T20:16:20.000Z", "max_issues_repo_issues_event_max_datetime": "2020-04-09T20:16:20.000Z", "max_forks_repo_path": "train.ipynb", "max_forks_repo_name": "ainio1995/predictive-maintenance-lstm", "max_forks_repo_head_hexsha": "6580c16d1c63d8d69ff9e40d721b088b87921171", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 18, "max_forks_repo_forks_event_min_datetime": "2019-03-18T14:27:55.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-11T08:48:14.000Z", "avg_line_length": 88.1171230617, "max_line_length": 50320, "alphanum_fraction": 0.7501787834, "converted": true, "num_tokens": 28841, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.689305616785446, "lm_q2_score": 0.6297746213017459, "lm_q1q2_score": 0.43410718377222063}} {"text": "```python\n%matplotlib inline\n```\n\n\nSequence-to-Sequence Modeling with nn.Transformer and TorchText\n===============================================================\n\nThis is a tutorial on how to train a sequence-to-sequence model\nthat uses the\n`nn.Transformer `__ module.\n\nPyTorch 1.2 release includes a standard transformer module based on the\npaper `Attention is All You\nNeed `__. The transformer model\nhas been proved to be superior in quality for many sequence-to-sequence\nproblems while being more parallelizable. The ``nn.Transformer`` module\nrelies entirely on an attention mechanism (another module recently\nimplemented as `nn.MultiheadAttention `__) to draw global dependencies\nbetween input and output. The ``nn.Transformer`` module is now highly\nmodularized such that a single component (like `nn.TransformerEncoder `__\nin this tutorial) can be easily adapted/composed.\n\n\n\n\n\n\n\nDefine the model\n----------------\n\n\n\n\nIn this tutorial, we train ``nn.TransformerEncoder`` model on a\nlanguage modeling task. The language modeling task is to assign a\nprobability for the likelihood of a given word (or a sequence of words)\nto follow a sequence of words. A sequence of tokens are passed to the embedding\nlayer first, followed by a positional encoding layer to account for the order\nof the word (see the next paragraph for more details). The\n``nn.TransformerEncoder`` consists of multiple layers of\n`nn.TransformerEncoderLayer `__. Along with the input sequence, a square\nattention mask is required because the self-attention layers in\n``nn.TransformerEncoder`` are only allowed to attend the earlier positions in\nthe sequence. For the language modeling task, any tokens on the future\npositions should be masked. To have the actual words, the output\nof ``nn.TransformerEncoder`` model is sent to the final Linear\nlayer, which is followed by a log-Softmax function.\n\n\n\n\n``PositionalEncoding`` module injects some information about the\nrelative or absolute position of the tokens in the sequence. The\npositional encodings have the same dimension as the embeddings so that\nthe two can be summed. Here, we use ``sine`` and ``cosine`` functions of\ndifferent frequencies.\n\n\n\n\n\n```python\nfrom collections import Counter\nimport math\nimport time\n\nimport numpy as np\nimport torch\nimport torch.nn as nn\nfrom torch.utils.data import TensorDataset, DataLoader\nimport torch.nn.functional as F\n\nimport torchtext\nfrom torchtext.data.utils import get_tokenizer\n\n\ndevice = torch.device(\"cuda\" if torch.cuda.is_available() else \"cpu\")\n```\n\n\n```python\nclass PositionalEncoding(nn.Module):\n def __init__(self, dimension: int, dropout: int=0.1, max_length: int=5000):\n super().__init__()\n self.dropout = nn.Dropout(p=dropout)\n positional_values = self.__generate_position_values(dimension, max_length)\n self.register_buffer(\"positional_values\", positional_values)\n \n @staticmethod\n def __generate_position_values(dimension: int, max_length: int):\n values = torch.zeros(max_length, dimension)\n positions = torch.arange(0, max_length, dtype=torch.float)\n positions = positions.unsqueeze(1)\n \n scaling_steps = torch.arange(0, dimension, 2).float()\n scaling = torch.exp(scaling_steps * (-math.log(10000.0)/dimension))\n \n values[:, 0::2] = torch.sin(positions * scaling)\n values[:, 1::2] = torch.cos(positions * scaling)\n values = values.unsqueeze(0).transpose(0, 1)\n \n return values\n \n def forward(self, x):\n x = x + self.positional_values[:x.size(0), :]\n return self.dropout(x)\n```\n\n\n```python\nclass TransformerScriptGenerator(nn.Module):\n def __init__(self,\n vocabulary_size: int,\n embedding_dim: int,\n attention_head_count: int,\n encoder_fc_dim: int,\n encoder_layer_count: int,\n dropout: float=0.5) -> None:\n \n super().__init__()\n \n self.input_mask = None\n self.embedding = nn.Embedding(vocabulary_size, embedding_dim)\n self.embedding_scale = math.sqrt(embedding_dim)\n self.positional_encoder = PositionalEncoding(embedding_dim, dropout)\n encoder_layers = nn.TransformerEncoderLayer(embedding_dim, attention_head_count, encoder_fc_dim, dropout)\n self.transformer_encoder = nn.TransformerEncoder(encoder_layers, encoder_layer_count)\n self.decoder = nn.Linear(embedding_dim, vocabulary_size)\n \n self.__init_weights()\n \n def __init_weights(self) -> None:\n value_range = 0.1\n self.embedding.weight.data.uniform_(-value_range, value_range)\n self.decoder.bias.data.zero_()\n self.decoder.weight.data.uniform_(-value_range, value_range)\n \n \n @staticmethod\n def __generate_input_mask(mask_size: int) -> torch.Tensor():\n mask = torch.ones(mask_size, mask_size, dtype=bool)\n mask = torch.triu(mask).t().float()\n mask = mask.masked_fill(mask == 0, float('-inf'))\n mask = mask.masked_fill(mask == 1, 0.0)\n return mask\n \n @staticmethod\n def __get_output_for_last_word(full_output: torch.Tensor) -> torch.Tensor:\n return full_output[:,-1,:]\n \n def forward(self, input: torch.Tensor) -> torch.Tensor:\n mask_size = input.shape[0]\n if self.input_mask is None or self.input_mask.size(0) != mask_size:\n self.input_mask = self.__generate_input_mask(mask_size).to(device)\n \n input = self.embedding(input) * self.embedding_scale\n input = self.positional_encoder(input)\n output = self.transformer_encoder(input, self.input_mask)\n #output = self.__get_output_for_last_word(output)\n output = self.decoder(output)\n return output\n```\n\nLoad and batch data\n-------------------\n\n\n\n\nThe training process uses Wikitext-2 dataset from ``torchtext``. The\nvocab object is built based on the train dataset and is used to numericalize\ntokens into tensors. Starting from sequential data, the ``batchify()``\nfunction arranges the dataset into columns, trimming off any tokens remaining\nafter the data has been divided into batches of size ``batch_size``.\nFor instance, with the alphabet as the sequence (total length of 26)\nand a batch size of 4, we would divide the alphabet into 4 sequences of\nlength 6:\n\n\\begin{align}\\begin{bmatrix}\n \\text{A} & \\text{B} & \\text{C} & \\ldots & \\text{X} & \\text{Y} & \\text{Z}\n \\end{bmatrix}\n \\Rightarrow\n \\begin{bmatrix}\n \\begin{bmatrix}\\text{A} \\\\ \\text{B} \\\\ \\text{C} \\\\ \\text{D} \\\\ \\text{E} \\\\ \\text{F}\\end{bmatrix} &\n \\begin{bmatrix}\\text{G} \\\\ \\text{H} \\\\ \\text{I} \\\\ \\text{J} \\\\ \\text{K} \\\\ \\text{L}\\end{bmatrix} &\n \\begin{bmatrix}\\text{M} \\\\ \\text{N} \\\\ \\text{O} \\\\ \\text{P} \\\\ \\text{Q} \\\\ \\text{R}\\end{bmatrix} &\n \\begin{bmatrix}\\text{S} \\\\ \\text{T} \\\\ \\text{U} \\\\ \\text{V} \\\\ \\text{W} \\\\ \\text{X}\\end{bmatrix}\n \\end{bmatrix}\\end{align}\n\nThese columns are treated as independent by the model, which means that\nthe dependence of ``G`` and ``F`` can not be learned, but allows more\nefficient batch processing.\n\n\n\n\nFunctions to generate input and target sequence\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n\n\n\n``get_batch()`` function generates the input and target sequence for\nthe transformer model. It subdivides the source data into chunks of\nlength ``bptt``. For the language modeling task, the model needs the\nfollowing words as ``Target``. For example, with a ``bptt`` value of 2,\nwe\u2019d get the following two Variables for ``i`` = 0:\n\n\n\n\nIt should be noted that the chunks are along dimension 0, consistent\nwith the ``S`` dimension in the Transformer model. The batch dimension\n``N`` is along dimension 1.\n\n\n\n\n\n```python\ndef text_to_tensor(text_field: torchtext.data.Field, input_text) -> torch.Tensor:\n return text_field.numericalize([input_text.examples[0].text])\n\n\ndef load_and_split_data(): \n '''\n Returns: training_text, validation_text, test_text, vocabulary_object\n '''\n text_field = torchtext.data.Field(tokenize=get_tokenizer(\"basic_english\"),\n init_token='',\n eos_token='',\n lower=True)\n train, validation, test = torchtext.datasets.WikiText2.splits(text_field)\n text_field.build_vocab(train)\n train = text_to_tensor(text_field, train)\n validation = text_to_tensor(text_field, validation)\n test = text_to_tensor(text_field, test)\n return train, validation, test, text_field.vocab\n \n\ndef divide_into_parallel_data_streams(data: torch.Tensor, stream_count: int) -> torch.Tensor:\n stream_length = data.size(0) // stream_count\n data = data.narrow(0, 0, stream_length * stream_count)\n data = data.view(stream_count, -1).t().contiguous()\n return data.to(device)\n\n\ndef batch_loader(source, max_sequence_length: int) -> (torch.Tensor, torch.Tensor):\n total_row_count = source.size(0)\n # -1 to account for the target sequence shift\n full_batch_count = (total_row_count - 1) // max_sequence_length\n for batch_index in range(full_batch_count):\n first_row_index = batch_index * max_sequence_length\n last_row_index = first_row_index + max_sequence_length\n inputs = source[first_row_index: last_row_index]\n targets = source[first_row_index+1: last_row_index+1].view(-1)\n yield inputs, targets\n \n first_row_index = full_batch_count * max_sequence_length\n inputs = source[first_row_index:-1]\n targets = source[first_row_index+1:].view(-1)\n yield inputs, targets\n```\n\n\n```python\ntrain_data, val_data, test_data, vocab = load_and_split_data()\n\nbatch_row_count = 20\neval_batch_row_count = 10\n\ntrain_data = divide_into_parallel_data_streams(train_data, batch_row_count)\nval_data = divide_into_parallel_data_streams(val_data, eval_batch_row_count)\ntest_data = divide_into_parallel_data_streams(test_data, eval_batch_row_count)\n```\n\nInitiate an instance\n--------------------\n\n\n\n\nThe model is set up with the hyperparameter below. The vocab size is\nequal to the length of the vocab object.\n\n\n\n\n\n```python\nvocab_size = len(vocab.stoi)\ninput_sequence_length = 35\n\nmodel = TransformerScriptGenerator(vocabulary_size=vocab_size,\n embedding_dim=200,\n attention_head_count=4,\n encoder_fc_dim=200,\n encoder_layer_count=3,\n dropout=0.2).to(device)\n\ncriterion = nn.CrossEntropyLoss()\nlearn_rate = 5.0\noptimizer = torch.optim.SGD(model.parameters(), lr=learn_rate)\nscheduler = torch.optim.lr_scheduler.StepLR(optimizer, 1.0, gamma=0.95)\n```\n\nntokens = len(TEXT.vocab.stoi) # the size of vocabulary\nemsize = 200 # embedding dimension\nnhid = 200 # the dimension of the feedforward network model in nn.TransformerEncoder\nnlayers = 3 # the number of nn.TransformerEncoderLayer in nn.TransformerEncoder\nnhead = 4 # the number of heads in the multiheadattention models\ndropout = 0.2 # the dropout value\nmodel = TransformerScriptGenerator(ntokens, emsize, nhead, nhid, nlayers, dropout).to(device)\n\n| end of epoch 10 | time: 211.53s | valid loss 5.36 | valid ppl 212.82\n\nRun the model\n-------------\n\n\n\n\n`CrossEntropyLoss `__\nis applied to track the loss and\n`SGD `__\nimplements stochastic gradient descent method as the optimizer. The initial\nlearning rate is set to 5.0. `StepLR `__ is\napplied to adjust the learn rate through epochs. During the\ntraining, we use\n`nn.utils.clip_grad_norm\\_ `__\nfunction to scale all the gradient together to prevent exploding.\n\n\n\n\n\n```python\ndef evaluation_step(model: TransformerScriptGenerator) -> float:\n model.eval()\n total_loss = 0.\n with torch.no_grad():\n for inputs, targets in batch_loader(val_data, input_sequence_length):\n output = model(inputs)\n output_flat = output.view(-1, vocab_size)\n total_loss += len(val_data) * criterion(output_flat, targets).item()\n return math.floor(total_loss) / (len(val_data) - 1)\n\n\ndef log_step_results(batch_index: int, \n batch_count: int, \n learning_rate: float, \n batch_time: float,\n total_loss: float) -> None:\n \n log_interval = 200\n if batch_index % log_interval == 0 and batch_index > 0:\n current_loss = total_loss / batch_index\n print(f\"Batch {batch_index:5d} of {batch_count:5d} | \"\n f\"lr {learning_rate:1.3f} | \"\n f\"ms/batch {batch_time:5.2f} | \"\n f\"loss {current_loss:5.2f}\")\n\n\ndef training_step(model: TransformerScriptGenerator):\n model.train()\n total_loss = 0.\n start_time = time.time()\n \n for batch_index, (data, targets) in enumerate(batch_loader(train_data, input_sequence_length)):\n optimizer.zero_grad()\n output = model(data)\n loss = criterion(output.view(-1, vocab_size), targets)\n loss.backward()\n torch.nn.utils.clip_grad_norm_(model.parameters(), 0.5)\n optimizer.step()\n\n total_loss += loss.item()\n log_step_results(batch_index, \n len(train_data) // input_sequence_length, \n scheduler.get_lr()[0],\n time.time() - start_time,\n total_loss)\n start_time = time.time()\n```\n\nLoop over epochs. Save the model if the validation loss is the best\nwe've seen so far. Adjust the learning rate after each epoch.\n\n\n\n\n```python\nbest_val_loss = float(\"inf\")\nepochs = 5 # The number of epochs\nbest_model = None\n\nfor epoch in range(1, epochs + 1):\n print(f\"Epoch {epoch}\")\n epoch_start_time = time.time()\n training_step(model)\n val_loss = evaluation_step(model)\n print('-' * 89)\n print(val_loss)\n print('| end of epoch {:3d} | time: {:5.2f}s | valid loss {:5.2f}'\n .format(epoch, (time.time() - epoch_start_time),\n val_loss))\n print('-' * 89)\n\n if val_loss < best_val_loss:\n best_val_loss = val_loss\n best_model = model\n\n scheduler.step()\n```\n\nEvaluate the model with the test dataset\n-------------------------------------\n\nApply the best model to check the result with the test dataset.\n\n\n\n\n```python\ntest_loss = evaluate(best_model, test_data)\nprint('=' * 89)\nprint('| End of training | test loss {:5.2f} | test ppl {:8.2f}'.format(\n test_loss, math.exp(test_loss)))\nprint('=' * 89)\n```\n\n\n```python\nbest_model.eval();\nvocab_size = len(TEXT.vocab.stoi)\nprint(vocab_size)\ndata, targets = get_batch(train_data, 0)\nprint(f\"BPTT = {bptt}\")\nprint(f\"Input shape = {data.shape}\")\nprint(f\"Target shape = {targets.shape}\")\noutput = best_model(data)\nprint(f\"Output shape = {output.shape}\")\noutput_flat = output.view(-1, ntokens)\nprint(f\"Output flat shape = {output_flat.shape}\")\noutput_flat[0]\n```\n\n\n```python\ndef tensor_to_list(tensor: torch.Tensor) -> list:\n return tensor.cpu().detach().numpy().tolist()\n\n\ndef scores_to_top_tokens(scores: torch.Tensor) -> list:\n _, top_index_tensor = torch.topk(scores, k=1)\n top_index_tensor.squeeze_().t_()\n top_index_list = tensor_to_list(top_index_tensor)\n return top_index_list\n\n\ndef tokens_to_words(batches: list) -> list:\n decoded_batches = []\n for batch in batches:\n decoded_batch = [TEXT.vocab.itos[token] for token in batch]\n decoded_batches.append(decoded_batch)\n return decoded_batches\n\n\ndef decode_transformer_output(scores: torch.Tensor) -> list:\n top_tokens = scores_to_top_tokens(scores)\n predicted_words = tokens_to_words(top_tokens)\n return predicted_words\n\n\ndef decode_targets(targets: torch.Tensor, batch_size: int) -> list:\n unsquashed_targets = targets.view(-1, batch_size).t()\n target_tokens = tensor_to_list(unsquashed_targets)\n return tokens_to_words(target_tokens)\n\n\ndef decode_inputs(inputs: torch.Tensor) -> list:\n input_tokens = tensor_to_list(inputs.t())\n return tokens_to_words(input_tokens)\n\n\ndef join_sequences(decoded_sequences: list) -> list:\n return [\" \".join(sequence) for sequence in decoded_sequences]\n\n\ndef present_result(model, inputs: torch.Tensor, targets: torch.Tensor, index: int) -> None:\n model.eval()\n output = model(inputs)\n input_text = join_sequences(decode_inputs(inputs))[index]\n target_text = join_sequences(decode_targets(targets, 20))[index]\n output_text = join_sequences(decode_transformer_output(output))[index]\n print( \"INPUT\\n-----\\n\" \n f\"{input_text}\\n\\n\"\n \"TARGET\\n------\\n\"\n f\"{target_text}\\n\\n\"\n \"OUTPUT\\n------\\n\"\n f\"{output_text}\\n\")\n```\n\n\n```python\npresent_result(best_model, data, targets, 12)\n```\n\n\n```python\ndef predict_next_word(text: str) -> str:\n encoded_text = TEXT.numericalize([text.split()]).to(device)\n custom_out = best_model(encoded_text)\n custom_out.shape\n _, custom_top_tokens = torch.topk(custom_out, k=1)\n out_tokens = tensor_to_list(custom_top_tokens.squeeze())\n return [TEXT.vocab.itos[token] for token in out_tokens][-1]\n```\n\n\n```python\nsentence = \"that the primary schools , and the kakapo \"\nfor i in range(30):\n input = \" \".join(sentence.split()[-20:])\n print(input)\n sentence += predict_next_word(input) + \" \"\nprint(sentence)\n```\n\n\n```python\ndef evaluate(eval_model, data_source):\n eval_model.eval() # Turn on the evaluation mode\n total_loss = 0.\n ntokens = len(TEXT.vocab.stoi)\n with torch.no_grad():\n for i in range(0, data_source.size(0) - 1, input_sequence_length):\n data, targets = get_batch(data_source, i)\n output = eval_model(data)\n output_flat = output.view(-1, ntokens)\n total_loss += len(data) * criterion(output_flat, targets).item()\n return total_loss / (len(data_source) - 1)\n```\n", "meta": {"hexsha": "eb3ca5c3234d86e378ba67ea7449b474ad3ab5b1", "size": 26136, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "transformer_tutorial_reimplemented.ipynb", "max_stars_repo_name": "rogalskim/script-generation-transformer", "max_stars_repo_head_hexsha": "d1b3ba662845d90cfe10abe377f54752225d55c9", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "transformer_tutorial_reimplemented.ipynb", "max_issues_repo_name": "rogalskim/script-generation-transformer", "max_issues_repo_head_hexsha": "d1b3ba662845d90cfe10abe377f54752225d55c9", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "transformer_tutorial_reimplemented.ipynb", "max_forks_repo_name": "rogalskim/script-generation-transformer", "max_forks_repo_head_hexsha": "d1b3ba662845d90cfe10abe377f54752225d55c9", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 37.4441260745, "max_line_length": 189, "alphanum_fraction": 0.5614478114, "converted": true, "num_tokens": 4409, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056040203135, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.43410716615355494}} {"text": "# Lab Assignment 1\n\n\n## Important notes\n\n**Submission deadline:**\n* **Regular problems: last lab session before or on Monday, 21.10.19**\n* **Bonus problems: deadline for Lab Assignment 2**\n\n**Points: 11 + 5 bonus points**\n\nPlease note: some of the assignments are tedious or boring if you are already a NumPy ninja. The bonus problems were designed to give you a more satisfying alternative.\n\nThe assignment is in the form of a Jupyter notebook. We will be using [Google Colab](https://colab.research.google.com) to solve it. Below you will find a \"Setup\" section. Follow instructions from this paragraph to download the notebook and open it using [Google Colab](https://colab.research.google.com). \n\nYour goal is to solve problems posted below. Whenever possible, add your solutions to the notebook.\n\nPlease email us about any problems with it - we will try to correct them quickly. Also, please do not hesitate to use GitHub\u00e2\u20ac\u2122s pull requests to send us corrections!\n\n## Setup\n\n### 1. Open the notebook using Google Colab\n\n1. From Github: Click on \"View in Colaboratory\", then save to your Google Drive.\n2. Alternatively upload manually to Drive:\n 1. Download the notebook or clone https://github.com/janchorowski/ml_uwr.\n 2. Go to [Google Colab](https://colab.research.google.com).\n 3. Go to \"UPLOAD\" tab and select a local copy of the notebook that you downloaded in point 1.\n \nColab Tips:\n1. Set tab width to 4 spaces under `Tools` \u00e2\u2020\u2019 `Preferences`.\n \n### 2. Open the notebook offline using Jupyter/IPython\n\nThis notebook can be opened using Jupyter notebook. Simply install a scientific Python distribution on your computer (e.g. [Anaconda](https://www.anaconda.com/) or [WinPython](http://winpython.github.io/)), clone the repository https://github.com/janchorowski/nn_assignments and run `jupyter notebook`.\n\n### 3. Install required dependencies, download data and import packages\n\nRun cells below. To run a cell either click it and click a run button or press \"shift + enter\"\n\n\n\n```python\n# Please note that this code needs only to be run in a fresh runtime.\n# However, it can be rerun afterwards too.\n!pip install -q gdown httpimport\n![ -e mnist.npz ] || gdown 'https://drive.google.com/uc?id=1QPaC3IKB_5tX6yIZgRgkpcqFrfVqPTXU' -O mnist.npz\n```\n\n Downloading...\n From: https://drive.google.com/uc?id=1QPaC3IKB_5tX6yIZgRgkpcqFrfVqPTXU\n To: /Users/jdu/Workspace/ml/uwr-things/mnist.npz\n 55.4MB [00:03, 14.1MB/s]\n\n\n\n```python\n# Standard IPython notebook imports\n%matplotlib inline\n\nimport os\n\nimport httpimport\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport pandas as pd\nfrom tqdm import tqdm_notebook\nimport scipy.stats as sstats\n\nimport seaborn as sns\nfrom sklearn import datasets\n\n# In this way we can import functions straight from github\nwith httpimport.github_repo('janchorowski', 'nn_assignments', \n module='common', branch='nn18'):\n from common.plotting import plot_mat\n\nsns.set_style('whitegrid')\n```\n\n\n```python\n\n```\n\n### 4. Follow the notebook and solve problems posted below\n\n## Problems\n\n### Problem 0 [0p]\n\n \n1. To learn more about Jupyter, read [Jupyter tutorial from Data Analysis in Biological Sciences course at Caltech](http://bebi103.caltech.edu/2015/tutorials/t0b_intro_to_jupyter_notebooks.html) (which itself can be downloaded as a Jupyter notebook). Feel free to skip the tutorial if you have some prior experience with Jupyter notebook.\n2. To learn more about basic Google Colab features, go to [Google Colab](https://colab.research.google.com) and select \"Overview of Colaboratory Features\" in \"EXAMPLES\" tab. To learn more about / set up useful keyboard shortcuts (e.g. to add a new cell without clicking \"\"+ code\"), go to \"Tools --> Keyboard shortcuts\"\n\n### Problem 1: NumPy [2p]\n\nFirst, get familiar with Python at https://docs.python.org/3/tutorial/. Then, get\nto know the capabilities of NumPy, the prime numerical library of Python http://www.numpy.org/, for instance with the tutorial at http://wiki.scipy.org/Tentative_NumPy_Tutorial. Finally, look into Pandas at https://pandas.pydata.org/pandas-docs/stable/getting_started/tutorials.html.\n\nYou might also need:\n 1. another intro to NumPy,\nhttp://people.duke.edu/~ccc14/pcfb/numerics.html\n 2. a better interactive shell for Python\nhttp://ipython.org/\n 3. a plotting library for Python\nhttp://matplotlib.org/\n 4. nice statistical plots for matplotlib https://seaborn.pydata.org/.\n\n\n**a) Declare variables:**\n1. $a=10$,\n2. $b=2.5\\times 10^{23}$,\n3. $c=2+3i$, where $i$ is an imaginary unit,\n4. $d=e^{i2\\pi/3}$, where $i$ is an imaginary unit, $e$ is the Euler's number (use `exp`, `pi`).\n\n\n```python\n# TODO: Complete the declarations\nimport math\nimport cmath\n\na = 10\nb = 2.5 * 10 ** 23\nc = 2 + 3j\nd = cmath.exp(1j * (2 * math.pi) / 3)\n\nprint(a)\nprint(b)\nprint(c)\nprint(d)\n\n(2 + 3j) * (1 + 5j)\n```\n\n 10\n 2.5e+23\n (2+3j)\n (-0.4999999999999998+0.8660254037844388j)\n\n\n\n\n\n (-13+13j)\n\n\n\n**b) Declare vectors:**\n1. $aVec=\\begin{bmatrix} 3.14 & 15 & 9 & 26 \\end{bmatrix}$,\n2. $bVec=\\begin{bmatrix} 5 & 4.8 & \\cdots & -4.8 & -5 \\end{bmatrix}$ (vector of numbers from $5$ to $-5$ decreasing by $0.2$),\n3. $cVec=\\begin{bmatrix} 10^0 & 10^{0.01} & \\cdots & 10^{0.99} & 10^1 \\end{bmatrix}$ (logarithmically spaced numbers from 1 to 10, use `logspace` and make sure, that the result has correct length!),\n4. $dVec=Hello$ ($eVec$ is a string of characters, thus a vector).\n\n\n```python\naVec = \nbVec = \ncVec = \ndVec = \n```\n\n**c) Declare matrices:**\n1. $aMat=\\begin{bmatrix}\n 2 & \\cdots & 2 \\\\\n \\vdots & \\ddots & \\vdots \\\\\n 2 & \\cdots & 2\n \\end{bmatrix}$,\n
    \nmatrix $9\\times 9$ filled with 2s (use `ones` or `zeros`),\n2. $bMat=\\begin{bmatrix}\n 1 & 0 & \\cdots & & 0 \\\\\n 0 & \\ddots & 0 & & 0 \\\\\n \\vdots & 0 & 5 & 0 & \\vdots \\\\\n & & 0 & \\ddots & 0 \\\\\n 0 & & \\cdots & 0 & 1\n \\end{bmatrix}$,\n
    \nmatrix $9\\times 9$ filled with zeros, with $\\begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 4 & 3 & 2 & 1 \\end{bmatrix}$ on its diagonal (use `zeros`, `diag`),\n3. $cMat=\\begin{bmatrix}\n 1 & 11 & \\cdots & 91 \\\\\n 2 & 12 & \\cdots & 92 \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n 10 & 20 & \\cdots & 100\n \\end{bmatrix}$,\n
    \nmatrix $10\\times 10$, columns of which form the vector $1:100$ (use `reshape`),\n4. $dMat=\\begin{bmatrix}\n NaN & NaN & NaN & NaN \\\\\n NaN & NaN & NaN & NaN \\\\\n NaN & NaN & NaN & NaN\n \\end{bmatrix}$,\n
    \nmatrix $3\\times 4$ filled with `NaN`s (use... `NaN`),\n5. $eMat=\\begin{bmatrix}\n 13 & -1 & 5 \\\\\n -22 & 10 & -87\n \\end{bmatrix}$,\n6. $fMat$ of shape $3\\times 3$ filled with random natural numbers from $[-3,3]$ (use `rand` and `floor` or `ceil`).\n\n\n```python\naMat = \nbMat = \ncMat = \ndMat = \neMat = \nfMat = \n```\n\n**d) Declare a multiplication table**\nas a $10\\times 10$ matrix `mulMat`. Use matrix/vector multiplication.\n\n\n```python\nmulMat = \n```\n\n**e) Compute element-wise using values from b).**\n\nFor instance, the first element of $xVec[0]$ should be equal to\n\n\\begin{equation}\n1/(\\sqrt{2\\pi2.5^2}) e^{-cVec[0]^2 / (2\\cdot\\pi 2.5^2)}.\n\\end{equation}\n\n1. $xVec=1/(\\sqrt{2\\pi2.5^2}) e^{-cVec^2 / (2\\cdot\\pi 2.5^2)}$\n2. $yVec=\\log_{10}(1/cVec)$, using `log10`\n\n\n```python\nxVec = \nyVec = \n```\n\n**f) Compute with matrix/vector operations.**\n\n**NOTE:** Every multiplication (and power) in this subtask is a [matrix multiplication](https://en.wikipedia.org/wiki/Matrix_multiplication).\n1. $xMat=[0, 1, ..., 6][0, 10, 20, ..., 60]^T$,\n2. $yMat=[0, 10, 20, ..., 60]^T[0, 1, ..., 6]$\n
    \n(remember, that matrix multiplication is not commutative).\n\n\n```python\nxMat = \nyMat = \n```\n\n**g) Declare `ismagic(A)` function** \nwhich checks if matrix $A$ is a [magic square](https://en.wikipedia.org/wiki/Magic_square) and returns a boolean.\n\n\n```python\ndef ismagic(A):\n #TODO\n \nassert not ismagic(np.array([[1,1], [2,2]]))\nassert ismagic(np.array([[2,7,6],[9,5,1],[4,3,8]]))\n```\n\n### Problem 2: Pandas and Seaborn [2p]\n\n1. Load the IRIS Data into a `DataFrame`\n\n\n```python\niris_url = 'https://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data'\n# Use read_csv to load the data. Make sure you get 150 examples!\niris_df = \n\n# Set the column names to\n# 'sepal_length', 'sepal_width', 'petal_length', 'petal_width', 'target'\niris_df.columns = ['sepal_length', 'sepal_width', 'petal_length', 'petal_width', 'target']\n\n# Print the first 10 entries\niris_df.TODO\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    sepal_lengthsepal_widthpetal_lengthpetal_widthtarget
    05.13.51.40.2Iris-setosa
    14.93.01.40.2Iris-setosa
    24.73.21.30.2Iris-setosa
    34.63.11.50.2Iris-setosa
    45.03.61.40.2Iris-setosa
    55.43.91.70.4Iris-setosa
    64.63.41.40.3Iris-setosa
    75.03.41.50.2Iris-setosa
    84.42.91.40.2Iris-setosa
    94.93.11.50.1Iris-setosa
    \n
    \n\n\n\n\n```python\n# Show numerical summary of the data, using DataFrame.describe()\niris_df.TODO\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    sepal_lengthsepal_widthpetal_lengthpetal_width
    count150.000000150.000000150.000000150.000000
    mean5.8433333.0540003.7586671.198667
    std0.8280660.4335941.7644200.763161
    min4.3000002.0000001.0000000.100000
    25%5.1000002.8000001.6000000.300000
    50%5.8000003.0000004.3500001.300000
    75%6.4000003.3000005.1000001.800000
    max7.9000004.4000006.9000002.500000
    \n
    \n\n\n\n\n```python\n# Plot the data using seaborn's pairplot\nsns.TODO\n```\n\nThe Iris data is in a so-called 'wide' format, in which each column corresponds to a variable and each row of the DataFrame corresponds to one observation. Turn it into a 'long' format in which each row is a measurement. \n\nSpecifically, change the data layout of the IRIS dataFrame so that it has 3 columns:\n- variable (one of sepal_length, sepal_width, petal_length, petal_width)\n- value\n- target\n\nIf you would like to learn more, [Tidy Data](https://www.jstatsoft.org/index.php/jss/article/view/v059i10/v59i10.pdf) by [Hadley Wickham](http://hadley.nz/) provides a very nice explanation of best practices for data formating.\n\nHint: look at reshaping functions in http://pandas.pydata.org/Pandas_Cheat_Sheet.pdf\n\n\n```python\niris_df_long = \niris_df_long.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    targetvariablevalue
    0Iris-setosasepal_length5.1
    1Iris-setosasepal_length4.9
    2Iris-setosasepal_length4.7
    3Iris-setosasepal_length4.6
    4Iris-setosasepal_length5.0
    \n
    \n\n\n\nNow create a box-plot of values that each variable takes, split by the target species.\n\n\n```python\n# Hint: use a `catplot`\nsns.catplot(...)\n\n# TODO: create two more plots, using a boxenplot and a swarmplot.\n\nsns.catplot(...)\nsns.catplot(...)\n```\n\n### k-Nearest Neighbors\n\nWe will use the loaded Iris data describing iris flowers\nand shows relations between their length and petal width for three\nspecies (namely: setosa, versicolor, virginica).\n\nFor this exercise we will restrict our analysis to just two variables: **petal length** and **petal width**.\n\n\n```python\nunknown_df = pd.DataFrame(\n [[1.5, 0.3, 'unknown'],\n [4.5, 1.2, 'unknown'],\n [5.1, 1.7, 'unknown'],\n [5.5, 2.3, 'unknown']],\n columns=['petal_length', 'petal_width', 'target'])\n\nsns.scatterplot(x='petal_length', y='petal_width', hue='target', data=iris_df)\nsns.scatterplot(x='petal_length', y='petal_width', color='gray', marker='v',\n label='unknown', s=70, data=unknown_df)\n```\n\nBased on these two features, it is easy to distinguish iris setosa from the two remaining species. Yet iris versicolor and virginica remain mixed together. \n\nLooking closely at the plot, we might estimate the species of the selected unknown irises (gray triangles). For three of them the answer seems obvious \u00e2\u20ac\u201c they belong in uniformly-colored areas covered by one species only. Yet unknown iris flower in (5.1, 1.7) is troublesome \u00e2\u20ac\u201c it lays on the boundary of versicolor and virginica clusters. We can assume, that its species is the one of the closest one to it, coming from the training set (and so having a label). \n\nK-Nearest Neighbors method (http://en.wikipedia.org/wiki/K-nearest_neighbors_algorithm) solves the classification problem, i.e. sets class labels (species in case of irises) of a previously unseen sample by choosing the most common class among the top k neighbors of the sample in question (for instance according to the Euclidean distance). Thus, the k-Nearest Neighbors algorithm works as follows. For each unlabeled sample x:\n1. Find k nearest neighbors among the labeled samples.\n2. Set the most common label among them as label of x.\n\n#### Problem 3 [3p]\n\n##### Implement the k-Nearest Neighbors algorithm [1p].\n\nTake advantage of matrix calculus rather than using for loops.\n\n**Tip:** What is computed by \\begin{equation} \\sqrt{(X - Y)^T (X - Y)} \\end{equation} when both X and Y are vectors?\n\n**Tip:** Try to use broadcasting (NumPy: http://docs.scipy.org/doc/numpy/user/basics.broadcasting.html) and built-ins sort, numpy.sort, numpy.argsort (sorting), scipy.stats.mode (choosing the most common element of the set).\n\n\n```python\ndef KNN(train_X, train_Y, test_X, ks, verbose=False):\n \"\"\"\n Compute predictions for various k\n Args:\n train_X: array of shape Ntrain x D\n train_Y: array of shape Ntrain\n test_X: array of shape Ntest x D\n ks: list of integers\n Returns:\n preds: dict k: predictions for k\n \"\"\"\n # Cats data to float32\n train_X = train_X.astype(np.float32)\n test_X = test_X.astype(np.float32)\n\n # Alloc space for results\n preds = {}\n\n if verbose:\n print(\"Computing distances... \", end='')\n #\n # TODO: fill in an efficient distance matrix computation\n # \n dists = \n\n if verbose:\n print(\"Sorting... \", end='')\n \n # TODO: findes closest trainig points\n # Hint: use argsort\n closest = \n\n if verbose:\n print(\"Computing predictions...\", end='')\n \n targets = train_Y[closest]\n\n for k in ks:\n ... = sstats.mode(...)\n predictions = predictions.ravel()\n preds[k] = predictions\n if verbose:\n print(\"Done\")\n return preds\n```\n\n\n```python\n# Now classify the 4 unknown points\niris_x = np.array(iris_df[['petal_length', 'petal_width']])\niris_y = np.array(iris_df['target'])\n\nunknown_x = np.array(unknown_df[['petal_length', 'petal_width']])\n\nKNN(iris_x, iris_y, unknown_x, [1, 3, 5, 7])\n```\n\n\n\n\n {1: array(['Iris-setosa', 'Iris-versicolor', 'Iris-virginica',\n 'Iris-virginica'], dtype=object),\n 3: array(['Iris-setosa', 'Iris-versicolor', 'Iris-versicolor',\n 'Iris-virginica'], dtype=object),\n 5: array(['Iris-setosa', 'Iris-versicolor', 'Iris-virginica',\n 'Iris-virginica'], dtype=object),\n 7: array(['Iris-setosa', 'Iris-versicolor', 'Iris-virginica',\n 'Iris-virginica'], dtype=object)}\n\n\n\n##### Plot the Decision boundary [1p]\n\n\nUse meshgrid to generate the points in the space spanned by data.\nThen map the classes to numbers 0, 1, 2 and make a contour plot with the\ndecision boundary.\n\n\n```python\niris_x = np.array(iris_df[['petal_length', 'petal_width']])\niris_y = np.array(iris_df['target'])\n\n\nmesh_x, mesh_y = np.meshgrid(...)\n\n#use np.unique with suitable options to map the class names to numbers\ntarget_names, iris_y_ids = np.unique(...)\n\nmesh_data = np.hstack([mesh_x.reshape(-1, 1), mesh_y.reshape(-1, 1)])\n\npreds = KNN(iris_x, iris_y_ids, mesh_data, [1, 3, 5, 7])\nfor k, preds_k in preds.items():\n plt.figure()\n plt.title(f\"Decision boundary for k={k}\")\n plt.contourf(...)\n plt.scatter(...)\n\n```\n\n##### Estimate performance for various ks [1p]\nConsider the following experiment:\n1. We scramble the data and split it into two parts - training set (66.6% of all samples) and test set (33.4%).\n2. Based on the training set, we use the k-NN algorithm to predict the labels on the test set.\n3. We then check the number of errors and write it down.\n\nDo this 500 times for k \u00e2\u02c6\u02c6 {1, 3, 5, ..., 19}. Plot a function of the average number of errors as the function of k. It should be similar to the one below.\n\n\n```python\n#TODO: write a function to compute error rates\ndef err_rates(preds, test_Y):\n ret = {}\n for k, preds_k in preds.items():\n # TODO: fill in error count computation\n ret[k] = \n return ret\n```\n\n\n```python\niris_x = np.array(iris_df[['sepal_length', 'sepal_width', 'petal_length', 'petal_width']])\niris_y = np.array(iris_df['target'])\n\nks = range(1, 20, 2)\nresults = []\n\nfor _rep in tqdm_notebook(range(1000)):\n #TODO\n # Use np.split and np.permutation to get training and testing indices\n train_idx, test_idx = np.split(...)\n\n #TODO: apply your kNN classifier to data subset\n preds = KNN(...)\n errs = err_rates(preds, iris_y[test_idx])\n \n for k, errs_k in errs.items():\n results.append({'K':k, 'err_rate': errs_k})\n\n# results_df will be a data_frame in long format\nresults_df = pd.DataFrame(results)\n\nplt.figure()\nsns.regplot(...)\nplt.figure()\nsns.regplot(...)\n\n```\n\n#### Problem 5 [2p + 2p bonus] \n\nDownload a categorical dataset from UCI and try to find the most predictive variables\n\n\n```python\ncolumns = [\n \"target\", \"cap-shape\", \"cap-surface\", \"cap-color\", \"bruises?\", \"odor\", \n \"gill-attachment\", \"gill-spacing\", \"gill-size\", \"gill-color\", \"stalk-shape\", \n \"stalk-root\", \"stalk-surface-above-ring\", \"stalk-surface-below-ring\", \n \"stalk-color-above-ring\", \"stalk-color-below-ring\", \"veil-type\", \"veil-color\", \n \"ring-number\", \"ring-type\", \"spore-print-color\", \"population\", \"habitat\", ]\n\n# Use read_csv to load the data.\nurl = 'http://archive.ics.uci.edu/ml/machine-learning-databases/mushroom/agaricus-lepiota.data'\nmushroom_df = pd.read_csv(url, header=None, names=columns)\nmushroom_df.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    targetcap-shapecap-surfacecap-colorbruises?odorgill-attachmentgill-spacinggill-sizegill-colorstalk-shapestalk-rootstalk-surface-above-ringstalk-surface-below-ringstalk-color-above-ringstalk-color-below-ringveil-typeveil-colorring-numberring-typespore-print-colorpopulationhabitat
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    \n
    \n\n\n\nImplement the function `entropy` to compute the entropy of a columnt of the dataset.\n\nThe [entropy](https://en.wikipedia.org/wiki/Entropy_(information_theory)) of a discrete variable is defined to be:\n\n$$H(X) = -\\sum_x p_X(x) \\log_2 p_X(x)$$.\n\nA good in tutorial is given by Chris Olah: https://colah.github.io/posts/2015-09-Visual-Information/.\n\n\nWhen $X$ is a discrete random variables, we can estimate the probabilities with counts:\n\n$$p_X(x) = \\frac{\\text{number of instances where }X=x}{\\text{total number of instances}}$$.\n\n\nHint: the following `pandas` functions may be useful:\n- `count`\n- `value_count`\n\nThen use the datafranme's `apply` function to compute the entropy of all columns.\n\n\n```python\ndef entropy(series):\n pass # TODO\n\nmushroom_df.apply(...)\n```\n\n\n\n\n target 0.999068\n cap-shape 1.652889\n cap-surface 1.575486\n cap-color 2.510143\n bruises? 0.979327\n odor 2.319414\n gill-attachment 0.173129\n gill-spacing 0.637878\n gill-size 0.892256\n gill-color 3.030433\n stalk-shape 0.986927\n stalk-root 1.822922\n stalk-surface-above-ring 1.221348\n stalk-surface-below-ring 1.399135\n stalk-color-above-ring 1.936809\n stalk-color-below-ring 1.978163\n veil-type -0.000000\n veil-color 0.196238\n ring-number 0.420680\n ring-type 1.535121\n spore-print-color 2.203227\n population 2.003398\n habitat 2.274747\n dtype: float64\n\n\n\nImplement the conditional entropy computation\n\n$$H(Y|X) = \\sum_x p_X(x) H(Y|x) = \\sum_x p_X(x) \\sum_y p_Y(y|x) \\log_2 p_Y(y|x)$$\n\nHint 1: the above formlua can be computed as follows:\n1. split the data by the values of $X$\n2. for each value $x$ that $X$ takes, compute the entropy of $Y$\n3. average the entropies, weighting them by how frequent the $x$ value ocurred.\n\nHint 2: helpful pandas constructs are:\n- `groupby` and `agg`\n- you can aggregate a grouping using your own custom functions\n\n\n\n```python\ndef cond_entropy(df, X, Y):\n \"\"\"Compute the conditional H(X|Y) in dataframe df\n Args:\n df: a dataframe\n X: the name of the conditioning columt\n Y: the name of the column whose entropy we wish to compute\n \"\"\"\n pass # TODO\n\n```\n\n\n```python\n# Now for each column C compute the conditional entropy H(target|C)\n# Knowing which variable tells us the most about the target?\nfor cname in mushroom_df.columns:\n print(f\"{cname}: {cond_entropy(mushroom_df, cname, 'target')}\")\n```\n\n target: 0.0\n cap-shape: 0.9502711949370874\n cap-surface: 0.9704776640986876\n cap-color: 0.9630186138962565\n bruises?: 0.8066884111112408\n odor: 0.0929929194884606\n gill-attachment: 0.9849028696218441\n gill-spacing: 0.8981847128758901\n gill-size: 0.7689135217244143\n gill-color: 0.5820903734563291\n stalk-shape: 0.9915511243027961\n stalk-root: 0.8642502592451846\n stalk-surface-above-ring: 0.7143422976539759\n stalk-surface-below-ring: 0.7271734234797138\n stalk-color-above-ring: 0.7452227234102206\n stalk-color-below-ring: 0.7576523303448939\n veil-type: 0.9990678968724603\n veil-color: 0.9752508807515436\n ring-number: 0.9606152276293699\n ring-type: 0.6810463860789229\n spore-print-color: 0.518362979187545\n population: 0.797109877805608\n habitat: 0.8422342922673685\n\n\nBonus questions:\n- **[1p]** Implement computation of [Mutual Information ](https://en.wikipedia.org/wiki/Mutual_information)\n- **[1p]** Add an ID column, that assigns a unique ID to each observation (row). Comompute the $H(target|ID)$ and the mutual information between target and ID. How to interpret the results? Do you think the ID is important in predicting the target? \n\n#### Problem 6 [2p] \n\nApply the K-Nearest Neighbors (K-NN) algorithm to the MNIST dataset. \n\nThe MNIST (http://yann.lecun.com/exdb/mnist/) dataset consists of normalized (centered and stretched) scans of hand-written digits. Specifically, each element of the dataset is a 28 \u00c3\u2014 28 grayscale image, thus having 764 8-bit pixels. \n\n1. Display a few objects from each of the classes, paying attention to aesthetics and clarity of your presentation. **Note:** You already downloaded the dataset in \"Setup\" section. Please use the code below to get started.\n\n2. **[2p]** Apply a k-NN classifier to the MNIST dataset. First, divide the training set into two parts, which we will call training and validation. On MNIST use the first 50000 samples for training and the last 10000 for validation. Then find the optimal number of neighbors by assessing the accuracy on the validation set. You do not need to repeat this experiment multiple times. Finally, compute the accuracy on the test set obtained with the best previously chosen number of neighbors. On MNIST you should get about 3% errors. Pick a few mislabeled samples from the test dataset and plot them along with the correct ones. **Note:**\n * MNIST is much larger than the Iris dataset. A good implementation may need a few minutes depending on your runtime type. Please optimize your algorithm:\n * Compute the distances only once, then test for different values of k.\n * Use vectorized expressions to compute the distance. It is possible to compute all distances between the training and testing points in one expression. Hint: think about the vectorized expression \\begin{equation}(X - Y)^T (X - Y)\\end{equation}\n * You can use single precision numbers in computation.\n * If your code is taking a long time to execute, please save its results before the lab session.\n\n**Note:** in NumPy, matrices have its own data type (dtype), which is retained during\ncalculations. Please pay attention to it. I particular, do not subtract values of data types not\nhaving the sign bit, do not divide integers, etc. Results of such operations will not be\nautomatically casted to types having the required precision.\n\n\n```python\nwith np.load('mnist.npz') as data:\n mnist_full_train_data_uint8 = data['train_data']\n mnist_full_train_labels_int64 = data['train_labels']\n mnist_test_data_uint8 = data['test_data']\n mnist_test_labels_int64 = data['test_labels']\n \n# Split train data into train and validation sets\nmnist_train_data_uint8 = mnist_full_train_data_uint8[:50000]\nmnist_train_labels_int64 = mnist_full_train_labels_int64[:50000]\nmnist_valid_data_uint8 = mnist_full_train_data_uint8[50000:]\nmnist_valid_labels_int64 = mnist_full_train_labels_int64[50000:]\n```\n\n\n```python\nplot_mat(mnist_train_data_uint8[:20, None], cmap='gray')\n```\n\n\n```python\n# MNIST is large.\n# Implement a batched KNN classifier, which processes the test data in small batches\n# and returns the error rates\n\n# The code should not run for more than a couple of minutes on the Colab runtime, \n# If it is slower, optimize the distance computation in KNN\n\ndef batched_KNN(train_X, train_Y, test_X, ks, verbose=False, batch_size=200):\n all_preds = {k: [] for k in ks}\n for i in range(0, test_X.shape[0], batch_size):\n batch_X = test_X[i:i + batch_size]\n if verbose:\n print(f\"Processing batch {i}:{i + batch_X.shape[0]}... \", end='')\n\n # TODO: run KNN on the batch and save the predictions\n for k in all_preds.keys():\n # TODO: combine predictions from batches\n all_preds[k] = np.concatenate...\n return all_preds\n```\n\n\n```python\n# Now find the best k on the validation set\nks = [1, 3, 5, 7, 9]\nmnist_validation_preds = batched_KNN(\n mnist_train_data_uint8.astype('float32').reshape(-1, 28*28), mnist_train_labels_int64,\n mnist_valid_data_uint8.astype('float32').reshape(-1, 28*28),\n ks, verbose=True)\n\nmnist_validation_errs = err_rates(mnist_validation_preds, mnist_valid_labels_int64)\nplt.plot(ks, [mnist_validation_errs[k] for k in ks])\n```\n\n\n```python\n# Now use the best k to compute the test error\n\nbest_K = TODO\n\nmnist_test_preds = batched_KNN(\n mnist_full_train_data_uint8.astype('float32').reshape(-1, 28*28), \n mnist_full_train_labels_int64,\n mnist_test_data_uint8.astype('float32').reshape(-1, 28*28), \n [best_K], verbose=True)\n\nmnist_test_errs = err_rates(mnist_test_preds, mnist_test_labels_int64)\nprint(f\"\\n\\nWhen k={best_K} the test error rate is {mnist_test_errs[best_K] * 100.0:.1f}%%\")\n```\n\n### Locality sensitive hashing\n\nProblem 5 was about speeding up the inference using loops implicitly present in matrix multiplication instead of explicit loops in Python. In this problem, we will explore a strategy to truly reduce the total number of computations required to find nearest neighbors without sacrificing too much accuracy.\n\nTo speed up nearest neighbor search we will employ *Locality Sensitive Hashing (LSH)* functions. For a given distance metric, the locality sensitive hash should put items that are similar into the same bucket. Notice that this is essentially a design choice opposite to traditional cryptographic hash functions that should amplify the difference of similar inputs (typically we want that small perturbations of data result in large changes to the hash value).\n\nOne of the simplest implementations of LSH approximates the cosine distance. Let $x\\in \\mathbb{R}^N$ and $y\\in \\mathbb{R}^N$ be two vectors. Their cosine distance is defined as:\n\n\\begin{equation}\n d_\\text{cos}(x,y) = \\frac{x \\cdot y}{\\|x\\| \\|y\\|} = \\cos\\left(\\theta(x,y)\\right),\n\\end{equation}\nwhere $\\theta(x,y)$ is the unsigned angle between $x$ and $y$.\n\nWe will construct a family $H$ of hash functions that are an LSH for angle distances (an approximation to cosine distance). Assume $p\\in \\mathbb{R}^N$ is a random vector (components are sampled from the normal distribution) of length 1. Then define the hash function $h(x) = \\text{sgn}(x\\cdot p)$, where $\\text{sgn()}$ is the sign function. It can be proven that:\n\n\\begin{equation}\n p_{h\\in H}[h(x)=h(y)] = 1 - \\frac{\\theta(x,y)}{\\pi}.\n\\end{equation}\n\nThe equation means that the probability of a hash collision grows as the the angle between two vectors gets smaller. Therefore, vectors that are close according to the cosine distance will be put with high probability into the same bin (we use the fact that for small $\\theta$ we can approximate $\\cos(\\theta) = 1 - \\theta/\\pi$.\n\nWe will say that a family of randomly chosen hash functions $H$ is $(d_1, d_2, p_1, p_2)$-sensitive with respect to a distance metric $d$ if for any $x$ and $y$:\n1. If $d(x,y) \\leq d_1$ then $p_{h\\in H}[h(x)=h(y)] \\geq p_1$.\n2. If $d(x,y) \\geq d_2$ then $p_{h\\in H}[h(x)=h(y)] \\leq p_2$.\n\nFor example, our family of randomly chosen hyperplanes is $(d_1, d_2, (1-d_1)/\\pi, (1-d_2)/\\pi)$-sensitive.\n\nIdeally, vectors should be placed into the same bin with a high probability if their distance is smaller than a threshold, and with a low probability if their distance is larger that the threshold. By combining hashing functions we can get closer to this ideal sensitivity.\n\nGiven a family of hash functions $H$ with sensitivity $(d_1, d_2, p_1, p_2)$ we can construct a new family $H'$ by combining $r$ functions from $H$:\n1. AND: let $h=[h_1, h_2, \\ldots, h_r] \\in H'$ and $h(x)=h(y)$ if and only if $\\forall_i h_i(x)=h_i(y)$. Then $H'$ is $(d_1, d_2, (p_1)^r, (p_2)^r)$-sensitive.\n2. OR: let $h=[h_1, h_2, \\ldots, h_r] \\in H'$ and $h(x)=h(y)$ if and only if $\\exists_i h_i(x)=h_i(y)$. Then $H'$ is $(d_1, d_2, 1-(1-p_1)^r, 1-(1-p_2)^r)$-sensitive.\n\nAND makes all probabilities shrink, but properly choosing $r$ we can make the lower probability approach 0 while the higher does not. Conversely, OR makes all probabilities grow, we can make the upper probability approach 1 while the lower does not.\n\n#### Problem 7 [1-3p bonus] \n\n1. **[1bp for exercises list]** **Note:** you can show sketches of proofs for this assignment.\n 1. Show that angle between vectors is a metric (https://en.wikipedia.org/wiki/Metric_(mathematics)).\n \n 2. Show that $p_{h\\in H}[h(x)=h(y)] = 1 - \\frac{\\theta(x,y)}{\\pi}$ for $h$ computed using a randomly chosen hyperplane.\n\n 3. Show the properties of either AND or OR boosting of LSH.\n\n Please show the solution to this problem dirung the Session for Homework 1, the bonus point will also be added to the points from Homework 1.\n\n3. **[1-3bp]** Reimplement k-Nearest Neighbors for MNIST classification using the cosine distance instead of the Euclidean distance. Choose a sensible value of $k$. Use Locality Sensitive Hashing to achieve an error rate no greater than $150\\%$ of the original error rate with at least a $90\\%$ speedup (i.e., by considering on average at most 5000 training samples per query image). For a few settings plot the speedup-vs-accuracy relation.\n\n **Note:** points will be awarded based on ingenuity of your solution. Feel free to explore your own ideas!\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "a9213c75f75130f4c28fbd1007d4638776673d93", "size": 776651, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "uwr-things/Assignment1.ipynb", "max_stars_repo_name": "JacekDuszenko/ml", "max_stars_repo_head_hexsha": "e2f8bc74169f1ddea1df1c52500c1c67ac5fd817", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "uwr-things/Assignment1.ipynb", "max_issues_repo_name": "JacekDuszenko/ml", "max_issues_repo_head_hexsha": "e2f8bc74169f1ddea1df1c52500c1c67ac5fd817", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "uwr-things/Assignment1.ipynb", "max_forks_repo_name": "JacekDuszenko/ml", "max_forks_repo_head_hexsha": "e2f8bc74169f1ddea1df1c52500c1c67ac5fd817", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 346.5644801428, "max_line_length": 167150, "alphanum_fraction": 0.9209348858, "converted": true, "num_tokens": 11533, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.7371581626286834, "lm_q1q2_score": 0.4341044296288175}} {"text": "# Chapter 6\n\n---\n\nThis chapter of [Bayesian Methods for Hackers](https://github.com/CamDavidsonPilon/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers) focuses on the most debated and discussed part of Bayesian methodologies: how to choose an appropriate prior distribution. We also present how the prior's influence changes as our dataset increases, and an interesting relationship between priors and penalties on linear regression.\n\n## Getting our priorities straight\n\nUp until now, we have mostly ignored our choice of priors. This is unfortunate as we can be very expressive with our priors, but we also must be careful about choosing them. This is especially true if we want to be objective, that is, not to express any personal beliefs in the priors. \n\n### Subjective vs Objective priors\n\nBayesian priors can be classified into two classes: *objective* priors, which aim to allow the data to influence the posterior the most, and *subjective* priors, which allow the practitioner to express his or her views into the prior. \n\nWhat is an example of an objective prior? We have seen some already, including the *flat* prior, which is a uniform distribution over the entire possible range of the unknown. Using a flat prior implies that we give each possible value an equal weighting. Choosing this type of prior is invoking what is called \"The Principle of Indifference\", literally we have no prior reason to favor one value over another. Calling a flat prior over a restricted space an objective prior is not correct, though it seems similar. If we know $p$ in a Binomial model is greater than 0.5, then $\\text{Uniform}(0.5,1)$ is not an objective prior (since we have used prior knowledge) even though it is \"flat\" over [0.5, 1]. The flat prior must be flat along the *entire* range of possibilities. \n\nAside from the flat prior, other examples of objective priors are less obvious, but they contain important characteristics that reflect objectivity. For now, it should be said that *rarely* is a objective prior *truly* objective. We will see this later. \n\n#### Subjective Priors\n\nOn the other hand, if we added more probability mass to certain areas of the prior, and less elsewhere, we are biasing our inference towards the unknowns existing in the former area. This is known as a subjective, or *informative* prior. In the figure below, the subjective prior reflects a belief that the unknown likely lives around 0.5, and not around the extremes. The objective prior is insensitive to this.\n\n\n```python\n%matplotlib inline\nimport numpy as np\nimport scipy.stats as stats\nfrom IPython.core.pylabtools import figsize\nimport matplotlib.pyplot as plt\n\nfigsize(12.5,3)\ncolors = [\"#348ABD\", \"#A60628\", \"#7A68A6\", \"#467821\"]\n\nx = np.linspace(0,1)\ny1, y2 = stats.beta.pdf(x, 1,1), stats.beta.pdf(x, 10,10)\n\np = plt.plot(x, y1, \n label='An objective prior \\n(uninformative, \\n\"Principle of Indifference\")')\nplt.fill_between(x, 0, y1, color = p[0].get_color(), alpha = 0.3)\n\np = plt.plot(x,y2 ,\n label = \"A subjective prior \\n(informative)\")\nplt.fill_between(x, 0, y2, color = p[0].get_color(), alpha = 0.3)\n\np = plt.plot(x[25:], 2*np.ones(25), label = \"another subjective prior\")\nplt.fill_between(x[25:], 0, 2, color = p[0].get_color(), alpha = 0.3)\n\nplt.ylim(0,4)\n\nplt.ylim(0, 4)\nleg = plt.legend(loc = \"upper left\")\nleg.get_frame().set_alpha(0.4)\nplt.title(\"Comparing objective vs. subjective priors for an unknown probability\");\n```\n\nThe choice of a subjective prior does not always imply that we are using the practitioner's subjective opinion: more often the subjective prior was once a posterior to a previous problem, and now the practitioner is updating this posterior with new data. A subjective prior can also be used to inject *domain knowledge* of the problem into the model. We will see examples of these two situations later.\n\n### Decision, decisions...\n\nThe choice, either *objective* or *subjective* mostly depends on the problem being solved, but there are a few cases where one is preferred over the other. In instances of scientific research, the choice of an objective prior is obvious. This eliminates any biases in the results, and two researchers who might have differing prior opinions would feel an objective prior is fair. Consider a more extreme situation:\n\n> A tobacco company publishes a report with a Bayesian methodology that retreated 60 years of medical research on tobacco use. Would you believe the results? Unlikely. The researchers probably chose a subjective prior that too strongly biased results in their favor.\n\nUnfortunately, choosing an objective prior is not as simple as selecting a flat prior, and even today the problem is still not completely solved. The problem with naively choosing the uniform prior is that pathological issues can arise. Some of these issues are pedantic, but we delay more serious issues to the Appendix of this Chapter (TODO).\n\nWe must remember that choosing a prior, whether subjective or objective, is still part of the modeling process. To quote Gelman [5]:\n\n>... after the model has been fit, one should look at the posterior distribution\nand see if it makes sense. If the posterior distribution does not make sense, this implies\nthat additional prior knowledge is available that has not been included in the model,\nand that contradicts the assumptions of the prior distribution that has been used. It is\nthen appropriate to go back and alter the prior distribution to be more consistent with\nthis external knowledge.\n\nIf the posterior does not make sense, then clearly one had an idea what the posterior *should* look like (not what one *hopes* it looks like), implying that the current prior does not contain all the prior information and should be updated. At this point, we can discard the current prior and choose a more reflective one.\n\nGelman [4] suggests that using a uniform distribution with large bounds is often a good choice for objective priors. Although, one should be wary about using Uniform objective priors with large bounds, as they can assign too large of a prior probability to non-intuitive points. Ask yourself: do you really think the unknown could be incredibly large? Often quantities are naturally biased towards 0. A Normal random variable with large variance (small precision) might be a better choice, or an Exponential with a fat tail in the strictly positive (or negative) case. \n\nIf using a particularly subjective prior, it is your responsibility to be able to explain the choice of that prior, else you are no better than the tobacco company's guilty parties. \n\n### Empirical Bayes\n\nWhile not a true Bayesian method, *empirical Bayes* is a trick that combines frequentist and Bayesian inference. As mentioned previously, for (almost) every inference problem there is a Bayesian method and a frequentist method. The significant difference between the two is that Bayesian methods have a prior distribution, with hyperparameters $\\alpha$, while empirical methods do not have any notion of a prior. Empirical Bayes combines the two methods by using frequentist methods to select $\\alpha$, and then proceeds with Bayesian methods on the original problem. \n\nA very simple example follows: suppose we wish to estimate the parameter $\\mu$ of a Normal distribution, with $\\sigma = 5$. Since $\\mu$ could range over the whole real line, we can use a Normal distribution as a prior for $\\mu$. How to select the prior's hyperparameters, denoted ($\\mu_p, \\sigma_p^2$)? The $\\sigma_p^2$ parameter can be chosen to reflect the uncertainty we have. For $\\mu_p$, we have two options:\n\n1. Empirical Bayes suggests using the empirical sample mean, which will center the prior around the observed empirical mean:\n\n$$ \\mu_p = \\frac{1}{N} \\sum_{i=0}^N X_i $$\n\n2. Traditional Bayesian inference suggests using prior knowledge, or a more objective prior (zero mean and fat standard deviation).\n\nEmpirical Bayes can be argued as being semi-objective, since while the choice of prior model is ours (hence subjective), the parameters are solely determined by the data.\n\nPersonally, I feel that Empirical Bayes is *double-counting* the data. That is, we are using the data twice: once in the prior, which will influence our results towards the observed data, and again in the inferential engine of MCMC. This double-counting will understate our true uncertainty. To minimize this double-counting, I would only suggest using Empirical Bayes when you have *lots* of observations, else the prior will have too strong of an influence. I would also recommend, if possible, to maintain high uncertainty (either by setting a large $\\sigma_p^2$ or equivalent.)\n\nEmpirical Bayes also violates a theoretical axiom in Bayesian inference. The textbook Bayesian algorithm of:\n\n>*prior* $\\Rightarrow$ *observed data* $\\Rightarrow$ *posterior* \n\nis violated by Empirical Bayes, which instead uses \n\n>*observed data* $\\Rightarrow$ *prior* $\\Rightarrow$ *observed data* $\\Rightarrow$ *posterior*\n\nIdeally, all priors should be specified *before* we observe the data, so that the data does not influence our prior opinions (see the volumes of research by Daniel Kahneman *et. al* about [anchoring](http://en.wikipedia.org/wiki/Anchoring_and_adjustment)).\n\n## Useful priors to know about\n\n### The Gamma distribution\n\n\nA Gamma random variable, denoted $X \\sim \\text{Gamma}(\\alpha, \\beta)$, is a random variable over the positive real numbers. It is in fact a generalization of the Exponential random variable, that is:\n\n$$ \\text{Exp}(\\beta) \\sim \\text{Gamma}(1, \\beta) $$\n\nThis additional parameter allows the probability density function to have more flexibility, hence allowing the practitioner to express his or her subjective priors more accurately. The density function for a $\\text{Gamma}(\\alpha, \\beta)$ random variable is:\n\n$$ f_X(x \\mid \\alpha, \\beta) = \\frac{\\beta^{\\alpha}x^{\\alpha-1}e^{-\\beta x}}{\\Gamma(\\alpha)} $$\n\nwhere $\\Gamma(\\alpha)$ is the [Gamma function](http://en.wikipedia.org/wiki/Gamma_function), and for differing values of $(\\alpha, \\beta)$ looks like:\n\n\n```python\nfigsize(12.5, 5)\ngamma = stats.gamma\n\nparameters = [(1, 0.5), (9, 2), (3, 0.5), (7, 0.5)]\nx = np.linspace(0.001 ,20, 150)\nfor alpha, beta in parameters:\n y = gamma.pdf(x, alpha, scale=1./beta)\n lines = plt.plot(x, y, label = \"(%.1f,%.1f)\"%(alpha,beta), lw = 3)\n plt.fill_between(x, 0, y, alpha = 0.2, color = lines[0].get_color())\n plt.autoscale(tight=True)\n \nplt.legend(title=r\"$\\alpha, \\beta$ - parameters\");\n```\n\n### The Wishart distribution\n\nUntil now, we have only seen random variables that are scalars. Of course, we can also have *random matrices*! Specifically, the Wishart distribution is a distribution over all [positive semi-definite matrices](http://en.wikipedia.org/wiki/Positive-definite_matrix). Why is this useful to have in our arsenal? (Proper) covariance matrices are positive-definite, hence the Wishart is an appropriate prior for covariance matrices. We can't really visualize a distribution of matrices, so I'll plot some realizations from the $5 \\times 5$ (above) and $20 \\times 20$ (below) Wishart distribution:\n\n\n```python\nn = 4\nfor i in range(10):\n ax = plt.subplot(2, 5, i+1)\n if i >= 5:\n n = 15\n plt.imshow(stats.wishart.rvs(n+1, np.eye(n)), interpolation=\"none\", \n cmap = \"hot\")\n ax.axis(\"off\")\n \nplt.suptitle(\"Random matrices from a Wishart Distribution\");\n```\n\nOne thing to notice is the symmetry of these matrices. The Wishart distribution can be a little troubling to deal with, but we will use it in an example later.\n\n### The Beta distribution\n\nYou may have seen the term `beta` in previous code in this book. Often, I was implementing a Beta distribution. The Beta distribution is very useful in Bayesian statistics. A random variable $X$ has a $\\text{Beta}$ distribution, with parameters $(\\alpha, \\beta)$, if its density function is:\n\n$$f_X(x | \\; \\alpha, \\beta ) = \\frac{ x^{(\\alpha - 1)}(1-x)^{ (\\beta - 1) } }{B(\\alpha, \\beta) }$$\n\nwhere $B$ is the [Beta function](http://en.wikipedia.org/wiki/Beta_function) (hence the name). The random variable $X$ is only allowed in [0,1], making the Beta distribution a popular distribution for decimal values, probabilities and proportions. The values of $\\alpha$ and $\\beta$, both positive values, provide great flexibility in the shape of the distribution. Below we plot some distributions:\n\n\n```python\nfigsize(12.5, 5)\n\nparams = [(2, 5), (1, 1), (0.5, 0.5), (5, 5), (20, 4), (5, 1)]\n\nx = np.linspace(0.01, .99, 100)\nbeta = stats.beta\nfor a, b in params:\n y = beta.pdf(x, a, b)\n lines = plt.plot(x, y, label = \"(%.1f,%.1f)\"%(a,b), lw = 3)\n plt.fill_between(x, 0, y, alpha = 0.2, color = lines[0].get_color())\n plt.autoscale(tight=True)\nplt.ylim(0)\nplt.legend(loc = 'upper left', title=\"(a,b)-parameters\");\n```\n\nOne thing I'd like the reader to notice is the presence of the flat distribution above, specified by parameters $(1,1)$. This is the Uniform distribution. Hence the Beta distribution is a generalization of the Uniform distribution, something we will revisit many times.\n\nThere is an interesting connection between the Beta distribution and the Binomial distribution. Suppose we are interested in some unknown proportion or probability $p$. We assign a $\\text{Beta}(\\alpha, \\beta)$ prior to $p$. We observe some data generated by a Binomial process, say $X \\sim \\text{Binomial}(N, p)$, with $p$ still unknown. Then our posterior *is again a Beta distribution*, i.e. $p | X \\sim \\text{Beta}( \\alpha + X, \\beta + N -X )$. Succinctly, one can relate the two by \"a Beta prior with Binomial observations creates a Beta posterior\". This is a very useful property, both computationally and heuristically.\n\nIn light of the above two paragraphs, if we start with a $\\text{Beta}(1,1)$ prior on $p$ (which is a Uniform), observe data $X \\sim \\text{Binomial}(N, p)$, then our posterior is $\\text{Beta}(1 + X, 1 + N - X)$. \n\n\n## Example: Bayesian Multi-Armed Bandits\n\n\nKenny's Summary:\n\nOne-armed bandit is a slot machine. A multi-armed bandit is a row of slot machines, or a slot machine with multiple arms (all independent). \n\nEach arm has a fixed probability of a payout. Your goal is to go in blind, start pulling arms, and get the maximum payout in $T$ pulls. What's the best strategy for pulling?\n\nThe (impossible) perfect strategy is to always pull the best arm $T$ times. Say this arm's payout probability is $w_{opt}$. This gives you a total payout of:\n\n$$\\text{maximum possible payout} = T w_{opt}$$\n\nGiven some strategy, we can measure its performance with a metric called regret, which is the difference between the max payout and what you actually got.\n\n$$\\text{regret}(\\text{some strategy})= Tw_{opt} - \\sum_{i=1}^{T} \\; w_{B(i)}$$\n\nAn optimal strategy to the multi-armed bandit problem achieves logarithmically growing regret as $T$ increases. A suboptimal strategy achieves sub-linear regret. A random strategy or always choosing the worst arm (the worst strategy) achieves linear regret. \n\nThe natural Bayesian strategy is to track a posterior distribution of the payout probability of each arm. On each turn, you could:\n\n1. Sample from all distributions and pick the arm with the highest sampled probability\n2. Pick the arm with the highest posterior mean\n3. Pick the arm with the highest posterior 95% credible interval upper bound ([Bayesian-UCB](https://lilianweng.github.io/lil-log/2018/01/23/the-multi-armed-bandit-problem-and-its-solutions.html#bayesian-ucb))\n\nAfter we pick and pull an arm, we update that arm's posterior distribution. And repeat.\n\n---\n\n*Adapted from an example by Ted Dunning of MapR Technologies*\n\n> Suppose you are faced with $N$ slot machines (colourfully called multi-armed bandits). Each bandit has an unknown probability of distributing a prize (assume for now the prizes are the same for each bandit, only the probabilities differ). Some bandits are very generous, others not so much. Of course, you don't know what these probabilities are. By only choosing one bandit per round, our task is devise a strategy to maximize our winnings.\n\nOf course, if we knew the bandit with the largest probability, then always picking this bandit would yield the maximum winnings. So our task can be phrased as \"Find the best bandit, and as quickly as possible\". \n\nThe task is complicated by the stochastic nature of the bandits. A suboptimal bandit can return many winnings, purely by chance, which would make us believe that it is a very profitable bandit. Similarly, the best bandit can return many duds. Should we keep trying losers then, or give up? \n\nA more troublesome problem is, if we have found a bandit that returns *pretty good* results, do we keep drawing from it to maintain our *pretty good score*, or do we try other bandits in hopes of finding an *even-better* bandit? This is the exploration vs. exploitation dilemma.\n\n### Applications\n\nThe Multi-Armed Bandit problem at first seems very artificial, something only a mathematician would love, but that is only before we address some applications:\n\n- Internet display advertising: companies have a suite of potential ads they can display to visitors, but the company is not sure which ad strategy to follow to maximize sales. This is similar to A/B testing, but has the added advantage of naturally minimizing strategies that do not work (and generalizes to A/B/C/D... strategies)\n- Ecology: animals have a finite amount of energy to expend, and following certain behaviours has uncertain rewards. How does the animal maximize its fitness?\n- Finance: which stock option gives the highest return, under time-varying return profiles.\n- Clinical trials: a researcher would like to find the best treatment, out of many possible treatment, while minimizing losses. \n- Psychology: how does punishment and reward affect our behaviour? How do humans learn?\n\nMany of these questions above are fundamental to the application's field.\n\nIt turns out the *optimal solution* is incredibly difficult, and it took decades for an overall solution to develop. There are also many approximately-optimal solutions which are quite good. The one I wish to discuss is one of the few solutions that can scale incredibly well. The solution is known as *Bayesian Bandits*.\n\n### A Proposed Solution\n\nAny proposed strategy is called an *online algorithm* (not in the internet sense, but in the continuously-being-updated sense), and more specifically a reinforcement learning algorithm. The algorithm starts in an ignorant state, where it knows nothing, and begins to acquire data by testing the system. As it acquires data and results, it learns what the best and worst behaviours are (in this case, it learns which bandit is the best). With this in mind, perhaps we can add an additional application of the Multi-Armed Bandit problem:\n\n- Psychology: how does punishment and reward affect our behaviour? How do humans learn?\n\n\nThe Bayesian solution begins by assuming priors on the probability of winning for each bandit. In our vignette we assumed complete ignorance of these probabilities. So a very natural prior is the flat prior over 0 to 1. The algorithm proceeds as follows:\n\nFor each round:\n\n1. Sample a random variable $X_b$ from the prior of bandit $b$, for all $b$.\n2. Select the bandit with largest sample, i.e. select $B = \\text{argmax}\\;\\; X_b$.\n3. Observe the result of pulling bandit $B$, and update your prior on bandit $B$.\n4. Return to 1.\n\nThat's it. Computationally, the algorithm involves sampling from $N$ distributions. Since the initial priors are $\\text{Beta}(\\alpha=1,\\beta=1)$ (a uniform distribution), and the observed result $X$ (a win or loss, encoded 1 and 0 respectfully) is Binomial, the posterior is a $\\text{Beta}(\\alpha=1+X,\\beta=1+1\u2212X)$.\n\nTo answer our question from before, this algorithm suggests that we should not discard losers, but we should pick them at a decreasing rate as we gather confidence that there exist *better* bandits. This follows because there is always a non-zero chance that a loser will achieve the status of $B$, but the probability of this event decreases as we play more rounds (see figure below).\n\nBelow we implement Bayesian Bandits using two classes, `Bandits` that defines the slot machines, and `BayesianStrategy` which implements the above learning strategy.\n\n\n```python\nrand = np.random.rand\n\nclass Bandits(object):\n \"\"\"\n This class represents N bandits machines.\n\n parameters:\n p_array: a (n,) Numpy array of probabilities >0, <1.\n\n methods:\n pull( i ): return the results, 0 or 1, of pulling \n the ith bandit.\n \"\"\"\n def __init__(self, p_array):\n self.p = p_array\n self.optimal = np.argmax(p_array)\n \n def pull(self, i):\n #i is which arm to pull\n return np.random.rand() < self.p[i]\n \n def __len__(self):\n return len(self.p)\n\n \nclass BayesianStrategy(object):\n \"\"\"\n Implements a online, learning strategy to solve\n the Multi-Armed Bandit problem.\n \n parameters:\n bandits: a Bandit class with .pull method\n \n methods:\n sample_bandits(n): sample and train on n pulls.\n\n attributes:\n N: the cumulative number of samples\n choices: the historical choices as a (N,) array\n bb_score: the historical score as a (N,) array\n \"\"\"\n \n def __init__(self, bandits):\n \n self.bandits = bandits\n n_bandits = len(self.bandits)\n self.wins = np.zeros(n_bandits)\n self.trials = np.zeros(n_bandits)\n self.N = 0\n self.choices = []\n self.bb_score = []\n\n \n def sample_bandits(self, n=1):\n \n bb_score = np.zeros(n)\n choices = np.zeros(n)\n \n for k in range(n):\n #sample from the bandits's priors, and select the largest sample\n choice = np.argmax(np.random.beta(1 + self.wins, 1 + self.trials - self.wins))\n \n #sample the chosen bandit\n result = self.bandits.pull(choice)\n \n #update priors and score\n self.wins[choice] += result\n self.trials[choice] += 1\n bb_score[k] = result \n self.N += 1\n choices[k] = choice\n \n self.bb_score = np.r_[self.bb_score, bb_score]\n self.choices = np.r_[self.choices, choices]\n return \n```\n\nBelow we visualize the learning of the Bayesian Bandit solution.\n\n\n```python\nfigsize(11.0, 10)\n\nbeta = stats.beta\nx = np.linspace(0.001,.999,200)\n\ndef plot_priors(bayesian_strategy, prob, lw = 3, alpha = 0.2, plt_vlines = True):\n ## plotting function\n wins = bayesian_strategy.wins\n trials = bayesian_strategy.trials\n for i in range(prob.shape[0]):\n y = beta(1+wins[i], 1 + trials[i] - wins[i])\n p = plt.plot(x, y.pdf(x), lw = lw)\n c = p[0].get_markeredgecolor()\n plt.fill_between(x,y.pdf(x),0, color = c, alpha = alpha, \n label=\"underlying probability: %.2f\" % prob[i])\n if plt_vlines:\n plt.vlines(prob[i], 0, y.pdf(prob[i]) ,\n colors = c, linestyles = \"--\", lw = 2)\n plt.autoscale(tight = \"True\")\n plt.title(\"Posteriors After %d pull\" % bayesian_strategy.N +\\\n \"s\"*(bayesian_strategy.N > 1))\n plt.autoscale(tight=True)\n return\n```\n\n\n```python\nhidden_prob = np.array([0.85, 0.60, 0.75])\nbandits = Bandits(hidden_prob)\nbayesian_strat = BayesianStrategy(bandits)\n\ndraw_samples = [1, 1, 3, 10, 10, 25, 50, 100, 200, 600]\n\nfor j,i in enumerate(draw_samples):\n plt.subplot(5, 2, j+1) \n bayesian_strat.sample_bandits(i)\n plot_priors(bayesian_strat, hidden_prob)\n #plt.legend()\n plt.autoscale(tight = True)\nplt.tight_layout()\n```\n\nNote that we don't really care how accurate we become about the inference of the hidden probabilities — for this problem we are more interested in choosing the best bandit (or more accurately, becoming *more confident* in choosing the best bandit). For this reason, the distribution of the red bandit is very wide (representing ignorance about what that hidden probability might be) but we are reasonably confident that it is not the best, so the algorithm chooses to ignore it.\n\nFrom the above, we can see that after 1000 pulls, the majority of the \"blue\" function leads the pack, hence we will almost always choose this arm. This is good, as this arm is indeed the best.\n\nBelow is a D3 app that demonstrates our algorithm updating/learning three bandits. The first figure are the raw counts of pulls and wins, and the second figure is a dynamically updating plot. I encourage you to try to guess which bandit is optimal, prior to revealing the true probabilities, by selecting the `arm buttons`.\n\n\n```python\nfrom IPython.core.display import HTML\n\n#try executing the below command twice if the first time doesn't work\nHTML(filename = \"BanditsD3.html\")\n```\n\n\n\n\n\n \n \n\n\n\n\n\n\n
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    \n\n\n\n\n\n\nDeviations of the observed ratio from the highest probability is a measure of performance. For example,in the long run, optimally we can attain the reward/pull ratio of the maximum bandit probability. Long-term realized ratios less than the maximum represent inefficiencies. (Realized ratios larger than the maximum probability is due to randomness, and will eventually fall below). \n\n### A Measure of *Good*\n\nWe need a metric to calculate how well we are doing. Recall the absolute *best* we can do is to always pick the bandit with the largest probability of winning. Denote this best bandit's probability by $w_{opt}$. Our score should be relative to how well we would have done had we chosen the best bandit from the beginning. This motivates the *total regret* of a strategy, defined:\n\n\\begin{align}\nR_T & = \\sum_{i=1}^{T} \\left( w_{opt} - w_{B(i)} \\right)\\\\\\\\\n& = Tw_{opt} - \\sum_{i=1}^{T} \\; w_{B(i)} \n\\end{align}\n\n\nwhere $w_{B(i)}$ is the probability of a prize of the chosen bandit in the $i$ round. A total regret of 0 means the strategy is matching the best possible score. This is likely not possible, as initially our algorithm will often make the wrong choice. Ideally, a strategy's total regret should flatten as it learns the best bandit. (Mathematically, we achieve $w_{B(i)}=w_{opt}$ often)\n\n\nBelow we plot the total regret of this simulation, including the scores of some other strategies:\n\n1. Random: randomly choose a bandit to pull. If you can't beat this, just stop. \n2. Largest Bayesian credible bound: pick the bandit with the largest upper bound in its 95% credible region of the underlying probability. \n3. Bayes-UCB algorithm: pick the bandit with the largest *score*, where score is a dynamic quantile of the posterior (see [4] )\n3. Mean of posterior: choose the bandit with the largest posterior mean. This is what a human player (sans computer) would likely do. \n3. Largest proportion: pick the bandit with the current largest observed proportion of winning. \n1. Bayesian Bandit (from earlier): sample a probability of winning from your prior for each bandit. Pick the largest.\n\nKenny: 4 & 5 are the same? The name of (5) is `max_mean()` in the code....\n\nKenny: 3 is just 2 but you can tweak the 95%? It seems like that [from this post](https://lilianweng.github.io/lil-log/2018/01/23/the-multi-armed-bandit-problem-and-its-solutions.html#bayesian-ucb).\n\nKenny: 2 & 3 actually encourage exploration. Less explored arms have wider posteriors, so those will likely have the highest 95% CI upper bound!\n\n\nThe code for these are in the `other_strats.py`, where you can implement your own very easily.\n\n\n```python\nfigsize(12.5, 5)\nfrom other_strats import *\n\n#define a harder problem\nhidden_prob = np.array([0.15, 0.2, 0.1, 0.05])\nbandits = Bandits(hidden_prob)\n\n#define regret\ndef regret(probabilities, choices):\n w_opt = probabilities.max()\n return (w_opt - probabilities[choices.astype(int)]).cumsum()\n\n#create new strategies\nstrategies= [upper_credible_choice, \n bayesian_bandit_choice, \n ucb_bayes ,] \n# max_mean,\n# random_choice]\nalgos = []\nfor strat in strategies:\n algos.append(GeneralBanditStrat(bandits, strat))\n```\n\n\n```python\n# Kenny: Do 10000 pulls for each strategy\nfor strat in algos:\n strat.sample_bandits(10000)\n \n#test and plot\nfor i,strat in enumerate(algos):\n _regret = regret(hidden_prob, strat.choices)\n plt.plot(_regret, label = strategies[i].__name__, lw = 3)\n\nplt.title(\"Total Regret of Bayesian Bandits Strategy vs. Random guessing\")\nplt.xlabel(\"Number of pulls\")\nplt.ylabel(\"Regret after $n$ pulls\");\nplt.legend(loc = \"upper left\");\n```\n\nLike we wanted, Bayesian bandits and other strategies have decreasing rates of regret, representing we are achieving optimal choices. To be more scientific so as to remove any possible luck in the above simulation, we should instead look at the *expected total regret*:\n\n$$\\bar{R}_T = E[ R_T ] $$\n\nIt can be shown that any *sub-optimal* strategy's expected total regret is bounded below logarithmically. Formally,\n\n$$ E[R_T] = \\Omega \\left( \\;\\log(T)\\; \\right) $$\n\nThus, any strategy that matches logarithmic-growing regret is said to \"solve\" the Multi-Armed Bandit problem [3].\n\nUsing the Law of Large Numbers, we can approximate Bayesian Bandit's expected total regret by performing the same experiment many times (500 times, to be fair):\n\n\n```python\ntrials = 10\nexpected_total_regret = np.zeros((10000, 3))\n\nfor i_strat, strat in enumerate(strategies):\n for i in range(trials):\n general_strat = GeneralBanditStrat(bandits, strat)\n general_strat.sample_bandits(10000)\n _regret = regret(hidden_prob, general_strat.choices)\n expected_total_regret[:,i_strat] += _regret\n plt.plot(expected_total_regret[:,i_strat]/trials, lw =3, label = strat.__name__)\n \nplt.title(\"Expected Total Regret of Multi-armed Bandit strategies\")\nplt.xlabel(\"Number of pulls\")\nplt.ylabel(\"Exepected Total Regret \\n after $n$ pulls\");\nplt.legend(loc = \"upper left\");\n```\n\n\n```python\nplt.figure()\n[pl1, pl2, pl3] = plt.plot(expected_total_regret[:, [0,1,2]], lw = 3)\nplt.xscale(\"log\")\nplt.legend([pl1, pl2, pl3], \n [\"Upper Credible Bound\", \"Bayesian Bandit\", \"UCB-Bayes\"],\n loc=\"upper left\")\nplt.ylabel(\"Exepected Total Regret \\n after $\\log{n}$ pulls\");\nplt.title( \"log-scale of above\" );\nplt.ylabel(\"Exepected Total Regret \\n after $\\log{n}$ pulls\");\n```\n\n### Extending the algorithm \n\nBecause of the Bayesian Bandits algorithm's simplicity, it is easy to extend. Some possibilities:\n\n- If interested in the *minimum* probability (eg: where prizes are a bad thing), simply choose $B = \\text{argmin} \\; X_b$ and proceed.\n\n- Adding learning rates: Suppose the underlying environment may change over time. Technically the standard Bayesian Bandit algorithm would self-update itself (awesome) by noting that what it thought was the best is starting to fail more often. We can motivate the algorithm to learn changing environments quicker by simply adding a *rate* term upon updating:\n\n self.wins[choice] = rate*self.wins[choice] + result\n self.trials[choice] = rate*self.trials[choice] + 1\n\n If `rate < 1`, the algorithm will *forget* its previous wins quicker and there will be a downward pressure towards ignorance. Conversely, setting `rate > 1` implies your algorithm will act more risky, and bet on earlier winners more often and be more resistant to changing environments. \n\n- Hierarchical algorithms: We can setup a Bayesian Bandit algorithm on top of smaller bandit algorithms. Suppose we have $N$ Bayesian Bandit models, each varying in some behavior (for example different `rate` parameters, representing varying sensitivity to changing environments). On top of these $N$ models is another Bayesian Bandit learner that will select a sub-Bayesian Bandit. This chosen Bayesian Bandit will then make an internal choice as to which machine to pull. The super-Bayesian Bandit updates itself depending on whether the sub-Bayesian Bandit was correct or not. \n\n- Extending the rewards, denoted $y_a$ for bandit $a$, to random variables from a distribution $f_{y_a}(y)$ is straightforward. More generally, this problem can be rephrased as \"Find the bandit with the largest expected value\", as playing the bandit with the largest expected value is optimal. In the case above, $f_{y_a}$ was Bernoulli with probability $p_a$, hence the expected value for a bandit is equal to $p_a$, which is why it looks like we are aiming to maximize the probability of winning. If $f$ is not Bernoulli, and it is non-negative, which can be accomplished apriori by shifting the distribution (we assume we know $f$), then the algorithm behaves as before:\n\n For each round, \n \n 1. Sample a random variable $X_b$ from the prior of bandit $b$, for all $b$.\n 2. Select the bandit with largest sample, i.e. select bandit $B = \\text{argmax}\\;\\; X_b$.\n 3. Observe the result,$R \\sim f_{y_a}$, of pulling bandit $B$, and update your prior on bandit $B$.\n 4. Return to 1\n\n The issue is in the sampling of $X_b$ drawing phase. With Beta priors and Bernoulli observations, we have a Beta posterior — this is easy to sample from. But now, with arbitrary distributions $f$, we have a non-trivial posterior. Sampling from these can be difficult.\n\n- There has been some interest in extending the Bayesian Bandit algorithm to commenting systems. Recall in Chapter 4, we developed a ranking algorithm based on the Bayesian lower-bound of the proportion of upvotes to total votes. One problem with this approach is that it will bias the top rankings towards older comments, since older comments naturally have more votes (and hence the lower-bound is tighter to the true proportion). This creates a positive feedback cycle where older comments gain more votes, hence are displayed more often, hence gain more votes, etc. This pushes any new, potentially better comments, towards the bottom. J. Neufeld proposes a system to remedy this that uses a Bayesian Bandit solution.\n\nHis proposal is to consider each comment as a Bandit, with the number of pulls equal to the number of votes cast, and number of rewards as the number of upvotes, hence creating a $\\text{Beta}(1+U,1+D)$ posterior. As visitors visit the page, samples are drawn from each bandit/comment, but instead of displaying the comment with the $\\max$ sample, the comments are ranked according to the ranking of their respective samples. From J. Neufeld's blog [7]:\n\n > [The] resulting ranking algorithm is quite straightforward, each new time the comments page is loaded, the score for each comment is sampled from a $\\text{Beta}(1+U,1+D)$, comments are then ranked by this score in descending order... This randomization has a unique benefit in that even untouched comments $(U=1,D=0)$ have some chance of being seen even in threads with 5000+ comments (something that is not happening now), but, at the same time, the user is not likely to be inundated with rating these new comments. \n\nJust for fun, though the colors explode, we watch the Bayesian Bandit algorithm learn 15 different options. \n\n\n```python\n# figsize(12.0, 8)\n# beta = stats.beta\n# hidden_prob = beta.rvs(1,13, size = 35)\n# print(hidden_prob)\n# bandits = Bandits(hidden_prob)\n# bayesian_strat = BayesianStrategy(bandits)\n\n# for j,i in enumerate([100, 200, 500, 1300]):\n# plt.subplot(2, 2, j+1) \n# bayesian_strat.sample_bandits(i)\n# plot_priors(bayesian_strat, hidden_prob, lw = 2, alpha = 0.0, plt_vlines=False)\n# #plt.legend()\n# plt.xlim(0, 0.5)\n\n```\n\n## Eliciting expert prior\n\n\nSpecifying a subjective prior is how practitioners incorporate domain knowledge about the problem into our mathematical framework. Allowing domain knowledge is useful for many reasons:\n\n- Aids speeds of MCMC convergence. For example, if we know the unknown parameter is strictly positive, then we can restrict our attention there, hence saving time that would otherwise be spent exploring negative values.\n- More accurate inference. By weighing prior values near the true unknown value higher, we are narrowing our eventual inference (by making the posterior tighter around the unknown) \n- Express our uncertainty better. See the *Price is Right* problem in Chapter 5.\n\nplus many other reasons. Of course, practitioners of Bayesian methods are not experts in every field, so we must turn to domain experts to craft our priors. We must be careful with how we elicit these priors though. Some things to consider:\n\n1. From experience, I would avoid introducing Betas, Gammas, etc. to non-Bayesian practitioners. Furthermore, non-statisticians can get tripped up by how a continuous probability function can have a value exceeding one.\n\n2. Individuals often neglect the rare *tail-events* and put too much weight around the mean of distribution. \n\n3. Related to above is that almost always individuals will under-emphasize the uncertainty in their guesses.\n\nEliciting priors from non-technical experts is especially difficult. Rather than introduce the notion of probability distributions, priors, etc. that may scare an expert, there is a much simpler solution. \n\n### Trial roulette method \n\nThe *trial roulette method* [8] focuses on building a prior distribution by placing counters (think casino chips) on what the expert thinks are possible outcomes. The expert is given $N$ counters (say $N=20$) and is asked to place them on a pre-printed grid, with bins representing intervals. Each column would represent their belief of the probability of getting the corresponding bin result. Each chip would represent an $\\frac{1}{N} = 0.05$ increase in the probability of the outcome being in that interval. For example [9]:\n\n> A student is asked to predict the mark in a future exam. The figure below shows a completed grid for the elicitation of a subjective probability distribution. The horizontal axis of the grid shows the possible bins (or mark intervals) that the student was asked to consider. The numbers in top row record the number of chips per bin. The completed grid (using a total of 20 chips) shows that the student believes there is a 30% chance that the mark will be between 60 and 64.9.\n\nFrom this, we can fit a distribution that captures the expert's choice. Some reasons in favor of using this technique are:\n\n1. Many questions about the shape of the expert's subjective probability distribution can be answered without the need to pose a long series of questions to the expert - the statistician can simply read off density above or below any given point, or that between any two points.\n\n2. During the elicitation process, the experts can move around the chips if unsatisfied with the way they placed them initially - thus they can be sure of the final result to be submitted.\n\n3. It forces the expert to be coherent in the set of probabilities that are provided. If all the chips are used, the probabilities must sum to one.\n\n4. Graphical methods seem to provide more accurate results, especially for participants with modest levels of statistical sophistication.\n\n## Example: Stock Returns\n\nTake note stock brokers: you're doing it wrong. When choosing which stocks to pick, an analyst will often look at the *daily return* of the stock. Suppose $S_t$ is the price of the stock on day $t$, then the daily return on day $t$ is :\n\n$$r_t = \\frac{ S_t - S_{t-1} }{ S_{t-1} } $$\n\nThe *expected daily return* of a stock is denoted $\\mu = E[ r_t ]$. Obviously, stocks with high expected returns are desirable. Unfortunately, stock returns are so filled with noise that it is very hard to estimate this parameter. Furthermore, the parameter might change over time (consider the rises and falls of AAPL stock), hence it is unwise to use a large historical dataset. \n\nHistorically, the expected return has been estimated by using the sample mean. This is a bad idea. As mentioned, the sample mean of a small sized dataset has enormous potential to be very wrong (again, see Chapter 4 for full details). Thus Bayesian inference is the correct procedure here, since we are able to see our uncertainty along with probable values.\n\nFor this exercise, we will be examining the daily returns of the AAPL, GOOG, MSFT and AMZN. Before we pull in the data, suppose we ask our a stock fund manager (an expert in finance, but see [10] ), \n\n> What do you think the return profile looks like for each of these companies?\n\nOur stock broker, without needing to know the language of Normal distributions, or priors, or variances, etc. creates four distributions using the trial roulette method above. Suppose they look enough like Normals, so we fit Normals to them. They may look like: \n\n\n```python\nfigsize(11., 5)\ncolors = [\"#348ABD\", \"#A60628\", \"#7A68A6\", \"#467821\"]\n\nnormal = stats.norm\nx = np.linspace(-0.15, 0.15, 100)\n\nexpert_prior_params = {\"AAPL\":(0.05, 0.03),\n \"GOOG\":(-0.03, 0.04), \n \"TSLA\": (-0.02, 0.01), \n \"AMZN\": (0.03, 0.02), \n }\n\nfor i, (name, params) in enumerate(expert_prior_params.items()):\n plt.subplot(2, 2, i+1)\n y = normal.pdf(x, params[0], scale = params[1])\n #plt.plot( x, y, c = colors[i] )\n plt.fill_between(x, 0, y, color = colors[i], linewidth=2,\n edgecolor = colors[i], alpha = 0.6)\n plt.title(name + \" prior\")\n plt.vlines(0, 0, y.max(), \"k\",\"--\", linewidth = 0.5)\n plt.xlim(-0.15, 0.15)\nplt.tight_layout()\n```\n\nNote that these are subjective priors: the expert has a personal opinion on the stock returns of each of these companies, and is expressing them in a distribution. He's not wishful thinking -- he's introducing domain knowledge.\n\nIn order to better model these returns, we should investigate the *covariance matrix* of the returns. For example, it would be unwise to invest in two stocks that are highly correlated, since they are likely to tank together (hence why fund managers suggest a diversification strategy). We will use the *Wishart distribution* for this, introduced earlier.\n\nLet's get some historical data for these stocks. We will use the covariance of the returns as a starting point for our Wishart random variable. This is not empirical bayes (as we will go over later) because we are only deciding the starting point, not influencing the parameters.\n\n\n```python\n# # I wish I could have used Pandas as a prereq for this book, but oh well.\n# import datetime\n# import collections\n# import ystockquote as ysq\n# import pandas as pd\n\n# n_observations = 100 # we will truncate the the most recent 100 days.\n\n# stocks = [\"AAPL\", \"GOOG\", \"TSLA\", \"AMZN\"]\n\n# enddate = \"2015-04-27\"\n# startdate = \"2012-09-01\"\n\n# CLOSE = 6\n\n# stock_closes = pd.DataFrame()\n\n# for stock in stocks:\n# x = np.array(ysq.get_historical_prices(stock, startdate, enddate))\n# stock_series = pd.Series(x[1:,CLOSE].astype(float), name=stock)\n# stock_closes[stock] = stock_series\n\n# stock_closes = stock_closes[::-1]\n# stock_returns = stock_closes.pct_change()[1:][-n_observations:]\n \n# dates = list(map(lambda x: datetime.datetime.strptime(x, \"%Y-%m-%d\"), x[1:n_observations+1,0]))\n```\n\nAnd here let's form our basic model:\n\n\n```python\nimport pymc3 as pm\nimport theano.tensor as tt\nfrom theano.tensor.nlinalg import matrix_inverse, diag, matrix_dot\n\nprior_mu = np.array([x[0] for x in expert_prior_params.values()])\nprior_std = np.array([x[1] for x in expert_prior_params.values()])\n\ninit = stock_returns.cov()\n\nwith pm.Model() as model:\n cov_matrix = pm.WishartBartlett(\"covariance\", np.diag(prior_std**2), 10, testval = init)\n\n mu = pm.Normal(\"returns\", mu=prior_mu, sd=1, shape=4)\n```\n\n Applied log-transform to c and added transformed c_log_ to model.\n Added new variable c to model diagonal of Wishart.\n Added new variable z to model off-diagonals of Wishart.\n\n\nHere are the returns for our chosen stocks:\n\n\n```python\nfigsize(12.5, 4)\n\ncum_returns = np.cumprod(1 + stock_returns) - 1\ncum_returns.index = dates[::-1]\ncum_returns.plot()\n\nplt.legend(loc = \"upper left\")\nplt.title(\"Return space\")\nplt.ylabel(\"Return of $1 on first date, x100%\");\n```\n\n\n```python\nfigsize(11., 5 )\n\nfor i, _stock in enumerate(stocks):\n plt.subplot(2,2,i+1)\n plt.hist(stock_returns[_stock], bins=20,\n normed = True, histtype=\"stepfilled\",\n color=colors[i], alpha=0.7)\n plt.title(_stock + \" returns\")\n plt.xlim(-0.15, 0.15)\n\nplt.tight_layout()\nplt.suptitle(\"Histogram of daily returns\", size =14);\n```\n\nBelow we perform the inference on the posterior mean return and posterior covariance matrix. \n\n\n```python\nwith model:\n obs = pm.MvNormal(\"observed returns\", mu=mu, cov=cov_matrix, observed=stock_returns)\n step = pm.NUTS()\n trace = pm.sample(5000, step=step)\n```\n\n [-------100%-------] 5000 of 5000 in 40.4 sec. | SPS: 123.8 | ETA: 0.0\n\n\n```python\nfigsize(12.5,4)\n\n#examine the mean return first.\nmu_samples = trace[\"returns\"]\n\nfor i in range(4):\n plt.hist(mu_samples[:,i], alpha = 0.8 - 0.05*i, bins = 30,\n histtype=\"stepfilled\", normed=True, \n label = \"%s\" % stock_returns.columns[i])\n\nplt.vlines(mu_samples.mean(axis=0), 0, 500, linestyle=\"--\", linewidth = .5)\n\nplt.title(\"Posterior distribution of $\\mu$, daily stock returns\")\nplt.legend();\n```\n\n(Plots like these are what inspired the book's cover.)\n\nWhat can we say about the results above? Clearly TSLA has been a strong performer, and our analysis suggests that it has an almost 1% daily return! Similarly, most of the distribution of AAPL is negative, suggesting that its *true daily return* is negative.\n\n\nYou may not have immediately noticed, but these variables are a whole order of magnitude *less* than our priors on them. For example, to put these one the same scale as the above prior distributions:\n\n\n```python\nfigsize(11.0,3)\nfor i in range(4):\n plt.subplot(2,2,i+1)\n plt.hist(mu_samples[:,i], alpha = 0.8 - 0.05*i, bins = 30,\n histtype=\"stepfilled\", normed=True, color = colors[i],\n label = \"%s\" % stock_returns.columns[i])\n plt.title(\"%s\" % stock_returns.columns[i])\n plt.xlim(-0.15, 0.15)\n \nplt.suptitle(\"Posterior distribution of daily stock returns\")\nplt.tight_layout()\n```\n\nWhy did this occur? Recall how I mentioned that finance has a very very low signal to noise ratio. This implies an environment where inference is much more difficult. One should be careful about over-interpreting these results: notice (in the first figure) that each distribution is positive at 0, implying that the stock may return nothing. Furthermore, the subjective priors influenced the results. From the fund managers point of view, this is good as it reflects his updated beliefs about the stocks, whereas from a neutral viewpoint this can be too subjective of a result. \n\nBelow we show the posterior correlation matrix, and posterior standard deviations. An important caveat to know is that the Wishart distribution models the *inverse covariance matrix*, so we must invert it to get the covariance matrix. We also normalize the matrix to acquire the *correlation matrix*. Since we cannot plot hundreds of matrices effectively, we settle by summarizing the posterior distribution of correlation matrices by showing the *mean posterior correlation matrix* (defined on line 2).\n\n\n```python\ncov_samples = trace[\"covariance\"]\nmean_covariance_matrix = cov_samples.mean(axis=0)\n\ndef cov2corr(A):\n \"\"\"\n covariance matrix to correlation matrix.\n \"\"\"\n d = np.sqrt(A.diagonal())\n A = ((A.T/d).T)/d\n #A[ np.diag_indices(A.shape[0]) ] = np.ones( A.shape[0] )\n return A\n\n\nplt.subplot(1,2,1)\nplt.imshow(cov2corr(mean_covariance_matrix) , interpolation=\"none\", \n cmap = \"hot\") \nplt.xticks(np.arange(4), stock_returns.columns)\nplt.yticks(np.arange(4), stock_returns.columns)\nplt.colorbar(orientation=\"vertical\")\nplt.title(\"(mean posterior) Correlation Matrix\")\n\nplt.subplot(1,2,2)\nplt.bar(np.arange(4), np.sqrt(np.diag(mean_covariance_matrix)),\n color = \"#348ABD\", alpha = 0.7)\nplt.xticks(np.arange(4) + 0.5, stock_returns.columns);\nplt.title(\"(mean posterior) standard deviations of daily stock returns\")\n\nplt.tight_layout();\n\n```\n\nLooking at the above figures, we can say that likely TSLA has an above-average volatility (looking at the return graph this is quite clear). The correlation matrix shows that there are not strong correlations present, but perhaps GOOG and AMZN express a higher correlation (about 0.30). \n\nWith this Bayesian analysis of the stock market, we can throw it into a Mean-Variance optimizer (which I cannot stress enough, do not use with frequentist point estimates) and find the minimum. This optimizer balances the tradeoff between a high return and high variance.\n\n$$ w_{opt} = \\max_{w} \\frac{1}{N}\\left( \\sum_{i=0}^N \\mu_i^T w - \\frac{\\lambda}{2}w^T\\Sigma_i w \\right)$$\n\nwhere $\\mu_i$ and $\\Sigma_i$ are the $i$th posterior estimate of the mean returns and the covariance matrix. This is another example of loss function optimization.\n\n### Protips for the Wishart distribution\n\nIf you plan to be using the Wishart distribution, read on. Else, feel free to skip this. \n\nIn the problem above, the Wishart distribution behaves pretty nicely. Unfortunately, this is rarely the case. The problem is that estimating an $NxN$ covariance matrix involves estimating $\\frac{1}{2}N(N-1)$ unknowns. This is a large number even for modest $N$. Personally, I've tried performing a similar simulation as above with $N = 23$ stocks, and ended up giving considering that I was requesting my MCMC simulation to estimate at least $\\frac{1}{2}23*22 = 253$ additional unknowns (plus the other interesting unknowns in the problem). This is not easy for MCMC. Essentially, you are asking you MCMC to traverse 250+ dimensional space. And the problem seemed so innocent initially! Below are some tips, in order of supremacy:\n\n1. Use conjugancy if it applies. See section below.\n\n2. Use a good starting value. What might be a good starting value? Why, the data's sample covariance matrix is! Note that this is not empirical Bayes: we are not touching the prior's parameters, we are modifying the starting value of the MCMC. Due to numerical instability, it is best to truncate the floats in the sample covariance matrix down a few degrees of precision (e.g. instability can cause unsymmetrical matrices, which can cause PyMC3 to cry.). \n\n3. Provide as much domain knowledge in the form of priors, if possible. I stress *if possible*. It is likely impossible to have an estimate about each $\\frac{1}{2}N(N-1)$ unknown. In this case, see number 4.\n\n4. Use empirical Bayes, i.e. use the sample covariance matrix as the prior's parameter.\n\n5. For problems where $N$ is very large, nothing is going to help. Instead, ask, do I really care about *every* correlation? Probably not. Further ask yourself, do I really really care about correlations? Possibly not. In finance, we can set an informal hierarchy of what we might be interested in the most: first a good estimate of $\\mu$, the variances along the diagonal of the covariance matrix are secondly important, and finally the correlations are least important. So, it might be better to ignore the $\\frac{1}{2}(N-1)(N-2)$ correlations and instead focus on the more important unknowns.\n\n**Another thing** to note is that the implementation of the Wishart distribution has changed in from PyMC to PyMC3. Wishart distribution matrices are required to have certain mathematical characteristics that are very restrictive. This makes it so that it is impossible for MCMC methods to propose matrices that will be accepted in our sampling procedure. With our model here we sample the Bartlett decomposition of a Wishart distribution matrix and use that to calculate our samples for the covariance matrix (http://en.wikipedia.org/wiki/Wishart_distribution#Bartlett_decomposition).\n\n## Conjugate Priors\n\nRecall that a $\\text{Beta}$ prior with $\\text{Binomial}$ data implies a $\\text{Beta}$ posterior. Graphically:\n\n$$ \\underbrace{\\text{Beta}}_{\\text{prior}} \\cdot \\overbrace{\\text{Binomial}}^{\\text{data}} = \\overbrace{\\text{Beta}}^{\\text{posterior} } $$ \n\nNotice the $\\text{Beta}$ on both sides of this equation (no, you cannot cancel them, this is not a *real* equation). This is a really useful property. It allows us to avoid using MCMC, since the posterior is known in closed form. Hence inference and analytics are easy to derive. This shortcut was the heart of the Bayesian Bandit algorithm above. Fortunately, there is an entire family of distributions that have similar behaviour. \n\nSuppose $X$ comes from, or is believed to come from, a well-known distribution, call it $f_{\\alpha}$, where $\\alpha$ are possibly unknown parameters of $f$. $f$ could be a Normal distribution, or Binomial distribution, etc. For particular distributions $f_{\\alpha}$, there may exist a prior distribution $p_{\\beta}$, such that:\n\n$$ \\overbrace{p_{\\beta}}^{\\text{prior}} \\cdot \\overbrace{f_{\\alpha}(X)}^{\\text{data}} = \\overbrace{p_{\\beta'}}^{\\text{posterior} } $$ \n\nwhere $\\beta'$ is a different set of parameters *but $p$ is the same distribution as the prior*. A prior $p$ that satisfies this relationship is called a *conjugate prior*. As I mentioned, they are useful computationally, as we can avoided approximate inference using MCMC and go directly to the posterior. This sounds great, right?\n\nUnfortunately, not quite. There are a few issues with conjugate priors.\n\n1. The conjugate prior is not objective. Hence only useful when a subjective prior is required. It is not guaranteed that the conjugate prior can accommodate the practitioner's subjective opinion.\n\n2. There typically exist conjugate priors for simple, one dimensional problems. For larger problems, involving more complicated structures, hope is lost to find a conjugate prior. For smaller models, Wikipedia has a nice [table of conjugate priors](http://en.wikipedia.org/wiki/Conjugate_prior#Table_of_conjugate_distributions).\n\nReally, conjugate priors are only useful for their mathematical convenience: it is simple to go from prior to posterior. I personally see conjugate priors as only a neat mathematical trick, and offer little insight into the problem at hand. \n\n## Jefferys Priors\n\nEarlier, we talked about objective priors rarely being *objective*. Partly what we mean by this is that we want a prior that doesn't bias our posterior estimates. The flat prior seems like a reasonable choice as it assigns equal probability to all values. \n\nBut the flat prior is not transformation invariant. What does this mean? Suppose we have a random variable $\\textbf X$ from Bernoulli($\\theta$). We define the prior on $p(\\theta) = 1$. \n\n\n```python\nfigsize(12.5, 5)\n\nx = np.linspace(0.000 ,1, 150)\ny = np.linspace(1.0, 1.0, 150)\nlines = plt.plot(x, y, color=\"#A60628\", lw = 3)\nplt.fill_between(x, 0, y, alpha = 0.2, color = lines[0].get_color())\nplt.autoscale(tight=True)\nplt.ylim(0, 2);\n```\n\nNow, let's transform $\\theta$ with the function $\\psi = log \\frac{\\theta}{1-\\theta}$. This is just a function to stretch $\\theta$ across the real line. Now how likely are different values of $\\psi$ under our transformation.\n\n\n```python\nfigsize(12.5, 5)\n\npsi = np.linspace(-10 ,10, 150)\ny = np.exp(psi) / (1 + np.exp(psi))**2\nlines = plt.plot(psi, y, color=\"#A60628\", lw = 3)\nplt.fill_between(psi, 0, y, alpha = 0.2, color = lines[0].get_color())\nplt.autoscale(tight=True)\nplt.ylim(0, 1);\n```\n\nOh no! Our function is no longer flat. It turns out flat priors do carry information in them after all. The point of Jeffreys Priors is to create priors that don't accidentally become informative when you transform the variables you placed them originally on.\n\nJeffreys Priors are defined as:\n\n$$p_J(\\theta) \\propto \\mathbf{I}(\\theta)^\\frac{1}{2}$$\n$$\\mathbf{I}(\\theta) = - \\mathbb{E}\\bigg[\\frac{d^2 \\text{ log } p(X|\\theta)}{d\\theta^2}\\bigg]$$\n\n$\\mathbf{I}$ being the *Fisher information*\n\n## Effect of the prior as $N$ increases\n\nIn the first chapter, I proposed that as the amount of our observations or data increases, the influence of the prior decreases. This is intuitive. After all, our prior is based on previous information, and eventually enough new information will shadow our previous information's value. The smothering of the prior by enough data is also helpful: if our prior is significantly wrong, then the self-correcting nature of the data will present to us a *less wrong*, and eventually *correct*, posterior. \n\nWe can see this mathematically. First, recall Bayes Theorem from Chapter 1 that relates the prior to the posterior. The following is a sample from [What is the relationship between sample size and the influence of prior on posterior?](http://stats.stackexchange.com/questions/30387/what-is-the-relationship-between-sample-size-and-the-influence-of-prior-on-poste)[1] on CrossValidated.\n\n>The posterior distribution for a parameter $\\theta$, given a data set ${\\textbf X}$ can be written as \n\n$$p(\\theta | {\\textbf X}) \\propto \\underbrace{p({\\textbf X} | \\theta)}_{{\\textrm likelihood}} \\cdot \\overbrace{ p(\\theta) }^{ {\\textrm prior} } $$\n\n\n\n>or, as is more commonly displayed on the log scale, \n\n$$ \\log( p(\\theta | {\\textbf X}) ) = c + L(\\theta;{\\textbf X}) + \\log(p(\\theta)) $$\n\n>The log-likelihood, $L(\\theta;{\\textbf X}) = \\log \\left( p({\\textbf X}|\\theta) \\right)$, **scales with the sample size**, since it is a function of the data, while the prior density does not. Therefore, as the sample size increases, the absolute value of $L(\\theta;{\\textbf X})$ is getting larger while $\\log(p(\\theta))$ stays fixed (for a fixed value of $\\theta$), thus the sum $L(\\theta;{\\textbf X}) + \\log(p(\\theta))$ becomes more heavily influenced by $L(\\theta;{\\textbf X})$ as the sample size increases. \n\nKenny: It's not clear to me why $L(\\theta;{\\textbf X})$ increases with the sample size. Why?\n\nThere is an interesting consequence not immediately apparent. As the sample size increases, the chosen prior has less influence. Hence inference converges regardless of chosen prior, so long as the areas of non-zero probabilities are the same. \n\nBelow we visualize this. We examine the convergence of two posteriors of a Binomial's parameter $\\theta$, one with a flat prior and the other with a biased prior towards 0. As the sample size increases, the posteriors, and hence the inference, converge.\n\n\n```python\nfigsize(12.5, 15)\n\np = 0.6\nbeta1_params = np.array([1.,1.])\nbeta2_params = np.array([2,10])\nbeta = stats.beta\n\nx = np.linspace(0.00, 1, 125)\ndata = stats.bernoulli.rvs(p, size=500)\n\nplt.figure()\nfor i,N in enumerate([0,4,8, 32,64, 128, 500]):\n s = data[:N].sum() \n plt.subplot(8,1,i+1)\n params1 = beta1_params + np.array([s, N-s])\n params2 = beta2_params + np.array([s, N-s])\n y1,y2 = beta.pdf(x, *params1), beta.pdf( x, *params2)\n plt.plot(x,y1, label = r\"flat prior\", lw =3)\n plt.plot(x, y2, label = \"biased prior\", lw= 3)\n plt.fill_between(x, 0, y1, color =\"#348ABD\", alpha = 0.15) \n plt.fill_between(x, 0, y2, color =\"#A60628\", alpha = 0.15) \n plt.legend(title = \"N=%d\" % N)\n plt.vlines(p, 0.0, 7.5, linestyles = \"--\", linewidth=1)\n #plt.ylim( 0, 10)#\n\n```\n\nKeep in mind, not all posteriors will \"forget\" the prior this quickly. This example was just to show that *eventually* the prior is forgotten. The \"forgetfulness\" of the prior as we become awash in more and more data is the reason why Bayesian and Frequentist inference eventually converge as well.\n\n## Bayesian perspective of Penalized Linear Regressions\n\nThere is a very interesting relationship between a penalized least-squares regression and Bayesian priors. A penalized linear regression is a optimization problem of the form:\n\n$$ \\text{argmin}_{\\beta} \\;\\; (Y - X\\beta)^T(Y - X\\beta) + f(\\beta)$$\n\nfor some function $f$ (typically a norm like $|| \\cdot ||_p^p$). \n\nWe will first describe the probabilistic interpretation of least-squares linear regression. Denote our response variable $Y$, and features are contained in the data matrix $X$. The standard linear model is:\n\n\\begin{equation}\nY = X\\beta + \\epsilon\n\\end{equation}\n\nwhere $\\epsilon \\sim \\text{Normal}( {\\textbf 0}, \\sigma{\\textbf I })$. Simply, the observed $Y$ is a linear function of $X$ (with coefficients $\\beta$) plus some noise term. Our unknown to be determined is $\\beta$. We use the following property of Normal random variables:\n\n$$ \\mu' + \\text{Normal}( \\mu, \\sigma ) \\sim \\text{Normal}( \\mu' + \\mu , \\sigma ) $$\n\nto rewrite the above linear model as:\n\n\\begin{align}\n& Y = X\\beta + \\text{Normal}( {\\textbf 0}, \\sigma{\\textbf I }) \\\\\\\\\n& Y = \\text{Normal}( X\\beta , \\sigma{\\textbf I }) \\\\\\\\\n\\end{align}\n\nIn probabilistic notation, denote $f_Y(y \\; | \\; \\beta )$ the probability distribution of $Y$, and recalling the density function for a Normal random variable (see [here](http://en.wikipedia.org/wiki/Normal_distribution) ):\n\n$$ f_Y( Y \\; |\\; \\beta, X) = L(\\beta|\\; X,Y)= \\frac{1}{\\sqrt{ 2\\pi\\sigma} } \\exp \\left( \\frac{1}{2\\sigma^2} (Y - X\\beta)^T(Y - X\\beta) \\right) $$\n\nThis is the likelihood function for $\\beta$. Taking the $\\log$:\n\n$$ \\ell(\\beta) = K - c(Y - X\\beta)^T(Y - X\\beta) $$\n\nwhere $K$ and $c>0$ are constants. Maximum likelihood techniques wish to maximize this for $\\beta$, \n\n$$\\hat{ \\beta } = \\text{argmax}_{\\beta} \\;\\; - (Y - X\\beta)^T(Y - X\\beta) $$\n\nEquivalently we can *minimize the negative* of the above:\n\n$$\\hat{ \\beta } = \\text{argmin}_{\\beta} \\;\\; (Y - X\\beta)^T(Y - X\\beta) $$\n\nThis is the familiar least-squares linear regression equation. Therefore we showed that the solution to a linear least-squares is the same as the maximum likelihood assuming Normal noise. Next we extend this to show how we can arrive at penalized linear regression by a suitable choice of prior on $\\beta$. \n\n#### Penalized least-squares\n\nIn the above, once we have the likelihood, we can include a prior distribution on $\\beta$ to derive to the equation for the posterior distribution:\n\n$$P( \\beta | Y, X ) = L(\\beta|\\;X,Y)p( \\beta )$$\n\nwhere $p(\\beta)$ is a prior on the elements of $\\beta$. What are some interesting priors? \n\n1\\. If we include *no explicit* prior term, we are actually including an uninformative prior, $P( \\beta ) \\propto 1$, think of it as uniform over all numbers. \n\n2\\. If we have reason to believe the elements of $\\beta$ are not too large, we can suppose that *a priori*:\n\n$$ \\beta \\sim \\text{Normal}({\\textbf 0 }, \\lambda {\\textbf I } ) $$\n\nThe resulting posterior density function for $\\beta$ is *proportional to*:\n\n$$ \\exp \\left( \\frac{1}{2\\sigma^2} (Y - X\\beta)^T(Y - X\\beta) \\right) \\exp \\left( \\frac{1}{2\\lambda^2} \\beta^T\\beta \\right) $$\n\nand taking the $\\log$ of this, and combining and redefining constants, we arrive at:\n\n$$ \\ell(\\beta) \\propto K - (Y - X\\beta)^T(Y - X\\beta) - \\alpha \\beta^T\\beta $$\n\nwe arrive at the function we wish to maximize (recall the point that maximizes the posterior distribution is the MAP, or *maximum a posterior*):\n\n$$\\hat{ \\beta } = \\text{argmax}_{\\beta} \\;\\; -(Y - X\\beta)^T(Y - X\\beta) - \\alpha \\;\\beta^T\\beta $$\n\nEquivalently, we can minimize the negative of the above, and rewriting $\\beta^T \\beta = ||\\beta||_2^2$:\n\n$$\\hat{ \\beta } = \\text{argmin}_{\\beta} \\;\\; (Y - X\\beta)^T(Y - X\\beta) + \\alpha \\;||\\beta||_2^2$$\n\nThis above term is exactly Ridge Regression. Thus we can see that ridge regression corresponds to the MAP of a linear model with Normal errors and a Normal prior on $\\beta$.\n\n3\\. Similarly, if we assume a *Laplace* prior on $\\beta$, ie. \n\n$$ f_\\beta( \\beta) \\propto \\exp \\left(- \\lambda ||\\beta||_1 \\right)$$\n\nand following the same steps as above, we recover:\n\n$$\\hat{ \\beta } = \\text{argmin}_{\\beta} \\;\\; (Y - X\\beta)^T(Y - X\\beta) + \\alpha \\;||\\beta||_1$$\n\nwhich is LASSO regression. Some important notes about this equivalence. The sparsity that is a result of using a LASSO regularization is not a result of the prior assigning high probability to sparsity. Quite the opposite actually. It is the combination of the $|| \\cdot ||_1$ function and using the MAP that creates sparsity on $\\beta$: [purely a geometric argument](http://camdp.com/blogs/least-squares-regression-l1-penalty). The prior does contribute to an overall shrinking of the coefficients towards 0 though. An interesting discussion of this can be found in [2].\n\nFor an example of Bayesian linear regression, see Chapter 4's example on financial losses.\n\n## References\n\n\n1. Macro, . \"What is the relationship between sample size and the influence of prior on posterior?.\" 13 Jun 2013. StackOverflow, Online Posting to Cross-Validated. Web. 25 Apr. 2013.\n\n2. Starck, J.-L., , et al. \"Sparsity and the Bayesian Perspective.\" Astronomy & Astrophysics. (2013): n. page. Print.\n\n3. Kuleshov, Volodymyr, and Doina Precup. \"Algorithms for the multi-armed bandit problem.\" Journal of Machine Learning Research. (2000): 1-49. Print.\n\n4. Gelman, Andrew. \"Prior distributions for variance parameters in hierarchical models.\" Bayesian Analysis. 1.3 (2006): 515-533. Print.\n\n5. Gelman, Andrew, and Cosma R. Shalizi. \"Philosophy and the practice of Bayesian statistics.\" British Journal of Mathematical and Statistical Psychology. (2012): n. page. Web. 17 Apr. 2013.\n\n6. http://jmlr.csail.mit.edu/proceedings/papers/v22/kaufmann12/kaufmann12.pdf\n\n7. James, Neufeld. \"Reddit's \"best\" comment scoring algorithm as a multi-armed bandit task.\" Simple ML Hacks. Blogger, 09 Apr 2013. Web. 25 Apr. 2013.\n\n8. Oakley, J. E., Daneshkhah, A. and O\u2019Hagan, A. Nonparametric elicitation using the roulette method. Submitted to Bayesian Analysis.\n\n9. \"Eliciting priors from experts.\" 19 Jul 2010. StackOverflow, Online Posting to Cross-Validated. Web. 1 May. 2013. .\n\n10. Taleb, Nassim Nicholas (2007), The Black Swan: The Impact of the Highly Improbable, Random House, ISBN 978-1400063512\n", "meta": {"hexsha": "333acab0e75c5b10be05d81fb6fe2b05596ad7e4", "size": 1002315, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Chapter6_Priorities/Ch6_Priors_PyMC3.ipynb", "max_stars_repo_name": "kennysong/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers", "max_stars_repo_head_hexsha": "343a0f9ccacc6051689a18ee28bc82dfbdd7ccb4", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Chapter6_Priorities/Ch6_Priors_PyMC3.ipynb", "max_issues_repo_name": "kennysong/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers", "max_issues_repo_head_hexsha": "343a0f9ccacc6051689a18ee28bc82dfbdd7ccb4", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Chapter6_Priorities/Ch6_Priors_PyMC3.ipynb", "max_forks_repo_name": "kennysong/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers", "max_forks_repo_head_hexsha": "343a0f9ccacc6051689a18ee28bc82dfbdd7ccb4", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 490.1295843521, "max_line_length": 173568, "alphanum_fraction": 0.9278949233, "converted": true, "num_tokens": 17173, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5156199157230157, "lm_q2_score": 0.8418256512199033, "lm_q1q2_score": 0.4340620713354793}} {"text": "# Taxi demand prediction in New York City\n\n\n\n\n\n```python\n#Importing Libraries\n!pip3 install graphviz\n!pip3 install dask\n!pip install \"dask[complete]\" \n!pip3 install toolz\n!pip3 install cloudpickle\n# https://www.youtube.com/watch?v=ieW3G7ZzRZ0\n# https://github.com/dask/dask-tutorial\n# please do go through this python notebook: https://github.com/dask/dask-tutorial/blob/master/07_dataframe.ipynb\nimport dask.dataframe as dd#similar to pandas\n\nimport pandas as pd#pandas to create small dataframes \n\n!pip3 install folium\n# if this doesnt work refere install_folium.JPG in drive\nimport folium #open street map\n\n# unix time: https://www.unixtimestamp.com/\nimport datetime #Convert to unix time\n\nimport time #Convert to unix time\n\n# if numpy is not installed already : pip3 install numpy\nimport numpy as np#Do aritmetic operations on arrays\n\n# matplotlib: used to plot graphs\nimport matplotlib\n# matplotlib.use('nbagg') : matplotlib uses this protocall which makes plots more user intractive like zoom in and zoom out\nmatplotlib.use('nbagg')\nimport matplotlib.pylab as plt\nimport seaborn as sns#Plots\nfrom matplotlib import rcParams#Size of plots \n\n# this lib is used while we calculate the stight line distance between two (lat,lon) pairs in miles\n!pip install gpxpy\nimport gpxpy.geo #Get the haversine distance\n\nfrom sklearn.cluster import MiniBatchKMeans, KMeans#Clustering\nimport math\nimport pickle\nimport os\n\n# download migwin: https://mingw-w64.org/doku.php/download/mingw-builds\n# install it in your system and keep the path, migw_path ='installed path'\nmingw_path = 'C:\\\\Program Files\\\\mingw-w64\\\\x86_64-5.3.0-posix-seh-rt_v4-rev0\\\\mingw64\\\\bin'\nos.environ['PATH'] = mingw_path + ';' + os.environ['PATH']\n\n# to install xgboost: pip3 install xgboost\n# if it didnt happen check install_xgboost.JPG\nimport xgboost as xgb\n\n# to install sklearn: pip install -U scikit-learn\nfrom sklearn.ensemble import RandomForestRegressor\nfrom sklearn.metrics import mean_squared_error\nfrom sklearn.metrics import mean_absolute_error\nimport warnings\nwarnings.filterwarnings(\"ignore\")\n\n\n```\n\n Requirement already satisfied: graphviz in d:\\ml_projects\\myenv\\lib\\site-packages (0.19.1)\n Requirement already satisfied: dask in d:\\ml_projects\\myenv\\lib\\site-packages (2022.2.0)\n Requirement already satisfied: toolz>=0.8.2 in d:\\ml_projects\\myenv\\lib\\site-packages (from dask) (0.11.2)\n Requirement already satisfied: partd>=0.3.10 in d:\\ml_projects\\myenv\\lib\\site-packages (from dask) (1.2.0)\n Requirement already satisfied: packaging>=20.0 in d:\\ml_projects\\myenv\\lib\\site-packages (from dask) (21.3)\n Requirement already satisfied: fsspec>=0.6.0 in d:\\ml_projects\\myenv\\lib\\site-packages (from dask) (2022.1.0)\n Requirement already satisfied: pyyaml>=5.3.1 in 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Requirement already satisfied: gpxpy in d:\\ml_projects\\myenv\\lib\\site-packages (1.5.0)\n\n\n# Data Information\n\n

    \nGe the data from : http://www.nyc.gov/html/tlc/html/about/trip_record_data.shtml (2016 data)\nThe data used in the attached datasets were collected and provided to the NYC Taxi and Limousine Commission (TLC) \n

    \n\n## Information on taxis:\n\n
    Yellow Taxi: Yellow Medallion Taxicabs
    \n

    These are the famous NYC yellow taxis that provide transportation exclusively through street-hails. The number of taxicabs is limited by a finite number of medallions issued by the TLC. You access this mode of transportation by standing in the street and hailing an available taxi with your hand. The pickups are not pre-arranged.

    \n\n
    For Hire Vehicles (FHVs)
    \n

    FHV transportation is accessed by a pre-arrangement with a dispatcher or limo company. These FHVs are not permitted to pick up passengers via street hails, as those rides are not considered pre-arranged.

    \n\n
    Green Taxi: Street Hail Livery (SHL)
    \n

    The SHL program will allow livery vehicle owners to license and outfit their vehicles with green borough taxi branding, meters, credit card machines, and ultimately the right to accept street hails in addition to pre-arranged rides.

    \n

    Credits: Quora

    \n\n
    Footnote:
    \nIn the given notebook we are considering only the yellow taxis for the time period between Jan - Mar 2015 & Jan - Mar 2016\n\n# Data Collection\nWe Have collected all yellow taxi trips data from jan-2015 to dec-2016(Will be using only 2015 data)\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
    file name file name size number of records number of features
    yellow_tripdata_2016-01 1. 59G 10906858 19
    yellow_tripdata_2016-02 1. 66G 11382049 19
    yellow_tripdata_2016-03 1. 78G 12210952 19
    yellow_tripdata_2016-04 1. 74G 11934338 19
    yellow_tripdata_2016-05 1. 73G 11836853 19
    yellow_tripdata_2016-06 1. 62G 11135470 19
    yellow_tripdata_2016-07 884Mb 10294080 17
    yellow_tripdata_2016-08 854Mb 9942263 17
    yellow_tripdata_2016-09 870Mb 10116018 17
    yellow_tripdata_2016-10 933Mb 10854626 17
    yellow_tripdata_2016-11 868Mb 10102128 17
    yellow_tripdata_2016-12 897Mb 10449408 17
    yellow_tripdata_2015-01 1.84Gb 12748986 19
    yellow_tripdata_2015-02 1.81Gb 12450521 19
    yellow_tripdata_2015-03 1.94Gb 13351609 19
    yellow_tripdata_2015-04 1.90Gb 13071789 19
    yellow_tripdata_2015-05 1.91Gb 13158262 19
    yellow_tripdata_2015-06 1.79Gb 12324935 19
    yellow_tripdata_2015-07 1.68Gb 11562783 19
    yellow_tripdata_2015-08 1.62Gb 11130304 19
    yellow_tripdata_2015-09 1.63Gb 11225063 19
    yellow_tripdata_2015-10 1.79Gb 12315488 19
    yellow_tripdata_2015-11 1.65Gb 11312676 19
    yellow_tripdata_2015-12 1.67Gb 11460573 19
    \n\n\n```python\n#Looking at the features\n# dask dataframe : # https://github.com/dask/dask-tutorial/blob/master/07_dataframe.ipynb\n\n```\n\n\n```python\n# !gdown --id 1kcIZlf-LQiQhqfSCZb719Nh6Rqkp2zKK\n\n```\n\n\n```python\n# However unlike Pandas, operations on dask.dataframes don't trigger immediate computation, \n# instead they add key-value pairs to an underlying Dask graph. Recall that in the diagram below, \n# circles are operations and rectangles are results.\n\n# to see the visulaization you need to install graphviz\n# pip3 install graphviz if this doesnt work please check the install_graphviz.jpg in the drive\n# month.visualize()\n```\n\n\n```python\n# month.fare_amount.sum().visualize()\n```\n\n## Features in the dataset:\n\n\n\n\t\n\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\n\t\n\t\n\t\t\n \n \n\t\n\n\t\n\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\n\t\n \n\t\n\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\n\t\n
    Field NameDescription
    VendorID\n\t\tA code indicating the TPEP provider that provided the record.\n\t\t
      \n\t\t\t
    1. Creative Mobile Technologies
      2. \n\t\t\t
    2. VeriFone Inc.
      3. \n\t\t
    \n\t\t
    tpep_pickup_datetimeThe date and time when the meter was engaged.
    tpep_dropoff_datetimeThe date and time when the meter was disengaged.
    Passenger_countThe number of passengers in the vehicle. This is a driver-entered value.
    Trip_distanceThe elapsed trip distance in miles reported by the taximeter.
    Pickup_longitudeLongitude where the meter was engaged.
    Pickup_latitudeLatitude where the meter was engaged.
    RateCodeIDThe final rate code in effect at the end of the trip.\n\t\t
      \n\t\t\t
    1. Standard rate
      2. \n\t\t\t
    2. JFK
      3. \n\t\t\t
    3. Newark
      4. \n\t\t\t
    4. Nassau or Westchester
      5. \n\t\t\t
    5. Negotiated fare
      6. \n\t\t\t
    6. Group ride
      7. \n\t\t
    \n\t\t
    Store_and_fwd_flagThis flag indicates whether the trip record was held in vehicle memory before sending to the vendor,
    aka \u201cstore and forward,\u201d because the vehicle did not have a connection to the server.\n\n\n\t\t
    Dropoff_longitudeLongitude where the meter was disengaged.
    Dropoff_ latitudeLatitude where the meter was disengaged.
    Payment_typeA numeric code signifying how the passenger paid for the trip.\n\t\t
      \n\t\t\t
    1. Credit card
      2. \n\t\t\t
    2. Cash
      3. \n\t\t\t
    3. No charge
      4. \n\t\t\t
    4. Dispute
      5. \n\t\t\t
    5. Unknown
      6. \n\t\t\t
    6. Voided trip
      7. \n\t\t
    \n\t\t
    Fare_amountThe time-and-distance fare calculated by the meter.
    ExtraMiscellaneous extras and surcharges. Currently, this only includes. the $0.50 and $1 rush hour and overnight charges.
    MTA_tax0.50 MTA tax that is automatically triggered based on the metered rate in use.
    Improvement_surcharge0.30 improvement surcharge assessed trips at the flag drop. the improvement surcharge began being levied in 2015.
    Tip_amountTip amount \u2013 This field is automatically populated for credit card tips.Cash tips are not included.
    Tolls_amountTotal amount of all tolls paid in trip.
    Total_amountThe total amount charged to passengers. Does not include cash tips.
    \n\n# ML Problem Formulation\n

    Time-series forecasting and Regression

    \n
    \n- To find number of pickups, given location cordinates(latitude and longitude) and time, in the query reigion and surrounding regions.\n

    \nTo solve the above we would be using data collected in Jan - Mar 2015 to predict the pickups in Jan - Mar 2016.\n

    \n\n# Performance metrics\n1. Mean Absolute percentage error.\n2. Mean Squared error.\n\n## Data Cleaning\n\nIn this section we will be doing univariate analysis and removing outlier/illegitimate values which may be caused due to some error\n\n\n```python\n#table below shows few datapoints along with all our features\nmonth = dd.read_csv('yellow_tripdata_2015-01.csv')\nprint(month.columns)\nmonth.head(5)\n```\n\n Index(['VendorID', 'tpep_pickup_datetime', 'tpep_dropoff_datetime',\n 'passenger_count', 'trip_distance', 'pickup_longitude',\n 'pickup_latitude', 'RateCodeID', 'store_and_fwd_flag',\n 'dropoff_longitude', 'dropoff_latitude', 'payment_type', 'fare_amount',\n 'extra', 'mta_tax', 'tip_amount', 'tolls_amount',\n 'improvement_surcharge', 'total_amount'],\n dtype='object')\n\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    VendorIDtpep_pickup_datetimetpep_dropoff_datetimepassenger_counttrip_distancepickup_longitudepickup_latitudeRateCodeIDstore_and_fwd_flagdropoff_longitudedropoff_latitudepayment_typefare_amountextramta_taxtip_amounttolls_amountimprovement_surchargetotal_amount
    022015-01-15 19:05:392015-01-15 19:23:4211.59-73.99389640.7501111N-73.97478540.750618112.01.00.53.250.00.317.05
    112015-01-10 20:33:382015-01-10 20:53:2813.30-74.00164840.7242431N-73.99441540.759109114.50.50.52.000.00.317.80
    212015-01-10 20:33:382015-01-10 20:43:4111.80-73.96334140.8027881N-73.95182040.82441329.50.50.50.000.00.310.80
    312015-01-10 20:33:392015-01-10 20:35:3110.50-74.00908740.7138181N-74.00432640.71998623.50.50.50.000.00.34.80
    412015-01-10 20:33:392015-01-10 20:52:5813.00-73.97117640.7624281N-74.00418140.742653215.00.50.50.000.00.316.30
    \n
    \n\n\n\n### 1. Pickup Latitude and Pickup Longitude\n\nIt is inferred from the source https://www.flickr.com/places/info/2459115 that New York is bounded by the location cordinates(lat,long) - (40.5774, -74.15) & (40.9176,-73.7004) so hence any cordinates not within these cordinates are not considered by us as we are only concerned with pickups which originate within New York.\n\n\n```python\n# Plotting pickup cordinates which are outside the bounding box of New-York \n# we will collect all the points outside the bounding box of newyork city to outlier_locations\noutlier_locations = month[((month.pickup_longitude <= -74.15) | (month.pickup_latitude <= 40.5774)| \\\n (month.pickup_longitude >= -73.7004) | (month.pickup_latitude >= 40.9176))]\n\n# creating a map with the a base location\n# read more about the folium here: http://folium.readthedocs.io/en/latest/quickstart.html\n\n# note: you dont need to remember any of these, you dont need indeepth knowledge on these maps and plots\n\nmap_osm = folium.Map(location=[40.734695, -73.990372], tiles='Stamen Toner')\n\n# we will spot only first 100 outliers on the map, plotting all the outliers will take more time\nsample_locations = outlier_locations.head(10000)\nfor i,j in sample_locations.iterrows():\n if int(j['pickup_latitude']) != 0:\n folium.Marker(list((j['pickup_latitude'],j['pickup_longitude']))).add_to(map_osm)\nmap_osm\n```\n\n\n\n\n
    Make this Notebook Trusted to load map: File -> Trust Notebook
    \n\n\n\nObservation:- As you can see above that there are some points just outside the boundary but there are a few that are in either South america, Mexico or Canada\n\n### 2. Dropoff Latitude & Dropoff Longitude\n\nIt is inferred from the source https://www.flickr.com/places/info/2459115 that New York is bounded by the location cordinates(lat,long) - (40.5774, -74.15) & (40.9176,-73.7004) so hence any cordinates not within these cordinates are not considered by us as we are only concerned with dropoffs which are within New York.\n\n\n```python\n# Plotting dropoff cordinates which are outside the bounding box of New-York \n# we will collect all the points outside the bounding box of newyork city to outlier_locations\noutlier_locations = month[((month.dropoff_longitude <= -74.15) | (month.dropoff_latitude <= 40.5774)| \\\n (month.dropoff_longitude >= -73.7004) | (month.dropoff_latitude >= 40.9176))]\n\n# creating a map with the a base location\n# read more about the folium here: http://folium.readthedocs.io/en/latest/quickstart.html\n\n# note: you dont need to remember any of these, you dont need indeepth knowledge on these maps and plots\n\nmap_osm = folium.Map(location=[40.734695, -73.990372], tiles='Stamen Toner')\n\n# we will spot only first 100 outliers on the map, plotting all the outliers will take more time\nsample_locations = outlier_locations.head(10000)\nfor i,j in sample_locations.iterrows():\n if int(j['pickup_latitude']) != 0:\n folium.Marker(list((j['dropoff_latitude'],j['dropoff_longitude']))).add_to(map_osm)\nmap_osm\n```\n\n\n\n\n
    Make this Notebook Trusted to load map: File -> Trust Notebook
    \n\n\n\nObservation:- The observations here are similar to those obtained while analysing pickup latitude and longitude\n\n### 3. Trip Durations:\n\n

    According to NYC Taxi & Limousine Commision Regulations the maximum allowed trip duration in a 24 hour interval is 12 hours.

    \n\n\n```python\n#The timestamps are converted to unix so as to get duration(trip-time) & speed also pickup-times in unix are used while binning \n\n# in out data we have time in the formate \"YYYY-MM-DD HH:MM:SS\" we convert thiss sting to python time formate and then into unix time stamp\n# https://stackoverflow.com/a/27914405\ndef convert_to_unix(s):\n return time.mktime(datetime.datetime.strptime(s, \"%Y-%m-%d %H:%M:%S\").timetuple())\n\n\n\n# we return a data frame which contains the columns\n# 1.'passenger_count' : self explanatory\n# 2.'trip_distance' : self explanatory\n# 3.'pickup_longitude' : self explanatory\n# 4.'pickup_latitude' : self explanatory\n# 5.'dropoff_longitude' : self explanatory\n# 6.'dropoff_latitude' : self explanatory\n# 7.'total_amount' : total fair that was paid\n# 8.'trip_times' : duration of each trip\n# 9.'pickup_times : pickup time converted into unix time \n# 10.'Speed' : velocity of each trip\ndef return_with_trip_times(month):\n duration = month[['tpep_pickup_datetime','tpep_dropoff_datetime']].compute()\n #pickups and dropoffs to unix time\n duration_pickup = [convert_to_unix(x) for x in duration['tpep_pickup_datetime'].values]\n duration_drop = [convert_to_unix(x) for x in duration['tpep_dropoff_datetime'].values]\n #calculate duration of trips\n durations = (np.array(duration_drop) - np.array(duration_pickup))/float(60)\n\n #append durations of trips and speed in miles/hr to a new dataframe\n new_frame = month[['passenger_count','trip_distance','pickup_longitude','pickup_latitude','dropoff_longitude','dropoff_latitude','total_amount']].compute()\n \n new_frame['trip_times'] = durations\n new_frame['pickup_times'] = duration_pickup\n new_frame['Speed'] = 60*(new_frame['trip_distance']/new_frame['trip_times'])\n \n return new_frame\n\n# print(frame_with_durations.head())\n# passenger_count\ttrip_distance\tpickup_longitude\tpickup_latitude\tdropoff_longitude\tdropoff_latitude\ttotal_amount\ttrip_times\tpickup_times\tSpeed\n# 1 1.59\t -73.993896 \t40.750111 \t-73.974785 \t40.750618 \t17.05 \t 18.050000\t1.421329e+09\t5.285319\n# 1 \t3.30 \t-74.001648 \t40.724243 \t-73.994415 \t40.759109 \t17.80 \t19.833333\t1.420902e+09\t9.983193\n# 1 \t1.80 \t-73.963341 \t40.802788 \t-73.951820 \t40.824413 \t10.80 \t10.050000\t1.420902e+09\t10.746269\n# 1 \t0.50 \t-74.009087 \t40.713818 \t-74.004326 \t40.719986 \t4.80 \t1.866667\t1.420902e+09\t16.071429\n# 1 \t3.00 \t-73.971176 \t40.762428 \t-74.004181 \t40.742653 \t16.30 \t19.316667\t1.420902e+09\t9.318378\nframe_with_durations = return_with_trip_times(month)\n```\n\n\n```python\n# the skewed box plot shows us the presence of outliers \nsns.boxplot(y=\"trip_times\", data =frame_with_durations)\nplt.show()\n```\n\n\n \n\n\n\n\n\n\n\n```python\n#calculating 0-100th percentile to find a the correct percentile value for removal of outliers\nfor i in range(0,100,10):\n var =frame_with_durations[\"trip_times\"].values\n var = np.sort(var,axis = None)\n print(\"{} percentile value is {}\".format(i,var[int(len(var)*(float(i)/100))]))\nprint (\"100 percentile value is \",var[-1])\n```\n\n 0 percentile value is -1211.0166666666667\n 10 percentile value is 3.8333333333333335\n 20 percentile value is 5.383333333333334\n 30 percentile value is 6.816666666666666\n 40 percentile value is 8.3\n 50 percentile value is 9.95\n 60 percentile value is 11.866666666666667\n 70 percentile value is 14.283333333333333\n 80 percentile value is 17.633333333333333\n 90 percentile value is 23.45\n 100 percentile value is 548555.6333333333\n\n\n\n```python\n#looking further from the 99th percecntile\nfor i in range(90,100):\n var =frame_with_durations[\"trip_times\"].values\n var = np.sort(var,axis = None)\n print(\"{} percentile value is {}\".format(i,var[int(len(var)*(float(i)/100))]))\nprint (\"100 percentile value is \",var[-1])\n```\n\n 90 percentile value is 23.45\n 91 percentile value is 24.35\n 92 percentile value is 25.383333333333333\n 93 percentile value is 26.55\n 94 percentile value is 27.933333333333334\n 95 percentile value is 29.583333333333332\n 96 percentile value is 31.683333333333334\n 97 percentile value is 34.46666666666667\n 98 percentile value is 38.71666666666667\n 99 percentile value is 46.75\n 100 percentile value is 548555.6333333333\n\n\n\n```python\n#removing data based on our analysis and TLC regulations\nframe_with_durations_modified=frame_with_durations[(frame_with_durations.trip_times>1) & (frame_with_durations.trip_times<720)]\n```\n\n\n```python\n#box-plot after removal of outliers\nsns.boxplot(y=\"trip_times\", data =frame_with_durations_modified)\nplt.show()\n```\n\n\n \n\n\n\n\n\n\n\n```python\n#pdf of trip-times after removing the outliers\nsns.FacetGrid(frame_with_durations_modified,size=6) \\\n .map(sns.kdeplot,\"trip_times\") \\\n .add_legend();\nplt.show();\n```\n\n\n```python\n#converting the values to log-values to chec for log-normal\nimport math\nframe_with_durations_modified['log_times']=[math.log(i) for i in frame_with_durations_modified['trip_times'].values]\n```\n\n\n```python\n#pdf of log-values\nsns.FacetGrid(frame_with_durations_modified,size=6) \\\n .map(sns.kdeplot,\"log_times\") \\\n .add_legend();\nplt.show();\n```\n\n\n```python\n#Q-Q plot for checking if trip-times is log-normal\nscipy.stats.probplot(frame_with_durations_modified['log_times'].values, plot=plt)\nplt.show()\n```\n\n### 4. Speed\n\n\n```python\n# check for any outliers in the data after trip duration outliers removed\n# box-plot for speeds with outliers\nframe_with_durations_modified['Speed'] = 60*(frame_with_durations_modified['trip_distance']/frame_with_durations_modified['trip_times'])\nsns.boxplot(y=\"Speed\", data =frame_with_durations_modified)\nplt.show()\n```\n\n\n```python\n#calculating speed values at each percntile 0,10,20,30,40,50,60,70,80,90,100 \nfor i in range(0,100,10):\n var =frame_with_durations_modified[\"Speed\"].values\n var = np.sort(var,axis = None)\n print(\"{} percentile value is {}\".format(i,var[int(len(var)*(float(i)/100))]))\nprint(\"100 percentile value is \",var[-1])\n```\n\n\n```python\n#calculating speed values at each percntile 90,91,92,93,94,95,96,97,98,99,100\nfor i in range(90,100):\n var =frame_with_durations_modified[\"Speed\"].values\n var = np.sort(var,axis = None)\n print(\"{} percentile value is {}\".format(i,var[int(len(var)*(float(i)/100))]))\nprint(\"100 percentile value is \",var[-1])\n```\n\n\n```python\n#calculating speed values at each percntile 99.0,99.1,99.2,99.3,99.4,99.5,99.6,99.7,99.8,99.9,100\nfor i in np.arange(0.0, 1.0, 0.1):\n var =frame_with_durations_modified[\"Speed\"].values\n var = np.sort(var,axis = None)\n print(\"{} percentile value is {}\".format(99+i,var[int(len(var)*(float(99+i)/100))]))\nprint(\"100 percentile value is \",var[-1])\n```\n\n\n```python\n#removing further outliers based on the 99.9th percentile value\nframe_with_durations_modified=frame_with_durations[(frame_with_durations.Speed>0) & (frame_with_durations.Speed<45.31)]\n```\n\n\n```python\n#avg.speed of cabs in New-York\nsum(frame_with_durations_modified[\"Speed\"]) / float(len(frame_with_durations_modified[\"Speed\"]))\n```\n\nThe avg speed in Newyork speed is 12.45miles/hr, so a cab driver can travel 2 miles per 10min on avg. \n\n### 4. Trip Distance\n\n\n```python\n# up to now we have removed the outliers based on trip durations and cab speeds\n# lets try if there are any outliers in trip distances\n# box-plot showing outliers in trip-distance values\nsns.boxplot(y=\"trip_distance\", data =frame_with_durations_modified)\nplt.show()\n```\n\n\n```python\n#calculating trip distance values at each percntile 0,10,20,30,40,50,60,70,80,90,100 \nfor i in range(0,100,10):\n var =frame_with_durations_modified[\"trip_distance\"].values\n var = np.sort(var,axis = None)\n print(\"{} percentile value is {}\".format(i,var[int(len(var)*(float(i)/100))]))\nprint(\"100 percentile value is \",var[-1])\n```\n\n\n```python\n#calculating trip distance values at each percntile 90,91,92,93,94,95,96,97,98,99,100\nfor i in range(90,100):\n var =frame_with_durations_modified[\"trip_distance\"].values\n var = np.sort(var,axis = None)\n print(\"{} percentile value is {}\".format(i,var[int(len(var)*(float(i)/100))]))\nprint(\"100 percentile value is \",var[-1])\n```\n\n\n```python\n#calculating trip distance values at each percntile 99.0,99.1,99.2,99.3,99.4,99.5,99.6,99.7,99.8,99.9,100\nfor i in np.arange(0.0, 1.0, 0.1):\n var =frame_with_durations_modified[\"trip_distance\"].values\n var = np.sort(var,axis = None)\n print(\"{} percentile value is {}\".format(99+i,var[int(len(var)*(float(99+i)/100))]))\nprint(\"100 percentile value is \",var[-1])\n```\n\n\n```python\n#removing further outliers based on the 99.9th percentile value\nframe_with_durations_modified=frame_with_durations[(frame_with_durations.trip_distance>0) & (frame_with_durations.trip_distance<23)]\n```\n\n\n```python\n#box-plot after removal of outliers\nsns.boxplot(y=\"trip_distance\", data = frame_with_durations_modified)\nplt.show()\n```\n\n### 5. Total Fare\n\n\n```python\n# up to now we have removed the outliers based on trip durations, cab speeds, and trip distances\n# lets try if there are any outliers in based on the total_amount\n# box-plot showing outliers in fare\nsns.boxplot(y=\"total_amount\", data =frame_with_durations_modified)\nplt.show()\n```\n\n\n```python\n#calculating total fare amount values at each percntile 0,10,20,30,40,50,60,70,80,90,100 \nfor i in range(0,100,10):\n var = frame_with_durations_modified[\"total_amount\"].values\n var = np.sort(var,axis = None)\n print(\"{} percentile value is {}\".format(i,var[int(len(var)*(float(i)/100))]))\nprint(\"100 percentile value is \",var[-1])\n```\n\n\n```python\n#calculating total fare amount values at each percntile 90,91,92,93,94,95,96,97,98,99,100\nfor i in range(90,100):\n var = frame_with_durations_modified[\"total_amount\"].values\n var = np.sort(var,axis = None)\n print(\"{} percentile value is {}\".format(i,var[int(len(var)*(float(i)/100))]))\nprint(\"100 percentile value is \",var[-1])\n```\n\n\n```python\n#calculating total fare amount values at each percntile 99.0,99.1,99.2,99.3,99.4,99.5,99.6,99.7,99.8,99.9,100\nfor i in np.arange(0.0, 1.0, 0.1):\n var = frame_with_durations_modified[\"total_amount\"].values\n var = np.sort(var,axis = None)\n print(\"{} percentile value is {}\".format(99+i,var[int(len(var)*(float(99+i)/100))]))\nprint(\"100 percentile value is \",var[-1])\n```\n\nObservation:- As even the 99.9th percentile value doesnt look like an outlier,as there is not much difference between the 99.8th percentile and 99.9th percentile, we move on to do graphical analyis\n\n\n```python\n#below plot shows us the fare values(sorted) to find a sharp increase to remove those values as outliers\n# plot the fare amount excluding last two values in sorted data\nplt.plot(var[:-2])\nplt.show()\n```\n\n\n```python\n# a very sharp increase in fare values can be seen \n# plotting last three total fare values, and we can observe there is share increase in the values\nplt.plot(var[-3:])\nplt.show()\n```\n\n\n```python\n#now looking at values not including the last two points we again find a drastic increase at around 1000 fare value\n# we plot last 50 values excluding last two values\nplt.plot(var[-50:-2])\nplt.show()\n```\n\n## Remove all outliers/erronous points.\n\n\n```python\n#removing all outliers based on our univariate analysis above\ndef remove_outliers(new_frame):\n\n \n a = new_frame.shape[0]\n print (\"Number of pickup records = \",a)\n temp_frame = new_frame[((new_frame.dropoff_longitude >= -74.15) & (new_frame.dropoff_longitude <= -73.7004) &\\\n (new_frame.dropoff_latitude >= 40.5774) & (new_frame.dropoff_latitude <= 40.9176)) & \\\n ((new_frame.pickup_longitude >= -74.15) & (new_frame.pickup_latitude >= 40.5774)& \\\n (new_frame.pickup_longitude <= -73.7004) & (new_frame.pickup_latitude <= 40.9176))]\n b = temp_frame.shape[0]\n print (\"Number of outlier coordinates lying outside NY boundaries:\",(a-b))\n\n \n temp_frame = new_frame[(new_frame.trip_times > 0) & (new_frame.trip_times < 720)]\n c = temp_frame.shape[0]\n print (\"Number of outliers from trip times analysis:\",(a-c))\n \n \n temp_frame = new_frame[(new_frame.trip_distance > 0) & (new_frame.trip_distance < 23)]\n d = temp_frame.shape[0]\n print (\"Number of outliers from trip distance analysis:\",(a-d))\n \n temp_frame = new_frame[(new_frame.Speed <= 65) & (new_frame.Speed >= 0)]\n e = temp_frame.shape[0]\n print (\"Number of outliers from speed analysis:\",(a-e))\n \n temp_frame = new_frame[(new_frame.total_amount <1000) & (new_frame.total_amount >0)]\n f = temp_frame.shape[0]\n print (\"Number of outliers from fare analysis:\",(a-f))\n \n \n new_frame = new_frame[((new_frame.dropoff_longitude >= -74.15) & (new_frame.dropoff_longitude <= -73.7004) &\\\n (new_frame.dropoff_latitude >= 40.5774) & (new_frame.dropoff_latitude <= 40.9176)) & \\\n ((new_frame.pickup_longitude >= -74.15) & (new_frame.pickup_latitude >= 40.5774)& \\\n (new_frame.pickup_longitude <= -73.7004) & (new_frame.pickup_latitude <= 40.9176))]\n \n new_frame = new_frame[(new_frame.trip_times > 0) & (new_frame.trip_times < 720)]\n new_frame = new_frame[(new_frame.trip_distance > 0) & (new_frame.trip_distance < 23)]\n new_frame = new_frame[(new_frame.Speed < 45.31) & (new_frame.Speed > 0)]\n new_frame = new_frame[(new_frame.total_amount <1000) & (new_frame.total_amount >0)]\n \n print (\"Total outliers removed\",a - new_frame.shape[0])\n print (\"---\")\n return new_frame\n```\n\n\n```python\nprint (\"Removing outliers in the month of Jan-2015\")\nprint (\"----\")\nframe_with_durations_outliers_removed = remove_outliers(frame_with_durations)\nprint(\"fraction of data points that remain after removing outliers\", float(len(frame_with_durations_outliers_removed))/len(frame_with_durations))\n```\n\n# Data-preperation\n## Clustering/Segmentation\n\n\n```python\n#trying different cluster sizes to choose the right K in K-means\ncoords = frame_with_durations_outliers_removed[['pickup_latitude', 'pickup_longitude']].values\nneighbours=[]\n\ndef find_min_distance(cluster_centers, cluster_len):\n nice_points = 0\n wrong_points = 0\n less2 = []\n more2 = []\n min_dist=1000\n for i in range(0, cluster_len):\n nice_points = 0\n wrong_points = 0\n for j in range(0, cluster_len):\n if j!=i:\n distance = gpxpy.geo.haversine_distance(cluster_centers[i][0], cluster_centers[i][1],cluster_centers[j][0], cluster_centers[j][1])\n min_dist = min(min_dist,distance/(1.60934*1000))\n if (distance/(1.60934*1000)) <= 2:\n nice_points +=1\n else:\n wrong_points += 1\n less2.append(nice_points)\n more2.append(wrong_points)\n neighbours.append(less2)\n print (\"On choosing a cluster size of \",cluster_len,\"\\nAvg. Number of Clusters within the vicinity (i.e. intercluster-distance < 2):\", np.ceil(sum(less2)/len(less2)), \"\\nAvg. Number of Clusters outside the vicinity (i.e. intercluster-distance > 2):\", np.ceil(sum(more2)/len(more2)),\"\\nMin inter-cluster distance = \",min_dist,\"\\n---\")\n\ndef find_clusters(increment):\n kmeans = MiniBatchKMeans(n_clusters=increment, batch_size=10000,random_state=42).fit(coords)\n frame_with_durations_outliers_removed['pickup_cluster'] = kmeans.predict(frame_with_durations_outliers_removed[['pickup_latitude', 'pickup_longitude']])\n cluster_centers = kmeans.cluster_centers_\n cluster_len = len(cluster_centers)\n return cluster_centers, cluster_len\n\n# we need to choose number of clusters so that, there are more number of cluster regions \n#that are close to any cluster center\n# and make sure that the minimum inter cluster should not be very less\nfor increment in range(10, 100, 10):\n cluster_centers, cluster_len = find_clusters(increment)\n find_min_distance(cluster_centers, cluster_len) \n```\n\n### Inference:\n- The main objective was to find a optimal min. distance(Which roughly estimates to the radius of a cluster) between the clusters which we got was 40\n\n\n```python\n# if check for the 50 clusters you can observe that there are two clusters with only 0.3 miles apart from each other\n# so we choose 40 clusters for solve the further problem\n\n# Getting 40 clusters using the kmeans \nkmeans = MiniBatchKMeans(n_clusters=40, batch_size=10000,random_state=0).fit(coords)\nframe_with_durations_outliers_removed['pickup_cluster'] = kmeans.predict(frame_with_durations_outliers_removed[['pickup_latitude', 'pickup_longitude']])\n```\n\n### Plotting the cluster centers:\n\n\n```python\n# Plotting the cluster centers on OSM\ncluster_centers = kmeans.cluster_centers_\ncluster_len = len(cluster_centers)\nmap_osm = folium.Map(location=[40.734695, -73.990372], tiles='Stamen Toner')\nfor i in range(cluster_len):\n folium.Marker(list((cluster_centers[i][0],cluster_centers[i][1])), popup=(str(cluster_centers[i][0])+str(cluster_centers[i][1]))).add_to(map_osm)\nmap_osm\n```\n\n### Plotting the clusters:\n\n\n```python\n#Visualising the clusters on a map\ndef plot_clusters(frame):\n city_long_border = (-74.03, -73.75)\n city_lat_border = (40.63, 40.85)\n fig, ax = plt.subplots(ncols=1, nrows=1)\n ax.scatter(frame.pickup_longitude.values[:100000], frame.pickup_latitude.values[:100000], s=10, lw=0,\n c=frame.pickup_cluster.values[:100000], cmap='tab20', alpha=0.2)\n ax.set_xlim(city_long_border)\n ax.set_ylim(city_lat_border)\n ax.set_xlabel('Longitude')\n ax.set_ylabel('Latitude')\n plt.show()\n\nplot_clusters(frame_with_durations_outliers_removed)\n```\n\n## Time-binning\n\n\n```python\n#Refer:https://www.unixtimestamp.com/\n# 1420070400 : 2015-01-01 00:00:00 \n# 1422748800 : 2015-02-01 00:00:00 \n# 1425168000 : 2015-03-01 00:00:00\n# 1427846400 : 2015-04-01 00:00:00 \n# 1430438400 : 2015-05-01 00:00:00 \n# 1433116800 : 2015-06-01 00:00:00\n\n# 1451606400 : 2016-01-01 00:00:00 \n# 1454284800 : 2016-02-01 00:00:00 \n# 1456790400 : 2016-03-01 00:00:00\n# 1459468800 : 2016-04-01 00:00:00 \n# 1462060800 : 2016-05-01 00:00:00 \n# 1464739200 : 2016-06-01 00:00:00\n\ndef add_pickup_bins(frame,month,year):\n unix_pickup_times=[i for i in frame['pickup_times'].values]\n unix_times = [[1420070400,1422748800,1425168000,1427846400,1430438400,1433116800],\\\n [1451606400,1454284800,1456790400,1459468800,1462060800,1464739200]]\n \n start_pickup_unix=unix_times[year-2015][month-1]\n # https://www.timeanddate.com/time/zones/est\n # (int((i-start_pickup_unix)/600)+33) : our unix time is in gmt to we are converting it to est\n tenminutewise_binned_unix_pickup_times=[(int((i-start_pickup_unix)/600)+33) for i in unix_pickup_times]\n frame['pickup_bins'] = np.array(tenminutewise_binned_unix_pickup_times)\n return frame\n```\n\n\n```python\n# clustering, making pickup bins and grouping by pickup cluster and pickup bins\nframe_with_durations_outliers_removed['pickup_cluster'] = kmeans.predict(frame_with_durations_outliers_removed[['pickup_latitude', 'pickup_longitude']])\njan_2015_frame = add_pickup_bins(frame_with_durations_outliers_removed,1,2015)\njan_2015_groupby = jan_2015_frame[['pickup_cluster','pickup_bins','trip_distance']].groupby(['pickup_cluster','pickup_bins']).count()\n```\n\n\n```python\n# we add two more columns 'pickup_cluster'(to which cluster it belogns to) \n# and 'pickup_bins' (to which 10min intravel the trip belongs to)\njan_2015_frame.head()\n```\n\n\n```python\n# hear the trip_distance represents the number of pickups that are happend in that particular 10min intravel\n# this data frame has two indices\n# primary index: pickup_cluster (cluster number)\n# secondary index : pickup_bins (we devid whole months time into 10min intravels 24*31*60/10 =4464bins)\njan_2015_groupby.head()\n```\n\n\n```python\n# upto now we cleaned data and prepared data for the month 2015,\n\n# now do the same operations for months Jan, Feb, March of 2016\n# 1. get the dataframe which inlcudes only required colums\n# 2. adding trip times, speed, unix time stamp of pickup_time\n# 4. remove the outliers based on trip_times, speed, trip_duration, total_amount\n# 5. add pickup_cluster to each data point\n# 6. add pickup_bin (index of 10min intravel to which that trip belongs to)\n# 7. group by data, based on 'pickup_cluster' and 'pickuo_bin'\n\n# Data Preparation for the months of Jan,Feb and March 2016\ndef datapreparation(month,kmeans,month_no,year_no):\n \n print (\"Return with trip times..\")\n\n frame_with_durations = return_with_trip_times(month)\n \n print (\"Remove outliers..\")\n frame_with_durations_outliers_removed = remove_outliers(frame_with_durations)\n \n print (\"Estimating clusters..\")\n frame_with_durations_outliers_removed['pickup_cluster'] = kmeans.predict(frame_with_durations_outliers_removed[['pickup_latitude', 'pickup_longitude']])\n #frame_with_durations_outliers_removed_2016['pickup_cluster'] = kmeans.predict(frame_with_durations_outliers_removed_2016[['pickup_latitude', 'pickup_longitude']])\n\n print (\"Final groupbying..\")\n final_updated_frame = add_pickup_bins(frame_with_durations_outliers_removed,month_no,year_no)\n final_groupby_frame = final_updated_frame[['pickup_cluster','pickup_bins','trip_distance']].groupby(['pickup_cluster','pickup_bins']).count()\n \n return final_updated_frame,final_groupby_frame\n \nmonth_jan_2016 = dd.read_csv('yellow_tripdata_2016-01.csv')\nmonth_feb_2016 = dd.read_csv('yellow_tripdata_2016-02.csv')\nmonth_mar_2016 = dd.read_csv('yellow_tripdata_2016-03.csv')\n\njan_2016_frame,jan_2016_groupby = datapreparation(month_jan_2016,kmeans,1,2016)\nfeb_2016_frame,feb_2016_groupby = datapreparation(month_feb_2016,kmeans,2,2016)\nmar_2016_frame,mar_2016_groupby = datapreparation(month_mar_2016,kmeans,3,2016)\n```\n\n## Smoothing\n\n\n```python\n# Gets the unique bins where pickup values are present for each each reigion\n\n# for each cluster region we will collect all the indices of 10min intravels in which the pickups are happened\n# we got an observation that there are some pickpbins that doesnt have any pickups\ndef return_unq_pickup_bins(frame):\n values = []\n for i in range(0,40):\n new = frame[frame['pickup_cluster'] == i]\n list_unq = list(set(new['pickup_bins']))\n list_unq.sort()\n values.append(list_unq)\n return values\n```\n\n\n```python\n# for every month we get all indices of 10min intravels in which atleast one pickup got happened\n\n#jan\njan_2015_unique = return_unq_pickup_bins(jan_2015_frame)\njan_2016_unique = return_unq_pickup_bins(jan_2016_frame)\n\n#feb\nfeb_2016_unique = return_unq_pickup_bins(feb_2016_frame)\n\n#march\nmar_2016_unique = return_unq_pickup_bins(mar_2016_frame)\n```\n\n\n```python\n# for each cluster number of 10min intravels with 0 pickups\nfor i in range(40):\n print(\"for the \",i,\"th cluster number of 10min intavels with zero pickups: \",4464 - len(set(jan_2015_unique[i])))\n print('-'*60)\n```\n\nthere are two ways to fill up these values\n
      \n
    • Fill the missing value with 0's
      • \n
    • Fill the missing values with the avg values\n
        \n
      • Case 1:(values missing at the start)
        Ex1: \\_ \\_ \\_ x =>ceil(x/4), ceil(x/4), ceil(x/4), ceil(x/4)
        Ex2: \\_ \\_ x => ceil(x/3), ceil(x/3), ceil(x/3)
        • \n
      • Case 2:(values missing in middle)
        Ex1: x \\_ \\_ y => ceil((x+y)/4), ceil((x+y)/4), ceil((x+y)/4), ceil((x+y)/4)
        Ex2: x \\_ \\_ \\_ y => ceil((x+y)/5), ceil((x+y)/5), ceil((x+y)/5), ceil((x+y)/5), ceil((x+y)/5)
        • \n
      • Case 3:(values missing at the end)
        Ex1: x \\_ \\_ \\_ => ceil(x/4), ceil(x/4), ceil(x/4), ceil(x/4)
        Ex2: x \\_ => ceil(x/2), ceil(x/2)
        • \n
      \n
      • \n
    \n\n\n```python\n# Fills a value of zero for every bin where no pickup data is present \n# the count_values: number pickps that are happened in each region for each 10min intravel\n# there wont be any value if there are no picksups.\n# values: number of unique bins\n\n# for every 10min intravel(pickup_bin) we will check it is there in our unique bin,\n# if it is there we will add the count_values[index] to smoothed data\n# if not we add 0 to the smoothed data\n# we finally return smoothed data\ndef fill_missing(count_values,values):\n smoothed_regions=[]\n ind=0\n for r in range(0,40):\n smoothed_bins=[]\n for i in range(4464):\n if i in values[r]:\n smoothed_bins.append(count_values[ind])\n ind+=1\n else:\n smoothed_bins.append(0)\n smoothed_regions.extend(smoothed_bins)\n return smoothed_regions\n```\n\n\n```python\n# Fills a value of zero for every bin where no pickup data is present \n# the count_values: number pickps that are happened in each region for each 10min intravel\n# there wont be any value if there are no picksups.\n# values: number of unique bins\n\n# for every 10min intravel(pickup_bin) we will check it is there in our unique bin,\n# if it is there we will add the count_values[index] to smoothed data\n# if not we add smoothed data (which is calculated based on the methods that are discussed in the above markdown cell)\n# we finally return smoothed data\ndef smoothing(count_values,values):\n smoothed_regions=[] # stores list of final smoothed values of each reigion\n ind=0\n repeat=0 \n smoothed_value=0\n for r in range(0,40):\n smoothed_bins=[] #stores the final smoothed values\n repeat=0\n for i in range(4464):\n if repeat!=0: # prevents iteration for a value which is already visited/resolved\n repeat-=1\n continue\n if i in values[r]: #checks if the pickup-bin exists \n smoothed_bins.append(count_values[ind]) # appends the value of the pickup bin if it exists\n else:\n if i!=0:\n right_hand_limit=0\n for j in range(i,4464):\n if j not in values[r]: #searches for the left-limit or the pickup-bin value which has a pickup value\n continue\n else:\n right_hand_limit=j\n break\n if right_hand_limit==0:\n #Case 1: When we have the last/last few values are found to be missing,hence we have no right-limit here\n smoothed_value=count_values[ind-1]*1.0/((4463-i)+2)*1.0 \n for j in range(i,4464): \n smoothed_bins.append(math.ceil(smoothed_value))\n smoothed_bins[i-1] = math.ceil(smoothed_value)\n repeat=(4463-i)\n ind-=1\n else:\n #Case 2: When we have the missing values between two known values\n smoothed_value=(count_values[ind-1]+count_values[ind])*1.0/((right_hand_limit-i)+2)*1.0 \n for j in range(i,right_hand_limit+1):\n smoothed_bins.append(math.ceil(smoothed_value))\n smoothed_bins[i-1] = math.ceil(smoothed_value)\n repeat=(right_hand_limit-i)\n else:\n #Case 3: When we have the first/first few values are found to be missing,hence we have no left-limit here\n right_hand_limit=0\n for j in range(i,4464):\n if j not in values[r]:\n continue\n else:\n right_hand_limit=j\n break\n smoothed_value=count_values[ind]*1.0/((right_hand_limit-i)+1)*1.0\n for j in range(i,right_hand_limit+1):\n smoothed_bins.append(math.ceil(smoothed_value))\n repeat=(right_hand_limit-i)\n ind+=1\n smoothed_regions.extend(smoothed_bins)\n return smoothed_regions\n\n```\n\n\n```python\n#Filling Missing values of Jan-2015 with 0\n# here in jan_2015_groupby dataframe the trip_distance represents the number of pickups that are happened\njan_2015_fill = fill_missing(jan_2015_groupby['trip_distance'].values,jan_2015_unique)\n\n#Smoothing Missing values of Jan-2015\njan_2015_smooth = smoothing(jan_2015_groupby['trip_distance'].values,jan_2015_unique)\n```\n\n\n```python\n# number of 10min indices for jan 2015= 24*31*60/10 = 4464\n# number of 10min indices for jan 2016 = 24*31*60/10 = 4464\n# number of 10min indices for feb 2016 = 24*29*60/10 = 4176\n# number of 10min indices for march 2016 = 24*30*60/10 = 4320\n# for each cluster we will have 4464 values, therefore 40*4464 = 178560 (length of the jan_2015_fill)\nprint(\"number of 10min intravels among all the clusters \",len(jan_2015_fill))\n```\n\n\n```python\n# Smoothing vs Filling\n# sample plot that shows two variations of filling missing values\n# we have taken the number of pickups for cluster region 2\nplt.figure(figsize=(10,5))\nplt.plot(jan_2015_fill[4464:8920], label=\"zero filled values\")\nplt.plot(jan_2015_smooth[4464:8920], label=\"filled with avg values\")\nplt.legend()\nplt.show()\n```\n\n\n```python\n# why we choose, these methods and which method is used for which data?\n\n# Ans: consider we have data of some month in 2015 jan 1st, 10 _ _ _ 20, i.e there are 10 pickups that are happened in 1st \n# 10st 10min intravel, 0 pickups happened in 2nd 10mins intravel, 0 pickups happened in 3rd 10min intravel \n# and 20 pickups happened in 4th 10min intravel.\n# in fill_missing method we replace these values like 10, 0, 0, 20\n# where as in smoothing method we replace these values as 6,6,6,6,6, if you can check the number of pickups \n# that are happened in the first 40min are same in both cases, but if you can observe that we looking at the future values \n# wheen you are using smoothing we are looking at the future number of pickups which might cause a data leakage.\n\n# so we use smoothing for jan 2015th data since it acts as our training data\n# and we use simple fill_misssing method for 2016th data.\n```\n\n\n```python\n# Jan-2015 data is smoothed, Jan,Feb & March 2016 data missing values are filled with zero\njan_2015_smooth = smoothing(jan_2015_groupby['trip_distance'].values,jan_2015_unique)\njan_2016_smooth = fill_missing(jan_2016_groupby['trip_distance'].values,jan_2016_unique)\nfeb_2016_smooth = fill_missing(feb_2016_groupby['trip_distance'].values,feb_2016_unique)\nmar_2016_smooth = fill_missing(mar_2016_groupby['trip_distance'].values,mar_2016_unique)\n\n# Making list of all the values of pickup data in every bin for a period of 3 months and storing them region-wise \nregions_cum = []\n\n# a =[1,2,3]\n# b = [2,3,4]\n# a+b = [1, 2, 3, 2, 3, 4]\n\n# number of 10min indices for jan 2015= 24*31*60/10 = 4464\n# number of 10min indices for jan 2016 = 24*31*60/10 = 4464\n# number of 10min indices for feb 2016 = 24*29*60/10 = 4176\n# number of 10min indices for march 2016 = 24*31*60/10 = 4464\n# regions_cum: it will contain 40 lists, each list will contain 4464+4176+4464 values which represents the number of pickups \n# that are happened for three months in 2016 data\n\nfor i in range(0,40):\n regions_cum.append(jan_2016_smooth[4464*i:4464*(i+1)]+feb_2016_smooth[4176*i:4176*(i+1)]+mar_2016_smooth[4464*i:4464*(i+1)])\n\n# print(len(regions_cum))\n# 40\n# print(len(regions_cum[0]))\n# 13104\n```\n\n## Time series and Fourier Transforms\n\n\n```python\ndef uniqueish_color():\n \"\"\"There're better ways to generate unique colors, but this isn't awful.\"\"\"\n return plt.cm.gist_ncar(np.random.random())\nfirst_x = list(range(0,4464))\nsecond_x = list(range(4464,8640))\nthird_x = list(range(8640,13104))\nfor i in range(40):\n plt.figure(figsize=(10,4))\n plt.plot(first_x,regions_cum[i][:4464], color=uniqueish_color(), label='2016 Jan month data')\n plt.plot(second_x,regions_cum[i][4464:8640], color=uniqueish_color(), label='2016 feb month data')\n plt.plot(third_x,regions_cum[i][8640:], color=uniqueish_color(), label='2016 march month data')\n plt.legend()\n plt.show()\n```\n\n\n```python\n# getting peaks: https://blog.ytotech.com/2015/11/01/findpeaks-in-python/\n# read more about fft function : https://docs.scipy.org/doc/numpy/reference/generated/numpy.fft.fft.html\nY = np.fft.fft(np.array(jan_2016_smooth)[0:4460])\n# read more about the fftfreq: https://docs.scipy.org/doc/numpy/reference/generated/numpy.fft.fftfreq.html \nfreq = np.fft.fftfreq(4460, 1)\nn = len(freq)\nplt.figure()\nplt.plot( freq[:int(n/2)], np.abs(Y)[:int(n/2)] )\nplt.xlabel(\"Frequency\")\nplt.ylabel(\"Amplitude\")\nplt.show()\n```\n\n\n```python\n#Preparing the Dataframe only with x(i) values as jan-2015 data and y(i) values as jan-2016\nratios_jan = pd.DataFrame()\nratios_jan['Given']=jan_2015_smooth\nratios_jan['Prediction']=jan_2016_smooth\nratios_jan['Ratios']=ratios_jan['Prediction']*1.0/ratios_jan['Given']*1.0\n```\n\n## Modelling: Baseline Models\n\nNow we get into modelling in order to forecast the pickup densities for the months of Jan, Feb and March of 2016 for which we are using multiple models with two variations \n1. Using Ratios of the 2016 data to the 2015 data i.e $\\begin{align} R_{t} = P^{2016}_{t} / P^{2015}_{t} \\end{align}$\n2. Using Previous known values of the 2016 data itself to predict the future values\n\n### Simple Moving Averages\nThe First Model used is the Moving Averages Model which uses the previous n values in order to predict the next value
    \n\nUsing Ratio Values - $\\begin{align}R_{t} = ( R_{t-1} + R_{t-2} + R_{t-3} .... R_{t-n} )/n \\end{align}$\n\n\n```python\ndef MA_R_Predictions(ratios,month):\n predicted_ratio=(ratios['Ratios'].values)[0]\n error=[]\n predicted_values=[]\n window_size=3\n predicted_ratio_values=[]\n for i in range(0,4464*40):\n if i%4464==0:\n predicted_ratio_values.append(0)\n predicted_values.append(0)\n error.append(0)\n continue\n predicted_ratio_values.append(predicted_ratio)\n predicted_values.append(int(((ratios['Given'].values)[i])*predicted_ratio))\n error.append(abs((math.pow(int(((ratios['Given'].values)[i])*predicted_ratio)-(ratios['Prediction'].values)[i],1))))\n if i+1>=window_size:\n predicted_ratio=sum((ratios['Ratios'].values)[(i+1)-window_size:(i+1)])/window_size\n else:\n predicted_ratio=sum((ratios['Ratios'].values)[0:(i+1)])/(i+1)\n \n \n ratios['MA_R_Predicted'] = predicted_values\n ratios['MA_R_Error'] = error\n mape_err = (sum(error)/len(error))/(sum(ratios['Prediction'].values)/len(ratios['Prediction'].values))\n mse_err = sum([e**2 for e in error])/len(error)\n return ratios,mape_err,mse_err\n```\n\nFor the above the Hyperparameter is the window-size (n) which is tuned manually and it is found that the window-size of 3 is optimal for getting the best results using Moving Averages using previous Ratio values therefore we get $\\begin{align}R_{t} = ( R_{t-1} + R_{t-2} + R_{t-3})/3 \\end{align}$\n\nNext we use the Moving averages of the 2016 values itself to predict the future value using $\\begin{align}P_{t} = ( P_{t-1} + P_{t-2} + P_{t-3} .... P_{t-n} )/n \\end{align}$\n\n\n```python\ndef MA_P_Predictions(ratios,month):\n predicted_value=(ratios['Prediction'].values)[0]\n error=[]\n predicted_values=[]\n window_size=1\n predicted_ratio_values=[]\n for i in range(0,4464*40):\n predicted_values.append(predicted_value)\n error.append(abs((math.pow(predicted_value-(ratios['Prediction'].values)[i],1))))\n if i+1>=window_size:\n predicted_value=int(sum((ratios['Prediction'].values)[(i+1)-window_size:(i+1)])/window_size)\n else:\n predicted_value=int(sum((ratios['Prediction'].values)[0:(i+1)])/(i+1))\n \n ratios['MA_P_Predicted'] = predicted_values\n ratios['MA_P_Error'] = error\n mape_err = (sum(error)/len(error))/(sum(ratios['Prediction'].values)/len(ratios['Prediction'].values))\n mse_err = sum([e**2 for e in error])/len(error)\n return ratios,mape_err,mse_err\n```\n\nFor the above the Hyperparameter is the window-size (n) which is tuned manually and it is found that the window-size of 1 is optimal for getting the best results using Moving Averages using previous 2016 values therefore we get $\\begin{align}P_{t} = P_{t-1} \\end{align}$\n\n### Weighted Moving Averages\nThe Moving Avergaes Model used gave equal importance to all the values in the window used, but we know intuitively that the future is more likely to be similar to the latest values and less similar to the older values. Weighted Averages converts this analogy into a mathematical relationship giving the highest weight while computing the averages to the latest previous value and decreasing weights to the subsequent older ones
    \n\nWeighted Moving Averages using Ratio Values - $\\begin{align}R_{t} = ( N*R_{t-1} + (N-1)*R_{t-2} + (N-2)*R_{t-3} .... 1*R_{t-n} )/(N*(N+1)/2) \\end{align}$\n\n\n```python\ndef WA_R_Predictions(ratios,month):\n predicted_ratio=(ratios['Ratios'].values)[0]\n alpha=0.5\n error=[]\n predicted_values=[]\n window_size=5\n predicted_ratio_values=[]\n for i in range(0,4464*40):\n if i%4464==0:\n predicted_ratio_values.append(0)\n predicted_values.append(0)\n error.append(0)\n continue\n predicted_ratio_values.append(predicted_ratio)\n predicted_values.append(int(((ratios['Given'].values)[i])*predicted_ratio))\n error.append(abs((math.pow(int(((ratios['Given'].values)[i])*predicted_ratio)-(ratios['Prediction'].values)[i],1))))\n if i+1>=window_size:\n sum_values=0\n sum_of_coeff=0\n for j in range(window_size,0,-1):\n sum_values += j*(ratios['Ratios'].values)[i-window_size+j]\n sum_of_coeff+=j\n predicted_ratio=sum_values/sum_of_coeff\n else:\n sum_values=0\n sum_of_coeff=0\n for j in range(i+1,0,-1):\n sum_values += j*(ratios['Ratios'].values)[j-1]\n sum_of_coeff+=j\n predicted_ratio=sum_values/sum_of_coeff\n \n ratios['WA_R_Predicted'] = predicted_values\n ratios['WA_R_Error'] = error\n mape_err = (sum(error)/len(error))/(sum(ratios['Prediction'].values)/len(ratios['Prediction'].values))\n mse_err = sum([e**2 for e in error])/len(error)\n return ratios,mape_err,mse_err\n```\n\nFor the above the Hyperparameter is the window-size (n) which is tuned manually and it is found that the window-size of 5 is optimal for getting the best results using Weighted Moving Averages using previous Ratio values therefore we get $\\begin{align} R_{t} = ( 5*R_{t-1} + 4*R_{t-2} + 3*R_{t-3} + 2*R_{t-4} + R_{t-5} )/15 \\end{align}$\n\nWeighted Moving Averages using Previous 2016 Values - $\\begin{align}P_{t} = ( N*P_{t-1} + (N-1)*P_{t-2} + (N-2)*P_{t-3} .... 1*P_{t-n} )/(N*(N+1)/2) \\end{align}$\n\n\n```python\ndef WA_P_Predictions(ratios,month):\n predicted_value=(ratios['Prediction'].values)[0]\n error=[]\n predicted_values=[]\n window_size=2\n for i in range(0,4464*40):\n predicted_values.append(predicted_value)\n error.append(abs((math.pow(predicted_value-(ratios['Prediction'].values)[i],1))))\n if i+1>=window_size:\n sum_values=0\n sum_of_coeff=0\n for j in range(window_size,0,-1):\n sum_values += j*(ratios['Prediction'].values)[i-window_size+j]\n sum_of_coeff+=j\n predicted_value=int(sum_values/sum_of_coeff)\n\n else:\n sum_values=0\n sum_of_coeff=0\n for j in range(i+1,0,-1):\n sum_values += j*(ratios['Prediction'].values)[j-1]\n sum_of_coeff+=j\n predicted_value=int(sum_values/sum_of_coeff)\n \n ratios['WA_P_Predicted'] = predicted_values\n ratios['WA_P_Error'] = error\n mape_err = (sum(error)/len(error))/(sum(ratios['Prediction'].values)/len(ratios['Prediction'].values))\n mse_err = sum([e**2 for e in error])/len(error)\n return ratios,mape_err,mse_err\n```\n\nFor the above the Hyperparameter is the window-size (n) which is tuned manually and it is found that the window-size of 2 is optimal for getting the best results using Weighted Moving Averages using previous 2016 values therefore we get $\\begin{align} P_{t} = ( 2*P_{t-1} + P_{t-2} )/3 \\end{align}$\n\n### Exponential Weighted Moving Averages\n https://en.wikipedia.org/wiki/Moving_average#Exponential_moving_average\nThrough weighted averaged we have satisfied the analogy of giving higher weights to the latest value and decreasing weights to the subsequent ones but we still do not know which is the correct weighting scheme as there are infinetly many possibilities in which we can assign weights in a non-increasing order and tune the the hyperparameter window-size. To simplify this process we use Exponential Moving Averages which is a more logical way towards assigning weights and at the same time also using an optimal window-size.\n\nIn exponential moving averages we use a single hyperparameter alpha $\\begin{align}(\\alpha)\\end{align}$ which is a value between 0 & 1 and based on the value of the hyperparameter alpha the weights and the window sizes are configured.
    \nFor eg. If $\\begin{align}\\alpha=0.9\\end{align}$ then the number of days on which the value of the current iteration is based is~$\\begin{align}1/(1-\\alpha)=10\\end{align}$ i.e. we consider values 10 days prior before we predict the value for the current iteration. Also the weights are assigned using $\\begin{align}2/(N+1)=0.18\\end{align}$ ,where N = number of prior values being considered, hence from this it is implied that the first or latest value is assigned a weight of 0.18 which keeps exponentially decreasing for the subsequent values.\n\n$\\begin{align}R^{'}_{t} = \\alpha*R_{t-1} + (1-\\alpha)*R^{'}_{t-1} \\end{align}$\n\n\n```python\ndef EA_R1_Predictions(ratios,month):\n predicted_ratio=(ratios['Ratios'].values)[0]\n alpha=0.6\n error=[]\n predicted_values=[]\n predicted_ratio_values=[]\n for i in range(0,4464*40):\n if i%4464==0:\n predicted_ratio_values.append(0)\n predicted_values.append(0)\n error.append(0)\n continue\n predicted_ratio_values.append(predicted_ratio)\n predicted_values.append(int(((ratios['Given'].values)[i])*predicted_ratio))\n error.append(abs((math.pow(int(((ratios['Given'].values)[i])*predicted_ratio)-(ratios['Prediction'].values)[i],1))))\n predicted_ratio = (alpha*predicted_ratio) + (1-alpha)*((ratios['Ratios'].values)[i])\n \n ratios['EA_R1_Predicted'] = predicted_values\n ratios['EA_R1_Error'] = error\n mape_err = (sum(error)/len(error))/(sum(ratios['Prediction'].values)/len(ratios['Prediction'].values))\n mse_err = sum([e**2 for e in error])/len(error)\n return ratios,mape_err,mse_err\n```\n\n$\\begin{align}P^{'}_{t} = \\alpha*P_{t-1} + (1-\\alpha)*P^{'}_{t-1} \\end{align}$\n\n\n```python\ndef EA_P1_Predictions(ratios,month):\n predicted_value= (ratios['Prediction'].values)[0]\n alpha=0.3\n error=[]\n predicted_values=[]\n for i in range(0,4464*40):\n if i%4464==0:\n predicted_values.append(0)\n error.append(0)\n continue\n predicted_values.append(predicted_value)\n error.append(abs((math.pow(predicted_value-(ratios['Prediction'].values)[i],1))))\n predicted_value =int((alpha*predicted_value) + (1-alpha)*((ratios['Prediction'].values)[i]))\n \n ratios['EA_P1_Predicted'] = predicted_values\n ratios['EA_P1_Error'] = error\n mape_err = (sum(error)/len(error))/(sum(ratios['Prediction'].values)/len(ratios['Prediction'].values))\n mse_err = sum([e**2 for e in error])/len(error)\n return ratios,mape_err,mse_err\n```\n\n\n```python\nmean_err=[0]*10\nmedian_err=[0]*10\nratios_jan,mean_err[0],median_err[0]=MA_R_Predictions(ratios_jan,'jan')\nratios_jan,mean_err[1],median_err[1]=MA_P_Predictions(ratios_jan,'jan')\nratios_jan,mean_err[2],median_err[2]=WA_R_Predictions(ratios_jan,'jan')\nratios_jan,mean_err[3],median_err[3]=WA_P_Predictions(ratios_jan,'jan')\nratios_jan,mean_err[4],median_err[4]=EA_R1_Predictions(ratios_jan,'jan')\nratios_jan,mean_err[5],median_err[5]=EA_P1_Predictions(ratios_jan,'jan')\n```\n\n## Comparison between baseline models\nWe have chosen our error metric for comparison between models as MAPE (Mean Absolute Percentage Error) so that we can know that on an average how good is our model with predictions and MSE (Mean Squared Error) is also used so that we have a clearer understanding as to how well our forecasting model performs with outliers so that we make sure that there is not much of a error margin between our prediction and the actual value\n\n\n```python\nprint (\"Error Metric Matrix (Forecasting Methods) - MAPE & MSE\")\nprint (\"--------------------------------------------------------------------------------------------------------\")\nprint (\"Moving Averages (Ratios) - MAPE: \",mean_err[0],\" MSE: \",median_err[0])\nprint (\"Moving Averages (2016 Values) - MAPE: \",mean_err[1],\" MSE: \",median_err[1])\nprint (\"--------------------------------------------------------------------------------------------------------\")\nprint (\"Weighted Moving Averages (Ratios) - MAPE: \",mean_err[2],\" MSE: \",median_err[2])\nprint (\"Weighted Moving Averages (2016 Values) - MAPE: \",mean_err[3],\" MSE: \",median_err[3])\nprint (\"--------------------------------------------------------------------------------------------------------\")\nprint (\"Exponential Moving Averages (Ratios) - MAPE: \",mean_err[4],\" MSE: \",median_err[4])\nprint (\"Exponential Moving Averages (2016 Values) - MAPE: \",mean_err[5],\" MSE: \",median_err[5])\n```\n\nPlese Note:- The above comparisons are made using Jan 2015 and Jan 2016 only\n\nFrom the above matrix it is inferred that the best forecasting model for our prediction would be:-\n$\\begin{align}P^{'}_{t} = \\alpha*P_{t-1} + (1-\\alpha)*P^{'}_{t-1} \\end{align}$ i.e Exponential Moving Averages using 2016 Values\n\n## Regression Models\n\n### Train-Test Split\nBefore we start predictions using the tree based regression models we take 3 months of 2016 pickup data and split it such that for every region we have 70% data in train and 30% in test, ordered date-wise for every region\n\n\n```python\n# Preparing data to be split into train and test, The below prepares data in cumulative form which will be later split into test and train\n# number of 10min indices for jan 2015= 24*31*60/10 = 4464\n# number of 10min indices for jan 2016 = 24*31*60/10 = 4464\n# number of 10min indices for feb 2016 = 24*29*60/10 = 4176\n# number of 10min indices for march 2016 = 24*31*60/10 = 4464\n# regions_cum: it will contain 40 lists, each list will contain 4464+4176+4464 values which represents the number of pickups \n# that are happened for three months in 2016 data\n\n# print(len(regions_cum))\n# 40\n# print(len(regions_cum[0]))\n# 12960\n\n# we take number of pickups that are happened in last 5 10min intravels\nnumber_of_time_stamps = 5\n\n# output varaible\n# it is list of lists\n# it will contain number of pickups 13099 for each cluster\noutput = []\n\n\n# tsne_lat will contain 13104-5=13099 times lattitude of cluster center for every cluster\n# Ex: [[cent_lat 13099times],[cent_lat 13099times], [cent_lat 13099times].... 40 lists]\n# it is list of lists\ntsne_lat = []\n\n\n# tsne_lon will contain 13104-5=13099 times logitude of cluster center for every cluster\n# Ex: [[cent_long 13099times],[cent_long 13099times], [cent_long 13099times].... 40 lists]\n# it is list of lists\ntsne_lon = []\n\n# we will code each day \n# sunday = 0, monday=1, tue = 2, wed=3, thur=4, fri=5,sat=6\n# for every cluster we will be adding 13099 values, each value represent to which day of the week that pickup bin belongs to\n# it is list of lists\ntsne_weekday = []\n\n# its an numbpy array, of shape (523960, 5)\n# each row corresponds to an entry in out data\n# for the first row we will have [f0,f1,f2,f3,f4] fi=number of pickups happened in i+1th 10min intravel(bin)\n# the second row will have [f1,f2,f3,f4,f5]\n# the third row will have [f2,f3,f4,f5,f6]\n# and so on...\ntsne_feature = []\n\n\ntsne_feature = [0]*number_of_time_stamps\nfor i in range(0,40):\n tsne_lat.append([kmeans.cluster_centers_[i][0]]*13099)\n tsne_lon.append([kmeans.cluster_centers_[i][1]]*13099)\n # jan 1st 2016 is thursday, so we start our day from 4: \"(int(k/144))%7+4\"\n # our prediction start from 5th 10min intravel since we need to have number of pickups that are happened in last 5 pickup bins\n tsne_weekday.append([int(((int(k/144))%7+4)%7) for k in range(5,4464+4176+4464)])\n # regions_cum is a list of lists [[x1,x2,x3..x13104], [x1,x2,x3..x13104], [x1,x2,x3..x13104], [x1,x2,x3..x13104], [x1,x2,x3..x13104], .. 40 lsits]\n tsne_feature = np.vstack((tsne_feature, [regions_cum[i][r:r+number_of_time_stamps] for r in range(0,len(regions_cum[i])-number_of_time_stamps)]))\n output.append(regions_cum[i][5:])\ntsne_feature = tsne_feature[1:]\n```\n\n\n```python\nlen(tsne_lat[0])*len(tsne_lat) == tsne_feature.shape[0] == len(tsne_weekday)*len(tsne_weekday[0]) == 40*13099 == len(output)*len(output[0])\n```\n\n\n```python\n# Getting the predictions of exponential moving averages to be used as a feature in cumulative form\n\n# upto now we computed 8 features for every data point that starts from 50th min of the day\n# 1. cluster center lattitude\n# 2. cluster center longitude\n# 3. day of the week \n# 4. f_t_1: number of pickups that are happened previous t-1th 10min intravel\n# 5. f_t_2: number of pickups that are happened previous t-2th 10min intravel\n# 6. f_t_3: number of pickups that are happened previous t-3th 10min intravel\n# 7. f_t_4: number of pickups that are happened previous t-4th 10min intravel\n# 8. f_t_5: number of pickups that are happened previous t-5th 10min intravel\n\n# from the baseline models we said the exponential weighted moving avarage gives us the best error\n# we will try to add the same exponential weighted moving avarage at t as a feature to our data\n# exponential weighted moving avarage => p'(t) = alpha*p'(t-1) + (1-alpha)*P(t-1) \nalpha=0.3\n\n# it is a temporary array that store exponential weighted moving avarage for each 10min intravel, \n# for each cluster it will get reset\n# for every cluster it contains 13104 values\npredicted_values=[]\n\n# it is similar like tsne_lat\n# it is list of lists\n# predict_list is a list of lists [[x5,x6,x7..x13104], [x5,x6,x7..x13104], [x5,x6,x7..x13104], [x5,x6,x7..x13104], [x5,x6,x7..x13104], .. 40 lsits]\npredict_list = []\ntsne_flat_exp_avg = []\nfor r in range(0,40):\n for i in range(0,13104):\n if i==0:\n predicted_value= regions_cum[r][0]\n predicted_values.append(0)\n continue\n predicted_values.append(predicted_value)\n predicted_value =int((alpha*predicted_value) + (1-alpha)*(regions_cum[r][i]))\n predict_list.append(predicted_values[5:])\n predicted_values=[]\n```\n\n\n```python\n\n```\n\n\n```python\n# train, test split : 70% 30% split\n# Before we start predictions using the tree based regression models we take 3 months of 2016 pickup data \n# and split it such that for every region we have 70% data in train and 30% in test,\n# ordered date-wise for every region\nprint(\"size of train data :\", int(13099*0.7))\nprint(\"size of test data :\", int(13099*0.3))\n```\n\n\n```python\n# extracting first 9169 timestamp values i.e 70% of 13099 (total timestamps) for our training data\ntrain_features = [tsne_feature[i*13099:(13099*i+9169)] for i in range(0,40)]\n# temp = [0]*(12955 - 9068)\ntest_features = [tsne_feature[(13099*(i))+9169:13099*(i+1)] for i in range(0,40)]\n```\n\n\n```python\nprint(\"Number of data clusters\",len(train_features), \"Number of data points in trian data\", len(train_features[0]), \"Each data point contains\", len(train_features[0][0]),\"features\")\nprint(\"Number of data clusters\",len(train_features), \"Number of data points in test data\", len(test_features[0]), \"Each data point contains\", len(test_features[0][0]),\"features\")\n```\n\n\n```python\n# extracting first 9169 timestamp values i.e 70% of 13099 (total timestamps) for our training data\ntsne_train_flat_lat = [i[:9169] for i in tsne_lat]\ntsne_train_flat_lon = [i[:9169] for i in tsne_lon]\ntsne_train_flat_weekday = [i[:9169] for i in tsne_weekday]\ntsne_train_flat_output = [i[:9169] for i in output]\ntsne_train_flat_exp_avg = [i[:9169] for i in predict_list]\n```\n\n\n```python\n# extracting the rest of the timestamp values i.e 30% of 12956 (total timestamps) for our test data\ntsne_test_flat_lat = [i[9169:] for i in tsne_lat]\ntsne_test_flat_lon = [i[9169:] for i in tsne_lon]\ntsne_test_flat_weekday = [i[9169:] for i in tsne_weekday]\ntsne_test_flat_output = [i[9169:] for i in output]\ntsne_test_flat_exp_avg = [i[9169:] for i in predict_list]\n```\n\n\n```python\n# the above contains values in the form of list of lists (i.e. list of values of each region), here we make all of them in one list\ntrain_new_features = []\nfor i in range(0,40):\n train_new_features.extend(train_features[i])\ntest_new_features = []\nfor i in range(0,40):\n test_new_features.extend(test_features[i])\n```\n\n\n```python\n# converting lists of lists into sinle list i.e flatten\n# a = [[1,2,3,4],[4,6,7,8]]\n# print(sum(a,[]))\n# [1, 2, 3, 4, 4, 6, 7, 8]\n\ntsne_train_lat = sum(tsne_train_flat_lat, [])\ntsne_train_lon = sum(tsne_train_flat_lon, [])\ntsne_train_weekday = sum(tsne_train_flat_weekday, [])\ntsne_train_output = sum(tsne_train_flat_output, [])\ntsne_train_exp_avg = sum(tsne_train_flat_exp_avg,[])\n```\n\n\n```python\n# converting lists of lists into sinle list i.e flatten\n# a = [[1,2,3,4],[4,6,7,8]]\n# print(sum(a,[]))\n# [1, 2, 3, 4, 4, 6, 7, 8]\n\ntsne_test_lat = sum(tsne_test_flat_lat, [])\ntsne_test_lon = sum(tsne_test_flat_lon, [])\ntsne_test_weekday = sum(tsne_test_flat_weekday, [])\ntsne_test_output = sum(tsne_test_flat_output, [])\ntsne_test_exp_avg = sum(tsne_test_flat_exp_avg,[])\n```\n\n\n```python\n# Preparing the data frame for our train data\ncolumns = ['ft_5','ft_4','ft_3','ft_2','ft_1']\ndf_train = pd.DataFrame(data=train_new_features, columns=columns) \ndf_train['lat'] = tsne_train_lat\ndf_train['lon'] = tsne_train_lon\ndf_train['weekday'] = tsne_train_weekday\ndf_train['exp_avg'] = tsne_train_exp_avg\n\nprint(df_train.shape)\n```\n\n\n```python\n# Preparing the data frame for our train data\ndf_test = pd.DataFrame(data=test_new_features, columns=columns) \ndf_test['lat'] = tsne_test_lat\ndf_test['lon'] = tsne_test_lon\ndf_test['weekday'] = tsne_test_weekday\ndf_test['exp_avg'] = tsne_test_exp_avg\nprint(df_test.shape)\n```\n\n\n```python\ndf_test.head()\n```\n\n### Using Linear Regression\n\n\n```python\n# find more about LinearRegression function here http://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LinearRegression.html\n# -------------------------\n# default paramters\n# sklearn.linear_model.LinearRegression(fit_intercept=True, normalize=False, copy_X=True, n_jobs=1)\n\n# some of methods of LinearRegression()\n# fit(X, y[, sample_weight])\tFit linear model.\n# get_params([deep])\tGet parameters for this estimator.\n# predict(X)\tPredict using the linear model\n# score(X, y[, sample_weight])\tReturns the coefficient of determination R^2 of the prediction.\n# set_params(**params)\tSet the parameters of this estimator.\n# -----------------------\n# video link: https://www.appliedaicourse.com/course/applied-ai-course-online/lessons/geometric-intuition-1-2-copy-8/\n# -----------------------\n\nfrom sklearn.linear_model import LinearRegression\nlr_reg=LinearRegression().fit(df_train, tsne_train_output)\n\ny_pred = lr_reg.predict(df_test)\nlr_test_predictions = [round(value) for value in y_pred]\ny_pred = lr_reg.predict(df_train)\nlr_train_predictions = [round(value) for value in y_pred]\n```\n\n### Using Random Forest Regressor\n\n\n```python\n# Training a hyper-parameter tuned random forest regressor on our train data\n# find more about LinearRegression function here http://scikit-learn.org/stable/modules/generated/sklearn.ensemble.RandomForestRegressor.html\n# -------------------------\n# default paramters\n# sklearn.ensemble.RandomForestRegressor(n_estimators=10, criterion=\u2019mse\u2019, max_depth=None, min_samples_split=2, \n# min_samples_leaf=1, min_weight_fraction_leaf=0.0, max_features=\u2019auto\u2019, max_leaf_nodes=None, min_impurity_decrease=0.0, \n# min_impurity_split=None, bootstrap=True, oob_score=False, n_jobs=1, random_state=None, verbose=0, warm_start=False)\n\n# some of methods of RandomForestRegressor()\n# apply(X)\tApply trees in the forest to X, return leaf indices.\n# decision_path(X)\tReturn the decision path in the forest\n# fit(X, y[, sample_weight])\tBuild a forest of trees from the training set (X, y).\n# get_params([deep])\tGet parameters for this estimator.\n# predict(X)\tPredict regression target for X.\n# score(X, y[, sample_weight])\tReturns the coefficient of determination R^2 of the prediction.\n# -----------------------\n# video link1: https://www.appliedaicourse.com/course/applied-ai-course-online/lessons/regression-using-decision-trees-2/\n# video link2: https://www.appliedaicourse.com/course/applied-ai-course-online/lessons/what-are-ensembles/\n# -----------------------\n\nregr1 = RandomForestRegressor(max_features='sqrt',min_samples_leaf=4,min_samples_split=3,n_estimators=40, n_jobs=-1)\nregr1.fit(df_train, tsne_train_output)\n```\n\n\n```python\n# Predicting on test data using our trained random forest model \n\n# the models regr1 is already hyper parameter tuned\n# the parameters that we got above are found using grid search\n\ny_pred = regr1.predict(df_test)\nrndf_test_predictions = [round(value) for value in y_pred]\ny_pred = regr1.predict(df_train)\nrndf_train_predictions = [round(value) for value in y_pred]\n```\n\n\n```python\n#feature importances based on analysis using random forest\nprint (df_train.columns)\nprint (regr1.feature_importances_)\n```\n\n### Using XgBoost Regressor\n\n\n```python\n# Training a hyper-parameter tuned Xg-Boost regressor on our train data\n\n# find more about XGBRegressor function here http://xgboost.readthedocs.io/en/latest/python/python_api.html?#module-xgboost.sklearn\n# -------------------------\n# default paramters\n# xgboost.XGBRegressor(max_depth=3, learning_rate=0.1, n_estimators=100, silent=True, objective='reg:linear', \n# booster='gbtree', n_jobs=1, nthread=None, gamma=0, min_child_weight=1, max_delta_step=0, subsample=1, colsample_bytree=1, \n# colsample_bylevel=1, reg_alpha=0, reg_lambda=1, scale_pos_weight=1, base_score=0.5, random_state=0, seed=None, \n# missing=None, **kwargs)\n\n# some of methods of RandomForestRegressor()\n# fit(X, y, sample_weight=None, eval_set=None, eval_metric=None, early_stopping_rounds=None, verbose=True, xgb_model=None)\n# get_params([deep])\tGet parameters for this estimator.\n# predict(data, output_margin=False, ntree_limit=0) : Predict with data. NOTE: This function is not thread safe.\n# get_score(importance_type='weight') -> get the feature importance\n# -----------------------\n# video link1: https://www.appliedaicourse.com/course/applied-ai-course-online/lessons/regression-using-decision-trees-2/\n# video link2: https://www.appliedaicourse.com/course/applied-ai-course-online/lessons/what-are-ensembles/\n# -----------------------\n\nx_model = xgb.XGBRegressor(\n learning_rate =0.1,\n n_estimators=1000,\n max_depth=3,\n min_child_weight=3,\n gamma=0,\n subsample=0.8,\n reg_alpha=200, reg_lambda=200,\n colsample_bytree=0.8,nthread=4)\nx_model.fit(df_train, tsne_train_output)\n```\n\n\n```python\n#predicting with our trained Xg-Boost regressor\n# the models x_model is already hyper parameter tuned\n# the parameters that we got above are found using grid search\n\ny_pred = x_model.predict(df_test)\nxgb_test_predictions = [round(value) for value in y_pred]\ny_pred = x_model.predict(df_train)\nxgb_train_predictions = [round(value) for value in y_pred]\n```\n\n\n```python\n#feature importances\nx_model.booster().get_score(importance_type='weight')\n```\n\n### Calculating the error metric values for various models\n\n\n```python\ntrain_mape=[]\ntest_mape=[]\n\ntrain_mape.append((mean_absolute_error(tsne_train_output,df_train['ft_1'].values))/(sum(tsne_train_output)/len(tsne_train_output)))\ntrain_mape.append((mean_absolute_error(tsne_train_output,df_train['exp_avg'].values))/(sum(tsne_train_output)/len(tsne_train_output)))\ntrain_mape.append((mean_absolute_error(tsne_train_output,rndf_train_predictions))/(sum(tsne_train_output)/len(tsne_train_output)))\ntrain_mape.append((mean_absolute_error(tsne_train_output, xgb_train_predictions))/(sum(tsne_train_output)/len(tsne_train_output)))\ntrain_mape.append((mean_absolute_error(tsne_train_output, lr_train_predictions))/(sum(tsne_train_output)/len(tsne_train_output)))\n\ntest_mape.append((mean_absolute_error(tsne_test_output, df_test['ft_1'].values))/(sum(tsne_test_output)/len(tsne_test_output)))\ntest_mape.append((mean_absolute_error(tsne_test_output, df_test['exp_avg'].values))/(sum(tsne_test_output)/len(tsne_test_output)))\ntest_mape.append((mean_absolute_error(tsne_test_output, rndf_test_predictions))/(sum(tsne_test_output)/len(tsne_test_output)))\ntest_mape.append((mean_absolute_error(tsne_test_output, xgb_test_predictions))/(sum(tsne_test_output)/len(tsne_test_output)))\ntest_mape.append((mean_absolute_error(tsne_test_output, lr_test_predictions))/(sum(tsne_test_output)/len(tsne_test_output)))\n```\n\n\n```python\nprint (\"Error Metric Matrix (Tree Based Regression Methods) - MAPE\")\nprint (\"--------------------------------------------------------------------------------------------------------\")\nprint (\"Baseline Model - Train: \",train_mape[0],\" Test: \",test_mape[0])\nprint (\"Exponential Averages Forecasting - Train: \",train_mape[1],\" Test: \",test_mape[1])\nprint (\"Linear Regression - Train: \",train_mape[3],\" Test: \",test_mape[3])\nprint (\"Random Forest Regression - Train: \",train_mape[2],\" Test: \",test_mape[2])\n```\n\n### Error Metric Matrix\n\n\n```python\nprint (\"Error Metric Matrix (Tree Based Regression Methods) - MAPE\")\nprint (\"--------------------------------------------------------------------------------------------------------\")\nprint (\"Baseline Model - Train: \",train_mape[0],\" Test: \",test_mape[0])\nprint (\"Exponential Averages Forecasting - Train: \",train_mape[1],\" Test: \",test_mape[1])\nprint (\"Linear Regression - Train: \",train_mape[4],\" Test: \",test_mape[4])\nprint (\"Random Forest Regression - Train: \",train_mape[2],\" Test: \",test_mape[2])\nprint (\"XgBoost Regression - Train: \",train_mape[3],\" Test: \",test_mape[3])\nprint (\"--------------------------------------------------------------------------------------------------------\")\n```\n\n# Assignments\n\n \n\n\n```python\n'''\nTask 1: Incorporate Fourier features as features into Regression models and measure MAPE.
    \n\nTask 2: Perform hyper-parameter tuning for Regression models.\n 2a. Linear Regression: Grid Search\n 2b. Random Forest: Random Search \n 2c. Xgboost: Random Search\nTask 3: Explore more time-series features using Google search/Quora/Stackoverflow\nto reduce the MAPE to < 12%\n'''\n```\n", "meta": {"hexsha": "323182fe23713b15c4fe331ed06fae442775f6d8", "size": 952354, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "NYC Final.ipynb", "max_stars_repo_name": "Abhinav1004/Taxi-Demand-Prediction", "max_stars_repo_head_hexsha": "00a60527254444473a00f9953590112066f3932f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "NYC Final.ipynb", "max_issues_repo_name": "Abhinav1004/Taxi-Demand-Prediction", "max_issues_repo_head_hexsha": "00a60527254444473a00f9953590112066f3932f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "NYC Final.ipynb", "max_forks_repo_name": "Abhinav1004/Taxi-Demand-Prediction", "max_forks_repo_head_hexsha": "00a60527254444473a00f9953590112066f3932f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 171.8120151542, "max_line_length": 631929, "alphanum_fraction": 0.7057102716, "converted": true, "num_tokens": 25475, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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Requirement already satisfied: mpmath>=0.19 in /usr/local/lib/python3.7/dist-packages (from sympy>=1.3->qiskit-terra==0.19.2->qiskit) (1.2.1)\n Building wheels for collected packages: qiskit, python-constraint\n Building wheel for qiskit (setup.py) ... \u001b[?25l\u001b[?25hdone\n Created wheel for qiskit: filename=qiskit-0.34.2-py3-none-any.whl size=11805 sha256=86bf40597bf6a339a290906585d04d02b59ae72f513308d837421b64aa1ea56b\n Stored in directory: /root/.cache/pip/wheels/62/77/65/cda6eedfdd2a525bd3f479a4386930ae3088a1eb01f8c944ed\n Building wheel for python-constraint (setup.py) ... \u001b[?25l\u001b[?25hdone\n Created wheel for python-constraint: filename=python_constraint-1.4.0-py2.py3-none-any.whl size=24081 sha256=e58230bcfdca1f0a25d09b7977b1ccb74deef36e568a28b35fcdb3841e3c2b58\n Stored in directory: /root/.cache/pip/wheels/07/27/db/1222c80eb1e431f3d2199c12569cb1cac60f562a451fe30479\n Successfully built qiskit python-constraint\n Installing collected packages: pbr, tweedledum, symengine, stevedore, scipy, retworkx, python-constraint, ply, ntlm-auth, cryptography, websocket-client, requests-ntlm, qiskit-terra, qiskit-ignis, qiskit-ibmq-provider, qiskit-aer, qiskit\n Attempting uninstall: scipy\n Found existing installation: scipy 1.4.1\n Uninstalling scipy-1.4.1:\n Successfully uninstalled scipy-1.4.1\n \u001b[31mERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.\n albumentations 0.1.12 requires imgaug<0.2.7,>=0.2.5, but you have imgaug 0.2.9 which is incompatible.\u001b[0m\n Successfully installed cryptography-36.0.2 ntlm-auth-1.5.0 pbr-5.8.1 ply-3.11 python-constraint-1.4.0 qiskit-0.34.2 qiskit-aer-0.10.3 qiskit-ibmq-provider-0.18.3 qiskit-ignis-0.7.0 qiskit-terra-0.19.2 requests-ntlm-1.1.0 retworkx-0.11.0 scipy-1.7.3 stevedore-3.5.0 symengine-0.9.2 tweedledum-1.1.1 websocket-client-1.3.1\n\n\n\n```python\nfrom qiskit import QuantumCircuit, assemble, Aer\n```\n\n\n```python\nfrom qiskit.visualization import plot_histogram\n```\n\n**Encoding The Input**\n\n\n\n```python\n##Using the Bit-Flip gate as 'NOT' gate of classical\nqc = QuantumCircuit(8)\n\n##We bit-flipped the 7th bit\nqc.x(7)\n```\n\n\n\n\n \n\n\n\n\n```python\n!pip install pylatexenc\n```\n\n Collecting pylatexenc\n Downloading pylatexenc-2.10.tar.gz (162 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 162 kB 5.2 MB/s \n \u001b[?25hBuilding wheels for collected packages: pylatexenc\n Building wheel for pylatexenc (setup.py) ... \u001b[?25l\u001b[?25hdone\n Created wheel for pylatexenc: filename=pylatexenc-2.10-py3-none-any.whl size=136835 sha256=1a9f16a6b036def16b4542707665be45c7324ae1e80499ada536db9e97025179\n Stored in directory: /root/.cache/pip/wheels/f1/8a/f5/33ee79d4473eb201b519fa40f989b842e373237395a3421f52\n Successfully built pylatexenc\n Installing collected packages: pylatexenc\n Successfully installed pylatexenc-2.10\n\n\n\n```python\nqc.draw()\n```\n\n\n\n\n
              \nq_0: \u2500\u2500\u2500\u2500\u2500\n\nq_1: \u2500\u2500\u2500\u2500\u2500\n\nq_2: \u2500\u2500\u2500\u2500\u2500\n\nq_3: \u2500\u2500\u2500\u2500\u2500\n\nq_4: \u2500\u2500\u2500\u2500\u2500\n\nq_5: \u2500\u2500\u2500\u2500\u2500\n\nq_6: \u2500\u2500\u2500\u2500\u2500\n     \u250c\u2500\u2500\u2500\u2510\nq_7: \u2524 X \u251c\n     \u2514\u2500\u2500\u2500\u2518
    \n\n\n\n\n```python\nqc.measure_all()\nqc.draw(initial_state=True)\n```\n\n\n\n\n
                    \u2591 \u250c\u2500\u2510                     \n  q_0: |0>\u2500\u2500\u2500\u2500\u2500\u2500\u2591\u2500\u2524M\u251c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                \u2591 \u2514\u2565\u2518\u250c\u2500\u2510                  \n  q_1: |0>\u2500\u2500\u2500\u2500\u2500\u2500\u2591\u2500\u2500\u256b\u2500\u2524M\u251c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                \u2591  \u2551 \u2514\u2565\u2518\u250c\u2500\u2510               \n  q_2: |0>\u2500\u2500\u2500\u2500\u2500\u2500\u2591\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2524M\u251c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                \u2591  \u2551  \u2551 \u2514\u2565\u2518\u250c\u2500\u2510            \n  q_3: |0>\u2500\u2500\u2500\u2500\u2500\u2500\u2591\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2524M\u251c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                \u2591  \u2551  \u2551  \u2551 \u2514\u2565\u2518\u250c\u2500\u2510         \n  q_4: |0>\u2500\u2500\u2500\u2500\u2500\u2500\u2591\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2524M\u251c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                \u2591  \u2551  \u2551  \u2551  \u2551 \u2514\u2565\u2518\u250c\u2500\u2510      \n  q_5: |0>\u2500\u2500\u2500\u2500\u2500\u2500\u2591\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2524M\u251c\u2500\u2500\u2500\u2500\u2500\u2500\n                \u2591  \u2551  \u2551  \u2551  \u2551  \u2551 \u2514\u2565\u2518\u250c\u2500\u2510   \n  q_6: |0>\u2500\u2500\u2500\u2500\u2500\u2500\u2591\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2524M\u251c\u2500\u2500\u2500\n          \u250c\u2500\u2500\u2500\u2510 \u2591  \u2551  \u2551  \u2551  \u2551  \u2551  \u2551 \u2514\u2565\u2518\u250c\u2500\u2510\n  q_7: |0>\u2524 X \u251c\u2500\u2591\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2524M\u251c\n          \u2514\u2500\u2500\u2500\u2518 \u2591  \u2551  \u2551  \u2551  \u2551  \u2551  \u2551  \u2551 \u2514\u2565\u2518\nmeas: 0 8/\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2569\u2550\u2550\u2569\u2550\u2550\u2569\u2550\u2550\u2569\u2550\u2550\u2569\u2550\u2550\u2569\u2550\u2550\u2569\u2550\u2550\u2569\u2550\n                   0  1  2  3  4  5  6  7
    \n\n\n\n\n```python\nsimul = Aer.get_backend('aer_simulator')\n```\n\n\n```python\nresult = simul.run(qc, validate=True).result()\n```\n\n\n```python\ncounts = result.get_counts()\nplot_histogram(counts)\n```\n\nThe output value 1 comes from the 7th qubit, as we had applied the bit-flip gate at the 7th qubit itself, and initially the values of all qubits are zero.\n\n\n```python\n##Using the CNOT gate to extract the output i.e. to check wheather the bits are different or the same\nqc_cnot = QuantumCircuit(2)\n\n##the CNOT gate is applied to the 2 qubits\nqc_cnot.cx(0,1)\nqc_cnot.draw()\n```\n\n\n\n\n
              \nq_0: \u2500\u2500\u25a0\u2500\u2500\n     \u250c\u2500\u2534\u2500\u2510\nq_1: \u2524 X \u251c\n     \u2514\u2500\u2500\u2500\u2518
    \n\n\n\nHere, q_0 is the control bit, and q_1 is the target bit\n\n\n```python\nqc_cnot1 = QuantumCircuit(2,2)\nqc_cnot1.x(0)\nqc_cnot1.cx(0,1)\nqc_cnot1.measure_all(1,1)\nqc_cnot1.measure_all(0,0)\nqc_cnot1.draw()\n```\n\n\n\n\n
            \u250c\u2500\u2500\u2500\u2510      \u2591 \u250c\u2500\u2510   \n   q_0: \u2524 X \u251c\u2500\u2500\u25a0\u2500\u2500\u2500\u2591\u2500\u2524M\u251c\u2500\u2500\u2500\n        \u2514\u2500\u2500\u2500\u2518\u250c\u2500\u2534\u2500\u2510 \u2591 \u2514\u2565\u2518\u250c\u2500\u2510\n   q_1: \u2500\u2500\u2500\u2500\u2500\u2524 X \u251c\u2500\u2591\u2500\u2500\u256b\u2500\u2524M\u251c\n             \u2514\u2500\u2500\u2500\u2518 \u2591  \u2551 \u2514\u2565\u2518\n   c: 2/\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256c\u2550\u2550\u256c\u2550\n                      \u2551  \u2551 \nmeas: 2/\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2569\u2550\u2550\u2569\u2550\n                      0  1
    \n\n\n\nInput (q1 q0)\tOutput (q1 q0)\n\n 00\t00\n 01\t11\n 10\t10\n 11\t01\n\nThe Output is 1 1, as q0 = 1 because we had initially introduced a bit flip operaor\n and q1 = 0\n\n\n```python\n\n```\n", "meta": {"hexsha": "fb2af86fb58efa043368c8a7c611a108cbb47302", "size": 35108, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "CNOT_gate.ipynb", "max_stars_repo_name": "parth1614/QuantumComputing-Qiskit", "max_stars_repo_head_hexsha": "24f67748266581ed02ebfbb6d9394e9cf27bc7c1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "CNOT_gate.ipynb", "max_issues_repo_name": "parth1614/QuantumComputing-Qiskit", "max_issues_repo_head_hexsha": "24f67748266581ed02ebfbb6d9394e9cf27bc7c1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "CNOT_gate.ipynb", "max_forks_repo_name": "parth1614/QuantumComputing-Qiskit", "max_forks_repo_head_hexsha": "24f67748266581ed02ebfbb6d9394e9cf27bc7c1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-03-20T19:35:25.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-20T19:35:25.000Z", "avg_line_length": 64.0656934307, "max_line_length": 10993, "alphanum_fraction": 0.6027401162, "converted": true, "num_tokens": 4611, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.626124191181315, "lm_q2_score": 0.6926419894793246, "lm_q1q2_score": 0.433679905440959}} {"text": "\n\n# Lambda School Data Science Module 132\n## Sampling, Confidence Intervals, and Hypothesis Testing\n\n## Prepare - examine other available hypothesis tests\n\nIf you had to pick a single hypothesis test in your toolbox, t-test would probably be the best choice - but the good news is you don't have to pick just one! Here's some of the others to be aware of:\n\n\n```\nimport numpy as np\nfrom scipy.stats import chisquare # One-way chi square test\n\n# Chi square can take any crosstab/table and test the independence of rows/cols\n# The null hypothesis is that the rows/cols are independent -> low chi square\n# The alternative is that there is a dependence -> high chi square\n# Be aware! Chi square does *not* tell you direction/causation\n\nind_obs = np.array([[1, 1], [2, 2]]).T\nprint(ind_obs)\nprint(chisquare(ind_obs, axis=None))\n\ndep_obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T\nprint(dep_obs)\nprint(chisquare(dep_obs, axis=None))\n```\n\n [[1 2]\n [1 2]]\n Power_divergenceResult(statistic=0.6666666666666666, pvalue=0.8810148425137847)\n [[16 32]\n [18 24]\n [16 16]\n [14 28]\n [12 20]\n [12 24]]\n Power_divergenceResult(statistic=23.31034482758621, pvalue=0.015975692534127565)\n\n\n\n```\n# Distribution tests:\n# We often assume that something is normal, but it can be important to *check*\n\n# For example, later on with predictive modeling, a typical assumption is that\n# residuals (prediction errors) are normal - checking is a good diagnostic\n\nfrom scipy.stats import normaltest\n# Poisson models arrival times and is related to the binomial (coinflip)\nsample = np.random.poisson(5, 1000)\nprint(normaltest(sample)) # Pretty clearly not normal\n```\n\n NormaltestResult(statistic=22.273646023922907, pvalue=1.4565963588062096e-05)\n\n\n\n```\n# Kruskal-Wallis H-test - compare the median rank between 2+ groups\n# Can be applied to ranking decisions/outcomes/recommendations\n# The underlying math comes from chi-square distribution, and is best for n>5\nfrom scipy.stats import kruskal\n\nx1 = [1, 3, 5, 7, 9]\ny1 = [2, 4, 6, 8, 10]\nprint(kruskal(x1, y1)) # x1 is a little better, but not \"significantly\" so\n\nx2 = [1, 1, 1]\ny2 = [2, 2, 2]\nz = [2, 2] # Hey, a third group, and of different size!\nprint(kruskal(x2, y2, z)) # x clearly dominates\n```\n\n KruskalResult(statistic=0.2727272727272734, pvalue=0.6015081344405895)\n KruskalResult(statistic=7.0, pvalue=0.0301973834223185)\n\n\nAnd there's many more! `scipy.stats` is fairly comprehensive, though there are even more available if you delve into the extended world of statistics packages. As tests get increasingly obscure and specialized, the importance of knowing them by heart becomes small - but being able to look them up and figure them out when they *are* relevant is still important.\n\n## T-test Assumptions\n\n\n```\nfrom scipy.stats import ttest_ind\n\n?ttest_ind\n```\n\n\n\n- Independence of means\n\nAre the means of our voting data independent (do not affect the outcome of one another)?\n \nThe best way to increase thel likelihood of our means being independent is to randomly sample (which we did not do).\n\n- \"Homogeneity\" of Variance? \n\nIs the magnitude of the variance between the two roughly the same?\n\nI think we're OK on this one for the voting data, although it probably could be better, one party was larger than the other.\n\nIf we suspect this to be a problem then we can use Welch's T-test\n\n- \"Dependent Variable\" (sample means) are Distributed Normally\n\n\n\nLots of statistical tests depend on normal distributions. We can test for normality using Scipy as was shown above.\n\nThis assumption is often assumed even if the assumption is a weak one. If you strongly suspect that things are not normally distributed, you can transform your data to get it looking more normal and then run your test. This problem typically goes away for large sample sizes (yay Central Limit Theorem) and is often why you don't hear it brought up. People declare the assumption to be satisfied either way. \n\n\n\n## Central Limit Theorem\n\n\n\n\n```\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\n\nN = 3000\nsample_means = []\nfor x in range(0, N):\n coinflips = np.random.binomial(n=1, p=.5, size=100)\n one_sample = coinflips\n sample_means.append(coinflips.mean())\n\nprint(len(sample_means))\nprint(sample_means)\n```\n\n 3000\n [0.45, 0.52, 0.45, 0.51, 0.45, 0.58, 0.49, 0.49, 0.48, 0.4, 0.41, 0.49, 0.44, 0.49, 0.49, 0.49, 0.44, 0.45, 0.41, 0.48, 0.42, 0.49, 0.53, 0.41, 0.51, 0.57, 0.54, 0.54, 0.47, 0.56, 0.49, 0.54, 0.45, 0.55, 0.46, 0.51, 0.59, 0.51, 0.59, 0.56, 0.59, 0.47, 0.59, 0.41, 0.41, 0.46, 0.56, 0.49, 0.4, 0.44, 0.37, 0.51, 0.51, 0.53, 0.54, 0.4, 0.47, 0.39, 0.53, 0.51, 0.44, 0.55, 0.43, 0.55, 0.52, 0.44, 0.43, 0.45, 0.56, 0.54, 0.47, 0.46, 0.54, 0.53, 0.61, 0.52, 0.56, 0.5, 0.49, 0.55, 0.51, 0.5, 0.5, 0.52, 0.51, 0.46, 0.49, 0.42, 0.52, 0.46, 0.48, 0.43, 0.54, 0.47, 0.59, 0.43, 0.56, 0.47, 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0.53, 0.56, 0.48, 0.52, 0.42, 0.49, 0.5, 0.49, 0.53, 0.55, 0.57, 0.56, 0.46, 0.52, 0.54, 0.55, 0.5, 0.49, 0.6, 0.48, 0.44, 0.41, 0.48, 0.46, 0.52, 0.49, 0.51, 0.51, 0.56, 0.51, 0.45, 0.51, 0.47, 0.53, 0.54, 0.49, 0.48, 0.53, 0.42, 0.52, 0.5, 0.42, 0.64, 0.53, 0.52, 0.48, 0.44, 0.5, 0.51, 0.54, 0.5, 0.37, 0.52, 0.56, 0.51, 0.54, 0.56, 0.5, 0.52, 0.52, 0.57, 0.5, 0.52, 0.52, 0.47, 0.57, 0.48, 0.44, 0.45, 0.5, 0.47, 0.53, 0.43, 0.5, 0.62, 0.49, 0.51, 0.47, 0.49, 0.57, 0.51, 0.4, 0.57, 0.49, 0.43, 0.54, 0.39, 0.48, 0.53, 0.49, 0.57, 0.49, 0.52, 0.56, 0.48, 0.47, 0.46, 0.51, 0.51, 0.5, 0.61, 0.53, 0.51, 0.45, 0.48, 0.44, 0.59, 0.42, 0.52, 0.6, 0.51, 0.53, 0.5, 0.43, 0.58, 0.56, 0.57, 0.51, 0.53, 0.49, 0.53, 0.57, 0.54, 0.56, 0.44, 0.52, 0.45, 0.48, 0.49, 0.41, 0.46, 0.47, 0.6, 0.53, 0.47, 0.56, 0.52, 0.56, 0.49, 0.49, 0.52, 0.52, 0.49, 0.59, 0.43, 0.61, 0.53, 0.43, 0.55, 0.45, 0.54, 0.57, 0.49, 0.51, 0.49, 0.53, 0.59, 0.47, 0.45, 0.53, 0.41, 0.46, 0.48, 0.47, 0.53, 0.56, 0.56, 0.55, 0.53, 0.48, 0.45, 0.5, 0.54, 0.5, 0.52, 0.36, 0.47, 0.53, 0.41, 0.49, 0.45, 0.48, 0.49, 0.5, 0.54, 0.43, 0.43, 0.45, 0.48, 0.49, 0.49, 0.54, 0.41, 0.47, 0.56, 0.49, 0.55, 0.39, 0.57, 0.46, 0.47, 0.53, 0.46, 0.45, 0.59, 0.45, 0.48, 0.52, 0.46, 0.5, 0.5, 0.52, 0.51, 0.52, 0.55, 0.49, 0.53, 0.42, 0.58, 0.47, 0.47, 0.56, 0.44, 0.55, 0.39, 0.5, 0.42, 0.59, 0.46, 0.56, 0.52, 0.45, 0.56, 0.52, 0.6, 0.44, 0.49, 0.5, 0.52, 0.47, 0.54, 0.46, 0.56, 0.48, 0.54, 0.45, 0.48, 0.5, 0.54, 0.5, 0.48, 0.5, 0.41, 0.46, 0.54, 0.57, 0.54, 0.5, 0.4, 0.49, 0.5, 0.44, 0.48, 0.5, 0.5, 0.47, 0.5, 0.51, 0.46, 0.53, 0.52, 0.51, 0.48, 0.52, 0.57, 0.42, 0.55, 0.48, 0.5, 0.55, 0.45, 0.53, 0.44, 0.53, 0.39, 0.44, 0.58, 0.44, 0.44, 0.39, 0.47, 0.52, 0.5, 0.49, 0.46, 0.46, 0.52, 0.51, 0.47, 0.6, 0.44, 0.49, 0.56, 0.54, 0.6, 0.52, 0.51, 0.56, 0.47, 0.54, 0.54, 0.49, 0.49, 0.55, 0.51, 0.48, 0.49, 0.43, 0.38, 0.45, 0.6, 0.54, 0.55, 0.46, 0.43, 0.48, 0.46, 0.48, 0.52, 0.52, 0.54, 0.56, 0.45, 0.52, 0.43, 0.58, 0.57, 0.44, 0.51, 0.5, 0.48, 0.46, 0.52, 0.53, 0.44, 0.49, 0.43, 0.52, 0.5, 0.46, 0.51, 0.51, 0.41, 0.53, 0.44, 0.48, 0.55, 0.49, 0.57, 0.54, 0.56, 0.54, 0.5, 0.48, 0.51, 0.5, 0.44]\n\n\n\n```\n# Create dataframe with single coin flip\n\ndf = pd.DataFrame({'one-samp': one_sample})\ndf.head()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    one-samp
    01
    11
    21
    30
    41
    \n
    \n\n\n\n\n```\n# Plot histogram to look at distribution of a single coin flip \n\ndf.hist();\n```\n\n\n```\n# Plot histogram to look at distribution of all coin flips\n\nax = plt.hist(sample_means, bins=30)\nplt.title(f'Distribution of {N} sample means \\n (of 30 coinflips each)');\n```\n\nWhat does the Central Limit Theorem State? That no matter the initial distribution of the population, the distribution of sample means taken will approximate a normal distribution as $n \\rightarrow \\infty$.\n\nThis has very important implications for hypothesis testing and is precisely the reason why the t-distribution begins to approximate the normal distribution as our sample size increases. \n\n## Standard Error of the Mean\n\nWhat does it mean to \"estimate\"? the Population mean?\n\n\n```\nimport numpy as np\nimport pandas as pd\n\n# Average Height\nmu = 70\nsigma = 3\n\nlambda_heights = np.random.normal(mu, sigma, 2000)\nprint(len(lambda_heights))\nlambda_heights\n```\n\n 2000\n\n\n\n\n\n array([71.51294399, 66.75282069, 67.59159409, ..., 71.10525723,\n 71.94880877, 69.13737422])\n\n\n\n\n```\nimport seaborn as sns\n\nsns.distplot(lambda_heights)\nplt.title('Distribution of Heights (in inches)');\n```\n\n\n```\nprint(\"Population Mean:\", lambda_heights.mean())\nprint(\"Population Standard Deviation:\", lambda_heights.std())\n```\n\n Population Mean: 70.0041524330643\n Population Standard Deviation: 3.0172901619538237\n\n\n\n```\npopulation = pd.DataFrame({'heights': lambda_heights})\nprint(population.shape)\npopulation.head()\n```\n\n (2000, 1)\n\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    heights
    071.512944
    166.752821
    267.591594
    370.164178
    476.079145
    \n
    \n\n\n\n\n```\n# Take a random sample and print sample mean\n\nsample1 = population.sample(100)\nprint(sample1.shape)\nsample1.head()\n```\n\n (100, 1)\n\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    heights
    10068.989625
    121776.369801
    29670.072318
    68770.283488
    8571.055414
    \n
    \n\n\n\n\n```\nprint('Sample Mean #1: ', sample1['heights'].mean())\n```\n\n Sample Mean #1: 70.2468623482625\n\n\n\n```\n# Take a different random sample and print sample mean\n\nsample2 = population.sample(100)\nprint(sample1.shape)\nsample2.head()\n```\n\n (100, 1)\n\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    heights
    53774.441399
    183267.811208
    134670.763693
    147971.480101
    22969.303825
    \n
    \n\n\n\n\n```\nprint('Sample Mean #2: ', sample2['heights'].mean())\n```\n\n Sample Mean #2: 70.25036871624675\n\n\n## Build and Interpret a Confidence Interval\n\n\n\n\n```\ncoinflips_100 = np.random.binomial(n=1, p=0.5, size=10000)\n\nsample_std = np.std(coinflips_100)\nprint('Sample St Dev: ', sample_std)\nsample_size = len(coinflips_100)\nprint('Sample Size: ', sample_size)\n```\n\n Sample St Dev: 0.4999963899869677\n Sample Size: 10000\n\n\n\n```\nstandard_error = sample_std/np.sqrt(sample_size)\nprint(standard_error)\n```\n\n 0.0049999638998696775\n\n\n### What confidence level do we want our confidence interval to represent?\n\n95% confidence Interval? 99% confidence interval? \n\n\n```\nimport scipy.stats as stats\n```\n\n\n```\nt = stats.t.ppf(0.975, sample_size-1)\nt\n```\n\n\n\n\n 1.9602012636213575\n\n\n\n\n```\nsample_mean = coinflips_100.mean()\nconfidence_interval = (sample_mean - t*standard_error, sample_mean + t*standard_error)\nmargin_of_error = t*standard_error\n\nprint('Sample Mean: ', sample_mean)\nprint('Margin of Error: ', margin_of_error)\nprint('Confidence Interval: ', confidence_interval)\n```\n\n Sample Mean: 0.4981\n Margin of Error: 0.009800935554585713\n Confidence Interval: (0.48829906444541427, 0.5079009355545857)\n\n\n## Graphically Represent a Confidence Interval\n\n\n```\nimport seaborn as sns\n\nsns.kdeplot(coinflips_100)\nplt.axvline(x=sample_mean, color='k')\nplt.axvline(x=confidence_interval[0], color='r')\nplt.axvline(x=confidence_interval[1], color='r')\n```\n\n## Relationship between Confidence Intervals and T-tests\n\nConfidence Interval == Bounds of statistical significance for our t-test\n\nA sample mean that falls inside of our confidence interval will \"FAIL TO REJECT\" our null hypothesis\n\nA sample mean that falls outside of our confidence interval will \"REJECT\" our null hypothesis\n\n\n```\nfrom scipy.stats import t, ttest_1samp\n```\n\n\n```\nimport numpy as np\n\ncoinflip_means = []\nfor x in range(0,100):\n coinflips = np.random.binomial(n=1, p=.5, size=30)\n coinflip_means.append(coinflips.mean())\n\nprint(coinflip_means)\n```\n\n [0.6333333333333333, 0.4666666666666667, 0.43333333333333335, 0.5333333333333333, 0.6333333333333333, 0.5333333333333333, 0.5, 0.4666666666666667, 0.4666666666666667, 0.5, 0.4, 0.6, 0.5333333333333333, 0.5666666666666667, 0.3333333333333333, 0.5333333333333333, 0.4, 0.5, 0.4666666666666667, 0.5, 0.6333333333333333, 0.3333333333333333, 0.5666666666666667, 0.5666666666666667, 0.43333333333333335, 0.5333333333333333, 0.5, 0.6666666666666666, 0.5333333333333333, 0.5333333333333333, 0.6, 0.36666666666666664, 0.5333333333333333, 0.5333333333333333, 0.43333333333333335, 0.43333333333333335, 0.43333333333333335, 0.5666666666666667, 0.3, 0.43333333333333335, 0.43333333333333335, 0.43333333333333335, 0.5333333333333333, 0.4666666666666667, 0.4, 0.6333333333333333, 0.4, 0.5666666666666667, 0.5666666666666667, 0.5333333333333333, 0.5666666666666667, 0.5333333333333333, 0.4666666666666667, 0.5333333333333333, 0.5, 0.5666666666666667, 0.43333333333333335, 0.4666666666666667, 0.5, 0.43333333333333335, 0.5666666666666667, 0.5, 0.6, 0.6, 0.4666666666666667, 0.5, 0.36666666666666664, 0.4666666666666667, 0.5666666666666667, 0.5333333333333333, 0.4666666666666667, 0.43333333333333335, 0.5333333333333333, 0.5, 0.6333333333333333, 0.6333333333333333, 0.43333333333333335, 0.43333333333333335, 0.4, 0.43333333333333335, 0.6, 0.5666666666666667, 0.3, 0.6333333333333333, 0.36666666666666664, 0.5666666666666667, 0.43333333333333335, 0.43333333333333335, 0.4666666666666667, 0.43333333333333335, 0.6666666666666666, 0.4666666666666667, 0.6666666666666666, 0.4666666666666667, 0.6, 0.36666666666666664, 0.5333333333333333, 0.43333333333333335, 0.5666666666666667, 0.5]\n\n\n\n```\nnp.mean(coinflip_means)\n```\n\n\n\n\n 0.5003333333333333\n\n\n\n\n```\n# 95% confidence interval\n\nt_stat = stats.t.ppf(0.975, 99)\nprint('T statistic: ', t_stat)\n\nstd_sample = np.std(coinflip_means)\nstd_err = std_sample/np.sqrt(len(coinflip_means))\n\nCI = stats.t.interval(0.95, 99, loc=np.mean(coinflip_means), scale=std_err)\nprint('95% confidence interval: ', CI)\n```\n\n T statistic: 1.9842169515086827\n 95% confidence interval: (0.48379832435242015, 0.5168683423142464)\n\n\nA null hypothesis that's just inside of our confidence interval == fail to reject\n\n\n\n\n```\nttest_1samp(coinflip_means, 0.51686)\n```\n\n\n\n\n Ttest_1sampResult(statistic=-1.973274871531969, pvalue=0.05125222761123825)\n\n\n\nA null hypothesis that's just outside of our confidence interval == reject\n\n\n\n\n```\nttest_1samp(coinflip_means, 0.52686)\n```\n\n\n\n\n Ttest_1sampResult(statistic=-3.1672693480539342, pvalue=0.0020463163356837315)\n\n\n\n\n```\ndef confidence_interval(data, confidence=0.95):\n \"\"\"\n Calculate a confidence interval around a sample mean for given data.\n Using t-distribution and two-tailed test, default 95% confidence. \n \n Arguments:\n data - iterable (list or numpy array) of sample observations\n confidence - level of confidence for the interval\n \n Returns:\n tuple of (mean, lower bound, upper bound)\n \"\"\"\n data = np.array(data)\n mean = np.mean(data)\n n = len(data)\n stderr = stats.sem(data)\n interval = stderr * stats.t.ppf((1 + confidence) / 2.0, n - 1)\n return (mean, mean - interval, mean + interval)\n```\n\n## Run a $\\chi^{2}$ Test \"by hand\" (Using Numpy)\n\n\n```\ndf = pd.read_csv('https://raw.githubusercontent.com/ryanleeallred/datasets/master/adult.csv', na_values=\" ?\")\nprint(df.shape)\ndf.head()\n```\n\n (32561, 15)\n\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    ageworkclassfnlwgteducationeducation-nummarital-statusoccupationrelationshipracesexcapital-gaincapital-losshours-per-weekcountrysalary
    039State-gov77516Bachelors13Never-marriedAdm-clericalNot-in-familyWhiteMale2174040United-States<=50K
    150Self-emp-not-inc83311Bachelors13Married-civ-spouseExec-managerialHusbandWhiteMale0013United-States<=50K
    238Private215646HS-grad9DivorcedHandlers-cleanersNot-in-familyWhiteMale0040United-States<=50K
    353Private23472111th7Married-civ-spouseHandlers-cleanersHusbandBlackMale0040United-States<=50K
    428Private338409Bachelors13Married-civ-spouseProf-specialtyWifeBlackFemale0040Cuba<=50K
    \n
    \n\n\n\n\n```\ndf.corr()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    agefnlwgteducation-numcapital-gaincapital-losshours-per-week
    age1.000000-0.0766460.0365270.0776740.0577750.068756
    fnlwgt-0.0766461.000000-0.0431950.000432-0.010252-0.018768
    education-num0.036527-0.0431951.0000000.1226300.0799230.148123
    capital-gain0.0776740.0004320.1226301.000000-0.0316150.078409
    capital-loss0.057775-0.0102520.079923-0.0316151.0000000.054256
    hours-per-week0.068756-0.0187680.1481230.0784090.0542561.000000
    \n
    \n\n\n\n\n```\ndf['hours-per-week'].hist()\n```\n\n\n```\ndf.describe(exclude='number')\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    workclasseducationmarital-statusoccupationrelationshipracesexcountrysalary
    count307253256132561307183256132561325613197832561
    unique816714652412
    topPrivateHS-gradMarried-civ-spouseProf-specialtyHusbandWhiteMaleUnited-States<=50K
    freq22696105011497641401319327816217902917024720
    \n
    \n\n\n\n\n```\ncut_points = [0, 9, 19, 29, 39, 49, 500]\nlabel_names = ['0-9', '10-19', '20-29', '30-39', '40-49', '50+']\ndf['hours_per_week_categories'] = pd.cut(df['hours-per-week'], cut_points, labels=label_names)\ndf.hours_per_week_categories.value_counts()\n```\n\n\n\n\n 40-49 18336\n 50+ 6462\n 30-39 3667\n 20-29 2392\n 10-19 1246\n 0-9 458\n Name: hours_per_week_categories, dtype: int64\n\n\n\n\n```\ndf.sex.value_counts()\n```\n\n\n\n\n Male 21790\n Female 10771\n Name: sex, dtype: int64\n\n\n\n\n```\ndf = df.sort_values(by='hours_per_week_categories')\ncontingency_table = pd.crosstab(df['sex'], df['hours_per_week_categories'], margins=True)\ncontingency_table\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    hours_per_week_categories0-910-1920-2930-3940-4950+All
    sex
    Female235671128719145636102810771
    Male2235751105175312700543421790
    All45812462392366718336646232561
    \n
    \n\n\n\n## Expected Value Calculation\n\\begin{align}\nexpected_{i,j} =\\frac{(row_{i} \\text{total})(column_{j} \\text{total}) }{(\\text{total observations})} \n\\end{align}\n\n\n```\nrow_sums = contingency_table.iloc[0:2, 6].values\ncol_sums = contingency_table.iloc[2, 0:6].values\n\nprint(row_sums)\nprint(col_sums)\n```\n\n [10771 21790]\n [ 458 1246 2392 3667 18336 6462]\n\n\n\n```\ntotal = contingency_table.loc['All', 'All']\ntotal\n```\n\n\n\n\n 32561\n\n\n\n\n```\nexpected = []\nfor i in range(len(row_sums)):\n expected_row = []\n for column in col_sums:\n expected_val = column*row_sums[i]/total\n expected_row.append(expected_val)\n expected.append(expected_row)\n \nexpected = np.array(expected)\nprint(expected.shape)\nprint(expected)\n```\n\n (2, 6)\n [[ 151.50388502 412.16995793 791.26046497 1213.02346365\n 6065.44811277 2137.59411566]\n [ 306.49611498 833.83004207 1600.73953503 2453.97653635\n 12270.55188723 4324.40588434]]\n\n\n\n```\ncontingency_table\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    hours_per_week_categories0-910-1920-2930-3940-4950+All
    sex
    Female235671128719145636102810771
    Male2235751105175312700543421790
    All45812462392366718336646232561
    \n
    \n\n\n\n## Chi-Squared Statistic with Numpy\n\n\\begin{align}\n\\chi^2 = \\sum \\frac{(observed_{i}-expected_{i})^2}{(expected_{i})}\n\\end{align}\n\nFor the $observed$ values we will just use a version of our contingency table without the margins as a numpy array. In this way, if our observed values array and our expected values array are the same shape, then we can subtract them and divide them directly which makes the calculations a lot cleaner. No for loops!\n\n\n```\nobserved = pd.crosstab(df['sex'], df['hours_per_week_categories']).values\nprint(observed.shape)\nobserved\n```\n\n (2, 6)\n\n\n\n\n\n array([[ 235, 671, 1287, 1914, 5636, 1028],\n [ 223, 575, 1105, 1753, 12700, 5434]])\n\n\n\n\n```\nchi_square = ((observed - expected)**2/(expected)).sum()\nchi_square\n```\n\n\n\n\n 2287.190943926107\n\n\n\n## Run a $\\chi^{2}$ Test using Scipy\n\n\n```\nchi_squared, p_value, dof, expected = stats.chi2_contingency(observed)\nprint(chi_squared, p_value, dof, expected)\n```\n\n 2287.190943926107 0.0 5 [[ 151.50388502 412.16995793 791.26046497 1213.02346365\n 6065.44811277 2137.59411566]\n [ 306.49611498 833.83004207 1600.73953503 2453.97653635\n 12270.55188723 4324.40588434]]\n\n\nNull Hypothesis: Hours worked per week bins is **independent** of sex. \n\nDue to a p-value of 0, we REJECT the null hypothesis that hours worked per week and sex are independent, and conclude that there is an association between hours worked per week and sex. \n\n## Assignment - Build a confidence interval\n\nA confidence interval refers to a neighborhood around some point estimate, the size of which is determined by the desired p-value. For instance, we might say that 52% of Americans prefer tacos to burritos, with a 95% confidence interval of +/- 5%.\n\n52% (0.52) is the point estimate, and +/- 5% (the interval $[0.47, 0.57]$) is the confidence interval. \"95% confidence\" means a p-value $\\leq 1 - 0.95 = 0.05$.\n\nIn this case, the confidence interval includes $0.5$ - which is the natural null hypothesis (that half of Americans prefer tacos and half burritos, thus there is no clear favorite). So in this case, we could use the confidence interval to report that we've failed to reject the null hypothesis.\n\nBut providing the full analysis with a confidence interval, including a graphical representation of it, can be a helpful and powerful way to tell your story. Done well, it is also more intuitive to a layperson than simply saying \"fail to reject the null hypothesis\" - it shows that in fact the data does *not* give a single clear result (the point estimate) but a whole range of possibilities.\n\nHow is a confidence interval built, and how should it be interpreted? It does *not* mean that 95% of the data lies in that interval - instead, the frequentist interpretation is \"if we were to repeat this experiment 100 times, we would expect the average result to lie in this interval ~95 times.\"\n\nFor a 95% confidence interval and a normal(-ish) distribution, you can simply remember that +/-2 standard deviations contains 95% of the probability mass, and so the 95% confidence interval based on a given sample is centered at the mean (point estimate) and has a range of +/- 2 (or technically 1.96) standard deviations.\n\nDifferent distributions/assumptions (90% confidence, 99% confidence) will require different math, but the overall process and interpretation (with a frequentist approach) will be the same.\n\nYour assignment - using the data from the prior module ([congressional voting records](https://archive.ics.uci.edu/ml/datasets/Congressional+Voting+Records)):\n\n\n### Confidence Intervals:\n1. Generate and numerically represent a confidence interval\n2. Graphically (with a plot) represent the confidence interval\n3. Interpret the confidence interval - what does it tell you about the data and its distribution?\n\n### Chi-squared tests:\n4. Take a dataset that we have used in the past in class that has **categorical** variables. Pick two of those categorical variables and run a chi-squared tests on that data\n - By hand using Numpy\n - In a single line using Scipy\n\nStretch goals:\n\n1. Write a summary of your findings, mixing prose and math/code/results. *Note* - yes, this is by definition a political topic. It is challenging but important to keep your writing voice *neutral* and stick to the facts of the data. Data science often involves considering controversial issues, so it's important to be sensitive about them (especially if you want to publish).\n2. Apply the techniques you learned today to your project data or other data of your choice, and write/discuss your findings here.\n3. Refactor your code so it is elegant, readable, and can be easily run for all issues.\n\n\n```\n# TODO - your code!\n```\n\n## Resources\n\n- [Interactive visualize the Chi-Squared test](https://homepage.divms.uiowa.edu/~mbognar/applets/chisq.html)\n- [Calculation of Chi-Squared test statistic](https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test)\n- [Visualization of a confidence interval generated by R code](https://commons.wikimedia.org/wiki/File:Confidence-interval.svg)\n- [Expected value of a squared standard normal](https://math.stackexchange.com/questions/264061/expected-value-calculation-for-squared-normal-distribution) (it's 1 - which is why the expected value of a Chi-Squared with $n$ degrees of freedom is $n$, as it's the sum of $n$ squared standard normals)\n", "meta": {"hexsha": "1442cc5c29414f476693bae2d982d7c28cf7b833", "size": 157189, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "module2-sampling-confidence-intervals-and-hypothesis-testing/LS_DS_132_Sampling_Confidence_Intervals_and_Hypothesis_Testing_Updated.ipynb", "max_stars_repo_name": "deanhadzi/DS-Unit-1-Sprint-3-Statistical-Tests-and-Experiments", "max_stars_repo_head_hexsha": "067845d974700b261c6d02532bc06c6f73c79932", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "module2-sampling-confidence-intervals-and-hypothesis-testing/LS_DS_132_Sampling_Confidence_Intervals_and_Hypothesis_Testing_Updated.ipynb", "max_issues_repo_name": "deanhadzi/DS-Unit-1-Sprint-3-Statistical-Tests-and-Experiments", "max_issues_repo_head_hexsha": "067845d974700b261c6d02532bc06c6f73c79932", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "module2-sampling-confidence-intervals-and-hypothesis-testing/LS_DS_132_Sampling_Confidence_Intervals_and_Hypothesis_Testing_Updated.ipynb", "max_forks_repo_name": "deanhadzi/DS-Unit-1-Sprint-3-Statistical-Tests-and-Experiments", "max_forks_repo_head_hexsha": "067845d974700b261c6d02532bc06c6f73c79932", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 64.4481344813, "max_line_length": 18564, "alphanum_fraction": 0.6278047446, "converted": true, "num_tokens": 24959, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381667555713, "lm_q2_score": 0.7931059462938815, "lm_q1q2_score": 0.43362129111965936}} {"text": "```c++\n#pragma cling add_include_path(\"../../include\")\n#pragma cling add_include_path(\"../feltor/inc\") // Feltor path\n#define THRUST_DEVICE_SYSTEM THRUST_DEVICE_SYSTEM_CPP\n#include \n#include \"dg/algorithm.h\"\n// include json and netcdf?\n```\n\n In file included from input_line_8:2:\n In file included from ../feltor/inc/dg/algorithm.h:8:\n \u001b[1m../feltor/inc/dg/backend/config.h:20:9: \u001b[0m\u001b[0;1;35mwarning: \u001b[0m\u001b[1mNOTE: Fast std::fma(a,b,c) not activated! Using a*b+c instead! [-W#pragma-messages]\u001b[0m\n #pragma message( \"NOTE: Fast std::fma(a,b,c) not activated! Using a*b+c instead!\")\n \u001b[0;1;32m ^\n \u001b[0mIn file included from input_line_8:2:\n In file included from ../feltor/inc/dg/algorithm.h:11:\n In file included from ../feltor/inc/dg/topology/split_and_join.h:4:\n In file included from ../feltor/inc/dg/backend/blas1_dispatch_shared.h:12:\n In file included from ../feltor/inc/dg/backend/blas1_serial.h:6:\n In file included from ../feltor/inc/dg/backend/exblas/exdot_serial.h:25:\n In file included from ../feltor/inc/dg/backend/exblas/accumulate.h:19:\n \u001b[1m../feltor/inc/dg/backend/exblas/config.h:31:9: \u001b[0m\u001b[0;1;35mwarning: \u001b[0m\u001b[1mWARNING: Instruction set below SSE4.1! Deactivating vectorization!\n [-W#pragma-messages]\u001b[0m\n #pragma message(\"WARNING: Instruction set below SSE4.1! Deactivating vectorization!\")\n \u001b[0;1;32m ^\n \u001b[0mIn file included from input_line_8:2:\n In file included from ../feltor/inc/dg/algorithm.h:11:\n In file included from ../feltor/inc/dg/topology/split_and_join.h:4:\n In file included from ../feltor/inc/dg/backend/blas1_dispatch_shared.h:12:\n In file included from ../feltor/inc/dg/backend/blas1_serial.h:6:\n In file included from ../feltor/inc/dg/backend/exblas/exdot_serial.h:25:\n \u001b[1m../feltor/inc/dg/backend/exblas/accumulate.h:93:43: \u001b[0m\u001b[0;1;35mwarning: \u001b[0m\u001b[1mshifting a negative signed value is undefined [-Wshift-negative-value]\u001b[0m\n carrybit = (s ? 1ll << KRX : -1ll << KRX);\n \u001b[0;1;32m ~~~~ ^\n \u001b[0mIn file included from input_line_8:2:\n In file included from ../feltor/inc/dg/algorithm.h:11:\n In file included from ../feltor/inc/dg/topology/split_and_join.h:4:\n In file included from ../feltor/inc/dg/backend/blas1_dispatch_shared.h:12:\n In file included from ../feltor/inc/dg/backend/blas1_serial.h:6:\n In file included from ../feltor/inc/dg/backend/exblas/exdot_serial.h:26:\n \u001b[1m../feltor/inc/dg/backend/exblas/ExSUM.FPE.hpp:143:46: \u001b[0m\u001b[0;1;35mwarning: \u001b[0m\u001b[1munknown attribute 'optimize' ignored [-Wunknown-attributes]\u001b[0m\n template UNROLL_ATTRIBUTE\n \u001b[0;1;32m ^\n \u001b[0m\u001b[1m../feltor/inc/dg/backend/exblas/config.h:42:41: \u001b[0m\u001b[0;1;30mnote: \u001b[0mexpanded from macro 'UNROLL_ATTRIBUTE'\u001b[0m\n #define UNROLL_ATTRIBUTE __attribute__((optimize(\"unroll-loops\")))\n \u001b[0;1;32m ^\n \u001b[0mIn file included from input_line_8:2:\n In file included from ../feltor/inc/dg/algorithm.h:11:\n In file included from ../feltor/inc/dg/topology/split_and_join.h:4:\n In file included from ../feltor/inc/dg/backend/blas1_dispatch_shared.h:12:\n In file included from ../feltor/inc/dg/backend/blas1_serial.h:6:\n In file included from ../feltor/inc/dg/backend/exblas/exdot_serial.h:26:\n \u001b[1m../feltor/inc/dg/backend/exblas/ExSUM.FPE.hpp:189:46: \u001b[0m\u001b[0;1;35mwarning: \u001b[0m\u001b[1munknown attribute 'optimize' ignored [-Wunknown-attributes]\u001b[0m\n template UNROLL_ATTRIBUTE\n \u001b[0;1;32m ^\n \u001b[0m\u001b[1m../feltor/inc/dg/backend/exblas/config.h:42:41: \u001b[0m\u001b[0;1;30mnote: \u001b[0mexpanded from macro 'UNROLL_ATTRIBUTE'\u001b[0m\n #define UNROLL_ATTRIBUTE __attribute__((optimize(\"unroll-loops\")))\n \u001b[0;1;32m ^\n \u001b[0mIn file included from input_line_8:2:\n In file included from ../feltor/inc/dg/algorithm.h:11:\n In file included from ../feltor/inc/dg/topology/split_and_join.h:4:\n In file included from ../feltor/inc/dg/backend/blas1_dispatch_shared.h:12:\n In file included from ../feltor/inc/dg/backend/blas1_serial.h:6:\n In file included from ../feltor/inc/dg/backend/exblas/exdot_serial.h:26:\n \u001b[1m../feltor/inc/dg/backend/exblas/ExSUM.FPE.hpp:221:46: \u001b[0m\u001b[0;1;35mwarning: \u001b[0m\u001b[1munknown attribute 'optimize' ignored [-Wunknown-attributes]\u001b[0m\n template UNROLL_ATTRIBUTE\n \u001b[0;1;32m ^\n \u001b[0m\u001b[1m../feltor/inc/dg/backend/exblas/config.h:42:41: \u001b[0m\u001b[0;1;30mnote: \u001b[0mexpanded from macro 'UNROLL_ATTRIBUTE'\u001b[0m\n #define UNROLL_ATTRIBUTE __attribute__((optimize(\"unroll-loops\")))\n \u001b[0;1;32m ^\n \u001b[0m\n\n(sec:pdes)=\n# Advanced Timesteppers\n\nWe now want to demonstrate how to use Feltor to solve partial differential equations.\nWe use the simple advection diffusion equation as a model equation\n\\begin{align}\n \\frac{\\partial \\omega}{\\partial t} &= -v\\cdot \\nabla\\omega + D \\Delta \\omega \\\\\n -\\Delta \\phi &= \\omega \\\\\n v_x &:= -\\partial_y \\phi \\\\\n v_y &:= \\partial_x \\phi\n\\end{align}\n\n## Explicit stepping\nAs long as the diffusion coefficient is small enough to not influence the CFL condition we can compute everything explicitly. In Feltor we simply need a functor implementing the right hand side like we did in the\nlast chapter. We here repeat the building blocks that regard the timestepper:\n\n```cpp\ntemplate\nstruct Equations\n{\n void operator()(double t, const Container& omega, Container& omegaDot)\n {\n // solve Poisson equation\n // implement advection term\n // implement diffusion term\n }\n};\n// Construct init condition\nomega = myproject::initial_conditions(grid, js[\"init\"] );\n\n// Construct Equations\nmyproject::Equations rhs( grid, js);\n\n// The timestepper\ndg::Adaptive< dg::ERKStep< dg::x::DVec>> adapt(tableau, omega);\n\n// The timeloop\ndg::AdaptiveTimeloop timeloop( adapt, rhs, \n dg::pid_control, dg::l2norm, rtol, atol);\nfor( unsigned u=1; u<=maxout; u++)\n{\n \n timeloop.integrate( time, omega, u*deltaT, omega,\n u < maxout ? dg::to::at_least : dg::to::exact);\n // ...\n}\n```\n\n## Explicit advection - implicit diffusion\nWe want to split the PDE into two parts: $E(\\phi, \\omega) = -\\vec v \\cdot \\nabla \\omega$ and $I(\\omega) = D\\Delta\\omega$ with $\\phi = S(\\omega) = \\Delta^{-1}\\omega$.\nWe intend to use a semi-implicit time integrator:\n```cpp\ntemplate\nstruct Explicit\n{\n Explicit(...){}\n void operator()( double t, const Container& omega,\n Container& k)\n {\n // Solve Phi = S(omega) with e.g. Multigrid\n // Compute k = E(phi,omega)\n }\n void implicit_part( double t, const Container& omega,\n Container& k)\n {\n // Compute k = D Delta omega\n }\n};\n```\n```{note}\nThere are several possibilities to partition the explicit, implicit and solver parts into functors (or lambdas in the main program). Generally, it is a good idea to keep all equation related functionality in one class. Since we cannot overload the `operator()` twice we reverted to a little trick, writing the `implicit_part` method that we later bind to the `operator()` of the Implicit class below.\n```\n```cpp\ntemplate\nstruct Implicit\n{\n Implicit( Explicit<...>& exp, ...): m_exp(exp) {...}\n void operator()( double t, const Container& omega, Container& k)\n {\n m_exp->implicit_part( t, omega, k); \n }\n```\n```{note}\nFor the solve method we here chose a PCG solver since the Laplace is self-adjoint. Note, how we used a small lambda wrapper to compute the implicit left hand side. \n\nTypically, the solver would also write some information about its performance to `std::cout` so that a user is kept informed about the status of the integration.\n```\n```cpp\n void operator()( double alpha, double t, Container& omega, const Container& rhs)\n {\n auto wrapper = [=]( const auto& x, auto& y){\n // x - a I (x,t)\n operator()( t, x, y); // calls the above operator\n dg::blas1::axpby( 1., x, -alpha, y);\n };\n dg::blas1::copy( rhs, omega); // use rhs as initial guess\n unsigned number = m_pcg.solve( wrapper, omega, rhs, 1., m_weights, m_eps_time);\n }\n private:\n Explicit& m_exp;\n dg::PCG< Container> m_pcg;\n Container m_weights;\n value_type m_eps_time;\n};\n\n// Construct equations\nExplicit<...> ex( ...);\nImplicit<...> im(ex, ...);\n\n// The timestepper\ndg::Adaptive< dg::ARKStep< dg::x::DVec>> adapt(tableau, omega);\n\n// The timeloop\ndg::AdaptiveTimeloop timeloop( adapt, std::tie( ex, im, im), \n dg::pid_control, dg::l2norm, rtol, atol);\n```\n\n## Implicit advection-diffusion solver\nIn order to solve the entire system implicitly we have to write both equations and Solvers.\nTo make it more clear let us reformulate the structure of equations that we have\n\\begin{align}\n \\dot \\omega &= I(\\omega, \\phi)\\\\\n 0 &= R(\\omega, \\phi)\n\\end{align}\nwith\n\\begin{align}\nI( \\omega, \\phi) &= -\\vec v \\cdot \\nabla \\omega + D\\Delta\\omega \\\\\nR( \\omega, \\phi) &= \\Delta \\phi + \\omega\n\\end{align}\nTo implement an implicit timestepper we need to solve the equation (the mass matrix is the identity)\n\\begin{align}\n\\begin{cases}\n \\omega - \\alpha I(\\omega, \\phi) &= \\omega^* \\\\\n \\omega+\\Delta\\phi &= 0\n\\end{cases}\n\\end{align}\nWe can solve this equation for $\\omega$ by first solving\n\\begin{align}\n -\\Delta\\phi - \\alpha I(-\\Delta\\phi, \\phi) = \\omega^*\n\\end{align}\nfor $\\phi$ and then using $\\omega = -\\Delta\\phi$.\n```{note}\nThe solution for $\\phi$ is the same for both variants but the solution for $\\omega$ is not (numerically). This can be seen by setting $\\alpha=0$. Then in the first version $\\omega=\\omega^*$ but in the second version $\\omega=-\\Delta\\phi = \\omega^* + \\mathcal O(\\epsilon)$, depends on how well the equation is solved. For this reason the accuracy of the implicit solver should be well higher than the accuracy of the timestepper.\n```\n\nSince these equations are non-linear the solver needs to be a non-linear solver. Our idea is to use a multigrid FAS solver. For this solver we need to implement both the operator as well as its inverse on multiple grids.\n\n```cpp\ntemplate\nstruct Equations //corresponds to I\n{\n void rhs( double t, const Container& omega, const Container& phi, Container& omegaDot)\n {\n // compute: omegaDot = I(omega, phi, t)\n // implement advection term\n // implement diffusion term\n }\n void compute_omega( const Container& phi, Container& omega)\n {\n // compute -Delta phi\n dg::blas2::symv( m_lapM, phi, omega);\n }\n};\ntemplate\nstruct Implicit\n{\n Implicit(...)\n {\n // constructed nested grids\n ...\n for ( unsigned u=0; u m_nested;\n Container m_phi;\n std::vector m_weights;\n std::vector> m_eqs;\n std::vector< std::function> m_imp, m_inv_imp;\n};\n\n// Construct init condition\nomega = myproject::initial_conditions(grid, js[\"init\"] );\n\n// Construct Equations\nmyproject::Equations rhs( grid, js);\n\n// The timestepper\ndg::Adaptive< dg::DIRKStep< dg::x::DVec>> adapt(tableau, omega);\n\n// The timeloop\ndg::AdaptiveTimeloop timeloop( adapt, std::tie(imp,imp), \n dg::pid_control, dg::l2norm, rtol, atol);\nfor( unsigned u=1; u<=maxout; u++)\n{\n \n timeloop.integrate( time, omega, u*deltaT, omega,\n u < maxout ? dg::to::at_least : dg::to::exact);\n // ...\n}\n```\n\n## Implicit mass-matrix - timestepper\n Consider the following general equation:\n\\begin{align}\n M(y,t)\\frac{d y}{d t} &= F(y,t)\n\\end{align}\nWe intend to solve the resulting implicit equation with multigrid nested iteration. We thus have to implement and solve the following operators on multiply grids:\n\\begin{align}\n M(y,t) k &= F(y,t) \\\\\n M(y,t) ( y - y^* ) - \\alpha F(y,t) &= 0 \n\\end{align}\nTo simplify the implementation we restrict ourselves to solvers that\nuse the implicit solve only (i.e. Multistep, symplectic DIRK and SDIRK timesteppers).\n\n```cpp\ntemplate<...>\nstruct Equations\n{\n Equations(...){}\n void mass_matrix( double a, double t, const Container& y,\n const Container& k, \n double b, Container& result)\n {\n // Compute result = a M(y,t) k + b result\n //...\n }\n void rhs( double t, const Container& y, Container& k)\n {\n // Compute k = F(y,t)\n }\n};\ntemplate< ...>\nclass ImplicitSolver\n{\n ImplicitSolver( ...){\n // Construct nested grids\n ...\n // Construct nested objects m_eqs\n ...\n // Construct nested Operators and Solvers:\n for( unsigend u=0; u m_nested;\n std::vector m_weights, m_ys;\n std::vector> m_eqs;\n std::vector< std::function> m_imp, m_inv_imp;\n};\n\n// in main\nImplicitSolver<...> imp(...);\ndg::ImplicitMultistep multistep(\"BDF-3-3\", y0);\nmultistep.init( std::tie( imp, imp), t0, y0, dt);\nmultistep.step( std::tie( imp, imp), t0, y0);\n```\n\n\n\n```c++\n\n```\n", "meta": {"hexsha": "d8993280c1e743bd17c2e31c26c86db1a55ad588", "size": 21751, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "timesteppers2.ipynb", "max_stars_repo_name": "feltor-dev/user-guide", "max_stars_repo_head_hexsha": "27b296a1e262c2bfcd95fb9c405ad01ccdfdfc74", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-01-13T20:16:48.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-13T20:16:48.000Z", "max_issues_repo_path": "timesteppers2.ipynb", "max_issues_repo_name": "feltor-dev/user-guide", "max_issues_repo_head_hexsha": "27b296a1e262c2bfcd95fb9c405ad01ccdfdfc74", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "timesteppers2.ipynb", "max_forks_repo_name": "feltor-dev/user-guide", "max_forks_repo_head_hexsha": "27b296a1e262c2bfcd95fb9c405ad01ccdfdfc74", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 45.7915789474, "max_line_length": 447, "alphanum_fraction": 0.5386878764, "converted": true, "num_tokens": 4701, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.658417487156366, "lm_q1q2_score": 0.43351359621959856}} {"text": "```python\n!pip install pennylane\r\nfrom IPython.display import clear_output\r\nclear_output()\n```\n\n\n```python\n# This cell is added by sphinx-gallery\n# It can be customized to whatever you like\n%matplotlib inline\n```\n\n\n\nData-reuploading classifier\n===========================\n*Author: Shahnawaz Ahmed (shahnawaz.ahmed95@gmail.com)*\n\n.. meta::\n :property=\"og:description\": Implement a single-qubit universal quantum classifier using PennyLane.\n :property=\"og:image\": https://pennylane.ai/qml/_images/universal_dnn1.png\n\n.. related::\n\n tutorial_variational_classifier Variational quantum classifier\n tutorial_multiclass_classification Multiclass margin classifier\n tutorial_expressivity_fourier_series Quantum models as Fourier series\n\nA single-qubit quantum circuit which can implement arbitrary unitary\noperations can be used as a universal classifier much like a single\nhidden-layered Neural Network. As surprising as it sounds,\n`P\u00e9rez-Salinas et al. (2019) `_\ndiscuss this with their idea of 'data\nreuploading'. It is possible to load a single qubit with arbitrary\ndimensional data and then use it as a universal classifier.\n\nIn this example, we will implement this idea with Pennylane - a\npython based tool for quantum machine learning, automatic\ndifferentiation, and optimization of hybrid quantum-classical\ncomputations.\n\nBackground\n----------\n\nWe consider a simple classification problem and will train a\nsingle-qubit variational quantum circuit to achieve this goal. The data\nis generated as a set of random points in a plane $(x_1, x_2)$ and\nlabeled as 1 (blue) or 0 (red) depending on whether they lie inside or\noutside a circle. The goal is to train a quantum circuit to predict the\nlabel (red or blue) given an input point\u2019s coordinate.\n\n.. figure:: ../demonstrations/data_reuploading/universal_circles.png\n :scale: 65%\n :alt: circles\n\n\nTransforming quantum states using unitary operations\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\nA single-qubit quantum state is characterized by a two-dimensional state\nvector and can be visualized as a point in the so-called Bloch sphere.\nInstead of just being a 0 (up) or 1 (down), it can exist in a\nsuperposition with say 30% chance of being in the $|0 \\rangle$ and\n70% chance of being in the $|1 \\rangle$ state. This is represented\nby a state vector $|\\psi \\rangle = 0.3|0 \\rangle + 0.7|1 \\rangle$ -\nthe probability \"amplitude\" of the quantum state. In general we can take\na vector $(\\alpha, \\beta)$ to represent the probabilities of a qubit\nbeing in a particular state and visualize it on the Bloch sphere as an\narrow.\n\n.. figure:: ../demonstrations/data_reuploading/universal_bloch.png\n :scale: 65%\n :alt: bloch\n\nData loading using unitaries\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\nIn order to load data onto a single qubit, we use a unitary operation\n$U(x_1, x_2, x_3)$ which is just a parameterized\nmatrix multiplication representing the rotation of the state vector in\nthe Bloch sphere. E.g., to load $(x_1, x_2)$ into the qubit, we\njust start from some initial state vector, $|0 \\rangle$,\napply the unitary operation $U(x_1, x_2, 0)$ and end up at a new\npoint on the Bloch sphere. Here we have padded 0 since our data is only\n2D. P\u00e9rez-Salinas et al. (2019) discuss how to load a higher\ndimensional data point ($[x_1, x_2, x_3, x_4, x_5, x_6]$) by\nbreaking it down in sets of three parameters\n($U(x_1, x_2, x_3), U(x_4, x_5, x_6)$).\n\nModel parameters with data re-uploading\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\nOnce we load the data onto the quantum circuit, we want to have some\ntrainable nonlinear model similar to a neural network as well as a way of\nlearning the weights of the model from data. This is again done with\nunitaries, $U(\\theta_1, \\theta_2, \\theta_3)$, such that we load the\ndata first and then apply the weights to form a single layer\n$L(\\vec \\theta, \\vec x) = U(\\vec \\theta)U(\\vec x)$. In principle,\nthis is just application of two matrix multiplications on an input\nvector initialized to some value. In order to increase the number of\ntrainable parameters (similar to increasing neurons in a single layer of\na neural network), we can reapply this layer again and again with new\nsets of weights,\n$L(\\vec \\theta_1, \\vec x) L(\\vec \\theta_2, , \\vec x) ... L(\\vec \\theta_L, \\vec x)$\nfor $L$ layers. The quantum circuit would look like the following:\n\n.. figure:: ../demonstrations/data_reuploading/universal_layers.png\n :scale: 75%\n :alt: Layers\n\n\nThe cost function and \"nonlinear collapse\"\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\nSo far, we have only performed linear operations (matrix\nmultiplications) and we know that we need to have some nonlinear\nsquashing similar to activation functions in neural networks to really\nmake a universal classifier (Cybenko 1989). Here is where things gets a\nbit quantum. After the application of the layers, we will end up at some\npoint on the Bloch sphere due to the sequence of unitaries implementing\nrotations of the input. These are still just linear transformations of\nthe input state. Now, the output of the model should be a class label\nwhich can be encoded as fixed vectors (Blue = $[1, 0]$, Red =\n$[0, 1]$) on the Bloch sphere. We want to end up at either of them\nafter transforming our input state through alternate applications of\ndata layer and weights.\n\nWe can use the idea of the \u201ccollapse\u201d of our quantum state into\none or other class. This happens when we measure the quantum state which\nleads to its projection as either the state 0 or 1. We can compute the\nfidelity (or closeness) of the output state to the class label making\nthe output state jump to either $| 0 \\rangle$ or\n$|1\\rangle$. By repeating this process several times, we can\ncompute the probability or overlap of our output to both labels and\nassign a class based on the label our output has a higher overlap. This\nis much like having a set of output neurons and selecting the one which\nhas the highest value as the label.\n\nWe can encode the output label as a particular quantum state that we want\nto end up in and use Pennylane to find the probability of ending up in that\nstate after running the circuit. We construct an observable corresponding to\nthe output label using the `Hermitian `_\noperator. The expectation value of the observable gives the overlap or fidelity.\nWe can then define the cost function as the sum of the fidelities for all\nthe data points after passing through the circuit and optimize the parameters\n$(\\vec \\theta)$ to minimize the cost.\n\n\\begin{align}\\texttt{Cost} = \\sum_{\\texttt{data points}} (1 - \\texttt{fidelity}(\\psi_{\\texttt{output}}(\\vec x, \\vec \\theta), \\psi_{\\texttt{label}}))\\end{align}\n\nNow, we can use our favorite optimizer to maximize the sum of the\nfidelities over all data points (or batches of datapoints) and find the\noptimal weights for classification. Gradient-based optimizers such as\nAdam (Kingma et. al., 2014) can be used if we have a good model of\nthe circuit and how noise might affect it. Or, we can use some\ngradient-free method such as L-BFGS (Liu, Dong C., and Nocedal, J., 1989)\nto evaluate the gradient and find the optimal weights where we can\ntreat the quantum circuit as a black-box and the gradients are computed\nnumerically using a fixed number of function evaluations and iterations.\nThe L-BFGS method can be used with the PyTorch interface for Pennylane.\n\nMultiple qubits, entanglement and Deep Neural Networks\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\nThe Universal Approximation Theorem declares that a neural network with\ntwo or more hidden layers can serve as a universal function approximator.\nRecently, we have witnessed remarkable progress of learning algorithms using\nDeep Neural Networks.\n\nP\u00e9rez-Salinas et al. (2019) make a connection to Deep Neural Networks by\ndescribing that in their approach the\n\u201clayers\u201d $L_i(\\vec \\theta_i, \\vec x )$ are analogous to the size\nof the intermediate hidden layer of a neural network. And the concept of\ndeep (multiple layers of the neural network) relates to the number\nof qubits. So, multiple qubits with entanglement between them could\nprovide some quantum advantage over classical neural networks. But here,\nwe will only implement a single qubit classifier.\n\n.. figure:: ../demonstrations/data_reuploading/universal_dnn.png\n :scale: 35%\n :alt: DNN\n\n\"Talk is cheap. Show me the code.\" - Linus Torvalds\n---------------------------------------------------\n\n\n\n```python\nimport pennylane as qml\nfrom pennylane import numpy as np\nfrom pennylane.optimize import AdamOptimizer, GradientDescentOptimizer\nqml.enable_tape()\n\nimport matplotlib.pyplot as plt\n\n\n# Set a random seed\nnp.random.seed(42)\n\n\n# Make a dataset of points inside and outside of a circle\ndef circle(samples, center=[0.0, 0.0], radius=np.sqrt(2 / np.pi)):\n \"\"\"\n Generates a dataset of points with 1/0 labels inside a given radius.\n\n Args:\n samples (int): number of samples to generate\n center (tuple): center of the circle\n radius (float: radius of the circle\n\n Returns:\n Xvals (array[tuple]): coordinates of points\n yvals (array[int]): classification labels\n \"\"\"\n Xvals, yvals = [], []\n\n for i in range(samples):\n x = 2 * (np.random.rand(2)) - 1\n y = 0\n if np.linalg.norm(x - center) < radius:\n y = 1\n Xvals.append(x)\n yvals.append(y)\n return np.array(Xvals), np.array(yvals)\n\n\ndef plot_data(x, y, fig=None, ax=None):\n \"\"\"\n Plot data with red/blue values for a binary classification.\n\n Args:\n x (array[tuple]): array of data points as tuples\n y (array[int]): array of data points as tuples\n \"\"\"\n if fig == None:\n fig, ax = plt.subplots(1, 1, figsize=(5, 5))\n reds = y == 0\n blues = y == 1\n ax.scatter(x[reds, 0], x[reds, 1], c=\"red\", s=20, edgecolor=\"k\")\n ax.scatter(x[blues, 0], x[blues, 1], c=\"blue\", s=20, edgecolor=\"k\")\n ax.set_xlabel(\"$x_1$\")\n ax.set_ylabel(\"$x_2$\")\n\n\nXdata, ydata = circle(500)\nfig, ax = plt.subplots(1, 1, figsize=(4, 4))\nplot_data(Xdata, ydata, fig=fig, ax=ax)\nplt.show()\n\n\n# Define output labels as quantum state vectors\ndef density_matrix(state):\n \"\"\"Calculates the density matrix representation of a state.\n\n Args:\n state (array[complex]): array representing a quantum state vector\n\n Returns:\n dm: (array[complex]): array representing the density matrix\n \"\"\"\n return state * np.conj(state).T\n\n\nlabel_0 = [[1], [0]]\nlabel_1 = [[0], [1]]\nstate_labels = [label_0, label_1]\n```\n\n\n```python\nXdata.shape, ydata.shape\n```\n\n\n\n\n ((500, 2), (500,))\n\n\n\nSimple classifier with data reloading and fidelity loss\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n\n\n\n```python\ndev = qml.device(\"default.qubit\", wires=1)\n# Install any pennylane-plugin to run on some particular backend\n\n@qml.qnode(dev)\ndef qcircuit(params, x=None, y=None):\n \"\"\"A variational quantum circuit representing the Universal classifier.\n\n Args:\n params (array[float]): array of parameters\n x (array[float]): single input vector\n y (array[float]): single output state density matrix\n\n Returns:\n float: fidelity between output state and input\n \"\"\"\n for i in range(len(params[0])):\n qml.Rot(*(params[0][i]*x + params[1][i]), wires=0)\n #qml.Rot(*params[1][i], wires=0)\n return qml.expval(qml.Hermitian(y, wires=[0]))\n\n\ndef cost(params, x, y, state_labels=None):\n \"\"\"Cost function to be minimized.\n\n Args:\n params (array[float]): array of parameters\n x (array[float]): 2-d array of input vectors\n y (array[float]): 1-d array of targets\n state_labels (array[float]): array of state representations for labels\n\n Returns:\n float: loss value to be minimized\n \"\"\"\n # Compute prediction for each input in data batch\n loss = 0.0\n dm_labels = [density_matrix(s) for s in state_labels]\n for i in range(len(x)):\n f = qcircuit(params, x=x[i], y=dm_labels[y[i]])\n loss = loss + (1 - f) ** 2\n return loss / len(x)\n```\n\nUtility functions for testing and creating batches\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n\n\n\n```python\ndef test(params, x, y, state_labels=None):\n \"\"\"\n Tests on a given set of data.\n\n Args:\n params (array[float]): array of parameters\n x (array[float]): 2-d array of input vectors\n y (array[float]): 1-d array of targets\n state_labels (array[float]): 1-d array of state representations for labels\n\n Returns:\n predicted (array([int]): predicted labels for test data\n output_states (array[float]): output quantum states from the circuit\n \"\"\"\n fidelity_values = []\n dm_labels = [density_matrix(s) for s in state_labels]\n predicted = []\n\n for i in range(len(x)):\n fidel_function = lambda y: qcircuit(params, x=x[i], y=y)\n fidelities = [fidel_function(dm) for dm in dm_labels]\n best_fidel = np.argmax(fidelities)\n\n predicted.append(best_fidel)\n fidelity_values.append(fidelities)\n\n return np.array(predicted), np.array(fidelity_values)\n\n\ndef accuracy_score(y_true, y_pred):\n \"\"\"Accuracy score.\n\n Args:\n y_true (array[float]): 1-d array of targets\n y_predicted (array[float]): 1-d array of predictions\n state_labels (array[float]): 1-d array of state representations for labels\n\n Returns:\n score (float): the fraction of correctly classified samples\n \"\"\"\n score = y_true == y_pred\n return score.sum() / len(y_true)\n\n\ndef iterate_minibatches(inputs, targets, batch_size):\n \"\"\"\n A generator for batches of the input data\n\n Args:\n inputs (array[float]): input data\n targets (array[float]): targets\n\n Returns:\n inputs (array[float]): one batch of input data of length `batch_size`\n targets (array[float]): one batch of targets of length `batch_size`\n \"\"\"\n for start_idx in range(0, inputs.shape[0] - batch_size + 1, batch_size):\n idxs = slice(start_idx, start_idx + batch_size)\n yield inputs[idxs], targets[idxs]\n```\n\nTrain a quantum classifier on the circle dataset\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n\n\n\n```python\n# Generate training and test data\nnum_training = 200\nnum_test = 2000\n\nXdata, y_train = circle(num_training)\nX_train = np.hstack((Xdata, np.zeros((Xdata.shape[0], 1))))\n\nXtest, y_test = circle(num_test)\nX_test = np.hstack((Xtest, np.zeros((Xtest.shape[0], 1))))\n\n\n# Train using Adam optimizer and evaluate the classifier\nnum_layers = 10\nlearning_rate = 0.6\nepochs = 10\nbatch_size = 32\n\nopt = AdamOptimizer(learning_rate, beta1=0.9, beta2=0.999)\n\n# initialize random weights\ntheta = np.random.uniform(size=(num_layers, 3))\nw = np.random.uniform(size=(num_layers, 3))\nparams = [w, theta]\n\npredicted_train, fidel_train = test(params, X_train, y_train, state_labels)\naccuracy_train = accuracy_score(y_train, predicted_train)\n\npredicted_test, fidel_test = test(params, X_test, y_test, state_labels)\naccuracy_test = accuracy_score(y_test, predicted_test)\n\n# save predictions with random weights for comparison\ninitial_predictions = predicted_test\n\nloss = cost(params, X_test, y_test, state_labels)\n\nprint(\n \"Epoch: {:2d} | Cost: {:3f} | Train accuracy: {:3f} | Test Accuracy: {:3f}\".format(\n 0, loss, accuracy_train, accuracy_test\n )\n)\n\nfor it in range(epochs):\n for Xbatch, ybatch in iterate_minibatches(X_train, y_train, batch_size=batch_size):\n params = opt.step(lambda v: cost(v, Xbatch, ybatch, state_labels), params)\n\n predicted_train, fidel_train = test(params, X_train, y_train, state_labels)\n accuracy_train = accuracy_score(y_train, predicted_train)\n loss = cost(params, X_train, y_train, state_labels)\n\n predicted_test, fidel_test = test(params, X_test, y_test, state_labels)\n accuracy_test = accuracy_score(y_test, predicted_test)\n res = [it + 1, loss, accuracy_train, accuracy_test]\n print(\n \"Epoch: {:2d} | Loss: {:3f} | Train accuracy: {:3f} | Test accuracy: {:3f}\".format(\n *res\n )\n )\n```\n\n Epoch: 0 | Cost: 0.397512 | Train accuracy: 0.440000 | Test Accuracy: 0.414000\n\n\n /usr/local/lib/python3.6/dist-packages/pennylane/tape/tapes/jacobian_tape.py:461: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray\n params = np.array(params)\n\n\n Epoch: 1 | Loss: 0.214109 | Train accuracy: 0.660000 | Test accuracy: 0.605500\n Epoch: 2 | Loss: 0.192650 | Train accuracy: 0.670000 | Test accuracy: 0.691500\n Epoch: 3 | Loss: 0.126039 | Train accuracy: 0.810000 | Test accuracy: 0.808500\n Epoch: 4 | Loss: 0.093838 | Train accuracy: 0.870000 | Test accuracy: 0.829000\n Epoch: 5 | Loss: 0.076994 | Train accuracy: 0.940000 | Test accuracy: 0.874500\n Epoch: 6 | Loss: 0.079456 | Train accuracy: 0.920000 | Test accuracy: 0.866000\n Epoch: 7 | Loss: 0.060256 | Train accuracy: 0.960000 | Test accuracy: 0.890000\n Epoch: 8 | Loss: 0.059333 | Train accuracy: 0.955000 | Test accuracy: 0.893500\n Epoch: 9 | Loss: 0.064725 | Train accuracy: 0.940000 | Test accuracy: 0.896000\n Epoch: 10 | Loss: 0.063304 | Train accuracy: 0.935000 | Test accuracy: 0.875000\n\n\nResults\n~~~~~~~\n\n\n\n\n```python\nprint(\n \"Cost: {:3f} | Train accuracy {:3f} | Test Accuracy : {:3f}\".format(\n loss, accuracy_train, accuracy_test\n )\n)\n\nprint(\"Learned weights\")\nfor i in range(num_layers):\n print(\"Layer {}: {}\".format(i, params[i]))\n\n\nfig, axes = plt.subplots(1, 3, figsize=(10, 3))\nplot_data(X_test, initial_predictions, fig, axes[0])\nplot_data(X_test, predicted_test, fig, axes[1])\nplot_data(X_test, y_test, fig, axes[2])\naxes[0].set_title(\"Predictions with random weights\")\naxes[1].set_title(\"Predictions after training\")\naxes[2].set_title(\"True test data\")\nplt.show()\n```\n\nReferences\n----------\n[1] P\u00e9rez-Salinas, Adri\u00e1n, et al. \u201cData re-uploading for a universal\nquantum classifier.\u201d arXiv preprint arXiv:1907.02085 (2019).\n\n[2] Kingma, Diederik P., and Ba, J. \"Adam: A method for stochastic\noptimization.\" arXiv preprint arXiv:1412.6980 (2014).\n\n[3] Liu, Dong C., and Nocedal, J. \"On the limited memory BFGS\nmethod for large scale optimization.\" Mathematical programming\n45.1-3 (1989): 503-528.\n\n\n", "meta": {"hexsha": "0582fdad0d1cbc77c9de8a3bbd71ba0ff97f2466", "size": 138491, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "PennyLane/Data Reuploading Classifier/tutorial_data_reuploading_classifier.ipynb", "max_stars_repo_name": "Graciaira/quantum_image_classifier", "max_stars_repo_head_hexsha": "1e6a8ec93f51dcbfd63c2e652be5d1fcbce283ce", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-06-08T12:32:09.000Z", "max_stars_repo_stars_event_max_datetime": "2021-06-08T12:32:09.000Z", "max_issues_repo_path": "PennyLane/Data Reuploading Classifier/tutorial_data_reuploading_classifier.ipynb", "max_issues_repo_name": "Graciaira/quantum_image_classifier", "max_issues_repo_head_hexsha": "1e6a8ec93f51dcbfd63c2e652be5d1fcbce283ce", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "PennyLane/Data Reuploading Classifier/tutorial_data_reuploading_classifier.ipynb", "max_forks_repo_name": "Graciaira/quantum_image_classifier", "max_forks_repo_head_hexsha": "1e6a8ec93f51dcbfd63c2e652be5d1fcbce283ce", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 138491.0, "max_line_length": 138491, "alphanum_fraction": 0.914066618, "converted": true, "num_tokens": 4873, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.6548947290421276, "lm_q1q2_score": 0.43348980242656787}} {"text": "# Enzyme Modules\n\nAn \"enzyme module\" is defined as a mechanistic description of a reaction consisting of mass action rate laws for all known reaction steps (Du et al., 2016). In **MASSpy**, enzyme modules are represented by the `EnzymeModule` object.\n\nTo demonstrate the utility of an `EnzymeModule` object and how it aids in constructing mechanistic models of enzyme behavior, an `EnzymeModule` of hexokinase$^{1, 2}$ is constructed and then merged with a model of glycolysis$^{3}$ for verification.\n\n## Constructing Enzyme Modules\n\nIn order to construct the `EnzymeModule` of hexokinase, the following information is necessary:\n\n1. The enzyme is a monomer.\n2. The enzyme binding of substrates follows a random sequential mechanism.\n3. The enzyme experiences product inhibtion and is competitively inhibited by 23DPG when complexed with D-glucose.\n\nTotal HEX1 Concentration$^2$: $\\text{[HEX1]}_{total} = 24 nM = 0.000024 mM$.\n\n\n```python\nfrom operator import attrgetter\n\nfrom mass import MassMetabolite\nfrom mass.enzyme_modules import EnzymeModule\nfrom mass.test import create_test_model\n\n# Load the glycolysis and hemoglobin models, then merge them\nglycolysis = create_test_model(\"Glycolysis\")\nhemoglobin = create_test_model(\"Hemoglobin\")\nglyc_hb = glycolysis.merge(hemoglobin, inplace=False)\n```\n\nThe `EnzymeModule` is a subclass of the `MassModel`, meaning that it inherits the methods and behaviors of the `MassModel` object. \nLike a `MassModel`, an `EnzymeModule` object requires a unique identifier in order to be created. Optionally, the `name` and `subsystem` attributes are set during initialization.\n\n\n```python\nHEX1 = EnzymeModule(\"HEX1\", name=\"Hexokinase (D-glucose:ATP)\",\n subsystem=\"Glycolysis\")\n```\n\n### Defining the enzyme ligands\n\nThe ligands that interact with the enzyme (e.g. as the substrates, activators, and inhibitors) are created as `MassMetabolite` objects and added to the model.\n\n\n```python\nglc__D_c = MassMetabolite(\n \"glc__D_c\",\n name=\"D-Glucose\",\n formula=\"C6H12O6\",\n charge=0,\n compartment=\"c\")\ng6p_c = MassMetabolite(\n \"g6p_c\",\n name=\"D-Glucose 6-phosphate\",\n formula=\"C6H11O9P\",\n charge=-2,\n compartment=\"c\")\natp_c = MassMetabolite(\n \"atp_c\",\n name=\"ATP\",\n formula=\"C10H12N5O13P3\",\n charge=-4,\n compartment=\"c\")\nadp_c = MassMetabolite(\n \"adp_c\",\n name=\"ADP\",\n formula=\"C10H12N5O10P2\",\n charge=-3,\n compartment=\"c\")\n_23dpg_c = MassMetabolite(\n \"_23dpg_c\", \n name=\"2,3-Disphospho-D-glycerate\", \n formula=\"C3H3O10P2\",\n charge=-5,\n compartment=\"c\")\nh_c = MassMetabolite(\n \"h_c\",\n name=\"H+\",\n formula=\"H\",\n charge=1,\n compartment=\"c\")\n\nHEX1.add_metabolites([glc__D_c, g6p_c, atp_c, adp_c, _23dpg_c, h_c])\n```\n\nOnce added to the `EnzymeModule`, ligands can be accessed using the `enzyme_module_ligands` attribute.\n\n\n```python\nHEX1.enzyme_module_ligands\n```\n\n\n\n\n [,\n ,\n ,\n ,\n ,\n ]\n\n\n\nTo keep track of the roles played by various ligands in the module, the `enzyme_module_ligands_categorized` attribute is set. The attribute takes a `dict`, with categories as keys and relevant `MassMetabolite` objects as values. Note that an object can be a part of multiple categories.\n\n\n```python\nHEX1.enzyme_module_ligands_categorized = {\n \"substrates\": glc__D_c,\n \"cofactors\": atp_c,\n \"inhibitors\": _23dpg_c,\n \"products\": [adp_c, g6p_c, h_c]}\nHEX1.enzyme_module_ligands_categorized\n```\n\n\n\n\n [,\n ,\n ,\n ]\n\n\n\nFor each category, a `cobra.Group` is created containing the relevant objects. Once set, the attribute returns a `cobra.DictList` that contains the categorized groups. The groups and their members are printed as follows:\n\n\n```python\nfor group in HEX1.enzyme_module_ligands_categorized:\n print(\"{0}: {1}\".format(\n group.id, str(sorted([m.id for m in group.members]))))\n```\n\n substrates: ['glc__D_c']\n cofactors: ['atp_c']\n inhibitors: ['_23dpg_c']\n products: ['adp_c', 'g6p_c', 'h_c']\n\n\n### Defining the enzyme module forms\n\nAfter adding `MassMetabolite` objects of ligands to the model, the various forms of the enzyme must be defined. These forms are represented by `EnzymeModuleForm` objects. \n\nThe `EnzymeModuleForm` object inherits from the `MassMetabolite` and is treated like any other metabolite in the model. However, the `EnzymeModuleForm` object contains the additional `bound_metabolites` attribute to assist in tracking metabolites bound to the enzyme form. \n\nThe `EnzymeModule.make_enzyme_module_form()` method allows for the creation of an `EnzymeModuleForm` object while assigning categories for the `EnzymeModuleForm` in the process. Using `make_enzyme_module_form()` also adds the species to the module upon creation, accessible via the `EnzymeModule.enzyme_module_forms` attribute.\n\n\n```python\nhex1_c = HEX1.make_enzyme_module_form(\n \"hex1_c\",\n name=\"automatic\",\n categories=\"Active\",\n compartment=\"c\")\n\nhex1_A_c = HEX1.make_enzyme_module_form(\n \"hex1_A_c\", # A stands complexted with ATP\n name=\"automatic\",\n categories=\"Active\",\n bound_metabolites={atp_c: 1},\n compartment=\"c\")\n\nhex1_G_c = HEX1.make_enzyme_module_form(\n \"hex1_G_c\", # G stands for complexed with Glucose\n name=\"automatic\",\n categories=\"Active\",\n bound_metabolites={glc__D_c: 1},\n compartment=\"c\")\n\nhex1_AG_c = HEX1.make_enzyme_module_form(\n \"hex1_AG_c\",\n name=\"automatic\",\n categories=\"Active\",\n bound_metabolites={glc__D_c: 1, atp_c: 1},\n compartment=\"c\")\n\nhex1_G_CI_c = HEX1.make_enzyme_module_form(\n \"hex1_G_CI_c\", # CI stands for competitive inhibition\n name=\"automatic\",\n categories=\"Inhibited\",\n bound_metabolites={glc__D_c: 1, _23dpg_c: 1},\n compartment=\"c\")\n\nhex1_A_PI_c = HEX1.make_enzyme_module_form(\n \"hex1_A_PI_c\", # PI stands for competitive inhibition\n name=\"automatic\",\n categories=\"Inhibited\",\n bound_metabolites={adp_c: 1},\n compartment=\"c\")\n\nhex1_G_PI_c = HEX1.make_enzyme_module_form(\n \"hex1_G_PI_c\", # PI stands for competitive inhibition\n name=\"automatic\",\n categories=\"Inhibited\",\n bound_metabolites={g6p_c: 1},\n compartment=\"c\")\n\nHEX1.enzyme_module_forms\n```\n\n\n\n\n [,\n ,\n ,\n ,\n ,\n ,\n ]\n\n\n\nThe `bound_metabolites` attribute represents the ligands bound to the site(s) of enzyme.\n\n\n```python\n# Print automatically generated names\nfor enzyme_form in HEX1.enzyme_module_forms:\n print(\"Bound to sites of {0}:\\n{1}\\n\".format(\n enzyme_form.id, {\n ligand.id: coeff\n for ligand, coeff in enzyme_form.bound_metabolites.items()}))\n```\n\n Bound to sites of hex1_c:\n {}\n \n Bound to sites of hex1_A_c:\n {'atp_c': 1}\n \n Bound to sites of hex1_G_c:\n {'glc__D_c': 1}\n \n Bound to sites of hex1_AG_c:\n {'glc__D_c': 1, 'atp_c': 1}\n \n Bound to sites of hex1_G_CI_c:\n {'glc__D_c': 1, '_23dpg_c': 1}\n \n Bound to sites of hex1_A_PI_c:\n {'adp_c': 1}\n \n Bound to sites of hex1_G_PI_c:\n {'g6p_c': 1}\n \n\n\nSetting the `bound_metabolites` attribute upon creation allow the `formula` and `charge` attributes of the various forms also to be set while ensuring mass and charge balancing is maintained. Note that the enzyme is represented as a moiety, and the ligands bound to the enzyme are represented in the chemical formula.\n\n\n```python\n# Get the elemental matrix for the enzyme\ndf = HEX1.get_elemental_matrix(array_type=\"DataFrame\")\n# Use iloc to only look at EnzymeModuleForms\ndf.iloc[:, 6:]\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    hex1_chex1_A_chex1_G_chex1_AG_chex1_G_CI_chex1_A_PI_chex1_G_PI_c
    C0.010.06.016.09.010.06.0
    H0.012.012.024.015.012.011.0
    O0.013.06.019.016.010.09.0
    P0.03.00.03.02.02.01.0
    N0.05.00.05.00.05.00.0
    S0.00.00.00.00.00.00.0
    q0.0-4.00.0-4.0-5.0-3.0-2.0
    [HEX]1.01.01.01.01.01.01.0
    \n
    \n\n\n\nSetting the `name` argument as \"automatic\" in the `EnzymeModule.make_enzyme_module_form()` method causes a name for the `EnzymeModuleForm` to be generated based on the metabolites in the `bound_metabolites` attribute.\n\n\n```python\n# Print automatically generated names\nfor enzyme_form in HEX1.enzyme_module_forms:\n print(enzyme_form.name)\n```\n\n HEX1\n HEX1-atp complex\n HEX1-glc__D complex\n HEX1-glc__D-atp complex\n HEX1-glc__D-_23dpg complex\n HEX1-adp complex\n HEX1-g6p complex\n\n\nThe `categories` argument allows for `EnzymeModuleForm` objects to be placed into `cobra.Group` objects representing those categories. As with the ligands, the categorized enzyme module forms are returned in a `DictList` of `Group` objects by the `enzyme_module_forms_categorized` attribute.\n\n\n```python\nfor group in HEX1.enzyme_module_forms_categorized:\n print(\"{0}: {1}\".format(\n group.id, str(sorted([m.id for m in group.members]))))\n```\n\n Active: ['hex1_AG_c', 'hex1_A_c', 'hex1_G_c', 'hex1_c']\n Inhibited: ['hex1_A_PI_c', 'hex1_G_CI_c', 'hex1_G_PI_c']\n\n\nAlternatively, the `enzyme_module_forms_categorized` attribute can be set using a `dict`:\n\n\n```python\nHEX1.enzyme_module_forms_categorized = {\n \"competitively_inhibited\": hex1_G_CI_c}\n\nfor group in HEX1.enzyme_module_forms_categorized:\n print(\"{0}: {1}\".format(\n group.id, str(sorted([m.id for m in group.members]))))\n```\n\n Active: ['hex1_AG_c', 'hex1_A_c', 'hex1_G_c', 'hex1_c']\n Inhibited: ['hex1_A_PI_c', 'hex1_G_CI_c', 'hex1_G_PI_c']\n competitively_inhibited: ['hex1_G_CI_c']\n\n\n### Defining enzyme module reactions\n\nThe next step is to define all of the reaction steps that represent the catalytic mechanism and regulation of the enzyme module. These reactions are represented as `EnzymeModuleReaction` objects. \n\nThe `EnzymeModuleReaction` object inherits from the `MassReaction` and is treated like any other reaction in the model. Like the `make_enzyme_module_form()` method, the `make_enzyme_module_reaction()` method allows for the creation of an `EnzymeModuleReaction` object while assigning categories for the `EnzymeModuleReaction` in the process.\n\nSpecies that exist in the model can also be added to the reaction by providing a dictionary of metabolites and their stoichiometric coefficients to the `metabolites_to_add` argument.\n\n\n```python\nHEX1_1 = HEX1.make_enzyme_module_reaction(\n \"HEX1_1\",\n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=True,\n categories=\"product_inhibition\",\n metabolites_to_add={\n \"hex1_c\": -1,\n \"adp_c\": -1,\n \"hex1_A_PI_c\": 1})\n\nHEX1_2 = HEX1.make_enzyme_module_reaction(\n \"HEX1_2\",\n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=True,\n categories=\"product_inhibition\",\n metabolites_to_add={\n \"hex1_c\": -1,\n \"g6p_c\": -1,\n \"hex1_G_PI_c\": 1})\n\nHEX1_3 = HEX1.make_enzyme_module_reaction(\n \"HEX1_3\",\n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=True,\n categories=\"glc__D_c_binding\",\n metabolites_to_add={\n \"hex1_c\": -1,\n \"glc__D_c\": -1,\n \"hex1_G_c\": 1})\n\nHEX1_4 = HEX1.make_enzyme_module_reaction(\n \"HEX1_4\",\n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=True,\n categories=\"atp_c_binding\",\n metabolites_to_add={\n \"hex1_c\": -1,\n \"atp_c\": -1,\n \"hex1_A_c\": 1})\n\nHEX1_5 = HEX1.make_enzyme_module_reaction(\n \"HEX1_5\",\n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=True,\n categories=\"competitive_inhibition\",\n metabolites_to_add={\n \"hex1_G_c\": -1,\n \"_23dpg_c\": -1,\n \"hex1_G_CI_c\": 1})\n\nHEX1_6 = HEX1.make_enzyme_module_reaction(\n \"HEX1_6\",\n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=True,\n categories=\"atp_c_binding\",\n metabolites_to_add={\n \"hex1_G_c\": -1,\n \"atp_c\": -1,\n \"hex1_AG_c\": 1})\n\nHEX1_7 = HEX1.make_enzyme_module_reaction(\n \"HEX1_7\",\n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=True,\n categories=\"glc__D_c_binding\",\n metabolites_to_add={\n \"hex1_A_c\": -1,\n \"glc__D_c\": -1,\n \"hex1_AG_c\": 1})\n\nHEX1_8 = HEX1.make_enzyme_module_reaction(\n \"HEX1_8\",\n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=True,\n categories=\"catalyzation\",\n metabolites_to_add={\n \"hex1_AG_c\": -1,\n \"hex1_c\": 1,\n \"adp_c\": 1,\n \"g6p_c\": 1,\n \"h_c\": 1})\n\nfor reaction in HEX1.enzyme_module_reactions:\n print(reaction)\n```\n\n HEX1_1: adp_c + hex1_c <=> hex1_A_PI_c\n HEX1_2: g6p_c + hex1_c <=> hex1_G_PI_c\n HEX1_3: glc__D_c + hex1_c <=> hex1_G_c\n HEX1_4: atp_c + hex1_c <=> hex1_A_c\n HEX1_5: _23dpg_c + hex1_G_c <=> hex1_G_CI_c\n HEX1_6: atp_c + hex1_G_c <=> hex1_AG_c\n HEX1_7: glc__D_c + hex1_A_c <=> hex1_AG_c\n HEX1_8: hex1_AG_c <=> adp_c + g6p_c + h_c + hex1_c\n\n\nThe `categories` argument allows for `EnzymeModuleReactions` objects to be placed into `cobra.Group` objects representing those categories. As with the ligands and enzyme forms, a `DictList` of the relevant groups are returned with the `enzyme_module_reactions_categorized` attribute.\n\n\n```python\nHEX1.enzyme_module_reactions_categorized\n```\n\n\n\n\n [,\n ,\n ,\n ,\n ]\n\n\n\n#### Unifying rate parameters\n\nFor this `EnzymeModule`, the reactions representing glucose binding to the enzyme and ATP binding to the enzyme have the same forward rate and equilibrium constants. Instead of defining the parameter values for each individual reaction, the `unify_rate_parameters()` method can be used to create custom rate laws for the given reactions that all depend on the same rate parameters.\n\nThe `unify_rate_parameters()` method takes a list of reactions and an identifier to use for the unified parameter. The `enzyme_prefix` flag can be set to `True` to prefix the new parameter identifier with the identifier of the `EnzymeModule`, ensuring that any existing custom parameters are not overwritten.\n\n\n```python\nfor ligand, pid in zip([glc__D_c, atp_c],[\"G\", \"A\"]):\n # Get the group of reactions corresponding to the ligand\n category = \"_\".join((ligand.id, \"binding\"))\n group = HEX1.enzyme_module_reactions_categorized.get_by_id(category)\n \n # Unify the parameters\n HEX1.unify_rate_parameters(\n group.members, new_parameter_id=pid, enzyme_prefix=True)\n\n # Print the new reaction rates\n print(\"\\n\" + category + \"\\n\" + \"-\" * len(category))\n for reaction in sorted(group.members, key=attrgetter(\"id\")):\n print(reaction.id + \": \" + str(reaction.rate))\n```\n\n \n glc__D_c_binding\n ----------------\n HEX1_3: kf_HEX1_G*(glc__D_c(t)*hex1_c(t) - hex1_G_c(t)/Keq_HEX1_G)\n HEX1_7: kf_HEX1_G*(glc__D_c(t)*hex1_A_c(t) - hex1_AG_c(t)/Keq_HEX1_G)\n \n atp_c_binding\n -------------\n HEX1_4: kf_HEX1_A*(atp_c(t)*hex1_c(t) - hex1_A_c(t)/Keq_HEX1_A)\n HEX1_6: kf_HEX1_A*(atp_c(t)*hex1_G_c(t) - hex1_AG_c(t)/Keq_HEX1_A)\n\n\n## Determining Enzyme Form Concentrations and Rate Constants\n\nThe next step is to solve for the steady state concentrations for the various forms of the enzyme symbolically using **SymPy**. Because the numerical values for the dissociation constants have been defined, these equations are solved in terms of the rate constants. The rate constants can be approximated using the total enzyme concentration as a constraint and substituted back into the equations to calculate the numerical values of the steady state concentrations.\n\n\n```python\nfrom sympy import Eq, Symbol, lambdify, simplify, solveset\n\nfrom mass import strip_time\nfrom mass.util.matrix import matrix_rank\n```\n\n### Solving steady state concentrations symbolically\n\nTo get the symbolic solutions for the individual enzyme forms, the ODEs are first collected in a `dict`. Keys are the enzyme forms, and values are their ODEs with the time dependency stripped via the `strip_time` function.\n\n\n```python\node_dict = {\n enzyme_form.id: Eq(strip_time(enzyme_form.ode), 0)\n for enzyme_form in HEX1.enzyme_module_forms}\n# Matrix rank of enzyme stoichiometric matrix without substrates\nrank = matrix_rank(HEX1.S[6:])\nprint(\"Rank Deficiency: {0}\".format(len(ode_dict) - rank))\n```\n\n Rank Deficiency: 1\n\n\nBecause the stoichiometric matrix (without ligands) has a rank deficiency of one, there is a dependent variable in the system unless another equation is added. Therefore, the completely free enzyme form is treated as the dependent variable, and all of the enzyme forms are solved in terms of the free enzyme form.\n\n\n```python\nenzyme_solutions = {}\nfor enzyme_form in HEX1.enzyme_module_forms:\n # Skip dependent variable\n if enzyme_form.id == \"hex1_c\":\n continue\n # Get the ODE for the enzyme form from the ODE dict\n equation = ode_dict[enzyme_form.id]\n # Solve the equation for the enzyme form, substituting \n # previously found enzyme form solutions into the equation\n solution = solveset(equation.subs(enzyme_solutions),\n enzyme_form.id)\n # Store the solution\n enzyme_solutions[enzyme_form.id] = list(solution)[0]\n # Substitute the new solution into existing solutions\n enzyme_solutions.update({\n enzyme_form: sol.subs(enzyme_solutions) \n for enzyme_form, sol in enzyme_solutions.items()})\n \nargs = set()\nfor solution in enzyme_solutions.values():\n args.update(solution.atoms(Symbol))\n```\n\n#### Defining the Rate Equation\n\nTo make up for the rank deficiency, an additional equation is needed. Typically, the rate of the enzyme is the summation of the rates for the catalyzation reaction step(s) of the enzyme. The `make_enzyme_rate_equation()` method can be used to create the rate equation from a list of reactions. If `use_rates=True`, the rate expressions of the reactions are added together. If `update_enzyme=True`, the rate equation is set as a symbolic expression for the `enzyme_rate_equation` attribute.\n\n\n```python\n# Get the catalyzation reactions\ncatalyzation_group = HEX1.enzyme_module_reactions_categorized.get_by_id(\n \"catalyzation\")\n\nHEX1.make_enzyme_rate_equation(catalyzation_group.members,\n use_rates=True,\n update_enzyme=True)\n\nprint(HEX1.enzyme_rate_equation)\n```\n\n kf_HEX1_8*(Keq_HEX1_8*hex1_AG_c(t) - adp_c(t)*g6p_c(t)*hex1_c(t))/Keq_HEX1_8\n\n\nWith the rate equation defined, the `enzyme_rate_error()` method is used to get the equation as the difference between the flux value and the rate equation.\n\n\n```python\nenzyme_rate_equation = strip_time(HEX1.enzyme_rate_error(use_values=False))\nprint(enzyme_rate_equation)\n```\n\n v_HEX1 - kf_HEX1_8*(Keq_HEX1_8*hex1_AG_c - adp_c*g6p_c*hex1_c)/Keq_HEX1_8\n\n\nThe solutions for the enzyme forms are substituted into the rate equation, and the equation is solved for the free enzyme form. The solutions are subsequently updated, resulting in symbolic equations that do not depend on any enzyme form.\n\n\n```python\n# Solve for last unknown concentration symbolically\nsolution = solveset(enzyme_rate_equation.subs(enzyme_solutions),\n \"hex1_c\")\n\n# Update solution dictionary with the new solution\nenzyme_solutions[\"hex1_c\"] = list(solution)[0]\n\n# Update solutions with free variable solutions\nenzyme_solutions = {\n enzyme_form: simplify(solution.subs(enzyme_solutions))\n for enzyme_form, solution in enzyme_solutions.items()}\n\nargs = set()\nfor solution in enzyme_solutions.values():\n args.update(solution.atoms(Symbol))\nprint(args)\n```\n\n {g6p_c, kf_HEX1_8, kf_HEX1_A, v_HEX1, Keq_HEX1_A, atp_c, glc__D_c, _23dpg_c, kf_HEX1_G, Keq_HEX1_5, Keq_HEX1_8, adp_c, Keq_HEX1_2, Keq_HEX1_1, Keq_HEX1_G}\n\n\nNumerical values for known quantities are substituted into the equations. For this `EnzymeModule` of Hexokinase, the following dissociation constants are used:\n\n$$\\begin{align}\nK_{d,\\ \\text{GLC-D}} &= 0.038\\ \\text{mM} \\\\\nK_{d,\\ \\text{ATP}} &= 2.06\\ \\text{mM} \\\\\nK_{i,\\ \\text{23DPG}} &= 5.5\\ \\text{mM} \\\\\nK_{i,\\ \\text{ADP}} &= 1\\ \\text{mM} \\\\\nK_{i,\\ \\text{G6P}} &= 66.67\\ \\text{mM} \\\\\n\\end{align}$$\n\nA value of $K_{\\text{HEX1}}= 313.12$ is used for the catalyzation step. Note that the inverse of the dissociation constant is used for reactions that form complexes. \n\n\n```python\nnumerical_values = {\n \"Keq_HEX1_1\": 1,\n \"Keq_HEX1_2\": 1 / 66.67,\n \"Keq_HEX1_G\": 1 / 0.038, \n \"Keq_HEX1_A\": 1 / 2.06,\n \"Keq_HEX1_5\": 1 / 5.5,\n \"Keq_HEX1_8\": 313.12}\n# Update the model with the parameters\nHEX1.update_parameters(numerical_values)\n```\n\nThe ligand concentrations and the rate for the enzyme are extracted from the merged glycolysis and hemoglobin model.\n\n\n```python\n# Get steady state flux for EnzymeModule\nHEX1.enzyme_rate = glyc_hb.reactions.get_by_id(\"HEX1\").steady_state_flux\nnumerical_values[HEX1.enzyme_flux_symbol_str] = HEX1.enzyme_rate\n\n# Get the ligand concentrations\nfor met in HEX1.enzyme_module_ligands:\n concentration = glyc_hb.metabolites.get_by_id(met.id).initial_condition\n # Set the ligand initial condition and add to numercal values dictionary\n met.initial_condition = concentration\n numerical_values[met.id] = concentration\n```\n\nThe numerical values are substituted into the symbolic equations, resulting in the steady state concentrations that depend only on the rate constants.\n\n\n```python\nenzyme_solutions = {\n enzyme_form: simplify(sol.subs(numerical_values))\n for enzyme_form, sol in enzyme_solutions.items()}\n\nargs = set()\nfor solution in enzyme_solutions.values():\n args.update(solution.atoms(Symbol))\nprint(args)\n```\n\n {kf_HEX1_A, kf_HEX1_G, kf_HEX1_8}\n\n\n### Approximating Rate Constants\n\nTo determine the set of rate constants for the enzyme module, the absolute error between the total hexokinase concentration value (found in literature) and the computed hexokinase concentration is minimized. For this example, the `minimize()` function of the **SciPy** package is utilized to find a feasible set of rate constants. \n\n\n```python\nfrom scipy.optimize import minimize\n```\n\nThe objective function for the minimization is first made symbolically. The `enzyme_total_symbol_str` property can be used to represent the total enzyme concentration, while the `enzyme_concentration_total_equation` property creates a symbolic expression for the sum of all enzyme forms.\n\n\n```python\nenzyme_total_error = abs(\n Symbol(HEX1.enzyme_total_symbol_str)\n - strip_time(HEX1.enzyme_concentration_total_equation))\nprint(enzyme_total_error)\n```\n\n Abs(-HEX1_Total + hex1_AG_c + hex1_A_PI_c + hex1_A_c + hex1_G_CI_c + hex1_G_PI_c + hex1_G_c + hex1_c)\n\n\nThe `enzyme_concentration_total` attribute stores the total amount of enzyme in the model and substituted into the expression. The total HEX1 concentration is $24 * 10^{-6} \\text{mM}$.\n\n\n```python\nHEX1.enzyme_concentration_total = 24e-6\nenzyme_total_error = enzyme_total_error.subs({\n HEX1.enzyme_total_symbol_str: HEX1.enzyme_concentration_total})\nprint(enzyme_total_error)\n```\n\n Abs(hex1_AG_c + hex1_A_PI_c + hex1_A_c + hex1_G_CI_c + hex1_G_PI_c + hex1_G_c + hex1_c - 2.4e-5)\n\n\nFinally, the symbolic equations for the enzyme forms are substituted into the enzyme total error equation, resulting in an expression that represents the objective function with the only unknown variables being rate constants. The `lambdify()` function of the **SymPy** package converts the symbolic objective into a lambda function that can be used with the `minimize()` function of **SciPy**.\n\n\n```python\nenzyme_total_error = simplify(enzyme_total_error.subs(enzyme_solutions))\n\n# Sort the arguments to ensure input format remains consistent\nargs = sorted(list(map(str, args)))\n# Use lambdify to make objective function as a lambda function\nobj_fun = lambda x: lambdify(args, enzyme_total_error)(*x)\n```\n\nThe `minimize()` function is now used to approximate the rate constants. The optimization problems for enzyme rate constants are typically nonlinear, and require nonlinear optimization routines to find feasible solutions.\n\n\n```python\n# Minimize the objective function, initial guess based on publication values\ninitial_guess = [1e8, 9376585, 52001]\nvariable_bounds = ((0, 1e9), (0, 1e9), (0, 1e9))\nsolution = minimize(obj_fun, x0=initial_guess,\n method=\"trust-constr\",\n bounds=variable_bounds)\n# Map solution array to variables\nrate_constants = dict(zip(args, solution.x))\nprint(rate_constants)\n```\n\n {'kf_HEX1_8': 100000000.0025878, 'kf_HEX1_A': 9376585.030755484, 'kf_HEX1_G': 52006.59981223971}\n\n\nBecause the rate constants associated with the inhibition of the enzyme forms are not necessary for computing the concentrations, a rapid binding assumption is made for the inhibition reactions. Therefore, a large number is set for the rate constants. The parameters are set using the `update_parameters()` method.\n\n\n```python\nrate_constants[\"kf_HEX1_1\"] = 1e6\nrate_constants[\"kf_HEX1_2\"] = 1e6\nrate_constants[\"kf_HEX1_5\"] = 1e6\nHEX1.update_parameters(rate_constants)\n```\n\n### Calculating numerical values for concentrations\n\nOnce the rate constants have been estimated, they are substituted back into the symbolic concentration equations in order to obtain their numerical values.\n\n\n```python\nfor enzyme_form, solution in enzyme_solutions.items():\n # Get the enzyme form object, determine the steady state concentration\n enzyme_form = HEX1.enzyme_module_forms.get_by_id(enzyme_form)\n enzyme_form.initial_condition = float(solution.subs(rate_constants))\n print(\"{0}: {1:e}\".format(enzyme_form.id,\n enzyme_form.initial_condition))\n```\n\n hex1_A_c: 9.401421e-06\n hex1_G_c: 5.718872e-08\n hex1_AG_c: 1.174630e-08\n hex1_G_CI_c: 3.223364e-08\n hex1_A_PI_c: 3.519706e-06\n hex1_G_PI_c: 8.847367e-09\n hex1_c: 1.213692e-05\n\n\n#### Error values\n\nAs a quality assurance check, the `enzyme_concentration_total_error()` method can be used to get the error between the `enzyme_concentration_total` attribute and the sum of the enzyme form concentrations. A positive value indicates the `enzyme_concentration_total` attribute is greater than the sum of the individual enzyme form concentrations that were computed.\n\n\n```python\nprint(\"Total Enzyme Concentration Error: {0}\".format(\n HEX1.enzyme_concentration_total_error(use_values=True)))\n```\n\n Total Enzyme Concentration Error: -1.1680622689093244e-06\n\n\nSimilarly, the error between the `enzyme_rate` attribute and the computed value from the `enzyme_rate_equation` can be also checked using the `enzyme_rate_error()` method, in which a positive value indicates that the `enzyme_rate` attribute is greater than the value computed when using the rate equation.\n\n\n```python\nprint(\"Enzyme Rate Error: {0}\".format(\n HEX1.enzyme_rate_error(use_values=True)))\n```\n\n Enzyme Rate Error: 4.440892098500626e-16\n\n\n## Adding EnzymeModules to Models\n\nWith the `EnzymeModule` built, it can be integrated into a larger network and simulated. To add an `EnzymeModule` to an existing `MassModel`, the `merge()` method is used. After merging, the `remove_reactions()` method is used to remove the reaction replaced with the enzyme module. The `EnzymeModule` should always be merged into the `MassModel` as demonstrated below:\n\n\n```python\nglyc_hb_HEX1 = glyc_hb.merge(HEX1, inplace=False)\nglyc_hb_HEX1.remove_reactions([\n glyc_hb_HEX1.reactions.get_by_id(\"HEX1\")])\n```\n\nAll objects, numerical values, and certain attributes of the `EnzymeModule` are transferred into the `MassModel` upon merging. This includes all enzyme forms, reactions steps, initial conditions, rate parameters, and category groups.\n\n\n```python\nglyc_hb_HEX1\n```\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    NameGlycolysis_Hemoglobin_HEX1
    Memory address0x07ff42b8e6d90
    Stoichiometric Matrix35x37
    Matrix Rank32
    Number of metabolites35
    Initial conditions defined35/35
    Number of reactions37
    Number of genes0
    Number of enzyme modules1
    Number of groups12
    Objective expression0
    CompartmentsCytosol
    \n\n\n\n\n### The EnzymeModuleDict object\n\nDuring the merge process, an `EnzymeModuleDict` is created from the `EnzymeModule` and added to the `MassModel.enzyme_modules` attribute.\n\n\n```python\nprint(glyc_hb_HEX1.enzyme_modules)\nHEX1_dict = glyc_hb_HEX1.enzyme_modules.get_by_id(\"HEX1\")\nHEX1_dict\n```\n\n []\n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    NameHEX1
    Memory address0x07ff42bab8950
    Stoichiometric Matrix13x8
    Matrix Rank7
    SubsystemGlycolysis
    Number of Ligands6
    Number of EnzymeForms7
    Number of EnzymeModuleReactions8
    Enzyme Concentration Total2.4e-05
    Enzyme Net Flux1.12
    \n\n\n\n\nThe `EnzymeModuleDict` inherits from an `OrderedDict`, thereby inheriting ordered dictionary methods such as `keys()`:\n\n\n```python\nprint(\"\\n\".join(HEX1_dict.keys()))\n```\n\n id\n name\n subsystem\n enzyme_module_ligands\n enzyme_module_forms\n enzyme_module_reactions\n enzyme_module_ligands_categorized\n enzyme_module_forms_categorized\n enzyme_module_reactions_categorized\n enzyme_concentration_total\n enzyme_rate\n enzyme_concentration_total_equation\n enzyme_rate_equation\n S\n model\n\n\nThe `EnzymeModuleDict` stores several of the enzyme-specific attributes so that they are still accessible after integrating the enzyme module into a larger network. The keys of the `EnzymeModuleDict` also can be treated as attribute accessors:\n\n\n```python\nprint(\"Enzyme Rate:\\n{0} = {1}\".format(\n HEX1_dict[\"enzyme_rate\"], # Returned using dict key\n HEX1_dict.enzyme_rate_equation # Returned using attribute accessor\n))\n```\n\n Enzyme Rate:\n 1.12 = kf_HEX1_8*(Keq_HEX1_8*hex1_AG_c(t) - adp_c(t)*g6p_c(t)*hex1_c(t))/Keq_HEX1_8\n\n\n### Steady State Validation\nThe last step is to ensure that a steady state is reached with the completed enzyme module within a larger network context.\n\n\n```python\nimport matplotlib.pyplot as plt\n\nfrom mass import Simulation\nfrom mass.visualization import plot_time_profile\n```\n\nHere, the model is simulated, and the enzyme's ability to reach steady state is graphically verified: \n\n\n```python\n# Setup simulation object\nsim = Simulation(glyc_hb_HEX1, verbose=True)\n# Simulate from 0 to 1000 with 10001 points in the output\nconc_sol, flux_sol = sim.simulate(\n glyc_hb_HEX1, time=(0, 1e3, 1e4 + 1))\n\nfig, ax = plt.subplots(nrows=1, ncols=1, figsize=(6, 4))\nplot_time_profile(\n conc_sol, observable=HEX1_dict.enzyme_module_forms, ax=ax,\n legend=\"right outside\", plot_function=\"loglog\",\n xlabel=\"Time [hr]\", ylabel=\"Concentration [mM]\",\n title=\"TIme profile of Concentrations for Enzyme Forms\");\n```\n\nThe plot shows that the enzyme can reach a steady state when integrated into a larger network, meaning the enzyme module that represents hexokinase in this system is complete!\n\n## Additional Examples\nFor additional examples of analyzing and visualizing systems with enzyme modules, see the following: \n\n* [Visualizing Catalytic Potentials of Glycolytic Regulatory Kinases](../gallery/visualization/catalytic_potential_visualizations.ipynb)\n\n$^1$ Procedure outlined in Du et al., 2016\n\n$^2$ Hexokinase based on Yurkovich et al., 2018, Du et al., 2016, and Mulquiney and Kuchel, 1999.\n\n$^3$ Glycolysis model based on Yurkovich et al., 2018 and Chapter 10 of Systems Biology: Simulation of Dynamic Network States\n", "meta": {"hexsha": "75e61faca7cad594fd2ecfb0a412fde7b40023a5", "size": 92875, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/tutorials/enzyme_modules.ipynb", "max_stars_repo_name": "SBRG/MASSpy", "max_stars_repo_head_hexsha": "1315c1d40be8feb8731c8143dbc9ba43bf8c78ff", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 13, "max_stars_repo_stars_event_min_datetime": "2020-07-13T00:48:13.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-27T15:42:15.000Z", "max_issues_repo_path": "docs/tutorials/enzyme_modules.ipynb", "max_issues_repo_name": "SBRG/MASSpy", "max_issues_repo_head_hexsha": "1315c1d40be8feb8731c8143dbc9ba43bf8c78ff", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 10, "max_issues_repo_issues_event_min_datetime": "2021-02-17T18:07:43.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-23T16:22:14.000Z", "max_forks_repo_path": "docs/tutorials/enzyme_modules.ipynb", "max_forks_repo_name": "SBRG/MASSpy", "max_forks_repo_head_hexsha": "1315c1d40be8feb8731c8143dbc9ba43bf8c78ff", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2020-01-15T00:48:49.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-04T07:01:17.000Z", "avg_line_length": 47.4336057201, "max_line_length": 26224, "alphanum_fraction": 0.6729582773, "converted": true, "num_tokens": 10280, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947155710234, "lm_q2_score": 0.66192288918838, "lm_q1q2_score": 0.4334898022449742}} {"text": "\n\n\n```python\nfrom google.colab import drive\ndrive.mount('/content/drive')\n```\n\n Go to this URL in a browser: https://accounts.google.com/o/oauth2/auth?client_id=947318989803-6bn6qk8qdgf4n4g3pfee6491hc0brc4i.apps.googleusercontent.com&redirect_uri=urn%3aietf%3awg%3aoauth%3a2.0%3aoob&response_type=code&scope=email%20https%3a%2f%2fwww.googleapis.com%2fauth%2fdocs.test%20https%3a%2f%2fwww.googleapis.com%2fauth%2fdrive%20https%3a%2f%2fwww.googleapis.com%2fauth%2fdrive.photos.readonly%20https%3a%2f%2fwww.googleapis.com%2fauth%2fpeopleapi.readonly\n \n Enter your authorization code:\n \u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\n Mounted at /content/drive\n\n\n# Neuromatch Academy: Week 3, Day 4, Tutorial 1\n# Deep Learning: Decoding Neural Responses\n\n**Content creators**: Jorge A. Menendez, Carsen Stringer\n\n**Content reviewers**: Roozbeh Farhoodi, Madineh Sarvestani, Kshitij Dwivedi, Spiros Chavlis, Ella Batty, Michael Waskom\n\n\n---\n# Tutorial Objectives\nIn this tutorial, we'll use deep learning to decode stimulus information from the responses of sensory neurons. Specifically, we'll look at the activity of ~20,000 neurons in mouse primary visual cortex responding to oriented gratings recorded in [this study](https://www.biorxiv.org/content/10.1101/679324v2.abstract). Our task will be to decode the orientation of the presented stimulus from the responses of the whole population of neurons. We could do this in a number of ways, but here we'll use deep learning. Deep learning is particularly well-suited to this problem for a number of reasons:\n* The data are very high-dimensional: the neural response to a stimulus is a ~20,000 dimensional vector. Many machine learning techniques fail in such high dimensions, but deep learning actually thrives in this regime, as long as you have enough data (which we do here!).\n* As you'll be able to see below, different neurons can respond quite differently to stimuli. This complex pattern of responses will, therefore, require non-linear methods to be decoded, which we can easily do with non-linear activation functions in deep networks.\n* Deep learning architectures are highly flexible, meaning we can easily adapt the architecture of our decoding model to optimize decoding. Here, we'll focus on a single architecture, but you'll see that it can easily be modified with few changes to the code.\n\nMore concretely, our goal will be learn how to:\n* Build a deep feed-forward network using PyTorch\n* Evaluate the network's outputs using PyTorch built-in loss functions\n* Compute gradients of the loss with respect to each parameter of the network using automatic differentiation\n* Implement gradient descent to optimize the network's parameters\n\nThis tutorial will take up the first full session (equivalent to two tutorials on other days).\n\n\n```python\n#@title Video 1: Decoding from neural data using feed-forward networks in pytorch\nfrom IPython.display import YouTubeVideo\nvideo = YouTubeVideo(id=\"SlrbMvvBOzM\", width=854, height=480, fs=1)\nprint(\"Video available at https://youtu.be/\" + video.id)\nvideo\n```\n\n Video available at https://youtu.be/SlrbMvvBOzM\n\n\n\n\n\n\n\n\n\n\n\n---\n# Setup\n\n\n\n```python\nimport os\nimport numpy as np\n\nimport torch\nfrom torch import nn\nfrom torch import optim\n\nimport matplotlib as mpl\nfrom matplotlib import pyplot as plt\n```\n\n\n```python\n#@title Data retrieval and loading\nimport hashlib\nimport requests\n\nfname = \"W3D4_stringer_oribinned1.npz\"\nurl = \"https://osf.io/683xc/download\"\nexpected_md5 = \"436599dfd8ebe6019f066c38aed20580\"\n\nif not os.path.isfile(fname):\n try:\n r = requests.get(url)\n except requests.ConnectionError:\n print(\"!!! Failed to download data !!!\")\n else:\n if r.status_code != requests.codes.ok:\n print(\"!!! Failed to download data !!!\")\n elif hashlib.md5(r.content).hexdigest() != expected_md5:\n print(\"!!! Data download appears corrupted !!!\")\n else:\n with open(fname, \"wb\") as fid:\n fid.write(r.content)\n```\n\n\n```python\n#@title Figure Settings\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/course-content/master/nma.mplstyle\")\n```\n\n\n```python\n#@title Helper Functions\n\ndef load_data(data_name=fname, bin_width=1):\n \"\"\"Load mouse V1 data from Stringer et al. (2019)\n\n Data from study reported in this preprint:\n https://www.biorxiv.org/content/10.1101/679324v2.abstract\n\n These data comprise time-averaged responses of ~20,000 neurons\n to ~4,000 stimulus gratings of different orientations, recorded\n through Calcium imaginge. The responses have been normalized by\n spontanous levels of activity and then z-scored over stimuli, so\n expect negative numbers. They have also been binned and averaged\n to each degree of orientation.\n\n This function returns the relevant data (neural responses and\n stimulus orientations) in a torch.Tensor of data type torch.float32\n in order to match the default data type for nn.Parameters in\n Google Colab.\n\n This function will actually average responses to stimuli with orientations\n falling within bins specified by the bin_width argument. This helps\n produce individual neural \"responses\" with smoother and more\n interpretable tuning curves.\n\n Args:\n bin_width (float): size of stimulus bins over which to average neural\n responses\n\n Returns:\n resp (torch.Tensor): n_stimuli x n_neurons matrix of neural responses,\n each row contains the responses of each neuron to a given stimulus.\n As mentioned above, neural \"response\" is actually an average over\n responses to stimuli with similar angles falling within specified bins.\n stimuli: (torch.Tensor): n_stimuli x 1 column vector with orientation\n of each stimulus, in degrees. This is actually the mean orientation\n of all stimuli in each bin.\n\n \"\"\"\n with np.load(data_name) as dobj:\n data = dict(**dobj)\n resp = data['resp']\n stimuli = data['stimuli']\n\n if bin_width > 1:\n # Bin neural responses and stimuli\n bins = np.digitize(stimuli, np.arange(0, 360 + bin_width, bin_width))\n stimuli_binned = np.array([stimuli[bins == i].mean() for i in np.unique(bins)])\n resp_binned = np.array([resp[bins == i, :].mean(0) for i in np.unique(bins)])\n else:\n resp_binned = resp\n stimuli_binned = stimuli\n\n # Return as torch.Tensor\n resp_tensor = torch.tensor(resp_binned, dtype=torch.float32)\n stimuli_tensor = torch.tensor(stimuli_binned, dtype=torch.float32).unsqueeze(1) # add singleton dimension to make a column vector\n\n return resp_tensor, stimuli_tensor\n\n\ndef plot_data_matrix(X, ax):\n \"\"\"Visualize data matrix of neural responses using a heatmap\n\n Args:\n X (torch.Tensor or np.ndarray): matrix of neural responses to visualize\n with a heatmap\n ax (matplotlib axes): where to plot\n\n \"\"\"\n\n cax = ax.imshow(X, cmap=mpl.cm.pink, vmin=np.percentile(X, 1), vmax=np.percentile(X, 99))\n cbar = plt.colorbar(cax, ax=ax, label='normalized neural response')\n\n ax.set_aspect('auto')\n ax.set_xticks([])\n ax.set_yticks([])\n\n\ndef identityLine():\n \"\"\"\n Plot the identity line y=x\n \"\"\"\n ax = plt.gca()\n lims = np.array([ax.get_xlim(), ax.get_ylim()])\n minval = lims[:, 0].min()\n maxval = lims[:, 1].max()\n equal_lims = [minval, maxval]\n ax.set_xlim(equal_lims)\n ax.set_ylim(equal_lims)\n line = ax.plot([minval, maxval], [minval, maxval], color=\"0.7\")\n line[0].set_zorder(-1)\n\ndef get_data(n_stim, train_data, train_labels):\n \"\"\" Return n_stim randomly drawn stimuli/resp pairs\n\n Args:\n n_stim (scalar): number of stimuli to draw\n resp (torch.Tensor):\n train_data (torch.Tensor): n_train x n_neurons tensor with neural\n responses to train on\n train_labels (torch.Tensor): n_train x 1 tensor with orientations of the\n stimuli corresponding to each row of train_data, in radians\n\n Returns:\n (torch.Tensor, torch.Tensor): n_stim x n_neurons tensor of neural responses and n_stim x 1 of orientations respectively\n \"\"\"\n n_stimuli = train_labels.shape[0]\n istim = np.random.choice(n_stimuli, n_stim)\n r = train_data[istim] # neural responses to this stimulus\n ori = train_labels[istim] # true stimulus orientation\n\n return r, ori\n\ndef stimulus_class(ori, n_classes):\n \"\"\"Get stimulus class from stimulus orientation\n\n Args:\n ori (torch.Tensor): orientations of stimuli to return classes for\n n_classes (int): total number of classes\n\n Returns:\n torch.Tensor: 1D tensor with the classes for each stimulus\n\n \"\"\"\n bins = np.linspace(0, 360, n_classes + 1)\n return torch.tensor(np.digitize(ori.squeeze(), bins)) - 1 # minus 1 to accomodate Python indexing\n\ndef plot_decoded_results(train_loss, test_labels, predicted_test_labels):\n \"\"\" Plot decoding results in the form of network training loss and test predictions\n\n Args:\n train_loss (list): training error over iterations\n test_labels (torch.Tensor): n_test x 1 tensor with orientations of the\n stimuli corresponding to each row of train_data, in radians\n predicted_test_labels (torch.Tensor): n_test x 1 tensor with predicted orientations of the\n stimuli from decoding neural network\n\n \"\"\"\n\n # Plot results\n fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(16, 6))\n\n # Plot the training loss over iterations of GD\n ax1.plot(train_loss)\n\n # Plot true stimulus orientation vs. predicted class\n ax2.plot(stimuli_test.squeeze(), predicted_test_labels, '.')\n\n ax1.set_xlim([0, None])\n ax1.set_ylim([0, None])\n ax1.set_xlabel('iterations of gradient descent')\n ax1.set_ylabel('negative log likelihood')\n ax2.set_xlabel('true stimulus orientation ($^o$)')\n ax2.set_ylabel('decoded orientation bin')\n ax2.set_xticks(np.linspace(0, 360, n_classes + 1))\n ax2.set_yticks(np.arange(n_classes))\n class_bins = [f'{i * 360 / n_classes: .0f}$^o$ - {(i + 1) * 360 / n_classes: .0f}$^o$' for i in range(n_classes)]\n ax2.set_yticklabels(class_bins);\n\n # Draw bin edges as vertical lines\n ax2.set_ylim(ax2.get_ylim()) # fix y-axis limits\n for i in range(n_classes):\n lower = i * 360 / n_classes\n upper = (i + 1) * 360 / n_classes\n ax2.plot([lower, lower], ax2.get_ylim(), '-', color=\"0.7\", linewidth=1, zorder=-1)\n ax2.plot([upper, upper], ax2.get_ylim(), '-', color=\"0.7\", linewidth=1, zorder=-1)\n\n plt.tight_layout()\n```\n\n---\n# Section 1: Load and visualize data\n\nIn the next cell, we have provided code to load the data and plot the matrix of neural responses.\n\nNext to it, we plot the tuning curves of three randomly selected neurons.\n\n\n```python\n#@title\n\n#@markdown Execute this cell to load and visualize data\n\n# Load data\nresp_all, stimuli_all = load_data() # argument to this function specifies bin width\nn_stimuli, n_neurons = resp_all.shape\n\nprint(f'{n_neurons} neurons in response to {n_stimuli} stimuli')\n\nfig, (ax1, ax2) = plt.subplots(1, 2, figsize=(2 * 6, 5))\n\n# Visualize data matrix\nplot_data_matrix(resp_all[:100, :].T, ax1) # plot responses of first 100 neurons\nax1.set_xlabel('stimulus')\nax1.set_ylabel('neuron')\n\n# Plot tuning curves of three random neurons\nineurons = np.random.choice(n_neurons, 3, replace=False) # pick three random neurons\nax2.plot(stimuli_all, resp_all[:, ineurons])\nax2.set_xlabel('stimulus orientation ($^o$)')\nax2.set_ylabel('neural response')\nax2.set_xticks(np.linspace(0, 360, 5))\n\nplt.tight_layout()\n```\n\nWe will split our data into a training set and test set. In particular, we will have a training set of orientations (`stimuli_train`) and the corresponding responses (`resp_train`). Our testing set will have held-out orientations (`stimuli_test`) and the corresponding responses (`resp_test`).\n\n\n```python\n#@title\n#@markdown Execute this cell to split into training and test sets\n\n# Set random seeds for reproducibility\nnp.random.seed(4)\ntorch.manual_seed(4)\n\n# Split data into training set and testing set\nn_train = int(0.6 * n_stimuli) # use 60% of all data for training set\nishuffle = torch.randperm(n_stimuli)\nitrain = ishuffle[:n_train] # indices of data samples to include in training set\nitest = ishuffle[n_train:] # indices of data samples to include in testing set\nstimuli_test = stimuli_all[itest]\nresp_test = resp_all[itest]\nstimuli_train = stimuli_all[itrain]\nresp_train = resp_all[itrain]\n```\n\n---\n# Section 2: Deep feed-forward networks in *pytorch* \n\nWe'll now build a simple deep neural network that takes as input a vector of neural responses and outputs a single number representing the decoded stimulus orientation.\n\nTo keep things simple, we'll build a deep network with **one** hidden layer. See the appendix for a deeper discussion of what this choice entails, and when one might want to use deeper/shallower and wider/narrower architectures.\n\nLet $\\mathbf{r}^{(n)} = \\begin{bmatrix} r_1^{(n)} & r_2^{(n)} & \\ldots & r_N^{(n)} \\end{bmatrix}^T$ denote the vector of neural responses (of neurons $1, \\ldots, N$) to the $n$th stimulus. The network we will use is described by the following set of equations:\n\\begin{align}\n \\mathbf{h}^{(n)} &= \\mathbf{W}^{in} \\mathbf{r}^{(n)} + \\mathbf{b}^{in}, && [\\mathbf{W}^{in}: M \\times N], \\\\\n y^{(n)} &= \\mathbf{W}^{out} \\mathbf{h}^{(n)} + \\mathbf{b}^{out}, && [\\mathbf{W}^{out}: 1 \\times M],\n\\end{align}\nwhere $y^{(n)}$ denotes the scalar output of the network: the decoded orientation of the $n$th stimulus. \n\nThe $M$-dimensional vector $\\mathbf{h}^{(n)}$ denotes the activations of the **hidden layer** of the network. \n\n

    \n \n

    \n\nThe blue components of this diagram denote the **parameters** of the network, which we will later optimize with gradient descent. These include all the weights and biases $\\mathbf{W}^{in}, \\mathbf{b}^{in}, \\mathbf{W}^{out}, \\mathbf{b}^{out}$.\n\n\n\n### Section 2.1: Introduction to PyTorch\n\nHere, we'll use the **PyTorch** package to build, run, and train deep networks of this form in Python. There are two core components to the PyTorch package: \n\n1. The first is the `torch.Tensor` data type used in PyTorch. `torch.Tensor`'s are effectively just like a `numpy` arrays, except that they have some important attributes and methods needed for automatic differentiation (to be discussed below). They also come along with infrastructure for easily storing and computing with them on GPU's, a capability we won't touch on here but which can be really useful in practice.\n\n2. The second core ingredient is the PyTorch `nn.Module` class. This is the class we'll use for constructing deep networks, so that we can then easily train them using built-in PyTorch functions. Keep in my mind that `nn.Module` classes can actually be used to build, run, and train any model -- not just deep networks!\n\n The next cell contains code for building the deep network we defined above using the `nn.Module` class. It contains three key ingredients:\n\n * `__init__()` method to initialize its parameters, like in any other Python class. In this case, it takes two arguments:\n * `n_inputs`: the number of input units. This should always be set to the number of neurons whose activities are being decoded (i.e. the dimensionality of the input to the network). \n * `n_hidden`: the number of hidden units. This is a parameter that we are free to vary in deciding how to build our network. See the appendix for a discussion of how this architectural choice affects the computations the network can perform.\n\n * `nn.Linear` modules, which are built-in PyTorch classes containing all the weights and biases for a given network layer (documentation [here](https://pytorch.org/docs/master/generated/torch.nn.Linear.html)). This class takes two arguments to initialize:\n * \\# of inputs to that layer\n * \\# of outputs from that layer\n\n For the input layer, for example, we have:\n * \\# of inputs = \\# of neurons whose responses are to be decoded ($N$, specified by `n_inputs`)\n * \\# of outputs = \\# of hidden layer units ($M$, specified by `n_hidden`)\n \n PyTorch will initialize all weights and biases randomly.\n\n * `forward()` method, which takes as argument an input to the network and returns the network output. In our case, this comprises computing the output $y$ from a given input $\\mathbf{r}$ using the above two equations. See the next cell for code implementing this computation using the built-in PyTorch `nn.Linear` classes.\n\n\n```python\nclass DeepNet(nn.Module):\n \"\"\"Deep Network with one hidden layer\n\n Args:\n n_inputs (int): number of input units\n n_hidden (int): number of units in hidden layer\n\n Attributes:\n in_layer (nn.Linear): weights and biases of input layer\n out_layer (nn.Linear): weights and biases of output layer\n\n \"\"\"\n\n def __init__(self, n_inputs, n_hidden):\n super().__init__() # needed to invoke the properties of the parent class nn.Module\n self.in_layer = nn.Linear(n_inputs, n_hidden) # neural activity --> hidden units\n self.out_layer = nn.Linear(n_hidden, 1) # hidden units --> output\n\n def forward(self, r):\n \"\"\"Decode stimulus orientation from neural responses\n\n Args:\n r (torch.Tensor): vector of neural responses to decode, must be of\n length n_inputs. Can also be a tensor of shape n_stimuli x n_inputs,\n containing n_stimuli vectors of neural responses\n\n Returns:\n torch.Tensor: network outputs for each input provided in r. If\n r is a vector, then y is a 1D tensor of length 1. If r is a 2D\n tensor then y is a 2D tensor of shape n_stimuli x 1.\n\n \"\"\"\n h = self.in_layer(r) # hidden representation\n y = self.out_layer(h)\n return y\n```\n\nThe next cell contains code for initializing and running this network. We use it to decode stimulus orientation from a vector of neural responses to the very first stimulus. Note that when the initialized network class is called as a function on an input (e.g. `net(r)`), its `.forward()` method is called. This is a special property of the `nn.Module` class.\n\nNote that the decoded orientations at this point will be nonsense, since the network has been initialized with random weights. Below, we'll learn how to optimize these weights for good stimulus decoding.\n\n\n```python\n# Set random seeds for reproducibility\nnp.random.seed(1)\ntorch.manual_seed(1)\n\n# Initialize a deep network with M=200 hidden units\nnet = DeepNet(n_neurons, 200)\n\n# Get neural responses (r) to and orientation (ori) to one stimulus in dataset\nr, ori = get_data(1, resp_train, stimuli_train) # using helper function get_data\n\n# Decode orientation from these neural responses using initialized network\nout = net(r) # compute output from network, equivalent to net.forward(r)\n\nprint('decoded orientation: %.2f degrees' % out)\nprint('true orientation: %.2f degrees' % ori)\n```\n\n decoded orientation: 0.08 degrees\n true orientation: 139.00 degrees\n\n\n---\n### Section 2.2: Activation functions\n\n\n```python\n#@title Video 2: Nonlinear activation functions\nfrom IPython.display import YouTubeVideo\nvideo = YouTubeVideo(id=\"JAdukDCQALA\", width=854, height=480, fs=1)\nprint(\"Video available at https://youtu.be/\" + video.id)\nvideo\n```\n\n Video available at https://youtu.be/JAdukDCQALA\n\n\n\n\n\n\n\n\n\n\n\nNote that the deep network we constructed above comprises solely **linear** operations on each layer: each layer is just a weighted sum of the elements in the previous layer. It turns out that linear hidden layers like this aren't particularly useful, since a sequence of linear transformations is actually essentially the same as a single linear transformation. We can see this from the above equations by plugging in the first one into the second one to obtain\n\\begin{equation}\n y^{(n)} = \\mathbf{W}^{out} \\left( \\mathbf{W}^{in} \\mathbf{r}^{(n)} + \\mathbf{b}^{in} \\right) + \\mathbf{b}^{out} = \\mathbf{W}^{out}\\mathbf{W}^{in} \\mathbf{r}^{(n)} + \\left( \\mathbf{W}^{out}\\mathbf{b}^{in} + \\mathbf{b}^{out} \\right)\n\\end{equation}\nIn other words, the output is still just a weighted sum of elements in the input -- the hidden layer has done nothing to change this.\n\nTo extend the set of computable input/output transformations to more than just weighted sums, we'll incorporate a **non-linear activation function** in the hidden units. This is done by simply modifying the equation for the hidden layer activations to be\n\\begin{equation}\n \\mathbf{h}^{(n)} = \\phi(\\mathbf{W}^{in} \\mathbf{r}^{(n)} + \\mathbf{b}^{in})\n\\end{equation}\nwhere $\\phi$ is referred to as the activation function. Using a non-linear activation function will ensure that the hidden layer performs a non-linear transformation of the input, which will make our network much more powerful (or *expressive*, cf. appendix). In practice, deep networks *always* use non-linear activation functions.\n\n\n\n#### Exercise 1: Nonlinear Activations \n\nCreate a new class `DeepNetReLU` by modifying our above deep network model to use a non-linear activation function. We'll use the linear rectification function:\n\\begin{equation}\n \\phi(x) = \n \\begin{cases}\n x & \\text{if } x > 0 \\\\\n 0 & \\text{else}\n \\end{cases}\n\\end{equation}\nwhich can be implemented in PyTorch using `torch.relu()`. Hidden layers with this activation function are typically referred to as \"**Re**ctified **L**inear **U**nits\", or **ReLU**'s.\n\nInitialize this network with 20 hidden units and run on an example stimulus.\n\n**Hint**: you only need to modify the `forward()` method of the above `DeepNet()` class.\n\n\n\n```python\nclass DeepNetReLU(nn.Module):\n\n def __init__(self, n_inputs, n_hidden):\n super().__init__() # needed to invoke the properties of the parent class nn.Module\n self.in_layer = nn.Linear(n_inputs, n_hidden) # neural activity --> hidden units\n self.out_layer = nn.Linear(n_hidden, 1) # hidden units --> output\n\n def forward(self, r):\n\n ############################################################################\n ## TO DO for students: write code for computing network output using a\n ## rectified linear activation function for the hidden units\n # Fill out function and remove\n #raise NotImplementedError(\"Student exercise: complete DeepNetReLU forward\")\n ############################################################################\n\n h = torch.relu(self.in_layer(r))\n y = self.out_layer(h)\n\n return y\n\n\n# Set random seeds for reproducibility\nnp.random.seed(1)\ntorch.manual_seed(1)\n\n# Get neural responses (r) to and orientation (ori) to one stimulus in dataset\nr, ori = get_data(1, resp_train, stimuli_train)\n\n# Uncomment to test your class\n\n# Initialize deep network with M=20 hidden units and uncomment lines below\nnet = DeepNetReLU(n_neurons,20)\n\n# Decode orientation from these neural responses using initialized network\n#net(r) is equivalent to net.forward(r)\nout = net(r)\n\nprint('decoded orientation: %.2f degrees' % out)\nprint('true orientation: %.2f degrees' % ori)\n```\n\n decoded orientation: 0.13 degrees\n true orientation: 139.00 degrees\n\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W3D4_DeepLearning1/solutions/W3D4_Tutorial1_Solution_5bdc2033.py)\n\n\n\nYou should see that the decoded orientation is 0.13 $^{\\circ}$ while the true orientation is 139.00 $^{\\circ}$.\n\n---\n# Section 3: Loss functions and gradient descent\n\n\n\n```python\n#@title Video 3: Loss functions & gradient descent\nfrom IPython.display import YouTubeVideo\nvideo = YouTubeVideo(id=\"aEtKpzEuviw\", width=854, height=480, fs=1)\nprint(\"Video available at https://youtu.be/\" + video.id)\nvideo\n```\n\n Video available at https://youtu.be/aEtKpzEuviw\n\n\n\n\n\n\n\n\n\n\n\n### Section 3.1: Loss functions\n\nBecause the weights of the network are currently randomly chosen, the outputs of the network are nonsense: the decoded stimulus orientation is nowhere close to the true stimulus orientation. We'll shortly write some code to change these weights so that the network does a better job of decoding.\n\nBut to do so, we first need to define what we mean by \"better\". One simple way of defining this is to use the squared error\n\\begin{equation}\n L = (y - \\tilde{y})^2\n\\end{equation}\nwhere $y$ is the network output and $\\tilde{y}$ is the true stimulus orientation. When the decoded stimulus orientation is far from the true stimulus orientation, $L$ will be large. We thus refer to $L$ as the **loss function**, as it quantifies how *bad* the network is at decoding stimulus orientation.\n\nPyTorch actually carries with it a number of built-in loss functions. The one corresponding to the squared error is called `nn.MSELoss()`. This will take as arguments a **batch** of network outputs $y_1, y_2, \\ldots, y_P$ and corresponding target outputs $\\tilde{y}_1, \\tilde{y}_2, \\ldots, \\tilde{y}_P$, and compute the **mean squared error (MSE)**\n\\begin{equation}\n L = \\frac{1}{P}\\sum_{n=1}^P \\left(y^{(n)} - \\tilde{y}^{(n)}\\right)^2\n\\end{equation}\n\n\n\n#### Exercise 2: Computing MSE \n\n\nEvaluate the mean squared error for a deep network with $M=20$ rectified linear units, on the decoded orientations from neural responses to 20 random stimuli.\n\n\n```python\n# Set random seeds for reproducibility\nnp.random.seed(1)\ntorch.manual_seed(1)\n\n# Initialize a deep network with M=20 hidden units\nnet = DeepNetReLU(n_neurons, 20)\n\n# Get neural responses to first 20 stimuli in the data set\nr, ori = get_data(20, resp_train, stimuli_train)\n\n# Decode orientation from these neural responses\nout = net(r)\n\n###################################################\n## TO DO for students: evaluate mean squared error\n###################################################\n\n# Initialize PyTorch mean squared error loss function (Hint: look at nn.MSELoss)\nloss_fn = nn.MSELoss()\n\n# Evaluate mean squared error\nloss = loss_fn(out, ori)\n\n# Uncomment once above is filled in\nprint('mean squared error: %.2f' % loss)\n```\n\n mean squared error: 42943.75\n\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W3D4_DeepLearning1/solutions/W3D4_Tutorial1_Solution_0e539ef5.py)\n\n\n\nYou should see a mean squared error of 42943.75.\n\n---\n### Section 3.2: Optimization with gradient descent\n\nOur goal is now to modify the weights to make the mean squared error loss $L$ as small as possible over the whole data set. To do this, we'll use the **gradient descent (GD)** algorithm, which consists of iterating three steps:\n1. **Evaluate the loss** on the training data,\n```\nout = net(train_data)\nloss = loss_fn(out, train_labels)\n```\nwhere `train_data` are the network inputs in the training data (in our case, neural responses), and `train_labels` are the target outputs for each input (in our case, true stimulus orientations).\n2. **Compute the gradient of the loss** with respect to each of the network weights. In PyTorch, we can do this with one line of code:\n```\nloss.backward()\n```\nThis command tells PyTorch to compute the gradients of the quantity stored in the variable `loss` with respect to each network parameter using [automatic differentiation](https://en.wikipedia.org/wiki/Automatic_differentiation). These gradients are then stored behind the scenes (see appendix for more details).\n3. **Update the network weights** by descending the gradient. In Pytorch, we can do this using built-in optimizers. We'll use the `optim.SGD` optimizer (documentation [here](https://pytorch.org/docs/stable/optim.html#torch.optim.SGD)) which updates parameters along the negative gradient, scaled by a learning rate (see appendix for details). To initialize this optimizer, we have to tell it\n * which parameters to update, and\n * what learning rate to use\n\n For example, to optimize *all* the parameters of a network `net` using a learning rate of .001, the optimizer would be initialized as follows\n ```\n optimizer = optim.SGD(net.parameters(), lr=.001)\n ```\n where `.parameters()` is a method of the `nn.Module` class that returns a [Python generator object](https://wiki.python.org/moin/Generators) over all the parameters of that `nn.Module` class (in our case, $\\mathbf{W}^{in}, \\mathbf{b}^{in}, \\mathbf{W}^{out}, \\mathbf{b}^{out}$).\n \n After computing all the parameter gradients in step 2, we can then update each of these parameters using the `.step()` method of this optimizer,\n ```\n optimizer.step()\n ```\n This single line of code will extract all the gradients computed with `.backward()` and execute the SGD updates for each parameter given to the optimizer. Note that this is true no matter how big/small the network is, allowing us to use the same two lines of code to perform the gradient descent updates for any deep network model built using PyTorch.\n\nFinally, an important detail to remember is that the gradients of each parameter need to be cleared before calling `.backward()`, or else PyTorch will try to accumulate gradients across iterations. This can again be done using built-in optimizers via the method `zero_grad()`, as follows:\n```\noptimizer.zero_grad()\n```\n\nPutting all this together, each iteration of the GD algorith will contain a block of code that looks something like this:\n```\nGet outputs from network\nEvaluate loss\n\n# Compute gradients\noptimizer.zero_grad() # clear gradients\nloss.backward()\n\n# Update weights\noptimizer.step()\n```\n\nIn the next exercise, we'll give you a code skeleton for implementing the GD algorithm. Your job will be to fill in the blanks.\n\nFor the mathematical details of the GD algorithm, see the appendix. Note, in particular, that here we using the gradient descent algorithm, rather than the more commonly used *stochastic* gradient descent algorithm. See the appendix for a more detailed discussion of how these differ and when one might need to use the stochastic variant.\n\n#### Exercise 3: Gradient descent in PyTorch\n\nComplete the function `train()` that uses the gradient descent algorithm to optimize the weights of a given network. This function takes as input arguments\n* `net`: the PyTorch network whose weights to optimize\n* `loss_fn`: the PyTorch loss function to use to evaluate the loss\n* `train_data`: the training data to evaluate the loss on (i.e. neural responses to decode)\n* `train_labels`: the target outputs for each data point in `train_data` (i.e. true stimulus orientations)\n\nWe will then train a neural network on our data and plot the loss (mean squared error) over time. When we run this function, behind the scenes PyTorch is actually changing the parameters inside this network to make the network better at decoding, so its weights will now be different than they were at initialization.\n\n\n**Hint:** all the code you need for doing this is provided in the above description of the GD algorithm.\n\n\n```python\ndef train(net, loss_fn, train_data, train_labels, n_iter=50, learning_rate=1e-4):\n \"\"\"Run gradient descent to opimize parameters of a given network\n\n Args:\n net (nn.Module): PyTorch network whose parameters to optimize\n loss_fn: built-in PyTorch loss function to minimize\n train_data (torch.Tensor): n_train x n_neurons tensor with neural\n responses to train on\n train_labels (torch.Tensor): n_train x 1 tensor with orientations of the\n stimuli corresponding to each row of train_data, in radians\n n_iter (int): number of iterations of gradient descent to run\n learning_rate (float): learning rate to use for gradient descent\n\n Returns:\n (list): training loss over iterations\n\n \"\"\"\n\n # Initialize PyTorch SGD optimizer\n optimizer = optim.SGD(net.parameters(), lr=learning_rate)\n\n # Placeholder to save the loss at each iteration\n track_loss = []\n\n # Loop over epochs (cf. appendix)\n for i in range(n_iter):\n\n ######################################################################\n ## TO DO for students: fill in missing code for GD iteration\n #raise NotImplementedError(\"Student exercise: write code for GD iterations\")\n ######################################################################\n\n # Evaluate loss using loss_fn\n out = net(train_data) # compute network output from inputs in train_data\n loss = loss_fn(out,train_labels) # evaluate loss function\n\n # Compute gradients\n optimizer.zero_grad()\n loss.backward()\n\n # Update weights\n optimizer.step()\n\n # Store current value of loss\n track_loss.append(loss.item()) # .item() needed to transform the tensor output of loss_fn to a scalar\n\n # Track progress\n if (i + 1) % (n_iter // 5) == 0:\n print(f'iteration {i + 1}/{n_iter} | loss: {loss.item():.3f}')\n\n return track_loss\n\n# Set random seeds for reproducibility\nnp.random.seed(1)\ntorch.manual_seed(1)\n\n# Initialize network\nnet = DeepNetReLU(n_neurons, 20)\n\n# Initialize built-in PyTorch MSE loss function\nloss_fn = nn.MSELoss()\n\n# Run GD on data\ntrain_loss = train(net, loss_fn, resp_train, stimuli_train)\n\n# Plot the training loss over iterations of GD\nplt.plot(train_loss)\nplt.xlim([0, None])\nplt.ylim([0, None])\nplt.xlabel('iterations of gradient descent')\nplt.ylabel('mean squared error')\nplt.show()\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W3D4_DeepLearning1/solutions/W3D4_Tutorial1_Solution_8f827dbe.py)\n\n*Example output:*\n\n\n\n\n\n---\n# Section 4: Evaluating model performance\n\n\n\n## Section 4.1: Generalization performance with test data\n\nNote that gradient descent is essentially an algorithm for fitting the network's parameters to a given set of training data. Selecting this training data is thus crucial for ensuring that the optimized parameters **generalize** to unseen data they weren't trained on. In our case, for example, we want to make sure that our trained network is good at decoding stimulus orientations from neural responses to any orientation, not just those in our data set.\n\nTo ensure this, we have split up the full data set into a **training set** and a **testing set**. In Exercise 3, we trained a deep network by optimizing the parameters on a training set. We will now evaluate how good the optimized parameters are by using the trained network to decode stimulus orientations from neural responses in the testing set. Good decoding performance on this testing set should then be indicative of good decoding performance on the neurons' responses to any other stimulus orientation. This procedure is commonly used in machine learning (not just in deep learning)and is typically referred to as **cross-validation**.\n\nWe will compute the MSE on the test data and plot the decoded stimulus orientations as a function of the true stimulus.\n\n\n\n```python\n#@title\n#@markdown Execute this cell to evaluate and plot test error\n\nout = net(resp_test) # decode stimulus orientation for neural responses in testing set\nori = stimuli_test # true stimulus orientations\ntest_loss = loss_fn(out, ori) # MSE on testing set (Hint: use loss_fn initialized in previous exercise)\n\nplt.plot(ori, out.detach(), '.') # N.B. need to use .detach() to pass network output into plt.plot()\nidentityLine() # draw the identity line y=x; deviations from this indicate bad decoding!\nplt.title('MSE on testing set: %.2f' % test_loss.item()) # N.B. need to use .item() to turn test_loss into a scalar\nplt.xlabel('true stimulus orientation ($^o$)')\nplt.ylabel('decoded stimulus orientation ($^o$)')\naxticks = np.linspace(0, 360, 5)\nplt.xticks(axticks)\nplt.yticks(axticks)\nplt.show()\n```\n\n**PyTorch Note**:\n\nAn important thing to note in the code snippet for plotting the decoded orientations is the `.detach()` method. The PyTorch `nn.Module` class is special in that, behind the scenes, each of the variables inside it are linked to each other in a computational graph, for the purposes of automatic differentiation (the algorithm used in `.backward()` to compute gradients). As a result, if you want to do anything that is not a `torch` operation to the parameters or outputs of an `nn.Module` class, you'll need to first \"detach\" it from its computational graph. This is what the `.detach()` method does. In this hidden code above, we need to call it on the outputs of the network so that we can plot them with the `plt.plot()` function.\n\n---\n## (Bonus) Section 4.2: Model criticism\n\nPlease move to the Summary and visit this section only if you have time after completing all non-bonus material! \n\nLet's now take a step back and think about how our model is succeeding/failing and how to improve it.\n\n\n```python\n#@title\n#@markdown Execute this cell to plot decoding error\n\nout = net(resp_test) # decode stimulus orientation for neural responses in testing set\nori = stimuli_test # true stimulus orientations\nerror = out - ori # decoding error\n\n\nplt.plot(ori, error.detach(), '.') # plot decoding error as a function of true orientation (make sure all arguments to plt.plot() have been detached from PyTorch network!)\n\n# Plotting\nplt.xlabel('true stimulus orientation ($^o$)')\nplt.ylabel('decoding error ($^o$)')\nplt.xticks(np.linspace(0, 360, 5))\nplt.yticks(np.linspace(-360, 360, 9))\nplt.show()\n```\n\n### Think \n\nIn the cell below, we will plot the *decoding error* for each neural response in the testing set. The decoding error is defined as the decoded stimulus orientation minus true stimulus orientation\n\\begin{equation}\n \\text{decoding error} = y^{(n)} - \\tilde{y}^{(n)}\n\\end{equation}\n\nIn particular, we plot decoding error as a function of the true stimulus orientation.\n\n\n * Are some stimulus orientations harder to decode than others?\n * If so, in what sense? Are the decoded orientations for these stimuli more variable and/or are they biased?\n * Can you explain this variability/bias? What makes these stimulus orientations different from the others?\n * (Will be addressed in next exercise) Can you think of a way to modify the deep network in order to avoid this?\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W3D4_DeepLearning1/solutions/W3D4_Tutorial1_Solution_3ccf8501.py)\n\n\n\n### (Advanced Bonus) Exercise 4: Improving the loss function \nAs illustrated in the previous exercise, the squared error is not a good loss function for circular quantities like angles, since two angles that are very close (e.g. $1^o$ and $359^o$) might actually have a very large squared error.\n\nHere, we'll avoid this problem by changing our loss function to treat our decoding problem as a **classification problem**. Rather than estimating the *exact* angle of the stimulus, we'll now aim to construct a decoder that classifies the stimulus into one of $C$ classes, corresponding to different bins of angles of width $b = \\frac{360}{C}$. The true class $\\tilde{y}^{(n)}$ of stimulus $i$ is now given by\n\\begin{equation}\n \\tilde{y}^{(n)} =\n \\begin{cases}\n 1 &\\text{if angle of stimulus $n$ is in the range } [0, b] \\\\\n 2 &\\text{if angle of stimulus $n$ is in the range } [b, 2b] \\\\\n 3 &\\text{if angle of stimulus $n$ is in the range } [2b, 3b] \\\\\n \\vdots \\\\\n C &\\text{if angle of stimulus $n$ is in the range } [(C-1)b, 360]\n \\end{cases}\n\\end{equation}\n\nWe have a helper function `stimulus_class` that will extract `n_classes` stimulus classes for us from the stimulus orientations.\n\nTo decode the stimulus class from neural responses, we'll use a deep network that outputs a $C$-dimensional vector of probabilities $\\mathbf{p} = \\begin{bmatrix} p_1, p_2, \\ldots, p_C \\end{bmatrix}^T$, corresponding to the estimated probabilities of the stimulus belonging to each class $1, 2, \\ldots, C$. \n\nTo ensure the network's outputs are indeed probabilities (i.e. they are positive numbers between 0 and 1, and sum to 1), we'll use a [softmax function](https://en.wikipedia.org/wiki/Softmax_function) to transform the real-valued outputs from the hidden layer into probabilities. Letting $\\sigma(\\cdot)$ denote this softmax function, the equations describing our network are\n\\begin{align}\n \\mathbf{h}^{(n)} &= \\phi(\\mathbf{W}^{in} \\mathbf{r}^{(n)} + \\mathbf{b}^{in}), && [\\mathbf{W}^{in}: M \\times N], \\\\\n \\mathbf{p}^{(n)} &= \\sigma(\\mathbf{W}^{out} \\mathbf{h}^{(n)} + \\mathbf{b}^{out}), && [\\mathbf{W}^{out}: C \\times M],\n\\end{align}\nThe decoded stimulus class is then given by that assigned the highest probability by the network:\n\\begin{equation}\n y^{(n)} = \\underset{i}{\\arg\\max} \\,\\, p_i\n\\end{equation}\nThe softmax function can be implemented in PyTorch simply using `torch.softmax()`.\n\nOften *log* probabilities are easier to work with than actual probabilities, because probabilities tend to be very small numbers that computers have trouble representing. We'll therefore actually use the logarithm of the softmax as the output of our network,\n\\begin{equation}\n \\mathbf{l}^{(n)} = \\log \\left( \\mathbf{p}^{(n)} \\right)\n\\end{equation}\nwhich can implemented in PyTorch together with the softmax via an `nn.LogSoftmax` layer. The nice thing about the logarithmic function is that it's *monotonic*, so if one probability is larger/smaller than another, then its logarithm is also larger/smaller than the other's. We therefore have that\n\\begin{equation}\n y^{(n)} = \\underset{i}{\\arg\\max} \\,\\, p_i^{(n)} = \\underset{i}{\\arg\\max} \\, \\log p_i^{(n)} = \\underset{i}{\\arg\\max} \\,\\, l_i^{(n)}\n\\end{equation}\n\nSee the next cell for code for constructing a deep network with one hidden layer that of ReLU's that outputs a vector of log probabilities.\n\n\n```python\n# Deep network for classification\nclass DeepNetSoftmax(nn.Module):\n \"\"\"Deep Network with one hidden layer, for classification\n\n Args:\n n_inputs (int): number of input units\n n_hidden (int): number of units in hidden layer\n n_classes (int): number of outputs, i.e. number of classes to output\n probabilities for\n\n Attributes:\n in_layer (nn.Linear): weights and biases of input layer\n out_layer (nn.Linear): weights and biases of output layer\n\n \"\"\"\n\n def __init__(self, n_inputs, n_hidden, n_classes):\n super().__init__() # needed to invoke the properties of the parent class nn.Module\n self.in_layer = nn.Linear(n_inputs, n_hidden) # neural activity --> hidden units\n self.out_layer = nn.Linear(n_hidden, n_classes) # hidden units --> outputs\n self.logprob = nn.LogSoftmax(dim=1) # probabilities across columns should sum to 1 (each output row corresponds to a different input)\n\n def forward(self, r):\n \"\"\"Predict stimulus orientation bin from neural responses\n\n Args:\n r (torch.Tensor): n_stimuli x n_inputs tensor with neural responses to n_stimuli\n\n Returns:\n torch.Tensor: n_stimuli x n_classes tensor with predicted class probabilities\n\n \"\"\"\n h = torch.relu(self.in_layer(r))\n logp = self.logprob(self.out_layer(h))\n return logp\n```\n\nWhat should our loss function now be? Ideally, we want the probabilities outputted by our network to be such that the probability of the true stimulus class is high. One way to formalize this is to say that we want to maximize the *log* probability of the true stimulus class $\\tilde{y}^{(n)}$ under the class probabilities predicted by the network,\n\\begin{equation}\n \\log \\left( \\text{predicted probability of stimulus } n \\text{ being of class } \\tilde{y}^{(n)} \\right) = \\log p^{(n)}_{\\tilde{y}^{(n)}} = l^{(n)}_{\\tilde{y}^{(n)}}\n\\end{equation}\nTo turn this into a loss function to be *minimized*, we can then simply multiply it by -1: maximizing the log probability is the same as minimizing the *negative* log probability. Summing over a batch of $P$ inputs, our loss function is then given by\n\\begin{equation}\n L = -\\sum_{n=1}^P \\log p^{(n)}_{\\tilde{y}^{(n)}} = -\\sum_{n=1}^P l^{(n)}_{\\tilde{y}^{(n)}}\n\\end{equation}\nIn the deep learning community, this loss function is typically referred to as the **cross-entropy**, or **negative log likelihood**. The corresponding built-in loss function in PyTorch is `nn.NLLLoss()` (documentation [here](https://pytorch.org/docs/master/generated/torch.nn.CrossEntropyLoss.html)).\n\nIn the next cell, we've provided most of the code to train and test a network to decode stimulus orientations via classification, by minimizing the negative log likelihood. Fill in the missing pieces.\n\nOnce you've done this, have a look at the plotted results. Does changing the loss function from mean squared error to a classification loss solve our problems? Note that errors may still occur -- but are these errors as bad as the ones that our network above was making?\n\n\n```python\ndef decode_orientation(n_classes, train_data, train_labels, test_data, test_labels):\n \"\"\" Initialize, train, and test deep network to decode binned orientation from neural responses\n\n Args:\n n_classes (scalar): number of classes in which to bin orientation\n train_data (torch.Tensor): n_train x n_neurons tensor with neural\n responses to train on\n train_labels (torch.Tensor): n_train x 1 tensor with orientations of the\n stimuli corresponding to each row of train_data, in radians\n test_data (torch.Tensor): n_test x n_neurons tensor with neural\n responses to train on\n test_labels (torch.Tensor): n_test x 1 tensor with orientations of the\n stimuli corresponding to each row of train_data, in radians\n\n Returns:\n (list, torch.Tensor): training loss over iterations, n_test x 1 tensor with predicted orientations of the\n stimuli from decoding neural network\n \"\"\"\n\n # Bin stimulus orientations in training set\n train_binned_labels = stimulus_class(train_labels, n_classes)\n\n ##############################################################################\n ## TODO for students: fill out missing pieces below to initialize, train, and\n # test network\n # Fill out function and remove\n raise NotImplementedError(\"Student exercise: complete decode_orientation function\")\n ##############################################################################\n\n # Initialize network\n net = ... # use M=20 hidden units\n\n # Initialize built-in PyTorch MSE loss function\n loss_fn = nn.NLLLoss()\n\n # Run GD on training set data, using learning rate of 0.1\n train_loss = ...\n\n # Decode neural responses in testing set data\n out = ...\n out_labels = np.argmax(out.detach(), axis=1) # predicted classes\n\n return train_loss, out_labels\n\n\n# Set random seeds for reproducibility\nnp.random.seed(1)\ntorch.manual_seed(1)\n\nn_classes = 12 # start with 12, then (bonus) try making this as big as possible! does decoding get worse?\n\n# Uncomment below to test your function\n\n# Initialize, train, and test network\n#train_loss, predicted_test_labels = decode_orientation(n_classes, resp_train, stimuli_train, resp_test, stimuli_test)\n\n# Plot results\n#plot_decoded_results(train_loss, stimuli_test, predicted_test_labels)\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W3D4_DeepLearning1/solutions/W3D4_Tutorial1_Solution_e2e84634.py)\n\n*Example output:*\n\n\n\n\n\n---\n# Summary\n\nWe have now covered a number of common and powerful techniques for applying deep learning to decoding from neural data, some of which are common to almost any machine learning problem:\n* Building and training deep networks using the **PyTorch** `nn.Module` class and built-in **optimizers**\n* Choosing and evaluating **loss functions**\n* Testing a trained model on unseen data via **cross-validation**, by splitting the data into a **training set and testing set**\n\nAn important aspect of this tutorial was the `train()` function we wrote in exercise 6. Note that it can be used to train *any* network to minimize *any* loss function (cf. advanced exercise) on *any* training data. This is the power of using PyTorch to train neural networks and, for that matter, **any other model**! There is nothing in the `nn.Module` class that forces us to use `nn.Linear` layers that implement neural network operations. You can actually put anything you want inside the `.__init__()` and `.forward()` methods of this class. As long as its parameters and computations involve only `torch.Tensor`'s, and the model is differentiable, you'll then be able to optimize the parameters of this model in exactly the same way we optimized the deep networks here.\n\nWhat kinds of conclusions can we draw from these sorts of analyses? If we can decode the stimulus well from visual cortex activity, that means that there is information about this stimulus available in visual cortex. Whether or not the animal uses that information to make decisions is not determined from an analysis like this. In fact mice perform poorly in orientation discrimination tasks compared to monkeys and humans, even though they have information about these stimuli in their visual cortex. Why do you think they perform poorly in orientation discrimination tasks?\n\nSee this paper for some potential hypotheses (https://www.biorxiv.org/content/10.1101/679324v2), but this is totally an open question!\n\n---\n# Appendix\n\n## Neural network *depth*, *width* and *expressivity*\n\nTwo important architectural choices that always have to be made when constructing deep feed-forward networks like those used here are\n* the number of hidden layers, or the network's *depth*\n* the number of units in each layer, or the layer *widths*\n\nHere, we restricted ourselves to networks with a single hidden layer with a width of $M$ units, but it is easy to see how this code could be adapted to arbitrary depths. Adding another hidden layer simply requires adding another `nn.Linear` module to the `__init__()` method and incorporating it into the `.forward()` method.\n\nThe depth and width of a network determine the set of input/output transormations that it can perform, often referred to as its *expressivity*. The deeper and wider the network, the more *expressive* it is; that is, the larger the class of input/output transformations it can compute. In fact, it turns out that an infinitely wide *or* infinitely deep networks can in principle [compute (almost) *any* input/output transformation](https://en.wikipedia.org/wiki/Universal_approximation_theorem).\n\nA classic mathematical demonstration of the power of depth is given by the so-called [XOR problem](https://medium.com/@jayeshbahire/the-xor-problem-in-neural-networks-50006411840b#:~:text=The%20XOr%2C%20or%20%E2%80%9Cexclusive%20or,value%20if%20they%20are%20equal.). This toy problem demonstrates how even a single hidden layer can drastically expand the set of input/output transformations a network can perform, relative to a shallow network with no hidden layers. The key intuition is that the hidden layer allows you to represent the input in a new format, which can then allow you to do almost anything you want with it. The *wider* this hidden layer, the more flexibility you have in this representation. In particular, if you have more hidden units than input units, then the hidden layer representation of the input is higher-dimensional than the raw data representation. This higher dimensionality effectively gives you more \"room\" to perform arbitrary computations in. It turns out that even with just this one hidden layer, if you make it wide enough you can actually approximate any input/output transformation you want. See [here](http://neuralnetworksanddeeplearning.com/chap4.html) for a neat visual demonstration of this.\n\nIn practice, however, it turns out that increasing depth seems to grant more expressivity with fewer units than increasing width does (for reasons that are not well understood). It is for this reason that truly *deep* networks are almost always used in machine learning, which is why this set of techniques is often referred to as *deep* learning.\n\nThat said, there is a cost to making networks deeper and wider. The bigger your network, the more parameters (i.e. weights and biases) it has, which need to be optimized! The extra expressivity afforded by higher width and/or depth thus carries with it (at least) two problems:\n* optimizing more parameters usually requires more data\n* a more highly parameterized network is more prone to overfit to the training data, so requires more sophisticated optimization algorithms to ensure generalization\n\n## Gradient descent equations\n\nHere we provide the equations for the three steps of the gradient descent algorithm, as applied to our decoding problem:\n\n1. **Evaluate the loss** on the training data. For a mean squared error loss, this is given by\n\\begin{equation}\n L = \\frac{1}{P}\\sum_{n=1}^P (y^{(n)} - \\tilde{y}^{(n)})^2\n\\end{equation}\nwhere $y^{(n)}$ denotes the stimulus orientation decoded from the population response $\\mathbf{r}^{(n)}$ to the $n$th stimulus in the training data, and $\\tilde{y}^{(n)}$ is the true orientation of that stimulus. $P$ denotes the total number of data samples in the training set. In the syntax of our `train()` function above, $\\mathbf{r}^{(n)}$ is given by `train_data[n, :]` and $\\tilde{y}^{(n)}$ by `train_labels[n]`.\n\n2. **Compute the gradient of the loss** with respect to each of the network weights. In our case, this entails computing the quantities\n\\begin{equation}\n \\frac{\\partial L}{\\partial \\mathbf{W}^{in}}, \\frac{\\partial L}{\\partial \\mathbf{b}^{in}}, \\frac{\\partial L}{\\partial \\mathbf{W}^{out}}, \\frac{\\partial L}{\\partial \\mathbf{b}^{out}}\n\\end{equation}\nUsually, we would require lots of math in order to derive each of these gradients, and lots of code to compute them. But this is where PyTorch comes to the rescue! Using a cool technique called [automatic differentiation](https://en.wikipedia.org/wiki/Automatic_differentiation), PyTorch automatically calculates these gradients when the `.backward()` function is called.\n\n More specifically, when this function is called on a particular variable (e.g. `loss`, as above), PyTorch will compute the gradients with respect to each network parameter. These are computed and stored behind the scenes, and can be accessed through the `.grad` attribute of each of the network's parameters. As we saw above, however, we actually never need to look at or call these gradients when implementing gradient descent, as this can be taken care of by PyTorch's built-in optimizers, like `optim.SGD`.\n\n3. **Update the network weights** by descending the gradient:\n\\begin{align}\n \\mathbf{W}^{in} &\\leftarrow \\mathbf{W}^{in} - \\alpha \\frac{\\partial L}{\\partial \\mathbf{W}^{in}} \\\\\n \\mathbf{b}^{in} &\\leftarrow \\mathbf{b}^{in} - \\alpha \\frac{\\partial L}{\\partial \\mathbf{b}^{in}} \\\\\n \\mathbf{W}^{out} &\\leftarrow \\mathbf{W}^{out} - \\alpha \\frac{\\partial L}{\\partial \\mathbf{W}^{out}} \\\\\n \\mathbf{b}^{out} &\\leftarrow \\mathbf{b}^{out} - \\alpha \\frac{\\partial L}{\\partial \\mathbf{b}^{out}}\n\\end{align}\nwhere $\\alpha$ is called the **learning rate**. This **hyperparameter** of the SGD algorithm controls how far we descend the gradient on each iteration. It should be as large as possible so that fewer iterations are needed, but not too large so as to avoid parameter updates from skipping over minima in the loss landscape.\n\nWhile the equations written down here are specific to the network and loss function considered in this tutorial, the code provided above for implementing these three steps is completely general: no matter what loss function or network you are using, exactly the same commands can be used to implement these three steps.\n\n## *Stochastic* gradient descent (SGD) vs. gradient descent (GD)\n\nIn this tutorial, we used the gradient descent algorithm, which differs in a subtle yet very important way from the more commonly used **stochastic gradient descent (SGD)** algorithm. The key difference is in the very first step of each iteration, where in the GD algorithm we evaluate the loss *at every data sample in the training set*. In SGD, on the other hand, we evaluate the loss only at a random subset of data samlpes from the full training set, called a **mini-batch**. At each iteration, we randomly sample a mini-batch to perform steps 1-3 on. All the above equations still hold, but now the $P$ data samples $\\mathbf{r}^{(n)}, \\tilde{y}^{(n)}$ denote a mini-batch of $P$ random samples from the training set, rather than the whole training set.\n\nThere are several reasons why one might want to use SGD instead of GD. The first is that the training set might be too big, so that we actually can't actually evaluate the loss on every single data sample in it. In this case, GD is simply infeasible, so we have no choice but to turn to SGD, which bypasses the restrictive memory demands of GD by sub-sampling the training set into smaller mini-batches.\n\nBut, even when GD is feasible, SGD turns out to be generally better. The stochasticity induced by the extra random sampling step in SGD effectively adds some noise in the search for local minima of the loss function. This can be really useful for avoiding potential local minima, and enforce that whatever minimum is converged to is a good one. This is particularly important when networks are wider and/or deeper, in which case the large number of parameters can lead to overfitting.\n\nHere, we used only GD because (1) it is simpler, and (2) it suffices for the problem being considered here. Because we have so many neurons in our data set, decoding is not too challenging and doesn't require a particularly deep or wide network. The small number of parameters in our deep networks therefore can be optimized without a problem using GD.\n", "meta": {"hexsha": "3d41be7f0d0714b57624e0ec634c4ac331610ee7", "size": 901482, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W3D4_DeepLearning1/student/W3D4_Tutorial1.ipynb", "max_stars_repo_name": "hnoamany/course-content", "max_stars_repo_head_hexsha": "d89047537e57854c62cb9536a9c768b235fe4bf8", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tutorials/W3D4_DeepLearning1/student/W3D4_Tutorial1.ipynb", "max_issues_repo_name": "hnoamany/course-content", "max_issues_repo_head_hexsha": "d89047537e57854c62cb9536a9c768b235fe4bf8", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/W3D4_DeepLearning1/student/W3D4_Tutorial1.ipynb", "max_forks_repo_name": "hnoamany/course-content", "max_forks_repo_head_hexsha": "d89047537e57854c62cb9536a9c768b235fe4bf8", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 901482.0, "max_line_length": 901482, "alphanum_fraction": 0.94628068, "converted": true, "num_tokens": 13995, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.6619228625116081, "lm_q1q2_score": 0.43348979369132906}} {"text": "\n\n\n\n\n```python\n%%html\n\n\n

    Code is hidden for ease of viewing. Click the Show/Hide button to see. \n

    \n```\n\n\n\n\n

    Code is hidden for ease of viewing. Click the Show/Hide button to see. \n

    \n\n\n\n\n```python\nfrom IPython.display import display, Math, Latex, HTML\nfrom helper import *\n%matplotlib inline\n```\n\n\n\n\n\n\n# Graphing two variables \n\n** or **\n\n## Understanding the Relationship Between Two Variables Using Graphs \n***\n\n\n\n\n\n
    *GIF taken from https://giphy.com/gifs/reaction-BmmfETghGOPrW, June 11th, 2018.*
    \n\n***\n\n## **Introduction**\n\nYou may have heard the old saying, \"you get what you give\". It means what you get is worth the same as what you give up for it. But have you ever noticed that if you *change* what you give, it can *change* what you get? Like when you eat more food, you get more energy, or when you spend more time on homework, you get better grades.\n\nIn these examples, the things you get (or give) are like **variables**\u2014they are amounts that are allowed to change. Sometimes, changing the value of one variable can affect the value of another. Think about this example:\n\n> ### Example \n> Say you have a job that pays you 10 dollars per hour. The amount of money you earn after working 5 hours depends on this wage (5 hours)x(10 dollars per hour) = 50 dollars. If your wage goes up to 12 dollars per hour, the amount of money you would make in 5 hours would change as well (5 hours)x(12 dollars per hour) = 60 dollars. So, your wage is a *variable* that determines how much money you are going to make. \n\nVariables are a part of our lives every day, but their exact effect can be hard to understand on their own. A useful tool to help us see how variables affect results is called a **graph**, which is simply a visual way to show how variables are related.\n\nThis lesson will focus on how we can use graphs to improve our understanding of how two variables interact with each other. There are many different types of graphs, but we will focus on line graphs in this lesson to show you how to work with **linear functions**. We will go through:\n- how to create a graph of a linear function \n- how to analyze some important features of a graph\n\nTake a look at the graph shown below. This is a line graph that shows the relationship between a person's wage and how much money they earn after working 5 hours. You can use the toolbar at the top of the graph to play with it.\n\n\n```python\nwages = np.linspace(0,50,11)\nmoney = []\nfor item in wages:\n y = item*5\n money.append(y)\n\nlayout = go.Layout(\n title = \"Money Earned After Working 5 Hours\",\n yaxis = dict(\n title=\"Money Earned (Dollars)\"\n ),\n xaxis = dict(\n title=\"Wage (Dollars per hour)\",\n ),\n)\n\ntrace1 = go.Scatter(\n x = wages,\n y = money,\n mode = \"lines+markers\",\n name = \"Plot 1\",\n line=dict(\n color = ('rgb(205, 12, 24)'),\n shape = \"spline\",\n dash = \"dot\"\n )\n \n)\n\nfig = go.Figure(data = [trace1], layout = layout)\npy.offline.iplot(fig)\n```\n\n\n
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    Figure 1: This graph shows the relationship between a person's wage, and the money they will make after working for 5 hours.
    \n\n\n***\n\n## **Background**\n\n### **Algebraic Equations**\n\nIn mathematics, the relationship between variables can be written as an **equation**. An equation is like a sentence\u2014it uses an 'equals' sign to connect two expressions, like how a sentence uses a verb to connect a subject and an object. For example, the relationship between wage and the amount of money made after working for 5 hours, as described above, can be written like this:\n\n\\begin{equation} \n\\label{1}\n\\rm MONEY\\: MADE = 5\\: HOURS \\times WAGE\n\\end{equation}\n\nHere is the same equation, only now it is expressed using **algebra** (with letters in place of the words):\n\n\\begin{equation} \n\\label{2}\nM = 5W\n\\end{equation}\n\nThis is an example of an **algebraic equation**. It compresses all the information about the variables into a very small space that's easy to work with. The $W$ corresponds to wage, and the $M$ corresponds to the resulting amount of money made.\n\nWhen creating an algebraic expression, the letters that you use for variables really doesn't matter. A good choice is to use the first letter of the word (like how we used $M$ for money and $W$ for wage). However, in mathematics, the default letters are $x$ and $y$, especially when a graph will be used.\n\nIn most of the equations you'll see in class, the variables may not have any real-world meaning, but that's okay. Take a look at the examples below\u2014each one is an algebraic equation.\n\n>#### **Examples**\n>\n>\\begin{equation}\ny = 3x + 9\n\\end{equation}\n>\n>\\begin{equation}\ny = -5x\n\\end{equation}\n>\n>\\begin{equation}\ny = \\frac{x}{4} + 3\n\\end{equation}\n>\n>\\begin{equation}\nx = \\frac{y + 7}{2}\n\\end{equation}\n***\n\n### **Linear Functions**\n\nIf an equation says a variable is equal to something that doesn't involve that variable, that equation is called a **function**. For example, the equation $y=2x-3$ says that $y$ (a variable) is equal to $2x-3$ (which uses $x$, but not $y$). In this case, we say $y$ is a *function* of $x$. A **linear function** has the form\n\n\\begin{equation}\ny = mx + b\n\\end{equation}\n\nwhere both $x$ and $y$ are variables, while $m$ and $b$ are **constants**. A *constant* is a number in an equation that does not change\u2014in other words, it is not a variable. A constant can be any number\u2014positive, negative, a fraction, etc. It can even be zero!\n\nTake another look at the equations shown in the section above, these are all linear functions! When a linear function is used to create a graph, the graph will look like a straight line (hence the name *linear* function).\n\nUse the section below to better your understanding of the difference between a *linear* function, and a *non-linear* function.\n\n\n\n\n\n\n```python\ndisplay(\"Is this function linear or non-linear?\")\ndisplay(Math('y = 12x -3'))\n\ninteract(lin_or_non1, val = widgets.Dropdown(options=[' ', 'Linear', 'Non-Linear'],value = ' ',description = 'Choose One:',disabled = False));\n\n```\n\n\n 'Is this function linear or non-linear?'\n\n\n\n$\\displaystyle y = 12x -3$\n\n\n\n interactive(children=(Dropdown(description='Choose One:', options=(' ', 'Linear', 'Non-Linear'), value=' '), O\u2026\n\n\n\n```python\ndisplay(\"Is this function linear or non-linear?\")\ndisplay(Math('y = 2x^2 + 17'))\n \ninteract(lin_or_non2, val = widgets.Dropdown(options=[' ', 'Linear', 'Non-Linear'],value = ' ',description = 'Choose One:',disabled = False));\n\n```\n\n\n 'Is this function linear or non-linear?'\n\n\n\n$\\displaystyle y = 2x^2 + 17$\n\n\n\n interactive(children=(Dropdown(description='Choose One:', options=(' ', 'Linear', 'Non-Linear'), value=' '), O\u2026\n\n\n\n```python\nx = np.linspace(-10,10,21)\ny = []\nfor num in x:\n y_val = num**2\n y.append(y_val)\n\nlayout = go.Layout(\n title = \"Y as a Function of X\",\n yaxis = dict(\n title=\"Y Values\"\n ),\n xaxis = dict(\n title=\"X Values\"\n ),\n)\n\ntrace1 = go.Scatter(\n x = x,\n y = y,\n mode = \"lines+markers\",\n name = \"Plot 1\",\n line=dict(\n color = ('rgb(22, 96, 167)'),\n shape = \"spline\",\n dash = \"dot\"\n )\n \n)\n\nfig = go.Figure(data = [trace1], layout = layout)\npy.offline.iplot(fig)\n```\n\n\n
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    \n\n\n\n```python\ndisplay(\"Is the above function linear or non-linear?\")\n\ninteract(lin_or_non3, val = widgets.Dropdown(options=[' ', 'Linear', 'Non-Linear'],value = ' ',description = 'Choose One:',disabled = False));\n```\n\n\n 'Is the above function linear or non-linear?'\n\n\n\n interactive(children=(Dropdown(description='Choose One:', options=(' ', 'Linear', 'Non-Linear'), value=' '), O\u2026\n\n\n\n```python\nx = np.linspace(-10,4,10)\ny = []\nfor num in x:\n y_val = -num - 6\n y.append(y_val)\n \n \nlayout = go.Layout(\n title = \"Y as a Function of X\",\n yaxis = dict(\n title=\"Y Values\"\n ),\n xaxis = dict(\n title=\"X Values\"\n ),\n)\n\ntrace1 = go.Scatter(\n x = x,\n y = y,\n mode = \"lines+markers\",\n name = \"Plot 1\",\n line=dict(\n color = ('rgb(100, 100, 100)'),\n shape = \"spline\",\n dash = \"dot\"\n )\n \n)\n\nfig = go.Figure(data = [trace1], layout = layout)\npy.offline.iplot(fig)\n \n```\n\n\n
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    \n\n\n\n```python\ndisplay(\"Is the above function linear or non-linear?\")\n \ninteract(lin_or_non4, val = widgets.Dropdown(options=[' ', 'Linear', 'Non-Linear'],value = ' ',description = 'Choose One:',disabled = False));\n```\n\n\n 'Is the above function linear or non-linear?'\n\n\n\n interactive(children=(Dropdown(description='Choose One:', options=(' ', 'Linear', 'Non-Linear'), value=' '), O\u2026\n\n\nNow that we know how linear functions can look, let's talk about how we can draw them on a graph!\n\n## **Creating a Graph**\n\n### **Independent and Dependent Variables**\n\nWhen you want to graph a linear function, you need to figure out which variable is **independent** and which one is **dependent**. The *independent* variable is the one that is allowed to change freely; the *dependent* variable is the one that responds to those changes\u2014in other words, it *depends* on the independent variable. The two variables form an \"if-then\" pair: \"*IF* the independent variable is this, *THEN* the dependent variable must be that.\" The difference between the two types of variables may be difficult to understand at first, so here is an example that may help.\n***\n>#### **Example**\nLet's say you want to measure how many push-ups you can do in a certain amount of time. There are two variables here: the amount of time, and the number of push-ups you do in that time. In this scenario, you are able to freely change how much time you have to do push-ups. You could give yourself 30 seconds, an hour, or a day! Therefore, *time is the independent variable*. \nBut the number of push-ups you are actually able to do *depends* on the amount of time you have to do them. If you give yourself more time, you can do more push-ups. Therefore, the *dependent variable is the number of push-ups you do*. \n***\nWhen the two variables in an equation are $x$ and $y$, we usually treat $x$ as the independent variable, and $y$ as the dependent variable.\n***\n\n\n\n```python\ndisplay(\"From the wage example shown in the section above, which variable do you think is the independent variable?\")\ndisplay(Math('M = 5W'))\n \ninteract(wage, val = widgets.Dropdown(options=[' ', 'W (Wage)', 'M (Money Made)'],value = ' ',description = 'Choose One:',disabled = False));\n\n```\n\n\n 'From the wage example shown in the section above, which variable do you think is the independent variable?'\n\n\n\n$\\displaystyle M = 5W$\n\n\n\n interactive(children=(Dropdown(description='Choose One:', options=(' ', 'W (Wage)', 'M (Money Made)'), value='\u2026\n\n\n***\n### **The Cartesian Plane**\n\nOnce we have established the independent and dependent variables in our linear function, we are ready to set up a graph of their relationship. When graphing two variables, we use what is called the **Cartesian Plane**. You have probably seen a graph on the Cartesian Plane before, but here is what it looks like:\n\n\n```python\nlayout = go.Layout(\n title = \"The Cartesian Plane\",\n yaxis = dict(\n title=\"Y Axis\",\n range = [-10,10],\n dtick = 1.00\n ),\n xaxis = dict(\n title=\"X Axis\",\n range = [-10,10],\n dtick = 1.00\n ),\n)\n\ntrace1 = go.Scatter(\n x = [5,-5,-5,5],\n y = [5,5,-5,-5],\n mode = \"markers+text\",\n name = \"Plot 1\",\n text=['Quadrant 1', 'Quadrant 2', 'Quadrant 3', 'Quadrant 4'],\n textposition='bottom center'\n)\n\nfig = go.Figure(data = [trace1], layout = layout)\npy.offline.iplot(fig)\n```\n\n\n
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    Figure 2: This is the standard layout for the Cartesian Plane, where you will plot all linear functions. The point where $x = 0$ and $y = 0$ is called the origin.
    \n\nOn the Cartesian Plane, there are two **axes**: the $x$ axis (horizontal), and the $y$ axis (vertical). When we want to graph a certain function, we label the $y$ axis with the dependent variable, and the $x$ axis with the independent variable. When you aren't given a specific name for what the independent and dependent variable represent, it is okay to just label the $x$ axis \"$X$\" and the $y$ axis \"$Y$\". \n\nWhen making a title for the graph, just describe what is being shown. Your title should allow whoever is looking at the graph to understand what is being shown. In most cases, you will describe the dependent variable as a function of the independent variable, or \"$Y$ as a Function of $X$\". \n\nThe Cartesian Plane consists of four **quadrants**, or sections. The top right section, where both $x$ and $y$ are positive (+,+), is quadrant 1. Then going counter-clockwise, quadrant 2 is the section where $x$ is negative and $y$ is positive (-,+), in quadrant 3 both $x$ and $y$ are negative (-,-), and in quadrant 4, $x$ is positive and $y$ is negative (+,-).\n\nNow that we have established how a graph should look, let's talk about how we decide which values we want to plot on the graph!\n\n### **Finding the Data Points**\n\nThe data points that are shown on a graph don't just appear out of nowhere; they can come from scientific observations, measurements, or mathematical relationships. When graphing on the Cartesian Plane, each data point has an $x$-value and a $y$-value (sometimes called an $x$-coordinate and a $y$-coordinate), and these values are what determine where the point will appear on the graph.\n\nThe best way to describe how we find the $x$-value and $y$-value for each data point is through an example.\n\n***\n#### **Example:** Plot a graph of $y = 3x + 2$ with 5 data points, starting from $x = 1$ up to $x = 5$.\n\nTo begin, we pick 5 $x$-values between $1$ and $5$. \n\nFor simplicity, let's pick 1, 2, 3, 4, and 5.\n\n\\begin{array}{| c | c |}\n\\hline\n X\\: Value & Y\\: Value \\\\\n \\hline\n 1 & ? \\\\\\hline\n 2 & ? \\\\\\hline\n 3 & ? \\\\\\hline\n 4 & ? \\\\\\hline\n 5 & ? \\\\\\hline\n\\end{array}\n\nNow we need to find the value of $y$ at each one of the chosen values for $x$. We can do this by plugging our chosen $x$-values into the equation that we have been given: $y = 3x + 2$.\n\nLet's take our first $x$-value, which is 1, and plug it into the equation:\n\n\\begin{equation} \ny = 3x + 2\n\\end{equation}\n\n\n\\begin{equation} \ny = 3(1) + 2\n\\end{equation}\n\\begin{equation}\ny = 3 + 2\n\\end{equation}\n\\begin{equation}\ny = 5\n\\end{equation}\n\nWe now have the $y$-value of our first data point! When $x = 1$, $y = 5$. We would write this data point down as (1,5). Let's fill in the table so we can keep track of the data points we find:\n\n\\begin{array}{| c | c |}\n\\hline\n X\\: Value & Y\\: Value \\\\\n \\hline\n 1 & 5 \\\\\\hline\n 2 & ? \\\\\\hline\n 3 & ? \\\\\\hline\n 4 & ? \\\\\\hline\n 5 & ? \\\\\\hline\n\\end{array}\n\n\nIn the section below, use the above method to find the remaining $y$ values.\n\n\n```python\ncheck_input_numbers()\n```\n\n The equation is: y = 3x + 2\n\n\n\n IntText(value=5, description='When x = 1, y = ', style=DescriptionStyle(description_width='initial'))\n\n\n\n IntText(value=0, description='When x = 2, y = ', style=DescriptionStyle(description_width='initial'))\n\n\n\n IntText(value=0, description='When x = 3, y = ', style=DescriptionStyle(description_width='initial'))\n\n\n\n IntText(value=0, description='When x = 4, y = ', style=DescriptionStyle(description_width='initial'))\n\n\n\n IntText(value=0, description='When x = 5, y = ', style=DescriptionStyle(description_width='initial'))\n\n\n\n Button(description='Check results', style=ButtonStyle())\n\n\n\n HTML(value='')\n\n\nGood job! Now that we have all of the data points, we are ready to plot them on the graph.\n\n\n### **Plotting the Data Points on the Graph**\n\nTo plot our data points on a graph, we will need our Cartesian Plane, with the $x$-axis and $y$-axis labelled as follows:\n\n\n```python\nlayout = go.Layout(\n title = \"Y as a Function of X\",\n yaxis = dict(\n title=\"Y Values\",\n range = [-1,20],\n dtick = 1.00\n ),\n xaxis = dict(\n title=\"X Values\",\n range = [-1,10],\n dtick = 1.00\n )\n)\n\ntrace1 = go.Scatter(\n x = [],\n y = [],\n mode = \"lines+markers\",\n name = \"Plot 1\",\n line=dict(\n color = ('rgb(205, 12, 24)'),\n shape = \"spline\",\n dash = \"dot\"\n )\n \n)\n\nfig = go.Figure(data = [trace1], layout = layout)\npy.offline.iplot(fig)\n\n```\n\n\n
    \n\n\n
    \n \n
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    Figure 3: The layout of our graph. As all of our data points have positive $x$ and $y$ values, we only need to focus on the first quadrant (top right corner).
    \n\nNow, using our chart of data points, let's plot our first point. The $x$-value of our first point is 1, and the $y$-value is 5. So let's find the point on the graph where $x=1$ and $y=5$ and mark this point\n\n\n\n```python\nlayout = go.Layout(\n title = \"Y as a Function of X\",\n yaxis = dict(\n title=\"Y Values\",\n range = [-1,20],\n dtick = 1.00\n ),\n xaxis = dict(\n title=\"X Values\",\n range = [-1,10],\n dtick = 1.00\n )\n)\n\ntrace1 = go.Scatter(\n x = [1],\n y = [5],\n mode = \"lines+markers\",\n name = \"Plot 1\",\n line=dict(\n color = ('rgb(205, 12, 24)'),\n shape = \"spline\",\n dash = \"dot\"\n )\n \n)\n\nfig = go.Figure(data = [trace1], layout = layout)\npy.offline.iplot(fig)\n```\n\n\n
    \n\n\n
    \n \n
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    Figure 4: Here is the location of our first data point.
    \n\nThat is your first data point!\n\nIn the section below, use the above methods to create a graph and plot all of the data points.\n\n\\begin{array}{| c | c |}\n\\hline\n X\\: Value & Y\\: Value \\\\\n \\hline\n 1 & 5 \\\\\\hline\n 2 & 8 \\\\\\hline\n 3 & 11 \\\\\\hline\n 4 & 14 \\\\\\hline\n 5 & 17 \\\\\\hline\n\\end{array}\n\n\n```python\ng = User_Graph([-1,10],[-1,20])\ng.run_it()\n```\n\n Enter the x-values as a list of the form [a,b,c,d,...]: \n\n\n\n Text(value='')\n\n\n Enter the y-values as a list of the form [a,b,c,d,...]: \n\n\n\n Text(value='')\n\n\n\n Button(description='Generate plot', style=ButtonStyle())\n\n\n\n HTML(value='')\n\n\nIf done correctly, your graph should look something like this:\n\n\n```python\nlayout = go.Layout(\n title = \"Y as a Function of X\",\n yaxis = dict(\n title=\"Y Values\",\n range = [-1,20]\n ),\n xaxis = dict(\n title=\"X Values\",\n range = [-1,10]\n )\n)\n\ntrace1 = go.Scatter(\n x = [1,2,3,4,5],\n y = [5,8,11,14,17],\n mode = \"lines+markers\",\n name = \"Plot 1\",\n line=dict(\n color = ('rgb(205, 12, 24)'),\n shape = \"spline\",\n dash = \"dot\"\n )\n \n)\n\nfig = go.Figure(data = [trace1], layout = layout)\npy.offline.iplot(fig)\n```\n\n\n
    \n\n\n
    \n \n
    \n\n\nIf the graph you made looks significantly different, try running the code segment again to re-try!\n\n\nNow that you know how to create a graph, let's talk about some of the important things that a graph can tell us.\n\n***\n\n**Features of a Linear Graph: Slope**\n\nAn important characteristic of a linear function is **slope**. The *slope* of a function can be thought of as the steepness or the angle of the function as it appears on a graph. What it really represents is the **rate of change** as you move along the graph. \n\nThat may sound complicated at first, but here is an example that might make it easier to understand what slope is.\n\n***\n\n> ### **Example**\n\nSay you have been given a set of data points describing how far a car has travelled in a given amount of time:\n\n\n\\begin{array}{| c | c |}\n\\hline\n Time(seconds) & Distance Travelled(meters) \\\\\n \\hline\n 0 & 0 \\\\\\hline\n 20 & 400 \\\\\\hline\n 40 & 800 \\\\\\hline\n 60 & 1200 \\\\\\hline\n 80 & 1600 \\\\\\hline\n 100 & 2000 \\\\\\hline\n 120 & 2400 \\\\\\hline\n 140 & 2800 \\\\\\hline\n 160 & 3200 \\\\\\hline\n 180 & 3600 \\\\\\hline\n 200 & 4000 \\\\\\hline\n\\end{array}\n\n\nIf you want to find out how fast the car is travelling, you are really trying to find out how quickly the distance travelled increases, or rather, the **rate** at which it is changing. \n\nLet's set up a graph of this data to take a closer look.\n\n\n\n\n\n\n```python\ntime = np.linspace(0,200,11)\ndistance = np.linspace(0,4000,11)\n\nlayout = go.Layout(\n title = \"Distance Travelled as a Function of Time\",\n yaxis = dict(\n title=\"Distance Travelled (Meters)\"\n ),\n xaxis = dict(\n title=\"Time (Seconds)\"\n ),\n)\n\ntrace1 = go.Scatter(\n x = time,\n y = distance,\n mode = \"lines+markers\",\n name = \"Plot 1\",\n line=dict(\n color = ('rgb(22, 96, 167)'),\n shape = \"spline\",\n dash = \"dot\"\n )\n \n)\n\nfig = go.Figure(data = [trace1], layout = layout)\npy.offline.iplot(fig)\n```\n\n\n
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    Figure 6: This graph shows the distance a car travels after a given amount of time.
    \n\nWe want to find how fast the car is travelling. Knowing that speed is a measurement of distance over time, let's use the total distance travelled and the total amount of time to find the average speed of the car.\nFrom the graph, we can see that the car initially started at position 0 meters, and travelled a total of 4000 meters. We also know that the car started moving at time 0 seconds, and stopped moving after 200 seconds. Using this information, we can set up the following equations to find the speed of the car:\n\n\\begin{equation}\nSpeed = \\frac{Total\\: Distance}{Total\\: Time}\n\\end{equation}\n\nand we know that:\n\n\\begin{equation}\nTotal\\: Distance = Final\\: Distance - Initial\\: Distance\n\\end{equation}\n\\begin{equation}\nTotal\\: Time = Final\\: Time - Initial\\: Time\n\\end{equation}\n\nso:\n\n\\begin{equation} \nSpeed = \\frac{Final\\: Distance - Initial\\: Distance}{Final\\: Time - Initial\\: Time}\n\\end{equation}\n\n\\begin{equation} \nSpeed = \\frac{4000m - 0m}{200s - 0s}\n\\end{equation}\n\n\\begin{equation}\nSpeed = \\frac{4000m}{200s}\n\\end{equation}\n\n\\begin{equation} \n Speed = 20 m/s\n\\end{equation}\n\n\n\nFrom the equations above, we can see that the car travels 20 meters every second. While it may not be obvious, we just found the slope of the graph! By finding the average speed of the car, we identified the rate at which the dependent variable (Distance Travelled) changes with respect to the independent variable (Time).\n\n***\n\n\nWhen looking at a graph, the general equation used to find the slope of a linear function from one point: ($x_1$,$y_1$), to a second point: ($x_2$,$y_2$), is given as follows:\n\n\\begin{equation} \nSlope = \\frac{y_2 - y_1}{x_2 - x_1}\n\\end{equation}\n\nWhen looking at a linear function that has the form:\n\\begin{equation} \ny = mx + b\n\\end{equation}\nthe value of the slope is the same as the value of $m$.\n\nUse the section below to see the effect that slope can have on a function.\n\n\n```python\ndef slider(slope):\n \n x_list = [0,1,2,3,4,5,6,7,8,9,10]\n y_list = []\n for i in x_list:\n y_val = slope*(i)\n y_list.append(y_val)\n \n \n layout = go.Layout(\n showlegend = True,\n title ='y = ' + str(slope) + 'x',\n yaxis = dict(\n title=\"X Values\",\n range = [-1,20]\n ),\n xaxis = dict(\n title=\"Y Values\",\n range = [-1,20]\n ),\n )\n\n trace1 = go.Scatter(\n x = x_list,\n y = y_list,\n mode = \"lines+markers\",\n name = str('$y = ' + str(slope) + 'x$'),\n line=dict(\n color = ('rgb(22, 96, 167)'),\n shape = \"spline\",\n dash = \"dot\"\n )\n \n )\n\n fig = go.Figure(data = [trace1], layout = layout)\n py.offline.iplot(fig, filename = 'show-legend')\n \ninteractive(slider, slope=(0,5,0.5))\n\n\n```\n\n\n interactive(children=(FloatSlider(value=2.0, description='slope', max=5.0, step=0.5), Output()), _dom_classes=\u2026\n\n\n
    Figure 7: This graph shows the way in which slope can affect the shape of a function. The larger the slope, the \"steeper\" the graph is.
    \n\nWhen dealing with a single linear function, the slope will be constant across the entire graph. However, if a graph consists of more than one linear function, the slope may change at some point. \nTake a look at the following graph to see what a graph with more than one linear function may look like:\n\n\n```python\nx1 = [0,1,2,3,4,5]\ny1 = []\n\nfor num in x1:\n y1_val = num + 1\n y1.append(y1_val)\n \nx2 = [6,7,8,9,10]\ny2 = []\n\nfor num in x2:\n y2_val = 3*(num) - 9\n y2.append(y2_val)\n\nx_list = x1 + x2\ny_list = y1 + y2\n \nlayout = go.Layout(\n title = \"Y as a Function of X\",\n yaxis = dict(\n title=\"Y Values\"\n ),\n xaxis = dict(\n title=\"X Values\"\n ),\n)\n\ntrace1 = go.Scatter(\n x = x_list,\n y = y_list,\n mode = \"lines+markers\",\n name = \"Plot 1\",\n line=dict(\n color = ('rgb(22, 96, 167)'),\n dash = \"dot\"\n )\n \n)\n\nfig = go.Figure(data = [trace1], layout = layout)\npy.offline.iplot(fig)\n\n \n```\n\n\n
    \n\n\n
    \n \n
    \n\n\n
    Figure 8: On this plot, notice the distinct change in steepness at $x = 5$. The slope increases at this point.
    \n\nAs you can see above, the graph has two distinct sections: one from $x = 0$ to $x = 5$, and the other from $x = 5$ to $x = 10$. As we have seen from the equations of linear functions, we know that the slope of one function has a constant value. Because these sections each have a different slope, they must represent two distinct linear functions.\n\nYou may be wondering how one graph can show two different functions at the same time, but consider our example of the car shown above. What if halfway through the trip, the car began to move at a faster speed? If this were the case, the slope of the graph would increase at the halfway point, much like the graph shown above.\n\n## **Questions**\n\nThe following section is meant to test your understanding and improve your familiarity with the concepts we have gone over in this lesson.\n\n\n### **Question 1.**\n\nUse the following table of data points describing the motion of a car driving to a destination and complete the following tasks:\n- Plot a graph (be sure to label the axes, and to give the graph a title!)\n- Find the slope of the graph\n- Create an equation for the linear function\n\n\\begin{array}{| c | c |}\n\\hline\n Time\\: (seconds) & Distance\\: Away\\: from\\: Destination\\: (meters) \\\\\n \\hline\n 0 & 10 \\\\\\hline\n 1 & 8 \\\\\\hline\n 2 & 6 \\\\\\hline\n 3 & 4 \\\\\\hline\n 4 & 2 \\\\\\hline\n 5 & 0 \\\\\\hline\n 6 & -2 \\\\\\hline\n 7 & -4 \\\\\\hline\n 8 & -6 \\\\\\hline\n 9 & -8 \\\\\\hline\n\\end{array}\n\n\nUse the section below to create the graph:\n\n\n```python\nq1 = User_Graph([-1,10], [-10,12])\nq1.run_it()\n```\n\n Enter the x-values as a list of the form [a,b,c,d,...]: \n\n\n\n Text(value='')\n\n\n Enter the y-values as a list of the form [a,b,c,d,...]: \n\n\n\n Text(value='')\n\n\n\n Button(description='Generate plot', style=ButtonStyle())\n\n\n\n HTML(value='')\n\n\nUse the following sections to answer some questions about the graph you just made:\n\n\n```python\nask_slope()\n```\n\n What is the value of the slope of the above graph?\n\n\n\n IntText(value=0)\n\n\n\n Button(description='Check results', style=ButtonStyle())\n\n\n\n HTML(value='')\n\n\n\n```python\nask_equation()\n```\n\n What is the equation of the linear function shown on the graph? (Enter your answer as: y = mx + b)\n\n\n\n Text(value='')\n\n\n\n Button(description='Check results', style=ButtonStyle())\n\n\n\n HTML(value='')\n\n\n\n```python\nask_question1()\n```\n\n At what point did the person walk past their destination? (Enter your answer as (x,y)) \n\n\n\n Text(value='')\n\n\n\n Button(description='Check results', style=ButtonStyle())\n\n\n\n HTML(value='')\n\n\n###### Question 2.\n\nQuentin and Isabelle both earn their allowance by doing exercise. Quentin's parents pay him $\\$8.00$ each time he goes for a run, plus an additional $\\$0.50$ for every kilometer he runs.\n\nIsabelle gets paid $\\$5.00$ every time she goes for a run, and an additional $\\$1.00$ for every kilometer she runs.\n\nAt what distance do Quentin and Isabelle make the same amount of money?\nWho makes the most money after running for 10 kilometers?\n\nTo help answer these questions, here are the linear equations that describe Quentin's and Isabelle's allowance:\n\nQuentin: \n\\begin{equation}\ny = \\frac{1}{2}x + 8\n\\end{equation}\n\nIsabelle:\n\\begin{equation}\ny = x + 5\n\\end{equation}\n\nCreate a graph that shows both of these functions at the points: \n\n\\begin{array}{| c | c |}\n\\hline\n Distance\\: (km) & Money\\: Made\\: (Dollars)\\\\\n \\hline\n 0 & ? \\\\\\hline\n 1 & ? \\\\\\hline\n 2 & ? \\\\\\hline\n 3 & ? \\\\\\hline\n 4 & ? \\\\\\hline\n 5 & ? \\\\\\hline\n 6 & ? \\\\\\hline\n 7 & ? \\\\\\hline\n\\end{array}\n\n*Hint*: Create one table for each of the equations and find the corresponding $y$-values, then use the section below to enter the points.\n\n\n```python\nD = Double_Graph([0,8],[0,15])\nD.run_it()\n```\n\n Please enter data points for Quentin:\n Enter the x-values as a list of the form [a,b,c,d,...]: \n\n\n\n Text(value='')\n\n\n Enter the y-values as a list of the form [a,b,c,d,...]: \n\n\n\n Text(value='')\n\n\n Please enter data points for Isabelle:\n Enter the x-values as a list of the form [a,b,c,d,...]: \n\n\n\n Text(value='')\n\n\n Enter the y-values as a list of the form [a,b,c,d,...]: \n\n\n\n Text(value='')\n\n\n\n Button(description='Generate plot', style=ButtonStyle())\n\n\n\n HTML(value='')\n\n\n\n```python\nask_question2()\n```\n\n At what distance do Quentin and Isabelle make the same amount of money? (Enter your answer as a whole number) \n\n\n\n IntText(value=0)\n\n\n\n Button(description='Check results', style=ButtonStyle())\n\n\n\n HTML(value='')\n\n\n\n```python\nask_question3()\n```\n\n Who makes the most money after running 10 kilometers? (Enter: Quentin, or Isabelle) \n\n\n\n Text(value='')\n\n\n\n Button(description='Check results', style=ButtonStyle())\n\n\n\n HTML(value='')\n\n\n## **Conclusion**\n\nIn this lesson, we have highlighted some of the many ways that a graph can help us understand the relationship between two variables in a linear function. Some of the important methods that you should remember include:\n\n- Defining a Linear Function\n- Identifying the Independent and Dependent Variables\n- Setting up a Graph on the Cartesian Plane\n- Creating and Plotting Data Points\n- Finding the slope of a graph\n\nThroughout your life, you will be presented with all sorts of statistics, data, charts and graphs. It is important that you have the skills required to analyze what a graph is trying to tell you, not just so you can pass your math class, but to enable you to make informed decisions when you need to consider how different variables can interact and affect results.\n\n\n\n[](https://github.com/callysto/curriculum-notebooks/blob/master/LICENSE.md)\n", "meta": {"hexsha": "0c1418968a77393b5df171f6f490794dad0ecb90", "size": 330114, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "_build/jupyter_execute/curriculum-notebooks/Mathematics/GraphingTwoVariables/graphing-two-variables.ipynb", "max_stars_repo_name": "BryceHaley/curriculum-jbook", "max_stars_repo_head_hexsha": "d1246799ddfe62b0cf5c389394a18c2904383437", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-03-18T18:19:40.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-18T18:19:40.000Z", "max_issues_repo_path": "_build/jupyter_execute/curriculum-notebooks/Mathematics/GraphingTwoVariables/graphing-two-variables.ipynb", "max_issues_repo_name": "callysto/curriculum-jbook", "max_issues_repo_head_hexsha": "ffb685901e266b0ae91d1250bf63e05a87c456d9", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "_build/jupyter_execute/curriculum-notebooks/Mathematics/GraphingTwoVariables/graphing-two-variables.ipynb", "max_forks_repo_name": "callysto/curriculum-jbook", "max_forks_repo_head_hexsha": "ffb685901e266b0ae91d1250bf63e05a87c456d9", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 31.8274199769, "max_line_length": 8630, "alphanum_fraction": 0.3805170335, "converted": true, "num_tokens": 9142, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6654105587468139, "lm_q2_score": 0.6513548714339145, "lm_q1q2_score": 0.43341840894330014}} {"text": "\n\n# Tutorial 2: Effects of Input Correlation\n**Week 2, Day 3: Biological Neuron Models**\n\n**By Neuromatch Academy**\n\n__Content creators:__ Qinglong Gu, Songtin Li, John Murray, Richard Naud, Arvind Kumar\n\n__Content reviewers:__ Maryam Vaziri-Pashkam, Ella Batty, Lorenzo Fontolan, Richard Gao, Matthew Krause, Spiros Chavlis, Michael Waskom\n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n

    \n\n---\n# Tutorial Objectives\n\n*Estimated timing of tutorial: 50 minutes*\n\nIn this tutorial, we will use the leaky integrate-and-fire (LIF) neuron model (see Tutorial 1) to study how they transform input correlations to output properties (transfer of correlations). In particular, we are going to write a few lines of code to:\n\n- inject correlated GWN in a pair of neurons\n\n- measure correlations between the spiking activity of the two neurons\n\n- study how the transfer of correlation depends on the statistics of the input, i.e. mean and standard deviation.\n\n\n```python\n# @title Tutorial slides\n\n# @markdown These are the slides for the videos in all tutorials today\nfrom IPython.display import IFrame\nIFrame(src=f\"https://mfr.ca-1.osf.io/render?url=https://osf.io/8djsm/?direct%26mode=render%26action=download%26mode=render\", width=854, height=480)\n```\n\n---\n# Setup\n\n\n```python\n# Imports\n\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport time\n```\n\n\n```python\n# @title Figure Settings\nimport ipywidgets as widgets # interactive display\n%config InlineBackend.figure_format = 'retina'\n# use NMA plot style\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/course-content/master/nma.mplstyle\")\nmy_layout = widgets.Layout()\n```\n\n\n```python\n# @title Plotting Functions\n\ndef example_plot_myCC():\n pars = default_pars(T=50000, dt=.1)\n\n c = np.arange(10) * 0.1\n r12 = np.zeros(10)\n for i in range(10):\n I1gL, I2gL = correlate_input(pars, mu=20.0, sig=7.5, c=c[i])\n r12[i] = my_CC(I1gL, I2gL)\n\n plt.figure()\n plt.plot(c, r12, 'bo', alpha=0.7, label='Simulation', zorder=2)\n plt.plot([-0.05, 0.95], [-0.05, 0.95], 'k--', label='y=x',\n dashes=(2, 2), zorder=1)\n plt.xlabel('True CC')\n plt.ylabel('Sample CC')\n plt.legend(loc='best')\n\n\n\n# the function plot the raster of the Poisson spike train\ndef my_raster_Poisson(range_t, spike_train, n):\n \"\"\"\n Generates poisson trains\n\n Args:\n range_t : time sequence\n spike_train : binary spike trains, with shape (N, Lt)\n n : number of Poisson trains plot\n\n Returns:\n Raster plot of the spike train\n \"\"\"\n\n # find the number of all the spike trains\n N = spike_train.shape[0]\n\n # n should smaller than N:\n if n > N:\n print('The number n exceeds the size of spike trains')\n print('The number n is set to be the size of spike trains')\n n = N\n\n # plot rater\n i = 0\n while i < n:\n if spike_train[i, :].sum() > 0.:\n t_sp = range_t[spike_train[i, :] > 0.5] # spike times\n plt.plot(t_sp, i * np.ones(len(t_sp)), 'k|', ms=10, markeredgewidth=2)\n i += 1\n plt.xlim([range_t[0], range_t[-1]])\n plt.ylim([-0.5, n + 0.5])\n plt.xlabel('Time (ms)', fontsize=12)\n plt.ylabel('Neuron ID', fontsize=12)\n\ndef plot_c_r_LIF(c, r, mycolor, mylabel):\n z = np.polyfit(c, r, deg=1)\n c_range = np.array([c.min() - 0.05, c.max() + 0.05])\n plt.plot(c, r, 'o', color=mycolor, alpha=0.7, label=mylabel, zorder=2)\n plt.plot(c_range, z[0] * c_range + z[1], color=mycolor, zorder=1)\n```\n\n\n```python\n# @title Helper Functions\ndef default_pars(**kwargs):\n pars = {}\n\n ### typical neuron parameters###\n pars['V_th'] = -55. # spike threshold [mV]\n pars['V_reset'] = -75. # reset potential [mV]\n pars['tau_m'] = 10. # membrane time constant [ms]\n pars['g_L'] = 10. # leak conductance [nS]\n pars['V_init'] = -75. # initial potential [mV]\n pars['V_L'] = -75. # leak reversal potential [mV]\n pars['tref'] = 2. # refractory time (ms)\n\n ### simulation parameters ###\n pars['T'] = 400. # Total duration of simulation [ms]\n pars['dt'] = .1 # Simulation time step [ms]\n\n ### external parameters if any ###\n for k in kwargs:\n pars[k] = kwargs[k]\n\n pars['range_t'] = np.arange(0, pars['T'], pars['dt']) # Vector of discretized\n # time points [ms]\n return pars\n\n\ndef run_LIF(pars, Iinj):\n \"\"\"\n Simulate the LIF dynamics with external input current\n\n Args:\n pars : parameter dictionary\n Iinj : input current [pA]. The injected current here can be a value or an array\n\n Returns:\n rec_spikes : spike times\n rec_v : mebrane potential\n \"\"\"\n\n # Set parameters\n V_th, V_reset = pars['V_th'], pars['V_reset']\n tau_m, g_L = pars['tau_m'], pars['g_L']\n V_init, V_L = pars['V_init'], pars['V_L']\n dt, range_t = pars['dt'], pars['range_t']\n Lt = range_t.size\n tref = pars['tref']\n\n # Initialize voltage and current\n v = np.zeros(Lt)\n v[0] = V_init\n Iinj = Iinj * np.ones(Lt)\n tr = 0.\n\n # simulate the LIF dynamics\n rec_spikes = [] # record spike times\n for it in range(Lt - 1):\n if tr > 0:\n v[it] = V_reset\n tr = tr - 1\n elif v[it] >= V_th: # reset voltage and record spike event\n rec_spikes.append(it)\n v[it] = V_reset\n tr = tref / dt\n\n # calculate the increment of the membrane potential\n dv = (-(v[it] - V_L) + Iinj[it] / g_L) * (dt / tau_m)\n\n # update the membrane potential\n v[it + 1] = v[it] + dv\n\n rec_spikes = np.array(rec_spikes) * dt\n\n return v, rec_spikes\n\n\ndef my_GWN(pars, sig, myseed=False):\n \"\"\"\n Function that calculates Gaussian white noise inputs\n\n Args:\n pars : parameter dictionary\n mu : noise baseline (mean)\n sig : noise amplitute (standard deviation)\n myseed : random seed. int or boolean\n the same seed will give the same random number sequence\n\n Returns:\n I : Gaussian white noise input\n \"\"\"\n\n # Retrieve simulation parameters\n dt, range_t = pars['dt'], pars['range_t']\n Lt = range_t.size\n\n # Set random seed. You can fix the seed of the random number generator so\n # that the results are reliable however, when you want to generate multiple\n # realization make sure that you change the seed for each new realization\n if myseed:\n np.random.seed(seed=myseed)\n else:\n np.random.seed()\n\n # generate GWN\n # we divide here by 1000 to convert units to sec.\n I_GWN = sig * np.random.randn(Lt) * np.sqrt(pars['tau_m'] / dt)\n\n return I_GWN\n\n\ndef LIF_output_cc(pars, mu, sig, c, bin_size, n_trials=20):\n \"\"\" Simulates two LIF neurons with correlated input and computes output correlation\n\n Args:\n pars : parameter dictionary\n mu : noise baseline (mean)\n sig : noise amplitute (standard deviation)\n c : correlation coefficient ~[0, 1]\n bin_size : bin size used for time series\n n_trials : total simulation trials\n\n Returns:\n r : output corr. coe.\n sp_rate : spike rate\n sp1 : spike times of neuron 1 in the last trial\n sp2 : spike times of neuron 2 in the last trial\n \"\"\"\n\n r12 = np.zeros(n_trials)\n sp_rate = np.zeros(n_trials)\n for i_trial in range(n_trials):\n I1gL, I2gL = correlate_input(pars, mu, sig, c)\n _, sp1 = run_LIF(pars, pars['g_L'] * I1gL)\n _, sp2 = run_LIF(pars, pars['g_L'] * I2gL)\n\n my_bin = np.arange(0, pars['T'], bin_size)\n\n sp1_count, _ = np.histogram(sp1, bins=my_bin)\n sp2_count, _ = np.histogram(sp2, bins=my_bin)\n\n r12[i_trial] = my_CC(sp1_count[::20], sp2_count[::20])\n sp_rate[i_trial] = len(sp1) / pars['T'] * 1000.\n\n return r12.mean(), sp_rate.mean(), sp1, sp2\n```\n\nThe helper function contains the:\n\n- Parameter dictionary: `default_pars( **kwargs)` from Tutorial 1\n- LIF simulator: `run_LIF` from Tutorial 1\n- Gaussian white noise generator: `my_GWN(pars, sig, myseed=False)` from Tutorial 1\n- Poisson type spike train generator: `Poisson_generator(pars, rate, n, myseed=False)`\n- Two LIF neurons with correlated inputs simulator: `LIF_output_cc(pars, mu, sig, c, bin_size, n_trials=20)`\n\n\n---\n# Section 1: Correlations (Synchrony)\nCorrelation or synchrony in neuronal activity can be described for any readout of brain activity. Here, we are concerned with the spiking activity of neurons. \n\nIn the simplest way, correlation/synchrony refers to coincident spiking of neurons, i.e., when two neurons spike together, they are firing in **synchrony** or are **correlated**. Neurons can be synchronous in their instantaneous activity, i.e., they spike together with some probability. However, it is also possible that spiking of a neuron at time $t$ is correlated with the spikes of another neuron with a delay (time-delayed synchrony). \n\n## Origin of synchronous neuronal activity:\n- Common inputs, i.e., two neurons are receiving input from the same sources. The degree of correlation of the shared inputs is proportional to their output correlation.\n- Pooling from the same sources. Neurons do not share the same input neurons but are receiving inputs from neurons which themselves are correlated.\n- Neurons are connected to each other (uni- or bi-directionally): This will only give rise to time-delayed synchrony. Neurons could also be connected via gap-junctions.\n- Neurons have similar parameters and initial conditions.\n\n## Implications of synchrony\nWhen neurons spike together, they can have a stronger impact on downstream neurons. Synapses in the brain are sensitive to the temporal correlations (i.e., delay) between pre- and postsynaptic activity, and this, in turn, can lead to the formation of functional neuronal networks - the basis of unsupervised learning (we will study some of these concepts in a forthcoming tutorial).\n\nSynchrony implies a reduction in the dimensionality of the system. In addition, correlations, in many cases, can impair the decoding of neuronal activity.\n\n\n```python\n# @title Video 1: Input & output correlations\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = 'https://player.bilibili.com/player.html?bvid={0}&page={1}'.format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=\"BV1Bh411o7eV\", width=854, height=480, fs=1)\n print('Video available at https://www.bilibili.com/video/{0}'.format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=\"nsAYFBcAkes\", width=854, height=480, fs=1, rel=0)\n print('Video available at https://youtube.com/watch?v=' + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nA simple model to study the emergence of correlations is to inject common inputs to a pair of neurons and measure the output correlation as a function of the fraction of common inputs. \n\nHere, we are going to investigate the transfer of correlations by computing the correlation coefficient of spike trains recorded from two unconnected LIF neurons, which received correlated inputs.\n\n\nThe input current to LIF neuron $i$ $(i=1,2)$ is:\n\n\\begin{equation}\n\\frac{I_i}{g_L} =\\mu_i + \\sigma_i (\\sqrt{1-c}\\xi_i + \\sqrt{c}\\xi_c) \\quad (1)\n\\end{equation}\n\nwhere $\\mu_i$ is the temporal average of the current. The Gaussian white noise $\\xi_i$ is independent for each neuron, while $\\xi_c$ is common to all neurons. The variable $c$ ($0\\le c\\le1$) controls the fraction of common and independent inputs. $\\sigma_i$ shows the variance of the total input.\n\nSo, first, we will generate correlated inputs.\n\n\n```python\n# @markdown Execute this cell to get a function `correlate_input` for generating correlated GWN inputs\ndef correlate_input(pars, mu=20., sig=7.5, c=0.3):\n \"\"\"\n Args:\n pars : parameter dictionary\n mu : noise baseline (mean)\n sig : noise amplitute (standard deviation)\n c. : correlation coefficient ~[0, 1]\n\n Returns:\n I1gL, I2gL : two correlated inputs with corr. coe. c\n \"\"\"\n\n # generate Gaussian whute noise xi_1, xi_2, xi_c\n xi_1 = my_GWN(pars, sig)\n xi_2 = my_GWN(pars, sig)\n xi_c = my_GWN(pars, sig)\n\n # Generate two correlated inputs by Equation. (1)\n I1gL = mu + np.sqrt(1. - c) * xi_1 + np.sqrt(c) * xi_c\n I2gL = mu + np.sqrt(1. - c) * xi_2 + np.sqrt(c) * xi_c\n\n return I1gL, I2gL\n\nhelp(correlate_input)\n```\n\n Help on function correlate_input in module __main__:\n \n correlate_input(pars, mu=20.0, sig=7.5, c=0.3)\n Args:\n pars : parameter dictionary\n mu : noise baseline (mean)\n sig : noise amplitute (standard deviation)\n c. : correlation coefficient ~[0, 1]\n \n Returns:\n I1gL, I2gL : two correlated inputs with corr. coe. c\n \n\n\n## Coding Exercise 1A: Compute the correlation\n\nThe _sample correlation coefficient_ between two input currents $I_i$ and $I_j$ is defined as the sample covariance of $I_i$ and $I_j$ divided by the square root of the sample variance of $I_i$ multiplied with the square root of the sample variance of $I_j$. In equation form: \n\n\\begin{align}\nr_{ij} &= \\frac{cov(I_i, I_j)}{\\sqrt{var(I_i)} \\sqrt{var(I_j)}}\\\\\ncov(I_i, I_j) &= \\sum_{k=1}^L (I_i^k -\\bar{I}_i)(I_j^k -\\bar{I}_j) \\\\\nvar(I_i) &= \\sum_{k=1}^L (I_i^k -\\bar{I}_i)^2\n\\end{align}\n\nwhere $\\bar{I}_i$ is the sample mean, k is the time bin, and L is the length of $I$. This means that $I_i^k$ is current i at time $k\\cdot dt$. Note that the equations above are not accurate for sample covariances and variances as they should be additionally divided by L-1 - we have dropped this term because it cancels out in the sample correlation coefficient formula.\n\nThe _sample correlation coefficient_ may also be referred to as the _sample Pearson correlation coefficient_. Here, is a beautiful paper that explains multiple ways to calculate and understand correlations [Rodgers and Nicewander 1988](https://www.stat.berkeley.edu/~rabbee/correlation.pdf).\n\nIn this exercise, we will create a function, `my_CC` to compute the sample correlation coefficient between two time series. Note that while we introduced this computation here in the context of input currents, the sample correlation coefficient is used to compute the correlation between any two time series - we will use it later on binned spike trains. \n\nWe then check our method is accurate by generating currents with a certain correlation (using `correlate_input`), computing the correlation coefficient using `my_CC`, and plotting the true vs sample correlation coefficients.\n\n\n```python\ndef my_CC(i, j):\n \"\"\"\n Args:\n i, j : two time series with the same length\n\n Returns:\n rij : correlation coefficient\n \"\"\"\n ########################################################################\n ## TODO for students: compute rxy, then remove the NotImplementedError #\n # Tip1: array([a1, a2, a3])*array([b1, b2, b3]) = array([a1*b1, a2*b2, a3*b3])\n # Tip2: np.sum(array([a1, a2, a3])) = a1+a2+a3\n # Tip3: square root, np.sqrt()\n # Fill out function and remove\n #raise NotImplementedError(\"Student exercise: compute the sample correlation coefficient\")\n ########################################################################\n\n # Calculate the covariance of i and j\n cov = np.sum((i-np.mean(i))*(j-np.mean(j)))\n\n # Calculate the variance of i\n var_i = np.sum((i-np.mean(i))**2)\n\n # Calculate the variance of j\n var_j = np.sum((i-np.mean(i))**2)\n\n # Calculate the correlation coefficient\n rij = cov/(np.sqrt(var_i)*np.sqrt(var_j))\n\n return rij\n\nexample_plot_myCC()\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W2D3_BiologicalNeuronModels/solutions/W2D3_Tutorial2_Solution_313f41e4.py)\n\n*Example output:*\n\n\n\n\n\nThe sample correlation coefficients (computed using `my_CC`) match the ground truth correlation coefficient!\n\nIn the next exercise, we will use the Poisson distribution to model spike trains. Remember that you have seen the Poisson distribution used in this way in the [pre-reqs math day on Statistics](https://compneuro.neuromatch.io/tutorials/W0D5_Statistics/student/W0D5_Tutorial1.html#section-2-2-poisson-distribution). Remember that a Poisson spike train has the following properties:\n- The ratio of the mean and variance of spike count is 1\n- Inter-spike-intervals are exponentially distributed\n- Spike times are irregular i.e. \ud835\udc36\ud835\udc49ISI=1\n- Adjacent spike intervals are independent of each other.\n\nIn the following cell, we provide a helper function `Poisson_generator` and then use it to produce a Poisson spike train.\n\n\n```python\n# @markdown Execute this cell to get helper function `Poisson_generator`\ndef Poisson_generator(pars, rate, n, myseed=False):\n \"\"\"\n Generates poisson trains\n\n Args:\n pars : parameter dictionary\n rate : noise amplitute [Hz]\n n : number of Poisson trains\n myseed : random seed. int or boolean\n\n Returns:\n pre_spike_train : spike train matrix, ith row represents whether\n there is a spike in ith spike train over time\n (1 if spike, 0 otherwise)\n \"\"\"\n\n # Retrieve simulation parameters\n dt, range_t = pars['dt'], pars['range_t']\n Lt = range_t.size\n\n # set random seed\n if myseed:\n np.random.seed(seed=myseed)\n else:\n np.random.seed()\n\n # generate uniformly distributed random variables\n u_rand = np.random.rand(n, Lt)\n\n # generate Poisson train\n poisson_train = 1. * (u_rand < rate * (dt / 1000.))\n\n return poisson_train\n\nhelp(Poisson_generator)\n```\n\n Help on function Poisson_generator in module __main__:\n \n Poisson_generator(pars, rate, n, myseed=False)\n Generates poisson trains\n \n Args:\n pars : parameter dictionary\n rate : noise amplitute [Hz]\n n : number of Poisson trains\n myseed : random seed. int or boolean\n \n Returns:\n pre_spike_train : spike train matrix, ith row represents whether\n there is a spike in ith spike train over time\n (1 if spike, 0 otherwise)\n \n\n\n\n```python\n# @markdown Execute this cell to visualize Poisson spike train\n\npars = default_pars()\npre_spike_train = Poisson_generator(pars, rate=10, n=100, myseed=2020)\nmy_raster_Poisson(pars['range_t'], pre_spike_train, 100)\n```\n\n## Coding Exercise 1B: Measure the correlation between spike trains\n\nAfter recording the spike times of the two neurons, how can we estimate their correlation coefficient? \n\nIn order to find this, we need to bin the spike times and obtain two time series. Each data point in the time series is the number of spikes in the corresponding time bin. You can use `np.histogram()` to bin the spike times.\n\nComplete the code below to bin the spike times and calculate the correlation coefficient for two Poisson spike trains. Note that `c` here is the ground-truth correlation coefficient that we define.\n\n\n\n\n```python\n# @markdown Execute this cell to get a function for generating correlated Poisson inputs (`generate_corr_Poisson`)\n\n\ndef generate_corr_Poisson(pars, poi_rate, c, myseed=False):\n \"\"\"\n function to generate correlated Poisson type spike trains\n Args:\n pars : parameter dictionary\n poi_rate : rate of the Poisson train\n c. : correlation coefficient ~[0, 1]\n\n Returns:\n sp1, sp2 : two correlated spike time trains with corr. coe. c\n \"\"\"\n\n range_t = pars['range_t']\n\n mother_rate = poi_rate / c\n mother_spike_train = Poisson_generator(pars, rate=mother_rate,\n n=1, myseed=myseed)[0]\n sp_mother = range_t[mother_spike_train > 0]\n\n L_sp_mother = len(sp_mother)\n sp_mother_id = np.arange(L_sp_mother)\n L_sp_corr = int(L_sp_mother * c)\n\n np.random.shuffle(sp_mother_id)\n sp1 = np.sort(sp_mother[sp_mother_id[:L_sp_corr]])\n\n np.random.shuffle(sp_mother_id)\n sp2 = np.sort(sp_mother[sp_mother_id[:L_sp_corr]])\n\n return sp1, sp2\n\nprint(help(generate_corr_Poisson))\n```\n\n Help on function generate_corr_Poisson in module __main__:\n \n generate_corr_Poisson(pars, poi_rate, c, myseed=False)\n function to generate correlated Poisson type spike trains\n Args:\n pars : parameter dictionary\n poi_rate : rate of the Poisson train\n c. : correlation coefficient ~[0, 1]\n \n Returns:\n sp1, sp2 : two correlated spike time trains with corr. coe. c\n \n None\n\n\n\n```python\ndef corr_coeff_pairs(pars, rate, c, trials, bins):\n \"\"\"\n Calculate the correlation coefficient of two spike trains, for different\n realizations\n\n Args:\n pars : parameter dictionary\n rate : rate of poisson inputs\n c : correlation coefficient ~ [0, 1]\n trials : number of realizations\n bins : vector with bins for time discretization\n\n Returns:\n r12 : correlation coefficient of a pair of inputs\n \"\"\"\n\n r12 = np.zeros(n_trials)\n\n for i in range(n_trials):\n ##############################################################\n ## TODO for students\n # Note that you can run multiple realizations and compute their r_12(diff_trials)\n # with the defined function above. The average r_12 over trials can get close to c.\n # Note: change seed to generate different input per trial\n # Fill out function and remove\n #raise NotImplementedError(\"Student exercise: compute the correlation coefficient\")\n ##############################################################\n\n # Generate correlated Poisson inputs\n sp1, sp2 = generate_corr_Poisson(pars, rate, c, myseed=2020+i)\n\n # Bin the spike times of the first input\n sp1_count, _ = np.histogram(sp1, bins=bins)\n\n # Bin the spike times of the second input\n sp2_count, _ = np.histogram(sp2, bins=bins)\n\n # Calculate the correlation coefficient\n r12[i] = my_CC(sp1_count, sp2_count)\n\n return r12\n\n\npoi_rate = 20.\nc = 0.2 # set true correlation\npars = default_pars(T=10000)\n\n# bin the spike time\nbin_size = 20 # [ms]\nmy_bin = np.arange(0, pars['T'], bin_size)\nn_trials = 100 # 100 realizations\n\nr12 = corr_coeff_pairs(pars, rate=poi_rate, c=c, trials=n_trials, bins=my_bin)\nprint(f'True corr coe = {c:.3f}')\nprint(f'Simu corr coe = {r12.mean():.3f}')\n```\n\n True corr coe = 0.200\n Simu corr coe = 0.199\n\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W2D3_BiologicalNeuronModels/solutions/W2D3_Tutorial2_Solution_e5eaac3e.py)\n\n\n\nSample output\n\n```\nTrue corr coe = 0.200\nSimu corr coe = 0.197\n```\n\n---\n# Section 2: Investigate the effect of input correlation on the output correlation\n\n*Estimated timing to here from start of tutorial: 20 min*\n\nNow let's combine the aforementioned two procedures. We first generate the correlated inputs. Then we inject the correlated inputs $I_1, I_2$ into a pair of neurons and record their output spike times. We continue measuring the correlation between the output and \ninvestigate the relationship between the input correlation and the output correlation.\n\n\nIn the following, you will inject correlated GWN in two neurons. You need to define the mean (`gwn_mean`), standard deviation (`gwn_std`), and input correlations (`c_in`).\n\nWe will simulate $10$ trials to get a better estimate of the output correlation. Change the values in the following cell for the above variables (and then run the next cell) to explore how they impact the output correlation.\n\n\n```python\n# Play around with these parameters\n\npars = default_pars(T=80000, dt=1.) # get the parameters\nc_in = 1. # set input correlation value\ngwn_mean = 10.\ngwn_std = 4.\n```\n\n\n```python\n# @title\n\n# @markdown Do not forget to execute this cell to simulate the LIF\n\n\nbin_size = 10. # ms\n\nstarttime = time.perf_counter() # time clock\n\nr12_ss, sp_ss, sp1, sp2 = LIF_output_cc(pars, mu=gwn_mean, sig=gwn_std, c=c_in,\n bin_size=bin_size, n_trials=10)\n\n# just the time counter\nendtime = time.perf_counter()\ntimecost = (endtime - starttime) / 60.\nprint(f\"Simulation time = {timecost:.2f} min\")\n\nprint(f\"Input correlation = {c_in}\")\nprint(f\"Output correlation = {r12_ss}\")\n\nplt.figure(figsize=(12, 6))\nplt.plot(sp1, np.ones(len(sp1)) * 1, '|', ms=20, label='neuron 1')\nplt.plot(sp2, np.ones(len(sp2)) * 1.1, '|', ms=20, label='neuron 2')\nplt.xlabel('time (ms)')\nplt.ylabel('neuron id.')\nplt.xlim(1000, 8000)\nplt.ylim(0.9, 1.2)\nplt.legend()\nplt.show()\n```\n\n## Think! 2: Input and Output Correlations\n- Is the output correlation always smaller than the input correlation? If yes, why?\n- Should there be a systematic relationship between input and output correlations? \n\nYou will explore these questions in the next figure but try to develop your own intuitions first!\n\nLets vary `c_in` and plot the relationship between the `c_in` and output correlation. This might take some time depending on the number of trials. \n\n\n```python\n#@title\n\n#@markdown Don't forget to execute this cell!\n\npars = default_pars(T=80000, dt=1.) # get the parameters\nbin_size = 10.\nc_in = np.arange(0, 1.0, 0.1) # set the range for input CC\nr12_ss = np.zeros(len(c_in)) # small mu, small sigma\n\nstarttime = time.perf_counter() # time clock\nfor ic in range(len(c_in)):\n r12_ss[ic], sp_ss, sp1, sp2 = LIF_output_cc(pars, mu=10.0, sig=10.,\n c=c_in[ic], bin_size=bin_size,\n n_trials=10)\n\nendtime = time.perf_counter()\ntimecost = (endtime - starttime) / 60.\nprint(f\"Simulation time = {timecost:.2f} min\")\n\nplt.figure(figsize=(7, 6))\nplot_c_r_LIF(c_in, r12_ss, mycolor='b', mylabel='Output CC')\nplt.plot([c_in.min() - 0.05, c_in.max() + 0.05],\n [c_in.min() - 0.05, c_in.max() + 0.05],\n 'k--', dashes=(2, 2), label='y=x')\n\nplt.xlabel('Input CC')\nplt.ylabel('Output CC')\nplt.legend(loc='best', fontsize=16)\nplt.show()\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W2D3_BiologicalNeuronModels/solutions/W2D3_Tutorial2_Solution_71e76f4d.py)\n\n\n\n---\n# Section 3: Correlation transfer function\n\n*Estimated timing to here from start of tutorial: 30 min*\n\nThe above plot of input correlation vs. output correlation is called the __correlation transfer function__ of the neurons. \n\n## Section 3.1: How do the mean and standard deviation of the Gaussian white noise (GWN) affect the correlation transfer function?\n\nThe correlations transfer function appears to be linear. The above can be taken as the input/output transfer function of LIF neurons for correlations, instead of the transfer function for input/output firing rates as we had discussed in the previous tutorial (i.e., F-I curve).\n\nWhat would you expect to happen to the slope of the correlation transfer function if you vary the mean and/or the standard deviation of the GWN of the inputs ?\n\n\n```python\n# @markdown Execute this cell to visualize correlation transfer functions\n\npars = default_pars(T=80000, dt=1.) # get the parameters\nno_trial = 10\nbin_size = 10.\nc_in = np.arange(0., 1., 0.2) # set the range for input CC\nr12_ss = np.zeros(len(c_in)) # small mu, small sigma\nr12_ls = np.zeros(len(c_in)) # large mu, small sigma\nr12_sl = np.zeros(len(c_in)) # small mu, large sigma\n\nstarttime = time.perf_counter() # time clock\nfor ic in range(len(c_in)):\n r12_ss[ic], sp_ss, sp1, sp2 = LIF_output_cc(pars, mu=10.0, sig=10.,\n c=c_in[ic], bin_size=bin_size,\n n_trials=no_trial)\n r12_ls[ic], sp_ls, sp1, sp2 = LIF_output_cc(pars, mu=18.0, sig=10.,\n c=c_in[ic], bin_size=bin_size,\n n_trials=no_trial)\n r12_sl[ic], sp_sl, sp1, sp2 = LIF_output_cc(pars, mu=10.0, sig=20.,\n c=c_in[ic], bin_size=bin_size,\n n_trials=no_trial)\nendtime = time.perf_counter()\ntimecost = (endtime - starttime) / 60.\nprint(f\"Simulation time = {timecost:.2f} min\")\n\nplt.figure(figsize=(7, 6))\nplot_c_r_LIF(c_in, r12_ss, mycolor='b', mylabel=r'Small $\\mu$, small $\\sigma$')\nplot_c_r_LIF(c_in, r12_ls, mycolor='y', mylabel=r'Large $\\mu$, small $\\sigma$')\nplot_c_r_LIF(c_in, r12_sl, mycolor='r', mylabel=r'Small $\\mu$, large $\\sigma$')\nplt.plot([c_in.min() - 0.05, c_in.max() + 0.05],\n [c_in.min() - 0.05, c_in.max() + 0.05],\n 'k--', dashes=(2, 2), label='y=x')\nplt.xlabel('Input CC')\nplt.ylabel('Output CC')\nplt.legend(loc='best', fontsize=14)\nplt.show()\n```\n\n### Think! 3.1: GWN and the Correlation Transfer Function\nWhy do both the mean and the standard deviation of the GWN affect the slope of the correlation transfer function? \n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W2D3_BiologicalNeuronModels/solutions/W2D3_Tutorial2_Solution_2deb4ccb.py)\n\n\n\n## Section 3.2: What is the rationale behind varying $\\mu$ and $\\sigma$?\nThe mean and the variance of the synaptic current depends on the spike rate of a Poisson process. We can use something called [Campbell's theorem](https://en.wikipedia.org/wiki/Campbell%27s_theorem_(probability)) to estimate the mean and the variance of the synaptic current:\n\n\\begin{align}\n\\mu_{\\rm syn} = \\lambda J \\int P(t) \\\\\n\\sigma_{\\rm syn} = \\lambda J \\int P(t)^2 dt\\\\\n\\end{align}\n\nwhere $\\lambda$ is the firing rate of the Poisson input, $J$ the amplitude of the postsynaptic current and $P(t)$ is the shape of the postsynaptic current as a function of time. \n\nTherefore, when we varied $\\mu$ and/or $\\sigma$ of the GWN, we mimicked a change in the input firing rate. Note that, if we change the firing rate, both $\\mu$ and $\\sigma$ will change simultaneously, not independently. \n\nHere, since we observe an effect of $\\mu$ and $\\sigma$ on correlation transfer, this implies that the input rate has an impact on the correlation transfer function.\n\n\n# Think!: Correlations and Network Activity\n\n- What are the factors that would make output correlations smaller than input correlations? (Notice that the colored lines are below the black dashed line)\n- What does the fact that output correlations are smaller mean for the correlations throughout a network?\n- Here we have studied the transfer of correlations by injecting GWN. But in the previous tutorial, we mentioned that GWN is unphysiological. Indeed, neurons receive colored noise (i.e., Shot noise or OU process). How do these results obtained from injection of GWN apply to the case where correlated spiking inputs are injected in the two LIFs? Will the results be the same or different?\n\nReference\n- De La Rocha, Jaime, et al. \"Correlation between neural spike trains increases with firing rate.\" Nature (2007) (https://www.nature.com/articles/nature06028/)\n\n- Bujan AF, Aertsen A, Kumar A. Role of input correlations in shaping the variability and noise correlations of evoked activity in the neocortex. Journal of Neuroscience. 2015 Jun 3;35(22):8611-25. (https://www.jneurosci.org/content/35/22/8611)\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W2D3_BiologicalNeuronModels/solutions/W2D3_Tutorial2_Solution_be65591c.py)\n\n\n\n---\n# Summary\n\n*Estimated timing of tutorial: 50 minutes*\n\nIn this tutorial, we studied how the input correlation of two LIF neurons is mapped to their output correlation. Specifically, we:\n\n- injected correlated GWN in a pair of neurons,\n\n- measured correlations between the spiking activity of the two neurons, and\n\n- studied how the transfer of correlation depends on the statistics of the input, i.e., mean and standard deviation.\n\nHere, we were concerned with zero time lag correlation. For this reason, we restricted estimation of correlation to instantaneous correlations. If you are interested in time-lagged correlation, then we should estimate the cross-correlogram of the spike trains and find out the dominant peak and area under the peak to get an estimate of output correlations. \n\nWe leave this as a future to-do for you if you are interested.\n\nIf you have time, check out the bonus video to think about responses of ensembles of neurons to time-varying input.\n\n---\n# Bonus\n\n---\n## Bonus Section 1: Ensemble Response\n\nFinally, there is a short BONUS lecture video on the firing response of an ensemble of neurons to time-varying input. There are no associated coding exercises - just enjoy.\n\n\n```python\n# @title Video 2: Response of ensemble of neurons to time-varying input\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = 'https://player.bilibili.com/player.html?bvid={0}&page={1}'.format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=\"BV18K4y1x7Pt\", width=854, height=480, fs=1)\n print('Video available at https://www.bilibili.com/video/{0}'.format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=\"78_dWa4VOIo\", width=854, height=480, fs=1, rel=0)\n print('Video available at https://youtube.com/watch?v=' + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n", "meta": {"hexsha": "bbfc861ad58781cf3721c394dcd9b1e558f0e81b", "size": 383389, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W2D3_BiologicalNeuronModels/student/W2D3_Tutorial2.ipynb", "max_stars_repo_name": "luisarai/NMA2021", "max_stars_repo_head_hexsha": "d6cd66bf32d929f3030d0d66c2c92de55bd2d886", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tutorials/W2D3_BiologicalNeuronModels/student/W2D3_Tutorial2.ipynb", "max_issues_repo_name": "luisarai/NMA2021", "max_issues_repo_head_hexsha": "d6cd66bf32d929f3030d0d66c2c92de55bd2d886", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/W2D3_BiologicalNeuronModels/student/W2D3_Tutorial2.ipynb", "max_forks_repo_name": "luisarai/NMA2021", "max_forks_repo_head_hexsha": "d6cd66bf32d929f3030d0d66c2c92de55bd2d886", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 216.6039548023, "max_line_length": 102886, "alphanum_fraction": 0.8832934696, "converted": true, "num_tokens": 9086, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.6513548511303336, "lm_q1q2_score": 0.43341839543308314}} {"text": "```python\n%matplotlib inline\n```\n\n\nAdversarial Example Generation\n==============================\n\n**Author:** `Nathan Inkawhich `__\n\nIf you are reading this, hopefully you can appreciate how effective some\nmachine learning models are. Research is constantly pushing ML models to\nbe faster, more accurate, and more efficient. However, an often\noverlooked aspect of designing and training models is security and\nrobustness, especially in the face of an adversary who wishes to fool\nthe model.\n\nThis tutorial will raise your awareness to the security vulnerabilities\nof ML models, and will give insight into the hot topic of adversarial\nmachine learning. You may be surprised to find that adding imperceptible\nperturbations to an image *can* cause drastically different model\nperformance. Given that this is a tutorial, we will explore the topic\nvia example on an image classifier. Specifically we will use one of the\nfirst and most popular attack methods, the Fast Gradient Sign Attack\n(FGSM), to fool an MNIST classifier.\n\n\n\n\nThreat Model\n------------\n\nFor context, there are many categories of adversarial attacks, each with\na different goal and assumption of the attacker\u2019s knowledge. However, in\ngeneral the overarching goal is to add the least amount of perturbation\nto the input data to cause the desired misclassification. There are\nseveral kinds of assumptions of the attacker\u2019s knowledge, two of which\nare: **white-box** and **black-box**. A *white-box* attack assumes the\nattacker has full knowledge and access to the model, including\narchitecture, inputs, outputs, and weights. A *black-box* attack assumes\nthe attacker only has access to the inputs and outputs of the model, and\nknows nothing about the underlying architecture or weights. There are\nalso several types of goals, including **misclassification** and\n**source/target misclassification**. A goal of *misclassification* means\nthe adversary only wants the output classification to be wrong but does\nnot care what the new classification is. A *source/target\nmisclassification* means the adversary wants to alter an image that is\noriginally of a specific source class so that it is classified as a\nspecific target class.\n\nIn this case, the FGSM attack is a *white-box* attack with the goal of\n*misclassification*. With this background information, we can now\ndiscuss the attack in detail.\n\nFast Gradient Sign Attack\n-------------------------\n\nOne of the first and most popular adversarial attacks to date is\nreferred to as the *Fast Gradient Sign Attack (FGSM)* and is described\nby Goodfellow et. al.\u00a0in `Explaining and Harnessing Adversarial\nExamples `__. The attack is remarkably\npowerful, and yet intuitive. It is designed to attack neural networks by\nleveraging the way they learn, *gradients*. The idea is simple, rather\nthan working to minimize the loss by adjusting the weights based on the\nbackpropagated gradients, the attack *adjusts the input data to maximize\nthe loss* based on the same backpropagated gradients. In other words,\nthe attack uses the gradient of the loss w.r.t the input data, then\nadjusts the input data to maximize the loss.\n\nBefore we jump into the code, let\u2019s look at the famous\n`FGSM `__ panda example and extract\nsome notation.\n\n.. figure:: /_static/img/fgsm_panda_image.png\n :alt: fgsm_panda_image\n\nFrom the figure, $\\mathbf{x}$ is the original input image\ncorrectly classified as a \u201cpanda\u201d, $y$ is the ground truth label\nfor $\\mathbf{x}$, $\\mathbf{\\theta}$ represents the model\nparameters, and $J(\\mathbf{\\theta}, \\mathbf{x}, y)$ is the loss\nthat is used to train the network. The attack backpropagates the\ngradient back to the input data to calculate\n$\\nabla_{x} J(\\mathbf{\\theta}, \\mathbf{x}, y)$. Then, it adjusts\nthe input data by a small step ($\\epsilon$ or $0.007$ in the\npicture) in the direction (i.e.\n$sign(\\nabla_{x} J(\\mathbf{\\theta}, \\mathbf{x}, y))$) that will\nmaximize the loss. The resulting perturbed image, $x'$, is then\n*misclassified* by the target network as a \u201cgibbon\u201d when it is still\nclearly a \u201cpanda\u201d.\n\nHopefully now the motivation for this tutorial is clear, so lets jump\ninto the implementation.\n\n\n\n\n\n```python\nfrom __future__ import print_function\nimport torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport torch.optim as optim\nfrom torchvision import datasets, transforms\nimport numpy as np\nimport matplotlib.pyplot as plt\n```\n\nImplementation\n--------------\n\nIn this section, we will discuss the input parameters for the tutorial,\ndefine the model under attack, then code the attack and run some tests.\n\nInputs\n~~~~~~\n\nThere are only three inputs for this tutorial, and are defined as\nfollows:\n\n- **epsilons** - List of epsilon values to use for the run. It is\n important to keep 0 in the list because it represents the model\n performance on the original test set. Also, intuitively we would\n expect the larger the epsilon, the more noticeable the perturbations\n but the more effective the attack in terms of degrading model\n accuracy. Since the data range here is $[0,1]$, no epsilon\n value should exceed 1.\n\n- **pretrained_model** - path to the pretrained MNIST model which was\n trained with\n `pytorch/examples/mnist `__.\n For simplicity, download the pretrained model `here `__.\n\n- **use_cuda** - boolean flag to use CUDA if desired and available.\n Note, a GPU with CUDA is not critical for this tutorial as a CPU will\n not take much time.\n\n\n\n\n\n```python\nepsilons = [0, .05, .1, .15, .2, .25, .3]\npretrained_model = \"lenet_mnist_model.pth\"\nuse_cuda=True\n```\n\n\n```python\npretrained_model\n```\n\n\n\n\n 'lenet_mnist_model.pth'\n\n\n\nModel Under Attack\n~~~~~~~~~~~~~~~~~~\n\nAs mentioned, the model under attack is the same MNIST model from\n`pytorch/examples/mnist `__.\nYou may train and save your own MNIST model or you can download and use\nthe provided model. The *Net* definition and test dataloader here have\nbeen copied from the MNIST example. The purpose of this section is to\ndefine the model and dataloader, then initialize the model and load the\npretrained weights.\n\n\n\n\n\n```python\n# LeNet Model definition\nclass Net(nn.Module):\n def __init__(self):\n super(Net, self).__init__()\n self.conv1 = nn.Conv2d(1, 10, kernel_size=5)\n self.conv2 = nn.Conv2d(10, 20, kernel_size=5)\n self.conv2_drop = nn.Dropout2d()\n self.fc1 = nn.Linear(320, 50)\n self.fc2 = nn.Linear(50, 10)\n\n def forward(self, x):\n x = F.relu(F.max_pool2d(self.conv1(x), 2))\n x = F.relu(F.max_pool2d(self.conv2_drop(self.conv2(x)), 2))\n x = x.view(-1, 320)\n x = F.relu(self.fc1(x))\n x = F.dropout(x, training=self.training)\n x = self.fc2(x)\n return F.log_softmax(x, dim=1)\n\n# MNIST Test dataset and dataloader declaration\ntest_loader = torch.utils.data.DataLoader(\n datasets.MNIST('datamnist', train=False, download=True, transform=transforms.Compose([\n transforms.ToTensor(),\n ])), \n batch_size=1, shuffle=True)\n\n# Define what device we are using\nprint(\"CUDA Available: \",torch.cuda.is_available())\ndevice = torch.device(\"cuda\" if (use_cuda and torch.cuda.is_available()) else \"cpu\")\n\n# Initialize the network\nmodel = Net().to(device)\n\n# Load the pretrained model\nmodel.load_state_dict(torch.load(pretrained_model, map_location='cpu'))\n\n# Set the model in evaluation mode. In this case this is for the Dropout layers\nmodel.eval()\n```\n\n CUDA Available: True\n\n\n\n\n\n Net(\n (conv1): Conv2d(1, 10, kernel_size=(5, 5), stride=(1, 1))\n (conv2): Conv2d(10, 20, kernel_size=(5, 5), stride=(1, 1))\n (conv2_drop): Dropout2d(p=0.5, inplace=False)\n (fc1): Linear(in_features=320, out_features=50, bias=True)\n (fc2): Linear(in_features=50, out_features=10, bias=True)\n )\n\n\n\nFGSM Attack\n~~~~~~~~~~~\n\nNow, we can define the function that creates the adversarial examples by\nperturbing the original inputs. The ``fgsm_attack`` function takes three\ninputs, *image* is the original clean image ($x$), *epsilon* is\nthe pixel-wise perturbation amount ($\\epsilon$), and *data_grad*\nis gradient of the loss w.r.t the input image\n($\\nabla_{x} J(\\mathbf{\\theta}, \\mathbf{x}, y)$). The function\nthen creates perturbed image as\n\n\\begin{align}perturbed\\_image = image + epsilon*sign(data\\_grad) = x + \\epsilon * sign(\\nabla_{x} J(\\mathbf{\\theta}, \\mathbf{x}, y))\\end{align}\n\nFinally, in order to maintain the original range of the data, the\nperturbed image is clipped to range $[0,1]$.\n\n\n\n\n\n```python\n# FGSM attack code\ndef fgsm_attack(image, epsilon, data_grad):\n # Collect the element-wise sign of the data gradient\n sign_data_grad = data_grad.sign()\n # Create the perturbed image by adjusting each pixel of the input image\n perturbed_image = image + epsilon*sign_data_grad\n # Adding clipping to maintain [0,1] range\n perturbed_image = torch.clamp(perturbed_image, 0, 1)\n # Return the perturbed image\n return perturbed_image\n```\n\nTesting Function\n~~~~~~~~~~~~~~~~\n\nFinally, the central result of this tutorial comes from the ``test``\nfunction. Each call to this test function performs a full test step on\nthe MNIST test set and reports a final accuracy. However, notice that\nthis function also takes an *epsilon* input. This is because the\n``test`` function reports the accuracy of a model that is under attack\nfrom an adversary with strength $\\epsilon$. More specifically, for\neach sample in the test set, the function computes the gradient of the\nloss w.r.t the input data ($data\\_grad$), creates a perturbed\nimage with ``fgsm_attack`` ($perturbed\\_data$), then checks to see\nif the perturbed example is adversarial. In addition to testing the\naccuracy of the model, the function also saves and returns some\nsuccessful adversarial examples to be visualized later.\n\n\n\n\n\n```python\ndef test( model, device, test_loader, epsilon ):\n\n # Accuracy counter\n correct = 0\n adv_examples = []\n\n # Loop over all examples in test set\n for data, target in test_loader:\n\n # Send the data and label to the device\n data, target = data.to(device), target.to(device)\n\n # Set requires_grad attribute of tensor. Important for Attack\n data.requires_grad = True\n\n # Forward pass the data through the model\n output = model(data)\n init_pred = output.max(1, keepdim=True)[1] # get the index of the max log-probability\n\n # If the initial prediction is wrong, dont bother attacking, just move on\n if init_pred.item() != target.item():\n continue\n\n # Calculate the loss\n loss = F.nll_loss(output, target)\n\n # Zero all existing gradients\n model.zero_grad()\n\n # Calculate gradients of model in backward pass\n loss.backward()\n\n # Collect datagrad\n data_grad = data.grad.data\n\n # Call FGSM Attack\n perturbed_data = fgsm_attack(data, epsilon, data_grad)\n\n # Re-classify the perturbed image\n output = model(perturbed_data)\n\n # Check for success\n final_pred = output.max(1, keepdim=True)[1] # get the index of the max log-probability\n if final_pred.item() == target.item():\n correct += 1\n # Special case for saving 0 epsilon examples\n if (epsilon == 0) and (len(adv_examples) < 5):\n adv_ex = perturbed_data.squeeze().detach().cpu().numpy()\n adv_examples.append( (init_pred.item(), final_pred.item(), adv_ex) )\n else:\n # Save some adv examples for visualization later\n if len(adv_examples) < 5:\n adv_ex = perturbed_data.squeeze().detach().cpu().numpy()\n adv_examples.append( (init_pred.item(), final_pred.item(), adv_ex) )\n\n # Calculate final accuracy for this epsilon\n final_acc = correct/float(len(test_loader))\n print(\"Epsilon: {}\\tTest Accuracy = {} / {} = {}\".format(epsilon, correct, len(test_loader), final_acc))\n\n # Return the accuracy and an adversarial example\n return final_acc, adv_examples\n```\n\nRun Attack\n~~~~~~~~~~\n\nThe last part of the implementation is to actually run the attack. Here,\nwe run a full test step for each epsilon value in the *epsilons* input.\nFor each epsilon we also save the final accuracy and some successful\nadversarial examples to be plotted in the coming sections. Notice how\nthe printed accuracies decrease as the epsilon value increases. Also,\nnote the $\\epsilon=0$ case represents the original test accuracy,\nwith no attack.\n\n\n\n\n\n```python\naccuracies = []\nexamples = []\n\n# Run test for each epsilon\nfor eps in epsilons:\n acc, ex = test(model, device, test_loader, eps)\n accuracies.append(acc)\n examples.append(ex)\n```\n\n Epsilon: 0\tTest Accuracy = 9810 / 10000 = 0.981\n Epsilon: 0.05\tTest Accuracy = 9423 / 10000 = 0.9423\n Epsilon: 0.1\tTest Accuracy = 8514 / 10000 = 0.8514\n Epsilon: 0.15\tTest Accuracy = 6839 / 10000 = 0.6839\n Epsilon: 0.2\tTest Accuracy = 4338 / 10000 = 0.4338\n Epsilon: 0.25\tTest Accuracy = 2102 / 10000 = 0.2102\n Epsilon: 0.3\tTest Accuracy = 878 / 10000 = 0.0878\n\n\nResults\n-------\n\nAccuracy vs Epsilon\n~~~~~~~~~~~~~~~~~~~\n\nThe first result is the accuracy versus epsilon plot. As alluded to\nearlier, as epsilon increases we expect the test accuracy to decrease.\nThis is because larger epsilons mean we take a larger step in the\ndirection that will maximize the loss. Notice the trend in the curve is\nnot linear even though the epsilon values are linearly spaced. For\nexample, the accuracy at $\\epsilon=0.05$ is only about 4% lower\nthan $\\epsilon=0$, but the accuracy at $\\epsilon=0.2$ is 25%\nlower than $\\epsilon=0.15$. Also, notice the accuracy of the model\nhits random accuracy for a 10-class classifier between\n$\\epsilon=0.25$ and $\\epsilon=0.3$.\n\n\n\n\n\n```python\nplt.figure(figsize=(5,5))\nplt.plot(epsilons, accuracies, \"*-\")\nplt.yticks(np.arange(0, 1.1, step=0.1))\nplt.xticks(np.arange(0, .35, step=0.05))\nplt.title(\"Accuracy vs Epsilon\")\nplt.xlabel(\"Epsilon\")\nplt.ylabel(\"Accuracy\")\nplt.show()\n```\n\nSample Adversarial Examples\n~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\nRemember the idea of no free lunch? In this case, as epsilon increases\nthe test accuracy decreases **BUT** the perturbations become more easily\nperceptible. In reality, there is a tradeoff between accuracy\ndegredation and perceptibility that an attacker must consider. Here, we\nshow some examples of successful adversarial examples at each epsilon\nvalue. Each row of the plot shows a different epsilon value. The first\nrow is the $\\epsilon=0$ examples which represent the original\n\u201cclean\u201d images with no perturbation. The title of each image shows the\n\u201coriginal classification -> adversarial classification.\u201d Notice, the\nperturbations start to become evident at $\\epsilon=0.15$ and are\nquite evident at $\\epsilon=0.3$. However, in all cases humans are\nstill capable of identifying the correct class despite the added noise.\n\n\n\n\n\n```python\n# Plot several examples of adversarial samples at each epsilon\ncnt = 0\nplt.figure(figsize=(8,10))\nfor i in range(len(epsilons)):\n for j in range(len(examples[i])):\n cnt += 1\n plt.subplot(len(epsilons),len(examples[0]),cnt)\n plt.xticks([], [])\n plt.yticks([], [])\n if j == 0:\n plt.ylabel(\"Eps: {}\".format(epsilons[i]), fontsize=14)\n orig,adv,ex = examples[i][j]\n plt.title(\"{} -> {}\".format(orig, adv))\n plt.imshow(ex, cmap=\"gray\")\nplt.tight_layout()\nplt.show()\n```\n\nWhere to go next?\n-----------------\n\nHopefully this tutorial gives some insight into the topic of adversarial\nmachine learning. There are many potential directions to go from here.\nThis attack represents the very beginning of adversarial attack research\nand since there have been many subsequent ideas for how to attack and\ndefend ML models from an adversary. In fact, at NIPS 2017 there was an\nadversarial attack and defense competition and many of the methods used\nin the competition are described in this paper: `Adversarial Attacks and\nDefences Competition `__. The work\non defense also leads into the idea of making machine learning models\nmore *robust* in general, to both naturally perturbed and adversarially\ncrafted inputs.\n\nAnother direction to go is adversarial attacks and defense in different\ndomains. Adversarial research is not limited to the image domain, check\nout `this `__ attack on\nspeech-to-text models. But perhaps the best way to learn more about\nadversarial machine learning is to get your hands dirty. Try to\nimplement a different attack from the NIPS 2017 competition, and see how\nit differs from FGSM. Then, try to defend the model from your own\nattacks.\n\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "d86b111714aea1879b9c21d53c58dd56cce0d3d3", "size": 110645, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorial-examples/eric-fiala-wmlce-notebooks-master/2-pytorch-fgsm_tutorial.ipynb", "max_stars_repo_name": "davidbau/getting-started", "max_stars_repo_head_hexsha": "ec703bffac04d2d67437bafab5a287624f9ff8c5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 22, "max_stars_repo_stars_event_min_datetime": "2019-12-20T23:37:01.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-11T02:11:47.000Z", "max_issues_repo_path": "tutorial-examples/eric-fiala-wmlce-notebooks-master/2-pytorch-fgsm_tutorial.ipynb", "max_issues_repo_name": "davidbau/getting-started", "max_issues_repo_head_hexsha": "ec703bffac04d2d67437bafab5a287624f9ff8c5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2020-01-22T15:01:40.000Z", "max_issues_repo_issues_event_max_datetime": "2020-06-22T17:14:20.000Z", "max_forks_repo_path": "tutorial-examples/eric-fiala-wmlce-notebooks-master/2-pytorch-fgsm_tutorial.ipynb", "max_forks_repo_name": "davidbau/getting-started", "max_forks_repo_head_hexsha": "ec703bffac04d2d67437bafab5a287624f9ff8c5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 26, "max_forks_repo_forks_event_min_datetime": "2019-11-24T19:30:31.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-08T20:55:02.000Z", "avg_line_length": 173.9701257862, "max_line_length": 69856, "alphanum_fraction": 0.8846039134, "converted": true, "num_tokens": 4236, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.4332994406916829}} {"text": "# Exploring molecular interactions using electronic structure methods.\n\nFor today's activity we will use the package of programs psi4, so we will need to install it first. \nThe psi4 python module can be installed using the anaconda python platform. To do this download anaconda in:\n\nhttps://www.anaconda.com/download/#linux\n\nYou can download either the 2.7 or 3.6 verison depending on your preference. \n\nOnce downloaded and installed it is recomended to create a psi4 environment form which\nthe psi4 module can be loaded. Run the comand:\n\n**conda create -n p4env python=2.7 psi4 psi4-rt -c psi4/label/dev -c psi4**\n\nThis will install all the psi4 binaries and a python module which can be imported from the notebook.\nTo finish the installation you need to provide a scratch directory in your .bashrc, for example:\n\n**export PSI_SCRATCH=/home/user_name/scratch/psi4**\n\nFinally source your .bashrc and activate the environment\n\n**source activate p4env**\n\nNow you may open jupyter-notebook and install psi4.\n\n\n\n\n```python\nimport psi4 \nimport numpy as np\n```\n\n# 1. Compute the energy of a diatomic molecules.\n\n\nAs a first example we will compute the scf energy of the diatomic molecule hydrogen fluoride (HF):\n\n\n\n```python\n# ==> Opciones B\u00e1sicas Psi4 <==\n# Memoria\npsi4.set_memory(int(5e8))\nnumpy_memory = 500\n\n# Output\npsi4.core.set_output_file('output.dat', False)\n\n# Geometr\u00eda\nhf_mol = psi4.geometry(\"\"\"\n0 1\nH\nF 1 0.917\n\"\"\")\n\n#psi4.optimize('scf/cc-pvtz')\n#di_mat = hf_mol.distance_matrix()\n#print np.asarray(di_mat)\nenergy_hf_mol , wfn_hf_mol = psi4.energy('mp2/cc-pvtz', return_wfn=True)\nprint energy_hf_mol\n```\n\nThis corresponds to the Born-Oppenheimer scf energy of HF. Now it's your turn. Compute the single point energy \nof F$_2$ and N$_2$ using the cc-pVDZ and cc-pVTZ. \n\n# 2. Compute the dipole and quadrupole moment of diatomic molecules\n\nSince we are interested in studying long range molecular interactions using classical electrodynamics, it is necessary \nto compute the dipole and quadrupole moments. Quantum mechanically the dipole can be computed using the one electron dipole operator:\n\n\\begin{equation}\n\\hat{\\mu} = \\sum_i q_i r_i \n\\end{equation}\n\nwhere $q_i$ is the charge of the particle and $r_i$ is the position vector of the particle. The dipole moment can be computed using the wavefunction through the expectation value of the operator $\\mu$.\n\n\\begin{equation}\n\\mu = <\\psi|\\hat{\\mu}|\\psi> \n\\end{equation}\n\nIn psi4 we can obtain the dipole moment from the wavefunction object that was defined above\n\n\n```python\npsi4.oeprop(wfn_hf_mol, 'DIPOLE', 'QUADRUPOLE', title='HF SCF')\n\nmux = psi4.core.get_variable('HF SCF DIPOLE X') # in debye\nmuy = psi4.core.get_variable('HF SCF DIPOLE Y')\nmuz = psi4.core.get_variable('HF SCF DIPOLE Z')\nquad_zz = psi4.core.get_variable('HF SCF QUADRUPOLE ZZ')\n```\n\n\n```python\nprint muz\nprint quad_zz\nmu = (np.sqrt(mux**2 + muy**2 + muz**2))\nprint mu\n```\n\n# 3. Compute a potential energy surface of HF dimer.\n\nIn order to study the physical interactions between two molecules it is convenient to draw \na potential energy surface along the interaction coordinate. In this section we will \nobtain a potential energy profile for the most favorable dipole-dipole interaction, which is the \nhorizontal orientation with opposing dipole vectors, HF---HF. First we need to define a list containing \nthe distances between both dimers for which the energy will be obtained.\n\n\n```python\nhf_dimer = psi4.geometry(\"\"\"\n 0 1\n H\n F 1 0.917\n H 2 R 1 180.0\n F 3 0.917 2 180.0 1 0.0\n \"\"\")\n```\n\nNext, we write a loop and in each step of the loop we compute the energy at the mp4 level of theory. \n\n\n```python\nenergy = []\ndist = []\n\nRval = np.arange(1.5,10.0,0.1)\n\nfor d in Rval:\n hf_dimer.R = d\n psi4.set_options({'freeze_core': 'True'})\n en = psi4.energy('scf/cc-pvtz')\n print en\n print d\n energy.append(en)\n dist.append(d)\n\n```\n\nNow we are ready to plot the potential energy profile. We will use the matplotlib python library for this \npurpose. The function ref_cero_kcal transforms the energy which is in hartee to kcal/mol and takes the \nenergy of the dimer with the farthest separation as the reference energy.\n\n\n```python\nimport matplotlib.pyplot as plt\n%matplotlib inline\n```\n\n\n```python\ndef ref_cero_kcal(en_list):\n energy_kcal = []\n for x in range(len(en_list)):\n energy_kcal.append((en_list[x] - en_list[-1])*627.51)\n return energy_kcal\n\n\nenergy_kcal = ref_cero_kcal(energy) \nplt.plot(dist,energy_kcal)\n\n```\n\n**Exercise:** Using the data from the plot, make an estimate of the interaction energy in kcal/mol. Corroborate your results computing the interaction energy using the super-molecule approach. ( Hint: 1 hartree = 627.51 kcal/mol). What percentage of the total energy is the intermolecular interaction energy? \n\n\n```python\nhf_dimer_min = psi4.geometry(\"\"\"\n 0 1\n H\n F 1 0.917\n H 2 1.980 1 180.0\n F 3 0.917 2 180.0 1 0.0\n \"\"\")\n\nenergy_hf_dimer_min , wfn_hf_dimer_min = psi4.energy('mp2/cc-pvtz', return_wfn=True)\n```\n\n\n```python\nprint energy_hf_dimer_min\n\nint_energy = (energy_hf_dimer_min - 2*energy_hf_mol)\n\nprint int_energy*627.51\n\nprint(\"Percentage: \"+str((int_energy/energy_hf_dimer_min)*100))\n\n```\n\n# 4. The effect of electron correlation\n\nSince we know that the dispersion energy arises only when the motion of the electrons on the different molecular fragments are correlated, it is possible to evalute the importance of disperison efects by using\nmethods that include diferent levels of electron correlation. \n\n**Exercices**: Construct a potential energy curve at the SCF, MP2 and MP4 level of theory and \nasses the importance of including electron correlation along the intermolecular coordinate.\n\n\n```python\nenergy_scf = []\nenergy_mp2 = []\nenergy_mp4 = []\ndist = []\n\nfor d in Rval:\n hf_dimer.R = d\n psi4.set_options({'freeze_core': 'True'})\n en_scf = psi4.energy('scf/cc-pvtz')\n en_mp2 = psi4.energy('mp2/cc-pvtz')\n en_mp4 = psi4.energy('mp4/cc-pvtz')\n print en_scf\n print d\n energy_scf.append(en_scf)\n energy_mp2.append(en_mp2)\n energy_mp4.append(en_mp4)\n dist.append(d)\n```\n\n\n```python\nenergy_scf_kcal = ref_cero_kcal(energy_scf) \nenergy_mp2_kcal = ref_cero_kcal(energy_mp2) \nenergy_mp4_kcal = ref_cero_kcal(energy_mp4) \n\nplt.plot(dist,energy_scf_kcal)\nplt.plot(dist,energy_mp2_kcal)\nplt.plot(dist,energy_mp4_kcal)\n\n\n```\n\n# 5. Energy decomposition with SAPT\n\nSo far we have only computed the interaction energy as a whole, however we know that we can\ndecompose the interaction energy into contributions stemming from different physical phenomena, \nin particular, electrostatic, induction, dispersion and exchange interactions. First we will run a \nSAPT0 calculation to obtain the total interaction energy and compare it to the results we obtained \nabove.\n\n\n```python\nhf_dimer_sapt = psi4.geometry('''\n 0 1\n H 0.0 0.0 0.0\n F 0.0 0.0 0.917\n --\n 0 1\n H 0.0 0.0 2.951\n F 0.0 0.0 3.868\n symmetry c1\n''')\n\npsi4.set_options({'basis': 'jun-cc-pVDZ',\n 'e_convergence': 1e-8,\n 'd_convergence': 1e-8})\n\npsi4.energy('sapt0')\n```\n\nCheck the output file for energy decomposition analysis. Which contribution is dominant, and what do you think\nis the physical reason for it? \n\n**Exercise:** Repeat the above analysis with the Argon dimer $Ar_2$. What can you say about the importance \nof the correlation energy in this case? Which is the dominant contribution in the case of the Ar dimer?\n\n\n```python\n\n```\n", "meta": {"hexsha": "88b78b6fcb88da45a88e5aad65d303c7129896f4", "size": 12393, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Theory_electronic_structure_day3/m_inter.ipynb", "max_stars_repo_name": "QCMM/workshop2017", "max_stars_repo_head_hexsha": "29b9f58046e4e85daee816995fb5b6944a333106", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-03-11T01:46:12.000Z", "max_stars_repo_stars_event_max_datetime": "2019-03-11T01:46:12.000Z", "max_issues_repo_path": "Theory_electronic_structure_day3/m_inter.ipynb", "max_issues_repo_name": "QCMM/workshop2017", "max_issues_repo_head_hexsha": "29b9f58046e4e85daee816995fb5b6944a333106", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2017-11-29T01:19:21.000Z", "max_issues_repo_issues_event_max_datetime": "2017-12-02T06:52:08.000Z", "max_forks_repo_path": "Theory_electronic_structure_day3/m_inter.ipynb", "max_forks_repo_name": "QCMM/workshop2017", "max_forks_repo_head_hexsha": "29b9f58046e4e85daee816995fb5b6944a333106", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2017-12-01T13:18:25.000Z", "max_forks_repo_forks_event_max_datetime": "2017-12-05T20:57:52.000Z", "avg_line_length": 28.8209302326, "max_line_length": 315, "alphanum_fraction": 0.5741144194, "converted": true, "num_tokens": 2115, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982043529715, "lm_q2_score": 0.6688802669716106, "lm_q1q2_score": 0.43329943587134556}} {"text": "```python\nimport pandas as pd\nimport matplotlib as mpl\nimport numpy as np\nimport scipy.stats\nmpl.rcParams['figure.dpi'] = 600\n```\n\n\n```python\nfrom tcga_dicts import *\n```\n\n# Part 0: Load Data\n\n## Immune Cell Fractions\n\nCIBERSORT result from [*The Immune Landscape of Cancer*](https://www.sciencedirect.com/science/article/pii/S1074761318301213) downloaded from [NIH PanCanAtlas](https://gdc.cancer.gov/about-data/publications/panimmune) includes are SampleID ([TCGA barcode](https://docs.gdc.cancer.gov/Encyclopedia/pages/TCGA_Barcode/)), Cancer Type ([TCGA Study Abbreviation](https://gdc.cancer.gov/resources-tcga-users/tcga-code-tables/tcga-study-abbreviations)), and the abundance of 22 imuune cell types.\n\nNote that the abundances in each row sum to 1. In other words, the abundance is the proportion of a cell type in the leukocyte compartment (see [below](#lf)), but not all cells (stroma, tumor, etc.).\n\n\n```python\ndata = pd.read_csv(\"TCGA.Kallisto.fullIDs.cibersort.relative.tsv\", sep=\"\\t\")\ndata[\"SampleID\"] = data[\"SampleID\"].apply(lambda x: x.replace('.', '-'))\ndata\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    SampleIDCancerTypeB.cells.naiveB.cells.memoryPlasma.cellsT.cells.CD8T.cells.CD4.naiveT.cells.CD4.memory.restingT.cells.CD4.memory.activatedT.cells.follicular.helper...Macrophages.M2Dendritic.cells.restingDendritic.cells.activatedMast.cells.restingMast.cells.activatedEosinophilsNeutrophilsP.valueCorrelationRMSE
    0TCGA-OR-A5JG-01A-11R-A29S-07ACC0.0000000.0485290.0160520.0460990.0270370.2906820.00.000000...0.3638610.0027150.0261250.0327880.0000000.0102900.0096070.1120.0957971.047142
    1TCGA-OR-A5LG-01A-11R-A29S-07ACC0.0071690.0111250.0079820.1398420.0000000.1427420.00.001614...0.4482430.0000000.0074640.1262370.0000000.0000000.0000000.1040.1033451.046163
    2TCGA-OR-A5JD-01A-11R-A29S-07ACC0.0000230.0146070.0000000.1048880.0000000.1748950.00.017928...0.3295520.0000000.0093300.0000000.1907300.0000000.0000000.0680.1432591.039812
    3TCGA-OR-A5LH-01A-11R-A29S-07ACC0.0472990.0381800.0000000.1985910.0000000.0652400.00.043310...0.2879390.0000000.0005760.0000000.0647250.0000000.0000000.3880.0287631.080553
    4TCGA-OR-A5KY-01A-11R-A29S-07ACC0.0000000.0247950.0044180.0515060.0000000.2258920.00.000000...0.4864000.0000000.0428270.0000000.0174180.0147690.0000000.2460.0556211.073474
    ..................................................................
    11368TCGA-V4-A9EQ-01A-11R-A405-07UVM0.0000000.0289390.0088940.0000000.0232940.0156760.00.002279...0.5213530.0091280.0000000.2080360.0000000.0000000.0000000.0900.1115381.060529
    11369TCGA-V4-A9EV-01A-11R-A405-07UVM0.0000000.0132990.0250690.3153560.0000000.0173610.00.095233...0.1636710.0155580.0000000.0617070.0000000.0000000.0000000.0020.3471250.952021
    11370TCGA-V4-A9EY-01A-11R-A405-07UVM0.0000000.0325070.0761600.0614000.0343590.0000000.00.007778...0.4315680.0045500.0000000.2277970.0000000.0000000.0000000.0860.1138881.041876
    11371TCGA-V4-A9EU-01A-11R-A405-07UVM0.0000000.0212280.0102250.0679850.0034340.0066750.00.004884...0.6276310.0000000.0000000.1474890.0000000.0000000.0000000.0960.1078521.068155
    11372TCGA-V4-A9EM-01A-11R-A405-07UVM0.0000000.0896360.0547500.0596440.0000000.0399690.00.005501...0.2053020.0000000.0000000.2713390.0000000.0000000.0000000.3180.0415081.064184
    \n

    11373 rows \u00d7 27 columns

    \n
    \n\n\n\n\n```python\ntcga_study_abbr_dict = {\n \"LAML\": \"Acute Myeloid Leukemia\",\n \"ACC\": \"Adrenocortical carcinoma\",\n \"BLCA\": \"Bladder Urothelial Carcinoma\",\n \"LGG\": \"Brain Lower Grade Glioma\",\n \"BRCA\": \"Breast invasive carcinoma\",\n \"CESC\": \"Cervical squamous cell carcinoma and endocervical adenocarcinoma\",\n \"CHOL\": \"Cholangiocarcinoma\",\n \"LCML\": \"Chronic Myelogenous Leukemia\",\n \"COAD\": \"Colon adenocarcinoma\",\n \"CNTL\": \"Controls\",\n \"ESCA\": \"Esophageal carcinoma\",\n \"FPPP\": \"FFPE Pilot Phase II\",\n \"GBM\": \"Glioblastoma multiforme\",\n \"HNSC\": \"Head and Neck squamous cell carcinoma\",\n \"KICH\": \"Kidney Chromophobe\",\n \"KIRC\": \"Kidney renal clear cell carcinoma\",\n \"KIRP\": \"Kidney renal papillary cell carcinoma\",\n \"LIHC\": \"Liver hepatocellular carcinoma\",\n \"LUAD\": \"Lung adenocarcinoma\",\n \"LUSC\": \"Lung squamous cell carcinoma\",\n \"DLBC\": \"Lymphoid Neoplasm Diffuse Large B-cell Lymphoma\",\n \"MESO\": \"Mesothelioma\",\n \"MISC\": \"Miscellaneous\",\n \"OV\": \"Ovarian serous cystadenocarcinoma\",\n \"PAAD\": \"Pancreatic adenocarcinoma\",\n \"PCPG\": \"Pheochromocytoma and Paraganglioma\",\n \"PRAD\": \"Prostate adenocarcinoma\",\n \"READ\": \"Rectum adenocarcinoma\",\n \"SARC\": \"Sarcoma\",\n \"SKCM\": \"Skin Cutaneous Melanoma\",\n \"STAD\": \"Stomach adenocarcinoma\",\n \"TGCT\": \"Testicular Germ Cell Tumors\",\n \"THYM\": \"Thymoma\",\n \"THCA\": \"Thyroid carcinoma\",\n \"UCS\": \"Uterine Carcinosarcoma\",\n \"UCEC\": \"Uterine Corpus Endometrial Carcinoma\",\n \"UVM\": \"Uveal Melanoma\"\n}\n```\n\n\n```python\ntcga_study_abbr_dict2 = {k: v + ' (' + k + ')' for k, v in tcga_study_abbr_dict.items()}\ntcga_study_abbr_dict2\n```\n\n\n\n\n {'LAML': 'Acute Myeloid Leukemia (LAML)',\n 'ACC': 'Adrenocortical carcinoma (ACC)',\n 'BLCA': 'Bladder Urothelial Carcinoma (BLCA)',\n 'LGG': 'Brain Lower Grade Glioma (LGG)',\n 'BRCA': 'Breast invasive carcinoma (BRCA)',\n 'CESC': 'Cervical squamous cell carcinoma and endocervical adenocarcinoma (CESC)',\n 'CHOL': 'Cholangiocarcinoma (CHOL)',\n 'LCML': 'Chronic Myelogenous Leukemia (LCML)',\n 'COAD': 'Colon adenocarcinoma (COAD)',\n 'CNTL': 'Controls (CNTL)',\n 'ESCA': 'Esophageal carcinoma (ESCA)',\n 'FPPP': 'FFPE Pilot Phase II (FPPP)',\n 'GBM': 'Glioblastoma multiforme (GBM)',\n 'HNSC': 'Head and Neck squamous cell carcinoma (HNSC)',\n 'KICH': 'Kidney Chromophobe (KICH)',\n 'KIRC': 'Kidney renal clear cell carcinoma (KIRC)',\n 'KIRP': 'Kidney renal papillary cell carcinoma (KIRP)',\n 'LIHC': 'Liver hepatocellular carcinoma (LIHC)',\n 'LUAD': 'Lung adenocarcinoma (LUAD)',\n 'LUSC': 'Lung squamous cell carcinoma (LUSC)',\n 'DLBC': 'Lymphoid Neoplasm Diffuse Large B-cell Lymphoma (DLBC)',\n 'MESO': 'Mesothelioma (MESO)',\n 'MISC': 'Miscellaneous (MISC)',\n 'OV': 'Ovarian serous cystadenocarcinoma (OV)',\n 'PAAD': 'Pancreatic adenocarcinoma (PAAD)',\n 'PCPG': 'Pheochromocytoma and Paraganglioma (PCPG)',\n 'PRAD': 'Prostate adenocarcinoma (PRAD)',\n 'READ': 'Rectum adenocarcinoma (READ)',\n 'SARC': 'Sarcoma (SARC)',\n 'SKCM': 'Skin Cutaneous Melanoma (SKCM)',\n 'STAD': 'Stomach adenocarcinoma (STAD)',\n 'TGCT': 'Testicular Germ Cell Tumors (TGCT)',\n 'THYM': 'Thymoma (THYM)',\n 'THCA': 'Thyroid carcinoma (THCA)',\n 'UCS': 'Uterine Carcinosarcoma (UCS)',\n 'UCEC': 'Uterine Corpus Endometrial Carcinoma (UCEC)',\n 'UVM': 'Uveal Melanoma (UVM)'}\n\n\n\n\n```python\ndata[\"CancerType\"].apply(tcga_study_abbr_dict.__getitem__).value_counts().plot(kind='bar')\n```\n\n\n```python\ndata['P.value'].plot(kind=\"hist\", bins=100)\n```\n\n\n```python\n{'all': data['P.value'].apply(1.00.__ge__).sum(),\n'0.05': data['P.value'].apply(0.05.__ge__).sum(),\n'0.01': data['P.value'].apply(0.01.__ge__).sum()\n}\n```\n\n\n\n\n {'all': 11373, '0.05': 6077, '0.01': 2893}\n\n\n\n\n```python\ndata['SampleID'].value_counts().where(lambda x: x > 1.0).dropna()\n```\n\n\n\n\n TCGA-A7-A13G-01B-04R-A22O-07 3.0\n TCGA-AC-A2QH-01B-04R-A22O-07 3.0\n TCGA-AC-A3OD-01B-06R-A22O-07 3.0\n TCGA-A7-A26I-01B-06R-A22O-07 3.0\n TCGA-A7-A26F-01B-04R-A22O-07 3.0\n TCGA-A7-A0DC-01B-04R-A22O-07 3.0\n TCGA-A7-A0DC-01A-11R-A00Z-07 3.0\n TCGA-AK-3425-01A-02R-1277-07 2.0\n TCGA-AC-A2QH-01A-11R-A18M-07 2.0\n TCGA-37-4132-01A-01R-1100-07 2.0\n TCGA-A6-5665-01B-03R-2302-07 2.0\n TCGA-AZ-4615-01A-01R-1410-07 2.0\n TCGA-FI-A2F8-01A-12R-A17B-07 2.0\n TCGA-37-4133-01A-01R-1100-07 2.0\n TCGA-HC-8261-01B-05R-2302-07 2.0\n TCGA-AZ-4614-01A-01R-1410-07 2.0\n TCGA-AK-3454-01A-02R-1277-07 2.0\n TCGA-AZ-4315-01A-01R-1410-07 2.0\n TCGA-A6-2684-01A-01R-1410-07 2.0\n TCGA-38-4625-01A-01R-1206-07 2.0\n TCGA-AK-3453-01A-02R-1277-07 2.0\n TCGA-CA-5256-01A-01R-1410-07 2.0\n TCGA-CM-4747-01A-01R-1410-07 2.0\n TCGA-A6-2685-01A-01R-1410-07 2.0\n TCGA-AW-A1PO-01A-12R-A157-07 2.0\n TCGA-HC-7740-01B-04R-2302-07 2.0\n TCGA-AK-3426-01A-02R-1325-07 2.0\n TCGA-AA-3502-01A-01R-1410-07 2.0\n TCGA-AA-A01Z-01A-11R-A083-07 2.0\n TCGA-AA-3492-01A-01R-1410-07 2.0\n TCGA-AA-3506-01A-01R-1410-07 2.0\n TCGA-A7-A13G-01A-11R-A13Q-07 2.0\n TCGA-AX-A1C7-01A-11R-A137-07 2.0\n TCGA-A6-2682-01A-01R-1410-07 2.0\n TCGA-AA-A01X-01A-21R-A083-07 2.0\n TCGA-A7-A0DC-11A-41R-A089-07 2.0\n TCGA-A7-A26F-01A-21R-A169-07 2.0\n TCGA-AZ-4313-01A-01R-1410-07 2.0\n TCGA-AC-A3OD-01A-11R-A21T-07 2.0\n TCGA-AZ-4684-01A-01R-1410-07 2.0\n TCGA-A6-2672-01B-03R-2302-07 2.0\n TCGA-A2-A0EM-01A-11R-A034-07 2.0\n TCGA-BH-A0B2-01A-11R-A10J-07 2.0\n TCGA-AA-3509-01A-01R-1410-07 2.0\n TCGA-HC-8258-01B-05R-2302-07 2.0\n TCGA-AA-A01P-01A-21R-A083-07 2.0\n TCGA-HC-8265-01B-04R-2302-07 2.0\n TCGA-AA-3495-01A-01R-1410-07 2.0\n TCGA-A7-A26I-01A-11R-A169-07 2.0\n TCGA-AC-A3QQ-01B-06R-A22O-07 2.0\n TCGA-CK-4951-01A-01R-1410-07 2.0\n TCGA-AC-A3QQ-01A-11R-A22K-07 2.0\n TCGA-AJ-A23O-01A-11R-A157-07 2.0\n TCGA-A6-5661-01B-05R-2302-07 2.0\n Name: SampleID, dtype: float64\n\n\n\n\n```python\ndata[data['SampleID'] == 'TCGA-A7-A0DC-01B-04R-A22O-07']\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    SampleIDCancerTypeB.cells.naiveB.cells.memoryPlasma.cellsT.cells.CD8T.cells.CD4.naiveT.cells.CD4.memory.restingT.cells.CD4.memory.activatedT.cells.follicular.helper...Macrophages.M2Dendritic.cells.restingDendritic.cells.activatedMast.cells.restingMast.cells.activatedEosinophilsNeutrophilsP.valueCorrelationRMSE
    1762TCGA-A7-A0DC-01B-04R-A22O-07BRCA0.00.0955660.0451220.0181110.1304550.1725050.00.0...0.1155550.00.0675550.2515920.00.0000000.00.1080.0990621.040622
    1763TCGA-A7-A0DC-01B-04R-A22O-07BRCA0.00.1139390.0403570.0057430.0674930.2149650.00.0...0.1256610.00.0589620.2780530.00.0000000.00.1380.0783721.048953
    1764TCGA-A7-A0DC-01B-04R-A22O-07BRCA0.00.1337030.0468570.0185520.0315060.2257820.00.0...0.1042920.00.0784650.2111480.00.0013340.00.2440.0455921.056600
    \n

    3 rows \u00d7 27 columns

    \n
    \n\n\n\n## Leukocyte Fractions (LFs)\n\n\nLeukocyte fractions from [The Immune Landscape of Cancer](https://www.sciencedirect.com/science/article/pii/S1074761318301213) downloaded from [NIH PanCanAtlas](https://gdc.cancer.gov/about-data/publications/panimmune).\n\n\n```python\nleuk = pd.read_csv(\"TCGA_all_leuk_estimate.masked.20170107.tsv\", sep=\"\\t\", header=None)\nleuk.columns = ['CancerType', 'SampleID', 'LF']\nleuk\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    CancerTypeSampleIDLF
    0ACCTCGA-OR-A5J1-01A-11D-A29J-050.046374
    1ACCTCGA-OR-A5J2-01A-11D-A29J-050.057859
    2ACCTCGA-OR-A5J3-01A-11D-A29J-050.048460
    3ACCTCGA-OR-A5J4-01A-11D-A29J-050.043988
    4ACCTCGA-OR-A5J5-01A-11D-A29J-050.016759
    ............
    10812TGCTTCGA-ZM-AA0D-01A-11D-A436-050.578000
    10813TGCTTCGA-ZM-AA0E-01A-12D-A436-050.512000
    10814TGCTTCGA-ZM-AA0F-01A-21D-A436-050.625000
    10815TGCTTCGA-ZM-AA0H-01A-11D-A436-050.431000
    10816TGCTTCGA-ZM-AA0N-01A-21D-A436-050.390000
    \n

    10817 rows \u00d7 3 columns

    \n
    \n\n\n\n\n```python\ndisplay(leuk.LF.min())\ndisplay(leuk.LF.max())\n```\n\n\n -9.29722165644888e-05\n\n\n\n 0.9642001238693829\n\n\n\n```python\nleuk.LF[leuk.LF < 0] = 0\n```\n\n :1: SettingWithCopyWarning: \n A value is trying to be set on a copy of a slice from a DataFrame\n \n See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n leuk.LF[leuk.LF < 0] = 0\n\n\n## Merge CIBERSORT and LF\n\nWe first investigate the number of samples available.\nRemember that the format of TCGA barcode is:\n\n\n| Label | Project | TSS | Participant | Sample & Vial | Portion & Analyte | Plate | Center |\n|--------------|---------|-----|-------------|---------------|-------------------|-------|--------|\n| Subscription | 0 | 1 | 2 | 3 | 4 | 5 | 6 |\n| Example | TCGA | OR | A5J1 | 01A | 11D | A29J | 05 |\n\nIf we use Label-Project-TSS-Participant-Sample&Vial as the identifier, there are 10103 matched ones. It goes down to zero if we use up to Portion&Analyte. In fact, the CIBERSORT is based on RNA, while the LF is based on the DNA sample (by methylation analysis).\n\nFinally, we use Label-Project-TSS-Participant-Sample&Vial-Portion, but exclude the Analyter, Plate, and Center as our identifier to match the samples.\n\nAlso note that there are \"duplicates\" in the original tables. For CIBERSORT, we keep the one with lowest p-value. For LF, we take the average.\n\nIt is worth noting that \"match\" here shall not be confused with \"matched samples\" in \"paired test\".\n\n\n```python\nid1 = set(data['SampleID'].apply(lambda x: '-'.join(x.split('-')[0:4])))\nid2 = set(leuk['SampleID'].apply(lambda x: '-'.join(x.split('-')[0:4])))\n{\n '1': len(id1),\n '2': len(id2),\n 'intersection': len(id1.intersection(id2))\n}\n```\n\n\n\n\n {'1': 11273, '2': 10770, 'intersection': 10103}\n\n\n\n\n```python\nid1 = set(data['SampleID'].apply(lambda x: '-'.join(x.split('-')[0:5])))\nid2 = set(leuk['SampleID'].apply(lambda x: '-'.join(x.split('-')[0:5])))\n{\n '1': len(id1),\n '2': len(id2),\n 'intersection': len(id1.intersection(id2))\n}\n```\n\n\n\n\n {'1': 11275, '2': 10782, 'intersection': 0}\n\n\n\n\n```python\nid1 = set(data['SampleID'].apply(lambda x: '-'.join(x.split('-')[0:4] + [x.split('-')[4][0:2]])))\nid2 = set(leuk['SampleID'].apply(lambda x: '-'.join(x.split('-')[0:4] + [x.split('-')[4][0:2]])))\ncommon_id = id1.intersection(id2)\n{\n '1': len(id1),\n '2': len(id2),\n 'intersection': len(common_id)\n}\n```\n\n\n\n\n {'1': 11275, '2': 10782, 'intersection': 10039}\n\n\n\n\n```python\nmerged = pd.DataFrame()\nfor id in common_id:\n temp = data[data['SampleID'].apply(lambda x: id in x)]\n temp.loc[:, 'LF'] = leuk.loc[leuk.loc[:, 'SampleID'].apply(lambda x: id in x), 'LF'].mean()\n merged = merged.append(temp.loc[temp.loc[:, 'P.value'].idxmax()])\n```\n\n /home/sliang3/miniconda3/envs/basic/lib/python3.8/site-packages/pandas/core/indexing.py:845: SettingWithCopyWarning: \n A value is trying to be set on a copy of a slice from a DataFrame.\n Try using .loc[row_indexer,col_indexer] = value instead\n \n See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n self.obj[key] = _infer_fill_value(value)\n /home/sliang3/miniconda3/envs/basic/lib/python3.8/site-packages/pandas/core/indexing.py:966: SettingWithCopyWarning: \n A value is trying to be set on a copy of a slice from a DataFrame.\n Try using .loc[row_indexer,col_indexer] = value instead\n \n See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n self.obj[item] = s\n\n\n\n```python\nmerged = merged[data.columns.tolist() + ['LF']]\n```\n\n\n```python\nmerged\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    SampleIDCancerTypeB.cells.naiveB.cells.memoryPlasma.cellsT.cells.CD8T.cells.CD4.naiveT.cells.CD4.memory.restingT.cells.CD4.memory.activatedT.cells.follicular.helper...Dendritic.cells.restingDendritic.cells.activatedMast.cells.restingMast.cells.activatedEosinophilsNeutrophilsP.valueCorrelationRMSELF
    6674TCGA-92-7341-01A-31R-2045-07LUSC0.1478730.0000000.0876860.0334310.00.1708790.0000000.000000...0.0000000.0197670.0185380.0010420.00.0046510.0060.2872520.9696590.216818
    7939TCGA-CH-5754-01A-11R-1580-07PRAD0.0653310.0000000.0973790.1458300.00.1441640.0000000.135648...0.0114410.0000000.0238680.0000000.00.0000000.0140.2294211.0069180.248506
    11182TCGA-BS-A0V7-01A-21R-A118-07UCEC0.0000000.1294410.1124270.1669600.00.0663480.0079720.108577...0.0101740.0047340.0819790.0000000.00.0002850.0040.3664370.9363390.138962
    1419TCGA-A2-A0CQ-01A-21R-A034-07BRCA0.0000000.0446070.0000000.1983270.00.0412380.0000000.018438...0.0066610.0000000.0592810.0000000.00.0029020.0120.2219091.0016120.120546
    6205TCGA-05-5420-01A-01R-1628-07LUAD0.0316640.0000000.0132260.0874010.00.1155600.0524060.055071...0.1404230.0000000.0207370.0000000.00.0146430.2160.0489651.0682500.571093
    ..................................................................
    3172TCGA-CV-5432-01A-02R-1686-07HNSC0.1270650.0000000.2610890.1051920.00.0239600.0000000.042882...0.0575830.0000000.0394490.0000000.00.0021970.0020.4373820.9035110.179283
    2824TCGA-L5-A8NQ-01A-11R-A36D-31ESCA0.0000000.0389280.0113050.1142160.00.0368880.0000000.100739...0.0000000.0000000.0534880.0000000.00.0000000.0960.1015061.0591700.392468
    9431TCGA-FP-8210-01A-11R-2343-13STAD0.1870190.0999270.0001760.1463840.00.1158630.0000000.066405...0.0000000.0182630.0726360.0000000.00.0000000.0020.3379420.9499990.488038
    8279TCGA-KK-A8I8-01A-11R-A36G-07PRAD0.0027270.0927360.0195800.2029590.00.1777910.0000000.036568...0.0145360.0000000.0641180.0000000.00.0000000.2220.0652461.0633350.066327
    1305TCGA-C8-A130-01A-31R-A115-07BRCA0.0843780.0000000.0473900.1805250.00.1192890.0000000.061659...0.0025190.0000000.0923240.0000000.00.0000000.0120.2114120.9972380.143928
    \n

    10039 rows \u00d7 28 columns

    \n
    \n\n\n\nMultiply abundance of cell types with the leukocyte fraction.\n\n\n```python\nmerged_adjusted = merged.copy()\nfor i in range(merged_adjusted.shape[0]):\n merged_adjusted.iloc[i, cell_types] *= merged.iloc[i, 27]\nmerged_adjusted\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    SampleIDCancerTypeB.cells.naiveB.cells.memoryPlasma.cellsT.cells.CD8T.cells.CD4.naiveT.cells.CD4.memory.restingT.cells.CD4.memory.activatedT.cells.follicular.helper...Dendritic.cells.restingDendritic.cells.activatedMast.cells.restingMast.cells.activatedEosinophilsNeutrophilsP.valueCorrelationRMSELF
    6674TCGA-92-7341-01A-31R-2045-07LUSC0.0320620.0000000.0190120.0072480.00.0370500.0000000.000000...0.0000000.0042860.0040190.0002260.00.0010080.0060.2872520.9696590.216818
    7939TCGA-CH-5754-01A-11R-1580-07PRAD0.0162350.0000000.0241990.0362400.00.0358250.0000000.033709...0.0028430.0000000.0059310.0000000.00.0000000.0140.2294211.0069180.248506
    11182TCGA-BS-A0V7-01A-21R-A118-07UCEC0.0000000.0179870.0156230.0232010.00.0092200.0011080.015088...0.0014140.0006580.0113920.0000000.00.0000400.0040.3664370.9363390.138962
    1419TCGA-A2-A0CQ-01A-21R-A034-07BRCA0.0000000.0053770.0000000.0239070.00.0049710.0000000.002223...0.0008030.0000000.0071460.0000000.00.0003500.0120.2219091.0016120.120546
    6205TCGA-05-5420-01A-01R-1628-07LUAD0.0180830.0000000.0075530.0499140.00.0659950.0299290.031451...0.0801950.0000000.0118430.0000000.00.0083630.2160.0489651.0682500.571093
    ..................................................................
    3172TCGA-CV-5432-01A-02R-1686-07HNSC0.0227810.0000000.0468090.0188590.00.0042960.0000000.007688...0.0103240.0000000.0070730.0000000.00.0003940.0020.4373820.9035110.179283
    2824TCGA-L5-A8NQ-01A-11R-A36D-31ESCA0.0000000.0152780.0044370.0448260.00.0144770.0000000.039537...0.0000000.0000000.0209920.0000000.00.0000000.0960.1015061.0591700.392468
    9431TCGA-FP-8210-01A-11R-2343-13STAD0.0912720.0487680.0000860.0714410.00.0565460.0000000.032408...0.0000000.0089130.0354490.0000000.00.0000000.0020.3379420.9499990.488038
    8279TCGA-KK-A8I8-01A-11R-A36G-07PRAD0.0001810.0061510.0012990.0134620.00.0117920.0000000.002425...0.0009640.0000000.0042530.0000000.00.0000000.2220.0652461.0633350.066327
    1305TCGA-C8-A130-01A-31R-A115-07BRCA0.0121440.0000000.0068210.0259820.00.0171690.0000000.008874...0.0003630.0000000.0132880.0000000.00.0000000.0120.2114120.9972380.143928
    \n

    10039 rows \u00d7 28 columns

    \n
    \n\n\n\n## Label samples\nLabel samples with its type and patient ID (i.e., Label-Project-TSS-Participant).\n\n\n```python\ntcga_sample_type_dict = {\n \"01\": \"Primary Solid Tumor\",\n \"02\": \"Recurrent Solid Tumor\",\n \"03\": \"Primary Blood Derived Cancer - Peripheral Blood\",\n \"04\": \"Recurrent Blood Derived Cancer - Bone Marrow\",\n \"05\": \"Additional - New Primary\",\n \"06\": \"Metastatic\",\n \"07\": \"Additional Metastatic\",\n \"08\": \"Human Tumor Original Cells\",\n \"09\": \"Primary Blood Derived Cancer - Bone Marrow\",\n \"10\": \"Blood Derived Normal\",\n \"11\": \"Solid Tissue Normal\",\n \"12\": \"Buccal Cell Normal\",\n \"13\": \"EBV Immortalized Normal\",\n \"14\": \"Bone Marrow Normal\",\n \"15\": \"sample type 15\",\n \"16\": \"sample type 16\",\n \"20\": \"Control Analyte\",\n \"40\": \"Recurrent Blood Derived Cancer - Peripheral Blood\",\n \"50\": \"Cell Lines\",\n \"60\": \"Primary Xenograft Tissue\",\n \"61\": \"Cell Line Derived Xenograft Tissue\",\n \"99\": \"sample type 99\"\n}\n```\n\n\n```python\nf = lambda x: tcga_sample_type_dict[x.split('-')[3][0:2]]\nmerged_adjusted['SampleType'] = merged_adjusted[\"SampleID\"].apply(f)\n```\n\n\n```python\nf = lambda x: '-'.join(x.split('-')[0:3])\nmerged_adjusted['PatientID'] = merged_adjusted[\"SampleID\"].apply(f)\n```\n\n\n```python\nmerged_adjusted\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    SampleIDCancerTypeB.cells.naiveB.cells.memoryPlasma.cellsT.cells.CD8T.cells.CD4.naiveT.cells.CD4.memory.restingT.cells.CD4.memory.activatedT.cells.follicular.helper...Mast.cells.restingMast.cells.activatedEosinophilsNeutrophilsP.valueCorrelationRMSELFSampleTypePatientID
    6674TCGA-92-7341-01A-31R-2045-07LUSC0.0320620.0000000.0190120.0072480.00.0370500.0000000.000000...0.0040190.0002260.00.0010080.0060.2872520.9696590.216818Primary Solid TumorTCGA-92-7341
    7939TCGA-CH-5754-01A-11R-1580-07PRAD0.0162350.0000000.0241990.0362400.00.0358250.0000000.033709...0.0059310.0000000.00.0000000.0140.2294211.0069180.248506Primary Solid TumorTCGA-CH-5754
    11182TCGA-BS-A0V7-01A-21R-A118-07UCEC0.0000000.0179870.0156230.0232010.00.0092200.0011080.015088...0.0113920.0000000.00.0000400.0040.3664370.9363390.138962Primary Solid TumorTCGA-BS-A0V7
    1419TCGA-A2-A0CQ-01A-21R-A034-07BRCA0.0000000.0053770.0000000.0239070.00.0049710.0000000.002223...0.0071460.0000000.00.0003500.0120.2219091.0016120.120546Primary Solid TumorTCGA-A2-A0CQ
    6205TCGA-05-5420-01A-01R-1628-07LUAD0.0180830.0000000.0075530.0499140.00.0659950.0299290.031451...0.0118430.0000000.00.0083630.2160.0489651.0682500.571093Primary Solid TumorTCGA-05-5420
    ..................................................................
    3172TCGA-CV-5432-01A-02R-1686-07HNSC0.0227810.0000000.0468090.0188590.00.0042960.0000000.007688...0.0070730.0000000.00.0003940.0020.4373820.9035110.179283Primary Solid TumorTCGA-CV-5432
    2824TCGA-L5-A8NQ-01A-11R-A36D-31ESCA0.0000000.0152780.0044370.0448260.00.0144770.0000000.039537...0.0209920.0000000.00.0000000.0960.1015061.0591700.392468Primary Solid TumorTCGA-L5-A8NQ
    9431TCGA-FP-8210-01A-11R-2343-13STAD0.0912720.0487680.0000860.0714410.00.0565460.0000000.032408...0.0354490.0000000.00.0000000.0020.3379420.9499990.488038Primary Solid TumorTCGA-FP-8210
    8279TCGA-KK-A8I8-01A-11R-A36G-07PRAD0.0001810.0061510.0012990.0134620.00.0117920.0000000.002425...0.0042530.0000000.00.0000000.2220.0652461.0633350.066327Primary Solid TumorTCGA-KK-A8I8
    1305TCGA-C8-A130-01A-31R-A115-07BRCA0.0121440.0000000.0068210.0259820.00.0171690.0000000.008874...0.0132880.0000000.00.0000000.0120.2114120.9972380.143928Primary Solid TumorTCGA-C8-A130
    \n

    10039 rows \u00d7 30 columns

    \n
    \n\n\n\n# Basic statistics\n\n\n```python\nf = lambda x: tcga_sample_type_dict[x.split('-')[3][0:2]]\nall_data = merged_adjusted.copy()\nall_data[\"SampleType\"] = all_data[\"SampleID\"].apply(f)\nall_data.groupby([\"CancerType\", \"SampleType\"]).SampleID.count().to_frame()\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    SampleID
    CancerTypeSampleType
    ACCPrimary Solid Tumor79
    BLCAPrimary Solid Tumor411
    BRCAMetastatic7
    Primary Solid Tumor1084
    CESCMetastatic2
    Primary Solid Tumor304
    CHOLPrimary Solid Tumor36
    COADMetastatic1
    Primary Solid Tumor466
    Recurrent Solid Tumor1
    ESCAMetastatic1
    Primary Solid Tumor184
    GBMPrimary Solid Tumor127
    Recurrent Solid Tumor13
    HNSCMetastatic2
    Primary Solid Tumor520
    KICHPrimary Solid Tumor66
    KIRCAdditional - New Primary1
    Primary Solid Tumor490
    KIRPAdditional - New Primary1
    Primary Solid Tumor280
    LGGPrimary Solid Tumor513
    Recurrent Solid Tumor18
    LIHCPrimary Solid Tumor371
    Recurrent Solid Tumor3
    LUADPrimary Solid Tumor525
    Recurrent Solid Tumor2
    LUSCPrimary Solid Tumor501
    MESOPrimary Solid Tumor87
    OVPrimary Solid Tumor419
    Recurrent Solid Tumor7
    PAADMetastatic1
    Primary Solid Tumor178
    PCPGAdditional - New Primary3
    Metastatic2
    Primary Solid Tumor179
    PRADMetastatic1
    Primary Solid Tumor501
    READPrimary Solid Tumor164
    Recurrent Solid Tumor1
    SARCMetastatic1
    Primary Solid Tumor259
    Recurrent Solid Tumor3
    SKCMAdditional Metastatic1
    Metastatic367
    Primary Solid Tumor103
    STADPrimary Solid Tumor416
    TGCTAdditional - New Primary4
    Primary Solid Tumor133
    THCAMetastatic8
    Primary Solid Tumor505
    UCECPrimary Solid Tumor549
    Recurrent Solid Tumor1
    UCSPrimary Solid Tumor57
    UVMPrimary Solid Tumor80
    \n
    \n\n\n\n\n```python\nall_data.SampleType.unique().tolist()\n```\n\n\n\n\n ['Primary Solid Tumor',\n 'Metastatic',\n 'Recurrent Solid Tumor',\n 'Additional Metastatic',\n 'Additional - New Primary']\n\n\n\n\n```python\nsample_size = pd.crosstab(all_data[\"CancerType\"], all_data[\"SampleType\"])\nsample_size.T.style\n```\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    CancerType ACC BLCA BRCA CESC CHOL COAD ESCA GBM HNSC KICH KIRC KIRP LGG LIHC LUAD LUSC MESO OV PAAD PCPG PRAD READ SARC SKCM STAD TGCT THCA UCEC UCS UVM
    SampleType
    Additional - New Primary000000000011000000030000040000
    Additional Metastatic000000000000000000000001000000
    Metastatic00720110200000000012101367008000
    Primary Solid Tumor7941110843043646618412752066490280513371525501874191781795011642591034161335055495780
    Recurrent Solid Tumor00000101300001832007000130000100
    \n\n\n\n\n```python\nall_data\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    SampleIDCancerTypeB.cells.naiveB.cells.memoryPlasma.cellsT.cells.CD8T.cells.CD4.naiveT.cells.CD4.memory.restingT.cells.CD4.memory.activatedT.cells.follicular.helper...Mast.cells.restingMast.cells.activatedEosinophilsNeutrophilsP.valueCorrelationRMSELFSampleTypePatientID
    6674TCGA-92-7341-01A-31R-2045-07LUSC0.0320620.0000000.0190120.0072480.00.0370500.0000000.000000...0.0040190.0002260.00.0010080.0060.2872520.9696590.216818Primary Solid TumorTCGA-92-7341
    7939TCGA-CH-5754-01A-11R-1580-07PRAD0.0162350.0000000.0241990.0362400.00.0358250.0000000.033709...0.0059310.0000000.00.0000000.0140.2294211.0069180.248506Primary Solid TumorTCGA-CH-5754
    11182TCGA-BS-A0V7-01A-21R-A118-07UCEC0.0000000.0179870.0156230.0232010.00.0092200.0011080.015088...0.0113920.0000000.00.0000400.0040.3664370.9363390.138962Primary Solid TumorTCGA-BS-A0V7
    1419TCGA-A2-A0CQ-01A-21R-A034-07BRCA0.0000000.0053770.0000000.0239070.00.0049710.0000000.002223...0.0071460.0000000.00.0003500.0120.2219091.0016120.120546Primary Solid TumorTCGA-A2-A0CQ
    6205TCGA-05-5420-01A-01R-1628-07LUAD0.0180830.0000000.0075530.0499140.00.0659950.0299290.031451...0.0118430.0000000.00.0083630.2160.0489651.0682500.571093Primary Solid TumorTCGA-05-5420
    ..................................................................
    3172TCGA-CV-5432-01A-02R-1686-07HNSC0.0227810.0000000.0468090.0188590.00.0042960.0000000.007688...0.0070730.0000000.00.0003940.0020.4373820.9035110.179283Primary Solid TumorTCGA-CV-5432
    2824TCGA-L5-A8NQ-01A-11R-A36D-31ESCA0.0000000.0152780.0044370.0448260.00.0144770.0000000.039537...0.0209920.0000000.00.0000000.0960.1015061.0591700.392468Primary Solid TumorTCGA-L5-A8NQ
    9431TCGA-FP-8210-01A-11R-2343-13STAD0.0912720.0487680.0000860.0714410.00.0565460.0000000.032408...0.0354490.0000000.00.0000000.0020.3379420.9499990.488038Primary Solid TumorTCGA-FP-8210
    8279TCGA-KK-A8I8-01A-11R-A36G-07PRAD0.0001810.0061510.0012990.0134620.00.0117920.0000000.002425...0.0042530.0000000.00.0000000.2220.0652461.0633350.066327Primary Solid TumorTCGA-KK-A8I8
    1305TCGA-C8-A130-01A-31R-A115-07BRCA0.0121440.0000000.0068210.0259820.00.0171690.0000000.008874...0.0132880.0000000.00.0000000.0120.2114120.9972380.143928Primary Solid TumorTCGA-C8-A130
    \n

    10039 rows \u00d7 30 columns

    \n
    \n\n\n\n\n```python\ncell_types = ['B.cells.naive', 'B.cells.memory', 'Plasma.cells', 'T.cells.CD8',\n 'T.cells.CD4.naive', 'T.cells.CD4.memory.resting',\n 'T.cells.CD4.memory.activated', 'T.cells.follicular.helper',\n 'T.cells.regulatory..Tregs.', 'T.cells.gamma.delta', 'NK.cells.resting',\n 'NK.cells.activated', 'Monocytes', 'Macrophages.M0', 'Macrophages.M1',\n 'Macrophages.M2', 'Dendritic.cells.resting',\n 'Dendritic.cells.activated', 'Mast.cells.resting',\n 'Mast.cells.activated', 'Eosinophils', 'Neutrophils']\n```\n\n\n```python\nall_data['Leukocytes.all'] = all_data[cell_types].sum(1)\n\nall_data['T.cells.all'] = all_data[['T.cells.CD8',\n 'T.cells.CD4.naive',\n 'T.cells.CD4.memory.resting',\n 'T.cells.CD4.memory.activated',\n 'T.cells.follicular.helper',\n 'T.cells.regulatory..Tregs.',\n 'T.cells.gamma.delta']].sum(1)\n\nall_data['B.cells.all'] = all_data[['B.cells.naive', 'B.cells.memory']].sum(1)\n\nall_data['Nk.cells.all'] = all_data[['NK.cells.resting', 'NK.cells.activated']].sum(1)\n\nall_data['Macrophages.all'] = all_data[['Macrophages.M0', 'Macrophages.M1', 'Macrophages.M2']].sum(1)\n\nall_data['Dendritic.cells.all'] = all_data[['Dendritic.cells.resting', 'Dendritic.cells.activated']].sum(1)\n\nall_data['Mast.cells.all'] = all_data[['Mast.cells.resting', 'Mast.cells.activated']].sum(1)\n\n\n\naugmented_cell_types = cell_types + ['T.cells.all', 'B.cells.all', 'Nk.cells.all', 'Macrophages.all', \n 'Dendritic.cells.all', 'Mast.cells.all', 'Leukocytes.all']\n```\n\n## Mean and variance\n\nSample mean $\\bar{x}$, confidence interval $(\\bar{x}_L, \\bar{x}_U)$.\nSample standard deviation $s$, confidence interval $(s_L, s_U)$.\n\n\\begin{align}\n\\bar{x} & = \\frac{\\sum{}x_i}{n} \\\\\ns & = \\sqrt{\\frac{\\sum{}(x_i - \\bar{x})^2}{n}} \\\\\n\\bar{x}_L, \\bar{x}_U & = \\bar{x} \\pm t_{1-\\alpha/2, \\nu}\\frac{s}{\\sqrt{n}} \\\\\ns_L & = \\sqrt{\\frac{(n-1)s^2}{\\chi^2_{1-\\alpha/2, \\nu}}} \\\\\ns_U & = \\sqrt{\\frac{(n-1)s^2}{\\chi^2_{\\alpha/2, \\nu}}} \\\\\n\\end{align}\n\nIn this context, degree of freedom $\\nu = n - 1$.\n\n\n```python\nsample_size.columns\n```\n\n\n\n\n Index(['Additional - New Primary', 'Additional Metastatic', 'Metastatic',\n 'Primary Solid Tumor', 'Recurrent Solid Tumor'],\n dtype='object', name='SampleType')\n\n\n\n\n```python\na = 0.05\n\nnum_data = all_data[augmented_cell_types]\n\nsanitize = lambda x: 0.0 if x < 0 else 1.0 if x > 1 else x\n\nfor sample_type in sample_size.columns:\n sample_type_mean = pd.DataFrame(index = augmented_cell_types, columns = sample_size.index)\n sample_type_mean_lower = pd.DataFrame(index = augmented_cell_types, columns = sample_size.index)\n sample_type_mean_upper = pd.DataFrame(index = augmented_cell_types, columns = sample_size.index)\n sample_type_sd = pd.DataFrame(index = augmented_cell_types, columns = sample_size.index)\n sample_type_sd_lower = pd.DataFrame(index = augmented_cell_types, columns = sample_size.index)\n sample_type_sd_upper = pd.DataFrame(index = augmented_cell_types, columns = sample_size.index)\n\n for j in sample_type_mean.columns:\n n = ((all_data.CancerType == j) & (all_data.SampleType == sample_type)).sum()\n\n # Mean\n sample_type_mean.loc[:, j] = num_data[(all_data.CancerType == j) & (all_data.SampleType == sample_type)].mean(axis=0)\n\n # Standard deviation\n sample_type_sd.loc[:, j] = all_data[(all_data.CancerType == j) & (all_data.SampleType == sample_type)].std(ddof=1, axis=0)\n \n # Mean CI\n err = scipy.stats.t.ppf(1 - a / 2, n - 1) * sample_type_sd.loc[:, j] / np.sqrt(n)\n sample_type_mean_lower.loc[:, j] = sample_type_mean.loc[:, j] - err\n sample_type_mean_upper.loc[:, j] = sample_type_mean.loc[:, j] + err\n \n # Standard deviation CI\n sample_type_sd_lower.loc[:, j] = np.sqrt((n - 1) * sample_type_sd.loc[:, j] ** 2 / scipy.stats.chi2.ppf(1 - a / 2, n - 1))\n sample_type_sd_upper.loc[:, j] = np.sqrt((n - 1) * sample_type_sd.loc[:, j] ** 2 / scipy.stats.chi2.ppf(a / 2, n - 1))\n\n index_str_len = sample_type_mean.index.to_series().apply(lambda x: len(x)).max()\n index_str_len\n \n with pd.ExcelWriter(sample_type + \" all.xlsx\", engine=\"xlsxwriter\") as writer:\n sample_type_mean.to_excel(writer, sheet_name=\"mean\")\n sample_type_mean_lower.applymap(sanitize).to_excel(writer, sheet_name=\"mean 95% CI lower\")\n sample_type_mean_upper.applymap(sanitize).to_excel(writer, sheet_name=\"mean 95% CI upper\")\n \n sample_type_sd.to_excel(writer, sheet_name=\"sd\")\n sample_type_sd_lower.to_excel(writer, sheet_name=\"sd 95% CI lower\")\n sample_type_sd_upper.to_excel(writer, sheet_name=\"sd 95% CI upper\")\n for i in writer.sheets:\n writer.sheets[i].set_column('A:A', index_str_len)\n```\n\n\n```python\nwith pd.ExcelWriter(\"all.xlsx\", engine=\"xlsxwriter\") as writer:\n sample_size.T.to_excel(writer, sheet_name=\"summary\")\n```\n\n\n```python\nscipy.stats.t.ppf(1 - a / 2, n - 1) * sample_type_sd.loc[:, j] / n\n```\n\n\n```python\na = 0.05\n\nloglog = lambda x: np.log(-np.log(x))\nexpexp = lambda x: np.exp(-np.exp(x))\n\nnum_data = all_data[augmented_cell_types]\n\nloglog_num_data = loglog(num_data)\n\nfor sample_type in sample_size.columns:\n sample_type_mean = pd.DataFrame(index = augmented_cell_types, columns = sample_size.index)\n sample_type_loglog_mean = pd.DataFrame(index = augmented_cell_types, columns = sample_size.index)\n sample_type_loglog_sd = pd.DataFrame(index = augmented_cell_types, columns = sample_size.index)\n sample_type_mean_lower = pd.DataFrame(index = augmented_cell_types, columns = sample_size.index)\n sample_type_mean_upper = pd.DataFrame(index = augmented_cell_types, columns = sample_size.index)\n sample_type_sd = pd.DataFrame(index = augmented_cell_types, columns = sample_size.index)\n sample_type_sd_lower = pd.DataFrame(index = augmented_cell_types, columns = sample_size.index)\n sample_type_sd_upper = pd.DataFrame(index = augmented_cell_types, columns = sample_size.index)\n\n for j in sample_type_mean.columns:\n n = ((all_data.CancerType == j) & (all_data.SampleType == sample_type)).sum()\n\n # Mean\n sample_type_mean.loc[:, j] = num_data[(all_data.CancerType == j) & (all_data.SampleType == sample_type)].mean(axis=0)\n sample_type_loglog_mean.loc[:, j] = loglog_num_data[(all_data.CancerType == j) & (all_data.SampleType == sample_type)].mean(axis=0)\n sample_type_loglog_sd.loc[:, j] = loglog_num_data[(all_data.CancerType == j) & (all_data.SampleType == sample_type)].std(ddof=1, axis=0)\n\n # Mean CI\n err = scipy.stats.t.ppf(1 - a / 2, n - 1) * sample_type_loglog_sd.loc[:, j] / np.sqrt(n)\n sample_type_mean_lower.loc[:, j] = (sample_type_loglog_mean.loc[:, j] + err).apply(expexp)\n sample_type_mean_upper.loc[:, j] = (sample_type_loglog_mean.loc[:, j] - err).apply(expexp)\n\n # Standard deviation\n sample_type_sd.loc[:, j] = all_data[(all_data.CancerType == j) & (all_data.SampleType == sample_type)].std(ddof=1, axis=0)\n\n \n # Standard deviation CI\n sample_type_sd_lower.loc[:, j] = np.sqrt((n - 1) * sample_type_sd.loc[:, j] ** 2 / scipy.stats.chi2.ppf(1 - a / 2, n - 1))\n sample_type_sd_upper.loc[:, j] = np.sqrt((n - 1) * sample_type_sd.loc[:, j] ** 2 / scipy.stats.chi2.ppf(a / 2, n - 1))\n\n index_str_len = sample_type_mean.index.to_series().apply(lambda x: len(x)).max()\n index_str_len\n \n with pd.ExcelWriter(sample_type + \".xlsx\", engine=\"xlsxwriter\") as writer:\n sample_type_mean.to_excel(writer, sheet_name=\"mean\")\n sample_type_mean_lower.to_excel(writer, sheet_name=\"mean 95% CI lower\")\n sample_type_mean_upper.to_excel(writer, sheet_name=\"mean 95% CI upper\")\n (sample_type_mean_upper - sample_type_mean_lower).to_excel(writer, sheet_name=\"mean 95% CI size\")\n \n sample_type_sd.to_excel(writer, sheet_name=\"sd\")\n sample_type_sd_lower.to_excel(writer, sheet_name=\"sd 95% CI lower\")\n sample_type_sd_upper.to_excel(writer, sheet_name=\"sd 95% CI upper\")\n (sample_type_sd_upper - sample_type_sd_lower).to_excel(writer, sheet_name=\"sd 95% CI size\")\n for i in writer.sheets:\n writer.sheets[i].set_column('A:A', index_str_len)\n```\n\n\n```python\nloglog_num_data[(all_data.CancerType == j) & (all_data.SampleType == sample_type)].mean(axis=0)\n```\n\n\n\n\n B.cells.naive inf\n B.cells.memory inf\n Plasma.cells inf\n T.cells.CD8 inf\n T.cells.CD4.naive inf\n T.cells.CD4.memory.resting inf\n T.cells.CD4.memory.activated inf\n T.cells.follicular.helper inf\n T.cells.regulatory..Tregs. inf\n T.cells.gamma.delta inf\n NK.cells.resting inf\n NK.cells.activated inf\n Monocytes inf\n Macrophages.M0 inf\n Macrophages.M1 inf\n Macrophages.M2 1.497889\n Dendritic.cells.resting inf\n Dendritic.cells.activated inf\n Mast.cells.resting inf\n Mast.cells.activated inf\n Eosinophils inf\n Neutrophils inf\n T.cells.all 1.576024\n B.cells.all inf\n Nk.cells.all inf\n Macrophages.all 1.441598\n Dendritic.cells.all inf\n Mast.cells.all inf\n Leukocytes.all 1.180608\n dtype: float64\n\n\n\n\n```python\nnum_data.min()\n```\n\n\n\n\n B.cells.naive 0.0\n B.cells.memory 0.0\n Plasma.cells 0.0\n T.cells.CD8 0.0\n T.cells.CD4.naive 0.0\n T.cells.CD4.memory.resting 0.0\n T.cells.CD4.memory.activated 0.0\n T.cells.follicular.helper 0.0\n T.cells.regulatory..Tregs. 0.0\n T.cells.gamma.delta 0.0\n NK.cells.resting 0.0\n NK.cells.activated 0.0\n Monocytes 0.0\n Macrophages.M0 0.0\n Macrophages.M1 0.0\n Macrophages.M2 0.0\n Dendritic.cells.resting 0.0\n Dendritic.cells.activated 0.0\n Mast.cells.resting 0.0\n Mast.cells.activated 0.0\n Eosinophils 0.0\n Neutrophils 0.0\n T.cells.all 0.0\n B.cells.all 0.0\n Nk.cells.all 0.0\n Macrophages.all 0.0\n Dendritic.cells.all 0.0\n Mast.cells.all 0.0\n Leukocytes.all 0.0\n dtype: float64\n\n\n", "meta": {"hexsha": "56b2a118dfd7eab1c36d96f30fbc4ae72c1abee3", "size": 248722, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "python/statistics/TCGA-immune-against-all.ipynb", "max_stars_repo_name": "KChen-lab/sensei", "max_stars_repo_head_hexsha": "591f5214b598f60e2ea21bb8b9955cc529f67eee", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "python/statistics/TCGA-immune-against-all.ipynb", "max_issues_repo_name": "KChen-lab/sensei", "max_issues_repo_head_hexsha": "591f5214b598f60e2ea21bb8b9955cc529f67eee", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "python/statistics/TCGA-immune-against-all.ipynb", "max_forks_repo_name": "KChen-lab/sensei", "max_forks_repo_head_hexsha": "591f5214b598f60e2ea21bb8b9955cc529f67eee", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-01-08T16:34:49.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-08T16:34:49.000Z", "avg_line_length": 63.1915650407, "max_line_length": 55628, "alphanum_fraction": 0.566821592, "converted": true, "num_tokens": 35278, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.668880247169804, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.43329943213996613}} {"text": "# **Modelagem**\n\n\n\n\n\n\n## **Instala\u00e7\u00e3o e importa\u00e7\u00e3o**\n\n\n```python\n%reset -f\n```\n\n\n```python\n#Instalar e importar bibliotecas:\n\ntry:\n import CoolProp\nexcept ImportError:\n !pip install CoolProp\n\n\nimport matplotlib.pyplot as plt\nfrom matplotlib.ticker import AutoMinorLocator\nimport CoolProp.CoolProp as cp\nfrom CoolProp.CoolProp import PropsSI as ps\nfrom CoolProp.CoolProp import State as st\nfrom tabulate import tabulate as tab\nimport numpy as np\nfrom sympy import *\nimport sympy as sp\nimport pandas as pd\nimport ipywidgets as wd\nfrom IPython.display import display\n\nfrom IPython.core.display import HTML\nHTML(\"\"\"\n\n\"\"\")\n```\n\n\n\n\n\n\n\n\n\n\n## **Escala de temperatura**\n\n\n```python\n# Selecionar escala de temperatura:\n\nTemperatura = input('Temperatura em \u00b0C ou em K? (Digite c ou k): ')\n```\n\n Temperatura em \u00b0C ou em K? (Digite c ou k): C\n\n\n## **Declara\u00e7\u00e3o de vari\u00e1veis**\n\n\n```python\n# Declarar estados:\n\nn = 12\n\nfld = \"Water\"\ncp.set_reference_state(fld,'DEF')\nif Temperatura == 'k':\n Tnn = \"Temperatura [K]\"\nelse:\n Tnn = \"Temperatura [\u00b0C]\"\n\nmnn = \"Vaz\u00e3o m\u00e1ssica [kg/s]\"\nhnn = \"Entalpia [kJ/kg]\"\nvnn = \"Volume espec\u00edfico [m\u00b3/kg]\"\nPnn = \"Press\u00e3o [kPa]\"\nsnn = \"Entropia [kJ/kg*K]\"\nXnn = \"T\u00edtulo\"\n\nestado = np.arange(1, n+1, 1)\n\nT = np.linspace(0,0,n)\nm = np.linspace(0,0,n)\nh = np.linspace(0,0,n)\nv = np.linspace(0,0,n)\nP = np.linspace(0,0,n)\ns = np.linspace(0,0,n)\nX = np.linspace(0,0,n)\n```\n\n## **Fornalha**\n\n\n```python\n# Combust\u00edvel:\n\nP_CH4 = 0.8897\nP_C2H6 = 0.0592\nP_C3H8 = 0.0191\nP_C4H10 = 0.0109\nP_C02 = 0.0121\nP_N2 = 0.0089\nP_O2 = 0.0001\nP_N2_Ar = 0.79\nP_O2_Ar = 0.21\n\n# Excesso de ar:\n\ne = 1.5\n```\n\n\n```python\n#C\u00e1lculos:\n\nCH4 = np.array([1,4,0,0])\nC2H6 = np.array([2,6,0,0])\nC3H8 = np.array([3,8,0,0])\nC4H10 = np.array([4,10,0,0])\nCO2 = np.array([1,0,2,0])\nN2 = np.array([0,0,0,2])\nO2 = np.array([0,0,2,0])\nH2O = np.array([0,2,1,0])\n\nMM = np.array([12.011,1.008,15.999,14.007])\nP_GN = [P_CH4,P_C2H6,P_C3H8,P_C4H10,P_C02,P_N2,P_O2]\n\nM_CH4 = np.sum(CH4*MM)\nM_C2H6 = np.sum(C2H6*MM)\nM_C3H8 = np.sum(C3H8*MM)\nM_C4H10 = np.sum(C4H10*MM)\nM_CO2 = np.sum(CO2*MM)\nM_N2 = np.sum(N2*MM)\nM_O2 = np.sum(O2*MM)\nM_H2O = np.sum(H2O*MM)\nM_Air = M_N2*P_N2_Ar+M_O2*P_O2_Ar\n\nM = [M_CH4,M_C2H6,M_C3H8,M_C4H10,M_CO2,M_N2,M_O2,M_N2,M_H2O]\n\nF_M = []\nfor i in range(0,4):\n F_M.append(P_GN)\n\nnp.array(F_M)\nquant_atomos_comb = np.array([CH4,C2H6,C3H8,C4H10,CO2,N2,O2])\nm_comp_comb = np.sum(F_M*quant_atomos_comb.T,axis=1)\nm_comp_comb[2] = m_comp_comb[2]*(-1)\n\nP_GN.append(P_N2_Ar/P_O2_Ar) #acrecentar partes de O2\nP_GN.append(P_O2_Ar/P_O2_Ar) ##acrecentar partes de N2\n\nb = m_comp_comb[0]\nc = m_comp_comb[1]/2\na = e*(m_comp_comb[2] + 2*b + c)/(2*P_GN[8])\nd = (m_comp_comb[3] + 2*a*P_GN[7])/2\nf = (2*e*(2*b+c)/(2*P_GN[8])-c-2*b)/2\n\nBLNC = [1,1,1,1,1,1,1,a,a]\n\nresposta = np.zeros((9,4))\nquant_atomos_molecula = np.array([CH4,C2H6,C3H8,C4H10,CO2,N2,O2,N2,O2])\n\nfor i in range(0,9):\n for j in range(0,4):\n resposta[i,j] = quant_atomos_molecula[i,j] * MM[j] * P_GN[i] * BLNC[i]\n\nfuel = resposta[0:7,0:4]\nair = resposta[7:9,0:4]\nsoma = np.sum(resposta)\nM_fuel = np.sum(fuel) #Massa total de combust\u00edvel\nsoma_air = np.sum(air) #Massa total de ar\ntotal_elem = np.sum(fuel, axis=0,keepdims=True) #C_tot, H_tot, O_tot, N_tot\n\nF_tot = total_elem/M_fuel\n\nPCS = 33900*F_tot[0,0] + 141800*(F_tot[0,1] - (F_tot[0,2]/8))\n\nw = PCS/2440 - 9*F_tot[0,1]\nF_w = w/M_fuel\n\nPCI = PCS - 2440*(9*F_tot[0,1] - F_w)\n\nA\u0307F\u0307 = a*(1+79/21)\nAF = A\u0307F\u0307*(M_Air/M_fuel)\n\nprint('A raz\u00e3o ar-combust\u00edvel \u00e9 de %.2f kg/kg\\n' %AF)\n\nelem = ['C', 'H', 'O', 'N']\n\nfor i in range(0,4):\n print(f'A raz\u00e3o {elem[i]}-combust\u00edvel \u00e9 de {round(F_tot[0,i],2)} kg/kg\\n')\n\nif e == 1:\n print('Mistura estequiom\u00e9trica (e = 1)\\n')\nelse:\n print(f'Mistura com {int((e-1)*100)}% de excesso de ar (e = {e})\\n')\n \nprint('O PCI \u00e9 de %.2f kJ/kg' %PCI)\n```\n\n A raz\u00e3o ar-combust\u00edvel \u00e9 de 24.22 kg/kg\n \n A raz\u00e3o C-combust\u00edvel \u00e9 de 0.74 kg/kg\n \n A raz\u00e3o H-combust\u00edvel \u00e9 de 0.23 kg/kg\n \n A raz\u00e3o O-combust\u00edvel \u00e9 de 0.02 kg/kg\n \n A raz\u00e3o N-combust\u00edvel \u00e9 de 0.01 kg/kg\n \n Mistura com 50% de excesso de ar (e = 1.5)\n \n O PCI \u00e9 de 54933.88 kJ/kg\n\n\n## **Equa\u00e7\u00e3o**\n\n

    \n ({{P_GN[0]}})$CH_{4}$ + ({{P_GN[1]}})$C_{2}H_{6}$ + ({{P_GN[2]}})$C_{3}H_{8}$ + ({{P_GN[3]}})$C_{4}H_{10}$ + ({{P_GN[4]}})$CO_{2}$ + ({{P_GN[5]}})$N_{2}$ + ({{P_GN[6]}})$O_{2}$ + ({{round(a,4)}})$\\cdot[$({{round(P_GN[7],4)}})$N_{2(Ar)}$ + ({{P_GN[8]}})$O_{2(Ar)}]$ $\\longrightarrow$ ({{round(b,4)}})$CO_{2}$ + ({{round(c,4)}})$H_{2}O$ + ({{round(d,4)}})$N_{2}$ + ({{round(f,4)}})$O_{2}$ \n

    \n\n## **Estados**\n\n\n```python\n# Estado 1 - Entrada da turbina:\n\nm[0] = 27.9 #kg/s\nP[0] = 6495 #kPa ---> 8000kPa (Teste); 6495kPa (Original)\nT[0] = 485 + 273.15 #K ---> 600\u00b0C (Teste); 485\u00b0C (Original)\nst1 = st(fld, {'P':P[0],'T':T[0]})\nh[0] = st1.h #entalpia\ns[0] = st1.s #entropia\nX[0] = st1.Q\nv[0] = 1/st1.rho\n```\n\n\n```python\n# Estado 2 - Primeira extra\u00e7\u00e3o da turbina:\n\nP[1] = 900 #kPa\nst_isen2 = st(fld,{'P':P[1],'S':s[0]}) #turbina/bomba isentr\u00f3pica ideal\nh_isen2 = st_isen2.h\n\u03b7_turb = 0.85 #efici\u00eancia da turbina\nh[1] = h[0] - (h[0] - h_isen2) * \u03b7_turb\nst2 = st(fld,{'P':P[1],'H':h[1]})\ns[1] = st2.s\nT[1] = st2.T\nX[1] = st2.Q\nv[1] = 1/st2.rho\n```\n\n\n```python\n# Estado 3 - Segunda extra\u00e7\u00e3o da turbina:\n\nP[2] = 250 #kPa\nst_isen3 = st(fld,{'P':P[2],'S':s[1]})\nh_isen3 = st_isen3.h\nh[2] = h[1] - (h[1] - h_isen3) * \u03b7_turb\nst3 = st(fld,{'P':P[2],'H':h[2]})\ns[2] = st3.s\nv[2] = 1/st3.rho\nT[2] = st3.T\nX[2] = st3.Q\n```\n\n\n```python\n# Estado 4 - Sa\u00edda final da turbina / entrada do condensador:\n\nT[3] = 51 + 273.15 #K ---> 30\u00b0C (Teste); 51\u00b0C (Original)\nst4 = st(fld, {'T': T[3], 'Q': 1})\nP[3] = st4.p\nst_isen4 = st(fld,{'P':P[3],'S':s[2]})\nh_isen4 = st_isen4.h\nh[3] = h[2] - (h[2] - h_isen4) * \u03b7_turb\nst4 = st(fld,{'P':P[3],'H':h[3]})\nX[3] = st4.Q #t\u00edtulo\ns[3] = st4.s\nv[3] = 1/st4.rho\n```\n\n\n```python\n# Estado 5 - Sa\u00edda do condensador / entrada da bomba:\n\nX[4] = 0\nP[4] = P[3]\nst5 = st(fld, {'P': P[4], 'Q': 0})\nh[4] = st5.h\ns[4] = st5.s\nT[4] = st5.T\nv[4] = 1/st5.rho\n```\n\n\n```python\n# Estado 6 - Entrada do desaerador / sa\u00edda da bomba:\n\nP[5] = P[2]\nst_isen6 = st(fld, {'P': P[5], 'S': s[4]})\nh_isen6 = st_isen6.h\n\u03b7_pump = 0.85 #efici\u00eancia das bombas\nh[5] = (h_isen6 - h[4]) / \u03b7_pump + h[4]\nst6 = st(fld, {'P':P[5], 'H': h[5]})\ns[5] = st6.s\nT[5] = st6.T\nX[5] = st6.Q\nv[5] = 1/st6.rho\n```\n\n\n```python\n# Estado 7 - Sa\u00edda do desaerador:\n\nP[6] = P[2]\nT[6] = 110 + 273.15\nst7 = st(fld, {'P': P[6], 'T': T[6]})\nh[6] = st7.h\ns[6] = st7.s\nX[6] = st7.Q\nv[6] = 1/st7.rho\n```\n\n\n```python\n# Estado 8 - Entrada da caldeira:\n\nP[7] = P[0]\nst_isen8 = st(fld, {'P': P[7], 'S': s[6]})\nh_isen8 = st_isen8.h\nh[7] = (h_isen8 - h[6]) / \u03b7_pump + h[6]\nst8 = st(fld, {'P': P[7], 'H': h[7]})\nT[7] = st8.T\ns[7] = st8.s\nX[7] = st8.Q\nv[7] = 1/st8.rho\n```\n\n\n```python\n# Estado 9 - Entrada da bomba:\n\nst9 = st7\nh[8] = st9.h\ns[8] = st9.s\nP[8] = st9.p\nT[8] = st9.T\nX[8] = st9.Q\nv[8] = 1/st9.rho\n```\n\n\n```python\n# Estado 10 - Sa\u00edda da bomba / entrada do trocador de calor:\n\nP[9] = P[1]\nst_isen10 = st(fld, {'P': P[9], 'S': s[8]})\nh_isen10 = st_isen10.h\nh[9] = (h_isen10 - h[8]) / \u03b7_pump + h[8]\nst10 = st(fld, {'P': P[9], 'H': h[9]})\ns[9] = st10.s\nT[9] = st10.T\nX[9] = st10.Q\nv[9] = 1/st10.rho\n```\n\n\n```python\n# Estado 11 - Entrada do processo industrial vizinho:\n\nX[10] = 1\nP[10] = P[9]\nst11 = st(fld, {'P': P[10], 'Q': X[10]})\nh[10] = st11.h\ns[10] = st11.s\nT[10] = st11.T\nv[10] = 1/st11.rho\n```\n\n\n```python\n# Estado 12 - Sa\u00edda do processo industrial vizinho:\n\nX[11] = 0\nP[11] = P[9]\nst12 = st(fld, {'P':P[11], 'Q': X[11]})\nh[11] = st12.h\ns[11] = st12.s\nT[11] = st12.T\nv[11] = 1/st12.rho\n```\n\n## **Tabela - Estados**\n\n\n```python\n# Tabela 1:\nheaders=[\"Estado\", Tnn, Pnn, Xnn, hnn, snn, vnn]\n\nfor i in range(0, n):\n if Temperatura == 'k':\n T[i] = round(float(T[i]),2)\n else:\n T[i] = round(float(T[i]-273.15),2)\n P[i] = round(float(P[i]),2)\n X[i] = round(float(X[i]),2)\n h[i] = round(float(h[i]),2)\n s[i] = round(float(s[i]),4)\n v[i] = round(float(v[i]),6)\n \nX_arrumado = []\n\nfor i in range(0, n):\n if X[i] == -1:\n X_arrumado.append('-')\n else:\n X_arrumado.append(X[i])\n\ndf1 = pd.DataFrame()\ntabela_estados = {Tnn: T, Pnn: P, Xnn: X_arrumado, hnn: h, snn: s, vnn: v}\ndf1 = pd.DataFrame(tabela_estados, index=[\"Estado 1\", \"Estado 2\", \"Estado 3\", \"Estado 4\", \"Estado 5\", \"Estado 6\", \"Estado 7\", \"Estado 8\", \"Estado 9\", \"Estado 10\", \"Estado 11\", \"Estado 12\"])\n\n#Voltar temperatura \u00e0 condi\u00e7\u00e3o inicial:\n\nfor i in range(0, n):\n if Temperatura == 'k':\n T[i] = round(float(T[i]),2)\n else:\n T[i] = round(float(T[i]+273.15),2)\n\nst_antes = [st1, st2, st3, st4, st5, st6, st7, st8, st9, st10, st11, st12]\n```\n\n{{df1}}\n\n## **C\u00e1lculos**\n\n\n```python\n# C\u00e1lculos:\nm1 = m[0]\nh1 = h[0]\nh2 = h[1]\nh3 = h[2]\nh4 = h[3]\nh5 = h[4]\nh6 = h[5]\nh7 = h[6]\nh8 = h[7]\nh9 = h[8]\nh10 = h[9]\nh11 = h[10]\nh12 = h[11]\n \n# Quantidade de itera\u00e7\u00f5es:\nquant = 10\n \nvariacao = np.zeros(quant + 1, dtype = float)\nresultados = np.zeros((n+1)*(quant + 1), dtype = float).reshape((n+1, quant + 1))\n \nT_hout_cond = []\n\u03b7_sys = []\nW_liq =[]\n\nSTT_T = []\nSTT_P = []\nSTT_X = []\nSTT_h = []\nSTT_s = []\nSTT_v = []\nTi = np.linspace(0,0,n)\nXi = list(range(0,n))\n\n# Vaz\u00f5es m\u00e1ssicas:\nfor i in range(quant + 1):\n processo = i/10\n \n m2, m3, m4, m5, m6, m7, m8, m9, m10, m11, m12 = var('m2, m3, m4, m5, m6, m7, m8, m9, m10, m11, m12')\n EqMassa = [m8-m1, m4-m5, m5-m6, m9-m10, m11-m12, m2+m10-m11, m2+m3+m4-m1, m12+m3+m6-m7, m2*h2+m10*h10-m11*h11, m12*h12+m3*h3+m6*h6-m7*h7, m12-6.94*processo]\n massa = linsolve(EqMassa, m2, m3, m4, m5, m6, m7, m8, m9, m10, m11, m12)\n (m2, m3, m4, m5, m6, m7, m8, m9, m10, m11, m12) = next(iter(massa))\n\n massa = np.array(list(massa))\n \n for j in range(quant + 1):\n massa[0,j] = round(float(massa[0,j]),2)\n \n variacao[i] = i*10\n resultados[0, i] = variacao[i]\n resultados[1, i] = m[0]\n resultados[2:(n+1), i] = massa\n\n # Sa\u00eddas de energia:\n W_p1 = -m5*(h5 - h6) #(Perde energia \"-\")\n W_p2 = -m8*(h7 - h8) #(Perde energia \"-\")\n W_p3 = -m9*(h9 - h10) #(Perde energia \"-\")\n W_pump = W_p1 + W_p2 + W_p3 #(Somat\u00f3rio dos trabalhos da bombas)\n\n # Efici\u00eancia da caldeira:\n \u03b7_cald = 0.85\n #PCI = 10000 # kJ/kg\n\n # Encontra o trabalho gerado pela turbina e a vaz\u00e3o de combust\u00edvel para a demanda:\n W_t = m1*h1 - m2*h2 - m3*h3 - m4*h4\n m_fuel = m1*(h1-h8)/(PCI*\u03b7_cald) #\u03b7_cald = m1*(h1-h8)/(m_fuel*PCI)\n\n # Entrada de energia:\n Q_in = m_fuel * PCI #(Ganha energia \"+\")\n\n \u03b7_sys.append(round((W_t - W_pump)/Q_in, 4))\n\n W_liq.append(round(\u03b7_sys[i]*Q_in,2))\n\n # Condensador:\n T_cout = 39.5 \n T_cin = 29\n T_hin = T[3] - 273.15\n\n \u03f5 = (T_cout - T_cin)/(T_hin - T_cin)\n NUT = -ln(1 - \u03f5)\n\n CT_in = st(fld, {'P': 101.325, 'T': T_cin + 273.15})\n CT_out = st(fld, {'P': 101.325, 'T': T_cout + 273.15})\n\n C_min = 4500/3600*(CT_in.rho + CT_out.rho)/2*(CT_in.cp + CT_out.cp)/2\n\n UA = NUT * C_min # Coeficiente global de tranfer\u00eancia de calor * \u00c1rea\n\n # Sistema n\u00e3o-linear:\n \u0394T_0, \u0394T_L, T_hout, \u0394T_ml, Q_out = var('\u0394T_0, \u0394T_L, T_hout, \u0394T_ml, Q_out')\n EqCond = [\u0394T_0-(T_hout-T_cin), \u0394T_L-(T_hout-T_cout), \u0394T_ml-(\u0394T_0-\u0394T_L)/ln(\u0394T_0/\u0394T_L), Q_out-UA*\u0394T_ml, Q_out-m4*(h4-h5)]\n EqCond_solve = nonlinsolve(EqCond, \u0394T_0, \u0394T_L, T_hout, \u0394T_ml, Q_out) \n (\u0394T_0, \u0394T_L, T_hout, \u0394T_ml, Q_out) = next(iter(EqCond_solve))\n\n # Redefinindo estados:\n T[3] = T_hout + 273.15 #K\n #P[3] = psi('P','T',T[3],'Q', 1, fld)/1000 #divide por 1000 pra sair em kPa\n st4 = st(fld, {'T': T[3], 'Q': 1})\n P[3] = st4.p\n st_isen4 = st(fld,{'P':P[3],'S':s[2]})\n h_isen4 = st_isen4.h\n h[3] = h[2] - (h[2] - h_isen4) * \u03b7_turb\n st4 = st(fld,{'P':P[3],'H':h[3]})\n X[3] = st4.Q #t\u00edtulo\n s[3]=st4.s\n v[3] = 1/st4.rho\n\n X[4] = 0\n P[4] = P[3]\n st5 = st(fld, {'P': P[4], 'Q': X[4]})\n h[4] = st5.h\n s[4] = st5.s\n T[4] = st5.T\n v[4] = 1/st5.rho\n\n P[5] = P[2]\n st_isen6 = st(fld, {'P': P[5], 'S': s[4]})\n h_isen6 = st_isen6.h\n \u03b7_pump = 0.85 #efici\u00eancia das bombas\n h[5] = (h_isen6 - h[4]) / \u03b7_pump + h[4]\n st6 = st(fld, {'P':P[5], 'H': h[5]})\n s[5] = st6.s\n T[5] = st6.T\n X[5] = st6.Q\n v[5] = 1/st6.rho\n\n P[6] = P[2]\n T[6] = 110 + 273.15\n st7 = st(fld, {'P': P[6], 'T': T[6]})\n h[6] = st7.h\n s[6] = st7.s\n X[6] = st7.Q\n v[6] = 1/st7.rho\n\n P[7] = P[0]\n st_isen8 = st(fld, {'P': P[7], 'S': s[6]})\n h_isen8 = st_isen8.h\n h[7] = (h_isen8 - h[6]) / \u03b7_pump + h[6]\n st8 = st(fld, {'P': P[7], 'H': h[7]})\n T[7] = st8.T\n s[7] = st8.s\n X[7] = st8.Q\n v[7] = 1/st8.rho\n\n st9 = st7\n h[8] = st9.h\n s[8] = st9.s\n P[8] = st9.p\n T[8] = st9.T\n X[8] = st9.Q\n v[8] = 1/st9.rho\n\n P[9] = P[1]\n st_isen10 = st(fld, {'P': P[9], 'S': s[8]})\n h_isen10 = st_isen10.h\n h[9] = (h_isen10 - h[8]) / \u03b7_pump + h[8]\n st10 = st(fld, {'P': P[9], 'H': h[9]})\n s[9] = st10.s\n T[9] = st10.T\n X[9] = st10.Q\n v[9] = 1/st10.rho\n\n X[10] = 1\n P[10] = P[9]\n st11 = st(fld, {'P': P[10], 'Q': X[10]})\n h[10] = st11.h\n s[10] = st11.s\n T[10] = st11.T\n v[10] = 1/st11.rho\n\n X[11] = 0\n P[11] = P[9]\n st12 = st(fld, {'P':P[11], 'Q': X[11]})\n h[11] = st12.h\n s[11] = st12.s\n T[11] = st12.T\n v[11] = 1/st12.rho\n \n for i in range(0,n):\n if Temperatura == 'k':\n Ti[i] = round(T[i],2)\n else:\n Ti[i] = round(T[i]-273.15,2)\n P[i] = round(P[i],2)\n X[i] = round(X[i],2)\n h[i] = round(h[i],2)\n s[i] = round(s[i],4)\n v[i] = round(v[i],6)\n if X[i] == -1:\n Xi[i] = '-'\n else:\n Xi[i] = X[i]\n\n STT_T.append([Tnn,processo*100,Ti[0],Ti[1],Ti[2],Ti[3],Ti[4],Ti[5],Ti[6],Ti[7],Ti[8],Ti[9],Ti[10],Ti[11]])\n STT_P.append([Pnn,processo*100,P[0],P[1],P[2],P[3],P[4],P[5],P[6],P[7],P[8],P[9],P[10],P[11]])\n STT_X.append([Xnn,processo*100,Xi[0],Xi[1],Xi[2],Xi[3],Xi[4],Xi[5],Xi[6],Xi[7],Xi[8],Xi[9],Xi[10],Xi[11]])\n STT_h.append([hnn,processo*100,h[0],h[1],h[2],h[3],h[4],h[5],h[6],h[7],h[8],h[9],h[10],h[11]])\n STT_s.append([snn,processo*100,s[0],s[1],s[2],s[3],s[4],s[5],s[6],s[7],s[8],s[9],s[10],s[11]])\n STT_v.append([vnn,processo*100,v[0],v[1],v[2],v[3],v[4],v[5],v[6],v[7],v[8],v[9],v[10],v[11]])\n\n T_hout = round(T_hout, 2) \n T_hout_cond.append(T_hout)\n \n'''# Adiciona linhas vazias entre as propriedades na tabela:\nSTT_T.append([\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\"])\nSTT_P.append([\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\"])\nSTT_X.append([\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\"])\nSTT_h.append([\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\"])\nSTT_s.append([\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\"])\nSTT_v.append([\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\",\"\"])\n'''\n# Incluir na tabela e gr\u00e1fico: \nnp.reshape(variacao, (1,n-1))\nnp.reshape(\u03b7_sys, (1,n-1))\nnp.reshape(T_hout_cond, (1,n-1))\nnp.reshape(W_liq, (1,n-1))\n\nfor i in range(0, n):\n T[i] = round(T[i],2)\n P[i] = round(P[i],2)\n X[i] = round(X[i],2)\n h[i] = round(h[i],2)\n s[i] = round(s[i],2)\n v[i] = round(v[i],2)\n\nx = []\ny = []\nz = []\nw = []\n\nfor i in range(0, quant+1):\n x.append(variacao[i])\n y.append(T_hout_cond[i])\n z.append(\u03b7_sys[i])\n w.append(W_liq[i])\n\ntable2 = resultados.T\n \nresultados2 = np.zeros((n+4)*(quant + 1), dtype = float).reshape((n+4, quant + 1))\n \nfor i in range(0, quant+1):\n variacao[i] = i*10\n resultados2[0, i] = resultados[0, i]\n resultados2[1, i] = resultados[1, i]\n resultados2[2:(n+1), i] = resultados[2:(n+1), i]\n resultados2[n+2,i] = round(float(w[i]),2)\n resultados2[n+3,i] = round(float(z[i]),4)\ntable2 = resultados2.T\n\nst_depois = [st1, st2, st3, st4, st5, st6, st7, st8, st9, st10, st11, st12]\n```\n\n## **Temperaturas da fornalha**\n\n\n\n\n```python\n# Balan\u00e7o de energia da caldeira:\n\nm_air = AF*m_fuel\nm_gas = m_air + m_fuel\nm13 = m8\nm14 = m_fuel\nm15 = 0.1*m_air\nm16 = 0.9*m_air\nm17 = m16\nm18 = m_gas\nm19 = m18\nm20 = m19\n\nfld2 = 'Air'\n\nT_amb = 20 + 273.15 #K\nP_amb = 101.325 #kPa\n\nT15 = T_amb \nT16 = T_amb\nT8 = T[7]\nT20 = 200 + 273.15 #K\nP20 = P_amb\nP8 = P[7]\n\ndef myfunction(\u0394T13, \u0394T17):\n\n from CoolProp.CoolProp import State as st\n\n T20 = 200 + 273.15 #K\n\n #Economizador:\n st13 = st(fld, {'P': P8, 'Q': 0})\n\n T13 = st13.T - \u0394T13 #chute\n\n st13_2 = st(fld, {'P': P8, 'T': T13})\n h13 = st13_2.h\n\n q13 = m13*(h13-h8)\n\n st20_1 = st('Water', {'P':P_amb, 'T':T20})\n st20_2 = st('CO2', {'P':P_amb, 'T':T20})\n st20_3 = st('Nitrogen', {'P':P_amb, 'T':T20})\n\n cp_20 = 0.0997*st20_1.cp+0.1771*st20_2.cp+0.7232*st20_3.cp\n\n #cachorrada\n cp_19 = cp_20\n cp_18 = cp_20\n\n T17 = T16 + 200 + \u0394T17 #chute\n\n st16 = st(fld2, {'P':P_amb, 'T':T_amb})\n st17 = st(fld2, {'P':P_amb, 'T':T17})\n h16 = st16.h\n h17 = st17.h\n\n st17_1 = st('Water', {'P':P_amb, 'T':T17})\n st17_2 = st('CO2', {'P':P_amb, 'T':T17})\n st17_3 = st('Nitrogen', {'P':P_amb, 'T':T17})\n\n cp_17 = 0.0997*st20_1.cp+0.1771*st20_2.cp+0.7232*st20_3.cp\n\n T18, T19, T20 = var('T18, T19, T20')\n EqTemp = [m19*cp_19*(T19-T_amb) + m8*h8 - h13*m13 - m20*cp_20*(T20-T_amb), m18*cp_18*(T18-T_amb) - m17*cp_17*(T17-T_amb) - m19*cp_19*(T19-T_amb), m_fuel*PCI + m17*cp_17*(T17-T_amb) + h13*m13 - m1*h1 - m18*cp_18*(T18-T_amb)]\n temp = linsolve(EqTemp, T18, T19, T20)\n (T18, T19, T20) = next(iter(temp))\n\n Q_disp = m_fuel*PCI + m17*cp_17*(T17-T_amb) \n Q_util = Q_disp - (m_gas*cp_18*(T18-T_amb))\n \u03b7_forn = round(Q_util/Q_disp,4)\n\n df = pd.DataFrame()\n Valores = [T13,T17,T18,T19,T20,\u03b7_forn*100]\n for i in range(0,5):\n Valores[i] = round((Valores[i]-273.15),2)\n\n data = {'Valores':Valores}\n df = pd.DataFrame(data, index=['T13 [\u00b0C]','T17 [\u00b0C]','T18 [\u00b0C]','T19 [\u00b0C]','T20 [\u00b0C]','\u03b7 fornalha [%]'])\n df.style.hide_index()\n return df\n\nslider_\u0394T13 = wd.IntSlider(min=30, max=100, step=10)\nslider_\u0394T17 = wd.IntSlider(min=-100, max=100, step=10)\nwd.interact(myfunction, \u0394T13=slider_\u0394T13, \u0394T17=slider_\u0394T17)\n```\n\n\n interactive(children=(IntSlider(value=30, description='\u0394T13', min=30, step=10), IntSlider(value=0, description\u2026\n\n\n\n\n\n \n\n\n\n## **Tabela - Vaz\u00f5es m\u00e1ssicas**\n\n\n```python\n# Tabela 2:\n\npercentagem = []\n\nfor i in range(0, quant+1):\n percentagem.append(str(int(resultados2[0,i])) + '%')\n if Temperatura == 'k':\n resultados2[n+1,i] = round(float(y[i]+273.15),2) #converter para K\n y[i] = resultados2[n+1,i]\n else:\n resultados2[n+1,i] = round(float(y[i]),2)\n\ndf2 = pd.DataFrame()\ntabela_m\u0307 = {\"m\u03071 [kg/s]\": resultados2[1,:], \"m\u03072 [kg/s]\": resultados2[2,:], \"m\u03073 [kg/s]\": resultados2[3,:], \"m\u03074 [kg/s]\": resultados2[4,:], \"m\u03075 [kg/s]\": resultados2[5,:], \"m\u03076 [kg/s]\": resultados2[6,:], \"m\u03077 [kg/s]\": resultados2[7,:], \"m\u03078 [kg/s]\": resultados2[8,:], \"m\u03079 [kg/s]\": resultados2[9,:], \"m\u030710 [kg/s]\": resultados2[10,:], \"m\u030711 [kg/s]\": resultados2[11,:], \"m\u030712 [kg/s]\": resultados2[12,:], \"T_hout [\u00b0C]\": resultados2[13,:], \"W_liq [kJ/s]\": resultados2[14,:], \"\u03b7_sistema\": resultados2[15,:]}\ndf2 = pd.DataFrame(tabela_m\u0307, index=percentagem)\n\nprint('A vaz\u00e3o m\u00e1ssica de combust\u00edvel \u00e9 de %.2f kg/s' %m_fuel)\nprint('A vaz\u00e3o m\u00e1ssica de ar \u00e9 de %.2f kg/s' %m_air)\n```\n\n A vaz\u00e3o m\u00e1ssica de combust\u00edvel \u00e9 de 1.74 kg/s\n A vaz\u00e3o m\u00e1ssica de ar \u00e9 de 42.15 kg/s\n\n\n{{df2}}\n\n\n```python\n# Gr\u00e1ficos:\n\nescala1 = 0.975\nplt.figure(dpi=100, figsize=[escala1*2*6.4, escala1*4.8])\nfig1 = plt.figure(dpi=100, figsize=[escala1*2*6.4, escala1*4.8])\nplt.subplot(1,2,1)\nplt.xlabel('Processo [%]', fontsize=12)\n\nif Temperatura == 'k':\n plt.ylabel('Temperatura de sa\u00edda do condensador [K]', fontsize=12)\n plt.plot(x, y, color='r', marker = 'o', linestyle = 'solid')\nelse:\n plt.ylabel('Temperatura de sa\u00edda do condensador [\u00b0C]', fontsize=12)\n plt.plot(x, y, color='r', marker = 'o', linestyle = 'solid')\n \nplt.grid()\nplt.subplot(1,2,2)\nplt.xlabel('Processo [%]', fontsize=12)\nplt.ylabel('$\u03b7_{sistema}$', fontsize=12)\nplt.plot(x, z, color='b', marker = 'o', linestyle = 'solid')\nplt.grid()\nplt.show()\n```\n\n## **Gr\u00e1fico completo Ciclo Rankine**\n\n\n```python\n# Gr\u00e1fico do Ciclo Rankine:\nescala2 = 1\nplt.figure(dpi=150, figsize=[escala2*6.4, escala2*4.8])\nfig2 = plt.figure(dpi=150, figsize=[escala2*6.4, escala2*4.8])\ny1 = 273.15\ny2 = 850\nx1 = -0.5\nx2 = 10\nplt.ylim(y1,y2)\nplt.xlim(x1,x2)\nplt.title('Ciclo Rankine',fontsize=20)\nplt.xlabel('Entropia, s[kJ/kg$\\cdot$K]')\nplt.ylabel('Temperatura, T[K]')\n\nTmin = ps('Tmin',fld )\nTcrit = ps('Tcrit',fld)\nPcrit = ps('Pcrit', fld)\n\nLarg_isolines = .25\nLarg_cycle = 1.5\nLarg_scatters = 15\n\n#Linhas de entalpia:\nT = np.linspace(Tmin,Tcrit,1000)\nQ = np.arange(0.1,1,0.1)\nfor i in Q:\n s = ps('S','T',T,'Q',i,'Water')/1000\n plt.plot(s,T,'#6ca8fa',alpha=0.7,lw=Larg_isolines)\n\n#Domo:\nT = np.linspace(Tmin,Tcrit,1000)\ns = ps('S','T',T,'Q',0,fld)/1000\nplt.plot(s,T,'black',lw=2*Larg_isolines)\n\nT = np.linspace(Tcrit,Tmin,1000)\ns = ps('S','T',T,'Q',1,fld)/1000\nplt.plot(s,T,'black',lw=2*Larg_isolines)\n\n#Linhas de press\u00e3o constante:\nT = np.linspace(Tmin,1000,1000)\nP = [st_antes[0].p,st_antes[1].p,st_antes[2].p,st_antes[3].p,st_depois[3].p] #P1, P2, P3, P4 e P4*\nfor i in P:\n s = ps('S','T',T,'P',i*1000,'Water')/1000\n plt.plot(s,T,'#cc00ff',alpha=.7,lw=Larg_isolines)\n\n#Ciclo:\n#(antes)\nT = np.linspace(st_antes[7].T,st_antes[0].T,1000)\ns = ps('S','P',st_antes[0].p * 1000,'T',T,'Water')/1000\nplt.plot(s,T,'r',lw=Larg_cycle)\n\nT = np.linspace(st_antes[9].T,st_antes[1].T,1000)\ns = ps('S','T',T,'P',st_antes[9].p*1000,'Water')/1000\nplt.plot(s,T,'r',lw=Larg_cycle)\nplt.plot([st_antes[3].s,st_antes[4].s],[st_antes[3].T,st_antes[4].T],'r',lw=Larg_cycle)\n\nplt.plot([st_antes[0].s,st_antes[1].s],[st_antes[0].T,st_antes[1].T],color='r',linestyle='dashed',lw=Larg_cycle)\nplt.plot([st_antes[1].s,st_antes[2].s],[st_antes[1].T,st_antes[2].T],color='r',linestyle='dashed',lw=Larg_cycle)\n\ns = np.linspace(st_antes[2].s,st_antes[6].s,1000)\nT = ps('T','S',s*1000,'P',st_antes[2].p*1000,'Water')\nplt.plot(s,T,'r',lw=Larg_cycle)\n\nT = np.linspace(st_antes[5].T,st_antes[6].T,1000)\ns = ps('S','P',st_antes[5].p*1000,'T',T,'Water')/1000\nplt.plot(s,T,'r',lw=Larg_cycle)\n\ns = np.linspace(st_antes[2].s,st_antes[3].s,1000)\nplt.plot([st_antes[3].s,st_antes[2].s],[st_antes[3].T,st_antes[2].T],color='r',linestyle='dashed',lw=Larg_cycle)\n\n#(depois)\nT = np.linspace(st_depois[7].T,st_depois[0].T,1000)\ns = ps('S','P',st_depois[0].p * 1000,'T',T,'Water')/1000\nplt.plot(s,T,'lime',lw=Larg_cycle)\n\nT = np.linspace(st_depois[9].T,st_depois[1].T,1000)\ns = ps('S','T',T,'P',st_depois[9].p*1000,'Water')/1000\nplt.plot(s,T,'lime',lw=Larg_cycle)\nplt.plot([st_depois[3].s,st_depois[4].s],[st_depois[3].T,st_depois[4].T],'lime',lw=Larg_cycle)\n\nplt.plot([st_depois[0].s,st_depois[1].s],[st_depois[0].T,st_depois[1].T],color='lime',linestyle='dashed',lw=Larg_cycle)\nplt.plot([st_depois[1].s,st_depois[2].s],[st_depois[1].T,st_depois[2].T],color='lime',linestyle='dashed',lw=Larg_cycle)\n\ns = np.linspace(st_depois[2].s,st_depois[6].s,1000)\nT = ps('T','S',s*1000,'P',st_depois[2].p*1000,'Water')\nplt.plot(s,T,'lime',lw=Larg_cycle)\n\nT = np.linspace(st_depois[5].T,st_depois[6].T,1000)\ns = ps('S','P',st_depois[5].p*1000,'T',T,'Water')/1000\nplt.plot(s,T,'lime',lw=Larg_cycle)\n\ns = np.linspace(st_depois[2].s,st_depois[3].s,1000)\nplt.plot([st_depois[3].s,st_depois[2].s],[st_depois[3].T,st_depois[2].T],color='lime',linestyle='dashed',lw=Larg_cycle)\n\n#Pontuar e nomear:\nfor i in range(0, n):\n plt.scatter(st_antes[i].s,st_antes[i].T,zorder=5,color='k',s=Larg_scatters)\n plt.scatter(st_depois[i].s,st_depois[i].T,zorder=5,color='k',s=Larg_scatters)\n\nplt.text(st_antes[0].s+.1,st_antes[0].T,'1',ha='left')\nplt.text(st_antes[1].s+.1,st_antes[1].T,'2',ha='left')\nplt.text(st_antes[2].s+.1,st_antes[2].T,'3',ha='left')\nplt.text(st_antes[3].s+.1,st_antes[3].T,'4',ha='left')\nplt.text(st_antes[4].s-.1,st_antes[4].T,'5, 6',ha='right')\nplt.text(st_antes[6].s-.1,st_antes[6].T,'7, 8, 9, 10',ha='right')\nplt.text(st_antes[10].s-.1,st_antes[10].T,'11',va='bottom',ha='right')\nplt.text(st_antes[11].s-.1,st_antes[11].T+.1,'12',ha='right')\nplt.text(st_depois[3].s+.1,st_depois[3].T,'4*',va='top',ha='left')\nplt.text(st_depois[4].s-.1,st_depois[4].T,'5*, 6*',va='top',ha='right')\n\n#plt.grid(lw=Larg_isolines)\nplt.twinx()\nplt.ylim(y1-273.15,y2-273.15)\nplt.ylabel('Temperatura, T[\u00b0C]')\n#plt.grid(lw=Larg_isolines)\nplt.show()\n\n# Recobrar os estados anteriores ao gr\u00e1fico:\n\ndel T\ndel P\ndel s\n\nT = [] \nP = []\ns = []\n\nfor i in range(0, n):\n T.append(round(st_depois[i].T,2))\n P.append(round(st_depois[i].p,2))\n s.append(round(st_depois[i].s,2))\n```\n\n## **Tabela relat\u00f3rio completo Ciclo Rankine**\n\n\n```python\n# Tabela 3:\n\ntable3 = np.array(STT_T + STT_P + STT_X + STT_h + STT_s + STT_v)\n\ndf3 = pd.DataFrame()\npd.set_option('display.max_rows', None)\ntabela_geral = {\"Propriedade\": table3[:,0], \"%\": table3[:,1], \"Estado 1\": table3[:,2], \"Estado 2\": table3[:,3], \"Estado 3\": table3[:,4], \"Estado 4\": table3[:,5], \"Estado 5\": table3[:,6], \"Estado 6\": table3[:,7], \"Estado 7\": table3[:,8], \"Estado 8\": table3[:,9], \"Estado 9\": table3[:,10], \"Estado 10\": table3[:,11], \"Estado 11\": table3[:,12], \"Estado 12\": table3[:,13]}\ndf3 = pd.DataFrame(tabela_geral)\n```\n\n{{df3}}\n\n## **Salvar dados e m\u00eddias**\n\n\n```python\n# Salvar tabelas em excel:\n\nresp = 0\nwhile resp != 's' or resp != 'S' or resp != 'n' or resp != 'N':\n resp = input('Salvar imagens dos gr\u00e1ficos? (S ou N) ')\n if resp == 's' or resp == 'S' or resp == 'n' or resp == 'N':\n break\n \nif resp == 's' or resp == 'S':\n print('Aguarde...\\n')\n fig1.savefig('Gr\u00e1ficos de temperatura e efic\u00eancia do conddensador.png', dpi = 500)\n fig2.savefig('Diagrama T-s.png', dpi = 500)\ndel resp\n\nresp = 0\nwhile resp != 's' or resp != 'S' or resp != 'n' or resp != 'N':\n resp = input('Salvar tabelas em Excel? (S ou N) ')\n if resp == 's' or resp == 'S' or resp == 'n' or resp == 'N':\n break\n\nif resp == 's' or resp == 'S':\n print('Aguarde...\\n')\n with pd.ExcelWriter('Tabelas.xlsx') as writer: \n df1.to_excel(writer, sheet_name='Tabela 1')\n df2.to_excel(writer, sheet_name='Tabela 2')\n df3.to_excel(writer, sheet_name='Tabela 3', index=None)\ndel resp\n\nclass color:\n PURPLE = '\\033[95m'\n CYAN = '\\033[96m'\n DARKCYAN = '\\033[36m'\n BLUE = '\\033[94m'\n GREEN = '\\033[92m'\n YELLOW = '\\033[93m'\n RED = '\\033[91m'\n BOLD = '\\033[1m'\n UNDERLINE = '\\033[4m'\n END = '\\033[0m'\n\nprint(color.GREEN + color.BOLD + '\\nPrograma executado com sucesso!')\n```\n\n Salvar imagens dos gr\u00e1ficos? (S ou N) N\n Salvar tabelas em Excel? (S ou N) N\n \u001b[92m\u001b[1m\n Programa executado com sucesso!\n\n", "meta": {"hexsha": "96a4849eeb9dcba79e778d01ed57f8e14d82db21", "size": 262792, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": ".ipynb_checkpoints/Modelagem_GDV_(final)-checkpoint.ipynb", "max_stars_repo_name": "AlePort/Projeto_GDV", "max_stars_repo_head_hexsha": "c4b8ae9d02e3fd981b4f6d942501df7fd7a20bb5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": ".ipynb_checkpoints/Modelagem_GDV_(final)-checkpoint.ipynb", "max_issues_repo_name": "AlePort/Projeto_GDV", "max_issues_repo_head_hexsha": "c4b8ae9d02e3fd981b4f6d942501df7fd7a20bb5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": ".ipynb_checkpoints/Modelagem_GDV_(final)-checkpoint.ipynb", "max_forks_repo_name": "AlePort/Projeto_GDV", "max_forks_repo_head_hexsha": "c4b8ae9d02e3fd981b4f6d942501df7fd7a20bb5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 129.3267716535, "max_line_length": 122944, "alphanum_fraction": 0.789403787, "converted": true, "num_tokens": 11976, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593171945416, "lm_q2_score": 0.6959583313396339, "lm_q1q2_score": 0.4332057477215211}} {"text": "# Aim of this notebook\n\n* To construct the singular curve of universal type to finalize the solution of the optimal control problem\n\n# Preamble\n\n\n```python\nfrom sympy import *\ninit_printing(use_latex='mathjax')\n\n# Plotting\n%matplotlib inline\n## Make inline plots raster graphics\nfrom IPython.display import set_matplotlib_formats\n## Import modules for plotting and data analysis\nimport matplotlib.pyplot as plt\nfrom matplotlib import gridspec,rc,colors\nimport matplotlib.ticker as plticker\n# Parameters for seaborn plots\nimport seaborn as sns\nclrs = sns.color_palette(\"Spectral\", 6)\ndef set_plot_style(usetex=False):\n sns.set_style('white', {'axes.linewidth': 0.5})\n sns.set(style='white', font_scale=1.1,#context='paper',\n rc={'xtick.major.size': 6, 'ytick.major.size': 6, 'legend.fontsize': 14,\n 'text.usetex': usetex, 'font.family': 'serif', 'font.serif': ['Verdana'],\n 'text.latex.preamble': r\"\\usepackage{type1cm}\"}) \n plt.rcParams['xtick.major.size'] = 6\n plt.rcParams['xtick.major.width'] = 1\n plt.rcParams['ytick.major.size'] = 6\n plt.rcParams['ytick.major.width'] = 1\n plt.rcParams['xtick.bottom'] = True\n plt.rcParams['ytick.left'] = True\n \nset_plot_style(True)\n\nimport pandas as pd\npd.set_option('mode.chained_assignment',None)\n\nimport numpy as np\nfrom scipy.optimize import fsolve, root\nfrom scipy.integrate import ode\nbackend = 'dopri5'\nimport warnings\n\n# Timer\nimport time\n\nfrom copy import deepcopy\n\nfrom itertools import cycle\npalette_size = 10;\nclrs = sns.color_palette(\"Reds\",palette_size)\niclrs = cycle(clrs) # iterated colors\n\nclrs0 = sns.color_palette(\"Set1\",palette_size)\n\n# Suppress warnings\nimport warnings\nwarnings.filterwarnings(\"ignore\")\n```\n\n# Parameter values\n\n* Birth rate and const of downregulation are defined below in order to fit some experim. data\n\n\n```python\nd = .13 # death rate\n\u03b1 = .3 # low equilibrium point at expression of the main pathway (high equilibrium is at one)\n\u03b8 = .45 # threshold value for the expression of the main pathway\n\u03ba = 40 # robustness parameter\n```\n\n* Symbolic variables - the list insludes \u03bc & \u03bcbar, because they will be varied later\n\n\n```python\n\u03c3, \u03c60, \u03c6, x, \u03bc, \u03bcbar = symbols('sigma, phi0, phi, x, mu, mubar')\n```\n\n* Main functions\n\n\n```python\nA = 1-\u03c3*(1-\u03b8)\nEminus = (\u03b1*A-\u03b8)**2/2\n\u0394E = A*(1-\u03b1)*((1+\u03b1)*A/2-\u03b8)\n\u0394Ef = lambdify(\u03c3,\u0394E)\n```\n\n* Birth rate and cost of downregulation\n\n\n```python\nb = (0.1*(exp(\u03ba*(\u0394Ef(1)))+1)-0.14*(exp(\u03ba*\u0394Ef(0))+1))/(exp(\u03ba*\u0394Ef(1))-exp(\u03ba*\u0394Ef(0))) # birth rate\n\u03c7 = 1-(0.14*(exp(\u03ba*\u0394Ef(0))+1)-b*exp(\u03ba*\u0394Ef(0)))/b\nb, \u03c7\n```\n\n\n\n\n$$\\left ( 0.140168330860362, \\quad 0.325961223954473\\right )$$\n\n\n\n\n```python\nc_relative = 0.1\nc = c_relative*(b-d)/b+(1-c_relative)*\u03c7/(exp(\u03ba*\u0394Ef(0))+1) # cost of resistance\nc\n```\n\n\n\n\n$$0.00833519849448376$$\n\n\n\n* Hamiltonian *H* and a part of it \u03c1 that includes the control variable \u03c3\n\n\n```python\nh = b*(\u03c7/(exp(\u03ba*\u0394E)+1)*(1-x)+c*x)\nH = -\u03c60 + \u03c6*(b*(\u03c7/(exp(\u03ba*\u0394E)+1)-c)*x*(1-x)+\u03bc*(1-x)/(exp(\u03ba*\u0394E)+1)-\u03bcbar*exp(-\u03ba*Eminus)*x) + h\n\u03c1 = (\u03c6*(b*\u03c7*x+\u03bc)+b*\u03c7)/(exp(\u03ba*\u0394E)+1)*(1-x)-\u03c6*\u03bcbar*exp(-\u03ba*Eminus)*x\n\u03c11 = (\u03c6*(b*\u03c7*x+\u03bc)+b*\u03c7)/(exp(\u03ba*\u0394E)+1)*(1-x)\n\u03c12 = \u03c6*\u03bcbar*exp(-\u03ba*Eminus)*x\nn = b*(1-\u03c7*(1-x)/(exp(\u03ba*\u0394E)+1)-c*x)-d\nH, \u03c1, n\n```\n\n\n\n\n$$\\left ( \\phi \\left(\\frac{\\mu \\left(- x + 1\\right)}{e^{40 \\left(- 0.385 \\sigma + 0.7\\right) \\left(- 0.3575 \\sigma + 0.2\\right)} + 1} - \\bar{\\mu} x e^{- 20 \\left(- 0.165 \\sigma - 0.15\\right)^{2}} + x \\left(-0.00116833086036159 + \\frac{0.045689440686899}{e^{40 \\left(- 0.385 \\sigma + 0.7\\right) \\left(- 0.3575 \\sigma + 0.2\\right)} + 1}\\right) \\left(- x + 1\\right)\\right) - \\phi_{0} + 0.00116833086036159 x + \\frac{0.045689440686899 \\left(- x + 1\\right)}{e^{40 \\left(- 0.385 \\sigma + 0.7\\right) \\left(- 0.3575 \\sigma + 0.2\\right)} + 1}, \\quad - \\bar{\\mu} \\phi x e^{- 20 \\left(- 0.165 \\sigma - 0.15\\right)^{2}} + \\frac{\\left(- x + 1\\right) \\left(\\phi \\left(\\mu + 0.045689440686899 x\\right) + 0.045689440686899\\right)}{e^{40 \\left(- 0.385 \\sigma + 0.7\\right) \\left(- 0.3575 \\sigma + 0.2\\right)} + 1}, \\quad - 0.00116833086036159 x - \\frac{0.140168330860362 \\left(- 0.325961223954473 x + 0.325961223954473\\right)}{e^{40 \\left(- 0.385 \\sigma + 0.7\\right) \\left(- 0.3575 \\sigma + 0.2\\right)} + 1} + 0.0101683308603616\\right )$$\n\n\n\n* Same but for no treatment (\u03c3 = 0)\n\n\n```python\nh0 = h.subs(\u03c3,0)\nH0 = H.subs(\u03c3,0)\n\u03c10 = \u03c1.subs(\u03c3,0)\nH0, \u03c10\n```\n\n\n\n\n$$\\left ( \\phi \\left(0.00368423989943599 \\mu \\left(- x + 1\\right) - 0.637628151621773 \\bar{\\mu} x - 0.001 x \\left(- x + 1\\right)\\right) - \\phi_{0} + 0.001 x + 0.000168330860361587, \\quad - 0.637628151621773 \\bar{\\mu} \\phi x + 0.00368423989943599 \\left(- x + 1\\right) \\left(\\phi \\left(\\mu + 0.045689440686899 x\\right) + 0.045689440686899\\right)\\right )$$\n\n\n\n* Machinery: definition of the Poisson brackets\n\n\n```python\nPoissonBrackets = lambda H1, H2: diff(H1,x)*diff(H2,\u03c6)-diff(H1,\u03c6)*diff(H2,x)\n```\n\n* Necessary functions and defining the right hand side of dynamical equations\n\n\n```python\n\u03c1f = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),\u03c1)\n\u03c11f = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),\u03c11)\n\u03c12f = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),\u03c12)\n\u03c10f = lambdify((x,\u03c6,\u03bc,\u03bcbar),\u03c10)\ndxd\u03c4 = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),-diff(H,\u03c6))\nd\u03c6d\u03c4 = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),diff(H,x))\n# dnd\u03c4 = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),-n)\ndVd\u03c4 = lambdify((x,\u03c3),h)\nd\u03c1d\u03c3 = lambdify((\u03c3,x,\u03c6,\u03bc,\u03bcbar),diff(\u03c1,\u03c3))\nd\u03b4\u03c1d\u03c4 = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),-PoissonBrackets(\u03c10-\u03c1,H))\ndef ode_rhs(t,state,\u03bc,\u03bcbar):\n x, \u03c6, V, \u03b4\u03c1 = state\n \u03c3s = [0,1]\n if (d\u03c1d\u03c3(1.,x,\u03c6,\u03bc,\u03bcbar)<0) and (d\u03c1d\u03c3(\u03b8,x,\u03c6,\u03bc,\u03bcbar)>0):\n \u03c3star = fsolve(d\u03c1d\u03c3,.8,args=(x,\u03c6,\u03bc,\u03bcbar,))[0]\n else:\n \u03c3star = 1.;\n if \u03c1f(x,\u03c6,\u03c3star,\u03bc,\u03bcbar) < \u03c10f(x,\u03c6,\u03bc,\u03bcbar):\n sgm = 0\n else:\n sgm = \u03c3star\n return [dxd\u03c4(x,\u03c6,sgm,\u03bc,\u03bcbar),d\u03c6d\u03c4(x,\u03c6,sgm,\u03bc,\u03bcbar),dVd\u03c4(x,sgm),d\u03b4\u03c1d\u03c4(x,\u03c6,\u03c3star,\u03bc,\u03bcbar)]\ndef \u03c3starf(x,\u03c6,\u03bc,\u03bcbar):\n if (d\u03c1d\u03c3(1.,x,\u03c6,\u03bc,\u03bcbar)<0) and (d\u03c1d\u03c3(\u03b8,x,\u03c6,\u03bc,\u03bcbar)>0):\n \u03c3star = fsolve(d\u03c1d\u03c3,.8,args=(x,\u03c6,\u03bc,\u03bcbar,))[0]\n else:\n \u03c3star = 1.;\n if \u03c1f(x,\u03c6,\u03c3star,\u03bc,\u03bcbar) < \u03c10f(x,\u03c6,\u03bc,\u03bcbar):\n sgm = 0\n else:\n sgm = \u03c3star\n return sgm\n```\n\n\n```python\ndef get_primary_field(name, experiment,\u03bc,\u03bcbar):\n solutions = {}\n solver = ode(ode_rhs).set_integrator(backend)\n \u03c40 = experiment['\u03c40']\n tms = np.linspace(\u03c40,experiment['T_end'],1e3+1)\n for x0 in experiment['x0']:\n \u03b4\u03c10 = \u03c10.subs(x,x0).subs(\u03c6,0)-\u03c1.subs(x,x0).subs(\u03c6,0).subs(\u03c3,1.)\n solver.set_initial_value([x0,0,0,\u03b4\u03c10],0.).set_f_params(\u03bc,\u03bcbar)\n sol = []; k = 0;\n while (solver.t < experiment['T_end']) and (solver.y[0]<=1.) and (solver.y[0]>=0.):\n solver.integrate(tms[k])\n sol.append([solver.t]+list(solver.y))\n k += 1\n solutions[x0] = {'solution': sol}\n for x0, entry in solutions.items():\n entry['\u03c4'] = [entry['solution'][j][0] for j in range(len(entry['solution']))]\n entry['x'] = [entry['solution'][j][1] for j in range(len(entry['solution']))]\n entry['\u03c6'] = [entry['solution'][j][2] for j in range(len(entry['solution']))]\n entry['V'] = [entry['solution'][j][3] for j in range(len(entry['solution']))]\n entry['\u03b4\u03c1'] = [entry['solution'][j][4] for j in range(len(entry['solution']))]\n return solutions\ndef get_\u03b4\u03c1_value(tme,x0,\u03bc,\u03bcbar):\n solver = ode(ode_rhs).set_integrator(backend)\n \u03b4\u03c10 = \u03c10.subs(x,x0).subs(\u03c6,0)-\u03c1.subs(x,x0).subs(\u03c6,0).subs(\u03c3,1.)\n solver.set_initial_value([x0,0,0,\u03b4\u03c10],0.).set_f_params(\u03bc,\u03bcbar)\n while (solver.t < tme) and (solver.y[0]<=1.) and (solver.y[0]>=0.):\n solver.integrate(tme)\n sol = [solver.t]+list(solver.y)\n return solver.y[3]\ndef get_\u03b4\u03c1_ending(params,\u03bc,\u03bcbar):\n tme, x0 = params\n solver = ode(ode_rhs).set_integrator(backend)\n \u03b4\u03c10 = \u03c10.subs(x,x0).subs(\u03c6,0)-\u03c1.subs(x,x0).subs(\u03c6,0).subs(\u03c3,1.)\n solver.set_initial_value([x0,0,0,\u03b4\u03c10],0.).set_f_params(\u03bc,\u03bcbar)\n \u03b4\u03c4 = 1.0e-8; tms = [tme,tme+\u03b4\u03c4]\n _k = 0; sol = []\n while (_k=0.):\n solver.integrate(tms[_k])\n sol.append(solver.y)\n _k += 1\n #print(sol)\n return(sol[0][3],(sol[1][3]-sol[0][3])/\u03b4\u03c4)\ndef get_state(tme,x0,\u03bc,\u03bcbar):\n solver = ode(ode_rhs).set_integrator(backend)\n \u03b4\u03c10 = \u03c10.subs(x,x0).subs(\u03c6,0)-\u03c1.subs(x,x0).subs(\u03c6,0).subs(\u03c3,1.)\n solver.set_initial_value([x0,0,0,\u03b4\u03c10],0.).set_f_params(\u03bc,\u03bcbar)\n \u03b4\u03c4 = 1.0e-8; tms = [tme,tme+\u03b4\u03c4]\n _k = 0; sol = []\n while (solver.t < tms[-1]) and (solver.y[0]<=1.) and (solver.y[0]>=0.):\n solver.integrate(tms[_k])\n sol.append(solver.y)\n _k += 1\n return(list(sol[0])+[(sol[1][3]-sol[0][3])/\u03b4\u03c4])\n```\n\n# Machinery for the universal line\n\n* To find the universal singular curve we need to define two parameters\n\n\n```python\n\u03b30 = PoissonBrackets(PoissonBrackets(H,H0),H)\n\u03b31 = PoissonBrackets(PoissonBrackets(H0,H),H0)\n```\n\n* The dynamics\n\n\n```python\ndxd\u03c4SingExpr = -(\u03b30*diff(H0,\u03c6)+\u03b31*diff(H,\u03c6))/(\u03b30+\u03b31)\nd\u03c6d\u03c4SingExpr = (\u03b30*diff(H0,x)+\u03b31*diff(H,x))/(\u03b30+\u03b31)\ndVd\u03c4SingExpr = (\u03b30*h0+\u03b31*h)/(\u03b30+\u03b31)\n\u03c3SingExpr = \u03b31*\u03c3/(\u03b30+\u03b31)\n```\n\n* Machinery for Python: lambdify the functions above\n\n\n```python\ndxd\u03c4Sing = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),dxd\u03c4SingExpr)\nd\u03c6d\u03c4Sing = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),d\u03c6d\u03c4SingExpr)\ndVd\u03c4Sing = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),dVd\u03c4SingExpr)\n\u03c3Sing = lambdify((x,\u03c6,\u03c3,\u03bc,\u03bcbar),\u03c3SingExpr)\n```\n\n\n```python\ndef ode_rhs_Sing(t,state,\u03bc,\u03bcbar):\n x, \u03c6, V = state\n if (d\u03c1d\u03c3(1.,x,\u03c6,\u03bc,\u03bcbar)<0) and (d\u03c1d\u03c3(\u03b8,x,\u03c6,\u03bc,\u03bcbar)>0):\n \u03c3star = fsolve(d\u03c1d\u03c3,.8,args=(x,\u03c6,\u03bc,\u03bcbar,))[0]\n else:\n \u03c3star = 1.;\n return [dxd\u03c4Sing(x,\u03c6,\u03c3star,\u03bc,\u03bcbar),d\u03c6d\u03c4Sing(x,\u03c6,\u03c3star,\u03bc,\u03bcbar),dVd\u03c4Sing(x,\u03c6,\u03c3star,\u03bc,\u03bcbar)]\ndef get_universal_curve(end_point,tmax,Nsteps,\u03bc,\u03bcbar):\n tms = np.linspace(end_point[0],tmax,Nsteps);\n solver = ode(ode_rhs_Sing).set_integrator(backend)\n solver.set_initial_value(end_point[1:4],end_point[0]).set_f_params(\u03bc,\u03bcbar)\n _k = 0; sol = []\n while (solver.t < tms[-1]):\n solver.integrate(tms[_k])\n sol.append([solver.t]+list(solver.y))\n _k += 1\n return sol\ndef get_\u03c3_universal(tme,end_point,\u03bc,\u03bcbar):\n \u03b4\u03c4 = 1.0e-8; tms = [tme,tme+\u03b4\u03c4]\n solver = ode(ode_rhs_Sing).set_integrator(backend)\n solver.set_initial_value(end_point[1:4],end_point[0]).set_f_params(\u03bc,\u03bcbar)\n _k = 0; sol = []\n while (solver.t < tme+\u03b4\u03c4):\n solver.integrate(tms[_k])\n sol.append([solver.t]+list(solver.y))\n _k += 1\n x, \u03c6 = sol[0][:2]\n sgm = fsolve(lambda \u03c3: dxd\u03c4(x,\u03c6,\u03c3,\u03bc,\u03bcbar)-(sol[1][0]-sol[0][0])/\u03b4\u03c4,\u03b8/2)[0]\n return sgm\ndef get_state_universal(tme,end_point,\u03bc,\u03bcbar):\n solver = ode(ode_rhs_Sing).set_integrator(backend)\n solver.set_initial_value(end_point[1:4],end_point[0]).set_f_params(\u03bc,\u03bcbar)\n solver.integrate(tme)\n return [solver.t]+list(solver.y)\n```\n\n\n```python\ndef ode_rhs_with_\u03c3star(t,state,\u03bc,\u03bcbar):\n x, \u03c6, V = state\n if (d\u03c1d\u03c3(1.,x,\u03c6,\u03bc,\u03bcbar)<0) and (d\u03c1d\u03c3(\u03b8,x,\u03c6,\u03bc,\u03bcbar)>0):\n \u03c3 = fsolve(d\u03c1d\u03c3,.8,args=(x,\u03c6,\u03bc,\u03bcbar,))[0]\n else:\n \u03c3 = 1.;\n return [dxd\u03c4(x,\u03c6,\u03c3,\u03bc,\u03bcbar),d\u03c6d\u03c4(x,\u03c6,\u03c3,\u03bc,\u03bcbar),dVd\u03c4(x,\u03c3)]\ndef ode_rhs_with_given_\u03c3(t,state,\u03c3,\u03bc,\u03bcbar):\n x, \u03c6, V = state\n return [dxd\u03c4(x,\u03c6,\u03c3,\u03bc,\u03bcbar),d\u03c6d\u03c4(x,\u03c6,\u03c3,\u03bc,\u03bcbar),dVd\u03c4(x,\u03c3)]\ndef get_trajectory_with_\u03c3star(starting_point,tmax,Nsteps,\u03bc,\u03bcbar):\n tms = np.linspace(starting_point[0],tmax,Nsteps)\n solver = ode(ode_rhs_with_\u03c3star).set_integrator(backend)\n solver.set_initial_value(starting_point[1:],starting_point[0]).set_f_params(\u03bc,\u03bcbar)\n sol = []; _k = 0;\n while solver.t < max(tms) and (solver.y[0]<=1.) and (solver.y[0]>=0.):\n solver.integrate(tms[_k])\n sol.append([solver.t]+list(solver.y))\n _k += 1\n return sol\ndef get_trajectory_with_given_\u03c3(starting_point,tmax,Nsteps,\u03c3,\u03bc,\u03bcbar):\n tms = np.linspace(starting_point[0],tmax,100)\n solver = ode(ode_rhs_with_given_\u03c3).set_integrator(backend)\n solver.set_initial_value(starting_point[1:],starting_point[0]).set_f_params(\u03c3,\u03bc,\u03bcbar)\n sol = []; _k = 0;\n while solver.t < max(tms) and (solver.y[0]<=1.) and (solver.y[0]>=0.):\n solver.integrate(tms[_k])\n sol.append([solver.t]+list(solver.y))\n _k += 1\n return sol\ndef get_state_with_\u03c3star(tme,starting_point,\u03bc,\u03bcbar):\n solver = ode(ode_rhs_with_\u03c3star).set_integrator(backend)\n solver.set_initial_value(starting_point[1:4],starting_point[0]).set_f_params(\u03bc,\u03bcbar)\n solver.integrate(tme)\n return [solver.t]+list(solver.y)\ndef get_finalizing_point_from_universal_curve(tme,tmx,end_point,\u03bc,\u03bcbar):\n unv_point = get_state_universal(tme,end_point,\u03bc,\u03bcbar)\n return get_state_with_\u03c3star(tmx,unv_point,\u03bc,\u03bcbar)[1]\n```\n\n# Field of optimal trajectories as the solution of the Bellman equation\n\n* \u03bc & \u03bcbar are varied by *T* and *T*bar ($\\mu=1/T$ and $\\bar\\mu=1/\\bar{T}$)\n\n\n```python\ntmx = 180.\nend_switching_curve = {'t': 12., 'x': .7} \n# for \u03a4, \u03a4bar in zip([28]*5,[14,21,28,35,60]):\n\u03a4 = 10.5; \u03a4bar = 14.0\n\u03bc = 1./\u03a4; \u03bcbar = 1./\u03a4bar\nprint(\"Parameters: \u03bc = %.5f, \u03bcbar = %.5f\"%(\u03bc,\u03bcbar))\nend_switching_curve['t'], end_switching_curve['x'] = fsolve(get_\u03b4\u03c1_ending,(end_switching_curve['t'],end_switching_curve['x']),args=(\u03bc,\u03bcbar),xtol=1.0e-12)\nend_point = [end_switching_curve['t']]+get_state(end_switching_curve['t'],end_switching_curve['x'],\u03bc,\u03bcbar)\nprint(\"Ending point for the switching line: \u03c4 = %.1f days, x = %.1f%%\" % (end_point[0], end_point[1]*100))\nprint(\"Checking the solution - should give zero values: \")\nprint(get_\u03b4\u03c1_ending([end_switching_curve['t'],end_switching_curve['x']],\u03bc,\u03bcbar))\nprint(\"* Constructing the primary field\")\nprimary_field1 = []\nexperiments = {\n 'sol1': { 'T_end': tmx, '\u03c40': 0., 'x0': list(np.linspace(0,end_switching_curve['x']-(1e-3),7)) } }\nfor name, values in experiments.items():\n primary_field1.append(get_primary_field(name,values,\u03bc,\u03bcbar))\nprimary_field2 = []\nexperiments = {\n 'sol1': { 'T_end': tmx, '\u03c40': 0., 'x0': list(np.linspace(end_switching_curve['x']+(3e-6),1.,7)) } }\nfor name, values in experiments.items():\n primary_field2.append(get_primary_field(name,values,\u03bc,\u03bcbar))\nprint(\"* Constructing the switching curve\")\nswitching_curve = []\nx0s = np.linspace(end_switching_curve['x'],1,21); _y = end_switching_curve['t']\n\nfor x0 in x0s:\n tme = fsolve(get_\u03b4\u03c1_value,_y,args=(x0,\u03bc,\u03bcbar))[0]\n if (tme>0):\n switching_curve = switching_curve+[[tme,get_state(tme,x0,\u03bc,\u03bcbar)[0]]]\n _y = tme\nprint(\"* Constructing the universal curve\")\nuniversal_curve = get_universal_curve(end_point,tmx,25,\u03bc,\u03bcbar)\nprint(\"* Finding the last characteristic\")\n#time0 = time.time()\ntuniv = fsolve(get_finalizing_point_from_universal_curve,tmx-40.,args=(tmx,end_point,\u03bc,\u03bcbar,))[0]\n#print(\"The proccess to find the last characteristic took %0.1f minutes\" % ((time.time()-time0)/60.))\nuniv_point = get_state_universal(tuniv,end_point,\u03bc,\u03bcbar)\nprint(\"The last point on the universal line:\")\nprint(univ_point)\nlast_trajectory = get_trajectory_with_\u03c3star(univ_point,tmx,50,\u03bc,\u03bcbar)\nprint(\"Final state:\")\nfinal_state = get_state_with_\u03c3star(tmx,univ_point,\u03bc,\u03bcbar)\nprint(final_state)\n```\n\n Parameters: \u03bc = 0.09524, \u03bcbar = 0.07143\n Ending point for the switching line: \u03c4 = 11.1 days, x = 62.9%\n Checking the solution - should give zero values: \n (-1.6853786659789397e-09, -4.3053518384272386e-10)\n * Constructing the primary field\n * Constructing the switching curve\n * Constructing the universal curve\n * Finding the last characteristic\n The last point on the universal line:\n [168.21662930295125, 0.628557975772097, -0.2400762227850881, 1.3270349109165358]\n Final state:\n [180.0, -3.68066688238855e-13, -0.3878368709539695, 1.6191478873005891]\n\n\n# Preparation for second figure\n\n\n```python\nfor idx, tend in enumerate(np.r_[np.arange(30,240,30),np.arange(240,30*48,120)]):\n tuniv = fsolve(get_finalizing_point_from_universal_curve,tend-20.,args=(tend,end_point,\u03bc,\u03bcbar,))[0]\n univ_point = get_state_universal(tuniv,end_point,\u03bc,\u03bcbar)\n trajectory = get_trajectory_with_\u03c3star(univ_point,tend,50,\u03bc,\u03bcbar)\n plt.plot([x[0] for x in trajectory],[x[1] for x in trajectory],linewidth=1,color=clrs0[4])\n sol_ = [[tend,\u03c4,\u03c3starf(x,\u03c6,\u03bc,\u03bcbar),x,exp((b-d)*\u03c4-V)] for \u03c4,x,\u03c6,V in trajectory]\n sol = sol+sol_ if idx else sol_\n universal_curve = get_universal_curve(end_point,univ_point[0],50,\u03bc,\u03bcbar)\n plt.plot([x[0] for x in universal_curve],[x[1] for x in universal_curve],linewidth=3,color=clrs0[0],zorder=3)\n sol = [[tend,\u03c4,get_\u03c3_universal(\u03c4,end_point,\u03bc,\u03bcbar),x,exp((b-d)*\u03c4-V)] for \u03c4,x,\u03c6,V in universal_curve] + sol\n trajectory = get_trajectory_with_\u03c3star([0,end_switching_curve['x'],0,0],end_point[0],50,\u03bc,\u03bcbar)\n sol = [[tend,\u03c4,\u03c3starf(x,\u03c6,\u03bc,\u03bcbar),x,exp((b-d)*\u03c4-V)] for \u03c4,x,\u03c6,V in trajectory] + sol\n\npd.DataFrame(sol,columns=['T','time','sigma','resistance','fold_change']).\\\n sort_values(['T','time']).to_csv('../figures/draft/Fig7X-trjs_optimal.csv',index=False,header=False)\n```\n\n\n```python\n\n```\n", 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YES", "lm_q1_score": 0.7217432182679957, "lm_q2_score": 0.6001883592602049, "lm_q1q2_score": 0.43318187797944824}} {"text": "```python\nimport pandas as pd\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport sympy\nsympy.init_printing(use_latex='mathjax')\n%matplotlib inline\n```\n\n# Sympy\ub97c \uc0ac\uc6a9\ud55c \ud568\uc218 \ubbf8\ubd84\n\n### Data \ubd84\uc11d\uc5d0\uc11c \ubbf8\ubd84\uc774 \ud544\uc694\ud55c \uc774\uc720\n- \ub370\uc774\ud130 \ubd84\uc11d\uc5d0\ub294 \ubbf8\ubd84(differentiation)\uc774 \ud544\uc218\uc801\uc774\ub2e4. \ub370\uc774\ud130 \ubd84\uc11d\uc758 \ubaa9\ud45c\ub294 \uac00\uc7a5 \ucd5c\uc801\uc758(optimal)\ubaa8\ud615\uc744 \uad6c\ud558\ub294 \uc77c\uc774\ub2e4. \uc608\uce21 \ubaa8\ud615\uc740 \uc785\ub825 \ub370\uc774\ud130 \uc774\uc678\uc5d0\ub3c4 \ubaa8\uc218(parameter)\ub77c\uace0 \ud558\ub294 \uc22b\uc790\ub97c \uac00\uc9c4\ub2e4. \uc120\ud615 \uc608\uce21 \ubaa8\ud615\uc5d0\uc11c\ub294 \uc785\ub825 \ub370\uc774\ud130\uc5d0 \ub300\ud55c \uac00\uc911\uce58\uac00 \ubaa8\uc218\uac00 \ub41c\ub2e4. \uc989 \ubaa8\uc218\ub97c \uc5b4\ub5a4 \uc22b\uc790\ub85c \uc815\ud558\ub290\ub0d0\uc5d0 \ub530\ub77c \uc608\uce21 \ubaa8\ud615\uc758 \uc131\ub2a5\uc774 \ub2ec\ub77c\uc9c4\ub2e4. \uc989, \ubaa8\uc218\ub97c \ud568\uc218\uc758 \uc785\ub825\uc774\ub77c\uace0 \ud55c\ub2e4\uba74 \uc131\ub2a5\uc740 \ud568\uc218\uc758 \ucd9c\ub825\uc774 \ub41c\ub2e4. \uc774 \ub54c\uc758 \ud568\uc218\ub294 \ubaa8\ud615\uc758 \ubaa8\uc218\ub97c \uc785\ub825\ubc1b\uc544 \ubaa8\ud615\uc758 \uc131\ub2a5\uc744 \ucd9c\ub825\ud558\ub294 \ud568\uc218\ub85c x\ub370\uc774\ud130\ub97c \ubc1b\uc544 y \ub370\uc774\ud130\ub97c \ucd9c\ub825\ud558\ub294 \ud568\uc218\uc640\ub294 \ub2e4\ub978 \ud568\uc218\ub77c\ub294 \uc810\uc5d0 \uc8fc\uc758\ud55c\ub2e4.\n\n- \uc6b0\ub9ac\uac00 \uc6d0\ud558\ub294 \uac83\uc740 \uc131\ub2a5\uc774 \uac00\uc7a5 \ub192\uac8c \ub9cc\ub4dc\ub294 \ubaa8\uc218\uc758 \uac12\uc774\ub2e4. \uc774\ub807\uac8c \ud568\uc218\uc758 \ucd9c\ub825\uc744 \uac00\uc7a5 \ud06c\uac8c \ub9cc\ub4dc\ub294 \uc785\ub825\uac12\uc744 \ucc3e\ub294 \uac83\uc744 \ucd5c\uc801\ud654(optimization) \uc791\uc5c5\uc774\ub77c\uace0 \ud558\ub294\ub370, \ucd5c\uc801\ud654\ub97c \ud558\ub824\uba74 \ubbf8\ubd84 \ud639\uc740 \ud3b8\ubbf8\ubd84\uc774 \uc720\uc6a9\ud558\uae30 \ub54c\ubb38\uc5d0 \ubbf8\ubd84\uc744 \uc798 \uc54c\uace0 \uc788\uc5b4\uc57c \ud55c\ub2e4.\n\n- \ub2e4\ud589\uc2a4\ub7ec\uc6b4 \uc810\uc740 \ub370\uc774\ud130 \ubd84\uc11d\uc5d0\uc11c \ud544\uc694\ud55c \ubbf8\ubd84\uc758 \uc218\uc900\uc740 \uadf8\ub2e4\uc9c0 \ub192\uc9c0 \uc54a\ub2e4\ub294 \uc810\uc774\ub2e4. \uc120\ud615 \ub2e4\ud56d\uc2dd\uc774\ub098 \uc9c0\uc218 \ud568\uc218\uc758 \ud3b8\ubbf8\ubd84 \uc815\ub3c4\ub9cc \uc54c\uace0 \uc788\uc73c\uba74 \ub418\uace0, \ub300\ubd80\ubd84\uc758 \uacbd\uc6b0 \ucd5c\uc801\ud654 \ub77c\uc774\ube0c\ub7ec\ub9ac\ub97c \uc0ac\uc6a9\ud558\uac70\ub098, theano, tensorflow\ub4f1\uc758 \ub77c\uc774\ube0c\ub7ec\ub9ac\uc5d0\ub3c4 \ub3c4\ud568\uc218\uc640 \ubbf8\ubd84\uac12\uc744 \uacc4\uc0b0\ud574 \uc8fc\uae30 \ub54c\ubb38\uc5d0 \uc2e4\uc81c\ub85c \uc9c1\uc811 \ubbf8\ubd84\uc744 \uacc4\uc0b0\ud558\ub294 \uc77c\uc740 \ub9ce\uc9c0 \uc54a\ub2e4.\n\n## \uae30\uc6b8\uae30 (Slope)\n\n- \ucd5c\uc801\ud654\ub97c \ud558\ub294 \uac00\uc7a5 \uc774\uc0c1\uc801\uc778 \ubc29\ubc95\uc740 \uac00\ub2a5\ud55c \ubaa8\ub4e0 x\uac12\uc744 \ub123\uc5b4\uc11c y\uac12\uc744 \uacc4\uc0b0\ud574 \ubcf4\uace0, \uc774 \uc911\uc5d0\uc11c \uac00\uc7a5 \ud070 \ud639\uc740 \uac00\uc7a5 \uc791\uc740 y\uac12\uc744 \ucd9c\ub825\ud558\uac8c \ub9cc\ub4dc\ub294 x\uc758 \uac12\uc744 \uace8\ub77c\ub0b4\ub294 \uac83\uc774\ub2e4. \uc774\ub7ec\ud55c \ubc29\uc2dd\uc744 \uadf8\ub9ac\ub4dc\uc11c\uce58(grid search)\ub77c\uace0 \ud55c\ub2e4. \ud558\uc9c0\ub9cc, \uadf8\ub9ac\ub4dc \uc11c\uce58\ub97c \ud558\ub824\uba74 \uc5c4\uccad\ub098\uac8c \ub9ce\uc740 \uc5f0\uc0b0\uc744 \ud574\uc57c \ud558\ubbc0\ub85c \ud604\uc2e4\uc801\uc73c\ub85c\ub294 \uc0ac\uc6a9\ud560 \uc218 \uc5c6\ub2e4. \ub300\uc2e0 x\uc758 \uac12\uc744 \uba87\uac00\uc9c0\ub9cc \uc2dc\ub3c4\ud574\uc11c \uadf8 \uc911 \uac00\uc7a5 \uc88b\uc740 \uac12\uc744 \uad6c\ud558\ub294 \uc218\uce58\uc801 \ucd5c\uc801\ud654\ub77c\ub294 \ubc29\ubc95\uc744 \uc0ac\uc6a9\ud558\uac8c \ub41c\ub2e4. \n\n- \uc218\uce58\uc801 \ucd5c\uc801\ud654\ub97c \ud558\ub824\uba74 \ud604\uc7ac \uc0ac\uc6a9\ud55c x\uc758 \uac12\uc774 \uc544\ub2cc \ub610 \ub2e4\ub978 x\uc758 \uac12\uc744 \uc2dc\ub3c4\ud560 \ub54c \uc5b4\ub5a4 \uac12\uc744 \uc2dc\ub3c4\ud574\uc57c \ud560 \uc9c0\ub97c \uacb0\uc815\ud574\uc57c \ud55c\ub2e4. \uadf8\ub7ec\ub824\uba74, x\ub97c \uc99d\uac00\uc2dc\ucf30\uc744 \ub54c y \uac12\uc774 \uc99d\uac00\ud558\ub294\uc9c0 \uac10\uc18c\ud558\ub294\uc9c0, \uc99d\uac00\ud55c\ub2e4\uba74 \uc5b4\ub290 \uc815\ub3c4 \uc99d\uac00\ud558\ub294 \uc9c0\ub97c \uc54c\uc544\uc57c \ud55c\ub2e4. \uc989 \ud568\uc218\uc5d0 \ub4e4\uc5b4\uac00\ub294 \uc785\ub825\ubcc0\uc218\uc758 \uac12\uc774 \ub2ec\ub77c\uc9c0\uba74 \ud568\uc218\uc758 \ucd9c\ub825 \uac12\uc774 \uc5b4\ub5bb\uac8c \ubcc0\ud558\ub294\uc9c0\ub97c \uc54c\uc544\uc57c \ud55c\ub2e4. \uc774\ub97c \uae30\uc6b8\uae30(slope) \ud639\uc740 \ubbfc\uac10\ub3c4(sensitivity)\ub77c\uace0 \ud55c\ub2e4. \n\n- \ub9cc\uc57d \uc785\ub825\ubcc0\uc218\uc758 \uac12\uc774 x\uc5d0\uc11c x2\ub85c \ub2ec\ub77c\uc84c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \ucd9c\ub825 \ubcc0\uc218\ub294 f(x)\uc774\ub77c\ub294 \uac12\uc5d0\uc11c f(x2)\ub77c\ub294 \uac12\uc73c\ub85c \ub2ec\ub77c\uc9c8 \uac83\uc774\ub2e4. \uc774\ub97c \ube44\uc728\ub85c \ub098\ud0c0\ub0b4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4. \uc5ec\uae30\uc5d0\uc11c dx= x2 - x\uc774\ub2e4.\n- y\uc758 \ubcc0\ud654\ub7c9 / x\uc758 \ubcc0\ud654\ub7c9 = f(x2) - f(x) / x2 - x = f(x+dx) - f(x) / dx\n\n- \uc774\ub7ec\ud55c \ubcc0\ud654\uc728\uc740 x2\ub97c x1\uc5d0\uc11c \uc5bc\ub9c8\ub098 \uba40\ub9ac \ub5a8\uc5b4\uc838 \uc788\ub294\uac00 \uc989, dx\uc758 \uac12\uc5d0 \ub530\ub77c \ub2ec\ub77c\uc9c4\ub2e4. \uc774\ub97c \ud574\uacb0\ud558\uae30 \uc704\ud574 ***\uae30\uc6b8\uae30(slope)***\ub77c\ub294 \uac1c\ub150\uc744 \uc0ac\uc6a9\ud55c\ub2e4. \uae30\uc6b8\uae30\ub294 dx\uac12\uc774 0\uc73c\ub85c \uadfc\uc811\ud560 \ub54c\uc758 \ubcc0\ud654\uc728\uc744 \ub9d0\ud55c\ub2e4. \uae30\ud638\ub85c\ub294 \ub2e4\uc74c\ucc98\ub7fc \uc4f4\ub2e4. \n- \uae30\uc6b8\uae30 = lim dx -> 0 (f(x+dx) - f(x) / dx)\n\n- \uc774\ubc88\uc5d0\ub294 \uae30\uc6b8\uae30\ub97c \ud568\uc218\uc758 \uadf8\ub798\ud504\uc5d0\uc11c \uc0b4\ud3b4\ubcf4\uc790. \ud568\uc218\uc758 \uadf8\ub798\ud504\ub294 \uc55e\uc5d0\uc11c \uadf8\ub9b0 \uac83 \ucc98\ub7fc \ubd80\ub4dc\ub7ec\uc6b4 \uace1\uc120(curve)\uc758 \ud615\ud0dc\ub85c \ub098\ud0c0\ub098\ub294 \uacbd\uc6b0\uac00 \ub9ce\ub2e4. \uc774 \uace1\uc120\uc5d0 \ub300\ud574 \ud55c \uc810\ub9cc \uacf5\ud1b5\uc73c\ub85c \uac00\uc9c0\ub294 \uc811\uc120(tangent)\uc744 \uadf8\ub9b4 \uc218 \uc788\ub294\ub370, \uc774 \uc811\uc120\uc774 \uc218\ud3c9\uc120\uacfc \uc774\ub8e8\ub294 \uae30\uc6b8\uae30\ub294 \uc811\uc120\uc774 x \ubc29\ud5a5\uc73c\ub85c \uc774\ub3d9\ud55c \uac70\ub9ac\uc640 y\ubc29\ud5a5\uc73c\ub85c \uc774\ub3d9\ud55c \uac70\ub9ac\uc758 \ube44\uc728\uc744 \ub9d0\ud55c\ub2e4. \n- \uae30\uc6b8\uae30 = \uc811\uc120\uc774y\ubc29\ud5a5\uc73c\ub85c \uc774\ub3d9\ud55c \uac70\ub9ac / \uc811\uc120\uc774 x\ubc29\ud5a5\uc73c\ub85c \uc774\ub3d9\ud55c \uac70\ub9ac\n- \ub2e4\uc74c \uadf8\ub9bc\uc5d0\uc11c x = 0\uacfc x = 1\uc5d0\uc11c\uc758 \uae30\uc6b8\uae30\ub294 \uac01\uac011, -2\uc774\ub2e4.\n- x = 0\uc5d0\uc11c\uc758 \uae30\uc6b8\uae30 1/1 = 1\n- x = 1\uc5d0\uc11c\uc758 \uae30\uc6b8\uae30 -2/1 = -2\n\n\n```python\ndef f(x):\n return x**3 - 3*x**2 + x\n\nx = np.linspace(-1, 3, 400)\ny = f(x)\n\nplt.plot(x, y)\nplt.plot(0, 0, 'ro') # (0,0) red circle\nplt.plot(x, x, 'r:') # \ube68\uac04 \uc810\uc120\nplt.plot(1, -1, 'go') # (1,-1) green circle\nplt.plot(x, (3*1 ** 2-6*1+1) * (x-1)-1, 'g--') #\ub179\uc0c9 \uc810\uc120\n\nplt.xlim(-3.5, 5.5)\nplt.ylim(-4, 2)\nplt.xticks(np.arange(-3, 6))\nplt.yticks(np.arange(-4, 2))\n\nplt.annotate('', xy=(1, 0), xytext=(0, 0), arrowprops=dict(facecolor='blue'))\nplt.annotate('', xy=(1, 1), xytext=(1, 0), arrowprops=dict(facecolor='blue'))\nplt.annotate('', xy=(2, -1), xytext=(1, -1), arrowprops=dict(facecolor='blue'))\nplt.annotate('', xy=(2, -3), xytext=(2, -1), arrowprops=dict(facecolor='blue'))\n\nplt.show()\n```\n\n### \uc218\uce58 \ubbf8\ubd84\n\n- scipy.misc.derivative \uba85\ub839\uc744 \uc0ac\uc6a9\ud558\uba74 \uc218\uce58\uc801\uc73c\ub85c \ub300\ub7b5\uc801\uc778 \ubbf8\ubd84\uac12\uc744 \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4. \uc778\uc218\ub85c\ub294 \uae30\uc6b8\uae30\ub97c \uad6c\ud558\uace0\uc790 \ud558\ub294 \ud568\uc218 f, \uae30\uc6b8\uae30\ub97c \uad6c\ud560 \uc704\uce58 x, \uae30\uc6b8\uae30\ub97c \uad6c\ud558\uae30 \uc704\ud574 \uc774\ub3d9\ud560 \uac70\ub9ac dx\ub97c \ubc1b\ub294\ub2e4. \uae30\uc6b8\uae30\ub294 \ub2e4\uc74c \uc218\uc2dd\uc73c\ub85c \uad6c\ud55c\ub2e4.\n- f(x+0.5dx) - f(x-0.5dx) / dx\n- dx \uc778\uc218\uc758 \uac12\uc774 \ub108\ubb34 \ud06c\uac70\ub098 \ub108\ubb34 \uc791\uc73c\uba74 \uc815\ud655\ud55c \uac12\uc744 \uad6c\ud558\uc9c0 \ubabb\ud560 \uc218 \uc788\ub2e4.\n\n\n```python\nfrom scipy.misc import derivative\nderivative(f, -0.5, dx=1e-6)\n```\n\n\n\n\n 4.749999999886789\n\n\n\n\n```python\nderivative(f, 0, dx=1e-6)\n```\n\n\n\n\n 1.000000000001\n\n\n\n\n```python\nderivative(f, 0.5, dx=1e-6)\n```\n\n\n\n\n -1.2499999999804334\n\n\n\n\n```python\nderivative(f, 1, dx=1e-6)\n```\n\n\n\n\n -2.000000000002\n\n\n\n\n```python\nderivative(f, 1.5, dx=1e-6)\n```\n\n\n\n\n -1.2500000001747225\n\n\n\n\n```python\nderivative(f, 2, dx=1e-6)\n```\n\n\n\n\n 1.000000000139778\n\n\n\n\n```python\nderivative(f, 2.5, dx=1e-6)\n```\n\n\n\n\n 4.749999999553722\n\n\n\n\n```python\n\n```\n\n### \ubbf8\ubd84\n\n- **\ubbf8\ubd84(differenciation)**\uc774\ub780 \uc5b4\ub5a4 \ud568\uc218\ub85c\ubd80\ud130 \uadf8 \ud568\uc218\uc640 \uc5f0\uad00\uc131\uc774 \uc788\ub294 \uc0c8\ub85c\uc6b4 \ud568\uc218\ub97c \ub9cc\ub4e4\uc5b4\ub0b4\ub294 \uc791\uc5c5\uc774\ub2e4. \ubbf8\ubd84\uc744 \ud1b5\ud574 \ub9cc\ub4e4\uc5b4\uc9c4 \uc0c8\ub85c\uc6b4 \ud568\uc218\ub294 \uc6d0\ub798 \ud568\uc218\uc758 \uae30\uc6b8\uae30(slope)\ub97c \ub098\ud0c0\ub0b8\ub2e4. \ubbf8\ubd84\uc73c\ub85c \ub9cc\ub4e4\uc5b4\uc9c4 \ud568\uc218\ub97c \uc6d0\ub798 \ud568\uc218\uc758 \ub3c4\ud568\uc218(derivative)\ub77c\uace0 \ud55c\ub2e4. \uc6d0\ub798\ub294 \uc218\ub834(converge)\uacfc \uadf9\ud55c(limit)\uc774\ub77c\ub294 \uc218\ud559\uc801\uc778 \uac1c\ub150\uc744 \uc0ac\uc6a9\ud558\uc5ec \ubbf8\ubd84\uc744 \uc815\uc758\ud558\uc9c0\ub9cc \uc5ec\uae30\uc5d0\uc11c\ub294 \uc790\uc138\ud55c \uc124\uba85\uc744 \uc0dd\ub7b5\ud55c\ub2e4.\n\n- \ub3c4\ud568\uc218\ub294 \uc6d0\ub798 \ud568\uc218\uc5d0 \ud504\ub77c\uc784(prime)\ub97c \ubd99\uc774\uac70\ub098 \uc6d0\ub798 \ud568\uc218\uc758 \uc55e\uc5d0 d/dx\ub97c \ubd99\uc5ec\uc11c \ud45c\uc2dc\ud55c\ub2e4. \ubd84\uc218\ucc98\ub7fc \ud45c\uae30\ud558\uae30\ub3c4 \ud558\ub294\ub370 \ubd84\ubaa8\uc758 \uc704\uce58\uc5d0\ub294 \ubbf8\ubd84\ud558\uace0\uc790 \ud558\ub294 \ubcc0\uc218\uac00 \uc624\uace0 \ubd84\uc790\uc758 \uc704\uce58\uc5d0\ub294 \ubbf8\ubd84\ud558\ub294 \ud568\uc218 \uc790\uccb4\uc758 \uae30\ud638\ub098 \ud639\uc740 \uacc4\uc0b0\uc758 \uacb0\uacfc\ub85c \uc5bb\uc5b4\uc9c0\ub294 \ucd9c\ub825 \ubcc0\uc218\ub97c \ub123\ub294\ub2e4. \uc608\ub97c \ub4e4\uc5b4 y=f(x)\ub77c\ub294 \ud568\uc218\ub97c \ubbf8\ubd84\ud558\uc5ec \uad6c\ud55c \ub3c4\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc5ec\ub7ec\uac00\uc9c0 \ubc29\ubc95\uc73c\ub85c \ud45c\uae30\ud560 \uc218 \uc788\ub2e4. \uc774 \uc2dd\uc5d0\uc11c\ub294 f'\ub294 \"f\ud504\ub77c\uc784(prime)\"\uc774\ub77c\uace0 \uc77d\uace0 df/dx\ub294 'df over dx'\ub77c\uace0 \uc77d\ub294\ub2e4.\n- f' = d/dx*f(x) = d/dx*f = df/dx = d/dx(y) = d/dx*y = dy/dx\n- \uc704\uce58 x\uc5d0\uc11c \ub3c4\ud568\uc218\uc758 \uac12 f'(x)\uc740 \uadf8 \uc704\uce58\uc5d0\uc11c\uc758 \ud568\uc218\uc758 \uae30\uc6b8\uae30\uc640 \uac19\uc73c\ubbc0\ub85c \ub2e4\uc74c\ucc98\ub7fc \uc4f8 \uc218 \uc788\ub2e4.\nf'(x) = f(x)\uc758 \uae30\uc6b8\uae30\n\n### \ubbf8\ubd84 \uacf5\uc2dd\n- \uba87\uac00\uc9c0 \ubbf8\ubd84 \uacf5\uc2dd\uc744 \uc870\ud569\ud558\uba74 \ubcf5\uc7a1\ud55c \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub3c4 \uc27d\uac8c \uad6c\ud560 \uc218 \uc788\ub2e4. \uc5ec\uae30\uc5d0\uc11c\ub294 \uac00\uc7a5 \ud575\uc2ec\uc801\uc778 4\uac00\uc9c0 \uacf5\uc2dd\ub9cc\uc744 \uc18c\uac1c\ud55c\ub2e4.\n - \uae30\ubcf8 \ubbf8\ubd84 \uacf5\uc2dd\n - \uc120\ud615 \uc870\ud569 \ubc95\uce59\n - \uacf1\uc148 \ubc95\uce59\n - \uc5f0\uc1c4 \ubc95\uce59\n\n#### \uae30\ubcf8 \ubbf8\ubd84 \uacf5\uc2dd\n- \uc0c1\uc218\n\uc0c1\uc218\ub97c \ubbf8\ubd84\ud558\uba74 0\uc774 \ub41c\ub2e4.\n- d/dx(c) = 0\n\n- \uac70\ub4ed\uc81c\uacf1\nx\uc758 n\uc81c\uacf1\uc744 \ubbf8\ubd84\ud558\uba74 n-1 \uc81c\uacf1\uc73c\ub85c \uc81c\uacf1\uc218\uac00 1\uc529 \uac10\uc18c\ud55c\ub2e4. \uc774 \uacf5\uc2dd\uc740 n\uc774 \uc790\uc5f0\uc218\uc774\uac70\ub098 \uc74c\uc758 \uc815\uc218 \uc77c \ub54c \uc131\ub9bd\ud55c\ub2e4. n = 0\uc77c \ub54c\ub294 \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4.\n- d/dx(x^n) = nx^(n-1)\n\n- log (\ub85c\uadf8)\n\ub85c\uadf8\ud568\uc218\ub97c \ubbf8\ubd84\ud558\uba74 x^-1\uc774 \ub41c\ub2e4.\n- d/dx(logx) = 1/x\n\n- \uc9c0\uc218(e)\n\ubc11\uc774 \uc624\uc77c\ub7ec \uc218\uc778 \uc9c0\uc218\ud568\uc218\ub294 \ubbf8\ubd84\ud574\ub3c4 \ubcc0\ud558\uc9c0 \uc54a\ub294\ub2e4.\n- d/dx(e^x) = e^x\n\n#### \uc120\ud615 \uc870\ud569 \ubc95\uce59\n- \uc5b4\ub5a4 \ud568\uc218\uc5d0 \uc0c1\uc218\ub97c \uacf1\ud55c \ud568\uc218\ub97c \ubbf8\ubd84\ud55c \uacb0\uacfc\ub294 \uc6d0\ub798 \ud568\uc218\uc758 \ub3c4\ud568\uc218\uc5d0 \uadf8 \uc0c1\uc218\ub97c \uacf1\ud55c \uac83\uacfc \uac19\ub2e4.\n- d/dx(cf) = c*d/df(f)\n\n- \uc5b4\ub5a4 \ub450 \ud568\uc218\ub97c \ub354\ud55c \ud568\uc218\ub97c \ubbf8\ubd84\ud55c \uacb0\uacfc\ub294 \uc6d0\ub798 \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \ub354\ud55c \uac83\uacfc \uac19\ub2e4.\n- d/dx(f1 + f2) = d/dx(f1) + d/dx(f2)\n\n- \uc704\uc758 \uacb0\uacfc\ub97c \ud569\uce58\uba74 \uc5b4\ub5a4 \ud568\uc218\uc5d0 \uac01\uac01 \uc0c1\uc218\ub97c \uacf1\ud55c \ud6c4 \ub354\ud55c **\uc120\ud615 \uc870\ud569(linear combination)**\uc740 \uac01 \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uc120\ud615 \uc870\ud569\ud55c \uac83\uacfc \uac19\ub2e4. \n- d/dx(c1f1+c2f2) = c1*d/df*f1 + c2*d/df*f2\n\n- \uc774\ub7ec\ud55c \uae30\ubcf8 \uacf5\uc2dd\uc744 \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c \ud568\uc218\ub97c \ubbf8\ubd84\ud558\uba74,\n- y = 1 + 2x + 3x^2 + 4 exp(x) + 5log(x)\n- y' = 2 + 6x + 4exp(x) + 5/x\n\n- \uc774 \ubc29\ubc95\uc73c\ub85c \uc704\uc5d0\uc11c \uadf8\ub798\ud504\ub97c \uadf8\ub838\ub358 \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n- f(x) = x^3 -3x^2 + x\n- f'(x) = 3*x^2 - 6*x + 1\n- \ub3c4\ud568\uc218\uc758 \uac12\uc774 \uae30\uc6b8\uae30\uc640 \uc77c\uce58\ud558\ub294 \uac83\uc744 \uc54c \uc218 \uc788\ub2e4.\n\n\n```python\ndef fprime(x):\n return 3*x**2 - 6*x+1\n\nx = np.linspace(-1, 3, 400)\n\nplt.figure(figsize=(10,15))\n\nplt.subplot(211)\nplt.plot(x, f(x))\nplt.xlim(-2,4)\nplt.xticks(np.arange(-1,4))\nplt.yticks(np.arange(-5,4))\nplt.title('f(x)')\n\nplt.subplot(212)\nplt.plot(x, fprime(x))\nplt.xlim(-2, 4)\nplt.xticks(np.arange(-1,4))\nplt.yticks(np.arange(-3,11))\nplt.title(\"f'(x)\")\n\nplt.show()\n```\n\n#### \uacf1\uc148 \ubc95\uce59\n- \uc5b4\ub5a4 \ud568\uc218\uc758 \ud615\ud0dc\uac00 \ub450 \uac1c\uc758 \ud568\uc218\ub97c \uacf1\ud55c \uac83\uacfc \uac19\uc744 \ub54c\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uac01 \uac1c\ubcc4 \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc6d0\ub798\uc758 \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud55c\ub2e4. \uc774\ub97c \uacf1\uc148\ubc95\uce59\uc774\ub77c\uace0 \ud55c\ub2e4.\n- d/dx*(f*g) = f* dg/dx + df/dx * g\n- \uacf1\uc148 \ubc95\uce59\uc744 \uc0ac\uc6a9\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \ud568\uc218\ub97c \ubbf8\ubd84\ud558\uc5ec,\n- f = x*e^x\n- \ub2e4\uc74c\uacfc \uac19\uc740 \ub3c4\ud568\uc218\ub97c \uad6c\ud55c\ub2e4\n- df/dx = xe^x + e^e\n\n#### \uc5f0\uc1c4\ubc95\uce59\n- \uc5f0\uc1c4 \ubc95\uce59(Chain Rule)\uc740 \ubbf8\ubd84\ud558\uace0\uc790 \ud558\ub294 \ud568\uc218\uc758 \uc785\ub825 \ubcc0\uc218\uac00 \ub2e4\ub978 \ud568\uc218\uc758 \ucd9c\ub825 \ubcc0\uc218\uc778 \uacbd\uc6b0 \uc801\uc6a9\ud560 \uc218 \uc788\ub2e4. \n- f(x) = h(g(x))\n- df/dx = dh/dg * dg/dx\n- for example) f = exp((x-u)^2/sigma^2)\n- f'= 2(x-u)/sigma^2 * exp((x-u)^2/sigma^2)\n\n#### \uc5f0\uc1c4\ubc95\uce59\uc758 \uc608: \ub85c\uadf8\ud568\uc218\uc758 \ubbf8\ubd84\n- d/dxlogf(x) = f'(x)/f(x)\n\n\n```python\n\n```\n\n\n```python\na, b, x = sympy.symbols('a, b, x')\nf = sympy.exp(a*x ** b)\nsympy.simplify(sympy.diff(f, x))\n```\n\n\n\n\n$$a b x^{b - 1} e^{a x^{b}}$$\n\n\n\n#### 2\ucc28 \ub3c4\ud568\uc218\n- \ub3c4\ud568\uc218\ub97c \ud55c \ubc88 \ub354 \ubbf8\ubd84\ud558\uc5ec \ub9cc\ub4e4\uc5b4\uc9c4 \ud568\uc218\ub97c 2\ucc28 \ub3c4\ud568\uc218 (second derivative)\ub77c\uace0 \ud55c\ub2e4. 2\ucc28 \ub3c4\ud568\uc218\ub294 2\uac1c\uc758 prime \uae30\ud638\ub97c \ubd99\uc774\uac70\ub098 d^2/dx^2 \uae30\ud638\ub85c \ud45c\uc2dc\ud55c\ub2e4.\n\n- \uc608\ub97c \ub4e4\uc5b4 y = f(x) \ub77c\ub294 \ud568\uc218\ub97c \ub450\ubc88 \ubbf8\ubd84\ud558\uc5ec \uad6c\ud55c 2\ucc28 \ub3c4\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \ud45c\uae30.\n- f'' = d^2/dx^2(f) = d^2/dx^2*f = d^2f/dx^2 = d^2/dx^2(y) = d^2/dx^2*y = d^2y/dx^2\n\n- 2\ucc28 \ub3c4\ud568\uc218\ub294 \ub3c4\ud568\uc218\uc758 \uae30\uc6b8\uae30\ub97c \ub098\ud0c0\ub0b8\ub2e4. \uc989 \ub3c4\ud568\uc218 \uac12\uc774 \uc99d\uac00\ud558\uba74 2\ucc28 \ub3c4\ud568\uc218 \uac12\uc74c \uc591\uc218\uc774\uace0, \ub3c4\ud568\uc218 \uac12\uc774 \uac10\uc18c\ud558\uba74 2\ucc28 \ub3c4\ud568\uc218 \uac12\uc740 \uc74c\uc218\uc774\ub2e4. 2\ucc28 \ub3c4\ud568\uc218 \uac12\uc774 \uc591\uc218\uc778 \uacbd\uc6b0\ub97c \ubcfc\ub85d (convex)\ud558\ub2e4\uace0 \ud558\uba70 2\ucc28 \ub3c4\ud568\uc218 \uac12\uc774 \uc74c\uc218\uc778 \uacbd\uc6b0\ub97c \uc624\ubaa9 (concave)\ud558\ub2e4\uace0 \ud55c\ub2e4. \uc774 \ub54c \ubcfc\ub85d\uacfc \uc624\ubaa9\uc740 \uc544\ub798\uc5d0\uc11c \ubc14\ub77c\ubcf8 \uad00\uc810\uc774\ub2e4. \n- \ub2e4\uc74c \uc608\uc81c\uc758 \ud568\uc218\ub294 f''(x)\uac00 \uc74c\uc218\uc778 \uad6c\uac04(x < 1)\uc5d0\uc11c \uc624\ubaa9(concave)\ud558\uace0 f''(x)\uac00 \uc591\uc218\uc778 \uad6c\uac04(x>1)\uc5d0\uc11c \ubcfc\ub85d(convex)\ud558\ub2e4 \n\n\n```python\ndef fprime2(x):\n return 6*x - 6\n\nx = np.linspace(-1, 3, 400)\n\nplt.figure(figsize=(10,15))\n\nplt.subplot(311)\nplt.plot(x, f(x))\nplt.xlim(-2, 4)\nplt.xticks(np.arange(-1, 4))\nplt.yticks(np.arange(-5, 4))\nplt.title('f(x)')\n\nplt.subplot(312)\nplt.plot(x, fprime(x))\nplt.xlim(-2, 4)\nplt.xticks(np.arange(-1, 4))\nplt.yticks(np.arange(-3, 11))\nplt.title(\"f'(x)\")\n\nplt.subplot(313)\nplt.plot(x, fprime2(x))\nplt.xlim(-2, 4)\nplt.xticks(np.arange(-1, 4))\nplt.title(\"f'(x)\")\n\nplt.show()\n```\n\n## \ud3b8\ubbf8\ubd84 (Partial Differentiation)\n\n- \ub9cc\uc57d \ud568\uc218\uac00 \ub450 \uac1c \uc774\uc0c1\uc758 \ub3c5\ub9bd\ubcc0\uc218\ub97c \uac00\uc9c0\ub294 \ub2e4\ubcc0\uc218 \ud568\uc218\uc778 \uacbd\uc6b0\uc5d0\ub3c4 \ubbf8\ubd84 \uc989, \uae30\uc6b8\uae30\ub294 \ud558\ub098\uc758 \ubcc0\uc218\uc5d0 \ub300\ud574\uc11c\ub9cc \uad6c\ud560 \uc218 \uc788\ub2e4. \uc774\ub97c \ud3b8\ubbf8\ubd84(partial differentiation)\uc774\ub77c\uace0 \ud55c\ub2e4. \ub530\ub77c\uc11c \ud3b8\ubbf8\ubd84\uc758 \uacb0\uacfc\ub85c \ud558\ub098\uc758 \ud568\uc218\uc5d0 \ub300\ud574 \uc5ec\ub7ec \uac1c\uc758 \ub3c4\ud568\uc218\uac00 \ub098\uc62c \uc218 \uc788\ub2e4.\n\n- \ud3b8\ubbf8\ubd84\uc758 \uacb0\uacfc \uc989 \ub3c4\ud568\uc218\ub294 \ub3c5\ub9bd \ubcc0\uc218\ub97c \ud568\uc218\uc758 \uc544\ub7ab\ucca8\uc790\ub85c \uc368\uc11c \ud45c\uae30\ud558\uac70\ub098 round\uae30\ud638\ub97c \uc0ac\uc6a9\ud558\uc5ec \ud45c\uae30\ud55c\ub2e4. x, y\ub450 \uac1c\uc758 \ub3c5\ub9bd \ubcc0\uc218\ub97c \uac00\uc9c0\ub294 \ud568\uc218 f(x,y)\uc758 \ud3b8\ubbf8\ubd84 \ub3c4\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \ud45c\uae30\ud55c\ub2e4.\n- fx(x,y) = round(f)/round(x)\n- fy(x,y) = round(f)/round(y)\n\n- \ud3b8\ubbf8\ubd84\uc744 \ud558\ub294 \ubc29\ubc95\uc740 \ubcc0\uc218\uac00 \ud558\ub098\uc778 \ud568\uc218\uc758 \ubbf8\ubd84\uacfc \uac19\ub2e4. \ub2e4\ub9cc, \uc5b4\ub5a4 \ud558\ub098\uc758 \ub3c5\ub9bd\ubcc0\uc218\uc5d0 \ub300\ud574 \ubbf8\ubd84\ud560 \ub54c\ub294 \ub2e4\ub978 \ub3c5\ub9bd \ubcc0\uc218\ub97c \uc0c1\uc218\ub85c \uc0dd\uac01\ud558\uba74 \ub41c\ub2e4. \uc608\ub97c \ub4e4\uc5b4 x,y\ub77c\ub294 \ub450 \uac1c\uc758 \ub3c5\ub9bd \ubcc0\uc218\ub97c \uac00\uc9c0\ub294 \ud568\uc218\uc5e3 x\ub85c \ud3b8\ubbf8\ubd84 \ud560 \ub54c\ub294 y\ub294 \ub3c5\ub9bd \ubcc0\uc218\uac00 \uc544\ub2cc \uc0c1\uc218\ub85c \uc0dd\uac01\ud55c\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c y\ub85c \ud3b8\ubbf8\ubd84\uc744 \ud560 \ub584\ub294 x\ub294 \ub3c5\ub9bd \ubcc0\uc218\uac00 \uc544\ub2cc \uc0c1\uc218\ub85c \uc0dd\uac01\ud55c\ub2e4.\n- f(x,y) = x^2 + 4xy + 4y^2\n- fx(x,y) = 2x + 4y\n- fy(x,y) = 4x + 8y\n\n- \ud3b8\ubbf8\ubd84\uc5d0 \ub300\ud574\uc11c\ub3c4 2\ucc28 \ub3c4\ud568\uc218\ub97c \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \ud3b8\ubbf8\ubd84\uc758 2\ucc28 \ub3c4\ud568\uc218\ub97c \uad6c\ud560 \ub54c\ub294 \uac01\uac01\uc758 \ubbf8\ubd84\uc5d0 \uc4f0\uc774\ub294 \ub3c5\ub9bd \ubcc0\uc218\ub97c \uc790\uc720\ub86d\uac8c \uc120\ud0dd\ud560 \uc218 \uc788\ub2e4.\n- \uccab\ubc88\uc9f8 \ubbf8\ubd84\uacfc \ub450\ubc88\uc9f8 \ubbf8\ubd84\uc5d0\uc11c \ubaa8\ub450 x\uc5d0 \ub300\ud574 \ubbf8\ubd84\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\uc774 \ud45c\uae30\ud55c\ub2e4.\n- fxx(x,y) = round^2f/roundx^2 = 2\n- fyy(x,y) = round^2f/roundy^2 = 8\n- fxy(x,y) = round^2f/roundyroundx = 4\n- fyx(x,y) = round^2f/roundxroundy = 4\n\n\n```python\nx, y = sympy.symbols('x y')\nf = sympy.exp((x**2) + 2 * (y ** 2))\nsympy.diff(sympy.diff(f,x), x)\n```\n\n\n\n\n$$4 x^{2} e^{x^{2} + 2 y^{2}} + 2 e^{x^{2} + 2 y^{2}}$$\n\n\n\n\n```python\nsympy.diff(sympy.diff(f,x), y)\n```\n\n\n\n\n$$8 x y e^{x^{2} + 2 y^{2}}$$\n\n\n\n\n```python\nsympy.diff(sympy.diff(f, y), x)\n```\n\n\n\n\n$$8 x y e^{x^{2} + 2 y^{2}}$$\n\n\n\n\n```python\nsympy.diff(sympy.diff(f, y), y)\n```\n\n\n\n\n$$16 y^{2} e^{x^{2} + 2 y^{2}} + 4 e^{x^{2} + 2 y^{2}}$$\n\n\n\n#### Sympy\n- SymPy\ub294 \uc2ec\ubcfc\ub9ad \uc5f0\uc0b0 (symbolic operation)\uc744 \uc9c0\uc6d0\ud558\uae30 \uc704\ud55c \ud30c\uc774\uc36c package. \uc2ec\ubcfc\ub9ad \uc5f0\uc0b0\uc774\ub780 \uc0ac\ub78c\uc774 \uc5f0\ud544\ub85c \uacc4\uc0b0\ud558\ub294 \ubbf8\ubd84/\uc801\ubd84\uacfc \ub3d9\uc77c\ud55c \ud615\ud0dc\uc758 \uc5f0\uc0b0\uc744 \ub9d0\ud55c\ub2e4. \uc989 X^2\uc758 \ubbf8\ubd84 \uc5f0\uc0b0\uc744 \uc218\ud589\ud558\uba70 \uadf8 \uacb0\uacfc\uac00 2x\ub77c\ub294 \ud615\ud0dc\ub85c \ucd9c\ub825\ub41c\ub2e4. Deep learning\ub4f1\uc5d0 \ub9ce\uc774 \uc0ac\uc6a9\ub418\ub294 \ud30c\uc774\uc36c\uc758 theano package\ub098 Tensorflow \ud328\ud0a4\uc9c0\ub3c4 \uae30\uc6b8\uae30 \ud568\uc218 \uacc4\uc0b0\uc744 \uc704\ud574 \uc774\ub7ec\ud55c \uc2ec\ubcfc\ub9ad \uc5f0\uc0b0 \uae30\ub2a5\uc744 \uac16\ucd94\uace0 \uc788\ub2e4.\n\n\n```python\nimport sympy\n# Jupyter \ub178\ud2b8\ubd81\uc5d0\uc11c \uc218\ud559\uc2dd\uc758 LaTex \ud45c\ud604\uc744 \uc704\ud574 \ud544\uc694\ud568\nsympy.init_printing(use_latex='mathjax')\n```\n\n- \uc2ec\ubcfc\ub9ad \uc5f0\uc0b0\uc5d0\uc11c \uc0ac\uc6a9\ud558\ub294 \uc2ec\ubcfc\ub9ad \ubcc0\uc218(symbolic variable)\ub294 \uc77c\ubc18 \ud504\ub85c\uadf8\ub798\ubc0d\uc5d0\uc11c \uc0ac\uc6a9\ud558\ub294 \ubcc0\uc218\uc640 \ub2e4\ub974\ub2e4 \uc77c\ubc18 \ud504\ub85c\uadf8\ub798\ubc0d\uc5d0\uc11c \uc0ac\uc6a9\ud558\ub294 \ubcc0\uc218\ub294 \uc774\ubbf8 \uba54\ubaa8\ub9ac\uc5d0 \uc50c\uc5ec\uc788\ub294 \uc5b4\ub5a4 \uc22b\uc790\ub97c \uae30\ud638\ub85c \uc4f4 \uac83\uc5d0 \uc9c0\ub098\uce58\uc9c0 \uc54a\uc9c0\ub9cc \uc2ec\ubcfc\ub9ad \ubcc0\uc218\ub294 \uc544\ubb34\ub7f0 \uc22b\uc790\ub3c4 \ub300\uc785\uc774 \ub418\uc5b4 \uc788\uc9c0 \uc54a\ub2e4. \ub530\ub77c\uc11c x^2\uc758 \ubbf8\ubd84 \uc5f0\uc0b0\uc744 \uc218\ud589\ud558\uae30 \uc704\ud574\uc11c\ub294 \uc6b0\uc120 Sympy\uc758 symbols\uba85\ub839\uc744 \uc0ac\uc6a9\ud558\uc5ec x\ub77c\ub294 \uae30\ud638\uac00 \ub2e8\uc21c\ud55c \uc22b\uc790\ub098 \ubca1\ud130 \ubcc0\uc218\uac00 \uc544\ub2cc \uc2ec\ubcfc(symbol)\uc784\uc744 \uc54c\ub824\uc8fc\uc5b4\uc57c \ud55c\ub2e4. \uc774\ub807\uac8c \uc815\uc758\ub41c \uc2ec\ubcfc \ubcc0\uc218 symbol\ud074\ub798\uc2a4 \uc790\ub8cc\ud615\uc774 \ub41c\ub2e4.\n\n\n```python\nx = sympy.symbols('x')\nx\n```\n\n\n\n\n$$x$$\n\n\n\n\n```python\ntype(x)\n```\n\n\n\n\n sympy.core.symbol.Symbol\n\n\n\n- \uc77c\ub2e8 \uc2ec\ubcfc \ubcc0\uc218\ub97c \uc815\uc758\ud558\uba74 \uc774\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \ud568\uc218\ub97c \uc815\uc758\ud55c\ub2e4. \uc774 \ub54c\ub294 \uc218\ud559 \ud568\uc218\ub294 Sympy\uc804\uc6a9 \ud568\uc218\ub97c \uc0ac\uc6a9\ud574\uc57c \ud55c\ub2e4.\n\n\n```python\nf = x * sympy.exp(x)\nf\n```\n\n\n\n\n$$x e^{x}$$\n\n\n\n- \ud568\uc218\uac00 \uc815\uc758\ub418\uba74 diff\ud568\uc218\ub85c \ubbf8\ubd84\uc744 \ud560 \uc218 \uc788\ub2e4. \ub610\ud55c simplify\ud568\uc218\ub97c \uc368\uc11c \uc18c\uc778\uc218 \ubd84\ud574 \ub4f1\uc744 \ud1b5\ud55c \uc218\uc2dd \uc815\ub9ac\uac00 \uac00\ub2a5\ud558\ub2e4\n\n\n```python\nsympy.diff(f)\n```\n\n\n\n\n$$x e^{x} + e^{x}$$\n\n\n\n\n```python\nsympy.simplify(sympy.diff(f))\n```\n\n\n\n\n$$\\left(x + 1\\right) e^{x}$$\n\n\n\n- \ud3b8\ubbf8\ubd84\uc744 \ud558\ub294 \uacbd\uc6b0\uc5d0\ub294 \uc5b4\ub5a4 \ubcc0\uc218\ub85c \ubbf8\ubd84\ud558\ub294\uc9c0\ub97c diff\ud568\uc218\ub85c \uba85\uc2dc\ud574\uc57c \ud55c\ub2e4. Symbols\uba85\ub839\uc744 \uc0ac\uc6a9\ud560 \ub584\ub294 \uc778\uc218\ub85c \uc8fc\ub294 \ubb38\uc790\uc5f4\uc5d0 \uc5ec\ub7ec\uac1c\uc758 \uc2ec\ubcfc \ubcc0\uc218\ub97c \ub3d9\uc2dc\uc5d0 \ub123\uc744 \uc218\ub3c4 \uc788\ub2e4.\n\n\n```python\nx, y = sympy.symbols('x y')\nf = x ** 2 + 4 * x * y + 4 * y ** 2\nf\n```\n\n\n\n\n$$x^{2} + 4 x y + 4 y^{2}$$\n\n\n\n\n```python\nsympy.diff(f, x)\n```\n\n\n\n\n$$2 x + 4 y$$\n\n\n\n\n```python\nsympy.diff(f, y)\n```\n\n\n\n\n$$4 x + 8 y$$\n\n\n\n- \uc0c1\uc218 \uc2ec\ubcfc\uc744 \ud3ec\ud568\ud558\ub294 \ud568\uc218\ub97c \ubbf8\ubd84\ud558\ub294 \uacbd\uc6b0, Sympy\ub294 \uc5b4\ub5a4 \uc2ec\ubcfc\uc774 \uc0c1\uc218\uc774\uace0 \uc5b4\ub5a4 \uc2ec\ubcfc\uc774 \ubcc0\uc218\uc778\uc9c0 \uc54c \uc218 \uc5c6\uae30 \ub54c\ubb38\uc5d0 \ud3b8\ubbf8\ubd84\uc778 \uac83\ucc98\ub7fc \uc785\ub825 \ubcc0\uc218\ub97c \uc9c0\uc815\ud574\uc57c \ud55c\ub2e4.\n\n\n```python\nx, mu, sigma = sympy.symbols('x mu sigma')\nf = sympy.exp((-mu + x) ** 2 / sigma ** 2)\nf\n```\n\n\n\n\n$$e^{\\frac{1}{\\sigma^{2}} \\left(- \\mu + x\\right)^{2}}$$\n\n\n\n\n```python\nsympy.diff(f, x)\n```\n\n\n\n\n$$\\frac{1}{\\sigma^{2}} \\left(- 2 \\mu + 2 x\\right) e^{\\frac{1}{\\sigma^{2}} \\left(- \\mu + x\\right)^{2}}$$\n\n\n\n\n```python\nsympy.simplify(sympy.diff(f, x))\n```\n\n\n\n\n$$\\frac{2}{\\sigma^{2}} \\left(- \\mu + x\\right) e^{\\frac{1}{\\sigma^{2}} \\left(\\mu - x\\right)^{2}}$$\n\n\n\n- \uc774\ucc28 \ub3c4\ud568\uc218\ub294 \ub2e4\uc74c\ucc98\ub7fc \uad6c\ud55c\ub2e4.\n\n\n```python\nsympy.diff(f, x, x)\n```\n\n\n\n\n$$\\frac{2}{\\sigma^{2}} \\left(1 + \\frac{2}{\\sigma^{2}} \\left(\\mu - x\\right)^{2}\\right) e^{\\frac{1}{\\sigma^{2}} \\left(\\mu - x\\right)^{2}}$$\n\n\n\n#### \uc5f0\uc2b5 \ubb38\uc81c 5\n- \ub2e4\uc74c \ud568\uc218\ub97c \ubbf8\ubd84\ud55c \ub3c4\ud568\uc218\ub97c sympy\ub97c \uc0ac\uc6a9\ud558\uc5ec \uad6c\ud558\ub77c. \uc5ec\uae30\uc11c k, a, b\ub294 \ubcc0\uc218\uac00 \uc544\ub2c8\ub77c \uc0c1\uc218\uc774\ub2e4. \n- 1. f(x) = x^3 - 1\n- 2. f(x) = log(x^2 - 3k)\n- 3. f(x) = exp(ax^b)\n\n\n```python\n# 1. f(x) = x^3 - 1\nimport sympy\nsympy.init_printing(use_latex='mathjax') # \uae30\ubcf8 \uc124\uc815\uac12\nx = sympy.symbols('x')\nf = x ** 3 - 1\nsympy.diff(f, x)\n```\n\n\n\n\n$$3 x^{2}$$\n\n\n\n\n```python\n# 2. f(x) = log(x^2 - 3k)\nimport sympy\nsympy.init_printing(use_latex='mathjax')\nx, k = sympy.symbols('x k')\nf = sympy.log(x ** 2 - 3 * k)\nsympy.diff(f, x)\n```\n\n\n\n\n$$\\frac{2 x}{- 3 k + x^{2}}$$\n\n\n\n\n```python\n# 3. f(x) = exp(ax^b)\nimport sympy\nsympy.init_printing(use_latex='mathjax')\nx, a, b = sympy.symbols('x a b')\nf = sympy.exp(a*x**b)\nsympy.simplify(sympy.diff(f, x))\n```\n\n\n\n\n$$a b x^{b - 1} e^{a x^{b}}$$\n\n\n\n\n```python\n# \uc5f0\uc2b5\ubb38\uc81c 6\n# \ub2e4\uc74c \ud568\uc218\uc5d0 \ub300\ud55c 1\ucc28, 2\ucc28 \ud3b8\ubbf8\ubd84 Sympy\ub97c \uad6c\ud558\ub77c\n# fx, fy, fxx, fxy, fyx, fyy\ub97c Sympy\ub85c \ud65c\uc6a9\n# f(x, y) = exp(x^2 + 2y^2)\n\n# 1) fx\nimport sympy\nsympy.init_printing(use_latex='mathjax')\nx, y = sympy.symbols('x y')\nf = sympy.exp(x ** 2 + 2 * y ** 2)\nf\n```\n\n\n\n\n$$e^{x^{2} + 2 y^{2}}$$\n\n\n\n\n```python\n#1 fx\nsympy.diff(f, x)\n```\n\n\n\n\n$$2 x e^{x^{2} + 2 y^{2}}$$\n\n\n\n\n```python\n#2 fy\nsympy.diff(f, y)\n```\n\n\n\n\n$$4 y e^{x^{2} + 2 y^{2}}$$\n\n\n\n\n```python\n#3 fxx\nsympy.diff(f, x, x)\n```\n\n\n\n\n$$2 \\left(2 x^{2} + 1\\right) e^{x^{2} + 2 y^{2}}$$\n\n\n\n\n```python\n#4 fxy\nsympy.diff(f, x, y)\n```\n\n\n\n\n$$8 x y e^{x^{2} + 2 y^{2}}$$\n\n\n\n\n```python\n#5 fyx\nsympy.diff(f, y, x)\n```\n\n\n\n\n$$8 x y e^{x^{2} + 2 y^{2}}$$\n\n\n\n\n```python\n#6 fyy\nsympy.diff(f, y, y)\n```\n\n\n\n\n$$4 \\left(4 y^{2} + 1\\right) e^{x^{2} + 2 y^{2}}$$\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "25bf3d93b8e38713f6ad89542496d6f0d5135437", "size": 139038, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "2018_05_26_Sympy_Differentiation_review.ipynb", "max_stars_repo_name": "jaykim-asset/datascience_review", "max_stars_repo_head_hexsha": "c55782f5d4226e179088346da399e299433c6ca6", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2018-05-30T10:39:47.000Z", "max_stars_repo_stars_event_max_datetime": "2018-11-10T15:39:53.000Z", "max_issues_repo_path": "2018_05_26_Sympy_Differentiation_review.ipynb", "max_issues_repo_name": "jaykim-asset/datascience_review", "max_issues_repo_head_hexsha": "c55782f5d4226e179088346da399e299433c6ca6", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "2018_05_26_Sympy_Differentiation_review.ipynb", "max_forks_repo_name": "jaykim-asset/datascience_review", "max_forks_repo_head_hexsha": "c55782f5d4226e179088346da399e299433c6ca6", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 97.8451794511, "max_line_length": 47788, "alphanum_fraction": 0.8454882838, "converted": true, "num_tokens": 7395, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.7217432122827968, "lm_q1q2_score": 0.4331818743872016}} {"text": "```python\n# This code block is for automatic testing purposes, please ignore.\ntry:\n import openfermion\nexcept:\n import os\n os.chdir('../src/')\n```\n\n# Introduction to OpenFermion\nNote that all the examples below must be run sequentially within a section.\n\n## Initializing the FermionOperator data structure\n\nFermionic systems are often treated in second quantization where arbitrary operators can be expressed using the fermionic creation and annihilation operators, $a^\\dagger_k$ and $a_k$. The fermionic ladder operators play a similar role to their qubit ladder operator counterparts, $\\sigma^+_k$ and $\\sigma^-_k$ but are distinguished by the canonical fermionic anticommutation relations, $\\{a^\\dagger_i, a^\\dagger_j\\} = \\{a_i, a_j\\} = 0$ and $\\{a_i, a_j^\\dagger\\} = \\delta_{ij}$. Any weighted sums of products of these operators are represented with the FermionOperator data structure in OpenFermion. The following are examples of valid FermionOperators:\n\n$$\n\\begin{align}\n& a_1 \\nonumber \\\\\n& 1.7 a^\\dagger_3 \\nonumber \\\\\n&-1.7 \\, a^\\dagger_3 a_1 \\nonumber \\\\\n&(1 + 2i) \\, a^\\dagger_4 a^\\dagger_3 a_9 a_1 \\nonumber \\\\\n&(1 + 2i) \\, a^\\dagger_4 a^\\dagger_3 a_9 a_1 - 1.7 \\, a^\\dagger_3 a_1 \\nonumber\n\\end{align}\n$$\n\nThe FermionOperator class is contained in $\\textrm{ops/_fermion_operators.py}$. In order to support fast addition of FermionOperator instances, the class is implemented as hash table (python dictionary). The keys of the dictionary encode the strings of ladder operators and values of the dictionary store the coefficients. The strings of ladder operators are encoded as a tuple of 2-tuples which we refer to as the \"terms tuple\". Each ladder operator is represented by a 2-tuple. The first element of the 2-tuple is an int indicating the tensor factor on which the ladder operator acts. The second element of the 2-tuple is Boole: 1 represents raising and 0 represents lowering. For instance, $a^\\dagger_8$ is represented in a 2-tuple as $(8, 1)$. Note that indices start at 0 and the identity operator is an empty list. Below we give some examples of operators and their terms tuple:\n\n$$\n\\begin{align}\nI & \\mapsto () \\nonumber \\\\\na_1 & \\mapsto ((1, 0),) \\nonumber \\\\\na^\\dagger_3 & \\mapsto ((3, 1),) \\nonumber \\\\\na^\\dagger_3 a_1 & \\mapsto ((3, 1), (1, 0)) \\nonumber \\\\\na^\\dagger_4 a^\\dagger_3 a_9 a_1 & \\mapsto ((4, 1), (3, 1), (9, 0), (1, 0)) \\nonumber\n\\end{align}\n$$\n\nNote that when initializing a single ladder operator one should be careful to add the comma after the inner pair. This is because in python ((1, 2)) = (1, 2) whereas ((1, 2),) = ((1, 2),). The \"terms tuple\" is usually convenient when one wishes to initialize a term as part of a coded routine. However, the terms tuple is not particularly intuitive. Accordingly, OpenFermion also supports another user-friendly, string notation below. This representation is rendered when calling \"print\" on a FermionOperator.\n\n$$\n\\begin{align}\nI & \\mapsto \\textrm{\"\"} \\nonumber \\\\\na_1 & \\mapsto \\textrm{\"1\"} \\nonumber \\\\\na^\\dagger_3 & \\mapsto \\textrm{\"3^\"} \\nonumber \\\\\na^\\dagger_3 a_1 & \\mapsto \\textrm{\"3^}\\;\\textrm{1\"} \\nonumber \\\\\na^\\dagger_4 a^\\dagger_3 a_9 a_1 & \\mapsto \\textrm{\"4^}\\;\\textrm{3^}\\;\\textrm{9}\\;\\textrm{1\"} \\nonumber\n\\end{align}\n$$\n\nLet's initialize our first term! We do it two different ways below.\n\n\n```python\nfrom openfermion.ops import FermionOperator\n\nmy_term = FermionOperator(((3, 1), (1, 0)))\nprint(my_term)\n\nmy_term = FermionOperator('3^ 1')\nprint(my_term)\n```\n\n 1.0 [3^ 1]\n 1.0 [3^ 1]\n\n\nThe preferred way to specify the coefficient in openfermion is to provide an optional coefficient argument. If not provided, the coefficient defaults to 1. In the code below, the first method is preferred. The multiplication in the second method actually creates a copy of the term, which introduces some additional cost. All inplace operands (such as +=) modify classes whereas binary operands such as + create copies. Important caveats are that the empty tuple FermionOperator(()) and the empty string FermionOperator('') initializes identity. The empty initializer FermionOperator() initializes the zero operator.\n\n\n```python\ngood_way_to_initialize = FermionOperator('3^ 1', -1.7)\nprint(good_way_to_initialize)\n\nbad_way_to_initialize = -1.7 * FermionOperator('3^ 1')\nprint(bad_way_to_initialize)\n\nidentity = FermionOperator('')\nprint(identity)\n\nzero_operator = FermionOperator()\nprint(zero_operator)\n```\n\n -1.7 [3^ 1]\n -1.7 [3^ 1]\n 1.0 []\n 0\n\n\nNote that FermionOperator has only one attribute: .terms. This attribute is the dictionary which stores the term tuples.\n\n\n```python\nmy_operator = FermionOperator('4^ 1^ 3 9', 1. + 2.j)\nprint(my_operator)\nprint(my_operator.terms)\n```\n\n (1+2j) [4^ 1^ 3 9]\n {((4, 1), (1, 1), (3, 0), (9, 0)): (1+2j)}\n\n\n## Manipulating the FermionOperator data structure\nSo far we have explained how to initialize a single FermionOperator such as $-1.7 \\, a^\\dagger_3 a_1$. However, in general we will want to represent sums of these operators such as $(1 + 2i) \\, a^\\dagger_4 a^\\dagger_3 a_9 a_1 - 1.7 \\, a^\\dagger_3 a_1$. To do this, just add together two FermionOperators! We demonstrate below.\n\n\n```python\nfrom openfermion.ops import FermionOperator\n\nterm_1 = FermionOperator('4^ 3^ 9 1', 1. + 2.j)\nterm_2 = FermionOperator('3^ 1', -1.7)\nmy_operator = term_1 + term_2\nprint(my_operator)\n\nmy_operator = FermionOperator('4^ 3^ 9 1', 1. + 2.j)\nterm_2 = FermionOperator('3^ 1', -1.7)\nmy_operator += term_2\nprint('')\nprint(my_operator)\n```\n\n (1+2j) [4^ 3^ 9 1] +\n -1.7 [3^ 1]\n \n (1+2j) [4^ 3^ 9 1] +\n -1.7 [3^ 1]\n\n\nThe print function prints each term in the operator on a different line. Note that the line my_operator = term_1 + term_2 creates a new object, which involves a copy of term_1 and term_2. The second block of code uses the inplace method +=, which is more efficient. This is especially important when trying to construct a very large FermionOperator. FermionOperators also support a wide range of builtins including, str(), repr(), ==, !=, *=, *, /, /=, +, +=, -, -=, - and **. Note that since FermionOperators involve floats, == and != check for (in)equality up to numerical precision. We demonstrate some of these methods below.\n\n\n```python\nterm_1 = FermionOperator('4^ 3^ 9 1', 1. + 2.j)\nterm_2 = FermionOperator('3^ 1', -1.7)\n\nmy_operator = term_1 - 33. * term_2\nprint(my_operator)\n\nmy_operator *= 3.17 * (term_2 + term_1) ** 2\nprint('')\nprint(my_operator)\n\nprint('')\nprint(term_2 ** 3)\n\nprint('')\nprint(term_1 == 2.*term_1 - term_1)\nprint(term_1 == my_operator)\n```\n\n (1+2j) [4^ 3^ 9 1] +\n 56.1 [3^ 1]\n \n (9.161299999999999+18.322599999999998j) [4^ 3^ 9 1 3^ 1 3^ 1] +\n (16.166999999999998-21.555999999999997j) [4^ 3^ 9 1 3^ 1 4^ 3^ 9 1] +\n (16.166999999999998-21.555999999999997j) [4^ 3^ 9 1 4^ 3^ 9 1 3^ 1] +\n (-34.87-6.34j) [4^ 3^ 9 1 4^ 3^ 9 1 4^ 3^ 9 1] +\n 513.9489299999999 [3^ 1 3^ 1 3^ 1] +\n (-302.32289999999995-604.6457999999999j) [3^ 1 3^ 1 4^ 3^ 9 1] +\n (-302.32289999999995-604.6457999999999j) [3^ 1 4^ 3^ 9 1 3^ 1] +\n (-533.511+711.348j) [3^ 1 4^ 3^ 9 1 4^ 3^ 9 1]\n \n -4.912999999999999 [3^ 1 3^ 1 3^ 1]\n \n True\n False\n\n\nAdditionally, there are a variety of methods that act on the FermionOperator data structure. We demonstrate a small subset of those methods here.\n\n\n```python\nfrom openfermion.ops import normal_ordered\nfrom openfermion.utils import commutator, count_qubits, hermitian_conjugated\n\n# Get the Hermitian conjugate of a FermionOperator, count its qubit, check if it is normal-ordered.\nterm_1 = FermionOperator('4^ 3 3^', 1. + 2.j)\nprint(hermitian_conjugated(term_1))\nprint(term_1.is_normal_ordered())\nprint(count_qubits(term_1))\n\n# Normal order the term.\nterm_2 = normal_ordered(term_1)\nprint('')\nprint(term_2)\nprint(term_2.is_normal_ordered())\n\n# Compute a commutator of the terms.\nprint('')\nprint(commutator(term_1, term_2))\n```\n\n (1-2j) [3 3^ 4]\n False\n 5\n \n (1+2j) [4^] +\n (-1-2j) [4^ 3^ 3]\n True\n \n (-3+4j) [4^ 3 3^ 4^] +\n (3-4j) [4^ 3 3^ 4^ 3^ 3] +\n (3-4j) [4^ 4^ 3 3^] +\n (-3+4j) [4^ 3^ 3 4^ 3 3^]\n\n\n## The QubitOperator data structure\nThe QubitOperator data structure is another essential part of openfermion. As the name suggests, QubitOperator is used to store qubit operators in almost exactly the same way that FermionOperator is used to store fermion operators. For instance $X_0 Z_3 Y_4$ is a QubitOperator. The internal representation of this as a terms tuple would be $((0, \\textrm{\"X\"}), (3, \\textrm{\"Z\"}), (4, \\textrm{\"Y\"}))$. Note that one important difference between QubitOperator and FermionOperator is that the terms in QubitOperator are always sorted in order of tensor factor. In some cases, this enables faster manipulation. We initialize some QubitOperators below.\n\n\n```python\nfrom openfermion.ops import QubitOperator\n\nmy_first_qubit_operator = QubitOperator('X1 Y2 Z3')\nprint(my_first_qubit_operator)\nprint(my_first_qubit_operator.terms)\n\noperator_2 = QubitOperator('X3 Z4', 3.17)\noperator_2 -= 77. * my_first_qubit_operator\nprint('')\nprint(operator_2)\n```\n\n 1.0 X1 Y2 Z3\n {((1, 'X'), (2, 'Y'), (3, 'Z')): 1.0}\n \n 3.17 X3 Z4 +\n -77.0 X1 Y2 Z3\n\n\n## Jordan-Wigner and Bravyi-Kitaev\nopenfermion provides functions for mapping FermionOperators to QubitOperators.\n\n\n```python\nfrom openfermion.ops import FermionOperator\nfrom openfermion.transforms import jordan_wigner, bravyi_kitaev\nfrom openfermion.utils import eigenspectrum, hermitian_conjugated\n\n# Initialize an operator.\nfermion_operator = FermionOperator('2^ 0', 3.17)\nfermion_operator += hermitian_conjugated(fermion_operator)\nprint(fermion_operator)\n\n# Transform to qubits under the Jordan-Wigner transformation and print its spectrum.\njw_operator = jordan_wigner(fermion_operator)\nprint('')\nprint(jw_operator)\njw_spectrum = eigenspectrum(jw_operator)\nprint(jw_spectrum)\n\n# Transform to qubits under the Bravyi-Kitaev transformation and print its spectrum.\nbk_operator = bravyi_kitaev(fermion_operator)\nprint('')\nprint(bk_operator)\nbk_spectrum = eigenspectrum(bk_operator)\nprint(bk_spectrum)\n```\n\n 3.17 [2^ 0] +\n 3.17 [0^ 2]\n \n (1.585+0j) X0 Z1 X2 +\n (1.585+0j) Y0 Z1 Y2\n [-3.17 -3.17 0. 0. 0. 0. 3.17 3.17]\n \n (-1.585+0j) Y0 Y1 +\n (-1.585+0j) X0 X1 Z2\n [-3.17 -3.17 0. 0. 0. 0. 3.17 3.17]\n\n\nWe see that despite the different representation, these operators are iso-spectral. We can also apply the Jordan-Wigner transform in reverse to map arbitrary QubitOperators to FermionOperators. Note that we also demonstrate the .compress() method (a method on both FermionOperators and QubitOperators) which removes zero entries.\n\n\n```python\nfrom openfermion.transforms import reverse_jordan_wigner\n\n# Initialize QubitOperator.\nmy_operator = QubitOperator('X0 Y1 Z2', 88.)\nmy_operator += QubitOperator('Z1 Z4', 3.17)\nprint(my_operator)\n\n# Map QubitOperator to a FermionOperator.\nmapped_operator = reverse_jordan_wigner(my_operator)\nprint('')\nprint(mapped_operator)\n\n# Map the operator back to qubits and make sure it is the same.\nback_to_normal = jordan_wigner(mapped_operator)\nback_to_normal.compress()\nprint('')\nprint(back_to_normal)\n```\n\n 88.0 X0 Y1 Z2 +\n 3.17 Z1 Z4\n \n -88j [1^ 0^] +\n 88j [1^ 0] +\n 88j [1 0^] +\n -88j [1 0] +\n 176j [2^ 2 1^ 0^] +\n -176j [2^ 2 1^ 0] +\n -176j [2^ 2 1 0^] +\n 176j [2^ 2 1 0] +\n 3.17 [] +\n -6.34 [1^ 1] +\n -6.34 [4^ 4] +\n 12.68 [4^ 4 1^ 1]\n \n 88.0 X0 Y1 Z2 +\n 3.17 Z1 Z4\n\n\n## Sparse matrices and the Hubbard model\nOften, one would like to obtain a sparse matrix representation of an operator which can be analyzed numerically. There is code in both openfermion.transforms and openfermion.utils which facilitates this. The function get_sparse_operator converts either a FermionOperator, a QubitOperator or other more advanced classes such as InteractionOperator to a scipy.sparse.csc matrix. There are numerous functions in openfermion.utils which one can call on the sparse operators such as \"get_gap\", \"get_hartree_fock_state\", \"get_ground_state\", etc. We show this off by computing the ground state energy of the Hubbard model. To do that, we use code from the openfermion.hamiltonians module which constructs lattice models of fermions such as Hubbard models.\n\n\n```python\nfrom openfermion.hamiltonians import fermi_hubbard\nfrom openfermion.transforms import get_sparse_operator, jordan_wigner\nfrom openfermion.utils import get_ground_state\n\n# Set model.\nx_dimension = 2\ny_dimension = 2\ntunneling = 2.\ncoulomb = 1.\nmagnetic_field = 0.5\nchemical_potential = 0.25\nperiodic = 1\nspinless = 1\n\n# Get fermion operator.\nhubbard_model = fermi_hubbard(\n x_dimension, y_dimension, tunneling, coulomb, chemical_potential,\n magnetic_field, periodic, spinless)\nprint(hubbard_model)\n\n# Get qubit operator under Jordan-Wigner.\njw_hamiltonian = jordan_wigner(hubbard_model)\njw_hamiltonian.compress()\nprint('')\nprint(jw_hamiltonian)\n\n# Get scipy.sparse.csc representation.\nsparse_operator = get_sparse_operator(hubbard_model)\nprint('')\nprint(sparse_operator)\nprint('\\nEnergy of the model is {} in units of T and J.'.format(\n get_ground_state(sparse_operator)[0]))\n```\n\n -0.25 [0^ 0] +\n 1.0 [0^ 0 1^ 1] +\n -2.0 [0^ 1] +\n -2.0 [1^ 0] +\n 1.0 [0^ 0 2^ 2] +\n -2.0 [0^ 2] +\n -2.0 [2^ 0] +\n -0.25 [1^ 1] +\n 1.0 [1^ 1 3^ 3] +\n -2.0 [1^ 3] +\n -2.0 [3^ 1] +\n -0.25 [2^ 2] +\n 1.0 [2^ 2 3^ 3] +\n -2.0 [2^ 3] +\n -2.0 [3^ 2] +\n -0.25 [3^ 3]\n \n 0.5 I +\n -0.375 Z0 +\n -0.375 Z1 +\n 0.25 Z0 Z1 +\n -1.0 Y0 Y1 +\n -1.0 X0 X1 +\n -0.375 Z2 +\n 0.25 Z0 Z2 +\n -1.0 Y0 Z1 Y2 +\n -1.0 X0 Z1 X2 +\n -0.375 Z3 +\n 0.25 Z1 Z3 +\n -1.0 Y1 Z2 Y3 +\n -1.0 X1 Z2 X3 +\n 0.25 Z2 Z3 +\n -1.0 Y2 Y3 +\n -1.0 X2 X3\n \n (1, 1)\t(-0.25+0j)\n (2, 1)\t(-2+0j)\n (4, 1)\t(-2+0j)\n (1, 2)\t(-2+0j)\n (2, 2)\t(-0.25+0j)\n (8, 2)\t(-2+0j)\n (3, 3)\t(0.5+0j)\n (6, 3)\t(2+0j)\n (9, 3)\t(-2+0j)\n (1, 4)\t(-2+0j)\n (4, 4)\t(-0.25+0j)\n (8, 4)\t(-2+0j)\n (5, 5)\t(0.5+0j)\n (6, 5)\t(-2+0j)\n (9, 5)\t(-2+0j)\n (3, 6)\t(2+0j)\n (5, 6)\t(-2+0j)\n (6, 6)\t(-0.5+0j)\n (10, 6)\t(-2+0j)\n (12, 6)\t(2+0j)\n (7, 7)\t(1.25+0j)\n (11, 7)\t(-2+0j)\n (13, 7)\t(2+0j)\n (2, 8)\t(-2+0j)\n (4, 8)\t(-2+0j)\n (8, 8)\t(-0.25+0j)\n (3, 9)\t(-2+0j)\n (5, 9)\t(-2+0j)\n (9, 9)\t(-0.5+0j)\n (10, 9)\t(-2+0j)\n (12, 9)\t(-2+0j)\n (6, 10)\t(-2+0j)\n (9, 10)\t(-2+0j)\n (10, 10)\t(0.5+0j)\n (7, 11)\t(-2+0j)\n (11, 11)\t(1.25+0j)\n (14, 11)\t(2+0j)\n (6, 12)\t(2+0j)\n (9, 12)\t(-2+0j)\n (12, 12)\t(0.5+0j)\n (7, 13)\t(2+0j)\n (13, 13)\t(1.25+0j)\n (14, 13)\t(-2+0j)\n (11, 14)\t(2+0j)\n (13, 14)\t(-2+0j)\n (14, 14)\t(1.25+0j)\n (15, 15)\t(3+0j)\n \n Energy of the model is -4.250000000000004 in units of T and J.\n\n\n## Hamiltonians in the plane wave basis\nA user can write plugins to openfermion which allow for the use of, e.g., third-party electronic structure package to compute molecular orbitals, Hamiltonians, energies, reduced density matrices, coupled cluster amplitudes, etc using Gaussian basis sets. We may provide scripts which interface between such packages and openfermion in future but do not discuss them in this tutorial.\n\nWhen using simpler basis sets such as plane waves, these packages are not needed. openfermion comes with code which computes Hamiltonians in the plane wave basis. Note that when using plane waves, one is working with the periodized Coulomb operator, best suited for condensed phase calculations such as studying the electronic structure of a solid. To obtain these Hamiltonians one must choose to study the system without a spin degree of freedom (spinless), one must the specify dimension in which the calculation is performed (n_dimensions, usually 3), one must specify how many plane waves are in each dimension (grid_length) and one must specify the length scale of the plane wave harmonics in each dimension (length_scale) and also the locations and charges of the nuclei. One can generate these models with plane_wave_hamiltonian() found in openfermion.hamiltonians. For simplicity, below we compute the Hamiltonian in the case of zero external charge (corresponding to the uniform electron gas, aka jellium). We also demonstrate that one can transform the plane wave Hamiltonian using a Fourier transform without effecting the spectrum of the operator.\n\n\n```python\nfrom openfermion.hamiltonians import jellium_model\nfrom openfermion.utils import eigenspectrum, fourier_transform, Grid\nfrom openfermion.transforms import jordan_wigner\n\n# Let's look at a very small model of jellium in 1D.\ngrid = Grid(dimensions=1, length=3, scale=1.0)\nspinless = True\n\n# Get the momentum Hamiltonian.\nmomentum_hamiltonian = jellium_model(grid, spinless)\nmomentum_qubit_operator = jordan_wigner(momentum_hamiltonian)\nmomentum_qubit_operator.compress()\nprint(momentum_qubit_operator)\n\n# Fourier transform the Hamiltonian to the position basis.\nposition_hamiltonian = fourier_transform(momentum_hamiltonian, grid, spinless)\nposition_qubit_operator = jordan_wigner(position_hamiltonian)\nposition_qubit_operator.compress()\nprint('')\nprint (position_qubit_operator)\n\n# Check the spectra to make sure these representations are iso-spectral.\nspectral_difference = eigenspectrum(momentum_qubit_operator) - eigenspectrum(position_qubit_operator)\nprint('')\nprint(spectral_difference)\n```\n\n 19.50047638754088 I +\n -9.71044945799746 Z0 +\n 0.15915494309189535 Z1 +\n -9.71044945799746 Z2 +\n -0.07957747154594767 Z0 Z1 +\n -0.07957747154594767 Z1 Z2 +\n -0.07957747154594767 Z0 Z2\n \n 19.500476387540854 I +\n -6.420581324301009 Z0 +\n -3.289868133696451 Y0 Y1 +\n -3.289868133696451 X0 X1 +\n -3.289868133696454 Y0 Z1 Y2 +\n -3.289868133696454 X0 Z1 X2 +\n -6.4205813243010095 Z1 +\n -3.289868133696451 Y1 Y2 +\n -3.289868133696451 X1 X2 +\n -6.420581324301009 Z2 +\n -0.07957747154594766 Z0 Z1 +\n -0.07957747154594763 Z0 Z2 +\n -0.07957747154594766 Z1 Z2\n \n [ 2.74502643e-14 2.66869860e-14 2.84217094e-14 7.10542736e-15\n 2.84217094e-14 2.13162821e-14 2.13162821e-14 0.00000000e+00]\n\n\n## Basics of MolecularData class\n\nData from electronic structure calculations can be saved in an OpenFermion data structure called MolecularData, which makes it easy to access within our library. Often, one would like to analyze a chemical series or look at many different Hamiltonians and sometimes the electronic structure calculations are either expensive to compute or difficult to converge (e.g. one needs to mess around with different types of SCF routines to make things converge). Accordingly, we anticipate that users will want some way to automatically database the results of their electronic structure calculations so that important data (such as the SCF integrals) can be looked up on-the-fly if the user has computed them in the past. OpenFermion supports a data provenance strategy which saves key results of the electronic structure calculation (including pointers to files containing large amounts of data, such as the molecular integrals) in an HDF5 container.\n\nThe MolecularData class stores information about molecules. One initializes a MolecularData object by specifying parameters of a molecule such as its geometry, basis, multiplicity, charge and an optional string describing it. One can also initialize MolecularData simply by providing a string giving a filename where a previous MolecularData object was saved in an HDF5 container. One can save a MolecularData instance by calling the class's .save() method. This automatically saves the instance in a data folder specified during OpenFermion installation. The name of the file is generated automatically from the instance attributes and optionally provided description. Alternatively, a filename can also be provided as an optional input if one wishes to manually name the file.\n\nWhen electronic structure calculations are run, the data files for the molecule can be automatically updated. If one wishes to later use that data they either initialize MolecularData with the instance filename or initialize the instance and then later call the .load() method.\n\nBasis functions are provided to initialization using a string such as \"6-31g\". Geometries can be specified using a simple txt input file (see geometry_from_file function in molecular_data.py) or can be passed using a simple python list format demonstrated below. Atoms are specified using a string for their atomic symbol. Distances should be provided in angstrom. Below we initialize a simple instance of MolecularData without performing any electronic structure calculations.\n\n\n```python\nfrom openfermion.hamiltonians import MolecularData\n\n# Set parameters to make a simple molecule.\ndiatomic_bond_length = .7414\ngeometry = [('H', (0., 0., 0.)), ('H', (0., 0., diatomic_bond_length))]\nbasis = 'sto-3g'\nmultiplicity = 1\ncharge = 0\ndescription = str(diatomic_bond_length)\n\n# Make molecule and print out a few interesting facts about it.\nmolecule = MolecularData(geometry, basis, multiplicity,\n charge, description)\nprint('Molecule has automatically generated name {}'.format(\n molecule.name))\nprint('Information about this molecule would be saved at:\\n{}\\n'.format(\n molecule.filename))\nprint('This molecule has {} atoms and {} electrons.'.format(\n molecule.n_atoms, molecule.n_electrons))\nfor atom, atomic_number in zip(molecule.atoms, molecule.protons):\n print('Contains {} atom, which has {} protons.'.format(\n atom, atomic_number))\n```\n\n Molecule has automatically generated name H2_sto-3g_singlet_0.7414\n Information about this molecule would be saved at:\n /home/kjs/Projects/OpenFermion/src/openfermion/data/H2_sto-3g_singlet_0.7414\n \n This molecule has 2 atoms and 2 electrons.\n Contains H atom, which has 1 protons.\n Contains H atom, which has 1 protons.\n\n\nIf we had previously computed this molecule using an electronic structure package, we can call molecule.load() to populate all sorts of interesting fields in the data structure. Though we make no assumptions about what electronic structure packages users might install, we assume that the calculations are saved in OpenFermion's MolecularData objects. Currently plugins are available for [Psi4](http://psicode.org/) [(OpenFermion-Psi4)](http://github.com/quantumlib/OpenFermion-Psi4) and [PySCF](https://github.com/sunqm/pyscf) [(OpenFermion-PySCF)](http://github.com/quantumlib/OpenFermion-PySCF), and there may be more in the future. For the purposes of this example, we will load data that ships with OpenFermion to make a plot of the energy surface of hydrogen. Note that helper functions to initialize some interesting chemical benchmarks are found in openfermion.utils.\n\n\n```python\n# Set molecule parameters.\nbasis = 'sto-3g'\nmultiplicity = 1\nbond_length_interval = 0.1\nn_points = 25\n\n# Generate molecule at different bond lengths.\nhf_energies = []\nfci_energies = []\nbond_lengths = []\nfor point in range(3, n_points + 1):\n bond_length = bond_length_interval * point\n bond_lengths += [bond_length]\n description = str(round(bond_length,2))\n print(description)\n geometry = [('H', (0., 0., 0.)), ('H', (0., 0., bond_length))]\n molecule = MolecularData(\n geometry, basis, multiplicity, description=description)\n \n # Load data.\n molecule.load()\n\n # Print out some results of calculation.\n print('\\nAt bond length of {} angstrom, molecular hydrogen has:'.format(\n bond_length))\n print('Hartree-Fock energy of {} Hartree.'.format(molecule.hf_energy))\n print('MP2 energy of {} Hartree.'.format(molecule.mp2_energy))\n print('FCI energy of {} Hartree.'.format(molecule.fci_energy))\n print('Nuclear repulsion energy between protons is {} Hartree.'.format(\n molecule.nuclear_repulsion))\n for orbital in range(molecule.n_orbitals):\n print('Spatial orbital {} has energy of {} Hartree.'.format(\n orbital, molecule.orbital_energies[orbital]))\n hf_energies += [molecule.hf_energy]\n fci_energies += [molecule.fci_energy]\n\n# Plot.\nimport matplotlib.pyplot as plt\n%matplotlib inline\n\nplt.figure(0)\nplt.plot(bond_lengths, fci_energies, 'x-')\nplt.plot(bond_lengths, hf_energies, 'o-')\nplt.ylabel('Energy in Hartree')\nplt.xlabel('Bond length in angstrom')\nplt.show()\n```\n\n## InteractionOperator and InteractionRDM for efficient numerical representations\n\nFermion Hamiltonians can be expressed as $H = h_0 + \\sum_{pq} h_{pq}\\, a^\\dagger_p a_q + \\frac{1}{2} \\sum_{pqrs} h_{pqrs} \\, a^\\dagger_p a^\\dagger_q a_r a_s$ where $h_0$ is a constant shift due to the nuclear repulsion and $h_{pq}$ and $h_{pqrs}$ are the famous molecular integrals. Since fermions interact pairwise, their energy is thus a unique function of the one-particle and two-particle reduced density matrices which are expressed in second quantization as $\\rho_{pq} = \\left \\langle p \\mid a^\\dagger_p a_q \\mid q \\right \\rangle$ and $\\rho_{pqrs} = \\left \\langle pq \\mid a^\\dagger_p a^\\dagger_q a_r a_s \\mid rs \\right \\rangle$, respectively.\n\nBecause the RDMs and molecular Hamiltonians are both compactly represented and manipulated as 2- and 4- index tensors, we can represent them in a particularly efficient form using similar data structures. The InteractionOperator data structure can be initialized for a Hamiltonian by passing the constant $h_0$ (or 0), as well as numpy arrays representing $h_{pq}$ (or $\\rho_{pq}$) and $h_{pqrs}$ (or $\\rho_{pqrs}$). Importantly, InteractionOperators can also be obtained by calling MolecularData.get_molecular_hamiltonian() or by calling the function get_interaction_operator() (found in openfermion.transforms) on a FermionOperator. The InteractionRDM data structure is similar but represents RDMs. For instance, one can get a molecular RDM by calling MolecularData.get_molecular_rdm(). When generating Hamiltonians from the MolecularData class, one can choose to restrict the system to an active space.\n\nThese classes inherit from the same base class, PolynomialTensor. This data structure overloads the slice operator [] so that one can get or set the key attributes of the InteractionOperator: $\\textrm{.constant}$, $\\textrm{.one_body_coefficients}$ and $\\textrm{.two_body_coefficients}$ . For instance, InteractionOperator[(p, 1), (q, 1), (r, 0), (s, 0)] would return $h_{pqrs}$ and InteractionRDM would return $\\rho_{pqrs}$. Importantly, the class supports fast basis transformations using the method PolynomialTensor.rotate_basis(rotation_matrix).\nBut perhaps most importantly, one can map the InteractionOperator to any of the other data structures we've described here.\n\nBelow, we load MolecularData from a saved calculation of LiH. We then obtain an InteractionOperator representation of this system in an active space. We then map that operator to qubits. We then demonstrate that one can rotate the orbital basis of the InteractionOperator using random angles to obtain a totally different operator that is still iso-spectral.\n\n\n```python\nfrom openfermion.hamiltonians import MolecularData\nfrom openfermion.transforms import get_fermion_operator, get_sparse_operator, jordan_wigner\nfrom openfermion.utils import get_ground_state\nimport numpy\nimport scipy\nimport scipy.linalg\n\n# Load saved file for LiH.\ndiatomic_bond_length = 1.45\ngeometry = [('Li', (0., 0., 0.)), ('H', (0., 0., diatomic_bond_length))]\nbasis = 'sto-3g'\nmultiplicity = 1\n\n# Set Hamiltonian parameters.\nactive_space_start = 1\nactive_space_stop = 3\n\n# Generate and populate instance of MolecularData.\nmolecule = MolecularData(geometry, basis, multiplicity, description=\"1.45\")\nmolecule.load()\n\n# Get the Hamiltonian in an active space.\nmolecular_hamiltonian = molecule.get_molecular_hamiltonian(\n occupied_indices=range(active_space_start),\n active_indices=range(active_space_start, active_space_stop))\n\n# Map operator to fermions and qubits.\nfermion_hamiltonian = get_fermion_operator(molecular_hamiltonian)\nqubit_hamiltonian = jordan_wigner(fermion_hamiltonian)\nqubit_hamiltonian.compress()\nprint('The Jordan-Wigner Hamiltonian in canonical basis follows:\\n{}'.format(qubit_hamiltonian))\n\n# Get sparse operator and ground state energy.\nsparse_hamiltonian = get_sparse_operator(qubit_hamiltonian)\nenergy, state = get_ground_state(sparse_hamiltonian)\nprint('Ground state energy before rotation is {} Hartree.\\n'.format(energy))\n\n# Randomly rotate.\nn_orbitals = molecular_hamiltonian.n_qubits // 2\nn_variables = int(n_orbitals * (n_orbitals - 1) / 2)\nnumpy.random.seed(1)\nrandom_angles = numpy.pi * (1. - 2. * numpy.random.rand(n_variables))\nkappa = numpy.zeros((n_orbitals, n_orbitals))\nindex = 0\nfor p in range(n_orbitals):\n for q in range(p + 1, n_orbitals):\n kappa[p, q] = random_angles[index]\n kappa[q, p] = -numpy.conjugate(random_angles[index])\n index += 1\n\n # Build the unitary rotation matrix.\n difference_matrix = kappa + kappa.transpose()\n rotation_matrix = scipy.linalg.expm(kappa)\n\n # Apply the unitary.\n molecular_hamiltonian.rotate_basis(rotation_matrix)\n\n# Get qubit Hamiltonian in rotated basis.\nqubit_hamiltonian = jordan_wigner(molecular_hamiltonian)\nqubit_hamiltonian.compress()\nprint('The Jordan-Wigner Hamiltonian in rotated basis follows:\\n{}'.format(qubit_hamiltonian))\n\n# Get sparse Hamiltonian and energy in rotated basis.\nsparse_hamiltonian = get_sparse_operator(qubit_hamiltonian)\nenergy, state = get_ground_state(sparse_hamiltonian)\nprint('Ground state energy after rotation is {} Hartree.'.format(energy))\n```\n\n The Jordan-Wigner Hamiltonian in canonical basis follows:\n -7.49894690201071 I +\n 0.16199475388004184 Z0 +\n 0.01291078027311749 Y0 Z1 Y2 +\n 0.01291078027311749 X0 Z1 X2 +\n 0.16199475388004186 Z1 +\n 0.012910780273117487 Y1 Z2 Y3 +\n 0.012910780273117487 X1 Z2 X3 +\n -0.013243698330265966 Z2 +\n -0.013243698330265952 Z3 +\n 0.12444770133137588 Z0 Z1 +\n 0.011536413200774975 Y0 Y2 +\n 0.011536413200774975 X0 X2 +\n 0.011536413200774975 Z0 Y1 Z2 Y3 +\n 0.011536413200774975 Z0 X1 Z2 X3 +\n 0.0029329964409502266 Y0 X1 X2 Y3 +\n -0.0029329964409502266 Y0 Y1 X2 X3 +\n -0.0029329964409502266 X0 X1 Y2 Y3 +\n 0.0029329964409502266 X0 Y1 Y2 X3 +\n 0.054130445793298836 Z0 Z2 +\n 0.05706344223424907 Z0 Z3 +\n -0.0013743761078958677 Y0 Z1 Y2 Z3 +\n -0.0013743761078958677 X0 Z1 X2 Z3 +\n 0.05706344223424907 Z1 Z2 +\n -0.0013743761078958677 Y1 Y3 +\n -0.0013743761078958677 X1 X3 +\n 0.054130445793298836 Z1 Z3 +\n 0.08479609543670981 Z2 Z3\n Ground state energy before rotation is -7.8627731630279865 Hartree.\n \n The Jordan-Wigner Hamiltonian in rotated basis follows:\n -7.498946902010708 I +\n 0.04248358003893336 Z0 +\n 0.042483580038933336 Z1 +\n 0.0823812751073221 Z0 Z1 +\n -0.005524724636333855 X0 X2 +\n -0.08262397170394276 X0 Z1 X2 +\n -0.005524724636333855 Y0 Y2 +\n -0.08262397170394276 Y0 Z1 Y2 +\n -0.005524724636333857 Z0 X1 Z2 X3 +\n -0.08262397170394277 X1 Z2 X3 +\n -0.005524724636333857 Z0 Y1 Z2 Y3 +\n -0.08262397170394277 Y1 Z2 Y3 +\n -0.024260054468246924 X0 X1 Y2 Y3 +\n 0.024260054468246924 X0 Y1 Y2 X3 +\n 0.024260054468246924 Y0 X1 X2 Y3 +\n -0.024260054468246924 Y0 Y1 X2 X3 +\n 0.1062674755108427 Z2 +\n 0.054130445793298815 Z0 Z2 +\n 0.1062674755108427 Z3 +\n 0.07839050026154575 Z0 Z3 +\n -0.016734989174013656 X0 Z1 X2 Z3 +\n -0.016734989174013656 Y0 Z1 Y2 Z3 +\n 0.07839050026154575 Z1 Z2 +\n -0.016734989174013656 X1 X3 +\n -0.016734989174013656 Y1 Y3 +\n 0.054130445793298815 Z1 Z3 +\n 0.08420840560617016 Z2 Z3\n Ground state energy after rotation is -7.862773163027971 Hartree.\n\n\n## Quadratic Hamiltonians and Slater determinants\n\nThe general electronic structure Hamiltonian\n$H = h_0 + \\sum_{pq} h_{pq}\\, a^\\dagger_p a_q + \\frac{1}{2} \\sum_{pqrs} h_{pqrs} \\, a^\\dagger_p a^\\dagger_q a_r a_s$ contains terms that act on up to 4 sites, or\nis quartic in the fermionic creation and annihilation operators. However, in many situations\nwe may fruitfully approximate these Hamiltonians by replacing these quartic terms with\nterms that act on at most 2 fermionic sites, or quadratic terms, as in mean-field approximation theory. \nThese Hamiltonians have a number of\nspecial properties one can exploit for efficient simulation and manipulation of the Hamiltonian, thus\nwarranting a special data structure. We refer to Hamiltonians which\nonly contain terms that are quadratic in the fermionic creation and annihilation operators\nas quadratic Hamiltonians, and include the general case of non-particle conserving terms as in\na general Bogoliubov transformation. Eigenstates of quadratic Hamiltonians can be prepared\nefficiently on both a quantum and classical computer, making them amenable to initial guesses for\nmany more challenging problems.\n\nA general quadratic Hamiltonian takes the form\n$$H = \\sum_{p, q} (M_{pq} - \\mu \\delta_{pq}) a^\\dagger_p a_q + \\frac{1}{2} \\sum_{p, q} (\\Delta_{pq} a^\\dagger_p a^\\dagger_q + \\Delta_{pq}^* a_q a_p) + \\text{constant},$$\nwhere $M$ is a Hermitian matrix, $\\Delta$ is an antisymmetric matrix,\n$\\delta_{pq}$ is the Kronecker delta symbol, and $\\mu$ is a chemical\npotential term which we keep separate from $M$ so that we can use it\nto adjust the expectation of the total number of particles.\nIn OpenFermion, quadratic Hamiltonians are conveniently represented and manipulated\nusing the QuadraticHamiltonian class, which stores $M$, $\\Delta$, $\\mu$ and the constant. It is specialized to exploit the properties unique to quadratic Hamiltonians. Like InteractionOperator and InteractionRDM, it inherits from the PolynomialTensor class.\n\nThe BCS mean-field model of superconductivity is a quadratic Hamiltonian. The following code constructs an instance of this model as a FermionOperator, converts it to a QuadraticHamiltonian, and then computes its ground energy:\n\n\n```python\nfrom openfermion.hamiltonians import mean_field_dwave\nfrom openfermion.transforms import get_quadratic_hamiltonian\n\n# Set model.\nx_dimension = 2\ny_dimension = 2\ntunneling = 2.\nsc_gap = 1.\nperiodic = True\n\n# Get FermionOperator.\nmean_field_model = mean_field_dwave(\n x_dimension, y_dimension, tunneling, sc_gap, periodic=periodic)\n\n# Convert to QuadraticHamiltonian\nquadratic_hamiltonian = get_quadratic_hamiltonian(mean_field_model)\n\n# Compute the ground energy\nground_energy = quadratic_hamiltonian.ground_energy()\nprint(ground_energy)\n```\n\n -10.0\n\n\nAny quadratic Hamiltonian may be rewritten in the form\n$$H = \\sum_p \\varepsilon_p b^\\dagger_p b_p + \\text{constant},$$\nwhere the $b_p$ are new annihilation operators that satisfy the fermionic anticommutation relations, and which are linear combinations of the old creation and annihilation operators. This form of $H$ makes it easy to deduce its eigenvalues; they are sums of subsets of the $\\varepsilon_p$, which we call the orbital energies of $H$. The following code computes the orbital energies and the constant:\n\n\n```python\norbital_energies, constant = quadratic_hamiltonian.orbital_energies()\nprint(orbital_energies)\nprint()\nprint(constant)\n```\n\n [ 1. 1. 1. 1. 4. 4. 4. 4.]\n \n -10.0\n\n\nEigenstates of quadratic hamiltonians are also known as fermionic Gaussian states, and they can be prepared efficiently on a quantum computer. One can use OpenFermion to obtain circuits for preparing these states. The following code obtains the description of a circuit which prepares the ground state (operations that can be performed in parallel are grouped together), along with a description of the starting state to which the circuit should be applied:\n\n\n```python\nfrom openfermion.utils import gaussian_state_preparation_circuit\n\ncircuit_description, start_orbitals = gaussian_state_preparation_circuit(quadratic_hamiltonian)\nfor parallel_ops in circuit_description:\n print(parallel_ops)\nprint('')\nprint(start_orbitals)\n```\n\n ('pht',)\n ((6, 7, 1.5707963267948966, 0.0),)\n ('pht', (5, 6, 1.5707963267948966, 0.0))\n ((4, 5, 1.0471975511965979, -3.1415926535897927), (6, 7, 1.0471975511965981, -3.1415926535897927))\n ('pht', (3, 4, 1.5707963267948966, 0.0), (5, 6, 1.5707963267948966, 0.0))\n ((2, 3, 1.2309594173407741, -8.3266726846886753e-17), (4, 5, 1.2309594173407747, 8.8817841970012523e-16), (6, 7, 1.1071487177940902, 3.1415926535897927))\n ('pht', (1, 2, 1.5707963267948966, 0.0), (3, 4, 1.5707963267948966, 0.0), (5, 6, 1.5707963267948966, 0.0))\n ((0, 1, 1.0471975511965979, -3.1415926535897927), (2, 3, 1.0471975511965976, -3.1415926535897927), (4, 5, 1.3181160716528177, -3.1415926535897927), (6, 7, 1.3181160716528177, -1.3322676295501878e-15))\n ('pht', (1, 2, 1.5707963267948966, 0.0), (3, 4, 1.5707963267948966, 0.0), (5, 6, 1.5707963267948966, 0.0))\n ((2, 3, 0.95531661812450908, 6.0368376963992877e-16), (4, 5, 0.95531661812450885, 0.0), (6, 7, 1.1071487177940904, -1.27675647831893e-15))\n ('pht', (3, 4, 1.5707963267948966, 0.0), (5, 6, 1.5707963267948966, 0.0))\n ((4, 5, 0.78539816339744806, 3.1415926535897922), (6, 7, 0.7853981633974485, -6.106226635438361e-16))\n ((5, 6, 1.5707963267948966, 0.0),)\n \n []\n\n\nIn the circuit description, each elementary operation is either a tuple of the form $(i, j, \\theta, \\varphi)$, indicating the operation $\\exp[i \\varphi a_j^\\dagger a_j]\\exp[\\theta (a_i^\\dagger a_j - a_j^\\dagger a_i)]$, which is a Givens rotation of modes $i$ and $j$, or the string 'pht', indicating the particle-hole transformation on the last fermionic mode, which is the operator $\\mathcal{B}$ such that $\\mathcal{B} a_N \\mathcal{B}^\\dagger = a_N^\\dagger$ and leaves the rest of the ladder operators unchanged. Operations that can be performed in parallel are grouped together.\n\nIn the special case that a quadratic Hamiltonian conserves particle number ($\\Delta = 0$), its eigenstates take the form\n$$\\lvert \\Psi_S \\rangle = b^\\dagger_{1}\\cdots b^\\dagger_{N_f}\\lvert \\text{vac} \\rangle,\\qquad\nb^\\dagger_{p} = \\sum_{k=1}^N Q_{pq}a^\\dagger_q,$$\nwhere $Q$ is an $N_f \\times N$ matrix with orthonormal rows. These states are also known as Slater determinants. OpenFermion also provides functionality to obtain circuits for preparing Slater determinants starting with the matrix $Q$ as the input.\n", "meta": {"hexsha": "4404e2aa27312b8b3669f372890521ad5e31bf38", "size": 82978, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/openfermion_tutorial.ipynb", "max_stars_repo_name": "quid256/OpenFermion", "max_stars_repo_head_hexsha": "562a03abf501885ee5a792ec3d7d10d91581b938", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "examples/openfermion_tutorial.ipynb", "max_issues_repo_name": "quid256/OpenFermion", "max_issues_repo_head_hexsha": "562a03abf501885ee5a792ec3d7d10d91581b938", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "examples/openfermion_tutorial.ipynb", "max_forks_repo_name": "quid256/OpenFermion", "max_forks_repo_head_hexsha": "562a03abf501885ee5a792ec3d7d10d91581b938", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 61.1030927835, "max_line_length": 21490, "alphanum_fraction": 0.7136710935, "converted": true, "num_tokens": 11991, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883592602049, "lm_q2_score": 0.7217432003123989, "lm_q1q2_score": 0.43318186720270807}} {"text": "[](https://colab.research.google.com/github.com/Rishit-dagli/TFUG-Mysuru-2020/blob/master/TFQuantum_starter.ipynb)\n\n# MNIST classification with QNNs\n\nIn this notebook you will build a quantum neural network (QNN) to classify a simplified version of MNIST with \n[Cirq](https://cirq.readthedocs.io/en/stable/) and TensorFlow Quantum (TFQ). The tutorial in general shows using a simple quantum neural network on a classical data problem. We here follow the approach used in the [Classification with Quantum Neural Networks on Near Term Processors](https://arxiv.org/pdf/1802.06002.pdf) paper by Edward Farhi and Hartmut Neven.\n\n\n\n> Note: This notebook is designed to be run in Google Colab if you want to run it locally or on a Jupyter notebook you \nwould skip the code cells with the `Colab only` comment.\n\n## Setup\n\n### Install TensorFlow 2.x (Colab only)\n\n\n```python\n# Colab only\npip install -q tensorflow==2.1.0\n```\n\n### Install TensorFlow Quantum (Colab only)\n\n\n```python\n# Colab only\npip install -q tensorflow-quantum\n```\n\n## Imports\n\nNow import TensorFlow and the module dependencies:\n\n\n```python\nimport tensorflow as tf\nimport tensorflow_quantum as tfq\n\nimport cirq\nimport sympy\nimport numpy as np\nimport seaborn as sns\nimport collections\nimport random\n\n# visualization tools\n%matplotlib inline\nimport matplotlib.pyplot as plt\nfrom cirq.contrib.svg import SVGCircuit\n```\n\n## Preprocessing the data\n\nIn the last tutorial you saw how to make a model to do hybrid quantum-classical binary classification. As this is meant to be an easy to follow and reproducoble tutorial, we will perform binary classification with MNIST. We will perform binary classification for this purpose with just two labels here `1` and `5` (Choosen as they look a lot different unlike something like 1 and 7)\n\n### Load the raw data\n\nLoad the MNIST dataset distributed with Keras.\n\n\n```python\n(x_train, y_train), (x_test, y_test) = tf.keras.datasets.mnist.load_data()\n```\n\n\n```python\nnum_of_samples = []\n\ncols = 5\nnum_classes = 10\n\nfig, axs = plt.subplots(nrows=num_classes, ncols = cols, figsize=(5, 8))\nfig.tight_layout()\nfor i in range(cols):\n for j in range(num_classes):\n x_selected = x_train[y_train == j]\n axs[j][i].imshow(x_selected[random.randint(0, len(x_selected - 1)), :, :], cmap=plt.get_cmap(\"gray\"))\n axs[j][i].axis(\"off\")\n if i == 2:\n axs[j][i].set_title(str(j))\n num_of_samples.append(len(x_selected))\n```\n\n\n```python\nx_train, x_test = x_train[..., np.newaxis]/255.0, x_test[..., np.newaxis]/255.0\n\nprint(\"Number of original training examples:\", len(x_train))\nprint(\"Number of original test examples:\", len(x_test))\n```\n\n Number of original training examples: 60000\n Number of original test examples: 10000\n\n\nFilter the dataset to keep just the 1s and 5s, remove the other classes. At the same time convert the label, `y`, to boolean: `True` for 1 and `False` for 5.\n\n\n```python\ndef filter_1_5(x, y):\n keep = (y == 1) | (y == 5)\n x, y = x[keep], y[keep]\n y = y == 1\n return x,y\n```\n\n\n```python\nx_train, y_train = filter_1_5(x_train, y_train)\nx_test, y_test = filter_1_5(x_test, y_test)\n\nprint(\"Number of filtered training examples:\", len(x_train))\nprint(\"Number of filtered test examples:\", len(x_test))\n```\n\n Number of filtered training examples: 12163\n Number of filtered test examples: 2027\n\n\nDisplay a random image\n\n\n```python\nplt.imshow(x_train[random.randint(0,12000), :, :, 0])\nplt.colorbar()\n```\n\n### Downscale the images\n\nAn image size of 28x28 is much too large for current quantum computers. Resize the image down to 4x4:\n\n\n```python\nx_train_small = tf.image.resize(x_train, (4,4)).numpy()\nx_test_small = tf.image.resize(x_test, (4,4)).numpy()\n```\n\n### Contradictory examples\n\n\"Learning to Distinguish Digits\", filter the dataset to remove images that are labeled as belonging to both classes mentioned in [this paper](https://arxiv.org/pdf/1802.06002.pdf).\n\nThis is not a standard machine-learning procedure, but is included in the interest of following the paper.\n\n\n```python\ndef remove_contradicting(xs, ys):\n mapping = collections.defaultdict(set)\n for x,y in zip(xs,ys):\n mapping[tuple(x.flatten())].add(y)\n \n new_x = []\n new_y = []\n for x,y in zip(xs, ys):\n labels = mapping[tuple(x.flatten())]\n if len(labels) == 1:\n new_x.append(x)\n new_y.append(list(labels)[0])\n else:\n # Do not include images that match more than one label.\n pass\n \n num_1 = sum(1 for value in mapping.values() if True in value)\n num_5 = sum(1 for value in mapping.values() if False in value)\n num_both = sum(1 for value in mapping.values() if len(value) == 2)\n\n print(\"Number of unique images:\", len(mapping.values()))\n print(\"Number of 1s: \", num_1)\n print(\"Number of 5s: \", num_5)\n print(\"Number of contradictory images: \", num_both)\n print()\n print(\"Initial number of examples: \", len(xs))\n print(\"Remaining non-contradictory examples: \", len(new_x))\n \n return np.array(new_x), np.array(new_y)\n```\n\n\n```python\nx_train_nocon, y_train_nocon = remove_contradicting(x_train_small, y_train)\n```\n\n Number of unique images: 6588\n Number of 1s: 2176\n Number of 5s: 4626\n Number of contradictory images: 214\n \n Initial number of examples: 12163\n Remaining non-contradictory examples: 7949\n\n\n### Data as quantum circuits\n\nWe will represent each pixel with a qubit, with the state depending on the value of the pixel. The first step is to convert to a binary encoding.\n\n\n```python\nTHRESHOLD = 0.5\n\nx_train_bin = np.array(x_train_nocon > THRESHOLD, dtype=np.float32)\nx_test_bin = np.array(x_test_small > THRESHOLD, dtype=np.float32)\n```\n\nThe qubits at pixel indices with values that exceed a threshold, are rotated through an `X gate`\n\n\n```python\ndef convert_to_circuit(image):\n values = np.ndarray.flatten(image)\n qubits = cirq.GridQubit.rect(4, 4)\n circuit = cirq.Circuit()\n for i, value in enumerate(values):\n if value:\n circuit.append(cirq.X(qubits[i]))\n return circuit\n\nx_train_circ = [convert_to_circuit(x) for x in x_train_bin]\nx_test_circ = [convert_to_circuit(x) for x in x_test_bin]\n```\n\nLets now visualize the circuit\n\n\n```python\nSVGCircuit(x_train_circ[random.randint(0,7949)])\n```\n\n\n\n\n \n\n \n\n\n\nConvert these Cirq circuits to tensors for `tfq`. This is made very easy with `convert_to_tensor()` in `tfq`.\n\n\n```python\nx_train_tfcirc = tfq.convert_to_tensor(x_train_circ)\nx_test_tfcirc = tfq.convert_to_tensor(x_test_circ)\n```\n\n## Quantum neural Network\n\n[The paper](https://arxiv.org/pdf/1802.06002.pdf) propose using two qubit gates, with the readout qubit always acted upon. This is in a way similar to running a [Unitary RNN](https://arxiv.org/abs/1511.06464) in some ways.\n\n### Build the model circuit\n\nEach layer uses `n` instances of the same gate, with each of the data qubits acting on the readout qubit.\n\nStart with a simple class that will add a layer of these gates to a circuit:\n\n\n```python\nclass CircuitLayerBuilder():\n def __init__(self, data_qubits, readout):\n self.data_qubits = data_qubits\n self.readout = readout\n \n def add_layer(self, circuit, gate, prefix):\n for i, qubit in enumerate(self.data_qubits):\n symbol = sympy.Symbol(prefix + '-' + str(i))\n circuit.append(gate(qubit, self.readout)**symbol)\n```\n\nBuild an example circuit layer:\n\n\n```python\ndemo_builder = CircuitLayerBuilder(data_qubits = cirq.GridQubit.rect(4,1),\n readout=cirq.GridQubit(-1,-1))\n\ncircuit = cirq.Circuit()\ndemo_builder.add_layer(circuit, gate = cirq.XX, prefix='xx')\nSVGCircuit(circuit)\n```\n\n\n\n\n \n\n \n\n\n\nNow build a two-layered model, matching the data-circuit size, and include the preparation and readout operations.\n\n\n```python\ndef create_quantum_model():\n \n # Place data qunits on the grid\n data_qubits = cirq.GridQubit.rect(4, 4) \n readout = cirq.GridQubit(-1, -1)\n circuit = cirq.Circuit()\n \n # Prepare the readout qubit.\n circuit.append(cirq.X(readout))\n circuit.append(cirq.H(readout))\n \n builder = CircuitLayerBuilder(\n data_qubits = data_qubits,\n readout=readout)\n\n # Then add layers (experiment by adding more).\n builder.add_layer(circuit, cirq.XX, \"xx1\")\n builder.add_layer(circuit, cirq.ZZ, \"zz1\")\n\n # Finally, prepare the readout qubit.\n circuit.append(cirq.H(readout))\n\n return circuit, cirq.Z(readout)\n```\n\n\n```python\nmodel_circuit, model_readout = create_quantum_model()\n```\n\n### Wrap the model in Keras layer\n\nBuild the Keras model with the quantum components. This model is fed the \"quantum data\", from x_train_circ, that encodes the classical data. It uses a Parametrized Quantum Circuit layer, `tfq.layers.PQC`, to train the model circuit, on the quantum data.\n\n\n```python\nmodel = tf.keras.Sequential([\n # The input is the data-circuit, encoded as a tf.string\n tf.keras.layers.Input(shape=(), dtype=tf.string),\n # The PQC layer returns the expected value of the readout gate, range [-1,1].\n tfq.layers.PQC(model_circuit, model_readout),\n])\n```\n\nNow we will compile the model. Since this is a binary classification problem we here use `BinaryCrossentropy`. We shift the output range to $[0,1]$, and treat it as the probability the model assigns to class 3. This could be used with a standard a tf.losses.BinaryCrossentropy loss.\n\n> Note: Another valid approach would be to use the Hinge Loss. Since the the expected readout is in the range $[-1,1]$, optimizing the hinge loss is also valid. To use the hinge loss here you need to make two small adjustments. First convert the `labels`, `y_train_nocon`, from boolean to $[-1,1]$, as expected by the hinge loss.\n\n\n```python\nmodel.compile(\n loss= tf.keras.losses.BinaryCrossentropy(),\n optimizer=tf.keras.optimizers.Adam(),)\n```\n\n\n```python\nEPOCHS = 3\nBATCH_SIZE = 32\n\nNUM_EXAMPLES = 500#len(x_train_tfcirc)\nx_train_tfcirc_sub = x_train_tfcirc[:NUM_EXAMPLES]\ny_train_hinge_sub = y_train_nocon[:NUM_EXAMPLES]\n```\n\n\n```python\nprint(model.summary())\n```\n\n Model: \"sequential\"\n _________________________________________________________________\n Layer (type) Output Shape Param # \n =================================================================\n pqc (PQC) (None, 1) 32 \n =================================================================\n Total params: 32\n Trainable params: 32\n Non-trainable params: 0\n _________________________________________________________________\n None\n\n\n\n```python\nqnn_history = model.fit(\n x_train_tfcirc_sub, y_train_hinge_sub,\n batch_size=32,\n epochs=EPOCHS,\n verbose=1,\n validation_data=(x_test_tfcirc, y_test))\n\nqnn_results = model.evaluate(x_test_tfcirc, y_test)\n```\n\n Train on 500 samples, validate on 2027 samples\n Epoch 1/3\n 500/500 [==============================] - 140s 281ms/sample - loss: 3.7780 - val_loss: 6.6636\n Epoch 2/3\n 500/500 [==============================] - 140s 280ms/sample - loss: 3.6805 - val_loss: 6.6183\n Epoch 3/3\n 500/500 [==============================] - 140s 281ms/sample - loss: 3.6135 - val_loss: 6.5709\n 2027/2027 [==============================] - 16s 8ms/sample - loss: 6.5709\n\n\n## References\n\n* [Classification with Quantum Neural Networks on Near Term Processors](https://arxiv.org/pdf/1802.06002.pdf)\n* [TensorFlow Quantum Whitepaper](https://arxiv.org/abs/2003.02989)\n* [TensorFlow Quantum Docs](https://www.tensorflow.org/quantum/api_docs/python/tfq)\n* [Announcing TensorFlow Quantum Dev Summit 2020](https://youtu.be/-o9AhIz1uvo)\n* [Programming a Quantum Computer with Cirq](https://youtu.be/16ZfkPRVf2w)\n* [Announcing TensorFlow Quantum](https://ai.googleblog.com/2020/03/announcing-tensorflow-quantum-open.html)\n* [Unitary RNNs](https://arxiv.org/abs/1511.06464)\n", "meta": {"hexsha": "f650c26ec9afe33ba92b4c4ccaa623311bacf820", "size": 80198, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Quantum_MNIST.ipynb", "max_stars_repo_name": "Rishit-dagli/TFUG-Mysuru-2020", "max_stars_repo_head_hexsha": "c7bdaf2bede52a8b9560db4de4c75f0fafb7bd74", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2020-07-24T03:47:25.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-01T00:49:04.000Z", "max_issues_repo_path": "Quantum_MNIST.ipynb", "max_issues_repo_name": "Rishit-dagli/TFUG-Mysuru-2020", "max_issues_repo_head_hexsha": "c7bdaf2bede52a8b9560db4de4c75f0fafb7bd74", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Quantum_MNIST.ipynb", "max_forks_repo_name": "Rishit-dagli/TFUG-Mysuru-2020", "max_forks_repo_head_hexsha": "c7bdaf2bede52a8b9560db4de4c75f0fafb7bd74", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 109.2615803815, "max_line_length": 46548, "alphanum_fraction": 0.8364048979, "converted": true, "num_tokens": 3154, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217431943271999, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.4331818636104614}} {"text": "# \u30a8\u30f3\u30b3\u30fc\u30c0\u3092\u5b66\u7fd2\u3059\u308b\n\n__\u76ee\u6b21\uff1a__\n\n- \u8996\u899a\u523a\u6fc0\u304b\u3089\u7279\u5fb4\u91cf\u3092\u9020\u308b\n- \u30e2\u30c7\u30eb\u3092\u521d\u671f\u5316\u3057\u3066\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u8d70\u3089\u305b\u308b\n- \u5b66\u7fd2\u6a5f\u306e\u51fa\u6765\u3092\u8a55\u4fa1\u3059\u308b\n- \u8ab2\u984c\u4e00\u89a7\n\n___\n\n\u3053\u3053\u3067\u306e\u76ee\u7684\u306f\u3001\u3053\u308c\u307e\u3067\u306b\u7fd2\u5f97\u3057\u3066\u304d\u305f\u6280\u6cd5\u3084\u77e5\u898b\u3092\u878d\u5408\u3055\u305b\u3001\u6b63\u5e38\u306b\u6a5f\u80fd\u3059\u308b\u5b66\u7fd2\u6a5f\u3092\u9020\u308b\u3053\u3068\u3067\u3042\u308b\u3002\u305d\u306e\u5fdc\u7528\u5148\u306f\u8996\u899a\u523a\u6fc0\u304b\u3089\u8133\u6d3b\u52d5\u3078\u306e\u30a8\u30f3\u30b3\u30fc\u30c7\u30a3\u30f3\u30b0\u3067\u3042\u308b\u3002\u307e\u305a\u306f\u3053\u308c\u307e\u3067\u306b\u7528\u610f\u3057\u3066\u304d\u305f\u95a2\u6570\u3084\u30af\u30e9\u30b9\u3092`import`\u3057\u3066\u304a\u304f\u3002\n\n\n```python\n\nimport scripts.FilterBank as fb\nimport scripts.AlgoIntro as ai\nimport scripts.AlgoSparseReg as asr\n\n```\n\n\u4f5c\u696d\u306e\u6d41\u308c\u306f\u4e0b\u8a18\u306e\u901a\u308a\u306b\u306a\u308b\uff1a\n\n \u8996\u899a\u523a\u6fc0\u306e\u8aad\u307f\u8fbc\u307f --> \u30ac\u30dc\u30fc\u30eb\u30d5\u30a3\u30eb\u30bf\u3067\u7279\u5fb4\u91cf\u3092\u4f5c\u3063\u3066\u4fdd\u5b58 --> \u30b9\u30d1\u30fc\u30b9\u306a\u7dda\u5f62\u56de\u5e30\n\n\u3053\u308c\u3089\u306e\u30bf\u30b9\u30af\u3092\u4e00\u3064\u305a\u3064\u3053\u306a\u3057\u3066\u3044\u304f\u3002\u3084\u308a\u65b9\u3092\u660e\u78ba\u306b\u4f1d\u3048\u308b\u3079\u304f\u3001\u975e\u5e38\u306b\u5358\u7d14\u306a\u30d7\u30ed\u30c8\u30bf\u30a4\u30d7\u3092\u4f5c\u3063\u3066\u304a\u304f\u3002\u305d\u306e\u5f8c\u306e\u8ab2\u984c\u3067\u306f\u3001\u3053\u306e\u30d7\u30ed\u30c8\u30bf\u30a4\u30d7\u3092\u30d9\u30fc\u30b9\u306b\u3057\u3066\u3001\u30a8\u30f3\u30b3\u30fc\u30c0\u3068\u3057\u3066\u3061\u3083\u3093\u3068\u6a5f\u80fd\u3059\u308b\u3088\u3046\u306b\u6539\u5584\u3057\u3066\u3082\u3089\u3046\u3053\u3068\u306b\u306a\u308b\u3002\n\n\n## \u8996\u899a\u523a\u6fc0\u304b\u3089\u7279\u5fb4\u91cf\u3092\u9020\u308b\n\n\n```python\nimport numpy as np\nimport math\n\n# Import the vim-2 data from the Python binary format we saved earlier.\n# Assumptions: that the data is saved in the vim-2 directory,\n# already of 96x96 size, with dtype of np.float32.\n\nPIX_W = 96\nPIX_H = 96\ndtype=np.float32\n\n# Read the raw training data.\nshape=(PIX_W,PIX_H,3,108000) # (downsized px, downsized px, rgd channels, time steps)\n# Index for temporal down-sampling. Alternatives: do after the feature-building and aggregate.\nidx_ds = np.arange(15//2, 108000+1, 15) # need length 7200.\nfname = \"data/vim-2/X_tr.dat\"\nwith open(fname, mode=\"br\") as fbin:\n print(\"Reading...\", end=\" \")\n raw_tr = np.fromfile(file=fbin,dtype=dtype).reshape(shape)[:,:,:,idx_ds] # temporally down-sampled.\n print(\"OK.\")\n\n# Check a few frames.\n#num_frames = raw_tr.shape[3]\n#frames_to_play = 5\n#for t in range(frames_to_play):\n# plt.imshow(raw_tr[:,:,:,t])\n# plt.show()\n\n\n# Read the raw testing data.\nshape=(PIX_W,PIX_H,3,8100) # (downsized px, downsized px, rgd channels, time steps)\n# Index for temporal down-sampling. Alternatives: do after the feature-building and aggregate.\nidx_ds = np.arange(15//2, 8100+1, 15) # need length 540.\nfname = \"data/vim-2/X_te.dat\"\nwith open(fname, mode=\"br\") as fbin:\n print(\"Reading...\", end=\" \")\n raw_te = np.fromfile(file=fbin, dtype=dtype).reshape(shape)[:,:,:,idx_ds] # temporally down-sampled.\n print(\"OK.\")\n\n# Check a few frames.\n#num_frames = raw_tr.shape[3]\n#frames_to_play = 5\n#for t in range(frames_to_play):\n# plt.imshow(raw_tr[:,:,:,t])\n# plt.show()\n\n```\n\n Reading... OK.\n Reading... OK.\n\n\n\n```python\n\n# Set up the parameters that specify the first filter bank.\nmyparas = {\"freqs\": 32/max(PIX_W,PIX_H),\n \"dir\": math.pi/2,\n \"amp\": 0.1,\n \"sdev\": max(PIX_W,PIX_H)/20,\n \"phase\": 0}\nmygrid_h = 3\nmygrid_w = 3\n\nprint(\"Getting features (tr)...\", end=\" \")\nX_tr = fb.G2_getfeatures(ims=raw_tr,\n fil_paras=myparas,\n gridshape=(mygrid_h,mygrid_w),\n mode=\"reflect\", cval=0)\nprint(\"OK.\")\n\nprint(\"Getting features (te)...\", end=\" \")\nX_te = fb.G2_getfeatures(ims=raw_te,\n fil_paras=myparas,\n gridshape=(mygrid_h,mygrid_w),\n mode=\"reflect\", cval=0)\nprint(\"OK.\")\n\n```\n\n Getting features (tr)... OK.\n Getting features (te)... OK.\n\n\n\n```python\nprint(\"Shape of the produced features:\")\nprint(\"tr:\", X_tr.shape)\nprint(\"te:\", X_te.shape)\n```\n\n Shape of the produced features:\n tr: (7200, 9)\n te: (540, 9)\n\n\n\u4e0a\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u3060\u3051\u3067\u306f\u3001\u7279\u5fb4\u91cf\u304c\u975e\u5e38\u306b\u5c11\u306a\u304f\u3001\u8aac\u660e\u80fd\u529b\u304c\u5230\u5e95\u8db3\u308a\u306a\u3044\u3002\u5897\u3084\u3059\u65b9\u6cd5\u306f\u3044\u304f\u3089\u3067\u3082\u3042\u308b\u304c\u3001\u5225\u306edict\u3092\u4e00\u3064\u7528\u610f\u3057\u3001\u7279\u5fb4\u91cf\u30d9\u30af\u30c8\u30eb\u3092\u9023\u7d50\u3055\u305b\u308b\u4e8b\u4f8b\u3092\u6b21\u306b\u793a\u3057\u3066\u304a\u304f\u3002\n\n\n```python\n# Set up the parameters that specify the second filter bank.\nmyparas = {\"freqs\": 32/max(PIX_W,PIX_H),\n \"dir\": 0,\n \"amp\": 0.1,\n \"sdev\": max(PIX_W,PIX_H)/20,\n \"phase\": 0}\nmygrid_h = 9\nmygrid_w = 9\n\nprint(\"Getting features (tr)...\", end=\" \")\ntmp_X = fb.G2_getfeatures(ims=raw_tr,\n fil_paras=myparas,\n gridshape=(mygrid_h,mygrid_w),\n mode=\"reflect\", cval=0)\nprint(\"OK.\")\nX_tr = np.concatenate((X_tr, tmp_X), axis=1) # concatenate!\n\nprint(\"Getting features (te)...\", end=\" \")\ntmp_X = fb.G2_getfeatures(ims=raw_te,\n fil_paras=myparas,\n gridshape=(mygrid_h,mygrid_w),\n mode=\"reflect\", cval=0)\nprint(\"OK.\")\nX_te = np.concatenate((X_te, tmp_X), axis=1) # concatenate!\n\nprint(\"Shape of the produced features:\")\nprint(\"tr:\", X_tr.shape)\nprint(\"te:\", X_te.shape)\n```\n\n Getting features (tr)... OK.\n Getting features (te)... OK.\n Shape of the produced features:\n tr: (7200, 90)\n te: (540, 90)\n\n\n\u4e0a\u304b\u3089\u660e\u3089\u304b\u306a\u3088\u3046\u306b\u3001\u9023\u7d50\u3055\u305b\u308b\u3053\u3068\u3067\u3001\u30c7\u30fc\u30bf\u884c\u5217\u306b\u5217\u3092\u8ffd\u52a0\u3057\u3066\u3044\u308b\u3060\u3051\u3067\u3042\u308b\u3002\u884c\u6570\uff08\u3064\u307e\u308a\u30b5\u30f3\u30d7\u30eb\u6570\uff09\u306f\u4e00\u5b9a\u3067\u3042\u308b\u3002\n\n\u3053\u308c\u307e\u3067\u3068\u540c\u69d8\u306b\u3001Nishimoto et al. (2011)\u306b\u3057\u305f\u304c\u3063\u3066\u3001\u7279\u5fb4\u91cf\u306eZ\u30b9\u30b3\u30a2\u3092\u8a08\u7b97\u3059\u308b\uff08\u5e73\u5747\u30bc\u30ed\u3001\u5206\u65631.0\u306e\u6a19\u6e96\u5316\uff09\u3002\u3053\u308c\u3092\u3057\u305f\u4e0a\u3067\u3001\u300c\u5916\u308c\u5024\u300d\u3068\u898b\u306a\u3059\u3079\u304d\u70b9\u306e\u95be\u5024\u3092\u5b9a\u3081\u3001\u5fc5\u8981\u306b\u5fdc\u3058\u3066\u5207\u65ad\u3059\u308b\u3002\u516c\u5e73\u306a\u5b66\u7fd2\u8ab2\u984c\u306b\u306a\u308b\u3088\u3046\u306b\u3001\u8a13\u7df4\u30fb\u691c\u8a3c\u3092\u5225\u3005\u306b\u6271\u3046\u3002\n\n\n```python\n\nX_tr = X_tr - np.mean(X_tr, axis=0)\nX_tr = X_tr / np.std(X_tr, axis=0)\nprint(\"Mean =\", np.mean(X_tr, axis=0), \"StdDev =\", np.std(X_tr, axis=0))\n\nX_te = X_te - np.mean(X_te, axis=0)\nX_te = X_te / np.std(X_te, axis=0)\nprint(\"Mean =\", np.mean(X_te, axis=0), \"StdDev =\", np.std(X_te, axis=0))\n\nfor j in range(X_tr.shape[1]):\n stdval = np.std(X_tr[:,j])\n X_tr[:,j] = np.clip(X_tr[:,j], a_min=(-stdval), a_max=stdval)\n stdval = np.std(X_te[:,j])\n X_te[:,j] = np.clip(X_te[:,j], a_min=(-stdval), a_max=stdval)\n\n```\n\n Mean = [ 2.58357159e-06 -8.92488515e-07 5.57957435e-07 -4.96258338e-07\n -1.69749057e-06 9.48549996e-07 -2.33410134e-07 -1.02121794e-06\n 1.77970367e-06 -1.18050307e-08 8.10722497e-07 -1.43411262e-06\n 3.32246231e-07 5.24578809e-07 1.41411192e-06 6.26593817e-08\n -1.32421647e-07 -1.81612037e-07 9.25486290e-07 1.96074453e-07\n 1.67777966e-06 5.08626314e-08 1.45783030e-08 4.33185022e-07\n 4.19616697e-07 1.14723207e-06 1.72787239e-07 3.96553020e-07\n 1.54698887e-06 -9.43500140e-07 3.42387295e-07 -1.26283203e-06\n 6.70444649e-07 -7.34544471e-07 1.22170479e-06 1.41645467e-06\n -6.07909442e-07 2.06748655e-06 1.42157910e-06 -1.08564893e-06\n 1.37013694e-06 -1.70287166e-07 -2.96019834e-07 -5.49770078e-08\n -4.36430184e-07 -8.49920866e-07 6.96124289e-07 -2.97485116e-07\n -1.17873981e-06 -1.08703972e-07 -3.01607770e-07 -1.50195433e-07\n -3.78158347e-07 1.52405761e-07 9.67168177e-08 8.73878605e-07\n -4.91572756e-08 3.38041133e-07 -1.45385670e-07 -4.44245018e-07\n -8.24688186e-07 4.60081623e-07 2.73320410e-07 1.42743602e-06\n -1.08632776e-06 6.17264050e-07 -1.17643015e-06 -8.23777555e-07\n 1.83092220e-06 4.18970984e-07 -7.49511855e-07 1.39700046e-06\n 3.56568222e-07 -1.55260170e-06 -1.30837577e-06 7.06145329e-07\n 1.28703812e-06 8.49184062e-07 7.94892117e-07 2.63253845e-08\n -7.78573281e-07 -3.04786681e-07 6.02404270e-07 1.40218685e-06\n -1.28369368e-06 -5.52998642e-08 -2.83420093e-07 4.07103869e-07\n 1.77071740e-06 -2.07353793e-07] StdDev = [ 0.99999827 0.99999785 1.0000006 1.0000025 0.99999905 0.99999976\n 1.00000107 1.00000155 1.00000024 0.99999952 0.99999857 1.00000072\n 1.00000131 0.99999893 0.99999881 0.99999958 0.99999976 0.99999708\n 1.00000072 1.00000012 0.99999785 0.99999976 0.99999958 1.0000006\n 0.9999994 0.99999803 0.9999997 0.99999958 0.99999887 0.99999976\n 1.00000048 0.99999952 1.00000107 0.99999976 0.9999997 0.99999911\n 0.99999982 0.99999833 0.99999928 1.00000036 0.9999997 1. 1.0000006\n 1.00000012 0.99999911 0.99999839 0.99999803 1.00000083 1.00000143\n 1. 0.99999869 0.99999952 0.99999988 1.00000083 0.99999928\n 1. 1.00000024 1.00000048 1.00000286 0.99999952 0.99999982\n 1.00000048 1.00000024 0.99999958 1.00000012 0.99999893 0.99999893\n 0.99999958 1.00000179 1. 1.00000036 0.9999997 1.00000024\n 0.99999934 0.99999917 1.00000155 0.99999976 0.9999994 1.00000036\n 1.00000012 1.00000012 0.99999982 0.99999881 1.00000119 1.00000107\n 0.9999994 1.0000006 0.99999899 0.99999923 1.00000072]\n Mean = [ 2.64688765e-07 -1.96474573e-07 4.01779445e-08 -4.15245694e-07\n -2.53016196e-07 6.95387525e-09 2.20426813e-07 -3.15256131e-07\n 3.20305986e-08 -3.60276971e-07 1.32234007e-07 1.12586553e-08\n -2.55858453e-07 -4.05201206e-07 -1.42830388e-07 1.97357608e-07\n -1.58724959e-07 -9.86788038e-08 6.62273847e-10 -4.85667471e-08\n 3.49018308e-07 3.86988688e-07 2.03456040e-07 7.66581934e-07\n 2.22303242e-07 -1.28591509e-07 -1.60049510e-07 -1.65347700e-07\n 2.77934248e-07 1.52433358e-07 -2.35521131e-08 4.64474709e-07\n 1.13028065e-07 1.00003348e-07 -2.53319740e-07 1.90845242e-07\n 9.09522697e-08 -3.40629498e-07 1.62698598e-07 1.52709305e-07\n -2.16673925e-07 2.63926466e-07 -4.03987030e-08 9.91203137e-08\n -1.52322983e-07 -7.49362812e-07 -4.57851968e-07 -3.92397261e-08\n 6.59100436e-08 -4.70490363e-07 -1.64409485e-07 3.10123511e-07\n 1.83202573e-07 -5.87436887e-07 1.88306529e-07 -5.52998642e-08\n -1.99785944e-07 5.36441803e-07 -6.27614838e-07 2.08064364e-07\n 4.77940958e-08 1.56075870e-07 -5.82800972e-08 2.58949058e-07\n 2.79479565e-07 -2.71201145e-07 1.81904539e-07 9.44843990e-08\n 1.35655753e-07 1.62036329e-07 2.83011673e-07 2.40184647e-07\n 3.85222627e-08 1.01824602e-08 3.68693350e-07 3.67672357e-07\n 3.37649283e-07 -2.20413014e-07 1.21748002e-07 5.80814174e-07\n 4.48138621e-07 -1.71087411e-08 2.11265359e-07 2.53154184e-07\n 1.85491857e-07 -1.17829551e-07 -2.74733253e-07 5.75626338e-08\n 1.35103861e-07 -2.75505926e-07] StdDev = [ 0.99999988 0.99999952 0.99999988 0.99999988 0.99999988 1.\n 0.99999988 0.9999994 1. 1. 1. 1.\n 0.99999994 1.00000012 1. 1.0000006 0.99999982 0.99999988\n 0.99999988 1.00000024 1.00000024 0.9999997 1.00000012 0.9999997 1.\n 1. 1.00000036 1.00000024 0.99999988 0.9999997 0.99999964\n 1. 1.00000036 1. 0.99999994 1. 1.\n 0.99999976 0.99999988 0.9999997 0.9999997 1.00000012 1.00000012\n 1. 1. 1.00000036 1.0000006 1.00000048 1.\n 1.00000036 1.00000012 0.99999982 1.00000036 1. 0.99999994\n 0.9999997 1. 0.99999994 0.99999988 0.99999976 1.00000012\n 0.99999934 0.99999988 0.99999982 0.99999964 1. 0.99999994\n 0.99999976 1. 0.99999982 1. 1.00000012 1.00000012\n 1.00000012 0.99999958 1.00000012 0.9999997 1.00000036 1. 1.\n 1.00000036 0.99999988 1.00000012 0.99999994 1. 0.99999976\n 1.00000036 1.00000012 1. 0.99999994]\n\n\n\u601d\u3044\u901a\u308a\u306b\u3053\u308c\u307e\u3067\u306e\u8a08\u7b97\u304c\u3067\u304d\u3066\u3044\u308b\u306e\u3067\u3042\u308c\u3070\u3001\u6574\u5217\u3055\u308c\u305f\u5168\u7279\u5fb4\u91cf\u3092\u30c7\u30a3\u30b9\u30af\u306b\u66f8\u304d\u8fbc\u307f\u3001\u4ed8\u5c5edata info\u30aa\u30d6\u30b8\u30a7\u30af\u30c8\u3082\u4f75\u305b\u3066\u7528\u610f\u3059\u308b\u3002\n\n\n```python\nimport support.classes as classes\n\n# Make a data_info file for this data, and save as data/encoder/info.dat.\ndinfo = classes.DataInfo()\ndinfo.mname = \"Encoder\"\n\nfname = \"data/encoder/X_tr.dat\"\ndtype = raw_tr.dtype\nshape = raw_tr.shape\nwith open(fname, mode=\"bw\") as fbin:\n X_tr.tofile(fbin)\n print(\"Saved to file.\")\ndinfo.X_tr[\"shape\"] = X_tr.shape\ndinfo.X_tr[\"path\"] = \"data/encoder/X_tr.dat\"\ndinfo.X_tr[\"dtype\"] = X_tr.dtype\n \nfname = \"data/encoder/X_te.dat\"\ndtype = raw_te.dtype\nshape = raw_te.shape\nwith open(fname, mode=\"bw\") as fbin:\n X_te.tofile(fbin)\n print(\"Saved to file.\")\ndinfo.X_te[\"shape\"] = X_te.shape\ndinfo.X_te[\"path\"] = \"data/encoder/X_te.dat\"\ndinfo.X_te[\"dtype\"] = X_te.dtype\n```\n\n Saved to file.\n Saved to file.\n\n\n\n```python\nimport support.classes as classes\n\n# Make a data_info file for this data, and save as data/encoder/info.dat.\ndinfo = classes.DataInfo()\ndinfo.mname = \"Encoder\"\n\nfname = \"data/encoder/X_tr.dat\"\ndtype = raw_tr.dtype\nshape = raw_tr.shape\nwith open(fname, mode=\"bw\") as fbin:\n X_tr.tofile(fbin)\n print(\"Saved to file.\")\ndinfo.X_tr[\"shape\"] = X_tr.shape\ndinfo.X_tr[\"path\"] = \"data/encoder/X_tr.dat\"\ndinfo.X_tr[\"dtype\"] = X_tr.dtype\n\nfname = \"data/encoder/X_te.dat\"\ndtype = raw_te.dtype\nshape = raw_te.shape\nwith open(fname, mode=\"bw\") as fbin:\n X_te.tofile(fbin)\n print(\"Saved to file.\")\ndinfo.X_te[\"shape\"] = X_te.shape\ndinfo.X_te[\"path\"] = \"data/encoder/X_te.dat\"\ndinfo.X_te[\"dtype\"] = X_te.dtype\n\n# Clear the raw data from memory.\ndel [raw_te, raw_tr]\n\n```\n\n Saved to file.\n Saved to file.\n\n\n\u5fdc\u7b54\u306b\u95a2\u3057\u3066\u306f\u3001\u7279\u5225\u306a\u51e6\u7406\u306f\u4f55\u3082\u8981\u3089\u306a\u3044\u306e\u3067\u3001`encoder`\u30c7\u30a3\u30ec\u30af\u30c8\u30ea\u3078\u79fb\u3059\u3060\u3051\u3067\u5341\u5206\u3067\u3042\u308b\u3002\n\n```\n$ mv data/vim-2/y_tr.dat data/vim-2/y_te.dat ./data/encoder/\n$ mv data/vim-2/cleanidx_tr.dat data/vim-2/cleanidx_te.dat ./data/encoder/\n\n```\n\n`dinfo`\u306e\u6b8b\u308a\u306e\u7a7a\u6b04\u3092\u57cb\u3081\u3066\u304b\u3089\u30c7\u30a3\u30b9\u30af\u306b\u66f8\u304d\u8fbc\u3080\u3002\n\n\n```python\n\nimport pickle\n\nfname = \"data/encoder/y_tr.dat\"\ndinfo.y_tr[\"shape\"] = (73728, 7200)\ndinfo.y_tr[\"path\"] = \"data/encoder/y_tr.dat\"\ndinfo.y_tr[\"dtype\"] = np.float32\n\nfname = \"data/encoder/y_te.dat\"\ndinfo.y_te[\"shape\"] = (73728, 540)\ndinfo.y_te[\"path\"] = \"data/encoder/y_te.dat\"\ndinfo.y_te[\"dtype\"] = np.float32\n\ndinfo.misc = {\"voxidx\": None} # to be filled in later.\n\nwith open(\"data/encoder/info.dat\", mode=\"bw\") as fbin:\n pickle.dump(dinfo, fbin)\n print(\"Saved to file.\")\n\n```\n\n Saved to file.\n\n\n\n## \u30e2\u30c7\u30eb\u3092\u521d\u671f\u5316\u3057\u3066\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u8d70\u3089\u305b\u308b\n\n\u7279\u5fb4\u91cf\u306e\u6e96\u5099\u304c\u6574\u3063\u3066\u3044\u308b\u306a\u3089\u3001Jupyter\u306e\u30ab\u30fc\u30cd\u30eb\u3092\u30ea\u30bb\u30c3\u30c8\u3057\u3001\u3053\u3053\u304b\u3089\u65b0\u305f\u306a\u4f5c\u696d\u3092\u59cb\u3081\u308b\u3002\n\n\u524d\u56de\u3067\u4f5c\u3063\u305f`Algo_LASSO_CD`\u3092\u3053\u3053\u3067\u672c\u9818\u767a\u63ee\u3057\u3066\u3082\u3089\u3046\u3002\u524d\u56de\u306e\u30c6\u30b9\u30c8\u3068\u307e\u3063\u305f\u304f\u540c\u69d8\u306b\u3001\u591a\u6570\u306e$\\lambda$\u5019\u88dc\u3092\u7528\u610f\u3057\u3001warm start\u3092\u751f\u304b\u3057\u306a\u304c\u3089\u5b66\u7fd2\u3055\u305b\u3066\u3044\u304f\u3002\n\n\u30d1\u30d5\u30a9\u30fc\u30de\u30f3\u30b9\u8a55\u4fa1\u6307\u6a19\u3068\u3057\u3066\u3001Nishimoto et al. (2011)\u3088\u308a\uff1a\n\n> *\"Prediction accuracy was defined as the correlation between predicted and observed BOLD signals. The averaged accuracy across subjects and voxels in early visual areas (V1, V2, V3, V3A, and V3B) was 0.24, 0.39, and 0.40 for the static, nondirectional, and directional encoding models, respectively.\"*\n\n\u6559\u80b2\u76ee\u7684\u306a\u306e\u3067\u6211\u3005\u306e\u5b66\u7fd2\u6a5f\u304c\u3042\u3089\u3086\u308b\u610f\u5473\u3067\u5f7c\u3089\u306e\u3082\u306e\u306b\u52a3\u3063\u3066\u3044\u308b\u306e\u3060\u304c\u3001\u3081\u3056\u3059\u57fa\u6e96\u3068\u3057\u3066\u306f\u3053\u306e\u6570\u5024\u306b\u306f\u610f\u7fa9\u304c\u3042\u308b\u3002\u7279\u306b\u76f8\u95a2\u4fc2\u65700.24\u306f\u30012\u6b21\u5143\u306e\u30ac\u30dc\u30fc\u30eb\u30d5\u30a3\u30eb\u30bf\u3092\u4f7f\u3063\u305f\u3068\u304d\u306b\u51fa\u305f\u6570\u5b57\u306a\u306e\u3067\u3001\u3088\u308a\u73fe\u5b9f\u7684\u306a\u57fa\u6e96\u3068\u898b\u3066\u3082\u3088\u304b\u308d\u3046\u3002\n\n\u7b97\u51fa\u65b9\u6cd5\u306f\u6b21\u306e\u901a\u308a\u3067\u3042\u308b\u3002\u8a13\u7df4\u30c7\u30fc\u30bf\u3092\u4f7f\u3063\u3066\u3001\u5b66\u7fd2\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u4e00\u901a\u308a\u8d70\u3089\u305b\u308b\u3068\u3001$\\widehat{w}$\u306b\u5bfe\u5fdc\u3059\u308b`w_est`\u304c\u5b9a\u307e\u308b\u3002\u7dda\u5f62\u30e2\u30c7\u30eb\u3092\u4f7f\u3063\u3066\u3044\u308b\u306e\u3067\u3001\u65b0\u3057\u3044\u5165\u529b$X$\u304b\u3089$\\widehat{y} = X\\widehat{w}$\u3092\u51fa\u3057\u3066\u3001$\\widehat{y} \\approx y$\u3067\u8fd1\u4f3c\u3057\u3066\u307f\u308b\u3002\u3053\u306e$X$\u3068$y$\u304c`X_te`\u3068`y_te`\u306b\u5bfe\u5fdc\u3059\u308b\u3002\u4e88\u6e2c\u4fe1\u53f7\u3068\u672c\u5f53\u306e\u4fe1\u53f7\u306e\u76f8\u95a2\u304c\u5f37\u3044\u307b\u3069\u3088\u3044\u3068\u3044\u3046\u3053\u3068\u306a\u306e\u3067\u3001\u4e0b\u8a18\u306e\u901a\u308a\u306b\u76f8\u95a2\u4fc2\u6570\u3092\u51fa\u3059\uff1a\n\n\\begin{align}\n\\text{corr}\\,(\\widehat{y},y) = \\frac{\\text{cov}\\,(\\widehat{y},y)}{\\sqrt{\\text{var}\\,(\\widehat{y})\\text{var}\\,(y)}},\n\\end{align}\n\n\u3053\u308c\u306f`scipy.stats.pearsonr`\u3067\u8a08\u7b97\u3067\u304d\u308b\u3002\u307e\u305f\u3001\u4e88\u6e2c\u30fb\u771f\u306e\u5e73\u5747\u7684\u306a2\u4e57\u8aa4\u5dee\uff08root mean squared error; RMSE\uff09\u3092\u6c42\u3081\u308b\u3053\u3068\u3082\u3067\u304d\u308b\uff1a\n\n\\begin{align}\n\\text{RMSE}\\,(\\widehat{y},y) = \\left( \\frac{1}{m} \\sum_{i=1}^{m} (\\widehat{y}_{i}-y_{i})^2 \\right)^{1/2},\n\\end{align}\n\n\u3053\u308c\u306f\u30e2\u30c7\u30eb\u30aa\u30d6\u30b8\u30a7\u30af\u30c8\u306e`mod.eval`\u3068\u3044\u3046\u30e1\u30bd\u30c3\u30c9\u3067\u5b9f\u88c5\u3057\u3066\u3044\u308b\u3002$m$\u306f\u691c\u8a3c\u30c7\u30fc\u30bf\u306e\u30b5\u30f3\u30d7\u30eb\u6570\uff08\u3053\u3053\u3067\u306f$m=540$\uff09\u3002\n\n\n\n```python\n\nimport numpy as np\nimport math\nimport pickle\nimport support.parse_model as mp\nimport scripts.AlgoSparseReg as asr\n\n# Load the general data info object.\nwith open(\"data/encoder/info.dat\", mode=\"br\") as fbin:\n dinfo = pickle.load(fbin)\n\n# Load the clean index, extracting indices from binary indicators.\nwith open(\"data/encoder/cleanidx_tr.dat\", mode=\"br\") as fbin:\n #cleanidx_tr_RAW = np.fromfile(file=fbin, dtype=np.uint32)\n cleanidx_tr = np.flatnonzero(np.fromfile(file=fbin, dtype=np.uint32))\n print(\"length:\", cleanidx_tr.size)\nwith open(\"data/encoder/cleanidx_te.dat\", mode=\"br\") as fbin:\n #cleanidx_te_RAW = np.fromfile(file=fbin, dtype=np.uint32)\n cleanidx_te = np.flatnonzero(np.fromfile(file=fbin, dtype=np.uint32))\n print(\"length:\", cleanidx_te.size)\n \n# Take the intersection of the clean voxel indices and sort.\ncleanidx = np.intersect1d(cleanidx_tr,cleanidx_te)\nprint(cleanidx_tr)\nprint(cleanidx_te)\nprint(\"cleanidx length:\", cleanidx.size)\n\n# Load the data info object. We shall modify its voxel index on the fly.\nwith open(\"data/encoder/info.dat\", mode=\"br\") as fbin:\n dinfo = pickle.load(fbin)\n print(dinfo)\n\n# Initialize model and weights for an individual voxel.\ndinfo.misc[\"voxidx\"] = cleanidx[0]\nmod = mp.model(dinfo)\nw_init = mod.w_initialize()\n\n\nlam_min = 1 / mod.n\n\n# TODO: set the START and STOP guys with proper computations using mod.X_tr and mod.y_tr.\ntodo_lambda = np.flipud(np.logspace(start=math.log10(lam_min),\n stop=math.log10(1),\n num=150))\n\n# Store performance metric statistics for each lambda setting.\nerr_overlam = np.zeros(todo_lambda.size, dtype=np.float32)\ncorr_overlam = np.zeros(todo_lambda.size, dtype=np.float32)\nspar_overlam = np.zeros(todo_lambda.size, dtype=np.float32)\n\n\n```\n\n length: 59928\n length: 63050\n [ 4355 4356 4357 ..., 73567 73568 73569]\n [ 300 301 302 ..., 73725 73726 73727]\n cleanidx length: 59928\n X_tr: {'shape': (7200, 90), 'path': 'data/encoder/X_tr.dat', 'dtype': dtype('float32')}\n X_te: {'shape': (540, 90), 'path': 'data/encoder/X_te.dat', 'dtype': dtype('float32')}\n y_tr: {'shape': (73728, 7200), 'path': 'data/encoder/y_tr.dat', 'dtype': }\n y_te: {'shape': (73728, 540), 'path': 'data/encoder/y_te.dat', 'dtype': }\n mname: Encoder\n misc: {'voxidx': None}\n\n\n\n```python\n# Iterate over the lambda values once, for a single candidate.\nprint(\"Working...\")\nfor i in range(todo_lambda.size):\n \n # Initialize and execute the algorithm.\n al = asr.Algo_LASSO_CD(w_init=w_init,\\\n t_max=20*w_init.size,\\\n lam_l1=todo_lambda[i],\\\n verbose=False)\n \n for mystep in al:\n al.update(model=mod)\n \n # Record performance.\n w_est = al.w\n err_overlam[i] = mod.eval(w_est)[0]\n if np.std(np.dot(mod.X_te, w_est)) > 0:\n corrval = mod.corr_te(w_est)\n else:\n corrval = 0 # watch out for zero-variance case.\n corr_overlam[i] = corrval\n spar_overlam[i] = np.count_nonzero(w_est)\n \n # Update the initializer to the most current observation.\n w_init = w_est\n\nprint(\"Done.\")\n```\n\n Working...\n Done.\n\n\n\u5b66\u7fd2\u304c\u7d42\u308f\u308b\u3068\u3001\u305d\u306e\u751f\u306e\u6210\u7e3e\u3092`raw/encoder`\u3068\u3044\u3046\u30c7\u30a3\u30ec\u30af\u30c8\u30ea\u306b\u4fdd\u5b58\u3057\u3066\u304a\u304f\u3002\n\n\n```python\nwith open(\"raw/encoder/lasso01_lam.raw\", mode=\"bw\") as fbin:\n pickle.dump(todo_lambda, fbin)\n print(\"Saved to file.\")\n\nwith open(\"raw/encoder/lasso01_err.raw\", mode=\"bw\") as fbin:\n pickle.dump(err_overlam, fbin)\n print(\"Saved to file.\")\n \nwith open(\"raw/encoder/lasso01_corr.raw\", mode=\"bw\") as fbin:\n pickle.dump(corr_overlam, fbin)\n print(\"Saved to file.\")\n \nwith open(\"raw/encoder/lasso01_spar.raw\", mode=\"bw\") as fbin:\n pickle.dump(spar_overlam, fbin)\n print(\"Saved to file.\")\n```\n\n Saved to file.\n Saved to file.\n Saved to file.\n Saved to file.\n\n\n\n## \u5b66\u7fd2\u6a5f\u306e\u51fa\u6765\u3092\u8a55\u4fa1\u3059\u308b\n\n\u5b66\u7fd2\u304c\u7d42\u308f\u3063\u3066\u3001\u3042\u3068\u306f\u6210\u7e3e\u3092\u96c6\u8a08\u3057\u305f\u308a\u3001\u53ef\u8996\u5316\u3057\u305f\u308a\u3059\u308b\u3060\u3051\u3067\u3042\u308b\u3002\nJupyter\u306e\u30ab\u30fc\u30cd\u30eb\u3092\u30ea\u30bb\u30c3\u30c8\u3057\u3001\u3053\u3053\u304b\u3089\u65b0\u305f\u306a\u4f5c\u696d\u3092\u59cb\u3081\u308b\u3002\n\n\n```python\nimport pickle\nimport matplotlib\nimport matplotlib.pyplot as plt\n\nwith open(\"raw/encoder/lasso01_lam.raw\", mode=\"br\") as fbin:\n todo_lambda = pickle.load(fbin)\n\nwith open(\"raw/encoder/lasso01_err.raw\", mode=\"br\") as fbin:\n err_overlam = pickle.load(fbin)\n \nwith open(\"raw/encoder/lasso01_corr.raw\", mode=\"br\") as fbin:\n corr_overlam = pickle.load(fbin)\n \nwith open(\"raw/encoder/lasso01_spar.raw\", mode=\"br\") as fbin:\n spar_overlam = pickle.load(fbin)\n\nmyfig = plt.figure(figsize=(18,4))\nax_err = myfig.add_subplot(1, 3, 1)\nplt.ylabel(\"Root mean squared error (RMSE)\")\nplt.xlabel(\"Lambda values\")\nax_err.set_yscale('log')\nax_err.set_xscale('log')\nax_err.plot(todo_lambda, err_overlam)\n\nax_corr = myfig.add_subplot(1, 3, 2)\nplt.ylabel(\"Correlation coefficient\")\nplt.xlabel(\"Lambda values\")\nax_corr.set_xscale('log')\nax_corr.plot(todo_lambda, corr_overlam)\n\nax_spar = myfig.add_subplot(1, 3, 3)\nplt.ylabel(\"Number of non-zero weights\")\nplt.xlabel(\"Lambda values\")\nax_spar.set_xscale('log')\nax_spar.plot(todo_lambda, spar_overlam)\n\nplt.show()\n```\n\n\u660e\u3089\u304b\u306a\u3088\u3046\u306b\u3001\u3053\u308c\u307b\u3069\u5358\u7d14\u306a\u30e2\u30c7\u30eb\u3068\u6063\u610f\u7684\u306a\u5b66\u7fd2\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u8a2d\u5b9a\u3067\u306f\u3001\u307e\u3063\u305f\u304f\u4e88\u6e2c\u304c\u3067\u304d\u306a\u3044\u3002\n\n\u3055\u307e\u3056\u307e\u306a\u8981\u56e0\u306f\u3042\u308b\u304c\u3001\u7279\u306b\u6ce8\u8996\u3059\u3079\u304d\u6539\u5584\u70b9\u306f\u4e0b\u8a18\u306e\u901a\u308a\u3067\u3042\u308b\u3002\n\n- \u521d\u671f\u5024`w_init`\u306e\u6c7a\u3081\u65b9\u3002\n\n- $\\lambda$\u306e\u5019\u88dc\u306e\u7bc4\u56f2\u3068\u5bc6\u5ea6\u3002\n\n- \u5b66\u7fd2\u6a5f\u306e\u53cd\u5fa9\u56de\u6570\u306e\u5236\u9650`t_max`\u3002\n\n- \u30d5\u30a3\u30eb\u30bf\u30d0\u30f3\u30af\u3092\u5b9a\u3081\u308b\u3042\u3089\u3086\u308b\u30d1\u30e9\u30e1\u30fc\u30bf\u304c\u5927\u4e8b\u3060\u304c\u3001\u7279\u306b\u91cd\u8981\u306a\u306e\u306f`freqs`\u3001`dir`\u3001`sdev`\u3067\u3042\u308d\u3046\u3002\n\n- \u30d5\u30a3\u30eb\u30bf\u30d0\u30f3\u30af\u306e\u8c4a\u5bcc\u3055\u3002\u30b0\u30ea\u30c3\u30c9\u306e\u89e3\u50cf\u5ea6\u306e\u9ad8\u4f4e\u3001\u591a\u69d8\u306a\u7a7a\u9593\u5468\u6ce2\u6570\u3001\u5411\u304d\u306a\u3069\u3092\u307e\u3093\u3079\u3093\u306a\u304f\u542b\u3080\u3053\u3068\u304c\u5fc5\u9808\u3002\n\n\u3053\u308c\u3089\u306e\u6539\u5584\u70b9\u3092\u5ff5\u982d\u306b\u304a\u3044\u3066\u3001\u6b21\u306e\u300c\u8ab2\u984c\u4e00\u89a7\u300d\u306e\u7df4\u7fd2\u554f\u984c\u306b\u53d6\u308a\u7d44\u3093\u3067\u304f\u3060\u3055\u3044\u3002\n\n\n## \u8ab2\u984c\u4e00\u89a7\n\n0. \u3059\u3079\u3066\u306e\u30dc\u30af\u30bb\u30eb\u5206\u306e\u5b66\u7fd2\u3092\u3059\u308b\u5fc5\u8981\u306f\u306a\u304f\u3001Nishimoto et al.\u306e\u3044\u3046\u300cearly visual areas\u300d\u306b\u7126\u70b9\u3092\u5f53\u3066\u308b\u3053\u3068\u3002\u305d\u308c\u306f\u4e21\u534a\u7403\u306b\u304a\u3044\u3066V1\u3001V2\u3001V3\u3001V3A\u3001V3B\u3068\u3044\u3046ROI\u306e\u3053\u3068\u3092\u6307\u3059\u3002\u8a13\u7df4\u30c7\u30fc\u30bf\u3092\u4f7f\u3044\u3001\u3053\u308c\u3089\u306e\u3059\u3079\u3066\u306e\u30af\u30ea\u30fc\u30f3\u306a\u30dc\u30af\u30bb\u30eb\u3092\u5bfe\u8c61\u306b\u5b66\u7fd2\u3055\u305b\u3001\u691c\u8a3c\u30c7\u30fc\u30bf\u306b\u304a\u3044\u3066\u3001\u8a13\u7df4\u30c7\u30fc\u30bf\u3067\u3082\u3063\u3068\u3082\u826f\u304b\u3063\u305f$\\lambda$\u306e\u3068\u304d\u306e\u5b66\u7fd2\u7d50\u679c\u306e\u6210\u7e3e\u3092\u7b97\u51fa\u3059\u308b\u3053\u3068\u3002ROI\u3054\u3068\u306b\u3001\u3053\u308c\u3089\u306e\u6210\u7e3e\uff08\u76f8\u95a2\u4fc2\u6570\u7b49\uff09\u306e\u30dc\u30af\u30bb\u30eb\u306e\u4e0a\u3067\u306e\u5e73\u5747\u3068\u5206\u6563\u3092\u6c42\u3081\u308b\u3053\u3068\u3002\n\n0. \u7279\u5fb4\u91cf\u306e\u914d\u5217\u3092\u9023\u7d50\u3055\u305b\u308b\u3053\u3068\u3067\u3001\u69d8\u3005\u306a\u30d5\u30a3\u30eb\u30bf\u304b\u3089\u306e\u7279\u5fb4\u91cf\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u5b9f\u969b\u3001\u826f\u3044\u8aac\u660e\u80fd\u529b\u3092\u5b9f\u73fe\u3057\u3088\u3046\u3068\u601d\u3048\u3070\u3001\u304a\u305d\u3089\u304f\u4e0d\u53ef\u6b20\u306a\u4f5c\u696d\u3067\u3042\u308d\u3046\u3002\u5f15\u7528\u3057\u3066\u3044\u308b\u8ad6\u6587\u3067\u3082\u3001\u591a\u6570\u306e\u7a7a\u9593\u5468\u6ce2\u6570\u3068\u5411\u304d\u306e\u7d44\u307f\u5408\u308f\u305b\u304c\u4f7f\u308f\u308c\u3066\u3044\u308b\u3002\u8ab2\u984c\u3068\u3057\u3066\u3001\u4e92\u3044\u306b\u7570\u306a\u308b\u7279\u5fb4\u91cf\u3092\u6349\u3048\u308b\u30d5\u30a3\u30eb\u30bf\u30d0\u30f3\u30af\u3092\uff12\u3064\u4ee5\u4e0a\u7528\u610f\u3057\u3001\u305d\u308c\u305e\u308c\u3092\u5b66\u7fd2\u7528\u306e\u4f7f\u3046\u3053\u3068\u3002\u305d\u306eROI\u3054\u3068\u306b\u7d50\u679c\u3092\u8e0f\u307e\u3048\u3066\u3001\u3069\u306e\u9818\u57df\u3067\u3069\u306e\u3088\u3046\u306a\u60c5\u5831\u304c\u5fc5\u8981\u304b\u3068\u601d\u308f\u308c\u308b\u304b\u3002\u307e\u305f\u3001\u8133\u306e\u90e8\u4f4d\u306e\u300c\u9078\u629e\u6027\u300d\u306b\u3064\u3044\u3066\u4f55\u304c\u3044\u3048\u308b\u304b\u3002\n\n0. \u5148\u306e\u8ab2\u984c\u3092\u3001\u5168\u88ab\u9a13\u8005\u5206\u306e\u30c7\u30fc\u30bf\u3092\u4f7f\u3063\u3066\u884c\u306a\u3046\u3053\u3068\u3002\u88ab\u9a13\u8005\u306e\u9593\u3001\u540c\u3058\u30e2\u30c7\u30eb\u3068\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u4f7f\u3063\u3066\u3001\u30d1\u30d5\u30a9\u30fc\u30de\u30f3\u30b9\u306b\u5dee\u7570\u304c\u3042\u308b\u304b\u3002\u88ab\u9a13\u8005\u306e\u4e0a\u3067\u3082\u5e73\u5747\u7684\u306a\u6210\u7e3e\u3092\u6c42\u3081\u3066\u3001\u5f15\u7528\u3057\u3066\u3044\u308b\u8ad6\u6587\u306e\u6210\u7e3e\u3068\u52dd\u8ca0\u3067\u304d\u308b\u304b\u3002\n\n0. \u624b\u6cd5\u306e\u66f4\u306a\u308b\u5f37\u5316\u3092\u6e2c\u308b\u305f\u3081\u3001\u6642\u9593\u306e\u9045\u5ef6\u3092\u5229\u7528\u3057\u3066\u7279\u5fb4\u91cf\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3064\u307e\u308a\u3001\u524d\u306e\u6642\u70b9\u306e\u7279\u5fb4\u91cf\u30d9\u30af\u30c8\u30eb\u3092\u3001\u73fe\u6642\u70b9\u306e\u7279\u5fb4\u91cf\u30d9\u30af\u30c8\u30eb\u306b\u9023\u7d50\u3055\u305b\u308b\u3068\u3044\u3046\u65b9\u6cd5\u3002\u73fe\u6642\u70b9\u306e\u7279\u5fb4\u91cf\u3092$x_{i}$\u3068\u3059\u308b\u3068\u3001$x_{i-1}$\u304c\u524d\u306e\u6642\u70b9\u306b\u76f8\u5f53\u3059\u308b\u306e\u3067\u3001\u9023\u7d50\u3057\u305f\u7d50\u679c\u304c$\\widetilde{x}_{i} = (x_{i},x_{i-1})$\u306b\u306a\u3063\u3066\u3001\u3053\u306e$\\widetilde{x}_{i}$\u3092\u5b66\u7fd2\u306b\u4f7f\u3046\u3053\u3068\u306b\u306a\u308b\u3002\u9045\u5ef6\u304c$k$\u6642\u70b9\u306a\u3089\u3001$\\widetilde{x}_{i} = (x_{i},x_{i-1},\\ldots,x_{i-k})$\u3068\u306a\u308b\u3002\u6570\u30b9\u30c6\u30c3\u30d7\u3092\u8a66\u3057\u3066\u3001\u30d1\u30d5\u30a9\u30fc\u30de\u30f3\u30b9\u304c\u3082\u3063\u3068\u3082\u826f\u3044\u9045\u5ef6\u306f\u3044\u304f\u3089\u304b\u3002\n\n0. \u5b66\u7fd2\u7d50\u679c\u306e\u826f\u3057\u60aa\u3057\u304cROI\u306b\u3088\u3063\u3066\u3069\u306e\u7a0b\u5ea6\u5909\u308f\u308b\u304b\u3002\u7279\u6bb5\u826f\u3044\u30fb\u60aa\u3044\u6210\u7e3e\u304c\u898b\u3089\u308c\u305fROI\u304c\u3042\u308c\u3070\u3001\u305d\u308c\u306f\u3069\u308c\u304b\u3002\n\n0. \u30e2\u30c7\u30eb\u306f\u5927\u4e8b\u3060\u304c\u3001\u5b66\u7fd2\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3082\u304d\u308f\u3081\u3066\u5927\u304d\u306a\u5f79\u5272\u3092\u679c\u305f\u3059\u3002\u30d9\u30b9\u30c8\u306a\u8a2d\u5b9a\u3092\u63a2\u308b\u3079\u304f\u3001\u53cd\u5fa9\u56de\u6570\u306a\u3069\u306e\u7d42\u4e86\u6761\u4ef6\u3084$\\lambda$\u306e\u5019\u88dc\u7bc4\u56f2\u3001\u305d\u306e\u4ed6\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u5de5\u592b\u3092\u8abf\u3079\u308b\u3053\u3068\u3002\u3069\u306e\u3088\u3046\u306a\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u8a2d\u5b9a\u304c\u3082\u3063\u3068\u3082\u826f\u3044\u6210\u7e3e\u306b\u3064\u306a\u304c\u3063\u305f\u304b\u3002\u72ec\u81ea\u306e\u6539\u9020\u3092\u884c\u3063\u305f\u5834\u5408\u3001\u305d\u308c\u3082\u4f75\u305b\u3066\u8aac\u660e\u3059\u308b\u3053\u3068\u3002\n\n0. \uff08\u304a\u307e\u3051\uff09\u3053\u308c\u307e\u3067\u306f\u540c\u4e00\u88ab\u9a13\u8005\u3092\u524d\u63d0\u3068\u3057\u3066\u3001\u300c\u6c4e\u5316\u80fd\u529b\u300d\u3092\u8a55\u4fa1\u3057\u3066\u304d\u305f\u3002\u5f53\u7136\u306a\u304c\u3089\u3001\u88ab\u9a13\u8005\u9593\u306e\u6c4e\u5316\u80fd\u529b\u3082\u6ce8\u76ee\u306b\u5024\u3059\u308b\u3002\u3064\u307e\u308a\u30012\u540d\u306e\u88ab\u9a13\u8005\u306e\u30c7\u30fc\u30bf\u3092\u8db3\u3057\u5408\u308f\u305b\u3066\u3001\u5927\u304d\u306a\u30c7\u30fc\u30bf\u30bb\u30c3\u30c8\u306b\u307e\u3068\u3081\u3001\u8a13\u7df4\u30c7\u30fc\u30bf\u3068\u3057\u3066\u4f7f\u3063\u3066\u5b66\u7fd2\u3057\u3066\u304b\u3089\u3001\u6b8b\u308a1\u540d\u306e\u88ab\u9a13\u8005\u306e\u30c7\u30fc\u30bf\u3092\u691c\u8a3c\u30c7\u30fc\u30bf\u3068\u3059\u308b\u3002\u540c\u3058\u30e2\u30c7\u30eb\u3068\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3067\u3082\u5341\u5206\u306a\u306e\u304b\u3002\u305d\u308c\u3068\u3082\u3001\u65b0\u305f\u306a\u5de5\u592b\u304c\u5fc5\u8981\u3060\u3063\u305f\u304b\u3002\u307e\u305f\u3001\u3053\u306e\u3068\u304d\u306e\u6c4e\u5316\u80fd\u529b\u306e\u89e3\u91c8\u304c\u3001\u88ab\u9a13\u8005\u3054\u3068\u306b\u5b66\u7fd2\u3059\u308b\u5834\u5408\u3068\u6bd4\u3079\u3066\u3001\u3069\u3046\u9055\u3046\u304b\u3002\n\n\n## \u53c2\u8003\u6587\u732e\uff1a\n\n - Nishimoto, Shinji, et al. \"Reconstructing visual experiences from brain activity evoked by natural movies.\" Current Biology 21.19 (2011): 1641-1646.\n - Description of dataset vim-2 (visual imaging 2), at CRCNS - Collaborative Research in Computational Neuroscience. https://crcns.org/data-sets/vc/vim-2/about-vim-2\n", "meta": {"hexsha": "5f9b5753492ba2b03e2cc86234f338a80c0bca6b", "size": 65654, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "FinishEncoderJPN.ipynb", "max_stars_repo_name": "ytakzk/learnml-exercises", "max_stars_repo_head_hexsha": "db436b0aa44ffbe77374c8a2d3076bc1703dec42", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "FinishEncoderJPN.ipynb", "max_issues_repo_name": "ytakzk/learnml-exercises", "max_issues_repo_head_hexsha": "db436b0aa44ffbe77374c8a2d3076bc1703dec42", 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YES\n2. YES", "lm_q1_score": 0.6723316991792861, "lm_q2_score": 0.6442250996557036, "lm_q1q2_score": 0.4331329559054642}} {"text": "#
    Models and Pricing of Financial Derivativs HW_01
    \n\n**
    11510691 \u7a0b\u8fdc\u661f
    **\n\n\n\n## Question 1\n\n$\\DeclareMathOperator*{\\argmin}{argmin}\n\\DeclareMathOperator*{\\argmax}{argmax}\n\\newcommand{\\using}[1]{\\stackrel{\\mathrm{#1}}{=}}\n\\newcommand{\\ffrac}{\\displaystyle \\frac}\n\\newcommand{\\space}{\\text{ }}\n\\newcommand{\\bspace}{\\;\\;\\;\\;}\n\\newcommand{\\QQQ}{\\boxed{?\\:}}\n\\newcommand{\\void}{\\left.\\right.}\n\\newcommand{\\CB}[1]{\\left\\{ #1 \\right\\}}\n\\newcommand{\\SB}[1]{\\left[ #1 \\right]}\n\\newcommand{\\P}[1]{\\left( #1 \\right)}\n\\newcommand{\\dd}{\\mathrm{d}}\n\\newcommand{\\Tran}[1]{{#1}^{\\mathrm{T}}}\n\\newcommand{\\d}[1]{\\displaystyle{#1}}\n\\newcommand{\\EE}[2][\\,\\!]{\\mathbb{E}_{#1}\\left[#2\\right]}\n\\newcommand{\\Var}[2][\\,\\!]{\\mathrm{Var}_{#1}\\left[#2\\right]}\n\\newcommand{\\Cov}[2][\\,\\!]{\\mathrm{Cov}_{#1}\\left(#2\\right)}\n\\newcommand{\\Corr}[2][\\,\\!]{\\mathrm{Corr}_{#1}\\left(#2\\right)}\n\\newcommand{\\I}[1]{\\mathrm{I}\\left( #1 \\right)}\n\\newcommand{\\N}[1]{\\mathrm{N} \\left( #1 \\right)}\n\\newcommand{\\ow}{\\text{otherwise}}\\bspace$Selling a call option: As the writer of the call option, I give the holder the right to buy an asset at a specified time $T$ for a specified price $K$. My payoff would be $-\\max\\P{S_T - K,0}$ for european call options. If I sold an american call option, the holder can exercise at any time before $T$.\n\n$\\bspace$Buying a put option: As the holder of the put option, I actually was granted the right to sell an asset at a specified time $T$ for a specified price $K$. My payoff would be $\\max\\P{K - S_T, 0}$ Or before $T$ if what I bought is an american put option.\n\n## Question 2\n\n$\\bspace$We can write their profit function on the stock price $S_t$.\n\n- Stock: $100\\P{S_t - 94}$\n- Option: $2000\\big(\\max\\P{S_t - 95,0} - 4.7\\big)$\n\n$\\bspace$They intersect at two points, $\\P{0,0}$ and $\\P{100,600}$. It's generally acknowledge that it's of higher possibility that the stock price moves less than more, thus I personally think that holding the stocks rather than the options have a better chance to profit.\n\n$\\bspace$As for the second question, since we've already acquire their intersection, we can say that when the stock price goes higher than $100$, options will win more.\n\n## Question 3\n\n$\\bspace$The trader now in the call holder's position. He has paid $c$ to buy the right that he can use $K$ to buy the underlying asset as time $T$. Also in the put writer's position. He has received $p$ and given the right to someone else selling him the asset at price $K$ at time $T$.\n\n$\\bspace$To let the prices equal, by the **Put-call parity**, we have $S_0 = Ke^{-rT}$, the time value of $K$ is equal to the initial price of the asset.\n\n## Question 4\n\n$\\bspace$We first write its payoff function on the stock price:\n\n$$\\begin{align}\np &= 100\\P{S_T - 40} + 100\\SB{5 - \\max\\P{S_T - 50, 0}} + 100\\SB{\\max\\P{30 - S_T,0} - 7}\\\\\n&= \\begin{cases}\n800, &\\text{if } S_T \\geq 50 \\\\\n100S_T - 4200, &\\text{if } 50 \\geq S_T \\geq 30 \\\\\n-1200, &\\text{if } 30 \\geq S_T \\geq 0\n\\end{cases}\n\\end{align}\n$$\n\n\n\nAfter that, the payoff would change to:\n\n$$\\begin{align}\np &= 100\\P{S_T - 40} + 200\\SB{5 - \\max\\P{S_T - 50, 0}} + 200\\SB{\\max\\P{30 - S_T,0} - 7}\\\\\n&= \\begin{cases}\n5600 - 100S_T, &\\text{if } S_T \\geq 50 \\\\\n100S_T - 4400, &\\text{if } 50 \\geq S_T \\geq 30 \\\\\n1600 - 100S_T, &\\text{if } 30 \\geq S_T \\geq 0\n\\end{cases}\n\\end{align}\n$$\n\n\n\n## Question 5\n\n$\\bspace$The lower bound of the option can be obtained using the formula\n\n$\\bspace\\begin{align}\nc &\\geq K e^{-rT} - S_0 \\\\\n&= 15 \\cdot e^{-6\\% \\times 1/12} - 12 \\\\\n&\\approx 2.93\n\\end{align}$\n\n## Qustion 6\n\n$\\bspace$The early exercise of an American put option is to sell the stock to the writer at the Strike price $K$ before the expiration date $T$. Suppose he exercised at time $t$ thus his time value of money is $Ke^{-r\\P{T-t}}$. But then he can not sell the stock at $K$ at time $T$ any more.\n\n## Question 7\n\n$\\bspace$By the put-call parity, we have: $1 + 20 \\times e^{-4\\% \\times 0.25} = p + 19$ thus $p = 1.80$\n\n## Question 8\n\n$\\bspace$Based on the put-call parity, $c + Ke^{-rT} = S_0 + p \\Longrightarrow c + 49.75 = 47+2.5$. Thus there's always a chance for arbitraging. He can use the same strategy that is to buy a stock and a put option using the borrowed money, $49.5$ with interest rate $6\\%$.\n\n$\\bspace$Then he can win $50 - 49.5e^{0.06\\times 1/12} \\approx 0.25 $ if the stock price goes lower than $50$ or more if the stock price goes higher than $50$.\n\n## Question 9\n\n$\\P{1}$\n\n$\\bspace P \\geq p = c + Ke^{-rT}-S_0 = C + Ke^{-rT} - S_0 = 4 + 30 e^{-8\\%\\times1/4} -31 \\approx 2.41$\n\n$\\bspace$And to find something that keeps over an American Put, we can use $K$ cash at the beginning, thus $c + K \\geq P+ S_0$ always holds. Therefore, $P \\leq c + K - S_0 = C + K - S_0 = 4 + 30 - 31 = 3$\n\n$\\P{2}$\n\n$\\bspace$If the American Put price is greater than $3$ that is to say that $P \\geq C + K - S_0$, then we write an American put option and sell it to somebody, then use the money to buy a American call option, to borrow a stock and sell it to gain $S_0$. Send $K$ cash to the bank. Then when the American put option holder want to exercise, we can instantly use $K = 30$ to buy the stock and return to the stock lender. Up to now, we start from nothing to a American call option and a positive payoff and some interest.\n\n## Question 10\n\n$\\P{1}$\n\n$\\bspace$If not, then $2c_2 > c_1 + c_3$. So that we have the arbitrage chance. First write two call option with strike price $K_2$ and then use the money gained to buy two call option with strike price $K_1$ and $K_3$. We already have some money left now.\n\n$\\bspace$Then we the exercise time comes, we can exercise all three at the same time since $2K_2 = K_1 + K_3$, meaning that we gain money from nothing. Thus $2c_2 \\leq c_1 + c_3$.\n\n$\\P{2}$\n\n$$p_2 \\leq 0.5\\P{p_1 + p_3}$$\n\n$\\bspace$The proof is obvious, similar to the preceding one.\n", "meta": {"hexsha": "c5d83b3e2267ccae91545c44f6fbcb05242e54ed", "size": 8982, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "FinMath/Models and Pricing of Financial Derivatives/HW/HW_01.ipynb", "max_stars_repo_name": "XavierOwen/Notes", "max_stars_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2018-11-27T10:31:08.000Z", "max_stars_repo_stars_event_max_datetime": "2019-01-20T03:11:58.000Z", "max_issues_repo_path": "FinMath/Models and Pricing of Financial Derivatives/HW/HW_01.ipynb", "max_issues_repo_name": "XavierOwen/Notes", "max_issues_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "FinMath/Models and Pricing of Financial Derivatives/HW/HW_01.ipynb", "max_forks_repo_name": "XavierOwen/Notes", "max_forks_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-07-14T19:57:23.000Z", "max_forks_repo_forks_event_max_datetime": "2020-07-14T19:57:23.000Z", "avg_line_length": 38.5493562232, "max_line_length": 526, "alphanum_fraction": 0.5557782231, "converted": true, "num_tokens": 2028, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166047041654, "lm_q2_score": 0.8459424334245618, "lm_q1q2_score": 0.4328827898071962}} {"text": "\n# Week 2 January 11-15: Introduction to the course and start Variational Monte Carlo\n\n \n**Morten Hjorth-Jensen Email morten.hjorth-jensen@fys.uio.no**, Department of Physics and Center fo Computing in Science Education, University of Oslo, Oslo, Norway and Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, East Lansing, Michigan, USA\n\nDate: **Feb 11, 2021**\n\nCopyright 1999-2021, Morten Hjorth-Jensen Email morten.hjorth-jensen@fys.uio.no. Released under CC Attribution-NonCommercial 4.0 license\n\n\n\n\n## Overview of week 2\n**Topics.**\n\n* Introduction to the course and overview of topics to be covered\n\n* Introduction to Variational Monte Carlo methods, Metropolis Algorithm, statistics and Markov Chain theory\n\n\n\n**Teaching Material, videos and written material.**\n\n* Asynchronuous vidoes\n\n* Lecture notes and reading assignments\n\n* Additional (often recommended) background material\n\n\n\n## Textbook\n\nThere are no unique textbooks which cover the material to be discussed. For each week however, we will, in addition to our own lecture notes, send links to additional literature. This can be articles or chapters from other textbooks.\nA useful textbook is however \n\n* [Bernd A. Berg, *Markov Chain Monte Carlo Simulations and their Statistical Analysis*, World Scientific, 2004](https://www.worldscientific.com/worldscibooks/10.1142/5602), chapters 1, 2\n\nThis book has its main focus on spin-models, but many of the concepts are general. Chapters 1 and 2 contain a good discussion of the statistical foundation. \n\n## Aims\n* Be able to apply central many-particle methods like the Variational Monte Carlo method to properties of many-fermion systems and many-boson systems.\n\n* Understand how to simulate quantum mechanical systems with many interacting particles. The methods are relevant for atomic, molecular, solid state, materials science, nanotechnology, quantum chemistry and nuclear physics. \n\n* Learn to manage and structure larger projects, with unit tests, object orientation and writing clean code\n\n* Learn about a proper statistical analysis of large data sets\n\n* Learn to optimize with convex optimization methods functions that depend on many variables.\n\n* Parallelization and code optimizations\n\n\n\n\n\n## Lectures and ComputerLab\n\n * Lectures: Thursday (2.15pm-4pm). First time January 14. Last lecture May 6.\n\n * Computerlab: Thursday (4.15pm-7pm), first time January 14, last lab session May 6.\n\n * Weekly plans and all other information are on the webpage of the course\n\n * **First project to be handed in March 26**.\n\n * **Second and final project to be handed in May 31.**\n\n * There is no final exam, only project work.\n\n\n\n## Course Format\n\n * Two compulsory projects. Electronic reports only. You are free to choose your format. We use devilry to hand in the projects.\n\n * Evaluation and grading: The two projects count 1/2 each of the final mark. No exam.\n\n * The computer lab (room 397 in the Physics buidling) has no PCs, so please bring your own laptops. C/C++ is the default programming language, but programming languages like Fortran2008, Rust, Julia, and/or Python can also be used. All source codes discussed during the lectures can be found at the webpage of the course.\n\n\n\n\n## Topics covered in this course\n * Parallelization (MPI and OpenMP), high-performance computing topics. Choose between Python, Fortran2008 and/or C++ as programming languages. \n\n * Algorithms for Monte Carlo Simulations (multidimensional integrals), Metropolis-Hastings and importance sampling algorithms. Improved Monte Carlo methods.\n\n * Statistical analysis of data from Monte Carlo calculations, bootstrapping, jackknife and blocking methods. \n\n * Eigenvalue solvers\n\n * For project 2 there will be at least three variants:\n\na. Variational Monte Carlo for fermions\n\nb. Hartree-Fock theory for fermions\n\nc. Coupled cluster theory for fermions (iterative methods)\n\nd. Neural networks and Machine Learning to solve the same problems as in project 1\n\ne. Eigenvalue problems with deep learning methods\n\nf. Possible project on quantum computing\n\n\n\n## Topics covered in this course\n * Search for minima in multidimensional spaces (conjugate gradient method, steepest descent method, quasi-Newton-Raphson, Broyden-Jacobian). Convex optimization, gradient methods\n\n * Iterative methods for solutions of non-linear equations.\n\n * Object orientation\n\n * Data analysis and resampling techniques\n\n * Variational Monte Carlo (VMC) for 'ab initio' studies of quantum mechanical many-body systems.\n\n * Simulation of two- and three-dimensional systems like quantum dots or atoms and molecules or systems from solid state physics\n\n * **Simulation of trapped bosons using VMC (project 1, default)**\n\n * **Machine learning and neural networks (project 2, default, same system as in project 1)**\n\n * Extension of project 1 to fermionic systems (project 2)\n\n * Coupled cluster theory (project 2, depends on interest)\n\n * Other quantum-mechanical methods and systems can be tailored to one's interests (Hartree-Fock Theory, Many-body perturbation theory, time-dependent theories and more).\n\n\n\n\n\n## Quantum Monte Carlo Motivation\n\nMost quantum mechanical problems of interest in for example atomic, molecular, nuclear and solid state \nphysics consist of a large number of interacting electrons and ions or nucleons. \n\nThe total number of particles $N$ is usually sufficiently large\nthat an exact solution cannot be found. \n\nTypically, \nthe expectation value for a chosen hamiltonian for a system of $N$ particles is\n\n$$\n\\langle H \\rangle =\n \\frac{\\int d\\boldsymbol{R}_1d\\boldsymbol{R}_2\\dots d\\boldsymbol{R}_N\n \\Psi^{\\ast}(\\boldsymbol{R_1},\\boldsymbol{R}_2,\\dots,\\boldsymbol{R}_N)\n H(\\boldsymbol{R_1},\\boldsymbol{R}_2,\\dots,\\boldsymbol{R}_N)\n \\Psi(\\boldsymbol{R_1},\\boldsymbol{R}_2,\\dots,\\boldsymbol{R}_N)}\n {\\int d\\boldsymbol{R}_1d\\boldsymbol{R}_2\\dots d\\boldsymbol{R}_N\n \\Psi^{\\ast}(\\boldsymbol{R_1},\\boldsymbol{R}_2,\\dots,\\boldsymbol{R}_N)\n \\Psi(\\boldsymbol{R_1},\\boldsymbol{R}_2,\\dots,\\boldsymbol{R}_N)},\n$$\n\nan in general intractable problem.\n\n This integral is actually the starting point in a Variational Monte Carlo calculation. **Gaussian quadrature: Forget it**! Given 10 particles and 10 mesh points for each degree of freedom\nand an\n ideal 1 Tflops machine (all operations take the same time), how long will it take to compute the above integral? The lifetime of the universe is of the order of $10^{17}$ s.\n\n\n\n\n## Quantum Monte Carlo Motivation\nAs an example from the nuclear many-body problem, we have Schroedinger's equation as a differential equation\n\n$$\n\\hat{H}\\Psi(\\boldsymbol{r}_1,..,\\boldsymbol{r}_A,\\alpha_1,..,\\alpha_A)=E\\Psi(\\boldsymbol{r}_1,..,\\boldsymbol{r}_A,\\alpha_1,..,\\alpha_A)\n$$\n\nwhere\n\n$$\n\\boldsymbol{r}_1,..,\\boldsymbol{r}_A,\n$$\n\nare the coordinates and\n\n$$\n\\alpha_1,..,\\alpha_A,\n$$\n\nare sets of relevant quantum numbers such as spin and isospin for a system of $A$ nucleons ($A=N+Z$, $N$ being the number of neutrons and $Z$ the number of protons).\n\n\n\n\n## Quantum Monte Carlo Motivation\nThere are\n\n$$\n2^A\\times \\left(\\begin{array}{c} A\\\\ Z\\end{array}\\right)\n$$\n\ncoupled second-order differential equations in $3A$ dimensions.\n\nFor a nucleus like beryllium-10 this number is **215040**.\nThis is a truely challenging many-body problem.\n\nMethods like partial differential equations can at most be used for 2-3 particles.\n\n\n\n\n## Various many-body methods\n* Monte-Carlo methods\n\n* Renormalization group (RG) methods, in particular density matrix RG\n\n* Large-scale diagonalization (Iterative methods, Lanczo's method, dimensionalities $10^{10}$ states)\n\n* Coupled cluster theory, favoured method in quantum chemistry, molecular and atomic physics. Applications to ab initio calculations in nuclear physics as well for large nuclei.\n\n* Perturbative many-body methods \n\n* Green's function methods\n\n* Density functional theory/Mean-field theory and Hartree-Fock theory\n\nThe physics of the system hints at which many-body methods to use.\n\n\n\n\n\n## Quantum Monte Carlo Motivation\n**Pros and Cons of Monte Carlo.**\n\n* Is physically intuitive.\n\n* Allows one to study systems with many degrees of freedom. Diffusion Monte Carlo (DMC) and Green's function Monte Carlo (GFMC) yield in principle the exact solution to Schroedinger's equation.\n\n* Variational Monte Carlo (VMC) is easy to implement but needs a reliable trial wave function, can be difficult to obtain. This is where we will use Hartree-Fock theory to construct an optimal basis.\n\n* DMC/GFMC for fermions (spin with half-integer values, electrons, baryons, neutrinos, quarks) has a sign problem. Nature prefers an anti-symmetric wave function. PDF in this case given distribution of random walkers.\n\n* The solution has a statistical error, which can be large. \n\n* There is a limit for how large systems one can study, DMC needs a huge number of random walkers in order to achieve stable results. \n\n* Obtain only the lowest-lying states with a given symmetry. Can get excited states with extra labor.\n\n\n\n\n\n## Quantum Monte Carlo Motivation\n**Where and why do we use Monte Carlo Methods in Quantum Physics.**\n\n* Quantum systems with many particles at finite temperature: Path Integral Monte Carlo with applications to dense matter and quantum liquids (phase transitions from normal fluid to superfluid). Strong correlations.\n\n* Bose-Einstein condensation of dilute gases, method transition from non-linear PDE to Diffusion Monte Carlo as density increases.\n\n* Light atoms, molecules, solids and nuclei. \n\n* Lattice Quantum-Chromo Dynamics. Impossible to solve without MC calculations. \n\n* Simulations of systems in solid state physics, from semiconductors to spin systems. Many electrons active and possibly strong correlations.\n\n\n\n## Quantum Monte Carlo Motivation\nWe start with the variational principle.\nGiven a hamiltonian $H$ and a trial wave function $\\Psi_T$, the variational principle states that the expectation value of $\\langle H \\rangle$, defined through\n\n$$\nE[H]= \\langle H \\rangle =\n \\frac{\\int d\\boldsymbol{R}\\Psi^{\\ast}_T(\\boldsymbol{R})H(\\boldsymbol{R})\\Psi_T(\\boldsymbol{R})}\n {\\int d\\boldsymbol{R}\\Psi^{\\ast}_T(\\boldsymbol{R})\\Psi_T(\\boldsymbol{R})},\n$$\n\nis an upper bound to the ground state energy $E_0$ of the hamiltonian $H$, that is\n\n$$\nE_0 \\le \\langle H \\rangle .\n$$\n\nIn general, the integrals involved in the calculation of various expectation values are multi-dimensional ones. Traditional integration methods such as the Gauss-Legendre will not be adequate for say the computation of the energy of a many-body system.\n\n\n\n## Quantum Monte Carlo Motivation\nThe trial wave function can be expanded in the eigenstates of the hamiltonian since they form a complete set, viz.,\n\n$$\n\\Psi_T(\\boldsymbol{R})=\\sum_i a_i\\Psi_i(\\boldsymbol{R}),\n$$\n\nand assuming the set of eigenfunctions to be normalized one obtains\n\n$$\n\\frac{\\sum_{nm}a^*_ma_n \\int d\\boldsymbol{R}\\Psi^{\\ast}_m(\\boldsymbol{R})H(\\boldsymbol{R})\\Psi_n(\\boldsymbol{R})}\n {\\sum_{nm}a^*_ma_n \\int d\\boldsymbol{R}\\Psi^{\\ast}_m(\\boldsymbol{R})\\Psi_n(\\boldsymbol{R})} =\\frac{\\sum_{n}a^2_n E_n}\n {\\sum_{n}a^2_n} \\ge E_0,\n$$\n\nwhere we used that $H(\\boldsymbol{R})\\Psi_n(\\boldsymbol{R})=E_n\\Psi_n(\\boldsymbol{R})$.\nIn general, the integrals involved in the calculation of various expectation\nvalues are multi-dimensional ones. \nThe variational principle yields the lowest state of a given symmetry.\n\n\n\n\n## Quantum Monte Carlo Motivation\nIn most cases, a wave function has only small values in large parts of \nconfiguration space, and a straightforward procedure which uses\nhomogenously distributed random points in configuration space \nwill most likely lead to poor results. This may suggest that some kind\nof importance sampling combined with e.g., the Metropolis algorithm \nmay be a more efficient way of obtaining the ground state energy.\nThe hope is then that those regions of configurations space where\nthe wave function assumes appreciable values are sampled more \nefficiently.\n\n\n\n\n## Quantum Monte Carlo Motivation\nThe tedious part in a VMC calculation is the search for the variational\nminimum. A good knowledge of the system is required in order to carry out\nreasonable VMC calculations. This is not always the case, \nand often VMC calculations \nserve rather as the starting\npoint for so-called diffusion Monte Carlo calculations (DMC). DMC is a way of\nsolving exactly the many-body Schroedinger equation by means of \na stochastic procedure. A good guess on the binding energy\nand its wave function is however necessary. \nA carefully performed VMC calculation can aid in this context.\n\n\n\n\n## Quantum Monte Carlo Motivation\n* Construct first a trial wave function $\\psi_T(\\boldsymbol{R},\\boldsymbol{\\alpha})$, for a many-body system consisting of $N$ particles located at positions $\\boldsymbol{R}=(\\boldsymbol{R}_1,\\dots ,\\boldsymbol{R}_N)$. The trial wave function depends on $\\alpha$ variational parameters $\\boldsymbol{\\alpha}=(\\alpha_1,\\dots ,\\alpha_M)$.\n\n* Then we evaluate the expectation value of the hamiltonian $H$\n\n$$\nE[H]=\\langle H \\rangle =\n \\frac{\\int d\\boldsymbol{R}\\Psi^{\\ast}_{T}(\\boldsymbol{R},\\boldsymbol{\\alpha})H(\\boldsymbol{R})\\Psi_{T}(\\boldsymbol{R},\\boldsymbol{\\alpha})}\n {\\int d\\boldsymbol{R}\\Psi^{\\ast}_{T}(\\boldsymbol{R},\\boldsymbol{\\alpha})\\Psi_{T}(\\boldsymbol{R},\\boldsymbol{\\alpha})}.\n$$\n\n* Thereafter we vary $\\alpha$ according to some minimization algorithm and return to the first step.\n\n\n\n\n## Quantum Monte Carlo Motivation\n**Basic steps.**\n\nChoose a trial wave function\n$\\psi_T(\\boldsymbol{R})$.\n\n$$\nP(\\boldsymbol{R})= \\frac{\\left|\\psi_T(\\boldsymbol{R})\\right|^2}{\\int \\left|\\psi_T(\\boldsymbol{R})\\right|^2d\\boldsymbol{R}}.\n$$\n\nThis is our new probability distribution function (PDF).\nThe approximation to the expectation value of the Hamiltonian is now\n\n$$\nE[H(\\boldsymbol{\\alpha})] = \n \\frac{\\int d\\boldsymbol{R}\\Psi^{\\ast}_T(\\boldsymbol{R},\\boldsymbol{\\alpha})H(\\boldsymbol{R})\\Psi_T(\\boldsymbol{R},\\boldsymbol{\\alpha})}\n {\\int d\\boldsymbol{R}\\Psi^{\\ast}_T(\\boldsymbol{R},\\boldsymbol{\\alpha})\\Psi_T(\\boldsymbol{R},\\boldsymbol{\\alpha})}.\n$$\n\n## Quantum Monte Carlo Motivation\nDefine a new quantity\n\n\n
    \n\n$$\nE_L(\\boldsymbol{R},\\boldsymbol{\\alpha})=\\frac{1}{\\psi_T(\\boldsymbol{R},\\boldsymbol{\\alpha})}H\\psi_T(\\boldsymbol{R},\\boldsymbol{\\alpha}),\n\\label{eq:locale1} \\tag{1}\n$$\n\ncalled the local energy, which, together with our trial PDF yields\n\n\n
    \n\n$$\nE[H(\\boldsymbol{\\alpha})]=\\int P(\\boldsymbol{R})E_L(\\boldsymbol{R}) d\\boldsymbol{R}\\approx \\frac{1}{N}\\sum_{i=1}^N E_L(\\boldsymbol{R_i},\\boldsymbol{\\alpha})\n\\label{eq:vmc1} \\tag{2}\n$$\n\nwith $N$ being the number of Monte Carlo samples.\n\n\n\n\n\n\n\n\n## Quantum Monte Carlo\nThe Algorithm for performing a variational Monte Carlo calculations runs thus as this\n\n * Initialisation: Fix the number of Monte Carlo steps. Choose an initial $\\boldsymbol{R}$ and variational parameters $\\alpha$ and calculate $\\left|\\psi_T^{\\alpha}(\\boldsymbol{R})\\right|^2$. \n\n * Initialise the energy and the variance and start the Monte Carlo calculation.\n\n * Calculate a trial position $\\boldsymbol{R}_p=\\boldsymbol{R}+r*step$ where $r$ is a random variable $r \\in [0,1]$.\n\n * Metropolis algorithm to accept or reject this move $w = P(\\boldsymbol{R}_p)/P(\\boldsymbol{R})$.\n\n * If the step is accepted, then we set $\\boldsymbol{R}=\\boldsymbol{R}_p$. \n\n * Update averages\n\n\n * Finish and compute final averages.\n\nObserve that the jumping in space is governed by the variable *step*. This is Called brute-force sampling.\nNeed importance sampling to get more relevant sampling, see lectures below.\n\n\n\n## Quantum Monte Carlo: hydrogen atom\nThe radial Schroedinger equation for the hydrogen atom can be\nwritten as\n\n$$\n-\\frac{\\hbar^2}{2m}\\frac{\\partial^2 u(r)}{\\partial r^2}-\n\\left(\\frac{ke^2}{r}-\\frac{\\hbar^2l(l+1)}{2mr^2}\\right)u(r)=Eu(r),\n$$\n\nor with dimensionless variables\n\n\n
    \n\n$$\n-\\frac{1}{2}\\frac{\\partial^2 u(\\rho)}{\\partial \\rho^2}-\n\\frac{u(\\rho)}{\\rho}+\\frac{l(l+1)}{2\\rho^2}u(\\rho)-\\lambda u(\\rho)=0,\n\\label{eq:hydrodimless1} \\tag{3}\n$$\n\nwith the hamiltonian\n\n$$\nH=-\\frac{1}{2}\\frac{\\partial^2 }{\\partial \\rho^2}-\n\\frac{1}{\\rho}+\\frac{l(l+1)}{2\\rho^2}.\n$$\n\nUse variational parameter $\\alpha$ in the trial\nwave function\n\n\n
    \n\n$$\nu_T^{\\alpha}(\\rho)=\\alpha\\rho e^{-\\alpha\\rho}. \n\\label{eq:trialhydrogen} \\tag{4}\n$$\n\n## Quantum Monte Carlo: hydrogen atom\nInserting this wave function into the expression for the\nlocal energy $E_L$ gives\n\n$$\nE_L(\\rho)=-\\frac{1}{\\rho}-\n \\frac{\\alpha}{2}\\left(\\alpha-\\frac{2}{\\rho}\\right).\n$$\n\nA simple variational Monte Carlo calculation results in\n\n\n\n\n\n\n\n\n\n\n\n\n\n
    $\\alpha$ $\\langle H \\rangle $ $\\sigma^2$ $\\sigma/\\sqrt{N}$
    7.00000E-01 -4.57759E-01 4.51201E-02 6.71715E-04
    8.00000E-01 -4.81461E-01 3.05736E-02 5.52934E-04
    9.00000E-01 -4.95899E-01 8.20497E-03 2.86443E-04
    1.00000E-00 -5.00000E-01 0.00000E+00 0.00000E+00
    1.10000E+00 -4.93738E-01 1.16989E-02 3.42036E-04
    1.20000E+00 -4.75563E-01 8.85899E-02 9.41222E-04
    1.30000E+00 -4.54341E-01 1.45171E-01 1.20487E-03
    \n\n\n\n\n## Quantum Monte Carlo: hydrogen atom\n\nWe note that at $\\alpha=1$ we obtain the exact\nresult, and the variance is zero, as it should. The reason is that \nwe then have the exact wave function, and the action of the hamiltionan\non the wave function\n\n$$\nH\\psi = \\mathrm{constant}\\times \\psi,\n$$\n\nyields just a constant. The integral which defines various \nexpectation values involving moments of the hamiltonian becomes then\n\n$$\n\\langle H^n \\rangle =\n \\frac{\\int d\\boldsymbol{R}\\Psi^{\\ast}_T(\\boldsymbol{R})H^n(\\boldsymbol{R})\\Psi_T(\\boldsymbol{R})}\n {\\int d\\boldsymbol{R}\\Psi^{\\ast}_T(\\boldsymbol{R})\\Psi_T(\\boldsymbol{R})}=\n\\mathrm{constant}\\times\\frac{\\int d\\boldsymbol{R}\\Psi^{\\ast}_T(\\boldsymbol{R})\\Psi_T(\\boldsymbol{R})}\n {\\int d\\boldsymbol{R}\\Psi^{\\ast}_T(\\boldsymbol{R})\\Psi_T(\\boldsymbol{R})}=\\mathrm{constant}.\n$$\n\n**This gives an important information: the exact wave function leads to zero variance!**\nVariation is then performed by minimizing both the energy and the variance.\n\n\n\n\n\n## [Quantum Monte Carlo for bosons](https://github.com/mortele/variational-monte-carlo-fys4411)\n\nFor bosons in a harmonic oscillator-like trap we will use is a spherical (S)\n or an elliptical (E) harmonic trap in one, two and finally three\n dimensions, with the latter given by\n\n\n
    \n\n$$\n\\begin{equation}\n V_{ext}(\\mathbf{r}) = \\Bigg\\{\n \\begin{array}{ll}\n\t \\frac{1}{2}m\\omega_{ho}^2r^2 & (S)\\\\\n \\strut\n\t \\frac{1}{2}m[\\omega_{ho}^2(x^2+y^2) + \\omega_z^2z^2] & (E)\n\\label{trap_eqn} \\tag{5}\n \\end{array}\n \\end{equation}\n$$\n\nwhere (S) stands for symmetric and\n\n\n
    \n\n$$\n\\begin{equation}\n \\hat{H} = \\sum_i^N \\left(\n\t \\frac{-\\hbar^2}{2m}\n\t { \\bigtriangledown }_{i}^2 +\n\t V_{ext}({\\bf{r}}_i)\\right) +\n\t \\sum_{i\n
    \n\n$$\n\\begin{equation}\n V_{int}(|\\mathbf{r}_i-\\mathbf{r}_j|) = \\Bigg\\{\n \\begin{array}{ll}\n\t \\infty & {|\\mathbf{r}_i-\\mathbf{r}_j|} \\leq {a}\\\\\n\t 0 & {|\\mathbf{r}_i-\\mathbf{r}_j|} > {a}\n \\end{array}\n\\label{_auto2} \\tag{7}\n\\end{equation}\n$$\n\nwhere $a$ is the so-called hard-core diameter of the bosons.\n Clearly, $V_{int}(|\\mathbf{r}_i-\\mathbf{r}_j|)$ is zero if the bosons are\n separated by a distance $|\\mathbf{r}_i-\\mathbf{r}_j|$ greater than $a$ but\n infinite if they attempt to come within a distance $|\\mathbf{r}_i-\\mathbf{r}_j| \\leq a$.\n\n\n\n## [Quantum Monte Carlo for bosons](https://github.com/mortele/variational-monte-carlo-fys4411)\n Our trial wave function for the ground state with $N$ atoms is given by\n\n\n
    \n\n$$\n\\begin{equation}\n \\Psi_T(\\mathbf{R})=\\Psi_T(\\mathbf{r}_1, \\mathbf{r}_2, \\dots \\mathbf{r}_N,\\alpha,\\beta)=\\prod_i g(\\alpha,\\beta,\\mathbf{r}_i)\\prod_{i\n
    \n\n$$\n\\begin{equation}\n g(\\alpha,\\beta,\\mathbf{r}_i)= \\exp{[-\\alpha(x_i^2+y_i^2+\\beta z_i^2)]}.\n\\label{_auto3} \\tag{9}\n\\end{equation}\n$$\n\n## [Quantum Monte Carlo for bosons](https://github.com/mortele/variational-monte-carlo-fys4411)\nFor spherical traps we have $\\beta = 1$ and for non-interacting\nbosons ($a=0$) we have $\\alpha = 1/2a_{ho}^2$. The correlation wave\n function is\n\n\n
    \n\n$$\n\\begin{equation}\n f(a,|\\mathbf{r}_i-\\mathbf{r}_j|)=\\Bigg\\{\n \\begin{array}{ll}\n\t 0 & {|\\mathbf{r}_i-\\mathbf{r}_j|} \\leq {a}\\\\\n\t (1-\\frac{a}{|\\mathbf{r}_i-\\mathbf{r}_j|}) & {|\\mathbf{r}_i-\\mathbf{r}_j|} > {a}.\n \\end{array}\n\\label{_auto4} \\tag{10}\n\\end{equation}\n$$\n\n### Simple example, the hydrogen atom\n\nThe radial Schroedinger equation for the hydrogen atom can be\nwritten as (when we have gotten rid of the first derivative term in the kinetic energy and used $rR(r)=u(r)$)\n\n$$\n-\\frac{\\hbar^2}{2m}\\frac{d^2 u(r)}{d r^2}-\n\\left(\\frac{ke^2}{r}-\\frac{\\hbar^2l(l+1)}{2mr^2}\\right)u(r)=Eu(r).\n$$\n\nWe will specialize to the case with $l=0$ and end up with\n\n$$\n-\\frac{\\hbar^2}{2m}\\frac{d^2 u(r)}{d r^2}-\n\\left(\\frac{ke^2}{r}\\right)u(r)=Eu(r).\n$$\n\nThen we introduce a dimensionless variable $\\rho=r/a$ where $a$ is a constant with dimension length.\nMultiplying with $ma^2/\\hbar^2$ we can rewrite our equations as\n\n$$\n-\\frac{1}{2}\\frac{d^2 u(\\rho)}{d \\rho^2}-\n\\frac{ke^2ma}{\\hbar^2}\\frac{u(\\rho)}{\\rho}-\\lambda u(\\rho)=0.\n$$\n\nSince $a$ is just a parameter we choose to set\n\n$$\n\\frac{ke^2ma}{\\hbar^2}=1,\n$$\n\nwhich leads to $a=\\hbar^2/mke^2$, better known as the Bohr radius with value $0.053$ nm. Scaling the equations this way does not only render our numerical treatment simpler since we avoid carrying with us all physical parameters, but we obtain also a **natural** length scale. We will see this again and again. In our discussions below with a harmonic oscillator trap, the **natural** lentgh scale with be determined by the oscillator frequency, the mass of the particle and $\\hbar$. We have also defined a dimensionless 'energy' $\\lambda = Ema^2/\\hbar^2$. \nWith the rescaled quantities, the ground state energy of the hydrogen atom is $1/2$. \nThe equation we want to solve is now defined by the Hamiltonian\n\n$$\nH=-\\frac{1}{2}\\frac{d^2 }{d \\rho^2}-\\frac{1}{\\rho}.\n$$\n\nAs trial wave function we peep now into the analytical solution for\nthe hydrogen atom and use (with $\\alpha$ as a variational parameter)\n\n$$\nu_T^{\\alpha}(\\rho)=\\alpha\\rho \\exp{-(\\alpha\\rho)}.\n$$\n\nInserting this wave function into the expression for the\nlocal energy $E_L$ gives\n\n$$\nE_L(\\rho)=-\\frac{1}{\\rho}-\n \\frac{\\alpha}{2}\\left(\\alpha-\\frac{2}{\\rho}\\right).\n$$\n\nTo have analytical local energies saves us from computing numerically\nthe second derivative, a feature which often increases our numerical\nexpenditure with a factor of three or more. Integratng up the local energy (recall to bring back the PDF in the integration) gives $\\overline{E}[\\boldsymbol{\\alpha}]=\\alpha(\\alpha/2-1)$. \n\n\n\n\n### Second example, the harmonic oscillator in one dimension\n\nWe present here another well-known example, the harmonic oscillator in\none dimension for one particle. This will also serve the aim of\nintroducing our next model, namely that of interacting electrons in a\nharmonic oscillator trap.\n\nHere as well, we do have analytical solutions and the energy of the\nground state, with $\\hbar=1$, is $1/2\\omega$, with $\\omega$ being the\noscillator frequency. We use the following trial wave function\n\n$$\n\\psi_T(x;\\alpha) = \\exp{-(\\frac{1}{2}\\alpha^2x^2)},\n$$\n\nwhich results in a local energy\n\n$$\n\\frac{1}{2}\\left(\\alpha^2+x^2(1-\\alpha^4)\\right).\n$$\n\nWe can compare our numerically calculated energies with the exact energy as function of $\\alpha$\n\n$$\n\\overline{E}[\\alpha] = \\frac{1}{4}\\left(\\alpha^2+\\frac{1}{\\alpha^2}\\right).\n$$\n\nSimilarly, with the above ansatz, we can also compute the exact variance which reads\n\n$$\n\\sigma^2[\\alpha]=\\frac{1}{4}\\left(1+(1-\\alpha^4)^2\\frac{3}{4\\alpha^4}\\right)-\\overline{E}^2.\n$$\n\nOur code for computing the energy of the ground state of the harmonic oscillator follows here. We start by defining directories where we store various outputs.\n\n\n```\n# Common imports\nimport os\n\n# Where to save the figures and data files\nPROJECT_ROOT_DIR = \"Results\"\nFIGURE_ID = \"Results/FigureFiles\"\nDATA_ID = \"Results/VMCHarmonic\"\n\nif not os.path.exists(PROJECT_ROOT_DIR):\n os.mkdir(PROJECT_ROOT_DIR)\n\nif not os.path.exists(FIGURE_ID):\n os.makedirs(FIGURE_ID)\n\nif not os.path.exists(DATA_ID):\n os.makedirs(DATA_ID)\n\ndef image_path(fig_id):\n return os.path.join(FIGURE_ID, fig_id)\n\ndef data_path(dat_id):\n return os.path.join(DATA_ID, dat_id)\n\ndef save_fig(fig_id):\n plt.savefig(image_path(fig_id) + \".png\", format='png')\n\noutfile = open(data_path(\"VMCHarmonic.dat\"),'w')\n```\n\nWe proceed with the implementation of the Monte Carlo algorithm but list first the ansatz for the wave function and the expression for the local energy\n\n\n```\n%matplotlib inline\n\n# VMC for the one-dimensional harmonic oscillator\n# Brute force Metropolis, no importance sampling and no energy minimization\nfrom math import exp, sqrt\nfrom random import random, seed\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom numba import jit\nfrom decimal import *\n# Trial wave function for the Harmonic oscillator in one dimension\ndef WaveFunction(r,alpha):\n return exp(-0.5*alpha*alpha*r*r)\n\n# Local energy for the Harmonic oscillator in one dimension\ndef LocalEnergy(r,alpha):\n return 0.5*r*r*(1-alpha**4) + 0.5*alpha*alpha\n```\n\nNote that in the Metropolis algorithm there is no need to compute the\ntrial wave function, mainly since we are just taking the ratio of two\nexponentials. It is then from a computational point view, more\nconvenient to compute the argument from the ratio and then calculate\nthe exponential. Here we have refrained from this purely of\npedagogical reasons.\n\n\n```\n# The Monte Carlo sampling with the Metropolis algo\n# The jit decorator tells Numba to compile this function.\n# The argument types will be inferred by Numba when the function is called.\n@jit\ndef MonteCarloSampling():\n\n NumberMCcycles= 100000\n StepSize = 1.0\n # positions\n PositionOld = 0.0\n PositionNew = 0.0\n\n # seed for rng generator\n seed()\n # start variational parameter\n alpha = 0.4\n for ia in range(MaxVariations):\n alpha += .05\n AlphaValues[ia] = alpha\n energy = energy2 = 0.0\n #Initial position\n PositionOld = StepSize * (random() - .5)\n wfold = WaveFunction(PositionOld,alpha)\n #Loop over MC MCcycles\n for MCcycle in range(NumberMCcycles):\n #Trial position \n PositionNew = PositionOld + StepSize*(random() - .5)\n wfnew = WaveFunction(PositionNew,alpha)\n #Metropolis test to see whether we accept the move\n if random() <= wfnew**2 / wfold**2:\n PositionOld = PositionNew\n wfold = wfnew\n DeltaE = LocalEnergy(PositionOld,alpha)\n energy += DeltaE\n energy2 += DeltaE**2\n #We calculate mean, variance and error\n energy /= NumberMCcycles\n energy2 /= NumberMCcycles\n variance = energy2 - energy**2\n error = sqrt(variance/NumberMCcycles)\n Energies[ia] = energy \n Variances[ia] = variance \n outfile.write('%f %f %f %f \\n' %(alpha,energy,variance,error))\n return Energies, AlphaValues, Variances\n```\n\nFinally, the results are presented here with the exact energies and variances as well.\n\n\n```\n#Here starts the main program with variable declarations\nMaxVariations = 20\nEnergies = np.zeros((MaxVariations))\nExactEnergies = np.zeros((MaxVariations))\nExactVariance = np.zeros((MaxVariations))\nVariances = np.zeros((MaxVariations))\nAlphaValues = np.zeros(MaxVariations)\n(Energies, AlphaValues, Variances) = MonteCarloSampling()\noutfile.close()\nExactEnergies = 0.25*(AlphaValues*AlphaValues+1.0/(AlphaValues*AlphaValues))\nExactVariance = 0.25*(1.0+((1.0-AlphaValues**4)**2)*3.0/(4*(AlphaValues**4)))-ExactEnergies*ExactEnergies\n\n#simple subplot\nplt.subplot(2, 1, 1)\nplt.plot(AlphaValues, Energies, 'o-',AlphaValues, ExactEnergies,'r-')\nplt.title('Energy and variance')\nplt.ylabel('Dimensionless energy')\nplt.subplot(2, 1, 2)\nplt.plot(AlphaValues, Variances, '.-',AlphaValues, ExactVariance,'r-')\nplt.xlabel(r'$\\alpha$', fontsize=15)\nplt.ylabel('Variance')\nsave_fig(\"VMCHarmonic\")\nplt.show()\n#nice printout with Pandas\nimport pandas as pd\nfrom pandas import DataFrame\ndata ={'Alpha':AlphaValues, 'Energy':Energies,'Exact Energy':ExactEnergies,'Variance':Variances,'Exact Variance':ExactVariance,}\nframe = pd.DataFrame(data)\nprint(frame)\n```\n\nFor $\\alpha=1$ we have the exact eigenpairs, as can be deduced from the\ntable here. With $\\omega=1$, the exact energy is $1/2$ a.u. with zero\nvariance, as it should. We see also that our computed variance follows rather well the exact variance.\nIncreasing the number of Monte Carlo cycles will improve our statistics (try to increase the number of Monte Carlo cycles).\n\nThe fact that the variance is exactly equal to zero when $\\alpha=1$ is that \nwe then have the exact wave function, and the action of the hamiltionan\non the wave function\n\n$$\nH\\psi = \\mathrm{constant}\\times \\psi,\n$$\n\nyields just a constant. The integral which defines various \nexpectation values involving moments of the hamiltonian becomes then\n\n$$\n\\langle H^n \\rangle =\n \\frac{\\int d\\boldsymbol{R}\\Psi^{\\ast}_T(\\boldsymbol{R})H^n(\\boldsymbol{R})\\Psi_T(\\boldsymbol{R})}\n {\\int d\\boldsymbol{R}\\Psi^{\\ast}_T(\\boldsymbol{R})\\Psi_T(\\boldsymbol{R})}=\n\\mathrm{constant}\\times\\frac{\\int d\\boldsymbol{R}\\Psi^{\\ast}_T(\\boldsymbol{R})\\Psi_T(\\boldsymbol{R})}\n {\\int d\\boldsymbol{R}\\Psi^{\\ast}_T(\\boldsymbol{R})\\Psi_T(\\boldsymbol{R})}=\\mathrm{constant}.\n$$\n\n**This gives an important information: the exact wave function leads to zero variance!**\nAs we will see below, many practitioners perform a minimization on both the energy and the variance.\n", "meta": {"hexsha": "ea76de6c1baf62a0d8e28d8d28806243e5108637", "size": 47261, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "doc/pub/week1/ipynb/week1.ipynb", "max_stars_repo_name": "Schoyen/ComputationalPhysics2", "max_stars_repo_head_hexsha": "9cf10ffb2557cc73c4e6bab060d53690ee39426f", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 87, "max_stars_repo_stars_event_min_datetime": "2015-01-21T08:29:56.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-28T07:11:53.000Z", "max_issues_repo_path": "doc/pub/week1/ipynb/week1.ipynb", "max_issues_repo_name": "Schoyen/ComputationalPhysics2", "max_issues_repo_head_hexsha": "9cf10ffb2557cc73c4e6bab060d53690ee39426f", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2020-01-18T10:43:38.000Z", "max_issues_repo_issues_event_max_datetime": "2020-02-08T13:15:42.000Z", "max_forks_repo_path": "doc/pub/week1/ipynb/week1.ipynb", "max_forks_repo_name": "Schoyen/ComputationalPhysics2", "max_forks_repo_head_hexsha": "9cf10ffb2557cc73c4e6bab060d53690ee39426f", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 54, "max_forks_repo_forks_event_min_datetime": "2015-02-09T10:02:00.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-07T10:44:14.000Z", "avg_line_length": 34.7252020573, "max_line_length": 570, "alphanum_fraction": 0.571866867, "converted": true, "num_tokens": 9205, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117165898111866, "lm_q2_score": 0.8459424431344437, "lm_q1q2_score": 0.4328827821773012}} {"text": "# Probabilistic Programming and Bayesian Methods for Hackers Chapter 2\n\n\n \n \n
    \n Run in Google Colab\n \n View source on GitHub\n
    \n
    \n
    \n
    \n\nOriginal content ([this Jupyter notebook](https://nbviewer.jupyter.org/github/CamDavidsonPilon/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers/blob/master/Chapter2_MorePyMC/Ch2_MorePyMC_PyMC2.ipynb)) created by Cam Davidson-Pilon ([`@Cmrn_DP`](https://twitter.com/Cmrn_DP))\n\nPorted to [Tensorflow Probability](https://www.tensorflow.org/probability/) by Matthew McAteer ([`@MatthewMcAteer0`](https://twitter.com/MatthewMcAteer0)), with help from Bryan Seybold, Mike Shwe ([`@mikeshwe`](https://twitter.com/mikeshwe)), Josh Dillon, and the rest of the TFP team at Google ([`tfprobability@tensorflow.org`](mailto:tfprobability@tensorflow.org)).\n\nWelcome to Bayesian Methods for Hackers. The full Github repository is available at [github/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers](https://github.com/CamDavidsonPilon/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers). The other chapters can be found on the project's [homepage](https://camdavidsonpilon.github.io/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers/). We hope you enjoy the book, and we encourage any contributions!\n___\n\n- [Dependencies & Prerequisites](#scrollTo=AIIO6GhdH89m)\n- [A little more on TFP](#scrollTo=24368dz9IAwM)\n - [TFP Variables](#scrollTo=ZlGWIiPLIAwo)\n - [Initializing Stochastic Variables](#scrollTo=NdKiqWtWIAwy)\n - [Deterministic variables](#scrollTo=UPt9k8YrIAwz)\n - [Combining with Tensorflow Core](#scrollTo=t_3wV3RwIAw9)\n - [Including observations in the Model](#scrollTo=IMNdtRTtIAxB)\n- [Modeling approaches](#scrollTo=8oxo5VcbIAxP)\n - [Same story; different ending](#scrollTo=3RJEK_yjIAxR)\n - [Example: Bayesian A/B testing](#scrollTo=2lU8C4C-IAxw)\n - [A Simple Case](#scrollTo=2lU8C4C-IAxw)\n - [Execute the TF graph to sample from the posterior](#scrollTo=yUVnbqhDVfAx)\n - [A and B together](#scrollTo=LdLJ2iriIAyI)\n - [Execute the TF graph to sample from the posterior](#scrollTo=beUUmGMbdrRr)\n- [An algorithm for human deceit](#scrollTo=f-jxOi70IAyl)\n - [The Binomial Distribution](#scrollTo=f-jxOi70IAyl)\n - [Example: Cheating among students](#scrollTo=9HmW-50PIAyv)\n - [Execute the TF graph to sample from the posterior](#scrollTo=eJYLS8EysHqj)\n - [Alternative TFP Model](#scrollTo=Y0bK5tMAIAze)\n - [Execute the TF graph to sample from the posterior](#scrollTo=eJYLS8EysHqj)\n - [More TFP Tricks](#scrollTo=X3NKe5vUIAzv)\n - [Example: Challenger Space Shuttle Disaster](#scrollTo=KMoiodMmIAzy)\n - [Normal Distributions](#scrollTo=_52Ml-KhIAz9)\n - [Execute the TF graph to sample from the posterior](#scrollTo=eNkhSXDkthRs)\n - [What about the day of the Challenger disaster?](#scrollTo=lesq_3oIIA0Y)\n - [Is our model appropriate?](#scrollTo=WjAFZ8W9IA0c)\n - [Execute the TF graph to sample from the posterior](#scrollTo=_pKu5j76wxlY)\n - [Exercises](#scrollTo=_I_3h7lsIA0v)\n - [References](#scrollTo=e6RYS1_jIA0y)\n___\n\nThis chapter introduces more TFP syntax and variables and ways to think about how to model a system from a Bayesian perspective. It also contains tips and data visualization techniques for assessing goodness-of-fit for your Bayesian model.\n\n### Dependencies & Prerequisites\n\n\n```python\n#@title Tensorflow Probability Installation (make sure to run this cell) { display-mode: \"form\" }\nTFP_Installation = \"Stable TFP\" #@param [\"Most Recent TFP\", \"Stable TFP\", \"Stable TFP-GPU\", \"Most Recent TFP-GPU\", \"TFP Already Installed\"]\n\nif TFP_Installation == \"Most Recent TFP\":\n !pip3 install -q tfp-nightly\n print(\"Most recent TFP version installed\")\nelif TFP_Installation == \"Stable TFP\":\n !pip3 install -q --upgrade tensorflow-probability\n print(\"Up-to-date, stable TFP version installed\")\nelif TFP_Installation == \"Stable TFP-GPU\":\n !pip3 install -q --upgrade tensorflow-probability-gpu\n print(\"Up-to-date, stable TFP-GPU version installed\")\n print(\"(make sure GPU is properly configured)\")\nelif TFP_Installation == \"Most Recent TFP-GPU\":\n !pip3 install -q tfp-nightly-gpu\n print(\"Most recent TFP-GPU version installed\")\n print(\"(make sure GPU is properly configured)\")\nelif TFP_Installation == \"TFP Already Installed\":\n print(\"TFP already instaled in this environment\")\n pass\nelse:\n print(\"Installation Error: Please select a viable TFP installation option.\")\n!pip3 install -q wget\n\n```\n\n Up-to-date, stable TFP version installed\n\n\n\n```python\n#@title Imports and Global Variables { display-mode: \"form\" }\n\"\"\"\nThe book uses a custom matplotlibrc file, which provides the unique styles for\nmatplotlib plots. If executing this book, and you wish to use the book's\nstyling, provided are two options:\n 1. Overwrite your own matplotlibrc file with the rc-file provided in the\n book's styles/ dir. See http://matplotlib.org/users/customizing.html\n 2. Also in the styles is bmh_matplotlibrc.json file. This can be used to\n update the styles in only this notebook. Try running the following code:\n\n import json\n s = json.load(open(\"../styles/bmh_matplotlibrc.json\"))\n matplotlib.rcParams.update(s)\n\"\"\"\nfrom __future__ import absolute_import, division, print_function\n\nwarning_status = \"ignore\" #@param [\"ignore\", \"always\", \"module\", \"once\", \"default\", \"error\"]\nimport warnings\nwarnings.filterwarnings(warning_status)\nwith warnings.catch_warnings():\n warnings.filterwarnings(warning_status, category=DeprecationWarning)\n warnings.filterwarnings(warning_status, category=UserWarning)\n\nimport numpy as np\nimport os\nmatplotlib_style = 'fivethirtyeight' #@param ['fivethirtyeight', 'bmh', 'ggplot', 'seaborn', 'default', 'Solarize_Light2', 'classic', 'dark_background', 'seaborn-colorblind', 'seaborn-notebook']\nimport matplotlib.pyplot as plt; plt.style.use(matplotlib_style)\nimport matplotlib.axes as axes;\nfrom matplotlib.patches import Ellipse\n%matplotlib inline\nimport seaborn as sns; sns.set_context('notebook')\nfrom IPython.core.pylabtools import figsize\nnotebook_screen_res = 'png' #@param ['retina', 'png', 'jpeg', 'svg', 'pdf']\n%config InlineBackend.figure_format = notebook_screen_res\n\nimport tensorflow as tf\ntfe = tf.contrib.eager\n\n# Eager Execution\nuse_tf_eager = False #@param {type:\"boolean\"}\n\n# Use try/except so we can easily re-execute the whole notebook.\nif use_tf_eager:\n try:\n tf.enable_eager_execution()\n except:\n pass\n\nimport tensorflow_probability as tfp\ntfd = tfp.distributions\ntfb = tfp.bijectors\n\n \ndef evaluate(tensors):\n \"\"\"Evaluates Tensor or EagerTensor to Numpy `ndarray`s.\n Args:\n tensors: Object of `Tensor` or EagerTensor`s; can be `list`, `tuple`,\n `namedtuple` or combinations thereof.\n\n Returns:\n ndarrays: Object with same structure as `tensors` except with `Tensor` or\n `EagerTensor`s replaced by Numpy `ndarray`s.\n \"\"\"\n if tf.executing_eagerly():\n return tf.contrib.framework.nest.pack_sequence_as(\n tensors,\n [t.numpy() if tf.contrib.framework.is_tensor(t) else t\n for t in tf.contrib.framework.nest.flatten(tensors)])\n return sess.run(tensors)\n\nclass _TFColor(object):\n \"\"\"Enum of colors used in TF docs.\"\"\"\n red = '#F15854'\n blue = '#5DA5DA'\n orange = '#FAA43A'\n green = '#60BD68'\n pink = '#F17CB0'\n brown = '#B2912F'\n purple = '#B276B2'\n yellow = '#DECF3F'\n gray = '#4D4D4D'\n def __getitem__(self, i):\n return [\n self.red,\n self.orange,\n self.green,\n self.blue,\n self.pink,\n self.brown,\n self.purple,\n self.yellow,\n self.gray,\n ][i % 9]\nTFColor = _TFColor()\n\ndef session_options(enable_gpu_ram_resizing=True, enable_xla=True):\n \"\"\"\n Allowing the notebook to make use of GPUs if they're available.\n \n XLA (Accelerated Linear Algebra) is a domain-specific compiler for linear \n algebra that optimizes TensorFlow computations.\n \"\"\"\n config = tf.ConfigProto()\n config.log_device_placement = True\n if enable_gpu_ram_resizing:\n # `allow_growth=True` makes it possible to connect multiple colabs to your\n # GPU. Otherwise the colab malloc's all GPU ram.\n config.gpu_options.allow_growth = True\n if enable_xla:\n # Enable on XLA. https://www.tensorflow.org/performance/xla/.\n config.graph_options.optimizer_options.global_jit_level = (\n tf.OptimizerOptions.ON_1)\n return config\n\n\ndef reset_sess(config=None):\n \"\"\"\n Convenience function to create the TF graph & session or reset them.\n \"\"\"\n if config is None:\n config = session_options()\n global sess\n tf.reset_default_graph()\n try:\n sess.close()\n except:\n pass\n sess = tf.InteractiveSession(config=config)\n\nreset_sess()\n```\n\n## A little more on TensorFlow and TensorFlow Probability\n\nTo explain TensorFlow Probability, it's worth going into the various methods of working with Tensorflow tensors. Here, we introduce the notion of Tensorflow graphs and how we can use certain coding patterns to make our tensor-processing workflows much faster and more elegant. \n\n### TensorFlow Graph and Eager Modes\n\nTFP accomplishes most of its heavy lifting via the main `tensorflow` library. The `tensorflow` library also contains many of the familiar computational elements of NumPy and uses similar notation. While NumPy directly executes computations (e.g. when you run `a + b`), `tensorflow` in graph mode instead builds up a \"compute graph\" that tracks that you want to perform the `+` operation on the elements `a` and `b`. Only when you evaluate a `tensorflow` expression does the computation take place--`tensorflow` is lazy evaluated. The benefit of using Tensorflow over NumPy is that the graph enables mathematical optimizations (e.g. simplifications), gradient calculations via automatic differentiation, compiling the entire graph to C to run at machine speed, and also compiling it to run on a GPU or TPU. \n\nFundamentally, TensorFlow uses [graphs](https://www.tensorflow.org/guide/graphs) for computation, wherein the graphs represent computation as dependencies among individual operations. In the programming paradigm for Tensorflow graphs, we first define the dataflow graph, and then create a TensorFlow session to run parts of the graph. A Tensorflow [`tf.Session()`](https://www.tensorflow.org/api_docs/python/tf/Session) object runs the graph to get the variables we want to model. In the example below, we are using a global session object `sess`, which we created above in the \"Imports and Global Variables\" section. \n\nTo avoid the sometimes confusing aspects of lazy evaluation, Tensorflow's eager mode does immediate evaluation of results to give an even more similar feel to working with NumPy. With Tensorflow [eager](https://www.tensorflow.org/guide/eager) mode, you can evaluate operations immediately, without explicitly building graphs: operations return concrete values instead of constructing a computational graph to run later. If we're in eager mode, we are presented with tensors that can be converted to numpy array equivalents immediately. Eager mode makes it easy to get started with TensorFlow and debug models.\n\n\nTFP is essentially:\n\n* a collection of tensorflow symbolic expressions for various probability distributions that are combined into one big compute graph, and\n* a collection of inference algorithms that use that graph to compute probabilities and gradients.\n\nFor practical purposes, what this means is that in order to build certain models we sometimes have to use core Tensorflow. This simple example for Poisson sampling is how we might work with both graph and eager modes:\n\n\n```python\nparameter = tfd.Exponential(rate=1., name=\"poisson_param\").sample()\ndata_generator = tfd.Poisson(parameter, name=\"data_generator\")\ndata_generator_samples = data_generator.sample()\n\nif tf.executing_eagerly():\n data_gen_samps_ = tf.contrib.framework.nest.pack_sequence_as(\n data_generator_samples,\n [t.numpy() if tf.contrib.framework.is_tensor(t) else t\n for t in tf.contrib.framework.nest.flatten(data_generator_samples)])\nelse:\n data_gen_samps_ = sess.run(data_generator_samples)\n \nprint(\"Value of sample from data generator:\", data_gen_samps_)\n```\n\n Value of sample from data generator: 0.0\n\n\nIn graph mode, Tensorflow will automatically assign any variables to a graph; they can then be evaluated in a session or made available in eager mode. If you try to define a variable when the session is already closed or in a finalized state, you will get an error. In the \"Imports and Global Variables\" section, we defined a particular type of session, called [`InteractiveSession`](https:///www.tensorflow.org/api_docs/python/tf/InteractiveSession). \nThis defnition of a global `InteractiveSession` allows us to access our session variables interactively via a shell or notebook.\n\nUsing the pattern of a global session, we can incrementally build a graph and run subsets of it to get the results.\n\nEager execution further simplifies our code, eliminating the need to call session functions explicitly. In fact, if you try to run graph mode semantics in eager mode, you will get an error message like this:\n\n```\nAttributeError: Tensor.graph is meaningless when eager execution is enabled.\n```\n\nAs mentioned in the previous chapter, we have a nifty tool that allows us to create code that's usable in both graph mode and eager mode. The custom `evaluate()` function allows us to evaluate tensors whether we are operating in TF graph or eager mode. The function looks like the following:\n\n```python\n\ndef evaluate(tensors):\n if tf.executing_eagerly():\n return tf.contrib.framework.nest.pack_sequence_as(\n tensors,\n [t.numpy() if tf.contrib.framework.is_tensor(t) else t\n for t in tf.contrib.framework.nest.flatten(tensors)])\n with tf.Session() as sess:\n return sess.run(tensors)\n\n```\n\nEach of the tensors corresponds to a NumPy-like output. To distinguish the tensors from their NumPy-like counterparts, we will use the convention of appending an underscore to the version of the tensor that one can use NumPy-like arrays on. In other words, the output of `evaluate()` gets named as `variable` + `_` = `variable_` . Now, we can do our Poisson sampling using both the `evaluate()` function and this new convention for naming Python variables in TFP.\n\n\n```python\n# Defining our Assumptions\nparameter = tfd.Exponential(rate=1., name=\"poisson_param\").sample()\n\n# Converting our TF to Numpy\n[ parameter_ ] = evaluate([ parameter ])\n\nprint(\"Sample from exponential distribution before evaluation: \", parameter)\nprint(\"Evaluated sample from exponential distribution: \", parameter_)\n```\n\n Sample from exponential distribution before evaluation: Tensor(\"poisson_param_1/sample/Reshape:0\", shape=(), dtype=float32)\n Evaluated sample from exponential distribution: 3.3942702\n\n\nMore generally, we can use our `evaluate()` function to convert between the Tensorflow `tensor` data type and one that we can run operations on:\n\n\n```python\n[ \n parameter_,\n data_generator_sample_,\n] = evaluate([ \n parameter, \n data_generator.sample(),\n])\n\nprint(\"'parameter_' evaluated Tensor :\", parameter_)\nprint(\"'data_generator_' sample evaluated Tensor :\", data_generator_sample_)\n\n```\n\n 'parameter_' evaluated Tensor : 4.2022924\n 'data_generator_' sample evaluated Tensor : 1.0\n\n\n\nA general rule of thumb for programming in TensorFlow is that if you need to do any array-like calculations that would require NumPy functions, you should use their equivalents in TensorFlow. This practice is necessary because NumPy can produce only constant values but TensorFlow tensors are a dynamic part of the computation graph. If you mix and match these the wrong way, you will typically get an error about incompatible types.\n\n### TFP Distributions\n\nLet's look into how [`tfp.distributions`](https://www.tensorflow.org/probability/api_docs/python/tfp/distributions) work.\n\nTFP uses distribution subclasses to represent *stochastic*, random variables. A variable is stochastic when the following is true: even if you knew all the values of the variable's parameters and components, it would still be random. Included in this category are instances of classes [`Poisson`](https://www.tensorflow.org/probability/api_docs/python/tfp/distributions/Poisson), [`Uniform`](https://www.tensorflow.org/probability/api_docs/python/tfp/distributions/Uniform), and [`Exponential`](https://www.tensorflow.org/probability/api_docs/python/tfp/distributions/Exponential).\n\nYou can draw random samples from a stochastic variable. When you draw samples, those samples become [`tensorflow.Tensors`](https://www.tensorflow.org/api_docs/python/tf/Tensor) that behave deterministically from that point on. A quick mental check to determine if something is *deterministic* is: *If I knew all of the inputs for creating the variable `foo`, I could calculate the value of `foo`.* You can add, subtract, and otherwise manipulate the tensors in a variety of ways discussed below. These operations are almost always deterministic.\n\n\n#### Initializing a Distribution\n\nInitializing a stochastic, or random, variable requires a few class-specific parameters that describe the Distribution's shape, such as the location and scale. For example:\n\n```python\nsome_distribution = tfd.Uniform(0., 4.)\n```\n\ninitializes a stochastic, or random, [`Uniform`](https://www.tensorflow.org/probability/api_docs/python/tfp/distributions/Uniform) distribution with the lower bound at 0 and upper bound at 4. Calling `sample()` on the distribution returns a tensor that will behave deterministically from that point on:\n\n```python\nsampled_tensor = some_distribution.sample()\n```\n\nThe next example demonstrates what we mean when we say that distributions are stochastic but tensors are deterministic:\n\n```\nderived_tensor_1 = 1 + sampled_tensor\nderived_tensor_2 = 1 + sampled_tensor # equal to 1\n\nderived_tensor_3 = 1 + some_distribution.sample()\nderived_tensor_4 = 1 + some_distribution.sample() # different from 3\n```\n\nThe first two lines produce the same value because they refer to the same sampled tensor. The last two lines likely produce different values because they refer to independent samples drawn from the same distribution.\n\nTo define a multiviariate distribution, just pass in arguments with the shape you want the output to be when creating the distribution. For example:\n\n```python\nbetas = tfd.Uniform([0., 0.], [1., 1.])\n```\n\nCreates a Distribution with batch_shape (2,). Now, when you call betas.sample(),\ntwo values will be returned instead of one. You can read more about TFP shape semantics in the [TFP docs](https://github.com/tensorflow/probability/blob/master/tensorflow_probability/examples/jupyter_notebooks/Understanding_TensorFlow_Distributions_Shapes.ipynb), but most uses in this book should be self-explanatory.\n\n#### Deterministic variables\n\nWe can create a deterministic distribution similarly to how we create a stochastic distribution. We simply call up the [`Deterministic`](https://www.tensorflow.org/probability/api_docs/python/tfp/distributions/Deterministic) class from Tensorflow Distributions and pass in the deterministic value that we desire\n```python\ndeterministic_variable = tfd.Deterministic(name=\"deterministic_variable\", loc=some_function_of_variables)\n```\n\nCalling `tfd.Deterministic` is useful for creating distributions that always have the same value. However, the much more common pattern for working with deterministic variables in TFP is to create a tensor or sample from a distribution:\n\n\n```python\nlambda_1 = tfd.Exponential(rate=1., name=\"lambda_1\") #stochastic variable\nlambda_2 = tfd.Exponential(rate=1., name=\"lambda_2\") #stochastic variable\ntau = tfd.Uniform(name=\"tau\", low=0., high=10.) #stochastic variable\n\n# deterministic variable since we are getting results of lambda's after sampling \nnew_deterministic_variable = tfd.Deterministic(name=\"deterministic_variable\", \n loc=(lambda_1.sample() + lambda_2.sample()))\n\n```\n\nThe use of the deterministic variable was seen in the previous chapter's text-message example. Recall the model for $\\lambda$ looked like: \n\n$$\n\\lambda = \n\\begin{cases}\\lambda_1 & \\text{if } t \\lt \\tau \\cr\n\\lambda_2 & \\text{if } t \\ge \\tau\n\\end{cases}\n$$\n\nAnd in TFP code:\n\n\n```python\nn_data_points = 5 # in CH1 we had ~70 data points\nidx = np.arange(n_data_points)\n\nlambda_deterministic = tfd.Deterministic(tf.gather([lambda_1.sample(), lambda_2.sample()],\n indices=tf.to_int32(\n tau.sample() >= idx)))\n[lambda_deterministic_] = evaluate([lambda_deterministic.sample()])\n\nprint(\"{} samples from our deterministic lambda model: \\n\".format(n_data_points), lambda_deterministic_ )\n```\n\n 5 samples from our deterministic lambda model: \n [1.705 1.705 1.705 1.705 1.705]\n\n\nClearly, if $\\tau, \\lambda_1$ and $\\lambda_2$ are known, then $\\lambda$ is known completely, hence it is a deterministic variable. We use indexing here to switch from $\\lambda_1$ to $\\lambda_2$ at the appropriate time. \n\n### Including observations in the model\n\nAt this point, it may not look like it, but we have fully specified our priors. For example, we can ask and answer questions like \"What does my prior distribution of $\\lambda_1$ look like?\" \n\nTo do this, we will sample from the distribution. The method `.sample()` has a very simple role: get data points from the given distribution. We can then evaluate the resulting tensor to get a NumPy array-like object. \n\n\n```python\n# Define our observed samples\nlambda_1 = tfd.Exponential(rate=1., name=\"lambda_1\")\nsamples = lambda_1.sample(sample_shape=20000)\n \n# Execute graph, convert TF to NumPy\n[ samples_ ] = evaluate([ samples ])\n\n# Visualize our stepwise prior distribution\nplt.figure(figsize(12.5, 5))\nplt.hist(samples_, bins=70, normed=True, histtype=\"stepfilled\")\nplt.title(r\"Prior distribution for $\\lambda_1$\")\nplt.xlim(0, 8);\n```\n\nTo frame this in the notation of the first chapter, though this is a slight abuse of notation, we have specified $P(A)$. Our next goal is to include data/evidence/observations $X$ into our model. \n\nSometimes we may want to match a property of our distribution to a property of observed data. To do so, we get the parameters for our distribution fom the data itself. In this example, the Poisson rate (average number of events) is explicitly set to one over the average of the data:\n\n\n```python\ndata = tf.constant([10., 5.], dtype=tf.float32)\npoisson = tfd.Poisson(rate=1./tf.reduce_mean(data))\n\n\n# Execute graph\n[ data_, poisson_sample_, ] = evaluate([ data, poisson.sample() ])\n\nprint(\"two predetermined data points: \", data_)\nprint(\"\\n mean of our data: \", np.mean(data_))\n\n\nprint(\"\\n random sample from poisson distribution \\n with the mean as the poisson's rate: \\n\", poisson_sample_)\n```\n\n two predetermined data points: [10. 5.]\n \n mean of our data: 7.5\n \n random sample from poisson distribution \n with the mean as the poisson's rate: \n 0.0\n\n\n## Modeling approaches\n\nA good starting thought to Bayesian modeling is to think about *how your data might have been generated*. Position yourself in an omniscient position, and try to imagine how *you* would recreate the dataset. \n\nIn the last chapter we investigated text message data. We begin by asking how our observations may have been generated:\n\n1. We started by thinking \"what is the best random variable to describe this count data?\" A Poisson random variable is a good candidate because it can represent count data. So we model the number of sms's received as sampled from a Poisson distribution.\n\n2. Next, we think, \"Ok, assuming sms's are Poisson-distributed, what do I need for the Poisson distribution?\" Well, the Poisson distribution has a parameter $\\lambda$. \n\n3. Do we know $\\lambda$? No. In fact, we have a suspicion that there are *two* $\\lambda$ values, one for the earlier behaviour and one for the later behaviour. We don't know when the behaviour switches though, but call the switchpoint $\\tau$.\n\n4. What is a good distribution for the two $\\lambda$s? The exponential is good, as it assigns probabilities to positive real numbers. Well the exponential distribution has a parameter too, call it $\\alpha$.\n\n5. Do we know what the parameter $\\alpha$ might be? No. At this point, we could continue and assign a distribution to $\\alpha$, but it's better to stop once we reach a set level of ignorance: whereas we have a prior belief about $\\lambda$, (\"it probably changes over time\", \"it's likely between 10 and 30\", etc.), we don't really have any strong beliefs about $\\alpha$. So it's best to stop here. \n\n What is a good value for $\\alpha$ then? We think that the $\\lambda$s are between 10-30, so if we set $\\alpha$ really low (which corresponds to larger probability on high values) we are not reflecting our prior well. Similar, a too-high alpha misses our prior belief as well. A good idea for $\\alpha$ as to reflect our belief is to set the value so that the mean of $\\lambda$, given $\\alpha$, is equal to our observed mean. This was shown in the last chapter.\n\n6. We have no expert opinion of when $\\tau$ might have occurred. So we will suppose $\\tau$ is from a discrete uniform distribution over the entire timespan.\n\n\nBelow we give a graphical visualization of this, where arrows denote `parent-child` relationships. (provided by the [Daft Python library](http://daft-pgm.org/) )\n\n\n\n\nTFP and other probabilistic programming languages have been designed to tell these data-generation *stories*. More generally, B. Cronin writes [2]:\n\n> Probabilistic programming will unlock narrative explanations of data, one of the holy grails of business analytics and the unsung hero of scientific persuasion. People think in terms of stories - thus the unreasonable power of the anecdote to drive decision-making, well-founded or not. But existing analytics largely fails to provide this kind of story; instead, numbers seemingly appear out of thin air, with little of the causal context that humans prefer when weighing their options.\n\n### Same story; different ending.\n\nInterestingly, we can create *new datasets* by retelling the story.\nFor example, if we reverse the above steps, we can simulate a possible realization of the dataset.\n\n1\\. Specify when the user's behaviour switches by sampling from $\\text{DiscreteUniform}(0, 80)$:\n\n\n```python\ntau = tf.random_uniform(shape=[1], minval=0, maxval=80, dtype=tf.int32)[0]\n\n[ tau_ ] = evaluate([ tau ])\n\nprint(\"Value of Tau (randomly taken from DiscreteUniform(0, 80)):\", tau_)\n```\n\n Value of Tau (randomly taken from DiscreteUniform(0, 80)): 44\n\n\n2\\. Draw $\\lambda_1$ and $\\lambda_2$ from a $\\text{Gamma}(\\alpha)$ distribution:\n\nNote: A gamma distribution is a generalization of the exponential distribution. A gamma distribution with shape parameter $\u03b1 = 1$ and scale parameter $\u03b2$ is an exponential ($\u03b2$) distribution. Here, we use a gamma distribution to have more flexibility than we would have had were we to model with an exponential. Rather than returning values between $0$ and $1$, we can return values much larger than $1$ (i.e., the kinds of numbers one would expect to show up in a daily SMS count).\n\n\n```python\nalpha = 1./8.\n\nlambdas = tfd.Gamma(concentration=1/alpha, rate=0.3).sample(sample_shape=[2]) \n[ lambda_1_, lambda_2_ ] = evaluate( lambdas )\nprint(\"Lambda 1 (randomly taken from Gamma(\u03b1) distribution): \", lambda_1_)\nprint(\"Lambda 2 (randomly taken from Gamma(\u03b1) distribution): \", lambda_2_)\n```\n\n Lambda 1 (randomly taken from Gamma(\u03b1) distribution): 17.011105\n Lambda 2 (randomly taken from Gamma(\u03b1) distribution): 46.05179\n\n\n3\\. For days before $\\tau$, represent the user's received SMS count by sampling from $\\text{Poi}(\\lambda_1)$, and sample from $\\text{Poi}(\\lambda_2)$ for days after $\\tau$. For example:\n\n\n```python\ndata = tf.concat([tfd.Poisson(rate=lambda_1_).sample(sample_shape=tau_),\n tfd.Poisson(rate=lambda_2_).sample(sample_shape= (80 - tau_))], axis=0)\ndays_range = tf.range(80)\n[ data_, days_range_ ] = evaluate([ data, days_range ])\nprint(\"Artificial day-by-day user SMS count created by sampling: \\n\", data_)\n```\n\n Artificial day-by-day user SMS count created by sampling: \n [20. 17. 16. 13. 22. 17. 13. 18. 22. 15. 27. 19. 13. 15. 14. 14. 15. 30.\n 16. 18. 21. 22. 13. 17. 18. 22. 7. 20. 17. 21. 16. 13. 24. 20. 25. 20.\n 13. 19. 21. 16. 19. 10. 23. 7. 54. 44. 46. 49. 36. 50. 44. 45. 51. 44.\n 44. 55. 41. 33. 51. 60. 52. 48. 50. 47. 42. 47. 40. 47. 49. 46. 44. 39.\n 45. 40. 51. 41. 52. 33. 40. 57.]\n\n\n4\\. Plot the artificial dataset:\n\n\n```python\nplt.bar(days_range_, data_, color=TFColor[3])\nplt.bar(tau_ - 1, data_[tau_ - 1], color=\"r\", label=\"user behaviour changed\")\nplt.xlabel(\"Time (days)\")\nplt.ylabel(\"count of text-msgs received\")\nplt.title(\"Artificial dataset\")\nplt.xlim(0, 80)\nplt.legend();\n```\n\nIt is okay that our fictional dataset does not look like our observed dataset: the probability is incredibly small it indeed would. TFP's engine is designed to find good parameters, $\\lambda_i, \\tau$, that maximize this probability. \n\n\nThe ability to generate an artificial dataset is an interesting side effect of our modeling, and we will see that this ability is a very important method of Bayesian inference. We produce a few more datasets below:\n\n\n```python\ndef plot_artificial_sms_dataset(): \n tau = tf.random_uniform(shape=[1], \n minval=0, \n maxval=80,\n dtype=tf.int32)[0]\n alpha = 1./8.\n lambdas = tfd.Gamma(concentration=1/alpha, rate=0.3).sample(sample_shape=[2]) \n [ lambda_1_, lambda_2_ ] = evaluate( lambdas )\n data = tf.concat([tfd.Poisson(rate=lambda_1_).sample(sample_shape=tau),\n tfd.Poisson(rate=lambda_2_).sample(sample_shape= (80 - tau))], axis=0)\n days_range = tf.range(80)\n \n [ \n tau_,\n data_,\n days_range_,\n ] = evaluate([ \n tau,\n data,\n days_range,\n ])\n \n plt.bar(days_range_, data_, color=TFColor[3])\n plt.bar(tau_ - 1, data_[tau_ - 1], color=\"r\", label=\"user behaviour changed\")\n plt.xlim(0, 80);\n\n\nplt.figure(figsize(12.5, 5))\nplt.title(\"More example of artificial datasets\")\nfor i in range(4):\n plt.subplot(4, 1, i+1)\n plot_artificial_sms_dataset()\n\n```\n\nLater we will see how we use this to make predictions and test the appropriateness of our models.\n\n### Example: Bayesian A/B testing\n\nA/B testing is a statistical design pattern for determining the difference of effectiveness between two different treatments. For example, a pharmaceutical company is interested in the effectiveness of drug A vs drug B. The company will test drug A on some fraction of their trials, and drug B on the other fraction (this fraction is often 1/2, but we will relax this assumption). After performing enough trials, the in-house statisticians sift through the data to determine which drug yielded better results. \n\nSimilarly, front-end web developers are interested in which design of their website yields more sales or some other metric of interest. They will route some fraction of visitors to site A, and the other fraction to site B, and record if the visit yielded a sale or not. The data is recorded (in real-time), and analyzed afterwards. \n\nOften, the post-experiment analysis is done using something called a hypothesis test like *difference of means test* or *difference of proportions test*. This involves often misunderstood quantities like a \"Z-score\" and even more confusing \"p-values\" (please don't ask). If you have taken a statistics course, you have probably been taught this technique (though not necessarily *learned* this technique). And if you were like me, you may have felt uncomfortable with their derivation -- good: the Bayesian approach to this problem is much more natural. \n\n\n### A Simple Case\n\nAs this is a hacker book, we'll continue with the web-dev example. For the moment, we will focus on the analysis of site A only. Assume that there is some true $0 \\lt p_A \\lt 1$ probability that users who, upon shown site A, eventually purchase from the site. This is the true effectiveness of site A. Currently, this quantity is unknown to us. \n\nSuppose site A was shown to $N$ people, and $n$ people purchased from the site. One might conclude hastily that $p_A = \\frac{n}{N}$. Unfortunately, the *observed frequency* $\\frac{n}{N}$ does not necessarily equal $p_A$ -- there is a difference between the *observed frequency* and the *true frequency* of an event. The true frequency can be interpreted as the probability of an event occurring. For example, the true frequency of rolling a 1 on a 6-sided die is $\\frac{1}{6}$. Knowing the true frequency of events like:\n\n- fraction of users who make purchases, \n- frequency of social attributes, \n- percent of internet users with cats etc. \n\nare common requests we ask of Nature. Unfortunately, often Nature hides the true frequency from us and we must *infer* it from observed data.\n\nThe *observed frequency* is then the frequency we observe: say rolling the die 100 times you may observe 20 rolls of 1. The observed frequency, 0.2, differs from the true frequency, $\\frac{1}{6}$. We can use Bayesian statistics to infer probable values of the true frequency using an appropriate prior and observed data.\n\n\nWith respect to our A/B example, we are interested in using what we know, $N$ (the total trials administered) and $n$ (the number of conversions), to estimate what $p_A$, the true frequency of buyers, might be. \n\nTo setup a Bayesian model, we need to assign prior distributions to our unknown quantities. *A priori*, what do we think $p_A$ might be? For this example, we have no strong conviction about $p_A$, so for now, let's assume $p_A$ is uniform over [0,1]:\n\n\n```python\nreset_sess()\n\n# The parameters are the bounds of the Uniform.\np = tfd.Uniform(low=0., high=1., name='p')\n\n```\n\nHad we had stronger beliefs, we could have expressed them in the prior above.\n\nFor this example, consider $p_A = 0.05$, and $N = 1500$ users shown site A, and we will simulate whether the user made a purchase or not. To simulate this from $N$ trials, we will use a *Bernoulli* distribution: if $X\\ \\sim \\text{Ber}(p)$, then $X$ is 1 with probability $p$ and 0 with probability $1 - p$. Of course, in practice we do not know $p_A$, but we will use it here to simulate the data. We can assume then that we can use the following generative model:\n\n$$\\begin{align*}\np &\\sim \\text{Uniform}[\\text{low}=0,\\text{high}=1) \\\\\nX\\ &\\sim \\text{Bernoulli}(\\text{prob}=p) \\\\\n\\text{for } i &= 1\\ldots N:\\text{# Users} \\\\\n X_i\\ &\\sim \\text{Bernoulli}(p_i)\n\\end{align*}$$\n\n\n```python\nreset_sess()\n\n#set constants\nprob_true = 0.05 # remember, this is unknown.\nN = 1500\n\n# sample N Bernoulli random variables from Ber(0.05).\n# each random variable has a 0.05 chance of being a 1.\n# this is the data-generation step\n\noccurrences = tfd.Bernoulli(probs=prob_true).sample(sample_shape=N, seed=6.45)\n\n[ \n occurrences_,\n occurrences_sum_,\n occurrences_mean_,\n] = evaluate([ \n occurrences, \n tf.reduce_sum(occurrences),\n tf.reduce_mean(tf.to_float(occurrences))\n])\n\nprint(\"Array of {} Occurences:\".format(N), occurrences_) \nprint(\"(Remember: Python treats True == 1, and False == 0)\")\nprint(\"Sum of (True == 1) Occurences:\", occurrences_sum_)\n```\n\n Array of 1500 Occurences: [0 0 0 ... 0 0 0]\n (Remember: Python treats True == 1, and False == 0)\n Sum of (True == 1) Occurences: 76\n\n\nThe observed frequency is:\n\n\n```python\n# Occurrences.mean is equal to n/N.\nprint(\"What is the observed frequency in Group A? %.4f\" % occurrences_mean_)\nprint(\"Does this equal the true frequency? %s\" % (occurrences_mean_ == prob_true))\n```\n\n What is the observed frequency in Group A? 0.0507\n Does this equal the true frequency? False\n\n\nWe combine our Bernoulli distribution and our observed occurrences into a log probability function based on the two.\n\n\n```python\ndef joint_log_prob(occurrences, prob_A):\n \"\"\"\n Joint log probability optimization function.\n \n Args:\n occurrences: An array of binary values (0 & 1), representing \n the observed frequency\n prob_A: scalar estimate of the probability of a 1 appearing \n Returns: \n Joint log probability optimization function.\n \"\"\" \n \n rv_prob_A = tfd.Uniform(low=0., high=1.)\n \n rv_occurrences = tfd.Bernoulli(probs=prob_A)\n \n return (\n rv_prob_A.log_prob(prob_A)\n + tf.reduce_sum(rv_occurrences.log_prob(occurrences_))\n )\n```\n\nThe goal of probabilistic inference is to find model parameters that may explain\ndata you have observed. TFP performs probabilistic inference by evaluating the\nmodel parameters using a `joint_log_prob` function. The arguments to `joint_log_prob` are data and model parameters\u2014for the model defined in the `joint_log_prob` function itself. The function returns the log of the joint probability that the model parameterized as such generated the observed data per the input arguments.\n\nAll `joint_log_prob` functions have a common structure:\n\n1. The function takes a set of **inputs** to evaluate. Each input is either an\nobserved value or a model parameter.\n\n1. The `joint_log_prob` function uses probability distributions to define a **model** for evaluating the inputs. These distributions measure the likelihood of the input values. (By convention, the distribution that measures the likelihood of the variable `foo` will be named `rv_foo` to note that it is a random variable.) We use two types of distributions in `joint_log_prob` functions:\n\n a. **Prior distributions** measure the likelihood of input values.\nA prior distribution never depends on an input value each prior distribution measures the\nlikelihood of a single input value. Each unknown variable\u2014one that has not been\nobserved directly\u2014needs a corresponding prior. Beliefs about which values could\nbe reasonable determine the prior distribution. Choosing a prior can be tricky,\nso we will cover it in depth in Chapter 6.\n\n b. **Conditional distributions** measure the likelihood of an input value given\nother input values. Typically, the conditional\ndistributions return the likelihood of observed data given the current guess of parameters in the model, p(observed_data | model_parameters).\n\n1. Finally, we calculate and return the **joint log probability** of the inputs.\nThe joint log probability is the sum of the log probabilities from all of the\nprior and conditional distributions. (We take the sum of log probabilities\ninstead of multiplying the probabilities directly for reasons of numerical\nstability: floating point numbers in computers cannot represent the very small\nvalues necessary to calculate the joint log probability unless they are in \nlog space.) The sum of probabilities is actually an unnormalized density; although the total sum of probabilities over all possible inputs might not sum to one, the sum of probabilities is proportional to the true probability density. This proportional distribution is sufficient to estimate the distribution of likely inputs.\n\nLet's map these terms onto the code above. In this example, the input values\nare the observed values in `occurrences` and the unknown value for `prob_A`. The `joint_log_prob` takes the current guess for `prob_A`\nand answers, how likely is the data if `prob_A` is the probability of\n`occurrences`. The answer depends on two distributions:\n1. The prior distribution, `rv_prob_A`, indicates how likely the current value of `prob_A` is by itself.\n2. The conditional distribution, `rv_occurrences`, indicates the likelihood of `occurrences` if `prob_A` were the probability for the Bernoulli distribution.\n\nThe sum of the log of these probabilities is the\njoint log probability. \n\nThe `joint_log_prob` is particularly useful in conjunction with the [`tfp.mcmc`](https://www.tensorflow.org/probability/api_docs/python/tfp/mcmc)\nmodule. Markov chain Monte Carlo (MCMC) algorithms proceed by making educated guesses about the unknown\ninput values and\ncomputing what the likelihood of this set of arguments is. (We\u2019ll talk about how it makes those guesses in Chapter 3.) By repeating this process\nmany times, MCMC builds a distribution of likely parameters. Constructing this\ndistribution is the goal of probabilistic inference.\n\nThen we run our inference algorithm:\n\n\n```python\nnumber_of_steps = 18000\nburnin = 1000\n\n# Set the chain's start state.\ninitial_chain_state = [\n tf.reduce_mean(tf.to_float(occurrences)) * tf.ones([], dtype=tf.float32, name=\"init_prob_A\")\n]\n\n# Since HMC operates over unconstrained space, we need to transform the\n# samples so they live in real-space.\nunconstraining_bijectors = [\n tfp.bijectors.Identity() # Maps R to R. \n]\n\n# Define a closure over our joint_log_prob.\n# The closure makes it so the HMC doesn't try to change the `occurrences` but\n# instead determines the distributions of other parameters that might generate\n# the `occurrences` we observed.\nunnormalized_posterior_log_prob = lambda *args: joint_log_prob(occurrences, *args)\n\n# Initialize the step_size. (It will be automatically adapted.)\nwith tf.variable_scope(tf.get_variable_scope(), reuse=tf.AUTO_REUSE):\n step_size = tf.get_variable(\n name='step_size',\n initializer=tf.constant(0.5, dtype=tf.float32),\n trainable=False,\n use_resource=True\n )\n\n# Defining the HMC\nhmc = tfp.mcmc.TransformedTransitionKernel(\n inner_kernel=tfp.mcmc.HamiltonianMonteCarlo(\n target_log_prob_fn=unnormalized_posterior_log_prob,\n num_leapfrog_steps=6,\n step_size=step_size,\n step_size_update_fn=tfp.mcmc.make_simple_step_size_update_policy(),\n state_gradients_are_stopped=True),\n bijector=unconstraining_bijectors)\n\n# Sampling from the chain.\n[\n posterior_prob_A\n], kernel_results = tfp.mcmc.sample_chain(\n num_results=number_of_steps,\n num_burnin_steps=burnin,\n current_state=initial_chain_state,\n kernel=hmc)\n\n# Initialize any created variables.\ninit_g = tf.global_variables_initializer()\ninit_l = tf.local_variables_initializer()\n```\n\n#### Execute the TF graph to sample from the posterior\n\n\n```python\nevaluate(init_g)\nevaluate(init_l)\n[\n posterior_prob_A_,\n kernel_results_,\n] = evaluate([\n posterior_prob_A,\n kernel_results,\n])\n\n \nprint(\"acceptance rate: {}\".format(\n kernel_results_.inner_results.is_accepted.mean()))\n\nburned_prob_A_trace_ = posterior_prob_A_[burnin:]\n```\n\n acceptance rate: 0.5849444444444445\n\n\nWe plot the posterior distribution of the unknown $p_A$ below:\n\n\n```python\nplt.figure(figsize(12.5, 4))\nplt.title(\"Posterior distribution of $p_A$, the true effectiveness of site A\")\nplt.vlines(prob_true, 0, 90, linestyle=\"--\", label=\"true $p_A$ (unknown)\")\nplt.hist(burned_prob_A_trace_, bins=25, histtype=\"stepfilled\", normed=True)\nplt.legend();\n```\n\nOur posterior distribution puts most weight near the true value of $p_A$, but also some weights in the tails. This is a measure of how uncertain we should be, given our observations. Try changing the number of observations, `N`, and observe how the posterior distribution changes.\n\n### *A* and *B* Together\n\nA similar anaylsis can be done for site B's response data to determine the analogous $p_B$. But what we are really interested in is the *difference* between $p_A$ and $p_B$. Let's infer $p_A$, $p_B$, *and* $\\text{delta} = p_A - p_B$, all at once. We can do this using TFP's deterministic variables. (We'll assume for this exercise that $p_B = 0.04$, so $\\text{delta} = 0.01$, $N_B = 750$ (signifcantly less than $N_A$) and we will simulate site B's data like we did for site A's data ). Our model now looks like the following:\n\n$$\\begin{align*}\np_A &\\sim \\text{Uniform}[\\text{low}=0,\\text{high}=1) \\\\\np_B &\\sim \\text{Uniform}[\\text{low}=0,\\text{high}=1) \\\\\nX\\ &\\sim \\text{Bernoulli}(\\text{prob}=p) \\\\\n\\text{for } i &= 1\\ldots N: \\\\\n X_i\\ &\\sim \\text{Bernoulli}(p_i)\n\\end{align*}$$\n\n\n```python\nreset_sess()\n\n#these two quantities are unknown to us.\ntrue_prob_A_ = 0.05\ntrue_prob_B_ = 0.04\n\n#notice the unequal sample sizes -- no problem in Bayesian analysis.\nN_A_ = 1500\nN_B_ = 750\n\n#generate some observations\nobservations_A = tfd.Bernoulli(name=\"obs_A\", \n probs=true_prob_A_).sample(sample_shape=N_A_, seed=6.45)\nobservations_B = tfd.Bernoulli(name=\"obs_B\", \n probs=true_prob_B_).sample(sample_shape=N_B_, seed=6.45)\n[ \n observations_A_,\n observations_B_,\n] = evaluate([ \n observations_A, \n observations_B, \n])\n\nprint(\"Obs from Site A: \", observations_A_[:30], \"...\")\nprint(\"Observed Prob_A: \", np.mean(observations_A_), \"...\")\nprint(\"Obs from Site B: \", observations_B_[:30], \"...\")\nprint(\"Observed Prob_B: \", np.mean(observations_B_))\n```\n\n Obs from Site A: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] ...\n Observed Prob_A: 0.050666666666666665 ...\n Obs from Site B: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] ...\n Observed Prob_B: 0.04\n\n\nBelow we run inference over the new model:\n\n\n```python\ndef delta(prob_A, prob_B):\n \"\"\"\n Defining the deterministic delta function. This is our unknown of interest.\n \n Args:\n prob_A: scalar estimate of the probability of a 1 appearing in \n observation set A\n prob_B: scalar estimate of the probability of a 1 appearing in \n observation set B\n Returns: \n Difference between prob_A and prob_B\n \"\"\"\n return prob_A - prob_B\n\n \ndef double_joint_log_prob(observations_A_, observations_B_, \n prob_A, prob_B):\n \"\"\"\n Joint log probability optimization function.\n \n Args:\n observations_A: An array of binary values representing the set of \n observations for site A\n observations_B: An array of binary values representing the set of \n observations for site B \n prob_A: scalar estimate of the probability of a 1 appearing in \n observation set A\n prob_B: scalar estimate of the probability of a 1 appearing in \n observation set B \n Returns: \n Joint log probability optimization function.\n \"\"\"\n tfd = tfp.distributions\n \n rv_prob_A = tfd.Uniform(low=0., high=1.)\n rv_prob_B = tfd.Uniform(low=0., high=1.)\n \n rv_obs_A = tfd.Bernoulli(probs=prob_A)\n rv_obs_B = tfd.Bernoulli(probs=prob_B)\n \n return (\n rv_prob_A.log_prob(prob_A)\n + rv_prob_B.log_prob(prob_B)\n + tf.reduce_sum(rv_obs_A.log_prob(observations_A_))\n + tf.reduce_sum(rv_obs_B.log_prob(observations_B_))\n )\n\n```\n\n\n```python\nnumber_of_steps = 20000\nburnin = 1000\n\n# Set the chain's start state.\ninitial_chain_state = [ \n tf.reduce_mean(tf.to_float(observations_A)) * tf.ones([], dtype=tf.float32, name=\"init_prob_A\"),\n tf.reduce_mean(tf.to_float(observations_B)) * tf.ones([], dtype=tf.float32, name=\"init_prob_B\")\n]\n\n# Since HMC operates over unconstrained space, we need to transform the\n# samples so they live in real-space.\nunconstraining_bijectors = [\n tfp.bijectors.Identity(), # Maps R to R.\n tfp.bijectors.Identity() # Maps R to R.\n]\n\n# Define a closure over our joint_log_prob.\nunnormalized_posterior_log_prob = lambda *args: double_joint_log_prob(observations_A_, observations_B_, *args)\n\n# Initialize the step_size. (It will be automatically adapted.)\nwith tf.variable_scope(tf.get_variable_scope(), reuse=tf.AUTO_REUSE):\n step_size = tf.get_variable(\n name='step_size',\n initializer=tf.constant(0.5, dtype=tf.float32),\n trainable=False,\n use_resource=True\n )\n\n# Defining the HMC\nhmc=tfp.mcmc.TransformedTransitionKernel(\n inner_kernel=tfp.mcmc.HamiltonianMonteCarlo(\n target_log_prob_fn=unnormalized_posterior_log_prob,\n num_leapfrog_steps=3,\n step_size=step_size,\n step_size_update_fn=tfp.mcmc.make_simple_step_size_update_policy(),\n state_gradients_are_stopped=True),\n bijector=unconstraining_bijectors)\n\n# Sample from the chain.\n[\n posterior_prob_A,\n posterior_prob_B\n], kernel_results = tfp.mcmc.sample_chain(\n num_results=number_of_steps,\n num_burnin_steps=burnin,\n current_state=initial_chain_state,\n kernel=hmc)\n\n# Initialize any created variables.\ninit_g = tf.global_variables_initializer()\ninit_l = tf.local_variables_initializer()\n```\n\n#### Execute the TF graph to sample from the posterior\n\n\n```python\nevaluate(init_g)\nevaluate(init_l)\n[\n posterior_prob_A_,\n posterior_prob_B_,\n kernel_results_\n] = evaluate([\n posterior_prob_A,\n posterior_prob_B,\n kernel_results\n])\n \nprint(\"acceptance rate: {}\".format(\n kernel_results_.inner_results.is_accepted.mean()))\n\nburned_prob_A_trace_ = posterior_prob_A_[burnin:]\nburned_prob_B_trace_ = posterior_prob_B_[burnin:]\nburned_delta_trace_ = (posterior_prob_A_ - posterior_prob_B_)[burnin:]\n\n```\n\n acceptance rate: 0.6215\n\n\nBelow we plot the posterior distributions for the three unknowns: \n\n\n```python\nplt.figure(figsize(12.5, 12.5))\n\n#histogram of posteriors\n\nax = plt.subplot(311)\n\nplt.xlim(0, .1)\nplt.hist(burned_prob_A_trace_, histtype='stepfilled', bins=25, alpha=0.85,\n label=\"posterior of $p_A$\", color=TFColor[0], normed=True)\nplt.vlines(true_prob_A_, 0, 80, linestyle=\"--\", label=\"true $p_A$ (unknown)\")\nplt.legend(loc=\"upper right\")\nplt.title(\"Posterior distributions of $p_A$, $p_B$, and delta unknowns\")\n\nax = plt.subplot(312)\n\nplt.xlim(0, .1)\nplt.hist(burned_prob_B_trace_, histtype='stepfilled', bins=25, alpha=0.85,\n label=\"posterior of $p_B$\", color=TFColor[2], normed=True)\nplt.vlines(true_prob_B_, 0, 80, linestyle=\"--\", label=\"true $p_B$ (unknown)\")\nplt.legend(loc=\"upper right\")\n\nax = plt.subplot(313)\nplt.hist(burned_delta_trace_, histtype='stepfilled', bins=30, alpha=0.85,\n label=\"posterior of delta\", color=TFColor[6], normed=True)\nplt.vlines(true_prob_A_ - true_prob_B_, 0, 60, linestyle=\"--\",\n label=\"true delta (unknown)\")\nplt.vlines(0, 0, 60, color=\"black\", alpha=0.2)\nplt.legend(loc=\"upper right\");\n```\n\nNotice that as a result of `N_B < N_A`, i.e. we have less data from site B, our posterior distribution of $p_B$ is fatter, implying we are less certain about the true value of $p_B$ than we are of $p_A$. \n\nWith respect to the posterior distribution of $\\text{delta}$, we can see that the majority of the distribution is above $\\text{delta}=0$, implying there site A's response is likely better than site B's response. The probability this inference is incorrect is easily computable:\n\n\n```python\n# Count the number of samples less than 0, i.e. the area under the curve\n# before 0, represent the probability that site A is worse than site B.\nprint(\"Probability site A is WORSE than site B: %.3f\" % \\\n np.mean(burned_delta_trace_ < 0))\n\nprint(\"Probability site A is BETTER than site B: %.3f\" % \\\n np.mean(burned_delta_trace_ > 0))\n```\n\n Probability site A is WORSE than site B: 0.138\n Probability site A is BETTER than site B: 0.862\n\n\nIf this probability is too high for comfortable decision-making, we can perform more trials on site B (as site B has less samples to begin with, each additional data point for site B contributes more inferential \"power\" than each additional data point for site A). \n\nTry playing with the parameters `true_prob_A`, `true_prob_B`, `N_A`, and `N_B`, to see what the posterior of $\\text{delta}$ looks like. Notice in all this, the difference in sample sizes between site A and site B was never mentioned: it naturally fits into Bayesian analysis.\n\nI hope the readers feel this style of A/B testing is more natural than hypothesis testing, which has probably confused more than helped practitioners. Later in this book, we will see two extensions of this model: the first to help dynamically adjust for bad sites, and the second will improve the speed of this computation by reducing the analysis to a single equation. \n\n## An algorithm for human deceit\n\nSocial data has an additional layer of interest as people are not always honest with responses, which adds a further complication into inference. For example, simply asking individuals \"Have you ever cheated on a test?\" will surely contain some rate of dishonesty. What you can say for certain is that the true rate is less than your observed rate (assuming individuals lie *only* about *not cheating*; I cannot imagine one who would admit \"Yes\" to cheating when in fact they hadn't cheated). \n\nTo present an elegant solution to circumventing this dishonesty problem, and to demonstrate Bayesian modeling, we first need to introduce the binomial distribution.\n\n\n## The Binomial Distribution\n\nThe binomial distribution is one of the most popular distributions, mostly because of its simplicity and usefulness. Unlike the other distributions we have encountered thus far in the book, the binomial distribution has 2 parameters: $N$, a positive integer representing $N$ trials or number of instances of potential events, and $p$, the probability of an event occurring in a single trial. Like the Poisson distribution, it is a discrete distribution, but unlike the Poisson distribution, it only weighs integers from $0$ to $N$. The mass distribution looks like:\n\n$$P( X = k ) = {{N}\\choose{k}} p^k(1-p)^{N-k}$$\n\nIf $X$ is a binomial random variable with parameters $p$ and $N$, denoted $X \\sim \\text{Bin}(N,p)$, then $X$ is the number of events that occurred in the $N$ trials (obviously $0 \\le X \\le N$). The larger $p$ is (while still remaining between 0 and 1), the more events are likely to occur. The expected value of a binomial is equal to $Np$. Below we plot the mass probability distribution for varying parameters. \n\n\n```python\nk_values = tf.range(start=0, limit=(N + 1), dtype=tf.float32)\nrandom_var_probs_1 = tfd.Binomial(total_count=10., probs=.4).prob(k_values)\nrandom_var_probs_2 = tfd.Binomial(total_count=10., probs=.9).prob(k_values)\n\n# Execute graph\n[\n k_values_,\n random_var_probs_1_,\n random_var_probs_2_,\n] = evaluate([\n k_values,\n random_var_probs_1,\n random_var_probs_2,\n])\n\n# Display results\nplt.figure(figsize=(12.5, 4))\ncolors = [TFColor[3], TFColor[0]] \n\nplt.bar(k_values_ - 0.5, random_var_probs_1_, color=colors[0],\n edgecolor=colors[0],\n alpha=0.6,\n label=\"$N$: %d, $p$: %.1f\" % (10., .4),\n linewidth=3)\nplt.bar(k_values_ - 0.5, random_var_probs_2_, color=colors[1],\n edgecolor=colors[1],\n alpha=0.6,\n label=\"$N$: %d, $p$: %.1f\" % (10., .9),\n linewidth=3)\n\nplt.legend(loc=\"upper left\")\nplt.xlim(0, 10.5)\nplt.xlabel(\"$k$\")\nplt.ylabel(\"$P(X = k)$\")\nplt.title(\"Probability mass distributions of binomial random variables\");\n```\n\nThe special case when $N = 1$ corresponds to the Bernoulli distribution. There is another connection between Bernoulli and Binomial random variables. If we have $X_1, X_2, ... , X_N$ Bernoulli random variables with the same $p$, then $Z = X_1 + X_2 + ... + X_N \\sim \\text{Binomial}(N, p )$.\n\nThe expected value of a Bernoulli random variable is $p$. This can be seen by noting the more general Binomial random variable has expected value $Np$ and setting $N=1$.\n\n## Example: Cheating among students\n\nWe will use the binomial distribution to determine the frequency of students cheating during an exam. If we let $N$ be the total number of students who took the exam, and assuming each student is interviewed post-exam (answering without consequence), we will receive integer $X$ \"Yes I did cheat\" answers. We then find the posterior distribution of $p$, given $N$, some specified prior on $p$, and observed data $X$. \n\nThis is a completely absurd model. No student, even with a free-pass against punishment, would admit to cheating. What we need is a better *algorithm* to ask students if they had cheated. Ideally the algorithm should encourage individuals to be honest while preserving privacy. The following proposed algorithm is a solution I greatly admire for its ingenuity and effectiveness:\n\n> In the interview process for each student, the student flips a coin, hidden from the interviewer. The student agrees to answer honestly if the coin comes up heads. Otherwise, if the coin comes up tails, the student (secretly) flips the coin again, and answers \"Yes, I did cheat\" if the coin flip lands heads, and \"No, I did not cheat\", if the coin flip lands tails. This way, the interviewer does not know if a \"Yes\" was the result of a guilty plea, or a Heads on a second coin toss. Thus privacy is preserved and the researchers receive honest answers. \n\nI call this the Privacy Algorithm. One could of course argue that the interviewers are still receiving false data since some *Yes*'s are not confessions but instead randomness, but an alternative perspective is that the researchers are discarding approximately half of their original dataset since half of the responses will be noise. But they have gained a systematic data generation process that can be modeled. Furthermore, they do not have to incorporate (perhaps somewhat naively) the possibility of deceitful answers. We can use TFP to dig through this noisy model, and find a posterior distribution for the true frequency of liars. \n\nSuppose 100 students are being surveyed for cheating, and we wish to find $p$, the proportion of cheaters. There are a few ways we can model this in TFP. I'll demonstrate the most explicit way, and later show a simplified version. Both versions arrive at the same inference. In our data-generation model, we sample $p$, the true proportion of cheaters, from a prior. Since we are quite ignorant about $p$, we will assign it a $\\text{Uniform}(0,1)$ prior.\n\n\n```python\nreset_sess()\n\nN = 100\np = tfd.Uniform(name=\"freq_cheating\", low=0., high=1.)\n```\n\nAgain, thinking of our data-generation model, we assign Bernoulli random variables to the 100 students: 1 implies they cheated and 0 implies they did not. \n\n\n```python\nN = 100\nreset_sess()\np = tfd.Uniform(name=\"freq_cheating\", low=0., high=1.)\ntrue_answers = tfd.Bernoulli(name=\"truths\", \n probs=p.sample()).sample(sample_shape=N, \n seed=5)\n# Execute graph\n[\n true_answers_,\n] = evaluate([\n true_answers,\n])\n\nprint(true_answers_)\n```\n\n [1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1\n 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1\n 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1]\n\n\nIf we carry out the algorithm, the next step that occurs is the first coin-flip each student makes. This can be modeled again by sampling 100 Bernoulli random variables with $p=1/2$: denote a 1 as a *Heads* and 0 a *Tails*.\n\n\n```python\nN = 100\nfirst_coin_flips = tfd.Bernoulli(name=\"first_flips\", \n probs=0.5).sample(sample_shape=N, \n seed=5)\n# Execute graph\n[\n first_coin_flips_,\n] = evaluate([\n first_coin_flips,\n])\n\nprint(first_coin_flips_)\n```\n\n [1 0 1 0 0 1 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 1\n 0 1 1 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0\n 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 1 0 1 0 0 1 1]\n\n\nAlthough *not everyone* flips a second time, we can still model the possible realization of second coin-flips:\n\n\n```python\nN = 100\nsecond_coin_flips = tfd.Bernoulli(name=\"second_flips\", \n probs=0.5).sample(sample_shape=N, \n seed=5)\n# Execute graph\n[\n second_coin_flips_,\n] = evaluate([\n second_coin_flips,\n])\n\nprint(second_coin_flips_)\n```\n\n [1 0 1 0 0 1 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 1\n 0 1 1 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0\n 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 1 0 1 0 0 1 1]\n\n\nUsing these variables, we can return a possible realization of the *observed proportion* of \"Yes\" responses. \n\n\n```python\ndef observed_proportion_calc(t_a = true_answers, \n fc = first_coin_flips,\n sc = second_coin_flips):\n \"\"\"\n Unnormalized log posterior distribution function\n \n Args:\n t_a: array of binary variables representing the true answers\n fc: array of binary variables representing the simulated first flips \n sc: array of binary variables representing the simulated second flips\n Returns: \n Observed proportion of coin flips\n Closure over: N\n \"\"\"\n observed = fc * t_a + (1 - fc) * sc\n observed_proportion = tf.to_float(tf.reduce_sum(observed)) / tf.to_float(N)\n \n return tf.to_float(observed_proportion)\n\n```\n\nThe line `fc*t_a + (1-fc)*sc` contains the heart of the Privacy algorithm. Elements in this array are 1 *if and only if* i) the first toss is heads and the student cheated or ii) the first toss is tails, and the second is heads, and are 0 else. Finally, the last line sums this vector and divides by `float(N)`, producing a proportion. \n\n\n```python\nobserved_proportion_val = observed_proportion_calc(t_a=true_answers_,\n fc=first_coin_flips_,\n sc=second_coin_flips_)\n# Execute graph\n[\n observed_proportion_val_,\n] = evaluate([\n observed_proportion_val,\n])\n\nprint(observed_proportion_val_)\n```\n\n 0.48\n\n\nNext we need a dataset. After performing our coin-flipped interviews the researchers received 35 \"Yes\" responses. To put this into a relative perspective, if there truly were no cheaters, we should expect to see on average 1/4 of all responses being a \"Yes\" (half chance of having first coin land Tails, and another half chance of having second coin land Heads), so about 25 responses in a cheat-free world. On the other hand, if *all students cheated*, we should expected to see approximately 3/4 of all responses be \"Yes\". \n\nThe researchers observe a Binomial random variable, with `N = 100` and `total_yes = 35`: \n\n\n```python\ntotal_count = 100\ntotal_yes = 35\n```\n\n\n```python\ndef coin_joint_log_prob(total_yes, total_count, lies_prob):\n \"\"\"\n Joint log probability optimization function.\n \n Args:\n headsflips: Integer for total number of observed heads flips\n N: Integer for number of total observation\n lies_prob: Test probability of a heads flip (1) for a Binomial distribution\n Returns: \n Joint log probability optimization function.\n \"\"\"\n \n rv_lies_prob = tfd.Uniform(name=\"rv_lies_prob\",low=0., high=1.)\n\n cheated = tfd.Bernoulli(probs=tf.to_float(lies_prob)).sample(total_count)\n first_flips = tfd.Bernoulli(probs=0.5).sample(total_count)\n second_flips = tfd.Bernoulli(probs=0.5).sample(total_count)\n observed_probability = tf.reduce_sum(tf.to_float(\n cheated * first_flips + (1 - first_flips) * second_flips)) / total_count\n\n rv_yeses = tfd.Binomial(name=\"rv_yeses\",\n total_count=float(total_count),\n probs=observed_probability)\n \n return (\n rv_lies_prob.log_prob(lies_prob)\n + tf.reduce_sum(rv_yeses.log_prob(tf.to_float(total_yes)))\n )\n```\n\nBelow we add all the variables of interest to our Metropolis-Hastings sampler and run our black-box algorithm over the model. It's important to note that we're using a Metropolis-Hastings MCMC instead of a Hamiltonian since we're sampling inside.\n\n\n```python\nburnin = 15000\nnum_of_steps = 40000\ntotal_count=100\n\n# Set the chain's start state.\ninitial_chain_state = [\n 0.4 * tf.ones([], dtype=tf.float32, name=\"init_prob\")\n]\n\n# Define a closure over our joint_log_prob.\nunnormalized_posterior_log_prob = lambda *args: coin_joint_log_prob(total_yes, total_count, *args)\n\n# Defining the Metropolis-Hastings\n# We use a Metropolis-Hastings method here instead of Hamiltonian method\n# because the coin flips in the above example are non-differentiable and cannot\n# bue used with HMC.\nmetropolis=tfp.mcmc.RandomWalkMetropolis(\n target_log_prob_fn=unnormalized_posterior_log_prob,\n seed=54)\n\n# Sample from the chain.\n[\n posterior_p\n], kernel_results = tfp.mcmc.sample_chain(\n num_results=num_of_steps,\n num_burnin_steps=burnin,\n current_state=initial_chain_state,\n kernel=metropolis,\n parallel_iterations=1,\n name='Metropolis-Hastings_coin-flips')\n```\n\n##### Executing the TF graph to sample from the posterior\n\n\n```python\n# Content Warning: This cell can take up to 5 minutes in Graph Mode\n[\n posterior_p_,\n kernel_results_\n] = evaluate([\n posterior_p,\n kernel_results,\n])\n \nprint(\"acceptance rate: {}\".format(\n kernel_results_.is_accepted.mean()))\n# print(\"prob_p trace: \", posterior_p_)\n# print(\"prob_p burned trace: \", posterior_p_[burnin:])\nburned_cheating_freq_samples_ = posterior_p_[burnin:]\n```\n\n acceptance rate: 0.1056\n\n\nAnd finally we can plot the results.\n\n\n```python\nplt.figure(figsize(12.5, 6))\np_trace_ = burned_cheating_freq_samples_\nplt.hist(p_trace_, histtype=\"stepfilled\", normed=True, alpha=0.85, bins=30, \n label=\"posterior distribution\", color=TFColor[3])\nplt.vlines([.1, .40], [0, 0], [5, 5], alpha=0.3)\nplt.xlim(0, 1)\nplt.legend();\n```\n\nWith regards to the above plot, we are still pretty uncertain about what the true frequency of cheaters might be, but we have narrowed it down to a range between 0.1 to 0.4 (marked by the solid lines). This is pretty good, as *a priori* we had no idea how many students might have cheated (hence the uniform distribution for our prior). On the other hand, it is also pretty bad since there is a .3 length window the true value most likely lives in. Have we even gained anything, or are we still too uncertain about the true frequency? \n\nI would argue, yes, we have discovered something. It is implausible, according to our posterior, that there are *no cheaters*, i.e. the posterior assigns low probability to $p=0$. Since we started with an uniform prior, treating all values of $p$ as equally plausible, but the data ruled out $p=0$ as a possibility, we can be confident that there were cheaters. \n\nThis kind of algorithm can be used to gather private information from users and be *reasonably* confident that the data, though noisy, is truthful. \n\n\n\n### Alternative TFP Model\n\nGiven a value for $p$ (which from our god-like position we know), we can find the probability the student will answer yes: \n\n\\begin{align}\nP(\\text{\"Yes\"}) &= P( \\text{Heads on first coin} )P( \\text{cheater} ) + P( \\text{Tails on first coin} )P( \\text{Heads on second coin} ) \\\\\\\\\n&= \\frac{1}{2}p + \\frac{1}{2}\\frac{1}{2}\\\\\\\\\n&= \\frac{p}{2} + \\frac{1}{4}\n\\end{align}\n\nThus, knowing $p$ we know the probability a student will respond \"Yes\". In TFP, we can create a deterministic function to evaluate the probability of responding \"Yes\", given $p$:\n\n\n```python\nreset_sess()\n\np_new = tfd.Uniform(name=\"new_freq_cheating\", \n low=0., \n high=1.)\np_new_skewed = tfd.Deterministic(name=\"p_skewed\", \n loc=(0.5 * p_new.sample(seed=0.5) + 0.25)).sample(seed=0.5)\n```\n\nI could have typed `p_skewed = 0.5*p + 0.25` instead for a one-liner, as the elementary operations of addition and scalar multiplication will implicitly create a deterministic variable, but I wanted to make the determinism explicit for clarity's sake. \n\nIf we know the probability of respondents saying \"Yes\", which is `p_skewed`, and we have $N=100$ students, the number of \"Yes\" responses is a binomial random variable with parameters `N` and `p_skewed`.\n\nThis is where we include our observed 35 \"Yes\" responses out of a total of 100 which are passed to the `joint_log_prob`.\n\n\n```python\nN = 100.\ntotal_yes = 35.\n\ndef alt_joint_log_prob(yes_responses, N, prob_cheating):\n \"\"\"\n Alternative joint log probability optimization function.\n \n Args:\n yes_responses: Integer for total number of affirmative responses\n N: Integer for number of total observation\n prob_cheating: Test probability of a student actually cheating\n Returns: \n Joint log probability optimization function.\n \"\"\"\n tfd = tfp.distributions\n \n rv_prob = tfd.Uniform(name=\"rv_new_freq_cheating\", low=0., high=1.)\n prob_skewed = 0.5 * prob_cheating + 0.25\n rv_yes_responses = tfd.Binomial(name=\"rv_yes_responses\",\n total_count=tf.to_float(N), \n probs=prob_skewed)\n\n return (\n rv_prob.log_prob(prob_cheating)\n + tf.reduce_sum(rv_yes_responses.log_prob(tf.to_float(yes_responses)))\n )\n```\n\n\nBelow we add all the variables of interest to our HMC component-defining cell and run our black-box algorithm over the model. \n\n\n```python\nnumber_of_steps = 25000\nburnin = 2500\n\n# Set the chain's start state.\ninitial_chain_state = [\n 0.2 * tf.ones([], dtype=tf.float32, name=\"init_skewed_p\")\n]\n\n# Since HMC operates over unconstrained space, we need to transform the\n# samples so they live in real-space.\nunconstraining_bijectors = [\n tfp.bijectors.Sigmoid(), # Maps [0,1] to R.\n]\n\n# Define a closure over our joint_log_prob.\n# unnormalized_posterior_log_prob = lambda *args: alt_joint_log_prob(headsflips, total_yes, N, *args)\nunnormalized_posterior_log_prob = lambda *args: alt_joint_log_prob(total_yes, N, *args)\n\n# Initialize the step_size. (It will be automatically adapted.)\nwith tf.variable_scope(tf.get_variable_scope(), reuse=tf.AUTO_REUSE):\n step_size = tf.get_variable(\n name='skewed_step_size',\n initializer=tf.constant(0.5, dtype=tf.float32),\n trainable=False,\n use_resource=True\n ) \n\n# Defining the HMC\nhmc=tfp.mcmc.TransformedTransitionKernel(\n inner_kernel=tfp.mcmc.HamiltonianMonteCarlo(\n target_log_prob_fn=unnormalized_posterior_log_prob,\n num_leapfrog_steps=2,\n step_size=step_size,\n step_size_update_fn=tfp.mcmc.make_simple_step_size_update_policy(),\n state_gradients_are_stopped=True),\n bijector=unconstraining_bijectors)\n\n# Sample from the chain.\n[\n posterior_skewed_p\n], kernel_results = tfp.mcmc.sample_chain(\n num_results=number_of_steps,\n num_burnin_steps=burnin,\n current_state=initial_chain_state,\n kernel=hmc)\n\n# Initialize any created variables.\n# This prevents a FailedPreconditionError\ninit_g = tf.global_variables_initializer()\ninit_l = tf.local_variables_initializer()\n```\n\n#### Execute the TF graph to sample from the posterior\n\n\n```python\n# This cell may take 5 minutes in Graph Mode\nevaluate(init_g)\nevaluate(init_l)\n[\n posterior_skewed_p_,\n kernel_results_\n] = evaluate([\n posterior_skewed_p,\n kernel_results\n])\n\n \nprint(\"acceptance rate: {}\".format(\n kernel_results_.inner_results.is_accepted.mean()))\n# print(\"final step size: {}\".format(\n# kernel_results_.inner_results.extra.step_size_assign[-100:].mean()))\n\n# print(\"p_skewed trace: \", posterior_skewed_p_)\n# print(\"p_skewed burned trace: \", posterior_skewed_p_[burnin:])\nfreq_cheating_samples_ = posterior_skewed_p_[burnin:]\n\n```\n\n acceptance rate: 0.60368\n\n\nNow we can plot our results\n\n\n```python\nplt.figure(figsize(12.5, 6))\np_trace_ = freq_cheating_samples_\nplt.hist(p_trace_, histtype=\"stepfilled\", normed=True, alpha=0.85, bins=30, \n label=\"posterior distribution\", color=TFColor[3])\nplt.vlines([.1, .40], [0, 0], [5, 5], alpha=0.2)\nplt.xlim(0, 1)\nplt.legend();\n```\n\nThe remainder of this chapter examines some practical examples of TFP and TFP modeling:\n\n## Example: Challenger Space Shuttle Disaster \n\nOn January 28, 1986, the twenty-fifth flight of the U.S. space shuttle program ended in disaster when one of the rocket boosters of the Shuttle Challenger exploded shortly after lift-off, killing all seven crew members. The presidential commission on the accident concluded that it was caused by the failure of an O-ring in a field joint on the rocket booster, and that this failure was due to a faulty design that made the O-ring unacceptably sensitive to a number of factors including outside temperature. Of the previous 24 flights, data were available on failures of O-rings on 23, (one was lost at sea), and these data were discussed on the evening preceding the Challenger launch, but unfortunately only the data corresponding to the 7 flights on which there was a damage incident were considered important and these were thought to show no obvious trend. The data are shown below (see [1]):\n\n\n\n\n\n\n\n```python\nreset_sess()\n\nimport wget\nurl = 'https://raw.githubusercontent.com/CamDavidsonPilon/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers/master/Chapter2_MorePyMC/data/challenger_data.csv'\nfilename = wget.download(url)\nfilename\n```\n\n\n\n\n 'challenger_data.csv'\n\n\n\n\n```python\nplt.figure(figsize(12.5, 3.5))\nnp.set_printoptions(precision=3, suppress=True)\nchallenger_data_ = np.genfromtxt(\"challenger_data.csv\", skip_header=1,\n usecols=[1, 2], missing_values=\"NA\",\n delimiter=\",\")\n#drop the NA values\nchallenger_data_ = challenger_data_[~np.isnan(challenger_data_[:, 1])]\n\n#plot it, as a function of tempature (the first column)\nprint(\"Temp (F), O-Ring failure?\")\nprint(challenger_data_)\n\nplt.scatter(challenger_data_[:, 0], challenger_data_[:, 1], s=75, color=\"k\",\n alpha=0.5)\nplt.yticks([0, 1])\nplt.ylabel(\"Damage Incident?\")\nplt.xlabel(\"Outside temperature (Fahrenheit)\")\nplt.title(\"Defects of the Space Shuttle O-Rings vs temperature\");\n\n```\n\nIt looks clear that *the probability* of damage incidents occurring increases as the outside temperature decreases. We are interested in modeling the probability here because it does not look like there is a strict cutoff point between temperature and a damage incident occurring. The best we can do is ask \"At temperature $t$, what is the probability of a damage incident?\". The goal of this example is to answer that question.\n\nWe need a function of temperature, call it $p(t)$, that is bounded between 0 and 1 (so as to model a probability) and changes from 1 to 0 as we increase temperature. There are actually many such functions, but the most popular choice is the *logistic function.*\n\n$$p(t) = \\frac{1}{ 1 + e^{ \\;\\beta t } } $$\n\nIn this model, $\\beta$ is the variable we are uncertain about. Below is the function plotted for $\\beta = 1, 3, -5$.\n\n\n```python\ndef logistic(x, beta):\n \"\"\"\n Logistic Function\n \n Args:\n x: independent variable\n beta: beta term\n Returns: \n Logistic function\n \"\"\"\n return 1.0 / (1.0 + tf.exp(beta * x))\n\nx_vals = tf.linspace(start=-4., stop=4., num=100)\nlog_beta_1 = logistic(x_vals, 1.)\nlog_beta_3 = logistic(x_vals, 3.)\nlog_beta_m5 = logistic(x_vals, -5.)\n\n[\n x_vals_,\n log_beta_1_,\n log_beta_3_,\n log_beta_m5_,\n] = evaluate([\n x_vals,\n log_beta_1,\n log_beta_3,\n log_beta_m5,\n])\n\nplt.figure(figsize(12.5, 3))\nplt.plot(x_vals_, log_beta_1_, label=r\"$\\beta = 1$\", color=TFColor[0])\nplt.plot(x_vals_, log_beta_3_, label=r\"$\\beta = 3$\", color=TFColor[3])\nplt.plot(x_vals_, log_beta_m5_, label=r\"$\\beta = -5$\", color=TFColor[6])\nplt.legend();\n```\n\nBut something is missing. In the plot of the logistic function, the probability changes only near zero, but in our data above the probability changes around 65 to 70. We need to add a *bias* term to our logistic function:\n\n$$p(t) = \\frac{1}{ 1 + e^{ \\;\\beta t + \\alpha } } $$\n\nSome plots are below, with differing $\\alpha$.\n\n\n```python\ndef logistic(x, beta, alpha=0):\n \"\"\"\n Logistic Function with offset\n \n Args:\n x: independent variable\n beta: beta term \n alpha: alpha term\n Returns: \n Logistic function\n \"\"\"\n return 1.0 / (1.0 + tf.exp((beta * x) + alpha))\n\nx_vals = tf.linspace(start=-4., stop=4., num=100)\nlog_beta_1_alpha_1 = logistic(x_vals, 1, 1)\nlog_beta_3_alpha_m2 = logistic(x_vals, 3, -2)\nlog_beta_m5_alpha_7 = logistic(x_vals, -5, 7)\n\n[\n x_vals_,\n log_beta_1_alpha_1_,\n log_beta_3_alpha_m2_,\n log_beta_m5_alpha_7_,\n] = evaluate([\n x_vals,\n log_beta_1_alpha_1,\n log_beta_3_alpha_m2,\n log_beta_m5_alpha_7,\n])\n\nplt.figure(figsize(12.5, 3))\nplt.plot(x_vals_, log_beta_1_, label=r\"$\\beta = 1$\", ls=\"--\", lw=1, color=TFColor[0])\nplt.plot(x_vals_, log_beta_3_, label=r\"$\\beta = 3$\", ls=\"--\", lw=1, color=TFColor[3])\nplt.plot(x_vals_, log_beta_m5_, label=r\"$\\beta = -5$\", ls=\"--\", lw=1, color=TFColor[6])\nplt.plot(x_vals_, log_beta_1_alpha_1_, label=r\"$\\beta = 1, \\alpha = 1$\", color=TFColor[0])\nplt.plot(x_vals_, log_beta_3_alpha_m2_, label=r\"$\\beta = 3, \\alpha = -2$\", color=TFColor[3])\nplt.plot(x_vals_, log_beta_m5_alpha_7_, label=r\"$\\beta = -5, \\alpha = 7$\", color=TFColor[6])\nplt.legend(loc=\"lower left\");\n```\n\nAdding a constant term $\\alpha$ amounts to shifting the curve left or right (hence why it is called a *bias*).\n\nLet's start modeling this in TFP. The $\\beta, \\alpha$ parameters have no reason to be positive, bounded or relatively large, so they are best modeled by a *Normal random variable*, introduced next.\n\n### Normal distributions\n\nA Normal random variable, denoted $X \\sim N(\\mu, 1/\\tau)$, has a distribution with two parameters: the mean, $\\mu$, and the *precision*, $\\tau$. Those familiar with the Normal distribution already have probably seen $\\sigma^2$ instead of $\\tau^{-1}$. They are in fact reciprocals of each other. The change was motivated by simpler mathematical analysis and is an artifact of older Bayesian methods. Just remember: the smaller $\\tau$, the larger the spread of the distribution (i.e. we are more uncertain); the larger $\\tau$, the tighter the distribution (i.e. we are more certain). Regardless, $\\tau$ is always positive. \n\nThe probability density function of a $N( \\mu, 1/\\tau)$ random variable is:\n\n$$ f(x | \\mu, \\tau) = \\sqrt{\\frac{\\tau}{2\\pi}} \\exp\\left( -\\frac{\\tau}{2} (x-\\mu)^2 \\right) $$\n\nWe plot some different density functions below. \n\n\n```python\nrand_x_vals = tf.linspace(start=-8., stop=7., num=150)\n\ndensity_func_1 = tfd.Normal(loc=float(-2.), scale=float(1./.7)).prob(rand_x_vals)\ndensity_func_2 = tfd.Normal(loc=float(0.), scale=float(1./1)).prob(rand_x_vals)\ndensity_func_3 = tfd.Normal(loc=float(3.), scale=float(1./2.8)).prob(rand_x_vals)\n\n[\n rand_x_vals_,\n density_func_1_,\n density_func_2_,\n density_func_3_,\n] = evaluate([\n rand_x_vals,\n density_func_1,\n density_func_2,\n density_func_3,\n])\n\ncolors = [TFColor[3], TFColor[0], TFColor[6]]\n\nplt.figure(figsize(12.5, 3))\nplt.plot(rand_x_vals_, density_func_1_,\n label=r\"$\\mu = %d, \\tau = %.1f$\" % (-2., .7), color=TFColor[3])\nplt.fill_between(rand_x_vals_, density_func_1_, color=TFColor[3], alpha=.33)\nplt.plot(rand_x_vals_, density_func_2_, \n label=r\"$\\mu = %d, \\tau = %.1f$\" % (0., 1), color=TFColor[0])\nplt.fill_between(rand_x_vals_, density_func_2_, color=TFColor[0], alpha=.33)\nplt.plot(rand_x_vals_, density_func_3_,\n label=r\"$\\mu = %d, \\tau = %.1f$\" % (3., 2.8), color=TFColor[6])\nplt.fill_between(rand_x_vals_, density_func_3_, color=TFColor[6], alpha=.33)\n\nplt.legend(loc=r\"upper right\")\nplt.xlabel(r\"$x$\")\nplt.ylabel(r\"density function at $x$\")\nplt.title(r\"Probability distribution of three different Normal random variables\");\n```\n\nA Normal random variable can be take on any real number, but the variable is very likely to be relatively close to $\\mu$. In fact, the expected value of a Normal is equal to its $\\mu$ parameter:\n\n$$ E[ X | \\mu, \\tau] = \\mu$$\n\nand its variance is equal to the inverse of $\\tau$:\n\n$$Var( X | \\mu, \\tau ) = \\frac{1}{\\tau}$$\n\n\n\nBelow we continue our modeling of the Challenger space craft:\n\n\n```python\nreset_sess()\n\ntemperature_ = challenger_data_[:, 0]\ntemperature = tf.convert_to_tensor(temperature_, dtype=tf.float32)\nD_ = challenger_data_[:, 1] # defect or not?\nD = tf.convert_to_tensor(D_, dtype=tf.float32)\n\nbeta = tfd.Normal(name=\"beta\", loc=0.3, scale=1000.).sample()\nalpha = tfd.Normal(name=\"alpha\", loc=-15., scale=1000.).sample()\np_deterministic = tfd.Deterministic(name=\"p\", loc=1.0/(1. + tf.exp(beta * temperature_ + alpha))).sample()\n\n[\n prior_alpha_,\n prior_beta_,\n p_deterministic_,\n D_,\n] = evaluate([\n alpha,\n beta,\n p_deterministic,\n D,\n])\n\n```\n\nWe have our probabilities, but how do we connect them to our observed data? A *Bernoulli* random variable with parameter $p$, denoted $\\text{Ber}(p)$, is a random variable that takes value 1 with probability $p$, and 0 else. Thus, our model can look like:\n\n$$ \\text{Defect Incident, $D_i$} \\sim \\text{Ber}( \\;p(t_i)\\; ), \\;\\; i=1..N$$\n\nwhere $p(t)$ is our logistic function and $t_i$ are the temperatures we have observations about. Notice in the code below we set the values of `beta` and `alpha` to 0 in `initial_chain_state`. The reason for this is that if `beta` and `alpha` are very large, they make `p` equal to 1 or 0. Unfortunately, `tfd.Bernoulli` does not like probabilities of exactly 0 or 1, though they are mathematically well-defined probabilities. So by setting the coefficient values to `0`, we set the variable `p` to be a reasonable starting value. This has no effect on our results, nor does it mean we are including any additional information in our prior. It is simply a computational caveat in TFP. \n\n\n```python\ndef challenger_joint_log_prob(D, temperature_, alpha, beta):\n \"\"\"\n Joint log probability optimization function.\n \n Args:\n D: The Data from the challenger disaster representing presence or \n absence of defect\n temperature_: The Data from the challenger disaster, specifically the temperature on \n the days of the observation of the presence or absence of a defect\n alpha: one of the inputs of the HMC\n beta: one of the inputs of the HMC\n Returns: \n Joint log probability optimization function.\n \"\"\"\n rv_alpha = tfd.Normal(loc=0., scale=1000.)\n rv_beta = tfd.Normal(loc=0., scale=1000.)\n logistic_p = 1.0/(1. + tf.exp(beta * tf.to_float(temperature_) + alpha))\n rv_observed = tfd.Bernoulli(probs=logistic_p)\n \n return (\n rv_alpha.log_prob(alpha)\n + rv_beta.log_prob(beta)\n + tf.reduce_sum(rv_observed.log_prob(D))\n )\n```\n\n\n```python\nnumber_of_steps = 60000\nburnin = 50000\n\n# Set the chain's start state.\ninitial_chain_state = [\n 0. * tf.ones([], dtype=tf.float32, name=\"init_alpha\"),\n 0. * tf.ones([], dtype=tf.float32, name=\"init_beta\")\n]\n\n# Since HMC operates over unconstrained space, we need to transform the\n# samples so they live in real-space.\nunconstraining_bijectors = [\n tfp.bijectors.Identity(),\n tfp.bijectors.Identity()\n]\n\n# Define a closure over our joint_log_prob.\nunnormalized_posterior_log_prob = lambda *args: challenger_joint_log_prob(D, temperature_, *args)\n\n# Initialize the step_size. (It will be automatically adapted.)\nwith tf.variable_scope(tf.get_variable_scope(), reuse=tf.AUTO_REUSE):\n step_size = tf.get_variable(\n name='step_size',\n initializer=tf.constant(0.5, dtype=tf.float32),\n trainable=False,\n use_resource=True\n )\n\n# Defining the HMC\nhmc=tfp.mcmc.TransformedTransitionKernel(\n inner_kernel=tfp.mcmc.HamiltonianMonteCarlo(\n target_log_prob_fn=unnormalized_posterior_log_prob,\n num_leapfrog_steps=2,\n step_size=step_size,\n step_size_update_fn=tfp.mcmc.make_simple_step_size_update_policy(),\n state_gradients_are_stopped=True),\n bijector=unconstraining_bijectors)\n\n# Sampling from the chain.\n[\n posterior_alpha,\n posterior_beta\n], kernel_results = tfp.mcmc.sample_chain(\n num_results = number_of_steps,\n num_burnin_steps = burnin,\n current_state=initial_chain_state,\n kernel=hmc)\n\n# Initialize any created variables for preconditions\ninit_g = tf.global_variables_initializer()\n```\n\n#### Execute the TF graph to sample from the posterior\n\n\n```python\n# In Graph Mode, this cell can take up to 36 Minutes\nevaluate(init_g)\n[\n posterior_alpha_,\n posterior_beta_,\n kernel_results_\n] = evaluate([\n posterior_alpha,\n posterior_beta,\n kernel_results\n])\n \nprint(\"acceptance rate: {}\".format(\n kernel_results_.inner_results.is_accepted.mean()))\nprint(\"final step size: {}\".format(\n kernel_results_.inner_results.extra.step_size_assign[-100:].mean()))\n\nalpha_samples_ = posterior_alpha_[burnin::8]\nbeta_samples_ = posterior_beta_[burnin::8]\n```\n\n acceptance rate: 0.6111833333333333\n final step size: 0.0149030527099967\n\n\nWe have trained our model on the observed data, now we can sample values from the posterior. Let's look at the posterior distributions for $\\alpha$ and $\\beta$:\n\n\n```python\nplt.figure(figsize(12.5, 6))\n\n#histogram of the samples:\nplt.subplot(211)\nplt.title(r\"Posterior distributions of the variables $\\alpha, \\beta$\")\nplt.hist(beta_samples_, histtype='stepfilled', bins=35, alpha=0.85,\n label=r\"posterior of $\\beta$\", color=TFColor[6], normed=True)\nplt.legend()\n\nplt.subplot(212)\nplt.hist(alpha_samples_, histtype='stepfilled', bins=35, alpha=0.85,\n label=r\"posterior of $\\alpha$\", color=TFColor[0], normed=True)\nplt.legend();\n```\n\nAll samples of $\\beta$ are greater than 0. If instead the posterior was centered around 0, we may suspect that $\\beta = 0$, implying that temperature has no effect on the probability of defect. \n\nSimilarly, all $\\alpha$ posterior values are negative and far away from 0, implying that it is correct to believe that $\\alpha$ is significantly less than 0. \n\nRegarding the spread of the data, we are very uncertain about what the true parameters might be (though considering the low sample size and the large overlap of defects-to-nondefects this behaviour is perhaps expected). \n\nNext, let's look at the *expected probability* for a specific value of the temperature. That is, we average over all samples from the posterior to get a likely value for $p(t_i)$.\n\n\n```python\nalpha_samples_1d_ = alpha_samples_[:, None] # best to make them 1d\nbeta_samples_1d_ = beta_samples_[:, None]\n\n\nbeta_mean = tf.reduce_mean(beta_samples_1d_.T[0])\nalpha_mean = tf.reduce_mean(alpha_samples_1d_.T[0])\n[ beta_mean_, alpha_mean_ ] = evaluate([ beta_mean, alpha_mean ])\n\n\nprint(\"beta mean:\", beta_mean_)\nprint(\"alpha mean:\", alpha_mean_)\ndef logistic(x, beta, alpha=0):\n \"\"\"\n Logistic function with alpha and beta.\n \n Args:\n x: independent variable\n beta: beta term \n alpha: alpha term\n Returns: \n Logistic function\n \"\"\"\n return 1.0 / (1.0 + tf.exp((beta * x) + alpha))\n\nt_ = np.linspace(temperature_.min() - 5, temperature_.max() + 5, 2500)[:, None]\np_t = logistic(t_.T, beta_samples_1d_, alpha_samples_1d_)\nmean_prob_t = logistic(t_.T, beta_mean_, alpha_mean_)\n[ \n p_t_, mean_prob_t_\n] = evaluate([ \n p_t, mean_prob_t\n])\n```\n\n beta samples shape: (1250, 1)\n alpha samples shape: (1250, 1)\n beta mean: 0.2165582\n alpha mean: -13.908086\n [48. 48.015 48.03 ... 85.97 85.985 86. ]\n [[0.971 0.971 0.971 ... 0.003 0.003 0.003]\n [0.977 0.977 0.977 ... 0.005 0.005 0.005]\n [0.987 0.987 0.987 ... 0.014 0.014 0.014]\n ...\n [0.976 0.976 0.976 ... 0.002 0.002 0.002]\n [0.963 0.963 0.963 ... 0.001 0.001 0.001]\n [0.976 0.976 0.976 ... 0.002 0.002 0.002]]\n [0.384 0.416 0.502 ... 0.384 0.325 0.381]\n [[0.971 0.971 0.971 ... 0.009 0.009 0.009]]\n\n\n\n```python\nplt.figure(figsize(12.5, 4))\n\nplt.plot(t_, mean_prob_t_.T, lw=3, label=\"average posterior \\nprobability \\\n#of defect\")\nplt.plot(t_, p_t_.T[:, 0], ls=\"--\", label=\"realization from posterior\")\nplt.plot(t_, p_t_.T[:, -8], ls=\"--\", label=\"realization from posterior\")\nplt.scatter(temperature_, D_, color=\"k\", s=50, alpha=0.5)\nplt.title(\"Posterior expected value of probability of defect; \\\nplus realizations\")\nplt.legend(loc=\"lower left\")\nplt.ylim(-0.1, 1.1)\nplt.xlim(t_.min(), t_.max())\nplt.ylabel(\"probability\")\nplt.xlabel(\"temperature\");\n```\n\nAbove we also plotted two possible realizations of what the actual underlying system might be. Both are equally likely as any other draw. The blue line is what occurs when we average all the 20000 possible dotted lines together.\n\n\n\n```python\nalpha_samples_means_ = np.array(alpha_samples_1d_.mean(axis=1))\nbeta_samples_means_ = np.array(beta_samples_1d_.mean(axis=1))\nsorted_alpha_means_ = np.sort(alpha_samples_means_)\nsorted_beta_means_ = np.sort(beta_samples_means_)\nalpha_index_ = sorted_alpha_means_.shape[0]\nbeta_index_ = sorted_beta_means_.shape[0]\nupper_alpha_quantile_ix_ = int(alpha_index_ * float(0.975))\nlower_alpha_quantile_ix_ = int(alpha_index_ * float(0.025))\n\nupper_beta_quantile_ix_ = int(beta_index_ * float(0.975))\nlower_beta_quantile_ix_ = int(beta_index_ * float(0.025))\n\ndef find_nearest(array, value):\n array = np.asarray(array)\n idx = (np.abs(array - value)).argmin()\n return idx\n\nalpha_upper_quantile_ix_ = find_nearest(alpha_samples_means_, sorted_alpha_means_[upper_alpha_quantile_ix_])\nalpha_lower_quantile_ix_ = find_nearest(alpha_samples_means_, sorted_alpha_means_[lower_alpha_quantile_ix_])\nbeta_upper_quantile_ix_ = find_nearest(beta_samples_means_, sorted_beta_means_[upper_beta_quantile_ix_])\nbeta_lower_quantile_ix_ = find_nearest(beta_samples_means_, sorted_beta_means_[lower_beta_quantile_ix_])\n\np_t_low = logistic(t_.T, beta_samples_1d_[beta_lower_quantile_ix_], alpha_samples_1d_[alpha_lower_quantile_ix_])\np_t_high = logistic(t_.T, beta_samples_1d_[beta_upper_quantile_ix_], alpha_samples_1d_[alpha_upper_quantile_ix_])\n\n[ \n p_t_low_, p_t_high_\n] = evaluate([ \n p_t_low, p_t_high\n])\nqs = np.stack([p_t_low_[0][::50], p_t_high_[0][::50]], axis=0)\n\nfig, (ax1) = plt.subplots(1, 1, sharex=True)\n\nax1.fill_between(t_[::50].T[0], qs[0], qs[1], alpha=0.7, color=TFColor[6])\nplt.plot(t_[::50].T[0], qs[0], label=\"95% CI\", color=TFColor[6], alpha=0.7)\n\nplt.plot(t_.T[0][::50], mean_prob_t_[0][::50], lw=1, ls=\"--\", color=\"k\",\n label=\"average posterior \\nprobability of defect\")\n\n\n\nplt.xlim(t_.min(), t_.max())\nplt.ylim(-0.02, 1.02)\nplt.legend(loc=\"lower left\")\nplt.scatter(temperature_, D_, color=\"k\", s=50, alpha=0.5)\nplt.xlabel(\"temp, $t$\")\n\nplt.ylabel(\"probability estimate\")\nplt.title(\"Posterior probability estimates given temp. $t$\");\n```\n\nMore generally, we can see that as the temperature nears 60 degrees, the CI's spread out over [0,1] quickly. As we pass 70 degrees, the CI's tighten again. This can give us insight about how to proceed next: we should probably test more O-rings around 60-65 temperature to get a better estimate of probabilities in that range. Similarly, when reporting to scientists your estimates, you should be very cautious about simply telling them the expected probability, as we can see this does not reflect how *wide* the posterior distribution is.\n\n### What about the day of the Challenger disaster?\n\nOn the day of the Challenger disaster, the outside temperature was 31 degrees Fahrenheit. What is the posterior distribution of a defect occurring, given this temperature? The distribution is plotted below. It looks almost guaranteed that the Challenger was going to be subject to defective O-rings.\n\n\n```python\nplt.figure(figsize(12.5, 3))\n\nprob_31 = logistic(31, beta_samples_, alpha_samples_)\n\n[ prob_31_ ] = evaluate([ prob_31 ])\nprint(prob_31_)\n\nplt.xlim(0.995, 1) # This should be changed to plt.xlim(0.995, 1), but illustrates the error\nplt.hist(prob_31_, bins=10, normed=True, histtype='stepfilled')\nplt.title(\"Posterior distribution of probability of defect, given $t = 31$\")\nplt.xlabel(\"probability of defect occurring in O-ring\");\n```\n\n### Is our model appropriate?\n\nThe skeptical reader will say \"You deliberately chose the logistic function for $p(t)$ and the specific priors. Perhaps other functions or priors will give different results. How do I know I have chosen a good model?\" This is absolutely true. To consider an extreme situation, what if I had chosen the function $p(t) = 1,\\; \\forall t$, which guarantees a defect always occurring: I would have again predicted disaster on January 28th. Yet this is clearly a poorly chosen model. On the other hand, if I did choose the logistic function for $p(t)$, but specified all my priors to be very tight around 0, likely we would have very different posterior distributions. How do we know our model is an expression of the data? This encourages us to measure the model's **goodness of fit**.\n\nWe can think: *how can we test whether our model is a bad fit?* An idea is to compare observed data with artificial dataset which we can simulate. The rationale is that if the simulated dataset does not appear similar, statistically, to the observed dataset, then likely our model is not accurately represented the observed data. \n\nPreviously in this Chapter, we simulated an artificial dataset for the SMS example. To do this, we sampled values from the priors. We saw how varied the resulting datasets looked like, and rarely did they mimic our observed dataset. In the current example, we should sample from the *posterior* distributions to create *very plausible datasets*. Luckily, our Bayesian framework makes this very easy. We only need to gather samples from the distribution of choice, and specify the number of samples, the shape of the samples (we had 21 observations in our original dataset, so we'll make the shape of each sample 21), and the probability we want to use to determine the ratio of 1 observations to 0 observations.\n\n\nHence we create the following:\n\n```python\nsimulated_data = tfd.Bernoulli(name=\"simulation_data\", probs=p).sample(sample_shape=N)\n```\nLet's simulate 10 000:\n\n\n```python\nalpha = alpha_mean_ # We're basing these values on the outputs of our model above\nbeta = beta_mean_\np_deterministic = tfd.Deterministic(name=\"p\", loc=1.0/(1. + tf.exp(beta * temperature_ + alpha))).sample()#seed=6.45)\nsimulated_data = tfd.Bernoulli(name=\"bernoulli_sim\", \n probs=p_deterministic_).sample(sample_shape=10000)\n[ \n bernoulli_sim_samples_,\n p_deterministic_\n] =evaluate([\n simulated_data,\n p_deterministic\n])\n```\n\n [[0 1 0 ... 1 0 1]\n [1 0 0 ... 0 1 1]\n [1 1 0 ... 0 0 0]\n ...\n [0 0 0 ... 0 0 1]\n [0 0 0 ... 0 0 1]\n [0 0 1 ... 0 0 1]] (10000, 23)\n\n\n\n```python\nsimulations_ = bernoulli_sim_samples_\nprint(\"Number of simulations: \", simulations_.shape[0])\nprint(\"Number data points per simulation: \", simulations_.shape[1])\n\nplt.figure(figsize(12.5, 12))\nplt.title(\"Simulated dataset using posterior parameters\")\nfor i in range(4):\n ax = plt.subplot(4, 1, i+1)\n plt.scatter(temperature_, simulations_[1000*i, :], color=\"k\",\n s=50, alpha=0.6)\n \n```\n\nNote that the above plots are different (if you can think of a cleaner way to present this, please send a pull request and answer [here](http://stats.stackexchange.com/questions/53078/how-to-visualize-bayesian-goodness-of-fit-for-logistic-regression)!).\n\nWe wish to assess how good our model is. \"Good\" is a subjective term of course, so results must be relative to other models. \n\nWe will be doing this graphically as well, which may seem like an even less objective method. The alternative is to use *Bayesian p-values*. These are still subjective, as the proper cutoff between good and bad is arbitrary. Gelman emphasises that the graphical tests are more illuminating [3] than p-value tests. We agree.\n\nThe following graphical test is a novel data-viz approach to logistic regression. The plots are called *separation plots*[4]. For a suite of models we wish to compare, each model is plotted on an individual separation plot. I leave most of the technical details about separation plots to the very accessible [original paper](http://mdwardlab.com/sites/default/files/GreenhillWardSacks.pdf), but I'll summarize their use here.\n\nFor each model, we calculate the proportion of times the posterior simulation proposed a value of 1 for a particular temperature, i.e. compute $P( \\;\\text{Defect} = 1 | t, \\alpha, \\beta )$ by averaging. This gives us the posterior probability of a defect at each data point in our dataset. For example, for the model we used above:\n\n\n```python\nposterior_probability_ = simulations_.mean(axis=0)\nprint(\"posterior prob of defect | realized defect \")\nfor i in range(len(D_)):\n print(\"%.2f | %d\" % (posterior_probability_[i], D_[i]))\n```\n\n posterior prob of defect | realized defect \n 0.41 | 0\n 0.22 | 1\n 0.25 | 0\n 0.31 | 0\n 0.36 | 0\n 0.16 | 0\n 0.13 | 0\n 0.22 | 0\n 0.82 | 1\n 0.57 | 1\n 0.22 | 1\n 0.04 | 0\n 0.35 | 0\n 0.92 | 1\n 0.35 | 0\n 0.09 | 0\n 0.23 | 0\n 0.03 | 0\n 0.07 | 0\n 0.04 | 0\n 0.09 | 1\n 0.07 | 0\n 0.79 | 1\n\n\nNext we sort each column by the posterior probabilities:\n\n\n```python\nix_ = np.argsort(posterior_probability_)\nprint(\"probb | defect \")\nfor i in range(len(D_)):\n print(\"%.2f | %d\" % (posterior_probability_[ix_[i]], D_[ix_[i]]))\n```\n\n probb | defect \n 0.03 | 0\n 0.04 | 0\n 0.04 | 0\n 0.07 | 0\n 0.07 | 0\n 0.09 | 1\n 0.09 | 0\n 0.13 | 0\n 0.16 | 0\n 0.22 | 0\n 0.22 | 1\n 0.22 | 1\n 0.23 | 0\n 0.25 | 0\n 0.31 | 0\n 0.35 | 0\n 0.35 | 0\n 0.36 | 0\n 0.41 | 0\n 0.57 | 1\n 0.79 | 1\n 0.82 | 1\n 0.92 | 1\n\n\nWe can present the above data better in a figure: we've creates a `separation_plot` function.\n\n\n```python\nimport matplotlib.pyplot as plt\n\ndef separation_plot( p, y, **kwargs ):\n \"\"\"\n This function creates a separation plot for logistic and probit classification. \n See http://mdwardlab.com/sites/default/files/GreenhillWardSacks.pdf\n \n p: The proportions/probabilities, can be a nxM matrix which represents M models.\n y: the 0-1 response variables.\n \n \"\"\" \n assert p.shape[0] == y.shape[0], \"p.shape[0] != y.shape[0]\"\n n = p.shape[0]\n\n try:\n M = p.shape[1]\n except:\n p = p.reshape( n, 1 )\n M = p.shape[1]\n\n colors_bmh = np.array( [\"#eeeeee\", \"#348ABD\"] )\n\n\n fig = plt.figure( )\n \n for i in range(M):\n ax = fig.add_subplot(M, 1, i+1)\n ix = np.argsort( p[:,i] )\n #plot the different bars\n bars = ax.bar( np.arange(n), np.ones(n), width=1.,\n color = colors_bmh[ y[ix].astype(int) ], \n edgecolor = 'none')\n ax.plot( np.arange(n+1), np.append(p[ix,i], p[ix,i][-1]), \"k\",\n linewidth = 1.,drawstyle=\"steps-post\" )\n #create expected value bar.\n ax.vlines( [(1-p[ix,i]).sum()], [0], [1] )\n plt.xlim( 0, n)\n \n plt.tight_layout()\n \n return\n\nplt.figure(figsize(11., 3))\nseparation_plot(posterior_probability_, D_)\n```\n\nThe snaking-line is the sorted probabilities, blue bars denote defects, and empty space (or grey bars for the optimistic readers) denote non-defects. As the probability rises, we see more and more defects occur. On the right hand side, the plot suggests that as the posterior probability is large (line close to 1), then more defects are realized. This is good behaviour. Ideally, all the blue bars *should* be close to the right-hand side, and deviations from this reflect missed predictions. \n\nThe black vertical line is the expected number of defects we should observe, given this model. This allows the user to see how the total number of events predicted by the model compares to the actual number of events in the data.\n\nIt is much more informative to compare this to separation plots for other models. Below we compare our model (top) versus three others:\n\n1. the perfect model, which predicts the posterior probability to be equal 1 if a defect did occur.\n2. a completely random model, which predicts random probabilities regardless of temperature.\n3. a constant model: where $P(D = 1 \\; | \\; t) = c, \\;\\; \\forall t$. The best choice for $c$ is the observed frequency of defects, in this case 7/23. \n\n\n\n```python\nplt.figure(figsize(11., 2))\n\n# Our temperature-dependent model\nseparation_plot(posterior_probability_, D_)\nplt.title(\"Temperature-dependent model\")\n\n# Perfect model\n# i.e. the probability of defect is equal to if a defect occurred or not.\np_ = D_\nseparation_plot(p_, D_)\nplt.title(\"Perfect model\")\n\n# random predictions\np_ = np.random.rand(23)\nseparation_plot(p_, D_)\nplt.title(\"Random model\")\n\n# constant model\nconstant_prob_ = 7./23 * np.ones(23)\nseparation_plot(constant_prob_, D_)\nplt.title(\"Constant-prediction model\");\n```\n\nIn the random model, we can see that as the probability increases there is no clustering of defects to the right-hand side. Similarly for the constant model.\n\nIn the perfect model, the probability line is not well shown, as it is stuck to the bottom and top of the figure. Of course the perfect model is only for demonstration, and we cannot infer any scientific inference from it.\n\n## Exercises\n\n1\\. Try putting in extreme values for our observations in the cheating example. What happens if we observe 25 affirmative responses? 10? 50? \n\n\n```python\n#type your code here.\n```\n\n2\\. Try plotting $\\alpha$ samples versus $\\beta$ samples. Why might the resulting plot look like this?\n\n\n```python\n#type your code here.\nplt.figure(figsize(12.5, 4))\n\nplt.scatter(alpha_samples_, beta_samples_, alpha=0.1)\nplt.title(\"Why does the plot look like this?\")\nplt.xlabel(r\"$\\alpha$\")\nplt.ylabel(r\"$\\beta$\");\n```\n\n## References\n\n[1] Dalal, Fowlkes and Hoadley (1989),JASA, 84, 945-957.\n\n[2] Cronin, Beau. \"Why Probabilistic Programming Matters.\" 24 Mar 2013. Google, Online Posting to Google . Web. 24 Mar. 2013. .\n\n[3] Gelman, Andrew. \"Philosophy and the practice of Bayesian statistics.\" British Journal of Mathematical and Statistical Psychology. (2012): n. page. Web. 2 Apr. 2013.\n\n[4] Greenhill, Brian, Michael D. Ward, and Audrey Sacks. \"The Separation Plot: A New Visual Method for Evaluating the Fit of Binary Models.\" American Journal of Political Science. 55.No.4 (2011): n. page. Web. 2 Apr. 2013.\n\n\n\n```python\nfrom IPython.core.display import HTML\ndef css_styling():\n styles = open(\"../styles/custom.css\", \"r\").read()\n return HTML(styles)\ncss_styling()\n\n```\n", "meta": {"hexsha": "4172b511be8e607e5682444a27514d58d4d753c1", "size": 693672, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Chapter2_MorePyMC/Ch2_MorePyMC_TFP.ipynb", "max_stars_repo_name": "AwaSamake/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers", "max_stars_repo_head_hexsha": "33ba3c5f5446cf89f4b791c5fe52f79923e04013", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-01-13T18:14:31.000Z", "max_stars_repo_stars_event_max_datetime": "2019-01-13T18:14:31.000Z", "max_issues_repo_path": "Chapter2_MorePyMC/Ch2_MorePyMC_TFP.ipynb", "max_issues_repo_name": "AwaSamake/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers", "max_issues_repo_head_hexsha": "33ba3c5f5446cf89f4b791c5fe52f79923e04013", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Chapter2_MorePyMC/Ch2_MorePyMC_TFP.ipynb", "max_forks_repo_name": "AwaSamake/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers", "max_forks_repo_head_hexsha": "33ba3c5f5446cf89f4b791c5fe52f79923e04013", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 162.3384039317, "max_line_length": 102248, "alphanum_fraction": 0.8720389464, "converted": true, "num_tokens": 27022, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631556226292, "lm_q2_score": 0.7122321781307374, "lm_q1q2_score": 0.4327972528989024}} {"text": "```python\nfrom decodes.core import *\nfrom decodes.io.jupyter_out import JupyterOut\nout = JupyterOut.unit_square( )\n```\n\n# Coordinate Systems\n\nOur experience as users of CAD has trained us to look at coordinate systems in a particular way. We understand that they are generally made up of **three axes**, and are visually represented by **three vectors** that are **mutually perpendicular**.\n\nMost CAD packages maintain a single **world coordinate system** that defines the underlying space of the model, but also allows **local coordinate systems** defined at arbitrary positions and orientations within this world space which are typically positioned for convenient drafting or modeling.\n\nOur experience in CAD suggests an intimate link between coordinate systems, vectors, and the space they inhabit. In this section, we establish the mathematical foundations that unpacks this relationship: by using two critical concepts:\n\n* **span**: the spatial extent that may be \u201creached\u201d using a set of vectors and a limited means of manipulation. \n\n* **basis**: any minimal set of vectors which spans a given space.\n\n\n## Basis and Coordinates\n\nWe already know that we may add and scalar multiply any set of vectors $\\vec{v_{1}},\\ \\vec{v_{2}},\\ \\ldots \\ , \\vec{v_{n}}$, in any order of operation, and that these operations will result in another vector. If the scalars are known, then the vector \n\n\\begin{align}\n\\vec{w} = c_{1}\\vec{v_{1}} \\ + \\ c_{2}\\vec{v_{2}} \\ + \\ \\ldots \\ c_{2}\\vec{v_{2}} \n\\end{align}\n\ncan be drawn easily using the \u201chead-to-tail\u201d method. This procedure is called a linear combination of $\\vec{v_{1}},\\ \\vec{v_{2}},\\ \\ldots \\ , \\vec{v_{n}}$.\n\nConsider for the moment **all possible linear combinations**. In other words, given some set of vectors, imagine all the locations that can be \u201creached\u201d by vectors resulting from a combination of both addition and scalar multiplication. Mathematically, this is termed the **span** of $\\vec{v_{1}},\\ \\vec{v_{2}},\\ \\ldots \\ , \\vec{v_{n}}$. \n\nTo develop our intuition for what the vector span looks like, we might consider three two-dimensional cases:\n\n* The span of a single vector\n* The span of any two (non-paralell) vectors\n* The span of three (mutually non-paralell) vectors\n\n\n\nThe span of a single vector $\\vec{v}$ is the set of scalar multiples $c\\vec{v}$. Since the scalars can be positive or negative, we may scale this vector to reach out infinitely in either direction. \n\n***The span of $\\vec{v}$, then, is the entire line through the origin determined by $\\vec{v}$***.\n\n\n\n\nAssuming $\\vec{v_{1}}$ and $\\vec{v_{2}}$ are not parallel, we find that any linear combination $c2\\vec{v_{1}} + c2\\vec{v_{2}}$ may be produced by applying the parallelogram rule and adjusting the scalar multipliers.\n\nThe span of *any* two vectors in the plane allow us to reach any location in two dimensions, so we say that the span of this set is the ***entirety of the plane***.\n\nThe third case, more than two vectors, is not necessary. There is only one parallelogram with sides in the directions of v1 and v2 for which the far corner coincides with the desired vector.\n\n\n\n\nThis second case is special. \n\nFor any two non-parallel vectors: \n* their span is the entire space\n* any other vector in the space can be expressed as a unique linear combination of this set of vectors. \n\nAny set of vectors that exhibits both of these properties is called a **basis** for the space, and ***the number of vectors*** in a basis is the **dimension** of the space. \n\nThe most standard of bases in R2 contains the vectors (1,0) and (0,1). Notice that any vector (x,y) is easily expressed as a linear combination of these basis vectors. Here we find a formal mathematical\naccount of a common instrument in CAD software, that of ***coordinates***.\n\n\n\n### Frames\n\nJust as two perpendicular unit vectors make up a special basis for two-dimensions, three mutually perpendicular unit vectors is a special basis in three dimensions. A coordinate system in three dimensions made up of such a basis, together with an origin, is called a **frame**.\n\nFurther refining this term, an **orthonormal basis** is one in which the contained vectors are unit vectors, and are mutually perpendicular. This is a particularly convenient format in computational geometry.\n\n\n\n\\begin{align}\n\\vec{w} = L_{1}\\vec{u_{1}} + L_{2}\\vec{u_{2}} + L_{3}\\vec{u_{3}}\n\\end{align}\n\n\n\n\n\n**span**\n\nThe spatial extent to which a given set of vectors may \u201creach\u201d by application\nof vector addition and scalar multiplication.\n\n**basis**\n\nA set of vectors that spans the entire space in which they are described,\nwith a number of vectors in the set that matches the dimension of the\nspace. A basis in R2 must contain two vectors, and a basis in R3 contains\nthree. A basis can be used to express any desired vector in the space as a\nunique linear combination of basis vectors.\n\n**orthonormal basis**\n\nA special kind of basis comprised of a set of mutually perpendicular unit\nvectors. Orthonormal bases are prevalent in computational geometry, as\na basis in this format makes the evaluation of coordinates particularly\nconvenient.\n\n**frame**\n\nA coordinate system in three dimensions made up of an orthonormal\nbasis (with three vectors) together with a position in space. Throughout\nthis text, we will refer to the mathematical concept as a frame, and the\nimplementation in code or in software as a coordinate system.\n\n**coordinate system**\n\nA frame implemented in software or in code. Frames implemented in these\ncontexts go by many names, including \u201cconstruction plane\u201d (Rhino), \u201cuser\ncoordinate system\u201d (AutoCAD), \u201cdrawing axis\u201d (SketchUp), and, fittingly,\n\u201cframe\u201d (Grasshopper).\n\n\n\n## CS Objects in Decodes\n\n\n\n\n \n\n \n\n\n\n \n \n \n\n\n\n \n \n \n\n\n
    *CS Members*
    `cs.origin`PointThe local origin of this coordinate system.
    `cs.x_axis`
    `cs.y_axis`
    `cs.z_axis`    		VecVectors that represent the axes of this coordinate system. Constrained upon construction to ensure orthonormality.
    \n\n\n```python\n\"\"\"\nCoordinate System Initialization\nShown here is an initialization by a Point and two Vecs which represent \nthe desired orientation of the resulting CS. The first of the given vectors \nis assigned to the x-axis. The second influences the direction of the \ny-axis, but is not used to set it directly as to ensure perpendicularity.\n\"\"\"\nclass CS(Geometry):\n\n def __init__(self, pt, vec_a, vec_b):\n self.origin = pt\n # set the x-axis to the first given vector, normalized\n self.x_axis = vec_a.normalized()\n # set the z-axis to a vector perpendicular to both given vectors\n self.z_axis = self.x_axis.cross(vec_b).normalized()\n # set the y-axis to a vector perpendicular to the x- and z-axes\n self.y_axis = self.z_axis.cross(self.x_axis).normalized()\n```\n\n\n\n### Coordinate System Evaluation\n\n\n```python\n\"\"\"\nCS Evaluation\nReturns a Point in \"world\" space that corresponds to the given u,v,w \ncoordinates that are described in the \"local\" space of this CS.\n\"\"\"\ndef eval(self,u,v,w):\n offset_vec = (self.x_axis*u) + (self.y_axis*v) + (self.z_axis*w)\n return Point(self.origin + offset_vec)\n\n```\n\n### Coordinate System Devaluation\n\n\n```python\n\"\"\"\nCS Devaluation\nReturns a Vec containing coordinates in the \"local\" space of this CS that \ncorrespond with the given x,y,z coordinates that are described in \"world\" \nspace.\n\"\"\" \ndef deval(self,x,y,z):\n pt = Point(x,y,z)\n # project the given point onto an axis line, store the distance\n xx = Line(self.origin,self.x_axis).near(pt)[1]\n yy = Line(self.origin,self.y_axis).near(pt)[1]\n zz = Line(self.origin,self.z_axis).near(pt)[1]\n return Vec(xx,yy,zz)\n\n\n```\n", "meta": {"hexsha": "ce0a96344acfca8977830838c0d2493e6a2a92fa", "size": 12688, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "103 - Points, Vectors, and Coordinate Systems/100 - Coordinate Systems.ipynb", "max_stars_repo_name": "ksteinfe/decodes_ipynb", "max_stars_repo_head_hexsha": "2e4bb6b398472fc61ef8b88dad7babbdeb2a5754", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-05-15T14:31:23.000Z", "max_stars_repo_stars_event_max_datetime": "2018-05-15T14:31:23.000Z", "max_issues_repo_path": "103 - Points, Vectors, and Coordinate Systems/100 - Coordinate Systems.ipynb", "max_issues_repo_name": "ksteinfe/decodes_ipynb", "max_issues_repo_head_hexsha": "2e4bb6b398472fc61ef8b88dad7babbdeb2a5754", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "103 - Points, Vectors, and Coordinate Systems/100 - Coordinate Systems.ipynb", "max_forks_repo_name": "ksteinfe/decodes_ipynb", "max_forks_repo_head_hexsha": "2e4bb6b398472fc61ef8b88dad7babbdeb2a5754", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2020-05-19T05:40:18.000Z", "max_forks_repo_forks_event_max_datetime": "2020-06-28T02:18:08.000Z", "avg_line_length": 37.7619047619, "max_line_length": 354, "alphanum_fraction": 0.6056116015, "converted": true, "num_tokens": 1933, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.8267117898012104, "lm_q1q2_score": 0.43271777348752255}} {"text": "# Pumas.jl Workshop Exercises\n### Chris Rackauckas, Vijay Ivaturi\n\nThese exercises teach common workflows which involve Pumas.jl. This exercise\nworksheet is meant to be a living document leading new users through a deep dive\nof the Pumas.jl feature set. If you have further suggestions or want to contribute\nnew problems, please open an issue or PR at the\n[PumasTutorials.jl](https://github.com/PumasAI/PumasTutorials.jl) repository.\n\nThe exercises are described as follows:\n\n- Exercise 1 takes you through simulating a compartmental system in Pumas. The\n user will gain experience in writing a model using the Pumas Domain Specific\n Language (DSL), setting up a dosing regimen and population, simulating the\n model into the designed population and finally plotting the results of the\n simulation.\n- Exercise 2 takes the user through performing an Non-compartmental analysis\n (NCA). The user will get familiarized with reading data sets in a spread\n sheet format (e.g, .csv) or as the simulated output as in exercise 1 and\n perform a simple NCA analysis.\n- Exercise 3 introduces the user to perform a non-linear mixed effects modeling\n estimation (NLME) on pharmacokinetic data. The user will learn how to read and\n set up the data for NLME estimation, fit the model, infer and inspect the\n results of the model fit.\n- Exercise 4 introduces the user to various post-processing steps of model\n fitting, including using the final\n model fits to peform simulation into an alternate dosing regimen or populaion\n of interest.\n\n# Problem 1: Simulate a first-order absorption model with linear elimination after a 100 mg oral dose in 24 subjects\n\nIn this poblem, we will walk through the basics of writing a model with Pumas.jl.\nThe pharmacokinetics after an oral dose are commonly described by a first-order\nprocess. In our example we will use a linear one-compartmental system for\nelimination of the drugs.\n\n## Part 1: Understand the system\n\nLet's understand the system by setting up the differential equation or the\nanalytical solution that describes the systems\n\n$$\\begin{align}\n\\frac{dDepot}{dt} &= -K_a.Depot\\\\\n\\frac{dCentral}{dt} &= K_a.Depot - (CL/V).Central\\end{align}$$\n\nThe analytical form of this equation to calculate the concentration, `Cp` at\nany given time can be written as\n\n$$Cp = \\frac{F \\times Dose \\times K_a}{V (K_a-K_{el})}(e^{-K_{el} t} - e^{-K_a t})$$\n\nwhere the parameters are: `Ka = 1 hr-1`, `CL = 1 L/hr`, `V = 20 L/hr`.\n\n## Part 2: Setup the population\n\nSetting up a dosing regimen and population of subjects. Go through the section on [Generating and simulating populations](https://tutorials.pumas.ai/html/introduction/simulating_populations.html) in the\ntutorials to set up a population of 24 subjects that receives 100 mg dose via the mouth and assign a random body weight to each subjects\n\n## Part 3: Write the model\n\nWrite up a first-order absorption model where body weight is a covariate on `CL` and `V`.\nunderstand the different steps of writing up a model by referring to the\ndocumentation on [models](https://docs.pumas.ai/dev/basics/models/).\n\n## Part 4: Simulate\n\nPerform the simulation by using the `simobs` function. The details on the use of `simobs` can be seen in the documentation on [simulation](https://docs.pumas.ai/dev/basics/simulation/) and by\nlooking at one of the [tutorials](https://tutorials.pumas.ai/html/introduction/introduction.html). Finish by plotting the result of the simulation.\n\n# Problem 2: Peform Non-compartmental analysis\n\nUse the dataset generated from Problem 1 that is stored as a CSV file [here]()\nto read in the data using the `read_nca` function and\ngenerate a NCA report. You can know more about how to do this by looking at\none of the [NCA tutorials](https://tutorials.pumas.ai/html/nca/basic_nca.html)\n\n# Problem 3: Estimate using Non-linear mixed effects\n\nThe same dataset that was read in for NCA analysis will be used for fitting a\nNLME model. You can learn more on how to read the data in for NLME estimation\nusing `read_pumas` by following the tutorial in the\n[readme](https://github.com/PumasAI/Pumas.jl/blob/master/README.md#fit) or the\n[documentation](https://docs.pumas.ai/dev/basics/doses_subjects_populations/)\n\n## Part 1: Read datasets for NLME estimation\n\nRead the dataset and evaluate the Population\n\n## Part 2: Perform a model fit\n\nFit the model using FOCEI() estimation\n\n## Part 3: Infer the results\n\nInfer the results of your `fit`\n\n## Part 4: Inspect the results\n\nInspect the results of your `fit`\n", "meta": {"hexsha": "8e1fded0800bda1adf1d6f28547464d88a7d022f", "size": 5056, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebook/exercises/workshop_exercises.ipynb", "max_stars_repo_name": "chriselrod/PumasTutorials.jl", "max_stars_repo_head_hexsha": "b9c7cc164400dbe337e81757df57ea88799cf79d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 42, "max_stars_repo_stars_event_min_datetime": "2019-07-20T01:15:20.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-04T08:20:19.000Z", "max_issues_repo_path": "notebook/exercises/workshop_exercises.ipynb", "max_issues_repo_name": "noilreed/PumasTutorials.jl", "max_issues_repo_head_hexsha": "52dc68dc5d303310a79cd24f3bf7dfd66f57053a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 53, "max_issues_repo_issues_event_min_datetime": "2019-07-20T01:29:50.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-08T12:08:22.000Z", "max_forks_repo_path": "notebook/exercises/workshop_exercises.ipynb", "max_forks_repo_name": "noilreed/PumasTutorials.jl", "max_forks_repo_head_hexsha": "52dc68dc5d303310a79cd24f3bf7dfd66f57053a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 20, "max_forks_repo_forks_event_min_datetime": "2019-07-22T18:56:52.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-23T00:38:37.000Z", "avg_line_length": 187.2592592593, "max_line_length": 4618, "alphanum_fraction": 0.7452531646, "converted": true, "num_tokens": 1126, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.679178699175393, "lm_q1q2_score": 0.4326577463298537}} {"text": "# The Laplace Transform\n\n*This Jupyter notebook is part of a [collection of notebooks](../index.ipynb) in the bachelors module Signals and Systems, Communications Engineering, Universit\u00e4t Rostock. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*\n\n## Summary of Properties, Theorems and Transforms\n\nThe [properties](properties.ipynb), [theorems](theorems.ipynb) and transforms of the two-sided Laplace transform as derived in the previous sections are summarized in the following. The corresponding tables serve as a reference for the application of the Laplace transform in the theory of signals and systems. Please refer to the respective sections for details.\n\n### Definition\n\nThe two-sided Laplace transform and its inverse are defined as\n\n\\begin{align}\nX(s) &= \\int_{-\\infty}^{\\infty} x(t) \\, e^{- s t} \\; dt \\\\\nx(t) &= \\frac{1}{2 \\pi j} \\int_{\\sigma - j \\infty}^{\\sigma + j \\infty} X(s) \\, e^{s t} \\; ds\n\\end{align}\n\nwhere $s \\in \\text{ROC} \\{ x(t) \\}$.\n\n### Properties and Theorems\n\nThe properties and theorems of the two-sided Laplace transform are given as\n\n| | $x(t) \\qquad \\qquad \\qquad \\qquad$ | $X(s) = \\mathcal{L} \\{ x(t) \\} \\qquad \\qquad \\qquad$ | ROC $\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad$ |\n|:---|:---:|:---:|:---|\n| [Linearity](properties.ipynb#Linearity) | $A \\, x_1(t) + B \\, x_2(t)$ | $A \\, X_1(s) + B \\, X_2(s)$ | $\\supseteq \\text{ROC}\\{x_1(t)\\} \\cap \\text{ROC}\\{x_2(t)\\}$ |\n| [Real-valued signal](properties.ipynb#Symmetry-for-Real-Valued-Signals) | $x(t) = x^*(t)$ | $X(s) = X^*(s^*)$ | | \n| [Scaling](theorems.ipynb#Temporal-Scaling-Theorem) | $x(a t)$ | $\\frac{1}{\\lvert a \\rvert} X\\left( \\frac{s}{a} \\right)$ | $s: \\frac{s}{a} \\in \\text{ROC}\\{x(t)\\}$ |\n| [Convolution](theorems.ipynb#Convolution-Theorem) | $x(t) * h(t)$ | $X(s) \\cdot H(s)$ | $\\supseteq \\text{ROC}\\{x(t)\\} \\cap \\text{ROC}\\{h(t)\\}$ |\n| [Shift](theorems.ipynb#Temporal-Shift-Theorem) | $x(t - \\tau)$ | $e^{-s \\tau} \\cdot X(s)$ | $\\text{ROC}\\{x(t)\\}$ |\n| [Differentiation](theorems.ipynb#Differentiation-Theorem) (causal signal) | $\\frac{d}{dt} x(t)$ | $s \\cdot X(s) - x(0+)$ | $\\supseteq \\text{ROC}\\{x(t)\\}$ |\n| [Integration](theorems.ipynb#Integration-Theorem) | $\\int_{-\\infty}^{t} x(t) \\; dt$ | $\\frac{1}{s} \\cdot X(s)$ | $\\supseteq \\text{ROC}\\{x(t)\\} \\cap \\{s: \\Re \\{s\\} > 0 \\}$ |\n| [Modulation](theorems.ipynb#Modulation-Theorem) | $e^{s_0 t}\\cdot x(t)$ | $X(s - s_0)$ | $s: s - \\Re \\{s_0\\} \\in \\text{ROC}\\{x(t)\\}$ |\n\nwhere $A, B, s_0 \\in \\mathbb{C}$, $a \\in \\mathbb{R} \\setminus \\{0\\}$ and $\\tau \\in \\mathbb{R}$\n\n### Selected Transforms\n\nTwo-sided Laplace transforms which are frequently used are given as\n\n| $x(t) \\qquad \\qquad \\qquad \\qquad$ | $X(s) = \\mathcal{L} \\{ x(t) \\} \\qquad \\qquad \\qquad$ | ROC $\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad$ |\n|:---:|:---:|:---|\n| $\\delta(t)$ | $1$ | $\\mathbb{C}$ |\n| $\\epsilon(t)$ | $\\frac{1}{s}$ | $\\Re \\{s\\} > 0$ |\n| $t \\epsilon(t)$ | $\\frac{1}{s^2}$ | $\\Re \\{s\\} > 0$ |\n| $e^{- s_0 t} \\epsilon(t)$ | $\\frac{1}{s + s_0}$ | $\\Re \\{s\\} > \\text{Re}\\{-s_0\\}$ |\n| $\\sin(\\omega_0 t) \\epsilon(t)$ | $\\frac{\\omega_0}{s^2 + \\omega_0^2}$ | $\\Re \\{s\\} > 0$ |\n| $\\cos(\\omega_0 t) \\epsilon(t)$ | $\\frac{s}{s^2 + \\omega_0^2}$ | $\\Re \\{s\\} > 0$ |\n| $t^n e^{-s_0 t} \\epsilon(t)$ | $\\frac{n!}{(s+s_0)^{n+1}}$ | $\\Re \\{s\\} > \\text{Re}\\{-s_0\\}$ |\n| $e^{-s_0 t} \\cos(\\omega_0 t) \\epsilon(t)$ | $\\frac{s + s_0}{(s+s_0)^2 + \\omega_0^2}$ | $\\Re \\{s\\} > \\Re \\{-s_0\\}$ |\n| $e^{-s_0 t} \\sin(\\omega_0 t) \\epsilon(t)$ | $\\frac{\\omega_0}{(s+s_0)^2 + \\omega_0^2}$ | $\\Re \\{s\\} > \\Re \\{-s_0\\}$ |\n| $t \\cos(\\omega_0 t) \\epsilon(t)$ | $\\frac{s^2 - \\omega_0^2}{(s^2 + \\omega_0^2)^2}$ | $\\Re \\{s\\} > 0$ |\n| $t \\sin(\\omega_0 t) \\epsilon(t)$ | $\\frac{2 \\omega_0 s}{(s^2 + \\omega_0^2)^2}$ | $\\Re \\{s\\} > 0$ |\n\nwhere $s_0 \\in \\mathbb{C}$, $\\omega_0 \\in \\mathbb{R}$ and $n \\in \\mathbb{N}$. More one- and two-sided transforms may be found in the literature or [online](https://en.wikipedia.org/wiki/List_of_Laplace_transforms).\n\n**Copyright**\n\nThe notebooks are provided as [Open Educational Resource](https://de.wikipedia.org/wiki/Open_Educational_Resources). Feel free to use the notebooks for your own educational purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Lecture Notes on Signals and Systems* by Sascha Spors.\n", "meta": {"hexsha": "992589623cb8144e65814906d66d4c06b2d19823", "size": 6177, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "laplace_transform/table_theorems_transforms.ipynb", "max_stars_repo_name": "xushoucai/signals-and-systems-lecture", "max_stars_repo_head_hexsha": "30dbbf9226d93b454639955f5462d57546a921c5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-01-11T02:04:18.000Z", "max_stars_repo_stars_event_max_datetime": "2019-01-11T02:04:18.000Z", "max_issues_repo_path": "laplace_transform/table_theorems_transforms.ipynb", "max_issues_repo_name": "xushoucai/signals-and-systems-lecture", "max_issues_repo_head_hexsha": "30dbbf9226d93b454639955f5462d57546a921c5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "laplace_transform/table_theorems_transforms.ipynb", "max_forks_repo_name": "xushoucai/signals-and-systems-lecture", "max_forks_repo_head_hexsha": "30dbbf9226d93b454639955f5462d57546a921c5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 50.631147541, "max_line_length": 486, "alphanum_fraction": 0.5293831957, "converted": true, "num_tokens": 1749, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269796369904, "lm_q2_score": 0.7745833841649233, "lm_q1q2_score": 0.43247080135780025}} {"text": "\n\n\n```\n!pip install daft seaborn PyDrive\n```\n\n Requirement already satisfied: daft in /usr/local/lib/python3.6/dist-packages (0.0.4)\n Requirement already satisfied: seaborn in /usr/local/lib/python3.6/dist-packages (0.7.1)\n Collecting PyDrive\n \u001b[?25l Downloading https://files.pythonhosted.org/packages/52/e0/0e64788e5dd58ce2d6934549676243dc69d982f198524be9b99e9c2a4fd5/PyDrive-1.3.1.tar.gz (987kB)\n \u001b[K 100% |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 993kB 18.5MB/s \n \u001b[?25hRequirement already satisfied: numpy in /usr/local/lib/python3.6/dist-packages (from daft) (1.14.6)\n Requirement already satisfied: matplotlib in /usr/local/lib/python3.6/dist-packages (from daft) (2.1.2)\n Requirement already satisfied: google-api-python-client>=1.2 in /usr/local/lib/python3.6/dist-packages (from PyDrive) (1.6.7)\n Requirement already satisfied: oauth2client>=4.0.0 in /usr/local/lib/python3.6/dist-packages (from PyDrive) (4.1.3)\n Requirement already satisfied: PyYAML>=3.0 in /usr/local/lib/python3.6/dist-packages (from PyDrive) (3.13)\n Requirement already satisfied: pytz in /usr/local/lib/python3.6/dist-packages (from matplotlib->daft) (2018.7)\n Requirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.6/dist-packages (from matplotlib->daft) (0.10.0)\n Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib->daft) (2.3.0)\n Requirement already satisfied: six>=1.10 in /usr/local/lib/python3.6/dist-packages (from matplotlib->daft) (1.11.0)\n Requirement already satisfied: python-dateutil>=2.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib->daft) (2.5.3)\n Requirement already satisfied: httplib2<1dev,>=0.9.2 in /usr/local/lib/python3.6/dist-packages (from google-api-python-client>=1.2->PyDrive) (0.11.3)\n Requirement already satisfied: uritemplate<4dev,>=3.0.0 in /usr/local/lib/python3.6/dist-packages (from google-api-python-client>=1.2->PyDrive) (3.0.0)\n Requirement already satisfied: rsa>=3.1.4 in /usr/local/lib/python3.6/dist-packages (from oauth2client>=4.0.0->PyDrive) (4.0)\n Requirement already satisfied: pyasn1-modules>=0.0.5 in /usr/local/lib/python3.6/dist-packages (from oauth2client>=4.0.0->PyDrive) (0.2.2)\n Requirement already satisfied: pyasn1>=0.1.7 in /usr/local/lib/python3.6/dist-packages (from oauth2client>=4.0.0->PyDrive) (0.4.4)\n Building wheels for collected packages: PyDrive\n Running setup.py bdist_wheel for PyDrive ... \u001b[?25l-\b \b\\\b \bdone\n \u001b[?25h Stored in directory: /root/.cache/pip/wheels/fa/d2/9a/d3b6b506c2da98289e5d417215ce34b696db856643bad779f4\n Successfully built PyDrive\n Installing collected packages: PyDrive\n Successfully installed PyDrive-1.3.1\n\n\n\n```\n# Prepare the libraries for the exercise\nimport matplotlib\nfrom matplotlib import pyplot as plt\nimport daft\nimport seaborn\n\nimport numpy as np\nimport tensorflow as tf\ntf.enable_eager_execution()\n\nimport tensorflow_probability as tfp\ntfd = tfp.distributions\nnp.random.seed(5218)\n```\n\n#### If you have trouble with mathematic mode in latex, following links are very good for reference!\n\n[Basic](https://en.wikibooks.org/wiki/LaTeX/Mathematics)\n\n[Advance](https://en.wikibooks.org/wiki/LaTeX/Mathematics)\n\n# Question 1 \n\nConsidering the directed graphical of Gaussian mixture model explained in the lectures. By making use of the d-separation criterion, show that posterior distribution of the latent variables factorizes with respect to the different data points. i.e.\n\n\\begin{align}\np(\\mathbf{Z} \\mid \\mathbf{X}, \\mathbf{\\mu}, \\sigma^2, \\pi) = \n\\prod_{n=1}^N p(z_n \\mid x_n, \\mathbf{\\mu}, \\sigma^2, \\pi)\n\\end{align}\n\n*Figure 9.6*, page 433, \"Pattern Recognition and Machine Learning\", M.Bishop \n\nGraphical representation of a Gaussian mixture model for a set of $N$ i.i.d. data points $\\{x_n\\}$, with corresponding latent points $\\{z_n\\}$, where $n = 1,...,N$.\n\n\n\n\n\n# Question 2\n\nImplement Gibbs sampler for Gaussian mixture models as defined in the lectures. \n\nTest it using your previously generated synthetic data.\n\n### The following code is used for generating synthetic data\n\nThe `gmm` method will generate 1-D data for a given value of `alpha` and `sigma0`\n\nYour task is to implement Gibbs sampling to infer the value of `alpha` and `sigma0` from returned data `X`\n\nNOTE: only `X` should be used, `Z` is returned only for plotting\n\n\n```\ndef gmm(batch_size, n_clusters, alpha, sigma0):\n \"\"\" This is the solution for the process in 1.3 (only for 1-D data) \"\"\"\n # parameters for Dirichlet distribution\n alpha = np.full(shape=(n_clusters,), fill_value=alpha, dtype='float32')\n\n # step 1: generate the assignment probability\n dirichlet = tfd.Dirichlet(concentration=alpha)\n theta = dirichlet.sample()\n\n # step 2: generate the centroid for each cluster\n normal_1 = tfd.Normal(loc=[0], scale=sigma0)\n # sampling `n_clusters` time, hence, mean for each cluster\n mu_k = normal_1.sample(n_clusters) # (n_clusters, n_dim)\n\n # ====== Now for the assignment, need 1 indicator for each\n # examples, hence we need `batch_size` amount of indicator\n # (step: 3(a))====== #\n categorical = tfd.OneHotCategorical(probs=theta)\n z = categorical.sample(batch_size) # (batch_size, n_clusters)\n z = tf.cast(z, tf.bool)\n\n # ====== sampling the data points (step: 3(b)) ====== #\n normal_2 = tfd.Normal(loc=mu_k, scale=1)\n # this make each draw sample will generate sample for\n # all 4 components\n normal_2 = tfd.Independent(normal_2, reinterpreted_batch_ndims=2)\n x_all_components = normal_2.sample(batch_size) # (batch_size, n_clusters, n_dim)\n # ====== selecting the right component for each sample (step: 3(b)) ====== #\n # (batch_size, n_clusters, n_dim) * (batch_size, n_clusters)\n # = (batch_size, n_dim)\n x = tf.boolean_mask(x_all_components, z)\n\n # ====== Return: X, Z, mu_k, theta ====== #\n return (x.numpy(),\n np.argmax(z.numpy(), axis=-1),\n mu_k.numpy(),\n theta.numpy())\n\n```\n\n\n```\nX, Z, mu_k, theta = gmm(2500, n_clusters=3, alpha=1, sigma0=40)\n\nplt.figure(figsize=(12, 6))\ncolors = seaborn.color_palette(palette='Set2', n_colors=len(np.unique(Z)))\nplt.subplot(1, 2, 1)\n# plotting the scatter points\nplt.scatter(np.arange(len(X)), X, c=[colors[int(z)] for z in Z],\n s=4, alpha=0.6)\n# plotting the mean\nfor i, mu in enumerate(mu_k):\n plt.axhline(mu, 0, len(X),\n label=r\"$\\mu=%.2f;\\theta=%.2f$\" % (mu_k[i], theta[i]),\n color=colors[i], linestyle='--', linewidth=2)\nplt.grid(True)\nplt.legend()\nplt.subplot(1, 2, 2)\n_ = plt.hist(X, bins=80)\n```\n\n### Gibbs sampling for GMM\n\nFrom the above figure, $\\mu$ and $\\theta$ are the unknown quantities you need to infer using Gibbs sampling\n\nThe algorithm in outline:\n\n* sample $\\mu$ from $\\mu \\mid x,z,\\theta$\n* sample $\\theta$ from $\\theta \\mid x,z,\\mu$\n* sample $z$ from $z \\mid x,\\theta,\\mu$\n\n\n# Question 3\n\nPerform Gaussian mixture model inference on the fish data. \n\nFishes have been kept in different tanks (tankki) and in one tank same type feeding is used. \n\nAll fishes are supposed to be same species, but some might be wild and others grown in the tanks. \n\nSo can we see two different modes in the length distributions? And are there differences between tanks? \n\nSo fit 2-component GMM to fish length (pituus) in each tank (tankki) separately. \n\nUse your favourite GMM estimator, preferably Gibbs sampler (you can get posterior of means). \n\nWhat do you conclude? \n\nData is described in the following paper \n(Harkonen, L., Hyvarinen, P., Mehtatalo, L. Vainikka, A. 2017. Growth, survival and social learning in the first hatchery generation of Eurasian perch (Perca fluviatilis). Aquaculture 466: 6471. http://dx.doi.org/10.1016/j.aquaculture.2016.09.027. )\n\n### Read the csv dataset from Google Drive\n\nLink to the data from Google Drive:\n\nhttps://drive.google.com/open?id=1t-vN62A2oO3HkcZKq1cgYu8m5tuooU6-\n\n\n```\n# Here: all the package we need to connect to Google Drive using PyDrive\nfrom pydrive.auth import GoogleAuth\nfrom pydrive.drive import GoogleDrive\nfrom google.colab import auth\nfrom oauth2client.client import GoogleCredentials\n```\n\n\n```\n# 1. Authenticate and create the PyDrive client.\nauth.authenticate_user()\ngauth = GoogleAuth()\ngauth.credentials = GoogleCredentials.get_application_default()\ndrive = GoogleDrive(gauth)\n# 2. Load a file by ID and print its contents.\ndownloaded = drive.CreateFile({'id': \"1t-vN62A2oO3HkcZKq1cgYu8m5tuooU6-\"})\ndownloaded.GetContentFile('fish.csv')\n```\n\n /usr/local/lib/python3.6/dist-packages/google/colab/auth.py:140: ResourceWarning: unclosed \n if _check_adc():\n\n\n### Use `pandas` package to load and do data analysis.\n\nImportant detail when you are preprocessing the data:\n\n* Only get sample with MITTAUSAIKA = \"LOPETUS\"\n* Remove all columns contain NaN (not-a-number) values\n\n\n\n```\nimport pandas as pd\n\nds = pd.read_csv('fish.csv',\n sep=\";\", decimal=',', encoding=\"Latin-1\")\n# remove all column that contain NaN values\nids = ds.apply(lambda x: np.all(x.notna()), axis=0)\nds = ds.iloc[:, ids.tolist()]\n# we only take sample with \"MITTAUSAIKA\" = \"LOPETUS\"\nselected_row = ds.MITTAUSAIKA == \"LOPETUS\"\nds = ds[selected_row]\nprint(ds.describe())\n\n_ = ds.hist(figsize=(8, 8), bins=25)\n_ = seaborn.pairplot(ds)\n```\n\n### Fitting GMM model\n\nNOTE: the *curve blue line* is fitted using KDE (kernel density estimation), a non-parametric algorithm for estimating the density of input data. \n\nKDE is different from GMM (a parametric algorithm), and should not be mistaken as GMM.\n\n\n```\n# ====== Getting the tank and fish length data ====== #\n# Pituus: length\n# Paino : weight\ndata = ds[['Allas', 'Pituus']]\n# ====== grouping the data by the Tank ====== #\nfrom sklearn.mixture import GaussianMixture\nn_components = 2\n# selecting random color for each component\ncolors = seaborn.color_palette(\n palette='Set2', n_colors=n_components)\n# we need this to draw Gaussian distribution\nimport matplotlib.mlab as mlab\n\nfor pool_id in data.Allas.unique():\n # select all data from given tank\n pool_data = data.Pituus[data.Allas == pool_id]\n\n # Fitting Gaussian on Pool data\n gmm = GaussianMixture(n_components=int(n_components),\n covariance_type='diag', n_init=8,\n random_state=5218)\n # the input data must be at least 2D, so we\n # need to do some preprocessing\n pool_data = np.atleast_2d(pool_data.values).T\n gmm.fit(pool_data)\n\n # Plotting the histogram\n plt.figure(figsize=(8, 2))\n seaborn.distplot(pool_data, bins=18)\n\n # Visualizing the GMM\n mean = gmm.means_.ravel()\n precision = gmm.precisions_.ravel()\n xmin, xmax = plt.gca().get_xlim()\n X = np.linspace(start=xmin, stop=xmax,\n num=1000)\n ax = plt.gca().twinx()\n for n in range(n_components):\n Y = mlab.normpdf(X, mean[n], np.sqrt(1 / precision[n]))\n _, = ax.plot(X, Y,\n label='Component:%d' % n,\n color=colors[n], linewidth=3, linestyle='--')\n\n # show extra info\n ax.set_xlim((np.min(pool_data) - 20,\n np.max(pool_data) + 20))\n plt.legend()\n plt.title(\"Pool #%s\" % str(pool_id))\n```\n", "meta": {"hexsha": "0950f731c917fdce37eefef89d7fcad51b31982e", "size": 622674, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ex4_sampler/ex5_tutorial.ipynb", "max_stars_repo_name": "trungnt13/uef_bay1_2018", "max_stars_repo_head_hexsha": "48a0f684eb4d18777d9f03998233774baa0524a8", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "ex4_sampler/ex5_tutorial.ipynb", "max_issues_repo_name": "trungnt13/uef_bay1_2018", "max_issues_repo_head_hexsha": "48a0f684eb4d18777d9f03998233774baa0524a8", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2018-11-30T16:36:40.000Z", "max_issues_repo_issues_event_max_datetime": "2018-11-30T16:36:40.000Z", "max_forks_repo_path": "ex4_sampler/ex5_tutorial.ipynb", "max_forks_repo_name": "trungnt13/uef_bay1_2018", "max_forks_repo_head_hexsha": "48a0f684eb4d18777d9f03998233774baa0524a8", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 890.8068669528, "max_line_length": 288646, "alphanum_fraction": 0.9333359029, "converted": true, "num_tokens": 3303, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352403, "lm_q2_score": 0.7853085859124002, "lm_q1q2_score": 0.4323966916214315}} {"text": "```python\nimport numpy as np\nimport keras\nimport pandas\nfrom keras_tqdm import TQDMNotebookCallback\nfrom sklearn import preprocessing\n\ndata = np.array(pandas.read_csv(\"~/trainingdata.csv\", header=0))\n\nprint(data.shape)\n\n```\n\n (72, 7)\n\n\n\n```python\nfrom sympy import *\ninit_printing(use_latex=True)\n\nimport matplotlib.pyplot as plt\n%matplotlib inline\n\nX = data[:,0:6]\nX = preprocessing.scale(X)\nprint(X.shape)\ndisplay(X)\n\nlabels = data[:,6]\nprint(labels.shape)\ndisplay(labels)\n\nY = keras.utils.to_categorical(labels, len(np.unique(labels)))\n```\n\n (72, 6)\n\n\n\n array([[-1.0233435 , -0.37716553, -0.34893403, 0.32330268, -0.05200758,\n 0.37994185],\n [-0.7046986 , -0.45616456, 0.97833202, -0.01720953, -0.93794118,\n -0.68779499],\n [-0.66929361, -1.40415296, 1.04153517, 1.00432709, -1.61125071,\n -0.80643241],\n [-0.59848363, -1.32515393, 1.48395719, 0.89082302, -1.18600258,\n -1.6368944 ],\n [-0.74010359, -0.29816649, -0.53854347, 0.32330268, -0.09925737,\n 0.6172167 ],\n [ 1.48155945, 0.17582771, -0.58594583, 1.91235964, 0.9717379 ,\n 1.12142576],\n [ 1.56122067, 0.41282481, -0.60174661, 1.23133522, 1.18829945,\n -0.80643241],\n [-0.77550858, 0.33382578, -0.79135605, -0.24421767, -0.00475779,\n -0.56915756],\n [ 1.27798076, 0.80781998, -0.55961118, -0.4712258 , 1.12923721,\n 1.09176641],\n [-0.66929361, -0.77216069, 0.67811708, -0.4712258 , -1.56400092,\n -0.68779499],\n [ 1.45500571, 1.28181418, 0.28309742, -1.03874615, 1.71985961,\n 1.56631611],\n [-0.66929361, -0.85115973, 1.02046745, -0.4712258 , -1.51675113,\n -0.92506984],\n [ 1.20717079, 1.51881128, -0.34893403, -0.58472987, 1.36548617,\n 1.92222839],\n [ 1.95067555, 2.15080355, -0.8545592 , -1.15225021, 0.89298825,\n 1.32904126],\n [ 1.39304698, 1.04481708, -0.60174661, -1.37925835, 1.01111273,\n 0.97312898],\n [-0.49226867, -1.56215103, 1.73676977, 0.66381488, -0.80800425,\n -0.80643241],\n [-0.59848363, -1.2461549 , 0.83612495, 1.23133522, -0.85525404,\n -0.80643241],\n [ 1.52581568, 0.49182384, -0.66494976, 2.0258637 , 0.89298825,\n 0.6172167 ],\n [-0.71650026, -0.61416263, -0.66494976, 0.13412923, -0.09925737,\n -0.21324528],\n [-0.66929361, -0.61416263, 1.67356662, -0.24421767, -1.37500175,\n -0.65813563],\n [ 1.38419573, 1.20281515, -0.60174661, -0.58472987, 1.12923721,\n 1.09176641],\n [-0.88172354, -0.14016842, -1.23377807, -1.15225021, 0.042492 ,\n -1.10302598],\n [-0.59848363, -1.32515393, 1.10473831, 1.00432709, -0.94975363,\n -0.80643241],\n [-0.49226867, -1.2461549 , 1.92637921, 0.32330268, -1.18600258,\n -1.51825697],\n [-0.6869961 , -0.45616456, -1.48659065, -0.10233758, -0.00475779,\n -0.21324528],\n [ 1.03899709, 1.61756007, -0.19092617, -1.15225021, 1.54267289,\n 2.45609681],\n [-0.63388862, -0.69316166, 1.48395719, -0.35772173, -1.46950133,\n -0.80643241],\n [ 0.99474085, 0.72882095, -0.72815291, 1.79885557, 1.176487 ,\n 1.41801933],\n [-0.63388862, -1.40415296, 1.3575509 , 0.32330268, -1.42225154,\n -0.68779499],\n [-0.53947532, -0.06116939, -0.91776234, -0.95361809, -0.00475779,\n 0.26130442],\n [-0.73125234, -0.06116939, -0.34893403, 0.70164957, -0.05200758,\n 0.49857928],\n [-0.77550858, -0.77216069, 1.04153517, -0.35772173, -0.80800425,\n -0.68779499],\n [ 1.49041069, 1.04481708, 0.28309742, -0.81173801, 1.83798408,\n 1.12142576],\n [-0.480467 , 0.01782964, -1.04416863, -0.10233758, -0.05200758,\n -0.45052013],\n [-0.7046986 , -0.25866697, -0.91776234, -0.58472987, -0.19375696,\n -0.33188271],\n [-0.6869961 , -0.15991818, -0.53854347, -0.24421767, -0.00475779,\n 0.02402957],\n [-0.59848363, -0.69316166, 1.2311446 , -0.24421767, -1.46950133,\n -1.04370727],\n [-0.77550858, -0.06116939, -1.10737178, -0.10233758, -0.09925737,\n -0.21324528],\n [ 1.13636081, 1.43981225, 0.17775885, -1.15225021, 1.48361065,\n 0.67653542],\n [ 1.45500571, 1.20281515, -0.66494976, -1.60626649, 1.12923721,\n 1.32904126],\n [-1.07055015, -0.14016842, -0.8545592 , 0.32330268, 0.13699158,\n 0.37994185],\n [-0.81091356, -0.15991818, -0.66494976, 0.20979861, 0.08974179,\n 0.6172167 ],\n [-0.81091356, 0.01782964, -1.04416863, -0.81173801, -0.09925737,\n -0.45052013],\n [ 1.52581568, 1.28181418, -0.42793796, 2.36637591, 1.05048756,\n 0.37994185],\n [-0.66929361, -1.483152 , 0.83612495, 1.00432709, -1.75300008,\n -0.92506984],\n [-0.77550858, -0.37716553, -0.98096549, -0.10233758, -0.09925737,\n 0.142667 ],\n [-0.71650026, 0.23507699, -0.72815291, -0.52797784, -0.05200758,\n 0.49857928],\n [ 1.34879074, 1.20281515, -0.77028834, -0.69823394, 1.36548617,\n 1.7146129 ],\n [-0.33294622, -0.37716553, -0.98096549, -0.24421767, -0.05200758,\n 0.02402957],\n [-0.95253352, -0.15991818, -1.10737178, 0.32330268, -0.00475779,\n -0.56915756],\n [ 1.52581568, 2.15080355, -1.92901267, -0.92524208, 0.30236585,\n 1.92222839],\n [ 1.45500571, 0.96581805, -0.34893403, 2.82039218, 1.24736169,\n 0.85449156],\n [-0.59848363, -0.65366214, 1.04153517, 0.32330268, -0.94975363,\n -1.16234469],\n [ 1.34879074, 1.91380645, -0.34893403, -1.71977056, 1.176487 ,\n 1.09176641],\n [ 0.95933587, 1.04481708, 0.36210135, -0.81173801, 1.77892185,\n 0.8248322 ],\n [-0.56307864, -0.45616456, 0.59911315, 0.20979861, -0.71350467,\n -0.80643241],\n [-0.63388862, -1.40415296, 1.15214067, 0.89082302, -1.42225154,\n -0.9547292 ],\n [-0.88172354, -1.16715586, 1.98958235, 0.89082302, -0.66625488,\n -1.04370727],\n [-0.52767366, -1.08815683, 1.54716033, -0.01720953, -0.66625488,\n -0.80643241],\n [ 1.56122067, 0.96581805, -0.60174661, 2.36637591, 0.75123888,\n 0.142667 ],\n [-0.66929361, -0.61416263, 1.3575509 , -0.24421767, -1.6585005 ,\n -0.80643241],\n [-0.480467 , -0.25866697, -0.98096549, -0.43339111, -0.05200758,\n -1.6368944 ],\n [-0.64273987, -0.37716553, -0.90196156, -0.81173801, 0.042492 ,\n -1.39961954],\n [-0.74010359, -1.483152 , 1.67356662, 0.55031081, -1.04425321,\n -1.2018905 ],\n [ 1.34879074, 1.51881128, -0.66494976, -1.26575428, 1.03473763,\n 0.6172167 ],\n [-0.59848363, -1.16715586, 1.3575509 , 0.77731895, -0.99700342,\n -1.04370727],\n [-0.7046986 , -1.32515393, 1.79997291, 0.43680675, -0.71350467,\n -0.92506984],\n [-0.95253352, -0.37716553, -0.34893403, 0.66381488, -0.05200758,\n 0.49857928],\n [-0.88172354, 0.17582771, -0.72815291, -0.62256456, -0.09925737,\n -1.16234469],\n [ 1.45500571, 1.59781032, -0.22252774, -2.06028276, 1.41273596,\n 0.97312898],\n [-0.88172354, 0.17582771, -0.34893403, 0.32330268, -0.05200758,\n 0.26130442],\n [ 1.56122067, 1.20281515, -0.72815291, -1.49276242, 1.07017497,\n 1.92222839]])\n\n\n (72,)\n\n\n\n array([0., 1., 1., 1., 0., 2., 2., 0., 2., 1., 2., 1., 2., 2., 2., 1., 1.,\n 2., 0., 1., 2., 0., 1., 1., 0., 2., 1., 2., 1., 0., 0., 1., 2., 0.,\n 0., 0., 1., 0., 2., 2., 0., 0., 0., 2., 1., 0., 0., 2., 0., 0., 2.,\n 2., 1., 2., 2., 1., 1., 1., 1., 2., 1., 0., 0., 1., 2., 1., 1., 0.,\n 0., 2., 0., 2.])\n\n\n\n```python\ninput_size = X.shape[1]\noutput_size = Y.shape[1]\ndisplay(X.shape[1])\n```\n\n\n```python\nmodel = keras.models.Sequential()\n\nmodel.add(keras.layers.Dense(100,input_dim=6,activation='relu', bias_initializer=keras.initializers.Constant(value=0.01)))\nmodel.add(keras.layers.Dense(100,input_dim=6,activation='relu', bias_initializer=keras.initializers.Constant(value=0.01)))\nmodel.add(keras.layers.Dense(100,input_dim=6,activation='relu', bias_initializer=keras.initializers.Constant(value=0.01)))\n\nmodel.add(keras.layers.Dense(3,activation='softmax'))\n#binary_crossentropy\nmodel.compile(loss='binary_crossentropy',optimizer='adam',metrics=['accuracy'])\n\nprint(model.summary())\n```\n\n _________________________________________________________________\n Layer (type) Output Shape Param # \n =================================================================\n dense_29 (Dense) (None, 100) 700 \n _________________________________________________________________\n dense_30 (Dense) (None, 100) 10100 \n _________________________________________________________________\n dense_31 (Dense) (None, 100) 10100 \n _________________________________________________________________\n dense_32 (Dense) (None, 3) 303 \n =================================================================\n Total params: 21,203\n Trainable params: 21,203\n Non-trainable params: 0\n _________________________________________________________________\n None\n\n\n\n```python\nhistory = model.fit(X, Y,\n batch_size=56, \n epochs=100, \n verbose=0,\n callbacks=[TQDMNotebookCallback()],\n validation_split = 0.25)\n\n```\n\n\n HBox(children=(IntProgress(value=0, description='Training', style=ProgressStyle(description_width='initial')),\u2026\n\n\n\n HBox(children=(IntProgress(value=0, description='Epoch 0', max=54, style=ProgressStyle(description_width='init\u2026\n\n\n \n\n\n\n```python\nplt.figure(1)\n\nplt.subplot(211)\nplt.plot(history.history['acc'])\nplt.plot(history.history['val_acc'])\nplt.title('model accuracy')\nplt.ylabel('accuracy')\nplt.xlabel('epoch')\nplt.legend(['train', 'test'], loc='upper left')\n\nplt.subplot(212)\nplt.plot(history.history['loss'])\nplt.plot(history.history['val_loss'])\nplt.title('model loss')\nplt.ylabel('loss')\nplt.xlabel('epoch')\nplt.legend(['train', 'test'], loc='upper left')\nplt.tight_layout()\nplt.show()\n\nscore = model.evaluate(X, Y, verbose=1)\nprint('Test loss:', score[0])\nprint('Test accuracy:', score[1])\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "7759a18f090cc1b195b65219f0c1eec07cb9b61a", "size": 36009, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": ".ipynb_checkpoints/Project_Multilayer_Net-checkpoint.ipynb", "max_stars_repo_name": "holypolarpanda7/S19-team2-project", "max_stars_repo_head_hexsha": "09b51f07849e3288dfa4ba91cf5d8d13909e35e2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": ".ipynb_checkpoints/Project_Multilayer_Net-checkpoint.ipynb", "max_issues_repo_name": "holypolarpanda7/S19-team2-project", "max_issues_repo_head_hexsha": "09b51f07849e3288dfa4ba91cf5d8d13909e35e2", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": ".ipynb_checkpoints/Project_Multilayer_Net-checkpoint.ipynb", "max_forks_repo_name": "holypolarpanda7/S19-team2-project", "max_forks_repo_head_hexsha": "09b51f07849e3288dfa4ba91cf5d8d13909e35e2", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 83.3541666667, "max_line_length": 19676, "alphanum_fraction": 0.7514232553, "converted": true, "num_tokens": 4267, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.785308580887758, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.4323966888548266}} {"text": "# Una mirada al concepto de Regresi\u00f3n lineal\n\n\n\n\n### Tabla de contenido\n\n* [1. Introducci\u00f3n](#Introducci\u00f3n)\n* [2. Antecedentes hist\u00f3ricos](#Antecedentes)\n* [3. Presentaci\u00f3n del algoritmo](#Algoritmo)\n* [4. Ejercicio de aplicaci\u00f3n](#Ejercicio)\n* [5. Conclusiones](#Conclusiones)\n* [Referencias](#Referencias)\n\n# Introducci\u00f3n\n\nEn el an\u00e1lisis y modelamiento de datos es importante el uso de diferentes herramientas matem\u00e1ticas, esto con el fin de describir el comportamiento de los mismos. Con esto en mente, uno de los m\u00e9todos m\u00e1s populares en medio de la estad\u00edstica y de la ciencia de datos es la [*regresi\u00f3n lineal*](https://es.wikipedia.org/wiki/Regresi%C3%B3n_lineal). La idea central de este algoritmo es relacionar un conjunto de [variables respuestas o dependientes](https://en.wikipedia.org/wiki/Dependent_and_independent_variables) con un conjunto de [variables explicativas o independientes](https://en.wikipedia.org/wiki/Dependent_and_independent_variables), ambas de car\u00e1cter cuantitativo, ya sea para hacer sociaci\u00f3n, control de calidad, calibraci\u00f3n, clasificaci\u00f3n y predicci\u00f3n; por medio de una relaci\u00f3n lineal entre las variables respuesta y los par\u00e1metros de inter\u00e9s. En caso que no sean cuantitativas se usan otros modelos como la regresi\u00f3n log\u00edstica (Hocking,2003)\n\nEn lo que respecta al aprendizaje de m\u00e1quina, la regresi\u00f3n lineal es de vital relevancia para clasificar y predicir. Para ello, se recurren a diferentes estrategias de optimizaci\u00f3n para estimar los par\u00e1metros como lo son [m\u00ednimos cuadrados](https://es.wikipedia.org/wiki/M%C3%ADnimos_cuadrados) o el m\u00e9todo de [Gradiente Descendiente](https://www.iartificial.net/gradiente-descendiente-para-aprendizaje-automatico/); adem\u00e1s de m\u00faltiples formas de validaci\u00f3n (Bowles, 2015) (M\u00fcller, A. C., & Guido, S, 2016). Sin embargo, podemos tener datos que est\u00e1n relacionados de diferentes formas y debemos ser cuidadosos a la hora de plantear nuestros modelos; pues podemos implementar el modelo y tener malos ajustes. \n\n\n\nCon lo anterior en mente, este cuaderno tiene como objetivo presentar el formalismo matem\u00e1tico de la regresi\u00f3n lineal junto con un ejemplo de aplicaci\u00f3n, en el cual se encuentra la relaci\u00f3n entre la potencia el\u00e9ctrica de climatizaci\u00f3n y temperatura exterior de una planta. Este ejemplo y los datos usados se pueden encontrar en el blog de machine learning [Koldopina](https://koldopina.com/regresion-lineal-simple/).\n\n# Antecedentes hist\u00f3ricos \n\nLos primeros desarrollos te\u00f3ricas sobre regresi\u00f3n lineal se remontan a comienzos del siglo XIX cuando [Legendre](https://es.wikipedia.org/wiki/Adrien-Marie_Legendre), en 1805, y [Gauss](https://es.wikipedia.org/wiki/Carl_Friedrich_Gauss), en 1809, implementaron este algoritmo para solucionar el movimientos de los planetas. En el planteamiento original se planteaba una relaci\u00f3n lineal entre las variables explicativas y los par\u00e1metros de inter\u00e9s, es decir $y = \\beta_1 x + \\beta_0$, donde $y$ es la variable respuesta, $x$ es la variable explicativa y, $\\beta_1$ y $\\beta_0$ los par\u00e1metros a determinar. A partir de esto, determinaron los par\u00e1metros usando m\u00ednimos cuadrados (Stigler, S. M. 1986).\n\n\n*El estudio del movimiento planetario contribuy\u00f3 al desarrollo de la regresi\u00f3n lineal y a la posterior implementaci\u00f3n de los modelos lineales*\n\n# Presentaci\u00f3n del algoritmo \n\nSi bien es cierto que hay cursos universitarios completos de regresi\u00f3n lineal, lo que vamos a abordar constituye las bases para modelos m\u00e1s elaborados. Si el lector se quiere familiarizar m\u00e1s con esto, puede ver los [usos de la regresi\u00f3n en el aprendizaje profundo](https://www.youtube.com/watch?v=E5RjzSK0fvY&ab_channel=edureka%21), [lecciones sobre el formalismo matem\u00e1tico mas detallado](https://www.youtube.com/watch?v=4b4MUYve_U8&ab_channel=stanfordonline) o [una explicaci\u00f3n en v\u00eddeo](https://www.youtube.com/watch?v=SsFBnvkoZa4&ab_channel=EstimadosEstadisticos).\n\nPara empezar tomemos una serie de variables explicativas $Y_i$ con $i=1,...,n$. Tales que \n\n$$\nY_{i}=g_{i}(\\theta)+\\varepsilon_{i}=\\alpha+b x_{2}+\\varepsilon_{i}, \\quad 1 \\leq i \\leq n\n$$ \n\ncon $\\pmb{\\theta}=(a, b)^{t}$, el vector de par\u00e1metros, y $x_{i}, i=1,2,3, \\ldots, n,$ valores conocidos (variables explicativas)y $\\varepsilon_{i}$ es una variable aleatoria que tiene una distribuci\u00f3n normal con media cero y varianza $\\sigma^2$. Adem\u00e1s, \n\n\n$$\ng_{i}(\\pmb{\\theta})=a+b x_{i}, \\quad i=1,2,3, \\ldots, n\n$$\n\n\nAhora, se quiere minimizar la distancia de los valores $y_i$ que predice nuestro modelo y los datos observados. As\u00ed\n\n$$\nQ=\\sum_{i=1}^{n} \\left[ y_{i}-g_{i}\\left(\\theta_{1}, \\ldots, \\theta_{x}\\right)\\right]^{2}=\\sum_{i=1}^{n} \\left[ y_{i}-a -b x_{i} \\right]^{2}\n$$\n\nse deriva $Q$ con respecto a $\\pmb{\\theta}, \\frac{d Q_f}{{d \\theta}}$ y se iguala a cero $\\frac{d Q}{d \\theta}=0$\n\n\\begin{align}\n \\left( \\begin{matrix} \\frac{\\partial Q}{\\partial a} \\\\ \\frac{\\partial Q}{\\partial b} \\end{matrix}\\right) & = \\left( \\begin{matrix} (-1)\\sum_{i=1}^n 2(y_i -a -bx_i) \\\\ \\sum_{i=1}^n 2(y_i -a -bx_i)(-x_i) \\frac{\\partial Q}{\\partial b} \\end{matrix}\\right) \n\\end{align}\n\nAs\u00ed, \n\\begin{align}\n \\left( \\begin{matrix} (-1)\\sum_{i=1}^n 2(y_i -a -bx_i) \\\\ \\sum_{i=1}^n 2(y_i -a -bx_i)(-x_i) \\frac{\\partial Q}{\\partial b} \\end{matrix}\\right) & = \\left( \\begin{matrix} 0 \\\\ 0 \\end{matrix}\\right) \n\\end{align}\n\n\n\nDespejando se obtiene\n\n\\begin{align}\na & = \\frac{\\bar{y} \\sum_{i=1}^n x_i^2 -\\bar{x} \\sum_{i=1}^n x_i y_i }{\\sum_{i=1}^n x_i^2 -n \\bar{x}\u00b2} \\\\\nb & = \\frac{\\sum_{i=1}^n x_i y_i - n \\bar{x}\\bar{y}}{\\sum_{i=1}^n x_i^2 -n \\bar{x}\u00b2} \\\\\n\\end{align}\n\nDe esta manera, obtenemos los par\u00e1metros de la regresi\u00f3n lineal simple. \n\nAdem\u00e1s, podemos calcular el coeficiente de correlaci\u00f3n como \n\\begin{equation}\nr_{x y}=\\frac{\\sum_{i=1}^{n}\\left(x_{i}-\\bar{x}\\right)\\left(y_{i}-\\bar{y}\\right)}{\\sqrt{\\sum_{i=1}^{n}\\left(x_{i}-\\bar{x}\\right)^{2} \\sum_{i=1}^{n}\\left(y_{i}-\\bar{y}\\right)^{2}}}\n\\end{equation}\n\n# Ejercicio de aplicaci\u00f3n \n\nComo se mencion\u00f3 anteriormente, este ejercicio se puede consultar en el blog [Koldopina](https://koldopina.com/regresion-lineal-simple/). La idea es entrenar un modelo de machine learning para predecir una relaci\u00f3n de la forma $y = \\beta_1 x + \\beta_0$ donde $y$ es la potencia el\u00e9ctrica de climatizaci\u00f3n y $x$ es la temperatura exterior de una planta. Para empezar, vamos a importar las librer\u00edas necesarias para cargar los datos y podemos procesar matem\u00e1ticamente\n\n\n```python\n#Importamos las librerias\nimport pandas as pd\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport seaborn as sns\nimport random\nimport math\nimport csv\n%matplotlib inline\n```\n\nAhora, extraemos los datos del archivo [data.csv](https://github.com/daagonzalezgu/Curso-Big-Data-2021-I/blob/20d22c8d16a70e9235c2cf9ea509cfc04716b398/EnsayoJupyter/data.csv) y los organizamos para poder procesarlos, creando listas para cada columna. \n\n\n```python\nwith open('data.csv', 'r') as f:\n reader = csv.reader(f, delimiter='|')\n OUTDOOR_TEMP = []\n ELECTRIC_POWER = []\n for index, row in enumerate(reader):\n if index == 0:\n header = row\n else:\n try:\n outNum = float(row[0].replace(\",\", \".\"))\n except:\n outNum = None\n OUTDOOR_TEMP.append(outNum)\n try:\n elecNum = float(row[1].replace(\",\", \".\"))\n except:\n elecNum = None\n ELECTRIC_POWER.append(elecNum)\n```\n\nAhora, procedemos a visualizar los datos \n\n\n```python\ndef info(header, data_list):\n \"\"\"\n :param header: lista con los encabezados de las columnas\n :param data_list: lista con las listas de datos de las columnas: [lista1, lista2, etc...]\n :return: diccionario de diccionarios con n\u00famero de registros, tipo de dato, n\u00famero de na,\n media, std, min, max de cada columna.\n \"\"\"\n\n from collections import defaultdict\n\n header = header\n columns = data_list\n values = defaultdict()\n for index, head in enumerate(header):\n aux = defaultdict()\n aux['len'] = len(columns[index])\n aux['clases'] = set([type(ele) for ele in columns[index]])\n aux['na'] = sum(1 for ele in columns[index] if ele == None)\n # media\n media = sum(ele for ele in columns[index] if ele != None) / len(columns[index])\n aux['media'] = media\n # std\n n = sum(1 for ele in columns[index] if ele != None)\n std = ((1 / (n - 1)) * sum((ele - media) ** 2 for ele in columns[index] if ele != None)) ** 0.5\n aux['std'] = std\n # minimo\n aux['min'] = min(ele for ele in columns[index] if ele != None)\n # maximo\n aux['max'] = max(ele for ele in columns[index] if ele != None)\n values[head] = aux\n return values\n```\n\n\n```python\nprint('_'*60 + 'COLUMNS')\nprint (header)\n\n```\n\n ____________________________________________________________COLUMNS\n ['OUTDOOR_TEMP', 'ELECTRIC_POWER']\n\n\n\n```python\nprint ('_'*60 + 'INFO') \nprint (info(header, [OUTDOOR_TEMP, ELECTRIC_POWER]))\n```\n\n ____________________________________________________________INFO\n defaultdict(None, {'OUTDOOR_TEMP': defaultdict(None, {'len': 1496, 'clases': {}, 'na': 0, 'media': 28.544429085561454, 'std': 6.364374947698145, 'min': 0.0, 'max': 38.310135}), 'ELECTRIC_POWER': defaultdict(None, {'len': 1496, 'clases': {, }, 'na': 2, 'media': 285.83182486631, 'std': 41.51823016424719, 'min': 0.0, 'max': 391.71})})\n\n\nLo anterior nos dice que tenemos dos columnas, una llamada OUTDOOR_TEMP, que es la temperatura exterior y ELECTRIC_POWER, es la potencia el\u00e9ctrica y tiene dos valores no definidos (na). Con ello, procedemos a limpiar los datos quitando esos valores (na)\n\n\n```python\n#Quitamos los nas\nindex_to_drop = [index for index, val in enumerate(ELECTRIC_POWER) if val is None]\nELECTRIC_POWER = [val for index, val in enumerate(ELECTRIC_POWER) if index not in index_to_drop]\nOUTDOOR_TEMP = [val for index, val in enumerate(OUTDOOR_TEMP) if index not in index_to_drop]\n```\n\nAhora, verificamos que esos valores no est\u00e9n \n\n\n```python\nprint (info(header, [OUTDOOR_TEMP, ELECTRIC_POWER]))\n```\n\n defaultdict(None, {'OUTDOOR_TEMP': defaultdict(None, {'len': 1494, 'clases': {}, 'na': 0, 'media': 28.54050408902272, 'std': 6.36757237847448, 'min': 0.0, 'max': 38.310135}), 'ELECTRIC_POWER': defaultdict(None, {'len': 1494, 'clases': {}, 'na': 0, 'media': 286.21446452476556, 'std': 41.51646570695531, 'min': 0.0, 'max': 391.71})})\n\n\nSeguido a ello, graficamos los datos para identificar valores at\u00edpicos que pueden afectar los valores que va a predecir nuestro modelo\n\n\n```python\ndef visual(header, X, y):\n \"\"\"\n :param header: Lista con los nombres de los encabezados\n :param X: Lista con los valores de la columna a colocar en el eje X\n :param y: Lista con los valores de la columna a colocar en el eje y\n :return: matplotlib figure plot\n \"\"\"\n\n fs = 10 # fontsize\n fig, axs = plt.subplots(3, 2, figsize=(6, 6))\n plt.subplots_adjust(top=0.9, bottom=0.1, hspace=0.5, wspace=0.2, left=0.125, right=0.9)\n axs[0, 0].scatter(X, y, c='r', edgecolors=(0, 0, 0), alpha=0.2)\n axs[0, 0].set_title('Scatter %s vs %s' %(header[1], header[0]), fontsize=fs)\n axs[1, 0].hist(X, color='red')\n axs[1, 0].set_title('Hist %s' %header[0], fontsize=fs)\n axs[0, 1].hist2d(X, y)\n axs[0, 1].set_title('Hist 2D', fontsize=fs)\n axs[1, 1].hist(y, color='blue')\n axs[1, 1].set_title('Hist %s' %header[1], fontsize=fs)\n axs[2, 0].boxplot(X)\n axs[2, 0].set_title('Box %s' %header[0], fontsize=fs)\n axs[2, 1].boxplot(y)\n axs[2, 1].set_title('Box %s' %header[1], fontsize=fs)\n plt.show()\n```\n\n\n```python\n#Ploteamos la gr\u00e1fica\nvisual(header, OUTDOOR_TEMP, ELECTRIC_POWER)\n```\n\n\n```python\n#Quitamos los outlier o valores at\u00edpicos\nindex_to_drop = [index for index, val in enumerate(OUTDOOR_TEMP) if val == 0]\nELECTRIC_POWER = [val for index, val in enumerate(ELECTRIC_POWER) if index not in index_to_drop]\nOUTDOOR_TEMP = [val for index, val in enumerate(OUTDOOR_TEMP) if index not in index_to_drop]\n```\n\n\n```python\nprint (info(header, [OUTDOOR_TEMP, ELECTRIC_POWER]))\n```\n\n defaultdict(None, {'OUTDOOR_TEMP': defaultdict(None, {'len': 1449, 'clases': {}, 'na': 0, 'media': 29.42685514768802, 'std': 3.963016838173277, 'min': 18.765305, 'max': 38.310135}), 'ELECTRIC_POWER': defaultdict(None, {'len': 1449, 'clases': {}, 'na': 0, 'media': 287.81427881297435, 'std': 35.88844333632387, 'min': 183.12, 'max': 391.71})})\n\n\n\n```python\nvisual(header, OUTDOOR_TEMP, ELECTRIC_POWER)\n```\n\nDividimos los datos para entrenar y para validar\n\n\n```python\n#creamos las listas para training y test aleatoriamente\nall_data = [[x_val, y_val] for x_val, y_val in zip(OUTDOOR_TEMP, ELECTRIC_POWER)]\nprint (all_data[:5])\nrandom.shuffle(all_data)\ndiv = math.ceil(len(all_data)*0.3)\ndata_train = all_data[:div]\ndata_test = all_data[div:]\n\ndata_train_X = [ele[0] for ele in data_train]\ndata_train_y = [ele[1] for ele in data_train]\ndata_test_X = [ele[0] for ele in data_test]\ndata_test_y = [ele[1] for ele in data_test]\n```\n\n [[31.321108, 324.54], [24.938467, 252.7], [33.316906, 331.1], [26.947517, 270.8], [34.769539, 288.66]]\n\n\nSe crea el modelo y se procede a entrenarlo\n\n\n```python\nclass Lin_reg():\n\n def __init__(self, X, Y):\n \"\"\"\n :param X: lista con los valores de la variable de las abscisas\n :param y: lista con los valores de la variable de las ordenadas\n \"\"\"\n self.X = X\n self.y = Y\n self.N = len(self.X)\n self.X_mean = sum(self.X) / len(self.X)\n self.y_mean = sum(self.y) / len(self.y)\n self.X_std = (1 / (self.N - 1) * sum((ele - self.X_mean) ** 2\n for ele in self.X)) ** 0.5\n self.y_std = (1 / (self.N - 1) * sum((ele - self.y_mean) ** 2\n for ele in self.y)) ** 0.5\n self.X_var = self.X_std ** 2\n self.y_var = self.y_std ** 2\n self.cov = sum([i * j for (i, j) in zip([ele - self.X_mean for ele in self.X],\n [ele - self.y_mean for ele in self.y])]) / (self.N)\n\n self.r = self.cov / (self.X_std * self.y_std)\n\n def Coeficientes(self):\n if len(self.X) != len(self.y):\n raise ValueError('unequal length')\n self.b = self.cov / self.X_var\n self.a = self.y_mean - (self.b * self.X_mean)\n return self.a, self.b\n\n def predict(self, X):\n yp = []\n for x in X:\n yp.append(self.a + self.b * x)\n return yp\n```\n\n\n```python\nmylinreg=Lin_reg(data_train_X,data_train_y)\n```\n\n\n```python\na, b = mylinreg.Coeficientes()\nprint('La recta de regresi\u00f3n es: y = %f + %f * X'%(mylinreg.Coeficientes()))\nprint('El coeficiente de correlaci\u00f3n es: r = %f' %mylinreg.r)\n```\n\n La recta de regresi\u00f3n es: y = 54.837434 + 7.871854 * X\n El coeficiente de correlaci\u00f3n es: r = 0.904796\n\n\nAhora se gr\u00e1fican los datos de entrenamiento y los resultados de la regresi\u00f3n\n\n\n```python\nplt.scatter(data_train_X, data_train_y, c='r', edgecolors=(0, 0, 0), alpha=0.5)\nplt.plot(data_train_X, [a + b * x for x in data_train_X], c=\"b\")\nplt.xlabel('Temperatura exterior (C)')\nplt.ylabel('Potencia el\u00e9ctrica kW')\nplt.show()\n```\n\nAhora, predecimos los valores y comparamos con los valores observados en los datos de validaci\u00f3n\n\n\n```python\npredictions = mylinreg.predict(data_test_X)\n```\n\n\n```python\nplt.scatter(data_test_y, predictions, c='r', edgecolors=(0, 0, 0), alpha=0.5)\nplt.title('Valores predichos vs valores reales', fontsize=10)\nplt.xlabel('Valores reales')\nplt.ylabel('Valores predichos')\nplt.show()\n```\n\nCon esto, calculamos el error promedio, el error cuadr\u00e1tico, el coeficiente de correlaci\u00f3n y la desviaci\u00f3n est\u00e1ndar\n\n\n```python\n#Metricas\n#Mean Error - Desviaci\u00f3n media\nME = sum(y_pred - y_test for y_pred, y_test in zip(predictions,data_test_y)) / len(predictions)\n#Mean Absolute Error (error absoluto medio)\nMAE = sum(abs(y_pred - y_test) for y_pred, y_test in zip(predictions,data_test_y)) / len(predictions)\n#Mean Square Error (error cuadr\u00e1tico medio)\nMSE = sum((y_pred - y_test)**2 for y_pred, y_test in zip(predictions, data_test_y)) / len(predictions)\n#Root Mean Square Error - error de la ra\u00edz cuadrada de la media RMSE\nRMSE = MSE ** 0.5\n#Standard Deviation of Residuals . Desviaci\u00f3n t\u00edpica de los residuos\nSDR = (1 / (len(data_test_y) - 1) * sum((y_test - y_pred) ** 2\n for y_pred, y_test in zip(predictions, data_test_y))) ** 0.5\n\nprint ('Mean Error: %f' %ME)\nprint ('Mean Absolute Error: %f' %MAE)\nprint ('Mean Square Error: %f' %MSE)\nprint ('Root Mean Square Error: %f' %RMSE)\nprint ('Standard Desviation of Residuals: %f' %SDR)\n```\n\n Mean Error: -1.904745\n Mean Absolute Error: 12.997670\n Mean Square Error: 265.198086\n Root Mean Square Error: 16.284904\n Standard Desviation of Residuals: 16.292940\n\n\n\n```python\ndata_test_mean = sum(ele\n for ele in data_test_y) / len(data_test_y)\npredictions_mean = sum(ele\n for ele in predictions) / len(predictions)\n\ndata_test_std = (1 / (len(data_test_y) - 1) * sum((ele - data_test_mean) ** 2\n for ele in data_test_y)) ** 0.5\npredictions_std = (1 / (len(predictions) - 1) * sum((ele - predictions_mean) ** 2\n for ele in predictions)) ** 0.5\ncov = sum([i * j\n for (i, j) in zip([ele - data_test_mean\n for ele in data_test_y], \n [ele - predictions_mean\n for ele in predictions])\n]) / (len(predictions))\n\nprint('El coeficiente de correlaci\u00f3n es: R2 = %f' % (cov**2 / (data_test_std ** 2 * predictions_std ** 2)))\n```\n\n El coeficiente de correlaci\u00f3n es: R2 = 0.799761\n\n\nPor \u00faltimo, gr\u00e1ficamos los residuos para comprobar su distribuci\u00f3n normal con media cero. Esto nos da una idea de que tan bien el modelo se reproduce los datos. \n\n\n```python\n#Distribuci\u00f3n de los Residuos\nsns.distplot((np.asarray(data_test_y) - np.asarray(predictions)), bins = 50)\nplt.show()\n```\n\n# Conclusiones \n\nA partir de lo mostrado, se tiene que la regresi\u00f3n lineal es uno de los m\u00e9todos m\u00e1s importantes en el modelamiento y predicci\u00f3n de fen\u00f3menos, debido a su simplicidad matem\u00e1tica comparada con otros modelos. Adem\u00e1s, que sirve de base para algoritmos m\u00e1s complejos tales como las redes neuranales o en modelos lineales generalizados. Por \u00faltimo, constituye un campo de estudio con muchas aplicaciones en la industria, demograf\u00eda, ciencias exactas y dem\u00e1s; aunque tampoco constituye una receta m\u00e1gica para todos nuestros problemas con datos. \n\n# Referencias \n\n* Bowles, M. (2015). Machine learning in Python: essential techniques for predictive analysis. John Wiley & Sons.\n* Hocking, R. R. (2003), Methods and Applications of Linear Models, segunda edn, John Wiley and Sons, New Jersey.\n* M\u00fcller, A. C., & Guido, S. (2016). Introduction to machine learning with Python: a guide for data scientists. \" O'Reilly Media, Inc.\".\n* Ravishanker, N. & Dey, D. K. (2002), A First Course in Linear Model Theory, Chapman & Hall/CRC., New York.\n* Stigler, S. M. (1986). The history of statistics: The measurement of uncertainty before 1900. Harvard University Press.\n\n", "meta": {"hexsha": "20832be169b0697376e1ed064915654a1bf969de", "size": 212738, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "EnsayoJupyter/Ensayo.ipynb", "max_stars_repo_name": "daagonzalezgu/Curso-Big-Data-2021-I", "max_stars_repo_head_hexsha": "db593a8891cb676e2cada87ca84f7fb11e95fb14", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "EnsayoJupyter/Ensayo.ipynb", "max_issues_repo_name": "daagonzalezgu/Curso-Big-Data-2021-I", "max_issues_repo_head_hexsha": "db593a8891cb676e2cada87ca84f7fb11e95fb14", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "EnsayoJupyter/Ensayo.ipynb", "max_forks_repo_name": "daagonzalezgu/Curso-Big-Data-2021-I", "max_forks_repo_head_hexsha": "db593a8891cb676e2cada87ca84f7fb11e95fb14", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 236.1132075472, "max_line_length": 61212, "alphanum_fraction": 0.9105660484, "converted": true, "num_tokens": 5943, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857982, "lm_q2_score": 0.7690802264851919, "lm_q1q2_score": 0.43235883062590236}} {"text": "# Demonstration book of WaveGlow\n\nThis demonstration book will:\n \n1. Define components in the model\n2. Load pre-trained model and generate waveform samples\n\n\nA few notes:\n* Official implementation is in https://github.com/NVIDIA/DeepLearningExamples/tree/master/PyTorch/SpeechSynthesis/Tacotron2/waveglow\n* Model here is re-implemented for tutorial purpose\n* It is NOT intended to surpass the official implementation\n* Post-processing such as de-noising is not include in this notebook\n* Your contribution is welcome to improve it\n\nModules for WaveGlow are defined in `../sandbox/block_waveglow.py`. For convenience, I copy those modules to this notebook and demonstrate the usage.\n\nThe project to train and run a WaveGlow on CMU arctic database is available in `../project/05-nn-vocoder/waveglow`.\n\n\n## 1. Define a WaveGlow\n\nWaveGlow in the paper has a fixed model structure\n\n* **Condition module**: process and up-sample input conditional features\n* **Squeeze module**: squeeze the length of target waveform and input conditional features\n* WaveGlow core: **3 WaveGlow blocks**, each block contains **4 flow steps**, and each flow steps contains **8 dilated conv layers**.\n\nWaveGlow paper simply says **12 coupling layers and 12 invertible 1x1 convolutions**, **output 2 of the channels after every 4 coupling layers**. \n\nBut it is more convienient to define the casacade of one coulpling layer and one 1x1 conv layer as one **flow step**; then **4 flow steps** makes one WaveGlow block. The early outputs will be extracted from the output of the 1st and 2nd WaveGlow blocks.\n\n('Flow step' may not be the best name here. I will use it here)\n\n**During training**\n\n* input feature is in shape (B, N, D), i.e., (Batch, Num_of_frame, Dimension)\n* target waveorm is in shape (B, T, 1), i.e., (Batch, Time length, 1)\n* maximize the likelihood $\\sum_{b=1}^{3} \\log \\mathcal{N}(\\boldsymbol{z}_{b}; \\boldsymbol{0}, \\boldsymbol{I}) + \\log(\\det|Jac|_b)$\n\n$z_1, z_2$ are referred to as 'early output' -- latent z extracted from the 1st and 2nd WaveGlow block. This is also called multi-scale in Glow (Kingma, D. P. & Dhariwal, P. Glow: Generative Flow with Invertible 1x1 Convolutions. arXiv Prepr. arXiv1807.03039 (2018)).\n\n```sh\n.\n =============================================== \n | WaveGlow Core | \n | |---------------------------------------> log_detJac1\n | |---------------------------------------> z1 -> N(0, I) (B, T/8, 2)\n | | | \n | | |------------------------> log_detJac2\n | | |------------------------> z2 -> N(0, I) (B, T/8, 2)\n | | | |\n --------- (B, T/8, 8)| ----------- ----------- ----------- ---> log_detJac3 \nWaveform -->|squeeze| ------------> |WGBlock 1| -> |WGBlock 2| -> |WGBlock 3| ---> z3 -> N(0, I) (B, T/8, 4)\n(B, T, 1) --------- | ----------- ----------- ----------- |\n | ^ ^ ^ |\n ========|==============|==============|========\n --------- ---------------|--------------- \n |squeeze| -----------------------------------\n --------- (B, T/8, 8D)\n ^ \n | up-sampled features (B, T, D)\n -----------\ninput_feat->|condition|\n(B, N, D) -----------\n``` \n\n**During generation**\n\n* input feature is in shape (Batch, Num_of_frame, Dimension)\n* target waveorm is in shape (Batch, Time length, 1)\n* Draw random noise $\\{\\boldsymbol{z}_1, \\boldsymbol{z}_2, \\boldsymbol{z}_3\\}$ and do reverse transformation \n\n```sh\n.\n =============================================== \n | WaveGlow Core | \n | |\n | |--------------------------------------- z1 <- N(0, I) (B, T/8, 2)\n | | | \n | | |\n | | |------------------------ z2 <- N(0, I) (B, T/8, 2)\n | v v |\n --------- (B, T/8, 8)| ----------- ----------- ----------- |\nWaveform <--|de-sque| <----------- |WGBlock 1| <- |WGBlock 2| <- |WGBlock 3| <-- z3 <- N(0, I) (B, T/8, 4)\n(B, T, 1) --------- | ----------- ----------- ----------- |\n | ^ ^ ^ |\n ========|==============|==============|========\n --------- ---------------|--------------- \n |squeeze| -----------------------------------\n --------- (B, T/8, 8D)\n ^ \n | up-sampled features (B, T, D)\n -----------\ninput_feat->|condition|\n(B, N, D) -----------\n\n```\n\nDetails of each module or block are illustrated in the following sections\n\n### 1.1 Preparation\n\n\n```python\n# load packages \nfrom __future__ import absolute_import\nfrom __future__ import print_function\nimport os\nimport sys\nimport numpy as np\nimport torch\nimport torch.nn as torch_nn\nimport torch.nn.functional as torch_nn_func\n\n# basic nn blocks\nimport sandbox.block_nn as nii_nn\nimport sandbox.util_dsp as nii_dsp\nimport sandbox.block_glow as nii_glow\nimport core_scripts.data_io.conf as nii_io_conf\n\n# misc functions for this demonstration book\n\nfrom plot_tools import plot_API\nfrom plot_tools import plot_lib\nimport tool_lib\nimport plot_lib as plot_lib_legacy\nimport matplotlib\nimport matplotlib.pyplot as plt\nmatplotlib.rcParams['figure.figsize'] = (10, 5)\n```\n\n### 1.1 Condition module\n\nIt transforms and up-samples the input acoustic features (e.g., Mel-spec)\n```sh\n.\n ===================================\n | condition module |\n input_feat | ---------------------------- | up-sampled features \n(Batch, frame_num, dimension) -> | | transposed convolution 1d| | -> (Batch, waveform_length, dimension)\n | ---------------------------- |\n ===================================\n```\n\n\nSimilar to the condition modules in WaveNet and many other vocoders, the waveform length = frame_num * up-samplg rate. The up-sampling rate is decided by the waveform sampling rate and the frame-shift when extracting the input features. For example, 5ms frame-shift on 16kHz waveform -> 16 * 5 = 80. Each frame must be up-sampled to a factor of 80.\n\nA condition module can be in numerous ways. Here we try transposed convolution (https://pytorch.org/docs/stable/generated/torch.nn.ConvTranspose1d.html)\n\n\n```python\nclass upsampleByTransConv(torch_nn.Module):\n \"\"\"upsampleByTransConv\n Upsampling layer using transposed convolution\n \"\"\"\n def __init__(self, feat_dim, upsample_rate, window_ratio=5):\n \"\"\"upsampleByTransConv(feat_dim, upsample_rate, window_ratio=5)\n \n Args\n ----\n feat_dim: int, input feature should be (batch, length, feat_dim)\n upsample_rate, int, output feature will be \n (batch, length*upsample_rate, feat_dim)\n window_ratio: int, default 5, window length of transconv will be \n upsample_rate * window_ratio\n \"\"\"\n super(upsampleByTransConv, self).__init__()\n window_l = upsample_rate * window_ratio\n self.m_layer = torch_nn.ConvTranspose1d(\n feat_dim, feat_dim, window_l, stride=upsample_rate)\n self.m_uprate = upsample_rate\n return\n \n def forward(self, x):\n \"\"\" y = upsampleByTransConv(x)\n \n input\n -----\n x: tensor, (batch, length, feat_dim)\n \n output\n ------\n y: tensor, (batch, length*upsample_rate, feat_dim)\n \"\"\"\n l = x.shape[1] * self.m_uprate\n y = self.m_layer(x.permute(0, 2, 1))[:, :, 0:l]\n return y.permute(0, 2, 1).contiguous()\n```\n\n\n```python\n# Example\n\nbatch = 2\nframe_num = 10\ndimension = 5\nupsample_rate = 80\n\nm_cond = upsampleByTransConv(dimension, upsample_rate)\n```\n\n\n```python\ninput_data = torch.randn([batch, frame_num, dimension])\nwith torch.no_grad():\n output_data = m_cond(input_data)\n \n\nprint(\"Input feature batch {:d}, frame {:d}, dim {:d} \".format(*input_data.shape))\nprint(\"Output feature batch {:d}, frame {:d}, dim {:d} \".format(*output_data.shape))\n\nplot_API.plot_API([input_data[0, :, 0].numpy(), output_data[0, :, 0].numpy()], plot_lib.plot_signal, 'v',\n {'sub': [{'title': \"Input feature in 1st dimension of 1st data in the data\", 'xlabel': 'Frame index'},\n {'title': \"Up-sampled feature\", 'xlabel': \"Time index\"}],\n 'hspace': 0.5})\n\n# The conv layer is randomly initialized, the up-sampled feature may be quite random.\n```\n\n### 1.2 Squeeze\n\nSqueeze module changes the shape of the input feature.\n\nPay attention to the following points:\n* How to align the elements in the last dimension when the last dimension is >1?\n* Reverse operation should be implemented\n* WaveGlow squeeze by a factor of 8\n\n```sh\n.\n --------- \n Waveform <-> |squeeze| <-> Squeezed waveform \n (B, T, 1) --------- (B, T/8, 8)\n \n --------- \n Feature <-> |squeeze| <-> Squeezed Feature\n (B, T, D) --------- (B, T/8, 8D)\n```\n\n\n```python\nclass SqueezeForWaveGlow(torch_nn.Module):\n \"\"\"SqueezeForWaveGlow\n Squeeze layer for WaveGlow\n \"\"\"\n def __init__(self, mode = 1, mode_1_para=8):\n \"\"\"SqueezeForGlow(mode=1)\n Args\n ----\n mode: int, mode of this squeeze layer\n mode_1_para: int, factor of squeeze (default 8)\n \n mode == 1: original squeeze method by squeezing 8 points\n \n \"\"\"\n super(SqueezeForWaveGlow, self).__init__()\n self.m_mode = mode\n self.m_mode_1_para = mode_1_para\n return\n \n def get_expected_squeeze_length(self, orig_length):\n # return expected length after squeezing\n if self.m_mode == 1:\n return orig_length//self.m_mode_1_para\n \n def get_squeeze_factor(self):\n # return the configuration for squeezing\n if self.m_mode == 1:\n return self.m_mode_1_para\n \n def forward(self, x):\n \"\"\"SqueezeForWaveGlow(x)\n \n input\n -----\n x: tensor, (batch, length, feat_dim)\n \n output\n ------\n y: tensor, (batch, length // squeeze, feat_dim * squeeze)\n \"\"\"\n if self.m_mode == 1:\n # squeeze, the 8 points should be the last dimension\n squeeze_len = x.shape[1] // self.m_mode_1_para\n # trim length first\n trim_len = squeeze_len * self.m_mode_1_para\n x_tmp = x[:, 0:trim_len, :]\n \n # (batch, time//squeeze_size, squeeze_size, dim)\n x_tmp = x_tmp.view(x_tmp.shape[0], squeeze_len, \n self.m_mode_1_para, -1)\n \n # (batch, time//squeeze_size, dim, squeeze_size)\n x_tmp = x_tmp.permute(0, 1, 3, 2).contiguous()\n \n # (batch, time//squeeze_size, dim * squeeze_size)\n return x_tmp.view(x_tmp.shape[0], squeeze_len, -1)\n else:\n print(\"SqueezeForWaveGlow not implemented\")\n return x_squeezed\n\n def reverse(self, x_squeezed):\n if self.m_mode == 1:\n # (batch, time//squeeze_size, dim * squeeze_size)\n batch, squeeze_len, squeeze_dim = x_squeezed.shape\n \n # (batch, time//squeeze_size, dim, squeeze_size)\n x_tmp = x_squeezed.view(\n batch, squeeze_len, squeeze_dim // self.m_mode_1_para, \n self.m_mode_1_para)\n \n # (batch, time//squeeze_size, squeeze_size, dim)\n x_tmp = x_tmp.permute(0, 1, 3, 2).contiguous()\n \n # (batch, time, dim)\n x = x_tmp.view(batch, squeeze_len * self.m_mode_1_para, -1)\n else:\n print(\"SqueezeForWaveGlow not implemented\")\n return x\n```\n\nLet's use example to show it.\n\nFor explanation, we set the squeeze factor to 3.\n\n\n```python\nm_squeeze = SqueezeForWaveGlow(mode_1_para=3)\n```\n\n\n```python\n# First example\n# last dimension size is 1, like waveform, (B, T, 1)\n\n# create input (B=1, T=6, 1)\nlength = 6\ninput_data = torch.tensor([np.arange(length)+1]).T\ninput_data = input_data.unsqueeze(0)\n\n# squeeze\nwith torch.no_grad():\n squeezed_data = m_squeeze(input_data)\n \nplot_lib_legacy.plot_tensor(input_data, title=\"Input data batch {:d}, length {:d}, dim {:d} \".format(*input_data.shape), color_on_value=True)\nplot_lib_legacy.plot_tensor(squeezed_data, title=\"Squeezed data batch {:d}, length {:d}, dim {:d} \".format(*squeezed_data.shape), color_on_value=True)\n```\n\nNote that, **the heigth of the matrix in the figure corresponds to the time axis**\n\n(For a tensor of shape (B, T, D), the height of the matrix in the figure corresponds to length T, the width of the matrix in the figure corresponds to dimension D.)\n\n\n\n```python\n# Second example, \n# data has shape (B=2, T=6, 2)\n\n# create a data of shape (batch=2, length=6, dimension=2)\nlength = 6\ninput_data = torch.tensor([np.arange(length)+1, np.arange(length)*-1-1]).T\ninput_data = torch.stack([input_data, input_data], dim=0)\n\n# squeeze\nwith torch.no_grad():\n squeezed_data = m_squeeze(input_data)\n \nplot_lib_legacy.plot_tensor(input_data, title=\"Input data batch {:d}, length {:d}, dim {:d} \".format(*input_data.shape), color_on_value=True)\nplot_lib_legacy.plot_tensor(squeezed_data, title=\"Squeezed data batch {:d}, length {:d}, dim {:d} \".format(*squeezed_data.shape), color_on_value=True)\n```\n\nIn the above example, the input data has shape (2, 6, 2). The squeezed data has shape (2, 2, 6)\n\ninput_data[0, 0:3, 0]=[1.0, 2.0, 3.0] and input_data[0, 0:3, 1] =[-1.0, -2.0, -3.0] are squeezed into squeezed_data[0, 0, 0:6]\n\n\n```python\nprint(input_data[0, 0:3, 0])\nprint(input_data[0, 0:3, 1])\nprint(squeezed_data[0, 0, 0:6])\n```\n\n tensor([1, 2, 3])\n tensor([-1, -2, -3])\n tensor([ 1, 2, 3, -1, -2, -3])\n\n\n**How to align the elements in the last dimension when the last dimension is >1?**\n\nAs the example shows, elements adjacent in time are adjace in the squeezed tensor. Therefore, squeezed_data[0, 0, 0:6] is [1, 2, 3, -1, -2, -3], **NOT** [1, -1, 2, -2, 3, -3]\n\n**Reverse (de-squeeze)**\n\nIt is straightforward to de-squeeze\n\n\n```python\n# de-squeeze\nwith torch.no_grad():\n de_squeezed_data = m_squeeze.reverse(squeezed_data)\n\nplot_lib_legacy.plot_tensor(de_squeezed_data, title=\"Recovered data data batch {:d}, length {:d}, dim {:d} \".format(*de_squeezed_data.shape), color_on_value=True)\n```\n\n### 1.3 First glimpse on WaveGlow core part\n\nThe WaveGlow core module contains **3 WaveGlow blocks (WGBlocks)**, each WGBlock contains **4 WaveGlow flow steps**.\n\n```sh\n. =============================================== \n | WaveGlow Core | \n | |---------------------------------------> log_detJac1\n | |---------------------------------------> z1 -> N(0, I) (B, T/8, 2)\n | | | \n | | |------------------------> log_detJac2\n | | |------------------------> z2 -> N(0, I) (B, T/8, 2)\n | | | 3 |\n | ----------- 1 ----------- 4 ----------- ---> log_detJac3 \nSqueezed ---> |WGBlock 1| -> |WGBlock 2| -> |WGBlock 3| ---> z3 -> N(0, I) (B, T/8, 4)\n wave | ----------- ----------- ----------- |\n(B, T/8, 8)| ^ ^ 2 ^ |\n ========|==============|==============|========\n ---------------|--------------- \n up-sampled and squeezed feature\n (B, T/8, 8D)\n```\n\n**WaveGlow block (WGBlock)**\n```sh\n. |-----------> log_detJac\n 3 | |---> early output z \n | | (B, T/8, d)\n =====================================================|========== \n | WaveGlow block | | | \n | -------------> + -----------> + ------------ + | | \n | | ^ ^ ^ | | \n | | | | | | |\n | ----------- ----------- ----------- ----------- | | 4 \n 1 --> |Flowstep1| -> |Flowstep2| -> |Flowstep3| -> |Flowstep4| ---|-> input to next block x\n(B, T/8, P)| ----------- ----------- ----------- ----------- | (B, T/8, P-d)\n | ^ ^ ^ ^ |\n ========|==============|==============|==============|==========\n | | | |\n ---------------|------------------------------\n 2 (B, T/8, 8D)\n```\nwhere input is\n1. output of previous block or squeezed waveform (if this is the 1st block), shape (B, T/8, P)\n2. up-sampled and squeezed condition features, shape (B, T/8, 8D)\n\noutput is:\n\n3. early output z, (B, T/8, d) and log_det|Jac| (scalar)\n4. input to the next WaveGlow block (B, T/8, P-d)\n\nThe input and output tensor shape for the three WaveGlow blocks are\n\n| | 1 input | 2 condition | 3 early output z | 4 output latent x |\n|-----------------|---|---|---|---|\n| WaveGlow Block1 | (B, T/8, 8) | (B, T/8, 8D) | (B, T/8, 2) | (B, T/8, 6) |\n| WaveGlow Block2 | (B, T/8, 6) | (B, T/8, 8D) | (B, T/8, 2) | (B, T/8, 4) |\n| WaveGlow Block3 | (B, T/8, 4) | (B, T/8, 8D) | (B, T/8, 4) | -|\n\n\nDetails of flow step, WaveGlow block will be defined in the following sectioins\n\n### 1.4. One flowstep\n\n**An Flowstep block** looks like this:\n\nIt contains an invertiable 1x1 conv and an affine transformation layer. \n\nThe parameters for affine transformation are produced by a WaveNet block.\n\n```sh\n. |---------------> log_detJac\n ==============================================================================================|============== \n | Flow step of WaveGlow | |\n | r (B, T/8, P/2) ---------------------- |\n | |----------------------------------------------------------------->| Affine transform | |\n | | | ra + b | | \n | | ============================================================= | / a (B, T/8, P/2) | |\n | ------------ | |WaveNet blocks|--------------> + --------------> + --|FC|--|--->| | |\n | |invertible| / | | | | | | \\ b (B, T/8, P/2) | |\n input --> | 1x1 conv | \\ | ---- --------------- --------------- --------------- | ---------------------- |\n(B, T/8, P)| ------------ |---|->|FC|->|WaveNetBlock1|->|WaveNetBlock2|...|WaveNetBlock8| | | |\n | | | ---- --------------- --------------- --------------- | | |\n | | | ^ ^ ^ | | ra + b | \n | | ===============|================|=================|========== v |\n | | | | | q ----------- |\n | |------------------|----------------|-----------------|---------------->| Concate | -------|--> output\n | q (B, T/8, P/2) | | | ----------- | (B, T/8, P)\n ====================================|================|=================|=====================================\n | | |\n ------------------------------------\n 2 \n (B, T/8, 8D)\n```\n\n\n\n\n#### 1.4.1 Invertible 1x1 conv\n\nThe name \"invertible 1x1 conv\" may be hard to understand. But it contains two things:\n* 1x1 conv: for 1D data, I prefer to naming it as a \"fully-connected\" layer -- indeed it is implemented as `torch.matmul(data, weight)` not `torch.conv1d`;\n* invertible: the transformation matrix `weight` is a square matrix, and it should be invertible. It should be inverible even after updating its parameter through model training\n\nIn all, \"invertible 1x1 conv\" means $y=xA$, and its reverse transformation is $x=yA^{-1}$. Both $x$ and $y$ has shape $(B, T/8, P)$, while $A$ is a matrix of size $(P, P)$.\n\n\n##### **How can it shuffle dimension? A toy example**\n\n\n```python\n# create a data of shape (batch=1, length=3, dimension=4)\nlength = 3\ninput_data = torch.tensor([np.arange(length)+1, np.arange(length)*-1-1, np.arange(length)+10, np.arange(length)*-1-10], dtype=torch.float32).T\ninput_data = input_data.unsqueeze(0)\n\n# create a transformation matrix\nweight_mat = torch.tensor([[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [1, 0, 0, 0]], dtype=input_data.dtype)\n\n# transform\nwith torch.no_grad():\n output_data = torch.matmul(input_data, weight_mat)\n \n\nfrom plot_tools import table_API\nprint(\"Input data x:\")\ntable_API.print_table(input_data[0].numpy(), None, None, print_latex_table=False, print_format=\"3.1f\")\nprint(\"Transformation matrix A:\")\ntable_API.print_table(weight_mat.numpy(), None, None, print_latex_table=False, print_format=\"3.1f\")\nprint(\"Transformed data y=xA:\")\ntable_API.print_table(output_data[0].numpy(), None, None, print_latex_table=False, print_format=\"3.1f\")\n\n\n# inverse transformation\nwith torch.no_grad():\n weight_mat_inv = torch.inverse(weight_mat)\n reversed_data = torch.matmul(output_data, weight_mat_inv)\n \nprint(\"Inverted transformation matrix A^-1:\")\ntable_API.print_table(weight_mat_inv.numpy(), None, None, print_latex_table=False, print_format=\"3.1f\")\nprint(\"Revere transformed data x=yA^-1:\")\ntable_API.print_table(reversed_data[0].numpy(), None, None, print_latex_table=False, print_format=\"3.1f\")\n```\n\n Input data x:\n \n 1.0 -1.0 10.0 -10.0\n 2.0 -2.0 11.0 -11.0\n 3.0 -3.0 12.0 -12.0\n \n Transformation matrix A:\n \n 0.0 1.0 0.0 0.0\n 0.0 0.0 1.0 0.0\n 0.0 0.0 0.0 1.0\n 1.0 0.0 0.0 0.0\n \n Transformed data y=xA:\n \n -10.0 1.0 -1.0 10.0 \n -11.0 2.0 -2.0 11.0 \n -12.0 3.0 -3.0 12.0 \n \n Inverted transformation matrix A^-1:\n \n 0.0 0.0 -0.0 1.0 \n 1.0 0.0 -0.0 0.0 \n 0.0 1.0 -0.0 0.0 \n 0.0 0.0 1.0 0.0 \n \n Revere transformed data x=yA^-1:\n \n 1.0 -1.0 10.0 -10.0\n 2.0 -2.0 11.0 -11.0\n 3.0 -3.0 12.0 -12.0\n \n\n\nNote that the input example data has shape (1, 3, 4). \n\nBy using transformation matrix, we can see how the rows (i.e., last dimension of input data) is shuffled.\n\nThe transformation matrix here is a simple Permutation matrix https://en.wikipedia.org/wiki/Permutation_matrix. Such a permutation matrix is of course invertible.\n\n\n##### **How can it shuffle dimension? A practical example in Glow**\n\nWe can use permutation matrix, generalized permutation matrix (https://en.wikipedia.org/wiki/Generalized_permutation_matrix), rotation matrix (https://en.wikipedia.org/wiki/Rotation_matrix) or any invertible matrix.\n\n\nIn neural network, the randomly initialized matrix may not be invertible. We have to manually get an invertible matrix from the randomly initialized one.\nAlso, we need to make sure that the updated matrix after model training is also invertible.\n\nFor all these requirements, I like the idea in the original Glow paper (Eq.(10) in Kingma, D. P. & Dhariwal, P. Glow: Generative Flow with Invertible 1x1 Convolutions. in Proc. NIPS (2018).)\n\n$ \\boldsymbol{A} = \\boldsymbol{P}\\boldsymbol{L}(\\boldsymbol{U} + \\text{diag}(\\boldsymbol{s})) \n= \\boldsymbol{P}\\boldsymbol{L}\\Big(\\boldsymbol{U} + \\text{sign}(\\text{diag}(\\boldsymbol{s}))\\exp(\\log|\\text{diag}(\\boldsymbol{s})|)\\Big)$\n\nwhere \"P is a permutation matrix, L is a lower triangular matrix with ones on the diagonal, U is an upper triangular matrix with zeros on the diagonal, and s is a vector. ... In this parameterization, we initialize the parameters by first sampling a random rotation matrix W, then computing the corresponding value of P (which remains fixed) and the corresponding initial values of L and U and s (which are optimized).\" (Kingma 2018)\n\nThe advatanges:\n1. Easy to invert and will be intertible\n2. Easy to compute the determinant of Jacobian matrix\n\nLet's show how it works:\n\n\n```python\n# How to compose matrix A\n\n# I will use scipy and numpy\nimport scipy.linalg\n\nfeat_dim = 4\n\n# step1. create an initial matrix that is invertible (a unitary matrix) \n# https://en.wikipedia.org/wiki/Unitary_matrix\n# https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.qr.html\nseed_mat = np.random.randn(feat_dim, feat_dim)\n# use QR decomposition to get the rotation_mat, which is a unitary matrix\nrotation_mat, _ = scipy.linalg.qr(seed_mat)\n\n# step2. decompose it into \n# https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.lu.html\npermute_mat, lower_mat, upper_mat = scipy.linalg.lu(rotation_mat)\n\n# step3. deal with the diagonal line of lower and upper mat\nu_mask = np.triu(np.ones_like(seed_mat), k=1)\nd_mask = u_mask.T\neye_mat = np.eye(feat_dim)\n\n# get the diag(s) from upper_mat\ntmp_diag_line = upper_mat.diagonal().copy()\n# decompose tmp_diag_line into sign(s) * exp(log(|s|)), this makes it easy to compute Determinant Jacobian\nsign_tmp_diag_line = np.sign(tmp_diag_line)\nlogabs_tmp_diag_line = np.log(np.abs(tmp_diag_line))\n\n# upper triangle mat \nupper_mat_new = upper_mat * u_mask\n\n\n\nprint(\"Randomly initialized invertible matrix\\n A = PL(U+diag(s)):\")\ntable_API.print_table(rotation_mat, None, None, print_latex_table=False, print_format=\"3.3f\")\n\nprint(\"P:\")\ntable_API.print_table(permute_mat, None, None, print_latex_table=False, print_format=\"3.3f\")\n\nprint(\"L:\")\ntable_API.print_table(lower_mat, None, None, print_latex_table=False, print_format=\"3.3f\")\n\nprint(\"U:\")\ntable_API.print_table(upper_mat_new, None, None, print_latex_table=False, print_format=\"3.3f\")\n\nprint(\"diag(s):\")\ntable_API.print_table(np.diag(tmp_diag_line), None, None, print_latex_table=False, print_format=\"3.3f\")\n\n# You can test whether the decomposing is correct\nA_computed = np.dot(np.dot(permute_mat, lower_mat), upper_mat_new + np.diag(tmp_diag_line) )\nprint(\"Verify the value of PL(U+diag(s)):\")\ntable_API.print_table(A_computed, None, None, print_latex_table=False, print_format=\"3.3f\")\n\n\n\n\nprint(\"\\n\\nIn Glow, the lower matrix L is manully modified so that its diagonal line is 1.\")\nprint(\"This is merely a practical choice -- the determinant of the Jacobian matrix will only be determined by diag(s)\")\n\n# change lower triangle mat\nlower_mat_new = lower_mat * d_mask + eye_mat\nprint(\"Modified L:\")\ntable_API.print_table(lower_mat_new, None, None, print_latex_table=False, print_format=\"3.3f\")\n```\n\n Randomly initialized invertible matrix\n A = PL(U+diag(s)):\n \n -0.650 0.273 0.570 -0.422\n 0.261 -0.394 0.784 0.402 \n -0.521 0.249 -0.116 0.808 \n 0.488 0.841 0.217 0.086 \n \n P:\n \n 1.000 0.000 0.000 0.000\n 0.000 0.000 1.000 0.000\n 0.000 0.000 0.000 1.000\n 0.000 1.000 0.000 0.000\n \n L:\n \n 1.000 0.000 0.000 0.000 \n -0.750 1.000 0.000 0.000 \n -0.402 -0.272 1.000 0.000 \n 0.802 0.029 -0.498 1.000 \n \n U:\n \n -0.000 0.273 0.570 -0.422\n 0.000 0.000 0.645 -0.230\n 0.000 0.000 0.000 0.170 \n 0.000 0.000 0.000 0.000 \n \n diag(s):\n \n -0.650 0.000 0.000 0.000 \n 0.000 1.046 0.000 0.000 \n 0.000 0.000 1.188 0.000 \n 0.000 0.000 0.000 1.238 \n \n Verify the value of PL(U+diag(s)):\n \n -0.650 0.273 0.570 -0.422\n 0.261 -0.394 0.784 0.402 \n -0.521 0.249 -0.116 0.808 \n 0.488 0.841 0.217 0.086 \n \n \n \n In Glow, the lower matrix L is manully modified so that its diagonal line is 1.\n This is merely a practical choice -- the determinant of the Jacobian matrix will only be determined by diag(s)\n Modified L:\n \n 1.000 0.000 0.000 0.000 \n -0.750 1.000 0.000 0.000 \n -0.402 -0.272 1.000 0.000 \n 0.802 0.029 -0.498 1.000 \n \n\n\n\n```python\n# Let's try to \"shuffle\" the input data\n# We will use the modified L\n\n# step1. always remember to compose the transformation matrix by \n# P L (U + diag( sign(s)exp(log|s|)))\nA_computed = np.dot(np.dot(permute_mat, lower_mat_new), upper_mat_new + np.diag(sign_tmp_diag_line * np.exp(logabs_tmp_diag_line)))\nweight_mat = torch.tensor(A_computed, dtype=torch.float32)\nprint(\"Transformation matrix A:\")\ntable_API.print_table(weight_mat.numpy(), None, None, print_latex_table=False, print_format=\"3.3f\")\n\n# create a data of shape (batch=1, length=3, dimension=4)\nlength = 3\ninput_data = torch.tensor([np.arange(length)+1, np.arange(length)*-1-1, np.arange(length)+10, np.arange(length)*-1-10], dtype=torch.float32).T\ninput_data = input_data.unsqueeze(0)\nfeat_dim = input_data.shape[-1]\n\n# transform\nwith torch.no_grad():\n output_data = torch.matmul(input_data, weight_mat)\n\nfrom plot_tools import table_API\nprint(\"Input data x:\")\ntable_API.print_table(input_data[0].numpy(), None, None, print_latex_table=False, print_format=\"3.1f\")\nprint(\"Transformed data y=xA:\")\ntable_API.print_table(output_data[0].numpy(), None, None, print_latex_table=False, print_format=\"3.3f\")\n\n\n# inverse transformation\nwith torch.no_grad():\n weight_mat_inv = torch.inverse(weight_mat)\n reversed_data = torch.matmul(output_data, weight_mat_inv)\n \nprint(\"Inverted transformation matrix A^-1:\")\ntable_API.print_table(weight_mat_inv.numpy(), None, None, print_latex_table=False, print_format=\"3.3f\")\nprint(\"Revere transformed data x=yA^-1:\")\ntable_API.print_table(reversed_data[0].numpy(), None, None, print_latex_table=False, print_format=\"3.1f\")\n```\n\n Transformation matrix A:\n \n -0.650 0.273 0.570 -0.422\n 0.261 -0.394 0.784 0.402 \n -0.521 0.249 -0.116 0.808 \n 0.488 0.841 0.217 0.086 \n \n Input data x:\n \n 1.0 -1.0 10.0 -10.0\n 2.0 -2.0 11.0 -11.0\n 3.0 -3.0 12.0 -12.0\n \n Transformed data y=xA:\n \n -11.000 -5.256 -3.534 6.393 \n -12.920 -5.181 -4.080 6.291 \n -14.840 -5.106 -4.626 6.189 \n \n Inverted transformation matrix A^-1:\n \n -0.650 0.261 -0.521 0.488 \n 0.273 -0.394 0.249 0.841 \n 0.570 0.784 -0.116 0.217 \n -0.422 0.402 0.808 0.086 \n \n Revere transformed data x=yA^-1:\n \n 1.0 -1.0 10.0 -10.0\n 2.0 -2.0 11.0 -11.0\n 3.0 -3.0 12.0 -12.0\n \n\n\nIt is hard to see how the dimensions are shuffled. But it is a way to mix information from different dimensions.\n\n##### **How to compute the Determinant of Jacobian**\n\nThis has been explained in Table 1 of Glow paper (Kingma 2018):\n\nFor $\\boldsymbol{y} = \\boldsymbol{x}\\boldsymbol{A}$ where $\\boldsymbol{y}, \\boldsymbol{x}\\in\\mathbb{R}^{B\\times T\\times D}$ and $\\boldsymbol{A} = \\boldsymbol{P}\\boldsymbol{L}(\\boldsymbol{U} + \\text{diag}(\\boldsymbol{s})) \n= \\boldsymbol{P}\\boldsymbol{L}\\Big(\\boldsymbol{U} + \\text{sign}(\\text{diag}(\\boldsymbol{s}))\\exp(\\log|\\text{diag}(\\boldsymbol{s})|)\\Big)$\n\nThe log Determinant of Jacobian is\n\n$B\\cdot{T}\\cdot\\text{sum}(\\log|\\text{diag}(\\boldsymbol{s})|)$\n\n\nThe $B$ and $T$ are there because the transformation is conducted for every time step and every data in the mini-batch. In other words, we are transforming $BT$ vectors simultaneously.\n\n(In practise, we can ignore the $B$ if we assign the same value $T\\cdot\\text{sum}(\\log|\\text{diag}(\\boldsymbol{s})|)$ to each data in the mini-batch and sum the values later. )\n\n\n```python\ndata_factor = np.prod(input_data.shape[1:-1])\nprint(\"logDetJac is: \", data_factor * np.sum(logabs_tmp_diag_line))\n```\n\n logDetJac is: 9.992007221626409e-16\n\n\n##### **Pytorch API**\n\nThe Pytorch API for Glow-style 1x1 invertible transformation is defined in `../sandbox/block_glow.py`.\n\nIt wrapps the explanations above into a single module\n\nIn Official WaveGlow implementation, the invertible 1x1 conv is in a different flavor from Glow:\n* It simply compute the initial invertible matrix W through QR decomposition (https://pytorch.org/docs/stable/generated/torch.qr.html)\n* It did NOT decompose W further\n\n\n```python\nclass Invertible1x1ConvWaveGlow(torch.nn.Module):\n def __init__(self, feat_dim, flag_detjac=False):\n \"\"\"\n Args\n ----\n feat_dim: int, dimension of the input feature, \n flag_detjac: bool, whether compute the Log DetJacobian in forward()\n \n input data should have shape (batch, length, feat_dim)\n \"\"\"\n super(Invertible1x1ConvWaveGlow, self).__init__()\n \n torch.manual_seed(100)\n \n with torch.no_grad():\n # QR decomposition\n W = torch.qr(torch.FloatTensor(feat_dim, feat_dim).normal_())[0]\n \n # Ensure determinant is 1.0 not -1.0\n if torch.det(W) < 0:\n W[:,0] = -1*W[:,0]\n \n # not necessary\n W = W.transpose(0, 1)\n \n self.weight = torch_nn.Parameter(W)\n self.weight_inv = torch_nn.Parameter(W.clone())\n self.weight_inv_flag = False\n self.flag_detjac = flag_detjac\n return\n \n def forward(self, y, factor):\n \"\"\"\n input\n -----\n y: tensor, (batch, length, dim)\n factor: int, the factor related to the mini-match size\n \"\"\"\n batch_size, length, feat_dim = y.size()\n\n # Forward computation\n log_det_W = length / factor * torch.logdet(self.weight)\n z = torch.matmul(y, self.weight)\n if self.flag_detjac:\n return z, log_det_W\n else:\n return z\n \n def reverse(self, x):\n \n self.weight_inv.data = torch.inverse(self.weight.data)\n self.weight_inv_flag = True\n return torch.matmul(x, self.weight_inv)\n```\n\nIn the above API method forward(), I added a scaling factor as argument.\n\nThe reason is that, although we compute ${T}\\cdot\\text{sum}(\\log|\\text{diag}(\\boldsymbol{s})|)$ for each mini-batch, \nThe $T$ may be quite large for speech data and its value may explode.\n\nSince at the end of the model training loop, we may divide the total likelihood by the number of data elements in the mini-batch, why not do the division inside each module in advance? For example, ${T}\\cdot\\text{sum}(\\log|\\text{diag}(\\boldsymbol{s})|) / {T}$ may be less likely to explode.\n\n\nWhen training the model, I set the factor to be `factor = np.prod([dim for dim in waveglow_input_data.shape])`\n\n\n```python\n# create a data of shape (batch=1, length=3, dimension=4)\nlength = 3\ninput_data = torch.tensor([np.arange(length)+1, np.arange(length)*-1-1, np.arange(length)+10, np.arange(length)*-1-10], dtype=torch.float32).T\ninput_data = input_data.unsqueeze(0)\nfeat_dim = input_data.shape[-1]\n\n\nm_inv1x1 = Invertible1x1ConvWaveGlow(feat_dim, flag_detjac=True)\n\nwith torch.no_grad():\n transformed_data, logDetJac = m_inv1x1(input_data, 1)\n recovered_data = m_inv1x1.reverse(transformed_data)\n \n\nprint(\"Input data x:\")\ntable_API.print_table(input_data[0].numpy(), None, None, print_latex_table=False, print_format=\"3.3f\")\nprint(\"Transformed data y=xA:\")\ntable_API.print_table(transformed_data[0].numpy(), None, None, print_latex_table=False, print_format=\"3.3f\")\nprint(\"Revere transformed data x=yA^-1:\")\ntable_API.print_table(reversed_data[0].numpy(), None, None, print_latex_table=False, print_format=\"3.3f\")\nprint(\"Log Determinant Jacobian:\\n\", logDetJac.item())\nprint(\"\\n\\n\")\n```\n\n Input data x:\n \n 1.000 -1.000 10.000 -10.000\n 2.000 -2.000 11.000 -11.000\n 3.000 -3.000 12.000 -12.000\n \n Transformed data y=xA:\n \n -8.050 10.385 4.613 -2.841\n -8.110 11.678 5.721 -3.888\n -8.171 12.971 6.829 -4.936\n \n Revere transformed data x=yA^-1:\n \n 1.000 -1.000 10.000 -10.000\n 2.000 -2.000 11.000 -11.000\n 3.000 -3.000 12.000 -12.000\n \n Log Determinant Jacobian:\n -6.705522537231445e-08\n \n \n \n\n\n\n#### 1.4.2 Bipartite affine transformation in WaveGlow\n\n$\\begin{align}\n[\\boldsymbol{y1}, \\boldsymbol{y2}] &= \\text{split}(\\boldsymbol{y}) \\\\\n[\\log\\boldsymbol{a}, \\boldsymbol{b}] &= \\text{NN}(\\boldsymbol{y1}) \\\\\n\\boldsymbol{x2} &= (\\boldsymbol{y2} + \\boldsymbol{b})\\odot\\exp(\\log\\boldsymbol{a}) \\\\\n\\boldsymbol{x1} &= \\boldsymbol{y1} \\\\\n\\boldsymbol{x} &= [\\boldsymbol{x1}, \\boldsymbol{x2}]\n\\end{align}$\n\nNotes on affine transformation in WaveGlow:\n* It is bipartite: input $\\boldsymbol{y}$ of shape (B, T, P) is decomposed into $\\boldsymbol{y1}$ (B, T, P/2) and $\\boldsymbol{y2}$ (B, T, P/2)\n* Affine transformation parameters are computed by $\\text{NN}()$ with 8 WaveNet blocks\n\n\n\n##### **Splitting the tensor**\n\nWe use `torch.chunk` https://pytorch.org/docs/stable/generated/torch.chunk.html to split a tensor (B, T, P) into a tensor (B, T, P/2) and another tensor (B, T, P/2)\n\n\n```python\n# (B=1, T=2, P=4)\ndata = torch.randn([1, 2, 4])\n# split along the last dimension\ndata1, data2 = data.chunk(2, -1)\nprint(data)\nprint(data1)\nprint(data2)\n```\n\n tensor([[[ 1.7482, -0.2759, -0.9755, 0.4790],\n [-2.3652, -0.8047, 0.6587, -0.2586]]])\n tensor([[[ 1.7482, -0.2759],\n [-2.3652, -0.8047]]])\n tensor([[[-0.9755, 0.4790],\n [ 0.6587, -0.2586]]])\n\n\n##### **WaveNet block for WaveGlow**\n\nWaveNet block is explained in s3_demonstration_wavenet. For convenience, a special module is designed to wrap the fully-connected layers and the 8 WaveNet blocks\n\nNotice that the last FC layer in the WaveNet block is initialized with weight 0 and bias 0. This helps the model training.\n```python\ntmp.weight.data.zero_()\ntmp.bias.data.zero_()\n```\nIn this notebook, we comment out the two lines for demonstration\n\n\n```python\nclass WaveNetModuleForNonAR(torch_nn.Module):\n \"\"\"WaveNetModuleWaveGlow\n Casecade of multiple WaveNet blocks:\n x -> ExpandDim -> conv1 -> gated -> res -> conv1 -> gated -> res ...\n ^ |\n | v\n cond skip\n output = sum(skip_channels)\n \"\"\"\n def __init__(self, input_dim, cond_dim, out_dim, n_blocks, \n gate_dim, res_ch, skip_ch, kernel_size=3):\n super(WaveNetModuleForNonAR, self).__init__()\n \n self.m_block_num = n_blocks\n self.m_res_ch_dim = res_ch\n self.m_skip_ch_dim = skip_ch\n self.m_gate_dim = gate_dim\n self.m_kernel_size = kernel_size\n self.m_n_blocks = n_blocks\n if self.m_gate_dim % 2 != 0:\n self.m_gate_dim = self.m_gate_dim // 2 * 2\n \n # input dimension expanding\n tmp = torch_nn.Conv1d(input_dim, res_ch, 1)\n self.l_expand = torch_nn.utils.weight_norm(tmp, name='weight')\n \n # end dimension compressing\n tmp = torch_nn.Conv1d(skip_ch, out_dim, 1)\n \n # Here we comment out these two lines\n #tmp.weight.data.zero_()\n #tmp.bias.data.zero_()\n \n self.l_compress = tmp\n \n # dilated convolution and residual-skip-channel transformation\n self.l_conv1 = []\n self.l_resskip = []\n for idx in range(n_blocks):\n dilation = 2 ** idx\n padding = int((kernel_size * dilation - dilation)/2)\n conv1 = torch_nn.Conv1d(\n res_ch, gate_dim, self.m_kernel_size, \n dilation = dilation, padding=padding)\n conv1 = torch_nn.utils.weight_norm(conv1, name='weight')\n self.l_conv1.append(conv1)\n \n if idx < n_blocks - 1:\n outdim = self.m_res_ch_dim + self.m_skip_ch_dim\n else:\n outdim = self.m_skip_ch_dim\n resskip = torch_nn.Conv1d(self.m_gate_dim//2, outdim, 1)\n resskip = torch_nn.utils.weight_norm(resskip, name='weight')\n self.l_resskip.append(resskip) \n self.l_conv1 = torch_nn.ModuleList(self.l_conv1)\n self.l_resskip = torch_nn.ModuleList(self.l_resskip)\n \n # a single conditional feature transformation layer\n cond_layer = torch_nn.Conv1d(cond_dim, gate_dim * n_blocks, 1)\n cond_layer = torch_nn.utils.weight_norm(cond_layer, name='weight')\n self.l_cond = cond_layer\n return\n \n def forward(self, x, cond):\n \"\"\"\n \"\"\"\n \n # input feature expansion\n # change the format to (batch, dimension, length)\n x_expanded = self.l_expand(x.permute(0, 2, 1))\n # condition feature transformation\n cond_proc = self.l_cond(cond.permute(0, 2, 1))\n\n # skip-channel accumulation\n skip_ch_out = 0\n \n conv_input = x_expanded\n for idx, (l_conv1, l_resskip) in \\\n enumerate(zip(self.l_conv1, self.l_resskip)):\n \n tmp_dim = idx * self.m_gate_dim\n # condition feature of this layer\n cond_tmp = cond_proc[:, tmp_dim : tmp_dim + self.m_gate_dim, :]\n # conv transformed\n conv_tmp = l_conv1(conv_input)\n \n # gated activation\n gated_tmp = cond_tmp + conv_tmp\n t_part = torch.tanh(gated_tmp[:, :self.m_gate_dim//2, :])\n s_part = torch.sigmoid(gated_tmp[:, self.m_gate_dim//2:, :])\n gated_tmp = t_part * s_part\n \n # transformation into skip / residual channels\n resskip_tmp = l_resskip(gated_tmp)\n \n # reschannel \n if idx == self.m_n_blocks - 1:\n skip_ch_out = skip_ch_out + resskip_tmp\n else:\n conv_input = conv_input + resskip_tmp[:, 0:self.m_res_ch_dim, :]\n skip_ch_out = skip_ch_out + resskip_tmp[:, self.m_res_ch_dim:,:]\n output = self.l_compress(skip_ch_out)\n \n # permute back to (batch, length, dimension)\n return output.permute(0, 2, 1)\n\n```\n\n\n```python\n# input data y of shape (B=2, T=100, P=16)\ninput_dim = 32\ny = torch.randn([2, 100, input_dim])\n\n# up-sampled condition features of shape (B=2, T=100, P=8)\ncond_dim = 8\ncond_feat = torch.randn([2, 100, cond_dim])\n\n# we should be get two tensors\n# log a and b should be (B=2, T=100, P=16)\noutput_dim = input_dim // 2 * 2\n\n\n\n# 8 wavenet layers\nn_blocks = 8\n\n# free to choose the dimension for gated activation, residual channel and skip channel\nm_wavenetb = WaveNetModuleForNonAR(input_dim // 2, cond_dim, output_dim, n_blocks, \n gate_dim=16, res_ch=16, skip_ch=16)\n\nwith torch.no_grad():\n y1, y2 = y.chunk(2, -1)\n loga, b = m_wavenetb(y1, cond_feat).chunk(2, -1)\n\n\nprint(\"Input feature y1 batch {:d}, length {:d}, dim {:d} \".format(*y1.shape))\n\nprint(\"Affine paramter log a batch {:d}, length {:d}, dim {:d} \".format(*loga.shape))\nprint(\"Affine paramter b batch {:d}, length {:d}, dim {:d} \".format(*b.shape))\n```\n\n Input feature y1 batch 2, length 100, dim 16 \n Affine paramter log a batch 2, length 100, dim 16 \n Affine paramter b batch 2, length 100, dim 16 \n\n\n##### **Affine transformation**\n\nGiven the parameter from WaveNet block, it is straightforward to do the transformation. \n\n\n```python\n# forward (do the WaveNet block again)\n\nwith torch.no_grad():\n y1, y2 = y.chunk(2, -1)\n loga, b = m_wavenetb(y1, cond_feat).chunk(2, -1)\n \n x2 = (y2 + b) * torch.exp(loga)\n logdetjac = torch.sum(loga)\n x = torch.cat([y1, x2], dim=-1)\n\n # reverse\n x1, x2 = x.chunk(2, -1)\n loga, b = m_wavenetb(x1, cond_feat).chunk(2, -1)\n\n y2 = x2 / torch.exp(loga) - b\n y_reverse = torch.cat([x1, y2], dim=-1)\n\n# The difference should be small\nprint(torch.std(y - y_reverse))\n```\n\n tensor(2.9801e-08)\n\n\n#### 1.4.3 Wrap up for one flow step\n\nBased on the above explanation, we can wrap the flow step into more module.\n\nBefore that we define affine transformation in a module\n\n\n```python\nclass AffineCouplingWaveGlow(torch_nn.Module):\n \"\"\"AffineCouplingWaveGlow\n \n AffineCoupling block in WaveGlow\n \n Example:\n m_tmp = AffineCouplingWaveGlow(10, 10, 8, 512, 3, True, True)\n data1 = torch.randn([2, 100, 10])\n cond = torch.randn([2, 100, 10])\n output, log_det = m_tmp(data1, cond)\n data1_re = m_tmp.reverse(output, cond)\n torch.std(data1 - data1_re)\n \"\"\"\n def __init__(self, in_dim, cond_dim, \n wn_num_conv1d, wn_dim_channel, wn_kernel_size, \n flag_affine=True, flag_detjac=False):\n \"\"\"AffineCouplingWaveGlow(in_dim, cond_dim, \n wn_num_conv1d, wn_dim_channel, wn_kernel_size, \n flag_affine=True, flag_detjac=False)\n \n Args:\n -----\n in_dim: int, dim of input audio data (batch, length, in_dim)\n cond_dim, int, dim of condition feature (batch, length, cond_dim)\n wn_num_conv1d: int, number of dilated conv WaveNet blocks\n wn_dim_channel: int, dime of the WaveNet residual & skip channels\n wn_kernel_size: int, kernel size of the dilated convolution layers\n flag_affine: bool, whether use affine or additive transformation?\n default True\n flag_detjac: bool, whether return the determinant of Jacobian,\n default False\n \n y -> split() -> y1, y2 -> concate([y1, (y2+bias) * scale])\n When flag_affine == True, y1 -> H() -> scale, bias\n When flag_affine == False, y1 -> H() -> bias, scale=1 \n Here, H() is WaveNet blocks (dilated conv + gated activation)\n \"\"\"\n super(AffineCouplingWaveGlow, self).__init__()\n \n self.flag_affine = flag_affine\n self.flag_detjac = flag_detjac\n \n if in_dim % 2 > 0:\n print(\"AffineCoulingGlow(feat_dim), feat_dim is an odd number?!\")\n sys.exit(1)\n \n if self.flag_affine:\n # scale and bias\n self.m_nn_outdim = in_dim // 2 * 2\n else:\n # only bias\n self.m_nn_outdim = in_dim // 2\n \n # WaveNet blocks (dilated conv, gated activation functions)\n self.m_wn = WaveNetModuleForNonAR(\n in_dim // 2, cond_dim, self.m_nn_outdim, wn_num_conv1d,\n wn_dim_channel * 2, wn_dim_channel, wn_dim_channel, \n wn_kernel_size \n )\n \n return\n \n def _detjac(self, log_scale, factor=1):\n # (batch, dim1, dim2, ..., feat_dim) -> (batch)\n # sum over dim1, ... feat_dim\n return nii_glow.sum_over_keep_batch(log_scale / factor)\n \n def _nn_trans(self, y1, cond):\n \"\"\"_nn_trans(self, y1, cond)\n \n input\n -----\n y1: tensor, input feature, (batch, lengh, input_dim//2)\n cond: tensor, condition feature, (batch, length, cond_dim)\n \n output\n ------\n scale: tensor, (batch, lengh, input_dim // 2)\n bias: tensor, (batch, lengh, input_dim // 2)\n log_scale: tensor, (batch, lengh, input_dim // 2)\n \n Affine transformaiton can be done by scale * feature + bias\n log_scale is used for det Jacobian computation\n \"\"\"\n y1_tmp = self.m_wn(y1, cond)\n \n if self.flag_affine:\n log_scale, bias = y1_tmp.chunk(2, -1)\n scale = torch.exp(log_scale)\n else:\n bias = y1_tmp\n scale = torch.ones_like(y1)\n log_scale = torch.zeros_like(y1)\n return scale, bias, log_scale\n \n def forward(self, y, cond, factor=1):\n \"\"\"AffineCouplingWaveGlow.forward(y, cond)\n \n input\n -----\n y: tensor, input feature, (batch, lengh, input_dim)\n cond: tensor, condition feature , (batch, lengh, cond_dim)\n \n output\n ------\n x: tensor, input feature, (batch, lengh, input_dim)\n detjac: tensor, det of jacobian, (batch,)\n \n y1, y2 = split(y)\n scale, bias = WN(y1)\n x2 = y2 * scale + bias or (y2 + bias) * scale\n return [y1, x2]\n \"\"\"\n # split\n y1, y2 = y.chunk(2, -1)\n scale, bias, log_scale = self._nn_trans(y1, cond)\n \n # transform\n x1 = y1\n x2 = (y2 + bias) * scale\n\n # concatenate\n x = torch.cat([x1, x2], dim=-1)\n if self.flag_detjac:\n return x, self._detjac(log_scale, factor)\n else:\n return x\n \n \n def reverse(self, x, cond):\n \"\"\"AffineCouplingWaveGlow.reverse(y, cond)\n \n input\n -----\n x: tensor, input feature, (batch, lengh, input_dim)\n cond: tensor, condition feature , (batch, lengh, cond_dim)\n \n output\n ------\n y: tensor, input feature, (batch, lengh, input_dim)\n \n x1, x2 = split(x)\n scale, bias = WN(x1)\n y2 = x2 / scale - bias\n return [x1, y2]\n \"\"\"\n # split\n x1, x2 = x.chunk(2, -1)\n # reverse transform\n y1 = x1\n scale, bias, log_scale = self._nn_trans(y1, cond)\n y2 = x2 / scale - bias\n return torch.cat([y1, y2], dim=-1)\n```\n\nThen one Flow step\n\n\n```python\nclass FlowStepWaveGlow(torch_nn.Module):\n \"\"\"FlowStepWaveGlow\n One flow step for waveglow\n y -> intertical_1x1() -> AffineCoupling -> x\n \n Example\n m_tmp = FlowStepWaveGlow(10, 10, 8, 512, 3, flag_affine=True)\n output, log_det = m_tmp(data1, cond)\n data1_re = m_tmp.reverse(output, cond)\n\n torch.std(data1 - data1_re)\n \"\"\"\n def __init__(self, in_dim, cond_dim, \n wn_num_conv1d, wn_dim_channel, wn_kernel_size, flag_affine,\n flag_affine_block_legacy=False):\n \"\"\"FlowStepWaveGlow(in_dim, cond_dim, \n wn_num_conv1d, wn_dim_channel, wn_kernel_size, flag_affine,\n flag_affine_block_legacy=False)\n \n Args\n ----\n in_dim: int, input feature dim, (batch, length, in_dim)\n cond_dim:, int, conditional feature dim, (batch, length, cond_dim)\n wn_num_conv1d: int, number of 1Dconv WaveNet block in this flow step\n wn_dim_channel: int, dim of the WaveNet residual and skip channels\n wn_kernel_size: int, kernel size of the dilated convolution layers\n flag_affine: bool, whether use affine or additive transformation?\n default True\n flag_affine_block_legacy, bool, whether use AffineCouplingWaveGlow or \n AffineCouplingWaveGlow_legacy.\n\n For wn_dim_channel and wn_kernel_size, see AffineCouplingWaveGlow\n For flag_affine == False, scale will be 1.0\n \"\"\"\n super(FlowStepWaveGlow, self).__init__()\n \n # Invertible transformation layer\n #self.m_invtrans = nii_glow.InvertibleTrans(in_dim, flag_detjac=True)\n self.m_invtrans = Invertible1x1ConvWaveGlow(in_dim, flag_detjac=True)\n \n # Coupling layer\n if flag_affine_block_legacy:\n self.m_coupling = AffineCouplingWaveGlow_legacy(\n in_dim, cond_dim, wn_num_conv1d, wn_dim_channel, wn_kernel_size,\n flag_affine, flag_detjac=True)\n else:\n self.m_coupling = AffineCouplingWaveGlow(\n in_dim, cond_dim, wn_num_conv1d, wn_dim_channel, wn_kernel_size,\n flag_affine, flag_detjac=True)\n return\n \n def forward(self, y, cond, factor=1):\n \"\"\"FlowStepWaveGlow.forward(y, cond, factor=1)\n \n input\n -----\n y: tensor, input feature, (batch, lengh, input_dim)\n cond: tensor, condition feature , (batch, lengh, cond_dim)\n factor: int, this is used to divde likelihood, default 1\n if we directly sum all detjac, they will become very large\n however, we cannot average them directly on y because y\n may have a different shape from the actual data y\n output\n ------\n x: tensor, input feature, (batch, lengh, input_dim)\n detjac: tensor, det of jacobian, (batch,)\n \"\"\"\n # 1x1 transform\n x_tmp, log_det_1 = self.m_invtrans(y, factor)\n # coupling\n x_tmp, log_det_2 = self.m_coupling(x_tmp, cond, factor) \n return x_tmp, log_det_1 + log_det_2\n \n def reverse(self, x, cond):\n \"\"\"FlowStepWaveGlow.reverse(y, cond)\n \n input\n -----\n x: tensor, input feature, (batch, lengh, input_dim)\n cond: tensor, condition feature , (batch, lengh, cond_dim)\n \n output\n ------\n y: tensor, input feature, (batch, lengh, input_dim)\n \"\"\"\n y_tmp = self.m_coupling.reverse(x, cond) \n y_tmp = self.m_invtrans.reverse(y_tmp)\n return y_tmp\n```\n\n\n```python\n# Try the example again\n\n# input data y of shape (B=2, T=100, P=16)\ninput_dim = 32\ny = torch.randn([2, 100, input_dim])\n\n# up-sampled condition features of shape (B=2, T=100, P=8)\ncond_dim = 8\ncond_feat = torch.randn([2, 100, cond_dim])\n\n# 8 wavenet layers\nn_blocks = 8\n# dimension of wavenet channels (same value for res, skip and gated channels)\nn_wn_dim = 64\n# kernel size of conv in wavenet\nn_wn_kernel_size =3\n# \nm_flowstep = FlowStepWaveGlow(input_dim, cond_dim, n_blocks, n_wn_dim, n_wn_kernel_size, flag_affine=True)\n\nwith torch.no_grad():\n # do the affine transformation \n x, log_det = m_flowstep.forward(y, cond_feat)\n\n # do the reverse transformation\n y_reversed = m_flowstep.reverse(x, cond_feat)\n\n \n \nprint(\"Input y batch {:d}, length {:d}, dim {:d} \".format(*y.shape))\nprint(\"x = Affine(y) batch {:d}, length {:d}, dim {:d} \".format(*x.shape))\nprint(\"y = Affine^-1(x) batch {:d}, length {:d}, dim {:d} \".format(*y_reversed.shape))\nprint(\"Log-det-Jacobian: \", log_det)\n\nprint(\"Difference between y and Affine^(-1)(x) is: \", end=\"\")\n# the difference should be small\nprint(torch.std(y_reversed - y).item())\nprint(\"\\n\\n\")\n```\n\n Input y batch 2, length 100, dim 32 \n x = Affine(y) batch 2, length 100, dim 32 \n y = Affine^-1(x) batch 2, length 100, dim 32 \n Log-det-Jacobian: tensor([-17.2288, -41.1602])\n Difference between y and Affine^(-1)(x) is: 3.795781537974108e-07\n \n \n \n\n\n\nBy running the examples multiple times, you will see how the Log-det-Jacobian change dramatically.\n\nThis is reason to initialize the weight and bias of the last FC layer after the WaveNet blocks with zero. The affine transformation will do nothing at the beginning of the model training.\n```python\ntmp.weight.data.zero_()\ntmp.bias.data.zero_()\n```\n\n### 1.5 WaveGlow Block\n\nTo recap, one WaveGlow block is like this:\n```sh\n. |-----------> log_detJac\n 3 | |---> early output z \n | | (B, T/8, d)\n =====================================================|========== \n | WaveGlow block | | | \n | -------------> + -----------> + ------------ + | | \n | | ^ ^ ^ | | \n | | | | | | |\n | ----------- ----------- ----------- ----------- | | 4 \n 1 --> |Flowstep1| -> |Flowstep2| -> |Flowstep3| -> |Flowstep4| ---|-> input to next block x\n(B, T/8, P)| ----------- ----------- ----------- ----------- | (B, T/8, P-d)\n | ^ ^ ^ ^ |\n ========|==============|==============|==============|==========\n | | | |\n ---------------|------------------------------\n 2 (B, T/8, 8D)\n```\nwhere input is\n1. output of previous block or squeezed waveform (if this is the 1st block), shape (B, T/8, P)\n2. up-sampled and squeezed condition features, shape (B, T/8, 8D)\n\noutput is:\n\n3. early output z, (B, T/8, d) and log_det|Jac| (scalar)\n4. input to the next WaveGlow block (B, T/8, P-d)\n\n\n\n```python\nclass WaveGlowBlock(torch_nn.Module):\n \"\"\"WaveGlowBlock\n A WaveGlowBlock includes multiple steps of flow.\n \n The Nvidia WaveGlow does not define WaveGlowBlock but directly\n defines 12 flow steps. However, after every 4 flow steps, two\n dimension of z will be extracted (multi-scale approach).\n It is not convenient to decide when to extract z.\n \n Here, we define a WaveGlowBlock as the casecade of multiple flow\n steps, and this WaveGlowBlock can extract the two dimensions from\n the output of final flow step. \n \n Example:\n data1 = torch.randn([2, 10, 10])\n cond = torch.randn([2, 10, 16])\n m_block = WaveGlowBlock(10, 16, 5, 8, 512, 3)\n x, z, log_det = m_block(data1, cond)\n data_re = m_block.reverse(x, z, cond)\n print(torch.std(data_re - data1))\n \"\"\"\n def __init__(self, in_dim, cond_dim, n_flow_steps,\n wn_num_conv1d, wn_dim_channel, wn_kernel_size,\n flag_affine=True, \n flag_split = False, \n flag_final_block=False,\n split_dim = 2, \n flag_affine_block_legacy=False):\n \"\"\"WaveGlowBlock(in_dim, cond_dim, n_flow_steps,\n wn_num_conv1d, wn_dim_channel, wn_kernel_size,\n flag_affine=True, flag_split = False, split_dim = 2,\n flag_affine_block_legacy=False)\n Args\n ----\n in_dim: int, input feature dim, (batch, length, in_dim)\n cond_dim:, int, conditional feature dim, (batch, length, cond_dim)\n n_flow_steps: int, number of flow steps in one block\n wn_num_conv1d: int, number of dilated conv WaveNet blocks\n wn_dim_channel: int, dim of the WaveNet residual and skip channels\n wn_kernel_size: int, kernel size of the dilated convolution layers\n flag_affine: bool, whether use affine or additive transformation?\n default True\n flag_split: bool, whether split output z for multi-scale structure\n default True\n flag_final_block: bool, whether this block is the final block\n default False\n split_dim: int, if flag_split==True, z[:, :, :split_dim] will be\n extracted, z[:, :, split_dim:] can be used for the next\n WaveGlowBlock\n flag_affine_block_legacy, bool, whether use the legacy implementation\n of wavenet-based affine transformaiton layer\n default False. \n \n For wn_dim_channel and wn_kernel_size, see AffineCouplingWaveGlow\n For flag_affine, see AffineCouplingWaveGlow\n \"\"\"\n super(WaveGlowBlock, self).__init__()\n \n tmp_flows = []\n for i in range(n_flow_steps):\n tmp_flows.append(\n FlowStepWaveGlow(\n in_dim, cond_dim,\n wn_num_conv1d, wn_dim_channel, wn_kernel_size,\n flag_affine, flag_affine_block_legacy))\n self.m_flows = torch_nn.ModuleList(tmp_flows)\n\n self.flag_split = flag_split\n self.flag_final_block = flag_final_block\n self.split_dim = split_dim\n \n if self.flag_split and self.flag_final_block:\n print(\"WaveGlowBlock: flag_split and flag_final_block are True\")\n print(\"This is unexpected. Please check model definition\")\n sys.exit(1)\n if self.flag_split and self.split_dim <= 0:\n print(\"WaveGlowBlock: split_dim should be > 0\")\n sys.exit(1)\n \n return\n \n def forward(self, y, cond, factor=1):\n \"\"\"x, z, log_detjac = WaveGlowBlock(y) \n \n y -> H() -> [z, x], log_det_jacobian\n H() consists of multiple flow steps (1x1conv + AffineCoupling)\n \n input\n -----\n y: tensor, (batch, length, dim)\n cond, tensor, (batch, length, cond_dim)\n factor, None or int, this is used to divde likelihood, default 1\n\n output\n ------\n log_detjac: tensor or scalar\n \n if self.flag_split:\n x: tensor, (batch, length, in_dim - split_dim), \n z: tensor, (batch, length, split_dim), \n else:\n if self.flag_final_block:\n x: None, no input to the next block\n z: tensor, (batch, length, dim), for N(z; 0, I)\n else:\n x: tensor, (batch, length, dim), \n z: None, no latent for N(z; 0, I) from this block\n concate([x,z]) should have the same size as y\n \"\"\"\n # flows\n log_detjac = 0\n\n x_tmp = y\n for l_flow in self.m_flows:\n x_tmp, log_detjac_tmp = l_flow(x_tmp, cond, factor)\n log_detjac = log_detjac + log_detjac_tmp\n \n if self.flag_split:\n z = x_tmp[:, :, :self.split_dim]\n x = x_tmp[:, :, self.split_dim:]\n else:\n if self.flag_final_block:\n z = x_tmp\n x = None\n else:\n z = None\n x = x_tmp\n return x, z, log_detjac\n \n def reverse(self, x, z, cond):\n \"\"\"y = WaveGlowBlock.reverse(x, z, cond) \n \n [z, x] -> H^{-1}() -> y\n \n input\n -----\n if self.flag_split:\n x: tensor, (batch, length, in_dim - split_dim), \n z: tensor, (batch, length, split_dim), \n else:\n if self.flag_final_block:\n x: None\n z: tensor, (batch, length, in_dim)\n else:\n x: tensor, (batch, length, in_dim)\n z: None\n output\n ------\n y: tensor, (batch, length, in_dim) \n \"\"\"\n if self.flag_split:\n if x is None or z is None:\n print(\"WaveGlowBlock.reverse: x and z should not be None\")\n sys.exit(1)\n y_tmp = torch.cat([z, x], dim=-1)\n else:\n if self.flag_final_block:\n if z is None: \n print(\"WaveGlowBlock.reverse: z should not be None\")\n sys.exit(1)\n y_tmp = z\n else:\n if x is None: \n print(\"WaveGlowBlock.reverse: x should not be None\")\n sys.exit(1)\n y_tmp = x\n \n for l_flow in self.m_flows[::-1]:\n # affine\n y_tmp = l_flow.reverse(y_tmp, cond)\n return y_tmp\n```\n\nWe simply wrap the definition around the flow-step module. \n\nTo make this module to be configuratable, I considered three cases when splitting the early output\n* No early output, `self.flag_split=False`. This is not used in this notebook\n* Early output, and this is not the last WaveGlow block, `self.flag_split=True` and `self.flag_final_block=False`\n* Early output, and this is the last WaveGlow block, `self.flag_split=False` and `self.flag_final_block=True`\n\n| | 1 input | 2 condition | 3 early output z | 4 output latent x | flag_split | flag_final_block\n|-----------------|---|---|---|---|---|---|\n| WaveGlow Block1 | (B, T/8, 8) | (B, T/8, 8D) | (B, T/8, 2) | (B, T/8, 6) | True | False\n| WaveGlow Block2 | (B, T/8, 6) | (B, T/8, 8D) | (B, T/8, 2) | (B, T/8, 4) | True | False\n| WaveGlow Block3 | (B, T/8, 4) | (B, T/8, 8D) | (B, T/8, 4) | - | False | True\n\n\n\n```python\n# Try one example \n\n# squeezed waveform has 8 dimensions\ninput_dim = 8\ny = torch.randn([2, 100, input_dim])\n\n# up-sampled condition features of shape (B=2, T=100, P=8)\ncond_dim = 8\ncond_feat = torch.randn([2, 100, cond_dim])\n\n# 4 flow steps\nn_flowsteps = 4\nn_wn_num = 8\nn_wn_dim = 64\nn_wn_kernel_size =3\nn_early_output_dim = 2\n\n# m_block1 \nm_block1 = WaveGlowBlock(input_dim, cond_dim, n_flowsteps, n_wn_num, n_wn_dim, n_wn_kernel_size, flag_affine=True, \n flag_split = True, flag_final_block = False, split_dim=n_early_output_dim)\n# m_block2\ninput_new_dim = input_dim - n_early_output_dim\nm_block2 = WaveGlowBlock(input_new_dim, cond_dim, n_flowsteps, n_wn_num, n_wn_dim, n_wn_kernel_size, flag_affine=True, \n flag_split = True, flag_final_block = False, split_dim=n_early_output_dim)\n# m_block3\ninput_new_dim = input_new_dim - n_early_output_dim\nm_block3 = WaveGlowBlock(input_new_dim, cond_dim, n_flowsteps, n_wn_num, n_wn_dim, n_wn_kernel_size, flag_affine=True, \n flag_split = False, flag_final_block = True, split_dim=n_early_output_dim)\n\nwith torch.no_grad():\n x1, z1, log_det1 = m_block1.forward(y, cond_feat)\n x2, z2, log_det2 = m_block2.forward(x1, cond_feat)\n x3, z3, log_det3 = m_block3.forward(x2, cond_feat)\n\n # do the reverse transformation\n x2_reverse = m_block3.reverse(x3, z3, cond_feat)\n x1_reverse = m_block2.reverse(x2_reverse, z2, cond_feat)\n y_reverse = m_block1.reverse(x1_reverse, z1, cond_feat)\n \n \nprint(\"Input y batch {:d}, length {:d}, dim {:d} \".format(*y.shape))\nprint(\"\\nx1 from block1 batch {:d}, length {:d}, dim {:d} \".format(*x1.shape))\nprint(\"z1 from block1 batch {:d}, length {:d}, dim {:d} \".format(*z1.shape))\nprint(\"\\nx2 from block2 batch {:d}, length {:d}, dim {:d} \".format(*x2.shape))\nprint(\"z2 from block3 batch {:d}, length {:d}, dim {:d} \".format(*z2.shape))\nif x3 is None:\n print(\"\\nx3 from block3 is None\")\nprint(\"z3 from block3 batch {:d}, length {:d}, dim {:d} \".format(*z3.shape))\n\n\nprint(\"\\nDifference between y and reversed y is: \", end=\"\")\n\n# the difference should be small\nprint(torch.std(y_reverse - y).item())\nprint(\"\\n\\n\")\n```\n\n Input y batch 2, length 100, dim 8 \n \n x1 from block1 batch 2, length 100, dim 6 \n z1 from block1 batch 2, length 100, dim 2 \n \n x2 from block2 batch 2, length 100, dim 4 \n z2 from block3 batch 2, length 100, dim 2 \n \n x3 from block3 is None\n z3 from block3 batch 2, length 100, dim 4 \n \n Difference between y and reversed y is: 8.11803033684555e-07\n \n \n \n\n\n### 1.6 WaveGlow in one Module\n\nWe can now wrap everything in a single Module\n\nThere is example code in the doc string. You can try it\n\n\n```python\nclass WaveGlow(torch_nn.Module):\n \"\"\"WaveGlow\n \n Example\n cond_dim = 4\n upsample = 80\n num_blocks = 4\n num_flows_inblock = 5\n wn_num_conv1d = 8\n wn_dim_channel = 512\n wn_kernel_size = 3\n\n # waveforms of length 1600\n wave1 = torch.randn([2, 1600, 1]) \n # condition feature\n cond = torch.randn([2, 1600//upsample, cond_dim])\n\n # model\n m_model = nii_waveglow.WaveGlow(\n cond_dim, upsample, \n num_blocks, num_flows_inblock, wn_num_conv1d, \n wn_dim_channel, wn_kernel_size)\n\n # forward computation, neg_log = -(logp + log_detjac)\n # neg_log.backward() can be used for backward\n z, neg_log, logp, log_detjac = m_model(wave1, cond) \n\n # recover the signal\n wave2 = m_model.reverse(z, cond)\n \n # check difference between original wave and recovered wave\n print(torch.std(wave1 - wave2)) \n \"\"\"\n def __init__(self, cond_dim, upsample_rate, \n num_blocks, num_flows_inblock, \n wn_num_conv1d, wn_dim_channel, wn_kernel_size,\n flag_affine = True,\n early_hid_dim=2, \n flag_affine_block_legacy=False):\n \"\"\"WaveGlow(cond_dim, upsample_rate,\n num_blocks, num_flows_inblock, \n wn_num_conv1d, wn_dim_channel, wn_kernel_size,\n flag_affine = True,\n early_hid_dim=2,\n flag_affine_block_legacy=False)\n \n Args\n ----\n cond_dim:, int, conditional feature dim, (batch, length, cond_dim)\n upsample_rate: int, up-sampling rate for condition features\n num_blocks: int, number of WaveGlowBlocks\n num_flows_inblock: int, number of flow steps in one WaveGlowBlock\n wn_num_conv1d: int, number of 1Dconv WaveNet block in this flow step\n wn_dim_channel: int, dim of the WaveNet residual and skip channels\n wn_kernel_size: int, kernel size of the dilated convolution layers\n flag_affine: bool, whether use affine or additive transformation?\n default True\n early_hid_dim: int, dimension for z_1, z_2 ... , default 2\n flag_affine_block_legacy, bool, whether use the legacy implementation\n of wavenet-based affine transformaiton layer\n default False. The difference is on the WaveNet part\n Please configure AffineCouplingWaveGlow and \n AffineCouplingWaveGlow_legacy\n \n\n This model defines:\n \n cond -> upsample/squeeze -> | ------> | --------> | \n v v v \n y -> squeeze -> WaveGlowBlock -> WGBlock ... WGBlock -> z\n |-> z_1 |-> z_2 \n \n z_1, z_2, ... are the extracted z from a multi-scale flow structure\n concate([z_1, z_2, z]) is expected to be the white Gaussian noise\n \n If early_hid_dim == 0, z_1 and z_2 will not be extracted\n \"\"\"\n super(WaveGlow, self).__init__()\n \n # input is assumed to be waveform\n self.m_input_dim = 1\n self.m_early_hid_dim = early_hid_dim\n \n # squeeze layer\n self.m_squeeze = SqueezeForWaveGlow()\n \n # up-sampling layer\n #self.m_upsample = nii_nn.UpSampleLayer(cond_dim, upsample_rate, True)\n self.m_upsample = upsampleByTransConv(cond_dim, upsample_rate)\n \n # wavenet-based flow blocks\n # squeezed input dimension\n squeezed_in_dim = self.m_input_dim * self.m_squeeze.get_squeeze_factor()\n # squeezed condition feature dimension\n squeezed_cond_dim = cond_dim * self.m_squeeze.get_squeeze_factor()\n \n # save the dimension for get_z_noises\n self.m_feat_dim = []\n \n # define blocks\n tmp_squeezed_in_dim = squeezed_in_dim\n tmp_flow_blocks = []\n for i in range(num_blocks):\n # if this is not the last block and early_hid_dim >0\n flag_split = (i < (num_blocks-1)) and early_hid_dim > 0\n flag_final_block = i == (num_blocks-1)\n \n # save the dimension for get_z_noises\n if flag_final_block:\n self.m_feat_dim.append(tmp_squeezed_in_dim)\n else:\n self.m_feat_dim.append(early_hid_dim if flag_split else 0)\n \n tmp_flow_blocks.append(\n WaveGlowBlock(\n tmp_squeezed_in_dim, squeezed_cond_dim, num_flows_inblock,\n wn_num_conv1d, wn_dim_channel, wn_kernel_size, flag_affine,\n flag_split = flag_split, flag_final_block=flag_final_block,\n split_dim = early_hid_dim, \n flag_affine_block_legacy = flag_affine_block_legacy))\n \n # multi-scale approach will extract a few dimensions for next flow\n # thus, input dimension to the next block will be this \n tmp_squeezed_in_dim = tmp_squeezed_in_dim - early_hid_dim\n \n self.m_flowblocks = torch_nn.ModuleList(tmp_flow_blocks)\n \n # done\n return\n \n \n def _normal_lh(self, noise):\n # likelihood of normal distribution on the given noise\n return -0.5 * np.log(2 * np.pi) - 0.5 * noise ** 2\n \n def forward(self, y, cond):\n \"\"\"z, neg_logp_y, logp_z, logdet = WaveGlow.forward(y, cond) \n \n cond -> upsample/squeeze -> | ------> | --------> | \n v v v \n y -> squeeze -> WaveGlowBlock -> WGBlock ... WGBlock -> z\n |-> z_1 |-> z_2 \n \n input\n -----\n y: tensor, (batch, waveform_length, 1)\n cond: tensor, (batch, cond_length, 1)\n \n output\n ------\n z: list of tensors, [z_1, z_2, ... ,z ] in figure above\n neg_logp_y: scalar, - log p(y)\n logp_z: scalar, -log N(z), summed over one data sequence, but averaged\n over batch.\n logdet: scalar, -|det dH(.)/dy|, summed over one data sequence, \n but averaged\n over batch.\n \n If self.early_hid_dim == 0, z_1, z_2 ... will be None\n \"\"\"\n \n # Rather than summing the likelihood and divide it by the number of \n # data in the final step, we divide this factor from the likelihood\n # caculating by each flow step and sum the scaled likelihood. \n # Two methods are equivalent, but the latter may prevent numerical \n # overflow of the likelihood value for long sentences\n factor = np.prod([dim for dim in y.shape])\n \n # waveform squeeze (batch, squeezed_length, squeezed_dim)\n y_squeezed = self.m_squeeze(y)\n squeezed_dim = y_squeezed.shape[-1]\n \n # condition feature upsampling and squeeze\n # (batch, squeezed_length, squeezed_dim_cond)\n cond_up_squeezed = self.m_squeeze(self.m_upsample(cond))\n \n # flows\n z_bags = []\n log_detjac = 0\n log_pz = 0\n\n x_tmp = y_squeezed\n for m_block in self.m_flowblocks:\n x_tmp, z_tmp, log_detjac_tmp = m_block(\n x_tmp, cond_up_squeezed, factor)\n \n # accumulate log det jacobian\n log_detjac += log_detjac_tmp\n \n # compute N(z; 0, I)\n # save z_tmp (even if it is None)\n z_bags.append(z_tmp)\n # accumulate log_N(z; 0, I) only if it is valid\n if z_tmp is not None:\n log_pz += nii_glow.sum_over_keep_batch2(\n self._normal_lh(z_tmp), factor)\n \n # average over batch and data points\n neg_logp_y = -(log_pz + log_detjac).sum()\n return z_bags, neg_logp_y, \\\n log_pz.sum(), log_detjac.sum()\n \n def reverse(self, z_bags, cond):\n \"\"\"y = WaveGlow.reverse(z_bags, cond) \n \n cond -> upsample/squeeze -> | ------> | --------> | \n v v v \n y <- unsqueeze <- WaveGlowBlock -> WGBlock ... WGBlock <- z\n |<- z_1 |<- z_2 \n \n input\n -----\n z: list of tensors, [z_1, z_2, ... ,z ] in figure above\n cond: tensor, (batch, cond_length, 1)\n \n output\n ------\n y: tensor, (batch, waveform_length, 1)\n \n If self.early_hid_dim == 0, z_1, z_2 ... should be None\n \"\"\"\n # condition feature upsampling and squeeze\n # (batch, squeezed_length, squeezed_dim_cond)\n cond_up_sqe = self.m_squeeze(self.m_upsample(cond))\n \n # initial\n y_tmp = None\n for z, m_block in zip(z_bags[::-1], self.m_flowblocks[::-1]):\n y_tmp = m_block.reverse(y_tmp, z, cond_up_sqe)\n y = self.m_squeeze.reverse(y_tmp)\n return y\n \n def get_z_noises(self, length, noise_std=0.7, batchsize=1):\n \"\"\"z_bags = WaveGlow.get_z_noises(length, noise_std=0.7, batchsize=1)\n Return a list of random noises for random sampling\n \n input\n -----\n length: int, length of target waveform (without squeeze)\n noise_std: float, std of Gaussian noise, default 0.7\n batchsize: int, batch size of this random data, default 1\n \n output\n ------\n z_bags: list of tensors\n \n Shape of tensor in z_bags is decided by WaveGlow configuration.\n WaveGlow.reverse(z_bags, cond) can be used to generate waveform\n \"\"\"\n squeeze_length = self.m_squeeze.get_expected_squeeze_length(length)\n \n device = next(self.parameters()).device\n z_bags = []\n \n # generate the z for each WaveGlowBlock\n for feat_dim in self.m_feat_dim:\n if feat_dim is not None and feat_dim > 0:\n z_tmp = torch.randn(\n [batchsize, squeeze_length, feat_dim], \n dtype=nii_io_conf.d_dtype, \n device=device)\n z_bags.append(z_tmp * noise_std)\n else:\n z_bags.append(None)\n return z_bags\n```\n\nIn the above API, we have a `get_z_noises` method to sample a bag of random noises $(z_1, z_2, z_3)$ given a required waveform length.\n\nThis is handy during generation: we only don't need to care about dimension of the early output z. The API will handle that for us.\n\n## 2. Using pre-trained model for waveform generation\n\nHere we load pre-trained Wavenet_v1 model and generate a sample.\n\n\n### 2.1 Meta-Model wrapper\n\nFor my pytorch script, I further wrap around the `WaveGlow` with a meta-model Module. This is convenient for methods irrelavant to a specific model, for example, input and output feature normalization.\n\nThis is merely a practical choise. Nothing new for WaveGlow.\n\n\n```python\n# just for convenience\nclass PrjConfig:\n def __init__(self, wav_samp_rate, up_sample_rate):\n self.wav_samp_rate = wav_samp_rate\n self.input_reso = [up_sample_rate]\n\n# \nclass Model(torch_nn.Module):\n \"\"\" Model definition\n \"\"\"\n def __init__(self, in_dim, out_dim, args, prj_conf, mean_std=None):\n super(Model, self).__init__()\n\n #################\n ## must-have\n #################\n # mean std of input and output\n in_m, in_s, out_m, out_s = self.prepare_mean_std(in_dim,out_dim,\\\n args, mean_std)\n self.input_mean = torch_nn.Parameter(in_m, requires_grad=False)\n self.input_std = torch_nn.Parameter(in_s, requires_grad=False)\n self.output_mean = torch_nn.Parameter(out_m, requires_grad=False)\n self.output_std = torch_nn.Parameter(out_s, requires_grad=False)\n self.input_dim = in_dim\n self.output_dim = out_dim\n \n # a flag for debugging (by default False) \n self.model_debug = False\n \n #################\n ## model config\n ################# \n # waveform sampling rate\n self.sample_rate = prj_conf.wav_samp_rate\n # up-sample rate\n self.up_sample = prj_conf.input_reso[0]\n \n # configuration for WaveGlow\n self.num_waveglow_blocks = 3\n self.num_flow_steps_perblock = 4\n self.num_wn_blocks_perflow = 8\n self.num_wn_channel_size = 256\n self.num_wn_conv_kernel = 3\n self.flag_affine = True\n self.early_z_feature_dim = 2\n self.flag_affine_legacy_implementation = False\n\n self.m_waveglow = WaveGlow(\n in_dim, self.up_sample, \n self.num_waveglow_blocks,\n self.num_flow_steps_perblock,\n self.num_wn_blocks_perflow, \n self.num_wn_channel_size,\n self.num_wn_conv_kernel,\n self.flag_affine,\n self.early_z_feature_dim,\n self.flag_affine_legacy_implementation)\n \n # done\n return\n \n def prepare_mean_std(self, in_dim, out_dim, args, data_mean_std=None):\n \"\"\"\n \"\"\"\n if data_mean_std is not None:\n in_m = torch.from_numpy(data_mean_std[0])\n in_s = torch.from_numpy(data_mean_std[1])\n out_m = torch.from_numpy(data_mean_std[2])\n out_s = torch.from_numpy(data_mean_std[3])\n if in_m.shape[0] != in_dim or in_s.shape[0] != in_dim:\n print(\"Input dim: {:d}\".format(in_dim))\n print(\"Mean dim: {:d}\".format(in_m.shape[0]))\n print(\"Std dim: {:d}\".format(in_s.shape[0]))\n print(\"Input dimension incompatible\")\n sys.exit(1)\n if out_m.shape[0] != out_dim or out_s.shape[0] != out_dim:\n print(\"Output dim: {:d}\".format(out_dim))\n print(\"Mean dim: {:d}\".format(out_m.shape[0]))\n print(\"Std dim: {:d}\".format(out_s.shape[0]))\n print(\"Output dimension incompatible\")\n sys.exit(1)\n else:\n in_m = torch.zeros([in_dim])\n in_s = torch.ones([in_dim])\n out_m = torch.zeros([out_dim])\n out_s = torch.ones([out_dim])\n \n return in_m, in_s, out_m, out_s\n \n def normalize_input(self, x):\n \"\"\" normalizing the input data\n \"\"\"\n return (x - self.input_mean) / self.input_std\n\n def normalize_target(self, y):\n \"\"\" normalizing the target data\n \"\"\"\n return (y - self.output_mean) / self.output_std\n\n def denormalize_output(self, y):\n \"\"\" denormalizing the generated output from network\n \"\"\"\n return y * self.output_std + self.output_mean\n\n def forward(self, input_feat, wav):\n \"\"\"loss = forward(self, input_feat, wav)\n\n input\n -----\n input_feat: tensor, input features (batchsize, length1, input_dim)\n wav: tensor, target waveform (batchsize, length2, 1)\n it should be raw waveform, flot valued, between (-1, 1)\n the code will do mu-law conversion\n output\n ------\n loss: tensor / scalar\n \n Note: returned loss can be directly used as the loss value\n no need to write Loss()\n \"\"\"\n # normalize conditiona feature\n #input_feat = self.normalize_input(input_feat)\n # compute \n z_bags, neg_logp, logp_z, log_detjac = self.m_waveglow(wav, input_feat)\n return neg_logp\n\n def inference(self, input_feat):\n \"\"\"wav = inference(mels)\n\n input\n -----\n input_feat: tensor, input features (batchsize, length1, input_dim)\n\n output\n ------\n wav: tensor, target waveform (batchsize, length2, 1)\n\n Note: length2 will be = length1 * self.up_sample\n \"\"\" \n #normalize input\n #input_feat = self.normalize_input(input_feat)\n \n length = input_feat.shape[1] * self.up_sample\n noise = self.m_waveglow.get_z_noises(length, noise_std=0.6, \n batchsize=input_feat.shape[0])\n output = self.m_waveglow.reverse(noise, input_feat)\n return output\n```\n\n\n```python\n# sampling rate of waveform (Hz)\nsampling_rate = 16000\n# up-sampling rate for the pre-trained model is 80\nupsampe_rate = 80\n\nprj_config = PrjConfig(16000, 80)\n\n\n# input feature dim (80 dimension Mel-spec)\nmel_dim = 80\ninput_dim = mel_dim\n\n# output dimension = 1 for waveform\noutput_dim = 1\n\n# declare the model\nm_waveglow = Model(input_dim, output_dim, None, prj_config)\n```\n\n\n```python\n# load pre-trained model\ndevice=torch.device(\"cpu\")\nm_waveglow.to(device, dtype=torch.float32)\n\npretrained_file = \"data_models/pre_trained_waveglow/__pre_trained/trained_network.pt\"\nif os.path.isfile(pretrained_file):\n checkpoint = torch.load(pretrained_file, map_location=\"cpu\")\n m_waveglow.load_state_dict(checkpoint)\nelse:\n print(\"Cannot find pre-trained model {:s}\".format(pretrained_file))\n print(\"Please run 00_download_model.sh and download the pre-trained model\")\n```\n\n### 2.2 Load waveform, extract feature, and generate waveform\n\n\nLet's use CPU.\n\n#### Extract input feature\n\n\n```python\n# If you want to do copy-synthesis\n# The tools to extract are in data_models/scripts\nimport sys\nimport os\nimport data_models.scripts.sub_get_mel as nii_mel_tk\n\nsampling_rate = 16000\nfftl_n = 1024\nframe_length = 400\n# This must be compatible with the WaveNet up-sampling rate configuration\nframe_shift = 80\nframe_length_in_ms = int(frame_length*1000/sampling_rate)\nframe_shift_in_ms = int(frame_shift*1000/sampling_rate)\n\n# use waveform from this folder\ninput_waveform_path = \"data_models/acoustic_features/hn_nsf/slt_arctic_b0474.wav\"\ninput_mel = nii_mel_tk.get_melsp(input_waveform_path, fftl=fftl_n, fl=frame_length, fs=frame_shift)\n\n# If you have problem running the code above, please just load the pre-extracted features\n#input_mel = tool_lib.read_raw_mat(\"data_models/acoustic_features/hn_nsf/slt_arctic_b0474.mfbsp\", mel_dim)\n\n# trim the mel for quick demonstration\n#input_mel = input_mel[:290, :]\n```\n\n\n```python\nprint(\"Input Mel shape:\" + str(input_mel.shape))\n# compose the input tensor\ninput_tensor = torch.tensor(input_mel, dtype=torch.float32).unsqueeze(0)\nprint(\"Input data tensor shape:\" + str(input_tensor.shape))\n```\n\n Input Mel shape:(554, 80)\n Input data tensor shape:torch.Size([1, 554, 80])\n\n\n#### Generate waveform\nThis may be slow because WaveGlow requires a huge amount of computation.\n\nPlease be patient! If it is too slow, please reduce the length of input mel-spec further.\n\n\n```python\nwith torch.no_grad():\n output_waveform = m_waveglow.inference(input_tensor)\n```\n\n\n```python\nimport IPython.display\nIPython.display.Audio(output_waveform[0, :, 0].numpy(), rate=sampling_rate, normalize=False)\n```\n\n\n\n\n\n\n\n\n\n\n\n```python\n# plot the spectrogram of the generated waveform\nplot_API.plot_API(output_waveform[0, :, 0].numpy(), plot_lib.plot_spec, 'single')\n```\n\nYou may also try other waveforms in `data_models/acoustic_features/hn_nsf`. \n\n\nNote that waveforms I used to train this WaveNet have normalized amplitude. The normalization tool is the sv56 https://github.com/openitu/STL. If you try part3 of this tutorial and run the script `../project/01-nsf/*/00_demo.sh`, you will download the normalized waveforms of CMU-arctic. There will also be a script to use sv56 `../project/01-nsf/DATA/cmu-arctic-data-set/scripts/wav`. But please compile the sv56 and sox first. \n\n## Final note\n\nProject to train a new WaveNet using CMU-arctic database is available in `../project/05-nn-vocoders/waveglow`. The pre-trained model was trained on CMU arctic data, which may not be sufficient for WaveGlow. You may try the script on other database. Furthermore, no post-processing is included in this notebook. \n\nThere is another variant `../project/05-nn-vocoders/waveglow-2` that uses a legacy version of WaveNet block for WaveGlow. You may also try that model as well.\n\nThat's all\n\n\n```python\n\n```\n", "meta": {"hexsha": "5090d5bcac4bc7b559e014dd8effe3265500d24d", "size": 668415, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/s4_demonstration_waveglow.ipynb", "max_stars_repo_name": "nii-yamagishilab/project-NN-Pytorch-scripts", "max_stars_repo_head_hexsha": "910815196bf265f9f849d4bedd1c1a342a1d7b5c", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 150, "max_stars_repo_stars_event_min_datetime": "2020-06-04T00:02:50.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-30T03:32:56.000Z", "max_issues_repo_path": "tutorials/s4_demonstration_waveglow.ipynb", "max_issues_repo_name": "nii-yamagishilab/project-NN-Pytorch-scripts", "max_issues_repo_head_hexsha": "910815196bf265f9f849d4bedd1c1a342a1d7b5c", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 15, "max_issues_repo_issues_event_min_datetime": "2020-06-17T04:08:33.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-01T03:42:25.000Z", "max_forks_repo_path": "tutorials/s4_demonstration_waveglow.ipynb", "max_forks_repo_name": "nii-yamagishilab/project-NN-Pytorch-scripts", "max_forks_repo_head_hexsha": "910815196bf265f9f849d4bedd1c1a342a1d7b5c", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 23, "max_forks_repo_forks_event_min_datetime": "2020-06-16T03:28:35.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-15T03:46:13.000Z", "avg_line_length": 221.8436773979, "max_line_length": 345052, "alphanum_fraction": 0.853459303, "converted": true, "num_tokens": 104420, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802264851919, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.4323588306259023}} {"text": " Trusted Notebook\" width=\"500 px\" align=\"left\">\n\n# Getting Started with Qiskit Terra\n\nHere, we provide an overview of working with Qiskit Terra. Qiskit Terra provides the basic building blocks necessary to program quantum computers. The basic concept of Qiskit Terra is an array of quantum circuits. A workflow using Terra consists of two stages: Build and Execute. Build allows you to make different quantum circuits that represent the problem you are solving, and Execute allows you to run them on different backends. After the jobs have been run, the data is collected. There are methods for putting this data together, depending on the program. This either gives you the answer you wanted, or allows you to make a better program for the next instance (e.g., when running the variational quantum eigensolver (VQE) algorithm).\n\n\n**Contents**\n\n[Circuit basics](#circuit_basics)\n\n[Simulating circuits with Qiskit Aer](#aer_simulation)\n\n[Running circuits using the IBMQ provider](#ibmq_provider)\n\n**Code imports**\n\n\n```python\nimport numpy as np\nfrom qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister\nfrom qiskit import execute \n```\n\n\n\nCircuit Basics\n================\n\n## Building the circuit \n\nThe basic elements needed for your first program are the QuantumCircuit, and QuantumRegister.\n\n\n```python\n# Create a Quantum Register with 3 qubits.\nq = QuantumRegister(3, 'q')\n\n# Create a Quantum Circuit acting on the q register\ncirc = QuantumCircuit(q)\n```\n\n
    \nNote: Naming the QuantumRegister is optional and not required.\n
    \n\nAfter you create the circuit with its registers, you can add gates (\"operations\") to manipulate the registers. (See [Summary of Quantum Operations](../terra/summary_of_quantum_operations.ipynb) for a thorough discussion of quantum gates.) As you proceed through the documentation you will find more gates and circuits but the below is an example of a quantum circuit that makes a three-qubit GHZ state\n\n$$|\\psi\\rangle = \\left(|000\\rangle+|111\\rangle\\right)/\\sqrt{2}.$$\n\nTo create such a state, we start with a 3-qubit quantum register. By default, each qubit in the register is initialized to $|0\\rangle$. To make the GHZ state, we apply the following gates:\n* A Hadamard gate $H$ on qubit 0, which puts it into a superposition state.\n* A controlled-X operation ($C_{X}$) between qubit 0 and qubit 1.\n* A controlled-X operation between qubit 0 and qubit 2.\n\nOn an ideal quantum computer, the state produced by running this circuit would be the GHZ state above.\n\nIn Qiskit Terra, operations can be added to the circuit one-by-one, as shown below.\n\n\n```python\n# Add a H gate on qubit 0, putting this qubit in superposition.\ncirc.h(q[0])\n# Add a CX (CNOT) gate on control qubit 0 and target qubit 1, putting\n# the qubits in a Bell state.\ncirc.cx(q[0], q[1])\n# Add a CX (CNOT) gate on control qubit 0 and target qubit 2, putting\n# the qubits in a GHZ state.\ncirc.cx(q[0], q[2])\n```\n\n\n\n\n \n\n\n\n## Visualize Circuit\n\nYou can visualize your circuit using Qiskit Terra `circuit_drawer`, which plots circuit in the form found in many textbooks.\n\n\n```python\nfrom qiskit.tools.visualization import circuit_drawer\n\ncircuit_drawer(circ)\n```\n\nIn this circuit, the qubits are put in order with qubit zero at the top and qubit three at the bottom. The circuit is read left-to-right (meaning that gates which are applied earlier in the circuit show up further to the left).\n\n\n\n## Simulating circuits using Qiskit Aer\n\nQiskit Aer is our package for simulating quantum circuits. It provides many different backends for doing a simulation.\n\n### Statevector backend\n\nThe most common backend in Qiskit Aer is the `statevector_simulator`. This simulator returns the quantum \nstate which is a complex vector of dimensions $2^n$ where $n$ is the number of qubits \n(so be careful using this as it will quickly get too large to run on your machine).\n\n
    \n\n\nWhen representing the state of a multi-qubit system, the tensor order used in qiskit is different than that use in most physics textbooks. Suppose there are $n$ qubits, and qubit $j$ is labeled as $Q_{j}$. In most textbooks (such as Nielsen and Chuang's \"Quantum Computation and Information\"), the basis vectors for the $n$-qubit state space would be labeled as $Q_{0}\\otimes Q_{1} \\otimes \\cdots \\otimes Q_{n}$. **This is not the ordering used by qiskit!** Instead, qiskit uses an ordering in which the $n^{\\mathrm{th}}$ qubit is on the _left_ side of the tesnsor product, so that the basis vectors are labeled as $Q_n\\otimes \\cdots \\otimes Q_1\\otimes Q_0$.\n\nFor example, if qubit zero is in state 0, qubit 1 is in state 0, and qubit 2 is in state 1, qiskit would represent this state as $|100\\rangle$, whereas most physics textbooks would represent it as $|001\\rangle$.\n\nThis difference in labeling affects the way multi-qubit operations are represented as matrices. For example, qiskit represents a controlled-X ($C_{X}$) operation with qubit 0 being the control and qubit 1 being the target as\n\n$$C_X = \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & 0 \\\\\\end{pmatrix}.$$\n\n
    \n\nTo run the above circuit using the statevector simulator first you need to import Aer and then set the backend to `statevector_simulator`.\n\n\n```python\n# Import Aer\nfrom qiskit import Aer\n\n# Run the quantum circuit on a statevector simulator backend\nbackend = Aer.get_backend('statevector_simulator')\n```\n\nNow we have chosen the backend its time to compile and run the quantum circuit. In Qiskit Terra we provide the `execute` function for this. ``execute`` returns a ``job`` object that encapsulates information about the job submitted to the backend.\n\n\n
    \nTip: You can obtain the above parameters in Jupyter. Simply place the text cursor on a function and press Shift+Tab.\n
    \n\n\n```python\n# Create a Quantum Program for execution \njob = execute(circ, backend)\n```\n\nWhen you run a program a job object is made that has the following two useful methods \n`job.status()` and `job.result()` which returns the status of the job a result object respectiviely.\n\n
    \nNote: Jobs run asynchronous but when the result method is called it switches to synchronous and waits for it to finish before moving on to another task.\n
    \n\n\n```python\nresult = job.result()\n```\n\nThe results object contains the data and Qiskit Terra provides the method \n`result.get_statevector(circ)` to return the statevector for the quantum circuit.\n\n\n```python\noutputstate = result.get_statevector(circ)\nprint(\"simulation: \", result )\nprint(np.around(outputstate,3))\n```\n\n simulation: COMPLETED\n [0.707+0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j\n 0.707+0.j]\n\n\nQiskit Terra also provides a visualization toolbox to allow you to view these results. See [Qiskit visualizations](qiskit_visualizations.ipynb) for more information.\n\nBelow, we use the visualization function to plot the real and imaginary components of the state vector.\n\n\n```python\nfrom qiskit.tools.visualization import plot_state\nplot_state(outputstate)\n```\n\n### Unitary backend\n\nQiskit Aer also includes a `unitary_simulator` that works _provided all the elements in the circuit are unitary operations_. This backend calculates the $2^n \\times 2^n$ matrix representing the gates in the quantum circuit. \n\n\n```python\n# Run the quantum circuit on a unitary simulator backend\nbackend = Aer.get_backend('unitary_simulator')\njob = execute(circ, backend)\nresult = job.result()\n\n# Show the results\nprint(\"simulation: \", result )\nprint(np.around(result.get_unitary(circ), 3))\n```\n\n simulation: COMPLETED\n [[ 0.707+0.j 0.707-0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j\n 0. +0.j 0. +0.j]\n [ 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j\n 0.707+0.j -0.707+0.j]\n [ 0. +0.j 0. +0.j 0.707+0.j 0.707-0.j 0. +0.j 0. +0.j\n 0. +0.j 0. +0.j]\n [ 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0.707+0.j -0.707+0.j\n 0. +0.j 0. +0.j]\n [ 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0.707+0.j 0.707-0.j\n 0. +0.j 0. +0.j]\n [ 0. +0.j 0. +0.j 0.707+0.j -0.707+0.j 0. +0.j 0. +0.j\n 0. +0.j 0. +0.j]\n [ 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j\n 0.707+0.j 0.707-0.j]\n [ 0.707+0.j -0.707+0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j\n 0. +0.j 0. +0.j]]\n\n\n### OpenQASM backend\n\nThe simulators above are useful because they provide information about the state output by the ideal circuit and the matrix representation of the circuit. However, a real experiment terminates by _measuring_ each qubit (usually in the computational $|0\\rangle, |1\\rangle$ basis). Without measurement, we cannot gain information about the state. Measurements cause the quantum system to collapse into classical bits. \n\nFor example, suppose we make independent measurements on each qubit of the three-qubit GHZ state\n$$|\\psi\\rangle = |000\\rangle +|111\\rangle)/\\sqrt{2},$$\nand let $xyz$ denote the bitstring that results. Recall that, under the qubit labeling used by Qiskit, $x$ would correspond to the outcome on qubit 2, $y$ to the outcome on qubit 1, and $z$ to the outcome on qubit 0. This representation of the bitstring puts the most significant bit (MSB) on the left, and the least significant bit (LSB) on the right. This is the standard ordering of binary bitstrings. We order the qubits in the same way, which is why Qiskit uses a non-standard tensor product order.\n\nThe probability of obtaining outcome $xyz$ is given by\n$$\\mathrm{Pr}(xyz) = |\\langle xyz | \\psi \\rangle |^{2}.$$\nBy explicit computation, we see there are only two bitstrings that will occur: $000$ and $111$. If the bitstring $000$ is obtained, the state of the qubits is $|000\\rangle$, and if the bitstring is $111$, the qubits are left in the state $|111\\rangle$. The probability of obtaining 000 or 111 is the same; namely, 1/2:\n$$\\begin{align}\n\\mathrm{Pr}(000) &= |\\langle 000 | \\psi \\rangle |^{2} = \\frac{1}{2}\\\\\n\\mathrm{Pr}(111) &= |\\langle 111 | \\psi \\rangle |^{2} = \\frac{1}{2}.\n\\end{align}$$\n\nTo simulate a circuit that includes measurement, we need to add measurements to the original circuit above, and use a different Aer backend.\n\n\n```python\n# Create a Classical Register with 3 bits.\nc = ClassicalRegister(3, 'c')\n# Create a Quantum Circuit\nmeas = QuantumCircuit(q, c)\nmeas.barrier(q)\n# map the quantum measurement to the classical bits\nmeas.measure(q,c)\n\n# The Qiskit circuit object supports composition using\n# the addition operator.\nqc = circ+meas\n\n#drawing the circuit\ncircuit_drawer(qc)\n```\n\nThis circuit adds a classical register, and three measurements that are used to map the outcome of qubits to the classical bits. \n\nTo simulate this circuit, we use the ``qasm_simulator`` in Qiskit Aer. Each run of this circuit will yield either the bitstring 000 or 111. To build up statistics about the distribution of the bitstrings (to, e.g., estimate $\\mathrm{Pr}(000)$), we need to repeat the circuit many times. The number of times the circuit is repeated can be specified in the ``execute`` function, via the ``shots`` keyword.\n\n\n```python\n# Use Aer's qasm_simulator\nbackend_sim = Aer.get_backend('qasm_simulator')\n\n# Execute the circuit on the qasm simulator.\n# We've set the number of repeats of the circuit\n# to be 1024, which is the default.\njob_sim = execute(qc, backend_sim, shots=1024)\n\n# Grab the results from the job.\nresult_sim = job_sim.result()\n```\n\nOnce you have a result object, you can access the counts via the function `get_counts(circuit)`. This gives you the _aggregated_ binary outcomes of the circuit you submitted.\n\n\n```python\ncounts = result_sim.get_counts(qc)\nprint(counts)\n```\n\n {'111': 524, '000': 500}\n\n\nApproximately 50 percent of the time the output bitstring is 000. Qiskit Terra also provides a function `plot_histogram` which allows you to view the outcomes. \n\n\n```python\nfrom qiskit.tools.visualization import plot_histogram\nplot_histogram(counts)\n```\n\nThe estimated outcome probabilities $\\mathrm{Pr}(000)$ and $\\mathrm{Pr}(111)$ are computed by taking the aggregate counts and dividing by the number of shots (times the circuit was repeated). Try changing the ``shots`` keyword in the ``execute`` function and see how the estimated probabilities change.\n\n\n\nRunning circuits using the IBMQ provider\n=======================\n\nTo faciliate access to real quantum computing hardware, we have provided a simple API interface.\nTo access IBMQ devices, you'll need an API token. For the public IBM Q devices, you can generate an API token [here](https://quantumexperience.ng.bluemix.net/qx/account/advanced) (create an account if you don't already have one). For Q Network devices, login to the q-console, click your hub, group, and project, and expand \"Get Access\" to generate your API token and access url.\n\nOur IBMQ provider lets you run your circuit on real devices or on our HPC simulator. Currently, this provider exists within Qiskit, and can be imported as shown below. For details on the provider, see [The IBMQ Provider](the_ibmq_provider.ipynb).\n\n\n```python\nfrom qiskit import IBMQ\n```\n\nAfter generating your API token, call, `IBMQ.save_account('MY_TOKEN')`. For Q Network users, you'll also need to include your access url: `IBMQ.save_account('MY_TOKEN', 'URL')`\n\nThis will store your IBMQ credentials in a local file. Unless your registration information has changed, you only need to do this once. You may now load your accounts by calling,\n\n\n```python\nIBMQ.load_accounts()\n```\n\nOnce your account has been loaded, you can view the list of backends available to you.\n\n\n```python\nprint(\"Available backends:\")\nIBMQ.backends()\n```\n\n Available backends:\n\n\n\n\n\n [,\n ,\n ,\n ,\n ]\n\n\n\n### Running circuits on real devices\n\nToday's quantum information processors are small and noisy, but are advancing at a fast pace. They provide a great opportunity to explore what [noisy, intermediate-scale quantum (NISQ)](https://arxiv.org/abs/1801.00862) computers can do.\n\nThe IBMQ provider uses a queue to allocate the devices to users. We now choose a device with the least busy queue which can support our program (has at least 3 qubits).\n\n\n```python\nfrom qiskit.backends.ibmq import least_busy\n\nlarge_enough_devices = IBMQ.backends(filters=lambda x: x.configuration()['n_qubits'] > 3 and\n not x.configuration()['simulator'])\nbackend = least_busy(large_enough_devices)\nprint(\"The best backend is \" + backend.name())\n```\n\n The best backend is ibmqx4\n\n\nTo run the circuit on the backend, we need to specify the number of shots and the number of credits we are willing to spend to run the circuit. Then, we execute the circuit on the backend using the ``execute`` function.\n\n\n```python\nshots = 1024 # Number of shots to run the program (experiment); maximum is 8192 shots.\nmax_credits = 3 # Maximum number of credits to spend on executions. \n\njob_exp = execute(qc, backend=backend, shots=shots, max_credits=max_credits)\n```\n\n``job_exp`` has a ``.result()`` method that lets us get the results from running our circuit.\n\n
    \nNote: When the .result() method is called, the code block will wait until the job has finished before releasing the cell.\n
    \n\n\n```python\nresult_exp = job_exp.result()\n```\n\nLike before, the counts from the execution can be obtained using ```get_counts(qc)``` \n\n\n```python\ncounts_exp = result_exp.get_counts(qc)\nplot_histogram(counts_exp)\n```\n\n### Simulating circuits using a HPC simulator\n\nThe IBMQ provider also comes with a remote optimized simulator called ``ibmq_qasm_simulator``. This remote simulator is capable of simulating up to 32 qubits. It can be used the \nsame way as the remote real backends. \n\n\n```python\nbackend = IBMQ.get_backend('ibmq_qasm_simulator')\n```\n\n\n```python\nshots = 1024 # Number of shots to run the program (experiment); maximum is 8192 shots.\nmax_credits = 3 # Maximum number of credits to spend on executions. \n\njob_hpc = execute(qc, backend=backend, shots=shots, max_credits=max_credits)\n```\n\n\n```python\nresult_hpc = job_hpc.result()\n```\n\n\n```python\ncounts_hpc = result_hpc.get_counts(qc)\nplot_histogram(counts_hpc)\n```\n\n### Retrieving a previously ran job\n\nIf your experiment takes longer to run then you have time to wait around, or if you simply want to retrieve old jobs back, the IBMQ backends allow you to do that.\nFirst you would need to note your job's ID:\n\n\n```python\njobID = job_exp.job_id()\n\nprint('JOB ID: {}'.format(jobID)) \n```\n\n JOB ID: 5be8ae5e17436b0052751909\n\n\nGiven a job ID, that job object can be later reconstructed from the backend using retrieve_job:\n\n\n```python\njob_get=backend.retrieve_job(jobID)\n```\n\nand then the results can be obtained from the new job object. \n\n\n```python\njob_get.result().get_counts(qc)\n```\n\n\n\n\n {'00000': 367,\n '00001': 10,\n '00010': 30,\n '00011': 27,\n '00100': 22,\n '00101': 83,\n '00110': 50,\n '00111': 435}\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "a66335b7b56f887b295617f60add23e371790747", "size": 224616, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "qiskit/basics/getting_started_with_qiskit_terra.ipynb", "max_stars_repo_name": "antoniomezzacapo/qiskit-tutorial", "max_stars_repo_head_hexsha": "0d9c17bff8efa37d6e2cfb47c9430fee67837f95", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2018-08-29T20:55:57.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-04T01:38:23.000Z", "max_issues_repo_path": "qiskit/basics/getting_started_with_qiskit_terra.ipynb", "max_issues_repo_name": "Rahulmisal27/qiskit-tutorials", "max_issues_repo_head_hexsha": "31ea17ed50f8af83b6c3fa31c10a3ea326d03f8b", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "qiskit/basics/getting_started_with_qiskit_terra.ipynb", "max_forks_repo_name": "Rahulmisal27/qiskit-tutorials", "max_forks_repo_head_hexsha": "31ea17ed50f8af83b6c3fa31c10a3ea326d03f8b", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-03-08T12:13:31.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-08T12:13:31.000Z", "avg_line_length": 233.246105919, "max_line_length": 123204, "alphanum_fraction": 0.9170317342, "converted": true, "num_tokens": 4757, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.4323492090936783}} {"text": "# CS109B Data Science 2: Advanced Topics in Data Science \n\n## Lab 4 - Bayesian Analysis\n\n**Harvard University**
    \n**Spring 2020**
    \n**Instructors:** Mark Glickman, Pavlos Protopapas, and Chris Tanner
    \n**Lab Instructors:** Chris Tanner and Eleni Angelaki Kaxiras
    \n**Content:** Eleni Angelaki Kaxiras\n\n---\n\n\n```python\n## RUN THIS CELL TO PROPERLY HIGHLIGHT THE EXERCISES\nimport requests\nfrom IPython.core.display import HTML\nstyles = requests.get(\"https://raw.githubusercontent.com/Harvard-IACS/2019-CS109B/master/content/styles/cs109.css\").text\nHTML(styles)\n```\n\n\n\n\n\n\n\n\n\n\n\n```python\nimport pymc3 as pm\nfrom pymc3 import summary\n```\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport scipy.stats as stats\nimport pandas as pd\n%matplotlib inline \n\nimport warnings\nwarnings.filterwarnings('ignore')\n```\n\n\n```python\nprint('Running on PyMC3 v{}'.format(pm.__version__))\n```\n\n Running on PyMC3 v3.8\n\n\n\n```javascript\n%%javascript\nIPython.OutputArea.auto_scroll_threshold = 20000;\n```\n\n\n \n\n\n\n\n## Learning Objectives\n\nBy the end of this lab, you should be able to:\n* Understand how probability distributions work.\n* Apply Bayes Rule in calculating probabilities.\n* Understand how to apply Bayesian analysis using PyMC3\n* Avoid getting fired when talking to your Bayesian employer.\n\n**This lab corresponds to Lectures 6, 7, and 8, and maps to Homework 3.**\n\n## Table of Contents\n\n1. The Bayesian Way of Thinking or Is this a Fair Coin?\n2. [Intro to `pyMC3`](#pymc3). \n3. [Bayesian Linear Regression](#blr).\n4. [Try this at Home: Example on Mining Disasters](#no4).\n\n## 1. The Bayesian way of Thinking\n\n```\nHere is my state of knowledge about the situation. Here is some data, I am now going to revise my state of knowledge.\n```\n\n
    Table Exercise: Discuss the statement above with your table mates and make sure everyone understands what it means and what constitutes Bayesian way of thinking. Finally, count the Bayesians among you.
    \n\n### A. Bayes Rule\n\n\\begin{equation}\n\\label{eq:bayes} \nP(A|\\textbf{B}) = \\frac{P(\\textbf{B} |A) P(A) }{P(\\textbf{B})} \n\\end{equation}\n\n$P(A|\\textbf{B})$ is the **posterior** distribution, prob(hypothesis | data) \n\n$P(\\textbf{B} |A)$ is the **likelihood** function, how probable is my data **B** for different values of the parameters\n\n$P(A)$ is the marginal probability to observe the data, called the **prior**, this captures our belief about the data before observing it.\n\n$P(\\textbf{B})$ is the marginal distribution (sometimes called marginal likelihood)\n\n
    \n
    Table Exercise: Solve the Monty Hall Paradox using Bayes Rule.
    \n\n\n\nYou are invited to play a game. There are 3 doors behind **one** of which are the keys to a brand new red Tesla. There is a goat behind each of the other two. \n\nYou are asked to pick one door, and let's say you pick **Door1**. The host knows where the keys are. Of the two remaining closed doors, he will always open the door that has a goat behind it. He'll say \"I will do you a favor and open **Door2**\". So he opens Door2 inside which there is, of course, a goat. He now asks you, do you want to open the initial Door you chose or change to **Door3**? Generally, in this game, when you are presented with this choice should you swap the doors?\n\n**Initial Steps:**\n- Start by defining the `events` of this probabilities game. One definition is:\n \n - $A_i$: car is behind door $i$ \n \n - $B_i$ host opens door $i$\n \n$i\\in[1,2,3]$\n \n- In more math terms, the question is: is the probability that the price is behind **Door 1** higher than the probability that the price is behind **Door2**, given that an event **has occured**?\n\n$P(A_1|B_2) = \\frac{P(B_2|A_1)P(A_1)}{P(B_2)} = \\frac{P(B_2|A_1)P(A_1)}{P(B_2|A_1)P(A_1) + P(B_2|A_2)P(A_2) + P(B_2|A_3)P(A_3)} = \\frac{\\frac{1}{2} \\cdot \\frac{1}{3}}{\\frac{1}{2}} = \\frac{1}{3}$\n\n$P(B_2|A_2) = 0$ because the host won't open a door with a prize behind it.\n\n$P(B_2|A_3) = 1$ because the host won't open door 2, which is the one you picked.\n\n$P(B_2) = P(B_2|A_1)P(A_1) + P(B_2|A_2)P(A_2) + P(B_2|A_3)P(A_3) = \\frac{1}{2} \\cdot \\frac{1}{3} + 0 + 1\\cdot \\frac{1}{3} = \\frac{1}{2}$\n\n### B. Bayes Rule written with Probability Distributions\n\nWe have data that we believe come from an underlying distribution of unknown parameters. If we find those parameters, we know everything about the process that generated this data and we can make inferences (create new data).\n\n$\\begin{equation}\n\\label{eq:bayes} \nP(\\theta|\\textbf{D}) = \\frac{P(\\textbf{D} |\\theta) P(\\theta) }{P(\\textbf{D})} \n\\end{equation}$\n\n#### But what is $\\theta \\;$?\n\n$\\theta$ is an unknown yet fixed set of parameters. In Bayesian inference we express our belief about what $\\theta$ might be and instead of trying to guess $\\theta$ exactly, we look for its **probability distribution**. What that means is that we are looking for the **parameters** of that distribution. For example, for a Poisson distribution our $\\theta$ is only $\\lambda$. In a normal distribution, our $\\theta$ is often just $\\mu$ and $\\sigma$.\n\n### C. A review of Common Probability Distributions\n\n#### Discrete Distributions\n\nThe random variable has a **probability mass function (pmf)** which measures the probability that our random variable will take a specific value $y$, denoted $P(Y=y)$.\n\n- **Bernoulli** (binary outcome, success has probability $\\theta$, $one$ trial):\n$\nP(Y=k) = \\theta^k(1-\\theta)^{1-k}\n$\n
    \n- **Binomial** (binary outcome, success has probability $\\theta$, $n$ trials):\n\\begin{equation}\nP(Y=k) = {{n}\\choose{k}} \\cdot \\theta^k(1-\\theta)^{n-k}\n\\end{equation}\n\n*Note*: Binomial(1,$p$) = Bernouli($p$)\n
    \n- **Negative Binomial**\n
    \n- **Poisson** (counts independent events occurring at a rate)\n\\begin{equation}\nP\\left( Y=y|\\lambda \\right) = \\frac{{e^{ - \\lambda } \\lambda ^y }}{{y!}}\n\\end{equation}\ny = 0,1,2,...\n
    \n- **Discrete Uniform** \n
    \n- **Categorical, or Multinulli** (random variables can take any of K possible categories, each having its own probability; this is a generalization of the Bernoulli distribution for a discrete variable with more than two possible outcomes, such as the roll of a die)\n
    \n- **Dirichlet-multinomial** (a generalization of the beta distribution for many variables)\n\n#### Continuous Distributions\n\nThe random variable has a **probability density function (pdf)**.\n- **Uniform** (variable equally likely to be near each value in interval $(a,b)$)\n\\begin{equation}\nP(X = x) = \\frac{1}{b - a}\n\\end{equation}\nanywhere within the interval $(a, b)$, and zero elsewhere.\n
    \n- **Normal** (a.k.a. Gaussian)\n\\begin{equation}\nX \\sim \\mathcal{N}(\\mu,\\,\\sigma^{2})\n\\end{equation} \n\n A Normal distribution can be parameterized either in terms of precision $\\tau$ or standard deviation ($\\sigma^{2}$. The link between the two is given by\n\\begin{equation}\n\\tau = \\frac{1}{\\sigma^{2}}\n\\end{equation}\n - Mean $\\mu$\n - Variance $\\frac{1}{\\tau}$ or $\\sigma^{2}$\n - Parameters: `mu: float`, `sigma: float` or `tau: float`\n
    \n- **Beta** (variable ($\\theta$) taking on values in the interval $[0,1]$, and parametrized by two positive parameters, $\\alpha$ and $\\beta$ that control the shape of the distribution. \n \n*Note:*Beta is a good distribution to use for priors (beliefs) because its range is $[0,1]$ which is the natural range for a probability and because we can model a wide range of functions by changing the $\\alpha$ and $\\beta$ parameters.\n\n\\begin{equation}\n\\label{eq:beta} \nP(\\theta) = \\frac{1}{B(\\alpha, \\beta)} {\\theta}^{\\alpha - 1} (1 - \\theta)^{\\beta - 1} \\propto {\\theta}^{\\alpha - 1} (1 - \\theta)^{\\beta - 1}\n\\end{equation}\n\n\nwhere the normalisation constant, $B$, is a beta function of $\\alpha$ and $\\beta$,\n\n\n\\begin{equation}\nB(\\alpha, \\beta) = \\int_{t=0}^1 t^{\\alpha - 1} (1 - t)^{\\beta - 1} dt.\n\\end{equation}\n
    \n- **Exponential**\n
    \n- **Gamma**\n\n\n\n #### Code Resources:\n - Statistical Distributions in numpy/scipy: [scipy.stats](https://docs.scipy.org/doc/scipy/reference/stats.html)\n - Statistical Distributions in pyMC3: [distributions in PyMC3](https://docs.pymc.io/api/distributions.html) (we will see those below).\n\n
    Exercise: Plot a Discrete variable
    \n\nChange the value of $\\mu$ in the Poisson PMF and see how the plot changes. Remember that the y-axis in a discrete probability distribution shows the probability of the random variable having a specific value in the x-axis.\n\n\\begin{equation}\nP\\left( X=k \\right) = \\frac{{e^{ - \\mu } \\mu ^k }}{{k!}}\n\\end{equation}\n\n**stats.poisson.pmf(x, mu)** $\\mu$(mu) is our $\\theta$ in this case.\n\n\n```python\nplt.style.use('seaborn-darkgrid')\nx = np.arange(0, 30)\nfor m in [0.5, 3, 8]:\n pmf = stats.poisson.pmf(x, m)\n plt.plot(x, pmf, 'o', alpha=0.5, label='$\\mu$ = {}'.format(m))\nplt.xlabel('random variable', fontsize=12)\nplt.ylabel('probability', fontsize=12)\nplt.legend(loc=1)\nplt.ylim=(-0.1)\nplt.show()\n```\n\n\n```python\npmf\n```\n\n\n\n\n array([3.35462628e-04, 2.68370102e-03, 1.07348041e-02, 2.86261442e-02,\n 5.72522885e-02, 9.16036616e-02, 1.22138215e-01, 1.39586532e-01,\n 1.39586532e-01, 1.24076917e-01, 9.92615338e-02, 7.21902064e-02,\n 4.81268043e-02, 2.96164949e-02, 1.69237114e-02, 9.02597941e-03,\n 4.51298971e-03, 2.12375986e-03, 9.43893272e-04, 3.97428746e-04,\n 1.58971498e-04, 6.05605708e-05, 2.20220258e-05, 7.65983504e-06,\n 2.55327835e-06, 8.17049071e-07, 2.51399714e-07, 7.44888042e-08,\n 2.12825155e-08, 5.87103876e-09])\n\n\n\n\n```python\n# same for binomial\nplt.style.use('seaborn-darkgrid')\nx = np.arange(0, 22)\nns = [10, 17]\nps = [0.5, 0.7]\nfor n, p in zip(ns, ps):\n pmf = stats.binom.pmf(x, n, p)\n plt.plot(x, pmf, 'o', alpha=0.5, label='n = {}, p = {}'.format(n, p))\nplt.xlabel('x', fontsize=14)\nplt.ylabel('f(x)', fontsize=14)\nplt.legend(loc=1)\nplt.show()\n```\n\n\n```python\n# discrete uniform\nplt.style.use('seaborn-darkgrid')\nls = [0]\nus = [3] # watch out, this number can only be integer!\nfor l, u in zip(ls, us):\n x = np.arange(l, u+1)\n pmf = [1.0 / (u - l + 1)] * len(x)\n plt.plot(x, pmf, '-o', label='lower = {}, upper = {}'.format(l, u))\nplt.xlabel('x', fontsize=12)\nplt.ylabel('probability P(x)', fontsize=12)\nplt.legend(loc=1)\nplt.show()\n```\n\n
    Exercise: Plot a continuous variable
    \n\nChange the value of $\\mu$ in the Uniform PDF and see how the plot changes.\n \nRemember that the y-axis in a continuous probability distribution does not shows the actual probability of the random variable having a specific value in the x-axis because that probability is zero!. Instead, to see the probability that the variable is within a small margin we look at the integral below the curve of the PDF.\n\nThe uniform is often used as a noninformative prior.\n\n```\nUniform - numpy.random.uniform(a=0.0, b=1.0, size)\n```\n\n$\\alpha$ and $\\beta$ are our parameters. `size` is how many tries to perform.\nOur $\\theta$ is basically the combination of the parameters a,b. We can also call it \n\\begin{equation}\n\\mu = (a+b)/2\n\\end{equation}\n\n\n```python\nfrom scipy.stats import uniform\n\nr = uniform.rvs(size=1000) # random numbers from a uniform distribution\nplt.plot(r, uniform.pdf(r),'r-', lw=5, alpha=0.6, label='uniform pdf')\nplt.hist(r, density=True, histtype='stepfilled', alpha=0.2)\nplt.ylabel(r'probability density')\nplt.xlabel(f'random variable')\nplt.legend(loc='best', frameon=False)\nplt.show()\n```\n\n\n```python\nfrom scipy.stats import beta\n\nalphas = [0.5, 1.5, 3.0]\nbetas = [0.5, 1.5, 3.0]\nx = np.linspace(0, 1, 1000) \ncolors = ['red', 'green', 'blue']\n\nfig, ax = plt.subplots(figsize=(8, 5))\n\nfor a, b, colors in zip(alphas, betas, colors):\n dist = beta(a, b)\n plt.plot(x, dist.pdf(x), c=colors,\n label=f'a={a}, b={b}')\n\nax.set_ylim(0, 3)\n\nax.set_xlabel(r'$\\theta$')\nax.set_ylabel(r'$p(\\theta|\\alpha,\\beta)$')\nax.set_title('Beta Distribution')\n\nax.legend(loc='best')\nfig.show();\n```\n\nNormal distribution\n\n\n```python\nplt.style.use('seaborn-darkgrid')\nx = np.linspace(-5, 5, 1000)\nmus = [0., 0., 0., -2.]\nsigmas = [0.4, 1., 2., 0.4]\nfor mu, sigma in zip(mus, sigmas):\n pdf = stats.norm.pdf(x, mu, sigma)\n plt.plot(x, pdf, label=r'$\\mu$ = '+ f'{mu},' + r'$\\sigma$ = ' + f'{sigma}') \nplt.xlabel('random variable', fontsize=12)\nplt.ylabel('probability density', fontsize=12)\nplt.legend(loc=1)\nplt.show()\n```\n\nUniform\n\n\n```python\nplt.style.use('seaborn-darkgrid')\nx = np.linspace(-5, 5, 1000)\nmus = [0., 0., 0., -2.] # mean\nsigmas = [0.4, 1., 2., 0.4] # std\nfor mu, sigma in zip(mus, sigmas):\n plt.plot(x, uniform.pdf(x, mu, sigma), lw=5, alpha=0.4, \\\n label=r'$\\mu$ = '+ f'{mu},' + r'$\\sigma$ = ' + f'{sigma}')\nplt.xlabel('random variable', fontsize=12)\nplt.ylabel('probability density', fontsize=12)\nplt.legend(loc=1)\nplt.show()\n```\n\n### D. Is this a Fair Coin?\n\nWe do not want to promote gambling but let's say you visit the casino in **Monte Carlo**. You want to test your theory that casinos are dubious places where coins have been manipulated to have a larger probability for tails. So you will try to estimate how fair a coin is based on 100 flips.
    \nYou begin by flipping the coin. You get either Heads ($H$) or Tails ($T$) as our observed data and want to see if your posterior probabilities change as you obtain more data, that is, more coin flips. A nice way to visualize this is to plot the posterior probabilities as we observe more flips (data). \n\nWe will be using Bayes rule. $\\textbf{D}$ is our data.\n\n$\\begin{equation}\n\\label{eq:bayes} \nP(\\theta|\\textbf{D}) = \\frac{P(\\textbf{D} |\\theta) P(\\theta) }{P(\\textbf{D})} \n\\end{equation}$\n\nbeta-binomial conjugacy.\n\nprior: $Beta(\\alpha, \\beta)$, $\\alpha$ is number of successes, $\\beta$ is the number of failures\n\nData: out of n tosses, we see k heads\n\nposterior: $Beta(\\alpha + k, \\beta + n - k)$\n\nIn the case of a coin toss when we observe $k$ heads in $n$ tosses:\n$\\begin{equation}\n\\label{eq:bayes} \nP(\\theta|\\textbf{k}) = Beta(\\alpha + \\textbf{k}, \\beta + n - \\textbf{k}) \n\\end{equation}$\n\nwe can say that $\\alpha$ and $\\beta$ play the roles of a \"prior number of heads\" and \"prior number of tails\".\n\n\n```python\n# play with the priors - here we manually set them but we could be sampling from a separate Beta\ntrials = np.array([0, 1, 3, 5, 10, 15, 20, 100, 200, 300])\nheads = np.array([0, 1, 2, 4, 8, 10, 10, 50, 180, 150])\nx = np.linspace(0, 1, 100)\n\n# for simplicity we set a,b=1\n\nplt.figure(figsize=(10,8))\nfor k, N in enumerate(trials):\n sx = plt.subplot(len(trials)/2, 2, k+1)\n posterior = stats.beta.pdf(x, 1 + heads[k], 1 + trials[k] - heads[k]) \n plt.plot(x, posterior, alpha = 0.5, label=f'{trials[k]} tosses\\n {heads[k]} heads');\n plt.fill_between(x, 0, posterior, color=\"#348ABD\", alpha=0.4) \n plt.legend(loc='upper left', fontsize=10)\n plt.legend()\n plt.autoscale(tight=True)\n \nplt.suptitle(\"Posterior probabilities for coin flips\", fontsize=15);\nplt.tight_layout()\nplt.subplots_adjust(top=0.88)\n```\n\nP(head) = 0.5 for 20 tosses and 10 heads (as the center of posterior distribution is centered at 0.5). MLE = 10/20 = 0.5\n\nAs we increase number of trials, heads is around half, so this coin is fair.\n\n [Top](#top)\n\n## 2. Introduction to `pyMC3`\n \nPyMC3 is a Python library for programming Bayesian analysis, and more specifically, data creation, model definition, model fitting, and posterior analysis. It uses the concept of a `model` which contains assigned parametric statistical distributions to unknown quantities in the model. Within models we define random variables and their distributions. A distribution requires at least a `name` argument, and other `parameters` that define it. You may also use the `logp()` method in the model to build the model log-likelihood function. We define and fit the model.\n\nPyMC3 includes a comprehensive set of pre-defined statistical distributions that can be used as model building blocks. Although they are not meant to be used outside of a `model`, you can invoke them by using the prefix `pm`, as in `pm.Normal`. \n\n#### Markov Chain Monte Carlo (MCMC) Simulations\n\nPyMC3 uses the **No-U-Turn Sampler (NUTS)** and the **Random Walk Metropolis**, two Markov chain Monte Carlo (MCMC) algorithms for sampling in posterior space. Monte Carlo gets into the name because when we sample in posterior space, we choose our next move via a pseudo-random process. NUTS is a sophisticated algorithm that can handle a large number of unknown (albeit continuous) variables.\n\n`pm.distributions` will have two important methods: `random()` and `logp()` with the following signatures:\n\n`logp()` method to return a log-probability evaluated at the passed value argument. This method is used internally by all of the inference methods to calculate the model log-probability that is used for fitting models. The `random()` method is used to simulate values from the variable, and is used internally for posterior predictive checks\n\n\n```python\nwith pm.Model() as model:\n # a scalar variable follows normal distribution\n z = pm.Normal('z', mu=0., sigma=5.) # name, params \n x = pm.Normal('x', mu=z, sigma=1., observed=5.) \nprint(x.logp({'z': 2.5})) \nprint(z.random(10, 100)[:10]) \n```\n\n -4.043938533204672\n [ 7.17624957 1.88396497 -3.84377077 -4.51319065 2.15195803 -0.74173189\n -0.06213591 -1.40335086 -3.67668208 0.46645415]\n\n\n\n```python\nz.random(10, 100) # point, size\n```\n\n\n\n\n array([ 4.20299819, 6.67756286, 1.86397596, 4.69772266,\n -1.74075718, -6.1587605 , 4.01425601, 1.47212497,\n -0.95908539, -0.75957457, -0.9965861 , 4.40711468,\n 0.86476568, -1.78865136, 2.43830109, 7.48101036,\n -3.83437749, 0.97882911, -4.60062897, 3.02594797,\n -7.87301058, -2.37757347, -5.25710143, -2.86329763,\n 4.04072601, -10.79232415, -0.17469766, 6.95349508,\n 5.65213913, -0.74215824, -0.26543113, 4.99100215,\n 3.86390824, -2.460854 , 7.62924736, 1.20906703,\n 4.80720307, 0.91501121, -4.11779673, -6.50301219,\n -0.53290222, -1.55428558, 4.79833462, 1.57705118,\n -1.6667158 , 8.146217 , 2.22696357, -0.45245175,\n -1.71733788, -6.82240357, 1.4551234 , -1.83267745,\n 11.78864822, -2.17962573, 4.63795644, -0.63709271,\n 1.15939193, 1.33588019, 3.33174281, -2.39045536,\n -4.36408123, -1.5982427 , 7.23550659, -0.48821265,\n 0.73843087, 3.23945607, 1.63729074, -1.45938923,\n 4.91725175, -2.90260851, -3.29998453, -13.40968767,\n -3.12695605, 3.36211579, -7.19234678, 3.37934005,\n 2.07161948, 1.68049247, -3.56791466, -2.78515778,\n -0.77515644, 8.75054104, 5.93503912, 7.73719646,\n -3.03921893, 0.6343688 , 6.26068286, -5.8339571 ,\n 2.25379384, -5.4811024 , 3.324151 , -0.62969373,\n 8.16379873, 0.04000131, -2.90106212, 5.08299257,\n 2.87763086, 3.48026559, 1.31247562, 2.81561549])\n\n\n\n\n```python\nmodel\n```\n\n\n\n\n$$\n \\begin{array}{rcl}\n \\text{z} &\\sim & \\text{Normal}(\\mathit{mu}=0.0,~\\mathit{sigma}=5.0)\\\\\\text{x} &\\sim & \\text{Normal}(\\mathit{mu}=\\text{z},~\\mathit{sigma}=1.0)\n \\end{array}\n $$\n\n\n\n**References**:\n\n- *Salvatier J, Wiecki TV, Fonnesbeck C. 2016. Probabilistic programming in Python using PyMC3. PeerJ Computer Science 2:e55* [(https://doi.org/10.7717/peerj-cs.55)](https://doi.org/10.7717/peerj-cs.55)\n- [Distributions in PyMC3](https://docs.pymc.io/api/distributions.html)\n- [More Details on Distributions](https://docs.pymc.io/developer_guide.html)\n\nInformation about PyMC3 functions including descriptions of distributions, sampling methods, and other functions, is available via the `help` command.\n\n\n```python\n#help(pm.Poisson)\n```\n\n [Top](#top)\n\n## 3. Bayesian Linear Regression\n\nLet's say we want to predict outcomes Y as normally distributed observations with an expected value $mu$ that is a linear function of two predictor variables, $\\bf{x}_1$ and $\\bf{x}_2$.\n\n\\begin{equation}\n\\mu = \\alpha + \\beta_1 \\bf{x}_1 + \\beta_2 x_2 \n\\end{equation}\n\n\\begin{equation}\nY \\sim \\mathcal{N}(\\mu,\\,\\sigma^{2})\n\\end{equation} \n\nwhere $\\sigma^2$ represents the measurement error. \n\nIn this example, we will use $\\sigma^2 = 10$\n\nWe also choose the parameters as normal distributions:\n\n\\begin{eqnarray}\n\\alpha \\sim \\mathcal{N}(0,\\,10) \\\\\n\\beta_i \\sim \\mathcal{N}(0,\\,10) \\\\\n\\sigma^2 \\sim |\\mathcal{N}(0,\\,10)|\n\\end{eqnarray} \n\nWe will artificially create the data to predict on. We will then see if our model predicts them correctly.\n\n\n```python\n# Initialize random number generator\nnp.random.seed(123)\n\n# True parameter values\nalpha, sigma = 1, 1 # in bayesian world, the answer is the mean\nbeta = [1, 2.5]\n\n# Size of dataset\nsize = 100\n\n# Predictor variable\nX1 = np.linspace(0, 1, size)\nX2 = np.linspace(0,.2, size)\n\n# Simulate outcome variable\nY = alpha + beta[0]*X1 + beta[1]*X2 + np.random.randn(size)*sigma\n\nfig, ax = plt.subplots(1,2, figsize=(10,6), sharex=True)\nax[0].scatter(X1,Y)\nax[1].scatter(X2,Y)\nax[0].set_xlabel(r'$x_1$', fontsize=14) \nax[0].set_ylabel(r'$Y$', fontsize=14)\nax[1].set_xlabel(r'$x_2$', fontsize=14) \nax[1].set_ylabel(r'$Y$', fontsize=14)\n```\n\n`observed`: observed data to be loaded into the distribution\n\nGiven distributions of $\\alpha$, $\\beta_i$ and $\\sigma$\n\nprior: prior distribution of $\\mu = \\alpha + \\beta_1 \\bf{x}_1 + \\beta_2 x_2$\n\nlikelihood: Ys follows a normal distribution with mean $\\mu = \\alpha + \\beta_1 \\bf{x}_1 + \\beta_2 x_2$ and variance $\\sigma^2$\n\nposterior: posterior distribution of $\\mu$ after update of observed data\n\n\n```python\nfrom pymc3 import Model, Normal, HalfNormal\n\nbasic_model = Model()\n\nwith basic_model:\n\n # Priors for unknown model parameters, specifically create stochastic random variables \n # with Normal prior distributions for the regression coefficients,\n # and a half-normal distribution for the standard deviation of the observations, \u03c3.\n alpha = Normal('alpha', mu=0, sd=10)\n beta = Normal('beta', mu=0, sd=10, shape=2)\n sigma = HalfNormal('sigma', sd=1)\n\n # Expected value of outcome - posterior\n mu = alpha + beta[0]*X1 + beta[1]*X2\n\n # Likelihood (sampling distribution) of observations\n # observed=Y: takes into account my data (simulated response variable in the previous chunk)\n Y_obs = Normal('Y_obs', mu=mu, sd=sigma, observed=Y)\n```\n\nThe maximum a posteriori (MAP) estimate for a model, is the mode of the posterior distribution and is generally found using numerical optimization methods. This is often fast and easy to do, but only gives a point estimate for the parameters and can be biased if the mode isn\u2019t representative of the distribution. PyMC3 provides this functionality with the find_MAP function.\n\n\n```python\n# model fitting with sampling\nfrom pymc3 import NUTS, sample, find_MAP\nfrom scipy import optimize\n\nwith basic_model:\n\n # obtain starting values via MAP (maximum a posterior probability (MAP))\n start = find_MAP(fmin=optimize.fmin_powell)\n\n # instantiate sampler\n step = NUTS(scaling=start)\n\n # draw 2000 posterior samples\n trace = sample(2000, step, start=start)\n```\n\n logp = -164.5: 5%|\u258c | 271/5000 [00:00<00:02, 1579.74it/s] \n\n\n Optimization terminated successfully.\n Current function value: 164.496957\n Iterations: 6\n Function evaluations: 271\n\n\n Multiprocess sampling (2 chains in 2 jobs)\n NUTS: [sigma, beta, alpha]\n Sampling 2 chains, 0 divergences: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 5000/5000 [00:14<00:00, 339.94draws/s]\n The number of effective samples is smaller than 10% for some parameters.\n\n\nalpha centered around 1 (true parameter)\n\nall pararams have centered around the true value\n\n\n```python\nfrom pymc3 import traceplot\n\ntraceplot(trace);\n```\n\n\n```python\nresults = pm.summary(trace, \n var_names=['alpha', 'beta', 'sigma'])\nresults\n```\n\n\n\n\n
    \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    meansdhpd_3%hpd_97%mcse_meanmcse_sdess_meaness_sdess_bulkess_tailr_hat
    alpha1.0100.2250.6061.4520.0050.0032248.02248.02246.02149.01.0
    beta[0]1.3511.983-2.3295.0060.1040.078362.0320.0361.0606.01.0
    beta[1]0.8989.812-16.50519.7380.5050.358377.0377.0375.0583.01.0
    sigma1.1450.0820.9941.2980.0010.0013573.03542.03586.02889.01.0
    \n
    \n\n\n\nThis linear regression example is from the original paper on PyMC3: *Salvatier J, Wiecki TV, Fonnesbeck C. 2016. Probabilistic programming in Python using PyMC3. PeerJ Computer Science 2:e55 https://doi.org/10.7717/peerj-cs.55*\n\n [Top](#top)\n\n## 4. Try this at Home: Example on Mining Disasters\nWe will go over the classical `mining disasters from 1851 to 1962` dataset. \n\nThis example is from the [pyMC3 Docs](https://docs.pymc.io/notebooks/getting_started.html).\n\n\n```python\nimport pandas as pd\ndisaster_data = pd.Series([4, 5, 4, 0, 1, 4, 3, 4, 0, 6, 3, 3, 4, 0, 2, 6,\n 3, 3, 5, 4, 5, 3, 1, 4, 4, 1, 5, 5, 3, 4, 2, 5,\n 2, 2, 3, 4, 2, 1, 3, np.nan, 2, 1, 1, 1, 1, 3, 0, 0,\n 1, 0, 1, 1, 0, 0, 3, 1, 0, 3, 2, 2, 0, 1, 1, 1,\n 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 1, 0, 2,\n 3, 3, 1, np.nan, 2, 1, 1, 1, 1, 2, 4, 2, 0, 0, 1, 4,\n 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1])\nfontsize = 12\nyears = np.arange(1851, 1962)\nplt.figure(figsize=(10,5))\n#plt.scatter(years, disaster_data); \nplt.bar(years, disaster_data)\nplt.ylabel('Disaster count', size=fontsize)\nplt.xlabel('Year', size=fontsize);\nplt.title('Was there a Turning Point in Mining disasters from 1851 to 1962?', size=15);\n```\n\n#### Building the model\n\n**Step1:** We choose the probability model for our experiment. Occurrences of disasters in the time series is thought to follow a **Poisson** process with a large **rate** parameter in the early part of the time series, and from one with a smaller **rate** in the later part. We are interested in locating the change point in the series, which perhaps is related to changes in mining safety regulations. \n\n```\ndisasters = pm.Poisson('disasters', rate, observed=disaster_data)\n```\n\nWe have two rates, `early_rate` if $t<=s$, and `late_rate` if $t>s$, where $s$ is the year the switch was made (a.k.a. the `switchpoint`). \n\n**Step2:** Choose a prior distributions of the two rates, what we believe the rates were before we observed the data, and the switchpoint. We choose Exponential.\n```\nearly_rate = pm.Exponential('early_rate', 1)\n```\n\nThe parameters of this model are: \n\n\n**Note:** Watch for missing values. Missing values are handled transparently by passing a MaskedArray or a pandas.DataFrame. Behind the scenes, another random variable, disasters.missing_values is created to model the missing values. If you pass a np.array with missing values you will get an error.\n\n\n```python\nwith pm.Model() as disaster_model:\n\n # discrete\n switchpoint = pm.DiscreteUniform('switchpoint', lower=years.min(), upper=years.max(), testval=1900)\n\n # Priors for pre- and post-switch rates number of disasters\n early_rate = pm.Exponential('early_rate', 1)\n late_rate = pm.Exponential('late_rate', 1)\n\n # our theta - allocate appropriate Poisson rates to years before and after current\n # switch is an `if` statement in puMC3\n rate = pm.math.switch(switchpoint >= years, early_rate, late_rate)\n\n # our observed data as a likelihood function of the `rate` parameters\n # shows how we think our data is distributed\n disasters = pm.Poisson('disasters', rate, observed=disaster_data)\n```\n\n#### Model Fitting\n\n\n```python\n# there are defaults but we can also more explicitly set the sampling algorithms\nwith disaster_model:\n \n # for continuous variables\n step1 = pm.NUTS([early_rate, late_rate])\n \n # for discrete variables\n step2 = pm.Metropolis([switchpoint, disasters.missing_values[0]] )\n\n trace = pm.sample(10000, step=[step1, step2])\n # try different number of samples\n #trace = pm.sample(5000, step=[step1, step2])\n```\n\n#### Posterior Analysis\n\nOn the left side plots we notice that our early rate is between 2.5 and 3.5 disasters a year. In the late period it seems to be between 0.6 and 1.2 so definitely lower.\n\nThe right side plots show the samples we drew to come to our conclusion.\n\n\n```python\npm.traceplot(trace, ['early_rate', 'late_rate', 'switchpoint'], figsize=(20,10));\n```\n\n\n```python\nresults = pm.summary(trace, \n var_names=['early_rate', 'late_rate', 'switchpoint'])\nresults\n```\n", "meta": {"hexsha": "50005b115bb9ea647a62f1d076969f77093d059d", "size": 440524, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "content/labs/lab4 bayes/cs109b_lab04_bayes.ipynb", "max_stars_repo_name": "Ruby122/2020-CS109B", "max_stars_repo_head_hexsha": "c185e17d5c9e100275d950b6ff48acc7e5fc45bb", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "content/labs/lab4 bayes/cs109b_lab04_bayes.ipynb", "max_issues_repo_name": "Ruby122/2020-CS109B", "max_issues_repo_head_hexsha": "c185e17d5c9e100275d950b6ff48acc7e5fc45bb", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "content/labs/lab4 bayes/cs109b_lab04_bayes.ipynb", "max_forks_repo_name": "Ruby122/2020-CS109B", "max_forks_repo_head_hexsha": "c185e17d5c9e100275d950b6ff48acc7e5fc45bb", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 296.8490566038, "max_line_length": 155636, "alphanum_fraction": 0.91635189, "converted": true, "num_tokens": 10042, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.734119521083126, "lm_q1q2_score": 0.4323150066503392}} {"text": "## Practical Data Science - Classroom to Careers\n\n### Session-1 \n\nThe main objective of this session is to explain the pre-requisites before we deal with any kind of Data in general.\n\n#### What is data analysis?\n\nData Analysis is a subset of Data Science which helps us to understand the underlying process that generates the data\n\n#### What are the different kinds of data?\n\nData can be mainly categorized into 2 main types - \n\n - Structured Data \n - Unstructured data\n\nExamples of **Structured data** include - Tabular data, and mainly easily interpretable by a machine (numeric or can be converted to numeric)\n\nExamples of **Unstructured data** include - Audio, Image or text data etc. which cannot be directly understood by a machine. The way we can transform this data to be understood by machines will be covered in the later part of the course\n\n\n\n\n\n### Tools we will be using for the course\n\n**Python version 3.6 or higher with Anaconda4** distribution (includes Jupyter/IPython Notebook)\n\nTo install Anaconda please visit the following link and choose the version compatible with your Windows OS. Right-click on My Computer/PC and select properties to get the version of your Windows Operating System - \n\nhttps://repo.anaconda.com/archive/Anaconda3-2020.02-Windows-x86_64.exe (64 bit version)\n\nhttps://repo.anaconda.com/archive/Anaconda3-2020.02-Windows-x86.exe (32 bit version)\n\n### Course Contents\n\n**1. Basic Probability and Statistics**\n \n - 1.1 Understanding and Representing Data\n - 1.2 What are probability distributions?\n - 1.3 Bayes Theorem\n \n\n**2. Probability and Data Distributions**\n \n - 2.1 What are Random Variables?\n - 2.2 What is a Probability Density Function?\n - 2.3 Normal Distribution\n - 2.4 Binomial Distribution\n - 2.5 Poisson Distribution\n - 2.6 Exponential Distribution\n - 2.7 Geometric Distribution\n\n\n**3. Introduction to Linear Algebra and Calculus**\n\n - 3.1 Matrix Addition and Multiplication\n - 3.2 Finding the determinant of a matrix\n - 3.3 Eigenvalues and Eigenvectors\n - 3.4 Functions\n - 3.5 Differentiation of commonly used functions\n \n \n**4. Statistical Estimation Methods**\n\n - 4.1 Maximum Likelihood Estimates \n - 4.2 Information Gain\n - 4.3 Parametric Estimation\n - 4.4 Non-parametric estimation\n - 4.5 Supervised vs. Unsupervised Learning\n \n \n**5. Regression**\n\n - 5.1 Linear Regression\n - 5.2 Logistic regression\n - 5.3 Determining the goodness of fit\n - 5.4 Decision Trees\n - 5.5 Ridge Regression\n - 5.6 Lasso Regression\n - 5.7 Regularization\n \n \n**6. Unsupervised Learning**\n\n - 6.1 Hierarchical Clustering\n - 6.2 K-Means Clustering\n - 6.3 Association Rule mining\n - 6.4 Principal Component Analysis (PCA)\n - 6.5 Singular Value Decomposition (SVD)\n - 6.6 One-class SVM (Anomaly detection)\n \n \n**7. Supervised Learning in ML**\n\n - 7.1 Bagging/Random Forests\n - 7.2 Support Vector Machines (SVM)\n - 7.3 Boosting/ Gradient Boosting\n - 7.4 Multi-output classification\n\n\n**8. Introduction to Neural Networks and Cost function**\n\n - 8.1 Artificial Neural Networks\n - 8.2 Activation Fucntions\n - 8.3 Backpropagation\n - 8.4 Gradient Descent\n - 8.5 Cost function vs. Loss function\n\n\n**9. Text Classification**\n\n - 9.1 Bag-of-Words\n - 9.2 Named Entity Recognition (NER)\n - 9.3 Stemming and Lemmatization\n - 9.2 Feature Extraction from Text\n - 9.3 Pre-trained Word Embeddings\n - 9.4 Text Clustering\n - 9.5 Topic Modeling\n\n\n**10. Sequence Models**\n\n - 10.1 Introduction to Recurrent Neural Networks (RNNs)\n - 10.2 Long Short Term Memory Networks (LSTMs)\n - 10.3 Gated Recurrent Units\n - 10.4 Attention Layers \n - 10.5 Multi-Output Classification\n \n**11 and 12 - To be decided** \n\n### Normal Distribution\n\nAlso known as Gaussian Distribution, is the most popular Distribution one would hear in Statistics and ML world.\n\nThis distribution has a Curve centered at mean and characterized byt the following -\n\n1) Mean \n\n2) Standard Deviation (sigma)\n\nThe equation of Normal Distribution is -\n\n\\begin{align}\nP(X=x) = Y = \\frac{1}{\\sigma\\sqrt(2\\pi)} * e^{-(x-\\mu)^2/2\\sigma^2}\\\\\n\\end{align}\n\nWhere x is a Random Variable and **mu** is the Mean and **sigma** is the Standard deviation\n\nRegression techniques have assumptions built around the data coming from a Normal Distribution\n\n\n```python\nimport numpy as np\nimport scipy\nimport seaborn as sns\n\n\n# generate random numbers from N(0,1)\ndata_normal = np.random.normal(size=1000,loc=0,scale=100)\n\n# Plot the normal distribution\nax = sns.distplot(data_normal,\n bins=100,\n kde=False,\n color='blue',\n hist_kws={\"linewidth\": 15,'alpha':1})\nax.set(xlabel='Normal', ylabel='Frequency')\n```\n\n#### Properties of normal distribution\n\n - The mean, mode and median are all equal.\n - The curve is symmetric at the center (i.e. around the mean, \u03bc).\n - Exactly half of the values are to the left of center and exactly half the values are to the right.\n - The total area under the curve is 1.\n\n#### Applications of Normal Distribution\n\n**Problem 1**\n\nAn average light bulb manufactured by the Acme Corporation lasts 300 days with a standard deviation of 50 days. Assuming that bulb life is normally distributed, what is the probability that an Acme light bulb will last at most 365 days?\n\n\n**Problem 2**\n\nSuppose scores on an IQ test are normally distributed. If the test has a mean of 100 and a standard deviation of 10, what is the probability that a person who takes the test will score between 90 and 110?\n\n### Binomial Distribution\n\nA binomial distribution is characterized by the following -\n\n - There are **n** repeated trials (where n is a positive integer)\n - The trial can result in only 2 outcomes (bi = 2)\n - The probability of success for each trial **p** is same and does not change (0<=p<=1)\n - The outcome of the trials is independent. i.e., the out come of one trial does not influence the outcome of any other trial\n \n **Example - Flipping a fair coin**\n \n - The probability of success of head/tail is always 0.5\n - The probability of success remains same\n - The flipping is independent\n - The outcome is either a head or a tail (only 2 outcomes)\n \n**Binomial Distribution Equation**\n\n\\begin{equation*}\nP(E) = {n \\choose k} p^k (1-p)^{ n-k}\n\\end{equation*}\n \n**k**: The number of successes that result from the binomial experiment.\n\n**n**: The number of trials in the binomial experiment.\n\n**p**: The probability of success on an individual trial.\n\n**q**: The probability of failure on an individual trial (1-p, since thre are only 2 outcomes)\n \n\n\n```python\nfrom scipy.stats import binom\nimport warnings\n\nwarnings.filterwarnings(action='ignore')\n\nbinom.rvs(size=5,n=10,p=0.5)\n\ndata_binom = binom.rvs(n=10,p=0.5,size=10000)\n\nax = sns.distplot(data_binom,\n kde=False,\n color='blue',\n hist_kws={\"linewidth\": 15,'alpha':1})\n\nax.set(xlabel='Binomial', ylabel='Frequency')\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "b580fdb257a6560d0cd39d1ea8526819ab44d98f", "size": 25144, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Session1_PDS.ipynb", "max_stars_repo_name": "rajivbits/IPython-Notebooks", "max_stars_repo_head_hexsha": "6ff78b85d18c79d4ca0b7aeb949c89bb1ada8679", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Session1_PDS.ipynb", "max_issues_repo_name": "rajivbits/IPython-Notebooks", "max_issues_repo_head_hexsha": "6ff78b85d18c79d4ca0b7aeb949c89bb1ada8679", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Session1_PDS.ipynb", "max_forks_repo_name": "rajivbits/IPython-Notebooks", "max_forks_repo_head_hexsha": "6ff78b85d18c79d4ca0b7aeb949c89bb1ada8679", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 67.9567567568, "max_line_length": 7120, "alphanum_fraction": 0.7846404709, "converted": true, "num_tokens": 1761, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891163376235, "lm_q2_score": 0.7341195152660687, "lm_q1q2_score": 0.4323149926312397}} {"text": "```python\n%matplotlib inline\nimport pandas as pd\n\nimport numpy as np\nfrom __future__ import division\nimport itertools\n\nimport matplotlib.pyplot as plt\nimport seaborn as sns\n\nimport logging\nlogger = logging.getLogger()\n```\n\n17 Amortized Analysis\n============\n\nIn an __amortized analysis__, we averget the time required to perform a sequence of data-structure operations over all the operations performed.\n\nAmortized analsis differs from average-case analysis in that probability is not involved; an amortized analysis guarantees the _average performance of each operation in the worst case__.\n\nBear in mind that the charges assigned during an amortized analysis are for analysis purposes only.\n\nWhen we perform an amortized analysis, we often gain insight into a particular data structure, and this insight can help us optimize the design.\n\n### 17.1 Aggregate analysis\nWe show that for all $n$, a sequence of $n$ operations takes worst-case time $T(n)$ in total.\n\n##### Stack operations\nPUSH $geq$ POP + MULTIPOP \n$1 \\times O(n) = O(n)$\n\n##### Incrementing a binary counter\n\n\n```python\nplt.imshow(plt.imread('./res/fig17_2.png'))\n```\n\nIn general, for $i = 0, 1, \\dotsc, k-1$, bit $A[i]$ flips $\\lfloor \\frac{n}{2^i} \\rfloor$ times in a sequence of $n$ INCREMENT operations on an initially zero counter.\n\n\\begin{align}\n \\sum_{i=0}^{k-1} \\lfloor \\frac{n}{2^i} \\rfloor &< n \\sum_{i=0}^{\\infty} \\frac{1}{2^i} \\\\\n &= 2n\n\\end{align}\n\n### 17.2 The accounting method\n**credit**: the cost that an operation's amortized cost $\\hat{c_i}$ exceeds its actual cost $c_i$.\n\nrequriments: $$\\sum_{i=1}^{n} \\hat{c_i} \\geq \\sum_{i=1}^{n} c_i$$\n\n\n##### Stack operations\n| | $c_i$ | $\\hat{c_i}$ |\n|------|-------|-------------|\n| PUSH | 1 | 2 |\n| POP | 1 | 0 |\n|MULTIPOP| min(k,s) | 0 |\n\n$2 \\times O(n) = O(n)$\n\n\n##### Incrementing a binary counter\nset a bit to 1: 2 \nset a bit to 0: 0\n\nThe INCREMENT procedure sets at most one bit, $2 \\times O(n) = O(n)$\n\n### 17.3 The potential method\nLet $D_i$ be the data structure that results after applying the $i$th operation to data structure $D_{i-1}$.\n\n**potential function $\\phi$**: maps each data structure $D_i$ to a real number $\\phi(D_i)$.\n\n$\\hat{c_i} = c_i + \\phi(D_i) - \\phi(D_{i-1})$ \nhence, the total amortized cost of the $n$ operations is:\n$$\\sum_{i=1}^n \\hat{c_i} = \\sum_{i=1}^n c_i + \\phi(D_n) - \\phi(D_0)$$\n\nDifferent potential functions may yield different amortized costs yet still be upper bounds on the actual costs. The best potential function to use depends on the disired time bounds.\n\n\n##### Stack operations\ndefine: $\\phi$ to be the number of objects in the stack.\n\nfor PUSH: \n\\begin{align}\n \\hat{c_i} &= c_i + \\phi(D_i) - \\phi(D_{i-1}) \\\\\n &= 1 + (s+1) - s \\\\\n &= 2\n\\end{align}\n\nfor POP: \n\\begin{align}\n \\hat{c_i} &= c_i + \\phi(D_i) - \\phi(D_{i-1}) \\\\\n &= 1 + (s-1) - s \\\\\n &= 0\n\\end{align}\n\nfor MULTIPOP: \n\\begin{align}\n \\hat{c_i} &= c_i + \\phi(D_i) - \\phi(D_{i-1}) \\\\\n &= k + (s-k) - s \\\\\n &= 0\n\\end{align}\n\n\n##### Incrementing a binary counter\ndefine: $\\phi$ to be $b_i$, the number of 1s in the counter after the $i$th operation.\n\nSuppose: the $i$th INCREMENT operation reset $t_i$ bits.\n\nfor INCREMENT:\n\\begin{align}\n \\hat{c_i} &= c_i + \\phi(D_i) - \\phi(D_{i-1}) \\\\\n &= (t_i + 1) + (b_{i-1} - t_i + 1) - b_{i-1} \\\\\n &= 2 \n\\end{align}\n\n### 17.4 Dynamic tables\n**load factor**: $$\\alpha(T) = \\frac{\\|\\text{items of T}\\|}{\\|T\\|}$$ \n\n#### 17.4.1 Table expansion\ninsert an item into a full table, we expand the table with twice spaces.\n\nThe cost of the $i$th operation is: \n\\begin{equation}\n c_i = \\begin{cases}\n i \\quad & \\text{expand: if i - 1 is an exact power of 2} \\\\\n 1 \\quad & \\text{otherwise} \n \\end{cases}\n\\end{equation}\n\nThe total cost of $n$ TABLE-INSERT operations is therefore:\n\\begin{align}\n \\sum_{i=1}^{n} c_i &\\leq n + \\sum_{j=0}^{\\lfloor \\lg n \\rfloor} 2^j \\\\\n &< n + 2n \\\\\n &= 3n\n\\end{align}\n\n#### 17.4.2 Table expansion and contraction\nHalve the table size when deleting an item causes the table to become less than 1/4 full, rather than 1/2 full as before(\u5f15\u8d77\u632f\u8361).\n\n", "meta": {"hexsha": "65eaf649666c6628fdb2244edb38299f5fee624f", "size": 80123, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Introduction_to_Algorithms/17_Amortized_Analysis/note.ipynb", "max_stars_repo_name": "ningchi/book_notes", "max_stars_repo_head_hexsha": "c6f8001f7d5f873896c4b3a8b1409b21ef33c328", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2017-12-31T12:10:42.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-10T15:49:34.000Z", "max_issues_repo_path": "Introduction_to_Algorithms/17_Amortized_Analysis/note.ipynb", "max_issues_repo_name": "ningchi/book_notes", "max_issues_repo_head_hexsha": "c6f8001f7d5f873896c4b3a8b1409b21ef33c328", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2017-12-05T13:04:14.000Z", "max_issues_repo_issues_event_max_datetime": "2017-12-07T16:24:50.000Z", "max_forks_repo_path": "Introduction_to_Algorithms/17_Amortized_Analysis/note.ipynb", "max_forks_repo_name": "ningchi/book_notes", "max_forks_repo_head_hexsha": "c6f8001f7d5f873896c4b3a8b1409b21ef33c328", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2017-06-27T07:19:28.000Z", "max_forks_repo_forks_event_max_datetime": "2017-11-19T08:57:35.000Z", "avg_line_length": 345.3577586207, "max_line_length": 73036, "alphanum_fraction": 0.9128090561, "converted": true, "num_tokens": 1364, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381667555713, "lm_q2_score": 0.7905303285397349, "lm_q1q2_score": 0.43221310259049417}} {"text": "```python\n%matplotlib inline\n```\n\n\nDCGAN \u6559\u7a0b\n==============\n\n**\u7ffb\u8bd1\u8005**: `Antares\u535a\u58eb `_\n\n\n\n\n\u4ecb\u7ecd\n------------\n\n\u672c\u6559\u7a0b\u5c06\u901a\u8fc7\u4e00\u4e2a\u793a\u4f8b\u4ecb\u7ecdDCGANs\u3002\u6211\u4eec\u5c06\u8bad\u7ec3\u4e00\u4e2a\u751f\u6210\u5bf9\u6297\u7f51\u7edc(generative adversarial network, GAN)\uff0c\n\u5728\u7ed9\u5b83\u5c55\u793a\u8bb8\u591a\u540d\u6d41\u7684\u7167\u7247\u4e4b\u540e\uff0c\u4ea7\u751f\u65b0\u7684\u540d\u4eba\u3002\u8fd9\u91cc\u7684\u5927\u90e8\u5206\u4ee3\u7801\u90fd\u6765\u81ea `pytorch/examples `__ \u7684\u5b9e\u73b0\uff0c\n\u672c\u6587\u6863\u5c06\u8be6\u7ec6\u89e3\u91ca\u5b9e\u73b0\uff0c\u5e76\u9610\u660e\u8be5\u6a21\u578b\u662f\u5982\u4f55\u5de5\u4f5c\u7684\u548c\u4e3a\u4ec0\u4e48\u5de5\u4f5c\u7684\u3002\u4f46\u522b\u62c5\u5fc3\uff0c\u4e0d\u9700\u8981\u4e8b\u5148\u77e5\u9053GANs\uff0c\n\u4f46\u5b83\u53ef\u80fd\u9700\u8981\u7b2c\u4e00\u6b21\u82b1\u4e00\u4e9b\u65f6\u95f4\u6765\u63a8\u7406\u5728\u8868\u8c61\u7684\u4e0b\u9762\u771f\u6b63\u53d1\u751f\u4e86\u4ec0\u4e48\u3002\u6b64\u5916\uff0c\u4e3a\u4e86\u65f6\u95f4\uff0c\u6709\u4e00\u4e2a\u6216\u4e24\u4e2aGPU\u53ef\u80fd\u662f\u4e2a\u597d\u4e8b\u513f\u3002\n\u8ba9\u6211\u4eec\u4ece\u5934\u5f00\u59cb\u3002\n\n\u751f\u6210\u5bf9\u6297\u7f51\u7edc\n-------------------------------\n\n\u4ec0\u4e48\u662f GAN?\n~~~~~~~~~~~~~~\n\nGANS\u662f\u4e00\u4e2a\u6846\u67b6\uff0c\u5b83\u6559\u6388DL\u6a21\u578b\u4ee5\u6355\u83b7\u8bad\u7ec3\u6570\u636e\u7684\u5206\u5e03\uff0c\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u4ece\u76f8\u540c\u7684\u5206\u5e03\u751f\u6210\u65b0\u7684\u6570\u636e\u3002\nGANs \u662f\u7531\u4f0a\u6069\u00b7\u53e4\u5fb7\u8d39\u7f57\u4e8e2014\u5e74\u53d1\u660e\u7684\uff0c\u5e76\u9996\u6b21\u5728\u8bba\u6587 \n`Generative Adversarial Nets `__ \n\u4e2d\u8fdb\u884c\u4e86\u63cf\u8ff0\u3002\u5b83\u4eec\u7531\u4e24\u79cd\u4e0d\u540c\u7684\u6a21\u578b\u7ec4\u6210\uff0c\u4e00\u79cd\u662f\u751f\u6210\u5668(*generator*)\uff0c\u53e6\u4e00\u79cd\u662f\u5224\u522b\u5668(*discriminator*)\u3002\n\u751f\u6210\u5668\u7684\u5de5\u4f5c\u662f\u751f\u6210\u770b\u8d77\u6765\u50cf\u8bad\u7ec3\u56fe\u50cf\u7684\u201c\u5047\u201d\u56fe\u50cf\u3002\u5224\u522b\u5668\u7684\u5de5\u4f5c\u662f\u67e5\u770b\u56fe\u50cf\u5e76\u8f93\u51fa\u5b83\u662f\u771f\u5b9e\u7684\u8bad\u7ec3\u56fe\u50cf\u8fd8\u662f\u6765\u81ea\u751f\u6210\u5668\u7684\u5047\u56fe\u50cf\u3002\n\u5728\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\uff0c\u751f\u6210\u5668\u4e0d\u65ad\u5730\u8bd5\u56fe\u901a\u8fc7\u751f\u6210\u8d8a\u6765\u8d8a\u597d\u7684\u4f2a\u56fe\u50cf\u6765\u80dc\u8fc7\u5224\u522b\u5668\uff0c\u800c\u5224\u522b\u5668\u6b63\u5728\u52aa\u529b\u6210\u4e3a\u4e00\u540d\u66f4\u597d\u7684\u4fa6\u63a2\uff0c\n\u5e76\u6b63\u786e\u5730\u5bf9\u771f\u5047\u56fe\u50cf\u8fdb\u884c\u5206\u7c7b\u3002\u8fd9\u4e2a\u6e38\u620f\u7684\u5747\u8861\u662f\u5f53\u751f\u6210\u5668\u751f\u6210\u770b\u8d77\u6765\u50cf\u662f\u76f4\u63a5\u6765\u81ea\u8bad\u7ec3\u6570\u636e\u7684\u5b8c\u7f8e\u5047\u8c61\u65f6\uff0c\n\u5224\u522b\u5668\u603b\u662f\u4ee550%\u7684\u4fe1\u5fc3\u731c\u6d4b\u751f\u6210\u5668\u8f93\u51fa\u662f\u771f\u662f\u5047\u7684\u3002\n\n\u73b0\u5728\uff0c\u8ba9\u6211\u4eec\u4ece\u5224\u522b\u5668\u5f00\u59cb\uff0c\u5728\u6574\u4e2a\u6559\u7a0b\u4e2d\u5b9a\u4e49\u4e00\u4e9b\u8981\u4f7f\u7528\u7684\u7b26\u53f7\u3002\u5047\u8bbe $x$ \u662f\u8868\u793a\u56fe\u50cf\u7684\u6570\u636e\u3002 \n$D(x)$ \u662f\u5224\u522b\u5668\u7f51\u7edc\uff0c\u5b83\u8f93\u51fa $x$ \u6765\u81ea\u8bad\u7ec3\u6570\u636e\u800c\u4e0d\u662f\u751f\u6210\u5668\u7684(\u6807\u91cf)\u6982\u7387\u3002\u8fd9\u91cc\uff0c\n\u7531\u4e8e\u6211\u4eec\u5904\u7406\u7684\u662f\u56fe\u50cf\uff0c$D(x)$ \u7684\u8f93\u5165\u662fHWC\u5927\u5c0f\u4e3a3x64x64\u7684\u56fe\u50cf\u3002\n\u76f4\u89c9\u4e0a\uff0c\u5f53 $x$ \u6765\u81ea\u8bad\u7ec3\u6570\u636e\u65f6\uff0c $D(x)$ \u5e94\u8be5\u662f\u9ad8\u7684\uff0c\n\u5f53 $x$ \u6765\u81ea\u751f\u6210\u5668\u65f6\uff0c$D(x)$ \u5e94\u8be5\u662f\u4f4e\u7684\u3002\n$D(x)$ \u4e5f\u53ef\u4ee5\u770b\u4f5c\u662f\u4e00\u79cd\u4f20\u7edf\u7684\u4e8c\u5143\u5206\u7c7b\u5668\u3002\n\n\u5bf9\u4e8e\u751f\u6210\u5668\u7684\u8868\u793a\u6cd5\uff0c\u8bbe $z$ \u662f\u4ece\u6807\u51c6\u6b63\u6001\u5206\u5e03\u4e2d\u91c7\u6837\u7684\u6f5c\u5728\u7a7a\u95f4\u5411\u91cf(latent space vector)\u3002\n$G(z)$ \u8868\u793a\u751f\u6210\u51fd\u6570\uff0c\u5b83\u5c06\u6f5c\u5728\u5411\u91cf $z$ \u6620\u5c04\u5230\u6570\u636e\u7a7a\u95f4\u3002 \n$G$ \u7684\u76ee\u6807\u662f\u4f30\u8ba1\u8bad\u7ec3\u6570\u636e\u7684\u5206\u5e03 ($p_{data}$) \uff0c\u4ece\u800c\u4ece\u4f30\u8ba1\u51fa\u7684\u5206\u5e03($p_g$)\u4e2d\u751f\u6210\u5047\u6837\u672c\u3002\n\n\u56e0\u6b64, $D(G(z))$ \u662f\u751f\u6210\u5668 $G$ \u8f93\u51fa\u7684\u56fe\u50cf\u4e3a\u771f\u5b9e\u56fe\u50cf\u7684\u6982\u7387(\u6807\u91cf)\u3002\n\u6b63\u5982 `\u53e4\u5fb7\u8d39\u7f57\u7684\u8bba\u6587 `__, \u6240\u63cf\u8ff0\u7684\u90a3\u6837\uff0c\n$D$ \u548c $G$ \u73a9\u4e86\u4e00\u4e2a\u6781\u5c0f\u6781\u5927\u7684\u535a\u5f08(minimax game)\uff0c\u5176\u4e2d $D$ \n\u8bd5\u56fe\u6700\u5927\u5316\u5b83\u6b63\u786e\u5730\u5206\u7c7b\u771f\u56fe\u50cf\u548c\u5047\u56fe\u50cf\u7684\u6982\u7387($logD(x)$)\uff0c$G$ \u8bd5\u56fe\u6700\u5c0f\u5316 $D$ \n\u9884\u6d4b\u5176\u8f93\u51fa\u662f\u5047\u7684\u7684\u6982\u7387 ($log(1-D(G(x)))$) \u3002\u6587\u4e2d\u7ed9\u51fa\u4e86GAN\u635f\u5931\u51fd\u6570:\n\n\\begin{align}\\underset{G}{\\text{min}} \\underset{D}{\\text{max}}V(D,G) = \\mathbb{E}_{x\\sim p_{data}(x)}\\big[logD(x)\\big] + \\mathbb{E}_{z\\sim p_{z}(z)}\\big[log(1-D(G(x)))\\big]\\end{align}\n\n\u7406\u8bba\u4e0a\uff0c\u8fd9\u4e2a\u6781\u5c0f\u6781\u5927\u535a\u5f08\u7684\u89e3\u662f \u5728 $p_g = p_{data}$ \u65f6\uff0c\u5224\u522b\u5668\u53ea\u80fd\u968f\u673a\u731c\u6d4b\u8f93\u5165\u662f\u771f\u8fd8\u662f\u5047\u3002\n\u7136\u800c\uff0cGANS\u7684\u6536\u655b\u7406\u8bba\u4ecd\u5728\u79ef\u6781\u7814\u7a76\u4e4b\u4e2d\uff0c\u800c\u5728\u73b0\u5b9e\u4e2d\uff0c\u6a21\u578b\u5e76\u4e0d\u603b\u662f\u8bad\u7ec3\u5230\u8fd9\u4e00\u70b9\u3002\n\n\u4ec0\u4e48\u53c8\u662f DCGAN?\n~~~~~~~~~~~~~~~~\n\nDCGAN\u662f\u4e0a\u8ff0GANs\u7684\u76f4\u63a5\u6269\u5c55\uff0c\u53ea\u662f\u5b83\u5728\u9274\u522b\u5668\u548c\u751f\u6210\u5668\u4e2d\u5206\u522b\u663e\u5f0f\u5730\u4f7f\u7528\u5377\u79ef\u548c\u5377\u79ef\u8f6c\u7f6e\u5c42\u3002\n\u5b83\u9996\u5148\u7531Radford\u5728\u6587\u7ae0 `Unsupervised Representation Learning With\nDeep Convolutional Generative Adversarial Networks `__ \n\u63d0\u51fa\u4e86\u4e00\u79cd\u57fa\u4e8e\u6df1\u5c42\u5377\u79ef\u751f\u6210\u5bf9\u6297\u7f51\u7edc\u7684\u65e0\u76d1\u7763\u8868\u793a\u5b66\u4e60\u65b9\u6cd5\u3002\n\u5224\u522b\u5668\u7531\u8de8\u6b65\u5377\u79ef\u5c42(`strided convolution layers `__ )\u3001\n\u6279\u5f52\u4e00\u5316\u5c42(`batch norm layers `__)\n\u548c `LeakyReLU `__ \u6fc0\u6d3b\u51fd\u6570\u6784\u6210\u3002\n\u8f93\u5165\u662f3x64x64\u56fe\u50cf\uff0c\u8f93\u51fa\u662f \u8f93\u5165\u6765\u81ea\u771f\u5b9e\u6570\u636e\u5206\u5e03\u7684 \u6807\u91cf\u6982\u7387\u3002\n\u751f\u6210\u5668\u7531\u5377\u79ef\u8f6c\u7f6e\u5c42(`convolutional-transpose `__)\u3001\n\u6279\u5f52\u4e00\u5316\u5c42\u548c `ReLU `__ \u6fc0\u6d3b\u5c42\u7ec4\u6210\u3002\n\u8f93\u5165\u662f\u4ece\u6807\u51c6\u6b63\u6001\u5206\u5e03\u4e2d\u63d0\u53d6\u7684\u6f5c\u5728\u77e2\u91cf(latent vector) $z$ \uff0c\u8f93\u51fa\u662f 3x64x64 \u7684RGB\u56fe\u50cf\u3002\n\u8de8\u6b65\u5377\u79ef\u8f6c\u7f6e\u5c42(strided conv-transpose layers)\u5141\u8bb8\u5c06\u6f5c\u5728\u77e2\u91cf(latent vector)\u53d8\u6362\u4e3a\u5177\u6709\u4e0e\u56fe\u50cf\u76f8\u540c\u7684shape\u3002\n\u4f5c\u8005\u8fd8\u5c31\u5982\u4f55\u8bbe\u7f6e\u4f18\u5316\u5668\u3001\u5982\u4f55\u8ba1\u7b97\u635f\u5931\u51fd\u6570\u4ee5\u53ca\u5982\u4f55\u521d\u59cb\u5316\u6a21\u578b\u7684\u6743\u91cd\u7b49\u65b9\u9762\u7ed9\u51fa\u4e86\u4e00\u4e9b\u63d0\u793a\uff0c\u8fd9\u4e9b\u90fd\u5c06\u5728\u540e\u9762\u7684\u7ae0\u8282\u4e2d\u52a0\u4ee5\u8bf4\u660e\u3002\n\n\n\n\n\n```python\nfrom __future__ import print_function\n#%matplotlib inline\nimport argparse\nimport os\nimport random\nimport torch\nimport torch.nn as nn\nimport torch.nn.parallel\nimport torch.backends.cudnn as cudnn\nimport torch.optim as optim\nimport torch.utils.data\nimport torchvision.datasets as dset\nimport torchvision.transforms as transforms\nimport torchvision.utils as vutils\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport matplotlib.animation as animation\nfrom IPython.display import HTML\n\n# Set random seem for reproducibility\nmanualSeed = 999\n#manualSeed = random.randint(1, 10000) # use if you want new results\nprint(\"Random Seed: \", manualSeed)\nrandom.seed(manualSeed)\ntorch.manual_seed(manualSeed)\n```\n\n\u8f93\u5165\n------\n\n\u6211\u4eec\u5148\u6765\u5b9a\u4e49\u4e00\u4e9b\u8f93\u5165:\n\n- **dataroot** - dataset \u6587\u4ef6\u5939\u6839\u76ee\u5f55\u7684\u8def\u5f84\u3002\u6211\u4eec\u5c06\u5728\u4e0b\u4e00\u8282\u4e2d\u66f4\u591a\u5730\u8ba8\u8bba\u6570\u636e\u96c6\u3002\n- **workers** - \u7528\u4e8e\u7528 DataLoader \u52a0\u8f7d\u6570\u636e\u7684\u5de5\u4f5c\u7ebf\u7a0b\u6570\u3002\n- **batch_size** - \u8bad\u7ec3\u4e2d\u4f7f\u7528\u7684\u6279\u6b21\u5927\u5c0f\u3002DCGAN \u4f7f\u7528\u7684\u6279\u6b21\u5927\u5c0f\u4e3a128\u3002\n- **image_size** - \u7528\u4e8e\u8bad\u7ec3\u7684\u56fe\u50cf\u7684\u7a7a\u95f4\u5927\u5c0f\u3002\u6b64\u5b9e\u73b0\u9ed8\u8ba4\u4e3a64x64\u3002\n \u5982\u679c\u9700\u8981\u53e6\u4e00\u4e2a\u5c3a\u5bf8\uff0c\u5219\u5fc5\u987b\u6539\u53d8D\u548cG\u7684\u7ed3\u6784\u3002\u6709\u5173\u66f4\u591a\u7ec6\u8282\uff0c\u8bf7\u53c2\u9605 `\u8fd9\u91cc `__ \u3002\n- **nc** - \u8f93\u5165\u56fe\u50cf\u7684\u989c\u8272\u901a\u9053\u6570. \u5f69\u8272\u56fe\u50cf\u662f3\u901a\u9053\u7684\u3002\n- **nz** - \u6f5c\u5728\u5411\u91cf(latent vector)\u7684\u957f\u5ea6\n- **ngf** - \u4e0e\u901a\u8fc7\u751f\u6210\u5668\u8fdb\u884c\u7684\u7279\u5f81\u6620\u5c04\u7684\u6df1\u5ea6\u6709\u5173\u3002\n- **ndf** - \u8bbe\u7f6e\u901a\u8fc7\u9274\u522b\u5668\u4f20\u64ad\u7684\u7279\u5f81\u6620\u5c04\u7684\u6df1\u5ea6\u3002\n- **num_epochs** - \u8981\u8fd0\u884c\u7684\u8bad\u7ec3\u56de\u5408(epoch)\u6570\u3002\u957f\u671f\u7684\u8bad\u7ec3\u53ef\u80fd\u4f1a\u5e26\u6765\u66f4\u597d\u7684\u6548\u679c\uff0c\u4f46\u4e5f\u9700\u8981\u66f4\u957f\u7684\u65f6\u95f4\u3002\n- **lr** - \u7528\u4e8e\u8bad\u7ec3\u7684\u5b66\u4e60\u7387. \u5c31\u50cf\u5728 DCGAN \u8bba\u6587\u4e2d\u5efa\u8bae\u7684, \u8fd9\u4e2a\u53c2\u6570\u8bbe\u4e3a 0.0002 \u3002\n- **beta1** - Adam \u4f18\u5316\u5668\u7684beta1\u8d85\u53c2\u6570\u3002 \u5c31\u50cf\u5728 DCGAN \u8bba\u6587\u4e2d\u5efa\u8bae\u7684, \u8fd9\u4e2a\u53c2\u6570\u8bbe\u4e3a 0.5 \u3002\n- **ngpu** - \u53ef\u7528\u7684 GPUs \u6570\u91cf\u3002 \u5982\u679c\u6ca1\u6709GPU, \u4ee3\u7801\u5c06\u4f1a\u5728 CPU \u6a21\u5f0f\u4e0b\u8fd0\u884c\u3002 \u5982\u679c\u6709\u591a\u4e2aGPU,\u90a3\u5c31\u53ef\u4ee5\u52a0\u901f\u8ba1\u7b97\u4e86\u3002\n\n\n\n\n\n```python\n# Root directory for dataset\ndataroot = \"./data/celeba\"\n\n# Number of workers for dataloader\nworkers = 2\n\n# Batch size during training\nbatch_size = 128\n\n# Spatial size of training images. All images will be resized to this\n# size using a transformer.\nimage_size = 64\n\n# Number of channels in the training images. For color images this is 3\nnc = 3\n\n# Size of z latent vector (i.e. size of generator input)\nnz = 100\n\n# Size of feature maps in generator\nngf = 64\n\n# Size of feature maps in discriminator\nndf = 64\n\n# Number of training epochs\nnum_epochs = 5\n\n# Learning rate for optimizers\nlr = 0.0002\n\n# Beta1 hyperparam for Adam optimizers\nbeta1 = 0.5\n\n# Number of GPUs available. Use 0 for CPU mode.\nngpu = 1\n```\n\n\u6570\u636e\n----\n\n\u5728\u672c\u6559\u7a0b\u4e2d\uff0c\u6211\u4eec\u5c06\u4f7f\u7528 `Celeb-A Faces `__ \u6570\u636e\u96c6\uff0c\n\u8be5\u6570\u636e\u96c6\u53ef\u4ee5\u5728\u94fe\u63a5\u7684\u7ad9\u70b9\u4e0a\u4e0b\u8f7d\uff0c\u4e5f\u53ef\u4ee5\u5728GoogleDrive\u4e2d\u4e0b\u8f7d\u3002dataset\u5c06\u4f5c\u4e3a\u4e00\u4e2a\u540d\u4e3a *img_align_celeba.zip* \u7684\u6587\u4ef6\u4e0b\u8f7d\u3002\n\u4e0b\u8f7d\u5b8c\u540e\uff0c\u521b\u5efa\u4e00\u4e2a\u540d\u4e3a *celeba* \u7684\u76ee\u5f55\uff0c\u5e76\u5c06zip\u6587\u4ef6\u89e3\u538b\u7f29\u5230\u8be5\u76ee\u5f55\u4e2d\u3002\n\u7136\u540e\uff0c\u5c06\u6b64\u7b14\u8bb0\u672c\u7684 *dataroot* \u8f93\u5165\u8bbe\u7f6e\u4e3a\u60a8\u521a\u521a\u521b\u5efa\u7684renarba\u76ee\u5f55\u3002\u7531\u6b64\u4ea7\u751f\u7684\u76ee\u5f55\u7ed3\u6784\u5e94\u8be5\u662f\uff1a\n\n::\n\n /path/to/celeba\n -> img_align_celeba \n -> 188242.jpg\n -> 173822.jpg\n -> 284702.jpg\n -> 537394.jpg\n ...\n\n\u8fd9\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u6b65\u9aa4\uff0c\u56e0\u4e3a\u6211\u4eec\u5c06\u4f7f\u7528 ImageFolder \u7c7b\uff0c\u5b83\u9700\u8981\u5728dataset\u7684\u6839\u6587\u4ef6\u5939\u4e2d\u6709\u5b50\u76ee\u5f55\u3002\n\u73b0\u5728\uff0c\u6211\u4eec\u53ef\u4ee5\u521b\u5efa dataset \uff0cdataloader \uff0c\u8bbe\u7f6e\u8bbe\u5907\u8fd0\u884c\uff0c\u5e76\u6700\u7ec8\u53ef\u89c6\u5316\u4e00\u4e9b\u8bad\u7ec3\u6570\u636e\u3002\n\n\n\n\n\n```python\n# We can use an image folder dataset the way we have it setup.\n# Create the dataset\ndataset = dset.ImageFolder(root=dataroot,\n transform=transforms.Compose([\n transforms.Resize(image_size),\n transforms.CenterCrop(image_size),\n transforms.ToTensor(),\n transforms.Normalize((0.5, 0.5, 0.5), (0.5, 0.5, 0.5)),\n ]))\n# Create the dataloader\ndataloader = torch.utils.data.DataLoader(dataset, batch_size=batch_size,\n shuffle=True, num_workers=workers)\n\n# Decide which device we want to run on\ndevice = torch.device(\"cuda:0\" if (torch.cuda.is_available() and ngpu > 0) else \"cpu\")\n\n# Plot some training images\nreal_batch = next(iter(dataloader))\nplt.figure(figsize=(8,8))\nplt.axis(\"off\")\nplt.title(\"Training Images\")\nplt.imshow(np.transpose(vutils.make_grid(real_batch[0].to(device)[:64], padding=2, normalize=True).cpu(),(1,2,0)))\n```\n\n\u5b9e\u73b0\n--------------\n\n\u5728\u8bbe\u7f6e\u4e86\u8f93\u5165\u53c2\u6570\u5e76\u51c6\u5907\u597d\u6570\u636e\u96c6\u4e4b\u540e\uff0c\u6211\u4eec\u73b0\u5728\u53ef\u4ee5\u8fdb\u5165\u5b9e\u73b0\u4e86\u3002\u6211\u4eec\u5c06\u4ecewigthts\u521d\u59cb\u5316\u7b56\u7565\u5f00\u59cb\uff0c\n\u7136\u540e\u8be6\u7ec6\u8ba8\u8bba\u751f\u6210\u5668\u3001\u5224\u522b\u5668\u3001\u635f\u5931\u51fd\u6570\u548c\u8bad\u7ec3\u5faa\u73af\u3002\n\n\u6743\u91cd\u521d\u59cb\u5316\n~~~~~~~~~~~~~~~~~~~~~\n\n\u4eceDCGAN\u7684\u6587\u732e\u4e2d\uff0c\u4f5c\u8005\u6307\u51fa\u6240\u6709\u6a21\u578b\u7684\u6743\u91cd\u90fd\u5e94\u4ece\u5747\u503c=0\uff0cstdev=0.2\u7684\u6b63\u6001\u5206\u5e03\u4e2d\u968f\u673a\u521d\u59cb\u5316\u3002\n\u6743\u503c\u51fd\u6570\u4ee5\u521d\u59cb\u5316\u6a21\u578b\u4f5c\u4e3a\u8f93\u5165\uff0c\u5e76\u91cd\u65b0\u521d\u59cb\u5316\u6240\u6709\u5377\u79ef\u3001\u5377\u79ef-\u8f6c\u7f6e\u548c\u6279\u5904\u7406\u5f52\u4e00\u5316\u5c42\uff0c\u4ee5\u6ee1\u8db3\u8fd9\u4e00\u6807\u51c6\u3002\n\u8be5\u51fd\u6570\u5728\u521d\u59cb\u5316\u540e\u7acb\u5373\u5e94\u7528\u4e8e\u6a21\u578b\u3002\n\n\n\n\n\n```python\n# custom weights initialization called on netG and netD\ndef weights_init(m):\n classname = m.__class__.__name__\n if classname.find('Conv') != -1:\n nn.init.normal_(m.weight.data, 0.0, 0.02)\n elif classname.find('BatchNorm') != -1:\n nn.init.normal_(m.weight.data, 1.0, 0.02)\n nn.init.constant_(m.bias.data, 0)\n```\n\n\u751f\u6210\u5668(Generator)\n~~~~~~~~~~~~~~~~~~~~~~\n\n\u751f\u6210\u5668 $G$ \u88ab\u8bbe\u8ba1\u7528\u4e8e\u5c06\u6f5c\u5728\u7a7a\u95f4\u77e2\u91cf($z$)\u6620\u5c04\u5230\u6570\u636e\u7a7a\u95f4\u3002\u7531\u4e8e\u6211\u4eec\u7684\u6570\u636e\u662f\u56fe\u50cf\uff0c\n\u5c06 $z$ \u8f6c\u6362\u4e3a\u6570\u636e\u7a7a\u95f4\u610f\u5473\u7740\u6700\u7ec8\u521b\u5efa\u4e00\u4e2a\u4e0e\u8bad\u7ec3\u56fe\u50cf(\u53733x64x64)\u76f8\u540c\u5927\u5c0f\u7684RGB\u56fe\u50cf\u3002\n\u5728\u5b9e\u8df5\u4e2d\uff0c\u8fd9\u662f\u901a\u8fc7\u4e00\u7cfb\u5217strided 2d convolutional transpose layers \u6765\u5b9e\u73b0\u7684\uff0c\n\u6bcf\u4e2a\u5c42\u4e0e\u4e00\u4e2a2d batch norm layer\u548c\u4e00\u4e2arelu activation\u5c42\u914d\u5bf9\u3002\n\u751f\u6210\u5668\u7684\u8f93\u51fa\u9001\u5165\u5230\u4e00\u4e2atanh\u51fd\u6570\uff0c\u5c06\u5176\u8f93\u51fa\u503c\u538b\u7f29\u5728 $[-1,1]$ \u7684\u8303\u56f4\u3002\n\u503c\u5f97\u6ce8\u610f\u7684\u662fbatch norm functions\u662f\u5728conv-transpose layers\u4e4b\u540e\u7684\uff0c\n\u56e0\u4e3a\u8fd9\u662fDCGAN\u8bba\u6587\u7684\u4e00\u4e2a\u5173\u952e\u8d21\u732e\u3002\u8fd9\u4e9b\u5c42\u6709\u52a9\u4e8e\u8bad\u7ec3\u671f\u95f4\u7684\u68af\u5ea6\u6d41\u3002\nDCGAN\u6587\u7ae0\u4e2d\u7ed9\u51fa\u7684\u751f\u6210\u5668\u7684\u7ed3\u6784\u5982\u4e0b\u6240\u793a\u3002\n\n.. figure:: /_static/img/dcgan_generator.png\n :alt: dcgan_generator\n\n\u6ce8\u610f\uff0c\u6211\u4eec\u5728\u8f93\u5165\u90e8\u5206(*nz*, *ngf*, \u548c *nc*) \u4e2d\u8bbe\u7f6e\u7684\u8f93\u5165\u5982\u4f55\u5f71\u54cd\u4ee3\u7801\u4e2d\u7684\u751f\u6210\u5668\u4f53\u7cfb\u7ed3\u6784\u3002\n*nz* \u662f z \u8f93\u5165\u5411\u91cf\u7684\u957f\u5ea6\uff0c *ngf* \u4e0e\u901a\u8fc7\u751f\u6210\u5668\u4f20\u64ad\u7684\u7279\u5f81\u56fe\u7684\u5927\u5c0f\u6709\u5173\uff0c\n*nc* \u662f\u8f93\u51fa\u56fe\u50cf\u4e2d\u7684\u901a\u9053\u6570(\u5bf9\u4e8eRGB\u56fe\u50cf\u8bbe\u7f6e\u4e3a3)\u3002\u4e0b\u9762\u662f\u751f\u6210\u5668\u7684\u4ee3\u7801\u3002\n\n\n\n\n\n```python\n# Generator Code\n\nclass Generator(nn.Module):\n def __init__(self, ngpu):\n super(Generator, self).__init__()\n self.ngpu = ngpu\n self.main = nn.Sequential(\n # input is Z, going into a convolution\n nn.ConvTranspose2d( nz, ngf * 8, 4, 1, 0, bias=False),\n nn.BatchNorm2d(ngf * 8),\n nn.ReLU(True),\n # state size. (ngf*8) x 4 x 4\n nn.ConvTranspose2d(ngf * 8, ngf * 4, 4, 2, 1, bias=False),\n nn.BatchNorm2d(ngf * 4),\n nn.ReLU(True),\n # state size. (ngf*4) x 8 x 8\n nn.ConvTranspose2d( ngf * 4, ngf * 2, 4, 2, 1, bias=False),\n nn.BatchNorm2d(ngf * 2),\n nn.ReLU(True),\n # state size. (ngf*2) x 16 x 16\n nn.ConvTranspose2d( ngf * 2, ngf, 4, 2, 1, bias=False),\n nn.BatchNorm2d(ngf),\n nn.ReLU(True),\n # state size. (ngf) x 32 x 32\n nn.ConvTranspose2d( ngf, nc, 4, 2, 1, bias=False),\n nn.Tanh()\n # state size. (nc) x 64 x 64\n )\n\n def forward(self, input):\n return self.main(input)\n```\n\n\u73b0\u5728\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9e\u4f8b\u5316\u751f\u6210\u5668\u5e76\u5e94\u7528 ``weights_init`` \u51fd\u6570\u3002\n\u67e5\u770b\u6253\u5370\u7684\u6a21\u578b\uff0c\u770b\u770b\u751f\u6210\u5668\u5bf9\u8c61\u662f\u5982\u4f55\u6784\u9020\u7684\u3002\n\n\n\n\n\n```python\n# \u521b\u5efa\u751f\u6210\u5668\u5bf9\u8c61\nnetG = Generator(ngpu).to(device)\n\n# Handle multi-gpu if desired\nif (device.type == 'cuda') and (ngpu > 1):\n netG = nn.DataParallel(netG, list(range(ngpu)))\n\n# \u5e94\u7528 weights_init \u51fd\u6570 \u6765\u968f\u673a\u521d\u59cb\u5316 \u6240\u6709\u6743\u91cd\u5230 mean=0, stdev=0.2.\nnetG.apply(weights_init)\n\n# \u6253\u5370\u8f93\u51fa\u6a21\u578b\nprint(netG)\n```\n\n\u5224\u522b\u5668(Discriminator)\n~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n\u5982\u4e0a\u6240\u8ff0\uff0c\u5224\u522b\u5668 $D$ \u662f\u4e00\u79cd\u4e24\u7c7b\u5206\u7c7b\u7f51\u7edc\uff0c\u5b83\u4ee5\u56fe\u50cf\u4e3a\u8f93\u5165\uff0c\u8f93\u51fa \u8f93\u5165\u56fe\u50cf\u4e3a\u771f(\u800c\u4e0d\u662f\u5047)\u7684\u6807\u91cf\u6982\u7387\u3002\n\u8fd9\u91cc\uff0c$D$ \u63a5\u53d7\u4e00\u4e2a 3x64x64 \u8f93\u5165\u56fe\u50cf\uff0c\u901a\u8fc7\u4e00\u7cfb\u5217Conv2d\u3001BatchNorm2d\u548cLeakyReLU\u5c42\u5904\u7406\u5b83\uff0c\n\u5e76\u901a\u8fc7 sigmoid \u6fc0\u6d3b\u51fd\u6570\u8f93\u51fa\u6700\u7ec8\u7684\u6982\u7387\u3002\u5982\u679c\u6709\u5fc5\u8981\u7684\u8bdd\uff0c\u53ef\u4ee5\u7528\u66f4\u591a\u7684\u5c42\u6765\u6269\u5c55\u8fd9\u4e2a\u4f53\u7cfb\u7ed3\u6784\uff0c\n\u4f46\u662f\u4f7f\u7528strided convolution\u3001BatchNorm\u548cLeakyReLU\u662f\u5f88\u6709\u610f\u4e49\u7684\u3002DCGAN\u7684\u8bba\u6587\u63d0\u5230\uff0c\n\u4f7f\u7528strided convolution\u800c\u4e0d\u662fpooling\u6765\u964d\u91c7\u6837\u662f\u4e00\u79cd\u5f88\u597d\u7684\u505a\u6cd5\uff0c\n\u56e0\u4e3a\u5b83\u8ba9\u7f51\u7edc\u5b66\u4e60\u81ea\u5df1\u7684\u6c60\u5316\u51fd\u6570\u3002\u6b64\u5916\uff0cbatch norm \u548cleaky relu\u51fd\u6570\u4fc3\u8fdb\u4e86\u5065\u5eb7\u7684\u68af\u5ea6\u6d41\uff0c\n\u8fd9\u5bf9\u4e8e $G$ \u548c $D$ \u7684\u5b66\u4e60\u8fc7\u7a0b\u90fd\u662f\u81f3\u5173\u91cd\u8981\u7684\u3002\n\n\n\n\nDiscriminator Code\n\n\n\n\n```python\nclass Discriminator(nn.Module):\n def __init__(self, ngpu):\n super(Discriminator, self).__init__()\n self.ngpu = ngpu\n self.main = nn.Sequential(\n # input is (nc) x 64 x 64\n nn.Conv2d(nc, ndf, 4, 2, 1, bias=False),\n nn.LeakyReLU(0.2, inplace=True),\n # state size. (ndf) x 32 x 32\n nn.Conv2d(ndf, ndf * 2, 4, 2, 1, bias=False),\n nn.BatchNorm2d(ndf * 2),\n nn.LeakyReLU(0.2, inplace=True),\n # state size. (ndf*2) x 16 x 16\n nn.Conv2d(ndf * 2, ndf * 4, 4, 2, 1, bias=False),\n nn.BatchNorm2d(ndf * 4),\n nn.LeakyReLU(0.2, inplace=True),\n # state size. (ndf*4) x 8 x 8\n nn.Conv2d(ndf * 4, ndf * 8, 4, 2, 1, bias=False),\n nn.BatchNorm2d(ndf * 8),\n nn.LeakyReLU(0.2, inplace=True),\n # state size. (ndf*8) x 4 x 4\n nn.Conv2d(ndf * 8, 1, 4, 1, 0, bias=False),\n nn.Sigmoid()\n )\n\n def forward(self, input):\n return self.main(input)\n```\n\n\u73b0\u5728\uff0c\u548c\u751f\u6210\u5668\u4e00\u6837\uff0c\u6211\u4eec\u53ef\u4ee5\u521b\u5efa\u5224\u522b\u5668\uff0c\u5e94\u7528 ``weights_init`` \u51fd\u6570\uff0c\u5e76\u6253\u5370\u6a21\u578b\u7684\u7ed3\u6784\u3002\n\n\n\n\n\n```python\n# \u521b\u5efa Discriminator\nnetD = Discriminator(ngpu).to(device)\n\n# Handle multi-gpu if desired\nif (device.type == 'cuda') and (ngpu > 1):\n netD = nn.DataParallel(netD, list(range(ngpu)))\n \n# \u5e94\u7528 weights_init \u51fd\u6570\uff0c\u968f\u673a\u521d\u59cb\u5316\u6240\u6709\u6743\u91cd\u5230 mean=0, stdev=0.2.\nnetD.apply(weights_init)\n\n# \u6253\u5370\u8f93\u51fa\u6a21\u578b\nprint(netD)\n```\n\n\u635f\u5931\u51fd\u6570\u548c\u4f18\u5316\u5668\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n\u5f53 $D$ \u548c $G$ \u8bbe\u7f6e\u597d\u4ee5\u540e, \u6211\u4eec\u53ef\u4ee5\u6307\u5b9a\u5b83\u4eec\u5982\u4f55\u901a\u8fc7\u635f\u5931\u51fd\u6570\u548c\u4f18\u5316\u5668\u5b66\u4e60\u3002\n\u6211\u4eec\u5c06\u4f7f\u7528\u4e8c\u503c\u4ea4\u53c9\u71b5\u635f\u5931(Binary Cross Entropy loss (`BCELoss `__))\n\u51fd\u6570\uff0c\u5728 PyTorch \u4e2d\u662f\u5982\u4e0b\u5b9a\u4e49\u7684:\n\n\\begin{align}\\ell(x, y) = L = \\{l_1,\\dots,l_N\\}^\\top, \\quad l_n = - \\left[ y_n \\cdot \\log x_n + (1 - y_n) \\cdot \\log (1 - x_n) \\right]\\end{align}\n\n\u6ce8\u610f\u8fd9\u4e2a\u51fd\u6570\u63d0\u4f9b\u76ee\u6807\u51fd\u6570\u4e2d\u7684\u4e24\u4e2a\u5bf9\u6570\u7ec4\u4ef6\u7684\u8ba1\u7b97 (i.e. $log(D(x))$ \u548c $log(1-D(G(z)))$) \u3002\n\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528 $y$ \u6307\u5b9a BCE \u7b49\u5f0f\u7684\u54ea\u4e00\u90e8\u5206\u5c06\u88ab\u8ba1\u7b97\u3002 \u8fd9\u5c06\u5728\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u5b8c\u6210\uff0c\u7a0d\u540e\u4f1a\u8bb2\u5230\u3002\u4f46\u662f\u7406\u89e3\u6211\u4eec\u5982\u4f55\u901a\u8fc7\n\u6539\u53d8 $y$ \u7684\u503c(i.e.\u00a0GT labels) \u53bb\u9009\u62e9\u6211\u4eec\u60f3\u8981\u8ba1\u7b97\u7684\u635f\u5931\u51fd\u6570\u7684\u4e00\u90e8\u5206\u662f\u975e\u5e38\u91cd\u8981\u7684\u3002\n\n\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u771f\u6807\u7b7e\u5b9a\u4e49\u4e3a1\uff0c\u5047\u6807\u7b7e\u5b9a\u4e49\u4e3a0\u3002\u8fd9\u4e9b\u6807\u7b7e\u5c06\u7528\u4e8e\u8ba1\u7b97 $D$ \u548c $G$ \u7684\u635f\u5931\uff0c\n\u8fd9\u4e5f\u662f\u5728\u539f\u59cbGAN\u6587\u7ae0\u4e2d\u4f7f\u7528\u7684\u7ea6\u5b9a\u3002\u6700\u540e\uff0c\u6211\u4eec\u5efa\u7acb\u4e86\u4e24\u4e2a\u5206\u5f00\u7684\u4f18\u5316\u5668\uff0c\u4e00\u4e2a\u7528\u4e8e $D$ \uff0c\n\u4e00\u4e2a\u7528\u4e8e $G$ \u3002\u6b63\u5982DCGAN\u8bba\u6587\u6240\u6307\u51fa\u7684\uff0c\u4e24\u8005\u90fd\u662fAdam\u4f18\u5316\u5668\uff0c\u5176\u5b66\u4e60\u901f\u7387\u4e3a0.0002\uff0cBeta1=0.5\u3002\n\u4e3a\u4e86\u8ddf\u8e2a\u751f\u6210\u5668\u7684\u5b66\u4e60\u8fc7\u7a0b\uff0c\u6211\u4eec\u5c06\u4ece\u9ad8\u65af\u5206\u5e03(\u5373\u56fa\u5b9a\u566a\u58f0)\u4e2d\u751f\u6210\u56fa\u5b9a\u6279\u6b21\u7684\u6f5c\u5728\u5411\u91cf(latent vectors)\u3002\n\u5728\u8bad\u7ec3\u5faa\u73af\u4e2d\uff0c\u6211\u4eec\u5c06\u5468\u671f\u6027\u5730\u5c06\u8fd9\u4e2a\u56fa\u5b9a\u7684\u566a\u58f0\u8f93\u5165\u5230 $G$ \u4e2d\u3002\u5728\u8fed\u4ee3\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u5c06\u770b\u5230\u56fe\u50cf\u4ece\u566a\u58f0\u4e2d\u5f62\u6210\u3002\n\n\n\n\n\n```python\n# \u521d\u59cb\u5316 BCELoss \u51fd\u6570\ncriterion = nn.BCELoss()\n\n# \u521b\u5efa\u4e00\u6279 latent vectors \u7528\u4e8e\u53ef\u89c6\u5316\u751f\u6210\u5668\u7684\u8fdb\u5ea6\u8fc7\u7a0b\nfixed_noise = torch.randn(64, nz, 1, 1, device=device)\n\n# \u4e3a\u5728\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u7684\u771f\u5047\u6807\u7b7e\u5efa\u7acb\u7ea6\u5b9a\nreal_label = 1\nfake_label = 0\n\n# \u4e3a G \u548c D \u8bbe\u7f6e Adam optimizers \noptimizerD = optim.Adam(netD.parameters(), lr=lr, betas=(beta1, 0.999))\noptimizerG = optim.Adam(netG.parameters(), lr=lr, betas=(beta1, 0.999))\n```\n\n\u8bad\u7ec3\n~~~~~~~~\n\n\u6700\u540e\uff0c\u73b0\u5728\u6211\u4eec\u5df2\u7ecf\u5b9a\u4e49\u4e86GAN\u6846\u67b6\u7684\u6240\u6709\u90e8\u5206\uff0c\u6211\u4eec\u53ef\u4ee5\u5bf9\u5176\u8fdb\u884c\u8bad\u7ec3\u3002\u8bf7\u6ce8\u610f\uff0c\n\u8bad\u7ec3GANs\u662f\u4e00\u79cd\u827a\u672f\uff0c\u56e0\u4e3a\u4e0d\u6b63\u786e\u7684\u8d85\u53c2\u6570\u8bbe\u7f6e\u4f1a\u5bfc\u81f4\u6a21\u5f0f\u5d29\u6e83\uff0c\n\u800c\u5bf9\u9519\u8bef\u7684\u539f\u56e0\u51e0\u4e4e\u6ca1\u6709\u89e3\u91ca\u3002\u5728\u8fd9\u91cc\uff0c\u6211\u4eec\u5c06\u5bc6\u5207\u9075\u5faa\u53e4\u5fb7\u8d39\u7f57\u8bba\u6587\u4e2d\u7684\u7b97\u6cd51\uff0c\n\u540c\u65f6\u9075\u5faa\u5728 `ganhacks `__ \u4e2d\u663e\u793a\u7684\u4e00\u4e9b\u6700\u4f73\u5b9e\u8df5\u3002\n\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u5c06\u201c\u4e3a\u771f\u5047\u56fe\u50cf\u6784\u9020\u4e0d\u540c\u7684\u5c0f\u6279\u91cf\u201d\u56fe\u50cf\uff0c\n\u5e76\u8c03\u6574G\u7684\u76ee\u6807\u51fd\u6570\uff0c\u4f7f $logD(G(z))$ \u6700\u5927\u5316\u3002\u8bad\u7ec3\u5206\u4e3a\u4e24\u4e2a\u4e3b\u8981\u90e8\u5206\u3002\n\u7b2c1\u90e8\u5206\u66f4\u65b0\u5224\u522b\u5668\uff0c\u7b2c2\u90e8\u5206\u66f4\u65b0\u751f\u6210\u5668\u3002\n\n**Part 1 - \u8bad\u7ec3\u5224\u522b\u5668(Discriminator) **\n\n\u56de\u60f3\u4e00\u4e0b\uff0c\u8bad\u7ec3\u5224\u522b\u5668\u7684\u76ee\u6807\u662f\u6700\u5927\u5316\u5c06\u7ed9\u5b9a\u7684\u8f93\u5165\u6b63\u786e\u5206\u7c7b\u4e3a\u771f\u6216\u5047\u7684\u6982\u7387\u3002\n\u6211\u4eec\u5e0c\u671b\u201c\u901a\u8fc7\u63d0\u5347\u5224\u522b\u5668\u7684\u968f\u673a\u68af\u5ea6\u6765\u66f4\u65b0\u5224\u522b\u5668\u201d\u3002\n\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u5e0c\u671b\u6700\u5927\u5316 $log(D(x)) + log(1-D(G(z)))$ \u3002\n\u7531\u4e8e\u6765\u81ea\u4e8eganhacks \u7684separate mini-batch\u7684\u5efa\u8bae\uff0c\n\u6211\u4eec\u5c06\u7528\u4e24\u4e2a\u6b65\u9aa4\u6765\u5b9e\u73b0\u4e0a\u8ff0\u6700\u5927\u5316\u7684\u8ba1\u7b97\u8fc7\u7a0b\u3002\u9996\u5148\u4ece\u8bad\u7ec3\u96c6\u6784\u9020\u4e00\u6279\u771f\u5b9e\u6837\u672c\uff0c\u524d\u5411\u901a\u8fc7 $D$ \uff0c\n\u8ba1\u7b97\u635f\u5931($log(D(x))$) \uff0c\u7136\u540e\u8ba1\u7b97\u540e\u4f20\u68af\u5ea6\u3002 \n\u5176\u6b21\uff0c\u7528\u5f53\u524d\u751f\u6210\u5668\u6784\u9020\u4e00\u6279\u5047\u6837\u672c\uff0c\u901a\u8fc7 $D$ \u5411\u524d\u4f20\u9012\u8be5\u6279\u6837\u672c\uff0c\n\u8ba1\u7b97\u635f\u5931 ($log(1-D(G(z)))$) \uff0c\u5e76\u7528\u53cd\u5411\u4f20\u9012\u7d2f\u79ef\u68af\u5ea6\u3002\n\u73b0\u5728\uff0c\u6709\u4e86\u5168\u771f\u548c\u5168\u5047\u6279\u6b21\u6837\u672c\u4e2d\u79ef\u7d2f\u7684\u68af\u5ea6\uff0c\u6211\u4eec\u518d\u8c03\u7528\u5224\u522b\u5668\u7684\u4f18\u5316\u5668\u8fdb\u884c\u4e00\u6b65\u4f18\u5316\u3002\n\n**Part 2 - \u8bad\u7ec3\u751f\u6210\u5668(Generator) **\n\n\u6b63\u5982\u5728\u6700\u521d\u7684\u8bba\u6587\u4e2d\u6240\u8ff0\uff0c\u6211\u4eec\u5e0c\u671b\u901a\u8fc7\u6700\u5c0f\u5316 $log(1-D(G(z)))$ \u6765\u8bad\u7ec3\u751f\u6210\u5668\uff0c\u4ee5\u4ea7\u751f\u66f4\u597d\u7684\u5047\u6837\u672c\u3002\n\u6b63\u5982\u524d\u9762\u63d0\u5230\u7684\uff0cGoodfellow\u6ca1\u6709\u63d0\u4f9b\u8db3\u591f\u7684\u68af\u5ea6\uff0c\u7279\u522b\u662f\u5728\u5b66\u4e60\u8fc7\u7a0b\u7684\u65e9\u671f\u3002\u4f5c\u4e3a\u4fee\u6b63\uff0c\n\u6211\u4eec\u5e0c\u671b\u6700\u5927\u5316 $log(D(G(z)))$ \u3002\u5728\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u901a\u8fc7\u4ee5\u4e0b\u65b9\u6cd5\u5b9e\u73b0\u4e86\u8fd9\u4e00\u70b9\uff1a\n\u7528\u7b2c1\u90e8\u5206\u7684\u5224\u522b\u5668\u5bf9\u751f\u6210\u5668\u7684\u8f93\u51fa\u8fdb\u884c\u5206\u7c7b\uff0c\u4f7f\u7528\u771f\u6807\u7b7e\u4f5c\u4e3aGroundTruth\u8ba1\u7b97G\u7684\u635f\u5931, \n\uff0c\u968f\u540e\u5728\u5411\u540e\u4f20\u9012\u4e2d\u8ba1\u7b97G\u7684\u68af\u5ea6\uff0c\u6700\u540e\u7528\u4f18\u5316\u5668\u7684 ``step`` \u65b9\u6cd5\u66f4\u65b0G\u7684\u53c2\u6570\u3002\n\u4f7f\u7528\u771f\u6807\u7b7e\u4f5c\u4e3aGT\u6807\u7b7e\u7528\u4e8e\u635f\u5931\u51fd\u6570\u7684\u8ba1\u7b97\u4f3c\u4e4e\u6709\u8fdd\u76f4\u89c9\uff0c\u4f46\u8fd9\u5141\u8bb8\u6211\u4eec\u4f7f\u7528BCELoss\u7684 $log(x)$ \u90e8\u5206\n(\u800c\u4e0d\u662f $log(1-x)$ \u90e8\u5206)\uff0c\u8fd9\u6b63\u662f\u6211\u4eec\u60f3\u8981\u7684\u3002\n\n\u6700\u540e\uff0c\u6211\u4eec\u5c06\u505a\u4e00\u4e9b\u7edf\u8ba1\u62a5\u544a\uff0c\u5e76\u5728\u6bcf\u4e2aepoch\u7ed3\u675f\u65f6\uff0c\u6211\u4eec\u5c06\u628a\u56fa\u5b9a\u6279\u6b21\u566a\u58f0\u63a8\u5230\u751f\u6210\u5668\u4e2d\n\u4ee5\u53ef\u89c6\u5316\u5730\u8ddf\u8e2aG\u7684\u8bad\u7ec3\u8fdb\u5ea6\u3002\u6240\u62a5\u544a\u7684\u8bad\u7ec3\u7edf\u8ba1\u6570\u5b57\u5982\u4e0b\uff1a\n\n- **Loss_D** - \u5224\u522b\u5668\u635f\u5931\uff0c\u662f\u6240\u6709\u771f\u6279\u6b21\u548c\u6240\u6709\u5047\u6279\u6b21\u6837\u672c\u4e0a\u7684\u635f\u5931\u4e4b\u548c ($log(D(x)) + log(D(G(z)))$)\u3002\n- **Loss_G** - \u751f\u6210\u5668\u635f\u5931\uff0c\u7528 $log(D(G(z)))$ \u8ba1\u7b97\u3002\n- **D(x)** - \u6240\u6709\u6279\u6b21\u7684\u771f\u6837\u672c\u4e0a\u5224\u522b\u5668\u7684\u5e73\u5747\u8f93\u51fa(\u8de8batch)\u3002\u8fd9\u4e2a\u503c\u5e94\u8be5\u5f00\u59cb\u63a5\u8fd11\uff0c\u7136\u540e\u5f53G\u53d8\u5f97\u66f4\u597d\u65f6\uff0c\u7406\u8bba\u4e0a\u6536\u655b\u52300.5\u3002\u60f3\u60f3\u8fd9\u662f\u4e3a\u4ec0\u4e48\u3002\n- **D(G(z))** - \u6240\u6709\u6279\u6b21\u7684\u5047\u6837\u672c\u4e0a\u5224\u522b\u5668\u7684\u5e73\u5747\u8f93\u51fa\u3002\u8fd9\u4e2a\u503c\u5e94\u8be5\u5f00\u59cb\u63a5\u8fd10\uff0c\u540e\u9762\u968f\u7740\u751f\u6210\u5668\u8d8a\u6765\u8d8a\u597d\u5c31\u6536\u655b\u52300.5\u3002\u60f3\u60f3\u8fd9\u662f\u4e3a\u4ec0\u4e48\u3002\n\n**Note:** \u8fd9\u4e00\u6b65\u53ef\u80fd\u4f1a\u82b1\u70b9\u65f6\u95f4, \u8fd9\u53d6\u51b3\u4e8e\u4f60\u8981\u8fd0\u884c\u591a\u5c11\u4e2aepoch\u4ee5\u53ca\u5982\u679c\u4f60\u4ece\u6570\u636e\u96c6\u79fb\u9664\u4e00\u4e9b\u6570\u636e\u3002\n\n\n\n\n\n```python\n# Training Loop\n\n# Lists to keep track of progress\nimg_list = []\nG_losses = []\nD_losses = []\niters = 0\n\nprint(\"Starting Training Loop...\")\n# For each epoch\nfor epoch in range(num_epochs):\n # For each batch in the dataloader\n for i, data in enumerate(dataloader, 0):\n \n ############################\n # (1) Update D network: maximize log(D(x)) + log(1 - D(G(z)))\n ###########################\n ## Train with all-real batch\n netD.zero_grad()\n # Format batch\n real_cpu = data[0].to(device)\n b_size = real_cpu.size(0)\n label = torch.full((b_size,), real_label, device=device)\n # Forward pass real batch through D\n output = netD(real_cpu).view(-1)\n # Calculate loss on all-real batch\n errD_real = criterion(output, label)\n # Calculate gradients for D in backward pass\n errD_real.backward()\n D_x = output.mean().item()\n\n ## Train with all-fake batch\n # Generate batch of latent vectors\n noise = torch.randn(b_size, nz, 1, 1, device=device)\n # Generate fake image batch with G\n fake = netG(noise)\n label.fill_(fake_label)\n # Classify all fake batch with D\n output = netD(fake.detach()).view(-1)\n # Calculate D's loss on the all-fake batch\n errD_fake = criterion(output, label)\n # Calculate the gradients for this batch\n errD_fake.backward()\n D_G_z1 = output.mean().item()\n # Add the gradients from the all-real and all-fake batches\n errD = errD_real + errD_fake\n # Update D\n optimizerD.step()\n\n ############################\n # (2) Update G network: maximize log(D(G(z)))\n ###########################\n netG.zero_grad()\n label.fill_(real_label) # fake labels are real for generator cost\n # Since we just updated D, perform another forward pass of all-fake batch through D\n output = netD(fake).view(-1)\n # Calculate G's loss based on this output\n errG = criterion(output, label)\n # Calculate gradients for G\n errG.backward()\n D_G_z2 = output.mean().item()\n # Update G\n optimizerG.step()\n \n # Output training stats\n if i % 50 == 0:\n print('[%d/%d][%d/%d]\\tLoss_D: %.4f\\tLoss_G: %.4f\\tD(x): %.4f\\tD(G(z)): %.4f / %.4f'\n % (epoch, num_epochs, i, len(dataloader),\n errD.item(), errG.item(), D_x, D_G_z1, D_G_z2))\n \n # Save Losses for plotting later\n G_losses.append(errG.item())\n D_losses.append(errD.item())\n \n # Check how the generator is doing by saving G's output on fixed_noise\n if (iters % 500 == 0) or ((epoch == num_epochs-1) and (i == len(dataloader)-1)):\n with torch.no_grad():\n fake = netG(fixed_noise).detach().cpu()\n img_list.append(vutils.make_grid(fake, padding=2, normalize=True))\n \n iters += 1\n```\n\n\u7ed3\u679c\n-------\n\n\u6700\u540e\uff0c\u8ba9\u6211\u4eec\u6765\u770b\u770b\u6211\u4eec\u662f\u5982\u4f55\u505a\u5230\u7684\u3002\u5728\u8fd9\u91cc\uff0c\u6211\u4eec\u5c06\u770b\u5230\u4e09\u4e2a\u4e0d\u540c\u7684\u7ed3\u679c\u3002\n\u9996\u5148\uff0c\u6211\u4eec\u5c06\u770b\u5230D\u548cG\u5728\u8bad\u7ec3\u4e2d\u7684\u635f\u5931\u662f\u5982\u4f55\u53d8\u5316\u7684\u3002\u7b2c\u4e8c\uff0c\u6211\u4eec\u5c06\u5728\u6bcf\u4e2aepoch\u7684\u56fa\u5b9a\u566a\u58f0\u6279\u6b21\u4e0a\u53ef\u89c6\u5316G\u7684\u8f93\u51fa\u3002\n\u7b2c\u4e09\uff0c\u6211\u4eec\u5c06\u770b\u5230\u4e00\u6279\u771f\u6570\u636e\uff0c\u65c1\u8fb9\u662f\u4e00\u6279\u6765\u81eaG\u7684\u5047\u6570\u636e\u3002\n\n**Loss versus training iteration**\n\n\u4e0b\u9762\u662f\u8fed\u4ee3\u8fc7\u7a0b\u4e2d D \u4e0e G \u7684\u635f\u5931\u5bf9\u6bd4\u56fe\u3002 \n\n\n\n\n\n```python\nplt.figure(figsize=(10,5))\nplt.title(\"Generator and Discriminator Loss During Training\")\nplt.plot(G_losses,label=\"G\")\nplt.plot(D_losses,label=\"D\")\nplt.xlabel(\"iterations\")\nplt.ylabel(\"Loss\")\nplt.legend()\nplt.show()\n```\n\n**G\u7684\u8fdb\u5ea6\u7684\u53ef\u89c6\u5316**\n\n\u8bb0\u4f4f\uff0c\u5728\u6bcf\u4e2a\u8bad\u7ec3\u56de\u5408(epoch)\u4e4b\u540e\uff0c\u6211\u4eec\u662f\u5982\u4f55\u5c06generator\u7684\u8f93\u51fa\u4fdd\u5b58\u5728\u56fa\u5b9a\u566a\u58f0\u6279\u6b21\u4e0a\u7684\u3002\n\u73b0\u5728\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u52a8\u753b\u6765\u53ef\u89c6\u5316G\u7684\u8bad\u7ec3\u8fdb\u5ea6\u3002\u6309\u201c\u64ad\u653e\u201d\u6309\u94ae\u542f\u52a8\u52a8\u753b\u3002\n\n\n\n\n\n```python\n#%%capture\nfig = plt.figure(figsize=(8,8))\nplt.axis(\"off\")\nims = [[plt.imshow(np.transpose(i,(1,2,0)), animated=True)] for i in img_list]\nani = animation.ArtistAnimation(fig, ims, interval=1000, repeat_delay=1000, blit=True)\n\nHTML(ani.to_jshtml())\n```\n\n**\u771f\u56fe\u50cf(Real Images) vs.\u00a0\u5047\u56fe\u50cf(Fake Images)**\n\n\u6700\u540e, \u8ba9\u6211\u4eec\u770b\u770b\u771f\u56fe\u50cf\u548c\u5047\u56fe\u50cf\u5427\uff01\n\n\n\n\n\n```python\n# \u4ece dataloader \u4e2d\u6293\u53d6\u4e00\u4e2a\u6279\u6b21\u7684\u771f\u56fe\u50cf\nreal_batch = next(iter(dataloader))\n\n# Plot the real images\nplt.figure(figsize=(15,15))\nplt.subplot(1,2,1)\nplt.axis(\"off\")\nplt.title(\"Real Images\")\nplt.imshow(np.transpose(vutils.make_grid(real_batch[0].to(device)[:64], padding=5, normalize=True).cpu(),(1,2,0)))\n\n# \u7ed8\u5236\u6700\u540e\u4e00\u4e2aepoch\u7684\u5047\u56fe\u50cf\nplt.subplot(1,2,2)\nplt.axis(\"off\")\nplt.title(\"Fake Images\")\nplt.imshow(np.transpose(img_list[-1],(1,2,0)))\nplt.show()\n```\n\n\u4e0b\u4e00\u6b65\u53bb\u54ea\u91cc\n----------------\n\n\u6211\u4eec\u7684\u65c5\u7a0b\u5df2\u7ecf\u5230\u4e86\u5c3d\u5934\uff0c\u4f46\u662f\u6709\u51e0\u4e2a\u5730\u65b9\u4f60\u53ef\u4ee5\u4ece\u8fd9\u91cc\u53bb\u3002\u4f60\u53ef\u4ee5\uff1a\n\n- \u8bad\u7ec3\u66f4\u957f\u7684\u65f6\u95f4\u770b\u770b\u5f97\u5230\u7684\u7ed3\u679c\u6709\u591a\u597d\n- \u4fee\u6539\u6b64\u6a21\u578b\u8ba9\u5176\u63a5\u6536\u4e0d\u540c\u7684\u6570\u636e\u96c6 \u548c \u53ef\u80fd\u6539\u53d8\u7684\u56fe\u50cf\u5927\u5c0f\u4e0e\u6a21\u578b\u67b6\u6784\n- \u68c0\u67e5\u5176\u4ed6\u4e00\u4e9b\u5f88\u9177\u7684 GAN \u9879\u76ee `\u8fd9\u91cc `__ \u3002\n- \u521b\u5efa\u4e00\u4e2a GANs \u8ba9\u5b83\u4ea7\u751f `\u97f3\u4e50 `__\n\n\n\n", "meta": {"hexsha": "e10a812654e00b9da9cae091c4a196606bfdb4dc", "size": 43409, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "build/_downloads/4d64c2efc340400ae87f18d12c816067/dcgan_faces_tutorial.ipynb", "max_stars_repo_name": "ScorpioDoctor/antares02", "max_stars_repo_head_hexsha": "631b817d2e98f351d1173b620d15c4a5efed11da", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, 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YES\n2. YES", "lm_q1_score": 0.7905303137346444, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.4322130828190965}} {"text": "```python\ntry:\n import openmdao.api as om\n import dymos as dm\nexcept ImportError:\n !python -m pip install openmdao[notebooks]\n !python -m pip install dymos[docs]\n import openmdao.api as om\n import dymos as dm\n```\n\n# Two-Burn Orbit Raise\n\nThis example demonstrates the use of a Trajectory to encapsulate a\nthree-phase orbit raising maneuver with a burn-coast-burn phase\nsequence. This example is based on the problem provided in\nEnright {cite}`enright1991optimal`.\n\nThe dynamics are given by\n\n\\begin{align}\n \\frac{dr}{dt} &= v_r \\\\\n \\frac{d\\theta}{dt} &= \\frac{v_\\theta}{r} \\\\\n \\frac{dv_r}{dt} &= \\frac{v^2_\\theta}{r} - \\frac{1}{r^2} + a_{thrust} \\sin u_1 \\\\\n \\frac{dv_\\theta}{dt} &= - \\frac{v_r v_\\theta}{r} + a_{thrust} \\cos u_1 \\\\\n \\frac{da_{thrust}}{dt} &= \\frac{a^2_{thrust}}{c} \\\\\n \\frac{d \\Delta v}{dt} &= a_{thrust}\n\\end{align}\n\nThe initial conditions are\n\n\\begin{align}\n r &= 1 \\rm{\\,DU} \\\\\n \\theta &= 0 \\rm{\\,rad} \\\\\n v_r &= 0 \\rm{\\,DU/TU}\\\\\n v_\\theta &= 1 \\rm{\\,DU/TU}\\\\\n a_{thrust} &= 0.1 \\rm{\\,DU/TU^2}\\\\\n \\Delta v &= 0 \\rm{\\,DU/TU}\n\\end{align}\n\nand the final conditions are\n\n\\begin{align}\n r &= 3 \\rm{\\,DU} \\\\\n \\theta &= \\rm{free} \\\\\n v_r &= 0 \\rm{\\,DU/TU}\\\\\n v_\\theta &= \\sqrt{\\frac{1}{3}} \\rm{\\,DU/TU}\\\\\n a_{thrust} &= \\rm{free}\\\\\n \\Delta v &= \\rm{free}\n\\end{align}\n\n## Building and running the problem\n\nThe following code instantiates our problem, our trajectory, three\nphases, and links them accordingly. The spacecraft initial position,\nvelocity, and acceleration magnitude are fixed. The objective is to\nminimize the delta-V needed to raise the spacecraft into a circular\norbit at 3 Earth radii.\n\nNote the call to _link\\_phases_ which provides time,\nposition, velocity, and delta-V continuity across all phases, and\nacceleration continuity between the first and second burn phases.\nAcceleration is 0 during the coast phase. Alternatively, we could have\nspecified a different ODE for the coast phase, as in the example.\n\nThis example runs inconsistently with SLSQP but is solved handily by\nSNOPT.\n\n\n```python\nom.display_source(\"dymos.examples.finite_burn_orbit_raise.finite_burn_eom\")\n```\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport openmdao.api as om\n\nimport dymos as dm\nfrom dymos.examples.finite_burn_orbit_raise.finite_burn_eom import FiniteBurnODE\n\np = om.Problem(model=om.Group())\n\np.driver = om.pyOptSparseDriver()\np.driver.options['optimizer'] = 'IPOPT'\np.driver.declare_coloring()\n\ntraj = dm.Trajectory()\n\ntraj.add_parameter('c', opt=False, val=1.5, units='DU/TU',\n targets={'burn1': ['c'], 'coast': ['c'], 'burn2': ['c']})\n\n# First Phase (burn)\n\nburn1 = dm.Phase(ode_class=FiniteBurnODE,\n transcription=dm.GaussLobatto(num_segments=5, order=3, compressed=False))\n\nburn1 = traj.add_phase('burn1', burn1)\n\nburn1.set_time_options(fix_initial=True, duration_bounds=(.5, 10), units='TU')\nburn1.add_state('r', fix_initial=True, fix_final=False, defect_scaler=100.0,\n rate_source='r_dot', units='DU')\nburn1.add_state('theta', fix_initial=True, fix_final=False, defect_scaler=100.0,\n rate_source='theta_dot', units='rad')\nburn1.add_state('vr', fix_initial=True, fix_final=False, defect_scaler=100.0,\n rate_source='vr_dot', units='DU/TU')\nburn1.add_state('vt', fix_initial=True, fix_final=False, defect_scaler=100.0,\n rate_source='vt_dot', units='DU/TU')\nburn1.add_state('accel', fix_initial=True, fix_final=False,\n rate_source='at_dot', units='DU/TU**2')\nburn1.add_state('deltav', fix_initial=True, fix_final=False,\n rate_source='deltav_dot', units='DU/TU')\nburn1.add_control('u1', rate_continuity=True, rate2_continuity=True, units='deg',\n scaler=0.01, rate_continuity_scaler=0.001, rate2_continuity_scaler=0.001,\n lower=-30, upper=30)\n# Second Phase (Coast)\ncoast = dm.Phase(ode_class=FiniteBurnODE,\n transcription=dm.GaussLobatto(num_segments=5, order=3, compressed=False))\n\ncoast.set_time_options(initial_bounds=(0.5, 20), duration_bounds=(.5, 50), duration_ref=50,\n units='TU')\ncoast.add_state('r', fix_initial=False, fix_final=False, defect_scaler=100.0,\n rate_source='r_dot', targets=['r'], units='DU')\ncoast.add_state('theta', fix_initial=False, fix_final=False, defect_scaler=100.0,\n rate_source='theta_dot', targets=['theta'], units='rad')\ncoast.add_state('vr', fix_initial=False, fix_final=False, defect_scaler=100.0,\n rate_source='vr_dot', targets=['vr'], units='DU/TU')\ncoast.add_state('vt', fix_initial=False, fix_final=False, defect_scaler=100.0,\n rate_source='vt_dot', targets=['vt'], units='DU/TU')\ncoast.add_state('accel', fix_initial=True, fix_final=True,\n rate_source='at_dot', targets=['accel'], units='DU/TU**2')\ncoast.add_state('deltav', fix_initial=False, fix_final=False,\n rate_source='deltav_dot', units='DU/TU')\n\ncoast.add_parameter('u1', opt=False, val=0.0, units='deg', targets=['u1'])\n\n# Third Phase (burn)\nburn2 = dm.Phase(ode_class=FiniteBurnODE,\n transcription=dm.GaussLobatto(num_segments=5, order=3, compressed=False))\n\ntraj.add_phase('coast', coast)\ntraj.add_phase('burn2', burn2)\n\nburn2.set_time_options(initial_bounds=(0.5, 50), duration_bounds=(.5, 10), initial_ref=10,\n units='TU')\nburn2.add_state('r', fix_initial=False, fix_final=True, defect_scaler=100.0,\n rate_source='r_dot', units='DU')\nburn2.add_state('theta', fix_initial=False, fix_final=False, defect_scaler=100.0,\n rate_source='theta_dot', units='rad')\nburn2.add_state('vr', fix_initial=False, fix_final=True, defect_scaler=1000.0,\n rate_source='vr_dot', units='DU/TU')\nburn2.add_state('vt', fix_initial=False, fix_final=True, defect_scaler=1000.0,\n rate_source='vt_dot', units='DU/TU')\nburn2.add_state('accel', fix_initial=False, fix_final=False, defect_scaler=1.0,\n rate_source='at_dot', units='DU/TU**2')\nburn2.add_state('deltav', fix_initial=False, fix_final=False, defect_scaler=1.0,\n rate_source='deltav_dot', units='DU/TU')\n\nburn2.add_objective('deltav', loc='final', scaler=100.0)\n\nburn2.add_control('u1', rate_continuity=True, rate2_continuity=True, units='deg',\n scaler=0.01, lower=-90, upper=90)\n\nburn1.add_timeseries_output('pos_x')\ncoast.add_timeseries_output('pos_x')\nburn2.add_timeseries_output('pos_x')\n\nburn1.add_timeseries_output('pos_y')\ncoast.add_timeseries_output('pos_y')\nburn2.add_timeseries_output('pos_y')\n\n# Link Phases\ntraj.link_phases(phases=['burn1', 'coast', 'burn2'],\n vars=['time', 'r', 'theta', 'vr', 'vt', 'deltav'])\n\ntraj.link_phases(phases=['burn1', 'burn2'], vars=['accel'])\n\np.model.add_subsystem('traj', subsys=traj)\n\n# Finish Problem Setup\n\n# Needed to move the direct solver down into the phases for use with MPI.\n# - After moving down, used fewer iterations (about 30 less)\n\np.driver.add_recorder(om.SqliteRecorder('two_burn_orbit_raise_example.db'))\n\np.setup(check=True, mode='fwd')\n\n# Set Initial Guesses\np.set_val('traj.parameters:c', value=1.5, units='DU/TU')\n\nburn1 = p.model.traj.phases.burn1\nburn2 = p.model.traj.phases.burn2\ncoast = p.model.traj.phases.coast\n\np.set_val('traj.burn1.t_initial', value=0.0)\np.set_val('traj.burn1.t_duration', value=2.25)\np.set_val('traj.burn1.states:r', value=burn1.interp('r', [1, 1.5]))\np.set_val('traj.burn1.states:theta', value=burn1.interp('theta', [0, 1.7]))\np.set_val('traj.burn1.states:vr', value=burn1.interp('vr', [0, 0]))\np.set_val('traj.burn1.states:vt', value=burn1.interp('vt', [1, 1]))\np.set_val('traj.burn1.states:accel', value=burn1.interp('accel', [0.1, 0]))\np.set_val('traj.burn1.states:deltav', value=burn1.interp('deltav', [0, 0.1]))\np.set_val('traj.burn1.controls:u1', value=burn1.interp('u1', [-3.5, 13.0]))\n\np.set_val('traj.coast.t_initial', value=2.25)\np.set_val('traj.coast.t_duration', value=3.0)\n\np.set_val('traj.coast.states:r', value=coast.interp('r', [1.3, 1.5]))\np.set_val('traj.coast.states:theta', value=coast.interp('theta', [2.1767, 1.7]))\np.set_val('traj.coast.states:vr', value=coast.interp('vr', [0.3285, 0]))\np.set_val('traj.coast.states:vt', value=coast.interp('vt', [0.97, 1]))\np.set_val('traj.coast.states:accel', value=coast.interp('accel', [0, 0]))\n\np.set_val('traj.burn2.t_initial', value=5.25)\np.set_val('traj.burn2.t_duration', value=1.75)\n\np.set_val('traj.burn2.states:r', value=burn2.interp('r', [1, 3.]))\np.set_val('traj.burn2.states:theta', value=burn2.interp('theta', [0, 4.0]))\np.set_val('traj.burn2.states:vr', value=burn2.interp('vr', [0, 0]))\np.set_val('traj.burn2.states:vt', value=burn2.interp('vt', [1, np.sqrt(1 / 3.)]))\np.set_val('traj.burn2.states:deltav', value=burn2.interp('deltav', [0.1, 0.2]))\np.set_val('traj.burn2.states:accel', value=burn2.interp('accel', [0.1, 0]))\n\np.set_val('traj.burn2.controls:u1', value=burn2.interp('u1', [0, 0]))\n\ndm.run_problem(p)\n\n#\n# Plot results\n#\ntraj = p.model.traj\nexp_out = traj.simulate()\n\nfig = plt.figure(figsize=(8, 4))\nfig.suptitle('Two Burn Orbit Raise Solution')\nax_u1 = plt.subplot2grid((2, 2), (0, 0))\nax_deltav = plt.subplot2grid((2, 2), (1, 0))\nax_xy = plt.subplot2grid((2, 2), (0, 1), rowspan=2)\n\nspan = np.linspace(0, 2 * np.pi, 100)\nax_xy.plot(np.cos(span), np.sin(span), 'k--', lw=1)\nax_xy.plot(3 * np.cos(span), 3 * np.sin(span), 'k--', lw=1)\nax_xy.set_xlim(-4.5, 4.5)\nax_xy.set_ylim(-4.5, 4.5)\n\nax_xy.set_xlabel('x ($R_e$)')\nax_xy.set_ylabel('y ($R_e$)')\n\nax_u1.set_xlabel('time ($TU$)')\nax_u1.set_ylabel('$u_1$ ($deg$)')\nax_u1.grid(True)\n\nax_deltav.set_xlabel('time ($TU$)')\nax_deltav.set_ylabel('${\\Delta}v$ ($DU/TU$)')\nax_deltav.grid(True)\n\nt_sol = dict((phs, p.get_val('traj.{0}.timeseries.time'.format(phs)))\n for phs in ['burn1', 'coast', 'burn2'])\nx_sol = dict((phs, p.get_val('traj.{0}.timeseries.pos_x'.format(phs)))\n for phs in ['burn1', 'coast', 'burn2'])\ny_sol = dict((phs, p.get_val('traj.{0}.timeseries.pos_y'.format(phs)))\n for phs in ['burn1', 'coast', 'burn2'])\ndv_sol = dict((phs, p.get_val('traj.{0}.timeseries.states:deltav'.format(phs)))\n for phs in ['burn1', 'coast', 'burn2'])\nu1_sol = dict((phs, p.get_val('traj.{0}.timeseries.controls:u1'.format(phs), units='deg'))\n for phs in ['burn1', 'burn2'])\n\nt_exp = dict((phs, exp_out.get_val('traj.{0}.timeseries.time'.format(phs)))\n for phs in ['burn1', 'coast', 'burn2'])\nx_exp = dict((phs, exp_out.get_val('traj.{0}.timeseries.pos_x'.format(phs)))\n for phs in ['burn1', 'coast', 'burn2'])\ny_exp = dict((phs, exp_out.get_val('traj.{0}.timeseries.pos_y'.format(phs)))\n for phs in ['burn1', 'coast', 'burn2'])\ndv_exp = dict((phs, exp_out.get_val('traj.{0}.timeseries.states:deltav'.format(phs)))\n for phs in ['burn1', 'coast', 'burn2'])\nu1_exp = dict((phs, exp_out.get_val('traj.{0}.timeseries.controls:u1'.format(phs),\n units='deg'))\n for phs in ['burn1', 'burn2'])\n\nfor phs in ['burn1', 'coast', 'burn2']:\n try:\n ax_u1.plot(t_exp[phs], u1_exp[phs], '-', marker=None, color='C0')\n ax_u1.plot(t_sol[phs], u1_sol[phs], 'o', mfc='C1', mec='C1', ms=3)\n except KeyError:\n pass\n\n ax_deltav.plot(t_exp[phs], dv_exp[phs], '-', marker=None, color='C0')\n ax_deltav.plot(t_sol[phs], dv_sol[phs], 'o', mfc='C1', mec='C1', ms=3)\n\n ax_xy.plot(x_exp[phs], y_exp[phs], '-', marker=None, color='C0', label='explicit')\n ax_xy.plot(x_sol[phs], y_sol[phs], 'o', mfc='C1', mec='C1', ms=3, label='implicit')\n\nplt.show()\n```\n\n\n```python\nfrom openmdao.utils.assert_utils import assert_near_equal\n\nassert_near_equal(p.get_val('traj.burn2.states:deltav')[-1], 0.3995,\n tolerance=2.0E-3)\n```\n\n## References\n\n```{bibliography}\n:filter: docname in docnames\n```\n", "meta": {"hexsha": "3ecf089d359ebde31b21fc4c14d9837a82da803b", "size": 16134, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/examples/finite_burn_orbit_raise/finite_burn_orbit_raise.ipynb", "max_stars_repo_name": "kaushikponnapalli/dymos", "max_stars_repo_head_hexsha": "3fba91d0fc2c0e8460717b1bec80774676287739", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 104, "max_stars_repo_stars_event_min_datetime": "2018-09-08T16:52:27.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-10T23:35:30.000Z", "max_issues_repo_path": "docs/examples/finite_burn_orbit_raise/finite_burn_orbit_raise.ipynb", "max_issues_repo_name": "kaushikponnapalli/dymos", "max_issues_repo_head_hexsha": "3fba91d0fc2c0e8460717b1bec80774676287739", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 628, "max_issues_repo_issues_event_min_datetime": "2018-06-27T20:32:59.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-31T19:24:32.000Z", "max_forks_repo_path": "docs/examples/finite_burn_orbit_raise/finite_burn_orbit_raise.ipynb", "max_forks_repo_name": "kaushikponnapalli/dymos", "max_forks_repo_head_hexsha": "3fba91d0fc2c0e8460717b1bec80774676287739", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 46, "max_forks_repo_forks_event_min_datetime": "2018-06-27T20:54:07.000Z", "max_forks_repo_forks_event_max_datetime": "2021-12-19T07:23:32.000Z", "avg_line_length": 41.0534351145, "max_line_length": 100, "alphanum_fraction": 0.5542332961, "converted": true, "num_tokens": 3858, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.702530051167069, "lm_q1q2_score": 0.43211770260794496}} {"text": "# Optimization of a dissipative state-to-state transfer in a Lambda system\n\n\n```python\n# NBVAL_IGNORE_OUTPUT\n%load_ext watermark\nimport qutip\nimport numpy as np\nimport scipy\nimport matplotlib\nimport matplotlib.pylab as plt\nimport krotov\nimport qutip\nfrom qutip import Qobj\nimport pickle\n%watermark -v --iversions\n```\n\n qutip 4.3.1\n numpy 1.15.4\n scipy 1.1.0\n matplotlib 3.0.2\n matplotlib.pylab 1.15.4\n krotov 0.0.1\n CPython 3.6.7\n IPython 7.2.0\n\n\n$\\newcommand{tr}[0]{\\operatorname{tr}}\n\\newcommand{diag}[0]{\\operatorname{diag}}\n\\newcommand{abs}[0]{\\operatorname{abs}}\n\\newcommand{pop}[0]{\\operatorname{pop}}\n\\newcommand{aux}[0]{\\text{aux}}\n\\newcommand{opt}[0]{\\text{opt}}\n\\newcommand{tgt}[0]{\\text{tgt}}\n\\newcommand{init}[0]{\\text{init}}\n\\newcommand{lab}[0]{\\text{lab}}\n\\newcommand{rwa}[0]{\\text{rwa}}\n\\newcommand{bra}[1]{\\langle#1\\vert}\n\\newcommand{ket}[1]{\\vert#1\\rangle}\n\\newcommand{Bra}[1]{\\left\\langle#1\\right\\vert}\n\\newcommand{Ket}[1]{\\left\\vert#1\\right\\rangle}\n\\newcommand{Braket}[2]{\\left\\langle #1\\vphantom{#2} \\mid\n#2\\vphantom{#1}\\right\\rangle}\n\\newcommand{Ketbra}[2]{\\left\\vert#1\\vphantom{#2}\n\\right\\rangle \\hspace{-0.2em} \\left\\langle #2\\vphantom{#1} \\right\\vert}\n\\newcommand{op}[1]{\\hat{#1}}\n\\newcommand{Op}[1]{\\hat{#1}}\n\\newcommand{dd}[0]{\\,\\text{d}}\n\\newcommand{Liouville}[0]{\\mathcal{L}}\n\\newcommand{DynMap}[0]{\\mathcal{E}}\n\\newcommand{identity}[0]{\\mathbf{1}}\n\\newcommand{Norm}[1]{\\lVert#1\\rVert}\n\\newcommand{Abs}[1]{\\left\\vert#1\\right\\vert}\n\\newcommand{avg}[1]{\\langle#1\\rangle}\n\\newcommand{Avg}[1]{\\left\\langle#1\\right\\rangle}\n\\newcommand{AbsSq}[1]{\\left\\vert#1\\right\\vert^2}\n\\newcommand{Re}[0]{\\operatorname{Re}}\n\\newcommand{Im}[0]{\\operatorname{Im}}\n\\newcommand{toP}[0]{\\omega_{12}}\n\\newcommand{toS}[0]{\\omega_{23}}$\n\nThe aim of this example is to test the\nbehaviour of Krotov's algorithm when\ndealing with non-Hermitian Hamiltonians.\nThis notebook is heavily based on the\nexample \"Optimization of a state-to-state\ntransfer in a lambda system with RWA\",\nand therefore it is recommended that the\nreader become familiar with the system\nbefore hand. The main change in the\nresults will be represented by the loss of\nnorm during the propagation.\n\n## Define the Hamiltonian\n\nWe start with the usual 3-level lambda system such that\n$E_1 < E_2$ and $E_3 <\nE_2$. The states $\\Ket{1}$ and $\\Ket{2}$ are coupled\ncoupled through a pump\nlaser with frequency $\\omega_{P}=\\omega_{P}(t)$ and\nsimilarly for the states\n$\\Ket{2}$ and $\\Ket{3}$ through a Stokes laser with\nfrequency\n$\\omega_{S}=\\omega_{S}(t)$.\n\nThe level $\\Ket{2}$ is assumed to\nexperience a\nspontaneous decay into a reservoir that is out of the Hilbert\nspace. This can be\nreproduced with a simple model that adds a dissipative\ncoefficient $\\gamma > 0$,\nsuch that the original eigenvalue for the second level\nis modified into $E_2\n\\rightarrow E_2 - i \\gamma$.\n\nWe perform the same\nrotating wave approximation as\ndescribed in the aforementioned example and\nobtain the time independent\nHamiltonian\n\n\\begin{equation}\n \\op{H}_{0} =\n \\Delta_{P}\\Ketbra{1}{1} -i\n\\gamma \\Ketbra{2}{2} +\\Delta_{S} \\Ketbra{3}{3}\n\\end{equation}\n\nwith the\ndetunings $\\Delta_{P}=E_{1} + \\omega_{P} - E_{2}$ and\n$\\Delta_{S} = E_{3} +\n\\omega_{S} -E_{2}$.\n\nThe control Hamiltonian is again given by\n\n\\begin{equation}\n\\op{H}_{1}(t)\n = \\op{H}_{1,P}(t) + \\op{H}_{1,S}(t)\n = \\Omega_{P}(t)\n\\Ketbra{1}{2} +\n \\Omega_{S}(t)\\Ketbra{2}{3} + \\text{H.c.}\\,,\n\\end{equation}\nwhere $\\Omega_{P} = \\Omega_{P}(t) = \\frac{\\mu_{21} \\varepsilon_{P}(t)}{2}\ne^{-i\\Phi_{S}(t) t}$\nand $\\Omega_{S} = \\Omega_{S}(t) = \\frac{\\mu_{23}\n\\varepsilon_{S}(t)}{2} e^{-i\\Phi_{P}(t) t}$\nwith the phases $\\Phi_{P}(t) = \\toP\n- \\omega_{P}(t)$ and\n$\\Phi_{S}(t) = \\toS - \\omega_{S}(t)$ and $\\mu_{ij}$ the\n$ij^{\\text{th}}$ dipole-transition moment.\n\nFor optimizing the complex pulses we\nonce again\nseparate them into their real and imaginary parts, i.e.,\n$\\Omega_{P}(t) = \\Omega_{P}^\\text{Re}(t) + i\\Omega_{P}^\\text{Im}(t)$\nand\n$\\Omega_{S}(t) = \\Omega_{S}^\\text{Re}(t) + i\\Omega_{S}^\\text{Im}(t)$,\nso we can\noptimize four real pulses.\n\n\n```python\ndef ham_and_states():\n \"\"\"Lambda-system Hamiltonian\"\"\"\n E1 = 0.0\n E2 = 10.0\n E3 = 5.0\n \u03c9_P = 9.5\n \u03c9_S = 4.5\n gamma = 0.5\n \u03a9_init = 5.0\n H0 = Qobj(\n [\n [E1 + \u03c9_P - E2, 0.0, 0.0],\n [0.0, -gamma * 1.0j, 0.0],\n [0.0, 0.0, E3 + \u03c9_S - E2],\n ]\n )\n\n H1P_re = Qobj([[0.0, -1.0, 0.0], [-1.0, 0.0, 0.0], [0.0, 0.0, 0.0]])\n H1P_im = Qobj([[0.0, -1.0j, 0.0], [1.0j, 0.0, 0.0], [0.0, 0.0, 0.0]])\n \u03a9P_re = lambda t, args: \u03a9_init\n \u03a9P_im = lambda t, args: \u03a9_init\n\n H1S_re = Qobj([[0.0, 0.0, 0.0], [0.0, 0.0, 1.0], [0.0, 1.0, 0.0]])\n H1S_im = Qobj([[0.0, 0.0, 0.0], [0.0, 0.0, 1.0j], [0.0, -1.0j, 0.0]])\n \u03a9S_re = lambda t, args: \u03a9_init\n \u03a9S_im = lambda t, args: \u03a9_init\n\n \"\"\"Initial and target states\"\"\"\n psi0 = qutip.Qobj(np.array([1.0, 0.0, 0.0]))\n psi1 = qutip.Qobj(np.array([0.0, 0.0, 1.0]))\n\n return (\n [\n H0,\n [H1P_re, \u03a9P_re],\n [H1P_im, \u03a9P_im],\n [H1S_re, \u03a9S_re],\n [H1S_im, \u03a9S_im],\n ],\n psi0,\n psi1,\n )\n\n\nH, psi0, psi1 = ham_and_states()\n```\n\nWe check whether our Hamiltonians are Hermitian:\n\n\n```python\nprint(\"H0 is Hermitian: \" + str(H[0].isherm))\nprint(\"H1 is Hermitian: \"+ str(\n H[1][0].isherm\n and H[2][0].isherm\n and H[3][0].isherm\n and H[4][0].isherm))\n```\n\n H0 is Hermitian: False\n H1 is Hermitian: True\n\n\nWe introduce projectors for each of the three energy levels\n$\\op{P}_{i} =\n\\Ketbra{i}{i}$\n\n\n```python\nproj1 = Qobj([[1.,0.,0.],[0.,0.,0.],[0.,0.,0.]])\nproj2 = Qobj([[0.,0.,0.],[0.,1.,0.],[0.,0.,0.]])\nproj3 = Qobj([[0.,0.,0.],[0.,0.,0.],[0.,0.,1.]])\n```\n\n## Define the optimization target\n\nIt is necessary to create a time grid to work\nwith. In this example we choose a time interval starting at $t_0 = 0$ and ending\nat $t_f = 5$ with $n_t = 500$ equidistant time steps.\n\n\n```python\nt0 = 0.\ntf = 5.\nnt = 500\ntlist = np.linspace(t0, tf, nt)\n```\n\nWe define the objective to be a state to state transfer from the initial state\n$\\Ket{\\Psi_{\\init}} = \\Ket{1}$ into the final state $\\Ket{\\Psi_{\\tgt}} =\n\\Ket{3}$ at the\nfinal time $t_{f}$.\n\n\n```python\nobjectives = [ krotov.Objective(initial_state=psi0, target=psi1, H=H) ]\n```\n\n## Initial guess shapes\n\n\"stimulated Raman adiabatic passage\" (STIRAP) is a\nprocess in which population in $\\Ket{1}$ is transferred into\n$\\Ket{3}$ without\nhaving to pass through $\\Ket{2}$ (which could for instance be a rapidly decaying\nlevel).\nIn order for this process to occur, a temporally finite Stokes pulse of\nsufficient amplitude driving the $\\Ket{2} \\leftrightarrow \\Ket{3}$ transition is\napplied first, whilst second pump pulse of similar intensity follows some time\nlater such that the pulses still have a partial temporal overlap.\n\nIn order to\ndemonstrate the Krotov's optimization method however, we choose an initial guess\nconsisting of two low intensity and real Blackman pulses which are temporally\ndisjoint.\n\nFor the real components of the matrix elements, we supply our guess\npulses shaped as Blackman window functions `S(t,offset)`, with an offset\nensuring that the two pulses don't overlap.\nThe imaginary components are coupled\nto pulses that are zero at all times.\n\n\n```python\ndef S(t,offset):\n \"\"\"Shape envelope function for the field update\"\"\"\n return krotov.shapes.blackman(t,1.+offset,4.+offset)\n\ndef shape_field_real(eps,offset):\n \"\"\"Applies the total pulse shape to the real part of a guess pulse\"\"\"\n field_shaped = lambda t, args: eps(t, args)*S(t,offset)\n return field_shaped\n\ndef shape_field_imag(eps,offset):\n \"\"\"Initializes the imaginary parts of the guess pulses to zero\"\"\"\n field_shaped = lambda t, args: eps(t, args)*0.\n return field_shaped\n\nH[1][1] = shape_field_real(H[1][1],1.)\nH[2][1] = shape_field_imag(H[2][1],1.)\nH[3][1] = shape_field_real(H[3][1],-1.)\nH[4][1] = shape_field_imag(H[4][1],-1.)\n```\n\nWe choose an appropriate update factor $\\lambda_{a}$ for the problem at hand and\nmake sure Krotov considers pulses which start and end with zero amplitude.\n\n\n```python\ndef update_shape(t):\n \"\"\"Scales the Krotov methods update of the pulse value at the time t\"\"\"\n return krotov.shapes.flattop(t,0.,5.,0.3,func='sinsq')\n```\n\n\n```python\nopt_lambda = 2.0\npulse_options = {\n H[1][1]: krotov.PulseOptions(lambda_a=opt_lambda, shape=update_shape),\n H[2][1]: krotov.PulseOptions(lambda_a=opt_lambda, shape=update_shape),\n H[3][1]: krotov.PulseOptions(lambda_a=opt_lambda, shape=update_shape),\n H[4][1]: krotov.PulseOptions(lambda_a=opt_lambda, shape=update_shape),\n}\n```\n\nIt is possible to keep track of the fidelity during optimization by printing it\nafter every iteration:\n\n\n```python\ndef print_fidelity(**args):\n F_re = np.average(np.array(args['tau_vals']).real)\n print(\" F = %f\" % F_re)\n return F_re\n```\n\n## Simulate dynamics of the guess field\n\n\n```python\ndef plot_pulse(pulse, tlist, plottitle=None):\n fig, ax = plt.subplots()\n if callable(pulse):\n pulse = np.array([pulse(t, args=None) for t in tlist])\n ax.plot(tlist, pulse)\n ax.set_xlabel('time')\n ax.set_ylabel('pulse amplitude')\n if(isinstance(plottitle, str)):\n ax.set_title(plottitle, fontsize = 15)\n plt.show(fig)\n```\n\n\n```python\nplot_pulse(H[1][1], tlist, plottitle='Re($\\Omega_P$)')\nplot_pulse(H[2][1], tlist, plottitle='Im($\\Omega_P$)')\nplot_pulse(H[3][1], tlist, plottitle='Re($\\Omega_S$)')\nplot_pulse(H[4][1], tlist, plottitle='Im($\\Omega_S$)')\n```\n\nAfter assuring ourselves that our guess pulses appear as expected, we propagate\nthe system using our guess. Since the pulses are temporally disjoint, we expect\nthe first pulse to have no effect, whilst the second merely transfers population\nout of $\\Ket{1}$ into $\\Ket{2}$ and back again.\n\n\n```python\nguess_dynamics = objectives[0].propagate(\n tlist, propagator=krotov.propagators.expm, e_ops=[proj1, proj2, proj3]\n)\nguess_states = objectives[0].propagate(\n tlist, propagator=krotov.propagators.expm\n)\n```\n\n\n```python\ndef plot_population(result):\n fig, ax = plt.subplots()\n ax.axhline(y=1.0, color='black', lw=0.5, ls='dashed')\n ax.axhline(y=0.0, color='black', lw=0.5, ls='dashed')\n ax.plot(result.times, result.expect[0], label='1')\n ax.plot(result.times, result.expect[1], label='2')\n ax.plot(result.times, result.expect[2], label='3')\n ax.legend()\n ax.set_title('Expected values', fontsize = 15)\n ax.set_xlabel('time')\n ax.set_ylabel('population')\n plt.show(fig)\n \ndef plot_norm(result):\n \n state_norm = lambda i: result.states[i].norm()\n states_norm=np.vectorize(state_norm)\n \n fig, ax = plt.subplots()\n ax.plot(result.times, states_norm(np.arange(len(result.states))))\n ax.set_title('Norm loss', fontsize = 15)\n ax.set_xlabel('time')\n ax.set_ylabel('state norm')\n plt.show(fig) \n```\n\n\n```python\nplot_population(guess_dynamics)\nplot_norm(guess_states)\n```\n\n## Optimize\n\nWe now use all the information that we have gathered to initialize\nthe optimization routine. That is: \n\n* The `objectives`: transferring population\nfrom $\\ket{1}$ to $\\ket{3}$ at $t_f$. \n\n* The `pulse_options`: initial pulses\nand their shapes restrictions. \n\n* The `propagator`: in our example we will\nchoose a simple matrix exponential.\n\n* The `chi_constructor`: the optimization\nfunctional to use. \n\n* The `info_hook`: all processes taking place inbetween\niterations, for example printing the fidelity in each step. \n\n* And the\n`iter_stop`: the number of iterations to perform the optimization.\n\n\n```python\noct_result = krotov.optimize_pulses(\n objectives, pulse_options, tlist,\n propagator=krotov.propagators.expm,\n chi_constructor=krotov.functionals.chis_re,\n info_hook=krotov.info_hooks.chain(\n #krotov.info_hooks.print_debug_information,\n print_fidelity),\n iter_stop=20\n)\n```\n\n F = -0.023471\n F = 0.252788\n F = 0.482934\n F = 0.645488\n F = 0.747822\n F = 0.808026\n F = 0.842348\n F = 0.861826\n F = 0.873076\n F = 0.879839\n F = 0.884167\n F = 0.887170\n F = 0.889446\n F = 0.891318\n F = 0.892959\n F = 0.894461\n F = 0.895876\n F = 0.897232\n F = 0.898543\n F = 0.899818\n F = 0.901063\n\n\nWe can check that the algorithm takes into account the non Hermitian behaviour\nwhich leads to a non unitary final state. To improve the fidelity it is\nnecessary to avoid these dissipative effects as much as possible.\n\n\n```python\ndef plot_pulse_amplitude_and_phase(pulse_real, pulse_imaginary,tlist):\n ax1 = plt.subplot(211)\n ax2 = plt.subplot(212)\n amplitudes = [np.sqrt(x*x + y*y) for x,y in zip(pulse_real,pulse_imaginary)]\n phases = [np.arctan2(y,x)/np.pi for x,y in zip(pulse_real,pulse_imaginary)]\n ax1.plot(tlist,amplitudes)\n ax1.set_xlabel('time')\n ax1.set_ylabel('pulse amplitude')\n ax2.plot(tlist,phases)\n ax2.set_xlabel('time')\n ax2.set_ylabel('pulse phase (\u03c0)') \n plt.show()\n \nprint(\"pump pulse amplitude and phase:\")\nplot_pulse_amplitude_and_phase(\n oct_result.optimized_controls[0], oct_result.optimized_controls[1], tlist)\nprint(\"Stokes pulse amplitude and phase:\")\nplot_pulse_amplitude_and_phase(\n oct_result.optimized_controls[2], oct_result.optimized_controls[3], tlist)\n```\n\nWe check the evolution of the population due to our optimized pulses.\n\n\n```python\nopt_dynamics = oct_result.optimized_objectives[0].propagate(\n tlist, propagator=krotov.propagators.expm, e_ops=[proj1, proj2, proj3])\nopt_states = oct_result.optimized_objectives[0].propagate(\n tlist, propagator=krotov.propagators.expm)\n```\n\n\n```python\nplot_population(opt_dynamics)\nplot_norm(opt_states)\n```\n\nAs we can see the algorithm takes into account the decay in level $\\ket{2}$ and\nminimizes the population in this level during the process.\n\nHowever, convergence\nis relatively slow, and after 20 iterations, we have only\nachieved a 90%\nfidelity. As we can see from the population dynamics, we do not\nfully transfer\nthe population in state $\\ket{3}$, and there is still non-\nnegligible population\nin state $\\ket{2}$. If we were to continue up to iteration\n2000, the\noptimization converges much farther:\n\n\n```python\noct_result = krotov.result.Result.load(\n './non_herm_oct_result.dump', objectives\n)\n```\n\n\n```python\nprint(\"Final fidelity: %.3f\" % oct_result.info_vals[-1])\n```\n\n Final fidelity: 0.986\n\n\n\n```python\ndef plot_convergence(result):\n fig, ax = plt.subplots()\n ax.semilogy(result.iters, 1-np.array(result.info_vals))\n ax.set_xlabel('OCT iteration')\n ax.set_ylabel('error')\n plt.show(fig)\n```\n\n\n```python\nplot_convergence(oct_result)\n```\n\nThe dynamics now show a very good population transfer and negligible population\nin state $\\ket{2}$.\n\n\n```python\nopt_dynamics = oct_result.optimized_objectives[0].propagate(\n tlist, propagator=krotov.propagators.expm, e_ops=[proj1, proj2, proj3])\nopt_states = oct_result.optimized_objectives[0].propagate(\n tlist, propagator=krotov.propagators.expm)\n```\n\n\n```python\nprint(\"pump pulse amplitude and phase:\")\nplot_pulse_amplitude_and_phase(\n oct_result.optimized_controls[0], oct_result.optimized_controls[1], tlist)\nprint(\"Stokes pulse amplitude and phase:\")\nplot_pulse_amplitude_and_phase(\n oct_result.optimized_controls[2], oct_result.optimized_controls[3], tlist)\n```\n\n\n```python\nplot_population(opt_dynamics)\nplot_norm(opt_states)\n```\n", "meta": {"hexsha": "f719ba608c468cabb57809999c9e824dd362f03f", "size": 280473, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/notebooks/03_example_lambda_system_rwa_non_hermitian.ipynb", "max_stars_repo_name": "TejasAvinashShetty/krotov", "max_stars_repo_head_hexsha": "e2dd0fad2f07f41004d7beef53e8ebc75b0d6d9b", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/notebooks/03_example_lambda_system_rwa_non_hermitian.ipynb", "max_issues_repo_name": "TejasAvinashShetty/krotov", "max_issues_repo_head_hexsha": "e2dd0fad2f07f41004d7beef53e8ebc75b0d6d9b", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/notebooks/03_example_lambda_system_rwa_non_hermitian.ipynb", "max_forks_repo_name": "TejasAvinashShetty/krotov", "max_forks_repo_head_hexsha": "e2dd0fad2f07f41004d7beef53e8ebc75b0d6d9b", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 235.889823381, "max_line_length": 25140, "alphanum_fraction": 0.9179279289, "converted": true, "num_tokens": 5040, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494678483918, "lm_q2_score": 0.629774621301746, "lm_q1q2_score": 0.43199356634635516}} {"text": "```python\n# CSS formatting for the Notebook. Run to apply format\nfrom IPython.core.display import HTML; \nHTML(\"\"\"\"\"\")\n```\n\n\n\n\n\n\n\n\n
    \n

    Estimating Financial Risk through Monte Carlo Simulation
    2017 Edition

    \n
    26/04/2017
    \n
    Authors Ole Andreas Hansen |\n Alberto Ibarrondo Luis
    \n
    \n\nRisk analysis is part of every decision we make when faced with uncertainty, ambiguity, and variability. Indeed, even though we have unprecedented access to information, we can't accurately predict the future. In finance, there is a fair amount of uncertainty and risk involved with estimating the future value of financial products, due to the wide variety of potential outcomes. Monte Carlo simulation (also known as the Monte Carlo Method) allows inspecting many possible outcomes of the decision making process, and can be used to assess the impact of risk: this, in turns, allows for better decision-making under uncertainty.\n\n## Goals\nThe main objectives we set for this Notebook are as follows:\n1. Develop fundamental knowledge about Risk analysis\n2. Understand Monte Carlo Simulation (MCS)\n3. Apply Monte Carlo Simulation for predicting risk\n\n\n## Steps\n1. First, in section 1, we introduce the basics of MCS\n2. In section 2, we work on a simple example to where we apply the MCS method\n3. In section 3, we briefly summarize the main characteristics of the Monte Carlo Simulation (MCS) technique\n4. In section 4, we overview the common distributions which are often used in MCS\n5. In section 5, we work on a real use case, that focuses on estimating financial risk. We will use techniques such as featurization (that is, generating additional features to improve model accuracy), linear regression, kernel density estimation, sampling distributions and so on ...\n\n## Reference\nThis Notebook is inspired by Chapter 9 of the book [Advanced Analytics with Spark](http://shop.oreilly.com/product/0636920035091.do) by Josh Wills, Sandy Ryza, Sean Owen, and Uri Laserson. It is strongly suggested to read this Chapter to get a general idea of the topic of this Notebook.\n\n# 1. Introduction\n\n## 1.1. Monte Carlo Simulation (MCS)\nMonte Carlo simulation is a computerized mathematical technique that can be applied such that it is possible to account for risk in quantitative analysis and decision making. This technique is used in many different fields, such as R&D, risk management, portfolio management, pricing derivatives, strategic planning, project planning, cost modeling and many more.\n\nIn general, MCS is a technique that \"converts\" uncertainty on input variables of a model into **probability distributions**. By combining the distributions and randomly selecting values from them, it recalculates the simulated model many times, to determine the probability of the output.\n\nHistorically, this technique was first used by scientists working on the atomic bomb: it was named after Monte Carlo, the Monaco resort town renowned for its casinos. Since its introduction in World War II, Monte Carlo simulation has been used to model a variety of physical and conceptual systems.\n\n## 1.2. How does it work?\nMonte Carlo simulation performs risk analysis by building models of possible results by *substituting a range of possible input values, that constitute uncertainty, into a statistical distribution*. It then computes possible outcomes repeatedly, each time using a different set of random values from the probability functions that \"model\" the input. Depending upon the number of random input variables and their distribution, a Monte Carlo simulation could involve thousands or tens of thousands of \"rounds\" before it is complete. When complete, *Monte Carlo simulation produces distributions of possible outcome values*.\n\nBy using probability distributions instead of actual input samples, it is possible to model more accurately uncertainty: different choices of distributions will yield different outputs.\n\n# 2. Illustrative example\n\nImagine you are the marketing manager for a firm that is planning to introduce a new product. You need to estimate the first-year net profit from this product, which might depend on:\n\n- Sales volume in units\n- Price per unit (also called \"Selling price\")\n- Unit cost\n- Fixed costs\n\nNet profit will be calculated as $Net Profit = Sales Volume* (Selling Price - Unit cost) - Fixed costs$. Fixed costs (accounting for various overheads, advertising budget, etc.) are known to be \\$ 120,000, which we assume to be deterministic. All other factors, instead, involve some uncertainty: *sales volume* (in units) can cover quite a large range, the *selling price* per unit will depend on competitor actions, which are hard to predict, and *unit costs* will also vary depending on vendor prices and production experience, for example.\n\nNow, to build a risk analysis model, we must first identify the uncertain variables -- which are essentially random variables. While there's some uncertainty in almost all variables in a business model, we want to focus on variables where the range of values is significant.\n\n## 2.1. Unit sales and unit price\n\nBased on a hypothetical market research you have done, you have beliefs that there are equal chances for the market to be `slow`, `normal`, or `hot`:\n\n- In a \"slow\" market, you expect to sell 50,000 units at an average selling price of \\$11.00 per unit\n- In a \"normal\" market, you expect to sell 75,000 units, but you'll likely realize a lower average selling price of \\$10.00 per unit\n- In a \"hot\" market, you expect to sell 100,000 units, but this will bring in competitors, who will drive down the average selling price to \\$8.00 per unit\n\n\n\n### Question 1\n
    \nCalculate the average units and the unit price that you expect to sell, which depend on the market state. Use the assumptions above to compute the expected quantity of products and their expected unit price. \n
    \n\n\n\n```python\nprices = [11, 10, 8]\nunits = [50000, 75000, 100000]\n\naverage_unit = round(sum(units)/3, 2)\naverage_price = round(sum(prices)/3, 8)\n\nprint(\"average unit:\", average_unit)\nprint(\"average_price:\", average_price)\n\n#Clearly the second option is the closest one to the average (75k units and 10\u20ac per unit)\n```\n\n average unit: 75000.0\n average_price: 9.66666667\n\n\n## 2.2. Unit Cost\n\nAnother uncertain variable is Unit Cost. In our illustrative example, we assume that your firm's production manager advises you that unit costs may be anywhere from \\$5.50 to \\$7.50, with a most likely expected cost of \\$6.50. In this case, the most likely cost can be considered as the average cost.\n\n## 2.3. A Flawed Model: using averages to represent our random variables\nOur next step is to identify uncertain functions -- also called functions of a random variable. Recall that Net Profit is calculated as $Net Profit = Sales Volume * (Selling Price - Unit cost) - Fixed costs$. However, Sales Volume, Selling Price and Unit Cost are all uncertain variables, so Net Profit is an uncertain function.\n\nThe simplest model to predict the Net Profit is using average of sales volume, average of selling price and average of unit cost for calculating. So, if only consider averages, we can say that the $Net Profit = 75,000*(9.66666666 - 6.5) - 120,000 \\sim 117,500$.\n\nHowever, as [Dr. Sam Savage](http://web.stanford.edu/~savage/faculty/savage/) warns, \"Plans based on average assumptions will be wrong on average.\" The calculated result is far from the actual value: indeed, the **true average Net Profit** is roughly \\$93,000, as we will see later in the example.\n\n\n\n### Question 2\n#### Question 2.1\n
    \nWrite a function named `calNetProfit` to calculate the Net Profit using the average of sales volume, the average of selling price and the average of unit cost.\n
    \n\n\n```python\ndef calNetProfit(average_unit, average_price, average_unitcost, fixed_cost):\n return average_unit * (average_price - average_unitcost) - fixed_cost\n\naverage_unitcost = 6.5\nfixed_cost = 120000\nNetProfit = calNetProfit(average_unit, average_price, average_unitcost, fixed_cost)\nprint(\"Net profit:\", NetProfit)\n```\n\n Net profit: 117500.00024999998\n\n\n#### Question 2.2\n
    \nVerify the warning message of Dr. Sam Savage by calculating the error of our estimated Net Profit using averages only. Recall that the true value is roughly \\$93,000, so we are interested in:\n
      \n\n$$ error = \\frac{your\\_value - true\\_value}{true\\_value}$$\n\n
        \nNote also we are interested in displaying the error as a percentage.\n
          \nLooking at the error we make, do you think that we can use the current model that only relies on averages?\n
          \n\n\n```python\ntrueNetProfit = 93000\nerror = (NetProfit - trueNetProfit) / trueNetProfit\nprint(\"Error: %.2f %%\"% (error*100))\n```\n\n Error: 26.34 %\n\n\n
          \nCOMMENT
          We see that our naive model gives an error of 26%. A deviation of 25% in the estimated profit over a year can definitely kill a company. This situation calls for a new upgraded model.\n\n
          \n\n## 2.4. Using the Monte Carlo Simulation method to improve our model\nAs discussed before, the selling price and selling volume both depend on the state of the market scenario (slow/normal/hot). So, the net profit is the result of two random variables: `market scenario` (which in turn determines `sales volumes` and `selling price`) and `unit cost`.\n\nNow, let's assume (this is an *a-priori* assumption we make) that `market scenario` follows a discrete, uniform distribution and that `unit cost` also follows a uniform distribution. Then, we can compute directly the values for selling price and selling volumes based on the outcome of the random variable `market scenario`, as shown in Section 2.1.\n\nFrom these a-priori distributions, in each run (or trial) of our Monte Carlo simulation, we can generate the sample value for each random variable and use it to calculate the Net Profit. The more simulation runs, the more accurate our results will be. For example, if we run the simulation 100,000 times, the average net profit will amount to roughly \\$92,600. Every time we run the simulation, a different prediction will be output: the average of such predictions will consistently be less than \\$117,500, which we predicted using averages only.\n\nNote also that in this simple example, we generate values for the `market scenario` and `unit cost` independently: we consider them to be **independent random variables**. This means that the eventual (and realistic!) correlation between the `market scenario` and `unit cost` variables is ignored. Later, we will learn how to be more precise and account for dependency between random variables.\n\n\n\n\n### Question 3\n#### Question 3.1\n
          \nWrite a function named `get_sales_volume_price` that returns the sales volume and price based on the market scenario. In particular, the scenario can get one of three values:\n
            \n
          • 0: Slow market
            • \n
          • 1: Normal market
            • \n
          • 2: Hot market
            • \n
          \n\nThe return value is a tuple in the form: `(sales_volume, price)`\n
          \n\n\n```python\n# Get sales volume and price based on market scenario\n# the function returns a tuple of (sales_volume, price)\ndef get_sales_volume_price(scenario):\n # Slow market\n if scenario == 0:\n return (50000, 11)\n # Normal market\n if scenario == 1:\n return (75000, 10)\n # Hot market\n if scenario == 2:\n return(100000, 8)\n\n```\n\n#### Question 3.2\n
          \nRun 100,000 Monte Carlo simulations and calculate the average net profit they produce. Then, compare the result to the \"average model\" we used in the previous questions (the one we called \"flawed\" model). Put your comments about the discrepancies between a simplistic model, and the more accurate MCS approach. \n
            \nNote that in each iteration, the `unit_cost` and `market_scenario` are generated according to their distributions. Also, recall what we have seen in Section 2.2: your firm account manager helped you with some research, to determine the variability of your random variables. \n
            \n\n\n
            HINT
            \n\nFunction `uniform(a,b)` in module `random` generates a number $a<=c<=b$, which is drawn from a uniform distribution. \n\nFunction `randint(a,b)` helps you generating an integer number $a<=c<=b$\n\n\n```python\nimport random\n\ntotal = 0.0\nnum_simulation = 1000000\nfor i in range(0,num_simulation):\n unit_cost = random.uniform(5.50, 7.50)\n market_scenario = random.randint(0, 2)\n sales_volume, price = get_sales_volume_price(market_scenario)\n netProfit = calNetProfit(sales_volume, price, unit_cost, fixed_cost)\n total += netProfit \n\nprint(\"average net profit:\", total/num_simulation)\nprint(\"Error: %.2f %% \"%((total/num_simulation - trueNetProfit) / trueNetProfit*100))\n```\n\n average net profit: 92445.65606423779\n Error: -0.60 % \n\n\n
            \nCOMMENT
            \nWe have jumped from 24% error to 0.5%! This is clearly manageable by the company as a deviation from the true results without setting the firm into bankrupcy. Nevertheless, there is still room for improvement...\n
            \n\n\n# 3. A brief summary of the Monte Carlo Simulation (MCS) technique\n\n- A MCS allows several inputs to be used at the same time to compute the probability distribution of one or more outputs\n- Different types of probability distributions can be assigned to the inputs of the model, depending on any *a-priori* information that is available. When the distribution is completely unknown, a common technique is to use a distribution computed by finding the best fit to the data you have\n- The MCS method is also called a **stochastic method** because it uses random variables. Note also that the general assumption is for input random variables to be independent from each other. When this is not the case, there are techniques to account for correlation between random variables.\n- A MCS generates the output as a range instead of a fixed value and shows how likely the output value is to occur in that range. In other words, the model outputs a probability distribution.\n\n# 4. Common distributions used in MCS\nIn what follows, we summarize the most common probability distributions that are used as *a-priori* distributions for input random variables:\n\n- *Normal/Gaussian Distribution*: this is a continuous distribution applied in situations where the mean and the standard deviation of a given input variable are given, and the mean represents the most probable value of the variable. In other words, values \"near\" the mean are most likely to occur. This is symmetric distribution, and it is not bounded in its co-domain. It is very often used to describe natural phenomena, such as people\u2019s heights, inflation rates, energy prices, and so on and so forth. An illustration of a normal distribution is given below:\n\n\n- *Lognormal Distribution*: this is a distribution which is appropriate for variables taking values in the range $[0, \\infty]$. Values are positively skewed, not symmetric like a normal distribution. Examples of variables described by some lognormal distributions include, for example, real estate property values, stock prices, and oil reserves. An illustration of a lognormal distribution is given below:\n \n\n- *Triangular Distribution*: this is a continuous distribution with fixed minimum and maximum values. It is bounded by the minimum and maximum values and can be either symmetrical (the most probable value = mean = median) or asymmetrical. Values around the most likely value (e.g. the mean) are more likely to occur. Variables that could be described by a triangular distribution include, for example, past sales history per unit of time and inventory levels. An illustration of a triangular distribution is given below:\n\n\n- *Uniform Distribution*: this is a continuous distribution bounded by known minimum and maximum values. In contrast to the triangular distribution, the likelihood of occurrence of the values between the minimum and maximum is the same. In other words, all values have an equal chance of occurring, and the distribution is simply characterized by the minimum and maximum values. Examples of variables that can be described by a uniform distribution include manufacturing costs or future sales revenues for a new product. An illustration of the uniform distribution is given below:\n\n\n- *Exponential Distribution*: this is a continuous distribution used to model the time that pass between independent occurrences, provided that the rate of occurrences is known. An example of the exponential distribution is given below:\n\n\n- *Discrete Distribution* : for this kind of distribution, the \"user\" defines specific values that may occur and the likelihood of each of them. An example might be the results of a lawsuit: 20% chance of positive verdict, 30% change of negative verdict, 40% chance of settlement, and 10% chance of mistrial.\n\n\n# 5. A real use case: estimating the financial risk of a portfolio of stocks\nWe hope that by now you have a good understanding about Monte Carlo simulation. Next, we apply this method to a real use case: *financial risk estimation*.\n\nImagine that you are an investor on the stock market. You plan to buy some stocks and you want to estimate the maximum loss you could incur after two weeks of investing. This is the quantity that the financial statistic \"Value at Risk\" (VaR) seeks to measure. [VaR](https://en.wikipedia.org/wiki/Value_at_risk) is defined as a measure of investment risk that can be used as a reasonable estimate of the maximum probable loss for a value of an investment portfolio, over a particular time period. A VaR statistic depends on three parameters: a portfolio, a time period, and a confidence level. A VaR of 1 million dollars with a 95% confidence level over two weeks, indicates the belief that the portfolio stands only a 5% chance of losing more than 1 million dollars over two weeks. VaR has seen widespread use across financial services organizations. This statistic plays a vital role in determining how much cash investors must hold to meet the credit ratings that they seek. In addition, it is also used to understand the risk characteristics of large portfolios: it is a good idea to compute the VaR before executing trades, such that it can help take informed decisions about investments. \n\nOur goal is calculating VaR of two weeks interval with 95% confidence level and the associated [VaR confidence interval](http://www.investopedia.com/ask/answers/041615/whats-difference-between-confidence-level-and-confidence-interval-value-risk-var.asp).\n\n\n## 5.1. Terminology\nIn this use case, we will use some terms that might require a proper definition, given the domain. This is what we call the *Domain Knowledge*.\n\n- **Instrument**: A tradable asset, such as a bond, loan, option, or stock investment. At any particular time, an instrument is considered to have a value, which is the price for which it can be sold. In the use case of this notebook, instruments are stock investments.\n- **Portfolio**: A collection of instruments owned by a financial institution. \n- **Return**: The change in an instrument or portfolio\u2019s value over a time period. \n- **Loss**: A negative return. \n- **Index**: An imaginary portfolio of instruments. For example, the NASDAQ Composite index includes about 3,000 stocks and similar instruments for major US and international companies. \n- **Market factor**: A value that can be used as an indicator of macro aspects of the financial climate at a particular time. For example, the value of an index, the Gross Domestic Product of the United States, or the exchange rate between the dollar and the euro. We will often refer to market factors as just factors.\n\n## 5.2. The context of our use case\nWe have a list of instruments that we plan to invest in. The historical data of each instrument has been collected for you. For simplicity, assume that the returns of instruments at a given time, depend on 4 market factors only: \n\n- GSPC value\n- IXIC value \n- The return of crude oil\n- The return of treasury bonds\n\nOur goal is building a model to predict the loss after two weeks' time interval with confidence level set to 95%.\n\nAs a side note, it is important to realize that the approach presented in this Notebook is a simplified version of what would happen in a real Financial firm. For example, the returns of instruments at a given time often depend on more than 4 market factors only! Moreover, the choice of what constitute an appropriate market factor is an art!\n\n\n\n## 5.3. The Data\nThe stock data can be downloaded (or scraped) from Yahoo! by making a series of REST calls. The data includes multiple files. Each file contains the historical information of each instrument that we want to invest in. The data is in the following format (with some samples):\n```\nDate, Open, High, Low, Close, Volume, Adj Close\n2016-01-22,66.239998,68.07,65.449997,67.860001,137400,67.860001\n2016-01-21,65.410004,66.18,64.459999,65.050003,148000,65.050003\n2016-01-20,64.279999,66.32,62.77,65.389999,141300,65.389999\n2016-01-19,67.720001,67.989998,64.720001,65.379997,178400,65.379997\n```\n\nThe data of GSPC and IXIC values (our two first market factors) are also available on Yahoo! and use the very same format. \n\nThe crude oil and treasure bonds data is collected from investing.com, and has a different format, as shown below (with some samples):\n```\nDate Price Open High Low Vol. Change %\nJan 25, 2016 32.17 32.36 32.44 32.10 - -0.59%\nJan 24, 2016 32.37 32.10 32.62 31.99 - 0.54%\nJan 22, 2016 32.19 29.84 32.35 29.53 - 9.01%\nJan 21, 2016 29.53 28.35 30.25 27.87 694.04K 11.22%\nJan 20, 2016 26.55 28.33 28.58 26.19 32.11K -6.71%\nJan 19, 2016 28.46 29.20 30.21 28.21 188.03K -5.21%\n```\n\nIn our use case, the factors' data will be used jointly to build a statistical model: as a consequence, we first need to preprocess the data to proceed.\n\n## 5.4. Data preprocessing\nIn this Notebook, all data files have been downloaded for you, such that you can focus on pre-processing. Next, we will:\n\n - Read the factor data files which are in two different formats, process and merge them together\n - Read the stock data and pre-process it\n - Trim all data into a specific time region\n - Fill in the missing values\n - Generate the data of returns in each two weeks' time interval window\n \n### Factor data pre-processing\n\nWe need two functions to read and parse data from Yahoo! and Investing.com respectively. We are interested only in information about the time and the corresponding returns of a factor or an instrument: as a consequence, we will project away many columns of our RAW data, and keep only the information we are interested in.\n\nThe 3000-instrument and the 4-factor history are small enough to be read and processed locally: we do not need to use the power of parallel computing to proceed. Note that this is true also for larger cases with hundreds of thousands of instruments and thousands of factors. The need for a distributed system like Spark comes in when actually **running** the Monte Carlo simulations, which can require massive amounts of computation on each instrument. \n\n\n\n### Question 4\n#### Question 4.1\n
            \nWrite a function named `readInvestingDotComHistory` to parse data from investing.com based on the format specified above (see Section 5.3). Recall that we use two factors here: one that is related to the price of crude oil, one that is related to some specific US bonds. \n\n
              \n\nPrint the first 5 entries of the first factor (crude oil price) in the parsed data.\n\n
                \n\nNote that we are only interested in the date and price of stocks.\n\n
                \n\n
                HINT
                \nYou can parse a string to `datetime` object by using the function `strptime(, )`. In this case, the datetime format is `\"%b %d, %Y\"`. For more information, please follow this [link](https://docs.python.org/2/library/datetime.html#strftime-and-strptime-behavior).\n\nIn the next cell, we simply copy data from our HDFS cluster (that contains everything we need for this Notebook) to the instance (a Docker container) running your Notebook. This means that you will have \"local\" data that you can process without using Spark. Note the folder location: find and verify that you have correctly downloaded the files!\n\n\n```python\n! [ -d monte-carlo-risk ] || (echo \"Downloading prepared data from HDFS. Please wait...\" ; hdfs dfs -copyToLocal /datasets/monte-carlo-risk . ; echo \"Done!\";)\n```\n\n\n```python\nfrom datetime import datetime\nfrom datetime import timedelta\nfrom itertools import islice\n%matplotlib inline\nimport numpy as np\nimport statsmodels.api as sm\n\nbase_folder = \"monte-carlo-risk/\"\n\nfactors_folder= base_folder + \"factors/\"\n\n# read data from local disk\ndef readInvestingDotComHistory(fname):\n def process_line(line):\n cols = line.split('\\t')\n date = datetime.strptime(cols[0], \"%b %d, %Y\")\n value = float(cols[2]) # open value\n return (date, value)\n \n with open(fname) as f:\n content_w_header = f.readlines()\n # remove the first line \n # and reverse lines to sort the data by date, in ascending order\n content = content_w_header[1:]\n return sorted(list(map(process_line , content)), key=lambda x: x[0])\n\nfactor1_files = ['crudeoil.tsv', 'us30yeartreasurybonds.tsv']\nfactor1_files = map(lambda fn: factors_folder + fn, factor1_files)\nfactors1 = [readInvestingDotComHistory(f) for f in factor1_files]\n\n\nprint('First factor first values:', factors1[0][:5])\nprint('First factor last values:', factors1[0][-5:])\n\nprint('Second factor first values:', factors1[1][:5])\nprint('Second factor last values:', factors1[1][-5:])\n```\n\n First factor first values: [(datetime.datetime(2006, 1, 26, 0, 0), 65.85), (datetime.datetime(2006, 1, 27, 0, 0), 66.49), (datetime.datetime(2006, 1, 30, 0, 0), 67.85), (datetime.datetime(2006, 1, 31, 0, 0), 68.4), (datetime.datetime(2006, 2, 1, 0, 0), 67.8)]\n First factor last values: [(datetime.datetime(2016, 1, 20, 0, 0), 28.33), (datetime.datetime(2016, 1, 21, 0, 0), 28.35), (datetime.datetime(2016, 1, 22, 0, 0), 29.84), (datetime.datetime(2016, 1, 24, 0, 0), 32.1), (datetime.datetime(2016, 1, 25, 0, 0), 32.36)]\n Second factor first values: [(datetime.datetime(2008, 2, 12, 0, 0), 4.401), (datetime.datetime(2008, 2, 13, 0, 0), 4.46), (datetime.datetime(2008, 2, 14, 0, 0), 4.533), (datetime.datetime(2008, 2, 15, 0, 0), 4.626), (datetime.datetime(2008, 2, 19, 0, 0), 4.583)]\n Second factor last values: [(datetime.datetime(2016, 1, 20, 0, 0), 2.825), (datetime.datetime(2016, 1, 21, 0, 0), 2.766), (datetime.datetime(2016, 1, 22, 0, 0), 2.81), (datetime.datetime(2016, 1, 24, 0, 0), 2.828), (datetime.datetime(2016, 1, 25, 0, 0), 2.825)]\n\n\nNow, the data structure `factors1` is a list, containing data that pertains to two (out of a total of four) factors that influence the market, as obtained by investing.com. Each element in the list is a tuple, containing some sort of timestamp, and the value of one of the two factors discussed above. From now on, we call these elements \"**records**\" or \"**entries**\". Visually, `factors1` looks like this:\n\n| 0 (crude oil) | 1 (US bonds)|\n| --- | --- |\n| time_stamp, value | time_stamp, value |\n| ... | ... |\n| time_stamp, value | time_stamp, value |\n| ... | ... |\n\n\n#### Question 4.2\n
                \nWrite a function named `readYahooHistory` to parse data from yahoo.com based on its format, as described in Section 5.3. \n
                  \nPrint the first 5 entries of the first factor (namely GSPC). Comment the time range of the second batch of data we use in our Notebook. \n
                    \n\nNote that we are only interested in the date and price of stocks.\n
                    \n\n
                    NOTE
                    The datetime format now is in a different format than the previous one.\n\n
                    HINT
                    Use a terminal (or put the bash commands inline in your Notebook) to list filenames in your local working directory to find and have a look at your local files.\n\n\n```python\n# read data from local disk\ndef readYahooHistory(fname):\n def process_line(line):\n cols = line.split(',')\n date = datetime.strptime(cols[0], '%Y-%m-%d')\n value = float(cols[1]) # open value\n return (date, value)\n \n with open(fname) as f:\n content_w_header = f.readlines()\n # remove the first line \n # and reverse lines to sort the data by date, in ascending order\n content = content_w_header[1:]\n return sorted(list(map(process_line , content)), key=lambda x: x[0])\n#'COPPER.csv', 'LEAD.csv', 'IRON.csv', , 'USD-EUR.csv'\n\nfactor2_files = ['GSPC.csv', 'IXIC.csv']\nfactor2_files = map(lambda fn: factors_folder + fn, factor2_files)\n\nfactors2 = [readYahooHistory(f) for f in factor2_files]\n\nprint(factors2[0][:5])\nprint(factors2[0][-5:])\n\nprint(factors2[1][:5])\nprint(factors2[1][-5:])\n```\n\n [(datetime.datetime(1950, 1, 3, 0, 0), 16.66), (datetime.datetime(1950, 1, 4, 0, 0), 16.85), (datetime.datetime(1950, 1, 5, 0, 0), 16.93), (datetime.datetime(1950, 1, 6, 0, 0), 16.98), (datetime.datetime(1950, 1, 9, 0, 0), 17.08)]\n [(datetime.datetime(2016, 1, 15, 0, 0), 1916.680054), (datetime.datetime(2016, 1, 19, 0, 0), 1888.660034), (datetime.datetime(2016, 1, 20, 0, 0), 1876.180054), (datetime.datetime(2016, 1, 21, 0, 0), 1861.459961), (datetime.datetime(2016, 1, 22, 0, 0), 1877.400024)]\n [(datetime.datetime(1971, 2, 5, 0, 0), 100.0), (datetime.datetime(1971, 2, 8, 0, 0), 100.839996), (datetime.datetime(1971, 2, 9, 0, 0), 100.760002), (datetime.datetime(1971, 2, 10, 0, 0), 100.690002), (datetime.datetime(1971, 2, 11, 0, 0), 101.449997)]\n [(datetime.datetime(2016, 1, 15, 0, 0), 4464.370117), (datetime.datetime(2016, 1, 19, 0, 0), 4548.049805), (datetime.datetime(2016, 1, 20, 0, 0), 4405.220215), (datetime.datetime(2016, 1, 21, 0, 0), 4480.700195), (datetime.datetime(2016, 1, 22, 0, 0), 4557.390137)]\n\n\n
                    \nCOMMENT:
                    The factors GSPC.csv and IXIC.csv has a history of 66 and 45 years respectively. The two other factors has much less history, with 10 and 8 years. Since we need a complete history for our estimations we need to limit the data to the factor with the shortest history, ie 8 years.\n
                    \n\nNow, the data structure `factors2` is again list, containing data that pertains to the next two (out of a total of four) factors that influence the market, as obtained by Yahoo!. Each element in the list is a tuple, containing some sort of timestamp, and the value of one of the two factors discussed above. Visually, `factors2` looks like this:\n\n| 0 (GSPC) | 1 (IXIC)|\n| --- | --- |\n| time_stamp, value | time_stamp, value |\n| ... | ... |\n| time_stamp, value | time_stamp, value |\n| ... | ... |\n\n\n### Stock data pre-processing\n\nNext, we prepare the data for the instruments we consider in this Notebook (i.e., the stocks we want to invest in). \n\n#### Question 4.3\n\n
                    \nIn this Notebook, we assume that we want to invest on the first 35 stocks out of the total 3000 stocks present in our datasets.\n\n
                      \n\nLoad and prepare all the data for the considered instruments (the first 35 stocks) which have historical information for more than 5 years. This means that all instruments with less than 5 years of history should be removed.\n\n
                      \n\n
                      HINT
                      we suggest to open a terminal window (not on your local machine, but the Notebook terminal that you can find on the Jupyter dashboard) and visually check the contents of the directories holding our dataset, if you didn't do this before! Have a look at how stock data is organized!\n\n\n```python\nfrom os import listdir\nfrom os.path import isfile, join\n\nstock_folder = base_folder + 'stocks'\nnum_stocks = 100\n\ndef process_stock_file(fname):\n try:\n yahooData = readYahooHistory(fname)\n return yahooData\n except Exception as e:\n raise e\n return None\n\n# select path of all stock data files in \"stock_folder\"\nfiles = [join(stock_folder, f) for f in listdir(stock_folder) if isfile(join(stock_folder, f))]\n\n# assume that we invest only the first 35 stocks (for faster computation)\nfiles = files[:num_stocks]\n\n# read each line in each file, convert it into the format: (date, value)\nrawStocks = [process_stock_file(f) for f in files]\n\n# select only instruments which have more than 5 years of history\n# Note: the number of business days in a year is 260\nnumber_of_years = 5\nrawStocks = list(filter(lambda instrument: len(instrument)>number_of_years*260 , rawStocks))\n\n# For testing, print the first 5 entry of the first stock\n#print(rawStocks[0][:10])\n\n#for s in rawStocks[0][:1000]:\n# print(s)\n\nprint(\"\\nInstruments with more that 5 years:\", len(rawStocks))\n\n# What the fuck happend here?\n#(datetime.datetime(1998, 5, 15, 0, 0), 56.375)\n#(datetime.datetime(1998, 5, 18, 0, 0), 28.75)\n```\n\n \n Instruments with more that 5 years: 75\n\n\n### Time alignment for our data\nDifferent types of instruments may trade on different days, or the data may have missing values for other reasons, so it is important to make sure that our different histories align. First, we need to trim all of our time series to the same region in time. Then, we need to fill in missing values. To deal with time series that have missing values at the start and end dates in the time region, we simply fill in those dates with nearby values in the time region.\n\n#### Question 4.4\n
                      \nAssume that we only focus on the data from 23/01/2009 to 23/01/2014. Write a function named `trimToRegion` to select only the records in that time interval. \n\n
                        \n\n**Requirements**: after processing, each instrument $i$ has a list of records: $[r_0, r_2,...,r_{m_i}]$ such that $r_0$ and $r_{m_i}$ are assigned, respectively, the first and the last values corresponding to the extremes of the given time interval. For example: $r_0$ should contain the value at date 23/01/2009.\n
                        \n\n\n```python\n# note that the data of crude oild and treasury is only available starting from 26/01/2006 \nstart = datetime(year=2009, month=1, day=23)\nend = datetime(year=2014, month=1, day=23)\n\ndef trimToRegion(history, start, end):\n def isInTimeRegion(entry):\n (date, value) = entry\n return date >= start and date <= end\n\n # only select entries which are in the time region\n trimmed = list(filter(isInTimeRegion, history))\n \n # if the data has incorrect time boundaries, add time boundaries\n if trimmed[0][0] != start:\n trimmed.insert(0, (start, trimmed[0][1]))\n if trimmed[-1][0] != end:\n trimmed.append((end, trimmed[-1][1]))\n return trimmed\n \n# test our function\ntrimmedStock0 = trimToRegion(rawStocks[0], start, end)\n# the first 5 records of stock 0\nprint(trimmedStock0[:5])\n# the last 5 records of stock 0\nprint(trimmedStock0[-5:])\n\nassert (trimmedStock0[0][0] == start), \"the first record must contain the price in the first day of time interval\"\nassert (trimmedStock0[-1][0] == end), \"the last record must contain the price in the last day of time interval\"\n```\n\n [(datetime.datetime(2009, 1, 23, 0, 0), 19.4), (datetime.datetime(2009, 1, 26, 0, 0), 19.67), (datetime.datetime(2009, 1, 27, 0, 0), 19.809999), (datetime.datetime(2009, 1, 28, 0, 0), 20.469999), (datetime.datetime(2009, 1, 29, 0, 0), 21.41)]\n [(datetime.datetime(2014, 1, 16, 0, 0), 37.369999), (datetime.datetime(2014, 1, 17, 0, 0), 37.470001), (datetime.datetime(2014, 1, 21, 0, 0), 37.73), (datetime.datetime(2014, 1, 22, 0, 0), 37.779999), (datetime.datetime(2014, 1, 23, 0, 0), 37.59)]\n\n\n### Dealing with missing values\nWe expect that we have the price of instruments and factors **in each business day**. Unfortunately, there are many missing values in our data: this means that we miss data for some days, e.g. we have data for the Monday of a certain week, but not for the subsequent Tuesday. So, we need a function that helps filling these missing values.\n\nNext, we provide to you the function to fill missing value: read it carefully!\n\n\n```python\ndef fillInHistory(history, start, end):\n curr = history\n filled = []\n idx = 0\n curDate = start\n numEntries = len(history)\n while curDate < end:\n \n # if the next entry is in the same day\n # or the next entry is at the weekend\n # but the curDate has already skipped it and moved to the next monday\n # (only in that case, curr[idx + 1][0] < curDate )\n # then move to the next entry\n while idx + 1 < numEntries and curr[idx + 1][0] == curDate:\n idx +=1\n\n # only add the last value of instrument in a single day\n # check curDate is weekday or not\n # 0: Monday -> 5: Saturday, 6: Sunday\n if curDate.weekday() < 5:\n \n filled.append((curDate, curr[idx][1]))\n # move to the next business day\n curDate += timedelta(days=1)\n \n # skip the weekends\n if curDate.weekday() >= 5:\n # if curDate is Sat, skip 2 days, otherwise, skip 1 day\n curDate += timedelta(days=(7-curDate.weekday()))\n\n return filled\n```\n\n#### Question 4.5\n
                        \nTrim data of stocks and factors into the given time interval.\n
                        \n\n\n```python\n#print rawStocks[0]\n\n# trim into a specific time region\n# and fill up the missing values\nstocks = list(map(lambda stock: \\\n fillInHistory(\n trimToRegion(stock, start, end), \n start, end), \n rawStocks))\n\n\n\n# merge two factors, trim each factor into a time region\n# and fill up the missing values\nallfactors = factors1 + factors2\nfactors = list(map(lambda factor: \\\n fillInHistory(\n trimToRegion(factor, start, end), \n start, end), \n allfactors))\n \n# test our code\nprint(\"the first 5 records of stock 0:\", stocks[0][:5], \"\\n\")\nprint(\"the last 5 records of stock 0:\", stocks[0][-5:], \"\\n\")\nprint(\"the first 5 records of factor 0:\", factors[0][:5], \"\\n\")\nprint(\"the first 5 records of factor 0:\", factors[0][-5:], \"\\n\")\n```\n\n the first 5 records of stock 0: [(datetime.datetime(2009, 1, 23, 0, 0), 19.4), (datetime.datetime(2009, 1, 26, 0, 0), 19.67), (datetime.datetime(2009, 1, 27, 0, 0), 19.809999), (datetime.datetime(2009, 1, 28, 0, 0), 20.469999), (datetime.datetime(2009, 1, 29, 0, 0), 21.41)] \n \n the last 5 records of stock 0: [(datetime.datetime(2014, 1, 16, 0, 0), 37.369999), (datetime.datetime(2014, 1, 17, 0, 0), 37.470001), (datetime.datetime(2014, 1, 20, 0, 0), 37.470001), (datetime.datetime(2014, 1, 21, 0, 0), 37.73), (datetime.datetime(2014, 1, 22, 0, 0), 37.779999)] \n \n the first 5 records of factor 0: [(datetime.datetime(2009, 1, 23, 0, 0), 43.26), (datetime.datetime(2009, 1, 26, 0, 0), 46.05), (datetime.datetime(2009, 1, 27, 0, 0), 45.65), (datetime.datetime(2009, 1, 28, 0, 0), 41.99), (datetime.datetime(2009, 1, 29, 0, 0), 42.22)] \n \n the first 5 records of factor 0: [(datetime.datetime(2014, 1, 16, 0, 0), 94.29), (datetime.datetime(2014, 1, 17, 0, 0), 94.17), (datetime.datetime(2014, 1, 20, 0, 0), 94.31), (datetime.datetime(2014, 1, 21, 0, 0), 94.0), (datetime.datetime(2014, 1, 22, 0, 0), 95.2)] \n \n\n\nRecall that Value at Risk (VaR) deals with **losses over a particular time horizon**. We are not concerned with the absolute prices of instruments, but how those prices **change over** a given period of time. In our project, we will set that length to two weeks: we use the sliding window method to transform time series of prices into an overlapping sequence of price change over two-week intervals.\n\nThe figure below illustrates this process. The returns of market factors after each two-week interval is calculated in the very same way.\n\n\n\n\n```python\ndef buildWindow(seq, k=2):\n \"Returns a sliding window (of width k) over data from iterable data structures\"\n \" s -> (s0,s1,...s[k-1]), (s1,s2,...,sk), ... \"\n it = iter(seq)\n result = tuple(islice(it, k))\n if len(result) == k:\n yield result \n for elem in it:\n result = result[1:] + (elem,)\n yield result\n```\n\n#### Question 4.6\n
                        \nCompute the returns of the stocks after each two-week time window.\n
                        \n\n\n```python\ndef calculateReturn(window):\n # return the change of value after two weeks\n return window[-1][1] - window[0][1]\n\ndef twoWeekReturns(history):\n # we use 10 instead of 14 to define the window\n # because financial data does not include weekends\n return [calculateReturn(entry) for entry in buildWindow(history, 10)]\n\nstocksReturns = list(map(twoWeekReturns, stocks))\nfactorsReturns = list(map(twoWeekReturns, factors))\n\n# test our functions\nprint(\"the first 5 returns of stock 0:\", stocksReturns[0][:5])\nprint(\"the last 5 returns of stock 0:\", stocksReturns[0][-5:])\n```\n\n the first 5 returns of stock 0: [0.8000010000000017, 1.0, 0.7200019999999974, -0.27999800000000263, -1.5800000000000018]\n the last 5 returns of stock 0: [-1.1599999999999966, -1.4599989999999963, -0.8399999999999963, -0.4599990000000034, 0.0]\n\n\nAlright! Now we have data that is properly aligned to start the training process: stocks' returns and factors' returns, per time windows of two weeks. Next, we will apply the MCS method.\n\n## 5.5. Summary guidelines to apply the MCS method on the data we prepared\nNext, we overview the steps that you have to follow to build a model of your data, and then use Monte Carlo simulations to produce output distributions:\n\n- **Step 1**: Defining the relationship between the market factors and the instrument's returns. This relationship takes the form of a model fitted to historical data.\n- **Step 2**: Defining the distributions for the market conditions (particularly, the returns of factors) that are straightforward to sample from. These distributions are fitted to historical data. \n- **Step 3**: Generate the data for each trial of a Monte Carlo run: this amount to generating the random values for market conditions along with these distributions.\n- **Step 4**: For each trial, from the above values of market conditions, and using the relationship built in step 1, we calculate the return for each instrument and the total return. We use the returns to define an empirical distribution over losses. This means that, if we run 100 trials and want to estimate the 5% VaR, we would choose it as the loss from the trial with the fifth greatest loss.\n- **Step 5**: Evaluating the result\n\n## 5.6. Applying MCS\n\n### Step 1: Defining relationship between market factors and instrument's returns\n\nIn our simulation, we will use a simple linear model. By our definition of return, a factor return is a **change** in the value of a market factor **over a particular time period**, e.g. if the value of the S&P 500 moves from 2000 to 2100 over a time interval, its return would be 100.\n\nA vector that contains the return of 4 market factors is called a *market factor vector*. Generally, instead of using this vector as features, we derive a set of features from simple transformation of it. In particular, a vector of 4 values is transformed into a vector of length $m$ by function $F$. In the simplest case $F(v) = v$.\n\nDenote $v_t$ the market factor vector, and $f_t$ the transformed features of $v_t$ at time $t$.\n\n$f_{tj}$ is the value of feature $j$ in $f_t$.\n\nDenote $r_{it}$ the return of instrument $i$ at time $t$ and $c_i$ the [intercept term](http://blog.minitab.com/blog/adventures-in-statistics/regression-analysis-how-to-interpret-the-constant-y-intercept) of instrument $i$.\n\nWe will use a simple linear function to calculate $r_{it}$ from $f_t$:\n\n$$\nr_{it} = c_i + \\sum_{j=1}^{m}{w_{ij}*f_{tj}}\n$$\n\nwhere $w_{ij}$ is the weight of feature $j$ for instrument $i$.\n\nAll that above means that given a market factor vector, we have to apply featurization and then use the result as a surrogate for calculating the return of the instruments, using the above linear function.\n\nThere are two questions that we should consider: **how we apply featurization to a factor vector?** and **how to pick values for $w_{ij}$?**\n\n**How we apply featurization to a factor vector?**\nIn fact, the instruments' returns may be non-linear functions of the factor returns. So, we should not use factor returns as features in the above linear function. Instead, we transform them into a set of features with different size. In this Notebook, we can include some additional features in our model that we derive from non-linear transformations of the factor returns. We will try adding two more features for each factor return: its square and its square root values. So, we can still assume that our model is a linear model in the sense that the response variable is a linear function of the new features. *Note that the particular feature transformation described here is meant to be an illustrative example of some of the options that are available: it shouldn't be considered as the state of the art in predictive financial modeling!!*.\n\n**How to pick values for $w_{ij}$?**\n\nFor all the market factor vectors in our historical data, we transform them to feature vectors. Now, we have feature vectors in many two-week intervals and the corresponding instrument's returns in these intervals. We can use Ordinary Least Square (OLS) regression model to estimate the weights for each instrument such that our linear function can fit to the data. The parameters for OLS function are:\n\n- `x`: The collection of columns where **each column** is the value of **a feature** in many two-week interval\n- `y`: The return of an instrument in the corresponding time interval of x.\n\nThe figure below shows the basic idea of the process to build a statistical model for predicting the returns of stock X.\n\n\n\n\n\n\n### Question 5\n#### Question 5.1\n\n
                        \nCurrently, our data is in form of: \n\n$$\nfactorsReturns=\n\\begin{bmatrix}\n r_{00} & r_{01} & r_{02} & ... & r_{0k} \\\\\n r_{10} & r_{11} & r_{12} & ... & r_{1k} \\\\\n ... & ... & ... & ... & ... \\\\\n r_{n0} & r_{n1} & r_{n2} & ... & r_{nk}\\\\\n\\end{bmatrix}\n$$\n\n
                          \n\n$$\nstocksReturns=\n\\begin{bmatrix}\n s_{00} & s_{01} & s_{02} & ... & s_{0k} \\\\\n s_{10} & s_{11} & s_{12} & ... & s_{1k} \\\\\n ... & ... & ... & ... & ... \\\\\n s_{n0} & s_{n1} & s_{n2} & ... & s_{nk}\\\\\n\\end{bmatrix}\n$$\n\n
                            \n\nWhere, $r_{ij}$ is the return of factor $i^{th}$ in time window $j^{th}$, $k$ is the number of time windows, and $n$ is the number of factors. A similar definition goes for $s_{ij}$.\n\n
                              \n\nIn order to use OLS, the parameter must be in form of:\n\n
                                \n\n$$\nx=factorsReturns^T =\n\\begin{bmatrix}\n r_{00} & r_{10} & ... & r_{n0} \\\\\n r_{01} & r_{11} & ... & r_{n1} \\\\\n r_{02} & r_{12} & ... & r_{n2}\\\\\n ... & ... & ... & ... \\\\\n r_{0k} & r_{1k} & ... & r_{nk}\\\\\n\\end{bmatrix}\n$$\n\n
                                  \n\nWhereas, $y$ can be any row in `stocksReturns`.\n\n
                                    \n\nSo, we need a function to transpose a matrix. Write a function named `transpose` to do just that.\n
                                    \n\n\n```python\ndef transpose(matrix):\n m_tr = [list(x) for x in zip(*matrix)]\n return m_tr\n \n# test function\nassert (transpose([[1,2,3], [4,5,6], [7,8,9]]) == [[1, 4, 7], [2, 5, 8], [3, 6, 9]]), \"Function transpose runs incorrectly\"\n```\n\n#### Question 5.2\n
                                    \nWrite a function named `featurize` that takes a list factor's returns $[x_1, x_2,...,x_k]$ and transform it into a new list of features $[u_1,u_2,..,u_k, v_1, v_2,..,v_k, x_1,x_2,...,x_k]$.\n\n
                                      \n\nWhere, \n\n\n$u_i$ = $\\left\\{\n\t\\begin{array}{ll}\n\t\tx_i^2 & \\mbox{if } x_i \\geq 0 \\\\\n\t\t-x_i^2 & \\mbox{if } x_i < 0\n\t\\end{array}\n\\right.\n$\n\n
                                        \n\nand \n\n$v_i$ = $\\left\\{\n\t\\begin{array}{ll}\n\t\t\\sqrt{x_i} & \\mbox{if } x_i \\geq 0 \\\\\n\t\t-\\sqrt{x_i} & \\mbox{if } x_i < 0\n\t\\end{array}\n\\right.\n$ \n\n
                                        \n\n\n```python\nimport math\ndef featurize(factorReturns):\n squaredReturns = [(x**2)*(1 if x>0 else -1) for x in factorReturns]\n squareRootedReturns = [math.sqrt(abs(x))*(1 if x>0 else -1) for x in factorReturns]\n # concat new features\n return squaredReturns + squareRootedReturns + factorReturns\n\n# test our function\nassert (featurize([4, -9, 25]) == [16, -81, 625, 2, -3, 5, 4, -9, 25]), \"Function runs incorrectly\"\n```\n\n#### Question 5.3\n
                                        \nUsing OLS, estimate the weights for each feature on each stock. What is the shape of `weights` (size of each dimension)? \n\nExplain it.\n
                                        \n\n\n```python\nfactorMat[0]\n```\n\n\n\n\n [-2.960000000000001,\n 0.41800000000000015,\n 9.590026999999964,\n 57.80993699999999,\n 44.25,\n 0.0,\n 0.0]\n\n\n\n\n```python\nfactor_columns[0]\n```\n\n\n\n\n array([ 1.00000000e+00, -8.76160000e+00, 1.74724000e-01,\n 9.19686179e+01, 3.34198882e+03, 1.95806250e+03,\n -0.00000000e+00, -0.00000000e+00, -1.72046505e+00,\n 6.46529195e-01, 3.09677687e+00, 7.60328462e+00,\n 6.65206735e+00, -0.00000000e+00, -0.00000000e+00,\n -2.96000000e+00, 4.18000000e-01, 9.59002700e+00,\n 5.78099370e+01, 4.42500000e+01, 0.00000000e+00,\n 0.00000000e+00])\n\n\n\n\n```python\ndef estimateParams(y, x):\n return sm.OLS(y, x).fit().params\n\n# transpose factorsReturns\nfactorMat = transpose(factorsReturns)\n\n# featurize each row of factorMat\nfactorFeatures = list(map(featurize,factorMat))\n\n# OLS require parameter is a numpy array\nfactor_columns = np.array(factorFeatures)\n\n#add a constant - the intercept term for each instrument i.\nfactor_columns = sm.add_constant(factor_columns, prepend=True)\n\n# estimate weights\nweights = [estimateParams(stockRet, factor_columns) for stockRet in stocksReturns]\n\n\n\nprint(\"factor_columns:\", len(factor_columns),len(factor_columns[0]))\nprint(\"stocksReturns:\", len(stocksReturns),len(stocksReturns[0]))\n\nprint(\"weights length:\", len(weights),len(weights[0]))\nprint(\"weights first row:\", weights[0])\n```\n\n factor_columns: 1295 13\n stocksReturns: 75 1295\n weights length: 75 13\n weights first row: [ -4.35466694e-03 -3.64753449e-04 -8.03349863e+00 7.93239441e-05\n -2.17409832e-05 8.81875154e-02 -1.40571325e+00 -8.49863338e-03\n -4.99535942e-02 -3.69387457e-02 4.72385267e+00 7.23621329e-03\n 1.06892611e-02]\n\n\n
                                        \nCOMMENT:
                                        \nThe two dimensions of our weight vector are the number of instruments (75, section 4.3) after filtering out those with less than 5 years of history and the number of features after feature expansion: Four initial features, with their respective squares and square roots, and an extra interception feature to reduce the bias (4+4+4+1 = 13).\n\n
                                        \n\n### Step 2: Defining the distributions for the market conditions\nSince we cannot define the distributions for the market factors directly, we can only approximate their distribution.\nThe best way to do that, is plotting their value. However, these values may fluctuate quite a lot. \n\nNext, we show how to use the Kernel density estimation (KDE) technique to approximate such distributions. In brief, kernel density estimation is a way of smoothing out a histogram: this is achieved by assigning (or centering) a probability distribution (usually a normal distribution) to each data point, and then summing. So, a set of two-week-return samples would result in a large number of \"super-imposed\" normal distributions, each with a different mean. \n\nTo estimate the probability density at a given point, KDE evaluates the PDFs of all the normal distributions at that point and takes their average. The smoothness of a kernel density plot depends on its *bandwidth*, and the standard deviation of each of the normal distributions. For a brief introduction on KDE, please refer to this [link](https://en.wikipedia.org/wiki/Kernel_density_estimation).\n\n\n```python\nfrom statsmodels.nonparametric.kernel_density import KDEMultivariate\nfrom statsmodels.nonparametric.kde import KDEUnivariate\nimport matplotlib.pyplot as plt\nimport scipy\n\n\nf, axarr = plt.subplots(2, 2)\nf.set_figheight(15)\nf.set_figwidth(15)\n\ndef plotDistribution(samples, a, b):\n vmin = min(samples)\n vmax = max(samples)\n stddev = np.std(samples)\n \n domain = np.arange(vmin, vmax, (vmax-vmin)/100)\n \n # a simple heuristic to select bandwidth\n bandwidth = 1.06 * stddev * pow(len(samples), -.2)\n \n # estimate density\n kde = KDEUnivariate(samples)\n kde.fit(bw=bandwidth)\n density = kde.evaluate(domain)\n \n # plot\n axarr[a, b].plot(domain, density)\n \nplotDistribution(factorsReturns[0], 0, 0)\nplotDistribution(factorsReturns[1], 0, 1)\nplotDistribution(factorsReturns[2], 1, 0)\nplotDistribution(factorsReturns[3], 1, 1)\nplt.show()\n```\n\nFor the sake of simplicity, we can say that our smoothed versions of the returns of each factor can be represented quite well by a normal distribution. Of course, more exotic distributions, perhaps with fatter tails, could fit more closely the data, but it is outside the scope of this Notebook to proceed in this way.\n\nNow, the simplest way to sample factors returns is to use a normal distribution for each of the factors, and sample from these distributions independently. However, this approach ignores the fact that market factors are often correlated. For example, when the price of crude oil is down, the price of treasury bonds is down too. We can check our data to verify about the correlation.\n\n\n\n### Question 6\n\n#### Question 6.1\n
                                        \n\nCalculate the correlation between market factors and explain the result.\n\n
                                        \n\n
                                        HINT
                                        function `np.corrcoef` might be useful.\n\n\n```python\ncorrelation = np.corrcoef(factorsReturns)\nfactor_names = ['Crudeoil', 'USTreasury', 'GSPC', 'IXIC']\ncorrelation\nsns.heatmap(correlation, \n xticklabels=factor_names,\n yticklabels=factor_names, annot=True)\n```\n\n
                                        \nCOMMENT:
                                        \nIt's a symmetric matrix, as expected from correlation. Values range between 0 and 1, being 0 the absolute independency and 1 the absoulte linear correlation. As expected, the cross-correlation between each feature and himself is 1 (hence the diagonal filled with ones).

                                        \nOther than that, we see a very strong correlation between GSPC and IXIC, which was expected since they are two stock indicators: S&P500 and Nasdaq. USTreasury is expected to be much more stable, and crudeoil doesn't affect all stock values equally (e.g.: a company based on clean energies will be affected inversely to one based on transportation)\n
                                        \n\n\n```python\nimport pandas as pd\nimport seaborn as sns\nimport matplotlib.pyplot as plt\nimport warnings\nwarnings.filterwarnings('ignore')\n\n# Generate a mask for the upper triangle\nmask = np.zeros_like(correlation, dtype=np.bool)\nmask[np.triu_indices_from(mask)] = True\n\n# Set up the matplotlib figure\nf, ax = plt.subplots(figsize=[20,15])\n\nabsWeights = transpose(list(map(lambda x: list(map(abs ,x.tolist())), weights)))\n\n# Draw the heatmap with the mask and correct aspect ratio\nsns.heatmap(absWeights, cmap=\"YlGnBu\", vmax=1.0,square=True, xticklabels=5, yticklabels=2,linewidths=.5,\n cbar_kws={\"orientation\": \"horizontal\"}, ax=ax)\nax.set_title('Correlation between Factors (13, after expansion) and instruments (75)', fontsize=17)\nax.set_xlabel('Instruments', fontsize=15)\nax.set_ylabel('Factors', fontsize=15)\n```\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Factors
                                        InterceptionsquaresqrtLinear
                                        CrudeOilUSTreasGSCP
                                        		IXICCrudeOilUSTreasGSCPIXICCrudeOilUSTreasGSCPIXIC
                                        0123
                                        		456789101112
                                        \n\n
                                        \nCOMMENT:
                                        \nWe decided to analyze the weights between factors and instruments, in order to assess the relative importance of each factor. Let's analyze the results one by one, remembering the order of the factors:
                                        \n
                                        • **Crudeoil** -> Only the sqrt series gives away a bit of information; the rest could be discarded
                                          • \n
                                        • **USTreasury** -> With a great difference among the rest, this is the most influential feature in all its expansions. Squared is the most significant value, linear is a bit less significant and sqrt is the least of the three (nevertheless it's much more important than any other features)
                                          • \n
                                        • **GSPC** -> Almost insignificant.
                                          • \n
                                        • **IXIC** -> Almost insignificant.
                                          • \n
                                        • **Interception** -> Doesn't affect almost at all
                                          • \n
                                        \n
                                        \nWe should consider getting rid of the instrument number **41** since it displays some strange behaviour when compared to the rest.\n

                                        \nIf we would have an arsenal of one hundred factors, we could use this tool/technique to determine which factors should we keep and which ones don't impact the value of the instruments at all, and thus the VaR. \n
                                        \n\nThe multivariate normal distribution can help here by taking the correlation information between the factors into account. Each sample from a multivariate normal distribution can be thought of as a vector. Given values for all of the dimensions but one, the distribution of values along that dimension is normal. But, in their joint distribution, the variables are not independent.\n\nFor this use case, we can write:\n\n$$\n\\left(\\begin{array}{c}f_{1}\\\\f_{2}\\\\f_{3}\\\\f_{4} \\end{array}\\right)\n\\sim N \n\\left[\n \\left(\n \\begin{array}{c}\n \\mu_1\\\\ \\mu_2 \\\\ \\mu_3 \\\\ \\mu_4 \n \\end{array}\n \\right), \n \\left(\n \\begin{array}{cccc}\n \\sigma^2_1 & \\rho_{12} \\sigma_1\\sigma_2 & \\rho_{13} \\sigma_1\\sigma_3 & \\rho_{14} \\sigma_1\\sigma_4 \\\\ \n \\rho_{12}\\sigma_2\\sigma_1 & \\sigma^2_2 & \\rho_{23} \\sigma_2\\sigma_3 & \\rho_{24} \\sigma_2\\sigma_4\\\\\n \\rho_{13} \\sigma_3\\sigma_1 & \\rho_{23} \\sigma_3\\sigma_2 & \\sigma^2_3 & \\rho_{34} \\sigma_3\\sigma_4 \\\\ \n \\rho_{14} \\sigma_4\\sigma_1 & \\rho_{24} \\sigma_4\\sigma_2 & \\rho_{34} \\sigma_3\\sigma_4 & \\sigma_4^2 \\\\ \n \\end{array}\n \\right)\n\\right]\n$$\n\nOr,\n\n$$\nf_t \\sim N(\\mu, \\sum)\n$$\n\nWhere $f_1$, $f_2$, $f_3$ and $f_4$ are the market factors, $\\sigma_i$ is the standard deviation of factor $i$, $\\mu$ is a vector of the empirical means of the returns of the factors and $\\sum$ is the empirical covariance matrix of the returns of the factors.\n\nThe multivariate normal is parameterized with a mean along each dimension and a matrix describing the covariance between each pair of dimensions. When the covariance matrix is diagonal, the multivariate normal reduces to sampling along each dimension independently, but placing non-zero values in the off-diagonals helps capture the relationships between variables. Whenever having the mean of this multivariate normal distribution and its covariance matrix, we can generate the sample values for market factors.\n\nNext, we will calculate the mean and the covariance matrix of this multivariate normal distribution from the historical data.\n\n\n#### Question 6.2\n
                                        \n\nCalculate the covariance matrix $\\sum$ and the means $\\mu$ of factors' returns then generate a random vector of factors return that follows a multivariate normal distribution $\\sim N(\\mu, \\sum)$\n\n
                                        \n\n
                                        HINT
                                        \nFunction `np.cov` can help calculating covariance matrix. Function `np.random.multivariate_normal(, )` is often used for generating samples.\n\n\n```python\nfactorCov = np.cov(factorsReturns)\nfactorMeans = [sum(factorsReturns[i])/len(factorsReturns[i]) for i in range(len(factorsReturns))]\nsample = np.random.multivariate_normal(factorMeans, factorCov)\nprint('Factor Covariance:\\n', factorCov, \"\\n\")\nprint('Factor Means:\\n',factorMeans, \"\\n\")\nprint('Sample: \\n',sample)\n```\n\n Factor Covariance:\n [[ 1.99479662e+01 2.70103191e-01 7.70665225e+01 1.61846128e+02]\n [ 2.70103191e-01 2.22473953e-02 3.14993992e+00 6.81903796e+00]\n [ 7.70665225e+01 3.14993992e+00 1.29672509e+03 2.71521187e+03]\n [ 1.61846128e+02 6.81903796e+00 2.71521187e+03 6.70934736e+03]] \n \n Factor Means:\n [0.3532664092664094, -0.0030517374517374466, 6.970339789189207, 18.737721800000003] \n \n Sample: \n [ 5.16111351 -0.0844882 -10.85362392 8.33807266]\n\n\n### Step 3&4: Generating samples, running simulation and calculating the VaR\n\nWe define some functions that helps us calculating VaR 5%. You will see that the functions below are pretty complicated! This is why we provide a solution for you: however, study them well!!\n\nThe basic idea of calculating VaR 5% is that we need to find a value such that only 5% of the losses are bigger than it. That means the 5th percentile of the losses should be VaR 5%.\n\nVaR can sometimes be problematic though, since it does give any information on the extent of the losses which can exceed the VaR estimate. CVar is an extension of VaR that is introduced to deal with this problem. Indeed, CVaR measures the expected value of the loss in those cases where VaR estimate has been exceeded.\n\n\n```python\ndef fivePercentVaR(trials):\n numTrials = trials.count()\n topLosses = trials.takeOrdered(max(round(numTrials/20.0), 1))\n return topLosses[-1]\n\n# an extension of VaR\ndef fivePercentCVaR(trials):\n numTrials = trials.count()\n topLosses = trials.takeOrdered(max(round(numTrials/20.0), 1))\n return sum(topLosses)/len(topLosses)\n\ndef bootstrappedConfidenceInterval(\n trials, computeStatisticFunction,\n numResamples, pValue):\n stats = []\n for i in range(0, numResamples):\n resample = trials.sample(True, 1.0)\n stats.append(computeStatisticFunction(resample))\n sorted(stats)\n lowerIndex = int(numResamples * pValue / 2 - 1)\n upperIndex = int(np.ceil(numResamples * (1 - pValue / 2)))\n return (stats[lowerIndex], stats[upperIndex])\n```\n\nNext, we will run the Monte Carlo simulation 10,000 times, in parallel using Spark. Since your cluster has 12 cores (two Spark worker nodes, each with 6 cores), we can set `parallelism = 12` to dispatch simulation on these cores, across the two machines (remember, those are not really \"physical machines\", they are Docker containers running in our infrastructure).\n\n\n\n### Question 7\n
                                        \nComplete the code below to define the simulation process and calculate VaR 5%.\n
                                        \n\n\n```python\n# RUN SILMULATION\ndef simulateTrialReturns(numTrials, factorMeans, factorCov, weights):\n trialReturns = []\n for i in range(0, numTrials):\n # generate sample of factors' returns\n trialFactorReturns = np.random.multivariate_normal(factorMeans, factorCov)\n \n # featurize the factors' returns\n trialFeatures = featurize(trialFactorReturns.tolist())\n \n # insert weight for intercept term\n trialFeatures.insert(0,1)\n \n trialTotalReturn = 0\n \n # calculate the return of each instrument\n # then calulate the total of return for this trial features\n \n #trialTotalReturn = sum(np.dot(weights,trialFeatures).tolist())\n trialreturns = np.array(weights).dot(trialFeatures)\n for ret in trialreturns:\n trialTotalReturn += ret\n \n trialReturns.append(trialTotalReturn)\n return trialReturns\n\n\n \nparallelism = 12\nnumTrials = 10000\ntrial_indexes = list(range(0, parallelism))\nseedRDD = sc.parallelize(trial_indexes, parallelism)\nbFactorWeights = sc.broadcast(weights)\n\ntrials = seedRDD.flatMap(lambda idx: \\\n simulateTrialReturns(\n max(int(numTrials/parallelism), 1), \n factorMeans, factorCov,\n bFactorWeights.value\n ))\ntrials.cache()\n```\n\n\n\n\n PythonRDD[1242] at RDD at PythonRDD.scala:48\n\n\n\n\n```python\nvalueAtRisk = fivePercentVaR(trials)\nconditionalValueAtRisk = fivePercentCVaR(trials)\n\nprint(\"Value at Risk(VaR) 5: %.3f\"% valueAtRisk)\nprint(\"Conditional Value at Risk(CVaR) 5: %.3f\"% conditionalValueAtRisk)\n```\n\n Value at Risk(VaR) 5: -586.548\n Conditional Value at Risk(CVaR) 5: -988.260\n\n\nThe value of VaR depends on how many invested stocks and the chosen distribution of random variables. Assume that we get VaR 5% = -2.66, that means that there is a 0.05 probability that the portfolio will fall in value by more than \\$2.66 over a two weeks' period if there is no trading. In other words, the loses are less than \\$2.66 over two weeks' period with 95% confidence level. When a loss over two weeks is more than \\$2.66, we call it **failure** (or **exception**). Informally, because of 5% probability, we expect that there are only $0.05*W$ failures out of total $W$ windows.\n\n### Step 5: Evaluating the results using backtesting method\nIn general, the error in a Monte Carlo simulation should be proportional to 1/sqrt(n), where n is the number of trials. This means, for example, that quadrupling the number of trials should approximately cut the error in half. A good way to check the quality of a result is backtesting on historical data. Backtesting is a statistical procedure where actual losses are compared to the estimated VaR. For instance, if the confidence level used to calculate VaR is 95% (or VaR 5%), we expect only 5 failures over 100 two-week time windows.\n\nThe most common test of a VaR model is counting the number of VaR failures, i.e., in how many windows, the losses exceed VaR estimate. If the number of exceptions is less than selected confidence level would indicate, the VaR model overestimates the risk. On the contrary, if there are too many exceptions, the risk is underestimated. However, it's very hard to observe the amount of failures suggested by the confidence level exactly. Therefore, people try to study whether the number of failures is reasonable or not, or will the model be accepted or rejected.\n\nOne common test is Kupiec's proportion-of-failures (POF) test. This test considers how the portfolio performed at many historical time intervals and counts the number of times that the losses exceeded the VaR. The null hypothesis is that the VaR is reasonable, and a sufficiently extreme test statistic means that the VaR estimate does not accurately describe the data. The test statistic is computed as:\n\n$$\n-2ln\\Bigg(\\frac{(1-p)^{T-x}p^x}{(1-\\frac{x}{T})^{T-x}(\\frac{x}{T})^x}\\Bigg)\n$$\n\nwhere:\n\n$p$ is the quantile-of-loss of the VaR calculation (e.g., in VaR 5%, p=0.05),\n\n$x$ (the number of failures) is the number of historical intervals over which the losses exceeded the VaR \n\n$T$ is the total number of historical intervals considered\n\nOr we can expand out the log for better numerical stability:\n\n$$\n\\begin{equation}\n-2\\Big((T-x)ln(1-p)+x*ln(p)-(T-x)ln(1-\\frac{x}{T})-x*ln(\\frac{x}{T})\\Big)\n\\end{equation}\n$$\n\nIf we assume the null hypothesis that the VaR is reasonable, then this test statistic is drawn from a chi-squared distribution with a single degree of freedom. By using Chi-squared distribution, we can find the `p-value` accompanying our test statistic value. If `p-value` exceeds the critical value of the Chi-squared distribution, we do have sufficient evidence to reject the null hypothesis that the model is reasonable. Or we can say, in that case, the model is considered as inaccurate.\n\nFor example, assume that we calculate VaR 5% (the confidence level of the VaR model is 95%) and get value VaR = 2.26. We also observed 50 exceptions over 500 time windows. Using the formula above, the test statistic `p-value` is calculated and equal to `8.08`. Compared to `3.84`, the critical value of Chi-squared distribution with one degree of freedom at probability 5%, the test statistic is larger. So, the model is rejected. The critical values of Chi-squared can be found by following [this link](https://people.richland.edu/james/lecture/m170/tbl-chi.html).\nHowever, in this Notebook, it's not a good idea to find the corresponding critical value by looking in a \"messy\" table, especially when we need to change the confidence level. Instead, from `p-value`, we will calculate the probability of the test statistic in Chi-square thanks to some functions in package `scipy`. If the calculated probability is smaller than the quantile of loss (e.g, 0.05), the model is rejected and vice versa.\n\n\n\n\n### Question 8\n\n#### Question 8.1\n
                                        \n\nWrite a function to calculate the number of failures, that is when the losses (in the original data) exceed the VaR.\n\n
                                        \n\n
                                        HINT
                                        \n
                                          \n
                                        • First, we need to calculate the total loss in each 2-week time interval
                                          • \n
                                        • If the total loss of a time interval exceeds VaR, then we say that our VaR fails to estimate the risk in that time interval
                                          • \n
                                        • Return the number of failures
                                          • \n
                                        \n\n
                                        NOTE
                                        The loss is often having negative value, so, be careful when compare it to VaR.\n\n\n```python\nfrom scipy import stats\nimport math\n\ndef countFailures(stocksReturns, valueAtRisk):\n failures = 0\n # iterate over time intervals\n for i in range(0, len(stocksReturns[0])):\n # calculate the losses in each time interval\n loss = sum([value[i] for value in stocksReturns])\n \n # if the loss exceeds VaR\n if loss < valueAtRisk:\n failures += 1\n return failures\n```\n\n#### Question 8.2\n
                                        \n\nWrite a function named `kupiecTestStatistic` to calculate the test statistic which was described in the above equation.\n\n
                                        \n\n\n```python\ndef kupiecTestStatistic(total, failures, confidenceLevel):\n failureRatio = failures/total\n logNumer = (total - failures) * np.log(1 - confidenceLevel) + failures * np.log(confidenceLevel)\n logDenom = (total - failures) * np.log(1 - failureRatio) + failures * np.log(failureRatio)\n return -2 * (logNumer - logDenom)\n \n# test the function\nassert (round(kupiecTestStatistic(250, 36, 0.1), 2) == 4.80), \"function kupiecTestStatistic runs incorrectly\"\n```\n\nNow we can find the p-value accompanying our test statistic value.\n\n\n```python\ndef kupiecTestPValue(stocksReturns, valueAtRisk, confidenceLevel):\n failures = countFailures(stocksReturns, valueAtRisk)\n print(\"num failures:\", failures)\n if failures == 0:\n # the model is very good\n return 1\n total = len(stocksReturns[0])\n testStatistic = kupiecTestStatistic(total, failures, confidenceLevel)\n #return 1 - stats.chi2.cdf(testStatistic, 1.0)\n return stats.chisqprob(testStatistic, 1.0)\n\nvarConfidenceInterval = bootstrappedConfidenceInterval(trials, fivePercentVaR, 100, 0.05)\ncvarConfidenceInterval = bootstrappedConfidenceInterval(trials, fivePercentCVaR, 100, .05)\nprint(\"VaR confidence interval: \" , varConfidenceInterval)\nprint(\"CVaR confidence interval: \" , cvarConfidenceInterval)\nprint(\"Kupiec test p-value: \" , kupiecTestPValue(stocksReturns, valueAtRisk, 0.05))\n```\n\n VaR confidence interval: (-615.64631098428345, -624.82407827385191)\n CVaR confidence interval: (-999.73198389203014, -963.96432441143213)\n num failures: 241\n Kupiec test p-value: 7.55594138964e-69\n\n\n#### Question 8.3\n
                                        \n\nDiscuss the results you have obtained\n\n
                                        \n\n
                                        \nCOMMENT:
                                        \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Investment in 35 stocks
                                        VaR confidence interval(-19.954194600895239, -19.936524213023823)
                                        CVaR confidence interval(-24.93940415592181, -25.805224770328739)
                                        Num failures104 - 1.04%
                                        Kupiec test p-value3.87023503002e-06
                                        \n
                                        \n*Now, what does this tell us?*
                                        \nThe range in the VaR confidence interval is 0.0177 meaning that the 95% of the values will be close to our predictions.

                                        \n\nHowever, since the p-value is far from 0.05 (the minimum value required for a good model) probably because of simplifications, our model has a high bias, and thus it doesn't represent the real world results with accuracy.\n
                                        \n\n\n\n### Question 9\n
                                        \nAssume that we invest in more than 100 stocks. Use the same market factors as for the previous questions to estimate VaR by running MCS, then validate your result. \n\nWhat is the main observation you have, once you answer this question? When you plan to invest in more instruments, how is your ability to predict the risk going to be affected?\n
                                        \n\n\n
                                        \nCOMMENT:
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Investment in 100 stocks
                                        VaR confidence interval(-696.91913934748061, -696.12843994443017)
                                        CVaR confidence interval(-1012.7641215060884, -977.57349530421345)
                                        Num failures214 - 2.14%
                                        Kupiec test p-value2.14454989582e-52
                                        \n
                                        \n*Now, what does this tell us?*
                                        \nThe Var increased in -21 to -696, and all the other values are also increased substantially -> much higher error rates. In short, increasing our portfolio size really affects our variance negatively,increasing the risk considerably. If we already had a bad model, using it masively is a definitely a bad idea.\n

                                        \nNOTE: we re-ran all the sections in the notebook with a hundred instruments. This changes slightly the results in some sections, which we modified accordingly.\n\n
                                        \n\n\n\n### Question 10\n
                                        \n\nIn the previous questions, we used the normal distributions to sample the factors returns. \n\nTry to study how results vary when selecting other probability distributions: our goal is to improve the result of our MCS.\n
                                        \n\n
                                        \nCOMMENT:
                                        \nThere are several combinations to be considered:\n
                                          \n
                                        • Triangular: oversimplifies the variance, thus it's hard to fit distributions in it. Not very suitable.
                                          • \n
                                        • Uniform: Never! You will be losing all the information from the distribution. Only usable in case we had absolutely no information from factors.
                                          • \n
                                        • Normal: since most of the factors come from values ranging 0 to infinite, it's more corect to ose lognormal instead (otherwise we would have to filter negative values).
                                          • \n
                                        \n

                                        \nAlso there is a problem with most random variables implemented in numpy: they don't have multivariate option!\n

                                        \nOur best solution would be to implement a linear combination of two (or more) gaussians. In order to do that, the best approach is to look at the distributions on the factors and try to emmulate what we see (check section 6 for details). Our best candidate for a sumation of two gaussians is the US Treasury data, with a small-variation gaussian and a high-variation one. It would be as simple as having two models, one with the first low covariance and the other with the highest covariance, and then do the average of both values.\n

                                        \nRegrettably, we didn't have the time to implement it. Nevertheless, we leave a comment on where should it be changed.\n
                                        \n\n\n```python\ndef simulateTrialReturnsTriang(numTrials, factorMeans, factorCov, weights):\n trialReturns = []\n signFactorMeans = np.sign(factorMeans)\n absFactorMeans = np.absolute(factorMeans)\n \n for i in range(0, numTrials):\n # generate sample of factors' returns\n trialFactorReturns = np.exp(np.random.triangular(np.log(absFactorMeans), factorCov))\n # HERE WE SHOULD ADD A SECOND LINE WITH A CHANGE IN THE factorCov only for the USTreasury\n # HERE WE SHOULD AVERAGE BOTH MODELS.\n trialFactorReturns = trialFactorReturns*signFactorMeans\n \n # featurize the factors' returns\n trialFeatures = featurize(trialFactorReturns.tolist())\n \n # insert weight for intercept term\n trialFeatures.insert(0,1)\n \n # calculate the return of each instrument\n # then calulate the total of return for this trial features\n trialTotalReturn = sum(np.dot(weights,trialFeatures).tolist())\n \n trialReturns.append(trialTotalReturn)\n return trialReturns\n\n```\n\n# 6 Adding Extra Factors\nWe wanted to exsperimebt with several other factors. We went for this three:\n* Copper\n* Gold \n* Lead\n\nBeneath you can see the different distributions and the correlation+impact on the instruments. Finally, we analyze the VaR with all of them.\n\n\n```python\nf, axarr = plt.subplots(4, 2)\nf.set_figheight(15)\nf.set_figwidth(15)\n\ndef plotDistribution(samples, a, b):\n vmin = min(samples)\n vmax = max(samples)\n stddev = np.std(samples)\n \n domain = np.arange(vmin, vmax, (vmax-vmin)/100)\n \n # a simple heuristic to select bandwidth\n bandwidth = 1.06 * stddev * pow(len(samples), -.2)\n \n # estimate density\n kde = KDEUnivariate(samples)\n kde.fit(bw=bandwidth)\n density = kde.evaluate(domain)\n \n # plot\n axarr[a, b].plot(domain, density)\n \nplotDistribution(factorsReturns[0], 0, 0)\nplotDistribution(factorsReturns[1], 0, 1)\nplotDistribution(factorsReturns[2], 1, 0)\nplotDistribution(factorsReturns[3], 1, 1)\nplotDistribution(factorsReturns[4], 2, 0)\nplotDistribution(factorsReturns[5], 2, 1)\nplotDistribution(factorsReturns[6], 3, 0)\nplt.show()\n```\n\n
                                        \nCOMMENT:
                                        \nThe distributions are in this order (left to right, up to bottom): 'Crudeoil', 'USTreasury', 'GSPC', 'IXIC', 'GOLD', 'COPPER', 'LEAD'\n

                                        \nClearly the distribution that can be turned into a double gaussian model is the USTreasury (for previous section).\n
                                        \n\n\n```python\ncorrelation = np.corrcoef(factorsReturns)\nfactor_names = ['Crudeoil', 'USTreasury', 'GSPC', 'IXIC', 'GOLD', 'COPPER', 'LEAD']\ncorrelation\nplt.figure(figsize=[12,10])\nsns.heatmap(correlation, \n xticklabels=factor_names,\n yticklabels=factor_names, annot=True)\n\n```\n\n
                                        \nCOMMENT:
                                        \nMost of the values are uncorrelated with each other, except copper and lead that seem to have a high correlation.\n
                                        \n\n\n```python\nimport pandas as pd\nimport seaborn as sns\nimport matplotlib.pyplot as plt\nimport warnings\nwarnings.filterwarnings('ignore')\n\n# Generate a mask for the upper triangle\nmask = np.zeros_like(correlation, dtype=np.bool)\nmask[np.triu_indices_from(mask)] = True\n\n# Set up the matplotlib figure\nf, ax = plt.subplots(figsize=[20,15])\n\nabsWeights = transpose(list(map(lambda x: list(map(abs ,x.tolist())), weights)))\n\n# Draw the heatmap with the mask and correct aspect ratio\nsns.heatmap(absWeights, cmap=\"YlGnBu\", vmax=1.0,square=True, xticklabels=5, yticklabels=2,linewidths=.5,\n cbar_kws={\"orientation\": \"horizontal\"}, ax=ax)\nax.set_title('Correlation between Factors (13, after expansion) and instruments (75)', fontsize=17)\nax.set_xlabel('Instruments', fontsize=15)\nax.set_ylabel('Factors', fontsize=15)\n```\n\n
                                        \nCOMMENT:
                                        \nAs we can see, most of them have very low correlation to the actual instruments we are considering. Nevertheless, the squared version of them does affect some certain instruments.

                                        \nOur Var results are lower than before:
                                        \nValue at Risk(VaR) 5: -1056.852
                                        \nConditional Value at Risk(CVaR) 5: -1396.144
                                        \nVaR confidence interval: (-1059.0618529045107, -1042.1359638067859)
                                        \nCVaR confidence interval: (-1351.1185455516584, -1416.9473957553221)
                                        \nnum failures: 117
                                        \nKupiec test p-value: 1.78525505493e-09
                                        \n
                                        \n\n# 7. Summary\nIn this lecture, we studied the Monte Carlo Simulation method and its application to estimate financial risk. To apply it, first, we needed to define the relationship between market factors and the instruments' returns. In such step, you must define the model which maps the market factors' values to the instruments' values: in our use case, we used a linear regression function for building our model. Next, we also had to find the parameters of our model, which are the weights of the factors we considered. Then, we had to study the distribution of each market factor. A good way to do that is using Kernel density estimation to smooth the distribution and plot it. Depending on the shape of each figure, we had to guess the best fit distribution for each factor: in our use case, we used a very simple approach, and decided that our smoothed distributions all looked normal distributions. \n\nThen, the idea of Monte Carlo simulation was to generate many possible values for each factor and calculate the corresponding outcomes by a well-defined model in each trial. After many trials, we were able to calculate VaR from the sequences of outcome's values. When the number of trials is large enough, the VaR converges to reasonable values, that we could validate using well-known statistical hypothesis. \n\n# References\n- The example in section 2 is inspired from [this article](http://www.solver.com/monte-carlo-simulation-example).\n- [Backtesting Value-at-Risk models](https://aaltodoc.aalto.fi/bitstream/handle/123456789/181/hse_ethesis_12049.pdf?sequence=1) (Kansantaloustiede, 2009) - (A good reference to study Backtesting).\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "ed1233c7b4222e2166f0bbcaf658c0f0305cdb2a", "size": 415054, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "3_MonteCarlo_FinancialRisk/MonteCarlo_FinancialRisk.ipynb", "max_stars_repo_name": "AlbertoIbarrondoLuis/AlgorithmicMachineLearning", "max_stars_repo_head_hexsha": "3f9abdad93b5e7df19be6a9ed4c72a88a94c485b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2019-02-08T15:07:22.000Z", "max_stars_repo_stars_event_max_datetime": "2020-05-12T22:18:27.000Z", "max_issues_repo_path": "3_MonteCarlo_FinancialRisk/MonteCarlo_FinancialRisk.ipynb", "max_issues_repo_name": "ibarrond/AlgorithmicMachineLearning", "max_issues_repo_head_hexsha": "3f9abdad93b5e7df19be6a9ed4c72a88a94c485b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "3_MonteCarlo_FinancialRisk/MonteCarlo_FinancialRisk.ipynb", "max_forks_repo_name": "ibarrond/AlgorithmicMachineLearning", "max_forks_repo_head_hexsha": "3f9abdad93b5e7df19be6a9ed4c72a88a94c485b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2019-01-06T16:42:17.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-02T06:13:08.000Z", "avg_line_length": 165.2285031847, "max_line_length": 101518, "alphanum_fraction": 0.8631310625, "converted": true, "num_tokens": 22114, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.7956581024858786, "lm_q1q2_score": 0.43193356979828973}} {"text": "###### Text provided under a Creative Commons Attribution license, CC-BY. All code is made available under the FSF-approved MIT license. (c) Daniel Koehn based on Jupyter notebooks by Marc Spiegelman [Dynamical Systems APMA 4101](https://github.com/mspieg/dynamical-systems) and Kyle Mandli from his course [Introduction to numerical methods](https://github.com/mandli/intro-numerical-methods), notebook style sheet by L.A. Barba, N.C. Clementi [Engineering Computations](https://github.com/engineersCode)\n\n\n```python\n# Execute this cell to load the notebook's style sheet, then ignore it\nfrom IPython.core.display import HTML\ncss_file = '../style/custom.css'\nHTML(open(css_file, \"r\").read())\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n# Stationary Solutions of Time-Dependent Problems\n\nOften, we do not want to compute the whole time evolution of a dynamical problem, but are only interested in **stationary solutions**, where the system reached an equilibrium state. These stationary solutions are estimated by setting all time derivatives in the differential equation to zero. As examples, we will compute the stationary solutions for streamlines from [Exercise: How to sail without wind](https://nbviewer.jupyter.org/github/daniel-koehn/Differential-equations-earth-system/blob/master/02_finite_difference_intro/3_fd_ODE_example_sailing_wo_wind.ipynb) and the Lorenz equations.\n\n## Stationary Solutions of Streamlines\n\nIn [Exercise: How to sail without wind](https://nbviewer.jupyter.org/github/daniel-koehn/Differential-equations-earth-system/blob/master/02_finite_difference_intro/3_fd_ODE_example_sailing_wo_wind.ipynb), we computed streamlines for a given velocity vector field ${\\bf{V}} = (v_x,v_y)^T$, starting from an initial position ${\\bf{x_0}} = (x_0,y_0)^T$. The governing equations related componentwise the velocity field to the spatial coordinates ${\\bf{x}} = (x,y)^T$ by \n\n\\begin{equation}\nv_x = \\frac{dx}{dt},\\; v_y = \\frac{dy}{dt} \\tag{1}\n\\end{equation}\n\nTo find the stationary solutions of differential eqs. (1), we have to set the time derivatives on the RHS to zero $\\frac{dx}{dt}=0$, $\\frac{dy}{dt}=0$ leading to \n\n\\begin{equation}\nv_x = 0,\\; v_y = 0 \\tag{2}\n\\end{equation}\n\nEqs. (2) simply state that a stationary solution is located at points, where the velocity field is zero or we can not change the system state, because no fluid flow exists. To explicitly calculate the stationary solutions of eqs.(1), also known as **Fixpoints**, we have to explicitly define the velocity field ${\\bf{V}} = (v_x,v_y)^T$. \n\n##### Exercise 1\n\nCompute the fixpoints ${\\bf{x_{fix}}} = (x_{fix},y_{fix})^T$ for the velocity fields \n\n\\begin{equation}\n{\\bf{V_1}} = (y,-x)^T \\notag\n\\end{equation}\n\nand\n\n\\begin{equation}\n{\\bf{V_2}}=(cos((x+1y)/500),sin((x-1y)/500))^T \\notag\n\\end{equation}\n\n- How many fixpoints exist for each velocity field $\\bf{V_1}$, $\\bf{V_2}$? \n- Mark all fixpoint positions on the square $-1000 \\le x \\le 1000;\\; -1000 \\le y \\le 1000$ by red dots in the vector plots below.\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport matplotlib as mpl\nfrom pylab import rcParams\n\n# Ignore Warning Messages\n# -----------------------\nimport warnings\nwarnings.filterwarnings(\"ignore\")\n\n# define font size\nFSize = 18\nfont = {'color': 'black',\n 'weight': 'normal',\n 'size': FSize}\nmpl.rc('xtick', labelsize=FSize) \nmpl.rc('ytick', labelsize=FSize)\n```\n\n\n```python\n# define functions to compute the vector fields V1 and V2\n# compute velocity components V = (vx,vy)^T at position x,y\ndef vel_xy_1(x,y):\n \n vx = y / 1000.\n vy = -x / 1000.\n \n return vx, vy\n\ndef vel_xy_2(x,y):\n \n vx = np.cos((x+1*y)/500)\n vy = np.sin((x-1*y)/500) \n \n return vx, vy\n```\n\n\n```python\n# Plot velocity vector fields\n# Define figure size\nrcParams['figure.figsize'] = 9, 4\n\n# Define coordinates\ndh = 100.\nx1 = -1000.\nx2 = 1000.\nX, Y = np.meshgrid(np.arange(x1, x2, dh), np.arange(x1, x2, dh))\n\n# Plot vector field V1\n# --------------------\nax1 = plt.subplot(1, 2, 1)\n\n# Define vector field components for coordinates X,Y\nVX1,VY1 = vel_xy_1(X,Y)\n\nax1.set_title(r'$V=(y,-x)^T/1000$')\nplt.axis('equal')\nQ = plt.streamplot(X,Y,VX1,VY1)\n# PLOT FIXPOINT POSITIONS HERE!\n\nplt.xlabel('x [m]')\nplt.ylabel('y [m]')\n\n# Plot vector field V2\n# --------------------\nax2 = plt.subplot(1, 2, 2)\n\n# Define vector field components for coordinates X,Y\nVX2,VY2 = vel_xy_2(X,Y)\n\nax2.set_title(r'$V=(cos((x+1y)/500),sin((x-1y)/500))^T$')\nplt.axis('equal')\nQ = plt.quiver(X,Y,VX2,VY2)\n# PLOT FIXPOINT POSITIONS HERE!\n\nplt.xlabel('x [m]')\nplt.ylabel('y [m]')\n\nplt.tight_layout()\nplt.show()\n```\n\n##### Exercise 2\n\nCompute the fixpoints ${\\bf{x_{fix}}} = (X_{fix},Y_{fix},Z_{fix})^T$ for the Lorenz equations:\n\n\\begin{equation}\n\\begin{split}\n\\frac{\\partial X}{\\partial t} &= \\sigma( Y - X)\\\\\n\\frac{\\partial Y}{\\partial t} &= rX - Y - XZ \\\\\n\\frac{\\partial Z}{\\partial t} &= XY -b Z\n\\end{split}\n\\notag\n\\end{equation}\n\nwhere $\\sigma$ denotes the \"Prandtl number\", $r = \\mathrm{Ra}/\\mathrm{Ra}_c$ is a scaled \"Rayleigh number\" and $b$ is a parameter that is related to the the aspect ratio of a convecting cell in the original derivation.\n\n- How many fixpoints exist? \n- Use the fixpoint solutions $X_{fix}$, $Y_{fix}$ and $Z_{fix}$ to plot the corresponding streamfunction $\\psi(x,z)$, temperature field $T(x,z)$ and velocity vector field ${\\bf{v}}$\n\n\\begin{equation}\n\\begin{split}\n\\psi(x,z) &= X_{fix} \\sin(a\\pi x)\\sin(\\pi z)\\\\\nT(x,z) &= Y_{fix} \\cos(a\\pi x)\\sin(\\pi z) - Z_{fix} \\sin(2\\pi z) + (1-z)\\\\\n{\\bf{v}} &=(u,0,w) = {\\bf{\\nabla\\times\\psi\\hat{j}}}=(-\\frac{\\partial\\psi}{\\partial z}, 0, \\frac{\\partial\\psi}{\\partial x})^T\\\\\n\\end{split}\n\\notag\n\\end{equation}\n\nfor $b=8/3$ and scaled Rayleigh numbers $r=0.5$ and $r=28$, respectively.\n\n- Describe and interpret the different Fixpoint solutions\n\n\n```python\n# Define coordinates\na = np.sqrt(0.5)\nx0 = np.linspace(0,1./a,20)\nz0 = np.linspace(0.,1.,20)\nx,z = np.meshgrid(x0,z0)\n\n# Define spatial part of streamfunction and temperature field\npsi = np.sin(a*np.pi*x)*np.sin(np.pi*z)\ntheta0 = np.cos(a*np.pi*x)*np.sin(np.pi*z)\ntheta1 = -np.sin(2.*np.pi*z)\n```\n\n## Fixpoints of Lorenz Equations (r = 0.5)\n\nPLOT VECTORFIELD $V = (U,V)^T$ AND TEMPERATURE $T$ FOR ALL FIXPOINT SOLUTIONS WITH $r=0.5$ HERE! PRODUCE A NEW PLOT FOR EACH FIXPOINT!\n\n\n```python\n# Define figure size\nplt.figure()\nplt.figure(figsize=(20,20))\n\n# DEFINE FIXPOINT HERE!\nb = 8/3\nr = 0.5\n\nXfix = \nYfix = \nZfix = \n\n# Initial Streamfunction psi and velocity field\nplt.subplot(1,2,1)\nplt.contourf(x,z,Xfix*psi,cmap='viridis_r')\n\n# Velocity field\nU = - Xfix * np.pi * np.sin(a*np.pi*x) * np.cos(np.pi*z)\nV = Xfix * a * np.pi * np.cos(a*np.pi*x) * np.sin(np.pi*z)\nplt.quiver(x,z,U,V)\nplt.gca().set_aspect('equal')\nplt.title('Streamfunction $\\psi$', fontdict=font)\n\n# Initial temperature field \nplt.subplot(1,2,2)\nplt.contourf(x,z,Yfix*theta0 + Zfix*theta1 + (1-z),cmap='magma')\nplt.gca().set_aspect('equal')\nplt.title(r'Temperature $T$', fontdict=font)\n\nplt.tight_layout()\nplt.show()\n```\n\n## Fixpoints of Lorenz Equations (r = 28)\n\nPLOT VECTORFIELD $V = (U,V)^T$ AND TEMPERATURE $T$ FOR ALL FIXPOINT SOLUTIONS WITH $r=28$ HERE! PRODUCE A NEW PLOT FOR EACH FIXPOINT!\n\n\n```python\n# Define figure size\nplt.figure()\nplt.figure(figsize=(20,20))\n\n# DEFINE FIXPOINT HERE!\nb = 8/3\nr = 28\n\nXfix =\nYfix =\nZfix =\n\n# Initial Streamfunction psi and velocity field\nplt.subplot(1,2,1)\nplt.contourf(x,z,Xfix*psi,cmap='viridis_r')\n\n# Velocity field\nU = - Xfix * np.pi * np.sin(a*np.pi*x) * np.cos(np.pi*z)\nV = Xfix * a * np.pi * np.cos(a*np.pi*x) * np.sin(np.pi*z)\nplt.quiver(x,z,U,V)\nplt.gca().set_aspect('equal')\nplt.title('Streamfunction $\\psi$', fontdict=font)\n\n# Initial temperature field \nplt.subplot(1,2,2)\nplt.contourf(x,z,Yfix*theta0 + Zfix*theta1 + (1-z),cmap='magma')\nplt.gca().set_aspect('equal')\nplt.title(r'Temperature $T$', fontdict=font)\n\nplt.tight_layout()\nplt.show()\n```\n\n## What we learned:\n\n- How to compute stationary solutions (fixpoints) of differential equations. \n", "meta": {"hexsha": "35aa7f0d37c7ecc8882ac98e46ecd9fabceefe6b", "size": 140711, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "03_Lorenz_equations/02_Stationary_solutions_of_DE.ipynb", "max_stars_repo_name": "daniel-koehn/Differential-equations-earth-system", "max_stars_repo_head_hexsha": "3916cbc968da43d0971b7139476350c1dd798746", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 30, "max_stars_repo_stars_event_min_datetime": "2019-10-16T19:07:36.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-10T03:48:44.000Z", "max_issues_repo_path": "03_Lorenz_equations/02_Stationary_solutions_of_DE.ipynb", "max_issues_repo_name": "daniel-koehn/Differential-equations-earth-system", "max_issues_repo_head_hexsha": "3916cbc968da43d0971b7139476350c1dd798746", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "03_Lorenz_equations/02_Stationary_solutions_of_DE.ipynb", "max_forks_repo_name": "daniel-koehn/Differential-equations-earth-system", "max_forks_repo_head_hexsha": "3916cbc968da43d0971b7139476350c1dd798746", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2020-11-19T08:21:55.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-10T09:33:37.000Z", "avg_line_length": 263.0112149533, "max_line_length": 122884, "alphanum_fraction": 0.911357321, "converted": true, "num_tokens": 3392, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166047041654, "lm_q2_score": 0.8438950947024555, "lm_q1q2_score": 0.4318351325876406}} {"text": "# Paralelismo en Julia\n\n## 1. Introducci\u00f3n\n\nJulia proporciona distintos mecanismos para paralelizar c\u00f3digo. Algunas de las estrategias y desaf\u00edos para escribir algoritmos paralelos son los siguientes: \n\n* Estrategias de paralelismo\n * SIMD\n * Multi-hilo\n * Tareas\n * Multiproceso\n * Memoria compartida\n * Memoria distribuida\n * Programaci\u00f3n de GPU\n\n* Desaf\u00edos de la computaci\u00f3n paralela\n * Orden de ejecuci\u00f3n\n * ejecuci\u00f3n de fuera de orden de posibilidad\n * acceso y mutaci\u00f3n simult\u00e1neos\n * Acceso y movimiento de datos\n * C\u00f3digo de acceso y movimiento\n * Adaptaci\u00f3n adecuada de la estrategia de paralelismo a las capacidades de su m\u00e1quina\n * Hacer coincidir adecuadamente la estrategia de paralelismo con el problema en cuesti\u00f3n\n\n## \u00bfQu\u00e9 es lo que est\u00e1 sucediendo con nuestras computadoras?\n\n\n\n\n## Lo dif\u00edcil de la computaci\u00f3n paralela\n * No pensamos en paralelo\n * Aprendemos a escribir y razonar sobre programas en serie.\n * El deseo de paralelismo a menudo surge _despu\u00e9s_ de haber escrito su algoritmo (\u00a1y lo encontr\u00f3 demasiado lento!)\n\n## Resumen:\n * Las arquitecturas computacionales actuales nos empujan hacia la programaci\u00f3n paralela para un rendimiento m\u00e1ximo, \u00a1incluso si no estamos en un cl\u00faster!\n * Pero es dif\u00edcil dise\u00f1ar buenos algoritmos paralelos\n * Es dif\u00edcil de expresar y razonar sobre esos algoritmos.\n\n## 2. SIMD: El paralelismo que puede (a veces) suceder autom\u00e1ticamente\n\nSIMD: Instrucci\u00f3n \u00fanica, datos m\u00faltiples (Single Instruction Multiple Data)\n\n**Nota:** Tambi\u00e9n llamado confusamente vectorizaci\u00f3n\n\n### Arquitectura\n\nEn lugar de calcular cuatro sumas secuencialmente:\n\n\\begin{align}\nx_1 + y_1 &\\rightarrow z_1 \\\\\nx_2 + y_2 &\\rightarrow z_2 \\\\\nx_3 + y_3 &\\rightarrow z_3 \\\\\nx_4 + y_4 &\\rightarrow z_4\n\\end{align}\n\nProcesadores modernos tienen unidades de procesamiento vectorial que pueden hacer lo anterior a la vez:\n\n$$\n\\left(\\begin{array}{cc}\nx_1 \\\\\nx_2 \\\\\nx_3 \\\\\nx_4\n\\end{array}\\right)\n+\n\\left(\\begin{array}{cc}\ny_1 \\\\\ny_2 \\\\\ny_3 \\\\\ny_4\n\\end{array}\\right)\n\\rightarrow\n\\left(\\begin{array}{cc}\nz_1 \\\\\nz_2 \\\\\nz_3 \\\\\nz_4\n\\end{array}\\right)\n$$\n\n### \u00bfC\u00f3mo se logra?\n\n\n```julia\nusing BenchmarkTools\n```\n\n\n```julia\nA = rand(100_000)\nfunction simplesum(A)\n result = zero(eltype(A))\n for i in eachindex(A)\n @inbounds result += A[i]\n end\n return result\nend\n\nsimplesum(A)\n```\n\nComo muchos lenguajes de programaci\u00f3n modernos, Julia utiliza la verificaci\u00f3n de l\u00edmites ([_boundchecking_](https://docs.julialang.org/en/v1/devdocs/boundscheck/)) para garantizar la seguridad del programa al acceder a arreglos.\nEn bucles internos u otras situaciones cr\u00edticas de rendimiento, es posible que se desee omitir estas comprobaciones de l\u00edmites para mejorar el rendimiento en tiempo de ejecuci\u00f3n.\n\nEn consecuencia, Julia incluye la macro `@inbounds(...)` para decirle al compilador que omita dichas comprobaciones de l\u00edmites dentro del bloque dado.\n\n\n```julia\n@btime simplesum($A)\n```\n\n\u00bfese tiempo es bueno?\n\n\n```julia\n@btime sum($A)\n```\n\nDise\u00f1amos una funci\u00f3n m\u00e1s lenta que la suma predise\u00f1ada `sum()`, \u00a1y tambi\u00e9n estamos obteniendo una respuesta diferente! Veamos qu\u00e9 sucede con un flotante de 32 bits en lugar de uno de 64 bits. Cada elemento tiene la mitad del n\u00famero de bits, por lo que tambi\u00e9n permite duplicar la longitud (para que el n\u00famero total de bits procesados permanezca constante).\n\n\n```julia\nA32 = rand(Float32, length(A)*2)\n@btime simplesum($A32)\n@btime sum($A32)\n```\n\n\u00a1Eso es aun peor! \u00bfQue est\u00e1 pasando aqui? \nEstamos viendo m\u00faltiples diferencias en el desempe\u00f1o: \u00bfquiz\u00e1s la suma incorporada de Julia est\u00e1 usando alg\u00fan paralelismo?\n\nIntentemos usar SIMD nosotros mismos:\n\n\n```julia\nfunction simdsum(A)\n result = zero(eltype(A))\n @simd for i in eachindex(A)\n @inbounds result += A[i]\n end\n return result\nend\n@btime simdsum($A)\n@btime simdsum($A32)\n```\n\n\u00bfQu\u00e9 hizo y por qu\u00e9 no siempre usamos (usa Julia pues) `@simd` para cada bucle **for** autom\u00e1ticamente?\n\nVeamos los resultados:\n\n\n```julia\nsimplesum(A), simdsum(A), sum(A)\n```\n\n\n```julia\nsimplesum(A32), simdsum(A32), sum(A32)\n```\n\n\u00bfPor qu\u00e9 no son iguales?\n\nSin `@simd`, Julia est\u00e1 haciendo _exactamente_ lo que le dijimos que hiciera: est\u00e1 tomando cada elemento de nuestro arreglo y lo agrega a una gran pila secuencialmente. Nuestra respuesta es m\u00e1s peque\u00f1a de lo que la \"suma\" incorporada de Julia cree que es: eso es porque, como la pila se hace m\u00e1s grande, comenzamos a perder las partes inferiores de cada elemento que estamos sumando, \u00a1y esas peque\u00f1as p\u00e9rdidas comienzan a acumularse!\n\nLa macro `@simd` le dice a Julia que puede reorganizar las adiciones de punto flotante -\nincluso si cambiara la respuesta. Dependiendo de su CPU, esto puede llevar a 2x o 4x\no incluso un paralelismo 8x. B\u00e1sicamente, Julia est\u00e1 calculando sumas independientes para\nlos \u00edndices pares y los \u00edndices impares simult\u00e1neamente:\n\n\\begin{align}\nodds &\\leftarrow 0 \\\\\nevens &\\leftarrow 0 \\\\\n\\text{loop}&\\ \\text{odd}\\ i: \\\\\n &\\left(\\begin{array}{cc}\nodds \\\\\nevens\n\\end{array}\\right)\n\\leftarrow\n\\left(\\begin{array}{cc}\nodds \\\\\nevens\n\\end{array}\\right)\n+\n\\left(\\begin{array}{cc}\nx_{i} \\\\\nx_{i+1}\n\\end{array}\\right) \\\\\ntotal &\\leftarrow evens + odds\n\\end{align}\n\n\nEn muchos casos, Julia puede y sabe que un bucle for puede ser vectorizado (SIMD-ed) y aprovechar\u00e1 esto por defecto.\n\n\n```julia\nB = rand(1:10, 100_000)\n@btime simplesum($B)\n@btime sum($B)\nB32 = rand(Int32(1):Int32(10), length(B)*2)\n@btime simplesum($B32)\n@btime simdsum($B32)\n```\n\n\u00bfC\u00f3mo inspeccionamos que se est\u00e1 vectorizando?\n\n\n```julia\n@code_llvm simdsum(A32)\n```\n\nEntonces, \u00bfcu\u00e1les son los desaf\u00edos?:\n\n- El mayor obst\u00e1culo es que tienes que convencer a Julia y LLVM de que puede usar instrucciones SIMD para tu algoritmo dado. Eso no siempre es posible.\n- Hay muchas limitaciones de lo que se puede y no se puede vectorizar\n- Es necesario pensar en las consecuencias de reordenar su algoritmo\n\n## Resumen\n\nSIMD:\n- Explota el paralelismo integrado en un procesador\n- Ideal para bucles internos peque\u00f1os (y estrechos)\n- A menudo ocurre autom\u00e1ticamente si tienes cuidado\n - Sigue las [mejores pr\u00e1cticas de rendimiento](https://docs.julialang.org/en/v1/manual/performance-tips/)\n - Usa `@inbounds` para cualquier acceso a un arreglo\n - Evita ramas o llamadas a funciones\n- Dependiendo del procesador y los tipos involucrados, puede producir ganancias de 2-16x con una sobrecarga extraordinariamente peque\u00f1a\n\n\n\n```julia\n\n```\n", "meta": {"hexsha": "8da742f0afcc8abb6a54fd4ac5e1141a8f04698b", "size": 10859, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Curso Julia/Github - Material/src/Introduccion/S8-OperacionesVectorizadas.ipynb", "max_stars_repo_name": "antonyayal/ProyectoFinalCursoJuliaAgo21", "max_stars_repo_head_hexsha": "79d199175ae9fbadb8bf90e685f34bc808efdff5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Curso Julia/Github - Material/src/Introduccion/S8-OperacionesVectorizadas.ipynb", "max_issues_repo_name": "antonyayal/ProyectoFinalCursoJuliaAgo21", "max_issues_repo_head_hexsha": "79d199175ae9fbadb8bf90e685f34bc808efdff5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Curso Julia/Github - Material/src/Introduccion/S8-OperacionesVectorizadas.ipynb", "max_forks_repo_name": "antonyayal/ProyectoFinalCursoJuliaAgo21", "max_forks_repo_head_hexsha": "79d199175ae9fbadb8bf90e685f34bc808efdff5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 28.7275132275, "max_line_length": 444, "alphanum_fraction": 0.5757436228, "converted": true, "num_tokens": 1932, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.8311430457670241, "lm_q1q2_score": 0.4317965338681733}} {"text": "#
                                        Models and Pricing of Financial Derivatives HW_02
                                        \n\n**
                                        11510691 \u7a0b\u8fdc\u661f$\\DeclareMathOperator*{\\argmin}{argmin}\n\\DeclareMathOperator*{\\argmax}{argmax}\n\\DeclareMathOperator*{\\plim}{plim}\n\\newcommand{\\using}[1]{\\stackrel{\\mathrm{#1}}{=}}\n\\newcommand{\\ffrac}{\\displaystyle \\frac}\n\\newcommand{\\asim}{\\overset{\\text{a}}{\\sim}}\n\\newcommand{\\space}{\\text{ }}\n\\newcommand{\\bspace}{\\;\\;\\;\\;}\n\\newcommand{\\QQQ}{\\boxed{?\\:}}\n\\newcommand{\\void}{\\left.\\right.}\n\\newcommand{\\Tran}[1]{{#1}^{\\mathrm{T}}}\n\\newcommand{\\d}[1]{\\displaystyle{#1}}\n\\newcommand{\\CB}[1]{\\left\\{ #1 \\right\\}}\n\\newcommand{\\SB}[1]{\\left[ #1 \\right]}\n\\newcommand{\\P}[1]{\\left( #1 \\right)}\n\\newcommand{\\abs}[1]{\\left| #1 \\right|}\n\\newcommand{\\norm}[1]{\\left\\| #1 \\right\\|}\n\\newcommand{\\dd}{\\mathrm{d}}\n\\newcommand{\\Exp}{\\mathrm{E}}\n\\newcommand{\\RR}{\\mathbb{R}}\n\\newcommand{\\EE}{\\mathbb{E}}\n\\newcommand{\\NN}{\\mathbb{N}}\n\\newcommand{\\ZZ}{\\mathbb{Z}}\n\\newcommand{\\QQ}{\\mathbb{Q}}\n\\newcommand{\\AcA}{\\mathscr{A}}\n\\newcommand{\\FcF}{\\mathscr{F}}\n\\newcommand{\\Var}[2][\\,\\!]{\\mathrm{Var}_{#1}\\left[#2\\right]}\n\\newcommand{\\Avar}[2][\\,\\!]{\\mathrm{Avar}_{#1}\\left[#2\\right]}\n\\newcommand{\\Cov}[2][\\,\\!]{\\mathrm{Cov}_{#1}\\left(#2\\right)}\n\\newcommand{\\Corr}[2][\\,\\!]{\\mathrm{Corr}_{#1}\\left(#2\\right)}\n\\newcommand{\\I}[1]{\\mathrm{I}\\left( #1 \\right)}\n\\newcommand{\\N}[1]{\\mathcal{N} \\left( #1 \\right)}\n\\newcommand{\\ow}{\\text{otherwise}}\n\\void^\\dagger$
                                        **\n\n## Question 1\n\nA stock price is currently $\\$40$. It is known that at the end of $3$ months it will be either $\\$45$ or $\\$35$. The risk-free rate of interest with ***quarterly compounding*** is $8\\%$ per annum. Calculate the value of a $3$-month European put option on the stock with an exercise price of $\\$40$. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.\n\n$\\bspace Answer$\n\n>Consider a portfolio with $x$ much bounds and $y$ units of asset which is used to replicate the put, then\n>\n>$$\\begin{cases}\n1.02x + 45y = \\P{40 - 45}^+ = 0\\\\\n1.02x + 35y = \\P{40 - 35}^+ = 5\n\\end{cases}$$\n>\n>$$x = \\ffrac{5\\times45-0\\times35} {1.02\\times\\P{45 - 35}}=22.059,y = \\ffrac{0-5} {45-35} =-0.5$$\n>\n>Or, first find the up and down rate, $u=45/40 = 1.125$ and $d = 35/40 = 0.857$, then\n>\n>$$x = \\ffrac{1.125\\times5 - 0.875\\times0} {\\P{1.125-0.875} 1.02}=22.059,y = \\ffrac{0-5} {\\P{1.125-0.875}40} =-0.5$$\n>\n>So the price of the put option: $p = x+yS_0 = 22.059-0.5\\times40 = 2.059$\n>\n>***\n>\n>And using the riskless hedging principle, suppose to hedge, we need to *short* $\\Delta$ shares, then the terminal payoff would be either $-35\\Delta+5$ or $-45\\Delta$. If\n>\n>$$-35\\Delta + 5 = -45\\Delta$$\n>\n>the portfolio would be riskless and the terminal payoff would be $22.5$ with $\\Delta = -0.5$. And discount that to time $0$, so we have\n>\n>$$-40\\Delta + p = \\ffrac{22.5}{1.02} \\Rightarrow p = 22.5/1.02 + 40\\P{-0.5} = 2.059$$\n>\n>Or by risk-neutral method, first we find the risk-neutral probability $q$:\n>\n>$$q = \\ffrac{1.02-0.875}{1.125-0.875} =0.58$$\n>\n>Then the put option price: $p = \\ffrac{1}{1.02}\\P{0\\times 0.58+5\\times 0.42}=2.059$\n\n$Remark$\n\n>The risk-free rate of interest with quarterly compounding is $8\\%$ per annum, meaning that $e^{rT} = 1.02$.\n\n## Question 2\n\nConsider the situation in which stock price movements during the life of a European option are governed by a *two-step binomial tree*. Explain why it is not possible to set up a position in the stock and the option that remains *riskless* for the *whole life* of the option.\n\n$\\bspace Answer$\n\n>When the stock price enters the second phase, the stock price will change, usually, so that the riskless\u00a0hedging\u00a0principle fails after that. To continue doing so, we need to find the new value of the number of shares to hold.\n\n## Question 3\n\nA stock price is currently $\\$50$. Over each of the next two $3$-month periods it is expected to go up by $6\\%$ or down by $5\\%$. The risk-free interest rate is $5\\%$ per annum with ***continuous compounding***. What is the value of a $6$-month European call option with a strike price of $\\$51$?\n\n$\\bspace Answer$\n\n>The risk-neutral probability of an *up move*, $q$, is given by\n>\n>$$q = \\ffrac{e^{rT} - d}{u-d} = \\ffrac{e^{0.05\\times 0.25-0.95}} {1.06-0.95} = 0.569$$\n>\n>And its value keeped for each perioed since the parameters never change. So at the end of the first phase, the option price would be \n>\n>$$c_1\\P{1} = e^{-0.05\\times0.25}\\SB{\\P{50\\times1.06^2-51}\\times q} = 2.911$$\n>\n>And then the first phase, we have\n>\n>$$c_0\\P{0} = e^{-0.05\\times0.25}\\SB{c_1\\P{1}\\times q} = 1.636$$\n>\n>And about the formula for $q$, here's the proof:\n>\n>$$\\begin{cases}\nxe^{rT} + yuS_0 = \\P{yuS_0 - K}^+ = c_u\\\\\nxe^{rT} + ydS_0 = \\P{ydS_0 - K}^+ = c_d\n\\end{cases}\\Rightarrow \\begin{cases}\nx = \\ffrac{uc_d - dc_u}{\\P{u-d}e^{rT}}\\leq 0\\\\\ny = \\ffrac{c_u-c_d}{\\P{u-d}S_0}\\geq 0\n\\end{cases}$$\n>\n>Then $c = x+yS_0 = e^{-rT}\\P{q\\cdot c_u + \\P{1-q}c_d}$ where $q = \\ffrac{e^{rT} - d}{u-d}$.\n\n## Question 4\n\nFor the situation considered in HW_2.3, what is the value of a $6$-month European *put* option with a strike price of $\\$51$? Verify that the *European call* and *European put* prices satisfy put-call parity. If the put option were *American*, would it ever be optimal to exercise it early at any of the nodes on the tree?\n\n$\\bspace Answer$\n\n>We now share the $q$ with the last question so we can directly use the path probability method:\n>\n>$$p = e^{-0.05\\times0.5}\\SB{\\P{1-q}^2\\P{51 - 50\\times0.95^2} + \\binom{2} {1} q\\P{1-q}\\P{51-50 \\times 0.95 \\times 1.06}} = 1.375$$\n>\n>And to test the put-call parity:\n>\n>$$1.636 + 51 \\times e^{-0.05\\times0.5}-1.375-50 \\approx 0$$\n>\n>That's it. And if it's American call options (without dividend), we won't early exercise cause it's never optimal while if it's American put option, first, we won't do that if the price goes up, or at the begining. Second, if the price goes down in the first period, at that time the payoff from early exercise is $51-50\\times 0.95 = 3.5$ while the American put option value at that time is\n>\n>$$P_1\\P{0} = e^{-0.05\\times0.25}\\SB{\\P{1-q}\\P{51 - 50\\times0.95^2}+q\\times\\P{51-50 \\times 0.95 \\times 1.06}} = 2.866 < 3.5$$\n>\n>That's to say that it's better to early exercise the American Put option if the stock price goes down in the first phase.\n\n## Question 5\n\nA stock price is currently $25$ bucks. It is known that at the end of $2$ months it will be either $\\$23$ or $\\$27$. The risk-free interest rate is $10\\%$ per annum with continuous compounding. Suppose $S_T$ is the stock price at the end of $2$ months. What is the value of a derivative that pays off $S^2_T$ at this time?\n\n$\\bspace Answer$\n\n>We replicate that portfolio with a claim which is priced as $V_t^h$, with $x$ amount of cash and $y$ amount of the stocks. Then using no arbitrage pricing, we have\n>\n>$$\\begin{cases}\nx\\cdot e^{0.1\\times 1/6} + y \\times 27 = 27^2 \\\\\nx\\cdot e^{0.1\\times 1/6} + y \\times 23 = 23^2 \n\\end{cases} \\\\[2em]\n\\Rightarrow x= \\ffrac{-27\\times23} {e^{0.1 \\times 1/6}},\\bspace y = 50$$\n>\n>So that the price of that claim would be\n>\n>$$\\Pi\\P{\\ffrac{1} {6}, U} = V_{1/6}^{h} = x + y\\cdot S_0 = \\ffrac{-27\\times23} {e^{0.1 \\times 1/6}} + 50 \\times 25 = 639.264$$\n>\n>***\n>\n>Or using the riskless hedging principle, consider portfolio that longs this derivative and short $\\Delta$ shares. The value at time $T$ is $27^2 - 27\\Delta = 23^2 - 23\\Delta$ and thus $\\Delta = 50$, the value is $-621$. Let $f$ be the price of the derivative at time $0$, then\n>\n>$$\\P{f - 25\\Delta}e^{rT} = \\P{f- 50\\times 25}e^{0.1\\times 1/6} = -621 \\Rightarrow f = 639.264$$\n\n## Question 6\n\nA stock price is currently $\\$40$. Over each of the next two $3$-month periods it is expected to go up by $10\\%$ or down by $10\\%$. The risk-free interest rate is $12\\%$ per annum with continuous compounding.\n\n$\\P{\\text a}$ What is the value of a $6$-month European put option with a strike price of $\\$42$?\n\n$\\bspace Answer$\n\n>$$q = \\ffrac{e^{0.12 \\times 0.25} - 0.9} {1.1-0.9} = 0.652$$\n>\n>The European put option can be find similarly:\n>\n>$$p = e^{-0.12\\times 0.5} \\SB{\\P{1-q}^2 \\P{42-40\\times0.9^2} + \\binom{2} {1} \\P{1-q}q\\P{42-40 \\times 0.9 \\times 0.1}}=2.121$$\n\n$\\P{\\text b}$ What is the value of a $6$-month American put option with a strike price of $\\$42$?\n\n$\\bspace Answer$\n\n>We will calculate the American Put option backwards.\n>\n>$$\\begin{align}\nP_{1}\\P{1} &= \\max\\P{e^{-0.12\\times 0.25}\\SB{\\P{1-q}\\P{42-40 \\times 0.9 \\times 1.1}},0} = 0.811 \\\\\nP_{1}\\P{0} &= \\max\\P{e^{-0.12\\times 0.25}\\SB{q\\P{42-40 \\times 0.9 \\times 1.1} + \\P{1-q}\\P{42-40\\times0.9^2}},42-40\\times0.9} = 6 \\\\\nP_{0}\\P{0} &= \\max\\P{e^{-0.12\\times 0.25}\\SB{q\\cdot P_{1}\\P{1} + \\P{1-q}\\cdot P_{1}\\P{0}},42-40} = 2.539\n\\end{align}\n$$\n\n## Question 7\n\nA stock price is currently $30$ bucks. During each $2$-month period for the next $4$ months it will increase by $8\\%$ or reduce by $10\\%$. The risk-free interest rate is $5\\%$. Use a two-step tree to calculate the value of a derivative that pays off $\\P{\\max\\P{30 \u2212 S_T , 0}}^2$ , where $S_T$ is the stock price in $4$ months. If the derivative is American-style, should it be exercised early?\n\n$\\bspace Answer$\n\n>It's not hard to find the value of the tree in the last column: $0$, $\\P{30-29.16}^2 = 0.7056$ and $\\P{30-24.3}^2 = 32.49$. Then we calculate the probability $q$:\n>\n>$$q = \\ffrac{e^{0.05 \\times 1/6}-0.9} {1.08-0.9} = 0.602$$\n>\n>Then for the European styled derivatives, the option value is:\n>\n>$$e^{-0.05\\times1/3}\\SB{\\P{1-q}^2\\times 32.49+\\binom{2} {1}\\P{1-q}q\\times0.7056} = 5.394$$\n>\n>While for the American styled, firstly, like the same, it won't happen at the end of second phase, nor the begining. Then at the end of the first phase, if the stock price goes up in the first period, then still nothing should be done. If it goes down, then the payoff from early exercise is $\\P{30-30\\times0.9}^2=9$ however the value of American Put option at that time is\n>\n>$$P_{1}\\P{0} = \\max\\P{e^{-0.05\\times1/6}\\SB{q\\times 0.7056 + \\P{1-q}\\times 32.49},9} = 13.245>9$$\n>\n>So, even if it's an American styled derivatives, at no point of time should you early exercise it.\n", "meta": {"hexsha": "056fb49d604cd9e26849186e31f591e46c2de4d5", "size": 13846, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "FinMath/Models and Pricing of Financial Derivatives/HW/HW_02_11510691_fixed.ipynb", "max_stars_repo_name": "XavierOwen/Notes", "max_stars_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2018-11-27T10:31:08.000Z", "max_stars_repo_stars_event_max_datetime": "2019-01-20T03:11:58.000Z", "max_issues_repo_path": "FinMath/Models and Pricing of Financial Derivatives/HW/HW_02_11510691_fixed.ipynb", "max_issues_repo_name": "XavierOwen/Notes", "max_issues_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "FinMath/Models and Pricing of Financial Derivatives/HW/HW_02_11510691_fixed.ipynb", "max_forks_repo_name": "XavierOwen/Notes", "max_forks_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-07-14T19:57:23.000Z", "max_forks_repo_forks_event_max_datetime": "2020-07-14T19:57:23.000Z", "avg_line_length": 46.6195286195, "max_line_length": 408, "alphanum_fraction": 0.5517116857, "converted": true, "num_tokens": 3789, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5078118642792044, "lm_q2_score": 0.8499711832583696, "lm_q1q2_score": 0.431625451154034}} {"text": "# Create Spark Cluster\n\n\n```python\nfrom pyspark.sql import SparkSession\nspark = ( \n SparkSession\n .builder\n .config(\"spark.master\", \"local[*]\")\n .config(\"spark.driver.memory\", \"120g\")\n .config(\"spark.driver.maxResultSize\", \"0\")\n .enableHiveSupport()\n .getOrCreate()\n)\n```\n\n\n```python\nimport socket\nport = spark.sparkContext.uiWebUrl.split(\":\")[-1]\nhostname = socket.gethostname()\nprint(f\"https://{hostname}/jupyter/user/stevengs/proxy/{port}/jobs/\")\n```\n\n https://epyc.astro.washington.edu/jupyter/user/stevengs/proxy/4043/jobs/\n\n\n# Load current cut of lightcurves\n\n\n```python\nimport axs\ncatalog = axs.AxsCatalog(spark)\nwtf = catalog.load(\"stevengs_cut_wtf\")\n```\n\nLook at the light curve/dib schema\n\n\n```python\nprint(wtf.columns)\n```\n\n ['ps1_objid', 'ra', 'dec', 'mean_mag_g', 'mean_mag_r', 'mean_mag_i', 'ra_stddev', 'dec_stddev', 'ps1_gMeanPSFMag', 'ps1_rMeanPSFMag', 'ps1_iMeanPSFMag', 'ra_detections', 'dec_detections', 'mjd_g', 'mag_g', 'magerr_g', 'psfflux_g', 'psffluxerr_g', 'catflags_g', 'expid_g', 'rcID_g', 'fieldID_g', 'xpos_g', 'ypos_g', 'nobs_g', 'mjd_r', 'mag_r', 'magerr_r', 'psfflux_r', 'psffluxerr_r', 'catflags_r', 'expid_r', 'rcID_r', 'fieldID_r', 'xpos_r', 'ypos_r', 'nobs_r', 'mjd_i', 'mag_i', 'magerr_i', 'psfflux_i', 'psffluxerr_i', 'catflags_i', 'expid_i', 'rcID_i', 'fieldID_i', 'xpos_i', 'ypos_i', 'nobs_i', 'dip', 'zone', 'dup']\n\n\n\n```python\n# dip schema\nwtf.select(\"dip\").head(1)[0]\n```\n\n\n\n\n Row(dip=Row(center_mjd=58282.18359375, start_mjd=58271.09765625, end_mjd=58293.70703125, length=22.608280181884766, guess_start_mjd=58274.85546875, guess_end_mjd=58286.29296875, guess_significance=7.1239423751831055, window_start_mjd=58263.421875, window_end_mjd=58297.73046875, significant_observation_count=5, significant_length=9.91579818725586, integral=0.730926513671875, integral_uncertainty=0.13865260779857635, significance=5.271639347076416, max_gap=10.001840591430664, max_gap_fraction=0.44239720702171326, core_count=6, core_start_mjd=58276.35546875, core_end_mjd=58286.26953125, core_length=9.91579818725586, core_significant_count=5, core_not_significant_count=1, core_not_significant_fraction=0.1666666716337204, ref_observation_count_before=10, ref_observation_count_after=218, ref_observation_count=228, ref_pull_std=1.4057750701904297, ref_large_pull_fraction=0.048245612531900406, ref_length_after=573.8453369140625, ref_length_before=57.98050308227539, ref_length=631.8258056640625, ref_length_fraction_after=16.726078033447266, ref_length_fraction_before=1.689978837966919, ref_length_fraction=18.416057586669922, deep_count=22, deep_length=15.858842849731445, deep_gap_fraction=0.6306790709495544, flip_count=2))\n\n\n\n\n```python\nfrom fit_utils import (\n fit_band_around_dip, plot_fit_result, make_udf_from_annotated_function,\n evaluate_in_dip, evaluate_around_dip, evaluate\n)\n```\n\n# Test simple quadratic model\n\n\n```python\ndef quadratic(x, a, b, c):\n return a * x**2 + b * x + c\n\n# fit model to data in window around the dip\n# fit r-band data\n# fit in window that is expanded by 4.0x the dip width from dip.start_mjd and dip.end_mjd\nquadratic_fit_df = fit_band_around_dip(wtf, quadratic, \"r\", 4.0)\n\n# select a small subset of columns\nquadratic_fit_df_slim = quadratic_fit_df.select(\n \"ps1_objid\",\n \"ra\",\n \"dec\",\n \"zone\",\n \"mjd_r\",\n \"mag_r\",\n \"magerr_r\",\n \"dip\",\n \"window_r\",\n \"fit_r\",\n)\n\n# only include fits that converged\nquadratic_fit_df_slim_good = quadratic_fit_df_slim.where(\n quadratic_fit_df_slim['fit_r']['info']['good']\n)\n# evaluate sum square error inside the dip\nfits_evaluated_df = evaluate_in_dip(quadratic_fit_df_slim_good, quadratic, \"r\")\n# around the dip\nfits_evaluated_df = evaluate_around_dip(fits_evaluated_df, quadratic, \"r\", 4.0)\n# over all data\nfits_evaluated_df = evaluate(fits_evaluated_df, quadratic, \"r\")\n```\n\n\n```python\nlc = fits_evaluated_df.head(1)[0]\n```\n\n## in-dip\n\n\n```python\nplot_fit_result(lc['dip_window_r']['x'], lc['dip_window_r']['y'], lc['dip_window_r']['yerr'], lc['fit_r'], quadratic)\nprint(lc['model_error_in_dip_r'])\n```\n\n## around-dip\n\n\n```python\nplot_fit_result(lc['window_r']['x'], lc['window_r']['y'], lc['window_r']['yerr'], lc['fit_r'], quadratic)\nprint(lc['model_error_around_dip_r'])\n```\n\n## all-data\n\n\n```python\nplot_fit_result(lc['mjd_r'], lc['mag_r'], lc['magerr_r'], lc['fit_r'], quadratic)\nprint(lc['model_error_r'])\n```\n\n# Skew Normal\n\nDescription: https://en.wikipedia.org/wiki/Skew_normal_distribution\n\nA combination of the standard normal PDF $\\phi(x)$ and the standard normal CDF $\\Phi(x)$:\n\n$$\nf(x) = \\frac{2}{\\omega} \\phi\\left(\\frac{x - \\xi}{\\omega} \\right) \\Phi\\left(\\alpha \\left(\\frac{x - \\xi}{\\omega} \\right) \\right)\n$$\n$\\xi$ is the location, $\\omega$ is the scale, $\\alpha$ is the skew.\n\n\\begin{align*}\n\\phi(x) &= \\frac{1}{2\\pi} \\exp\\left( -\\frac{x^2}{2} \\right) \\\\\n\\Phi(x) &= \\int_{-\\infty}^{x} \\phi(t) dt = \\frac{1}{2} \\left[ 1 + \\text{erf}\\left( \\frac{x}{\\sqrt{2}} \\right) \\right]\n\\end{align*}\n\n\n```python\nfrom models import skew_normal, skew_normal_p0, skew_normal_p0_udf\n```\n\n\n```python\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nloc = 0\nxscale = 1\nyscale = 1\noffset = 0\nplt.rc(\"figure\", figsize=[10, 8])\nplt.rc(\"font\", size=18)\nfor skew in [-2.5, -1, 0, 1, 2.5]:\n _x = np.linspace(-5, 5, 100)\n _y = skew_normal(_x, skew, loc, xscale, yscale, offset)\n plt.plot(_x, _y, label=f\"skew = {skew}\")\nplt.legend()\nplt.show()\n```\n\nFit r-band data\n\n\n```python\nband = \"r\"\nwiggle = 4.0\n\nskew_normal_p0_column = skew_normal_p0_udf(\n wtf['mjd_r'], wtf['mag_r'], wtf['magerr_r'], wtf['dip']['start_mjd'], \n wtf['dip']['end_mjd'], wtf['dip']['integral']\n)\n\nskew_norm_fit_df = fit_band_around_dip(wtf, skew_normal, band, wiggle, p0=skew_normal_p0_column)\n\n# select a small subset of columns\nskew_norm_fit_df_slim = skew_norm_fit_df.select(\n \"ps1_objid\",\n \"ra\",\n \"dec\",\n \"zone\",\n \"mjd_r\",\n \"mag_r\",\n \"magerr_r\",\n \"dip\",\n \"window_r\",\n \"fit_r\",\n)\n\n# only include fits that converged\nskew_norm_fit_df_slim_good = skew_norm_fit_df_slim.where(\n skew_norm_fit_df_slim['fit_r']['info']['good']\n)\n# evaluate sum square error inside the dip\nskew_norm_fits_evaluated_df = evaluate_in_dip(skew_norm_fit_df_slim_good, skew_normal, band)\n# around the dip\nskew_norm_fits_evaluated_df = evaluate_around_dip(skew_norm_fits_evaluated_df, skew_normal, band, wiggle)\n# over all data\nskew_norm_fits_evaluated_df = evaluate(skew_norm_fits_evaluated_df, skew_normal, band)\n```\n\n\n```python\n_lc = skew_norm_fits_evaluated_df.head(1)[0]\n```\n\n\n```python\nplot_fit_result(\n _lc['dip_window_r']['x'], _lc['dip_window_r']['y'], _lc['dip_window_r']['yerr'], \n _lc['fit_r'], skew_normal, with_p0=True\n)\nprint(_lc['model_error_in_dip_r'])\nprint(_lc['model_error_around_dip_r'])\nprint(_lc['model_error_r'])\n```\n\n\n```python\ncatalog.save_axs_table(skew_norm_fits_evaluated_df, \"6_4_20_stevengs_skew_normal_fits_r_band\")\n```\n\n# Top hat\n\n\\begin{align}\nf(x) = \\begin{cases}\nd + c&\\text{for $x\\in[l - w/2, l + w/2]$}\\\\\nc&\\text{otherwise}\\\\\n\\end{cases}\n\\end{align}\nwhere $d$ is the depth, $c$ is the offset, $l$ is the location, and $w$ is the width\n\n\n```python\nfrom models import top_hat, top_hat_p0, top_hat_p0_udf\n```\n\n\n```python\nloc = 0\nwidth = 1\ndepth = 1\noffset = 0\nplt.rc(\"figure\", figsize=[10, 8])\nplt.rc(\"font\", size=18)\nfor width in [1, 2, 5, 6]:\n _x = np.linspace(-5, 5, 100)\n _y = top_hat(_x, loc, width, depth, offset)\n plt.plot(_x, _y, label=f\"width = {width}\")\nplt.legend()\nplt.show()\n```\n\n\n```python\nband = \"r\"\nwiggle = 4.0\n\ntop_hat_p0_column = top_hat_p0_udf(\n wtf['mjd_r'], wtf['mag_r'], wtf['magerr_r'], \n wtf['dip']['start_mjd'], wtf['dip']['end_mjd'], wtf['dip']['integral']\n)\n\ntop_hat_fit_df = fit_band_around_dip(wtf, top_hat, band, wiggle, p0=top_hat_p0_column)\n\n# select a small subset of columns\ntop_hat_fit_df_slim = top_hat_fit_df.select(\n \"ps1_objid\",\n \"ra\",\n \"dec\",\n \"zone\",\n \"mjd_r\",\n \"mag_r\",\n \"magerr_r\",\n \"dip\",\n \"window_r\",\n \"fit_r\",\n)\n\n# only include fits that converged\ntop_hat_fit_df_slim_good = top_hat_fit_df_slim.where(\n top_hat_fit_df_slim['fit_r']['info']['good']\n)\n# evaluate sum square error inside the dip\ntop_hat_fits_evaluated_df = evaluate_in_dip(top_hat_fit_df_slim_good, top_hat, band)\n# around the dip\ntop_hat_fits_evaluated_df = evaluate_around_dip(top_hat_fits_evaluated_df, top_hat, band, wiggle)\n# over all data\ntop_hat_fits_evaluated_df = evaluate(top_hat_fits_evaluated_df, top_hat, band)\n```\n\n\n```python\ncatalog.save_axs_table(top_hat_fits_evaluated_df, \"6_4_20_stevengs_top_hat_fits_r_band\")\n```\n\n\n```python\ntop_hat_lcs = catalog.load(\"6_4_20_stevengs_top_hat_fits_r_band\")\ntop_hat_lc = top_hat_lcs.head(1)[0]\n```\n\n\n```python\nplot_fit_result(\n top_hat_lc['window_r']['x'], top_hat_lc['window_r']['y'], top_hat_lc['window_r']['yerr'], \n top_hat_lc['fit_r'], top_hat\n)\nprint(top_hat_lc['model_error_in_dip_r'])\nprint(top_hat_lc['model_error_around_dip_r'])\nprint(top_hat_lc['model_error_r'])\n```\n\n\n```python\ntop_hat_lc_min_chi_square = top_hat_lcs.sort(\n top_hat_lcs['model_error_around_dip_r.reduced_sum_square_error'], ascending=True\n).head(1)[0]\n```\n\n\n```python\nplot_fit_result(\n top_hat_lc_min_chi_square['around_dip_window_r']['x'], \n top_hat_lc_min_chi_square['around_dip_window_r']['y'], \n top_hat_lc_min_chi_square['around_dip_window_r']['yerr'], \n top_hat_lc_min_chi_square['fit_r'], top_hat\n)\nprint(top_hat_lc_min_chi_square['model_error_in_dip_r'])\nprint(top_hat_lc_min_chi_square['model_error_around_dip_r'])\nprint(top_hat_lc_min_chi_square['model_error_r'])\n```\n", "meta": {"hexsha": "ad2928248b93229e4dbc7047369d624cc6415b2d", "size": 243245, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "fitting/Fit Light Curves.ipynb", "max_stars_repo_name": "dirac-institute/ZTF_Boyajian", "max_stars_repo_head_hexsha": "edf9e644ce2dfa74514e207b90526d9660ad66e2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-08-01T16:02:45.000Z", "max_stars_repo_stars_event_max_datetime": "2021-06-03T23:35:39.000Z", "max_issues_repo_path": "fitting/Fit Light Curves.ipynb", "max_issues_repo_name": "dirac-institute/ZTF_Boyajian", "max_issues_repo_head_hexsha": "edf9e644ce2dfa74514e207b90526d9660ad66e2", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 8, "max_issues_repo_issues_event_min_datetime": "2019-08-20T21:57:13.000Z", "max_issues_repo_issues_event_max_datetime": "2021-04-29T22:46:27.000Z", "max_forks_repo_path": "fitting/Fit Light Curves.ipynb", "max_forks_repo_name": "dirac-institute/ZTF_Boyajian", "max_forks_repo_head_hexsha": "edf9e644ce2dfa74514e207b90526d9660ad66e2", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2019-08-06T22:18:49.000Z", "max_forks_repo_forks_event_max_datetime": "2019-08-06T22:18:49.000Z", "avg_line_length": 353.5537790698, "max_line_length": 71416, "alphanum_fraction": 0.9333799256, "converted": true, "num_tokens": 3185, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056040203136, "lm_q2_score": 0.6261241632752915, "lm_q1q2_score": 0.4315908945581883}} {"text": "# **CS224W - Colab 4**\n\nIn Colab 2 we constructed GNN models by using PyTorch Geometric's built in GCN layer, `GCNConv`. In Colab 3 we implemented the **GraphSAGE** ([Hamilton et al. (2017)](https://arxiv.org/abs/1706.02216)) layer. In this colab you'll use what you've learned and implement a more powerful layer: **GAT** ([Veli\u010dkovi\u0107 et al. (2018)](https://arxiv.org/abs/1710.10903)). Then we will run our models on the CORA dataset, which is a standard citation network benchmark dataset.\n\n**Note**: Make sure to **sequentially run all the cells in each section** so that the intermediate variables / packages will carry over to the next cell\n\nHave fun and good luck on Colab 4 :)\n\n# Device\nWe recommend using a GPU for this Colab.\n\nPlease click `Runtime` and then `Change runtime type`. Then set the `hardware accelerator` to **GPU**.\n\n## Installation\n\n\n```python\n# Install torch geometric\nimport os\nif 'IS_GRADESCOPE_ENV' not in os.environ:\n !pip install torch-scatter -f https://data.pyg.org/whl/torch-1.10.0+cu113.html\n !pip install torch-sparse -f https://data.pyg.org/whl/torch-1.10.0+cu113.html\n !pip install torch-geometric\n !pip install -q git+https://github.com/snap-stanford/deepsnap.git\n```\n\n Looking in links: https://data.pyg.org/whl/torch-1.10.0+cu113.html\n Requirement already satisfied: torch-scatter in /usr/local/lib/python3.7/dist-packages (2.0.9)\n Looking in links: https://data.pyg.org/whl/torch-1.10.0+cu113.html\n Requirement already satisfied: torch-sparse in /usr/local/lib/python3.7/dist-packages (0.6.12)\n Requirement already satisfied: scipy in /usr/local/lib/python3.7/dist-packages (from torch-sparse) (1.4.1)\n Requirement already satisfied: numpy>=1.13.3 in /usr/local/lib/python3.7/dist-packages (from scipy->torch-sparse) (1.19.5)\n Requirement already satisfied: torch-geometric in /usr/local/lib/python3.7/dist-packages (2.0.3)\n Requirement already satisfied: networkx in /usr/local/lib/python3.7/dist-packages (from torch-geometric) (2.6.3)\n Requirement already satisfied: PyYAML in /usr/local/lib/python3.7/dist-packages (from torch-geometric) (3.13)\n Requirement already satisfied: rdflib in /usr/local/lib/python3.7/dist-packages (from torch-geometric) (6.1.1)\n Requirement already satisfied: numpy in /usr/local/lib/python3.7/dist-packages (from torch-geometric) (1.19.5)\n Requirement already satisfied: pandas in /usr/local/lib/python3.7/dist-packages (from torch-geometric) (1.1.5)\n Requirement already satisfied: pyparsing in /usr/local/lib/python3.7/dist-packages (from torch-geometric) (3.0.6)\n Requirement already satisfied: yacs in /usr/local/lib/python3.7/dist-packages (from torch-geometric) (0.1.8)\n Requirement already satisfied: scipy in /usr/local/lib/python3.7/dist-packages (from torch-geometric) (1.4.1)\n Requirement already satisfied: tqdm in /usr/local/lib/python3.7/dist-packages (from torch-geometric) (4.62.3)\n Requirement already satisfied: jinja2 in /usr/local/lib/python3.7/dist-packages (from torch-geometric) (2.11.3)\n Requirement already satisfied: scikit-learn in /usr/local/lib/python3.7/dist-packages (from torch-geometric) (1.0.2)\n Requirement already satisfied: googledrivedownloader in /usr/local/lib/python3.7/dist-packages (from torch-geometric) (0.4)\n Requirement already satisfied: requests in /usr/local/lib/python3.7/dist-packages (from torch-geometric) (2.23.0)\n Requirement already satisfied: MarkupSafe>=0.23 in /usr/local/lib/python3.7/dist-packages (from jinja2->torch-geometric) (2.0.1)\n Requirement already satisfied: pytz>=2017.2 in /usr/local/lib/python3.7/dist-packages (from pandas->torch-geometric) (2018.9)\n Requirement already satisfied: python-dateutil>=2.7.3 in /usr/local/lib/python3.7/dist-packages (from pandas->torch-geometric) (2.8.2)\n Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.7/dist-packages (from python-dateutil>=2.7.3->pandas->torch-geometric) (1.15.0)\n Requirement already satisfied: importlib-metadata in /usr/local/lib/python3.7/dist-packages (from rdflib->torch-geometric) (4.10.0)\n Requirement already satisfied: setuptools in /usr/local/lib/python3.7/dist-packages (from rdflib->torch-geometric) (57.4.0)\n Requirement already satisfied: isodate in /usr/local/lib/python3.7/dist-packages (from rdflib->torch-geometric) (0.6.1)\n Requirement already satisfied: typing-extensions>=3.6.4 in /usr/local/lib/python3.7/dist-packages (from importlib-metadata->rdflib->torch-geometric) (3.10.0.2)\n Requirement already satisfied: zipp>=0.5 in /usr/local/lib/python3.7/dist-packages (from importlib-metadata->rdflib->torch-geometric) (3.7.0)\n Requirement already satisfied: certifi>=2017.4.17 in /usr/local/lib/python3.7/dist-packages (from requests->torch-geometric) (2021.10.8)\n Requirement already satisfied: urllib3!=1.25.0,!=1.25.1,<1.26,>=1.21.1 in /usr/local/lib/python3.7/dist-packages (from requests->torch-geometric) (1.24.3)\n Requirement already satisfied: chardet<4,>=3.0.2 in /usr/local/lib/python3.7/dist-packages (from requests->torch-geometric) (3.0.4)\n Requirement already satisfied: idna<3,>=2.5 in /usr/local/lib/python3.7/dist-packages (from requests->torch-geometric) (2.10)\n Requirement already satisfied: threadpoolctl>=2.0.0 in /usr/local/lib/python3.7/dist-packages (from scikit-learn->torch-geometric) (3.0.0)\n Requirement already satisfied: joblib>=0.11 in /usr/local/lib/python3.7/dist-packages (from scikit-learn->torch-geometric) (1.1.0)\n\n\n\n```python\nimport torch_geometric\ntorch_geometric.__version__\n```\n\n\n\n\n '2.0.3'\n\n\n\n# 1) GNN Layers\n\n## Implementing Layer Modules\n\nIn Colab 2, we implemented a GCN model for node and graph classification tasks. However, for that notebook we took advantage of PyG's built in GCN module. For Colabs 3 and 4, we provide a build upon a general Graph Neural Network Stack, into which we will be able to plugin our own module implementations: GraphSAGE and GAT.\n\nWe will then use our layer implemenations to complete node classification on the CORA dataset, a standard citation network benchmark. In this dataset, nodes correspond to documents and edges correspond to undirected citations. Each node or document in the graph is assigned a class label and features based on the documents binarized bag-of-words representation. Specifically, the Cora graph has 2708 nodes, 5429 edges, 7 prediction classes, and 1433 features per node. \n\n## GNN Stack Module\n\nBelow is the implementation of a general GNN stack, where we can plugin any GNN layer, such as **GraphSage**, **GAT**, etc. This module is provided for you. Your implementations of the **GraphSage** and **GAT** layers will function as components in the GNNStack Module.\n\n\n```python\nimport torch\nimport torch_scatter\nimport torch.nn as nn\nimport torch.nn.functional as F\n\nimport torch_geometric.nn as pyg_nn\nimport torch_geometric.utils as pyg_utils\n\nfrom torch import Tensor\nfrom typing import Union, Tuple, Optional\nfrom torch_geometric.typing import (OptPairTensor, Adj, Size, NoneType,\n OptTensor)\n\nfrom torch.nn import Parameter, Linear\nfrom torch_sparse import SparseTensor, set_diag\nfrom torch_geometric.nn.conv import MessagePassing\nfrom torch_geometric.utils import remove_self_loops, add_self_loops, softmax\n\nclass GNNStack(torch.nn.Module):\n\n def __init__(\n self,\n input_dim,\n hidden_dim,\n output_dim,\n args,\n emb=False\n ):\n\n # Parentizing\n super(GNNStack, self).__init__()\n \n # What type of GNN we are talking about\n conv_model = self.build_conv_model(args.model_type)\n\n # List of conv layers\n self.convs = nn.ModuleList()\n\n # Append input layer\n self.convs.append(conv_model(input_dim, hidden_dim))\n\n # Check if the number of layers is more than 1 \n assert (args.num_layers >= 1), 'Number of layers is not >=1'\n \n # Go through all layers\n for l in range(args.num_layers-1):\n self.convs.append(conv_model(args.heads * hidden_dim, hidden_dim))\n\n # Post-message-passing\n self.post_mp = nn.Sequential(\n nn.Linear(args.heads * hidden_dim, hidden_dim),\n nn.Dropout(args.dropout), \n nn.Linear(hidden_dim, output_dim)\n )\n \n # Define dropout\n self.dropout = args.dropout\n\n # Store number of layers for future use into the methods\n self.num_layers = args.num_layers\n\n # Storing boolean about embeddings (True dont apply a classification \n # layer in the end)\n self.emb = emb\n\n def build_conv_model(self, model_type):\n\n if model_type == 'GraphSage':\n return GraphSage\n \n elif model_type == 'GAT':\n # When applying GAT with num heads > 1, you need to modify the \n # input and output dimension of the conv layers (self.convs),\n # to ensure that the input dim of the next layer is num heads\n # multiplied by the output dim of the previous layer.\n # HINT: In case you want to play with multiheads, you need to change the for-loop that builds up self.convs to be\n # self.convs.append(conv_model(hidden_dim * num_heads, hidden_dim)), \n # and also the first nn.Linear(hidden_dim * num_heads, hidden_dim) in post-message-passing.\n return GAT\n\n def forward(self, data):\n\n # Extract from batch\n x, edge_index, batch = data.x, data.edge_index, data.batch\n \n # Go through all layers\n for i in range(self.num_layers):\n x = self.convs[i](x, edge_index)\n x = F.relu(x)\n x = F.dropout(x, p=self.dropout, training=self.training)\n\n x = self.post_mp(x)\n\n if self.emb == True:\n return x\n\n return F.log_softmax(x, dim=1)\n\n def loss(self, pred, label):\n return F.nll_loss(pred, label)\n```\n\n## Creating Our Own Message Passing Layer\n\nNow let's start implementing our own message passing layers! Working through this part will help us become acutely familiar with the behind the scenes work of implementing Pytorch Message Passing Layers, allowing us to build our own GNN models. To do so, we will work with and implement 3 critcal functions needed to define a PyG Message Passing Layer: `forward`, `message`, and `aggregate`.\n\nBefore diving head first into the coding details, let us quickly review the key components of the message passing process. To do so, we will focus on a single round of messsage passing with respect to a single central node $x$. Before message passing, $x$ is associated with a feature vector $x^{l-1}$, and the goal of message passing is to update this feature vector as $x^l$. To do so, we implement the following steps: 1) each neighboring node $v$ passes its current message $v^{l-1}$ across the edge $(x, v)$ - 2) for the node $x$, we aggregate all of the messages of the neighboring nodes (for example through a sum or mean) - and 3) we transform the aggregated information by for example applying linear and non-linear transformations. Altogether, the message passing process is applied such that every node $u$ in our graph updates its embedding by acting as the central node $x$ in step 1-3 described above. \n\nNow, we extending this process to that of a single message passing layer, the job of a message passing layer is to update the current feature representation or embedding of each node in a graph by propagating and transforming information within the graph. Overall, the general paradigm of a message passing layers is: 1) pre-processing -> 2) **message passing** / propagation -> 3) post-processing. \n\nThe `forward` fuction that we will implement for our message passing layer captures this execution logic. Namely, the `forward` function handles the pre and post-processing of node features / embeddings, as well as initiates message passing by calling the `propagate` function. \n\n\nThe `propagate` function encapsulates the message passing process! It does so by calling three important functions: 1) `message`, 2) `aggregate`, and 3) `update`. Our implementation will vary slightly from this, as we will not explicitly implement `update`, but instead place the logic for updating node embeddings after message passing and within the `forward` function. To be more specific, after information is propagated (message passing), we can further transform the node embeddings outputed by `propagate`. Therefore, the output of `forward` is exactly the node embeddings after one GNN layer.\n\nLastly, before starting to implement our own layer, let us dig a bit deeper into each of the functions described above:\n\n1. \n\n```\ndef propagate(edge_index, x=(x_i, x_j), extra=(extra_i, extra_j), size=size):\n```\nCalling `propagate` initiates the message passing process. Looking at the function parameters, we highlight a couple of key parameters. \n\n - `edge_index` is passed to the forward function and captures the edge structure of the graph.\n - `x=(x_i, x_j)` represents the node features that will be used in message passing. In order to explain why we pass the tuple `(x_i, x_j)`, we first look at how our edges are represented. For every edge $(i, j) \\in \\mathcal{E}$, we can differentiate $i$ as the source or central node ($x_{central}$) and j as the neighboring node ($x_{neighbor}$). \n \n Taking the example of message passing above, for a central node $u$ we will aggregate and transform all of the messages associated with the nodes $v$ s.t. $(u, v) \\in \\mathcal{E}$ (i.e. $v \\in \\mathcal{N}_{u}$). Thus we see, the subscripts `_i` and `_j` allow us to specifcally differenciate features associated with central nodes (i.e. nodes recieving message information) and neighboring nodes (i.e. nodes passing messages). \n\n This is definitely a somewhat confusing concept; however, one key thing to remember / wrap your head around is that depending on the perspective, a node $x$ acts as a central node or a neighboring node. In fact, in undirected graphs we store both edge directions (i.e. $(i, j)$ and $(j, i)$). From the central node perspective, `x_i`, x is collecting neighboring information to update its embedding. From a neighboring node perspective, `x_j`, x is passing its message information along the edge connecting it to a different central node.\n\n - `extra=(extra_i, extra_j)` represents additional information that we can associate with each node beyond its current feature embedding. In fact, we can include as many additional parameters of the form `param=(param_i, param_j)` as we would like. Again, we highlight that indexing with `_i` and `_j` allows us to differentiate central and neighboring nodes. \n\n The output of the `propagate` function is a matrix of node embeddings after the message passing process and has shape $[N, d]$.\n\n2. \n```\ndef message(x_j, ...):\n```\nThe `message` function is called by propagate and constructs the messages from\nneighboring nodes $j$ to central nodes $i$ for each edge $(i, j)$ in *edge_index*. This function can take any argument that was initially passed to `propagate`. Furthermore, we can again differentiate central nodes and neighboring nodes by appending `_i` or `_j` to the variable name, .e.g. `x_i` and `x_j`. Looking more specifically at the variables, we have:\n\n - `x_j` represents a matrix of feature embeddings for all neighboring nodes passing their messages along their respective edge (i.e. all nodes $j$ for edges $(i, j) \\in \\mathcal{E}$). Thus, its shape is $[|\\mathcal{E}|, d]$!\n - In implementing GAT we will see how to access additional variables passed to propagate\n\n Critically, we see that the output of the `message` function is a matrix of neighboring node embeddings ready to be aggregated, having shape $[|\\mathcal{E}|, d]$.\n\n3. \n```\ndef aggregate(self, inputs, index, dim_size = None):\n```\nLastly, the `aggregate` function is used to aggregate the messages from neighboring nodes. Looking at the parameters we highlight:\n\n - `inputs` represents a matrix of the messages passed from neighboring nodes (i.e. the output of the `message` function).\n - `index` has the same shape as `inputs` and tells us the central node that corresponding to each of the rows / messages $j$ in the `inputs` matrix. Thus, `index` tells us which rows / messages to aggregate for each central node.\n\n The output of `aggregate` is of shape $[N, d]$.\n\n\nFor additional resources refer to the PyG documentation for implementing custom message passing layers: https://pytorch-geometric.readthedocs.io/en/latest/notes/create_gnn.html\n\n## GAT Implementation\n\nAttention mechanisms have become the state-of-the-art in many sequence-based tasks such as machine translation and learning sentence representations. One of the major benefits of attention-based mechanisms is their ability to focus on the most relevant parts of the input to make decisions. In this problem, we will see how attention mechanisms can be used to perform node classification over graph-structured data through the usage of Graph Attention Networks (GATs) ([Veli\u010dkovi\u0107 et al. (2018)](https://arxiv.org/abs/1710.10903)).\n\nThe building block of the Graph Attention Network is the graph attention layer, which is a variant of the aggregation function. Let $N$ be the number of nodes and $F$ be the dimension of the feature vector for each node. The input to each graph attentional layer is a set of node features: $\\mathbf{h} = \\{\\overrightarrow{h_1}, \\overrightarrow{h_2}, \\dots, \\overrightarrow{h_N}$\\}, $\\overrightarrow{h_i} \\in R^F$. The output of each graph attentional layer is a new set of node features, which may have a new dimension $F'$: $\\mathbf{h'} = \\{\\overrightarrow{h_1'}, \\overrightarrow{h_2'}, \\dots, \\overrightarrow{h_N'}\\}$, with $\\overrightarrow{h_i'} \\in \\mathbb{R}^{F'}$.\n\nWe will now describe how this transformation is performed for each graph attention layer. First, a shared linear transformation parametrized by the weight matrix $\\mathbf{W} \\in \\mathbb{R}^{F' \\times F}$ is applied to every node. \n\nNext, we perform self-attention on the nodes. We use a shared attention function $a$:\n\\begin{equation} \na : \\mathbb{R}^{F'} \\times \\mathbb{R}^{F'} \\rightarrow \\mathbb{R}.\n\\end{equation}\n\nthat computes the attention coefficients capturing the importance of node $j$'s features to node $i$:\n\\begin{equation}\ne_{ij} = a(\\mathbf{W_l}\\overrightarrow{h_i}, \\mathbf{W_r} \\overrightarrow{h_j})\n\\end{equation}\n\nThe most general formulation of self-attention allows every node to attend to all other nodes which drops all structural information. However, to utilize graph structure in the attention mechanisms, we use **masked attention**. In masked attention, we only compute attention coefficients $e_{ij}$ for nodes $j \\in \\mathcal{N}_i$ where $\\mathcal{N}_i$ is some neighborhood of node $i$ in the graph.\n\nTo easily compare coefficients across different nodes, we normalize the coefficients across $j$ using a softmax function:\n\\begin{equation}\n\\alpha_{ij} = \\text{softmax}_j(e_{ij}) = \\frac{\\exp(e_{ij})}{\\sum_{k \\in \\mathcal{N}_i} \\exp(e_{ik})}\n\\end{equation}\n\nFor this problem, our attention mechanism $a$ will be a single-layer feedforward neural network parametrized by a weight vectors $\\overrightarrow{a_l} \\in \\mathbb{R}^{F'}$ and $\\overrightarrow{a_r} \\in \\mathbb{R}^{F'}$, followed by a LeakyReLU nonlinearity (with negative input slope 0.2). Let $\\cdot^T$ represent transposition and $||$ represent concatenation. The coefficients computed by our attention mechanism may be expressed as:\n\n\\begin{equation}\n\\alpha_{ij} = \\frac{\\exp\\Big(\\text{LeakyReLU}\\Big(\\overrightarrow{a_l}^T \\mathbf{W_l} \\overrightarrow{h_i} + \\overrightarrow{a_r}^T\\mathbf{W_r}\\overrightarrow{h_j}\\Big)\\Big)}{\\sum_{k\\in \\mathcal{N}_i} \\exp\\Big(\\text{LeakyReLU}\\Big(\\overrightarrow{a_l}^T \\mathbf{W_l} \\overrightarrow{h_i} + \\overrightarrow{a_r}^T\\mathbf{W_r}\\overrightarrow{h_k}\\Big)\\Big)}\n\\end{equation}\n\nFor the following questions, we denote `alpha_l` = $\\alpha_l = [...,\\overrightarrow{a_l}^T \\mathbf{W_l} \\overrightarrow{h_i},...] \\in \\mathcal{R}^n$ and `alpha_r` = $\\alpha_r = [..., \\overrightarrow{a_r}^T \\mathbf{W_r} \\overrightarrow{h_j}, ...] \\in \\mathcal{R}^n$.\n\n\nAt every layer of GAT, after the attention coefficients are computed for that layer, the aggregation function can be computed by a weighted sum of neighborhood messages, where weights are specified by $\\alpha_{ij}$.\n\nNow, we use the normalized attention coefficients to compute a linear combination of the features corresponding to them. These aggregated features will serve as the final output features for every node.\n\n\\begin{equation}\nh_i' = \\sum_{j \\in \\mathcal{N}_i} \\alpha_{ij} \\mathbf{W_r} \\overrightarrow{h_j}.\n\\end{equation}\n\nAt this point, we have covered a lot of information! Before reading further about multi-head attention, we encourage you to go again through the excersize of thinking about what components of the attention mechanism correspond with the different functions: 1) `forward`, 2) `message`, and 3 `aggregate`. \n\n- Hint 1: Our aggregation is very similar to that of GraphSage except now we are using sum aggregation\n- Hint 2: The terms we aggregate over again represent the individual message that each neighbor node j sends. Thus, we see that $\\alpha_{ij}$ is part of the message each node sends and is thus computed during the message step. This makes sense since an attention weight is associated with each edge in the graph.\n- Hint 3: Look at the terms in the definition of $\\alpha_{ij}$. What values do we want to pre-process and pass as parameters to the `propagate` function. The parameters of `message(..., x_j, alpha_j, alpha_i, ...)` should give a good hint. \n\n### Multi-Head Attention\nTo stabilize the learning process of self-attention, we use multi-head attention. To do this we use $K$ independent attention mechanisms, or ``heads'' compute output features as in the above equations. Then, we concatenate these output feature representations:\n\n\\begin{equation}\n \\overrightarrow{h_i}' = ||_{k=1}^K \\Big(\\sum_{j \\in \\mathcal{N}_i} \\alpha_{ij}^{(k)} \\mathbf{W_r}^{(k)} \\overrightarrow{h_j}\\Big)\n\\end{equation}\n\nwhere $||$ is concentation, $\\alpha_{ij}^{(k)}$ are the normalized attention coefficients computed by the $k$-th attention mechanism $(a^k)$, and $\\mathbf{W}^{(k)}$ is the corresponding input linear transformation's weight matrix. Note that for this setting, $\\mathbf{h'} \\in \\mathbb{R}^{KF'}$.\n\n\n```python\nclass GAT(MessagePassing):\n\n def __init__(\n self,\n in_channels,\n out_channels,\n heads = 2,\n negative_slope = 0.2,\n dropout = 0.,\n **kwargs):\n \n super(GAT, self).__init__(node_dim=0, **kwargs)\n\n self.in_channels = in_channels\n self.out_channels = out_channels\n self.heads = heads\n self.negative_slope = negative_slope\n self.dropout = dropout\n\n self.lin_l = None\n self.lin_r = None\n self.att_l = None\n self.att_r = None\n\n ############################################################################\n # TODO: Your code here! \n # Define the layers needed for the message functions below.\n # self.lin_l is the linear transformation that you apply to embeddings \n # BEFORE message passing.\n # \n # Pay attention to dimensions of the linear layers, since we're using \n # multi-head attention.\n # Our implementation is ~1 lines, but don't worry if you deviate from this.\n self.lin_l = Linear(in_channels, heads * out_channels)\n\n ############################################################################\n\n self.lin_r = self.lin_l\n\n ############################################################################\n # TODO: Your code here! \n # Define the attention parameters \\overrightarrow{a_l/r}^T in the above intro.\n # You have to deal with multi-head scenarios.\n # Use nn.Parameter instead of nn.Linear\n # Our implementation is ~2 lines, but don't worry if you deviate from this.\n\n # The idea here is to initialize all the parameters involving the attention \n # mechanism, which is described by the vector of weight a of dimension \n # out_channels.\n\n # However, if we stack several heads together, we need such a vector\n # for each one of these heads\n #\n # Therefore, following the documentation: https://pytorch.org/docs/1.9.1/generated/torch.nn.parameter.Parameter.html\n self.att_l = Parameter(torch.zeros(heads, out_channels))\n self.att_r = Parameter(torch.zeros(heads, out_channels)) \n \n # PS: I almost used self.attention_l; but in the reset_parameters below \n # the guy is using att_l\n\n ############################################################################\n\n self.reset_parameters()\n\n def reset_parameters(self):\n nn.init.xavier_uniform_(self.lin_l.weight)\n nn.init.xavier_uniform_(self.lin_r.weight)\n nn.init.xavier_uniform_(self.att_l)\n nn.init.xavier_uniform_(self.att_r)\n\n def forward(self, x, edge_index, size=None):\n \"\"\"\n Implement message passing, as well as any post-processing (our update rule).\n\n Parameters\n ----------\n\n self : GAT object\n GAT.\n \n x : tensor of shape (num_nodes, in_channels)\n Data associated to each node.\n\n edge_index : tensor of shape (2, num_edges)\n Describe graph structure via edge list.\n \n size : tuple with shape\n Needed if edge_index is a matrix representing connections as \n (num_nodes, num_connected_nodes). \n\n See: https://github.com/pyg-team/pytorch_geometric/blob/master/torch_geometric/nn/conv/message_passing.py\n \n Returns\n -------\n\n out : tensor of shape (num_nodes, out_channels)\n Embedded data associated to each node.\n \"\"\"\n # Now what was called K is H, and F' is C... this guy is so confused\n H, C = self.heads, self.out_channels\n\n ############################################################################\n # TODO: Your code here! \n # Implement message passing, as well as any pre- and post-processing (our update rule).\n # 1. First apply linear transformation to node embeddings, and split that \n # into multiple heads. We use the same representations for source and\n # target nodes, but apply different linear weights (W_l and W_r)\n # 2. Calculate alpha vectors for central nodes (alpha_l) and neighbor nodes (alpha_r).\n # 3. Call propagate function to conduct the message passing. \n # 3.1 Remember to pass alpha = (alpha_l, alpha_r) as a parameter.\n # 3.2 See there for more information: https://pytorch-geometric.readthedocs.io/en/latest/notes/create_gnn.html\n # 4. Transform the output back to the shape of [N, H * C].\n # Our implementation is ~5 lines, but don't worry if you deviate from this.\n\n\n # 1. First apply linear transformation to node embeddings, and split that \n # into multiple heads. We use the same representations for source and\n # target nodes, but apply different linear weights (W_l and W_r)\n\n # Apply linear transformations\n x_i = self.lin_l(x) # (num_nodes, heads * out_channels)\n x_j = self.lin_r(x) # (num_nodes, heads * out_channels)\n\n # Split into multiple heads\n # I assume that split here means that we want to have a tensor of the \n # form \n # (num_nodes, heads, out_channels)\n # Then, we can treat heads independently, as if we are doing things in \n # parallel.\n #\n # Fortunately, tensor also has a .reshape method\n x_i = x_i.reshape(-1, H, C) # (num_nodes, heads, out_channels)\n x_j = x_j.reshape(-1, H, C) # (num_nodes, heads, out_channels)\n\n\n # 2. Calculate alpha vectors for central nodes (alpha_l) and neighbor nodes (alpha_r).\n # The guy defined it as \n # `alpha_l` = $\\alpha_l = [...,\\overrightarrow{a_l}^T \\mathbf{W_l} \\overrightarrow{h_i},...]\n # We already have the W h part... now, we neet to apply the attention coeff\n alpha_l = self.att_l * x_i # (num_nodes, heads, out_channels)\n alpha_r = self.att_r * x_j # (num_nodes, heads, out_channels)\n\n\n # 3. Call propagate function to conduct the message passing. \n # 3.1 Remember to pass alpha = (alpha_l, alpha_r) as a parameter.\n # 3.2 See there for more information: https://pytorch-geometric.readthedocs.io/en/latest/notes/create_gnn.html\n out = self.propagate(edge_index, x=(x_i, x_j), alpha=(alpha_l, alpha_r), size=size) # (num_nodes, heads, out_channels)\n\n\n # 4. Transform the output back to the shape of [N, H * C].\n out = out.reshape(-1, H * C) # (num_edges, heads * out_channels)\n\n ############################################################################\n\n return out\n\n\n def message(self, x_j, alpha_j, alpha_i, index, ptr, size_i):\n \"\"\"\n Messages from the neighbors. \n \n For GAT, the neighbors send their weighted raw data, where the weights\n are given by the attention mechanism.\n\n Parameters\n ----------\n\n x_j : tensor of shape (num_edges, out_channels)\n Data of the target node of each edge.\n\n alpha_j : tensor of shape (num_edges, heads, out_channels)\n Attention parameters for the target nodes.\n\n alpha_i : tensor of shape (num_edges, heads, out_channels)\n Attention parameters for the central nodes.\n\n index : tensor of shape (num_edges)\n List o indexes of the (target) nodes receiving the message:\n index = edge_list[1],\n since we are using the flow=\"source-to-target\".\n\n Returns\n -------\n\n out : tensor of shape (num_edges, head, out_channels)\n Messages sent to the central node from the target node of each edge.\n\n PS: Note how different is this message function from the one of Colab 3.\n\n \"\"\"\n out = None\n\n ############################################################################\n # TODO: Your code here! \n # Implement your message function. Putting the attention in message \n # instead of in update is a little tricky.\n # 1. Calculate the final attention weights using alpha_i and alpha_j,\n # and apply leaky Relu.\n # 2. Calculate softmax over the neighbor nodes for all the nodes. Use \n # torch_geometric.utils.softmax instead of the one in Pytorch.\n # 3. Apply dropout to attention weights (alpha).\n # 4. Multiply embeddings and attention weights. As a sanity check, the output\n # should be of shape [E, H, C].\n # 5. ptr (LongTensor, optional): If given, computes the softmax based on\n # sorted inputs in CSR representation. You can simply pass it to softmax.\n # Our implementation is ~4-5 lines, but don't worry if you deviate from this.\n\n # 1. Calculate the final attention weights using alpha_i and alpha_j,\n # and apply leaky Relu.\n #\n # Documentation on Leak ReLU: https://pytorch.org/docs/1.9.1/generated/torch.nn.LeakyReLU.html\n e_vector = F.leaky_relu(alpha_j + alpha_i, negative_slope=self.negative_slope) # (num_edges, heads, out_channels)\n\n # Following the notation of the guy, e_vector has the e_ij values before\n # normalization\n\n # 2. Calculate softmax over the neighbor nodes for all the nodes. Use \n # torch_geometric.utils.softmax instead of the one in Pytorch.\n if ptr:\n alpha = F.softmax(e_vector, dim=ptr) # (num_edges, heads, out_channels) \n # ptr figure outs how to compute the neighborhoods and so on\n # see: https://github.com/pyg-team/pytorch_geometric/blob/50b7bfc4a59b5b6f7ec547ff862985f3b2e22798/torch_geometric/nn/conv/message_passing.py#L336\n else:\n alpha = pyg_utils.softmax(e_vector, index=index) # (num_edges, heads, out_channels)\n # THIS FUNCTION IS ALREADY FIGURING OUT THE NEIGHBOR SETS FOR US TO COMPUTE\n # THE SOFTMAX FUNCTION, see: https://pytorch-geometric.readthedocs.io/en/1.3.2/_modules/torch_geometric/utils/softmax.html\n # Unsurprisingly, it uses the scatter function.\n\n # 3. Apply dropout to attention weights (alpha).\n alpha = F.dropout(alpha, p=self.dropout) # (num_edges, heads, out_channels)\n\n # 4. Multiply embeddings and attention weights. As a sanity check, the output\n # should be of shape [E, H, C].\n out = alpha * x_j # (num_edges, heads, out_channels)\n\n # This is denoted as h' in the explanation above, but without the sum\n # yet (aggreation); this will be realized in the aggreagation function\n\n # So, here we have this alpha Wr h\n\n\n # 5. ptr (LongTensor, optional): If given, computes the softmax based on\n # sorted inputs in CSR representation. You can simply pass it to softmax.\n #\n # - OK, why he didnt say this before.. lets go back to the softmax\n # implementation\n\n\n ############################################################################\n\n return out\n\n\n def aggregate(self, inputs, index, dim_size=None):\n \"\"\"\n Aggregates at the central node incoming messages from the neighbors.\n\n For GAT, the aggregation is a weighted sum over the data coming from\n the neighbors. The weights are defined by the attention mechanism.\n\n Parameters\n ----------\n\n inputs : tensor of shape (E, heads, out_channels)\n Messages sent to the central node from the target node of each edge.\n\n index : tensor of shape (num_edges)\n List o indexes of the (target) nodes receiving the message:\n index = edge_list[1],\n since we are using the flow=\"source-to-target\".\n\n dim_size : tuple with shape\n Tuple with shape of the aggregation.\n\n Returns\n -------\n\n out : tensor of shape (num_edges, heads, out_channels)\n Aggregated messages at the central nodes.\n \"\"\"\n\n ############################################################################\n # TODO: Your code here! \n # Implement your aggregate function here.\n # See here as how to use torch_scatter.scatter: https://pytorch-scatter.readthedocs.io/en/latest/_modules/torch_scatter/scatter.html\n # Pay attention to \"reduce\" parameter is different from that in GraphSage.\n # Our implementation is ~1 lines, but don't worry if you deviate from this.\n\n # Now, we just need to sum over the neighboorhood sets, see the h' equation\n # above!!\n out = torch_scatter.scatter(src=inputs, index=index, dim=self.node_dim, dim_size=dim_size, reduce=\"sum\")\n\n # dim needs to match the node dimension!! Using the node_dim\n # property that the MensagePassing class has; similar to Colab 3\n # see: https://github.com/pyg-team/pytorch_geometric/blob/50b7bfc4a59b5b6f7ec547ff862985f3b2e22798/torch_geometric/nn/conv/message_passing.py#L336\n\n ############################################################################\n\n return out\n```\n\n## Building Optimizers\n\nThis function has been implemented for you. **For grading purposes please use the default Adam optimizer**, but feel free to play with other types of optimizers on your own.\n\n\n```python\nimport torch.optim as optim\n\ndef build_optimizer(args, params):\n weight_decay = args.weight_decay\n filter_fn = filter(lambda p : p.requires_grad, params)\n if args.opt == 'adam':\n optimizer = optim.Adam(filter_fn, lr=args.lr, weight_decay=weight_decay)\n elif args.opt == 'sgd':\n optimizer = optim.SGD(filter_fn, lr=args.lr, momentum=0.95, weight_decay=weight_decay)\n elif args.opt == 'rmsprop':\n optimizer = optim.RMSprop(filter_fn, lr=args.lr, weight_decay=weight_decay)\n elif args.opt == 'adagrad':\n optimizer = optim.Adagrad(filter_fn, lr=args.lr, weight_decay=weight_decay)\n if args.opt_scheduler == 'none':\n return None, optimizer\n elif args.opt_scheduler == 'step':\n scheduler = optim.lr_scheduler.StepLR(optimizer, step_size=args.opt_decay_step, gamma=args.opt_decay_rate)\n elif args.opt_scheduler == 'cos':\n scheduler = optim.lr_scheduler.CosineAnnealingLR(optimizer, T_max=args.opt_restart)\n return scheduler, optimizer\n```\n\n## Training and Testing\n\nHere we provide you with the functions to train and test. **Please do not modify this part for grading purposes.**\n\n\n```python\nimport time\n\nimport networkx as nx\nimport numpy as np\nimport torch\nimport torch.optim as optim\nfrom tqdm import trange\nimport pandas as pd\nimport copy\n\nfrom torch_geometric.datasets import TUDataset\nfrom torch_geometric.datasets import Planetoid\nfrom torch_geometric.data import DataLoader\n\nimport torch_geometric.nn as pyg_nn\n\nimport matplotlib.pyplot as plt\n\n\ndef train(dataset, args):\n \n print(\"Node task. test set size:\", np.sum(dataset[0]['test_mask'].numpy()))\n print()\n test_loader = loader = DataLoader(dataset, batch_size=args.batch_size, shuffle=False)\n\n # build model\n model = GNNStack(dataset.num_node_features, args.hidden_dim, dataset.num_classes, \n args)\n scheduler, opt = build_optimizer(args, model.parameters())\n\n # train\n losses = []\n test_accs = []\n best_acc = 0\n best_model = None\n for epoch in trange(args.epochs, desc=\"Training\", unit=\"Epochs\"):\n total_loss = 0\n model.train()\n for batch in loader:\n opt.zero_grad()\n pred = model(batch)\n label = batch.y\n pred = pred[batch.train_mask]\n label = label[batch.train_mask]\n loss = model.loss(pred, label)\n loss.backward()\n opt.step()\n total_loss += loss.item() * batch.num_graphs\n total_loss /= len(loader.dataset)\n losses.append(total_loss)\n\n if epoch % 10 == 0:\n test_acc = test(test_loader, model)\n test_accs.append(test_acc)\n if test_acc > best_acc:\n best_acc = test_acc\n best_model = copy.deepcopy(model)\n else:\n test_accs.append(test_accs[-1])\n \n return test_accs, losses, best_model, best_acc, test_loader\n\ndef test(loader, test_model, is_validation=False, save_model_preds=False, model_type=None):\n test_model.eval()\n\n correct = 0\n # Note that Cora is only one graph!\n for data in loader:\n with torch.no_grad():\n # max(dim=1) returns values, indices tuple; only need indices\n pred = test_model(data).max(dim=1)[1]\n label = data.y\n\n mask = data.val_mask if is_validation else data.test_mask\n # node classification: only evaluate on nodes in test set\n pred = pred[mask]\n label = label[mask]\n\n if save_model_preds:\n print (\"Saving Model Predictions for Model Type\", model_type)\n\n data = {}\n data['pred'] = pred.view(-1).cpu().detach().numpy()\n data['label'] = label.view(-1).cpu().detach().numpy()\n\n df = pd.DataFrame(data=data)\n # Save locally as csv\n df.to_csv('CORA-Node-' + model_type + '.csv', sep=',', index=False)\n \n correct += pred.eq(label).sum().item()\n\n total = 0\n for data in loader.dataset:\n total += torch.sum(data.val_mask if is_validation else data.test_mask).item()\n\n return correct / total\n \nclass objectview(object):\n def __init__(self, d):\n self.__dict__ = d\n\n```\n\n## Let's Start the Training!\n\nWe will be working on the CORA dataset on node-level classification.\n\nThis part is implemented for you. **For grading purposes, please do not modify the default parameters.** However, feel free to play with different configurations just for fun!\n\n**Submit your best accuracy and loss on Gradescope.**\n\n\n```python\nif 'IS_GRADESCOPE_ENV' not in os.environ:\n for args in [{\n 'model_type': 'GAT',\n 'dataset': 'cora',\n 'num_layers': 2,\n 'heads': 1,\n 'batch_size': 32,\n 'hidden_dim': 32,\n 'dropout': 0.5,\n 'epochs': 500,\n 'opt': 'adam',\n 'opt_scheduler': 'none',\n 'opt_restart': 0,\n 'weight_decay': 5e-3,\n 'lr': 0.01},\n ]:\n args = objectview(args)\n for model in ['GAT']:\n args.model_type = model\n\n # Match the dimension.\n if model == 'GAT':\n args.heads = 2\n else:\n args.heads = 1\n\n if args.dataset == 'cora':\n dataset = Planetoid(root='/tmp/cora', name='Cora')\n else:\n raise NotImplementedError(\"Unknown dataset\") \n test_accs, losses, best_model, best_acc, test_loader = train(dataset, args) \n\n print(\"Maximum test set accuracy: {0}\".format(max(test_accs)))\n print(\"Minimum loss: {0}\".format(min(losses)))\n\n # Run test for our best model to save the predictions!\n test(test_loader, best_model, is_validation=False, save_model_preds=True, model_type=model)\n print()\n\n plt.title(dataset.name)\n plt.plot(losses, label=\"training loss\" + \" - \" + args.model_type)\n plt.plot(test_accs, label=\"test accuracy\" + \" - \" + args.model_type)\n plt.legend()\n plt.show()\n\n```\n\n## Question 1: What is the maximum accuracy obtained on test set for GAT? (10 points)\n\n\nRunning the training cell above will also save your best GAT model predictions as *CORA-Node-GAT.csv*. \n\nWhen you sumbit your assignment, you will have to download this file and attatch it to your submission. As with the other colabs, please zip this file (DON'T CHANGE ITS NAME) and the .csv file that's generated!\n\n", "meta": {"hexsha": "5292b73310efa402795071f7b7e345bc4581c8e5", "size": 80448, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Colab 4/CS224W - Colab4_victor.ipynb", "max_stars_repo_name": "victorcroisfelt/aau-cs224w-ml-with-graphs", "max_stars_repo_head_hexsha": "adb38651be8da98cc574f127763c785ed16dfb5a", "max_stars_repo_licenses": ["Unlicense", "MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Colab 4/CS224W - Colab4_victor.ipynb", "max_issues_repo_name": "victorcroisfelt/aau-cs224w-ml-with-graphs", "max_issues_repo_head_hexsha": "adb38651be8da98cc574f127763c785ed16dfb5a", "max_issues_repo_licenses": ["Unlicense", "MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Colab 4/CS224W - Colab4_victor.ipynb", "max_forks_repo_name": "victorcroisfelt/aau-cs224w-ml-with-graphs", "max_forks_repo_head_hexsha": "adb38651be8da98cc574f127763c785ed16dfb5a", "max_forks_repo_licenses": ["Unlicense", "MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 76.8366762178, "max_line_length": 22574, "alphanum_fraction": 0.6625894988, "converted": true, "num_tokens": 10208, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.7057850154599562, "lm_q1q2_score": 0.431504235473351}} {"text": "```bash\n%%bash -e\nif ! [[ -f ./linkern ]]; then\n wget http://www.math.uwaterloo.ca/tsp/concorde/downloads/codes/src/co031219.tgz\n echo 'c3650a59c8d57e0a00e81c1288b994a99c5aa03e5d96a314834c2d8f9505c724 co031219.tgz' | sha256sum -c\n tar xf co031219.tgz\n (cd concorde && CFLAGS='-O3 -march=native -mtune=native -fPIC' ./configure --build=x86_64)\n (cd concorde/LINKERN && make -j && cp linkern ../../)\n rm -rf concorde co031219.tgz\nfi\n```\n\n\n```python\nfrom concorde.tsp import TSPSolver\nfrom matplotlib import collections as mc\nimport numpy as np\nimport pandas as pd\nimport time\nimport pylab as pl\n```\n\n\n```python\ncities = pd.read_csv('cities.csv', index_col=['CityId'])\n```\n\n\n```python\ncities1k = cities * 1000\n```\n\n\n```python\ndef write_tsp(cities, filename, name='traveling-santa-2018-prime-paths'):\n with open(filename, 'w') as f:\n f.write('NAME : %s\\n' % name)\n f.write('COMMENT : %s\\n' % name)\n f.write('TYPE : TSP\\n')\n f.write('DIMENSION : %d\\n' % len(cities))\n f.write('EDGE_WEIGHT_TYPE : EUC_2D\\n')\n f.write('NODE_COORD_SECTION\\n')\n for row in cities.itertuples():\n f.write('%d %.11f %.11f\\n' % (row.Index+1, row.X, row.Y))\n f.write('EOF\\n')\n\nwrite_tsp(cities1k, 'cities1k.tsp')\n```\n\n\n```bash\n%%bash -e\ntime ./linkern -s 42 -S linkern.tour -R 1000000000 -t 18000 ./cities1k.tsp >linkern.log\n```\n\n\n```python\n!sed -Ene 's/([0-9]+) Steps.*Best: ([0-9]+).*/\\1,\\2/p' linkern.log >linkern.csv\npd.read_csv('linkern.csv', index_col=0, names=['TSP tour length']).plot();\n```\n\n\n```python\nimport sympy\ndef read_tour(filename):\n tour = open(filename).read().split()[1:]\n tour = list(map(int, tour))\n if tour[-1] == 0: tour.pop()\n return tour\n\ndef score_tour(tour):\n df = cities.reindex(tour + [0]).reset_index()\n primes = list(sympy.primerange(0, len(cities)))\n df['prime'] = df.CityId.isin(primes).astype(int)\n df['dist'] = np.hypot(df.X - df.X.shift(-1), df.Y - df.Y.shift(-1))\n df['penalty'] = df['dist'][9::10] * (1 - df['prime'][9::10]) * 0.1\n return df.dist.sum() + df.penalty.sum()\n\ndef write_submission(tour, filename):\n assert set(tour) == set(range(len(tour)))\n pd.DataFrame({'Path': list(tour) + [0]}).to_csv(filename, index=False)\n```\n\n\n```python\ntour = read_tour('linkern.tour')\nwrite_submission(tour, 'submission.csv')\n```\n\n\n```python\nscore_tour(tour)\n```\n\n\n\n\n 1517361.2529791819\n\n\n\n\n```python\ndef flip(lst, k):\n return lst[0:1] + lst[k+1:] + lst[1:k+1]\n```\n\n\n```python\nprint(\"Tour path (0-5):\",tour[0:5])\ntourflip = tour[::-1]\nprint(\"Flipped tour path (0-5):\", tourflip[0:5])\n```\n\n ('Tour path (0-5):', [78934, 111804, 52086, 89712, 81072])\n ('Flipped tour path (0-5):', [0, 48816, 40230, 75405, 153911])\n\n\n\n```python\noriginScore = score_tour(tour)\nprint(\"Score of original tour:\", originScore)\ntourflip = tour\nfor n in range(1,101):\n # And the flipped tour looks like:\n tourflip = flip(tourflip, 1)\n # The scores of our tours are:\n flippedScore = score_tour(tourflip)\n print(\"Score of flipped tour:\", flippedScore)\n\n # If the flipped tour is quicker, change our tour:\n if flippedScore < originScore:\n print(\"The total improvement was:\", abs(flippedScore - originScore))\n originScore = flippedScore\n tour = tourflip \n print(\"The better of the original/flipped tour is:\", tour[0:5])\n```\n\n ('Score of original tour:', 1517332.415616737)\n ('Score of flipped tour:', 1517356.0910889404)\n ('Score of flipped tour:', 1517356.9139951372)\n ('Score of flipped tour:', 1517413.2228083517)\n ('Score of flipped tour:', 1517403.1202388373)\n ('Score of flipped tour:', 1517403.8425916804)\n ('Score of flipped tour:', 1517390.3233206801)\n ('Score of flipped tour:', 1517426.1401942007)\n ('Score of flipped tour:', 1517471.5298787779)\n ('Score of flipped tour:', 1517440.7268941959)\n ('Score of flipped tour:', 1517424.4124984485)\n ('Score of flipped tour:', 1517426.253180478)\n ('Score of flipped tour:', 1517421.8727339443)\n ('Score of flipped tour:', 1517460.3421580733)\n ('Score of flipped tour:', 1517434.6743891994)\n ('Score of flipped tour:', 1517418.130972293)\n ('Score of flipped tour:', 1517393.934870807)\n ('Score of flipped tour:', 1517420.2643460205)\n ('Score of flipped tour:', 1517447.2864386647)\n ('Score of flipped tour:', 1517407.5226760462)\n ('Score of flipped tour:', 1517394.2942589528)\n ('Score of flipped tour:', 1517394.7993746104)\n ('Score of flipped tour:', 1517379.850304974)\n ('Score of flipped tour:', 1517414.5208543059)\n ('Score of flipped tour:', 1517387.1811606274)\n ('Score of flipped tour:', 1517381.9075844737)\n ('Score of flipped tour:', 1517369.7748633618)\n ('Score of flipped tour:', 1517413.2023676371)\n ('Score of flipped tour:', 1517456.4925552462)\n ('Score of flipped tour:', 1517417.9342675023)\n ('Score of flipped tour:', 1517390.2807072725)\n ('Score of flipped tour:', 1517375.7881616235)\n ('Score of flipped tour:', 1517354.4794941714)\n ('Score of flipped tour:', 1517391.7376516568)\n ('Score of flipped tour:', 1517373.4370702687)\n ('Score of flipped tour:', 1517382.8213495801)\n ('Score of flipped tour:', 1517374.1632063612)\n ('Score of flipped tour:', 1517415.894082329)\n ('Score of flipped tour:', 1517467.3961351814)\n ('Score of flipped tour:', 1517445.8441427345)\n ('Score of flipped tour:', 1517435.2475747936)\n ('Score of flipped tour:', 1517439.9246245627)\n ('Score of flipped tour:', 1517447.1559378831)\n ('Score of flipped tour:', 1517499.0819099913)\n ('Score of flipped tour:', 1517493.0927003776)\n ('Score of flipped tour:', 1517502.3384920147)\n ('Score of flipped tour:', 1517485.5076681615)\n ('Score of flipped tour:', 1517502.5371459988)\n ('Score of flipped tour:', 1517526.8949108373)\n ('Score of flipped tour:', 1517477.4368143654)\n ('Score of flipped tour:', 1517438.1595332872)\n ('Score of flipped tour:', 1517428.0130642096)\n ('Score of flipped tour:', 1517409.3595959663)\n ('Score of flipped tour:', 1517431.0013628749)\n ('Score of flipped tour:', 1517393.0333302217)\n ('Score of flipped tour:', 1517372.2586601183)\n ('Score of flipped tour:', 1517349.1398652336)\n ('Score of flipped tour:', 1517387.9553650559)\n ('Score of flipped tour:', 1517442.3627667939)\n ('Score of flipped tour:', 1517424.9399056523)\n ('Score of flipped tour:', 1517423.5400056166)\n ('Score of flipped tour:', 1517439.107014118)\n ('Score of flipped tour:', 1517455.0654943897)\n ('Score of flipped tour:', 1517519.386947802)\n ('Score of flipped tour:', 1517498.676528994)\n ('Score of flipped tour:', 1517498.4435672665)\n ('Score of flipped tour:', 1517492.0689992518)\n ('Score of flipped tour:', 1517525.8929229486)\n ('Score of flipped tour:', 1517580.3247003811)\n ('Score of flipped tour:', 1517556.868937073)\n ('Score of flipped tour:', 1517537.0343341671)\n ('Score of flipped tour:', 1517524.3500662614)\n ('Score of flipped tour:', 1517524.9141456622)\n ('Score of flipped tour:', 1517585.1937728256)\n ('Score of flipped tour:', 1517573.1172919387)\n ('Score of flipped tour:', 1517573.151211919)\n ('Score of flipped tour:', 1517550.4248782436)\n ('Score of flipped tour:', 1517581.9113448008)\n ('Score of flipped tour:', 1517626.8242998021)\n ('Score of flipped tour:', 1517599.5859399824)\n ('Score of flipped tour:', 1517586.2745984038)\n ('Score of flipped tour:', 1517579.5737715021)\n ('Score of flipped tour:', 1517570.7709022472)\n ('Score of flipped tour:', 1517621.9204671208)\n ('Score of flipped tour:', 1517610.5224881524)\n ('Score of flipped tour:', 1517608.9746056185)\n ('Score of flipped tour:', 1517594.5004874978)\n ('Score of flipped tour:', 1517625.8114939115)\n ('Score of flipped tour:', 1517667.3593493842)\n ('Score of flipped tour:', 1517638.9068630573)\n ('Score of flipped tour:', 1517618.783138324)\n ('Score of flipped tour:', 1517603.4920099739)\n ('Score of flipped tour:', 1517576.0349226668)\n ('Score of flipped tour:', 1517627.2630021616)\n ('Score of flipped tour:', 1517631.1658613831)\n ('Score of flipped tour:', 1517641.401856899)\n ('Score of flipped tour:', 1517611.6196710353)\n ('Score of flipped tour:', 1517620.6900758257)\n ('Score of flipped tour:', 1517642.3172794664)\n ('Score of flipped tour:', 1517612.9561278727)\n ('Score of flipped tour:', 1517606.5370181862)\n\n\n\n```python\nscore_tour(tour)\n```\n\n\n\n\n 1517332.415616737\n\n\n\n\n```python\nwrite_submission(tour, 'submission.csv')\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "2669973adcf4d07f5d42d8c6c2f1aaee1b83da8d", "size": 25480, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "linkern.ipynb", "max_stars_repo_name": "yarntime/TSP", "max_stars_repo_head_hexsha": "a6427c8f35e85fd3e432e72f897bbfa1e4389007", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "linkern.ipynb", "max_issues_repo_name": "yarntime/TSP", 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YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.7279754548076477, "lm_q1q2_score": 0.43144673765086616}} {"text": "# Problema de selecci\u00f3n de portafolio con preferencias media-varianza\n\n\n\nEn la clase pasada hablamos acerca de:\n- preferencias,\n- funciones de utilidad,\n- la actitud de los inversionistas de cara al riesgo,\n- la aversi\u00f3n al riesgo, entre otros.\n\nTodas ellas son piezas que necesitamos para responder la pregunta de \u00bfc\u00f3mo un inversionista toma la decisi\u00f3n \u00f3ptima de selecci\u00f3n de portafolio?\n\nEn esta clase al fin estamos listos para ensamblar estos conceptos y escribir el problema de selecci\u00f3n de portafolios. \n\nEn el camino aprenderemos acerca del concepto de **utilidad esperada**, que nos permite trabajar con incertidumbre en el modelado econ\u00f3mico (una de las ideas m\u00e1s importantes en econom\u00eda). Esta idea tiene m\u00e1s de 60 a\u00f1os, y b\u00e1sicamente dice que los individuos, cuando est\u00e1n de cara a incertidumbre, maximizan el valor esperado de su utilidad (solo cierto si somos homo economicus). \n\nAdem\u00e1s del concepto de utilidad esperada, aprenderemos acerca de **preferencias media-varianza**. Es decir, supondremos que los inversionistas toman decisiones basados en un tipo particular de preferencias.\n\nCon lo anterior, estableceremos el problema de selecci\u00f3n de portafolios.\n\n**Objetivos:**\n\n- \u00bfQu\u00e9 es utilidad esperada?\n- \u00bfQu\u00e9 son las preferencias media-varianza?\n- Funciones de utilidad media-varianza.\n- Enunciado y soluci\u00f3n del problema b\u00e1sico de selecci\u00f3n de portafolio.\n\n*Referencia:*\n- Notas del curso \"Portfolio Selection and Risk Management\", Rice University, disponible en Coursera.\n___\n\n## 1. Utilidad esperada\n- B\u00e1sicamente combina las probabilidades de los resultados con c\u00f3mo los inversionistas se sienten con dichos resultados.\n- En otras palabras, la utilidad esperada multiplica la probabilidad de suceso de un evento con la utilidad que genera dicho evento.\n\nRecordemos que las *funciones de utilidad* permiten a los inversionistas expresar c\u00f3mo se sienten con los resultados, especialmente en los malos ratos. \n\nEntonces la *utilidad esperada* es una herramienta que nos permite cuantificar c\u00f3mo nos sentimos en nuestros malos momentos econ\u00f3micos, capturando el riesgo con la probabilidad de ocurrencia de dichos malos momentos. \n\nDado este marco de trabajo, cualquier decisi\u00f3n se puede escribir como la maximizaci\u00f3n de la utilidad esperada:\n\\begin{align}\n\\max_{\\theta} & \\quad E[U(W)], \\\\\n\\end{align}\nmediante la escogencia de cierta variable $\\theta$ (gastos, planes de ahorro, compra de activos, planes de producci\u00f3n, etc.).\n\nPara nuestros prop\u00f3sitos, la variable de decisi\u00f3n ser\u00e1n los pesos o ponderaciones del portafolio.\n\nAdicionalmente, en el contexto de la decisi\u00f3n de distribuci\u00f3n de la riqueza entre activos, el problema de maximizaci\u00f3n tendr\u00e1 com\u00fanmente las siguientes restricciones:\n- universo de inversi\u00f3n,\n- posici\u00f3n en los activos dados.\n\n**Ejemplo.** \n\nSupongamos que un inversionista debe determinar la composici\u00f3n \u00f3ptima de su portafolio, que contiene activos y bonos. Supongamos que son los \u00fanicos instrumentos disponibles.\n\nSean:\n- $w_s$: peso o ponderaci\u00f3n de activos en el portafolio,\n- $w_b$: peso o ponderaci\u00f3n de bonos en el portafolio,\n- $r_s$: rendimiento de los activos, y\n- $r_b$: rendimiento de los bonos.\n\nDe manera que podemos escribir el problema de selecci\u00f3n de portafolios como la maximizaci\u00f3n de la utilidad esperade de nuestra riqueza futura, la cual depender\u00e1 de nuestros rendimientos:\n \n\\begin{align}\n\\max_{w_s,w_b} &\\quad E[U(W)]\\\\\n\\text{s. a.} &\\quad W=W_0(1+w_sr_s+w_br_b)\\\\\n &\\quad w_s+w_b=1\n\\end{align}\n\n\nPreguntas:\n- \u00bfQu\u00e9 significan las restricciones?\n- Ya que tenemos planteado este problema b\u00e1sico, \u00bfqu\u00e9 har\u00eda falta para empezar a resolverlo?\n___\n\n## 2. Preferencias media-varianza\n\n### 2.1. Utilidad media-varianza\n\nEntonces, \u00bfqu\u00e9 funciones de utilidad deber\u00edamos de usar en este problema de selecci\u00f3n de portafolios?\n\n- La respuesta es: **preferencias media-varianza**.\n- \u00c9stas ser\u00e1n representadas en t\u00e9rminos de funciones de utilidad como: **utilidad media-varianza**.\n\nUsamos la *utilidad media-varianza* en el problema de selecci\u00f3n de portafolios dado que \u00e9sta decribe el \"trade-off\" entre riesgo y rendimiento que enfrentan los inversionistas. La *utilidad media-varianza* est\u00e1 dada por la siguiente expresi\u00f3n:\n\n$$U=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2,$$\n\ndonde\n- $E[r_p]$ es el rendimiento esperado del portafolio,\n- $\\sigma_p^2$ es la varianza del portafolio, y\n- $\\gamma$ es el coeficiente de aversi\u00f3n al riesgo. \n\n#### Intuici\u00f3n acerca de la funci\u00f3n de utilidad media-varianza:\n- S\u00f3lo se preocupa por medias :) y varianzas :(.\n- Incrementa con: rendimiento esperado del portafolio.\n- Decrece con: varianza del portafolio.\n- Malos tiempos: rendimientos son bajos y las volatilidades son altas.\n- Conecta bastante bien con la teor\u00eda moderna de portafolios, la cual caracteriza los rendimientos con medias y varianzas \u00fanicamente.\n- Criticada por su limitaci\u00f3n: supone que los inversionistas s\u00f3lo se preocupan por medias y varianzas.\n\n### 2.2. Curvas de indiferencia\n\n*\u00bfRecuerdan las curvas de nivel que se ven en c\u00e1lculo de varias variables?*\n- Bien, ac\u00e1 nos servir\u00e1n para representar la utilidad media-varianza gr\u00e1ficamente.\n- En el contexto de utilidad media-varianza, las curvas de nivel se llaman **curvas de indiferencia**.\n\nDados ciertos niveles de utilidad $U_1>U_2>U_3$, las curvas de indiferencia relativas a estos niveles de utilidad, son los lugares geom\u00e9tricos en el espacio de rendimiento esperado vs. volatilidad representados por las siguientes expresiones\n\n$$U_1=E[r_p]-\\gamma\\sigma_p^2\\Rightarrow E[r_p]=\\gamma\\sigma_p^2+U_1,$$\n\n$$U_2=E[r_p]-\\gamma\\sigma_p^2\\Rightarrow E[r_p]=\\gamma\\sigma_p^2+U_2,$$\n\n$$U_3=E[r_p]-\\gamma\\sigma_p^2\\Rightarrow E[r_p]=\\gamma\\sigma_p^2+U_3.$$\n\n**Gr\u00e1ficamente**\n\n\n```python\n# Importar numpy y pyplot\nimport numpy as np\nimport matplotlib.pyplot as plt\n%matplotlib inline\n```\n\n\n```python\n# Coeficiente de aversi\u00f3n al riesgo\ng = 3\n# Niveles de utilidad\nU1, U2, U3 = 3, 2, 1\n# Vector de volatilidades\nsp = np.linspace(0.01, 0.7, 100)\n# Curvas de indiferencia\nErp1, Erp2, Erp3 = 0.5*g*sp**2+U1, 0.5*g*sp**2+U2, 0.5*g*sp**2+U3\n```\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(8,6))\nplt.plot(sp, Erp1, lw = 3, label=\"$U_1$\")\nplt.plot(sp, Erp2, lw = 3, label=\"$U_2$\")\nplt.plot(sp, Erp3, lw = 3, label=\"$U_3$\")\nplt.xlabel(\"Volatilidad $\\sigma$\")\nplt.ylabel(\"Rendimiento esperado $E[r]$\")\nplt.legend(loc=\"best\")\nplt.title(\"Curvas de indiferencia\")\nplt.grid()\nplt.show()\n```\n\nBueno, \u00bfy porqu\u00e9 se llaman curvas de indiferencia?, \u00bfqu\u00e9 representa una cuva de indiferencia?\n- Sobre las combinaciones de riesgo-rendimiento en una misma curva, la utilidad es indiferente.\n- Las combinaciones de riesgo-rendimiento, para las cuales la utilidad tiene un nivel dado.\n\nVolviendo al problema de selecci\u00f3n de portafolios, queremos la utilidad m\u00e1s alta.\n- \u00bfCu\u00e1l de las anteriores curvas de indiferencia corresponde a la utilidad m\u00e1s alta?\n- Intuitivamente, \u00bfporqu\u00e9?\n- Curvas de indiferencia para niveles de utilidad m\u00e1s altos, tendr\u00e1n...\n\nNotamos adem\u00e1s que las anteriores curvas de indiferencia son *paralelas* una con otra. Claro, las dibujamos con el mismo coeficiente de aversi\u00f3n al riesgo.\n\n\u00bfC\u00f3mo cambian estas curvas para coeficientes de aversi\u00f3n al riesgo m\u00e1s altos?\n\n\n```python\n# Coeficientes de aversi\u00f3n al riesgo\ng1, g2, g3 = 3, 5, 7\n# Nivel de utilidad\nU = 1\n# Vector de volatilidades\nsp = np.linspace(0.01, 0.7, 100)\n# Curvas de indiferencia\nErp1, Erp2, Erp3 = 0.5*g1*sp**2+U, 0.5*g2*sp**2+U, 0.5*g3*sp**2+U\n```\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(8,6))\nplt.plot(sp, Erp1, lw = 3, label=\"$\\gamma_1$\")\nplt.plot(sp, Erp2, lw = 3, label=\"$\\gamma_2$\")\nplt.plot(sp, Erp3, lw = 3, label=\"$\\gamma_3$\")\nplt.xlabel(\"Volatilidad $\\sigma$\")\nplt.ylabel(\"Rendimiento esperado $E[r]$\")\nplt.legend(loc=\"best\")\nplt.title(\"Curvas de indiferencia\")\nplt.grid()\nplt.show()\n```\n\n\u00bfC\u00f3mo interpretamos las anteriores gr\u00e1ficas?, \u00bfqu\u00e9 pasa con las personas m\u00e1s aversas al riesgo?\n- Para un nivel de volatilidad dado, una persona m\u00e1s aversa al riesgo requiere mayor rendimiento esperado para obtener el mismo nivel de utilidad.\n\nCon todo lo anterior, el problema de selecci\u00f3n de portafolios se puede plantear como *\"encontrar la curva de indeferencia m\u00e1s alta dado el conjunto de oportunidades de inversi\u00f3n y restricciones\"*.\n\n## 3. Problema de selecci\u00f3n de portafolios: una ilustraci\u00f3n\n\nAhora ilustraremos el problema de selecci\u00f3n de portafolios con algunos datos. \n- Por ahora solo queremos ilustrar gr\u00e1ficamente c\u00f3mo se resuelve este problema. Trabajar en la intuici\u00f3n.\n- En las siguientes dos clases nos enfocaremos en c\u00f3mo resolverlo anal\u00edticamente.\n\nAc\u00e1 tenemos el rendimiento medio anual y la volatilidad para dos instrumentos usando datos de EU: instrumentos de deuda (bonos) y acciones. Supondremos que el inversionista solo puede invertir en estas dos clases de instrumentos.\n\n\n```python\n# Importamos pandas\nimport pandas as pd\n```\n\n\n```python\n# Datos\ndata=pd.DataFrame(index=['Stocks','Bonds', 'CorrSB'], columns=['Mean', 'Std'])\ndata['Mean'] = np.array([0.119,0.0591,0.113])\ndata['Std'] = np.array([0.1915,0.0833,None])\ndata.round(4)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        MeanStd
                                        Stocks0.11900.1915
                                        Bonds0.05910.0833
                                        CorrSB0.1130None
                                        \n
                                        \n\n\n\nEntonces, \u00bfcu\u00e1l es la distribuci\u00f3n de riqueza \u00f3ptima?, o m\u00e1s bien, \u00bfcu\u00e1l es la composici\u00f3n \u00f3ptima del portafolio para un inversionista dado su nivel de aversi\u00f3n al riesgo?\n\n**Primero.** Recordamos que, para dos activos, podemos trazar la frontera de m\u00ednima varianza tomando todas las posibles combinaciones de los dos activos.\n\nDe nuevo, sean:\n- $w_s=w$: peso o ponderaci\u00f3n de activos en el portafolio,\n- $w_b=1-w$: peso o ponderaci\u00f3n de bonos en el portafolio,\n- $r_s$: rendimiento de los activos, y\n- $r_b$: rendimiento de los bonos.\n\nEntonces\n\n$$E[r_p]=wE[r_{s}]+(1-w)E[r_b]$$\n\n$$\\sigma_p^2=w^2\\sigma_{s}^2+(1-w)^2\\sigma_b^2+2w(1-w)\\rho_{s,b}\\sigma_s\\sigma_b$$\n\n\n```python\n# Vector de w variando entre 0 y 1 con n pasos\nn = 101\nw = np.linspace(0, 1, n)\n# Rendimientos esperados individuales\nErs, Erb = data.loc[\"Stocks\", \"Mean\"], data.loc[\"Bonds\", \"Mean\"]\n# Volatilidades individuales\nss, sb = data.loc[\"Stocks\", \"Std\"], data.loc[\"Bonds\", \"Std\"]\n# Correlacion\nrsb = data.loc[\"CorrSB\",\"Mean\"]\n```\n\n\n```python\n# Crear un DataFrame cuyas columnas sean rendimiento\n# y volatilidad del portafolio para cada una de las w\n# generadas\nportfolios = pd.DataFrame(columns=[\"Ret\", \"Vol\"], index=w)\nportfolios.Ret = w*Ers+(1-w)*Erb\nportfolios.Vol = ((w*ss)**2+((1-w)*sb)**2+2*w*(1-w)*rsb*ss*sb)**(1/2)\nportfolios\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        RetVol
                                        0.000.0591000.083300
                                        0.010.0596990.082705
                                        0.020.0602980.082155
                                        0.030.0608970.081650
                                        0.040.0614960.081191
                                        0.050.0620950.080779
                                        0.060.0626940.080415
                                        0.070.0632930.080099
                                        0.080.0638920.079832
                                        0.090.0644910.079614
                                        0.100.0650900.079446
                                        0.110.0656890.079328
                                        0.120.0662880.079261
                                        0.130.0668870.079244
                                        0.140.0674860.079277
                                        0.150.0680850.079361
                                        0.160.0686840.079495
                                        0.170.0692830.079679
                                        0.180.0698820.079913
                                        0.190.0704810.080195
                                        0.200.0710800.080527
                                        0.210.0716790.080907
                                        0.220.0722780.081334
                                        0.230.0728770.081808
                                        0.240.0734760.082327
                                        0.250.0740750.082892
                                        0.260.0746740.083501
                                        0.270.0752730.084153
                                        0.280.0758720.084847
                                        0.290.0764710.085582
                                        .........
                                        0.710.1016290.140756
                                        0.720.1022280.142414
                                        0.730.1028270.144080
                                        0.740.1034260.145755
                                        0.750.1040250.147437
                                        0.760.1046240.149128
                                        0.770.1052230.150826
                                        0.780.1058220.152532
                                        0.790.1064210.154244
                                        0.800.1070200.155964
                                        0.810.1076190.157690
                                        0.820.1082180.159422
                                        0.830.1088170.161161
                                        0.840.1094160.162905
                                        0.850.1100150.164656
                                        0.860.1106140.166412
                                        0.870.1112130.168173
                                        0.880.1118120.169940
                                        0.890.1124110.171712
                                        0.900.1130100.173489
                                        0.910.1136090.175271
                                        0.920.1142080.177057
                                        0.930.1148070.178848
                                        0.940.1154060.180643
                                        0.950.1160050.182443
                                        0.960.1166040.184246
                                        0.970.1172030.186054
                                        0.980.1178020.187866
                                        0.990.1184010.189681
                                        1.000.1190000.191500
                                        \n

                                        101 rows \u00d7 2 columns

                                        \n
                                        \n\n\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(8,6))\nplt.plot(portfolios.Vol, portfolios.Ret, lw=3, label=\"Portafolios\")\nplt.xlabel(\"Volatilidad $\\sigma$\")\nplt.ylabel(\"Rendimiento esperado $E[r]$\")\nplt.legend(loc=\"best\")\nplt.grid()\nplt.show()\n```\n\n**Segundo.** Graficamos en la misma ventana, curvas de indiferencia.\n\n\n```python\n# Niveles de utilidad 0.04, 0.05, 0.06\nU1, U2, U3 = 0.07, 0.08, 0.083\n# Coeficiente de aversi\u00f3n al riesgo\ng = 2\n# Curvas de indiferencia\nErp1 = 0.5*g*portfolios.Vol**2+U1\nErp2 = 0.5*g*portfolios.Vol**2+U2\nErp3 = 0.5*g*portfolios.Vol**2+U3\n```\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(8,6))\nplt.plot(portfolios.Vol, portfolios.Ret, lw=3, label=\"Portafolios\")\nplt.plot(portfolios.Vol, Erp1, '--', lw = 3, label=\"$U_1$\")\nplt.plot(portfolios.Vol, Erp2, '--', lw = 3, label=\"$U_2$\")\nplt.plot(portfolios.Vol, Erp3, '--', lw = 3, label=\"$U_3$\")\nplt.xlabel(\"Volatilidad $\\sigma$\")\nplt.ylabel(\"Rendimiento esperado $E[r]$\")\nplt.legend(loc=\"best\")\nplt.grid()\nplt.show()\n```\n\n**Tercero.** La elecci\u00f3n \u00f3ptima est\u00e1 dada por la curva de indiferencia para el nivel de utilidad m\u00e1s alto que es tangente a la frontera media-varianza.\n- Claramente, esta selecci\u00f3n depende del coeficiente de aversi\u00f3n al riesgo.\n\n\n```python\n# Gr\u00e1fica con zoom\nplt.figure(figsize=(8,6))\nplt.plot(portfolios.Vol, portfolios.Ret, lw=3, label=\"Portafolios\")\nplt.plot(portfolios.Vol, Erp1, '--', lw = 3, label=\"$U_1$\")\nplt.plot(portfolios.Vol, Erp2, '--', lw = 3, label=\"$U_2$\")\nplt.plot(portfolios.Vol, Erp3, '--', lw = 3, label=\"$U_3$\")\nplt.xlabel(\"Volatilidad $\\sigma$\")\nplt.ylabel(\"Rendimiento esperado $E[r]$\")\nplt.legend(loc=\"best\")\nplt.xlim([0.15, 0.19])\nplt.ylim([0.105, 0.12])\nplt.grid()\nplt.show()\n```\n\n# Anuncios parroquiales\n\n## 1. Quiz la siguiente clase.\n## 2. Un par de art\u00edculos del WSJ y el NYT que discuten herramientas disponibles para la medici\u00f3n de su propia tolerancia al riesgo:\n- [Art\u00edculo 1](https://www.nytimes.com/2016/02/13/your-money/as-stocks-fall-its-time-to-measure-your-risk-tolerance.html)\n- [Art\u00edculo 2](https://www.wsj.com/articles/check-your-tolerance-for-investment-risk-now-before-markets-sag-1405619939)\n\n\n\n
                                        \nCreated with Jupyter by Esteban Jim\u00e9nez Rodr\u00edguez.\n
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YES\n2. YES", "lm_q1_score": 0.5736783928749127, "lm_q2_score": 0.7520125737597972, "lm_q1q2_score": 0.43141336473624725}} {"text": "Note that python fsps is needed and not installed on the CITA machines by default so all results will be in the pdf\n\n\n```python\nimport fsps\nimport math\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom sympy import *\nimport emcee\nfrom IPython.display import display, Math\nimport corner\nfrom astropy.cosmology import WMAP9 as cosmo\n```\n\nPart 1\n\n\n```python\ndef create_plot(x, y, x_lim, x_label, y_lim, y_label, loglog=True):\n \"\"\"Creates plot of x vs. y with xlim of x_lim and ylim of y_lim and\n respective labels (log-log scale or log-y scale)\n\n Parameters:\n x - np array of x values\n y - np array of corresponding y values\n x_lim - Tuple containing leftmost and rightmost x value to be on graph\n y_lim - Tuple containing lowest and highest y value to be on graph\n x_label - Label for x-axis\n y_label - Label for y-axis\n loglog - By default, graphs with a log x and y axis, if true, graphs\n with a log y axis only\n \n\n Returns:\n Function has no return value, however, it shows the matplotlib graph\n \"\"\"\n plt.figure(figsize=(15, 9))\n \n if loglog:\n plt.loglog(x, y)\n else:\n plt.semilogy(x, y)\n \n plt.xlabel(x_label, fontsize=14)\n plt.xlim(x_lim)\n plt.ylabel(y_label, fontsize=14)\n plt.ylim(y_lim)\n plt.savefig(\"plot.pdf\")\n plt.show()\n```\n\n\n```python\ndef linear_interpolation(x_s, y_s, x_val):\n \"\"\"Determines y value at x_val by linearly interpolating y_s\n\n Parameters:\n x_s - np array of x values\n y_s - np array of corresponding y values\n x_val - x value at which y_s will be linearly interpolated\n \n\n Returns:\n y value linearly interpolated from y_s at x_val\n \"\"\"\n index = np.abs(x_s - x_val).argmin()\n \n if x_s[index] > x_val:\n l_index = index - 1\n r_index = index\n elif x_s[index] < x_val:\n l_index = index\n r_index = index + 1\n else:\n return y_s[index]\n \n \n m = (y_s[r_index] - y_s[l_index])/(x_s[r_index] - x_s[l_index])\n b = y_s[r_index] - m*x_s[r_index]\n \n return b + m*x_val\n```\n\nQuestion 3.\n\n\n```python\nsp = fsps.StellarPopulation(imf_type=1, dust_type=2, dust1=0.0, dust2=0.5, logzsol=math.log10(0.2),sfh=0)\n```\n\nQuestion 4.\n\n\n```python\nwave, spec = sp.get_spectrum(tage=0.01)\n\ncreate_plot(wave, spec, (10**2, 10**8), r\"$\\lambda$ ($\\AA$)\", (10**-17, 10**-3), r\"L$_v$ $(L_\\odot/Hz)$\")\n```\n\n\n```python\nsp.libraries\n```\n\n\n\n\n (b'mist', b'miles')\n\n\n\n\n```python\nsp.stellar_mass\n```\n\n\n\n\n 0.9187920115803033\n\n\n\nQuestion 5. (Incorrectly numbered as question 6 in assignment)\n\n\n```python\ndef get_spectral_density(wave, spec, z):\n \"\"\"Converts spectrum from (bolometric) luminosity in units of solar luminosities \n per hertz to flux density in units of microJansky\n\n Parameters:\n wave - np array of wavelengths (units don't matter)\n spec - np array of luminosities measured in solar luminosities per hertz\n z - Redshift of the galaxy\n\n Returns:\n New np array of wavelengths and the corresponding np array of flux densities\n measured in microJansky\n \"\"\"\n d = cosmo.luminosity_distance(z)\n spec = 4.02*(10**13)*((1 + z)*spec)/(4*math.pi*d.value**2)\n wave = wave*(1 + z)\n \n return wave, spec\n```\n\nQuestion 6. (Incorrectly numbered as question 5 in assignment)\n\n\n```python\nwave_1, spec_1 = get_spectral_density(wave, spec, 1)\n\ncreate_plot(wave_1, spec_1, (10**2, 10**8), r\"$\\lambda$ ($\\AA$)\", (10**-17, 10**-3), r\"f$_v$ $(\\mu Jy)$\")\n```\n\n\n```python\nwave_2, spec_2 = get_spectral_density(wave, spec, 2)\n\ncreate_plot(wave_2, spec_2, (10**2, 10**8), r\"$\\lambda$ ($\\AA$)\", (10**-17, 10**-3), r\"f$_v$ $(\\mu Jy)$\")\n```\n\n\n```python\nwave_3, spec_3 = get_spectral_density(wave, spec, 3)\n\ncreate_plot(wave_3, spec_3, (10**2, 10**8), r\"$\\lambda$ ($\\AA$)\", (10**-17, 10**-3), r\"f$_v$ $(\\mu Jy)$\")\n```\n\nQuestion 7. (Incorrectly numbered as question 6 in assignment)\n\n\n```python\nz_incr = 0.01\nz = 0.01\nz_s = []\nnew_spec = []\n\nwhile z <= 10:\n des_wave = 8000/(z + 1)\n \n close_spec = linear_interpolation(wave, spec, des_wave)\n \n np_close_wave, np_close_spec = get_spectral_density(np.array([-1]), np.array([close_spec]), z)\n \n z_s.append(z)\n new_spec.append(np_close_spec[0])\n z += z_incr\n\ncreate_plot(np.array(z_s), np.array(new_spec), (0, 10), r\"$z$\", (7.46*10**-13, 1.66*10**-5), r\"f$_v$ $(\\mu Jy)$\", loglog=False)\n```\n\nQuestion 8. (Incorrectly numbered as question 7 in assignment)\n\n\n```python\ndef generate_SED(age, z, plot=False, x_lim=(10**2, 10**8), y_lim=(10**-17, 10**-4)):\n \"\"\"Calculates and optionally plots SED of galaxy with a given age and redshift\n\n Parameters:\n age - Age of galaxy\n z - Redshift of galaxy\n plot - If true, draw an SED, otherwise \n\n Returns:\n np array of wavelengths and corresponding np array of flux densities in microJansky\n \"\"\"\n wave, spec = sp.get_spectrum(tage=age)\n new_wave, new_spec = get_spectral_density(wave, spec, z)\n \n if plot:\n create_plot(new_wave, new_spec, x_lim, r\"$\\lambda$ ($\\AA$)\", y_lim, y_label = r\"f$_v$ $(\\mu Jy)$\")\n \n return new_wave, new_spec\n\ngenerate_SED(0.01, 1, plot=True)\n```\n\n\n```python\ngenerate_SED(0.1, 1, plot=True)\n```\n\n\n```python\ngenerate_SED(1, 1, plot=True)\n```\n\n\n```python\ngenerate_SED(3, 1, plot=True)\n```\n\n\n```python\ngenerate_SED(10, 1, plot=True)\n```\n\nPart 2\n\n\n```python\ndef read_file(name):\n \"\"\"Creates 3 np arrays of data, read in from a file consisting of three columns\n\n Parameters:\n name - name of file\n\n Returns:\n 3 np arrays containing data in the three coloumns from the file\n \"\"\"\n lst_1 = []\n lst_2 = []\n lst_3 = []\n \n with open(name) as reader:\n for line in reader:\n list_str = line.split()\n if len(list_str) == 3:\n lst_1.append(float(list_str[0]))\n lst_2.append(float(list_str[1]))\n lst_3.append(float(list_str[2]))\n return np.array(lst_1), np.array(lst_2), np.array(lst_3)\n```\n\nQuestion 1.\n\n\n```python\ndef weighted_fitting(x_s, y_s, errs):\n \"\"\"Calculates y-int and slope (and uncertainties) of \n line that fits data using weighted linear least-square fitting\n\n Parameters:\n x_s - numpy array of x values\n y_s - numpy array of corresponding y values\n errs - numpy array of corresponding errors in the y values\n\n Returns:\n Tuple containing y-int, uncertainty in y-int, slope, and \n uncertainy in slope\n \"\"\"\n list_Y = []\n list_A = []\n list_C = []\n \n for i in range(len(x_s)):\n list_Y.append([y_s[i]])\n list_A.append([1, x_s[i]])\n \n C_row = []\n \n counter = 0\n while counter < len(x_s):\n if counter == i:\n C_row.append(errs[i]**2)\n else:\n C_row.append(0)\n \n counter += 1\n \n list_C.append(C_row)\n \n \n A = Matrix(list_A)\n Y = Matrix(list_Y)\n C = Matrix(list_C)\n \n cov = ((A.T*C.inv()*A).inv())\n \n b_error = cov[0, 0]\n m_error = cov[1, 1]\n \n X = cov*(A.T*C.inv()*Y)\n \n return X[0, 0], sqrt(b_error), X[1, 0], sqrt(m_error)\n```\n\n\n```python\ndef calc_chi(x_s, y_s, errs, b, m):\n \"\"\"Calculates chi^2, a measure of how closely a line fits\n the given data\n\n Parameters:\n x_s - np array of x values\n y_s - np array of corresponding y values\n errs - np array of corresponding errors in the y values\n b - y-int of line\n m - slope of line\n\n Returns:\n The value of chi^2\n \"\"\"\n return np.sum(((y_s - (b + m*x_s))**2)/(errs**2))\n```\n\n\n```python\nx_s, y_s, errs = read_file('fitting_simple_data.txt')\n \nb, b_err, m, m_err = weighted_fitting(x_s, y_s, errs)\n\ntxt = \"\\mathrm{{{3}}} = {0:.3f}_{{-{1:.3f}}}^{{{2:.3f}}}\"\ntxt = txt.format(b, b_err, b_err, 'b')\ndisplay(Math(txt))\ntxt = \"\\mathrm{{{3}}} = {0:.3f}_{{-{1:.3f}}}^{{{2:.3f}}}\"\ntxt = txt.format(m, m_err, m_err, 'm')\ndisplay(Math(txt))\n\nx_line = np.linspace(0, 12)\ny_line = b + m*x_line \n\nplt.figure(figsize=(15, 9))\nplt.scatter(x_s, y_s)\nplt.errorbar(x_s, y_s, yerr=errs, fmt=\".\", capsize=0)\nplt.xlabel('x')\nplt.ylabel('y')\n\nplt.plot(x_line, y_line)\n\nplt.show()\n\nprint('chi^2 = ', calc_chi(x_s, y_s, errs, b, m))\n```\n\nQuestion 2.\n\n\n```python\ndef spectrum_likelihood(wave_obs, spec_obs, errs, wave_theo, spec_theo):\n \"\"\"Calculates the likelihood that a theoretical SED matches\n an observed SED\n\n Parameters:\n wave_obs - np array of observed wavelength values\n spec_obs - np array of corresponding observed flux values\n errs - np array of errors in observed flux values\n wave_theo - np array of theoretical wavelength values\n spec_theo - np array of corresponding theoretical flux values\n\n Returns:\n The likelihood, note that it is always less than 0\n \"\"\"\n interpol_flux = []\n \n for i in range(len(wave_obs)):\n interpol_flux.append(linear_interpolation(wave_theo, spec_theo, wave_obs[i]))\n \n inter_flux = np.array(interpol_flux)\n \n return -np.sum(((spec_obs - inter_flux)**2)/(errs**2))\n```\n\n\n```python\nwave_obs, spec_obs, errs = read_file('fitting_spectrum_1.txt')\ncreate_plot(wave_obs, spec_obs, (10**3, 199600.0), r\"$\\lambda$ ($\\AA$)\", (10**-14, 10**-8), r\"f$_v$ $(\\mu Jy)$\")\n```\n\n\n```python\nwave_theo, spec_theo = generate_SED(0.1, 1)\nspectrum_likelihood(wave_obs, spec_obs, errs, wave_theo, spec_theo)\n```\n\n\n\n\n -3121221711061.225\n\n\n\n\n```python\nwave_obs, spec_obs, errs = read_file('fitting_spectrum_2.txt')\n\ncreate_plot(wave_obs, spec_obs, (10**3, 99960.0), r\"$\\lambda$ ($\\AA$)\", (10**-12, 10**-7), r\"f$_v$ $(\\mu Jy)$\")\n```\n\n\n```python\nspectrum_likelihood(wave_obs, spec_obs, errs, wave_theo, spec_theo)\n```\n\n\n\n\n -364732.5275292772\n\n\n\n\n```python\ndef find_optimal_z_age(z_start, z_incr, z_end, age_start, age_incr, age_end, name):\n \"\"\"Uses a brute force loop to estimate age and z that\n maximize the likelihood function\n\n Parameters:\n z_start - Starting z (the loop will start with z_start + z_incr)\n z_incr - Increment in z\n z_end - Value of z the loop will stop at\n age_start - Starting age (the loop will start with age_start + age_incr)\n age_incr - Increment in age\n age_end - Value of age the loop will stop at\n name - Name of file\n\n Returns:\n Returns tuple of the age and z that maximized the likelihood function\n \"\"\"\n wave_obs, spec_obs, errs = read_file(name)\n max_likelihood = -10**10\n max_z = -1\n max_age = -1\n z = z_start\n \n while z <= z_end:\n z += z_incr\n age = age_start + age_incr\n while age <= age_end:\n wave_theo, spec_theo = generate_SED(age, z)\n likelihood = round(spectrum_likelihood(wave_obs, spec_obs, errs, wave_theo, spec_theo), 3)\n \n if likelihood > max_likelihood:\n max_likelihood = likelihood\n max_z = round(z, 3)\n max_age = round(age, 3)\n \n age += age_incr\n \n return max_age, max_z\n```\n\n\n```python\nage, z = find_optimal_z_age(0.0, 0.2, 5, 0.0, 0.2, 10, 'fitting_spectrum_1.txt')\n```\n\n\n```python\ngenerate_SED(age, z, plot=True, x_lim=(10**3, 199600.0), y_lim=(10**-14, 10**-8))\n```\n\n\n```python\nfind_optimal_z_age(0.0, 0.2, 5, 0.0, 0.2, 10, 'fitting_spectrum_2.txt')\n```\n\n\n```python\ngenerate_SED(age, z, plot=True, x_lim=(10**3, 99960.0), y_lim=(10**-12, 10**-7))\n```\n\nStretch Goals\n\nQuestion 1.\n\n\n```python\ndef log_likelihood(theta, x, y, yerr):\n \"\"\"Calculates the log of the likelihood that the values\n match the model\n\n Parameters:\n theta - Tuple containing slope, y-int, and f parameter\n x - np array of x values\n y - np array of corresponding y values\n yerr - np array of corresponding errors in the y values\n\n Returns:\n Returns the log of the likelihood that the y values\n fit the model\n \"\"\"\n m, b, f = theta\n model = m*x + b\n sigma2 = yerr**2 + (model**2)*(f**2)\n return -0.5*np.sum(((y - model)**2)/sigma2 + np.log(sigma2))\n```\n\n\n```python\ndef create_rand_pos(sig_1, mu_1, sig_2, mu_2, num_walkers):\n \"\"\"Creates random position list for the walkers\n\n Parameters:\n sig_1 - Standard deviation for first parameter\n mu_1 - Mean value of first parameter\n sig_2 - Standard deviation for second parameter\n mu_2 - Mean value of second parameter\n num_walkers - Number of walkers\n\n Returns:\n 2D np array contaning initial position for the walkers\n \"\"\" \n array_1 = sig_1 * np.random.randn(num_walkers) + mu_1\n array_2 = sig_2 * np.random.randn(num_walkers) + mu_2\n \n lst_pos = []\n for i in range(len(array_1)):\n lst_pos.append([array_1[i], array_2[i]])\n \n return np.array(lst_pos)\n```\n\n\n```python\ndef prior(theta):\n \"\"\"Calculates the prior probability that the values\n matches the model\n\n Parameters:\n theta - Tuple containing slope, y-int, and f parameter\n\n Returns:\n Returns the 1.0 if the parameters are within a\n reasonable range and 0.0 otherwise\n \"\"\"\n m, b, f = theta\n if 2 < m < 12 and -15.0 < b < 15.0 and 0 < f < 100:\n return 1.0\n return 0.0\n```\n\n\n```python\ndef log_probability(theta, x, y, yerr):\n \"\"\"Calculates the log of the probability that \n the values matches the model\n\n Parameters:\n theta - Tuple containing slope, y-int, and f parameter\n x - np array of x values\n y - np array of corresponding y values\n yerr - np array of corresponding errors in the y values\n\n Returns:\n Returns the sum of the log of the likelihood and the \n log of the prior\n \"\"\" \n lp = np.log(prior(theta))\n if not np.isfinite(lp):\n return -np.inf\n return lp + log_likelihood(theta, x, y, yerr)\n```\n\n\n```python\npos = np.array([5.5597, -7.8453, 0.01]) + 1e-4 * np.random.randn(32, 3)\nnwalkers, ndim = pos.shape\n\nx_s, y_s, errs = read_file('fitting_simple_data.txt')\n\nsampler = emcee.EnsembleSampler(nwalkers, ndim, log_probability, args=(x_s, y_s, errs))\nsampler.run_mcmc(pos, 10000)\n```\n\n /opt/python/3.7.3/lib/python3.7/site-packages/ipykernel_launcher.py:15: RuntimeWarning: divide by zero encountered in log\n from ipykernel import kernelapp as app\n\n\n\n\n\n (array([[ 6.33362698e+00, -1.37200425e+01, 1.45149347e-01],\n [ 4.87724734e+00, -4.41345382e+00, 5.03339197e-04],\n [ 5.18677888e+00, -7.60882649e+00, 1.02507701e-01],\n [ 5.61930880e+00, -8.81661778e+00, 4.80961776e-02],\n [ 6.18197208e+00, -1.09961192e+01, 1.23418185e-01],\n [ 5.36009131e+00, -8.25458980e+00, 1.58321003e-01],\n [ 5.42370403e+00, -7.20127370e+00, 3.72172467e-02],\n [ 5.11090425e+00, -4.63823283e+00, 5.97144207e-02],\n [ 5.72088020e+00, -8.88197850e+00, 1.01835833e-01],\n [ 4.41551692e+00, -4.78118874e+00, 9.53740988e-02],\n [ 4.65483016e+00, -4.82975797e+00, 7.53092708e-02],\n [ 4.53186657e+00, -5.10960341e+00, 3.51295579e-02],\n [ 5.67700421e+00, -1.08648392e+01, 2.09555479e-02],\n [ 5.49891283e+00, -9.06146337e+00, 7.43670114e-02],\n [ 5.27988338e+00, -5.68981036e+00, 8.18228956e-02],\n [ 5.17034194e+00, -7.61044009e+00, 9.95636655e-02],\n [ 5.32340016e+00, -7.65984673e+00, 1.50052518e-02],\n [ 4.92387988e+00, -4.59903583e+00, 1.00486380e-01],\n [ 4.47771095e+00, -6.81105987e+00, 4.96699167e-02],\n [ 4.62659793e+00, -7.12618442e+00, 1.05026658e-01],\n [ 5.59304083e+00, -6.86061482e+00, 1.49411046e-02],\n [ 5.80274766e+00, -1.02258515e+01, 1.31419203e-01],\n [ 5.72478276e+00, -9.78969236e+00, 1.39142569e-02],\n [ 6.16144718e+00, -1.02565345e+01, 4.93534264e-02],\n [ 5.32537445e+00, -7.04244627e+00, 7.72369962e-03],\n [ 4.81763085e+00, -7.61623322e+00, 3.76363337e-02],\n [ 4.60289168e+00, -6.26443515e+00, 7.17386923e-02],\n [ 5.56017185e+00, -7.91880942e+00, 1.47532848e-01],\n [ 4.79036013e+00, -4.78856931e+00, 1.37557717e-01],\n [ 6.26156554e+00, -1.26786146e+01, 2.00439391e-02],\n [ 5.11061046e+00, -7.29126002e+00, 7.41676848e-02],\n [ 5.69084570e+00, -8.97090962e+00, 7.26647312e-02]]),\n array([-298.3533462 , -292.98943435, -292.94998687, -292.53217902,\n -295.6272552 , -294.19615993, -292.31759039, -293.51736239,\n -293.49378134, -294.43918774, -293.39940019, -293.49201054,\n -293.456015 , -292.80781155, -293.32402788, -292.92017929,\n -292.04459625, -293.71550348, -294.93025331, -294.69416996,\n -292.93474954, -294.52449995, -292.68577695, -294.21700491,\n -292.09275635, -293.44648112, -293.83224996, -294.26537973,\n -294.20880273, -295.21205074, -292.57673195, -292.94266497]),\n ('MT19937', array([1466182691, 606170754, 2543722276, 1826066639, 211646718,\n 865924328, 1923011466, 1839234061, 916819979, 4260035936,\n 2845415622, 645459043, 3225688882, 1881327150, 817593334,\n 3126063455, 4198567978, 55908045, 2133880683, 3084601428,\n 2142101267, 1615178036, 2336478754, 2496556596, 1678550166,\n 3587831194, 1985047484, 1507563058, 102112026, 304163543,\n 3346731023, 1042896468, 93032331, 1258582899, 3208792930,\n 2866889664, 2262710170, 2219459563, 2016095585, 2899209527,\n 3855688204, 1568946856, 1848240861, 1131279054, 1113999611,\n 208818819, 2913234401, 4142963675, 2129391538, 3591663181,\n 2654516368, 1054290052, 2516388144, 3478700974, 2321464322,\n 476471248, 3367747998, 2749007180, 3167541398, 2189248742,\n 3141446016, 434880195, 3705660660, 1504693304, 3949043189,\n 3376483493, 133683898, 1236067195, 2814292950, 1659130022,\n 3041097961, 1375848395, 2990490829, 761537995, 329951961,\n 4146046875, 1576702451, 2598946899, 2047060109, 1068245718,\n 979173567, 1584342334, 2707814960, 3297997261, 2909365097,\n 2445171886, 3398931861, 2232335257, 3980658073, 2930831251,\n 2607494522, 2721398140, 1522385699, 1160859637, 1427204942,\n 737473275, 2670961226, 646857846, 4089137396, 3592463882,\n 630889616, 2681995034, 3082470739, 605709536, 3243818020,\n 2503896657, 2580841657, 662645894, 2569216106, 910555551,\n 2771059411, 3333106127, 2434178306, 3949871461, 3305942518,\n 1858585655, 4071540626, 3834260109, 3451482896, 1255522083,\n 2837203424, 2932835008, 3655587551, 3546786190, 1621131403,\n 3689118226, 1899582298, 1309868446, 845238554, 3541042199,\n 2237872646, 909030060, 2876518755, 2755295097, 852496782,\n 841913933, 2529656646, 2308513285, 3602629451, 1268924281,\n 744560714, 412288199, 1417255557, 1024414430, 2282289772,\n 272930133, 715286195, 1825800419, 772141375, 2589319121,\n 972259032, 692100825, 4244190884, 2684020575, 995964388,\n 53605272, 1278253109, 163220217, 981154026, 698142553,\n 3225006354, 3251941213, 1297959761, 3102769641, 2342527166,\n 527680363, 40447491, 2067621853, 1307621064, 2302065349,\n 1008750178, 445480243, 2834936264, 479574329, 3187285566,\n 3286512906, 1195513341, 4022522346, 2541336011, 1875206631,\n 1737515658, 4153602119, 2189425946, 2405886706, 1114248276,\n 495137439, 2376896867, 411637893, 2269414888, 293560653,\n 3638048433, 1685788835, 3377341984, 763618187, 1396231621,\n 3494395605, 345257849, 1692789936, 3138188741, 1705908018,\n 2321843982, 2726702755, 1369499360, 3217977102, 1472697886,\n 1523393953, 1917009740, 1125051452, 4069327900, 4155468561,\n 2339637318, 2374355854, 3240076502, 472306082, 1022386677,\n 181392949, 2535154047, 49925435, 2221155113, 2004121065,\n 2674330689, 1195663042, 728833916, 3202574626, 4105086968,\n 3738319975, 3064272624, 1769115560, 756033159, 3126806902,\n 214611932, 141257427, 1427186813, 3002553276, 3354054165,\n 1625501757, 2763882227, 1059771794, 2699736342, 1829858487,\n 3841958985, 2062794342, 1583448537, 3016737353, 1623796879,\n 4109610571, 224636982, 2344136874, 964012439, 233414031,\n 1509631193, 3274878183, 447068140, 184005732, 97903540,\n 642563453, 1270511466, 2786408748, 989923568, 841676872,\n 3540316021, 1804748809, 32657308, 2280509636, 1630641053,\n 939836840, 3527214517, 3863507488, 3339213550, 941382,\n 2322999271, 2046507992, 3438003850, 2863089719, 2667425856,\n 1295826468, 2399353539, 1349708713, 4119348849, 1505474677,\n 3073986386, 3484090396, 4078862043, 152196580, 1751134293,\n 356529090, 556064050, 2105040800, 3465157277, 143803086,\n 2672695234, 566121117, 3590024711, 2530069007, 2482761180,\n 3994239247, 3353679149, 3760654881, 1524306228, 3015029269,\n 1397278947, 3816287004, 1914076492, 2941607778, 461463870,\n 1173025752, 3357474015, 3462898740, 1903357380, 4191812384,\n 2810661116, 1297955145, 3988829794, 1752937208, 2767959309,\n 726475778, 3509595818, 2186980455, 2669276576, 1176536384,\n 1787515012, 1283714058, 2744922150, 400963417, 2148735165,\n 1024472760, 2447894299, 3386271293, 1368080640, 1738155913,\n 1121519406, 280837696, 1525774707, 2629202650, 189396436,\n 1719059158, 3381191961, 599614686, 705612318, 201969062,\n 722502921, 2236291909, 3038822125, 1049026884, 3688283166,\n 2063214105, 201511604, 2300763253, 752140927, 2707231119,\n 2643080028, 1269799859, 2307301494, 3524799704, 2834810490,\n 1666365301, 2028387109, 693180019, 2474529056, 2417578980,\n 505760047, 1126753697, 1082642571, 1996047864, 1568732791,\n 2227817923, 920190655, 3757954098, 856781057, 433603288,\n 1883453647, 3406272663, 1404225140, 1607405375, 678115555,\n 3181799146, 221821424, 1827438536, 19791745, 1530694991,\n 1301440313, 647326599, 3876743577, 452706458, 3709937779,\n 73489335, 69062471, 3470900067, 1020946487, 333200848,\n 739780538, 2058130451, 2854485982, 2208734626, 915837296,\n 876655428, 3515638593, 1398805283, 2014262213, 1952760057,\n 1836802215, 764721801, 1844965973, 1739002064, 977365137,\n 1336533455, 2207204269, 738785288, 2158777903, 266003396,\n 3491923254, 400238618, 41066087, 667351290, 3143678066,\n 2755520370, 3822986875, 3143161056, 1088383330, 312013834,\n 2689470618, 4256426849, 2185829825, 583217254, 3940144317,\n 3114034826, 1665606372, 1156221216, 355532343, 3777073634,\n 1788669923, 2031113294, 3239022394, 3735384727, 1194753031,\n 1137969359, 2385064691, 2980715537, 195117552, 3909266176,\n 3888659671, 743909867, 262850182, 3876244862, 3077139311,\n 548140254, 4172980275, 3711627054, 3751678258, 1610174691,\n 1823803146, 2015592703, 2683096289, 3734234899, 3672140798,\n 3993390294, 3754106024, 2841641839, 4040541022, 3815592829,\n 2929493679, 2271613577, 785494353, 2449946924, 2588909047,\n 3413199132, 2400821800, 2555033918, 103013946, 3926938397,\n 2416448213, 4187159074, 2216088980, 901723346, 1262332777,\n 2783936431, 584134313, 700526710, 3651189482, 1033561285,\n 2429640874, 1618594851, 3468328169, 2383208401, 1196601962,\n 4286506431, 180479350, 1233779870, 3232011222, 4261282947,\n 1372057305, 615877629, 1574709694, 3479382295, 3415279017,\n 3163235975, 2745964691, 1996570682, 4231905913, 2146683743,\n 3358899482, 218900534, 2239251134, 2062544713, 1962893861,\n 3381663101, 4274733377, 2326174714, 2759016390, 3528031965,\n 3605919922, 959385630, 1476600028, 3753823040, 3448393790,\n 3175091973, 3151005953, 3987679569, 2475058839, 1370105074,\n 4229127035, 769468910, 2424946524, 3351931721, 1653320952,\n 421058743, 1571786486, 2693663119, 1784774671, 2295902906,\n 3532811565, 2622804814, 366965330, 2480800590, 1235256892,\n 2114409339, 4190380418, 2801789222, 1501754635, 2850720591,\n 2974479289, 647244148, 3238900487, 1930775693, 2477578861,\n 791112975, 4126575627, 4156752667, 3095764681, 4225473889,\n 1316502486, 1927232114, 1481380279, 3687426790, 1385164702,\n 1848597798, 339675871, 315069954, 1624400516, 3939577342,\n 3427544177, 2203896583, 2013227218, 3845191842, 1414893465,\n 3255730195, 792471328, 2768823846, 2032611874, 2952421264,\n 4122553428, 4244188936, 484243516, 2733975040, 3887781582,\n 2324468556, 3035986082, 1886541661, 2813810287, 4259512522,\n 2911925671, 732846059, 3256550575, 1355705379, 4236909828,\n 4093941084, 4244880699, 1222216488, 1918812273, 266520353,\n 1896919105, 1558844531, 3331476082, 2090296822, 1790625131,\n 681225801, 2002433872, 722386770, 1598803220, 2210191664,\n 1947817185, 1422107590, 1236098313, 1850597042, 2701341332,\n 2066026785, 782492487, 1337891038, 1431855099, 3608024889,\n 1083964300, 2009472203, 632723507, 2003154257, 2721402347,\n 2373400149, 2979117654, 2666297079, 3927960008, 1750217368,\n 2071979224, 2934660547, 4270754306, 1664533903], dtype=uint32), 63, 0, 0.0))\n\n\n\n\n```python\nfig, axes = plt.subplots(3, figsize=(10, 7), sharex=True)\nsamples = sampler.chain\nlabels = [\"m\", \"b\", \"f\"]\nfor i in range(ndim):\n ax = axes[i]\n ax.plot(samples[:, :, i], \"k\", alpha=0.3)\n ax.set_xlim(0, len(samples))\n ax.set_ylabel(labels[i])\n ax.yaxis.set_label_coords(-0.1, 0.5)\n\naxes[-1].set_xlabel(\"step number\")\n```\n\n\n```python\nprint(sampler.get_autocorr_time())\n\nflat_samples = sampler.chain[:, 40:, :].reshape((-1, ndim))\nfig = corner.corner(flat_samples, labels=labels)\n```\n\n\n```python\nfor i in range(ndim):\n mcmc = np.percentile(flat_samples[:, i], [16, 50, 84])\n q = np.diff(mcmc)\n txt = \"\\mathrm{{{3}}} = {0:.3f}_{{-{1:.3f}}}^{{{2:.3f}}}\"\n txt = txt.format(mcmc[1], q[0], q[1], labels[i])\n \n display(Math(txt))\n \n if i == 0:\n m_2 = mcmc[1]\n elif i == 1:\n b_2 = mcmc[1]\n```\n\n\n$\\displaystyle \\mathrm{m} = 5.261_{-0.455}^{0.444}$\n\n\n\n$\\displaystyle \\mathrm{b} = -7.391_{-2.108}^{2.172}$\n\n\n\n$\\displaystyle \\mathrm{f} = 0.052_{-0.037}^{0.056}$\n\n\n\n```python\nx_line = np.linspace(0, 12)\ny_line = b_2 + m_2*x_line\n\nplt.figure(figsize=(15, 9))\nplt.scatter(x_s, y_s)\nplt.errorbar(x_s, y_s, yerr=errs, fmt=\".\", capsize=0)\n\nplt.plot(x_line, y_line)\nplt.savefig(\"Data and line 2.pdf\")\nplt.show()\n\nprint('chi^2 = ', calc_chi(x_s, y_s, errs, b_2, m_2))\n```\n\nQuestion 3.\n\n\n```python\ndef galaxy_prior(theta):\n \"\"\"Calculates the prior probability that the values\n matches the model\n\n Parameters:\n theta - Tuple containing log(age) and z\n\n Returns:\n Returns the 1.0 if the parameters are within a\n reasonable range and 0.0 otherwise\n \"\"\" \n log_age, z = theta\n if 0 < log_age < 5 and 0 < z < 2.0:\n return 1.0\n return 0.0\n```\n\n\n```python\ndef galaxy_probability(theta, wave_obs, spec_obs, errs):\n \"\"\"Calculates the log of the probability that \n the values matches the model\n\n Parameters:\n theta - Tuple containing log(age) and z\n wave_obs - np array of observed wavelength values\n spec_obs - np array of corresponding observed flux values\n errs - np array of corresponding errors in the flux values\n\n Returns:\n Returns the sum of the log of the likelihood and the \n log of the prior\n \"\"\"\n lp = np.log(galaxy_prior(theta))\n if not np.isfinite(lp):\n return -np.inf\n \n log_age, z = theta\n wave_theo, spec_theo = generate_SED(np.exp(log_age), z)\n return lp + spectrum_likelihood(wave_obs, spec_obs, errs, wave_theo, spec_theo)\n```\n\n\n```python\npos = create_rand_pos(1.0, 1.86, 0.5, 0.8, 16)\n#pos = create_rand_pos(0.75, 1.22, 0.15, 0.2, 16)\nnwalkers, ndim = pos.shape\n\nwave, spec, yerr = read_file('fitting_spectrum_1.txt')\n#wave, spec, yerr = read_file('fitting_spectrum_2.txt')\n\nsampler = emcee.EnsembleSampler(nwalkers, ndim, galaxy_probability, args=(wave, spec, yerr))\n\nsampler.run_mcmc(pos, 2500)\n```\n\n\n```python\nfig, axes = plt.subplots(2, figsize=(10, 7), sharex=True)\nsamples = sampler.chain\nlabels = [\"log(age)\", \"z\"]\nfor i in range(ndim):\n ax = axes[i]\n ax.plot(samples[:, :, i], \"k\", alpha=0.3)\n ax.set_xlim(0, len(samples))\n ax.set_ylabel(labels[i])\n ax.yaxis.set_label_coords(-0.1, 0.5)\n\naxes[-1].set_xlabel(\"step number\")\n```\n\n\n```python\nflat_samples = sampler.chain[:, 5:, :].reshape((-1, ndim))\nfig = corner.corner(flat_samples, labels=labels)\n```\n\n\n```python\nfor i in range(ndim):\n mcmc = np.percentile(flat_samples[:, i], [16, 50, 84])\n q = np.diff(mcmc)\n txt = \"\\mathrm{{{3}}} = {0:.3f}_{{-{1:.3f}}}^{{{2:.3f}}}\"\n txt = txt.format(mcmc[1], q[0], q[1], labels[i])\n \n display(Math(txt))\n \n if i == 0:\n m_2 = mcmc[1]\n elif i == 1:\n b_2 = mcmc[1]\n```\n", "meta": {"hexsha": "df0a7e7bca004a896c215b7a14638262244f1ed3", "size": 729847, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Final Project/Computing Project Script.ipynb", "max_stars_repo_name": "CalebLammers/CTA200", "max_stars_repo_head_hexsha": "2b8e442f10479b8f82a9b8c4558a45aa9e791118", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Final Project/Computing Project Script.ipynb", "max_issues_repo_name": "CalebLammers/CTA200", "max_issues_repo_head_hexsha": "2b8e442f10479b8f82a9b8c4558a45aa9e791118", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Final Project/Computing Project Script.ipynb", "max_forks_repo_name": "CalebLammers/CTA200", 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YES\n2. YES", "lm_q1_score": 0.7185943925708561, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.43129198945068575}} {"text": "# Predict gene knockout strategies\n\nIn cameo we have two ways of predicting gene knockout targets: using evolutionary algorithms (OptGene) or linear programming (OptKnock)\n\n\n```python\nfrom cameo import models\n```\n\n\n\n\n\n\n\n\n```python\nmodel = models.bigg.e_coli_core.copy()\nmodel.solver = \"cplex\"\n```\n\n\n```python\nfrom cameo import phenotypic_phase_plane\n```\n\n\n```python\nppp = phenotypic_phase_plane(model, variables=[model.reactions.BIOMASS_Ecoli_core_w_GAM], objective=model.reactions.EX_ac_e)\nppp.plot()\n```\n\n\n\n\n
                                        \n\n\n\n## OptGene\n\nOptGene is an approach to search for gene or reaction knockouts that relies on evolutionary algorithms[1]. The following image from authors summarizes the OptGene workflow.\n\n\n\nEvery iteration we keep the best 50 individuals so we can generate a library of targets.\n\n\n```python\nfrom cameo.strain_design.heuristic.evolutionary_based import OptGene\n```\n\n\n```python\noptgene = OptGene(model)\n```\n\n\n```python\nresult = optgene.run(target=\"EX_ac_e\", \n biomass=model.reactions.BIOMASS_Ecoli_core_w_GAM,\n substrate=model.metabolites.glc__D_e,\n max_evaluations=5000,\n plot=False)\n```\n\n Starting optimization at Fri, 10 Jun 2016 06:38:55\n\n\n\n\n\n
                                        \n\n\n\n /opt/conda/envs/python3.4/lib/python3.4/site-packages/ipywidgets/widgets/widget_string.py:55: UserWarning:\n \n The Latex widget is deprecated. Use Label instead\n \n\n\n Finished after 00:00:11\n\n\n\n```python\nresult\n```\n\n /opt/conda/envs/python3.4/lib/python3.4/site-packages/ipywidgets/widgets/widget_string.py:55: UserWarning:\n \n The Latex widget is deprecated. Use Label instead\n \n\n\n\n\n\n\n

                                        OptGene Result

                                        \n
                                          \n
                                        • Simulation: fba
                                          • \n
                                        • Objective Function: $$bpcy = \\frac{(BIOMASS\\_Ecoli\\_core\\_w\\_GAM * EX\\_ac\\_e)}{EX\\_glc\\_\\_D\\_e}$$
                                          • \n
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                                        \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        reactionsgenessizefva_minfva_maxtarget_fluxbiomass_fluxyieldfitness
                                        0(SUCDi, GND)((b4090, b2296, b0723, b2029, b0724),)5.09.08353710.71573710.6219570.6170371.0621960.655414
                                        1(SUCDi, GND, GLUt2r)((b0721, b3962, b4077, b2029, b2133),)5.09.08353710.71573710.6219570.6170371.0621960.655414
                                        2(FRD7, SUCDi, PGL, ADK1)((b0474, b0767, b0118, b4154, b0721),)5.09.08353710.71573710.6219570.6170371.0621960.655414
                                        3(SUCDi, PGL, ADK1)((b0767, b2492, b0903, b2464, b0474, b0721),)6.09.08353710.71573710.6219570.6170371.0621960.655414
                                        4(FRD7, SUCDi, GND)((b0903, b4152, b4232, b0724, b0723, b2029),)6.09.08353710.71573710.6219570.6170371.0621960.655414
                                        5(SUCDi, GND, PPS)((b1702, b2464, b0902, b0733, b0721, b2029), (...6.09.08353710.71573710.6219570.6170371.0621960.655414
                                        6(SUCDi, GND, ADK1)((b2464, b0474, b4395, b0721, b2935, b2029),)6.09.08353710.71573710.6219570.6170371.0621960.655414
                                        7(FRD7, SUCDi, SUCOAS, GND)((b0729, b4152, b1621, b1241, b0723, b2029),)6.09.62760410.71573710.6219570.6170371.0621960.655414
                                        8(SUCDi, GND, ME1)((b1479, b1812, b0724, b1297, b0723, b2029),)6.09.08353710.71573710.6219570.6170371.0621960.655414
                                        9(GLUSy, SUCDi, GND)((b3952, b3212, b3925, b2464, b0721, b2029),)6.09.08353710.71573710.6219570.6170371.0621960.655414
                                        10(GLUt2r, SUCDi, GND)((b3386, b2464, b2133, b0721, b4077, b2029), (...6.09.08353710.71573710.6219570.6170371.0621960.655414
                                        11(SUCDi, PGL, GND)((b0767, b0722, b2464, b0118, b2935, b2029),)6.09.08353710.71573710.6219570.6170371.0621960.655414
                                        12(SUCDi, PGL)((b0767, b0734, b0903, b2464, b0118, b0721, b4...7.09.08353710.71573710.6219570.6170371.0621960.655414
                                        13(SUCDi, GND, PPS)((b1702, b3603, b2458, b0721, b0979, b0875, b2...7.09.08353710.71573710.6219570.6170371.0621960.655414
                                        14(SUCDi, GND, PGL, ADK1)((b0767, b0903, b2464, b0118, b0474, b0721, b2...7.09.08353710.71573710.6219570.6170371.0621960.655414
                                        15(SUCDi, GND, FRUpts2)((b1818, b0733, b4395, b0721, b0723, b2935, b2...7.09.08353710.71573710.6219570.6170371.0621960.655414
                                        16(SUCDi, GND, AKGDH, ME2)((b0727, b2463, b0724, b0721, b4090, b0723, b2...7.09.62760410.71573710.6219570.6170371.0621960.655414
                                        17(SUCDi, PGL, ADK1)((b0767, b0903, b2464, b0118, b0474, b0721, b3...7.09.08353710.71573710.6219570.6170371.0621960.655414
                                        18(SUCDi, GND, AKGDH)((b0727, b0451, b0724, b0721, b4090, b0723, b2...7.09.62760410.71573710.6219570.6170371.0621960.655414
                                        19(AKGDH, G6PDH2r, SUCDi, GND, PPS)((b1702, b0727, b1852, b0721, b0875, b2029, b0...7.09.62760410.71573710.6219570.6170371.0621960.655414
                                        20(SUCDi, GND, ADK1)((b3952, b0903, b2464, b0118, b0474, b0721, b2...7.09.08353710.71573710.6219570.6170371.0621960.655414
                                        21(PGL, SUCDi)((b0767, b0903, b2464, b0118, b1241, b0721, b4...7.09.08353710.71573710.6219570.6170371.0621960.655414
                                        22(ME1, SUCDi, GND, FRUpts2)((b3925, b1818, b1479, b4395, b0723, b3115, b2...7.09.08353710.71573710.6219570.6170371.0621960.655414
                                        23(GLUSy, ME1, SUCDi, GND)((b3212, b1479, b0721, b1297, b0723, b3115, b2...7.09.08353710.71573710.6219570.6170371.0621960.655414
                                        24(FRD7, SUCDi, GND, ME1)((b0903, b4152, b1479, b0724, b0721, b0723, b2...7.09.08353710.71573710.6219570.6170371.0621960.655414
                                        25(FRD7, SUCDi, GND, AKGDH)((b4152, b0727, b0724, b0721, b4090, b0723, b2...7.09.62760410.71573710.6219570.6170371.0621960.655414
                                        26(FRD7, SUCDi, GND)((b3603, b2133, b0451, b0721, b2417, b2029, b4...7.09.08353710.71573710.6219570.6170371.0621960.655414
                                        27(FRD7, ME1, SUCDi, GND)((b4152, b1479, b1241, b0724, b0721, b0723, b2...7.09.08353710.71573710.6219570.6170371.0621960.655414
                                        28(FRD7, GLUSy, ME1, SUCDi, GND)((b3212, b4152, b1479, b0721, b1297, b0723, b2...7.09.08353710.71573710.6219570.6170371.0621960.655414
                                        29(AKGDH, SUCDi, GND, PPS)((b1702, b3603, b0727, b0721, b0875, b2029, b0...7.09.62760410.71573710.6219570.6170371.0621960.655414
                                        30(SUCDi, PGL, GND, ADK1)((b0767, b2464, b0118, b0474, b0721, b2935, b2...7.09.08353710.71573710.6219570.6170371.0621960.655414
                                        31(SUCDi, PGL, GND)((b0351, b0767, b0903, b2464, b0118, b0721, b2...8.09.08353710.71573710.6219570.6170371.0621960.655414
                                        32(FRD7, AKGDH, SUCDi, GND, ADK1)((b0903, b4152, b2464, b0727, b0118, b0474, b0...8.09.62760410.71573710.6219570.6170371.0621960.655414
                                        33(SUCDi, GND, ADK1)((b0903, b2464, b0474, b0978, b0721, b4090, b3...8.09.08353710.71573710.6219570.6170371.0621960.655414
                                        \n
                                        \n\n\n\n\n\n```python\nresult.plot(0)\n```\n\n\n\n\n
                                        \n\n\n\n\n```python\nresult.display_on_map(0, \"e_coli_core.Core metabolism\")\n```\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n
                                        \n\n\n\n\n\n## OptKnock\n\nOptKnock uses a bi-level mixed integer linear programming approach to identify reaction knockouts[2]:\n\n$$\n\\begin{matrix}\nmaximize & \\mathit{v_{chemical}} & & (\\mathbf{OptKnock}) \\\\\n\\mathit{y_j} & & & \\\\\nsubject~to & maximize & \\mathit{v_{biomass}} & (\\mathbf{Primal}) \\\\\n& \\mathit{v_j} & & & & \\\\\n\\end{matrix}\\\\\n\\begin{bmatrix}\nsubject~to & \\sum_{j=1}^{M}S_{ij}v_{j} = 0,\\\\ \n& v_{carbon\\_uptake} = v_{carbon~target}\\\\ \n& v_{apt} \\ge v_{apt\\_main}\\\\ \n& v_{biomass} \\ge v_{target\\_biomass}\\\\ \n& v_{j}^{min} \\cdot y_j \\le v_j \\le v_{j}^{max} \\cdot y_j, \\forall j \\in \\boldsymbol{M} \\\\\n\\end{bmatrix}\\\\\n\\begin{align}\n & y_j = {0, 1}, & & \\forall j \\in \\boldsymbol{M} & \\\\\n & \\sum_{j \\in M} (1 - y_j) \\le K& & & \\\\\n\\end{align}\n$$\n\n\n\n\n```python\nfrom cameo.strain_design.deterministic.linear_programming import OptKnock\n```\n\n\n```python\noptknock = OptKnock(model, fraction_of_optimum=0.1)\n```\n\n Warning: File contains basis. Basis is loaded.\n\n\nRunning multiple knockouts with OptKnock can take a few hours or days...\n\n\n```python\nresult = optknock.run(max_knockouts=1, target=\"EX_ac_e\", biomass=\"BIOMASS_Ecoli_core_w_GAM\")\n```\n\n\n\n\n\n\n\n\n \n\n\n\n```python\nresult\n```\n\n /opt/conda/envs/python3.4/lib/python3.4/site-packages/ipywidgets/widgets/widget_string.py:55: UserWarning:\n \n The Latex widget is deprecated. Use Label instead\n \n\n\n\n\n\n\n

                                        OptKnock:

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                                        • Target: EX_ac_e
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                                        \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        reactionssizeEX_ac_ebiomassfva_minfva_max
                                        0{ATPS4r}1.014.3122670.374230.014.369145
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                                        \n\n\n\n\n```python\nresult.plot(0)\n```\n\n\n\n\n
                                        \n\n\n\n\n```python\nresult.display_on_map(0, \"e_coli_core.Core metabolism\")\n```\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n
                                        \n\n\n\n\n\n## References\n\n[1]Patil, K. R., Rocha, I., F\u00f6rster, J., & Nielsen, J. (2005). Evolutionary programming as a platform for in silico metabolic engineering. BMC Bioinformatics, 6, 308. doi:10.1186/1471-2105-6-308\n\n[2]Burgard, A.P., Pharkya, P., Maranas, C.D. (2003), \"OptKnock: A Bilevel Programming Framework for Identifying Gene Knockout Strategies for Microbial Strain Optimization,\" Biotechnology and Bioengineering, 84(6), 647-657.\n\n## Exercises\n\n* Use OptGene or OptKnock to find a growth coupled design for a product of your choice!\n\n\n```python\n\n```\n", "meta": {"hexsha": "0accd52337868ad128989aa99f48fdcb1afe504e", "size": 703996, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Advanced-SynBio-for-Cell-Factories-Course/Predict-growth-coupled-designs-with-OptGene.ipynb", "max_stars_repo_name": "biosustain/cameo-notebooks", "max_stars_repo_head_hexsha": "7f592077eecb10c8c91756ae6ca4ad2d40d283ca", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Advanced-SynBio-for-Cell-Factories-Course/Predict-growth-coupled-designs-with-OptGene.ipynb", "max_issues_repo_name": "biosustain/cameo-notebooks", "max_issues_repo_head_hexsha": "7f592077eecb10c8c91756ae6ca4ad2d40d283ca", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Advanced-SynBio-for-Cell-Factories-Course/Predict-growth-coupled-designs-with-OptGene.ipynb", "max_forks_repo_name": "biosustain/cameo-notebooks", "max_forks_repo_head_hexsha": "7f592077eecb10c8c91756ae6ca4ad2d40d283ca", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2018-06-06T13:20:34.000Z", "max_forks_repo_forks_event_max_datetime": "2018-06-06T13:20:34.000Z", "avg_line_length": 352.1740870435, "max_line_length": 194003, "alphanum_fraction": 0.6303601157, "converted": true, "num_tokens": 6630, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660542, "lm_q2_score": 0.6619228758499942, "lm_q1q2_score": 0.43114668521863836}} {"text": "```python\n!pip install python-snap7==0.5\n# va en version 1.1\nimport snap7.client as c\nfrom snap7.util import * #set_int set_bool\nfrom snap7.snap7types import * # areas\nimport time\n# !pip install keyboard \nimport keyboard # using module keyboard\nimport os\nimport math\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\n```\n\n Requirement already satisfied: python-snap7==0.5 in c:\\users\\usuario\\.conda\\envs\\sistdin\\lib\\site-packages (0.5)\n\n\n# Excitaci\u00f3n del m\u00f3dulo multivariable SIEMENS del PCJIC\n\nEn el laboratorio de autom\u00e1tica del Polit\u00e9cnico Colombiano Jaime Isaza Cadavid se encuentra un m\u00f3dulo multivariable que est\u00e1 controlado con un PLC SIEMENS. Por esta raz\u00f3n, se utiliza el m\u00f3dulo `SNAP7` para interactuar con el PLC desde el notebook.\n\nPara instalar `SNAP7`:\n- Use la instrucci\u00f3n \n~~~\n!pip install python-snap7==0.5\n~~~\n\n- Descargue la librer\u00eda completa de [https://sourceforge.net/projects/snap7/files/1.2.1/](https://sourceforge.net/projects/snap7/files/1.2.1/)\n\n- Copie `snap7.dll` y `snap7.lib` a su carpeta `System32`.\n\n\n\n## 1. Establezca conexi\u00f3n con el PLC.\n\nEl PLC tiene configurada la direcci\u00f3n IP **192.168.0.1**. Este funcionar\u00e1 como un servidor al cu\u00e1l debe conectar su PC configur\u00e1ndole una direcci\u00f3n est\u00e1tica dentro de la subred del PLC. Su direcci\u00f3n puede ser **192.168.0.2**, por ejemplo. Una vez su equipo haya establecido comunicaci\u00f3n con el PLC, establezca conexi\u00f3n desde el notebook con las siguientes instrucciones.\n\n\n```python\n#Conexi\u00f3n con el PLC\nplc = c.Client()\nIP_PLC = '192.168.0.1'\n```\n\n\n```python\ntry:\n plc.connect(IP_PLC,0,1)\n print(\"Conectado\")\nexcept:\n print(\"Algo no funcion\u00f3\")\n```\n\n Conectado\n\n\n## 2. Identifique actuadores y sensores.\n\nCon `SNAP7` puede manipular los actuadores del sistema y capturar los datos de los sensores. Se han construido algunas funciones para facilitar la interacci\u00f3n con el PLC. \n\nTenga en cuenta la siguiente informaci\u00f3n, que le orientar\u00e1 en c\u00f3mo se ha conectado el sistema al PLC.\n\n|Elemento | Punto |\n|-----------|---------|\n| Presi\u00f3n \t| IW96 |\n| Flujo\t\t| IW98 |\n| Nivel\t\t| IW100 |\n| Run Bomba\t| Q0.5 |\n| Bomba\t\t| QW80 |\n| V\u00e1lvula H20 | QW96 |\n\n\n```python\n#ESCRIBIR SALIDA BOOLEANA\ndef escr_sal_bool(byte, bit, valor):\n lectura = plc.ab_read(byte, bit)\n set_bool(lectura, byte, bit, valor)\n plc.ab_write(0, lectura)\n#escr_sal_bool(0,1,1)\n\n#ESCRIBIR SALIDA ENTERO\ndef escr_sal_ent(byte,valor):\n lectura = plc.read_area(areas['PA'], 0, byte, 2) #PA: salidas, 0: bloque de datos, direcci\u00f3n, # bytes.\n # print(lectura)\n set_int(lectura, 0, valor) # se da formato al valor deseado, en este caso entero\n plc.write_area(areas['PA'], 0, byte, lectura) # Escribe en la direcci\u00f3n definida\n#escr_sal_ent(90,9000)\n\n#LEER MARCA ENTERA\ndef leer_ent_ent(byte):\n leer = plc.read_area(areas['PE'],0,byte,2) #PE: entradas, 0: bloque de datos, direcci\u00f3n, # bytes.\n leer_ent = get_int(leer,0) #Comando get_int(_bytearray, byte_index)\n return leer_ent\n\ndef AbrirValvulaQW96(valvulap):\n # Abre la valvula en QW96 a un porcentaje determinado\n valvula = ((7800 / 71.5) * (valvulap - 7.4)) + 6200\n escr_sal_ent(96,valvula)\n \n\ndef BombaQW80(motobombaHz):\n #Poner bomba en 60Hz salida QW80\n motobomba = motobombaHz * (22118 / 60) + 5530\n escr_sal_ent(80,motobomba)\n\ndef leerNivelPLCIW100():\n nivelplc = leer_ent_ent(100)\n nivelcm = ((60 / 15105) * (nivelplc - 10125)) + 20\n # print(nivelplc, nivelcm)\n return nivelcm\n\ndef leerPresionPLCIW96():\n presionplc = leer_ent_ent(96)\n # 0.64 5679\n # 0.4 5623\n mp = (0.64-0.4)/(5679-5623)\n presion = mp* presionplc - mp *5623+0.4\n # print(presionplc, presion)\n return presion\n\n\ndef leerFlujoPLCIW98():\n flujoplc = leer_ent_ent(98)\n # 4.67 8773 \n # 16.52 16957 \n mf = (4.67-16.52)/(8773-16957)\n flujo = mf* flujoplc - mf *16957+16.52\n # print(flujoplc, flujo)\n return flujo\n\ndef leerNada():\n return np.random.rand(1)[0]\n```\n\n## 3. Inicie el experimento.\n\nEl m\u00f3dulo multivariable tiene:\n- 2 actuadores:\n - Motobomba\n - V\u00e1lvula\n- 3 sensores:\n - Flujo\n - Presi\u00f3n\n - Nivel\n\n3.1 Escoja una variable como salida y un actuador para su experimento.\n\n3.2 Ponga un punto de operaci\u00f3n para el otro actuador.\n\n**Establecer un punto para la bomba**\n~~~\nBombaQW80(60)\n~~~\n\n**Establecer un punto para la v\u00e1lvula**\n~~~\nAbrirValvulaQW96(100)\n~~~\n\n\n```python\nescr_sal_bool(0,5,1) #Habilitar RUN motobomba\n# Complete para establecer un punto de operaci\u00f3n\nBombaQW80(20)\nAbrirValvulaQW96(100)\n```\n\n bytearray(b']\\x99')\n bytearray(b'\\x15\\x10')\n\n\n## 5. Defina la entrada del sistema\n\nEsta vez se configura una se\u00f1al de referencia $r(t)$ en forma de un escal\u00f3n.\n\n\\begin{equation}\nr(t) = \\begin{cases} V_0 & \\forall t t_0 \\end{cases}\n\\end{equation}\n\nDefina:\n- Duraci\u00f3n para el experimento $t_f$\n- Valor inicial para el actuador $V_0$\n- Valor final para el actuador $V_f$\n\nRecuerde el efecto de discretizar las se\u00f1ales con un tiempo de muestreo $t_m$.\n\n**Lista de tiempos $t$**\n\n\n```python\ntm = 1\ntf = 20\nt = np.linspace(0.0 ,tf, round(tf/tm) + 1)\n\n```\n\n**Se\u00f1al de entrada $U(t)$**\n\n\n```python\nt0 = 30\nv0 = 20\nvf = 60\nr = np.where(t>=t0,vf,v0)\n\n```\n\n**Gr\u00e1fica de la se\u00f1al**\n\n\n```python\nplt.plot(t,r)\n```\n\n\n## 6. Realimente el sistema y var\u00ede $k$\n\nDefina un valor $k$.\n\n- Use los valores definidos para $U(t)$ y las lecturas $y(t)$ del sensor para calcular el error $e(t)$ como:\n\n$$e(t)=U(t)-y(t)$$\n\n- Env\u00ede hacia el actuador una se\u00f1al $u(t)$ generada a partir de la se\u00f1al $e(t)$ como:\n$$u(t) = k \\cdot e(t)$$\n\n- Guarde la informaci\u00f3n necesaria. Para esto puede crear un archivo `csv` a partir de un **dataframe**.\n\nPara leer los sensores se han definido las siguientes funciones:\n\n~~~\nleerNivelPLCIW100()\nleerPresionPLCIW96()\nleerFlujoPLCIW98()\n~~~\n\n\n```python\n# Define k\nk = 0.1\n\n##\n\ny = []\ne = []\nu = []\nprint('Para abortar el experimento presione ESPACIO')\nabortado = False\nfor idt,ti in enumerate(t):\n ref_t = r[idt] # Valor de referencia\n y_t = leerPresionPLCIW96() # Salda del sistema\n e_t = ref_t - y_t # error\n u_t = k*e_t # Se\u00f1al de salida del controlador\n BombaQW80(u_t) # Salida hacia actuador\n y.append(y_t)\n e.append(e_t)\n u.append(u_t)\n \n if keyboard.is_pressed(' '):\n print('Abortando el experimento a los ' + \n str(ti) +' segundos')\n abortado = True\n escr_sal_bool(0,5,0)\n BombaQW80(5)\n AbrirValvulaQW96(0)\n \n break\n time.sleep(tm)\n \nif abortado:\n y = y + [np.nan]*(len(r)-len(y))\n e = e + [np.nan]*(len(r)-len(e))\n u = u + [np.nan]*(len(r)-len(u))\nd = {'Tiempo': t, 'Referencia': r, 'Error': e, 'Se\u00f1al de control': u,'Salida': y }\n\ndf = pd.DataFrame(data=d)\ndf.head()\nescr_sal_bool(0,5,0)\nAbrirValvulaQW96(0)\n\n# Gr\u00e1fica de entrada y salida\nplt.plot(df[\"Tiempo\"],df[\"Referencia\"],\n df[\"Tiempo\"],10*df[\"Salida\"])\n```\n\n Para abortar el experimento presione ESPACIO\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'2f')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n 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bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n bytearray(b'l\\x00')\n Abortando el experimento a los 359.0 segundos\n bytearray(b'l\\x00')\n bytearray(b'?\\xad')\n bytearray(b'\\x15\\x10')\n\n\n\n```python\n# Guarda datos\narchivo = \"Datos/misDatos\" + str(k) + \".csv\"\ndf.to_csv(archivo, index=False)\n```\n\n\n## 6. Compare con el LGDR a partir del modelo\n\n- Grafique el **Lugar Geom\u00e9trico de las Ra\u00edces** a partir del modelo que obtuvo anteriormente para el sistema.\n- Obtenga los polos y ceros de lazo cerrado para los valores $k$ que utiliz\u00f3 en el punto anterior.\n- Compare las respuestas experimentales y las relacionadas con las ra\u00edces seg\u00fan el modelo.\n\n\n```python\n## Escriba su c\u00f3digo\n```\n\n## Conclusiones\n- Aqu\u00ed\n- Otra\n\n\n```python\n\n```\n", "meta": {"hexsha": "a89766196f5ffabb13d3282adccd5c357e41b3d1", "size": 31897, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "LGDR_PLC_Siemens.ipynb", "max_stars_repo_name": "pierrediazp/Control", "max_stars_repo_head_hexsha": "2a185eff5b5dc84045115009e62296174d072220", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "LGDR_PLC_Siemens.ipynb", "max_issues_repo_name": "pierrediazp/Control", "max_issues_repo_head_hexsha": "2a185eff5b5dc84045115009e62296174d072220", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "LGDR_PLC_Siemens.ipynb", "max_forks_repo_name": "pierrediazp/Control", "max_forks_repo_head_hexsha": "2a185eff5b5dc84045115009e62296174d072220", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-11-18T13:08:36.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-18T13:08:36.000Z", "avg_line_length": 35.920045045, "max_line_length": 6648, "alphanum_fraction": 0.5943819168, "converted": true, "num_tokens": 5101, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660542, "lm_q2_score": 0.6619228691808011, "lm_q1q2_score": 0.431146680874627}} {"text": "\n# **Click-Through Rate Prediction Lab**\n#### This lab covers the steps for creating a click-through rate (CTR) prediction pipeline. You will work with the [Criteo Labs](http://labs.criteo.com/) dataset that was used for a recent [Kaggle competition](https://www.kaggle.com/c/criteo-display-ad-challenge).\n#### ** This lab will cover: **\n+ ####*Part 1:* Featurize categorical data using one-hot-encoding (OHE)\n+ ####*Part 2:* Construct an OHE dictionary\n+ ####*Part 3:* Parse CTR data and generate OHE features\n + #### *Visualization 1:* Feature frequency\n+ ####*Part 4:* CTR prediction and logloss evaluation\n + #### *Visualization 2:* ROC curve\n+ ####*Part 5:* Reduce feature dimension via feature hashing\n + #### *Visualization 3:* Hyperparameter heat map\n \n#### Note that, for reference, you can look up the details of the relevant Spark methods in [Spark's Python API](https://spark.apache.org/docs/latest/api/python/pyspark.html#pyspark.RDD) and the relevant NumPy methods in the [NumPy Reference](http://docs.scipy.org/doc/numpy/reference/index.html)\n\n\n```python\nlabVersion = 'cs190_week4_v_1_3'\n```\n\n### ** Part 1: Featurize categorical data using one-hot-encoding **\n\n#### ** (1a) One-hot-encoding **\n#### We would like to develop code to convert categorical features to numerical ones, and to build intuition, we will work with a sample unlabeled dataset with three data points, with each data point representing an animal. The first feature indicates the type of animal (bear, cat, mouse); the second feature describes the animal's color (black, tabby); and the third (optional) feature describes what the animal eats (mouse, salmon).\n#### In a one-hot-encoding (OHE) scheme, we want to represent each tuple of `(featureID, category)` via its own binary feature. We can do this in Python by creating a dictionary that **maps each tuple to a distinct integer**, where the integer corresponds to a binary feature. To start, manually enter the entries in the OHE dictionary associated with the sample dataset by mapping the tuples to consecutive integers starting from zero, ordering the tuples first by featureID and next by category.\n#### Later in this lab, we'll use OHE dictionaries to transform data points into compact lists of features that can be used in machine learning algorithms.\n\n\n```python\n# Data for manual OHE\n# Note: the first data point does not include any value for the optional third feature\nsampleOne = [(0, 'mouse'), (1, 'black')]\nsampleTwo = [(0, 'cat'), (1, 'tabby'), (2, 'mouse')]\nsampleThree = [(0, 'bear'), (1, 'black'), (2, 'salmon')]\nsampleDataRDD = sc.parallelize([sampleOne, sampleTwo, sampleThree])\n```\n\n\n```python\n# TODO: Replace with appropriate code\nsampleOHEDictManual = {}\nsampleOHEDictManual[(0,'bear')] = 0\nsampleOHEDictManual[(0,'cat')] = 1\nsampleOHEDictManual[(0,'mouse')] = 2\nsampleOHEDictManual[(1, 'black')] = 3\nsampleOHEDictManual[(1, 'tabby')] = 4\nsampleOHEDictManual[(2, 'mouse')] = 5\nsampleOHEDictManual[(2, 'salmon')] = 6\n```\n\n\n```python\n# TEST One-hot-encoding (1a)\nfrom test_helper import Test\n\nTest.assertEqualsHashed(sampleOHEDictManual[(0,'bear')],\n 'b6589fc6ab0dc82cf12099d1c2d40ab994e8410c',\n \"incorrect value for sampleOHEDictManual[(0,'bear')]\")\nTest.assertEqualsHashed(sampleOHEDictManual[(0,'cat')],\n '356a192b7913b04c54574d18c28d46e6395428ab',\n \"incorrect value for sampleOHEDictManual[(0,'cat')]\")\nTest.assertEqualsHashed(sampleOHEDictManual[(0,'mouse')],\n 'da4b9237bacccdf19c0760cab7aec4a8359010b0',\n \"incorrect value for sampleOHEDictManual[(0,'mouse')]\")\nTest.assertEqualsHashed(sampleOHEDictManual[(1,'black')],\n '77de68daecd823babbb58edb1c8e14d7106e83bb',\n \"incorrect value for sampleOHEDictManual[(1,'black')]\")\nTest.assertEqualsHashed(sampleOHEDictManual[(1,'tabby')],\n '1b6453892473a467d07372d45eb05abc2031647a',\n \"incorrect value for sampleOHEDictManual[(1,'tabby')]\")\nTest.assertEqualsHashed(sampleOHEDictManual[(2,'mouse')],\n 'ac3478d69a3c81fa62e60f5c3696165a4e5e6ac4',\n \"incorrect value for sampleOHEDictManual[(2,'mouse')]\")\nTest.assertEqualsHashed(sampleOHEDictManual[(2,'salmon')],\n 'c1dfd96eea8cc2b62785275bca38ac261256e278',\n \"incorrect value for sampleOHEDictManual[(2,'salmon')]\")\nTest.assertEquals(len(sampleOHEDictManual.keys()), 7,\n 'incorrect number of keys in sampleOHEDictManual')\n```\n\n 1 test passed.\n 1 test passed.\n 1 test passed.\n 1 test passed.\n 1 test passed.\n 1 test passed.\n 1 test passed.\n 1 test passed.\n\n\n#### ** (1b) Sparse vectors **\n#### Data points can typically be represented with a small number of non-zero OHE features relative to the total number of features that occur in the dataset. By leveraging this sparsity and using sparse vector representations of OHE data, we can reduce storage and computational burdens. Below are a few sample vectors represented as dense numpy arrays. Use [SparseVector](https://spark.apache.org/docs/latest/api/python/pyspark.mllib.html#pyspark.mllib.linalg.SparseVector) to represent them in a sparse fashion, and verify that both the sparse and dense representations yield the same results when computing [dot products](http://en.wikipedia.org/wiki/Dot_product) (we will later use MLlib to train classifiers via gradient descent, and MLlib will need to compute dot products between SparseVectors and dense parameter vectors).\n#### Use `SparseVector(size, *args)` to create a new sparse vector where size is the length of the vector and args is either a dictionary, a list of (index, value) pairs, or two separate arrays of indices and values (sorted by index). You'll need to create a sparse vector representation of each dense vector `aDense` and `bDense`.\n\n\n```python\nimport numpy as np\nfrom pyspark.mllib.linalg import SparseVector\n```\n\n\n```python\n# TODO: Replace with appropriate code\naDense = np.array([0., 3., 0., 4.])\naSparse = SparseVector(len(aDense), {1:3, 3:4})\n\nbDense = np.array([0., 0., 0., 1.])\nbSparse = SparseVector(len(bDense), {3:1})\n\nw = np.array([0.4, 3.1, -1.4, -.5])\nprint aDense.dot(w)\nprint aSparse.dot(w)\nprint bDense.dot(w)\nprint bSparse.dot(w)\n```\n\n 7.3\n 7.3\n -0.5\n -0.5\n\n\n\n```python\n# TEST Sparse Vectors (1b)\nTest.assertTrue(isinstance(aSparse, SparseVector), 'aSparse needs to be an instance of SparseVector')\nTest.assertTrue(isinstance(bSparse, SparseVector), 'aSparse needs to be an instance of SparseVector')\nTest.assertTrue(aDense.dot(w) == aSparse.dot(w),\n 'dot product of aDense and w should equal dot product of aSparse and w')\nTest.assertTrue(bDense.dot(w) == bSparse.dot(w),\n 'dot product of bDense and w should equal dot product of bSparse and w')\n```\n\n 1 test passed.\n 1 test passed.\n 1 test passed.\n 1 test passed.\n\n\n#### **(1c) OHE features as sparse vectors **\n#### Now let's see how we can represent the OHE features for points in our sample dataset. Using the mapping defined by the OHE dictionary from Part (1a), manually define OHE features for the three sample data points using SparseVector format. Any feature that occurs in a point should have the value 1.0. For example, the `DenseVector` for a point with features 2 and 4 would be `[0.0, 0.0, 1.0, 0.0, 1.0, 0.0, 0.0]`.\n\n\n```python\n# Reminder of the sample features\n# sampleOne = [(0, 'mouse'), (1, 'black')]\n# sampleTwo = [(0, 'cat'), (1, 'tabby'), (2, 'mouse')]\n# sampleThree = [(0, 'bear'), (1, 'black'), (2, 'salmon')]\n```\n\n\n```python\n# TODO: Replace with appropriate code\nsampleOneOHEFeatManual = SparseVector(7, [2, 3], [1, 1])\nsampleTwoOHEFeatManual = SparseVector(7, [1, 4, 5], [1, 1, 1])\nsampleThreeOHEFeatManual = SparseVector(7, {0:1, 3:1, 6:1})\n```\n\n\n```python\n# TEST OHE Features as sparse vectors (1c)\nTest.assertTrue(isinstance(sampleOneOHEFeatManual, SparseVector),\n 'sampleOneOHEFeatManual needs to be a SparseVector')\nTest.assertTrue(isinstance(sampleTwoOHEFeatManual, SparseVector),\n 'sampleTwoOHEFeatManual needs to be a SparseVector')\nTest.assertTrue(isinstance(sampleThreeOHEFeatManual, SparseVector),\n 'sampleThreeOHEFeatManual needs to be a SparseVector')\nTest.assertEqualsHashed(sampleOneOHEFeatManual,\n 'ecc00223d141b7bd0913d52377cee2cf5783abd6',\n 'incorrect value for sampleOneOHEFeatManual')\nTest.assertEqualsHashed(sampleTwoOHEFeatManual,\n '26b023f4109e3b8ab32241938e2e9b9e9d62720a',\n 'incorrect value for sampleTwoOHEFeatManual')\nTest.assertEqualsHashed(sampleThreeOHEFeatManual,\n 'c04134fd603ae115395b29dcabe9d0c66fbdc8a7',\n 'incorrect value for sampleThreeOHEFeatManual')\n```\n\n 1 test passed.\n 1 test passed.\n 1 test passed.\n 1 test passed.\n 1 test passed.\n 1 test passed.\n\n\n#### **(1d) Define a OHE function **\n#### Next we will use the OHE dictionary from Part (1a) to programatically generate OHE features from the original categorical data. First write a function called `oneHotEncoding` that creates OHE feature vectors in `SparseVector` format. Then use this function to create OHE features for the first sample data point and verify that the result matches the result from Part (1c).\n\n\n```python\n# TODO: Replace with appropriate code\ndef oneHotEncoding(rawFeats, OHEDict, numOHEFeats):\n \"\"\"Produce a one-hot-encoding from a list of features and an OHE dictionary.\n\n Note:\n You should ensure that the indices used to create a SparseVector are sorted.\n\n Args:\n rawFeats (list of (int, str)): The features corresponding to a single observation. Each\n feature consists of a tuple of featureID and the feature's value. (e.g. sampleOne)\n OHEDict (dict): A mapping of (featureID, value) to unique integer.\n numOHEFeats (int): The total number of unique OHE features (combinations of featureID and\n value).\n\n Returns:\n SparseVector: A SparseVector of length numOHEFeats with indicies equal to the unique\n identifiers for the (featureID, value) combinations that occur in the observation and\n with values equal to 1.0.\n \"\"\"\n sparseIndex = np.sort(list(OHEDict[i] for i in rawFeats))\n sparseVal = np.ones(len(rawFeats))\n return SparseVector(numOHEFeats, sparseIndex, sparseVal)\n\n# Calculate the number of features in sampleOHEDictManual\nnumSampleOHEFeats = len(sampleOHEDictManual)\n\n# Run oneHotEnoding on sampleOne\nsampleOneOHEFeat = oneHotEncoding(sampleOne, sampleOHEDictManual, numSampleOHEFeats)\n\nprint sampleOneOHEFeat\n```\n\n (7,[2,3],[1.0,1.0])\n\n\n\n```python\n# TEST Define an OHE Function (1d)\nTest.assertTrue(sampleOneOHEFeat == sampleOneOHEFeatManual,\n 'sampleOneOHEFeat should equal sampleOneOHEFeatManual')\nTest.assertEquals(sampleOneOHEFeat, SparseVector(7, [2,3], [1.0,1.0]),\n 'incorrect value for sampleOneOHEFeat')\nTest.assertEquals(oneHotEncoding([(1, 'black'), (0, 'mouse')], sampleOHEDictManual,\n numSampleOHEFeats), SparseVector(7, [2,3], [1.0,1.0]),\n 'incorrect definition for oneHotEncoding')\n```\n\n 1 test passed.\n 1 test passed.\n 1 test passed.\n\n\n#### **(1e) Apply OHE to a dataset **\n#### Finally, use the function from Part (1d) to create OHE features for all 3 data points in the sample dataset.\n\n\n```python\n# TODO: Replace with appropriate code\nsampleOHEData = sampleDataRDD.map(lambda x: \n oneHotEncoding(x, \n sampleOHEDictManual, \n numSampleOHEFeats))\nprint sampleOHEData.collect()\n```\n\n [SparseVector(7, {2: 1.0, 3: 1.0}), SparseVector(7, {1: 1.0, 4: 1.0, 5: 1.0}), SparseVector(7, {0: 1.0, 3: 1.0, 6: 1.0})]\n\n\n\n```python\n# TEST Apply OHE to a dataset (1e)\nsampleOHEDataValues = sampleOHEData.collect()\nTest.assertTrue(len(sampleOHEDataValues) == 3, 'sampleOHEData should have three elements')\nTest.assertEquals(sampleOHEDataValues[0], SparseVector(7, {2: 1.0, 3: 1.0}),\n 'incorrect OHE for first sample')\nTest.assertEquals(sampleOHEDataValues[1], SparseVector(7, {1: 1.0, 4: 1.0, 5: 1.0}),\n 'incorrect OHE for second sample')\nTest.assertEquals(sampleOHEDataValues[2], SparseVector(7, {0: 1.0, 3: 1.0, 6: 1.0}),\n 'incorrect OHE for third sample')\n```\n\n 1 test passed.\n 1 test passed.\n 1 test passed.\n 1 test passed.\n\n\n### ** Part 2: Construct an OHE dictionary **\n\n#### **(2a) Pair RDD of `(featureID, category)` **\n#### To start, create an RDD of distinct `(featureID, category)` tuples. In our sample dataset, the 7 items in the resulting RDD are `(0, 'bear')`, `(0, 'cat')`, `(0, 'mouse')`, `(1, 'black')`, `(1, 'tabby')`, `(2, 'mouse')`, `(2, 'salmon')`. Notably `'black'` appears twice in the dataset but only contributes one item to the RDD: `(1, 'black')`, while `'mouse'` also appears twice and contributes two items: `(0, 'mouse')` and `(2, 'mouse')`. Use [flatMap](https://spark.apache.org/docs/latest/api/python/pyspark.html#pyspark.RDD.flatMap) and [distinct](https://spark.apache.org/docs/latest/api/python/pyspark.html#pyspark.RDD.distinct).\n\n\n```python\n# TODO: Replace with appropriate code\nsampleDistinctFeats = (sampleDataRDD\n .flatMap(lambda x: x).distinct())\n```\n\n\n```python\n# TEST Pair RDD of (featureID, category) (2a)\nTest.assertEquals(sorted(sampleDistinctFeats.collect()),\n [(0, 'bear'), (0, 'cat'), (0, 'mouse'), (1, 'black'),\n (1, 'tabby'), (2, 'mouse'), (2, 'salmon')],\n 'incorrect value for sampleDistinctFeats')\n```\n\n 1 test passed.\n\n\n#### ** (2b) OHE Dictionary from distinct features **\n#### Next, create an `RDD` of key-value tuples, where each `(featureID, category)` tuple in `sampleDistinctFeats` is a key and the values are distinct integers ranging from 0 to (number of keys - 1). Then convert this `RDD` into a dictionary, which can be done using the `collectAsMap` action. Note that there is no unique mapping from keys to values, as all we require is that each `(featureID, category)` key be mapped to a unique integer between 0 and the number of keys. In this exercise, any valid mapping is acceptable. Use [zipWithIndex](https://spark.apache.org/docs/latest/api/python/pyspark.html#pyspark.RDD.zipWithIndex) followed by [collectAsMap](https://spark.apache.org/docs/latest/api/python/pyspark.html#pyspark.RDD.collectAsMap).\n#### In our sample dataset, one valid list of key-value tuples is: `[((0, 'bear'), 0), ((2, 'salmon'), 1), ((1, 'tabby'), 2), ((2, 'mouse'), 3), ((0, 'mouse'), 4), ((0, 'cat'), 5), ((1, 'black'), 6)]`. The dictionary defined in Part (1a) illustrates another valid mapping between keys and integers.\n\n\n```python\n# TODO: Replace with appropriate code\nsampleOHEDict = (sampleDistinctFeats\n .zipWithIndex().collectAsMap())\nprint sampleOHEDict\n```\n\n {(2, 'mouse'): 0, (0, 'cat'): 1, (0, 'bear'): 2, (2, 'salmon'): 3, (1, 'tabby'): 4, (1, 'black'): 5, (0, 'mouse'): 6}\n\n\n\n```python\n# TEST OHE Dictionary from distinct features (2b)\nTest.assertEquals(sorted(sampleOHEDict.keys()),\n [(0, 'bear'), (0, 'cat'), (0, 'mouse'), (1, 'black'),\n (1, 'tabby'), (2, 'mouse'), (2, 'salmon')],\n 'sampleOHEDict has unexpected keys')\nTest.assertEquals(sorted(sampleOHEDict.values()), range(7), 'sampleOHEDict has unexpected values')\n```\n\n 1 test passed.\n 1 test passed.\n\n\n#### **(2c) Automated creation of an OHE dictionary **\n#### Now use the code from Parts (2a) and (2b) to write a function that takes an input dataset and outputs an OHE dictionary. Then use this function to create an OHE dictionary for the sample dataset, and verify that it matches the dictionary from Part (2b).\n\n\n```python\n# TODO: Replace with appropriate code\ndef createOneHotDict(inputData):\n \"\"\"Creates a one-hot-encoder dictionary based on the input data.\n\n Args:\n inputData (RDD of lists of (int, str)): An RDD of observations where each observation is\n made up of a list of (featureID, value) tuples.\n\n Returns:\n dict: A dictionary where the keys are (featureID, value) tuples and map to values that are\n unique integers.\n \"\"\"\n return inputData.flatMap(lambda x: x).distinct().zipWithIndex().collectAsMap()\n\nsampleOHEDictAuto = createOneHotDict(sampleDataRDD)\nprint sampleOHEDictAuto\n```\n\n {(2, 'mouse'): 0, (0, 'cat'): 1, (0, 'bear'): 2, (2, 'salmon'): 3, (1, 'tabby'): 4, (1, 'black'): 5, (0, 'mouse'): 6}\n\n\n\n```python\n# TEST Automated creation of an OHE dictionary (2c)\nTest.assertEquals(sorted(sampleOHEDictAuto.keys()),\n [(0, 'bear'), (0, 'cat'), (0, 'mouse'), (1, 'black'),\n (1, 'tabby'), (2, 'mouse'), (2, 'salmon')],\n 'sampleOHEDictAuto has unexpected keys')\nTest.assertEquals(sorted(sampleOHEDictAuto.values()), range(7),\n 'sampleOHEDictAuto has unexpected values')\n```\n\n 1 test passed.\n 1 test passed.\n\n\n### **Part 3: Parse CTR data and generate OHE features**\n\n#### Before we can proceed, you'll first need to obtain the data from Criteo. If you have already completed this step in the setup lab, just run the cells below and the data will be loaded into the `rawData` variable.\n#### Below is Criteo's data sharing agreement. After you accept the agreement, you can obtain the download URL by right-clicking on the \"Download Sample\" button and clicking \"Copy link address\" or \"Copy Link Location\", depending on your browser. Paste the URL into the `# TODO` cell below. The file is 8.4 MB compressed. The script below will download the file to the virtual machine (VM) and then extract the data.\n#### If running the cell below does not render a webpage, open the [Criteo agreement](http://labs.criteo.com/downloads/2014-kaggle-display-advertising-challenge-dataset/) in a separate browser tab. After you accept the agreement, you can obtain the download URL by right-clicking on the \"Download Sample\" button and clicking \"Copy link address\" or \"Copy Link Location\", depending on your browser. Paste the URL into the `# TODO` cell below.\n#### Note that the download could take a few minutes, depending upon your connection speed.\n\n\n```python\n# Run this code to view Criteo's agreement\nfrom IPython.lib.display import IFrame\n\nIFrame(\"http://labs.criteo.com/downloads/2014-kaggle-display-advertising-challenge-dataset/\",\n 600, 350)\n```\n\n\n\n\n\n\n\n\n\n\n\n```python\n# TODO: Replace with appropriate code\n# Just replace with the url for dac_sample.tar.gz\nimport glob\nimport os.path\nimport tarfile\nimport urllib\nimport urlparse\n\n# Paste url, url should end with: dac_sample.tar.gz\nurl = 'https://s3-eu-west-1.amazonaws.com/criteo-labs/dac.tar.gz'\n\nurl = url.strip()\nbaseDir = os.path.join('data')\ninputPath = os.path.join('cs190', 'dac_sample.txt')\nfileName = os.path.join(baseDir, inputPath)\ninputDir = os.path.split(fileName)[0]\n\ndef extractTar(check = False):\n # Find the zipped archive and extract the dataset\n tars = glob.glob('dac_sample*.tar.gz*')\n if check and len(tars) == 0:\n return False\n\n if len(tars) > 0:\n try:\n tarFile = tarfile.open(tars[0])\n except tarfile.ReadError:\n if not check:\n print 'Unable to open tar.gz file. Check your URL.'\n return False\n\n tarFile.extract('dac_sample.txt', path=inputDir)\n print 'Successfully extracted: dac_sample.txt'\n return True\n else:\n print 'You need to retry the download with the correct url.'\n print ('Alternatively, you can upload the dac_sample.tar.gz file to your Jupyter root ' +\n 'directory')\n return False\n\n\nif os.path.isfile(fileName):\n print 'File is already available. Nothing to do.'\nelif extractTar(check = True):\n print 'tar.gz file was already available.'\nelif not url.endswith('dac_sample.tar.gz'):\n print 'Check your download url. Are you downloading the Sample dataset?'\nelse:\n # Download the file and store it in the same directory as this notebook\n try:\n urllib.urlretrieve(url, os.path.basename(urlparse.urlsplit(url).path))\n except IOError:\n print 'Unable to download and store: {0}'.format(url)\n\n extractTar()\n```\n\n File is already available. Nothing to do.\n\n\n\n```python\nimport os.path\nbaseDir = os.path.join('data')\ninputPath = os.path.join('cs190', 'dac_sample.txt')\nfileName = os.path.join(baseDir, inputPath)\n\nif os.path.isfile(fileName):\n rawData = (sc\n .textFile(fileName, 2)\n .map(lambda x: x.replace('\\t', ','))) # work with either ',' or '\\t' separated data\n print rawData.take(1)\n```\n\n [u'0,1,1,5,0,1382,4,15,2,181,1,2,,2,68fd1e64,80e26c9b,fb936136,7b4723c4,25c83c98,7e0ccccf,de7995b8,1f89b562,a73ee510,a8cd5504,b2cb9c98,37c9c164,2824a5f6,1adce6ef,8ba8b39a,891b62e7,e5ba7672,f54016b9,21ddcdc9,b1252a9d,07b5194c,,3a171ecb,c5c50484,e8b83407,9727dd16']\n\n\n#### **(3a) Loading and splitting the data **\n#### We are now ready to start working with the actual CTR data, and our first task involves splitting it into training, validation, and test sets. Use the [randomSplit method](https://spark.apache.org/docs/latest/api/python/pyspark.html#pyspark.RDD.randomSplit) with the specified weights and seed to create RDDs storing each of these datasets, and then [cache](https://spark.apache.org/docs/latest/api/python/pyspark.html#pyspark.RDD.cache) each of these RDDs, as we will be accessing them multiple times in the remainder of this lab. Finally, compute the size of each dataset.\n\n\n```python\n# TODO: Replace with appropriate code\nweights = [.8, .1, .1]\nseed = 42\n# Use randomSplit with weights and seed\nrawTrainData, rawValidationData, rawTestData = rawData.randomSplit(weights, seed)\n# Cache the data\nrawTrainData.cache()\nrawValidationData.cache()\nrawTestData.cache()\n\nnTrain = rawTrainData.count()\nnVal = rawValidationData.count()\nnTest = rawTestData.count()\nprint nTrain, nVal, nTest, nTrain + nVal + nTest\nprint rawData.take(1)\n```\n\n 79911 10075 10014 100000\n [u'0,1,1,5,0,1382,4,15,2,181,1,2,,2,68fd1e64,80e26c9b,fb936136,7b4723c4,25c83c98,7e0ccccf,de7995b8,1f89b562,a73ee510,a8cd5504,b2cb9c98,37c9c164,2824a5f6,1adce6ef,8ba8b39a,891b62e7,e5ba7672,f54016b9,21ddcdc9,b1252a9d,07b5194c,,3a171ecb,c5c50484,e8b83407,9727dd16']\n\n\n\n```python\n# TEST Loading and splitting the data (3a)\nTest.assertTrue(all([rawTrainData.is_cached, rawValidationData.is_cached, rawTestData.is_cached]),\n 'you must cache the split data')\nTest.assertEquals(nTrain, 79911, 'incorrect value for nTrain')\nTest.assertEquals(nVal, 10075, 'incorrect value for nVal')\nTest.assertEquals(nTest, 10014, 'incorrect value for nTest')\n```\n\n 1 test passed.\n 1 test passed.\n 1 test passed.\n 1 test passed.\n\n\n#### ** (3b) Extract features **\n#### We will now parse the raw training data to create an RDD that we can subsequently use to create an OHE dictionary. Note from the `take()` command in Part (3a) that each raw data point is a string containing several fields separated by some delimiter. For now, we will ignore the first field (which is the 0-1 label), and parse the remaining fields (or raw features). To do this, complete the implemention of the `parsePoint` function.\n\n\n```python\n# TODO: Replace with appropriate code\ndef parsePoint(point):\n \"\"\"Converts a comma separated string into a list of (featureID, value) tuples.\n\n Note:\n featureIDs should start at 0 and increase to the number of features - 1.\n\n Args:\n point (str): A comma separated string where the first value is the label and the rest\n are features.\n\n Returns:\n list: A list of (featureID, value) tuples.\n \"\"\"\n features = point.split(',')[1:]\n return [(i, features[i]) for i in range(len(features))]\n\n\nparsedTrainFeat = rawTrainData.map(parsePoint)\n\nnumCategories = (parsedTrainFeat\n .flatMap(lambda x: x)\n .distinct()\n .map(lambda x: (x[0], 1))\n .reduceByKey(lambda x, y: x + y)\n .sortByKey()\n .collect())\n\nprint numCategories[2][1]\n```\n\n 855\n\n\n\n```python\n# TEST Extract features (3b)\nTest.assertEquals(numCategories[2][1], 855, 'incorrect implementation of parsePoint')\nTest.assertEquals(numCategories[32][1], 4, 'incorrect implementation of parsePoint')\n```\n\n 1 test passed.\n 1 test passed.\n\n\n#### **(3c) Create an OHE dictionary from the dataset **\n#### Note that parsePoint returns a data point as a list of `(featureID, category)` tuples, which is the same format as the sample dataset studied in Parts 1 and 2 of this lab. Using this observation, create an OHE dictionary using the function implemented in Part (2c). Note that we will assume for simplicity that all features in our CTR dataset are categorical.\n\n\n```python\n# TODO: Replace with appropriate code\nctrOHEDict = createOneHotDict(parsedTrainFeat)\nnumCtrOHEFeats = len(ctrOHEDict.keys())\nprint numCtrOHEFeats\nprint ctrOHEDict[(0, '')]\n```\n\n 233286\n 36177\n\n\n\n```python\n# TEST Create an OHE dictionary from the dataset (3c)\nTest.assertEquals(numCtrOHEFeats, 233286, 'incorrect number of features in ctrOHEDict')\nTest.assertTrue((0, '') in ctrOHEDict, 'incorrect features in ctrOHEDict')\n```\n\n 1 test passed.\n 1 test passed.\n\n\n#### ** (3d) Apply OHE to the dataset **\n#### Now let's use this OHE dictionary by starting with the raw training data and creating an RDD of [LabeledPoint](http://spark.apache.org/docs/1.3.1/api/python/pyspark.mllib.html#pyspark.mllib.regression.LabeledPoint) objects using OHE features. To do this, complete the implementation of the `parseOHEPoint` function. Hint: `parseOHEPoint` is an extension of the `parsePoint` function from Part (3b) and it uses the `oneHotEncoding` function from Part (1d).\n\n\n```python\nfrom pyspark.mllib.regression import LabeledPoint\n```\n\n\n```python\n# TODO: Replace with appropriate code\ndef parseOHEPoint(point, OHEDict, numOHEFeats):\n \"\"\"Obtain the label and feature vector for this raw observation.\n\n Note:\n You must use the function `oneHotEncoding` in this implementation or later portions\n of this lab may not function as expected.\n\n Args:\n point (str): A comma separated string where the first value is the label and the rest\n are features.\n OHEDict (dict of (int, str) to int): Mapping of (featureID, value) to unique integer.\n numOHEFeats (int): The number of unique features in the training dataset.\n\n Returns:\n LabeledPoint: Contains the label for the observation and the one-hot-encoding of the\n raw features based on the provided OHE dictionary.\n \"\"\"\n parsedPoint = point.split(',')\n label = parsedPoint[0]\n features = parsedPoint[1:]\n rawFeats = [(i, features[i]) for i in range(len(features))]\n return LabeledPoint(label, oneHotEncoding(rawFeats, OHEDict, numOHEFeats))\n\nOHETrainData = rawTrainData.map(lambda point: parseOHEPoint(point, ctrOHEDict, numCtrOHEFeats))\nOHETrainData.cache()\nprint OHETrainData.take(1)\n\n# Check that oneHotEncoding function was used in parseOHEPoint\nbackupOneHot = oneHotEncoding\noneHotEncoding = None\nwithOneHot = False\ntry: parseOHEPoint(rawTrainData.take(1)[0], ctrOHEDict, numCtrOHEFeats)\nexcept TypeError: withOneHot = True\noneHotEncoding = backupOneHot\n```\n\n [LabeledPoint(0.0, (233286,[382,3101,6842,8311,8911,11887,12893,16211,17631,18646,23513,29366,33157,39536,55820,61797,81485,82753,93671,96986,109720,110662,112139,120263,128571,132400,132805,140595,160666,185457,190322,191105,195902,202638,204242,206037,222753,225966,229941],[1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0]))]\n\n\n\n```python\n# TEST Apply OHE to the dataset (3d)\nnumNZ = sum(parsedTrainFeat.map(lambda x: len(x)).take(5))\nnumNZAlt = sum(OHETrainData.map(lambda lp: len(lp.features.indices)).take(5))\nTest.assertEquals(numNZ, numNZAlt, 'incorrect implementation of parseOHEPoint')\nTest.assertTrue(withOneHot, 'oneHotEncoding not present in parseOHEPoint')\n```\n\n 1 test passed.\n 1 test passed.\n\n\n#### **Visualization 1: Feature frequency **\n#### We will now visualize the number of times each of the 233,286 OHE features appears in the training data. We first compute the number of times each feature appears, then bucket the features by these counts. The buckets are sized by powers of 2, so the first bucket corresponds to features that appear exactly once ( $ \\scriptsize 2^0 $ ), the second to features that appear twice ( $ \\scriptsize 2^1 $ ), the third to features that occur between three and four ( $ \\scriptsize 2^2 $ ) times, the fifth bucket is five to eight ( $ \\scriptsize 2^3 $ ) times and so on. The scatter plot below shows the logarithm of the bucket thresholds versus the logarithm of the number of features that have counts that fall in the buckets.\n\n\n```python\ndef bucketFeatByCount(featCount):\n \"\"\"Bucket the counts by powers of two.\"\"\"\n for i in range(11):\n size = 2 ** i\n if featCount <= size:\n return size\n return -1\n\nfeatCounts = (OHETrainData\n .flatMap(lambda lp: lp.features.indices)\n .map(lambda x: (x, 1))\n .reduceByKey(lambda x, y: x + y))\nfeatCountsBuckets = (featCounts\n .map(lambda x: (bucketFeatByCount(x[1]), 1))\n .filter(lambda (k, v): k != -1)\n .reduceByKey(lambda x, y: x + y)\n .collect())\nprint featCountsBuckets\n```\n\n [(256, 748), (1024, 255), (2, 24076), (4, 16639), (32, 4755), (8, 11440), (64, 2627), (128, 1476), (16, 7752), (512, 414), (1, 162813)]\n\n\n\n```python\nimport matplotlib.pyplot as plt\n\nx, y = zip(*featCountsBuckets)\nx, y = np.log(x), np.log(y)\n\ndef preparePlot(xticks, yticks, figsize=(10.5, 6), hideLabels=False, gridColor='#999999',\n gridWidth=1.0):\n \"\"\"Template for generating the plot layout.\"\"\"\n plt.close()\n fig, ax = plt.subplots(figsize=figsize, facecolor='white', edgecolor='white')\n ax.axes.tick_params(labelcolor='#999999', labelsize='10')\n for axis, ticks in [(ax.get_xaxis(), xticks), (ax.get_yaxis(), yticks)]:\n axis.set_ticks_position('none')\n axis.set_ticks(ticks)\n axis.label.set_color('#999999')\n if hideLabels: axis.set_ticklabels([])\n plt.grid(color=gridColor, linewidth=gridWidth, linestyle='-')\n map(lambda position: ax.spines[position].set_visible(False), ['bottom', 'top', 'left', 'right'])\n return fig, ax\n\n# generate layout and plot data\nfig, ax = preparePlot(np.arange(0, 10, 1), np.arange(4, 14, 2))\nax.set_xlabel(r'$\\log_e(bucketSize)$'), ax.set_ylabel(r'$\\log_e(countInBucket)$')\nplt.scatter(x, y, s=14**2, c='#d6ebf2', edgecolors='#8cbfd0', alpha=0.75)\npass\n```\n\n#### **(3e) Handling unseen features **\n#### We naturally would like to repeat the process from Part (3d), e.g., to compute OHE features for the validation and test datasets. However, we must be careful, as some categorical values will likely appear in new data that did not exist in the training data. To deal with this situation, update the `oneHotEncoding()` function from Part (1d) to ignore previously unseen categories, and then compute OHE features for the validation data.\n\n\n```python\n# TODO: Replace with appropriate code\ndef oneHotEncoding(rawFeats, OHEDict, numOHEFeats):\n \"\"\"Produce a one-hot-encoding from a list of features and an OHE dictionary.\n\n Note:\n If a (featureID, value) tuple doesn't have a corresponding key in OHEDict it should be\n ignored.\n\n Args:\n rawFeats (list of (int, str)): The features corresponding to a single observation. Each\n feature consists of a tuple of featureID and the feature's value. (e.g. sampleOne)\n OHEDict (dict): A mapping of (featureID, value) to unique integer.\n numOHEFeats (int): The total number of unique OHE features (combinations of featureID and\n value).\n\n Returns:\n SparseVector: A SparseVector of length numOHEFeats with indicies equal to the unique\n identifiers for the (featureID, value) combinations that occur in the observation and\n with values equal to 1.0.\n \"\"\"\n sparseIndex = np.sort([OHEDict[i] for i in rawFeats if i in OHEDict.keys()])\n sparseVal = np.ones(len(sparseIndex))\n return SparseVector(numOHEFeats, sparseIndex, sparseVal)\n\n\nOHEValidationData = rawValidationData.map(lambda point: parseOHEPoint(point, ctrOHEDict, numCtrOHEFeats))\nOHEValidationData.cache()\nprint OHEValidationData.take(1)\n```\n\n \n KeyboardInterrupt\n\n\n\n```python\n# TEST Handling unseen features (3e)\nnumNZVal = (OHEValidationData\n .map(lambda lp: len(lp.features.indices))\n .sum())\nTest.assertEquals(numNZVal, 372080, 'incorrect number of features')\n```\n\n### ** Part 4: CTR prediction and logloss evaluation **\n\n#### ** (4a) Logistic regression **\n#### We are now ready to train our first CTR classifier. A natural classifier to use in this setting is logistic regression, since it models the probability of a click-through event rather than returning a binary response, and when working with rare events, probabilistic predictions are useful. First use [LogisticRegressionWithSGD](https://spark.apache.org/docs/latest/api/python/pyspark.mllib.html#pyspark.mllib.classification.LogisticRegressionWithSGD) to train a model using `OHETrainData` with the given hyperparameter configuration. `LogisticRegressionWithSGD` returns a [LogisticRegressionModel](https://spark.apache.org/docs/latest/api/python/pyspark.mllib.html#pyspark.mllib.regression.LogisticRegressionModel). Next, use the `LogisticRegressionModel.weights` and `LogisticRegressionModel.intercept` attributes to print out the model's parameters. Note that these are the names of the object's attributes and should be called using a syntax like `model.weights` for a given `model`.\n\n\n```python\nfrom pyspark.mllib.classification import LogisticRegressionWithSGD\n\n# fixed hyperparameters\nnumIters = 50\nstepSize = 10.\nregParam = 1e-6\nregType = 'l2'\nincludeIntercept = True\n```\n\n\n```python\n# TODO: Replace with appropriate code\nmodel0 = LogisticRegressionWithSGD.train(OHETrainData, \n iterations=numIters,\n step=stepSize,\n regParam=regParam,\n regType=regType,\n intercept=includeIntercept)\nsortedWeights = sorted(model0.weights)\nprint sortedWeights[:5], model0.intercept\n```\n\n\n```python\n# TEST Logistic regression (4a)\nTest.assertTrue(np.allclose(model0.intercept, 0.56455084025), 'incorrect value for model0.intercept')\nTest.assertTrue(np.allclose(sortedWeights[0:5],\n [-0.45899236853575609, -0.37973707648623956, -0.36996558266753304,\n -0.36934962879928263, -0.32697945415010637]), 'incorrect value for model0.weights')\n```\n\n#### ** (4b) Log loss **\n#### Throughout this lab, we will use log loss to evaluate the quality of models. Log loss is defined as: $$ \\begin{align} \\scriptsize \\ell_{log}(p, y) = \\begin{cases} -\\log (p) & \\text{if } y = 1 \\\\\\ -\\log(1-p) & \\text{if } y = 0 \\end{cases} \\end{align} $$ where $ \\scriptsize p$ is a probability between 0 and 1 and $ \\scriptsize y$ is a label of either 0 or 1. Log loss is a standard evaluation criterion when predicting rare-events such as click-through rate prediction (it is also the criterion used in the [Criteo Kaggle competition](https://www.kaggle.com/c/criteo-display-ad-challenge)). Write a function to compute log loss, and evaluate it on some sample inputs.\n\n\n```python\n# TODO: Replace with appropriate code\nfrom math import log\n\ndef computeLogLoss(p, y):\n \"\"\"Calculates the value of log loss for a given probabilty and label.\n\n Note:\n log(0) is undefined, so when p is 0 we need to add a small value (epsilon) to it\n and when p is 1 we need to subtract a small value (epsilon) from it.\n\n Args:\n p (float): A probabilty between 0 and 1.\n y (int): A label. Takes on the values 0 and 1.\n\n Returns:\n float: The log loss value.\n \"\"\"\n epsilon = 10e-12\n if p == 0:\n p += epsilon\n elif p == 1:\n p -= epsilon\n if y == 1:\n return -log(p)\n elif y == 0:\n return -log(1 - p)\n\nprint computeLogLoss(.5, 1)\nprint computeLogLoss(.5, 0)\nprint computeLogLoss(.99, 1)\nprint computeLogLoss(.99, 0)\nprint computeLogLoss(.01, 1)\nprint computeLogLoss(.01, 0)\nprint computeLogLoss(0, 1)\nprint computeLogLoss(1, 1)\nprint computeLogLoss(1, 0)\n```\n\n\n```python\n# TEST Log loss (4b)\nTest.assertTrue(np.allclose([computeLogLoss(.5, 1), computeLogLoss(.01, 0), computeLogLoss(.01, 1)],\n [0.69314718056, 0.0100503358535, 4.60517018599]),\n 'computeLogLoss is not correct')\nTest.assertTrue(np.allclose([computeLogLoss(0, 1), computeLogLoss(1, 1), computeLogLoss(1, 0)],\n [25.3284360229, 1.00000008275e-11, 25.3284360229]),\n 'computeLogLoss needs to bound p away from 0 and 1 by epsilon')\n```\n\n#### ** (4c) Baseline log loss **\n#### Next we will use the function we wrote in Part (4b) to compute the baseline log loss on the training data. A very simple yet natural baseline model is one where we always make the same prediction independent of the given datapoint, setting the predicted value equal to the fraction of training points that correspond to click-through events (i.e., where the label is one). Compute this value (which is simply the mean of the training labels), and then use it to compute the training log loss for the baseline model. The log loss for multiple observations is the mean of the individual log loss values.\n\n\n```python\n# TODO: Replace with appropriate code\n# Note that our dataset has a very high click-through rate by design\n# In practice click-through rate can be one to two orders of magnitude lower\nclassOneFracTrain = OHETrainData.map(lambda lp: lp.label).sum() / OHETrainData.count()\nprint classOneFracTrain\n\nlogLossTrBase = OHETrainData.map(lambda lp:\n computeLogLoss(classOneFracTrain, lp.label)).sum() / OHETrainData.count()\nprint 'Baseline Train Logloss = {0:.3f}\\n'.format(logLossTrBase)\n```\n\n\n```python\n# TEST Baseline log loss (4c)\nTest.assertTrue(np.allclose(classOneFracTrain, 0.22717773523), 'incorrect value for classOneFracTrain')\nTest.assertTrue(np.allclose(logLossTrBase, 0.535844), 'incorrect value for logLossTrBase')\n```\n\n#### ** (4d) Predicted probability **\n#### In order to compute the log loss for the model we trained in Part (4a), we need to write code to generate predictions from this model. Write a function that computes the raw linear prediction from this logistic regression model and then passes it through a [sigmoid function](http://en.wikipedia.org/wiki/Sigmoid_function) $ \\scriptsize \\sigma(t) = (1+ e^{-t})^{-1} $ to return the model's probabilistic prediction. Then compute probabilistic predictions on the training data.\n#### Note that when incorporating an intercept into our predictions, we simply add the intercept to the value of the prediction obtained from the weights and features. Alternatively, if the intercept was included as the first weight, we would need to add a corresponding feature to our data where the feature has the value one. This is not the case here.\n\n\n```python\n# TODO: Replace with appropriate code\nfrom math import exp # exp(-t) = e^-t\n\ndef getP(x, w, intercept):\n \"\"\"Calculate the probability for an observation given a set of weights and intercept.\n\n Note:\n We'll bound our raw prediction between 20 and -20 for numerical purposes.\n\n Args:\n x (SparseVector): A vector with values of 1.0 for features that exist in this\n observation and 0.0 otherwise.\n w (DenseVector): A vector of weights (betas) for the model.\n intercept (float): The model's intercept.\n\n Returns:\n float: A probability between 0 and 1.\n \"\"\"\n rawPrediction = 1 / (1 + exp(-x.dot(w)-intercept))\n\n # Bound the raw prediction value\n rawPrediction = min(rawPrediction, 20)\n rawPrediction = max(rawPrediction, -20)\n return rawPrediction\n\ntrainingPredictions = OHETrainData.map(lambda lp: getP(lp.features, model0.weights, model0.intercept))\n\nprint trainingPredictions.take(5)\n```\n\n\n```python\n# TEST Predicted probability (4d)\nTest.assertTrue(np.allclose(trainingPredictions.sum(), 18135.4834348),\n 'incorrect value for trainingPredictions')\n```\n\n#### ** (4e) Evaluate the model **\n#### We are now ready to evaluate the quality of the model we trained in Part (4a). To do this, first write a general function that takes as input a model and data, and outputs the log loss. Then run this function on the OHE training data, and compare the result with the baseline log loss.\n\n\n```python\n# TODO: Replace with appropriate code\ndef evaluateResults(model, data):\n \"\"\"Calculates the log loss for the data given the model.\n\n Args:\n model (LogisticRegressionModel): A trained logistic regression model.\n data (RDD of LabeledPoint): Labels and features for each observation.\n\n Returns:\n float: Log loss for the data.\n \"\"\"\n probabilityAndLabel = data.map(lambda lp: (getP(lp.features, model.weights, model.intercept), lp.label))\n logLoss = probabilityAndLabel.map(lambda (x,y): computeLogLoss(x, y)).sum() / probabilityAndLabel.count()\n return logLoss\n \n\nlogLossTrLR0 = evaluateResults(model0, OHETrainData)\nprint ('OHE Features Train Logloss:\\n\\tBaseline = {0:.3f}\\n\\tLogReg = {1:.3f}'\n .format(logLossTrBase, logLossTrLR0))\n```\n\n\n```python\n# TEST Evaluate the model (4e)\nTest.assertTrue(np.allclose(logLossTrLR0, 0.456903), 'incorrect value for logLossTrLR0')\n```\n\n#### ** (4f) Validation log loss **\n#### Next, following the same logic as in Parts (4c) and 4(e), compute the validation log loss for both the baseline and logistic regression models. Notably, the baseline model for the validation data should still be based on the label fraction from the training dataset.\n\n\n```python\n# TODO: Replace with appropriate code\nlogLossValBase = OHEValidationData.map(lambda lp: \n computeLogLoss(classOneFracTrain, lp.label)).sum() / OHEValidationData.count()\n\nlogLossValLR0 = evaluateResults(model0, OHEValidationData)\nprint ('OHE Features Validation Logloss:\\n\\tBaseline = {0:.3f}\\n\\tLogReg = {1:.3f}'\n .format(logLossValBase, logLossValLR0))\n```\n\n\n```python\n# TEST Validation log loss (4f)\nTest.assertTrue(np.allclose(logLossValBase, 0.527603), 'incorrect value for logLossValBase')\nTest.assertTrue(np.allclose(logLossValLR0, 0.456957), 'incorrect value for logLossValLR0')\n```\n\n#### **Visualization 2: ROC curve **\n#### We will now visualize how well the model predicts our target. To do this we generate a plot of the ROC curve. The ROC curve shows us the trade-off between the false positive rate and true positive rate, as we liberalize the threshold required to predict a positive outcome. A random model is represented by the dashed line.\n\n\n```python\nlabelsAndScores = OHEValidationData.map(lambda lp:\n (lp.label, getP(lp.features, model0.weights, model0.intercept)))\nlabelsAndWeights = labelsAndScores.collect()\nlabelsAndWeights.sort(key=lambda (k, v): v, reverse=True)\nlabelsByWeight = np.array([k for (k, v) in labelsAndWeights])\n\nlength = labelsByWeight.size\ntruePositives = labelsByWeight.cumsum()\nnumPositive = truePositives[-1]\nfalsePositives = np.arange(1.0, length + 1, 1.) - truePositives\n\ntruePositiveRate = truePositives / numPositive\nfalsePositiveRate = falsePositives / (length - numPositive)\n\n# Generate layout and plot data\nfig, ax = preparePlot(np.arange(0., 1.1, 0.1), np.arange(0., 1.1, 0.1))\nax.set_xlim(-.05, 1.05), ax.set_ylim(-.05, 1.05)\nax.set_ylabel('True Positive Rate (Sensitivity)')\nax.set_xlabel('False Positive Rate (1 - Specificity)')\nplt.plot(falsePositiveRate, truePositiveRate, color='#8cbfd0', linestyle='-', linewidth=3.)\nplt.plot((0., 1.), (0., 1.), linestyle='--', color='#d6ebf2', linewidth=2.) # Baseline model\npass\n```\n\n### **Part 5: Reduce feature dimension via feature hashing**\n\n#### ** (5a) Hash function **\n#### As we just saw, using a one-hot-encoding featurization can yield a model with good statistical accuracy. However, the number of distinct categories across all features is quite large -- recall that we observed 233K categories in the training data in Part (3c). Moreover, the full Kaggle training dataset includes more than 33M distinct categories, and the Kaggle dataset itself is just a small subset of Criteo's labeled data. Hence, featurizing via a one-hot-encoding representation would lead to a very large feature vector. To reduce the dimensionality of the feature space, we will use feature hashing.\n####Below is the hash function that we will use for this part of the lab. We will first use this hash function with the three sample data points from Part (1a) to gain some intuition. Specifically, run code to hash the three sample points using two different values for `numBuckets` and observe the resulting hashed feature dictionaries.\n\n\n```python\nfrom collections import defaultdict\nimport hashlib\n\ndef hashFunction(numBuckets, rawFeats, printMapping=False):\n \"\"\"Calculate a feature dictionary for an observation's features based on hashing.\n\n Note:\n Use printMapping=True for debug purposes and to better understand how the hashing works.\n\n Args:\n numBuckets (int): Number of buckets to use as features.\n rawFeats (list of (int, str)): A list of features for an observation. Represented as\n (featureID, value) tuples.\n printMapping (bool, optional): If true, the mappings of featureString to index will be\n printed.\n\n Returns:\n dict of int to float: The keys will be integers which represent the buckets that the\n features have been hashed to. The value for a given key will contain the count of the\n (featureID, value) tuples that have hashed to that key.\n \"\"\"\n mapping = {}\n for ind, category in rawFeats:\n featureString = category + str(ind)\n mapping[featureString] = int(int(hashlib.md5(featureString).hexdigest(), 16) % numBuckets)\n if(printMapping): print mapping\n sparseFeatures = defaultdict(float)\n for bucket in mapping.values():\n sparseFeatures[bucket] += 1.0\n return dict(sparseFeatures)\n\n# Reminder of the sample values:\n# sampleOne = [(0, 'mouse'), (1, 'black')]\n# sampleTwo = [(0, 'cat'), (1, 'tabby'), (2, 'mouse')]\n# sampleThree = [(0, 'bear'), (1, 'black'), (2, 'salmon')]\n```\n\n\n```python\n# TODO: Replace with appropriate code\n# Use four buckets\nsampOneFourBuckets = hashFunction(4, sampleOne, True)\nsampTwoFourBuckets = hashFunction(4, sampleTwo, True)\nsampThreeFourBuckets = hashFunction(4, sampleThree, True)\n\n# Use one hundred buckets\nsampOneHundredBuckets = hashFunction(100, sampleOne, True)\nsampTwoHundredBuckets = hashFunction(100, sampleTwo, True)\nsampThreeHundredBuckets = hashFunction(100, sampleThree, True)\n\nprint '\\t\\t 4 Buckets \\t\\t\\t 100 Buckets'\nprint 'SampleOne:\\t {0}\\t\\t {1}'.format(sampOneFourBuckets, sampOneHundredBuckets)\nprint 'SampleTwo:\\t {0}\\t\\t {1}'.format(sampTwoFourBuckets, sampTwoHundredBuckets)\nprint 'SampleThree:\\t {0}\\t {1}'.format(sampThreeFourBuckets, sampThreeHundredBuckets)\n```\n\n\n```python\n# TEST Hash function (5a)\nTest.assertEquals(sampOneFourBuckets, {2: 1.0, 3: 1.0}, 'incorrect value for sampOneFourBuckets')\nTest.assertEquals(sampThreeHundredBuckets, {72: 1.0, 5: 1.0, 14: 1.0},\n 'incorrect value for sampThreeHundredBuckets')\n```\n\n#### ** (5b) Creating hashed features **\n#### Next we will use this hash function to create hashed features for our CTR datasets. First write a function that uses the hash function from Part (5a) with numBuckets = $ \\scriptsize 2^{15} \\approx 33K $ to create a `LabeledPoint` with hashed features stored as a `SparseVector`. Then use this function to create new training, validation and test datasets with hashed features. Hint: `parsedHashPoint` is similar to `parseOHEPoint` from Part (3d).\n\n\n```python\n# TODO: Replace with appropriate code\ndef parseHashPoint(point, numBuckets):\n \"\"\"Create a LabeledPoint for this observation using hashing.\n\n Args:\n point (str): A comma separated string where the first value is the label and the rest are\n features.\n numBuckets: The number of buckets to hash to.\n\n Returns:\n LabeledPoint: A LabeledPoint with a label (0.0 or 1.0) and a SparseVector of hashed\n features.\n \"\"\"\n tokenizedPoint = point.split(',')\n lable = tokenizedPoint[0]\n features = tokenizedPoint[1:]\n rawFeats = [(i, features[i]) for i in range(len(features))]\n hashedFeats = hashFunction(numBuckets, rawFeats)\n return LabeledPoint(label, hashedFeats)\n\nnumBucketsCTR = 2 ** 15\nhashTrainData = rawTrainData.map(lambda x: parseHashPoint(x, numBucketsCTR))\nhashTrainData.cache()\nhashValidationData = rawValidationData.map(lambda x: parseHashPoint(x, numBucketsCTR))\nhashValidationData.cache()\nhashTestData = rawTestData.map(lambda x: parseHashPoint(x, numBucketsCTR))\nhashTestData.cache()\n\nprint hashTrainData.take(1)\n```\n\n\n```python\n# TEST Creating hashed features (5b)\nhashTrainDataFeatureSum = sum(hashTrainData\n .map(lambda lp: len(lp.features.indices))\n .take(20))\nhashTrainDataLabelSum = sum(hashTrainData\n .map(lambda lp: lp.label)\n .take(100))\nhashValidationDataFeatureSum = sum(hashValidationData\n .map(lambda lp: len(lp.features.indices))\n .take(20))\nhashValidationDataLabelSum = sum(hashValidationData\n .map(lambda lp: lp.label)\n .take(100))\nhashTestDataFeatureSum = sum(hashTestData\n .map(lambda lp: len(lp.features.indices))\n .take(20))\nhashTestDataLabelSum = sum(hashTestData\n .map(lambda lp: lp.label)\n .take(100))\n\nTest.assertEquals(hashTrainDataFeatureSum, 772, 'incorrect number of features in hashTrainData')\nTest.assertEquals(hashTrainDataLabelSum, 24.0, 'incorrect labels in hashTrainData')\nTest.assertEquals(hashValidationDataFeatureSum, 776,\n 'incorrect number of features in hashValidationData')\nTest.assertEquals(hashValidationDataLabelSum, 16.0, 'incorrect labels in hashValidationData')\nTest.assertEquals(hashTestDataFeatureSum, 774, 'incorrect number of features in hashTestData')\nTest.assertEquals(hashTestDataLabelSum, 23.0, 'incorrect labels in hashTestData')\n```\n\n#### ** (5c) Sparsity **\n#### Since we have 33K hashed features versus 233K OHE features, we should expect OHE features to be sparser. Verify this hypothesis by computing the average sparsity of the OHE and the hashed training datasets.\n#### Note that if you have a `SparseVector` named `sparse`, calling `len(sparse)` returns the total number of features, not the number features with entries. `SparseVector` objects have the attributes `indices` and `values` that contain information about which features are nonzero. Continuing with our example, these can be accessed using `sparse.indices` and `sparse.values`, respectively.\n\n\n```python\n# TODO: Replace with appropriate code\ndef computeSparsity(data, d, n):\n \"\"\"Calculates the average sparsity for the features in an RDD of LabeledPoints.\n\n Args:\n data (RDD of LabeledPoint): The LabeledPoints to use in the sparsity calculation.\n d (int): The total number of features.\n n (int): The number of observations in the RDD.\n\n Returns:\n float: The average of the ratio of features in a point to total features.\n \"\"\"\n return data.map(lambda x: len(x.features.values)).sum() / float(d * n)\n\naverageSparsityHash = computeSparsity(hashTrainData, numBucketsCTR, nTrain)\naverageSparsityOHE = computeSparsity(OHETrainData, numCtrOHEFeats, nTrain)\n\nprint 'Average OHE Sparsity: {0:.7e}'.format(averageSparsityOHE)\nprint 'Average Hash Sparsity: {0:.7e}'.format(averageSparsityHash)\n```\n\n\n```python\n# TEST Sparsity (5c)\nTest.assertTrue(np.allclose(averageSparsityOHE, 1.6717677e-04),\n 'incorrect value for averageSparsityOHE')\nTest.assertTrue(np.allclose(averageSparsityHash, 1.1805561e-03),\n 'incorrect value for averageSparsityHash')\n```\n\n#### ** (5d) Logistic model with hashed features **\n#### Now let's train a logistic regression model using the hashed features. Run a grid search to find suitable hyperparameters for the hashed features, evaluating via log loss on the validation data. Note: This may take a few minutes to run. Use `1` and `10` for `stepSizes` and `1e-6` and `1e-3` for `regParams`.\n\n\n```python\nnumIters = 500\nregType = 'l2'\nincludeIntercept = True\n\n# Initialize variables using values from initial model training\nbestModel = None\nbestLogLoss = 1e10\n```\n\n\n```python\n# TODO: Replace with appropriate code\nstepSizes = (1, 10)\nregParams = (1e-6, 1e-3)\nfor stepSize in stepSizes:\n for regParam in regParams:\n model = (LogisticRegressionWithSGD\n .train(hashTrainData, numIters, stepSize, regParam=regParam, regType=regType,\n intercept=includeIntercept))\n logLossVa = evaluateResults(model, hashValidationData)\n print ('\\tstepSize = {0:.1f}, regParam = {1:.0e}: logloss = {2:.3f}'\n .format(stepSize, regParam, logLossVa))\n if (logLossVa < bestLogLoss):\n bestModel = model\n bestLogLoss = logLossVa\n\nprint ('Hashed Features Validation Logloss:\\n\\tBaseline = {0:.3f}\\n\\tLogReg = {1:.3f}'\n .format(logLossValBase, bestLogLoss))\n```\n\n\n```python\n# TEST Logistic model with hashed features (5d)\nTest.assertTrue(np.allclose(bestLogLoss, 0.4481683608), 'incorrect value for bestLogLoss')\n```\n\n#### **Visualization 3: Hyperparameter heat map**\n#### We will now perform a visualization of an extensive hyperparameter search. Specifically, we will create a heat map where the brighter colors correspond to lower values of `logLoss`.\n#### The search was run using six step sizes and six values for regularization, which required the training of thirty-six separate models. We have included the results below, but omitted the actual search to save time.\n\n\n```python\nfrom matplotlib.colors import LinearSegmentedColormap\n\n# Saved parameters and results. Eliminate the time required to run 36 models\nstepSizes = [3, 6, 9, 12, 15, 18]\nregParams = [1e-7, 1e-6, 1e-5, 1e-4, 1e-3, 1e-2]\nlogLoss = np.array([[ 0.45808431, 0.45808493, 0.45809113, 0.45815333, 0.45879221, 0.46556321],\n [ 0.45188196, 0.45188306, 0.4518941, 0.4520051, 0.45316284, 0.46396068],\n [ 0.44886478, 0.44886613, 0.44887974, 0.44902096, 0.4505614, 0.46371153],\n [ 0.44706645, 0.4470698, 0.44708102, 0.44724251, 0.44905525, 0.46366507],\n [ 0.44588848, 0.44589365, 0.44590568, 0.44606631, 0.44807106, 0.46365589],\n [ 0.44508948, 0.44509474, 0.44510274, 0.44525007, 0.44738317, 0.46365405]])\n\nnumRows, numCols = len(stepSizes), len(regParams)\nlogLoss = np.array(logLoss)\nlogLoss.shape = (numRows, numCols)\n\nfig, ax = preparePlot(np.arange(0, numCols, 1), np.arange(0, numRows, 1), figsize=(8, 7),\n hideLabels=True, gridWidth=0.)\nax.set_xticklabels(regParams), ax.set_yticklabels(stepSizes)\nax.set_xlabel('Regularization Parameter'), ax.set_ylabel('Step Size')\n\ncolors = LinearSegmentedColormap.from_list('blue', ['#0022ff', '#000055'], gamma=.2)\nimage = plt.imshow(logLoss,interpolation='nearest', aspect='auto',\n cmap = colors)\npass\n```\n\n#### ** (5e) Evaluate on the test set **\n#### Finally, evaluate the best model from Part (5d) on the test set. Compare the resulting log loss with the baseline log loss on the test set, which can be computed in the same way that the validation log loss was computed in Part (4f).\n\n\n```python\n# TODO: Replace with appropriate code\n# Log loss for the best model from (5d)\nlogLossTest = evaluateResults(bestModel, hashTestData)\n\n# Log loss for the baseline model\nlogLossTestBaseline = hashTestData.map(lambda lp: \n computeLogLoss(classOneFracTrain, lp.label)).sum() / hashTestData.count()\n\nprint ('Hashed Features Test Log Loss:\\n\\tBaseline = {0:.3f}\\n\\tLogReg = {1:.3f}'\n .format(logLossTestBaseline, logLossTest))\n```\n\n\n```python\n# TEST Evaluate on the test set (5e)\nTest.assertTrue(np.allclose(logLossTestBaseline, 0.537438),\n 'incorrect value for logLossTestBaseline')\nTest.assertTrue(np.allclose(logLossTest, 0.455616931), 'incorrect value for logLossTest')\n```\n", "meta": {"hexsha": "e93e328b6c961429e29cea04f76228ba456c8a2c", "size": 110288, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Scalable-Machine-Learning/ML_lab4_ctr_student.ipynb", "max_stars_repo_name": "csyezheng/Course-Notes", "max_stars_repo_head_hexsha": "ec54fa1fd7b3ec41a0d581205d2c2a771bd7e3c2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Scalable-Machine-Learning/ML_lab4_ctr_student.ipynb", "max_issues_repo_name": "csyezheng/Course-Notes", "max_issues_repo_head_hexsha": "ec54fa1fd7b3ec41a0d581205d2c2a771bd7e3c2", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Scalable-Machine-Learning/ML_lab4_ctr_student.ipynb", "max_forks_repo_name": "csyezheng/Course-Notes", "max_forks_repo_head_hexsha": "ec54fa1fd7b3ec41a0d581205d2c2a771bd7e3c2", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 55.0614078882, "max_line_length": 30222, "alphanum_fraction": 0.6896761207, "converted": true, "num_tokens": 15473, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.6926419831347361, "lm_q1q2_score": 0.4311414655843741}} {"text": "_Lambda School Data Science \u2014\u00a0Linear Models_\n\n# Understanding Linear Regression\n\n#### Objectives\n- understand how ordinary least squares regression minimizes the sum of squared errors\n- understand how linear algebra can solve ordinary least squares regression\n- get and interpret coefficients of a linear model\n- visualize a line of best fit in 2D, and hyperplane in 3D\n- use regression metrics: MSE, RMSE, MAE, R^2\n\n#### Extra Links\n- [Statistics 101: Simple Linear Regression](https://www.youtube.com/watch?v=ZkjP5RJLQF4) (20 minute video)\n- [_An Introduction to Statistical Learning_](http://www-bcf.usc.edu/~gareth/ISL/ISLR%20Seventh%20Printing.pdf), Chapter 3.1, Simple Linear Regression, & 3.2, Multiple Linear Regression\n- Priceonomics, [The Discovery of Statistical Regression](https://priceonomics.com/the-discovery-of-statistical-regression/)\n- Priceonomics, [Why the Father of Modern Statistics Didn\u2019t Believe Smoking Caused Cancer](https://priceonomics.com/why-the-father-of-modern-statistics-didnt-believe/)\n- Harvard Business Review, [When to Act on a Correlation, and When Not To](https://hbr.org/2014/03/when-to-act-on-a-correlation-and-when-not-to)\n- [xkcd 552: Correlation](https://www.explainxkcd.com/wiki/index.php/552:_Correlation)\n- [xkcd 1725: Linear Regression](https://www.explainxkcd.com/wiki/index.php/1725:_Linear_Regression)\n\n## What is Linear Regression?\n\n[Linear Regression](https://en.wikipedia.org/wiki/Linear_regression) is a statistical model that seeks to describe the relationship between some y variable and one or more x variables. \n\nIn the simplest case, linear regression seeks to fit a straight line through a cloud of points. This line is referred to as the \"regression line\" or \"line of best fit.\" This line tries to summarize the relationship between our X and Y in a way that enables us to use the equation for that line to make predictions.\n\n\n```python\n%matplotlib inline\nimport seaborn as sns\nx = [10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5]\ny = [ 8, 6, 7, 8, 8, 9, 7, 4, 10, 4, 5]\nsns.regplot(x, y);\n```\n\n\n\n### Synonyms for \"y variable\" \n- Dependent Variable\n- Response Variable\n- Outcome Variable \n- Predicted Variable\n- Measured Variable\n- Explained Variable\n- Label\n- Target\n\n### Synonyms for \"x variable\"\n- Independent Variable\n- Explanatory Variable\n- Regressor\n- Covariate\n- Feature\n\n## Simple Linear Regresion\n\n#### Making Predictions\n\nSay that we were trying to create a model that captured the relationship between temperature outside and ice cream sales. In Machine Learning our goal is often different that of other flavors of Linear Regression Analysis, because we're trying to fit a model to this data with the intention of making **predictions** on new data (in the future) that we don't have yet.\n\n#### What are we trying to predict?\n\nSo if we had measured ice cream sales and the temprature outside on 11 different days, at the end of our modeling **what would be the thing that we would want to predict? - Ice Cream Sales or Temperature?**\n\nWe would probably want to be measuring temperature with the intention of using that to **forecast** ice cream sales. If we were able to successfully forecast ice cream sales from temperature, this might help us know beforehand how much ice cream to make or how many cones to buy or on which days to open our store, etc. Being able to make predictions accurately has a lot of business implications. This is why making accurate predictions is so valuable (And in large part is why data scientists are paid so well).\n\n#### y variable intuition\n\nWe want the thing that we're trying to predict to serve as our **y** variable. This is why it's sometimes called the \"predicted variable.\" We call it the \"dependent\" variable because our prediction for how much ice cream we're going to sell \"depends\" on the temperature outside. \n\n#### x variable intuition\n\nAll other variables that we use to predict our y variable (we're going to start off just using one) we call our **x** variables. These are called our \"independent\" variables because they don't *depend* on y, they \"explain\" y. Hence they are also referred to as our \"explanatory\" variables.\n\n## Example: Predict presidential election voting\n\n#### Douglas Hibbs, [Background Information on the \u2018Bread and Peace\u2019 Model of Voting in Postwar US Presidential Elections](https://douglas-hibbs.com/background-information-on-bread-and-peace-voting-in-us-presidential-elections/)\n\n> Aggregate two-party vote shares going to candidates of the party holding the presidency during the postwar era are well explained by just two fundamental determinants:\n\n> (1) Positively by weighted-average growth of per capita real disposable personal income over the term. \n> (2) Negatively by cumulative US military fatalities (scaled to population) owing to unprovoked, hostile deployments of American armed forces in foreign wars.\n\n#### Data sources\n- 1952-2012: Douglas Hibbs, [2014 lecture at Deakin University Melbourne](http://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf), Slide 40\n- 2016, Vote Share: [The American Presidency Project](https://www.presidency.ucsb.edu/statistics/elections)\n- 2016, Recent Growth in Personal Incomes: [The 2016 election economy: the \"Bread and Peace\" model final forecast](https://angrybearblog.com/2016/11/the-2016-election-economy-the-bread-and-peace-model-final-forecast.html)\n- 2016, US Military Fatalities: Assumption that Afghanistan War fatalities in 2012-16 occured at the same rate as 2008-12\n\n\n```python\nimport pandas as pd\n\ncolumns = ['Year','Incumbent Party Candidate','Other Candidate','Incumbent Party Vote Share']\n\ndata = [[1952,\"Stevenson\",\"Eisenhower\",44.6],\n [1956,\"Eisenhower\",\"Stevenson\",57.76],\n [1960,\"Nixon\",\"Kennedy\",49.91],\n [1964,\"Johnson\",\"Goldwater\",61.34],\n [1968,\"Humphrey\",\"Nixon\",49.60],\n [1972,\"Nixon\",\"McGovern\",61.79],\n [1976,\"Ford\",\"Carter\",48.95],\n [1980,\"Carter\",\"Reagan\",44.70],\n [1984,\"Reagan\",\"Mondale\",59.17],\n [1988,\"Bush, Sr.\",\"Dukakis\",53.94],\n [1992,\"Bush, Sr.\",\"Clinton\",46.55],\n [1996,\"Clinton\",\"Dole\",54.74],\n [2000,\"Gore\",\"Bush, Jr.\",50.27],\n [2004,\"Bush, Jr.\",\"Kerry\",51.24],\n [2008,\"McCain\",\"Obama\",46.32],\n [2012,\"Obama\",\"Romney\",52.00], \n [2016,\"Clinton\",\"Trump\",48.2]]\n \nvotes = pd.DataFrame(data=data, columns=columns)\n```\n\n\n```python\nvotes.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        YearIncumbent Party CandidateOther CandidateIncumbent Party Vote Share
                                        01952StevensonEisenhower44.60
                                        11956EisenhowerStevenson57.76
                                        21960NixonKennedy49.91
                                        31964JohnsonGoldwater61.34
                                        41968HumphreyNixon49.60
                                        \n
                                        \n\n\n\n\n```python\ncolumns = ['Year','income_growth']\n\ndata = [[1952,2.40],\n [1956,2.89],\n [1960, .85],\n [1964,4.21],\n [1968,3.02],\n [1972,3.62],\n [1976,1.08],\n [1980,-.39],\n [1984,3.86],\n [1988,2.27],\n [1992, .38],\n [1996,1.04],\n [2000,2.36],\n [2004,1.72],\n [2008, .10],\n [2012, .95], \n [2016, .10]]\n \ngrowth = pd.DataFrame(data=data, columns=columns)\n```\n\n\n```python\ngrowth.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Yearincome_growth
                                        019522.40
                                        119562.89
                                        219600.85
                                        319644.21
                                        419683.02
                                        \n
                                        \n\n\n\n\n```python\n\"\"\"\nFatalities denotes the cumulative number of American military\nfatalities per millions of US population the in Korea, Vietnam,\nIraq and Afghanistan wars during the presidential terms\npreceding the 1952, 1964, 1968, 1976 and 2004, 2008 and\n2012 elections.\n\nhttp://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf\n\"\"\"\n\ncolumns = ['Year','US MIL Deaths/million']\n\ndata = [[1952,190],\n [1956, 0],\n [1960, 0],\n [1964, 1],\n [1968,146],\n [1972, 0],\n [1976, 2],\n [1980, 0],\n [1984, 0],\n [1988, 0],\n [1992, 0],\n [1996, 0],\n [2000, 0],\n [2004, 4],\n [2008, 14],\n [2012, 5], \n [2016, 5]]\n \ndeaths = pd.DataFrame(data=data, columns=columns)\n```\n\n\n```python\ndeaths.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        YearUS MIL Deaths/million
                                        01952190
                                        119560
                                        219600
                                        319641
                                        41968146
                                        \n
                                        \n\n\n\n\n```python\ndf = votes.merge(growth).merge(deaths)\n```\n\n\n```python\ndf.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        YearIncumbent Party CandidateOther CandidateIncumbent Party Vote Shareincome_growthUS MIL Deaths/million
                                        01952StevensonEisenhower44.602.40190
                                        11956EisenhowerStevenson57.762.890
                                        21960NixonKennedy49.910.850
                                        31964JohnsonGoldwater61.344.211
                                        41968HumphreyNixon49.603.02146
                                        \n
                                        \n\n\n\n### Plot univariate correlations\n[Seaborn tutorial: Visualizing linear relationships](https://seaborn.pydata.org/tutorial/regression.html)\n\n\n```python\n%matplotlib inline\nimport seaborn as sns\n#take a logartihm of deaths in millions may be the better choice than standard scaling here\ntarget = 'Incumbent Party Vote Share'\nfeatures = ['income_growth', \n 'US MIL Deaths/million']\n\nfor feature in features:\n sns.lmplot(x=feature, y=target, data=df)\n```\n\nWe can see from the scatterplot that these data points seem to follow a somewhat linear relationship for the \"Average Recent Growth in Personal Incomes\" feature. This means that we could probably summarize their relationship well by fitting a line of best fit to these points. Lets do it.\n\n\n## The Equation for a Line\n\nA common equation for a line is:\n\n\\begin{align}\ny = mx + b\n\\end{align}\n\nWhere $m$ is the slope of our line and $b$ is the y-intercept. \n\nIf we want to plot a line through our cloud of points we figure out what these two values should be. Linear Regression seeks to **estimate** the slope and intercept values that describe a line that best fits the data points.\n\n## The Anatomy of Linear Regression\n\n- Intercept: The $b$ value in our line equation $y=mx+b$\n- Slope: The $m$ value in our line equation $y=mx+b$. These two values together define our regression line.\n- $\\hat{y}$ : A prediction\n- Line of Best Fit (Regression Line)\n- Predicted (fitted) Values: Points on our regression line\n- Observed Values: Points from our dataset\n- Error: The distance between predicted and observed values.\n\n\nOrdinary Least Squares Regression is a way to solve for $m$ and $b$.\n\nLet's start by seeing what would happen if we just guessed and checked some values for $m$ and $b$. \n\nWhat's the line of \"best\" fit look like? What's the error?\n\n\n\n```python\n# TODO\n\nx = df['income_growth']\ny = df['Incumbent Party Vote Share']\n\nm = 0\nb = y.mean()\npred = m*x+b\n```\n\n\n```python\nprint(pred)\n```\n\n 0 51.828235\n 1 51.828235\n 2 51.828235\n 3 51.828235\n 4 51.828235\n 5 51.828235\n 6 51.828235\n 7 51.828235\n 8 51.828235\n 9 51.828235\n 10 51.828235\n 11 51.828235\n 12 51.828235\n 13 51.828235\n 14 51.828235\n 15 51.828235\n 16 51.828235\n Name: income_growth, dtype: float64\n\n\n\n```python\nfrom sklearn.metrics import mean_absolute_error as mae\nfrom sklearn.metrics import r2_score\nimport matplotlib.pyplot as plt\n```\n\n\n```python\ndef plot_preds (x,y,y_pred):\n plt.scatter(x,y, label=\"Y_True\")\n plt.plot(x,y_pred, label=\"Y_PRed\")\n plt.legend\n mean_ae = mae(y,y_pred)\n r2 = r2_score(y,y_pred)\n print(f'Mean Absolute Error {mean_ae}')\n print(f'R2 Score {r2}')\n```\n\n\n```python\nplot_preds(x,y,pred)\n```\n\n\n```python\nm = 3.8\nb = 45\ny_pred = m*x+b\nplot_preds(x,y,y_pred)\nplt.title(\"Guessing Regression Values\");\n```\n\n## R Squared: $R^2$\n\nOne final attribute of linear regressions that we're going to talk about today is a measure of goodness of fit known as $R^2$ or R-squared. $R^2$ is a statistical measure of how close the data are fitted to our regression line. A helpful interpretation for the $R^2$ is the percentage of the dependent variable that is explained by the model.\n\nIn other words, the $R^2$ is the percentage of y that is explained by the x variables included in the model. For this reason the $R^2$ is also known as the \"coefficient of determination,\" because it explains how much of y is explained (or determined) by our x varaibles. We won't go into the calculation of $R^2$ today, just know that a higher $R^2$ percentage is nearly always better and indicates a model that fits the data more closely. \n\n## Residual Error \n\nThe residual error is the distance between points in our dataset and our regression line.\n\n\n```python\ndef regression_residuals(df, feature, target, m, b):\n x = df[feature]\n y = df[target]\n y_pred = m*x + b\n \n plt.scatter(x, y, label='y_true')\n plt.plot(x, y_pred, label='y_pred')\n plt.legend()\n \n # Plot residual errors\n for x, y1, y2 in zip(x, y, y_pred):\n plt.plot((x, x), (y1, y2), color='grey')\n \n mean_ae = mae(y, y_pred) \n r2 = r2_score(y, y_pred)\n print('Mean Absolute Error:', mean_ae)\n print('R^2:', r2)\n \nregression_residuals(df, feature='income_growth', \n target='Incumbent Party Vote Share', m=3, b=46)\n```\n\n\n```python\nfrom ipywidgets import interact, fixed\n\ninteract(regression_residuals, \n df=fixed(df), \n feature=fixed('income_growth'), \n target=fixed('Incumbent Party Vote Share'), \n m=(-10,10,0.01), \n b=(40,60,0.5));\n```\n\n\n interactive(children=(FloatSlider(value=0.0, description='m', max=10.0, min=-10.0, step=0.01), FloatSlider(val\u2026\n\n\n## More Formal Notation\n\n\n\nWe have talked about a line of regression being represented like a regular line $y=mx+b$ but as we get to more complicated versions we're going to need to extend this equation. So lets establish the proper terminology.\n\n**X** - Independent Variable, predictor variable, explanatory variable, regressor, covariate\n\n**Y** - Response variable, predicted variable, measured vairable, explained variable, outcome variable\n\n$\\beta_0$ - \"Beta Naught\" or \"Beta Zero\", the intercept value. This is how much of y would exist if X were zero. This is sometimes represented by the letter \"a\" but I hate that. So it's \"Beta 0\" during my lecture.\n\n$\\beta_1$ - \"Beta One\" The primary coefficient of interest. This values is the slope of the line that is estimated by \"minimizing the sum of the squared errors/residuals\" - We'll get to that. \n\n$\\epsilon$ - \"Epsilon\" The \"error term\", random noise, things outside of our model that affect y.\n\n## Minimizing the Sum of the Squared Error\n\nThe most common method of estimating our $\\beta$ parameters is what's known as \"Ordinary Least Squares\" (OLS). (There are different methods of arriving at a line of best fit). OLS estimates the parameters that minimize the squared distance between each point in our dataset and our line of best fit. \n\n\\begin{align}\nSSE = \\sum(y_i - \\hat{y})^2\n\\end{align}\n\n\n\n```python\nfrom matplotlib.patches import Rectangle\nimport numpy as np\nfrom sklearn.metrics import mean_squared_error\n\ndef regression_squared_errors(df, feature, target, m, b):\n x = df[feature]\n y = df[target]\n y_pred = m*x + b\n \n fig = plt.figure(figsize=(7,7))\n ax = plt.axes()\n ax.scatter(x, y, label='y_true')\n ax.plot(x, y_pred, label='y_pred')\n ax.legend()\n \n # Plot square errors\n xmin, xmax = ax.get_xlim()\n ymin, ymax = ax.get_ylim()\n scale = (xmax-xmin)/(ymax-ymin)\n for x, y1, y2 in zip(x, y, y_pred):\n bottom_left = (x, min(y1, y2))\n height = abs(y1 - y2)\n width = height * scale\n ax.add_patch(Rectangle(xy=bottom_left, width=width, height=height, alpha=0.1))\n \n mse = mean_squared_error(y, y_pred)\n rmse = np.sqrt(mse)\n mean_ae = mae(y, y_pred)\n r2 = r2_score(y, y_pred)\n print('Mean Squared Error:', mse)\n print('Root Mean Squared Error:', rmse)\n print('Mean Absolute Error:', mean_ae)\n print('R^2:', r2)\n \nregression_squared_errors(df, feature='income_growth', \n target='Incumbent Party Vote Share', m=3, b=46)\n```\n\n\n```python\ninteract(regression_squared_errors, \n df=fixed(df), \n feature=fixed('income_growth'), \n target=fixed('Incumbent Party Vote Share'), \n m=(-10,10,0.5), \n b=(40,60,0.5));\n```\n\n\n interactive(children=(FloatSlider(value=0.0, description='m', max=10.0, min=-10.0, step=0.5), FloatSlider(valu\u2026\n\n\n## Hypotheses\n\n\n```python\nb = 46\nms = np.arange(-10,10,0.5)\nsses = []\nfeature = 'Average Recent Growth in Personal Incomes'\n\nfor m in ms:\n predictions = m * df[feature] + b\n errors = predictions - df[target]\n square_errors = errors ** 2\n sse = square_errors.sum()\n sses.append(sse)\n \nhypotheses = pd.DataFrame({'Slope': ms})\nhypotheses['Intercept'] = b\nhypotheses['Sum of Square Errors'] = sses\n\nhypotheses\n```\n\n\n```python\nhypotheses.plot(x='Slope', y='Sum of Square Errors', \n title=f'Intercept={b}');\n```\n\n## Scikit-learn\n\n#### Jake VanderPlas, [Python Data Science Handbook, Chapter 5.2, Introducing Scikit-Learn](https://jakevdp.github.io/PythonDataScienceHandbook/05.02-introducing-scikit-learn.html), Scikit-Learn's Estimator API\n\n> Most commonly, the steps in using the Scikit-Learn estimator API are as follows (we will step through a handful of detailed examples in the sections that follow).\n\n> 1. Choose a class of model by importing the appropriate estimator class from Scikit-Learn. \n> 2. Choose model hyperparameters by instantiating this class with desired values. \n> 3. Arrange data into a features matrix and target vector following the discussion above.\n> 4. Fit the model to your data by calling the `fit()` method of the model instance.\n> 5. Apply the Model to new data: For supervised learning, often we predict labels for unknown data using the `predict()` method.\n\n\n```python\n# TODO\n```\n\n\n```python\nfrom sklearn.linear_model import LinearRegression\n\nlinr = LinearRegression()\nfeatures = ['income_growth']\ntarget = 'Incumbent Party Vote Share'\n\nX = df[features]\ny = df[target]\n\nlinr.fit(X,y)\n```\n\n\n\n\n LinearRegression(copy_X=True, fit_intercept=True, n_jobs=None,\n normalize=False)\n\n\n\n\n```python\nlinr.intercept_\n```\n\n\n\n\n 46.499209757741625\n\n\n\n\n```python\nlinr.coef_\n```\n\n\n\n\n array([2.97417709])\n\n\n\n\n```python\n\n```\n\n## Linear Algebra!\n\nThe same result that is found by minimizing the sum of the squared errors can be also found through a linear algebra process known as the \"Least Squares Solution:\"\n\n\\begin{align}\n\\hat{\\beta} = (X^{T}X)^{-1}X^{T}y\n\\end{align}\n\nBefore we can work with this equation in its linear algebra form we have to understand how to set up the matrices that are involved in this equation. \n\n### The $\\beta$ vector\n\nThe $\\beta$ vector represents all the parameters that we are trying to estimate, our $y$ vector and $X$ matrix values are full of data from our dataset. The $\\beta$ vector holds the variables that we are solving for: $\\beta_0$ and $\\beta_1$\n\nNow that we have all of the necessary parts we can set them up in the following equation:\n\n\\begin{align}\ny = X \\beta + \\epsilon\n\\end{align}\n\nSince our $\\epsilon$ value represents **random** error we can assume that it will equal zero on average.\n\n\\begin{align}\ny = X \\beta\n\\end{align}\n\nThe objective now is to isolate the $\\beta$ matrix. We can do this by pre-multiplying both sides by \"X transpose\" $X^{T}$.\n\n\\begin{align}\nX^{T}y = X^{T}X \\beta\n\\end{align}\n\nSince anything times its transpose will result in a square matrix, if that matrix is then an invertible matrix, then we should be able to multiply both sides by its inverse to remove it from the right hand side. (We'll talk tomorrow about situations that could lead to $X^{T}X$ not being invertible.)\n\n\\begin{align}\n(X^{T}X)^{-1}X^{T}y = (X^{T}X)^{-1}X^{T}X \\beta\n\\end{align}\n\nSince any matrix multiplied by its inverse results in the identity matrix, and anything multiplied by the identity matrix is itself, we are left with only $\\beta$ on the right hand side:\n\n\\begin{align}\n(X^{T}X)^{-1}X^{T}y = \\hat{\\beta}\n\\end{align}\n\nWe will now call it \"beta hat\" $\\hat{\\beta}$ because it now represents our estimated values for $\\beta_0$ and $\\beta_1$\n\n### Lets calculate our $\\beta$ coefficients with numpy!\n\n\n```python\nfrom statsmodels.api import add_constant\n\nX = add_constant(df[feature].values)\nprint('X')\nprint(X)\n\ny = df[target].values[:, np.newaxis]\nprint('y')\nprint(y)\n\nX_transpose = X.T\nprint('X Transpose')\nprint(X_transpose)\n\nX_transpose_X = X_transpose @ X\nprint('X Transpose X')\nprint(X_transpose_X)\n\nX_transpose_X_inverse = np.linalg.inv(X_transpose_X)\nprint('X Transpose X Inverse')\nprint(X_transpose_X_inverse)\n\nX_transpose_y = X_transpose @ y\nprint('X Transpose y')\nprint(X_transpose_y)\n\nbeta_hat = X_transpose_X_inverse @ X_transpose_y\nprint('Beta Hat')\nprint(beta_hat)\n```\n\n# Multiple Regression\n\nSimple or bivariate linear regression involves a single $x$ variable and a single $y$ variable. However, we can have many $x$ variables. A linear regression model that involves multiple x variables is known as **Multiple** Regression (NOT MULTIVARIATE).\n\n\\begin{align}\ny = \\beta_0 + \\beta_1 X_1 + \\beta_2 X_2 + ... + \\beta_n X_n + \\epsilon\n\\end{align}\n\n\n```python\n# TODO\n\nlinr = LinearRegression()\nfeatures = ['income_growth', \n 'US MIL Deaths/million']\n\ntarget = 'Incumbent Party Vote Share'\n\nX = df[features]\ny = df[target]\n\nlinr.fit(X,y)\n```\n\n\n\n\n LinearRegression(copy_X=True, fit_intercept=True, n_jobs=None,\n normalize=False)\n\n\n\n\n```python\nlinr.coef_\n```\n\n\n\n\n array([ 3.40621407, -0.05375223])\n\n\n\n## Visualize hyperplane of best fit in 3D\n\n\n```python\n# https://stackoverflow.com/a/47230966\n# Plotly notebook mode with google colaboratory\n# You need to define this function\n# And call it in each offline plotting cell\n\ndef configure_plotly_browser_state():\n import IPython\n display(IPython.core.display.HTML('''\n \n \n '''))\n```\n\n\n```python\nimport itertools\nimport plotly.graph_objs as go\nfrom plotly.offline import init_notebook_mode, iplot\ninit_notebook_mode(connected=True)\n\ndef viz3D(fitted_model, X, features, target='', num=100):\n \"\"\"\n Visualize model predictions in 3D, for regression model fit on 2 features\n \n Parameters\n ----------\n fitted_model : scikit-learn model, already fitted\n X : pandas dataframe, which was used to fit model\n features : list of strings, column names of the 2 features\n target : string, name of target\n num : int, number of grid points for each feature\n \n References\n ----------\n https://plot.ly/python/3d-charts/\n \"\"\"\n feature1, feature2 = features\n min1, max1 = X[feature1].min(), X[feature1].max()\n min2, max2 = X[feature2].min(), X[feature2].max()\n x1 = np.linspace(min1, max1, num)\n x2 = np.linspace(min2, max2, num)\n combos = list(itertools.product(x1, x2))\n Z = fitted_model.predict(combos).reshape(num, num)\n \n configure_plotly_browser_state()\n data = [go.Surface(x=x1, y=x2, z=Z)]\n layout = go.Layout(\n scene={'xaxis': {'title': feature1, 'range': [min1,max1], 'showticklabels': True}, \n 'yaxis': {'title': feature2, 'range': [min2,max2], 'showticklabels': True}, \n 'zaxis': {'title': target, 'showticklabels': True}}, \n )\n fig = go.Figure(data=data, layout=layout)\n iplot(fig)\n```\n\n\n```python\n# TODO\n```\n\n## Dimensionality in Linear Regression\n\nMuliple Regression is simply an extension of the bivariate case. The reason why we see the bivariate case demonstrated so often is simply because it's easier to graph and all of the intuition from the bivariate case is the same as we keep on adding explanatory variables.\n\nAs we increase the number of $x$ values in our model we are simply fitting a n-1-dimensional plane to an n-dimensional cloud of points within an n-dimensional hypercube. \n\n## Interpreting Coefficients\n\nOne of Linear Regression's strengths is that the parameters of the model (coefficients) are readily interpretable and useful. Not only do they describe the relationship between x and y but they put a number on just how much x is associated with y. We should be careful to not speak about this relationshiop in terms of causality because these coefficients are in fact correlative measures. We would need a host of additional techniques in order to estimate a causal effect using linear regression (econometrics).\n\n\\begin{align}\n\\hat{\\beta} = \\frac{Cov(x,y)}{Var(y)}\n\\end{align}\n\n## Why is Linear Regression so Important?\n\n### Popularity \n\nLinear Regression is an extremely popular technique that every data scientist **needs** to understand. It's not the most advanced technique and there are supervised learning techniques that will obtain a higher accuracy, but where it lacks in accuracy it makes up for it in interpretability and simplicity.\n\n### Interpretability\n\nFew other models possess coefficients that are so directly linked to their variables with a such a clear interpretation. Tomorrow we're going to learn about ways to make them even easier to interpret.\n\n### Simplicity\n\nA linear regression model can be communicated just by writing out its equation. It's kind of incredible that such high dimensional relationships can be described from just a linear combination of variables and coefficients. \n\n# Assignment\n- Continue to predict New York City apartment rents. This is your last assignment with this dataset.\n- You may select any number of features. You are encouraged to engineer new features.\n- Get and plot your model's coefficients.\n- Report your Root Mean Squared Error, Mean Absolute Error, and R^2 Score, for your Train and Test sets. Share your scores with your cohort on Slack!\n- Fit a model with 2 features, and visualize the plane of best fit in 3D.\n- Commit your notebook to your fork of the repo.\n\n## Stretch Goals\n\nStudy more about Linear Regression. Here are two helpful links. If you find more links, share your favorites with your cohort on Slack.\n\n1. Watch this 20 minute video that just hit 1 million views: Brandon Foltz, Statistics 101: Simple Linear Regression (https://www.youtube.com/watch?v=ZkjP5RJLQF4)\n2. Skim _An Introduction to Statistical Learning_, Chapter 3.1, Simple Linear Regression, & Chapter 3.2, Multiple Linear Regression (http://www-bcf.usc.edu/~gareth/ISL/ISLR%20Seventh%20Printing.pdf)\n\nIn your 3D visualization, can you include the actual datapoints, like in [this notebook](https://nbviewer.jupyter.org/urls/s3.amazonaws.com/datarobotblog/notebooks/multiple_regression_in_python.ipynb)? Can you also include the residual lines from the datapoints to the plane of the best fit, like in _An Introduction to Statistical Learning?_ This would be hard to do, but awesome!\n\n\nCan you get creative with feature engineering? Share with your cohort on Slack. We mentioned some feature ideas at the end of last lesson, but didn't demonstrate how to engineer them. So here are some example solutions:\n\n```python\n# Does apartment have a non-empty description?\ndf['description'] = df['description'].str.strip().fillna('')\ndf['has_description'] = df['description'] != ''\n\n# How long is the description?\ndf['description_length'] = df['description'].str.len()\n\n# How many total perks does each apartment have?\nperk_cols = ['elevator', 'cats_allowed', 'hardwood_floors', 'dogs_allowed',\n 'doorman', 'dishwasher', 'no_fee', 'laundry_in_building',\n 'fitness_center', 'pre-war', 'laundry_in_unit', 'roof_deck',\n 'outdoor_space', 'dining_room', 'high_speed_internet', 'balcony',\n 'swimming_pool', 'new_construction', 'exclusive', 'terrace', \n 'loft', 'garden_patio', 'common_outdoor_space', \n 'wheelchair_access']\ndf['perk_count'] = df[perk_cols].sum(axis=1)\n\n# Are pets allowed?\ndf['pets_allowed'] = (df['cats_allowed']==1) | (df['dogs_allowed']==1)\n```\n\n", "meta": {"hexsha": "b14f0d81beef4233ee6da0856d97fc029a8fb1dc", "size": 161466, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "module3-understanding-linear-regression/understanding_linear_regression.ipynb", "max_stars_repo_name": "GwenStacey/DS-Unit-2-Linear-Models", "max_stars_repo_head_hexsha": "e3e2bf3369e5926e72cff187dc3d23ec9ff8d440", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "module3-understanding-linear-regression/understanding_linear_regression.ipynb", "max_issues_repo_name": "GwenStacey/DS-Unit-2-Linear-Models", "max_issues_repo_head_hexsha": "e3e2bf3369e5926e72cff187dc3d23ec9ff8d440", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "module3-understanding-linear-regression/understanding_linear_regression.ipynb", "max_forks_repo_name": "GwenStacey/DS-Unit-2-Linear-Models", "max_forks_repo_head_hexsha": "e3e2bf3369e5926e72cff187dc3d23ec9ff8d440", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 92.3718535469, "max_line_length": 20556, "alphanum_fraction": 0.8091610618, "converted": true, "num_tokens": 8742, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593171945416, "lm_q2_score": 0.6926419894793248, "lm_q1q2_score": 0.4311414598315694}} {"text": "# Multiple Regression Analysis\n## Motivation for Multiple Regression\n### The Model with Two Independent Variables\n\nwe skip some similar definition and approaches$\\DeclareMathOperator*{\\argmin}{argmin}\n\\newcommand{\\using}[1]{\\stackrel{\\mathrm{#1}}{=}}\n\\newcommand{\\ffrac}{\\displaystyle \\frac}\n\\newcommand{\\space}{\\text{ }}\n\\newcommand{\\bspace}{\\;\\;\\;\\;}\n\\newcommand{\\QQQ}{\\boxed{?\\:}}\n\\newcommand{\\CB}[1]{\\left\\{ #1 \\right\\}}\n\\newcommand{\\SB}[1]{\\left[ #1 \\right]}\n\\newcommand{\\P}[1]{\\left( #1 \\right)}\n\\newcommand{\\dd}{\\mathrm{d}}\n\\newcommand{\\Tran}[1]{{#1}^{\\mathrm{T}}}\n\\newcommand{\\d}[1]{\\displaystyle{#1}}\n\\newcommand{\\EE}[2][\\,\\!]{\\mathbb{E}_{#1}\\left[#2\\right]}\n\\newcommand{\\Var}[2][\\,\\!]{\\mathrm{Var}_{#1}\\left[#2\\right]}\n\\newcommand{\\Cov}[2][\\,\\!]{\\mathrm{Cov}_{#1}\\left(#2\\right)}\n\\newcommand{\\Corr}[2][\\,\\!]{\\mathrm{Corr}_{#1}\\left(#2\\right)}\n\\newcommand{\\I}[1]{\\mathrm{I}\\left( #1 \\right)}\n\\newcommand{\\N}[1]{\\mathrm{N} \\left( #1 \\right)}\n\\newcommand{\\ow}{\\text{otherwise}}$. One thing to mention is that the key assumption about how $u$ related to $x_1$ and $x_2$ is $\\EE{u \\mid x_1 , x_2} = 0$.\n\n### The Model with $k$ Independent Variables\n\nNot too many things different from before, only one thing: $\\EE{u \\mid x_1 , x_2, \\dots, x_k}= 0$.\n\n\n\n## Mechanics and Interpretation of Ordinary Least Squares\n### Obtaining the OLS estimates\n\nOLS ***first order conditions***:\n\n$$\\left\\{\\begin{align}\n\\sum_{i=1}^{n} \\P{y_i - \\hat\\beta_0 - \\hat\\beta_1x_{i1} - \\cdots - \\hat\\beta_k x_{ik}} &= 0\\\\\n\\sum_{i=1}^{n} x_{i1}\\P{y_i - \\hat\\beta_0 - \\hat\\beta_1x_{i1} - \\cdots - \\hat\\beta_k x_{ik}} &= 0\\\\\n\\sum_{i=1}^{n} x_{i2}\\P{y_i - \\hat\\beta_0 - \\hat\\beta_1x_{i1} - \\cdots - \\hat\\beta_k x_{ik}} &= 0\\\\\n&\\vdots \\\\\n\\sum_{i=1}^{n} x_{ik}\\P{y_i - \\hat\\beta_0 - \\hat\\beta_1x_{i1} - \\cdots - \\hat\\beta_k x_{ik}} &= 0\\\\\n\\end{align}\\right.$$\n\nAlso these can be obtained by **moment methods**\u77e9\u65b9\u6cd5 cause these are equivalent to $\\EE{u} = 0$ and $\\EE{x_j u} = 0$ where $j = 1, 2, \\dots, k$.\n\n$Remark$\n\nFurther assumption about whether there's an unique solution about this equation set will be shown later\n\n### Interpreting the OLS Regression Equation\n\nOLS ***regression line***: $\\hat y = \\hat \\beta_0 + \\hat\\beta_1 x_1 + \\hat\\beta_2 x_2 + \\cdots \\hat\\beta_k x_k$\n\n### On the Meaning of \"Holding Other Factors Fixed\" in Multiple regression\n\nThe power of multiple regression analysis is that it allows us to do in nonexperimental environments what natural scientists are able to do in a controlled laboratory setting: keep other factors fixed.\n\n### Changing More than One Independent Variable Simultaneously\n### OLS Fitted Values and Residuals\n\n***Fitted value***: $\\hat y_i = \\hat \\beta_0 + \\hat\\beta_1 x_{i1} + \\cdots + \\hat\\beta_k x_{ik}$\n\n***Residual***: $\\hat u_i = y_i - \\hat y_i$\n\nHere're some properties inherited from the **SLR** method.\n\n- The sample average of the residuals is zero: $\\sum\\limits_{i=1}^{n} \\hat u_i = 0$ ($\\EE{\\hat u} = 0$) and so $\\bar y = \\bar{\\hat y}$\n- The sample covariance between each independent variable and the OLS residuals is zero: $\\sum\\limits_{i=1}^{n} x_{ij} \\hat u_i = 0$ and so: the sample covariance between the OLS fitted values and the OLS residuals is zero: $\\sum\\limits_{i=1}^{n}\\hat y_i \\hat u_i = 0$\n- $\\bar y = \\hat \\beta_0 + \\hat\\beta_1 \\bar x_1 + \\cdots + \\hat \\beta_k \\bar x_k$\n\n### A \"Partialling Out\" Interpretation of Multiple Regression\n\nWe now focus on the \"partialling Out\" Interpretation.\n\nConsider the case with $k=2$ independent variables, and the regression result is $\n\\hat y = \\hat\\beta_0 + \\hat\\beta_1 x_1 + \\hat\\beta_2 x_2$. Here's another expression for $\\hat\\beta_1$.\n\n$$\\hat\\beta_1 = \\ffrac{\\sum\\limits_{i=1}^{n} \\hat r_{i1}y_i} {\\sum\\limits_{i=1}^{n} \\hat r_{i1}^2}$$\n\nHere $\\hat r_{i1}$ is the residual from a simple regression of $x_1$ on $x_2$ (which has NOTHING to do with $y$). After that we do another regression of $y$ on $\\hat r_i$, so that to obtian $\\hat\\beta_1$.\n\nThis can be interpreted as we first partial out the correlated part of $x_1$ and $x_2$ where only $\\hat r_i$ left. So that $\\hat \\beta_1$ now can measure the sample relation ship between $y$ and $x_1$ after $x_2$ has been partialled out.\n\n$$\\hat\\beta_1 = \\ffrac{\\sum \\P{\\hat r_{i1} - \\bar{\\hat r}_{i1}}\\P{y_i - \\bar y}} {\\sum \\P{\\hat r_{i1} - \\bar{\\hat r}_{i1}}^2 }$$\n\nthen by the fact that $\\sum \\hat r_{i1} = 0$ (it's a residual anyway), we can see its final appearance. Also in the general model with $k$ explanatory variables, $\\hat\\beta_1$ can be written as the same.\n\n$Proof$\n\n>To derive the equation, we first follow the strategy and write: $x_{i1} = \\hat x_{i1} + \\hat r_{i1}$ from the regression of $x_1$ on $x_2, x_3, \\dots,x_k$ for all $i = 1,2,\\dots,n$. Then we plug it back into **OLS first order conditions** and obtain:\n>\n>$$\\sum\\nolimits_{i=1}^{n} \\P{\\hat x_{i1} + \\hat r_{i1}} \\P{y_i - \\hat\\beta_0 - \\hat\\beta_1 x_{i1} - \\cdots - \\hat\\beta_k x_{ik}} = 0$$\n>\n>At this time, $\\hat x_{i1}$ is the linear combination of the other explainatory variables $x_{12}, x_{i3}, \\dots, x_{ik}$ and thus $\\sum \\hat x_{i1}\\hat u_i = 0$. Therefore,\n\n>$$\\sum_{i=1}^{n} \\hat r_{i1} \\P{y_i - \\hat\\beta_0 - \\hat\\beta_1 x_{i1} - \\cdots - \\hat\\beta_k x_{ik}} = 0$$\n>\n>Then, on account of the fact that $\\hat r_{i1}$ are the residuals from regressing $x_1$ on $x_2, x_3, \\dots, x_k$ we have $\\sum_{i=1}^{n} x_{ij} \\hat r_{i1} = 0$ for all $j = 2,3,\\dots,k$. Therefore, the preceding equation is simplified to:\n>\n>$$\\sum_{i=1}^{n} \\hat r_{i1} \\P{y_i - \\hat\\beta_1 x_{i1}} = 0 \\\\\n\\Longrightarrow\n\\hat\\beta_1 = \\ffrac{\\d{\\sum_{i=1}^{n} \\hat r_{i1}y_i}} {\\d{\\sum_{i=1}^{n} \\hat r_{i1} x_{i1}}}$$\n>\n>Finally, use the fact that $\\sum_{i=1}^{n} \\hat x_{i1} \\hat r_{i1} = 0$, we have \n>\n>$$\\hat\\beta_1 = \\ffrac{\\d{\\sum_{i=1}^{n} \\hat r_{i1}y_i}} {\\d{\\sum_{i=1}^{n} \\hat r_{i1} \\P{x_{i1} - \\hat x_{i1} }}} = \\ffrac{\\d{\\sum_{i=1}^{n} \\hat r_{i1}y_i}} {\\d{\\sum_{i=1}^{n} \\hat r_{i1}^2}}$$\n\n### Comparison of Simple and Multiple Regression Estimates\n\nIf the model has two variables but we itentionally omit one, say $x_2$, then denote the simple regression result as $\\tilde y = \\tilde \\beta_0 + \\tilde \\beta_1 x_1$ while the one with full variable has the form: $\\hat y = \\hat \\beta_0 + \\hat\\beta_1 x_1 + \\hat\\beta_2 x_2$. And the relation between $\\tilde \\beta_0$ and $\\tilde \\beta_1$ can be expressed as: $\\boxed{\\tilde\\beta_1 = \\hat\\beta_1 + \\hat\\beta_2 \\cdot \\tilde\\delta_1}$, where $\\tilde\\delta_1$ is the slope coefficient from the simple regression of $x_{i2}$ on $x_{i1}$, $i = 1,2,\\dots, n$.\n\nInterpretation: $\\tilde\\beta_1$ is somewhat the sum of the partial effects of $x_1$ on $\\hat y$ and the partial effects of $x_2$ on $\\hat y$ times the slope in the sample regression of $x_2$ on $x_1$. ***3A.4***\n\nAnd only in two cases will they equal: \n\n1. $\\hat\\beta_2 = 0$, that the partial effect of $x_2$ on $\\hat y$ is $0$ in the sample$\\\\[0.5em]$\n2. $\\tilde\\delta_1 = 0$, that $x_1$ and $x_2$ are uncorrelated in the sample\n\nAnd the generalized one:\n\n1. the OLS coefficients on $x_2$ through $x_k$ are all $0$\n2. $x_1$ is uncorrelated with *each* of $x_2, x_3,\\dots,x_k$\n\n### Goodness-of-Fit\n\nThen we define the $\\text{SST} = \\sum \\P{y_i - \\bar y}^2$, $\\text{SSE} = \\sum \\P{\\hat y_i - \\bar y}^2$, and $\\text{SSR} = \\sum \\hat u_i^2$ and $R^2 = \\ffrac{\\text{SSE}} {\\text{SST}}$. And using the same argument in the SLR, we have $\\text{SSR} = \\text{SST} - \\text{SSE}$.\n\nAnd there's also an alternative expression of $R^2$, seemingly using an asymptotic way:\n\n$$\\begin{align}\nR^2 &= \\rho^2\\P{y_i, \\hat y_i} \\\\\n&= \\ffrac{\\Cov{y_i, \\hat y_i}^2} {\\Var{y_i}\\Var{\\hat y_i}} \\\\\n&= \\ffrac{\\EE{\\P{y_i - \\EE{y_i}}\\P{\\hat y_i - \\EE{\\hat y_i}}}^2} {\\EE{y_i - \\EE{y_i}}^2\\EE{\\hat y_i - \\EE{\\hat y_i}}^2} \\\\\n&\\approx \\ffrac{\\P{\\sum\\limits_{i=1}^{n} \\P{y_i - \\bar y}\\P{\\hat y - \\bar{\\hat y}}}^2} {\\P{\\sum\\limits_{i=1}^{n} \\P{y_i - \\bar{y}}^2}\\P{\\sum\\limits_{i=1}^{n} \\P{\\hat y - \\bar{\\hat y}}^2}}\n\\end{align}$$\n\n### Regression through the Origin\n\n## The Expected Value of the OLS Estimators\n\n$Assumption$ $\\text{MLR}.1$ to $\\text{MLR}.4$\n\n- Linear in parameters: In the population, the relationship between $y$ and the explanatory variables is linear: $y = \\beta_0 + \\beta_1 x_1 + \\cdots + \\beta_k x_k + u$. This model is called the ***population model*** or ***true model***\n- Random Sampling: The data is a random sample drawn from the population: $\\CB{\\P{x_{i1},x_{i2},\\dots,x_{ik}}:i=1,\\dots,n}$\n - and we write: $y_i = \\beta_0 + \\beta_1 x_{i1} + \\beta_2 x_{i2} + \\cdots + \\beta_kx_{ik} + u_i\\\\[0.5em]$\n- No perfect collinearity: In the sample (and therefore in the population), none of the independent variables is constant and there are no *exact linear* relationships among the independent variables\n - Later we will see that its variance will soar up if almost linear\n - And if yes, we say ***perfect collinearity*** occurs and can't be estimated using OLS.\n- Zero conditional mean: The value of the explanatory variables must contain no information about the mean of the unobserved factors: $\\EE{u \\mid x_{1} , x_{2}, \\dots, x_{k}} = 0$\n\n$Theorem.1$\n\nBy assumptions $\\text{MLR}.1$ to $\\text{MLR}.4$, we claim that $\\EE{\\hat\\beta_j} = \\beta_j$\n\n$Proof$\n\n> Using matrices is a better way, but here we just focus on one slope parameter.\n>\n>First under $\\text{MLR}.3$ we have $\\hat\\beta_1 = \\ffrac{\\sum\\limits_{i=1}^{n} \\hat r_{i1} y_i} {\\sum\\limits_{i = 1}^{n} \\hat{r}^2_{i1}}$.\n>\n>Then under $\\text{MLR}.1$, we have $y_i = \\beta_0 + \\beta_1 x_{i1} + \\beta_2 x_{i2} + \\cdots + \\beta_k x_{ik} + u_i$; we can substitute this $y_i$ back and obtain $\\hat\\beta_1 = \\ffrac{\\sum\\limits_{i=1}^{n} \\hat r_{i1} \\P{\\beta_0 + \\beta_1 x_{i1} + \\beta_2 x_{i2} + \\cdots + \\beta_k x_{ik} + u_i}} {\\sum\\limits_{i = 1}^{n} \\hat{r}^2_{i1}}$.\n>\n>We now deal with the terms separately.\n>\n>$\\sum \\hat r_{i1} = 0$, since it's a residual; $\\sum x_{ij}\\hat r_{i1} = 0$, since that's the Covariance of residual and explanatory variable; they hold true for all $j = 2,3,\\dots,k$. And $\\sum x_{i1}\\hat r_{i1} = \\sum \\hat r_{i1}^2$ since $x_{i1} = \\text{linear function}\\P{x_{i2}, x_{i3}, \\dots, x_{ik}} + \\hat r_{i1}$.\n\n>Finally, $\\hat \\beta_1 = \\beta_1 + \\ffrac{\\sum\\limits_{i=1}^{n} \\hat r_{i1} u_i} {\\sum\\limits_{i = 1}^{n} \\hat{r}^2_{i1}}$\n>\n>Next step, under assumption $\\text{MLR}.2$ and $\\text{MLR}.4$ we consider the expecation of $\\hat\\beta_1$ conditioned on $\\mathbf{X} = \\P{X_1, X_2, \\dots, X_k}$:\n>\n>$$\\begin{align}\n\\EE{\\hat\\beta_1 \\mid \\mathbf{X}} &= \\beta_1 + \\ffrac{\\sum\\limits_{i=1}^{n} \\hat r_{i1} \\EE{u_i\\mid \\mathbf{X}}} {\\sum\\limits_{i = 1}^{n} \\hat{r}^2_{i1}} \\\\\n&= \\beta_1 + \\ffrac{\\sum\\limits_{i=1}^{n} \\hat r_{i1} \\cdot 0} {\\sum\\limits_{i = 1}^{n} \\hat{r}^2_{i1}} \\\\\n&= \\beta_1 = \\EE{\\hat\\beta_1}\n\\end{align}$$\n\n### Including Irrelevant Variables in a Regression Model\n\nMore variables are included in the model while they have no partial effects on $y$ in the population. A simple example, say the model is $y = \\beta_0 + \\beta_1 x_1 + \\beta_2 x_2 + \\beta_3 x_3 + u$ and $x_3$ is useless here. Then in terms of conditional expectations, $\\EE{y \\mid x_1,x_2,x_3} = \\EE{y \\mid x_1,x_2} = \\beta_0 + \\beta_1 x_1 + \\beta_2 x_2$\n\nThen, $\\EE{\\hat\\beta_3} = 0$. Though $\\hat\\beta_3$ might not be exactly $0$, it's expectation will. And our conclusion is including one or more *irrelevant variables* in a multiple regression model, or overspecifying the model, does not affect the unbiasedness of the OLS estimators. However, variances will suffer the harm.\n\n### Omitted Variable Bias: The Simple Case\n\nSee this from a simple case. We suppose the true model is $y = \\beta_0 + \\beta_1 x_1 + \\beta_2 x_2 + u$ while we assume it to be in another form $y = \\alpha_0 + \\alpha_1 x_1 + w$. What's the bias from this?\n\nFirst we assume that $x_2 = \\delta_0 + \\delta_1 x_1 + v$, thus the model changes to $y = \\P{\\beta_0 + \\beta_2 \\delta_0} + \\P{\\beta_1 + \\beta_2 \\delta_1}x_1 + \\P{\\beta_2 v + u}$. Here the estimated intercept is $\\P{\\beta_0 + \\beta_2 \\delta_0}$, altered, from $\\beta_0$; the estimated slope on $x_1$ will be $\\P{\\beta_1 + \\beta_2 \\delta_1}$, also altered, from $\\beta_1$; and the error term, which was changed to $\\P{\\beta_2 v + u}$ from a simple $u$. **All estimated coefficients will be biased now**.\n\nIf we do the sample regression of $y$ only on $x_1$, we will have $\\tilde y = \\tilde\\beta_0 + \\tilde\\beta_1 x_1$. An interesting algebraic relationship is $\\tilde\\beta_1 = \\hat\\beta_1 + \\hat\\beta_2 \\tilde\\delta_1$. Thus, $\\EE{\\tilde\\beta_1} = \\beta_1 + \\beta_2 \\tilde\\delta_1$ and $\\text{Bias}\\P{\\tilde\\beta_1} = \\EE{\\tilde\\beta_1} - \\beta_1 = \\beta_2 \\tilde\\delta_1$, called the ***omitted variable***.\n\n1. $\\beta_2 = 0$: when it just really not a variable in the **true model**.\n2. $\\tilde\\delta_1 = 0$: since $\\tilde\\delta_1$ is the sample covariance between $x_1$ and $x_2$ over the sample variance of $x_1$, it's $0$ $iff$ $x_1$ and $x_2$ are *uncorrelated* in the sample.\n\n### Omitted Variable Bias: More general Cases\n\nSuppose the population model: $y = \\beta_0 + \\beta_1 x_1 + \\beta_2 x_2 + \\beta_3 x_3 + u$ satisfies the Assumptions $\\text{MLR}.1$ to $\\text{MLR}.4$. However if we omit the variable $x_3$, the estimated model is $\\tilde y = \\tilde\\beta_0 + \\tilde\\beta_1 x_1 + \\tilde\\beta_2 x_2$.\n\nTo see how $\\EE{\\tilde\\beta_1}$ and $\\EE{\\tilde\\beta_2}$ are biased, we write them out. To obtain a value for this, we first need to assume that $x_1$ and $x_2$ are uncorrelated, then\n\n$$\\begin{align}\n\\EE{\\tilde\\beta_1} &= \\EE{\\hat\\beta_1 + \\hat\\beta_3 \\tilde\\delta_1} & \\EE{\\tilde\\beta_2} &= \\EE{\\hat\\beta_2 + \\hat\\beta_3 \\tilde\\delta_2}\\\\\n&= \\beta_1 + \\beta_3 \\cdot \\ffrac{\\d{\\sum_{i=1}^{n} \\P{x_{i1} - \\bar{x}_1}x_{i3}}} {\\d{\\sum_{i=1}^{n} \\P{x_{i1} - \\bar{x}_1}^2}} &&= \\beta_2 + \\beta_3 \\cdot \\ffrac{\\d{\\sum_{i=1}^{n} \\P{x_{i2} - \\bar{x}_2}x_{i3}}} {\\d{\\sum_{i=1}^{n} \\P{x_{i2} - \\bar{x}_2}^2}}\n\\end{align}$$\n\n## The Variance of the OLS Estimators\n\n$Assumption$ $\\text{MLR}.5$\n- Homoscedasticity: The value of the explanatory variables must contain no information about the variance of the unobserved factors: $\\Var{u \\mid x_{1},x_{2},\\dots, x_{k}} = \\sigma^2$\n\n$Remark$\n\n$\\text{MLR}.1$ to $\\text{MLR}.5$ are collectively known as the ***Gauss-Markov assumptions*** (for cross-sectional regression).\n\n$Theorem.2$\n\nBy assumptions $\\text{MLR}.1$ to $\\text{MLR}.5$, we claim that $\\Var{\\hat\\beta_j} = \\ffrac{\\sigma^2} {\\text{SST}_j \\P{1-R_j^2}}$ for $j = 1,2,\\dots,k$. Here $\\text{SST}_j$ is the **Total sample variation** in explanatory variable $x_j$: $\\sum_{i=1}^{n}\\P{x_{ij} - \\bar x_j}^2$ and $R_j^2 = \\rho^2\\P{x_j,\\hat x_j} = \\ffrac{\\P{\\sum\\limits_{i=1}^{n} \\P{x_{ij} - \\bar x_j}\\P{\\hat x_{ij} - \\bar{\\hat x_j}}}^2} {\\P{\\sum\\limits_{i=1}^{n} \\P{x_{ij} - \\bar{x_j}}^2}\\P{\\sum\\limits_{i=1}^{n} \\P{\\hat x_{ij} - \\bar{\\hat x_j}}^2}}$. This $R_j^2$ is from regressing $x_j$ on all other independent variables (and including an intercept).\n\n### The Components of The OLS Variances: Multicollinearity\n\n- The Error Variance: $\\sigma^2$.\n - Bigger error variance, bigger sampling variance, less imprecise the estimation\n- The total Sample Variation in $x_j$: $\\text{SST}$ \n - More sample, higher $\\text{SST}$, more accurate\n - No sample variance is so rare and not allowed by $\\text{MLR}.4$\n - ***micronumerosity***: Small sample size can lead to large sampling variances $\\text{SST}_j$\n- the Linear relationships among the Independent Variables: $R_j^2$\n - If two are correlated, then $R_j \\to 1$ which greatly magnify the variance\n - ***multicollinearity***: high (but not perfect) correlation between two or more independent variables\n\nHere we call $1/\\P{1-R_j^2}$ the ***Variance Inflation Factor***. And the conlusion is: dropping some variables will reduce the multicollinearity while lead to omitted variable bias.\n\n### Variances in Misspecified Models\n\n- True Model: $y = \\beta_0 + \\beta_1 x_1 + \\beta_2 x_2 + u$\n- Estimated Model: $\\hat y = \\hat \\beta_0 + \\hat \\beta_1 x_1 + \\hat \\beta_2 x_2$\n- Estimated Model with $\\beta_2$ omitted: $\\tilde y = \\tilde \\beta_0 + \\tilde \\beta_1 x_1$\n\nTheir variances are: $\\Var{\\hat \\beta_1} = \\ffrac{\\sigma^2} {\\text{SST}_1 \\P{1-R_1^2}}$ and $\\Var{\\tilde \\beta_1} = \\ffrac{\\sigma^2} {\\text{SST}_1}$. And we can divine this into two cases:\n\n1. for $\\beta_2 = 0$, $\\EE{\\hat \\beta_1} = \\beta_1$ and $\\EE{\\tilde\\beta_1} = \\beta_1$, besides, $\\Var{\\tilde\\beta_1} < \\Var{\\hat\\beta_1}$\n2. for $\\beta_2 \\neq 0$, $\\EE{\\hat \\beta_1} = \\beta_1$ and $\\EE{\\tilde\\beta_1} \\neq \\beta_1$, but still $\\Var{\\tilde\\beta_1} < \\Var{\\hat\\beta_1}$\n\n### Estimating $\\sigma^2$: Standard Errors of the OLS Estimators\n\nAnalogy to the simple regresion, $\\hat\\sigma^2 = \\ffrac{1} {n-k-1} \\sum \\hat u_i^2 = \\ffrac{\\text{SSR}} {n-k-1}$, unbiased. Here's the theorem\n\n$Remark$\n\n$n-k-1$ is the ***degrees of freedom***: $\\text{number of observations} - \\text{number of estimated parameters}$\n\n$Theorem.3$\n\nUnder $\\text{MLR}.1$ through $\\text{MLR}.5$, $\\EE{\\hat\\sigma^2} = \\sigma^2$\n\nHere, $\\hat\\sigma^2$ is called the ***standard error of the regression (SER)***.\n\nThen we can use this to estimate the sampling variation. The ***standard deviation*** of $\\hat\\beta_j$: \n\n$$\\text{sd}\\P{\\hat\\beta_j} = \\sqrt{\\Var{\\hat\\beta_j}} = \\sqrt{\\ffrac{1} {\\text{SST}_j \\P{1-R_j^2}}}\\sigma$$\n\nand then the estimated one, ***standard error*** of $\\hat\\beta_j$, by replacing the $\\sigma$ in the last expression with $\\hat\\sigma$:\n\n$$\\text{se}\\P{\\hat\\beta_j} = \\sqrt{\\widehat{\\Var{\\hat\\beta_j}}} = \\sqrt{\\ffrac{1} {\\text{SST}_j \\P{1-R_j^2}}}\\hat\\sigma$$\n\n## Efficiency of OLS: The Gauss-Markov Theorem\n\n1. Under Assumption $\\text{MLR}.1$ to $\\text{MLR}.4$, OLS is unbiased\n2. And then $\\text{MLR}.5$, it becomes the one with the smallest variance\n\nThus we call this estimation the ***best linear unbiased estimators (BLUE)*** of the regression coefficients.\n\n$Remark$\n\nLinear here means that the estimator can be expressed as a weighted sum of dependent variables:\n\n$$\\tilde\\beta_j = \\sum_{i=1}^{n} w_{ij}y_i$$\n\nAnd best means the least variance among all others.\n\n$Theorem.4$ GAUSS-MARKOV THEOREM\n\nUnder Assumption $\\text{MLR}.1$ through $\\text{MLR}.5$, $\\hat\\beta_1, \\hat\\beta_2,\\dots,\\hat\\beta_k$, the OLS estimators, are the ***best linear unbiased estimators (BLUEs)*** of $\\beta_1, \\beta_2,\\dots,\\beta_k$, respectively.\n\n***\n", "meta": {"hexsha": "53c46f4d4110bf6489f3733edcc3e8479ce092e1", "size": 23559, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "FinMath/Econometrics/Chap_03.ipynb", "max_stars_repo_name": "XavierOwen/Notes", "max_stars_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2018-11-27T10:31:08.000Z", "max_stars_repo_stars_event_max_datetime": "2019-01-20T03:11:58.000Z", "max_issues_repo_path": "FinMath/Econometrics/Chap_03.ipynb", "max_issues_repo_name": "XavierOwen/Notes", "max_issues_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "FinMath/Econometrics/Chap_03.ipynb", "max_forks_repo_name": "XavierOwen/Notes", "max_forks_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-07-14T19:57:23.000Z", "max_forks_repo_forks_event_max_datetime": "2020-07-14T19:57:23.000Z", "avg_line_length": 59.045112782, "max_line_length": 670, "alphanum_fraction": 0.5663652956, "converted": true, "num_tokens": 6821, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984137988772, "lm_q2_score": 0.805632181981183, "lm_q1q2_score": 0.43109250268345944}} {"text": "```python\nimport torch\nimport torch.nn.functional as F\n\nimport torchsde\nimport math\nimport matplotlib.pyplot as plt\n\nimport numpy as np\n\nfrom tqdm.notebook import tqdm\n# from torch import datasets\n\nfrom torch import _vmap_internals\nfrom torchvision import datasets, transforms\n# import torch.nn.functional as F\n\nimport pandas as pd\n```\n\n\n```python\nfrom cfollmer.objectives import log_g, relative_entropy_control_cost, stl_relative_entropy_control_cost_xu\nfrom cfollmer.sampler_utils import FollmerSDE\nfrom cfollmer.drifts import *\nfrom cfollmer.trainers import basic_batched_trainer\n```\n\n# The Model\n\n\\begin{align}\n\\theta &\\sim \\mathcal{N}(\\theta | 0, \\sigma_w^2 \\mathbb{I}) \\\\\ny_i | x_i, \\theta &\\sim \\mathrm{Bernouli}\\left[\\mathrm{NN}_{\\theta}\\left(x_i \\right)\\right]\n\\end{align}\n\nWe want samples from $p(\\theta | \\{(y_i, x_i)\\})$. Note $f(x; \\theta)$ is a neural net with params $\\theta$\n\n## Loading the iris dataset\n\n\n```python\nimages_train = datasets.MNIST(\"../data/mnist/\", download=True, train=True)\nimages_test = datasets.MNIST(\"../data/mnist/\", download=True, train=False)\n\ntransform = torch.nn.Sequential(transforms.Normalize((0.1307,), (0.3081)))\n```\n\n\n```python\nX_train, y_train = images_train.data, images_train.targets\nX_test, y_test = images_test.data, images_test.targets\n\nX_train = torch.flatten(transform(X_train.float()), 1)\nX_test = torch.flatten(transform(X_test.float()), 1)\n\ny_train = F.one_hot(y_train)\ny_test = F.one_hot(y_test)\n\n# X_train = np.concatenate((X_train, np.ones((X_train.shape[0],X_train.shape[1]))), axis=1)\n# X_test = np.concatenate((X_test, np.ones((X_test.shape[0],X_train.shape[1]))), axis=1)\n```\n\n\n```python\ndevice = \"cuda\" if torch.cuda.is_available() else \"cpu\"\n\nX_train, X_test, y_train, y_test = \\\n torch.tensor(X_train, dtype=torch.float32, device=device), \\\n torch.tensor(X_test, dtype=torch.float32, device=device), \\\n torch.tensor(y_train, dtype=torch.float32, device=device), \\\n torch.tensor(y_test, dtype=torch.float32, device=device) \n```\n\n :4: UserWarning: To copy construct from a tensor, it is recommended to use sourceTensor.clone().detach() or sourceTensor.clone().detach().requires_grad_(True), rather than torch.tensor(sourceTensor).\n torch.tensor(X_train, dtype=torch.float32, device=device), \\\n :5: UserWarning: To copy construct from a tensor, it is recommended to use sourceTensor.clone().detach() or sourceTensor.clone().detach().requires_grad_(True), rather than torch.tensor(sourceTensor).\n torch.tensor(X_test, dtype=torch.float32, device=device), \\\n :6: UserWarning: To copy construct from a tensor, it is recommended to use sourceTensor.clone().detach() or sourceTensor.clone().detach().requires_grad_(True), rather than torch.tensor(sourceTensor).\n torch.tensor(y_train, dtype=torch.float32, device=device), \\\n :7: UserWarning: To copy construct from a tensor, it is recommended to use sourceTensor.clone().detach() or sourceTensor.clone().detach().requires_grad_(True), rather than torch.tensor(sourceTensor).\n torch.tensor(y_test, dtype=torch.float32, device=device)\n\n\n\n```python\nX_train.shape\n```\n\n\n\n\n torch.Size([60000, 784])\n\n\n\n$$\\DeclareMathOperator*{\\argmin}{arg\\,min}$$\n$$\\def\\E{{\\mathbb{E}}}$$\n$$\\def\\rvu{{\\mathbf{u}}}$$\n$$\\def\\rvTheta{{\\bm{\\Theta}}}$$\n$$\\def\\gU{{\\mathcal{U}}}$$\n$$\\def\\mX{{\\mathbf{X}}}$$\n\n## Controlled Schrodinger Follmer Sampler\n\nThe objevtive we are trying to implement is:\n\n\\begin{align}\n \\mathbf{u}_t^{*}= \\argmin_{\\rvu_t \\in \\mathcal{U}}\\mathbb{E}\\left[\\frac{1}{2\\gamma}\\int_0^1||\\rvu(t, \\Theta_t)||^2 dt - \\ln\\left(\\frac{ p(\\mX | \\Theta_1)p(\\Theta_1)}{\\mathcal{N}(\\Theta_1|\\mathbf{0}, \\gamma \\mathbb{I} )}\\right)\\right] \\\n\\end{align}\n\nWhere:\n\\begin{align}\nd\\Theta_t = \\rvu(t, \\Theta_t)dt + \\sqrt{\\gamma} dB_t\n\\end{align}\n\nTo do so we use the EM discretisation.\n\n\n```python\nimport torch.nn.functional as F\n\n\nclass ClassificationNetwork(object):\n \n def __init__(\n self, input_dim=1, output_dim=1, depth=None,\n width=20, width_seq=None, device=\"cpu\", activation=F.relu\n ):\n \n self.device = device\n self.output_dim = output_dim\n self.input_dim = input_dim \n self.activation = activation\n \n self.depth = depth\n if not self.depth:\n self.depth = 1\n if not width_seq:\n self.width = width\n self.width_seq = [self.width] * (self.depth + 1)\n self.shapes = [(self.width_seq[i-1], self.width_seq[i]) for i in range(1,self.depth)]\n self.shapes += [(self.width_seq[-1], self.output_dim)]\n self.shapes = [(self.input_dim, self.width_seq[0])] + self.shapes\n \n self.dim = sum([wx * wy + wy for wx, wy in self.shapes])\n \n def forward(self, x, \u0398):\n index = 0\n n, d = x.shape\n \n# dim_bl = sum([wx * wy + wy for wx, wy in self.shapes[:-1]])\n# \u0398[:dim_bl] = (\u0398[:dim_bl] - \u0398[:dim_bl].mean()) / \u0398[:dim_bl].std()\n# \u03c3_\u0398, \u03bc_\u0398 = \u0398.std(), \u0398.mean()\n# \u0398 = (\u0398 - \u03bc_\u0398) / \u03c3_\u0398\n\n for wx, wy in self.shapes[:-1]:\n x = F.linear(\n x,\n \u0398[index: index + wx * wy].reshape(wy, wx),\n \u0398[index + wx * wy: index + wx * wy + wy].reshape(1,wy)\n )\n x = self.activation(x)\n index += wx * wy + wy\n wx, wy = self.shapes[-1]\n x = F.linear(\n x,\n \u0398[index: index + wx * wy].reshape(wy, wx), #* \u03c3_\u0398 + \u03bc_\u0398,\n \u0398[index + wx * wy: index + wx * wy + wy].reshape(1,wy) # * \u03c3_\u0398 + \u03bc_\u0398\n )\n return x.to(self.device)\n \n def map_forward(self, x, \u0398):\n preds_func = lambda \u03b8: self.forward(x, \u03b8)\n batched_preds = torch._vmap_internals.vmap(preds_func)\n preds = torch.hstack(list(map(preds_func, \u0398)))\n return preds\n```\n\n\n```python\ndim = X_train.shape[1]\nout_dim = y_train.shape[1]\n\nnet = ClassificationNetwork(\n dim, out_dim, device=device, depth=1, width=50, activation=F.tanh\n)\n\n\ndef gaussian_prior(\u0398, \u03c3_w=3.8):\n \"\"\"\n Logistic regresion bayesian prior\n \"\"\"\n return -0.5 * (\u0398**2).sum(axis=1) / \u03c3_w\n\n\ndef log_likelihood_vmap_nn(\u0398, X, y, net=net):\n \"\"\"\n Hoping this implementation is less buggy / faster\n \n still feels a bit slow.\n \"\"\"\n \n def loss(\u03b8):\n preds = net.forward(X, \u03b8)\n cel = torch.nn.CrossEntropyLoss(reduction=\"sum\")\n# import pdb; pdb.set_trace()\n ll_cel = -1.0 * cel(preds, y.argmax(dim=1))\n return ll_cel\n \n batched_loss = torch._vmap_internals.vmap(loss)\n\n return batched_loss(\u0398)\n```\n\n\n```python\nnet.dim\n```\n\n\n\n\n 39760\n\n\n\n\n```python\nclass SimpleForwardNetBN_larger(AbstractDrift):\n\n def __init__(self, input_dim=1, width=300, activation=torch.nn.Softplus):\n super(SimpleForwardNetBN_larger, self).__init__()\n \n self.nn = torch.nn.Sequential(\n torch.nn.Linear(input_dim + 1, width), torch.nn.BatchNorm1d(width, affine=False), activation(),\n torch.nn.Linear(width, width), torch.nn.BatchNorm1d(width, affine=False), activation(),\n torch.nn.Linear(width, width), torch.nn.BatchNorm1d(width, affine=False), activation(),\n torch.nn.Linear(width, width), torch.nn.BatchNorm1d(width, affine=False), activation(),\n torch.nn.Linear(width, input_dim )\n )\n \n self.nn[-1].weight.data.fill_(0.0)\n\n\n\u03b3 = 0.1**2\n\u0394t=0.01\n\ndim= net.dim\n\nprior = gaussian_prior\n\nsde, losses = basic_batched_trainer(\n \u03b3, \u0394t, prior, log_likelihood_vmap_nn, dim, X_train, y_train,\n method=\"euler\", stl=\"stl_xu\", adjoint=False, optimizer=None,\n num_steps=79, batch_size_data=int(X_train.shape[0] // 5), batch_size_\u0398=30,\n batchnorm=True, device=device, lr=0.0001, drift=SimpleForwardNetBN_larger, schedule=\"uniform\",\n \u03b3_min= 0.1**2, \u03b3_max= 0.4**2\n)\n```\n\n\n\n /local/scratch/home/fav25/ControlledFollmerDrift/cfollmer/objectives.py:143: UserWarning: torch.vmap is an experimental prototype that is subject to change and/or deletion. Please use at your own risk. There may be unexpected performance cliffs due to certain operators not being implemented. To see detailed performance warnings please use `torch._C._debug_only_display_vmap_fallback_warnings(True) before the call to `vmap`.\n f = _vmap_internals.vmap(f_)\n /local/scratch/home/fav25/ControlledFollmerDrift/cfollmer/objectives.py:144: UserWarning: torch.vmap is an experimental prototype that is subject to change and/or deletion. Please use at your own risk. There may be unexpected performance cliffs due to certain operators not being implemented. To see detailed performance warnings please use `torch._C._debug_only_display_vmap_fallback_warnings(True) before the call to `vmap`.\n f_detached = _vmap_internals.vmap(sde.f_detached)\n /local/scratch/home/fav25/ControlledFollmerDrift/cfollmer/objectives.py:152: UserWarning: torch.vmap is an experimental prototype that is subject to change and/or deletion. Please use at your own risk. There may be unexpected performance cliffs due to certain operators not being implemented. To see detailed performance warnings please use `torch._C._debug_only_display_vmap_fallback_warnings(True) before the call to `vmap`.\n g = _vmap_internals.vmap(sde.g)\n :30: UserWarning: torch.vmap is an experimental prototype that is subject to change and/or deletion. Please use at your own risk. There may be unexpected performance cliffs due to certain operators not being implemented. To see detailed performance warnings please use `torch._C._debug_only_display_vmap_fallback_warnings(True) before the call to `vmap`.\n batched_loss = torch._vmap_internals.vmap(loss)\n /home/fav25/.local/lib/python3.8/site-packages/torch/nn/functional.py:1795: UserWarning: nn.functional.tanh is deprecated. Use torch.tanh instead.\n warnings.warn(\"nn.functional.tanh is deprecated. Use torch.tanh instead.\")\n\n\n 2.3770670890808105\n 2.1863701343536377\n 2.005516529083252\n 1.6569232940673828\n 1.3457823991775513\n 1.146380066871643\n 0.9544356465339661\n 0.828987717628479\n 0.7359426021575928\n 0.6586539149284363\n 0.5723750591278076\n 0.5308206081390381\n 0.4964335560798645\n 0.4613948464393616\n 0.41332441568374634\n 0.39699238538742065\n 0.3962780833244324\n 0.3923579752445221\n 0.35848891735076904\n 0.3654201626777649\n 0.32917773723602295\n 0.3504791557788849\n 0.33872053027153015\n 0.3236846923828125\n 0.3243337571620941\n 0.3130742311477661\n 0.3227761685848236\n 0.3165150284767151\n 0.31513333320617676\n 0.31661343574523926\n 0.29742196202278137\n 0.2974570095539093\n 0.30057328939437866\n 0.30576732754707336\n 0.3074263334274292\n 0.28385505080223083\n 0.29548871517181396\n 0.3094801902770996\n 0.2861082851886749\n 0.28496497869491577\n 0.2873198091983795\n 0.2715984582901001\n 0.2797081768512726\n 0.29713472723960876\n 0.2939736843109131\n 0.29375120997428894\n 0.28445959091186523\n 0.2784840762615204\n 0.28768420219421387\n 0.27820685505867004\n 0.27226054668426514\n 0.28426581621170044\n 0.28468215465545654\n 0.2703218460083008\n 0.27081453800201416\n 0.2609834372997284\n 0.2778591215610504\n 0.27768397331237793\n 0.26419878005981445\n 0.2887670397758484\n 0.24598759412765503\n 0.26529061794281006\n 0.2722141146659851\n 0.2683166563510895\n 0.27142342925071716\n 0.26191475987434387\n 0.269707053899765\n 0.27216488122940063\n 0.2489825189113617\n 0.2507188022136688\n 0.24342603981494904\n 0.2559380531311035\n 0.2484971135854721\n 0.23990465700626373\n 0.2531616687774658\n 0.24644878506660461\n 0.25310108065605164\n 0.2627120912075043\n 0.2561471462249756\n 0.2561856806278229\n 0.2667142450809479\n 0.2590910494327545\n 0.24467585980892181\n 0.2725905179977417\n 0.25489065051078796\n 0.24542665481567383\n 0.24539409577846527\n 0.23968243598937988\n 0.23511043190956116\n 0.25360390543937683\n 0.24650028347969055\n 0.23169977962970734\n 0.24455244839191437\n 0.24662987887859344\n 0.24624189734458923\n 0.23428931832313538\n 0.24767740070819855\n 0.23768506944179535\n 0.2472761869430542\n 0.23279839754104614\n 0.24041056632995605\n 0.2530815899372101\n 0.24753203988075256\n 0.22979888319969177\n 0.2543557584285736\n 0.23379823565483093\n 0.23657512664794922\n 0.24298308789730072\n 0.21724991500377655\n 0.2442023903131485\n 0.23337283730506897\n 0.2285902053117752\n 0.2123374193906784\n 0.2410821169614792\n 0.20786207914352417\n 0.23221848905086517\n 0.22075426578521729\n 0.22609378397464752\n 0.22128520905971527\n 0.21948060393333435\n 0.23802122473716736\n 0.2361726313829422\n 0.21666672825813293\n 0.2261905074119568\n 0.24219931662082672\n 0.24513652920722961\n 0.23348087072372437\n 0.22907604277133942\n 0.23224423825740814\n 0.22439652681350708\n 0.22613592445850372\n 0.22645197808742523\n 0.23348374664783478\n 0.24152936041355133\n 0.24003289639949799\n 0.23111993074417114\n 0.2491626739501953\n 0.23659995198249817\n 0.22733712196350098\n 0.24321267008781433\n 0.2534855008125305\n 0.21976731717586517\n 0.24374708533287048\n 0.24605804681777954\n 0.2256333827972412\n 0.2384244203567505\n 0.22401297092437744\n 0.23391099274158478\n 0.22404631972312927\n 0.24439045786857605\n 0.2421136051416397\n 0.22506213188171387\n 0.2355397343635559\n 0.225271075963974\n 0.2387121170759201\n 0.23744675517082214\n 0.23890596628189087\n 0.2433251142501831\n 0.23942258954048157\n 0.2403792440891266\n 0.23808282613754272\n 0.2377949208021164\n 0.22013503313064575\n 0.24291060864925385\n 0.22773082554340363\n 0.21763896942138672\n 0.23886115849018097\n 0.22507654130458832\n 0.22411797940731049\n 0.21453329920768738\n 0.22563707828521729\n 0.22145503759384155\n 0.20734599232673645\n 0.20742680132389069\n 0.20916259288787842\n 0.2244996726512909\n 0.22623026371002197\n 0.21211287379264832\n 0.21153806149959564\n 0.23802340030670166\n 0.24202489852905273\n 0.2214781939983368\n 0.2111789584159851\n 0.23414115607738495\n 0.22392886877059937\n 0.24190984666347504\n 0.20013433694839478\n 0.22895900905132294\n 0.21900513768196106\n 0.21849389374256134\n 0.21820752322673798\n 0.22375166416168213\n 0.20359684526920319\n 0.22696225345134735\n 0.2061384618282318\n 0.2187628149986267\n 0.20472930371761322\n 0.22114497423171997\n 0.22017987072467804\n 0.23507489264011383\n 0.21708005666732788\n 0.21247068047523499\n 0.2210300862789154\n 0.22727221250534058\n 0.23032468557357788\n 0.1979924440383911\n 0.21653865277767181\n 0.23370163142681122\n 0.24236881732940674\n 0.23169346153736115\n 0.2510874271392822\n 0.23820054531097412\n 0.19949331879615784\n 0.22931447625160217\n 0.23242343962192535\n 0.22658228874206543\n 0.23572611808776855\n 0.23893143236637115\n 0.21720652282238007\n 0.21717922389507294\n 0.21199148893356323\n 0.22558334469795227\n 0.22055932879447937\n 0.21420429646968842\n 0.24341493844985962\n 0.2080707848072052\n 0.22922423481941223\n 0.21393057703971863\n 0.2212357521057129\n 0.2179993838071823\n 0.21821030974388123\n 0.2253318428993225\n 0.22530899941921234\n 0.22054241597652435\n 0.21719351410865784\n 0.19995954632759094\n 0.2111138552427292\n 0.22463010251522064\n 0.24538905918598175\n 0.20477385818958282\n 0.20941229164600372\n 0.22261181473731995\n 0.25003132224082947\n 0.2224467396736145\n 0.21941380202770233\n 0.23637056350708008\n 0.23289692401885986\n 0.2278335988521576\n 0.22571961581707\n 0.2225751280784607\n 0.22287991642951965\n 0.22273682057857513\n 0.2220742404460907\n\n\n\n```python\nlosses\n```\n\n\n```python\nplt.plot(losses[:])\n```\n\n\n```python\nX_train.shape\n```\n\n\n```python\nt_size = int(math.ceil(1.0/\u0394t))\nts = torch.linspace(0, 1, t_size).to(device)\nno_posterior_samples = 100\n\u0398_0 = torch.zeros((no_posterior_samples, net.dim)).to(device)\n\n\u0398_1 = torchsde.sdeint(sde, \u0398_0, ts, dt=\u0394t)[-1,...]\n```\n\n\n```python\nfig, (ax1,ax2,ax3) = plt.subplots(1,3)\n\nax1.hist(\u0398_1[:,0].cpu().detach().numpy())\nax2.hist(\u0398_1[:,1].cpu().detach().numpy())\nax3.hist(\u0398_1[:,2].cpu().detach().numpy())\n```\n\n\n```python\ndef predc(X, \u0398):\n return torch.vstack([(net.forward(X, \u03b8)[None,...]).softmax(dim=-1) for \u03b8 in \u0398]).mean(dim=0)\n```\n\n\n```python\npred = predc(X_train, \u0398_1)\n```\n\n\n```python\npred.shape\n```\n\n\n```python\n\n((pred.argmax(dim=-1)).float().flatten()== y_train.argmax(dim=-1)).float().mean()\n```\n\n\n```python\npred_test = predc(X_test.float(), \u0398_1)\n```\n\n\n```python\n((pred_test.argmax(dim=-1)).float().flatten()== y_test.argmax(dim=-1)).float().mean()\n```\n\n## MAP Baseline\n\nWe run the point estimate approximation (Maximum a posteriori) to double check what the learned weights look like. We get the exact same training accuracy as with the controlled model and similarly large weights for the non bias weights. \n\n\n```python\n\u0398_map = torch.zeros((1, dim), requires_grad=True, device=device)\noptimizer_map = torch.optim.Adam([\u0398_map], lr=0.05)\n# optimizer = torch.optim.LBFGS(gpr.parameters(), lr=0.01)\n\nlosses_map = []\nnum_steps = 1000\nfor i in tqdm(range(num_steps)):\n optimizer_map.zero_grad()\n\n if isinstance(optimizer_map, torch.optim.LBFGS):\n def closure_map():\n loss_map = log_likelihood_vmap()\n optimizer_map.zero_grad()\n loss_map.backward()\n return loss\n\n optimizer_map.step(closure_map)\n losses_map.append(closure_map().item())\n else:\n loss_map = -(log_likelihood_vmap(\u0398_map, X_train, y_train) + gaussian_prior(\u0398_map))\n optimizer_map.zero_grad()\n loss_map.backward()\n print(loss_map.item())\n optimizer_map.step()\n losses_map.append(loss_map.item())\n\n\u0398_map\npred_map = torch.sigmoid(X_train.mm(\u0398_map.T)).mean(axis=1)\n((pred_map < 0.5).float() == y_train).float().mean(), \u0398_map\n```\n", "meta": {"hexsha": "fb3414c5d0708466cca4852aafb0bcdf6184a7f3", "size": 26567, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/outdated_notebooks/mnist_nn_classifier.ipynb", "max_stars_repo_name": "franciscovargas/ControlledFollmerDrift", "max_stars_repo_head_hexsha": "608d20912cf6503502322fc759abb5216b9e1227", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-12-07T14:53:43.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-23T13:27:11.000Z", "max_issues_repo_path": "notebooks/outdated_notebooks/mnist_nn_classifier.ipynb", "max_issues_repo_name": "franciscovargas/ControlledFollmerDrift", "max_issues_repo_head_hexsha": "608d20912cf6503502322fc759abb5216b9e1227", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/outdated_notebooks/mnist_nn_classifier.ipynb", "max_forks_repo_name": "franciscovargas/ControlledFollmerDrift", "max_forks_repo_head_hexsha": "608d20912cf6503502322fc759abb5216b9e1227", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.47799511, "max_line_length": 437, "alphanum_fraction": 0.5704821771, "converted": true, "num_tokens": 6201, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.4310517741627041}} {"text": "### EXAMEN DEL M\u00d3DULO 2: \n#### G1: CLASE DE REPASO 11 DE NOVIEMBRE. ENTREGA EL DOMINGO 15 DE NOVIEMBRE. \n#### G2: CLASE DE REPASO 10 DE NOVIEMBRE. ENTREGA EL DOMINGO 14 DE NOVIEMBRE. \n\n\n### PROYECTO DE M\u00d3DULO 2 \n#### G1: PRESENTACI\u00d3N 23 DE NOVIEMBRE\n#### G2: PRESENTACI\u00d3N 20 DE NOVIEMBRE\n\n\n### REFERENTE AL M\u00d3DULO 3: ECUACIONES DIFERENCIALES. NO HABR\u00c1 EXAMEN, SOLO PROYECTO PARA ENTREGAR EL D\u00cdA:\n\n#### G1: 2 DE DICIEMBRE\n#### G2: 1 DE DICIEMBRE\n\n# Aplicando Python para an\u00e1lisis de precios: simulaci\u00f3n de escenarios futuros de precios\n\n\n\n> En la clase anterior vimos como importar datos de activos de la base de datos de Yahoo Finance usando el paquete pandas-datareader. En esta clase, veremos como pronosticar escenarios de evoluci\u00f3n de precios, suponiendo que los rendimientos diarios se distribuyen normalmente. Como esta evoluci\u00f3n de precios es aleatoria, utilizaremos la simulaci\u00f3n montecarlo (hacer muchas simulaciones de escenarios de evoluci\u00f3n de precios) para obtener probabilidades de que los precios de cierre est\u00e9n encima de un valor umbral y tomar decisiones con base en estas probabilidades.\n\n**Referencias:**\n- http://pandas.pydata.org/\n- http://www.learndatasci.com/python-finance-part-yahoo-finance-api-pandas-matplotlib/\n\n## 1. Recordemos como descargar datos...\n\nAntes que nada, para poder hacer simular escenarios de predicci\u00f3n de precios, vamos a recordar lo que hicimos en la clase pasada de descargar los datos de Yahoo Finance, utilizando el paquete `data` de la librer\u00eda `pandas_datareader`.\n\nEsta vez, utilizaremos los datos de precios de cierre ajustados de activos de la compa\u00f1\u00eda Apple en el a\u00f1o 2016 para nuestra aplicaci\u00f3n.\n\n\n```python\n# Importamos librer\u00edas\nimport pandas as pd\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas_datareader as web\n```\n\n\n```python\n# Funci\u00f3n para descargar precios de cierre ajustados de varios activos a la vez:\ndef get_closes(names, start, end):\n datos = web.DataReader(names,'yahoo',start,end)\n closes = datos['Adj Close']\n return closes \n```\n\n\n```python\n# Descargamos datos...\n# Instrumento: \nnames = ['CEMEXCPO.MX','AAPL','FB']\n\n# Fechas de inter\u00e9s (inicio y fin): 2015-2019\nstart = '2015-01-01'\nend = '2019-12-31'\n\n# Funci\u00f3n DataReader\nprecios_NIO = get_closes(names, start, end)\n```\n\n\n```python\n# Graficamos\nprecios_NIO.plot(figsize=(15,8))\n```\n\n\n```python\nprecios_NIO.shift()\n```\n\n\n\n\n Date\n 2018-09-12 NaN\n 2018-09-13 6.60\n 2018-09-14 11.60\n 2018-09-17 9.90\n 2018-09-18 8.50\n ... \n 2019-12-24 2.67\n 2019-12-26 2.53\n 2019-12-27 2.51\n 2019-12-30 2.42\n 2019-12-31 3.72\n Name: Adj Close, Length: 328, dtype: float64\n\n\n\n## 2. Simulaci\u00f3n de rendimientos diarios\n\nRecordemos que los precios diarios de cierre ajustados no son un proceso estoc\u00e1stico estacionario, pero los rendimientos diarios si lo son. Por tanto calculamos los rendimientos a partir de los precios de cierre, obtenemos sus propiedades estad\u00edsticas muestrales y proyectamos los rendimientos. Luego, obtenemos la proyecci\u00f3n de los precios.\n\nPara una sucesi\u00f3n de precios $\\{S_t\\}_{t=0}^{n}$, el rendimiento simple $R_t$ se define como el cambio porcentual\n\n$$\nR_t=\\frac{S_t-S_{t-1}}{S_{t-1}}\\approx \\ln\\left(\\frac{S_t}{S_{t-1}}\\right)=r_t.\n$$\npara $t=1,\\ldots,n$.\n\nPara el ejemplo en curso, \u00bfc\u00f3mo calcular esto?\n\nAdem\u00e1s, supusimos que los rendimientos diarios eran una variable aleatoria con distribuci\u00f3n normal (que se caracteriza con su media y varianza). Por tanto obtenemos la media y desviaci\u00f3n estandar muestrales. Hagamos una funci\u00f3n que retorne lo anterior.\n\n\n```python\n# Calcular rendimientos diarios y graficarlos\nret_NIO = (precios_NIO - precios_NIO.shift())/precios_NIO.shift()\nret_NIO.plot(figsize=(15,8))\n```\n\nEntonces, suponemos que el cambio porcentual de los precios (rendimientos diarios) tiene una distribuci\u00f3n normal.\n\n\u00bfC\u00f3mo se caracteriza una [distribuci\u00f3n normal](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_normal)?\n\n\n```python\n# Calculamos media y desviaci\u00f3n est\u00e1ndar\nmu = ret_NIO.mean()\nsigma = ret_NIO.std()\nmu, sigma\n```\n\n\n\n\n (0.0009845917169846305, 0.07651331373467106)\n\n\n\nHabiendo caracterizado los rendimientos diarios como una variable aleatoria normal con la media y la varianza muestral obtenida de los datos del 2019, podemos generar n\u00fameros aleatorios con estas caracter\u00edsticas para simular el comportamiento de los precios de cierre de las acciones en el 2020 (hay un supuesto de que las cosas no cambiar\u00e1n fundamentalmente).\n\nSin embargo, cada simulaci\u00f3n que hagamos nos conducir\u00e1 a distintos resultados (los precios siguen evolucionando aleatoriamente). Entonces, lo que haremos es simular varios escenarios para as\u00ed ver alguna tendencia y tomar decisiones.\n\nHagamos una una funci\u00f3n que simule varios escenarios de rendimientos diarios y que devuelva un dataframe con esta simulaci\u00f3n.\n\n\n```python\n# Ayuda en la funci\u00f3n np.random.randn\nhelp(np.random.randn)\n```\n\n Help on built-in function randn:\n \n randn(...) method of numpy.random.mtrand.RandomState instance\n randn(d0, d1, ..., dn)\n \n Return a sample (or samples) from the \"standard normal\" distribution.\n \n .. note::\n This is a convenience function for users porting code from Matlab,\n and wraps `standard_normal`. That function takes a\n tuple to specify the size of the output, which is consistent with\n other NumPy functions like `numpy.zeros` and `numpy.ones`.\n \n .. note::\n New code should use the ``standard_normal`` method of a ``default_rng()``\n instance instead; see `random-quick-start`.\n \n If positive int_like arguments are provided, `randn` generates an array\n of shape ``(d0, d1, ..., dn)``, filled\n with random floats sampled from a univariate \"normal\" (Gaussian)\n distribution of mean 0 and variance 1. A single float randomly sampled\n from the distribution is returned if no argument is provided.\n \n Parameters\n ----------\n d0, d1, ..., dn : int, optional\n The dimensions of the returned array, must be non-negative.\n If no argument is given a single Python float is returned.\n \n Returns\n -------\n Z : ndarray or float\n A ``(d0, d1, ..., dn)``-shaped array of floating-point samples from\n the standard normal distribution, or a single such float if\n no parameters were supplied.\n \n See Also\n --------\n standard_normal : Similar, but takes a tuple as its argument.\n normal : Also accepts mu and sigma arguments.\n Generator.standard_normal: which should be used for new code.\n \n Notes\n -----\n For random samples from :math:`N(\\mu, \\sigma^2)`, use:\n \n ``sigma * np.random.randn(...) + mu``\n \n Examples\n --------\n >>> np.random.randn()\n 2.1923875335537315 # random\n \n Two-by-four array of samples from N(3, 6.25):\n \n >>> 3 + 2.5 * np.random.randn(2, 4)\n array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random\n [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random\n \n\n\n\n```python\n# Funci\u00f3n que simula varios escenarios de rendimientos diarios\ndef ret_sim(mu,std,ndays,nscen,start):\n dates = pd.date_range(start=start,periods=ndays)\n simulados = std*np.random.randn(ndays,nscen) + mu\n datos = pd.DataFrame(data=simulados,index=dates)\n return datos\n```\n\n\n```python\n# Simulamos 100 escenarios para todo el 2020\nndays = 365\nnscen = 100\nstart = '2020-01-01'\nret_sim_NIO = ret_sim(mu,sigma,ndays,nscen,start)\n```\n\n\n```python\n# Mostrar\nret_sim_NIO\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        0123456789...90919293949596979899
                                        2020-01-010.0658980.110085-0.0575020.0263400.0077550.087987-0.009491-0.010940-0.061202-0.093377...0.0262830.131707-0.1374120.032258-0.0224230.0799560.063081-0.116615-0.031812-0.003118
                                        2020-01-020.0867080.1866540.0977200.040814-0.0044560.1057450.2206130.017709-0.0799500.063618...-0.054647-0.002310-0.0490160.056004-0.0005570.079252-0.087690-0.003558-0.086108-0.166188
                                        2020-01-03-0.017570-0.0047140.0047620.2268130.062640-0.098609-0.0580840.026110-0.029590-0.017806...0.0788750.031291-0.0380880.010499-0.0971200.017551-0.016013-0.0850350.010837-0.001291
                                        2020-01-040.0527290.093592-0.028358-0.1029130.108923-0.0667470.083230-0.1089130.0191480.032749...-0.024760-0.023625-0.082270-0.183516-0.0078400.0030550.0476210.063813-0.041140-0.148213
                                        2020-01-05-0.0276850.004496-0.020785-0.0258140.027429-0.056043-0.0277160.0664230.100174-0.156343...0.126394-0.0241370.0200320.032588-0.078944-0.149992-0.038563-0.100643-0.029856-0.079198
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                                        2020-12-260.0284310.006592-0.016664-0.0072580.098845-0.0590620.042655-0.0474630.0528570.133505...0.108280-0.0567730.038862-0.044612-0.1109700.0321640.0878300.0581490.097467-0.029266
                                        2020-12-27-0.110557-0.012235-0.034524-0.0441370.015505-0.0579010.119489-0.061956-0.014230-0.014533...-0.027747-0.073917-0.072987-0.056499-0.050299-0.0166570.046081-0.1008660.0685190.031324
                                        2020-12-28-0.061543-0.044171-0.0693660.0725390.0500920.1529630.1459510.0708890.0093980.052022...0.0256820.0115410.0699540.0471480.0346830.0799880.046632-0.0547450.012158-0.117194
                                        2020-12-290.1273360.016501-0.059412-0.090356-0.039909-0.0296650.0349570.160847-0.0713970.029767...0.1865620.182057-0.0685850.0705980.063003-0.057641-0.058878-0.067135-0.0803600.034025
                                        2020-12-300.0808380.1266370.0386120.0228610.013793-0.0242000.0085500.0368240.029335-0.086587...0.113710-0.048215-0.008837-0.008893-0.0673710.0104430.004200-0.0713450.0052610.102711
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                                        \n\n\n\n## 3. Proyecci\u00f3n de precios de cierre\n\nPor tanto, para calcular los precios, tenemos:\n\n$$\\begin{align}\np_i&=p_{i-1}(R_i+1)\\\\\np_{i+1}&=p_i(R_{i+1}+1)=p_{i-1}(R_i+1)(R_{i+1}+1)\\\\\n&\\vdots\\\\\np_{i+k}&=p_{i-1}(R_i+1)\\cdots(R_{i+k}+1).\n\\end{align}$$\n\nSi hacemos $i=0$ en la \u00faltima ecuaci\u00f3n, tenemos que $p_{k}=p_{-1}(R_0+1)\\cdots(R_{k}+1)$, donde $p_{-1}$ es el \u00faltimo precio reportado en el 2019.\n\nCon los rendimientos, calculamos los precios de cierre...\n\n\n```python\n# Obtenemos los precios. Transformar los rendimientos simulados del 2020 a precios de acci\u00f3n simulados del 2019.\nprecios_sim_NIO = precios_NIO.iloc[-1]*((1+ret_sim_NIO).cumprod())\nprecios_sim_NIO\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
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                                        2020-01-014.2849114.4625433.7888444.1258874.0511744.3737093.9818453.9760213.7739673.644623...4.1256594.5494643.4676024.1496763.9298584.3414234.2735843.5512093.8921164.007465
                                        2020-01-024.6564455.2954944.1590884.2942814.0331214.8362054.8602934.0464333.4722403.876486...3.9002034.5389533.2976354.3820753.9276684.6854913.8988323.5385733.5569733.341470
                                        2020-01-034.5746315.2705304.1788945.2682784.2857574.3593104.5779894.1520853.3694953.807461...4.2078324.6809813.1720354.4280813.5462134.7677283.8363983.2376723.5955203.337156
                                        2020-01-044.8158485.7638104.0603884.7261054.7525774.0683384.9590163.6998703.4340153.932151...4.1036484.5703942.9110723.6154583.5184124.7822934.0190923.4442783.4476022.842546
                                        2020-01-054.6825225.7897263.9759954.6041044.8829353.8403354.8215723.9456253.7780163.317388...4.6223264.4600772.9693883.7332783.2406564.0649883.8641033.0976343.3446702.617422
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                                        2020-12-260.3013541.2119452.8872811.48557019.6970632.0536870.3185551.0808869.1374220.699969...0.6591960.4944995.3305291.6698391.7919301.6889723.1834133.1755700.2319390.135195
                                        2020-12-270.2680371.1971172.7875991.42000220.0024721.9347760.3566191.0139199.0073990.689796...0.6409050.4579474.9414681.5754961.7017971.6608383.3301072.8552630.2478310.139430
                                        2020-12-280.2515411.1442392.5942341.52300821.0044402.2307250.4086681.0857959.0920480.725681...0.6573650.4632335.2871431.6497771.7608211.7936863.4853962.6989530.2508440.123089
                                        2020-12-290.2835711.1631202.4401061.38539520.1661752.1645510.4229541.2604428.4429040.747283...0.7800040.5475674.9245251.7662481.8717571.6902953.2801852.5177590.2306860.127277
                                        2020-12-300.3064951.3104142.5343231.41706620.4443302.1121690.4265701.3068568.6905790.682577...0.8686990.5211664.8810071.7505401.7456551.7079473.2939612.3381290.2319000.140350
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                                        365 rows \u00d7 100 columns

                                        \n
                                        \n\n\n\n\n```python\n# Graficar\nprecios_sim_NIO.plot(figsize=(15,8),legend=False)\n```\n\n\n```python\nK = precios_NIO.iloc[-1]*1.2\nK\n```\n\n\n\n\n 4.823999977111816\n\n\n\n\n```python\n\n```\n\n## 4. Probabilidad Precio-Umbral\n\nYa que tenemos muchos escenarios de precios proyectados, podemos ver varias cosas. Por ejemplo, \u00bfcu\u00e1l es la probabilidad de que el precio de cierre sobrepase alg\u00fan valor umbral en alg\u00fan momento?\n\n\n```python\n# Umbral de 120% del ultimo precio\nK = precios_NIO.iloc[-1]*1.2\n```\n\n\n```python\n# Comparar cada escenario en cada fecha\nTF = precios_sim_NIO > K\n\n# Sumamos para cada fecha y dividimos entre el n\u00famero de escenarios\nprob = TF.sum(axis=1)/nscen\n\n# Gr\u00e1fico de probabilidad\nprob.plot(figsize=(15,8),legend=False)\n```\n\n\n```python\n# Descargamos datos...\n# Instrumento: \nname = 'NIO'\n\n# Fechas de inter\u00e9s (inicio y fin): 2015-2019\nstart='2020-01-01'\nend = '2020-11-10'\n\n# Funci\u00f3n DataReader\nprecios_reales = get_closes(name,start,end)\n```\n\n\n```python\nprecios_reales.plot(figsize=(15,8))\n```\n\n___\nEntonces, ya aprendimos a bajar datos con pandas-datareader. En espec\u00edfico, a partir de los precios de cierre ajustados obtuvimos los rendimientos diarios.\n\nSuponiendo que los rendimientos diarios son un proceso estoc\u00e1stico estacionario de distribuci\u00f3n normal, pudimos caracaterizarlo y proyectar varios escenarios de evoluci\u00f3n de los precios (montecarlo).\n\nCon estas proyecciones pudimos calcular probabilidades de sobrepasar cierto precio umbral: toma de decisiones.\n\n\n\n
                                        \nCreated with Jupyter by Cristian Camilo Zapata Zuluaga.\n
                                        \n", "meta": {"hexsha": "5fb0586e76cf9943011be9b0f0cd0fb978c1a25c", "size": 457434, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "M\u00f3dulo 2/Clase13_ProbabilidadPrecioUmbral .ipynb", "max_stars_repo_name": "zapatacc/SimMat2020", "max_stars_repo_head_hexsha": "80a2bc927348235abb3fd64d44fdcaeece4fb11f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "M\u00f3dulo 2/Clase13_ProbabilidadPrecioUmbral .ipynb", "max_issues_repo_name": "zapatacc/SimMat2020", "max_issues_repo_head_hexsha": "80a2bc927348235abb3fd64d44fdcaeece4fb11f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "M\u00f3dulo 2/Clase13_ProbabilidadPrecioUmbral .ipynb", "max_forks_repo_name": "zapatacc/SimMat2020", "max_forks_repo_head_hexsha": "80a2bc927348235abb3fd64d44fdcaeece4fb11f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 48, "max_forks_repo_forks_event_min_datetime": "2020-08-26T15:22:24.000Z", "max_forks_repo_forks_event_max_datetime": "2020-09-03T14:32:14.000Z", "avg_line_length": 327.2060085837, "max_line_length": 211404, "alphanum_fraction": 0.9124288968, "converted": true, "num_tokens": 8773, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.4310517741627041}} {"text": "```python\n#from pandas_datareader import data as pdr\n#import fix_yahoo_finance as yf\n#yf.pdr_override() # <== that's all it takes :-)\nimport quandl\nimport pandas as pd\nimport numpy as np\nimport seaborn as sns\n\n\nimport matplotlib.pyplot as plt\nimport matplotlib.dates as mdates\n%matplotlib inline\nsns.set(style='darkgrid', context='talk', palette='Dark2')\nmy_year_month_fmt = mdates.DateFormatter('%m/%y')\n\n\n```\n\n\n```python\n# Define the instruments to download. We would like to see Apple, Microsoft and the S&P500 index.\ntickers = ['AAPL', 'MSFT', '^GSPC']\n\n# We would like all available data from 01/01/2000 until 12/31/2016.\nstart_date = '2010-01-01'\nend_date = '2016-12-31'\n\n```\n\n\n```python\ns = \"AAPL\"\napple = quandl.get(\"WIKI/\" + s, start_date=start_date, end_date=end_date)\n\n```\n\n\n```python\napple\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        OpenHighLowCloseVolumeEx-DividendSplit RatioAdj. OpenAdj. HighAdj. LowAdj. CloseAdj. Volume
                                        Date
                                        2010-01-04213.430214.5000212.380214.01017633200.00.01.027.42873027.56624027.29379027.503268123432400.0
                                        2010-01-05214.600215.5900213.250214.38021496600.00.01.027.57909127.70632027.40559727.550818150476200.0
                                        2010-01-06214.380215.2300210.750210.97019720000.00.01.027.55081827.66005527.08431227.112585138040000.0
                                        2010-01-07211.750212.0000209.050210.58017040400.00.01.027.21282627.24495526.86583927.062465119282800.0
                                        2010-01-08210.300212.0000209.060211.98015986100.00.01.027.02648127.24495526.86712427.242385111902700.0
                                        2010-01-11212.800213.0000208.450210.11016508200.00.01.027.34776627.37346926.78873027.002063115557400.0
                                        2010-01-12209.190209.7700206.420207.72021230700.00.01.026.88383126.95836926.52784726.694915148614900.0
                                        2010-01-13207.870210.9300204.100210.65021639000.00.01.026.71419227.10744526.22969527.071461151473000.0
                                        2010-01-14210.110210.4600209.020209.43015460500.00.01.027.00206327.04704326.86198326.914674108223500.0
                                        2010-01-15210.930211.6000205.870205.93021216700.00.01.027.10744527.19354926.45716426.464875148516900.0
                                        2010-01-19208.330215.1900207.240215.04026071700.00.01.026.77330927.65491426.63322827.635637182501900.0
                                        2010-01-20214.910215.5500209.500211.72521862600.00.01.027.61893027.70117926.92367027.209613153038200.0
                                        2010-01-21212.080213.3138207.210208.07221719800.00.01.027.25523627.41379626.62937326.740152152038600.0
                                        2010-01-22206.780207.5000197.160197.75031491700.00.01.026.57411226.66664225.33780825.413631220441900.0
                                        2010-01-25202.510204.7000200.190203.07538060700.00.01.026.02535826.30680325.72720526.097968266424900.0
                                        2010-01-26205.950213.7100202.580205.94066682500.00.01.026.46744527.46471426.03435326.466160466777500.0
                                        2010-01-27206.850210.5800199.531207.88461520300.00.01.026.58310827.06246525.64251426.715991430642100.0
                                        2010-01-28204.930205.5000198.700199.29041910800.00.01.026.33636126.40961425.53571925.611543293375600.0
                                        2010-01-29201.080202.2000190.250192.06344498300.00.01.025.84158325.98551824.44977724.682772311488100.0
                                        2010-02-01192.370196.0000191.300194.73026781300.00.01.024.72222625.18873224.58471625.025519187469100.0
                                        2010-02-02195.910196.3200193.380195.86024940800.00.01.025.17716625.22985624.85202525.170740174585600.0
                                        2010-02-03195.170200.2000194.420199.23021976000.00.01.025.08206525.72849024.98568025.603832153832000.0
                                        2010-02-04196.730198.3700191.570192.05027059000.00.01.025.28254725.49331024.61941524.681102189413000.0
                                        2010-02-05192.625196.0000190.850195.46030368100.00.01.024.75499725.18873224.52688525.119334212576700.0
                                        2010-02-08195.690197.8800194.000194.12017081100.00.01.025.14889225.43033824.93170424.947126119567700.0
                                        2010-02-09196.420197.5000194.750196.19022603100.00.01.025.24270825.38150325.02808925.213149158221700.0
                                        2010-02-10195.890196.6000194.260195.11613227200.00.01.025.17459525.26584024.96511825.07512592590400.0
                                        2010-02-11194.880199.7500194.060198.67019655200.00.01.025.04479625.67065924.93941525.531864137586400.0
                                        2010-02-12198.110201.6400195.500200.38023409600.00.01.025.45989625.91355025.12447525.751623163867200.0
                                        2010-02-16201.940203.6900201.520203.40019419200.00.01.025.95210526.17700425.89812926.139735135934400.0
                                        .......................................
                                        2016-11-17109.810110.3500108.830109.95027632003.00.01.0108.453686108.987016107.485790108.59195727632003.0
                                        2016-11-18109.720110.5400109.660110.06028428917.00.01.0108.364798109.174669108.305539108.70059828428917.0
                                        2016-11-21110.120111.9900110.010111.73029264571.00.01.0108.759857110.606760108.651216110.34997129264571.0
                                        2016-11-22111.950112.4200111.400111.80025965534.00.01.0110.567254111.031449110.024047110.41910725965534.0
                                        2016-11-23111.360111.5100110.330111.23027426394.00.01.0109.984541110.132689108.967263109.85614727426394.0
                                        2016-11-25111.130111.8700110.950111.79011475922.00.01.0109.757382110.488242109.579605110.40923011475922.0
                                        2016-11-28111.430112.4650111.390111.57027193983.00.01.0110.053677111.075893110.014171110.19194727193983.0
                                        2016-11-29110.780112.0300110.070111.46028528750.00.01.0109.411705110.646266108.710475110.08330628528750.0
                                        2016-11-30111.560112.2000110.270110.52036162258.00.01.0110.182071110.814166108.908004109.15491736162258.0
                                        2016-12-01110.365110.9400109.030109.49037086862.00.01.0109.001831109.569729107.683320108.13763937086862.0
                                        2016-12-02109.170110.0900108.850109.90026527997.00.01.0107.821591108.730228107.505543108.54257426527997.0
                                        2016-12-05110.000110.0300108.250109.11034324540.00.01.0108.641339108.670969106.912954107.76233234324540.0
                                        2016-12-06109.500110.3600109.190109.95026195462.00.01.0108.147515108.996893107.841344108.59195726195462.0
                                        2016-12-07109.260111.1900109.160111.03029998719.00.01.0107.910479109.816641107.811715109.65861729998719.0
                                        2016-12-08110.860112.4300110.600112.12027068316.00.01.0109.490717111.041325109.233928110.73515427068316.0
                                        2016-12-09112.310114.7000112.310113.95034402627.00.01.0110.922807113.283287110.922807112.54255134402627.0
                                        2016-12-12113.290115.0000112.490113.30026374377.00.01.0111.890703113.579582111.100584111.90057926374377.0
                                        2016-12-13113.840115.9200113.750115.19043733811.00.01.0112.433910114.488219112.345021113.76723543733811.0
                                        2016-12-14115.040116.2000114.980115.19034031834.00.01.0113.619088114.764760113.559829113.76723534031834.0
                                        2016-12-15115.380116.7300115.230115.82046524544.00.01.0113.954888115.288214113.806741114.38945446524544.0
                                        2016-12-16116.470116.5000115.645115.97044351134.00.01.0115.031425115.061055114.216615114.53760144351134.0
                                        2016-12-19115.800117.3800115.750116.64027779423.00.01.0114.369701115.930186114.320318115.19932627779423.0
                                        2016-12-20116.740117.5000116.680116.95021424965.00.01.0115.298090116.048703115.238832115.50549721424965.0
                                        2016-12-21116.800117.4000116.780117.06023783165.00.01.0115.357349115.949938115.337596115.61413823783165.0
                                        2016-12-22116.350116.5100115.640116.29026085854.00.01.0114.912908115.070931114.211677114.85364926085854.0
                                        2016-12-23115.590116.5200115.590116.52014249484.00.01.0114.162295115.080808114.162295115.08080814249484.0
                                        2016-12-27116.520117.8000116.490117.26018296855.00.01.0115.080808116.344998115.051178115.81166818296855.0
                                        2016-12-28117.520118.0166116.200116.76020905892.00.01.0116.068456116.558923114.764760115.31784320905892.0
                                        2016-12-29116.450117.1095116.400116.73015039519.00.01.0115.011672115.663027114.962290115.28821415039519.0
                                        2016-12-30116.650117.2000115.430115.82030586265.00.01.0115.209202115.752409114.004271114.38945430586265.0
                                        \n

                                        1762 rows \u00d7 12 columns

                                        \n
                                        \n\n\n\n\n```python\n\n```\n\n\n```python\n# view of panel data as dataframe\npanel_data.to_frame().head(9)\n```\n\n\n```python\n# Getting just the adjusted closing prices. This will return a Pandas DataFrame\n# The index in this DataFrame is the major index of the panel_data.\nclose = panel_data['Close']\n\n# Getting all weekdays between 01/01/2000 and 12/31/2016\nall_weekdays = pd.date_range(start=start_date, end=end_date, freq='B')\n\n# How do we align the existing prices in adj_close with our new set of dates?\n# All we need to do is reindex close using all_weekdays as the new index\nclose = close.reindex(all_weekdays)\n\n# Reindexing will insert missing values (NaN) for the dates that were not present\n# in the original set. To cope with this, we can fill the missing by replacing them\n# with the latest available price for each instrument.\nclose = close.fillna(method='ffill')\n```\n\n\n```python\n#showing the series for all_weekday\nprint(all_weekdays)\n```\n\n\n```python\nclose.head(10)\n```\n\n\n```python\n# cleaned up dataset free of missing values\nclose.describe()\n\n```\n\n\n```python\n# Get the MSFT timeseries. This now returns a Pandas Series object indexed by date.\nmsft = close.loc[:, 'MSFT']\n\n# Calculate the 20 and 100 days moving averages of the closing prices\nshort_rolling_msft = msft.rolling(window=20).mean()\nlong_rolling_msft = msft.rolling(window=100).mean()\n\n# Plot everything by leveraging the very powerful matplotlib package\nfig, ax = plt.subplots(figsize=(16,9))\n\nax.plot(msft.index, msft, label='MSFT')\nax.plot(short_rolling_msft.index, short_rolling_msft, label='20 days rolling')\nax.plot(long_rolling_msft.index, long_rolling_msft, label='100 days rolling')\n\nax.set_xlabel('Date')\nax.set_ylabel('Adjusted closing price ($)')\nax.legend()\n```\n\n\n```python\ndata = pd.read_pickle('./data.pkl')\ndata.head(10)\n```\n\n\n```python\n# Calculating the short-window moving average\nshort_rolling = data.rolling(window=20).mean()\nshort_rolling.head()\n```\n\n\n```python\n# Calculating the short-window moving average\nlong_rolling = data.rolling(window=100).mean()\nlong_rolling.tail()\n```\n\n\n```python\n# Relative returns\nreturns = data.pct_change(1)\n```\n\n\\begin{equation} r_{\\text{relative}}\\left(t\\right) = \\frac{p\\left(t\\right) - p\\left(t-1\\right)}{p\\left(t-1\\right)} \\end{equation}\n\n\n```python\n# Log returns - First the logarithm of the prices is taken and the the difference of consecutive (log) observations\nlog_returns = np.log(data).diff()\nlog_returns.head()\n```\n\n\\begin{equation} r\\left(t\\right) = \\log\\left( \\frac{p\\left(t\\right)}{p\\left(t-1\\right)} \\right) \\end{equation}\n\nSince we have log returns we can use the sum to generate a cumulitive return over time; then we convert it back to relative return to make it interpreable\n\nlog-return of 1 != 100% return and money dobule but relative return = 1 does \n\n\\begin{equation} c\\left(t\\right) = \\sum_{k=1}^t r\\left(t\\right) \\end{equation}\n\n\n\\begin{equation} c_{\\text{relative}}\\left(t\\right) = e^{c\\left(t\\right)} - 1 \\end{equation}\n\n\n```python\nfig, (ax1, ax2) = plt.subplots(2, 1, figsize=(16,12))\n\nfor c in log_returns:\n ax1.plot(log_returns.index, log_returns[c].cumsum(), label=str(c))\n\nax1.set_ylabel('Cumulative log returns')\nax1.legend(loc='best')\n\nfor c in log_returns:\n ax2.plot(log_returns.index, 100*(np.exp(log_returns[c].cumsum()) - 1), label=str(c))\n\nax2.set_ylabel('Total relative returns (%)')\nax2.legend(loc='best')\n\nplt.show()\n```\n\n## Creating a trading strategy and optimisation of allocation\n\n\\begin{equation} w_i\\left(t\\right) \\in \\mathbb{R} \\ \\text{and} \\ \\sum_{i=1}^K w_i\\left(t\\right) \\leq 1 \\end{equation}\n\n\n```python\n# Last day returns. Make this a column vector\nr_t = log_returns.tail(1).transpose()\nr_t\n```\n\n\n```python\n# Weights as defined above\nweights_vector = pd.DataFrame(1 / 3, index=r_t.index, columns=r_t.columns)\nweights_vector\n```\n\n\n```python\n# Total log_return for the portfolio is:\nportfolio_log_return = weights_vector.transpose().dot(r_t)\nportfolio_log_return\n```\n\nIf computer memory is not an issue, a very fast way of computing the portfolio returns for all days, t=1,\u2026,Tt=1,\u2026,T is the following:\n\nAssume that R\u2208RT\u00d7KR\u2208RT\u00d7K is a matrix, the ttth row of which is the row vector r\u20d7 (t)Tr\u2192(t)T. Similarly, W\u2208RT\u00d7KW\u2208RT\u00d7K is a matrix, the ttth row of which is the row vector w\u20d7 (t)Tw\u2192(t)T. Then if r\u20d7 p=[rp(1),\u2026,rp(T)]T\u2208RT\u00d71r\u2192p=[rp(1),\u2026,rp(T)]T\u2208RT\u00d71 is a column vector of all portfolio returns, we have\n\nr\u20d7 p=diag{WRT}\nr\u2192p=diag{WRT}\nwhere diag{A}diag{A} is the diagonal of a matrix AA. The diagonal extraction is required because only in the diagonal the weights and log-returns vectors are properly time-aligned.\n\n\n```python\n#This should be computated using matrix manipulation on a face PC\n\nweights_matrix = pd.DataFrame(1 / 3, index=data.index, columns=data.columns)\nweights_matrix.tail()\n```\n\n\n```python\nlog_returns.head()\n\n```\n\n\n```python\n# Initially the two matrices are multiplied. Note that we are only interested in the diagonal, \n# which is where the dates in the row-index and the column-index match.\ntemp_var = weights_matrix.dot(log_returns.transpose())\ntemp_var.head().iloc[:, 0:5]\n```\n\n\n```python\n# The numpy np.diag function is used to extract the diagonal and then\n# a Series is constructed using the time information from the log_returns index\nportfolio_log_returns = pd.Series(np.diag(temp_var), index=log_returns.index)\nportfolio_log_returns.tail()\n```\n\n\n```python\ntotal_relative_returns = (np.exp(portfolio_log_returns.cumsum()) - 1)\n\nfig, (ax1, ax2) = plt.subplots(2, 1, figsize=(16,12))\n\nax1.plot(portfolio_log_returns.index, portfolio_log_returns.cumsum())\nax1.set_ylabel('Portfolio cumulative log returns')\n\nax2.plot(total_relative_returns.index, 100 * total_relative_returns)\nax2.set_ylabel('Portfolio total relative returns (%)')\n\nplt.show()\n```\n\n\n```python\n# Calculating the time-related parameters of the simulation\ndays_per_year = 52 * 5\ntotal_days_in_simulation = data.shape[0]\nnumber_of_years = total_days_in_simulation / days_per_year\n\n# The last data point will give us the total portfolio return\ntotal_portfolio_return = total_relative_returns[-1]\n# Average portfolio return assuming compunding of returns\naverage_yearly_return = (1 + total_portfolio_return)**(1 / number_of_years) - 1\n\nprint('Total portfolio return is: ' +\n '{:5.2f}'.format(100 * total_portfolio_return) + '%')\nprint('Average yearly return is: ' +\n '{:5.2f}'.format(100 * average_yearly_return) + '%')\n```\n\n\n```python\ndata = pd.read_pickle('./data.pkl')\ndata.head(10)\n```\n\n\n```python\n# Calculating the short-window simple moving average\nshort_rolling = data.rolling(window=20).mean()\nshort_rolling.head(20)\n```\n\n\n```python\n# Calculating the long-window simple moving average\nlong_rolling = data.rolling(window=100).mean()\nlong_rolling.tail()\n```\n\n\n```python\nstart_date = '2015-01-01'\nend_date = '2016-12-31'\n\nfig, ax = plt.subplots(figsize=(16,9))\n\nax.plot(data.loc[start_date:end_date, :].index, data.loc[start_date:end_date, 'MSFT'], label='Price')\nax.plot(long_rolling.loc[start_date:end_date, :].index, long_rolling.loc[start_date:end_date, 'MSFT'], label = '100-days SMA')\nax.plot(short_rolling.loc[start_date:end_date, :].index, short_rolling.loc[start_date:end_date, 'MSFT'], label = '20-days SMA')\n\nax.legend(loc='best')\nax.set_ylabel('Price in $')\nax.xaxis.set_major_formatter(my_year_month_fmt)\n```\n\n\n```python\n# to reduce the lag on moving averages; we use the EMA approach to weight the various points\n# Using Pandas to calculate a 20-days span EMA. adjust=False specifies that we are interested in the recursive calculation mode.\nema_short = data.ewm(span=20, adjust=False).mean()\n\nfig, ax = plt.subplots(figsize=(15,9))\n\nax.plot(data.loc[start_date:end_date, :].index, data.loc[start_date:end_date, 'MSFT'], label='Price')\nax.plot(ema_short.loc[start_date:end_date, :].index, ema_short.loc[start_date:end_date, 'MSFT'], label = 'Span 20-days EMA')\nax.plot(short_rolling.loc[start_date:end_date, :].index, short_rolling.loc[start_date:end_date, 'MSFT'], label = '20-days SMA')\n\nax.legend(loc='best')\nax.set_ylabel('Price in $')\nax.xaxis.set_major_formatter(my_year_month_fmt)\n```\n\n\n```python\n# Taking the sign of the difference to determine whether the price or the EMA is greater and then multiplying by 1/3\ntrading_positions = trading_positions_raw.apply(np.sign) * 1/3\ntrading_positions.tail()\n```\n\n\n```python\n# Lagging our trading signals by one day.\ntrading_positions_final = trading_positions.shift(1)\n\n```\n\n\n```python\nfig, (ax1, ax2) = plt.subplots(2, 1, figsize=(16,9))\n\nax1.plot(data.loc[start_date:end_date, :].index, data.loc[start_date:end_date, 'MSFT'], label='Price')\nax1.plot(ema_short.loc[start_date:end_date, :].index, ema_short.loc[start_date:end_date, 'MSFT'], label = 'Span 20-days EMA')\n\nax1.set_ylabel('$')\nax1.legend(loc='best')\nax1.xaxis.set_major_formatter(my_year_month_fmt)\n\nax2.plot(trading_positions_final.loc[start_date:end_date, :].index, trading_positions_final.loc[start_date:end_date, 'MSFT'], \n label='Trading position')\n\nax2.set_ylabel('Trading position')\nax2.xaxis.set_major_formatter(my_year_month_fmt)\n```\n\n\n```python\n# Log returns - First the logarithm of the prices is taken and the the difference of consecutive (log) observations\nasset_log_returns = np.log(data).diff()\nasset_log_returns.head()\n```\n\n\n```python\nstrategy_asset_log_returns = trading_positions_final * asset_log_returns\nstrategy_asset_log_returns.tail()\n```\n\n\n```python\n# Get the cumulative log-returns per asset\ncum_strategy_asset_log_returns = strategy_asset_log_returns.cumsum()\n\n# Transform the cumulative log returns to relative returns\ncum_strategy_asset_relative_returns = np.exp(cum_strategy_asset_log_returns) - 1\n\nfig, (ax1, ax2) = plt.subplots(2, 1, figsize=(16,9))\n\nfor c in asset_log_returns:\n ax1.plot(cum_strategy_asset_log_returns.index, cum_strategy_asset_log_returns[c], label=str(c))\n\nax1.set_ylabel('Cumulative log-returns')\nax1.legend(loc='best')\nax1.xaxis.set_major_formatter(my_year_month_fmt)\n\nfor c in asset_log_returns:\n ax2.plot(cum_strategy_asset_relative_returns.index, 100*cum_strategy_asset_relative_returns[c], label=str(c))\n\nax2.set_ylabel('Total relative returns (%)')\nax2.legend(loc='best')\nax2.xaxis.set_major_formatter(my_year_month_fmt)\n```\n\n\n```python\n# Total strategy relative returns. This is the exact calculation.\ncum_relative_return_exact = cum_strategy_asset_relative_returns.sum(axis=1)\n\n# Get the cumulative log-returns per asset\ncum_strategy_log_return = cum_strategy_asset_log_returns.sum(axis=1)\n\n# Transform the cumulative log returns to relative returns. This is the approximation\ncum_relative_return_approx = np.exp(cum_strategy_log_return) - 1\n\nfig, ax = plt.subplots(figsize=(16,9))\n\nax.plot(cum_relative_return_exact.index, 100*cum_relative_return_exact, label='Exact')\nax.plot(cum_relative_return_approx.index, 100*cum_relative_return_approx, label='Approximation')\n\nax.set_ylabel('Total cumulative relative returns (%)')\nax.legend(loc='best')\nax.xaxis.set_major_formatter(my_year_month_fmt)\n```\n", "meta": {"hexsha": "6dfa8a680a6ad3949be4f2b66dcfe9c7485af1d6", "size": 67978, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Tuts/Python for Finance.ipynb", "max_stars_repo_name": "atsnova/JunkJuice", "max_stars_repo_head_hexsha": "c6aea30ea810db0cf58c96c55eae2d250dd01a8b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Tuts/Python for Finance.ipynb", "max_issues_repo_name": "atsnova/JunkJuice", "max_issues_repo_head_hexsha": "c6aea30ea810db0cf58c96c55eae2d250dd01a8b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Tuts/Python for Finance.ipynb", "max_forks_repo_name": "atsnova/JunkJuice", "max_forks_repo_head_hexsha": "c6aea30ea810db0cf58c96c55eae2d250dd01a8b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 37.2686403509, "max_line_length": 1103, "alphanum_fraction": 0.4402600841, "converted": true, "num_tokens": 12582, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.6688802669716107, "lm_q1q2_score": 0.43090946121641727}} {"text": "# Expectation Values\n\nGiven a circuit generating a quantum state $\\lvert \\psi \\rangle$, it is very common to have an operator $H$ and ask for the expectation value $\\langle \\psi \\vert H \\vert \\psi \\rangle$. A notable example is in quantum computational chemistry, where $\\lvert \\psi \\rangle$ encodes the wavefunction for the electronic state of a small molecule, and the energy of the molecule can be derived from the expectation value with respect to the molecule's Hamiltonian operator $H$.
                                        \n
                                        \nThis example uses this chemistry scenario to demonstrate the overall procedure for using `pytket` to perform advanced high-level procedures. We build on top of topics covered by several other example notebooks, including circuit generation, optimisation, and using different backends.
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                                        \nThere is limited built-in functionality in `pytket` for obtaining expectation values from circuits. This is designed to encourage users to consider their needs for parallelising the processing of circuits, manipulating results (e.g. filtering, adjusting counts to mitigate errors, and other forms of data processing), or more advanced schemes for grouping the terms of the operator into measurement circuits. For this example, suppose that we want to focus on reducing the queueing time for IBM device backends, and filter our shots to eliminate some detected errors.
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                                        \nThis notebook makes use of the Qiskit and ProjectQ backend modules `pytket_qiskit` and `pytket_projectq`, as well as the electronic structure module `openfermion`, all three of which should first be installed via `pip`.
                                        \n
                                        \nWe will start by generating an ansatz and Hamiltonian for the chemical of interest. Here, we are just using a simple model of $\\mathrm{H}_2$ with four qubits representing the occupation of four spin orbitals.\n\n\n```python\nfrom pytket import Circuit, Qubit, Bit\nfrom pytket.utils.operators import QubitPauliOperator\nfrom sympy import symbols\nfrom openfermion import QubitOperator\n```\n\nGenerate ansatz and Hamiltonian:\n\n\n```python\nansatz = Circuit()\nqubits = ansatz.add_q_register(\"q\", 4)\nargs = symbols(\"a0 a1 a2 a3 a4 a5 a6 a7\")\nfor i in range(4):\n ansatz.Ry(args[i], qubits[i])\nfor i in range(3):\n ansatz.CX(qubits[i], qubits[i + 1])\nfor i in range(4):\n ansatz.Ry(args[4 + i], qubits[i])\nansatz.measure_all()\n```\n\n\n```python\nfor command in ansatz:\n print(command)\n```\n\nIn reality, you would use an expectation value calculation as the objective function for a classical optimisation routine to determine the parameter values for the ground state. For the purposes of this notebook, we will use some predetermined values for the ansatz, already optimised for $\\mathrm{H}_2$.\n\n\n```python\narg_values = [\n 7.17996183e-02,\n 2.95442468e-08,\n 1.00000015e00,\n 1.00000086e00,\n 9.99999826e-01,\n 1.00000002e00,\n 9.99999954e-01,\n 1.13489747e-06,\n]\n```\n\n\n```python\nansatz.symbol_substitution(dict(zip(args, arg_values)))\n```\n\n\n```python\nhamiltonian = (\n -0.0970662681676282 * QubitOperator(\"\")\n + -0.045302615503799284 * QubitOperator(\"X0 X1 Y2 Y3\")\n + 0.045302615503799284 * QubitOperator(\"X0 Y1 Y2 X3\")\n + 0.045302615503799284 * QubitOperator(\"Y0 X1 X2 Y3\")\n + -0.045302615503799284 * QubitOperator(\"Y0 Y1 X2 X3\")\n + 0.17141282644776884 * QubitOperator(\"Z0\")\n + 0.16868898170361213 * QubitOperator(\"Z0 Z1\")\n + 0.12062523483390425 * QubitOperator(\"Z0 Z2\")\n + 0.16592785033770352 * QubitOperator(\"Z0 Z3\")\n + 0.17141282644776884 * QubitOperator(\"Z1\")\n + 0.16592785033770352 * QubitOperator(\"Z1 Z2\")\n + 0.12062523483390425 * QubitOperator(\"Z1 Z3\")\n + -0.22343153690813597 * QubitOperator(\"Z2\")\n + 0.17441287612261608 * QubitOperator(\"Z2 Z3\")\n + -0.22343153690813597 * QubitOperator(\"Z3\")\n)\n```\n\nWe can simulate this exactly using a statevector simulator like ProjectQ. This has a built-in method for fast calculations of expectation values that works well for small examples like this.\n\n\n```python\nfrom pytket.extensions.projectq import ProjectQBackend\n```\n\n\n```python\nbackend = ProjectQBackend()\nideal_energy = backend.get_operator_expectation_value(\n ansatz, QubitPauliOperator.from_OpenFermion(hamiltonian)\n)\nprint(ideal_energy)\n```\n\nIdeally the state generated by this ansatz will only span the computational basis states with exactly two of the four qubits in state $\\lvert 1 \\rangle$. This is because these basis states correspond to two electrons being present in the molecule.
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                                        \nThis ansatz is a hardware-efficient model that is designed to explore a large portion of the Hilbert space with relatively few entangling gates. Unfortunately, with this much freedom, it will regularly generate states that have no physical interpretation such as states spanning multiple basis states corresponding to different numbers of electrons in the system (which we assume is fixed and conserved).
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                                        \nWe can mitigate this by using a syndrome qubit that calculates the parity of the other qubits. Post-selecting this syndrome with $\\langle 0 \\rvert$ will project the remaining state onto the subspace of basis states with even parity, increasing the likelihood the observed state will be a physically admissible state.
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                                        \nEven if the ansatz parameters are tuned to give a physical state, real devices have noise and imperfect gates, so in practice we may also measure bad states with a small probability. If this syndrome qubit is measured as 1, it means an error has definitely occurred, so we should discard the shot.\n\n\n```python\nsyn = Qubit(\"synq\", 0)\nsyn_res = Bit(\"synres\", 0)\nansatz.add_qubit(syn)\nansatz.add_bit(syn_res)\nfor qb in qubits:\n ansatz.CX(qb, syn)\nansatz.Measure(syn, syn_res)\n```\n\nUsing this, we can define a filter function which removes the shots which the syndrome qubit detected as erroneous. `BackendResult` objects allow retrieval of shots in any bit order, so we can retrieve the `synres` results separately and use them to filter the shots from the remaining bits. The Backends example notebook describes this in more detail.\n\n\n```python\ndef filter_shots(backend_result, syn_res_bit):\n bits = sorted(backend_result.get_bitlist())\n bits.remove(syn_res_bit)\n syn_shots = backend_result.get_shots([syn_res])[:, 0]\n main_shots = backend_result.get_shots(bits)\n return main_shots[syn_shots == 0]\n filtered_rows = shot_table[shot_table[:, syn_res_index] == 0]\n return np.delete(filtered_rows, syn_res_index, axis=1)\n```\n\nDepending on which backend we will be using, we will need to compile each circuit we run to conform to the gate set and connectivity constraints. We can define a compilation pass for each backend that optimises the circuit and maps it onto the backend's gate set and connectivity constraints. We don't expect this to change our circuit too much as it is already near-optimal.\n\n\n```python\nfrom pytket.passes import OptimisePhaseGadgets, SequencePass\n```\n\n\n```python\ndef compiler_pass(backend):\n return SequencePass([OptimisePhaseGadgets(), backend.default_compilation_pass()])\n```\n\nThe OpenFermion `QubitOperator` class represents the operator by its decomposition into a linear combination of Pauli operators (tensor products of the $I$, $X$, $Y$, and $Z$ matrices).
                                        \n
                                        \nGiven the full statevector, the expectation value can be calculated simply by matrix multiplication. However, with a real quantum system, we cannot observe the full statevector directly. Fortunately, the Pauli decomposition of the operator gives us a sequence of measurements we should apply to obtain the relevant information to reconstruct the expectation value.
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                                        \nThe utility method `append_pauli_measurement` takes a single term of a `QubitPauliOperator` (a `QubitPauliString`) and appends measurements in the corresponding bases to obtain the expectation value for that particular Pauli operator. We will want to make a new `Circuit` object for each of the measurements we wish to observe.
                                        \n
                                        \nA `QubitPauliString` is a sparse representation of a Pauli operator with support over some subset of qubits.
                                        \n
                                        \nFirst we need a little utility function to generate a `QubitPauliString` from OpenFermion's representation.\n\n\n```python\nfrom pytket.pauli import Pauli, QubitPauliString\nfrom pytket.predicates import CompilationUnit\nfrom pytket.utils import append_pauli_measurement\n```\n\n\n```python\npauli_sym = {\"I\": Pauli.I, \"X\": Pauli.X, \"Y\": Pauli.Y, \"Z\": Pauli.Z}\n```\n\n\n```python\ndef qps_from_openfermion(paulis):\n # translate from openfermion format to a QubitPauliString\n qlist = []\n plist = []\n for q, p in paulis:\n qlist.append(Qubit(q))\n plist.append(pauli_sym[p])\n return QubitPauliString(qlist, plist)\n```\n\n\n```python\ndef gen_pauli_measurement_circuits(state_circuit, compiler_pass, operator):\n # compile main circuit once\n state_cu = CompilationUnit(state_circuit)\n compiler_pass.apply(state_cu)\n compiled_state = state_cu.circuit\n final_map = state_cu.final_map\n # make a measurement circuit for each pauli\n pauli_circuits = []\n coeffs = []\n energy = 0\n for p, c in operator.terms.items():\n if p == ():\n # constant term\n energy += c\n else:\n # make measurement circuits and compile them\n pauli_circ = Circuit(state_circuit.n_qubits - 1) # ignore syndrome qubit\n append_pauli_measurement(qps_from_openfermion(p), pauli_circ)\n pauli_cu = CompilationUnit(pauli_circ)\n compiler_pass.apply(pauli_cu)\n pauli_circ = pauli_cu.circuit\n init_map = pauli_cu.initial_map\n # map measurements onto the placed qubits from the state\n rename_map = {\n i: final_map[o] for o, i in init_map.items() if o in final_map\n }\n pauli_circ.rename_units(rename_map)\n state_and_measure = compiled_state.copy()\n state_and_measure.append(pauli_circ)\n pauli_circuits.append(state_and_measure)\n coeffs.append(c)\n return pauli_circuits, coeffs, energy\n```\n\nWe can now start composing these together to get our generalisable expectation value function. Passing all of our circuits to `process_circuits` allows them to be submitted to IBM Quantum devices at the same time, giving substantial savings in overall queueing time. Since the backend will cache any results from `Backend.process_circuits`, we will remove the results when we are done with them to prevent memory bloating when this method is called many times.\n\n\n```python\nfrom pytket.utils import expectation_from_shots\n```\n\n\n```python\ndef expectation_value(state_circuit, operator, backend, n_shots):\n if backend.supports_expectation:\n compiled_circuit = state_circuit.copy()\n backend.compile_circuit(compiled_circuit)\n return backend.get_operator_expectation_value(\n compiled_circuit, QubitPauliOperator.from_OpenFermion(operator)\n )\n elif backend.supports_shots:\n syn_res_index = state_circuit.bit_readout[syn_res]\n pauli_circuits, coeffs, energy = gen_pauli_measurement_circuits(\n state_circuit, compiler_pass(backend), operator\n )\n handles = backend.process_circuits(pauli_circuits, n_shots=n_shots)\n for handle, coeff in zip(handles, coeffs):\n res = backend.get_result(handle)\n filtered = filter_shots(res, syn_res)\n energy += coeff * expectation_from_shots(filtered)\n backend.pop_result(handle)\n return energy\n else:\n raise NotImplementedError(\"Implementation for state and counts to be written\")\n```\n\n...and then run it for our ansatz. `AerBackend` supports faster expectation value from snapshopts (using the `AerBackend.get_operator_expectation_value` method), but this only works when all the qubits in the circuit are default register qubits that go up from 0. So we will need to rename `synq`.\n\n\n```python\nfrom pytket.extensions.qiskit import IBMQEmulatorBackend, AerBackend\n```\n\n\n```python\nansatz.rename_units({Qubit(\"synq\", 0): Qubit(\"q\", 4)})\n```\n\n\n```python\nprint(expectation_value(ansatz, hamiltonian, AerBackend(), 8000))\n# Try replacing IBMQEmulatorBackend with IBMQBackend to submit the circuits to a real IBM Quantum device.\nprint(expectation_value(ansatz, hamiltonian, IBMQEmulatorBackend(\"ibmq_santiago\"), 8000))\n```\n\nFor basic practice with using pytket backends and their results, try editing the code here to:
                                        \n* Extend `expectation_value` to work with statevector backends (e.g. `AerStateBackend`)
                                        \n* Remove the row filtering from `filter_shots` and see the effect on the expectation value on a noisy simulation/device
                                        \n* Adapt `filter_shots` to be able to filter a counts dictionary and adapt `expectation_value` to calulate the result using the counts summary from the backend (`pytket.utils.expectation_from_counts` will be useful here)\n", "meta": {"hexsha": "a1c76c3cfdd5861a449dcd63617bba038e8ab7a4", "size": 16365, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/expectation_value_example.ipynb", "max_stars_repo_name": "CQCL/pytket", "max_stars_repo_head_hexsha": "44fa95eb060afc8c45598f89afda993aa2d06634", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 249, "max_stars_repo_stars_event_min_datetime": "2018-07-20T03:04:52.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T08:45:58.000Z", "max_issues_repo_path": "examples/expectation_value_example.ipynb", "max_issues_repo_name": "CQCL/pytket", "max_issues_repo_head_hexsha": "44fa95eb060afc8c45598f89afda993aa2d06634", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 67, "max_issues_repo_issues_event_min_datetime": "2018-08-03T09:38:15.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-22T09:39:45.000Z", "max_forks_repo_path": "examples/expectation_value_example.ipynb", "max_forks_repo_name": "CQCL/pytket", "max_forks_repo_head_hexsha": "44fa95eb060afc8c45598f89afda993aa2d06634", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 69, "max_forks_repo_forks_event_min_datetime": "2019-02-26T15:15:30.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-15T14:47:24.000Z", "avg_line_length": 16365.0, "max_line_length": 16365, "alphanum_fraction": 0.6966086159, "converted": true, "num_tokens": 3184, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251201477016, "lm_q2_score": 0.668880247169804, "lm_q1q2_score": 0.4309094575973913}} {"text": "# Model 1 ~ What sport are we most likely to succeed in?\n\nIn order to answer this question we will split this up in two smaller questions which we will be tackling individually:\n\n1. **Which sport should we participate in the Olympics in?:**\n\n2. **What probability do we have of winning a medal at a given sport?:**\n\nFirst step however is to do some basic data pre-processing steps to adjust the data set to our benefit and obtain better final results (most of these data processing steps are done based on our own logic and considering that these steps would improve our models).\n\nWe will be going step by step throughout this notebook, explaining at every step what we are doing and why.\n\nWe would also like to highlight that we have leveraged on a kaggle notebook which also looked into the first subquestion. We have used a lot of his/her ideas and code, but have extended the model based on two main things:\n\n- Include additional feature (medals won per country per sport)\n- Divide models based on sex\n\nThe kaggle notebook can be found under: https://www.kaggle.com/deanpower/what-sport-will-you-compete-in\n\nReviewing this notebook has allowed us to better understand how to do a \"real\" machine learning notebook and we consider we have been able to learn a lot from it despite having copied a lot of it's ideas.\n\n##### Import libraries\n\n\n```python\n%load_ext autoreload\n%autoreload 2\n```\n\n\n```python\nimport model1_funcs as M1\nimport pandas as pd\nfrom sklearn.preprocessing import LabelEncoder\nfrom sklearn.metrics import (\n classification_report,\n f1_score,\n make_scorer,\n precision_recall_fscore_support,\n precision_score,\n recall_score,\n)\nfrom sklearn.neighbors import KNeighborsClassifier\nfrom sklearn.svm import LinearSVC\nfrom sklearn.tree import DecisionTreeClassifier\nfrom sklearn.dummy import DummyClassifier\nfrom sklearn.ensemble import RandomForestClassifier\nfrom sklearn.linear_model import LogisticRegression\npd.options.mode.chained_assignment = None # default='warn'\n```\n\n\n```python\n\n```\n\n##### Read data\n\n\n```python\ndf = pd.read_csv('data/athlete_events.csv', index_col=0)\n```\n\n\n```python\ndf.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        NameSexAgeHeightWeightTeamNOCGamesYearSeasonCitySportEventMedal
                                        ID
                                        1A DijiangM24.0180.080.0ChinaCHN1992 Summer1992SummerBarcelonaBasketballBasketball Men's BasketballNaN
                                        2A LamusiM23.0170.060.0ChinaCHN2012 Summer2012SummerLondonJudoJudo Men's Extra-LightweightNaN
                                        3Gunnar Nielsen AabyM24.0NaNNaNDenmarkDEN1920 Summer1920SummerAntwerpenFootballFootball Men's FootballNaN
                                        4Edgar Lindenau AabyeM34.0NaNNaNDenmark/SwedenDEN1900 Summer1900SummerParisTug-Of-WarTug-Of-War Men's Tug-Of-WarGold
                                        5Christine Jacoba AaftinkF21.0185.082.0NetherlandsNED1988 Winter1988WinterCalgarySpeed SkatingSpeed Skating Women's 500 metresNaN
                                        \n
                                        \n\n\n\n\n```python\n\n```\n\n##### Basic explanatory analysis\n\n\n```python\ndf.describe()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        AgeHeightWeightYear
                                        count261642.000000210945.000000208241.000000271116.000000
                                        mean25.556898175.33897070.7023931978.378480
                                        std6.39356110.51846214.34802029.877632
                                        min10.000000127.00000025.0000001896.000000
                                        25%21.000000168.00000060.0000001960.000000
                                        50%24.000000175.00000070.0000001988.000000
                                        75%28.000000183.00000079.0000002002.000000
                                        max97.000000226.000000214.0000002016.000000
                                        \n
                                        \n\n\n\n\n```python\ndf.isna().sum()\n```\n\n\n\n\n Name 0\n Sex 0\n Age 9474\n Height 60171\n Weight 62875\n Team 0\n NOC 0\n Games 0\n Year 0\n Season 0\n City 0\n Sport 0\n Event 0\n Medal 231333\n dtype: int64\n\n\n\n\n```python\n# Show missing evolution of Age, Height and Weight\nM1.show_missing_evolution(df)\n```\n\n\n```python\n\n```\n\n##### Data Pre-processing\n\nWe will now apply all the data pre-processing steps in one single function. For transparency purposes we will however explain what this function does over the input dataframe:\n\n- Remove observations before 1960 based on the previous plot.\n- Split Athletics sport into sub-sports based on its events. This is due to the fact that athletics covers a very wide range of sport events which are probably not comparable.\n- Replace missing values with the average feature values at Sport, Sex and Year level. This can give us an approximation of the height, age and year of each athlete that has missing data.\n- We will only foocus on one olympic game season. For this analysis we have choosen the summer games since none of our team-mates is especially keen of winter sports.\n- We will only focus our analysis on sports that have competed during the last Olympic games. This is motivated due to the fact that our test data will be the last Olympic games and we want both data set (train and test) to have the same sports.\n- Construct one of the target variables (1 if an athlete has won a medal, 0 otherwise). This will be used to identify team sports as well.\n- Convert team sports it one athlete. This is done to reduce the number of medals won by sport, and that it does not depend on the number of team members.\n- Add column with the number of medals won per country at sport and sex level.\n\n\n```python\ndf_clean = M1.data_preprocessing(df)\n```\n\n\n```python\n\n```\n\n## 1. What probability do we have of winning a medal at a given sport?\n\n##### Generate Dictionary that maps country and sport to number of medals won\n\nFor later on, it will be very comfortable to have a dictionary where we can quickly access the number of medals won by a country, sport and sex.\n\n\n```python\nmedals_country_dict = M1.generate_mapping_table_country_sport_sex(df_clean)\n```\n\n\n```python\n\n```\n\n##### Standardize features\n\nWe have standardized our data since the main features we will be using throughout our analyis have different units and magnituude\n\n\n```python\ndf_standardized = M1.standardize_features(df_clean)\n```\n\n\n```python\n\n```\n\n##### Generate training and testing dataframe samples\n\nWe have decided to split the data as follows:\n\n- **Training Sample:** All observations from 1960 until 2012\n- **Testing Sample:** All observations from 2016\n\nThe motivation for this decision is that our model should be able to do predictions over the upcoming olympic games given a set of historical olympic game features. Any other random split would not have made too much sense from our point of view. \n\n\n```python\ndf_train, df_test = M1.generate_sample_split(df_standardized)\n```\n\n\n```python\n\n```\n\n##### Generate a set of additional columns to increase the possible degrees of our logistic regression fit\n\nIn order to train the logistic regression model (to predict the probability of winning a medal or not) we will also investigate how well the model performs if we consider higher degree orders of our classification model\n\n\n```python\ntemp_list = M1.extend_dataset_features([df_train, df_test])\ndf_train_extended = temp_list[0]\ndf_test_extended = temp_list[1]\n```\n\n\n```python\n\n```\n\n### Using all the different features (including the new ones we have just generated), which gives the best performances at sport and sex level? \n\n1. Compute the optimized set of theta values for the given input training set\n2. From all these potential theta values, which one gives the best results over the test data?\n\n##### 1. Compute the optimized set of theta values for the given input training set\n\n\n```python\nfeature_permutations = {'featureBag_1': ['Age', 'Height', 'Weight', 'Medals_per_country', 'Age^2', 'Height^2', 'Weight^2', \n 'Medals_per_country^2', 'Age*Height', 'Age*Weight', 'Age*Medals_per_country', \n 'Height*Weight', 'Height*Medals_per_country', 'Weight*Medals_per_country', 'Age^3', \n 'Height^3', 'Weight^3', 'Medals_per_country^3'], \n 'featureBag_2': ['Age', 'Height', 'Weight', 'Medals_per_country'], \n 'featureBag_3': ['Age', 'Height', 'Weight', 'Medals_per_country', 'Age^2', 'Height^2', 'Weight^2', \n 'Medals_per_country^2', 'Age*Height', 'Age*Weight', 'Age*Medals_per_country', \n 'Height*Weight', 'Height*Medals_per_country', 'Weight*Medals_per_country'], \n 'featureBag_4': ['Age^2', 'Height^2', 'Weight^2', 'Medals_per_country^2'], \n 'featureBag_5': ['Height', 'Weight', 'Medals_per_country'], \n 'featureBag_6': ['Age', 'Height', 'Weight', 'Age^2', 'Height^2', 'Weight^2'], \n 'featureBag_7': ['Age*Height', 'Age*Weight', 'Age*Medals_per_country', \n 'Height*Weight', 'Height*Medals_per_country', 'Weight*Medals_per_country'], \n 'featureBag_8': ['Age^3', 'Height^3', 'Weight^3', 'Medals_per_country^3'], \n 'featureBag_9': ['Medals_per_country', 'Medals_per_country^2', 'Medals_per_country^3']}\n```\n\n\n```python\nhyperparameter_thetas = M1.optimal_feature_combinations(df_train_extended, feature_permutations)\n```\n\n Model 0\n Features inspected: ['Age', 'Height', 'Weight', 'Medals_per_country', 'Age^2', 'Height^2', 'Weight^2', 'Medals_per_country^2', 'Age*Height', 'Age*Weight', 'Age*Medals_per_country', 'Height*Weight', 'Height*Medals_per_country', 'Weight*Medals_per_country', 'Age^3', 'Height^3', 'Weight^3', 'Medals_per_country^3']\n Sex: F\n Sex: M\n ====================\n Model 1\n Features inspected: ['Age', 'Height', 'Weight', 'Medals_per_country']\n Sex: F\n Sex: M\n ====================\n Model 2\n Features inspected: ['Age', 'Height', 'Weight', 'Medals_per_country', 'Age^2', 'Height^2', 'Weight^2', 'Medals_per_country^2', 'Age*Height', 'Age*Weight', 'Age*Medals_per_country', 'Height*Weight', 'Height*Medals_per_country', 'Weight*Medals_per_country']\n Sex: F\n Sex: M\n ====================\n Model 3\n Features inspected: ['Age^2', 'Height^2', 'Weight^2', 'Medals_per_country^2']\n Sex: F\n Sex: M\n ====================\n Model 4\n Features inspected: ['Height', 'Weight', 'Medals_per_country']\n Sex: F\n Sex: M\n ====================\n Model 5\n Features inspected: ['Age', 'Height', 'Weight', 'Age^2', 'Height^2', 'Weight^2']\n Sex: F\n Sex: M\n ====================\n Model 6\n Features inspected: ['Age*Height', 'Age*Weight', 'Age*Medals_per_country', 'Height*Weight', 'Height*Medals_per_country', 'Weight*Medals_per_country']\n Sex: F\n Sex: M\n ====================\n Model 7\n Features inspected: ['Age^3', 'Height^3', 'Weight^3', 'Medals_per_country^3']\n Sex: F\n Sex: M\n ====================\n Model 8\n Features inspected: ['Medals_per_country', 'Medals_per_country^2', 'Medals_per_country^3']\n Sex: F\n Sex: M\n ====================\n\n\n\n```python\nhyperparameter_thetas['featureBag_2']['M']\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        InterceptAgeHeightWeightMedals_per_country
                                        Archery-1.263956-0.338999-0.0978710.0909824.022280
                                        Athletics Endurance-2.873725-0.0601900.0260150.1432871.020414
                                        Athletics Sprints-3.316379-0.0309520.2641650.3660350.449914
                                        Athletics Throws-3.0086920.0605490.4614480.0533481.513966
                                        Badminton-1.069912-0.063780-0.031614-0.1922274.040050
                                        Basketball-3.326468-1.0184281.0229950.7643903.796012
                                        Beach Volleyball-3.0950740.2158850.2528130.5407100.606925
                                        Boxing-1.9550680.0642360.1057020.1019001.147090
                                        Canoeing-2.316584-0.003184-0.1501590.7092921.543597
                                        Cycling-3.037151-0.0647250.0124760.9001091.766603
                                        Diving-1.9409290.1255160.153418-0.3451672.660342
                                        Equestrianism-2.3921210.0939660.128429-0.3056292.866382
                                        Fencing-2.919638-0.0211630.2941580.1027741.467790
                                        Football-0.3629401.0560660.4101310.8132412.611478
                                        Gymnastics-3.707569-0.100438-0.3068580.1219071.244677
                                        Handball-2.4002300.6745720.6014620.5814651.492534
                                        Hockey-0.388756-0.0088190.4921130.3404573.016240
                                        Judo-1.949442-0.0755570.0246170.1052971.953922
                                        Modern Pentathlon-1.4823270.2669810.0722330.2974563.777742
                                        Rowing-2.1576990.4535720.0365680.4298432.081328
                                        Sailing-2.1733010.0541260.1690970.0785411.978570
                                        Shooting-2.545168-0.1413280.0528230.0760701.643497
                                        Swimming-3.715960-0.0137410.5671820.0984430.383196
                                        Table Tennis-1.434151-0.4122830.253649-0.0107134.474477
                                        Taekwondo-0.416472-0.2729440.862232-0.4277083.541313
                                        Tennis-2.098668-0.0621020.3183480.2579202.628818
                                        Trampolining-0.950576-0.4436200.3528580.0481020.907100
                                        Triathlon-2.472077-0.2716900.1441100.4163220.733031
                                        Volleyball-0.5946381.455339-0.4152850.7047361.901708
                                        Water Polo-1.657350-0.2771800.4904280.7799113.065504
                                        Weightlifting-2.287220-0.064086-0.3222420.2737382.136439
                                        Wrestling-2.2302060.0293240.0259200.0430090.854385
                                        \n
                                        \n\n\n\n##### 2. From all these potential theta values, which one gives the best results over the test data?\n\n##### We now have the theta values for each sport. Several observations can already be made from the previous:\n\n- Ideal theta values will be estimated at sport level (for some sports the country might have more influence, in other sport we might prefer giving more weight to the variable height, etc.)\n- In general, the variable that seems to have most weight on the prediction of winning a gold medal or not is Medals_per_country (this theta has the highest weight).\n- The other features which should intuitevely also be relevant (age, height, weight) seem to be less relevant. This is also an expected (although undesired) result since already the best althelets arrive at the olympics, regardless of their physical features. Another relevant and probably more insightful variable would be the number of training hours, age at which they started practicing the sport, etc.\n\n\n##### Apply model on test data and see how well is predicts medal winners (we will compute performance using f1-score, which considers both recall and precision)\n\n\n```python\ndf_female_performance, df_male_performance = M1.model_performance(df_test_extended, hyperparameter_thetas, \n print_evolution=True)\ndf_performance = {'F': df_female_performance,\n 'M': df_male_performance}\n```\n\n featureBag_1\n Sex: M\n Sex: F\n ===========================\n featureBag_2\n Sex: M\n Sex: F\n ===========================\n featureBag_3\n Sex: M\n Sex: F\n ===========================\n featureBag_4\n Sex: M\n Sex: F\n ===========================\n featureBag_5\n Sex: M\n Sex: F\n ===========================\n featureBag_6\n Sex: M\n Sex: F\n ===========================\n featureBag_7\n Sex: M\n Sex: F\n ===========================\n featureBag_8\n Sex: M\n Sex: F\n ===========================\n featureBag_9\n Sex: M\n Sex: F\n ===========================\n\n\n\n```python\ndf_performance['M']\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Model featuresArcheryAthletics EnduranceAthletics SprintsAthletics ThrowsBadmintonBasketballBeach VolleyballBoxingCanoeing...SwimmingTable TennisTaekwondoTennisTrampoliningTriathlonVolleyballWater PoloWeightliftingWrestling
                                        0featureBag_10.4444440.0666670.0555560.0000000.20.3333330.40.0952380.108108...0.0727270.6666670.1052630.0000000.0000000.3333330.6666670.3333330.1481480.039216
                                        1featureBag_20.2222220.0666670.0000000.1333330.20.6666670.80.0952380.054054...0.0727270.6666670.2105260.2857140.0000000.0000000.6666670.3333330.1481480.039216
                                        2featureBag_30.4444440.0000000.1111110.0000000.20.3333330.80.0476190.054054...0.0727270.6666670.1052630.0000000.3333330.6666670.6666670.3333330.1481480.000000
                                        3featureBag_40.2222220.0666670.0000000.1333330.20.3333330.80.0476190.108108...0.1090910.0000000.2105260.0000000.3333330.3333330.6666670.3333330.0740740.000000
                                        4featureBag_50.2222220.0666670.0000000.1333330.20.6666670.80.0952380.054054...0.1090910.6666670.2105260.2857140.0000000.0000001.0000000.3333330.1481480.039216
                                        5featureBag_60.2222220.0000000.0000000.0000000.00.3333330.80.0952380.054054...0.0000000.2222220.2105260.0000000.3333330.0000000.6666670.3333330.0740740.000000
                                        6featureBag_70.0000000.0666670.0000000.1333330.00.6666670.80.0952380.054054...0.0363640.0000000.2105260.2857140.3333330.0000000.6666670.3333330.1481480.000000
                                        7featureBag_80.0000000.0666670.0000000.1333330.40.3333330.80.0952380.108108...0.1090910.0000000.2105260.0000000.3333330.0000000.6666670.3333330.1481480.039216
                                        8featureBag_90.4444440.0000000.0555560.0000000.20.6666670.40.0952380.108108...0.1090910.6666670.3157890.0000000.6666670.6666671.0000000.3333330.0740740.000000
                                        \n

                                        9 rows \u00d7 33 columns

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                                        \n\n\n\n\n```python\n\n```\n\n##### Evaluate Model over a set of input variables \n\n\n```python\nsex = 'M'\ncountry = 'ESP'\nsport = 'Basketball'\nage = 20\nheight = 200\nweight = 100\n\nM1.compute_probability_medal(sex, country, sport, age, height, weight, medals_country_dict, df_performance,\n hyperparameter_thetas, df_clean=df_clean)\n```\n\n\n\n\n 0.6429739404621343\n\n\n\n\n```python\n\n```\n\n##### Inspect features against target variable\n\n\n```python\nM1.plot_sport_features('Swimming', df_test, x_feature='Age', y_feature='Height')\n```\n\n\n```python\n\n```\n\n## Which sport should we participate in the Olympics in?\n\nWe will now start looking into the models that can tell us what sport we are best suited for a given athlete. For this we have investigated the following models using a 5-fold cross validation approach, allowing us to decide which of the hyperparameters of each model would be best suited over the training sample (metric to choose which hyperparameter is best is the F1-score). From these best models we have then evaluated them over the test data sample (2016 Olympic games) to see which of the models would outperform the rest.\n\n1. **Decision Tree Classifier:** Basic Classifier which sets up a set of decisions (controlled by the depth of the tree) which converts input features (branches) into categories or target values (leaves) through a set of boolean if statements. One of the advantages os such models is that it is easily reproducible and therefore can be validated easily. \n2. **Random Forest Classifier:** The basic principal is very similar to the decision tree classifier. The main difference is that the random forest classifies based on some set of bootstraps over the initial data set. This allows our model to be more flexible when predicting unseen data since it trains over more data sets than the basic decision tree classifier.\n3. **Logisitic Regression:** The main difference between our model and this logistic regression is that this model will be used to classify between multiple categories. In the previous model we were classifying a athelete between winning or not a medal. The method used in this case is the one-vs-rest approach, i.e. basically the logistic regression classifies a given catgegory into 1 and all the rest into 0 and generates a normal logisit regression over these binary categories.\n4. **K-Nearest Neighbours:** One of the most popular classification models used throughout the data science community. The principal is very straight forward and easy: Given a set of clusters (choosen by the user if the \"real\" number of clusters is not known, or inferred by the data, in our case the number of groups should ideally be equal to the number of sport types), look for groups over which a specific distance measure is minimized.\n5. **Linear SVC:** Algorithm to classify observations into categories according to some features by maximizing the margin between two different category examples.\n\nThese models will be evaluated similarly to the previous model, i.e. using the following metrics:\n\n1. **Accuracy:** Percentage of atheletes correctly classified\n\n\\begin{align}\nAccuracy = \\frac{True Positives + True Negatives}{True Positives + True Negatives + False Positives + False Negatives} \n\\end{align}\n\n2. **Weighted precision:** Percentage of athelets correctly classified out of all positive results\n\n\\begin{align}\nPrecision = \\frac{True Positives}{True Positives + False Positives} \n\\end{align}\n\n3. **Weighted recall:** Percentage of athlets correctly classified from all the athletes that have been assigned to a specific class\n\n\\begin{align}\nRecall = \\frac{True Positives}{True Positives + False Negatives} \n\\end{align}\n\n4. **Weighted F1:** Similarly as in the previous model selection, this will be the utlimate metric over which we will decide which model is the best. This metric combines the recall and precision, and for this reason is sometimes considered to be the most accurate metric to use.\n\n\\begin{align}\nF1 = 2 \\cdot \\frac{recall \\cdot precision}{recall + precision}\n\\end{align}\n\n\n```python\n\n```\n\n##### Convert sports into integer labels\n\n\n```python\nle = LabelEncoder()\ndf_standardized['Sport_label'] = le.fit_transform(df_standardized.Sport)\n```\n\n\n```python\n\n```\n\n##### Generate training and testing samples (we continue following the same approach as before):\n\nAs in the previous models, we will train our model over all the olympic game history and then apply the best hyper-parameters over the unseen 2016 Olympic Games\n\n\n```python\ndf_train_Female_X, df_train_Male_X, df_train_Female_Y, df_train_Male_Y, \\\ndf_test_Female_X, df_test_Male_X, df_test_Female_Y, df_test_Male_Y = M1.generate_sample_split_advanced(df_standardized)\n\ndata_sets = {'F': {'Train': [df_train_Female_X, df_train_Female_Y],\n 'Test': [df_test_Female_X, df_test_Female_Y]},\n 'M': {'Train': [df_train_Male_X, df_train_Male_Y],\n 'Test': [df_test_Male_X, df_test_Male_Y]}}\n```\n\n\n```python\n## Count sport labels\n#train_counts_Female = df_train_Female_Y.value_counts()\n#train_counts_Male = df_train_Male_Y.value_counts()\n#\n#test_counts_Female = df_test_Female_Y.value_counts()\n#test_counts_Male = df_test_Male_Y.value_counts()\n#\n## Assign labels to actual sport names\n#train_counts_Female.index = le.inverse_transform(train_counts_Female.index)\n#train_counts_Male.index = le.inverse_transform(train_counts_Male.index)\n#\n#test_counts_Female.index = le.inverse_transform(test_counts_Female.index)\n#test_counts_Male.index = le.inverse_transform(test_counts_Male.index)\n```\n\n\n```python\n# Define scoring dict to make loop easier to read afterwards\nscoring = {\n 'accuracy': 'accuracy',\n 'weighted_precision': make_scorer(precision_score, average='weighted'),\n 'weighted_recall': make_scorer(recall_score, average='weighted'),\n 'weighted_F1': make_scorer(f1_score, average='weighted')\n}\n```\n\n\n```python\n# Define model and hyperparameter list to make loop easier to read afterwards\n\nclassifiers = [(\"DT\", DecisionTreeClassifier(), \n {\"max_depth\": [3, 5, 10, None]}),\n \n (\"RF\", RandomForestClassifier(),\n {\"max_depth\": [3, 5, 10, None]}),\n \n (\"LOGREG\", LogisticRegression(),\n {\"C\": np.logspace(-5, 5, 5 + 5 + 1, base=10)}), \n \n (\"KNN\", KNeighborsClassifier(), \n {\"n_neighbors_male\": {\"n_neighbors\": np.append(np.logspace(0, 3, 3 + 0 + 1, base=10).astype(\"int\"),\n np.sqrt(len(df_train_Male_X)).astype(\"int\"))},\n \"n_neighbors_female\": {\"n_neighbors\": np.append(np.logspace(0, 3, 3 + 0 + 1, base=10).astype(\"int\"),\n np.sqrt(len(df_train_Female_X)).astype(\"int\"))}}), \n #(\"SVM\", LinearSVC(), \n # {\"C\": np.logspace(-5, 5, 5 + 5 + 1, base=10)})\n ]\n\n\n```\n\n\n```python\nmodels_Female, results_Female, models_Male, results_Male = M1.train_models(data_sets, classifiers, scoring)\n```\n\n F\n ===================\n Model: DT\n Fitting 5 folds for each of 4 candidates, totalling 20 fits\n\n\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 4 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 20 out of 20 | elapsed: 1.3s remaining: 0.0s\n [Parallel(n_jobs=-1)]: Done 20 out of 20 | elapsed: 1.3s finished\n\n\n Model: RF\n Fitting 5 folds for each of 4 candidates, totalling 20 fits\n\n\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 4 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 20 out of 20 | elapsed: 41.5s remaining: 0.0s\n [Parallel(n_jobs=-1)]: Done 20 out of 20 | elapsed: 41.5s finished\n\n\n Model: LOGREG\n Fitting 5 folds for each of 11 candidates, totalling 55 fits\n\n\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 4 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 24 tasks | elapsed: 1.7min\n [Parallel(n_jobs=-1)]: Done 55 out of 55 | elapsed: 4.0min finished\n C:\\Users\\juanm\\Anaconda3\\envs\\data_science_general\\lib\\site-packages\\sklearn\\linear_model\\_logistic.py:764: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG)\n\n\n Model: KNN\n Fitting 5 folds for each of 5 candidates, totalling 25 fits\n\n\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 4 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 25 out of 25 | elapsed: 15.5s finished\n\n\n M\n ===================\n Model: DT\n Fitting 5 folds for each of 4 candidates, totalling 20 fits\n\n\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 4 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 20 out of 20 | elapsed: 3.1s remaining: 0.0s\n [Parallel(n_jobs=-1)]: Done 20 out of 20 | elapsed: 3.1s finished\n\n\n Model: RF\n Fitting 5 folds for each of 4 candidates, totalling 20 fits\n\n\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 4 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 20 out of 20 | elapsed: 1.8min finished\n [Parallel(n_jobs=-1)]: Done 20 out of 20 | elapsed: 1.8min remaining: 0.0s\n\n\n Model: LOGREG\n Fitting 5 folds for each of 11 candidates, totalling 55 fits\n\n\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 4 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 24 tasks | elapsed: 3.3min\n [Parallel(n_jobs=-1)]: Done 55 out of 55 | elapsed: 7.5min finished\n C:\\Users\\juanm\\Anaconda3\\envs\\data_science_general\\lib\\site-packages\\sklearn\\linear_model\\_logistic.py:764: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG)\n\n\n Model: KNN\n Fitting 5 folds for each of 5 candidates, totalling 25 fits\n\n\n [Parallel(n_jobs=-1)]: Using backend LokyBackend with 4 concurrent workers.\n [Parallel(n_jobs=-1)]: Done 25 out of 25 | elapsed: 40.0s finished\n\n\n\n```python\nresults_Male\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        paramsaccuracyweighted_precisionweighted_recallweighted_F1Modeldummy_accuracydummy_weighted_precisiondummy_weighted_recalldummy_weighted_F1
                                        0{'max_depth': 3}0.2632600.1819400.2632600.198626DT0.0310350.0666090.0310350.039431
                                        1{'max_depth': 5}0.2926690.2134660.2926690.220856DT0.0310350.0666090.0310350.039431
                                        2{'max_depth': 10}0.3297720.3252750.3297720.304810DT0.0310350.0666090.0310350.039431
                                        3{'max_depth': None}0.2665840.3402100.2665840.281883DT0.0310350.0666090.0310350.039431
                                        4{'max_depth': 3}0.2901000.2148100.2901000.198988RF0.0310350.0666090.0310350.039431
                                        5{'max_depth': 5}0.3336770.2920480.3336770.257399RF0.0310350.0666090.0310350.039431
                                        6{'max_depth': 10}0.3834950.4053800.3834950.343647RF0.0310350.0666090.0310350.039431
                                        7{'max_depth': None}0.3460300.4246260.3460300.353108RF0.0310350.0666090.0310350.039431
                                        8{'C': 1e-05}0.1217590.0783990.1217590.034903LOGREG0.0310350.0666090.0310350.039431
                                        9{'C': 0.0001}0.2516620.1392220.2516620.134024LOGREG0.0310350.0666090.0310350.039431
                                        10{'C': 0.001}0.2742630.1481930.2742630.170356LOGREG0.0310350.0666090.0310350.039431
                                        11{'C': 0.01}0.2827260.1812150.2827260.190921LOGREG0.0310350.0666090.0310350.039431
                                        12{'C': 0.1}0.2809410.1606810.2809410.187746LOGREG0.0310350.0666090.0310350.039431
                                        13{'C': 1.0}0.2840760.1763800.2840760.195517LOGREG0.0310350.0666090.0310350.039431
                                        14{'C': 10.0}0.2830740.1661500.2830740.190986LOGREG0.0310350.0666090.0310350.039431
                                        15{'C': 100.0}0.2820150.1661190.2820150.192997LOGREG0.0310350.0666090.0310350.039431
                                        16{'C': 1000.0}0.2820290.1649650.2820290.191891LOGREG0.0310350.0666090.0310350.039431
                                        17{'C': 10000.0}0.2817100.1648490.2817100.190716LOGREG0.0310350.0666090.0310350.039431
                                        18{'C': 100000.0}0.2820580.1683260.2820580.191111LOGREG0.0310350.0666090.0310350.039431
                                        19{'n_neighbors': 1}0.1012920.2056110.1012920.106831KNN0.0310350.0666090.0310350.039431
                                        20{'n_neighbors': 10}0.1045430.1770150.1045430.101749KNN0.0310350.0666090.0310350.039431
                                        21{'n_neighbors': 100}0.0885910.1373390.0885910.076270KNN0.0310350.0666090.0310350.039431
                                        22{'n_neighbors': 1000}0.0831470.0508240.0831470.043962KNN0.0310350.0666090.0310350.039431
                                        23{'n_neighbors': 262}0.0769050.1126480.0769050.059310KNN0.0310350.0666090.0310350.039431
                                        \n
                                        \n\n\n\n##### Evaluate best hyper-parameters\n\nWe now have run all the models and computed their metric values using a 5-fold cross validation approach. As we can observe the results are not ideal (most models are not able to classify corrcetly). However we are always able to improve the results of the dummy classification which randomly assigns sports to athlets without looking into aby of their features. \n\nIt is important to highlight that an accuracy of ~30% is already pretty good since we are trying to classify athletes to some sport (we have more than 40 sports to classify between) solely based on a few features.\n\n\n```python\nbest_models_Female = results_Female.loc[results_Female.groupby('Model')['weighted_F1'].idxmax()]\nbest_models_Male = results_Male.loc[results_Male.groupby('Model')['weighted_F1'].idxmax()]\n```\n\n\n```python\nbest_models_Female\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        paramsaccuracyweighted_precisionweighted_recallweighted_F1Modeldummy_accuracydummy_weighted_precisiondummy_weighted_recalldummy_weighted_F1
                                        2{'max_depth': 10}0.3494740.3595940.3494740.309587DT0.0294190.0999660.0294190.038228
                                        19{'n_neighbors': 1}0.1309350.2304870.1309350.132123KNN0.0294190.0999660.0294190.038228
                                        13{'C': 1.0}0.3476460.2186270.3476460.246400LOGREG0.0294190.0999660.0294190.038228
                                        7{'max_depth': None}0.3629520.4268470.3629520.355706RF0.0294190.0999660.0294190.038228
                                        \n
                                        \n\n\n\n\n```python\nbest_models_Male\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        paramsaccuracyweighted_precisionweighted_recallweighted_F1Modeldummy_accuracydummy_weighted_precisiondummy_weighted_recalldummy_weighted_F1
                                        2{'max_depth': 10}0.3297720.3252750.3297720.304810DT0.0310350.0666090.0310350.039431
                                        19{'n_neighbors': 1}0.1012920.2056110.1012920.106831KNN0.0310350.0666090.0310350.039431
                                        13{'C': 1.0}0.2840760.1763800.2840760.195517LOGREG0.0310350.0666090.0310350.039431
                                        7{'max_depth': None}0.3460300.4246260.3460300.353108RF0.0310350.0666090.0310350.039431
                                        \n
                                        \n\n\n\n\n```python\n\n```\n\n##### Let's now see how our champion models predict based on our test data at sport and sex level \n\n\n```python\nmodels = {'F': models_Female, \n 'M': models_Male}\n```\n\n\n```python\nevaluation_df = M1.evaluate_models(data_sets, models, le)\n```\n\n\n```python\nevaluation_df\n```\n\n C:\\Users\\juanm\\Anaconda3\\envs\\data_science_general\\lib\\site-packages\\ipykernel\\ipkernel.py:287: DeprecationWarning: `should_run_async` will not call `transform_cell` automatically in the future. Please pass the result to `transformed_cell` argument and any exception that happen during thetransform in `preprocessing_exc_tuple` in IPython 7.17 and above.\n and should_run_async(code)\n\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        precisionrecallf1-scoresupportSexModel
                                        Archery0.0000000.0000000.00000076.0FDT
                                        Athletics Endurance0.4380280.6775600.532079459.0FDT
                                        Athletics Sprints0.2057970.5546880.300211512.0FDT
                                        Athletics Throws0.4850000.7293230.582583133.0FDT
                                        Badminton0.0000000.0000000.00000059.0FDT
                                        .....................
                                        Volleyball0.1052630.1666670.12903212.0MKNN
                                        Water Polo0.1000000.0833330.09090912.0MKNN
                                        Weightlifting0.3631280.4276320.392749152.0MKNN
                                        Wrestling0.2163120.2606840.236434234.0MKNN
                                        overall0.2557440.2401920.243197NaNMKNN
                                        \n

                                        288 rows \u00d7 6 columns

                                        \n
                                        \n\n\n\nFrom the previous dataframe we can see that depending on the sport and sex, the evaulation metrics can behave better or worse. The champion model is evaluated over the overall performance of the model over all sport types\n\n\n```python\n\n```\n\n##### Decide which is the champion model \n\n\n```python\nevaluation_df.loc['overall'][['precision', 'recall', 'f1-score', 'Sex', 'Model']]\n```\n\n C:\\Users\\juanm\\Anaconda3\\envs\\data_science_general\\lib\\site-packages\\ipykernel\\ipkernel.py:287: DeprecationWarning: `should_run_async` will not call `transform_cell` automatically in the future. Please pass the result to `transformed_cell` argument and any exception that happen during thetransform in `preprocessing_exc_tuple` in IPython 7.17 and above.\n and should_run_async(code)\n\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        precisionrecallf1-scoreSexModel
                                        overall0.2740930.3414340.283723FDT
                                        overall0.3912140.4133660.397887FRF
                                        overall0.1446360.2364520.148133FLOGREG
                                        overall0.2593700.2420410.247230FKNN
                                        overall0.2987680.3407690.303445MDT
                                        overall0.4016940.4217310.407771MRF
                                        overall0.1464880.2707690.184091MLOGREG
                                        overall0.2557440.2401920.243197MKNN
                                        \n
                                        \n\n\n\n### From the previous dataframe we can conclude that the best model to predict which sport an athlete participated in is the Random Forest, for both male and female\n\nApply such models over our FDS team members we would get the following results:\n\n1. Apply random forest to obtain the ideal sport (without setting a fixed max_depth of the branches)\n2. Applying a logisitic regression to compute the probability of winning a medal\n\n\n```python\ndf_us = pd.DataFrame(index=['Tansel', 'Alessandro', 'Leonardo', 'Juan'], \n columns=['Year', 'Age', 'Height', 'Weight', 'Country', 'Sex'])\n\ndf_us['Year'] = [2016, 2016, 2016, 2016]\ndf_us['Age'] = [23, 26, 24, 25]\ndf_us['Height'] = [172, 173, 165, 180]\ndf_us['Weight'] = [57, 69, 55, 80]\ndf_us['Country'] = ['TUR', 'ITA', 'ITA', 'ESP']\ndf_us['Sex'] = ['F', 'M', 'M', 'M']\n```\n\n\n```python\ndf = M1.predict_ideal_sport_and_prob(df_us, models, le, medals_country_dict, df_clean, df_performance, \n hyperparameter_thetas)\ndf\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        YearAgeHeightWeightCountrySexSport_predictP_Medal
                                        Tansel20162317257TURFTaekwondo0.321129
                                        Alessandro20162617369ITAMSailing0.111271
                                        Leonardo20162416555ITAMWrestling0.101105
                                        Juan20162518080ESPMAthletics Sprints0.039841
                                        \n
                                        \n\n\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "9ebb3ec46089f72a62965565aa3a039e96dce904", "size": 404686, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Olympics - Final Project/Model1.ipynb", "max_stars_repo_name": "tanselsimsek/Fundamental-of-Data-Science", "max_stars_repo_head_hexsha": "8330114487033d617dc5d57e19151da28d15122d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Olympics - Final Project/Model1.ipynb", "max_issues_repo_name": "tanselsimsek/Fundamental-of-Data-Science", "max_issues_repo_head_hexsha": "8330114487033d617dc5d57e19151da28d15122d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Olympics - Final Project/Model1.ipynb", "max_forks_repo_name": "tanselsimsek/Fundamental-of-Data-Science", "max_forks_repo_head_hexsha": "8330114487033d617dc5d57e19151da28d15122d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 126.3853841349, "max_line_length": 239676, "alphanum_fraction": 0.8117577579, "converted": true, "num_tokens": 19964, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.6688802537704063, "lm_q1q2_score": 0.4309094527118699}} {"text": "# The *grmpy* package \nThis notebook demonstrates the current capabilities of the *grmpy* package. *grmpy* is an open source package for the programming language python that enables researchers to simulate datasets andestimate parameters using already existing data within the structure of the generalized Roy model. Currently the package serves as a teaching tool for a course on the econometrics of policy evaluation at the University of Bonn. The corresponding lecture materials can be found on [GitHub](https://github.com/HumanCapitalAnalysis/econometrics). Morover it is thought of as a promotion for the conceptual framework as well as a showcase for basic software engineering practices.\n\nFor a more detailed overview on the economic background as well as the installation routine feel free to take a look on the [online documentation](https://grmpy.readthedocs.io/en/develop/).\n\nThe notebook itself is divided in three parts. Firstly we provide a basic outline on how to use the package and introduce the core features. Next we will show that the results obtained by the package's estimation process withstand a critical examination by comparing its performance in the presence of essential heterogeneity with several other estimation approaches like Ordinary Least Squares and Instrumental variables. We conclude by conducting a replication of results from \n\nCarneiro, P., Heckman, J. J., & Vytlacil, E. J. (2011). [Estimating marginal returns to education.](https://pubs.aeaweb.org/doi/pdfplus/10.1257/aer.101.6.2754)\n*American Economic Review, 101*(6), 2754-81.\n\n\n# The Framework\n\nAs mentioned before the package makes use of the normal-linear-in-parameters version generalized Roy model. In addition we assume that the unobservable terms $\\{U_1, U_0, V\\}$ are normally distributed according to the covariance matrix $\\Sigma$. The following set of equations characterize the underlying model:\n\n\\begin{align*}\n &\\textbf{Potential Outcomes} & & \\textbf{Choice} &\\\\\n & Y_1 = \\beta_1 X + U_{1} & &D_i = \\mathbf{1}\\{\\Phi(\\gamma Z) > u_D\\} &\\\\\n & Y_0 = \\beta_0 X + U_{0} & & \\text{with $u_D = \\Phi(V)$} &\\\\\n&&&&\\\\\n&\\textbf{Distributional Characteristics}&&&\\\\\n&\\{U_{1}, U_{0}, V\\} \\sim \\mathcal{N}\\left(0, \\Sigma\\right)&&\\Sigma = \\begin{bmatrix}\n \\sigma_1^{2} & \\sigma_{1,0} & \\sigma_{1,V} \\\\\n \\sigma_{1,0} & \\sigma_0^{2} & \\sigma_{0,V} \\\\\n \\sigma_{1,V} & \\sigma_{0,V} & \\sigma_V^{2} \\\\\n \\end{bmatrix}&\\\\\n&&&&\\\\\n& \\textbf{Observed Outcome} &&&\\\\\n& Y = D Y_1 + (1-D) Y_0 &&&\n\\end{align*}\n\n\n# Preliminaries\n\nThe first step consists of some style and import improving settings.\n\n\n```python\nfrom IPython.core.display import display, HTML\ndisplay(HTML(\"\"))\n%load_ext autoreload\n%autoreload 2\n```\n\n\n\n\n\nBefore we can start with the application examples we have to import several libaries that we rely on during this presentation.\n\n\n```python\nimport matplotlib.pyplot as plt\nimport statsmodels.api as sm\nimport pandas as pd\nimport numpy as np\nimport json\n\nimport grmpy\n```\n\nIn addition we will import some custom functions for plotting several figures of interest as well as some data management purposes. The code is provided in the auxiliary.py file.\n\n\n```python\nfrom auxiliary import plot_joint_distribution_unobservables\nfrom auxiliary import plot_joint_distribution_potential\nfrom auxiliary import plot_marginal_effects\nfrom auxiliary import process_data\nfrom auxiliary import plot_est_mte\nfrom auxiliary import monte_carlo\nfrom auxiliary import effects\n```\n\n# Part I - Introduction\n\nThis part provides some basic information about the general application of the package's core methods. \n\n## The initialization file\n\nCurrently the package has two core features. The first one is the simulation process. It enables users to simulate data sets along a pre-specified parameterization. The data generating process follows the structure of the previously introduced parametric Roy model framework. Therefore users specify their desired parametrization in an initialization file which allows altering any aspects of the model according to their preferences. The ability to estimate parameters on already existing data sets is the second core feature. As before the initialization file is the starting point for the estimation process. \n\nThe initialization file is structured as follows:\n\n* **Simulation**: The simulation section contains information for the estimation process. For instance users are able to change the number of individuals for which the program runs a simulation through setting the agents option to a specific value.\n\n\n* **Estimation**: Accordingly information regarding the estimation process are stored in the estimation section. Here, users can set the type of optimizer which should be used for running the process as well as the maximum number of iterations that the process is allowed to perform. In addition the flags *dependent* and *indicator* determine which variable of the input data frame is seen as the dependent variable and the treatment indicator variable, respectively.\n\n\n* **Treated, Untreated, Choice**: These sections are essential for both methods. They contain the parameterization which is used for the simulation process. Additionally the second column indicates the variable labels well as for the simulated data set. This column is also essential for the estimation process because it will only include variables that are pre-specified in a specific section for the following optimization.\n\n\n* **Dist**: The Dist section determines the distributional characteristics of the unobservable variables. Therefore the section reports the upper triangle of the covariance matrix of $(U_1, U_0,V)$.\n\n\n* **SCIPY-BFGS/SCIPY-POWELL**: The SCIPY-related sections contain options for the relevant optimization algorithms.\n\n\n```python\n%%file files/tutorial.grmpy.yml\n---\nSIMULATION:\n agents: 10000\n seed: 2356\n source: data\nESTIMATION:\n file: data.grmpy.txt\n output_file: output/est.grmpy.info\n optimizer: SCIPY-BFGS\n start: auto\n maxiter: 6383\n agents: 165\n dependent: Y\n indicator: D\n comparison: 1\n print_output: 0\nTREATED:\n params:\n - 1.0\n - 0.555\n order:\n - const\n - X2\nUNTREATED:\n params:\n - 0.5\n - 0.25\n order:\n - const\n - X2\nCHOICE:\n params:\n - 0.378\n - -0.39\n order:\n - const\n - X3\nDIST:\n params:\n - 0.1\n - 0.0\n - 0.0\n - 0.1\n - 0.0\n - 1.0\nVARTYPES:\n const: nonbinary\n X2: nonbinary\n X3: nonbinary\nSCIPY-BFGS:\n gtol: 1.0e-05\n eps: 1.4901161193847655e-08\nSCIPY-POWELL:\n xtol: 9.147777614048603e-05\n ftol: 9.749582129043358e-05\n```\n\n Overwriting files/tutorial.grmpy.yml\n\n\n## The Simulation Process\n\nFor simulating a dataset we only have to provide the package with the related inititalization file.\n\n\n```python\ndata = grmpy.simulate('files/tutorial.grmpy.yml')\ndata.head(20)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        constX2X3U1U0VY1Y0DY
                                        01.0-0.0147100.0234390.0026760.0614520.7673100.9945120.5577750.00.557775
                                        11.01.941447-1.3758910.0870870.066451-0.5679792.1645901.0518131.02.164590
                                        21.0-0.2206681.5554530.0785570.008140-0.3078280.9560870.4529731.00.956087
                                        31.00.5318990.192161-0.0055740.055052-1.1292021.2896300.6880271.01.289630
                                        41.0-0.2951611.663835-0.1115340.077755-0.0247870.7246520.5039650.00.503965
                                        51.02.0306130.845409-0.096886-0.0861800.1177102.0301040.9214730.00.921473
                                        61.0-0.189934-0.8306590.019401-0.045266-0.0371560.9139880.4072501.00.913988
                                        71.00.5908191.731056-0.0870720.0326170.7197591.2408330.6803220.00.680322
                                        81.00.7904661.1332870.049938-0.2009240.7498261.4886470.4966920.00.496692
                                        91.0-1.3784290.0200670.1516130.015832-1.1195190.3865850.1712251.00.386585
                                        101.0-0.089817-0.2790000.062335-0.0534590.0492701.0124870.4240871.01.012487
                                        111.00.2120422.2170550.009479-0.0247190.3913821.1271620.5282910.00.528291
                                        121.00.140421-0.9498040.0500720.0028390.6585551.1280050.5379451.01.128005
                                        131.0-0.1524880.9977280.191663-0.072238-0.6080021.1070320.3896391.01.107032
                                        141.0-0.0081470.812770-0.0609110.0251550.7325110.9345670.5231180.00.523118
                                        151.00.962076-1.2100270.2604560.008918-1.2091361.7944080.7494371.01.794408
                                        161.0-2.2129430.6131030.0021010.0841700.483983-0.2260820.0309340.00.030934
                                        171.0-0.243372-0.2568260.065807-0.0075840.9065690.9307360.4315730.00.431573
                                        181.00.882084-0.6199580.059393-0.044658-1.7198711.5489500.6758631.01.548950
                                        191.00.3635491.1374130.1502810.109781-0.4213981.3520510.7006681.01.352051
                                        \n
                                        \n\n\n\nThe simulation process generates an info file that provides additional information about the distribution of outcomes and effects, respectively. Additionally it contains the criterion function value, information about the corresponding marginal treatment effects as well as the paramterization.\n\n\n```python\n!cat \"data.grmpy.info\"\n```\n\n \r\n \r\n Number of Observations \r\n \r\n Count\r\n \r\n All 10000\r\n Treated 6391\r\n Untreated 3609\r\n \r\n \r\n Distribution of Outcomes\r\n \r\n Mean Std-Dev. 25% 50% 75%\r\n \r\n All 0.8196 0.5328 0.4401 0.7467 1.1652\r\n Treated 1.0018 0.5580 0.6305 1.0059 1.3721\r\n Untreated 0.4968 0.2689 0.3218 0.5001 0.6782\r\n \r\n \r\n Distribution of Effects\r\n \r\n Mean Std-Dev. 25% 50% 75%\r\n \r\n All 0.4998 0.3340 0.2739 0.4974 0.7208\r\n Treated 0.5007 0.3354 0.2731 0.5016 0.7233\r\n Untreated 0.4982 0.3315 0.2761 0.4905 0.7177\r\n \r\n \r\n Criterion Function \r\n \r\n Value -0.270418786930 \r\n \r\n \r\n \r\n Marginal Treatment Effect \r\n \r\n Quantile Value\r\n \r\n 1% 0.5006\r\n 5% 0.5006\r\n 10% 0.5006\r\n 15% 0.5006\r\n 20% 0.5006\r\n 25% 0.5006\r\n 30% 0.5006\r\n 35% 0.5006\r\n 40% 0.5006\r\n 45% 0.5006\r\n 50% 0.5006\r\n 55% 0.5006\r\n 60% 0.5006\r\n 65% 0.5006\r\n 70% 0.5006\r\n 75% 0.5006\r\n 80% 0.5006\r\n 85% 0.5006\r\n 90% 0.5006\r\n 95% 0.5006\r\n 99% 0.5006\r\n \r\n \r\n Parameterization \r\n \r\n Section Identifier Coef\r\n \r\n TREATED \r\n const 1.0000\r\n X2 0.5550\r\n \r\n UNTREATED \r\n const 0.5000\r\n X2 0.2500\r\n \r\n CHOICE \r\n const 0.3780\r\n X3 -0.3900\r\n \r\n DIST \r\n sigma1 0.1000\r\n sigma10 0.0000\r\n sigma1v 0.0000\r\n sigma0 0.1000\r\n sigma0v 0.0000\r\n sigmaV 1.0000\r\n\n\nThe generated data set is specififed so that the treatment selection process is not affected by essential heterogeneity. Therefore the conventional effects are nearly identical. \n\n\n```python\neffects(data)\n```\n\n### Essential Heterogeneity\nFor providing an example on how essential heterogeneity biases the results obtained by naive comparison between treated and untreated individuals, we alter the initialization file. Specifically we will introduce correlation between the unobservable terms. This leads to the situation where individuals select into treatment based on unobservable gains. Therefore the treatment decision $D$ is no longer independent from the outcomes $Y_1$ and $Y_0$.\n\n\\begin{align}\nY_1,Y_0\\;\\; {\\perp\\!\\!\\!\\!\\!\\!\\diagup\\!\\!\\!\\!\\!\\!\\!\\perp} \\;\\; D\n\\end{align}\n\nUnlike in the absence of essential heterogeneity individuals who select themselves into treatment differ from\nindividuals who do not with respect to their unobservable characteristics. From this follows that\n\n\\begin{align}\nB^{ATE} \\neq B^{TT} \\neq B^{TUT}\n\\end{align}\n\n\n```python\n%%file files/tutorial_eh.grmpy.yml\n---\nSIMULATION:\n seed: 2356\n agents: 10000\n source: data_eh\nESTIMATION:\n file: data_eh.grmpy.txt\n start: auto\n agents: 165\n optimizer: SCIPY-BFGS\n maxiter: 6383\n dependent: Y\n indicator: D\n output_file: output/est.grmpy.info\n comparison: 1\n print_output: 0\n\nTREATED:\n params:\n - 1.0\n - 0.555\n order:\n - const\n - X2\nUNTREATED:\n params:\n - 0.5\n - 0.25\n order:\n - const\n - X2\nCHOICE:\n params:\n - 0.378\n - -0.39\n order:\n - const\n - X3\nDIST:\n params:\n - 0.1\n - 0.0\n - 0.0524\n - 0.1\n - -0.0216\n - 1.0\nVARTYPES:\n const: nonbinary\n X2: nonbinary\n X3: nonbinary\nSCIPY-BFGS:\n gtol: 1.0e-05\n eps: 1.4901161193847655e-08\nSCIPY-POWELL:\n xtol: 9.147777614048603e-05\n ftol: 9.749582129043358e-05\n```\n\n Overwriting files/tutorial_eh.grmpy.yml\n\n\n\n```python\ndata_eh = grmpy.simulate('files/tutorial_eh.grmpy.yml')\ndata_eh.head(20)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        constX2X3U1U0VY1Y0DY
                                        01.0-0.0147100.023439-0.0150290.072663-0.7674270.9768070.5689861.00.976807
                                        11.01.941447-1.3758910.1214090.0218290.5638922.1989121.0071911.02.198912
                                        21.0-0.2206681.5554530.078986-0.0237540.3041430.9565150.4210790.00.421079
                                        31.00.5318990.1921610.0763210.0280851.1294501.3715250.6610600.00.661060
                                        41.0-0.2951611.663835-0.0537390.1062550.0300130.7824460.5324640.00.532464
                                        51.02.0306130.845409-0.112613-0.046794-0.1131682.0143770.9608591.02.014377
                                        61.0-0.189934-0.830659-0.000561-0.0487300.0362470.8940260.4037861.00.894026
                                        71.00.5908191.731056-0.0916490.073059-0.7156711.2362550.7207641.01.236255
                                        81.00.7904661.133287-0.078217-0.185083-0.7521581.3604920.5125331.01.360492
                                        91.0-1.3784290.0200670.180205-0.0571581.1124020.4151770.0982340.00.098234
                                        101.0-0.089817-0.2790000.024354-0.067862-0.0521910.9745060.4096841.00.974506
                                        111.00.2120422.217055-0.022872-0.017309-0.3918221.0948110.5357020.00.535702
                                        121.00.140421-0.9498040.0043550.001276-0.6608951.0822890.5363811.01.082289
                                        131.0-0.1524880.9977280.150062-0.1399930.5990141.0654310.3218850.00.321885
                                        141.0-0.0081470.812770-0.0753040.058249-0.7296480.9201750.5562121.00.920175
                                        151.00.962076-1.2100270.264935-0.0995641.1969181.7988880.6409550.00.640955
                                        161.0-2.2129430.6131030.0081390.087685-0.484076-0.2200440.0344501.0-0.220044
                                        171.0-0.243372-0.256826-0.000754-0.007892-0.9096430.8641740.4312651.00.864174
                                        181.00.882084-0.6199580.118808-0.0972781.7170691.6083640.6232430.00.623243
                                        191.00.3635491.1374130.1781680.0452990.4143511.3799380.6361860.00.636186
                                        \n
                                        \n\n\n\n\n```python\n!cat \"data_eh.grmpy.info\"\n```\n\n \r\n \r\n Number of Observations \r\n \r\n Count\r\n \r\n All 10000\r\n Treated 6384\r\n Untreated 3616\r\n \r\n \r\n Distribution of Outcomes\r\n \r\n Mean Std-Dev. 25% 50% 75%\r\n \r\n All 0.7945 0.5279 0.4237 0.7200 1.1301\r\n Treated 0.9721 0.5557 0.6012 0.9698 1.3457\r\n Untreated 0.4809 0.2674 0.3072 0.4807 0.6615\r\n \r\n \r\n Distribution of Effects\r\n \r\n Mean Std-Dev. 25% 50% 75%\r\n \r\n All 0.4987 0.3328 0.2705 0.5020 0.7210\r\n Treated 0.4607 0.3279 0.2391 0.4578 0.6831\r\n Untreated 0.5658 0.3308 0.3415 0.5696 0.7857\r\n \r\n \r\n Criterion Function \r\n \r\n Value -0.322419326705 \r\n \r\n \r\n \r\n Marginal Treatment Effect \r\n \r\n Quantile Value\r\n \r\n 1% 0.3285\r\n 5% 0.3789\r\n 10% 0.4058\r\n 15% 0.4239\r\n 20% 0.4383\r\n 25% 0.4507\r\n 30% 0.4618\r\n 35% 0.4721\r\n 40% 0.4819\r\n 45% 0.4913\r\n 50% 0.5006\r\n 55% 0.5099\r\n 60% 0.5194\r\n 65% 0.5291\r\n 70% 0.5394\r\n 75% 0.5505\r\n 80% 0.5629\r\n 85% 0.5773\r\n 90% 0.5955\r\n 95% 0.6223\r\n 99% 0.6728\r\n \r\n \r\n Parameterization \r\n \r\n Section Identifier Coef\r\n \r\n TREATED \r\n const 1.0000\r\n X2 0.5550\r\n \r\n UNTREATED \r\n const 0.5000\r\n X2 0.2500\r\n \r\n CHOICE \r\n const 0.3780\r\n X3 -0.3900\r\n \r\n DIST \r\n sigma1 0.1000\r\n sigma10 0.0000\r\n sigma1v 0.0524\r\n sigma0 0.1000\r\n sigma0v -0.0216\r\n sigmaV 1.0000\r\n\n\n\n```python\neffects(data_eh)\n```\n\nNext we illustrate how essential heterogeneity is refelected in the joint distribution of the error term. For that purpose we created two figures that show the relationship between $U_1$ and $V$ for both data sets which we simulated before.\n\n\n```python\nplot_joint_distribution_unobservables(data, data_eh)\n```\n\nSimulating data sets enables us to explore more facets of the impact of essential heterogeneity on the marginal effect of treatment. The marginal effect of treatment, henceforth denoted as $B^{MTE}$ is defined as the effect of treatment for individuals that are indifferent between taking the treatment or not along the distribution of the unobservable variable $V$. This means that instead of assigning explicitly one value as the effect of treatment the $B^{MTE}$ provides a continuum of effects along the distribution of the unobservable $V$. More formally:\n\n\\begin{align*}\nB^{MTE} = E[Y_1-Y_0|X=x, U_D = u_D] \\\\\n\\end{align*}\n\nIn the abscence of essential heterogeneity $B^{MTE}$ provides a constant value over the whole distribution of $V$. This implies that the individual's decision for treatment participation is independent of their unobservable benefits so that each individual on the margin faces the same average benefit of treatment. This can be seen in the figure below. The flat line captures the $B^{MTE}$ for the simulated data without essential heterogeneity, wheras the increasing cure illustrates the marginal effect for the one that is affected by essential heterogeneity.\n\n\n```python\nplot_marginal_effects('data.grmpy.info', 'data_eh.grmpy.info')\n```\n\nUsing simulated data allows us to tackle issues and explore additional objects of interest for which we are not able to obtain reliable information if we use empirical data sets. For instance we are able to construct the joint distribution of potential outcomes from the previously simulated data sets because our simulated data allows us to evade the evaluation problem. This means that we have information about the whole space of potential outcomes for each individual. \n\n\n```python\nplot_joint_distribution_potential(data_eh)\n```\n\n## The Estimation Process\n\ngrmpy enables users to estimate parameters on data sets. Executing an estimation is simple. Just setup your estimation specifications in the initialization file and provide the estimation process with the resulting initialization file.\n\n\n```python\nrslt = grmpy.fit('files/tutorial_eh.grmpy.yml')\n```\n\nThe estimation process returns a detailed overview of the results via an output file\n\n\n```python\n!cat output/est.grmpy.info\n```\n\n \r\n \r\n Optimization Information\r\n \r\n Optimizer: SCIPY-BFGS \r\n Start Values: auto \r\n Success: 1 \r\n Status: 0 \r\n Message: Optimization terminated successfully. \r\n Number of Evaluations: 276 \r\n Criterion: -0.3230 \r\n Observations: 10000 \r\n Warning: The optimization algorithm has failed to provide the parametrization that leads to the minimal criterion function value. \r\n The estimation output is automatically adjusted and provides the parameterization with the smallest criterion function value \r\n that was reached during the optimization.\r\n \r\n \r\n \r\n Criterion Function\r\n \r\n Start Finish\r\n \r\n \r\n -0.3187406802544943 -0.3229750696988651\r\n \r\n \r\n \r\n Economic Parameters\r\n \r\n \r\n Start Finish\r\n \r\n Section Identifier Coef Coef Std err t P>|t| 95% Conf. Int.\r\n \r\n TREATED \r\n const 0.9708 0.9981 0.0031 320.4145 0.0000 0.9920 1.0042\r\n X2 0.5533 0.5533 0.0012 470.7484 0.0000 0.5510 0.5556\r\n \r\n UNTREATED \r\n const 0.4805 0.4953 0.0064 77.2435 0.0000 0.4827 0.5078\r\n X2 0.2477 0.2477 0.0017 148.9133 0.0000 0.2445 0.2510\r\n \r\n CHOICE \r\n const 0.3795 0.3796 0.0133 28.5424 0.0000 0.3535 0.4056\r\n X3 -0.4175 -0.4145 0.0141 -29.4912 0.0000 -0.4420 -0.3870\r\n \r\n DIST \r\n sigma1 0.0936 0.0998 0.0016 63.0250 0.0000 0.0967 0.1029\r\n rho1 0.0000 0.5055 0.0457 11.0522 0.0000 0.4159 0.5952\r\n sigma0 0.1002 0.1009 0.0013 75.1643 0.0000 0.0983 0.1036\r\n rho0 0.0000 -0.1528 0.0631 -2.4197 0.0156 -0.2765 -0.0290\r\n\n\nIn addition the process is capable of simulating a new data set according to the estimation results. This enables users to verify their obtained results easily.\n\n\n```python\n!cat comparison.grmpy.txt\n```\n\n \r\n \r\n Number of Observations \r\n \r\n Sample Observed Simulated (finish) Simulated (start)\r\n \r\n \r\n All 10000 10000 10000\r\n Treated 6384 6374 6398\r\n Untreated 3616 3626 3602\r\n \r\n \r\n Distribution of Outcomes\r\n \r\n \r\n \r\n All \r\n \r\n Mean Std-Dev. 25% 50% 75%\r\n \r\n Observed Sample 0.7945 0.5279 0.4237 0.7200 1.1301\r\n Simulated Sample (finish) 0.7941 0.5261 0.4249 0.7196 1.1309\r\n Simulated Sample (start) 0.7944 0.5281 0.4204 0.7235 1.1358\r\n \r\n \r\n Treated \r\n \r\n Mean Std-Dev. 25% 50% 75%\r\n \r\n Observed Sample 0.9721 0.5557 0.6012 0.9698 1.3457\r\n Simulated Sample (finish) 0.9721 0.5538 0.5985 0.9700 1.3458\r\n Simulated Sample (start) 0.9726 0.5546 0.6072 0.9752 1.3414\r\n \r\n \r\n Untreated \r\n \r\n Mean Std-Dev. 25% 50% 75%\r\n \r\n Observed Sample 0.4809 0.2674 0.3072 0.4807 0.6615\r\n Simulated Sample (finish) 0.4812 0.2658 0.3086 0.4804 0.6620\r\n Simulated Sample (start) 0.4778 0.2673 0.3040 0.4816 0.6576\r\n \r\n \r\n MTE Information \r\n \r\n Quantile Value\r\n \r\n 1% 0.3503\r\n 5% 0.3951\r\n 10% 0.4191\r\n 15% 0.4352\r\n 20% 0.4480\r\n 25% 0.4590\r\n 30% 0.4689\r\n 35% 0.4781\r\n 40% 0.4868\r\n 45% 0.4952\r\n 50% 0.5035\r\n 55% 0.5117\r\n 60% 0.5201\r\n 65% 0.5288\r\n 70% 0.5380\r\n 75% 0.5479\r\n 80% 0.5589\r\n 85% 0.5717\r\n 90% 0.5878\r\n 95% 0.6118\r\n 99% 0.6566\r\n\n\n## Software Engineering Practices\n\n\n# Part II - Monte Carlo Simulation\n\nFor illustrating the advantages of grmpy's estimation process in the presence of essential heterogeneity we conduct a Monte Carlo exercise. As before the starting point of the exericise is an initialization file over which we iterate several times during the process. The distributional characteristics are such that the unobservable variables are distributed according to the following covariance matrix\n\n\\begin{align}\n\\Sigma = \\begin{bmatrix}\n 0.01 & 0 & \\frac{\\rho_{1,V}}{0.1} \\\\\n 0 & 0.01 & 0 \\\\\n \\frac{\\rho_{1,V}}{0.1} & 0 & 1 \\\\\n \\end{bmatrix}\n\\end{align}\n\nDuring each step of the iteration we increase the correlation between $U_1$ and $V$. We will start from a value of $\\rho_1 =0.0$ and end at $\\rho_1 = 0.99$. This increase is equivalent to the observation of incremental reverse selection behavior, because individuals with a low value of $V$ which are most likely to select into treatment have on average a lower value of $U_1$ than individuals that have larger values of $V$. in addition we estimate the average effect of treatment during each step. For this purpose we use the grmpy estimation process, an ordinary least squares regression and an instrumental variables approach as well as a naive comparison of outputs between treated and untreated individuals.\n\n\n\n### The Initialization file\n\n\n```python\n%%file files/mc.grmpy.yml\n---\nSIMULATION:\n seed: 5133\n agents: 10000\n source: mc\nESTIMATION:\n file: mc.grmpy.txt\n start: auto\n agents: 165\n optimizer: SCIPY-BFGS\n maxiter: 6383\n dependent: wage\n indicator: state\n output_file: mc_rslt.grmpy.info\n comparison: 0\n print_output: 0\n\nTREATED:\n params:\n - 0.99\n - 0.555\n - -0.555\n - 0.755\n - 0.155\n order:\n - const\n - X2\n - X3\n - X4\n - X5\nUNTREATED:\n params:\n - 0.5\n - 0.255\n - -0.255\n - 0.1768\n - 0.0987\n \n order:\n - const\n - X2\n - X3\n - X4\n - X5\nCHOICE:\n params:\n - 0.28\n - -0.39\n - 0.59\n - -0.89\n - -0.73\n\n order:\n - const\n - X6\n - X7\n - X8\n - X9\n\nDIST:\n params:\n - 0.2\n - 0.0\n - 0.0\n - 0.2\n - 0.0\n - 1.0\nVARTYPES:\n const: nonbinary\n X2: nonbinary\n X3: nonbinary\n X4: nonbinary\n X5: nonbinary\n X6: nonbinary\n X7: nonbinary\n X8: nonbinary\n X9: nonbinary\nSCIPY-BFGS:\n gtol: 1.0e-05\n eps: 1.4901161193847655e-08\nSCIPY-POWELL:\n xtol: 9.147777614048603e-05\n ftol: 9.749582129043358e-05\n```\n\n Overwriting files/mc.grmpy.yml\n\n\n\n```python\ngrmpy.simulate('files/mc.grmpy.yml')\nmonte_carlo('files/mc.grmpy.yml', 10)\n```\n\nAs can be seen from the figure, the OLS estimator and the naive comparison of outcomes between the treated and untreated subpopulation underestimate the effect significantly. The stronger the correlation between the unobservable variables the more or less stronger bias. Moreover the IV estimates become upward biased as soon the impact of essential heterogeneity increases. Conversely to the other estimation approaches the grmpy estimate of the average effect is close to the true value even if the unobservables are almost perfectly correlated.\n\n# Part III - Replication Carneiro & Heckman & Vytlacil 2011\n\nSince the current version of grmpy is not capable of estimating non-parametric versions of the Roy models, our\nreplication of Carneiro et al. (2011) will focus on reproducing the results for the marginal treatment effect of the parametric selection model. Due to reasons of privacy regarding local variables, we are not able to merge the data provided by the authors so that they fully coincide with the original data set. Therefore our replication setup makes use of a mock data set. For this purpose we randomly merge the individual specific data with the local characteristics.\n\n\n```python\nbasic = pd.read_stata('data/basicvariables.dta')\nlocal = pd.read_stata('data/localvariables.dta') \ndf = pd.concat([basic, local], axis = 1)\nprocess_data(df,'data/aer-replication-mock')\n```\n\nIn the next step we have to create a inititalization file that fully coincides with the setup by Carneiro et. al. (2011). Therefore we use the information that the authors provide in their appendix to create the following init file:\n\n\n```python\n%%file files/replication.grmpy.yml\n---\nSIMULATION:\n seed: 5062\n agents: 991\n source: 8EF73AA0\nESTIMATION:\n file: data/aer-replication-mock.pkl\n start: auto\n agents: 1000\n optimizer: SCIPY-BFGS\n maxiter: 80000\n dependent: wage\n indicator: state\n output_file: replication.grmp.info\n comparison: 0\n print_output: 0\n\nTREATED:\n params:\n - 1.0\n order:\n - const\n - exp\n - expsq\n - lwage5\n - lurate\n - cafqt\n - cafqtsq\n - mhgc\n - mhgcsq\n - numsibs\n - numsibssq\n - urban14\n - lavlocwage17\n - lavlocwage17sq\n - avurate\n - avuratesq\n - d57\n - d58\n - d59\n - d60\n - d61\n - d62\n - d63\nUNTREATED:\n params:\n - 1.0\n order:\n - const\n - exp\n - expsq\n - lwage5\n - lurate\n - cafqt\n - cafqtsq\n - mhgc\n - mhgcsq\n - numsibs\n - numsibssq\n - urban14\n - lavlocwage17\n - lavlocwage17sq\n - avurate\n - avuratesq\n - d57\n - d58\n - d59\n - d60\n - d61\n - d62\n - d63\nCHOICE:\n params:\n - 1.0\n order:\n - const\n - cafqt\n - cafqtsq\n - mhgc\n - mhgcsq\n - numsibs\n - numsibssq\n - urban14\n - lavlocwage17\n - lavlocwage17sq\n - avurate\n - avuratesq\n - d57\n - d58\n - d59\n - d60\n - d61\n - d62\n - d63\n - lwage5_17numsibs\n - lwage5_17mhgc\n - lwage5_17cafqt\n - lwage5_17\n - lurate_17\n - lurate_17numsibs\n - lurate_17mhgc\n - lurate_17cafqt\n - tuit4c\n - tuit4cnumsibs\n - tuit4cmhgc\n - tuit4ccafqt\n - pub4\n - pub4numsibs\n - pub4mhgc\n - pub4cafqt\nDIST:\n params:\n - 0.1\n - 0.0\n - 0.0\n - 0.1\n - 0.0\n - 1.0\nVARTYPES:\n const: nonbinary\nSCIPY-BFGS:\n gtol: 1.0e-05\n eps: 1.4901161193847656e-08\nSCIPY-POWELL:\n xtol: 0.0001\n ftol: 0.0001\n```\n\n Overwriting files/replication.grmpy.yml\n\n\nWe then conduct an estimation based on the initialization file.\n\n\n```python\nrslt = grmpy.fit('files/replication.grmpy.yml')\n```\n\nNext we plot $B^{MTE}$ based on our estimation results. As shown in the figure below the results are really close to the original results. The deviation seems to be negligible because we use a mock dataset.\n\n\n```python\nmte = plot_est_mte(rslt, 'files/replication.grmpy.yml')\n```\n\n\nFor a detailed overview on the theoretical economic background, more application examples as well as contact informations see the [online documentation](https://grmpy.readthedocs.io/en/latest/index.html). In addition, the most current code is available on [GitHub](https://github.com/OpenSourceEconomics/grmpy).\n\n\n\n", "meta": {"hexsha": "c410935436e92c97b57fd08bfb11dca7ee3bf67b", "size": 374213, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "promotion/grmpy_tutorial_notebook/grmpy_tutorial_notebook.ipynb", "max_stars_repo_name": "SeBecker/grmpy", "max_stars_repo_head_hexsha": "3ff5ec9cd108582c23cb61e6b8d87f4db6ceaee1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-05-28T06:11:14.000Z", "max_stars_repo_stars_event_max_datetime": "2018-05-28T06:11:14.000Z", "max_issues_repo_path": "promotion/grmpy_tutorial_notebook/grmpy_tutorial_notebook.ipynb", "max_issues_repo_name": "SeBecker/grmpy", "max_issues_repo_head_hexsha": "3ff5ec9cd108582c23cb61e6b8d87f4db6ceaee1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2018-05-14T20:44:51.000Z", "max_issues_repo_issues_event_max_datetime": "2018-07-04T16:36:49.000Z", "max_forks_repo_path": "promotion/grmpy_tutorial_notebook/grmpy_tutorial_notebook.ipynb", "max_forks_repo_name": "SeBecker/grmpy", "max_forks_repo_head_hexsha": "3ff5ec9cd108582c23cb61e6b8d87f4db6ceaee1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 182.8998044966, "max_line_length": 78812, "alphanum_fraction": 0.8644542012, "converted": true, "num_tokens": 13399, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.6688802537704063, "lm_q1q2_score": 0.4309094527118699}} {"text": "# The *grmpy* package \nThis notebook demonstrates the current capabilities of the *grmpy* package. *grmpy* is an open source package for the programming language python that enables researchers to simulate datasets andestimate parameters using already existing data within the structure of the generalized Roy model. Currently the package serves as a teaching tool for a course on the econometrics of policy evaluation at the University of Bonn. The corresponding lecture materials can be found on [GitHub](https://github.com/HumanCapitalAnalysis/econometrics). Morover it is thought of as a promotion for the conceptual framework as well as a showcase for basic software engineering practices.\n\nFor a more detailed overview on the economic background as well as the installation routine feel free to take a look on the [online documentation](https://grmpy.readthedocs.io/en/develop/).\n\nThe notebook itself is divided in three parts. Firstly we provide a basic outline on how to use the package and introduce the core features. Next we will show that the results obtained by the package's estimation process withstand a critical examination by comparing its performance in the presence of essential heterogeneity with several other estimation approaches like Ordinary Least Squares and Instrumental variables. We conclude by conducting a replication of results from \n\nCarneiro, P., Heckman, J. J., & Vytlacil, E. J. (2011). [Estimating marginal returns to education.](https://pubs.aeaweb.org/doi/pdfplus/10.1257/aer.101.6.2754)\n*American Economic Review, 101*(6), 2754-81.\n\n\n# The Framework\n\nAs mentioned before the package makes use of the normal-linear-in-parameters version generalized Roy model. In addition we assume that the unobservable terms $\\{U_1, U_0, V\\}$ are normally distributed according to the covariance matrix $\\Sigma$. The following set of equations characterize the underlying model:\n\n\\begin{align*}\n &\\textbf{Potential Outcomes} & & \\textbf{Choice} &\\\\\n & Y_1 = \\beta_1 X + U_{1} & &D_i = \\mathbf{1}\\{\\Phi(\\gamma Z) > u_D\\} &\\\\\n & Y_0 = \\beta_0 X + U_{0} & & \\text{with $u_D = \\Phi(V)$} &\\\\\n&&&&\\\\\n&\\textbf{Distributional Characteristics}&&&\\\\\n&\\{U_{1}, U_{0}, V\\} \\sim \\mathcal{N}\\left(0, \\Sigma\\right)&&\\Sigma = \\begin{bmatrix}\n \\sigma_1^{2} & \\sigma_{1,0} & \\sigma_{1,V} \\\\\n \\sigma_{1,0} & \\sigma_0^{2} & \\sigma_{0,V} \\\\\n \\sigma_{1,V} & \\sigma_{0,V} & \\sigma_V^{2} \\\\\n \\end{bmatrix}&\\\\\n&&&&\\\\\n& \\textbf{Observed Outcome} &&&\\\\\n& Y = D Y_1 + (1-D) Y_0 &&&\n\\end{align*}\n\n\n# Preliminaries\n\nThe first step consists of some style and import improving settings.\n\n\n```python\nfrom IPython.core.display import display, HTML\ndisplay(HTML(\"\"))\n%load_ext autoreload\n%autoreload 2\n```\n\n\n\n\n\nBefore we can start with the application examples we have to import several libaries that we rely on during this presentation.\n\n\n```python\nimport matplotlib.pyplot as plt\nimport statsmodels.api as sm\nimport pandas as pd\nimport numpy as np\nimport json\n\nimport grmpy\n```\n\nIn addition we will import some custom functions for plotting several figures of interest as well as some data management purposes. The code is provided in the auxiliary.py file.\n\n\n```python\nfrom auxiliary import plot_joint_distribution_unobservables\nfrom auxiliary import plot_joint_distribution_potential\nfrom auxiliary import plot_marginal_effects\nfrom auxiliary import process_data\nfrom auxiliary import plot_est_mte\nfrom auxiliary import monte_carlo\nfrom auxiliary import effects\n```\n\n# Part I - Introduction\n\nThis part provides some basic information about the general application of the package's core methods. \n\n## The initialization file\n\nCurrently the package has two core features. The first one is the simulation process. It enables users to simulate data sets along a pre-specified parameterization. The data generating process follows the structure of the previously introduced parametric Roy model framework. Therefore users specify their desired parametrization in an initialization file which allows altering any aspects of the model according to their preferences. The ability to estimate parameters on already existing data sets is the second core feature. As before the initialization file is the starting point for the estimation process. \n\nThe initialization file is structured as follows:\n\n* **Simulation**: The simulation section contains information for the estimation process. For instance users are able to change the number of individuals for which the program runs a simulation through setting the agents option to a specific value.\n\n\n* **Estimation**: Accordingly information regarding the estimation process are stored in the estimation section. Here, users can set the type of optimizer which should be used for running the process as well as the maximum number of iterations that the process is allowed to perform. In addition the flags *dependent* and *indicator* determine which variable of the input data frame is seen as the dependent variable and the treatment indicator variable, respectively.\n\n\n* **Treated, Untreated, Choice**: These sections are essential for both methods. They contain the parameterization which is used for the simulation process. Additionally the second column indicates the variable labels well as for the simulated data set. This column is also essential for the estimation process because it will only include variables that are pre-specified in a specific section for the following optimization.\n\n\n* **Dist**: The Dist section determines the distributional characteristics of the unobservable variables. Therefore the section reports the upper triangle of the covariance matrix of $(U_1, U_0,V)$.\n\n\n* **SCIPY-BFGS/SCIPY-POWELL**: The SCIPY-related sections contain options for the relevant optimization algorithms.\n\n\n```python\n%%file files/tutorial.grmpy.yml\n---\nSIMULATION:\n agents: 10000\n seed: 2356\n source: data\nESTIMATION:\n file: data.grmpy.txt\n output_file: output/est.grmpy.info\n optimizer: BFGS\n start: auto\n maxiter: 6383\n agents: 165\n dependent: Y\n indicator: D\n comparison: 1\n print_output: 0\nTREATED:\n params:\n - 1.0\n - 0.555\n order:\n - const\n - X2\nUNTREATED:\n params:\n - 0.5\n - 0.25\n order:\n - const\n - X2\nCHOICE:\n params:\n - 0.378\n - -0.39\n order:\n - const\n - X3\nDIST:\n params:\n - 0.1\n - 0.0\n - 0.0\n - 0.1\n - 0.0\n - 1.0\nVARTYPES:\n const: nonbinary\n X2: nonbinary\n X3: nonbinary\nSCIPY-BFGS:\n gtol: 1.0e-05\n eps: 1.4901161193847655e-08\nSCIPY-POWELL:\n xtol: 9.147777614048603e-05\n ftol: 9.749582129043358e-05\n```\n\n Overwriting files/tutorial.grmpy.yml\n\n\n## The Simulation Process\n\nFor simulating a dataset we only have to provide the package with the related inititalization file.\n\n\n```python\ndata = grmpy.simulate('files/tutorial.grmpy.yml')\ndata.head(20)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        constX2X3U1U0VY1Y0DY
                                        01.0-0.0147100.0234390.0026760.0614520.7673100.9945120.5577750.00.557775
                                        11.01.941447-1.3758910.0870870.066451-0.5679792.1645901.0518131.02.164590
                                        21.0-0.2206681.5554530.0785570.008140-0.3078280.9560870.4529731.00.956087
                                        31.00.5318990.192161-0.0055740.055052-1.1292021.2896300.6880271.01.289630
                                        41.0-0.2951611.663835-0.1115340.077755-0.0247870.7246520.5039650.00.503965
                                        51.02.0306130.845409-0.096886-0.0861800.1177102.0301040.9214730.00.921473
                                        61.0-0.189934-0.8306590.019401-0.045266-0.0371560.9139880.4072501.00.913988
                                        71.00.5908191.731056-0.0870720.0326170.7197591.2408330.6803220.00.680322
                                        81.00.7904661.1332870.049938-0.2009240.7498261.4886470.4966920.00.496692
                                        91.0-1.3784290.0200670.1516130.015832-1.1195190.3865850.1712251.00.386585
                                        101.0-0.089817-0.2790000.062335-0.0534590.0492701.0124870.4240871.01.012487
                                        111.00.2120422.2170550.009479-0.0247190.3913821.1271620.5282910.00.528291
                                        121.00.140421-0.9498040.0500720.0028390.6585551.1280050.5379451.01.128005
                                        131.0-0.1524880.9977280.191663-0.072238-0.6080021.1070320.3896391.01.107032
                                        141.0-0.0081470.812770-0.0609110.0251550.7325110.9345670.5231180.00.523118
                                        151.00.962076-1.2100270.2604560.008918-1.2091361.7944080.7494371.01.794408
                                        161.0-2.2129430.6131030.0021010.0841700.483983-0.2260820.0309340.00.030934
                                        171.0-0.243372-0.2568260.065807-0.0075840.9065690.9307360.4315730.00.431573
                                        181.00.882084-0.6199580.059393-0.044658-1.7198711.5489500.6758631.01.548950
                                        191.00.3635491.1374130.1502810.109781-0.4213981.3520510.7006681.01.352051
                                        \n
                                        \n\n\n\nThe simulation process generates an info file that provides additional information about the distribution of outcomes and effects, respectively. Additionally it contains the criterion function value, information about the corresponding marginal treatment effects as well as the paramterization.\n\n\n```python\n!cat \"data.grmpy.info\"\n```\n\n \r\n \r\n Number of Observations \r\n \r\n Count\r\n \r\n All 10000\r\n Treated 6391\r\n Untreated 3609\r\n \r\n \r\n Distribution of Outcomes\r\n \r\n Mean Std-Dev. 25% 50% 75%\r\n \r\n All 0.8196 0.5328 0.4401 0.7467 1.1652\r\n Treated 1.0018 0.5580 0.6305 1.0059 1.3721\r\n Untreated 0.4968 0.2689 0.3218 0.5001 0.6782\r\n \r\n \r\n Distribution of Effects\r\n \r\n Mean Std-Dev. 25% 50% 75%\r\n \r\n All 0.4998 0.3340 0.2739 0.4974 0.7208\r\n Treated 0.5007 0.3354 0.2731 0.5016 0.7233\r\n Untreated 0.4982 0.3315 0.2761 0.4905 0.7177\r\n \r\n \r\n Criterion Function \r\n \r\n Value -0.270418786930 \r\n \r\n \r\n \r\n Marginal Treatment Effect \r\n \r\n Quantile Value\r\n \r\n 1% 0.5006\r\n 5% 0.5006\r\n 10% 0.5006\r\n 15% 0.5006\r\n 20% 0.5006\r\n 25% 0.5006\r\n 30% 0.5006\r\n 35% 0.5006\r\n 40% 0.5006\r\n 45% 0.5006\r\n 50% 0.5006\r\n 55% 0.5006\r\n 60% 0.5006\r\n 65% 0.5006\r\n 70% 0.5006\r\n 75% 0.5006\r\n 80% 0.5006\r\n 85% 0.5006\r\n 90% 0.5006\r\n 95% 0.5006\r\n 99% 0.5006\r\n \r\n \r\n Parameterization \r\n \r\n Section Identifier Coef\r\n \r\n TREATED \r\n const 1.0000\r\n X2 0.5550\r\n \r\n UNTREATED \r\n const 0.5000\r\n X2 0.2500\r\n \r\n CHOICE \r\n const 0.3780\r\n X3 -0.3900\r\n \r\n DIST \r\n sigma1 0.1000\r\n sigma10 0.0000\r\n sigma1v 0.0000\r\n sigma0 0.1000\r\n sigma0v 0.0000\r\n sigmaV 1.0000\r\n\n\nThe generated data set is specififed so that the treatment selection process is not affected by essential heterogeneity. Therefore the conventional effects are nearly identical. \n\n\n```python\neffects(data)\n```\n\n### Essential Heterogeneity\nFor providing an example on how essential heterogeneity biases the results obtained by naive comparison between treated and untreated individuals, we alter the initialization file. Specifically we will introduce correlation between the unobservable terms. This leads to the situation where individuals select into treatment based on unobservable gains. Therefore the treatment decision $D$ is no longer independent from the outcomes $Y_1$ and $Y_0$.\n\n\\begin{align}\nY_1,Y_0\\;\\; {\\perp\\!\\!\\!\\!\\!\\!\\diagup\\!\\!\\!\\!\\!\\!\\!\\perp} \\;\\; D\n\\end{align}\n\nUnlike in the absence of essential heterogeneity individuals who select themselves into treatment differ from\nindividuals who do not with respect to their unobservable characteristics. From this follows that\n\n\\begin{align}\nB^{ATE} \\neq B^{TT} \\neq B^{TUT}\n\\end{align}\n\n\n```python\n%%file files/tutorial_eh.grmpy.yml\n---\nSIMULATION:\n seed: 2356\n agents: 10000\n source: data_eh\nESTIMATION:\n file: data_eh.grmpy.txt\n start: auto\n agents: 165\n optimizer: BFGS\n maxiter: 6383\n dependent: Y\n indicator: D\n output_file: output/est.grmpy.info\n comparison: 1\n print_output: 0\n\nTREATED:\n params:\n - 1.0\n - 0.555\n order:\n - const\n - X2\nUNTREATED:\n params:\n - 0.5\n - 0.25\n order:\n - const\n - X2\nCHOICE:\n params:\n - 0.378\n - -0.39\n order:\n - const\n - X3\nDIST:\n params:\n - 0.1\n - 0.0\n - 0.0524\n - 0.1\n - -0.0216\n - 1.0\nVARTYPES:\n const: nonbinary\n X2: nonbinary\n X3: nonbinary\nSCIPY-BFGS:\n gtol: 1.0e-05\n eps: 1.4901161193847655e-08\nSCIPY-POWELL:\n xtol: 9.147777614048603e-05\n ftol: 9.749582129043358e-05\n```\n\n Overwriting files/tutorial_eh.grmpy.yml\n\n\n\n```python\ndata_eh = grmpy.simulate('files/tutorial_eh.grmpy.yml')\ndata_eh.head(20)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        constX2X3U1U0VY1Y0DY
                                        01.0-0.0147100.023439-0.0150290.072663-0.7674270.9768070.5689861.00.976807
                                        11.01.941447-1.3758910.1214090.0218290.5638922.1989121.0071911.02.198912
                                        21.0-0.2206681.5554530.078986-0.0237540.3041430.9565150.4210790.00.421079
                                        31.00.5318990.1921610.0763210.0280851.1294501.3715250.6610600.00.661060
                                        41.0-0.2951611.663835-0.0537390.1062550.0300130.7824460.5324640.00.532464
                                        51.02.0306130.845409-0.112613-0.046794-0.1131682.0143770.9608591.02.014377
                                        61.0-0.189934-0.830659-0.000561-0.0487300.0362470.8940260.4037861.00.894026
                                        71.00.5908191.731056-0.0916490.073059-0.7156711.2362550.7207641.01.236255
                                        81.00.7904661.133287-0.078217-0.185083-0.7521581.3604920.5125331.01.360492
                                        91.0-1.3784290.0200670.180205-0.0571581.1124020.4151770.0982340.00.098234
                                        101.0-0.089817-0.2790000.024354-0.067862-0.0521910.9745060.4096841.00.974506
                                        111.00.2120422.217055-0.022872-0.017309-0.3918221.0948110.5357020.00.535702
                                        121.00.140421-0.9498040.0043550.001276-0.6608951.0822890.5363811.01.082289
                                        131.0-0.1524880.9977280.150062-0.1399930.5990141.0654310.3218850.00.321885
                                        141.0-0.0081470.812770-0.0753040.058249-0.7296480.9201750.5562121.00.920175
                                        151.00.962076-1.2100270.264935-0.0995641.1969181.7988880.6409550.00.640955
                                        161.0-2.2129430.6131030.0081390.087685-0.484076-0.2200440.0344501.0-0.220044
                                        171.0-0.243372-0.256826-0.000754-0.007892-0.9096430.8641740.4312651.00.864174
                                        181.00.882084-0.6199580.118808-0.0972781.7170691.6083640.6232430.00.623243
                                        191.00.3635491.1374130.1781680.0452990.4143511.3799380.6361860.00.636186
                                        \n
                                        \n\n\n\n\n```python\n!cat \"data_eh.grmpy.info\"\n```\n\n \r\n \r\n Number of Observations \r\n \r\n Count\r\n \r\n All 10000\r\n Treated 6384\r\n Untreated 3616\r\n \r\n \r\n Distribution of Outcomes\r\n \r\n Mean Std-Dev. 25% 50% 75%\r\n \r\n All 0.7945 0.5279 0.4237 0.7200 1.1301\r\n Treated 0.9721 0.5557 0.6012 0.9698 1.3457\r\n Untreated 0.4809 0.2674 0.3072 0.4807 0.6615\r\n \r\n \r\n Distribution of Effects\r\n \r\n Mean Std-Dev. 25% 50% 75%\r\n \r\n All 0.4987 0.3328 0.2705 0.5020 0.7210\r\n Treated 0.4607 0.3279 0.2391 0.4578 0.6831\r\n Untreated 0.5658 0.3308 0.3415 0.5696 0.7857\r\n \r\n \r\n Criterion Function \r\n \r\n Value -0.322419326705 \r\n \r\n \r\n \r\n Marginal Treatment Effect \r\n \r\n Quantile Value\r\n \r\n 1% 0.3285\r\n 5% 0.3789\r\n 10% 0.4058\r\n 15% 0.4239\r\n 20% 0.4383\r\n 25% 0.4507\r\n 30% 0.4618\r\n 35% 0.4721\r\n 40% 0.4819\r\n 45% 0.4913\r\n 50% 0.5006\r\n 55% 0.5099\r\n 60% 0.5194\r\n 65% 0.5291\r\n 70% 0.5394\r\n 75% 0.5505\r\n 80% 0.5629\r\n 85% 0.5773\r\n 90% 0.5955\r\n 95% 0.6223\r\n 99% 0.6728\r\n \r\n \r\n Parameterization \r\n \r\n Section Identifier Coef\r\n \r\n TREATED \r\n const 1.0000\r\n X2 0.5550\r\n \r\n UNTREATED \r\n const 0.5000\r\n X2 0.2500\r\n \r\n CHOICE \r\n const 0.3780\r\n X3 -0.3900\r\n \r\n DIST \r\n sigma1 0.1000\r\n sigma10 0.0000\r\n sigma1v 0.0524\r\n sigma0 0.1000\r\n sigma0v -0.0216\r\n sigmaV 1.0000\r\n\n\n\n```python\neffects(data_eh)\n```\n\nNext we illustrate how essential heterogeneity is refelected in the joint distribution of the error term. For that purpose we created two figures that show the relationship between $U_1$ and $V$ for both data sets which we simulated before.\n\n\n```python\nplot_joint_distribution_unobservables(data, data_eh)\n```\n\nSimulating data sets enables us to explore more facets of the impact of essential heterogeneity on the marginal effect of treatment. The marginal effect of treatment, henceforth denoted as $B^{MTE}$ is defined as the effect of treatment for individuals that are indifferent between taking the treatment or not along the distribution of the unobservable variable $V$. This means that instead of assigning explicitly one value as the effect of treatment the $B^{MTE}$ provides a continuum of effects along the distribution of the unobservable $V$. More formally:\n\n\\begin{align*}\nB^{MTE} = E[Y_1-Y_0|X=x, U_D = u_D] \\\\\n\\end{align*}\n\nIn the abscence of essential heterogeneity $B^{MTE}$ provides a constant value over the whole distribution of $V$. This implies that the individual's decision for treatment participation is independent of their unobservable benefits so that each individual on the margin faces the same average benefit of treatment. This can be seen in the figure below. The flat line captures the $B^{MTE}$ for the simulated data without essential heterogeneity, wheras the increasing cure illustrates the marginal effect for the one that is affected by essential heterogeneity.\n\n\n```python\nplot_marginal_effects('data.grmpy.info', 'data_eh.grmpy.info')\n```\n\nUsing simulated data allows us to tackle issues and explore additional objects of interest for which we are not able to obtain reliable information if we use empirical data sets. For instance we are able to construct the joint distribution of potential outcomes from the previously simulated data sets because our simulated data allows us to evade the evaluation problem. This means that we have information about the whole space of potential outcomes for each individual. \n\n\n```python\nplot_joint_distribution_potential(data_eh)\n```\n\n## The Estimation Process\n\ngrmpy enables users to estimate parameters on data sets. Executing an estimation is simple. Just setup your estimation specifications in the initialization file and provide the estimation process with the resulting initialization file.\n\n\n```python\nrslt = grmpy.fit('files/tutorial_eh.grmpy.yml')\n```\n\nThe estimation process returns a detailed overview of the results via an output file\n\n\n```python\n!cat output/est.grmpy.info\n```\n\n Optimization Results\r\n ================================================================================\r\n Dep. Variable: Y Optimizer: BFGS\r\n Choice Var: D No. Evaluations: 30\r\n Date: Wed, 14 Apr 2021 Success: 1\r\n Time: 20:00:21 Status: 0\r\n Observations: 10000 Message: Optimization terminated\r\n Start Values: auto successfully.\r\n Criterion Func: \r\n Start: -0.1862\r\n Finish: -0.3230\r\n ================================================================================\r\n coef std err t P>|t| [0.025 0.975]\r\n --------------------------------------------------------------------------------\r\n TREATED \r\n \r\n const 0.9981 0.003 320.417 0.000 0.992 1.004\r\n X2 0.5533 0.001 470.749 0.000 0.551 0.556\r\n \r\n UNTREATED \r\n \r\n const 0.4953 0.006 77.244 0.000 0.483 0.508\r\n X2 0.2477 0.002 148.914 0.000 0.244 0.251\r\n \r\n CHOICE \r\n \r\n const 0.3796 0.013 28.543 0.000 0.354 0.406\r\n X3 -0.4145 0.014 -29.491 0.000 -0.442 -0.387\r\n \r\n DIST \r\n \r\n sigma1 0.0998 0.002 63.025 0.000 0.097 0.103\r\n rho1 0.5055 0.046 11.053 0.000 0.416 0.595\r\n sigma0 0.1009 0.001 75.167 0.000 0.098 0.104\r\n rho0 -0.1528 0.063 -2.419 0.016 -0.277 -0.029\r\n ================================================================================\r\n \r\n Warning:\r\n \r\n ---\r\n\n\n## Software Engineering Practices\n\n\n# Part II - Monte Carlo Simulation\n\nFor illustrating the advantages of grmpy's estimation process in the presence of essential heterogeneity we conduct a Monte Carlo exercise. As before the starting point of the exericise is an initialization file over which we iterate several times during the process. The distributional characteristics are such that the unobservable variables are distributed according to the following covariance matrix\n\n\\begin{align}\n\\Sigma = \\begin{bmatrix}\n 0.01 & 0 & \\frac{\\rho_{1,V}}{0.1} \\\\\n 0 & 0.01 & 0 \\\\\n \\frac{\\rho_{1,V}}{0.1} & 0 & 1 \\\\\n \\end{bmatrix}\n\\end{align}\n\nDuring each step of the iteration we increase the correlation between $U_1$ and $V$. We will start from a value of $\\rho_1 =0.0$ and end at $\\rho_1 = 0.99$. This increase is equivalent to the observation of incremental reverse selection behavior, because individuals with a low value of $V$ which are most likely to select into treatment have on average a lower value of $U_1$ than individuals that have larger values of $V$. in addition we estimate the average effect of treatment during each step. For this purpose we use the grmpy estimation process, an ordinary least squares regression and an instrumental variables approach as well as a naive comparison of outputs between treated and untreated individuals.\n\n\n\n### The Initialization file\n\n\n```python\n%%file files/mc.grmpy.yml\n---\nSIMULATION:\n seed: 5133\n agents: 10000\n source: mc\nESTIMATION:\n file: mc.grmpy.txt\n start: auto\n agents: 165\n optimizer: BFGS\n maxiter: 6383\n dependent: wage\n indicator: state\n output_file: mc_rslt.grmpy.info\n comparison: 0\n print_output: 0\n\nTREATED:\n params:\n - 0.99\n - 0.555\n - -0.555\n - 0.755\n - 0.155\n order:\n - const\n - X2\n - X3\n - X4\n - X5\nUNTREATED:\n params:\n - 0.5\n - 0.255\n - -0.255\n - 0.1768\n - 0.0987\n \n order:\n - const\n - X2\n - X3\n - X4\n - X5\nCHOICE:\n params:\n - 0.28\n - -0.39\n - 0.59\n - -0.89\n - -0.73\n\n order:\n - const\n - X6\n - X7\n - X8\n - X9\n\nDIST:\n params:\n - 0.2\n - 0.0\n - 0.0\n - 0.2\n - 0.0\n - 1.0\nVARTYPES:\n const: nonbinary\n X2: nonbinary\n X3: nonbinary\n X4: nonbinary\n X5: nonbinary\n X6: nonbinary\n X7: nonbinary\n X8: nonbinary\n X9: nonbinary\nSCIPY-BFGS:\n gtol: 1.0e-05\n eps: 1.4901161193847655e-08\nSCIPY-POWELL:\n xtol: 9.147777614048603e-05\n ftol: 9.749582129043358e-05\n```\n\n Overwriting files/mc.grmpy.yml\n\n\n\n```python\ngrmpy.simulate('files/mc.grmpy.yml')\nmonte_carlo('files/mc.grmpy.yml', 10)\n```\n\nAs can be seen from the figure, the OLS estimator and the naive comparison of outcomes between the treated and untreated subpopulation underestimate the effect significantly. The stronger the correlation between the unobservable variables the more or less stronger bias. Moreover the IV estimates become upward biased as soon the impact of essential heterogeneity increases. Conversely to the other estimation approaches the grmpy estimate of the average effect is close to the true value even if the unobservables are almost perfectly correlated.\n\n# Part III - Replication Carneiro & Heckman & Vytlacil 2011\n\nSince the current version of grmpy is not capable of estimating non-parametric versions of the Roy models, our\nreplication of Carneiro et al. (2011) will focus on reproducing the results for the marginal treatment effect of the parametric selection model. Due to reasons of privacy regarding local variables, we are not able to merge the data provided by the authors so that they fully coincide with the original data set. Therefore our replication setup makes use of a mock data set. For this purpose we randomly merge the individual specific data with the local characteristics.\n\n\n```python\nbasic = pd.read_stata('data/basicvariables.dta')\nlocal = pd.read_stata('data/localvariables.dta') \ndf = pd.concat([basic, local], axis = 1)\nprocess_data(df,'data/aer-replication-mock')\n```\n\nIn the next step we have to create a inititalization file that fully coincides with the setup by Carneiro et. al. (2011). Therefore we use the information that the authors provide in their appendix to create the following init file:\n\n\n```python\n%%file files/replication.grmpy.yml\n---\nSIMULATION:\n seed: 5062\n agents: 991\n source: 8EF73AA0\nESTIMATION:\n file: data/aer-replication-mock.pkl\n start: auto\n agents: 1000\n optimizer: BFGS\n maxiter: 80000\n dependent: wage\n indicator: state\n output_file: replication.grmpy.info\n comparison: 0\n print_output: True\n\nTREATED:\n params:\n - 1.0\n order:\n - const\n - exp\n - expsq\n - lwage5\n - lurate\n - cafqt\n - cafqtsq\n - mhgc\n - mhgcsq\n - numsibs\n - numsibssq\n - urban14\n - lavlocwage17\n - lavlocwage17sq\n - avurate\n - avuratesq\n - d57\n - d58\n - d59\n - d60\n - d61\n - d62\n - d63\nUNTREATED:\n params:\n - 1.0\n order:\n - const\n - exp\n - expsq\n - lwage5\n - lurate\n - cafqt\n - cafqtsq\n - mhgc\n - mhgcsq\n - numsibs\n - numsibssq\n - urban14\n - lavlocwage17\n - lavlocwage17sq\n - avurate\n - avuratesq\n - d57\n - d58\n - d59\n - d60\n - d61\n - d62\n - d63\nCHOICE:\n params:\n - 1.0\n order:\n - const\n - cafqt\n - cafqtsq\n - mhgc\n - mhgcsq\n - numsibs\n - numsibssq\n - urban14\n - lavlocwage17\n - lavlocwage17sq\n - avurate\n - avuratesq\n - d57\n - d58\n - d59\n - d60\n - d61\n - d62\n - d63\n - lwage5_17numsibs\n - lwage5_17mhgc\n - lwage5_17cafqt\n - lwage5_17\n - lurate_17\n - lurate_17numsibs\n - lurate_17mhgc\n - lurate_17cafqt\n - tuit4c\n - tuit4cnumsibs\n - tuit4cmhgc\n - tuit4ccafqt\n - pub4\n - pub4numsibs\n - pub4mhgc\n - pub4cafqt\nDIST:\n params:\n - 0.1\n - 0.0\n - 0.0\n - 0.1\n - 0.0\n - 1.0\nVARTYPES:\n const: nonbinary\nSCIPY-BFGS:\n gtol: 1.0e-05\n eps: 1.4901161193847656e-08\nSCIPY-POWELL:\n xtol: 0.0001\n ftol: 0.0001\n```\n\n Overwriting files/replication.grmpy.yml\n\n\nWe then conduct an estimation based on the initialization file.\n\n\n```python\nrslt = grmpy.fit('files/replication.grmpy.yml')\n```\n\n Optimization Results\n ================================================================================\n Dep. Variable: wage Optimizer: BFGS\n Choice Var: state No. Evaluations: 289\n Date: Wed, 14 Apr 2021 Success: 1\n Time: 20:00:48 Status: 0\n Observations: 1747 Message: Optimization terminated\n Start Values: auto successfully.\n Criterion Func: \n Start: +1.2580\n Finish: +1.0666\n ================================================================================\n coef std err t P>|t| [0.025 0.975]\n --------------------------------------------------------------------------------\n TREATED \n \n const -35.0684 39.324 -0.892 0.373 -112.141 42.004\n exp 0.0883 0.019 4.732 0.000 0.052 0.125\n expsq -0.0042 0.001 -3.395 0.001 -0.007 -0.002\n lwage5 -0.0803 0.116 -0.693 0.488 -0.307 0.147\n lurate 0.0019 0.018 0.107 0.915 -0.033 0.036\n cafqt 0.1532 0.048 3.178 0.002 0.059 0.248\n cafqtsq 0.0593 0.019 3.046 0.002 0.021 0.097\n mhgc 0.0068 0.048 0.141 0.888 -0.088 0.101\n mhgcsq 0.0013 0.002 0.718 0.473 -0.002 0.005\n numsibs -0.0169 0.029 -0.583 0.560 -0.074 0.040\n numsibssq 0.0010 0.004 0.268 0.789 -0.006 0.008\n urban14 0.1060 0.042 2.546 0.011 0.024 0.188\n lavlocwage17 7.0468 7.672 0.919 0.358 -7.990 22.083\n lavlocwage17sq -0.3360 0.374 -0.898 0.369 -1.069 0.397\n avurate 0.1116 0.187 0.598 0.550 -0.254 0.478\n avuratesq -0.0081 0.014 -0.562 0.574 -0.037 0.020\n d57 0.1432 0.082 1.750 0.080 -0.017 0.304\n d58 0.1880 0.075 2.495 0.013 0.040 0.336\n d59 0.0304 0.077 0.396 0.692 -0.120 0.181\n d60 0.0385 0.070 0.554 0.580 -0.098 0.175\n d61 0.0711 0.067 1.060 0.289 -0.060 0.203\n d62 0.0410 0.062 0.664 0.507 -0.080 0.162\n d63 0.0604 0.063 0.959 0.338 -0.063 0.184\n \n UNTREATED \n \n const 29.8616 36.596 0.816 0.415 -41.864 101.588\n exp 0.0612 0.021 2.902 0.004 0.020 0.103\n expsq 0.0001 0.001 0.101 0.920 -0.002 0.002\n lwage5 0.1378 0.100 1.381 0.167 -0.058 0.333\n lurate -0.0103 0.014 -0.746 0.456 -0.037 0.017\n cafqt 0.0754 0.052 1.436 0.151 -0.027 0.178\n cafqtsq -0.0327 0.025 -1.319 0.187 -0.081 0.016\n mhgc -0.0402 0.033 -1.228 0.220 -0.104 0.024\n mhgcsq 0.0022 0.002 1.275 0.202 -0.001 0.005\n numsibs -0.0006 0.019 -0.030 0.976 -0.038 0.037\n numsibssq -0.0003 0.002 -0.148 0.882 -0.004 0.003\n urban14 0.0780 0.032 2.417 0.016 0.015 0.141\n lavlocwage17 -5.8732 7.120 -0.825 0.410 -19.828 8.081\n lavlocwage17sq 0.2887 0.348 0.831 0.406 -0.392 0.970\n avurate 0.1532 0.136 1.126 0.260 -0.113 0.420\n avuratesq -0.0123 0.011 -1.166 0.244 -0.033 0.008\n d57 -0.2638 0.078 -3.394 0.001 -0.416 -0.111\n d58 -0.2043 0.077 -2.648 0.008 -0.356 -0.053\n d59 -0.2194 0.062 -3.516 0.000 -0.342 -0.097\n d60 -0.0809 0.059 -1.379 0.168 -0.196 0.034\n d61 -0.0621 0.055 -1.120 0.263 -0.171 0.047\n d62 -0.0517 0.054 -0.966 0.334 -0.157 0.053\n d63 0.0118 0.052 0.225 0.822 -0.091 0.115\n \n CHOICE \n \n const 157.2465 88.098 1.785 0.074 -15.422 329.915\n cafqt -4.2036 2.788 -1.508 0.132 -9.669 1.262\n cafqtsq 0.1932 0.039 4.993 0.000 0.117 0.269\n mhgc -0.3172 1.245 -0.255 0.799 -2.756 2.122\n mhgcsq 0.0104 0.004 2.720 0.007 0.003 0.018\n numsibs -0.5652 1.406 -0.402 0.688 -3.321 2.190\n numsibssq 0.0013 0.007 0.195 0.845 -0.012 0.014\n urban14 0.2011 0.082 2.441 0.015 0.040 0.362\n lavlocwage17 -28.8785 16.882 -1.711 0.087 -61.967 4.210\n lavlocwage17sq 1.4301 0.824 1.736 0.083 -0.184 3.044\n avurate -0.1106 0.371 -0.298 0.765 -0.837 0.616\n avuratesq 0.0116 0.029 0.407 0.684 -0.044 0.068\n d57 0.2389 0.148 1.611 0.107 -0.052 0.530\n d58 0.2255 0.147 1.536 0.125 -0.062 0.513\n d59 -0.0826 0.146 -0.566 0.571 -0.368 0.203\n d60 0.0645 0.138 0.468 0.640 -0.206 0.335\n d61 0.0561 0.137 0.408 0.683 -0.213 0.326\n d62 0.2000 0.131 1.531 0.126 -0.056 0.456\n d63 -0.0068 0.138 -0.049 0.961 -0.278 0.265\n lwage5_17numsibs 0.0410 0.138 0.296 0.767 -0.230 0.312\n lwage5_17mhgc 0.0206 0.124 0.166 0.868 -0.222 0.263\n lwage5_17cafqt 0.4865 0.276 1.760 0.079 -0.055 1.028\n lwage5_17 -1.1619 1.624 -0.716 0.474 -4.344 2.020\n lurate_17 -0.0234 0.137 -0.170 0.865 -0.293 0.246\n lurate_17numsibs -0.0022 0.011 -0.195 0.845 -0.025 0.020\n lurate_17mhgc 0.0000 0.011 0.001 0.999 -0.021 0.021\n lurate_17cafqt -0.0155 0.026 -0.596 0.551 -0.066 0.035\n tuit4c 0.0049 0.034 0.145 0.885 -0.061 0.071\n tuit4cnumsibs 0.0024 0.003 0.877 0.381 -0.003 0.008\n tuit4cmhgc -0.0004 0.003 -0.156 0.876 -0.006 0.005\n tuit4ccafqt 0.0002 0.006 0.031 0.975 -0.011 0.011\n pub4 0.1398 0.520 0.269 0.788 -0.880 1.159\n pub4numsibs 0.0297 0.043 0.684 0.494 -0.055 0.115\n pub4mhgc -0.0125 0.043 -0.289 0.772 -0.097 0.072\n pub4cafqt -0.0278 0.096 -0.289 0.773 -0.216 0.161\n \n DIST \n \n sigma1 0.4837 0.020 24.478 0.000 0.445 0.522\n rho1 -0.4135 0.153 -2.700 0.007 -0.714 -0.113\n sigma0 0.3967 0.010 41.477 0.000 0.378 0.415\n rho0 0.0229 0.344 0.066 0.947 -0.652 0.697\n ================================================================================\n \n Warning:\n \n ---\n \n\n\nNext we plot the MTE based on our estimation results. As shown in the figure below the results are really close to the original results. The deviation seems to be negligible because we use a mock dataset.\n\n\n```python\nmte = plot_est_mte(rslt, 'files/replication.grmpy.yml')\n```\n\n\nFor a detailed overview on the theoretical economic background, more application examples as well as contact informations see the [online documentation](https://grmpy.readthedocs.io/en/latest/index.html). In addition, the most current code is available on [GitHub](https://github.com/OpenSourceEconomics/grmpy).\n\n\n\n", "meta": {"hexsha": "d2c9f2a0159f554e0e659520d117932f040fb04d", "size": 549185, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "promotion/grmpy_tutorial_notebook/grmpy_tutorial_notebook.ipynb", "max_stars_repo_name": "OpenSourceEconomics/grmpy", "max_stars_repo_head_hexsha": "13a262fb615c79829eb4869cbb6693c9c51fb101", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 18, "max_stars_repo_stars_event_min_datetime": "2018-04-10T01:08:22.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-23T02:37:24.000Z", "max_issues_repo_path": "promotion/grmpy_tutorial_notebook/grmpy_tutorial_notebook.ipynb", "max_issues_repo_name": "grmToolbox/grmpy", "max_issues_repo_head_hexsha": "13a262fb615c79829eb4869cbb6693c9c51fb101", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 127, "max_issues_repo_issues_event_min_datetime": "2017-08-02T13:29:26.000Z", "max_issues_repo_issues_event_max_datetime": "2018-03-27T19:42:07.000Z", "max_forks_repo_path": "promotion/grmpy_tutorial_notebook/grmpy_tutorial_notebook.ipynb", "max_forks_repo_name": "OpenSourceEconomics/grmpy", "max_forks_repo_head_hexsha": "13a262fb615c79829eb4869cbb6693c9c51fb101", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 13, "max_forks_repo_forks_event_min_datetime": "2018-04-28T09:46:22.000Z", "max_forks_repo_forks_event_max_datetime": "2020-11-06T09:32:27.000Z", "avg_line_length": 265.4349927501, "max_line_length": 96924, "alphanum_fraction": 0.8913917897, "converted": true, "num_tokens": 15715, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442250928250375, "lm_q2_score": 0.668880247169804, "lm_q1q2_score": 0.430909439321801}} {"text": "# Nor operator modeling and characterization\nIn this notebook we will construct a genetic network to model a Nor operator, a device that is repressed by either of two inputs, upload the simulated data to Flapjack, and then show how to characterize the operator based on this data. The GeneticNetwork will use two Receiver operators to drive the input repressors, and a Nor operator to produce the output based on these inputs.\n\nNOTE: In order to run this notebook, and characterize the Nor operator, you must first run Receiver1.ipynb and Receiver2.ipynb to generate data for the Receivers used in the network.\n\n## Import required packages\n\n\n```python\nfrom loica import *\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport getpass\n```\n\n## Make a connection to Flapjack\nNote here you should specify which instance of Flapjack you will use, whether it is local or the public instance for example.\n\n\n```python\nfrom flapjack import *\n#fj = Flapjack(url_base='flapjack.rudge-lab.org:8000')\nfj = Flapjack(url_base='localhost:8000')\nfj.log_in(username=input('Flapjack username: '), password=getpass.getpass('Password: '))\n```\n\n## Get or create Flapjack objects\nTo associate with the components of the genetic network and the simulated data with Flapjack we need the Ids of the appropriate objects. Note that if the objects already exist you will be prompted and can simply hit return to use the existing objects.\n\n\n```python\nreceiver1_vector = fj.get('vector', name='receiver1')\nreceiver2_vector = fj.get('vector', name='receiver2')\n```\n\n\n```python\nstudy = fj.create('study', name='Loica testing', description='Test study for demonstrating Loica')\n```\n\n\n```python\ndna = fj.create('dna', name='nor')\nvector = fj.create('vector', name='nor', dnas=dna.id)\n```\n\n\n```python\nsfp = fj.create('signal', name='SFP', color='green', description='Simulated fluorescent protein')\n```\n\n## Create the network with measurable reporter\nFirst we create a GeneticNetwork object and associate it with a Flapjack Vector (collection of DNA). The connection to Flapjack is optional, but we will use it here to upload data and characterize our components.\n\n\n```python\nnetwork = GeneticNetwork(vector=vector.id[0])\n```\n\n\n```python\nreporter = Reporter(name='SFP', color='green', degradation_rate=0, init_concentration=0, signal_id=sfp.id[0])\n```\n\n\n```python\nnetwork.add_reporter(reporter)\n```\n\n## Create and add the Receiver operators\nThe receiver operator responds to a signal $s$ to produce an output expression rate $\\phi(s)$ modeled as follows:\n\n\\begin{equation}\n \\phi(s)\n =\n \\frac\n {\n \\alpha_0 + \\alpha_1 (\\frac{s}{K})^n\n }\n {\n 1 + (\\frac{s}{K})^n\n }\n\\end{equation}\n\nHere we must create two Supplement objects to represent the signals, in this case modeling acyl-homoserine lactones (AHLs). The Receivers drive the input repressors, which then are the inputs to the Nor operator.\n\n\n```python\nahl1 = Supplement(name='AHL1')\nrepressor1 = Regulator('LacI')\nrec1 = Receiver(input=ahl1, output=repressor1, alpha=[0,100], K=1, n=2)\n\nahl2 = Supplement(name='AHL2')\nrepressor2 = Regulator('TetR')\nrec2 = Receiver(input=ahl2, output=repressor2, alpha=[0,100], K=1, n=2)\n\nnetwork.add_operators([rec1,rec2])\nnetwork.add_regulators([repressor1,repressor2])\n```\n\n## Create and add the Nor operator\nThe Nor operator represents a device which can be repressed by either of two repressors $r_1$ and $r_2$, and is modeled as follows, where $\\phi(r_1, r_2)$ is the output expression rate:\n\n\\begin{equation}\n \\phi(r_1, r_2)\n =\n \\frac{\n \\alpha_0 \n + \n \\alpha_1 (\\frac{r1}{K_1})^{n_1} \n + \n \\alpha_2 (\\frac{r2}{K_2})^{n_2}\n +\n \\alpha_3 (\\frac{r1}{K_1})^{n_1} (\\frac{r2}{K_2})^{n_2}\n }\n {\n 1 \n + \n (\\frac{r1}{K_1})^{n_1}\n + \n (\\frac{r2}{K_2})^{n_2}\n +\n (\\frac{r1}{K_1})^{n_1} (\\frac{r2}{K_2})^{n_2}\n }\n\\end{equation}\n\n\n\n```python\nnor = Hill2(input=[repressor1, repressor2], output=reporter, alpha=[1,0.1,0.1,0.1], K=[100,1], n=[4,2])\n```\n\n\n```python\nnetwork.add_operator(nor)\n```\n\n## Draw the GeneticNetwork as a graph\nWe can now make a visual representation of our GeneticNetwork to check it is wired up correctly.\n\n\n```python\nplt.figure(figsize=(3,3), dpi=150)\nnetwork.draw()\n```\n\n## Simulate the GeneticNetwork\nIn order to simulate the GeneticNetwork behaviour we need to specify the growth conditions in which it will operate. To do this we create a SimulatedMetabolism object which specifies growth functions.\n\n\n```python\ndef growth_rate(t):\n return gompertz_growth_rate(t, 0.05, 1, 1, 1)\n\ndef biomass(t):\n return gompertz(t, 0.05, 1, 1, 1)\n \nmetab = SimulatedMetabolism(biomass, growth_rate)\n\nmedia = fj.create('media', name='loica', description='Simulated loica media')\nstrain = fj.create('strain', name='loica', description='Loica test strain')\n```\n\nNow we can create Samples that contain our GeneticNetwork driven by the SimulatedMetabolism. We also need to specify the Media and Strain, in order to link to the Flapjack data model. To test the inverter behaviour we must also add the signals (ahls) at a range of concentrations.\n\n\n```python\n# Create list of samples \nsamples = []\nfor conc1 in np.append(0, np.logspace(-2, 2, 12)):\n for conc2 in np.append(0, np.logspace(-3, 1, 12)):\n for _ in range(1):\n sample = Sample(genetic_network=network, \n metabolism=metab,\n media=media.id[0],\n strain=strain.id[0])\n # Add AHL to samples at given concentration\n sample.set_supplement(ahl1, conc1)\n sample.set_supplement(ahl2, conc2)\n samples.append(sample)\n```\n\nGiven our Samples, we can now create an Assay which will simulate an experiment containing them. We need to specify the biomass signal in order to link to the Flapjack data model for later upload. Running the assay will simulate the behaviour of the GeneticNetwork.\n\n\n```python\nbiomass_signal = fj.create('signal', name='SOD', description='Simulated OD', color='black')\n```\n\n\n```python\nassay = Assay(samples, \n n_measurements=100, \n interval=0.24,\n name='Loica NOR',\n description='Simulated NOR generated by loica',\n biomass_signal_id=biomass_signal.id[0]\n )\nassay.run()\n```\n\n## Upload simulated data to Flapjack\n\n\n```python\nassay.upload(fj, study.id[0])\n```\n\nNow we can check that the simulation worked by plotting a heatmap using the PyFlapjack package to connect to the Flapjack API. This also allows us to see if we have covered the dynamic range of the Nor operator, in order to correctly characterize it.\n\n\n```python\nahl1_id = fj.get('chemical', name='AHL1').id[0]\nahl2_id = fj.get('chemical', name='AHL2').id[0]\n```\n\n\n```python\nfig = fj.plot(study=study.id, \n vector=vector.id,\n signal=sfp.id,\n type='Heatmap',\n analyte1=ahl1_id,\n analyte2=ahl2_id,\n function='Mean Expression',\n biomass_signal=biomass_signal.id[0],\n normalize='None',\n subplots='Signal',\n markers='Vector',\n plot='All data points'\n )\nfig\n```\n\n## Characterize the Nor operator from the uploaded data\n\n\n```python\nnor.characterize(fj, \n receiver1_vector.id,\n receiver2_vector.id, \n ahl1_id,\n ahl2_id,\n vector.id, \n media.id, \n strain.id, \n sfp.id, \n biomass_signal.id,\n gamma=0,\n init_x=[1,1,1,1,1,1,1,1]\n )\n```\n\n\n```python\nnor.alpha0, nor.alpha1, nor.alpha2, nor.alpha3\n```\n\n\n```python\nnor.rep1_K, nor.rep1_n\n```\n\n\n```python\nnor.rep2_K, nor.rep2_n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "92536ace719b233184c444e3fa41b66e66e2cf06", "size": 14040, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/Hill2.ipynb", "max_stars_repo_name": "RudgeLab/LOICA", "max_stars_repo_head_hexsha": "ea7f203ccd9642a6793537184dbccc764521f6fc", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-01-16T22:00:24.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-16T22:00:24.000Z", "max_issues_repo_path": "notebooks/Hill2.ipynb", "max_issues_repo_name": "RudgeLab/LOICA", "max_issues_repo_head_hexsha": "ea7f203ccd9642a6793537184dbccc764521f6fc", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 18, "max_issues_repo_issues_event_min_datetime": "2021-12-03T13:26:47.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-23T00:57:46.000Z", "max_forks_repo_path": "notebooks/Hill2.ipynb", "max_forks_repo_name": "RudgeLab/LOICA", "max_forks_repo_head_hexsha": "ea7f203ccd9642a6793537184dbccc764521f6fc", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 27.801980198, "max_line_length": 390, "alphanum_fraction": 0.5391737892, "converted": true, "num_tokens": 2080, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.746138993030751, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.4308918102596674}} {"text": "```python\nclean_up=True # removes gams-related files in work-folder if true\n%run StdPackages.ipynb\nos.chdir(py['main'])\nimport global_settings, ReadData, ShockFunction, Production, Household, GE, Invest,Trade,Government\nos.chdir(curr)\ndata_folder = os.getcwd()+'\\\\Data\\\\IO'\ngams_folder = os.getcwd()+'\\\\gamsmodels\\\\GE'\n```\n\n The file_gams_py_gdb0.gdx is still active and was not deleted.\n The file_gams_py_gdb1.gdx is still active and was not deleted.\n\n\n# Set up a dynamic, general equilibrium model\n\n*The current general equilibrium model is a small open economy that features exogenous long run interest-, inflation-, and growth rates. These settings are defined in the global settings:*\n\n\n```python\nname = 'GE'\ngs_v = 'gs_v1'\ntindex = range(1,4)\ngs = getattr(global_settings,gs_v)(kwargs_vals={'t':tindex})\n```\n\n## *1: Equilibrium*\n\nThe equilibrium clearing conditions are in general that:\n1. Goods markets clear: For $n\\in d\\_vS[s,n]$ the clearing entails\n$$\\begin{align}\n\\sum_{s\\in d\\_vS[s,n]} qS[t,s,n] = \\sum_{s\\in d\\_vD[s,n]} qD[t,s,n].\n\\end{align}$$\n2. Prices clear: The standard setup allows for tax wedges on outputs and inputs. The mapping from output prices $(PbT)$ to equilibrium prices $(Peq)$ differs across sectors and setups, and is thus left for the individual modules. This module simply includes the price conditions, for $(s,n)\\in d\\_tauD[s,n]$:\n$$\\begin{align}\n PwT[t,s,n] = Peq[t,n] + \\tau D[t,s,n]\n\\end{align}$$ \n\nWe note that the dummies $d\\_vS[s,n]$ and $d\\_tauD[s,n]$ does not cover all combinations of the $(s,n)$. In the case of the goods markets clearing, several markets are left out, simply as these are implicitly modelled, or we don't in fact care about the equilibrium. This includes:\n* The clearing of demand for foreignly produced goods (we don't track the total supply here), \n* the equilibrium of durables in the production sector (these are in equilibrium by default when investment goods are in equilibrium), \n* clearing of the assets market (in the simple small open economy, this is not needed. *NB: Add conditions for other types of global settings later.*).\n\n## *2: Data*\n\n### *2.1: IO-data*\n\nThe IO data should cover a single baseline year on the following:\n* Equilibrium prices on all final goods in the economy: $Peq[n]$ for all $n\\in fg[n]$. \n* Value of demand/supply on sector-goods level (s,n): $vD[s,n]$ and $vS[s,n]$.\n\nThe following reads in the IO data and defines a number of default subsets:\n\n\n```python\ndsheets = {'Production_v': data_folder+'\\\\IO_v.xlsx', 'Production_p': data_folder+'\\\\IO_p.xlsx'}\nGE_data = ReadData.read_data.main(dsheets,name='GE_data',components=['domstic','trade','HH','tax','invest'],balanced_data_check=False)\n```\n\n### *2.2: Technical data, production sectors*\n\nFor production sectors, we need data on the nesting structure, and technical parameters (CES parameters). For each production sector, we should have data on:\n* Nesting structure: Defined as mappings $map\\_all[s,n,nn]$, between branches/knots in the nesting tree. The *nesting\\_tree* class includes methods for reading these from excel-structured data.\n* ... \n\n## *3: Get components*\n\n*init model:*\n\n\n```python\ngm = GE.GE_v1(work_folder=work_folder,**{'data_folder': gams_folder,'name':name})\n```\n\n*Load individual modules:*\n\n\n```python\nmodules = {'p': Production.pr_dynamic(pickle_path=gm.model.settings.data_folder+'\\\\p'),\n 'HH': Household.hh_dynamic(pickle_path=gm.model.settings.data_folder+'\\\\HH'),\n 'inv': Invest.inv_dynamic(pickle_path=gm.model.settings.data_folder+'\\\\inv'),\n 'itory': Invest.itoryD(pickle_path=gm.model.settings.data_folder+'\\\\itory'),\n 'trade': Trade.trade_dynamic(pickle_path=gm.model.settings.data_folder+'\\\\trade'),\n 'G': Government.g_dynamic(pickle_path=gm.model.settings.data_folder+'\\\\G')}\n```\n\n*Create a merged model:*\n\n\n```python\nfrom gmspython import gmspython_i\n```\n\n\n```python\ngm_i = gmspython_i(work_folder=work_folder,**{'name':'int_model'})\n[gm_i.add_module(m) for n,m in modules.items()];\ngm_i.merge_settings()\n```\n\nAdd the equilibrium module.\n\n\n```python\nctree_kwargs = {'qS_endo': {'not': gm_i.g('exo',module='hh_dyn')}}\ngm.init_from_model_i(gm_i,ctree_kwargs=ctree_kwargs)\ngm.write()\ngm_i.add_module(gm)\ngm_i.merge_settings()\n```\n\n\n```python\ngm_i.write_and_run(write=False)\n```\n\n## *4: Run in calibration mode*\n\nSet state to 'DC', and self.write for all modules:\n\n\n```python\ngm_i.setstate('DC')\nfor m in gm_i.modules.values():\n m.write()\ngm_i.merge_settings()\n```\n\nCreate database with target exogenous values in baseline year:\n\n\n```python\nGE_t = DataBase.GPM_database()\nfor var in ('qS','qD','Peq'):\n GE_t[var] = Production.pr_static.add_t_to_variable(GE_data.get(var),gm_i.get('t0'))\nkwargs = {'n_steps': 10,'diff':True}\nkwargs_write = {'end': DB2Gams.run_text(g_exo = gm_i.exo_groups.keys(), g_endo = gm_i.endo_groups.keys(),blocks=gm_i.model.settings.get_conf('blocks'), name=gm_i.model.settings.get_conf('name'))}\ngm_i.setstate('B')\nfor m in gm_i.modules.values():\n m.write()\ngm_i.merge_settings()\ngm_i.write_and_run(write=False,overwrite=True,kwargs_write=kwargs_write,add_checkpoint='baseline')\n```\n\n\n```python\nshock_db,kwargs_shock = ShockFunction.sneaky_db(gm_i.model_instances['baseline'].out_db,GE_t,**{'n_steps':100})\ngm_i.model_instances['baseline'].solve_sneakily(from_cp=True,cp_init=gm_i.checkpoints['baseline'],shock_db=shock_db,kwargs_shock=kwargs_shock,model_name=gm_i.model.settings.conf['DC']['name'])\n```\n\nInspect solution:\n\n\n```python\ndb = gm_i.model_instances['baseline'].out_db\n```\n\n\n```python\nvar,s = 'PwT','b'\nplot_series(db.get(var).xs(s,level='s').unstack())\n```\n", "meta": {"hexsha": "8fb9d585bc2f92184b39e2a81695bd3980ca0bd4", "size": 136654, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/gmspython_i_tests.ipynb", "max_stars_repo_name": "ChampionApe/GPM_v05", "max_stars_repo_head_hexsha": "fa44f7db58d30002726daf014c63734091c85860", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-05-18T07:11:15.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-18T07:11:15.000Z", "max_issues_repo_path": "examples/gmspython_i_tests.ipynb", "max_issues_repo_name": "ChampionApe/GPM_v05", "max_issues_repo_head_hexsha": "fa44f7db58d30002726daf014c63734091c85860", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "examples/gmspython_i_tests.ipynb", "max_forks_repo_name": "ChampionApe/GPM_v05", "max_forks_repo_head_hexsha": "fa44f7db58d30002726daf014c63734091c85860", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 377.4972375691, "max_line_length": 126700, "alphanum_fraction": 0.9356550119, "converted": true, "num_tokens": 1510, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.43084481430802074}} {"text": "# Reinforcement Learning with GridWorld\n\nThis is the first iPython notebook from my series on Reinforcement Learning (RL).\nIn this notebook I will try to explain some notions of Tabular Solution Methods for RL. In order to do so I will closely follow the second edition of an amazing book by Sutton and Barto on Reinforcement Learning; a draft of the book can be downloaded [Here](http://incompleteideas.net/book/the-book-2nd.html). The content of this notebook roughly corresponds to the main ideas presented in much more detail in the first part, Tabular Solution Methods, of the above book. Some of the presented algorithms are implemented in the examples, usually following the theoretical explanation. For the examples to run you need to download GridWorld.py and StateSpace.py files, and place them under the same directory as this notebook. The dependencies are NumPy, PyGame, and of course iPython.\n\n## Basics\n\nI assume that my environment is a finite **Marcov Decision Process** (MDP); in case you don't recall what MDP is, skimming through the third chapter of the above mentioned book should be more than enough to get you going. The assumption of MDP is a reasonable one since I will work only with the GridWorld example.\n\nIn the RL, there are six main notions I will need going forward, namely agent, environment, policy, reward, value function, and model. The first two don't need much of an explanation. The next one, **policy**, is formally a set of principles guiding one's actions to achieve desired outcomes. Or how a mathematician would put it, it is a map from the set of states of the environment to the agent's actions. Rewards lie at the heart of the RL, they define the goals of an agent. To be precise, a **reward** is a number returned from the environment to the agent after performing an action. The goal of an agent is to maximise the total reward it receives over the long run. The **value function**, a map from the set of states (and possibly actions of the states) to the real numbers, helps an agent to do precisely this; it indicates the long run desirability of being in a certain state. As opposed to the reward, which indicates the immediate desirability of being in that state. The last element of the above list, **model**, is an agent's internal model of the environment. Models are used for planning and will perhaps be considered much later when I come to planning (TODO).\n\nThere are two types of value functions:\n* **state-value function**, $v_{\\pi}(s)$, a map from the space of states to the real numbers, indicating the possible future (discounted) rewards obtained when following a particular policy $\\pi$.\n* **action-value function**, $q_{\\pi}(s, a)$, a map from the space of states and their actions to the real numbers, indicating the possible future (discounted) rewards obtained when first selecting an action $a$ and thereafter following a particular policy $\\pi$.\n\nThe next notion, needing our attention, is the **Bellman optimality equation** for:\n\n* state-value function:\n\\begin{equation}\n\\begin{split}\nv_{*}(s) =& \\max_{a} \\mathbb{E} \\big[R_{t+1} + \\gamma v_{*}(S_{t+1}) \\,\\big|\\, S_{t} = s, A_{t} = a \\big] \\\\\n =& \\max_{a} \\sum_{s^{\\prime}, r} p(s^{\\prime}, r \\,|\\, s, a) \\big[r + \\gamma v_{*}(s^{\\prime}) \\big] \\,,\n\\end{split}\n\\end{equation}\n\n* action-value function:\n\\begin{equation}\n\\begin{split}\nq_{*}(s,a) =& \\mathbb{E} \\big[R_{t+1} + \\gamma \\max_{a^{\\prime}} q_{*}(S_{t+1}, a^{\\prime}) \\,\\big|\\, S_{t} = s, A_{t} = a \\big] \\\\\n =& \\sum_{s^{\\prime}, r} p(s^{\\prime}, r \\,|\\, s, a) \\big[r + \\gamma \\max_{a^{\\prime}} q_{*}(s^{\\prime}, a^{\\prime}) \\big] \\,.\n\\end{split}\n\\end{equation}\n\nWhere $\\mathbb{E}[\\cdots]$ denotes the expected value of the expression in brackets, $p(s^{\\prime}, r \\,|\\, s, a)$ indicates the transition probability to the state $s^{\\prime}$ with reward $r$ when in state $s$ one chooses the action $a$; $\\gamma$ is the discount factor. The reward received at the time step $t+1$ is represented by $R_{t+1}$; I use a capital character since this is a statistical variable as opposed to $r$ which denotes a concrete reward value. Both these equations are recursive; and as argued in the Sutton and Barto book, the Bellman optimality equation says that the value of a state under an optimal policy must be equal to the expected return for the best action from that state. Or in other words, if one knows the optimal value function for a state $s^{\\prime}$, then the optimal value function for a preceeding state $s$ is exactly the above Bellman optimality equation.\n\nSo far so good; now, let me move to dynamic programming.\n\n## Dynamic Programming\n\nThe general idea for solving RL problems, also known as **optimal control problems**, is to use a value function to organise the search for desirable policies. The first such a method usually presented is called **dynamic programming** (DP). DP denotes a set of methods for solving RL problems by solving the Bellman equation (defined below). Although DP methods are not very suitable for RL because they assume a perfect model of the environment as an MDP, they provide essential foundations for understanding more suitable methods.\n\nThe problem of finding the optimal policy in terms of DP methods splits into two separate sub-problems. The first, **policy evaluation** or prediction problem, handles the computation of the state-value function $v_{\\pi}$ for an arbitrary policy $\\pi$. The other problem, **policy improvement**, takes care of improving the current policy. In order to find the optimal policy, one alternates between these two processes, completing each before the other begins. This process is called **generalised policy iteration**. In general, to achieve the optimal policy one does not need to wait for a process to complete before the other starts.\n\n### Policy Evaluation\n\nLet me start with the policy evaluation algorithm, recall that for all states $s \\in \\mathcal{S}$ and a policy $\\pi$ one has the Bellman equation for $v_{\\pi}(s)$ (not to be confused with the Bellman optimality equation)\n\\begin{equation}\nv_{\\pi}(s) = \\sum_{a} \\pi(a\\,|\\,s) \\sum_{s^{\\prime}, r} p(s^{\\prime}, r \\,|\\, s,a) \\big[ r+ \\gamma v_{\\pi}(s^{\\prime}) \\big] \\,.\n\\end{equation}\n$\\pi(a\\,|\\,s)$ above denotes the probability of selecting action $a$ in state $s$ when following policy $\\pi$. The existence and uniqueness of $v_{\\pi}$ are guaranteed as long as $\\gamma < 1$ or eventual termination of an episode is guaranteed from all states under the policy $\\pi$.\n\nThere is a number of possibilities to solve these equations. The most preferred here is the iterative method called **iterative policy evaluation**. To this end one turns the Bellman equation for $v_{\\pi}(s)$ into an update rule, i.e.\n\\begin{equation}\nv_{k+1}(s) = \\sum_{a} \\pi(a\\,|\\,s) \\sum_{s^{\\prime}, r} p(s^{\\prime}, r \\,|\\, s,a) \\big[ r+ \\gamma v_{k}(s^{\\prime}) \\big] \\,.\n\\end{equation}\nThe initial approximation, $v_{0}(s)$, can be chosen arbitrarily for all non-terminal states $s$. The terminal states must be assigned value 0. Notice that this algorithm uses a general idea of **bootstrapping**, i.e. updating estimates of values of states based on estimates of the values of successor states. From the Bellman equation for $v_{\\pi}(s)$, one can clearly see that $v_{k} = v_{\\pi}$ is a fixed point. The convergence to this fixed point is guaranteed under the same conditions that guarantee the existence of $v_{\\pi}$.\n\n### Policy Improvement\n\nThe algorithm for policy improvement hinges on a result known as **policy improvement theorem** which states that for any pair of policies $\\pi$ and $\\pi^{\\prime}$ such that\n\\begin{equation}\nq_{\\pi}(s, \\pi^{\\prime}(s)) \\geq v_{\\pi}(s) \\,, \\qquad \\forall s \\in \\mathcal{S} \\,,\n\\end{equation}\nthe policy $\\pi^{\\prime}$ must be as good as or better than $\\pi$. From this it is straightforward to infer the algorithm for policy improvement; one just needs to construct a new policy, $\\pi^{\\prime}$, which selects actions more greedily with respect to $v_{\\pi}$ than the original policy. This new policy clearly fulfils the conditions of the policy improvement theorem and thus is better than or the same as the original policy.\n\nAs an example one can take the greedy policy, which picks the best possible action with respect to the current value function, defined by\n\\begin{equation}\n\\pi^{\\prime}(s) = {\\arg\\max}_{a} q_{\\pi}(s, a) \\,, \\qquad \\forall s \\in \\mathcal{S} \\,.\n\\end{equation}\nNow, consider the case in which this greedy policy is as good as but not better than the old one. Then the value function for these two policies must be equal, i.e. $v_{\\pi^{\\prime}} = v_{\\pi}$, which leads to the Bellman optimality equation\n\\begin{equation}\n\\begin{split}\nv_{\\pi^{\\prime}}(s) &= \\sum_{a} \\pi^{\\prime}(a\\,|\\,s) q_{\\pi^{\\prime}}(s, a) \\\\\n &= \\sum_{a} \\pi^{\\prime}(a\\,|\\,s) q_{\\pi}(s, a) \\\\\n &= \\max_{a} q_{\\pi}(s, a) \\\\\n &=\\max_{a} \\sum_{s^{\\prime}, r} p(s^{\\prime}, r \\,|\\, s,a) \\big[ r + \\gamma v_{\\pi}(s^{\\prime}) \\big] \\\\\n &=\\max_{a} \\sum_{s^{\\prime}, r} p(s^{\\prime}, r \\,|\\, s,a) \\big[ r + \\gamma v_{\\pi^{\\prime}}(s^{\\prime}) \\big] \\,,\n\\end{split}\n\\end{equation}\nfor all $s \\in \\mathcal{S}$. Hence, both policies must be optimal.\n\n### Policy Iteration\n\nNow one can combine the two above algorithms to build **policy iteration** algorithm.\n- start with an arbitrary policy $\\pi$\n- calculate its the value function, $v_{\\pi}$, with the policy evaluation algorithm (only approximately, since the precise calculation would require an infinite number of iteration steps)\n- improve $\\pi$ with the policy improvement algorithm to get a better policy $\\pi^{\\prime}$\n- calculate $v_{\\pi^{\\prime}}$\n\nContinue until a desirable (good enough) policy is obtained. This algorithm converges to the optimal policy $\\pi^{*}$.\n\n### Value Iteration\n\nAs was already suggested, one does not need to wait until the policy evaluation converges close enough to the value function. The process can be stopped right after the first iteration, giving an algorithm called **value iteration**. This has a particularly simple update step, namely\n\\begin{equation}\nv_{k+1}(s) = \\max_{a} \\sum_{s^{\\prime}, r} p(s^{\\prime}, r \\,|\\, s,a) \\big[ r + \\gamma v_{k}(s^{\\prime}) \\big] \\,.\n\\end{equation}\n\nSo far so good; let me now I present a short intro into the GridWorld environment used in all examples of this notebook.\n\n## Digression: GridWorld Environment\n\nBefore starting with the examples, let me first introduce the GridWorld environment I will be using in the examples of this notebook.\n\nI will use two classes from the file GridWorld.py, namely Worlds and GridWorld. The **class Worlds** contains several pre-defined worlds (cliff, mirror, maze, empty, track). Each of which is a numpy array of characters denoting the states. The constructor of this class does not take any arguments; and for getting a world one needs to use get member function which takes one argument, the name of the world.\n\nThere are 6 possible states and their corresponding characters:\n- d - default state;\n- s - start state;\n- t - trap state, upon accessing the agent receives the trap reward;\n- c - cliff state, upon accessing the agent receives the cliff reward and is put at one of the start states;\n- g - goal state, results in termination of the episode and the agent receives the goal reward;\n- w - wall state, the agent can't access this state; actions which would lead to this state do not change agent's position and return a penalty for hitting the wall.\n\nThe precise values of rewards and penalties are defined in the constructor of GridWorld. More precisely, the **GridWorld class** constructor takes four arguments:\n- world - one of the worlds returned by the class Worlds;\n- rewards - 5-element 1D array containing the goal, cliff, trap rewards, the step cost, and penalty for hitting the wall; the default values are [100,-50,-10,-1,-5];\n- useGUI - this is a boolean variable to indicate if a GUI should be used, it can be changed later;\n- name - a string to be displayed in the window's bar.\n\nA set of wall states is automatically added around the whole world in the constructor. The member functions for interacting with the GridWorld are:\n- reset - Resets the environment, i.e. move the agent to one of the start states, and sets the attribute done to False.\n- step - Performs selected action which must be one of the following characters: N, S, W, E. It returns agent's new coordinates, reward, done, and info.\n- render - If useGUI is True it displays the GUI; to see all agent's moves one needs to call it after every step. If action-value function is provided, the method displays agent's preferred actions, and state values in shades of green.\n- set_useGUI - Sets the boolean attribute useGUI.\n- get_shape - Returns the shape of the world array.\n- get_agent_coos - Returns agent's coordinates.\n- is_done - Returns a boolean variable indicating if the episode is over.\n\nThe last class to be used is the **StateSpace class** defined in StateSpace.py. This class will make our life easier when working with action-value functions, and when choosing actions or policies to follow. Its constructor takes 2 arguments the shape of the world and a list of possible actions, the default is ['N', 'S', 'W', 'E']. The most useful member functions are:\n- reset - which sets all the state-value and action-value functions to arrays of zeros.\n- eps_greedy_policy - returns an action or a list of probabilities for actions based on the action-value function.\n- update_actionVF - updates action-value function, either the whole function or at a particular state.\n- get_actionVF - returns the action-value function.\n\n### Example 1: cliff-walking GridWorld - Benchmarking\n\nThe Idea behind this first example is to see how well the agent will perform in an environment if he chooses a random policy.\n\nFirst, let me introduce the environment, cliff-walking GridWorld, I will be using in many examples. It consists of 12x4 accessible states for agent, with the start state and the goal state in the lower left and lower right corner, respectively. In between these two states there are 10 cliff states which upon accessing return agent on the start state and give him a large negative reward. In this setting the cliff reward is -100 points, every move costs agent -1 point, and accessing the goal states yields 13 points (so that following the optimal path gives +1). The slight modification compared to the original setting from the Sutton and Barto book is penalty for hitting a wall, -5 points, resulting in faster learning and helping to avoid useless moves.\n\nAs one can imagine, the results of the random policy will be quite bad; the reason is that the agent falls many times off the cliff, and takes a long time to find the goal state.\nRunning this policy for 1000 runs gives an average return of -71k with standard deviation of 73k.\n\nI don't recommend wasting too much time on this example. It is here just for the sake of completeness, especially if one would like to benchmark other environments in the GridWorld class. But if you really want to, go ahead and play around.\n\n\n```python\n# In case you restart the kernel and wish to continue with later examples run this cell again.\nimport numpy as np\nfrom StateSpace import StateSpace\nfrom GridWorld import GridWorld, Worlds\nfrom HelperFunctions_TSM import printTrainingReturns, printReturns, printHighestLowest\n\nactions = np.array(['N', 'S', 'W', 'E'])\nworlds = Worlds()\n```\n\n\n```python\ngw = GridWorld(world=worlds.get('cliff'), rewards=[13,-100,-5,-1,-5],\n name='Random', useGUI=False)\n\n# To get shape use gw.get_shape() instead of worlds.get('cliff').shape.\n# This is because GridWorld adds walls around the world.\nworldShape = gw.get_shape()\nstateSpace = StateSpace(worldShape, actions=actions)\n```\n\n\n```python\nepisodes = 2\nreturns = np.zeros(episodes)\ngw.set_useGUI(False) # in case you would like it with visualization, set this to True\n\nfor i in range(episodes):\n gw.reset()\n coos1 = gw.get_agent_coos()\n done = False\n cum_reward = 0\n while not done:\n action = stateSpace.eps_greedy_policy(coos=coos1, eps=1, ret_probs=False)\n coos2, reward, done, _ = gw.step(action)\n cum_reward = cum_reward + reward\n returns[i] = cum_reward\n\nprintReturns(returns)\nprintHighestLowest(returns)\n```\n\n Performance measure: the return per episode averaged over 2 episodes was -32493.0 with the standard deviation 28812.0.\n The highest return was -3681.0, and the lowest return was -61305.0.\n\n\n## Temporal-Difference Learning\n\nOften, to solve the problem of RL, especially when a complete knowledge of the environment is not known, it is desirable to use experience, i.e. sample sequences of states, actions, and rewards from actual or simulated interaction with an environment, as the main source of learning. **Temporal-difference** (TD) methods use this experience as well as bootstrapping. They are very similar to what is known as Monte Carlo methods, which will be considered as a special case of TD. As outlined above, TD also uses the principle of generalised policy iteration. I will start with TD prediction; I will also follow the notation outlined in the Sutton and Barto book, namely the use of capital letters for statistical variables.\n\n### TD(0), one-step TD\n\nNow, I would like to describe the simplest TD method. It is a prediction method, meaning that it iteratively evaluates the value function $V$ where a policy $\\pi$ is being followed (to prevent a cluttered notation I got rid of the subscript $\\pi$). Essentially, the algorithm instructs the agent to take an action, $A_{t}$, given by the policy, $\\pi$, which moves the agent into a new state, $S_{t+1}$, and returns the reward $R_{t+1}$.\n\nAt each time-step, this simplest TD method updates its estimates of state-value function as follows\n\\begin{equation}\nV(S_{t}) \\leftarrow V(S_{t}) + \\alpha \\big[ R_{t+1} + \\gamma V(S_{t+1}) - V(S_{t}) \\big] \\,,\n\\end{equation}\nwhere $\\alpha$ is known as learning rate. Notice the term in the square bracket, the first two terms are the reward just returned and the discounted value of the new state. The last term is negative of value of the old state. Hence, the bracket contains the error between an updated version of $V(S_{t})$ and the old version; the whole update moves $V(S_{t})$ in the direction of minimising the error. This TD method is called **TD(0)** or **one-step TD**, because it looks only one state ahead in order to update the state-value function. The error in the square bracket is usually referred to as the **TD error** and is denoted by\n\\begin{equation}\n\\delta_{t} = R_{t+1} + \\gamma V(S_{t+1}) - V(S_{t}) \\,.\n\\end{equation}\n\nThis prediction method can be combined with the policy improvement method to obtain a full algorithm for solving RL problems.\n\n### SARSA; on-policy, one-step TD control\n\nAll algorithms considered so far used for learning state-value function rather than action-value function. To solve the full RL problem (policy evaluation and policy improvement) with the state-value function requires the knowledge of all transition probabilities between states. All this si in principle known for GridWorld, however, it is quite cumbersome to obtain in explicit forms (and I'm also lazy :D). It is much easier to work with algorithms using the action-value function; SARSA is the first such algorithm I present.\n\nIt is basically identical to the TD(0) algorithm with the only difference that it uses transitions from one state-action pair to another rather than transitions from a state to another state. Hence, the update rule for action-value function is\n\\begin{equation}\nQ(S_{t}, A_{t}) \\leftarrow Q(S_{t}, A_{t}) + \\alpha \\big[ R_{t+1} + \\gamma Q(S_{t+1}, A_{t+1}) - Q(S_{t}, A_{t}) \\big] \\,.\n\\end{equation}\nAgain the quantity in the square brackets is called TD error. The name for this algorithm comes from the sequence of states, actions, and the corresponding reward, i.e. $(S_{t}, A_{t}, R_{t+1}, S_{t+1}, A_{t+1})$.\n\nThis is also the first algorithm I will fully implement in the example 2 below.\n\n### Example 2: cliff-walking with SARSA; on-policy, one-step TD control\n\nIn the previous exercise, I benchmarked the cliff-walking GridWorld environment with a random policy. Here, I will implement the SARSA algorithm. Let's see how much better it gets compared to the random policy.\n\nI encourage the reader to play around with the parameters for rewards, learning rate, discount or greediness epsilon. Since the visualisation imposes a huge overhead, I use GUI to display the agent's moves only for each 200th episode (starting with the first one).\n\n\n```python\ngw = GridWorld(world=worlds.get('cliff'), rewards=[13,-100,-5,-1,-5],\n name='SARSA; on-policy, one-step TD', useGUI=False)\nworldShape = gw.get_shape()\nstateSpace = StateSpace(worldShape, actions=actions)\n```\n\n\n```python\ndiscount = 0.9\nepsilon = 0.3\nepisodes = 2000\nreturns = np.zeros(episodes)\n\nfor i in range(episodes):\n if i%200 == 0: gw.set_useGUI(True) # render every 200-th episode, start with the first\n else: gw.set_useGUI(False) \n learning_rate = 0.2 / 2**(i//(episodes/5)) # divide by 2 after each 20% of training\n cum_reward = 0\n gw.reset()\n done = gw.is_done()\n coos_1 = gw.get_agent_coos()\n action_1 = stateSpace.eps_greedy_policy(coos=coos_1, eps=epsilon, ret_probs=False)\n\n while not done:\n coos_2, reward, done, info = gw.step(action_1)\n action_2 = stateSpace.eps_greedy_policy(coos=coos_2, eps=epsilon, ret_probs=False)\n cum_reward = cum_reward + reward\n\n actionValue_1 = stateSpace.get_actionVF()[coos_1][action_1 == actions]\n actionValue_2 = stateSpace.get_actionVF()[coos_2][action_2 == actions]\n updatedAV_1 = actionValue_1 + learning_rate*(reward + discount*actionValue_2 - actionValue_1)\n stateSpace.update_actionVF(value=updatedAV_1, coos=coos_1, action=action_1)\n\n coos_1, action_1 = coos_2, action_2\n gw.render(actionVF=stateSpace.get_actionVF())\n returns[i] = cum_reward\n\nprintTrainingReturns(returns)\nprintHighestLowest(returns)\n```\n\n Performance measure: the return per episode averaged over: \n - the first 10% of the training was -79.5 with the standard deviation 219.0;\n - 10% to 50% of the training was -38.2 with the standard deviation 54.4;\n - 50% to 90% of the training was -30.6 with the standard deviation 42.4;\n - the last 10% of the training was -35.0 with the standard deviation 53.6.\n The highest return was -1.0, and the lowest return was -2764.0.\n\n\nIn the next cell, the agent chooses its actions according to the greedy policy rather than the $\\epsilon$-greedy policy. This will demonstrate, that the agent learned the safest possible path.\n\n\n```python\nepsilon = 0\nepisodes = 100\nreturns = np.zeros(episodes)\n\nfor i in range(episodes):\n if i%20 == 0: gw.set_useGUI(True) # render every 20-th episode, start with the first\n else: gw.set_useGUI(False)\n cum_reward = 0\n gw.reset()\n done = gw.is_done()\n\n while not done:\n coos_1 = gw.get_agent_coos()\n action_1 = stateSpace.eps_greedy_policy(coos_1, epsilon)\n coos_2, reward, done, info = gw.step(action_1)\n cum_reward = cum_reward + reward\n gw.render(actionVF=stateSpace.get_actionVF())\n returns[i] = cum_reward\n\nprintReturns(returns)\nprintHighestLowest(returns)\n```\n\n Performance measure: the return per episode averaged over 100 episodes was -3.0 with the standard deviation 0.0.\n The highest return was -3.0, and the lowest return was -3.0.\n\n\n### Q-learning; off-policy, one-step TD control\n\nLet me now introduce one of the off-policy algorithms for TD control. A general difference between on-policy and off-policy methods is that an on-policy method evaluates or improves the policy being followed. Whereas, on the other hand, an off-policy method attempts to evaluate or improve a policy other than the one generating the data.\n\nThe Q-learning algorithm is defined by the following update\n\\begin{equation}\nQ(S_{t}, A_{t}) \\leftarrow Q(S_{t}, A_{t}) + \\alpha \\big[ R_{t+1} + \\gamma \\max_{a} Q(S_{t+1}, a) - Q(S_{t}, A_{t}) \\big] \\,.\n\\end{equation}\nThe maximum over actions hints that this algorithm will directly approximate the optimal policy $q^{*}$, independent of the policy followed. The convergence is guaranteed if all state-action pairs continue to be visited and updated. For more details I again refer the reader to the book mentioned at the beginning of this notebook.\n\n### Example 3: cliff-walking with Q-learning; off-policy, one-step TD control\n\nThis example implements the above Q-learning algorithm; it is insightful to compare the results obtained here with those obtained in the previous example (SARSA), both in the cliff-walking world.\n\nI again encourage the reader to play around with the parameters for rewards, learning rate, discount or greediness epsilon.\n\n\n```python\ngw = GridWorld(world=worlds.get('cliff'), rewards=[13,-100,-5,-1,-5],\n name='Q-learning; off-policy, one-step TD', useGUI=True)\nworldShape = gw.get_shape()\nstateSpace = StateSpace(worldShape, actions=actions)\n```\n\n\n```python\ndiscount = 0.9\nepsilon = 0.3\nepisodes = 1000\nreturns = np.zeros(episodes)\n\nfor i in range(episodes):\n if i%100 == 0: gw.set_useGUI(True) # render every 100-th episode, start with the first\n else: gw.set_useGUI(False)\n \n learning_rate = 0.2 / 2**(i//(episodes/5)) # divide by 2 after each 20% of training\n cum_reward = 0\n\n gw.reset()\n done = gw.is_done()\n while not done:\n coos_1 = gw.get_agent_coos()\n action_1 = stateSpace.eps_greedy_policy(coos_1, epsilon)\n coos_2, reward, done, info = gw.step(action_1)\n cum_reward = cum_reward + reward\n \n actionValue_1 = stateSpace.get_actionVF()[coos_1][action_1 == actions]\n max_AV_2 = np.amax(stateSpace.get_actionVF()[coos_2])\n updatedAV_1 = actionValue_1 + learning_rate*(reward + discount*max_AV_2 - actionValue_1)\n stateSpace.update_actionVF(updatedAV_1, coos_1, action_1)\n \n gw.render(actionVF=stateSpace.get_actionVF())\n returns[i] = cum_reward\n \nprintTrainingReturns(returns)\nprintHighestLowest(returns)\n```\n\n Performance measure: the return per episode averaged over: \n - the first 10% of the training was -237.7 with the standard deviation 419.2;\n - 10% to 50% of the training was -230.8 with the standard deviation 285.1;\n - 50% to 90% of the training was -203.1 with the standard deviation 248.0;\n - the last 10% of the training was -219.8 with the standard deviation 219.2.\n The highest return was 1.0, and the lowest return was -3842.0.\n\n\nThe result is approximately 5-times worse than SARSA. This is because the agent learned the optimal policy which is walking along the cliff; unfortunately the followed policy is $\\epsilon$-greedy which sometimes makes the agent jump off the cliff.\nIt is also instructive to see the agent following the greedy policy to confirm what I stated above.\n\n\n```python\nepsilon = 0\nepisodes = 100\nreturns = np.zeros(episodes)\n\nfor i in range(episodes):\n if i%20 == 0: gw.set_useGUI(True) # render every 20-th episode, start with the first\n else: gw.set_useGUI(False)\n cum_reward = 0\n gw.reset()\n done = gw.is_done()\n\n while not done:\n coos_1 = gw.get_agent_coos()\n action_1 = stateSpace.eps_greedy_policy(coos_1, epsilon)\n coos_2, reward, done, info = gw.step(action_1)\n cum_reward = cum_reward + reward\n gw.render(actionVF=stateSpace.get_actionVF())\n returns[i] = cum_reward\n\nprintReturns(returns)\nprintHighestLowest(returns)\n```\n\n Performance measure: the return per episode averaged over 100 episodes was 1.0 with the standard deviation 0.0.\n The highest return was 1.0, and the lowest return was 1.0.\n\n\n### Expected SARSA; one-step TD control\n\nA slight modification to the SARSA yields an algorithm called expected SARSA. Its update rule is as follows\n\\begin{equation}\nQ(S_{t}, A_{t}) \\leftarrow Q(S_{t}, A_{t}) + \\alpha \\big[ R_{t+1} + \\gamma \\sum_{a} \\pi(a \\,|\\, S_{t+1})Q(S_{t+1}, a) - Q(S_{t}, A_{t}) \\big] \\,.\n\\end{equation}\n\n### Example 4: cliff-walking with Expected SARSA; one-step TD control\n\nAlthough, this algorithm is more computationally complex, it seems that an agent is able to learn different policies based on the parameter $\\epsilon$. I encourage the reader to experiment with different values for $\\epsilon$.\n\n\n```python\ngw = GridWorld(world=worlds.get('cliff'), rewards=[13,-100,-5,-1,-5],\n name='Expected SARSA', useGUI=True)\nworldShape = gw.get_shape()\nstateSpace = StateSpace(worldShape, actions=actions)\n```\n\n\n```python\ndiscount = 0.9\nepsilon = 0.3\nepisodes = 1000\nreturns = np.zeros(episodes)\n\nfor i in range(episodes):\n if i%100 == 0: gw.set_useGUI(True) # render every 100-th episode, start with the first\n else: gw.set_useGUI(False)\n \n learning_rate = 0.2 / 2**(i//(episodes/5)) # divide by 2 after each 20% of training\n cum_reward = 0\n \n gw.reset()\n done = gw.is_done()\n while not done:\n coos_1 = gw.get_agent_coos()\n action_1 = stateSpace.eps_greedy_policy(coos_1, epsilon)\n coos_2, reward, done, info = gw.step(action_1)\n probs_2 = stateSpace.eps_greedy_policy(coos_2, epsilon, ret_probs=True)\n cum_reward = cum_reward + reward\n \n actionValue_1 = stateSpace.get_actionVF()[coos_1][action_1 == actions]\n expected_AV_2 = np.dot(probs_2, stateSpace.get_actionVF()[coos_2])\n updatedAV_1 = actionValue_1 + learning_rate*(reward + discount*expected_AV_2 - actionValue_1)\n stateSpace.update_actionVF(updatedAV_1, coos_1, action_1)\n \n gw.render(actionVF=stateSpace.get_actionVF())\n returns[i] = cum_reward\n \nprintTrainingReturns(returns)\nprintHighestLowest(returns)\n```\n\n Performance measure: the return per episode averaged over: \n - the first 10% of the training was -88.3 with the standard deviation 242.0;\n - 10% to 50% of the training was -34.1 with the standard deviation 49.7;\n - 50% to 90% of the training was -37.2 with the standard deviation 56.1;\n - the last 10% of the training was -27.4 with the standard deviation 36.8.\n The highest return was -1.0, and the lowest return was -2343.0.\n\n\nFrom the 'greedy' visualization in the next cell, one can see that for $\\epsilon=0.3$ the agent learns an intermediate policy between the optimal and the safe. A little bit of experimentation gives different policies for different values of $\\epsilon$. One can approximately write:\n* $0.00 < \\epsilon \\leq 0.02$ gives the greedy policy,\n* $0.02 < \\epsilon \\leq 0.32$ gives the intermediate policy,\n* $0.32 < \\epsilon \\leq 0.05$ gives the safe policy.\n\n\n```python\nepsilon = 0\nepisodes = 100\nreturns = np.zeros(episodes)\n\nfor i in range(episodes):\n if i%20 == 0: gw.set_useGUI(True) # render every 20-th episode, start with the first\n else: gw.set_useGUI(False)\n cum_reward = 0\n gw.reset()\n done = gw.is_done()\n\n while not done:\n coos_1 = gw.get_agent_coos()\n action_1 = stateSpace.eps_greedy_policy(coos_1, epsilon)\n coos_2, reward, done, info = gw.step(action_1)\n cum_reward = cum_reward + reward\n gw.render(actionVF=stateSpace.get_actionVF())\n returns[i] = cum_reward\n\nprintReturns(returns)\nprintHighestLowest(returns)\n```\n\n Performance measure: the return per episode averaged over 100 episodes was -1.0 with the standard deviation 0.0.\n The highest return was -1.0, and the lowest return was -1.0.\n\n\n## Eligibility Traces a.k.a. $\\lambda$-bootstrapping\n\nSo far all the algorithms used were one-step; meaning that they looked only one step ahead before performing an update to the value function. The efficiency of a learning method can be dramatically improved by using **n-step updates**. The update to a state-value function using n-step method is defined as follows\n\\begin{equation}\n\\begin{split}\nV_{t+n}(S_{t}) &= V_{t+n-1}(S_{t}) + \\alpha \\big[ G_{t:t+n} - V_{t+n-1}(S_{t}) \\big] \\,, \\\\\nG_{t:t+n} &= \\sum_{i=1}^{n} \\gamma^{i-1} R_{t+i} + \\gamma^{n} V_{t+n-1}(S_{t+n}) \\,,\n\\end{split}\n\\end{equation}\nwhere $G_{t:t+n}$ is called **n-step return**. Now, instead of performing only one n-step update, one can update the value function using a weighted sum of many n-step returns with weights summing up to 1. A particularly useful choice of weights is $(1-\\lambda)\\lambda$, where $\\lambda$ is called **decay factor**; and the return corresponding to this choice of weights,\n\\begin{equation}\nG_{t}^{\\lambda} = (1-\\lambda) \\sum_{n=1}^{\\infty} \\lambda^{n-1} G_{t:t+n} \\,,\n\\end{equation}\nis called **$\\lambda$-return**. It can easily be shown that if an episode terminates at a terminal state $S_{t}$, and all subsequent n-step returns are equal to $G_{t}$, the above equation takes the form of\n\\begin{equation}\nG_{t}^{\\lambda} = (1-\\lambda) \\sum_{n=1}^{T-t-1} \\lambda^{n-1} G_{t:t+n} + \\lambda^{T-t-1}G_{t} \\,.\n\\end{equation}\n\n### SARSA($\\lambda$)\n\nLet me present an algorithms called **SARSA($\\lambda$)**, which uses the machinery I recalled in the above paragraph as well as the concept of **eligibility traces**. The eligibility trace is a vector, $z$, of the same shape as the value function.\n\nAt the beginning of the episode, eligibility trace vector (ETV) is initialised to zero. Then, at each time-step, $t$, the element of ETV corresponding to the state-action pair just visited is either\n- incremented by $1$, called **accumulating trace** or\n- set to $1$, called **replacing trace**;\n\nand then fades away by $\\gamma\\lambda$ (a product of the discount and decay factors), more precisely\n\\begin{equation}\n\\begin{split}\nz_{-1} &= 0 \\,,\\\\\nz_{t} &= \\gamma\\lambda z_{t-1} \\,,\\\\\nz_{t}(S_{t}, A_{t}) &= z_{t-1}(S_{t}, A_{t}) + 1 \\quad \\text{or}\\,,\\\\\nz_{t}(S_{t}, A_{t}) &= 1 \\,.\n\\end{split}\n\\end{equation}\nAt each time step $t$, the action-value function is updated as follows\n\\begin{equation}\n\\begin{split}\n\\delta_{t} &= R_{t+1} + \\gamma Q(S_{t+1}, A_{t+1}) - Q(S_{t}, A_{t}) \\,,\\\\\nQ(\\cdot, \\cdot) &\\leftarrow Q(\\cdot, \\cdot) + \\alpha\\delta_{t} z_{t}(\\cdot, \\cdot) \\,,\n\\end{split}\n\\end{equation}\nwhere, as usual, $\\delta_{t}$ denotes the TD error at the time-step, $t$. If one chooses $V(\\cdot)$ rather than $Q(\\cdot, \\cdot)$, and eliminates the actions, $A_{t}$ and $A_{t+1}$, the resulting algorithm is the one known as **TD($\\lambda$)**.\n\nTo better understand this algorithm one might consider what happens at different values of the decay parameter $\\lambda$.\n- if $\\lambda = 0$, then the eligibility trace has precisely one non-zero component at each time, namely $z_{t}(S_{t}, A_{t})$; and only the corresponding component of the action-value function is updated, namely $Q_{t}(S_{t}, A_{t})$. Hence, the whole algorithm reduces to the one-step TD method or TD(0).\n- if $0 < \\lambda < 1$, then more of the preceding states are updated; but due to the decay factor, the states further back in time are updated less and less.\n- if $\\lambda = 1$, then the resulting algorithm precisely corresponds to discounted every-visit Monte Carlo for accumulating trace, and first-visit Monte Carlo for replacing trace.\n\n#### Digression:\nEvery-visit **Monte Carlo** is an algorithm where a full episode is generated upfront, and only then the value function for each state (state-action pair) is updated according to actual returns seen during the episode. I.e. for each state, $S$ appearing in the episode compute the discounted return, $G$, following it; and then update the state-value function as follows\n\\begin{equation}\nV(S) \\leftarrow V(S) + \\alpha \\big[ G - V(S) \\big] \\,.\n\\end{equation}\nThe first-visit is analogous with the only difference that the return is calculated only for the states visited for the first time in the episode, all the other visits use the return calculated when visited for the first time.\n\n### Example 5: mirror GridWorld with SARSA($\\lambda$); replacing trace\n\nThis example puts the theoretical discussion above in work. But since the cliff-walking world was solved with one step TD methods, the more sophisticated methods wouldn't add much value, and one would have hard times to judge how much better they are.\n\nTherefore, in this and the next example I will use mirror GridWorld rather than cliff-walking because it has much larger state space. The mirror GridWorld also contains so-called trap states, these are accessible for the agent but results in a penalty; I set the penalty to be -10. All this requires a bit more sophisticated approach than just one-step algorithms.\n\n\n```python\ngw = GridWorld(world=worlds.get('mirror'), rewards=[100,-100,-10,-1,-5],\n name='SARSA with replacing trace', useGUI=True)\nworldShape = gw.get_shape()\nstateSpace = StateSpace(worldShape, actions=actions)\n```\n\n\n```python\ndiscount = 0.95\ndecay = 0.9\nepisodes = 3000\nreturns = np.zeros(episodes)\n\nfor i in range(episodes):\n if i%300 == 9: gw.set_useGUI(True) # render every 100-th episode, start with the 10th\n else: gw.set_useGUI(False)\n cum_reward = 0\n gw.reset()\n done = gw.is_done()\n \n learning_rate = 0.2 / 2**(i//(episodes/5)) # divide by 2 after each 20% of training\n epsilon = 0.3 * (1 - (i/episodes)**2) # diminishes from 0.3 to 0\n replacing_trace = np.zeros( (worldShape[0], worldShape[1], len(actions)) )\n \n coos_1 = gw.get_agent_coos()\n action_1 = stateSpace.eps_greedy_policy(coos_1, epsilon)\n \n while not done:\n coos_2, reward, done, info = gw.step(action_1)\n action_2 = stateSpace.eps_greedy_policy(coos_2, epsilon)\n cum_reward = cum_reward + reward\n \n replacing_trace = discount*decay*replacing_trace\n replacing_trace[coos_1][action_1 == actions] = 1\n \n actionVF = stateSpace.get_actionVF()\n td_error = reward + discount*actionVF[coos_2][action_2 == actions] - actionVF[coos_1][action_1 == actions]\n actionVF = actionVF + learning_rate*td_error*replacing_trace\n stateSpace.update_actionVF(value=actionVF)\n\n coos_1, action_1 = coos_2, action_2\n \n gw.render(actionVF=stateSpace.get_actionVF())\n returns[i] = cum_reward\n \nprintTrainingReturns(returns)\nprintHighestLowest(returns)\n```\n\n Performance measure: the return per episode averaged over: \n - the first 10% of the training was -63.5 with the standard deviation 726.1;\n - 10% to 50% of the training was 40.2 with the standard deviation 36.7;\n - 50% to 90% of the training was 58.1 with the standard deviation 17.3;\n - the last 10% of the training was 71.3 with the standard deviation 8.5.\n The highest return was 80.0, and the lowest return was -11002.0.\n\n\nIf you try to find the optimal path yourself, you will see that each start state has precisely one optimal path with return of 82. Here the agent learns a policy with an average return being roughly between 65 and 75 per episode, this indicates that it is actually a decent solution. However, one can do better!\n\n### True online SARSA($\\lambda$) (dutch trace)\n\nThe **true online TD($\\lambda$)** is currently the best performing TD algorithm (this holds only for tabular methods and for linear function approximation). To derive the full theoretical understanding of this algorithm is a bit messy. (Alright, you guessed it. I'm way too lazy to write it here. :D ) Hence, I will only state the update rules below and leave the experimenting and comparing with other algorithms presented in this notebook to the reader. For further explanation and references the reader is welcomed to consult Sutton and Barto book or the actual [paper](https://arxiv.org/abs/1512.04087) by Seijen et al. where this algorithm was introduced.\n\nAs usual, ETV is initialized to zero at the beginning of each episode; then at each time step, $t$, it is updated as follows\n\\begin{equation}\n\\begin{split}\nz_{t} &= \\gamma\\lambda z_{t-1} \\,,\\\\\nz_{t}(S_{t}, A_{t}) &= 1 - \\alpha\\gamma\\lambda z_{t-1}(S_{t}, A_{t})\\,.\\\\\n\\end{split}\n\\end{equation}\nThe update rule for the action-value function reads\n\\begin{equation}\n\\begin{split}\nQ_{t+1} &= Q_{t} + \\alpha \\delta_{t} z_{t} + \\alpha \\big[ Q_{t}(S_{t}, A_{t}) - Q_{t-1}(S_{t}, A_{t}) \\big] z_{t} \\,,\\\\\nQ_{t+1}(S_{t}, A_{t}) &= Q_{t+1}(S_{t}, A_{t}) - \\alpha \\big[ Q_{t}(S_{t}, A_{t}) - Q_{t-1}(S_{t}, A_{t}) \\big] \\,,\n\\end{split}\n\\end{equation}\nwhere $\\delta_{t}$ is as usual.\n\n### Example 6: mirror GridWorld with true online SARSA($\\lambda$); dutch trace\n\nAnd the actual implementation follows.\n\n\n```python\ngw = GridWorld(world=worlds.get('mirror'), rewards=[100,-100,-10,-1,-5],\n name='True online SARSA', useGUI=True)\nworldShape = gw.get_shape()\nstateSpace = StateSpace(worldShape, actions=actions)\n```\n\n\n```python\ndiscount = 0.95\ndecay = 0.9\nepisodes = 3000\nreturns = np.zeros(episodes)\n\nfor i in range(episodes):\n if i%300 == 9: gw.set_useGUI(True) # render every 100-th episode, start with the 10th\n else: gw.set_useGUI(False)\n cum_reward = 0\n\n gw.reset()\n done = gw.is_done()\n \n learning_rate = 0.2 / 2**(i//(episodes/5)) # divide by 2 after each 20% of training\n epsilon = 0.3 * (1 - (i/episodes)**2) # diminishes from 0.3 to 0\n\n dutch_trace = np.zeros( (worldShape[0], worldShape[1], len(actions)) )\n old_actionVF = stateSpace.get_actionVF()\n \n coos_1 = gw.get_agent_coos()\n action_1 = stateSpace.eps_greedy_policy(coos_1, epsilon)\n \n while not done:\n coos_2, reward, done, info = gw.step(action_1)\n action_2 = stateSpace.eps_greedy_policy(coos_2, epsilon)\n cum_reward = cum_reward + reward\n \n # dutch trace update\n temp1 = learning_rate*discount*decay*dutch_trace[coos_1][action_1 == actions]\n dutch_trace = discount*decay*dutch_trace\n dutch_trace[coos_1][action_1 == actions] = 1 - temp1\n \n # actionVF update\n actionVF = stateSpace.get_actionVF()\n td_error = reward + discount*actionVF[coos_2][action_2 == actions] \\\n -actionVF[coos_1][action_1 == actions]\n temp2 = actionVF[coos_1][action_1 == actions] \\\n -old_actionVF[coos_1][action_1 == actions]\n actionVF = actionVF + learning_rate*(td_error + temp2)*dutch_trace\n actionVF[coos_1][action_1 == actions] = actionVF[coos_1][action_1 == actions] - learning_rate*temp2\n \n # save the current actionVF to the old one and update it\n old_actionVF = stateSpace.get_actionVF()\n stateSpace.update_actionVF(value=actionVF)\n \n coos_1, action_1 = coos_2, action_2\n \n gw.render(actionVF=stateSpace.get_actionVF())\n returns[i] = cum_reward\n\nprintTrainingReturns(returns)\nprintHighestLowest(returns)\n```\n\n Performance measure: the return per episode averaged over: \n - the first 10% of the training was -37.2 with the standard deviation 357.9;\n - 10% to 50% of the training was 34.1 with the standard deviation 58.5;\n - 50% to 90% of the training was 60.3 with the standard deviation 23.9;\n - the last 10% of the training was 74.0 with the standard deviation 10.4.\n The highest return was 82.0, and the lowest return was -4654.0.\n\n\nThis algorithm gives a slightly better result for the final policy with an average return being roughly between 68-78 per episode; but more importantly it gives much better results during training. To see that, one can compare the intermediate results, namely after first 10%, between 10%-50%, and between 50%-90% of this algorithm (true online SARSA) and the plain SARSA.\n\n### Example 7: The Maze GridWorld.\n\nIt is also insightful to play around with the other worlds, especially with the maze. I recommend to change discount from 1.0 to e.g. 0.8 and see what effect it has on delayed rewards.\n\n\n```python\ngw = GridWorld(world=worlds.get('maze'), rewards=[100,-100,-10,-1,-5],\n name='The Maze', useGUI=True)\nworldShape = gw.get_shape()\nstateSpace = StateSpace(worldShape, actions=actions)\n```\n\n\n```python\ndiscount = 1\ndecay = 0.5\nepisodes = 2500\n\nreturns = np.zeros(episodes)\n\nfor i in range(episodes):\n if i%250 == 149: gw.set_useGUI(True) # render every 250-th episode, start with the 150th\n else: gw.set_useGUI(False)\n cum_reward = 0\n\n gw.reset()\n done = gw.is_done()\n\n learning_rate = 0.2 / 2**(i//(episodes/5)) # divide by 2 after each 20% of training\n epsilon = 0.5 * (1.0 - (i/episodes)**2) # diminishes from 0.3 to 0\n\n\n dutch_trace = np.zeros( (worldShape[0], worldShape[1], len(actions)) )\n old_actionVF = stateSpace.get_actionVF()\n \n coos_1 = gw.get_agent_coos()\n action_1 = stateSpace.eps_greedy_policy(coos_1, epsilon)\n \n while not done:\n coos_2, reward, done, info = gw.step(action_1)\n action_2 = stateSpace.eps_greedy_policy(coos_2, epsilon)\n cum_reward = cum_reward + reward\n \n # dutch trace update\n temp1 = learning_rate*discount*decay*dutch_trace[coos_1][action_1 == actions]\n dutch_trace = discount*decay*dutch_trace\n dutch_trace[coos_1][action_1 == actions] = 1 - temp1\n \n # actionVF update\n actionVF = stateSpace.get_actionVF()\n td_error = reward + discount*actionVF[coos_2][action_2 == actions] \\\n -actionVF[coos_1][action_1 == actions]\n temp2 = actionVF[coos_1][action_1 == actions] \\\n -old_actionVF[coos_1][action_1 == actions]\n actionVF = actionVF + learning_rate*(td_error + temp2)*dutch_trace\n actionVF[coos_1][action_1 == actions] = actionVF[coos_1][action_1 == actions] - learning_rate*temp2\n \n # save the current actionVF to the old one and update it\n old_actionVF = stateSpace.get_actionVF()\n stateSpace.update_actionVF(value=actionVF)\n \n coos_1, action_1 = coos_2, action_2\n \n gw.render(actionVF=stateSpace.get_actionVF())\n returns[i] = cum_reward\n \nprintTrainingReturns(returns)\nprintHighestLowest(returns)\n```\n\n Performance measure: the return per episode averaged over: \n - the first 10% of the training was -995.7 with the standard deviation 1688.6;\n - 10% to 50% of the training was -245.5 with the standard deviation 371.9;\n - 50% to 90% of the training was -40.2 with the standard deviation 123.6;\n - the last 10% of the training was 26.8 with the standard deviation 58.4.\n The highest return was 80.0, and the lowest return was -19474.0.\n\n\n***The end of the notebook.***\n", "meta": {"hexsha": "56ad192c9ad1d81351932b06c3ae648b7aad5087", "size": 57986, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Tabular_Solution_Methods/RL_with_GridWorld.ipynb", "max_stars_repo_name": "JurajX/Reinforcement_Learning", "max_stars_repo_head_hexsha": "c746c99d7f7442b2c7db542e0973b13773695bba", "max_stars_repo_licenses": ["Unlicense"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Tabular_Solution_Methods/RL_with_GridWorld.ipynb", "max_issues_repo_name": "JurajX/Reinforcement_Learning", "max_issues_repo_head_hexsha": "c746c99d7f7442b2c7db542e0973b13773695bba", "max_issues_repo_licenses": ["Unlicense"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Tabular_Solution_Methods/RL_with_GridWorld.ipynb", "max_forks_repo_name": "JurajX/Reinforcement_Learning", "max_forks_repo_head_hexsha": "c746c99d7f7442b2c7db542e0973b13773695bba", "max_forks_repo_licenses": ["Unlicense"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 54.0914179104, "max_line_length": 1190, "alphanum_fraction": 0.6222364019, "converted": true, "num_tokens": 12457, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936430859597, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.43079227892109545}} {"text": "```python\n# @hidden_cell\n# The project token is an authorization token that is used to access project resources like data sources, connections, and used by platform APIs.\nfrom project_lib import Project\nproject = Project(project_id='7bb82b7e-8a0f-4854-aac0-182e777a31bb', project_access_token='p-261bcd06faa60bd899ecbd4f9c1746c366285a8b')\n\n```\n\n# Time Series Forecasting using the NOAA Weather Data of JFK Airport (New York)\n\nThis notebook relates to the NOAA Weather Dataset - JFK Airport (New York). The dataset contains 114,546 hourly observations of 12 local climatological variables (such as temperature and wind speed) collected at JFK airport. This dataset can be obtained for free from the IBM Developer [Data Asset Exchange](https://developer.ibm.com/exchanges/data/all/jfk-weather-data/).\n\nIn this notebook we explore approaches to predicting future temperatures by using the time-series dataset.\n\n### Table of Contents:\n* [0. Prerequisites](#cell0)\n* [1. Read the Cleaned Data](#cell1)\n* [2. Explore Baseline Models](#cell2)\n* [3. Train Statistical Time-series Analysis Models](#cell3)\n* [Authors](#cell4)\n\n\n### 0. Prerequisites\n\nBefore you run this notebook complete the following steps:\n- Insert a project token\n- Import required modules\n\n#### Insert a project token\n\nWhen you import this project from the Watson Studio Gallery, a token should be automatically generated and inserted at the top of this notebook as a code cell such as the one below:\n\n```python\n# @hidden_cell\n# The project token is an authorization token that is used to access project resources like data sources, connections, and used by platform APIs.\nfrom project_lib import Project\nproject = Project(project_id='YOUR_PROJECT_ID', project_access_token='YOUR_PROJECT_TOKEN')\npc = project.project_context\n```\n\nIf you do not see the cell above, follow these steps to enable the notebook to access the dataset from the project's resources:\n\n* Click on `More -> Insert project token` in the top-right menu section\n\n\n\n* This should insert a cell at the top of this notebook similar to the example given above.\n\n > If an error is displayed indicating that no project token is defined, follow [these instructions](https://dataplatform.cloud.ibm.com/docs/content/wsj/analyze-data/token.html?audience=wdp&context=data).\n\n* Run the newly inserted cell before proceeding with the notebook execution below\n\n#### Import required modules\n\nImport and configure the required modules.\n\n\n```python\nimport numpy as np\nimport pandas as pd\nfrom sklearn.metrics import mean_squared_error\nfrom statsmodels.tsa.statespace.sarimax import SARIMAX\nfrom matplotlib import pyplot as plt\n```\n\n\n\n### 1. Read the Cleaned Data\n\nWe start by reading the cleaned dataset that was created in the project notebook `Part 1 - Data Cleaning`. \n\n**Note:** if you haven't yet run this notebook, run it first; otherwise the cells below will not work.\n\n\n```python\ndef get_file_handle(fname):\n # Project data path for the raw data file\n data_path = project.get_file(fname)\n data_path.seek(0)\n return data_path\n\n# Using pandas to read the data \n# Since the `DATE` column consists date-time information, we use Pandas parse_dates keyword for easier data processing\ndata_path = get_file_handle('jfk_weather_cleaned.csv')\ndata = pd.read_csv(data_path, parse_dates=['DATE'])\n# Set date index\ndata = data.set_index(pd.DatetimeIndex(data['DATE']))\ndata.drop(['DATE'], axis=1, inplace=True)\ndata.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        visibilitydry_bulb_temp_fwet_bulb_temp_fdew_point_temp_frelative_humiditywind_speedstation_pressuresea_level_pressureprecipaltimeter_settingwind_direction_sinwind_direction_cos
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                                        2010-01-01 01:00:006.033.032.031.092.00.00.029.97829.990.0129.99
                                        2010-01-01 02:00:006.033.033.032.096.00.00.029.97829.990.0229.99
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                                        2010-01-01 04:00:005.033.033.032.096.00.00.029.95829.970.0029.97
                                        2010-01-01 05:00:005.033.032.031.092.00.00.029.93829.960.0029.95
                                        \n
                                        \n\n\n\nFor purposes of time-series modeling, we will restrict our analysis to a 2-year sample of the dataset to avoid overly long model-training times. \n\n\n```python\nsample = data['2016-01-01':'2018-01-01']\nsample.info()\n```\n\n \n DatetimeIndex: 17568 entries, 2016-01-01 00:00:00 to 2018-01-01 23:00:00\n Data columns (total 12 columns):\n visibility 17568 non-null float64\n dry_bulb_temp_f 17568 non-null float64\n wet_bulb_temp_f 17568 non-null float64\n dew_point_temp_f 17568 non-null float64\n relative_humidity 17568 non-null float64\n wind_speed 17568 non-null float64\n station_pressure 17568 non-null float64\n sea_level_pressure 17568 non-null float64\n precip 17568 non-null int64\n altimeter_setting 17568 non-null float64\n wind_direction_sin 17568 non-null float64\n wind_direction_cos 17568 non-null float64\n dtypes: float64(11), int64(1)\n memory usage: 1.7 MB\n\n\n#### Create Training/Validation/Test Splits\n\nBefore we attempt any time-series analysis and prediction, we should split the dataset into training, validation and test sets. We use a portion of the data for training, and a portion of _future_ data for our validation and test sets.\n\nIf we instead trained a model on the full dataset, the model would learn to be very good at making predictions on that particular dataset, essentially just copying the answers it knows. However, when presented with data the model _has not seen_ , it would perform poorly since it has not learned how to generalize its answers.\n\nBy training on a portion of the dataset and testing the model's performance on another portion of the dataset (which data the model has not seen in training), we try to avoid our models \"over-fitting\" the dataset and make them better at predicting temperatures given unseen, future data. This process of splitting the dataset and evaluating a model's performance on the validation and test sets is commonly known as [cross-validation](https://en.wikipedia.org/wiki/Cross-validation_(statistics)).\n\nBy default here we use 80% of the data for the training set and 10% each for validation and test sets.\n\n\n```python\ndef split_data(data, val_size=0.1, test_size=0.1):\n \"\"\"\n Splits data to training, validation and testing parts\n \"\"\"\n ntest = int(round(len(data) * (1 - test_size)))\n nval = int(round(len(data) * (1 - test_size - val_size)))\n\n df_train, df_val, df_test = data.iloc[:nval], data.iloc[nval:ntest], data.iloc[ntest:]\n \n return df_train, df_val, df_test\n\n\n# Create data split\ndf_train, df_val, df_test = split_data(sample)\n\nprint('Total data size: {} rows'.format(len(sample)))\nprint('Training set size: {} rows'.format(len(df_train)))\nprint('Validation set size: {} rows'.format(len(df_val)))\nprint('Test set size: {} rows'.format(len(df_test)))\n```\n\n Total data size: 17568 rows\n Training set size: 14054 rows\n Validation set size: 1757 rows\n Test set size: 1757 rows\n\n\n\n\n### 2. Explore Baseline Models\n\nIn this section, we'll create a few simple predictive models of temperature, using shifting and rolling averages. These will serve as a baseline against which we can compare more sophisticated models.\n\nUsing values at recent timesteps (such as the most recent timestep `t-1` and second-most recent timestep `t-2`) to predict the current value at time `t` is what's known as persistence modeling, or using the last observed value to predict the next following value. These preceding timesteps are often referred to in time-series analysis as `lags`. So, the value at time `t-1` is known as the `1st lag` and the value at time `t-2` is the `2nd lag`.\n\nWe can also create baselines based on rolling (or moving) averages. This is a time-series constructed by averaging each lagged value up to the selected lag. For example, a 6-period (or 6-lag) rolling avearge is the average of the previous 6 hourly lags `t-6` to `t-1`.\n\nOur baseline models will be:\n1. `1st lag` - i.e. values at `t-1`\n2. `2nd lag` - i.e. values at `t-2`\n3. `6-lag` rolling average\n4. `12-lag` rolling average\n\n\n```python\n# define the column containing the data we wish to model - in this case Dry Bulb Temperature (F)\nY_COL = 'dry_bulb_temp_f'\n\n# Use shifting and rolling averages to predict Y_COL (t)\nn_in = 2\nn_out = 1\nfeatures = [Y_COL]\nn_features = len(features)\n\n# create the baseline on the entire sample dataset.\n# we will evaluate the prediction error on the validation set\nbaseline = sample[[Y_COL]].loc[:]\nbaseline['{} (t-1)'.format(Y_COL)] = baseline[Y_COL].shift(1)\nbaseline['{} (t-2)'.format(Y_COL)] = baseline[Y_COL].shift(2)\nbaseline['{} (6hr rollavg)'.format(Y_COL)] = baseline[Y_COL].rolling('6H').mean()\nbaseline['{} (12hr rollavg)'.format(Y_COL)] = baseline[Y_COL].rolling('12H').mean()\nbaseline.dropna(inplace=True)\nbaseline.head(10)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        dry_bulb_temp_fdry_bulb_temp_f (t-1)dry_bulb_temp_f (t-2)dry_bulb_temp_f (6hr rollavg)dry_bulb_temp_f (12hr rollavg)
                                        DATE
                                        2016-01-01 02:00:0043.043.044.043.33333343.333333
                                        2016-01-01 03:00:0042.043.043.043.00000043.000000
                                        2016-01-01 04:00:0043.042.043.043.00000043.000000
                                        2016-01-01 05:00:0042.043.042.042.83333342.833333
                                        2016-01-01 06:00:0041.042.043.042.33333342.571429
                                        2016-01-01 07:00:0041.041.042.042.00000042.375000
                                        2016-01-01 08:00:0041.041.041.041.66666742.222222
                                        2016-01-01 09:00:0040.041.041.041.33333342.000000
                                        2016-01-01 10:00:0041.040.041.041.00000041.909091
                                        2016-01-01 11:00:0041.041.040.040.83333341.833333
                                        \n
                                        \n\n\n\nNext, we will plot data from our validation dataset to get a sense for how well these baseline models predict the next hourly temperature. Note thatd we only use a few days of data in order to make the plot easier to view.\n\n\n```python\n# plot first 7 days of the validation set, 168 hours \nstart = df_val.index[0]\nend = df_val.index[167]\nsliced = baseline[start:end]\n```\n\n\n```python\n# Plot baseline predictions sample\ncols = ['dry_bulb_temp_f', 'dry_bulb_temp_f (t-1)', 'dry_bulb_temp_f (t-2)', 'dry_bulb_temp_f (6hr rollavg)', 'dry_bulb_temp_f (12hr rollavg)']\nsliced[cols].plot()\n\nplt.legend(['t', 't-1', 't-2', '6hr', '12hr'], loc=2, ncol=3)\nplt.title('Baselines for First 7 Days of Validation Set')\nplt.ylabel('Temperature (F)')\nplt.tight_layout()\nplt.rcParams['figure.dpi'] = 100\nplt.show()\n```\n\n#### Evaluate baseline models\n\nAs you can perhaps see from the graph above, the _lagged_ baselines appear to do a better job of forecasting temperatures than the _rolling average_ baselines.\n\nIn order to evaluate our baseline models more precisely, we need to answer the question _\"how well do our models predict future temperature?\"_. In regression problems involving prediction of a numerical value, we often use a measure of the difference between our predicted value and the actual value. This is referred to as an error measure or error metric. A common measure is the Mean Squared Error (MSE):\n\n\\begin{equation}\nMSE = \\frac{1}{n} \\sum_{i=1}^{n}{(y_i - \\hat y_i)^{2}}\n\\end{equation}\n\nThis is the average of the squared differences between predicted values $ \\hat y $ and actual values $ y $.\n\nBecause the MSE is in \"units squared\" it can be difficult to interpet, hence the Root Mean Squared Error (RMSE) is often used:\n\n\\begin{equation}\nRMSE = \\sqrt {MSE} \n\\end{equation}\n\nThis is the square root of the MSE, and is in the same units as the values $ y $. We can compare the RMSE (and MSE) values for different models and say that the model that has the lower MSE is better at predicting temperatures, all things equal. Note that MSE and RMSE will grow large quickly if the differences between predicted and actual values are large. This may or may not be a desired quality of your error measure. In this case, it is probably a good thing, since a model that makes large mistakes in temperature prediction will be much less useful than one which makes small mistakes.\n\nNext, we calculate the RMSE measure for each of our baseline models, on the full validation set.\n\n\n```python\n# Calculating baseline RMSE\nstart_val = df_val.index[0]\nend_val = df_val.index[-1]\nbaseline_val = baseline[start_val:end_val]\n\nbaseline_y = baseline_val[Y_COL]\nbaseline_t1 = baseline_val['dry_bulb_temp_f (t-1)']\nbaseline_t2 = baseline_val['dry_bulb_temp_f (t-2)']\nbaseline_avg6 = baseline_val['dry_bulb_temp_f (6hr rollavg)']\nbaseline_avg12 = baseline_val['dry_bulb_temp_f (12hr rollavg)']\n\nrmse_t1 = round(np.sqrt(mean_squared_error(baseline_y, baseline_t1)), 2)\nrmse_t2 = round(np.sqrt(mean_squared_error(baseline_y, baseline_t2)), 2)\nrmse_avg6 = round(np.sqrt(mean_squared_error(baseline_y, baseline_avg6)), 2)\nrmse_avg12 = round(np.sqrt(mean_squared_error(baseline_y, baseline_avg12)), 2)\n\nprint('Baseline t-1 RMSE: {0:.3f}'.format(rmse_t1))\nprint('Baseline t-2 RMSE: {0:.3f}'.format(rmse_t2))\nprint('Baseline 6hr rollavg RMSE: {0:.3f}'.format(rmse_avg6))\nprint('Baseline 12hr rollavg RMSE: {0:.3f}'.format(rmse_avg12))\n```\n\n Baseline t-1 RMSE: 1.690\n Baseline t-2 RMSE: 2.880\n Baseline 6hr rollavg RMSE: 3.080\n Baseline 12hr rollavg RMSE: 5.010\n\n\nThe RMSE results confirm what we saw in the graph above. It is clear that the _rolling average_ baselines perform poorly. In fact, the `t-2` lagged baseline is also not very good. It appears that the best baseline model is to simply use the current hour's temperature to predict the next hour's temperature!\n\nCan we do better than this simple baseline using more sophisticated models?\n\n\n\n### 2. Train Statistical Time-series Analysis Models\n\n\nIn the previous section, we saw that a simple `lag-1` baseline model performed reasonably well at forecasting temperature for the next hourly time step. This is perhaps not too surprising, given what we know about hourly temperatures. Generally, the temperature in a given hour will be quite closely related to the temperature in the previous hour. This phenomenon is very common in time-series analysis and is known as [autocorrelation](https://en.wikipedia.org/wiki/Autocorrelation) - that is, the time series is `correlated` with previous values of itself. More precisely, the values at time `t` are correlated with lagged values (which could be `t-1`, `t-2` and so on).\n\nAnother thing we saw previously is the concept of _moving averages_. In this case the moving-average baseline was not that good at prediction. However it is common in many time-series for a moving average to capture some of the underlying structure and be useful for prediction.\n\nIn order to make our model better at predicting temperature, ideally we would want to take these aspects into account. Fortunately, the statistical community has a long history of analyzing time series and has created many different forecasting models.\n\nHere, we will explore one called SARIMAX - the **S**easonal **A**uto**R**egressive **I**ntegrated **M**oving **A**verage with e**X**ogenous regressors model. \n\nThis sounds like a very complex name, but if we look at the components of the name, we see that it includes `autocorrelation` (this is what auto regressive means) and `moving averages`, which are the components mentioned above. \n\nThe SARIMAX model also allows including a _seasonal_ model component as well as handling *exogenous* variables, which are external to the time-series value itself. For example, for temperature prediction we may wish to take into account not just previous temperature values, but perhaps other weather features which may have an effect on temperature (such as humidity, rainfall, wind, and so on).\n\nFor the purposes of this notebook, we will not explore modeling of seasonal components or exogenous variables.\n\nIf we drop the \"S\" and \"X\" from the model, we are left with an [ARIMA model](https://en.wikipedia.org/wiki/Autoregressive_integrated_moving_average) (Auto-regressive Integrated Moving Average). This is a very commonly used model for time-series analysis and we will use it in this notebook by only specifying the relevant model components of the full SARIMAX model.\n\n#### 2.1 Replicating a baseline model\n\nAs a starting point, we will see how we can use SARIMAX to create a simple model that in fact replicates one of the baselines we created previously. Auto-regression, as we have seen, means using values from preceding time periods to predict the current value. Recall that one of our baseline models was the `1st lag` or `t-1` model. In time-series analysis this is referred to as an **AR(1)** model, meaning an **A**uto-**R**egressive model for `lag 1`.\n\nTechnically, the AR(1) model is not exactly the same as our baseline model. A statistical time series model like SARIMAX learns a set of `weights` to apply to each component of the model. These weights are set so as to best fit the dataset. We can think of our baseline as setting the `weight` for the `t-1` lag to be exactly `1`. In practice, our time-series model will not have a weight of exactly `1` (though it will likely be very close to that), hence the predictions will be slightly different.\n\nNow, lets fit our model to the dataset. First, we will set up the model inputs by taking the temperature column of our dataframe. We do this for training and validation sets.\n\n\n```python\nX_train = df_train[Y_COL]\nX_val = df_val[Y_COL]\nX_both = np.hstack((X_train, X_val))\n```\n\nHere we created a variable called `X_both` to cover both the training and validation data. This is required later when we forecast values for our SARIMAX model, in order to give the model access to all the datapoints for which it must create forecasts. Note that the forecasts themselves will only be based on the _model weights_ learned from the training data (this is important for over-fitting as we have seen above)!\n\nThe SARIMAX model takes an argument called `order`: this specifies the components of the model and itself has 3 parts: `(p, d, q)`. `p` denotes the lags for the AR model and `q` denotes the lags for the MA model. We will not cover the `d` parameter here. Taken together this specifies the parameters of the [ARIMA](https://en.wikipedia.org/wiki/Autoregressive_integrated_moving_average) model portion of SARIMAX.\n\nTo create an AR(1) model, we set the `order` to be `(1, 0, 0)`. This sets up the AR model to be a `lag 1` model. Then, we fit our model on the training data and inspect a summary of the trained model. \n\n\n```python\norder = (1, 0, 0)\nmodel_ar1 = SARIMAX(X_train, order=order)\nresults_ar1 = model_ar1.fit()\nresults_ar1.summary()\n```\n\n /opt/conda/envs/Python36/lib/python3.6/site-packages/statsmodels/tsa/base/tsa_model.py:171: ValueWarning: No frequency information was provided, so inferred frequency H will be used.\n % freq, ValueWarning)\n /opt/conda/envs/Python36/lib/python3.6/site-packages/statsmodels/tsa/base/tsa_model.py:191: FutureWarning: Creating a DatetimeIndex by passing range endpoints is deprecated. Use `pandas.date_range` instead.\n start=index[0], end=index[-1], freq=freq)\n\n\n\n\n\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        Statespace Model Results
                                        Dep. Variable: dry_bulb_temp_f No. Observations: 14054
                                        Model: SARIMAX(1, 0, 0) Log Likelihood -27650.431
                                        Date: Wed, 25 Mar 2020 AIC 55304.863
                                        Time: 11:06:13 BIC 55319.964
                                        Sample: 01-01-2016 HQIC 55309.889
                                        - 08-08-2017
                                        Covariance Type: opg
                                        \n\n\n \n\n\n \n\n\n \n\n
                                        coef std err z P>|z| [0.025 0.975]
                                        ar.L1 0.9996 0.000 4277.653 0.000 0.999 1.000
                                        sigma2 2.9937 0.024 125.518 0.000 2.947 3.040
                                        \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        Ljung-Box (Q): 16360.93 Jarque-Bera (JB): 3656.68
                                        Prob(Q): 0.00 Prob(JB): 0.00
                                        Heteroskedasticity (H): 0.97 Skew: 0.15
                                        Prob(H) (two-sided): 0.26 Kurtosis: 5.48


                                        Warnings:
                                        [1] Covariance matrix calculated using the outer product of gradients (complex-step).\n\n\n\nThere's quite a lot of information printed out in the model summary above. Much of it is related to the statistical properties of our model.\n\nThe most important thing for now is to look at the second table, where we can see a `coef` value of `0.9996` for the weight `ar.L1`. This tells us the model has set a weight for the `1st lag` component of the AR model to be `0.9996`. This is almost `1` and hence we should expect the prediction results to indeed be close to our `t-1` baseline.\n\nLet's create our model forecast on the validation dataset. We will then plot a few data points like we did with our baseline models (using 7 days of validation data) and compute the RMSE value based on the full validation set.\n\n\n```python\nfull_data_ar1 = SARIMAX(X_both, order=order)\nmodel_forecast_ar1 = full_data_ar1.filter(results_ar1.params)\n```\n\n\n```python\nstart = len(X_train)\nend = len(X_both)\nforecast_ar1 = model_forecast_ar1.predict(start=start, end=end - 1, dynamic=False)\n\n# plot actual vs predicted values for the same 7-day window for easier viewing\nplt.plot(sliced[Y_COL].values)\nplt.plot(forecast_ar1[:168], color='r', linestyle='--')\nplt.legend(['t', 'AR(1)'], loc=2)\nplt.title('AR(1) Model Predictions for First 7 Days of Validation Set')\nplt.ylabel('Temperature (F)')\nplt.tight_layout()\nplt.show()\n```\n\nWe can see that the plot looks almost identical to the plot above, for the `t` and `t-1 baseline` values.\n\nNext, we compute the RMSE values.\n\n\n```python\n# compute print RMSE values\nrmse_ar1 = np.sqrt(mean_squared_error(baseline_val[Y_COL], forecast_ar1))\nprint('AR(1) RMSE: {0:.3f}'.format(rmse_ar1))\nprint('Baseline t-1 RMSE: {0:.3f}'.format(rmse_t1))\n```\n\n AR(1) RMSE: 1.692\n Baseline t-1 RMSE: 1.690\n\n\nWe can see that the RMSE values for the validation set also almost identical.\n\n#### 2.2 Create a more complex model\n\nOne of our baseline models was a `lag 2` model, i.e. `t-2`. We saw that it performed a lot worse than the `t-1` baseline. Intuitively, this makes sense, since we are throwing away a lot of information about the most recent lag `t-1`. However, the `t-2` lag still provides some useful information. In fact, for temperature prediction it's likely that the last few hours can provide some value.\n\nFortunately, our ARIMA model framework provides an easy way to incorporate further lag information. We can construct a model that includes _both_ the `t-1` and `t-2` lags. This is an **AR(2)** model (meaning an auto-regressive model up to lag `2`). We can specify this with the model order parameter `p=2`.\n\n\n```python\norder = (2, 0, 0)\nmodel_ar2 = SARIMAX(X_train, order=order)\nresults_ar2 = model_ar2.fit()\nresults_ar2.summary()\n```\n\n /opt/conda/envs/Python36/lib/python3.6/site-packages/statsmodels/tsa/base/tsa_model.py:171: ValueWarning: No frequency information was provided, so inferred frequency H will be used.\n % freq, ValueWarning)\n /opt/conda/envs/Python36/lib/python3.6/site-packages/statsmodels/tsa/base/tsa_model.py:191: FutureWarning: Creating a DatetimeIndex by passing range endpoints is deprecated. Use `pandas.date_range` instead.\n start=index[0], end=index[-1], freq=freq)\n\n\n\n\n\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        Statespace Model Results
                                        Dep. Variable: dry_bulb_temp_f No. Observations: 14054
                                        Model: SARIMAX(2, 0, 0) Log Likelihood -26867.232
                                        Date: Wed, 25 Mar 2020 AIC 53740.465
                                        Time: 11:06:14 BIC 53763.117
                                        Sample: 01-01-2016 HQIC 53748.005
                                        - 08-08-2017
                                        Covariance Type: opg
                                        \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        coef std err z P>|z| [0.025 0.975]
                                        ar.L1 1.3242 0.006 220.175 0.000 1.312 1.336
                                        ar.L2 -0.3248 0.006 -53.852 0.000 -0.337 -0.313
                                        sigma2 2.6779 0.020 136.374 0.000 2.639 2.716
                                        \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        Ljung-Box (Q): 4350.76 Jarque-Bera (JB): 7579.33
                                        Prob(Q): 0.00 Prob(JB): 0.00
                                        Heteroskedasticity (H): 0.99 Skew: -0.01
                                        Prob(H) (two-sided): 0.75 Kurtosis: 6.60


                                        Warnings:
                                        [1] Covariance matrix calculated using the outer product of gradients (complex-step).\n\n\n\nThis time, the results table indicates a weight for variable `ar.L1` _and_ `ar.L2`. Note the values are now quite different from `1` (or `0.5` say, for a simple equally-weighted model). Next, we compute the RMSE on the validation set. \n\n\n```python\nfull_data_ar2 = SARIMAX(X_both, order=order)\nmodel_forecast_ar2 = full_data_ar2.filter(results_ar2.params)\n\nstart = len(X_train)\nend = len(X_both)\nforecast_ar2 = model_forecast_ar2.predict(start=start, end=end - 1, dynamic=False)\n\n# compute print RMSE values\nrmse_ar2 = np.sqrt(mean_squared_error(baseline_val[Y_COL], forecast_ar2))\nprint('AR(2) RMSE: {0:.3f}'.format(rmse_ar2))\nprint('AR(1) RMSE: {0:.3f}'.format(rmse_ar1))\nprint('Baseline t-1 RMSE: {0:.3f}'.format(rmse_t1))\n```\n\n AR(2) RMSE: 1.526\n AR(1) RMSE: 1.692\n Baseline t-1 RMSE: 1.690\n\n\nWe've improved the RMSE value by including information from the first two lags.\n\nIn fact, you will see that if you continue to increase the `p` parameter value, the RMSE will continue to decrease, indicating that a few recent lags provide useful information to our model.\n\n#### 2.3 Incorporate moving averages\n\nFinally, what if we also include moving average information in our model? The ARIMA framework makes this easy to do, by setting the order parameter `q`. A value of `q=1` specifies a **MA(1)** model (including the first lag `t-1`), while `q=6` would include all the lags from `t-1` to `t-6`.\n\nNote that the moving average model component is a little different from the simple moving or rolling averages computed in the baseline models. The [definition of the MA model](https://en.wikipedia.org/wiki/Moving-average_model) is rather technical, but conceptually you can think of it as using a form of weighted moving average (compared to our baseline which would be a simple, unweighted average).\n\nLet's add an MA(1) component to our AR(2) model.\n\n\n```python\norder = (2, 0, 1)\nmodel_ar2ma1 = SARIMAX(X_train, order=order)\nresults_ar2ma1 = model_ar2ma1.fit()\nresults_ar2ma1.summary()\n```\n\n /opt/conda/envs/Python36/lib/python3.6/site-packages/statsmodels/tsa/base/tsa_model.py:171: ValueWarning: No frequency information was provided, so inferred frequency H will be used.\n % freq, ValueWarning)\n /opt/conda/envs/Python36/lib/python3.6/site-packages/statsmodels/tsa/base/tsa_model.py:191: FutureWarning: Creating a DatetimeIndex by passing range endpoints is deprecated. Use `pandas.date_range` instead.\n start=index[0], end=index[-1], freq=freq)\n\n\n\n\n\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        Statespace Model Results
                                        Dep. Variable: dry_bulb_temp_f No. Observations: 14054
                                        Model: SARIMAX(2, 0, 1) Log Likelihood -26667.060
                                        Date: Wed, 25 Mar 2020 AIC 53342.120
                                        Time: 11:06:15 BIC 53372.322
                                        Sample: 01-01-2016 HQIC 53352.172
                                        - 08-08-2017
                                        Covariance Type: opg
                                        \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        coef std err z P>|z| [0.025 0.975]
                                        ar.L1 1.6697 0.015 114.448 0.000 1.641 1.698
                                        ar.L2 -0.6700 0.015 -45.898 0.000 -0.699 -0.641
                                        ma.L1 -0.3825 0.016 -23.348 0.000 -0.415 -0.350
                                        sigma2 2.6026 0.019 139.868 0.000 2.566 2.639
                                        \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        Ljung-Box (Q): 2762.35 Jarque-Bera (JB): 8834.61
                                        Prob(Q): 0.00 Prob(JB): 0.00
                                        Heteroskedasticity (H): 1.00 Skew: -0.04
                                        Prob(H) (two-sided): 0.92 Kurtosis: 6.88


                                        Warnings:
                                        [1] Covariance matrix calculated using the outer product of gradients (complex-step).\n\n\n\nWe see the results table shows an additional weight value for `ma.L1`, our MA(1) component. Next, we compare the RMSE to the other models and finally plot all the model forecasts together - _note_ we use a much smaller 48-hour window to make the plot readable for illustrative purposes. \n\n\n```python\nfull_data_ar2ma1 = SARIMAX(X_both, order=order)\nmodel_forecast_ar2ma1 = full_data_ar2ma1.filter(results_ar2ma1.params)\n\nstart = len(X_train)\nend = len(X_both)\nforecast_ar2ma1 = model_forecast_ar2ma1.predict(start=start, end=end - 1, dynamic=False)\n\n# compute print RMSE values\nrmse_ar2ma1 = np.sqrt(mean_squared_error(baseline_val[Y_COL], forecast_ar2ma1))\nprint('AR(2) MA(1) RMSE: {0:.3f}'.format(rmse_ar2ma1))\nprint('AR(2) RMSE: {0:.3f}'.format(rmse_ar2))\nprint('AR(1) RMSE: {0:.3f}'.format(rmse_ar1))\nprint('Baseline t-1 RMSE: {0:.3f}'.format(rmse_t1))\n```\n\n AR(2) MA(1) RMSE: 1.491\n AR(2) RMSE: 1.526\n AR(1) RMSE: 1.692\n Baseline t-1 RMSE: 1.690\n\n\n\n```python\n# plot actual vs predicted values for a smaller 2-day window for easier viewing\nhrs = 48\nplt.plot(sliced[Y_COL][:hrs].values)\nplt.plot(forecast_ar1[:hrs], color='r', linestyle='--')\nplt.plot(forecast_ar2[:hrs], color='g', linestyle='--')\nplt.plot(forecast_ar2ma1[:hrs], color='c', linestyle='--')\nplt.legend(['t', 'AR(1)', 'AR(2)', 'AR(2) MA(1)'], loc=2, ncol=1)\nplt.title('ARIMA Model Predictions for First 48 hours of Validation Set')\nplt.ylabel('Temperature (F)')\nplt.tight_layout()\nplt.show()\n```\n\nWe've again managed to reduce the RMSE value for our model, indicating that adding the MA(1) component has improved our forecast!\n\nCongratulations! You've applied the basics of time-series analysis for forecasting hourly temperatures. See if you can further improve the RMSE values by exploring the different values for the model parameters `p`, `q` and even `d`!\n\n \n### Authors\n\nThis notebook was created by the [Center for Open-Source Data & AI Technologies](http://codait.org).\n\nCopyright \u00a9 2019 IBM. This notebook and its source code are released under the terms of the MIT License.\n\n
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                                        \n", "meta": {"hexsha": "ce8fd17c97b3e58d11ce4cfead42ab920a091b13", "size": 274620, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "DAX-Weather-Project/notebooks/Part 3 - Time Series Forecasting.ipynb", "max_stars_repo_name": "mariusmos/coursera-data-sci", "max_stars_repo_head_hexsha": "7400d0373b847895e794d311425e1633f651b452", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "DAX-Weather-Project/notebooks/Part 3 - Time Series Forecasting.ipynb", "max_issues_repo_name": "mariusmos/coursera-data-sci", "max_issues_repo_head_hexsha": "7400d0373b847895e794d311425e1633f651b452", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "DAX-Weather-Project/notebooks/Part 3 - Time Series Forecasting.ipynb", "max_forks_repo_name": "mariusmos/coursera-data-sci", "max_forks_repo_head_hexsha": "7400d0373b847895e794d311425e1633f651b452", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 274620.0, "max_line_length": 274620, "alphanum_fraction": 0.8964132255, "converted": true, "num_tokens": 11610, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358411176238, "lm_q2_score": 0.672331705744791, "lm_q1q2_score": 0.4307197878198609}} {"text": "In this tutorial we show how to run ULTRAFAST for the ground state optimization of the Heisenberg model\n\n\\begin{equation}\n\\hat{H} = J_\\text{ex}\\sum_{\\langle ij \\rangle}\\hat{S}_i \\cdot \\hat{S}_j\n\\end{equation}\ndefined on a $4\\times 4$ lattice with $J_\\text{ex}=1$.\n\nThe input data, such as network size, the optimization hyperparameters and model-dependent quantities used in this tutorial are set in the file \"Examples/groundstate/main.jl\". In particular, we initizilize an RBM with $\\alpha=2$ and we perform 300 iterations with learning rate $\\eta = 0.005$. We choose 2000 Monte Carlo samples and we parallelize the simulation across 3 workers (plus the master worker).\n\n\n```julia\ninclude(\"src/main/main.jl\")\n```\n\n # Starting ground state optimization for the Heisenberg Hamiltonian, with 16 = 4 x 4 spins and \u03b1=2\n # Number of workers = 4\n # The following hyper-parameters are used:\n # Number of sweeps = 2000, iteration step = 300, learning rate = 0.005\n Iteration step #10\n Energy: -36.50576815538164 +- 107.69156695555571\n \n Iteration step #20\n Energy: -36.330318828100246 +- 107.18255386312377\n \n Iteration step #30\n Energy: -36.142818099093866 +- 98.68939285039563\n \n Iteration step #40\n Energy: -38.04390026971374 +- 82.4028073226377\n \n Iteration step #50\n Energy: -41.89800004524652 +- 27.135194063022084\n \n Iteration step #60\n Energy: -43.71165762441394 +- 11.772478931051854\n \n Iteration step #70\n Energy: -43.94130136350545 +- 6.158103582545235\n \n Iteration step #80\n Energy: -44.32638214465312 +- 6.616803390444755\n \n Iteration step #90\n Energy: -44.412975348332935 +- 5.928300888612564\n \n Iteration step #100\n Energy: -44.755645434557486 +- 3.420937057571623\n \n Iteration step #110\n Energy: -44.77479382254967 +- 2.762591616085437\n \n Iteration step #120\n Energy: -44.7758901611388 +- 2.408306905896985\n \n Iteration step #130\n Energy: -44.80120779656822 +- 2.244388427219228\n \n Iteration step #140\n Energy: -44.788500466187905 +- 2.5240727771716465\n \n Iteration step #150\n Energy: -44.80465694275794 +- 2.551059596625919\n \n Iteration step #160\n Energy: -44.86724104900637 +- 2.1813866529625274\n \n Iteration step #170\n Energy: -44.841138965996244 +- 2.2846891196629358\n \n Iteration step #180\n Energy: -44.76202341704043 +- 2.6582511173400554\n \n Iteration step #190\n Energy: -44.84649384785449 +- 2.2696634822557344\n \n Iteration step #200\n Energy: -44.78228724780536 +- 3.213952585669619\n \n Iteration step #210\n Energy: -44.83497505313653 +- 2.6799449977684104\n \n Iteration step #220\n Energy: -44.80729166848035 +- 2.475411577003716\n \n Iteration step #230\n Energy: -44.83909684866624 +- 2.382773640950612\n \n Iteration step #240\n Energy: -44.86376308487456 +- 2.7860708973746116\n \n Iteration step #250\n Energy: -44.83605617226564 +- 2.4201719845434946\n \n Iteration step #260\n Energy: -44.767410362939344 +- 2.248737502315309\n \n Iteration step #270\n Energy: -44.79746298852114 +- 2.490243265289373\n \n Iteration step #280\n Energy: -44.83713052475309 +- 2.2806388620753015\n \n Iteration step #290\n Energy: -44.88188847163767 +- 2.867579206036322\n \n Iteration step #300\n Energy: -44.848658953988746 +- 2.4643962459364794\n \n # ground state optimization completed\n \n 60.231748 seconds (11.35 M allocations: 794.996 MiB, 0.50% gc time, 3.44% compilation time)\n\n\nThe optiization looks successful! To better visualize this, let us plot the evolution of the energy and energy variance (stored respectively in the variables Energy_ and Variance_) during the ground state optimization. To this purpose we first import the Plots package. \n\n\n```julia\nusing Plots\n```\n\n\n```julia\n#Let's plot the evolution of the variational energy during the training\nplot(collect(1:GS_HP.n_iter),Energy_,title = \"Energy optimization\", label = \"Energy\")\n```\n\n\n\n\n \n\n \n\n\n\n\n```julia\n# Let's plot the evolution of the energy variance during the training\nplot(collect(1:GS_HP.n_iter),Variance_,title = \"Variance optimization\", label = \"Variance\")\n```\n\n\n\n\n \n\n \n\n\n\nThe (independent) network parameters obtained at the end of the ground state optimization are stored in the folder \"src/output\" and are termed \"W_RBM_N_alpha_real.jl\" for their real part, and \"W_RBM_N_alpha_imag.jl\" for their imaginary part. Now we upload them for later use.\n\n\n```julia\n#Upload trained network parameters\nW_RBM = readdlm(\"W_RBM_16_2_real.jl\") .+ im*readdlm(\"W_RBM_16_2_imag.jl\");\n```\n\nWith ULTRAFAST it is possible to evaluate the ground state spin-spin correlations\n\n\\begin{equation}\nC(i,j) = \\langle \\hat{S}_i \\cdot \\hat{S}_j \\rangle,\n\\end{equation}\n\nfor any $i,j$, with $i,j<=N$. This can be done by calling the function Spincorr_GS(), which returns the ground state value of $C(i,j)$ estimated over nSample states. Here, we choose nSample = 10000 and $i=1$, $j=2$.\n\n\n```julia\n#Let's calculate spin-spin correlation between spin 1 and spin 2 of the 4x4 lattice sampling nSample states\nnSample = 10000\nSpincorr_GS(W_RBM[:],nSample,1,2)\n```\n\n\n\n\n -1.3815571659993242\n\n\n", "meta": {"hexsha": "007c83fd3cd85dadf9b18d37f82b9ecd843e6c9a", "size": 88707, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Examples/groundstate/tutorial_gs.ipynb", "max_stars_repo_name": "GiamFbn/MyULTRAFAST", "max_stars_repo_head_hexsha": "719654bf1692ae3d55bef2cbae3a424e623a55ff", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Examples/groundstate/tutorial_gs.ipynb", "max_issues_repo_name": "GiamFbn/MyULTRAFAST", "max_issues_repo_head_hexsha": "719654bf1692ae3d55bef2cbae3a424e623a55ff", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Examples/groundstate/tutorial_gs.ipynb", "max_forks_repo_name": "GiamFbn/MyULTRAFAST", "max_forks_repo_head_hexsha": "719654bf1692ae3d55bef2cbae3a424e623a55ff", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 171.580270793, "max_line_length": 17892, "alphanum_fraction": 0.6868792767, "converted": true, "num_tokens": 1631, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583376458153, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.43064540203612645}} {"text": "# Chemical Reactions\nThe simulation procedure described in the previous chapter is now applied to a series of simple examples that represent chemical reactions. We first remind the reader of some key properties of chemical reactions that will show up in dynamic simulations and determine characteristics of network dynamic responses. We then go through a set of examples of chemical reactions that occur in a _closed system_. A closed system is isolated from its environment. No molecules enter or leave the system. Reactions being carried out in the laboratory in a sealed container represent an example of closed systems. In this chapter we assign numerical values to all the parameters for illustration purposes. We start by importing **MASSpy**:\n\n\n```python\nfrom mass import (\n MassModel, MassMetabolite, MassReaction, Simulation)\nfrom mass.visualization import plot_time_profile, plot_phase_portrait\n```\n\nOther useful packages are also imported at this time.\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\n```\n\n## Basic Properties of Reactions\nLinks between molecular components in a biochemical reaction network are given by chemical reactions or associations between chemical components. These links are therefore characterized and constrained by basic chemical rules.\n\n### Bi-linear reactions are prevalent in biology\nAlthough there are linear reactions found in biological reaction networks, the prototypical transformations in living systems at the molecular level are bi-linear. This association involves two compounds coming together to either be chemically transformed through the breakage and formation of a new covalent bond, as is typical of metabolic reactions or macromolecular synthesis, \n\n$$\\begin{equation*} \\text{X} +\\text{Y} \\rightleftharpoons \\text{X-Y}\\ \\text{covalent bonds} \\end{equation*}$$\n\nor two molecules associated together to form a complex that may be held together by hydrogen bonds and/or other physical association forces to form a complex that has a different functionality than individual components, \n\n$$\\begin{equation*}\\text{X} +\\text{Y} \\rightleftharpoons \\text{X:Y}\\ \\text{association of molecules} \\end{equation*}$$\n\nSuch association, for instance, could designate the binding of a transcription factor to DNA to form an activated site to which an activated polymerase binds. Such bi-linear association between two molecules might also involve the binding of an allosteric regulator to an allosteric enzyme that induces a conformational change in the enzyme. \n\n### Properties of biochemical reactions \nChemical transformations have three key properties that will influence the dynamic features of reaction networks and how we interpret dynamic states: \n\n#### Stoichiometry\nThe stoichiometry of chemical reactions is fixed and is described by integral numbers counting the molecules that react and that form as a consequence of the chemical reaction. Thus, stoichiometry basically represents \"digital information.\" Chemical transformations are constrained by elemental and charge balancing, as well as other features. Stoichiometry is invariant between organisms for the same reactions and does not change with pressure, temperature, or other conditions. Stoichiometry gives the primary topological properties of a biochemical reaction network. \n\n#### Thermodynamics\nAll reactions inside a cell are governed by thermodynamics that determine the equilibrium state of a reaction. The relative rates of the forward and reverse reactions are therefore fixed by basic thermodynamic properties. Unlike stoichiometry, thermodynamic properties do change with physico-chemical conditions such as pressure and temperature. Thus the thermodynamics of transformation between small molecules in cells are fixed but condition-dependent. The thermodynamic properties of associations between macromolecules can be changed by altering the amino acid sequence of a protein or by phosphorylation of amino acids in the interface region, or by conformational change induced by the binding of a small molecule ligand. \n\n#### Absolute Rates\nIn contrast to stoichiometry and thermodynamics, the absolute rates of chemical reactions inside cells are highly manipulable. Highly evolved enzymes are very specific in catalyzing particular chemical transformations. Cells can thus extensively manipulate the absolute rates of reactions through changes in their DNA sequence. \n\nAll biochemical transformations are subject to the basic rules of chemistry and thermodynamics. \n\n## The Reversible Linear Reaction\nWe start with the reversible linear reaction:\n\n$$\\begin{equation} x_1 \\underset{v_{-1}}{\\stackrel{v_1}{\\rightleftharpoons}} x_2 \\tag{4.1} \\end{equation}$$ \n\nHere we have that\n\n$$\\begin{equation*} \\textbf{S} = \\begin{pmatrix} {-1} & {1} \\\\ {1} & {-1} \\\\ \\end{pmatrix}, \\textbf{v}(\\textbf{x}) = \\begin{pmatrix} {v_1(x_1)} \\\\ {v_{-1}(x_2)} \\end{pmatrix} = \\begin{pmatrix} {k_1x_1} \\\\ {k_{-1}x_2} \\end{pmatrix} \\end{equation*}$$\n\nand thus the differential equations that we will need to simulate are: \n\n$$\\begin{equation} \\frac{dx_1}{dt} = -k_1x_1 + k_{-1}x_2, \\frac{dx_2}{dt} = k_1x_1 - k_{-1}x_2 = -\\frac{dx_1}{dt} \\tag{4.2} \\end{equation}$$ \n\nwith the reaction rate given as the difference between two elementary reaction rates \n\n$$\\begin{equation} v_{1, net} = v_1 - v_{-1} = k_1x_1 - k_{-1}x_2 = k_1(x_1 - x_2/K_1) \\tag{4.3} \\end{equation}$$\n\nwhere $K_1 = k_1/k_{-1} = x_{2, eq}/x_{1, eq}$ or the ratio of the product to reactant concentrations at equilibrium, the conventional definition of an equilibrium constant in chemistry. Note that in Eq. (4.3), $k_1$ represents the kinetics, or the rate of change, while $(x_1 - x_2/K_1)$ represents the thermodynamics measuring how far from equilibrium the system is, i.e.,$(x_{1, eq} - x_{2, eq}/K_1) = 0$. \n\nBelow, a sample solution is shown for $k_1 = 1$ and $k_{-1} = 2$. These simulation results can be examined further, and they reveal three important observations; 1) the existence of a conservation quantity, 2) a thermodynamic driving force, and 3) the pooling of variables based on chemistry and thermodynamics. \n\n\n```python\n# Create MassModel\nmodel = MassModel('Linear_Reversible')\n# Generate the MassMetabolites \nx1 = MassMetabolite(\"x1\")\nx2 = MassMetabolite(\"x2\")\n# Generate the MassReactions \nv1 = MassReaction(\"v1\")\n# Add metabolites to the reaction, add reaction to the model\nv1.add_metabolites({x1: -1, x2: 1})\nmodel.add_reactions([v1])\n# Set parameters and initial conditions\nv1.kf = 1\nv1.kr = 2\nmodel.update_initial_conditions({x1: 1, x2: 0})\n# Utilize type 2 rate law for kf and kr parameters defined\nmodel.get_rate_expressions(rate_type=2, update_reactions=True)\nmodel\n```\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        NameLinear_Reversible
                                        Memory address0x07fd1902d7f10
                                        Stoichiometric Matrix2x1
                                        Matrix Rank1
                                        Number of metabolites2
                                        Initial conditions defined2/2
                                        Number of reactions1
                                        Number of genes0
                                        Number of enzyme modules0
                                        Number of groups0
                                        Objective expression0
                                        Compartments
                                        \n\n\n\n\n\n```python\nt0 = 0\ntf = 2\nsim = Simulation(model, verbose=True)\nconc_sol, flux_sol = sim.simulate(model, time=(t0, tf), verbose=True)\n\n# Define pools\npools = [\"x1 - x2 / Keq_v1\", \"x1 + x2\"]\n\nfor i, equation_str in enumerate(pools):\n pool_id = \"p\" + str(i + 1)\n conc_sol.make_aggregate_solution(\n pool_id, equation=equation_str, \n parameters={v1.Keq_str: v1.kf/v1.kr}, update=True)\n```\n\n \u001b[93mWARNING:\u001b[0m \u001b[93mNo compartments found in model. Therefore creating compartment 'compartment' for entire model.\u001b[0m\n\n\n Successfully loaded MassModel 'Linear_Reversible' into RoadRunner.\n Getting time points\n Setting output selections\n Setting simulation values for 'Linear_Reversible'\n Simulating 'Linear_Reversible'\n Simulation for 'Linear_Reversible' successful\n Adding 'Linear_Reversible' simulation solutions to output\n Updating stored solutions\n\n\n\n```python\nfig_4_1 = plt.figure(figsize=(9, 6))\ngs = fig_4_1.add_gridspec(nrows=2, ncols=2, width_ratios=[1, 1.5],\n height_ratios=[1, 1])\n\nax1 = fig_4_1.add_subplot(gs[0, 0])\nax2 = fig_4_1.add_subplot(gs[0, 1])\nax3 = fig_4_1.add_subplot(gs[1, 1])\n\nplot_phase_portrait(\n conc_sol, x=x1, y=x2, ax=ax1,\n xlabel=x1.id, ylabel=x2.id,\n xlim=(-0.05, 1.05), ylim=(-0.05, 1.05),\n title=(\"(a) Phase portrait\", {\"size\":\"large\"}),\n annotate_time_points=\"endpoints\",\n annotate_time_points_labels=True);\n\nplot_time_profile(\n conc_sol, ax=ax2, observable=model.metabolites, legend=\"right outside\",\n xlabel=\"Time\", ylabel=\"Concentrations\",\n title=(\"(b) Time Profiles of Species\", {\"size\": \"large\"}));\n\nplot_time_profile(\n conc_sol, ax=ax3, observable=[\"p1\", \"p2\"],\n legend=\"right outside\",\n xlabel=\"Time\", ylabel=\"Concentrations\",\n title=(\"(c) Time Profiles of Pools\", {\"size\": \"large\"}));\nfig_4_1.tight_layout()\n```\n\n**Figure 4.1:** Dynamic simulation of the reaction $x_1 \\underset{v_{-1}}{\\stackrel{v_1}{\\rightleftharpoons}} x_2$ for $k_1 =1$ and $k_{-1} = 2$, and $x_1(0)=1$, $x_2(0)=0$. (a) The phase portrait. (b) The time profiles. (c) The time profile of the pooled variables $p_1 = x_1 - x_2/K_1$ and $p_1 = x_1 + x_2$.\n\n### Mass conservation: \nThe time profiles in Figure\u00a04.1b show $x_1$ fall and $x_2$ rise to their equilibrium values. The phase portrait (Figure\u00a04.1a) is a straight line of slope -1. This implies that\n\n$$\\begin{equation} p_1 = x_1 + x_2 = \\big \\langle (1, 1), (x_1, x_2)^T \\big \\rangle \\tag{4.4} \\end{equation}$$\n\nis a constant. This summation represents a conservation quantity that stems from the fact that as $x_1$ reacts, $x_2$ appears in an equal and opposite amount. The stoichiometric matrix is singular with a rank of 1, showing that this is a one-dimensional dynamic system. It has a left null space that is spanned by the vector $(1, 1), i.e., (1, 1) \\centerdot \\textbf{S} = 0$, thus $p_2$ is in the left null space of $\\textbf{S}$. \n\nWe also note that since $x_1 + x_2$ is a constant, we can describe the concentration of $x_1$ as a fraction of the total mass, i.e., \n\n$$\\begin{equation*} f_1 = \\frac{x_1}{x_1 + x_2} = \\frac{x_1}{p_2} \\end{equation*}$$\n\nPool sizes and the fraction of molecules in a particular state will be used later in the text to define physiologically useful quantities. \n\n### Disequilibrium and the thermodynamic driving force:\nA pooled variable: \n\n$$\\begin{equation} p_1 = x_1 - x_2/K_1 \\tag{4.5} \\end{equation}$$\n\ncan be formed (see Figure\u00a04.1c). Combination of the differential equations for $x_1$ and $x_2$ leads to \n\n$$\\begin{equation} \\frac{dp_1}{dt} = -(k_1 + k_{-1}) p_1 \\tag{4.6} \\end{equation}$$\n\nand thus the time constant for this reaction is \n\n$$\\begin{equation} \\tau_{1} = \\frac{1}{k_1 + k_{-1}} \\tag{4.7} \\end{equation}$$\n\nNote that when $t \\rightarrow \\infty, p_1 \\rightarrow 0$ and then\n\n$$\\begin{equation} \\frac{x_2}{x_1} \\rightarrow \\frac{k_1}{k_{-1}} = K_1 = \\frac{x_{2,eq}}{x_{1, eq}} \\tag{4.8} \\end{equation}$$\n\nthe reaction has reached equilibrium. The pool $p_1$ thus represents a disequilibrium quantity and represents the thermodynamic driver for the reaction, see Eq.\u00a0(4.3). With an initial condition of $x_{1, 0} = 1$ and $K_1 = 1/2$, the eventual concentrations $(t \\rightarrow \\infty)$ will be $x_{1, eq} = 2/3$ and $x_{2, eq} = 1/3$. \n\n### Representing dynamics with pools for reversible linear reactions\nThese considerations show that we can think about the dynamics of reaction\u00a0(4.1) in terms of two pooled variables rather than the concentrations themselves. Thus, a useful pool transforming matrix for this reaction would be \n\n$$\\begin{equation} \\textbf{P} = \\begin{pmatrix} {1} & {-1/K_1} \\\\ {1} & {1} \\\\ \\end{pmatrix} \\end{equation} \\tag{4.9}$$\n\nleading to disequilibrium $(p_1)$ and conservation $(p_2)$ quantities associated with the reaction in Eq. (4.1). The former quantity moves on the time scale given by $\\tau_1$ while the latter is time invariant. For practical purposes, the dynamics of the reaction have relaxed within a time duration of three to five times $\\tau_1$ (see Figure\u00a04.1b). \n\nThe differential equations for the pools can be obtained as \n\n$$\\begin{equation} \\textbf{P} \\frac{d\\textbf{x}}{dt} = \\frac{d}{dt}\\begin{pmatrix} {p_1} \\\\ {p_2} \\\\ \\end{pmatrix} = -(k_1 + k_{-1}) \\begin{pmatrix} {x_1 - x_2/K_1} \\\\ {0} \\\\ \\end{pmatrix} = -(k_1 + k_{-1})\\begin{pmatrix} {p_1} \\\\ {0} \\\\ \\end{pmatrix} \\end{equation}$$\n\nTherefore, the conservation quantity is a constant (time derivative is zero) and the disequilibrium pool is driven by a thermodynamic driving force that is itself multiplied by $-(k_1 + k_{-1})$, that is the inverse of the time constant for the reaction. Thus, the three key features of chemical reactions, the stoichiometry, thermodynamics, and kinetics, are separately accounted for. \n\n## The Reversible Bi-Linear Reaction\nThe reaction mechanism for the reversible bi-linear reaction is: \n\n$$\\begin{equation} x_1 + x_2 \\underset{v_{-1}}{\\stackrel{v_1}{\\rightleftharpoons}} x_3 \\tag{4.10} \\end{equation}$$ \n\nwhere the elementary reaction rates are \n\n$$\\begin{equation} v_1 = k_1x_1x_2, \\ v_{-1} = k_{-1}x_3 \\tag{4.11} \\end{equation}$$\n\nThe forward rate, $v_1$, is a non-linear function, or more specifically, a bi-linear function. The variable \n\n$$\\begin{equation} p_1 = x_1x_2 - x_3/K_1 \\tag{4.12} \\end{equation}$$\n\nrepresents a disequilibrium quantity. The dynamic states of this system can be computed from \n\n$$\\begin{equation} \\frac{dx_1}{dt} = -v_1 + v_{-1} = -k_1x_1x_2 + k_{-1}x_3 = -k_1(x_1x_2 - x_3/K_1) = \\frac{dx_2}{dt} = -\\frac{dx_3}{dt} \\tag{4.13} \\end{equation}$$\n\nThis example will be used to illustrate the essential features of a bi-linear reaction; 1) That there are two conservation quantities associated with it, 2) How to compute the equilibrium state, 3) The use of linearization and deviation variables from the equilibrium state, 4) The derivation of a single linear disequilibrium quantity, and 5) Formation of pools.\n\n### Conservation quantities for reversible bi-linear reactions\nThe stoichiometric matrix is \n\n$$\\begin{equation} S = \\begin{pmatrix} {-1} & {1} \\\\ {-1} & {1} \\\\ {1} & {-1} \\\\ \\end{pmatrix} \\end{equation}$$\n\nThe stoichiometric matrix has a rank of 1, and thus the dynamic dimension of this system is 1. Two vectors that span the left null space of **S** are (1,0,1) and (0,1,1) and the corresponding conservation quantities are: \n\n$$\\begin{equation} p_2 = x_1 + x_3, \\ p_3 = x_2 + x_3 \\tag{4.14} \\end{equation}$$\n\nThis selection of conservation quantities is not unique, as one can find other sets of two vectors that span the left null space. \n\n### The equilibrium state for reversible bi-linear reactions\nWe can examine the equilibrium state for the specific parameter values to be used for numerical simulation below, Figure\u00a04.2. At equilibrium, $p_1 \\rightarrow 0$ and we have that $(K_1 = 1)$\n\n$$\\begin{equation} x_{1, eq}x_{2, eq} = x_{3, eq} \\tag{4.15} \\end{equation}$$\n\nand that \n\n$$\\begin{equation} x_1(0) = 3 = x_{1, eq} + x_{3, eq}, \\ \\ \\ x_2(0) = 2 = x_{2, eq} + x_{3, eq} \\tag{4.16} \\end{equation}$$\n\nThese three equations can be combined to give a second order algebraic equation \n\n$$\\begin{equation} x_{3, eq}^2 - 6x_{3, eq} + 6 = 0 \\tag{4.17} \\end{equation}$$\n\nthat has a positive root that yields \n\n$$\\begin{equation} x_{1, eq} = 1.73,\\ \\ x_{2, eq} = 0.73, \\ \\ x_{3, eq} = 1.27 \\tag{4.18} \\end{equation}$$\n\n### Linearization and deviation variables for reversible bi-linear reactions\nEquation\u00a0(13) can be linearized around the equilibrium point $\\textbf{x}_{eq}$==(1.73,0.73,1.27) to give \n\n$$\\begin{equation} \\frac{dx_1}{dt} = x_1x_2 - x_3 \\tag{4.19} \\end{equation}$$\n$$\\begin{equation} \\rightarrow \\frac{dx_1}{dt} = 0.73(x_1 -1.73) + 1.73(x_2 - 0.73) - (x_3 - 1.27) \\tag{4.20} \\end{equation}$$\n\nwhere a numerical value of $k_1$ used is unity. \n\n### The disequilibrium and conservation quantities for reversible bi-linear reactions\nEquation (20) can be written in terms of deviation variables from the equilibrium state, i.e., \n\n$$\\begin{equation} x_i' x_i - x_{i, eq} \\tag{4.21} \\end{equation}$$\n\nas\n\n$$\\begin{equation} \\frac{dx_1'}{dt} = 0.73x_1' + 1.73x_2' - x_3' = p_1' \\tag{4.22} \\end{equation}$$\n\nwhich simply is the linearized version of the disequilibrium quantity in Eq.(4.12), and we have that \n\n$$\\begin{equation} \\frac{dx_2'}{dt} =\\frac{dx_1'}{dt} \\ \\ and \\ \\ \\frac{dx_3'}{dt} = -\\frac{dx_1'}{dt} \\tag{4.23} \\end{equation}$$\n\n### Representing dynamics with pools for reversible bi-linear reactions\nWe can therefore form a pool transformation matrix as: \n\n$$\\begin{equation} \\textbf{P} = \\begin{pmatrix} {0.73} & {1.73} & {-1} \\\\ {1} & {0} & {1} \\\\ {0} & {1} & {1} \\\\ \\end{pmatrix} \\end{equation} \\tag{4.24}$$\n\nwhere the first pool represents the disequilibrium quantity and the second and third are conservation quantities. Now we transform the deviation variables with this matrix, i.e., $\\textbf{p'} = \\textbf{Px'}$ and can look at the time derivatives of the pools \n\n$$\\begin{align} \\frac{d\\textbf{p'}}{dt} &= \\textbf{P}\\frac{d\\textbf{x'}}{dt} \\\\ &= \\begin{pmatrix} {-3.46} & {3.46} \\\\ {0} & {0} \\\\ {0} & {0} \\\\ \\end{pmatrix}(0.73x_1' + 1.73x_2' - x_3') \\\\ &= 3.46\\begin{pmatrix} {1} \\\\ {0} \\\\ {0} \\\\ \\end{pmatrix} p_1' \\end{align} \\tag{4.25}$$\n\nThis result is similar to that obtained above for the linear reversible reaction. There are two conservation pools and a disequilibrium pool that is moved by itself multiplied by a characteristic rate constant. We note that the conservation quantities, for both the linear and bi-linear reaction, do not change if the reactions are irreversible (i.e., if $K_{eq} \\rightarrow \\infty$) \n\n### Numerical simulation of reversible bi-linear reactions\nThe dynamic response of this reaction can readily be computed and the results graphed; see Figure\u00a04.2.\n\n\n```python\n# Create MassModel\nmodel = MassModel('BiLinear_Reversible')\n# Generate the MassMetabolites \nx1 = MassMetabolite(\"x1\")\nx2 = MassMetabolite(\"x2\")\nx3 = MassMetabolite(\"x3\")\n# Generate the MassReactions \nv1 = MassReaction(\"v1\")\n# Add metabolites to the reaction, add reaction to the model\nv1.add_metabolites({x1: -1, x2: -1, x3: 1})\nmodel.add_reactions([v1])\n# Set parameters and initial conditions\nv1.kf = 1\nv1.kr = 1\nmodel.update_initial_conditions({x1: 3, x2: 2, x3: 0})\n# Utilize type 2 rate law for kf and kr parameters defined\nmodel.get_rate_expressions(rate_type=2, update_reactions=True)\nmodel\n```\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        NameBiLinear_Reversible
                                        Memory address0x07fd19052fd10
                                        Stoichiometric Matrix3x1
                                        Matrix Rank1
                                        Number of metabolites3
                                        Initial conditions defined3/3
                                        Number of reactions1
                                        Number of genes0
                                        Number of enzyme modules0
                                        Number of groups0
                                        Objective expression0
                                        Compartments
                                        \n\n\n\n\n\n```python\nt0 = 0\ntf = 5\nsim = Simulation(model, verbose=True)\nconc_sol, flux_sol = sim.simulate(model, time=(t0, tf), verbose=True)\n\n# Define pools\npools = ['x1*x2 - x3 / Keq_v1', 'x1 + x3', 'x2 + x3']\n\nfor i, equation_str in enumerate(pools):\n pool_id = \"p\" + str(i + 1)\n conc_sol.make_aggregate_solution(\n pool_id, equation=equation_str, \n parameters={v1.Keq_str: v1.kf/v1.kr}, update=True)\n```\n\n \u001b[93mWARNING:\u001b[0m \u001b[93mNo compartments found in model. Therefore creating compartment 'compartment' for entire model.\u001b[0m\n\n\n Successfully loaded MassModel 'BiLinear_Reversible' into RoadRunner.\n Getting time points\n Setting output selections\n Setting simulation values for 'BiLinear_Reversible'\n Simulating 'BiLinear_Reversible'\n Simulation for 'BiLinear_Reversible' successful\n Adding 'BiLinear_Reversible' simulation solutions to output\n Updating stored solutions\n\n\n\n```python\nfig_4_2 = plt.figure(figsize=(12, 4))\ngs = fig_4_2.add_gridspec(nrows=1, ncols=2)\n\nax1 = fig_4_2.add_subplot(gs[0, 0])\nax2 = fig_4_2.add_subplot(gs[0, 1])\n\nplot_time_profile(\n conc_sol, ax=ax1, observable=model.metabolites,\n legend=\"lower outside\",\n xlabel=\"Time\", ylabel=\"Concentrations\",\n title=(\"(a) Time Profiles of Species\", {\"size\": \"large\"}));\n\nplot_time_profile(\n conc_sol, ax=ax2, observable=[\"p1\", \"p2\", \"p3\"],\n legend=\"lower outside\",\n xlabel=\"Time\", ylabel=\"Concentrations\",\n title=(\"(b) Time Profiles of Pools\", {\"size\": \"large\"}));\nfig_4_2.tight_layout()\n```\n\n**Figure 4.2:** The concentration time profiles for the reaction $x_1 + x_2 {\\rightleftharpoons} x_3$ for $k_1$ = $k_{-1} = 1$ and $x_1(0)=3$, $x_2(0)=2$, and $x_3(0)=0$. (a) The concentrations as a function of time. (b) The pools as a function of time.\n\n## Connected Reversible Linear Reactions\nNow we consider more than one reaction working simultaneously. We will consider two reversible first order reactions that are connected by an irreversible reaction; \n\n$$\\begin{equation} x_1 \\underset{v_{-1}}{\\stackrel{v_1}{\\rightleftharpoons}} x_2 \\stackrel{v_2} \\rightarrow x_3 \\underset{v_{-3}}{\\stackrel{v_3}{\\rightleftharpoons}} x_4 \\tag{4.26} \\end{equation}$$ \n\nThe stoichiometric matrix and the reaction vector, are \n\n$$\\begin{equation} \\textbf{S} = \\begin{pmatrix} {-1} & {1} & {0} & {0} & {0} \\\\ {1} & {-1} & {-1} & {0} & {0} \\\\ {0} & {0} & {1} & {-1} & {1} \\\\ {0} & {0} & {0} & {1} & {-1} \\\\ \\end{pmatrix}, \\ \\textbf{v}(\\textbf{x}) = \\begin{pmatrix} {k_1x_1} \\\\ {k_{-1}x_2} \\\\ {k_2x_2} \\\\ {k_3x_3} \\\\ {k_{-3}x_4} \\\\\\end{pmatrix}\\end{equation} \\tag{4.27}$$\n \nand thus the dynamic mass balances are; \n\n$$\\begin{align} \\frac{dx_1}{dt} &= -k_1x_1 + k_{-1}x_2 \\\\ \\frac{dx_2}{dt} &= k_1x_1 - k_{-1}x_2 - k_2x_2 \\\\ \\frac{dx_3}{dt} &= k_2x_2 - k_3x_3 + k_{-3}x_4 \\\\ \\frac{dx_4}{dt} &= k_3x_3 - k_{-3}x_4 \\\\ \\end {align} \\tag{4.28}$$\n\nThe net reaction rates are: \n\n$$\\begin{equation} v_{1, net} = k_1x_1 - k_{-1}x_2 = k_1(x_1 - x_2/K_1) \\tag{4.29} \\end{equation}$$ \n\nand\n\n$$\\begin{equation} v_{3, net} = k_3x_3 - k_{-3}x_4 = k_3(x_3 - x_4/K_3) \\tag{4.30} \\end{equation}$$ \n\nwhere $K_1 = k_1/k_{-1}$ and $K_3 = k_3/k_{-3}$ are the equilibrium constants. This example can be used to illustrate three concepts: 1) dynamic decoupling, 2) stoichiometric decoupling, and 3) formation of multi-reaction pools. \n\n### Dynamic decoupling through seperated time scales:\nThis linear system can be described by \n\n$$\\begin{equation} \\frac{d\\textbf{x}}{dt} = \\textbf{Jx} \\tag{4.31} \\end{equation}$$\n\nwhere the Jacobian matrix for this system is obtained directly from the equations in (4.28): \n\n$$\\begin{equation} \\textbf{J} = \\begin{pmatrix} {-k_1} & {k_{-1}} & {0} & {0} \\\\ {k_1} & {-k_{-1} - k_2} & {0} & {0} \\\\ {0} & {k_2} & {-k_3} & {-k_{-3}} \\\\ {0} & {0} & {k_3} & {k_{-3}} \\\\ \\end{pmatrix} \\end{equation} \\tag{4.32}$$\n\nNote that for linear systems, $\\textbf{x} = \\textbf{x'}$. Observe that the second column in $\\textbf{J}$ is a combination of the second and third column in $\\textbf{S}$; \n\n$$\\begin{equation} \\begin{pmatrix} {j_{12}} \\\\ {j_{22}} \\\\ {j_{32}} \\\\ {j_{42}} \\end{pmatrix} = \\begin{pmatrix} {1} \\\\ {-1} \\\\ {0} \\\\ {0} \\end{pmatrix} + \\begin{pmatrix} {0} \\\\ {-1} \\\\ {1} \\\\ {0} \\end{pmatrix} k_2 = k_{-1}\\textbf{s}_2 + k_2\\textbf{s}_3 \\end{equation} \\tag{4.33}$$\n\n\n```python\n# Create MassModel\nmodel = MassModel('Connected_Linear_Reversible')\n# Generate the MassMetabolites \nx1 = MassMetabolite(\"x1\")\nx2 = MassMetabolite(\"x2\")\nx3 = MassMetabolite(\"x3\")\nx4 = MassMetabolite(\"x4\")\n# Generate the MassReactions \nv1 = MassReaction(\"v1\")\nv2 = MassReaction(\"v2\", reversible=False)\nv3 = MassReaction(\"v3\")\n# Add metabolites to the reaction, add reaction to the model\nv1.add_metabolites({x1: -1, x2: 1})\nv2.add_metabolites({x2: -1, x3: 1})\nv3.add_metabolites({x3: -1, x4: 1})\nmodel.add_reactions([v1, v2, v3])\n# Set parameters and initial conditions\nv1.kf = 1\nv1.Keq = 1\nv2.kf = 1\nv3.kf = 1\nv3.Keq = 1\nmodel.update_initial_conditions({x1: 1, x2: 0, x3: 0, x4: 0})\nmodel\n```\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        NameConnected_Linear_Reversible
                                        Memory address0x07fd1902a1ad0
                                        Stoichiometric Matrix4x3
                                        Matrix Rank3
                                        Number of metabolites4
                                        Initial conditions defined4/4
                                        Number of reactions3
                                        Number of genes0
                                        Number of enzyme modules0
                                        Number of groups0
                                        Objective expression0
                                        Compartments
                                        \n\n\n\n\n\n```python\nt0 = 0\ntf = 10\n\nsim = Simulation(model, verbose=True)\nconc_sol, flux_sol = sim.simulate(model, time=(t0, tf), verbose=True)\n```\n\n \u001b[93mWARNING:\u001b[0m \u001b[93mNo compartments found in model. Therefore creating compartment 'compartment' for entire model.\u001b[0m\n\n\n Successfully loaded MassModel 'Connected_Linear_Reversible' into RoadRunner.\n Getting time points\n Setting output selections\n Setting simulation values for 'Connected_Linear_Reversible'\n Simulating 'Connected_Linear_Reversible'\n Simulation for 'Connected_Linear_Reversible' successful\n Adding 'Connected_Linear_Reversible' simulation solutions to output\n Updating stored solutions\n\n\n\n```python\nfig_4_3 = plt.figure(figsize=(6, 4))\ngs = fig_4_3.add_gridspec(nrows=1, ncols=1)\n\nax1 = fig_4_3.add_subplot(gs[0, 0])\n\nplot_time_profile(\n conc_sol, ax=ax1, legend=\"right outside\",\n xlabel=\"Time\", ylabel=\"Concentrations\",\n title=(\"Time Profiles of Species\", {\"size\": \"large\"}));\nfig_4_3.tight_layout()\n```\n\n**Figure 4.3:** The dynamic response of the reactions in Eq. (4.17) for $K_1=K_3=1$ and $k_1=k_2=k_3=1$. The graphs show the concentrations varying with time for $x_1(0)=1, x_2(0)=x_3(0)=x_4(0)=0$.\n\nThe kinetic effects of $x_2$ are thus felt through both reactions 2 and 3 (i.e., the second and third column in $\\textbf{S}$ that are the corresponding reaction vectors), $\\textbf{s}_2$ and $\\textbf{s}_3$. These two reaction vectors are weighted by the rate constants (reciprocal of the time constants). Therefore, we expect that if $k_2$ is numerically much smaller than $k_{-1}$ then the dynamic coupling is 'weak.' We consider two sets of parameter values. \n\n* First, we simulate this system with all the rate constants being equal, Figure\u00a04.3. All the concentrations are moving on the same time scale. For a series of reactions, the overall dynamics are expected to unfold on a time scale that is the sum of the individual time constants. Here, this sum is three, and the dynamics have relaxed after a time period of three to five times this value. \n\n* Next, we make the second reaction ten times slower compared to the other two by decreasing $k_2$. We see that the two faster reactions come to a quasi-equilibrium state relatively quickly (relative to reaction 3), and form two conservation pools that exchange mass slowly. The sum of the rate constants for the three reactions in series is now seven, and the dynamics unfold on this time scale. \n\n#### Stoichiometric decoupling\nReaction 3 does not influence reaction 1 at all. They are separated by the irreversible reaction 2. Thus, changes in the kinetics of reaction 3 will not influence the progress of reaction 1. This can be illustrated through simulation by changing the rate constants for reaction 3 and observing what happens to reaction 1. \n\n#### Formation of multi-reaction pools:\nWe can form the following pooled variables based on the properties of the individual reversible reactions\n\n$$\\begin{align} &p_1 = x_1 - x_2/K_1\\ && disequilibrium\\ quantity\\ for\\ reaction\\ 1 \\\\ &p_2 = x_1 + x_2 \\ && conservation\\ quantity\\ for\\ reaction\\ 1 \\\\ &p_3 = x_3 - x_4/K_3\\ && disequilibrium\\ quantity\\ for\\ reaction\\ 3 \\\\ &p_4 = x_3 + x_4 \\ && conservation\\ quantity\\ for\\ reaction\\ 3\\end{align} \\tag{4.34}$$\n\nA representation of the dynamics of this reaction system can be obtained by plotting these pools as a function of time; Figure 4.4a. To prepare this plot, we use the pooling matrix \n\n$$\\begin{equation} \\textbf{P} = \\begin{pmatrix} {1} & {-1/K_1} & {0} & {0} \\\\ {1} & {1} & {0} & {0} \\\\ {0} & {0} & {1} & {-1/K_3} \\\\ {0} & {0} & {1} & {1} \\\\ \\end{pmatrix} \\end{equation} \\tag{4.35}$$\n \nto post-process the output. However, we note that the conservation quantities associated with the individual reactions are no longer time-invariant. \n\nThe rank of $\\textbf{S}$ is 3 and its one-dimensional left null space is spanned by (1,1,1,1); thus the conservation quantity is $x_1 + x_2 + x_3 + x_4$. Therefore an alternative pooling matrix may be formulated as \n\n$$\\begin{equation} \\textbf{P} = \\begin{pmatrix} {1} & {-1/K_1} & {0} & {0} \\\\ {0} & {1} & {0} & {0} \\\\ {0} & {0} & {1} & {-1/K_3} \\\\ {1} & {1} & {1} & {1} \\\\ \\end{pmatrix}\\end{equation} \\tag{4.36}$$\n \nwhere we use $x_2$ as the coupling variable and the overall conservation pool instead of the conservation pools associated with the individual reactions. The two conservation pools are combined into one overall mass conservation pool. \n\nWe can now derive the dynamic mass balances on the pools as \n\n$$\\begin{align} \\frac{d\\textbf{p}}{dt} = \\textbf{PSv} &= \\begin{pmatrix} {-(k_1 + k_{-1})(x_1 - x_2/K_1) +\\frac{k_2}{K_1}x_2} \\\\ {k_1(x_1 - x_2/K_1) - k_2x_2} \\\\ {-(k_3 + k_{-3})(x_3 - x_4/K_3) +k_2x_2} \\\\ {0}\\end{pmatrix} \\\\ &= \\begin{pmatrix} {-(k_1 + k_{-1})p_1 +\\frac{k_2}{K_1}x_2} \\\\ {k_1p_1 - k_2p_2} \\\\ {-(k_3 + k_{-3})p_3 +k_2p_2} \\\\{0}\\end{pmatrix} \\\\ &= \\begin{pmatrix} {-(k_1 + k_{-1})} \\\\ {k_1} \\\\ {0} \\\\ {0} \\end{pmatrix}p_1 + \\begin{pmatrix} {\\frac{k_2}{K_1}} \\\\ {-k_2} \\\\ {k_2} \\\\ {0} \\end{pmatrix}p_2 + \\begin{pmatrix} {0} \\\\ {0} \\\\ {-(k_3 + k_{-3})} \\\\ {0} \\end{pmatrix}p_3 \\end{align} \\tag{4.37}$$\n \nThis equation shows that $p_1$ and $p_3$ create fast motion compared to $p_2$ given the relative numerical values of the rate constants; $p_2$ creates a slow drift in this system for the numerical values used in Figure\u00a04.4b. \n\n\n```python\n# Define pools\nreg_pools = ['x1 - x2 / Keq_v1', 'x1 + x2', \n 'x3 - x4 / Keq_v3', 'x3 + x4']\n\nalt_pools = ['x1 - x2 / Keq_v1', 'x2', \n 'x3 - x4 / Keq_v3', 'x1 + x2 + x3 + x4']\n\nfor prefix, pools in zip([\"p\", \"alt_p\"], [reg_pools, alt_pools]):\n for i, equation_str in enumerate(pools):\n pool_id = prefix + str(i + 1)\n conc_sol.make_aggregate_solution(\n pool_id, equation=equation_str,\n parameters={v1.Keq_str: v1.Keq, v3.Keq_str: v3.Keq},\n update=True)\n```\n\n\n```python\nfig_4_4 = plt.figure(figsize=(12, 8))\ngs = fig_4_4.add_gridspec(nrows=2, ncols=2)\n\nax1 = fig_4_4.add_subplot(gs[0, 0])\nax2 = fig_4_4.add_subplot(gs[1, 0])\nax3 = fig_4_4.add_subplot(gs[1, 1])\n\nplot_time_profile(\n conc_sol, ax=ax1, observable=model.metabolites,\n legend=\"right outside\", xlabel=\"Time\", ylabel=\"Concentrations\",\n title=(\"(a) Time Profiles of Species\", {\"size\": \"large\"}));\n\nplot_time_profile(\n conc_sol, observable=[\"p1\", \"p2\", \"p3\"], ax=ax2,\n legend=\"lower outside\", xlabel=\"Time\", ylabel=\"Concentrations\",\n title=(\"(b) Time Profiles of Pools\", {\"size\": \"large\"}));\n\nplot_time_profile(\n conc_sol, observable=[\"alt_p1\", \"alt_p2\", \"alt_p3\"], ax=ax3,\n legend=\"lower outside\", xlabel=\"Time\", ylabel=\"Concentrations\",\n title=(\"(c) Time Profiles of Alternate Pools\", {\"size\": \"large\"}));\nfig_4_4.tight_layout()\n```\n\n**Figure 4.4:** (a) The time profiles (b) The conservation pools $p_2 = x_1 + x_2$ and $p_4 = x_3 + x_4$ and the disequilibrium pools $p_1 = x_1 - x_2/K_1$ and $p_3 = x_3 - x_4/K_3$ for the individual reactions. The disequilibrium pools move quickly towards a quasi-equilibrium state, while the conservation pools move more slowly. These pools are defined in Eq (4.35). (c) The dynamic response with alternative pools; $p_2 = x_2$, and $p_4 = x_1 + x_2 + x_3 + x_4$. These pools are defined in Eq (4.36).\n\n## Connected Reversible Bi-linear Reactions\nAn important case of connected bi-linear reactions is represented by the reaction mechanism \n\n$$\\begin{equation} x_1 + x_2 \\underset{v_{-1}}{\\stackrel{v_1}{\\rightleftharpoons}} x_3 \\underset{v_{-2}}{\\stackrel{v_2}{\\rightleftharpoons}} x_4 + x_5 \\tag{4.38} \\end{equation}$$\n\nThis reaction network is similar to reaction mechanisms for enzymes, and thus leads us into the treatment of enzyme kinetics (Chapter\u00a05). The elementary reaction rates are: \n\n$$\\begin{equation} v_1 = k_1x_1x_2, \\ \\ v_{-1} = k_{-1}x_3 \\ \\ v_2 = k_2x_3, \\text{and} \\ \\ v_{-2} = k_{-2}x_4x_5 \\tag{4.39} \\end{equation}$$\n\nand the equilibrium constants are $K_1 = k_1/k_{-1}$ and $K_2 = k_2/k_{-2}$. There are two disequilibrium quantities. \n\n$$\\begin{equation} p_1 = x_1x_2 - x_3 / K_1 \\tag{4.40} \\end{equation}$$\n$$\\begin{equation} p_2 = x_3 - x_4x_5 / K_2 \\tag{4.41} \\end{equation}$$\n\nWe now explore the same features of this coupled system of bi-linear reactions as we did for the single reversible bi-linear reaction. \n\n### Conservation quantities for connected reversible bi-linear reactions\nThe (5x4) stoichiometric matrix \n\n$$\\begin{equation} \\textbf{S} = \\begin{pmatrix} {-1} & {1} & {0} & {0} \\\\ {-1} & {1} & {0} & {0} \\\\ {1} & {-1} & {-1} & {1} \\\\ {0} & {0} & {1} & {-1} \\\\ {0} & {0} & {1} & {-1} \\\\ \\end{pmatrix} \\end{equation} \\tag{4.42}$$\n\nhas a rank of 2 and thus there are three conservation variables and two independent dynamic variables. \n\nThe conservation quantities are not unique and which one we will use depends on the reaction chemistry that is being studied. An example is \n\n$$\\begin{equation} AB + C \\underset{v_{-1}}{\\stackrel{v_1}{\\rightleftharpoons}} ABC \\underset{v_{-2}}{\\stackrel{v_2}{\\rightleftharpoons}} A + BC \\tag{4.43} \\end{equation}$$\n\n\nin which case the three independent conservation quantities would be: \n\n$$\\begin{equation} \\text{Conservation of A}:\\ p_3 = x_1 + x_3 + x_4 \\\\ \\text{Conservation of B}:\\ p_4 = x_1 + x_3 + x_5 \\\\ \\text{Conservation of C}:\\ p_5 = x_2 + x_3 + x_5 \\end{equation}$$ $$\\tag{4.44}$$\n\nThese are convex quantities as all the coefficients are non-negative (the concentrations are $x_i>0$). The individual bi-linear reactions have two each, but once coupled, the number of conservation quantities drops by one. \n\n### The equilibrium state for connected reversible bi-linear reactions\nThe computation of the equilibrium state involves setting the net fluxes to zero and combining those equations with the conservation quantities to get a set of independent equations. For convenience of illustration, we pick $K_1 = K_2 = 1$ and the equilibrium equations become \n\n$$\\begin{equation} x_{1, eq}x_{2, eq} = x_{3, eq} = x_{4, eq}x_{5, eq} \\tag{4.45} \\end{equation}$$\n\nand if we pick $p_3 = p_4 = p_5 = 3$ then the solution for the equilibrium state is simple: $x_{1, eq}x_{2, eq} = x_{3, eq} = x_{4, eq}x_{5, eq} = 1$. These equations can also be solved for arbitrary parameter values. \n\n### Linearization and deviation variables for connected reversible bi-linear reactions\nBy linearizing the differential equations around the steady state we obtain \n\n$$\\begin{align} \\frac{dx_1'}{dt} &= \\frac{dx_2'}{dt} = -(k_1x_{1,eq})x_2' -(k_1x_{2,eq})x_1' + k_{-1}x_3' \\\\ \\frac{dx_3'}{dt} &= (k_1x_{2,eq})x_1' + (k_1x_{1,eq})x_2' - (k_{-1} + k_2)x_3' + (k_{-2}x_{5,eq})x_4'+ (k_{-2}x_{4,eq})x_5'\\\\ \\frac{dx_4'}{dt} &= \\frac{dx_5'}{dt} = k_{2}x_3' - (k_{-2}x_{5,eq})x_4' -(k_{-2}x_{4,eq})x_5' \\end{align} \\tag{4.46}$$\n\nwhere $x_i' = x_i - x_{i, eq}$ represent the concentration deviation around equilibrium, $i$ =1,2,3,4 and 5.\n\n### The disequilibrium and conservation quantities for connected reversible bi-linear reactions\nSimilar to the reversible bi-linear reaction, we obtain two pools that represent the disequilibrium driving forces of the two reactions, \n\n$$\\begin{align} p_1 &= x_1x_2 - x_3 / K_1 \\approx (x_{2, eq})x_1' + (x_{1, eq})x_2' - (1/K_1)x_3' = p_1'\\\\ p_2 &= x_3 - x_4x_5 / K_2 \\approx x_3' - (x_{5, eq}/K_2)x_4' - (x_{4, eq}/K_2)x_5' = p_2' \\end{align} \\tag{4.47}$$\n\nand the three pools that represent conservative quantities do not change: \n\n$$\\begin{align} p_3 &= x_1 + x_3 + x_4 \\\\ p_4 &= x_1 + x_3 + x_5 \\\\ p_5 &= x_2 + x_3 + x_5 \\end{align} \\tag{4.48}$$\n\nWe thus can define the pooling matrix as: \n\n$$\\begin{align} \\textbf{P} &= \\begin{pmatrix} {x_{2, eq}} & {x_{1, eq}} & {1/K_1} & {0} & {0} \\\\ {0} & {0} & {1} & {-x_{5, eq}/K_2} & {-x_{4, eq}/K_2} \\\\ {1} & {0} & {1} & {1} & {0} \\\\ {1} & {0} & {1} & {0} & {1} \\\\ {0} & {1} & {1} & {0} & {1} \\\\ \\end{pmatrix} \\\\ &= \\begin{pmatrix} {1} & {1} & {-1} & {0} & {0} \\\\ {0} & {0} & {1} & {-1} & {-1} \\\\ {1} & {0} & {1} & {1} & {0} \\\\ {1} & {0} & {1} & {0} & {1} \\\\ {0} & {1} & {1} & {0} & {1} \\\\ \\end{pmatrix} \\end{align} \\tag{4.49}$$\n\nfor the particular equilibrium constants and concentrations values given above. \n\nThe differential equations for the pools are then formed by (and at this stage we remove the conservation pools as they are always constant): \n\n$$\\begin{equation} \\frac{d\\textbf{p'}}{dt} = \\textbf{PSv(x)} \\ \\text{where} \\ \\textbf{v(x)} \\approx \\begin{pmatrix} {k_1p_1'} \\\\ {k_2p_2'} \\\\ \\end{pmatrix} = \\begin{pmatrix} {k_1} & {0} \\\\ {0} & {k_2} \\\\ \\end{pmatrix} \\begin{pmatrix} {p_1'} \\\\ {p_2'} \\\\ \\end{pmatrix} \\end{equation} \\tag{4.50}$$\n\nwhich gives \n\n$$\\begin{equation} \\frac{d\\textbf{p'}}{dt} = \\begin{pmatrix} {-(x_{2, eq} + x_{1, eq} + 1/K_1)} \\\\ {1} \\\\ \\end{pmatrix}k_1p_1' + \\begin{pmatrix} {1/K_1} \\\\ {-(1 + (x_{5, eq} + x_{4, eq})/K_2} \\\\ \\end{pmatrix}k_2p_2' \\end{equation} \\tag{4.51}$$\n\n### Numerical simulation of connected reversible bi-linear reactions\nThese equations can be simulated once parameter values and initial conditions are specified. In order to illustrate the dynamic behavior in terms of the pools, we consider the particular situation where $K_1 = K_2 = x_{1, eq} = x_{2, eq} = x_{3, eq} = x_{4, eq} = x_{5, eq} = 1$, see Figure\u00a04.5. \n\n\n```python\n# Create MassModel\nmodel = MassModel('Connected_BiLinear_Reversible')\n# Generate the MassMetabolites \nx1 = MassMetabolite(\"x1\")\nx2 = MassMetabolite(\"x2\")\nx3 = MassMetabolite(\"x3\")\nx4 = MassMetabolite(\"x4\")\nx5 = MassMetabolite(\"x5\")\n# Generate the MassReactions \nv1 = MassReaction(\"v1\")\nv2 = MassReaction(\"v2\")\n# Add metabolites to the reaction, add reaction to the model\nv1.add_metabolites({x1: -1, x2: -1, x3: 1})\nv2.add_metabolites({x3: -1, x4: 1, x5: 1})\nmodel.add_reactions([v1, v2])\n# Set parameters and initial conditions\nv1.kf = 1\nv1.kr = 1\nv2.kf = 1\nv2.kr = 1\nmodel.update_initial_conditions({x1: 3, x2: 3, x3: 0, x4: 0, x5: 0})\n\n# Utilize type 2 rate law for kf and kr parameters defined\nmodel.get_rate_expressions(rate_type=2, update_reactions=True)\nmodel\n```\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        NameConnected_BiLinear_Reversible
                                        Memory address0x07fd190e33b50
                                        Stoichiometric Matrix5x2
                                        Matrix Rank2
                                        Number of metabolites5
                                        Initial conditions defined5/5
                                        Number of reactions2
                                        Number of genes0
                                        Number of enzyme modules0
                                        Number of groups0
                                        Objective expression0
                                        Compartments
                                        \n\n\n\n\n\n```python\nt0 = 0\ntf = 10\nsim = Simulation(model, verbose=True)\nconc_sol, flux_sol = sim.simulate(model, time=(t0, tf), verbose=True)\n\n# Define pools\npools = ['x1*x2 - x3 / Keq_v1', 'x3 - x4*x5 / Keq_v2', \n 'x1 + x3 + x4', 'x1 + x3 + x5', 'x2 + x3 + x4']\n\nfor i, equation_str in enumerate(pools):\n pool_id = \"p\" + str(i + 1)\n conc_sol.make_aggregate_solution(\n pool_id, equation=equation_str, \n parameters={v1.Keq_str: v1.kf/v1.kr,\n v2.Keq_str: v2.kf/v2.kr}, update=True)\n```\n\n \u001b[93mWARNING:\u001b[0m \u001b[93mNo compartments found in model. Therefore creating compartment 'compartment' for entire model.\u001b[0m\n\n\n Successfully loaded MassModel 'Connected_BiLinear_Reversible' into RoadRunner.\n Getting time points\n Setting output selections\n Setting simulation values for 'Connected_BiLinear_Reversible'\n Simulating 'Connected_BiLinear_Reversible'\n Simulation for 'Connected_BiLinear_Reversible' successful\n Adding 'Connected_BiLinear_Reversible' simulation solutions to output\n Updating stored solutions\n\n\n\n```python\nfig_4_5 = plt.figure(figsize=(12, 4))\ngs = fig_4_5.add_gridspec(nrows=1, ncols=2)\n\nax1 = fig_4_5.add_subplot(gs[0, 0])\nax2 = fig_4_5.add_subplot(gs[0, 1])\n\nplot_time_profile(\n conc_sol, observable=model.metabolites, ax=ax1, legend=\"left outside\",\n xlabel=\"Time\", ylabel=\"Concentrations\",\n title=(\"(a) Time Profiles of Species\", {\"size\": \"large\"}));\n\nplot_time_profile(\n conc_sol, ax=ax2, observable=[\"p1\", \"p2\", \"p3\", \"p4\", \"p5\"],\n legend=\"right outside\",\n xlabel=\"Time\", ylabel=\"Concentrations\",\n title=(\"(a) Time Profiles of Pools\", {\"size\": \"large\"}));\nfig_4_5.tight_layout()\n```\n\n**Figure 4.5:** The concentration time profiles for the reaction system $x_1 + x_2 \\rightleftharpoons x_3 \\rightleftharpoons x_4 + x_5$ for $k_1 = k_{-1} = k_2 = k_{-2} = 1$\nand $x_1(0)=3, x_2(0)=3, x_3(0)=0, x_4(0)=0, x_5(0)=0$ (a) The concentrations as a function of time. (b) The pools as a function of time.\n\nIn this situation, the dynamic equation for the linearized pools becomes \n\n$$\\begin{equation} \\frac{d\\textbf{p'}}{dt} = \\begin{pmatrix} {-3} \\\\ {1} \\\\ \\end{pmatrix}k_1p_1' + \\begin{pmatrix} {1} \\\\ {-3} \\\\ \\end{pmatrix}k_2p_2' \\end{equation} \\tag{4.52}$$\n\nWe can solve this equation and present the results with a dynamic phase portrait, Figure\u00a04.6. The dynamic behavior of the non-equilibrium pools is shown for a range of parameters. We make three observations here. \n\n1. Figure\u00a04.6 shows that the dynamics for the pools can be decomposed to consider a fast equilibration of the two disequilibrium pools followed by the slow decay of the slower disequilibrium pool (Change values for $k_1$ and $k_2$ to visualize). \n\n2. When reaction 1 is 10 times faster than reaction 2, then initial motion is along the vector $(-3,1)^T$, and when reaction 1 is 10 times slower than reaction 2, then initial motion is along the vector $(1,-3)^T$, \n\n3. The linearized pools move in a similar fashion to the bi-linear disequilibrium pools, see Figure\u00a04.6. The bi-linear and linear simulation do not change that much even though $x_1$, $x_2$, $x_4$, $x_5$ are 25% from their equilibrium value, and $x_3$ is 50% away from equilibrium. \n\n\n```python\n# Set new initial conditions\nmodel.update_initial_conditions({x1: 0.75, x2: 0.75, x3: 1.5, \n x4: 0.75, x5: 0.75})\nsim.update_model_simulation_values(model)\nconc_sol, flux_sol = sim.simulate(model, time=(t0, tf))\n\nfor i, equation_str in enumerate(pools):\n pool_id = \"p\" + str(i + 1)\n conc_sol.make_aggregate_solution(\n pool_id, equation=equation_str, \n parameters={v1.Keq_str: v1.kf/v1.kr,\n v2.Keq_str: v2.kf/v2.kr}, update=True)\n```\n\n\n```python\n# Visualize solution\nfig_4_6 = plt.figure(figsize=(5, 5))\ngs = fig_4_6.add_gridspec(nrows=1, ncols=1)\n\nax1 = fig_4_6.add_subplot(gs[0, 0])\n\nplot_phase_portrait(\n conc_sol, x=\"p1\", y=\"p2\", ax=ax1, legend=\"best\",\n xlabel=\"p1\", ylabel=\"p2\", xlim=(-1.5, .5), ylim=(-.5, 1.5),\n title=(\"Phase portrait of p1 vs p2\", {\"size\": \"large\"}),\n annotate_time_points=\"endpoints\",\n annotate_time_points_labels=True);\nfig_4_6.tight_layout()\n```\n\n**Figure 4.6:** The dynamic response of $x_1 + x_2 \\rightleftharpoons x_3 \\rightleftharpoons x_4 + x_5$ for $K_1 = K_{-1} = 1$. The graphs show the concentrations varying with time for $x_3(0)=1.5, \\ x_1(0) = x_2(0) = x_4(0) = x_5(0) =0.75$ The disequilibrium pools $p_1 = x_1x_2 - x_3 / K_1$ (x-axis) and $p_2 = x_3 - x_4x_5 / K_2$ (y-axis) shown in a phase portrait.\n\n## Summary \n\n* Chemical properties associated with chemical reactions are; stoichiometry thermodynamics, and kinetics. The first two are physico-chemical properties, while the third can be biologically altered through enzyme action. \n\n* Each net reaction can be described by pooled variables that represent a dis-equilibrium quantity and a mass conservation quantity that is associated with the reaction. \n\n* If a reaction is fast compared to its network environment, its disequilibrium variable can be relaxed and then described by the conservation quantity associated with the reaction. \n\n* Linearizing bi-linear rate laws does not create much error for small changes around the reference state. \n\n* Removing a time scale from a model corresponds to reducing the dynamic dimension of the transient response by one. \n\n* As the number of reactions grow, the number of conservation quantities may change. \n\n* Irreversibility of reactions does not change the number of conservation quantities for a system. \n\n$\\tiny{\\text{\u00a9 B. \u00d8. Palsson 2011;}\\ \\text{This publication is in copyright.}\\\\ \\text{Subject to statutory exception and to the provisions of relevant collective licensing agreements,}\\\\ \\text{no reproduction of any part may take place without the written permission of Cambridge University Press.}}$\n", "meta": {"hexsha": "4f7a1c9110f5f65b3062495f5d8e8a8c78d84a8b", "size": 265567, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/education/sb2/chapters/sb2_chapter4.ipynb", "max_stars_repo_name": "SBRG/MASSpy", "max_stars_repo_head_hexsha": "1315c1d40be8feb8731c8143dbc9ba43bf8c78ff", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 13, "max_stars_repo_stars_event_min_datetime": "2020-07-13T00:48:13.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-27T15:42:15.000Z", "max_issues_repo_path": "docs/education/sb2/chapters/sb2_chapter4.ipynb", "max_issues_repo_name": "SBRG/MASSpy", "max_issues_repo_head_hexsha": "1315c1d40be8feb8731c8143dbc9ba43bf8c78ff", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 10, "max_issues_repo_issues_event_min_datetime": "2021-02-17T18:07:43.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-23T16:22:14.000Z", "max_forks_repo_path": "docs/education/sb2/chapters/sb2_chapter4.ipynb", "max_forks_repo_name": "SBRG/MASSpy", "max_forks_repo_head_hexsha": "1315c1d40be8feb8731c8143dbc9ba43bf8c78ff", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2020-01-15T00:48:49.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-04T07:01:17.000Z", "avg_line_length": 189.1502849003, "max_line_length": 66080, "alphanum_fraction": 0.8619331468, "converted": true, "num_tokens": 15410, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.7310585786300049, "lm_q1q2_score": 0.43051245090979035}} {"text": "> Heavily adopted code and format from greilliam\n\n# Simulated Classification\n\n1. State assumptions\n2. Formally define classification/regression problem\n3. provide algorithm for solving problem (including choosing hyperparameters as appropriate)\n4. sample data from a simulation setting inspired by your data (from both null and alternative as defined before)\n5. compute accuracy\n6. plot accuracy vs. sample size in simulation\n7. apply method directly on real data\n8. explain the degree to which you believe the result and why\n\n## Step 1: State assumptions\n\n\nHistogram Equalization \n\\begin{align}\nF(k) = floor((L-1) \\sum_{n=0}^k p_n ) \\\nn = 0, 1..L-1 \\\nL=256 \\\n\\end{align}\n\n\\begin{align}\n\\newline\nF(x,y) | ~~ Histogram~Data\\\nF_{x|0} & = Norm(\\mu0)\\\nF_{x|1} & = Norm(\\mu1)\\\nF_{x|2} & = Norm(\\mu2)\\\n\\mu0 \\neq \\mu1 \\neq \\mu2 \\\n\\end{align}\n\nAdditionally we assume the elements correspond to features 1-4.\n\n## Step 2: Formally define classification/regression problem\n\n\n\\begin{align}\n\\newline\nX = {\\mu0, \\mu1, \\mu2}\\\nY = { 0, 1, 2} \\\n\\end{align}\n\nClassification is applied to reduce the estimated error.\nObjective is to minimize Error:\n$E[l] = \\sum \\Theta(\\hat{Y}_i \\neq Y_i)$\n\n## Step 3: Provide algorithm for solving problem (including choosing hyperparameters as appropriate)\n\n\"Machine Learning\" is usually classification or prediction\n - predictive is subject specific\n \nClarity brains have $(X, Y)$ ~iid $F_{xy}$ which is some distribution\n - X is subject \n - Y is $\\{0, 1\\}$\n - Function g(x) spits out a class label (thus g is a classifier function)\n - G = {g: map from reals to {0, 1}}\n - Classifier takes a single x \n - If $x>k$ but $<0$ is one classifier\n - Best clasisifier is statistical decision theory\n - Need to define a loss function that tells us how wrong we are\n - We need to choose classifier that minimizes loss\n - G* = $\\underset{g \\in G}{argmin} l(g(x), y)$\n - Squared error is a good option $(g(x)-y)^2$\n - Problem is that $(0-1)^2 = (1-0)^2$ so you don't know which side of the \"wrong\" you are\n - Absolute error is $|g(x)-y|$\n - Zero one error \n - If $g(x)=y$ then $l=0$\n - If $g(y)!=y$ then $l=1$\n - If L is the set of loss functions \n - L = {l: yxy -> Real+}\n - Here we are finding which scores are the best \n - Definitions: Voxels, Priors, Baye's rule F_(x,y) = F_(x|y)F_y=F_(y|x)F_x -> \n - $F_(x|y)=N(M_y. 1)$\n - $F_y = Bern(pi)$\n - Next need to fit the joint distribution \n - After fitting the Bayes optimal is called the Bayes plugin\n - MLE - minimizing squared error\n - Sample n train sample (xi, yi) ~iid F_xx generate training data i\u2208[n_train]\n - estimate classifier theta\n - sample iidF_xy i\u2208[n_test]\n \n \n - Best classifier is called the Bayes Optimal\n - g* = argmax F_((x=(x)|(y)=y))\n - Can use a posterari if priors are not equal\n - F_(x=x, y=y) = F_(x|y)F_y\n - compute argmax for y\u2208y\n - Let y = 0 is .99 y-1 is .01\n \n - Next need to relect get change level accuracy which will almost definitely ahppen if you use a regular loss function \n - Use histogram instead of image data\n - Classifier list:\n - LDA\n - Variances are the same \n - Made by Fischer\n - Finds optimal linear classifier (optimal line) under the assumptions that we have made\n - Advantages: Very interpretable, Very fast, Linear\n - Random Forest\n - Decision tree thresholds are created\n - Choose a loss function and then try to do a greedy search\n - Find the optimal thresholds to maximize purity\n - Change thresholds to maximize purity so that most of one group is in one part and the others are in the others\n - Random Forest uses decision trees on subsets of your data, since each tree is noisy and can overfit, so averageing over many different classifiers it will be much more effective\n - This is an ensemble method \n - Every single classifier is on a different point on the bias variance tradeoff so when you average everything it will be more consistent\n - SVM\n - Logistic\n - Neural Network\n - Uses linear algebra, runs on GPU\n - Takes in more information and is very useful for computer vision techniques\n - Natively do the classificiation\n - KNN\n - K nearest neighbor \n - specify apriori k and find the distance between the points and K \n - Assuming K is big enough, it will always converge irrespectively\n - Doesn't care about the distributions, but it is universally consistent\n - QDA\n - Quadratic descriminatory analysis\n - Optimal discriminatory boundary is curved\n - Covariance matrices\n\nClassification Techniques :\n\nK Nearest Neighbors(3) - Default Parameters \\\nSupport Vector Machines - Linear , C= 0.5 - Default\\\nRandom Forest - Max depth = 5 , N Estimators = 10, Max Features =1 - Default\\\nLinear Discriminant Analysis \\\nQuadratic Discriminant Analysis \\\nWe ran into errors with QDA, for which reason it was ignored in the assignment.\n\n\n```python\nimport os\nPATH=\"/Users/david/Desktop/CourseWork/TheArtOfDataScience/claritycontrol/code/scripts/\" # use your own path\nos.chdir(PATH)\n\nimport clarity as cl # I wrote this module for easier operations on data\nimport clarity.resources as rs\nimport csv,gc # garbage memory collection :)\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport jgraph as ig\n%matplotlib inline\n\n# settings for histogram\nBINS=32 # histogram bins\nRANGE=(10.0,300.0)\n```\n\n### Histogram data preparation\n\n> Skip this step if the data are already baked, just eat them!\n\nThis and next step will load raw image datasets (which are pretty large) and extract histograms from these dataset. The datasets are not included in the repository (Because they are too large), but you can skip this and next step, use the histogram data that already generated in the repository.\n\n1. Set suitable data value range to get histogram from the majority of the datasets.\n\n\n```python\nfor token in rs.TOKENS:\n c = cl.Clarity(token)\n fname = rs.HIST_DATA_PATH+token+\".csv\"\n hist, bin_edges = c.loadImg().getHistogram(bins=BINS,range=RANGE,density=False)\n np.savetxt(fname,hist,delimiter=',')\n print fname,\"saved.\"\n del c\n gc.collect()\n```\n\n Image Loaded: ../data/raw/Cocaine174.img\n ../data/hist/Cocaine174.csv saved.\n Image Loaded: ../data/raw/Cocaine175.img\n ../data/hist/Cocaine175.csv saved.\n Image Loaded: ../data/raw/Cocaine178.img\n ../data/hist/Cocaine178.csv saved.\n Image Loaded: ../data/raw/Control181.img\n ../data/hist/Control181.csv saved.\n Image Loaded: ../data/raw/Control182.img\n ../data/hist/Control182.csv saved.\n Image Loaded: ../data/raw/Control189.img\n ../data/hist/Control189.csv saved.\n Image Loaded: ../data/raw/Control239.img\n ../data/hist/Control239.csv saved.\n Image Loaded: ../data/raw/Control258.img\n ../data/hist/Control258.csv saved.\n Image Loaded: ../data/raw/Fear187.img\n ../data/hist/Fear187.csv saved.\n Image Loaded: ../data/raw/Fear197.img\n ../data/hist/Fear197.csv saved.\n Image Loaded: ../data/raw/Fear199.img\n ../data/hist/Fear199.csv saved.\n Image Loaded: ../data/raw/Fear200.img\n ../data/hist/Fear200.csv saved.\n\n\n### Scale data\n\n\n```python\nimport numpy as np\nimport clarity.resources as rs\nfeatures = np.empty(shape=(1,BINS))\nfor token in rs.TOKENS:\n fname = rs.HIST_DATA_PATH+token+\".csv\"\n data = np.loadtxt(fname,delimiter=',')\n features = np.vstack([features,data])\nfeatures = features[1:,]\nminc = np.min(features)\nmaxc = np.max(features)\nfeatures = (features-minc)/(maxc-minc)\nprint features\nnp.savetxt(rs.HIST_DATA_PATH+\"features.csv\",features,delimiter=',')\n```\n\n [[ 3.02680054e-08 5.44824096e-08 2.05822436e-07 5.87199304e-07\n 1.83726793e-06 6.61658597e-06 2.59366538e-05 1.53237831e-04\n 5.70115436e-03 4.51718743e-01 1.00000000e+00 1.35286291e-01\n 8.49485827e-02 7.86618544e-02 7.35613535e-02 6.39701853e-02\n 5.84338221e-02 4.16227176e-02 3.39392057e-02 2.77592599e-02\n 2.26080025e-02 1.85846279e-02 1.54385957e-02 1.29580479e-02\n 1.10633858e-02 9.58203931e-03 8.43465187e-03 7.51167143e-03\n 6.77831398e-03 6.13509768e-03 5.61766007e-03 5.66908239e-03]\n [ 5.75092102e-08 1.63447229e-07 3.29921258e-07 7.65780536e-07\n 2.70898648e-06 6.04452067e-06 2.48560860e-05 1.17918095e-04\n 3.74111033e-03 2.92376673e-01 5.84049388e-01 1.55948727e-01\n 1.33041235e-01 1.10344462e-01 8.33999686e-02 5.94000858e-02\n 4.60053584e-02 2.86529742e-02 2.09942094e-02 1.57893807e-02\n 1.21508849e-02 9.69784168e-03 7.96022813e-03 6.71293811e-03\n 5.74635963e-03 4.98288249e-03 4.34869513e-03 3.79060457e-03\n 3.33360309e-03 2.91697611e-03 2.57125192e-03 2.48431616e-03]\n [ 1.51340027e-08 3.66242865e-07 2.99653253e-07 4.78234485e-07\n 1.74949071e-06 5.71157261e-06 2.15841146e-05 1.47459669e-04\n 4.75858749e-03 3.65928895e-01 5.59457624e-01 9.14320651e-02\n 7.04332368e-02 6.77734601e-02 6.61138926e-02 6.02482195e-02\n 5.87921771e-02 4.49651953e-02 3.86687359e-02 3.28632205e-02\n 2.74471362e-02 2.28967502e-02 1.90871041e-02 1.59889346e-02\n 1.34611687e-02 1.14588645e-02 9.82884766e-03 8.54004507e-03\n 7.50793030e-03 6.61872592e-03 5.89751502e-03 5.80544883e-03]\n [ 3.63216064e-08 6.96164123e-08 2.30036841e-07 5.81145703e-07\n 3.06826770e-05 5.18793612e-06 1.32725204e-05 7.93142812e-05\n 2.81480040e-03 2.27715179e-01 3.88426713e-01 4.86363048e-02\n 3.30274910e-02 3.20018384e-02 3.28574847e-02 3.29833875e-02\n 3.59319392e-02 3.01235029e-02 2.75277914e-02 2.42873199e-02\n 2.08061179e-02 1.76493012e-02 1.49466922e-02 1.27984447e-02\n 1.11091631e-02 9.75884987e-03 8.67802177e-03 7.79015826e-03\n 7.11130139e-03 6.48077343e-03 5.96284851e-03 6.07441638e-03]\n [ 9.08040161e-09 3.08733655e-07 9.08040161e-08 2.72412048e-07\n 1.01700498e-06 3.18722096e-06 1.19074333e-05 7.10420354e-05\n 2.30429719e-03 1.88612606e-01 3.07727452e-01 6.41034917e-02\n 6.12421027e-02 6.40454376e-02 6.18823649e-02 5.43280337e-02\n 4.93737908e-02 3.35422378e-02 2.49486728e-02 1.82312308e-02\n 1.34255372e-02 1.02467338e-02 8.07892109e-03 6.57214043e-03\n 5.49152724e-03 4.67500239e-03 4.07299598e-03 3.58319912e-03\n 3.20458875e-03 2.88046684e-03 2.60133832e-03 2.60068453e-03]\n [ 6.35628113e-08 1.48313226e-07 2.84519250e-07 7.99075341e-07\n 3.58675864e-06 8.91090078e-06 2.40842519e-05 1.47432427e-04\n 4.54237405e-03 3.57057751e-01 5.48353403e-01 1.44579656e-01\n 1.03936638e-01 7.37345530e-02 4.85449711e-02 3.20946643e-02\n 2.47973903e-02 1.60047890e-02 1.22824932e-02 9.62169040e-03\n 7.62616016e-03 6.16654008e-03 5.07482156e-03 4.24220018e-03\n 3.58879265e-03 3.05773746e-03 2.62668777e-03 2.26323562e-03\n 1.96884900e-03 1.70269032e-03 1.49130765e-03 1.42861050e-03]\n [ 6.96164123e-08 1.18045221e-07 3.72296466e-07 7.89994940e-07\n 1.88872353e-06 6.16861949e-06 2.43203423e-05 1.57578263e-04\n 5.39432759e-03 4.22281610e-01 6.68446346e-01 1.20708070e-01\n 8.04814670e-02 6.31055404e-02 5.22423896e-02 4.33928239e-02\n 3.99923679e-02 2.89120350e-02 2.35104855e-02 1.89281002e-02\n 1.51471511e-02 1.22824206e-02 1.01160638e-02 8.46310077e-03\n 7.12237039e-03 6.07994029e-03 5.20926296e-03 4.50099768e-03\n 3.92865392e-03 3.42491864e-03 3.02073785e-03 2.94322755e-03]\n [ 2.11876038e-08 8.17236145e-08 2.14902838e-07 5.81145703e-07\n 3.54741023e-06 8.78982876e-06 2.38723758e-05 1.79274369e-04\n 5.95812368e-03 4.64601472e-01 6.34477707e-01 8.05992186e-02\n 6.43304381e-02 6.39353075e-02 6.14997047e-02 5.54860483e-02\n 5.27806062e-02 3.82177487e-02 3.09508758e-02 2.50019384e-02\n 2.01594753e-02 1.64956997e-02 1.36656594e-02 1.14503289e-02\n 9.70391646e-03 8.28194071e-03 7.10695793e-03 6.15469016e-03\n 5.40964926e-03 4.74080806e-03 4.20901132e-03 4.10615458e-03]\n [ 1.24098822e-07 2.17929639e-07 3.60189264e-07 6.78003320e-07\n 1.89175033e-06 5.66919740e-06 2.02674564e-05 1.37934327e-04\n 4.31221613e-03 3.23488629e-01 4.72462055e-01 9.56688382e-02\n 7.78883222e-02 7.01059247e-02 5.98414599e-02 4.71343913e-02\n 4.03152942e-02 2.79142745e-02 2.27308120e-02 1.87224594e-02\n 1.53413506e-02 1.26528919e-02 1.04781902e-02 8.81702456e-03\n 7.56055728e-03 6.58286136e-03 5.78614087e-03 5.12447016e-03\n 4.59966533e-03 4.10518298e-03 3.68900698e-03 3.64269088e-03]\n [ 5.75092102e-08 9.68576172e-08 1.78581232e-07 5.44824096e-07\n 1.59815068e-06 5.24241853e-06 1.89144765e-05 1.22225232e-04\n 4.28175139e-03 3.32084395e-01 5.21520490e-01 8.18890443e-02\n 6.44157788e-02 5.91305219e-02 5.30068656e-02 4.60999217e-02\n 4.42667762e-02 3.33409010e-02 2.82204747e-02 2.37420660e-02\n 1.98499335e-02 1.66562594e-02 1.39673678e-02 1.17270996e-02\n 9.87607785e-03 8.36676982e-03 7.17462508e-03 6.21617658e-03\n 5.47604213e-03 4.82562809e-03 4.30427986e-03 4.25544546e-03]\n [ 1.48313226e-07 1.75554431e-07 4.08618072e-07 9.05013360e-07\n 2.22772519e-06 7.56700134e-06 2.32034529e-05 1.40591858e-04\n 4.59383571e-03 3.52431895e-01 5.96727000e-01 1.51405639e-01\n 1.13602783e-01 8.30500614e-02 5.40600529e-02 3.30028922e-02\n 2.26985584e-02 1.32255959e-02 9.49274264e-03 7.10924921e-03\n 5.44678507e-03 4.31920502e-03 3.49370873e-03 2.88800357e-03\n 2.42458830e-03 2.04189175e-03 1.74202965e-03 1.48730017e-03\n 1.27906537e-03 1.10062943e-03 9.60428024e-04 9.17032785e-04]\n [ 0.00000000e+00 1.33179224e-07 1.33179224e-07 2.87546051e-07\n 1.38930145e-06 2.94507692e-06 9.83407494e-06 6.18950442e-05\n 2.06246794e-03 1.63011485e-01 2.58921671e-01 4.43054648e-02\n 3.46486605e-02 3.27022038e-02 3.08151722e-02 2.81701844e-02\n 2.80819501e-02 2.16081807e-02 1.83370235e-02 1.52709412e-02\n 1.25290171e-02 1.02423268e-02 8.38137413e-03 6.85885411e-03\n 5.64271290e-03 4.68338360e-03 3.92759151e-03 3.32991342e-03\n 2.88653255e-03 2.52064382e-03 2.24798660e-03 2.22261293e-03]]\n\n\n### Setup Step\n\n\n```python\nfrom sklearn import cross_validation\nfrom sklearn.cross_validation import LeaveOneOut\nfrom sklearn.neighbors import KNeighborsClassifier\nfrom sklearn.ensemble import RandomForestClassifier\nfrom sklearn.svm import SVC\nfrom sklearn.discriminant_analysis import LinearDiscriminantAnalysis\nfrom sklearn.discriminant_analysis import QuadraticDiscriminantAnalysis\n\n%matplotlib inline\n\nnp.random.seed(12345678) # for reproducibility, set random seed\n\n# Cocaine = [\"Cocaine174\",\"Cocaine175\",\"Cocaine178\"]\n# Control = [\"Control181\",\"Control182\",\"Control189\",\"Control239\",\"Control258\"]\n# Fear = [\"Fear187\",\"Fear197\",\"Fear199\",\"Fear200\"]\n\nfeatures = np.loadtxt(rs.HIST_DATA_PATH+\"features.csv\",delimiter=',')\ntemp_mu = np.mean(features,axis=1)\ntemp_std = np.std(features,axis=1)\n\nmu = [np.mean(temp_mu[0:3]),np.mean(temp_mu[3:8]),np.mean(temp_mu[8:12])]\nstd = [np.mean(temp_std[0:3]),np.mean(temp_std[3:8]),np.mean(temp_std[8:12])]\nprint mu\nprint std\nstd=[1,1,1]\n\n# define number of subjects per class\nS = np.array((9, 21, 30, 39, 45, 63, 81, 96, 108, 210, 333))\n\nnames = [\"Nearest Neighbors\", \"Linear SVM\", \"Random Forest\",\n \"Linear Discriminant Analysis\", \"Quadratic Discriminant Analysis\"]\n\nclassifiers = [\n KNeighborsClassifier(3),\n SVC(kernel=\"linear\", C=0.5),\n RandomForestClassifier(max_depth=5, n_estimators=10, max_features=1),\n LinearDiscriminantAnalysis()]\n# QuadraticDiscriminantAnalysis()]\n```\n\n [0.056280388503550105, 0.042498942485078253, 0.038805754482162662]\n [0.13726273492760607, 0.1031691019233711, 0.093076772034756644]\n\n\n## Steps 4 & 5: Sample data from setting similar to data and record classification accuracy\n\n\n```python\naccuracy = np.zeros((len(S), len(classifiers), 2), dtype=np.dtype('float64'))\nfor idx1, s in enumerate(S):\n s0=s/3\n s1=s/3\n s2=s/3\n \n x0 = np.random.normal(mu[0],std[0],(s0,BINS))\n x1 = np.random.normal(mu[1],std[1],(s1,BINS))\n x2 = np.random.normal(mu[2],std[2],(s2,BINS))\n X = x0\n X = np.vstack([X,x1])\n X = np.vstack([X,x2])\n y = np.append(np.append(np.zeros(s0), np.ones(s1)),np.ones(s2)*2)\n for idx2, cla in enumerate(classifiers):\n X_train, X_test, y_train, y_test = cross_validation.train_test_split(X, y, test_size=0.4, random_state=0)\n clf = cla.fit(X_train, y_train)\n loo = LeaveOneOut(len(X))\n scores = cross_validation.cross_val_score(clf, X, y, cv=loo)\n accuracy[idx1, idx2,] = [scores.mean(), scores.std()]\n print(\"Accuracy of %s: %0.2f (+/- %0.2f)\" % (names[idx2], scores.mean(), scores.std() * 2))\n \nprint accuracy\n```\n\n Accuracy of Nearest Neighbors: 0.33 (+/- 0.94)\n Accuracy of Linear SVM: 0.44 (+/- 0.99)\n Accuracy of Random Forest: 0.11 (+/- 0.63)\n Accuracy of Linear Discriminant Analysis: 0.22 (+/- 0.83)\n Accuracy of Nearest Neighbors: 0.24 (+/- 0.85)\n Accuracy of Linear SVM: 0.24 (+/- 0.85)\n Accuracy of Random Forest: 0.14 (+/- 0.70)\n\n /usr/local/lib/python2.7/site-packages/sklearn/discriminant_analysis.py:387: UserWarning: Variables are collinear.\n warnings.warn(\"Variables are collinear.\")\n /usr/local/lib/python2.7/site-packages/sklearn/discriminant_analysis.py:453: UserWarning: The priors do not sum to 1. Renormalizing\n UserWarning)\n\n\n \n Accuracy of Linear Discriminant Analysis: 0.19 (+/- 0.79)\n Accuracy of Nearest Neighbors: 0.30 (+/- 0.92)\n Accuracy of Linear SVM: 0.27 (+/- 0.88)\n Accuracy of Random Forest: 0.40 (+/- 0.98)\n Accuracy of Linear Discriminant Analysis: 0.37 (+/- 0.96)\n Accuracy of Nearest Neighbors: 0.28 (+/- 0.90)\n Accuracy of Linear SVM: 0.28 (+/- 0.90)\n Accuracy of Random Forest: 0.26 (+/- 0.87)\n Accuracy of Linear Discriminant Analysis: 0.33 (+/- 0.94)\n Accuracy of Nearest Neighbors: 0.27 (+/- 0.88)\n Accuracy of Linear SVM: 0.27 (+/- 0.88)\n Accuracy of Random Forest: 0.40 (+/- 0.98)\n Accuracy of Linear Discriminant Analysis: 0.36 (+/- 0.96)\n Accuracy of Nearest Neighbors: 0.44 (+/- 0.99)\n Accuracy of Linear SVM: 0.37 (+/- 0.96)\n Accuracy of Random Forest: 0.32 (+/- 0.93)\n Accuracy of Linear Discriminant Analysis: 0.30 (+/- 0.92)\n Accuracy of Nearest Neighbors: 0.32 (+/- 0.93)\n Accuracy of Linear SVM: 0.26 (+/- 0.88)\n Accuracy of Random Forest: 0.22 (+/- 0.83)\n Accuracy of Linear Discriminant Analysis: 0.40 (+/- 0.98)\n Accuracy of Nearest Neighbors: 0.32 (+/- 0.94)\n Accuracy of Linear SVM: 0.25 (+/- 0.87)\n Accuracy of Random Forest: 0.32 (+/- 0.94)\n Accuracy of Linear Discriminant Analysis: 0.24 (+/- 0.85)\n Accuracy of Nearest Neighbors: 0.29 (+/- 0.90)\n Accuracy of Linear SVM: 0.47 (+/- 1.00)\n Accuracy of Random Forest: 0.28 (+/- 0.90)\n Accuracy of Linear Discriminant Analysis: 0.38 (+/- 0.97)\n Accuracy of Nearest Neighbors: 0.27 (+/- 0.88)\n Accuracy of Linear SVM: 0.38 (+/- 0.97)\n Accuracy of Random Forest: 0.35 (+/- 0.96)\n Accuracy of Linear Discriminant Analysis: 0.37 (+/- 0.96)\n Accuracy of Nearest Neighbors: 0.39 (+/- 0.97)\n Accuracy of Linear SVM: 0.34 (+/- 0.94)\n Accuracy of Random Forest: 0.35 (+/- 0.95)\n Accuracy of Linear Discriminant Analysis: 0.32 (+/- 0.93)\n [[[ 0.33333333 0.47140452]\n [ 0.44444444 0.49690399]\n [ 0.11111111 0.31426968]\n [ 0.22222222 0.41573971]]\n \n [[ 0.23809524 0.42591771]\n [ 0.23809524 0.42591771]\n [ 0.14285714 0.34992711]\n [ 0.19047619 0.39267673]]\n \n [[ 0.3 0.45825757]\n [ 0.26666667 0.44221664]\n [ 0.4 0.48989795]\n [ 0.36666667 0.48189441]]\n \n [[ 0.28205128 0.44999817]\n [ 0.28205128 0.44999817]\n [ 0.25641026 0.43665093]\n [ 0.33333333 0.47140452]]\n \n [[ 0.26666667 0.44221664]\n [ 0.26666667 0.44221664]\n [ 0.4 0.48989795]\n [ 0.35555556 0.47868132]]\n \n [[ 0.44444444 0.49690399]\n [ 0.36507937 0.48145241]\n [ 0.31746032 0.4654882 ]\n [ 0.3015873 0.45894706]]\n \n [[ 0.32098765 0.46685606]\n [ 0.25925926 0.43822813]\n [ 0.22222222 0.41573971]\n [ 0.39506173 0.48886395]]\n \n [[ 0.32291667 0.46759116]\n [ 0.25 0.4330127 ]\n [ 0.32291667 0.46759116]\n [ 0.23958333 0.42682919]]\n \n [[ 0.28703704 0.45237902]\n [ 0.47222222 0.4992278 ]\n [ 0.27777778 0.44790321]\n [ 0.37962963 0.48529473]]\n \n [[ 0.26666667 0.44221664]\n [ 0.38095238 0.48562091]\n [ 0.35238095 0.47771186]\n [ 0.36666667 0.48189441]]\n \n [[ 0.38738739 0.48715336]\n [ 0.33633634 0.47245551]\n [ 0.34834835 0.47644703]\n [ 0.31831832 0.46582375]]]\n\n\n## Step 6: Plot Accuracy versus N\n\n\n```python\nplt.errorbar(S, accuracy[:,0,0], yerr = accuracy[:,0,1], hold=True, label=names[0])\nplt.errorbar(S, accuracy[:,1,0], yerr = accuracy[:,1,1], color='green', hold=True, label=names[1])\nplt.errorbar(S, accuracy[:,2,0], yerr = accuracy[:,2,1], color='red', hold=True, label=names[2])\nplt.errorbar(S, accuracy[:,3,0], yerr = accuracy[:,3,1], color='black', hold=True, label=names[3])\n# plt.errorbar(S, accuracy[:,4,0], yerr = accuracy[:,4,1], color='brown', hold=True, label=names[4])\nplt.xscale('log')\nplt.xlabel('number of samples')\nplt.ylabel('accuracy')\nplt.title('Accuracy of classification under simulated data')\nplt.axhline(1, color='red', linestyle='--')\nplt.legend(loc='center left', bbox_to_anchor=(1, 0.5))\nplt.show()\n```\n\n## Step 7: Apply technique to data\n\n\n```python\ny=np.array([0,0,0,1,1,1,1,1,2,2,2,2])\nfeatures = np.loadtxt(rs.HIST_DATA_PATH+\"features.csv\",delimiter=',')\n```\n\n\n```python\naccuracy=np.zeros((len(classifiers),2))\nfor idx, cla in enumerate(classifiers):\n X_train, X_test, y_train, y_test = cross_validation.train_test_split(features, y, test_size=0.4, random_state=0)\n clf = cla.fit(X_train, y_train)\n loo = LeaveOneOut(len(features))\n scores = cross_validation.cross_val_score(clf, features, y, cv=loo)\n accuracy[idx,] = [scores.mean(), scores.std()]\n print(\"Accuracy of %s: %0.2f (+/- %0.2f)\" % (names[idx], scores.mean(), scores.std() * 2))\n```\n\n Accuracy of Nearest Neighbors: 0.08 (+/- 0.55)\n Accuracy of Linear SVM: 0.25 (+/- 0.87)\n Accuracy of Random Forest: 0.25 (+/- 0.87)\n Accuracy of Linear Discriminant Analysis: 0.25 (+/- 0.87)\n\n\n## Step 8: Reflect on result\n\nOur results are highly unsatisfactory with very low accuracy rates and very high error bar values. This is roughly what we expected however due to the nature of our data being rather unsuited for this kind of analysis. We will do some statistical reconfigurations in order to analyze the data to a more satisfactory state.\n", "meta": {"hexsha": "a5f11f2d52826cffa81e34fb2a9a5153ed2a0fee", "size": 65976, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "assignments/a05_classification_simulation.ipynb", "max_stars_repo_name": "Upward-Spiral-Science/claritycontrol", "max_stars_repo_head_hexsha": "3da44a35f4eb8746c408ad34e7f433d14c031323", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2016-02-04T20:32:20.000Z", "max_stars_repo_stars_event_max_datetime": "2016-02-21T15:44:01.000Z", "max_issues_repo_path": "code/a05_classification_simulation.ipynb", "max_issues_repo_name": "Upward-Spiral-Science/claritycontrol", "max_issues_repo_head_hexsha": "3da44a35f4eb8746c408ad34e7f433d14c031323", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2016-02-04T20:24:34.000Z", "max_issues_repo_issues_event_max_datetime": "2016-04-28T10:08:32.000Z", "max_forks_repo_path": "code/a05_classification_simulation.ipynb", "max_forks_repo_name": "Upward-Spiral-Science/claritycontrol", "max_forks_repo_head_hexsha": "3da44a35f4eb8746c408ad34e7f433d14c031323", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 93.5829787234, "max_line_length": 35374, "alphanum_fraction": 0.7718715897, "converted": true, "num_tokens": 8988, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.6370307944803832, "lm_q1q2_score": 0.4304828606688831}} {"text": "# [ATM 623: Climate Modeling](../index.ipynb)\n[Brian E. J. Rose](http://www.atmos.albany.edu/facstaff/brose/index.html), University at Albany\n# Lecture 7: Grey radiation modeling with climlab\n\n### About these notes:\n\nThis document uses the interactive [`IPython notebook`](http://ipython.org/notebook.html) format (now also called [`Jupyter`](https://jupyter.org)). The notes can be accessed in several different ways:\n\n- The interactive notebooks are hosted on `github` at https://github.com/brian-rose/ClimateModeling_courseware\n- The latest versions can be viewed as static web pages [rendered on nbviewer](http://nbviewer.ipython.org/github/brian-rose/ClimateModeling_courseware/blob/master/index.ipynb)\n- A complete snapshot of the notes as of May 2015 (end of spring semester) are [available on Brian's website](http://www.atmos.albany.edu/facstaff/brose/classes/ATM623_Spring2015/Notes/index.html).\n\nMany of these notes make use of the `climlab` package, available at https://github.com/brian-rose/climlab\n\n## Contents\n\n1. [Introducing `climlab`](#section1)\n2. [Using `climlab` to implement the two-layer leaky greenhouse model](#section2)\n3. [The observed annual, global mean temperature profile](#section3)\n4. [A 30-layer model using the observed temperatures](#section4)\n5. [Radiative forcing in the 30-layer model](#section5)\n6. [Radiative equilibrium in the 30-layer model](#section6)\n7. [Radiative-Convective Equilibrium in the 30-layer model](#section7)\n8. [Putting stratospheric ozone in the grey-gas model](#section8)\n\n____________\n\n\n## 1. Introducing `climlab`\n____________\n\n``climlab`` is a flexible engine for process-oriented climate modeling.\nIt is based on a very general concept of a model as a collection of individual, \ninteracting processes. ``climlab`` defines a base class called ``Process``, which\ncan contain an arbitrarily complex tree of sub-processes (each also some \nsub-class of ``Process``). Every climate process (radiative, dynamical, \nphysical, turbulent, convective, chemical, etc.) can be simulated as a stand-alone\nprocess model given appropriate input, or as a sub-process of a more complex model. \nNew classes of model can easily be defined and run interactively by putting together an\nappropriate collection of sub-processes.\n\n``climlab`` is a work-in-progress, and the code base will evolve substantially over the course of this semester.\nThe latest code can always be found on ``github``:\n\nhttps://github.com/brian-rose/climlab\n\nYou are strongly encouraged to clone the ``climlab`` repository and use ``git`` to keep your local copy up-to-date.\n\nRunning this notebook requires that ``climlab`` is already installed on your system.\n\n\n```python\n%matplotlib inline\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport netCDF4 as nc\nimport climlab\n```\n\n____________\n\n\n## 2. Using `climlab` to implement the two-layer leaky greenhouse model\n____________\n\nOne of the things that ``climlab`` is set up to do is the grey-radiation modeling we have already been discussing.\n\nSince we already derived a [complete analytical solution to the two-layer leaky greenhouse model](Lecture06 -- Elementary greenhouse models.ipynb), we will use this to validate the `climlab` code.\n\n\n\n### Validation\n\nWe want to verify that the model reproduces the observed OLR given observed temperatures, and the absorptivity that we tuned in the analytical model. The target numbers are:\n\n\\begin{align}\nT_s &= 288 \\text{ K} \\\\\nT_0 &= 275 \\text{ K} \\\\\nT_1 &= 230 \\text{ K} \\\\\n\\end{align}\n\n$$ \\epsilon = 0.58377 $$\n\n$$ OLR = 239 \\text{ W m}^{-2} $$\n\n\n### Initialize a model in `climlab`\nThe first thing we do is create a new model.\n\nThe following example code is sparsely commented but will hopefully orient you on the basics of defining and working with a `climlab Process` object.\n\n\n```python\n# Test in a 2-layer atmosphere\ncol = climlab.GreyRadiationModel(num_lev=2)\nprint col\n```\n\n climlab Process of type . \n State variables and domain shapes: \n Tatm: (2,) \n Ts: (1,) \n The subprocess tree: \n top: \n LW: \n SW: \n surface: \n insolation: \n \n\n\n\n```python\ncol.subprocess\n```\n\n\n\n\n {'LW': ,\n 'SW': ,\n 'insolation': ,\n 'surface': }\n\n\n\n\n```python\ncol.state\n```\n\n\n\n\n {'Tatm': Field([ 278., 200.]), 'Ts': Field([ 288.])}\n\n\n\n\n```python\ncol.Ts\n```\n\n\n\n\n Field([ 288.])\n\n\n\n\n```python\ncol.Ts[:] = 288.\ncol.Tatm[:] = np.array([275., 230.])\ncol.state\n```\n\n\n\n\n {'Tatm': Field([ 275., 230.]), 'Ts': Field([ 288.])}\n\n\n\n\n```python\nLW = col.subprocess['LW']\nprint LW\n```\n\n climlab Process of type . \n State variables and domain shapes: \n Tatm: (2,) \n Ts: (1,) \n The subprocess tree: \n top: \n \n\n\n\n```python\nLW.absorptivity\n```\n\n\n\n\n Field([ 0.47737425, 0.47737425])\n\n\n\n\n```python\nLW.absorptivity = 0.58377\nLW.absorptivity\n```\n\n\n\n\n Field([ 0.58377, 0.58377])\n\n\n\n\n```python\n# This does all the calculations that would be performed at each time step, \n# but doesn't actually update the temperatures\ncol.compute_diagnostics()\ncol.diagnostics\n```\n\n\n\n\n {'ASR': Field([ 239.2513]),\n 'LW_absorbed_atm': array([-96.82138041, 20.03935568]),\n 'LW_absorbed_sfc': Field([-162.23386935]),\n 'LW_down_sfc': array([ 227.87116061]),\n 'LW_emission': Field([ 189.31461699, 92.63278385]),\n 'LW_up_sfc': Field([ 390.10502995]),\n 'OLR': array([ 239.01589408]),\n 'SW_absorbed_atm': array([-0., -0.]),\n 'SW_absorbed_sfc': Field([ 239.2513]),\n 'SW_absorbed_total': 239.25130000000001,\n 'SW_down_TOA': Field([ 341.3]),\n 'SW_down_sfc': array([ 341.3]),\n 'SW_emission': Field([ 0., 0.]),\n 'SW_up_TOA': array([ 102.0487]),\n 'SW_up_sfc': Field([ 0.]),\n 'insolation': Field([ 341.3]),\n 'planetary_albedo': Field([ 0.299])}\n\n\n\n\n```python\n# Check OLR against our analytical solution\ncol.diagnostics['OLR']\n```\n\n\n\n\n array([ 239.01589408])\n\n\n\n\n```python\ncol.state\n```\n\n\n\n\n {'Tatm': Field([ 275., 230.]), 'Ts': Field([ 288.])}\n\n\n\n\n```python\n# perform a single time step\ncol.step_forward()\n```\n\n\n```python\ncol.state\n```\n\n\n\n\n {'Tatm': Field([ 273.36692033, 230.33800245]), 'Ts': Field([ 289.59144429])}\n\n\n\n\n```python\n# integrate out to radiative equilibrium\ncol.integrate_years(2.)\n```\n\n Integrating for 730 steps, 730.4844 days, or 2.0 years.\n Total elapsed time is 2.00962539378 years.\n\n\n\n```python\n# Check for equilibrium\ncol.diagnostics['ASR'] - col.diagnostics['OLR']\n```\n\n\n\n\n Field([ -5.20579960e-07])\n\n\n\n\n```python\n# Compare these temperatures against our analytical solutions for radiative equilibrium\ncol.state\n```\n\n\n\n\n {'Tatm': Field([ 262.08988341, 233.62925798]), 'Ts': Field([ 296.20384538])}\n\n\n\nSo it looks like `climlab` agrees with our analytical results. That's good.\n\n____________\n\n\n## 3. The observed annual, global mean temperature profile\n____________\n\nWe want to model the OLR in a column whose temperatures match observations. As we've done before, we'll calculate the global, annual mean air temperature from the NCEP Reanalysis data.\n\n\n```python\n# This will try to read the data over the internet.\n# If you have a local copy of the data, just use the local path to the .nc file instead of the URL\nncep_url = \"http://www.esrl.noaa.gov/psd/thredds/dodsC/Datasets/ncep.reanalysis.derived/\"\nncep_air = nc.Dataset( ncep_url + \"pressure/air.mon.1981-2010.ltm.nc\" )\nlevel = ncep_air.variables['level'][:]\nlat = ncep_air.variables['lat'][:]\n# A log-pressure height coordinate\nzstar = -np.log(level/1000)\n```\n\n\n```python\nTzon = np.mean(ncep_air.variables['air'][:],axis=(0,3))\nTglobal = np.average( Tzon , weights=np.cos(np.deg2rad(lat)), axis=1) + climlab.constants.tempCtoK\n# Note the useful conversion factor. climlab.constants has lots of commonly used constant pre-defined\n```\n\n\n```python\n# Here we are plotting with respect to log(pressure) but labeling the axis in pressure units\nfig = plt.figure( figsize=(8,6) )\nax = fig.add_subplot(111)\nax.plot( Tglobal , zstar )\nyticks = np.array([1000., 750., 500., 250., 100., 50., 20., 10.])\nax.set_yticks(-np.log(yticks/1000.))\nax.set_yticklabels(yticks)\nax.set_xlabel('Temperature (K)', fontsize=16)\nax.set_ylabel('Pressure (hPa)', fontsize=16 )\nax.set_title('Global, annual mean sounding from NCEP Reanalysis', fontsize = 24)\nax.grid()\n```\n\n____________\n\n\n## 4. A 30-layer model using the observed temperatures\n____________\n\n\n\n\n```python\n# initialize a grey radiation model with 30 levels\ncol = climlab.GreyRadiationModel()\nprint col\n```\n\n climlab Process of type . \n State variables and domain shapes: \n Tatm: (30,) \n Ts: (1,) \n The subprocess tree: \n top: \n LW: \n SW: \n surface: \n insolation: \n \n\n\n\n```python\n# interpolate to 30 evenly spaced pressure levels\nlev = col.lev\nTinterp = np.flipud(np.interp(np.flipud(lev), np.flipud(level), np.flipud(Tglobal)))\nTinterp\n# Need to 'flipud' because the interpolation routine needs the pressure data to be in increasing order\n```\n\n\n\n\n array([ 287.4463874 , 285.6810201 , 283.98269653, 282.48550415,\n 280.98831177, 279.29533895, 277.60236613, 275.90939331,\n 274.21642049, 272.25855509, 270.03579712, 267.81303914,\n 265.29490662, 262.48139954, 259.66789246, 256.48087056,\n 252.92033386, 249.35979716, 245.279658 , 240.67991638,\n 236.08017476, 231.30422974, 226.3520813 , 221.78252157,\n 217.19397481, 212.58644104, 208.29142761, 206.96233453,\n 211.66334534, 224.34736633])\n\n\n\n\n```python\n# Initialize model with observed temperatures\ncol.Ts[:] = Tglobal[0]\ncol.Tatm[:] = Tinterp\n```\n\n\n```python\n# A handy re-usable routine for making a plot of the temperature profiles\n# We will plot temperatures with respect to log(pressure) to get a height-like coordinate\ndef plot_sounding(collist):\n color_cycle=['r', 'g', 'b', 'y']\n # col is either a column model object or a list of column model objects\n if isinstance(collist, climlab.Process):\n # make a list with a single item\n collist = [collist]\n fig = plt.figure()\n ax = fig.add_subplot(111)\n for i, col in enumerate(collist):\n zstar = -np.log(col.lev/climlab.constants.ps)\n ax.plot(col.Tatm, zstar, color=color_cycle[i])\n ax.plot(col.Ts, 0, 'o', markersize=12, color=color_cycle[i])\n #ax.invert_yaxis()\n yticks = np.array([1000., 750., 500., 250., 100., 50., 20., 10.])\n ax.set_yticks(-np.log(yticks/1000.))\n ax.set_yticklabels(yticks)\n ax.set_xlabel('Temperature (K)')\n ax.set_ylabel('Pressure (hPa)')\n ax.grid()\n return ax\n```\n\n\n```python\n# This should look just like the observations\nplot_sounding(col)\n```\n\n### Tune absorptivity to get observed OLR\n\n\n```python\ncol.compute_diagnostics()\ncol.diagnostics['OLR']\n```\n\n\n\n\n array([ 263.15000167])\n\n\n\n\n```python\n# Need to tune absorptivity to get OLR = 239\nepsarray = np.linspace(0.01, 0.1, 100)\nOLRarray = np.zeros_like(epsarray)\n```\n\n\n```python\nfor i in range(epsarray.size):\n col.subprocess['LW'].absorptivity = epsarray[i]\n col.compute_diagnostics()\n OLRarray[i] = col.diagnostics['OLR']\n\nplt.plot(epsarray, OLRarray)\nplt.grid()\n```\n\nThe necessary value seems to lie near 0.055 or so.\n\nWe can be more precise with a numerical root-finder.\n\n\n```python\ndef OLRanom(eps):\n col.subprocess['LW'].absorptivity = eps\n col.compute_diagnostics()\n return col.diagnostics['OLR'] - 239.\n```\n\n\n```python\n# Use numerical root-finding to get the equilibria\nfrom scipy.optimize import brentq\n# brentq is a root-finding function\n# Need to give it a function and two end-points\n# It will look for a zero of the function between those end-points\neps = brentq(OLRanom, 0.01, 0.1)\nprint eps\n```\n\n 0.0534031813092\n\n\n\n```python\ncol.subprocess['LW'].absorptivity = eps\ncol.subprocess['LW'].absorptivity\n```\n\n\n\n\n Field([ 0.05340318, 0.05340318, 0.05340318, 0.05340318, 0.05340318,\n 0.05340318, 0.05340318, 0.05340318, 0.05340318, 0.05340318,\n 0.05340318, 0.05340318, 0.05340318, 0.05340318, 0.05340318,\n 0.05340318, 0.05340318, 0.05340318, 0.05340318, 0.05340318,\n 0.05340318, 0.05340318, 0.05340318, 0.05340318, 0.05340318,\n 0.05340318, 0.05340318, 0.05340318, 0.05340318, 0.05340318])\n\n\n\n\n```python\ncol.compute_diagnostics()\ncol.diagnostics['OLR']\n```\n\n\n\n\n array([ 239.])\n\n\n\n____________\n\n\n## 5. Radiative forcing in the 30-layer model\n____________\n\nLet's compute radiative forcing for a **2% increase in absorptivity**.\n\n\n```python\n# clone our model using a built-in climlab function\ncol2 = climlab.process_like(col)\nprint col2\n```\n\n climlab Process of type . \n State variables and domain shapes: \n Tatm: (30,) \n Ts: (1,) \n The subprocess tree: \n top: \n LW: \n SW: \n insolation: \n surface: \n \n\n\n\n```python\ncol2.subprocess['LW'].absorptivity *= 1.02\ncol2.subprocess['LW'].absorptivity\n```\n\n\n\n\n Field([ 0.05447124, 0.05447124, 0.05447124, 0.05447124, 0.05447124,\n 0.05447124, 0.05447124, 0.05447124, 0.05447124, 0.05447124,\n 0.05447124, 0.05447124, 0.05447124, 0.05447124, 0.05447124,\n 0.05447124, 0.05447124, 0.05447124, 0.05447124, 0.05447124,\n 0.05447124, 0.05447124, 0.05447124, 0.05447124, 0.05447124,\n 0.05447124, 0.05447124, 0.05447124, 0.05447124, 0.05447124])\n\n\n\n\n```python\n# Radiative forcing by definition is the change in TOA radiative flux, HOLDING THE TEMPERATURES FIXED.\ncol2.Ts - col.Ts\n```\n\n\n\n\n Field([ 0.])\n\n\n\n\n```python\ncol2.Tatm - col.Tatm\n```\n\n\n\n\n Field([ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,\n 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,\n 0., 0., 0., 0.])\n\n\n\n\n```python\ncol2.compute_diagnostics()\ncol2.diagnostics['OLR']\n```\n\n\n\n\n array([ 237.15483194])\n\n\n\nThe OLR decreased after we added the extra absorbers, as we expect. Now we can calculate the Radiative Forcing:\n\n\n```python\nRF = -(col2.diagnostics['OLR'] - col.diagnostics['OLR'])\nprint 'The radiative forcing is %f W/m2.' %RF\n```\n\n The radiative forcing is 1.845168 W/m2.\n\n\n____________\n\n\n## 6. Radiative equilibrium in the 30-layer model\n____________\n\n\n\n```python\nre = climlab.process_like(col)\n```\n\n\n```python\n# To get to equilibrium, we just time-step the model forward long enough\nre.integrate_years(2.)\n```\n\n Integrating for 730 steps, 730.4844 days, or 2.0 years.\n Total elapsed time is 2.91039753895 years.\n\n\n\n```python\n# Check for energy balance\nre.diagnostics['ASR'] - re.diagnostics['OLR']\n```\n\n\n\n\n Field([ -8.67412012e-07])\n\n\n\n\n```python\nplot_sounding([col, re])\n```\n\nSome properties of the **radiative equilibrium** temperature profile:\n\n- The surface is warmer than observed.\n- The lower troposphere is colder than observed.\n- Very cold air is sitting immediately above the warm surface.\n- There is no tropopause, no stratosphere.\n\n____________\n\n\n## 7. Radiative-Convective Equilibrium in the 30-layer model\n____________\n\nWe recognize that the large drop in temperature just above the surface is unphysical. Parcels of air in direct contact with the ground will be warmed by mechansisms other than radiative transfer.\n\nThese warm air parcels will then become buoyant, and will convect upward, mixing their heat content with the environment.\n\nWe **parameterize** the statistical effects of this mixing through a **convective adjustment**. \n\nAt each timestep, our model checks for any locations at which the **lapse rate** exceeds some threshold. Unstable layers are removed through an energy-conserving mixing formula.\n\nThis process is assumed to be fast relative to radiative heating. In the model, it is instantaneous.\n\n\n```python\nrce = climlab.RadiativeConvectiveModel(adj_lapse_rate=6.)\nprint rce\n```\n\n climlab Process of type . \n State variables and domain shapes: \n Tatm: (30,) \n Ts: (1,) \n The subprocess tree: \n top: \n convective adjustment: \n LW: \n SW: \n surface: \n insolation: \n \n\n\nThis model is exactly like our previous models, except for one additional subprocess called ``convective adjustment``. \n\nWe passed a parameter ``adj_lapse_rate`` (in K / km) that sets the neutrally stable lapse rate -- in this case, 6 K / km.\n\nThis number is chosed to very loosely represent the net effect of **moist convection**. We'll look at this in more detail later.\n\n\n```python\n# Set our tuned absorptivity value\nrce.subprocess['LW'].absorptivity = eps\n```\n\n\n```python\n# Run out to equilibrium\nrce.integrate_years(2.)\n```\n\n Integrating for 730 steps, 730.4844 days, or 2.0 years.\n Total elapsed time is 1.99867375676 years.\n\n\n\n```python\n# Check for energy balance\nrce.diagnostics['ASR'] - rce.diagnostics['OLR']\n```\n\n\n\n\n Field([ 6.23502038e-06])\n\n\n\n\n```python\n# Make a plot to compare observations, Radiative Equilibrium, and Radiative-Convective Equilibrium\nplot_sounding([col, re, rce])\n```\n\nIntroducing convective adjustment into the model cools the surface quite a bit (compared to Radiative Equilibrium, in green here) -- and warms the lower troposphere. It gives us a MUCH better fit to observations.\n\nBut of course we still have no stratosphere.\n\n____________\n\n\n## 8. Putting stratospheric ozone in the grey-gas model\n____________\n\nOur model has no equivalent of the stratosphere, where temperature increases with height. That's because our model has been completely transparent to shortwave radiation up until now.\n\nWe can load the observed ozone climatology from the input files for the CESM model:\n\n\n```python\ndatapath = \"http://ramadda.atmos.albany.edu:8080/repository/opendap/latest/Top/Users/Brian+Rose/CESM+runs/\"\nendstr = \"/entry.das\"\n\nozone = nc.Dataset( datapath + 'som_input/ozone_1.9x2.5_L26_2000clim_c091112.nc' + endstr )\n```\n\n\n```python\nprint ozone.variables['O3']\n```\n\n \n float32 O3(time, lev, lat, lon)\n units: mol/mol\n long_name: O3 concentration\n cell_method: time: mean\n unlimited dimensions: time\n current shape = (12, 26, 96, 144)\n filling off\n \n\n\n\n```python\nlat_O3 = ozone.variables['lat'][:]\nlon_O3 = ozone.variables['lon'][:]\nlev_O3 = ozone.variables['lev'][:]\n```\n\nThe pressure levels in this dataset are:\n\n\n```python\nprint lev_O3\n```\n\n [ 3.544638 7.3888135 13.967214 23.944625 37.23029 53.114605\n 70.05915 85.439115 100.514695 118.250335 139.115395 163.66207\n 192.539935 226.513265 266.481155 313.501265 368.81798 433.895225\n 510.455255 600.5242 696.79629 787.70206 867.16076 929.648875\n 970.55483 992.5561 ]\n\n\n### Take the global average of the ozone climatology, and plot it as a function of pressure (or height)\n\n\n```python\nO3_zon = np.mean( ozone.variables['O3'][:],axis=(0,3) )\nprint O3_zon.shape\n```\n\n (26, 96)\n\n\n\n```python\nO3_global = np.average( O3_zon, axis=1, weights=np.cos(np.deg2rad(lat_O3)))\nprint O3_global\n```\n\n [ 7.82792904e-06 8.64150570e-06 7.58940041e-06 5.24567122e-06\n 3.17761578e-06 1.82320025e-06 9.80756909e-07 6.22870516e-07\n 4.47620522e-07 3.34481172e-07 2.62570325e-07 2.07898125e-07\n 1.57074552e-07 1.12425546e-07 8.06005076e-08 6.27826466e-08\n 5.42990612e-08 4.99506108e-08 4.60075675e-08 4.22977777e-08\n 3.80559086e-08 3.38768551e-08 3.12171622e-08 2.97807148e-08\n 2.87980981e-08 2.75429919e-08]\n\n\n\n```python\nax = plt.figure(figsize=(10,8)).add_subplot(111)\nax.plot( O3_global * 1.E6, -np.log(lev_O3/climlab.constants.ps) )\nax.set_xlabel('Ozone (ppm)', fontsize=16)\nax.set_ylabel('Pressure (hPa)', fontsize=16 )\nax.set_yticks( -np.log(yticks/1000.) )\nax.set_yticklabels( yticks )\nax.grid()\nax.set_title('Global, annual mean ozone concentration', fontsize = 24);\n```\n\nThis shows that most of the ozone is indeed in the stratosphere, and peaks near the top of the stratosphere.\n\nNow create a new column model object **on the same pressure levels as the ozone data**. We are also going set an adjusted lapse rate of 6 K / km.\n\n\n```python\noz_col = climlab.RadiativeConvectiveModel(lev = lev_O3, adj_lapse_rate=6)\nprint oz_col\n```\n\n climlab Process of type . \n State variables and domain shapes: \n Tatm: (26,) \n Ts: (1,) \n The subprocess tree: \n top: \n convective adjustment: \n LW: \n SW: \n surface: \n insolation: \n \n\n\nNow we will do something new: let the column absorb some shortwave radiation. We will assume that the shortwave absorptivity is proportional to the ozone concentration we plotted above. \n\nFirst we have to deal with a little inconsistency:\n\n\n```python\nprint lev_O3\nprint oz_col.lev\n```\n\n [ 3.544638 7.3888135 13.967214 23.944625 37.23029 53.114605\n 70.05915 85.439115 100.514695 118.250335 139.115395 163.66207\n 192.539935 226.513265 266.481155 313.501265 368.81798 433.895225\n 510.455255 600.5242 696.79629 787.70206 867.16076 929.648875\n 970.55483 992.5561 ]\n [ 992.5561 970.55483 929.648875 867.16076 787.70206 696.79629\n 600.5242 510.455255 433.895225 368.81798 313.501265\n 266.481155 226.513265 192.539935 163.66207 139.115395\n 118.250335 100.514695 85.439115 70.05915 53.114605 37.23029\n 23.944625 13.967214 7.3888135 3.544638 ]\n\n\nThe two arrays are in reverse order!\n\nSo we need to flip the ozone data before using it:\n\n\n```python\nO3_flipped = np.flipud(O3_global)\n```\n\nNow we need to weight the absorptivity by the pressure (mass) of each layer.\n\n\n```python\n# This number is an arbitrary parameter that scales how absorptive we are making the ozone\n# in our grey gas model\nozonefactor = 75\ndp = oz_col.Tatm.domain.lev.delta\nepsSW = np.flipud(O3_global) * dp * ozonefactor\n```\n\nWe want to use the field `epsSW` as the absorptivity for our SW radiation model.\n\nLet's see what the absorptivity is current set to:\n\n\n```python\nprint oz_col.subprocess['SW'].absorptivity\n```\n\n [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0.]\n\n\nIt defaults to zero.\n\nBefore changing this (putting in the ozone), let's take a look at the shortwave absorption in the column:\n\n\n```python\noz_col.compute_diagnostics()\n```\n\n\n```python\noz_col.diagnostics['SW_absorbed_atm']\n```\n\n\n\n\n array([-0., -0., -0., -0., -0., -0., -0., -0., -0., -0., -0., -0., -0.,\n -0., -0., -0., -0., -0., -0., -0., -0., -0., -0., -0., -0., -0.])\n\n\n\nLet's now put in the ozone:\n\n\n```python\noz_col.subprocess['SW'].absorptivity = epsSW\nprint oz_col.subprocess['SW'].absorptivity\n```\n\n [ 3.81013258e-05 6.79353164e-05 1.15468099e-04 1.66169128e-04\n 2.16427967e-04 2.67120883e-04 2.95567938e-04 2.87482268e-04\n 2.65307565e-04 2.45147963e-04 2.40936627e-04 2.62922886e-04\n 3.11733077e-04 3.70212125e-04 4.16507314e-04 4.47141525e-04\n 4.84170276e-04 5.50761578e-04 7.11369789e-04 1.18884324e-03\n 2.24450947e-03 3.47591208e-03 4.57614181e-03 4.71182560e-03\n 3.37750312e-03 3.20948559e-03]\n\n\nLet's check how this changes the SW absorption:\n\n\n```python\noz_col.compute_diagnostics()\noz_col.diagnostics['SW_absorbed_atm']\n```\n\n\n\n\n array([ 0.01641658, 0.02927233, 0.04975715, 0.07161233, 0.09328389,\n 0.11515129, 0.12743552, 0.12396838, 0.11442202, 0.1057411 ,\n 0.10393805, 0.11343945, 0.13452308, 0.1597929 , 0.17981756,\n 0.19309136, 0.20913838, 0.23797741, 0.30750441, 0.51429087,\n 0.97239294, 1.50916588, 1.99236556, 2.05685577, 1.47671292,\n 1.40571625])\n\n\n\nIt is now non-zero, and largest near the top of the column (bottom of array) where the ozone concentration is highest.\n\nNow it's time to run the model out to radiative-convective equilibrium\n\n\n```python\noz_col.integrate_years(1.)\n```\n\n Integrating for 365 steps, 365.2422 days, or 1.0 years.\n Total elapsed time is 1.01576433391 years.\n\n\n\n```python\nprint oz_col.diagnostics['ASR'] - oz_col.diagnostics['OLR']\n```\n\n [-0.00553705]\n\n\nAnd let's now see what we got!\n\n\n```python\n# Make a plot to compare observations, Radiative Equilibrium, Radiative-Convective Equilibrium, and RCE with ozone!\nplot_sounding([col, re, rce, oz_col])\n```\n\nAnd we finally have something that looks looks like the tropopause, with temperature increasing above at approximately the correct rate. \n\nThere are still plenty of discrepancies between this model solution and the observations, including:\n\n- Tropopause temperature is too warm, by about 15 degrees.\n- Surface temperature is too cold\n\nThere are a number of parameters we might adjust if we wanted to improve the fit, including:\n\n- Longwave absorptivity\n- Surface albedo\n\nFeel free to experiment! (That's what models are for, after all).\n\n### The take home message\n\nThe dominant effect of stratospheric ozone is to vastly increase the radiative equilibrium temperature in the ozone layer. The temperature needs to be higher so that the longwave emission can balance the shortwave absorption.\n\nWithout ozone to absorb incoming solar radiation, the **temperature does not increase with height**.\n\nThis simple grey-gas model illustrates this principle very clearly.\n\n
                                        \n[Back to ATM 623 notebook home](../index.ipynb)\n
                                        \n\n____________\n## Credits\n\nThe author of this notebook is [Brian E. J. Rose](http://www.atmos.albany.edu/facstaff/brose/index.html), University at Albany.\n\nIt was developed in support of [ATM 623: Climate Modeling](http://www.atmos.albany.edu/facstaff/brose/classes/ATM623_Spring2015/), a graduate-level course in the [Department of Atmospheric and Envionmental Sciences](http://www.albany.edu/atmos/index.php), offered in Spring 2015.\n____________\n\n____________\n## Version information\n____________\n\n\n\n\n```python\n%install_ext http://raw.github.com/jrjohansson/version_information/master/version_information.py\n%load_ext version_information\n%version_information numpy, climlab\n```\n\n Installed version_information.py. To use it, type:\n %load_ext version_information\n\n\n\n\n\n
                                        SoftwareVersion
                                        Python2.7.9 64bit [GCC 4.2.1 (Apple Inc. build 5577)]
                                        IPython3.1.0
                                        OSDarwin 14.3.0 x86_64 i386 64bit
                                        numpy1.9.2
                                        climlab0.2.11
                                        Thu May 14 16:04:11 2015 EDT
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                                        \n

                                        \nPropuesta para un Framework Basado En Software Libre
                                        \npara facilitar el Proceso de Ense\u00f1anza-Aprendizaje
                                        \nen Materias de Ciencias Exactas en Carreras de Grado
                                        \n

                                        \n
                                        \n\n\n```python\nfrom IPython.display import Javascript, display\nfrom ipywidgets.widgets import Layout\nfrom ipywidgets import widgets\n\ndef run_all(ev):\n display(Javascript('IPython.notebook.execute_cells_below()'))\n\nbutton = widgets.Button(description=\"Ejecutar Todas las Celdas\", layout=Layout(width='99%', height=\"50px\"))\nbutton.on_click(run_all)\n```\n\n### Ejecutar todas las celdas\n\n\n```python\ndisplay(button)\n```\n\n\n Button(description='Ejecutar Todas las Celdas', layout=Layout(height='50px', width='99%'), style=ButtonStyle()\u2026\n\n\n## \u00bfQui\u00e9n Soy? - Ezequiel Leonardo Casta\u00f1o\n\n- Estudiante de ISI en UTN Facultad Regional Rosario\n- Programo en Python por m\u00e1s de 5 a\u00f1os como hobby\n\n**Me interesa**\n- Inteligencia Artificial\n- Data Visualization\n- Simulaci\u00f3n y modelado\n- Aplicaci\u00f3n de inform\u00e1tica en Educaci\u00f3n\n\n## Agenda\n\n- \u00bfPor qu\u00e9?\n- \u00bfD\u00f3nde?\n- \u00bfQui\u00e9n?\n- \u00bfQu\u00e9?\n- \u00bfC\u00f3mo?\n- \u00bfPara qui\u00e9n?\n- \u00bfJunto a qu\u00e9?\n- \u00bfAntes de qu\u00e9?\n\n## \u00bfPor qu\u00e9 y D\u00f3nde? - Software Privativo vs Software Libre\n\n
                                        \n\n\n```python\nfrom IPython.display import IFrame\n```\n\n# Jupyter Education Map\n\n\n```python\nIFrame('https://elc.github.io/jupyter-map', width=\"100%\", height=600)\n```\n\n\n\n\n\n\n\n\n\n\n## \u00bfQui\u00e9n? - Universidades que ya lo implementan\n\n- 85 Cursos ya lo implementan\n- 64 Tienen el material disponible de manera p\u00fablica\n- Algunas de las universidades:\n - University of Notre Dame\n - University of Amsterdam\n - National Institutes of Health (NIH)\n - Universitat de Barcelona\n - Stanford University\n - California Institute of Technology\n\n## \u00bfQu\u00e9? - Pasos para implementar la propuesta\n\n1. **Material de estudio**\n2. Experimentaci\u00f3n en clase\n3. Trabajos pr\u00e1cticos\n4. Tareas y asignaciones\n\n## Primer Paso - Material de estudio\n\n- Din\u00e1mico\n- Editable\n- Entendible\n- Documentado\n\n## \u00bfC\u00f3mo? Tecnolog\u00edas\n\n
                                        \n\n### Demostraci\u00f3n\n\n### Correcci\u00f3n de Errores en Vivo\n\n$$ \\int_1^\\infty \\!\\frac{1}{x^2}\\, dx=\\left[\\frac{1}{x}\\right]_1^\\infty=1 $$\n\n### Graficar Funciones y ver como var\u00edan - Funci\u00f3n Cuadr\u00e1tica\n\n\n```python\nimport matplotlib.pyplot as plt\nfrom ipywidgets import interact\nimport numpy as np\nfrom matplotlib import animation, rc\nfrom matplotlib import style\n\n# Jupyter Specifics\nimport matplotlib as mpl\nfrom IPython.display import HTML\nfrom ipywidgets.widgets import interact, IntSlider, FloatSlider, Layout\n```\n\n\n```python\nplt.style.use('bmh')\n\n%matplotlib inline\n\nmpl.rcParams['figure.figsize'] = (16.0, 6.0)\nrc('animation', html='html5')\n```\n\n\n```python\ndef f1(a, b, c):\n mpl.rcParams['figure.figsize'] = (16.0, 6.0)\n x = np.linspace(-5,5,100)\n y = a*x**2+b*x+c\n plt.title(f\"Expresion: $ax^2 + bx + c$ \\n $a = {a}, b = {b}, c = {c}$\")\n plt.ylim(-1,20)\n plt.xlim(-5,5)\n plt.grid(color='grey', linewidth=0.5)\n plt.plot(x, y)\n```\n\n\n```python\ninteract(f1, a=FloatSlider(min=-5, max=6, step=0.25, value=1, layout=Layout(width='99%')), b=FloatSlider(min=-5, max=6, step=1, value=0, layout=Layout(width='99%')), c=FloatSlider(min=-5, max=6, step=1, value=1, layout=Layout(width='99%')),);\n```\n\n\n interactive(children=(FloatSlider(value=1.0, description='a', layout=Layout(width='99%'), max=6.0, min=-5.0, s\u2026\n\n\n### Graficar Funciones y ver como var\u00edan - Funci\u00f3n Cuadr\u00e1tica Can\u00f3nica\n\n\n```python\ndef f2(a, b, c):\n mpl.rcParams['figure.figsize'] = (16.0, 6.0)\n x = np.linspace(-5,5,1000)\n y = (a*x+b)**2+c\n plt.title(\"Expresion: $(ax+b)^2 + c$ \\n a = {}, b = {}, c = {}\".format(a,b,c))\n plt.ylim(-1,20)\n plt.xlim(-5,5)\n plt.grid(color='grey', linewidth=0.5)\n plt.plot(x, y)\n```\n\n\n```python\ninteract(f2, a=FloatSlider(min=-5, max=6, step=0.25, value=1, layout=Layout(width='99%')), b=FloatSlider(min=-5, max=6, step=1, value=0, layout=Layout(width='99%')), c=FloatSlider(min=-5, max=6, step=1, value=1, layout=Layout(width='99%')),);\n```\n\n\n interactive(children=(FloatSlider(value=1.0, description='a', layout=Layout(width='99%'), max=6.0, min=-5.0, s\u2026\n\n\n### Integraci\u00f3n Num\u00e9rica y Graficaci\u00f3n\n\n\n```python\nfrom matplotlib.patches import Polygon\nimport scipy.integrate as integrate\n\n\ndef func(x):\n return (x - 3) * (x - 5) * (x - 7) + 85\n\n\ndef f3(a, b):\n mpl.rcParams['figure.figsize'] = (16.0, 6.0)\n x = np.linspace(0, 10)\n y = func(x)\n\n fig, ax = plt.subplots()\n plt.plot(x, y, linewidth=2)\n plt.ylim(ymin=0)\n\n # Make the shaded region\n ix = np.linspace(a, b)\n iy = func(ix)\n verts = [(a, 0)] + list(zip(ix, iy)) + [(b, 0)]\n poly = Polygon(verts, facecolor='0.8', edgecolor='0.5')\n ax.add_patch(poly)\n\n inte = int(integrate.quad(func, a, b)[0])\n \n plt.text(0.5 * (a + b), 30, r\"$\\int_a^b f(x)\\mathrm{d}x\" + f\" = {inte}$\",\n horizontalalignment='center', fontsize=20)\n\n ax.set_xticks((a, b))\n ax.set_xticklabels(('$a$', '$b$'))\n\n plt.title(f\"Funci\u00f3n: $f(x) = (x - 3)(x - 5)(x - 7) + 85$ \\n $a = {a}, b= {b}$\")\n plt.show()\n```\n\n\n```python\ninteract(f3, a=FloatSlider(min=-5, max=10, step=0.25, value=2, layout=Layout(width='99%')), b=FloatSlider(min=-5, max=10, step=0.25, value=9, layout=Layout(width='99%')));\n```\n\n\n interactive(children=(FloatSlider(value=2.0, description='a', layout=Layout(width='99%'), max=10.0, min=-5.0, \u2026\n\n\n### Polinomio de Taylor\n\n\n```python\nimport sympy as sy\nimport numpy as np\nfrom sympy.functions import sin,cos\nimport matplotlib.pyplot as plt\n\n# Factorial function\ndef factorial(n):\n if n <= 0:\n return 1\n else:\n return n*factorial(n-1)\n\n# Taylor approximation at x0 of the function 'function'\ndef taylor(function,x0,n):\n i = 0\n p = 0\n while i <= n:\n p = p + (function.diff(x, i).subs(x, x0)) / (factorial(i)) * (x - x0) ** i\n i += 1\n return p\n```\n\n\n```python\nx = sy.Symbol('x')\nf = sin(x) * x**2\n\ndef animate(j):\n if j % 2 == 0:\n return []\n \n x_lims = [-5,5]\n x1 = np.linspace(x_lims[0],x_lims[1],800)\n \n plt.xlim(x_lims)\n plt.ylim([-5,5])\n \n if j == 1:\n plt.plot(x1, np.sin(x1) * x1**2, label='$sin(x) * x^2$')\n return []\n y1 = []\n func = taylor(f,0,j)\n print(j, 'Polinomio de Taylor para n='+str(j), func)\n for k in x1:\n y1.append(func.subs(x,k))\n plt.plot(x1,y1,label='Orden '+str(j))\n\n plt.xlim(x_lims)\n plt.ylim([-5,5])\n plt.xlabel('x')\n plt.ylabel('y')\n plt.legend()\n plt.grid(True)\n plt.title('Aproximaci\u00f3n por serie de Taylor')\n return []\n\n# Plot results\ndef plot():\n mpl.rcParams['figure.figsize'] = (12.0, 6.0)\n fig, ax = plt.subplots(); \n anim = animation.FuncAnimation(fig, animate, frames=10, interval=500, blit=True);\n return anim\n```\n\n\n```python\nanim = plot()\nHTML(anim.to_html5_video())\n```\n\n### Polinomio de Taylor interactivo\n\n\n```python\nx = sy.Symbol('x')\nf = sin(x)\n\ndef f4(order):\n mpl.rcParams['figure.figsize'] = (16.0, 6.0)\n x_lims = [-10, 10]\n x1 = np.linspace(x_lims[0],x_lims[1],800)\n plt.plot(x1, np.sin(x1), label='sin of x')\n y1 = []\n func = taylor(f,0,order)\n for k in x1:\n y1.append(func.subs(x,k))\n plt.plot(x1,y1,label='order '+str(order))\n plt.xlim(x_lims)\n plt.ylim([-5,5])\n plt.legend()\n plt.grid(True)\n plt.title('Taylor series approximation')\n plt.show()\n```\n\n\n```python\ninteract(f4, order=IntSlider(min=1, max=15, step=2, value=1, layout=Layout(width='99%')),);\n```\n\n\n interactive(children=(IntSlider(value=1, description='order', layout=Layout(width='99%'), max=15, min=1, step=\u2026\n\n\n### C\u00f3nicas\n\n\n```python\nimport sympy as sy\nfrom sympy import plot_implicit, Eq\nx = sy.Symbol('x')\ny = sy.Symbol('y')\n\ndef plot_conic(a, b, h, k):\n if a == 0 or b == 0:\n return []\n mpl.rcParams['figure.figsize'] = (10.0, 10.0)\n plot_implicit(Eq((x + h)**2 / a + (y + k)**2 / b, 1), (x, -np.pi, np.pi), (y, -np.pi, np.pi), title=\"Ecuaci\u00f3n: $\\\\frac{(x+h)^2}{a} + \\\\frac{(y+k)^2}{b} = 1$\")\n```\n\n\n```python\ninteract(plot_conic, a=FloatSlider(min=-5, max=5, step=1, value=2, layout=Layout(width='99%')), \n b=FloatSlider(min=-5, max=5, step=1, value=2, layout=Layout(width='99%')),\n h=FloatSlider(min=-5, max=5, step=1, value=0, layout=Layout(width='99%')), \n k=FloatSlider(min=-5, max=5, step=1, value=0, layout=Layout(width='99%')));\n```\n\n\n interactive(children=(FloatSlider(value=2.0, description='a', layout=Layout(width='99%'), max=5.0, min=-5.0, s\u2026\n\n\n### Ventajas\n\n- Visualizaci\u00f3n Integrada\n- Visualizaci\u00f3n de los datos\n- Problemas de mayor dificultad\n- Material interactivo\n- Habilidades de computaci\u00f3n\n- Acompa\u00f1amiento\n- Filosof\u00eda Open Source y Open Science\n- Colaboraci\u00f3n\n- Instalaci\u00f3n sencilla\n- Versatilidad\n\n### Desventajas\n\n- Carga Cognitiva\n- Testing\n- Orden de Ejecuci\u00f3n\n- Variables globales\n- Borrado accidental\n- Errores intimidantes\n\n## \u00bfPara qui\u00e9n? - Docentes objetivos\n\n- Docentes de ciencias exactas\n- Docentes de materias del ciclo b\u00e1sico\n- Docentes que quieran implementar la tecnolog\u00eda en clase\n\n### Desaf\u00edos\n\n- Esfuerzo inicial para el aprendizaje\n- Migraci\u00f3n del material existente\n- Resistencia a la tecnolog\u00eda\n\n## \u00bfJunto a qu\u00e9? - Tecnolog\u00edas complementarias\n\n### Tradicionales\n\n
                                        \n\n### Basadas en Video\n\n
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                                        \n\n## \u00bfAntes de qu\u00e9? - Pr\u00f3ximos Pasos\n\n- Realizar Pruebas Piloto\n- Desarrollar metodolog\u00edas para medir resultados\n- Generar documentaci\u00f3n\n\n
                                        \n\n

                                        Muchas gracias

                                        \n\n

                                        Ezequiel Leonardo Casta\u00f1o

                                        \n\n

                                        Blog Personal: elc.github.io

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                                        \n", "meta": {"hexsha": "8b5679c1138574030726c98b20c39ba41de7a4ef", "size": 155347, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Jupyter-CONAIISI.ipynb", "max_stars_repo_name": "ELC/CONAIISI2018", "max_stars_repo_head_hexsha": "16fcf91985ae015fe7714dccc941ec33034bf074", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Jupyter-CONAIISI.ipynb", "max_issues_repo_name": "ELC/CONAIISI2018", "max_issues_repo_head_hexsha": "16fcf91985ae015fe7714dccc941ec33034bf074", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Jupyter-CONAIISI.ipynb", "max_forks_repo_name": "ELC/CONAIISI2018", "max_forks_repo_head_hexsha": "16fcf91985ae015fe7714dccc941ec33034bf074", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 81.7615789474, "max_line_length": 65012, "alphanum_fraction": 0.831989031, "converted": true, "num_tokens": 3126, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.7217432062975979, "lm_q1q2_score": 0.43047157132719494}} {"text": "# Optimizaci\u00f3n media-varianza\n\n\n\n\nLa **teor\u00eda de portafolios** es uno de los avances m\u00e1s importantes en las finanzas modernas e inversiones.\n- Apareci\u00f3 por primera vez en un [art\u00edculo corto](https://www.math.ust.hk/~maykwok/courses/ma362/07F/markowitz_JF.pdf) llamado \"Portfolio Selection\" en la edici\u00f3n de Marzo de 1952 de \"the Journal of Finance\".\n- Escrito por un desconocido estudiante de la Universidad de Chicago, llamado Harry Markowitz.\n- Escrito corto (s\u00f3lo 14 p\u00e1ginas), poco texto, f\u00e1cil de entender, muchas gr\u00e1ficas y unas cuantas referencias.\n- No se le prest\u00f3 mucha atenci\u00f3n hasta los 60s.\n\nFinalmente, este trabajo se convirti\u00f3 en una de las m\u00e1s grandes ideas en finanzas, y le di\u00f3 a Markowitz el Premio Nobel casi 40 a\u00f1os despu\u00e9s.\n- Markowitz estaba incidentalmente interesado en los mercados de acciones e inversiones.\n- Estaba m\u00e1s bien interesado en entender c\u00f3mo las personas tomaban sus mejores decisiones cuando se enfrentaban con \"trade-offs\".\n- Principio de conservaci\u00f3n de la miseria. O, dir\u00edan los instructores de gimnasio: \"no pain, no gain\".\n- Si queremos m\u00e1s de algo, tenemos que perder en alg\u00fan otro lado.\n- El estudio de este fen\u00f3meno era el que le atra\u00eda a Markowitz.\n\nDe manera que nadie se hace rico poniendo todo su dinero en la cuenta de ahorros. La \u00fanica manera de esperar altos rendimientos es si se toma bastante riesgo. Sin embargo, riesgo significa tambi\u00e9n la posibilidad de perder, tanto como ganar.\n\nPero, \u00bfqu\u00e9 tanto riesgo es necesario?, y \u00bfhay alguna manera de minimizar el riesgo mientras se maximizan las ganancias?\n- Markowitz b\u00e1sicamente cambi\u00f3 la manera en que los inversionistas pensamos acerca de esas preguntas.\n- Alter\u00f3 completamente la pr\u00e1ctica de la administraci\u00f3n de inversiones.\n- Incluso el t\u00edtulo de su art\u00edculo era innovador. Portafolio: una colecci\u00f3n de activos en lugar de tener activos individuales.\n- En ese tiempo, un portafolio se refer\u00eda a una carpeta de piel.\n- En el resto de este m\u00f3dulo, nos ocuparemos de la parte anal\u00edtica de la teor\u00eda de portafolios, la cual puede ser resumida en dos frases:\n - No pain, no gain.\n - No ponga todo el blanquillo en una sola bolsa.\n \n\n**Objetivos:**\n- \u00bfQu\u00e9 es la l\u00ednea de asignaci\u00f3n de capital?\n- \u00bfQu\u00e9 es el radio de Sharpe?\n- \u00bfC\u00f3mo deber\u00edamos asignar nuestro capital entre un activo riesgoso y un activo libre de riesgo?\n\n*Referencia:*\n- Notas del curso \"Portfolio Selection and Risk Management\", Rice University, disponible en Coursera.\n___ \n\n## 1. L\u00ednea de asignaci\u00f3n de capital\n\n### 1.1. Motivaci\u00f3n\n\nEl proceso de construcci\u00f3n de un portafolio tiene entonces los siguientes dos pasos:\n1. Escoger un portafolio de activos riesgosos.\n2. Decidir qu\u00e9 tanto de tu riqueza invertir\u00e1s en el portafolio y qu\u00e9 tanto invertir\u00e1s en activos libres de riesgo.\n\nAl paso 2 lo llamamos **decisi\u00f3n de asignaci\u00f3n de activos**.\n\nPreguntas importantes:\n1. \u00bfQu\u00e9 es el portafolio \u00f3ptimo de activos riesgosos?\n - \u00bfCu\u00e1l es el mejor portafolio de activos riesgosos?\n - Es un portafolio eficiente en media-varianza.\n2. \u00bfQu\u00e9 es la distribuci\u00f3n \u00f3ptima de activos?\n - \u00bfC\u00f3mo deber\u00edamos distribuir nuestra riqueza entre el portafolo riesgoso \u00f3ptimo y el activo libre de riesgo?\n - Concepto de **l\u00ednea de asignaci\u00f3n de capital**.\n - Concepto de **radio de Sharpe**.\n\nDos suposiciones importantes:\n- Funciones de utilidad media-varianza.\n- Inversionista averso al riesgo.\n\nLa idea sorprendente que saldr\u00e1 de este an\u00e1lisis, es que cualquiera que sea la actitud del inversionista de cara al riesgo, el mejor portafolio de activos riesgosos es id\u00e9ntico para todos los inversionistas.\n\nLo que nos importar\u00e1 a cada uno de nosotros en particular, es simplemente la desici\u00f3n \u00f3ptima de asignaci\u00f3n de activos.\n___\n\n### 1.2. L\u00ednea de asignaci\u00f3n de capital\n\nSean:\n- $r_s$ el rendimiento del activo riesgoso,\n- $r_f$ el rendimiento libre de riesgo, y\n- $w$ la fracci\u00f3n invertida en el activo riesgoso.\n\n Realizar deducci\u00f3n de la l\u00ednea de asignaci\u00f3n de capital en el tablero.\n\n**Tres doritos despu\u00e9s...**\n\n#### L\u00ednea de asignaci\u00f3n de capital (LAC):\n$E[r_p]$ se relaciona con $\\sigma_p$ de manera af\u00edn. Es decir, mediante la ecuaci\u00f3n de una recta:\n\n$$E[r_p]=r_f+\\frac{E[r_s-r_f]}{\\sigma_s}\\sigma_p.$$\n\n- La pendiente de la LAC es el radio de Sharpe $\\frac{E[r_s-r_f]}{\\sigma_s}=\\frac{E[r_s]-r_f}{\\sigma_s}$,\n- el cual nos dice qu\u00e9 tanto rendimiento obtenemos por unidad de riesgo asumido en la tenencia del activo (portafolio) riesgoso.\n\nAhora, la pregunta es, \u00bfd\u00f3nde sobre esta l\u00ednea queremos estar?\n___\n\n### 1.3. Resolviendo para la asignaci\u00f3n \u00f3ptima de capital\n\nRecapitulando de la clase pasada, tenemos las curvas de indiferencia: **queremos estar en la curva de indiferencia m\u00e1s alta posible, que sea tangente a la LAC**.\n\n Ver en el tablero.\n\nAnal\u00edticamente, el problema es\n\n$$\\max_{w} \\quad E[U(r_p)]\\equiv\\max_{w} \\quad E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2,$$\n\ndonde los puntos $(\\sigma_p,E[r_p])$ se restringen a estar en la LAC, esto es $E[r_p]=r_f+\\frac{E[r_s-r_f]}{\\sigma_s}\\sigma_p$ y $\\sigma_p=w\\sigma_s$. Entonces el problema anterior se puede escribir de la siguiente manera:\n\n$$\\max_{w} \\quad r_f+wE[r_s-r_f]-\\frac{1}{2}\\gamma w^2\\sigma_s^2.$$\n\n Encontrar la $w$ que maximiza la anterior expresi\u00f3n en el tablero.\n\n**Tres doritos despu\u00e9s...**\n\nLa soluci\u00f3n es entonces:\n\n$$w^\\ast=\\frac{E[r_s]-r_f}{\\gamma\\sigma_s^2}.$$\n\nDe manera intuitiva:\n- $w^\\ast\\propto E[r_s-r_f]$: a m\u00e1s exceso de rendimiento que se obtenga del activo riesgoso, m\u00e1s querremos invertir en \u00e9l.\n- $w^\\ast\\propto \\frac{1}{\\gamma}$: mientras m\u00e1s averso al riesgo seas, menos querr\u00e1s invertir en el activo riesgoso.\n- $w^\\ast\\propto \\frac{1}{\\sigma_s^2}$: mientras m\u00e1s riesgoso sea el activo, menos querr\u00e1s invertir en \u00e9l.\n___\n\n## 2. Ejemplo de asignaci\u00f3n \u00f3ptima de capital: acciones y billetes de EU\n\nPongamos algunos n\u00fameros con algunos datos, para ilustrar la derivaci\u00f3n que acabamos de hacer.\n\nEn este caso, consideraremos:\n- **Portafolio riesgoso**: mercado de acciones de EU (representados en alg\u00fan \u00edndice de mercado como el S&P500).\n- **Activo libre de riesgo**: billetes del departamento de tesorer\u00eda de EU (T-bills).\n\nTenemos los siguientes datos:\n\n$$E[r_{US}]=11.9\\%,\\quad \\sigma_{US}=19.15\\%, \\quad r_f=1\\%.$$\n\nRecordamos que podemos escribir la expresi\u00f3n de la LAC como:\n\n\\begin{align}\nE[r_p]&=r_f+\\left[\\frac{E[r_{US}-r_f]}{\\sigma_{US}}\\right]\\sigma_p\\\\\n &=0.01+\\text{S.R.}\\sigma_p,\n\\end{align}\n\ndonde $\\text{S.R}=\\frac{0.119-0.01}{0.1915}\\approx0.569$ es el radio de Sharpe (\u00bfqu\u00e9 es lo que es esto?).\n\nGrafiquemos la LAC con estos datos reales:\n\n\n```python\n# Importamos librer\u00edas que vamos a utilizar\nimport numpy as np\nfrom matplotlib import pyplot as plt\n%matplotlib inline\n```\n\n\n```python\n# Datos\nErs = 0.119\nss = 0.1915\nrf = 0.01\n# Radio de Sharpe para este activo\nRS = (Ers - rf) / ss\n# Vector de volatilidades del portafolio (sugerido: 0% a 50%)\nsp = np.linspace(0, 0.5)\n# LAC\nErp = RS * sp + rf\n```\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(6, 4))\nplt.plot(sp, Erp, lw=1, label='LAC')\nplt.plot(0, rf, 'ob', ms=5, label='Libre de riesgo')\nplt.plot(ss, Ers, 'or', ms=5, label='Portafolio/activo riesgoso')\nplt.legend(loc='best')\nplt.xlabel('Volatilidad $\\sigma$')\nplt.ylabel('Rendimiento esperado $E[r]$')\nplt.grid()\n```\n\nBueno, y \u00bfen qu\u00e9 punto de esta l\u00ednea querr\u00edamos estar?\n- Pues ya vimos que depende de tus preferencias.\n- En particular, de tu actitud de cara al riesgo, medido por tu coeficiente de aversi\u00f3n al riesgo.\n\nSoluci\u00f3n al problema de asignaci\u00f3n \u00f3ptima de capital:\n\n$$\\max_{w} \\quad E[U(r_p)]$$\n\n$$w^\\ast=\\frac{E[r_s-r_f]}{\\gamma\\sigma_s^2}$$\n\nDado que ya tenemos datos, podemos intentar para varios coeficientes de aversi\u00f3n al riesgo:\n\n\n```python\n# importar pandas\nimport pandas as pd\n```\n\n\n```python\n# Crear un DataFrame con los pesos, rendimiento\n# esperado y volatilidad del portafolio \u00f3ptimo \n# entre los activos riesgoso y libre de riesgo\n# cuyo \u00edndice sean los coeficientes de aversi\u00f3n\n# al riesgo del 1 al 10 (enteros)\ngamma = np.arange(1, 11)\ndist_cap = pd.DataFrame({'$\\gamma$': gamma,\n r'$w^{\\ast}$': (Ers - rf) / (gamma * ss**2)\n })\ndist_cap\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        $\\gamma$$w^{\\ast}$
                                        012.972275
                                        121.486137
                                        230.990758
                                        340.743069
                                        450.594455
                                        560.495379
                                        670.424611
                                        780.371534
                                        890.330253
                                        9100.297227
                                        \n
                                        \n\n\n\n\n```python\ng = 4.5\nw_ac = (Ers - rf) / (g * ss**2)\nw_ac\n```\n\n\n\n\n 0.6605054836346889\n\n\n\n\u00bfC\u00f3mo se interpreta $w^\\ast>1$?\n- Cuando $01$, tenemos $1-w^\\ast<0$. Lo anterior implica una posici\u00f3n corta en el activo libre de riesgo (suponiendo que se puede) y una posici\u00f3n larga (de m\u00e1s del 100%) en el mercado de activos: apalancamiento.\n\n# Anuncios parroquiales.\n\n## 1. Quiz la siguiente clase.\n\n## 2. Pueden consultar sus calificaciones en el siguiente [enlace](https://docs.google.com/spreadsheets/d/1BwI1Mm7B3xxJ-jQIQEDQ_WdRHyehZrQBpHGd0hY9fU4/edit?usp=sharing)\n\n\n\n\n
                                        \nCreated with Jupyter by Esteban Jim\u00e9nez Rodr\u00edguez.\n
                                        \n", "meta": {"hexsha": "6ca757060a7acfa1850bf463c13105b9b4ca5521", "size": 37907, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Modulo3/Clase12_OptimizacionMediaVarianza.ipynb", "max_stars_repo_name": "HKael/porinvv2020", "max_stars_repo_head_hexsha": "f6f56a516a25786018321c4537b4680b307d28a9", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Modulo3/Clase12_OptimizacionMediaVarianza.ipynb", "max_issues_repo_name": "HKael/porinvv2020", "max_issues_repo_head_hexsha": "f6f56a516a25786018321c4537b4680b307d28a9", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Modulo3/Clase12_OptimizacionMediaVarianza.ipynb", "max_forks_repo_name": "HKael/porinvv2020", "max_forks_repo_head_hexsha": "f6f56a516a25786018321c4537b4680b307d28a9", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 72.7581573896, "max_line_length": 20988, "alphanum_fraction": 0.7747117947, "converted": true, "num_tokens": 3265, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.8221891392358014, "lm_q1q2_score": 0.4303505261504591}} {"text": "```\n#Enable LaTeX printing\nfrom IPython.core.display import *\n\nglobal MathJax \nMathJax = True\ndef MDPL(string): display(Math(string)) if MathJax else display(Latex(string))\n\nimport IPython.core.display \nIPython.core.display.HTML(\"\"\"\n \ndiv.input {\nwidth: 105ex; /* about 80 chars + buffer */\n}\n \ndiv.text_cell {\nwidth: 105ex /* instead of 100%, */\n}\n \ndiv.text_cell_render {\n/*font-family: \"Helvetica Neue\", Arial, Helvetica, Geneva, sans-serif;*/\nfont-family: \"Charis SIL\", serif; /* Make non-code text serif. */\nline-height: 145%; /* added for some line spacing of text. */\nwidth: 105ex; /* instead of 'inherit' for shorter lines */\n}\n \n/* Set the size of the headers */\ndiv.text_cell_render h1 {\nfont-size: 18pt;\n}\n \ndiv.text_cell_render h2 {\nfont-size: 14pt;\n}\n \n.CodeMirror {\n font-family: Consolas, monospace;\n }\n \n \n\"\"\")\n```\n\n\n\n\n\n\ndiv.input {\nwidth: 105ex; /* about 80 chars + buffer */\n}\n\ndiv.text_cell {\nwidth: 105ex /* instead of 100%, */\n}\n\ndiv.text_cell_render {\n/*font-family: \"Helvetica Neue\", Arial, Helvetica, Geneva, sans-serif;*/\nfont-family: \"Charis SIL\", serif; /* Make non-code text serif. */\nline-height: 145%; /* added for some line spacing of text. */\nwidth: 105ex; /* instead of 'inherit' for shorter lines */\n}\n\n/* Set the size of the headers */\ndiv.text_cell_render h1 {\nfont-size: 18pt;\n}\n\ndiv.text_cell_render h2 {\nfont-size: 14pt;\n}\n\n.CodeMirror {\n font-family: Consolas, monospace;\n }\n\n\n\n\n\n\n\n```\n#(Extended) Euclid Algorithm\n#Code Block\n#Derived from Algorithm by Robert W. Sebesta\ndef Euclid(a,b,ans=0,pt=0,retA=1):\n \"\"\"Finds the greatest common divisor of a and b and x,y\n such that a*x+b*y=gcd(a,b)\n Three flags vary the output (Any number may be true):\n ans: If true will print the equation a*x+b*y=gcd(a,b)\n pt: If true will show the table method for finding the solution\n retA: If true will return the list [a,x,b,y,gcd(a,b)]\n \"\"\"\n Temp=_Euc([1,0,a],[0,1,b],[[['A1','A2','A3'],['B1','B2','B3']],[[1,0,a],[0,1,b]]])\n T=Temp[0]\n if ans:\n print '\\n({0})({1})+({2})({3})={4}'.format(a,T[0],b,T[1],T[2])\n if pt: \n for x in Temp[1]: print '{0}\\t{1}\\t{2}\\t{3}\\t{4}\\t{5}'.format(x[0][0],x[0][1],x[0][2],x[1][0],x[1][1],x[1][2])\n print\n if retA:\n return [a,T[0],b,T[1],T[2]]\ndef _Euc(a,b,s):\n \"\"\"Main recursive loop for Extended Euclidean Algorithm\"\"\"\n if b[2]==0: return [b[0],b[1],a[2]],s\n if b[2]==1: return b,s\n else:\n Q=a[2]/b[2]\n t=[a[n]-Q*b[n] for n in range(3)]\n a=b\n b=t\n return _Euc(a,b,s+[[a,b]])\ndef MI(a,b):\n \"\"\"Returns the multiplicative inverse of a mod b\"\"\"\n temp = Euclid(a,b,ans=0,pt=0,retA=1)[1]\n if temp<0:temp+=b\n return temp\n```\n\n\n```\n#Chinese Remainder Theorum\n#Code Block\npprint=MDPL\ndef CRT(A,ptable=0):\n \"\"\"A is a list of lists such that [[a1,a2],[b1,b2],...] is equivalent to \n x = a1 mod a2, x = b1 mod b2,... where a,b,... are integers\n The flag ptable if true will output the work to find the solution\"\"\"\n M=reduce(lambda x,y:x*y,[i[1] for i in A])\n if ptable: pprint( 'M={0}'.format(M))\n count=1;Ans=[];outS='';sumS=0\n for i in A:\n Ans=Ans+[[count,i[0],i[1],M/i[1],Euclid(M/i[1],i[1],retA=1)[1]%i[1]]]\n #print 'a({0})={1} m({0})={2} M/M({0})={3} c({0})={4}'.format(Ans[count-1][0],Ans[count-1][1],Ans[count-1][2],Ans[count-1][3],Ans[count-1][4])\n if ptable: pprint('a_{{{0}}}={1},\\ m_{{{0}}}={2},\\ \\\\frac{{M}}{{M_{{{0}}}}}={3},\\ c_{{{0}}}={4}'.format(Ans[count-1][0],Ans[count-1][1],Ans[count-1][2],Ans[count-1][3],Ans[count-1][4]))\n outS=outS+'({0}*{1}*{2})+'.format(Ans[count-1][3],Ans[count-1][4],Ans[count-1][1])\n sumS=sumS+(Ans[count-1][3]*Ans[count-1][4]*Ans[count-1][1])\n count=count+1\n if ptable: \n pprint( outS[:-1] + ' = {0}'.format(sumS) )\n pprint( '= {0}\\ mod\\ {1}'.format(sumS%M,M) )\n return sumS%M\n#ChRem([[5,7],[7,11],[3,13]]);print\n#CRT([[5,8],[3,5]],ptable=1);print\n#ChRem([[7,11],[9,7],[2,3],[1,2]])\n```\n\n\n```\n#Basic Encryption/Decryption Algorithms\n#Code Block\nfrom binascii import * #Hex conversion\nfrom collections import deque #Queue\nfrom collections import defaultdict as dd #Dynamic Dictionary\nfrom string import Template #Output formatting\nfrom heapq import heappush, heappop #Minheap\nimport re\n\ndef type2(a):\n try:\n b=type(a)\n except:\n b='NULL'\n return b\n\ndef remDups(seq):\n seen = set()\n seen_add = seen.add\n return reduce(lambda x,y:x+y,[ x for x in seq if x not in seen and not seen_add(x)])\n\nclass Cipher:\n def __init__(self):\n pass\n def enc(self,s):\n print \"No Encryption Algorithm Added\"\n def dec(self,s):\n print \"No Decryption Algorithm Added\"\n\nclass MAS(Cipher):\n \"\"\"Monoalphabetic (Affine) Substitution Cipher\"\"\"\n def __init__(self,a,b):\n self.a=a\n self.b=b\n self.an=Euclid(a,95,retA=1)[1]\n def enc(self,s):\n return reduce(lambda x,y:x+y,[chr(((ord(p)-32)*self.a+self.b)%95+32) for p in s])#if (ord(p)>=97 and ord(p)<=122) else p for p in s])\n def dec(self,s):\n return reduce(lambda x,y:x+y,[chr((((ord(p)-32)-self.b)*self.an)%95+32) for p in s])#if (ord(p)>=97 and ord(p)<=122) else p for p in s])\n \nclass PF(Cipher):\n \"\"\"Playfair Cipher\"\"\"\n def __init__(self,s):\n Temp=remDups(s.replace('q','')+'abcdefghijklmnoprstuvwxyz')\n Temp2=[]\n for i in range(5):\n Temp2=Temp2+[Temp[i*5:(i+1)*5]]\n self.Table=Temp2\n def printTable(self):\n for i in self.Table:\n print '{0}'.format(str(i))\n\n```\n\n\n```\n#Primality Tests\n##Code Block\nfrom random import randint as ri\nppt=MDPL\nclass PTest:\n def __init__(self):\n pass\n def TryDiv(self,a):\n return [(i,a/i) for i in range(2,int(a**.5)) if a%i==0]\n def Fermat(self,a):\n pass\n def Carmic(self,a):\n pass\ndef MilRab(n,rep=5,showwork=0):\n if n<=3: return 1\n if n%2==0: \n if showwork: ppt(\"{0}\\ is\\ even\".format(n))\n return 0\n n1=q=(n-1)\n k=0\n while q%2==0:\n q=(q)/2\n k+=1\n c=pow(2,q,n)\n d,s=q,k #Other explanations use this notation\n if showwork: ppt(\"{2}-1 = 2^{0} {1}\".format(k,q,n)) #Show n-1 factored\n if q==1: return 1\n def _MRHelp(a):\n if pow(a,d,n)==1: return 0\n for i in range(s): #Witness Loop \n temp2=pow(a,2**i*d,n)\n if temp2==n-1: return 0\n return 1\n for temp in range(rep):\n a=ri(2,n-2)\n if _MRHelp(a): return 0\n return 1 #Since nothing says otherwise, return probably prime\n```\n\n\n```\n#Factorization\n#Code Block\nclass Factorizer:\n def __init__(self):\n pass\n def TryDiv(self,a):\n return [(i,a/i) for i in range(2,int(a**.5)) if a%i==0]\n def Fermat(self,a):\n ret=[]\n for i in range(1,100):\n temp=(a+i**2)**.5\n if temp==int(temp):\n temp=int(temp)\n ret+=[[temp+i,temp-i]]\n return ret\n def Pollard(self,a):\n pass\n def QuadSieve(self,a):\n pass\n```\n\n\n```\n#Pohlig Hellman Discrete Log\n#Code Block\n#Source: http://www-math.ucdenver.edu/~wcherowi/courses/m5410/phexam.html\nfrom sympy import factorint as fc\nppt=MDPL #LaTeX-style print function\ndef DLog(g,p,y,ShowWork=0):\n \"\"\"Prompt for a generator g, prime p, and y=g^x value.\n Essentially finds x\n Compute x using the Pohlig Hellman algorithm.\"\"\"\n pf=fc(p-1) #Factor p-1\n if ShowWork: \n ppt(\"Pollig-Hellman\\ Discrete\\ Log\\ Example\")\n ppt(\"Given\\ p={0},g={1},\\ and\\ y={2}\\ such\\ that\\ {1}^x\\equiv {2}\\mod {0} \\equiv {3}\\mod {0},\\ find\\ x:\".format(p,g,y,y%p))\n ppt(\"$p-1={0}-1={1}$\".format(p,\"\".join([str(i[0])+\"^\"+str(i[1])+\" * \" for i in pf.iteritems()]))[:-3])\n ppt(\"Now\\ we\\ need\\ to\\ find\\ a\\ congruency\\ for\\ each\\ factor:\")\n x=[]\n for i in pf.iteritems(): #For each factor\n q=i[0] #Factor\n r=i[1] #Power\n if ShowWork: \n print\n ppt(\"Find\\ x_{{{0}}} \\mod {1}\\ [x_{{{0}}}={2}]:\".format(q,q**r,\"\".join([\"c_{{{0}}}*({{{1}}})+\".format(ii,q**ii) for ii in range(r)])[:-1]))\n yy=y #Initially set base to y for each factor\n NewCLast=dict((pow(g,ii*(p-1)/q,p),ii%q) for ii in range(1,q+1)) #Dictionary of possible values for c_? to hit\n if ShowWork: ppt(\"y^{{ k \\\\frac{{p-1}}{{q^{{c_0}}}} }}\\ must\\ be\\ one\\ of\\ these:\\ {0}\".format(\"\".join([\"{2}^{{{0}*\\\\frac{{p-1}}{{{3}}}}} \\equiv {1},\\ \".format(NewCLast[ii],ii,g,q) for ii in NewCLast.keys()])[:-3] ) )\n xTemp=0 #Using cumulative sum for finding each c_?\n for k in range(r): #For each c_?\n for j in range(1,q+1): #Try values until temp is in the NewCLast dictionary\n j=j%q #Since it would normally start with 0, making 0 last by wrapping with mod\n trypow=j*(p-1)/(q**(k+1)) #Generate power to raise yy by\n temp=pow(yy,trypow,p) \n if temp in NewCLast.keys():\n j=NewCLast[temp] #Find the power of yy that makes temp\n xTemp+=(q**k)*j #Add the cumulating sum for x\n if ShowWork: ppt(\"$ y^{{ k \\\\frac{{p-1}}{{q^{{{9}}}}} }} \\equiv {0}^{{ {7} \\\\frac{{{1}}}{{{2}}} }} \\equiv {0}^{{{3}}} \\equiv {4} \\mod {6},\\ so\\ c_{{{8}}} = {7} $\".format(yy,p-1,q**(k+1),trypow,temp,temp-p,p,j,k,k+1))\n #cLast=[j]\n if k\n# PHY321: Introduction to Classical Mechanics\n\n \n**[Morten Hjorth-Jensen](http://mhjgit.github.io/info/doc/web/)**, Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University, USA and Department of Physics, University of Oslo, Norway \n\n **[Scott Pratt](https://pa.msu.edu/profile/pratts/)**, Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University, USA \n\n **[Carl Schmidt](https://pa.msu.edu/profile/schmidt/)**, Department of Physics and Astronomy, Michigan State University, USA\n\nDate: **Jan 6, 2020**\n\nCopyright 1999-2020, [Morten Hjorth-Jensen](http://mhjgit.github.io/info/doc/web/). Released under CC Attribution-NonCommercial 4.0 license\n\n\n\n\n## Introduction\n\nClassical mechanics is a topic which has been taught intensively over\nseveral centuries. It is, with its many variants and ways of\npresenting the educational material, normally the first **real** physics\ncourse many of us meet and it lays the foundation for further physics\nstudies. Many of the equations and ways of reasoning about the\nunderlying laws of motion and pertinent forces, shape our approaches and understanding\nof the scientific method and discourse, as well as the way we develop our insights\nand deeper understanding about physical systems. \n\nThere is a wealth of\nwell-tested (from both a physics point of view and a pedagogical\nstandpoint) exercises and problems which can be solved\nanalytically. However, many of these problems represent idealized and\nless realistic situations. The large majority of these problems are\nsolved by paper and pencil and are traditionally aimed\nat what we normally refer to as continuous models from which we may find an analytical solution. As a consequence,\nwhen teaching mechanics, it implies that we can seldomly venture beyond an idealized case\nin order to develop our understandings and insights about the\nunderlying forces and laws of motion.\n\n## Numerical Elements\nOn the other hand, numerical algorithms call for approximate discrete\nmodels and much of the development of methods for continuous models\nare nowadays being replaced by methods for discrete models in science and\nindustry, simply because **much larger classes of problems can be addressed** with discrete models, often by simpler and more\ngeneric methodologies.\n\nAs we will see below, when properly scaling the equations at hand,\ndiscrete models open up for more advanced abstractions and the possibility to\nstudy real life systems, with the added bonus that we can explore and\ndeepen our basic understanding of various physical systems\n\nAnalytical solutions are as important as before. In addition, such\nsolutions provide us with invaluable benchmarks and tests for our\ndiscrete models. Such benchmarks, as we will see below, allow us \nto discuss possible sources of errors and their behaviors. And\nfinally, since most of our models are based on various algorithms from\nnumerical mathematics, we have a unique oppotunity to gain a deeper\nunderstanding of the mathematical approaches we are using.\n\n\n\nWith computing and data science as important elements in essentially\nall aspects of a modern society, we could then try to define Computing as\n**solving scientific problems using all possible tools, including\nsymbolic computing, computers and numerical algorithms, and analytical\npaper and pencil solutions**. \nComputing provides us with the tools to develope our own understanding of the scientific method by enhancing algorithmic thinking.\n\n## Computations and the Scientific Method\n\nThe way we will teach this course reflects\nthis definition of computing. The course contains both classical paper\nand pencil exercises as well as computational projects and exercises. The\nhope is that this will allow you to explore the physics of systems\ngoverned by the degrees of freedom of classical mechanics at a deeper\nlevel, and that these insights about the scientific method will help\nyou to develop a better understanding of how the underlying forces and\nequations of motion and how they impact a given system. Furthermore, by introducing various numerical methods\nvia computational projects and exercises, we aim at developing your competences and skills about these topics.\n\n\nThese competences will enable you to\n\n* understand how algorithms are used to solve mathematical problems,\n\n* derive, verify, and implement algorithms,\n\n* understand what can go wrong with algorithms,\n\n* use these algorithms to construct reproducible scientific outcomes and to engage in science in ethical ways, and\n\n* think algorithmically for the purposes of gaining deeper insights about scientific problems.\n\nAll these elements are central for maturing and gaining a better understanding of the modern scientific process *per se*.\n\nThe power of the scientific method lies in identifying a given problem\nas a special case of an abstract class of problems, identifying\ngeneral solution methods for this class of problems, and applying a\ngeneral method to the specific problem (applying means, in the case of\ncomputing, calculations by pen and paper, symbolic computing, or\nnumerical computing by ready-made and/or self-written software). This\ngeneric view on problems and methods is particularly important for\nunderstanding how to apply available, generic software to solve a\nparticular problem.\n\n*However, verification of algorithms and understanding their limitations requires much of the classical knowledge about continuous models.*\n\n\n## A well-known examples to illustrate many of the above concepts\n\nBefore we venture into a reminder on Python and mechanics relevant applications, let us briefly outline some of the\nabovementioned topics using an example many of you may have seen before in for example CMSE201. \nA simple algorithm for integration is the Trapezoidal rule. \nIntegration of a function $f(x)$ by the Trapezoidal Rule is given by following algorithm for an interval $x \\in [a,b]$\n\n$$\n\\int_a^b(f(x) dx = \\frac{1}{2}\\left [f(a)+2f(a+h)+\\dots+2f(b-h)+f(b)\\right] +O(h^2),\n$$\n\nwhere $h$ is the so-called stepsize defined by the number of integration points $N$ as $h=(b-a)/(n)$.\nPython offers an extremely versatile programming environment, allowing for\nthe inclusion of analytical studies in a numerical program. Here we show an\nexample code with the **trapezoidal rule**. We use also **SymPy** to evaluate the exact value of the integral and compute the absolute error\nwith respect to the numerically evaluated one of the integral\n$\\int_0^1 dx x^2 = 1/3$.\nThe following code for the trapezoidal rule allows you to plot the relative error by comparing with the exact result. By increasing to $10^8$ points one arrives at a region where numerical errors start to accumulate.\n\n\n```python\n%matplotlib inline\n\nfrom math import log10\nimport numpy as np\nfrom sympy import Symbol, integrate\nimport matplotlib.pyplot as plt\n# function for the trapezoidal rule\ndef Trapez(a,b,f,n):\n h = (b-a)/float(n)\n s = 0\n x = a\n for i in range(1,n,1):\n x = x+h\n s = s+ f(x)\n s = 0.5*(f(a)+f(b)) +s\n return h*s\n# function to compute pi\ndef function(x):\n return x*x\n# define integration limits\na = 0.0; b = 1.0;\n# find result from sympy\n# define x as a symbol to be used by sympy\nx = Symbol('x')\nexact = integrate(function(x), (x, a, b))\n# set up the arrays for plotting the relative error\nn = np.zeros(9); y = np.zeros(9);\n# find the relative error as function of integration points\nfor i in range(1, 8, 1):\n npts = 10**i\n result = Trapez(a,b,function,npts)\n RelativeError = abs((exact-result)/exact)\n n[i] = log10(npts); y[i] = log10(RelativeError);\nplt.plot(n,y, 'ro')\nplt.xlabel('n')\nplt.ylabel('Relative error')\nplt.show()\n```\n\n## Analyzing the above example\nThis example shows the potential of combining numerical algorithms with symbolic calculations, allowing us to \n\n* Validate and verify their algorithms. \n\n* Including concepts like unit testing, one has the possibility to test and test several or all parts of the code.\n\n* Validation and verification are then included *naturally* and one can develop a better attitude to what is meant with an ethically sound scientific approach.\n\n* The above example allows the student to also test the mathematical error of the algorithm for the trapezoidal rule by changing the number of integration points. The students get **trained from day one to think error analysis**. \n\n* With a Jupyter notebook you can keep exploring similar examples and turn them in as your own notebooks. \n\nIn this process we can easily bake in\n1. How to structure a code in terms of functions\n\n2. How to make a module\n\n3. How to read input data flexibly from the command line\n\n4. How to create graphical/web user interfaces\n\n5. How to write unit tests (test functions or doctests)\n\n6. How to refactor code in terms of classes (instead of functions only)\n\n7. How to conduct and automate large-scale numerical experiments\n\n8. How to write scientific reports in various formats (LaTeX, HTML)\n\nThe conventions and techniques outlined here will save you a lot of time when you incrementally extend software over time from simpler to more complicated problems. In particular, you will benefit from many good habits:\n1. New code is added in a modular fashion to a library (modules)\n\n2. Programs are run through convenient user interfaces\n\n3. It takes one quick command to let all your code undergo heavy testing \n\n4. Tedious manual work with running programs is automated,\n\n5. Your scientific investigations are reproducible, scientific reports with top quality typesetting are produced both for paper and electronic devices.\n\n## Teaching team, grading and other practicalities\n\n\n\n\n\n\n\n\n\n\n
                                        Lectures Location
                                        Monday 3:00-3:50pm Wednesday 3:00-3:50pm Friday 3:00-3:50pm Room 1420 BPS
                                        \n\n\n\n\n\n\n\n
                                        Instructor Email Office Office phone/cellphone
                                        [Morten Hjorth-Jensen](https://github.com/mhjensen) hjensen@msu.edu Office: NSCL/FRIB 2131 5179087290/5172491375
                                        \n\n\n\n\n\n\n\n
                                        Office Hours
                                        Monday/Wednesday 4-5:00pm, Room 2131 NSCL/FRIB or immediately after class
                                        \n\n\n\n\n\n\n\n
                                        Homework Grader Email
                                        Kasun Senanayaka senanaya@msu.edu
                                        \n\n\n\n\n\n\n\n\n
                                        Learning Assistant Email
                                        Dylan R. Smith smithdy6@msu.edu
                                        \n## Grading and dates\n\n\n\n\n\n\n\n\n\n\n\n\n
                                        Activity Percentage of total score
                                        Homeworks, 10 in total and due Wednesdays the week after 20%
                                        First Midterm Project, due Friday February 28 25%
                                        Second Midterm Project, due Friday April 10 25%
                                        Final Exam, April 29, 5:45-7:45pm 30%
                                        Extra Credit Assignment (Due Friday April 24) 10%
                                        \n\n\n\n\n\n\n\n\n
                                        Grading scale
                                        4.0(90%) 3.5(80%) 3.0(70%) 2.5(60%) 2.0(50%) 1.5(40%) 1.0(30%)
                                        \n\n## Possible textbooks and lecture notes\n\n**Recommended textbook**:\n* [John R. Taylor, Classical Mechanics (Univ. Sci. Books 2005)](https://www.uscibooks.com/taylor2.htm), see also [the GitHub link of the course](https://github.com/mhjensen/Physics321/tree/master/doc/Literature)\n\n**Additional textbooks**:\n* [Anders Malthe-S\u00f8renssen, Elementary Mechanics using Python (Springer 2015)](https://www.springer.com/gp/book/9783319195957) and [the GitHub link of the course](https://github.com/mhjensen/Physics321/tree/master/doc/Literature)\n\n* [Alessandro Bettini, A Course in Classical Physics 1, Mechanics (Springer 2017)](https://www.springer.com/gp/book/9783319292564) and the [GitHub link of the course](https://github.com/mhjensen/Physics321/tree/master/doc/Literature).\n\nThe books from Springer can be downloaded for free (pdf or ebook format) from any MSU IP address. \n\n**Lecture notes**:\nPosted lecture notes are in the doc/pub folder here or at for easier viewing. They are not meant to be a replacement for textbook. These notes are updated on a weekly basis and a **git pull** should thus always give you the latest update. \n\n## Teaching schedule with links to material (This will be updated asap)\n\nWeekly mails (Wednesdays or Thursdays) with updates, plans for lectures etc will sent to everybody. We use also Piazza as a discussion forum. Please use this sign-up link . The class link is \n### Week 2, January 6-10, 2020\n\n1. Monday: Introduction to the course and start discussion of vectors, space, time and motion, Taylor chapter 1.2 and lecture notes (https://mhjensen.github.io/Physics321/doc/pub/Introduction/html/Introduction.html)\n\n2. Wednesday: More on time,space, vectors and motion, Taylor chapters 1.2 and 1.3 and lecture notes (https://mhjensen.github.io/Physics321/doc/pub/Introduction/html/Introduction.html), first homework available\n\n3. Friday: Forces and Newton's laws of motion. Taylor chapter 1.4 and lecture notes (https://mhjensen.github.io/Physics321/doc/pub/Introduction/html/Introduction.html). Introduction to Git and GitHub and getting started with numerical exercises. Installing software (anaconda) and first homework due January 15.\n\n### Week 3, January 13-17, 2020\n\n1. Monday:\n\n2. Wednesday: \n\n3. Friday: 2nd homework, due January 22\n\n### Week 4, January 20-24, 2020\n\n1. Monday: MLK day, no lectures\n\n2. Wednesday: \n\n3. Friday: 3rd homework, due January 29\n\n### Week 5, January 27-31, 2020\n\n1. Monday:\n\n2. Wednesday: \n\n3. Friday: 4th homework, due February 5\n\n### Week 6, February 3-7, 2020\n\n1. Monday:\n\n2. Wednesday: \n\n3. Friday: 5th homework, due February 12\n\n### Week 7, February 10-14, 2020\n\n1. Monday:\n\n2. Wednesday: \n\n3. Friday: 6th homework, due February 19\n\n### Week 8, February 17-21, 2020\n\n1. Monday:\n\n2. Wednesday: \n\n3. Friday: **First midterm project, due February 28, 2020** \n\n### Week 9, February 24-28, 2020\n\n1. Monday:\n\n2. Wednesday: \n\n3. Friday: \n\n### Week 10, March 2-6, 2020, Spring break\n\n1. Monday: No lectures\n\n2. Wednesday: No lectures\n\n3. Friday: No lectures \n\n### Week 11, March 9-13, 2020\n\n1. Monday:\n\n2. Wednesday: \n\n3. Friday: 7th homework, due March 18\n\n### Week 12, March 16-20, 2020\n\n1. Monday:\n\n2. Wednesday: \n\n3. Friday: 8th homework, due March 25\n\n### Week 13, March 23-27, 2020\n\n1. Monday:\n\n2. Wednesday: \n\n3. Friday: 9th homework, due April 1\n\n### Week 14, March 30-April 3, 2020\n\n1. Monday:\n\n2. Wednesday: \n\n3. Friday: **Second midterm project, due April 10, 2020**\n\n### Week 15, April 13-17, 2020\n\n1. Monday:\n\n2. Wednesday: \n\n3. Friday: 10th homework and extra assignments, due April 24 \n\n### Week 16, April 20-24, 2020\n\n1. Monday:\n\n2. Wednesday: \n\n3. Friday: Summary and discussions of finals exams \n\n### Week 17, April 27- May 1, 2020, Finals week\n\n1. Final Exam: April 29, 5:45pm - 7:45pm in 1420 Biomedical & Physical Sciences\n\n## Space, Time, Motion and Reminder on vectors and other mathematical quantities\n\n## Newton's Laws of Motion\n", "meta": {"hexsha": "9b5b4cb4d2352e31d2ab91bae2db3b5dc3b00dc3", "size": 22601, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "doc/pub/Introduction/ipynb/.ipynb_checkpoints/Introduction-checkpoint.ipynb", "max_stars_repo_name": "Shield94/Physics321", "max_stars_repo_head_hexsha": "9875a3bf840b0fa164b865a3cb13073aff9094ca", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 20, "max_stars_repo_stars_event_min_datetime": "2020-01-09T17:41:16.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-09T00:48:58.000Z", "max_issues_repo_path": "doc/pub/Introduction/ipynb/.ipynb_checkpoints/Introduction-checkpoint.ipynb", "max_issues_repo_name": "Shield94/Physics321", "max_issues_repo_head_hexsha": "9875a3bf840b0fa164b865a3cb13073aff9094ca", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2020-01-08T03:47:53.000Z", "max_issues_repo_issues_event_max_datetime": "2020-12-15T15:02:57.000Z", "max_forks_repo_path": "doc/pub/Introduction/ipynb/.ipynb_checkpoints/Introduction-checkpoint.ipynb", "max_forks_repo_name": "Shield94/Physics321", "max_forks_repo_head_hexsha": "9875a3bf840b0fa164b865a3cb13073aff9094ca", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 33, "max_forks_repo_forks_event_min_datetime": "2020-01-10T20:40:55.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-11T20:28:41.000Z", "avg_line_length": 47.6814345992, "max_line_length": 320, "alphanum_fraction": 0.6054156896, "converted": true, "num_tokens": 4566, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5389832354982645, "lm_q2_score": 0.7981867801399695, "lm_q1q2_score": 0.43020929329178265}} {"text": "```python\nimport axelrod as axl\nimport numpy as np\nimport pandas as pd\nimport scipy as sp\nfrom sympy import *\nimport random\nfrom itertools import groupby\nimport difflib\n\nfrom pyecharts import options as opts\nfrom pyecharts.charts import Pie\nfrom pyecharts.render import make_snapshot\nfrom snapshot_selenium import snapshot\nimport matplotlib.pyplot as plt\n\nfrom matplotlib.colors import LinearSegmentedColormap\n\nfrom IPython.display import display, HTML\n```\n\n\n```python\n#axl.ZDExtortion(phi = 0.1, s = 0.5)\n```\n\n\n```python\n#me = axl.Human(name='me')\n#players = [axl.TitForTat(), me]\n#match = axl.Match(players, turns = 3)\n```\n\n\n```python\n_Image_PATH_ = './images/'\n_Figure_PATH_ = './figures/'\n```\n\n\n```python\narmyrose = ['#798234', '#a3ad62', '#d0d3a2', '#fdfbe4', '#f0c6c3', '#df91a3', '#d46780']\nfall = ['#3d5941', '#778868', '#b5b991', '#f6edbd', '#edbb8a', '#de8a5a', '#ca562c']\ngeyser = ['#008080', '#70a494', '#b4c8a8', '#f6edbd', '#edbb8a', '#de8a5a', '#ca562c']\ntemps = ['#009392', '#39b185', '#9ccb86', '#e9e29c', '#eeb479', '#e88471', '#cf597e']\ntealrose = ['#009392', '#72aaa1', '#b1c7b3', '#f1eac8', '#e5b9ad', '#d98994', '#d0587e']\ntropic = ['#009B9E', '#42B7B9', '#A7D3D4', '#F1F1F1', '#E4C1D9', '#D691C1', '#C75DAB']\nearth = ['#A16928', '#bd925a', '#d6bd8d', '#edeac2', '#b5c8b8', '#79a7ac', '#2887a1']\n```\n\n\n```python\n# Define symbols\nR, S, T, P = symbols('R, S, T, P', real = True) # payoffs\nchi, phi = symbols('chi, phi', real = True) # parameters for the probabilities\nQ_1 = 1 - (R - P)*phi*(chi - 1)\nQ_2 = 1 - phi*((T - P)*chi + (P - S))\nQ_3 = phi*((P - S)*chi + (T - P))\nQ_4 = phi*(P - P)\n\nphi_upper_con = 1/(chi*(T - P) + (P - S)) # the range of phi is 0 < phi < phi_upper \nphi_upper_abn = 1/(chi*(P - S) + (T - P)) # the range of phi is 0 < phi < phi_upper\n\npayoff_con_subs = [(R, 3), (S, 0), (T, 5), (P, 1)] # payments of the conventional IPD\n\nhuman_score_dict = {('C', 'C'): R, ('C', 'D'): S, ('D', 'C'): T, ('D', 'D'): P}\nZD_score_dict = {('C', 'C'): R, ('C', 'D'): T, ('D', 'C'): S, ('D', 'D'): P}\n```\n\n\n```python\nstrategies = ['Cooperator', 'Defector', 'Random', 'Tit For Tat', 'Win-Stay Lose-Shift', 'Human']\ndescriptions = ['A player who only ever cooperates. ' + r'$p = [1, 1, 1, 1]$', 'A player who only ever defects. ' + r'$p = [0, 0, 0, 0]$', \n 'A player who randomly chooses between cooperating and defecting. ' + r'$p = [0.5, 0.5, 0.5, 0.5]$', \n 'A player who starts by cooperating and then repeats the opponent\u2019s previous move. '+ r'$p = [1, 0, 1, 0]$',\n 'A player who repeats the previous move if the resulting payoff has met its aspiration level and changes otherwise. ' + r'$p = [1, 0, 0, 1]$',\n 'A player does whatever you want.']\np_0_dict = {'Cooperator': 1, 'Defector': 0, 'Random': 0.5, 'Tit For Tat': 1, 'Win-Stay Lose-Shift': 1, 'Human': np.nan}\np_dict = {'Cooperator': [1, 1, 1, 1], 'Defector': [0, 0, 0, 0], 'Random': [0.5, 0.5, 0.5, 0.5], \n 'Tit For Tat': [1, 0, 1, 0], 'Win-Stay Lose-Shift': [1, 0, 0, 1], 'Human': [np.nan, np.nan, np.nan, np.nan]}\n\ncmaps = [armyrose, earth, fall, tealrose, tropic, temps]\nfig, axes = plt.subplots(2, 3, figsize = (3*5, 2*2), sharey = False)\nfig.subplots_adjust(hspace = -0.25, wspace = 0.1)\ngradient = np.linspace(0, 1, 256)\ngradient = np.vstack((gradient, gradient))\nfor i, title in enumerate(strategies):\n ix = np.unravel_index(i, axes.shape)\n ax = axes[ix]\n cm = LinearSegmentedColormap.from_list('tealrose', cmaps[i], N = 100)\n ax.imshow(gradient, aspect = 5, cmap = cm)\n ax.axis(\"off\")\n ax.set_title(title, fontsize = 15) \nfig.suptitle('iPrisoner Game: ZDGo', fontsize = 20, y = 0.95)\nfig.savefig(_Image_PATH_ + 'cover.png', dpi = 400)\n\ndef ZDGo(printing = True):\n \n player = input(\"Enter your player: \")\n while player not in strategies:\n # find closest match\n options = difflib.get_close_matches(player, strategies)\n if len(options) == 1:\n player = input(\"Do you mean \" + options[0] + '? If yes, enter ' + options[0] + \", otherwise, choose another player: \")\n else:\n player = input(\"Unjustified player, re-enter your player: \")\n \n chi_value = input(\"Enter chi (chi is the extortion factor and is a number no less than 1): \")\n while True:\n try:\n val = float(chi_value)\n if val < 1:\n chi_value = input(\"Input is less than 1. Re-enter chi: \")\n continue\n break\n except ValueError:\n chi_value = input(\"Input is not a number. Re-enter chi: \")\n chi_value = float(chi_value)\n \n n = input(\"Enter number of rounds n (n is a positive integer): \")\n while True:\n try:\n val = int(n)\n if val < 0: # if not a positive int print message and ask for input again\n n = input(\"Input is not a positive integer. Re-enter n: \")\n continue\n break\n except ValueError:\n try:\n val = float(n)\n n = input(\"Input is a float. Re-enter n: \")\n except ValueError:\n n = input(\"Input is not a number. Re-enter n: \")\n n = int(n)\n \n df_information = pd.DataFrame(data = {'your strategy': [player], 'extortion factor': [chi_value], 'number of rounds': [n]})\n df_information.index = ['']\n display(HTML(df_information.to_html()))\n \n chi_subs = [(chi, chi_value)]\n payoff_subs, phi_upper = payoff_con_subs, phi_upper_con\n phi_value = 1/2*phi_upper.subs(payoff_subs).subs(chi_subs)\n phi_subs = [(phi, phi_value)]\n q_1 = Q_1.subs(payoff_subs).subs(chi_subs + phi_subs)\n q_2 = Q_2.subs(payoff_subs).subs(chi_subs + phi_subs)\n q_3 = Q_3.subs(payoff_subs).subs(chi_subs + phi_subs)\n q_4 = Q_4.subs(payoff_subs).subs(chi_subs + phi_subs)\n \n # count the frequencies of different outcomes\n def freqs(outcomes):\n freq_cc = outcomes.count(('C', 'C'))\n freq_cd = outcomes.count(('C', 'D'))\n freq_dc = outcomes.count(('D', 'C'))\n freq_dd = outcomes.count(('D', 'D'))\n return freq_cc, freq_cd, freq_dc, freq_dd\n \n # calculate the average scores of human player and ZD extorter\n def scores(outcomes):\n outcomes = outcomes.copy()\n outcomes.sort()\n # frequencies of (C, C), (C, D), (D, C) and (D, D)\n frequencies = [(key, len(list(group))) for key, group in groupby(outcomes)]\n human_score = ZD_score = 0\n for item in frequencies:\n outcome = item[0]\n freq = item[1]\n human_score += human_score_dict[outcome].subs(payoff_subs)*freq\n ZD_score += ZD_score_dict[outcome].subs(payoff_subs)*freq\n # take the average\n human_score /= n\n ZD_score /= n\n return float(human_score), float(ZD_score)\n \n # decide the action of ZD strategy according to the outcome of the last round\n def ZDAction(last_round):\n if last_round == ('C', 'C'):\n return (lambda x: 'C' if x <= q_1 else 'D')(random.uniform(0, 1))\n elif last_round == ('C', 'D'):\n return (lambda x: 'C' if x <= q_3 else 'D')(random.uniform(0, 1))\n elif last_round == ('D', 'C'):\n return (lambda x: 'C' if x <= q_2 else 'D')(random.uniform(0, 1))\n else:\n return (lambda x: 'C' if x <= q_4 else 'D')(random.uniform(0, 1))\n \n # decide the action of player according to the outcome of the last round\n def Action(last_round, p):\n p_1, p_2, p_3, p_4 = p\n if last_round == ('C', 'C'):\n return (lambda x: 'C' if x <= p_1 else 'D')(random.uniform(0, 1))\n elif last_round == ('C', 'D'):\n return (lambda x: 'C' if x <= p_2 else 'D')(random.uniform(0, 1))\n elif last_round == ('D', 'C'):\n return (lambda x: 'C' if x <= p_3 else 'D')(random.uniform(0, 1))\n else:\n return (lambda x: 'C' if x <= p_4 else 'D')(random.uniform(0, 1))\n \n def NonHuman(p_0, p):\n i = 0\n outcomes = []\n while i < n:\n if i == 0:\n action = (lambda x: 'C' if x <= p_0 else 'D')(random.uniform(0, 1))\n ZD_action = 'C'\n else:\n action = Action(outcomes[-1], p)\n ZD_action = ZDAction(outcomes[-1])\n \n outcomes.append((action, ZD_action))\n if printing == True:\n print(\"round \" + str(i + 1) + '| outcome: ' + '(' + action + ', ' + ZD_action + ')')\n i += 1\n return outcomes\n \n def Human():\n i = 0\n outcomes = []\n while i < n:\n human_action = input(\"Round \" + str(i + 1) + '| enter your action (C or D): ')\n while human_action not in ['C', 'D']:\n human_action = input('Unjustified action, re-enter your action: ')\n if i == 0:\n ZD_action = 'C'\n else:\n ZD_action = ZDAction(outcomes[-1])\n outcomes.append((human_action, ZD_action))\n if printing == True:\n print(\"round \" + str(i + 1) + '| outcome: ' + '(' + human_action + ', ' + ZD_action + ')')\n i += 1\n return outcomes\n \n if player == 'Human':\n outcomes = Human()\n else:\n outcomes = NonHuman(p_0_dict[player], p_dict[player])\n \n \n df_information[\"player's score\"] = [scores(outcomes)[0]]\n df_information[\"ZD's score\"] = [scores(outcomes)[1]]\n freq_cc, freq_cd, freq_dc, freq_dd = freqs(outcomes)\n df_information['(C, C) pair'] = [freq_cc]\n df_information['(C, D) pair'] = [freq_cd]\n df_information['(D, C) pair'] = [freq_dc]\n df_information['(D, D) pair'] = [freq_dd]\n \n display(HTML(df_information.to_html()))\n \n # change data type\n columns = df_information.columns.tolist()\n columns = columns[1:]\n for col in columns:\n df_information[col] = df_information[col].astype(float)\n \n return df_information, outcomes\n```\n\n\n```python\ndef figure_pie(df_information):\n \n title = 'ZDGo'\n subtitle = df_information['your strategy'].tolist()[0] + ' against ZD strategy'\n chi_value = df_information['extortion factor'].tolist()[0]\n rounds = int(df_information['number of rounds'].tolist()[0])\n scores = [df_information[\"player's score\"].tolist()[0], df_information[\"ZD's score\"].tolist()[0]]\n freqs = df_information.iloc[0].tolist()[-4:]\n \n pie = Pie(init_opts = opts.InitOpts(width='600px', height='500px'))\n v = ['(C, C)', '(C, D)', '(D, C)', '(D, D)']\n d = freqs\n pie.add(\"frequency\", [list(z) for z in zip(v, d)],\n radius = [\"35%\", \"60%\"], \n center = [\"50%\", \"55%\"],\n rosetype = \"radius\",\n label_opts = opts.LabelOpts(\n position = \"outside\",\n formatter = \"{b|{b}}{abg|}\\n{hr|}\\n {c|{c} } {per|{d}%} \", # formatter = \"{a|{a}}{abg|}\\n{hr|}\\n {b|{b}: }{c} {per|{d}%} \",\n background_color = \"#eee\",\n border_color = \"#aaa\",\n border_width = 1,\n border_radius = 4,\n rich = {\n \"b\": {\"color\": \"#999\", \"fontSize\": 12, \"lineHeight\": 20, \"align\": \"center\"},\n \"abg\": {\n \"backgroundColor\": \"#e3e3e3\",\n \"width\": \"100%\",\n \"align\": \"right\",\n \"height\": 20,\n \"borderRadius\": [4, 4, 0, 0],\n },\n \"hr\": {\n \"borderColor\": \"#aaa\",\n \"width\": \"100%\",\n \"borderWidth\": 0.5,\n \"height\": 0,\n },\n \"c\": {\"fontSize\": 12, \"lineHeight\": 25},\n \"per\": {\n \"fontSize\": 12, \n \"color\": \"#eee\",\n \"backgroundColor\": \"#334455\",\n \"height\": 10,\n \"padding\": [2, 2],\n \"borderRadius\": 4,\n },\n \"d\": {\"fontSize\": 4},\n },\n ),\n )\n pie.set_global_opts(title_opts = opts.TitleOpts(title = title, subtitle = subtitle + '\\n\\n' + 'chi: ' + str(chi_value) + ' ' + 'n: ' + str(rounds) + ' ' + 'scores: ' + '(' + str(scores[0]) + ', ' + str(scores[1]) + ')',\n title_textstyle_opts = opts.TextStyleOpts(font_size = 20),\n subtitle_textstyle_opts = opts.TextStyleOpts(font_size = 16)),\n legend_opts = opts.LegendOpts(is_show = False, type_ = \"plain\", \n textstyle_opts = opts.TextStyleOpts(font_size = 12), \n pos_top = \"30%\", pos_left = \"1%\", orient = \"vertical\"),\n tooltip_opts = opts.TooltipOpts(formatter = \"{b}: {c} ({d}%)\"))\n \n make_snapshot(snapshot, pie.render(_Figure_PATH_ + 'ZDGo/' + df_information['your strategy'].tolist()[0].replace(\" \", \"_\") + \".html\"),\n _Figure_PATH_ + 'ZDGo/' + df_information['your strategy'].tolist()[0].replace(\" \", \"_\") + \".png\")\n \n \n return pie\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "ccb297ede9f10394d3c150de3cd21362a1c1c9f4", "size": 32547, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "scripts/utils.ipynb", "max_stars_repo_name": "fudab/iPrisoner", "max_stars_repo_head_hexsha": "a00a0ec66174f5de13158a42e4593901f2ac2033", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-10-27T18:40:10.000Z", "max_stars_repo_stars_event_max_datetime": "2020-10-27T18:40:10.000Z", "max_issues_repo_path": "scripts/utils.ipynb", "max_issues_repo_name": "fudab/iPrisoner", "max_issues_repo_head_hexsha": "a00a0ec66174f5de13158a42e4593901f2ac2033", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "scripts/utils.ipynb", "max_forks_repo_name": "fudab/iPrisoner", "max_forks_repo_head_hexsha": "a00a0ec66174f5de13158a42e4593901f2ac2033", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 80.5618811881, "max_line_length": 14404, "alphanum_fraction": 0.6749316373, "converted": true, "num_tokens": 3913, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.4301987589290581}} {"text": "\n\n\n# Tutorial-IllinoisGRMHD: harm_utoprim_2d.c\n\n## Authors: Leo Werneck & Zach Etienne\n\n**This module is currently under development**\n\n## In this tutorial module we explain the conservative-to-primitive algorithm used by `HARM`. This module will likely be absorbed by another one once we finish documenting the code.\n\n### Required and recommended citations:\n\n* **(Required)** Etienne, Z. B., Paschalidis, V., Haas R., M\u00f6sta P., and Shapiro, S. L. IllinoisGRMHD: an open-source, user-friendly GRMHD code for dynamical spacetimes. Class. Quantum Grav. 32 (2015) 175009. ([arxiv:1501.07276](http://arxiv.org/abs/1501.07276)).\n* **(Required)** Noble, S. C., Gammie, C. F., McKinney, J. C., Del Zanna, L. Primitive Variable Solvers for Conservative General Relativistic Magnetohydrodynamics. Astrophysical Journal, 641, 626 (2006) ([astro-ph/0512420](https://arxiv.org/abs/astro-ph/0512420)).\n* **(Recommended)** Del Zanna, L., Bucciantini N., Londrillo, P. An efficient shock-capturing central-type scheme for multidimensional relativistic flows - II. Magnetohydrodynamics. A&A 400 (2) 397-413 (2003). DOI: 10.1051/0004-6361:20021641 ([astro-ph/0210618](https://arxiv.org/abs/astro-ph/0210618)).\n\n\n\n# Table of Contents\n$$\\label{toc}$$\n\nThis module is organized as follows\n\n0. [Step 0](#src_dir): **Source directory creation**\n1. [Step 1](#introduction): **Introduction**\n1. [Step 2](#harm_utoprim_2d__c__eos_indep): **EOS independent routines**\n 1. [Step 2.a](#utoprim_2d): *The `Utoprim_2d()` function*\n 1. [Step 2.a.i](#utoprim_2d__bi_and_alpha): Setting $B^{i}_{\\rm HARM}$ and $\\alpha$\n 1. [Step 2.a.ii](#utoprim_2d__converting): Preparing the variables to be used by the `Utoprim_new_body()` function\n 1. [Step 2.b](#utoprim_new_body): *The `Utoprim_new_body()` function*\n 1. [Step 2.b.i](#utoprim_new_body__basic_quantities): Computing basic quantities\n 1. [Step 2.b.ii](#utoprim_new_body__wlast): Determining $W$ from the previous iteration, $W_{\\rm last}$\n 1. [Step 2.b.iii](#utoprim_new_body__vsqlast_and_recompute_w_and_vsq): Compute $v^{2}_{\\rm last}$, then update $v^{2}$ and $W$\n 1. [Step 2.b.iv](#utoprim_new_body__compute_prims): Computing the primitive variables\n 1. [Step 2.c](#vsq_calc): *The `vsq_calc()` function*\n 1. [Step 2.d](#x1_of_x0): *The `x1_of_x0()` function*\n 1. [Step 2.e](#validate_x): *The `validate_x()` function*\n 1. [Step 2.f](#general_newton_raphson): *The `general_newton_raphson()` function*\n 1. [Step 2.g](#func_vsq): *The `func_vsq()` function*\n1. [Step 3](#harm_utoprim_2d__c__eos_dep): **EOS dependent routines**\n 1. [Step 3.a](#pressure_w_vsq): *The `pressure_W_vsq()` function*\n 1. [Step 3.b](#dpdw_calc_vsq): *The `dpdW_calc_vsq()` function*\n 1. [Step 3.c](#dpdvsq_calc): *The `dpdvsq_calc()` function*\n 1. [Step 3.c.i](#dpdvsq_calc__basic_quantities): Setting basic quantities and computing $P_{\\rm cold}$ and $\\epsilon_{\\rm cold}$\n 1. [Step 3.c.ii](#dpdvsq_calc__dpcolddvsq): Computing $\\frac{\\partial P_{\\rm cold}}{\\partial\\left(v^{2}\\right)}$\n 1. [Step 3.c.iii](#dpdvsq_calc__depscolddvsq): Computing $\\frac{\\partial \\epsilon_{\\rm cold}}{\\partial\\left(v^{2}\\right)}$\n 1. [Step 3.c.iv](#dpdvsq_calc__dpdvsq): Computing $\\frac{\\partial p_{\\rm hybrid}}{\\partial\\left(v^{2}\\right)}$\n1. [Step 4](#code_validation): **Code validation**\n1. [Step 5](#latex_pdf_output): **Output this notebook to $\\LaTeX$-formatted PDF file**\n\n\n\n# Step 0: Source directory creation \\[Back to [top](#toc)\\]\n$$\\label{src_dir}$$\n\nWe will now use the [cmdline_helper.py NRPy+ module](Tutorial-Tutorial-cmdline_helper.ipynb) to create the source directory within the `IllinoisGRMHD` NRPy+ directory, if it does not exist yet.\n\n\n```python\n# Step 0: Creation of the IllinoisGRMHD source directory\n# Step 0a: Add NRPy's directory to the path\n# https://stackoverflow.com/questions/16780014/import-file-from-parent-directory\nimport os,sys\nnrpy_dir_path = os.path.join(\"..\",\"..\")\nif nrpy_dir_path not in sys.path:\n sys.path.append(nrpy_dir_path)\n\n# Step 0b: Load up cmdline_helper and create the directory\nimport cmdline_helper as cmd\nIGM_src_dir_path = os.path.join(\"..\",\"src\")\ncmd.mkdir(IGM_src_dir_path)\n\n# Step 0c: Create the output file path \noutfile_path__harm_utoprim_2d__c = os.path.join(IGM_src_dir_path,\"harm_utoprim_2d.c\")\n```\n\n\n\n# Step 1: Introduction \\[Back to [top](#toc)\\]\n$$\\label{introduction}$$\n\nComment on license: `HARM` uses GPL, while `IllinoisGRMHD` uses BSD.\n\n\n\n# Step 2: EOS independent routines \\[Back to [top](#toc)\\]\n$$\\label{harm_utoprim_2d__c__eos_indep}$$\n\nLet us now start documenting the `harm_utoprim_2d.c`, which is a part of the `Harm` code. Our main reference throughout this discussion will be the required citation [Noble *et al.* (2006)](https://arxiv.org/abs/astro-ph/0512420).\n\nWe will start with the code's required preamble.\n\n\n```python\n%%writefile $outfile_path__harm_utoprim_2d__c\n#ifndef __HARM_UTOPRIM_2D__C__\n#define __HARM_UTOPRIM_2D__C__\n/***********************************************************************************\n Copyright 2006 Charles F. Gammie, Jonathan C. McKinney, Scott C. Noble, \n Gabor Toth, and Luca Del Zanna\n\n HARM version 1.0 (released May 1, 2006)\n\n This file is part of HARM. HARM is a program that solves hyperbolic \n partial differential equations in conservative form using high-resolution\n shock-capturing techniques. This version of HARM has been configured to \n solve the relativistic magnetohydrodynamic equations of motion on a \n stationary black hole spacetime in Kerr-Schild coordinates to evolve\n an accretion disk model. \n\n You are morally obligated to cite the following two papers in his/her \n scientific literature that results from use of any part of HARM:\n\n [1] Gammie, C. F., McKinney, J. C., \\& Toth, G.\\ 2003, \n Astrophysical Journal, 589, 444.\n\n [2] Noble, S. C., Gammie, C. F., McKinney, J. C., \\& Del Zanna, L. \\ 2006, \n Astrophysical Journal, 641, 626.\n\n \n Further, we strongly encourage you to obtain the latest version of \n HARM directly from our distribution website:\n http://rainman.astro.uiuc.edu/codelib/\n\n\n HARM is free software; you can redistribute it and/or modify\n it under the terms of the GNU General Public License as published by\n the Free Software Foundation; either version 2 of the License, or\n (at your option) any later version.\n\n HARM is distributed in the hope that it will be useful,\n but WITHOUT ANY WARRANTY; without even the implied warranty of\n MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the\n GNU General Public License for more details.\n\n You should have received a copy of the GNU General Public License\n along with HARM; if not, write to the Free Software\n Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA\n\n***********************************************************************************/\n\n/*************************************************************************************/\n/*************************************************************************************/\n/*************************************************************************************\n\nutoprim_2d.c: \n---------------\n\n Uses the 2D method: \n -- solves for two independent variables (W,v^2) via a 2D\n Newton-Raphson method \n -- can be used (in principle) with a general equation of state. \n\n -- Currently returns with an error state (>0) if a negative rest-mass\n density or internal energy density is calculated. You may want \n to change this aspect of the code so that it still calculates the \n velocity and so that you can floor the densities. If you want to \n change this aspect of the code please comment out the \"return(retval)\"\n statement after \"retval = 5;\" statement in Utoprim_new_body();\n\n******************************************************************************/\n\nstatic const int NEWT_DIM=2;\n\n// Declarations: \nstatic CCTK_REAL vsq_calc(CCTK_REAL W,CCTK_REAL &Bsq,CCTK_REAL &QdotBsq,CCTK_REAL &Qtsq,CCTK_REAL &Qdotn,CCTK_REAL &D);\nstatic int Utoprim_new_body(eos_struct eos, CCTK_REAL U[], CCTK_REAL gcov[NDIM][NDIM], CCTK_REAL gcon[NDIM][NDIM], CCTK_REAL gdet, CCTK_REAL prim[],long &n_iter);\nstatic int general_newton_raphson( eos_struct eos, CCTK_REAL x[], int n, long &n_iter, void (*funcd) (eos_struct, CCTK_REAL [], CCTK_REAL [], CCTK_REAL [], CCTK_REAL [][NEWT_DIM], CCTK_REAL *, CCTK_REAL *, int,CCTK_REAL &,CCTK_REAL &,CCTK_REAL &,CCTK_REAL &,CCTK_REAL &),CCTK_REAL &Bsq,CCTK_REAL &QdotBsq,CCTK_REAL &Qtsq,CCTK_REAL &Qdotn,CCTK_REAL &D);\nstatic void func_vsq( eos_struct eos, CCTK_REAL [], CCTK_REAL [], CCTK_REAL [], CCTK_REAL [][NEWT_DIM], CCTK_REAL *f, CCTK_REAL *df, int n,CCTK_REAL &Bsq,CCTK_REAL &QdotBsq,CCTK_REAL &Qtsq,CCTK_REAL &Qdotn,CCTK_REAL &D);\nstatic CCTK_REAL x1_of_x0(CCTK_REAL x0, CCTK_REAL &Bsq, CCTK_REAL &QdotBsq, CCTK_REAL &Qtsq, CCTK_REAL &Qdotn, CCTK_REAL &D ) ;\nstatic CCTK_REAL pressure_W_vsq(eos_struct eos, CCTK_REAL W, CCTK_REAL vsq, CCTK_REAL &D) ;\nstatic CCTK_REAL dpdW_calc_vsq(CCTK_REAL W, CCTK_REAL vsq);\nstatic CCTK_REAL dpdvsq_calc(eos_struct eos, CCTK_REAL W, CCTK_REAL vsq, CCTK_REAL &D);\n\n/**********************************************************************/\n/******************************************************************\n\n Utoprim_2d():\n \n -- Driver for new prim. var. solver. The driver just translates\n between the two sets of definitions for U and P. The user may \n wish to alter the translation as they see fit. Note that Greek\n indices run 0,1,2,3 and Latin indices run 1,2,3 (spatial only).\n\n\n / rho u^t \\\n U = | T^t_t + rho u^t | sqrt(-det(g_{\\mu\\nu}))\n | T^t_i |\n \\ B^i /\n\n / rho \\\n P = | uu |\n | \\tilde{u}^i |\n \\ B^i /\n\n\n Arguments:\n U[NPR] = conserved variables (current values on input/output);\n gcov[NDIM][NDIM] = covariant form of the metric ;\n gcon[NDIM][NDIM] = contravariant form of the metric ;\n gdet = sqrt( - determinant of the metric) ;\n prim[NPR] = primitive variables (guess on input, calculated values on\n output if there are no problems);\n \n -- NOTE: for those using this routine for special relativistic MHD and are\n unfamiliar with metrics, merely set \n gcov = gcon = diag(-1,1,1,1) and gdet = 1. ;\n\n******************************************************************/\n```\n\n Writing ../src/harm_utoprim_2d.c\n\n\n\n\n## Step 2.a: The `Utoprim_2d()` function \\[Back to [top](#toc)\\]\n$$\\label{utoprim_2d}$$\n\nThe `Utoprim_2d()` function is the driver function of the `HARM` conservative-to-primitive algorithm. We remind you from the definitions of primitive and conservative variables used in the code:\n\n$$\n\\begin{align}\n\\boldsymbol{P}_{\\rm HARM} &= \\left\\{\\rho_{b},u,\\tilde{u}^{i},B^{i}_{\\rm HARM}\\right\\}\\ ,\\\\\n\\boldsymbol{C}_{\\rm HARM} &= \\left\\{\\sqrt{-g}\\rho_{b}u^{0},\\sqrt{-g}\\left(T^{0}_{\\ 0}+\\rho_{b}u^{0}\\right),\\sqrt{-g}T^{0}_{\\ i},\\sqrt{-g}B^{i}_{\\rm HARM}\\right\\}\\ .\n\\end{align}\n$$\n\n\n\n### Step 2.a.i: Setting $B^{i}_{\\rm HARM}$ and $\\alpha$ \\[Back to [top](#toc)\\]\n$$\\label{utoprim_2d__bi_and_alpha}$$\n\nLet\n\n$$\n\\tilde{B}^{i}_{\\rm HARM} \\equiv \\sqrt{-g}B^{i}_{\\rm HARM}\\ .\n$$\n\nThe code starts by relating\n\n$$\n\\boxed{B^{i}_{\\rm HARM} = \\frac{\\tilde{B}^{i}_{\\rm HARM}}{\\sqrt{-g}}}\\ ,\n$$\n\nand setting\n\n$$\n\\boxed{\\alpha = \\frac{1}{\\sqrt{-g^{00}}}} \\ .\n$$\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n\nint Utoprim_2d(eos_struct eos, CCTK_REAL U[NPR], CCTK_REAL gcov[NDIM][NDIM], CCTK_REAL gcon[NDIM][NDIM], \n CCTK_REAL gdet, CCTK_REAL prim[NPR], long &n_iter)\n{\n\n CCTK_REAL U_tmp[NPR], prim_tmp[NPR];\n int i, ret; \n CCTK_REAL alpha;\n\n if( U[0] <= 0. ) { \n return(-100);\n }\n\n /* First update the primitive B-fields */\n for(i = BCON1; i <= BCON3; i++) prim[i] = U[i] / gdet ;\n\n /* Set the geometry variables: */\n alpha = 1.0/sqrt(-gcon[0][0]);\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n### Step 2.a.ii: Preparing the variables to be used by the `Utoprim_new_body()` function \\[Back to [top](#toc)\\]\n$$\\label{utoprim_2d__converting}$$\n\nThe conservative-to-primitive algorithm uses the `Utoprim_new_body()` function. However, this function assumes a *different* set of primitive/conservative variables. Thus, we must perform the proper conversion. First, let us ease on the notation a bit by defining:\n\n$$\n\\boldsymbol{C} \\equiv \\left\\{\\rho_{\\star},u_{\\star},\\tilde{S}_{i},\\tilde{B}^{i}_{\\rm HARM}\\right\\} \\equiv \\left\\{\\sqrt{-g}\\rho_{b}u^{0},\\sqrt{-g}\\left(T^{0}_{\\ 0}+\\rho_{b}u^{0}\\right),\\sqrt{-g}T^{0}_{\\ i},\\sqrt{-g}B^{i}_{\\rm HARM}\\right\\}\\ .\n$$\n\n\n\nBelow we list the main differences in the conservative variables:\n\n| `Utoprim_2d()` | `Utoprim_new_body()` |\n|------------------------------------------|---------------------------------------------------------------------------|\n| $\\color{blue}{\\textbf{Conservatives}}$ | $\\color{red}{\\textbf{Conservatives}}$ |\n| $\\color{blue}{\\rho_{\\star}}$ | $\\color{red}{\\frac{\\alpha}{\\sqrt{-g}}\\rho_{\\star}}$ |\n| $\\color{blue}{u_{\\star}}$ | $\\color{red}{\\frac{\\alpha}{\\sqrt{-g}}\\left(u_{\\star}-\\rho_{\\star}\\right)}$|\n| $\\color{blue}{\\tilde{S}_{i}}$ | $\\color{red}{\\frac{\\alpha}{\\sqrt{-g}}\\tilde{S}_{i}}$ |\n| $\\color{blue}{\\tilde{B}^{i}_{\\rm HARM}}$ | $\\color{red}{\\frac{\\alpha}{\\sqrt{-g}}\\tilde{B}^{i}_{\\rm HARM}}$ |\n\nThese are necessary conversions because while `Utoprim_2d()` assumes the set of conservatives above, `Utoprim_new_body()` assumes\n\n$$\n\\left\\{\\gamma\\rho_{b},\\alpha T^{0}_{\\ \\ 0}, \\alpha T^{0}_{\\ \\ i}, \\alpha B^{i}_{\\rm HARM}\\right\\}\\ .\n$$\n\nLet us first pause to understand the table above. From definition (15) in [Noble *et al.* (2006)](https://arxiv.org/abs/astro-ph/0512420) and the discussion just below it, we know that $\\gamma = \\alpha u^{0}$. Thus\n\n$$\n\\rho_{\\star} = \\sqrt{-g}\\rho_{b}u^{0} = \\sqrt{-g}\\left(\\frac{\\gamma}{\\alpha}\\rho_{b}\\right)\\implies\\boxed{\\gamma \\rho_{b} = \\frac{\\alpha}{\\sqrt{-g}}\\rho_{\\star}}\\ .\n$$\n\nThen we have\n\n$$\nu_{\\star} = \\sqrt{-g}\\left(T^{0}_{\\ \\ 0} + \\rho_{b}u^{0}\\right)= \\sqrt{-g}\\left(T^{0}_{\\ \\ 0} + \\frac{\\rho_{\\star}}{\\sqrt{-g}}\\right) = \\sqrt{-g}T^{0}_{\\ \\ 0} + \\rho_{\\star} \\implies \\boxed{\\alpha T^{0}_{\\ \\ 0} = \\frac{\\alpha}{\\sqrt{-g}}\\left(u_{\\star}-\\rho_{\\star}\\right)}\\ .\n$$\n\nThe other two relations are more straightforward. We have\n\n$$\n\\tilde{S}_{i} = \\sqrt{-g}T^{0}_{\\ \\ i} \\implies \\boxed{\\alpha T^{0}_{\\ \\ i} = \\frac{\\alpha}{\\sqrt{-g}}\\tilde{S}_{i}}\\ ,\n$$\n\nand\n\n$$\n\\tilde{B}^{i}_{\\rm HARM} = \\sqrt{-g}B^{i}_{\\rm HARM}\\implies \\boxed{\\alpha B^{i}_{\\rm HARM} = \\frac{\\alpha}{\\sqrt{-g}}\\tilde{B}^{i}_{\\rm HARM}}\\ .\n$$\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n \n /* Transform the CONSERVED variables into the new system */\n U_tmp[RHO] = alpha * U[RHO] / gdet;\n U_tmp[UU] = alpha * (U[UU] - U[RHO]) / gdet ;\n for( i = UTCON1; i <= UTCON3; i++ ) {\n U_tmp[i] = alpha * U[i] / gdet ;\n }\n for( i = BCON1; i <= BCON3; i++ ) {\n U_tmp[i] = alpha * U[i] / gdet ;\n }\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\nBelow we list the necessary transformations on the primitive variables:\n\n| `Utoprim_2d()` | `Utoprim_new_body()` |\n|-------------------------------------|----------------------------------------|\n| $\\color{blue}{\\textbf{Primitives}}$ | $\\color{red}{\\textbf{Primitives}}$ |\n| $\\color{blue}{\\rho_{b}}$ | $\\color{red}{\\rho_{b}}$ |\n| $\\color{blue}{u}$ | $\\color{red}{u}$ |\n| $\\color{blue}{\\tilde{u}^{i}}$ | $\\color{red}{\\tilde{u}^{i}}$ |\n| $\\color{blue}{B^{i}_{\\rm HARM}}$ | $\\color{red}{\\alpha B^{i}_{\\rm HARM}}$ |\n\nAfter this slight modification we call the `Utoprim_new_body()` function. If it returns without errors, than the variables ${\\rm prim\\_tmp}$ will now contain the values of the primitives. We then update the ${\\rm prim}$ variables with these newly computed values.\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n\n /* Transform the PRIMITIVE variables into the new system */\n for( i = 0; i < BCON1; i++ ) {\n prim_tmp[i] = prim[i];\n }\n for( i = BCON1; i <= BCON3; i++ ) {\n prim_tmp[i] = alpha*prim[i];\n }\n\n ret = Utoprim_new_body(eos, U_tmp, gcov, gcon, gdet, prim_tmp,n_iter);\n\n /* Transform new primitive variables back if there was no problem : */ \n if( ret == 0 || ret == 5 || ret==101 ) {\n for( i = 0; i < BCON1; i++ ) {\n prim[i] = prim_tmp[i];\n }\n }\n\n return( ret ) ;\n\n}\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n## Step 2.b: The `Utoprim_new_body()` function \\[Back to [top](#toc)\\]\n$$\\label{utoprim_new_body}$$\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n\n\n/**********************************************************************/\n/**********************************************************************************\n\n Utoprim_new_body():\n\n -- Attempt an inversion from U to prim using the initial guess prim.\n\n -- This is the main routine that calculates auxiliary quantities for the \n Newton-Raphson routine. \n\n -- assumes that \n / rho gamma \\\n U = | alpha T^t_\\mu |\n \\ alpha B^i /\n\n\n\n / rho \\\n prim = | uu |\n | \\tilde{u}^i |\n \\ alpha B^i /\n\n\nreturn: (i*100 + j) where \n i = 0 -> Newton-Raphson solver either was not called (yet or not used) \n or returned successfully;\n 1 -> Newton-Raphson solver did not converge to a solution with the \n given tolerances;\n 2 -> Newton-Raphson procedure encountered a numerical divergence \n (occurrence of \"nan\" or \"+/-inf\" ;\n \n j = 0 -> success \n 1 -> failure: some sort of failure in Newton-Raphson; \n 2 -> failure: utsq<0 w/ initial p[] guess;\n 3 -> failure: W<0 or W>W_TOO_BIG\n 4 -> failure: v^2 > 1 \n 5 -> failure: rho,uu <= 0 ;\n\n**********************************************************************************/\n\nstatic int Utoprim_new_body(eos_struct eos, CCTK_REAL U[NPR], CCTK_REAL gcov[NDIM][NDIM], \n CCTK_REAL gcon[NDIM][NDIM], CCTK_REAL gdet, CCTK_REAL prim[NPR], long &n_iter)\n{\n\n CCTK_REAL x_2d[NEWT_DIM];\n CCTK_REAL QdotB,Bcon[NDIM],Bcov[NDIM],Qcov[NDIM],Qcon[NDIM],ncov[NDIM],ncon[NDIM],Qsq,Qtcon[NDIM];\n CCTK_REAL rho0,u,p,w,gammasq,gamma,gtmp,W_last,W,utsq,vsq;\n int i,j, n, retval, i_increase;\n\n n = NEWT_DIM ;\n\n // Assume ok initially:\n retval = 0;\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n## Step 2.b.i: Computing basic quantities \\[Back to [top](#toc)\\]\n$$\\label{utoprim_new_body__basic_quantities}$$\n\nWe start by computing basic quantities from the input variables. Notice that this conservative-to-primitive algorithm does not need to update the magnetic field, thus\n\n$$\n\\boxed{B_{\\rm prim}^{i} = B_{\\rm conserv}^{i}}\\ .\n$$\n\nSince they are both equal, we will not distinguish between prim and conserv in what follows. We also set $B^{0} = 0$. Then we define\n\n$$\n\\boxed{Q_{\\mu} \\equiv \\alpha T^{0}_{\\ \\ \\mu}}\\ .\n$$\n\nFrom these, the following quantities are then computed:\n\n$$\n\\boxed{\n\\begin{align}\nB_{i} &= g_{i\\mu}B^{\\mu}\\\\\nQ^{\\mu} &= g^{\\mu\\nu}Q_{\\nu}\\\\\nB^{2} &= B_{i}B^{i}\\\\\nQ\\cdot B &= Q_{\\mu}B^{\\mu}\\\\\n\\left(Q\\cdot B\\right)^{2} &= \\left(Q\\cdot B\\right)\\left(Q\\cdot B\\right)\\\\\nn_{\\mu} &= \\left(-\\alpha,0,0,0\\right)\\\\\nn^{\\mu} &= g^{\\mu\\nu}n_{\\nu}\\\\\n\\left(Q\\cdot n\\right) &= Q^{\\mu}n_{\\mu}\\\\\nQ^{2} &= Q_{\\mu}Q^{\\mu}\\\\\n\\tilde{Q}^{2} &= Q^{2} + \\left(Q\\cdot n\\right)\\left(Q\\cdot n\\right)\\\\\nD &\\equiv \\gamma \\rho_{b}\n\\end{align}\n}\\ .\n$$\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n\n for(i = BCON1; i <= BCON3; i++) prim[i] = U[i] ;\n\n // Calculate various scalars (Q.B, Q^2, etc) from the conserved variables:\n Bcon[0] = 0. ;\n for(i=1;i<4;i++) Bcon[i] = U[BCON1+i-1] ;\n\n lower_g(Bcon,gcov,Bcov) ;\n\n for(i=0;i<4;i++) Qcov[i] = U[QCOV0+i] ;\n raise_g(Qcov,gcon,Qcon) ;\n\n\n CCTK_REAL Bsq = 0. ;\n for(i=1;i<4;i++) Bsq += Bcon[i]*Bcov[i] ;\n\n QdotB = 0. ;\n for(i=0;i<4;i++) QdotB += Qcov[i]*Bcon[i] ;\n CCTK_REAL QdotBsq = QdotB*QdotB ;\n\n ncov_calc(gcon,ncov) ;\n // FIXME: The exact form of n^{\\mu} can be found\n // in eq. (2.116) and implementing it\n // directly is a lot more efficient than\n // performing n^{\\mu} = g^{\\mu\\nu}n_{nu}\n raise_g(ncov,gcon,ncon);\n\n CCTK_REAL Qdotn = Qcon[0]*ncov[0] ;\n\n Qsq = 0. ;\n for(i=0;i<4;i++) Qsq += Qcov[i]*Qcon[i] ;\n\n CCTK_REAL Qtsq = Qsq + Qdotn*Qdotn ;\n\n CCTK_REAL D = U[RHO] ;\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n## Step 2.b.ii: Determining $W$ from the previous iteration, $W_{\\rm last}$ \\[Back to [top](#toc)\\]\n$$\\label{utoprim_new_body__wlast}$$\n\nThe quantity $W$ is defined as\n\n$$\nW \\equiv w\\gamma^{2}\\ ,\n$$\n\nwhere\n\n$$\n\\begin{align}\nw &= \\rho_{b} + u + p\\ ,\\\\\n\\gamma^{2} &= 1 + g_{ij}\\tilde{u}^{i}\\tilde{u}^{j}\\ .\n\\end{align}\n$$\n\nThus the quantities $g_{ij}\\tilde{u}^{i}\\tilde{u}^{j}$ and then $\\gamma^{2}$ and $\\gamma$. Thus, by computing $\\rho_{b}$ and $p$ from the input variables, i.e. $D$, one can determine $w$ and then compute the value of $W$ from the input values (previous iteration), which we denote by $W_{\\rm last}$.\n\n**Dependecy note:** Note that this function depends on the `pressure_rho0_u()` function, which is *not* EOS independent.\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n\n /* calculate W from last timestep and use for guess */\n utsq = 0. ;\n for(i=1;i<4;i++)\n for(j=1;j<4;j++) utsq += gcov[i][j]*prim[UTCON1+i-1]*prim[UTCON1+j-1] ;\n\n\n if( (utsq < 0.) && (fabs(utsq) < 1.0e-13) ) { \n utsq = fabs(utsq);\n }\n if(utsq < 0. || utsq > UTSQ_TOO_BIG) {\n retval = 2;\n return(retval) ;\n }\n\n gammasq = 1. + utsq ;\n gamma = sqrt(gammasq);\n \n // Always calculate rho from D and gamma so that using D in EOS remains consistent\n // i.e. you don't get positive values for dP/d(vsq) . \n rho0 = D / gamma ;\n u = prim[UU] ;\n p = pressure_rho0_u(eos, rho0,u) ;\n w = rho0 + u + p ;\n\n W_last = w*gammasq ;\n\n\n // Make sure that W is large enough so that v^2 < 1 : \n i_increase = 0;\n while( (( W_last*W_last*W_last * ( W_last + 2.*Bsq ) \n - QdotBsq*(2.*W_last + Bsq) ) <= W_last*W_last*(Qtsq-Bsq*Bsq))\n && (i_increase < 10) ) {\n W_last *= 10.;\n i_increase++;\n }\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n## Step 2.b.iii: Compute $v^{2}_{\\rm last}$, then update $v^{2}$ and $W$ \\[Back to [top](#toc)\\]\n$$\\label{utoprim_new_body__vsqlast_and_recompute_w_and_vsq}$$\n\nThen we use equation (28) in [Noble *et al.* (2006)](https://arxiv.org/abs/astro-ph/0512420) to determine $v^{2}$:\n\n$$\n\\boxed{v^{2} = \\frac{\\tilde{Q}^{2}W^{2} + \\left(Q\\cdot B\\right)^{2}\\left(B^{2}+2W\\right)}{\\left(B^{2}+W\\right)^{2}W^{2}}}\\ .\n$$\n\nThis is done by calling the `x1_of_x0()` function, where $x_{0} = W$ and $x_{1} = v^{2}$, which itself calls the `vsq_calc()` function which implements the boxed equation above.\n\nAfter we have $\\left\\{W_{\\rm last},v^{2}_{\\rm last}\\right\\}$ we use them as the initial guess for the `general_newton_raphson()`, which returns the updated values $\\left\\{W,v^{2}\\right\\}$.\n\nAll functions mentioned above are documented in this tutorial notebook, so look at the [Table of Contents](#toc) for more information.\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n \n // Calculate W and vsq: \n x_2d[0] = fabs( W_last );\n x_2d[1] = x1_of_x0( W_last , Bsq,QdotBsq,Qtsq,Qdotn,D) ;\n retval = general_newton_raphson( eos, x_2d, n, n_iter, func_vsq, Bsq,QdotBsq,Qtsq,Qdotn,D) ; \n\n W = x_2d[0];\n vsq = x_2d[1];\n \n /* Problem with solver, so return denoting error before doing anything further */\n if( (retval != 0) || (W == FAIL_VAL) ) {\n retval = retval*100+1;\n return(retval);\n }\n else{\n if(W <= 0. || W > W_TOO_BIG) {\n retval = 3;\n return(retval) ;\n }\n }\n\n // Calculate v^2:\n if( vsq >= 1. ) {\n vsq = 1.-2.e-16;\n //retval = 4;\n //return(retval) ;\n }\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n## Step 2.b.iv: Computing the primitive variables \\[Back to [top](#toc)\\]\n$$\\label{utoprim_new_body__compute_prims}$$\n\nNow that we have $\\left\\{W,v^{2}\\right\\}$, we recompute the primitive variables. We start with\n\n$$\n\\left\\{\n\\begin{align}\n\\tilde{g} &\\equiv \\sqrt{1-v^{2}}\\\\\n\\gamma &= \\frac{1}{\\tilde{g}}\n\\end{align}\n\\right.\n\\implies\n\\boxed{\\rho_{b} = D\\tilde{g}}\\ .\n$$\n\nThen, we determine the pressure $p$ using the `pressure_rho0_w()` function and\n\n$$\nw = W\\left(1-v^{2}\\right)\n\\implies\n\\boxed{u = w - \\left(\\rho_{b} + p\\right)}\\ .\n$$\n\n**Dependecy note:** Note that this function depends on the `pressure_rho0_w()` function, which is *not* EOS independent.\n\nFinally, we can obtain $\\tilde{u}^{i}$ using eq. 31 in [Noble *et al.* (2006)](https://arxiv.org/abs/astro-ph/0512420)\n\n$$\n\\boxed{\n\\tilde{u}^{i} = \\frac{\\gamma}{\\left(W+B^{2}\\right)}\\left[\\tilde{Q}^{i} + \\frac{\\left(Q\\cdot B\\right)}{W}B^{i}\\right]\n}\\ ,\n$$\n\nwhere\n\n$$\n\\tilde{Q}^{i} = Q^{i} + \\left(Q\\cdot n\\right)n^{i}\\ .\n$$\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n\n // Recover the primitive variables from the scalars and conserved variables:\n gtmp = sqrt(1. - vsq);\n gamma = 1./gtmp ;\n rho0 = D * gtmp;\n\n w = W * (1. - vsq) ;\n p = pressure_rho0_w(eos, rho0,w) ;\n u = w - (rho0 + p) ; // u = rho0 eps, w = rho0 h\n\n if( (rho0 <= 0.) || (u <= 0.) ) {\n // User may want to handle this case differently, e.g. do NOT return upon \n // a negative rho/u, calculate v^i so that rho/u can be floored by other routine:\n\n retval = 5;\n //return(retval) ;\n }\n\n /*\n if(retval==5 && fabs(u)<1e-16) {\n u = fabs(u);\n CCTK_VInfo(CCTK_THORNSTRING,\"%e\\t%e\\t%e\",1.0-w/(rho0 + p),rho0,p);\n retval=0;\n }\n */\n\n prim[RHO] = rho0 ;\n prim[UU] = u ;\n\n for(i=1;i<4;i++) Qtcon[i] = Qcon[i] + ncon[i] * Qdotn;\n for(i=1;i<4;i++) prim[UTCON1+i-1] = gamma/(W+Bsq) * ( Qtcon[i] + QdotB*Bcon[i]/W ) ;\n \n /* set field components */\n for(i = BCON1; i <= BCON3; i++) prim[i] = U[i] ;\n\n\n /* done! */\n return(retval) ;\n\n}\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n## Step 2.c: The `vsq_calc()` function \\[Back to [top](#toc)\\]\n$$\\label{vsq_calc}$$\n\nThis function implements eq. (28) in [Noble *et al.* (2006)](https://arxiv.org/abs/astro-ph/0512420) to determine $v^{2}$:\n\n$$\n\\boxed{v^{2} = \\frac{\\tilde{Q}^{2}W^{2} + \\left(Q\\cdot B\\right)^{2}\\left(B^{2}+2W\\right)}{\\left(B^{2}+W\\right)^{2}W^{2}}}\\ .\n$$\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n\n\n/**********************************************************************/ \n/****************************************************************************\n vsq_calc(): \n \n -- evaluate v^2 (spatial, normalized velocity) from \n W = \\gamma^2 w \n\n****************************************************************************/\nstatic CCTK_REAL vsq_calc(CCTK_REAL W,CCTK_REAL &Bsq,CCTK_REAL &QdotBsq,CCTK_REAL &Qtsq,CCTK_REAL &Qdotn,CCTK_REAL &D)\n{\n CCTK_REAL Wsq,Xsq;\n \n Wsq = W*W ;\n Xsq = (Bsq + W) * (Bsq + W);\n\n return( ( Wsq * Qtsq + QdotBsq * (Bsq + 2.*W)) / (Wsq*Xsq) );\n}\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n## Step 2.d: The `x1_of_x0()` function \\[Back to [top](#toc)\\]\n$$\\label{x1_of_x0}$$\n\nThis function computes $v^{2}$, as described [above](#vsq_calc), then performs physical checks on $v^{2}$ (i.e. whether or not it is superluminal). This function assumes $W$ is physical.\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n\n\n/********************************************************************\n\n x1_of_x0(): \n \n -- calculates v^2 from W with some physical bounds checking;\n -- asumes x0 is already physical\n -- makes v^2 physical if not;\n\n*********************************************************************/\n\nstatic CCTK_REAL x1_of_x0(CCTK_REAL x0, CCTK_REAL &Bsq, CCTK_REAL &QdotBsq, CCTK_REAL &Qtsq, CCTK_REAL &Qdotn, CCTK_REAL &D ) \n{\n CCTK_REAL vsq;\n CCTK_REAL dv = 1.e-15;\n \n vsq = fabs(vsq_calc(x0,Bsq,QdotBsq,Qtsq,Qdotn,D)) ; // guaranteed to be positive \n\n\n return( ( vsq > 1. ) ? (1.0 - dv) : vsq ); \n\n}\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n## Step 2.e: The `validate_x()` function \\[Back to [top](#toc)\\]\n$$\\label{validate_x}$$\n\nThis function performs physical tests on $\\left\\{W,v^{2}\\right\\}$ based on their definitions.\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n\n/********************************************************************\n\n validate_x(): \n \n -- makes sure that x[0,1] have physical values, based upon \n their definitions:\n \n*********************************************************************/\n\nstatic void validate_x(CCTK_REAL x[2], CCTK_REAL x0[2] ) \n{\n \n CCTK_REAL dv = 1.e-15;\n\n /* Always take the absolute value of x[0] and check to see if it's too big: */ \n x[0] = fabs(x[0]);\n x[0] = (x[0] > W_TOO_BIG) ? x0[0] : x[0];\n \n\n x[1] = (x[1] < 0.) ? 0. : x[1]; /* if it's too small */\n x[1] = (x[1] > 1.) ? (1. - dv) : x[1]; /* if it's too big */\n\n return;\n\n}\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n## Step 2.f: The `general_newton_raphson()` function \\[Back to [top](#toc)\\]\n$$\\label{general_newton_raphson}$$\n\nThis function implements a [multidimensional Newton-Raphson method](https://en.wikipedia.org/wiki/Newton%27s_method#k_variables,_k_functions). We will not make the effort of explaining the algorithm exhaustively since it is pretty standard, so we will settle for a summary of the method.\n\nGiven a system of $N$ non-linear of equations and $N$ variables, $\\left\\{\\vec{F}\\!\\left(\\vec{x}\\right),\\vec{x}\\right\\}$, the Newton-Raphson method attempts to determine the root vector, $\\vec{x}_{\\star}$, iteratively through\n\n$$\n\\begin{align}\n\\vec{x}_{n+1} = \\vec{x}_{n} - J^{-1}_{F}\\!\\left(\\vec{x}_{n}\\right)\\vec{F}\\!\\left(\\vec{x}\\right)\\ ,\n\\end{align}\n$$\n\nwhere $J^{-1}_{F}$ is the Jacobian matrix\n\n$$\n\\left(J_{F}\\right)^{i}_{\\ \\ j} = \\frac{\\partial F^{i}}{\\partial x^{j}}\\ .\n$$\n\nThe index $n$ above is an *iteration* index and $\\vec{x}_{n+1}$ represents an improved approximation to $\\vec{x}_{\\star}$ when compared to $\\vec{x}_{n}$.\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n\n/************************************************************\n\n general_newton_raphson(): \n\n -- performs Newton-Rapshon method on an arbitrary system.\n\n -- inspired in part by Num. Rec.'s routine newt();\n\n*****************************************************************/\nstatic int general_newton_raphson( eos_struct eos, CCTK_REAL x[], int n, long &n_iter,\n void (*funcd) (eos_struct, CCTK_REAL [], CCTK_REAL [], CCTK_REAL [], \n CCTK_REAL [][NEWT_DIM], CCTK_REAL *, \n CCTK_REAL *, int,CCTK_REAL &,CCTK_REAL &,CCTK_REAL &,CCTK_REAL &,CCTK_REAL &),CCTK_REAL &Bsq,CCTK_REAL &QdotBsq,CCTK_REAL &Qtsq,CCTK_REAL &Qdotn,CCTK_REAL &D)\n{\n CCTK_REAL f, df, dx[NEWT_DIM], x_old[NEWT_DIM];\n CCTK_REAL resid[NEWT_DIM], jac[NEWT_DIM][NEWT_DIM];\n CCTK_REAL errx, x_orig[NEWT_DIM];\n int id, i_extra, doing_extra;\n\n int keep_iterating;\n\n\n // Initialize various parameters and variables:\n errx = 1. ; \n df = f = 1.;\n i_extra = doing_extra = 0;\n for( id = 0; id < n ; id++) x_old[id] = x_orig[id] = x[id] ;\n\n n_iter = 0;\n\n /* Start the Newton-Raphson iterations : */\n keep_iterating = 1;\n while( keep_iterating ) { \n\n (*funcd) (eos, x, dx, resid, jac, &f, &df, n, Bsq,QdotBsq,Qtsq,Qdotn,D); /* returns with new dx, f, df */\n \n\n /* Save old values before calculating the new: */\n errx = 0.;\n for( id = 0; id < n ; id++) {\n x_old[id] = x[id] ;\n }\n\n /* Make the newton step: */\n for( id = 0; id < n ; id++) {\n x[id] += dx[id] ;\n }\n\n /****************************************/\n /* Calculate the convergence criterion */\n /****************************************/\n errx = (x[0]==0.) ? fabs(dx[0]) : fabs(dx[0]/x[0]);\n\n\n /****************************************/\n /* Make sure that the new x[] is physical : */\n /****************************************/\n validate_x( x, x_old ) ;\n\n\n /*****************************************************************************/\n /* If we've reached the tolerance level, then just do a few extra iterations */\n /* before stopping */\n /*****************************************************************************/\n \n if( (fabs(errx) <= NEWT_TOL) && (doing_extra == 0) && (EXTRA_NEWT_ITER > 0) ) {\n doing_extra = 1;\n }\n\n if( doing_extra == 1 ) i_extra++ ;\n\n if( ((fabs(errx) <= NEWT_TOL)&&(doing_extra == 0)) \n || (i_extra > EXTRA_NEWT_ITER) || (n_iter >= (MAX_NEWT_ITER-1)) ) {\n keep_iterating = 0;\n }\n\n n_iter++;\n\n } // END of while(keep_iterating)\n\n /* Check for bad untrapped divergences : */\n if( (finite(f)==0) || (finite(df)==0) ) {\n return(2);\n }\n\n\n if( fabs(errx) > MIN_NEWT_TOL){\n //CCTK_VInfo(CCTK_THORNSTRING,\"%d %e %e %e %e\",n_iter,f,df,errx,MIN_NEWT_TOL);\n return(1);\n } \n if( (fabs(errx) <= MIN_NEWT_TOL) && (fabs(errx) > NEWT_TOL) ){\n return(0);\n }\n if( fabs(errx) <= NEWT_TOL ){\n return(0);\n }\n\n return(0);\n\n}\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n## Step 2.g: The `func_vsq()` function \\[Back to [top](#toc)\\]\n$$\\label{func_vsq}$$\n\nThis function is used by the `general_newton_raphson()` function to compute the residuals and stepping. We will again not describe it in great detail since the method itself is relatively straightforward.\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n\n\n\n/**********************************************************************/\n/*********************************************************************************\n func_vsq(): \n\n -- calculates the residuals, and Newton step for general_newton_raphson();\n -- for this method, x=W,vsq here;\n\n Arguments:\n x = current value of independent var's (on input & output);\n dx = Newton-Raphson step (on output);\n resid = residuals based on x (on output);\n jac = Jacobian matrix based on x (on output);\n f = resid.resid/2 (on output)\n df = -2*f; (on output)\n n = dimension of x[];\n*********************************************************************************/\n\nstatic void func_vsq(eos_struct eos, CCTK_REAL x[], CCTK_REAL dx[], CCTK_REAL resid[], \n CCTK_REAL jac[][NEWT_DIM], CCTK_REAL *f, CCTK_REAL *df, int n,\n CCTK_REAL &Bsq,CCTK_REAL &QdotBsq,CCTK_REAL &Qtsq,CCTK_REAL &Qdotn,CCTK_REAL &D)\n{\n\n \n CCTK_REAL W, vsq, Wsq, p_tmp, dPdvsq, dPdW;\n CCTK_REAL t11;\n CCTK_REAL t16;\n CCTK_REAL t18;\n CCTK_REAL t2;\n CCTK_REAL t21;\n CCTK_REAL t23;\n CCTK_REAL t24;\n CCTK_REAL t25;\n CCTK_REAL t3;\n CCTK_REAL t35;\n CCTK_REAL t36;\n CCTK_REAL t4;\n CCTK_REAL t40;\n CCTK_REAL t9;\n\n // vv TESTING vv\n // CCTK_REAL D,gtmp,gamma,rho0,w,p,u;\n // ^^ TESTING ^^\n\n W = x[0];\n vsq = x[1];\n \n Wsq = W*W;\n\n // vv TESTING vv\n /*\n D = U[RHO] ;\n gtmp = sqrt(1. - vsq);\n gamma = 1./gtmp ;\n rho0 = D * gtmp;\n\n w = W * (1. - vsq) ;\n p = pressure_rho0_w(eos, rho0,w) ;\n u = w - (rho0 + p) ;\n\n if(u<=0 && 1==1) {\n vsq = 0.9999999 * (1.0-(rho0+p)/W);\n\n w = W * (1. - vsq) ;\n p = pressure_rho0_w(eos, rho0,w) ;\n u = w - (rho0 + p) ;\n\n //CCTK_VInfo(CCTK_THORNSTRING,\"%e check\",u);\n }\n */\n // ^^ TESTING ^^\n\n \n p_tmp = pressure_W_vsq( eos, W, vsq , D);\n dPdW = dpdW_calc_vsq( W, vsq );\n dPdvsq = dpdvsq_calc( eos, W, vsq, D );\n\n // These expressions were calculated using Mathematica, but made into efficient \n // code using Maple. Since we know the analytic form of the equations, we can \n // explicitly calculate the Newton-Raphson step: \n\n t2 = -0.5*Bsq+dPdvsq;\n t3 = Bsq+W;\n t4 = t3*t3;\n t9 = 1/Wsq;\n t11 = Qtsq-vsq*t4+QdotBsq*(Bsq+2.0*W)*t9;\n t16 = QdotBsq*t9;\n t18 = -Qdotn-0.5*Bsq*(1.0+vsq)+0.5*t16-W+p_tmp;\n t21 = 1/t3;\n t23 = 1/W;\n t24 = t16*t23;\n t25 = -1.0+dPdW-t24;\n t35 = t25*t3+(Bsq-2.0*dPdvsq)*(QdotBsq+vsq*Wsq*W)*t9*t23;\n t36 = 1/t35;\n dx[0] = -(t2*t11+t4*t18)*t21*t36;\n t40 = (vsq+t24)*t3;\n dx[1] = -(-t25*t11-2.0*t40*t18)*t21*t36;\n //detJ = t3*t35; // <- set but not used...\n jac[0][0] = -2.0*t40;\n jac[0][1] = -t4;\n jac[1][0] = t25;\n jac[1][1] = t2;\n resid[0] = t11;\n resid[1] = t18;\n\n \n\n *df = -resid[0]*resid[0] - resid[1]*resid[1];\n\n *f = -0.5 * ( *df );\n\n}\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n# Step 3: EOS dependent routines \\[Back to [top](#toc)\\]\n$$\\label{harm_utoprim_2d__c__eos_dep}$$\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n\n\n/********************************************************************** \n ********************************************************************** \n \n The following routines specify the equation of state. All routines \n above here should be indpendent of EOS. If the user wishes \n to use another equation of state, the below functions must be replaced \n by equivalent routines based upon the new EOS. \n\n **********************************************************************\n **********************************************************************/\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n## Step 3.a: The `pressure_W_vsq()` function \\[Back to [top](#toc)\\]\n$$\\label{pressure_w_vsq}$$\n\nThis function computes $p\\left(W,v^{2}\\right)$. For a $\\Gamma$-law equation of state,\n\n$$\np_{\\Gamma} = \\left(\\Gamma-1\\right)u\\ ,\n$$\n\nand with the definitions\n\n$$\n\\begin{align}\n\\gamma^{2} &= \\frac{1}{1-v^{2}}\\ ,\\\\\nW &= \\gamma^{2}w\\ ,\\\\\nD &= \\gamma\\rho_{b}\\ ,\\\\\nw &= \\rho_{b} + u + p\\ ,\n\\end{align}\n$$\n\nwe have\n\n$$\n\\begin{align}\np_{\\Gamma} &= \\left(\\Gamma-1\\right)u\\\\\n &= \\left(\\Gamma-1\\right)\\left(w - \\rho_{b} - p_{\\Gamma}\\right)\\\\\n &= \\left(\\Gamma-1\\right)\\left(\\frac{W}{\\gamma^{2}} - \\frac{D}{\\gamma}\\right) - \\left(\\Gamma-1\\right)p_{\\Gamma}\\\\\n\\implies\n&\\boxed{\np_{\\Gamma} = \\frac{\\left(\\Gamma-1\\right)}{\\Gamma}\\left(\\frac{W}{\\gamma^{2}} - \\frac{D}{\\gamma}\\right)\n}\\ .\n\\end{align}\n$$\n\nThus, the pre-PPEOS Patch version of this function was\n\n```c\n/**********************************************************************/\n/********************************************************************** \n pressure_W_vsq(): \n \n -- Gamma-law equation of state;\n -- pressure as a function of W, vsq, and D:\n**********************************************************************/\nstatic CCTK_REAL pressure_W_vsq(CCTK_REAL W, CCTK_REAL vsq, CCTK_REAL &D) \n{\n\n CCTK_REAL gtmp;\n gtmp = 1. - vsq;\n \n return( (GAMMA - 1.) * ( W * gtmp - D * sqrt(gtmp) ) / GAMMA );\n\n}\n```\n\nWe are now, however, interested in the hybrid EOS of the form\n\n$$\np_{\\rm hybrid} = P_{\\rm cold} + P_{\\rm th}\\ ,\n$$\n\nwhere $P_{\\rm cold}$ is given by a single or piecewise polytrope EOS,\n\n$$\nP_{\\rm cold} = K_{i}\\rho_{b}^{\\Gamma_{i}}\\ ,\n$$\n\n$P_{\\rm th}$ accounts for thermal effects and is given by\n\n$$\nP_{\\rm th} = \\left(\\Gamma_{\\rm th} - 1\\right)\\epsilon_{\\rm th}\\ ,\n$$\n\nand\n\n$$\n\\begin{align}\n\\epsilon \\equiv \\frac{u}{\\rho_{b}} &= \\epsilon_{\\rm th}+\\epsilon_{\\rm cold}\\ ,\\\\\n\\epsilon_{\\rm cold} &= \\int d\\rho \\frac{P_{\\rm cold}(\\rho)}{\\rho^{2}}\\ .\n\\end{align}\n$$\n\nWe then have\n\n$$\n\\begin{align}\np_{\\rm hybrid} &= P_{\\rm cold} + P_{\\rm th}\\\\\n &= P_{\\rm cold} + \\left(\\Gamma_{\\rm th}-1\\right)\\rho_{b}\\epsilon_{\\rm th}\\\\\n &= P_{\\rm cold} + \\left(\\Gamma_{\\rm th}-1\\right)\\rho_{b}\\left(\\epsilon - \\epsilon_{\\rm cold}\\right)\\\\\n &= P_{\\rm cold} + \\left(\\Gamma_{\\rm th}-1\\right)\\left(u - \\frac{D}{\\gamma}\\epsilon_{\\rm cold}\\right)\\\\\n &= P_{\\rm cold} + \\left(\\Gamma_{\\rm th}-1\\right)\\left(w - \\rho_{b} - p_{\\rm hybrid} - \\frac{D}{\\gamma}\\epsilon_{\\rm cold}\\right)\\\\\n &= P_{\\rm cold} + \\left(\\Gamma_{\\rm th}-1\\right)\\left(\\frac{W}{\\gamma^{2}} - \\frac{D}{\\gamma} - \\frac{D}{\\gamma}\\epsilon_{\\rm cold}\\right)-\\left(\\Gamma_{\\rm th}-1\\right)p_{\\rm hybrid}\\\\\n &= P_{\\rm cold} + \\left(\\Gamma_{\\rm th}-1\\right)\\left[\\frac{W}{\\gamma^{2}} - \\frac{D}{\\gamma}\\left(1+\\epsilon_{\\rm cold}\\right)\\right]-\\left(\\Gamma_{\\rm th}-1\\right)p_{\\rm hybrid}\\\\\n\\implies\n&\\boxed{ p_{\\rm hybrid} = \\frac{P_{\\rm cold}}{\\Gamma_{\\rm th}} + \\frac{\\left(\\Gamma_{\\rm th}-1\\right)}{\\Gamma_{\\rm th}}\\left[\\frac{W}{\\gamma^{2}} - \\frac{D}{\\gamma}\\left(1+\\epsilon_{\\rm cold}\\right)\\right] }\n\\end{align}\n$$\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n\n/**********************************************************************/\n/********************************************************************** \n pressure_W_vsq(): \n \n -- Hybrid single and piecewise polytropic equation of state;\n -- pressure as a function of P_cold, eps_cold, W, vsq, and D:\n**********************************************************************/\nstatic CCTK_REAL pressure_W_vsq(eos_struct eos, CCTK_REAL W, CCTK_REAL vsq, CCTK_REAL &D) \n{\n\n#ifndef ENABLE_STANDALONE_IGM_C2P_SOLVER\n DECLARE_CCTK_PARAMETERS;\n#endif\n\n // Compute gamma^{-2} = 1 - v^{2} and gamma^{-1}\n CCTK_REAL inv_gammasq = 1.0 - vsq;\n CCTK_REAL inv_gamma = sqrt(inv_gammasq);\n\n // Compute rho_b = D / gamma\n CCTK_REAL rho_b = D*inv_gamma;\n\n // Compute P_cold and eps_cold\n CCTK_REAL P_cold, eps_cold;\n compute_P_cold__eps_cold(eos,rho_b, P_cold,eps_cold);\n\n // Compute p = P_{cold} + P_{th}\n return( ( P_cold + (Gamma_th - 1.0)*( W*inv_gammasq - D*inv_gamma*( 1.0 + eps_cold ) ) )/Gamma_th );\n\n}\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n## Step 3.b: The `dpdW_calc_vsq()` function \\[Back to [top](#toc)\\]\n$$\\label{dpdw_calc_vsq}$$\n\nThis function computes $\\frac{\\partial p\\left(W,v^{2}\\right)}{\\partial W}$. For a $\\Gamma$-law equation of state, remember that\n\n$$\np_{\\Gamma} = \\frac{\\left(\\Gamma-1\\right)}{\\Gamma}\\left(\\frac{W}{\\gamma^{2}} - \\frac{D}{\\gamma}\\right)\\ ,\n$$\n\nwhich then implies\n\n$$\n\\boxed{\\frac{\\partial p_{\\Gamma}}{\\partial W} = \\frac{\\Gamma-1}{\\Gamma \\gamma^{2}} = \\frac{\\left(\\Gamma-1\\right)\\left(1-v^{2}\\right)}{\\Gamma}}\\ .\n$$\n\nThus, the pre-PPEOS Patch version of this function was\n\n```c\n/**********************************************************************/\n/********************************************************************** \n dpdW_calc_vsq(): \n \n -- partial derivative of pressure with respect to W;\n**********************************************************************/\nstatic CCTK_REAL dpdW_calc_vsq(CCTK_REAL W, CCTK_REAL vsq)\n{\n\n return( (GAMMA - 1.) * (1. - vsq) / GAMMA ) ;\n\n}\n```\n\nFor the case of a hybrid, single or piecewise polytropic EOS, we have\n\n$$\np_{\\rm hybrid} = \\frac{P_{\\rm cold}}{\\Gamma_{\\rm th}} + \\frac{\\left(\\Gamma_{\\rm th}-1\\right)}{\\Gamma_{\\rm th}}\\left[\\frac{W}{\\gamma^{2}} - \\frac{D}{\\gamma}\\left(1+\\epsilon_{\\rm cold}\\right)\\right]\\ .\n$$\n\nIt is important to notice that the cold components of $p_{\\rm hybrid}$ are *not* functions of $W$, but instead functions of $D$: $P_{\\rm cold} = P_{\\rm cold}(\\rho_{b}) = P_{\\rm cold}(D)$ and $\\epsilon_{\\rm cold} = \\epsilon_{\\rm cold}(\\rho_{b}) = \\epsilon_{\\rm cold}(D)$. Thus\n\n$$\n\\boxed{\\frac{\\partial p_{\\rm hybrid}}{\\partial W} = \\frac{\\Gamma_{\\rm th}-1}{\\Gamma_{\\rm th} \\gamma^{2}} = \\frac{\\left(\\Gamma_{\\rm th}-1\\right)\\left(1-v^{2}\\right)}{\\Gamma_{\\rm th}}}\\ .\n$$\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n\n\n/**********************************************************************/\n/********************************************************************** \n dpdW_calc_vsq(): \n \n -- partial derivative of pressure with respect to W;\n**********************************************************************/\nstatic CCTK_REAL dpdW_calc_vsq(CCTK_REAL W, CCTK_REAL vsq)\n{\n\n#ifndef ENABLE_STANDALONE_IGM_C2P_SOLVER\n DECLARE_CCTK_PARAMETERS;\n#endif\n\n return( (Gamma_th - 1.0) * (1.0 - vsq) / Gamma_th ) ;\n\n}\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n## Step 3.c: The `dpdvsq_calc()` function \\[Back to [top](#toc)\\]\n$$\\label{dpdvsq_calc}$$\n\nThis function computes $\\frac{\\partial p\\left(W,v^{2}\\right)}{\\partial W}$. For a $\\Gamma$-law equation of state, remember that\n\n$$\np_{\\Gamma} = \\frac{\\left(\\Gamma-1\\right)}{\\Gamma}\\left(\\frac{W}{\\gamma^{2}} - \\frac{D}{\\gamma}\\right) = \\frac{\\left(\\Gamma-1\\right)}{\\Gamma}\\left[W\\left(1-v^{2}\\right) - D\\sqrt{1-v^{2}}\\right]\\ ,\n$$\n\nwhich then implies\n\n$$\n\\boxed{\\frac{\\partial p_{\\Gamma}}{\\partial\\left(v^{2}\\right)} = \\frac{\\Gamma-1}{\\Gamma}\\left(\\frac{D}{2\\sqrt{1-v^{2}}}-W\\right)} \\ .\n$$\n\nThus, the pre-PPEOS Patch version of this function was\n\n```c\n/**********************************************************************/\n/********************************************************************** \n dpdvsq_calc(): \n \n -- partial derivative of pressure with respect to vsq\n**********************************************************************/\nstatic CCTK_REAL dpdvsq_calc(CCTK_REAL W, CCTK_REAL vsq, CCTK_REAL &D)\n{\n return( (GAMMA - 1.) * ( 0.5 * D / sqrt(1.-vsq) - W ) / GAMMA ) ;\n}\n```\n\n\n\n### Step 3.c.i: Setting basic quantities and computing $P_{\\rm cold}$ and $\\epsilon_{\\rm cold}$ \\[Back to [top](#toc)\\]\n$$\\label{dpdvsq_calc__basic_quantities}$$\n\nFor the case of a hybrid, single or piecewise polytropic EOS, we have\n\n$$\np_{\\rm hybrid} = \\frac{P_{\\rm cold}}{\\Gamma_{\\rm th}} + \\frac{\\left(\\Gamma_{\\rm th}-1\\right)}{\\Gamma_{\\rm th}}\\left[\\frac{W}{\\gamma^{2}} - \\frac{D}{\\gamma}\\left(1+\\epsilon_{\\rm cold}\\right)\\right]\\ .\n$$\n\nLet us thus begin by setting the necessary parameters from the hybrid EOS.\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n\n/**********************************************************************/\n/********************************************************************** \n dpdvsq_calc(): \n \n -- partial derivative of pressure with respect to vsq\n**********************************************************************/\nstatic CCTK_REAL dpdvsq_calc(eos_struct eos, CCTK_REAL W, CCTK_REAL vsq, CCTK_REAL &D)\n{\n\n // This sets Gamma_th\n#ifndef ENABLE_STANDALONE_IGM_C2P_SOLVER\n DECLARE_CCTK_PARAMETERS;\n#endif\n\n\n // Set gamma and rho\n CCTK_REAL gamma = 1.0/sqrt(1.0 - vsq);\n CCTK_REAL rho_b = D/gamma;\n \n // Compute P_cold and eps_cold\n CCTK_REAL P_cold, eps_cold;\n compute_P_cold__eps_cold(eos,rho_b, P_cold,eps_cold);\n\n // Set basic polytropic quantities\n int polytropic_index = find_polytropic_K_and_Gamma_index(eos,rho_b);\n CCTK_REAL Gamma_ppoly_tab = eos.Gamma_ppoly_tab[polytropic_index];\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n### Step 3.c.ii: Computing $\\frac{\\partial P_{\\rm cold}}{\\partial\\left(v^{2}\\right)}$ \\[Back to [top](#toc)\\]\n$$\\label{dpdvsq_calc__dpcolddvsq}$$\n\nNext, remember that $P_{\\rm cold} = P_{\\rm cold}(\\rho_{b}) = P_{\\rm cold}(D,v^{2})$ and also $\\epsilon_{\\rm cold} = \\epsilon_{\\rm cold}(D,v^{2})$. Therefore, we must start by finding the derivatives of $P_{\\rm cold}$ and $\\epsilon_{\\rm cold}$ with respect to $v^{2}$.\n\nLet us first notice that\n\n$$\n\\frac{\\partial\\gamma}{\\partial\\left(v^{2}\\right)} = \\frac{\\partial}{\\partial\\left(v^{2}\\right)}\\left[\\frac{1}{\\sqrt{1-v^{2}}}\\right] = \\frac{1}{2}\\left(1-v^{2}\\right)^{-3/2} = \\frac{\\gamma^{3}}{2}\\ .\n$$\n\nThus, for a general power\n\n$$\n\\frac{\\partial\\gamma^{a}}{\\partial\\left(v^{2}\\right)} = a\\gamma^{a-1}\\frac{\\partial\\gamma}{\\partial\\left(v^{2}\\right)} = a\\gamma^{a-1}\\left(\\frac{\\gamma^{3}}{2}\\right) = \\frac{a}{2}\\gamma^{a+2}\n$$\n\nThus we have\n\n$$\n\\begin{align}\n\\frac{\\partial P_{\\rm cold}}{\\partial \\left(v^{2}\\right)}\n&= \\frac{\\partial}{\\partial\\left(v^{2}\\right)}\\left(K_{\\rm poly}\\rho_{b}^{\\Gamma_{\\rm poly}}\\right)\\\\\n&= \\frac{\\partial}{\\partial\\left(v^{2}\\right)}\\left[K_{\\rm poly}\\left(\\frac{D}{\\gamma}\\right)^{\\Gamma_{\\rm poly}}\\right]\\\\\n&= K_{\\rm poly}D^{\\Gamma_{\\rm poly}}\\frac{\\partial}{\\partial\\left(v^{2}\\right)}\\left[\\gamma^{-\\Gamma_{\\rm poly}/2}\\right]\\\\\n&=K_{\\rm poly}D^{\\Gamma_{\\rm poly}}\\left[\\frac{-\\Gamma_{\\rm poly}/2}{2}\\gamma^{-\\Gamma_{\\rm poly}/2 + 2}\\right]\\\\\n&=K_{\\rm poly}\\left(\\frac{D}{\\gamma}\\right)^{\\Gamma_{\\rm poly}}\\gamma^{-\\frac{\\Gamma_{\\rm poly}}{2} + 2 + \\Gamma_{\\rm poly}}\\\\\n\\implies &\\boxed{ \\frac{\\partial P_{\\rm cold}}{\\partial \\left(v^{2}\\right)} = \\gamma^{2+\\frac{\\Gamma_{\\rm poly}}{2}}P_{\\rm cold}}\\ .\n\\end{align}\n$$\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n\n /* Now we implement the derivative of P_cold with respect\n * to v^{2}, given by\n * ----------------------------------------------------\n * | dP_cold/dvsq = gamma^{2 + Gamma_{poly}/2} P_{cold} |\n * ----------------------------------------------------\n */\n CCTK_REAL dPcold_dvsq = P_cold * pow(gamma,2.0 + 0.5*Gamma_ppoly_tab);\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n### Step 3.c.iii: Computing $\\frac{\\partial \\epsilon_{\\rm cold}}{\\partial\\left(v^{2}\\right)}$ \\[Back to [top](#toc)\\]\n$$\\label{dpdvsq_calc__depscolddvsq}$$\n\nNow, obtaining $\\epsilon_{\\rm cold}$ from $P_{\\rm cold}$ requires an integration and, therefore, generates an integration constant. Since we are interested in a *derivative* of $\\epsilon_{\\rm cold}$, however, we will simply drop the constant altogether. Remember that:\n\n$$\n\\epsilon_{\\rm cold} = K_{\\rm poly}\\int d\\rho_{b} \\rho_{b}^{\\Gamma_{\\rm poly}-2} = \\frac{K_{\\rm poly}\\rho_{b}^{\\Gamma_{\\rm poly}-1}}{\\Gamma_{\\rm poly}-1} = \\frac{P_{\\rm cold}}{\\rho_{b}\\left(\\Gamma_{\\rm poly}-1\\right)} = \\frac{\\gamma P_{\\rm cold}}{D\\left(\\Gamma_{\\rm poly}-1\\right)}\\ .\n$$\n\nThus\n\n$$\n\\begin{align}\n\\frac{\\partial \\epsilon_{\\rm cold}}{\\partial \\left(v^{2}\\right)}\n&= \\frac{1}{D\\left(\\Gamma_{\\rm poly}-1\\right)}\\left[\\gamma\\frac{\\partial P_{\\rm cold}}{\\partial \\left(v^{2}\\right)} + P_{\\rm cold}\\frac{\\partial\\gamma}{\\partial \\left(v^{2}\\right)}\\right]\\\\\n&=\\frac{1}{D\\left(\\Gamma_{\\rm poly}-1\\right)}\\left[\\gamma\\frac{\\partial P_{\\rm cold}}{\\partial \\left(v^{2}\\right)} + P_{\\rm cold}\\left(\\frac{\\gamma^{3}}{2}\\right)\\right]\\\\\n\\implies &\\boxed{\n\\frac{\\partial \\epsilon_{\\rm cold}}{\\partial \\left(v^{2}\\right)} = \\frac{\\gamma}{D\\left(\\Gamma_{\\rm poly}-1\\right)}\\left[\\frac{\\partial P_{\\rm cold}}{\\partial \\left(v^{2}\\right)} + \\frac{\\gamma^{2} P_{\\rm cold}}{2}\\right]\\ .\n}\n\\end{align}\n$$\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n\n /* Now we implement the derivative of eps_cold with respect\n * to v^{2}, given by\n * -----------------------------------------------------------------------------------\n * | deps_cold/dvsq = gamma/(D*(Gamma_ppoly_tab-1)) * (dP_cold/dvsq + gamma^{2} P_cold / 2) |\n * -----------------------------------------------------------------------------------\n */\n CCTK_REAL depscold_dvsq = ( gamma/(D*(Gamma_ppoly_tab-1.0)) ) * ( dPcold_dvsq + 0.5*gamma*gamma*P_cold );\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n### Step 3.c.iv: Computing $\\frac{\\partial p_{\\rm hybrid}}{\\partial\\left(v^{2}\\right)}$ \\[Back to [top](#toc)\\]\n$$\\label{dpdvsq_calc__dpdvsq}$$\n\nFinally, remembering that\n\n$$\n\\begin{align}\np_{\\rm hybrid} &= \\frac{P_{\\rm cold}}{\\Gamma_{\\rm th}} + \\frac{\\left(\\Gamma_{\\rm th}-1\\right)}{\\Gamma_{\\rm th}}\\left[\\frac{W}{\\gamma^{2}} - \\frac{D}{\\gamma}\\left(1+\\epsilon_{\\rm cold}\\right)\\right]\\ ,\\\\\n\\frac{\\partial\\gamma^{a}}{\\partial\\left(v^{2}\\right)} &= \\frac{a}{2}\\gamma^{a+2}\\ ,\n\\end{align}\n$$\n\nwe have\n\n$$\n\\boxed{\n\\frac{\\partial p_{\\rm hybrid}}{\\partial\\left(v^{2}\\right)}\n= \\frac{1}{\\Gamma_{\\rm th}}\\left\\{\\frac{\\partial P_{\\rm cold}}{\\partial\\left(v^{2}\\right)} + \\left(\\Gamma_{\\rm th}-1\\right)\\left[-W + \\frac{D\\gamma}{2}\\left(1+\\epsilon_{\\rm cold}\\right) - \\frac{D}{\\gamma}\\frac{\\partial \\epsilon_{\\rm cold}}{\\partial\\left(v^{2}\\right)}\\right]\\right\\}\\ .\n}\n$$\n\n\n```python\n%%writefile -a $outfile_path__harm_utoprim_2d__c\n\n /* Now we implement the derivative of p_hybrid with respect\n * to v^{2}, given by\n * -----------------------------------------------------------------------------\n * | dp/dvsq = Gamma_th^{-1}( dP_cold/dvsq |\n * | + (Gamma_{th}-1)*(-W |\n * | + D gamma (1 + eps_cold)/2 |\n * | - (D/gamma) * deps_cold/dvsq) ) |\n * -----------------------------------------------------------------------------\n */\n return( ( dPcold_dvsq + (Gamma_th-1.0)*( -W + D*gamma*(1+eps_cold)/2.0 - D*depscold_dvsq/gamma ) )/Gamma_th );\n}\n\n\n/****************************************************************************** \n END OF UTOPRIM_2D.C\n******************************************************************************/\n#endif\n\n\n\n\n```\n\n Appending to ../src/harm_utoprim_2d.c\n\n\n\n\n# Step 4: Code validation \\[Back to [top](#toc)\\]\n$$\\label{code_validation}$$\n\nFirst we download the original `IllinoisGRMHD` source code and then compare it to the source code generated by this tutorial notebook.\n\n\n```python\n# Verify if the code generated by this tutorial module\n# matches the original IllinoisGRMHD source code\n\n# First download the original IllinoisGRMHD source code\nimport urllib\nfrom os import path\n\noriginal_IGM_file_url = \"https://bitbucket.org/zach_etienne/wvuthorns/raw/5611b2f0b17135538c9d9d17c7da062abe0401b6/IllinoisGRMHD/src/harm_utoprim_2d.c\"\noriginal_IGM_file_name = \"harm_utoprim_2d-original.c\"\noriginal_IGM_file_path = os.path.join(IGM_src_dir_path,original_IGM_file_name)\n\n# Then download the original IllinoisGRMHD source code\n# We try it here in a couple of ways in an attempt to keep\n# the code more portable\ntry:\n original_IGM_file_code = urllib.request.urlopen(original_IGM_file_url).read().decode(\"utf-8\")\n # Write down the file the original IllinoisGRMHD source code\n with open(original_IGM_file_path,\"w\") as file:\n file.write(original_IGM_file_code)\nexcept:\n try:\n original_IGM_file_code = urllib.urlopen(original_IGM_file_url).read().decode(\"utf-8\")\n # Write down the file the original IllinoisGRMHD source code\n with open(original_IGM_file_path,\"w\") as file:\n file.write(original_IGM_file_code)\n except:\n # If all else fails, hope wget does the job\n !wget -O $original_IGM_file_path $original_IGM_file_url\n\n# Perform validation\nValidation__harm_utoprim_2d__c = !diff $original_IGM_file_path $outfile_path__harm_utoprim_2d__c\n\nif Validation__harm_utoprim_2d__c == []:\n # If the validation passes, we do not need to store the original IGM source code file\n !rm $original_IGM_file_path\n print(\"Validation test for harm_utoprim_2d.c: PASSED!\")\nelse:\n # If the validation fails, we keep the original IGM source code file\n print(\"Validation test for harm_utoprim_2d.c: FAILED!\")\n # We also print out the difference between the code generated\n # in this tutorial module and the original IGM source code\n print(\"Diff:\")\n for diff_line in Validation__harm_utoprim_2d__c:\n print(diff_line)\n```\n\n Validation test for harm_utoprim_2d.c: FAILED!\n Diff:\n 0a1,2\n > #ifndef __HARM_UTOPRIM_2D__C__\n > #define __HARM_UTOPRIM_2D__C__\n 70,72c72,74\n < static int Utoprim_new_body(CCTK_REAL U[], CCTK_REAL gcov[NDIM][NDIM], CCTK_REAL gcon[NDIM][NDIM], CCTK_REAL gdet, CCTK_REAL prim[],long &n_iter);\n < static int general_newton_raphson( CCTK_REAL x[], int n, long &n_iter, void (*funcd) (CCTK_REAL [], CCTK_REAL [], CCTK_REAL [], CCTK_REAL [][NEWT_DIM], CCTK_REAL *, CCTK_REAL *, int,CCTK_REAL &,CCTK_REAL &,CCTK_REAL &,CCTK_REAL &,CCTK_REAL &),CCTK_REAL &Bsq,CCTK_REAL &QdotBsq,CCTK_REAL &Qtsq,CCTK_REAL &Qdotn,CCTK_REAL &D);\n < static void func_vsq( CCTK_REAL [], CCTK_REAL [], CCTK_REAL [], CCTK_REAL [][NEWT_DIM], CCTK_REAL *f, CCTK_REAL *df, int n,CCTK_REAL &Bsq,CCTK_REAL &QdotBsq,CCTK_REAL &Qtsq,CCTK_REAL &Qdotn,CCTK_REAL &D);\n ---\n > static int Utoprim_new_body(eos_struct eos, CCTK_REAL U[], CCTK_REAL gcov[NDIM][NDIM], CCTK_REAL gcon[NDIM][NDIM], CCTK_REAL gdet, CCTK_REAL prim[],long &n_iter);\n > static int general_newton_raphson( eos_struct eos, CCTK_REAL x[], int n, long &n_iter, void (*funcd) (eos_struct, CCTK_REAL [], CCTK_REAL [], CCTK_REAL [], CCTK_REAL [][NEWT_DIM], CCTK_REAL *, CCTK_REAL *, int,CCTK_REAL &,CCTK_REAL &,CCTK_REAL &,CCTK_REAL &,CCTK_REAL &),CCTK_REAL &Bsq,CCTK_REAL &QdotBsq,CCTK_REAL &Qtsq,CCTK_REAL &Qdotn,CCTK_REAL &D);\n > static void func_vsq( eos_struct eos, CCTK_REAL [], CCTK_REAL [], CCTK_REAL [], CCTK_REAL [][NEWT_DIM], CCTK_REAL *f, CCTK_REAL *df, int n,CCTK_REAL &Bsq,CCTK_REAL &QdotBsq,CCTK_REAL &Qtsq,CCTK_REAL &Qdotn,CCTK_REAL &D);\n 74c76\n < static CCTK_REAL pressure_W_vsq(CCTK_REAL W, CCTK_REAL vsq, CCTK_REAL &D) ;\n ---\n > static CCTK_REAL pressure_W_vsq(eos_struct eos, CCTK_REAL W, CCTK_REAL vsq, CCTK_REAL &D) ;\n 76c78\n < static CCTK_REAL dpdvsq_calc(CCTK_REAL W, CCTK_REAL vsq, CCTK_REAL &D);\n ---\n > static CCTK_REAL dpdvsq_calc(eos_struct eos, CCTK_REAL W, CCTK_REAL vsq, CCTK_REAL &D);\n 89,97c91,99\n < / rho u^t \\\n < U = | T^t_t + rho u^t | sqrt(-det(g_{\\mu\\nu}))\n < | T^t_i |\n < \\ B^i /\n < \n < / rho \\\n < P = | uu |\n < | \\tilde{u}^i |\n < \\ B^i /\n ---\n > / rho u^t \\\n > U = | T^t_t + rho u^t | sqrt(-det(g_{\\mu\\nu}))\n > | T^t_i |\n > \\ B^i /\n > \n > / rho \\\n > P = | uu |\n > | \\tilde{u}^i |\n > \\ B^i /\n 101c103\n < U[NPR] = conserved variables (current values on input/output);\n ---\n > U[NPR] = conserved variables (current values on input/output);\n 105,106c107,108\n < prim[NPR] = primitive variables (guess on input, calculated values on \n < output if there are no problems);\n ---\n > prim[NPR] = primitive variables (guess on input, calculated values on\n > output if there are no problems);\n 114c116,117\n < int Utoprim_2d(CCTK_REAL U[NPR], CCTK_REAL gcov[NDIM][NDIM], CCTK_REAL gcon[NDIM][NDIM], \n ---\n > \n > int Utoprim_2d(eos_struct eos, CCTK_REAL U[NPR], CCTK_REAL gcov[NDIM][NDIM], CCTK_REAL gcon[NDIM][NDIM], \n 130a134\n > \n 141a146\n > \n 150c155\n < ret = Utoprim_new_body(U_tmp, gcov, gcon, gdet, prim_tmp,n_iter);\n ---\n > ret = Utoprim_new_body(eos, U_tmp, gcov, gcon, gdet, prim_tmp,n_iter);\n 163a169\n > \n 175,177c181,183\n < / rho gamma \\\n < U = | alpha T^t_\\mu |\n < \\ alpha B^i /\n ---\n > / rho gamma \\\n > U = | alpha T^t_\\mu |\n > \\ alpha B^i /\n 181,184c187,190\n < / rho \\\n < prim = | uu |\n < | \\tilde{u}^i |\n < \\ alpha B^i /\n ---\n > / rho \\\n > prim = | uu |\n > | \\tilde{u}^i |\n > \\ alpha B^i /\n 198c204\n < 3 -> failure: W<0 or W>W_TOO_BIG\n ---\n > 3 -> failure: W<0 or W>W_TOO_BIG\n 204c210\n < static int Utoprim_new_body(CCTK_REAL U[NPR], CCTK_REAL gcov[NDIM][NDIM], \n ---\n > static int Utoprim_new_body(eos_struct eos, CCTK_REAL U[NPR], CCTK_REAL gcov[NDIM][NDIM], \n 217a224\n > \n 237a245,248\n > // FIXME: The exact form of n^{\\mu} can be found\n > // in eq. (2.116) and implementing it\n > // directly is a lot more efficient than\n > // performing n^{\\mu} = g^{\\mu\\nu}n_{nu}\n 248a260\n > \n 270c282\n < p = pressure_rho0_u(rho0,u) ;\n ---\n > p = pressure_rho0_u(eos, rho0,u) ;\n 283a296\n > \n 288c301\n < retval = general_newton_raphson( x_2d, n, n_iter, func_vsq, Bsq,QdotBsq,Qtsq,Qdotn,D) ; \n ---\n > retval = general_newton_raphson( eos, x_2d, n, n_iter, func_vsq, Bsq,QdotBsq,Qtsq,Qdotn,D) ; \n 311a325\n > \n 318c332\n < p = pressure_rho0_w(rho0,w) ;\n ---\n > p = pressure_rho0_w(eos, rho0,w) ;\n 352a367\n > \n 371a387\n > \n 393a410\n > \n 419a437\n > \n 429,430c447,448\n < static int general_newton_raphson( CCTK_REAL x[], int n, long &n_iter,\n < void (*funcd) (CCTK_REAL [], CCTK_REAL [], CCTK_REAL [], \n ---\n > static int general_newton_raphson( eos_struct eos, CCTK_REAL x[], int n, long &n_iter,\n > void (*funcd) (eos_struct, CCTK_REAL [], CCTK_REAL [], CCTK_REAL [], \n 454c472\n < (*funcd) (x, dx, resid, jac, &f, &df, n, Bsq,QdotBsq,Qtsq,Qdotn,D); /* returns with new dx, f, df */\n ---\n > (*funcd) (eos, x, dx, resid, jac, &f, &df, n, Bsq,QdotBsq,Qtsq,Qdotn,D); /* returns with new dx, f, df */\n 522a541\n > \n 540c559\n < static void func_vsq(CCTK_REAL x[], CCTK_REAL dx[], CCTK_REAL resid[], \n ---\n > static void func_vsq(eos_struct eos, CCTK_REAL x[], CCTK_REAL dx[], CCTK_REAL resid[], \n 579c598\n < p = pressure_rho0_w(rho0,w) ;\n ---\n > p = pressure_rho0_w(eos, rho0,w) ;\n 586c605\n < p = pressure_rho0_w(rho0,w) ;\n ---\n > p = pressure_rho0_w(eos, rho0,w) ;\n 595c614\n < p_tmp = pressure_W_vsq( W, vsq , D);\n ---\n > p_tmp = pressure_W_vsq( eos, W, vsq , D);\n 597c616\n < dPdvsq = dpdvsq_calc( W, vsq, D );\n ---\n > dPdvsq = dpdvsq_calc( eos, W, vsq, D );\n 635a655\n > \n 646a667\n > \n 651,652c672,673\n < -- Gamma-law equation of state;\n < -- pressure as a function of W, vsq, and D:\n ---\n > -- Hybrid single and piecewise polytropic equation of state;\n > -- pressure as a function of P_cold, eps_cold, W, vsq, and D:\n 654c675\n < static CCTK_REAL pressure_W_vsq(CCTK_REAL W, CCTK_REAL vsq, CCTK_REAL &D) \n ---\n > static CCTK_REAL pressure_W_vsq(eos_struct eos, CCTK_REAL W, CCTK_REAL vsq, CCTK_REAL &D) \n 655a677,678\n > \n > #ifndef ENABLE_STANDALONE_IGM_C2P_SOLVER\n 656a680\n > #endif\n 658,661c682,694\n < CCTK_REAL gtmp;\n < gtmp = 1. - vsq;\n < \n < return( (gamma_th /* <- Should be local polytropic Gamma factor */ - 1.) * ( W * gtmp - D * sqrt(gtmp) ) / gamma_th /* <- Should be local polytropic Gamma factor */ );\n ---\n > // Compute gamma^{-2} = 1 - v^{2} and gamma^{-1}\n > CCTK_REAL inv_gammasq = 1.0 - vsq;\n > CCTK_REAL inv_gamma = sqrt(inv_gammasq);\n > \n > // Compute rho_b = D / gamma\n > CCTK_REAL rho_b = D*inv_gamma;\n > \n > // Compute P_cold and eps_cold\n > CCTK_REAL P_cold, eps_cold;\n > compute_P_cold__eps_cold(eos,rho_b, P_cold,eps_cold);\n > \n > // Compute p = P_{cold} + P_{th}\n > return( ( P_cold + (Gamma_th - 1.0)*( W*inv_gammasq - D*inv_gamma*( 1.0 + eps_cold ) ) )/Gamma_th );\n 665a699\n > \n 673a708,709\n > \n > #ifndef ENABLE_STANDALONE_IGM_C2P_SOLVER\n 675c711,713\n < return( (gamma_th /* <- Should be local polytropic Gamma factor */ - 1.) * (1. - vsq) / gamma_th /* <- Should be local polytropic Gamma factor */ ) ;\n ---\n > #endif\n > \n > return( (Gamma_th - 1.0) * (1.0 - vsq) / Gamma_th ) ;\n 678a717\n > \n 685c724\n < static CCTK_REAL dpdvsq_calc(CCTK_REAL W, CCTK_REAL vsq, CCTK_REAL &D)\n ---\n > static CCTK_REAL dpdvsq_calc(eos_struct eos, CCTK_REAL W, CCTK_REAL vsq, CCTK_REAL &D)\n 686a726,728\n > \n > // This sets Gamma_th\n > #ifndef ENABLE_STANDALONE_IGM_C2P_SOLVER\n 688c730,772\n < return( (gamma_th /* <- Should be local polytropic Gamma factor */ - 1.) * ( 0.5 * D / sqrt(1.-vsq) - W ) / gamma_th /* <- Should be local polytropic Gamma factor */ ) ;\n ---\n > #endif\n > \n > \n > // Set gamma and rho\n > CCTK_REAL gamma = 1.0/sqrt(1.0 - vsq);\n > CCTK_REAL rho_b = D/gamma;\n > \n > // Compute P_cold and eps_cold\n > CCTK_REAL P_cold, eps_cold;\n > compute_P_cold__eps_cold(eos,rho_b, P_cold,eps_cold);\n > \n > // Set basic polytropic quantities\n > int polytropic_index = find_polytropic_K_and_Gamma_index(eos,rho_b);\n > CCTK_REAL Gamma_ppoly_tab = eos.Gamma_ppoly_tab[polytropic_index];\n > \n > \n > /* Now we implement the derivative of P_cold with respect\n > * to v^{2}, given by\n > * ----------------------------------------------------\n > * | dP_cold/dvsq = gamma^{2 + Gamma_{poly}/2} P_{cold} |\n > * ----------------------------------------------------\n > */\n > CCTK_REAL dPcold_dvsq = P_cold * pow(gamma,2.0 + 0.5*Gamma_ppoly_tab);\n > \n > \n > /* Now we implement the derivative of eps_cold with respect\n > * to v^{2}, given by\n > * -----------------------------------------------------------------------------------\n > * | deps_cold/dvsq = gamma/(D*(Gamma_ppoly_tab-1)) * (dP_cold/dvsq + gamma^{2} P_cold / 2) |\n > * -----------------------------------------------------------------------------------\n > */\n > CCTK_REAL depscold_dvsq = ( gamma/(D*(Gamma_ppoly_tab-1.0)) ) * ( dPcold_dvsq + 0.5*gamma*gamma*P_cold );\n > \n > /* Now we implement the derivative of p_hybrid with respect\n > * to v^{2}, given by\n > * -----------------------------------------------------------------------------\n > * | dp/dvsq = Gamma_th^{-1}( dP_cold/dvsq |\n > * | + (Gamma_{th}-1)*(-W |\n > * | + D gamma (1 + eps_cold)/2 |\n > * | - (D/gamma) * deps_cold/dvsq) ) |\n > * -----------------------------------------------------------------------------\n > */\n > return( ( dPcold_dvsq + (Gamma_th-1.0)*( -W + D*gamma*(1+eps_cold)/2.0 - D*depscold_dvsq/gamma ) )/Gamma_th );\n 694a779\n > #endif\n\n\n\n\n# Step 5: Output this notebook to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n$$\\label{latex_pdf_output}$$\n\nThe following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n[Tutorial-IllinoisGRMHD__harm_utoprim_2d.pdf](Tutorial-IllinoisGRMHD__harm_utoprim_2d.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means).\n\n\n```python\nlatex_nrpy_style_path = os.path.join(nrpy_dir_path,\"latex_nrpy_style.tplx\")\n#!jupyter nbconvert --to latex --template $latex_nrpy_style_path --log-level='WARN' Tutorial-IllinoisGRMHD__harm_utoprim_2d.ipynb\n#!pdflatex -interaction=batchmode Tutorial-IllinoisGRMHD__harm_utoprim_2d.tex\n#!pdflatex -interaction=batchmode Tutorial-IllinoisGRMHD__harm_utoprim_2d.tex\n#!pdflatex -interaction=batchmode Tutorial-IllinoisGRMHD__harm_utoprim_2d.tex\n!rm -f Tut*.out Tut*.aux Tut*.log\n```\n", "meta": {"hexsha": "7907094f839f6f8bab815bf6c366dcc5bbc45e4b", "size": 92391, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "IllinoisGRMHD/doc/Tutorial-IllinoisGRMHD__harm_utoprim_2d.ipynb", "max_stars_repo_name": "Steve-Hawk/nrpytutorial", "max_stars_repo_head_hexsha": "42d7450dba8bf43aa9c2d8f38f85f18803de69b7", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-12-23T05:31:25.000Z", "max_stars_repo_stars_event_max_datetime": "2019-12-23T05:31:25.000Z", "max_issues_repo_path": "IllinoisGRMHD/doc/Tutorial-IllinoisGRMHD__harm_utoprim_2d.ipynb", "max_issues_repo_name": "Steve-Hawk/nrpytutorial", "max_issues_repo_head_hexsha": "42d7450dba8bf43aa9c2d8f38f85f18803de69b7", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "IllinoisGRMHD/doc/Tutorial-IllinoisGRMHD__harm_utoprim_2d.ipynb", "max_forks_repo_name": "Steve-Hawk/nrpytutorial", "max_forks_repo_head_hexsha": "42d7450dba8bf43aa9c2d8f38f85f18803de69b7", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-03-02T12:51:56.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-02T12:51:56.000Z", "avg_line_length": 39.944228275, "max_line_length": 365, "alphanum_fraction": 0.4734010889, "converted": true, "num_tokens": 22458, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.754914997895581, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.4301903131431097}} {"text": "```python\n%%HTML\n\n\n```\n\n\n\n\n\n\n\n\n```python\n%matplotlib notebook\nimport numpy as np\nimport scipy.fftpack\nimport matplotlib.pyplot as plt\nfrom matplotlib import animation, patches\nfrom IPython.display import display, Audio, HTML\nplt.rcParams['image.cmap'] = 'gray'\nfrom ipywidgets import interact, FloatSlider, IntSlider, SelectionSlider, Layout\nfrom functools import partial\nslider_layout = Layout(width='600px', height='20px')\nslider_style = {'description_width': 'initial'}\nFloatSlider_nice = partial(FloatSlider, style=slider_style, layout=slider_layout, continuous_update=False)\nIntSlider_nice = partial(IntSlider, style=slider_style, layout=slider_layout, continuous_update=False)\nSelectionSlider_nice = partial(SelectionSlider, style=slider_style, layout=slider_layout, continuous_update=False)\n\ndef color2bw(img):\n return np.dot(img, [0.299, 0.587, 0.114]) \n```\n\n***\n### Universidad Austral de Chile\n## Computaci\u00f3n de alto rendimiento\n# Fast Fourier Transform\n\n### Crist\u00f3bal Navarro y Pablo Huijse\n \n***\n\n# Serie trigonom\u00e9trica\n\nPodemos descomponer una se\u00f1al peri\u00f3dica $P=1/f$ en una suma de sinusoides\n\n$$\n\\begin{align}\ns(t)\n&= \\sum_{k=0}^\\infty A_k \\cos(2\\pi k f t + \\phi_k) \\nonumber \\\\\n&= \\sum_{k=0}^\\infty a_k \\cos(2\\pi k f t) + b_k \\sin(2\\pi k f t), \\nonumber\n\\end{align}\n$$\n\ndonde\n\n$$\na_k = f \\int_{-\\frac{1}{2f}}^{\\frac{1}{2f}} s(t) \\cos(2\\pi k f t) \\,dt \\quad b_k = f \\int_{-\\frac{1}{2f}}^{\\frac{1}{2f}} s(t) \\sin(2\\pi k f t) \\,dt \n$$\n\n\n```python\nplt.close('all'); fig, ax = plt.subplots(figsize=(6, 4))\nf = 1.51518; t = np.linspace(-2/f, 2/f, num=5000); \nline = ax.plot(t, np.zeros_like(t))\nax.set_xticks([-2/f, -1/f, 0, 1/f, 2/f]); \nax.set_xticklabels([r\"$-2/f_0$\", r\"$-1/f_0$\", \"0\", r\"$1/f_0$\", r\"$2/f_0$\"]);\na, b = np.random.randn(101), np.random.randn(101)\ndef update(K):\n y = np.zeros_like(t)\n for k in range(1, K+1):\n #y += (1/k)*np.sin(2.0*np.pi*k*f*t)\n #y += (2*np.sin(np.pi*k/2)/(np.pi*k))*np.cos(2.0*np.pi*k*f*t)\n y += a[k]*np.cos(2.0*np.pi*k*f*t)/k**2 + b[k]*np.sin(2.0*np.pi*k*f*t)/k**2\n line[0].set_ydata(y); ax.set_ylim([np.amin(y)*1.1, np.amax(y)*1.1])\ninteract(update, K=SelectionSlider_nice(options=[1, 2, 3, 4, 5, 10, 20, 30, 50, 100]));\n```\n\n\n \n\n\n\n\n\n\n\n interactive(children=(SelectionSlider(continuous_update=False, description='K', layout=Layout(height='20px', w\u2026\n\n\nVisualizaci\u00f3n interactiva de la FS: https://bl.ocks.org/jinroh/7524988\n\nSerie de Fourier interesante: https://www.wolframalpha.com/input/?i=longcat+curve\n\n# Transformada de Fourier\n***\n- El concepto de frecuencia puede aplicarse tambi\u00e9n a se\u00f1ales no-peri\u00f3dicas\n- **Joseph Fourier:** Una se\u00f1al no-peri\u00f3dica puede ser vista como una se\u00f1al peri\u00f3dica **con un per\u00edodo infinito**\n- El \u00fanico requisito es que ahora las frecuencias son un continuo, con un espaciado infinitesimal\n\n\nDirecta:\n$$\nS(\\omega) = \\mathbb{FT}[s(t)] = \\int_{-\\infty}^{\\infty} s(t) e^{-j\\omega t } dt,\n$$\n\n***\n\nInversa:\n$$\ns(t) = \\mathbb{FT}^{-1}[S(\\omega)] = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} S(\\omega) e^{j \\omega t } d\\omega,\n$$\n\n***\n\n# Discrete Fourier Transform (DFT)\n\n***\n\n- Nos interesa trabajar con se\u00f1ales digitales que est\u00e1n muestreadas en el tiempo\n- Asumimos que la se\u00f1al fue observada en un ventana de tiempo de ancho $T$ [s]\n- Sea un sistema muestreador con frecuencia de muestreo $F_s$ [Hz] tal que\n$$\ns(t) = \\sum_{n=0}^{N-1} s[n] \\delta(t - n/F_s),\n$$\ndel cual hemos tomado $N = T F_s$ muestras de $s(t)$\n\n\n\n- Reemplazando en la transformada de Fourier\n$$\n\\begin{align}\nS(\\omega) &= \\int s(t) e^{-j\\omega t} dt \\nonumber \\\\\n&= \\int \\sum_{n=0}^{N-1} s[n] \\delta(t - n/F_s) e^{-j\\omega t} dt \\nonumber \\\\\n&= \\sum_{n=0}^{N-1} s[n] \\int \\delta(t - n/F_s) e^{-j\\omega t} dt \\nonumber \\\\\n&= \\sum_{n=0}^{N-1} s[n] e^{-j\\omega n/F_s} \\nonumber \n\\end{align}\n$$\n- Definiendo entonces $\\omega = 2 \\pi f = 2 \\pi k \\Delta f$ donde $\\Delta f = \\frac{1}{T} = \\frac{F_s}{N}$ y reemplazando \n$$\nS[k] = \\sum_{n=0}^{N-1} s[n] e^{-j \\frac{2 \\pi}{N} k n},\n$$\ndonde $k = [0, 1, \\ldots N-1]$, \u00bfA qu\u00e9 frecuencias corresponden estos \u00edndices?\n\n\n***\n\n## DFT como producto matricial\n\nSea $\\{s_n\\}_{n=0,\\ldots,N-1}$ y definiendo \n\n$$\nW_N = e^{-j \\frac{2\\pi}{N}} = \\cos \\left(\\frac{2\\pi}{N}\\right) - j \\sin \\left(\\frac{2\\pi}{N}\\right)\n$$\n\npodemos expresar la transformada de Fourier discreta como\n\n$$\nS[k] = \\sum_{n=0}^{N-1} s[n] W_N^{kn}, \\quad k = [0, 1, \\ldots N-1],\n$$\n\nque tambi\u00e9n puede ser expresado matricialmente como\n\n$$\n\\begin{align}\n\\begin{pmatrix} \nS[0] \\\\\nS[1] \\\\\nS[2] \\\\\n\\vdots \\\\\nS[N-1] \\\\\n\\end{pmatrix} &=\n\\begin{pmatrix}\n1 & 1 & 1 & \\cdots & 1 \\\\\n1 & W_N^1 & W_N^2 & \\cdots & W_N^{N-1} \\\\\n1 & W_N^2 & W_N^4 & \\cdots & W_N^{N-2} \\\\\n\\vdots & \\dots & \\dots & \\ddots & \\vdots \\\\\n1 & W_N^{N-1} & W_N^{N-2} & \\cdots & W_N^1\\\\\n\\end{pmatrix} \n\\begin{pmatrix} \ns[0] \\\\\ns[1] \\\\\ns[2] \\\\\n\\vdots \\\\\ns[N-1] \\\\\n\\end{pmatrix} \\nonumber \\\\\nS &= \\Omega s,\n\\end{align}\n$$\n\n\nNotemos que:\n- Por definici\u00f3n $W_N^{kn} = \\left(e^{-j \\frac{2\\pi}{N}}\\right)^{kn} = e^{-j \\frac{2\\pi}{N}kn}$\n- Por periodicidad $W_N^{2(N-1)} = W_N^{2(N-1) - N} = W_N^{N-2}$\n- Tambi\u00e9n se tiene simetr\u00eda herm\u00edtica: $W_N^{k(-n)} = W_N^{-kn} = (W_N^{kn})^*$\n- $\\Omega$ es una matriz cuadrada y sim\u00e9trica \n\n***\nLa DFT tiene complejidad cuadr\u00e1tica: $N^2$ multiplicaciones y $N$ sumas\n***\n\n\n```python\nF_s = 100\nt = np.linspace(0, 2, 2*F_s, endpoint=False)\nx = np.sin(2*np.pi*t*5) + 0.5*np.cos(2*np.pi*t*12 + np.pi/4)\n\nfig, ax = plt.subplots(figsize=(7, 3), tight_layout=True)\nax.plot(t, x)\nax.set_xlabel('Tiempo [s]'); ax.set_ylabel('Amplitud');\n\n\ndef DFT(x):\n N = len(x)\n W_N = np.exp(-1j*2*np.pi/N)\n n = np.arange(N)\n Omega = W_N**(n*n.reshape(1,-1).T)\n return np.dot(Omega, x)\n\nfig, ax = plt.subplots(figsize=(7, 3), tight_layout=True)\nax.plot(np.linspace(0, F_s, num=len(t)), np.abs(DFT(x)))\nax.set_xlabel('Frecuencia [Hz]'); \ndisplay(np.angle(DFT(x))[np.abs(DFT(x)) > 10]*180/np.pi);\n```\n\n***\n\n# Transformada R\u00e1pida de Fourier (FFT)\n\n***\n\n- La computaci\u00f3n de la DFT tiene complejidad $\\mathcal{O}(N^2)$\n- Existe una aproximaci\u00f3n num\u00e9rica con complejidad $\\mathcal{O}(N\\log N)$: la Fast Fourier Transform (FFT). \n\nEl algoritmo *radix-2* obtiene una FFT recursiva que explota las simetr\u00edas en la DFT\n\n$$\n\\begin{align}\nS[k] &= \\sum_{n=0}^{N-1} s[n] W_N^{kn} \\nonumber \\\\\n&= \\sum_{n=0}^{N/2-1} s[2n] W_N^{k 2n} + \\sum_{n=0}^{N/2-1} s[2n+1] W_N^{k(2n+1)} \\nonumber \\\\\n&= \\sum_{n=0}^{N/2-1} s[2n] W_{N/2}^{kn} + W_N^{k} \\sum_{n=0}^{N/2-1} s[2n+1] W_{N/2}^{kn} \\nonumber \\\\\n&= S_E[k] + W_N^{k} S_O[k] ~~ \\forall k \\in [0,N/2] \\nonumber \n\\end{align} \n$$\n\nNotar que se calculan dos \"medias\" DFT\n\nPor periodicidad de la DFT tenemos que\n$$\n\\begin{align}\nS_E[k + N/2] &= \\sum_{n=0}^{N/2-1} s[2n] W_{N/2}^{(k+N/2)n} \\nonumber \\\\\n&= \\sum_{n=0}^{N/2-1} s[2n] W_{N/2}^{kn} \\exp \\left(-j2\\pi n \\right) = S_E[k], \\nonumber\n\\end{align}\n$$\n\ne igualmente\n\n$$\nS_O[k + N/2] = S_O[k],\n$$\n\njuntando ambos tenemos que\n$$\n\\begin{align}\nS[k + N/2] &= S_E[k + N/2] + W_{N}^{(k+N/2)} S_O[k + N/2] \\nonumber \\\\\n&= S_E[k] + W_{N}^{k} \\exp \\left(-j\\pi\\right) S_O[k] \\nonumber \\\\\n&= S_E[k] - W_{N}^{k} S_O[k] \\nonumber \n\\end{align}\n$$\n\nes decir\n\n$$\n\\begin{align}\nS[k] &= S_E[k] + W_{N}^{k} S_O[k] \\nonumber \\\\\nS[k + N/2] &= S_E[k] - W_{N}^{k} S_O[k] \\quad \\forall k \\in [0,N/2] \\nonumber \n\\end{align}\n$$\n\n- La DFT de $k$ y $k+N/2$ difieren en un signo\n- Se han explotado las simetr\u00edas de la DFT para reducir el costo computacional\n\nDiagramas de mariposa: http://www.themobilestudio.net/the-fourier-transform-part-14\n\n\n```python\nt = np.linspace(0, 10, num=20)\nx = np.random.randn(20)\nplt.figure(figsize=(7, 4))\nplt.plot(t, x, c='k', alpha=0.5)\nplt.scatter(t[::2], x[::2], marker='x', zorder=100)\nplt.scatter(t[1::2], x[1::2], marker='o', zorder=100);\n```\n\n\n```python\nnp.set_printoptions(precision=3)\ndisplay(scipy.fftpack.fft(x),\n DFT(x))\n```\n\n\n```python\nS = DFT(x) # N*N multiplicaciones\nSe = DFT(x[0::2]) # N/2*N/2 \nSo = DFT(x[1::2]) # N/2*N/2 \n# Se y So = N*N/2 multiplicaciones\ndisplay(S[:10],\n Se + np.exp(-1j*2*np.pi*np.arange(10)/len(x))*So,\n S[10:],\n Se - np.exp(-1j*2*np.pi*np.arange(10)/len(x))*So)\n```\n\n# Trabajo pr\u00e1ctico\n\nGenere una se\u00f1al y calcule su transformada de Fourier escribiendo una rutina en C++\n\nOptimize su rutina usando el algoritmo FFT\n\nParalelice el algoritmo FFT usando OpenMP y CUDA\n\n\n```python\ntimes, dt = np.linspace(0, 10, num=2048, retstep=True)\nfreqs = 0.01 + 20*np.random.rand(5)\nphases = np.pi*np.random.randn(5)\nsignal = np.zeros(shape=(len(times),))\nfor freq, phase in zip(freqs, phases):\n signal += np.cos(2.0*np.pi*times*freq + phase)\n \nfig, ax = plt.subplots()\nax.plot(times, signal)\nnp.savetxt('signal.dat', signal)\n```\n\n\n \n\n\n\n\n\n\n\n```python\nfig, ax = plt.subplots(2, figsize=(7, 4))\nSIGNAL = scipy.fftpack.fft(signal, n=len(times))\nSIGNAL_plot = scipy.fftpack.fftshift(SIGNAL)\nfreqs_plot = scipy.fftpack.fftshift(scipy.fftpack.fftfreq(n=len(times), d=dt))\nax[0].plot(freqs_plot, np.abs(SIGNAL_plot))\nax[1].plot(freqs_plot, np.angle(SIGNAL_plot));\n```\n\n\n \n\n\n\n\n\n\n\n```python\n#%%bash\n!make\n%timeit -n1 !./prog 0\n%timeit -n1 !./prog 1\n#!./prog 1\n!head spectrum.dat\n!make clean\n```\n\n\n```python\nSIGNAL_c = np.loadtxt(\"spectrum.dat\")\nSIGNAL_c_complex = SIGNAL_c[:, 0] + 1j*SIGNAL_c[:, 1]\nnp.allclose(SIGNAL, SIGNAL_c_complex, rtol=1e-9)\n```\n\n\n```python\nfig, ax = plt.subplots(2, figsize=(7, 4), tight_layout=True)\nax[0].plot(freqs_plot, np.abs(SIGNAL_plot), alpha=0.5)\nSIGNAL_c_plot = scipy.fftpack.fftshift(SIGNAL_c_complex)\nax[0].plot(freqs_plot, np.abs(SIGNAL_c_plot), alpha=0.5)\nax[1].plot(freqs_plot, np.abs(SIGNAL_plot) - np.abs(SIGNAL_c_plot));\n```\n\n# Ap\u00e9ndices\n\n## Transformada de Fourier bidimensional\n\nLa DFT se puede aplicar a funciones multi-dimensionales. En el caso discreto de una se\u00f1al bidimensional $g[n_1, n_2]$ con \u00edndices $n_1 \\in [0, N_1-1]$ y $n_2 \\in [0, N_2-1]$ tenemos\n\n$$\nG[k_1, k_2] = \\sum_{n_1=0}^{N_1-1} \\sum_{n_2=0}^{N_2-1} g[n_1, n_2] \\exp \\left ( -j2\\pi \\left[\\frac{n_1 k_1}{N_1} + \\frac{n_2 k_2}{N_2} \\right] \\right)\n$$\ny su inversa\n\n$$\ng[n_1, n_2] = \\frac{1}{N_1 N_2}\\sum_{k_1=0}^{N_1-1} \\sum_{k_2=0}^{N_2-1} G[k_1, k_2] \\exp \\left ( j2\\pi \\left[\\frac{n_1 k_1}{N_1} + \\frac{n_2 k_2}{N_2} \\right] \\right)\n$$\n\nNotemos que\n\n\\begin{align}\nG[k_1, k_2] &= \\sum_{n_1=0}^{N_1-1} \\left(\\sum_{n_2=0}^{N_2-1} g[n_1, n_2] \\exp \\left (-j2\\pi \\frac{n_2 k_2}{N_2}\\right) \\right) \\exp \\left (-j2\\pi \\frac{n_1 k_1}{N_1}\\right) \\\\\n&= \\sum_{n_1=0}^{N_1-1} \\gamma_{n_1}[n_2] \\exp \\left (-j2\\pi \\frac{n_1 k_1}{N_1}\\right),\n\\end{align}\n\n*i.e.* se descompone como dos DFT de una dimensi\u00f3n. En cada paso podemos usar la FFT\n\n\n```python\nx = np.arange(0, 32, step=1)\nX, Y = np.meshgrid(x, x)\nfig, ax = plt.subplots(9, 9, figsize=(11, 11), tight_layout=False)\nfor n in range(9):\n for m in range(9):\n ax[n, m].matshow(np.cos(2.0*np.pi*X*m/len(x) + 2.0*np.pi*Y*n/len(x)), \n cmap=plt.cm.RdBu_r, vmin=-1, vmax=1)\n ax[n, m].axis('off')\n```\n\n\n \n\n\n\n\n\n\n\n```python\nimg_doge = color2bw(plt.imread('doge.jpg')) \n\nplt.figure(figsize=(8, 5))\nplt.imshow(img_doge)\nplt.colorbar(orientation='horizontal');\n```\n\n\n```python\nfrom scipy import fftpack\nfig, ax = plt.subplots(1, 2, figsize=(10, 5), tight_layout=True)\nS_img = fftpack.fft2(img_doge)\nim = ax[0].imshow(fftpack.fftshift(np.log(1.+np.abs(S_img))))\nfig.colorbar(im, ax=ax[0], orientation='horizontal')\nim = ax[1].imshow(fftpack.fftshift(np.angle(S_img))) # arctan(imag/real)\nfig.colorbar(im, ax=ax[1], orientation='horizontal');\n```\n\n\n```python\nplt.close('all'); fig, ax = plt.subplots(1, 2, figsize=(6, 3), tight_layout=True);\ncy, cx = img_doge.shape[0]/2, img_doge.shape[1]/2\nx = np.arange(0, img_doge.shape[1]); y = np.arange(0, img_doge.shape[0]);\nX, Y = np.meshgrid(x, y)\n\ndef update(sigma1=1, sigma2=1):\n for ax_ in ax:\n ax_.cla()\n ax_.axis('off')\n mask1 = np.exp(-(((X-cx)/sigma1)**2 + ((Y-cy)/sigma1)**2)) \n mask2 = np.exp(-(((X-cx)/sigma2)**2 + ((Y-cy)/sigma2)**2)) \n im = ax[0].imshow(fftpack.fftshift(np.log(1.0+np.abs(S_img)))*(mask1-mask2))\n im = ax[1].imshow(np.real(fftpack.ifft2(fftpack.ifftshift(fftpack.fftshift(S_img)*(mask1-mask2)))))\ninteract(update, sigma1=FloatSlider_nice(min=1, max=200.0, value=200, description=\"$\\sigma_1$\"),\n sigma2=FloatSlider_nice(min=1, max=200.0, value=1, description=\"$\\sigma_2$\"));\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "719b9f0e4d558e048d82e07d1e9a82c2f122ddd6", "size": 795032, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "labs/lab07-FFT/Fast Fourier Transform.ipynb", "max_stars_repo_name": "magister-informatica-uach/INFO335", "max_stars_repo_head_hexsha": "da55bdd2feef66f72bad51d812a1e7b8d467d000", "max_stars_repo_licenses": ["Unlicense"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-07-05T02:42:49.000Z", "max_stars_repo_stars_event_max_datetime": "2021-07-05T02:42:49.000Z", "max_issues_repo_path": "labs/lab07-FFT/Fast Fourier Transform.ipynb", "max_issues_repo_name": "magister-informatica-uach/INFO335", "max_issues_repo_head_hexsha": "da55bdd2feef66f72bad51d812a1e7b8d467d000", "max_issues_repo_licenses": ["Unlicense"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "labs/lab07-FFT/Fast Fourier Transform.ipynb", "max_forks_repo_name": "magister-informatica-uach/INFO335", "max_forks_repo_head_hexsha": "da55bdd2feef66f72bad51d812a1e7b8d467d000", "max_forks_repo_licenses": ["Unlicense"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 210.2147012163, "max_line_length": 267335, "alphanum_fraction": 0.8693461395, "converted": true, "num_tokens": 4976, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.7549149813536518, "lm_q1q2_score": 0.4301903037166474}} {"text": "```python\n# \uadf8\ub798\ud504, \uc218\ud559 \uae30\ub2a5 \ucd94\uac00\n# Add graph and math features\nimport pylab as py\nimport numpy as np\nimport numpy.linalg as nl\n# \uae30\ud638 \uc5f0\uc0b0 \uae30\ub2a5 \ucd94\uac00\n# Add symbolic operation capability\nimport sympy as sy\n\n\n```\n\n# \uad6c\uae00 \ud398\uc774\uc9c0\ub7ad\ud06c \uc54c\uace0\ub9ac\ub4ec
                                        The PageRank Algorithm of Google\n\n\n\n* \uc704\ud0a4\ubc31\uacfc \uae30\uc5ec\uc790, '\ud398\uc774\uc9c0\ub7ad\ud06c', \uc704\ud0a4\ubc31\uacfc, , 2017\ub144 9\uc6d4 1\uc77c, 02:44 UTC, [2018\ub144 7\uc6d4 31\uc77c\uc5d0 \uc811\uadfc] \n* Wikipedia contributors, 'PageRank', Wikipedia, The Free Encyclopedia, 16 July 2018, 19:43 UTC, [accessed 31 July 2018] \n\n\n\n\ub2e4\uc74c \uadf8\ub9bc\uc744 \uc0b4\ud3b4 \ubcf4\uc790.
                                        \nLet's take a look a the following figure.\n\n\n\n\n\n\n\n\uc704 \uc5f0\uacb0 \uc0c1\ud0dc\ub97c \ub2e4\uc74c\uacfc \uac19\uc740 \ud589\ub82c\ub85c \ud45c\uc2dc\ud560 \uc218 \uc788\ub2e4.
                                        The connectivity status can be represented as a matrix as follows.\n\n\n\n| m | A | B | C | D | E | F | G1 | G2 | G3 | G4 | G5 |\n|:---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|\n| A | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |\n| B | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |\n| C | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |\n| D | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |\n| E | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |\n| F | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |\n| G1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |\n| G2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |\n| G3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |\n| G4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |\n| G5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |\n\n\n\n\uc608\ub97c \ub4e4\uc5b4 \uccab \ud589\uc5d0\uc11c A \uc808\uc810\uc73c\ub85c \ub4e4\uc5b4\uc624\ub294 \uc5f0\uacb0\uc120\uc740 D \uc808\uc810\uc5d0\uc11c \uc624\ub294 \uac83 \ud558\ub098 \ubfd0\uc774\ub2e4.
                                        \nFor example, in the first row, node A has only one incoming link from node D.\n\n\n\n\n```python\nm = np.array(\n [\n [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],\n [0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0],\n [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],\n [0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1],\n [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],\n [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n ]\n)\n\n\n```\n\n\uac01 \uc5f4\uc758 \ud569\uc774 1\uc774 \ub418\ub3c4\ub85d \ub9cc\ub4e0\ub2e4.
                                        Make sum of each row to one.\n\n\n\n\n```python\ns = m.sum(axis=0)\n\n\n```\n\n\n```python\nfor j in range(m.shape[1]):\n if 1 < s[j]:\n si = 1.0 / s[j]\n for i in range(m.shape[0]):\n m[i, j] *= si\n\n\n```\n\n\uc774\uc81c \ud589\ub82c\uc758 \uace0\uc720\ubca1\ud130\ub97c \uad6c\ud55c\ub2e4.
                                        Now let's find the eigenvector of the matrix.\n\n\n\n\n```python\neval, evac = nl.eig(m)\n\n\n```\n\n\n```python\neval\n\n\n```\n\n\n```python\npy.matshow(evac)\npy.colorbar()\npy.axis('equal')\n\n\n```\n\n## Final Bell
                                        \ub9c8\uc9c0\ub9c9 \uc885\n\n\n\n\n```python\n# stackoverfow.com/a/24634221\nimport os\nos.system(\"printf '\\a'\");\n\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "b9c80a3961ab13a017aca9e3c93df4732cc06718", "size": 5742, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "200_Google_PageRank.ipynb", "max_stars_repo_name": "kangwonlee/19ECA-60-lin-alg-2", "max_stars_repo_head_hexsha": "2885b92d4a8cc4c31fb575931e693a3c82390e41", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "200_Google_PageRank.ipynb", "max_issues_repo_name": "kangwonlee/19ECA-60-lin-alg-2", "max_issues_repo_head_hexsha": "2885b92d4a8cc4c31fb575931e693a3c82390e41", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "200_Google_PageRank.ipynb", "max_forks_repo_name": "kangwonlee/19ECA-60-lin-alg-2", "max_forks_repo_head_hexsha": "2885b92d4a8cc4c31fb575931e693a3c82390e41", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.2278481013, "max_line_length": 200, "alphanum_fraction": 0.4188436085, "converted": true, "num_tokens": 1594, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.4301396890660718}} {"text": "# Basics of Wick&d\n\n## Loading the module\n\nTo use wick&d you will have to first import the module `wicked`. Here we abbreviate it with `w` for convenience. We also define a function (`latex`) to display objects in LaTeX format.\n\n\n```python\nimport wicked as w\nfrom IPython.display import display, Math, Latex\n\ndef latex(expr):\n \"\"\"Function to render any object that has a member latex() function\"\"\"\n display(Math(expr.latex()))\n```\n\n## Defining orbital spaces\n\nTo use wick&d, you need to specify what type of reference state is used as a vacuum state. This information is stored in the `OrbitalsSpaceInfo` class, which holds information about the orbital space defined. We get access to this object (a singleton) via the function `osi()`. Calling the print function we can the see information about the orbital spaces defined. When wick&d is initialized, no orbital space is defined and no text is printed:\n\n\n```python\nosi = w.osi()\nprint(str(osi))\n```\n\n## Defining orbital spaces\n\nTo define an orbital space one must specify:\n- The label (a single character, e.g., 'o' for occupied)\n- The type of operator field (a string, currently the only option is `\"fermion\"`).\n- The type of reference state associated with this space (a string, one of `[\"occupied\", \"unoccupied\", \"general\"]`)\n- The *pretty* indices that we associate with this space (e.g., `['i','j','k']` for occupied orbitals)\n\nWick&d defines three types of occupations associated to each space:\n- **Occupied** (`occupied`): orbitals that are occupied in the vacuum (applies to fermions)\n- **Unoccupied** (`unoccupied`): orbitals that are unoccupied in the vacuum\n- **General** (`general`): orbitals that are partially occupied in the vacuum\n\n## Examples of different reference states/orbital spaces\n\n### Physical vacuum\nThe reference state is the physical vacuum state\n\\begin{equation}\n| - \\rangle\n\\end{equation}\nIn this case all orbitals are unoccupied. The following code initializes a single orbital space with label `p` and unoccupied orbitals:\n\n\n```python\nw.reset_space()\nw.add_space(\"p\", \"fermion\", \"unoccupied\", ['a','b','c','d','e','f'])\n```\n\n### Single determinant (Fermi vacuum)\n\nThe reference state is the determinant\n\\begin{equation}\n| \\Phi \\rangle = |\\psi_1 \\cdots \\psi_N \\rangle\n\\end{equation}\nIn this case you need to specify only two spaces, occupied and unoccupied orbitals. The following code initializes two spaces: 1) occupied (`o`) and 2) virtual (`v`) orbitals,\n\n\n```python\nw.reset_space()\nw.add_space(\"o\", \"fermion\", \"occupied\", ['i','j','k','l','m'])\nw.add_space(\"v\", \"fermion\", \"unoccupied\", ['a','b','c','d','e','f'])\n```\n\n### Linear combination of determinants (Generalized vacuum)\n\nThis case requires three orbital spaces: core (`occupied`), active (`general`), and virtual (`unoccupied`).\nThe reference is a linear combination of determinants\n\\begin{equation}\n| \\Psi \\rangle = \\sum_\\mu c_\\mu | \\Phi_\\mu \\rangle \n\\end{equation}\nwhere each determinant $| \\Phi_\\mu \\rangle $ is \n\\begin{equation}\n| \\Phi_\\mu \\rangle = \\underbrace{\\hat{a}^\\dagger_u \\hat{a}^\\dagger_v \\cdots}_{\\text{active}} | \\Phi_\\mathrm{c} \\rangle\n\\end{equation}\nwhere the core determinant $\\Phi_\\mathrm{c} \\rangle$ is defined as\n\\begin{equation}\n| \\Phi_\\mathrm{c} \\rangle = |\\psi_1 \\cdots \\psi_{N_\\mathrm{c}} \\rangle.\n\\end{equation}\nTo specify this reference you can use the code:\n\n\n```python\nw.reset_space()\nw.add_space(\"c\", \"fermion\", \"occupied\", ['i','j','k','l','m'])\nw.add_space(\"a\", \"fermion\", \"general\", ['u','v','w','x','y','z'])\nw.add_space(\"v\", \"fermion\", \"unoccupied\", ['a','b','c','d','e','f'])\n```\n\nWe can now verify that the orbital spaces have been updated\n\n\n```python\nprint(str(osi))\n```\n", "meta": {"hexsha": "b1892d33faa3431303ab4841dc7fc0af10c9272a", "size": 6074, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/01-Basics.ipynb", "max_stars_repo_name": "fevangelista/pyWicked", "max_stars_repo_head_hexsha": "9bc0e13f6e45c86222ea95fdadf1cb66eb59862f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tutorials/01-Basics.ipynb", "max_issues_repo_name": "fevangelista/pyWicked", "max_issues_repo_head_hexsha": "9bc0e13f6e45c86222ea95fdadf1cb66eb59862f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/01-Basics.ipynb", "max_forks_repo_name": "fevangelista/pyWicked", "max_forks_repo_head_hexsha": "9bc0e13f6e45c86222ea95fdadf1cb66eb59862f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 31.9684210526, "max_line_length": 451, "alphanum_fraction": 0.5735923609, "converted": true, "num_tokens": 1005, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321720225278, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.43013968168818734}} {"text": "# Construction of the Model of Adaptive Resistance in Melanoma (MARM1)\n\nIn this notebook we will provide a step-by-step construction of the MARM model. The model is constructed using the rule-based modeling tool PySB . The model definition files generated by this script in PySB (*MARM1.py*), BGN (*MARM1.bng*) and SBML (*MARM1.sbml*) formats will replace the current ones in the MARM1 supplement directory and can be used to simulate different versions of the MARM1 model. Please see the Jupyter Notebooks *MARM1_simulation_single_run.ipynb* and *MARM1_simulation_multiple_conditions.ipynb* to run simulations using the generated model.To start, we import all required pysb classes and instantiate the model:\n\n\n```python\nfrom pysb import Model, Monomer, Parameter, Expression, Rule, Observable, Initial, Annotation, EnergyPattern, ANY\nfrom pysb.bng import generate_equations\nfrom pysb.export import export\nfrom sympy import exp, log\n\nModel();\n```\n\nIn the following, we split the model into individual submodules of strongly interacting components, which we will describe individually. The model describes protein and drug species as intracellular concentrations in uM at a volume of `1pL = 1e-12L`. For ligands and drugs we assume that the extracellular compartment is much bigger than the intracellular compartement and that equilibration is fast such that concentrations can be assumed to be constanct over time (implemented by specifying `fixed=True` in the initialization). For EGF, the parameter that specifies the abundance, `EGF_0` is assumed to have units [ng/ml] and is accordingly transformed into [uM] using the appropriate molecular weight `m_Da_EGF`. For the drugs Vemurafenib (`RAFi`) and Cobimetinib (`MEKi`), the respective parameters, `RAFi_0` and `MEKi_0`, are assumed to be in uM. The unit of time is hours.\n\n\n```python\nExpression('N_Avogadro', 6.02214076000000e+23)\nExpression('volume', 1.00000000000000e-12)\nExpression('m_Da_EGF', 6200.00000000000)\n\nMonomer('RAFi', ['raf'])\nAnnotation(RAFi, 'http://identifiers.org/chebi/63637', 'is')\nParameter('RAFi_0', 0.0)\nExpression('initRAFi', RAFi_0)\nInitial(RAFi(raf=None), initRAFi, fixed=True)\n\nMonomer('MEKi', ['mek'])\nAnnotation(MEKi, 'http://identifiers.org/chebi/90851', 'is')\nParameter('MEKi_0', 0.0)\nExpression('initMEKi', MEKi_0)\nInitial(MEKi(mek=None), initMEKi, fixed=True)\n\nMonomer('EGF', ['rtk'])\nAnnotation(EGF, 'http://identifiers.org/uniprot/P01133', 'is')\nParameter('EGF_0', 0.0)\nExpression('initEGF', 6.02214085854916e+23*EGF_0*(m_Da_EGF*N_Avogadro)**(-1))\nInitial(EGF(rtk=None), initEGF, fixed=True);\n```\n\n## EGFR Signaling\nThe EGFR submodule describes the interaction between Egfr, Egf, Grb2, Sos1 and Ras (which describes the action K-Ras, N-Ras and H-Ras). Here, we instantiate\nthe respective monomers and provide uniprot IDs as monomer annotations.\n\n\n```python\nMonomer('EGFR', ['SH2', 'rtk', 'rtkf'])\nAnnotation(EGFR, 'http://identifiers.org/uniprot/P00533', 'is')\n\nMonomer('GRB2', ['SH2', 'SH3'])\nAnnotation(GRB2, 'http://identifiers.org/uniprot/P62993', 'is')\n\nMonomer('SOS1', ['S1134', 'SH3', 'ras'], {'S1134': ['p', 'u']})\nAnnotation(SOS1, 'http://identifiers.org/uniprot/Q07889', 'is')\n\nMonomer('RAS', ['raf', 'sos1', 'state'], {'state': ['gdp', 'gtp']})\nAnnotation(RAS, 'http://identifiers.org/uniprot/P01111', 'is')\nAnnotation(RAS, 'http://identifiers.org/uniprot/P01116', 'is')\nAnnotation(RAS, 'http://identifiers.org/uniprot/P01112', 'is');\n```\n\n## EGF activation\nTo describe activation of Egfr through Egf, we implement base binding rates between Egf and Egfr. We implement these rules as energy rules, which allow us to later use the energy-based formalism in BioNetGen to describe allosteric interactions between the two molecules. Energy rules\nare parameterised using three distinct parameters: the `kD` parameter which describes the affinity between the two molecules, the `kf` parameter which describes the timescale of the interaction and the `phi` parameter which takes values between 0 and 1 and balance whether allosteric interactions modulate the affinity by changing the association rate (`kf`, only association if `phi=0`) or the dissociation rate (`kr = kf*kD`, only dissociation if `phi=1`).\n\nWe implement the ligand-receptor binding using the `rtk` binding site on the ligand and the `rtkf` binding site on the receptor. We implement the receptor-receptor binding using the `rtk` binding site on the receptor. This formalism simplifies future extensions of the model to additional ligands or receptors by ensuring that each receptor can only bind one ligand and one receptor.\n\n\n```python\n# binding Egf-Egfr\nParameter('bind_EGF_EGFR_kf', 10.0)\nParameter('bind_EGF_EGFR_kD', 0.01)\nParameter('bind_EGF_EGFR_phi', 1.0)\nExpression('bind_EGF_EGFR_kr', bind_EGF_EGFR_kD*bind_EGF_EGFR_kf)\nExpression('Gf_bind_EGF_EGFR', log(bind_EGF_EGFR_kD))\nExpression('Ea0_bind_EGF_EGFR', -bind_EGF_EGFR_phi*log(bind_EGF_EGFR_kD) - log(bind_EGF_EGFR_kf))\nRule('EGF_and_EGFR_bind_and_dissociate',\n EGF(rtk=None) + EGFR(rtkf=None) | EGF(rtk=1) % EGFR(rtkf=1),\n bind_EGF_EGFR_phi, Ea0_bind_EGF_EGFR, energy=True)\nEnergyPattern('ep_bind_EGF_EGFR', EGF(rtk=1) % EGFR(rtkf=1), Gf_bind_EGF_EGFR)\n\n# binding Egfr-Egfr\nParameter('bind_EGFR_EGFR_kf', 10.0)\nParameter('bind_EGFR_EGFR_kD', 100.0)\nParameter('bind_EGFR_EGFR_phi', 1.0)\nExpression('bind_EGFR_EGFR_kr', bind_EGFR_EGFR_kD*bind_EGFR_EGFR_kf)\nExpression('Gf_bind_EGFR_EGFR', log(bind_EGFR_EGFR_kD))\nExpression('Ea0_bind_EGFR_EGFR', -bind_EGFR_EGFR_phi*log(bind_EGFR_EGFR_kD) - log(bind_EGFR_EGFR_kf))\nRule('EGFR_and_EGFR_bind_and_dissociate',\n EGFR(rtk=None) + EGFR(rtk=None) | EGFR(rtk=1) % EGFR(rtk=1),\n bind_EGFR_EGFR_phi, Ea0_bind_EGFR_EGFR, energy=True)\nEnergyPattern('ep_bind_EGFR_EGFR', EGFR(rtk=1) % EGFR(rtk=1), Gf_bind_EGFR_EGFR);\n```\n\nTo implement allosteric interaction between Egf ligands and Egfr , we add two energy patterns that allow modulation of the affinities of Egfr-Egf and Egfr-Egfr interactions through binding of ligands.\n\n\n```python\n# Egf mediated affinity modulation\nParameter('ep_EGFR_EGFR_mod_EGF_single_deltaG', 0.001)\nParameter('ep_EGFR_EGFR_mod_EGF_double_deltaG', 0.001)\nExpression('ep_EGFR_EGFR_mod_EGF_single_Gf', log(ep_EGFR_EGFR_mod_EGF_single_deltaG))\nExpression('ep_EGFR_EGFR_mod_EGF_double_Gf', \n log(ep_EGFR_EGFR_mod_EGF_double_deltaG) + log(ep_EGFR_EGFR_mod_EGF_single_deltaG))\n\nEnergyPattern('ep_EGFR_EGFR_mod_EGF_single',\n EGF(rtk=2) % EGFR(rtk=1, rtkf=2) % EGFR(rtk=1, rtkf=None),\n ep_EGFR_EGFR_mod_EGF_single_Gf)\nEnergyPattern('ep_EGFR_EGFR_mod_EGF_double',\n EGF(rtk=2) % EGF(rtk=3) % EGFR(rtk=1, rtkf=2) % EGFR(rtk=1, rtkf=3),\n ep_EGFR_EGFR_mod_EGF_double_Gf)\n\nAnnotation(EGF_and_EGFR_bind_and_dissociate, 'http://identifiers.org/pubmed/16946702', 'isDescribedBy')\nAnnotation(EGFR_and_EGFR_bind_and_dissociate, 'http://identifiers.org/pubmed/16946702', 'isDescribedBy');\n```\n\n## GRB2 recruitement\nRecruitement of Grb2 to receptor requires transactivation of multiple Egfr phosphorylation sites that are recognized by the Grb2 Src2 homology 2 (SH2) domain . To simplify these interactions we require Egfr to be bound to an activating ligand-Egfr complex to recruit Grb2 to the plasma membrane. To simplify notation the binding site on the Egfr monomer is also called `SH2`, although Egfr does not harbor any SH2 domain.\n\nThe dissociation is implemented as unconditional rule to avoid the creation of irreversible states (e.g, dissociation of receptor dimers could protect from dissociation of Grb2). Accordingly, this rule is not implemented as energy rule, as energy rules always require the same conditions for association and dissociation rules. Nevertheless, we parameterize the rule using `kf` and `kD` parameters, although, no `phi` parameter is required here as the rule kinetics are never modulated through energy patterns. \n\n\n\n```python\n# EGFR-Grb2 binding\nParameter('bind_EGFR_GRB2_kf', 10.0)\nParameter('bind_EGFR_GRB2_kD', 0.01)\nExpression('bind_EGFR_GRB2_kr', bind_EGFR_GRB2_kD*bind_EGFR_GRB2_kf)\nRule('transactivated_EGFR_dimers_bind_GRB2',\n EGFR(SH2=None, rtk=1) % EGFR(rtk=1, rtkf=ANY) + GRB2(SH2=None)\n >>\n EGFR(SH2=2, rtk=1) % EGFR(rtk=1, rtkf=ANY) % GRB2(SH2=2), \n bind_EGFR_GRB2_kf)\n\nRule('GRB2_dissociates_from_EGFR',\n EGFR(SH2=1) % GRB2(SH2=1) >> EGFR(SH2=None) + GRB2(SH2=None),\n bind_EGFR_GRB2_kr)\n\nAnnotation(transactivated_EGFR_dimers_bind_GRB2, 'http://identifiers.org/pubmed/16777603', 'isDescribedBy');\n```\n\nThe recruitement of Grb2 to Egfr is not oly important for signal transduction, but also regulates the balance between slower, clathrin-independent and faster, clathrin-dependent (Grb2 dependent) receptor endocytosis . \n\nWe describe the baseline Egfr expression (determined by clathrin-independent endocytosis) using basal Egfr expression and synthesis rules. Similar to binding rules, synthesis rules are parameterized equilibrium abundance parameter `[]_eq`, which regulates steady-state behaviour, and degradation rate `[]_kdeg`, which regulates the timescale of the rule. The equilibrium abundance can be modulated by the `[]_crispr` factor by up or down-regulating the synthesis rate `[]_ksyn`. To simplify specification of estimation boundaries for the `[...]_eq` parameter, we assume that the parameter is specified in [molecules/cell] and accordingly rescale the respective synthesis rate to [uM/h].\n\nWe implement clathrin-dependent endocytosis as three seperate rules for Egfr dimers with one Grb2 bound, with two Grb2 bound and Grb2 bound Egfr monomers. The `delete_molecules=True` parameter encodes that only Egfr molecules are degradede and all other molecules are remain as possibly fragmented comlexes. The formulation as three seperate rules ensures that Egfr endocytosis rate constant with respect to Grb2 stochiometry in the complex and that dissociation of Egfr homodimers does not protect from endocytosis.\n\n\n```python\n# basal synthesis + degradation\nParameter('EGFR_eq', 10000.0)\nParameter('synthesize_ERKphosphop_EGFR_kdeg', 10.0)\nParameter('EGFR_crispr', 1.0)\nExpression('synthesize_ERKphosphop_EGFR_ksyn', \n 1000000.0*EGFR_crispr*EGFR_eq*synthesize_ERKphosphop_EGFR_kdeg/(N_Avogadro*volume))\n\nRule('basal_synthesis_EGFR',\n None >> EGFR(SH2=None, rtk=None, rtkf=None),\n synthesize_ERKphosphop_EGFR_ksyn)\nRule('basal_degradation_EGFR',\n EGFR() >> None,\n synthesize_ERKphosphop_EGFR_kdeg, delete_molecules=True)\n\n# Grb2 mediated degradation\nParameter('catalyze_GRB2_RTKSH2ANY_deg_kcat', 10.0)\nRule('GRB2_degrades_activeRTK_dimers',\n EGFR(SH2=ANY, rtk=1) % EGFR(SH2=None, rtk=1) >> None,\n catalyze_GRB2_RTKSH2ANY_deg_kcat, delete_molecules=True)\nRule('GRB2_degrades_doubleactiveRTK_dimers',\n EGFR(SH2=ANY, rtk=1) % EGFR(SH2=ANY, rtk=1) >> None,\n catalyze_GRB2_RTKSH2ANY_deg_kcat, delete_molecules=True)\nRule('GRB2_degrades_activeRTK_monomers',\n EGFR(SH2=ANY, rtk=None) >> None,\n catalyze_GRB2_RTKSH2ANY_deg_kcat, delete_molecules=True);\n```\n\n## SOS1 and Ras activation\nGrb2 recruitement by Egfr also induces association of Sos1 to the plasma-membrane, where it can activate Ras by inducing nucleotide exchange . In the model we implement that Grb2 and Sos1 preform complexes in the cytoplasm and that Sos1 is activated through association to the plasma-membrane, which is mediated recruitement of Grb2 by Egfr. Given that the training data does not include any data or perturbations at the Ras level, we implement a simple catalysis rule for activation of Ras, where SOS1 only binds GDP bound Ras and catalysis is implemented without homonucleotide exchange, allosteric feedback mechanisms or interactions with cdc25 . Similarly, we implement inactivation of Ras through GTPase activation proteins, such as NF1, as a simple linear reaction with constant rate.\n\nAs previous binding reactions, the rules are paramterized using `kf` and `kD` values. As for the Grb2-Egfr interaction, the binding domains on Sos1 and Grb2 are implemented as `SH3` sites on both molecules, simplifying the much more intricate binding dynamics . The translocation of Grb2 and Sos1 to the plasma membrane is not explicitely implemented in the model but implicitely encoded by requiring Grb2 and Egfr association for binding to Ras. To reduce correlations between parameter estimates, we implement the inactivation rate `[...]_gdp_kcat` of Ras as product of the activation rate `[...]_gtp_kcat` and a scaling factor `[...]_gdp_kcatr`. The requirement of `raf=None` in the inactivation rule will be explained in the Section on [Raf activation](#Raf-activation).\n\n\n```python\n# binding Grb2-Sos1\nParameter('bind_GRB2_SOS1_kf', 10.0)\nParameter('bind_GRB2_SOS1_kD', 0.01)\nParameter('bind_GRB2_SOS1_phi', 1.0)\nExpression('bind_GRB2_SOS1_kr', bind_GRB2_SOS1_kD*bind_GRB2_SOS1_kf)\nExpression('Gf_bind_GRB2_SOS1', log(bind_GRB2_SOS1_kD))\nExpression('Ea0_bind_GRB2_SOS1', \n -bind_GRB2_SOS1_phi*log(bind_GRB2_SOS1_kD) - log(bind_GRB2_SOS1_kf))\nEnergyPattern('ep_bind_GRB2_SOS1', GRB2(SH3=1) % SOS1(SH3=1), Gf_bind_GRB2_SOS1)\nRule('GRB2_and_SOS1_bind_and_dissociate',\n GRB2(SH3=None) + SOS1(SH3=None) | GRB2(SH3=1) % SOS1(SH3=1),\n bind_GRB2_SOS1_phi, Ea0_bind_GRB2_SOS1, energy=True)\n\n\nAnnotation(GRB2_and_SOS1_bind_and_dissociate, 'http://identifiers.org/pubmed/7893993', 'isDescribedBy')\n\n# binding Sos1-Ras\nParameter('bind_SOS1_RAS_kf', 10.0)\nParameter('bind_SOS1_RAS_kD', 0.01)\nExpression('bind_SOS1_RAS_kr', bind_SOS1_RAS_kD*bind_SOS1_RAS_kf)\nRule('RTK_and_GRB2_bound_SOS1_binds_RASgdp',\n GRB2(SH2=ANY, SH3=1) % SOS1(SH3=1, ras=None) + RAS(sos1=None, state='gdp')\n >>\n GRB2(SH2=ANY, SH3=1) % RAS(sos1=2, state='gdp') % SOS1(SH3=1, ras=2),\n bind_SOS1_RAS_kf)\n\nRule('SOS1_dissociates_from_RAS',\n RAS(sos1=1) % SOS1(ras=1) >> SOS1(ras=None) + RAS(sos1=None),\n bind_SOS1_RAS_kr)\n\nAnnotation(RTK_and_GRB2_bound_SOS1_binds_RASgdp, 'http://identifiers.org/pubmed/26565026', 'isDescribedBy')\n\n# activation + inactivation Ras\nParameter('catalyze_SOS1_RAS_gtp_kcat', 0.01)\nParameter('catalyze_NF1_RAS_gdp_kcatr', 100.0)\nExpression('catalyze_NF1_RAS_gdp_kcat', catalyze_NF1_RAS_gdp_kcatr*catalyze_SOS1_RAS_gtp_kcat)\nRule('SOS1_catalyzes_RAS_guanosine_exchange',\n RAS(sos1=1, state='gdp') % SOS1(ras=1) >> SOS1(ras=None) + RAS(sos1=None, state='gtp'),\n catalyze_SOS1_RAS_gtp_kcat)\n\nRule('RAS_hydrolisis_GTP', \n RAS(raf=None, state='gtp') >> RAS(raf=None, state='gdp'), \n catalyze_NF1_RAS_gdp_kcat);\n```\n\n# ERK signaling\nThe Erk submodule describes the interaction between the two Rafs c-Raf and B-Raf, Mek (which describes Mek1 and Mek2) and Erk (which describes Erk1 and Erk2). Again, we instantiate the respective monomers and provide uniprot IDs as monomer annotations.\n\n\n```python\nMonomer('BRAF', ['AA600', 'RBD', 'mek', 'raf', 'rafi'], {'AA600': ['E']})\nAnnotation(BRAF, 'http://identifiers.org/uniprot/P15056', 'is')\n\nMonomer('CRAF', ['RBD', 'mek', 'raf', 'rafi'])\nAnnotation(CRAF, 'http://identifiers.org/uniprot/P04049', 'is')\n\nMonomer('MEK', ['Dsite', 'meki', 'phospho', 'raf'], {'phospho': ['p', 'u']})\nAnnotation(MEK, 'http://identifiers.org/uniprot/Q02750', 'is')\nAnnotation(MEK, 'http://identifiers.org/uniprot/P36507', 'is')\n\nMonomer('ERK', ['CD', 'phospho'], {'phospho': ['p', 'u']})\nAnnotation(ERK, 'http://identifiers.org/uniprot/P27361', 'is')\nAnnotation(ERK, 'http://identifiers.org/uniprot/P28482', 'is');\n```\n\n## Raf activation\nGTP-bound Ras activates Raf by recruiting it to the membrane where it is phosphorylated and dimerizes. As understanding of the exact mechanisms of this activation process are still incomplete, we implemented a simplistic model of this process that captures essential features of the interaction. We implement a base dimerization rate between Raf monomers and only allow binding of Raf monomers to GTP-bound Ras. To implement the Ras-mediated dimerization of Raf, we implement an energy pattern for Ras-Raf tetramers modulates both binding affinities.\n\nWe implement both binding rules as energy rules, where we assume that affinities for B-Raf and C-Raf are the same for all interactions. As energy rules require symmetric forward and backward reactions, inactivation of Ras would protect Raf from unbinding from Ras. Accordingly, we add two additional inactivation reactions that dissociate Raf from Ras and required that no Raf was bound in the baseline inactivation.\n\n\n```python\n# Raf-Raf binding\nParameter('bind_RAF_RAF_kf', 10.0)\nParameter('bind_RAF_RAF_kD', 0.01)\nParameter('bind_RAF_RAF_phi', 1.0)\nExpression('bind_BRAF_BRAF_kr', bind_RAF_RAF_kD*bind_RAF_RAF_kf)\nExpression('Gf_bind_BRAF_BRAF', log(bind_RAF_RAF_kD))\nExpression('Ea0_bind_BRAF_BRAF', -bind_RAF_RAF_phi*log(bind_RAF_RAF_kD) - log(bind_RAF_RAF_kf))\nExpression('bind_BRAF_CRAF_kr', bind_RAF_RAF_kD*bind_RAF_RAF_kf)\nExpression('Gf_bind_BRAF_CRAF', log(bind_RAF_RAF_kD))\nExpression('Ea0_bind_BRAF_CRAF', -bind_RAF_RAF_phi*log(bind_RAF_RAF_kD) - log(bind_RAF_RAF_kf))\nExpression('bind_CRAF_CRAF_kr', bind_RAF_RAF_kD*bind_RAF_RAF_kf)\nExpression('Gf_bind_CRAF_CRAF', log(bind_RAF_RAF_kD))\nExpression('Ea0_bind_CRAF_CRAF', -bind_RAF_RAF_phi*log(bind_RAF_RAF_kD) - log(bind_RAF_RAF_kf))\nRule('BRAF_and_BRAF_bind_and_dissociate',\n BRAF(raf=None) + BRAF(raf=None) | BRAF(raf=1) % BRAF(raf=1),\n bind_RAF_RAF_phi, Ea0_bind_BRAF_BRAF, energy=True)\nRule('BRAF_and_CRAF_bind_and_dissociate',\n BRAF(raf=None) + CRAF(raf=None) | BRAF(raf=1) % CRAF(raf=1),\n bind_RAF_RAF_phi, Ea0_bind_BRAF_CRAF, energy=True)\nRule('CRAF_and_CRAF_bind_and_dissociate',\n CRAF(raf=None) + CRAF(raf=None) | CRAF(raf=1) % CRAF(raf=1),\n bind_RAF_RAF_phi, Ea0_bind_CRAF_CRAF, energy=True)\n\nEnergyPattern('ep_bind_BRAF_BRAF', BRAF(raf=1) % BRAF(raf=1), Gf_bind_BRAF_BRAF)\nEnergyPattern('ep_bind_BRAF_CRAF', BRAF(raf=1) % CRAF(raf=1), Gf_bind_BRAF_CRAF)\nEnergyPattern('ep_bind_CRAF_CRAF', CRAF(raf=1) % CRAF(raf=1), Gf_bind_CRAF_CRAF)\n\n# Ras_gtp-RAF binding\nParameter('bind_RASstategtp_RAF_kf', 10.0)\nParameter('bind_RASstategtp_RAF_kD', 0.01)\nParameter('bind_RASstategtp_RAF_phi', 1.0)\nExpression('bind_RASstategtp_BRAF_kr',\n bind_RASstategtp_RAF_kD*bind_RASstategtp_RAF_kf)\nExpression('Gf_bind_RASstategtp_BRAF', log(bind_RASstategtp_RAF_kD))\nExpression('Ea0_bind_RASstategtp_BRAF',\n -bind_RASstategtp_RAF_phi*log(bind_RASstategtp_RAF_kD) - log(bind_RASstategtp_RAF_kf))\nExpression('bind_RASstategtp_CRAF_kr', \n bind_RASstategtp_RAF_kD*bind_RASstategtp_RAF_kf)\nExpression('Gf_bind_RASstategtp_CRAF', log(bind_RASstategtp_RAF_kD))\nExpression('Ea0_bind_RASstategtp_CRAF',\n -bind_RASstategtp_RAF_phi*log(bind_RASstategtp_RAF_kD) - log(bind_RASstategtp_RAF_kf))\n\nRule('RASgtp_and_BRAF_bind_and_dissociate',\n RAS(raf=None, state='gtp') + BRAF(RBD=None) | BRAF(RBD=1) % RAS(raf=1, state='gtp'),\n bind_RASstategtp_RAF_phi, Ea0_bind_RASstategtp_BRAF, energy=True)\nRule('RASgtp_and_CRAF_bind_and_dissociate',\n RAS(raf=None, state='gtp') + CRAF(RBD=None) | CRAF(RBD=1) % RAS(raf=1, state='gtp'),\n bind_RASstategtp_RAF_phi, Ea0_bind_RASstategtp_CRAF, energy=True)\n\nEnergyPattern('ep_bind_RASstategtp_BRAF', BRAF(RBD=1) % RAS(raf=1, state='gtp'), Gf_bind_RASstategtp_BRAF)\nEnergyPattern('ep_bind_RASstategtp_CRAF', CRAF(RBD=1) % RAS(raf=1, state='gtp'), Gf_bind_RASstategtp_CRAF)\n\nAnnotation(RASgtp_and_BRAF_bind_and_dissociate, 'http://identifiers.org/pubmed/7969158', 'isDescribedBy')\nAnnotation(RASgtp_and_CRAF_bind_and_dissociate, 'http://identifiers.org/pubmed/7969158', 'isDescribedBy')\n\n# Ras-Raf inactivation\nRule('GTP_hydrolysis_dissociates_BRAF_from_RAS',\n BRAF(RBD=1) % RAS(raf=1, state='gtp') >> RAS(raf=None, state='gdp') + BRAF(RBD=None),\n catalyze_NF1_RAS_gdp_kcat)\nRule('GTP_hydrolysis_dissociates_CRAF_from_RAS',\n CRAF(RBD=1) % RAS(raf=1, state='gtp') >> RAS(raf=None, state='gdp') + CRAF(RBD=None),\n catalyze_NF1_RAS_gdp_kcat)\n\n# Ras induces Raf dimerization\nParameter('ep_RAF_RAF_mod_RASstategtp_double_deltaG', 1000.0)\nExpression('ep_RAF_RAF_mod_RASstategtp_double_Gf', log(ep_RAF_RAF_mod_RASstategtp_double_deltaG))\nEnergyPattern('ep_BRAF_BRAF_mod_RAS_double',\n BRAF(RBD=2, raf=1) % BRAF(RBD=3, raf=1) % RAS(raf=2, state='gtp') % RAS(raf=3, state='gtp'),\n ep_RAF_RAF_mod_RASstategtp_double_Gf)\nEnergyPattern('ep_BRAF_CRAF_mod_RAS_double',\n BRAF(RBD=2, raf=1) % CRAF(RBD=3, raf=1) % RAS(raf=2, state='gtp') % RAS(raf=3, state='gtp'),\n ep_RAF_RAF_mod_RASstategtp_double_Gf)\nEnergyPattern('ep_CRAF_CRAF_mod_RAS_double',\n CRAF(RBD=2, raf=1) % CRAF(RBD=3, raf=1) % RAS(raf=2, state='gtp') % RAS(raf=3, state='gtp'),\n ep_RAF_RAF_mod_RASstategtp_double_Gf);\n```\n\n## Raf inhibition\nAlthough Vemurafenib is an ATP-competitive inihibitor, it is known to exert allosteric effects. Here, we adapt a previously published model for these allosteric interactions between C-Raf, B-Raf and Vemurafenib . The model implements binding between Vemurafenib (`RAFi`) and C-Raf and B-Raf (`RAF`) as energy rule as well as two sets of energy patterns that modulate `RAF-RAF` and `RAF-RAFi` affinities in single and double drug bound RAF-RAF homo and heterodimers.\n\nIn this implementation, C-Raf and B-Raf both correspond to kinase monomers `R` of the Kholodenko Core Model (KCM). `K1` of the KCM is equivalent to the parameter `bind_RAF_RAF_kD`, which was introduced in the previous Section. `K2` in the KCM is equivalent to `bind_RAFi_RAF_kD`, `f` to `ep_RAF_RAF_mod_RAFi_single_deltaG` and `g` to `ep_RAF_RAF_mod_RAFi_double_deltaG`.\n\n\n```python\n# Binding RAFi-Raf\nParameter('bind_RAFi_RAF_kf', 10.0)\nParameter('bind_RAFi_RAF_kD', 0.01)\nParameter('bind_RAFi_RAF_phi', 1.0)\nExpression('bind_BRAFi_RAF_kr', bind_RAFi_RAF_kD*bind_RAFi_RAF_kf)\nExpression('Gf_bind_BRAFi_RAF', log(bind_RAFi_RAF_kD))\nExpression('Ea0_bind_BRAFi_RAF', -bind_RAFi_RAF_phi*log(bind_RAFi_RAF_kD) - log(bind_RAFi_RAF_kf))\nExpression('bind_CRAFi_RAF_kr', bind_RAFi_RAF_kD*bind_RAFi_RAF_kf)\nExpression('Gf_bind_CRAFi_RAF', log(bind_RAFi_RAF_kD))\nExpression('Ea0_bind_CRAFi_RAF', -bind_RAFi_RAF_phi*log(bind_RAFi_RAF_kD) - log(bind_RAFi_RAF_kf))\nRule('RAFi_and_BRAF_bind_and_dissociate',\n RAFi(raf=None) + BRAF(rafi=None) | BRAF(rafi=1) % RAFi(raf=1),\n bind_RAFi_RAF_phi, Ea0_bind_BRAFi_RAF, energy=True)\nRule('RAFi_and_CRAF_bind_and_dissociate',\n RAFi(raf=None) + CRAF(rafi=None) | CRAF(rafi=1) % RAFi(raf=1),\n bind_RAFi_RAF_phi, Ea0_bind_CRAFi_RAF, energy=True)\n\nEnergyPattern('ep_bind_BRAFi_RAF', BRAF(rafi=1) % RAFi(raf=1), Gf_bind_BRAFi_RAF)\nEnergyPattern('ep_bind_CRAFi_RAF', CRAF(rafi=1) % RAFi(raf=1), Gf_bind_CRAFi_RAF)\n\n# RAFi mediated affinity modulation\nParameter('ep_RAF_RAF_mod_RAFi_single_deltaG', 0.001)\nParameter('ep_RAF_RAF_mod_RAFi_double_deltaG', 1000.0)\nExpression('ep_RAF_RAF_mod_RAFi_single_Gf', log(ep_RAF_RAF_mod_RAFi_single_deltaG))\nExpression('ep_RAF_RAF_mod_RAFi_double_Gf',\n log(ep_RAF_RAF_mod_RAFi_double_deltaG) + log(ep_RAF_RAF_mod_RAFi_single_deltaG))\n\n\n## Single Vemurafenib bound Raf dimers\nEnergyPattern('ep_BRAF_BRAF_mod_RAFi_single',\n BRAF(raf=1, rafi=2) % BRAF(raf=1, rafi=None) % RAFi(raf=2),\n ep_RAF_RAF_mod_RAFi_single_Gf)\nEnergyPattern('ep_BRAF_CRAF_mod_RAFi_single',\n BRAF(raf=1, rafi=None) % CRAF(raf=1, rafi=2) % RAFi(raf=2),\n ep_RAF_RAF_mod_RAFi_single_Gf)\nEnergyPattern('ep_CRAF_BRAF_mod_RAFi_single',\n BRAF(raf=1, rafi=2) % CRAF(raf=1, rafi=None) % RAFi(raf=2),\n ep_RAF_RAF_mod_RAFi_single_Gf)\nEnergyPattern('ep_CRAF_CRAF_mod_RAFi_single',\n CRAF(raf=1, rafi=2) % CRAF(raf=1, rafi=None) % RAFi(raf=2),\n ep_RAF_RAF_mod_RAFi_single_Gf)\n\n\n## Double Vemurafenib bound Raf dimers\nEnergyPattern('ep_BRAF_BRAF_mod_RAFi_double',\n BRAF(raf=1, rafi=2) % BRAF(raf=1, rafi=3) % RAFi(raf=2) % RAFi(raf=3),\n ep_RAF_RAF_mod_RAFi_double_Gf)\nEnergyPattern('ep_BRAF_CRAF_mod_RAFi_double',\n BRAF(raf=1, rafi=2) % CRAF(raf=1, rafi=3) % RAFi(raf=2) % RAFi(raf=3),\n ep_RAF_RAF_mod_RAFi_double_Gf)\nEnergyPattern('ep_CRAF_CRAF_mod_RAFi_double',\n CRAF(raf=1, rafi=2) % CRAF(raf=1, rafi=3) % RAFi(raf=2) % RAFi(raf=3),\n ep_RAF_RAF_mod_RAFi_double_Gf);\n```\n\n## Mek activation\n\nRaf activation of Mek is mediated through phosphorylation at to S218 and S222 . As neither mass-spectrometry nor immunofluorescence measurements resolved individual phosphorylations on these sites, we decided to combine both into a single `phospho` site in the model. We implement the respective Mek phosphorylation as two step catalytic process including binding and phosphorylation. In agreement with previous studies, we assume that binding of Raf to Mek does not require prior activation of Raf, but is specific for unphosphorylated Mek . \n\n\n```python\n# Raf-Mek binding\nParameter('bind_RAF_MEKphosphou_kf', 10.0)\nParameter('bind_RAF_MEKphosphou_kD', 0.01)\nParameter('bind_RAF_MEKphosphou_phi', 1.0)\nExpression('bind_BRAF_MEKphosphou_kr', bind_RAF_MEKphosphou_kD*bind_RAF_MEKphosphou_kf)\nExpression('Gf_bind_BRAF_MEKphosphou', log(bind_RAF_MEKphosphou_kD))\nExpression('Ea0_bind_BRAF_MEKphosphou',\n -bind_RAF_MEKphosphou_phi*log(bind_RAF_MEKphosphou_kD) - log(bind_RAF_MEKphosphou_kf))\nExpression('bind_CRAF_MEKphosphou_kr', bind_RAF_MEKphosphou_kD*bind_RAF_MEKphosphou_kf)\nExpression('Gf_bind_CRAF_MEKphosphou', log(bind_RAF_MEKphosphou_kD))\nExpression('Ea0_bind_CRAF_MEKphosphou',\n -bind_RAF_MEKphosphou_phi*log(bind_RAF_MEKphosphou_kD) - log(bind_RAF_MEKphosphou_kf))\n\nRule('BRAF_and_uMEK_bind_and_dissociate',\n BRAF(mek=None) + MEK(phospho='u', raf=None) | BRAF(mek=1) % MEK(phospho='u', raf=1),\n bind_RAF_MEKphosphou_phi, Ea0_bind_BRAF_MEKphosphou, energy=True)\nRule('CRAF_and_uMEK_bind_and_dissociate',\n CRAF(mek=None) + MEK(phospho='u', raf=None) | CRAF(mek=1) % MEK(phospho='u', raf=1),\n bind_RAF_MEKphosphou_phi, Ea0_bind_CRAF_MEKphosphou, energy=True)\n\nEnergyPattern('ep_bind_BRAF_MEKphosphou', BRAF(mek=1) % MEK(phospho='u', raf=1), Gf_bind_BRAF_MEKphosphou)\nEnergyPattern('ep_bind_CRAF_MEKphosphou', CRAF(mek=1) % MEK(phospho='u', raf=1), Gf_bind_CRAF_MEKphosphou)\n\nAnnotation(BRAF_and_uMEK_bind_and_dissociate, 'http://identifiers.org/pubmed/25155755', 'isDescribedBy')\nAnnotation(CRAF_and_uMEK_bind_and_dissociate, 'http://identifiers.org/pubmed/25155755', 'isDescribedBy');\n```\n\nFor the activation step, we implement a physiological activation through Raf-Ras tetramers, which is implemented in four rules that account for all possible Raf homo- and heterodimer configurations, as well as oncogenic activation through B-Raf V600E mutation, which is implemented in four rules that account for B-Raf V600E molecules complexes that do not signal as dimers. In both cases, we require that the activating Raf molecule is not bound by an inhibitor and assume that kinetic rates are the same in both cases. \n\nExperimental data presented in this study indicated phospho-Mek accumulation for physiological signaling but not for oncogenic signaling, which is consistent with previous studies . we required Mek not to be bound by an inhibitor. The exact molecular mechanisms for this difference between dimer mediated and V600E mediated are unknown. The exact molecular mechanisms for this difference between dimer mediated and V600E mediated are unknown but previous studies have suggested the involvement of drug-induced effects on RAF:MEK stabiliziation, MEK dimerization or involvement of KSR scaffolding proteins .\n\n\n```python\n# phosphorylation\nParameter('catalyze_RAFrafiNone_MEK_p_kcat', 10.0)\n## Raf-dimer mediated\nRule('BRAF_BRAF_phosphorylates_MEK',\n BRAF(RBD=ANY, mek=1, raf=2, rafi=None) % BRAF(RBD=ANY, raf=2) % MEK(phospho='u', raf=1)\n >>\n MEK(phospho='p', raf=None) + BRAF(RBD=ANY, mek=None, raf=2, rafi=None) % BRAF(RBD=ANY, raf=2),\n catalyze_RAFrafiNone_MEK_p_kcat)\nRule('BRAF_CRAF_phosphorylates_MEK',\n BRAF(RBD=ANY, mek=1, raf=2, rafi=None) % CRAF(RBD=ANY, raf=2) % MEK(phospho='u', raf=1)\n >>\n MEK(phospho='p', raf=None) + BRAF(RBD=ANY, mek=None, raf=2, rafi=None) % CRAF(RBD=ANY, raf=2),\n catalyze_RAFrafiNone_MEK_p_kcat)\nRule('CRAF_BRAF_phosphorylates_MEK',\n BRAF(RBD=ANY, raf=2) % CRAF(RBD=ANY, mek=1, raf=2, rafi=None) % MEK(phospho='u', raf=1)\n >>\n MEK(phospho='p', raf=None) + BRAF(RBD=ANY, raf=2) % CRAF(RBD=ANY, mek=None, raf=2, rafi=None),\n catalyze_RAFrafiNone_MEK_p_kcat)\nRule('CRAF_CRAF_phosphorylates_MEK',\n CRAF(RBD=ANY, mek=1, raf=2, rafi=None) % CRAF(RBD=ANY, raf=2) % MEK(phospho='u', raf=1)\n >>\n MEK(phospho='p', raf=None) + CRAF(RBD=ANY, mek=None, raf=2, rafi=None) % CRAF(RBD=ANY, raf=2),\n catalyze_RAFrafiNone_MEK_p_kcat)\n\n## B-Raf V600E mediated\nRule('BRAFV600E_phosphorylates_MEK_1',\n BRAF(AA600='E', mek=1, raf=None, rafi=None) % MEK(meki=None, phospho='u', raf=1)\n >>\n MEK(meki=None, phospho='p', raf=None) + BRAF(AA600='E', mek=None, raf=None, rafi=None),\n catalyze_RAFrafiNone_MEK_p_kcat)\nRule('BRAFV600E_phosphorylates_MEK_2',\n BRAF(AA600='E', RBD=None, mek=1, raf=ANY, rafi=None) % MEK(meki=None, phospho='u', raf=1)\n >>\n MEK(meki=None, phospho='p', raf=None) + BRAF(AA600='E', RBD=None, mek=None, raf=ANY, rafi=None),\n catalyze_RAFrafiNone_MEK_p_kcat)\nRule('BRAFV600E_phosphorylates_MEK_3',\n BRAF(AA600='E', RBD=ANY, mek=1, raf=2, rafi=None) % BRAF(RBD=None, raf=2) % MEK(meki=None, phospho='u', raf=1)\n >>\n MEK(meki=None, phospho='p', raf=None) + BRAF(AA600='E', RBD=ANY, mek=None, raf=2, rafi=None) % BRAF(RBD=None, raf=2),\n catalyze_RAFrafiNone_MEK_p_kcat)\nRule('BRAFV600E_phosphorylates_MEK_4',\n BRAF(AA600='E', RBD=ANY, mek=1, raf=2, rafi=None) % CRAF(RBD=None, raf=2) % MEK(meki=None, phospho='u', raf=1)\n >>\n MEK(meki=None, phospho='p', raf=None) + BRAF(AA600='E', RBD=ANY, mek=None, raf=2, rafi=None) % CRAF(RBD=None, raf=2),\n catalyze_RAFrafiNone_MEK_p_kcat);\n```\n\nPP2A and PPP1 have been suggested as putative phosphatases of Mek1, but exact regulatory mechanisms remain poorly understood. Accordingly we implement MEK dephosphorylation as conversion rule with constant rate.\n\n\n```python\n# dephosphorylation\nParameter('catalyze_PP2A_MEK_u_kcatr', 1.0)\nExpression('catalyze_PP2A_MEK_u_kcat', catalyze_PP2A_MEK_u_kcatr*catalyze_RAFrafiNone_MEK_p_kcat)\nRule('MEK_is_dephosphorylated', MEK(phospho='p') >> MEK(phospho='u'), catalyze_PP2A_MEK_u_kcat);\n```\n\n## Mek inhibition\nCobimetinib is an allosteric (type III) inhibitor that potently and selectively inhibits Erk activation when binding Mek. Recent studies reported a distinct potency of allosteric Mek inhibitor for Ras-GTP versus B-Raf V600E driven Mek signaling. One component of this difference in potency is the accumulation of phosphorylated Mek in Ras-GTP driven signaling, but not in B-Raf V600E driven signaling that was already described in the previous Section. A second component is the differential affinity of allosteric Mek inihibitors towards phosphorylated MEK . Accordingly, we implement binding as energy rule and add an energy pattern that allows Mek phosphorylation to modulation the respective affinity.\n\n\n```python\n# MEKi binding\nParameter('bind_MEKi_MEK_kf', 10.0)\nParameter('bind_MEKi_MEK_kD', 0.01)\nParameter('bind_MEKi_MEK_phi', 1.0)\nExpression('bind_MEKi_MEK_kr', bind_MEKi_MEK_kD*bind_MEKi_MEK_kf)\nExpression('Gf_bind_MEKi_MEK', log(bind_MEKi_MEK_kD))\nExpression('Ea0_bind_MEKi_MEK', -bind_MEKi_MEK_phi*log(bind_MEKi_MEK_kD) - log(bind_MEKi_MEK_kf))\nRule('MEKi_and_MEK_bind_and_dissociate',\n MEKi(mek=None) + MEK(meki=None) | MEK(meki=1) % MEKi(mek=1),\n bind_MEKi_MEK_phi, Ea0_bind_MEKi_MEK, energy=True)\n\nEnergyPattern('ep_bind_MEKi_MEK', MEK(meki=1) % MEKi(mek=1), Gf_bind_MEKi_MEK)\n\n# Modulated pMEK affinity \nParameter('ep_MEKphosphop_MEKi_deltaG', 100.0)\nExpression('ep_MEKphosphop_MEKi_Gf', log(ep_MEKphosphop_MEKi_deltaG))\nEnergyPattern('ep_MEKphosphop_MEKi_single', MEK(meki=1, phospho='p') % MEKi(mek=1), ep_MEKphosphop_MEKi_Gf);\n```\n\n## Erk activation\nMek phosphorylates Erk1 at T202 and T185 and Erk2 at Y204 and Y187. As for Mek phosphorylation sites, neither mass-spectrometry nor immunofluorescence measurements resolved individual phosphorylations, both sites were combined into a single `phospho` site in the model. \n\nAs for the Mek phosphorylation we implement Erk phosphorylation as two step catalysis reaction. The binding between Mek and Erk is mediated through interaction of the MAPK-Docking site on the N-Terminus of MEK (`Dsite` site in the model) with the common docking domain on Erk (`CD` site in the model) . In combination with Mek's nuclear export sequence, this interaction allows Mek to act as a cytosolic anchor for inactive Erk , which suggests that this interaction is not dependent of Mek phosphorylation. After Mek activation, Erk releases from Mek and shuttles in and out of the nucleus, which suggests that affinity between Mek and Erk depends on Erk phosphorylation . In the model, we implement the dependence on Erk phosphorylation as binary interaction, where Mek only binds unphosphorylated Erk. Nuclear shuttling of Mek and Erk was not implemented in the model.\n\n\n```python\n# Mek-Erk binding\nParameter('bind_MEK_ERKphosphou_kf', 10.0)\nParameter('bind_MEK_ERKphosphou_kD', 0.01)\nExpression('bind_MEK_ERKphosphou_kr', bind_MEK_ERKphosphou_kD*bind_MEK_ERKphosphou_kf)\nRule('MEK_binds_uERK',\n MEK(Dsite=None) + ERK(CD=None, phospho='u') >> ERK(CD=1, phospho='u') % MEK(Dsite=1),\n bind_MEK_ERKphosphou_kf)\nRule('MEK_dissociates_from_ERK',\n ERK(CD=1) % MEK(Dsite=1) >> MEK(Dsite=None) + ERK(CD=None),\n bind_MEK_ERKphosphou_kr)\n\n\nAnnotation(MEK_dissociates_from_ERK, 'http://identifiers.org/pubmed/10567369', 'isDescribedBy')\nAnnotation(MEK_dissociates_from_ERK, 'http://identifiers.org/pubmed/10655591', 'isDescribedBy')\nAnnotation(MEK_dissociates_from_ERK, 'http://identifiers.org/pubmed/11157753', 'isDescribedBy')\nAnnotation(MEK_dissociates_from_ERK, 'http://identifiers.org/pubmed/15979847', 'isDescribedBy')\n\n# Erk phosphorylation\nParameter('catalyze_MEKmekiNone_phosphop_ERK_p_kcat', 10.0)\nRule('pMEK_phosphorylates_ERK',\n ERK(CD=1, phospho='u') % MEK(Dsite=1, meki=None, phospho='p')\n >>\n ERK(CD=None, phospho='p') + MEK(Dsite=None, meki=None, phospho='p'),\n catalyze_MEKmekiNone_phosphop_ERK_p_kcat)\n\nAnnotation(pMEK_phosphorylates_ERK, 'http://identifiers.org/pubmed/19406201', 'isDescribedBy');\n```\n\n# Feedback mechanisms\nActivation of Erk through phosphorylation induces a variety of different negative feedback mechanisms that inhibit signal transduction in Egfr and Erk pathways . In this module we specifically implement four of these mechanisms that, specifically the increase in Spry, Egfr and Dusp expression levels as well as phosphorylation of Sos1. For this purpose we introduce two new monomers, namely Dusp (which describes Dusp4 and Dusp6) and Spry (which describes Spry2 and Spry4).\n\nSome of the negative feedback mechanisms that were described in previous studies are omitted in this model, including Erk negative feedback phosphorylations on Mek, and Raf. As the timescale of phosphorylation reactions is in the order of seconds, a high inhibitory potency of these feedback mechanisms should give rise to phospho-Erk pulses on a timescale of seconds. Yet the experimentally observed timescale of phospho-Erk pulses was in the order of minutes, which suggests that it is unlikely that phosphorylations feedbacks are the primary determinant of the pulse shape. This hypothesis is backed by previous studies on phospho-turnover in Egfr mediated signaling . Accordingly we only expect a subtle effect of the negative feedback mechanisms, which would be difficult to be resolve in the model, given that no experimental data on respective phosphorylation sites or Ras-GTP levels were available. \n\n\n```python\nMonomer('DUSP', ['erk'])\nAnnotation(DUSP, 'http://identifiers.org/uniprot/Q13115', 'is')\nAnnotation(DUSP, 'http://identifiers.org/uniprot/Q16828', 'is')\n\nMonomer('SPRY', ['SH3'])\nAnnotation(SPRY, 'http://identifiers.org/uniprot/O43597', 'is')\nAnnotation(SPRY, 'http://identifiers.org/uniprot/Q9C004', 'is');\n```\n\n## Spry expression\nThe proteomic data collected in this study confirmed previous findings that phosphorylation of Erk enhances expression of Spry. Increase in Spry expression inhibits Egfr signaling and is mediated by regulation of downstream transcription factors . Specifically, Spry competes with Sos1 for the binding of Grb2 . Accordingly higher levels of Spry can antagonize Sos1 signaling by sequestering Grb2 in Spry:Grb2 complexes. \n\nHere, we implement basal synthesis and degradation reactions for Spry, which are parameterised with `[...]_eq` and `[...]_kdeg` parameters as done for Egfr expression in the Section on [Grb2 activation](#GRB2-Activation). On top of this basal expression, we implement an additional expression rule for which the expression rate linearly depends on the abundance of phospho-Erk. To avoid correlations of parameter estimates, we normalize this expression rate with the basal Spry degradation rate. The binding between Spry and Grb2 is parametrized using `[...]_kD` and `[...]_kf` parameters as previously described. Competitive binding between Spry and Sos1 for Grb2 is implemented by specifying the same binding site on Grb2 (`SH3`) for the binding of both reactions.\n\n\n```python\n# basal expression and degradation\nParameter('SPRY_eq', 10000.0)\nParameter('synthesize_ERKphosphop_SPRY_kdeg', 10.0)\nExpression('synthesize_ERKphosphop_SPRY_ksyn',\n 1000000.0*SPRY_eq*synthesize_ERKphosphop_SPRY_kdeg/(N_Avogadro*volume))\n\nRule('basal_synthesis_SPRY',\n None >> SPRY(SH3=None), synthesize_ERKphosphop_SPRY_ksyn)\nRule('basal_degradation_SPRY',\n SPRY() >> None, synthesize_ERKphosphop_SPRY_kdeg, delete_molecules=True)\n\n# Erk mediated expression\nParameter('synthesize_ERKphosphop_SPRY_ERK_gexpslope', 1000.0)\nExpression('synthesize_ERKphosphop_SPRY_kmodslope',\n synthesize_ERKphosphop_SPRY_ERK_gexpslope*synthesize_ERKphosphop_SPRY_kdeg)\nRule('ERK_synthesizes_SPRY',\n None + ERK(phospho='p') >> SPRY(SH3=None) + ERK(phospho='p'),\n synthesize_ERKphosphop_SPRY_kmodslope)\n\n\n# Spry-Grb2 binding\nParameter('bind_SPRY_GRB2_kf', 10.0)\nParameter('bind_SPRY_GRB2_kD', 0.01)\nExpression('bind_SPRY_GRB2_kr', bind_SPRY_GRB2_kD*bind_SPRY_GRB2_kf)\nRule('SPRY_binds_GRB2',\n SPRY(SH3=None) + GRB2(SH3=None) >> GRB2(SH3=1) % SPRY(SH3=1),\n bind_SPRY_GRB2_kf)\nRule('SPRY_dissociates_from_GRB2',\n GRB2(SH3=1) % SPRY(SH3=1) >> SPRY(SH3=None) + GRB2(SH3=None),\n bind_SPRY_GRB2_kr);\n```\n\n## Sos1 phosphorylation\nPhospho-proteomic data collected in this study confirmed previous findings that phosphorylation of Erk enhances phosphorylation of Sos1 at S1134. Phosphorylation of Sos1 inhibits Egfr signaling and is mediated through upregulation of Rsk expression and phosphorylation . Specifically it creates a docking site for 14-3-3 on Sos1 and thereby sequestering it from Grb2 .\n\nWe do not include Rsk or 14-3-3 in the model as absolute abundances were measured for neither of proteins. Instead we decided to implement basal phosphorylation and dephosphorylation rules as well as an phospho-Erk dependent phosphorylation rule for which the phosphorylation linearly depends on Erk phosphorylation. The baseline rules were implemented using constant rate `[...]_kbase` and `[...]_kcat` respectively. To avoid parameter correlations, the phospho-Erk dependent phosphorylation rate was normalized by the baseline phosphorylation rate.\n\n\n```python\n# base phosphorylation rate\nParameter('catalyze_ERKphosphop_SOS1_pS1134_kbase', 0.0)\nParameter('catalyze_ERKphosphop_SOS1_pS1134_kcat', 10.0)\nParameter('catalyze_phosphatase_SOS1_uS1134_kcatr', 1.0)\nExpression('catalyze_phosphatase_SOS1_uS1134_kcat',\n catalyze_ERKphosphop_SOS1_pS1134_kcat*catalyze_phosphatase_SOS1_uS1134_kcatr)\n\nRule('SOS1_is_phosphorylated',\n SOS1(S1134='u') >> SOS1(S1134='p'),\n catalyze_ERKphosphop_SOS1_pS1134_kbase)\n\nRule('SOS1_is_dephosphorylated',\n SOS1(S1134='p') >> SOS1(S1134='u'),\n catalyze_phosphatase_SOS1_uS1134_kcat)\n\nRule('pERK_phosphorylates_SOS1',\n ERK(phospho='p') + SOS1(S1134='u') >> ERK(phospho='p') + SOS1(S1134='p'),\n catalyze_ERKphosphop_SOS1_pS1134_kcat)\n\n# SPRY modulates binding between GRB2 and SOS1\nParameter('ep_SOS1S1134p_GRB2_deltaG', 100.0)\nExpression('ep_SOS1S1134p_GRB2_Gf', log(ep_SOS1S1134p_GRB2_deltaG))\nEnergyPattern('ep_SOS1S1134p_GRB2_single', GRB2(SH3=1) % SOS1(S1134='p', SH3=1), ep_SOS1S1134p_GRB2_Gf);\n```\n\nAs the binding between as Grb2 and Sos1 was implemented as energy rule, we use an energy pattern to implement the modulated affinity of Grb2 to Sos1. \n\n## Egfr expression\nProteomic data collected in this study confirmed previous findings that phosphorylation of Erk increases Egfr expression. Increased Egfr expression promotes Egfr signaling and is mediated through regulation of downstream transcription factors .\n\nAs baseline Egfr expression and degradation rules are already implemented in the Section on [GRB2 activation](#GRB2-Activation), we here simply add an expression rule for which the rate linearly depends on phospho-Erk. As for Spry expression, we normalized the phospho-Erk dependent expression rate by the baseline degradation rate.\n\n\n```python\nParameter('synthesize_ERKphosphop_EGFR_ERK_gexpslope', 1000.0)\nExpression('synthesize_ERKphosphop_EGFR_kmodslope',\n synthesize_ERKphosphop_EGFR_ERK_gexpslope*synthesize_ERKphosphop_EGFR_kdeg)\n\nRule('ERK_synthesizes_EGFR',\n None + ERK(phospho='p') >> EGFR(SH2=None, rtk=None, rtkf=None) + ERK(phospho='p'),\n synthesize_ERKphosphop_EGFR_kmodslope);\n```\n\n## Dusp expression\nThe proteomic data collected in this study confirmed previous findings that phosphorylation of Erk enhances expression of Dusp. Increase in Dusp expression inhibits Erk signaling and is mediated by regulation of downstream transcription factors . Specifically, Dusp are phosphatases that dephosphorylated Erk on the same residues that are phosphorylated by Mek. In the proteomic data we individually measured Dusp4 and Dusp6 abundances. Dusp4 has primarily nuclear localisation , while Dusp6 has primarily cytosolic localisation . As we did not implement shuttling of Erk between the nucleus and cytosol, we lumped Dusp4 and Dusp6 species.\n\nWe implement the dephosphorylation as two-step catalytic process. As both Dusps interact with Erk via the common docking domain that also mediates interaction between Mek to Erk , the binding was implemented using the same site on the Erk (`CD`) monomer. This could lead to competitive binding between Mek and Erk, yet as Dusp, in contrast to Mek, seem to preferentially bind the phosphorylated form of Erk , we constrained binding of Dusp to Erk on Erk phosphorylation, which makes the two binding reactions independent.\n\n\n```python\n# expression\nParameter('DUSP_eq', 10000.0)\nParameter('synthesize_ERKphosphop_DUSP_ERK_gexpslope', 1000.0)\nParameter('synthesize_ERKphosphop_DUSP_kdeg', 10.0)\nExpression('synthesize_ERKphosphop_DUSP_ksyn',\n 1000000.0*DUSP_eq*synthesize_ERKphosphop_DUSP_kdeg/(N_Avogadro*volume))\nExpression('synthesize_ERKphosphop_DUSP_kmodslope',\n synthesize_ERKphosphop_DUSP_ERK_gexpslope*synthesize_ERKphosphop_DUSP_kdeg)\n\nRule('basal_synthesis_DUSP',\n None >> DUSP(erk=None), synthesize_ERKphosphop_DUSP_ksyn)\nRule('basal_degradation_DUSP',\n DUSP() >> None, synthesize_ERKphosphop_DUSP_kdeg, delete_molecules=True)\nRule('ERK_synthesizes_DUSP',\n None + ERK(phospho='p') >> DUSP(erk=None) + ERK(phospho='p'), synthesize_ERKphosphop_DUSP_kmodslope)\n\n\n# Dusp-Erk binding\nParameter('bind_DUSP_ERKphosphop_kf', 10.0)\nParameter('bind_DUSP_ERKphosphop_kD', 0.01)\nExpression('bind_DUSP_ERKphosphop_kr', bind_DUSP_ERKphosphop_kD*bind_DUSP_ERKphosphop_kf)\nRule('DUSP_binds_pERK',\n DUSP(erk=None) + ERK(CD=None, phospho='p') >> DUSP(erk=1) % ERK(CD=1, phospho='p'),\n bind_DUSP_ERKphosphop_kf)\nRule('DUSP_dissociates_from_ERK',\n DUSP(erk=1) % ERK(CD=1) >> DUSP(erk=None) + ERK(CD=None),\n bind_DUSP_ERKphosphop_kr)\n\n\nAnnotation(DUSP_dissociates_from_ERK, 'http://identifiers.org/pubmed/10655591', 'isDescribedBy')\nAnnotation(DUSP_dissociates_from_ERK, 'http://identifiers.org/pubmed/11157753', 'isDescribedBy')\n\n# Erk dephosphorylation\nParameter('catalyze_DUSP_ERK_u_kcatr', 1.0)\nExpression('catalyze_DUSP_ERK_u_kcat', catalyze_DUSP_ERK_u_kcatr*catalyze_MEKmekiNone_phosphop_ERK_p_kcat)\nRule('DUSP_dephosphorylates_ERK',\n DUSP(erk=1) % ERK(CD=1, phospho='p') >> ERK(CD=None, phospho='u') + DUSP(erk=None),\n catalyze_DUSP_ERK_u_kcat);\n```\n\n# Initialization\nFor all monomers for which we did not implement expression and basal degradation rules we implemented initial abundances that were assumed to be constant in all experiments. To simplify specification of estimation boundaries for these parameters, these parameters were assumed to be specified in [molecules/cell] and then accordingly transformed to yield model initializations in [uM]. \n\n\n```python\nParameter('BRAF_0', 0.0)\nParameter('CRAF_0', 0.0)\nParameter('RAS_0', 0.0)\nParameter('MEK_0', 0.0)\nParameter('ERK_0', 0.0)\nParameter('GRB2_0', 0.0)\nParameter('SOS1_0', 0.0)\n\nExpression('initBRAF', 1000000.0*BRAF_0/(N_Avogadro*volume))\nExpression('initCRAF', 1000000.0*CRAF_0/(N_Avogadro*volume))\nExpression('initRAS', 1000000.0*RAS_0/(N_Avogadro*volume))\nExpression('initMEK', 1000000.0*MEK_0/(N_Avogadro*volume))\nExpression('initERK', 1000000.0*ERK_0/(N_Avogadro*volume))\nExpression('initGRB2', 1000000.0*GRB2_0/(N_Avogadro*volume))\nExpression('initSOS1', 1000000.0*SOS1_0/(N_Avogadro*volume))\n\nInitial(BRAF(AA600='E', RBD=None, mek=None, raf=None, rafi=None), initBRAF)\nInitial(CRAF(RBD=None, mek=None, raf=None, rafi=None), initCRAF)\nInitial(RAS(raf=None, sos1=None, state='gdp'), initRAS)\nInitial(MEK(Dsite=None, meki=None, phospho='u', raf=None), initMEK)\nInitial(ERK(CD=None, phospho='u'), initERK)\nInitial(GRB2(SH2=None, SH3=None), initGRB2)\nInitial(SOS1(S1134='u', SH3=None, ras=None), initSOS1);\n```\n\n# Link to Experimental Data\nTo calibrate the model to experimental data we create a series of expression that account for normalizations in data generation. For absolute proteomic measurements we create expressions `t[...]_obs` that convert the unit of model simulations from [uM] to [molecules/cell]. For phospho-proteomic measurements we create expressions `p[...]_obs` that are equal to phospho-abundances normalized by total abundances. For immuno-fluorescence measurements we created expressions `p[...]_IF_obs` that were first normalized by total abundances and then scaled by a mulitplicative factor `[...]_IF_scale` and an additive factor `[...]_IF_offset`. For species that describe multiple protein isoforms, absolute proteomic data was averaged over both isoforms.\n\n\n```python\n# proteomic measurements\nObservable('tBRAF', BRAF())\nObservable('tCRAF', CRAF())\nObservable('tRAS', RAS())\nObservable('tMEK', MEK())\nObservable('tERK', ERK())\nObservable('tDUSP', DUSP())\nObservable('tEGFR', EGFR())\nObservable('tGRB2', GRB2())\nObservable('tSPRY', SPRY())\nObservable('tSOS1', SOS1())\nExpression('tBRAF_obs', 1.0e-6*tBRAF*N_Avogadro*volume)\nExpression('tCRAF_obs', 1.0e-6*tCRAF*N_Avogadro*volume)\nExpression('tRAS_obs', 1.0e-6*tRAS*N_Avogadro*volume)\nExpression('tMEK_obs', 1.0e-6*tMEK*N_Avogadro*volume)\nExpression('tDUSP_obs', 1.0e-6*tDUSP*N_Avogadro*volume)\nExpression('tEGFR_obs', 1.0e-6*tEGFR*N_Avogadro*volume)\nExpression('tGRB2_obs', 1.0e-6*tGRB2*N_Avogadro*volume)\nExpression('tSPRY_obs', 1.0e-6*tSPRY*N_Avogadro*volume)\nExpression('tSOS1_obs', 1.0e-6*tSOS1*N_Avogadro*volume)\nExpression('tERK_obs', 1.0e-6*tERK*N_Avogadro*volume)\n\n# phospho-proteomic measurements\nObservable('pERK', ERK(phospho='p'))\nObservable('pS1134SOS1', SOS1(S1134='p'))\nExpression('pERK_obs', pERK/tERK)\nExpression('pS1134SOS1_obs', pS1134SOS1/tSOS1)\n\n# immunofluorescence measurements\nParameter('pMEK_IF_scale', 1.0)\nParameter('pMEK_IF_offset', 0.1)\nParameter('pERK_IF_scale', 1.0)\nParameter('pERK_IF_offset', 0.1)\nObservable('pMEK', MEK(phospho='p'))\nExpression('pMEK_obs', pMEK/tMEK)\nExpression('pMEK_IF_obs', pMEK_obs*pMEK_IF_scale + pMEK_IF_offset)\nExpression('pERK_IF_obs', pERK_obs*pERK_IF_scale + pERK_IF_offset);\n```\n\n# Observables\n\nWe add observables that are used to extract the biological activity of proteins of interest.\n\n\n```python\n#total EGF\nObservable('tEGF', EGF())\n#active SOS1 (SOS1 is bound to a GRB2 that is bound to EGFR)\nObservable('SOS1_active', GRB2(SH2=ANY, SH3=1) % SOS1(SH3=1))\n#signaling competent SOS1 (SOS1 is bond to GRB2)\nObservable('SOS1_signaling_competent', GRB2(SH3=1) % SOS1(SH3=1))\n#signaling inhibited GBR2 (bound by SPRY)\nObservable('GRB2_bound_by_SPRY', GRB2(SH3=1) % SPRY(SH3=1))\n#unphosphorylated SOS1\nObservable('uS1134SOS1', SOS1(S1134='u'))\n#Ras-GDP and -GTP\nObservable('RAS_gtp', RAS(state='gtp'))\nObservable('RAS_gdp', RAS(state='gdp'))\n#active RAF monomers that are not bound by RAFi\nObservable('active_RAF_monomers', BRAF(AA600='E', raf=None, rafi=None) + BRAF(AA600='E', RBD=None, raf=ANY, rafi=None) + BRAF(AA600='E', RBD=ANY, raf=2, rafi=None) % BRAF(RBD=None, raf=2) + BRAF(AA600='E', RBD=ANY, raf=2, rafi=None) % CRAF(RBD=None, raf=2))\n#active RAF monomers bound by RAFi\nObservable('inhibited_RAF_monomers', BRAF(AA600='E', raf=None, rafi=ANY) + BRAF(AA600='E', RBD=None, raf=ANY, rafi=ANY) + BRAF(AA600='E', RBD=ANY, raf=2, rafi=ANY) % BRAF(RBD=None, raf=2) + BRAF(AA600='E', RBD=ANY, raf=2, rafi=ANY) % CRAF(RBD=None, raf=2))\n#active RAF dimers that are not bound by RAFi\nObservable('active_RAF_dimers', BRAF(RBD=ANY, raf=1, rafi=None) % BRAF(RBD=ANY, raf=1) + BRAF(RBD=ANY, raf=1, rafi=None) % CRAF(RBD=ANY, raf=1) + CRAF(RBD=ANY, raf=1, rafi=None) % BRAF(RBD=ANY, raf=1) + CRAF(RBD=ANY, raf=1, rafi=None) % CRAF(RBD=ANY, raf=1))\n#active RAF dimers bound by RAFi\nObservable('inhibited_RAF_dimers', BRAF(RBD=ANY, raf=1, rafi=ANY) % BRAF(RBD=ANY, raf=1) + BRAF(RBD=ANY, raf=1, rafi=ANY) % CRAF(RBD=ANY, raf=1) + CRAF(RBD=ANY, raf=1, rafi=ANY) % BRAF(RBD=ANY, raf=1) + CRAF(RBD=ANY, raf=1, rafi=ANY) % CRAF(RBD=ANY, raf=1))\n#active MEK that are not bound by MEKi\nObservable('active_pMEK', MEK(meki=None, phospho='p'))\n#active MEK bound by MEKi\nObservable('inhibited_pMEK', MEK(meki=ANY, phospho='p'))\n#unphosphorylate MEK\nObservable('uMEK', MEK(phospho='u'))\n#unophosphorylated ERK\nObservable('uERK', ERK(phospho='u'));\n#total RAFi\nObservable('tRAFi', RAFi())\n#total RAFi\nObservable('tMEKi', MEKi());\n```\n\n# Save generated MARM1 in SBML, BNG and PySB formats\n\nWe save the generated MARM1 model as a SBML, BNG and PySB format. These files are identical to one provided in the Docker container.\n\nWe explicitly set the model name here so the result matches the provided files.\n\n\n```python\nmodel.name = 'MARM1'\n```\n\nWe first generate the PySB and BNGL formats. We do this before the SBML generation, as we do not want to include the extra expressions induced by the Energy BNG process that get added for the SBML export.\n\n\n```python\ngenerated_model_code = export(model, 'pysb_flat')\nwith open('MARM1.py', 'wt') as f:\n f.write(generated_model_code);\n\ngenerated_model_code = export(model, 'bngl')\nwith open('MARM1.bngl', 'wt') as f:\n f.write(generated_model_code);\n```\n\nNow generate the SBML format, which implicitly triggers construction of the extra energy expressions.\n\n\n```python\ngenerated_model_code = export(model, 'sbml')\nwith open('MARM1.sbml', 'wt') as f:\n f.write(generated_model_code);\n```\n\n# MARM1 information\n\nWe list here the attributes of the MARM1 PySB model constructed by the code above.\n\n\n```python\nprint ('MARM1 information')\nnexpress=len(model.expressions);\ngenerate_equations(model)\nprint ('Species:',len(model.species))\n#parameters include initial conditions that were estimated, apart the three that are perturbations (RAFi, MEKi, EGF) \nprint ('Parameters:',len(model.parameters)+len(model.initial_conditions)-3)\nprint ('Expressions:',nexpress)\nprint ('Observables:', len(model.observables))\nntotr=len(model.rules);\nnenergy=len([r for r in model.rules if r.energy]);\nprint ('Total Rules:', ntotr)\nprint ('Energy Rules:', nenergy)\nprint('Non-energy Rules:', ntotr-nenergy)\nprint('Energy Patterns:', len(model.energypatterns))\nprint('Reactions:',len(model.reactions))\n```\n\n MARM1 information\n Species: 1007\n Parameters: 82\n Expressions: 185\n Observables: 30\n Total Rules: 53\n Energy Rules: 13\n Non-energy Rules: 40\n Energy Patterns: 27\n Reactions: 13102\n\n\n# Bibliography\n
                                        \n\n\n```python\n\n```\n", "meta": {"hexsha": "0699bc470e9b6ff54f13918397b8b85335646a60", "size": 157491, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "resources/MARM1_construction.ipynb", "max_stars_repo_name": "labsyspharm/marm1-supplement", "max_stars_repo_head_hexsha": "52508584fb428d302ea287d441d0c192a561419a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2020-12-09T11:55:54.000Z", "max_stars_repo_stars_event_max_datetime": "2021-02-20T20:00:50.000Z", "max_issues_repo_path": "resources/MARM1_construction.ipynb", "max_issues_repo_name": "labsyspharm/marm1-supplement", "max_issues_repo_head_hexsha": "52508584fb428d302ea287d441d0c192a561419a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "resources/MARM1_construction.ipynb", "max_forks_repo_name": "labsyspharm/marm1-supplement", "max_forks_repo_head_hexsha": "52508584fb428d302ea287d441d0c192a561419a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2021-01-28T20:55:05.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-16T00:40:21.000Z", "avg_line_length": 51.518155054, "max_line_length": 1907, "alphanum_fraction": 0.663212501, "converted": true, "num_tokens": 17966, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.4301029023091603}} {"text": "\n\n# Tutorial 3: Confidence intervals and bootstrapping\n**Week 1, Day 3: Model Fitting**\n\n**By Neuromatch Academy**\n\n**Content creators**: Pierre-\u00c9tienne Fiquet, Anqi Wu, Alex Hyafil with help from Byron Galbraith\n\n**Content reviewers**: Lina Teichmann, Saeed Salehi, Patrick Mineault, Ella Batty, Michael Waskom \n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n

                                        \n\n# Tutorial Objectives\n\n*Estimated timing of tutorial: 23 minutes*\n\nThis is Tutorial 3 of a series on fitting models to data. We start with simple linear regression, using least squares optimization (Tutorial 1) and Maximum Likelihood Estimation (Tutorial 2). We will use bootstrapping to build confidence intervals around the inferred linear model parameters (Tutorial 3). We'll finish our exploration of regression models by generalizing to multiple linear regression and polynomial regression (Tutorial 4). We end by learning how to choose between these various models. We discuss the bias-variance trade-off (Tutorial 5) and Cross Validation for model selection (Tutorial 6).\n\nIn this tutorial, we will discuss how to gauge how good our estimated model parameters are. \n- Learn how to use bootstrapping to generate new sample datasets\n- Estimate our model parameter on these new sample datasets\n- Quantify the variance of our estimate using confidence intervals\n\n\n```python\n# @title Tutorial slides\n\n# @markdown These are the slides for the videos in all tutorials today\nfrom IPython.display import IFrame\nIFrame(src=f\"https://mfr.ca-1.osf.io/render?url=https://osf.io/2mkq4/?direct%26mode=render%26action=download%26mode=render\", width=854, height=480)\n```\n\n\n```python\n# @title Video 1: Confidence Intervals & Bootstrapping\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = 'https://player.bilibili.com/player.html?bvid={0}&page={1}'.format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=\"BV1vK4y1s7py\", width=854, height=480, fs=1)\n print('Video available at https://www.bilibili.com/video/{0}'.format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=\"hs6bVGQNSIs\", width=854, height=480, fs=1, rel=0)\n print('Video available at https://youtube.com/watch?v=' + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nUp to this point we have been finding ways to estimate model parameters to fit some observed data. Our approach has been to optimize some criterion, either minimize the mean squared error or maximize the likelihood while using the entire dataset. How good is our estimate really? How confident are we that it will generalize to describe new data we haven't seen yet?\n\nOne solution to this is to just collect more data and check the MSE on this new dataset with the previously estimated parameters. However this is not always feasible and still leaves open the question of how quantifiably confident we are in the accuracy of our model.\n\nIn Section 1, we will explore how to implement bootstrapping. In Section 2, we will build confidence intervals of our estimates using the bootstrapping method.\n\n---\n# Setup\n\n\n```python\n# Imports\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n```\n\n\n```python\n#@title Figure Settings\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/course-content/master/nma.mplstyle\")\n```\n\n\n```python\n# @title Plotting Functions\n\ndef plot_original_and_resample(x, y, x_, y_):\n \"\"\" Plot the original sample and the resampled points from this sample.\n\n Args:\n x (ndarray): An array of shape (samples,) that contains the input values.\n y (ndarray): An array of shape (samples,) that contains the corresponding\n measurement values to the inputs.\n x_ (ndarray): An array of shape (samples,) with a subset of input values from x\n y_ (ndarray): An array of shape (samples,) with a the corresponding subset\n of measurement values as x_ from y\n\n \"\"\"\n fig, (ax1, ax2) = plt.subplots(ncols=2, figsize=(12, 5))\n ax1.scatter(x, y)\n ax1.set(title='Original', xlabel='x', ylabel='y')\n\n ax2.scatter(x_, y_, color='c')\n\n ax2.set(title='Resampled', xlabel='x', ylabel='y',\n xlim=ax1.get_xlim(), ylim=ax1.get_ylim());\n```\n\n---\n# Section 1: Bootstrapping\n\n*Estimated timing to here from start of tutorial: 7 min*\n\n[Bootstrapping](https://en.wikipedia.org/wiki/Bootstrapping_(statistics)) is a widely applicable method to assess confidence/uncertainty about estimated parameters, it was originally [proposed](https://projecteuclid.org/euclid.aos/1176344552) by [Bradley Efron](https://en.wikipedia.org/wiki/Bradley_Efron). The idea is to generate many new synthetic datasets from the initial true dataset by randomly sampling from it, then finding estimators for each one of these new datasets, and finally looking at the distribution of all these estimators to quantify our confidence.\n\nNote that each new resampled datasets will be the same size as our original one, with the new data points sampled with replacement i.e. we can repeat the same data point multiple times. Also note that in practice we need a lot of resampled datasets, here we use 2000.\n\nTo explore this idea, we will start again with our noisy samples along the line $y_i = 1.2x_i + \\epsilon_i$, but this time only use half the data points as last time (15 instead of 30).\n\n\n```python\n#@title\n\n#@markdown Execute this cell to simulate some data\n\n# setting a fixed seed to our random number generator ensures we will always\n# get the same psuedorandom number sequence\nnp.random.seed(121)\n\n# Let's set some parameters\ntheta = 1.2\nn_samples = 15\n\n# Draw x and then calculate y\nx = 10 * np.random.rand(n_samples) # sample from a uniform distribution over [0,10)\nnoise = np.random.randn(n_samples) # sample from a standard normal distribution\ny = theta * x + noise\n\nfig, ax = plt.subplots()\nax.scatter(x, y) # produces a scatter plot\nax.set(xlabel='x', ylabel='y');\n```\n\n## Coding Exercise 1: Resample Dataset with Replacement\n\nIn this exercise you will implement a method to resample a dataset with replacement. The method accepts $\\mathbf{x}$ and $\\mathbf{y}$ arrays. It should return a new set of $\\mathbf{x}'$ and $\\mathbf{y}'$ arrays that are created by randomly sampling from the originals.\n\nWe will then compare the original dataset to a resampled dataset.\n\nTIP: The [numpy.random.choice](https://numpy.org/doc/stable/reference/random/generated/numpy.random.choice.html) method would be useful here.\n\n\n```python\ndef resample_with_replacement(x, y):\n \"\"\"Resample data points with replacement from the dataset of `x` inputs and\n `y` measurements.\n\n Args:\n x (ndarray): An array of shape (samples,) that contains the input values.\n y (ndarray): An array of shape (samples,) that contains the corresponding\n measurement values to the inputs.\n\n Returns:\n ndarray, ndarray: The newly resampled `x` and `y` data points.\n \"\"\"\n #######################################################\n ## TODO for students: resample dataset with replacement\n # Fill out function and remove\n # raise NotImplementedError(\"Student exercise: resample dataset with replacement\")\n #######################################################\n\n # Get array of indices for resampled points\n sample_idx = np.random.choice(len(x), size=len(x), replace=True)\n\n # Sample from x and y according to sample_idx\n x_ = x[sample_idx]\n y_ = y[sample_idx]\n\n return x_, y_\n\nx_, y_ = resample_with_replacement(x, y)\n\nplot_original_and_resample(x, y, x_, y_)\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W1D3_ModelFitting/solutions/W1D3_Tutorial3_Solution_81af3bd6.py)\n\n*Example output:*\n\n\n\n\n\nIn the resampled plot on the right, the actual number of points is the same, but some have been repeated so they only display once.\n\nNow that we have a way to resample the data, we can use that in the full bootstrapping process.\n\n## Coding Exercise 2: Bootstrap Estimates\n\nIn this exercise you will implement a method to run the bootstrap process of generating a set of $\\hat\\theta$ values from a dataset of inputs ($\\mathbf{x}$) and measurements ($\\mathbf{y}$). You should use `resample_with_replacement` here, and you may also invoke helper function `solve_normal_eqn` from Tutorial 1 to produce the MSE-based estimator.\n\nWe will then use this function to look at the theta_hat from different samples.\n\n\n\n```python\n# @markdown Execute this cell for helper function `solve_normal_eqn`\ndef solve_normal_eqn(x, y):\n \"\"\"Solve the normal equations to produce the value of theta_hat that minimizes\n MSE.\n\n Args:\n x (ndarray): An array of shape (samples,) that contains the input values.\n y (ndarray): An array of shape (samples,) that contains the corresponding\n measurement values to the inputs.\n thata_hat (float): An estimate of the slope parameter.\n\n Returns:\n float: the value for theta_hat arrived from minimizing MSE\n \"\"\"\n theta_hat = (x.T @ y) / (x.T @ x)\n return theta_hat\n```\n\n\n```python\ndef bootstrap_estimates(x, y, n=2000):\n \"\"\"Generate a set of theta_hat estimates using the bootstrap method.\n\n Args:\n x (ndarray): An array of shape (samples,) that contains the input values.\n y (ndarray): An array of shape (samples,) that contains the corresponding\n measurement values to the inputs.\n n (int): The number of estimates to compute\n\n Returns:\n ndarray: An array of estimated parameters with size (n,)\n \"\"\"\n theta_hats = np.zeros(n)\n\n ##############################################################################\n ## TODO for students: implement bootstrap estimation\n # Fill out function and remove\n # raise NotImplementedError(\"Student exercise: implement bootstrap estimation\")\n ##############################################################################\n\n # Loop over number of estimates\n for i in range(n):\n\n # Resample x and y\n x_, y_ = resample_with_replacement(x, y)\n\n # Compute theta_hat for this sample\n theta_hats[i] = solve_normal_eqn(x_, y_)\n\n return theta_hats\n\n\n# Set random seed\nnp.random.seed(123)\n\n# Get boostrap estimates\ntheta_hats = bootstrap_estimates(x, y, n=2000)\nprint(theta_hats[0:5])\n```\n\n [1.27550888 1.17317819 1.18198819 1.25329255 1.20714664]\n\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W1D3_ModelFitting/solutions/W1D3_Tutorial3_Solution_1b78959c.py)\n\n\n\nYou should see `[1.27550888 1.17317819 1.18198819 1.25329255 1.20714664]` as the first five estimates.\n\nNow that we have our bootstrap estimates, we can visualize all the potential models (models computed with different resampling) together to see how distributed they are.\n\n\n```python\n#@title\n#@markdown Execute this cell to visualize all potential models\n\nfig, ax = plt.subplots()\n\n# For each theta_hat, plot model\ntheta_hats = bootstrap_estimates(x, y, n=2000)\nfor i, theta_hat in enumerate(theta_hats):\n y_hat = theta_hat * x\n ax.plot(x, y_hat, c='r', alpha=0.01, label='Resampled Fits' if i==0 else '')\n\n# Plot observed data\nax.scatter(x, y, label='Observed')\n\n# Plot true fit data\ny_true = theta * x\nax.plot(x, y_true, 'g', linewidth=2, label='True Model')\n\nax.set(\n title='Bootstrapped Slope Estimation',\n xlabel='x',\n ylabel='y'\n)\n\n# Change legend line alpha property\nhandles, labels = ax.get_legend_handles_labels()\nhandles[0].set_alpha(1)\n\nax.legend();\n```\n\nThis looks pretty good! The bootstrapped estimates spread around the true model, as we would have hoped. Note that here we have the luxury to know the ground truth value for $\\theta$, but in applications we are trying to guess it from data. Therefore, assessing the quality of estimates based on finite data is a task of fundamental importance in data analysis.\n\n\n---\n# Section 2: Confidence Intervals\n\n*Estimated timing to here from start of tutorial: 17 min*\n\nLet us now quantify how uncertain our estimated slope is. We do so by computing [confidence intervals](https://en.wikipedia.org/wiki/Confidence_interval) (CIs) from our bootstrapped estimates. The most direct approach is to compute percentiles from the empirical distribution of bootstrapped estimates. Note that this is widely applicable as we are not assuming that this empirical distribution is Gaussian.\n\n\n```python\n#@title\n\n#@markdown Execute this cell to plot bootstrapped CI\n\ntheta_hats = bootstrap_estimates(x, y, n=2000)\nprint(f\"mean = {np.mean(theta_hats):.2f}, std = {np.std(theta_hats):.2f}\")\n\nfig, ax = plt.subplots()\nax.hist(theta_hats, bins=20, facecolor='C1', alpha=0.75)\nax.axvline(theta, c='g', label=r'True $\\theta$')\nax.axvline(np.percentile(theta_hats, 50), color='r', label='Median')\nax.axvline(np.percentile(theta_hats, 2.5), color='b', label='95% CI')\nax.axvline(np.percentile(theta_hats, 97.5), color='b')\nax.legend()\nax.set(\n title='Bootstrapped Confidence Interval',\n xlabel=r'$\\hat{{\\theta}}$',\n ylabel='count',\n xlim=[1.0, 1.5]\n);\n```\n\nLooking at the distribution of bootstrapped $\\hat{\\theta}$ values, we see that the true $\\theta$ falls well within the 95% confidence interval, wich is reinsuring. We also see that the value $\\theta = 1$ does not fall within the confidence interval. From this we would reject the hypothesis that the slope was 1.\n\n---\n# Summary\n\n*Estimated timing of tutorial: 23 minutes*\n\n- Bootstrapping is a resampling procedure that allows to build confidence intervals around inferred parameter values\n- it is a widely applicable and very practical method that relies on computational power and pseudo-random number generators (as opposed to more classical approaches than depend on analytical derivations)\n\n---\n# Notation\n\n\\begin{align}\n\\theta &\\quad \\text{parameter}\\\\\n\\hat{\\theta} &\\quad \\text{estimated parameter}\\\\\nx &\\quad \\text{input, independent variable}\\\\\ny &\\quad \\text{response measurement, dependent variable}\\\\\n\\mathbf{x} &\\quad \\text{vector of input values}\\\\\n\\mathbf{y} &\\quad \\text{vector of measurements}\\\\\n\\mathbf{x}' &\\quad \\text{vector of resampled input values }\\\\\n\\mathbf{y}' &\\quad \\text{vector of resampled measurement values}\\\\\n\\end{align}\n\n**Suggested readings** \n\nComputer Age Statistical Inference: Algorithms, Evidence and Data Science, by Bradley Efron and Trevor Hastie\n\n", "meta": {"hexsha": "66c42d3ff77e92a3edd97d61ebc8a0bb2b8f4049", "size": 309746, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W1D3_ModelFitting/student/W1D3_Tutorial3.ipynb", "max_stars_repo_name": "luisarai/NMA2021", "max_stars_repo_head_hexsha": "d6cd66bf32d929f3030d0d66c2c92de55bd2d886", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tutorials/W1D3_ModelFitting/student/W1D3_Tutorial3.ipynb", "max_issues_repo_name": "luisarai/NMA2021", "max_issues_repo_head_hexsha": "d6cd66bf32d929f3030d0d66c2c92de55bd2d886", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/W1D3_ModelFitting/student/W1D3_Tutorial3.ipynb", "max_forks_repo_name": "luisarai/NMA2021", "max_forks_repo_head_hexsha": "d6cd66bf32d929f3030d0d66c2c92de55bd2d886", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 281.5872727273, "max_line_length": 139330, "alphanum_fraction": 0.9082312605, "converted": true, "num_tokens": 3706, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544085240402, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.43010289459920736}} {"text": "# Symbolic Differentiation vs Automatic Differentiation\n\nConsider the function below that, at least computationally, is very simple.\n\n\n```python\nfrom math import sin, cos\n\ndef func(x):\n y = x\n for i in range(30):\n y = sin(x + y)\n\n return y\n```\n\nWe can compute a derivative symbolically, but it is of course horrendous (see below). Think of how much worse it would be if we chose a function with products, more dimensions, or iterated more than 20 times.\n\n\n```python\nfrom sympy import diff, Symbol, sin\nfrom __future__ import print_function\n\nx = Symbol('x')\ndexp = diff(func(x), x)\nprint(dexp)\n```\n\n (((((((((((((((((((((((((((((2*cos(2*x) + 1)*cos(x + sin(2*x)) + 1)*cos(x + sin(x + sin(2*x))) + 1)*cos(x + sin(x + sin(x + sin(2*x)))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(2*x))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(2*x)))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x)))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x)))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x)))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x))))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x)))))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x))))))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x)))))))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x))))))))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x)))))))))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x))))))))))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x)))))))))))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x))))))))))))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x)))))))))))))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x))))))))))))))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x)))))))))))))))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x))))))))))))))))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x)))))))))))))))))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x))))))))))))))))))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x)))))))))))))))))))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x))))))))))))))))))))))))))))) + 1)*cos(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(x + sin(2*x))))))))))))))))))))))))))))))\n\n\nWe can now evaluate the expression.\n\n\n```python\nxpt = 0.1\n\ndfdx = dexp.subs(x, xpt)\n\nprint('dfdx =', dfdx)\n```\n\n dfdx = 1.91770676038667\n\n\nLet's compare with automatic differentiation using operator overloading:\n\n\n```python\nfrom algopy import UTPM, sin\n\nx_algopy = UTPM.init_jacobian(xpt)\ny_algopy = func(x_algopy)\ndfdx = UTPM.extract_jacobian(y_algopy)\n \nprint('dfdx =', dfdx)\n```\n\n dfdx = [ 1.91770676]\n\n\nLet's also compare to AD using a source code transformation method (I used Tapenade in Fortran)\n\n\n```python\ndef funcad(x):\n xd = 1.0\n yd = xd\n y = x\n for i in range(30):\n yd = (xd + yd)*cos(x + y)\n y = sin(x + y)\n return yd\n\ndfdx = funcad(xpt)\n\nprint('dfdx =', dfdx)\n```\n\n dfdx = 1.91770676039\n\n\nFor a simple expression like this, symbolic differentiation is long but actually works reasonbly well, and both will give a numerically exact answer. But if we change the loop to 100 (go ahead and try this) or add other complications, the symbolic solver will fail. However, automatic differentiation will continue to work without issue (see the simple source code transformation version). Furthermore, if we add other dimensions to the problem, symbolic differentiation quickly becomes costly as lots of computations get repeated, whereas automatic differentiation is able to reuse a lot of calculations.\n", "meta": {"hexsha": "06ae00e07c224da84f8f91f90c2a1d14b93e7a27", "size": 8617, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "SymbolicVsAD.ipynb", "max_stars_repo_name": "BYUFLOWLab/MDOnotebooks", "max_stars_repo_head_hexsha": "49344cb874a52cd67cc04ebb728195fa025d5590", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2017-03-13T23:22:32.000Z", "max_stars_repo_stars_event_max_datetime": "2017-08-10T14:15:31.000Z", "max_issues_repo_path": "SymbolicVsAD.ipynb", "max_issues_repo_name": "BYUFLOWLab/MDOnotebooks", "max_issues_repo_head_hexsha": "49344cb874a52cd67cc04ebb728195fa025d5590", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "SymbolicVsAD.ipynb", "max_forks_repo_name": "BYUFLOWLab/MDOnotebooks", "max_forks_repo_head_hexsha": "49344cb874a52cd67cc04ebb728195fa025d5590", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2019-03-12T11:31:01.000Z", "max_forks_repo_forks_event_max_datetime": "2019-03-12T11:31:01.000Z", "avg_line_length": 45.3526315789, "max_line_length": 4370, "alphanum_fraction": 0.4876407102, "converted": true, "num_tokens": 2200, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.8947894604912848, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.42992725710805246}} {"text": "# 02 Data Structures and Libraries\n## CLASS MATERIAL\n\n
                                        1. Data Structures\n
                                        2. Libraries \n
                                        3. Installing Pygame\n
                                        4. Review Exercises\n\n# Update the new class notes.\n__Navigate to the directory where your files are stored.__\n\n__Update the course notes by downloading the changes__\n\n\n\n\n##### Windows\nSearch for __Git Bash__ in the programs menu.\n\nSelect __Git Bash__, a terminal will open.\n\nUse `cd` to navigate to *inside* the __ILAS_PyEv2019__ repository you downloaded. \n\nRun the command:\n>`./automerge`\n\n\n\n##### Mac\nOpen a terminal. \n\nUse `cd` to navigate to *inside* the __ILAS_PyEv2019__ repository you downloaded. \n\nRun the command:\n>`sudo ./automerge`\n\nEnter your password when prompted. \n\n\n# Primer Summary \nFor more information refer to the primer notebook for this class 02_DataStructures_Libraries__Primer.ipynb\n\n###### Data Structures\n\n- A data structure is used to assign a collection of values to a single collection name.\n - A Python list can store multiple items of data in sequentially numbered elements (numbering starts at zero)\n - Data stored in a list element can be referenced using the list name can be referenced using the list name followed by an index number in [] square brackets.\n - The `len()` function returns the length of a specified list. \n\n\n\n###### Libraries\n- Python has an extensive __standard library__ of built-in functions. \n- More specialised libraries of functions and constants are available. We call these __packages__. \n- Packages are imported using the keyword `import`\n- The function documentation tells is what it does and how to use it.\n- When calling a library function it must be prefixed with a __namespace__ is used to show from which package it should be called. \n\n\n## Lesson Goal\n\n- Build a guessing game.\n- Build the game 0 and Xs or tic-tac-toe.\n\n

                                        \n \n

                                        \n\n\n\n## Fundamental programming concepts\n - Importing existing libraries of code to use in your program\n - Storing and representing data e.g. a grid with 0 and X in each grid cell\n\n\n# 1. Data Structures\n\nIn the last seminar we learnt to generate a range of numbers for use in control flow of a program, using the function `range()`:\n\n\n for j in range(20):\n ...\n \n \nOften we want to manipulate data that is more meaningful than ranges of numbers.\n\nThese collections of variables might include:\n - the results of an experiment\n - a list of names\n - the components of a vector\n - a telephone directory with names and associated numbers.\n \n\nPython has different __data structures__ that can be used to store and manipulate these values.\n\nLike variable types (`string`, `int`,`float`...) different data structures behave in different ways.\n\nToday we will learn to use `list`s \n\nA list is a container with compartments in which we can store data:\n

                                        \n \n

                                        \n\nExample\n\nIf we want to store the names of students in a laboratory group, \nrather than representing each students using an individual string variable, we could use a list of names. \n\n\n\n\n```python\nlab_group0 = [\"Yukari\", \"Sajid\", \"Hemma\", \"Ayako\"]\nlab_group1 = [\"Sara\", \"Mari\", \"Quang\", \"Sam\", \"Ryo\", \"Nao\", \"Takashi\"]\n\nprint(lab_group0)\nprint(lab_group1)\n```\n\n ['Yukari', 'Sajid', 'Hemma', 'Ayako']\n ['Sara', 'Mari', 'Quang', 'Sam', 'Ryo', 'Nao', 'Takashi']\n\n\nThis is useful because we can perform operations on lists such as:\n - checking its length (number of students in a lab group)\n - sorting the names in the list into alphabetical order\n - making a list of lists (we call this a *nested list*):\n\n\n\n```python\nlab_groups = [lab_group0, lab_group1]\nprint(lab_groups)\n```\n\n [['Yukari', 'Sajid', 'Hemma', 'Ayako'], ['Sara', 'Mari', 'Quang', 'Sam', 'Ryo', 'Nao', 'Takashi']]\n\n\n\n### Example: Change in Position: (Representing Vectors using Lists)\n\n__Vector:__ A quantity with magnitude and direction.\n\nThe position of a point in 2D space (e.g. the position of a character in a game), can be expressed in terms of horizontal (x) and vertical (y) conrdinates. \n\nThe movement to a new position can be expressed as a change in x and y. \n\n\n\n[Daniel Schiffman, The Nature of Code]\n\n\nWe can conveniently express the position $\\mathbf{r}$ in matrix (or basis vector) form using the coefficients $x$ and $y$: \n$$\n\\mathbf{r} = [r_x, r_y]\n$$\n\n\n__...which looks a lot like a Python list!__\n\n\nWhen we move a character in a game, we change it's position. \n\nThe change in position with each time-step is the __velocity__ of the character.\n\n__Velocity__ : the magnitude and direction of a change in position per time increment.\n\n$$\n\\mathbf{v} = [v_x, v_y]\n$$\n\n\n\nTo get the position at the next time step we simply add the x and y component of the veclocity to the x and y component of the initial position vector:\n\n \n \n\n\n\\begin{align}\n {\\displaystyle {\\begin{aligned}\\ \n \\mathbf{w}\n &=\\mathbf{u} + \\mathbf{u}\\\\\n &=[(u_x+v_x),\\;\\; (u_y+v_y)] \\\\ \\end{aligned}}} \n\\end{align}\n\n\nFor example, let's find the position at the next timestep where:\n - initial position, $\\mathbf{u} = [5, 2]$\n - velocity, $\\mathbf{v} = [3, 4]$\n \n \n \n [Daniel Schiffman, The Nature of Code]\n\n\n```python\n# Example: Change in Position\n\n```\n\n\n```python\n# Example Solution: Change in Position\n\nu = [5, 2]\nv = [3, 4]\n\nu = [u[0] + v[0], \n u[1] + v[1]]\n\nprint(u)\n```\n\n [8, 6]\n\n\nArranging the code on seperate lines:\n - makes the code more readable\n - does not effect how the code works\n \nLine breaks can only be used within code that is enclosed by at elast one set of brackets (), []. \n\n__Check Your Solution:__ \n\n\n$ \\mathbf{u} = [5, 2]$\n
                                        $ \\mathbf{v} = [3, 4]$\n\n\n\\begin{align}\n {\\displaystyle {\\begin{aligned}\\ \n \\mathbf{u} + \\mathbf{v}\n &=[5, 2]+ [3, 4]\\\\\n &=[(5+3), \\quad (2+4)] \\\\\n & = [8, 6] \\end{aligned}}} \n\\end{align}\n\n\n# 2. Libraries\n\n
                                           __2.1 The Standard Library__ \n
                                           __2.2 Packages__ \n
                                           __2.3 Function Documentation__ \n
                                           __2.4 Using Package Functions to Optimise your Code__ \n \nOne of the most important concepts in good programming is to reuse code and avoid repetitions.\n\nPython, like other modern programming languages, has an extensive *library* of built-in functions. \n\nThese functions are designed, tested and optimised by the developers of the Python langauge. \n\nWe can use these functions to make our code shorter, faster and more reliable.\n\n \n\n\n## 2.1 The Standard Library\n\nPython has a large standard library. \n\ne.g. `print()` takes the __input__ in the parentheses and __outputs__ a visible representation.\n\nThey are listed on the Python website:\nhttps://docs.python.org/3/library/functions.html\n\nWe could write our own code to find the minimum of a group of numbers\n\n\n\n\n\n```python\nx0 = 1\nx1 = 2\nx2 = 4\n\nx_min = x0\nif x1 < x_min:\n x_min = x1\nif x2 < x_min:\n x_min = x2\n \nprint(x_min)\n```\n\n 1\n\n\nHowever, it is much faster to use the build in function:\n\n\n```python\nprint(min(1,2,4))\n```\n\n 1\n\n\nThe built-in functions can be found in (.py) files called 'modules'.\n\nThe files are neatly arranged into a system of __sub-packages__ (sub-folders) and __modules__ (files).\n\nThese files are stored on the computer you are using.\n\nA quick google search for \"python function to sum all the numbers in a list\"...\n\nhttps://www.google.co.jp/search?q=python+function+to+sum+all+the+numbers+in+a+list&rlz=1C5CHFA_enJP751JP751&oq=python+function+to+sum+&aqs=chrome.0.0j69i57j0l4.7962j0j7&sourceid=chrome&ie=UTF-8\n\n...returns the function `sum()`.\n\n`sum()` finds the sum of the values in a data structure.\n\n\n\n\n\n\n```python\nprint(sum([1,2,3,4,5]))\n\nprint(sum((1,2,3,4,5)))\n\na = [1,2,3,4,5]\nprint(sum(a))\n```\n\n 15\n 15\n 15\n\n\nThe function `max()` finds the maximum value in data structure.\n\n\n## 2.2 Packages\n\nThe standard library tools are available in any Python environment.\n\nMore specialised libraries, called packages, are available for more specific tasks \n
                                        e.g. solving trigonometric functions.\n\nPackages contain functions and constants. \n\nWe install the packages to use them. \n\n\n\n__Pygame__\n
                                        For a large part of this course we will use functions from a package called `Pygame`.\n
                                        `Pygame` is a set of Python modules designed for writing computer games, graphics and sound projects. \n
                                        Instructions for how install pygame will be given later in today's seminar.\n\n__math__\n
                                        `math` is already installed and allows you to use convenient mathematical functions and operators.\n\n \n\nA package is a collection of Python modules: \n- a __module__ is a single Python file\n- a __package__ is a directory of Python modules.
                                        (It contains an __init__.py file, to distinguish it from folders that are not libraries).\n\nThe files that are stored on your computer when Pygame is installed:\n
                                        https://github.com/pygame/pygame\n\n\n## Importing a Package\n\nTo use an installed package, we simply `import` it.\n\nWe only need to import the package once.\n
                                        The `import` statement must appear before the use of the package in the code so all packages are usually imported at the start of the program. \n\n import math\n\nWe can then use variable and functions defined within that package by prefixing them with the name of the package:\n\n\n```python\nimport math\n```\n\nAfter this, any constant, variable or function from the package can be used by prefixing it with the name of the package:\n\nAny constant in `math` can be called as:\n\n `math.constant`.\n \n\nAny function in `numpy` can be called as:\n\n `math.function()`\n \n\n\n\n\n\n```python\n# pi\nprint(math.pi)\n\nx = 1\n\n# Trigonometric functions e.g. cosine\ny = math.cos(x)\nprint(y)\n```\n\n 3.141592653589793\n 0.5403023058681398\n\n\n\n## Using Package Functions. \n\nLet's learn to use `math` functions in our programs...\n\n\n\n\n\n\n```python\n# Some examples math functions with their definitions (as given in the documentation)\n\nx = 1\n\n# Return the sine of x radians.\nprint(math.sin(x))\n\n# Return the tangent of x radians.\nprint(math.tan(x))\n\n# Return the inverse hyperbolic tangent of x.\nprint(math.atan(x))\n\n\n```\n\n 0.8414709848078965\n 1.557407724654902\n 0.7853981633974483\n\n\n\n```python\nx = 1\n\n# Convert angle x from radians to degrees.\ndegrees = math.degrees(x)\nprint(degrees)\n\n# Convert angle x from degrees to radians.\nradians = math.radians(degrees)\nprint(radians) \n```\n\n 57.29577951308232\n 1.0\n\n\n\n## 2.3 Function Documentation\n\nOnline documentation can be used to find out: \n- what to include in the () parentheses\n- allowable data types to use as arguments\n- the order in which arguments should be given \n\n\nA google search for 'python math documentation' returns:\n\nhttps://docs.python.org/3/library/math.html\n\n(this list is not exhaustive). \n\n### Try it yourself:\n
                                        Find a function in the Python math documentation (https://docs.python.org/3/library/math.html) that can be used to solve the following problem: \n\n##### Return, $\\left| x \\right|$, the absolute value of x \n\n\nWrite your answer in the cell below:\n\n\n```python\n# Return the absolute value of x.\n```\n\n\n### Example : math.pow ($x^y$)\nDocumentation : https://docs.python.org/3/library/math.html#hyperbolic-functions\n\n \n\nThe documentation tells us the following information...\n\n##### What arguments to input:\n\"math.pow(x, y)\"\n\n##### What the function returns:\n\"Return x raised to the power y.\"\n\n##### The format of the returned argument:\nmath.pow() converts both its arguments to type float\n\n\n\n\n## 2.4 Using Package Functions to Optimise your Code\nOne purpose of using imported functions is to make your code shorter and neater.\n\n\n
                                        For example, when designing a game it can be very useful to generate (pseudo) random numbers so that the challenges and problems for the user to solve are not identical every time the game is played.\n\nWriting your own algorithm to generate random numbers is unecessarily time-consuming.\n\n\n\n`random` is a python module that implements pseudo-random number generators for various distributions.\n\nThe documentation of the functions in this package can be found here:\n
                                        https://docs.python.org/3/library/random.html#\n\nImport random to use functions from this package:\n\n\n```python\nimport random\n```\n\nHere are some examples of functions from `random`...\n\n\n```python\n# random.randint(a, b) \n# Return a random integer N such that a <= N <= b. \nrandom.randint(1, 15)\n\n```\n\n\n\n\n 14\n\n\n\n\n```python\n# random.sample(population, k)\n# Return a k length list of unique elements chosen from the population sequence or set. \n# Used for random sampling without replacement.\nrandom.sample([1,2,3,4,5,6,7,8], 4)\n```\n\n\n\n\n [4, 7, 8, 1]\n\n\n\nRandom numbers can be used to introduce an element of uncertainty to your programs.\n\nThis makes a game more intersting as the outcome may be different every time it is played.\n\nFor example, an adventure game may have a different outcome depending on a random variable.\n\n\n```python\nprint(\"Inside the castle you see a giant spider.\")\nanswer = input(\"Do you fight the spider? (Yes/No) \")\n \n \nif answer == 'Yes':\n print(\"you defeat the spider. \\nYou win!\")\n \nelse:\n print(\"The spider eats you. \\nYou lose!\")\n```\n\n Inside the castle you see a giant spider.\n Do you fight the spider? (Yes/No) Yes\n you defeat the spider. \n You win!\n\n\nAn adventure game where the outcome is random can be more interesting.\n\n\n```python\nprint(\"Inside the castle you see a giant spider.\")\nanswer = input(\"Do you fight the spider? (Yes/No) \")\n \n \nif answer == 'Yes':\n number = int(random.randint(0, 2))\n if number < 2:\n print(\"The spider defeats you. \\nYou lose!\")\n else:\n print(\"you defeat the spider. \\nYou win!\")\n \n\nelse:\n print(\"The spider eats you. \\nYou lose!\")\n```\n\n Inside the castle you see a giant spider.\n Do you fight the spider? (Yes/No) \n The spider eats you. \n You lose!\n\n\nRandom, computer generated values can also be used to build a game of rock, paper scissors where the user plays against the computer.\n\nThe code section below shows the case that the player choses *rock*.\n\n```python\n\ncomputer = int(random.randint(0, 2))\n if computer == 0:\n c_choice = \"rock\"\n elif computer == 1:\n c_choice = \"paper\"\n else:\n c_choice = \"scissors\"\n```\n\n\nThe full version of the code is given in example program: Examples/02_RockPaperScissors.py\n\nTry running this from the terminal by typing `python3 02_RockPaperScissors.py` from within the __PyEv2019/Examples__ directory.\n\n\n## 3. Installing Pygame\nInstall Pygame now. We will begin using this package in next week's class.\n\n##### Windows \n\n1. Open the Anaconda Prompt from the terminal.\n

                                        \n \n

                                        \n\n1. The window that opens will look like the command line. In the window type the following code then press 'Enter':\n>`conda install -c anaconda pip`\n\n1. When the installation completes type the following code then press 'Enter':\n>`pip install pygame`\n\n##### Mac\n\n1. Open a terminal. \n\n1. Type the following code then press 'Enter':\n>`conda install -c anaconda pip`\n\n1. When the installation completes type the following code then press 'Enter':\n>`pip install pygame`\n\n\n\nTo check the installation has worked type:\n>`import pygame` \n\nin a Jupyter notebook cell and run the cell. If no error is generated you have installed pygame successfully. \n\n\n# 4. Review Exercises\n\nCompete the exercise below.\n\nSave your answers as .py files and email them to:\n
                                        philamore.hemma.5s@kyoto-u.ac.jp\n\n## Review Exercise 1 : Guessing Game\n__(A)__\n
                                        \nWrite a game that:\n- chooses a random number between 1 and 10\n- asks the user to guess what it is\n- exits when the user guesses the right number\n\nUse:\n- a while loop\n- a break statement (to break out of the while loop)\n- a random number generator (from the package, `random`)\n\n\n\n
                                        \nExample : The output from your game might look like this:\n\n```\n I'm thinking of a random number between 1 and 10.\n Can you guess what it is?\n\n Guess what number I am thinking of: 5\n\n Guess what number I am thinking of: 2\n\n Guess what number I am thinking of: 1\n\n Guess what number I am thinking of: 9\n\n You win! I was thinking of 9.\n```\n\n*Hint: Remember that the function `input` returns a string even when a numerical value is entered.* \n\n\n```python\n# Review Exercise A: Guessing Game\n\n\n```\n\n I'm thinking of a random number between 1 and 10.\n Can you guess what it is?\n Guess what number I am thinking of: 4\n Guess what number I am thinking of: 5\n Guess what number I am thinking of: 6\n You win! I was thinking of 6\n\n\n\n```python\n# Review Exercise : Guessing Game \n# Example Solution\n\nimport random \nimport math\n\nnum = random.randint(1, 10)\n\nprint(\"I'm thinking of a random number between 1 and 10.\")\nprint(\"Can you guess what it is?\")\n\nwhile(1):\n\n guess = int(input(\"Guess what number I am thinking of: \"))\n\n if guess == num:\n print(f\"You win! I was thinking of {num}\")\n break\n```\n\n__(B)__\n
                                        Use `if` and `else` to generate a __clue__ for the player gets the answer wrong \n\n
                                        \nThe output from your game might look like this:\n```\n I'm thinking of a random number between 1 and 10.\n Can you guess what it is?\n\n Guess what number I am thinking of: 1\n Too low.\n\n Guess what number I am thinking of: 5\n Too low.\n\n Guess what number I am thinking of: 9\n\n You win! I was thinking of 9.\n ```\n\n\n```python\n# Review Exercise B: Guessing Game with Clues\n```\n\n\n```python\n# Review Exercise : Guessing Game with Clues\n# Example Solution\n\nimport random \nimport math\n\nnum = random.randint(1, 10)\n\nprint(\"I'm thinking of a random number between 1 and 10.\")\nprint(\"Can you guess what it is?\")\n\nwhile(1):\n\n guess = int(input(\"Guess what number I am thinking of: \"))\n\n if guess == num:\n print(f\"You win! I was thinking of {num}\")\n break\n\n elif guess < num:\n print(\"Too low\")\n\n else: \n print(\"Too high\")\n \n```\n\n I'm thinking of a random number between 1 and 10.\n Can you guess what it is?\n Guess what number I am thinking of: 2\n Too low\n Guess what number I am thinking of: 5\n Too high\n Guess what number I am thinking of: 3\n Too low\n Guess what number I am thinking of: 4\n You win! I was thinking of 4\n\n\n__(C)__\n
                                        Use `break` to quit the game if the user makes three consecutive wrong guesses.\n\n
                                        \nThe output from your game might look something like this:\n\n I'm thinking of a random number between 1 and 10.\n Can you guess what it is? You have 3 guesses...\n\n Guess what number I am thinking of: 1\n Too low.\n\n Guess what number I am thinking of: 2\n Too low.\n\n Guess what number I am thinking of: 5\n \n You used all your lives! You lose! I was thinking of 9.\n\n\n```python\n# Review Exercise C: Guessing game with maximum number of tries\n\n```\n\n\n```python\n# Review Exercise : Guessing game with maximum number of tries\n# Example Solution\n\nimport random \nimport math\n\nnum = random.randint(1, 10)\n\nprint(\"I'm thinking of a random number between 1 and 10.\")\nprint(\"Can you guess what it is?\")\n\nfor i in range(3):\n guess = int(input(\"Guess what number I am thinking of: \"))\n \n if guess == num:\n print(f\"You win! I was thinking of {num}\")\n break\n \n elif guess < num:\n print(\"Too low\")\n \n else: \n print(\"Too high\")\n \nprint(f\"You used all your lives! You lose! I was thinking of {num}.\")\n```\n\n I'm thinking of a random number between 1 and 10.\n Can you guess what it is?\n Guess what number I am thinking of: 3\n Too low\n Guess what number I am thinking of: 4\n Too low\n Guess what number I am thinking of: 5\n Too low\n You used all your lives! You lose! I was thinking of 7.\n\n\n## Review Exercise 2: List with `for` loop.\nIn the cell below, use a `for` loop to print the first letter of each month in the list.\n\n\n\n\n```python\n# Print the first letter of each month in the list\n\nmonths = [\"January\",\n \"February\",\n \"March\",\n \"April\",\n \"May\",\n \"June\",\n \"July\",\n \"August\",\n \"September\",\n \"October\",\n \"November\",\n \"December\"]\n```\n\n\n```python\n# Review Exercise: List with for loop\n# Example Solution\n```\n\n\n## Review Exercise 3 : Tic-tac-toe\nA list of lists can be used to represent a discrete set of positions the a player can occupy.\n\nWe can write a program that plays the the game tic-tac-toe, using a list of list. \n\nThe program is shown in the cell below. \n\n

                                        \n \n

                                        \n\n__(A)__\n\nThe program decides whether to quit the game based on user input after each turn taken.\n\nCopy and paste the program into the cell provided below.\n \nEdit the program to quit the game if either:\n - one of the players places three marks in a row.\n - all positions have been marked but noone has won.\n\n
                                        *Hints:* \n- Use:\n - `if` and `else`\n - the boolean operators `and`, `or`, `not`\n- If the value of three colinear places is equal, the game has finished/been won.\n\n\n\n\n\n```python\n# tic-tac-toe \n#\n# Board positions:\n#\n# 00 01 02\n# 10 11 12\n# 20 21 22\n\n\n# Example solution\n\n# # use a for loop to set-up the 3x3 grid \nboard = []\nfor row in range(3):\n #board.append(['_', '_', '_'])\n board.append(['_']*3)\n \n \n# print the output nicely \nfor row in board:\n for column in row:\n print(column, end='\\t')\n # new line at end of row\n print() \n\n \n# choose who goes first, X or 0 \nplayer = 'X'\n\n\n# keep playing the game until told to quit\nwhile(1):\n # use if and else to take turns at adding input from each player to the board\n if player == 'X':\n position = input('Player X, choose position: ')\n board[int(position[0])][int(position[1])] = 'X'\n player = '0'\n \n else: \n position = input('Player 0, choose position: ')\n board[int(position[0])][int(position[1])] = '0'\n player = 'X'\n \n \n #board[int(position[0])][int(position[1])] = player\n \n \n # print the output nicely \n for row in board:\n for column in row:\n print(column, end='\\t')\n # new line at end of row\n print() \n \n print('\\nnext turn \\n')\n \n \n game_over = input('Game over?(Y/N): ')\n if game_over == 'Y':\n break\n```\n\n__(B)__\n\n
                                        Edit the program to prevent a player from choosing a place that is already occupied.\n
                                        For example, the program might ask the player to choose again.\n
                                        e.g.\n\n position already occupied, Player X choose a different position\n\n\n```python\n# Review Exercise: Tic-tac-toe \n```\n\n\n```python\n# Review Exercise: Tic-tac-toe \n# Example Solution\n\n# use a for loop to set-up the board\nboard = []\nfor row in range(3):\n #board.append(['_', '_', '_'])\n board.append(['_']*3)\n \n \n# display the output nicely \nfor lists in board:\n for i in lists:\n print(i,end='\\t')\n # leave gap before next row\n print()\n \n \n# choose who goes first, X or 0 \nplayer = 'X'\n\n# keep playing the game until told to quit\nwhile(1):\n # use if and else to take turns at adding input from each player to the board\n if player == 'X':\n position = input('Player X, choose position: ')\n while(board[int(position[0])][int(position[1])] != '_'):\n position = input('Position taken, Player X, choose again: ')\n board[int(position[0])][int(position[1])] = 'X'\n player = '0'\n \n else: \n position = input('Player 0, choose position: ')\n while(board[int(position[0])][int(position[1])] != '_'):\n position = input('Position taken, Player 0, choose again: ')\n board[int(position[0])][int(position[1])] = '0'\n player = 'X' \n \n \n #board[int(position[0])][int(position[1])] = player\n \n \n # display the output nicely \n for lists in board:\n for i in lists:\n print(i,end='\\t')\n print()\n \n \n # game over : one player gets 3 in a row\n if((board[0][0] == board[1][1] == board[2][2]== 'X')or\n (board[2][0] == board[1][1] == board[0][2]== 'X')or\n \n (board[0][0] == board[1][0] == board[2][0]== 'X')or\n (board[0][1] == board[1][1] == board[2][1]== 'X')or\n (board[0][2] == board[1][2] == board[2][2]== 'X')or\n \n (board[0][0] == board[0][1] == board[0][2]== 'X')or\n (board[1][0] == board[1][1] == board[1][2]== 'X')or\n (board[2][0] == board[2][1] == board[2][2]== 'X')or\n \n \n (board[0][0] == board[1][1] == board[2][2]== '0')or\n (board[2][0] == board[1][1] == board[0][2]== '0')or\n \n (board[0][0] == board[1][0] == board[2][0]== '0')or\n (board[0][1] == board[1][1] == board[2][1]== '0')or\n (board[0][2] == board[1][2] == board[2][2]== '0')or\n \n (board[0][0] == board[0][1] == board[0][2]== '0')or\n (board[1][0] == board[1][1] == board[1][2]== '0')or\n (board[2][0] == board[2][1] == board[2][2]== '0')):\n print(\"Game over! Player \" + player + \" wins!\")\n break\n \n # game over : no empty sapces remain\n gameover = True\n for lists in board:\n for i in lists:\n if i=='_':\n gameover = False\n if gameover:\n print(\"Game over! No winner.\") \n\n\n```\n\n _\t_\t_\t\n _\t_\t_\t\n _\t_\t_\t\n Player X, choose position: 20\n _\t_\t_\t\n _\t_\t_\t\n X\t_\t_\t\n Player 0, choose position: 11\n _\t_\t_\t\n _\t0\t_\t\n X\t_\t_\t\n Player X, choose position: 20\n Position taken, Player X, choose again: 00\n X\t_\t_\t\n _\t0\t_\t\n X\t_\t_\t\n Player 0, choose position: 11\n Position taken, Player 0, choose again: 21\n X\t_\t_\t\n _\t0\t_\t\n X\t0\t_\t\n\n\n\n```python\n# tic-tac-toe \n```\n", "meta": {"hexsha": "b93f4af6f7b7e84a7f069bcd37dbf8f2e2c5b9eb", "size": 54164, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "02_DataStructures_Libraries__ClassMaterial.ipynb", "max_stars_repo_name": "hphilamore/ILAS_PyEv2019", "max_stars_repo_head_hexsha": "b3c8ebe00d6795f67879a50ce6ef517353b069c1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "02_DataStructures_Libraries__ClassMaterial.ipynb", "max_issues_repo_name": "hphilamore/ILAS_PyEv2019", "max_issues_repo_head_hexsha": "b3c8ebe00d6795f67879a50ce6ef517353b069c1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "02_DataStructures_Libraries__ClassMaterial.ipynb", "max_forks_repo_name": "hphilamore/ILAS_PyEv2019", "max_forks_repo_head_hexsha": "b3c8ebe00d6795f67879a50ce6ef517353b069c1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 29.1204301075, "max_line_length": 1408, "alphanum_fraction": 0.5304999631, "converted": true, "num_tokens": 7071, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.8152324938410784, "lm_q1q2_score": 0.42988556439833187}} {"text": "# Effect of $\\sigma_0^2$\n\n## Packages\n\n\n```python\ntry:\n import os\n\n from google.colab import drive\n drive.mount('/content/gdrive')\n\n os.chdir( '/content/gdrive/My Drive/Colab Notebooks/GoA/02_binary/' )\n !ls\nexcept Exception:\n pass\n```\n\n\n```python\nfrom tensorflow.keras.layers import Dense, Input, BatchNormalization\nfrom tensorflow.keras.layers import Conv2D, Flatten, Lambda\nfrom tensorflow.keras.layers import Reshape, Conv2DTranspose\nfrom tensorflow.keras.models import Model\nfrom tensorflow.keras.datasets import mnist\nfrom tensorflow.keras.losses import mse, binary_crossentropy, categorical_crossentropy\nfrom tensorflow.keras.utils import plot_model\nfrom tensorflow.keras import backend as K\nfrom tensorflow.keras.models import model_from_json\n\nfrom CommVAEBinary import CommVAEBinary\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.spatial.distance import cdist # For calculating QPSK decoding\nfrom functools import reduce\n\nimport datetime, itertools, dill\n```\n\n\n```python\n# The one who steals the data\ndef robinhood(fig, filename, col_dtype=[float, float], col_fmt=None):\n assert (len(fig.axes) < 2), \"More than one axis not supported\"\n ax = fig.axes[0]\n \n header = []\n fmt = []\n \n # Don't modify the argument here, it will get updated for all the following calls\n if not col_fmt:\n _col_fmt = [ \"%d\" if d == int else \"%.5f\" for d in col_dtype ]\n else:\n _col_fmt = col_fmt.copy()\n \n n_lines = len(ax.lines)\n x_data = ax.lines[0].get_xdata()\n \n data = np.zeros((x_data.shape[0], 2 * n_lines))\n \n for(i, line) in enumerate(ax.lines):\n data[:, 2*i] = line.get_xdata()\n data[:, 2*i+1] = line.get_ydata()\n \n header += [\"x_\" + line.get_label(), \"y_\" + line.get_label()]\n fmt += _col_fmt\n \n if filename is not None:\n with open(filename, 'w') as f:\n f.write(\",\".join(header) + \"\\n\")\n np.savetxt(f, data, delimiter=\",\", fmt=\",\".join(fmt))\n```\n\n### Evaluate for models\n\n\n```python\ninVecDim = 5\nencDim = 2\nmodels = {\n# # \"AWGN_S0100_00\": \"./models_32x01_sigma0_2/AWGN_s0100_00\",\n# \"AWGN_S0100\": \"./models_32x01_sigma0_2/AWGN_s0100_01\", #!\n# # \"AWGN_S0100_02\": \"./models_32x01_sigma0_2/AWGN_s0100_02\",\n \n# # \"AWGN_S0200_00\": \"./models_32x01_sigma0_2/AWGN_s0200_00\",\n# # \"AWGN_S0200_01\": \"./models_32x01_sigma0_2/AWGN_s0200_01\",\n# \"AWGN_S0200\": \"./models_32x01_sigma0_2/AWGN_s0200_02\", #!\n \n# # \"AWGN_S0300_00\": \"./models_32x01_sigma0_2/AWGN_s0300_00\",\n# # \"AWGN_S0300_01\": \"./models_32x01_sigma0_2/AWGN_s0300_01\",\n# \"AWGN_S0300\": \"./models_32x01_sigma0_2/AWGN_s0300_02\", #!\n \n# # \"AWGN_S0400_00\": \"./models_32x01_sigma0_2/AWGN_s0400_00\",\n# \"AWGN_S0400\": \"./models_32x01_sigma0_2/AWGN_s0400_01\", #!\n# # \"AWGN_S0400_02\": \"./models_32x01_sigma0_2/AWGN_s0400_02\",\n \n # \"AWGN_S0500_00\": \"./models_32x01_sigma0_2/AWGN_s0500_00\",\n # \"AWGN_S0500_01\": \"./models_32x01_sigma0_2/AWGN_s0500_01\", #!\n # \"AWGN_S0500_02\": \"./models_32x01_sigma0_2/AWGN_s0500_02\",\n# \"AWGN_S0500\": \"./models_32x01/model_32symbols_gray_awgn_s050\", #!\n \n# # \"AWGN_S1000_00\": \"./models_32x01_sigma0_2/AWGN_s1000_00\",\n# \"AWGN_S1000\": \"./models_32x01_sigma0_2/AWGN_s1000_01\", #!\n# # \"AWGN_S1000_02\": \"./models_32x01_sigma0_2/AWGN_s1000_02\"\n \n# \"AWGN_S0750\": \"./models_32x01_sigma0_2/AWGN_s0750_00\", #!\n# # \"AWGN_S0750_01\": \"./models_32x01_sigma0_2/AWGN_s0750_01\",\n# # \"AWGN_S0750_02\": \"./models_32x01_sigma0_2/AWGN_s0750_02\"\n \n # \"S0010_00\": \"./models_32x01_sigma0_2/RBF_s0010_00\",\n \"S0010\": \"./models_32x01_sigma0_2/RBF_s0010_01\", #!\n # \"S0010_02\": \"./models_32x01_sigma0_2/RBF_s0010_02\",\n \n # \"S0050_00\": \"./models_32x01_sigma0_2/RBF_s0050_00\",\n \"S0050\": \"./models_32x01_sigma0_2/RBF_s0050_01\", #!\n # \"S0050_02\": \"./models_32x01_sigma0_2/RBF_s0050_02\",\n \n # \"S0075_00\": \"./models_32x01_sigma0_2/RBF_s0075_00\",\n \"S0075\": \"./models_32x01_sigma0_2/RBF_s0075_01\", #!\n # \"S0075_02\": \"./models_32x01_sigma0_2/RBF_s0075_02\",\n \n # \"S0100_00\": \"./models_32x01_sigma0_2/RBF_s0100_00\",\n \"S0100\": \"./models_32x01_sigma0_2/RBF_s0100_01\", #!\n # \"S0100_02\": \"./models_32x01_sigma0_2/RBF_s0100_02\",\n \n # \"S0200_00\": \"./models_32x01_sigma0_2/RBF_s0200_00\", # <-\n \"S0200\": \"./models_32x01_sigma0_2/RBF_s0200_01\", #!\n # \"S0200_02\": \"./models_32x01_sigma0_2/RBF_s0200_02\",\n \n # \"S0300_00\": \"./models_32x01_sigma0_2/RBF_s0300_00\",\n \"S0300\": \"./models_32x01_sigma0_2/RBF_s0300_01\", #!\n # \"S0300_02\": \"./models_32x01_sigma0_2/RBF_s0300_02\",\n \n # \"S0400_00\": \"./models_32x01_sigma0_2/RBF_s0400_00\",\n \"S0400\": \"./models_32x01_sigma0_2/RBF_s0400_01\", #!\n # \"S0400_02\": \"./models_32x01_sigma0_2/RBF_s0400_02\",\n \n # \"S0500_00\": \"./models_32x01_sigma0_2/RBF_s0500_00\",\n \"S0500\": \"./models_32x01_sigma0_2/RBF_s0500_01\", #!\n # \"S0500_02\": \"./models_32x01_sigma0_2/RBF_s0500_02\",\n # \"S0500_AA\": \"./models_32x01/model_32symbols_gray_awgn_s050\", #!\n}\n\nSNR_range_dB = np.arange( 0.0, 16.0, 1.0 )\nresults = {}\n```\n\n\n```python\n# fig, ax = plt.subplots(nrows=len(models), ncols=2, figsize=(3.0*2, 3.0*len(models)))\nfor idx, (model_label, model_file) in enumerate(models.items()):\n # Clear any old models\n try:\n K.clear_session()\n del model\n except:\n pass\n \n model = CommVAEBinary()\n model.load_model(model_file)\n m_points = model.get_constellation()\n m_pow = np.mean(np.sum(m_points*m_points,axis=1))\n# m_points = np.sqrt(1.0/m_pow) * m_points\n# m_pow = np.mean(np.sum(m_points*m_points,axis=1))\n\n # Plot constellation\n fig_const = plt.figure(figsize=(3.0, 3.0))\n chDim = model.latent_dim//2\n for i in range(chDim):\n plt.scatter(m_points[:,i], m_points[:,i+chDim], c=np.arange(2**model.in_dim), s=80)\n for j in range(2**model.in_dim):\n plt.annotate( j, (m_points[j,i],m_points[j,i+chDim]), size=16)\n # # trick to avoid overlap during cheating\n # ax1.annotate( \"{:2d}\".format(j) if j < 16 else \" {:2d}\".format(j), (m2_points[j,i],m2_points[j,i+chDim]), size=16)\n plt.grid()\n plt.xticks(np.arange(-4.0,4.1,1.0))\n plt.yticks(np.arange(-4.0,4.1,1.0))\n# plt.xlabel(\"I\", fontdict={'fontsize':16})\n# plt.ylabel(\"Q\", fontdict={'fontsize':16})\n# plt.title(model_label)\n plt.savefig(\"gray_const{:02d}x{:02d}_{}.pdf\".format(inVecDim, chDim, model_label), format='pdf', bbox_inches='tight')\n \n # Plot distance matrix\n fig_dist = plt.figure(figsize=(3.0, 3.0))\n plt.imshow(cdist(m_points,m_points), vmin=0, vmax=8.0)\n plt.savefig(\"gray_cdist{:02d}x{:02d}_{}.pdf\".format(inVecDim, chDim, model_label), format='pdf', bbox_inches='tight')\n \n ## Check BLER\n noisePower = m_pow * 10.0**(-SNR_range_dB/10.0)\n n0_per_comp = noisePower/model.latent_dim\n\n err = []\n for n0 in n0_per_comp:\n thisErr = 0\n thisCount = 0\n while thisErr < 5000:\n txBlk = np.random.randint(2, size=(1000,model.in_dim))\n txTest, _ = model.encode(txBlk)\n rxTest = txTest + np.random.normal(scale=np.sqrt(n0), size=txTest.shape)\n rxDecode = model.decode(rxTest)\n rxBlk = np.where(rxDecode>0.5, 1, 0 )\n # thisErr += txBlk.shape[0]-np.sum(np.prod(rxBlk==txBlk,axis=1))\n thisErr += np.sum(rxBlk!=txBlk)\n thisCount += (1000*model.in_dim)\n err.append(thisErr/thisCount)\n results[model_label] = { \n \"ber\": np.array(err),\n \"n0\": model.n0/model.latent_dim,\n \"sigma0_2\": int(model_label[1:])/100, # because we didn't save sigma_0^2 in model\n \"pow\": m_pow\n }\n```\n\n### Traditional Methods\nLoad the constellation data from prespecified files and find BLER.\n\n\n```python\nresults_traditional = {}\n```\n\n\n```python\nqam_map = np.genfromtxt(\"./../AWGN/sphere_data/{:03d}x{:03d}_qam_gray.csv\".format(2**inVecDim,encDim))\nqam_sym_pow = np.mean(np.sum(qam_map*qam_map,axis=1))\nqam_map = np.sqrt(1.0/qam_sym_pow) * qam_map\n# print( \"QAM Avg. Tx Power:\", qam_sym_pow )\n\n# noisePower = qam_sym_pow * 10.0**(-SNR_range_dB/10.0)\nnoisePower = 1.0 * 10.0**(-SNR_range_dB/10.0)\nn0_per_comp = noisePower/encDim\n\nqam_d_min = np.unique(cdist(qam_map,qam_map))[1]\nprint(\"d_min:\", qam_d_min )\n\n# qam_en = qam_sym_pow / (qam_d_min**2)\nqam_en = 1.0 / (qam_d_min**2)\nprint(\"En:\", qam_en)\n\n# Plot constellation\nfig_const = plt.figure(figsize=(3.0, 3.0))\nchDim = encDim//2\nfor i in range(chDim):\n plt.scatter(qam_map[:,i], qam_map[:,i+chDim], c=np.arange(2**inVecDim), s=80)\n for j in range(2**inVecDim):\n plt.annotate( j, (qam_map[j,i],qam_map[j,i+chDim]), size=16)\n# # trick to avoid overlap during cheating\n# ax1.annotate( \"{:2d}\".format(j) if j < 16 else \" {:2d}\".format(j), (m2_points[j,i],m2_points[j,i+chDim]), size=16)\nplt.grid()\nplt.xticks(np.arange(-2.0,2.1,1.0))\nplt.yticks(np.arange(-2.0,2.1,1.0))\nplt.xlabel(\"I\", fontdict={'fontsize':16})\nplt.ylabel(\"Q\", fontdict={'fontsize':16})\nplt.savefig(\"gray_const{:02d}x{:02d}_qam.pdf\".format(inVecDim, chDim), format='pdf', bbox_inches='tight')\n\n# Plot distance matrix\nfig_cdist = plt.figure(figsize=(3.0, 3.0))\nplt.imshow(cdist(qam_map,qam_map))\nplt.savefig(\"gray_cdist{:02d}x{:02d}_qam.pdf\".format(inVecDim, chDim), format='pdf', bbox_inches='tight')\n```\n\n\n```python\nerr = []\nfor n0 in n0_per_comp:\n thisErr = 0\n thisCount = 0\n\n while thisErr < 5000:\n txSym = np.random.randint(2**inVecDim, size=1000)\n txTest = qam_map[txSym]\n rxTest = txTest + np.random.normal(scale=np.sqrt(n0), size=txTest.shape)\n rxDecode = cdist(rxTest, qam_map)\n rxSym = np.argmin(rxDecode,axis=1)\n # thisErr += np.sum(rxSym!=txSym)\n thisErr += reduce(lambda x1, x2: x1 + x2, map(lambda x: bin(x).count(\"1\"), rxSym ^ txSym))\n thisCount += (1000 * inVecDim)\n err.append(thisErr/thisCount)\n\nresults_traditional[\"QAM\"] = {\n \"en\": qam_en,\n \"dmin\": qam_d_min,\n \"sym_pow\": qam_sym_pow,\n \"ber\": np.array(err)\n}\n```\n\n\n```python\nagrell_map = np.genfromtxt(\"./../AWGN/sphere_data/{:03d}x{:03d}_agrell.csv\".format(2**inVecDim,encDim))\nagrell_sym_pow = np.mean(np.sum(agrell_map*agrell_map,axis=1))\nagrell_map = np.sqrt(1.0/agrell_sym_pow) * agrell_map\nprint( \"Agrell Avg. Tx Power:\", agrell_sym_pow )\n\n# noisePower = agrell_sym_pow * 10.0**(-SNR_range_dB/10.0)\nnoisePower = 1.0 * 10.0**(-SNR_range_dB/10.0)\nn0_per_comp = noisePower/encDim\n\nagrell_d_min = np.unique(cdist(agrell_map,agrell_map))[1]\nprint(\"d_min:\", agrell_d_min )\n\n# agrell_en = agrell_sym_pow / (agrell_d_min**2)\nagrell_en = 1.0 / (agrell_d_min**2)\nprint(\"En:\", agrell_en)\n\n# Plot constellation\nfig, ax = plt.subplots(nrows=1, ncols=2, figsize=(3.0*2, 3.0))\nchDim = encDim//2\nfor i in range(chDim):\n ax[0].scatter(agrell_map[:,i], agrell_map[:,i+chDim], c=np.arange(2**inVecDim), s=80)\n for j in range(2**inVecDim):\n ax[0].annotate( j, (agrell_map[j,i],agrell_map[j,i+chDim]), size=16)\n# # trick to avoid overlap during cheating\n# ax1.annotate( \"{:2d}\".format(j) if j < 16 else \" {:2d}\".format(j), (m2_points[j,i],m2_points[j,i+chDim]), size=16)\nax[0].grid()\nax[0].set_xticks(np.arange(-2.0,2.1,1.0))\nax[0].set_yticks(np.arange(-2.0,2.1,1.0))\nax[0].set_xlabel(\"I\", fontdict={'fontsize':16})\nax[0].set_ylabel(\"Q\", fontdict={'fontsize':16})\n\n# Plot distance matrix\nax[1].imshow(cdist(agrell_map,agrell_map))\n```\n\n\n```python\nerr = []\nfor n0 in n0_per_comp:\n thisErr = 0\n thisCount = 0\n \n while thisErr < 5000:\n txSym = np.random.randint(2**inVecDim, size=1000)\n txTest = agrell_map[txSym]\n rxTest = txTest + np.random.normal(scale=np.sqrt(n0), size=txTest.shape)\n rxDecode = cdist(rxTest, agrell_map)\n rxSym = np.argmin(rxDecode,axis=1)\n # thisErr += np.sum(rxSym!=txSym)\n thisErr += reduce(lambda x1, x2: x1 + x2, map(lambda x: bin(x).count(\"1\"), rxSym ^ txSym))\n thisCount += (1000 * inVecDim)\n err.append(thisErr/thisCount)\n\nresults_traditional[\"Agrell\"] = {\n \"en\": agrell_en,\n \"d_min\": agrell_d_min,\n \"sym_pow\": agrell_sym_pow,\n \"ber\": np.array(err)\n}\n```\n\n### Plot Results\n\n\n```python\n# colors = cycle(['b', 'g', 'c', 'r', 'm', 'y'])\nfig = plt.figure(figsize=(8*1.5,6*1.5))\n\nfor (label, result) in results.items():\n plt.semilogy(SNR_range_dB, \n result[\"ber\"], \n label=label, \n linewidth=2,\n linestyle=\":\" if \"Oshea\" in label or \"[1]\" in label else \"-\")\n\nplt.semilogy(SNR_range_dB, results_traditional[\"QAM\"][\"ber\"], label=\"QAM\",linestyle=\"-.\")\n# plt.semilogy(SNR_range_dB, results_traditional[\"Agrell\"][\"bler\"], label=\"Agrell [17]\", color=next(colors), linestyle=\"-.\")\nplt.semilogy(SNR_range_dB, results_traditional[\"Agrell\"][\"ber\"], label=\"Agrell\", linestyle=\"-.\")\n\nplt.legend(loc=\"lower left\", prop={'size':14})\nplt.grid()\n# plt.title(\"Best observed BLER of trained models\", fontdict={'fontsize':18})\nplt.xlabel(\"SNR ($dB$)\", fontdict={'fontsize':16})\nplt.ylabel(\"BER\", fontdict={'fontsize':16})\nplt.ylim((1e-2,1e0))\n\nrobinhood(fig, \"gray_ber_{:02d}x{:02d}.csv\".format(inVecDim,chDim), col_dtype=[int, float])\n```\n\n## Mutual Information\n\nThe mutual information in AWGN channel is upper bounded as\n\\begin{align}\n I(\\textbf{X}, \\hat{\\textbf{Z}}) \n &\\leq \\mathbb{E}_{p(\\textbf{x})} \\left(\n \\frac{1}{2\\sigma_0^2} \\sum \\limits_{j=1}^{m} z_j^2\n - \\frac{m}{2} \\left( 1 - \\frac{\\sigma_n^2}{\\sigma_0^2} \n + \\log \\frac{\\sigma_n^2}{\\sigma_0^2} \\right) \\right)\n\\end{align}\n\nCompute the bound and plot for each $\\sigma_0^2$.\n\n\n```python\nm = 2\n\nplt.plot([result['sigma0_2'] for (_, result) in results.items()],\n [ 1.0/(2*result['sigma0_2']) * 1.0 - m/2.0 * (\n 1.0 - result['n0']/result['sigma0_2'] + np.log(result['n0']/result['sigma0_2'])) \n for (model_label, result) in results.items()], \n marker='d',\n label = \"from models\")\n\nplt.plot([result['sigma0_2'] for (_, result) in results.items()],\n [ 1.0/(2*result['sigma0_2']) * result['pow'] - m/2.0 * (\n 1.0 - result['n0']/result['sigma0_2'] + np.log(result['n0']/result['sigma0_2'])) \n for (model_label, result) in results.items()], \n marker='h',\n label = \"With unit energy\")\n\nplt.grid()\nplt.xlabel(\"$\\sigma_0^2$\")\nplt.ylabel(\"Upper bound on Mutual Information\")\nplt.legend()\n```\n\n\n```python\nsigma0_2 = np.linspace(0.1, 10.0, 100)\nconst_pow = 1.0\n\nn0 = 0.10\nplt.plot(sigma0_2,\n [ 1.0/(2*s) * const_pow - m/2.0 * (1.0 - n0/s + np.log(n0/s)) for s in sigma0_2], \n label = \"$n_0^2 = {:.2f}$\".format(n0))\n\nn0 = 0.20\nplt.plot(sigma0_2,\n [ 1.0/(2*s) * const_pow - m/2.0 * (1.0 - n0/s + np.log(n0/s)) for s in sigma0_2], \n label = \"$n_0^2 = {:.2f}$\".format(n0))\n\nn0 = 0.50\nplt.plot(sigma0_2,\n [ 1.0/(2*s) * const_pow - m/2.0 * (1.0 - n0/s + np.log(n0/s)) for s in sigma0_2], \n label = \"$n_0^2 = {:.2f}$\".format(n0))\n\nn0 = 1.00\nplt.plot(sigma0_2,\n [ 1.0/(2*s) * const_pow - m/2.0 * (1.0 - n0/s + np.log(n0/s)) for s in sigma0_2], \n label = \"$n_0^2 = {:.2f}$\".format(n0))\n\nn0 = 2.00\nplt.plot(sigma0_2,\n [ 1.0/(2*s) * const_pow - m/2.0 * (1.0 - n0/s + np.log(n0/s)) for s in sigma0_2], \n label = \"$n_0^2 = {:.2f}$\".format(n0))\n\nn0 = 3.00\nplt.plot(sigma0_2,\n [ 1.0/(2*s) * const_pow - m/2.0 * (1.0 - n0/s + np.log(n0/s)) for s in sigma0_2], \n label = \"$n_0^2 = {:.2f}$\".format(n0))\n\n\nplt.grid()\nplt.xlabel(\"$\\sigma_0^2$\")\nplt.ylabel(\"Upper bound on Mutual Information\")\nplt.legend()\nplt.ylim([0.0, 5.0])\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "a48402b7339b8fa0e082d08da491ae0923530fb2", "size": 475415, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "GrayCodes/diff_sigma0_eval.ipynb", "max_stars_repo_name": "v-i-s-h/dl-vi-comm", "max_stars_repo_head_hexsha": "80032588a5e6e13bdc397ce9bd51fed73a6045bf", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2020-07-30T10:45:03.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-21T01:03:53.000Z", "max_issues_repo_path": "GrayCodes/diff_sigma0_eval.ipynb", "max_issues_repo_name": "v-i-s-h/dl-vi-comm", "max_issues_repo_head_hexsha": "80032588a5e6e13bdc397ce9bd51fed73a6045bf", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "GrayCodes/diff_sigma0_eval.ipynb", "max_forks_repo_name": "v-i-s-h/dl-vi-comm", "max_forks_repo_head_hexsha": "80032588a5e6e13bdc397ce9bd51fed73a6045bf", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-02-26T10:55:27.000Z", "max_forks_repo_forks_event_max_datetime": "2021-02-26T10:55:27.000Z", "avg_line_length": 504.1516436903, "max_line_length": 85164, "alphanum_fraction": 0.9379110882, "converted": true, "num_tokens": 5558, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872131147276, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.4297351594626957}} {"text": "**Objective:** \n\nIn this tutorial we will create a simple gravity problem from scratch using the SimPEG framework.\n\nThe relation between density and the gravity field is well known, thanks to the classic work of Newton in 1686. Since we generally only measure the vertical component of the field, this relationship can be written as:\n$$G(r)_z = \\gamma \\int_{V} \\rho(r) \\left(\\frac{z - z_0}{{|\\vec r - \\vec r_0|}^3}\\right) \\; dV $$\nwhere $\\rho$ is the anomalous density and $\\gamma$ is the Newton's gravitational constant.\nOnce again, this integral can be evaluated analytically for simple prisms, giving rise to a linear system of equations relating a discrete Earth to the observed data:|\n$$ \\mathbf{d}_z = \\mathbf{G}_z \\; \\rho $$\n\n\n```python\n## Need to be on \\pf\\dev branch !!!\n%pylab inline \nimport SimPEG.PF as PF\nfrom SimPEG import *\nfrom SimPEG.Utils import io_utils\n```\n\n Populating the interactive namespace from numpy and matplotlib\n Efficiency Warning: Interpolation will be slow, use setup.py!\n \n python setup.py build_ext --inplace\n \n\n\n\n```python\nimport matplotlib\nmatplotlib.rcParams['font.size'] = 14\n```\n\n\n```python\n# We first need to create a susceptibility model.\n# Based on a set of parametric surfaces representing TKC, \n# we use VTK to discretize the 3-D space.\n\nmshfile = 'MEsh_TEst.msh'\n\nmodel_dir = '../../Geological_model/'\n\n# Load mesh file\nmesh = Mesh.TensorMesh.readUBC(model_dir+mshfile)\n\n# Create our own mesh!\ncsx, csy, csz = 10., 10., 10.\nncx, ncy, ncz = 55, 55, 30\nnpad = 10\nhx = [(csx,npad, -1.3),(csx,ncx),(csx,npad, 1.3)]\nhy = [(csy,npad, -1.3),(csy,ncy),(csy,npad, 1.3)]\nhz = [(csz,npad, -1.3),(csz,ncz), (10,20)]\nmesh = Mesh.TensorMesh([hx, hy, hz],x0=\"CCN\")\nxc = 300+5.57e5\nyc = 600+7.133e6\nzc = 450.\nx0_new = np.r_[mesh.x0[0]+xc, mesh.x0[1]+yc, mesh.x0[2]+zc]\nmesh._x0 = x0_new\n\n#Mesh.TensorMesh.writeUBC(mesh,model_dir+\"Mesh_mag.msh\")\n# Define no-data-value\nndv = -100\n\n# Define survey flight height\nZ_bird = 2.\n\n# Read in topo surface\ntopofile = model_dir+'TKCtopo.dat'\ngeosurf = [\n [model_dir+'Till.ts',True,True,0],\n [model_dir+'XVK.ts',True,True,1],\n [model_dir+'PK1.ts',True,True,2],\n [model_dir+'PK2.ts',True,True,3],\n [model_dir+'PK3.ts',True,True,4],\n [model_dir+'HK1.ts',True,True,5],\n [model_dir+'VK.ts',True,True,6]\n]\n\n\n\n```\n\n\n```python\nimport time as tm\nimport mpl_toolkits.mplot3d as a3\nimport matplotlib.colors as colors\nimport scipy as sp\n# from mayavi import mlab\n# mlab.close(all=True)\n# fig = plt.figure(figsize=(8,6))\n\n\n# mlab.figure(bgcolor=(1.,1.,1.))\nmodelInd = np.ones(mesh.nC)*ndv\nfor ii in range(len(geosurf)):\n tin = tm.time()\n print \"Computing indices with VTK: \" + geosurf[ii][0]\n T, S = io_utils.read_GOCAD_ts(geosurf[ii][0])\n indx = io_utils.surface2inds(T,S,mesh, boundaries=geosurf[ii][1], internal=geosurf[ii][2])\n print \"VTK operation completed in \" + str(tm.time() - tin) + \" sec\"\n modelInd[indx] = geosurf[ii][3]\n \n# S = S-1\n# mlab.triangular_mesh(T[:,0], T[:,1], T[:,2], S)\n# mlab.view(azimuth=-45.,distance=1000,elevation=75.)\n\n# arr = mlab.screenshot()\n\n# plt.imshow(arr)\n```\n\n Computing indices with VTK: ../../Geological_model/Till.ts\n Extracting indices from grid...\n VTK operation completed in 30.5199999809 sec\n Computing indices with VTK: ../../Geological_model/XVK.ts\n Extracting indices from grid...\n VTK operation completed in 5.8599998951 sec\n Computing indices with VTK: ../../Geological_model/PK1.ts\n Extracting indices from grid...\n VTK operation completed in 12.0950000286 sec\n Computing indices with VTK: ../../Geological_model/PK2.ts\n Extracting indices from grid...\n VTK operation completed in 5.27600002289 sec\n Computing indices with VTK: ../../Geological_model/PK3.ts\n Extracting indices from grid...\n VTK operation completed in 6.99499988556 sec\n Computing indices with VTK: ../../Geological_model/HK1.ts\n Extracting indices from grid...\n VTK operation completed in 13.6840000153 sec\n Computing indices with VTK: ../../Geological_model/VK.ts\n Extracting indices from grid...\n VTK operation completed in 15.2269999981 sec\n\n\n\n```python\n# Load topography file in UBC format and find the active cells\n#T, S = io_utils.read_GOCAD_ts(topsurf)\n#indx = io_utils.surface2inds(T,S, mesh, boundaries=True, internal=True) \n#actv = np.zeros(mesh.nC)\n#actv[indx] = 1\n\ntopo = np.genfromtxt(topofile,skip_header=1)\n# Find the active cells\nactv = Utils.surface2ind_topo(mesh, topo, gridLoc='N')\n# actv = PF.Magnetics.getActiveTopo(mesh,topo,'N')\n\n# Create active map to go from reduce set to full\nactvMap = Maps.InjectActiveCells(mesh, actv, ndv)\n\nprint \"Active cells created from topography!\"\n```\n\n Active cells created from topography!\n\n\n\n```python\n# Build model\n\ndef getModel(Till=-1e-2, XVK=2e-3, PK1=-0.4, PK2=-0.3, PK3=-0.2, HK1=-0.1, VK=-0.15, bkgr=0.):\n vals = [Till, XVK, PK1, PK2, PK3, HK1, VK]\n model= np.ones(mesh.nC) * bkgr\n\n for ii, den in zip(range(7),vals):\n model[modelInd == ii] = den\n return model\nmodel = getModel()\nmodel = model[actv]\n\n```\n\n\n```python\nfrom ipywidgets.widgets import interact, IntSlider\n```\n\n\n```python\n# Here you can visualize the current model\nm_true = actvMap*model\nMesh.TensorMesh.writeModelUBC(mesh,\"Synthetic_Grav.den\",m_true)\nairc = m_true==ndv\nm_true[airc] = np.nan\ndef slide(s,normal):\n colorbar(mesh.plotSlice(m_true, normal=normal, ind=s, clim=np.r_[model.min(), model.max()])[0])\n plt.gca().set_aspect('equal')\ninteract(slide, s=(0,60), normal=['X','Y','Z'])\n```\n\n**Forward system:**\n\nNow that we have all our spatial components, we can create our linear system. For a single location and single component of the data, \n#Need to add stuff\n\nthe system end up looking like this system:\n\n$$ d^{obs} = \\mathbf{F\\; \\rho}$$\n\nwhere $\\mathbf{F} \\in \\mathbb{R}^{nd \\times nc}$ is our $forward$ operator.\n\n\n```python\nfrom scipy.interpolate import NearestNDInterpolator\n# We need to define the direction of the inducing field\n# From old convention, field orientation is given as an azimuth from North \n# (positive clockwise) and dip from the horizontal (positive downward).\n# The field parameters at TKC are [H:60,308 nT, I:83.8 d D:25.4 d ]\nH0 = (60308.,83.8,25.4)\n\n# We create a synthetic survey with observations in cell center.\nX, Y = np.meshgrid(mesh.vectorCCx[npad:-npad:2], mesh.vectorCCy[npad:-npad:2])\n\n# Using our topography, we trape the survey and shift it up by the flight height\nFtopo = NearestNDInterpolator(topo[:,:2], topo[:,2])\nZ = Ftopo(Utils.mkvc(X.T),Utils.mkvc(Y.T)) + Z_bird\n\nrxLoc = np.c_[Utils.mkvc(X.T), Utils.mkvc(Y.T), Utils.mkvc(Z.T)]\nrxLoc = PF.BaseGrav.RxObs(rxLoc)\n\nsrcField = PF.BaseGrav.SrcField([rxLoc])\nsurvey = PF.BaseGrav.LinearSurvey(srcField)\n```\n\n\n```python\n# Now that we have a model and a survey we can build the linear system ...\nnactv = np.int(np.sum(actv))\n\n# Creat reduced identity map\nidenMap = Maps.IdentityMap(nP=nactv)\n\n# Create the forward model operator\nprob = PF.Gravity.GravityIntegral(mesh, mapping=idenMap, actInd=actv)\n\n# Pair the survey and problem\nsurvey.pair(prob)\n\n```\n\n\n```python\n# Fist time that we ask for predicted data,\n# the dense matrix T is calculated.\n# This is generally the bottleneck of the integral formulation, \n# in terms of time and memory.\nd = prob.fields(model)\n\n# Add noise to the data and assign uncertainties\ndata = d + randn(len(d))*0.01 # We add some random Gaussian noise (1nT)\nwd = np.ones(len(data))*np.sqrt(0.01) # Assign flat uncertainties\n\nsurvey.dobs = data\nsurvey.std = wd\n\nPF.Gravity.writeUBCobs('GRAV_Synthetic_data.obs',survey,data)\n\nd2D = data.reshape(X.shape)\ndat = plt.contourf(X-xc,Y-yc, d2D,40) \nplt.gca().set_aspect('equal')\nplt.plot(X.flatten()-xc,Y.flatten()-yc,'k.', ms=2)\nplt.colorbar(dat) \nplt.xlabel(\"Easting (m)\")\nplt.ylabel(\"Northing (m)\")\nplt.title(\"Gz (mGal)\")\nxlim(-500, 500)\nylim(-500, 500) \n```\n\n\n```python\n# d2Dx, d2Dy, d2Dz = d[0:survey.nRx].reshape(X.shape), d[survey.nRx:2*survey.nRx].reshape(X.shape), d[2*survey.nRx:].reshape(X.shape)\n\n\n# plt.figure(figsize=[5,5])\n# xc, yc = (X.min()+X.max())*0.5, (Y.min()+Y.max())*0.5\n# # colorbar(imshow(d2D,extent=[X.min()-xc, X.max()-xc, Y.min()-yc, Y.max()-yc],origin = 'lower'))\n# # plt.contour(X-xc,Y-yc, d2D,20,colors='k')\n# # dat = plt.contourf(X-xc,Y-yc, d2Dz,40)\n\n# # plt.gca().set_aspect('equal')\n\n\n# mlab.figure(bgcolor=(1.,1.,1.))\n# mlab.triangular_mesh(T[:,0], T[:,1], T[:,2], S)\n# mlab.quiver3d(X,Y,Z,d2Dx, d2Dy, d2Dz, color)\n# mlab.view(azimuth=-45.,distance=1500,elevation=75.)\n# arr = mlab.screenshot()\n\n# plt.imshow(arr)\n```\n\n**Inverse problem**\n\nWe have generated synthetic data, we now what to see if we can solve the inverse. Using the usual formulation, we seek a model that can reproduce the data, let\u2019s say a least-squares measure of the form:\n\n\\begin{equation}\n\\phi_d = \\|\\mathbf{W}_d \\left( \\mathbb{F}[\\mathbf{m}] - \\mathbf{d}^{obs} \\right)\\|_2^2\n\\end{equation}\n\nThe inverse problem is hard because we don\u2019t have great data coverage, and the Earth is big, and there is usually noise in the data. So we need to add something to regularize it.\nThe simplest way to do it is to penalize solutions that won\u2019t make sense geologically, for example to assume that the model is small.\nThe usual smooth inversion function use an l2-norm measure:\n\n\\begin{equation}\n\\phi_d = \\|\\mathbf{W}_d \\left( \\mathbb{F}[\\mathbf{m}] - \\mathbf{d}^{obs} \\right)\\|_2^2 \\\\\n\\phi_m = \\beta \\Big [ {\\| \\mathbf{W}_s \\;( \\mathbf{m - m^{ref}})\\|}^2_2 + \\sum_{i = x,y,z} {\\| \\mathbf{W}_i \\; \\mathbf{G}_i \\; \\mathbf{m}\\|}^2_2 \\Big ]\\;,\n\\end{equation}\n\nThe full objective function to be minimized can be written as:\n\\begin{equation}\n\\phi(m) = \\phi_d + \\beta \\phi_m\\;,\n\\end{equation}\nwhich will yield our usual *small* and *smooth* models. \n\nWe propose a fancier regularization function that can allow to recover *sparse* and *blocky* solutions.\nStarting with the well known Ekblom norm:\n\\begin{equation}\n\\phi_m = \\sum_{i=1}^{nc} {(x_i^2 + \\epsilon^2)}^{p/2} \\;,\n\\end{equation}\nwhere $x_i$ denotes some function of the model parameter, and $\\epsilon$ is a small value to avoid singularity as $m\\rightarrow0$.\nFor p=2, we get the usual least-squares measure and we recover the regularization presented above. For $p \\leq 1$, the function becomes non-linear which requires some tweaking.\n\nWe can linearize the function by updating the penality function iteratively, commonly known as an Iterative Re-weighted Least-Squares (IRLS) method:\n\\begin{equation} \n\\phi_m^{(k)} = \\frac{1}{2}\\sum_{i=1}^{nc} r_i \\; x_i^2\n\\end{equation}\nwhere we added the superscript $\\square^{(k)}$ to denote the IRLS iterations. The weights $r(x)$ are computed from model values obtained at a previous iteration such that:\n\\begin{equation}\n\t{r}_i ={\\Big( {({x_i}^{(k-1)})}^{2} + \\epsilon^2 \\Big)}^{p/2 - 1} \\;,\n\\end{equation}\nwhere ${r}(x) \\in \\mathbb{R}^{nc}$.\n\nIn matrix form, our objective function simply becomes:\n\\begin{equation}\n\\phi(m) = \\|\\mathbf{W}_d \\left( \\mathbb{F}[\\mathbf{m}] - \\mathbf{d}^{obs} \\right)\\|_2^2 + \\beta \\Big [ {\\| \\mathbf{W}_s \\;\\mathbf{R}_s\\;( \\mathbf{m - m^{ref}})\\|}^2_2 + \\sum_{i = x,y,z} {\\| \\mathbf{W}_i\\; \\mathbf{R}_i \\; \\mathbf{G}_i \\; \\mathbf{m}\\|}^2_2 \\Big ]\\;,\n\\end{equation}\nwhere the IRLS weights $\\mathbf{R}_s$ and $\\mathbf{R}_i$ are diagonal matrices defined as:\n\\begin{equation}\n\\begin{split}\n\t{R}_{s_{jj}} &= \\sqrt{\\eta_p}{\\Big[ {({m_j}^{(k-1)})}^{2} + \\epsilon_p^2 \\Big]}^{(p/2 - 1)/2} \\\\\n\t{R}_{i_{jj}} &= \\sqrt{\\eta_q}{\\Big[ {\\left ({{(G_i\\;m^{(k-1)})}_j }\\right)}^{2} + \\epsilon_q^2 \\Big]}^{(q/2 - 1)/2} \\\\\n\\eta_p &= {\\epsilon_p}^{(1-p/2)} \\\\\n\\eta_q &= {\\epsilon_q}^{(1-q/2)} \\;, \n\\end{split}\n\\end{equation}\n\nwe added two scaling parameters $\\eta_p$ and $\\eta_q$ for reasons that we won't dicuss here, but turn out to be important to get stable solves.\n\nIn order to initialize the IRLS and get an estimate for the stabilizing parameters $\\epsilon_p$ and $\\epsilon_q$, we first invert with the smooth $l_2$-norm. \nThe whole IRLS process is implemented with a directive added to the inversion workflow (see below).\n\n\n\n```python\n# It is potential fields, so we will need to push the inverison down\n# Create distance weights from our linera forward operator\n# rxLoc = survey.srcField.rxList[0].locs\n# wr = PF.Magnetics.get_dist_wgt(mesh, rxLoc, actv, 3., np.min(mesh.hx)/4.)\n# wr = wr**2.\nwr = np.sum(prob.G**2.,axis=0)**0.5\nwr = ( wr/np.max(wr) )\n \nreg = Regularization.Sparse(mesh, indActive = actv, mapping = idenMap)\nreg.cell_weights = wr\n\ndmis = DataMisfit.l2_DataMisfit(survey)\ndmis.Wd = 1/wd\n\n# Add directives to the inversion\nopt = Optimization.ProjectedGNCG(maxIter=100 ,lower=-1.,upper=1., maxIterLS = 20, maxIterCG= 10, tolCG = 1e-3)\ninvProb = InvProblem.BaseInvProblem(dmis, reg, opt)\nbetaest = Directives.BetaEstimate_ByEig()\n\n# Here is where the norms are applied\neps = [1e-2,1e-2] # Threshold parameters to penalize different model parameters\nnorms = [0,1,1,1] # Norms applies on the model and 3 gradients [p, qx, qy, qz]\n\n\nIRLS = Directives.Update_IRLS( norms=norms, eps=eps, f_min_change = 1e-1, minGNiter=6)\nupdate_Jacobi = Directives.Update_lin_PreCond()\ninv = Inversion.BaseInversion(invProb, directiveList=[IRLS,betaest,update_Jacobi])\n\nm0 = np.ones(idenMap.nP)*1e-4\n\n```\n\n\n```python\n# Run inversion...\nmrec = inv.run(m0)\n```\n\n SimPEG.InvProblem will set Regularization.mref to m0.\n SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.\n ***Done using same Solver and solverOpts as the problem***\n =============================== Projected GNCG ===============================\n # beta phi_d phi_m f |proj(x-g)-x| LS Comment \n -----------------------------------------------------------------------------\n 0 1.97e+05 1.10e+04 0.00e+00 1.10e+04 5.47e+02 0 \n 1 9.84e+04 1.05e+04 1.21e-03 1.06e+04 5.29e+02 0 \n 2 4.92e+04 9.85e+03 5.62e-03 1.01e+04 5.13e+02 0 Skip BFGS \n 3 2.46e+04 8.95e+03 1.91e-02 9.42e+03 5.01e+02 0 Skip BFGS \n 4 1.23e+04 7.56e+03 6.03e-02 8.30e+03 4.76e+02 0 Skip BFGS \n 5 6.15e+03 5.81e+03 1.62e-01 6.81e+03 4.24e+02 0 Skip BFGS \n 6 3.07e+03 4.12e+03 3.54e-01 5.21e+03 3.35e+02 0 Skip BFGS \n 7 1.54e+03 2.89e+03 6.26e-01 3.86e+03 2.34e+02 0 Skip BFGS \n 8 7.68e+02 2.05e+03 9.95e-01 2.82e+03 1.58e+02 0 Skip BFGS \n 9 3.84e+02 1.39e+03 1.59e+00 2.01e+03 1.21e+02 0 Skip BFGS \n 10 1.92e+02 8.30e+02 2.63e+00 1.33e+03 8.74e+01 0 Skip BFGS \n 11 9.60e+01 4.10e+02 4.13e+00 8.07e+02 6.20e+01 0 Skip BFGS \n Convergence with smooth l2-norm regularization: Start IRLS steps...\n L[p qx qy qz]-norm : [0, 1, 1, 1]\n eps_p: 0.01 eps_q: 0.01\n Regularization decrease: 0.000e+00\n 12 2.22e+02 1.69e+02 4.13e+00 1.09e+03 1.54e+02 0 Skip BFGS \n 13 2.22e+02 1.99e+02 2.46e+00 7.45e+02 8.83e+01 0 \n 14 2.22e+02 1.31e+02 2.33e+00 6.50e+02 7.37e+01 0 \n 15 2.22e+02 1.36e+02 2.10e+00 6.04e+02 5.68e+01 0 \n 16 2.22e+02 1.09e+02 2.10e+00 5.76e+02 5.24e+01 0 \n 17 2.22e+02 1.13e+02 2.00e+00 5.58e+02 4.22e+01 0 \n Regularization decrease: 6.548e-01\n 18 8.89e+02 9.80e+01 2.00e+00 1.88e+03 2.78e+02 0 \n 19 8.89e+02 2.59e+02 1.18e+00 1.31e+03 1.51e+02 0 \n 20 8.89e+02 2.48e+02 1.10e+00 1.22e+03 1.18e+02 0 \n 21 8.89e+02 2.77e+02 1.02e+00 1.19e+03 9.36e+01 0 \n 22 8.89e+02 2.48e+02 1.03e+00 1.16e+03 9.12e+01 0 \n 23 8.89e+02 2.61e+02 9.95e-01 1.15e+03 7.77e+01 0 \n Regularization decrease: 4.986e-01\n 24 1.48e+03 2.35e+02 9.95e-01 1.71e+03 2.12e+02 0 \n 25 1.48e+03 3.70e+02 7.54e-01 1.49e+03 1.16e+02 0 \n 26 1.48e+03 3.53e+02 7.32e-01 1.44e+03 1.05e+02 0 \n 27 1.48e+03 3.77e+02 6.97e-01 1.41e+03 8.30e+01 0 \n 28 1.48e+03 3.39e+02 7.08e-01 1.39e+03 8.89e+01 0 \n 29 1.48e+03 3.53e+02 6.87e-01 1.37e+03 7.26e+01 0 \n Regularization decrease: 3.063e-01\n 30 1.81e+03 3.21e+02 6.87e-01 1.56e+03 1.60e+02 0 \n 31 1.81e+03 4.03e+02 5.79e-01 1.45e+03 1.01e+02 0 \n 32 1.81e+03 3.72e+02 5.76e-01 1.41e+03 9.19e+01 0 \n 33 1.81e+03 3.88e+02 5.55e-01 1.39e+03 7.90e+01 0 \n 34 1.81e+03 3.54e+02 5.63e-01 1.37e+03 7.99e+01 0 \n 35 1.81e+03 3.62e+02 5.50e-01 1.36e+03 7.06e+01 0 \n Regularization decrease: 2.003e-01\n 36 2.13e+03 3.34e+02 5.50e-01 1.50e+03 1.37e+02 0 \n 37 2.13e+03 3.96e+02 4.87e-01 1.43e+03 9.30e+01 0 \n 38 2.13e+03 3.68e+02 4.87e-01 1.40e+03 8.41e+01 0 \n 39 2.13e+03 3.81e+02 4.73e-01 1.39e+03 7.56e+01 0 \n 40 2.13e+03 3.53e+02 4.79e-01 1.37e+03 7.41e+01 0 \n 41 2.13e+03 3.59e+02 4.71e-01 1.36e+03 6.84e+01 0 \n Regularization decrease: 1.471e-01\n 42 2.48e+03 3.36e+02 4.71e-01 1.50e+03 1.21e+02 0 \n 43 2.48e+03 3.86e+02 4.25e-01 1.44e+03 8.27e+01 0 \n 44 2.48e+03 3.65e+02 4.25e-01 1.42e+03 7.75e+01 0 \n 45 2.48e+03 3.75e+02 4.15e-01 1.40e+03 6.90e+01 0 \n 46 2.48e+03 3.53e+02 4.20e-01 1.39e+03 6.93e+01 0 \n 47 2.48e+03 3.59e+02 4.14e-01 1.38e+03 6.28e+01 0 \n Regularization decrease: 1.225e-01\n 48 2.86e+03 3.40e+02 4.14e-01 1.52e+03 1.12e+02 0 \n 49 2.86e+03 3.77e+02 3.82e-01 1.47e+03 7.99e+01 0 \n 50 2.86e+03 3.62e+02 3.81e-01 1.45e+03 7.95e+01 0 \n 51 2.86e+03 3.71e+02 3.74e-01 1.44e+03 6.68e+01 0 \n 52 2.86e+03 3.54e+02 3.77e-01 1.43e+03 7.18e+01 0 \n 53 2.86e+03 3.60e+02 3.73e-01 1.43e+03 6.09e+01 0 \n Regularization decrease: 1.016e-01\n 54 3.25e+03 3.45e+02 3.73e-01 1.56e+03 1.05e+02 0 \n 55 3.25e+03 3.79e+02 3.50e-01 1.52e+03 8.26e+01 0 \n 56 3.25e+03 3.65e+02 3.50e-01 1.50e+03 7.97e+01 0 \n 57 3.25e+03 3.75e+02 3.44e-01 1.49e+03 7.06e+01 0 \n 58 3.25e+03 3.61e+02 3.46e-01 1.49e+03 7.08e+01 0 \n 59 3.25e+03 3.67e+02 3.42e-01 1.48e+03 6.43e+01 0 \n Regularization decrease: 8.279e-02\n Minimum decrease in regularization. End of IRLS\n ------------------------- STOP! -------------------------\n 1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 1.0964e+03\n 0 : |xc-x_last| = 2.1686e-01 <= tolX*(1+|x0|) = 1.0567e-01\n 0 : |proj(x-g)-x| = 6.4312e+01 <= tolG = 1.0000e-01\n 0 : |proj(x-g)-x| = 6.4312e+01 <= 1e3*eps = 1.0000e-02\n 0 : maxIter = 100 <= iter = 60\n ------------------------- DONE! -------------------------\n\n\n\n```python\n# Get the final model back to full space\nm_lp = actvMap*mrec\nm_lp[airc] = np.nan\n\n# Get the smooth model aslo\nm_l2 = actvMap*reg.l2model\nm_l2[airc] = np.nan\n\n# Save both models to file\nMesh.TensorMesh.writeModelUBC(mesh,'SimPEG_GRAV_l2l2.den',m_l2)\nMesh.TensorMesh.writeModelUBC(mesh,'SimPEG_GRAV_lplq.den',m_lp)\n```\n\n\n```python\n# Plot the recoverd models \nvmin, vmax = -.3, 0.05\n\nmesh = Mesh.TensorMesh([mesh.hx, mesh.hy, mesh.hz],x0=\"CCN\")\n\ndef slide(s,normal):\n \n if normal == \"Z\":\n fig = plt.figure(figsize(10*1.2, 8))\n else:\n fig = plt.figure(figsize(10*1.2, 4))\n \n ax1 = plt.subplot(2,2,3)\n dat = mesh.plotSlice(m_lp, ax = ax1, normal=normal, ind=s, clim=np.r_[vmin, vmax],pcolorOpts={'cmap':'viridis'})\n# plt.colorbar(dat[0])\n plt.gca().set_aspect('equal')\n plt.title('Compact model')\n \n if normal == \"Z\":\n xlim(-600, 600)\n ylim(-600, 600.) \n else:\n xlim(-600, 600)\n ylim(-500, 0.) \n \n ax2 = plt.subplot(2,2,1)\n dat = mesh.plotSlice(m_l2, ax = ax2, normal=normal, ind=s, clim=np.r_[vmin, vmax],pcolorOpts={'cmap':'viridis'})\n# plt.colorbar(dat[0])\n plt.gca().set_aspect('equal')\n plt.title('Smooth model')\n \n if normal == \"Z\":\n xlim(-600, 600)\n ylim(-600, 600.) \n else:\n xlim(-600, 600)\n ylim(-500, 0.) \n \n ax2.set_xticklabels([])\n \n ax2 = plt.subplot(1,2,2)\n dat = mesh.plotSlice(m_true, ax = ax2, normal=normal, ind=s, clim=np.r_[vmin, vmax],pcolorOpts={'cmap':'viridis'})\n# plt.colorbar(dat[0])\n plt.gca().set_aspect('equal')\n plt.title('True model')\n \n pos = ax2.get_position()\n\n ax2.yaxis.set_visible(False)\n if normal == \"Z\":\n xlim(-600, 600)\n ylim(-600, 600.) \n ax2.set_position([pos.x0 -0.04 , pos.y0, pos.width, pos.height])\n else:\n xlim(-600, 600)\n ylim(-500, 0.) \n\n pos = ax2.get_position()\n cbarax = fig.add_axes([pos.x0 + 0.375 , pos.y0 + 0.05, pos.width*0.1, pos.height*0.75]) ## the parameters are the specified position you set\n cb = fig.colorbar(dat[0],cax=cbarax, orientation=\"vertical\", ax = ax2, ticks=np.linspace(vmin,vmax, 4))\n cb.set_label(\"Susceptibility (SI)\",size=12)\n fig.savefig('PF_Compact.png',dpi = 150)\n \ninteract(slide, s=(0,60), normal=['X','Y','Z'])\n\n# interact(lambda ind: viz(m_l2, ind, normal=\"Z\"), ind=IntSlider(min=0, max=32,step=1, value=28))\n\n```\n\n\n```python\n# Lets compare the distribution of model parameters and model gradients\nplt.figure(figsize=[15,5])\nax = plt.subplot(121)\nplt.hist(reg.l2model,100)\nplt.plot((eps[0],eps[0]),(0,mesh.nC),'r--')\nplt.yscale('log', nonposy='clip')\nplt.title('Hist model - l2-norm')\nax = plt.subplot(122)\nplt.hist(reg.regmesh.cellDiffxStencil*reg.l2model,100)\nplt.plot((eps[1],eps[1]),(0,mesh.nC),'r--')\nplt.yscale('log', nonposy='clip')\nplt.title('Hist model gradient - l2-norm')\n\n# Lets look at the distribution of model parameters and model gradients\nplt.figure(figsize=[15,5])\nax = plt.subplot(121)\nplt.hist(mrec,100)\nplt.plot((eps[0],eps[0]),(0,mesh.nC),'r--')\nplt.yscale('log', nonposy='clip')\nplt.title('Hist model - lp-norm')\nax = plt.subplot(122)\nplt.hist(reg.regmesh.cellDiffxStencil*mrec,100)\nplt.plot((eps[1],eps[1]),(0,mesh.nC),'r--')\nplt.yscale('log', nonposy='clip')\nplt.title('Hist model gradient - lp-norm')\n```\n\n\n```python\nMesh.MeshIO.TensorMeshIO.writeUBC(mesh,'MAG_mesh.msh')\n\n# import pickle\n# Results = {\"mesh\":mesh, \"model_true\":sus, \"model_pred\":m_IRLS, \"Obs\":data, \"XYZ\": xyz}\n# outputs = open(\"Magresults\", 'wb')\n# pickle.dump(Results, outputs)\n# outputs.close()\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "3109d9ff3867b23f9067792ded08620a419d4d60", "size": 180321, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "SciPy2016/notebooks/PF/Gravity over TKC.ipynb", "max_stars_repo_name": "simpeg/simpegExamples", "max_stars_repo_head_hexsha": "38b8064fb854d809f72b7f1ca8b8096bca696af1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-08-07T13:46:54.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-07T13:46:54.000Z", "max_issues_repo_path": "SciPy2016/notebooks/PF/Gravity over TKC.ipynb", "max_issues_repo_name": "simpeg/simpegExamples", "max_issues_repo_head_hexsha": "38b8064fb854d809f72b7f1ca8b8096bca696af1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2016-07-27T22:20:36.000Z", "max_issues_repo_issues_event_max_datetime": "2016-07-27T22:20:36.000Z", "max_forks_repo_path": "SciPy2016/notebooks/PF/Gravity over TKC.ipynb", "max_forks_repo_name": "simpeg/simpegExamples", "max_forks_repo_head_hexsha": "38b8064fb854d809f72b7f1ca8b8096bca696af1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 203.7525423729, "max_line_length": 42796, "alphanum_fraction": 0.874407307, "converted": true, "num_tokens": 8871, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872131147275, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.4297351594626956}} {"text": "```python\nfrom IPython.display import display, Math, Latex\nimport pystan\nimport matplotlib.pyplot as plt\nimport numpy as np\n```\n\n INFO:matplotlib.font_manager:generated new fontManager\n\n\n\n\n\n## Dr Peter Hurley\n\n

                                        What is Probability?

                                        \n
                                        \n\n

                                        Idea: Probability is the Long Run Expected Number of Occurances of an Event

                                        \n
                                        \n
                                        \n \n

                                        When you roll a fair dice once it will come up with only one answer, but after doing this many times each outcome will occur 1/6 of the time. So the probability of an outcome is 1/6.

                                        \n
                                        \n

                                        Another way of saying this: The probability of an event A is equal to the number of ways in which A can occur divided by the number of ways in which any event can happen.

                                        \n
                                        \n

                                        This then helps us answer questions such as, what is the probability of drawing three aces in a row from a shuffled deck of cards?

                                        \n
                                        \n

                                        There are four ways of picking the first ace, out of 52 possible outcomes, three ways of picking the first ace, out of 51 possible outcomes, and two ways of picking the first ace, out of 50 possible outcomes. Therefore:

                                        \n\n\n\\begin{equation}\nP(3\\ Aces) = \\frac{4}{52} * \\frac{3}{51} * \\frac{2}{50} \n\\end{equation}\n\n## ***QUESTION***\n

                                        We can also ask more nuanced questions, such as what is the probability of event A given event B. For example, what is the probability of a card is a King given it is a face card?

                                        \n
                                        \n\n\n\\begin{equation}\nP(A\\ |\\ B) = \\frac{P(A \\cap\tB)}{P(B)}\n\\end{equation}\n\n\n\n\n\\begin{equation}\nP(King\\ |\\ Face\\ Card) = \\frac{4}{12} = \\frac{1}{3}\n\\end{equation}\n\n## ***QUESTION***\n

                                        Suppose a family has two children and suppose one of the children is a boy. What is the probability that both children are boys?

                                        \n \n\n

                                        Total number of ways in which a family can have two children:

                                        \n \n
                                        \n\n

                                        {BB}, {BG}, {GB}, {GG}

                                        \n \n
                                        \n\n

                                        Of these three involve at least one boy, and of these only one pairing is two boys so:

                                        \n \n
                                        \n\n\n\\begin{equation}\nP(Two\\ Boys\\ |\\ One\\ child\\ definitely\\ a\\ boy) = \\frac{1}{3}\n\\end{equation}\n\n
                                        \n\n\n\n\n## ***QUESTION***\n\nThe Monte Hall Problem:\n \nSuppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, \"Do you want to pick door No. 2?\" Is it to your advantage to switch your choice?\n\n\n\\begin{equation}\nP(Prize\\ behind\\ any\\ random\\ door) = \\frac{1}{3}\n\\end{equation}\n\n
                                        \n\n\\begin{equation}\nP(Keep\\ Door\\ 1\\ and\\ Win) = \\frac{P(Prize\\ Door\\ 1\\ and\\ Host\\ Opens\\ Door\\ 3)}{Host\\ Opens\\ Door\\ 3} \n\\end{equation}\n\n
                                        \n\n\\begin{equation}\nP(Prize\\ Door\\ 1\\ and\\ Host\\ Opens\\ Door\\ 3) = \\frac{1}{3} * \\frac{1}{2} = \\frac{1}{6}\n\\end{equation}\n\n
                                        \n\n

                                        The probability the host opens door 3 depends on which door the prize is behind.

                                        \n\n\\begin{equation}\nP(Host\\ Door\\ 3) = P(Host\\ Door\\ 3\\ |\\ Prize\\ Door\\ 1) * P(Prize\\ Door\\ 1) + P(Host\\ Door\\ 3\\ |\\ Prize\\ Door\\ 2) * P(Prize\\ Door\\ 3)+ P(Host\\ Door\\ 3\\ |\\ Prize\\ Door\\ 3) * P(Prize\\ Door\\ 3) \n\\end{equation}\n\n
                                        \n\n\\begin{equation}\nP(Host\\ Door\\ 3) = \\frac{1}{2} * \\frac{1}{3} + 1 * \\frac{1}{3} + 0*\\frac{1}{3} = \\frac{1}{2}\n\\end{equation}\n\n
                                        \n\n

                                        So the probability of winning if we keep our door is:

                                        \n\n
                                        \n\n\\begin{equation}\nP(Keep\\ Door\\ 1\\ and\\ Win) = \\frac{\\frac{1}{6}}{\\frac{1}{2}} = \\frac{1}{3}\n\\end{equation}\n\n
                                        \n\nAnd so the probability of winning if we swap is: \n \n\\begin{equation}\nP(Swap\\ Door\\ 2\\ and\\ Win) = 1 - \\frac{1}{3} = \\frac{2}{3}\n\\end{equation}\n\n

                                        We can manipulate these equations as in other areas of maths as we did above

                                        \n
                                        \n\n\\begin{equation}\nP(A\\ |\\ B) = \\frac{P(A \\cap\tB)}{P(B)}\n\\end{equation}\n\n\\begin{equation}\nP(A \\cap\tB) = P(A) * P(B | A)\n\\end{equation}\n\n\\begin{equation}\nP(A \\cap\tB) = P(B) * P(A | B)\n\\end{equation}\n\n\\begin{equation}\nP(A) * P(B | A) = P(B) * P(A | B)\n\\end{equation}\n\n\\begin{equation}\nP(B | A) = \\frac{P(B) * P(A | B)}{P(A)}\n\\end{equation}\n\n

                                        This last equation is Bayes Theorem, a source of heated debate among statisticians throughout the 20th century.

                                        \n\n
                                        \n\n

                                        You may wonder what the fuss is as surely everyone accepts the above?

                                        \n\n
                                        \n\n

                                        The issue comes in when we consider how far we can apply this equation.

                                        \n\n## Probabilistic models \n* A model describes data that one could observe from a system\n* Uses probability theory to express all forms of uncertainty and noise associated with our model\n* Inverse probability (i.e. Bayes rule) to infer parameters, adapt models, make predictions and learn from data.\n\n\n
                                        \n\nA good example of how Bayesian probability is more intuitive is to think of a situation such as the probability of rain, given the sky is cloudy, or that there is a sprinkler nearby. We use prior information to decide if it has rained, rather than just looking at the number of times it has rained in past.\n\n
                                        \n\nAlternatively there can also be cases where a frequentist model breaks down. Suppose you are an astronomer looking at Saturn with an old telescope. Combining many measurements of the shape of the planet will not improve your result on resolving the shape. However given a model of how a planet with a ring or without you can test how well each model describes Saturn and infer the correct shape. \n\n
                                        \n\nOther key cases are those in which you can never repeat the experiment, e.g. modelling how the universe grew since the big bang or the diversification of songbirds in the Andes. \n\n\n\n\\begin{equation}\nP(B | A) = \\frac{P(B) * P(A | B)}{P(A)}\n\\end{equation}\n\n$P(B | A)$ = Posterior\n\n$P(B)$ = Prior\n\n$P(A | B)$ = Likelihood\n\n$P(A)$ = Marginal Likelihood\n\n\n
                                        \n\n

                                        Transparent way of including model prior information
                                        \nGives full probability distribution\n

                                        \n
                                        \n\n

                                        As the area under a posterior must equal one the marginal likelihood can be thought of as a normalistion value, and so the equation is often written as:

                                        \n\n
                                        \n\n\\begin{equation}\nP(B | A) \\propto P(B) * P(A | B)\n\\end{equation}\n\n
                                        \n\n

                                        In that case your posterior is your prior state of belief being updated by the new data.

                                        \n\n
                                        \n\nP(A) is actually an integral which is often difficult to solve, and must be done numerically, motivates the need for MCMC type sampling. MCMC-like algorithms will generate points proportionally to the posterior values in each region. A simplistic way of thinking about this is a sampler generates some initial starting parameter values, calculates how well a model with these parameters explains the data and then generates a new set of parameter values nearby. If these new values are an improvement we always accept the new point and move there, otherwise we have a small chance we move there. The exact way in which this works varies by method.\n\n\n\n

                                        So far so good, but there is a lot of debate and discomfort with how you choose an appropriate prior. However disagreements over what prior is used can at least be transparent, and the correct description of the current state of knowledge before new data is an important and necessary part of debate.

                                        \n\n### Why bother with all this Bayesian probabilistic stuff?\n* Prior information help us extract more information from data\n\n### Prob. model of straight line\n$\\mathbf{y}=m\\mathbf{x}+c + \\epsilon$\n\nlikelihood = $p(\\mathbf{y}|m,c,\\mathbf{x}) = \\mathcal{N}(m\\mathbf{x}+c,\\sigma^2)$\n\nprior = $p(m,c) = \\mathcal{N}(0,\\Sigma_p)$\n\nposterior (after a bit of maths)\n\n$p(m,c|\\mathbf{y},\\mathbf{x}) \\propto \\mathcal{N}(\\sigma^{-2}(\\sigma^{-2}\\mathbf{x}\\mathbf{x}^T + \\Sigma_{p}^{-1})^{-1}\\mathbf{x}\\mathbf{y},$\n$\\sigma^{-2}\\mathbf{x}\\mathbf{x}^T + \\Sigma_{p}^{-1})$\n\nSimple example straight line, error only in y, simple gaussian prior\n\nWhat happens if intercept is positive? gets complicated fast\n\n\n\n\n\n\n\nHave prior information on intercept\n\n$c \\sim \\mathcal{N}(0.3,0.1)$\n\n\n\n\n\n\n\n\n\n* The uncertainty in our inferred model parameters contains information.\n\n* We do not want to just know the best answer, we\nwant to know how uncertain we are.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n### The mode is an inappropriate measure of Probability mass for High D!!!\n\nThink about mass $m$ and density $\\rho$\n\n$$\nm=\\int \\rho dV\n$$\n\nProbability works in a similar way:\n\n$$\n\\mathbb{E}_{\\pi}[f]=\\int_{Q}\\pi(q)f(q)dq\n$$\n\nTo compute expectations efficiently with MCMC, we want to spend time in areas of the posterior that will contribute most to the expectation.\n\nThere is a balance between density and volume.. but how does volume change with number of dimensions??\n\n### 1D\n\n\n```python\nplt.hist(np.random.normal(0,1,50000),bins=np.arange(-4.25,4.25,0.5),normed=True,alpha=0.5);\nplt.axvline(x=-0.25,c='r')\nplt.axvline(x=0.25,c='r')\n\nplt.axvline(x=1.25,c='g')\nplt.axvline(x=1.75,c='g')\n```\n\nWhere is most of the mass?\n\nCentre (red): $0.4 \\times 0.5 = 0.2$\n\nEdge (green): $0.13 \\times 0.5 = 0.065$\n\n\n```python\npoints=np.random.multivariate_normal([0,0],[[1,0],[0,1]],50000)\nplt.hist2d(points[:,0],points[:,1],bins=np.arange(-4.25,4.25,0.5),normed=True);\nplt.colorbar()\n```\n\nWhere is most of the mass?\n\nCentre (red): $0.15\u00d70.5^2=0.0375$\n\nEdge (green): $\\approx 0.08 \\times 0.5^2 \\times 16=0.32$\n\nTo identify the regions of parameter space that dominate expectations we should instead consider the product of density and volume. In high dimensions, the volume behaves very differently from the density, resulting in a tension that pushes the important regions of parameter space away from the mode further out towards the tails of the distribution into a region known as the **typical set**. The move to a thin shell with higher dimensions is called the **concentration of measure**.\n\nFor more info, see this [blog post](https://mc-stan.org/users/documentation/case-studies/curse-dims.html)\n\n### Using Uncertianty in decision making means making better decisions\n\n* Business consider risk when making decisions\n* Uncertianty should be considered when making decisions based on predictions\n\nSee [Blog post on Bayesian Decision making](https://pdh21.github.io/jekyll/update/2019/06/06/FACYNation_BDT.html)\n\n\n\n## Calculating the Posterior\n* Analytical approximation (often impossible)\n* Grid approximation (not efficient)\n* Quadratic approximation (limited)\n* Markov chain Monte Carlo (intensive)\n\n### [Comparing other MCMC techniques](https://chi-feng.github.io/mcmc-demo/app.html?algorithm=NaiveNUTS&target=banana)\ne.g. \n* Metropolis Hastings\n* Gibbs\n* Standard NUTS\n* Nested Sampling\n\n\n
                                        \n\n## Probabilistic Programming\n*Define probability models & solve automatically*\n\n\n\n[Good blog post on philosophy of Probabilistic Programming](https://www.oreilly.com/content/probabilistic-programming/)\n\n\n\n# Stan \n\n\n\n* Language for building probabilistic models\n* Algorithms for fitting probabilistic models\n* Interfaces for using probabilistic models \n\n\n\n### Other PPLs:\n* [Tensorflow probability](https://www.tensorflow.org/probability), formally Edward\n* [PyMC3 (soon PyMC4)](https://docs.pymc.io/)\n* [Pyro](http://docs.pyro.ai/en/stable/)\n\n## Language\nSpecify probabilisitic model: the joint probability distribution function,\n\n$$p(\\theta,x)$$\n\n* $\\theta$ = parameters\n* $x$ = data\n* $p$ = model\n\n\nStan is a language:\n* Statically typed, imperative\n* User can define programs using parameters, data and $\\log p(\\theta,x)$\n* Any program can be written as long as $\\log p(\\theta,x)$ is differentiable\n\n\nStatically typed like C, C++. i.e. types declared at compile time rather than run time like Python.\n\nImperative, you tell the compiler what you want to happen, step by step\n\n\n\n## Algorithms\n* Bayesian Inference: No U-Turn Hamiltonian Monte Carlo,\n - $p(\\theta|x)$ approximated with $[\\theta^1,\\theta^2,..\\theta^N]$\n* Approximate Bayesian Inference: Variational Inference,\n - $\\hat{p}(\\theta|x) \\approx q(\\hat{\\phi})$, where $\\phi = argmin_{\\phi} D_{KL}(q(\\theta|\\phi)||p(\\theta,x))$\n* Optimisation: Max \n - $\\hat{\\theta} = argmax_{\\theta} p(\\theta,x)$\n \n\n\n### The advantages of Stan and HMC\n\n* Uses gradient to traverse posterior more efficiently\n* It knows when there is a problem\n* Stan automatically tunes the parameters of HMC\n\n\n## Interfaces\n* CmdStan,PyStan,RStan\n* C++ API\n* C++ auto-diff library\n* Software built with Stan: RStanArm, brms, prophet\n\n## Writing a model in Stan\n* A Stan program is organized into a sequence of named blocks\n\n\n\n```python\nexample_model=\"\"\"functions {\n// ... function declarations and definitions ...\n}\ndata {\n// ... declarations ...\n}\ntransformed data {\n// ... declarations ... statements ...\n}\nparameters {\n// ... declarations ...\n} \ntransformed parameters {\n// ... declarations ... statements ...}\nmodel {\n// ... declarations ... statements ...\n}\ngenerated quantities {\n// ... declarations ... statements ...\n}\n\"\"\"\n```\n\n### Other things to note:\n* Add comments with `\\\\`\n* Sampling statements with `~`\n* Can print output with `print()`\n* `;` at end of each line\n\n\nDocumentation for Stan: https://mc-stan.org/docs/2_23/stan-users-guide/index.html\n\n\n

                                        As an example lets consider a question of if a coin is fair or not. We begin with a flat prior and update it as we flip a coin.

                                        \n\n\n\n\n\n\n```python\ncoin_model =\"\"\"// Inferring a Rate\ndata { \n int n; \n int h;\n real mu;\n real sig;\n} \nparameters {\n real theta;\n} \nmodel {\n // Prior Distribution for Rate Theta\n theta ~ normal(mu, sig);\n \n // Observed Counts\n h ~ binomial(n, theta);\n}\"\"\"\n\nsm=pystan.StanModel(model_code=coin_model)\n```\n\n INFO:pystan:COMPILING THE C++ CODE FOR MODEL anon_model_83871b9f0690c538517d8bcb1cc4d317 NOW.\n\n\n\n```python\nmodel_data={\n 'n':100,\n 'h':50,\n 'mu':0.5,\n 'sig':0.5\n}\n\n```\n\n\n```python\nfit=sm.sampling(data=model_data,chains=4,iter=1000,seed=12329)\n```\n\n\n```python\nfit\n```\n\n\n\n\n Inference for Stan model: anon_model_83871b9f0690c538517d8bcb1cc4d317.\n 4 chains, each with iter=1000; warmup=500; thin=1; \n post-warmup draws per chain=500, total post-warmup draws=2000.\n \n mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat\n theta 0.5 2.0e-3 0.05 0.41 0.47 0.5 0.53 0.59 566 1.01\n lp__ -71.17 0.02 0.65 -73.06 -71.35 -70.92 -70.76 -70.7 1017 1.0\n \n Samples were drawn using NUTS at Thu Jul 30 10:54:32 2020.\n For each parameter, n_eff is a crude measure of effective sample size,\n and Rhat is the potential scale reduction factor on split chains (at \n convergence, Rhat=1).\n\n\n\n\n```python\n\nplt.plot(np.arange(0, fit['theta'].size), fit['theta'])\nplt.xlabel('Sample')\nplt.ylabel('Theta')\nplt.show()\n\nplt.hist(fit['theta'])\nplt.xlabel('Theta')\nplt.ylabel('Number of Samples')\nplt.show\n```\n\n\n```python\nmodel_data={\n 'n':10,\n 'h':7,\n 'mu':0.5,\n 'sig':0.5\n}\n\nfit=sm.sampling(data=model_data,chains=4,iter=1000,seed=12329)\n\nprint(fit)\n\nplt.hist(fit['theta'])\nplt.xlabel('Theta')\nplt.ylabel('Number of Samples')\nplt.show\n```\n\n### Exercise:\nUsing Python, generate some fake data for a coin tosses (i.e. ten flips) and fit the Stan model to sequential number of tosses and plot the results to see how the posterior changes as a function of the number of tosses.\n\n\n```python\n\n```\n\n### The double coin toss game\n\n

                                        Sometimes our data may come in a more complex form. For instance imagine if you have a game that involves coin tosses. You flip a coin until you flip a heads and a tails in a row. Your opponent flips a coin until they flip two heads in a row. All the information you have is who won each game. From this you want to understand if one coin was biased or not.

                                        \n\n

                                        As HT and HH in two coin flips are equally likely we may run the code above and see how close our theta value is to 0.5

                                        \n \n

                                        After 2,000 flips your opponent has won 747 times but you have won 1,253.

                                        \n \n \n\n\n```python\nmodel_data={\n 'n':2000,\n 'h':747,\n 'mu':0.5,\n 'sig':0.5}\n\nfit=sm.sampling(data=model_data,chains=4,iter=1000,seed=12329)\n\nprint(fit)\n\nplt.hist(fit['theta'])\nplt.xlabel('Theta')\nplt.ylabel('Number of Samples')\nplt.show\n```\n\n

                                        So can we conclusively say that our coin was biased?

                                        \n \n

                                        Once again this is a case where we should have questioned our underlying assumptions, i.e. that both opponents had a 50% chance of winning. In future this will be covered in part by prior predictive checks which should give us intuition on how our model behaves. For now let us first generate data in the way described assuming both players have a fair coin

                                        \n\n\n```python\nbob_wins = 0\nalice_wins = 0\n\nfor i in range(2000):\n bob_seq = np.array([])\n alice_seq = np.array([])\n bob_poss_win = 0\n alice_poss_win = 0\n while bob_wins + alice_wins < 2000:\n bob_seq = np.append(bob_seq, np.random.randint(2,size=1))\n alice_seq = np.append(alice_seq, np.random.randint(2,size=1))\n if np.prod((bob_seq.size > 1) and (bob_seq[-2:] == [0,0])):\n bob_poss_win = 1\n if np.prod((alice_seq.size > 1) and (alice_seq[-2:] == [0,1])):\n alice_poss_win = 1\n if (bob_poss_win == 1) and (alice_poss_win == 1):\n bob_seq = np.array([])\n alice_seq = np.array([])\n bob_poss_win = 0\n alice_poss_win = 0\n elif bob_poss_win == 1:\n bob_wins += 1 \n bob_seq = np.array([])\n alice_seq = np.array([])\n bob_poss_win = 0\n elif alice_poss_win == 1:\n alice_wins += 1 \n bob_seq = np.array([])\n alice_seq = np.array([])\n alice_poss_win = 0\n \nprint(bob_wins)\nprint(alice_wins)\n\n```\n\n 762\n 1238\n\n\n

                                        Understanding a problem deeply is important when building a model. Misunderstanding a model can lead to misinterpretation. In this case there is a subtle difference, if you need to flip HT and first flip a H you will not win if on the next coin toss you flip another H - but you are still only one flip away. If you need to flip HH and after one H toss flip a T, you are again two flips away from winning.

                                        \n\n\n\n## Exercise:\nCreate some fake data by sampling from a normal distribution with a known mean and standard deviation.\n\nBuild a Stan model that fits that data and estimates the mean and standard deviation.\n\n\n```python\nmodel=\"\"\"\ndata {\nint N; //number of datapoints\nreal x[N]; //x values\n}\n\n\nparameters {\nreal mu; //mean of normal distribution\nreal sigma;\n}\n\nmodel {\n\nsigma ~ cauchy(0,1);\nx ~ normal(mu,sigma);\n}\n\"\"\"\n```\n\n\n```python\nsm=pystan.StanModel(model_code=model)\n```\n\n INFO:pystan:COMPILING THE C++ CODE FOR MODEL anon_model_23840a14d34f49990979ae476b61c011 NOW.\n\n\n\n```python\nx=np.random.normal(10,2,10)\n```\n\n\n```python\ndata={\n 'N':10,\n 'x':x\n}\n```\n\n\n```python\nfit=sm.sampling(data=data,chains=4)\n```\n\n\n```python\nfit\n```\n\n\n\n\n Inference for Stan model: anon_model_23840a14d34f49990979ae476b61c011.\n 4 chains, each with iter=2000; warmup=1000; thin=1; \n post-warmup draws per chain=1000, total post-warmup draws=4000.\n \n mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat\n mu 10.16 0.02 0.79 8.57 9.66 10.16 10.67 11.69 2366 1.0\n sigma 2.45 0.01 0.61 1.6 2.02 2.35 2.75 3.94 2231 1.0\n lp__ -15.14 0.03 1.04 -17.92 -15.57 -14.81 -14.4 -14.11 1685 1.0\n \n Samples were drawn using NUTS at Thu Jul 30 12:31:05 2020.\n For each parameter, n_eff is a crude measure of effective sample size,\n and Rhat is the potential scale reduction factor on split chains (at \n convergence, Rhat=1).\n\n\n\n\n```python\nplt.hist(fit['sigma'],bins=np.arange(0,6,0.1))\n```\n\n\n```python\nplt.hist(fit['mu'],bins=np.arange(6,13,0.1))\n```\n\n* What did you choose as a prior?\n* How did it affect your posterior?\n\n### Exercise:\n* Create some fake data for a linear regression model (i.e. $y=mx +c$)\n\n* Build a Stan model to fit the data\n\n* Look at results, compare posterior for slope and intercept\n\n\n```python\nm=3\nc=2\nx=np.arange(0,10)\ny=np.random.normal(m*x+c,1)\n```\n\n\n```python\nplt.plot(x,y,'o')\nplt.xlabel('x')\nplt.ylabel('y')\n```\n\n\n```python\nmodel_code=\"\"\"\ndata {\nint N;\nreal x[N];\nreal y[N];\n}\n\nparameters {\nreal m;\nreal c;\nreal sigma;\n}\n\nmodel {\nsigma ~ cauchy(0,1);\n for( i in 1:N){\ny[i] ~ normal(m*x[i]+c,sigma);\n}\n}\n\"\"\"\n```\n\n\n```python\nsm=pystan.StanModel(model_code=model_code)\n```\n\n INFO:pystan:COMPILING THE C++ CODE FOR MODEL anon_model_d353f9caa8524d7e59c48efd239f8868 NOW.\n\n\n\n```python\ndata={\n 'N':x.size,\n 'x':x,\n 'y':y\n}\n```\n\n\n```python\nfit=sm.sampling(data=data,chains=4)\n```\n\n\n```python\nfit\n```\n\n\n\n\n Inference for Stan model: anon_model_d353f9caa8524d7e59c48efd239f8868.\n 4 chains, each with iter=2000; warmup=1000; thin=1; \n post-warmup draws per chain=1000, total post-warmup draws=4000.\n \n mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat\n m 3.12 3.0e-3 0.11 2.9 3.06 3.12 3.19 3.34 1380 1.0\n c 1.67 0.02 0.6 0.47 1.31 1.66 2.03 2.87 1316 1.0\n sigma 0.98 7.0e-3 0.28 0.61 0.79 0.93 1.11 1.63 1558 1.0\n lp__ -5.12 0.04 1.44 -8.71 -5.74 -4.79 -4.08 -3.51 1126 1.01\n \n Samples were drawn using NUTS at Thu Jul 30 12:56:23 2020.\n For each parameter, n_eff is a crude measure of effective sample size,\n and Rhat is the potential scale reduction factor on split chains (at \n convergence, Rhat=1).\n\n\n\n\n```python\nplt.hist(fit['m']);\n```\n\n\n```python\nplt.hist(fit['c'])\n```\n\n\n```python\nplt.hist(fit['sigma'])\n```\n\n\n```python\nplt.scatter(fit['m'],fit['c'])\nplt.xlabel('m')\nplt.ylabel('c')\n```\n\n\n```python\nplt.plot(fit['m'])\n```\n\n### So far:\n* Introduced Probability theory\n* Provided motivation as to we want to use for science and machine learning\n* Stan is a PPL that we can use to build probabilisitic models\n* Can build some simple Stan models\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "bdcdd5494e77f1f623dc30a0cd8c80b5d89ea1bf", "size": 179263, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Thursday/with_answers/DISCUS_Course_PartI.ipynb", "max_stars_repo_name": "DataJavelin/DISCUS-ProbProgwithStan", "max_stars_repo_head_hexsha": "0d5ccf37cca331a53822d8a8473c2c58ffbd7610", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2020-07-30T08:52:19.000Z", "max_stars_repo_stars_event_max_datetime": "2020-07-31T10:41:06.000Z", "max_issues_repo_path": "Thursday/with_answers/DISCUS_Course_PartI.ipynb", "max_issues_repo_name": "DataJavelin/DISCUS-ProbProgwithStan", "max_issues_repo_head_hexsha": "0d5ccf37cca331a53822d8a8473c2c58ffbd7610", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Thursday/with_answers/DISCUS_Course_PartI.ipynb", "max_forks_repo_name": "DataJavelin/DISCUS-ProbProgwithStan", "max_forks_repo_head_hexsha": "0d5ccf37cca331a53822d8a8473c2c58ffbd7610", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 104.1016260163, "max_line_length": 23584, "alphanum_fraction": 0.8451771977, "converted": true, "num_tokens": 6660, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166047041654, "lm_q2_score": 0.839733963661418, "lm_q1q2_score": 0.42970581273959174}} {"text": "```python\nimport torch\nimport torch.nn.functional as F\n\nimport torchsde\nimport math\nimport matplotlib.pyplot as plt\n\nfrom cfollmer.objectives import log_g, relative_entropy_control_cost\nfrom cfollmer.sampler_utils import FollmerSDE\nimport numpy as np\n\nfrom tqdm.notebook import tqdm\n\nfrom torch import _vmap_internals\n```\n\n## Generating Toy 2D Dataset.\n\n\n```python\nimport torch\nfrom sklearn.datasets import make_blobs\nfrom sklearn.model_selection import train_test_split\nfrom sklearn.preprocessing import StandardScaler\n\ndevice = \"cuda\" if torch.cuda.is_available() else \"cpu\"\nrandom_state = 170\n\nX, y = make_blobs(n_samples=100,\n cluster_std=[0.2, 0.2, 0.2],\n random_state=random_state)\n\n# X = X[y==0,:]\n# y = y[y==0]\n# # Scale data to have mean 0 and variance 1 \n# # which is importance for convergence of the neural network\nscaler = StandardScaler()\nX_scaled = scaler.fit_transform(X)\n\n# Split the data set into training and testing\nX_train, X_test, y_train, y_test = train_test_split(\n X_scaled, y, test_size=0.2, random_state=2)\n\n\nX_train, X_test, y_train, y_test = \\\n torch.tensor(X_train, dtype=torch.float32, device=device), \\\n torch.tensor(X_test, dtype=torch.float32, device=device), \\\n torch.tensor(y_train, dtype=torch.float32, device=device), \\\n torch.tensor(y_test, dtype=torch.float32, device=device) \n```\n\n\n```python\ny\n```\n\n\n\n\n array([0, 2, 2, 0, 0, 1, 1, 0, 2, 1, 1, 1, 0, 1, 0, 2, 1, 1, 1, 2, 2, 2,\n 2, 2, 1, 1, 0, 2, 0, 1, 2, 0, 1, 2, 1, 0, 0, 0, 0, 0, 2, 0, 2, 0,\n 0, 1, 2, 2, 0, 1, 2, 2, 1, 2, 0, 1, 2, 1, 0, 0, 1, 0, 0, 2, 0, 2,\n 2, 1, 1, 1, 2, 0, 1, 0, 0, 0, 2, 1, 1, 1, 1, 1, 2, 1, 0, 0, 1, 2,\n 2, 1, 2, 2, 1, 0, 2, 2, 0, 0, 0, 2])\n\n\n\n\n```python\nfig, ax1 = plt.subplots(1, 1, figsize=(16, 6))\n\nax1.plot(X_scaled[:, 0], X_scaled[:, 1], \n linestyle='none', \n marker='o')\n\nax1.axis('equal');\n```\n\n$$\\DeclareMathOperator*{\\argmin}{arg\\,min}$$\n$$\\def\\E{{\\mathbb{E}}}$$\n$$\\def\\rvu{{\\mathbf{u}}}$$\n$$\\def\\rvTheta{{\\bm{\\Theta}}}$$\n$$\\def\\gU{{\\mathcal{U}}}$$\n$$\\def\\mX{{\\mathbf{X}}}$$\n\n## Controlled Schrodinger Follmer Sampler\n\nThe objevtive we are trying to implement is:\n\n\\begin{align}\n \\mathbf{u}_t^{*}= \\argmin_{\\rvu_t \\in \\mathcal{U}}\\mathbb{E}\\left[\\frac{1}{2\\gamma}\\int_0^1||\\rvu(t, \\Theta_t)||^2 dt - \\ln\\left(\\frac{ p(\\mX | \\Theta_1)p(\\Theta_1)}{\\mathcal{N}(\\Theta_1|\\mathbf{0}, \\gamma \\mathbb{I} )}\\right)\\right] \\\n\\end{align}\n\nWhere:\n\\begin{align}\nd\\Theta_t = \\rvu(t, \\Theta_t)dt + \\sqrt{\\gamma} dB_t\n\\end{align}\n\nTo do so we use the EM discretisation.\n\n\n```python\ndef log_gaussian(x, mean=0):\n \"\"\"\n Returns the density of x under the supplied gaussian. Defaults to\n standard gaussian N(0, I)\n :param x: (*) torch.Tensor\n :param mean: float or torch.FloatTensor with dimensions (*)\n :param logvar: float or torch.FloatTensor with dimensions (*)\n :return: (*) elementwise log density\n \"\"\"\n \n log_norm_constant = -0.5 * np.log(2 * np.pi)\n \n var = torch.tensor(0.2)\n logvar = torch.log(var)\n logvar = x.new(1).fill_(logvar)\n \n A = (x - mean) ** 2\n log_p = -0.5 * (logvar + A / logvar.exp())\n log_p = log_p + log_norm_constant\n# import pdb; pdb.set_trace()\n return log_p.sum(dim=-1)\n\n\n\ndef ln_prior(\u0398, \u03c3_w=0.7):\n \"\"\"\n Prior for means in Bayesian GMM\n \"\"\"\n return -0.5 * (\u0398**2).sum(axis=1) / \u03c3_w**2\n\n\ndef log_likelihood_single(\u03bc, X, log=True, K=3):\n \"\"\"\n :param X: design matrix (examples, features)\n :param mu: the component means (K, features)\n :param logvar: the component log-variances (K, features)\n :param log: return value in log domain?\n\n \"\"\"\n \n n, d = X.shape\n # get feature-wise log-likelihoods (K, examples, features)\n log_likelihoods = log_gaussian(\n X[None, :, :], # (1, examples, features)\n \u03bc.reshape(K, d)[:, None, :], # (K, 1, features)\n )\n \n\n # sum over the feature dimension\n log_likelihoods = torch.logsumexp(log_likelihoods, dim=0) \n\n return log_likelihoods.sum()\n\n\ndef log_likelihood(\u0398, X, y=None, K=3):\n \"\"\"\n batching the above (hopefully its right)\n \"\"\"\n\n loss_ = lambda \u03bc: log_likelihood_single(\u03bc, X,K=K)\n \n batched_loss = torch._vmap_internals.vmap(loss_)\n\n return batched_loss(\u0398)\n```\n\n\n```python\n\u0394t=0.01\nK = 3\nt_size = int(math.ceil(1.0/\u0394t))\ndim = K * 2\n\nts = torch.linspace(0, 1, t_size).to(device)\nno_posterior_samples = 50\n\nsde = FollmerSDE(dim, dim, no_posterior_samples, 1.0, device=device).to(device)\n\u0398_0 = torch.zeros((no_posterior_samples, dim)).to(device) # \u0398_0 ~ \u03b4_0\n\n# Initial state y0, the SDE is solved over the interval [ts[0], ts[-1]].\n# ys will have shape (t_size, batch_size, state_size)\nys = torchsde.sdeint(sde, \u0398_0, ts, dt=\u0394t)\n```\n\n\n```python\n\u0398_0.shape\n```\n\n\n\n\n torch.Size([50, 6])\n\n\n\n\n```python\nrelative_entropy_control_cost(sde, \u0398_0, X_train, y_train, \n ln_prior, log_likelihood, \u03b3=1.0, device=device)\n```\n\n C:\\Users\\vargf\\AppData\\Local\\Temp/ipykernel_25376/1674092342.py:64: UserWarning: torch.vmap is an experimental prototype that is subject to change and/or deletion. Please use at your own risk. There may be unexpected performance cliffs due to certain operators not being implemented. To see detailed performance warnings please use `torch._C._debug_only_display_vmap_fallback_warnings(True) before the call to `vmap`.\n batched_loss = torch._vmap_internals.vmap(loss_)\n\n\n\n\n\n tensor(340.2701, device='cuda:0', grad_fn=)\n\n\n\n\n```python\n\u03b3 = 0.4\n\u0394t=0.0009\nt_size = int(math.ceil(1.0/\u0394t))\n# print(t_size)\nts = torch.linspace(0, 1, t_size).to(device)\n# print(ts)\nsde = FollmerSDE(dim, dim, no_posterior_samples , \u03b3=\u03b3, device=device).to(device)\noptimizer = torch.optim.Adam(sde.\u03bc.parameters(), lr=0.05, weight_decay =1)\n# optimizer = torch.optim.LBFGS(gpr.parameters(), lr=0.01)\nlosses = []\nnum_steps = 300\nfor i in tqdm(range(num_steps)):\n optimizer.zero_grad()\n\n if isinstance(optimizer, torch.optim.LBFGS):\n def closure():\n loss = relative_entropy_control_cost(\n sde, \u0398_0.float(),\n X_train.float(), y_train.float(),\n ln_prior, log_likelihood, \u03b3=\u03b3\n )\n optimizer.zero_grad()\n loss.backward()\n return loss\n\n optimizer.step(closure)\n losses.append(closure().item())\n else:\n loss = relative_entropy_control_cost(\n sde, \u0398_0,\n X_train, y_train,\n ln_prior, log_likelihood, \u03b3=\u03b3\n )\n optimizer.zero_grad()\n loss.backward()\n\n optimizer.step()\n losses.append(loss.item())\ntorch.cuda.empty_cache()\n```\n\n\n 0%| | 0/300 [00:00)\n\n\n\n\n```python\n\u0398_1.shape\n```\n\n\n\n\n torch.Size([50, 3, 2])\n\n\n\n\n```python\nfig, ax_1 = plt.subplots(1, 1, figsize=(16, 6))\n\u0398_plot = \u0398_1.cpu().detach() #.reshape(50*3,2)\n\n\nfor i in range(K):\n ax_1.plot(\u0398_plot[:,i, 0], \u0398_plot[:, i, 1], \n linestyle='none', \n marker='o', color=\"red\", label=\"posterior $\\mu$ samples\", alpha= (i+1)**2 * 0.1)\n\nax_1.plot(X_scaled[:, 0], X_scaled[:, 1], \n linestyle='none', \n marker='o', label=\"Observations $X$\")\n\nax_1.legend()\nax_1.axis('equal');\n```\n", "meta": {"hexsha": "c9a0c964d24ce3d849a9b46ceb8d9e16a91dd739", "size": 88853, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/outdated_notebooks/GMM_Example.ipynb", "max_stars_repo_name": "franciscovargas/ControlledFollmerDrift", "max_stars_repo_head_hexsha": "608d20912cf6503502322fc759abb5216b9e1227", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-12-07T14:53:43.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-23T13:27:11.000Z", "max_issues_repo_path": "notebooks/outdated_notebooks/GMM_Example.ipynb", "max_issues_repo_name": "franciscovargas/ControlledFollmerDrift", "max_issues_repo_head_hexsha": "608d20912cf6503502322fc759abb5216b9e1227", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/outdated_notebooks/GMM_Example.ipynb", "max_forks_repo_name": "franciscovargas/ControlledFollmerDrift", "max_forks_repo_head_hexsha": "608d20912cf6503502322fc759abb5216b9e1227", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 95.3358369099, "max_line_length": 28980, "alphanum_fraction": 0.8317220578, "converted": true, "num_tokens": 5795, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702642896702, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.4296063949443668}} {"text": "### Lab 3: Expectation Maximization and Variational Autoencoder\n\n### Machine Learning 2 (2019)\n\n* The lab exercises can be done in groups of two people, or individually.\n* The deadline is Tuesday, October 15th at 17:00.\n* Assignment should be submitted through Canvas! Make sure to include your and your teammates' names with the submission.\n* Attach the .IPYNB (IPython Notebook) file containing your code and answers. Naming of the file should be \"studentid1\\_studentid2\\_lab#\", for example, the attached file should be \"12345\\_12346\\_lab1.ipynb\". Only use underscores (\"\\_\") to connect ids, otherwise the files cannot be parsed.\n\nNotes on implementation:\n\n* You should write your code and answers in an IPython Notebook: http://ipython.org/notebook.html. If you have problems, please ask.\n* Use __one cell__ for code and markdown answers only!\n * Put all code in the cell with the ```# YOUR CODE HERE``` comment and overwrite the ```raise NotImplementedError()``` line.\n * For theoretical questions, put your solution using LaTeX style formatting in the YOUR ANSWER HERE cell.\n* Among the first lines of your notebook should be \"%pylab inline\". This imports all required modules, and your plots will appear inline.\n* Large parts of you notebook will be graded automatically. Therefore it is important that your notebook can be run completely without errors and within a reasonable time limit. To test your notebook before submission, select Kernel -> Restart \\& Run All.\n$\\newcommand{\\bx}{\\mathbf{x}} \\newcommand{\\bpi}{\\mathbf{\\pi}} \\newcommand{\\bmu}{\\mathbf{\\mu}} \\newcommand{\\bX}{\\mathbf{X}} \\newcommand{\\bZ}{\\mathbf{Z}} \\newcommand{\\bz}{\\mathbf{z}}$\n\n### Installing PyTorch\n\nIn this lab we will use PyTorch. PyTorch is an open source deep learning framework primarily developed by Facebook's artificial-intelligence research group. In order to install PyTorch in your conda environment go to https://pytorch.org and select your operating system, conda, Python 3.6, no cuda. Copy the text from the \"Run this command:\" box. Now open a terminal and activate your 'ml2labs' conda environment. Paste the text and run. After the installation is done you should restart Jupyter.\n\n### MNIST data\n\nIn this Lab we will use several methods for unsupervised learning on the MNIST dataset of written digits. The dataset contains digital images of handwritten numbers $0$ through $9$. Each image has 28x28 pixels that each take 256 values in a range from white ($= 0$) to black ($=1$). The labels belonging to the images are also included. \nFortunately, PyTorch comes with a MNIST data loader. The first time you run the box below it will download the MNIST data set. That can take a couple of minutes.\nThe main data types in PyTorch are tensors. For Part 1, we will convert those tensors to numpy arrays. In Part 2, we will use the torch module to directly work with PyTorch tensors.\n\n\n```python\n%pylab inline\nimport torch\nfrom torchvision import datasets, transforms\n\ntrain_dataset = datasets.MNIST('../data', train=True, download=True,\n transform=transforms.Compose([\n transforms.ToTensor(),\n transforms.Normalize((0.1307,), (0.3081,))\n ]))\n\ntrain_labels = train_dataset.train_labels.numpy()\ntrain_data = train_dataset.train_data.numpy()\n# For EM we will use flattened data\ntrain_data = train_data.reshape(train_data.shape[0], -1)\n\n```\n\n Populating the interactive namespace from numpy and matplotlib\n\n\n C:\\Users\\lintl\\Anaconda3\\envs\\rl2019\\lib\\site-packages\\torchvision\\datasets\\mnist.py:43: UserWarning: train_labels has been renamed targets\n warnings.warn(\"train_labels has been renamed targets\")\n C:\\Users\\lintl\\Anaconda3\\envs\\rl2019\\lib\\site-packages\\torchvision\\datasets\\mnist.py:53: UserWarning: train_data has been renamed data\n warnings.warn(\"train_data has been renamed data\")\n\n\n## Part 1: Expectation Maximization\nWe will use the Expectation Maximization (EM) algorithm for the recognition of handwritten digits in the MNIST dataset. The images are modelled as a Bernoulli mixture model (see Bishop $\\S9.3.3$):\n$$\np(\\bx|\\bmu, \\bpi) = \\sum_{k=1}^K \\pi_k \\prod_{i=1}^D \\mu_{ki}^{x_i}(1-\\mu_{ki})^{(1-x_i)}\n$$\nwhere $x_i$ is the value of pixel $i$ in an image, $\\mu_{ki}$ represents the probability that pixel $i$ in class $k$ is black, and $\\{\\pi_1, \\ldots, \\pi_K\\}$ are the mixing coefficients of classes in the data. We want to use this data set to classify new images of handwritten numbers.\n\n### 1.1 Binary data (5 points)\nAs we like to apply our Bernoulli mixture model, write a function `binarize` to convert the (flattened) MNIST data to binary images, where each pixel $x_i \\in \\{0,1\\}$, by thresholding at an appropriate level.\n\n\n```python\ndef binarize(X):\n return (X > 128).astype(np.float)\n```\n\n\n```python\n# Test test test!\nbin_train_data = binarize(train_data)\nassert bin_train_data.dtype == np.float\nassert bin_train_data.shape == train_data.shape\n\n```\n\nSample a few images of digits $2$, $3$ and $4$; and show both the original and the binarized image together with their label.\n\n\n```python\nnumbers = [2,3,4]\n\ndef plot_image(number, images, labels, sample):\n \n index = np.where(labels == number)[0][sample]\n \n title = \"Digit\" + str(labels[index]) + \":\"\n plt.imshow(images[index].reshape(28,28))\n plt.title(title + \" Non Binarized\")\n\ndef plot_bin_non_bin(numbers, train, train_labels, bin_train, sample):\n \n counter = 0\n for i in range(len(numbers)):\n fig = plt.figure(figsize=(15,10))\n\n plt.subplot(341)\n plot_image(numbers[i], train_data, train_labels, sample[0]) \n plt.subplot(341 + 1)\n plot_image(numbers[i], bin_train, train_labels, sample[0])\n plt.subplot(341 + 2)\n plot_image(numbers[i], train_data, train_labels, sample[1]) \n plt.subplot(341 + 3)\n plot_image(numbers[i], bin_train, train_labels, sample[1])\n plt.show()\n\nplot_bin_non_bin(numbers, train_data, train_labels, bin_train_data, [3,4])\n```\n\n### 1.2 Implementation (40 points)\nYou are going to write a function ```EM(X, K, max_iter)``` that implements the EM algorithm on the Bernoulli mixture model. \n\nThe only parameters the function has are:\n* ```X``` :: (NxD) array of input training images\n* ```K``` :: size of the latent space\n* ```max_iter``` :: maximum number of iterations, i.e. one E-step and one M-step\n\nYou are free to specify your return statement.\n\nMake sure you use a sensible way of terminating the iteration process early to prevent unnecessarily running through all epochs. Vectorize computations using ```numpy``` as much as possible.\n\nYou should implement the `E_step(X, mu, pi)` and `M_step(X, gamma)` separately in the functions defined below. These you can then use in your function `EM(X, K, max_iter)`.\n\n\n```python\ndef E_step(X, mu, pi):\n # get dimensions of gamma\n X_expand = np.expand_dims(X, axis=1)\n \n unnorm_gamma = pi * np.prod((mu ** X_expand) * ((1 - mu) ** (1 - X_expand)), axis=2)\n \n denominator = np.expand_dims(np.sum(unnorm_gamma, axis=1), axis=1)\n \n gamma = unnorm_gamma / denominator\n return gamma\n```\n\n\n```python\n# Let's test on 5 datapoints\nn_test = 5\nX_test = bin_train_data[:n_test]\nD_test, K_test = X_test.shape[1], 10\n\nnp.random.seed(2018)\nmu_test = np.random.uniform(low=.25, high=.75, size=(K_test,D_test))\npi_test = np.ones(K_test) / K_test\n\ngamma_test = E_step(X_test, mu_test, pi_test)\nassert gamma_test.shape == (n_test, K_test)\n\n```\n\n\n```python\ndef M_step(X, gamma):\n # we sum over the number of datapoints\n N_k = gamma.sum(axis=0)\n \n # pi is the fraction of N_k over the total sum ofer N_k\n pi = N_k / np.sum(N_k)\n \n # expanding with [None,:] and [:, None]\n tmp_mu = gamma[None,:].T * X[None, :]\n \n # unnormalized mu\n unnorm_mu = tmp_mu.sum(axis=1)\n \n # normalized mu\n mu = unnorm_mu / N_k[:, None]\n \n return mu, pi\n```\n\n\n```python\n# Oh, let's test again\nmu_test, pi_test = M_step(X_test, gamma_test)\n\nassert mu_test.shape == (K_test,D_test)\nassert pi_test.shape == (K_test, )\n\n```\n\n\n```python\ndef EM(X, K, max_iter, mu=None, pi=None):\n # dimensions\n N = X.shape[0]\n D = X.shape[1]\n \n # parameter initialization\n if mu is None:\n mu = np.random.uniform(0.25, 0.75, (K,D))\n if pi is None:\n pi = np.ones(K) / K\n \n conv_crit = 1e-4\n \n for i in range(max_iter):\n mu_old = mu\n pi_old = pi\n \n gamma = E_step(X, mu, pi)\n mu, pi = M_step(X, gamma)\n \n mu_update = np.linalg.norm(mu-mu_old)\n pi_update = np.linalg.norm(pi-pi_old)\n \n print(\"Iteration: {}, delta_mu: {}, delta_pi: {}\".format(i, np.round(mu_update, 5),\n np.round(pi_update, 5)))\n \n if mu_update < conv_crit and pi_update < conv_crit:\n print(\"After iteration {}, convergence is reached\".format(i))\n break\n \n return gamma, mu, pi \n\n```\n\n### 1.3 Three digits experiment (10 points)\nIn analogue with Bishop $\\S9.3.3$, sample a training set consisting of only __binary__ images of written digits $2$, $3$, and $4$. Run your EM algorithm and show the reconstructed digits.\n\n\n```python\ndef plot_mu(mu):\n \n K = mu.shape[0]\n fig, axes = plt.subplots(1, K, figsize=(15, 10))\n \n for k, ax in enumerate(axes):\n ax.imshow(mu[k].reshape((28,28)))\n plt.show()\n\nnumbers = [2,3,4]\nmax_iter = 100\n\nexample_train_labels = train_labels[np.isin(train_labels, numbers)]\nexample_trainbinary = bin_train_data[np.isin(train_labels, numbers), :]\n\n```\n\n\n```python\ngamma, mu, pi = EM(example_trainbinary, 3, 100)\n\n#show_mu(mu_3)\n```\n\n Iteration: 0, delta_mu: 20.62302, delta_pi: 0.47597\n Iteration: 1, delta_mu: 2.69712, delta_pi: 0.16751\n Iteration: 2, delta_mu: 1.76431, delta_pi: 0.12087\n Iteration: 3, delta_mu: 1.19617, delta_pi: 0.11609\n Iteration: 4, delta_mu: 0.6832, delta_pi: 0.06307\n Iteration: 5, delta_mu: 0.23769, delta_pi: 0.01967\n Iteration: 6, delta_mu: 0.06954, delta_pi: 0.00507\n Iteration: 7, delta_mu: 0.02194, delta_pi: 0.00145\n Iteration: 8, delta_mu: 0.00868, delta_pi: 0.00052\n Iteration: 9, delta_mu: 0.00348, delta_pi: 0.00019\n Iteration: 10, delta_mu: 0.00165, delta_pi: 6e-05\n Iteration: 11, delta_mu: 0.00181, delta_pi: 8e-05\n Iteration: 12, delta_mu: 0.00072, delta_pi: 5e-05\n Iteration: 13, delta_mu: 0.00044, delta_pi: 3e-05\n Iteration: 14, delta_mu: 0.00095, delta_pi: 4e-05\n Iteration: 15, delta_mu: 0.0011, delta_pi: 5e-05\n Iteration: 16, delta_mu: 0.00022, delta_pi: 2e-05\n Iteration: 17, delta_mu: 9e-05, delta_pi: 1e-05\n After iteration 17, convergence is reached\n\n\n\n```python\nplot_mu(mu)\nprint(\"Mixing coefficients: {}\".format(pi))\n```\n\nCan you identify which element in the latent space corresponds to which digit? What are the identified mixing coefficients for digits $2$, $3$ and $4$, and how do these compare to the true ones?\n\nConsidering the visualizations, it is obvious that the elements in the latent space illustrate the numbers of the classes, namely 2,3,4. So, in the above shown images, the activated regions illustrate probabilities represent if a pixel is active and therefore, the shape of the digits is detectable. \n\nThe mixing coefficients are uniformly distributed, which is in line with the assumption, that each number appears equally often. \n\n\n### 1.4 Experiments (20 points)\nPerform the follow-up experiments listed below using your implementation of the EM algorithm. For each of these, describe/comment on the obtained results and give an explanation. You may still use your dataset with only digits 2, 3 and 4 as otherwise computations can take very long.\n\n#### 1.4.1 Size of the latent space (5 points)\nRun EM with $K$ larger or smaller than the true number of classes. Describe your results.\n\n\n```python\ngamma_2, mu_2, pi_2 = EM(example_trainbinary, 2, 100)\nplot_mu(mu_2)\nprint(\"Mixing coefficients for 2 classes: {}\".format(pi_2))\n```\n\n\n```python\ngamma_5, mu_5, pi_5 = EM(example_trainbinary, 5, 100)\nplot_mu(mu_5)\nprint(\"Mixing coefficients for 5 classes: {}\".format(pi))\n```\n\nThe first case, a number smaller than the actual number of classes, the convergence of $\\mu_k$s to the actual classes is not guaranteed anymore, as seen in the picture, the first image is not clearly a three, but rather a mixture between three and two. The second image actually looks like it converged to a proper 4. \n\nFor the larger case, we see, that convergence in itself takes much longer. Also, it is notable that the digits 4 and 2 are represented doubly. Therefore, compared to the smaller number of classes each $\\mu_k$ seemingly actually converged to a true class, as none of the shown digits seems distorted or as a mixture of classes. It is also notable, that the digits of the same class vary in their look (the two 2s and 4s seem different), which might stemm from the fact that they reflect a different region of the possible shapes of the respective digits. \n\nThe mixture components seem to adapt, as they stay relatively uniform. \n\nSo, both a too small and a too large number of classes lead to acuracy loss. Once, the classes are mixed and once multiple representations appear. \n\n#### 1.4.2 Identify misclassifications (10 points)\nHow can you use the data labels to assign a label to each of the clusters/latent variables? Use this to identify images that are 'misclassified' and try to understand why they are. Report your findings.\n\n\n```python\n\n```\n\nYOUR ANSWER HERE\n\n#### 1.4.3 Initialize with true values (5 points)\nInitialize the three classes with the true values of the parameters and see what happens. Report your results.\n\n\n```python\n\n```\n\nYOUR ANSWER HERE\n\n## Part 2: Variational Auto-Encoder\n\nA Variational Auto-Encoder (VAE) is a probabilistic model $p(\\bx, \\bz)$ over observed variables $\\bx$ and latent variables and/or parameters $\\bz$. Here we distinguish the decoder part, $p(\\bx | \\bz) p(\\bz)$ and an encoder part $p(\\bz | \\bx)$ that are both specified with a neural network. A lower bound on the log marginal likelihood $\\log p(\\bx)$ can be obtained by approximately inferring the latent variables z from the observed data x using an encoder distribution $q(\\bz| \\bx)$ that is also specified as a neural network. This lower bound is then optimized to fit the model to the data. \n\nThe model was introduced by Diederik Kingma (during his PhD at the UVA) and Max Welling in 2013, https://arxiv.org/abs/1312.6114. \n\nSince it is such an important model there are plenty of well written tutorials that should help you with the assignment. E.g: https://jaan.io/what-is-variational-autoencoder-vae-tutorial/.\n\nIn the following, we will make heavily use of the torch module, https://pytorch.org/docs/stable/index.html. Most of the time replacing `np.` with `torch.` will do the trick, e.g. `np.sum` becomes `torch.sum` and `np.log` becomes `torch.log`. In addition, we will use `torch.FloatTensor()` as an equivalent to `np.array()`. In order to train our VAE efficiently we will make use of batching. The number of data points in a batch will become the first dimension of our data tensor, e.g. A batch of 128 MNIST images has the dimensions [128, 1, 28, 28]. To check check the dimensions of a tensor you can call `.size()`.\n\n### 2.1 Loss function\nThe objective function (variational lower bound), that we will use to train the VAE, consists of two terms: a log Bernoulli loss (reconstruction loss) and a Kullback\u2013Leibler divergence. We implement the two terms separately and combine them in the end.\nAs seen in Part 1: Expectation Maximization, we can use a multivariate Bernoulli distribution to model the likelihood $p(\\bx | \\bz)$ of black and white images. Formally, the variational lower bound is maximized but in PyTorch we are always minimizing therefore we need to calculate the negative log Bernoulli loss and Kullback\u2013Leibler divergence.\n\n### 2.1.1 Negative Log Bernoulli loss (5 points)\nThe negative log Bernoulli loss is defined as,\n\n\\begin{align}\nloss = - (\\sum_i^D \\bx_i \\log \\hat{\\bx_i} + (1 \u2212 \\bx_i) \\log(1 \u2212 \\hat{\\bx_i})).\n\\end{align}\n\nWrite a function `log_bernoulli_loss` that takes a D dimensional vector `x`, its reconstruction `x_hat` and returns the negative log Bernoulli loss. Make sure that your function works for batches of arbitrary size.\n\n\n```python\ndef log_bernoulli_loss(x_hat, x):\n \n loss = -torch.sum(x * torch.log(x_hat) + (1 - x) * torch.log(1 - x_hat))\n return loss\n\n```\n\n\n```python\n### Test test test\nx_test = torch.FloatTensor([[0.1, 0.2, 0.3, 0.4], [0.5, 0.6, 0.7, 0.8], [0.9, 0.9, 0.9, 0.9]])\nx_hat_test = torch.FloatTensor([[0.11, 0.22, 0.33, 0.44], [0.55, 0.66, 0.77, 0.88], [0.99, 0.99, 0.99, 0.99]])\n\nassert log_bernoulli_loss(x_hat_test, x_test) > 0.0\nassert log_bernoulli_loss(x_hat_test, x_test) < 10.0\n\n```\n\n### 2.1.2 Negative Kullback\u2013Leibler divergence (10 Points)\nThe variational lower bound (the objective to be maximized) contains a KL term $D_{KL}(q(\\bz)||p(\\bz))$ that can often be calculated analytically. In the VAE we assume $q = N(\\bz, \\mu, \\sigma^2I)$ and $p = N(\\bz, 0, I)$. Solve analytically!\n\nYOUR ANSWER HERE\n\nWrite a function `KL_loss` that takes two J dimensional vectors `mu` and `logvar` and returns the negative Kullback\u2013Leibler divergence. Where `logvar` is $\\log(\\sigma^2)$. Make sure that your function works for batches of arbitrary size.\n\n\n```python\ndef KL_loss(mu, logvar):\n \n loss = -(1/2) * torch.sum(1 + logvar - mu.pow(2) - logvar.exp())\n return loss\n\n```\n\n\n```python\n### Test test test\nmu_test = torch.FloatTensor([[0.1, 0.2], [0.3, 0.4], [0.5, 0.6]])\nlogvar_test = torch.FloatTensor([[0.01, 0.02], [0.03, 0.04], [0.05, 0.06]])\n\nassert KL_loss(mu_test, logvar_test) > 0.0\nassert KL_loss(mu_test, logvar_test) < 10.0\n\n```\n\n### 2.1.3 Putting the losses together (5 points)\nWrite a function `loss_function` that takes a D dimensional vector `x`, its reconstruction `x_hat`, two J dimensional vectors `mu` and `logvar` and returns the final loss. Make sure that your function works for batches of arbitrary size.\n\n\n```python\ndef loss_function(x_hat, x, mu, logvar):\n \n loss = log_bernoulli_loss(x_hat, x) + KL_loss(mu, logvar)\n return loss\n\n```\n\n\n```python\nx_test = torch.FloatTensor([[0.1, 0.2, 0.3], [0.4, 0.5, 0.6], [0.7, 0.8, 0.9]])\nx_hat_test = torch.FloatTensor([[0.11, 0.22, 0.33], [0.44, 0.55, 0.66], [0.77, 0.88, 0.99]])\nmu_test = torch.FloatTensor([[0.1, 0.2], [0.3, 0.4], [0.5, 0.6]])\nlogvar_test = torch.FloatTensor([[0.01, 0.02], [0.03, 0.04], [0.05, 0.06]])\n\nassert loss_function(x_hat_test, x_test, mu_test, logvar_test) > 0.0\nassert loss_function(x_hat_test, x_test, mu_test, logvar_test) < 10.0\n\n```\n\n### 2.2 The model\nBelow you see a data structure for the VAE. The modell itself consists of two main parts the encoder (images $\\bx$ to latent variables $\\bz$) and the decoder (latent variables $\\bz$ to images $\\bx$). The encoder is using 3 fully-connected layers, whereas the decoder is using fully-connected layers. Right now the data structure is quite empty, step by step will update its functionality. For test purposes we will initialize a VAE for you. After the data structure is completed you will do the hyperparameter search.\n\n\n\n```python\nfrom torch import nn\nfrom torch.nn import functional as F \n\nclass VAE(nn.Module):\n def __init__(self, fc1_dims, fc21_dims, fc22_dims, fc3_dims, fc4_dims):\n super(VAE, self).__init__()\n\n self.fc1 = nn.Linear(*fc1_dims)\n self.fc21 = nn.Linear(*fc21_dims)\n self.fc22 = nn.Linear(*fc22_dims)\n self.fc3 = nn.Linear(*fc3_dims)\n self.fc4 = nn.Linear(*fc4_dims)\n\n def encode(self, x):\n # To be implemented\n raise Exception('Method not implemented')\n\n def reparameterize(self, mu, logvar):\n # To be implemented\n raise Exception('Method not implemented')\n\n def decode(self, z):\n # To be implemented\n raise Exception('Method not implemented')\n\n def forward(self, x):\n # To be implemented\n raise Exception('Method not implemented')\n\nVAE_test = VAE(fc1_dims=(784, 4), fc21_dims=(4, 2), fc22_dims=(4, 2), fc3_dims=(2, 4), fc4_dims=(4, 784))\n\n```\n\n### 2.3 Encoding (10 points)\nWrite a function `encode` that gets a vector `x` with 784 elements (flattened MNIST image) and returns `mu` and `logvar`. Your function should use three fully-connected layers (`self.fc1()`, `self.fc21()`, `self.fc22()`). First, you should use `self.fc1()` to embed `x`. Second, you should use `self.fc21()` and `self.fc22()` on the embedding of `x` to compute `mu` and `logvar` respectively. PyTorch comes with a variety of activation functions, the most common calls are `F.relu()`, `F.sigmoid()`, `F.tanh()`. Make sure that your function works for batches of arbitrary size. \n\n\n```python\ndef encode(self, x):\n \n h1 = F.relu(self.fc1(x))\n mu = self.fc21(h1)\n logvar = self.fc22(h1)\n\n return mu, logvar\n\n```\n\n\n```python\n### Test, test, test\nVAE.encode = encode\n\nx_test = torch.ones((5,784))\nmu_test, logvar_test = VAE_test.encode(x_test)\n\nassert np.allclose(mu_test.size(), [5, 2])\nassert np.allclose(logvar_test.size(), [5, 2])\n\n```\n\n### 2.4 Reparameterization (10 points)\nOne of the major question that the VAE is answering, is 'how to take derivatives with respect to the parameters of a stochastic variable?', i.e. if we are given $\\bz$ that is drawn from a distribution $q(\\bz|\\bx)$, and we want to take derivatives. This step is necessary to be able to use gradient-based optimization algorithms like SGD.\nFor some distributions, it is possible to reparameterize samples in a clever way, such that the stochasticity is independent of the parameters. We want our samples to deterministically depend on the parameters of the distribution. For example, in a normally-distributed variable with mean $\\mu$ and standard deviation $\\sigma$, we can sample from it like this:\n\n\\begin{align}\n\\bz = \\mu + \\sigma \\odot \\epsilon,\n\\end{align}\n\nwhere $\\odot$ is the element-wise multiplication and $\\epsilon$ is sampled from $N(0, I)$.\n\n\nWrite a function `reparameterize` that takes two J dimensional vectors `mu` and `logvar`. It should return $\\bz = \\mu + \\sigma \\odot \\epsilon$.\n\n\n\n```python\ndef reparameterize(self, mu, logvar):\n \n return mu + torch.exp(1/2 * logvar) * torch.randn_like(mu)\n\n```\n\n\n```python\n### Test, test, test\nVAE.reparameterize = reparameterize\nVAE_test.train()\n\nmu_test = torch.FloatTensor([[0.1, 0.2], [0.3, 0.4], [0.5, 0.6]])\nlogvar_test = torch.FloatTensor([[0.01, 0.02], [0.03, 0.04], [0.05, 0.06]])\n\nz_test = VAE_test.reparameterize(mu_test, logvar_test)\n\nassert np.allclose(z_test.size(), [3, 2])\nassert z_test[0][0] < 5.0\nassert z_test[0][0] > -5.0\n\n```\n\n### 2.5 Decoding (10 points)\nWrite a function `decode` that gets a vector `z` with J elements and returns a vector `x_hat` with 784 elements (flattened MNIST image). Your function should use two fully-connected layers (`self.fc3()`, `self.fc4()`). PyTorch comes with a variety of activation functions, the most common calls are `F.relu()`, `F.sigmoid()`, `F.tanh()`. Make sure that your function works for batches of arbitrary size.\n\n\n```python\ndef decode(self, z):\n \n h1 = self.fc3(z)\n h1 = F.relu(h1)\n h1 = self.fc4(h1)\n x_hat = F.sigmoid(h1)\n return x_hat\n\n```\n\n\n```python\n# test test test\nVAE.decode = decode\n\nz_test = torch.ones((5,2))\nx_hat_test = VAE_test.decode(z_test)\n\nassert np.allclose(x_hat_test.size(), [5, 784])\nassert (x_hat_test <= 1).all()\nassert (x_hat_test >= 0).all()\n\n```\n\n C:\\Users\\lintl\\Anaconda3\\envs\\rl2019\\lib\\site-packages\\torch\\nn\\functional.py:1350: UserWarning: nn.functional.sigmoid is deprecated. Use torch.sigmoid instead.\n warnings.warn(\"nn.functional.sigmoid is deprecated. Use torch.sigmoid instead.\")\n\n\n### 2.6 Forward pass (10)\nTo complete the data structure you have to define a forward pass through the VAE. A single forward pass consists of the encoding of an MNIST image $\\bx$ into latent space $\\bz$, the reparameterization of $\\bz$ and the decoding of $\\bz$ into an image $\\bx$.\n\nWrite a function `forward` that gets a a vector `x` with 784 elements (flattened MNIST image) and returns a vector `x_hat` with 784 elements (flattened MNIST image), `mu` and `logvar`.\n\n\n```python\ndef forward(self, x):\n x = x.view(-1, 784)\n \n mu, logvar = self.encode(x)\n \n z = self.reparameterize(mu, logvar)\n \n x_hat = self.decode(z)\n \n return x_hat, mu, logvar\n\n```\n\n\n```python\n# test test test \nVAE.forward = forward\n\nx_test = torch.ones((5,784))\nx_hat_test, mu_test, logvar_test = VAE_test.forward(x_test)\n\nassert np.allclose(x_hat_test.size(), [5, 784])\nassert np.allclose(mu_test.size(), [5, 2])\nassert np.allclose(logvar_test.size(), [5, 2])\n\n```\n\n### 2.7 Training (15)\nWe will now train the VAE using an optimizer called Adam, https://arxiv.org/abs/1412.6980. The code to train a model in PyTorch is given below.\n\n\n```python\nfrom torch.autograd import Variable\n\ndef train(epoch, train_loader, model, optimizer):\n model.train()\n train_loss = 0\n for batch_idx, (data, _) in enumerate(train_loader):\n data = Variable(data)\n optimizer.zero_grad()\n recon_batch, mu, logvar = model(data)\n loss = loss_function(recon_batch, data.view(-1, 784), mu, logvar)\n loss.backward()\n train_loss += loss.data\n optimizer.step()\n if batch_idx % 100 == 0:\n print('Train Epoch: {} [{}/{} ({:.0f}%)]\\tLoss: {:.6f}'.format(\n epoch, batch_idx * len(data), len(train_loader.dataset),\n 100. * batch_idx / len(train_loader),\n loss.data / len(data)))\n\n print('====> Epoch: {} Average loss: {:.4f}'.format(\n epoch, train_loss / len(train_loader.dataset)))\n\n```\n\nLet's train. You have to choose the hyperparameters. Make sure your loss is going down in a reasonable amount of epochs (around 10).\n\n\n```python\n# Hyperparameters\nfc1_dims = (784, 400)\nfc21_dims = (400, 20)\nfc22_dims = (400, 20)\nfc3_dims = (20, 400)\nfc4_dims = (400, 784)\nlr = 1e-4\nbatch_size = 120\nepochs = 10\n\n```\n\n\n```python\n# This cell contains a hidden test, please don't delete it, thx\n```\n\nRun the box below to train the model using the hyperparameters you entered above.\n\n\n```python\nfrom torchvision import datasets, transforms\nfrom torch import nn, optim\n\n# Load data\ntrain_data = datasets.MNIST('../data', train=True, download=True,\n transform=transforms.ToTensor())\n\ntrain_loader = torch.utils.data.DataLoader(train_data,\n batch_size=batch_size, shuffle=True, **{})\n\n# Init model\nVAE_MNIST = VAE(fc1_dims=fc1_dims, fc21_dims=fc21_dims, fc22_dims=fc22_dims, fc3_dims=fc3_dims, fc4_dims=fc4_dims)\n\n# Init optimizer\noptimizer = optim.Adam(VAE_MNIST.parameters(), lr=lr)\n\n# Train\nfor epoch in range(1, epochs + 1):\n train(epoch, train_loader, VAE_MNIST, optimizer)\n\n```\n\n Train Epoch: 1 [0/60000 (0%)]\tLoss: 550.567566\n Train Epoch: 1 [12000/60000 (20%)]\tLoss: 268.343658\n Train Epoch: 1 [24000/60000 (40%)]\tLoss: 235.497818\n Train Epoch: 1 [36000/60000 (60%)]\tLoss: 212.567719\n Train Epoch: 1 [48000/60000 (80%)]\tLoss: 207.858917\n ====> Epoch: 1 Average loss: 254.7273\n Train Epoch: 2 [0/60000 (0%)]\tLoss: 192.445572\n Train Epoch: 2 [12000/60000 (20%)]\tLoss: 184.283844\n Train Epoch: 2 [24000/60000 (40%)]\tLoss: 174.772934\n Train Epoch: 2 [36000/60000 (60%)]\tLoss: 176.407883\n Train Epoch: 2 [48000/60000 (80%)]\tLoss: 168.189941\n ====> Epoch: 2 Average loss: 174.8130\n Train Epoch: 3 [0/60000 (0%)]\tLoss: 161.005066\n Train Epoch: 3 [12000/60000 (20%)]\tLoss: 158.476639\n Train Epoch: 3 [24000/60000 (40%)]\tLoss: 156.709290\n Train Epoch: 3 [36000/60000 (60%)]\tLoss: 158.456818\n Train Epoch: 3 [48000/60000 (80%)]\tLoss: 151.390640\n ====> Epoch: 3 Average loss: 155.7453\n Train Epoch: 4 [0/60000 (0%)]\tLoss: 151.145935\n Train Epoch: 4 [12000/60000 (20%)]\tLoss: 137.667313\n Train Epoch: 4 [24000/60000 (40%)]\tLoss: 143.940842\n Train Epoch: 4 [36000/60000 (60%)]\tLoss: 144.788986\n Train Epoch: 4 [48000/60000 (80%)]\tLoss: 138.139908\n ====> Epoch: 4 Average loss: 144.3600\n Train Epoch: 5 [0/60000 (0%)]\tLoss: 140.378143\n Train Epoch: 5 [12000/60000 (20%)]\tLoss: 138.112839\n Train Epoch: 5 [24000/60000 (40%)]\tLoss: 139.341797\n Train Epoch: 5 [36000/60000 (60%)]\tLoss: 140.853287\n Train Epoch: 5 [48000/60000 (80%)]\tLoss: 130.976883\n ====> Epoch: 5 Average loss: 136.7890\n Train Epoch: 6 [0/60000 (0%)]\tLoss: 131.719711\n Train Epoch: 6 [12000/60000 (20%)]\tLoss: 132.215408\n Train Epoch: 6 [24000/60000 (40%)]\tLoss: 131.544479\n Train Epoch: 6 [36000/60000 (60%)]\tLoss: 137.173676\n Train Epoch: 6 [48000/60000 (80%)]\tLoss: 134.207764\n ====> Epoch: 6 Average loss: 131.5690\n Train Epoch: 7 [0/60000 (0%)]\tLoss: 130.113449\n Train Epoch: 7 [12000/60000 (20%)]\tLoss: 134.579498\n Train Epoch: 7 [24000/60000 (40%)]\tLoss: 127.660347\n Train Epoch: 7 [36000/60000 (60%)]\tLoss: 130.052444\n Train Epoch: 7 [48000/60000 (80%)]\tLoss: 126.369286\n ====> Epoch: 7 Average loss: 127.7944\n Train Epoch: 8 [0/60000 (0%)]\tLoss: 127.497429\n Train Epoch: 8 [12000/60000 (20%)]\tLoss: 124.193436\n Train Epoch: 8 [24000/60000 (40%)]\tLoss: 124.477249\n Train Epoch: 8 [36000/60000 (60%)]\tLoss: 126.167824\n Train Epoch: 8 [48000/60000 (80%)]\tLoss: 124.408363\n ====> Epoch: 8 Average loss: 124.9271\n Train Epoch: 9 [0/60000 (0%)]\tLoss: 121.864845\n Train Epoch: 9 [12000/60000 (20%)]\tLoss: 120.650017\n Train Epoch: 9 [24000/60000 (40%)]\tLoss: 118.843796\n Train Epoch: 9 [36000/60000 (60%)]\tLoss: 121.871872\n Train Epoch: 9 [48000/60000 (80%)]\tLoss: 121.571045\n ====> Epoch: 9 Average loss: 122.5673\n Train Epoch: 10 [0/60000 (0%)]\tLoss: 127.931313\n Train Epoch: 10 [12000/60000 (20%)]\tLoss: 123.621147\n Train Epoch: 10 [24000/60000 (40%)]\tLoss: 119.833107\n Train Epoch: 10 [36000/60000 (60%)]\tLoss: 114.540863\n Train Epoch: 10 [48000/60000 (80%)]\tLoss: 118.989517\n ====> Epoch: 10 Average loss: 120.6853\n\n\nRun the box below to check if the model you trained above is able to correctly reconstruct images.\n\n\n```python\n### Let's check if the reconstructions make sense\n# Set model to test mode\nVAE_MNIST.eval()\n \n# Reconstructed\ntrain_data_plot = datasets.MNIST('../data', train=True, download=True,\n transform=transforms.ToTensor())\n\ntrain_loader_plot = torch.utils.data.DataLoader(train_data_plot,\n batch_size=1, shuffle=False, **{})\n\nfor batch_idx, (data, _) in enumerate(train_loader_plot):\n x_hat, mu, logvar = VAE_MNIST(data)\n plt.imshow(x_hat.view(1,28,28).squeeze().data.numpy(), cmap='gray')\n plt.title('%i' % train_data.train_labels[batch_idx])\n plt.show()\n if batch_idx == 3:\n break\n\n```\n\n### 2.8 Visualize latent space (20 points)\nNow, implement the auto-encoder now with a 2-dimensional latent space, and train again over the MNIST data. Make a visualization of the learned manifold by using a linearly spaced coordinate grid as input for the latent space, as seen in https://arxiv.org/abs/1312.6114 Figure 4.\n\n\n```python\nfc1_dims = (784, 256)\nfc21_dims = (256, 2)\nfc22_dims = (256, 2)\nfc3_dims = (2, 256)\nfc4_dims = (256, 784)\nlr = 1e-4\nbatch_size = 128\nepochs = 10\n\n# Load data\ntrain_data = datasets.MNIST('../data', train=True, download=True,\n transform=transforms.ToTensor())\n\ntrain_loader = torch.utils.data.DataLoader(train_data,\n batch_size=batch_size, shuffle=True, **{})\n\n# Init model\nVAE_MNIST = VAE(fc1_dims=fc1_dims,\n fc21_dims=fc21_dims,\n fc22_dims=fc22_dims,\n fc3_dims=fc3_dims,\n fc4_dims=fc4_dims)\n\n# Init optimizer\noptimizer = optim.Adam(VAE_MNIST.parameters(), lr=lr)\n\n# Train\nfor epoch in range(1, epochs + 1):\n train(epoch, train_loader, VAE_MNIST, optimizer)\n```\n\n Train Epoch: 1 [0/60000 (0%)]\tLoss: 551.562500\n Train Epoch: 1 [12800/60000 (21%)]\tLoss: 263.747467\n Train Epoch: 1 [25600/60000 (43%)]\tLoss: 238.770248\n Train Epoch: 1 [38400/60000 (64%)]\tLoss: 221.702301\n Train Epoch: 1 [51200/60000 (85%)]\tLoss: 223.060852\n ====> Epoch: 1 Average loss: 264.5905\n Train Epoch: 2 [0/60000 (0%)]\tLoss: 208.024170\n Train Epoch: 2 [12800/60000 (21%)]\tLoss: 211.439774\n Train Epoch: 2 [25600/60000 (43%)]\tLoss: 202.066025\n Train Epoch: 2 [38400/60000 (64%)]\tLoss: 198.744766\n Train Epoch: 2 [51200/60000 (85%)]\tLoss: 193.100998\n ====> Epoch: 2 Average loss: 199.6053\n Train Epoch: 3 [0/60000 (0%)]\tLoss: 190.765076\n Train Epoch: 3 [12800/60000 (21%)]\tLoss: 192.474442\n Train Epoch: 3 [25600/60000 (43%)]\tLoss: 187.244492\n Train Epoch: 3 [38400/60000 (64%)]\tLoss: 187.458191\n Train Epoch: 3 [51200/60000 (85%)]\tLoss: 190.158081\n ====> Epoch: 3 Average loss: 190.6890\n Train Epoch: 4 [0/60000 (0%)]\tLoss: 187.374985\n Train Epoch: 4 [12800/60000 (21%)]\tLoss: 180.420563\n Train Epoch: 4 [25600/60000 (43%)]\tLoss: 188.807617\n Train Epoch: 4 [38400/60000 (64%)]\tLoss: 187.241302\n Train Epoch: 4 [51200/60000 (85%)]\tLoss: 185.843384\n ====> Epoch: 4 Average loss: 185.7557\n Train Epoch: 5 [0/60000 (0%)]\tLoss: 178.010010\n Train Epoch: 5 [12800/60000 (21%)]\tLoss: 180.615494\n Train Epoch: 5 [25600/60000 (43%)]\tLoss: 186.501984\n Train Epoch: 5 [38400/60000 (64%)]\tLoss: 182.561722\n Train Epoch: 5 [51200/60000 (85%)]\tLoss: 176.204254\n ====> Epoch: 5 Average loss: 182.3810\n Train Epoch: 6 [0/60000 (0%)]\tLoss: 184.070938\n Train Epoch: 6 [12800/60000 (21%)]\tLoss: 186.254715\n Train Epoch: 6 [25600/60000 (43%)]\tLoss: 174.112152\n Train Epoch: 6 [38400/60000 (64%)]\tLoss: 181.852188\n Train Epoch: 6 [51200/60000 (85%)]\tLoss: 178.781525\n ====> Epoch: 6 Average loss: 179.9251\n Train Epoch: 7 [0/60000 (0%)]\tLoss: 185.035431\n Train Epoch: 7 [12800/60000 (21%)]\tLoss: 179.148438\n Train Epoch: 7 [25600/60000 (43%)]\tLoss: 170.358719\n Train Epoch: 7 [38400/60000 (64%)]\tLoss: 168.970840\n Train Epoch: 7 [51200/60000 (85%)]\tLoss: 173.702774\n ====> Epoch: 7 Average loss: 177.9628\n Train Epoch: 8 [0/60000 (0%)]\tLoss: 171.521820\n Train Epoch: 8 [12800/60000 (21%)]\tLoss: 171.705261\n Train Epoch: 8 [25600/60000 (43%)]\tLoss: 178.153992\n Train Epoch: 8 [38400/60000 (64%)]\tLoss: 172.498566\n Train Epoch: 8 [51200/60000 (85%)]\tLoss: 174.557022\n ====> Epoch: 8 Average loss: 176.2270\n Train Epoch: 9 [0/60000 (0%)]\tLoss: 173.551559\n Train Epoch: 9 [12800/60000 (21%)]\tLoss: 172.849670\n Train Epoch: 9 [25600/60000 (43%)]\tLoss: 174.010452\n Train Epoch: 9 [38400/60000 (64%)]\tLoss: 174.835938\n Train Epoch: 9 [51200/60000 (85%)]\tLoss: 183.921448\n ====> Epoch: 9 Average loss: 174.4494\n Train Epoch: 10 [0/60000 (0%)]\tLoss: 169.704361\n Train Epoch: 10 [12800/60000 (21%)]\tLoss: 183.385513\n Train Epoch: 10 [25600/60000 (43%)]\tLoss: 163.904984\n Train Epoch: 10 [38400/60000 (64%)]\tLoss: 171.295914\n Train Epoch: 10 [51200/60000 (85%)]\tLoss: 172.903488\n ====> Epoch: 10 Average loss: 172.6839\n\n\n### 2.8 Amortized inference (10 points)\nWhat is amortized inference? Where in the code of Part 2 is it used? What is the benefit of using it?\n\n\n\n", "meta": {"hexsha": "ef57f712f7781fc96fd6ccb5c9b5ad35d4007ea9", "size": 179066, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lab 3/12140910_12152498_lab3_.ipynb", "max_stars_repo_name": "PhilLint/ML2", "max_stars_repo_head_hexsha": "f5e19faa6809006066f0d513e0883a333881b9a8", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "lab 3/12140910_12152498_lab3_.ipynb", "max_issues_repo_name": "PhilLint/ML2", "max_issues_repo_head_hexsha": "f5e19faa6809006066f0d513e0883a333881b9a8", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "lab 3/12140910_12152498_lab3_.ipynb", "max_forks_repo_name": "PhilLint/ML2", "max_forks_repo_head_hexsha": "f5e19faa6809006066f0d513e0883a333881b9a8", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 75.7150105708, "max_line_length": 17488, "alphanum_fraction": 0.7901109088, "converted": true, "num_tokens": 10861, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819874558604, "lm_q2_score": 0.7025300449389326, "lm_q1q2_score": 0.4295142151222195}} {"text": " # Singular Matrices \n \nRecently I've been working a lot on coding models and functions 'from scratch'. The point is to force myself to understand not only how each model/method/algorithm works, but also really understand the implementations and parts of the models. \n\nOne of the problems I've hit with a few different statistical models is singular matrices. In this post I'm going to talk about:\n\n1. What is a singular matrix?\n2. Why are singular matrices a problem?\n3. How can I identify a singular matrix?\n4. How can I work with/around a singular matrix?\n\n## What is a singular matrix?\n\nA singular (or degenerate) matrix is a matrix that can not be inverted. A lot of this post will discuss the bridge between theory and practice, so I'll further specify that a singular amtrix is a matrix that can't theoretically be inverted. \n\nOf course, practically we can do all sorts of silly things. There are many functions that one can run on a computer that will try and 'succeed' at inverting a sigular matrix. I put succeed in quotes as any function that returns a result when trying to invert a singular matrix is returning nonsense. This is why it is important not to blindly trust functions for random packages you find on the internet! On the other hand, there may be matrices that are theoretically invertable but impossible to practically invert for a variety of reasons that I'll discuss in a bit. \n\nFirst, let's review the definition of 'invertable'. A matrix is invertable if there exists a square ($nxn$) matrix B such that $AB = BA = I$ where $I$ is the identity matrix. Matrix inversion is the process of finding matrix $B$ for matrix $A$ that satisfies the equation above. \n\n**Technical Asside:** I'd like to burrow a little deeper into what a singular matrix is, but it's a bit mathy. Feel free to skip this if you aren't a hard core math nerd. One can show that non-singular coefficient matrices lead to unique solutions for every vector of constants one could choose. Singular matrices, on the other hand, have non-trivial nullspaces (see proof NMTNS at bottom). For vector contraints b, the system $\\mathcal{LS}(A,b)$ could be inconsistant (i.e. no solution). However, if $\\mathcal{LS}(A,b)$ has at elast one solution $(w)$, then it will have infinitely many solutions (see proof PSPHS)! A system of equations with a singular coefficient matrix will never have a unique solution. \n\nWe'll also note that for singular matrices that there will often be a way to write one row of the matrix as a linear combination of the other rows (may also be true for the columns).\n\n## Why are singular matrices a problem?\n\nWhy are singular matrices a problem? Well, as it turns out, we often need to invert matrices. For example, what if we want to evaluate the probability density function of a multivariate gaussian distribution?\n\n$$\np(x;\\mu,\\Sigma)= \\frac{1}{(2\\pi)^{\\frac{n}{2}} \\left\\lvert \\Sigma \\right\\rvert ^\\frac{1}{2}} \\exp\\biggl( \\frac{-1}{2}(x-\\mu)^T \\Sigma^{-1} (x-\\mu) \\biggr)\n$$\n\nWe would need to find $\\Sigma^{-1}$, the inverse of the caovariance matrix. Or what if we wanted to evaulate the posterior of a Gaussian Process Model? \n\n$$\n\\overline{\\mathcal{f}}_* = k^T_*(K+\\sigma^2_nI)^-1y\n$$\n\n$$\n\\mathcal{V}[\\mathcal{f}_*] = k(x+*,x_*)-k^T_*(K+\\sigma^2_nI)^-1k_*\n$$\n\nI borrowed the notation above from Gaussian Processes for MAchine Learning Eq. 2.25-26. I could go on listing examples of important equations that require matrix inversions but I think you get the point. \n\nThe problem is, if we ever need to invert a singular matrix we are in big trouble! \n\n## How can I identify a singular matrix?\n\nIn many classrooms we teach that the simplest way to find out if a matrix is singular is to find the determinant. If the determinant is zero, then the matrix is singular. This would work fine if theory and practice always went hand in hand, but in the real world things can go terribly wrong with using the determinant to find out if a matrix is singular.\n\nHere is a good example (courtesy of anonymous user 85109 on stack exchange). Let's take the determinant of the matrix below. \n\n\n```python\nimport numpy as np\n\narr = np.array([\n [16, 2, 3, 13],\n [5, 11, 10, 8],\n [9, 7, 6, 12],\n [4, 14, 15, 1]\n ])\n\nnp.linalg.det(arr)\n```\n\n\n\n\n -1.449507180950607e-12\n\n\n\nWell that's not zero! Awesome, so the matrix must be non-singular, right? Nope. We can see that there is a way to write a row on this matrix as a linear combination of the other raws (and the same for the columns). This implies that the matrix is singular!\n\nLet's check the symbolic determinant to get a second opinion. \n\n\n```python\nimport sympy as sym\n\nM = sym.Matrix(arr)\nM.det()\n```\n\n\n\n\n 0\n\n\n\nWow! The symbolic determinant is exactly what we expect for a singular matrix (zero). So why did numpy give us a different answer?\n\nWell, calcualting the determinant of large matrices is very inefficient. A nice approximation that is commonly leveraged by apckages like numpy is to use the product of the diagonal elements of a specific matrix factorization of the array (LU factorization as of version 1.15). Let's look at this factorization below.\n\n\n```python\nimport scipy.linalg as la\n\nP, L, U = la.lu(arr)\n\nprint(L)\nprint(U)\nprint(P)\n```\n\n [[1. 0. 0. 0. ]\n [0.25 1. 0. 0. ]\n [0.3125 0.76851852 1. 0. ]\n [0.5625 0.43518519 1. 1. ]]\n [[ 1.60000000e+01 2.00000000e+00 3.00000000e+00 1.30000000e+01]\n [ 0.00000000e+00 1.35000000e+01 1.42500000e+01 -2.25000000e+00]\n [ 0.00000000e+00 0.00000000e+00 -1.88888889e+00 5.66666667e+00]\n [ 0.00000000e+00 0.00000000e+00 0.00000000e+00 3.55271368e-15]]\n [[1. 0. 0. 0.]\n [0. 0. 1. 0.]\n [0. 0. 0. 1.]\n [0. 1. 0. 0.]]\n\n\nThe diagonal of the lower triable (L) are all ones and the diagnonal of the upper triangle (U) are all on-zero! This makes for nice easy math then writing statistical/scientific computing packages. We can take the product of the diagonal of the upper triagle to approximate the determinant of the original matrix.\n\n\n```python\nnp.prod(np.diag(U))\n```\n\n\n\n\n -1.4495071809506048e-12\n\n\n\nWe got the same answer as when we called the determinant function from numpy! Neat. Now this LU decomposition technique is super fast, but it relies on floating point arithmatic. The product of the daigonal of the upper triangle is not quite zero as we would expect. This is why using standard functions that calcualte determinants to identify singualr matrices is a bad idea. \n\nHere are a few other weird examples where using the determinant misleads us! Now, the identity matrix is NOT singular,\n\n\n```python\nnp.linalg.det(np.eye(100))\n```\n\n\n\n\n 1.0\n\n\n\nBut by multiplying our matrix by a very samll number, we suddenly see a determinant value that is WAY closer to zero than the determinant value for the singular matrix above!\n\n\n```python\nnp.linalg.det(0.1*np.eye(100))\n```\n\n\n\n\n 1.00000000000033e-100\n\n\n\nNow this matrix is NOT singular (for any constant $c$ with identiy matrix $I$, $c*I=D$, matrix D is non-singular jsut like $I$), but with a determinant of $1e^{-100}$ we might easily be fooled into thinking that it is....just wait, it gets worse. Look at the example below.\n\n\n```python\nnp.linalg.det(.0001*np.eye(100))\n```\n\n\n\n\n 0.0\n\n\n\nThe determinant should jsut be the determinant of $I$ scaled by $.0001^{-100}$...but numpy can't represent that number! Instead the number underflows and becomes zero, thus tricking us into thinking that this matrix is singular. We could easily invert this matrix and get the correct inversion. We can try the same trick with a large constant to get overflow issues (at least this time numpy warns us!). \n\n\n```python\nnp.linalg.det(10000*np.eye(100))\n```\n\n /anaconda3/envs/squidward_env/lib/python3.6/site-packages/numpy/linalg/linalg.py:2022: RuntimeWarning: overflow encountered in det\n r = _umath_linalg.det(a, signature=signature)\n\n\n\n\n\n inf\n\n\n\nWhat other tests might we try for identifying if a matrix is singular? One common tool is using the matrix rank. If the rank of an NxM matrix is less than the minimum of N and M, then we call the matrix singular. \n\nThe [rank](https://stattrek.com/matrix-algebra/matrix-rank.aspx) of a matrix is defined as either 1) the maximum number of linearly independent column vectors in the matrix or 2) the maximum number of linearly independent row vectors in the matrix. Both definitions are equivalent.\n\n\n```python\nA = .0001*np.eye(100)\nrank = np.linalg.matrix_rank(A)\nsize_M = A.shape[0]\ndet = np.linalg.det(A)\n\nprint(\"rank {} = dimension {}\".format(rank, size_M))\nprint(\"determinant {}\".format(det))\n```\n\n rank 100 = dimension 100\n determinant 0.0\n\n\nThe scaled identity matrix from above still fails to pass the determinant test (due to underflow issues), passes the rank test. We can try this for our original array as well!\n\n\n```python\nrank = np.linalg.matrix_rank(arr)\nsize_M = arr.shape[0]\ndet = np.linalg.det(arr)\n\nprint(\"rank {} != dimension {}\".format(rank, size_M))\nprint(\"determinant {}\".format(det))\n```\n\n rank 3 != dimension 4\n determinant -1.449507180950607e-12\n\n\nThis array passes the determinant test (even though it is singular), but fails to pass the rank test. \n\nAnother test that we can try is the [condition](https://en.wikipedia.org/wiki/Condition_number) test. The condition of a matrix can be thought of as a measure of how easy the matrix is to invert. The best condition is one. The higher the condition number the harder a matrix is to invert and the more errors may propagate through to the inverted matrix. This is nice, because it not only gives us a clue as to whether a matrix is singular, but also whether the matrix is close enough to singular that we can expect errors when computing the inversion on a computer (due to floating point errors and what not).\n\nThe condition is technically the norm of a matrix times the norm of it's 'inverse' (or the matrix the computer gets when it tries to invert the matrix). If these two norms are very dissimilar (meaning the norm changed a lot when the matrix was inverted) then we say that the matrix is poorly (or ill) conditioned. The condition number will be high in this case.\n\nNow the computer may still invert ill conditioned matrices. In fact, it takes the same amount of steps to invert a matrix using Gaussian elimination no matter the condition. However, ill conditioned matrices will have many errors in their inverted counter parts (even to the point of being completely useless). The condition becomes a kind of error multiplier. \n\nWhen solving the linear system $Ax = b$, you might expect that a small error in $b$ would result in a small error in $x$. That\u2019s true if $A$ is well-conditioned. But small changes in $b$ could result in large changes in x if $A$ is ill-conditioned. Any error (like measurement error from real world observations) will be multiplied by poor conditioning (not just floating point errors).\n\nAs a rule of thumb in double precision, a condition greater than 1e15 is really bad.\n\n\n```python\nnp.linalg.cond(arr)\n```\n\n\n\n\n 8.147992566622624e+16\n\n\n\nOur original matrix (the tricky singular one) has a HUGE condition and is probably even singular based only on looking at the condition. Obviously it will be bad to try to invert this matrix without taking the proper precautions. \n\nOne good check is to see if the reciprocal of the condition is larger than than float epsilon. If it is clsot to epsilon then you are bound to run into some issues.\n\n\n```python\nimport sys\n\n1.0 / np.linalg.cond(arr) >= sys.float_info.epsilon\n```\n\n\n\n\n False\n\n\n\nFinally there is [svd](https://en.wikipedia.org/wiki/Singular_value_decomposition) (singular value decomposition). This is what rank and condition are based on! When an of the singular values of a matrix are small compared to the largest singular value...beware!\n\n\n```python\nnp.set_printoptions(suppress=True)\nsingular_values = np.linalg.svd(arr)[1]\nmax_sv = np.max(singular_values)\nmin_sv = np.min(singular_values)\n \nmin_sv / max_sv\n```\n\n\n\n\n 1.2272961613838391e-17\n\n\n\nWe notice that the ratio of the largest and smallest value is REALLY small...that's a bad sign. The svd can tell us if a amtrix is close to singularity. If multiple singualr values are really small it can tell us about the matrix rank. \n\nAll of the tools above are easy to use and pretty efficient. A careful scientific coder should always check if his/her matrices are invertable.\n\nSo what do we do if we find a singular matrix?!\n\n## How can I work with/around a singular matrix?\n\nSingular marticies are, as it turns out, a very small subset of the space of all possible square matricies. In fact, if you were to fill matricies with random uniform samples, you would almost NEVER get a singular matrix.\n\nSo the easiest trick to work with a singular amtrix is to add a very small value to the diagonal of the matrix to 'nudge' it out of the singular subset. \n\n\n```python\narr_nudged = arr + np.eye(arr.shape[0])*1e-10\nprint(\"Original Matrix Condition: {}\".format(np.linalg.cond(arr)))\nprint(\"Nudged Matrix Condition: {}\".format(np.linalg.cond(arr_nudged)))\n```\n\n Original Matrix Condition: 8.147992566622624e+16\n Nudged Matrix Condition: 339998354530.4601\n\n\nThe condition of our nudged matrix is still really big...but not NEARLY as bad as it's original condition! Adding a tiny value like 1e-10 to the diagonal of a covariance matrix (for example) might not change the matrix in any meaningful way from a scientific standpoint, but it can be many fewer errors when calculating the matrix's inverse. \n\nAnother good piece of advice to to looka t different methods of inverting matrices. Instead of using [Cramer's formula](https://en.wikipedia.org/wiki/Invertible_matrix#Analytic_solution) or standard funtions like `np.linal.inv`, try using SVD decomposition or LU decomposition. You can even find some very nice numerically stable methods leveraging cholesky decomposition (a favorite for Gaussian Process models). \n\n**Author's Note:** The observant reader may note that earlier I said that singular matrices are very rare...so why worry about them? Well, they are rare in the sense that you are unlikely to stumble arocss one when randomly sampling many common distributions from the exponential family. However, there are good reasons why we may run acorss them commonly in the real world. Covariances matrices, for example, are often build around multiple samples from a test set. Many data points/samples may be identical or very close resulting in rows/columns in the matrix that are identical/close to identical. This is why we regularize the matrix we want to invert by adding a very small number to the 'ridge' or 'principal diagonal' of the matrix (just like in [ridge regression](https://link.springer.com/content/pdf/10.3758/BF03208332.pdf)) in the same way that we might add a noise term in the noisy case of Gaussian Process regression! In layman's terms: this is why we add a small number to the matrix diagonal. If you'd like to read more about this in the case of Gaussian Processes, you can check out equation 3.26 on page 45 of Gaussian Processes for Machine Learning. \n\n## Closing Remarks\n\nWell now you know how to find hot singular matrices in your area and even how to work around them! My advice is to always check your matrices before you try to invert them and have a plan for how to treat the matrix if it is poorly conditioned. \n\n## Proofs\nI credit these to 'A First Course in Linear Algrebra' by Robert Breezer from which I took these proofs. I thank Robert for releasing this great reference for free under the GNU open source liscence!\n\n**Theorem NMTNS:** Nonsingular Matrices have Trivial Null Spaces.
                                        \nSuppose that $A$ is a square matrix. Then $A$ is nonsingular if and only if the null space of $A$ is the set containing only the zero vector, i.e. $\\mathcal{N}(A)=\\{0\\}$.\n\nProof: The null space of a square matrix, $A$, is equal to the set of solutions to the homogeneous system, $\\mathcal{LS}(A,0)$. A matrix is nonsingular if and only if the set of solutions to the homogeneous system, $\\mathcal{LS}(A,0)$, has only a trivial solution. These two observations may be chained together to construct the two proofs necessary for each half of this theorem.\n\n**Theorem PSPHS:** Particular Solution Plus Homogeneous Solutions.
                                        Suppose that $w$ s one solution to the linear system of equations $\\mathcal{LS}(A,b)$. Then $y$ is a solution to $\\mathcal{LS}(A,b)$ if and only if $y=w+z$ for some vector $z \\in \\mathcal{N}(A)$.\n\nProof: [PSPHS Proof](http://linear.ups.edu/html/section-LC.html)\n\n\n```python\n\n```\n", "meta": {"hexsha": "779bad5996adfeb5a36d5cc45784b901a1e4d1d7", "size": 23750, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "singular_matrices/Singular_Matrices.ipynb", "max_stars_repo_name": "James-Montgomery/blog_material", "max_stars_repo_head_hexsha": "7410858d3bbb4a1e9287f53bf109531f1f0aa951", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-06-20T23:04:52.000Z", "max_stars_repo_stars_event_max_datetime": "2021-06-20T23:04:52.000Z", "max_issues_repo_path": "singular_matrices/Singular_Matrices.ipynb", "max_issues_repo_name": "James-Montgomery/blog_material", "max_issues_repo_head_hexsha": "7410858d3bbb4a1e9287f53bf109531f1f0aa951", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "singular_matrices/Singular_Matrices.ipynb", "max_forks_repo_name": "James-Montgomery/blog_material", "max_forks_repo_head_hexsha": "7410858d3bbb4a1e9287f53bf109531f1f0aa951", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 39.9159663866, "max_line_length": 1176, "alphanum_fraction": 0.6290947368, "converted": true, "num_tokens": 4181, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.8267118004748677, "lm_q1q2_score": 0.4294944074560982}} {"text": "\n\n# Tutorial 1: Introduction to Reinforcement Learning\n**Week 3, Day 2: Basic Reinforcement Learning (RL)**\n\n**By Neuromatch Academy**\n\n__Content creators:__ Matthew Sargent, Anoop Kulkarni, Sowmya Parthiban, Feryal Behbahani, Jane Wang\n\n__Content reviewers:__ Ezekiel Williams, Mehul Rastogi, Lily Cheng, Roberto Guidotti, Arush Tagade\n\n__Content editors:__ Spiros Chavlis \n\n__Production editors:__ Spiros Chavlis\n\n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n

                                        \n\n---\n# Tutorial Objectives\n\nBy the end of the tutorial, you should be able to:\n\n1. Within the RL framework, be able to identify the different components: environment, agent, states, and actions. \n2. Understand the Bellman equation and components involved. \n3. Implement tabular value-based model-free learning (Q-learning and SARSA).\n4. Run a DQN agent and experiment with different hyperparameters.\n5. Have a high-level understanding of other (nonvalue-based) RL methods.\n6. Discuss real-world applications and ethical issues of RL.\n\n**Note:** There is an issue with some images not showing up if you're using a Safari browser. Please switch to Chrome if this is the case.\n\n\n```\n# @title Tutorial slides\n\n# @markdown These are the slides for the videos in this tutorial\nfrom IPython.display import IFrame\nIFrame(src=f\"https://mfr.ca-1.osf.io/render?url=https://osf.io/m3kqy/?direct%26mode=render%26action=download%26mode=render\", width=854, height=480)\n```\n\n---\n\n# Setup\n\nRun the following 5 cells in order to set up needed functions. Don't worry about the code for now!\n\n\n```\n# @title Install requirements\nfrom IPython.display import clear_output\n# @markdown we install the acme library, see [here](https://github.com/deepmind/acme) for more info\n\n# @markdown WARNING: There may be errors and warnings reported during the installation.\n# @markdown However, they should be ignored.\n!apt-get install -y xvfb ffmpeg --quiet\n!pip install --upgrade pip --quiet\n!pip install imageio --quiet\n!pip install imageio-ffmpeg\n!pip install gym --quiet\n!pip install enum34 --quiet\n!pip install dm-env --quiet\n!pip install pandas --quiet\n!pip install keras-nightly==2.5.0.dev2021020510 --quiet\n!pip install grpcio==1.34.0 --quiet\n!pip install tensorflow --quiet\n!pip install typing --quiet\n!pip install einops --quiet\n!pip install dm-acme --quiet\n!pip install dm-acme[reverb] --quiet\n!pip install dm-acme[tf] --quiet\n!pip install dm-acme[envs] --quiet\n!pip install dm-env --quiet\n\nclear_output()\n```\n\n\n```\n# Import modules\nimport gym\nimport enum\nimport copy\nimport time\nimport acme\nimport torch\nimport base64\nimport dm_env\nimport IPython\nimport imageio\nimport warnings\nimport itertools\nimport collections\n\nimport numpy as np\nimport pandas as pd\nimport torch.nn as nn\nimport torch.optim as optim\nimport torch.nn.functional as F\n\nimport matplotlib as mpl\nimport matplotlib.pyplot as plt\n\nimport tensorflow.compat.v2 as tf\n\nfrom acme import specs\nfrom acme import wrappers\nfrom acme.utils import tree_utils\nfrom acme.utils import loggers\nfrom torch.autograd import Variable\nfrom torch.distributions import Categorical\nfrom typing import Callable, Sequence\n\ntf.enable_v2_behavior()\nwarnings.filterwarnings('ignore')\nnp.set_printoptions(precision=3, suppress=1)\n```\n\n\n```\n# @title Figure settings\nimport ipywidgets as widgets # interactive display\n%matplotlib inline\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/course-content/master/nma.mplstyle\")\nmpl.rc('image', cmap='Blues')\n```\n\n\n```\n# @title Helper Functions\n# @markdown Implement helpers for value visualisation\n\nmap_from_action_to_subplot = lambda a: (2, 6, 8, 4)[a]\nmap_from_action_to_name = lambda a: (\"up\", \"right\", \"down\", \"left\")[a]\n\n\ndef plot_values(values, colormap='pink', vmin=-1, vmax=10):\n plt.imshow(values, interpolation=\"nearest\",\n cmap=colormap, vmin=vmin, vmax=vmax)\n plt.yticks([])\n plt.xticks([])\n plt.colorbar(ticks=[vmin, vmax])\n\n\ndef plot_state_value(action_values, epsilon=0.1):\n q = action_values\n fig = plt.figure(figsize=(4, 4))\n vmin = np.min(action_values)\n vmax = np.max(action_values)\n v = (1 - epsilon) * np.max(q, axis=-1) + epsilon * np.mean(q, axis=-1)\n plot_values(v, colormap='summer', vmin=vmin, vmax=vmax)\n plt.title(\"$v(s)$\")\n\n\ndef plot_action_values(action_values, epsilon=0.1):\n q = action_values\n fig = plt.figure(figsize=(8, 8))\n fig.subplots_adjust(wspace=0.3, hspace=0.3)\n vmin = np.min(action_values)\n vmax = np.max(action_values)\n dif = vmax - vmin\n for a in [0, 1, 2, 3]:\n plt.subplot(3, 3, map_from_action_to_subplot(a))\n\n plot_values(q[..., a], vmin=vmin - 0.05*dif, vmax=vmax + 0.05*dif)\n action_name = map_from_action_to_name(a)\n plt.title(r\"$q(s, \\mathrm{\" + action_name + r\"})$\")\n\n plt.subplot(3, 3, 5)\n v = (1 - epsilon) * np.max(q, axis=-1) + epsilon * np.mean(q, axis=-1)\n plot_values(v, colormap='summer', vmin=vmin, vmax=vmax)\n plt.title(\"$v(s)$\")\n\n\ndef plot_stats(stats, window=10):\n plt.figure(figsize=(16,4))\n plt.subplot(121)\n xline = range(0, len(stats.episode_lengths), window)\n plt.plot(xline, smooth(stats.episode_lengths, window=window))\n plt.ylabel('Episode Length')\n plt.xlabel('Episode Count')\n plt.subplot(122)\n plt.plot(xline, smooth(stats.episode_rewards, window=window))\n plt.ylabel('Episode Return')\n plt.xlabel('Episode Count')\n```\n\n\n```\n# @title Helper functions\ndef smooth(x, window=10):\n return x[:window*(len(x)//window)].reshape(len(x)//window, window).mean(axis=1)\n```\n\n\n```\n# @title Set random seed\n\n# @markdown Executing `set_seed(seed=seed)` you are setting the seed\n\n# for DL its critical to set the random seed so that students can have a\n# baseline to compare their results to expected results.\n# Read more here: https://pytorch.org/docs/stable/notes/randomness.html\n\n# Call `set_seed` function in the exercises to ensure reproducibility.\nimport random\nimport torch\n\ndef set_seed(seed=None, seed_torch=True):\n if seed is None:\n seed = np.random.choice(2 ** 32)\n random.seed(seed)\n np.random.seed(seed)\n if seed_torch:\n torch.manual_seed(seed)\n torch.cuda.manual_seed_all(seed)\n torch.cuda.manual_seed(seed)\n torch.backends.cudnn.benchmark = False\n torch.backends.cudnn.deterministic = True\n\n print(f'Random seed {seed} has been set.')\n\n\n# In case that `DataLoader` is used\ndef seed_worker(worker_id):\n worker_seed = torch.initial_seed() % 2**32\n np.random.seed(worker_seed)\n random.seed(worker_seed)\n```\n\n\n```\n# @title Set device (GPU or CPU). Execute `set_device()`\n# especially if torch modules used.\n\n# inform the user if the notebook uses GPU or CPU.\n\ndef set_device():\n device = \"cuda\" if torch.cuda.is_available() else \"cpu\"\n if device != \"cuda\":\n print(\"WARNING: For this notebook to perform best, \"\n \"if possible, in the menu under `Runtime` -> \"\n \"`Change runtime type.` select `GPU` \")\n else:\n print(\"GPU is enabled in this notebook.\")\n\n return device\n```\n\n\n```\nSEED = 2021\nset_seed(seed=SEED)\nDEVICE = set_device()\n```\n\n---\n# Section 1: Introduction to Reinforcement Learning\n\n\n```\n# @title Video 1: Introduction to RL\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV18V411p7iK\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"BWz3scQN50M\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n\n## Acme: a research framework for reinforcement learning\n\n**Acme** is a library of reinforcement learning (RL) agents and agent building blocks by Google DeepMind. Acme strives to expose simple, efficient, and readable agents, that serve both as reference implementations of popular algorithms and as strong baselines, while still providing enough flexibility to do novel research. The design of Acme also attempts to provide multiple points of entry to the RL problem at differing levels of complexity.\n\nFor more information see [github repository](https://github.com/deepmind/acme).\n\n---\n# Section 2: General Formulation of RL Problems and Gridworlds\n\n\n\n```\n# @title Video 2: General Formulation and MDPs\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1k54y1E7Zn\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"h6TxAALY5Fc\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n\nThe agent interacts with the environment in a loop corresponding to the following diagram. The environment defines a set of **actions** that an agent can take. The agent takes an action informed by the **observations** it receives, and will get a **reward** from the environment after each action. The goal in RL is to find an agent whose actions maximize the total accumulation of rewards obtained from the environment. \n\n\n
                                        \n\n\n\n\n## Section 2.1: The Environment\n\n\n\nFor this practical session we will focus on a **simple grid world** environment,which consists of a 9 x 10 grid of either wall or empty cells, depicted in black and white, respectively. The smiling agent starts from an initial location and needs to navigate to reach the goal square.\n\n
                                        \n\n
                                        \n\nBelow you will find an implementation of this Gridworld as a ```dm_env.Environment```.\n\nThere is no coding in this section, but if you want, you can look over the provided code so that you can familiarize yourself with an example of how to set up a **grid world** environment.\n\n \n\n\n\n```\n# @title Implement GridWorld { form-width: \"30%\" }\n# @markdown *Double-click* to inspect the contents of this cell.\n\nclass ObservationType(enum.IntEnum):\n STATE_INDEX = enum.auto()\n AGENT_ONEHOT = enum.auto()\n GRID = enum.auto()\n AGENT_GOAL_POS = enum.auto()\n\n\nclass GridWorld(dm_env.Environment):\n\n def __init__(self,\n layout,\n start_state,\n goal_state=None,\n observation_type=ObservationType.STATE_INDEX,\n discount=0.9,\n penalty_for_walls=-5,\n reward_goal=10,\n max_episode_length=None,\n randomize_goals=False):\n \"\"\"Build a grid environment.\n\n Simple gridworld defined by a map layout, a start and a goal state.\n\n Layout should be a NxN grid, containing:\n * 0: empty\n * -1: wall\n * Any other positive value: value indicates reward; episode will terminate\n\n Args:\n layout: NxN array of numbers, indicating the layout of the environment.\n start_state: Tuple (y, x) of starting location.\n goal_state: Optional tuple (y, x) of goal location. Will be randomly\n sampled once if None.\n observation_type: Enum observation type to use. One of:\n * ObservationType.STATE_INDEX: int32 index of agent occupied tile.\n * ObservationType.AGENT_ONEHOT: NxN float32 grid, with a 1 where the\n agent is and 0 elsewhere.\n * ObservationType.GRID: NxNx3 float32 grid of feature channels.\n First channel contains walls (1 if wall, 0 otherwise), second the\n agent position (1 if agent, 0 otherwise) and third goal position\n (1 if goal, 0 otherwise)\n * ObservationType.AGENT_GOAL_POS: float32 tuple with\n (agent_y, agent_x, goal_y, goal_x)\n discount: Discounting factor included in all Timesteps.\n penalty_for_walls: Reward added when hitting a wall (should be negative).\n reward_goal: Reward added when finding the goal (should be positive).\n max_episode_length: If set, will terminate an episode after this many\n steps.\n randomize_goals: If true, randomize goal at every episode.\n \"\"\"\n if observation_type not in ObservationType:\n raise ValueError('observation_type should be a ObservationType instace.')\n self._layout = np.array(layout)\n self._start_state = start_state\n self._state = self._start_state\n self._number_of_states = np.prod(np.shape(self._layout))\n self._discount = discount\n self._penalty_for_walls = penalty_for_walls\n self._reward_goal = reward_goal\n self._observation_type = observation_type\n self._layout_dims = self._layout.shape\n self._max_episode_length = max_episode_length\n self._num_episode_steps = 0\n self._randomize_goals = randomize_goals\n if goal_state is None:\n # Randomly sample goal_state if not provided\n goal_state = self._sample_goal()\n self.goal_state = goal_state\n\n def _sample_goal(self):\n \"\"\"Randomly sample reachable non-starting state.\"\"\"\n # Sample a new goal\n n = 0\n max_tries = 1e5\n while n < max_tries:\n goal_state = tuple(np.random.randint(d) for d in self._layout_dims)\n if goal_state != self._state and self._layout[goal_state] == 0:\n # Reachable state found!\n return goal_state\n n += 1\n raise ValueError('Failed to sample a goal state.')\n\n @property\n def layout(self):\n return self._layout\n\n @property\n def number_of_states(self):\n return self._number_of_states\n\n @property\n def goal_state(self):\n return self._goal_state\n\n @property\n def start_state(self):\n return self._start_state\n\n @property\n def state(self):\n return self._state\n\n def set_state(self, x, y):\n self._state = (y, x)\n\n @goal_state.setter\n def goal_state(self, new_goal):\n if new_goal == self._state or self._layout[new_goal] < 0:\n raise ValueError('This is not a valid goal!')\n # Zero out any other goal\n self._layout[self._layout > 0] = 0\n # Setup new goal location\n self._layout[new_goal] = self._reward_goal\n self._goal_state = new_goal\n\n def observation_spec(self):\n if self._observation_type is ObservationType.AGENT_ONEHOT:\n return specs.Array(\n shape=self._layout_dims,\n dtype=np.float32,\n name='observation_agent_onehot')\n elif self._observation_type is ObservationType.GRID:\n return specs.Array(\n shape=self._layout_dims + (3,),\n dtype=np.float32,\n name='observation_grid')\n elif self._observation_type is ObservationType.AGENT_GOAL_POS:\n return specs.Array(\n shape=(4,), dtype=np.float32, name='observation_agent_goal_pos')\n elif self._observation_type is ObservationType.STATE_INDEX:\n return specs.DiscreteArray(\n self._number_of_states, dtype=int, name='observation_state_index')\n\n def action_spec(self):\n return specs.DiscreteArray(4, dtype=int, name='action')\n\n def get_obs(self):\n if self._observation_type is ObservationType.AGENT_ONEHOT:\n obs = np.zeros(self._layout.shape, dtype=np.float32)\n # Place agent\n obs[self._state] = 1\n return obs\n elif self._observation_type is ObservationType.GRID:\n obs = np.zeros(self._layout.shape + (3,), dtype=np.float32)\n obs[..., 0] = self._layout < 0\n obs[self._state[0], self._state[1], 1] = 1\n obs[self._goal_state[0], self._goal_state[1], 2] = 1\n return obs\n elif self._observation_type is ObservationType.AGENT_GOAL_POS:\n return np.array(self._state + self._goal_state, dtype=np.float32)\n elif self._observation_type is ObservationType.STATE_INDEX:\n y, x = self._state\n return y * self._layout.shape[1] + x\n\n def reset(self):\n self._state = self._start_state\n self._num_episode_steps = 0\n if self._randomize_goals:\n self.goal_state = self._sample_goal()\n return dm_env.TimeStep(\n step_type=dm_env.StepType.FIRST,\n reward=None,\n discount=None,\n observation=self.get_obs())\n\n def step(self, action):\n y, x = self._state\n\n if action == 0: # up\n new_state = (y - 1, x)\n elif action == 1: # right\n new_state = (y, x + 1)\n elif action == 2: # down\n new_state = (y + 1, x)\n elif action == 3: # left\n new_state = (y, x - 1)\n else:\n raise ValueError(\n 'Invalid action: {} is not 0, 1, 2, or 3.'.format(action))\n\n new_y, new_x = new_state\n step_type = dm_env.StepType.MID\n if self._layout[new_y, new_x] == -1: # wall\n reward = self._penalty_for_walls\n discount = self._discount\n new_state = (y, x)\n elif self._layout[new_y, new_x] == 0: # empty cell\n reward = 0.\n discount = self._discount\n else: # a goal\n reward = self._layout[new_y, new_x]\n discount = 0.\n new_state = self._start_state\n step_type = dm_env.StepType.LAST\n\n self._state = new_state\n self._num_episode_steps += 1\n if (self._max_episode_length is not None and\n self._num_episode_steps >= self._max_episode_length):\n step_type = dm_env.StepType.LAST\n return dm_env.TimeStep(\n step_type=step_type,\n reward=np.float32(reward),\n discount=discount,\n observation=self.get_obs())\n\n def plot_grid(self, add_start=True):\n plt.figure(figsize=(4, 4))\n plt.imshow(self._layout <= -1, interpolation='nearest')\n ax = plt.gca()\n ax.grid(0)\n plt.xticks([])\n plt.yticks([])\n # Add start/goal\n if add_start:\n plt.text(\n self._start_state[1],\n self._start_state[0],\n r'$\\mathbf{S}$',\n fontsize=16,\n ha='center',\n va='center')\n plt.text(\n self._goal_state[1],\n self._goal_state[0],\n r'$\\mathbf{G}$',\n fontsize=16,\n ha='center',\n va='center')\n h, w = self._layout.shape\n for y in range(h - 1):\n plt.plot([-0.5, w - 0.5], [y + 0.5, y + 0.5], '-w', lw=2)\n for x in range(w - 1):\n plt.plot([x + 0.5, x + 0.5], [-0.5, h - 0.5], '-w', lw=2)\n\n def plot_state(self, return_rgb=False):\n self.plot_grid(add_start=False)\n # Add the agent location\n plt.text(\n self._state[1],\n self._state[0],\n u'\ud83d\ude03',\n # fontname='symbola',\n fontsize=18,\n ha='center',\n va='center',\n )\n if return_rgb:\n fig = plt.gcf()\n plt.axis('tight')\n plt.subplots_adjust(0, 0, 1, 1, 0, 0)\n fig.canvas.draw()\n data = np.fromstring(fig.canvas.tostring_rgb(), dtype=np.uint8, sep='')\n w, h = fig.canvas.get_width_height()\n data = data.reshape((h, w, 3))\n plt.close(fig)\n return data\n\n def plot_policy(self, policy):\n action_names = [\n r'$\\uparrow$', r'$\\rightarrow$', r'$\\downarrow$', r'$\\leftarrow$'\n ]\n self.plot_grid()\n plt.title('Policy Visualization')\n h, w = self._layout.shape\n for y in range(h):\n for x in range(w):\n # if ((y, x) != self._start_state) and ((y, x) != self._goal_state):\n if (y, x) != self._goal_state:\n action_name = action_names[policy[y, x]]\n plt.text(x, y, action_name, ha='center', va='center')\n\n def plot_greedy_policy(self, q):\n greedy_actions = np.argmax(q, axis=2)\n self.plot_policy(greedy_actions)\n\n\ndef build_gridworld_task(task,\n discount=0.9,\n penalty_for_walls=-5,\n observation_type=ObservationType.STATE_INDEX,\n max_episode_length=200):\n \"\"\"Construct a particular Gridworld layout with start/goal states.\n\n Args:\n task: string name of the task to use. One of {'simple', 'obstacle',\n 'random_goal'}.\n discount: Discounting factor included in all Timesteps.\n penalty_for_walls: Reward added when hitting a wall (should be negative).\n observation_type: Enum observation type to use. One of:\n * ObservationType.STATE_INDEX: int32 index of agent occupied tile.\n * ObservationType.AGENT_ONEHOT: NxN float32 grid, with a 1 where the\n agent is and 0 elsewhere.\n * ObservationType.GRID: NxNx3 float32 grid of feature channels.\n First channel contains walls (1 if wall, 0 otherwise), second the\n agent position (1 if agent, 0 otherwise) and third goal position\n (1 if goal, 0 otherwise)\n * ObservationType.AGENT_GOAL_POS: float32 tuple with\n (agent_y, agent_x, goal_y, goal_x).\n max_episode_length: If set, will terminate an episode after this many\n steps.\n \"\"\"\n tasks_specifications = {\n 'simple': {\n 'layout': [\n [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1],\n [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n [-1, 0, 0, 0, -1, -1, 0, 0, 0, -1],\n [-1, 0, 0, 0, -1, -1, 0, 0, 0, -1],\n [-1, 0, 0, 0, -1, -1, 0, 0, 0, -1],\n [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1],\n ],\n 'start_state': (2, 2),\n 'goal_state': (7, 2)\n },\n 'obstacle': {\n 'layout': [\n [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1],\n [-1, 0, 0, 0, 0, 0, -1, 0, 0, -1],\n [-1, 0, 0, 0, -1, 0, 0, 0, 0, -1],\n [-1, 0, 0, 0, -1, -1, 0, 0, 0, -1],\n [-1, 0, 0, 0, -1, -1, 0, 0, 0, -1],\n [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1],\n ],\n 'start_state': (2, 2),\n 'goal_state': (2, 8)\n },\n 'random_goal': {\n 'layout': [\n [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1],\n [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n [-1, 0, 0, 0, -1, -1, 0, 0, 0, -1],\n [-1, 0, 0, 0, -1, -1, 0, 0, 0, -1],\n [-1, 0, 0, 0, -1, -1, 0, 0, 0, -1],\n [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1],\n ],\n 'start_state': (2, 2),\n # 'randomize_goals': True\n },\n }\n return GridWorld(\n discount=discount,\n penalty_for_walls=penalty_for_walls,\n observation_type=observation_type,\n max_episode_length=max_episode_length,\n **tasks_specifications[task])\n\n\ndef setup_environment(environment):\n \"\"\"Returns the environment and its spec.\"\"\"\n\n # Make sure the environment outputs single-precision floats.\n environment = wrappers.SinglePrecisionWrapper(environment)\n\n # Grab the spec of the environment.\n environment_spec = specs.make_environment_spec(environment)\n\n return environment, environment_spec\n```\n\n\nWe will use two distinct tabular GridWorlds:\n* `simple` where the goal is at the bottom left of the grid, little navigation required.\n* `obstacle` where the goal is behind an obstacle the agent must avoid.\n\nYou can visualize the grid worlds by running the cell below. \n\nNote that **S** indicates the start state and **G** indicates the goal. \n\n\n\n```\n# Visualise GridWorlds\n\n# Instantiate two tabular environments, a simple task, and one that involves\n# the avoidance of an obstacle.\nsimple_grid = build_gridworld_task(\n task='simple', observation_type=ObservationType.GRID)\nobstacle_grid = build_gridworld_task(\n task='obstacle', observation_type=ObservationType.GRID)\n\n# Plot them.\nsimple_grid.plot_grid()\nplt.title('Simple')\n\nobstacle_grid.plot_grid()\nplt.title('Obstacle')\n```\n\nIn this environment, the agent has four possible **actions**: `up`, `right`, `down`, and `left`. The **reward** is `-5` for bumping into a wall, `+10` for reaching the goal, and `0` otherwise. The episode ends when the agent reaches the goal, and otherwise continues. The **discount** on continuing steps, is $\\gamma = 0.9$. \n\nBefore we start building an agent to interact with this environment, let's first look at the types of objects the environment either returns (e.g., **observations**) or consumes (e.g., **actions**). The `environment_spec` will show you the form of the **observations**, **rewards** and **discounts** that the environment exposes and the form of the **actions** that can be taken.\n\n\n\n```\n# @title Look at environment_spec { form-width: \"30%\" }\n\n# Note: setup_environment is implemented in the same cell as GridWorld.\nenvironment, environment_spec = setup_environment(simple_grid)\n\nprint('actions:\\n', environment_spec.actions, '\\n')\nprint('observations:\\n', environment_spec.observations, '\\n')\nprint('rewards:\\n', environment_spec.rewards, '\\n')\nprint('discounts:\\n', environment_spec.discounts, '\\n')\n```\n\n\nWe first set the environment to its initial state by calling the `reset()` method which returns the first observation and resets the agent to the starting location.\n\n\n\n```\nenvironment.reset()\nenvironment.plot_state()\n```\n\nNow we want to take an action to interact with the environment. We do this by passing a valid action to the `dm_env.Environment.step()` method which returns a `dm_env.TimeStep` namedtuple with fields `(step_type, reward, discount, observation)`.\n\nLet's take an action and visualise the resulting state of the grid-world. (You'll need to rerun the cell if you pick a new action.)\n\n**Note for kaggle users:** As kaggle does not render the forms automatically the students should be careful to notice the various instructions and manually play around with the values for the variables\n\n\n```\n# @title Pick an action and see the state changing\naction = \"left\" #@param [\"up\", \"right\", \"down\", \"left\"] {type:\"string\"}\n\naction_int = {'up': 0,\n 'right': 1,\n 'down': 2,\n 'left':3 }\naction = int(action_int[action])\ntimestep = environment.step(action) # pytype: dm_env.TimeStep\nenvironment.plot_state()\n```\n\n\n```\n# @title Run loop { form-width: \"30%\" }\n# @markdown This function runs an agent in the environment for a number of\n# @markdown episodes, allowing it to learn.\n\n# @markdown *Double-click* to inspect the `run_loop` function.\n\n\ndef run_loop(environment,\n agent,\n num_episodes=None,\n num_steps=None,\n logger_time_delta=1.,\n label='training_loop',\n log_loss=False,\n ):\n \"\"\"Perform the run loop.\n\n We are following the Acme run loop.\n\n Run the environment loop for `num_episodes` episodes. Each episode is itself\n a loop which interacts first with the environment to get an observation and\n then give that observation to the agent in order to retrieve an action. Upon\n termination of an episode a new episode will be started. If the number of\n episodes is not given then this will interact with the environment\n infinitely.\n\n Args:\n environment: dm_env used to generate trajectories.\n agent: acme.Actor for selecting actions in the run loop.\n num_steps: number of steps to run the loop for. If `None` (default), runs\n without limit.\n num_episodes: number of episodes to run the loop for. If `None` (default),\n runs without limit.\n logger_time_delta: time interval (in seconds) between consecutive logging\n steps.\n label: optional label used at logging steps.\n \"\"\"\n logger = loggers.TerminalLogger(label=label, time_delta=logger_time_delta)\n iterator = range(num_episodes) if num_episodes else itertools.count()\n all_returns = []\n\n num_total_steps = 0\n for episode in iterator:\n # Reset any counts and start the environment.\n start_time = time.time()\n episode_steps = 0\n episode_return = 0\n episode_loss = 0\n\n timestep = environment.reset()\n\n # Make the first observation.\n agent.observe_first(timestep)\n\n # Run an episode.\n while not timestep.last():\n # Generate an action from the agent's policy and step the environment.\n action = agent.select_action(timestep.observation)\n timestep = environment.step(action)\n\n # Have the agent observe the timestep and let the agent update itself.\n agent.observe(action, next_timestep=timestep)\n agent.update()\n\n # Book-keeping.\n episode_steps += 1\n num_total_steps += 1\n episode_return += timestep.reward\n\n if log_loss:\n episode_loss += agent.last_loss\n\n if num_steps is not None and num_total_steps >= num_steps:\n break\n\n # Collect the results and combine with counts.\n steps_per_second = episode_steps / (time.time() - start_time)\n result = {\n 'episode': episode,\n 'episode_length': episode_steps,\n 'episode_return': episode_return,\n }\n if log_loss:\n result['loss_avg'] = episode_loss/episode_steps\n\n all_returns.append(episode_return)\n\n # Log the given results.\n logger.write(result)\n\n if num_steps is not None and num_total_steps >= num_steps:\n break\n return all_returns\n```\n\n\n```\n# @title Implement the evaluation loop { form-width: \"30%\" }\n# @markdown This function runs the agent in the environment for a number of\n# @markdown episodes, without allowing it to learn, in order to evaluate it.\n\n# @markdown *Double-click* to inspect the `evaluate` function.\n\ndef evaluate(environment: dm_env.Environment,\n agent: acme.Actor,\n evaluation_episodes: int):\n frames = []\n\n for episode in range(evaluation_episodes):\n timestep = environment.reset()\n episode_return = 0\n steps = 0\n while not timestep.last():\n frames.append(environment.plot_state(return_rgb=True))\n\n action = agent.select_action(timestep.observation)\n timestep = environment.step(action)\n steps += 1\n episode_return += timestep.reward\n print(\n f'Episode {episode} ended with reward {episode_return} in {steps} steps'\n )\n return frames\n\ndef display_video(frames: Sequence[np.ndarray],\n filename: str = 'temp.mp4',\n frame_rate: int = 12):\n \"\"\"Save and display video.\"\"\"\n # Write the frames to a video.\n with imageio.get_writer(filename, fps=frame_rate) as video:\n for frame in frames:\n video.append_data(frame)\n\n # Read video and display the video.\n video = open(filename, 'rb').read()\n b64_video = base64.b64encode(video)\n video_tag = ('
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                                        \n\n## Model selection and evidence \n\n $\\newcommand{\\thetavec}{\\boldsymbol{\\theta}}$\n Determine the evidence for different models $M_1$ and $M_2$ via marginalization by integrating over all possible sets of parameters ${\\thetavec}$ in different models, with the same data $D$ and information $I$. \n\n\nThe evidence ratio for two different models:\n$$ \n \\frac{p(M_1\\mid D, I)}{p(M_2\\mid D, I)}\n = \\frac{p(D\\mid M_1, I)\\,p(M_1,I)}{p(D\\mid M_2, I)\\,p(M_2,I)}\n$$\n\nThe Bayes Ratio (implements Occam\u2019s Razor):\n$$\n\\frac{p(D\\mid M_1, I)}{p(D\\mid M_2, I)}\n = \\frac{\\int\\!d\\thetavec_1\\, p(D\\mid\\thetavec_1,M_1,I)\n \\,p(\\thetavec_1\\mid M_1,I)}\n {\\int\\!d\\thetavec_2\\, p(D\\mid\\thetavec_2,M_2,I)\n \\,p(\\thetavec_2\\mid M_2,I)} \n$$\n\nExample: what order polynomial underlies the noisy data?\n\n\n

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                                        \n\n
                                        \n\n## Gaussian processes (GPs) \n\n### Overview of GPs\n\nGP: the natural generalization of multivariate Gaussian random variables to infinite (countably or continuous) index sets. They look like random functions, but with characteristic degrees of smoothness, correlation lengths, and range. Here are some examples with different \"kernels\" and different parameters that dictate those features (figure from J. Melendez):\n\n\n\n\n#### Explanations of GPs from the web\n\nThe following is adapted from a blog entry from Kathleen Bailey at http://katbailey.github.io/post/gaussian-processes-for-dummies/.\n\n\"Here\u2019s how Kevin Murphy explains it in the excellent textbook Machine Learning: A Probabilistic Perspective:\" \n'A GP defines a prior over functions, which can be converted into a posterior over functions once we have seen some data. Although it might seem difficult to represent a distribution over a function, it turns out that we only need to be able to define a distribution over the function\u2019s values at a finite, but arbitrary, set of points, say $x_1, \\ldots, x_N$. A GP assumes that $p(f(x_1),\\ldots,f(x_N))$ is jointly Gaussian, with some mean $\\mu(x)$ and covariance $\\Sigma(x)$ given by $\\kappa_{ij} = \\Sigma(x_i,x_j)$, where $\\kappa$ is a positive definite kernel function. The key idea is that if $x_i$ and $x_j$ are deemed by the kernel to be similar, then we expect the output of the function at those points to be similar, too.'\n\nSo it is important to stress that we are really only dealing with a discrete set of points. Thus the physicist-friendly idea of a continuum limit of masses on springs may be preferred to more abstract notions in function space.\n\nIt should also be sufficient to consider the bivariate case, because the generalization from one to two variables is really where the new feature of correlation comes in. Generalizing further really doesn't introduce anything new.\n\n### Bivariate normal case\n\n$\\newcommand{\\xvec}{\\textbf{x}}$\n$\\newcommand{\\muvec}{\\boldsymbol{\\mu}}$\nThe general multivariate Gaussian distribution is\n$$\n p(\\xvec\\mid \\muvec,\\Sigma) = \\frac{1}{\\sqrt{\\det(2\\pi\\Sigma)}} e^{-\\frac12(\\xvec-\\muvec)^{\\rm T}\\Sigma^{-1}(\\xvec-\\muvec)}\n$$\n\nFor the bivariate case we can parameterize the mean vector and covariance matrix as\n$$\n \\muvec = \\left( \\begin{array}{c}\n \\mu_x \\\\ \\mu_y\n \\end{array} \\right)\n \\;, \\qquad\n \\Sigma = \\left( \\begin{array}{cc}\n \\sigma_x^2 & \\rho\\sigma_x\\sigma_y \\\\\n \\rho\\sigma_x\\sigma_y & \\sigma_y^2\n \\end{array}\n \\right) \n$$\nThe covariance matrix must be positive definite, which implies $\\color{red}{0\\lt\\rho^2\\lt 1}$.\n\nIf take $\\mu_x = \\mu_y = 0$ and $\\sigma_x = \\sigma_y = \\sigma$ for clarity,\nso that\n$$\n \\Sigma = \\sigma^2 \\left(\\begin{array}{cc}\n 1 & \\rho \\\\\n \\rho & 1\n \\end{array}\n \\right)\n$$\nand\n$$\n p(x,y\\mid \\sigma,\\rho) = \\frac{1}{2\\pi\\sigma^2} \n \\exp\\left(-\\frac{x^2 + y^2 - 2\\rho x y }{2\\sigma^2\\sqrt{1-\\rho^2}} \n \\right)\n \\;.\n$$\nIt's clear that contours of equal probability have $x^2 + y^2 - 2\\rho xy = \\mbox{constant}$, so they are ellipses. The value of $\\rho$ determines the eccentricity of the ellipse.\nIf $\\rho=0$, $x$ and $y$ are independent (uncorrelated) and we have a circle. As $\\rho$ approaches $+1$, $x$ and $y$ are increasingly correlated (toward straight line at $45^\\circ$), while for $\\rho$ approaching $-1$ they become increasingly anti-correlated (toward straight line at $-45^\\circ$).\n\nFor reference, the Cholesky decomposition of $\\Sigma$ is\n$$\n \\Sigma = \\sigma^2\\left( \\begin{array}{cc}\n 1 & \\rho \\\\\n \\rho & 1\n \\end{array}\n \\right)\n =\n \\sigma^2\\left( \\begin{array}{cc}\n 1 & 0 \\\\\n \\rho & \\sqrt{1-\\rho^2}\n \\end{array}\n \\right) \n \\left( \\begin{array}{cc}\n 1 & \\rho \\\\\n 0 & \\sqrt{1-\\rho^2}\n \\end{array}\n \\right) \n$$\n\n\n### Example code for generating and plotting GPs\n\nThe following code is adapted from a blog post by Katherine Bailey entitled Gaussian Processes for Dummies. First we generate several instances of draws from a Gaussian process with a squared exponential kernel function, which is the covariance between $x$ and $x'$:\n$$ \\kappa_{\\rm SE}(x,x') = \\sigma^2 e^{-(x-x')^2/2l^2} $$\n\n\n\nSo we can see that $\\sigma$ controls the vertical extent of the functions while $l$ controls\nhow rapidly they wiggle. Comparing to our expression above for the bivariate normal case,\nwe see that $\\rho$ is given by $e^{-(x-x')^2/2l^2}$. So when $x$ and $x'$ are close, \n$\\rho \\approx 1$ and the value of the function is highly correlated. When $x$ and $x'$ are far apart, $\\rho \\rightarrow 0$, and they become independent (thus $l$ plays the role of a correlation length).\n\nLet's generate some GPs with this kernel! For the function $f(x)$ we write draws as\n$$\n f(x) \\sim \\mathcal{GP[\\mu(x),\\kappa(x,x')]}\n$$\nwhere $\\mu(x)$ is the mean at each $x$ and $\\kappa(x,x')$ is the covariance between $x$ and $x'$. In practice we have a finite set of $N$ points $\\textbf{x} = \\{x_i\\}_{i=1}^{N}$ with corresponding function values $\\textbf{f}=\\{f(x_i)\\}_{i=1}^{N}$.\nWe form the mean vector $\\boldsymbol{\\mu} = m(\\textbf{x})$ and the covariance matrix $K_{ij} = \\kappa(x_i,x_j)$. Then\n$$ \\textbf{f} \\mid \\textbf{x} \\sim \\mathcal{N}(\\boldsymbol{\\mu},K)\n$$\nare draws from a multivariate normal distribution. Try it:\n\n\n```python\n%matplotlib inline\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# Define the squared exponential kernel function for the covariance\n# We take the variance to be 1.\ndef sqr_exp_kernel(a, b, length_param):\n sqdist = np.sum(a**2,1).reshape(-1,1) + np.sum(b**2,1) - 2*np.dot(a, b.T)\n return np.exp(-sqdist / (2*length_param**2))\n\n# Grid of x points\nnpts = 500\nx_min = -5\nx_max = +5\nXtest = np.linspace(x_min, x_max, npts).reshape(-1,1) \n\nlength_param = 1. # this is \"l\" (correlation length)\nK_ss = sqr_exp_kernel(Xtest, Xtest, length_param)\n\n# Get Cholesky decomposition (square root) of the covariance matrix\nnugget = 1e-12 # size of nugget will depend on how many points are used \n # 1e-12 for 500; 1e-13 for 100; 1e-15 for 50\nL = np.linalg.cholesky(K_ss + nugget*np.eye(npts))\n\n# Sample 3 sets of standard normals for our test points,\n# multiply them by the square root of the covariance matrix\n# Note: mean mu = 0 here implicitly.\nf_prior = np.dot(L, np.random.normal(size=(npts,3)))\n\n# Now let's plot the 3 sampled functions.\nplt.plot(Xtest, f_prior)\nplt.axis([-5, 5, -3, 3])\nplt.title('Three samples from the GP prior with l = {:1.1f}'.format(length_param))\nplt.show()\n```\n\nNow we train it on some data (see references for details):\n\n\n```python\n# Noiseless training data\nXtrain = np.array([-4, -3, -2, -1, 1]).reshape(5,1)\nytrain = np.sin(Xtrain)\n\n# Apply the same kernel function to our training points\nnugget_train = 5e-5\nK = sqr_exp_kernel(Xtrain, Xtrain, length_param)\nL = np.linalg.cholesky(K + nugget_train*np.eye(len(Xtrain)))\n\n# Compute the mean at our test points.\nK_s = sqr_exp_kernel(Xtrain, Xtest, length_param)\nLk = np.linalg.solve(L, K_s)\nmu = np.dot(Lk.T, np.linalg.solve(L, ytrain)).reshape((npts,))\n\n# Compute the standard deviation so we can plot it\ns2 = np.diag(K_ss) - np.sum(Lk**2, axis=0)\nstdv = np.sqrt(s2)\n\n# Draw samples from the posterior at our test points.\nnugget_test = 1e-6\nL = np.linalg.cholesky(K_ss + nugget_test*np.eye(npts) - np.dot(Lk.T, Lk))\nf_post = mu.reshape(-1,1) + np.dot(L, np.random.normal(size=(npts,3)))\n\nplt.plot(Xtrain, ytrain, 'bs', ms=8)\nplt.plot(Xtest, f_post)\nplt.gca().fill_between(Xtest.flat, mu-2*stdv, mu+2*stdv, color=\"#dddddd\")\nplt.plot(Xtest, mu, 'r--', lw=2)\nplt.axis([-5, 5, -3, 3])\nplt.title('Three samples from the GP posterior')\nplt.show()\n```\n\n### Other demos for Gaussian Processes (and other regression)\n\n
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                                        • \nGaussian process regression, where you can add data points, play with the hyperparameters, and then see the inference for the curve. It\u2019s by Tomi Peltola:\nhttp://www.tmpl.fi/gp/ \n\n
                                        • \nThis simulation shows how a GP prior is a distribution over functions, and how observing data conditions the prior to obtain the GP posterior.\nhttp://rpradeep.webhop.net/gpr/\n\n
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                                        \n\n## Appendices\n\n### References\n\nPlease suggest additional references (with links).\n\n\n\n### Physics-oriented pedagogical articles and texts\n\n \n\n### Standard statistics references\n\n\n\n### BUQEYE references\n\n \n\n\n\n### Github repositories\n\nPlease suggest more!\n\n
                                          \n
                                        • https://github.com/jakevdp/BayesianAstronomy Materials for the Bayesian Methods in Astronomy workshop at the 227th American Astronomical Society meeting. Includes Jupyter notebooks and useful exercises. \n \n
                                        • http://people.duke.edu/~ccc14/sta-663-2018/ STA 663: Computational Statistics and Statistical Computing (2018) at Duke University. Lots of good things here! \n
                                        \n\n### Vocabulary\n\nPlan: build up a good set of definitions with appropriate links. Please add more words/phrases!\n\n
                                        \n
                                        conjugate prior
                                        \n
                                        If the probability distribution family (e.g., beta distributions) for the posterior pdf is the same as for the prior pdf, the latter is said to be a conjugate prior. This means that the updating by Bayes' rule can be carried out analytically. Some Bayesian practitioners are strongly opposed to the use of conjugate priors (see comments here).
                                        \n\n\n\n
                                        credible vs. confidence interval
                                        \n
                                        This is a contrast between Bayesian and frequentist statistics. For a frequentist, a parameter has a true value, which is fixed and not a distribution. A 95% confidence interval mean that with a large number of repeated trials, 95% of the calculated confidence intervals would include the true value. This is clearly hard to think about! A Bayesian 95% credible interval is the range of the posterior for the parameter (which is treated as a random variable) that has 95% of the probability. So there is a 95% probability that the parameter is in that interval. \n
                                        \n\n
                                        evidence
                                        \n
                                        In the standard context of inferring parameters $\\boldsymbol{\\theta}$ given data $D$ and information $I$, the evidence is $p(D\\mid I) = \\int\\! d\\boldsymbol{\\theta}\\, p(D \\mid \\boldsymbol{\\theta},I)\\,p(\\boldsymbol{\\theta},I)$. This is also called the Fully Marginalized Likelihood or FML. The expression shows that it is the integral over all $\\boldsymbol{\\theta}$ weighted by the likelihood. This is typically an expensive integral to do. In the context of model fitting (i.e., parameter estimation), it acts as a normalization constant and in most cases can be ignored because the normalization can be found directly (or only relative probabilities are needed).
                                        \n\n
                                        gaussian process
                                        \n
                                        \nFrom [Wikipedia](https://en.wikipedia.org/wiki/Gaussian_process): \"In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space.\"\n
                                        \n\n
                                        hierarchical model
                                        \n
                                        A model with hyperparameters. See [Wikipedia](https://en.wikipedia.org/wiki/Bayesian_hierarchical_modeling).
                                        \n\n
                                        hyperparameter
                                        \n
                                        A parameter of a prior distribution.
                                        \n\n
                                        iid (independently and identically distributed)
                                        \n
                                        A set of random variables is iid (or i.i.d. or IID) if each random variable has the same probability distribution and all are mutually independent. \n
                                        \n\n
                                        likelihood
                                        \n
                                        Usually in the form $p(D\\mid \\boldsymbol{\\theta},I)$, where $\\boldsymbol{\\theta}$ are the parameters of our model, $D$ is the data, and $I$ is any other information we use. This is the probability of observing our actual data given the model (with the particular parameters $\\boldsymbol{\\theta}$). It is the same quantity that is maximized in frequentist maximum-likelihood approaches.
                                        \n\n
                                        MAP estimate
                                        \n
                                        Maximum a posteriori estimate. This is the mode (maximum) of the posterior distribution for the quantity of interest. If the prior is uniform, the MAP estimate equals the maximum likelihood estimate.
                                        \n\n
                                        maximum entropy
                                        \n
                                        A method used to determine priors.
                                        \n\n
                                        MCMC
                                        \n
                                        Markov-chain Monte Carlo. A generic name for stochastic sampling methods.
                                        \n\n
                                        model selection and model averaging
                                        \n
                                        \n\n
                                        nugget
                                        \n
                                        \nFor Gaussian process (GP) calculations or any sampling of a multivariate normal distribution, one typically needs to find the Cholesky decomposition of the covariance matrix. However, this matrix can become ill-conditioned (too-small or negative eigenvalues). A standard solution is to add a small number, called a nugget, to the diagonal of the covariance matrix. For GP regression, this is equivalent to adding (or increasing, if already present) the data noise.\n
                                        \n\n
                                        nuisance parameter
                                        \n
                                        A nuisance parameter is a parameter in your model whose value you don't care about for the posterior. So you integrate it out (marginalize).
                                        \n \n
                                        overfitting and underfitting
                                        \n
                                        This example from http://scikit-learn.org/stable/auto_examples/model_selection/plot_underfitting_overfitting.html of fitting polynomials to nonlinear functions illustrates overfitting and underfitting. The true function is a cosine with noise added. A polynomial of degree 1 is an inadequate model for this data; this is underfitting. The polynomial of degree 15 tries to fit the noise; this is overfitting. \n\n \n
                                        \n\n
                                        point estimate (cf. interval estimate)
                                        \n
                                        A point estimate is a single value to characterize a\nposterior. It could be the mode, mean, median or something\nelse. An interval estimate is more natural in Bayesian statistics, because the full posterior is the real target. Giving a series of credible intervals often conveys much of the information about the posterior.
                                        \n\n
                                        posterior
                                        \n
                                        This is the quantity of the left side of Bayes' rule, the thing we want to compute. Often in the form $p(\\boldsymbol{\\theta}\\mid D,I)$, where $\\boldsymbol{\\theta}$ are the parameters of our model, $D$ is the data, and $I$ is any other information we use. It is our knowledge of the model given the data and any relevant background knowledge (which include the choice of model).
                                        \n\n
                                        prior
                                        \n
                                        A pdf that encodes what is known about the answer (e.g., parameters) before any data is used. The notation consistent with our definitions of posterior and likelihood is $p(\\boldsymbol{\\theta}\\mid I)$, where $\\boldsymbol{\\theta}$ are the parameters of our model and $I$ is any other information we use (e.g., some of the parameters must be positive or less than a known magnitude because of physics reasons).\n See also conjugate prior and maximum entropy.\n
                                        \n\n
                                        residual
                                        \n
                                        The difference of theory prediction and experimental data.\n
                                        \n\n
                                        \n
                                        \n\n \n
                                        \n\n### Notation   [still coming . . .]\n\n\n\nPlan: build up a dictionary of notation with appropriate links and examples (with code).\n\nunivariate normal distribution\n$$\\mathcal{N}(\\mu,\\sigma^2)$$\n\n\n\n
                                        \n\n

                                        [Return to Contents]

                                        \n\n
                                        \n\n\n```python\n\n```\n", "meta": {"hexsha": "44ff2b80c94bf5c4d6a2759adbc142a016b77223", "size": 746684, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Bayesian_Statistics_for_Physicists/.ipynb_checkpoints/Bayesian_statistics_for_physicists_v3-checkpoint.ipynb", "max_stars_repo_name": "furnstahl/Bayes_for_physicists", "max_stars_repo_head_hexsha": "74e0cb4d706c44d1eaf546c2c5601d30a84554a1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2018-10-09T07:19:13.000Z", "max_stars_repo_stars_event_max_datetime": "2021-02-02T17:58:35.000Z", "max_issues_repo_path": "Bayesian_Statistics_for_Physicists/.ipynb_checkpoints/Bayesian_statistics_for_physicists_v3-checkpoint.ipynb", "max_issues_repo_name": "furnstahl/Bayes_for_physicists", "max_issues_repo_head_hexsha": "74e0cb4d706c44d1eaf546c2c5601d30a84554a1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Bayesian_Statistics_for_Physicists/.ipynb_checkpoints/Bayesian_statistics_for_physicists_v3-checkpoint.ipynb", "max_forks_repo_name": "furnstahl/Bayes_for_physicists", "max_forks_repo_head_hexsha": "74e0cb4d706c44d1eaf546c2c5601d30a84554a1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2018-10-09T13:59:24.000Z", "max_forks_repo_forks_event_max_datetime": "2019-12-08T09:13:13.000Z", "avg_line_length": 311.5077179808, "max_line_length": 205732, "alphanum_fraction": 0.911610534, "converted": true, "num_tokens": 20124, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.7279754430043072, "lm_q1q2_score": 0.4286968258511339}} {"text": "```python\n\"\"\"Header cell, contains modules and functions to make the whole notebook experience better\"\"\"\n%matplotlib inline \n# plots graphs within the notebook\n\nfrom IPython.display import display,Image, Latex\nfrom sympy.interactive import printing\nprinting.init_printing(use_latex='mathjax')\nfrom IPython.display import clear_output\n\nimport time\n\nfrom IPython.display import display,Image, Latex\n\nfrom IPython.display import clear_output\n\n\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport math\nimport scipy.constants as sc\n\n\nimport sympy as sym\n\n \nfont = {'family' : 'serif',\n #'color' : 'black',\n 'weight' : 'normal',\n 'size' : 12,\n }\nfontlabel = {'family' : 'serif',\n #'color' : 'black',\n 'weight' : 'normal',\n 'size' : 12,\n }\n\nfrom matplotlib.ticker import FormatStrFormatter\nplt.rc('font', **font)\n\nclass PDF(object):\n def __init__(self, pdf, size=(200,200)):\n self.pdf = pdf\n self.size = size\n\n def _repr_html_(self):\n return ''.format(self.pdf, self.size)\n\n def _repr_latex_(self):\n return r'\\includegraphics[width=1.0\\textwidth]{{{0}}}'.format(self.pdf)\n\nclass ListTable(list):\n \"\"\" Overridden list class which takes a 2-dimensional list of \n the form [[1,2,3],[4,5,6]], and renders an HTML Table in \n IPython Notebook. \"\"\"\n \n def _repr_html_(self):\n html = [\"\"]\n for row in self:\n html.append(\"\")\n \n for col in row:\n html.append(\"\".format(col))\n \n html.append(\"\")\n html.append(\"
                                        {0}
                                        \")\n return ''.join(html)\n \n```\n\n\n```python\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport math\nimport scipy.constants as sc\n\n\nimport sympy as sym\n\n\nfrom math import sin, cos\n\nclass Robot(object):\n \"\"\"Defines basic mobile robot properties\"\"\"\n def __init__(self):\n self.pos_x = 0.0\n self.pos_y = 0.0\n self.angle = 0.0\n self.plot = False\n self._delta = 0.01\n self.step_plot = int(5)\n self.mag_plot = 1.\n\n # Movement\n def step(self):\n \"\"\" updates the x,y and angle \"\"\"\n self.deltax()\n self.deltay()\n self.deltaa()\n\n def move(self, seconds):\n \"\"\" Moves the robot for an 's' amount of seconds\"\"\"\n for i in range(int(seconds/self._delta)):\n self.step()\n if i % self.step_plot == 0 and self.plot: # plot path every 3 steps\n self.plot_xya()\n\n # Printing-and-plotting:\n def print_xya(self):\n \"\"\" prints the x,y position and angle \"\"\"\n print (\"x = \" + str(self.pos_x) +\" \"+ \"y = \" + str(self.pos_y))\n print (\"a = \" + str(self.angle))\n\n def plot_robot(self):\n \"\"\" plots a representation of the robot \"\"\"\n plt.arrow(self.pos_x, self.pos_y, 0.001\n * cos(self.angle), 0.001 * sin(self.angle),\n head_width=self.mag_plot*self.length, head_length=self.mag_plot*self.length,\n fc='k', ec='k')\n\n def plot_xya(self):\n \"\"\" plots a dot in the position of the robot \"\"\"\n plt.scatter(self.pos_x, self.pos_y, c='r', edgecolors='r')\n\n\nclass DDRobot(Robot):\n \"\"\"Defines a differential drive robot\"\"\"\n\n def __init__(self):\n Robot.__init__(self)\n self.radius = 0.1\n self.length = 0.4\n\n self.rt_spd_left = 0.0\n self.rt_spd_right = 0.0\n\n def deltax(self):\n \"\"\" update x depending on l and r angular speeds \"\"\"\n self.pos_x += self._delta * (self.radius*0.5) \\\n * (self.rt_spd_right + self.rt_spd_left)*cos(self.angle)\n\n def deltay(self):\n \"\"\" update y depending on l and r angular speeds \"\"\"\n self.pos_y += self._delta * (self.radius*0.5) \\\n * (self.rt_spd_right + self.rt_spd_left)*sin(self.angle)\n\n def deltaa(self):\n \"\"\" update z depending on l and r angular speeds \"\"\"\n self.angle += self._delta * (self.radius/self.length) \\\n * (self.rt_spd_right - self.rt_spd_left)\n# function to convert degrees to radians\ndef D2R(a):\n return math.pi*a/180\ndef R2D(a):\n return 180*a/math.pi\n# mybot = DDRobot() # robot called 'enesbot'\n\n# mybot.angle = 3.1416/4 # 45 degrees\n# mybot.plot = True # plot the robot!\n# mybot.plot_robot()\n\n# mybot.rt_spd_left = 10\n# mybot.rt_spd_right = 10 # straight line\n# mybot.move(2) # move for 2 seconds\n\n# mybot.rt_spd_left = 12.5664\n# mybot.rt_spd_right = 18.8496 # (2m diameter circle)\n# mybot.move(1) # move for 1 second\n\n# mybot.rt_spd_left = 18.8496\n# mybot.rt_spd_right = 12.5664 # (2m diameter circle)\n# mybot.move(2.5) # move for 2.5 second\n\n# mybot.rt_spd_left = 12.5664\n# mybot.rt_spd_right = 18.8496 # (2m diameter circle)\n# mybot.move(3.5) # move for 2.5 second\n\n# mybot.plot_robot()\n\n# plt.xlim([-1, 6]) # axis limits\n# plt.ylim([-1, 6])\n\n# plt.show()\n```\n\n\n```python\nPDF('bot-sketch.pdf',size = (550,400))\n```\n\n\n\n\n\n\n\n\n# Differential Drive Robot - 2: Calibration\n\nAlphabot 2 is a differential drive robot: Two independent motors drive the left and right wheels and the differential speed between the two wheels control the speed and direction of the robot. The following code is copied from http://enesbot.me/kinematic-model-of-a-differential-drive-robot.html\n\n## Objective\nThe objective is to investigate methods of calibration of the robot. \n\n## Quick Theoretical Background\n\nThe robot parameters are:\n
                                          \n
                                        • Wheel radius: $r$.
                                          • \n
                                        • Length between wheels: $L$
                                          • \n
                                        • Angular velocity of the left and right wheels: $\\omega_L$, $\\omega_R$, respectively.
                                          • \n
                                        • Angle from horizontal: $\\alpha$.
                                          • \n
                                        • Position vector of the robot: $(x,y)$.
                                          • \n
                                        • Velocity vector of the robot: $V=(\\dot{x},\\dot{y})$.
                                          • \n
                                        \nHereafter $\\dot{a}$ refers to the time derivative of variable $a$:\n\n$$\n\\dot{a}=\\frac{da}{dt}\n$$\n\nThe velocities of the wheels are therefore defined as:\n\n$$\nV_L=\\omega_Lr\\text{ and }V_R=\\omega_Rr.\n$$\n\nThe velocity of the robot, taken at the center of the wheels, is simply:\n\n$$\n\\vec{V}=\\frac{V_R+V_L}{2}(\\cos(\\alpha)\\vec{e}_x+\\sin(\\alpha\\vec{e}_y),\n$$\n\nyielding the following equations of motions:\n\n$$\n\\dot{x}=r\\frac{\\omega_R+\\omega_L}{2}\\cos{\\alpha}\n$$\n\n$$\n\\dot{y}=r\\frac{\\omega_R+\\omega_L}{2}\\sin{\\alpha}\n$$\n\nand\n\n$$\n\\dot{\\alpha}=\\frac{r}{L}(\\omega_R-\\omega_L)\n$$\n\nIn the previous assignment, an equation was derived to determine the time $T$ needed to turn the robot an angle $\\alpha_o$ at given rotation speed $\\omega_L$ and $\\omega_R$ and robot dimensions $r$ and $L$. This equation started by intergrating the robot angle equation above:\n\n$$\n\\dot{\\alpha}=\\frac{d\\alpha}{dt}=\\frac{r}{L}(\\omega_R-\\omega_L)\n$$\n\nfrom $0$ to a time $T$ yields\n\n$$\n\\int_0^T\\frac{d\\alpha}{dt}dt=\\int_0^T\\frac{r}{L}(\\omega_R-\\omega_L)dt\n$$\n\nSince the left hand side function is constant with respect to time, the result is simply:\n\n$$\n\\alpha(T) - \\alpha(0) =Tr\\frac{\\omega_R-\\omega_L}{L}\n$$\n\nTherefore the time to execute a rotation of angle $\\alpha_o=\\alpha(T) - \\alpha(0)$ is\n\n$$\nT = \\frac{\\alpha_oL}{r(\\omega_R-\\omega_L)}.\\;\\;\\;\\mathrm{Eq. 1}\n$$\n\n\n## Position of the Problem\nIn a perfect world if your code sets the same rotation speed for both wheels, the robot travels on a straight line. In the present case of independent motors, this outcome is very unlikely. Actually one should not expect the robot to drive straight. Below is a model of drift. Here the drift has two components: a constant drift (it could be a signal offset issue), and complex mechanical friction modeled by a quadratic function of the rotation speed. You will assume that you don't know the equation of the drift.\n\n## Goal\nYou will derive a calibration curve using polynomial fit and test your calibration on a specific motion.\n\n\n## Example of drift\n\nBelow is a code showing the trajectory of a robot for a series of 5 runs at different rotation speeds $\\omega=1, 2.5, 5, 7.5, 10$. The robot starts from the same initial conditions, $x=0,y=0,\\alpha=0$, has no drift (perfect robot) and moves for 10 s for all runs.\n\n\n```python\nT = 10\nomega = np.array([1, 2.5, 5, 7.5, 10])\nend_angle = np.zeros(5)\nomega_R = omega \nomega_L = omega\nmybot = DDRobot()\nmybot.mag_plot = 2 #coefficient of magnification of the arrow\nmybot.step_plot = 100 # plot location every 100 iterations\n\nfor i in range(5):\n mybot.pos_x = 0 #x\n mybot.pos_y = 0 #y\n mybot.angle = D2R(0) #alpha\n mybot.length =0.4 #L\n mybot.radius = 0.1 #r\n mybot.rt_spd_left = omega_L[i]\n mybot.rt_spd_right = omega_R[i]\n\n\n mybot.plot = True #True if you want to plot the robot's trajectory\n mybot.plot_robot() #draw an arrow for the location of the robot at t=0\n\n mybot.move(T) #move the robot for 10s\n mybot.plot_robot() #draw the new location\n end_angle[i] = mybot.angle\n\nplt.xlim(-5,11)\nplt.ylim(-5,11)\nplt.xlabel(r\"$x$\")\nplt.ylabel(r\"$y$\")\nplt.show()\n```\n\nNow we introduce a constant random drift $\\delta$ and see its effect on the different trajectories. The function numpy.random.rand returns a random number between 0 and 1.\n\n\n```python\ndelta = np.random.rand(1)\nT = 10\nomega = np.array([1, 2.5, 5, 7.5, 10])\nend_angle = np.zeros(5)\nomega_R = omega + delta[0]\nomega_L = omega\nmybot = DDRobot()\nmybot.mag_plot = 2 #coefficient of magnification of the arrow\nmybot.step_plot = 100 # plot location every 100 iterations\n\nfor i in range(5):\n mybot.pos_x = 0 #x\n mybot.pos_y = 0 #y\n mybot.angle = D2R(0) #alpha\n mybot.length =0.4 #L\n mybot.radius = 0.1 #r\n mybot.rt_spd_left = omega_L[i]\n mybot.rt_spd_right = omega_R[i]\n\n\n mybot.plot = True #True if you want to plot the robot's trajectory\n mybot.plot_robot() #draw an arrow for the location of the robot at t=0\n\n mybot.move(T) #move the robot for 10s\n mybot.plot_robot() #draw the new location\n end_angle[i] = mybot.angle\n\nplt.xlim(-5,11)\nplt.ylim(-5,11)\nplt.xlabel(r\"$x$\")\nplt.ylabel(r\"$y$\")\nplt.show()\n```\n\nThe plot below shows the angle at the end of each run. Based on Eq. 1 above, does this plot make sense?\n\n\n```python\nplt.plot(omega,R2D(end_angle),'o-',label=\"experiment\")\nplt.xlabel(r\"$\\omega$\")\nplt.ylabel(r\"$\\alpha\\;(^\\circ)$ \")\nplt.legend(loc=3, bbox_to_anchor=[0, 1.], ncol=2, shadow=False, fancybox=True)\nplt.show()\n```\n\nThe correction is straightforward. For each run, the rotation speed of the right wheel must be corrected by $-\\delta$.\n\n\n```python\n\nT = 10\nomega = np.array([1, 2.5, 5, 7.5, 10])\nend_angle = np.zeros(5)\nmybot = DDRobot()\nmybot.mag_plot = 2 #coefficient of magnification of the arrow\nmybot.step_plot = 100 # plot location every 100 iterations\n\nfor i in range(5):\n mybot.pos_x = 0 #x\n mybot.pos_y = 0 #y\n mybot.angle = D2R(0) #alpha\n mybot.rt_spd_left = omega_L[i]\n mybot.rt_spd_right = omega_R[i] - delta[0]\n\n\n mybot.plot = True #True if you want to plot the robot's trajectory\n mybot.plot_robot() #draw an arrow for the location of the robot at t=0\n\n mybot.move(T) #move the robot for 10s\n mybot.plot_robot() #draw the new location\n end_angle[i] = mybot.angle\n\nplt.xlim(-5,11)\nplt.ylim(-5,11)\nplt.xlabel(r\"$x$\")\nplt.ylabel(r\"$y$\")\nplt.show()\n```\n\n## Proposed calibration\n\nYou will assume that the drift $\\delta$ is caused by the right motor and additive:\n\n$$\n\\omega_L = \\omega,\\;\\omega_R = \\omega+\\delta(\\omega).\n$$\n\nIf $\\delta(\\omega)$ is positive, the robot, when programmed to go straight, the robot will actually turn left.\n\nThe proposed calibration consists of programming the robot for a series of straight runs for various $\\omega$ and constant $T$, $r$ and $L$. Once the robot stops, the deviation from the straight motion is measured by the angle of the robot at time $T$. This angle is a function of $\\omega$, $\\alpha(\\omega)$ if the drift is a function of the rotation speed. Use Eq. 1 above to determine the drift $\\delta(\\omega)$ as a function of $\\alpha(\\omega)$, $L$, $T$ and $r$.\n\n\n```python\n\"\"\" random quadratic drift equation. Note that every time \nyou run this cell, the trajectories will change\"\"\"\na = np.random.random(2)# random drift amplitude\nprint(a)\ndef drift(omega):\n f = a[0] + a[1] * (omega/5)**2\n return f\n```\n\n [0.03285105 0.15091591]\n\n\n### Uncalibrated robot\n\n\n```python\nT = 10\nomega = np.array([1, 2.5, 5, 7.5, 10])\nend_angle = np.zeros(5)\nomega_R = omega + drift(omega)\nomega_L = omega\nmybot = DDRobot()\nmybot.mag_plot = 2 #coefficient of magnification of the arrow\nmybot.step_plot = 100 # plot location every 100 iterations\n\nfor i in range(5):\n mybot.pos_x = 0 #x\n mybot.pos_y = 0 #y\n mybot.angle = D2R(0) #alpha\n mybot.length =0.4 #L\n mybot.radius = 0.1 #r\n mybot.rt_spd_left = omega_L[i]\n mybot.rt_spd_right = omega_R[i]\n\n\n mybot.plot = True #True if you want to plot the robot's trajectory\n mybot.plot_robot() #draw an arrow for the location of the robot at t=0\n\n mybot.move(T) #move the robot for 10s\n mybot.plot_robot() #draw the new location\n end_angle[i] = mybot.angle\n\nplt.xlim(-5,11)\nplt.ylim(-5,11)\nplt.xlabel(r\"$x$\")\nplt.ylabel(r\"$y$\")\nplt.show()\n```\n\nNote in the code above, the angle at the end of the motion is stored in the array end_angle. As shown below, the drift appears to follow a parabola.\n\n\n```python\nplt.plot(omega,R2D(end_angle),'o',label=\"experiment\")\nplt.xlabel(r\"$\\omega$\")\nplt.ylabel(r\"$\\alpha$\")\nplt.legend(loc=3, bbox_to_anchor=[0, 1.], ncol=2, shadow=False, fancybox=True)\nplt.show()\n```\n\nLet's now fit a quadratic polynomial to these points. The creation of a polynomial in python is shown below using numpy polyfit function. p2_coef are the coefficients $p_i$ of the polynomial $p(x)=p_0x^2+p_1x+p_2$ (see https://docs.scipy.org/doc/numpy-1.14.0/reference/generated/numpy.polyfit.html).\n\n\n```python\np2_coef = np.polyfit(omega,end_angle,2)\nprint(p2_coef)\n```\n\n [ 1.50915907e-02 -4.78006585e-15 8.21276361e-02]\n\n\nTo generate a function, use the numpy function poly1d:\n\n\n```python\np2 = np.poly1d(p2_coef)\np2 = np.poly1d(np.polyfit(omega,end_angle,2)) #equivalent compact form\np2\n```\n\n\n\n\n poly1d([ 1.50915907e-02, -4.78006585e-15, 8.21276361e-02])\n\n\n\nYou can now generate a fit over a range of your choice, here $0\\leq\\omega\\leq10$\n\n\n```python\nN = 51\nom = np.linspace(0,10,N)\n\nplt.plot(omega,R2D(end_angle),'o',label=\"experiment\")\nplt.plot(om,R2D(p2(om)),label='fit')\nplt.xlabel(r\"$\\omega$\")\nplt.ylabel(r\"$\\alpha$\")\nplt.legend(loc=3, bbox_to_anchor=[0, 1.], ncol=2, shadow=False, fancybox=True)\nplt.show()\n```\n\n\nIn Eq. 1, introduce the drift at $\\delta$, i.e. $\\omega_L=\\omega$ and $\\omega_R=\\omega+\\delta$, and derive a correction for the drift at as a function of $\\alpha(\\omega)$, which here is the array end_angle or the polynomial p2. Using the knowledge of the drift for all 5 rotation speed in the array omega, create a polynomial to compute the drift for $\\omega\\in[0,10]$ and introduce this polynomial into the code below to correct for the drift.\n\n\n\n\n```python\ndelta = mybot.length/(mybot.radius*T)*end_angle\nprint(end_angle)\nprint(mybot.length/(mybot.radius*T))\nprint(delta)\nprint(R2D(p2(2)))\n```\n\n [ 0.34574835 0.9517579 4.88811036 5.34589337 7.02843155]\n [ 2. 1.33333333 1. 2. 1.33333333]\n [ 0.69149669 1.26901054 4.88811036 10.69178675 9.37124206]\n 107.370677846\n\n\n\n```python\nomega = np.array([1, 2.5, 5, 7.5, 10])\nend_angle = np.zeros(5)\nomega_L = omega \nomega_R = omega + drift(omega) # DO NOT MODIFY\nmybot = DDRobot()\nmybot.mag_plot = 2 #coefficient of magnification of the arrow\nmybot.step_plot = 100 # plot location every 100 iterations\n\nfor i in range(5):\n mybot.pos_x = 0 #x\n mybot.pos_y = 0 #y\n mybot.angle = D2R(0) #alpha\n mybot.length =0.4 #L\n mybot.radius = 0.1 #r\n mybot.rt_spd_left = omega_L[i]\n mybot.rt_spd_right = omega_R[i] - delta[i] #write your correction here\n\n\n mybot.plot = True #True if you want to plot the robot's trajectory\n mybot.plot_robot() #draw an arrow for the location of the robot at t=0\n\n mybot.move(10) #move the robot for 10s\n mybot.plot_robot() #draw the new location\n end_angle[i] = mybot.angle\n\nplt.xlim(-1,11)\nplt.ylim(-1,11)\nplt.xlabel(r\"$x$\")\nplt.ylabel(r\"$y$\")\nplt.show()\n```\n\nNow apply your correction to the following example. In this example the robot experiences 5 step motions of different durations and speed. The robot is supposed to move in a straight line. Print the angle of the robot at the end of each step. Does your correction work? What would it take the correction to work for positive AND negative rotation speed?\n\nWrite your answers here.\n\n\n```python\nomega = np.array([1.5, 2.4, 10, 3 , 7])\nT = np.array([2, 3, 4, 2, 3])\nend_angle = np.zeros(5)\nomega_L = omega \nomega_R = omega + drift(omega)\nmybot = DDRobot()\nmybot.mag_plot = 2 #coefficient of magnification of the arrow\nmybot.step_plot = 100 # plot location every 100 iterations\nmybot.pos_x = 0 #x\nmybot.pos_y = 0 #y\nmybot.angle = D2R(0) #alpha\nmybot.length =0.4 #L\nmybot.radius = 0.1 #r\nfor i in range(5):\n \n mybot.rt_spd_left = omega_L[i]\n mybot.rt_spd_right = omega_R[i] # write your correction here\n\n mybot.plot = True #True if you want to plot the robot's trajectory\n mybot.plot_robot() #draw an arrow for the location of the robot at t=0\n\n mybot.move(T[i]) #move the robot for 10s\n mybot.plot_robot() #draw the new location\n end_angle[i] = mybot.angle\n\nplt.xlim(-5,11)\nplt.ylim(-5,11)\nplt.xlabel(r\"$x$\")\nplt.ylabel(r\"$y$\")\nplt.show()\n```\n\n\n```python\nprint(R2D(end_angle))\n```\n\nCreate a complex motion (with rotations and speed variations). Show the result with and without correction. At minimum the motion should include one rotation and two different wheel rotation speed. See example below \n\n\n```python\nNmoves = 2 # make sure to increase this number if you increase the number of moves\nomega = np.array([6, 9])\nT = np.array([12,8])\nrotation_angle = np.array([90, 90])\nrotation_angle = D2R(rotation_angle)\nend_angle = np.zeros(Nmoves) \nomega_L = omega \nomega_R = omega + drift(omega)\nt_rotation = rotation_angle*mybot.length/(mybot.radius*(omega_R+omega_L))\nmybot = DDRobot()\nmybot.mag_plot = 2 #coefficient of magnification of the arrow\nmybot.step_plot = 100 # plot location every 100 iterations\nmybot.pos_x = 0 #x\nmybot.pos_y = 0 #y\nmybot.angle = D2R(0) #alpha\nmybot.length =0.4 #L\nmybot.radius = 0.1 #r\nmybot.plot = True #True if you want to plot the robot's trajectory\nmybot.plot_robot() #draw an arrow for the location of the robot at t=0\nfor i in range(Nmoves):\n \n mybot.rt_spd_left = omega_L[i]\n mybot.rt_spd_right = omega_R[i] # write your correction here\n\n mybot.move(T[i]) #move the robot\n # rotation \n mybot.rt_spd_left = -omega_L[i]\n mybot.rt_spd_right = omega_R[i] # write your correction here\n mybot.move(t_rotation[i]) #move the robot\n mybot.plot_robot() #draw the new location\n end_angle[i] = mybot.angle\n\nplt.xlim(-5,11)\nplt.ylim(-5,11)\nplt.xlabel(r\"$x$\")\nplt.ylabel(r\"$y$\")\nplt.show()\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "bf50c94d7da128cb22e1630fe3e9baf90f0afece", "size": 109023, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Course-Documents/SimRobot-2/ME003-SimBot-2.ipynb", "max_stars_repo_name": "yvesdubief/UVM-ME003-Intro-to-Robotics", "max_stars_repo_head_hexsha": "48643fe6147a72a66e8e6b9ab4782442d6cbd8be", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Course-Documents/SimRobot-2/ME003-SimBot-2.ipynb", "max_issues_repo_name": "yvesdubief/UVM-ME003-Intro-to-Robotics", "max_issues_repo_head_hexsha": "48643fe6147a72a66e8e6b9ab4782442d6cbd8be", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Course-Documents/SimRobot-2/ME003-SimBot-2.ipynb", "max_forks_repo_name": "yvesdubief/UVM-ME003-Intro-to-Robotics", "max_forks_repo_head_hexsha": "48643fe6147a72a66e8e6b9ab4782442d6cbd8be", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-03-29T01:58:02.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-29T01:58:02.000Z", "avg_line_length": 112.0483042138, "max_line_length": 12640, "alphanum_fraction": 0.8438402906, "converted": true, "num_tokens": 5654, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863698, "lm_q2_score": 0.6654105720171531, "lm_q1q2_score": 0.4286741966149067}} {"text": "**Nguy\u1ec5n Ti\u1ebfn D\u0169ng**\n\nKSTN To\u00e1n Tin - K62\n\n*20170062*\n\n\u0110\u1ea1i h\u1ecdc B\u00e1ch khoa H\u00e0 N\u1ed9i\n\n\n```python\nimport processviz as pvz\n# import sympy as sy\n\n# sy.init_printing()\n```\n\n**C\u00e2u 1:**\n\n*a.* T\u00ednh $h_1, h_2$\n\nTr\u01b0\u1edbc h\u1ebft ta c\u00f3 $h_{00} = 1$\n\nG\u1ecdi $h_{i0}$ l\u00e0 x\u00e1c su\u1ea5t xu\u1ea5t ph\u00e1t t\u1eeb tr\u1ea1ng th\u00e1i $i$ c\u00f3 th\u1ec3 \u0111\u1ebfn \u0111\u01b0\u1ee3c tr\u1ea1ng th\u00e1i $0$. Khi \u0111\u00f3, xu\u1ea5t ph\u00e1t t\u1eeb tr\u1ea1ng th\u00e1i $1$, nh\u1eadn th\u1ea5y r\u1eb1ng c\u00f3 $p_{1i}$ kh\u1ea3 n\u0103ng tr\u1ea1ng th\u00e1i 1 chuy\u1ec3n \u0111\u1ebfn tr\u1ea1ng th\u00e1i $i$, v\u00e0 t\u1eeb tr\u1ea1ng th\u00e1i $i$ \u0111\u00f3 chuy\u1ec3n \u0111\u1ebfn tr\u1ea1ng th\u00e1i 0. Do \u0111\u00f3 ta c\u00f3:\n\n$$\nh_{10} = 0.2h_{00} + 0.4h_{10}+0.2h_{20} + 0.2h_{30}\n$$\n\nT\u01b0\u01a1ng t\u1ef1, ta c\u00f3:\n\n$$\n\\left\\{\\begin{matrix} \nh_{10} = 0.2h_{00} + 0.4h_{10}+0.2h_{20} + 0.2h_{30} \\\\\nh_{20} = 0.3h_{00} + 0.3h_{10} + 0.2h_{20} + 0.2h_{30} \\\\\nh_{30} = 0.5h_{00} + 0.3h_{20} + 0.2h_{30} \\\\\nh_{00} = 1\n\\end{matrix}\\right.\n$$\n\nGi\u1ea3i h\u1ec7 tr\u00ean ta thu \u0111\u01b0\u1ee3c: \n\n$$\n\\left\\{ \\begin{matrix}\nh_{10} = 1\\\\\nh_{20} = 1 \\\\\nh_{30} = 1\n\\end{matrix}\\right.\n$$\n\n\n```python\nG = pvz.MarkovChain()\nG.from_file('./ass5/input_1.csv')\nG.generate_graph(1)\n```\n\n*b.* T\u00ednh $k_{10}, k_{20}$\n\nD\u1ec5 th\u1ea5y $k_{00} = 0$\n\nK\u00ed hi\u1ec7u $k_{i0}$ l\u00e0 s\u1ed1 b\u01b0\u1edbc trung b\u00ecnh \u0111\u1ec3 \u0111i t\u1eeb tr\u1ea1ng th\u00e1i $i$ \u0111\u1ebfn tr\u1ea1ng th\u00e1i 0. Th\u1ea5y r\u1eb1ng t\u1eeb tr\u1ea1ng th\u00e1i $i$, ta c\u00f3 th\u1ec3 \u0111\u1ebfn tr\u1ea1ng th\u00e1i $j$ m\u1ea5t 1 b\u01b0\u1edbc, r\u1ed3i t\u1eeb tr\u1ea1ng th\u00e1i $j$ m\u1ea5t trung b\u00ecnh $k_{j0}$ b\u01b0\u1edbc \u0111\u1ec3 \u0111\u1ebfn tr\u1ea1ng th\u00e1i 0. T\u1eeb \u0111\u00f3 ta c\u00f3:\n\n$$\nk_{10} = 1+0.4k_{10} + 0.2k_{20} + 0.2k_{30} + 0.2k_{00}\n$$\n\nT\u01b0\u01a1ng t\u1ef1 ta c\u00f3: \n\n$$\n\\left\\{\\begin{matrix}\nk_{00} = 0 \\\\\nk_{10} = 1+0.4k_{10} + 0.2k_{20} + 0.2k_{30} + 0.2k_{00} \\\\\nk_{20} = 1 + 0.3k_{00} + 0.3k_{10} + 0.2k_{20} + 0.2k_{30} \\\\\nk_{30} = 1 + 0.5k_{00} + 0.3k_{20} + 0.2k_{30}\n\\end{matrix}\\right.\n$$\n\nGi\u1ea3i h\u1ec7 tr\u00ean ta thu \u0111\u01b0\u1ee3c \n\n$$\n\\left\\{\\begin{matrix}\nk_{00} = 0\\\\\nk_{10} = \\frac{500}{141} \\sim 3.55 \\\\\nk_{20} = \\frac{150}{47} \\sim 3.19 \\\\\nk_{30} = \\frac{115}{47} \\sim 2.45\n\\end{matrix}\\right.\n$$\n\n**C\u00e2u 2:** \n*a.* T\u01b0 t\u01b0\u1edfng ch\u00ednh l\u00e0 x\u00e2y d\u1ef1ng b\u00e0i to\u00e1n s\u1eed d\u1ee5ng c\u00f4ng th\u1ee9c truy h\u1ed3i. \u0110\u1ea7u ti\u00ean, ta s\u1ebd t\u00ednh $P_2$.\n\nGi\u1ea3 s\u1eed ma tr\u1eadn $P$ c\u00f3 k\u00edch th\u01b0\u1edbc $(m+n) \\times (m+n)$, trong \u0111\u00f3 t\u01b0\u01a1ng \u1ee9ng c\u00e1c th\u00e0nh ph\u1ea7n c\u00f3 k\u00edch th\u01b0\u1edbc nh\u01b0 sau:\n\n- Ma tr\u1eadn \u0111\u01a1n v\u1ecb $I$: $m \\times m$\n- Ma tr\u1eadn O: $m \\times n$\n- Ma tr\u1eadn R: $n \\times m$\n- Ma tr\u1eadn Q: $n \\times n$\n\nMa tr\u1eadn $P^2$ hi\u1ec3n nhi\u00ean c\u0169ng c\u00f3 th\u1ec3 chia ra th\u00e0nh c\u00e1c ph\u1ea7n nh\u01b0 tr\u00ean. Gi\u1ea3 s\u1eed \n\n$$\nP^2 = \\begin{pmatrix}\nI_2 & |& O_2 \\\\ \n R_2 & |& Q_2\n\\end{pmatrix}\n$$\n\nTa s\u1ebd ch\u1ee9ng minh $I_2 = I$ v\u00e0 $O_2 = O$.\n\nTh\u1eadt v\u00e2\u1ef5, v\u1edbi $P_{ij}^{2} \\in I_2$ ta c\u00f3:\n\n$$\nP_{ij}^2 = \\sum_{k=1}^{m+n}P_{ik}P_{kj} = \\sum_{k=1}^{m}P_{ik}P_{kj} + \\sum_{k=m+1}^{m+n}P_{ik}P_{kj}\n$$\n\nDo $(i, j)$ n\u1eb1m trong $I_2$ n\u00ean $1 \\le i, j \\le m$. Th\u1ea5y r\u1eb1ng: \n\n- N\u1ebfu $i=k$ th\u00ec $P_{ik} = 1$, m\u1ecdi i\n- N\u1ebfu $i \\ne k$ th\u00ec $P_{ik} = 0$, m\u1ecdi i\n\nDo \u0111\u00f3 $\\sum_{k=1}^{m}P_{ik}P_{kj} = 1$ n\u1ebfu $i=j$ v\u00e0 b\u1eb1ng 0 n\u1ebfu $i \\ne j$\n\nX\u00e9t $\\sum_{k=m+1}^{m+n}P_{ik}P_{kj}$, d\u1ec5 th\u1ea5y t\u1ed5ng n\u00e0y b\u1eb1ng 0 do $P_{ik}$ l\u00fac n\u00e0y thu\u1ed9c ph\u1ea7n ma tr\u1eadn $O$.\n\nT\u1eeb 2 \u0111i\u1ec1u tr\u00ean, suy ra $I_2 = I$.\n\nCh\u1ee9ng minh ho\u00e0n to\u00e0n t\u01b0\u01a1ng t\u1ef1 ta c\u0169ng c\u00f3 $O_2 = O$.\n\n---\n\n> G\u1ecdi $X_{c(k)}$ l\u00e0 c\u1ed9t th\u1ee9 $k$ c\u1ee7a ma tr\u1eadn $X$; $X_{r(k)}$ l\u00e0 h\u00e0ng th\u1ee9 $k$ c\u1ee7a ma tr\u1eadn $X$. \n\n---\n\nTa t\u00ednh c\u00e1c v\u1ecb tr\u00ed $R_2$. \n\nTa c\u00f3:\n\n$$\nP_{ij}^2 = \\sum_{k=1}^{m+n}P_{ik}P_{kj} = \\sum_{k=1}^{m}P_{ik}P_{kj} + \\sum_{k=m+1}^{m+n}P_{ik}P_{kj}\n$$\n\nX\u00e9t $(i, j) \\in R_2$. \n\nX\u00e9t t\u1ed5ng $\\sum_{k=1}^{m}P_{ik}P_{kj}$. D\u1ec5 th\u1ea5y v\u1edbi $k = j$ th\u00ec $P_{kj} = 1$ v\u00e0 $P_{kj} = 0$ n\u1ebfu $k \\ne j$. Do \u0111\u00f3 $\\sum_{k=1}^{m}P_{ik}P_{kj} = P_{ij}$.\n\nX\u00e9t $\\sum_{k=m+1}^{m+n}P_{ik}P_{kj} = R_{c(k)}.Q_{r(k)}$. \u0110i\u1ec1u n\u00e0y ch\u1ee9ng t\u1ecf v\u1edbi $P_{ij}^2 \\in R_2$ th\u00ec $P_{ij}^2 = P_{ij} + R_{c(j)}Q_{r(i)}$\n\nDo \u0111\u00f3: $R_2 = R + Q.R$\n\n---\n\nT\u00ednh $Q_2$:\n\nTa c\u00f3:\n\n$$\nQ_{ij}^2 = \\sum_{k=1}^{m+n}P_{ik}P_{kj} = \\sum_{k=1}^{m}P_{ik}P_{kj} + \\sum_{k=m+1}^{m+n}P_{ik}P_{kj}\n$$\n\nX\u00e9t $(i, j) \\in Q_2$. \n\nX\u00e9t t\u1ed5ng $\\sum_{k=1}^{m}P_{ik}P_{kj}$. D\u1ec5 th\u1ea5y v\u1edbi $P_{kj} = 0$ v\u1edbi m\u1ecdi $k$. Do \u0111\u00f3 $\\sum_{k=1}^{m}P_{ik}P_{kj} = 0$.\n\nX\u00e9t $\\sum_{k=m+1}^{m+n}P_{ik}P_{kj} = Q.Q = Q^2$. \u0110i\u1ec1u n\u00e0y ch\u1ee9ng t\u1ecf v\u1edbi $P_{ij}^2 \\in Q_2$ th\u00ec $Q_{ij}^2 = Q_{r(i)}Q_{c(j)}$, hay $Q_2 = Q^2$\n\n---\n\nL\u1eadp lu\u1eadn ho\u00e0n to\u00e0n t\u01b0\u01a1ng t\u1ef1, ta c\u00f3 th\u1ec3 ch\u1ee9ng minh \u0111\u01b0\u1ee3c ma tr\u1eadn $P^n$ nh\u01b0 sau: \n\n$$\nP^n = \\begin{pmatrix}\nI & |& O \\\\ \n R_n & |& Q_n\n\\end{pmatrix}\n$$\n\ntrong \u0111\u00f3:\n- $Q_n = Q^n$\n- $R_n = R + Q^{n-1}*R$\n \n---\n\n**Remark:** \u0110\u1ebfn \u0111\u00e2y th\u00ec em c\u00f3 th\u1ec3 l\u1eadp tr\u00ecnh \u0111\u01b0\u1ee3c ch\u01b0\u01a1ng tr\u00ecnh t\u00ednh ma tr\u1eadn $P^n$ c\u00f3 d\u1ea1ng n\u00e0y.\n\n---\n\n**C\u00e2u 3:** M\u00ea cung c\u00f3 d\u1ea1ng nh\u01b0 sau:\n\n| | | |\n|---|---|---|\n| 1 | 2 | 3 |\n| 4 | 5 | 6 |\n| 7 | 8 | 9 |\n\n---\n\n*a.* X\u00e9t $$P(X_{n+1} = i_{n+1}|X_n = i_n,...,X_0 = i_0) = \\frac{P(X_{n+1} = i_{n+1},X_n = i_n,...,X_0 = i_0)}{P(X_n = i_n,...,X_0 = i_0)} = \\frac{P(X_{n+1} = i_{n+1},X_n = i_n|X_{n-1} = i_{n-1},...,X_0 = i_0)}{P(X_n = i_n|X_{n-1}=i_{n-1}...,X_0 = i_0)}$$\n$$\n= \\frac{P(X_{n+1} = i_{n+1}|X_n = i_n)P(X_n = i_n|X_{n-1} = i_{n-1},...,X_0 = i_0)}{P(X_n = i_n|X_{n-1}=i_{n-1}...,X_0 = i_0)} = P(X_{n+1} = i_{n+1}|X_n = i_n)\n$$\n\nch\u1ee9ng t\u1ecf $(X_n)$ l\u00e0 x\u00edch Markov.\n\n*b.* V\u1edbi s\u01a1 \u0111\u1ed3 m\u00ea cung nh\u01b0 tr\u00ean ta c\u00f3 ma tr\u1eadn x\u00e1c su\u1ea5t chuy\u1ec3n nh\u01b0 sau\n\n$$\nP = \\left(\\begin{matrix}\n0 & 0.5 & 0 &0.5&0&0&0&0&0 \\\\\n\\frac{1}{3} & 0& \\frac{1}{3} & 0 &\\frac{1}{3}&0&0&0& 0\\\\\n0 & 0.5 & 0 & 0 & 0 & 0.5&0&0&0 \\\\\n\\frac{1}{3}&\t0\t&0\t&0&\t\\frac{1}{3}&\t0\t&\\frac{1}{3}&\t0\t&0 \\\\\n0\t&0.25&\t0\t&0.25&\t0\t&0.25&\t0 &\t0.25&\t0 \\\\\n0&\t0&\t\\frac{1}{3}&\t0&\t\\frac{1}{3}&\t0&\t0&\t0&\t\\frac{1}{3} \\\\\n0&\t0&\t0&\t0.5&\t0&\t0&\t0&\t0.5&\t0 \\\\\n0&\t0&\t0\t&0&\t\\frac{1}{3}&\t0&\t\\frac{1}{3}&\t0&\t\\frac{1}{3} \\\\\n0&\t0&\t0&\t0&\t0&\t0&\t0&\t0&\t1\n\\end{matrix}\\right)\n$$\n\n---\n\n*c.* G\u1ecdi $k_i$ l\u00e0 th\u1eddi gian trung b\u00ecnh con chu\u1ed9t r\u01a1i v\u00e0o b\u1eaby. Ta c\u1ea7n t\u00ednh $k_1$.\n\nT\u1eeb tr\u1ea1ng th\u00e1i 1, con chu\u1ed9t c\u00f3 th\u1ec3 t\u1edbi \u0111\u01b0\u1ee3c c\u00e1c tr\u1ea1ng th\u00e1i $2, 4$ sau 1 b\u01b0\u1edbc r\u1ed3i t\u1eeb \u0111\u00f3 t\u1ed1n t\u01b0\u01a1ng \u1ee9ng trung b\u00ecnh $k_2, k_4$ b\u01b0\u1edbc \u0111\u1ec3 \u0111\u1ebfn tr\u1ea1ng th\u00e1i $9$.\n\nTa c\u00f3 quan h\u1ec7 $k_1 = 1 + 0.5(k_2 + k_4)$. L\u1eadp lu\u1eadn t\u01b0\u01a1ng t\u1ef1 ta thu \u0111\u01b0\u1ee3c: \n\n$$\n\\left\\{\\begin{matrix}\nk_1 = 1 + 0.5(k_2 + k_4) \\\\\nk_2 = 1 + \\frac{1}{3}(k_5 + k_1 + k_3) \\\\\nk_3 = 1 + 0.5(k_2 + k_6) \\\\\nk_4 = 1 + \\frac{1}{3}(k_1 + k_5 + k_7) \\\\\nk_5 = 1 + \\frac{1}{4}(k_2 + k_4 + k_6 + k_8) \\\\\nk_6 = 1 + \\frac{1}{3}(k_3 + k_5 + k_9) \\\\\nk_7 = 1 + 0.5(k_4 + k_8)\\\\\nk_8 = 1 + \\frac{1}{3}(k_5 + k_7 + k_9) \\\\\nk_9 = 0\n\\end{matrix}\\right.\n$$\n\nH\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh tr\u00ean c\u00f3 th\u1ec3 vi\u1ebft l\u1ea1i th\u00e0nh:\n\n$$\nAk = b\n$$\n\ntrong \u0111\u00f3 $A = I - P, k = (k_1, k_2,...,k_8,0)^T, b=(1,1,1,1,1,1,1,1,0)^T$\n\nGi\u1ea3i h\u1ec7 tr\u00ean ta \u0111\u01b0\u1ee3c $k = (17.88, 16.88, 14.90, 16.87, 14.90,\n 10.92, 14.90, 10.92, 0)^T$\n\n\n```python\nG3 = pvz.MarkovChain()\nG3.from_file('./ass5/input_2.csv')\nG3.generate_graph()\n```\n\n**C\u00e2u 4:** X\u00e9t $(X_n)$ v\u1edbi $X_n$ l\u00e0 s\u1ed1 \u0111i\u1ec3m \u1edf l\u1ea7n gieo th\u1ee9 $n$. D\u1ec5 th\u1ea5y $(X_n)$ l\u00e0 x\u00edch Markov.\n\nTa t\u00ednh x\u00e1c su\u1ea5t gieo \u0111\u01b0\u1ee3c $k$ \u0111i\u1ec3m, $2\\le k \\le 12$.\n\nG\u1ecdi s\u1ed1 \u0111i\u1ec3m tr\u00ean con th\u1ee9 nh\u1ea5t v\u00e0 th\u1ee9 hai l\u1ea7n l\u01b0\u1ee3t l\u00e0 $x_1, x_2$. Ta c\u00f3 \n\n$$\nx_1 + x_2 = k\n$$\n\nKh\u00f4ng gian m\u1eabu $6 \\times 6 = 36$. Kh\u00f4ng gian tr\u1ea1ng th\u00e1i $I = \\{2,3,4,...,12\\}$\n\nT\u1eeb \u0111\u00f3 ta c\u00f3 b\u1ea3ng ph\u00e2n ph\u1ed1i x\u00e1c su\u1ea5t nh\u01b0 sau:\n\n---\n\n\n```python\nG2 = pvz.MarkovChain()\nG2.from_file('./ass5/input_3.csv')\nG2.generate_graph()\nG2.data\n```\n\n\n```python\nG2.get_mean_time(['10'])\n```\n\n\n\n\n {'Mean time spent of 2': 12.0,\n 'Mean time spent of 3': 12.0,\n 'Mean time spent of 4': 12.0,\n 'Mean time spent of 5': 12.0,\n 'Mean time spent of 6': 12.0,\n 'Mean time spent of 7': 12.0,\n 'Mean time spent of 8': 12.0,\n 'Mean time spent of 9': 12.0,\n 'Mean time spent of 11': 12.0,\n 'Mean time spent of 12': 12.0}\n\n\n\n**C\u00e2u 5:**\n\nX\u00e9t ma tr\u1eadn x\u00e1c su\u1ea5t chuy\u1ec3n $P$, kh\u00f4ng gian tr\u1ea1ng th\u00e1i $I$ v\u00e0 t\u1eadp c\u00e1c tr\u1ea1ng th\u00e1i h\u00fat $S$. Ta c\u00f3:\n\n$$\nP_{ii} = 1, \\forall i \\in S\n$$\n\nTh\u1eddi gian trung b\u00ecnh \u0111\u01b0\u1ee3c x\u00e1c \u0111\u1ecbnh nh\u01b0 sau;\n\n$$\n\\left\\{\\begin{matrix}\nk_i = 0, \\forall i \\in S\\\\\nk_i = 1+\\sum_{j \\in I / S}{P_{ij}k_j}\n\\end{matrix}\\right.\n$$\n\nChuy\u1ec3n v\u1ebf ta \u0111\u01b0\u1ee3c \n\n$$\n\\left\\{\\begin{matrix}\nk_i = 0, \\forall i \\in S\\\\\n(1-P_{ii})k_i - \\sum_{j \\ne i \\in I / S}{P_{ij}k_j} = 1\n\\end{matrix}\\right.\n$$\n\nHay ta \u0111\u01b0\u1ee3c h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh vi\u1ebft l\u1ea1i d\u01b0\u1edbi d\u1ea1ng ma tr\u1eadn nh\u01b0 sau: \n\n$$\nA_1k = b_1\n$$\n\ntrong \u0111\u00f3 $A_1 = I - P, b_1 = (b_{1i})$ th\u1ecfa m\u00e3n $b_{1i} = 0$ n\u1ebfu $i \\in S$ v\u00e0 $b_{1i} = 1$ n\u1ebfu $i \\notin S$.\n\nD\u1ec5 th\u1ea5y r\u1eb1ng $A_1$ l\u00e0 m\u1ed9t `singular matrix`, do \u0111\u00f3 ta s\u1ebd lo\u1ea1i b\u1edbt t\u1ea5t c\u1ea3 nh\u1eefng h\u00e0ng $i$ c\u00f3 gi\u00e1 tr\u1ecb b\u1eb1ng 0 v\u00e0 c\u1ed9t $i$ t\u01b0\u01a1ng \u1ee9ng \u0111\u1ec3 \u0111\u01b0\u1ee3c ma tr\u1eadn $A$. \n\nTh\u1ef1c hi\u1ec7n t\u01b0\u01a1ng t\u1ef1 v\u1edbi $b_1$, ta x\u00f3a h\u1ebft t\u1ea5t c\u1ea3 c\u00e1c h\u00e0ng c\u00f3 s\u1ed1 $0$.\n\n```python\ndef get_mean_time(self, source=None, target=None, type='transient'):\n try:\n state = mt.get_transient_state(self.state, self.P)\n matrix = mt.get_mean_time_transient(self.state, self.P)\n if type == 'absoring':\n return state, (mt.get_mean_time_absoring(self.state, self.P)).tolist()\n elif type == 'transient':\n if source == None and target == None:\n return state, matrix\n elif source == None:\n return state, (np.transpose(matrix)).tolist()[state.index(target)]\n elif target == None:\n return state, (matrix[state.index(source)]).tolist()\n else:\n return state, matrix[state.index(source)][state.index(target)]\n except:\n return \"Invalid\"\n```\n\nTh\u1ef1c hi\u1ec7n gi\u1ea3i h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh tr\u00ean b\u1eb1ng c\u00e1ch nh\u00e2n v\u1edbi ngh\u1ecbch \u0111\u1ea3o c\u1ee7a ma tr\u1eadn $A$. K\u1ebft qu\u1ea3 tr\u1ea3 v\u1ec1 l\u00e0 dictionary t\u01b0\u01a1ng \u1ee9ng gi\u1eefa `mean time spent` v\u00e0 tr\u1ea1ng th\u00e1i.\n\n---\n\nTh\u1ef1c hi\u1ec7n ch\u1ea1y v\u1edbi v\u00ed d\u1ee5 c\u1ee7a **c\u00e2u 3**. Ta c\u00f3:\n\n\n```python\nG3.get_mean_time(['9'])\n```\n\n\n\n\n {'Mean time spent of 1': 17.87,\n 'Mean time spent of 2': 16.87,\n 'Mean time spent of 3': 14.9,\n 'Mean time spent of 4': 16.87,\n 'Mean time spent of 5': 14.9,\n 'Mean time spent of 6': 10.92,\n 'Mean time spent of 7': 14.9,\n 'Mean time spent of 8': 10.92}\n\n\n\n**Remark:** T\u1ea1m th\u1eddi em \u0111ang \u0111\u1ec3 l\u00e0m tr\u00f2n \u0111\u1ebfn 2 ch\u1eef s\u1ed1 th\u1eadp ph\u00e2n. \n", "meta": {"hexsha": "c492658605464667222dc75608401d6566077d93", "size": 62226, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Assignment_4.1.ipynb", "max_stars_repo_name": "jurgendn/processviz", "max_stars_repo_head_hexsha": "82808a92662962f04c48673c9cf159d7bc904ff7", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Assignment_4.1.ipynb", "max_issues_repo_name": "jurgendn/processviz", "max_issues_repo_head_hexsha": "82808a92662962f04c48673c9cf159d7bc904ff7", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Assignment_4.1.ipynb", "max_forks_repo_name": "jurgendn/processviz", "max_forks_repo_head_hexsha": "82808a92662962f04c48673c9cf159d7bc904ff7", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2020-03-19T11:14:13.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-14T14:24:08.000Z", "avg_line_length": 109.746031746, "max_line_length": 18764, "alphanum_fraction": 0.8223893549, "converted": true, "num_tokens": 4816, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804478040616, "lm_q2_score": 0.6926419831347361, "lm_q1q2_score": 0.4285933164920053}} {"text": "#### [**NICOLAS CACHANOSKY**](http://www.ncachanosky.com) | Department of Economics | Metropolitan State University of Denver | ncachano@msudenver.edu\n\n# AD-AS MODEL\n---\n\nThis note illustrates how to code a simple AD-S Model in Python. The purpose of the note is to walk through Python applications, not to offer a detailed discussion of the AD-AS Model or to show best coding practices. The note also assumes familiarity with the AD-AS model and a beginner experience with Python.\n\nFor a more complete and detailed discussion of Python applications see the material in [Quant Econ](https://quantecon.org/).\n\n---\n\n## TABLE OF CONTENTS\n1. [The AD-AS model](#1.-THE-AD-AS-MODEL)\n2. [Aggregate Demand (AD)](#2.-AGGREGATE-DEMAND-(AD))\n3. [Long-Run Aggregate Supply (LRAS)](#3.-LONG-RUN-AGGREGATE-SUPPLY-(LRAS))\n4. [Short-Run Aggregate Supply (SRAS)](#4.-SHORT-RUN-AGGREGATE-SUPPLY-(SRAS))\n5. [Putting all the pieces together](#5.-PUTTING-ALL-THE-PIECES-TOGETHER)\n6. [Shocks](#6.-SHOCKS)\n\n# 1. THE AD-AS MODEL\n\nThe AD-AS model shows the relationship between output $(Y)$ and the price level $(P)$. Different to the IS-LM model, in this case $P$ is endogenous and varies with different levels of $Y$. The AD-AS model has three components:\n\n1. AD: Aggregate demand\n2. LRAS: Long-run aggregate supply\n3. SRAR: Short-run aggregate supply\n\nThe AD-AS model is more *general* than the IS-LM model in the sense that it allows for the price level to change. Therea are two differences between these models. The first one is taht while the AD-AS model allows for the interest rate $(i)$ to change, it does not show up explicitly in the model as it does in the IS-LM framework. The second one, is that both, monetary and fiscal policy affect the same line (the $AD$), while in the IS-LM framework monetary and fiscal policy shift *different* lines.\n\n# 2. AGGREGATE DEMAND (AD)\n\nThe $AD$ line tracks all the output and price level combinations for whith the IS-LM model is in equilibrium. Graphically, it shows how equilibrum moves in the IS-LM graph when $P$ changes. A change in $P$ shifts the LM schedule giving a new equilibrium point on the IS schedule. In simple terms: $AD = Y = C + I + G + (X - Z)$ where $C$ is the household consumption, $I$ is investment, $G$ is govegnment spending, $X$ is exports, and $Z$ is imports.\n\nIn the [IS-LM model note](https://nbviewer.jupyter.org/github/ncachanosky/Macroeconomics-with-Python/blob/master/IS-LM%20Model.ipynb) there is consumption function, an investment function, and an imports function. The remainig variables $(G = \\bar{G} \\text{ and } X = \\bar{X})$ are treated as exogenous. The consumption, investment, imports, and money demadn functions are:\n\n\\begin{align}\n C &= a + b(Y-T) \\\\[10pt]\n I &= \\bar{I} - d \\cdot i \\\\[10pt]\n Z & = \\alpha + \\beta(Y-T) \\\\[10pt]\n M^d &= c_1 + c_2 Y - c_3 i\n\\end{align}\n\nWhere $a>0$ and $b \\in (0, 1)$ are the household level of autonomous consumption and the marginal propensity to consume respectively with $T$ representing the nominal value of taxes; \\bar{I} is the level of investment when $i = 0$ and $d >0$ is the slope of investment with respect to $i$; $\\alpha >0$ and $\\beta \\in (0, 1)$ are the autonomous level and the marginal propensity to import respectively; and $c_1>0, c_2>0, c_3>0$ capture the keynesian precautionary, transation, and especualtion reasons to demand money.\n\nThe $AD$ is the equilibrium level of output $(Y^*)$ from the IS-LM model, which is a function of $(P)$. From the [IS-LM model note](https://nbviewer.jupyter.org/github/ncachanosky/Macroeconomics-with-Python/blob/master/IS-LM%20Model.ipynb) (section 4):\n\n\\begin{align}\n Y^* &= \\frac{\\left[(a-\\alpha)-(b-\\beta)T+\\bar{I}+\\bar{G}+X\\right]/d + (1/c_3) \\left(M^S_0/P - c_1 \\right)}{(1-b+\\beta)/d - (c_2/c_3)} \\\\[10pt]\n Y^* &= \\underbrace{\\left[\\frac{(a-\\alpha)-(b-\\beta)T+\\bar{I}+\\bar{G} + X}{d} - \\frac{c_1}{c_3} \\right] \\left[\\frac{1-b+\\beta}{d} - \\frac{c_2}{c_3} \\right]^{-1}}_\\text{vertical level} + \\underbrace{\\frac{M^S_0}{c_3} \\left[\\frac{1-b+\\beta}{d} - \\frac{c_2}{c_3} \\right]^{-1}}_\\text{shape} \\cdot \\frac{1}{P}\n\\end{align}\n\nEven though the function looks complicated, note that the relationship between $Y$ and $P$ is hiperbolic. Note that an increaes in $M^S_0$ increases the level of Y but also changes the *shape* of $AD$.\n\n## 2.1 MONEY SUPPLY AND VELOCITY\n\nThe AD-AS model has a real variable $(Y)$ and a nominal variable $(P)$. Because $PY = NGDP$ the model can be framed in terms of the equation of exchange.\n\n\\begin{equation}\n MV_{Y} = P_{Y}Y\n\\end{equation}\n\nWhere $M$ is money supply (shown as $M^S_0$ above), $V_Y$ is the velocity of money circulation, and $P_{Y}$ is the GDP deflator of real output $Y$. Note this simple form, $Y = \\frac{MV_Y}{P_Y}$ also has the hiperbolic shape discussed above.\n\nTo add a layer of complexity, money supply can be open in base money $(B)$ times the money-multiplier $m$:\n\n\\begin{align}\n m &= \\frac{1 + \\lambda}{\\rho + \\lambda} \\\\[10pt]\n M &= B \\cdot m \\\\[10pt]\n \\left(Bm\\right) V_Y &= P_Y Y\n\\end{align}\n\nwhere $\\lambda \\in (0, 1)$ is the currency-drain ratio (cash-to-deposit ratio) and $\\rho \\in (0, 1)$ is the reserve ratio (desired plus required level of reserves).\n\nIf money demand $M^D$ is a $k$ proportion of nominal income $(P_YY)$, then, assuming equilibrium in the money market, money velocity is the inverse of money demand: $V_Y = 1/k$. As less (more) money is demanded to be hold as a cash-balance, the more (less) quickly money moves (in average) in the economy.\n\n\\begin{align}\n M^S_0V_Y &= P_YY \\\\[10pt]\n M^D &= k \\cdot \\left(P_YY \\right) \\\\[10pt]\n M^S_0 &= M^D \\\\[10pt]\n V_Y &= \\frac{1}{k}\n\\end{align}\n\n---\n\nWe can code $AD$ with a `class`. A `class` allows to build our own type of objects. In this case the code constructs a class called `AD` that has (1) the money multiplier, (2) the money supply, and (3) estimates $AD$ for a given $P$.\n\nThe code follows the following structure. The first section imports the required packages. The second section builds the $AD$ `class`. The third section show the values of the money multiplier and total money supply. The fourth section plots the $AD$.\n\nThe `class` is build the following way. The first element, `__init__` collects the model (or `class`) parameters. Note that the values of these parameters are defined **inside** the `class` (this does not need to be the case) and that these parameters exist **inside** the class (they are not global values). After the parameters are defined, the `class` continues to build the three components. Note that the first two (money multiplier and money supply) can be defined with the paramters already included in the `class`. The third component, the value of $AD$, requires an exogenous value, $P$.\n\n\n```python\n\"1|IMPORT PACKAGES\"\nimport numpy as np # Package for scientific computing with Python\nimport matplotlib.pyplot as plt # Matplotlib is a 2D plotting library\n\n\"2|BUILD AD CLASS\"\nclass class_AD:\n \"Define the parameters of the model\"\n def __init__(self, a = 20 , # AD: autonomous consumption \n b = 0.2 , # AD: marginal propensity to consume\n alpha = 5 , # AD: autonomous imports\n beta = 0.1 , # AD: marginal propensity to import\n T = 1 , # AD: Taxes\n I = 10 , # AD: Investment\n G = 8 , # AD: Government spending\n X = 2 , # AD: Exports\n d = 5 , # AD: Investment slope\n c1 = 175 , # AD: Precautionary money demand\n c2 = 2 , # AD: Transactions money demand\n c3 = 50 , # AD: Speculatio money demand\n B = 250 , # AD: Base money\n lmbda = 0.05 , # AD: Currency drain ratio\n rho = 0.10 ): # AD: Reserve requirement)\n\n \"Assign the parameter values\"\n self.a = a\n self.b = b\n self.alpha = alpha\n self.beta = beta\n self.T = T\n self.I = I\n self.G = G\n self.X = X\n self.d = d\n self.c1 = c1\n self.c2 = c2\n self.c3 = c3\n self.B = B\n self.lmbda = lmbda\n self.rho = rho\n\n \"Money multiplier\"\n def m(self):\n #Unpack the parameters (simplify notation)\n lmbda = self.lmbda\n rho = self.rho\n #Calculate m\n return ((1 + lmbda)/(rho + lmbda))\n \n \"Money supply\" \n def M(self):\n #Unpack the parameters (simplify notation)\n B = self.B\n #Calculate M\n return (B*self.m())\n \n \"AD: Aggregate demand\"\n def AD(self, P):\n #Unpack the parameters (simplify notation)\n a = self.a\n alpha = self.alpha\n b = self.b\n beta = self.beta\n T = self.T\n I = self.I\n G = self.G\n X = self.X\n d = self.d\n c1 = self.c1\n c2 = self.c2\n c3 = self.c3\n #Calculate AD\n AD_level1 = (((a-alpha)-(b-beta)*T+I+G+X)/d-c1/c3)\n AD_level2 = ((1-b+beta)/d-c2/c3)**(-1)\n AD_shape = self.M()/c3 * ((1-b+beta)/d - c2/c3)**(-1)\n return (AD_level1 * AD_level2 + AD_shape/P)\n\n\"3|SHOW RESULTS\"\nout = class_AD()\n\nprint(\"Money multiplier =\", round(out.m(), 2))\nprint(\"Money supply =\" , round(out.M(), 2))\n\nsize = 50\nP = np.arange(1, size)\nY = out.AD(P)\n\n\"4|PLOT AD\"\ny_max = np.max(P)\nx_max = np.max(Y)\nv = [0, x_max, 0, y_max]\nfig, ax = plt.subplots(figsize=(10, 8))\nax.set(title=\"AGREGGATE DEMAND\", xlabel=\"Y\", ylabel=\"P\")\nax.plot(Y, P, \"k-\")\nax.yaxis.set_major_locator(plt.NullLocator()) # Hide ticks\nax.xaxis.set_major_locator(plt.NullLocator()) # Hide ticks\nplt.show()\n```\n\n# 3. LONG-RUN AGGREGATE SUPPLY (LRAS)\n\nPrices are flexible in the long-run, therefore money is neutral and the $LRAS$ is a vertical line (a given value of $Y$ rather than a function). The valor of $Y$ in the long-run can be derived from the Solow model's steady-state. From the [Solow model notes](https://nbviewer.jupyter.org/github/ncachanosky/Macroeconomics-with-Python/blob/master/Solow%20Model.ipynb) (section 2) we know that there is an equilibrium level of capital $K^*$. Therefore, assuming a typical Cobb-Douglass production function with constant returns to scale, for any period of time $t$, output in the long run $(Y_{LR})$ equals:\n\n\\begin{align}\n K_{LR} &= N \\cdot \\left(\\frac{sA}{\\delta} \\right)^{\\frac{1}{1-\\theta}} \\\\\n Y_{LR} &= A \\cdot \\left(K_{LR}^{\\theta} N^{1-\\theta} \\right)\n\\end{align}\n\nWhere $A$ is techonology or total factor productivity (TFP), $N$ is labor, and $\\theta$ is the output elasticity of capital. In the long-run, the leve of output will grow at the rate of growth of population $(n)$ and technology $(\\gamma)$. This is shown as a right-ward movemet in the common pictorial depiction of the AD-AS model. Also note that since th $LRAS$ is a value (a vertical line), it can only shift **left** or **right**, but not up or down.\n\n# 4. SHORT-RUN AGGREGATE SUPPLY (SRAS)\n\nAs long as wages are sticky, then changes in $P$ do have short-run effects on the level of $Y$ thourgh changes in the real wage $(W/P)$. In the long-run $N$ is at full employment and the representative firm can change the size of $K$. But in the short-run, the firm can change $N$ but $K$ is fixed. Therefore, to construct the $SRAS$ we need the bheavior of labor supply in the short-run. For simplicity, this note takes a simple approach. It assumes a closed economy (you can try to add a foreign sector yourself) and lets the level of employment be defined by the following function with respect to the price level (you cantry to replace with a labor market as discussed in the [labor market notes](https://nbviewer.jupyter.org/github/ncachanosky/Macroeconomics-with-Python/blob/master/Labor%20Market.ipynb)):\n\n\\begin{equation}\n N = \\eta \\cdot P^{\\nu}; \\eta, \\nu > 0\n\\end{equation}\n\nThe function is (implicitly) assuming constant expectations by labor. Therefore, an increase in the price level reduces the real wage paid by the firms, but labor does not realize that the real wage has decreased. \n\n# 5. PUTTING ALL THE PIECES TOGETHER\n\nWe can now put all the pieces together. In this case, instead of using a `class` the sample code uses `functions`. The model assumes a given stock of capital $K = \\bar{K}$ (you can try to add a *steady-state* level of capital as discussed in the [Solow Model notes](https://nbviewer.jupyter.org/github/ncachanosky/Macroeconomics-with-Python/blob/master/Solow%20Model.ipynb). The code uses the `root` function to find the price level of equilibrium (section 4).\n\n\n```python\n\"1|IMPORT PACKAGES\"\nimport numpy as np # Package for scientific computing with Python\nimport matplotlib.pyplot as plt # Matplotlib is a 2D plotting library\nfrom scipy.optimize import root # Package to find the roots of a function\n\n\"2|DEFINE PARAMETERS\"\n# PRODUCTION FUNCTION\nA = 1 # Total Factor Productivity\nvarphi = 0.70 # Output elasticity of capital\n# CAPITAL STOCK\ns = 0.25 # Savings rate\ndelta = 0.20 # Depreciation rate\nK = 100 # Capital stock\n# LABOR SUPPLY\nv = 0.75 # Labor supply \"slope\" with respect to P\n# MONEY SUPPLY\nB = 100 # Base Money\nlmbda = 0.05 # Currency drain\nrho = 0.20 # Reserve ratio\n# AGGREGATE DEMAND\na = 40 # Autonomous household domestic consumption\nb = 0.3 # Marginal propensity to consume\nT = 1 # Taxes\nI = 4 # Investment with i = 0\nG = 2 # Government Spengin\nd = 2 # Slope of investment with respect to i\nc1 = 200 # Money demand: Precuationary\nc2 = 0.6 # Money demand: Transactions\nc3 = 10 # Money demand: Speculation\n\n\"3|FUNCTIONS\"\n# LABOR SUPPLY\ndef N(P):\n N = v * P\n return N\n\n# OUTPUT\ndef output(P):\n output = A * (K**(varphi)) * (N(P)**(1-varphi))\n return output\n\n# MONEY SUPPLY\nm = (1+lmbda)/(rho+lmbda) # Money Multiplier\nM = B*m # Money Supply\n\n# AGGREGATE DEMAND\ndef AD(P):\n AD_level1 = (a-b*T + I + G)/d - (c1/c3)\n AD_level2 = ((1-b)/d - (c2/c3))**(-1)\n AD_shape = M/c3 * AD_level2\n AD = AD_level1 * AD_level2 + AD_shape/P\n return AD\n\n\"4|EQUILIBRIUM: PRICE LEVEL\"\ndef equation(P):\n Eq1 = AD(P) - output(N(P))\n equation = Eq1\n return equation\n\nsol = root(equation, 10)\nPstar = sol.x\nNstar = N(Pstar)\n\n# #Define domain of the model\nsize = np.round(Pstar, 0)*2\nP_vector = np.linspace(1, size, 500) # 500 dots between 1 and size\n\n# Agregate Supply and Aggregate Demand\nLRAS = output(N(Pstar))\nSRAS = output(N(P_vector))\nAD_vector = AD(P_vector)\n\n\"5|PLOT AD-AS MODEL\"\nv = [0, 80, 0, size] # Axis range\nfig, ax = plt.subplots(figsize=(10, 8))\nax.set(title=\"AD-AS MODEL\", ylabel=r'$P$', xlabel=\"Output\")\nax.plot(AD_vector, P_vector, \"k-\", alpha = 0.7)\nax.plot(SRAS , P_vector, \"b-\", alpha = 0.7)\nplt.axvline(x = LRAS, color = 'r', alpha = 0.7)\nplt.axhline(y = Pstar, xmax = LRAS/80, ls=':', color='k')\nax.yaxis.set_major_locator(plt.NullLocator()) # Hide ticks\nax.xaxis.set_major_locator(plt.NullLocator()) # Hide ticks\nplt.text(75, 2.5, \"AD\")\nplt.text(45, 11.0, \"SRAS\", color='b')\nplt.text(30, 11.0, \"LRAS\", color='r')\nplt.axis(v)\nplt.show()\n```\n\n# 6. SHOCKS\n\nWe now add a real and a nominal positive shocks. In the case of the real shock, technology increases by 10-percent. In the case of the nominal shock, base money increases by 10-percent. Even though the output is not shown, the code inclides the inflation rate and the percent change in output calculations for each one of the two shocks. Note that the $SRAS$ shifts horizontally with shocks to the $LRAS$. \n\n\n```python\n\"1|IMPORT PACKAGES\"\nimport numpy as np # Package for scientific computing with Python\nimport matplotlib.pyplot as plt # Matplotlib is a 2D plotting library\nfrom scipy.optimize import root # Package to find the roots of a function\n\n\"2|DEFINE PARAMETERS\"\n# PRODUCTION FUNCTION\nA = 1 # Total Factor Productivity\nvarphi = 0.70 # Output elasticity of capital\n# CAPITAL STOCK\ns = 0.25 # Savings rate\ndelta = 0.20 # Depreciation rate\nK = 100 # Capital stock\n# LABOR SUPPLY\nv = 0.75 # Labor supply \"slope\" with respect to P\n# MONEY SUPPLY\nB = 100 # Base Money\nlmbda = 0.05 # Currency drain\nrho = 0.20 # Reserve ratio\n# AGGREGATE DEMAND\na = 40 # Autonomous household domestic consumption\nb = 0.3 # Marginal propensity to consume\nT = 1 # Taxes\nI = 4 # Investment with i = 0\nG = 2 # Government Spengin\nd = 2 # Slope of investment with respect to i\nc1 = 200 # Money demand: Precuationary\nc2 = 0.6 # Money demand: Transactions\nc3 = 10 # Money demand: Speculation\n\n\"3|FUNCTIONS\"\n# LABOR SUPPLY\ndef N(P):\n N = v * P\n return N\n\n# OUTPUT\ndef output(P):\n output = A * (K**(varphi)) * (N(P)**(1-varphi))\n return output\n\n# MONEY SUPPLY\nm = (1+lmbda)/(rho+lmbda) # Money Multiplier\nM = B*m # Money Supply\n\n# AGGREGATE DEMAND\ndef AD(P):\n AD_level1 = (a-b*T + I + G)/d - (c1/c3)\n AD_level2 = ((1-b)/d - (c2/c3))**(-1)\n AD_shape = M/c3 * AD_level2\n AD = AD_level1 * AD_level2 + AD_shape/P\n return AD\n\n\"4|EQUILIBRIUM: PRICE LEVEL\"\ndef equation(P):\n Eq1 = AD(P) - output(N(P))\n equation = Eq1\n return equation\n\nsol = root(equation, 10)\nPstar = sol.x\nNstar = N(Pstar)\n\n# #Define domain of the model\nsize = np.round(Pstar, 0)*2\nP_vector = np.linspace(1, size, 500) # 500 dots between 1 and size\n\n# Agregate Supply and Aggregate Demand\nLRAS = output(N(Pstar))\nSRAS = output(N(P_vector))\nAD_vector = AD(P_vector)\n\n\"5|CALCULATE SHOCK EFFECTS\"\nreal_shock = 1.10\nnominal_shock = 1.10\n\n# Real shock\ndef output2(P):\n A2 = A * real_shock\n output2 = A2 * (K**(varphi)) * (N(P)**(1-varphi))\n return output2\n\ndef equation_real(P):\n Eq1 = AD(P) - output2(N(P))\n equation_real = Eq1\n return equation_real\n\nsol_real = root(equation_real, 10)\nPstar2 = sol_real.x\nNstar2 = N(Pstar2)\nLRAS2 = output2(N(Pstar2))\nSRAS2 = output2(N(P_vector))\n\ngP2 = Pstar2/Pstar - 1 # Percent change in P\ngY2 = LRAS2/LRAS - 1 # Percent change in Y \n\n# Nominal shock\nM3 = (B * nominal_shock) * m\n\ndef AD3(P):\n AD_level1 = (a-b*T + I + G)/d - (c1/c3)\n AD_level2 = ((1-b)/d - (c2/c3))**(-1)\n AD_shape = M3/c3 * AD_level2\n AD3 = AD_level1 * AD_level2 + AD_shape/P\n return AD3\n\ndef equation_nominal(P):\n Eq1 = AD3(P) - output(N(P))\n equation_nominal = Eq1\n return equation_nominal\n\nsol_nominal = root(equation_nominal, 10)\nPstar3 = sol_nominal.x\nNstar3 = N(Pstar3)\nAD_vector3 = AD3(P_vector)\n\ngP3 = Pstar3/Pstar - 1 # Percent change in P\ngY3 = output(Pstar3)/output(Pstar) - 1 # Percent change in Y \n\n\"6|PLOT AD-AS MODEL WITH SHOCKS\"\nP3_stop = output(N(Pstar3))\n\nv = [0, 80, 0, size] # Axis range\nfig, ax = plt.subplots(nrows=2, figsize=(10, 20))\nax[0].set(title=\"AD-AS MODEL: REAL SHOCK\", ylabel=r'$P$', xlabel=\"Output\")\nax[0].plot(AD_vector , P_vector, \"k-\", alpha = 0.7)\nax[0].plot(SRAS , P_vector, \"b-\", alpha = 0.7)\nax[0].plot(SRAS2 , P_vector, \"b:\", alpha = 0.7)\nax[0].axvline(x = LRAS , ls=\"-\", color='r', alpha = 0.7)\nax[0].axvline(x = LRAS2 , ls=\":\", color=\"r\", alpha = 0.7)\nax[0].axhline(y = Pstar , xmax = LRAS/80 , ls=\":\", color=\"k\", alpha = 0.7)\nax[0].axhline(y = Pstar2, xmax = LRAS2/80, ls=\":\", color=\"k\", alpha = 0.7)\nax[0].text(75.0, 2.5, \"AD\")\nax[0].text(44.5, 11.5, \"SRAS\" , color = \"b\")\nax[0].text(48.0, 11.0 , \"SRAS'\", color = \"b\")\nax[0].text(30.0, 11.0, \"LRAS\" , color = \"r\")\nax[0].text(39.0, 11.0, \"LRAS'\", color = \"r\")\nax[0].yaxis.set_major_locator(plt.NullLocator()) # Hide ticks\nax[0].xaxis.set_major_locator(plt.NullLocator()) # Hide ticks\nax[0].axis(v)\nax[1].set(title=\"AD-AS MODEL: NOMINAL SHOCK\", ylabel=r'$P$', xlabel=\"Output\")\nax[1].plot(AD_vector , P_vector, \"k-\", alpha = 0.7)\nax[1].plot(AD_vector3 , P_vector, \"k:\", alpha = 0.7)\nax[1].plot(SRAS , P_vector, \"b-\", alpha = 0.7)\nax[1].axvline(x = LRAS , ls=\"-\", color=\"r\", alpha = 0.7)\nax[1].axhline(y = Pstar , xmax = LRAS/80, ls=\":\", color=\"k\", alpha = 0.7)\nax[1].axhline(y = Pstar3, xmax = P3_stop/80, ls=\":\", color=\"k\", alpha = 0.7)\nax[1].text(75, 1.7, \"AD\")\nax[1].text(75, 2.7, \"AD'\")\nax[1].text(45, 11.0, \"SRAS\" , color = \"b\")\nax[1].text(30, 11.0, \"LRAS\" , color = \"r\")\nax[1].yaxis.set_major_locator(plt.NullLocator()) # Hide ticks\nax[1].xaxis.set_major_locator(plt.NullLocator()) # Hide ticks\nax[1].axis(v)\nplt.show()\n```\n", "meta": {"hexsha": "67f94ddaf6fb4abe33d31c476a4e2ae348a986d9", "size": 140608, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "adas.ipynb", "max_stars_repo_name": "devmacro/nb-open", "max_stars_repo_head_hexsha": "98304713c9e6d4196edf4f28001f550c42b5bf58", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "adas.ipynb", "max_issues_repo_name": "devmacro/nb-open", "max_issues_repo_head_hexsha": "98304713c9e6d4196edf4f28001f550c42b5bf58", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "adas.ipynb", "max_forks_repo_name": "devmacro/nb-open", "max_forks_repo_head_hexsha": "98304713c9e6d4196edf4f28001f550c42b5bf58", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-08-14T10:21:50.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-14T10:21:50.000Z", "avg_line_length": 223.898089172, "max_line_length": 74872, "alphanum_fraction": 0.881109183, "converted": true, "num_tokens": 6561, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.6926419958239132, "lm_q1q2_score": 0.42859331460512695}} {"text": "# Reproducibility_Challenge_NeurIPS_2019\n\nThis is a blog explains method proposed in the paper Competitive gradient descent [(Sch\u00e4fer et al., 2019)](https://arxiv.org/abs/1905.12103). This has been written as a supplimentary to the reproducibility report for reproducibility challenge of NeurlIPS\u201919. The pdf format of the report is present [here](https://gopikishan14.github.io/Reproducibility_Challenge_NeurIPS_2019/) with this github [repository](https://github.com/GopiKishan14/Reproducibility_Challenge_NeurIPS_2019) as its source.\n\n# Paper Overview\nThe paper introduces a new algorithm for the numerical computation of Nash equilibria of competitive two-player games. The method is a natural generalization of gradient descent to the two-player setting where the update is given by the Nash equilibrium of a regularized bilinear local approximation of the underlying game. It avoids oscillatory and divergent behaviors seen in alternating gradient descent. Convergence and stability properties of the method are robust to strong interactions between the players, without adapting the stepsize, which is not the case with previous methods. The ability to choose larger stepsizes furthermore allows the algorithm to achieve faster convergence, as measured by the number of model evaluations (See the [report](https://gopikishan14.github.io/Reproducibility_Challenge_NeurIPS_2019/) experiments section).\n\n\n## Background\nThe traditional optimization is concerned with a single agent trying to optimize a cost function. It\ncan be seen as $\\min_{x \\in R^m} f(x)$ . The agent has a clear objective to find (\u201cGood local\u201d) minimum of\nf. Gradeint Descent (and its varients) are reliable Algorithmic Baseline for this purpose.\n\nThe paper talks about Competitive optimization. Competitive optimization extends this problem\nto the setting of multiple agents each trying to minimize their own cost function, which in general\ndepends on the actions of all agents.\n The paper deals with the case of two such agents:\n \\begin{align}\n &\\min_{x \\in R^m} f(x,y),\\ \\ \\ \\min_{y \\in R^n} g(x,y)\n \\end{align}\n for two functions $f,g: R^m \\times R^n \\longrightarrow R$.\n\nIn single agent optimization, the solution of the problem consists of the minimizer of the cost function.\nIn competitive optimization, the right definition of solution is less obvious, but often one is\ninterested in computing Nash\u2013 or strategic equilibria: Pairs of strategies, such that no player can\ndecrease their costs by unilaterally changing their strategies. If f and g are not convex, finding a\nglobal Nash equilibrium is typically impossible and instead we hope to find a \"good\" local Nash\nequilibrium\n\n## About the problem\n##### Gradient descent/ascent and the cycling problem:\n\nFor differentiable objective functions, the most naive approach to solving\n\\begin{align}\n \\label{eqn:game}\n &\\min_{x \\in R^m} f(x,y),\\ \\ \\ \\min_{y \\in R^n} g(x,y)\n \\end{align}\nis gradient descent ascent (GDA), whereby both players independently change their strategy in the direction of steepest descent of their cost function.\nUnfortunately, this procedure features oscillatory or divergent behavior even in the simple case of a bilinear game ($f(x,y) = x^{\\top} y = -g(x,y)$)\n\n\n\n## Solution approach\n\nTo motivate this algorithm, authors remind us that gradient descent with stepsize $\\eta$ applied to the function $f:R^m \\longrightarrow R$ can be written as\n\n\\begin{equation}\n x_{k+1} = argmin_{x \\in R^m} (x^T - x_{k}^T) \\nabla_x f(x_k) + \\frac{1}{2\\eta} \\|x - x_{k}\\|^2.\n \\end{equation}\n\nThis models a (single) player solving a local linear approximation of the (minimization) game, subject to a quadratic penalty that expresses her limited confidence in the global accuracy of the model. \n\n```The natural generalization of this idea to the competitive case should then be given by the two players solving a local approximation of the true game, both subject to a quadratic penalty that expresses their limited confidence in the accuracy of the local approximation.```\n\nIn order to implement this idea, we need to find the appropriate way to generalize the linear approximation in the single agent setting to the competitive setting. \n\nAuthors suggest to use a **bilinear** approximation in the two-player setting.\nSince the bilinear approximation is the lowest order approximation that can capture some interaction between the two players, they argue that the natural generalization of gradient descent to competitive optimization is not GDA, but rather the update rule $(x_{k+1},y_{k+1}) = (x_k,y_k) + (x,y)$, where $(x,y)$ is a Nash equilibrium of **the game**.\n\n\\begin{align}\n \\begin{split}\n \\label{eqn:localgame}\n \\min_{x \\in R^m} x^{\\top} \\nabla_x f &+ x^{\\top} D_{xy}^2 f y + y^{\\top} \\nabla_y f + \\frac{1}{2\\eta} x^{\\top} x \\\\\n \\min_{y \\in R^n} y^{\\top} \\nabla_y g &+ y^{\\top} D_{yx}^2 g x + x^{\\top} \\nabla_x g + \\frac{1}{2\\eta} y^{\\top} y.\n \\end{split}\n\\end{align}\n\nIndeed, the (unique) Nash equilibrium of the above Game can be computed in closed form.\n\n\n## Proposed method\n**Among all (possibly randomized) strategies with finite first moment, the only Nash equilibrium of `the Game` is given by\n\\begin{align}\n\\label{eqn:nash}\n&x = -\\eta \\left( Id - \\eta^2 D_{xy}^2f D_{yx}^2 g \\right)^{-1} \n \\left( \\nabla_{x} f - \\eta D_{xy}^2f \\nabla_{y} g \\right) \\\\\n&y = -\\eta \\left( Id - \\eta^2 D_{yx}^2g D_{xy}^2 f \\right)^{-1} \n \\left( \\nabla_{y} g - \\eta D_{yx}^2g \\nabla_{x} f \\right),\n\\end{align}\ngiven that the matrix inverses in the above expression exist.** \n\nNote that the matrix inverses exist for all but one value of $\\eta$, and for all $\\eta$ in the case of a zero sum game.\n\nAccording to the above Theorem, the Game has exactly one optimal pair of strategies, which is deter-ministic. Thus, we can use these strategies as an update rule, generalizing the idea of local optimalityfrom the single\u2013 to the multi agent setting and obtaining the following Algorithm.\n\n`Competitive Gradient Descent (CGD)`\n\\begin{align}\nfor\\ (0 <= k <= N-1)\\\\\n&x_{k+1} = x_{k} - \\eta \\left( Id - \\eta^2 D_{xy}^2f D_{yx}^2 g \\right)^{-1}\\left( \\nabla_{x} f - \\eta D_{xy}^2f \\nabla_{y} g \\right)\\\\\n&y_{k+1} = y_{k} - \\eta \\left( Id - \\eta^2 D_{yx}^2g D_{xy}^2 f \\right)^{-1} \n \\left( \\nabla_{y} g - \\eta D_{yx}^2g \\nabla_{x} f \\right)\\\\\n return\\ (x_{N},y_{N})\\;\n\\end{align}\n\n\n\n\n**What I think that they think that I think ... that they do**: Another game-theoretic interpretation of CGD follows from the observation that its update rule can be written as \n\n\\begin{equation}\n\\begin{pmatrix}\n \\Delta x\\\\\n \\Delta y\n\\end{pmatrix} = -\n\\begin{pmatrix}\n Id & \\eta D_{xy}^2 f \\\\\n \\eta D_{yx}^2 g & Id \n\\end{pmatrix}^{-1}\n\\begin{pmatrix}\n \\nabla_{x} f\\\\\n \\nabla_{y} g\n\\end{pmatrix}.\n\\end{equation}\n\nApplying the expansion $ \\lambda_{\\max} (A) < 1 \\Rightarrow \\left( Id - A \\right)^{-1} = \\lim_{N \\rightarrow \\infty} \\sum_{k=0}^{N} A^k$ to the above equation, we observe that: \\\\\n\n1. The first partial sum ($N = 0$) corresponds to the optimal strategy if the other player's strategy stays constant (GDA).\n2. The second partial sum ($N = 1$) corresponds to the optimal strategy if the other player thinks that the other player's strategy stays constant (LCGD).\n3. The third partial sum ($N = 2$) corresponds to the optimal strategy if the other player thinks that the other player thinks that the other player's strategy stays constant, and so forth, until the Nash equilibrium is recovered in the limit.\n\n\n\n\n## Comparison\nThese six algorithms amount to different subsets of the following four terms.\n\n\\begin{align*}\n & \\text{GDA: } &\\Delta x = &&&- \\nabla_x f&\\\\\n & \\text{LCGD: } &\\Delta x = &&&- \\nabla_x f& &-\\eta D_{xy}^2 f \\nabla_y f&\\\\\n & \\text{SGA: } &\\Delta x = &&&- \\nabla_x f& &- \\gamma D_{xy}^2 f \\nabla_y f& & & \\\\\n & \\text{ConOpt: } &\\Delta x = &&&- \\nabla_x f& &- \\gamma D_{xy}^2 f \\nabla_y f& &- \\gamma D_{xx}^2 f \\nabla_x f& \\\\\n & \\text{OGDA: } &\\Delta x \\approx &&&- \\nabla_x f& &-\\eta D_{xy}^2 f \\nabla_y f& &+\\eta D_{xx}^2 f \\nabla_x f& \\\\\n & \\text{CGD: } &\\Delta x = &\\left(Id + \\eta^2 D_{xy}^2 f D_{yx}^2 f\\right)^{-1}&\\bigl( &- \\nabla_x f& &-\\eta D_{xy}^2 f \\nabla_y f& & & \\bigr)\n \\end{align*}\n\n1. The **gradient term** $-\\nabla_{x}f$, $\\nabla_{y}f$ which corresponds to the most immediate way in which the players can improve their cost.\n\n\n\n2. The **competitive term** $-D_{xy}f \\nabla_yf$, $D_{yx}f \\nabla_x f$ which can be interpreted either as anticipating the other player to use the naive (GDA) strategy, or as decreasing the other players influence (by decreasing their gradient).\n\n\n\n3. The **consensus term** $ \\pm D_{xx}^2 \\nabla_x f$, $\\mp D_{yy}^2 \\nabla_y f$ that determines whether the players prefer to decrease their gradient ($\\pm = +$) or to increase it ($\\pm = -$). The former corresponds the players seeking consensus, whereas the latter can be seen as the opposite of consensus. (It also corresponds to an approximate Newton's method. \\footnote{Applying a damped and regularized Newton's method to the optimization problem of Player 1 would amount to choosing $x_{k+1} = x_{k} - \\eta(Id + \\eta D_{xx}^2)^{-1} f \\nabla_x f \\approx x_{k} - \\eta( \\nabla_xf - \\eta D_{xx}^{2}f \\nabla_x f)$, for $\\|\\eta D_{xx}^2f\\| \\ll 1$.)\n\n\n\n\n4. The **equilibrium term** $(Id + \\eta^2 D_{xy}^2 D_{yx}^2 f)^{-1}$, $(Id + \\eta^2 D_{yx}^2 D_{xy}^2 f)^{-1}$, which arises from the players solving for the Nash equilibrium. \n This term lets each player prefer strategies that are less vulnerable to the actions of the other player.\n\n\n## Code Implementation\n\nThe competitive gradeint descent algorithm contains gradient, competitive and equilibrium term. So, we need to efficiently calculat them. The equibrium term is a matrix inverse\n\n### Computing Hessian vector products\n\nThe algorithm requires products of the mixed Hessian $v \\mapsto D_{xy}f v$ and $v \\mapsto D_{yx}g v$, which we want to compute using automatic differentiation.\n\nMany AD frameworks, like Autograd (https://github.com/HIPS/autograd) and ForwardDiff(https://github.com/JuliaDiff/ForwardDiff.jl) together with ReverseDiff(https://github.com/JuliaDiff/ReverseDiff.jl) support this procedure. While the authors used the AD frameworks from Julia, I will be using Autograd from PyTorch (https://pytorch.org/docs/stable/autograd.html)\n\n### Matrix inversion for the equilibrium term\nAuthors propose to use iterative methods to approximate the inverse-matrix vector products arising in the *equilibrium term*.\nAuthors focus on zero-sum games, where the matrix is always symmetric positive definite, making the [conjugate gradient (CG)](https://en.wikipedia.org/wiki/Conjugate_gradient_method) algorithm the method of choice. \nThey also suggest terminating the iterative solver after a given relative decrease of the residual is achieved ($\\| M x - y \\| \\leq \\epsilon \\|x\\|$ for a small parameter $\\epsilon$, when solving the system $Mx = y$).\n\nBriefly, conjugate gradient (CG) iteratively solves the system $Mx = y$ for $x$ without calculating $M^{-1}$.\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\n\"\"\"\nSimple python implemetation of CG tested on an example\n\"\"\"\n\n# Problem setup\nA = np.matrix([[3.0, 2.0], \n [2.0, 6.0]]) # the matrix A in : Ax = b\nb = np.matrix([[2.0], \n [-8.0]]) # we will use the convention that a vector is a column vector\n\n\n# solution approach\nx = np.matrix([[-2.0],\n [-2.0]])\n\nsteps = [(-2.0, -2.0)] # modify according to x\ni = 0\nimax = 10\neps = 0.01\nr = b - A * x\nd = r\ndeltanew = r.T * r\ndelta0 = deltanew\nwhile i < imax and deltanew > eps**2 * delta0:\n alpha = float(deltanew / float(d.T * (A * d)))\n x = x + alpha * d\n steps.append((x[0, 0], x[1, 0]))\n r = b - A * x\n deltaold = deltanew\n deltanew = r.T * r\n beta = float(deltanew / float(deltaold))\n d = r + beta * d\n i += 1\n \nprint(\"Solution vector x* for Ax = b :\")\nprint(x)\n\nprint(\"And the steps taken by algorithm : \", steps)\nplt.plot(steps)\n```\n\n#### Now to solve our problem of CGD, the following equation\n\n\\begin{equation}\n\\begin{pmatrix}\n \\Delta x\\\\\n \\Delta y\n\\end{pmatrix} = -\n\\begin{pmatrix}\n Id & \\eta D_{xy}^2 f \\\\\n \\eta D_{yx}^2 g & Id \n\\end{pmatrix}^{-1}\n\\begin{pmatrix}\n \\nabla_{x} f\\\\\n \\nabla_{y} g\n\\end{pmatrix}.\n\\end{equation}\n\n#### can be written as \n\n\\begin{equation}\n\\begin{pmatrix}\n Id & \\eta D_{xy}^2 f \\\\\n \\eta D_{yx}^2 g & Id \n\\end{pmatrix}\n\\begin{pmatrix}\n \\Delta x\\\\\n \\Delta y\n\\end{pmatrix} = -\n\\begin{pmatrix}\n \\nabla_{x} f\\\\\n \\nabla_{y} g\n\\end{pmatrix}.\n\\end{equation}\n\n#### so that the conjugate gradient method can used to calculate $\\Delta x$ and $\\Delta y$ without inverting the matrix.\n\n\n## Conclusion\n\nIn the words of the authors of original paper,\n`We propose a novel and natural generalization of gradient descent to competitive optimization. Besides its attractive game-theoretic interpretation, the algorithm shows improved robustness properties compared to the existing methods, which we study using a combination of theoretical analysis and computational experiments.`\n\nLook out to the conclusion section of the [paper](https://arxiv.org/pdf/1905.12103.pdf) for extensive conclusion and future aspects.\nRefers to the Experiment and Conclusion section of the reproducibility [report](https://gopikishan14.github.io/Reproducibility_Challenge_NeurIPS_2019/index.html) for details on replication.\n", "meta": {"hexsha": "e7dbbf6780d4dd5ae8bf1ca0e53e548d7dc8f972", "size": 35119, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": ".ipynb_checkpoints/README-checkpoint.ipynb", "max_stars_repo_name": "GopiKishan14/Reproducibility_Challenge_NeurIPS_2019", "max_stars_repo_head_hexsha": "fccee3f5ac8894580a88fc178571107024dd1cfa", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2020-01-30T03:36:22.000Z", "max_stars_repo_stars_event_max_datetime": "2020-08-25T19:43:07.000Z", "max_issues_repo_path": "README.ipynb", "max_issues_repo_name": "GopiKishan14/Reproducibility_Challenge_NeurIPS_2019", "max_issues_repo_head_hexsha": "fccee3f5ac8894580a88fc178571107024dd1cfa", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "README.ipynb", "max_forks_repo_name": "GopiKishan14/Reproducibility_Challenge_NeurIPS_2019", "max_forks_repo_head_hexsha": "fccee3f5ac8894580a88fc178571107024dd1cfa", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 95.9535519126, "max_line_length": 17128, "alphanum_fraction": 0.7758763063, "converted": true, "num_tokens": 3959, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.7520125737597972, "lm_q1q2_score": 0.4285363590538048}} {"text": "# QG8 demo - Writing and reading a quantum graph in Python\n\nAuthor: S. Whitlock (whitlock@unistra.fr)\nVersion 1.0: 18 July 2021 \n\nThe latest version of this [IPython notebook] (http://ipython.org/ipython-doc/dev/interactive/htmlnotebook.html) is available at https://github.com/aQCess/QG8-Python.\n\n## Introduction\n\nQG8 is a lightweight data format for storing and exchanging numerical tensor data and data flow graphs in a binary format. It was created as way to store quantum models and data in a single file, including quantum gate and quantum circuit specifications, control pulse definitions, time independent and time-dependent model Hamiltonians, device characteristics, calibration data and noise models, user defined control flow instructions and simulation or measurement results. \n\nIn this notebook we will build a quantum graph representing a quantum gate protocol with time-dependent control pulses. The graph can be written to a QG8 file and read again. This allows for defining problem specifications in one language or one one system and executing the graph on another. In this example we show how a QG8 graph can be integrated with Python and the QuTip package in order to simulate the gate.\n\nFor more information see the QG8 project GitHub page at https://github.com/aQCess/QG8.\n\n### Installation\n\nTo use QG8 just copy the qg8 folder to the python path or put it in your project directory. \n\n> *For this demonstration we additionally make use of the Numpy and QuTip packages. To install QuTiP, download the latest release from https://qutip.org.*\n\n> *For an introduction to QuTip notebooks we recommend the tutorials page of the QuTip project https://qutip.org/tutorials.html and the guide to basic operations https://qutip.org/docs/latest/guide/guide-basics.html.*\n\n### Working principle\n\nThe core idea of the QG8 data format and graph based representations of quantum systems in general is to store quantum data and operations and as nodes of a data flow graph. Each node accepts information from its input nodes and edges represent the flow of information through the graph. The advantage of this approach is that it allows to store all the data required for a computation as well as the instructions on how to perform operations on the data in a single object which can then easily be exchanged between different systems. \n\nIn QG8 we classify nodes according to two main types: (i) *tensor nodes* which contain data. These act as input nodes or intermediate placeholder nodes for the graph. (ii) *op nodes* which do not contain data but indicate which operations should be performed on the data. In QG8 we do not provide an exhaustive set of possible operations, rather we allow the user to write their own operations which will be called by the graph when executed.\n\n## Getting started with a QG8 graph: `QuantumGraph()`\n\nLets first include the necessary modules.\n\n\n```python\n%matplotlib inline\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nimport qg8\n```\n\nThe starting point for our demonstration is the `QuantumGraph` class. This is the basic object which stores all the information for the quantum gate as a list of nodes.\n\nWe can create an empty `QuantumGraph` instance by calling `QuantumGraph()`.\n\n\n```python\nmygraph = qg8.QuantumGraph()\n```\n\nA quantum graph is a container for QG8 nodes. For the moment we can see that it is empty.\n\n\n```python\nmygraph.nodes\n```\n\n\n\n\n []\n\n\n\n### QG8 node: `QG8Node`\n\nGraph nodes are instances of the general `QG8Node` class (which in turn inherits from `qg8_chunk` in the core qg8 implementation). QG8 nodes can be either: (i) *tensor nodes*, containing input or placeholders for output data which can be represented as a scalar, vector, matrix or higher order tensors of numerical values; or (ii) *op nodes*, that describe operations that should be done on the tensors when the graph is run. \n\nThe node types that are implemented in this demo can be found in the `type_registry` dictionary. More custom node types can be defined and added in the `qg8.ops` library.\n\n\n```python\nqg8.node_types()\n```\n\n\n\n\n {'constant',\n 'expectationvalue',\n 'input',\n 'join',\n 'ket',\n 'matmul',\n 'observable',\n 'operator',\n 'solvequtip',\n 'time',\n 'track'}\n\n\n\nAs a simple example, lets create two generic `input` tensor nodes and fill them with data from a numpy array using the `QuantumGraph.input` method.\n\n\n```python\nx = np.array([1., 2., 3., 4., 5.])\nM = np.array([[0,0,1,1,0], [1,0,0,0,0], [1,0,1,0,0], [0,0,0,1,1], [1,0,0,0,1]]) \n\nvector = mygraph.input(x)\nmatrix = mygraph.input(M)\n```\n\nHere we have used the `np.array` method to construct a five element vector (a rank 1 tensor with shape [5]) and a 5x5 element matrix (a rank 2 tensor).\n\nQG8 tensor data are efficiently stored in a sparse format as a list of indice arrays `(i,j...)` and values `re` and `im` corresponding to the real and imaginary parts of non-zero elements. \n\n> *If you want to store all elements of the tensor, including zeros, you can use the argument `packing='full'` when creating the node.* \n\n> *By default QG8 stores data in a format depending on the data type of the numpy array. If you want to specify a particular data type you can use the `dtype='name'` keyword argument when defining tensor nodes, where 'name' is a numerical type string accepted as input to the `numpy.dtype()` class.*\n\n\nOne can see what is inside the tensor by calling `node.indices()`, `node.values()` or the `node.full()` method\n\n\n```python\nmatrix.indices(), matrix.values()\n```\n\n\n\n\n ([array([0, 0, 1, 2, 2, 3, 3, 4, 4], dtype=uint8),\n array([2, 3, 0, 0, 2, 3, 4, 0, 4], dtype=uint8)],\n array([1, 1, 1, 1, 1, 1, 1, 1, 1], dtype=int64))\n\n\n\n\n```python\nmatrix.full()\n```\n\n\n\n\n array([[0, 0, 1, 1, 0],\n [1, 0, 0, 0, 0],\n [1, 0, 1, 0, 0],\n [0, 0, 0, 1, 1],\n [1, 0, 0, 0, 1]])\n\n\n\nWe can now add a simple op node which performs a matrix product of two nodes. In this case the behavior is that of `numpy.matmul()`.\n\n\n```python\nproduct = mygraph.matmul(matrix, vector)\n```\n\nThe functions which are called to evaluate op nodes are defined in the `qg8.ops` library. Each op function takes a `QG8Node` as an input and processes them by setting the `.output` attribute depending on their assigned input nodes (`vector` and `matrix` in this example). These functions are evaluated according to their topological order in the graph when `QuantumGraph.run()` is called.\n\n> *While we have assigned the nodes `vector` and `matrix`, these are just a pointers to nodes in the graph. If you must remove a node from the graph one can use the `mygraph.remove(qg8_node)` method, but this will not remove dependencies of other nodes*.\n\nNow our quantum graph should contain three nodes. \n\n> *Beware that running the above code several times results in adding several nodes to the graph.*\n\n\n\nBefore we save the graph we must also store the edges between graph nodes (i.e. indicating that `matrix` and `vector` are input nodes to `product`). This can be done by building an adjacency matrix for the graph. Adjacency matrices are not regular graph nodes and will be stored in the graph along with the nodes as \"chunks\".\n\n\n```python\nA = mygraph.adjacency_matrix()\nmygraph.chunks\n```\n\n\n\n\n [,\n ,\n ,\n ]\n\n\n\nTo save a graph to file, call `qg8.save('filename',graph)`\n\n\n```python\nqg8.save('mygraph.qg8',mygraph);\n```\n\nIt can be read in again as a new graph.\n\n\n```python\nsavedgraph = qg8.load('mygraph.qg8')\nsavedgraph.chunks\n```\n\n\n\n\n [,\n ,\n ,\n ]\n\n\n\nWe can now execute the graph by calling the `run` method and verify the result. By default the graph is computed up to the last node added to the graph.\n\n\n```python\nresult = qg8.run(savedgraph)\n\nprint(result)\n\nall(result == np.matmul(M,x)) # verify result\n```\n\n [7. 1. 4. 9. 6.]\n\n\n\n\n\n True\n\n\n\n## Working example: iSWAP gate\n\nWe now show how to build a complete graph for simulating a quantum gate, including Hamiltonian terms, operators, observables and op nodes which connect them. \n\nFor illustration we consider a two-qubit iSWAP gate for Rydberg atom qubits, which is a special case of the XY gate family described in M. Morgardo & S. Whitlock, AVS Quantum Sci. 3, 023501 (2021) (open access link: https://arxiv.org/abs/2011.03031). This specific gate protocol works by coupling the qubit states $0$ and $1$ to two auxiliary states called $S$ and $P$ and by exploiting two-body exchange interactions $SP \\leftrightarrow PS$.\n\n### Building the Hamiltonian\n\n\n```python\niSWAP = qg8.QuantumGraph()\n```\n\nTo represent the Hamiltonian of the two-qubit system use a decomposition in terms of time-independent operators $\\hat h_j$ and time-dependent coefficients $c_j(t)$\n \n$$\\hat H = \\sum_j c_j(t) \\hat h_j + h.c.$$\n\nwhere $h.c.$ stands for Hermitian conjugate. \n\n> *With QG8 it is not necessary to save the Hermitian conjugate operators, since we can use the packing argument `packing=\"half-hermitian\"`. The hermitian conjugate terms will then be automatically added in the solver.*\n\n#### Operators\n\nFor this demonstration we define three operators describing the single particle couplings acting on both qubits and the two-particle exchange interaction operator:\n\\begin{align}\n\\hat h_1 &= |S\\rangle\\langle 1|\\otimes \\mathbb{1} + \\mathbb{1}\\otimes |S\\rangle\\langle 1|\\\\\n\\hat h_2 &= |P\\rangle\\langle 0|\\otimes \\mathbb{1} + \\mathbb{1}\\otimes |P\\rangle\\langle 0|\\\\\n\\hat h_3 &= |P S\\rangle\\langle S P|\n\\end{align}\n\nwhere the subscripts indicate which qubit the operator is acting on, and $|\\alpha \\beta\\rangle = |\\alpha\\rangle\\otimes|\\beta\\rangle$ represents a two qubit state, $\\otimes$ is the tensor product and $\\mathbb{1}$ is the identity operator in the single qubit basis.\n\nTo generate these operators as Numpy arrays, lets start by defining a set of single qubit basis vectors and some helper functions which return matrices representing single qubit and two qubit operators in the two-qubit basis. *We can also save a little bit of space by expressing these and the corresponding operators as 8 bit integers.*\n\n\n```python\n# single qubit basis vectors\nzero = np.array([[1],[0],[0],[0]],dtype='uint8')\none = np.array([[0],[1],[0],[0]],dtype='uint8')\nS = np.array([[0],[0],[1],[0]],dtype='uint8')\nP = np.array([[0],[0],[0],[1]],dtype='uint8')\n```\n\n\n```python\n# Helper functions for constructing single qubit and two-qubit operators as Numpy matrices \ndef O1(s1,s2,target=1):\n \"\"\"\n Function which returns a general one-qubit operator |s1>S for both atoms\n\nh2 = iSWAP.operator(O1(P, zero, target=1) + O1(P, zero, target=2), \n packing = \"half-hermitian\") # 0->P for both atoms\n```\n\nEach call of the operator() method adds a new node to the graph. You can also give the operator a label using the optional `string_id` keyword argument (when written to file this will be truncated to 16 characters).\n\n\n```python\nh3 = iSWAP.operator(O2(P, S, S, P), \n packing= \"half-hermitian\",\n string_id= \"interaction\", \n ) # SP->PS for both atoms\n```\n\n#### Time-dependent coefficients\n\nTo define the time-dependent coefficients lets first create a `time` tensor node which contains the time samples. \n\n> *QuTip currently only accepts a single set of equidistantly spaced time points, so we will use `np.linspace` to define the time grid and to save a little bit of space we can use integer values (samples).*\n\n\n```python\n# define time samples to be used in the simulation\nt_list = np.linspace(0,40,41, dtype ='uint8')\n\ntime = iSWAP.time(t_list) # from 0 to 1 in 41 steps\n```\n\nIt is also possible to integrate custom objects with QG8 graphs. In this case one should make sure that the object has a `.shape` attribute as well as `.index()` and `.values()` methods which return numpy arrays.\n\nWe will now define two `track` tensor nodes to contain the coefficients themselves which will be associated with the first two operator terms in the Hamiltonian. For this specific example we will use a custom `Track` class which includes two methods for generating pulses. The parameters of these pulses have been optimized by hand.\n\n\n```python\nch1 = qg8.Track(t_list)\nch1.add_sinepulse(start=0, duration=12, amplitude=0.2613)\nch1.add_sinepulse(start=28, duration=12, amplitude=0.2613)\n\nch2 = qg8.Track(t_list)\nch2.add_squarepulse(start=12, duration=8, amplitude=0.3912)\nch2.add_squarepulse(start=20, duration=8, amplitude=-0.3912)\n\ntrack1 = iSWAP.track(ch1, dtype='float32') \ntrack2 = iSWAP.track(ch2, dtype='float32')\n```\n\nWe can extract the full samples as a numpy array using `node.full()` and plot them.\n\n\n```python\nplt.plot(track1.full());\nplt.plot(track2.full());\n```\n\nThe third coefficient describing the exchange interactions will be time-independent. So in this case it is simplest to add a `constant` node.\n\n\n```python\ntrack3 = iSWAP.constant(0.1988)\n```\n\n### Setting up a simulation with Op nodes\n\nFor this demonstration we will use the QuTip solver wrapped as a `solvequtip` op node.\n\nThe `solvequtip` op node takes as inputs a `ket` tensor node defining the initial state of the simulation a `time` tensor node defining the time samples used to output the simulation results and an arbitrary number of Hamiltonian terms encoded as `(operator,track)` pairs, in that order. Upon running the graph, the output of the `solvequtip` returns an array of kets evaluated at each time step (a rank 2 tensor).\n\nThe initial state for the simulation will be $|01\\rangle$, expressed as a row vector (using `np.kron` to do the tensor product).\n\n\n```python\npsi0 = iSWAP.ket(np.kron(zero,one))\n```\n\nWe can create the `(operator,track)` pairs using join nodes.\n\n\n```python\nop1 = iSWAP.join(h1,track1); op2 = iSWAP.join(h2,track2); op3 = iSWAP.join(h3,track3)\n```\n\n> *Reminder: these operations are not actually joining anything at this stage. It is just adding three new `join` nodes to the graph which have `operators` and `tracks` as input nodes. The process of performing the operations will only be done when `QuantumGraph.run()` is called.*\n\nNow we can add the `solvequtip` op node to the graph.\n\n\n```python\npsif = iSWAP.solvequtip(psi0,time,op1,op2,op3)\n```\n\n#### Expectation values\n\nWhile we could just return these wavefunctions as the solution, it may be convenient to directly compute expectation values for certain observables. \n\nFor that we define two projection operators, one for the initial state $|01\\rangle$ and one for the desired target state $|10\\rangle$ and add them as inputs to nodes which compute the expectation value of the time-dependent wavefunction given by the `solvequtip` node. Finally we use a `join` node to combine their outputs into one object for plotting.\n\n\n```python\nP_01 = iSWAP.expectationvalue(psif,iSWAP.observable(O2(zero,one,zero,one)))\n\nP_10 = iSWAP.expectationvalue(psif,iSWAP.observable(O2(one,zero,one,zero)))\n\noutput = iSWAP.join(P_01,P_10)\n```\n\nThats it. Now we have completed our graph. Lets take a look at the list of nodes.\n\n\n```python\niSWAP.nodes\n```\n\n\n\n\n [,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ]\n\n\n\nFinally we must build the adjacency matrix which describes the connections between nodes and add it to the graph.\n\n\n```python\nA = iSWAP.adjacency_matrix()\n```\n\n### Write the graph to a QG8 file\n\n\n```python\nqg8.save(\"iSWAP.qg8\", iSWAP);\n```\n\nThe resulting file is a bit under 1kB in size.\n\n\n```python\niSWAP.datasize() # datasize in bytes\n```\n\n\n\n\n 914\n\n\n\n### Read in the graph\n\n\n```python\nsavedgate = qg8.load(\"iSWAP.qg8\")\nsavedgate.nodes\n```\n\n\n\n\n [,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ]\n\n\n\n### Simulate the graph\n\n\n```python\nresult = qg8.run(savedgate)\n```\n\n\n```python\nplt.plot(result[0],label=r'P_{01}');\nplt.plot(result[1],label=r'P_{10}');\nplt.ylabel(\"populations\")\nplt.xlabel(\"time (samples)\")\nplt.legend();\n```\n\nThis shows that starting from the $|01\\rangle$ state the system evolves to the $|10\\rangle$ state, thereby realizing a swap operation. At intermediate times the system evolves through auxilliary two-qubit states involving the $|S\\rangle$ and $|P\\rangle$ states (region where $P_{01}$ and $P_{10}$ are approximately zero), but returns back to the computational subspace at the end of the protocol. \n\nWe leave it as an exercise to confirm the full gate matrix for this protocol and to experiment with your own gate protocols or other quantum problems.\n", "meta": {"hexsha": "50e4f9430d861538aa6930462cce70732715d4ae", "size": 62084, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/Getting_started_with_QG8.ipynb", "max_stars_repo_name": "aQCess/QG8-Python", "max_stars_repo_head_hexsha": "581a92fb5e8b77787f692266e473407862ee460d", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tutorials/Getting_started_with_QG8.ipynb", "max_issues_repo_name": "aQCess/QG8-Python", "max_issues_repo_head_hexsha": "581a92fb5e8b77787f692266e473407862ee460d", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/Getting_started_with_QG8.ipynb", "max_forks_repo_name": "aQCess/QG8-Python", "max_forks_repo_head_hexsha": "581a92fb5e8b77787f692266e473407862ee460d", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-07-30T14:59:18.000Z", "max_forks_repo_forks_event_max_datetime": "2021-07-30T14:59:18.000Z", "avg_line_length": 61.347826087, "max_line_length": 15408, "alphanum_fraction": 0.7713903743, "converted": true, "num_tokens": 5375, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.6688802669716106, "lm_q1q2_score": 0.428508681616897}} {"text": "\n# Logbook\n\n\n```python\n# %load ../imports.py\n\"\"\"\nThese is the standard setup for the notebooks.\n\"\"\"\n\n%matplotlib inline\n%load_ext autoreload\n%autoreload 2\n\n#from jupyterthemes import jtplot\n#jtplot.style(theme='onedork', context='notebook', ticks=True, grid=False)\n\nimport pandas as pd\npd.options.display.max_rows = 999\npd.options.display.max_columns = 999\npd.set_option(\"display.max_columns\", None)\nimport numpy as np\nimport os\nimport matplotlib.pyplot as plt\n#plt.style.use('paper')\n\n#import data\nimport copy\nfrom rolldecay.bis_system import BisSystem\nfrom rolldecay import database\nfrom mdldb.tables import Run\n\nfrom sklearn.pipeline import Pipeline\nfrom rolldecayestimators.transformers import CutTransformer, LowpassFilterDerivatorTransformer, ScaleFactorTransformer, OffsetTransformer\nfrom rolldecayestimators.direct_estimator_cubic import EstimatorQuadraticB, EstimatorCubic\nfrom rolldecayestimators.ikeda_estimator import IkedaQuadraticEstimator\nimport rolldecayestimators.equations as equations\nimport rolldecayestimators.lambdas as lambdas\nfrom rolldecayestimators.substitute_dynamic_symbols import lambdify\nimport rolldecayestimators.symbols as symbols\nimport sympy as sp\n\nfrom sklearn.metrics import r2_score\n\n\n```\n\n## Nomenclature\n* **Ikeda** refer to the original ikeda method (require strip calculations and Lewis sections).\n* **Simplified Ikeda** or **SI** refer to the simplified Ikeda method (explicit formula).\n\n[Logbook](#logbook)\n## 2020-09-22\nI've been working with adding additional material to extend the paper for the [SAOC](https://www.tandfonline.com/toc/tsos20/current).\n\n### Ikeda for one ship\nI have implemented the original ikeda method and made calculations for one of the ships in the database: [01.01_ikeda_one_ship](06_ikeda/01.01_ikeda_one_ship.ipynb). I developed a python wrapper for SSPA strip calculation tool *ScoresII* here: [pyscores2](https://github.com/martinlarsalbert/pyscores2) and calculated the wave roll damping $B_W$.\nThe results were compared with the Simplified Ikeda (SI).\n* It was found that the $B_W$ was quite different. It was however hard to say which one was the better. \n\n* It was also found that the speed dependecy of the eddy component $B_E$ was not implemented as in .\n\n* It was also found that the $B_{BK}$ contribution was quite different: [Ikeda vs. SI](06_ikeda/01.01_ikeda_one_ship.ipynb#ikeda_vs_si). \n\n* For this ship $B_{BK}$ calculated with original ikeda implementation gave better results:\n[Ikeda vs. SI2](06_ikeda/01.01_ikeda_one_ship.ipynb#ikeda_vs_si2).\n\nI made a new comparison between SI and model test results where the influence of findings above were investigated ([07.2_si_compare](04_simplified_ikeda/07.2_si_compare.ipynb)). The $B_E$ speed dependency gave an improvement. Switching the $B_{BK}$ also gave an improved accuracy with maximum accuracy with fixed bilge radius $ \\frac{R}{beam}=0.08$ according to:[scores](04_simplified_ikeda/07.2_si_compare.ipynb#scores). \n\n### Validation \nI also made a validation study to reproduce results from Carl-Johan for the *S175* ship: [Ikeda S175](06_ikeda/01.02_ikeda_S175.ipynb).\n\n\n## 2020-09-23\n\n### Recalculating $B_E$\n[Sectional Lewis coefficients](06_ikeda/01.01_ikeda_one_ship.ipynb#lewis_coefficients) have been calculated. These are used to calculate the eddy damping: $B_E$ [here](06_ikeda/01.01_ikeda_one_ship.ipynb#eddy). This is also a translation of Carl-Johans matlab code which seams to be an implementation according to with some exceptions where changes according to . It was a bit unclear if the $\\overline{OG}$ should be possitive uppward (Journe) or into the water (Ikeda). It was also unclear if Journe or Ikeda was the best, this should be investigated!\n\n also proposes these formulas to estimate the bilge radius:\n$$\nR=\\left\\{\\begin{array}{ll}\n2 a \\sqrt{\\frac{a_{0}(\\sigma-1)}{\\pi-4}} & (R1\\right) \\\\\nB / 2 & \\left(H_{0} \\leq 1, R / d>H_{0}\\right)\n\\end{array}\\right.\n$$\n\n## 2020-09-24\n* I found that there might be a difference in the speed dependency between Ikeda and Journe:\n\n\n$$\n\\begin{aligned}\n\\frac{B_{W}}{B_{k 0}} &=0.5\\left[\\left(\\left(A_{2}+1\\right)+\\left(A_{2}-1\\right) \\tanh (20 \\tau-b)\\right\\}\\right.\\\\\n&\\left.+\\left(2 A_{1}-A_{2}-1\\right) \\exp \\left\\{-150(\\tau-0.25)^{2}\\right\\}\\right]\n\\end{aligned}\n$$\n\n\n\n$$\nB_{44 S}=B_{44} \\cdot\\left\\{0.5 \\cdot\\left(\\begin{array}{l}\nA_{2}+1+\\left(A_{2}-1\\right) \\cdot \\tanh [20 \\cdot(\\Omega-0.3)] \\\\\n+\\left(2 \\cdot A_{1}-A_{2}-1\\right) \\cdot e^{-150(\\omega-0.25)^{2}}\n\\end{array}\\right)-1.0\\right\\}\n$$\n\nThe difference is the $b$ coefficient which Journee has set to $20*0.3$. call this $b$ the extinction coefficient.\n\n* I found that the speed dependency correction for $B_W$ is not $1$ when the ship speed is $0$ which is quite strange. Is there an implementation error there?\n\nHere is the current implementation (which is clearly wrong according to the plot below, since it is not 1 at V=0). Have I, Carl-Johan or Journ\u00e9e made a misstaker here???\n\n\n```python\nfrom numpy import exp, tanh\ndef B_W_speed_correction_factor(w, V, d, g=9.81):\n \"\"\"\n Wave damping speed correction\n Parameters\n ----------\n w\n \"omega\" frequency of motion [rad/s]\n V\n ship speed [m/s]\n d\n ship draught [m]\n g\n gravity\n\n Returns\n -------\n Bw_div_Bw0\n Bw_div_Bw0 = B_W/B_W0\n\n \"\"\"\n OMEGA = w * V / g\n zeta_d = w ** 2 * d / g\n A1 = 1 + zeta_d ** (-1.2) * exp(-2 * zeta_d)\n A2 = 0.5 + zeta_d ** (-1) * exp(-2 * zeta_d)\n\n Bw_div_Bw0 = 0.5 * (\n ((A1 + 1) + (A2 - 1) * tanh(20 * (OMEGA - 0.3))) + (2 * A1 - A2 - 1) * exp(-150 * (OMEGA - 0.25) ** 2))\n\n return Bw_div_Bw0\n```\n\n\n```python\nw = 0.2\nlpp = 175\nomega_hat = 0.719\nbeam = 25.40\nd = 9.5\ng=9.81\nw = lambdas.omega(beam=beam, g=g, omega_hat=omega_hat)\nFn = np.linspace(0, 0.3, 100)\nV = Fn*np.sqrt(lpp*g)\n\n\nfactor = B_W_speed_correction_factor(w=w, V=V, d=d)\n\nfig,ax=plt.subplots()\nax.plot(Fn,factor)\nax.grid(True)\nax.set_ylabel(r'$\\frac{B_W}{B_{W0}}$')\nax.set_xlabel('Ship speed Fn [-]');\nax.set_title('$B_W$ Speed dependency factor');\n```\n\nThis plot should look something as:\n\n\n\nI would say that the Journee equation is even worse:\n\n\n```python\nfrom rolldecayestimators.ikeda_speed import B_W_speed_correction_factor_journee\n\nfactor = B_W_speed_correction_factor_journee(w=w, V=V, d=d)\n\nfig,ax=plt.subplots()\nax.plot(Fn,factor)\nax.grid(True)\nax.set_ylabel(r'$\\frac{B_W}{B_{W0}}$')\nax.set_xlabel('Ship speed Fn [-]');\nax.set_title('$B_W$ Speed dependency factor');\n```\n\nI went back to the original Ikeda paper and this implementation is in fact much more reasonable. The Himeno equation is the same (but the value of $b$) is not written out (it is $20*0.3$ in Ikeda). I found a typo in Carl-Johans implementation which was the root cause of this error. (But Journ\u00e9e is simply wrong still).\n\n\n```python\ndef B_W_speed_correction_factor_ikeda(w, V, d, b=20*0.3, g=9.81):\n \"\"\"\n Wave damping speed correction\n Parameters\n ----------\n w\n \"omega\" frequency of motion [rad/s]\n V\n ship speed [m/s]\n d\n ship draught [m]\n g\n gravity\n\n Returns\n -------\n Bw_div_Bw0\n Bw_div_Bw0 = B_W/B_W0\n\n \"\"\"\n tau=w*V/g\n zeta_d=w**2*d/g\n A1=1+zeta_d**(-1.2)*exp(-2*zeta_d);\n A2=0.5+zeta_d**(-1)*exp(-2*zeta_d);\n\n \n Bw_div_Bw0=0.5*((A2+1)+(A2-1)*tanh(20*tau-b)\n +(2*A1-A2-1)*exp(-150*(tau-0.25)**2))\n\n return Bw_div_Bw0\n```\n\n\n\n\n```python\nfactor = B_W_speed_correction_factor_ikeda(w=w, V=V, d=d)\n\nfig,ax=plt.subplots()\nax.plot(Fn,factor)\nax.grid(True)\nax.set_ylabel(r'$\\frac{B_W}{B_{W0}}$')\nax.set_xlabel('Ship speed Fn [-]');\nax.set_title('$B_W$ Speed dependency factor');\n```\n\n\nShould the speed dependence factor be modified, here is an example where $b$ is varied:\n\n\n```python\nfig,ax=plt.subplots()\nfig.set_size_inches(10,4)\n \nfor b in np.linspace(1,8,8):\n factor = B_W_speed_correction_factor_ikeda(w=w, V=V, d=d, b=b)\n\n ax.plot(Fn,factor, label='b=%0.1f'%b)\n\nax.grid(True)\nax.set_ylabel(r'$\\frac{B_W}{B_{W0}}$')\nax.set_xlabel('Ship speed Fn [-]');\nax.set_title('$B_W$ Speed dependency factor');\nax.set_ylim(1,3)\nax.legend();\n \n```\n\n\n```python\ndef f(b=6.0,omega_hat=0.719):\n \n w = lambdas.omega(beam=beam, g=g, omega_hat=omega_hat)\n factor = B_W_speed_correction_factor_ikeda(w=w, V=V, d=d, b=b)\n\n fig,ax=plt.subplots()\n fig.set_size_inches(10,4)\n ax.plot(Fn,factor)\n ax.grid(True)\n ax.set_ylabel(r'$\\frac{B_W}{B_{W0}}$')\n ax.set_xlabel('Ship speed Fn [-]');\n ax.set_title('$B_W$ Speed dependency factor');\n ax.set_ylim(1,4)\n```\n\n\n```python\nfrom ipywidgets import interactive\n\ninteractive_plot = interactive(f, b=(0.1,8,0.5),omega_hat=(0.2,1,0.1))\noutput = interactive_plot.children[-1]\noutput.layout.height = '350px'\ninteractive_plot\n```\n\n## 2020-09-25\n* Rerun the [04.2_maa_mdl_db_build.ipynb](rolldecay/02_roll_damping_DB/04.2_maa_mdl_db_build.ipynb).\n * With fixed $B_W$ speed dependency\n * Alternative bilge keel R/B=0.08\n* Rerun the cross validation and got ([quite much better score with SI](05_new_method/05.1_maa_new_method_and_correction.ipynb#score))\n\n* Made a new sensitivity study of the SI with some interesting plots: ([07.3_si_sensitivity2.ipynb](04_simplified_ikeda/07.3_si_sensitivity2.ipynb)).\n\n## 2020-09-28\n* Improved the plots in the ([07.3_si_sensitivity2.ipynb](04_simplified_ikeda/07.3_si_sensitivity2.ipynb)). It seems unlikely that the $\\hat{\\omega}$ and $F_n$ dependencies are correct in the SI.\n* Started to assemble more section data ([01.03_ikeda_many_dev](06_ikeda/01.03_ikeda_many_dev.ipynb)) to make a larger comparison with the Ikeda method.\n\n## 2020-09-30\n\nThere is equations to estimate the bilge radius according to :\n$$R_{b}=\\left\\{\\begin{array}{ll}\n2 D \\sqrt{\\frac{H_{0}(\\sigma-1)}{\\pi-4}} \\text { for } R_{b}1 \\\\\n\\frac{B}{2} & \\text { for } H_{0} \\leq 1, \\frac{R_{b}}{D}>H_{0}\n\\end{array}\\right\\}$$\nThis one has now been implemented (but I don't know if it is good).\n\n* Rerun this one ([ikeda_vs_si2](06_ikeda/01.01_ikeda_one_ship.ipynb#ikeda_vs_si2)) and realized that the frequency in scoresII is not scaled correctly... This needs to be investigated!\n* Also realized that I put a max constraint on the $\\hat{B_{W0}}$ in the SI method, this also needs to be investigated and reconsidered. (The constraint do however increase the accuracy).\n\n\n## 2020-10-01\n* The scaling of ScoresII is now solved.\n* Got quite good results with the Ikeda class now\n* Used the estimation of bilge keel radius (above) ([ikeda_r_ikeda](06_ikeda/01.01_ikeda_one_ship.ipynb#ikeda_r_ikeda)) which gave better results than the ([ikeda_r_guess](06_ikeda/01.01_ikeda_one_ship.ipynb#ikeda_r_guess)).\n* Finding a good value for bilge keel radius therefore seem to be important!\n* I investigated the [R-dependency](06_ikeda/01.01_ikeda_one_ship.ipynb#R_dependency) to confirm this. It seems that the bilge keel damping can differ 2-3 times depending on the bilge radius.\n\n\n## 2020-10-02\n* Got very good results with Ikeda for one of the ships: ([01.03_ikeda_many_dev](06_ikeda/01.03_ikeda_many_dev.ipynb))\n* I got very good results with the Ikeda method: ([01.04_ikeda_many](06_ikeda/01.04_ikeda_many.ipynb)).\n * Also made a comparison with the SI which had significantly worse results\n * The SI prediction error seem to occur mainly at high B/D.\n * It therefore seem that Ikedas method is working pretty well.\n * The SI is not, at least not outside its limits.\n\n\n## 2020-10-05\n* Investigated the residual between ikeda and si: [01.04_ikeda_many - residuals](06_ikeda/01.04_ikeda_many.ipynb#residuals) and it seems that the $B_W$ is the one to blame.\n\n## 2020-10-06\n* The wave damping error is not increasing with speed ([B_W-speed-factor-residual](06_ikeda/01.04_ikeda_many.ipynb#B_W_speed_factor_residual)).\n\n## 2020-10-14\nThere was a restriction on damping coefficients $B_1,...,B_3 >= 0$ in ```python rolldecayestimators.estimators_cubic ``` which seems to be a false assumption (based on results from ShipFlow Motions). This is however not supported by Ikeda ([07.3_si_sensitivity2.ipynb](04_simplified_ikeda/07.3_si_sensitivity2.ipynb)) where $B_2$ is always positive. This is an interesting deviation!\n\n* Droped this constraint and rebuild the roll damping database\n* Will need to rerun the analysis to see if this changes anything.\n\n## 2020-10-20\n* Looked a bit closer at the [08.1_Be_equation](04_simplified_ikeda/08.1_Be_equation.ipynb) and found it work well for some cases, but not for all. I'm not sure if I've made a misstake or if this is something interesting..?\n\n\n\n## 2020-10-21\nSpend the day rewriting the paper, based on feedback from Jonas. I'm beginning to doubt the approach of using limited/unlimited. Does it not make more sense to just look at accuracy within/outside the input limits of the SI?\n\n## 2020-10-22\n* Investigated the SI within its limits (also using the limited approach): [10.1_si_limits.ipynb](04_simplified_ikeda/10.1_si_limits.ipynb). And found that it gave quite good results within the limits (but the remaining tests are very few)\n* Also found that the **bBk/B** and **CMID** limits makes the other limits redundant for the present data\n* This analysis was made for the limited approach, does that makes sense or should the other one be used instead?\n* Calculated the correlation coefficient for the error, but Wengang said that it can only be used when the function is linear.\n* Made a version with the unlimited approach: [10.2_si_limits.ipynb](04_simplified_ikeda/10.2_si_limits.ipynb). Found some point with **B/d** > 4.5 but with low error.\n* Tried to find what other variable that is causing this behaviour, but haven't fully understood it. Hower made some 3D surface plots of the SI method, which reveals huge extrapolations!!!\n* Found for instance a huge [mountain](04_simplified_ikeda/10.2_si_limits.ipynb#mountain) in the polynom!\n\n\n## References\n
                                        \n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "9dd120b5417ebd0d88c1b8e2cecea8093d67f304", "size": 22821, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/rolldecay/logbook.ipynb", "max_stars_repo_name": "martinlarsalbert/rolldecay", "max_stars_repo_head_hexsha": "bee335c27e6519bd32ed9488751bc13e7f6fa9b3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/rolldecay/logbook.ipynb", "max_issues_repo_name": "martinlarsalbert/rolldecay", "max_issues_repo_head_hexsha": "bee335c27e6519bd32ed9488751bc13e7f6fa9b3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2021-02-02T23:07:50.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-13T03:27:41.000Z", "max_forks_repo_path": "notebooks/rolldecay/logbook.ipynb", "max_forks_repo_name": "martinlarsalbert/rolldecay", "max_forks_repo_head_hexsha": "bee335c27e6519bd32ed9488751bc13e7f6fa9b3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.927184466, "max_line_length": 652, "alphanum_fraction": 0.5912098506, "converted": true, "num_tokens": 4538, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.6688802603710086, "lm_q1q2_score": 0.42850867738831466}} {"text": "```python\n# -*- coding: utf-8 -*-\n\"\"\"\nSubsets of the Real Line\n\nThis module contains subsets of the real line that can be constructed\nas the union of a finite set of open and closed intervals.\n\nEXAMPLES::\n\n sage: RealSet(0,1)\n (0, 1)\n sage: RealSet((0,1), [2,3])\n (0, 1) \u222a [2, 3]\n sage: RealSet(-oo, oo)\n (-oo, +oo)\n\nBrackets must be balanced in Python, so the naive notation for\nhalf-open intervals does not work::\n\n sage: RealSet([0,1))\n Traceback (most recent call last):\n ...\n SyntaxError: ...\n\nInstead, you can use the following construction functions::\n\n sage: RealSet.open_closed(0,1)\n (0, 1]\n sage: RealSet.closed_open(0,1)\n [0, 1)\n sage: RealSet.point(1/2)\n {1/2}\n sage: RealSet.unbounded_below_open(0)\n (-oo, 0)\n sage: RealSet.unbounded_below_closed(0)\n (-oo, 0]\n sage: RealSet.unbounded_above_open(1)\n (1, +oo)\n sage: RealSet.unbounded_above_closed(1)\n [1, +oo)\n\nRelations containing symbols and numeric values or constants::\n\n sage: RealSet(x != 0)\n (-oo, 0) \u222a (0, +oo)\n sage: RealSet(x == pi)\n {pi}\n sage: RealSet(x < 1/2)\n (-oo, 1/2)\n sage: RealSet(1/2 < x)\n (1/2, +oo)\n sage: RealSet(1.5 <= x)\n [1.50000000000000, +oo)\n\nNote that multiple arguments are combined as union::\n\n sage: RealSet(x >= 0, x < 1)\n (-oo, +oo)\n sage: RealSet(x >= 0, x > 1)\n [0, +oo)\n sage: RealSet(x >= 0, x > -1)\n (-1, +oo)\n\nAUTHORS:\n\n- Laurent Claessens (2010-12-10): Interval and ContinuousSet, posted\n to sage-devel at\n http://www.mail-archive.com/sage-support@googlegroups.com/msg21326.html.\n\n- Ares Ribo (2011-10-24): Extended the previous work defining the\n class RealSet.\n\n- Jordi Saludes (2011-12-10): Documentation and file reorganization.\n\n- Volker Braun (2013-06-22): Rewrite\n\"\"\"\n\n#*****************************************************************************\n# Copyright (C) 2013 Volker Braun \n#\n# This program is free software: you can redistribute it and/or modify\n# it under the terms of the GNU General Public License as published by\n# the Free Software Foundation, either version 2 of the License, or\n# (at your option) any later version.\n# http://www.gnu.org/licenses/\n#*****************************************************************************\n\nfrom sage.structure.richcmp import richcmp, richcmp_method\nfrom sage.structure.parent import Parent\nfrom sage.structure.unique_representation import UniqueRepresentation\nfrom sage.categories.topological_spaces import TopologicalSpaces\nfrom sage.categories.sets_cat import Sets, EmptySetError\nfrom sage.sets.set import Set_base, Set_boolean_operators, Set_add_sub_operators\nfrom sage.rings.all import ZZ\nfrom sage.rings.real_lazy import LazyFieldElement, RLF\nfrom sage.rings.infinity import infinity, minus_infinity\n\nfrom sage.misc.superseded import deprecated_function_alias\n\n\n@richcmp_method\nclass InternalRealInterval(UniqueRepresentation, Parent):\n \"\"\"\n A real interval.\n\n You are not supposed to create :class:`RealInterval` objects\n yourself. Always use :class:`RealSet` instead.\n\n INPUT:\n\n - ``lower`` -- real or minus infinity; the lower bound of the\n interval.\n\n - ``lower_closed`` -- boolean; whether the interval is closed\n at the lower bound\n\n - ``upper`` -- real or (plus) infinity; the upper bound of the\n interval\n\n - ``upper_closed`` -- boolean; whether the interval is closed\n at the upper bound\n\n - ``check`` -- boolean; whether to check the other arguments\n for validity\n \"\"\"\n def __init__(self, lower, lower_closed, upper, upper_closed, check=True):\n \"\"\"\n Initialize ``self``.\n\n EXAMPLES::\n\n sage: RealSet([0, oo])\n Traceback (most recent call last):\n ...\n ValueError: interval cannot be closed at +oo\n \"\"\"\n self._lower = lower\n self._upper = upper\n self._lower_closed = lower_closed\n self._upper_closed = upper_closed\n if check:\n if not (isinstance(lower, LazyFieldElement) or lower is minus_infinity):\n raise ValueError('lower bound must be in RLF or minus infinity')\n if not (isinstance(upper, LazyFieldElement) or upper is infinity):\n raise ValueError('upper bound must be in RLF or plus infinity')\n if not isinstance(lower_closed, bool):\n raise ValueError('lower_closed must be boolean')\n if not isinstance(upper_closed, bool):\n raise ValueError('upper_closed must be boolean')\n # comparison of infinity with RLF is broken\n if not(lower is minus_infinity or upper is infinity) and lower > upper:\n raise ValueError('lower/upper bounds are not sorted')\n if (lower_closed and lower == minus_infinity):\n raise ValueError('interval cannot be closed at -oo')\n if (upper_closed and upper == infinity):\n raise ValueError('interval cannot be closed at +oo')\n\n def is_empty(self):\n \"\"\"\n Return whether the interval is empty\n\n The normalized form of :class:`RealSet` has all intervals\n non-empty, so this method usually returns ``False``.\n\n OUTPUT:\n\n Boolean.\n\n EXAMPLES::\n\n sage: I = RealSet(0, 1)[0]\n sage: I.is_empty()\n False\n \"\"\"\n return (self._lower == self._upper) and not (self._lower_closed and self._upper_closed)\n\n def is_point(self):\n \"\"\"\n Return whether the interval consists of a single point\n\n OUTPUT:\n\n Boolean.\n\n EXAMPLES::\n\n sage: I = RealSet(0, 1)[0]\n sage: I.is_point()\n False\n \"\"\"\n return (self._lower == self._upper) and self._lower_closed and self._upper_closed\n\n def lower(self):\n \"\"\"\n Return the lower bound\n\n OUTPUT:\n\n The lower bound as it was originally specified.\n\n EXAMPLES::\n\n sage: I = RealSet(0, 1)[0]\n sage: I.lower()\n 0\n sage: I.upper()\n 1\n \"\"\"\n if self._lower is minus_infinity:\n return minus_infinity\n else:\n return self._lower._value\n\n def upper(self):\n \"\"\"\n Return the upper bound\n\n OUTPUT:\n\n The upper bound as it was originally specified.\n\n EXAMPLES::\n\n sage: I = RealSet(0, 1)[0]\n sage: I.lower()\n 0\n sage: I.upper()\n 1\n \"\"\"\n if self._upper is infinity:\n return infinity\n else:\n return self._upper._value\n\n def lower_closed(self):\n \"\"\"\n Return whether the interval is open at the lower bound\n\n OUTPUT:\n\n Boolean.\n\n EXAMPLES::\n\n sage: I = RealSet.open_closed(0, 1)[0]; I\n (0, 1]\n sage: I.lower_closed()\n False\n sage: I.lower_open()\n True\n sage: I.upper_closed()\n True\n sage: I.upper_open()\n False\n \"\"\"\n return self._lower_closed\n\n def upper_closed(self):\n \"\"\"\n Return whether the interval is closed at the lower bound\n\n OUTPUT:\n\n Boolean.\n\n EXAMPLES::\n\n sage: I = RealSet.open_closed(0, 1)[0]; I\n (0, 1]\n sage: I.lower_closed()\n False\n sage: I.lower_open()\n True\n sage: I.upper_closed()\n True\n sage: I.upper_open()\n False\n \"\"\"\n return self._upper_closed\n\n def lower_open(self):\n \"\"\"\n Return whether the interval is closed at the upper bound\n\n OUTPUT:\n\n Boolean.\n\n EXAMPLES::\n\n sage: I = RealSet.open_closed(0, 1)[0]; I\n (0, 1]\n sage: I.lower_closed()\n False\n sage: I.lower_open()\n True\n sage: I.upper_closed()\n True\n sage: I.upper_open()\n False\n \"\"\"\n return not self._lower_closed\n\n def upper_open(self):\n \"\"\"\n Return whether the interval is closed at the upper bound\n\n OUTPUT:\n\n Boolean.\n\n EXAMPLES::\n\n sage: I = RealSet.open_closed(0, 1)[0]; I\n (0, 1]\n sage: I.lower_closed()\n False\n sage: I.lower_open()\n True\n sage: I.upper_closed()\n True\n sage: I.upper_open()\n False\n \"\"\"\n return not self._upper_closed\n\n def __richcmp__(self, other, op):\n \"\"\"\n Intervals are sorted by lower bound, then upper bound\n\n OUTPUT:\n\n `-1`, `0`, or `+1` depending on how the intervals compare.\n\n EXAMPLES::\n\n sage: I1 = RealSet.open_closed(1, 3)[0]; I1\n (1, 3]\n sage: I2 = RealSet.open_closed(0, 5)[0]; I2\n (0, 5]\n sage: I1 > I2\n True\n sage: sorted([I1, I2])\n [(0, 5], (1, 3]]\n\n TESTS:\n\n Check if a bug in sorting is fixed (:trac:`17714`)::\n\n sage: RealSet((0, 1),[1, 1],(1, 2))\n (0, 2)\n \"\"\"\n x = (self._lower, not self._lower_closed, self._upper, self._upper_closed)\n y = (other._lower, not other._lower_closed, other._upper, other._upper_closed)\n return richcmp(x, y, op)\n\n element_class = LazyFieldElement\n\n def _repr_(self):\n \"\"\"\n Return a string representation\n\n OUTPUT:\n\n String.\n\n EXAMPLES::\n\n sage: RealSet.open_closed(0, 1)\n (0, 1]\n sage: RealSet.point(0)\n {0}\n \"\"\"\n if self.is_point():\n return '{' + str(self.lower()) + '}'\n s = '[' if self._lower_closed else '('\n if self.lower() is minus_infinity:\n s += '-oo'\n else:\n s += str(self.lower())\n s += ', '\n if self.upper() is infinity:\n s += '+oo'\n else:\n s += str(self.upper())\n s += ']' if self._upper_closed else ')'\n return s\n\n def _latex_(self):\n \"\"\"\n Return a latex representation of ``self``.\n\n EXAMPLES::\n\n sage: RealSet.open_closed(1/2, pi)._latex_()\n '(\\\\frac{1}{2}, \\\\pi]'\n sage: (RealSet.point(sqrt(2)))._latex_()\n '\\\\{\\\\sqrt{2}\\\\}'\n \"\"\"\n from sage.misc.latex import latex\n if self.is_point():\n # Converting to str avoids the extra whitespace\n # that LatexExpr add on concenation. We do not need\n # the whitespace because we are wrapping it in\n # non-letter characters.\n return r'\\{' + str(latex(self.lower())) + r'\\}'\n s = '[' if self._lower_closed else '('\n s += str(latex(self.lower()))\n s += ', '\n s += str(latex(self.upper()))\n s += ']' if self._upper_closed else ')'\n return s\n\n def _sympy_condition_(self, variable):\n \"\"\"\n Convert to a sympy conditional expression.\n\n INPUT:\n\n - ``variable`` -- a symbolic variable\n\n EXAMPLES::\n\n sage: RealSet(0, 4)._sympy_condition_(x)\n (0 < x) & (x < 4)\n \"\"\"\n x = variable\n if self.is_point():\n return (x == self.lower())._sympy_()\n true = (x == 0)._sympy_() | True # trick to get sympy's True\n if self.lower() is not minus_infinity:\n if self._lower_closed:\n lower_condition = (self.lower() <= x)._sympy_()\n else:\n lower_condition = (self.lower() < x)._sympy_()\n else:\n lower_condition = true\n if self.upper() is not infinity:\n if self._upper_closed:\n upper_condition = (x <= self.upper())._sympy_()\n else:\n upper_condition = (x < self.upper())._sympy_()\n else:\n upper_condition = true\n return lower_condition & upper_condition\n\n def _sympy_(self):\n r\"\"\"\n Return the SymPy set corresponding to ``self``.\n\n EXAMPLES::\n\n sage: RealSet.open_closed(0, 1)[0]._sympy_()\n Interval.Lopen(0, 1)\n sage: RealSet.point(0)[0]._sympy_()\n FiniteSet(0)\n sage: RealSet.open(0,1)[0]._sympy_()\n Interval.open(0, 1)\n sage: RealSet.open(-oo,1)[0]._sympy_()\n Interval.open(-oo, 1)\n sage: RealSet.open(0, oo)[0]._sympy_()\n Interval.open(0, oo)\n \"\"\"\n from sympy import Interval\n from sage.interfaces.sympy import sympy_init\n sympy_init()\n return Interval(self.lower(), self.upper(),\n left_open=not self._lower_closed,\n right_open=not self._upper_closed)\n\n def closure(self):\n \"\"\"\n Return the closure\n\n OUTPUT:\n\n The closure as a new :class:`RealInterval`\n\n EXAMPLES::\n\n sage: RealSet.open(0,1)[0].closure()\n [0, 1]\n sage: RealSet.open(-oo,1)[0].closure()\n (-oo, 1]\n sage: RealSet.open(0, oo)[0].closure()\n [0, +oo)\n \"\"\"\n lower_closed = (self._lower != minus_infinity)\n upper_closed = (self._upper != infinity)\n return InternalRealInterval(self._lower, lower_closed, self._upper, upper_closed)\n\n def interior(self):\n \"\"\"\n Return the interior\n\n OUTPUT:\n\n The interior as a new :class:`RealInterval`\n\n EXAMPLES::\n\n sage: RealSet.closed(0, 1)[0].interior()\n (0, 1)\n sage: RealSet.open_closed(-oo, 1)[0].interior()\n (-oo, 1)\n sage: RealSet.closed_open(0, oo)[0].interior()\n (0, +oo)\n \"\"\"\n return InternalRealInterval(self._lower, False, self._upper, False)\n\n def boundary_points(self):\n \"\"\"\n Generate the boundary points of ``self``\n\n EXAMPLES::\n\n sage: list(RealSet.open_closed(-oo, 1)[0].boundary_points())\n [1]\n sage: list(RealSet.open(1, 2)[0].boundary_points())\n [1, 2]\n\n \"\"\"\n if self._lower != minus_infinity:\n yield self._lower\n if self._upper != infinity:\n yield self._upper\n\n def is_connected(self, other):\n \"\"\"\n Test whether two intervals are connected\n\n OUTPUT:\n\n Boolean. Whether the set-theoretic union of the two intervals\n has a single connected component.\n\n EXAMPLES::\n\n sage: I1 = RealSet.open(0, 1)[0]; I1\n (0, 1)\n sage: I2 = RealSet.closed(1, 2)[0]; I2\n [1, 2]\n sage: I1.is_connected(I2)\n True\n sage: I1.is_connected(I2.interior())\n False\n sage: I1.closure().is_connected(I2.interior())\n True\n sage: I2.is_connected(I1)\n True\n sage: I2.interior().is_connected(I1)\n False\n sage: I2.closure().is_connected(I1.interior())\n True\n sage: I3 = RealSet.closed(1/2, 3/2)[0]; I3\n [1/2, 3/2]\n sage: I1.is_connected(I3)\n True\n sage: I3.is_connected(I1)\n True\n \"\"\"\n # self is separated and below other\n if self._upper < other._lower:\n return False\n # self is adjacent and below other\n if self._upper == other._lower:\n return self._upper_closed or other._lower_closed\n # self is separated and above other\n if other._upper < self._lower:\n return False\n # self is adjacent and above other\n if other._upper == self._lower:\n return self._lower_closed or other._upper_closed\n # They are not separated\n return True\n\n def convex_hull(self, other):\n \"\"\"\n Return the convex hull of the two intervals\n\n OUTPUT:\n\n The convex hull as a new :class:`RealInterval`.\n\n EXAMPLES::\n\n sage: I1 = RealSet.open(0, 1)[0]; I1\n (0, 1)\n sage: I2 = RealSet.closed(1, 2)[0]; I2\n [1, 2]\n sage: I1.convex_hull(I2)\n (0, 2]\n sage: I2.convex_hull(I1)\n (0, 2]\n sage: I1.convex_hull(I2.interior())\n (0, 2)\n sage: I1.closure().convex_hull(I2.interior())\n [0, 2)\n sage: I1.closure().convex_hull(I2)\n [0, 2]\n sage: I3 = RealSet.closed(1/2, 3/2)[0]; I3\n [1/2, 3/2]\n sage: I1.convex_hull(I3)\n (0, 3/2]\n \"\"\"\n if self._lower < other._lower:\n lower = self._lower\n lower_closed = self._lower_closed\n elif self._lower > other._lower:\n lower = other._lower\n lower_closed = other._lower_closed\n else:\n lower = self._lower\n lower_closed = self._lower_closed or other._lower_closed\n if self._upper > other._upper:\n upper = self._upper\n upper_closed = self._upper_closed\n elif self._upper < other._upper:\n upper = other._upper\n upper_closed = other._upper_closed\n else:\n upper = self._upper\n upper_closed = self._upper_closed or other._upper_closed\n return InternalRealInterval(lower, lower_closed, upper, upper_closed)\n\n def intersection(self, other):\n \"\"\"\n Return the intersection of the two intervals\n\n INPUT:\n\n - ``other`` -- a :class:`RealInterval`\n\n OUTPUT:\n\n The intersection as a new :class:`RealInterval`\n\n EXAMPLES::\n\n sage: I1 = RealSet.open(0, 2)[0]; I1\n (0, 2)\n sage: I2 = RealSet.closed(1, 3)[0]; I2\n [1, 3]\n sage: I1.intersection(I2)\n [1, 2)\n sage: I2.intersection(I1)\n [1, 2)\n sage: I1.closure().intersection(I2.interior())\n (1, 2]\n sage: I2.interior().intersection(I1.closure())\n (1, 2]\n\n sage: I3 = RealSet.closed(10, 11)[0]; I3\n [10, 11]\n sage: I1.intersection(I3)\n (0, 0)\n sage: I3.intersection(I1)\n (0, 0)\n \"\"\"\n lower = upper = None\n lower_closed = upper_closed = None\n if self._lower < other._lower:\n lower = other._lower\n lower_closed = other._lower_closed\n elif self._lower > other._lower:\n lower = self._lower\n lower_closed = self._lower_closed\n else:\n lower = self._lower\n lower_closed = self._lower_closed and other._lower_closed\n if self._upper > other._upper:\n upper = other._upper\n upper_closed = other._upper_closed\n elif self._upper < other._upper:\n upper = self._upper\n upper_closed = self._upper_closed\n else:\n upper = self._upper\n upper_closed = self._upper_closed and other._upper_closed\n if lower > upper:\n lower = upper = RLF(0)\n lower_closed = upper_closed = False\n return InternalRealInterval(lower, lower_closed, upper, upper_closed)\n\n def contains(self, x):\n \"\"\"\n Return whether `x` is contained in the interval\n\n INPUT:\n\n - ``x`` -- a real number.\n\n OUTPUT:\n\n Boolean.\n\n EXAMPLES::\n\n sage: i = RealSet.open_closed(0,2)[0]; i\n (0, 2]\n sage: i.contains(0)\n False\n sage: i.contains(1)\n True\n sage: i.contains(2)\n True\n \"\"\"\n if self._lower < x < self._upper:\n return True\n if self._lower == x:\n return self._lower_closed\n if self._upper == x:\n return self._upper_closed\n return False\n\n def __mul__(self, right):\n r\"\"\"\n Scale an interval by a scalar on the left or right.\n\n If scaled with a negative number, the interval is flipped.\n\n EXAMPLES::\n\n sage: i = RealSet.open_closed(0,2)[0]; i\n (0, 2]\n sage: 2 * i\n (0, 4]\n sage: 0 * i\n {0}\n sage: (-2) * i\n [-4, 0)\n sage: i * (-3)\n [-6, 0)\n sage: i * 0\n {0}\n sage: i * 1\n (0, 2]\n\n TESTS::\n\n sage: from sage.sets.real_set import InternalRealInterval\n sage: i = InternalRealInterval(RLF(0), False, RLF(0), False)\n sage: (0 * i).is_empty()\n True\n \"\"\"\n if not isinstance(right, InternalRealInterval):\n right = RLF(right)\n if self.is_empty():\n return self\n lower = self._lower * right\n lower_closed = self._lower_closed\n upper = self._upper * right\n upper_closed = self._upper_closed\n scalar = right\n elif not isinstance(self, InternalRealInterval):\n self = RLF(self)\n if right.is_empty():\n return right\n lower = self * right._lower\n lower_closed = right._lower_closed\n upper = self * right._upper\n upper_closed = right._upper_closed\n scalar = self\n else:\n return NotImplemented\n if scalar == RLF(0):\n return InternalRealInterval(RLF(0), True, RLF(0), True)\n elif scalar < RLF(0):\n lower, lower_closed, upper, upper_closed = upper, upper_closed, lower, lower_closed\n if lower == -infinity:\n lower = -infinity\n if upper == infinity:\n upper = infinity\n return InternalRealInterval(lower, lower_closed,\n upper, upper_closed)\n\n def __rmul__(self, other):\n r\"\"\"\n Scale an interval by a scalar on the left.\n\n If scaled with a negative number, the interval is flipped.\n\n EXAMPLES::\n\n sage: i = RealSet.open_closed(0,2)[0]; i\n (0, 2]\n sage: 2 * i\n (0, 4]\n sage: 0 * i\n {0}\n sage: (-2) * i\n [-4, 0)\n \"\"\"\n return self * other\n\n\n@richcmp_method\nclass RealSet(UniqueRepresentation, Parent, Set_base,\n Set_boolean_operators, Set_add_sub_operators):\n r\"\"\"\n A subset of the real line, a finite union of intervals\n\n INPUT:\n\n - ``*args`` -- arguments defining a real set. Possibilities are either:\n\n - two extended real numbers ``a, b``, to construct the open interval `(a, b)`, or\n - a list/tuple/iterable of (not necessarily disjoint) intervals or real sets,\n whose union is taken. The individual intervals can be specified by either\n\n - a tuple ``(a, b)`` of two extended real numbers (constructing an open interval),\n - a list ``[a, b]`` of two real numbers (constructing a closed interval),\n - an :class:`InternalRealInterval`,\n - an :class:`~sage.manifolds.differentiable.examples.real_line.OpenInterval`.\n\n - ``structure`` -- (default: ``None``) if ``None``, construct the real set as an\n instance of :class:`RealSet`; if ``\"differentiable\"``, construct it as a subset of\n an instance of :class:`~sage.manifolds.differentiable.examples.real_line.RealLine`,\n representing the differentiable manifold `\\RR`.\n - ``ambient`` -- (default: ``None``) an instance of\n :class:`~sage.manifolds.differentiable.examples.real_line.RealLine`; construct\n a subset of it. Using this keyword implies ``structure='differentiable'``.\n - ``names`` or ``coordinate`` -- coordinate symbol for the canonical chart; see\n :class:`~sage.manifolds.differentiable.examples.real_line.RealLine`. Using these\n keywords implies ``structure='differentiable'``.\n - ``name``, ``latex_name``, ``start_index`` -- see\n :class:`~sage.manifolds.differentiable.examples.real_line.RealLine`.\n\n There are also specialized constructors for various types of intervals:\n\n ====================================== ====================\n Constructor Interval\n ====================================== ====================\n :meth:`RealSet.open` `(a, b)`\n :meth:`RealSet.closed` `[a, b]`\n :meth:`RealSet.point` `\\{a\\}`\n :meth:`RealSet.open_closed` `(a, b]`\n :meth:`RealSet.closed_open` `[a, b)`\n :meth:`RealSet.unbounded_below_closed` `(-\\infty, b]`\n :meth:`RealSet.unbounded_below_open` `(-\\infty, b)`\n :meth:`RealSet.unbounded_above_closed` `[a, +\\infty)`\n :meth:`RealSet.unbounded_above_open` `(a, +\\infty)`\n :meth:`RealSet.real_line` `(-\\infty, +\\infty)`\n :meth:`RealSet.interval` any\n ====================================== ====================\n\n EXAMPLES::\n\n sage: RealSet(0,1) # open set from two numbers\n (0, 1)\n sage: i = RealSet(0,1)[0]\n sage: RealSet(i) # interval\n (0, 1)\n sage: RealSet(i, (3,4)) # tuple of two numbers = open set\n (0, 1) \u222a (3, 4)\n sage: RealSet(i, [3,4]) # list of two numbers = closed set\n (0, 1) \u222a [3, 4]\n\n Initialization from manifold objects::\n\n sage: R = manifolds.RealLine(); R\n Real number line \u211d\n sage: RealSet(R)\n (-oo, +oo)\n sage: I02 = manifolds.OpenInterval(0, 2); I\n I\n sage: RealSet(I02)\n (0, 2)\n sage: I01_of_R = manifolds.OpenInterval(0, 1, ambient_interval=R); I01_of_R\n Real interval (0, 1)\n sage: RealSet(I01_of_R)\n (0, 1)\n sage: RealSet(I01_of_R.closure())\n [0, 1]\n sage: I01_of_I02 = manifolds.OpenInterval(0, 1, ambient_interval=I02); I01_of_I02\n Real interval (0, 1)\n sage: RealSet(I01_of_I02)\n (0, 1)\n sage: RealSet(I01_of_I02.closure())\n (0, 1]\n\n Real sets belong to a subcategory of topological spaces::\n\n sage: RealSet().category()\n Join of\n Category of finite sets and\n Category of subobjects of sets and\n Category of connected topological spaces\n sage: RealSet.point(1).category()\n Join of\n Category of finite sets and\n Category of subobjects of sets and\n Category of connected topological spaces\n sage: RealSet([1, 2]).category()\n Join of\n Category of infinite sets and\n Category of compact topological spaces and\n Category of subobjects of sets and\n Category of connected topological spaces\n sage: RealSet((1, 2), (3, 4)).category()\n Join of\n Category of infinite sets and\n Category of subobjects of sets and\n Category of topological spaces\n\n Constructing real sets as manifolds or manifold subsets by passing\n ``structure='differentiable'``::\n\n sage: RealSet(-oo, oo, structure='differentiable')\n Real number line \u211d\n\n sage: RealSet([0, 1], structure='differentiable')\n Subset [0, 1] of the Real number line \u211d\n sage: _.category()\n Category of subobjects of sets\n\n sage: RealSet.open_closed(0, 5, structure='differentiable')\n Subset (0, 5] of the Real number line \u211d\n\n This is implied when a coordinate name is given using the keywords ``coordinate``\n or ``names``::\n\n sage: RealSet(0, 1, coordinate='\u03bb')\n Open subset (0, 1) of the Real number line \u211d\n sage: _.category()\n Join of\n Category of smooth manifolds over Real Field with 53 bits of precision and\n Category of connected manifolds over Real Field with 53 bits of precision and\n Category of subobjects of sets\n\n It is also implied by assigning a coordinate name using generator notation::\n\n sage: R_xi.<\u03be> = RealSet.real_line(); R_xi\n Real number line \u211d\n sage: R_xi.canonical_chart()\n Chart (\u211d, (\u03be,))\n\n With the keyword ``ambient``, we can construct a subset of a previously\n constructed manifold::\n\n sage: P_xi = RealSet(0, oo, ambient=R_xi); P_xi\n Open subset (0, +oo) of the Real number line \u211d\n sage: P_xi.default_chart()\n Chart ((0, +oo), (\u03be,))\n sage: B_xi = RealSet(0, 1, ambient=P_xi); B_xi\n Open subset (0, 1) of the Real number line \u211d\n sage: B_xi.default_chart()\n Chart ((0, 1), (\u03be,))\n sage: R_xi.subset_family()\n Set {(0, +oo), (0, 1), \u211d} of open subsets of the Real number line \u211d\n\n sage: F = RealSet.point(0).union(RealSet.point(1)).union(RealSet.point(2)); F\n {0} \u222a {1} \u222a {2}\n sage: F_tau = RealSet(F, names=\"\u03c4\"); F_tau\n Subset {0} \u222a {1} \u222a {2} of the Real number line \u211d\n sage: F_tau.manifold().canonical_chart()\n Chart (\u211d, (\u03c4,))\n\n TESTS::\n\n sage: TestSuite(R_xi).run()\n sage: TestSuite(P_xi).run()\n sage: R_xi.point((1,)) in P_xi\n True\n sage: R_xi.point((-1,)) in P_xi\n False\n sage: TestSuite(B_xi).run()\n sage: p = B_xi.an_element(); p\n Point on the Real number line \u211d\n sage: p.coordinates()\n (1/2,)\n \"\"\"\n\n @staticmethod\n def __classcall__(cls, *args, **kwds):\n \"\"\"\n Normalize the input.\n\n INPUT:\n\n See :class:`RealSet`.\n\n OUTPUT:\n\n A :class:`RealSet`.\n\n EXAMPLES::\n\n sage: R = RealSet(RealSet.open_closed(0,1), RealSet.closed_open(2,3)); R\n (0, 1] \u222a [2, 3)\n\n ::\n\n sage: RealSet(x != 0)\n (-oo, 0) \u222a (0, +oo)\n sage: RealSet(x == pi)\n {pi}\n sage: RealSet(x < 1/2)\n (-oo, 1/2)\n sage: RealSet(1/2 < x)\n (1/2, +oo)\n sage: RealSet(1.5 <= x)\n [1.50000000000000, +oo)\n sage: RealSet(x >= -1)\n [-1, +oo)\n sage: RealSet(x > oo)\n {}\n sage: RealSet(x >= oo)\n {}\n sage: RealSet(x <= -oo)\n {}\n sage: RealSet(x < oo)\n (-oo, +oo)\n sage: RealSet(x > -oo)\n (-oo, +oo)\n sage: RealSet(x != oo)\n (-oo, +oo)\n sage: RealSet(x <= oo)\n Traceback (most recent call last):\n ...\n ValueError: interval cannot be closed at +oo\n sage: RealSet(x == oo)\n Traceback (most recent call last):\n ...\n ValueError: interval cannot be closed at +oo\n sage: RealSet(x >= -oo)\n Traceback (most recent call last):\n ...\n ValueError: interval cannot be closed at -oo\n \"\"\"\n manifold_keywords = ('structure', 'ambient', 'names', 'coordinate')\n if any(kwds.get(kwd, None)\n for kwd in manifold_keywords):\n # Got manifold keywords\n real_set = cls.__classcall__(cls, *args)\n ambient = kwds.pop('ambient', None)\n structure = kwds.pop('structure', 'differentiable')\n if structure != 'differentiable':\n # TODO\n raise NotImplementedError\n \n from sage.manifolds.differentiable.examples.real_line import RealLine\n if real_set.is_universe():\n if ambient is None:\n ambient = RealLine(**kwds)\n else:\n # TODO: Check that ambient makes sense\n pass\n return ambient\n\n name = kwds.pop('name', None)\n latex_name = kwds.pop('latex_name', None)\n\n if ambient is None:\n ambient = RealLine(**kwds)\n else:\n # TODO: Check that ambient makes sense\n pass\n\n if name is None:\n name = str(real_set)\n if latex_name is None:\n from sage.misc.latex import latex\n latex_name = latex(real_set)\n\n return ambient.manifold().canonical_chart().pullback(real_set, name=name, latex_name=latex_name)\n\n if kwds:\n raise TypeError(f'unless manifold keywords {manifold_keywords} are given, RealSet constructors take no keyword arguments')\n\n from sage.symbolic.expression import Expression\n if len(args) == 1 and isinstance(args[0], RealSet):\n return args[0] # common optimization\n intervals = []\n if len(args) == 2:\n # allow RealSet(0,1) interval constructor\n try:\n lower, upper = args\n lower.n()\n upper.n()\n args = (RealSet._prep(lower, upper), )\n except (AttributeError, ValueError, TypeError):\n pass\n for arg in args:\n if isinstance(arg, tuple):\n lower, upper = RealSet._prep(*arg)\n intervals.append(InternalRealInterval(lower, False, upper, False))\n elif isinstance(arg, list):\n lower, upper = RealSet._prep(*arg)\n intervals.append(InternalRealInterval(lower, True, upper, True))\n elif isinstance(arg, InternalRealInterval):\n intervals.append(arg)\n elif isinstance(arg, RealSet):\n intervals.extend(arg._intervals)\n elif isinstance(arg, Expression) and arg.is_relational():\n from operator import eq, ne, lt, gt, le, ge\n def rel_to_interval(op, val):\n \"\"\"\n Internal helper function.\n \"\"\"\n oo = infinity\n try:\n val = val.pyobject()\n except AttributeError:\n pass\n val = RLF(val)\n if op == eq:\n return [InternalRealInterval(val, True, val, True)]\n elif op == ne:\n return [InternalRealInterval(-oo, False, val, False),\n InternalRealInterval(val, False, oo, False)]\n elif op == gt:\n return [InternalRealInterval(val, False, oo, False)]\n elif op == ge:\n return [InternalRealInterval(val, True, oo, False)]\n elif op == lt:\n return [InternalRealInterval(-oo, False, val, False)]\n else:\n return [InternalRealInterval(-oo, False, val, True)]\n\n if (arg.lhs().is_symbol()\n and (arg.rhs().is_numeric() or arg.rhs().is_constant())\n and arg.rhs().is_real()):\n intervals.extend(rel_to_interval(arg.operator(), arg.rhs()))\n elif (arg.rhs().is_symbol()\n and (arg.lhs().is_numeric() or arg.lhs().is_constant())\n and arg.lhs().is_real()):\n op = arg.operator()\n if op == lt:\n op = gt\n elif op == gt:\n op = lt\n elif op == le:\n op = ge\n elif op == ge:\n op = le\n intervals.extend(rel_to_interval(op, arg.lhs()))\n else:\n raise ValueError(str(arg) + ' does not determine real interval')\n else:\n from sage.manifolds.differentiable.examples.real_line import OpenInterval\n from sage.manifolds.subsets.closure import ManifoldSubsetClosure\n if isinstance(arg, OpenInterval):\n lower, upper = RealSet._prep(arg.lower_bound(), arg.upper_bound())\n intervals.append(InternalRealInterval(lower, False, upper, False))\n elif (isinstance(arg, ManifoldSubsetClosure)\n and isinstance(arg._subset, OpenInterval)):\n interval = arg._subset\n lower, upper = RealSet._prep(interval.lower_bound(),\n interval.upper_bound())\n ambient = interval.manifold()\n ambient_lower, ambient_upper = RealSet._prep(ambient.lower_bound(),\n ambient.upper_bound())\n lower_closed = ambient_lower < lower\n upper_closed = upper < ambient_upper\n intervals.append(InternalRealInterval(lower, lower_closed,\n upper, upper_closed))\n else:\n raise ValueError(str(arg) + ' does not determine real interval')\n intervals = RealSet.normalize(intervals)\n return UniqueRepresentation.__classcall__(cls, *intervals)\n\n def __init__(self, *intervals):\n r\"\"\"\n TESTS::\n\n sage: Empty = RealSet(); Empty\n {}\n sage: TestSuite(Empty).run()\n sage: I1 = RealSet.open_closed(1, 3); I1\n (1, 3]\n sage: TestSuite(I1).run()\n sage: R = RealSet(RealSet.open_closed(0,1), RealSet.closed_open(2,3)); R\n (0, 1] \u222a [2, 3)\n sage: TestSuite(R).run()\n \"\"\"\n category = TopologicalSpaces()\n if len(intervals) <= 1:\n category = category.Connected()\n if all(i.is_point() for i in intervals):\n category = category.Subobjects().Finite()\n else:\n # Have at least one non-degenerate interval\n category = category.Infinite()\n inf = intervals[0].lower()\n sup = intervals[-1].upper()\n if not (len(intervals) == 1 and inf is minus_infinity and sup is infinity):\n category = category.Subobjects() # subobject of real line\n if inf is not minus_infinity and sup is not infinity:\n # Bounded\n if all(i.lower_closed() and i.upper_closed()\n for i in intervals):\n category = category.Compact()\n Parent.__init__(self, category=category)\n self._intervals = intervals\n\n def __richcmp__(self, other, op):\n r\"\"\"\n Intervals are sorted by lower bound, then upper bound\n\n OUTPUT:\n\n `-1`, `0`, or `+1` depending on how the intervals compare.\n\n EXAMPLES::\n\n sage: I1 = RealSet.open_closed(1, 3); I1\n (1, 3]\n sage: I2 = RealSet.open_closed(0, 5); I2\n (0, 5]\n sage: I1 > I2\n True\n sage: sorted([I1, I2])\n [(0, 5], (1, 3]]\n sage: I1 == I1\n True\n \"\"\"\n if not isinstance(other, RealSet):\n return NotImplemented\n # note that the interval representation is normalized into a\n # unique form\n return richcmp(self._intervals, other._intervals, op)\n\n def __iter__(self):\n r\"\"\"\n Iterate over the component intervals is ascending order\n\n OUTPUT:\n\n An iterator over the intervals.\n\n EXAMPLES::\n\n sage: s = RealSet(RealSet.open_closed(0,1), RealSet.closed_open(2,3))\n sage: i = iter(s)\n sage: next(i)\n (0, 1]\n sage: next(i)\n [2, 3)\n \"\"\"\n return iter(self._intervals)\n\n def n_components(self):\n r\"\"\"\n Return the number of connected components\n\n See also :meth:`get_interval`\n\n EXAMPLES::\n\n sage: s = RealSet(RealSet.open_closed(0,1), RealSet.closed_open(2,3))\n sage: s.n_components()\n 2\n \"\"\"\n return len(self._intervals)\n\n def cardinality(self):\n r\"\"\"\n Return the cardinality of the subset of the real line.\n\n OUTPUT:\n\n Integer or infinity. The size of a discrete set is the number\n of points; the size of a real interval is Infinity.\n\n EXAMPLES::\n\n sage: RealSet([0, 0], [1, 1], [3, 3]).cardinality()\n 3\n sage: RealSet(0,3).cardinality()\n +Infinity\n \"\"\"\n n = ZZ(0)\n for interval in self._intervals:\n if interval.is_point():\n n += 1\n else:\n return infinity\n return n\n\n def is_empty(self):\n r\"\"\"\n Return whether the set is empty\n\n EXAMPLES::\n\n sage: RealSet(0, 1).is_empty()\n False\n sage: RealSet(0, 0).is_empty()\n True\n \"\"\"\n return len(self._intervals) == 0\n\n def is_universe(self):\n r\"\"\"\n Return whether the set is the ambient space (the real line).\n\n EXAMPLES::\n\n sage: RealSet().ambient().is_universe()\n True\n \"\"\"\n return self == self.ambient()\n\n def get_interval(self, i):\n r\"\"\"\n Return the ``i``-th connected component.\n\n Note that the intervals representing the real set are always\n normalized, see :meth:`normalize`.\n\n INPUT:\n\n - ``i`` -- integer.\n\n OUTPUT:\n\n The $i$-th connected component as a :class:`RealInterval`.\n\n EXAMPLES::\n\n sage: s = RealSet(RealSet.open_closed(0,1), RealSet.closed_open(2,3))\n sage: s.get_interval(0)\n (0, 1]\n sage: s[0] # shorthand\n (0, 1]\n sage: s.get_interval(1)\n [2, 3)\n sage: s[0] == s.get_interval(0)\n True\n \"\"\"\n return self._intervals[i]\n\n __getitem__ = get_interval\n\n def __bool__(self):\n r\"\"\"\n A set is considered True unless it is empty, in which case it is\n considered to be False.\n\n EXAMPLES::\n\n sage: bool(RealSet(0, 1))\n True\n sage: bool(RealSet())\n False\n \"\"\"\n return not self.is_empty()\n\n # ParentMethods of Subobjects\n\n def ambient(self):\n r\"\"\"\n Return the ambient space (the real line).\n\n EXAMPLES::\n\n sage: s = RealSet(RealSet.open_closed(0,1), RealSet.closed_open(2,3))\n sage: s.ambient()\n (-oo, +oo)\n \"\"\"\n return self.__class__.real_line()\n\n def lift(self, x):\n r\"\"\"\n Lift ``x`` to the ambient space for ``self``.\n\n This version of the method just returns ``x``.\n\n EXAMPLES::\n\n sage: s = RealSet(0, 2); s\n (0, 2)\n sage: s.lift(1)\n 1\n \"\"\"\n return x\n\n def retract(self, x):\n r\"\"\"\n Retract ``x`` to ``self``.\n\n It raises an error if ``x`` does not lie in the set ``self``.\n\n EXAMPLES::\n\n sage: s = RealSet(0, 2); s\n (0, 2)\n sage: s.retract(1)\n 1\n sage: s.retract(2)\n Traceback (most recent call last):\n ...\n ValueError: 2 is not an element of (0, 2)\n \"\"\"\n if x not in self:\n raise ValueError(f'{x} is not an element of {self}')\n return x\n\n #\n\n @staticmethod\n def normalize(intervals):\n r\"\"\"\n Bring a collection of intervals into canonical form\n\n INPUT:\n\n - ``intervals`` -- a list/tuple/iterable of intervals.\n\n OUTPUT:\n\n A tuple of intervals such that\n\n * they are sorted in ascending order (by lower bound)\n\n * there is a gap between each interval\n\n * all intervals are non-empty\n\n EXAMPLES::\n\n sage: i1 = RealSet((0, 1))[0]\n sage: i2 = RealSet([1, 2])[0]\n sage: i3 = RealSet((2, 3))[0]\n sage: RealSet.normalize([i1, i2, i3])\n ((0, 3),)\n\n sage: RealSet((0, 1), [1, 2], (2, 3))\n (0, 3)\n sage: RealSet((0, 1), (1, 2), (2, 3))\n (0, 1) \u222a (1, 2) \u222a (2, 3)\n sage: RealSet([0, 1], [2, 3])\n [0, 1] \u222a [2, 3]\n sage: RealSet((0, 2), (1, 3))\n (0, 3)\n sage: RealSet(0,0)\n {}\n \"\"\"\n # sort by lower bound\n intervals = sorted(intervals)\n if not intervals:\n return tuple()\n merged = []\n curr = intervals.pop(0)\n while intervals:\n next = intervals.pop(0)\n if curr._upper > next._lower or (\n curr._upper == next._lower and\n (curr._upper_closed or next._lower_closed)):\n curr = curr.convex_hull(next)\n else:\n if not curr.is_empty():\n merged.append(curr)\n curr = next\n if not curr.is_empty():\n merged.append(curr)\n return tuple(merged)\n\n def _repr_(self):\n r\"\"\"\n Return a string representation of ``self``.\n\n OUTPUT:\n\n A string representation.\n\n EXAMPLES::\n\n sage: RealSet(0, 1)._repr_()\n '(0, 1)'\n \"\"\"\n if self.n_components() == 0:\n return '{}'\n else:\n return ' \u222a '.join(map(repr, self._intervals))\n\n def _latex_(self):\n r\"\"\"\n Return a latex representation of ``self``.\n\n EXAMPLES::\n\n sage: latex(RealSet(0, 1))\n (0, 1)\n sage: latex((RealSet(0, 1).union(RealSet.unbounded_above_closed(2))))\n (0, 1) \\cup [2, +\\infty)\n \"\"\"\n from sage.misc.latex import latex\n if self.n_components() == 0:\n return r'\\emptyset'\n else:\n return r' \\cup '.join(latex(i) for i in self._intervals)\n\n def _sympy_condition_(self, variable):\n r\"\"\"\n Convert to a sympy conditional expression.\n\n INPUT:\n\n - ``variable`` -- a symbolic variable\n\n EXAMPLES::\n\n sage: RealSet(0, 1)._sympy_condition_(x)\n (0 < x) & (x < 1)\n sage: RealSet((0,1), [2,3])._sympy_condition_(x)\n ((2 <= x) & (x <= 3)) | ((0 < x) & (x < 1))\n sage: RealSet.unbounded_below_open(0)._sympy_condition_(x)\n x < 0\n sage: RealSet.unbounded_above_closed(2)._sympy_condition_(x)\n 2 <= x\n\n TESTS::\n\n sage: RealSet(6,6)._sympy_condition_(x)\n False\n sage: RealSet([6,6])._sympy_condition_(x)\n Eq(x, 6)\n \"\"\"\n x = variable\n false = (x == 0)._sympy_() & False # trick to get sympy's False\n if self.n_components() == 0:\n return false\n else:\n cond = false\n for it in self._intervals:\n cond = cond | it._sympy_condition_(x)\n return cond\n\n @staticmethod\n def _prep(lower, upper=None):\n r\"\"\"\n Helper to prepare the lower and upper bound\n\n EXAMPLES::\n\n sage: RealSet._prep(1, 0)\n (0, 1)\n sage: RealSet._prep(oo)\n +Infinity\n \"\"\"\n if lower == minus_infinity:\n lower = minus_infinity\n if lower == infinity:\n lower = infinity\n else:\n lower = RLF(lower)\n if upper is None:\n return lower\n if upper == minus_infinity:\n upper = minus_infinity\n if upper == infinity:\n upper = infinity\n else:\n upper = RLF(upper)\n if upper is infinity or lower is minus_infinity:\n return lower, upper\n elif lower is infinity or upper is minus_infinity:\n return upper, lower\n elif upper < lower:\n return upper, lower\n else:\n return lower, upper\n\n @classmethod\n def interval(cls, lower, upper, *, lower_closed=None, upper_closed=None, **kwds):\n r\"\"\"\n Construct an interval\n\n INPUT:\n\n - ``lower``, ``upper`` -- two real numbers or infinity. They\n will be sorted if necessary.\n\n - ``lower_closed``, ``upper_closed`` -- boolean; whether the interval\n is closed at the lower and upper bound of the interval, respectively.\n\n - ``**kwds`` -- see :class:`RealSet`.\n\n OUTPUT:\n\n A new :class:`RealSet`.\n\n EXAMPLES::\n\n sage: RealSet.interval(1, 0, lower_closed=True, upper_closed=False)\n [0, 1)\n \"\"\"\n if lower_closed is None or upper_closed is None:\n raise ValueError('lower_closed and upper_closed must be explicitly given')\n lower, upper = RealSet._prep(lower, upper)\n return cls(InternalRealInterval(lower, lower_closed, upper, upper_closed), **kwds)\n\n @classmethod\n def open(cls, lower, upper, **kwds):\n r\"\"\"\n Construct an open interval\n\n INPUT:\n\n - ``lower``, ``upper`` -- two real numbers or infinity. They\n will be sorted if necessary.\n\n - ``**kwds`` -- see :class:`RealSet`.\n\n OUTPUT:\n\n A new :class:`RealSet`.\n\n EXAMPLES::\n\n sage: RealSet.open(1, 0)\n (0, 1)\n \"\"\"\n lower, upper = RealSet._prep(lower, upper)\n return cls(InternalRealInterval(lower, False, upper, False), **kwds)\n\n @classmethod\n def closed(cls, lower, upper, **kwds):\n r\"\"\"\n Construct a closed interval\n\n INPUT:\n\n - ``lower``, ``upper`` -- two real numbers or infinity. They\n will be sorted if necessary.\n\n - ``**kwds`` -- see :class:`RealSet`.\n\n OUTPUT:\n\n A new :class:`RealSet`.\n\n EXAMPLES::\n\n sage: RealSet.closed(1, 0)\n [0, 1]\n \"\"\"\n lower, upper = RealSet._prep(lower, upper)\n return cls(InternalRealInterval(lower, True, upper, True), **kwds)\n\n @classmethod\n def point(cls, p, **kwds):\n r\"\"\"\n Construct an interval containing a single point\n\n INPUT:\n\n - ``p`` -- a real number.\n\n - ``**kwds`` -- see :class:`RealSet`.\n\n OUTPUT:\n\n A new :class:`RealSet`.\n\n EXAMPLES::\n\n sage: RealSet.open(1, 0)\n (0, 1)\n \"\"\"\n p = RealSet._prep(p)\n return cls(InternalRealInterval(p, True, p, True), **kwds)\n\n @classmethod\n def open_closed(cls, lower, upper, **kwds):\n r\"\"\"\n Construct a half-open interval\n\n INPUT:\n\n - ``lower``, ``upper`` -- two real numbers or infinity. They\n will be sorted if necessary.\n\n - ``**kwds`` -- see :class:`RealSet`.\n\n OUTPUT:\n\n A new :class:`RealSet` that is open at the lower bound and\n closed at the upper bound.\n\n EXAMPLES::\n\n sage: RealSet.open_closed(1, 0)\n (0, 1]\n \"\"\"\n lower, upper = RealSet._prep(lower, upper)\n return cls(InternalRealInterval(lower, False, upper, True), **kwds)\n\n @classmethod\n def closed_open(cls, lower, upper, **kwds):\n r\"\"\"\n Construct an half-open interval\n\n INPUT:\n\n - ``lower``, ``upper`` -- two real numbers or infinity. They\n will be sorted if necessary.\n\n - ``**kwds`` -- see :class:`RealSet`.\n\n OUTPUT:\n\n A new :class:`RealSet` that is closed at the lower bound and\n open an the upper bound.\n\n EXAMPLES::\n\n sage: RealSet.closed_open(1, 0)\n [0, 1)\n \"\"\"\n lower, upper = RealSet._prep(lower, upper)\n return cls(InternalRealInterval(lower, True, upper, False), **kwds)\n\n @classmethod\n def unbounded_below_closed(cls, bound, **kwds):\n r\"\"\"\n Construct a semi-infinite interval\n\n INPUT:\n\n - ``bound`` -- a real number.\n\n OUTPUT:\n\n A new :class:`RealSet` from minus infinity to the bound (including).\n\n - ``**kwds`` -- see :class:`RealSet`.\n\n EXAMPLES::\n\n sage: RealSet.unbounded_below_closed(1)\n (-oo, 1]\n \"\"\"\n bound = RealSet._prep(bound)\n return cls(InternalRealInterval(minus_infinity, False, bound, True), **kwds)\n\n @classmethod\n def unbounded_below_open(cls, bound, **kwds):\n r\"\"\"\n Construct a semi-infinite interval\n\n INPUT:\n\n - ``bound`` -- a real number.\n\n OUTPUT:\n\n A new :class:`RealSet` from minus infinity to the bound (excluding).\n\n - ``**kwds`` -- see :class:`RealSet`.\n\n EXAMPLES::\n\n sage: RealSet.unbounded_below_open(1)\n (-oo, 1)\n \"\"\"\n bound = RealSet._prep(bound)\n return cls(InternalRealInterval(RLF(minus_infinity), False, RLF(bound), False), **kwds)\n\n @classmethod\n def unbounded_above_closed(cls, bound, **kwds):\n r\"\"\"\n Construct a semi-infinite interval\n\n INPUT:\n\n - ``bound`` -- a real number.\n\n - ``**kwds`` -- see :class:`RealSet`.\n\n OUTPUT:\n\n A new :class:`RealSet` from the bound (including) to plus\n infinity.\n\n EXAMPLES::\n\n sage: RealSet.unbounded_above_closed(1)\n [1, +oo)\n \"\"\"\n bound = RealSet._prep(bound)\n return cls(InternalRealInterval(RLF(bound), True, RLF(infinity), False), **kwds)\n\n @classmethod\n def unbounded_above_open(cls, bound, **kwds):\n r\"\"\"\n Construct a semi-infinite interval\n\n INPUT:\n\n - ``bound`` -- a real number.\n\n - ``**kwds`` -- see :class:`RealSet`.\n\n OUTPUT:\n\n A new :class:`RealSet` from the bound (excluding) to plus\n infinity.\n\n EXAMPLES::\n\n sage: RealSet.unbounded_above_open(1)\n (1, +oo)\n \"\"\"\n bound = RealSet._prep(bound)\n return cls(InternalRealInterval(RLF(bound), False, RLF(infinity), False), **kwds)\n\n @classmethod\n def real_line(cls, **kwds):\n r\"\"\"\n Construct the real line\n\n INPUT:\n\n - ``**kwds`` -- see :class:`RealSet`.\n\n EXAMPLES::\n\n sage: RealSet.real_line()\n (-oo, +oo)\n \"\"\"\n return cls(InternalRealInterval(RLF(minus_infinity), False, RLF(infinity), False), **kwds)\n\n def union(self, *other):\n \"\"\"\n Return the union of the two sets\n\n INPUT:\n\n - ``other`` -- a :class:`RealSet` or data that defines one.\n\n OUTPUT:\n\n The set-theoretic union as a new :class:`RealSet`.\n\n EXAMPLES::\n\n sage: s1 = RealSet(0,2)\n sage: s2 = RealSet(1,3)\n sage: s1.union(s2)\n (0, 3)\n sage: s1.union(1,3)\n (0, 3)\n sage: s1 | s2 # syntactic sugar\n (0, 3)\n sage: s1 + s2 # syntactic sugar\n (0, 3)\n \"\"\"\n other = RealSet(*other)\n intervals = self._intervals + other._intervals\n return self.__class__(*intervals)\n\n def intersection(self, *other):\n \"\"\"\n Return the intersection of the two sets\n\n INPUT:\n\n - ``other`` -- a :class:`RealSet` or data that defines one.\n\n OUTPUT:\n\n The set-theoretic intersection as a new :class:`RealSet`.\n\n EXAMPLES::\n\n sage: s1 = RealSet(0,2) + RealSet.unbounded_above_closed(10); s1\n (0, 2) \u222a [10, +oo)\n sage: s2 = RealSet(1,3) + RealSet.unbounded_below_closed(-10); s2\n (-oo, -10] \u222a (1, 3)\n sage: s1.intersection(s2)\n (1, 2)\n sage: s1 & s2 # syntactic sugar\n (1, 2)\n\n sage: s1 = RealSet((0, 1), (2, 3)); s1\n (0, 1) \u222a (2, 3)\n sage: s2 = RealSet([0, 1], [2, 3]); s2\n [0, 1] \u222a [2, 3]\n sage: s3 = RealSet([1, 2]); s3\n [1, 2]\n sage: s1.intersection(s2)\n (0, 1) \u222a (2, 3)\n sage: s1.intersection(s3)\n {}\n sage: s2.intersection(s3)\n {1} \u222a {2}\n \"\"\"\n other = RealSet(*other)\n # TODO: this can be done in linear time since the intervals are already sorted\n intervals = []\n for i1 in self._intervals:\n for i2 in other._intervals:\n intervals.append(i1.intersection(i2))\n return self.__class__(*intervals)\n\n def inf(self):\n \"\"\"\n Return the infimum\n\n OUTPUT:\n\n A real number or infinity.\n\n EXAMPLES::\n\n sage: s1 = RealSet(0,2) + RealSet.unbounded_above_closed(10); s1\n (0, 2) \u222a [10, +oo)\n sage: s1.inf()\n 0\n\n sage: s2 = RealSet(1,3) + RealSet.unbounded_below_closed(-10); s2\n (-oo, -10] \u222a (1, 3)\n sage: s2.inf()\n -Infinity\n \"\"\"\n if self.n_components() == 0:\n return infinity\n return self._intervals[0].lower()\n\n def sup(self):\n \"\"\"\n Return the supremum\n\n OUTPUT:\n\n A real number or infinity.\n\n EXAMPLES::\n\n sage: s1 = RealSet(0,2) + RealSet.unbounded_above_closed(10); s1\n (0, 2) \u222a [10, +oo)\n sage: s1.sup()\n +Infinity\n\n sage: s2 = RealSet(1,3) + RealSet.unbounded_below_closed(-10); s2\n (-oo, -10] \u222a (1, 3)\n sage: s2.sup()\n 3\n \"\"\"\n if self.n_components() == 0:\n return minus_infinity\n return self._intervals[-1].upper()\n\n def complement(self):\n \"\"\"\n Return the complement\n\n OUTPUT:\n\n The set-theoretic complement as a new :class:`RealSet`.\n\n EXAMPLES::\n\n sage: RealSet(0,1).complement()\n (-oo, 0] \u222a [1, +oo)\n\n sage: s1 = RealSet(0,2) + RealSet.unbounded_above_closed(10); s1\n (0, 2) \u222a [10, +oo)\n sage: s1.complement()\n (-oo, 0] \u222a [2, 10)\n\n sage: s2 = RealSet(1,3) + RealSet.unbounded_below_closed(-10); s2\n (-oo, -10] \u222a (1, 3)\n sage: s2.complement()\n (-10, 1] \u222a [3, +oo)\n \"\"\"\n n = self.n_components()\n if n == 0:\n return RealSet(minus_infinity, infinity)\n intervals = []\n if self.inf() != minus_infinity:\n first = self._intervals[0]\n intervals.append(InternalRealInterval(RLF(minus_infinity), False,\n first._lower, first.lower_open()))\n if self.sup() != infinity:\n last = self._intervals[-1]\n intervals.append(InternalRealInterval(last._upper, last.upper_open(),\n RLF(infinity), False))\n for i in range(1,n):\n prev = self._intervals[i-1]\n next = self._intervals[i]\n i = InternalRealInterval(prev._upper, prev.upper_open(),\n next._lower, next.lower_open())\n intervals.append(i)\n return self.__class__(*intervals)\n\n def difference(self, *other):\n \"\"\"\n Return ``self`` with ``other`` subtracted\n\n INPUT:\n\n - ``other`` -- a :class:`RealSet` or data that defines one.\n\n OUTPUT:\n\n The set-theoretic difference of ``self`` with ``other``\n removed as a new :class:`RealSet`.\n\n EXAMPLES::\n\n sage: s1 = RealSet(0,2) + RealSet.unbounded_above_closed(10); s1\n (0, 2) \u222a [10, +oo)\n sage: s2 = RealSet(1,3) + RealSet.unbounded_below_closed(-10); s2\n (-oo, -10] \u222a (1, 3)\n sage: s1.difference(s2)\n (0, 1] \u222a [10, +oo)\n sage: s1 - s2 # syntactic sugar\n (0, 1] \u222a [10, +oo)\n sage: s2.difference(s1)\n (-oo, -10] \u222a [2, 3)\n sage: s2 - s1 # syntactic sugar\n (-oo, -10] \u222a [2, 3)\n sage: s1.difference(1,11)\n (0, 1] \u222a [11, +oo)\n \"\"\"\n other = RealSet(*other)\n return self.intersection(other.complement())\n\n def symmetric_difference(self, *other):\n r\"\"\"\n Returns the symmetric difference of ``self`` and ``other``.\n\n INPUT:\n\n - ``other`` -- a :class:`RealSet` or data that defines one.\n\n OUTPUT:\n\n The set-theoretic symmetric difference of ``self`` and ``other``\n as a new :class:`RealSet`.\n\n EXAMPLES::\n\n sage: s1 = RealSet(0,2); s1\n (0, 2)\n sage: s2 = RealSet.unbounded_above_open(1); s2\n (1, +oo)\n sage: s1.symmetric_difference(s2)\n (0, 1] \u222a [2, +oo)\n \"\"\"\n other = RealSet(*other)\n return self.difference(other).union(other.difference(self))\n\n def contains(self, x):\n \"\"\"\n Return whether `x` is contained in the set\n\n INPUT:\n\n - ``x`` -- a real number.\n\n OUTPUT:\n\n Boolean.\n\n EXAMPLES::\n\n sage: s = RealSet(0,2) + RealSet.unbounded_above_closed(10); s\n (0, 2) \u222a [10, +oo)\n sage: s.contains(1)\n True\n sage: s.contains(0)\n False\n sage: 10 in s # syntactic sugar\n True\n \"\"\"\n x = RLF(x)\n for interval in self._intervals:\n if interval.contains(x):\n return True\n return False\n\n __contains__ = contains\n\n def is_subset(self, *other):\n r\"\"\"\n Return whether ``self`` is a subset of ``other``.\n\n INPUT:\n\n - ``*other`` -- a :class:`RealSet` or something that defines\n one.\n\n OUTPUT:\n\n Boolean.\n\n EXAMPLES::\n\n sage: I = RealSet((1,2))\n sage: J = RealSet((1,3))\n sage: K = RealSet((2,3))\n sage: I.is_subset(J)\n True\n sage: J.is_subset(K)\n False\n \"\"\"\n return RealSet(*other).intersection(self) == self\n\n is_included_in = deprecated_function_alias(31927, is_subset)\n\n def _an_element_(self):\n \"\"\"\n Return a point of the set\n\n OUTPUT:\n\n A real number.\n\n It raises an :class:`~sage.categories.sets_cat.EmptySetError` if the set is empty.\n\n EXAMPLES::\n\n sage: RealSet.open_closed(0, 1).an_element()\n 1\n sage: RealSet(0, 1).an_element()\n 1/2\n sage: RealSet(-oo,+oo).an_element()\n 0\n sage: RealSet(-oo,7).an_element()\n 6\n sage: RealSet(7,+oo).an_element()\n 8\n \"\"\"\n from sage.rings.infinity import AnInfinity\n if not self._intervals:\n raise EmptySetError\n i = self._intervals[0]\n if isinstance(i.lower(), AnInfinity):\n if isinstance(i.upper(), AnInfinity):\n return ZZ.zero()\n else:\n return i.upper() - 1\n if isinstance(i.upper(), AnInfinity):\n return i.lower() + 1\n if i.lower_closed():\n return i.lower()\n if i.upper_closed():\n return i.upper()\n return (i.lower() + i.upper())/ZZ(2)\n\n def is_open(self):\n \"\"\"\n Return whether ``self`` is an open set.\n\n EXAMPLES::\n\n sage: RealSet().is_open()\n True\n sage: RealSet.point(1).is_open()\n False\n sage: RealSet((1, 2)).is_open()\n True\n sage: RealSet([1, 2], (3, 4)).is_open()\n False\n\n \"\"\"\n return all(not i.lower_closed()\n and not i.upper_closed()\n for i in self._intervals)\n\n def is_closed(self):\n \"\"\"\n Return whether ``self`` is a closed set.\n\n EXAMPLES::\n\n sage: RealSet().is_closed()\n True\n sage: RealSet.point(1).is_closed()\n True\n sage: RealSet([1, 2]).is_closed()\n True\n sage: RealSet([1, 2], (3, 4)).is_closed()\n False\n \"\"\"\n return all((i.lower_closed() or i.lower() is minus_infinity)\n and (i.upper_closed() or i.upper() is infinity)\n for i in self._intervals)\n\n def closure(self):\n \"\"\"\n Return the topological closure of ``self``.\n\n EXAMPLES::\n\n sage: RealSet(-oo, oo).closure()\n (-oo, +oo)\n sage: RealSet((1, 2), (2, 3)).closure()\n [1, 3]\n \"\"\"\n return self.__class__(*[i.closure() for i in self._intervals])\n\n def interior(self):\n \"\"\"\n Return the topological interior of ``self``.\n\n EXAMPLES::\n\n sage: RealSet(-oo, oo).interior()\n (-oo, +oo)\n sage: RealSet.point(2).interior()\n {}\n sage: RealSet([1, 2], (3, 4)).interior()\n (1, 2) \u222a (3, 4)\n \"\"\"\n return self.__class__(*[i.interior() for i in self._intervals])\n\n def boundary(self):\n \"\"\"\n Return the topological boundary of ``self``.\n\n EXAMPLES::\n\n sage: RealSet(-oo, oo).boundary()\n {}\n sage: RealSet.point(2).boundary()\n {2}\n sage: RealSet([1, 2], (3, 4)).boundary()\n {1} \u222a {2} \u222a {3} \u222a {4}\n sage: RealSet((1, 2), (2, 3)).boundary()\n {1} \u222a {2} \u222a {3}\n\n \"\"\"\n return self.__class__(*[RealSet.point(x) for i in self._intervals for x in i.boundary_points()])\n\n def is_disjoint(self, *other):\n \"\"\"\n Test whether the two sets are disjoint\n\n INPUT:\n\n - ``other`` -- a :class:`RealSet` or data defining one.\n\n OUTPUT:\n\n Boolean.\n\n EXAMPLES::\n\n sage: s1 = RealSet((0, 1), (2, 3)); s1\n (0, 1) \u222a (2, 3)\n sage: s2 = RealSet([1, 2]); s2\n [1, 2]\n sage: s1.is_disjoint(s2)\n True\n sage: s1.is_disjoint([1, 2])\n True\n \"\"\"\n other = RealSet(*other)\n return self.intersection(other).is_empty()\n\n is_disjoint_from = deprecated_function_alias(31927, is_disjoint)\n\n @staticmethod\n def are_pairwise_disjoint(*real_set_collection):\n \"\"\"\n Test whether sets are pairwise disjoint\n\n INPUT:\n\n - ``*real_set_collection`` -- a list/tuple/iterable of\n :class:`RealSet`.\n\n OUTPUT:\n\n Boolean.\n\n EXAMPLES::\n\n sage: s1 = RealSet((0, 1), (2, 3))\n sage: s2 = RealSet((1, 2))\n sage: s3 = RealSet.point(3)\n sage: RealSet.are_pairwise_disjoint(s1, s2, s3)\n True\n sage: RealSet.are_pairwise_disjoint(s1, s2, s3, [10,10])\n True\n sage: RealSet.are_pairwise_disjoint(s1, s2, s3, [-1, 1/2])\n False\n \"\"\"\n sets = [RealSet(_) for _ in real_set_collection]\n for i in range(len(sets)):\n for j in range(i):\n si = sets[i]\n sj = sets[j]\n if not si.is_disjoint(sj):\n return False\n return True\n\n def _sage_input_(self, sib, coerced):\n \"\"\"\n Produce an expression which will reproduce this value when evaluated.\n\n TESTS::\n\n sage: sage_input(RealSet())\n RealSet()\n sage: sage_input(RealSet.open(-oo, +oo))\n RealSet(-oo, oo)\n sage: sage_input(RealSet.point(77))\n RealSet.point(77)\n sage: sage_input(RealSet.closed_open(0, +oo))\n RealSet.closed_open(0, oo)\n sage: sage_input(RealSet.open_closed(-oo, 0))\n RealSet.open_closed(-oo, 0)\n sage: sage_input(RealSet.open_closed(-1, 0))\n RealSet.open_closed(-1, 0)\n sage: sage_input(RealSet.closed_open(-1, 0))\n RealSet.closed_open(-1, 0)\n sage: sage_input(RealSet.closed(0, 1))\n RealSet.closed(0, 1)\n sage: sage_input(RealSet.open(0, 1))\n RealSet.open(0, 1)\n sage: sage_input(RealSet.open(0, 1) + RealSet.open(1, 2))\n RealSet.open(0, 1) + RealSet.open(1, 2)\n \"\"\"\n\n def interval_input(i):\n lower, upper = i.lower(), i.upper()\n if i.is_point():\n return sib.name('RealSet.point')(lower)\n elif lower == minus_infinity and upper == infinity:\n return sib.name('RealSet')(sib(minus_infinity), sib(infinity))\n else:\n if i.lower_closed():\n if i.upper_closed():\n t = 'RealSet.closed'\n else:\n t = 'RealSet.closed_open'\n else:\n if i.upper_closed():\n t = 'RealSet.open_closed'\n else:\n t = 'RealSet.open'\n return sib.name(t)(sib(lower), sib(upper))\n\n if self.is_empty():\n return sib.name('RealSet')()\n else:\n return sib.sum(interval_input(i) for i in self)\n\n def __mul__(self, right):\n r\"\"\"\n Scale a real set by a scalar on the left or right.\n\n EXAMPLES::\n\n sage: A = RealSet([0, 1/2], (2, infinity)); A\n [0, 1/2] \u222a (2, +oo)\n sage: 2 * A\n [0, 1] \u222a (4, +oo)\n sage: A * 100\n [0, 50] \u222a (200, +oo)\n sage: 1.5 * A\n [0.000000000000000, 0.750000000000000] \u222a (3.00000000000000, +oo)\n sage: (-2) * A\n (-oo, -4) \u222a [-1, 0]\n \"\"\"\n if not isinstance(right, self.__class__):\n return self.__class__(*[e * right for e in self])\n elif not isinstance(self, self.__class__):\n return self.__class__(*[self * e for e in right])\n else:\n return NotImplemented\n\n def __rmul__(self, other):\n r\"\"\"\n Scale a real set by a scalar on the left.\n\n TESTS::\n\n sage: A = RealSet([0, 1/2], RealSet.unbounded_above_closed(2)); A\n [0, 1/2] \u222a [2, +oo)\n sage: pi * A\n [0, 1/2*pi] \u222a [2*pi, +oo)\n \"\"\"\n return self * other\n\n def _sympy_(self):\n r\"\"\"\n Return the SymPy set corresponding to ``self``.\n\n EXAMPLES::\n\n sage: RealSet()._sympy_()\n EmptySet\n sage: RealSet.point(5)._sympy_()\n FiniteSet(5)\n sage: (RealSet.point(1).union(RealSet.point(2)))._sympy_()\n FiniteSet(1, 2)\n sage: (RealSet(1, 2).union(RealSet.closed(3, 4)))._sympy_()\n Union(Interval.open(1, 2), Interval(3, 4))\n sage: RealSet(-oo, oo)._sympy_()\n Reals\n\n Infinities are not elements::\n\n sage: import sympy\n sage: RealSet(-oo, oo)._sympy_().contains(sympy.oo)\n False\n \"\"\"\n from sympy import Reals, Union\n from sage.interfaces.sympy import sympy_init\n sympy_init()\n if self.is_universe():\n return Reals\n else:\n return Union(*[interval._sympy_()\n for interval in self._intervals])\n\n@richcmp_method\nclass RealSet_rtree(RealSet):\n def __init__(self, *intervals):\n '''\n A subset of the real line with rtree data structure.\n \n TEST::\n sage: from sage.sets.real_set import RealSet_rtree \n sage: A = RealSet_rtree(0,1); C = A.complement(); print(type(C), \"with\", C.rtree) \n with rtree.index.Index(bounds=[0.0, 0.0, -inf, inf], size=2)\n sage: A = RealSet_rtree(-infinity, infinity); C = A.complement(); print(type(C), \"with\", C.rtree) \n with None\n sage: type(RealSet_rtree(-infinity, infinity).complement()) \n \n \n '''\n \n super().__init__(*intervals)\n ###\n # adding rtree as a preliminary filter to overestimate the interval \n # ranges, so that it can give a quicker respond to the case when the \n # ntervals set does NOT contain a certain point.\n # No-list 1. intervals set is Empty, 2. intervals only have -infinity, +infinity, 0 as endpoints\n self.multiplier = 1\n self.rtree = None\n \n if len(self._intervals) == 0: #rtree is not needed, since it is empty!\n return \n # x_max is the largest but not infinity absolute value of intervals\n x_max = 0\n from sage.functions.log import log\n from sage.functions.other import floor\n from sage.functions.other import ceil\n\n\n for interval in self._intervals:\n lower, upper = abs(interval.lower()), abs(interval.upper())\n if (x_max < lower) and (lower != +infinity):\n x_max = lower\n if (x_max < upper) and (upper != +infinity):\n x_max = upper\n if x_max == 0: #rtree is not needed, since they are only +-infinity and zero!\n return \n # the multiplier should have: multiplier <= int(safebound / x_max)\n # for safe, we can use log2(x) to deal with it\n # that would be:\n # log2_multiplier <= log2_safebound - log2_x_max\n log2_safebound = 52\n log2_multiplier = log2_safebound - int(log(x_max,2)) - 1\n self.multiplier = 2**log2_multiplier\n\n from rtree import index\n \n p = index.Property()\n p.dimension = 2\n self.rtree = index.Index(properties = p, interleaved = False)\n for interval in self._intervals:\n # Since the floor() and ceil() can not deal with +-infinity, we have to make it seperately\n if interval.lower() in [-infinity, infinity]:\n lower = interval.lower()\n else: \n lower = floor(interval.lower()*self.multiplier)\n if interval.upper() in [-infinity, infinity]:\n upper = interval.upper()\n else: \n upper = ceil(interval.upper()*self.multiplier)\n self.rtree.insert(0,(0, 0, lower, upper))\n ###\n def contains(self, x):\n \"\"\"\n Return whether `x` is contained in the set.\n \n Class `RealSet_rtree` provides a fast path for __contains__ when the result will be False.\n\n INPUT:\n\n - ``x`` -- a real number.\n\n OUTPUT:\n\n Boolean.\n\n TESTS::\n sage: from sage.sets.real_set import RealSet_rtree\n sage: RealSet_rtree(0,1).contains(0.5) \n True\n sage: RealSet_rtree(0,1).contains(3) \n False\n sage: (RealSet_rtree(0,1)+RealSet_rtree(3,4)).contains(3) \n False\n sage: (RealSet_rtree(0,1)+RealSet_rtree([3,4])).contains(3) \n True\n \n \"\"\"\n ### rtree filters non-containing points\n if self.rtree != None:\n scaled_x = x * self.multiplier\n if self.rtree.count((0, 0, scaled_x, scaled_x)) == 0:\n return False\n ###\n\n# x = RLF(x)\n# for interval in self._intervals:\n# if interval.contains(x):\n# return True\n return super().contains(x)\n \n __contains__ = contains \n def intersection(self, *other):\n \"\"\"\n Return the intersection of the two sets with rtree data structure.\n\n Class `RealSet_rtree` provides a fast path for computing empty intersection when the result will be empty.\n \n INPUT:\n \n - ``other`` -- a :class: `RealSet`, `RealSet_rtree` or data that defines either.\n\n OUTPUT:\n \n The set-theoretic intersection as a new :class:`RealSet_rtree`.\n\n TESTS::\n sage: from sage.sets.real_set import RealSet_rtree \n sage: A = RealSet(0,1); B = RealSet_rtree(0.5,3); print(type(A.intersection(B)), type(B.intersection(A)))\n \n \"\"\"\n \n ###\n if self.rtree != None:\n other = RealSet(*other)\n # TODO: this can be done in linear time since the intervals are already sorted\n intervals = []\n for i2 in other._intervals:\n if self.rtree.count((0, 0, self.multiplier * i2.lower(), self.multiplier * i2.upper())) == 0:\n continue\n else:\n for i1 in self._intervals:\n intervals.append(i1.intersection(i2))\n return self.__class__(*intervals)\n \n else:\n ###\n return super().intersection(self, *other)\n\n __and__ = intersection\n```\n", "meta": {"hexsha": "0cb64e305989dceff57e5339f5e0c2731f9a48e5", "size": 101052, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Real_Set_modified.ipynb", "max_stars_repo_name": "DRKWang/cutgeneratingfunctions_new", "max_stars_repo_head_hexsha": "9044a5864d3f279bfd6ad11d0531173f746ab303", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-09-27T17:33:50.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-27T17:33:50.000Z", "max_issues_repo_path": "Real_Set_modified.ipynb", "max_issues_repo_name": "DRKWang/cutgeneratingfunctions_new", "max_issues_repo_head_hexsha": "9044a5864d3f279bfd6ad11d0531173f746ab303", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Real_Set_modified.ipynb", "max_forks_repo_name": "DRKWang/cutgeneratingfunctions_new", "max_forks_repo_head_hexsha": "9044a5864d3f279bfd6ad11d0531173f746ab303", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 38.9560524287, "max_line_length": 143, "alphanum_fraction": 0.4175968808, "converted": true, "num_tokens": 18997, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.6723316991792861, "lm_q1q2_score": 0.4282959964825267}} {"text": "```python\nfrom __future__ import print_function\nimport numpy as np\nimport pprint as pp\nfrom copy import deepcopy\nimport pickle\nfrom numbers import Number\nfrom collections import OrderedDict\nimport itertools\nimport torch\nimport torch.nn as nn\nfrom torch.autograd import Variable\nfrom torch.nn import functional as F\nimport torch.optim as optim\nfrom torch.optim.lr_scheduler import ReduceLROnPlateau, LambdaLR, CosineAnnealingLR, CosineAnnealingWarmRestarts\nfrom torch.distributions import constraints\nfrom torch.distributions.normal import Normal\nfrom torch.distributions.multivariate_normal import MultivariateNormal\nfrom torch.distributions.distribution import Distribution\nfrom torch.distributions.utils import broadcast_all\n\nimport sys, os\nsys.path.append(os.path.join(os.path.dirname(__file__), '..'))\nfrom pytorch_net.modules import get_Layer, load_layer_dict, Simple_2_Symbolic\nfrom pytorch_net.util import forward, get_epochs_T_mult, Loss_Fun, get_activation, get_criterion, get_criteria_value, get_optimizer, get_full_struct_param, plot_matrices, get_model_DL, PrecisionFloorLoss, get_list_DL, init_weight\nfrom pytorch_net.util import Early_Stopping, Performance_Monitor, record_data, to_np_array, to_Variable, make_dir, formalize_value, RampupLR, Transform_Label, view_item, load_model, save_model, to_cpu_recur, filter_kwargs\n```\n\n## Training functionality:\n\n\n```python\ndef train(\n model,\n X=None,\n y=None,\n train_loader=None,\n validation_data=None,\n validation_loader=None,\n criterion=nn.MSELoss(),\n inspect_interval=10,\n isplot=False,\n is_cuda=None,\n **kwargs\n ):\n \"\"\"Training function for generic models. \"model\" can be a single model or a ordered list of models\"\"\"\n def get_regularization(model, loss_epoch, **kwargs):\n \"\"\"Compute regularization.\"\"\"\n reg_dict = kwargs[\"reg_dict\"] if \"reg_dict\" in kwargs else None\n reg = to_Variable([0], is_cuda = is_cuda)\n if reg_dict is not None:\n for reg_type, reg_coeff in reg_dict.items():\n # Setting up regularization strength:\n if isinstance(reg_coeff, Number):\n reg_coeff_ele = reg_coeff\n else:\n if loss_epoch < len(reg_coeff):\n reg_coeff_ele = reg_coeff[loss_epoch]\n else:\n reg_coeff_ele = reg_coeff[-1]\n # Accumulate regularization:\n reg = reg + model.get_regularization(source=[reg_type], mode=reg_mode, **kwargs) * reg_coeff_ele\n return reg\n\n if is_cuda is None:\n if X is None and y is None:\n assert train_loader is not None\n is_cuda = train_loader.dataset.tensors[0].is_cuda\n else:\n is_cuda = X.is_cuda\n\n # Optimization kwargs:\n epochs = kwargs[\"epochs\"] if \"epochs\" in kwargs else 10000\n lr = kwargs[\"lr\"] if \"lr\" in kwargs else 5e-3\n lr_rampup_steps = kwargs[\"lr_rampup\"] if \"lr_rampup\" in kwargs else 200\n optim_type = kwargs[\"optim_type\"] if \"optim_type\" in kwargs else \"adam\"\n optim_kwargs = kwargs[\"optim_kwargs\"] if \"optim_kwargs\" in kwargs else {}\n scheduler_type = kwargs[\"scheduler_type\"] if \"scheduler_type\" in kwargs else \"ReduceLROnPlateau\"\n gradient_noise = kwargs[\"gradient_noise\"] if \"gradient_noise\" in kwargs else None\n data_loader_apply = kwargs[\"data_loader_apply\"] if \"data_loader_apply\" in kwargs else None\n\n # Inspection kwargs:\n inspect_step = kwargs[\"inspect_step\"] if \"inspect_step\" in kwargs else None # Whether to inspect each step\n inspect_items = kwargs[\"inspect_items\"] if \"inspect_items\" in kwargs else None\n inspect_items_train = get_inspect_items_train(inspect_items)\n inspect_functions = kwargs[\"inspect_functions\"] if \"inspect_functions\" in kwargs else None\n if inspect_functions is not None:\n for inspect_function_key in inspect_functions:\n if inspect_function_key not in inspect_items:\n inspect_items.append(inspect_function_key)\n inspect_items_interval = kwargs[\"inspect_items_interval\"] if \"inspect_items_interval\" in kwargs else 1000\n inspect_image_interval = kwargs[\"inspect_image_interval\"] if \"inspect_image_interval\" in kwargs else None\n inspect_loss_precision = kwargs[\"inspect_loss_precision\"] if \"inspect_loss_precision\" in kwargs else 4\n callback = kwargs[\"callback\"] if \"callback\" in kwargs else None\n\n # Saving kwargs:\n record_keys = kwargs[\"record_keys\"] if \"record_keys\" in kwargs else [\"loss\"]\n filename = kwargs[\"filename\"] if \"filename\" in kwargs else None\n if filename is not None:\n make_dir(filename)\n save_interval = kwargs[\"save_interval\"] if \"save_interval\" in kwargs else None\n save_step = kwargs[\"save_step\"] if \"save_step\" in kwargs else None\n logdir = kwargs[\"logdir\"] if \"logdir\" in kwargs else None\n data_record = {key: [] for key in record_keys}\n info_to_save = kwargs[\"info_to_save\"] if \"info_to_save\" in kwargs else None\n if info_to_save is not None:\n data_record.update(info_to_save)\n patience = kwargs[\"patience\"] if \"patience\" in kwargs else 20\n if patience is not None:\n early_stopping_epsilon = kwargs[\"early_stopping_epsilon\"] if \"early_stopping_epsilon\" in kwargs else 0\n early_stopping_monitor = kwargs[\"early_stopping_monitor\"] if \"early_stopping_monitor\" in kwargs else \"loss\"\n early_stopping = Early_Stopping(patience = patience, epsilon = early_stopping_epsilon, mode = \"max\" if early_stopping_monitor in [\"accuracy\"] else \"min\")\n if logdir is not None:\n from pytorch_net.logger import Logger\n batch_idx = 0\n logger = Logger(logdir)\n logimages = kwargs[\"logimages\"] if \"logimages\" in kwargs else None\n reg_mode = kwargs[\"reg_mode\"] if \"reg_mode\" in kwargs else \"L1\"\n\n if validation_loader is not None:\n assert validation_data is None\n X_valid, y_valid = None, None\n elif validation_data is not None:\n X_valid, y_valid = validation_data\n else:\n X_valid, y_valid = X, y\n\n # Setting up dynamic label noise:\n label_noise_matrix = kwargs[\"label_noise_matrix\"] if \"label_noise_matrix\" in kwargs else None\n transform_label = Transform_Label(label_noise_matrix = label_noise_matrix, is_cuda=is_cuda)\n\n # Setting up cotrain optimizer:\n co_kwargs = kwargs[\"co_kwargs\"] if \"co_kwargs\" in kwargs else None\n if co_kwargs is not None:\n co_optimizer = co_kwargs[\"co_optimizer\"]\n co_model = co_kwargs[\"co_model\"]\n co_criterion = co_kwargs[\"co_criterion\"] if \"co_criterion\" in co_kwargs else None\n co_multi_step = co_kwargs[\"co_multi_step\"] if \"co_multi_step\" in co_kwargs else 1\n\n # Get original loss:\n if len(inspect_items_train) > 0:\n loss_value_train = get_loss(model, train_loader, X, y, criterion=criterion, loss_epoch=-1, transform_label=transform_label, **kwargs)\n info_dict_train = prepare_inspection(model, train_loader, X, y, transform_label=transform_label, **kwargs)\n if \"loss\" in record_keys:\n record_data(data_record, [loss_value_train], [\"loss_tr\"])\n loss_original = get_loss(model, validation_loader, X_valid, y_valid, criterion=criterion, loss_epoch=-1, transform_label=transform_label, **kwargs)\n if \"loss\" in record_keys:\n record_data(data_record, [-1, loss_original], [\"iter\", \"loss\"])\n if \"reg\" in record_keys and \"reg_dict\" in kwargs and len(kwargs[\"reg_dict\"]) > 0:\n reg_value = get_regularization(model, loss_epoch=0, **kwargs)\n record_data(data_record, [reg_value], [\"reg\"])\n if \"param\" in record_keys:\n record_data(data_record, [model.get_weights_bias(W_source=\"core\", b_source=\"core\")], [\"param\"])\n if \"param_grad\" in record_keys:\n record_data(data_record, [model.get_weights_bias(W_source=\"core\", b_source=\"core\", is_grad=True)], [\"param_grad\"])\n if co_kwargs is not None:\n co_loss_original = get_loss(co_model, validation_loader, X_valid, y_valid, criterion=criterion, loss_epoch=-1, transform_label=transform_label, **co_kwargs)\n if \"co_loss\" in record_keys:\n record_data(data_record, [co_loss_original], [\"co_loss\"])\n if filename is not None and save_interval is not None:\n record_data(data_record, [{}], [\"model_dict\"])\n\n # Setting up optimizer:\n parameters = model.parameters()\n num_params = len(list(model.parameters()))\n if num_params == 0:\n print(\"No parameters to optimize!\")\n loss_value = get_loss(model, validation_loader, X_valid, y_valid, criterion = criterion, loss_epoch = -1, transform_label=transform_label, **kwargs)\n if \"loss\" in record_keys:\n record_data(data_record, [0, loss_value], [\"iter\", \"loss\"])\n if \"param\" in record_keys:\n record_data(data_record, [model.get_weights_bias(W_source = \"core\", b_source = \"core\")], [\"param\"])\n if \"param_grad\" in record_keys:\n record_data(data_record, [model.get_weights_bias(W_source = \"core\", b_source = \"core\", is_grad = True)], [\"param_grad\"])\n if co_kwargs is not None:\n co_loss_value = get_loss(co_model, validation_loader, X_valid, y_valid, criterion = criterion, loss_epoch = -1, transform_label=transform_label, **co_kwargs)\n record_data(data_record, [co_loss_value], [\"co_loss\"])\n return loss_original, loss_value, data_record\n optimizer = get_optimizer(optim_type, lr, parameters, **optim_kwargs) if \"optimizer\" not in kwargs or (\"optimizer\" in kwargs and kwargs[\"optimizer\"] is None) else kwargs[\"optimizer\"]\n\n # Initialize inspect_items:\n if inspect_items is not None:\n print(\"{}:\".format(-1), end = \"\")\n print(\"\\tlr: {0:.3e}\\t loss:{1:.{2}f}\".format(optimizer.param_groups[0][\"lr\"], loss_original, inspect_loss_precision), end = \"\")\n info_dict = prepare_inspection(model, validation_loader, X_valid, y_valid, transform_label=transform_label, **kwargs)\n if len(inspect_items_train) > 0:\n print(\"\\tloss_tr: {0:.{1}f}\".format(loss_value_train, inspect_loss_precision), end = \"\")\n info_dict_train = update_key_train(info_dict_train, inspect_items_train)\n info_dict.update(info_dict_train)\n if \"reg\" in record_keys and \"reg_dict\" in kwargs and len(kwargs[\"reg_dict\"]) > 0:\n print(\"\\treg:{0:.{1}f}\".format(to_np_array(reg_value), inspect_loss_precision), end=\"\")\n if len(info_dict) > 0:\n for item in inspect_items:\n if item in info_dict:\n print(\" \\t{0}: {1:.{2}f}\".format(item, info_dict[item], inspect_loss_precision), end = \"\")\n if item in record_keys and item not in [\"loss\", \"reg\"]:\n record_data(data_record, [to_np_array(info_dict[item])], [item])\n\n if co_kwargs is not None:\n co_info_dict = prepare_inspection(co_model, validation_loader, X_valid, y_valid, transform_label=transform_label, **co_kwargs)\n if \"co_loss\" in inspect_items:\n co_loss_value = get_loss(co_model, validation_loader, X_valid, y_valid, criterion=criterion, loss_epoch=-1, transform_label=transform_label, **co_kwargs)\n print(\"\\tco_loss: {}\".format(formalize_value(co_loss_value, inspect_loss_precision)), end=\"\")\n if len(co_info_dict) > 0:\n for item in inspect_items:\n if item in co_info_dict:\n print(\" \\t{0}: {1}\".format(item, formalize_value(co_info_dict[item], inspect_loss_precision)), end=\"\")\n if item in record_keys and item != \"loss\":\n record_data(data_record, [to_np_array(co_info_dict[item])], [item])\n print(\"\\n\")\n\n # Setting up gradient noise:\n if gradient_noise is not None:\n from pytorch_net.util import Gradient_Noise_Scale_Gen\n scale_gen = Gradient_Noise_Scale_Gen(epochs=epochs,\n gamma=gradient_noise[\"gamma\"], # decay rate\n eta=gradient_noise[\"eta\"], # starting variance\n gradient_noise_interval_epoch=1,\n )\n gradient_noise_scale = scale_gen.generate_scale(verbose=True)\n\n # Set up learning rate scheduler:\n if scheduler_type is not None:\n if scheduler_type == \"ReduceLROnPlateau\":\n scheduler_patience = kwargs[\"scheduler_patience\"] if \"scheduler_patience\" in kwargs else 40\n scheduler_factor = kwargs[\"scheduler_factor\"] if \"scheduler_factor\" in kwargs else 0.1\n scheduler_verbose = kwargs[\"scheduler_verbose\"] if \"scheduler_verbose\" in kwargs else False\n scheduler = ReduceLROnPlateau(optimizer, factor=scheduler_factor, patience=scheduler_patience, verbose=scheduler_verbose)\n elif scheduler_type == \"LambdaLR\":\n scheduler_lr_lambda = kwargs[\"scheduler_lr_lambda\"] if \"scheduler_lr_lambda\" in kwargs else (lambda epoch: 0.97 ** (epoch // 2))\n scheduler = LambdaLR(optimizer, lr_lambda=scheduler_lr_lambda)\n elif scheduler_type == \"cos\":\n scheduler = CosineAnnealingLR(optimizer, T_max=epochs)\n elif scheduler_type == \"coslr\":\n T_0 = max(min(25, epochs//31), 1)\n T_mult = kwargs[\"scheduler_T_mult\"] if \"scheduler_T_mult\" in kwargs else 2\n epochs = get_epochs_T_mul(epochs, T_0=T_0, T_mult=T_mult)\n scheduler = CosineAnnealingWarmRestarts(optimizer, T_0=T_0, T_mult=T_mult)\n else:\n raise\n # Ramping or learning rate for the first lr_rampup_steps steps:\n if lr_rampup_steps is not None and train_loader is not None:\n scheduler_rampup = RampupLR(optimizer, num_steps=lr_rampup_steps)\n if hasattr(train_loader, \"dataset\"):\n data_size = len(train_loader.dataset)\n else:\n data_size = kwargs[\"data_size\"]\n\n # Initialize logdir:\n if logdir is not None:\n if logimages is not None:\n for tag, image_fun in logimages[\"image_fun\"].items():\n image = image_fun(model, logimages[\"X\"], logimages[\"y\"])\n logger.log_images(tag, image, -1)\n\n # Training:\n to_stop = False\n\n for i in range(epochs + 1):\n model.train()\n\n # Updating gradient noise:\n if gradient_noise is not None:\n hook_handle_list = []\n if i % scale_gen.gradient_noise_interval_epoch == 0:\n for h in hook_handle_list:\n h.remove()\n hook_handle_list = []\n scale_idx = int(i / scale_gen.gradient_noise_interval_epoch)\n if scale_idx >= len(gradient_noise_scale):\n current_gradient_noise_scale = gradient_noise_scale[-1]\n else:\n current_gradient_noise_scale = gradient_noise_scale[scale_idx]\n for param_group in optimizer.param_groups:\n for param in param_group[\"params\"]:\n if param.requires_grad:\n h = param.register_hook(lambda grad: grad + Variable(torch.normal(mean=torch.zeros(grad.size()),\n std=current_gradient_noise_scale * torch.ones(grad.size()))))\n hook_handle_list.append(h)\n\n if X is not None and y is not None:\n if optim_type != \"LBFGS\":\n optimizer.zero_grad()\n reg = get_regularization(model, loss_epoch=i, **kwargs)\n loss = model.get_loss(X, transform_label(y), criterion=criterion, loss_epoch=i, **kwargs) + reg\n loss.backward()\n optimizer.step()\n else:\n # \"LBFGS\" is a second-order optimization algorithm that requires a slightly different procedure:\n def closure():\n optimizer.zero_grad()\n reg = get_regularization(model, loss_epoch=i, **kwargs)\n loss = model.get_loss(X, transform_label(y), criterion=criterion, loss_epoch=i, **kwargs) + reg\n loss.backward()\n return loss\n optimizer.step(closure)\n \n # Cotrain step:\n if co_kwargs is not None:\n if \"co_warmup_epochs\" not in co_kwargs or \"co_warmup_epochs\" in co_kwargs and i >= co_kwargs[\"co_warmup_epochs\"]:\n for _ in range(co_multi_step):\n co_optimizer.zero_grad()\n co_reg = get_regularization(co_model, loss_epoch=i, **co_kwargs)\n co_loss = co_model.get_loss(X, transform_label(y), criterion=co_criterion, loss_epoch=i, **co_kwargs) + co_reg\n co_loss.backward()\n co_optimizer.step()\n else:\n if inspect_step is not None:\n info_dict_step = {key: [] for key in inspect_items}\n\n if \"loader_process\" in kwargs and kwargs[\"loader_process\"] is not None:\n train_loader = kwargs[\"loader_process\"](\"train\")\n for k, data_batch in enumerate(train_loader):\n if isinstance(data_batch, tuple) or isinstance(data_batch, list):\n X_batch, y_batch = data_batch\n if data_loader_apply is not None:\n X_batch, y_batch = data_loader_apply(X_batch, y_batch)\n else:\n X_batch, y_batch = data_loader_apply(data_batch)\n if optim_type != \"LBFGS\":\n optimizer.zero_grad()\n reg = get_regularization(model, loss_epoch=i, **kwargs)\n loss = model.get_loss(X_batch, transform_label(y_batch), criterion=criterion, loss_epoch=i, loss_step=k, **kwargs) + reg\n loss.backward()\n if logdir is not None:\n batch_idx += 1\n if len(model.info_dict) > 0:\n for item in inspect_items:\n if item in model.info_dict:\n logger.log_scalar(item, model.info_dict[item], batch_idx)\n optimizer.step()\n else:\n def closure():\n optimizer.zero_grad()\n reg = get_regularization(model, loss_epoch=i, **kwargs)\n loss = model.get_loss(X_batch, transform_label(y_batch), criterion=criterion, loss_epoch=i, loss_step=k, **kwargs) + reg\n loss.backward()\n return loss\n if logdir is not None:\n batch_idx += 1\n if len(model.info_dict) > 0:\n for item in inspect_items:\n if item in model.info_dict:\n logger.log_scalar(item, model.info_dict[item], batch_idx)\n optimizer.step(closure)\n\n # Rampup scheduler:\n if lr_rampup_steps is not None and i * data_size // len(X_batch) + k < lr_rampup_steps:\n scheduler_rampup.step()\n\n # Cotrain step:\n if co_kwargs is not None:\n if \"co_warmup_epochs\" not in co_kwargs or \"co_warmup_epochs\" in co_kwargs and i >= co_kwargs[\"co_warmup_epochs\"]:\n for _ in range(co_multi_step):\n co_optimizer.zero_grad()\n co_reg = get_regularization(co_model, loss_epoch=i, **co_kwargs)\n co_loss = co_model.get_loss(X_batch, transform_label(y_batch), criterion=co_criterion, loss_epoch=i, loss_step=k, **co_kwargs) + co_reg\n co_loss.backward()\n if logdir is not None:\n if len(co_model.info_dict) > 0:\n for item in inspect_items:\n if item in co_model.info_dict:\n logger.log_scalar(item, co_model.info_dict[item], batch_idx)\n co_optimizer.step()\n\n # Inspect at each step:\n if inspect_step is not None:\n if k % inspect_step == 0:\n print(\"s{}:\".format(k), end = \"\")\n info_dict = prepare_inspection(model, validation_loader, X_valid, y_valid, transform_label=transform_label, **kwargs) \n if \"loss\" in inspect_items:\n info_dict_step[\"loss\"].append(loss.item())\n print(\"\\tloss: {0:.{1}f}\".format(loss.item(), inspect_loss_precision), end=\"\")\n if len(info_dict) > 0:\n for item in inspect_items:\n if item in info_dict:\n info_dict_step[item].append(info_dict[item])\n print(\" \\t{0}: {1}\".format(item, formalize_value(info_dict[item], inspect_loss_precision)), end = \"\")\n if co_kwargs is not None:\n if \"co_warmup_epochs\" not in co_kwargs or \"co_warmup_epochs\" in co_kwargs and i >= co_kwargs[\"co_warmup_epochs\"]:\n co_info_dict = prepare_inspection(co_model, validation_loader, X_valid, y_valid, transform_label=transform_label, **co_kwargs)\n if \"co_loss\" in inspect_items:\n print(\"\\tco_loss: {0:.{1}f}\".format(co_loss.item(), inspect_loss_precision), end=\"\")\n info_dict_step[\"co_loss\"].append(co_loss.item())\n if len(co_info_dict) > 0:\n for item in inspect_items:\n if item in co_info_dict and item != \"co_loss\":\n info_dict_step[item].append(co_info_dict[item])\n print(\" \\t{0}: {1}\".format(item, formalize_value(co_info_dict[item], inspect_loss_precision)), end=\"\")\n print()\n if k % save_step == 0:\n if filename is not None:\n pickle.dump(model.model_dict, open(filename[:-2] + \"_model.p\", \"wb\"))\n\n if logdir is not None:\n # Log values and gradients of the parameters (histogram summary)\n# for tag, value in model.named_parameters():\n# tag = tag.replace('.', '/')\n# logger.log_histogram(tag, to_np_array(value), i)\n# logger.log_histogram(tag + '/grad', to_np_array(value.grad), i)\n if logimages is not None:\n for tag, image_fun in logimages[\"image_fun\"].items():\n image = image_fun(model, logimages[\"X\"], logimages[\"y\"])\n logger.log_images(tag, image, i)\n\n if i % inspect_interval == 0:\n model.eval()\n if inspect_items is not None and i % inspect_items_interval == 0 and len(inspect_items_train) > 0:\n loss_value_train = get_loss(model, train_loader, X, y, criterion = criterion, loss_epoch = i, transform_label=transform_label, **kwargs)\n info_dict_train = prepare_inspection(model, train_loader, X, y, transform_label=transform_label, **kwargs)\n loss_value = get_loss(model, validation_loader, X_valid, y_valid, criterion = criterion, loss_epoch = i, transform_label=transform_label, **kwargs)\n reg_value = get_regularization(model, loss_epoch = i, **kwargs)\n if scheduler_type is not None:\n if lr_rampup_steps is None or train_loader is None or (lr_rampup_steps is not None and i * data_size // len(X_batch) + k >= lr_rampup_steps):\n if scheduler_type == \"ReduceLROnPlateau\":\n scheduler.step(loss_value)\n else:\n scheduler.step()\n if callback is not None:\n assert callable(callback)\n callback(model = model,\n X = X_valid,\n y = y_valid,\n iteration = i,\n loss = loss_value,\n )\n if patience is not None:\n if early_stopping_monitor == \"loss\":\n to_stop = early_stopping.monitor(loss_value)\n else:\n info_dict = prepare_inspection(model, validation_loader, X_valid, y_valid, transform_label=transform_label, **kwargs)\n to_stop = early_stopping.monitor(info_dict[early_stopping_monitor])\n if inspect_items is not None:\n if i % inspect_items_interval == 0:\n # Get loss:\n print(\"{}:\".format(i), end = \"\")\n print(\"\\tlr: {0:.3e}\\tloss: {1:.{2}f}\".format(optimizer.param_groups[0][\"lr\"], loss_value, inspect_loss_precision), end = \"\")\n info_dict = prepare_inspection(model, validation_loader, X_valid, y_valid, transform_label=transform_label, **kwargs)\n if len(inspect_items_train) > 0:\n print(\"\\tloss_tr: {0:.{1}f}\".format(loss_value_train, inspect_loss_precision), end = \"\")\n info_dict_train = update_key_train(info_dict_train, inspect_items_train)\n info_dict.update(info_dict_train)\n if \"reg\" in inspect_items and \"reg_dict\" in kwargs and len(kwargs[\"reg_dict\"]) > 0:\n print(\"\\treg:{0:.{1}f}\".format(to_np_array(reg_value), inspect_loss_precision), end=\"\")\n \n # Print and record:\n if len(info_dict) > 0:\n for item in inspect_items:\n if item + \"_val\" in info_dict:\n print(\" \\t{0}: {1}\".format(item, formalize_value(info_dict[item + \"_val\"], inspect_loss_precision)), end = \"\")\n if item in record_keys and item not in [\"loss\", \"reg\"]:\n record_data(data_record, [to_np_array(info_dict[item + \"_val\"])], [item])\n\n # logger:\n if logdir is not None:\n for item in inspect_items:\n if item + \"_val\" in info_dict:\n logger.log_scalar(item + \"_val\", info_dict[item + \"_val\"], i)\n\n # Co_model:\n if co_kwargs is not None:\n co_loss_value = get_loss(co_model, validation_loader, X_valid, y_valid, criterion = criterion, loss_epoch = i, transform_label=transform_label, **co_kwargs)\n co_info_dict = prepare_inspection(co_model, validation_loader, X_valid, y_valid, transform_label=transform_label, **co_kwargs)\n if \"co_loss\" in inspect_items:\n print(\"\\tco_loss: {0:.{1}f}\".format(co_loss_value, inspect_loss_precision), end=\"\")\n if len(co_info_dict) > 0:\n for item in inspect_items:\n if item + \"_val\" in co_info_dict:\n print(\" \\t{0}: {1}\".format(item, formalize_value(co_info_dict[item + \"_val\"], inspect_loss_precision)), end=\"\")\n if item in record_keys and item != \"co_loss\":\n record_data(data_record, [to_np_array(co_info_dict[item + \"_val\"])], [item])\n if \"co_loss\" in record_keys:\n record_data(data_record, [co_loss_value], [\"co_loss\"])\n\n # Training metrics:\n if inspect_step is not None:\n for item in info_dict_step:\n if len(info_dict_step[item]) > 0:\n print(\" \\t{0}_s: {1}\".format(item, formalize_value(np.mean(info_dict_step[item]), inspect_loss_precision)), end = \"\")\n if item in record_keys and item != \"loss\":\n record_data(data_record, [np.mean(info_dict_step[item])], [\"{}_s\".format(item)])\n\n # Record loss:\n if \"loss\" in record_keys:\n record_data(data_record, [i, loss_value], [\"iter\", \"loss\"])\n if \"reg\" in record_keys and \"reg_dict\" in kwargs and len(kwargs[\"reg_dict\"]) > 0:\n record_data(data_record, [reg_value], [\"reg\"])\n if \"param\" in record_keys:\n record_data(data_record, [model.get_weights_bias(W_source = \"core\", b_source = \"core\")], [\"param\"])\n if \"param_grad\" in record_keys:\n record_data(data_record, [model.get_weights_bias(W_source = \"core\", b_source = \"core\", is_grad = True)], [\"param_grad\"])\n print(\"\\n\")\n try:\n sys.stdout.flush()\n except:\n pass\n if isplot:\n if inspect_image_interval is not None and hasattr(model, \"plot\"):\n if i % inspect_image_interval == 0:\n if gradient_noise is not None:\n print(\"gradient_noise: {0:.9f}\".format(current_gradient_noise_scale))\n plot_model(model, data_loader = validation_loader, X = X_valid, y = y_valid, transform_label=transform_label, data_loader_apply=data_loader_apply)\n if co_kwargs is not None and \"inspect_image_interval\" in co_kwargs and co_kwargs[\"inspect_image_interval\"] and hasattr(co_model, \"plot\"):\n if i % co_kwargs[\"inspect_image_interval\"] == 0:\n plot_model(co_model, data_loader = validation_loader, X = X_valid, y = y_valid, transform_label=transform_label, data_loader_apply=data_loader_apply)\n if save_interval is not None:\n if i % save_interval == 0:\n record_data(data_record, [model.model_dict], [\"model_dict\"])\n if co_kwargs is not None:\n record_data(data_record, [co_model.model_dict], [\"co_model_dict\"])\n if filename is not None:\n pickle.dump(data_record, open(filename, \"wb\"))\n if to_stop:\n break\n\n loss_value = get_loss(model, validation_loader, X_valid, y_valid, criterion=criterion, loss_epoch=epochs, transform_label=transform_label, **kwargs)\n if isplot:\n import matplotlib.pylab as plt\n for key, item in data_record.items():\n if isinstance(item, Number) or len(data_record[\"iter\"]) != len(item):\n continue\n if key not in [\"iter\", \"model_dict\"]:\n if key in [\"accuracy\"]:\n plt.figure(figsize = (8,6))\n plt.plot(data_record[\"iter\"], data_record[key])\n plt.xlabel(\"epoch\")\n plt.ylabel(key)\n plt.title(key)\n plt.show()\n else:\n plt.figure(figsize = (8,6))\n plt.semilogy(data_record[\"iter\"], data_record[key])\n plt.xlabel(\"epoch\")\n plt.ylabel(key)\n plt.title(key)\n plt.show()\n return loss_original, loss_value, data_record\n\n\ndef train_simple(model, X, y, validation_data = None, inspect_interval = 5, **kwargs):\n \"\"\"minimal version of training. \"model\" can be a single model or a ordered list of models\"\"\"\n def get_regularization(model, **kwargs):\n reg_dict = kwargs[\"reg_dict\"] if \"reg_dict\" in kwargs else None\n reg = to_Variable([0], is_cuda = X.is_cuda)\n for model_ele in model:\n if reg_dict is not None:\n for reg_type, reg_coeff in reg_dict.items():\n reg = reg + model_ele.get_regularization(source = [reg_type], mode = \"L1\", **kwargs) * reg_coeff\n return reg\n if not(isinstance(model, list) or isinstance(model, tuple)):\n model = [model]\n epochs = kwargs[\"epochs\"] if \"epochs\" in kwargs else 2000\n lr = kwargs[\"lr\"] if \"lr\" in kwargs else 5e-3\n optim_type = kwargs[\"optim_type\"] if \"optim_type\" in kwargs else \"adam\"\n optim_kwargs = kwargs[\"optim_kwargs\"] if \"optim_kwargs\" in kwargs else {}\n loss_type = kwargs[\"loss_type\"] if \"loss_type\" in kwargs else \"mse\"\n early_stopping_epsilon = kwargs[\"early_stopping_epsilon\"] if \"early_stopping_epsilon\" in kwargs else 0\n patience = kwargs[\"patience\"] if \"patience\" in kwargs else 40\n record_keys = kwargs[\"record_keys\"] if \"record_keys\" in kwargs else [\"loss\", \"mse\", \"data_DL\", \"model_DL\"]\n scheduler_type = kwargs[\"scheduler_type\"] if \"scheduler_type\" in kwargs else \"ReduceLROnPlateau\"\n loss_precision_floor = kwargs[\"loss_precision_floor\"] if \"loss_precision_floor\" in kwargs else PrecisionFloorLoss\n autoencoder = kwargs[\"autoencoder\"] if \"autoencoder\" in kwargs else None\n data_record = {key: [] for key in record_keys}\n isplot = kwargs[\"isplot\"] if \"isplot\" in kwargs else False\n if patience is not None:\n early_stopping = Early_Stopping(patience = patience, epsilon = early_stopping_epsilon)\n \n if validation_data is not None:\n X_valid, y_valid = validation_data\n else:\n X_valid, y_valid = X, y\n \n # Get original loss:\n criterion = get_criterion(loss_type, loss_precision_floor = loss_precision_floor)\n DL_criterion = Loss_Fun(core = \"DLs\", loss_precision_floor = loss_precision_floor, DL_sum = True)\n DL_criterion_absolute = Loss_Fun(core = \"DLs\", loss_precision_floor = PrecisionFloorLoss, DL_sum = True)\n pred_valid = forward(model, X_valid, **kwargs)\n loss_original = to_np_array(criterion(pred_valid, y_valid))\n if \"loss\" in record_keys:\n record_data(data_record, [-1, loss_original], [\"iter\",\"loss\"])\n if \"mse\" in record_keys:\n record_data(data_record, [to_np_array(nn.MSELoss()(pred_valid, y_valid))], [\"mse\"])\n if \"data_DL\" in record_keys:\n record_data(data_record, [to_np_array(DL_criterion(pred_valid, y_valid))], [\"data_DL\"])\n if \"data_DL_absolute\" in record_keys:\n record_data(data_record, [to_np_array(DL_criterion_absolute(pred_valid, y_valid))], [\"data_DL_absolute\"])\n if \"model_DL\" in record_keys:\n record_data(data_record, [get_model_DL(model)], [\"model_DL\"])\n if \"param\" in record_keys:\n record_data(data_record, [model[0].get_weights_bias(W_source = \"core\", b_source = \"core\")], [\"param\"])\n if \"param_grad\" in record_keys:\n record_data(data_record, [model[0].get_weights_bias(W_source = \"core\", b_source = \"core\", is_grad = True)], [\"param_grad\"])\n if \"param_collapse_layers\" in record_keys:\n record_data(data_record, [simplify(deepcopy(model[0]), X, y, \"collapse_layers\", verbose = 0)[0]\\\n .get_weights_bias(W_source = \"core\", b_source = \"core\")], [\"param\"])\n\n # Setting up optimizer:\n parameters = itertools.chain(*[model_ele.parameters() for model_ele in model])\n num_params = np.sum([[len(list(model_ele.parameters())) for model_ele in model]])\n if num_params == 0:\n print(\"No parameters to optimize!\")\n pred_valid = forward(model, X_valid, **kwargs)\n loss_value = to_np_array(criterion(pred_valid, y_valid))\n if \"loss\" in record_keys:\n record_data(data_record, [0, loss_value], [\"iter\", \"loss\"])\n if \"mse\" in record_keys:\n record_data(data_record, [to_np_array(nn.MSELoss()(pred_valid, y_valid))], [\"mse\"])\n if \"data_DL\" in record_keys:\n record_data(data_record, [to_np_array(DL_criterion(pred_valid, y_valid))], [\"data_DL\"])\n if \"data_DL_absolute\" in record_keys:\n record_data(data_record, [to_np_array(DL_criterion_absolute(pred_valid, y_valid))], [\"data_DL_absolute\"])\n if \"model_DL\" in record_keys:\n record_data(data_record, [get_model_DL(model)], [\"model_DL\"])\n if \"param\" in record_keys:\n record_data(data_record, [model[0].get_weights_bias(W_source = \"core\", b_source = \"core\")], [\"param\"])\n if \"param_grad\" in record_keys:\n record_data(data_record, [model[0].get_weights_bias(W_source = \"core\", b_source = \"core\", is_grad = True)], [\"param_grad\"])\n if \"param_collapse_layers\" in record_keys:\n record_data(data_record, [simplify(deepcopy(model[0]), X, y, \"collapse_layers\", verbose = 0)[0]\\\n .get_weights_bias(W_source = \"core\", b_source = \"core\")], [\"param\"])\n return loss_original, loss_value, data_record\n optimizer = get_optimizer(optim_type, lr, parameters, **optim_kwargs)\n \n # Set up learning rate scheduler:\n if scheduler_type is not None:\n if scheduler_type == \"ReduceLROnPlateau\":\n scheduler_patience = kwargs[\"scheduler_patience\"] if \"scheduler_patience\" in kwargs else 10\n scheduler_factor = kwargs[\"scheduler_factor\"] if \"scheduler_factor\" in kwargs else 0.1\n scheduler = ReduceLROnPlateau(optimizer, factor = scheduler_factor, patience = scheduler_patience)\n elif scheduler_type == \"LambdaLR\":\n scheduler_lr_lambda = kwargs[\"scheduler_lr_lambda\"] if \"scheduler_lr_lambda\" in kwargs else (lambda epoch: 1 / (1 + 0.01 * epoch))\n scheduler = LambdaLR(optimizer, lr_lambda = scheduler_lr_lambda)\n else:\n raise\n\n # Training:\n to_stop = False\n for i in range(epochs + 1):\n if optim_type != \"LBFGS\":\n optimizer.zero_grad()\n pred = forward(model, X, **kwargs)\n reg = get_regularization(model, **kwargs)\n loss = criterion(pred, y) + reg\n loss.backward()\n optimizer.step()\n else:\n # \"LBFGS\" is a second-order optimization algorithm that requires a slightly different procedure:\n def closure():\n optimizer.zero_grad()\n pred = forward(model, X, **kwargs)\n reg = get_regularization(model, **kwargs)\n loss = criterion(pred, y) + reg\n loss.backward()\n return loss\n optimizer.step(closure)\n if i % inspect_interval == 0:\n pred_valid = forward(model, X_valid, **kwargs)\n loss_value = to_np_array(criterion(pred_valid, y_valid))\n if scheduler_type is not None:\n if scheduler_type == \"ReduceLROnPlateau\":\n scheduler.step(loss_value)\n else:\n scheduler.step()\n if \"loss\" in record_keys:\n record_data(data_record, [i, loss_value], [\"iter\", \"loss\"])\n if \"mse\" in record_keys:\n record_data(data_record, [to_np_array(nn.MSELoss()(pred_valid, y_valid))], [\"mse\"])\n if \"data_DL\" in record_keys:\n record_data(data_record, [to_np_array(DL_criterion(pred_valid, y_valid))], [\"data_DL\"])\n if \"data_DL_absolute\" in record_keys:\n record_data(data_record, [to_np_array(DL_criterion_absolute(pred_valid, y_valid))], [\"data_DL_absolute\"])\n if \"model_DL\" in record_keys:\n record_data(data_record, [get_model_DL(model)], [\"model_DL\"])\n if \"param\" in record_keys:\n record_data(data_record, [model[0].get_weights_bias(W_source = \"core\", b_source = \"core\")], [\"param\"])\n if \"param_grad\" in record_keys:\n record_data(data_record, [model[0].get_weights_bias(W_source = \"core\", b_source = \"core\", is_grad = True)], [\"param_grad\"])\n if \"param_collapse_layers\" in record_keys:\n record_data(data_record, [simplify(deepcopy(model[0]), X, y, \"collapse_layers\", verbose = 0)[0]\\\n .get_weights_bias(W_source = \"core\", b_source = \"core\")], [\"param\"])\n if patience is not None:\n to_stop = early_stopping.monitor(loss_value)\n if to_stop:\n break\n\n pred_valid = forward(model, X_valid, **kwargs)\n loss_value = to_np_array(criterion(pred_valid, y_valid))\n if isplot:\n import matplotlib.pylab as plt\n if \"mse\" in data_record:\n plt.semilogy(data_record[\"iter\"], data_record[\"mse\"])\n plt.xlabel(\"epochs\")\n plt.title(\"MSE\")\n plt.show()\n if \"loss\" in data_record:\n plt.plot(data_record[\"iter\"], data_record[\"loss\"])\n plt.xlabel(\"epochs\")\n plt.title(\"Loss\")\n plt.show()\n return loss_original, loss_value, data_record\n\n\ndef load_model_dict_net(model_dict, is_cuda = False):\n net_type = model_dict[\"type\"]\n if net_type.startswith(\"MLP\"):\n return MLP(input_size = model_dict[\"input_size\"],\n struct_param = model_dict[\"struct_param\"] if \"struct_param\" in model_dict else None,\n W_init_list = model_dict[\"weights\"] if \"weights\" in model_dict else None,\n b_init_list = model_dict[\"bias\"] if \"bias\" in model_dict else None,\n settings = model_dict[\"settings\"] if \"settings\" in model_dict else {},\n is_cuda = is_cuda,\n )\n elif net_type == \"Labelmix_MLP\":\n model = Labelmix_MLP(input_size=model_dict[\"input_size\"],\n struct_param=model_dict[\"struct_param\"],\n idx_label=model_dict[\"idx_label\"] if \"idx_label\" in model_dict else None,\n is_cuda=is_cuda,\n )\n if \"state_dict\" in model_dict:\n model.load_state_dict(model_dict[\"state_dict\"])\n return model\n elif net_type == \"Multi_MLP\":\n return Multi_MLP(input_size = model_dict[\"input_size\"],\n struct_param = model_dict[\"struct_param\"],\n W_init_list = model_dict[\"weights\"] if \"weights\" in model_dict else None,\n b_init_list = model_dict[\"bias\"] if \"bias\" in model_dict else None,\n settings = model_dict[\"settings\"] if \"settings\" in model_dict else {},\n is_cuda = is_cuda,\n )\n elif net_type == \"Branching_Net\":\n return Branching_Net(net_base_model_dict = model_dict[\"net_base_model_dict\"],\n net_1_model_dict = model_dict[\"net_1_model_dict\"],\n net_2_model_dict = model_dict[\"net_2_model_dict\"],\n is_cuda = is_cuda,\n )\n elif net_type == \"Fan_in_MLP\":\n return Fan_in_MLP(model_dict_branch1=model_dict[\"model_dict_branch1\"],\n model_dict_branch2=model_dict[\"model_dict_branch2\"],\n model_dict_joint=model_dict[\"model_dict_joint\"],\n is_cuda=is_cuda,\n )\n elif net_type == \"Net_reparam\":\n return Net_reparam(model_dict=model_dict[\"model\"],\n reparam_mode=model_dict[\"reparam_mode\"],\n is_cuda=is_cuda,\n )\n elif net_type == \"Wide_ResNet\":\n model = Wide_ResNet(depth=model_dict[\"depth\"],\n widen_factor=model_dict[\"widen_factor\"],\n input_channels=model_dict[\"input_channels\"],\n output_size=model_dict[\"output_size\"],\n dropout_rate=model_dict[\"dropout_rate\"],\n is_cuda=is_cuda,\n )\n if \"state_dict\" in model_dict:\n model.load_state_dict(model_dict[\"state_dict\"])\n return model\n elif net_type.startswith(\"ConvNet\"):\n return ConvNet(input_channels = model_dict[\"input_channels\"],\n struct_param = model_dict[\"struct_param\"],\n W_init_list = model_dict[\"weights\"] if \"weights\" in model_dict else None,\n b_init_list = model_dict[\"bias\"] if \"bias\" in model_dict else None,\n settings = model_dict[\"settings\"] if \"settings\" in model_dict else {},\n return_indices = model_dict[\"return_indices\"] if \"return_indices\" in model_dict else False,\n is_cuda = is_cuda,\n )\n elif net_type == \"Conv_Autoencoder\":\n model = Conv_Autoencoder(input_channels_encoder = model_dict[\"input_channels_encoder\"],\n input_channels_decoder = model_dict[\"input_channels_decoder\"],\n struct_param_encoder = model_dict[\"struct_param_encoder\"],\n struct_param_decoder = model_dict[\"struct_param_decoder\"],\n settings = model_dict[\"settings\"],\n is_cuda = is_cuda,\n )\n if \"encoder\" in model_dict:\n model.encoder.load_model_dict(model_dict[\"encoder\"])\n if \"decoder\" in model_dict:\n model.decoder.load_model_dict(model_dict[\"decoder\"])\n return model\n elif model_dict[\"type\"] == \"Conv_Model\":\n is_generative = model_dict[\"is_generative\"] if \"is_generative\" in model_dict else False\n return Conv_Model(encoder_model_dict = model_dict[\"encoder_model_dict\"] if not is_generative else None,\n core_model_dict = model_dict[\"core_model_dict\"],\n decoder_model_dict = model_dict[\"decoder_model_dict\"],\n latent_size = model_dict[\"latent_size\"],\n is_generative = model_dict[\"is_generative\"] if is_generative else False,\n is_res_block = model_dict[\"is_res_block\"] if \"is_res_block\" in model_dict else False,\n is_cuda = is_cuda,\n )\n else:\n raise Exception(\"net_type {} not recognized!\".format(net_type))\n \n\ndef load_model_dict(model_dict, is_cuda = False):\n net_type = model_dict[\"type\"]\n if net_type not in [\"Model_Ensemble\", \"LSTM\", \"Model_with_Uncertainty\", \"Mixture_Model\", \"Mixture_Gaussian\"]:\n return load_model_dict_net(model_dict, is_cuda = is_cuda)\n elif net_type == \"Model_Ensemble\":\n if model_dict[\"model_type\"] == \"MLP\":\n model_ensemble = Model_Ensemble(\n num_models = model_dict[\"num_models\"],\n input_size = model_dict[\"input_size\"],\n model_type = model_dict[\"model_type\"],\n output_size = model_dict[\"output_size\"],\n is_cuda = is_cuda,\n # Here we just create some placeholder network. The model will be overwritten in the next steps:\n struct_param = [[1, \"Simple_Layer\", {}]],\n )\n elif model_dict[\"model_type\"] == \"LSTM\":\n model_ensemble = Model_Ensemble(\n num_models = model_dict[\"num_models\"],\n input_size = model_dict[\"input_size\"],\n model_type = model_dict[\"model_type\"],\n output_size = model_dict[\"output_size\"],\n is_cuda = is_cuda,\n # Here we just create some placeholder network. The model will be overwritten in the next steps:\n hidden_size = 3,\n output_struct_param = [[1, \"Simple_Layer\", {}]],\n )\n else:\n raise\n for k in range(model_ensemble.num_models):\n setattr(model_ensemble, \"model_{}\".format(k), load_model_dict(model_dict[\"model_{}\".format(k)], is_cuda = is_cuda))\n return model_ensemble\n elif net_type == \"Model_with_Uncertainty\":\n return Model_with_Uncertainty(model_pred = load_model_dict(model_dict[\"model_pred\"], is_cuda = is_cuda),\n model_logstd = load_model_dict(model_dict[\"model_logstd\"], is_cuda = is_cuda))\n elif net_type == \"Mixture_Model\":\n return Mixture_Model(model_dict_list=model_dict[\"model_dict_list\"],\n weight_logits_model_dict=model_dict[\"weight_logits_model_dict\"],\n num_components=model_dict[\"num_components\"],\n is_cuda=is_cuda,\n )\n elif net_type == \"Mixture_Gaussian\":\n return load_model_dict_Mixture_Gaussian(model_dict, is_cuda = is_cuda)\n else:\n raise Exception(\"net_type {} not recognized!\".format(net_type))\n\n\n## Helper functions:\ndef get_accuracy(pred, target):\n \"\"\"Get accuracy from prediction and target\"\"\"\n assert len(pred.shape) == len(target.shape) == 1\n assert len(pred) == len(target)\n pred, target = to_np_array(pred, target)\n accuracy = ((pred == target).sum().astype(float) / len(pred))\n return accuracy\n\n\ndef flatten(*tensors):\n \"\"\"Flatten the tensor except the first dimension\"\"\"\n new_tensors = []\n for tensor in tensors:\n new_tensors.append(tensor.view(tensor.size(0), -1))\n if len(new_tensors) == 1:\n new_tensors = new_tensors[0]\n return new_tensors\n\n\ndef fill_triangular(vec, dim, mode=\"lower\"):\n \"\"\"Fill an lower or upper triangular matrices with given vectors\"\"\"\n# num_examples, size = vec.shape\n# assert size == dim * (dim + 1) // 2\n# matrix = torch.zeros(num_examples, dim, dim).to(vec.device)\n# if mode == \"lower\":\n# idx = (torch.tril(torch.ones(dim, dim)) == 1)[None]\n# elif mode == \"upper\":\n# idx = (torch.triu(torch.ones(dim, dim)) == 1)[None]\n# else:\n# raise Exception(\"mode {} not recognized!\".format(mode))\n# idx = idx.repeat(num_examples,1,1)\n# matrix[idx] = vec.contiguous().view(-1)\n num_examples, size = vec.shape\n assert size == dim * (dim + 1) // 2\n if mode == \"lower\":\n rows, cols = torch.tril_indices(dim, dim)\n elif mode == \"upper\":\n rows, cols = torch.triu_indices(dim, dim)\n else:\n raise Exception(\"mode {} not recognized!\".format(mode))\n matrix = torch.zeros(num_examples, dim, dim).type(vec.dtype).to(vec.device)\n matrix[:, rows, cols] = vec\n return matrix\n\n\ndef matrix_diag_transform(matrix, fun):\n \"\"\"Return the matrices whose diagonal elements have been executed by the function 'fun'.\"\"\"\n num_examples = len(matrix)\n idx = torch.eye(matrix.size(-1)).bool().unsqueeze(0)\n idx = idx.repeat(num_examples, 1, 1)\n new_matrix = matrix.clone()\n new_matrix[idx] = fun(matrix.diagonal(dim1=1, dim2=2).contiguous().view(-1))\n return new_matrix\n\n\ndef Zip(*data, **kwargs):\n \"\"\"Recursive unzipping of data structure\n Example: Zip(*[(('a',2), 1), (('b',3), 2), (('c',3), 3), (('d',2), 4)])\n ==> [[['a', 'b', 'c', 'd'], [2, 3, 3, 2]], [1, 2, 3, 4]]\n Each subtree in the original data must be in the form of a tuple.\n In the **kwargs, you can set the function that is applied to each fully unzipped subtree.\n \"\"\"\n import collections\n function = kwargs[\"function\"] if \"function\" in kwargs else None\n if len(data) == 1:\n return data[0]\n data = [list(element) for element in zip(*data)]\n for i, element in enumerate(data):\n if isinstance(element[0], tuple):\n data[i] = Zip(*element, **kwargs)\n elif isinstance(element, list):\n if function is not None:\n data[i] = function(element)\n return data\n\n\ndef get_loss(model, data_loader=None, X=None, y=None, criterion=None, transform_label=None, **kwargs):\n \"\"\"Get loss using the whole data or data_loader. Return the average validation loss with np.ndarray format\"\"\"\n max_validation_iter = kwargs[\"max_validation_iter\"] if \"max_validation_iter\" in kwargs else None\n if transform_label is None:\n transform_label = Transform_Label()\n if \"loader_process\" in kwargs and kwargs[\"loader_process\"] is not None:\n data_loader = kwargs[\"loader_process\"](\"test\")\n if data_loader is not None:\n assert X is None and y is None\n loss_record = 0\n count = 0\n # Taking the average of all metrics:\n for j, data_batch in enumerate(data_loader):\n if isinstance(data_batch, tuple) or isinstance(data_batch, list):\n X_batch, y_batch = data_batch\n if \"data_loader_apply\" in kwargs and kwargs[\"data_loader_apply\"] is not None:\n X_batch, y_batch = kwargs[\"data_loader_apply\"](X_batch, y_batch)\n else:\n X_batch, y_batch = kwargs[\"data_loader_apply\"](data_batch)\n loss_ele = to_np_array(model.get_loss(X_batch, transform_label(y_batch), criterion = criterion, **kwargs))\n if j == 0:\n all_info_dict = {key: 0 for key in model.info_dict.keys()}\n loss_record = loss_record + loss_ele\n count += 1\n for key in model.info_dict:\n all_info_dict[key] = all_info_dict[key] + model.info_dict[key]\n\n if max_validation_iter is not None and count > max_validation_iter:\n break\n\n for key in model.info_dict:\n all_info_dict[key] = all_info_dict[key] / count\n loss = loss_record / count\n model.info_dict = deepcopy(all_info_dict)\n else:\n assert X is not None and y is not None\n loss = to_np_array(model.get_loss(X, transform_label(y), criterion = criterion, **kwargs))\n return loss\n\n\ndef plot_model(model, data_loader=None, X=None, y=None, transform_label=None, **kwargs):\n data_loader_apply = kwargs[\"data_loader_apply\"] if \"data_loader_apply\" in kwargs else None\n max_validation_iter = kwargs[\"max_validation_iter\"] if \"max_validation_iter\" in kwargs else None\n if transform_label is None:\n transform_label = Transform_Label()\n if \"loader_process\" in kwargs and kwargs[\"loader_process\"] is not None:\n data_loader = kwargs[\"loader_process\"](\"test\")\n if data_loader is not None:\n assert X is None and y is None\n X_all = []\n y_all = []\n for i, data_batch in enumerate(data_loader):\n if isinstance(data_batch, tuple) or isinstance(data_batch, list):\n X_batch, y_batch = data_batch\n if data_loader_apply is not None:\n X_batch, y_batch = data_loader_apply(X_batch, y_batch)\n else:\n X_batch, y_batch = data_loader_apply(data_batch)\n X_all.append(X_batch)\n y_all.append(y_batch)\n if max_validation_iter is not None and i >= max_validation_iter:\n break\n if not isinstance(X_all[0], torch.Tensor):\n X_all = Zip(*X_all, function = torch.cat)\n else:\n X_all = torch.cat(X_all, 0)\n y_all = torch.cat(y_all)\n model.plot(X_all, transform_label(y_all))\n else:\n assert X is not None and y is not None\n model.plot(X, transform_label(y))\n\n\ndef prepare_inspection(model, data_loader=None, X=None, y=None, transform_label=None, **kwargs):\n inspect_functions = kwargs[\"inspect_functions\"] if \"inspect_functions\" in kwargs else None\n max_validation_iter = kwargs[\"max_validation_iter\"] if \"max_validation_iter\" in kwargs else None\n verbose = kwargs[\"verbose\"] if \"verbose\" in kwargs else False\n if transform_label is None:\n transform_label = Transform_Label()\n if \"loader_process\" in kwargs and kwargs[\"loader_process\"] is not None:\n data_loader = kwargs[\"loader_process\"](\"test\")\n if data_loader is None:\n assert X is not None and y is not None\n all_dict_summary = model.prepare_inspection(X, transform_label(y), **kwargs)\n if inspect_functions is not None:\n for inspect_function_key, inspect_function in inspect_functions.items():\n all_dict_summary[inspect_function_key] = inspect_function(model, X, y, **kwargs)\n else:\n assert X is None and y is None\n all_dict = {}\n for j, data_batch in enumerate(data_loader):\n if verbose is True:\n print(\"valid step: {}\".format(j))\n if isinstance(data_batch, tuple) or isinstance(data_batch, list):\n X_batch, y_batch = data_batch\n if \"data_loader_apply\" in kwargs and kwargs[\"data_loader_apply\"] is not None:\n X_batch, y_batch = kwargs[\"data_loader_apply\"](X_batch, y_batch)\n else:\n X_batch, y_batch = kwargs[\"data_loader_apply\"](data_batch)\n info_dict = model.prepare_inspection(X_batch, transform_label(y_batch), valid_step=j, **kwargs)\n for key, item in info_dict.items():\n if key not in all_dict:\n all_dict[key] = [item]\n else:\n all_dict[key].append(item)\n if inspect_functions is not None:\n for inspect_function_key, inspect_function in inspect_functions.items():\n inspect_function_result = inspect_function(model, X_batch, transform_label(y_batch), **kwargs)\n if inspect_function_key not in all_dict:\n all_dict[inspect_function_key] = [inspect_function_result]\n else:\n all_dict[inspect_function_key].append(inspect_function_result)\n if max_validation_iter is not None and j >= max_validation_iter:\n break\n all_dict_summary = {}\n for key, item in all_dict.items():\n all_dict_summary[key + \"_val\"] = np.mean(all_dict[key])\n return all_dict_summary\n\n\ndef get_inspect_items_train(inspect_items):\n if inspect_items is None:\n return []\n inspect_items_train = []\n for item in inspect_items:\n if item.endswith(\"_tr\"):\n inspect_items_train.append(\"_\".join(item.split(\"_\")[:-1]))\n return inspect_items_train\n\n\ndef update_key_train(info_dict_train, inspect_items_train):\n info_dict_train_new = {}\n for key, item in info_dict_train.items():\n if key in inspect_items_train:\n info_dict_train_new[key + \"_tr\"] = item\n return deepcopy(info_dict_train_new)\n```\n\n## Simplification functionality:\n\n\n```python\ndef simplify(\n model,\n X=None,\n y=None,\n mode=\"full\",\n isplot=False,\n target_name=None,\n validation_data=None,\n **kwargs\n):\n \"\"\"Simplify a neural network model in various ways. \"model\" can be a single model or a ordered list of models\"\"\"\n verbose = kwargs[\"verbose\"] if \"verbose\" in kwargs else 1\n if validation_data is None:\n X_valid, y_valid = X, y\n else:\n X_valid, y_valid = validation_data\n simplify_criteria = kwargs[\"simplify_criteria\"] if \"simplify_criteria\" in kwargs else (\"DLs\", 0.05, 3, \"relative\") # the first argument choose from \"DL\", \"loss\"\n simplify_epsilon = simplify_criteria[1]\n simplify_patience = simplify_criteria[2]\n simplify_compare_mode = simplify_criteria[3]\n performance_monitor = Performance_Monitor(patience = simplify_patience, epsilon = simplify_epsilon, compare_mode = simplify_compare_mode)\n record_keys = kwargs[\"record_keys\"] if \"record_keys\" in kwargs else [\"mse\"]\n loss_precision_floor = kwargs[\"loss_precision_floor\"] if \"loss_precision_floor\" in kwargs else PrecisionFloorLoss\n if X is not None:\n if y is None:\n y = Variable(forward(model, X, **kwargs).data, requires_grad = False)\n if not (isinstance(model, list) or isinstance(model, tuple)):\n model = [model]\n is_list = False\n else:\n is_list = True\n if mode == \"full\":\n mode = [\"collapse_layers\", \"snap\"]\n if not isinstance(mode, list):\n mode = [mode]\n\n # Obtain the original loss and setup criterion:\n loss_type = kwargs[\"loss_type\"] if \"loss_type\" in kwargs else \"mse\"\n criterion = get_criterion(loss_type, loss_precision_floor = loss_precision_floor)\n DL_criterion = Loss_Fun(core = \"DLs\", loss_precision_floor = loss_precision_floor, DL_sum = True)\n loss_dict = OrderedDict()\n\n for mode_ele in mode:\n if verbose >= 1:\n print(\"\\n\" + \"=\" * 48 + \"\\nSimplifying mode: {}\".format(mode_ele), end = \"\")\n if mode_ele == \"snap\":\n snap_mode = kwargs[\"snap_mode\"] if \"snap_mode\" in kwargs else \"integer\"\n print(\" {}\".format(snap_mode), end = \"\")\n if target_name is not None:\n print(\" for {}\".format(target_name))\n else:\n print()\n print(\"=\" * 48)\n \n # Record the loss before simplification:\n if X is not None:\n pred_valid = forward(model, X_valid, **kwargs)\n loss_original = to_np_array(criterion(pred_valid, y_valid))\n loss_list = [loss_original]\n if verbose >= 1:\n print(\"original_loss: {}\".format(loss_original))\n mse_record_whole = [to_np_array(nn.MSELoss()(pred_valid, y_valid))]\n data_DL_whole = [to_np_array(DL_criterion(pred_valid, y_valid))]\n model_DL_whole = [get_model_DL(model)]\n event_list = [\"before simplification\"]\n iter_end_whole = [1]\n is_accept_whole = []\n if \"param\" in record_keys:\n param_record_whole = [model[0].get_weights_bias(W_source = \"core\", b_source = \"core\")]\n if \"param_grad\" in record_keys:\n param_grad_record_whole = [model[0].get_weights_bias(W_source = \"core\", b_source = \"core\", is_grad = True)]\n \n # Begin simplification:\n if mode_ele == \"collapse_layers\":\n all_collapse_dict = {}\n for model_id, model_ele in enumerate(model):\n # Obtain activations for each layer:\n activation_list = []\n for k in range(len(model_ele.struct_param)):\n if \"activation\" in model_ele.struct_param[k][2]:\n activation_list.append(model_ele.struct_param[k][2][\"activation\"])\n elif \"activation\" in model_ele.settings:\n activation_list.append(model_ele.settings[\"activation\"])\n else:\n activation_list.append(\"default\")\n \n # Build the collapse_list that stipulates which layers to collapse:\n collapse_dict = {}\n current_start = None\n current_layer_type = None\n for k, activation in enumerate(activation_list):\n if activation == \"linear\" and k != len(activation_list) - 1:\n if k not in collapse_dict and current_start is None:\n # Create a new bunch:\n if model_ele.struct_param[k + 1][1] == model_ele.struct_param[k][1]: # The current layer must have the same layer_type as the next layer\n current_start = k\n collapse_dict[current_start] = [k]\n current_layer_type = model_ele.struct_param[k][1]\n else:\n # Adding to current bunch:\n if model_ele.struct_param[k + 1][1] == model_ele.struct_param[k][1] == current_layer_type:\n collapse_dict[current_start].append(k)\n else:\n collapse_dict[current_start].append(k)\n current_start = None\n else:\n if current_start is not None:\n collapse_dict[current_start].append(k)\n current_start = None\n\n # Build new layer:\n new_layer_info = {}\n for current_start, layer_ids in collapse_dict.items():\n for i, layer_id in enumerate(layer_ids):\n layer = getattr(model_ele, \"layer_{}\".format(layer_id))\n if i == 0:\n W_accum = layer.W_core\n b_accum = layer.b_core\n else:\n W_accum = torch.matmul(W_accum, layer.W_core)\n b_accum = torch.matmul(b_accum, layer.W_core) + layer.b_core\n if model_ele.is_cuda:\n W_accum = W_accum.cpu()\n b_accum = b_accum.cpu()\n last_layer_id = collapse_dict[current_start][-1]\n new_layer_info[current_start] = {\"W_init\": W_accum.data.numpy(), \"b_init\": b_accum.data.numpy(),\n \"layer_struct_param\": [b_accum.size(0), model_ele.struct_param[last_layer_id][1], deepcopy(model_ele.struct_param[last_layer_id][2])],\n }\n new_layer_info[current_start].pop(\"snap_dict\", None)\n if verbose >= 1:\n print(\"model_id {}, layers collapsed: {}\".format(model_id, collapse_dict))\n \n # Rebuild the Net:\n if len(collapse_dict) > 0:\n all_collapse_dict[model_id] = {\"collapse_dict\": collapse_dict, \n \"new_layer_info\": new_layer_info, \n \"collapse_layer_ids\": [idx for item in collapse_dict.values() for idx in item],\n }\n\n # Rebuild the list of models:\n if len(all_collapse_dict) > 0:\n model_new = []\n for model_id, model_ele in enumerate(model):\n if model_id in all_collapse_dict:\n W_list, b_list = model_ele.get_weights_bias(W_source = \"core\", b_source = \"core\")\n W_init_list = []\n b_init_list = []\n struct_param = []\n for k in range(len(model_ele.struct_param)):\n if k not in all_collapse_dict[model_id][\"collapse_layer_ids\"]:\n struct_param.append(model_ele.struct_param[k])\n W_init_list.append(W_list[k])\n b_init_list.append(b_list[k])\n else:\n if k in all_collapse_dict[model_id][\"collapse_dict\"].keys():\n struct_param.append(all_collapse_dict[model_id][\"new_layer_info\"][k][\"layer_struct_param\"])\n W_init_list.append(all_collapse_dict[model_id][\"new_layer_info\"][k][\"W_init\"])\n b_init_list.append(all_collapse_dict[model_id][\"new_layer_info\"][k][\"b_init\"])\n model_ele_new = MLP(input_size = model_ele.input_size,\n struct_param = struct_param,\n W_init_list = W_init_list,\n b_init_list = b_init_list,\n settings = model_ele.settings,\n is_cuda = model_ele.is_cuda,\n )\n else:\n model_ele_new = model_ele\n model_new.append(model_ele_new) \n model = model_new\n\n # Calculate the loss again:\n pred_valid = forward(model, X_valid, **kwargs)\n loss_new = to_np_array(criterion(pred_valid, y_valid))\n if verbose >= 1:\n print(\"after collapsing linear layers in all models, new loss {}\".format(loss_new))\n loss_list.append(loss_new)\n mse_record_whole.append(to_np_array(nn.MSELoss()(pred_valid, y_valid)))\n data_DL_whole.append(to_np_array(DL_criterion(pred_valid, y_valid)))\n model_DL_whole.append(get_model_DL(model))\n if \"param\" in record_keys:\n param_record_whole.append(model[0].get_weights_bias(W_source = \"core\", b_source = \"core\"))\n if \"param_grad\" in record_keys:\n param_grad_record_whole.append(model[0].get_weights_bias(W_source = \"core\", b_source = \"core\", is_grad = True))\n iter_end_whole.append(1)\n event_list.append({mode_ele: all_collapse_dict})\n\n elif mode_ele in [\"local\", \"snap\"]:\n # 'local': greedily try reducing the input dimension by removing input dimension from the beginning;\n # 'snap': greedily snap each float parameter into an integer or rational number. Set argument 'snap_mode' == 'integer' or 'rational'.\n if mode_ele == \"snap\":\n target_params = [[(model_id, layer_id), \"snap\"] for model_id, model_ele in enumerate(model) for layer_id in range(len(model_ele.struct_param))]\n elif mode_ele == \"local\":\n for model_id, model_ele in enumerate(model):\n if len(model_ele.struct_param) > 0:\n first_model_id = model_id\n break\n first_layer = getattr(model[first_model_id], \"layer_0\")\n target_params = [[(first_model_id, 0), [[((\"weight\", (i, j)), 0.) for j in range(first_layer.output_size)] for i in range(first_layer.input_size)]]]\n else:\n raise\n\n excluded_idx_dict = {item[0]: [] for item in target_params}\n target_layer_ids_exclude = []\n for (model_id, layer_id), target_list in target_params:\n layer = getattr(model[model_id], \"layer_{}\".format(layer_id))\n if isinstance(target_list, list):\n max_passes = len(target_list)\n elif target_list == \"snap\":\n max_passes = (layer.input_size + 1) * layer.output_size\n if \"max_passes\" in kwargs:\n max_passes = min(max_passes, kwargs[\"max_passes\"])\n else:\n raise Exception(\"target_list {} not recognizable!\".format(target_list))\n if verbose >= 2:\n print(\"\\n****starting model:****\")\n model[model_id].get_weights_bias(W_source = \"core\", b_source = \"core\", verbose = True)\n print(\"********\\n\" )\n \n \n performance_monitor.reset()\n criteria_value, criteria_result = get_criteria_value(model, X, y, criteria_type = simplify_criteria[0], criterion = criterion, **kwargs)\n to_stop, pivot_dict, log, _, pivot_id = performance_monitor.monitor(criteria_value, model_dict = model[model_id].model_dict, criteria_result = criteria_result)\n for i in range(max_passes):\n # Perform tentative simplification\n if isinstance(target_list, list):\n info = layer.simplify(mode = \"snap\", excluded_idx = excluded_idx_dict[(model_id, layer_id)], snap_targets = target_list[i], **kwargs)\n else:\n info = layer.simplify(mode = \"snap\", excluded_idx = excluded_idx_dict[(model_id, layer_id)], **kwargs)\n if len(info) == 0:\n target_layer_ids_exclude.append((model_id, layer_id))\n print(\"Pass {0}, (model {1}, layer {2}) has no parameters to snap. Revert to pivot model. Go to next layer\".format(i, model_id, layer_id))\n break\n excluded_idx_dict[(model_id, layer_id)] = excluded_idx_dict[(model_id, layer_id)] + info\n\n _, loss_new, data_record = train_simple(model, X, y, optim_type = \"adam\", validation_data = validation_data, **kwargs)\n if verbose >= 2:\n print(\"=\" * 8)\n model[model_id].get_weights_bias(W_source = \"core\", b_source = \"core\", verbose = True) \n criteria_value, criteria_result = get_criteria_value(model, X, y, criteria_type = simplify_criteria[0], criterion = criterion, **kwargs)\n to_stop, pivot_dict, log, is_accept, pivot_id = performance_monitor.monitor(criteria_value, model_dict = model[model_id].model_dict, criteria_result = criteria_result)\n is_accept_whole.append(is_accept)\n if is_accept:\n print('[Accepted] as pivot model!')\n print()\n\n # Check if the criterion after simplification and refit is worse. If it is worse than the simplify_epsilon, revert:\n if to_stop:\n target_layer_ids_exclude.append((model_id, layer_id))\n if verbose >= 1:\n print(\"Pass {0}, loss: {1}\\tDL: {2}. New snap {3} is do not improve by {4} = {5} for {6} steps. Revert the simplification to pivot model. Go to next layer.\".format(\n i, view_item(log, (\"criteria_result\", \"loss\")), view_item(log, (\"criteria_result\", \"DL\")), info, simplify_criteria[0], simplify_epsilon, simplify_patience))\n break\n mse_record_whole += data_record[\"mse\"]\n data_DL_whole += data_record[\"data_DL\"]\n model_DL_whole += data_record[\"model_DL\"]\n if \"param\" in record_keys:\n param_record_whole += data_record[\"param\"]\n if \"param_grad\" in record_keys:\n param_grad_record_whole += data_record[\"param_grad\"]\n iter_end_whole.append(len(data_record[\"mse\"]))\n\n model[model_id].reset_layer(layer_id, layer)\n loss_list.append(loss_new)\n event_list.append({mode_ele: ((model_id, layer_id), info)})\n if verbose >= 1:\n print(\"Pass {0}, snap (model {1}, layer {2}), snap {3}. \\tloss: {4}\\tDL: {5}\".format(\n i, model_id, layer_id, info, view_item(log, (\"criteria_result\", \"loss\")), view_item(log, (\"criteria_result\", \"DL\"))))\n\n # Update the whole model's struct_param and snap_dict:\n model[model_id].load_model_dict(pivot_dict[\"model_dict\"])\n model[model_id].synchronize_settings()\n if verbose >= 2:\n print(\"\\n****pivot model at {}th transformation:****\".format(pivot_id))\n model[model_id].get_weights_bias(W_source = \"core\", b_source = \"core\", verbose = True)\n print(\"********\\n\" )\n\n elif mode_ele == \"pair_snap\":\n model_new = []\n for model_id, model_ele in enumerate(model):\n for layer_id, layer_struct_param in enumerate(model_ele.struct_param):\n if layer_struct_param[1] == \"Symbolic_Layer\":\n layer = getattr(model_ele, \"layer_{}\".format(layer_id))\n max_passes = len(layer.get_param_dict()) - 1\n if \"max_passes\" in kwargs:\n max_passes = min(max_passes, kwargs[\"max_passes\"])\n if verbose > 1:\n print(\"original:\")\n print(\"symbolic_expression: \", layer.symbolic_expression)\n print(\"numerical_expression: \", layer.numerical_expression)\n print()\n\n performance_monitor.reset()\n criteria_value, criteria_result = get_criteria_value(model, X, y, criteria_type = simplify_criteria[0], criterion = criterion, **kwargs)\n to_stop, pivot_dict, log, _, pivot_id = performance_monitor.monitor(criteria_value, model_dict = model[model_id].model_dict, criteria_result = criteria_result)\n for i in range(max_passes):\n info = layer.simplify(mode = \"pair_snap\", **kwargs)\n if len(info) == 0:\n target_layer_ids_exclude.append((model_id, layer_id))\n print(\"Pass {0}, (model {1}, layer {2}) has no parameters to pair_snap. Revert to pivot model. Go to next layer\".format(i, model_id, layer_id))\n break\n _, loss, data_record = train_simple(model, X, y, optim_type = \"adam\", epochs = 1000, validation_data = validation_data, **kwargs)\n criteria_value, criteria_result = get_criteria_value(model, X, y, criteria_type = simplify_criteria[0], criterion = criterion, **kwargs)\n to_stop, pivot_dict, log, is_accept, pivot_id = performance_monitor.monitor(criteria_value, model_dict = model[model_id].model_dict, criteria_result = criteria_result)\n is_accept_whole.append(is_accept)\n if to_stop:\n if verbose >= 1:\n print(\"\\nPass {0}, loss: {1}\\tDL: {2}. New snap {3} is do not improve by {4} = {5} for {6} steps. Revert the simplification to pivot model. Go to next layer.\".format(\n i, view_item(log, (\"criteria_result\", \"loss\")), view_item(log, (\"criteria_result\", \"DL\")), info, simplify_criteria[0], simplify_epsilon, simplify_patience))\n break\n\n mse_record_whole += data_record[\"mse\"]\n data_DL_whole += data_record[\"data_DL\"]\n model_DL_whole += data_record[\"model_DL\"]\n if \"param\" in record_keys:\n param_record_whole += data_record[\"param\"]\n if \"param_grad\" in record_keys:\n param_grad_record_whole += data_record[\"param_grad\"]\n iter_end_whole.append(len(data_record[\"mse\"]))\n\n model[model_id].reset_layer(layer_id, layer)\n loss_list.append(loss)\n event_list.append({mode_ele: ((model_id, layer_id), info)})\n if verbose >= 1:\n print(\"\\nPass {0}, snap (model {1}, layer {2}), snap {3}. \\tloss: {4}\\tDL: {5}\".format(\n i, model_id, layer_id, info, view_item(log, (\"criteria_result\", \"loss\")), view_item(log, (\"criteria_result\", \"DL\"))))\n print(\"symbolic_expression: \", layer.symbolic_expression)\n print(\"numerical_expression: \", layer.numerical_expression)\n print()\n\n model[model_id].load_model_dict(pivot_dict[\"model_dict\"])\n print(\"final: \\nsymbolic_expression: \", getattr(model[model_id], \"layer_{0}\".format(layer_id)).symbolic_expression)\n print(\"numerical_expression: \", getattr(model[model_id], \"layer_{0}\".format(layer_id)).numerical_expression)\n print()\n\n elif mode_ele[:11] == \"to_symbolic\":\n from sympy import Symbol\n force_simplification = kwargs[\"force_simplification\"] if \"force_simplification\" in kwargs else False\n is_multi_model = True if len(model) > 1 else False\n for model_id, model_ele in enumerate(model):\n for layer_id, layer_struct_param in enumerate(model_ele.struct_param):\n prefix = \"L{}_\".format(layer_id)\n if layer_struct_param[1] == \"Simple_Layer\":\n # Obtain loss before simplification:\n layer = getattr(model_ele, \"layer_{}\".format(layer_id))\n if X is not None:\n criteria_prev, criteria_result_prev = get_criteria_value(model, X, y, criteria_type = simplify_criteria[0], criterion = criterion, **kwargs)\n \n if mode_ele.split(\"_\")[-1] == \"separable\":\n new_layer = Simple_2_Symbolic(layer, settings = model_ele.settings, mode = \"separable\", prefix = prefix)\n else:\n new_layer = Simple_2_Symbolic(layer, settings = model_ele.settings, prefix = prefix)\n model[model_id].reset_layer(layer_id, new_layer)\n\n if \"snap_dict\" in model_ele.settings and layer_id in model_ele.settings[\"snap_dict\"]:\n subs_targets = []\n for (pos, true_idx), item in model_ele.settings[\"snap_dict\"][layer_id].items():\n if pos == \"weight\":\n subs_targets.append((Symbol(\"W{0}{1}\".format(true_idx[0], true_idx[1])), item[\"new_value\"]))\n elif pos == \"bias\":\n subs_targets.append((Symbol(\"b{}\".format(true_idx)), item[\"new_value\"]))\n else:\n raise Exception(\"pos {} not recognized!\".format(pos))\n new_expression = [expression.subs(subs_targets) for expression in new_layer.symbolic_expression]\n new_layer.set_symbolic_expression(new_expression)\n model_ele.settings[\"snap_dict\"].pop(layer_id)\n model_ele.struct_param[layer_id][2].update(new_layer.struct_param[2])\n \n # Calculate the loss again:\n if X is not None:\n criteria_new, criteria_result_new = get_criteria_value(model, X, y, criteria_type = simplify_criteria[0], criterion = criterion, **kwargs)\n if verbose >= 1:\n print(\"Prev_loss: {0}, new loss: {1}\\tprev_DL: {2:.9f}, new DL: {3:.9f}\".format(\n criteria_result_prev[\"loss\"], criteria_result_new[\"loss\"], criteria_result_prev[\"DL\"], criteria_result_new[\"DL\"]))\n print()\n if criteria_new > criteria_prev * (1 + 0.05):\n print(\"to_symbolic DL increase more than 5%! \", end = \"\")\n if not force_simplification:\n print(\"Reset layer.\")\n model[model_id].reset_layer(layer_id, layer)\n else:\n print(\"Nevertheless, force simplification.\")\n\n loss_list.append(criteria_result_new[\"loss\"])\n print(\"{0} succeed. Prev_loss: {1}\\tnew_loss: {2}\\tprev_DL: {3:.9f}, new_DL: {4:.9f}\".format(\n mode_ele, criteria_result_prev[\"loss\"], criteria_result_new[\"loss\"],\n criteria_result_prev[\"DL\"], criteria_result_new[\"DL\"]))\n else:\n print(\"{0} succeed.\".format(mode_ele))\n event_list.append({mode_ele: (model_id, layer_id)})\n \n \n elif layer_struct_param[1] == \"Sneuron_Layer\":\n # Obtain loss before simplification:\n layer = getattr(model_ele, \"layer_{0}\".format(layer_id))\n criteria_prev, criteria_result_prev = get_criteria_value(model, X, y, criteria_type = simplify_criteria[0], criterion = criterion, **kwargs)\n \n new_layer = Sneuron_2_Symbolic(layer, prefix = prefix)\n model[model_id].reset_layer(layer_id, new_layer)\n \n # Calculate the loss again:\n criteria_new, criteria_result_new = get_criteria_value(model, X, y, criteria_type = simplify_criteria[0], criterion = criterion, **kwargs)\n if verbose >= 1:\n print(\"Prev_loss: {0}, new loss: {1}\\tprev_DL: {2:.9f}, new DL: {3:.9f}\".format(\n criteria_result_prev[\"loss\"], criteria_result_new[\"loss\"], criteria_result_prev[\"DL\"], criteria_result_new[\"DL\"]))\n print()\n if criteria_new > criteria_prev * (1 + 0.05): \n print(\"to_symbolic DL increase more than 5%! \", end = \"\")\n if not force_simplification:\n print(\"Reset layer.\")\n model[model_id].reset_layer(layer_id, layer)\n else:\n print(\"Nevertheless, force simplification.\")\n \n loss_list.append(criteria_result_new[\"loss\"])\n event_list.append({mode_ele: (model_id, layer_id)})\n print(\"{0} succeed. Prev_loss: {1}\\tnew_loss: {2}\\tprev_DL: {3:.9f}, new_DL: {4:.9f}\".format(\n mode_ele, criteria_result_prev[\"loss\"], criteria_result_new[\"loss\"],\n criteria_result_prev[\"DL\"], criteria_result_new[\"DL\"]))\n if X is not None:\n mse_record_whole.append(to_np_array(nn.MSELoss()(pred_valid, y_valid)))\n data_DL_whole.append(to_np_array(DL_criterion(pred_valid, y_valid)))\n model_DL_whole.append(get_model_DL(model))\n if \"param\" in record_keys:\n param_record_whole.append(model[0].get_weights_bias(W_source = \"core\", b_source = \"core\"))\n if \"param_grad\" in record_keys:\n param_grad_record_whole.append(model[0].get_weights_bias(W_source = \"core\", b_source = \"core\", is_grad = True))\n iter_end_whole.append(1)\n\n elif mode_ele == \"symbolic_simplification\":\n \"\"\"Collapse multi-layer symbolic expression\"\"\"\n from sympy import Symbol, Poly, expand, prod\n force_simplification = kwargs[\"force_simplification\"] if \"force_simplification\" in kwargs else False\n numerical_threshold = kwargs[\"numerical_threshold\"] if \"numerical_threshold\" in kwargs else None\n is_numerical = kwargs[\"is_numerical\"] if \"is_numerical\" in kwargs else False\n max_poly_degree = kwargs[\"max_poly_degree\"] if \"max_poly_degree\" in kwargs else None\n show_before_truncate = kwargs[\"show_before_truncate\"] if \"show_before_truncate\" in kwargs else False\n for model_id, model_ele in enumerate(model):\n is_all_symbolic = True\n for layer_id, layer_struct_param in enumerate(model_ele.struct_param):\n if layer_struct_param[1] != \"Symbolic_Layer\":\n is_all_symbolic = False\n if is_all_symbolic:\n criteria_prev, criteria_result_prev = get_criteria_value(model, X, y, criteria_type = simplify_criteria[0], criterion = criterion, **kwargs)\n variables = OrderedDict()\n for i in range(model[0].layer_0.input_size):\n variables[\"x{0}\".format(i)] = Symbol(\"x{0}\".format(i))\n expression = list(variables.values())\n param_dict_all = {}\n\n # Collapse multiple layers:\n for layer_id, layer_struct_param in enumerate(model_ele.struct_param):\n layer = getattr(model_ele, \"layer_{0}\".format(layer_id))\n layer_expression = deepcopy(layer.numerical_expression)\n layer_expression_new = []\n for expr in layer_expression:\n new_expr = expr.subs({\"x{0}\".format(i): \"t{0}\".format(i) for i in range(len(expression))}) # Use a temporary variable to prevent overriding\n new_expr = new_expr.subs({\"t{0}\".format(i): expression[i] for i in range(len(expression))})\n layer_expression_new.append(expand(new_expr))\n expression = layer_expression_new\n \n # Show full expression before performing truncation:\n if show_before_truncate:\n for i, expr in enumerate(expression):\n print(\"Full expression {0}:\".format(i))\n pp.pprint(Poly(expr, *list(variables.values())))\n print()\n\n model_ele_candidate = MLP(input_size = model[0].layer_0.input_size,\n struct_param = [[layer.output_size, \"Symbolic_Layer\", {\"symbolic_expression\": \"x0\"}]],\n settings = {},\n is_cuda = model_ele.is_cuda,\n )\n # Setting maximul degree for polynomial:\n if max_poly_degree is not None:\n new_expression = []\n for expr in expression:\n expr = Poly(expr, *list(variables.values()))\n degree_list = []\n coeff_list = []\n for degree, coeff in expr.terms():\n # Only use monomials with degree not larger than max_poly_degree:\n if sum(degree) <= max_poly_degree: \n degree_list.append(degree)\n coeff_list.append(coeff)\n\n new_expr = 0\n for degree, coeff in zip(degree_list, coeff_list):\n new_expr += prod([variables[\"x{0}\".format(i)] ** degree[i] for i in range(len(degree))]) * coeff\n new_expression.append(new_expr)\n expression = new_expression\n\n # Update symbolic expression for model_ele_candidate:\n if not is_numerical:\n param_dict_all = {}\n expression_new_all = []\n for expr in expression:\n expression_new, param_dict = numerical_2_parameter(expr, idx = len(param_dict_all), threshold = numerical_threshold)\n expression_new_all.append(expression_new)\n param_dict_all.update(param_dict)\n model_ele_candidate.layer_0.set_symbolic_expression(expression_new_all, p_init = param_dict_all)\n else:\n model_ele_candidate.layer_0.set_symbolic_expression(expression)\n model_ele_candidate.layer_0.set_numerical(True)\n \n criteria_new, criteria_result_new = get_criteria_value(model_ele_candidate, X, y, criteria_type = simplify_criteria[0], criterion = criterion, **kwargs)\n if criteria_new > criteria_prev * (1 + 0.05): \n print(\"to_symbolic DL increase more than 5%! \", end = \"\")\n if force_simplification:\n print(\"Nevertheless, force simplification.\")\n model[model_id] = model_ele_candidate\n else:\n print(\"Revert.\")\n else:\n model[model_id] = model_ele_candidate\n\n elif mode_ele == \"activation_snap\":\n from sympy import Function\n def get_sign_snap_candidate(layer, activation_source, excluded_neurons = None):\n coeff_dict = {}\n for i in range(len(layer.symbolic_expression)):\n current_expression = [layer.symbolic_expression[i]]\n func_names = layer.get_function_name_list(current_expression)\n if activation_source in func_names:\n coeff = [element for element in layer.get_param_name_list(current_expression) if element[0] == \"W\"]\n coeff_dict[i] = np.mean([np.abs(value) for key, value in layer.get_param_dict().items() if key in coeff])\n best_idx = None\n best_value = 0\n for key, value in coeff_dict.items():\n if value > best_value and key not in excluded_neurons:\n best_value = value\n best_idx = key\n return best_idx, best_value\n\n activation_source = kwargs[\"activation_source\"] if \"activation_source\" in kwargs else \"sigmoid\"\n activation_target = kwargs[\"activation_target\"] if \"activation_target\" in kwargs else \"heaviside\"\n activation_fun_source = Function(activation_source)\n activation_fun_target = Function(activation_target)\n\n for model_id, model_ele in enumerate(model):\n for layer_id, layer_struct_param in enumerate(model_ele.struct_param):\n if layer_struct_param[1] == \"Symbolic_Layer\":\n layer = getattr(model_ele, \"layer_{0}\".format(layer_id))\n excluded_neurons = []\n if activation_source not in layer.get_function_name_list():\n continue\n\n performance_monitor.reset()\n criteria_value, criteria_result = get_criteria_value(model, X, y, criteria_type = simplify_criteria[0], criterion = criterion, **kwargs)\n to_stop, pivot_dict, log, _, pivot_id = performance_monitor.monitor(criteria_value, model_dict = model[model_id].model_dict, criteria_result = criteria_result)\n for i in range(layer_struct_param[0]):\n # Obtain loss before simplification:\n layer = getattr(model_ele, \"layer_{0}\".format(layer_id))\n best_idx, _ = get_sign_snap_candidate(layer, activation_source, excluded_neurons = excluded_neurons)\n excluded_neurons.append(best_idx)\n \n new_expression = [expression.subs(activation_fun_source, activation_fun_target) if j == best_idx else expression for j, expression in enumerate(layer.symbolic_expression)]\n print(\"Pass {0}, candidate new expression: {1}\".format(i, new_expression))\n layer.set_symbolic_expression(new_expression)\n\n # Train:\n _, loss_new, data_record = train_simple(model, X, y, validation_data = validation_data, **kwargs)\n\n criteria_value, criteria_result = get_criteria_value(model, X, y, criteria_type = simplify_criteria[0], criterion = criterion, **kwargs)\n to_stop, pivot_dict, log, is_accept, pivot_id = performance_monitor.monitor(criteria_value, model_dict = model[model_id].model_dict, criteria_result = criteria_result)\n is_accept_whole.append(is_accept)\n # Check if the criterion after simplification and refit is worse. If it is worse than the simplify_epsilon, revert:\n if to_stop:\n model[model_id].load_model_dict(pivot_dict[\"model_dict\"])\n if verbose >= 1:\n print(\"Pass {0}, loss: {1}\\tDL: {2}. New snap {3} is do not improve by {4} = {5} for {6} steps. Revert the simplification to pivot model. Continue\".format(\n i, view_item(log, (\"criteria_result\", \"loss\")), view_item(log, (\"criteria_result\", \"DL\")), info, simplify_criteria[0], simplify_epsilon, simplify_patience))\n continue \n \n mse_record_whole += data_record[\"mse\"]\n data_DL_whole += data_record[\"data_DL\"]\n model_DL_whole += data_record[\"model_DL\"]\n if \"param\" in record_keys:\n param_record_whole += data_record[\"param\"]\n if \"param_grad\" in record_keys:\n param_grad_record_whole += data_record[\"param_grad\"]\n iter_end_whole.append(len(data_record[\"mse\"]))\n\n loss_list.append(loss_new)\n event_list.append({mode_ele: (model_id, layer_id)})\n if verbose >= 1:\n print(\"{0} succeed at (model {1}, layer {2}). loss: {3}\\tDL: {4}\".format(\n mode_ele, model_id, layer_id, view_item(log, (\"criteria_result\", \"loss\")), view_item(log, (\"criteria_result\", \"DL\"))))\n print(\"symbolic_expression: \", layer.symbolic_expression)\n print(\"numerical_expression: \", layer.numerical_expression)\n print()\n model[model_id].load_model_dict(pivot_dict[\"model_dict\"])\n \n elif mode_ele == \"ramping-L1\":\n loss_list_specific = []\n ramping_L1_list = kwargs[\"ramping_L1_list\"] if \"ramping_L1_list\" in kwargs else np.logspace(-7, -1, 30)\n ramping_mse_threshold = kwargs[\"ramping_mse_threshold\"] if \"ramping_mse_threshold\" in kwargs else 1e-5\n ramping_final_multiplier = kwargs[\"ramping_final_multiplier\"] if \"ramping_final_multiplier\" in kwargs else 1e-2\n layer_dict_dict = {}\n for i, L1_amp in enumerate(ramping_L1_list):\n reg_dict = {\"weight\": L1_amp, \"bias\": L1_amp, \"param\": L1_amp}\n _, loss_end, data_record = train_simple(model, X, y, reg_dict = reg_dict, patience = None, validation_data = validation_data, **kwargs)\n layer_dict_dict[i] = model[0].layer_0.layer_dict\n weight, bias = model[0].layer_0.get_weights_bias()\n print(\"L1-amp: {0}\\tloss: {1}\\tweight: {2}\\tbias: {3}\".format(L1_amp, loss_end, weight, bias))\n loss_list_specific.append(loss_end)\n if \"param\" in record_keys:\n param_record_whole.append((weight, bias))\n if loss_end > ramping_mse_threshold:\n if len(loss_list_specific) == 1:\n print(\"\\nThe MSE after the first L1-amp={0} is already larger than the ramping_mse_threshold. Stop and use current L1-amp. The figures will look empty.\".format(ramping_mse_threshold))\n else:\n print(\"\\nThe MSE {0} is larger than the ramping_mse_threshold {1}, stop ramping-L1 simplification\".format(loss_end, ramping_mse_threshold))\n break \n mse_record_whole.append(data_record[\"mse\"][-1])\n data_DL_whole.append(data_record[\"data_DL\"][-1])\n model_DL_whole.append(data_record[\"model_DL\"][-1])\n iter_end_whole.append(1)\n final_L1_amp = L1_amp * ramping_final_multiplier\n final_L1_idx = np.argmin(np.abs(np.array(ramping_L1_list) - final_L1_amp))\n layer_dict_final = layer_dict_dict[final_L1_idx]\n print(\"Final L1_amp used: {0}\".format(ramping_L1_list[final_L1_idx]))\n if \"param\" in record_keys:\n print(\"Final param value:\\nweights: {0}\\nbias{1}\".format(param_record_whole[final_L1_idx][0], param_record_whole[final_L1_idx][1]))\n model[0].layer_0.load_layer_dict(layer_dict_final)\n mse_record_whole = mse_record_whole[: final_L1_idx + 2]\n data_DL_whole = data_DL_whole[: final_L1_idx + 2]\n model_DL_whole = model_DL_whole[: final_L1_idx + 2]\n iter_end_whole = iter_end_whole[: final_L1_idx + 2]\n\n if isplot:\n def dict_to_list(Dict):\n return np.array([value for value in Dict.values()])\n weights_list = []\n bias_list = []\n for element in param_record_whole:\n if isinstance(element[0], dict):\n element_core = dict_to_list(element[0])\n weights_list.append(element_core)\n else:\n element_core = to_np_array(element[0]).squeeze(1)\n weights_list.append(element_core)\n bias_list.append(to_np_array(element[1]))\n weights_list = np.array(weights_list)\n bias_list = np.array(bias_list).squeeze(1)\n\n import matplotlib.pylab as plt\n plt.figure(figsize = (7,5))\n plt.loglog(ramping_L1_list[: len(loss_list_specific)], loss_list_specific)\n plt.xlabel(\"L1 amp\", fontsize = 16)\n plt.ylabel(\"mse\", fontsize = 16)\n plt.show()\n\n plt.figure(figsize = (7,5))\n plt.semilogx(ramping_L1_list[: len(loss_list_specific)], loss_list_specific)\n plt.xlabel(\"L1 amp\", fontsize = 16)\n plt.ylabel(\"mse\", fontsize = 16)\n plt.show()\n\n plt.figure(figsize = (7,5))\n for i in range(weights_list.shape[1]):\n plt.semilogx(ramping_L1_list[: len(loss_list_specific)], weights_list[:,i], label = \"weight_{0}\".format(i))\n if len(bias_list) > 0:\n plt.semilogx(ramping_L1_list[: len(loss_list_specific)], bias_list, label = \"bias\")\n plt.xlabel(\"L1 amp\", fontsize = 16)\n plt.ylabel(\"parameter_values\", fontsize = 16)\n plt.legend()\n plt.show()\n plt.clf()\n plt.close()\n else:\n raise Exception(\"mode {0} not recognized!\".format(mode_ele))\n\n loss_dict[mode_ele] = {}\n if X is not None:\n loss_dict[mode_ele][\"mse_record_whole\"] = mse_record_whole\n loss_dict[mode_ele][\"data_DL_whole\"] = data_DL_whole\n loss_dict[mode_ele][\"{0}_test\".format(loss_type)] = loss_list\n loss_dict[mode_ele][\"model_DL_whole\"] = model_DL_whole\n if \"param\" in record_keys:\n loss_dict[mode_ele][\"param_record_whole\"] = param_record_whole\n if \"param_grad\" in record_keys:\n loss_dict[mode_ele][\"param_grad_record_whole\"] = param_grad_record_whole\n loss_dict[mode_ele][\"iter_end_whole\"] = iter_end_whole\n loss_dict[mode_ele][\"event_list\"] = event_list\n loss_dict[mode_ele][\"is_accept_whole\"] = is_accept_whole\n if mode_ele == \"ramping-L1\":\n loss_dict[mode_ele][\"ramping_L1_list\"] = ramping_L1_list\n loss_dict[mode_ele][\"loss_list_specific\"] = loss_list_specific\n\n if not is_list:\n model = model[0]\n \n return model, loss_dict\n```\n\n## Model architectures:\n\n### MLP:\n\n\n```python\nclass MLP(nn.Module):\n def __init__(\n self,\n input_size,\n struct_param = None,\n W_init_list = None, # initialization for weights\n b_init_list = None, # initialization for bias\n settings = {}, # Default settings for each layer, if the settings for the layer is not provided in struct_param\n is_cuda = False,\n ):\n super(MLP, self).__init__()\n self.input_size = input_size\n self.is_cuda = is_cuda\n self.settings = deepcopy(settings)\n if struct_param is not None:\n self.num_layers = len(struct_param)\n self.W_init_list = W_init_list\n self.b_init_list = b_init_list\n self.info_dict = {}\n\n self.init_layers(deepcopy(struct_param))\n else:\n self.num_layers = 0\n\n\n @property\n def struct_param(self):\n return [getattr(self, \"layer_{0}\".format(i)).struct_param for i in range(self.num_layers)]\n\n \n @property\n def output_size(self):\n return self.get_layer(-1).output_size\n\n\n @property\n def structure(self):\n structure = OrderedDict()\n structure[\"input_size\"] = self.input_size\n structure[\"output_size\"] = self.output_size\n structure[\"struct_param\"] = self.struct_param if hasattr(self, \"struct_param\") else None\n return structure\n\n\n def init_layers(self, struct_param):\n res_forward = self.settings[\"res_forward\"] if \"res_forward\" in self.settings else False\n for k, layer_struct_param in enumerate(struct_param):\n if res_forward:\n num_neurons_prev = struct_param[k - 1][0] + self.input_size if k > 0 else self.input_size\n else:\n num_neurons_prev = struct_param[k - 1][0] if k > 0 else self.input_size\n num_neurons = layer_struct_param[0]\n W_init = self.W_init_list[k] if self.W_init_list is not None else None\n b_init = self.b_init_list[k] if self.b_init_list is not None else None\n\n # Get settings for the current layer:\n layer_settings = deepcopy(self.settings) if bool(self.settings) else {}\n layer_settings.update(layer_struct_param[2]) \n\n # Construct layer:\n layer = get_Layer(layer_type = layer_struct_param[1],\n input_size = num_neurons_prev,\n output_size = num_neurons,\n W_init = W_init,\n b_init = b_init,\n settings = layer_settings,\n is_cuda = self.is_cuda,\n )\n setattr(self, \"layer_{}\".format(k), layer)\n\n\n def forward(self, *input, p_dict=None, **kwargs):\n kwargs = filter_kwargs(kwargs, [\"res_forward\", \"is_res_block\", \"act_noise_scale\"]) # only allow certain kwargs to be passed\n if isinstance(input, tuple):\n input = torch.cat(input, -1)\n output = input\n res_forward = self.settings[\"res_forward\"] if \"res_forward\" in self.settings else False\n is_res_block = self.settings[\"is_res_block\"] if \"is_res_block\" in self.settings else False\n for k in range(len(self.struct_param)):\n p_dict_ele = p_dict[k] if p_dict is not None else None\n if res_forward and k > 0:\n output = getattr(self, \"layer_{}\".format(k))(torch.cat([output, input], -1), p_dict=p_dict_ele, **kwargs)\n else:\n output = getattr(self, \"layer_{}\".format(k))(output, p_dict=p_dict_ele, **kwargs)\n if is_res_block:\n output = output + input\n return output\n \n \n def copy(self):\n return deepcopy(self)\n\n\n def simplify(self, X=None, y=None, mode=\"full\", isplot=False, target_name=None, validation_data = None, **kwargs):\n new_model, _ = simplify(self, X, y, mode=mode, isplot=isplot, target_name=target_name, validation_data=validation_data, **kwargs)\n self.__dict__.update(new_model.__dict__)\n \n \n def snap(self, snap_mode=\"integer\", top=5, **kwargs):\n \"\"\"Generate a set of new models whose parameters are snapped, each model with a different number of snapped parameters.\"\"\"\n if not hasattr(self, \"num_layers\") or self.num_layers != 1:\n return False, [self]\n else:\n model_list = []\n top = top if snap_mode != \"unsnap\" else 1\n for top_ele in range(1, top + 1):\n new_model = self.copy()\n layer = new_model.layer_0\n info_list = layer.simplify(mode=\"snap\", top=top_ele, snap_mode=snap_mode)\n if len(info_list) > 0:\n new_model.reset_layer(0, layer)\n model_list.append(new_model)\n is_succeed = len(model_list) > 0\n return is_succeed, model_list\n\n\n def get_regularization(self, source = [\"weight\", \"bias\"], mode = \"L1\", **kwargs):\n reg = to_Variable([0], is_cuda=self.is_cuda)\n for k in range(len(self.struct_param)):\n layer = getattr(self, \"layer_{}\".format(k))\n reg = reg + layer.get_regularization(mode = mode, source = source)\n return reg\n\n\n def get_layer(self, layer_id):\n if layer_id < 0:\n layer_id += self.num_layers\n return getattr(self, \"layer_{}\".format(layer_id))\n\n\n def reset_layer(self, layer_id, layer):\n setattr(self, \"layer_{}\".format(layer_id), layer)\n\n\n def insert_layer(self, layer_id, layer):\n if layer_id < 0:\n layer_id += self.num_layers\n if layer_id < self.num_layers - 1:\n next_layer = getattr(self, \"layer_{}\".format(layer_id + 1))\n if next_layer.struct_param[1] == \"Simple_Layer\":\n assert next_layer.input_size == layer.output_size, \"The inserted layer's output_size {0} must be compatible with next layer_{1}'s input_size {2}!\"\\\n .format(layer.output_size, layer_id + 1, next_layer.input_size)\n for i in range(self.num_layers - 1, layer_id - 1, -1):\n setattr(self, \"layer_{}\".format(i + 1), getattr(self, \"layer_{}\".format(i)))\n setattr(self, \"layer_{}\".format(layer_id), layer)\n self.num_layers += 1\n \n \n def remove_layer(self, layer_id):\n if layer_id < 0:\n layer_id += self.num_layers\n if layer_id < self.num_layers - 1:\n num_neurons_prev = self.struct_param[layer_id - 1][0] if layer_id > 0 else self.input_size\n replaced_layer = getattr(self, \"layer_{}\".format(layer_id + 1))\n if replaced_layer.struct_param[1] == \"Simple_Layer\":\n assert replaced_layer.input_size == num_neurons_prev, \\\n \"After deleting layer_{0}, the replaced layer's input_size {1} must be compatible with previous layer's output neurons {2}!\"\\\n .format(layer_id, replaced_layer.input_size, num_neurons_prev)\n for i in range(layer_id, self.num_layers - 1):\n setattr(self, \"layer_{}\".format(i), getattr(self, \"layer_{}\".format(i + 1)))\n self.num_layers -= 1\n\n\n def prune_neurons(self, layer_id, neuron_ids):\n if layer_id == \"input\":\n layer = self.get_layer(0)\n layer.prune_input_neurons(neuron_ids)\n self.input_size = layer.input_size\n else:\n if layer_id < 0:\n layer_id = self.num_layers + layer_id\n layer = getattr(self, \"layer_{}\".format(layer_id))\n layer.prune_output_neurons(neuron_ids)\n self.reset_layer(layer_id, layer)\n if layer_id < self.num_layers - 1:\n next_layer = getattr(self, \"layer_{}\".format(layer_id + 1))\n next_layer.prune_input_neurons(neuron_ids)\n self.reset_layer(layer_id + 1, next_layer)\n\n\n def add_neurons(self, layer_id, num_neurons, mode = (\"imitation\", \"zeros\")):\n if not isinstance(mode, list) and not isinstance(mode, tuple):\n mode = (mode, mode)\n if layer_id < 0:\n layer_id = self.num_layers + layer_id\n layer = getattr(self, \"layer_{}\".format(layer_id))\n layer.add_output_neurons(num_neurons, mode = mode[0])\n self.reset_layer(layer_id, layer)\n if layer_id < self.num_layers - 1:\n next_layer = getattr(self, \"layer_{}\".format(layer_id + 1))\n next_layer.add_input_neurons(num_neurons, mode = mode[1])\n self.reset_layer(layer_id + 1, next_layer)\n if layer_id == 0:\n self.input_size = self.get_layer(0).input_size\n\n \n def inspect_operation(self, input, operation_between, p_dict = None, **kwargs):\n output = input\n res_forward = self.settings[\"res_forward\"] if \"res_forward\" in self.settings else False\n is_res_block = self.settings[\"is_res_block\"] if \"is_res_block\" in self.settings else False\n for k in range(*operation_between):\n p_dict_ele = p_dict[k] if p_dict is not None else None\n if res_forward and k > 0:\n output = getattr(self, \"layer_{}\".format(k))(torch.cat([output, input], -1), p_dict = p_dict_ele)\n else:\n output = getattr(self, \"layer_{}\".format(k))(output, p_dict = p_dict_ele)\n if is_res_block:\n output = output + input\n return output\n\n\n def get_weights_bias(self, W_source = \"core\", b_source = \"core\", layer_ids = None, is_grad = False, isplot = False, verbose = False, raise_error = True):\n if not hasattr(self, \"struct_param\"):\n return None, None\n layer_ids = range(len(self.struct_param)) if layer_ids is None else layer_ids\n W_list = []\n b_list = []\n if W_source is not None:\n for k in range(len(self.struct_param)):\n if k in layer_ids:\n if W_source == \"core\":\n try:\n W, _ = getattr(self, \"layer_{}\".format(k)).get_weights_bias(is_grad = is_grad)\n except Exception as e:\n if raise_error:\n raise\n else:\n print(e)\n W = np.array([np.NaN])\n else:\n raise Exception(\"W_source '{}' not recognized!\".format(W_source))\n W_list.append(W)\n \n if b_source is not None:\n for k in range(len(self.struct_param)):\n if k in layer_ids:\n if b_source == \"core\":\n try:\n _, b = getattr(self, \"layer_{}\".format(k)).get_weights_bias(is_grad = is_grad)\n except Exception as e:\n if raise_error:\n raise\n else:\n print(e)\n b = np.array([np.NaN])\n else:\n raise Exception(\"b_source '{}' not recognized!\".format(b_source))\n b_list.append(b)\n\n if verbose:\n import pprint as pp\n if W_source is not None:\n print(\"weight:\")\n pp.pprint(W_list)\n if b_source is not None:\n print(\"bias:\")\n pp.pprint(b_list)\n \n if isplot:\n if W_source is not None:\n print(\"weight {}:\".format(W_source))\n plot_matrices(W_list)\n if b_source is not None:\n print(\"bias {}:\".format(b_source))\n plot_matrices(b_list)\n\n return W_list, b_list\n\n\n def split_to_model_ensemble(self, mode = \"standardize\"):\n num_models = self.struct_param[-1][0]\n model_core = deepcopy(self)\n if mode == \"standardize\":\n last_layer = getattr(model_core, \"layer_{}\".format(model_core.num_layers - 1))\n last_layer.standardize(mode = \"b_mean_zero\")\n else:\n raise Exception(\"mode {} not recognized!\".format(mode))\n model_list = [deepcopy(model_core) for i in range(num_models)]\n for i, model in enumerate(model_list):\n to_prune = list(range(num_models))\n to_prune.pop(i)\n model.prune_neurons(-1, to_prune)\n return construct_model_ensemble_from_nets(model_list)\n \n \n @property\n def model_dict(self):\n model_dict = {\"type\": self.__class__.__name__}\n model_dict[\"input_size\"] = self.input_size\n model_dict[\"struct_param\"] = get_full_struct_param(self.struct_param, self.settings)\n model_dict[\"weights\"], model_dict[\"bias\"] = self.get_weights_bias(W_source = \"core\", b_source = \"core\")\n model_dict[\"settings\"] = deepcopy(self.settings)\n model_dict[\"net_type\"] = self.__class__.__name__\n return model_dict\n\n\n @property\n def DL(self):\n return np.sum([getattr(self, \"layer_{}\".format(i)).DL for i in range(self.num_layers)])\n\n\n def load_model_dict(self, model_dict):\n new_net = load_model_dict_net(model_dict, is_cuda = self.is_cuda)\n self.__dict__.update(new_net.__dict__)\n \n \n def load(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n model_dict = load_model(filename, mode=mode)\n self.load_model_dict(model_dict)\n\n\n def save(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n save_model(self.model_dict, filename, mode=mode)\n\n\n def get_loss(self, input, target, criterion, **kwargs):\n y_pred = self(input, **kwargs)\n return criterion(y_pred, target)\n\n\n def prepare_inspection(self, X, y, **kwargs):\n return {}\n\n\n def set_cuda(self, is_cuda):\n for k in range(self.num_layers):\n getattr(self, \"layer_{}\".format(k)).set_cuda(is_cuda)\n self.is_cuda = is_cuda\n\n\n def set_trainable(self, is_trainable):\n for k in range(self.num_layers):\n getattr(self, \"layer_{}\".format(k)).set_trainable(is_trainable)\n\n\n def get_snap_dict(self):\n snap_dict = {}\n for k in range(len(self.struct_param)):\n layer = getattr(self, \"layer_{}\".format(k))\n if hasattr(layer, \"snap_dict\"):\n recorded_layer_snap_dict = {}\n for key, item in layer.snap_dict.items():\n recorded_layer_snap_dict[key] = {\"new_value\": item[\"new_value\"]}\n if len(recorded_layer_snap_dict) > 0:\n snap_dict[k] = recorded_layer_snap_dict\n return snap_dict\n\n\n def synchronize_settings(self):\n snap_dict = self.get_snap_dict()\n if len(snap_dict) > 0:\n self.settings[\"snap_dict\"] = snap_dict\n return self.settings\n\n\n def get_sympy_expression(self, verbose = True):\n expressions = {i: {} for i in range(self.num_layers)}\n for i in range(self.num_layers):\n layer = getattr(self, \"layer_{}\".format(i))\n if layer.struct_param[1] == \"Symbolic_Layer\":\n if verbose:\n print(\"Layer {}, symbolic_expression: {}\".format(i, layer.symbolic_expression))\n print(\" numerical_expression: {}\".format(layer.numerical_expression))\n expressions[i][\"symbolic_expression\"] = layer.symbolic_expression\n expressions[i][\"numerical_expression\"] = layer.numerical_expression\n expressions[i][\"param_dict\"] = layer.get_param_dict()\n expressions[i][\"DL\"] = layer.DL\n else:\n if verbose:\n print(\"Layer {} is not a symbolic layer.\".format(i))\n expressions[i] = None\n return expressions\n```\n\n### Labelmix_MLP:\n\n\n```python\nclass Labelmix_MLP(nn.Module):\n def __init__(\n self,\n input_size,\n struct_param,\n idx_label=None,\n is_cuda=False,\n ):\n super(Labelmix_MLP, self).__init__()\n self.input_size = input_size\n self.struct_param = struct_param\n self.num_layers = len(struct_param)\n if idx_label is not None and len(idx_label) == input_size:\n idx_label = None\n if idx_label is not None:\n self.idx_label = torch.LongTensor(idx_label)\n idx_main = list(set(range(input_size)) - set(to_np_array(idx_label).astype(int).tolist()))\n self.idx_main = torch.LongTensor(idx_main)\n else:\n self.idx_label = None\n self.idx_main = torch.LongTensor(list(range(input_size)))\n num_neurons_prev = len(self.idx_main)\n for i, layer_struct_param in enumerate(struct_param):\n num_neurons = layer_struct_param[0]\n setattr(self, \"W_{}_main\".format(i), nn.Parameter(torch.randn(num_neurons_prev, num_neurons)))\n setattr(self, \"b_{}_main\".format(i), nn.Parameter(torch.zeros(num_neurons)))\n init_weight(getattr(self, \"W_{}_main\".format(i)), init=None)\n num_neurons_prev = num_neurons\n if self.idx_label is not None:\n setattr(self, \"W_{}_mul\".format(i), nn.Parameter(torch.randn(len(self.idx_label), num_neurons)))\n setattr(self, \"W_{}_add\".format(i), nn.Parameter(torch.randn(len(self.idx_label), num_neurons)))\n init_weight(getattr(self, \"W_{}_mul\".format(i)), init=None)\n init_weight(getattr(self, \"W_{}_add\".format(i)), init=None)\n setattr(self, \"b_{}_mul\".format(i), nn.Parameter(torch.zeros(num_neurons)))\n setattr(self, \"b_{}_add\".format(i), nn.Parameter(torch.zeros(num_neurons)))\n self.set_cuda(is_cuda)\n \n\n def forward(self, input):\n output = input[:, self.idx_main]\n if self.idx_label is not None:\n labels = input[:, self.idx_label]\n for i, layer_struct_param in enumerate(self.struct_param):\n output = torch.matmul(output, getattr(self, \"W_{}_main\".format(i))) + getattr(self, \"b_{}_main\".format(i))\n if \"activation\" in layer_struct_param[2]:\n output = get_activation(layer_struct_param[2][\"activation\"])(output)\n if self.idx_label is not None:\n A_mul = torch.matmul(labels, getattr(self, \"W_{}_mul\".format(i))) + getattr(self, \"b_{}_mul\".format(i))\n A_add = torch.matmul(labels, getattr(self, \"W_{}_add\".format(i))) + getattr(self, \"b_{}_add\".format(i))\n output = output * A_mul + A_add\n return output\n \n \n def get_loss(self, X, y, criterion, **kwargs):\n y_pred = self(X)\n return criterion(y_pred, y)\n\n\n def set_cuda(self, is_cuda):\n if isinstance(is_cuda, str):\n self.cuda(is_cuda)\n else:\n if is_cuda:\n self.cuda()\n else:\n self.cpu()\n self.is_cuda = is_cuda\n\n\n def get_regularization(self, source = [\"weight\", \"bias\"], mode = \"L1\", **kwargs):\n reg = to_Variable([0], is_cuda=self.is_cuda)\n return reg\n\n\n @property\n def model_dict(self):\n model_dict = {\"type\": \"Labelmix_MLP\"}\n model_dict[\"input_size\"] = self.input_size\n model_dict[\"struct_param\"] = self.struct_param\n if self.idx_label is not None:\n model_dict[\"idx_label\"] = to_np_array(self.idx_label).astype(int)\n model_dict[\"state_dict\"] = to_cpu_recur(self.state_dict())\n return model_dict\n```\n\n### Multi_MLP (MLPs in series):\n\n\n```python\nclass Multi_MLP(nn.Module):\n def __init__(\n self,\n input_size,\n struct_param,\n W_init_list = None, # initialization for weights\n b_init_list = None, # initialization for bias\n settings = None, # Default settings for each layer, if the settings for the layer is not provided in struct_param\n is_cuda = False,\n ):\n super(Multi_MLP, self).__init__()\n self.input_size = input_size\n self.num_layers = len(struct_param)\n self.W_init_list = W_init_list\n self.b_init_list = b_init_list\n self.settings = deepcopy(settings)\n self.num_blocks = len(struct_param)\n self.is_cuda = is_cuda\n \n for i, struct_param_ele in enumerate(struct_param):\n input_size_block = input_size if i == 0 else struct_param[i - 1][-1][0]\n setattr(self, \"block_{0}\".format(i), MLP(input_size = input_size_block,\n struct_param = struct_param_ele,\n W_init_list = W_init_list[i] if W_init_list is not None else None,\n b_init_list = b_init_list[i] if b_init_list is not None else None,\n settings = self.settings[i] if self.settings is not None else {},\n is_cuda = self.is_cuda,\n ))\n \n def forward(self, input):\n output = input\n for i in range(self.num_blocks):\n output = getattr(self, \"block_{0}\".format(i))(output)\n return output\n\n\n def get_loss(self, input, target, criterion, **kwargs):\n y_pred = self(input, **kwargs)\n return criterion(y_pred, target)\n\n\n def get_regularization(self, source = [\"weight\", \"bias\"], mode = \"L1\", **kwargs):\n reg = Variable(torch.FloatTensor([0]), requires_grad = False)\n if self.is_cuda:\n reg = reg.cuda()\n for i in range(self.num_blocks):\n reg = reg + getattr(self, \"block_{0}\".format(i)).get_regularization(mode = mode, source = source)\n return reg\n\n\n @property\n def struct_param(self):\n return [getattr(self, \"block_{0}\".format(i)).struct_param for i in range(self.num_blocks)]\n\n\n @property\n def model_dict(self):\n model_dict = {\"type\": self.__class__.__name__}\n model_dict[\"input_size\"] = self.input_size\n model_dict[\"struct_param\"] = self.struct_param\n model_dict[\"weights\"], model_dict[\"bias\"] = self.get_weights_bias(W_source = \"core\", b_source = \"core\")\n model_dict[\"settings\"] = deepcopy(self.settings)\n model_dict[\"net_type\"] = self.__class__.__name__\n return model_dict\n\n\n def load_model_dict(self, model_dict):\n new_net = load_model_dict_Multi_MLP(model_dict, is_cuda = self.is_cuda)\n self.__dict__.update(new_net.__dict__)\n\n\n def load(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n model_dict = load_model(filename, mode=mode)\n self.load_model_dict(model_dict)\n\n\n def save(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n save_model(self.model_dict, filename, mode=mode)\n\n\n def get_weights_bias(self, W_source = \"core\", b_source = \"core\"):\n W_list = []\n b_list = []\n for i in range(self.num_blocks):\n W, b = getattr(self, \"block_{0}\".format(i)).get_weights_bias(W_source = W_source, b_source = b_source)\n W_list.append(W)\n b_list.append(b)\n return deepcopy(W_list), deepcopy(b_list)\n\n\n def prepare_inspection(self, X, y, **kwargs):\n return {}\n\n\n def set_cuda(self, is_cuda):\n for i in range(self.num_blocks):\n getattr(self, \"block_{0}\".format(i)).set_cuda(is_cuda)\n self.is_cuda = is_cuda\n\n\n def set_trainable(self, is_trainable):\n for i in range(self.num_blocks):\n getattr(self, \"block_{0}\".format(i)).set_trainable(is_trainable)\n```\n\n### Branching_Net:\n\n\n```python\nclass Branching_Net(nn.Module):\n \"\"\"An MLP that consists of a base network, and net_1 and net_2 that branches off from the output of the base network.\"\"\"\n def __init__(\n self,\n net_base_model_dict,\n net_1_model_dict,\n net_2_model_dict,\n is_cuda = False,\n ):\n super(Branching_Net, self).__init__()\n self.net_base = load_model_dict(net_base_model_dict, is_cuda = is_cuda)\n self.net_1 = load_model_dict(net_1_model_dict, is_cuda = is_cuda)\n self.net_2 = load_model_dict(net_2_model_dict, is_cuda = is_cuda)\n self.info_dict = {}\n \n \n def forward(self, X, **kwargs):\n shared = self.net_base(X)\n shared = shared.max(0, keepdim = True)[0]\n return self.net_1(shared)[0], self.net_2(shared)[0]\n\n\n def get_regularization(self, source = [\"weights\", \"bias\"], mode = \"L1\"):\n reg = self.net_base.get_regularization(source = source, mode = mode) + \\\n self.net_1.get_regularization(source = source, mode = mode) + \\\n self.net_2.get_regularization(source = source, mode = mode)\n return reg\n\n\n def set_trainable(self, is_trainable):\n self.net_base.set_trainable(is_trainable)\n self.net_1.set_trainable(is_trainable)\n self.net_2.set_trainable(is_trainable)\n\n\n def prepare_inspection(self, X, y, **kwargs):\n return deepcopy(self.info_dict)\n\n\n @property\n def model_dict(self):\n model_dict = {\"type\": \"Branching_Net\"}\n model_dict[\"net_base_model_dict\"] = self.net_base.model_dict\n model_dict[\"net_1_model_dict\"] = self.net_1.model_dict\n model_dict[\"net_2_model_dict\"] = self.net_2.model_dict\n return model_dict\n\n\n def load(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n model_dict = load_model(filename, mode=mode)\n self.load_model_dict(model_dict)\n\n\n def save(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n save_model(self.model_dict, filename, mode=mode)\n\n \nclass Fan_in_MLP(nn.Module):\n def __init__(\n self,\n model_dict_branch1,\n model_dict_branch2,\n model_dict_joint,\n is_cuda=False,\n ):\n super(Fan_in_MLP, self).__init__()\n if model_dict_branch1 is not None:\n self.net_branch1 = load_model_dict(model_dict_branch1, is_cuda=is_cuda)\n else:\n self.net_branch1 = None\n if model_dict_branch2 is not None:\n self.net_branch2 = load_model_dict(model_dict_branch2, is_cuda=is_cuda)\n else:\n self.net_branch2 = None\n self.net_joint = load_model_dict(model_dict_joint, is_cuda=is_cuda)\n self.is_cuda = is_cuda\n self.info_dict = {}\n \n def forward(self, X1, X2, is_outer=False):\n if is_outer:\n X2 = X2[...,None,:]\n if self.net_branch1 is not None:\n X1 = self.net_branch1(X1)\n if self.net_branch2 is not None:\n X2 = self.net_branch2(X2)\n X1, X2 = broadcast_all(X1, X2)\n out = torch.cat([X1, X2], -1)\n # if is_outer=True, then output dimension: [..., X2dim, X1dim, out_dim]:\n return self.net_joint(out).squeeze(-1)\n \n def get_loss(self, input, target, criterion, **kwargs):\n X1, X2 = input\n y_pred = self(X1, X2)\n return criterion(y_pred, target)\n \n def get_regularization(self, source = [\"weight\", \"bias\"], mode = \"L1\", **kwargs):\n reg = Variable(torch.FloatTensor([0]), requires_grad = False)\n if self.is_cuda:\n reg = reg.cuda()\n if self.net_branch1 is not None:\n reg = reg + self.net_branch1.get_regularization(source=source, mode=mode)\n if self.net_branch2 is not None:\n reg = reg + self.net_branch2.get_regularization(source=source, mode=mode)\n return reg\n \n def prepare_inspection(self, X, y, **kwargs):\n return deepcopy(self.info_dict)\n \n @property\n def model_dict(self):\n model_dict = {'type': self.__class__.__name__}\n model_dict[\"model_dict_branch1\"] = self.net_branch1.model_dict if self.net_branch1 is not None else None\n model_dict[\"model_dict_branch2\"] = self.net_branch2.model_dict if self.net_branch2 is not None else None\n model_dict[\"model_dict_joint\"] = self.net_joint.model_dict\n return model_dict\n\n\n def load(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n model_dict = load_model(filename, mode=mode)\n self.load_model_dict(model_dict)\n\n\n def save(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n save_model(self.model_dict, filename, mode=mode)\n```\n\n### Mixture_Model:\n\n\n```python\nclass Mixture_Model(nn.Module):\n def __init__(\n self,\n model_dict_list,\n weight_logits_model_dict,\n num_components,\n is_cuda=False,\n ):\n super(Mixture_Model, self).__init__()\n self.num_components = num_components\n for i in range(self.num_components):\n if isinstance(model_dict_list, list):\n setattr(self, \"model_{}\".format(i), load_model_dict(model_dict_list[i], is_cuda=is_cuda))\n else:\n assert isinstance(model_dict_list, dict)\n setattr(self, \"model_{}\".format(i), load_model_dict(model_dict_list, is_cuda=is_cuda))\n self.weight_logits_model = load_model_dict(weight_logits_model_dict, is_cuda=is_cuda)\n self.is_cuda = is_cuda\n\n\n def forward(self, input):\n output_list = []\n for i in range(self.num_components):\n output = getattr(self, \"model_{}\".format(i))(input)\n output_list.append(output)\n output_list = torch.stack(output_list, -1)\n weight_logits = self.weight_logits_model(input)\n return output_list, weight_logits\n\n @property\n def model_dict(self):\n model_dict = {\"type\": \"Mixture_Model\",\n \"model_dict_list\": [getattr(self, \"model_{}\".format(i)).model_dict for i in range(self.num_components)],\n \"weight_logits_model_dict\": self.weight_logits_model.model_dict,\n \"num_components\": self.num_components,\n }\n return model_dict\n```\n\n### Model_Ensemble:\n\n\n```python\nclass Model_Ensemble(nn.Module):\n \"\"\"Model_Ensemble is a collection of models with the same architecture \n but independent parameters\"\"\"\n def __init__(\n self,\n num_models,\n input_size,\n struct_param,\n W_init_list = None,\n b_init_list = None,\n settings = None,\n net_type = \"MLP\",\n is_cuda = False,\n ):\n super(Model_Ensemble, self).__init__()\n self.num_models = num_models\n self.input_size = input_size\n self.net_type = net_type\n self.is_cuda = is_cuda\n for i in range(self.num_models):\n if settings is None:\n settings_model = {}\n elif isinstance(settings, list) or isinstance(settings, tuple):\n settings_model = settings[i]\n else:\n settings_model = settings\n if isinstance(struct_param, tuple):\n struct_param_model = struct_param[i]\n else:\n struct_param_model = struct_param\n if net_type == \"MLP\":\n net = MLP(input_size = self.input_size,\n struct_param = deepcopy(struct_param_model),\n W_init_list = deepcopy(W_init_list[i]) if W_init_list is not None else None,\n b_init_list = deepcopy(b_init_list[i]) if b_init_list is not None else None,\n settings = deepcopy(settings_model),\n is_cuda = is_cuda,\n )\n elif net_type == \"ConvNet\":\n net = ConvNet(input_channels = self.input_size,\n struct_param = deepcopy(struct_param_model),\n settings = deepcopy(settings_model),\n is_cuda = is_cuda,\n )\n else:\n raise Exception(\"Net_type {0} not recognized!\".format(net_type))\n setattr(self, \"model_{0}\".format(i), net)\n\n\n @property\n def struct_param(self):\n return tuple(getattr(self, \"model_{0}\".format(i)).struct_param for i in range(self.num_models))\n\n\n @property\n def settings(self):\n return [getattr(self, \"model_{0}\".format(i)).settings for i in range(self.num_models)]\n \n \n def get_all_models(self):\n return [getattr(self, \"model_{0}\".format(i)) for i in range(self.num_models)]\n\n\n def init_bias_with_input(self, input, mode = \"std_sqrt\", neglect_last_layer = True):\n for i in range(self.num_models):\n model = getattr(self, \"model_{0}\".format(i))\n model.init_bias_with_input(input, mode = mode, neglect_last_layer = neglect_last_layer)\n \n \n def initialize_param_freeze(self, update_values = True):\n for i in range(self.num_models):\n model = getattr(self, \"model_{0}\".format(i))\n model.initialize_param_freeze(update_values = update_values)\n \n \n def apply_model(self, input, model_id):\n return fetch_model(self, model_id)(input)\n\n\n def fetch_model(self, model_id):\n return getattr(self, \"model_{0}\".format(model_id))\n\n\n def set_trainable(self, is_trainable):\n for i in range(self.num_models):\n getattr(self, \"model_{0}\".format(i)).set_trainable(is_trainable)\n\n\n def forward(self, input):\n output_list = []\n for i in range(self.num_models):\n if self.net_type == \"MLP\":\n output = getattr(self, \"model_{0}\".format(i))(input)\n elif self.net_type == \"ConvNet\":\n output = getattr(self, \"model_{0}\".format(i))(input)[0]\n else:\n raise Exception(\"Net_type {0} not recognized!\".format(self.net_type))\n output_list.append(output)\n return torch.stack(output_list, 1)\n\n\n def get_regularization(self, source = [\"weight\", \"bias\"], mode = \"L1\", **kwargs):\n if not isinstance(source, list):\n source = [source]\n reg = Variable(torch.FloatTensor([0]), requires_grad = False)\n if self.is_cuda:\n reg = reg.cuda()\n model0 = self.model_0\n # Elastic_weight_reg:\n if \"elastic_weight\" in source or \"elastic_bias\" in source:\n # Setting up excluded layer:\n excluded_layer = kwargs[\"excluded_layer\"] if \"excluded_layer\" in kwargs else [-1]\n if not isinstance(excluded_layer, list):\n excluded_layer = [excluded_layer]\n excluded_layer = [element + model0.num_layers if element < 0 else element for element in excluded_layer]\n elastic_mode = kwargs[\"elastic_mode\"] if \"elastic_mode\" in kwargs else \"var\"\n \n # Compute the elastic_weight_reg:\n for k in range(model0.num_layers):\n if k in excluded_layer:\n continue\n W_accum_k = []\n b_accum_k = []\n num_neurons_prev = model0.struct_param[k - 1][0] if k > 0 else self.input_size\n num_neurons = model0.struct_param[k][0]\n for i in range(self.num_models):\n model = getattr(self, \"model_{0}\".format(i))\n assert model0.num_layers == model.num_layers\n assert num_neurons_prev == model.struct_param[k - 1][0] if k > 0 else model.input_size, \\\n \"all models' input/output size at each layer must be identical!\"\n assert num_neurons == model.struct_param[k][0], \\\n \"all models' input/output size at each layer must be identical!\"\n layer_k = getattr(model, \"layer_{0}\".format(k))\n if \"elastic_weight\" in source:\n W_accum_k.append(layer_k.W_core)\n if \"elastic_bias\" in source:\n b_accum_k.append(layer_k.b_core)\n if \"elastic_weight\" in source:\n if elastic_mode == \"var\":\n reg = reg + torch.stack(W_accum_k, -1).var(-1).sum()\n elif elastic_mode == \"std\":\n reg = reg + torch.stack(W_accum_k, -1).std(-1).sum()\n else:\n raise\n if \"elastic_bias\" in source:\n if elastic_mode == \"var\":\n reg = reg + torch.stack(b_accum_k, -1).var(-1).sum()\n elif elastic_mode == \"std\":\n reg = reg + torch.stack(b_accum_k, -1).std(-1).sum()\n else:\n raise\n source_core = deepcopy(source)\n if \"elastic_weight\" in source_core:\n source_core.remove(\"elastic_weight\")\n if \"elastic_bias\" in source_core:\n source_core.remove(\"elastic_bias\")\n else:\n source_core = source\n \n # Other regularizations:\n for k in range(self.num_models):\n reg = reg + getattr(self, \"model_{0}\".format(k)).get_regularization(source = source_core, mode = mode, **kwargs)\n return reg\n \n \n def get_weights_bias(self, W_source = None, b_source = None, verbose = False, isplot = False):\n W_list_dict = {}\n b_list_dict = {}\n for i in range(self.num_models):\n if verbose:\n print(\"\\nmodel {0}:\".format(i))\n W_list_dict[i], b_list_dict[i] = getattr(self, \"model_{0}\".format(i)).get_weights_bias(\n W_source = W_source, b_source = b_source, verbose = verbose, isplot = isplot)\n return W_list_dict, b_list_dict\n \n \n def combine_to_net(self, mode = \"mean\", last_layer_mode = \"concatenate\"):\n model0 = self.model_0\n if mode == \"mean\":\n struct_param = deepcopy(model0.struct_param)\n settings = deepcopy(model0.settings)\n W_init_list = []\n b_init_list = []\n for k in range(model0.num_layers):\n num_neurons_prev = model0.struct_param[k - 1][0] if k > 0 else self.input_size\n num_neurons = model0.struct_param[k][0]\n W_accum_k = []\n b_accum_k = []\n for i in range(self.num_models):\n model = getattr(self, \"model_{0}\".format(i))\n assert model0.num_layers == model.num_layers\n assert num_neurons_prev == model.struct_param[k - 1][0] if k > 0 else model.input_size, \\\n \"If mode == 'mean', all models' input/output size at each layer must be identical!\"\n assert num_neurons == model.struct_param[k][0], \\\n \"If mode == 'mean', all models' input/output size at each layer must be identical!\"\n layer_k = getattr(model, \"layer_{0}\".format(k))\n W_accum_k.append(layer_k.W_core)\n b_accum_k.append(layer_k.b_core)\n\n if k == model0.num_layers - 1:\n current_mode = last_layer_mode\n else:\n current_mode = mode\n\n if current_mode == \"mean\":\n W_accum_k = torch.stack(W_accum_k, -1).mean(-1)\n b_accum_k = torch.stack(b_accum_k, -1).mean(-1)\n elif current_mode == \"concatenate\":\n W_accum_k = torch.cat(W_accum_k, -1)\n b_accum_k = torch.cat(b_accum_k, -1)\n struct_param[-1][0] = sum([self.struct_param[i][-1][0] for i in range(self.num_models)])\n else:\n raise Exception(\"mode {0} not recognized!\".format(last_layer_mode))\n W_init_list.append(W_accum_k.data.numpy())\n b_init_list.append(b_accum_k.data.numpy())\n \n # Build the net:\n net = MLP(input_size = self.input_size,\n struct_param = struct_param,\n W_init_list = W_init_list, \n b_init_list = b_init_list,\n settings = settings,\n )\n else:\n raise Exception(\"mode {0} not recognized!\".format(mode))\n return net\n \n \n def remove_models(self, model_ids):\n if not isinstance(model_ids, list):\n model_ids = [model_ids]\n model_list = []\n k = 0\n for i in range(self.num_models):\n if i not in model_ids:\n if k != i:\n setattr(self, \"model_{0}\".format(k), getattr(self, \"model_{0}\".format(i)))\n k += 1\n num_models_new = k\n for i in range(num_models_new, self.num_models):\n delattr(self, \"model_{0}\".format(i))\n self.num_models = num_models_new\n\n\n def add_models(self, models):\n if not isinstance(models, list):\n models = [models]\n for i, model in enumerate(models):\n setattr(self, \"model_{0}\".format(i + self.num_models), model)\n self.num_models += len(models)\n\n\n def simplify(self, X, y, idx, mode = \"full\", validation_data = None, isplot = False, **kwargs):\n def process_idx(idx):\n idx = idx.byte()\n if len(idx.size()) == 1:\n idx = idx.unqueeze(1)\n if idx.size(1) == 1:\n idx = idx.repeat(1, self.num_models)\n return idx\n idx = process_idx(idx)\n if validation_data is not None:\n X_valid, y_valid, idx_valid = validation_data\n idx_valid = process_idx(idx_valid) \n \n loss_dict = {}\n for i in range(self.num_models):\n model = getattr(self, \"model_{0}\".format(i))\n X_chosen = torch.masked_select(X, idx[:, i:i+1]).view(-1, X.size(1))\n y_chosen = torch.masked_select(y, idx[:, i:i+1]).view(-1, y.size(1))\n if validation_data is not None:\n X_valid_chosen = torch.masked_select(X_valid, idx_valid[:, i:i+1]).view(-1, X_valid.size(1))\n y_valid_chosen = torch.masked_select(y_valid, idx_valid[:, i:i+1]).view(-1, y_valid.size(1))\n if len(X_valid_chosen) == 0:\n validation_data_chosen = None\n else:\n validation_data_chosen = (X_valid_chosen, y_valid_chosen)\n else:\n validation_data_chosen = None\n if len(X_chosen) == 0:\n print(\"The {0}'th model has no corresponding data to simplify with, skip.\".format(i))\n else:\n new_model, loss_dict[\"model_{0}\".format(i)] = simplify(model, X_chosen, y_chosen, mode = mode, validation_data = validation_data_chosen, isplot = isplot, target_name = \"model_{0}\".format(i), **kwargs)\n setattr(self, \"model_{0}\".format(i), new_model)\n return loss_dict\n \n \n def get_sympy_expression(self):\n expressions = {}\n for k in range(self.num_models):\n print(\"\\nmodel {0}:\".format(k))\n expressions[\"model_{0}\".format(k)] = getattr(self, \"model_{0}\".format(k)).get_sympy_expression()\n return expressions\n\n\n @property\n def DL(self):\n return np.sum([getattr(self, \"model_{0}\".format(i)).DL for i in range(self.num_models)])\n\n\n def get_weights_bias(self, W_source = None, b_source = None, verbose = False, isplot = False):\n W_list_dict = {}\n b_list_dict = {}\n for i in range(self.num_models):\n if verbose:\n print(\"\\nmodel {0}:\".format(i))\n W_list_dict[i], b_list_dict[i] = getattr(self, \"model_{0}\".format(i)).get_weights_bias(W_source = W_source, b_source = b_source, verbose = verbose, isplot = isplot)\n return W_list_dict, b_list_dict\n\n\n @property\n def model_dict(self):\n model_dict = {\"type\": \"Model_Ensemble\"}\n for i in range(self.num_models):\n model_dict[\"model_{0}\".format(i)] = getattr(self, \"model_{0}\".format(i)).model_dict\n model_dict[\"input_size\"] = self.input_size\n model_dict[\"struct_param\"] = self.struct_param\n model_dict[\"num_models\"] = self.num_models\n model_dict[\"net_type\"] = self.net_type\n return model_dict\n\n\n def load_model_dict(self, model_dict):\n new_model_ensemble = load_model_dict(model_dict, is_cuda = self.is_cuda)\n self.__dict__.update(new_model_ensemble.__dict__)\n\n\n def load(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n model_dict = load_model(filename, mode=mode)\n self.load_model_dict(model_dict)\n\n\n def save(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n save_model(self.model_dict, filename, mode=mode)\n\n \ndef load_model_dict_model_ensemble(model_dict, is_cuda = False):\n num_models = len([model_name for model_name in model_dict if model_name[:6] == \"model_\"])\n return Model_Ensemble(num_models = num_models,\n input_size = model_dict[\"input_size\"],\n struct_param = tuple([deepcopy(model_dict[\"model_{0}\".format(i)][\"struct_param\"]) for i in range(num_models)]),\n W_init_list = [deepcopy(model_dict[\"model_{0}\".format(i)][\"weights\"]) for i in range(num_models)],\n b_init_list = [deepcopy(model_dict[\"model_{0}\".format(i)][\"bias\"]) for i in range(num_models)],\n settings = [deepcopy(model_dict[\"model_{0}\".format(i)][\"settings\"]) for i in range(num_models)],\n net_type = model_dict[\"net_type\"] if \"net_type\" in model_dict else \"MLP\",\n is_cuda = is_cuda,\n )\n\n\ndef combine_model_ensembles(model_ensembles, input_size):\n model_ensembles = deepcopy(model_ensembles)\n model_ensemble_combined = None\n model_id = 0\n for k, model_ensemble in enumerate(model_ensembles):\n if model_ensemble.input_size == input_size:\n if model_ensemble_combined is None:\n model_ensemble_combined = model_ensemble\n else:\n continue \n for i in range(model_ensemble.num_models):\n model = getattr(model_ensemble, \"model_{0}\".format(i))\n setattr(model_ensemble_combined, \"model_{0}\".format(model_id), model)\n model_id += 1\n model_ensemble_combined.num_models = model_id\n return model_ensemble_combined\n\n\ndef construct_model_ensemble_from_nets(nets):\n num_models = len(nets)\n if num_models is None:\n return None\n input_size = nets[0].input_size\n struct_param = tuple(net.struct_param for net in nets)\n is_cuda = False\n for net in nets:\n if net.input_size != input_size:\n raise Exception(\"The input_size for all nets must be the same!\")\n if net.is_cuda:\n is_cuda = True\n model_ensemble = Model_Ensemble(num_models = num_models, input_size = input_size, struct_param = struct_param, is_cuda = is_cuda)\n for i, net in enumerate(nets):\n setattr(model_ensemble, \"model_{0}\".format(i), net)\n return model_ensemble\n```\n\n\n```python\nclass Model_with_uncertainty(nn.Module):\n def __init__(\n self,\n model_pred,\n model_logstd,\n ):\n super(Model_with_uncertainty, self).__init__()\n self.model_pred = model_pred\n self.model_logstd = model_logstd\n \n def forward(self, input, noise_amp = None, **kwargs):\n return self.model_pred(input, noise_amp = noise_amp, **kwargs), self.model_logstd(input, **kwargs)\n \n def get_loss(self, input, target, criterion, noise_amp = None, **kwargs):\n pred, log_std = self(input, noise_amp = noise_amp, **kwargs)\n return criterion(pred = pred, target = target, log_std = log_std)\n \n def get_regularization(self, source = [\"weight\", \"bias\"], mode = \"L1\", **kwargs):\n return self.model_pred.get_regularization(source = source, mode = mode, **kwargs) + self.model_logstd.get_regularization(source = source, mode = mode, **kwargs)\n \n @property\n def model_dict(self):\n model_dict = {}\n model_dict[\"type\"] = \"Model_with_Uncertainty\"\n model_dict[\"model_pred\"] = self.model_pred.model_dict\n model_dict[\"model_logstd\"] = self.model_logstd.model_dict\n return model_dict\n\n def load(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n model_dict = load_model(filename, mode=mode)\n self.load_model_dict(model_dict)\n\n def save(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n save_model(self.model_dict, filename, mode=mode)\n\n def set_cuda(self, is_cuda):\n self.model_pred.set_cuda(is_cuda)\n self.model_logstd.set_cuda(is_cuda)\n \n def set_trainable(self, is_trainable):\n self.model_pred.set_trainable(is_trainable)\n self.model_logstd.set_trainable(is_trainable)\n```\n\n### RNN:\n\n\n```python\nclass RNNCellBase(nn.Module):\n def extra_repr(self):\n s = '{input_size}, {hidden_size}'\n if 'bias' in self.__dict__ and self.bias is not True:\n s += ', bias={bias}'\n if 'nonlinearity' in self.__dict__ and self.nonlinearity != \"tanh\":\n s += ', nonlinearity={nonlinearity}'\n return s.format(**self.__dict__)\n\n def check_forward_input(self, input):\n if input.size(1) != self.input_size:\n raise RuntimeError(\n \"input has inconsistent input_size: got {}, expected {}\".format(\n input.size(1), self.input_size))\n\n def check_forward_hidden(self, input, hx, hidden_label=''):\n if input.size(0) != hx.size(0):\n raise RuntimeError(\n \"Input batch size {} doesn't match hidden{} batch size {}\".format(\n input.size(0), hidden_label, hx.size(0)))\n\n if hx.size(1) != self.hidden_size:\n raise RuntimeError(\n \"hidden{} has inconsistent hidden_size: got {}, expected {}\".format(\n hidden_label, hx.size(1), self.hidden_size))\n```\n\n### LSTM:\n\n\n```python\nclass LSTM(RNNCellBase):\n \"\"\"a LSTM class\"\"\"\n def __init__(\n self,\n input_size,\n hidden_size,\n output_struct_param,\n output_settings = {},\n bias = True,\n is_cuda = False,\n ):\n super(LSTM, self).__init__()\n self.input_size = input_size\n self.hidden_size = hidden_size\n self.bias = bias\n self.W_ih = nn.Parameter(torch.Tensor(4 * hidden_size, input_size))\n self.W_hh = nn.Parameter(torch.Tensor(4 * hidden_size, hidden_size))\n self.output_net = MLP(input_size = self.hidden_size, struct_param = output_struct_param, settings = output_settings, is_cuda = is_cuda)\n if bias:\n self.b_ih = nn.Parameter(torch.Tensor(4 * hidden_size))\n self.b_hh = nn.Parameter(torch.Tensor(4 * hidden_size))\n else:\n self.register_parameter('b_ih', None)\n self.register_parameter('b_hh', None)\n self.reset_parameters()\n self.is_cuda = is_cuda\n self.device = torch.device(self.is_cuda if isinstance(self.is_cuda, str) else \"cuda\" if self.is_cuda else \"cpu\")\n self.to(self.device)\n\n def reset_parameters(self):\n stdv = 1.0 / np.sqrt(self.hidden_size)\n for weight in self.parameters():\n weight.data.uniform_(-stdv, stdv)\n\n def forward_one_step(self, input, hx):\n self.check_forward_input(input)\n self.check_forward_hidden(input, hx[0], '[0]')\n self.check_forward_hidden(input, hx[1], '[1]')\n return self._backend.LSTMCell(\n input, hx,\n self.W_ih, self.W_hh,\n self.b_ih, self.b_hh,\n )\n \n def forward(self, input, hx = None):\n if hx is None:\n hx = [torch.randn(input.size(0), self.hidden_size).to(self.device),\n torch.randn(input.size(0), self.hidden_size).to(self.device),\n ]\n hhx, ccx = hx\n for i in range(input.size(1)):\n hhx, ccx = self.forward_one_step(input[:, i], (hhx, ccx))\n output = self.output_net(hhx)\n return output\n\n def get_regularization(self, source, mode = \"L1\", **kwargs):\n if not isinstance(source, list):\n source = [source]\n reg = self.output_net.get_regularization(source = source, mode = mode)\n for source_ele in source:\n if source_ele == \"weight\":\n if mode == \"L1\":\n reg = reg + self.W_ih.abs().sum() + self.W_hh.abs().sum()\n elif mode == \"L2\":\n reg = reg + (self.W_ih ** 2).sum() + (self.W_hh ** 2).sum()\n else:\n raise Exception(\"mode {0} not recognized!\".format(mode))\n elif source_ele == \"bias\":\n if self.bias:\n if mode == \"L1\":\n reg = reg + self.b_ih.abs().sum() + self.b_hh.abs().sum()\n elif mode == \"L2\":\n reg = reg + (self.b_ih ** 2).sum() + (self.b_hh ** 2).sum()\n else:\n raise Exception(\"mode {0} not recognized!\".format(mode))\n else:\n raise Exception(\"source {0} not recognized!\".format(source_ele))\n return reg\n \n def get_weights_bias(self, W_source = None, b_source = None, verbose = False, isplot = False):\n W_dict = OrderedDict()\n b_dict = OrderedDict()\n W_o, b_o = self.output_net.get_weights_bias(W_source = W_source, b_source = b_source)\n if W_source == \"core\":\n W_dict[\"W_ih\"] = self.W_ih.cpu().detach().numpy()\n W_dict[\"W_hh\"] = self.W_hh.cpu().detach().numpy()\n W_dict[\"W_o\"] = W_o\n if isplot:\n print(\"W_ih, W_hh:\")\n plot_matrices([W_dict[\"W_ih\"], W_dict[\"W_hh\"]])\n print(\"W_o:\")\n plot_matrices(W_o)\n if self.bias and b_source == \"core\":\n b_dict[\"b_ih\"] = self.b_ih.cpu().detach().numpy()\n b_dict[\"b_hh\"] = self.b_hh.cpu().detach().numpy()\n b_dict[\"b_o\"] = b_o\n if isplot:\n print(\"b_ih, b_hh:\")\n plot_matrices([b_dict[\"b_ih\"], b_dict[\"b_hh\"]])\n print(\"b_o:\")\n plot_matrices(b_o)\n return W_dict, b_dict\n \n def get_loss(self, input, target, criterion, hx = None, **kwargs):\n y_pred = self(input, hx = hx)\n return criterion(y_pred, target)\n \n def prepare_inspection(self, X, y, **kwargs):\n return {}\n\n def load(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n model_dict = load_model(filename, mode=mode)\n self.load_model_dict(model_dict)\n\n def save(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n save_model(self.model_dict, filename, mode=mode)\n```\n\n### Wide ResNet:\n\n\n```python\ndef conv3x3(in_planes, out_planes, stride=1):\n return nn.Conv2d(in_planes, out_planes, kernel_size=3, stride=stride, padding=1, bias=True)\n\ndef conv_init(m):\n classname = m.__class__.__name__\n if classname.find('Conv') != -1:\n init.xavier_uniform_(m.weight, gain=np.sqrt(2))\n init.constant_(m.bias, 0)\n elif classname.find('BatchNorm') != -1:\n init.constant_(m.weight, 1)\n init.constant_(m.bias, 0)\n\n\nclass wide_basic(nn.Module):\n def __init__(self, in_planes, planes, dropout_rate=None, stride=1):\n super(wide_basic, self).__init__()\n self.bn1 = nn.BatchNorm2d(in_planes)\n self.conv1 = nn.Conv2d(in_planes, planes, kernel_size=3, padding=1, bias=True)\n if dropout_rate is not None:\n self.dropout = nn.Dropout(p=dropout_rate)\n else:\n self.dropout = None\n self.bn2 = nn.BatchNorm2d(planes)\n self.conv2 = nn.Conv2d(planes, planes, kernel_size=3, stride=stride, padding=1, bias=True)\n\n self.shortcut = nn.Sequential()\n if stride != 1 or in_planes != planes:\n self.shortcut = nn.Sequential(\n nn.Conv2d(in_planes, planes, kernel_size=1, stride=stride, bias=True),\n )\n\n def forward(self, x):\n out = self.conv1(F.relu(self.bn1(x)))\n if self.dropout is not None:\n out = self.dropout(out)\n out = self.conv2(F.relu(self.bn2(out)))\n out += self.shortcut(x)\n return out\n\n\nclass Wide_ResNet(nn.Module):\n \"\"\"Adapted from https://github.com/meliketoy/wide-resnet.pytorch/blob/master/networks/wide_resnet.py\"\"\"\n def __init__(\n self,\n depth,\n widen_factor,\n input_channels,\n output_size,\n dropout_rate=None,\n is_cuda=False,\n ):\n super(Wide_ResNet, self).__init__()\n\n self.depth = depth\n self.widen_factor = widen_factor\n self.input_channels = input_channels\n self.dropout_rate = dropout_rate\n self.output_size = output_size\n\n assert ((depth-4)%6 ==0), 'Wide-resnet depth should be 6n+4'\n n = (depth-4)//6\n k = widen_factor\n\n nStages = [16*k, 16*k, 32*k, 64*k]\n self.in_planes = nStages[0]\n\n self.conv1 = conv3x3(self.input_channels,nStages[0])\n self.layer1 = self._wide_layer(wide_basic, nStages[1], n, dropout_rate, stride=1)\n self.layer2 = self._wide_layer(wide_basic, nStages[2], n, dropout_rate, stride=2)\n self.layer3 = self._wide_layer(wide_basic, nStages[3], n, dropout_rate, stride=2)\n self.bn1 = nn.BatchNorm2d(nStages[3], momentum=0.9)\n self.linear = nn.Linear(nStages[3], output_size)\n self.set_cuda(is_cuda)\n\n def _wide_layer(self, block, planes, num_blocks, dropout_rate, stride):\n strides = [stride] + [1]*(int(num_blocks)-1)\n layers = []\n\n for stride in strides:\n layers.append(block(self.in_planes, planes, dropout_rate, stride))\n self.in_planes = planes\n\n return nn.Sequential(*layers)\n\n def forward(self, x):\n out = self.conv1(x)\n out = self.layer1(out)\n out = self.layer2(out)\n out = self.layer3(out)\n out = F.relu(self.bn1(out))\n out = out.mean((-1,-2)) # replacing the out= F.avg_pool2d(out, 8) which is sensitive to the input shape.\n out = out.view(out.size(0), -1)\n out = self.linear(out)\n\n return out\n \n def set_cuda(self, is_cuda):\n if isinstance(is_cuda, str):\n self.cuda(is_cuda)\n else:\n if is_cuda:\n self.cuda()\n else:\n self.cpu()\n self.is_cuda = is_cuda\n\n \n @property\n def model_dict(self):\n model_dict = {\"type\": \"Wide_ResNet\"}\n model_dict[\"state_dict\"] = to_cpu_recur(self.state_dict())\n model_dict[\"depth\"] = self.depth\n model_dict[\"widen_factor\"] = self.widen_factor\n model_dict[\"input_channels\"] = self.input_channels\n model_dict[\"output_size\"] = self.output_size\n model_dict[\"dropout_rate\"] = self.dropout_rate\n return model_dict\n\n def load(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n model_dict = load_model(filename, mode=mode)\n self.load_model_dict(model_dict)\n\n def save(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n save_model(self.model_dict, filename, mode=mode)\n \n def get_regularization(self, *args, **kwargs):\n return to_Variable([0], is_cuda = self.is_cuda)\n \n def prepare_inspection(self, *args, **kwargs):\n return {}\n```\n\n### CNN:\n\n\n```python\nclass ConvNet(nn.Module):\n def __init__(\n self,\n input_channels,\n struct_param=None,\n W_init_list=None,\n b_init_list=None,\n settings={},\n return_indices=False,\n is_cuda=False,\n ):\n super(ConvNet, self).__init__()\n self.input_channels = input_channels\n if struct_param is not None:\n self.struct_param = struct_param\n self.W_init_list = W_init_list\n self.b_init_list = b_init_list\n self.settings = settings\n self.num_layers = len(struct_param)\n self.info_dict = {}\n self.param_available = [\"Conv2d\", \"ConvTranspose2d\", \"BatchNorm2d\", \"Simple_Layer\"]\n self.return_indices = return_indices\n for i in range(len(self.struct_param)):\n if i > 0:\n k = 1\n while self.struct_param[i - k][0] is None:\n k += 1\n num_channels_prev = self.struct_param[i - k][0]\n else:\n num_channels_prev = input_channels\n k = 0\n if self.struct_param[i - k][1] == \"Simple_Layer\" and isinstance(num_channels_prev, tuple) and len(num_channels_prev) == 3:\n num_channels_prev = num_channels_prev[0]\n num_channels = self.struct_param[i][0]\n layer_type = self.struct_param[i][1]\n layer_settings = self.struct_param[i][2]\n if \"layer_input_size\" in layer_settings and isinstance(layer_settings[\"layer_input_size\"], tuple):\n num_channels_prev = layer_settings[\"layer_input_size\"][0]\n if layer_type == \"Conv2d\":\n layer = nn.Conv2d(num_channels_prev, \n num_channels,\n kernel_size = layer_settings[\"kernel_size\"],\n stride = layer_settings[\"stride\"] if \"stride\" in layer_settings else 1,\n padding = layer_settings[\"padding\"] if \"padding\" in layer_settings else 0,\n dilation = layer_settings[\"dilation\"] if \"dilation\" in layer_settings else 1,\n )\n elif layer_type == \"ConvTranspose2d\":\n layer = nn.ConvTranspose2d(num_channels_prev,\n num_channels,\n kernel_size = layer_settings[\"kernel_size\"],\n stride = layer_settings[\"stride\"] if \"stride\" in layer_settings else 1,\n padding = layer_settings[\"padding\"] if \"padding\" in layer_settings else 0,\n output_padding = layer_settings[\"output_padding\"] if \"output_padding\" in layer_settings else 0,\n dilation = layer_settings[\"dilation\"] if \"dilation\" in layer_settings else 1,\n )\n elif layer_type == \"Simple_Layer\":\n layer = get_Layer(layer_type = layer_type,\n input_size = layer_settings[\"layer_input_size\"],\n output_size = num_channels,\n W_init = W_init_list[i] if self.W_init_list is not None and self.W_init_list[i] is not None else None,\n b_init = b_init_list[i] if self.b_init_list is not None and self.b_init_list[i] is not None else None,\n settings = layer_settings,\n is_cuda = is_cuda,\n )\n elif layer_type == \"MaxPool2d\":\n layer = nn.MaxPool2d(kernel_size = layer_settings[\"kernel_size\"],\n stride = layer_settings[\"stride\"] if \"stride\" in layer_settings else None,\n padding = layer_settings[\"padding\"] if \"padding\" in layer_settings else 0,\n return_indices = layer_settings[\"return_indices\"] if \"return_indices\" in layer_settings else False,\n )\n elif layer_type == \"MaxUnpool2d\":\n layer = nn.MaxUnpool2d(kernel_size = layer_settings[\"kernel_size\"],\n stride = layer_settings[\"stride\"] if \"stride\" in layer_settings else None,\n padding = layer_settings[\"padding\"] if \"padding\" in layer_settings else 0,\n )\n elif layer_type == \"Upsample\":\n layer = nn.Upsample(scale_factor = layer_settings[\"scale_factor\"],\n mode = layer_settings[\"mode\"] if \"mode\" in layer_settings else \"nearest\",\n )\n elif layer_type == \"BatchNorm2d\":\n layer = nn.BatchNorm2d(num_features = num_channels)\n elif layer_type == \"Dropout2d\":\n layer = nn.Dropout2d(p = 0.5)\n elif layer_type == \"Flatten\":\n layer = Flatten()\n else:\n raise Exception(\"layer_type {0} not recognized!\".format(layer_type))\n\n # Initialize using provided initial values:\n if self.W_init_list is not None and self.W_init_list[i] is not None and layer_type not in [\"Simple_Layer\"]:\n layer.weight.data = torch.FloatTensor(self.W_init_list[i])\n layer.bias.data = torch.FloatTensor(self.b_init_list[i])\n\n setattr(self, \"layer_{0}\".format(i), layer)\n self.set_cuda(is_cuda)\n\n\n def forward(self, input, indices_list = None, **kwargs):\n return self.inspect_operation(input, operation_between = (0, self.num_layers), indices_list = indices_list)\n \n \n def inspect_operation(self, input, operation_between, indices_list = None):\n output = input\n if indices_list is None:\n indices_list = []\n start_layer, end_layer = operation_between\n if end_layer < 0:\n end_layer += self.num_layers\n for i in range(start_layer, end_layer):\n if \"layer_input_size\" in self.struct_param[i][2]:\n output_size_last = output.shape[0]\n layer_input_size = self.struct_param[i][2][\"layer_input_size\"]\n if not isinstance(layer_input_size, tuple):\n layer_input_size = (layer_input_size,)\n output = output.view(-1, *layer_input_size)\n assert output.shape[0] == output_size_last, \"output_size reshaped to different length. Check shape!\"\n if \"Unpool\" in self.struct_param[i][1]:\n output_tentative = getattr(self, \"layer_{0}\".format(i))(output, indices_list.pop(-1))\n else:\n output_tentative = getattr(self, \"layer_{0}\".format(i))(output)\n if isinstance(output_tentative, tuple):\n output, indices = output_tentative\n indices_list.append(indices)\n else:\n output = output_tentative\n if \"activation\" in self.struct_param[i][2]:\n activation = self.struct_param[i][2][\"activation\"]\n else:\n if \"activation\" in self.settings:\n activation = self.settings[\"activation\"]\n else:\n activation = \"linear\"\n if \"Pool\" in self.struct_param[i][1] or \"Unpool\" in self.struct_param[i][1] or \"Upsample\" in self.struct_param[i][1]:\n activation = \"linear\"\n output = get_activation(activation)(output)\n if self.return_indices:\n return output, indices_list\n else:\n return output\n\n\n def get_loss(self, input, target, criterion, **kwargs):\n y_pred = self(input, **kwargs)\n if self.return_indices:\n y_pred = y_pred[0]\n return criterion(y_pred, target)\n\n\n def get_regularization(self, source = [\"weight\", \"bias\"], mode = \"L1\", **kwargs):\n if not isinstance(source, list):\n source = [source]\n reg = Variable(torch.FloatTensor([0]), requires_grad = False)\n if self.is_cuda:\n reg = reg.cuda()\n for k in range(self.num_layers):\n if self.struct_param[k][1] not in self.param_available:\n continue\n layer = getattr(self, \"layer_{0}\".format(k))\n for source_ele in source:\n if source_ele == \"weight\":\n if self.struct_param[k][1] not in [\"Simple_Layer\"]:\n item = layer.weight\n else:\n item = layer.W_core\n elif source_ele == \"bias\":\n if self.struct_param[k][1] not in [\"Simple_Layer\"]:\n item = layer.bias\n else:\n item = layer.b_core\n if mode == \"L1\":\n reg = reg + item.abs().sum()\n elif mode == \"L2\":\n reg = reg + (item ** 2).sum()\n else:\n raise Exception(\"mode {0} not recognized!\".format(mode))\n return reg\n\n\n def get_weights_bias(self, W_source = \"core\", b_source = \"core\"):\n W_list = []\n b_list = []\n for k in range(self.num_layers):\n if self.struct_param[k][1] == \"Simple_Layer\":\n layer = getattr(self, \"layer_{0}\".format(k))\n if W_source == \"core\":\n W_list.append(to_np_array(layer.W_core))\n if b_source == \"core\":\n b_list.append(to_np_array(layer.b_core))\n elif self.struct_param[k][1] in self.param_available:\n layer = getattr(self, \"layer_{0}\".format(k))\n if W_source == \"core\":\n W_list.append(to_np_array(layer.weight))\n if b_source == \"core\":\n b_list.append(to_np_array(layer.bias, full_reduce = False))\n else:\n if W_source == \"core\":\n W_list.append(None)\n if b_source == \"core\":\n b_list.append(None)\n return W_list, b_list\n\n\n @property\n def model_dict(self):\n model_dict = {\"type\": self.__class__.__name__}\n model_dict[\"net_type\"] = self.__class__.__name__\n model_dict[\"input_channels\"] = self.input_channels\n model_dict[\"struct_param\"] = self.struct_param\n model_dict[\"settings\"] = self.settings\n model_dict[\"weights\"], model_dict[\"bias\"] = self.get_weights_bias(W_source = \"core\", b_source = \"core\")\n model_dict[\"return_indices\"] = self.return_indices\n return model_dict\n\n \n @property\n def output_size(self):\n return self.struct_param[-1][0]\n \n \n @property\n def structure(self):\n structure = OrderedDict()\n structure[\"input_channels\"] = self.input_channels\n structure[\"output_size\"] = self.output_size\n structure[\"struct_param\"] = self.struct_param if hasattr(self, \"struct_param\") else None\n return structure\n \n\n\n def get_sympy_expression(self, verbose=True):\n expressions = {i: None for i in range(self.num_layers)}\n return expressions\n\n\n def load(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n model_dict = load_model(filename, mode=mode)\n self.load_model_dict(model_dict)\n\n\n def save(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n save_model(self.model_dict, filename, mode=mode)\n \n \n def DL(self):\n DL = 0\n for k in range(self.num_layers):\n layer_type = self.struct_param[k][1]\n if layer_type in self.param_available:\n layer = getattr(self, \"layer_{0}\".format(k))\n if layer_type == \"Simple_Layer\":\n DL += layer.DL\n else:\n DL += get_list_DL(to_np_array(layer.weight), \"non-snapped\")\n DL += get_list_DL(to_np_array(layer.bias), \"non-snapped\")\n return DL\n \n\n def load_model_dict(self, model_dict):\n new_net = load_model_dict_net(model_dict, is_cuda = self.is_cuda)\n self.__dict__.update(new_net.__dict__)\n\n\n def prepare_inspection(self, X, y, **kwargs):\n pred_prob = self(X)\n if self.return_indices:\n pred_prob = pred_prob[0]\n pred = pred_prob.max(1)[1]\n# self.info_dict[\"accuracy\"] = get_accuracy(pred, y)\n return deepcopy(self.info_dict)\n \n\n def set_cuda(self, is_cuda):\n if isinstance(is_cuda, str):\n self.cuda(is_cuda)\n else:\n if is_cuda:\n self.cuda()\n else:\n self.cpu()\n self.is_cuda = is_cuda\n\n\n\n def set_trainable(self, is_trainable):\n for k in range(self.num_layers):\n layer = getattr(self, \"layer_{0}\".format(k))\n if self.struct_param[k][1] == \"Simple_Layer\":\n layer.set_trainable(is_trainable)\n elif self.struct_param[k][1] in self.param_available:\n for param in layer.parameters():\n param.requires_grad = is_trainable\n\n\n\nclass Conv_Model(nn.Module):\n def __init__(\n self,\n encoder_model_dict,\n core_model_dict,\n decoder_model_dict,\n latent_size = 2,\n is_generative = True,\n is_res_block = True,\n is_cuda = False,\n ):\n \"\"\"Conv_Model consists of an encoder, a core and a decoder\"\"\"\n super(Conv_Model, self).__init__()\n self.latent_size = latent_size\n self.is_generative = is_generative\n if not is_generative:\n self.encoder = load_model_dict(encoder_model_dict, is_cuda = is_cuda)\n self.core = load_model_dict(core_model_dict, is_cuda = is_cuda)\n self.decoder = load_model_dict(decoder_model_dict, is_cuda = is_cuda)\n self.is_res_block = is_res_block\n self.is_cuda = is_cuda\n self.info_dict = {}\n\n\n @property\n def num_layers(self):\n if self.is_generative:\n return 1\n else:\n return len(self.core.model_dict[\"struct_param\"])\n\n\n def forward(\n self,\n X,\n latent = None,\n **kwargs\n ):\n if self.is_generative:\n if len(latent.shape) == 1:\n latent = latent.repeat(len(X), 1)\n latent = self.core(latent)\n else:\n p_dict = {k: latent if k == 0 else None for k in range(self.num_layers)}\n latent = self.encoder(X)\n latent = self.core(latent, p_dict = p_dict)\n output = self.decoder(latent)\n if self.is_res_block:\n output = (X + nn.Sigmoid()(output)).clamp(0, 1)\n return output\n \n \n def forward_multistep(self, X, latents, isplot = False, num_images = 1):\n assert len(latents.shape) == 1\n length = int(len(latents) / 2)\n output = X\n for i in range(length - 1):\n latent = latents[i * self.latent_size: (i + 2) * self.latent_size]\n output = self(output, latent = latent)\n if isplot:\n plot_matrices(output[:num_images,0])\n return output\n\n\n def get_loss(self, X, y, criterion, **kwargs):\n return criterion(self(X = X[0], latent = X[1]), y)\n \n \n def plot(self, X, y, num_images = 1):\n y_pred = self(X[0], latent = X[1])\n idx_list = np.random.choice(len(X[0]), num_images)\n for idx in idx_list:\n matrix = torch.cat([X[0][idx], y[idx], y_pred[idx]])\n plot_matrices(matrix, images_per_row = 8)\n \n \n def get_regularization(self, source = [\"weights\", \"bias\"], mode = \"L1\"):\n if self.is_generative:\n return self.core.get_regularization(source = source, mode = mode) + \\\n self.decoder.get_regularization(source = source, mode = mode)\n else:\n return self.encoder.get_regularization(source = source, mode = mode) + \\\n self.core.get_regularization(source = source, mode = mode) + \\\n self.decoder.get_regularization(source = source, mode = mode)\n\n\n def prepare_inspection(self, X, y, **kwargs):\n return deepcopy(self.info_dict)\n \n \n def set_trainable(self, is_trainable):\n if not self.is_generative:\n self.encoder.set_trainable(is_trainable)\n self.core.set_trainable(is_trainable)\n self.decoder.set_trainable(is_trainable)\n \n \n @property\n def model_dict(self):\n model_dict = {\"type\": \"Conv_Model\"}\n if not self.is_generative:\n model_dict[\"encoder_model_dict\"] = self.encoder.model_dict\n model_dict[\"latent_size\"] = self.latent_size\n model_dict[\"core_model_dict\"] = self.core.model_dict\n model_dict[\"decoder_model_dict\"] = self.decoder.model_dict\n model_dict[\"is_generative\"] = self.is_generative\n model_dict[\"is_res_block\"] = self.is_res_block\n return model_dict\n\n\n def load(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n model_dict = load_model(filename, mode=mode)\n self.load_model_dict(model_dict)\n\n\n def save(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n save_model(self.model_dict, filename, mode=mode)\n\n\n\nclass Conv_Autoencoder(nn.Module):\n def __init__(\n self,\n input_channels_encoder,\n input_channels_decoder,\n struct_param_encoder,\n struct_param_decoder,\n latent_size = (1,2),\n share_model_among_steps = False,\n settings = {},\n is_cuda = False,\n ):\n \"\"\"Conv_Autoencoder consists of an encoder and a decoder\"\"\"\n super(Conv_Autoencoder, self).__init__()\n self.input_channels_encoder = input_channels_encoder\n self.input_channels_decoder = input_channels_decoder\n self.struct_param_encoder = struct_param_encoder\n self.struct_param_decoder = struct_param_decoder\n self.share_model_among_steps = share_model_among_steps\n self.settings = settings\n self.encoder = ConvNet(input_channels = input_channels_encoder, struct_param = struct_param_encoder, settings = settings, is_cuda = is_cuda)\n self.decoder = ConvNet(input_channels = input_channels_decoder, struct_param = struct_param_decoder, settings = settings, is_cuda = is_cuda)\n self.is_cuda = is_cuda\n \n def encode(self, input):\n if self.share_model_among_steps:\n latent = []\n for i in range(input.shape[1]):\n latent_step = self.encoder(input[:, i:i+1])\n latent.append(latent_step)\n return torch.cat(latent, 1)\n else:\n return self.encoder(input)\n \n def decode(self, latent):\n if self.share_model_among_steps:\n latent_size = self.struct_param_encoder[-1][0]\n latent = latent.view(latent.size(0), -1, latent_size)\n output = []\n for i in range(latent.shape[1]):\n output_step = self.decoder(latent[:, i].contiguous())\n output.append(output_step)\n return torch.cat(output, 1)\n else:\n return self.decoder(latent)\n \n def set_trainable(self, is_trainable):\n self.encoder.set_trainable(is_trainable)\n self.decoder.set_trainable(is_trainable)\n \n def forward(self, input):\n return self.decode(self.encode(input))\n \n def get_loss(self, input, target, criterion, **kwargs):\n return criterion(self(input), target)\n \n def get_regularization(self, source = [\"weight\", \"bias\"], mode = \"L1\"):\n return self.encoder.get_regularization(source = source, mode = mode) + \\\n self.decoder.get_regularization(source = source, mode = mode)\n \n @property\n def model_dict(self):\n model_dict = {\"type\": \"Conv_Autoencoder\"}\n model_dict[\"net_type\"] = \"Conv_Autoencoder\"\n model_dict[\"input_channels_encoder\"] = self.input_channels_encoder\n model_dict[\"input_channels_decoder\"] = self.input_channels_decoder\n model_dict[\"struct_param_encoder\"] = self.struct_param_encoder\n model_dict[\"struct_param_decoder\"] = self.struct_param_decoder\n model_dict[\"share_model_among_steps\"] = self.share_model_among_steps\n model_dict[\"settings\"] = self.settings\n model_dict[\"encoder\"] = self.encoder.model_dict\n model_dict[\"decoder\"] = self.decoder.model_dict\n return model_dict\n \n def load_model_dict(self, model_dict):\n model = load_model_dict(model_dict, is_cuda = self.is_cuda)\n self.__dict__.update(model.__dict__)\n\n def load(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n model_dict = load_model(filename, mode=mode)\n self.load_model_dict(model_dict)\n\n def save(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n save_model(self.model_dict, filename, mode=mode)\n\n def DL(self):\n return self.encoder.DL + self.decoder.DL\n\n\n\nclass Flatten(nn.Module):\n def __init__(self):\n super(Flatten, self).__init__()\n\n def forward(self, x):\n return x.view(x.size(0), -1)\n```\n\n### VAE:\n\n\n```python\nclass VAE(nn.Module):\n def __init__(\n self,\n encoder_model_dict,\n decoder_model_dict,\n is_cuda = False,\n ):\n super(VAE, self).__init__()\n self.encoder = load_model_dict(encoder_model_dict, is_cuda = is_cuda)\n self.decoder = load_model_dict(decoder_model_dict, is_cuda = is_cuda)\n self.is_cuda = is_cuda\n self.info_dict = {}\n\n\n def encode(self, X):\n Z = self.encoder(X)\n latent_size = int(Z.shape[-1] / 2)\n mu = Z[..., :latent_size]\n logvar = Z[..., latent_size:]\n return mu, logvar\n\n\n def reparameterize(self, mu, logvar):\n std = torch.exp(0.5*logvar)\n eps = torch.randn_like(std)\n return eps.mul(std).add_(mu)\n\n\n def decode(self, Z):\n return self.decoder(Z)\n\n\n def forward(self, X):\n mu, logvar = self.encode(X)\n Z = self.reparameterize(mu, logvar)\n return self.decode(Z), mu, logvar\n\n\n def get_loss(self, X, y = None, **kwargs):\n recon_X, mu, logvar = self(X)\n BCE = F.binary_cross_entropy(recon_X.view(recon_X.shape[0], -1), X.view(X.shape[0], -1), reduction='sum')\n # see Appendix B from VAE paper:\n # Kingma and Welling. Auto-Encoding Variational Bayes. ICLR, 2014\n # https://arxiv.org/abs/1312.6114\n # 0.5 * sum(1 + log(sigma^2) - mu^2 - sigma^2)\n KLD = -0.5 * torch.sum(1 + logvar - mu.pow(2) - logvar.exp())\n loss = (BCE + KLD) / len(X)\n self.info_dict[\"KLD\"] = KLD.item() / len(X)\n self.info_dict[\"BCE\"] = BCE.item() / len(X)\n return loss\n\n\n def model_dict(self):\n model_dict = {\"type\": \"VAE\"}\n model_dict[\"encoder_model_dict\"] = self.encoder.model_dict\n model_dict[\"decoder_model_dict\"] = self.decoder.model_dict\n return model_dict\n\n\n def load(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n model_dict = load_model(filename, mode=mode)\n self.load_model_dict(model_dict)\n\n\n def save(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n save_model(self.model_dict, filename, mode=mode)\n\n\n def get_regularization(self, source = [\"weight\", \"bias\"], mode = \"L1\"):\n return self.encoder.get_regularization(source = source, mode = mode) + self.decoder.get_regularization(source = source, mode = mode)\n\n\n def prepare_inspection(self, X, y, **kwargs):\n return deepcopy(self.info_dict)\n```\n\n## Reparameterization toolkit:\n\n\n```python\nclass Net_reparam(nn.Module):\n \"\"\"Module that uses reparameterization to take into two inputs and gets a scaler\"\"\"\n def __init__(\n self,\n model_dict,\n reparam_mode,\n is_cuda=False,\n ):\n super(Net_reparam, self).__init__()\n self.model = load_model_dict(model_dict, is_cuda=is_cuda)\n self.reparam_mode = reparam_mode\n\n def forward(self, X, Z, is_outer=False):\n \"\"\"\n Obtaining single value using reparameterization.\n\n Args:\n X shape: [Bx, ...]\n Z shape: [S, Bz, Z]\n is_outer: whether to use outer product to get a tensor with shape [S, Bz, Bx].\n \n Returns:\n If is_outer==True, return log_prob of shape [S, Bz, Bx]\n If is_outer==False, return log_prob of shape [S, Bz] (where Bz=Bx)\n \"\"\"\n dist, _ = reparameterize(self.model, X, mode=self.reparam_mode)\n if is_outer:\n log_prob = dist.log_prob(Z[...,None,:])\n else:\n log_prob = dist.log_prob(Z)\n if self.reparam_mode == 'diag':\n log_prob = log_prob.sum(-1)\n return log_prob\n\n def get_regularization(self, source = [\"weight\", \"bias\"], mode = \"L1\", **kwargs):\n return self.model.get_regularization(source=source, model=mode, **kwargs)\n\n def prepare_inspection(self, X, y, **kwargs):\n return {}\n\n @property\n def model_dict(self):\n model_dict = {\"type\": \"Net_reparam\"}\n model_dict[\"model\"] = self.model.model_dict\n model_dict[\"reparam_mode\"] = self.reparam_mode\n return model_dict\n\n def load(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n model_dict = load_model(filename, mode=mode)\n self.load_model_dict(model_dict)\n\n def save(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n save_model(self.model_dict, filename, mode=mode)\n\n\ndef reparameterize(model, input, mode=\"full\", size=None):\n if mode.startswith(\"diag\"):\n if model is not None and model.__class__.__name__ == \"Mixture_Model\":\n return reparameterize_mixture_diagonal(model, input, mode=mode)\n else:\n return reparameterize_diagonal(model, input, mode=mode)\n elif mode == \"full\":\n return reparameterize_full(model, input, size=size)\n else:\n raise Exception(\"Mode {} is not valid!\".format(mode))\n\n\ndef reparameterize_diagonal(model, input, mode):\n if model is not None:\n mean_logit = model(input)\n else:\n mean_logit = input\n if mode.startswith(\"diagg\"):\n if isinstance(mean_logit, tuple):\n mean = mean_logit[0]\n else:\n mean = mean_logit\n std = torch.ones(mean.shape).to(mean.device)\n dist = Normal(mean, std)\n return dist, (mean, std)\n elif mode.startswith(\"diag\"):\n if isinstance(mean_logit, tuple):\n mean_logit = mean_logit[0]\n size = int(mean_logit.size(-1) / 2)\n mean = mean_logit[:, :size]\n std = F.softplus(mean_logit[:, size:], beta=1) + 1e-10\n dist = Normal(mean, std)\n return dist, (mean, std)\n else:\n raise Exception(\"mode {} is not valid!\".format(mode))\n\n\ndef reparameterize_mixture_diagonal(model, input, mode):\n mean_logit, weight_logits = model(input)\n if mode.startswith(\"diagg\"):\n mean_list = mean_logit\n scale_list = torch.ones(mean_list.shape).to(mean_list.device)\n else:\n size = int(mean_logit.size(-2) / 2)\n mean_list = mean_logit[:, :size]\n scale_list = F.softplus(mean_logit[:, size:], beta=1) + 0.01 # Avoid the std to go to 0\n dist = Mixture_Gaussian_reparam(mean_list=mean_list,\n scale_list=scale_list,\n weight_logits=weight_logits,\n )\n return dist, (mean_list, scale_list)\n\n\ndef reparameterize_full(model, input, size=None):\n if model is not None:\n mean_logit = model(input)\n else:\n mean_logit = input\n if isinstance(mean_logit, tuple):\n mean_logit = mean_logit[0]\n if size is None:\n dim = mean_logit.size(-1)\n size = int((np.sqrt(9 + 8 * dim) - 3) / 2)\n mean = mean_logit[:, :size]\n scale_tril = fill_triangular(mean_logit[:, size:], size)\n scale_tril = matrix_diag_transform(scale_tril, F.softplus)\n dist = MultivariateNormal(mean, scale_tril = scale_tril)\n return dist, (mean, scale_tril)\n\n\ndef sample(dist, n=None):\n \"\"\"Sample n instances from distribution dist\"\"\"\n if n is None:\n return dist.rsample()\n else:\n return dist.rsample((n,))\n```\n\n## Probability models:\n### Mixture of Gaussian:\n\n\n```python\nclass Mixture_Gaussian(nn.Module):\n def __init__(\n self,\n num_components,\n dim,\n param_mode = \"full\",\n is_cuda = False,\n ):\n super(Mixture_Gaussian, self).__init__()\n self.num_components = num_components\n self.dim = dim\n self.param_mode = param_mode\n self.is_cuda = is_cuda\n self.device = torch.device(self.is_cuda if isinstance(self.is_cuda, str) else \"cuda\" if self.is_cuda else \"cpu\")\n self.info_dict = {}\n\n\n def initialize(self, model_dict = None, input = None, num_samples = 100, verbose = False):\n if input is not None:\n neg_log_prob_min = np.inf\n loc_init_min = None\n scale_init_min = None\n for i in range(num_samples):\n neg_log_prob, loc_init_list, scale_init_list = self.initialize_ele(input)\n if verbose:\n print(\"{0}: neg_log_prob: {1:.4f}\".format(i, neg_log_prob))\n if neg_log_prob < neg_log_prob_min:\n neg_log_prob_min = neg_log_prob\n loc_init_min = self.loc_list.detach()\n scale_init_min = self.scale_list.detach()\n\n self.loc_list = nn.Parameter(loc_init_min.to(self.device))\n self.scale_list = nn.Parameter(scale_init_min.to(self.device))\n print(\"min neg_log_prob: {0:.6f}\".format(to_np_array(neg_log_prob_min)))\n else:\n if model_dict is None:\n self.weight_logits = nn.Parameter((torch.randn(self.num_components) * np.sqrt(2 / (1 + self.dim))).to(self.device))\n else:\n self.weight_logits = nn.Parameter((torch.FloatTensor(model_dict[\"weight_logits\"])).to(self.device))\n if self.param_mode == \"full\": \n size = self.dim * (self.dim + 1) // 2\n elif self.param_mode == \"diag\":\n size = self.dim\n else:\n raise\n \n if model_dict is None:\n self.loc_list = nn.Parameter(torch.randn(self.num_components, self.dim).to(self.device))\n self.scale_list = nn.Parameter((torch.randn(self.num_components, size) / self.dim).to(self.device))\n else:\n self.loc_list = nn.Parameter(torch.FloatTensor(model_dict[\"loc_list\"]).to(self.device))\n self.scale_list = nn.Parameter(torch.FloatTensor(model_dict[\"scale_list\"]).to(self.device))\n\n\n def initialize_ele(self, input):\n if self.param_mode == \"full\":\n size = self.dim * (self.dim + 1) // 2\n elif self.param_mode == \"diag\":\n size = self.dim\n else:\n raise\n length = len(input)\n self.weight_logits = nn.Parameter(torch.zeros(self.num_components).to(self.device))\n self.loc_list = nn.Parameter(input[torch.multinomial(torch.ones(length) / length, self.num_components)].detach())\n self.scale_list = nn.Parameter((torch.randn(self.num_components, size).to(self.device) * input.std() / 5).to(self.device))\n neg_log_prob = self.get_loss(input)\n return neg_log_prob\n\n\n def prob(self, input):\n if len(input.shape) == 1:\n input = input.unsqueeze(1)\n assert len(input.shape) in [0, 2, 3]\n input = input.unsqueeze(-2)\n if self.param_mode == \"diag\":\n scale_list = F.softplus(self.scale_list)\n logits = (- (input - self.loc_list) ** 2 / 2 / scale_list ** 2 - torch.log(scale_list * np.sqrt(2 * np.pi))).sum(-1)\n else:\n raise\n prob = torch.matmul(torch.exp(logits), nn.Softmax(dim = 0)(self.weight_logits))\n# prob_list = []\n# for i in range(self.num_components):\n# if self.param_mode == \"full\":\n# scale_tril = fill_triangular(getattr(self, \"scale_{0}\".format(i)), self.dim)\n# scale_tril = matrix_diag_transform(scale_tril, F.softplus)\n# dist = MultivariateNormal(getattr(self, \"loc_{0}\".format(i)), scale_tril = scale_tril)\n# log_prob = dist.log_prob(input)\n# elif self.param_mode == \"diag\":\n# dist = Normal(getattr(self, \"loc_{0}\".format(i)).unsqueeze(0), F.softplus(getattr(self, \"scale_{0}\".format(i))))\n# mu = getattr(self, \"loc_{0}\".format(i)).unsqueeze(0)\n# sigma = F.softplus(getattr(self, \"scale_{0}\".format(i)))\n# log_prob = (- (input - mu) ** 2 / 2 / sigma ** 2 - torch.log(sigma * np.sqrt(2 * np.pi))).sum(-1)\n# else:\n# raise\n# setattr(self, \"component_{0}\".format(i), dist)\n# prob = torch.exp(log_prob)\n# prob_list.append(prob)\n# prob_list = torch.stack(prob_list, -1)\n# prob = torch.matmul(prob_list, nn.Softmax(dim = 0)(self.weight_logits))\n return prob\n\n\n def log_prob(self, input):\n return torch.log(self.prob(input) + 1e-45)\n\n\n def get_loss(self, X, y = None, **kwargs):\n \"\"\"Optimize negative log-likelihood\"\"\"\n neg_log_prob = - self.log_prob(X).mean() / np.log(2)\n self.info_dict[\"loss\"] = to_np_array(neg_log_prob)\n return neg_log_prob\n\n\n def prepare_inspection(X, y, criterion, **kwargs):\n return deepcopy(self.info_dict)\n\n\n @property\n def model_dict(self):\n model_dict = {\"type\": \"Mixture_Gaussian\"}\n model_dict[\"num_components\"] = self.num_components\n model_dict[\"dim\"] = self.dim\n model_dict[\"param_mode\"] = self.param_mode\n model_dict[\"weight_logits\"] = to_np_array(self.weight_logits)\n model_dict[\"loc_list\"] = to_np_array(self.loc_list)\n model_dict[\"scale_list\"] = to_np_array(self.scale_list)\n return model_dict\n\n\n def load(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n model_dict = load_model(filename, mode=mode)\n self.load_model_dict(model_dict)\n\n\n def save(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n save_model(self.model_dict, filename, mode=mode)\n\n\n def get_param(self):\n weights = to_np_array(nn.Softmax(dim = 0)(self.weight_logits))\n loc_list = to_np_array(self.loc_list)\n scale_list = to_np_array(self.scale_list)\n print(\"weights: {0}\".format(weights))\n print(\"loc:\")\n pp.pprint(loc_list)\n print(\"scale:\")\n pp.pprint(scale_list)\n return weights, loc_list, scale_list\n\n\n def visualize(self, input):\n import scipy\n import matplotlib.pylab as plt\n std = to_np_array(input.std())\n X = np.arange(to_np_array(input.min()) - 0.2 * std, to_np_array(input.max()) + 0.2 * std, 0.1)\n Y_dict = {}\n weights = nn.Softmax(dim = 0)(self.weight_logits)\n plt.figure(figsize=(10, 4), dpi=100).set_facecolor('white')\n for i in range(self.num_components):\n Y_dict[i] = weights[0].item() * scipy.stats.norm.pdf((X - self.loc_list[i].item()) / self.scale_list[i].item())\n plt.plot(X, Y_dict[i])\n Y = np.sum([item for item in Y_dict.values()], 0)\n plt.plot(X, Y, 'k--')\n plt.plot(input.data.numpy(), np.zeros(len(input)), 'k*')\n plt.title('Density of {0}-component mixture model'.format(self.num_components))\n plt.ylabel('probability density');\n\n\n def get_regularization(self, source = [\"weights\", \"bias\"], mode = \"L1\", **kwargs):\n reg = to_Variable([0], requires_grad = False).to(self.device)\n return reg\n```\n\n### Mixture_Gaussian for reparameterization:\n\n\n```python\nclass Mixture_Gaussian_reparam(nn.Module):\n def __init__(\n self,\n # Use as reparamerization:\n mean_list=None,\n scale_list=None,\n weight_logits=None,\n # Use as prior:\n Z_size=None,\n n_components=None,\n mean_scale=0.1,\n scale_scale=0.1,\n # Mode:\n is_reparam=True,\n reparam_mode=\"diag\",\n is_cuda=False,\n ):\n super(Mixture_Gaussian_reparam, self).__init__()\n self.is_reparam = is_reparam\n self.reparam_mode = reparam_mode\n self.is_cuda = is_cuda\n self.device = torch.device(self.is_cuda if isinstance(self.is_cuda, str) else \"cuda\" if self.is_cuda else \"cpu\")\n\n if self.is_reparam:\n self.mean_list = mean_list # size: [B, Z, k]\n self.scale_list = scale_list # size: [B, Z, k]\n self.weight_logits = weight_logits # size: [B, k]\n self.n_components = self.weight_logits.shape[-1]\n self.Z_size = self.mean_list.shape[-2]\n else:\n self.n_components = n_components\n self.Z_size = Z_size\n self.mean_list = nn.Parameter((torch.rand(1, Z_size, n_components) - 0.5) * mean_scale)\n self.scale_list = nn.Parameter(torch.log(torch.exp((torch.rand(1, Z_size, n_components) * 0.2 + 0.9) * scale_scale) - 1))\n self.weight_logits = nn.Parameter(torch.zeros(1, n_components))\n if mean_list is not None:\n self.mean_list.data = to_Variable(mean_list)\n self.scale_list.data = to_Variable(scale_list)\n self.weight_logits.data = to_Variable(weight_logits)\n\n self.to(self.device)\n\n\n def log_prob(self, input):\n \"\"\"Obtain the log_prob of the input.\"\"\"\n input = input.unsqueeze(-1) # [S, B, Z, 1]\n if self.reparam_mode == \"diag\":\n if self.is_reparam:\n # logits: [S, B, Z, k]\n logits = - (input - self.mean_list) ** 2 / 2 / self.scale_list ** 2 - torch.log(self.scale_list * np.sqrt(2 * np.pi))\n else:\n scale_list = F.softplus(self.scale_list, beta=1)\n logits = - (input - self.mean_list) ** 2 / 2 / scale_list ** 2 - torch.log(scale_list * np.sqrt(2 * np.pi))\n else:\n raise\n # log_softmax(weight_logits): [B, k]\n # logits: [S, B, Z, k]\n # log_prob: [S, B, Z]\n log_prob = torch.logsumexp(logits + F.log_softmax(self.weight_logits, -1).unsqueeze(-2), axis=-1) # F(...).unsqueeze(-2): [B, 1, k]\n return log_prob\n\n\n def prob(self, Z):\n return torch.exp(self.log_prob(Z))\n\n\n def sample(self, n=None):\n if n is None:\n n_core = 1\n else:\n assert isinstance(n, tuple)\n n_core = n[0]\n weight_probs = F.softmax(self.weight_logits, -1) # size: [B, m]\n idx = torch.multinomial(weight_probs, n_core, replacement=True).unsqueeze(-2).expand(-1, self.mean_list.shape[-2], -1) # multinomial result: [B, S]; result: [B, Z, S]\n mean_list = torch.gather(self.mean_list, dim=-1, index=idx) # [B, Z, S]\n if self.is_reparam:\n scale_list = torch.gather(self.scale_list, dim=-1, index=idx) # [B, Z, S]\n else:\n scale_list = F.softplus(torch.gather(self.scale_list, dim=-1, index=idx), beta=1) # [B, Z, S]\n Z = torch.normal(mean_list, scale_list).permute(2, 0, 1)\n if n is None:\n Z = Z.squeeze(0)\n return Z\n\n\n def rsample(self, n=None):\n return self.sample(n=n)\n\n\n def __repr__(self):\n return \"Mixture_Gaussian_reparam({}, Z_size={})\".format(self.n_components, self.Z_size)\n\n\n @property\n def model_dict(self):\n model_dict = {\"type\": \"Mixture_Gaussian_reparam\"}\n model_dict[\"is_reparam\"] = self.is_reparam\n model_dict[\"reparam_mode\"] = self.reparam_mode\n model_dict[\"Z_size\"] = self.Z_size\n model_dict[\"n_components\"] = self.n_components\n model_dict[\"mean_list\"] = to_np_array(self.mean_list)\n model_dict[\"scale_list\"] = to_np_array(self.scale_list)\n model_dict[\"weight_logits\"] = to_np_array(self.weight_logits)\n return model_dict\n```\n\n### Triangular distribution:\n\n\n```python\nclass Triangular_dist(Distribution):\n \"\"\"Probability distribution with a Triangular shape.\"\"\"\n def __init__(self, loc, a, b, validate_args=None):\n self.loc, self.a, self.b = broadcast_all(loc, a, b)\n batch_shape = torch.Size() if isinstance(loc, Number) else self.loc.size()\n super(Triangular_dist, self).__init__(batch_shape, validate_args=validate_args)\n \n @property\n def mean(self):\n return self.loc + (self.b - self.a) / 3\n \n @property\n def variance(self):\n return (self.a ** 2 + self.b ** 2 + self.a * self.b) / 18\n \n @property\n def stddev(self):\n return torch.sqrt(self.variance)\n \n def expand(self, batch_shape, _instance=None):\n new = self._get_checked_instance(PieceWise, _instance)\n batch_shape = torch.Size(batch_shape)\n new.loc = self.loc.expand(batch_shape)\n new.a = self.a.expand(batch_shape)\n new.b = self.b.expand(batch_shape)\n super(Triangular_dist, new).__init__(batch_shape, validate_args=False)\n new._validate_args = self._validate_args\n return new\n \n @constraints.dependent_property\n def support(self):\n return constraints.interval(self.loc - self.a, self.loc + self.b)\n \n def sample(self, sample_shape=torch.Size()):\n shape = self._extended_shape(sample_shape)\n rand = torch.rand(shape, dtype=self.loc.dtype, device=self.loc.device)\n with torch.no_grad():\n return self.icdf(rand)\n \n def rsample(self, sample_shape=torch.Size()):\n \"\"\"Sample with reparameterization.\"\"\"\n shape = self._extended_shape(sample_shape)\n rand = torch.rand(shape, dtype=self.loc.dtype, device=self.loc.device)\n return self.icdf(rand)\n\n def icdf(self, value):\n \"\"\"Inverse cdf.\"\"\"\n if self._validate_args:\n self._validate_sample(value)\n assert value.min() >= 0 and value.max() <= 1\n value, loc, a, b = broadcast_all(value, self.loc, self.a, self.b)\n a_plus_b = a + b\n idx = value < a / a_plus_b\n iidx = ~idx\n out = torch.ones_like(value)\n out[idx] = loc[idx] - a[idx] + torch.sqrt(a[idx] * a_plus_b[idx] * value[idx])\n out[iidx] = loc[iidx] + b[iidx] - torch.sqrt(b[iidx] * a_plus_b[iidx] * (1 - value[iidx]) )\n return out\n\n def prob(self, value):\n \"\"\"Get probability.\"\"\"\n if self._validate_args:\n self._validate_sample(value)\n # compute the variance\n value, loc, a, b = broadcast_all(value, self.loc, self.a, self.b)\n idx1 = (loc - a <= value) & (value <= loc)\n idx2 = (loc < value) & (value <= loc + b)\n a_plus_b = a + b\n\n out = torch.zeros_like(value)\n out[idx1] = 2 * (value[idx1] - loc[idx1] + a[idx1]) / a[idx1] / a_plus_b[idx1]\n out[idx2] = -2 * (value[idx2] - loc[idx2] - b[idx2]) / b[idx2] / a_plus_b[idx2]\n return out\n\n def log_prob(self, value):\n \"\"\"Get log probability.\"\"\"\n return torch.log(self.prob(value))\n \n @property\n def model_dict(self):\n model_dict = {\"type\": \"Triangular_dist\"}\n model_dict[\"loc\"] = to_np_array(self.loc)\n model_dict[\"a\"] = to_np_array(self.a)\n model_dict[\"b\"] = to_np_array(self.b)\n return model_dict\n\n def load(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n model_dict = load_model(filename, mode=mode)\n self.load_model_dict(model_dict)\n\n def save(self, filename):\n mode = \"json\" if filename.endswith(\".json\") else \"pickle\"\n save_model(self.model_dict, filename, mode=mode)\n```\n\n\n```python\ndef load_model_dict_distribution(model_dict, is_cuda = False):\n if model_dict[\"type\"] == \"Mixture_Gaussian\":\n model = Mixture_Gaussian(\n num_components=model_dict[\"num_components\"],\n dim=model_dict[\"dim\"],\n param_mode=model_dict[\"param_mode\"],\n is_cuda=is_cuda,\n )\n model.initialize(model_dict = model_dict)\n elif model_dict[\"type\"] == \"Mixture_Gaussian_reparam\":\n model = Mixture_Gaussian_reparam(\n is_reparam=model_dict[\"is_reparam\"],\n reparam_mode=model_dict[\"reparam_mode\"],\n mean_list=model_dict[\"mean_list\"],\n scale_list=model_dict[\"scale_list\"],\n weight_logits=model_dict[\"weight_logits\"],\n Z_size=model_dict[\"Z_size\"],\n n_components=model_dict[\"n_components\"],\n is_cuda=is_cuda,\n )\n elif model_dict[\"type\"] == \"Triangular_dist\":\n model = Triangular_dist(\n loc=model_dict[\"loc\"],\n a=model_dict[\"a\"],\n b=model_dict[\"b\"],\n )\n else:\n raise Exception(\"Type {} is not valid!\".format(model_dict[\"type\"]))\n return model\n```\n", "meta": {"hexsha": "6978a0709265cf64c2a362684d5e6b819d9e5780", "size": 255131, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "net.ipynb", "max_stars_repo_name": "tailintalent/pytorch_net", "max_stars_repo_head_hexsha": "4846a3d0e7ad4d511d1a199058b4a275ae65841f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2018-07-20T04:03:35.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-16T06:20:43.000Z", "max_issues_repo_path": "net.ipynb", "max_issues_repo_name": "tailintalent/pytorch_net", "max_issues_repo_head_hexsha": "4846a3d0e7ad4d511d1a199058b4a275ae65841f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "net.ipynb", "max_forks_repo_name": "tailintalent/pytorch_net", "max_forks_repo_head_hexsha": "4846a3d0e7ad4d511d1a199058b4a275ae65841f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 55.3789884958, "max_line_length": 238, "alphanum_fraction": 0.4967918442, "converted": true, "num_tokens": 45068, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850933, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.42819922561362195}} {"text": "# This notebook\nThe purpose of this notebook is to generate all the plots (and some supplementary plots) for Becker, Gallo et al. (submitted). All the needed supplementary code and data files are included in this GitHub repository. \n\n\n\n```python\n%pylab inline\nfrom astropy.io import fits\nimport pandas as pd\nimport glob\nimport os\nimport trappist_machine\nimport scipy.interpolate\n```\n\n Populating the interactive namespace from numpy and matplotlib\n\n\n\n```python\n# set up file paths - for other users, these will need to be updated!\ncomputer_path = '/Users/jcbecker/Documents/GitHub/'\ncurrent_dir = computer_path + 'trappist1/'\nassorteddata = current_dir + 'AssortedData/'\nextraFigures = current_dir + 'ExtraFigures/'\npaperFigures = current_dir + 'PaperFigures/'\n\n# set up other variables to save time\nregen_Q_values = False # if true, re-create the file used for the k/Q constraints. Takes a while to run!\n```\n\n# Import data\n\nThe orbital parameters used in this analysis come from [Grimm et al. 2018](https://www.aanda.org/articles/aa/full_html/2018/05/aa32233-17/T3.html). We won't be running Monte Carlo sims over the errors in this paper, so we will import the best fit values of each parameter. \n\n\n\n\n```python\n## trappist system data\ntrappist1 = pd.read_csv(assorteddata + \"trappist_system_properties.csv\") # from Grimm+ 2018\ntrappist1\n\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        PlanetMass(Mearth)Radius(Rearth)Period(days)a(AU)ei(deg)
                                        0TRAPPIST-1 h0.3310.77318.7679530.0619350.0056789.80
                                        1TRAPPIST-1 g1.1481.14812.3544730.0468770.0020889.71
                                        2TRAPPIST-1 f0.9341.0469.2055850.0385340.0100789.68
                                        3TRAPPIST-1 e0.7720.9106.0990430.0292830.0051089.86
                                        4TRAPPIST-1 d0.2970.7844.0499590.0222800.0083789.75
                                        5TRAPPIST-1 c1.1561.0952.4218070.0158150.0065489.67
                                        6TRAPPIST-1 b1.0171.1211.5108760.0115480.0062289.65
                                        \n
                                        \n\n\n\n\n```python\n# define global truths about TRAPPIST-1\n\ndist = 12.1 # parsecs\nRearth_to_Mjup = 0.08911498 # 1 earth radii is 0.08 Rjup\ntrappist_lum = 0.000522 #solar luminosity, used value is bolometric. Visual luminosity is 0.00000373 L_sun. \nsolar_lum = 3.848 * 10**33.0 # erg/sec\nmstar = 0.089 # solar masses\n\n```\n\n# Plotting the lightcurve (UVOT)\nThe 300 ks of SWIFT data is split into epochs at which the data was taken, and a flux estimate or upper limit is derived using heasoft software XSelect at each of those epochs. \n\n\n```python\nmanual_lc = pd.read_csv(assorteddata + \"manual_LC.csv\")\nmanual_lc['newtime_start'] = 51910 + (manual_lc['time_start'] ) / (60.* 60. *24.) # convert time to JD, since time units in SWIFT headers is funny.\nmanual_lc['newtime_end'] = 51910 + (manual_lc['time_end'] ) / (60.* 60. *24.)\n\n```\n\n\n```python\n# print table for the paper. The latex printout can be copy&pasted into the .tex draft. \n# If more data is added, rerun this and replace table in paper. \nprint_cols = ['obs_ID', 'newtime_start', 'newtime_end','count']\nmanual_lc.sort_values('obs_ID')[print_cols].to_latex(index = False)\n```\n\n\n\n\n '\\\\begin{tabular}{rrrr}\\n\\\\toprule\\n obs\\\\_ID & newtime\\\\_start & newtime\\\\_end & count \\\\\\\\\\n\\\\midrule\\n 10283001.0 & 58010.348779 & 58010.772233 & 0.0200 \\\\\\\\\\n 10283002.0 & 58095.975270 & 58096.984039 & 0.0220 \\\\\\\\\\n 10283003.0 & 58097.648316 & 58097.718778 & 0.0200 \\\\\\\\\\n 10283004.0 & 58103.413035 & 58103.811122 & 0.0180 \\\\\\\\\\n 10283005.0 & 58105.003031 & 58105.010428 & 0.0230 \\\\\\\\\\n 10283006.0 & 58113.704544 & 58113.986122 & 0.0230 \\\\\\\\\\n 10283007.0 & 58114.700832 & 58114.852095 & 0.0150 \\\\\\\\\\n 10283008.0 & 58117.955622 & 58118.306262 & 0.0120 \\\\\\\\\\n 10283009.0 & 58124.928569 & 58125.000012 & 0.0240 \\\\\\\\\\n 10283010.0 & 58126.323775 & 58126.405568 & 0.0200 \\\\\\\\\\n 10283011.0 & 58131.504108 & 58131.843762 & 0.0210 \\\\\\\\\\n 10283012.0 & 58133.164500 & 58133.298580 & 0.0260 \\\\\\\\\\n 10283013.0 & 58136.635220 & 58136.957650 & 0.0410 \\\\\\\\\\n 10283014.0 & 58236.915500 & 58236.994455 & 0.0240 \\\\\\\\\\n 10283015.0 & 58237.506675 & 58237.914594 & 0.0210 \\\\\\\\\\n 10283016.0 & 58243.298947 & 58243.560428 & 0.0540 \\\\\\\\\\n 10283017.0 & 58244.294514 & 58244.306260 & 0.0370 \\\\\\\\\\n 10283018.0 & 58250.925080 & 58251.000012 & 0.0300 \\\\\\\\\\n 10283019.0 & 58251.520821 & 58251.807649 & 0.0280 \\\\\\\\\\n 10283020.0 & 58257.901041 & 58257.988900 & 0.0270 \\\\\\\\\\n 10283021.0 & 58258.705638 & 58258.719456 & 0.0270 \\\\\\\\\\n 10283022.0 & 58264.806395 & 58265.626402 & 0.0240 \\\\\\\\\\n 10283023.0 & 58271.511782 & 58271.870845 & 0.0150 \\\\\\\\\\n 10283024.0 & 58278.021731 & 58278.770845 & 0.0130 \\\\\\\\\\n 10283025.0 & 58279.746508 & 58279.765983 & 0.0430 \\\\\\\\\\n 10283026.0 & 58285.594931 & 58286.000012 & 0.0180 \\\\\\\\\\n 10283027.0 & 58286.137720 & 58286.809734 & 0.0170 \\\\\\\\\\n 10283028.0 & 58287.314815 & 58287.314815 & 0.0330 \\\\\\\\\\n 10283029.0 & 58293.316351 & 58294.052806 & 0.0260 \\\\\\\\\\n 10283030.0 & 58296.417096 & 58296.969456 & 0.0190 \\\\\\\\\\n 10283031.0 & 58297.758323 & 58297.891678 & 0.0300 \\\\\\\\\\n 10283032.0 & 58298.756438 & 58299.959056 & 0.0260 \\\\\\\\\\n 10283034.0 & 58305.782756 & 58306.995150 & 0.0190 \\\\\\\\\\n 10283035.0 & 58307.048495 & 58307.923622 & 0.0190 \\\\\\\\\\n NaN & 58264.806395 & 58264.806395 & 0.0072 \\\\\\\\\\n\\\\bottomrule\\n\\\\end{tabular}\\n'\n\n\n\n\n```python\n# Plot the lightcurve for the paper\n\nidx = (manual_lc['detection_type'] != 'SUM') & pd.isnull(manual_lc['counterr']) & pd.notnull(manual_lc['count'])\nidx_witherrs = (manual_lc['detection_type'] != 'SUM') & pd.notnull(manual_lc['counterr']) & pd.notnull(manual_lc['count'])\n\ntime_centers = np.true_divide(manual_lc.loc[idx]['time_end'].values - manual_lc.loc[idx]['time_start'].values, 2)\nidx2 = manual_lc['detection_type'] == 'SUM'\n\n\nxarr_sec = manual_lc.loc[idx]['time_start'].values + time_centers \nxarr = 51910 + (xarr_sec) / (60.* 60. *24.) # the 51910 figure is from: https://swift.gsfc.nasa.gov/analysis/suppl_uguide/time_guide.html\nxarr_err = time_centers / (60.* 60. *24.)\nyarr = manual_lc.loc[idx]['count'].values\nfigsize(12,5)\nfig, ax1 = subplots()\nax1.errorbar(xarr, yarr, xerr = xarr_err,\n yerr = [np.mean(yarr)*0.3] * len(yarr), \n uplims = 1, marker='.', color='grey', markersize = 15,linestyle = 'None')\nax1.plot(np.linspace(58000, 58325, 10), \n [manual_lc.loc[idx2]['count'].values[0]] * 10, linewidth=2.6, color='Orange')\nax1.fill_between(np.linspace(58000, 58325, 10), \n [manual_lc.loc[idx2]['count'].values[0] - manual_lc.loc[idx2]['counterr'].values[0]] * 10,\n [manual_lc.loc[idx2]['count'].values[0] + manual_lc.loc[idx2]['counterr'].values[0]] * 10,\n color='Orange', alpha=0.3)\n\n#\n#\ntime_centers = np.true_divide(manual_lc.loc[idx_witherrs]['time_end'].values - manual_lc.loc[idx_witherrs]['time_start'].values, 2)\nxarr_sec = manual_lc.loc[idx_witherrs]['time_start'].values + time_centers \nxarr = 51910 + (xarr_sec) / (60.* 60. *24.)\nxarr_err = time_centers / (60.* 60. *24.)\nyarr = manual_lc.loc[idx_witherrs]['count'].values\nyarr_err = manual_lc.loc[idx_witherrs]['counterr'].values\nax1.errorbar(xarr, yarr, xerr = xarr_err,\n yerr = yarr_err, \n marker='.', color='IndianRed', markersize = 15,linestyle = 'None')\n#\n#\n#\n\nax1.set_xlim(58000, 58325)\nax1.set_ylabel(\"SWIFT UVW2 \\n Corrected Rate (count/s)\", fontsize=17)\nax1.set_xlabel(\"Time, MJD (JD - 2400000.5)\", fontsize=17) #MJD not BJD - see SWIFT website above\n\n# make the limits and second axis\nylimz = [0, 0.065] # these will be the limits in count/sec for both y-axes. Set it once to make sure they both match. \nax1.set_ylim(ylimz) # Left y-axis. \nxticks(fontsize=15)\nyticks(fontsize=15)\nax2 = ax1.twinx()\nax2.set_ylim(np.multiply(ylimz, 5.77* 2033 / 100)) # right y-axis\nax2.set_ylabel(\"Flux at Earth \\n[10$^{-14}$ erg cm$^{-2}$ s$^{-1}$ ]\", fontsize=17)\nxticks(fontsize=15)\nyticks(fontsize=15)\nsavefig(paperFigures + \"lcdata_bestfit_mag.pdf\", dpi=100, bbox_inches = 'tight')\n\n\n```\n\n\n# Filter responses / normalization factor\nSWIFT's UVW2 band which we are using has a little different coverage compared to GALEX bands, which are typically used. Just to help guide our understanding, let's just check and look at how different the various NUV/FUV/UV filters are. The [filter profiles](http://svo2.cab.inta-csic.es/svo/theory/fps3/index.php?id=Swift/UVOT.UVW2&&mode=browse&gname=Swift&gname2=UVOT#filter) can be found online at the mission's document pages. I estimated the normalizations from published instrument papers. This plot won't go in the paper. \n\n\n```python\nuvw2_rsf = pd.read_csv(assorteddata + \"Swift_UVOT.UVW2.dat\", names = ['wavelength', 'eff_area_cm2'], delim_whitespace=True)\ngalex_nuv = pd.read_csv(assorteddata + \"GALEX_GALEX.NUV.dat\", names = ['wavelength', 'eff_area_cm2'], delim_whitespace=True)\ngalex_fuv = pd.read_csv(assorteddata + \"GALEX_GALEX.FUV.dat\", names = ['wavelength', 'eff_area_cm2'], delim_whitespace=True)\ngalex_vega = 4.531e-9 #(erg/cm2/s/A)\nuvw2_vega = 5.237e-9 #(erg/cm2/s/A)\ngalex_fuv_vega =6.491e-9\n\n# filter profiles: \n# http://svo2.cab.inta-csic.es/svo/theory/fps3/index.php?id=Swift/UVOT.UVW2&&mode=browse&gname=Swift&gname2=UVOT#filter\n# the coefficients 20 and 60 correct the file approximately to match with the effective area definitions I found in\n# the instrumentation documentation. This is not exact, but it's pretty close, and this plot is just illustrative anyway..\n\nfigsize(6,5)\nfill_between(galex_nuv['wavelength'], 60 * galex_nuv['eff_area_cm2'] / np.max(galex_nuv['eff_area_cm2'] ) , color='Purple', alpha=0.1)\nfill_between(uvw2_rsf['wavelength'], uvw2_rsf['eff_area_cm2'] , color='Salmon', alpha = 0.1)\nplot(uvw2_rsf['wavelength'], uvw2_rsf['eff_area_cm2'] , label=\"SWIFT UVW2\", color='Salmon')\nplot(galex_nuv['wavelength'], 60 * galex_nuv['eff_area_cm2'] / np.max(galex_nuv['eff_area_cm2'] ) , label=\"GALEX NUV\", color='Purple')\ngalex_eff_ar = 60 * galex_nuv['eff_area_cm2'] / np.max(galex_nuv['eff_area_cm2'] )\nfill_between(galex_fuv['wavelength'], 20 * galex_fuv['eff_area_cm2'] / np.max(galex_fuv['eff_area_cm2']), color='Green', alpha = 0.1)\nplot(galex_fuv['wavelength'], 20 * galex_fuv['eff_area_cm2']/ np.max(galex_fuv['eff_area_cm2']), label=\"GALEX FUV\", color='Green')\nswift_eff_ar = uvw2_rsf['eff_area_cm2']\nxlabel(\"Wavelength (Angstroms)\", fontsize=17)\nylabel(\"Effective Area (cm$^{2}$)\", fontsize=17)\nylim(0,62)\nxlim(1250, 3500)\nlegend()\nxticks(fontsize=12)\nyticks(fontsize=14)\nsavefig(extraFigures + \"filter_response_comparison.pdf\", dpi=100, bbox_inches = 'tight')\n\n```\n\n\n```python\ncountrate = manual_lc.loc[idx2]['count'].values[0] \ncounterr = manual_lc.loc[idx2]['counterr'].values[0]\nconvfactor = 5.77 # Brown 2016 multiply by count rate to give the flux density in units of 10\u221216 erg s\u22121 cm\u22122 \u00c5\u22121 - this (5.77) was the value Edmund suggested\nfit_val = countrate * convfactor * 10**-16* 2033. #correct for the width of the waveband - I got this value (2033) from Edmund\nfiterr = counterr * 5.77 * 10**-16* 2033.\n```\n\n# X-ray data: SWIFT XRT\n\nAt first, I thought this data was terrible, but it's actually not. This data requires stacking to get a reliable flux estimate, but we do have one if we stack the entire 300 ks. Edmund did this reduction, and here are his files:\n\ncombined_evt.fits \u2013 the full events list for the combined XRT data (can be opened as an image, but each event is tagged with time, energy, and position).\n\ntrappist_full_xrt_lc.fits \u2013 the light curve for the source region (columns include: time bin, count rate, error, and fractional exposure). The light curve was binned to 5000 s and the fractional exposure is the fraction of 5000 s covered by any one bin.\n\nbkg_full_xrt_lc.fits \u2013 the light curve for the background region\n\nbkg.reg, xmm.reg \u2013 the region files in ds9 format.\n\nThen, we also have: output.lc, created using lcmath and the two \\*fits light curves described above. \n\nThis data was taken in event mode (each photon hit is recorded as an event), so we will bin the data into 5000 second chunks instead of using the epochs that we used for the UVOT exposures (which were integrated over the exposure times). \n\n\n```python\nlcdata = fits.open(assorteddata + \"SWIFT_XRT/output.lc\") # read in background subtracted LC\nbkgdata = fits.open(assorteddata + \"SWIFT_XRT/bkg_full_xrt_lc.fits\") # read in background subtracted LC\n\n# format time column:\nlc_x = 51910 + 5.270701373846318E+08 / (60.* 60. *24.) + \\\n np.true_divide(np.add(0, list(zip(*lcdata[1].data))[0]),60.* 60. *24.)\nlc_y = list(zip(*lcdata[1].data))[1]\nlc_er = list(zip(*lcdata[1].data))[2]\nlc_ff = list(zip(*lcdata[1].data))[3]\nxrt_lc = pd.DataFrame({'time':lc_x,'flux':lc_y,'error': lc_er, 'filling_factor': lc_ff})\nsolid_detection = (xrt_lc['flux'] - xrt_lc['error']) > 0\nxrt_lc['detect'] = solid_detection\nxrt_lc['back_level']= list(zip(*bkgdata[1].data))[1]\nassert list(zip(*bkgdata[1].data))[0] == list(zip(*lcdata[1].data))[0] # source and background should have same time array; stop if not\n\ndetection_type = []\nerr = []\ntrue_flux = []\nfor item in xrt_lc.index: \n if xrt_lc.iloc[item]['flux'] < 0: #if less than background, we do not have reliable detection, so replace value with upper limit set at background level. \n # We already subtracted background using lcmath, so less than 0 means 'less than background' here. \n true_flux.append(xrt_lc.iloc[item]['back_level'])\n detection_type.append('upper_limit')\n err.append(NaN)\n elif (xrt_lc.iloc[item]['flux'] - xrt_lc.iloc[item]['error']) <= 0: # then, this data point is indistinguishable from zero. Also upper limit. \n true_flux.append(xrt_lc.iloc[item]['flux'])\n detection_type.append('upper_limit')\n err.append(NaN)\n else:\n true_flux.append(xrt_lc.iloc[item]['flux'])\n detection_type.append('detection')\n err.append(xrt_lc.iloc[item]['error'])\n\nxrt_lc['xrt_flux'] = true_flux\nxrt_lc['detection_type'] = detection_type\nxrt_lc['xrt_error'] = err\n\nxrt_lc.to_csv(\"AssortedData/SWIFT_XRT/xray_lightcurve_processed.csv\") # save this# default - don't include totally empty bins\nidxnull = (xrt_lc['filling_factor'] > 0.0)\n##\nidx_upplim = idxnull & (xrt_lc['detection_type'] =='upper_limit')\nfigsize(12,5)\nfig, ax1 = subplots()\nax1.errorbar(xrt_lc.loc[idx_upplim]['time'], xrt_lc.loc[idx_upplim]['xrt_flux'], alpha = 0.2, uplims = [1] * len(xrt_lc.loc[idx_upplim]['xrt_flux']), yerr = [0.002] * len(xrt_lc.loc[idx_upplim]['xrt_flux']), color='grey',marker='o',linestyle='None')\nidx_detect = idxnull &(xrt_lc['detection_type'] =='detection') # points with detections only\nax1.errorbar(xrt_lc.loc[idx_detect]['time'], xrt_lc.loc[idx_detect]['xrt_flux'], yerr = xrt_lc.loc[idx_detect]['xrt_error'], color='DarkRed',marker='o',linestyle='None')\nax1.plot(np.linspace(58000, 58325, 10), \n [0.0024] * 10, linewidth=2.6, color='HotPink', label = \"Detections\")\nax1.fill_between(np.linspace(58000, 58325, 10), \n [0.0024 - 0.00015] * 10,\n [0.0024 + 0.00015] * 10,\n color='Pink', alpha=0.3)\n\nax1.plot(np.linspace(58000, 58325, 10), \n [0.00026] * 10, linewidth=2.6, color='Orange')\nax1.fill_between(np.linspace(58000, 58325, 10), \n [0.00026 - 0.00003] * 10,\n [0.00026 + 0.00003] * 10,\n color='Orange', alpha=0.3)\nax1.set_ylabel(\"SWIFT XRT \\n Corrected Rate (count/s)\", fontsize=17)\nax1.set_xlabel(\"Date (MJD - 2400000.5)\", fontsize=17)\nxticks(fontsize=15)\nyticks(fontsize=15)\nax1.set_xlim(58000, 58325)\nax1.set_ylim(0, 25e-3)\n\nsavefig(paperFigures + \"lcdata_xray_subtracted.pdf\", dpi=100, bbox_inches = 'tight')\n\n\n```\n\n\n```python\nprint( np.average(xrt_lc.loc[idx_detect]['xrt_flux'].values, weights = 1./xrt_lc.loc[idx_detect]['xrt_error'].values))\nprint( np.sum(1./np.power(1./xrt_lc.loc[idx_detect]['xrt_error'].values,2.)))\n```\n\n 0.002387190936025226\n 0.00015375166280284414\n\n\nThe photon count estimate (0.00042 counts/sec $\\pm$ 0.00005 counts/sec) comes from the stacked 300 ks exposure (and can be computed in XSPEC). To convert that to a flux, we need to make assumptions about the spectrum shape. \n\nOne way to do this is using this online tool: https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3pimms/w3pimms.pl to convert the count rate into a flux. We use the 'Swift/XRT/PC count rate' option since this data was collected in Photon Collecting mode. The energy input range runs from 0.3 keV to 10 keV (put that in manually). We put in a nH column density, then choose a model for the spectrum. \n\nWe chose APEC (https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node134.html) as the model. \n[APEC](https://iopscience.iop.org/article/10.1086/322992/pdf) calculates emissivities for a hot, diffuse, optically thin plasma that is in collisionally-ionized equiliubrium. These are the conditions in the stellar corona creating the observed X-ray flux, so we choose the log(Temperature) we expect for the M8 to be log(T) = 6.5 (you can change this, it doesn't change the flux much). We choose the abundances to be 0.2 Solar (also you can change that without changing flux much). \n\nThe exact temperature we use for the corona temperature changes the count rate to flux conversion. \n\nThe referee suggested we use instead as done in Wheatley 2017 a two-temperature APEC model. For the Wheatley 2017 model, the Swift XRT count rate measured in the 0.3-10 keV band (or 0.3-2 keV, since the model produces almost no emission above 2 keV) translates to flux as follows: \n\n1 count/s -> 2.67e-11 erg/s/cm$^2$ (for a bandpass of 0.3-10 keV)\n\nUsing the parameters described above as well as our computed count rate and error on the count rate, we compute an x-ray flux and lower and upper limits of: (see next cell)\n\n\n\n```python\n# Old values, which we are not using anymore:\nflux_from_pimms = 1.063E-14 # yields erg/s/cm^2\nflux_from_pimmsmin = 9.360E-15 # yields erg/s/cm^2\nflux_from_pimmsmax = 1.189E-14 # yields erg/s/cm^2\n\n# New values:\nrawflux = 0.00042 # counts / sec\nrawfluxerr = 0.00005 # counts / sec\nflux_from_2param_APEC = (rawflux) * 2.67e-11 # yields erg/s/cm^2\nflux_from_2param_APECerr = (rawfluxerr) * 2.67e-11 # yields erg/s/cm^2\n\nprint(\"total flux (ergs/s/cm^2):\", flux_from_2param_APEC, \" and error:\", flux_from_2param_APECerr)\n```\n\n total flux (ergs/s/cm^2): 1.1214e-14 and error: 1.335e-15\n\n\n\n```python\n# Flaring count rate to flux, using APEC two parameter model:\nrawflux = 0.0024 # counts / sec\nrawfluxerr = 0.00015 # counts / sec\nflux_from_2param_APEC_flare = (rawflux) * 2.67e-11 # yields erg/s/cm^2\nflux_from_2param_APEC_flare_err = (rawfluxerr) * 2.67e-11 # yields erg/s/cm^2\nprint(\"loud flux (ergs/s/cm^2):\", flux_from_2param_APEC_flare, \" and error:\", flux_from_2param_APEC_flare_err)\n```\n\n loud flux (ergs/s/cm^2): 6.407999999999999e-14 and error: 4.0049999999999994e-15\n\n\n\n```python\n# Also, we can find the baseline flux (but we won't use this - except to add to table):\nraw_baseline, raw_baseline_err = 0.00026, 0.00003\nprint( \"quiescent flux (ergs/s/cm^2):\", 2.67e-11 * raw_baseline, \" and error:\", 2.67e-11 * raw_baseline_err)\n```\n\n quiescent flux (ergs/s/cm^2): 6.941999999999999e-15 and error: 8.01e-16\n\n\n# Convert between wavebands to compute the XUV flux. \nHere we are using the empirical scaling relations from a collection of literature papers. \n\nOriginally, I had to convert UV->X-ray->EUV, but since we have a reliable estimate of the average X-ray flux now, I can use that vaue (which comes from the online calculator) directly. I will also do the extrapolation from UV to X-ray, just to check what it is and how good the empirical models in the literature are. The models in the literature use mostly much more massive stars than M8Vs, so it's likely that they won't be 'that' good.\n\nThe papers that I used in figuring all this out are as follows:\n1. [Sanz-Forcada et al. 2011](https://arxiv.org/pdf/1105.0550.pdf) - Table 5 was helpful for seeing the types of stars for which measurements in different wavebands exist, and Equation 3 is the EUV <-> X-ray scaling relation\n2. [Chadney et al. 2015](https://arxiv.org/pdf/1412.3380.pdf) - Equations 2a and 2b, also see Figure 2\n3. [Stelzer et al. 2013](https://academic.oup.com/mnras/article/431/3/2063/1068594) - Equation 4 combined with Table 1, and Figure 14 which visually presents the empirical correlations. \n\n\n\n```python\nfit_valnew = fit_val\ndist_to_surf = 5.627e-4# AU\ndist_to_earth = 12.1 * 206265 # AU\nuv_at_surface = fit_valnew / np.power(dist_to_surf/dist_to_earth,2)\nuv_at_surfacemin = (fit_valnew - fiterr) / np.power(dist_to_surf/dist_to_earth,2)\nuv_at_surfacemax = (fit_valnew + fiterr) / np.power(dist_to_surf/dist_to_earth,2)\nuv_at_earth = uv_at_surface / np.power(dist_to_earth / dist_to_surf,2)\n\nuvw2lum = trappist_machine.flux_to_lum(fit_valnew, 12.1) # parsecs\nuvw2lumlow = trappist_machine.flux_to_lum(fit_valnew - fiterr, 12.1)\nuvw2lumup = trappist_machine.flux_to_lum(fit_valnew + fiterr, 12.1)\n\n\nprint( \" ----- \")\nprint( \" USING REAL XRAY FLUX \")\nprint( \" ----- \")\n\n## Define the x-ray flux from the values found from APEC two-parameter model\nres_xrayflux_direct = [flux_from_2param_APEC - flux_from_2param_APECerr, flux_from_2param_APEC, flux_from_2param_APEC + flux_from_2param_APECerr]\n\nres_xraylum_fraction_direct = np.true_divide(trappist_machine.flux_to_lum([res_xrayflux_direct, res_xrayflux_direct, res_xrayflux_direct],12.1), ( solar_lum* trappist_lum))\nxray_direct_at_surface = res_xrayflux_direct[1] / np.power(dist_to_surf/dist_to_earth,2)\neuv_over_Fx = 425 * (xray_direct_at_surface)**-0.42 # this is euv/Fx, fixed for measured value\n\n# EUV with true measured xray flux\neuvlum = trappist_machine.flux_to_lum(euv_over_Fx * res_xrayflux_direct[1], 12.1)\neuvlummin = trappist_machine.flux_to_lum(euv_over_Fx * res_xrayflux_direct[0], 12.1)\neuvlummax = trappist_machine.flux_to_lum(euv_over_Fx * res_xrayflux_direct[2], 12.1)\n\n# also x-ray, again:\nxraylum_direct = trappist_machine.flux_to_lum(res_xrayflux_direct[1], 12.1)\nxraylummin_direct = trappist_machine.flux_to_lum(res_xrayflux_direct[0], 12.1)\nxraylummax_direct = trappist_machine.flux_to_lum(res_xrayflux_direct[2], 12.1)\n\n## finally the sum:\nxuvmiddle = euvlum + xraylum_direct\nxuvmax = euvlummax + xraylummax_direct\nxuvmin = euvlummin +xraylummin_direct\n\nprint (\"euv luminosity level:\", euvlum)\nprint( \"euv flux level:\", trappist_machine.planet_flux(euvlum, 12.1 * 206265))\nprint (\"euv fractional luminosity:: %e\" % ( euvlum / ( solar_lum* trappist_lum)))\n\n\n\n```\n\n ----- \n USING REAL XRAY FLUX \n ----- \n euv luminosity level: 4.75770972222334e+26\n euv flux level: 2.71624586120575e-14\n euv fractional luminosity:: 2.368604e-04\n\n\n\n```python\n# get the values for the table AND their errors. This can be copy pasted into the stellar parameters table in the paper. \n#res_xrayflux = [xray_at_earthmin, xray_at_earth, xray_at_earthmax]\n\n#res_xraylum_fraction = np.true_divide(trappist_machine.flux_to_lum([xray_at_earthmin, xray_at_earth, xray_at_earthmax],12.1), ( solar_lum* trappist_lum))\nprint (\"$F_{UV, uvw2, Earth}$ & %0.1e $\\pm %0.1e$ & This Work \\\\\\ \" % (fit_val,fiterr))\nprint (\"$L_{UV, uvw2}$ / L$_{*}$ & %0.1e $\\pm %0.1e $ & This Work \\\\\\ \" % ( uvw2lum/ ( solar_lum* trappist_lum), (uvw2lumup - uvw2lum)/ ( solar_lum* trappist_lum)))\n#print \"$L_{X}$ / L$_{*}$ & %0.1e $^{+%0.1e}_{%0.1e}$ & This Work, extrapolated$ \\\\\\ \" % (xraylum/ ( solar_lum* trappist_lum), (xraylummax - xraylum)/ ( solar_lum* trappist_lum), (xraylum - xraylummin) / ( solar_lum* trappist_lum))\nprint (\"$L_{X}$ / L$_{*}$ (total) & %0.1e $\\pm %0.1e$ & This Work, measured \\\\\\ \" % (xraylum_direct/ ( solar_lum* trappist_lum), (xraylummax_direct - xraylum_direct)/ ( solar_lum* trappist_lum)))\nprint( \"$L_{X}$ / L$_{*}$ (detections) & %0.1e $\\pm %0.1e$ & This Work, measured \\\\\\ \" % ((trappist_machine.flux_to_lum(flux_from_2param_APEC_flare, 12.1) / ( solar_lum* trappist_lum)), (trappist_machine.flux_to_lum(flux_from_2param_APEC_flare_err, 12.1) / ( solar_lum* trappist_lum))))\nprint( \"$L_{X}$ / L$_{*}$ (baseline) & %0.1e $\\pm %0.1e$ & This Work, measured \\\\\\ \" % ((trappist_machine.flux_to_lum(2.67e-11 *raw_baseline, 12.1) / ( solar_lum* trappist_lum)), (trappist_machine.flux_to_lum(2.67e-11 *raw_baseline_err, 12.1) / ( solar_lum* trappist_lum))))\nprint( \"$F_{X}$ (total) & %0.1e $\\pm %0.1e$ & This Work, measured \\\\\\ \" % (res_xrayflux_direct[1], res_xrayflux_direct[2] - res_xrayflux_direct[1],))\nprint( \"$F_{X}$ (detections) & %0.1e $\\pm %0.1e$ & This Work, measured \\\\\\ \" % (flux_from_2param_APEC_flare, flux_from_2param_APEC_flare_err))\nprint( \"$F_{X}$ (baseline) & %0.1e $\\pm %0.1e$ & This Work, measured \\\\\\ \" % (2.67e-11 * raw_baseline, 2.67e-11 * raw_baseline_err))\nprint (\"$L_{EUV}$ / L$_{*}$ (total) & %0.1e $\\pm %0.1e$ & This Work, extrapolated \\\\\\ \" % ( euvlum/ ( solar_lum* trappist_lum), (euvlummax - euvlum)/ ( solar_lum* trappist_lum)))\n#print \"$F_{EUV}$ & %0.1e $^{+%0.1e}_{%0.1e}$ & This Work, extrapolated \\\\\\ \" % (trappist_machine.planet_flux(euvlum, 12.1 * 206265), trappist_machine.planet_flux(euvlummax, 12.1 * 206265), trappist_machine.planet_flux(euvlummin, 12.1 * 206265))\nprint (\"$L_{XUV}$ / L$_{*}$ (total) & %0.1e $\\pm %0.1e$ & This Work, extrapolated \\\\\\ \" % ((xuvmiddle) / ( solar_lum* trappist_lum), (xuvmax - xuvmiddle) / ( solar_lum* trappist_lum)))\n\n\n```\n\n $F_{UV, uvw2, Earth}$ & 8.4e-15 $\\pm 1.4e-15$ & This Work \\\\ \n $L_{UV, uvw2}$ / L$_{*}$ & 7.4e-05 $\\pm 1.2e-05 $ & This Work \\\\ \n $L_{X}$ / L$_{*}$ (total) & 9.8e-05 $\\pm 1.2e-05$ & This Work, measured \\\\ \n $L_{X}$ / L$_{*}$ (detections) & 5.6e-04 $\\pm 3.5e-05$ & This Work, measured \\\\ \n $L_{X}$ / L$_{*}$ (baseline) & 6.1e-05 $\\pm 7.0e-06$ & This Work, measured \\\\ \n $F_{X}$ (total) & 1.1e-14 $\\pm 1.3e-15$ & This Work, measured \\\\ \n $F_{X}$ (detections) & 6.4e-14 $\\pm 4.0e-15$ & This Work, measured \\\\ \n $F_{X}$ (baseline) & 6.9e-15 $\\pm 8.0e-16$ & This Work, measured \\\\ \n $L_{EUV}$ / L$_{*}$ (total) & 2.4e-04 $\\pm 2.8e-05$ & This Work, extrapolated \\\\ \n $L_{XUV}$ / L$_{*}$ (total) & 3.3e-04 $\\pm 4.0e-05$ & This Work, extrapolated \\\\ \n\n\nSo, we find that the measured value of the x-ray flux is a factor of a few lower than the extrapolated value. The extrapolations are right in line with the Wheatley+ estimates, but our measured values are a bit lower (despite the fact we seem to have caught at least one flare in our time series, which could be biasing us even higher depending on how frequent flares actually are). \n\nDoes this matter? Let's check how it affects the mass loss rates and the ocean retention.\n\n\n```python\nflux_xuv_average = trappist_machine.planet_flux(euvlum + xraylum_direct, trappist1['a(AU)'].values)\nflux_xuv_average\ntrappist1['XUV_flux(erg,cm-2,s-1)'] = flux_xuv_average\n# get the mass loss rates, put it in the pandas planet frame\nflux_scaled = trappist1['XUV_flux(erg,cm-2,s-1)'] #this needs to be scaled\nG = 6.674*10**(-8.0) # cm3g\u22121s\u22122\neta = 0.1 # 10-20 percent for low mass planets; Owen & Alvarez 2016\nalpha = 1 #planet cross section increases in the UV , less important for cooler planets\nK = 1\nmdot = (eta * np.pi * flux_scaled * alpha**2.0 * (np.multiply(trappist1['Radius(Rearth)'],1.0) * 6.371*10**8.0 )**3.0) / \\\n (G * 5.9721986*10**27.0 * np.multiply(trappist1['Mass(Mearth)'],1) * K)# eq 1 of Wheatley \n \ntrappist1['mass_loss(g/s)'] = mdot # create mass loss column of table\n\n```\n\n\n```python\ntrappist1\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        PlanetMass(Mearth)Radius(Rearth)Period(days)a(AU)ei(deg)XUV_flux(erg,cm-2,s-1)mass_loss(g/s)
                                        0TRAPPIST-1 h0.3310.77318.7679530.0619350.0056789.8062.3243961.772641e+07
                                        1TRAPPIST-1 g1.1481.14812.3544730.0468770.0020889.71108.7954022.922446e+07
                                        2TRAPPIST-1 f0.9341.0469.2055850.0385340.0100789.68161.0086784.021145e+07
                                        3TRAPPIST-1 e0.7720.9106.0990430.0292830.0051089.86278.8059225.547038e+07
                                        4TRAPPIST-1 d0.2970.7844.0499590.0222800.0083789.75481.5966891.592676e+08
                                        5TRAPPIST-1 c1.1561.0952.4218070.0158150.0065489.67955.8366902.212689e+08
                                        6TRAPPIST-1 b1.0171.1211.5108760.0115480.0062289.651792.8086295.061533e+08
                                        \n
                                        \n\n\n\n\n```python\n# solve optimization problem:\n# n-body sims stabilty (ecc damping), mass loss rates, tidal Qs, system age\n# we will leave this for a future work \n```\n\n# Tidal Q Analysis\n\nThe planets' orbits will evolve The magnitude of these effects can be estimated using the following expression from Goldreich 1963 (with expressions written in a functionally identical form to that used in Barnes 2008):\n\\begin{equation}\nda/dt = \\Big( \\frac{-63}{2}\\frac{1}{Q' m_p} \\sqrt{GM^3} r_p^5 e^2 + \\frac{9}{2}\\frac{\\sigma}{Q_{*}'} \\sqrt{G/M} R^5 m_{p} \\Big) a^{-11/2},\n\\end{equation}\nwhere $Q' = 3Q / 2k_{2}$, $m_p$ denotes a planetary mass, $r_p$ denotes a planetary radius, $G$ is the gravitational constant, $M$ denotes the stellar mass, $R$ the stellar radius, $a$ denotes the planetary semi-major axis, $e$ the planetary eccentricity, and the term $\\sigma = sign(2\\Omega - 3n)$ (where $\\Omega$ is the stellar spin rate and $n$ the planetary mean motion) described the relative frequencies of the stellar rotational period and planetary orbital period. \nIn Gillon 2016, the TRAPPIST-1 rotation rate was estimated using ground-based data to be $P_{rot}= 1.40$ days, but follow-up work using data from the K2 spacecraft (Luger 2017, Vida 207) provided an updated value of $P_{rot}= 3.3$ days, which we use in this analysis. \nThis value of $P_{rot}= 3.3$ days (giving the star a rotational period situated in between the orbital periods of planets c and d) means that the sign term $\\sigma$ cannot be neglected, as different planets will be affected differently by tides raised on the star. \n\nWithout a non-zero eccentricity, the orbital energy will not significantly change, and so $a$ will not change.\nA secondary paired equation can be written to describe the evolution of eccentricity over time \\citep{Goldreich1963},\n\\begin{equation}\nde/dt = \\Big(\\frac{-63}{4}\\frac{1}{Q' m_p} \\sqrt{GM^3} r_p^5 + \\frac{171}{16}\\frac{\\sigma}{Q_{*}'} \\sqrt{G/M} R^5 m_{p} \\Big) e\\ a^{-13/2},\n\\end{equation}\nwhere the first term is due to tides raised on the planet and the second term is due to tides raised on the star. \n\nAlso, we need an expression for the mass loss rate (Luger et al. 2015, Bourrier et al. 2017):\n\\begin{equation}\n \\dot{m_p} = \\epsilon ( r_{XUV} / r_p )^{2} ( 3 F_{XUV} / 4 G \\rho K )\n \\label{eq:mdot}\n\\end{equation}\nwhere $\\epsilon$ is the heating efficiency, $r_p$ the true radius of the planet, $r_{XUV}$ the effective planetary radius in the XUV waveband, $F_{XUV}$ the XUV flux received by the planet, $G$ the gravitational constant, and $K$ is a tidal enhancement factor, where:\n\\begin{equation}\n K = 1 - \\frac{3}{2} \\frac{r_{XUV}}{r_{roche}} + \\frac{1}{2}\\frac{r_{XUV}^{2}}{r_{roche}^2}\n\\end{equation}\nwhere $r_{roche}$ is the planetary Roche Lobe ($r_{roche}~=~(m_p / 2 M_{*})^{1/3} a$). \n\n\n```python\nfrom scipy.integrate import odeint, ode\n\nperz = np.copy(trappist1['Period(days)'].values)\nperz.sort()\ncolor_to_use = ['#6a11d6','#7dd611','#d6117d','#d6a111','#1125d6']\nlinez = [1,1.5,2,2.5,3] #line widths for plotting to make it color-blind accessible \ny0 = [0.011,0.00622, 0.00270 * 330.0] # initial conditions at y0\nt = np.logspace(1, 10, 10000)\nabserr = 1.0e-10\nrelerr = 1.0e-8\ndmdtval = 0\ne_zero = y0[1]\n\nfigsize(6,3)\nfor i, qval in enumerate(np.true_divide([1,10,100,1000,10000], 5.)):\n sol = odeint(trappist_machine.tidal_evo, y0, t, args = (qval,1e-5, perz[0], dmdtval), atol=abserr, rtol=relerr)\n asol,esol,msol = sol.T\n if i == 0:\n plot(t, esol / e_zero, label = r\"$Q' = (3 Q) / 2 k_{2}$ = %i\" % (3. * qval / (2 * 0.3)), linewidth=linez[i], color=color_to_use[i])\n else:\n plot(t, esol / e_zero, label = r\"$Q'$ = %i\" % (3. * qval / (2 * 0.3)), color=color_to_use[i], linewidth=linez[i])\nlegend() \nfor i, qval in enumerate(np.true_divide([1,10,100,1000,10000], 5.)):\n sol = odeint(trappist_machine.tidal_evo, y0, t, args = (qval,1e-5, perz[0], 0), atol=abserr, rtol=relerr)\n asol,esol,msol =sol.T\n if i == 0:\n plot(t, esol / e_zero, label = r\"$Q' = (3 Q) / 2 k_{2}$ = %i\" % (3. * qval / (2 * 0.3)), linewidth=linez[i],linestyle=\"--\", color=color_to_use[i])\n else:\n plot(t, esol / e_zero, label = r\"$Q'$ = %i\" % (3. * qval / (2 * 0.3)), color=color_to_use[i], linestyle=\"--\", linewidth=linez[i])\n\n#ylim(0.0108,0.0115)\nxlabel(\"Time (years) post disk dispersal\", fontsize=17)\n#xlabel(\"3/2 * Q/k\", fontsize=17)\nylabel(\"$e_b(t)$ /$e_b(0)$\", fontsize=17)\nxticks(fontsize=14)\nyticks(fontsize=14)\nxlim(1e4, 7.6e9)\nylim(0,1)\nxscale(\"log\")\nsavefig(paperFigures + \"old-trappist-ecc.pdf\", dpi=150, bbox_inches = 'tight')\n```\n\n\n```python\nfrom scipy.integrate import odeint, ode\n\nperz = np.copy(trappist1['Period(days)'].values)\nperz.sort()\ncolor_to_use = ['#6a11d6','#7dd611','#d6117d','#d6a111','#1125d6']\nlinez = [1,1.5,2,2.5,3] #line widths for plotting to make it color-blind accessible \ny0 = [0.011,0.00622, 0.00270 * 330.0] # initial conditions at y0\nt = np.logspace(1, 10, 10000)\nabserr = 1.0e-10\nrelerr = 1.0e-8\ndmdtval = trappist1.loc[trappist1['Planet'] == 'TRAPPIST-1 b']['mass_loss(g/s)'].values[0] * 5.2804674*10**(-21.)\ne_zero = trappist1.loc[trappist1['Planet'] == 'TRAPPIST-1 b']['e'].values[0] \n\nfigsize(6,6)\nsubplot(2,1,1)\nfor i, qval in enumerate(np.true_divide([1,10,100,1000,10000], 5.)):\n sol = odeint(trappist_machine.tidal_evo, y0, t, args = (qval,1e-5, perz[0], 1*dmdtval), atol=abserr, rtol=relerr)\n asol,esol,msol = sol.T\n sol2 = odeint(trappist_machine.tidal_evo, y0, t, args = (qval,1e-5, perz[0], 0), atol=abserr, rtol=relerr)\n asol2,esol2,msol2 = sol2.T\n subplot(2,1,1)\n\n if i == 0:\n plot(t, esol / e_zero, label = r\"$Q' = (3 Q) / 2 k_{2}$ = %i\" % (3. * qval / (2 * 0.3)), linewidth=linez[i], color=color_to_use[i])\n else:\n plot(t, esol / e_zero, label = r\"$Q'$ = %i\" % (3. * qval / (2 * 0.3)), color=color_to_use[i], linewidth=linez[i])\n \n subplot(2,1,2)\n if i == 0:\n plot(t, 100 * np.abs(np.subtract(esol,esol2)) / e_zero, label = r\"$Q' = (3 Q) / 2 k_{2}$ = %i\" % (3. * qval / (2 * 0.3)), linewidth=linez[i], color=color_to_use[i])\n else:\n plot(t, 100 *np.abs(np.subtract(esol,esol2)) / e_zero, label = r\"$Q'$ = %i\" % (3. * qval / (2 * 0.3)), color=color_to_use[i], linewidth=linez[i])\n\nfor i in [1,2]: #make plot parameters identical for each subplot\n subplot(2,1,i)\n #ylim(0.0108,0.0115)\n #xlabel(\"3/2 * Q/k\", fontsize=17)\n xlim(1e5, 1e10)\n xscale(\"log\")\n \n# top plot\nsubplot(2,1,1)\nylabel(\"$e_b(t)$ /$e_b(0)$\", fontsize=17)\nxticks([])\nylim(0,1)\nyticks(fontsize=14)\n\n#second plot\nsubplot(2,1,2)\nylabel(\"Deviation ($\\delta e_b(t)$ /$e_b(0)$) \\n from constant mass model\", fontsize=13)\nxlabel(\"Time (years) post disk dispersal\", fontsize=17)\nylim(0,0.33)\nlegend() \nxticks(fontsize=15)\nyticks([0.3,0.2,.1,0],['0.3 %','0.2 %','0.1 %','0 %'],fontsize=12)\n\n\n\nsubplots_adjust(hspace = 0)\nsavefig(paperFigures + \"trappist-ecc.pdf\", dpi=150, bbox_inches = 'tight')\n```\n\n# DE/DT DA/DT DL/DT\nWhat happens to the eccentricity of the planets as the orbits evolve, according to a simple tidal model? This can also be computed using the timescale expression and not the full time-series. \n\nIf we increase $dm/dt$ enough, it will start to change the solutions: for $de/dt$, this can even mean that the tide raised on the star becomes less and less important than the tide raised on the planet. However, the expected values of $dm/dt$ for this system are really too small to matter, and the imbalance between the sizes of the two terms if already extreme because $Q_{*}$ is so large. \n\n\n```python\n# make e plot with all planets\n# don't forget to make things radians. and convert back at the end. \nt = np.linspace(0, 7.6e9, 100000)\n#integrate it\nfrom scipy.integrate import odeint, ode\nabserr = 1.0e-10\nrelerr = 1.0e-8\n\ncolors_for_this = [\"#666666\",\"#004C70\",\"#0093D1\",\"#F2635F\",\"#F4D00C\",\"#E0A025\",\"k\"]\n## first look at planet b alone, its future evolution from this current point. \n\n#\nfigsize(4,10)\nqvals = [10./5.,100./5., 1000./5]\nfor knum in range(len(qvals)):\n subplot(len(qvals), 1,knum +1)\n counter =0\n for i, planet in enumerate(['TRAPPIST-1 b','TRAPPIST-1 c','TRAPPIST-1 d','TRAPPIST-1 e','TRAPPIST-1 f','TRAPPIST-1 g', 'TRAPPIST-1 h']):\n sma = trappist1.loc[trappist1['Planet'] == planet]['a(AU)'].values[0]\n ecc = trappist1.loc[trappist1['Planet'] == planet]['e'].values[0]\n mass = trappist1.loc[trappist1['Planet'] == planet]['Mass(Mearth)'].values[0] \n per = trappist1.loc[trappist1['Planet'] == planet]['Period(days)'].values[0] \n dmdtval = trappist1.loc[trappist1['Planet'] == planet]['mass_loss(g/s)'].values[0] * 5.2804674*10**(-21.) #convert units\n y0 = [sma, ecc, mass] # initial conditions at y0\n sol = odeint(trappist_machine.tidal_evo, y0, t, args = (qvals[knum],1e-5,per, dmdtval), atol=abserr, rtol=relerr) # solve this\n asol,esol,msol = sol.T\n plot(t /1e9, esol, label = \"%s\" % planet, color=colors_for_this[i])\n if esol[-1] < 0.001: # add text labels so we know which planet is which if seen without color\n text(7.7 + counter * 0.35,esol[-1], planet[-1], color = colors_for_this[i], fontsize=17)\n counter = counter + 1\n else:\n text(7.7,esol[-1], planet[-1], color = colors_for_this[i], fontsize=17)\n\n if knum != len(qvals) -1:\n xticks([],[],fontsize=14)\n yticks(fontsize=14)\n xlim(0,9.2)\n text(3.4,0.0085, \"Q' = %i\" % (3. * qvals[knum] / (0.3 * 2)), fontweight = 'bold', fontsize=20, alpha=0.7)\n ylabel(\"Eccentrcity ($e$)\", fontsize=15)\nxticks(fontsize=14)\nsubplots_adjust(hspace = 0)\nxlabel(\"Time (Gyr)\", fontsize=17)\nyticks(fontsize=14)\nsavefig(paperFigures + \"trappist-all-ecc-pumpedn_botherms.pdf\", dpi=150, bbox_inches = 'tight')\n```\n\n# Find the values of k/Q that keep eccentricites non-zero over system age. \nThis cell takes a little while to run (on the order of ~30 minutes), so the results will also be saved as \"timescale_plot_data.pkl\" and can be loaded rather than re-generated. We note that since $dm/dt$ is small, the derived timescales should not vary compared to the classic analytic expression; however, we solve the ODE in anticipation that for systems with larger $dm/dt$, the changing planetary mass could matter over long time scales. \n\n\n```python\n# don't forget to make things radians. and convert back at the end. \nt = np.linspace(0, 10e9, 1000000)\n\ncolors_for_this = [\"#666666\",\"#004C70\",\"#0093D1\",\"#F2635F\",\"#F4D00C\",\"#E0A025\",\"k\"]\n\n#\nfigsize(4,10)\nqvals = np.logspace(-1, 4, 60)\nanswers = pd.DataFrame()\nendtimes = pd.DataFrame()\n\n# using the errors from the Grimm paper, find the place that we couldn't see if the result is incompatable with zero or not:\nincomp_with_zero = [0.00622 - 2 * 0.00304, 0.00654 - 3 * 0.00188, 0.00837 - 8* 0.00093, 0.00510 - 8* 0.00058, 0.01007 - 7* 0.00068, \n 0.00208 - 3* 0.00058, 0.00567 - 4 * 0.00121] # comes from errors on each measured value of the ecc\nif regen_Q_values ==True:\n for knum in range(len(qvals)):\n tmpz =[]\n timez = []\n for i, planet in enumerate(['TRAPPIST-1 b','TRAPPIST-1 c','TRAPPIST-1 d','TRAPPIST-1 e','TRAPPIST-1 f','TRAPPIST-1 g', 'TRAPPIST-1 h']):\n sma = trappist1.loc[trappist1['Planet'] == planet]['a(AU)'].values[0]\n ecc = trappist1.loc[trappist1['Planet'] == planet]['e'].values[0]\n mass = trappist1.loc[trappist1['Planet'] == planet]['Mass(Mearth)'].values[0] \n per = trappist1.loc[trappist1['Planet'] == planet]['Period(days)'].values[0] \n dmdtval = trappist1.loc[trappist1['Planet'] == planet]['mass_loss(g/s)'].values[0] * 5.2804674*10**(-21.) #convert units\n\n y0 = [sma, ecc, mass] # initial conditions at y0\n sol = odeint(trappist_machine.tidal_evo, y0, t, args = (qvals[knum],1e-5,per, dmdtval), atol=abserr, rtol=relerr)\n asol,esol,msol = zip(*sol)\n tmpz.append(esol[-1])\n try:\n timeend = min(pd.Series(t).loc[pd.Series(esol) < (incomp_with_zero[i])])\n except ValueError:\n timeend = NaN\n timez.append(timeend)\n\n answers = answers.append({'mean': np.mean(tmpz), 'b':tmpz[0],'c':tmpz[1], 'd':tmpz[2], \n 'e':tmpz[3], 'f':tmpz[4], 'g':tmpz[5], 'h':tmpz[6],\n 'btime':timez[0],'ctime':timez[1], 'dtime':timez[2], \n 'etime':timez[3], 'ftime':timez[4], 'gtime':timez[5], 'htime':timez[6],\n 'q': qvals[knum]}, ignore_index=True)\n\n\n```\n\n\n```python\nif regen_Q_values ==True:\n answers['k/Q'] =0.3 / answers['q']\n answers.to_pickle(assorteddata + \"timescale_plot_data.pkl\") # save results\n```\n\n\n```python\nanswers = pd.read_pickle(assorteddata + \"timescale_plot_data.pkl\") # shortcut to make plot without generating results\n```\n\n\n```python\nassorteddata + \"timescale_plot_data.pkl\"\n```\n\n\n\n\n '/Users/jcbecker/Documents/GitHub/trappist1/AssortedData/timescale_plot_data.pkl'\n\n\n\n\n```python\nfigsize(6,4)\ncolors_for_this = [\"#666666\",\"#004C70\",\"#0093D1\",\"#F2635F\",\"#F4D00C\",\"#E0A025\",\"k\"]\n\nfor i, plano in enumerate(['b','c','d','e']):\n x = 3/2. * 1./answers['k/Q']\n plot(x, answers[plano +'time'] /1e9, color=colors_for_this[i], linewidth = 2, label=plano)\n fill_between(np.append(x.values, 10000), np.append((answers[plano +'time'].fillna(10e9).values/1e9),10), color= colors_for_this[i], alpha=0.03)\n\nytimes = np.linspace(0,9)\npap_est = 1.2*10**3*(ytimes/5)\nfill_between(np.append(pap_est, 10000), np.append(ytimes,10), color= '#458B00', alpha=0.08)\nplot(pap_est, ytimes, '#3B5323', linewidth=1.4,linestyle=\":\", alpha=0.6)\nhlines(7.6, 0, 5000, 'SlateGrey', linestyle=\":\")\nxscale('log')\nxlim(1, 5e3)\nylim(0.1, 8)\nxticks(fontsize=14)\nyticks(fontsize=14)\nxlabel(\"$Q' = 3 Q / (2 k_{2})$\", fontsize=17)\nylabel(\"$t_{ecc, damp}$ (Gyr)\", fontsize=17)\n\n\n# the arrows are added manually at the intersection point between curve and 7.6 Gyr line\n\n\nendt = [600, 125, 54, 3] # initial guesses\nfor i, plano in enumerate(['b','c','d','e']):\n x = 3/2. * 1./answers['k/Q']\n y = scipy.interpolate.interp1d(x, answers[plano +'time']/1e9)\n xtest = np.linspace(1,endt[i], 5000)\n ynew = y(xtest)\n plot(xtest[np.argmin(np.abs(ynew - 7.6))], 7.6, marker ='>', color=colors_for_this[i],markersize = 10,)\n \ntext(2000, 5.7,\"Constraint from \\nPapaloizou et al. 2018\", color = '#3B5323', fontsize=14, rotation=90) \nlegend(fontsize= 15, loc='lower center')\n \nsavefig(paperFigures + \"timescale_plot_t1.pdf\", dpi=150, bbox_inches = 'tight')\n```\n\n# Make TRAPPIST array ready to print to put in the paper\nWe have computed the mass loss rate due to the measured UV and extrapolated XUV flux. Now, we want to print a table of these values for the paper. We will also put the planetary properties in desired units.\n\n\n```python\nprint_cols = ['Planet', 'Period(days)','Radius(Rearth)', 'Mass(Mearth)', 'e', 'XUV_flux(erg,cm-2,s-1)', \n u'mass_loss(g/s)','mass_loss(mearth/Gyr)'] # these will go in paper, in order\ntrappist1['mass_loss(mearth/Gyr)'] = trappist1['mass_loss(g/s)'] * 5.2804674*10**(-21.) * 1.0e9# Earth masses per year * gyr/ yr\ntrappist1.sort_values('Period(days)')[print_cols].to_latex(index = False)\n```\n\n\n\n\n '\\\\begin{tabular}{lrrrrrrr}\\n\\\\toprule\\n Planet & Period(days) & Radius(Rearth) & Mass(Mearth) & e & XUV\\\\_flux(erg,cm-2,s-1) & mass\\\\_loss(g/s) & mass\\\\_loss(mearth/Gyr) \\\\\\\\\\n\\\\midrule\\n TRAPPIST-1 b & 1.510876 & 1.121 & 1.017 & 0.00622 & 1792.808629 & 5.061533e+08 & 0.002673 \\\\\\\\\\n TRAPPIST-1 c & 2.421807 & 1.095 & 1.156 & 0.00654 & 955.836690 & 2.212689e+08 & 0.001168 \\\\\\\\\\n TRAPPIST-1 d & 4.049959 & 0.784 & 0.297 & 0.00837 & 481.596689 & 1.592676e+08 & 0.000841 \\\\\\\\\\n TRAPPIST-1 e & 6.099043 & 0.910 & 0.772 & 0.00510 & 278.805922 & 5.547038e+07 & 0.000293 \\\\\\\\\\n TRAPPIST-1 f & 9.205585 & 1.046 & 0.934 & 0.01007 & 161.008678 & 4.021145e+07 & 0.000212 \\\\\\\\\\n TRAPPIST-1 g & 12.354473 & 1.148 & 1.148 & 0.00208 & 108.795402 & 2.922446e+07 & 0.000154 \\\\\\\\\\n TRAPPIST-1 h & 18.767953 & 0.773 & 0.331 & 0.00567 & 62.324396 & 1.772641e+07 & 0.000094 \\\\\\\\\\n\\\\bottomrule\\n\\\\end{tabular}\\n'\n\n\n\n\n```python\ntrappist1.to_csv(assorteddata + \"TRAPPIST_derived.csv\")\n```\n\n# Check luminosity evolution of a star of this mass from Baraffe models\n\n\n```python\n# Luminosity evolution: BHC 2015\nfigsize(6,4)\nnamez = ['M/Ms','log t(yr)','Teff','L/Ls','g','R/Rs','Log(Li/Li0)','log Tc','log ROc', 'Mrad', 'Rrad', 'k2conv', 'k2rad']\nmodel = pd.read_csv(assorteddata + \"BHAC15_tracks+structure.txt\", dtype ='float',names = namez, skiprows=50, delim_whitespace=True)\nidx = (model['M/Ms'] == 0.0900)\n# L/Ls: log luminosity in units of solar luminosity (value used Ls=3.839d+33)\nplot(np.power(10,model.loc[idx]['log t(yr)'].values) / 1e9, np.power(10,model.loc[idx]['L/Ls']),\n linewidth = 3, label = \"BHC 2015 Model\")\nxlabel(\"Time (Gyr)\", fontsize=17)\nylabel(\"$L_{bol}$ / L${\\odot}$\", fontsize=17)\nxticks(fontsize=14)\nyticks(fontsize=14)\nplot(7.6, trappist_lum, markersize = 15, color = \"Red\", linestyle='None',marker=\"*\", label=\"TRAPPIST-1\")\nyscale('log')\nxscale(\"log\")\nlegend()\ntitle(\"Baraffe 2015, M = 0.09 Msun\", fontsize=17)\ntitle(\"0.09 M$_{\\odot}$ Stellar Evolution\", fontsize = 17)\nsavefig(extraFigures + \"BHAC_stellarmodel.pdf\", dpi=150, bbox_inches = 'tight')\n```\n\n# Plot results from [VPlanet](https://github.com/VirtualPlanetaryLaboratory/vplanet) runs. \nWe know ocean retention will depend on the XUV flux. \nTo test how our derived XUV luminosity ratio affects the ocean evolution, we will run two integrations with VPlanet. We will also try running integrations with different values of Q, to see if that matters.\n\n\n\n```python\nplanets = ['b','c','d','e','f','g','h']\nnamz = ['Time','SurfWaterMass','OxygenMass','Ecc']\nfigsize(6,4)\nfor i, planet in enumerate(planets):\n file_loc = current_dir + \"VPlanetRuns/run01/trappist1.%s.forward\" % planet\n data = pd.read_csv(file_loc, names = namz, delim_whitespace=True)\n plot(data['Time'], data['SurfWaterMass'] / data['SurfWaterMass'].values[0], label='_nolegend_', linewidth = 4, alpha=0.3, c = colors_for_this[i])\n plot(data['Time'], data['SurfWaterMass'] / data['SurfWaterMass'].values[0], label=planet, linewidth = 2, c = colors_for_this[i])\nxscale(\"log\")\nlegend(fontsize=12, loc='upper left')\nxlabel(\"Time (years)\", fontsize=17)\nylabel(\"Fractional Ocean Retention\", fontsize=17)\nyticks(fontsize=14)\nxticks(fontsize=14)\nylim(0,1)\nannotate(\"$L_{XUV} = 3.7 \\cdot 10^{-4}$ L$_{bol}$\\n$Q'$ = 600\", (1e6, 0.05), fontsize = 14)\nsavefig(paperFigures + \"trappist_water_midQ_trueLXUV.pdf\", dpi=150, bbox_inches = 'tight')\n\n```\n\n\n```python\nplanets = ['b','c','d','e','f','g','h']\nnamz = ['Time','SurfWaterMass','OxygenMass','Ecc']\nfigsize(6,4)\nfor i, planet in enumerate(planets):\n file_loc = current_dir + \"VPlanetRuns/run02/trappist1.%s.forward\" % planet\n data = pd.read_csv(file_loc, names = namz, delim_whitespace=True)\n plot(data['Time'], data['SurfWaterMass'] / data['SurfWaterMass'].values[0], label='_nolegend_', linewidth = 4, alpha=0.3, c = colors_for_this[i])\n plot(data['Time'], data['SurfWaterMass'] / data['SurfWaterMass'].values[0], label=planet, linewidth = 2, c = colors_for_this[i])\nxscale(\"log\")\nlegend(fontsize=12, loc='upper left')\nxlabel(\"Time (years)\", fontsize=17)\nylabel(\"Fractional Ocean Retention\", fontsize=17)\nyticks(fontsize=14)\nxticks(fontsize=14)\nylim(0,1)\nannotate(\"$L_{XUV} = 3.7 \\cdot 10^{-4}$ L$_{bol}$\\n$Q'$ = 60\", (1e6, 0.05), fontsize = 14)\nsavefig(extraFigures + \"trappist_water_midQ_lowLXUV.pdf\", dpi=150, bbox_inches = 'tight')\n\n```\n\n\n```python\nplanets = ['b','c','d','e','f','g','h']\nnamz = ['Time','SurfWaterMass','OxygenMass','Ecc']\nfigsize(6,4)\nfor i, planet in enumerate(planets):\n file_loc = current_dir + \"VPlanetRuns/run03/trappist1.%s.forward\" % planet\n data = pd.read_csv(file_loc, names = namz, delim_whitespace=True)\n plot(data['Time'], data['SurfWaterMass'] / data['SurfWaterMass'].values[0], label='_nolegend_', linewidth = 4, alpha=0.3, c = colors_for_this[i])\n plot(data['Time'], data['SurfWaterMass'] / data['SurfWaterMass'].values[0], label=planet, linewidth = 2, c = colors_for_this[i])\nxscale(\"log\")\nlegend(fontsize=12, loc='upper left')\nxlabel(\"Time (years)\", fontsize=17)\nylabel(\"Fractional Ocean Retention\", fontsize=17)\nyticks(fontsize=14)\nxticks(fontsize=14)\nylim(0,1)\nannotate(\"$L_{XUV} = 1 \\cdot 10^{-4}$ L$_{bol}$\\n$Q'$ = 600\", (1e6, 0.05), fontsize = 14)\nsavefig(extraFigures + \"trappist_water_midQ_lowLXUV.pdf\", dpi=150, bbox_inches = 'tight')\n\n```\n\n# Check luminosity evolution computed by VPlanet\n\n\n```python\nplanets = ['b','c','d','e','f','g','h']\nnamz = ['Time', 'Luminosity', 'LXUVTot' ,'Temperature', 'HZLimRecVenus', 'HZLimRunaway', 'HZLimMaxGreenhouse', 'HZLimEarlyMars']\nfigsize(6,4)\nfile_loc = current_dir + \"VPlanetRuns/AtmEsc+EqTide/trappist1.star.forward\"\ndata = pd.read_csv(file_loc, names = namz, delim_whitespace=True)\nplot(data['Time'] / 1e9, data['LXUVTot'] / trappist_lum, linewidth = 2, c = 'Purple', label=\"$L_{XUV}$ (Total) Computed by VPlanet\")\nxscale(\"log\")\n#legend(fontsize=12, loc='upper left')\nxlabel(\"Time (Gyr)\", fontsize=17)\nylabel(\"$L_{XUV}$ / $L_{bol}$\", fontsize=17)\nyticks(fontsize=14)\nxticks(fontsize=14)\n#ylim(0,1)\nyscale(\"log\")\n#annotate(\"$L_{XUV} = 3 \\cdot 10^{-4}$ L$_{bol}$\\n$Q'$ = 600\", (1e6, 0.05), fontsize = 14)\nerrorbar(7.6, (xuvmiddle) / ( solar_lum* trappist_lum), yerr = (xuvmax - xuvmiddle) / ( solar_lum* trappist_lum), markersize = 10, color = \"Red\", linestyle='None',marker=\".\", label=\"Derived value for TRAPPIST-1\")\nlegend()\nsavefig(extraFigures + \"star_test.pdf\", dpi=150, bbox_inches = 'tight')\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "11ce305edc6651e8696782c6289615d6a7f76441", "size": 589229, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": ".ipynb_checkpoints/MakeFigures-checkpoint.ipynb", "max_stars_repo_name": "jxcbecker/trappist1", "max_stars_repo_head_hexsha": "cf99216928c92ffa308ecf9482d3cdf91db37d43", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": ".ipynb_checkpoints/MakeFigures-checkpoint.ipynb", "max_issues_repo_name": "jxcbecker/trappist1", "max_issues_repo_head_hexsha": "cf99216928c92ffa308ecf9482d3cdf91db37d43", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": ".ipynb_checkpoints/MakeFigures-checkpoint.ipynb", "max_forks_repo_name": "jxcbecker/trappist1", "max_forks_repo_head_hexsha": "cf99216928c92ffa308ecf9482d3cdf91db37d43", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 356.0296072508, "max_line_length": 62964, "alphanum_fraction": 0.9140809431, "converted": true, "num_tokens": 18912, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191214879992, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.4281992181870154}} {"text": "# PyFolding SI Notebook 8\n---\n\n[Authors] ERGM & ARL\n\n---\n\n## This is to Global fit Equilibrium Curves from a dimeric protein to dimeric equilibrium models ##\n\nIn this notebook we will show how equilbrium folding data from dimeric proteins can be imported into a notebook and fitted to dimeric Equilibrium denaturation models using global fits over multiple datasets.\n\nIf you are less script/computer orientated, you can simply change the data paths and other parameters for your proteins and re-run the jupyter notebook ( \"Kernal/Restart & Run all\" from the menu above).\n\n---\n\n## Data Format\n\nPlease see PyFolding SI Notebooks 1 and 2 for the format your data has to be in to enable this type of analysis.\nRemember for Ising Model Analysis here each protein dataset (equilibrium denaturation curve) must have its own .csv\n\n---\n\n#### First off lets load pyfolding & pyplot into this ipython notebook (pyplot allows us to plot more complex figures of our results):\n\n---\n\n\n```python\n# use this command to tell Jupyter to plot figures inline with the text\n%matplotlib inline\n\n# import pyfolding, the pyfolding models and ising models\nimport pyfolding\nfrom pyfolding import *\n\n# import the package for plotting, call it plt\nimport matplotlib.pyplot as plt\n\n# import numpy as well\nimport numpy as np\n```\n\n\n \n\n\n PyFolding: Jupyter autoscrolling has been disabled\n\n\n---\n\n#### Now, we need to load some data to analyse.\n\nI will import the equilibrium denaturations of dimeric tr-LcrH at a series of protein concentrations from:\n\n`Singh S.K., Boyle A.L. & Main E.R. (2013) \u201cLcrH, a class II chaperone from the type three secretion system, has a highly flexible native structure.\u201d J Biol Chem., 288, (6) 4048-55.`\n\nI will load 4 denaturation curves that correspond tr-LcrH at the following protein concentrations:\n\nProtein |Filename |Total Protein Concentration\n:--------: |:-------: |:-------:\ntr-LcrH | 50uM_1.csv | 50 uM\ntr-LcrH | 50uM_2.csv | 50 uM\ntr-LcrH | 80uM_1.csv | 80 uM\ntr-LcrH | 80uM_2.csv | 80 uM\n\n---\n\n\n\n```python\n# we will load all of the data together, as follows:\n\n# arguments are \"path\", \"filename\"\npth = \"../examples/LcrH\"\n\n# this is a set of commands to automate loading the data \n# for each denaturation \nfn = [\"50uM_1.csv\",\"50uM_2.csv\",\"80uM_1.csv\",\"80uM_2.csv\"]\n\n# Here we are loading all the curves in a list called \"proteins\" \n# and assigning them names\nproteins = [pyfolding.read_equilibrium_data(pth,f) for f in fn] \n\n# also store the total protein conentration for each denaturation\nPt = [50e-6, 50e-6, 80e-6, 80e-6]\n\n```\n\n\n```python\n# Set temperature to 25.00\u00b0C\n# (NOTE: Careful, this sets the temperature for all subsequent calculations)\npyfolding.set_temperature(25.)\n```\n\n Set temperature to 25.00\u00b0C\n (NOTE: Careful, this sets the temperature for all subsequent calculations)\n\n\n\n```python\n# the following commands plot all the data curves on one plot\n\nplt.figure(figsize=(10,6)) \nfor c in proteins: \n plt.plot(c.x, c.y, '.')\n\n# the following commands plot all the data curves on one plot, \n# where \"loc\" command determines where the legend goes. \nplt.legend([c.ID for c in proteins], loc='best')\nplt.title(\"LcrH Equilm Curves\")\nplt.xlim([0, 8]) # x axis from 0 to 8\nplt.show()\n\n\n```\n\n\n```python\n# Command imports pyfolding models\nfrom pyfolding.models import *\n\n# command lists models\nlist_models()\n\n# After the model name:\n#'Verified: True' model has been rigourously and it functions as expected.\n#'Verified:False' signifies that the model has not been rigourously tested.\n \n```\n\n\n\n\n [('ChevronPolynomialFit', 'Verified: True'),\n ('HeteropolymerIsingEquilibrium', 'Verified: False'),\n ('HomozipperIsingEquilibrium', 'Verified: True'),\n ('ParallelTwoStateChevron', 'Verified: False'),\n ('ParallelTwoStateUnfoldingChevron', 'Verified: False'),\n ('TemplateModel', 'Verified: False'),\n ('ThreeStateChevron', 'Verified: True'),\n ('ThreeStateDimericIEquilibrium', 'Verified: True'),\n ('ThreeStateEquilibrium', 'Verified: True'),\n ('ThreeStateFastPhaseChevron', 'Verified: True'),\n ('ThreeStateMonoIEquilibrium', 'Verified: True'),\n ('ThreeStateSequentialChevron', 'Verified: True'),\n ('TwoStateChevron', 'Verified: True'),\n ('TwoStateChevronMovingTransition', 'Verified: True'),\n ('TwoStateDimerEquilibrium', 'Verified: True'),\n ('TwoStateEquilibrium', 'Verified: True'),\n ('TwoStateEquilibriumSloping', 'Verified: True')]\n\n\n\n\n```python\n# We are going to fit this data to 3 state denaturation that unfolds via \n# a dimeric intermediate (as per the J.B.C. paper)\n# Lets print the equation & Info\n\nmodels.ThreeStateDimericIEquilibrium().info()\n```\n\n\n$$\\begin{equation} \\\n \\begin{aligned} \\\n & \\Upsilon_{rel} = \\Upsilon_N F_N + \\Upsilon_I F_I + \\Upsilon_D F_D \\\\ \\\n \\text{expanded:} \\\\ \\\n & \\Upsilon_{rel} = \\Upsilon_N \\cdot \\frac{2PtF_D^2} {K_1 K_2} + \\Upsilon_I \\frac{2PtF_D^2} {K_2} + \\Upsilon_D * (F_D) \\\\ \\\n \\\\ \\\n \\text{where:} \\\\ \\\n & F_D = \\frac {- K_1 K_2 + \\sqrt{((K_1 K_2)^2 + 8(1+K_1)(K_1 K_2)Pt)}} {4Pt (1 + K_1)} \\\\ \\\n & K_1 = \\exp \\frac{-\\Delta G_{H_20}^1 + m_1 x} {RT} \\\\ \\\n & K_2 = \\exp \\frac{-\\Delta G_{H_20}^2 + m_2 x} {RT}\\\n \\end{aligned}\\\n \\end{equation}$$\n\n\n Three State model for a dimer denaturation Equilibrium - Dimeric Intermediate.\n \n Folding Scheme:\n N2 <-> I2 <-> 2D\n \n Params:\n Y_rel = spectroscopic signal at a given concentration of urea\n Y_N = spectroscopic signal for native state\n Y_D = spectroscopic signal for denatured state\n Y_I = spectroscopic signal for intermediate state\n F_D = fraction denatured monomers\n F_N = fraction native dimers\n F_I = fraction intermediate dimers\n Pt = total protein concentration. This variable needs to be set per denaturation curve.\n K1 = equilibrium contstant of unfolding native to intermediate state\n K2 = equilibrium contstant of unfolding intermediate to denatured state\n DG1 = stability of native state relative to intermediate state\n m1 = m-value of native to intermediate transition\n DG2 = stability of intermediate state relative to denatured state\n m2 = m-value of intermediate to denatured transition\n x = denaturant concentration (M)\n R = Universal Gas Constant (kcal.mol-1.K-1)\n T = Temperature (Kelvin)\n \n Reference:\n Mallam and Jackson. Folding studies on a knotted protein.\n Journal of Molecular Biology (2005) vol. 346 (5) pp. 1409-1421\n \n\n\n---\n\n### Automatic global fitting to the Three State Dimeric Intermediate Equilibrium model\n\nWe could fit individual denaturations to this model. However, one aspect of the model is that it should be able to fit globally to a number of denaturations. We will try to fit this with the Three State Dimeric Intermediate Equilibrium model\n\n---\n\n\n```python\nglobal_fit = core.GlobalFit()\nglobal_fit.fit_funcs = [models.ThreeStateDimericIEquilibrium \n for i in xrange(len(proteins))]\nglobal_fit.constants = [(('Pt',c),) for c in Pt]\nglobal_fit.shared = ['DG1','DG2','m1','m2'] #\nglobal_fit.x = [p.x for p in proteins]\nglobal_fit.y = [p.y for p in proteins]\nglobal_fit.ID = [p.ID for p in proteins]\n\nglobal_fit.initialise()\n```\n\n\n```python\nprint global_fit.params.keys(), len(global_fit.params.keys())\n```\n\n ['DG2', 'DG1', 'm1', 'm2', 'Y_N_{50uM_1}', 'Y_I_{50uM_1}', 'Y_D_{50uM_1}', 'Y_N_{50uM_2}', 'Y_I_{50uM_2}', 'Y_D_{50uM_2}', 'Y_N_{80uM_1}', 'Y_I_{80uM_1}', 'Y_D_{80uM_1}', 'Y_N_{80uM_2}', 'Y_I_{80uM_2}', 'Y_D_{80uM_2}'] 16\n\n\n\n```python\n# this commands gives our initial parameters for fitting\np0 = [5.,1.7,1.,1.7,\n -200,-50,-10,\n -200,-50,-10,\n -200,-50,-10,\n -200,-100,-10]\n\n# this command gives fitting constraints\nb = bounds=((0.,0.,-3.,-3.,\n -300,-300,-300,\n -300,-300,-300,\n -300,-300,-300,\n -300,-300,-300),\n (20.,20.,3.,3.,\n 0,0,0,\n 0,0,0,\n 0,0,0,\n 0,0,0))\n\n# this commands checks that our inputs for the initial parameters and \n# constraints (bounds) above\n# match the number of our variables\nprint len(b[0]), len(b[1]), len(p0)\n\n#this commands tell pyfolding to fit our data\nout, covar = global_fit.fit( p0=p0, bounds=b )\n\n# print out the results of the fit\nfor result in global_fit.results:\n result.display()\n```\n\n 16 16 16\n ================================================================================\n Fitting results\n ================================================================================\n ID: 50uM_1\n Model: ThreeStateDimericIEquilibrium\n Optimiser: pyfolding.GlobalFit and scipy.optimize.curve_fit\n Temperature: 25.00\u00b0C\n \n (s) DG1 1.66967 \u00b1 0.19094 \t 95% CI[ 1.62134, 1.71800]\n (s) m1 1.07183 \u00b1 0.11736 \t 95% CI[ 1.04213, 1.10154]\n (s) DG2 13.91136 \u00b1 0.63253 \t 95% CI[ 13.75126, 14.07146]\n (s) m2 1.71686 \u00b1 0.11881 \t 95% CI[ 1.68679, 1.74694]\n (f) Y_N -133.91502 \u00b1 2.65244 \t 95% CI[ -134.58638, -133.24366]\n (f) Y_I -75.49314 \u00b1 3.07308 \t 95% CI[ -76.27097, -74.71531]\n (f) Y_D -6.57648 \u00b1 1.04424 \t 95% CI[ -6.84079, -6.31218]\n (c) Pt 0.00005\n --------------------------------------------------------------------------------\n R^2: \t0.99661\n DOF: \t133\n |SS|: \t1.81e+03\n ================================================================================\n \n \n ================================================================================\n Fitting results\n ================================================================================\n ID: 50uM_2\n Model: ThreeStateDimericIEquilibrium\n Optimiser: pyfolding.GlobalFit and scipy.optimize.curve_fit\n Temperature: 25.00\u00b0C\n \n (s) DG1 1.66967 \u00b1 0.19094 \t 95% CI[ 1.62134, 1.71800]\n (s) m1 1.07183 \u00b1 0.11736 \t 95% CI[ 1.04213, 1.10154]\n (s) DG2 13.91136 \u00b1 0.63253 \t 95% CI[ 13.75126, 14.07146]\n (s) m2 1.71686 \u00b1 0.11881 \t 95% CI[ 1.68679, 1.74694]\n (f) Y_N -135.06345 \u00b1 2.51075 \t 95% CI[ -135.69895, -134.42796]\n (f) Y_I -72.07032 \u00b1 3.03212 \t 95% CI[ -72.83778, -71.30285]\n (f) Y_D -5.47747 \u00b1 1.06953 \t 95% CI[ -5.74818, -5.20676]\n (c) Pt 0.00005\n --------------------------------------------------------------------------------\n R^2: \t0.99375\n DOF: \t133\n |SS|: \t1.81e+03\n ================================================================================\n \n \n ================================================================================\n Fitting results\n ================================================================================\n ID: 80uM_1\n Model: ThreeStateDimericIEquilibrium\n Optimiser: pyfolding.GlobalFit and scipy.optimize.curve_fit\n Temperature: 25.00\u00b0C\n \n (s) DG1 1.66967 \u00b1 0.19094 \t 95% CI[ 1.62134, 1.71800]\n (s) m1 1.07183 \u00b1 0.11736 \t 95% CI[ 1.04213, 1.10154]\n (s) DG2 13.91136 \u00b1 0.63253 \t 95% CI[ 13.75126, 14.07146]\n (s) m2 1.71686 \u00b1 0.11881 \t 95% CI[ 1.68679, 1.74694]\n (f) Y_N -226.85435 \u00b1 3.62134 \t 95% CI[ -227.77095, -225.93775]\n (f) Y_I -126.95009 \u00b1 4.24583 \t 95% CI[ -128.02476, -125.87542]\n (f) Y_D -10.23377 \u00b1 1.20873 \t 95% CI[ -10.53972, -9.92783]\n (c) Pt 0.00008\n --------------------------------------------------------------------------------\n R^2: \t0.99589\n DOF: \t133\n |SS|: \t1.81e+03\n ================================================================================\n \n \n ================================================================================\n Fitting results\n ================================================================================\n ID: 80uM_2\n Model: ThreeStateDimericIEquilibrium\n Optimiser: pyfolding.GlobalFit and scipy.optimize.curve_fit\n Temperature: 25.00\u00b0C\n \n (s) DG1 1.66967 \u00b1 0.19094 \t 95% CI[ 1.62134, 1.71800]\n (s) m1 1.07183 \u00b1 0.11736 \t 95% CI[ 1.04213, 1.10154]\n (s) DG2 13.91136 \u00b1 0.63253 \t 95% CI[ 13.75126, 14.07146]\n (s) m2 1.71686 \u00b1 0.11881 \t 95% CI[ 1.68679, 1.74694]\n (f) Y_N -207.47574 \u00b1 3.31055 \t 95% CI[ -208.31368, -206.63780]\n (f) Y_I -122.90644 \u00b1 3.83992 \t 95% CI[ -123.87836, -121.93451]\n (f) Y_D -16.04308 \u00b1 1.67988 \t 95% CI[ -16.46827, -15.61788]\n (c) Pt 0.00008\n --------------------------------------------------------------------------------\n R^2: \t0.99668\n DOF: \t133\n |SS|: \t1.81e+03\n ================================================================================\n \n \n\n\n---\n\n### To plot the fitted data we need to use this script:\n\n---\n\n\n```python\nresults = global_fit.results\nplt.figure(figsize=(14,8))\nfor i, p in enumerate(proteins):\n plt.plot(p.x, p.y, 'ko')\n plt.plot(results[i].x_fit, results[i].y_fit, '-', label=p.ID)\nplt.legend()\nplt.xlabel(p.denaturant_label, fontsize=constants.FONT_SIZE)\nplt.ylabel('Elipticity @ 222 nm', fontsize=constants.FONT_SIZE)\nplt.show()\n```\n\n### We can also simulate the data as follows:\n\nThis is also a good way to work out what initial parameters might work well\n\n\n```python\n#Lets first choose some values for simulation:\nDG1 = 1.7\nDG2 = 14.\nm1 = 1.\nm2 = 1.7\nY_N = -200\nY_I = -120\nY_D = -10\nPt = 80e-6\n\n# Then lets simulate what curve we would obtain with these values:\nx = np.linspace(0.,10.,100)\nm = models.ThreeStateDimericIEquilibrium()\ny = m(x, DG1, m1, DG2, m2, Y_N, Y_I, Y_D, Pt)\n\n#Then lets plot the simulation against the data we have:\nplt.figure()\nfor i, p in enumerate(proteins):\n plt.plot(p.x, p.y, 'ko')\nplt.plot(x,y,'r-')\nplt.show()\nprint y[0]\n```\n\n---\n\n#### End of this Notebook.\n\n---\n", "meta": {"hexsha": "5a2892177ef45bb3401f24b2ef63ca4b856b00ae", "size": 128475, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/SI8_FoldingModels3.ipynb", 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YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.4280755136733799}} {"text": "# \u7b2c\u4e8c\u5341\u8bb2\uff1a\u7b56\u7565\u641c\u7d22\u7b97\u6cd5\n\n\u56de\u5fc6[\u4e0a\u4e00\u8bb2](chapter19.ipynb)\u7684\u5185\u5bb9\uff0c\u5b9e\u9645\u4e0a\u4e0a\u4e00\u8bb2\u7684\u4f8b\u5b50\u5c31\u662f\u4e00\u4e2a\u90e8\u5206\u53ef\u89c2\u6d4bMDP\u7684\u7279\u4f8b\u3002\n\n\u5728\u539fLQR\u95ee\u9898\u4e2d\u6709\u6a21\u578b$s_{t+1}=As_t+Ba_t+w_t$\uff0c\u6211\u4eec\u6539\u53d8\u4e86\u6761\u4ef6\uff0c\u5047\u8bbe\u53ea\u80fd\u89c2\u6d4b\u5230$y_t=Cs_t+v_t$\uff08\u4e0d\u80fd\u76f4\u63a5\u89c2\u6d4b\u5230\u72b6\u6001\uff0c\u53ea\u80fd\u89c2\u6d4b\u5230\u67d0\u4e9b\u5173\u4e8e\u72b6\u6001\u7684\u51fd\u6570\uff0c\u5e76\u9700\u8981\u6839\u636e\u8fd9\u4e9b\u89c2\u6d4b\u503c\u5bf9\u6700\u4f18\u52a8\u4f5c\u4f5c\u51fa\u5224\u65ad\uff09\u3002\n\n\u5728\u72b6\u6001\u5b8c\u5168\u53ef\u89c2\u6d4b\u7684\u60c5\u5f62\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u6839\u636e$a_t=L_ts_t$\u9009\u62e9\u52a8\u4f5c\uff1b\u800c\u5728\u90e8\u5206\u53ef\u89c2\u6d4b\u7684\u4f8b\u5b50\u4e2d\uff0c\u6211\u4eec\u4f1a\u4f7f\u7528$s_{t\\mid t}$\u4f30\u8ba1$s_t$\uff0c\u7136\u540e\u4f7f\u7528Kalman\u6ee4\u6ce2\u7684\u601d\u60f3$s_{t\\mid y_0,\\cdots,y_t}\\sim\\mathcal N\\left(s_{t\\mid t},\\varSigma_{t\\mid t}\\right)$\u8ba1\u7b97\u8fd9\u4e2a\u4f30\u8ba1$s_{t\\mid t}$\u3002\n\n\u6700\u540e\uff0c\u6211\u4eec\u4f7f\u7528$a_t=L_ts_{t\\mid t}$\u9009\u62e9\u6700\u4f18\u52a8\u4f5c\u3002\u4e8b\u5b9e\u4e0a\u8fd9\u4e2a\u7b97\u6cd5\u8ba9\u6211\u4eec\u80fd\u591f\u5728\u4e0d\u5b8c\u5168\u6e05\u695a\u7cfb\u7edf\u6bcf\u4e00\u4e2a\u72b6\u6001\u7684\u524d\u63d0\u4e0b\u9009\u62e9\u5c3d\u53ef\u80fd\u597d\u7684\u7b56\u7565\uff0c\u800c\u5728\u4e00\u4e2a\u90e8\u5206\u53ef\u89c2\u6d4bMDP\u4e2d\u6c42\u6700\u4f18\u7b56\u7565\u662f\u4e00\u4e2aNP\u56f0\u96be\u95ee\u9898\u3002\n\n### 10.3 \u90e8\u5206\u53ef\u89c2\u6d4bMDP\uff08POMDP: Partial Observed MDP\uff09\n\n\u6211\u4eec\u73b0\u5728\u7ed9\u51faPOMDP\u7684\u6b63\u5f0f\u63cf\u8ff0\u2014\u2014\u4e03\u5143\u7ec4$(S,A,Y,\\{P_{sa}\\},\\{O_s\\},T,R)$\uff1a\n* $Y$\u662f\u53ef\u89c2\u6d4b\u5230\u7684\u503c\u7684\u96c6\u5408\uff1b\n* $O$\u662f\u53ef\u89c2\u6d4b\u5230\u7684\u503c\u7684\u5206\u5e03\uff1b\n \n \u4e5f\u5c31\u662f\u5728\u6bcf\u4e2a$s_t$\u4e0b\u90fd\u80fd\u591f\u89c2\u6d4b\u5230\u67d0\u4e2a\u670d\u4ece$y_t\\sim O_{s_t}$\u7684\u968f\u673a\u53d8\u91cf\u3002\u4e0a\u4e00\u8bb2\u4e2d\u6211\u4eec\u4ecb\u7ecd\u7684\u5728\u7ebf\u6027\u52a8\u529b\u5b66\u7cfb\u7edf\u4e2d\u7684\u7279\u4f8b\u53ef\u4ee5\u4f7f\u7528Kalman\u8fc7\u6ee4\u7b97\u6cd5\uff0c\u4ece\u89c2\u6d4b\u503c\u4e2d\u4f30\u8ba1\u72b6\u6001\uff0c\u518d\u901a\u8fc7\u72b6\u6001\u8ba1\u7b97\u6700\u4f18\u7b56\u7565\u3002\u7136\u800c\u8fd9\u4e2a\u7b97\u6cd5\u4ec5\u5bf9\u4e0a\u4e00\u8bb2\u90a3\u4e2a\u7684\u7279\u4f8b\uff08\u4e0a\u4e00\u8bb2\u7684\u4f8b\u5b50\u4e5f\u662fPOMDP\u7684\u7279\u4f8b\uff09\u6709\u6548\uff0c\u4e0d\u8fc7\u8fd9\u4e2a\u7b97\u6cd5\u5bf9\u666e\u901a\u7684POMDP\u95ee\u9898\u4e5f\u6709\u4e00\u5b9a\u7684\u6548\u679c\uff0c\u53ea\u662f\u5bf9\u8be5\u95ee\u9898\u8fd9\u4e2a\u7b97\u6cd5\u4e0d\u4e00\u5b9a\u662f\u6700\u4f18\u7684\u7b97\u6cd5\u3002\n\n## 12. \u7b56\u7565\u641c\u7d22\uff08Policy Search\uff09\n\n\u518d\u6765\u4ecb\u7ecd\u4e00\u4e2a\u4e0d\u540c\u7684\u5f3a\u5316\u5b66\u4e60\u7b97\u6cd5\u2014\u2014\u7b56\u7565\u641c\u7d22\u7b97\u6cd5\uff0c\u770b\u540d\u5b57\u5c31\u5927\u6982\u53ef\u4ee5\u731c\u51fa\uff0c\u5b83\u53ef\u4ee5\u88ab\u7528\u4e8e\u5b8c\u5168\u53ef\u89c2\u6d4bMDP\uff08Fully Observed MDP\uff09\u6216\u90e8\u5206\u53ef\u89c2\u6d4bMDP\u7684\u6700\u4f18\u7b56\u7565\u6c42\u89e3\u3002\u4e0b\u9762\u6211\u4eec\u5148\u6765\u4ecb\u7ecd\u7b56\u7565\u641c\u7d22\u7b97\u6cd5\u5728\u5b8c\u5168\u53ef\u89c2\u6d4bMDP\u4e2d\u7684\u5e94\u7528\u3002\uff08\u5728\u540e\u9762\u6211\u4eec\u4f1a\u4ecb\u7ecd\u5982\u4f55\u5c06\u7b56\u7565\u641c\u7d22\u5e94\u7528\u5728POMDP\u4e0a\uff0c\u4f46\u8fd9\u5e76\u4e0d\u80fd\u4fdd\u8bc1\u7b97\u6cd5\u4f1a\u627e\u5230\u5168\u5c40\u6700\u4f18\u89e3\uff0c\u7b56\u7565\u641c\u7d22\u901a\u5e38\u5728POMDP\u4e2d\u627e\u5230\u7684\u662f\u5c40\u90e8\u6700\u4f18\u7b56\u7565\u3002\u89e3\u51b3POMDP\u662f\u5f02\u5e38\u56f0\u96be\u7684\uff0c\u800c\u7b56\u7565\u641c\u7d22\u7b97\u6cd5\u5728\u5b9e\u8df5\u4e2d\u88ab\u8bc1\u660e\u662f\u6700\u597d\u7684POMDP\u89e3\u51b3\u7b97\u6cd5\u4e4b\u4e00\u3002\uff09\n\n* \u9996\u5148\u5b9a\u4e49$\\varPi$\u4e3a\u7b56\u7565\u7684\u96c6\u5408\uff0c\u800c\u6211\u4eec\u7684\u7b97\u6cd5\u5c31\u662f\u5728$\\varPi$\u4e2d\u641c\u7d22$\\pi\\in\\varPi$\u3002\u53ef\u4ee5\u7c7b\u6bd4\u524d\u9762\u5728\u76d1\u7763\u5b66\u4e60\u7b97\u6cd5\u4e2d\u7684\u6982\u5ff5\u2014\u2014\u4ece\u5047\u8bbe\u7c7b$\\mathcal H$\u4e2d\u641c\u7d22\u6700\u4f18\u5047\u8bbe$h$\uff08\u53c2\u89c1[\u7b2c\u4e5d\u8bb2](chapter09.ipynb)\uff09\u3002\n\n \u5728\u4e4b\u524d\u7684\u5173\u4e8eMDP\u7684\u7b97\u6cd5\u4e2d\uff0c\u6211\u4eec\u90fd\u662f\u901a\u8fc7\u89e3\u51fa\u6700\u4f18\u4ef7\u503c\u51fd\u6570$V^*$\u8fdb\u800c\u5f97\u5230\u6700\u4f18\u7b56\u7565$\\pi^*$\u7684\u3002\u800c\u5728\u7b56\u7565\u641c\u7d22\u7b97\u6cd5\u4e2d\u2014\u2014\u6709\u65f6\u88ab\u79f0\u4f5c\u76f4\u63a5\u641c\u7d22\u7b97\u6cd5\u2014\u2014\u6211\u4eec\u4f1a\u201c\u76f4\u63a5\u201d\u53bb\u5bfb\u627e\u6700\u4f18\u7b56\u7565\uff08\u4e0d\u518d\u901a\u8fc7\u4e2d\u95f4\u7684\u6b65\u9aa4\u2014\u2014\u8ba1\u7b97\u6700\u4f18\u4ef7\u503c\u51fd\u6570\uff09\u3002\n\n* \u63a5\u4e0b\u6765\u518d\u5b9a\u4e49\u968f\u673a\u7b56\u7565\u2014\u2014\u662f\u4e00\u4e2a\u51fd\u6570$\\pi:S\\times A\\to\\mathbb R$\uff0c$\\pi(s,a)$\u8868\u793a\u5728\u72b6\u6001$s$\u6267\u884c\u52a8\u4f5c$a$\u7684\u6982\u7387\u3002\u4ece\u5b9a\u4e49\u5c31\u53ef\u4ee5\u77e5\u9053\uff0c\u51fd\u6570$\\pi$\u5b9e\u9645\u4e0a\u662f\u4e8b\u4ef6\u201c\u5728\u72b6\u6001$s$\u6267\u884c\u52a8\u4f5c$a$\u7684\u6982\u7387\u201d\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\uff0c\u6ee1\u8db3$\\displaystyle\\sum_a\\pi(s,a)=1,\\pi(s,a)\\geq0$\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5bf9\u4e8e\u6bcf\u4e00\u4e2a\u72b6\u6001\uff0c\u51fd\u6570$\\pi$\u90fd\u786e\u5b9a\u4e86\u4e00\u4e2a\u5728$a$\u4e0a\u7684\u5206\u5e03\u3002\n\n \u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5047\u8bbe\u6211\u4eec\u8981\u6267\u884c\u968f\u673a\u7684\u7b56\u7565$\\pi$\uff0c\u800c\u5728\u5f53\u524d\u7684MDP\u4e2d\u6709\u4e09\u4e2a\u52a8\u4f5c$a_t\\in\\{a_1,a_2,a_3\\}$\uff0c\u90a3\u4e48\u6211\u4eec\u6267\u884c\u52a8\u4f5c$a_1$\u7684\u6982\u7387\u5c31\u662f$\\pi(s,a_1)$\u3001\u6267\u884c\u52a8\u4f5c$a_2$\u7684\u6982\u7387\u5c31\u662f$\\pi(s,a_2)$\u3001\u6267\u884c\u52a8\u4f5c$a_3$\u7684\u6982\u7387\u5c31\u662f$\\pi(s,a_3)$\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u5c31\u662f\u6267\u884c\u4e00\u4e2a\u968f\u673a\u7b56\u7565\u3002\n\n\u4e0b\u9762\u7ee7\u7eed\u4f7f\u7528\u5012\u7f6e\u949f\u6446\u7684\u4f8b\u5b50\u6765\u8bb2\u89e3\u7b56\u7565\u641c\u7d22\u7b97\u6cd5\uff0c\u6211\u4eec\u5b9a\u4e49$\\phi$\u4e3a\u949f\u6446\u504f\u79bb\u7ad6\u76f4\u4f4d\u7684\u89d2\u5ea6\uff0c$a_1$\u662f\u5411\u53f3\u7684\u52a0\u901f\u5ea6\uff0c$a_2$\u662f\u5411\u5de6\u7684\u52a0\u901f\u5ea6\u3002\u518d\u627e\u4e00\u4e2a\u5956\u52b1\u51fd\u6570\uff0c\u6746\u4e0d\u8bba\u5012\u5411\u90a3\u4e00\u8fb9\u90fd\u5bf9\u6a21\u578b\u505a\u51fa\u60e9\u7f5a\u3002\u63d0\u51fa\u4e00\u7c7b\u968f\u673a\u7b56\u7565\u610f\u5473\u7740\u63d0\u51fa\u4e00\u7c7b\u51fd\u6570\uff0c\u8fd9\u7c7b\u51fd\u6570\u88ab\u7528\u6765\u4f30\u8ba1\u5728\u72b6\u6001$s_t$\u4e0b\u5c06\u8981\u6267\u884c\u7684\u884c\u52a8$a_t$\u3002\u800c\u73b0\u5728\uff0c\u6211\u4eec\u9009\u62e9\u903b\u8f91\u51fd\u6570\u4f5c\u4e3a\u6211\u4eec\u7684\u7b56\u7565\uff0c\u56e0\u4e3a\u5b83\u662f\u6211\u4eec\u7ecf\u5e38\u4f7f\u7528\u7684\u4e00\u4e2a\u5f88\u65b9\u4fbf\u7684\u51fd\u6570\uff1a\n$$\n\\begin{align}\n\\pi_\\theta(s,a_1)&=\\frac{1}{1+e^{-\\theta^Ts}}\\\\\n\\pi_\\theta(s,a_2)&=1-\\frac{1}{1+e^{-\\theta^Ts}}\n\\end{align}\n$$\n\u56e0\u4e3a\u6211\u4eec\u4e0d\u662f\u9009\u62e9$a_1$\u5c31\u662f\u9009\u62e9$a_2$\uff0c\u6240\u4ee5\u9009\u62e9\u5b83\u4eec\u7684\u6982\u7387\u4e4b\u548c\u4e3a$1$\u3002\u4e3e\u4e2a\u4f8b\u5b50\u89e3\u91ca\u8fd9\u4e2a\u9009\u62e9\u7684\u5408\u7406\u6027\uff0c\u5047\u8bbe\u6211\u4eec\u7684\u72b6\u6001\u5411\u91cf\u4e3a$s=\\begin{bmatrix}1\\\\x\\\\\\dot x\\\\\\phi\\\\\\dot\\phi\\end{bmatrix}$\uff08\u8fd9\u91cc\u6dfb\u52a0\u4e86$1$\u4f5c\u4e3a\u622a\u8ddd\u9879\u7ed9\u903b\u8f91\u56de\u5f52\u4e00\u4e2a\u989d\u5916\u7684\u7279\u5f81\u503c\uff09\uff0c\u5982\u679c\u6211\u4eec\u9009\u62e9$\\theta=\\begin{bmatrix}0\\\\0\\\\0\\\\1\\\\0\\end{bmatrix}$\uff0c\u5219\u6709$P\\displaystyle\\left(a=\\textrm{\"right\"}\\right)=\\frac{1}{1+e^{-\\theta^Ts}}=\\frac{1}{1+e^{-\\phi}}$\uff0c\u8fd9\u8bf4\u660e\u201c\u5411\u53f3\u52a0\u901f\u201d\u7684\u6982\u7387\u53ea\u53d6\u51b3\u4e8e\u201c\u6746\u7684\u503e\u89d2$\\phi$\u201d\uff0c\u6240\u4ee5\u5f53\u7684\u6746\u5411\u53f3\u504f\u7684\u65f6\u5019\u5c0f\u8f66\u5c31\u4f1a\u5c1d\u8bd5\u5411\u53f3\u52a0\u901f\u4ee5\u63a5\u4f4f\u6746\u3002\u5f53\u7136\uff0c\u8fd9\u4e2a\u4f8b\u5b50\u91cc\u7684\u53c2\u6570\u9009\u62e9\u5e76\u4e0d\u662f\u6700\u4f18\u7684\uff0c\u56e0\u4e3a\u5b83\u5ffd\u7565\u7684\u522b\u7684\u53c2\u6570\u3002\u73b0\u5728\uff0c\u6211\u4eec\u7684\u76ee\u6807\u5c31\u662f\u8c03\u6574$\\theta$\uff0c\u4f7f\u5f97\u5f53\u6267\u884c$\\pi_\\theta$\u65f6\u80fd\u591f\u8ba9\u6746\u5c3d\u53ef\u80fd\u7684\u4fdd\u6301\u7ad6\u76f4\uff0c\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u9009\u62e9\u4e00\u4e2a\u53c2\u6570$\\theta$\u4f7f\u5f97\u6267\u884c$\\pi_\\theta$\u65f6\u7684\u9884\u671f\u603b\u6536\u76ca\u6700\u5927\u5316\u3002\u518d\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5982\u679c\u6709$d$\u4e2a\u79bb\u6563\u7684\u52a8\u4f5c\uff0c\u90a3\u4e48\u6211\u4eec\u4e5f\u53ef\u4ee5\u7528[\u7b2c\u56db\u8bb2](chapter04.ipynb)\u63d0\u5230\u7684softmax\u56de\u5f52$\\displaystyle\\theta_1,\\cdots,\\theta_d;\\ \\pi_{\\theta_i}(s,a_i)=\\frac{e^{\\theta_i^Ts}}{\\sum_je^{\\theta_j^Ts}}$\uff0c\u7b56\u7565\u7c7b\u578b\u7684\u9009\u62e9\u662f\u53ef\u4ee5\u81ea\u5df1\u786e\u5b9a\u7684\uff0c\u7ebf\u6027\u51fd\u6570\u3001\u4f7f\u7528\u4e8c\u6b21\u7279\u5f81\u503c\u7684\u7ebf\u6027\u51fd\u6570\u3001\u76d1\u7763\u5b66\u4e60\u7b97\u6cd5\u7b49\u7b49\u3002\n\n### 12.1 \u589e\u5f3a\u7b97\u6cd5\uff08Reinforce algorithm\uff09\n\n\u8bbe$s_0$\u662f\u4e00\u4e2a\u56fa\u5b9a\u7684\u521d\u59cb\u5316\u72b6\u6001\uff08\u5373\u4ece\u4e00\u4e2a\u786e\u5b9a\u7684\u521d\u59cb\u72b6\u6001\u5206\u5e03\u4e2d\u62bd\u6837\u5f97\u5230\uff09\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\uff1a\n\n$$\n\\begin{align}\n\\max\\ &\\mathrm E[R(s_0,a_0)+\\cdots+R(s_T,a_T]\\\\\n&=\\sum_{(s_0,a_0),\\cdots,(s_T,a_T)}P(s_0a_0\\cdots s_Ta_T)[R(s_0,a_0)+\\cdots+R(s_T,a_T)]\\tag{1}\\\\\n&=\\sum_{(s_0,a_0),\\cdots,(s_T,a_T)}P(s_0)\\pi_\\theta(s_0,a_0)P_{s_0a_0}(s_1)\\pi_\\theta(s_1,a_1)P_{s_1a_1}(s_2)\\cdots P_{s_{T-1}\\ a_{T-1}}\\ (s_T)\\pi_\\theta(s_T,a_T)\\cdot\\underbrace{[R(s_0,a_0)+\\cdots+R(s_T,a_T)]}_{\\textrm{payoff}}\\tag{2}\n\\end{align}\n$$\n\n\u5728\u540e\u9762\u6211\u4eec\u5c31\u5c06$R(s_0,a_0)+\\cdots+R(s_T,a_T)$\u90e8\u5206\u7b80\u79f0\u4e3a\u6536\u76ca\uff08payoff\uff09\uff0c\u5148\u5199\u51fa\u7b97\u6cd5\u6b65\u9aa4\uff1a\n\n* \u91cd\u590d\uff1a`{`\n * \u62bd\u6837\uff1a$s_0,a_0,s_1,a_1,\\cdots,s_T,a_t$\n * \u8ba1\u7b97\u6536\u76ca\uff1a$R(s_0,a_0)+\\cdots+R(s_T,a_T)$\n * \u66f4\u65b0\u53c2\u6570\uff1a$\\displaystyle\\theta:=\\theta+\\alpha\\left[\\frac{\\nabla_\\theta\\pi_\\theta(s_0,a_0)}{\\pi_\\theta(s_0,a_0)}+\\cdots+\\frac{\\nabla_\\theta\\pi_\\theta(s_T,a_T)}{\\pi_\\theta(s_T,a_T)}\\right]\\cdot[R(s_0,a_0)+\\cdots+R(s_T,a_T)]$\n \n `}`\n\n\u7b97\u6cd5\u7684\u7b2c\u4e00\u6b65\u5c31\u662f\u968f\u673a\u62bd\u6837\u5f97\u5230\u4e00\u4e2a\u72b6\u6001-\u52a8\u4f5c\u5e8f\u5217\uff0c\u4e5f\u5c31\u662f\u5728MDP\u4e2d\u6267\u884c\u5f53\u524d\u7684\u968f\u673a\u7b56\u7565\u2014\u2014\u4ece\u67d0\u4e2a$s_0$\u5f00\u59cb\uff0c\u6839\u636e\u968f\u673a\u7b56\u7565\u9009\u62e9\u4e00\u4e2a\u52a8\u4f5c$a_0$\uff0c\u7136\u540e\u770b\u72b6\u6001\u8f6c\u6362\u6982\u7387\u7ed9\u51fa\u7684\u4e0b\u4e00\u4e2a\u72b6\u6001\uff0c\u7ee7\u7eed\u91cd\u590d\u6267\u884c\u7b56\u7565\u3002\u7b2c\u4e8c\u6b65\u5c31\u662f\u8ba1\u7b97\u6536\u76ca\u3002\u7b2c\u4e09\u6b65\u66f4\u65b0\u53c2\u6570\u3002\n\n\u7531\u4e8e\u8fd9\u4e2a\u7b97\u6cd5\u6267\u884c\u7684\u662f\u968f\u673a\u7b56\u7565\uff0c\u6240\u4ee5\u6211\u4eec\u60f3\u77e5\u9053\u5b83\u4e3a\u4ec0\u4e48\u8fd9\u6837\u66f4\u65b0\u53c2\u6570\u3002\u6211\u4eec\u53ef\u4ee5\u8ba1\u7b97\u66f4\u65b0\u53c2\u6570\u6b65\u9aa4\u7684\u671f\u671b\uff0c\u53ea\u8981\u66f4\u65b0\u53c2\u6570\u540e\u5e26\u6765\u7684\u6536\u76ca\u5448\u589e\u957f\u8d8b\u52bf\uff0c\u6211\u4eec\u5c31\u5bf9\u7b97\u6cd5\u662f\u6ee1\u610f\u7684\u3002\u5b9e\u9645\u4e0a\u8fd9\u4e2a\u7b97\u6cd5\u7c7b\u4f3c\u968f\u673a\u68af\u5ea6\u4e0a\u5347\uff0c\u5b83\u5c06\u5728\u5c40\u90e8\u6700\u4f18\u89e3\u9644\u8fd1\u201c\u6e38\u8361\u201d\uff0c\u4f46\u603b\u8d8b\u52bf\u662f\u5411\u5c40\u90e8\u6700\u4f18\u89e3\u524d\u8fdb\u3002\n\n\u56de\u987e\u4e0a\u9762\u7684$(2)$\u5f0f\uff0c\u5b83\u8868\u793a\u6982\u7387\u4e0e\u6536\u76ca\u4e4b\u79ef\u5728\u6240\u6709\u72b6\u6001\u4e0a\u7684\u548c\uff0c\u6211\u4eec\u73b0\u5728\u8981\u505a\u7684\u5c31\u662f\u5bf9$(2)$\u5f0f\u6c42\u5173\u4e8e$\\theta$\u7684\u5bfc\u6570\uff0c\u56e0\u4e3a\u6211\u4eec\u60f3\u5bf9\u8fd9\u4e2a\u5f0f\u5b50\u505a\u68af\u5ea6\u4e0a\u5347\uff1a\n$$\n\\begin{align}\n\\nabla_\\theta\\mathrm E\\left[R(s_0,a_0)+\\cdots+R(s_T,a_T)\\right]\n=\\sum_{(s_0,a_0),\\cdots,(s_T,a_T)}\n&\\Bigg[\\ P(s_0)\\underline{(\\nabla_\\theta\\pi_\\theta(s_0,a_0))}P_{s_0a_0}(s_1)\\pi_\\theta(s_1,a_1)P_{s_1a_1}(s_2)\\cdots P_{s_{T-1}\\ a_{T-1}}\\ (s_T)\\pi_\\theta(s_T,a_T)\\\\\n&+P(s_0)\\pi_\\theta(s_0,a_0)P_{s_0a_0}(s_1)\\underline{(\\nabla_\\theta\\pi_\\theta(s_1,a_1))}P_{s_1a_1}(s_2)\\cdots P_{s_{T-1}\\ a_{T-1}}\\ (s_T)\\pi_\\theta(s_T,a_T)\\\\\n&+\\cdots\\\\\n&+P(s_0)\\pi_\\theta(s_0,a_0)P_{s_0a_0}(s_1)\\pi_\\theta(s_1,a_1)P_{s_1a_1}(s_2)\\cdots P_{s_{T-1}\\ a_{T-1}}\\ (s_T)\\underline{(\\nabla_\\theta\\pi_\\theta(s_T,a_T))}\\ \\Bigg]\\\\\n&\\times[R(s_0,a_0)+\\cdots+R(s_T,a_T)]\\\\\n=\\sum_{(s_0,a_0),\\cdots,(s_T,a_T)}&P(s_0)\\pi_\\theta(s_0,a_0)P_{s_0a_0}(s_1)\\pi_\\theta(s_1,a_1)P_{s_1a_1}(s_2)\\cdots P_{s_{T-1}\\ a_{T-1}}\\ (s_T)\\pi_\\theta(s_T,a_T)\\\\\n&\\times\\left[\\frac{\\nabla_\\theta\\pi_\\theta(s_0,a_0)}{\\pi_\\theta(s_0,a_0)}+\\frac{\\nabla_\\theta\\pi_\\theta(s_1,a_1)}{\\pi_\\theta(s_1,a_1)}+\\cdots+\\frac{\\nabla_\\theta\\pi_\\theta(s_T,a_T)}{\\pi_\\theta(s_T,a_T)}\\right]\\\\\n&\\times[R(s_0,a_0)+\\cdots+R(s_T,a_T)]\\\\\n=\\sum_{(s_0,a_0),\\cdots,(s_T,a_T)}&P(s_0a_0\\cdots s_Ta_T)\\cdot\\left[\\frac{\\nabla_\\theta\\pi_\\theta(s_0,a_0)}{\\pi_\\theta(s_0,a_0)}+\\cdots+\\frac{\\nabla_\\theta\\pi_\\theta(s_T,a_T)}{\\pi_\\theta(s_T,a_T)}\\right]\\\\\n&\\times[R(s_0,a_0)+\\cdots+R(s_T,a_T)]\\\\\n\\end{align}\\\\\n\\Downarrow\\\\\n\\nabla_\\theta\\mathrm E\\left[R(s_0,a_0)+\\cdots+R(s_T,a_T)\\right]=\\mathrm E\\left[\\left(\\frac{\\nabla_\\theta\\pi_\\theta(s_0,a_0)}{\\pi_\\theta(s_0,a_0)}+\\cdots+\\frac{\\nabla_\\theta\\pi_\\theta(s_T,a_T)}{\\pi_\\theta(s_T,a_T)}\\right)\\cdot\\left(R(s_0,a_0)+\\cdots+R(s_T,a_T)\\right)\\right]\n$$\n\u7b2c\u4e00\u6b65\u5e94\u7528\u4e86\u51e0\u4e2a\u76f8\u4e58\u51fd\u6570\u7684\u6c42\u5bfc\u6cd5\u5219$\\displaystyle\\frac{\\mathrm d}{\\mathrm d\\theta}f(\\theta)g(\\theta)h(\\theta)=f'(\\theta)g(\\theta)h(\\theta)+f(\\theta)g'(\\theta)h(\\theta)+f(\\theta)g(\\theta)h'(\\theta)$\u3002\n\n\u4f46\u662f\uff0c\u4f7f\u7528\u52a0\u5f3a\u7b97\u6cd5\u7684\u65f6\u5019\uff0c\u6211\u4eec\u901a\u5e38\u8ba4\u4e3a\u5bf9\u4e8e\u5f85\u89e3\u51b3\u7684\u95ee\u9898\uff0c\u5b58\u5728\u4e00\u4e2a\u7b80\u5355\u7684\u51fd\u6570\uff08\u5982\u7ebf\u6027\u51fd\u6570\u3001\u903b\u8f91\u51fd\u6570\u7b49\uff09\uff0c\u63cf\u8ff0\u4e86\u4ece\u72b6\u6001\u7a7a\u95f4\u5230\u52a8\u4f5c\u7a7a\u95f4\u7684\u6620\u5c04\u3002\u5bf9\u4e8e\u5012\u7f6e\u949f\u6446\u8fd9\u79cd\u7b80\u5355\u7684\u4efb\u52a1\uff0c\u4e8b\u5b9e\u53ef\u80fd\u5c31\u662f\u8fd9\u6837\u7684\uff08\u6746\u5411\u53f3\u504f\u65f6\u5c0f\u8f66\u5c31\u5411\u53f3\u52a0\u901f\u63a5\u4f4f\u6746\uff09\u3002\u5b9e\u9645\u4e0a\u5bf9\u4e8e\u5f88\u591a\u4f4e\u7ea7\u522b\u7684\u63a7\u5236\u4efb\u52a1\uff08\u5982\u5f00\u8f66\u65f6\u53f3\u8fb9\u6709\u969c\u788d\u5c31\u5e94\u8be5\u5411\u5de6\u8f6c\u5411\u4ee5\u907f\u8ba9\uff09\uff0c\u4eba\u7c7b\u4e5f\u662f\u6761\u4ef6\u53cd\u5c04\u5f0f\u7684\u505a\u51fa\u201c\u672c\u80fd\u201d\u52a8\u4f5c\u3002\u5bf9\u4e8e\u8fd9\u79cd\u8f83\u4e3a\u7b80\u5355\u7684\u3001\u6781\u77ed\u65f6\u95f4\u5185\u7684\u201c\u672c\u80fd\u201d\u5224\u65ad\uff0c\u901a\u5e38\u80fd\u591f\u627e\u5230\u4e00\u4e2a\u7531\u7b80\u5355\u51fd\u6570\u7ec4\u6210\u7684\u5408\u7406\u7684\u7b56\u7565\u7c7b\u3002\n\n\u4e0e\u5176\u76f8\u53cd\u7684\u662f\u590d\u6742\u4efb\u52a1\uff0c\u9700\u8981\u5f88\u957f\u7684\u591a\u6b65\u63a8\u7406\u7684\u4efb\u52a1\uff0c\u6bd4\u5982\u8c61\u68cb\u3001\u56f4\u68cb\u7b49\u3002\u5728\u8fd9\u79cd\u6d3b\u52a8\u4e2d\u505a\u51fa\u51b3\u7b56\u9700\u8981\u591a\u6b65\u4e25\u5bc6\u7684\u56e0\u679c\u63a8\u7406\uff0c\u8fd9\u662f\u4e00\u79cd\u9ad8\u7ea7\u522b\u7684\u63a7\u5236\u4efb\u52a1\uff0c\u6240\u4ee5\u5c31\u4e0d\u80fd\u4f7f\u7528\u201c\u672c\u80fd\u201d\u4e86\u3002\u5728\u8fd9\u79cd\u4efb\u52a1\u4e2d\u6211\u4eec\u6709\u65f6\u4f1a\u7528\u524d\u9762\u63d0\u5230\u7684\u4ef7\u503c\u51fd\u6570\u7684\u7ebf\u6027\u8fd1\u4f3c\u3002\u53e6\u5916\uff0c\u5728POMDP\u4e2d\u5982\u679c\u4f7f\u7528$\\hat s$\u8fd1\u4f3c\u771f\u5b9e\u72b6\u6001\uff08\u6bd4\u5982\u5728Kalman\u6ee4\u6ce2\u4e2d\u7684$\\hat s=s_{t\\mid t}$\uff09\uff0c\u5219\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u4f7f\u7528\u7b56\u7565\u641c\u7d22\u7b97\u6cd5$\\displaystyle\\pi_\\theta(\\hat s,a_t)=\\frac{1}{1+e^{-\\theta^T\\hat s}}$\u3002\n\n\u6700\u540e\uff0c\u5173\u4e8e\u52a0\u5f3a\u7b97\u6cd5\u2014\u2014\u8fd9\u4e2a\u7b97\u6cd5\u5728\u505a\u4e0a\u5347\u65f6\u672c\u8d28\u4e0a\u662f\u6ca1\u6709\u660e\u786e\u65b9\u5411\u7684\uff08\u867d\u7136\u5176\u603b\u4f53\u671f\u671b\u4e0a\u662f\u6b63\u786e\u7684\uff09\uff0c\u8fd9\u53ef\u80fd\u4f1a\u5bfc\u81f4\u52a0\u5f3a\u7b97\u6cd5\u9700\u8981\u5927\u91cf\u7684\u8fed\u4ee3\u624d\u5230\u8fbe\u5230\u6700\u4f18\u503c\u9644\u8fd1\uff1b\u800c\u4e14\uff0c\u5728\u7b97\u6cd5\u4e2d\uff0c\u6bcf\u6b21\u8fed\u4ee3\u90fd\u9700\u8981\u4e00\u6b21\u62bd\u6837\uff0c\u5982\u679c\u5bf9\u4e8e\u4e00\u4e2a\u7269\u7406\u5b9e\u4f53\u6765\u8bf4\u6210\u672c\u53ef\u80fd\u4f1a\u5f88\u9ad8\uff0c\u6bd4\u5982\u6211\u4eec\u5728\u7814\u7a76\u63a7\u5236\u4e00\u4e2a\u673a\u5668\u4eba\uff0c\u90a3\u4e48\u8fd9\u4e2a\u673a\u5668\u4eba\u53ef\u80fd\u5c31\u9700\u8981\u505a\u5f88\u591a\u6b21\u52a8\u4f5c\uff08\u591a\u6b21\u8fed\u4ee3\uff0c\u6bcf\u6b21\u90fd\u9700\u8981\u62bd\u6837\uff09\uff0c\u6240\u4ee5\u8fd9\u4e2a\u7b97\u6cd5\u901a\u5e38\u8fd0\u884c\u5728\u6a21\u62df\u5668\u4e2d\u3002\n\n### 12.2 Pegasus\u7b97\u6cd5\n\n\u5728[\u7b2c\u5341\u4e03\u8bb2](chapter17.ipynb)\u6211\u4eec\u63d0\u5230\u6a21\u62df\u5668\u7684\u5e94\u7528\u3002\u6a21\u62df\u5668\u53ef\u4ee5\u63a5\u53d7\u72b6\u6001$s_t$\u548c\u52a8\u4f5c$a_t$\uff0c\u8f93\u51fa\u4e0b\u4e00\u6b65\u7684\u72b6\u6001$s_{t+1}$\uff0c$s_{t+1}$\u901a\u5e38\u662f\u4e00\u4e2a\u4ece\u968f\u673a\u72b6\u6001\u8f6c\u6362\u6982\u7387\u4e2d\u62bd\u6837\u7684\u968f\u673a\u72b6\u6001\uff0c\u4e5f\u5c31\u662f\u8bf4\u5728\u6a21\u62df\u5668\u4e2d$s_{t+1}$\u662f\u4e00\u4e2a\u5173\u4e8e$s_t,a_t$\u7684\u968f\u673a\u51fd\u6570\u3002\u6bd4\u5982\u6211\u4eec\u5728\u76f4\u5347\u673a\u7684\u4f8b\u5b50\u4e2d\uff0c\u4f7f\u7528\u7ebf\u6027\u56de\u5f52\u3001\u5c40\u90e8\u52a0\u6743\u56de\u5f52\u7b49\u76d1\u7763\u5b66\u4e60\u7b97\u6cd5\uff0c\u5c31\u53ef\u4ee5\u6784\u5efa\u4e00\u4e2a\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u6a21\u578b\uff0c\u5728\u8fd9\u4e2a\u6a21\u578b\u4e2d\uff0c$s_{t+1}$\u5c31\u662f\u4e00\u4e2a\u5173\u4e8e$s_t,a_t$\u7684\u968f\u673a\u51fd\u6570\u3002\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u6a21\u62df\u5668\u4f30\u8ba1\u4efb\u610f\u7684\u72b6\u6001-\u52a8\u4f5c\u5e8f\u5217\u5bf9\u5e94\u7684\u9884\u671f\u603b\u6536\u76ca\uff08\u5c06\u6bcf\u4e2a\u72b6\u6001\u7684\u5956\u52b1\u51fd\u6570\u503c\u76f8\u52a0\u5373\u53ef\uff09\u3002\n\n\u540c\u6837\u662f\u5728[\u7b2c\u5341\u4e03\u8bb2](chapter17.ipynb)\u6211\u4eec\u5b9a\u4e49\u4e86\u7b56\u7565\uff0c\u5b83\u63a5\u53d7\u5f53\u524d\u72b6\u6001\uff0c\u8f93\u51fa\u5f53\u524d\u72b6\u6001\u4e0b\u5e94\u6267\u884c\u7684\u52a8\u4f5c\uff0c\u90a3\u4e48\u7ed3\u5408\u6a21\u62df\u5668\uff0c\u5c31\u53ef\u4ee5\u4f30\u8ba1\u4efb\u610f\u7b56\u7565\u7684\u9884\u671f\u603b\u6536\u76ca\u4e86\uff08\u9009\u5b9a\u521d\u59cb\u72b6\u6001$s_0$\u540e\uff0c\u8f93\u5165\u7b56\u7565$\\pi$\u5f97\u5230$a_0$\uff0c\u518d\u5c06$s_0,a_0$\u8f93\u5165\u6a21\u62df\u5668\u5f97\u5230$s_1$\uff1b\u5c06$s_1$\u5e26\u5165\u7b56\u7565$\\pi$\u5f97\u5230$a_1$\uff0c\u518d\u5c06$s_1,a_1$\u5e26\u5165\u6a21\u62df\u5668\u5f97\u5230$s_2$\u2026\u2026\u5c06\u6240\u6709\u72b6\u6001\u5e26\u5165\u5956\u52b1\u51fd\u6570\uff0c\u6c42\u548c\u5373\u53ef\u4f30\u8ba1\u51fa\u8be5\u7b56\u7565\u7684\u9884\u671f\u603b\u6536\u76ca\uff09\u3002\u90a3\u4e48\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5c1d\u8bd5\u627e\u5230\u4f7f\u8fd9\u4e2a\u7531\u6a21\u62df\u5668\u4f30\u8ba1\u51fa\u7684\u603b\u6536\u76ca\u6700\u5927\u7684\u7b56\u7565\uff0c\u4e5f\u5c31\u662f\u627e\u5230\u8be5\u7b56\u7565\u5bf9\u5e94\u7684\u53c2\u6570\u5373\u53ef\u3002\u4ece\u6982\u5ff5\u4e0a\u8bb2\uff0c\u8fd9\u662f\u4e00\u4e2a\u4e0d\u9519\u7684\u601d\u8def\uff0c\u4f46\u662f\u5b9e\u9645\u4e0a\u51e0\u4e4e\u4e0d\u53ef\u884c\uff1a\u7531\u4e8e\u6a21\u62df\u5668\u7684\u968f\u673a\u6027\uff0c\u5373\u4f7f\u6211\u4eec\u5bf9\u540c\u4e00\u4e2a\u7b56\u7565\u505a\u4f30\u8ba1\u65f6\uff0c\u4e5f\u4f1a\u5f97\u5230\u6709\u7ec6\u5fae\u7684\u5dee\u522b\u4e24\u4e2a\u4e0d\u540c\u7684\u7ed3\u679c\uff0c\u6240\u4ee5\u5728\u7b56\u7565\u7a7a\u95f4\u4e2d\u5bf9\u7b56\u7565\u8fdb\u884c\u4f30\u8ba1\u65f6\uff0c\u5982\u679c\u7b56\u7565\u7a7a\u95f4\u7ef4\u6570\u8f83\u9ad8\uff0c\u90a3\u4e48\u6211\u4eec\u5f88\u96be\u77e5\u9053\u7a7a\u95f4\u4e2d\u7b56\u7565\u5bf9\u5e94\u7684\u6536\u76ca\u7684\u5206\u5e03\u8d8b\u52bf\u3002\n\n\u6a21\u62df\u5668\u4f1a\u6839\u636e\u8f93\u5165\u7684\u72b6\u6001-\u52a8\u4f5c\u8f93\u51fa\u4e00\u4e2a\u5e26\u6709\u673a\u6027\u7684\u4e0b\u4e00\u6b65\u72b6\u6001\uff0c\u901a\u5e38\uff0c\u80fd\u591f\u505a\u5230\u8fd9\u4e00\u70b9\u7684\u6a21\u62df\u5668\u90fd\u4f1a\u8c03\u7528\u4e00\u4e2a\u968f\u673a\u6570\u751f\u6210\u5668\u3002\u90a3\u4e48\uff0c\u4e3a\u4e86\u964d\u4f4e\u4f7f\u7528\u6a21\u62df\u5668\u5bf9\u7b56\u7565\u7a7a\u95f4\u8fdb\u884c\u641c\u7d22\u7684\u96be\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u5728MDP\u4e2d\u6bcf\u4e00\u6b65\u8c03\u7528\u6a21\u62df\u5668\u65f6\uff0c\u56fa\u5b9a\u968f\u673a\u6570\u751f\u6210\u5668\u5f97\u5230\u7684\u503c\uff08\u4e5f\u5c31\u662f\u8ba9\u6bcf\u4e00\u6b65\u7684\u751f\u6210\u5668\u53ea\u751f\u6210\u5668\u53ea\u8fd0\u884c\u4e00\u6b21\uff0c\u7136\u540e\u5c31\u56fa\u5b9a\u8fd9\u4e2a\u503c\uff09\uff0c\u5982\u679c\u6211\u4eec\u6bcf\u6b21\u4f30\u8ba1\u7b56\u7565\u90fd\u4f7f\u7528\u540c\u4e00\u7ec4\u968f\u673a\u6570\u5e8f\u5217\uff0c\u5219\u6a21\u62df\u5668\u5c31\u4e0d\u518d\u662f\u968f\u673a\u7684\u4e86\uff0c\u6b64\u65f6\u5bf9\u540c\u4e00\u4e2a\u7b56\u7565\u8fdb\u884c\u591a\u6b21\u4f30\u8ba1\uff0c\u5f97\u5230\u7684\u4e5f\u5c06\u662f\u76f8\u540c\u7684\u503c\uff08\u53d6\u6d88\u4e86\u968f\u673a\u6570\u4e5f\u5c31\u610f\u5473\u7740\u786e\u5b9a\u4e86\u5728\u7b56\u7565\u7a7a\u95f4\u4e0a\u7684\u6536\u76ca\u51fd\u6570\uff0c\u8fd9\u6781\u5927\u7a0b\u5ea6\u4e0a\u964d\u4f4e\u4e86\u6700\u4f18\u7b56\u7565\u7684\u6c42\u89e3\u96be\u5ea6\uff09\u3002\u867d\u7136\u6211\u4eec\u73b0\u5728\u901a\u8fc7\u6a21\u62df\u5668\u53ea\u80fd\u5f97\u5230\u7684\u7b56\u7565\u9884\u671f\u603b\u6536\u76ca\u7684\u4f30\u8ba1\uff08\u56e0\u4e3a\u56fa\u5b9a\u4e86\u968f\u673a\u6570\uff0c\u6a21\u62df\u5668\u4e0d\u518d\u662f\u539f\u6765\u901a\u8fc7\u8bd5\u9a8c\u6570\u636e\u62df\u5408\u51fa\u6765\u7684\u7cfb\u7edf\u4e86\uff09\uff0c\u4f46\u8fd9\u4e2a\u4f30\u8ba1\u503c\u79bb\u5b9e\u9645\u7684\u9884\u671f\u603b\u6536\u76ca\u76f8\u4e58\u4e0d\u5927\uff0c\u6240\u4ee5\u73b0\u5728\u6211\u4eec\u5c31\u53ef\u4ee5\u4f7f\u7528\u8bf8\u5982\u68af\u5ea6\u4e0a\u5347\u7b49\u7b97\u6cd5\u5728\u7b56\u7565\u7a7a\u95f4\u4e2d\u641c\u7d22\u6700\u4f18\u7b56\u7565\u4e86\u3002\n\n\u5173\u4e8e\u8fd9\u4e2a\u968f\u673a\u6570\uff0c\u53ef\u4ee5\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5728\u76f4\u5347\u673a\u6a21\u578b\u4e2d\uff0c\u5047\u8bbe\u968f\u673a\u6570\u662f\u5bf9\u76f4\u5347\u673a\u5728\u98ce\u573a\u4e2d\u7684\u6a21\u62df\uff0c\u4e0d\u540c\u7684\u968f\u673a\u6570\u5bf9\u5e94\u4e0d\u540c\u7684\u98ce\u3002\u5728\u5bf9\u98ce\u573a\u7684\u6a21\u62df\u4e2d\uff0c\u6211\u4eec\u4e0d\u662f\u5bf9\u6bcf\u4e00\u79cd\u6a21\u5f0f\u7684\u98ce\u90fd\u8fdb\u884c\u5efa\u6a21\u4f30\u8ba1\uff0c\u800c\u662f\u91c7\u6837\u4e0d\u540c\u6a21\u5f0f\u7684\u98ce\uff0c\u5bf9\u5b83\u4eec\u505a\u5e73\u5747\u7684\u4f30\u8ba1\u3002\u5728\u6a21\u62df\u5668\u4e2d\uff0c\u6211\u4eec\u5bf9\u63a7\u5236\u7b56\u7565\u505a\u6536\u76ca\u4f30\u8ba1\uff0c\u5e76\u4e0d\u662f\u53ea\u505a\u4e00\u6b21\uff0c\u800c\u662f\u505a\u5f88\u591a\u6b21\uff0c\u7136\u540e\u53d6\u6240\u6709\u6d4b\u8bd5\u7684\u5e73\u5747\u6536\u76ca\u3002\u8fd9\u5c31\u76f8\u5f53\u4e8e\u5728\u4e0d\u540c\u7684\u98ce\u573a\u4e2d\u505a\u4e86\u591a\u6b21\u6d4b\u8bd5\uff0c\u7136\u540e\u5f97\u51fa\u63a7\u5236\u7b56\u7565\u5728\u4e0d\u540c\u98ce\u573a\u72b6\u51b5\u4e0b\u7684\u5e73\u5747\u9884\u671f\u6536\u76ca\u3002\n\n### 12.3 \u5f3a\u5316\u5b66\u4e60\u7b97\u6cd5\u5c0f\u7ed3\n\n\u6211\u4eec\u4e00\u5171\u4ecb\u7ecd\u4e86\u4e24\u79cd\u5f3a\u5316\u5b66\u4e60\u7b97\u6cd5\uff0c\u7b2c\u4e00\u79cd\u65b9\u6cd5\u662f\u901a\u8fc7\u6c42\u89e3\u6700\u4f18\u4ef7\u503c\u51fd\u6570\u63a8\u5bfc\u51fa\u6700\u4f18\u7b56\u7565\uff0c\u8fd9\u79cd\u65b9\u6cd5\u5e38\u88ab\u7528\u4e8e\u89e3\u51b3\u9700\u8981\u201c\u6df1\u601d\u719f\u8651\u201d\u624d\u80fd\u89e3\u51b3\u7684\u95ee\u9898\uff0c\u5b83\u6d89\u53ca\u7684\u95ee\u9898\u4e00\u822c\u90fd\u4f1a\u6709\u591a\u6b65\u63a8\u5bfc\uff0c\u4e14\u540e\u9762\u6b65\u9aa4\u7684\u51b3\u7b56\u4e5f\u4f1a\u5f71\u54cd\u524d\u9762\u7684\u6b65\u9aa4\uff0c\u5e38\u89c1\u4e8e\u8bf8\u5982\u4fc4\u7f57\u65af\u65b9\u5757\u3001\u8c61\u68cb\u7b49\u573a\u666f\uff1b\u7b2c\u4e8c\u79cd\u65b9\u6cd5\u662f\u901a\u8fc7\u7b56\u7565\u641c\u7d22\u76f4\u63a5\u627e\u5230\u6700\u4f18\u7b56\u7565\uff0c\u8fd9\u79cd\u65b9\u6cd5\u901a\u5e38\u88ab\u7528\u4e8e\u89e3\u51b3\u201c\u6761\u4ef6\u53cd\u5c04\u201d\u5f0f\u7684\u95ee\u9898\uff0c\u6bd4\u5982\u76f4\u5347\u673a\u60ac\u505c\u63a7\u5236\u3001\u6c7d\u8f66\u969c\u788d\u907f\u8ba9\u7b49\u573a\u666f\u3002\u5f3a\u5316\u5b66\u4e60\u7b97\u6cd5\u7684\u6838\u5fc3\u662f\u6839\u636e\u72b6\u6001\u505a\u51fa\u7684\u51b3\u7b56\u5e8f\u5217\uff0c\u5b83\u7684\u9002\u7528\u573a\u666f\u4e3a\u201c\u9700\u8981\u4f9d\u6b21\u6839\u636e\u72b6\u6001\u505a\u51fa\u51b3\u7b56\u7684\u95ee\u9898\uff0c\u800c\u4e14\u51b3\u7b56\u53ef\u80fd\u5177\u6709\u957f\u671f\u5f71\u54cd\u201d\u3002\u5217\u51fa\u4e00\u4e9bRL\u7b97\u6cd5\u5df2\u7ecf\u5e94\u7528\u7684\u573a\u666f\uff1a\u533b\u7597\u51b3\u7b56\u2014\u2014\u6839\u636e\u75c5\u4eba\u6240\u5904\u7684\u72b6\u6001\u9009\u62e9\u4e0d\u540c\u7684\u6cbb\u7597\u65b9\u6848\uff1b\u7528\u4e8e\u51cf\u5c11\u6392\u961f\u7b49\u5f85\u65f6\u95f4\u2014\u2014\u5176\u4e2d\u5305\u62ec\u6d41\u6c34\u7ebf\u4e0a\u7684\u6548\u7387\u63d0\u5347\u95ee\u9898\u7b49\uff1b\u7528\u4e8e\u535a\u5f08\u8bba\u4e2d\u67d0\u4e9b\u573a\u666f\u2014\u2014\u6bd4\u5982\u600e\u6837\u5927\u91cf\u629b\u552e\u80a1\u7968\u540c\u65f6\u53c8\u5c06\u5bf9\u5e02\u573a\u7684\u5f71\u54cd\u964d\u5230\u6700\u4f4e\uff1b\u4e5f\u5e94\u7528\u4e8e\u8fd0\u7b79\u5b66\u2014\u2014\u6700\u5927\u5316\u5de5\u5382\u7684\u751f\u4ea7\u6548\u7387\u540c\u65f6\u964d\u4f4e\u6210\u672c\u2026\u2026\u3002\n", 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YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.42803623411562536}} {"text": "```python\nimport holoviews as hv\nhv.extension('bokeh')\nhv.opts.defaults(hv.opts.Curve(width=500), \n hv.opts.Image(width=500, colorbar=True, cmap='Viridis'))\n```\n\n\n```python\nimport numpy as np\nimport scipy.signal\nimport scipy.fft\nfrom IPython.display import Audio\n```\n\n# Sistemas para el procesamiento de se\u00f1ales\n\n## Introducci\u00f3n\n\n### Definici\u00f3n de sistema\n\nHasta ahora hemos realizado *an\u00e1lisis de se\u00f1ales*, es decir el estudio de las se\u00f1ales y sus propiedades en el dominio del tiempo y frecuencia\n\nEn esta unidad nos enfocaremos en el *procesamiento de se\u00f1ales*, es decir el dise\u00f1o de **sistemas** que procesan una **se\u00f1al de entrada** y producen una **se\u00f1al de salida**\n\n\n\nUsaremos\n\n- $x[n]$ para denotar la se\u00f1al (discreta) de entrada y $X[k]$ su espectro\n- $y[n]$ para denotar la se\u00f1al (discreta) de salida e $Y[k]$ su espectro\n \n\n\n### Ejemplos de sistemas\n\nUtilizando sistemas podemos modificar una se\u00f1al para mejorar su calidad o remover efectos indeseados\n\n- Un sistema para reducir el ruido de una se\u00f1al de electroencefalograma (EEG)\n\n\n\n- Un sistema para mejorar una imagen fuera de foco (sharpening)\n\n\n\n- Un sistema para eliminar el eco de un audio\n\n\n\n\n## Propiedades y ejemplos de sistemas\n\n\n### Sistemas sin memoria\n\nDiremos que un sistema $\\Phi$ es un sistema **sin memoria** si \n\n$$\ny[n] = \\Phi(x[n]),\n$$\n\nes decir que la salida del sistema en un instante $n$ dado depende solo de la entrada en ese mismo instante\n\nVeamos algunos ejemplos\n\n**Sistema atenuador/amplificador ideal** \n\n$$\ny[n] = A x[n], \n$$\n\ndonde $A>0$ se llama *ganancia*\n\nEste sistema puede atenuar la entrada si $01$\n\n**Sistema saturador (clamp)**\n\n$$\ny[n] = \\begin{cases} B &x[n] > B \\\\x[n] & x[n] \\in [-B, B]\\\\ -B & x[n] < -B\\end{cases}\n$$\n\nEste sistema limita los valores de la se\u00f1al de entrada en un rango fijo\n\n**Sistema rectificador**\n\n$$\ny[n] = | x[n] |\n$$\n\nEste sistema eliminar la parte negativa de la se\u00f1al de entrada\n\n### Sistema Lineal\n\nDiremos que un sistema $\\Phi$ es lineal si cumple con las siguientes propiedades\n\n**Homogeneidad**\n\nUn cambio en la amplitud de la entrada produce un cambio equivalente en la salida\n\n$$\n\\Phi(cx[n]) = c \\Phi(x[n]) = c y[n]\n$$\n\n**Aditividad**\n\nSe\u00f1ales que se suman en la entrada producen se\u00f1ales que se suman en la salida\n\n$$\n\\Phi(x_1[n] + x_2[n]) = \\Phi(x_1[n]) + \\Phi(x_2[n]) = y_1[n] + y_2[n]\n$$\n\nEs decir que las se\u00f1ales pasan por el sistema sin interactuar entre ellas\n\n\n\n**Otras propiedades de los sistemas lineales**\n\nProducto de las propiedades anteriores se tiene que una cascada de sistemas lineales forma un sistema lineal equivalente y adem\u00e1s la cascada de sistemas es **conmutativa**, es decir que el orden de los sistemas en la cascada no altera el resultado final\n\nLa siguiente figura ejemplifica esta propiedad\n\n\n\n\nOtra propiedad interesante de los sistemas lineales es que cumplen el **Principio de superposici\u00f3n:**\n\n1. Si descomponemos una se\u00f1al en $M$ componentes: $x[n] = x_1[n] + x_2[n] + \\ldots + x_M[n]$\n1. Y aplicamos un **sistema lineal** a cada componente: $y_j[n] = \\Phi(x_j[n])$\n1. Podemos recuperar la salida total usando **aditividad**: $y_1[n] + y_2[n] + \\ldots + y_M[n] = y[n]$\n\nLa siguiente figura ejemplifica esta propiedad\n\n\n \n\n\n### Sistemas con memoria\n\nUn sistema $\\Phi$ es un sistema con memoria si su salida actual depende s\u00f3lo de la entrada actual, las entradas anteriores o las salidas anteriores\n\n$$\n\\begin{align}\ny[n] = \\Phi(x[n], & x[n-1], x[n-2], \\ldots, x[0], \\\\ \\nonumber\n& y[n-1], y[n-2], \\ldots, y[0]) \\nonumber\n\\end{align}\n$$\n\nesto tambi\u00e9n se conoce como **sistema causal**\n\nUn **sistema con memoria no-causal** usa entradas futuras (es decir $x[n+1]$, $x[n+2]$, ...) y por ende solo se puede implementar de forma offline, es decir una vez que sea ha observado toda la se\u00f1al\n\nA continuaci\u00f3n veremos algunos ejemplos de sistemas con memoria causales\n\n**Sistema con un retardo (delay)**\n\nDefinido como \n\n$$\ny[n] = x[n-m],\n$$\n\ndonde\n- la salida depende solo de \"una\" entrada anterior\n- el valor de m define que tan \"antigua\" es la entrada pasada\n\nEl *delay* o retarno no afecta la amplitud de los componentes frecuenciales de la se\u00f1al pero si su fase, como muestra la siguiente figura\n\n\n```python\nn = np.arange(0, 200, step=1)\nx = lambda m: np.sin(2.0*np.pi*0.05*(n-m)) \nf = scipy.fft.rfftfreq(d=1, n=len(n))\n\np = []\nfor m in [0, 4, 8]:\n x_delayed = x(m)\n p.append(hv.Curve((n, x_delayed), 'Tiempo [s]', 'Se\u00f1al'))\n X = scipy.fft.rfft(x_delayed)\n Xm = np.absolute(X) \n p.append(hv.Curve((f, Xm), 'Frecuencia [Hz]', 'Espectro de amplitud'))\n Xp = np.angle(X)\n p.append(hv.Curve((f, Xm*Xp/np.amax(Xm)), 'Frecuencia [Hz]', 'Espectro de fase'))\n```\n\n\n```python\nhv.Layout(p).cols(3).opts(hv.opts.Curve(width=250, height=200))\n```\n\n**Sistema reverberador o eco**\n\nDefinido como\n\n$$\ny[n] = x[n] + A x[n-m],\n$$\n\ndonde\n- la salida depende de una entrada \"pasada\" y la entrada actual\n- la ganancia controla si el \"eco\" se atenua o amplifica\n\n\nAl contrario del sistema anterior, el eco si puede modificar el espectro de amplitud. \n\nNotemos el efecto de interferencia constructiva y destructiva al modificar el retardo, como muestra la siguiente animaci\u00f3n \n\n\n```python\nn = np.arange(0, 200, step=1)\nx = lambda m, A=1: A*np.sin(2.0*np.pi*0.05*(n-m)) \nf = scipy.fft.rfftfreq(d=1, n=len(n))\n\nbuffer = {}\nfor m in range(0, 40, 2):\n xm = x(m)\n buffer[m] = (xm, np.abs(scipy.fft.rfft(x(0) + xm))) \n```\n\n\n```python\nhMap1 = hv.HoloMap(kdims='m')\nhMap2 = hv.HoloMap(kdims='m')\nhMap3 = hv.HoloMap(kdims='m')\n\nfor m, (xm, X) in buffer.items(): \n hMap1[m] = hv.Curve((n, xm), 'Tiempo', 'x', label='Ax[n-m]') \n hMap2[m] = hv.Curve((n, x(0) + xm), 'Tiempo', 'y')\n hMap3[m] = hv.Curve((f, X), 'Frecuencia', 'Espectro')\n\np_clean = hv.Curve((n, x(0)), 'Tiempo', 'x', label='x[n]').opts(width=500, height=200)\nplot = (p_clean * hMap1 + hMap2 + hMap3).cols(1).opts(hv.opts.Curve(height=200), \n hv.opts.Overlay(legend_position='top_right'))\n\nhv.output(plot, holomap='gif', fps=5)\n```\n\nEl siguiente video muestra un experimento que ejemplifica la interferencia destructiva en una onda mec\u00e1nica: https://www.youtube.com/watch?v=IU8xeJlJ0mk\n\n**Sistemas con m\u00faltiples ecos**\n\nPueden combinarse m\u00e1s retardos para hacer un sistema reverberante m\u00e1s complejo\n\nPor ejemplo el siguiente sistema\n\n$$\ny[n] = x[n] + A_1 x[n-m_1] + A_2 x[n-m_2] + A_3 x[n-m_3] + \\ldots,\n$$\n\nque lo implementamos como\n\n\n```python\nFs = 44100; \nn = np.arange(0, 4, step=1.0/Fs) \nx = lambda m: np.sin(2.0*np.pi*880*(n-m))*np.exp(-(n-m)**2/0.5**2)*np.heaviside(n-m, 0)\ny = x(0) + 0.5*x(1.) + 0.25*x(2.) + 0.125*x(3.)\n```\n\nDa como resultado gr\u00e1fico lo siguiente:\n\n\n```python\nhv.Curve((n, y), 'Tiempo [s]', 'Se\u00f1al con eco')\n```\n\ny el resultado sonoro es:\n\n\n```python\nAudio(y, rate=Fs, normalize=False)\n```\n\n## Sistema de respuesta finita al impulso (FIR)\n\n\nGeneralizando el ejemplo de sistema lineal reverberante a $L$ retardos llegamos a \n\n$$\n\\begin{align}\ny[n] &= h[0] x[n] + h[1] x[n-1] + h[2] x[n-2] + \\ldots + h[L] x[n-L] \\nonumber \\\\\n&= \\sum_{j=0}^{L} h[j] x[n-j] \\nonumber \\\\\n&= (h* x)[n] \\nonumber \n\\end{align}\n$$\n\nque se puede modelar como una convoluci\u00f3n discreta entre $h$ y $x$\n\nEste sistema se conoce como\n\n- Sistema FIR (finite impulse response)\n- Sistema MA (moving average)\n- Sistema todo-zeros \n\ny es de orden L (posee L+1 coeficientes)\n\n\n\n**Intepretaci\u00f3n como media movil (MA)**\n\nEl sistema FIR es equivalente a una media movil ponderada que se aplica sobre la entrada, donde los coeficientes del sistema son los ponderadores \n\nPor ejemplo sea un sistema de 3 coeficientes $h[0]=a$, $h[1]=b$ y $h[2]=c$\n\n$$\n\\begin{align}\ny[n] = (h*x)[n] &= \\sum_{j=0}^{2} h[j] x[n-j] \\nonumber \\\\\n&= a x[n] + b x[n-1] + c x[n-2] \\nonumber\n\\end{align}\n$$\n\ndonde cada salida se calcula a partir de \n\n$$\n\\overbrace{x[0], x[1], x[2]}^{y[2]} , x[3], x[4], \\ldots\n$$\n$$\nx[0], \\overbrace{x[1], x[2] , x[3]}^{y[3]}, x[4], \\ldots\n$$\n$$\nx[0], x[1], \\overbrace{x[2] , x[3], x[4]}^{y[4]}, \\ldots\n$$\n\nUn detalle es que para obtener el valor de $y[0]$ e $y[1]$ se deben establecer \"condiciones de borde\", como por ejemplo $x[-2] = x[-1]= 0$\n\nA continuaci\u00f3n veremos algunos ejemplos de aplicaciones usando filtros FIR sencillos\n\n### Eliminando ruido blanco aditivo\n\nEn ciertos problemas la se\u00f1al que observamos $x[n]$ no es exactamente la se\u00f1al que deseamos $s[n]$, sino una versi\u00f3n corrupta de la misma. \n\nUn ejemplo muy t\u00edpico es cuando la se\u00f1al que deseamos estudiar debe ser transmitida desde su lugar de origen hasta nosotros. El canal suele agregar ruido a la se\u00f1al y adicionalmente el ruido aumenta mientras m\u00e1s largo sea el recorrido de nuestra se\u00f1al.\n\n\n\nUn tipo particular de corrupci\u00f3n es el siguiente\n\n$$\nx[n] = s[n] + \\epsilon[n], \\quad \\epsilon[n] \\sim \\mathcal{N}(0,\\sigma^2)\n$$\n\nes decir una corrupci\u00f3n con ruido blanco aditivo y gaussiano, muy com\u00fan en las transmisiones satelitales y espaciales\n\nConsideremos a modo de ejemplo la siguiente se\u00f1al observada que fue contaminada por ruido blanco aditivo:\n\n\n```python\nnp.random.seed(0)\nn = np.arange(0, 200, step=1)\nC = 5*np.exp(-0.5*(n[:, np.newaxis] - n[:, np.newaxis].T)**2/10**2)\n# Se\u00f1al limpia (lo que no conocemos)\ns = np.random.multivariate_normal(np.zeros_like(n), C)/8\n# Datos: Se\u00f1al + ruido (lo que medimos)\nx = s + np.random.randn(len(n))/4\n\np2 = hv.Curve((n, s), 'Tiempo', 'Se\u00f1al', label='Deseada (s)').opts(color='k')\np1 = hv.Scatter((n, x), 'Tiempo', 'Se\u00f1al', label='Observada (x)').opts(width=500, height=250)\nhv.Overlay([p1, p2]).opts(legend_position='top')\n```\n\nPodemos usar un sistema FIR de tipo promediador para \"suavizar la contaminaci\u00f3n\" e intentar recuperar la se\u00f1al deseada\n\nSea un sistema FIR con $L$ coeficientes id\u00e9nticos e iguales a $1/L$\n\n\n```python\nL = 10 \nh = np.ones(shape=(L,))/L\ny = scipy.signal.convolve(x, h, mode='same', method='auto')\n```\n\nLa siguiente gr\u00e1fica interactiva muestra el resultado de convolucionar este sistema con los datos contaminados\n\n\n```python\nhMap1 = hv.HoloMap(kdims='Instante')\nhMap2 = hv.HoloMap(kdims='Instante')\n\nfor m in range(0, len(n)-L, 5): \n c = np.zeros_like(n, dtype=np.float64); \n c[m:m+L] = h\n hMap1[m] = hv.Curve((n, c), 'Tiempo', 'Entrada', label='h').opts(color='r')\n hMap2[m] = hv.Curve((n[:m], y[:m]), 'Tiempo', 'Salida', label='y')\n \np_obs = hv.Scatter((n, x), 'Tiempo', 'Entrada', label='x')\np_clean = hv.Curve((n, s), 'Tiempo', 'Salida', label='s').opts(color='k', height=200)\n(hMap1 * p_obs + hMap2 * p_clean).cols(1).opts(hv.opts.Curve(height=200), \n hv.opts.Overlay(legend_position='top_left'))\n```\n\n:::{note}\n\nEste filtro promedia los datos vecinos resultando una versi\u00f3n suavizada de los mismos. Esta versi\u00f3n suavizada se aproxima a la \"se\u00f1al limpia\" que est\u00e1 escondida en el ruido. \n\n:::\n\nEn general, mientras m\u00e1s \"largo\" sea el filtro mayor ser\u00e1 el efecto de suavizado\n\n### Encontrando cambios en una se\u00f1al\n\nSea la siguiente se\u00f1al escalonada\n\n\n```python\nn = np.arange(0, 100, step=1)\nx = np.zeros_like(n, dtype=np.float64)\nx[20:] += 1.\nx[40:] += 1.\nx[80:] -= 1.\n\nhv.Curve((n, x), 'Tiempo', 'x').opts(width=500, height=200)\n```\n\nSi nos interesa encontrar cambios en la se\u00f1al podemos usar un sistema de tipo diferenciador\n\n$$\ny[n] = (h * x)[n] = \\frac{1}{2}x[n] - \\frac{1}{2} x[n-1]\n$$\n\n\n```python\nh = np.array([0.5, -0.5])\ny = scipy.signal.convolve(x, h, mode='same', method='auto')\n```\n\nLa siguiente gr\u00e1fica interactiva muestra el resultado de convolucionar este sistema con la se\u00f1al escalonada\n\n\n```python\nhMap1 = hv.HoloMap(kdims='Instante')\nhMap2 = hv.HoloMap(kdims='Instante')\n\nfor m in range(0, len(n)-len(h), 5): \n c = np.zeros_like(n, dtype=np.float64); \n c[m:m+len(h)] = h\n hMap1[m] = hv.Curve((n, c), 'Tiempo', label='h')\n hMap2[m] = hv.Curve((n[:m+1], y[:m+1]), 'Tiempo', 'Salida') \n\np_obs = hv.Curve((n, x), 'Tiempo', 'Entrada', label='x')\n(p_obs * hMap1 + hMap2).cols(1).opts(hv.opts.Curve(height=200), \n hv.opts.Overlay(legend_position='top_left'))\n```\n\n:::{note}\n\nLos pulsos en la convoluci\u00f3n (salida) est\u00e1n asociados a un cambio (ascenso o descenso) en la se\u00f1al original\n \n:::\n\nEn un caso m\u00e1s general, este filtro nos da informaci\u00f3n de la derivada de la se\u00f1al, es decir de su velocidad o tasa de cambio\n\n### Remover una tendencia\n\nEn ciertas situaciones la se\u00f1al que nos interesa (se\u00f1al deseada) puede aparecer combinada con otras se\u00f1ales (interferencia)\n\nConsidere el siguiente ejemplo:\n\n\n```python\nnp.random.seed(0); \nn = np.arange(0, 150, step=1)\nC = np.exp(-0.5*(n[:, np.newaxis] - n[:, np.newaxis].T)**2/30**2)\nx_tendencia = 3*np.random.multivariate_normal(np.zeros_like(n), C)+2.5\nx_deseada = np.sin(2.0*np.pi*0.1*n) \nx = x_deseada + x_tendencia\n\np3=hv.Curve((n, x_deseada), 'Tiempo', 'Se\u00f1al', label='Deseada (s)').opts(color='k', alpha=0.75)\np2=hv.Curve((n, x_tendencia), 'Tiempo', 'Se\u00f1al', label='Tendencia').opts(alpha=0.75)\np1=hv.Curve((n, x), 'Tiempo', 'Se\u00f1al', label='Observada (x)').opts(height=250)\nhv.Overlay([p1,p2,p3]).opts(legend_position='bottom_right')\n```\n\nSupongamos que nuestro sensor retorna la se\u00f1al observada (azul) pero que lo que en realidad necesitamos es la se\u00f1al deseada (negra)\n\nEl filtro que se muestra a continuaci\u00f3n es capaz de separar la se\u00f1al deseada (negro) de la tendencia (rojo) a partir de la se\u00f1al observada (azul). En pr\u00f3ximas lecciones veremos como dise\u00f1ar este tipo de filtros\n\n\n```python\nh = np.array([ 0.00099858, 0.00172998, 0.00288273, 0.00437671, 0.00567733, 0.00580422, \n 0.00350188, -0.00245489, -0.01289227, -0.02790429, -0.04665419, -0.06738128,\n -0.08763411, -0.10469179, -0.11608356, 0.88063544, -0.11608356, -0.10469179,\n -0.08763411, -0.06738128, -0.04665419, -0.02790429, -0.01289227, -0.00245489,\n 0.00350188, 0.00580422, 0.00567733, 0.00437671, 0.00288273, 0.00172998,\n 0.00099858])\n\nhv.Curve((h), 'Tiempo', 'Filtro (h)').opts(height=200)\n```\n\n\n```python\ny = scipy.signal.convolve(x, h, mode='same', method='auto')\n```\n\n\n```python\nhMap1 = hv.HoloMap(kdims='Instante')\nhMap2 = hv.HoloMap(kdims='Instante')\n\nfor m in range(0, len(n)-len(h), 5): \n c = np.zeros_like(n, dtype=np.float64); \n c[m:m+len(h)] = h\n hMap1[m] = hv.Curve((n, c), 'Tiempo', 'Entrada', label='h') \n hMap2[m] = hv.Curve((n[:m], y[:m]), 'Tiempo', 'Salida', label='y').opts(ylim=(-1.2, 1.2))\n\np_obs = hv.Curve((n, x), 'Tiempo', 'Se\u00f1al', label='x')\np_target = hv.Curve((n, x_deseada), 'Tiempo', 'Se\u00f1al', label='s').opts(color='k', alpha=.75, width=4) \n(p_obs * hMap1 + hMap2 *p_target).cols(1).opts(hv.opts.Curve(height=200))\n```\n\n## Convoluci\u00f3n con scipy\n\nPodemos convolucionar una se\u00f1al en Python usando \n\n```python\nscipy.signal.convolve(in1, # Se\u00f1al de entrada\n in2, # Coeficientes del sistema\n mode='full', \n method='auto' \n )\n```\n\ndonde el argumento `method` puede ser\n\n- `direct`: Realiza la convoluci\u00f3n en el dominio del tiempo\n- `fft`: Realiza la convoluci\u00f3n multiplicando los espectros\n- `auto`: Se decide automaticamente en base al largo de las se\u00f1ales\n\n\ny el argumento `mode` indica donde se hace la convoluci\u00f3n\n\nPara ejemplificar la influencia de este argumento consideremos una se\u00f1al $x=[a,b,c]$ y un sistema $h=[d, e]$ \n\n- Si uso `mode=valid` el resultado ser\u00e1 $y=[ad+be, bd+ce]$\n- Si uso `mode=same` el resultado ser\u00e1 $y=[ae, ad+be, bd+ce]$, es decir se agregan ceros al principio de $x$ tal que $y$ sea del mismo largo que $x$\n- Si uso `mode=full` el resultado ser\u00e1 $y=[ae, ad+be, bd+ce, cd]$, es decir se agregan ceros al principio y al final de $x$ \n\n\n\n### Eliminando ruido versi\u00f3n 2.0\n\nConsiderando la misma se\u00f1al contaminada con ruido blanco que vimos anteriormente utilizaremos `scipy.signal.convolve` para convolucionar con un filtro que suavice el ruido.\n\nProbaremos dos sistemas FIR\n\n- coeficientes id\u00e9nticos e iguales a $1/L$, es decir una ventana rectangular\n- coeficientes que decaen suavemente a cero, como por ejemplo una ventana de Hamming\n\ncon distintos valores de $L$ (largo del filtro)\n\nEn este caso los valores los obtenemos usando la funci\u00f3n de scipy\n\n```python\n\nscipy.signal.get_window(window, # String con el nombre de la ventana\n Nx, # Entero con el largo de la ventana \n ...)\n```\n\nLa ventana rectangular se llama `boxcar` mientras que la de Hamming se llama `hamming`. En la documentaci\u00f3n de la funci\u00f3n se puede revisar otras ventanas disponibles\n\n\n```python\n# La se\u00f1al de ejemplo de la secci\u00f3n 7.3.1\nnp.random.seed(0)\nn = np.arange(0, 200, step=1)\nC = 5*np.exp(-0.5*(n[:, np.newaxis] - n[:, np.newaxis].T)**2/10**2)\ns = np.random.multivariate_normal(np.zeros_like(n), C)/8 \nx = s + np.random.randn(len(n))/4\n\n# La filtramos con distintas funciones y L para luego visualizar el resultado\nfilters = {}\nresults = {}\nfor L in [5, 10, 20, 30, 40]:\n for filter_name in ['boxcar', 'hamming']: \n h = scipy.signal.get_window(filter_name, L) \n h = h/np.sum(h)\n if not L in filters:\n filters[L] = {}\n results[L] = {}\n filters[L][filter_name] = h\n results[L][filter_name] = scipy.signal.convolve(x, h, mode='same', method='auto')\n```\n\n\n```python\nhMap1 = hv.HoloMap(kdims='L')\nhMap2 = hv.HoloMap(kdims='L')\n\nfor L, curves in filters.items():\n p = []\n for filter_name, h in curves.items():\n p.append(hv.Curve((range(L), h), 'Largo del filtro (L)', 'Filtro (h)', label=filter_name))\n hMap1[L] = hv.Overlay(p)\n\nfor L, curves in results.items():\n p = []\n for filter_name, y in curves.items():\n p.append(hv.Curve((n, y), 'Tiempo', 'Se\u00f1al', label=filter_name).opts(line_width=2, alpha=0.75))\n hMap2[L] = hv.Overlay(p)\n \np_clean = hv.Curve((n, s), 'Tiempo', 'Se\u00f1al', label='Deseada (s)').opts(color='k')\np_data = hv.Scatter((n, x), 'Tiempo', 'Datos (x)').opts(width=500, height=200)\n(p_data + hMap1 + hMap2 * p_clean).cols(1).opts(hv.opts.Curve(height=250), \n hv.opts.Overlay(legend_position='top'))\n```\n\n:::{note}\n\nMientras m\u00e1s larga es la ventana (L) mayor es el suavizado. Adicionalmente la ventana de Hamming produce un filtrado m\u00e1s suave que la rectangular\n\n:::\n\nEn la lecci\u00f3n siguiente veremos como dise\u00f1ar un filtro, es decir como calcular los valores de `h` para resolver una tarea en particular\n\n\n```python\n\n```\n", "meta": {"hexsha": "c466b2c029213ea408fbdd14c0fe7ce91f2d32a9", "size": 29836, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lectures/unit2/lecture2.ipynb", "max_stars_repo_name": "phuijse/UACH-INFO183", "max_stars_repo_head_hexsha": "0e1b6bef0bd80cda2753bd11e62016268f2de638", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 11, "max_stars_repo_stars_event_min_datetime": "2018-08-27T23:53:15.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-16T23:31:05.000Z", "max_issues_repo_path": "lectures/unit2/lecture2.ipynb", "max_issues_repo_name": "phuijse/UACH-INFO183", "max_issues_repo_head_hexsha": "0e1b6bef0bd80cda2753bd11e62016268f2de638", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "lectures/unit2/lecture2.ipynb", "max_forks_repo_name": "phuijse/UACH-INFO183", "max_forks_repo_head_hexsha": "0e1b6bef0bd80cda2753bd11e62016268f2de638", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 10, "max_forks_repo_forks_event_min_datetime": "2019-01-04T17:43:55.000Z", "max_forks_repo_forks_event_max_datetime": "2021-09-07T16:07:18.000Z", "avg_line_length": 30.2289766971, "max_line_length": 262, "alphanum_fraction": 0.5334830406, "converted": true, "num_tokens": 6057, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.531209388216861, "lm_q2_score": 0.8056321796478255, "lm_q1q2_score": 0.42795937727853767}} {"text": "\n\n\n# Generating C code for the right-hand sides of Maxwell's equations, in ***curvilinear*** coordinates, using a reference metric formalism\n\n## Author: Ian Ruchlin\n### Formatting improvements courtesy Brandon Clark\n\n[comment]: <> (Abstract: TODO)\n\n### The following formulations of Maxwell's equations, called System I and System II, are described in [Illustrating Stability Properties of Numerical Relativity in Electrodynamics](https://arxiv.org/abs/gr-qc/0201051) by Knapp et al.\n\n**Notebook Status:** In progress \n\n**Validation Notes:** This module has not yet undergone validation testing. Do ***not*** use it until after appropriate validation testing has been performed.\n\n## Introduction:\n[Maxwell's equations](https://en.wikipedia.org/wiki/Maxwell%27s_equations) are subject to the Gauss' law constraint\n$$\\mathcal{C} \\equiv \\hat{D}_{i} E^{i} - 4 \\pi \\rho = 0 \\; ,$$\nwhere $E^{i}$ is the electric vector field, $\\hat{D}_{i}$ is the [covariant derivative](https://en.wikipedia.org/wiki/Covariant_derivative) associated with the reference metric $\\hat{\\gamma}_{i j}$ (which is taken to represent flat space), and $\\rho$ is the electric charge density. We use $\\mathcal{C}$ as a measure of numerical error. Maxwell's equations are also required to satisfy $\\hat{D}_{i} B^{i} = 0$, where $B^{i}$ is the magnetic vector field. The magnetic constraint implies that the magnetic field can be expressed as\n$$B_{i} = \\epsilon_{i j k} \\hat{D}^{j} A^{k} \\; ,$$\nwhere $\\epsilon_{i j k}$ is the totally antisymmetric [Levi-Civita tensor](https://en.wikipedia.org/wiki/Levi-Civita_symbol) and $A^{i}$ is the vector potential field. Together with the scalar potential $\\psi$, the electric field can be expressed in terms of the potential fields as\n$$E_{i} = -\\hat{D}_{i} \\psi - \\partial_{t} A_{i} \\; .$$\nFor now, we work in vacuum, where the electric charge density and the electric current density vector both vanish ($\\rho = 0$ and $j_{i} = 0$).\n\nIn addition to the Gauss constraints, the electric and magnetic fields obey two independent [electromagnetic invariants](https://en.wikipedia.org/wiki/Classification_of_electromagnetic_fields#Invariants)\n\\begin{align}\n\\mathcal{P} &\\equiv B_{i} B^{i} - E_{i} E^{i} \\; , \\\\\n\\mathcal{Q} &\\equiv E_{i} B^{i} \\; .\n\\end{align}\nIn vacuum, these satisfy $\\mathcal{P} = \\mathcal{Q} = 0$.\n\n\n\n# Table of Contents:\n$$\\label{toc}$$\n\nThis notebook is organized as follows\n\n1. [Step 1](#sys1): System I\n1. [Step 2](#sys2): System II\n1. [Step 3](#latex_pdf_output): Output this notebook to $\\LaTeX$-formatted PDF file\n\n\n\n# Step 1: System I \\[Back to [top](#toc)\\]\n$$\\label{sys1}$$\n\nIn terms of the above definitions, the evolution Maxwell's equations take the form\n\\begin{align}\n\\partial_{t} A_{i} &= -E_{i} - \\hat{D}_{i} \\psi \\; , \\\\\n\\partial_{t} E_{i} &= -\\hat{D}_{j} \\hat{D}^{j} A_{i} + \\hat{D}_{i} \\hat{D}_{j} A^{j}\\; , \\\\\n\\partial_{t} \\psi &= -\\hat{D}_{i} A^{i} \\; .\n\\end{align}\nNote that this coupled system contains mixed second derivatives in the second term on the right hand side of the $E^{i}$ evolution equation. We will revisit this fact when building System II.\n\nIt can be shown that the Gauss constraint satisfies the evolution equation\n$$\\partial_{t} \\mathcal{C} = 0 \\; .$$\nThis implies that any constraint violating numerical error remains fixed in place during the evolution. This becomes problematic when the violations grow large and spoil the physics of the simulation.\n\n\n```python\nimport NRPy_param_funcs as par # NRPy+: parameter interface\nimport indexedexp as ixp # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support\nimport grid as gri # NRPy+: Functions having to do with numerical grids\nimport finite_difference as fin # NRPy+: Finite difference C code generation module\nimport reference_metric as rfm # NRPy+: Reference metric support\nfrom outputC import lhrh # NRPy+: Core C code output module\n\npar.set_parval_from_str(\"reference_metric::CoordSystem\", \"Spherical\")\npar.set_parval_from_str(\"grid::DIM\", 3)\n\nrfm.reference_metric()\n\n# The name of this module (\"maxwell\") is given by __name__:\nthismodule = __name__\n\n# Step 0: Read the spatial dimension parameter as DIM.\nDIM = par.parval_from_str(\"grid::DIM\")\n\n# Step 1: Set the finite differencing order to 4.\npar.set_parval_from_str(\"finite_difference::FD_CENTDERIVS_ORDER\", 4)\n\n# Step 2: Register gridfunctions that are needed as input.\npsi = gri.register_gridfunctions(\"EVOL\", [\"psi\"])\n\n# Step 3a: Declare the rank-1 indexed expressions E_{i}, A_{i},\n# and \\partial_{i} \\psi. Derivative variables like these\n# must have an underscore in them, so the finite\n# difference module can parse the variable name properly.\nED = ixp.register_gridfunctions_for_single_rank1(\"EVOL\", \"ED\")\nAD = ixp.register_gridfunctions_for_single_rank1(\"EVOL\", \"AD\")\npsi_dD = ixp.declarerank1(\"psi_dD\")\n\n# Step 3b: Declare the rank-2 indexed expression \\partial_{j} A_{i},\n# which is not symmetric in its indices.\n# Derivative variables like these must have an underscore\n# in them, so the finite difference module can parse the\n# variable name properly.\nAD_dD = ixp.declarerank2(\"AD_dD\", \"nosym\")\n\n# Step 3c: Declare the rank-3 indexed expression \\partial_{jk} A_{i},\n# which is symmetric in the two {jk} indices.\nAD_dDD = ixp.declarerank3(\"AD_dDD\", \"sym12\")\n\n# Step 4: Calculate first and second covariant derivatives, and the\n# necessary contractions.\n# First covariant derivative\n# D_{j} A_{i} = A_{i,j} - \\Gamma^{k}_{ij} A_{k}\nAD_dHatD = ixp.zerorank2()\nfor i in range(DIM):\n for j in range(DIM):\n AD_dHatD[i][j] = AD_dD[i][j]\n for k in range(DIM):\n AD_dHatD[i][j] -= rfm.GammahatUDD[k][i][j] * AD[k]\n\n# Second covariant derivative\n# D_{k} D_{j} A_{i} = \\partial_{k} D_{j} A_{i} - \\Gamma^{l}_{jk} D_{l} A_{i}\n# - \\Gamma^{l}_{ik} D_{j} A_{l}\n# = A_{i,jk}\n# - \\Gamma^{l}_{ij,k} A_{l}\n# - \\Gamma^{l}_{ij} A_{l,k}\n# - \\Gamma^{l}_{jk} A_{i;\\hat{l}}\n# - \\Gamma^{l}_{ik} A_{l;\\hat{j}}\nAD_dHatDD = ixp.zerorank3()\nfor i in range(DIM):\n for j in range(DIM):\n for k in range(DIM):\n AD_dHatDD[i][j][k] = AD_dDD[i][j][k]\n for l in range(DIM):\n AD_dHatDD[i][j][k] += - rfm.GammahatUDDdD[l][i][j][k] * AD[l] \\\n - rfm.GammahatUDD[l][i][j] * AD_dD[l][k] \\\n - rfm.GammahatUDD[l][j][k] * AD_dHatD[i][l] \\\n - rfm.GammahatUDD[l][i][k] * AD_dHatD[l][j]\n\n# Covariant divergence\n# D_{i} A^{i} = ghat^{ij} D_{j} A_{i}\nDivA = 0\n# Gradient of covariant divergence\n# DivA_dD_{i} = ghat^{jk} A_{k;\\hat{j}\\hat{i}}\nDivA_dD = ixp.zerorank1()\n# Covariant Laplacian\n# LapAD_{i} = ghat^{jk} A_{i;\\hat{j}\\hat{k}}\nLapAD = ixp.zerorank1()\nfor i in range(DIM):\n for j in range(DIM):\n DivA += rfm.ghatUU[i][j] * AD_dHatD[i][j]\n for k in range(DIM):\n DivA_dD[i] += rfm.ghatUU[j][k] * AD_dHatDD[k][j][i]\n LapAD[i] += rfm.ghatUU[j][k] * AD_dHatDD[i][j][k]\n\n# Step 5: Define right-hand sides for the evolution.\nAD_rhs = ixp.zerorank1()\nED_rhs = ixp.zerorank1()\nfor i in range(DIM):\n AD_rhs[i] = -ED[i] - psi_dD[i]\n ED_rhs[i] = -LapAD[i] + DivA_dD[i]\npsi_rhs = -DivA\n\n# Step 6: Generate C code for System I Maxwell's evolution equations,\n# print output to the screen (standard out, or stdout).\nlhrh_list = []\nfor i in range(DIM):\n lhrh_list.append(lhrh(lhs=gri.gfaccess(\"rhs_gfs\", \"AD\" + str(i)), rhs=AD_rhs[i]))\n lhrh_list.append(lhrh(lhs=gri.gfaccess(\"rhs_gfs\", \"ED\" + str(i)), rhs=ED_rhs[i]))\nlhrh_list.append(lhrh(lhs=gri.gfaccess(\"rhs_gfs\", \"psi\"), rhs=psi_rhs))\n\nfin.FD_outputC(\"stdout\", lhrh_list)\n```\n\n {\n /*\n * NRPy+ Finite Difference Code Generation, Step 1 of 2: Read from main memory and compute finite difference stencils:\n */\n /*\n * Original SymPy expressions:\n * \"[const double AD_dD00 = invdx0*(-2*AD0_i0m1_i1_i2/3 + AD0_i0m2_i1_i2/12 + 2*AD0_i0p1_i1_i2/3 - AD0_i0p2_i1_i2/12),\n * const double AD_dD01 = invdx1*(-2*AD0_i0_i1m1_i2/3 + AD0_i0_i1m2_i2/12 + 2*AD0_i0_i1p1_i2/3 - AD0_i0_i1p2_i2/12),\n * const double AD_dD02 = invdx2*(-2*AD0_i0_i1_i2m1/3 + AD0_i0_i1_i2m2/12 + 2*AD0_i0_i1_i2p1/3 - AD0_i0_i1_i2p2/12),\n * const double AD_dD10 = invdx0*(-2*AD1_i0m1_i1_i2/3 + AD1_i0m2_i1_i2/12 + 2*AD1_i0p1_i1_i2/3 - AD1_i0p2_i1_i2/12),\n * const double AD_dD11 = invdx1*(-2*AD1_i0_i1m1_i2/3 + AD1_i0_i1m2_i2/12 + 2*AD1_i0_i1p1_i2/3 - AD1_i0_i1p2_i2/12),\n * const double AD_dD12 = invdx2*(-2*AD1_i0_i1_i2m1/3 + AD1_i0_i1_i2m2/12 + 2*AD1_i0_i1_i2p1/3 - AD1_i0_i1_i2p2/12),\n * const double AD_dD20 = invdx0*(-2*AD2_i0m1_i1_i2/3 + AD2_i0m2_i1_i2/12 + 2*AD2_i0p1_i1_i2/3 - AD2_i0p2_i1_i2/12),\n * const double AD_dD21 = invdx1*(-2*AD2_i0_i1m1_i2/3 + AD2_i0_i1m2_i2/12 + 2*AD2_i0_i1p1_i2/3 - AD2_i0_i1p2_i2/12),\n * const double AD_dD22 = invdx2*(-2*AD2_i0_i1_i2m1/3 + AD2_i0_i1_i2m2/12 + 2*AD2_i0_i1_i2p1/3 - AD2_i0_i1_i2p2/12),\n * const double AD_dDD001 = invdx0*invdx1*(4*AD0_i0m1_i1m1_i2/9 - AD0_i0m1_i1m2_i2/18 - 4*AD0_i0m1_i1p1_i2/9 + AD0_i0m1_i1p2_i2/18 - AD0_i0m2_i1m1_i2/18 + AD0_i0m2_i1m2_i2/144 + AD0_i0m2_i1p1_i2/18 - AD0_i0m2_i1p2_i2/144 - 4*AD0_i0p1_i1m1_i2/9 + AD0_i0p1_i1m2_i2/18 + 4*AD0_i0p1_i1p1_i2/9 - AD0_i0p1_i1p2_i2/18 + AD0_i0p2_i1m1_i2/18 - AD0_i0p2_i1m2_i2/144 - AD0_i0p2_i1p1_i2/18 + AD0_i0p2_i1p2_i2/144),\n * const double AD_dDD002 = invdx0*invdx2*(4*AD0_i0m1_i1_i2m1/9 - AD0_i0m1_i1_i2m2/18 - 4*AD0_i0m1_i1_i2p1/9 + AD0_i0m1_i1_i2p2/18 - AD0_i0m2_i1_i2m1/18 + AD0_i0m2_i1_i2m2/144 + AD0_i0m2_i1_i2p1/18 - AD0_i0m2_i1_i2p2/144 - 4*AD0_i0p1_i1_i2m1/9 + AD0_i0p1_i1_i2m2/18 + 4*AD0_i0p1_i1_i2p1/9 - AD0_i0p1_i1_i2p2/18 + AD0_i0p2_i1_i2m1/18 - AD0_i0p2_i1_i2m2/144 - AD0_i0p2_i1_i2p1/18 + AD0_i0p2_i1_i2p2/144),\n * const double AD_dDD011 = invdx1**2*(-5*AD0/2 + 4*AD0_i0_i1m1_i2/3 - AD0_i0_i1m2_i2/12 + 4*AD0_i0_i1p1_i2/3 - AD0_i0_i1p2_i2/12),\n * const double AD_dDD022 = invdx2**2*(-5*AD0/2 + 4*AD0_i0_i1_i2m1/3 - AD0_i0_i1_i2m2/12 + 4*AD0_i0_i1_i2p1/3 - AD0_i0_i1_i2p2/12),\n * const double AD_dDD100 = invdx0**2*(-5*AD1/2 + 4*AD1_i0m1_i1_i2/3 - AD1_i0m2_i1_i2/12 + 4*AD1_i0p1_i1_i2/3 - AD1_i0p2_i1_i2/12),\n * const double AD_dDD101 = invdx0*invdx1*(4*AD1_i0m1_i1m1_i2/9 - AD1_i0m1_i1m2_i2/18 - 4*AD1_i0m1_i1p1_i2/9 + AD1_i0m1_i1p2_i2/18 - AD1_i0m2_i1m1_i2/18 + AD1_i0m2_i1m2_i2/144 + AD1_i0m2_i1p1_i2/18 - AD1_i0m2_i1p2_i2/144 - 4*AD1_i0p1_i1m1_i2/9 + AD1_i0p1_i1m2_i2/18 + 4*AD1_i0p1_i1p1_i2/9 - AD1_i0p1_i1p2_i2/18 + AD1_i0p2_i1m1_i2/18 - AD1_i0p2_i1m2_i2/144 - AD1_i0p2_i1p1_i2/18 + AD1_i0p2_i1p2_i2/144),\n * const double AD_dDD112 = invdx1*invdx2*(4*AD1_i0_i1m1_i2m1/9 - AD1_i0_i1m1_i2m2/18 - 4*AD1_i0_i1m1_i2p1/9 + AD1_i0_i1m1_i2p2/18 - AD1_i0_i1m2_i2m1/18 + AD1_i0_i1m2_i2m2/144 + AD1_i0_i1m2_i2p1/18 - AD1_i0_i1m2_i2p2/144 - 4*AD1_i0_i1p1_i2m1/9 + AD1_i0_i1p1_i2m2/18 + 4*AD1_i0_i1p1_i2p1/9 - AD1_i0_i1p1_i2p2/18 + AD1_i0_i1p2_i2m1/18 - AD1_i0_i1p2_i2m2/144 - AD1_i0_i1p2_i2p1/18 + AD1_i0_i1p2_i2p2/144),\n * const double AD_dDD122 = invdx2**2*(-5*AD1/2 + 4*AD1_i0_i1_i2m1/3 - AD1_i0_i1_i2m2/12 + 4*AD1_i0_i1_i2p1/3 - AD1_i0_i1_i2p2/12),\n * const double AD_dDD200 = invdx0**2*(-5*AD2/2 + 4*AD2_i0m1_i1_i2/3 - AD2_i0m2_i1_i2/12 + 4*AD2_i0p1_i1_i2/3 - AD2_i0p2_i1_i2/12),\n * const double AD_dDD202 = invdx0*invdx2*(4*AD2_i0m1_i1_i2m1/9 - AD2_i0m1_i1_i2m2/18 - 4*AD2_i0m1_i1_i2p1/9 + AD2_i0m1_i1_i2p2/18 - AD2_i0m2_i1_i2m1/18 + AD2_i0m2_i1_i2m2/144 + AD2_i0m2_i1_i2p1/18 - AD2_i0m2_i1_i2p2/144 - 4*AD2_i0p1_i1_i2m1/9 + AD2_i0p1_i1_i2m2/18 + 4*AD2_i0p1_i1_i2p1/9 - AD2_i0p1_i1_i2p2/18 + AD2_i0p2_i1_i2m1/18 - AD2_i0p2_i1_i2m2/144 - AD2_i0p2_i1_i2p1/18 + AD2_i0p2_i1_i2p2/144),\n * const double AD_dDD211 = invdx1**2*(-5*AD2/2 + 4*AD2_i0_i1m1_i2/3 - AD2_i0_i1m2_i2/12 + 4*AD2_i0_i1p1_i2/3 - AD2_i0_i1p2_i2/12),\n * const double AD_dDD212 = invdx1*invdx2*(4*AD2_i0_i1m1_i2m1/9 - AD2_i0_i1m1_i2m2/18 - 4*AD2_i0_i1m1_i2p1/9 + AD2_i0_i1m1_i2p2/18 - AD2_i0_i1m2_i2m1/18 + AD2_i0_i1m2_i2m2/144 + AD2_i0_i1m2_i2p1/18 - AD2_i0_i1m2_i2p2/144 - 4*AD2_i0_i1p1_i2m1/9 + AD2_i0_i1p1_i2m2/18 + 4*AD2_i0_i1p1_i2p1/9 - AD2_i0_i1p1_i2p2/18 + AD2_i0_i1p2_i2m1/18 - AD2_i0_i1p2_i2m2/144 - AD2_i0_i1p2_i2p1/18 + AD2_i0_i1p2_i2p2/144),\n * const double psi_dD0 = invdx0*(-2*psi_i0m1_i1_i2/3 + psi_i0m2_i1_i2/12 + 2*psi_i0p1_i1_i2/3 - psi_i0p2_i1_i2/12),\n * const double psi_dD1 = invdx1*(-2*psi_i0_i1m1_i2/3 + psi_i0_i1m2_i2/12 + 2*psi_i0_i1p1_i2/3 - psi_i0_i1p2_i2/12),\n * const double psi_dD2 = invdx2*(-2*psi_i0_i1_i2m1/3 + psi_i0_i1_i2m2/12 + 2*psi_i0_i1_i2p1/3 - psi_i0_i1_i2p2/12)]\"\n */\n const double psi_i0_i1_i2m2 = in_gfs[IDX4(PSIGF, i0,i1,i2-2)];\n const double psi_i0_i1_i2m1 = in_gfs[IDX4(PSIGF, i0,i1,i2-1)];\n const double psi_i0_i1m2_i2 = in_gfs[IDX4(PSIGF, i0,i1-2,i2)];\n const double psi_i0_i1m1_i2 = in_gfs[IDX4(PSIGF, i0,i1-1,i2)];\n const double psi_i0m2_i1_i2 = in_gfs[IDX4(PSIGF, i0-2,i1,i2)];\n const double psi_i0m1_i1_i2 = in_gfs[IDX4(PSIGF, i0-1,i1,i2)];\n const double psi_i0p1_i1_i2 = in_gfs[IDX4(PSIGF, i0+1,i1,i2)];\n const double psi_i0p2_i1_i2 = in_gfs[IDX4(PSIGF, i0+2,i1,i2)];\n const double psi_i0_i1p1_i2 = in_gfs[IDX4(PSIGF, i0,i1+1,i2)];\n const double psi_i0_i1p2_i2 = in_gfs[IDX4(PSIGF, i0,i1+2,i2)];\n const double psi_i0_i1_i2p1 = in_gfs[IDX4(PSIGF, i0,i1,i2+1)];\n const double psi_i0_i1_i2p2 = in_gfs[IDX4(PSIGF, i0,i1,i2+2)];\n const double ED0 = in_gfs[IDX4(ED0GF, i0,i1,i2)];\n const double ED1 = in_gfs[IDX4(ED1GF, i0,i1,i2)];\n const double ED2 = in_gfs[IDX4(ED2GF, i0,i1,i2)];\n const double AD0_i0m2_i1_i2m2 = in_gfs[IDX4(AD0GF, i0-2,i1,i2-2)];\n const double AD0_i0m1_i1_i2m2 = in_gfs[IDX4(AD0GF, i0-1,i1,i2-2)];\n const double AD0_i0_i1_i2m2 = in_gfs[IDX4(AD0GF, i0,i1,i2-2)];\n const double AD0_i0p1_i1_i2m2 = in_gfs[IDX4(AD0GF, i0+1,i1,i2-2)];\n const double AD0_i0p2_i1_i2m2 = in_gfs[IDX4(AD0GF, i0+2,i1,i2-2)];\n const double AD0_i0m2_i1_i2m1 = in_gfs[IDX4(AD0GF, i0-2,i1,i2-1)];\n const double AD0_i0m1_i1_i2m1 = in_gfs[IDX4(AD0GF, i0-1,i1,i2-1)];\n const double AD0_i0_i1_i2m1 = in_gfs[IDX4(AD0GF, i0,i1,i2-1)];\n const double AD0_i0p1_i1_i2m1 = in_gfs[IDX4(AD0GF, i0+1,i1,i2-1)];\n const double AD0_i0p2_i1_i2m1 = in_gfs[IDX4(AD0GF, i0+2,i1,i2-1)];\n const double AD0_i0m2_i1m2_i2 = in_gfs[IDX4(AD0GF, i0-2,i1-2,i2)];\n const double AD0_i0m1_i1m2_i2 = in_gfs[IDX4(AD0GF, i0-1,i1-2,i2)];\n const double AD0_i0_i1m2_i2 = in_gfs[IDX4(AD0GF, i0,i1-2,i2)];\n const double AD0_i0p1_i1m2_i2 = in_gfs[IDX4(AD0GF, i0+1,i1-2,i2)];\n const double AD0_i0p2_i1m2_i2 = in_gfs[IDX4(AD0GF, i0+2,i1-2,i2)];\n const double AD0_i0m2_i1m1_i2 = in_gfs[IDX4(AD0GF, i0-2,i1-1,i2)];\n const double AD0_i0m1_i1m1_i2 = in_gfs[IDX4(AD0GF, i0-1,i1-1,i2)];\n const double AD0_i0_i1m1_i2 = in_gfs[IDX4(AD0GF, i0,i1-1,i2)];\n const double AD0_i0p1_i1m1_i2 = in_gfs[IDX4(AD0GF, i0+1,i1-1,i2)];\n const double AD0_i0p2_i1m1_i2 = in_gfs[IDX4(AD0GF, i0+2,i1-1,i2)];\n const double AD0_i0m2_i1_i2 = in_gfs[IDX4(AD0GF, i0-2,i1,i2)];\n const double AD0_i0m1_i1_i2 = in_gfs[IDX4(AD0GF, i0-1,i1,i2)];\n const double AD0 = in_gfs[IDX4(AD0GF, i0,i1,i2)];\n const double AD0_i0p1_i1_i2 = in_gfs[IDX4(AD0GF, i0+1,i1,i2)];\n const double AD0_i0p2_i1_i2 = in_gfs[IDX4(AD0GF, i0+2,i1,i2)];\n const double AD0_i0m2_i1p1_i2 = in_gfs[IDX4(AD0GF, i0-2,i1+1,i2)];\n const double AD0_i0m1_i1p1_i2 = in_gfs[IDX4(AD0GF, i0-1,i1+1,i2)];\n const double AD0_i0_i1p1_i2 = in_gfs[IDX4(AD0GF, i0,i1+1,i2)];\n const double AD0_i0p1_i1p1_i2 = in_gfs[IDX4(AD0GF, i0+1,i1+1,i2)];\n const double AD0_i0p2_i1p1_i2 = in_gfs[IDX4(AD0GF, i0+2,i1+1,i2)];\n const double AD0_i0m2_i1p2_i2 = in_gfs[IDX4(AD0GF, i0-2,i1+2,i2)];\n const double AD0_i0m1_i1p2_i2 = in_gfs[IDX4(AD0GF, i0-1,i1+2,i2)];\n const double AD0_i0_i1p2_i2 = in_gfs[IDX4(AD0GF, i0,i1+2,i2)];\n const double AD0_i0p1_i1p2_i2 = in_gfs[IDX4(AD0GF, i0+1,i1+2,i2)];\n const double AD0_i0p2_i1p2_i2 = in_gfs[IDX4(AD0GF, i0+2,i1+2,i2)];\n const double AD0_i0m2_i1_i2p1 = in_gfs[IDX4(AD0GF, i0-2,i1,i2+1)];\n const double AD0_i0m1_i1_i2p1 = in_gfs[IDX4(AD0GF, i0-1,i1,i2+1)];\n const double AD0_i0_i1_i2p1 = in_gfs[IDX4(AD0GF, i0,i1,i2+1)];\n const double AD0_i0p1_i1_i2p1 = in_gfs[IDX4(AD0GF, i0+1,i1,i2+1)];\n const double AD0_i0p2_i1_i2p1 = in_gfs[IDX4(AD0GF, i0+2,i1,i2+1)];\n const double AD0_i0m2_i1_i2p2 = in_gfs[IDX4(AD0GF, i0-2,i1,i2+2)];\n const double AD0_i0m1_i1_i2p2 = in_gfs[IDX4(AD0GF, i0-1,i1,i2+2)];\n const double AD0_i0_i1_i2p2 = in_gfs[IDX4(AD0GF, i0,i1,i2+2)];\n const double AD0_i0p1_i1_i2p2 = in_gfs[IDX4(AD0GF, i0+1,i1,i2+2)];\n const double AD0_i0p2_i1_i2p2 = in_gfs[IDX4(AD0GF, i0+2,i1,i2+2)];\n const double AD1_i0_i1m2_i2m2 = in_gfs[IDX4(AD1GF, i0,i1-2,i2-2)];\n const double AD1_i0_i1m1_i2m2 = in_gfs[IDX4(AD1GF, i0,i1-1,i2-2)];\n const double AD1_i0_i1_i2m2 = in_gfs[IDX4(AD1GF, i0,i1,i2-2)];\n const double AD1_i0_i1p1_i2m2 = in_gfs[IDX4(AD1GF, i0,i1+1,i2-2)];\n const double AD1_i0_i1p2_i2m2 = in_gfs[IDX4(AD1GF, i0,i1+2,i2-2)];\n const double AD1_i0_i1m2_i2m1 = in_gfs[IDX4(AD1GF, i0,i1-2,i2-1)];\n const double AD1_i0_i1m1_i2m1 = in_gfs[IDX4(AD1GF, i0,i1-1,i2-1)];\n const double AD1_i0_i1_i2m1 = in_gfs[IDX4(AD1GF, i0,i1,i2-1)];\n const double AD1_i0_i1p1_i2m1 = in_gfs[IDX4(AD1GF, i0,i1+1,i2-1)];\n const double AD1_i0_i1p2_i2m1 = in_gfs[IDX4(AD1GF, i0,i1+2,i2-1)];\n const double AD1_i0m2_i1m2_i2 = in_gfs[IDX4(AD1GF, i0-2,i1-2,i2)];\n const double AD1_i0m1_i1m2_i2 = in_gfs[IDX4(AD1GF, i0-1,i1-2,i2)];\n const double AD1_i0_i1m2_i2 = in_gfs[IDX4(AD1GF, i0,i1-2,i2)];\n const double AD1_i0p1_i1m2_i2 = in_gfs[IDX4(AD1GF, i0+1,i1-2,i2)];\n const double AD1_i0p2_i1m2_i2 = in_gfs[IDX4(AD1GF, i0+2,i1-2,i2)];\n const double AD1_i0m2_i1m1_i2 = in_gfs[IDX4(AD1GF, i0-2,i1-1,i2)];\n const double AD1_i0m1_i1m1_i2 = in_gfs[IDX4(AD1GF, i0-1,i1-1,i2)];\n const double AD1_i0_i1m1_i2 = in_gfs[IDX4(AD1GF, i0,i1-1,i2)];\n const double AD1_i0p1_i1m1_i2 = in_gfs[IDX4(AD1GF, i0+1,i1-1,i2)];\n const double AD1_i0p2_i1m1_i2 = in_gfs[IDX4(AD1GF, i0+2,i1-1,i2)];\n const double AD1_i0m2_i1_i2 = in_gfs[IDX4(AD1GF, i0-2,i1,i2)];\n const double AD1_i0m1_i1_i2 = in_gfs[IDX4(AD1GF, i0-1,i1,i2)];\n const double AD1 = in_gfs[IDX4(AD1GF, i0,i1,i2)];\n const double AD1_i0p1_i1_i2 = in_gfs[IDX4(AD1GF, i0+1,i1,i2)];\n const double AD1_i0p2_i1_i2 = in_gfs[IDX4(AD1GF, i0+2,i1,i2)];\n const double AD1_i0m2_i1p1_i2 = in_gfs[IDX4(AD1GF, i0-2,i1+1,i2)];\n const double AD1_i0m1_i1p1_i2 = in_gfs[IDX4(AD1GF, i0-1,i1+1,i2)];\n const double AD1_i0_i1p1_i2 = in_gfs[IDX4(AD1GF, i0,i1+1,i2)];\n const double AD1_i0p1_i1p1_i2 = in_gfs[IDX4(AD1GF, i0+1,i1+1,i2)];\n const double AD1_i0p2_i1p1_i2 = in_gfs[IDX4(AD1GF, i0+2,i1+1,i2)];\n const double AD1_i0m2_i1p2_i2 = in_gfs[IDX4(AD1GF, i0-2,i1+2,i2)];\n const double AD1_i0m1_i1p2_i2 = in_gfs[IDX4(AD1GF, i0-1,i1+2,i2)];\n const double AD1_i0_i1p2_i2 = in_gfs[IDX4(AD1GF, i0,i1+2,i2)];\n const double AD1_i0p1_i1p2_i2 = in_gfs[IDX4(AD1GF, i0+1,i1+2,i2)];\n const double AD1_i0p2_i1p2_i2 = in_gfs[IDX4(AD1GF, i0+2,i1+2,i2)];\n const double AD1_i0_i1m2_i2p1 = in_gfs[IDX4(AD1GF, i0,i1-2,i2+1)];\n const double AD1_i0_i1m1_i2p1 = in_gfs[IDX4(AD1GF, i0,i1-1,i2+1)];\n const double AD1_i0_i1_i2p1 = in_gfs[IDX4(AD1GF, i0,i1,i2+1)];\n const double AD1_i0_i1p1_i2p1 = in_gfs[IDX4(AD1GF, i0,i1+1,i2+1)];\n const double AD1_i0_i1p2_i2p1 = in_gfs[IDX4(AD1GF, i0,i1+2,i2+1)];\n const double AD1_i0_i1m2_i2p2 = in_gfs[IDX4(AD1GF, i0,i1-2,i2+2)];\n const double AD1_i0_i1m1_i2p2 = in_gfs[IDX4(AD1GF, i0,i1-1,i2+2)];\n const double AD1_i0_i1_i2p2 = in_gfs[IDX4(AD1GF, i0,i1,i2+2)];\n const double AD1_i0_i1p1_i2p2 = in_gfs[IDX4(AD1GF, i0,i1+1,i2+2)];\n const double AD1_i0_i1p2_i2p2 = in_gfs[IDX4(AD1GF, i0,i1+2,i2+2)];\n const double AD2_i0_i1m2_i2m2 = in_gfs[IDX4(AD2GF, i0,i1-2,i2-2)];\n const double AD2_i0_i1m1_i2m2 = in_gfs[IDX4(AD2GF, i0,i1-1,i2-2)];\n const double AD2_i0m2_i1_i2m2 = in_gfs[IDX4(AD2GF, i0-2,i1,i2-2)];\n const double AD2_i0m1_i1_i2m2 = in_gfs[IDX4(AD2GF, i0-1,i1,i2-2)];\n const double AD2_i0_i1_i2m2 = in_gfs[IDX4(AD2GF, i0,i1,i2-2)];\n const double AD2_i0p1_i1_i2m2 = in_gfs[IDX4(AD2GF, i0+1,i1,i2-2)];\n const double AD2_i0p2_i1_i2m2 = in_gfs[IDX4(AD2GF, i0+2,i1,i2-2)];\n const double AD2_i0_i1p1_i2m2 = in_gfs[IDX4(AD2GF, i0,i1+1,i2-2)];\n const double AD2_i0_i1p2_i2m2 = in_gfs[IDX4(AD2GF, i0,i1+2,i2-2)];\n const double AD2_i0_i1m2_i2m1 = in_gfs[IDX4(AD2GF, i0,i1-2,i2-1)];\n const double AD2_i0_i1m1_i2m1 = in_gfs[IDX4(AD2GF, i0,i1-1,i2-1)];\n const double AD2_i0m2_i1_i2m1 = in_gfs[IDX4(AD2GF, i0-2,i1,i2-1)];\n const double AD2_i0m1_i1_i2m1 = in_gfs[IDX4(AD2GF, i0-1,i1,i2-1)];\n const double AD2_i0_i1_i2m1 = in_gfs[IDX4(AD2GF, i0,i1,i2-1)];\n const double AD2_i0p1_i1_i2m1 = in_gfs[IDX4(AD2GF, i0+1,i1,i2-1)];\n const double AD2_i0p2_i1_i2m1 = in_gfs[IDX4(AD2GF, i0+2,i1,i2-1)];\n const double AD2_i0_i1p1_i2m1 = in_gfs[IDX4(AD2GF, i0,i1+1,i2-1)];\n const double AD2_i0_i1p2_i2m1 = in_gfs[IDX4(AD2GF, i0,i1+2,i2-1)];\n const double AD2_i0_i1m2_i2 = in_gfs[IDX4(AD2GF, i0,i1-2,i2)];\n const double AD2_i0_i1m1_i2 = in_gfs[IDX4(AD2GF, i0,i1-1,i2)];\n const double AD2_i0m2_i1_i2 = in_gfs[IDX4(AD2GF, i0-2,i1,i2)];\n const double AD2_i0m1_i1_i2 = in_gfs[IDX4(AD2GF, i0-1,i1,i2)];\n const double AD2 = in_gfs[IDX4(AD2GF, i0,i1,i2)];\n const double AD2_i0p1_i1_i2 = in_gfs[IDX4(AD2GF, i0+1,i1,i2)];\n const double AD2_i0p2_i1_i2 = in_gfs[IDX4(AD2GF, i0+2,i1,i2)];\n const double AD2_i0_i1p1_i2 = in_gfs[IDX4(AD2GF, i0,i1+1,i2)];\n const double AD2_i0_i1p2_i2 = in_gfs[IDX4(AD2GF, i0,i1+2,i2)];\n const double AD2_i0_i1m2_i2p1 = in_gfs[IDX4(AD2GF, i0,i1-2,i2+1)];\n const double AD2_i0_i1m1_i2p1 = in_gfs[IDX4(AD2GF, i0,i1-1,i2+1)];\n const double AD2_i0m2_i1_i2p1 = in_gfs[IDX4(AD2GF, i0-2,i1,i2+1)];\n const double AD2_i0m1_i1_i2p1 = in_gfs[IDX4(AD2GF, i0-1,i1,i2+1)];\n const double AD2_i0_i1_i2p1 = in_gfs[IDX4(AD2GF, i0,i1,i2+1)];\n const double AD2_i0p1_i1_i2p1 = in_gfs[IDX4(AD2GF, i0+1,i1,i2+1)];\n const double AD2_i0p2_i1_i2p1 = in_gfs[IDX4(AD2GF, i0+2,i1,i2+1)];\n const double AD2_i0_i1p1_i2p1 = in_gfs[IDX4(AD2GF, i0,i1+1,i2+1)];\n const double AD2_i0_i1p2_i2p1 = in_gfs[IDX4(AD2GF, i0,i1+2,i2+1)];\n const double AD2_i0_i1m2_i2p2 = in_gfs[IDX4(AD2GF, i0,i1-2,i2+2)];\n const double AD2_i0_i1m1_i2p2 = in_gfs[IDX4(AD2GF, i0,i1-1,i2+2)];\n const double AD2_i0m2_i1_i2p2 = in_gfs[IDX4(AD2GF, i0-2,i1,i2+2)];\n const double AD2_i0m1_i1_i2p2 = in_gfs[IDX4(AD2GF, i0-1,i1,i2+2)];\n const double AD2_i0_i1_i2p2 = in_gfs[IDX4(AD2GF, i0,i1,i2+2)];\n const double AD2_i0p1_i1_i2p2 = in_gfs[IDX4(AD2GF, i0+1,i1,i2+2)];\n const double AD2_i0p2_i1_i2p2 = in_gfs[IDX4(AD2GF, i0+2,i1,i2+2)];\n const double AD2_i0_i1p1_i2p2 = in_gfs[IDX4(AD2GF, i0,i1+1,i2+2)];\n const double AD2_i0_i1p2_i2p2 = in_gfs[IDX4(AD2GF, i0,i1+2,i2+2)];\n const double FDPart1_Rational_2_3 = 2.0/3.0;\n const double FDPart1_Rational_1_12 = 1.0/12.0;\n const double FDPart1_Rational_4_9 = 4.0/9.0;\n const double FDPart1_Rational_1_18 = 1.0/18.0;\n const double FDPart1_Rational_1_144 = 1.0/144.0;\n const double FDPart1_Rational_5_2 = 5.0/2.0;\n const double FDPart1_Rational_4_3 = 4.0/3.0;\n const double FDPart1_1 = -AD0_i0_i1_i2p2;\n const double FDPart1_9 = -AD0*FDPart1_Rational_5_2;\n const double FDPart1_12 = -AD1*FDPart1_Rational_5_2;\n const double FDPart1_14 = -AD2*FDPart1_Rational_5_2;\n const double AD_dD00 = invdx0*(FDPart1_Rational_1_12*(AD0_i0m2_i1_i2 - AD0_i0p2_i1_i2) + FDPart1_Rational_2_3*(-AD0_i0m1_i1_i2 + AD0_i0p1_i1_i2));\n const double AD_dD01 = invdx1*(FDPart1_Rational_1_12*(AD0_i0_i1m2_i2 - AD0_i0_i1p2_i2) + FDPart1_Rational_2_3*(-AD0_i0_i1m1_i2 + AD0_i0_i1p1_i2));\n const double AD_dD02 = invdx2*(FDPart1_Rational_1_12*(AD0_i0_i1_i2m2 + FDPart1_1) + FDPart1_Rational_2_3*(-AD0_i0_i1_i2m1 + AD0_i0_i1_i2p1));\n const double AD_dD10 = invdx0*(FDPart1_Rational_1_12*(AD1_i0m2_i1_i2 - AD1_i0p2_i1_i2) + FDPart1_Rational_2_3*(-AD1_i0m1_i1_i2 + AD1_i0p1_i1_i2));\n const double AD_dD11 = invdx1*(FDPart1_Rational_1_12*(AD1_i0_i1m2_i2 - AD1_i0_i1p2_i2) + FDPart1_Rational_2_3*(-AD1_i0_i1m1_i2 + AD1_i0_i1p1_i2));\n const double AD_dD12 = invdx2*(FDPart1_Rational_1_12*(AD1_i0_i1_i2m2 - AD1_i0_i1_i2p2) + FDPart1_Rational_2_3*(-AD1_i0_i1_i2m1 + AD1_i0_i1_i2p1));\n const double AD_dD20 = invdx0*(FDPart1_Rational_1_12*(AD2_i0m2_i1_i2 - AD2_i0p2_i1_i2) + FDPart1_Rational_2_3*(-AD2_i0m1_i1_i2 + AD2_i0p1_i1_i2));\n const double AD_dD21 = invdx1*(FDPart1_Rational_1_12*(AD2_i0_i1m2_i2 - AD2_i0_i1p2_i2) + FDPart1_Rational_2_3*(-AD2_i0_i1m1_i2 + AD2_i0_i1p1_i2));\n const double AD_dD22 = invdx2*(FDPart1_Rational_1_12*(AD2_i0_i1_i2m2 - AD2_i0_i1_i2p2) + FDPart1_Rational_2_3*(-AD2_i0_i1_i2m1 + AD2_i0_i1_i2p1));\n const double AD_dDD001 = invdx0*invdx1*(FDPart1_Rational_1_144*(AD0_i0m2_i1m2_i2 - AD0_i0m2_i1p2_i2 - AD0_i0p2_i1m2_i2 + AD0_i0p2_i1p2_i2) + FDPart1_Rational_1_18*(-AD0_i0m1_i1m2_i2 + AD0_i0m1_i1p2_i2 - AD0_i0m2_i1m1_i2 + AD0_i0m2_i1p1_i2 + AD0_i0p1_i1m2_i2 - AD0_i0p1_i1p2_i2 + AD0_i0p2_i1m1_i2 - AD0_i0p2_i1p1_i2) + FDPart1_Rational_4_9*(AD0_i0m1_i1m1_i2 - AD0_i0m1_i1p1_i2 - AD0_i0p1_i1m1_i2 + AD0_i0p1_i1p1_i2));\n const double AD_dDD002 = invdx0*invdx2*(FDPart1_Rational_1_144*(AD0_i0m2_i1_i2m2 - AD0_i0m2_i1_i2p2 - AD0_i0p2_i1_i2m2 + AD0_i0p2_i1_i2p2) + FDPart1_Rational_1_18*(-AD0_i0m1_i1_i2m2 + AD0_i0m1_i1_i2p2 - AD0_i0m2_i1_i2m1 + AD0_i0m2_i1_i2p1 + AD0_i0p1_i1_i2m2 - AD0_i0p1_i1_i2p2 + AD0_i0p2_i1_i2m1 - AD0_i0p2_i1_i2p1) + FDPart1_Rational_4_9*(AD0_i0m1_i1_i2m1 - AD0_i0m1_i1_i2p1 - AD0_i0p1_i1_i2m1 + AD0_i0p1_i1_i2p1));\n const double AD_dDD011 = ((invdx1)*(invdx1))*(FDPart1_9 + FDPart1_Rational_1_12*(-AD0_i0_i1m2_i2 - AD0_i0_i1p2_i2) + FDPart1_Rational_4_3*(AD0_i0_i1m1_i2 + AD0_i0_i1p1_i2));\n const double AD_dDD022 = ((invdx2)*(invdx2))*(FDPart1_9 + FDPart1_Rational_1_12*(-AD0_i0_i1_i2m2 + FDPart1_1) + FDPart1_Rational_4_3*(AD0_i0_i1_i2m1 + AD0_i0_i1_i2p1));\n const double AD_dDD100 = ((invdx0)*(invdx0))*(FDPart1_12 + FDPart1_Rational_1_12*(-AD1_i0m2_i1_i2 - AD1_i0p2_i1_i2) + FDPart1_Rational_4_3*(AD1_i0m1_i1_i2 + AD1_i0p1_i1_i2));\n const double AD_dDD101 = invdx0*invdx1*(FDPart1_Rational_1_144*(AD1_i0m2_i1m2_i2 - AD1_i0m2_i1p2_i2 - AD1_i0p2_i1m2_i2 + AD1_i0p2_i1p2_i2) + FDPart1_Rational_1_18*(-AD1_i0m1_i1m2_i2 + AD1_i0m1_i1p2_i2 - AD1_i0m2_i1m1_i2 + AD1_i0m2_i1p1_i2 + AD1_i0p1_i1m2_i2 - AD1_i0p1_i1p2_i2 + AD1_i0p2_i1m1_i2 - AD1_i0p2_i1p1_i2) + FDPart1_Rational_4_9*(AD1_i0m1_i1m1_i2 - AD1_i0m1_i1p1_i2 - AD1_i0p1_i1m1_i2 + AD1_i0p1_i1p1_i2));\n const double AD_dDD112 = invdx1*invdx2*(FDPart1_Rational_1_144*(AD1_i0_i1m2_i2m2 - AD1_i0_i1m2_i2p2 - AD1_i0_i1p2_i2m2 + AD1_i0_i1p2_i2p2) + FDPart1_Rational_1_18*(-AD1_i0_i1m1_i2m2 + AD1_i0_i1m1_i2p2 - AD1_i0_i1m2_i2m1 + AD1_i0_i1m2_i2p1 + AD1_i0_i1p1_i2m2 - AD1_i0_i1p1_i2p2 + AD1_i0_i1p2_i2m1 - AD1_i0_i1p2_i2p1) + FDPart1_Rational_4_9*(AD1_i0_i1m1_i2m1 - AD1_i0_i1m1_i2p1 - AD1_i0_i1p1_i2m1 + AD1_i0_i1p1_i2p1));\n const double AD_dDD122 = ((invdx2)*(invdx2))*(FDPart1_12 + FDPart1_Rational_1_12*(-AD1_i0_i1_i2m2 - AD1_i0_i1_i2p2) + FDPart1_Rational_4_3*(AD1_i0_i1_i2m1 + AD1_i0_i1_i2p1));\n const double AD_dDD200 = ((invdx0)*(invdx0))*(FDPart1_14 + FDPart1_Rational_1_12*(-AD2_i0m2_i1_i2 - AD2_i0p2_i1_i2) + FDPart1_Rational_4_3*(AD2_i0m1_i1_i2 + AD2_i0p1_i1_i2));\n const double AD_dDD202 = invdx0*invdx2*(FDPart1_Rational_1_144*(AD2_i0m2_i1_i2m2 - AD2_i0m2_i1_i2p2 - AD2_i0p2_i1_i2m2 + AD2_i0p2_i1_i2p2) + FDPart1_Rational_1_18*(-AD2_i0m1_i1_i2m2 + AD2_i0m1_i1_i2p2 - AD2_i0m2_i1_i2m1 + AD2_i0m2_i1_i2p1 + AD2_i0p1_i1_i2m2 - AD2_i0p1_i1_i2p2 + AD2_i0p2_i1_i2m1 - AD2_i0p2_i1_i2p1) + FDPart1_Rational_4_9*(AD2_i0m1_i1_i2m1 - AD2_i0m1_i1_i2p1 - AD2_i0p1_i1_i2m1 + AD2_i0p1_i1_i2p1));\n const double AD_dDD211 = ((invdx1)*(invdx1))*(FDPart1_14 + FDPart1_Rational_1_12*(-AD2_i0_i1m2_i2 - AD2_i0_i1p2_i2) + FDPart1_Rational_4_3*(AD2_i0_i1m1_i2 + AD2_i0_i1p1_i2));\n const double AD_dDD212 = invdx1*invdx2*(FDPart1_Rational_1_144*(AD2_i0_i1m2_i2m2 - AD2_i0_i1m2_i2p2 - AD2_i0_i1p2_i2m2 + AD2_i0_i1p2_i2p2) + FDPart1_Rational_1_18*(-AD2_i0_i1m1_i2m2 + AD2_i0_i1m1_i2p2 - AD2_i0_i1m2_i2m1 + AD2_i0_i1m2_i2p1 + AD2_i0_i1p1_i2m2 - AD2_i0_i1p1_i2p2 + AD2_i0_i1p2_i2m1 - AD2_i0_i1p2_i2p1) + FDPart1_Rational_4_9*(AD2_i0_i1m1_i2m1 - AD2_i0_i1m1_i2p1 - AD2_i0_i1p1_i2m1 + AD2_i0_i1p1_i2p1));\n const double psi_dD0 = invdx0*(FDPart1_Rational_1_12*(psi_i0m2_i1_i2 - psi_i0p2_i1_i2) + FDPart1_Rational_2_3*(-psi_i0m1_i1_i2 + psi_i0p1_i1_i2));\n const double psi_dD1 = invdx1*(FDPart1_Rational_1_12*(psi_i0_i1m2_i2 - psi_i0_i1p2_i2) + FDPart1_Rational_2_3*(-psi_i0_i1m1_i2 + psi_i0_i1p1_i2));\n const double psi_dD2 = invdx2*(FDPart1_Rational_1_12*(psi_i0_i1_i2m2 - psi_i0_i1_i2p2) + FDPart1_Rational_2_3*(-psi_i0_i1_i2m1 + psi_i0_i1_i2p1));\n /*\n * NRPy+ Finite Difference Code Generation, Step 2 of 2: Evaluate SymPy expressions and write to main memory:\n */\n /*\n * Original SymPy expressions:\n * \"[rhs_gfs[IDX4(AD0GF, i0, i1, i2)] = -ED0 - psi_dD0,\n * rhs_gfs[IDX4(ED0GF, i0, i1, i2)] = (AD0 + AD_dD00*xx0 + AD_dDD101 - 2*(AD0*xx0 + AD_dD11)/xx0)/xx0**2 - (AD_dD00*xx0 - AD_dD11/xx0 + AD_dDD011 - (AD0*xx0 + AD_dD11)/xx0)/xx0**2 + (AD0*sin(xx1)**2 + AD_dD00*xx0*sin(xx1)**2 + AD_dD10*sin(2*xx1)/2 + AD_dDD202 - 2*(AD0*xx0*sin(xx1)**2 + AD1*sin(2*xx1)/2 + AD_dD22)/xx0)/(xx0**2*sin(xx1)**2) - (AD_dD00*xx0*sin(xx1)**2 - AD_dD22/xx0 + AD_dDD022 + (-AD1/xx0 + AD_dD01)*sin(2*xx1)/2 - (AD0*xx0*sin(xx1)**2 + AD1*sin(2*xx1)/2 + AD_dD22)/xx0)/(xx0**2*sin(xx1)**2),\n * rhs_gfs[IDX4(AD1GF, i0, i1, i2)] = -ED1 - psi_dD1,\n * rhs_gfs[IDX4(ED1GF, i0, i1, i2)] = -AD1/xx0**2 + AD_dD10/xx0 + AD_dDD001 - AD_dDD100 - (-AD1/xx0 + AD_dD01)/xx0 - (-AD_dD22*sin(2*xx1)/(2*sin(xx1)**2) + AD_dDD122 + xx0*(-AD1/xx0 + AD_dD10)*sin(xx1)**2 + (AD0*xx0 + AD_dD11)*sin(2*xx1)/2 - (AD0*xx0*sin(xx1)**2 + AD1*sin(2*xx1)/2 + AD_dD22)*sin(2*xx1)/(2*sin(xx1)**2))/(xx0**2*sin(xx1)**2) + (2*AD0*xx0*sin(xx1)*cos(xx1) + AD1*cos(2*xx1) + AD_dD01*xx0*sin(xx1)**2 + AD_dD11*sin(2*xx1)/2 + AD_dDD212 - (AD0*xx0*sin(xx1)**2 + AD1*sin(2*xx1)/2 + AD_dD22)*sin(2*xx1)/sin(xx1)**2)/(xx0**2*sin(xx1)**2),\n * rhs_gfs[IDX4(AD2GF, i0, i1, i2)] = -ED2 - psi_dD2,\n * rhs_gfs[IDX4(ED2GF, i0, i1, i2)] = -AD2/xx0**2 + AD_dD20/xx0 + AD_dDD002 - AD_dDD200 - (-AD2/xx0 + AD_dD02)/xx0 + (AD_dD02*xx0 + AD_dDD112 - (-AD2*sin(2*xx1)/(2*sin(xx1)**2) + AD_dD12)*sin(2*xx1)/(2*sin(xx1)**2) - (-AD2*sin(2*xx1)/(2*sin(xx1)**2) + AD_dD21)*sin(2*xx1)/(2*sin(xx1)**2))/xx0**2 - (AD2*(-cos(2*xx1)/sin(xx1)**2 + sin(2*xx1)*cos(xx1)/sin(xx1)**3) - AD_dD21*sin(2*xx1)/(2*sin(xx1)**2) + AD_dDD211 + xx0*(-AD2/xx0 + AD_dD20) - (-AD2*sin(2*xx1)/(2*sin(xx1)**2) + AD_dD21)*sin(2*xx1)/(2*sin(xx1)**2))/xx0**2,\n * rhs_gfs[IDX4(PSIGF, i0, i1, i2)] = -AD_dD00 - (AD0*xx0 + AD_dD11)/xx0**2 - (AD0*xx0*sin(xx1)**2 + AD1*sin(2*xx1)/2 + AD_dD22)/(xx0**2*sin(xx1)**2)]\"\n */\n const double FDPart3_0 = (1.0/((xx0)*(xx0)));\n const double FDPart3_1 = AD_dD00*xx0;\n const double FDPart3_2 = (1.0/(xx0));\n const double FDPart3_3 = AD0*xx0;\n const double FDPart3_4 = AD_dD11 + FDPart3_3;\n const double FDPart3_6 = sin(xx1);\n const double FDPart3_7 = ((FDPart3_6)*(FDPart3_6));\n const double FDPart3_9 = sin(2*xx1);\n const double FDPart3_10 = (1.0/2.0)*FDPart3_9;\n const double FDPart3_12 = AD1*FDPart3_10 + AD_dD22 + FDPart3_3*FDPart3_7;\n const double FDPart3_14 = (1.0/(FDPart3_7));\n const double FDPart3_15 = FDPart3_0*FDPart3_14;\n const double FDPart3_16 = -AD1*FDPart3_2;\n const double FDPart3_18 = cos(2*xx1);\n const double FDPart3_20 = cos(xx1);\n const double FDPart3_22 = FDPart3_10*FDPart3_14;\n const double FDPart3_23 = -AD2*FDPart3_2;\n const double FDPart3_24 = -AD2*FDPart3_22;\n const double FDPart3_25 = -FDPart3_22*(AD_dD21 + FDPart3_24);\n rhs_gfs[IDX4(AD0GF, i0, i1, i2)] = -ED0 - psi_dD0;\n rhs_gfs[IDX4(ED0GF, i0, i1, i2)] = FDPart3_0*(AD0 + AD_dDD101 + FDPart3_1 - 2*FDPart3_2*FDPart3_4) - FDPart3_0*(-AD_dD11*FDPart3_2 + AD_dDD011 + FDPart3_1 - FDPart3_2*FDPart3_4) + FDPart3_15*(AD0*FDPart3_7 + AD_dD10*FDPart3_10 + AD_dDD202 + FDPart3_1*FDPart3_7 - 2*FDPart3_12*FDPart3_2) - FDPart3_15*(-AD_dD22*FDPart3_2 + AD_dDD022 + FDPart3_1*FDPart3_7 + FDPart3_10*(AD_dD01 + FDPart3_16) - FDPart3_12*FDPart3_2);\n rhs_gfs[IDX4(AD1GF, i0, i1, i2)] = -ED1 - psi_dD1;\n rhs_gfs[IDX4(ED1GF, i0, i1, i2)] = -AD1*FDPart3_0 + AD_dD10*FDPart3_2 + AD_dDD001 - AD_dDD100 - FDPart3_15*(-AD_dD22*FDPart3_22 + AD_dDD122 - FDPart3_10*FDPart3_12*FDPart3_14 + FDPart3_10*FDPart3_4 + FDPart3_7*xx0*(AD_dD10 + FDPart3_16)) + FDPart3_15*(AD1*FDPart3_18 + AD_dD01*FDPart3_7*xx0 + AD_dD11*FDPart3_10 + AD_dDD212 - FDPart3_12*FDPart3_14*FDPart3_9 + 2*FDPart3_20*FDPart3_3*FDPart3_6) - FDPart3_2*(AD_dD01 + FDPart3_16);\n rhs_gfs[IDX4(AD2GF, i0, i1, i2)] = -ED2 - psi_dD2;\n rhs_gfs[IDX4(ED2GF, i0, i1, i2)] = -AD2*FDPart3_0 + AD_dD20*FDPart3_2 + AD_dDD002 - AD_dDD200 + FDPart3_0*(AD_dD02*xx0 + AD_dDD112 - FDPart3_22*(AD_dD12 + FDPart3_24) + FDPart3_25) - FDPart3_0*(AD2*(-FDPart3_14*FDPart3_18 + FDPart3_20*FDPart3_9/((FDPart3_6)*(FDPart3_6)*(FDPart3_6))) - AD_dD21*FDPart3_22 + AD_dDD211 + FDPart3_25 + xx0*(AD_dD20 + FDPart3_23)) - FDPart3_2*(AD_dD02 + FDPart3_23);\n rhs_gfs[IDX4(PSIGF, i0, i1, i2)] = -AD_dD00 - FDPart3_0*FDPart3_4 - FDPart3_12*FDPart3_15;\n }\n\n\n\n\n# Step 2: System II \\[Back to [top](#toc)\\]\n$$\\label{sys2}$$\n\nDefine the auxiliary variable\n$$\\Gamma \\equiv \\hat{D}_{i} A^{i} \\; .$$\nSubstituting this into Maxwell's equations yields the system\n\\begin{align}\n\\partial_{t} A_{i} &= -E_{i} - \\hat{D}_{i} \\psi \\; , \\\\\n\\partial_{t} E_{i} &= -\\hat{D}_{j} \\hat{D}^{j} A_{i} + \\hat{D}_{i} \\Gamma \\; , \\\\\n\\partial_{t} \\psi &= -\\Gamma \\; , \\\\\n\\partial_{t} \\Gamma &= -\\hat{D}_{i} \\hat{D}^{i} \\psi \\; .\n\\end{align}\n\n\n\nIt can be shown that the Gauss constraint now satisfies the wave equation\n$$\\partial_{t}^{2} \\mathcal{C} = \\hat{D}_{i} \\hat{D}^{i} \\mathcal{C} \\; .$$\nThus, any constraint violation introduced by numerical error propagates away at the speed of light. This property increases the stability of of the simulation, compared to System I above. A similar trick is used in the [BSSN formulation](Tutorial-BSSNCurvilinear.ipynb) of Einstein's equations.\n\n\n```python\n# We inherit here all of the definitions from System I, above\n\n# Step 7a: Register the scalar auxiliary variable \\Gamma\nGamma = gri.register_gridfunctions(\"EVOL\", [\"Gamma\"])\n\n# Step 7b: Declare the ordinary gradient \\partial_{i} \\Gamma\nGamma_dD = ixp.declarerank1(\"Gamma_dD\")\n\n# Step 8a: Construct the second covariant derivative of the scalar \\psi\n# \\psi_{;\\hat{i}\\hat{j}} = \\psi_{,i;\\hat{j}}\n# = \\psi_{,ij} - \\Gamma^{k}_{ij} \\psi_{,k}\npsi_dDD = ixp.declarerank2(\"psi_dDD\", \"sym01\")\npsi_dHatDD = ixp.zerorank2()\nfor i in range(DIM):\n for j in range(DIM):\n psi_dHatDD[i][j] = psi_dDD[i][j]\n for k in range(DIM):\n psi_dHatDD[i][j] += - rfm.GammahatUDD[k][i][j] * psi_dD[k]\n\n# Step 8b: Construct the covariant Laplacian of \\psi\n# Lappsi = ghat^{ij} D_{j} D_{i} \\psi\nLappsi = 0\nfor i in range(DIM):\n for j in range(DIM):\n Lappsi += rfm.ghatUU[i][j] * psi_dHatDD[i][j]\n\n# Step 9: Define right-hand sides for the evolution.\nAD_rhs = ixp.zerorank1()\nED_rhs = ixp.zerorank1()\nfor i in range(DIM):\n AD_rhs[i] = -ED[i] - psi_dD[i]\n ED_rhs[i] = -LapAD[i] + Gamma_dD[i]\npsi_rhs = -Gamma\nGamma_rhs = -Lappsi\n\n# Step 10: Generate C code for System II Maxwell's evolution equations,\n# print output to the screen (standard out, or stdout).\nlhrh_list = []\nfor i in range(DIM):\n lhrh_list.append(lhrh(lhs=gri.gfaccess(\"rhs_gfs\", \"AD\" + str(i)), rhs=AD_rhs[i]))\n lhrh_list.append(lhrh(lhs=gri.gfaccess(\"rhs_gfs\", \"ED\" + str(i)), rhs=ED_rhs[i]))\nlhrh_list.append(lhrh(lhs=gri.gfaccess(\"rhs_gfs\", \"psi\"), rhs=psi_rhs))\nlhrh_list.append(lhrh(lhs=gri.gfaccess(\"rhs_gfs\", \"Gamma\"), rhs=Gamma_rhs))\n\nfin.FD_outputC(\"stdout\", lhrh_list)\n```\n\n {\n /*\n * NRPy+ Finite Difference Code Generation, Step 1 of 2: Read from main memory and compute finite difference stencils:\n */\n /*\n * Original SymPy expressions:\n * \"[const double AD_dD00 = invdx0*(-2*AD0_i0m1_i1_i2/3 + AD0_i0m2_i1_i2/12 + 2*AD0_i0p1_i1_i2/3 - AD0_i0p2_i1_i2/12),\n * const double AD_dD01 = invdx1*(-2*AD0_i0_i1m1_i2/3 + AD0_i0_i1m2_i2/12 + 2*AD0_i0_i1p1_i2/3 - AD0_i0_i1p2_i2/12),\n * const double AD_dD02 = invdx2*(-2*AD0_i0_i1_i2m1/3 + AD0_i0_i1_i2m2/12 + 2*AD0_i0_i1_i2p1/3 - AD0_i0_i1_i2p2/12),\n * const double AD_dD10 = invdx0*(-2*AD1_i0m1_i1_i2/3 + AD1_i0m2_i1_i2/12 + 2*AD1_i0p1_i1_i2/3 - AD1_i0p2_i1_i2/12),\n * const double AD_dD11 = invdx1*(-2*AD1_i0_i1m1_i2/3 + AD1_i0_i1m2_i2/12 + 2*AD1_i0_i1p1_i2/3 - AD1_i0_i1p2_i2/12),\n * const double AD_dD12 = invdx2*(-2*AD1_i0_i1_i2m1/3 + AD1_i0_i1_i2m2/12 + 2*AD1_i0_i1_i2p1/3 - AD1_i0_i1_i2p2/12),\n * const double AD_dD20 = invdx0*(-2*AD2_i0m1_i1_i2/3 + AD2_i0m2_i1_i2/12 + 2*AD2_i0p1_i1_i2/3 - AD2_i0p2_i1_i2/12),\n * const double AD_dD21 = invdx1*(-2*AD2_i0_i1m1_i2/3 + AD2_i0_i1m2_i2/12 + 2*AD2_i0_i1p1_i2/3 - AD2_i0_i1p2_i2/12),\n * const double AD_dD22 = invdx2*(-2*AD2_i0_i1_i2m1/3 + AD2_i0_i1_i2m2/12 + 2*AD2_i0_i1_i2p1/3 - AD2_i0_i1_i2p2/12),\n * const double AD_dDD000 = invdx0**2*(-5*AD0/2 + 4*AD0_i0m1_i1_i2/3 - AD0_i0m2_i1_i2/12 + 4*AD0_i0p1_i1_i2/3 - AD0_i0p2_i1_i2/12),\n * const double AD_dDD011 = invdx1**2*(-5*AD0/2 + 4*AD0_i0_i1m1_i2/3 - AD0_i0_i1m2_i2/12 + 4*AD0_i0_i1p1_i2/3 - AD0_i0_i1p2_i2/12),\n * const double AD_dDD022 = invdx2**2*(-5*AD0/2 + 4*AD0_i0_i1_i2m1/3 - AD0_i0_i1_i2m2/12 + 4*AD0_i0_i1_i2p1/3 - AD0_i0_i1_i2p2/12),\n * const double AD_dDD100 = invdx0**2*(-5*AD1/2 + 4*AD1_i0m1_i1_i2/3 - AD1_i0m2_i1_i2/12 + 4*AD1_i0p1_i1_i2/3 - AD1_i0p2_i1_i2/12),\n * const double AD_dDD111 = invdx1**2*(-5*AD1/2 + 4*AD1_i0_i1m1_i2/3 - AD1_i0_i1m2_i2/12 + 4*AD1_i0_i1p1_i2/3 - AD1_i0_i1p2_i2/12),\n * const double AD_dDD122 = invdx2**2*(-5*AD1/2 + 4*AD1_i0_i1_i2m1/3 - AD1_i0_i1_i2m2/12 + 4*AD1_i0_i1_i2p1/3 - AD1_i0_i1_i2p2/12),\n * const double AD_dDD200 = invdx0**2*(-5*AD2/2 + 4*AD2_i0m1_i1_i2/3 - AD2_i0m2_i1_i2/12 + 4*AD2_i0p1_i1_i2/3 - AD2_i0p2_i1_i2/12),\n * const double AD_dDD211 = invdx1**2*(-5*AD2/2 + 4*AD2_i0_i1m1_i2/3 - AD2_i0_i1m2_i2/12 + 4*AD2_i0_i1p1_i2/3 - AD2_i0_i1p2_i2/12),\n * const double AD_dDD222 = invdx2**2*(-5*AD2/2 + 4*AD2_i0_i1_i2m1/3 - AD2_i0_i1_i2m2/12 + 4*AD2_i0_i1_i2p1/3 - AD2_i0_i1_i2p2/12),\n * const double Gamma_dD0 = invdx0*(-2*Gamma_i0m1_i1_i2/3 + Gamma_i0m2_i1_i2/12 + 2*Gamma_i0p1_i1_i2/3 - Gamma_i0p2_i1_i2/12),\n * const double Gamma_dD1 = invdx1*(-2*Gamma_i0_i1m1_i2/3 + Gamma_i0_i1m2_i2/12 + 2*Gamma_i0_i1p1_i2/3 - Gamma_i0_i1p2_i2/12),\n * const double Gamma_dD2 = invdx2*(-2*Gamma_i0_i1_i2m1/3 + Gamma_i0_i1_i2m2/12 + 2*Gamma_i0_i1_i2p1/3 - Gamma_i0_i1_i2p2/12),\n * const double psi_dD0 = invdx0*(-2*psi_i0m1_i1_i2/3 + psi_i0m2_i1_i2/12 + 2*psi_i0p1_i1_i2/3 - psi_i0p2_i1_i2/12),\n * const double psi_dD1 = invdx1*(-2*psi_i0_i1m1_i2/3 + psi_i0_i1m2_i2/12 + 2*psi_i0_i1p1_i2/3 - psi_i0_i1p2_i2/12),\n * const double psi_dD2 = invdx2*(-2*psi_i0_i1_i2m1/3 + psi_i0_i1_i2m2/12 + 2*psi_i0_i1_i2p1/3 - psi_i0_i1_i2p2/12),\n * const double psi_dDD00 = invdx0**2*(-5*psi/2 + 4*psi_i0m1_i1_i2/3 - psi_i0m2_i1_i2/12 + 4*psi_i0p1_i1_i2/3 - psi_i0p2_i1_i2/12),\n * const double psi_dDD11 = invdx1**2*(-5*psi/2 + 4*psi_i0_i1m1_i2/3 - psi_i0_i1m2_i2/12 + 4*psi_i0_i1p1_i2/3 - psi_i0_i1p2_i2/12),\n * const double psi_dDD22 = invdx2**2*(-5*psi/2 + 4*psi_i0_i1_i2m1/3 - psi_i0_i1_i2m2/12 + 4*psi_i0_i1_i2p1/3 - psi_i0_i1_i2p2/12)]\"\n */\n const double psi_i0_i1_i2m2 = in_gfs[IDX4(PSIGF, i0,i1,i2-2)];\n const double psi_i0_i1_i2m1 = in_gfs[IDX4(PSIGF, i0,i1,i2-1)];\n const double psi_i0_i1m2_i2 = in_gfs[IDX4(PSIGF, i0,i1-2,i2)];\n const double psi_i0_i1m1_i2 = in_gfs[IDX4(PSIGF, i0,i1-1,i2)];\n const double psi_i0m2_i1_i2 = in_gfs[IDX4(PSIGF, i0-2,i1,i2)];\n const double psi_i0m1_i1_i2 = in_gfs[IDX4(PSIGF, i0-1,i1,i2)];\n const double psi = in_gfs[IDX4(PSIGF, i0,i1,i2)];\n const double psi_i0p1_i1_i2 = in_gfs[IDX4(PSIGF, i0+1,i1,i2)];\n const double psi_i0p2_i1_i2 = in_gfs[IDX4(PSIGF, i0+2,i1,i2)];\n const double psi_i0_i1p1_i2 = in_gfs[IDX4(PSIGF, i0,i1+1,i2)];\n const double psi_i0_i1p2_i2 = in_gfs[IDX4(PSIGF, i0,i1+2,i2)];\n const double psi_i0_i1_i2p1 = in_gfs[IDX4(PSIGF, i0,i1,i2+1)];\n const double psi_i0_i1_i2p2 = in_gfs[IDX4(PSIGF, i0,i1,i2+2)];\n const double ED0 = in_gfs[IDX4(ED0GF, i0,i1,i2)];\n const double ED1 = in_gfs[IDX4(ED1GF, i0,i1,i2)];\n const double ED2 = in_gfs[IDX4(ED2GF, i0,i1,i2)];\n const double AD0_i0_i1_i2m2 = in_gfs[IDX4(AD0GF, i0,i1,i2-2)];\n const double AD0_i0_i1_i2m1 = in_gfs[IDX4(AD0GF, i0,i1,i2-1)];\n const double AD0_i0_i1m2_i2 = in_gfs[IDX4(AD0GF, i0,i1-2,i2)];\n const double AD0_i0_i1m1_i2 = in_gfs[IDX4(AD0GF, i0,i1-1,i2)];\n const double AD0_i0m2_i1_i2 = in_gfs[IDX4(AD0GF, i0-2,i1,i2)];\n const double AD0_i0m1_i1_i2 = in_gfs[IDX4(AD0GF, i0-1,i1,i2)];\n const double AD0 = in_gfs[IDX4(AD0GF, i0,i1,i2)];\n const double AD0_i0p1_i1_i2 = in_gfs[IDX4(AD0GF, i0+1,i1,i2)];\n const double AD0_i0p2_i1_i2 = in_gfs[IDX4(AD0GF, i0+2,i1,i2)];\n const double AD0_i0_i1p1_i2 = in_gfs[IDX4(AD0GF, i0,i1+1,i2)];\n const double AD0_i0_i1p2_i2 = in_gfs[IDX4(AD0GF, i0,i1+2,i2)];\n const double AD0_i0_i1_i2p1 = in_gfs[IDX4(AD0GF, i0,i1,i2+1)];\n const double AD0_i0_i1_i2p2 = in_gfs[IDX4(AD0GF, i0,i1,i2+2)];\n const double AD1_i0_i1_i2m2 = in_gfs[IDX4(AD1GF, i0,i1,i2-2)];\n const double AD1_i0_i1_i2m1 = in_gfs[IDX4(AD1GF, i0,i1,i2-1)];\n const double AD1_i0_i1m2_i2 = in_gfs[IDX4(AD1GF, i0,i1-2,i2)];\n const double AD1_i0_i1m1_i2 = in_gfs[IDX4(AD1GF, i0,i1-1,i2)];\n const double AD1_i0m2_i1_i2 = in_gfs[IDX4(AD1GF, i0-2,i1,i2)];\n const double AD1_i0m1_i1_i2 = in_gfs[IDX4(AD1GF, i0-1,i1,i2)];\n const double AD1 = in_gfs[IDX4(AD1GF, i0,i1,i2)];\n const double AD1_i0p1_i1_i2 = in_gfs[IDX4(AD1GF, i0+1,i1,i2)];\n const double AD1_i0p2_i1_i2 = in_gfs[IDX4(AD1GF, i0+2,i1,i2)];\n const double AD1_i0_i1p1_i2 = in_gfs[IDX4(AD1GF, i0,i1+1,i2)];\n const double AD1_i0_i1p2_i2 = in_gfs[IDX4(AD1GF, i0,i1+2,i2)];\n const double AD1_i0_i1_i2p1 = in_gfs[IDX4(AD1GF, i0,i1,i2+1)];\n const double AD1_i0_i1_i2p2 = in_gfs[IDX4(AD1GF, i0,i1,i2+2)];\n const double AD2_i0_i1_i2m2 = in_gfs[IDX4(AD2GF, i0,i1,i2-2)];\n const double AD2_i0_i1_i2m1 = in_gfs[IDX4(AD2GF, i0,i1,i2-1)];\n const double AD2_i0_i1m2_i2 = in_gfs[IDX4(AD2GF, i0,i1-2,i2)];\n const double AD2_i0_i1m1_i2 = in_gfs[IDX4(AD2GF, i0,i1-1,i2)];\n const double AD2_i0m2_i1_i2 = in_gfs[IDX4(AD2GF, i0-2,i1,i2)];\n const double AD2_i0m1_i1_i2 = in_gfs[IDX4(AD2GF, i0-1,i1,i2)];\n const double AD2 = in_gfs[IDX4(AD2GF, i0,i1,i2)];\n const double AD2_i0p1_i1_i2 = in_gfs[IDX4(AD2GF, i0+1,i1,i2)];\n const double AD2_i0p2_i1_i2 = in_gfs[IDX4(AD2GF, i0+2,i1,i2)];\n const double AD2_i0_i1p1_i2 = in_gfs[IDX4(AD2GF, i0,i1+1,i2)];\n const double AD2_i0_i1p2_i2 = in_gfs[IDX4(AD2GF, i0,i1+2,i2)];\n const double AD2_i0_i1_i2p1 = in_gfs[IDX4(AD2GF, i0,i1,i2+1)];\n const double AD2_i0_i1_i2p2 = in_gfs[IDX4(AD2GF, i0,i1,i2+2)];\n const double Gamma_i0_i1_i2m2 = in_gfs[IDX4(GAMMAGF, i0,i1,i2-2)];\n const double Gamma_i0_i1_i2m1 = in_gfs[IDX4(GAMMAGF, i0,i1,i2-1)];\n const double Gamma_i0_i1m2_i2 = in_gfs[IDX4(GAMMAGF, i0,i1-2,i2)];\n const double Gamma_i0_i1m1_i2 = in_gfs[IDX4(GAMMAGF, i0,i1-1,i2)];\n const double Gamma_i0m2_i1_i2 = in_gfs[IDX4(GAMMAGF, i0-2,i1,i2)];\n const double Gamma_i0m1_i1_i2 = in_gfs[IDX4(GAMMAGF, i0-1,i1,i2)];\n const double Gamma = in_gfs[IDX4(GAMMAGF, i0,i1,i2)];\n const double Gamma_i0p1_i1_i2 = in_gfs[IDX4(GAMMAGF, i0+1,i1,i2)];\n const double Gamma_i0p2_i1_i2 = in_gfs[IDX4(GAMMAGF, i0+2,i1,i2)];\n const double Gamma_i0_i1p1_i2 = in_gfs[IDX4(GAMMAGF, i0,i1+1,i2)];\n const double Gamma_i0_i1p2_i2 = in_gfs[IDX4(GAMMAGF, i0,i1+2,i2)];\n const double Gamma_i0_i1_i2p1 = in_gfs[IDX4(GAMMAGF, i0,i1,i2+1)];\n const double Gamma_i0_i1_i2p2 = in_gfs[IDX4(GAMMAGF, i0,i1,i2+2)];\n const double FDPart1_Rational_2_3 = 2.0/3.0;\n const double FDPart1_Rational_1_12 = 1.0/12.0;\n const double FDPart1_Rational_5_2 = 5.0/2.0;\n const double FDPart1_Rational_4_3 = 4.0/3.0;\n const double FDPart1_1 = -AD0_i0_i1p2_i2;\n const double FDPart1_9 = ((invdx0)*(invdx0));\n const double FDPart1_10 = -AD0*FDPart1_Rational_5_2;\n const double FDPart1_11 = ((invdx1)*(invdx1));\n const double FDPart1_12 = ((invdx2)*(invdx2));\n const double FDPart1_13 = -AD1*FDPart1_Rational_5_2;\n const double FDPart1_14 = -AD2*FDPart1_Rational_5_2;\n const double FDPart1_18 = -FDPart1_Rational_5_2*psi;\n const double AD_dD00 = invdx0*(FDPart1_Rational_1_12*(AD0_i0m2_i1_i2 - AD0_i0p2_i1_i2) + FDPart1_Rational_2_3*(-AD0_i0m1_i1_i2 + AD0_i0p1_i1_i2));\n const double AD_dD01 = invdx1*(FDPart1_Rational_1_12*(AD0_i0_i1m2_i2 + FDPart1_1) + FDPart1_Rational_2_3*(-AD0_i0_i1m1_i2 + AD0_i0_i1p1_i2));\n const double AD_dD02 = invdx2*(FDPart1_Rational_1_12*(AD0_i0_i1_i2m2 - AD0_i0_i1_i2p2) + FDPart1_Rational_2_3*(-AD0_i0_i1_i2m1 + AD0_i0_i1_i2p1));\n const double AD_dD10 = invdx0*(FDPart1_Rational_1_12*(AD1_i0m2_i1_i2 - AD1_i0p2_i1_i2) + FDPart1_Rational_2_3*(-AD1_i0m1_i1_i2 + AD1_i0p1_i1_i2));\n const double AD_dD11 = invdx1*(FDPart1_Rational_1_12*(AD1_i0_i1m2_i2 - AD1_i0_i1p2_i2) + FDPart1_Rational_2_3*(-AD1_i0_i1m1_i2 + AD1_i0_i1p1_i2));\n const double AD_dD12 = invdx2*(FDPart1_Rational_1_12*(AD1_i0_i1_i2m2 - AD1_i0_i1_i2p2) + FDPart1_Rational_2_3*(-AD1_i0_i1_i2m1 + AD1_i0_i1_i2p1));\n const double AD_dD20 = invdx0*(FDPart1_Rational_1_12*(AD2_i0m2_i1_i2 - AD2_i0p2_i1_i2) + FDPart1_Rational_2_3*(-AD2_i0m1_i1_i2 + AD2_i0p1_i1_i2));\n const double AD_dD21 = invdx1*(FDPart1_Rational_1_12*(AD2_i0_i1m2_i2 - AD2_i0_i1p2_i2) + FDPart1_Rational_2_3*(-AD2_i0_i1m1_i2 + AD2_i0_i1p1_i2));\n const double AD_dD22 = invdx2*(FDPart1_Rational_1_12*(AD2_i0_i1_i2m2 - AD2_i0_i1_i2p2) + FDPart1_Rational_2_3*(-AD2_i0_i1_i2m1 + AD2_i0_i1_i2p1));\n const double AD_dDD000 = FDPart1_9*(FDPart1_10 + FDPart1_Rational_1_12*(-AD0_i0m2_i1_i2 - AD0_i0p2_i1_i2) + FDPart1_Rational_4_3*(AD0_i0m1_i1_i2 + AD0_i0p1_i1_i2));\n const double AD_dDD011 = FDPart1_11*(FDPart1_10 + FDPart1_Rational_1_12*(-AD0_i0_i1m2_i2 + FDPart1_1) + FDPart1_Rational_4_3*(AD0_i0_i1m1_i2 + AD0_i0_i1p1_i2));\n const double AD_dDD022 = FDPart1_12*(FDPart1_10 + FDPart1_Rational_1_12*(-AD0_i0_i1_i2m2 - AD0_i0_i1_i2p2) + FDPart1_Rational_4_3*(AD0_i0_i1_i2m1 + AD0_i0_i1_i2p1));\n const double AD_dDD100 = FDPart1_9*(FDPart1_13 + FDPart1_Rational_1_12*(-AD1_i0m2_i1_i2 - AD1_i0p2_i1_i2) + FDPart1_Rational_4_3*(AD1_i0m1_i1_i2 + AD1_i0p1_i1_i2));\n const double AD_dDD111 = FDPart1_11*(FDPart1_13 + FDPart1_Rational_1_12*(-AD1_i0_i1m2_i2 - AD1_i0_i1p2_i2) + FDPart1_Rational_4_3*(AD1_i0_i1m1_i2 + AD1_i0_i1p1_i2));\n const double AD_dDD122 = FDPart1_12*(FDPart1_13 + FDPart1_Rational_1_12*(-AD1_i0_i1_i2m2 - AD1_i0_i1_i2p2) + FDPart1_Rational_4_3*(AD1_i0_i1_i2m1 + AD1_i0_i1_i2p1));\n const double AD_dDD200 = FDPart1_9*(FDPart1_14 + FDPart1_Rational_1_12*(-AD2_i0m2_i1_i2 - AD2_i0p2_i1_i2) + FDPart1_Rational_4_3*(AD2_i0m1_i1_i2 + AD2_i0p1_i1_i2));\n const double AD_dDD211 = FDPart1_11*(FDPart1_14 + FDPart1_Rational_1_12*(-AD2_i0_i1m2_i2 - AD2_i0_i1p2_i2) + FDPart1_Rational_4_3*(AD2_i0_i1m1_i2 + AD2_i0_i1p1_i2));\n const double AD_dDD222 = FDPart1_12*(FDPart1_14 + FDPart1_Rational_1_12*(-AD2_i0_i1_i2m2 - AD2_i0_i1_i2p2) + FDPart1_Rational_4_3*(AD2_i0_i1_i2m1 + AD2_i0_i1_i2p1));\n const double Gamma_dD0 = invdx0*(FDPart1_Rational_1_12*(Gamma_i0m2_i1_i2 - Gamma_i0p2_i1_i2) + FDPart1_Rational_2_3*(-Gamma_i0m1_i1_i2 + Gamma_i0p1_i1_i2));\n const double Gamma_dD1 = invdx1*(FDPart1_Rational_1_12*(Gamma_i0_i1m2_i2 - Gamma_i0_i1p2_i2) + FDPart1_Rational_2_3*(-Gamma_i0_i1m1_i2 + Gamma_i0_i1p1_i2));\n const double Gamma_dD2 = invdx2*(FDPart1_Rational_1_12*(Gamma_i0_i1_i2m2 - Gamma_i0_i1_i2p2) + FDPart1_Rational_2_3*(-Gamma_i0_i1_i2m1 + Gamma_i0_i1_i2p1));\n const double psi_dD0 = invdx0*(FDPart1_Rational_1_12*(psi_i0m2_i1_i2 - psi_i0p2_i1_i2) + FDPart1_Rational_2_3*(-psi_i0m1_i1_i2 + psi_i0p1_i1_i2));\n const double psi_dD1 = invdx1*(FDPart1_Rational_1_12*(psi_i0_i1m2_i2 - psi_i0_i1p2_i2) + FDPart1_Rational_2_3*(-psi_i0_i1m1_i2 + psi_i0_i1p1_i2));\n const double psi_dD2 = invdx2*(FDPart1_Rational_1_12*(psi_i0_i1_i2m2 - psi_i0_i1_i2p2) + FDPart1_Rational_2_3*(-psi_i0_i1_i2m1 + psi_i0_i1_i2p1));\n const double psi_dDD00 = FDPart1_9*(FDPart1_18 + FDPart1_Rational_1_12*(-psi_i0m2_i1_i2 - psi_i0p2_i1_i2) + FDPart1_Rational_4_3*(psi_i0m1_i1_i2 + psi_i0p1_i1_i2));\n const double psi_dDD11 = FDPart1_11*(FDPart1_18 + FDPart1_Rational_1_12*(-psi_i0_i1m2_i2 - psi_i0_i1p2_i2) + FDPart1_Rational_4_3*(psi_i0_i1m1_i2 + psi_i0_i1p1_i2));\n const double psi_dDD22 = FDPart1_12*(FDPart1_18 + FDPart1_Rational_1_12*(-psi_i0_i1_i2m2 - psi_i0_i1_i2p2) + FDPart1_Rational_4_3*(psi_i0_i1_i2m1 + psi_i0_i1_i2p1));\n /*\n * NRPy+ Finite Difference Code Generation, Step 2 of 2: Evaluate SymPy expressions and write to main memory:\n */\n /*\n * Original SymPy expressions:\n * \"[rhs_gfs[IDX4(AD0GF, i0, i1, i2)] = -ED0 - psi_dD0,\n * rhs_gfs[IDX4(ED0GF, i0, i1, i2)] = -AD_dDD000 + Gamma_dD0 - (AD_dD00*xx0 - AD_dD11/xx0 + AD_dDD011 - (AD0*xx0 + AD_dD11)/xx0)/xx0**2 - (AD_dD00*xx0*sin(xx1)**2 - AD_dD22/xx0 + AD_dDD022 + (-AD1/xx0 + AD_dD01)*sin(2*xx1)/2 - (AD0*xx0*sin(xx1)**2 + AD1*sin(2*xx1)/2 + AD_dD22)/xx0)/(xx0**2*sin(xx1)**2),\n * rhs_gfs[IDX4(AD1GF, i0, i1, i2)] = -ED1 - psi_dD1,\n * rhs_gfs[IDX4(ED1GF, i0, i1, i2)] = -AD1/xx0**2 + AD_dD10/xx0 - AD_dDD100 + Gamma_dD1 + (-AD1/xx0 + AD_dD10)/xx0 - (AD_dD01*xx0 + AD_dDD111 + xx0*(-AD1/xx0 + AD_dD01) + xx0*(-AD1/xx0 + AD_dD10))/xx0**2 - (-AD_dD22*sin(2*xx1)/(2*sin(xx1)**2) + AD_dDD122 + xx0*(-AD1/xx0 + AD_dD10)*sin(xx1)**2 + (AD0*xx0 + AD_dD11)*sin(2*xx1)/2 - (AD0*xx0*sin(xx1)**2 + AD1*sin(2*xx1)/2 + AD_dD22)*sin(2*xx1)/(2*sin(xx1)**2))/(xx0**2*sin(xx1)**2),\n * rhs_gfs[IDX4(AD2GF, i0, i1, i2)] = -ED2 - psi_dD2,\n * rhs_gfs[IDX4(ED2GF, i0, i1, i2)] = -AD2/xx0**2 + AD_dD20/xx0 - AD_dDD200 + Gamma_dD2 + (-AD2/xx0 + AD_dD20)/xx0 - (AD2*(-cos(2*xx1)/sin(xx1)**2 + sin(2*xx1)*cos(xx1)/sin(xx1)**3) - AD_dD21*sin(2*xx1)/(2*sin(xx1)**2) + AD_dDD211 + xx0*(-AD2/xx0 + AD_dD20) - (-AD2*sin(2*xx1)/(2*sin(xx1)**2) + AD_dD21)*sin(2*xx1)/(2*sin(xx1)**2))/xx0**2 - (AD_dD02*xx0*sin(xx1)**2 + AD_dD12*sin(2*xx1)/2 + AD_dDD222 + xx0*(-AD2/xx0 + AD_dD02)*sin(xx1)**2 + xx0*(-AD2/xx0 + AD_dD20)*sin(xx1)**2 + (-AD2*sin(2*xx1)/(2*sin(xx1)**2) + AD_dD12)*sin(2*xx1)/2 + (-AD2*sin(2*xx1)/(2*sin(xx1)**2) + AD_dD21)*sin(2*xx1)/2)/(xx0**2*sin(xx1)**2),\n * rhs_gfs[IDX4(PSIGF, i0, i1, i2)] = -Gamma,\n * rhs_gfs[IDX4(GAMMAGF, i0, i1, i2)] = -psi_dDD00 - (psi_dD0*xx0 + psi_dDD11)/xx0**2 - (psi_dD0*xx0*sin(xx1)**2 + psi_dD1*sin(2*xx1)/2 + psi_dDD22)/(xx0**2*sin(xx1)**2)]\"\n */\n const double FDPart3_0 = (1.0/((xx0)*(xx0)));\n const double FDPart3_2 = (1.0/(xx0));\n const double FDPart3_4 = AD0*xx0 + AD_dD11;\n const double FDPart3_5 = sin(xx1);\n const double FDPart3_6 = ((FDPart3_5)*(FDPart3_5));\n const double FDPart3_7 = -AD1*FDPart3_2;\n const double FDPart3_10 = sin(2*xx1);\n const double FDPart3_11 = (1.0/2.0)*FDPart3_10;\n const double FDPart3_12 = AD0*FDPart3_6*xx0 + AD1*FDPart3_11 + AD_dD22;\n const double FDPart3_13 = (1.0/(FDPart3_6));\n const double FDPart3_14 = FDPart3_0*FDPart3_13;\n const double FDPart3_16 = xx0*(AD_dD10 + FDPart3_7);\n const double FDPart3_17 = FDPart3_11*FDPart3_13;\n const double FDPart3_18 = -AD2*FDPart3_2;\n const double FDPart3_20 = xx0*(AD_dD20 + FDPart3_18);\n const double FDPart3_21 = -AD2*FDPart3_17;\n const double FDPart3_22 = FDPart3_11*(AD_dD21 + FDPart3_21);\n rhs_gfs[IDX4(AD0GF, i0, i1, i2)] = -ED0 - psi_dD0;\n rhs_gfs[IDX4(ED0GF, i0, i1, i2)] = -AD_dDD000 - FDPart3_0*(AD_dD00*xx0 - AD_dD11*FDPart3_2 + AD_dDD011 - FDPart3_2*FDPart3_4) - FDPart3_14*(AD_dD00*FDPart3_6*xx0 - AD_dD22*FDPart3_2 + AD_dDD022 + FDPart3_11*(AD_dD01 + FDPart3_7) - FDPart3_12*FDPart3_2) + Gamma_dD0;\n rhs_gfs[IDX4(AD1GF, i0, i1, i2)] = -ED1 - psi_dD1;\n rhs_gfs[IDX4(ED1GF, i0, i1, i2)] = -AD1*FDPart3_0 + AD_dD10*FDPart3_2 - AD_dDD100 - FDPart3_0*(AD_dD01*xx0 + AD_dDD111 + FDPart3_16 + xx0*(AD_dD01 + FDPart3_7)) - FDPart3_14*(-AD_dD22*FDPart3_17 + AD_dDD122 + FDPart3_11*FDPart3_4 - FDPart3_12*FDPart3_17 + FDPart3_16*FDPart3_6) + FDPart3_2*(AD_dD10 + FDPart3_7) + Gamma_dD1;\n rhs_gfs[IDX4(AD2GF, i0, i1, i2)] = -ED2 - psi_dD2;\n rhs_gfs[IDX4(ED2GF, i0, i1, i2)] = -AD2*FDPart3_0 + AD_dD20*FDPart3_2 - AD_dDD200 - FDPart3_0*(AD2*(FDPart3_10*cos(xx1)/((FDPart3_5)*(FDPart3_5)*(FDPart3_5)) - FDPart3_13*cos(2*xx1)) - AD_dD21*FDPart3_17 + AD_dDD211 - FDPart3_13*FDPart3_22 + FDPart3_20) - FDPart3_14*(AD_dD02*FDPart3_6*xx0 + AD_dD12*FDPart3_11 + AD_dDD222 + FDPart3_11*(AD_dD12 + FDPart3_21) + FDPart3_20*FDPart3_6 + FDPart3_22 + FDPart3_6*xx0*(AD_dD02 + FDPart3_18)) + FDPart3_2*(AD_dD20 + FDPart3_18) + Gamma_dD2;\n rhs_gfs[IDX4(PSIGF, i0, i1, i2)] = -Gamma;\n rhs_gfs[IDX4(GAMMAGF, i0, i1, i2)] = -FDPart3_0*(psi_dD0*xx0 + psi_dDD11) - FDPart3_14*(FDPart3_11*psi_dD1 + FDPart3_6*psi_dD0*xx0 + psi_dDD22) - psi_dDD00;\n }\n\n\n\n\n# Step 3: Output this notebook to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n$$\\label{latex_pdf_output}$$\n\nThe following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n[Tutorial-MaxwellCurvilinear.pdf](Tutorial-MaxwellCurvilinear.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means.)\n\n\n```python\nimport cmdline_helper as cmd # NRPy+: Multi-platform Python command-line interface\ncmd.output_Jupyter_notebook_to_LaTeXed_PDF(\"Tutorial-MaxwellCurvilinear\")\n```\n\n Created Tutorial-MaxwellCurvilinear.tex, and compiled LaTeX file to PDF\n file Tutorial-MaxwellCurvilinear.pdf\n\n", "meta": {"hexsha": "d34fdb9ad74cf668bf7772ad5c4a3af686102d44", "size": 63980, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Tutorial-MaxwellCurvilinear.ipynb", "max_stars_repo_name": "rhaas80/nrpytutorial", "max_stars_repo_head_hexsha": "4398cd6b5a071c8fb8b2b584a01f07a4591dd5f4", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Tutorial-MaxwellCurvilinear.ipynb", "max_issues_repo_name": "rhaas80/nrpytutorial", "max_issues_repo_head_hexsha": "4398cd6b5a071c8fb8b2b584a01f07a4591dd5f4", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Tutorial-MaxwellCurvilinear.ipynb", "max_forks_repo_name": "rhaas80/nrpytutorial", "max_forks_repo_head_hexsha": "4398cd6b5a071c8fb8b2b584a01f07a4591dd5f4", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 79.5771144279, "max_line_length": 636, "alphanum_fraction": 0.6380275086, "converted": true, "num_tokens": 26174, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7826624890918021, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.4279114429137299}} {"text": "\n\n# **MAC023 - Mec\u00e2nica das Estruturas**\n# ME-01 - Primeira Avalia\u00e7\u00e3o de Conhecimentos\n\n\n**ATEN\u00c7\u00c3O:** N\u00e3o edite este arquivo!!! Fa\u00e7a uma c\u00f3pia deste arquivo para o seu Drive. Para isso, no pr\u00f3prio colab, v\u00e1 em **File -> Save a copy in Drive**\n\n---\n\nAlunos:\n\nBrian Maia
                                        \nMathews Edwirds
                                        \n\n# Condi\u00e7\u00f5es Gerais\n\nEsta avalia\u00e7\u00e3o tem como objetivo avaliar os conhecimentos adquiridos na primeira parte da disciplina de Mec\u00e2nica das Estruturas.\n\n\n---\n\nAs condic\u00f5es abaixo devem ser observadas: \n\n1. Ser\u00e3o formadas equipes e cada uma delas com no m\u00ednimo **2** e no m\u00e1ximo **3** integrantes. \n\n2. A avalia\u00e7\u00e3o ser\u00e1 realizada por meio da entrega de uma c\u00f3pia deste notebook com as solu\u00e7\u00f5es desenvolvidas at\u00e9 a data estipulada de entrega.\n\n\n3. Da entrega da avalia\u00e7\u00e3o.\n * Os documentos necess\u00e1rios para a entrega do trabalho s\u00e3o (1) os c\u00f3digos desenvolvidos pela equipe e (2) v\u00eddeo com a descri\u00e7\u00e3o da solu\u00e7\u00e3o. \n * A equipe deve usar este modelo de notebook para desenvolver os c\u00f3digos. \n * Os c\u00f3digos podem ser desenvolvidos combinado a linguagem LaTeX e computa\u00e7\u00e3o simb\u00f3lica ou computa\u00e7\u00e3o num\u00e9rica quando necess\u00e1rio.\n * Os gr\u00e1ficos necess\u00e1rios para a apresenta\u00e7\u00e3o da solu\u00e7\u00e3o devem estar embutidos no notebook.\n\n4. Da distribui\u00e7\u00e3o das quest\u00f5es.\n * A quantidade de quest\u00f5es ser\u00e1 a mesma para cada grupo. \n * Ser\u00e3o atribu\u00eddas as mesmas quest\u00f5es para todos os grupos.\n * A pontuac\u00e3o referente a cada quest\u00e3o ser\u00e1 igualit\u00e1ria e o valor total da avalia\u00e7\u00e3o ser\u00e1 100 pontos.\n\n5. As equipes devem ser formadas at\u00e9 \u00e0s **23:59 horas o dia 10/12/2021** por meio do preenchimento da planilha [[MAC023] Forma\u00e7\u00e3o das Equipes](https://docs.google.com/spreadsheets/d/1Dlftymao970nnrE4mu958iP8nMqKqSuhHiiLH91BKpQ/edit?usp=sharing).\n\n6. A forma\u00e7\u00e3o das equipes pode ser acompanhada arquivo indicado acima. Cada equipe ser\u00e1 indentificada por uma letra em ordem alfab\u00e9tica seguida do n\u00famero 1 (A1, B1, C1, e assim por diante). O arquivo est\u00e1 aberto para edi\u00e7\u00e3o e pode ser alterado pelos alunos at\u00e9 a data estipulada.\n\n7. Equipes formadas ap\u00f3s a data estabelecida para a forma\u00e7\u00e3o das equipes ter\u00e3o a nota da avalia\u00e7\u00e3o multiplicada por um coeficiente de **0.80**.\n\n8. A equipe deve indicar no arquivo de indica\u00e7\u00e3o de equipes um respons\u00e1vel pela entrega do projeto. \n * Somente o respons\u00e1vel pela entrega deve fazer o upload do arquivo na plataforma\n\n9. A entrega dos projetos deve ocorrer at\u00e9 \u00e0s **23:59 do dia 21/12/2021** na plataforma da disciplina pelo respons\u00e1vel pela entrega. \n * Caso a entrega seja feita por outro integrante diferente daquele indicado pela pela equipe a avalia\u00e7\u00e3o ser\u00e1 desconsiderada e n\u00e3o ser\u00e1 corrigida at\u00e9 que a a condi\u00e7\u00e3o de entrega seja satisfeita.\n\n10. Quaisquer d\u00favidas ou esclarecimentos devem ser encaminhadas pela sala de aula virtual.\n\n\n# Quest\u00e3o Q1\n---\nConsidere a estrutura mostrada abaixo.\n\nAs barras s\u00e3o feitas com um material com m\u00f3dulo de elasticidade E e t\u00eam se\u00e7\u00f5es transversais com \u00e1rea A.\n\n---\n\n\n\n\n---\n\nRecorde-se que as condi\u00e7\u00f5es de compatibilidade entre deslocamentos e deforma\u00e7\u00f5es s\u00e3o condi\u00e7\u00f5es geom\u00e9tricas que devem ser satisfeitas para garantir que a estrutura, ao se deformar, permane\u00e7a cont\u00ednua (sem vazios ou sobreposi\u00e7\u00e3o de pontos) e compat\u00edvel com seus v\u00ednculos externos.\n\n\n## Implementa\u00e7\u00e3o\n---\n \nImplemente um modelo considerando as condi\u00e7\u00f5es b\u00e1sicas de an\u00e1lise estrutural e realize as seguintes entregas:\n \na. (10 pontos) Escreva as posi\u00e7\u00f5es dos n\u00f3s nas configura\u00e7\u00f5es indeformada e deformada.\n\nb. (10 pontos) Determine as condi\u00e7\u00f5es de compatibilidade para a estrutura na configura\u00e7\u00e3o deformada. Escreva as equa\u00e7\u00f5es das deforma\u00e7\u00f5es nas barras em fun\u00e7\u00e3o \n\n* (a) dos deslocamentos nos n\u00f3s livres para se deslocar e \n* (b) as informa\u00e7\u00f5es geom\u00e9tricas da estrutura. \n\nUse uma nota\u00e7\u00e3o vetorial para as condi\u00e7\u00f5es de compatibilidade.\n\nc. (20 pontos) Linearize as condi\u00e7\u00f5es de compatibilidade em fun\u00e7\u00e3o dos deslocamentos. Reescreve-as usando a matriz jacobiana da forma vetorial das condi\u00e7\u00f5es de compatibilidade. Neste caso, a transforma\u00e7\u00e3o linear representada pela Jacobiana ${\\bf J(p)}$ \u00e9 a aproxima\u00e7\u00e3o linear da fun\u00e7\u00e3o vetorial ${\\bf f(u)}$ nas proximidades do ponto ${\\bf p}$, e pode ser representada pela express\u00e3o $$ {\\bf f(u)} - {\\bf f(p)} = {\\bf J(p)} {\\bf (u-p)} $$\nonde ${\\bf u}$ \u00e9 um ponto de interesse na vizinhan\u00e7a de ${\\bf p}$. \n\nd. (10 pontos) Responda \u00e0 seguinte quest\u00e3o: Qual a hip\u00f3tese envolvida no processo de lineariza\u00e7\u00e3o e por que a lineariza\u00e7\u00e3o \u00e9 necess\u00e1ria?\n\ne. (20 pontos) Escreva as condi\u00e7\u00f5es de equil\u00edbrio na configura\u00e7\u00e3o **indeformada**. Use uma nota\u00e7\u00e3o vetorial para as equa\u00e7\u00f5es de equil\u00edbrio.\n\nf. (10 pontos) Usando as rela\u00e7\u00f5es constitutivas, reescreva as for\u00e7as internas em fun\u00e7\u00e3o dos deslocamentos e reescreva as equa\u00e7\u00f5es de equil\u00edbrio em fun\u00e7\u00e3o dos deslocamentos.\n\ng. (10 pontos) Resolva o sistema de equa\u00e7\u00f5es de equil\u00edbrio para os deslocamentos. Assuma valores unit\u00e1rios para as \u00e1reas das se\u00e7\u00f5es transversais (A). O m\u00f3dulo de elasticidade (E) tem valor igual 150000 N/m$^2$.\n\nh. (10 pontos) Compare o resultado encontrado com o pacote anastruct.\n \n---\n\n\n#Alternativa A) \nEscreva as posi\u00e7\u00f5es dos n\u00f3s nas configura\u00e7\u00f5es indeformada e deformada.\n\n\n```\n# Importa as bibliotecas gr\u00e1ficas, num\u00e9ricas e de computa\u00e7\u00e3o simb\u00f3lica\nimport pylab as pl\nimport numpy as np\nimport sympy as sp\n\nsp.init_printing(use_latex='mathjax')\na,b,l,P,u= sp.var('a b l P u', real=True, positive=True)\nP=-40000\n```\n\n\n```\n#--# Posi\u00e7\u00e3o dos n\u00f3s na configura\u00e7\u00e3o indeformada#--\nconf_indeformada=sp.Matrix([[0,0], [a,0], [a,b], [2*a,b]]) # A, B, C, D\n\nprint(\"Configura\u00e7\u00e3o Indeformada\")\nconf_indeformada\n```\n\n Configura\u00e7\u00e3o Indeformada\n\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\a & 0\\\\a & b\\\\2 a & b\\end{matrix}\\right]$\n\n\n\n\n```\n#--# Posi\u00e7\u00e3o dos n\u00f3s na configura\u00e7\u00e3o deformada #--\ndesloc_xb, desloc_yb, desloc_xc, desloc_yc = sp.var('desloc_xb desloc_yb desloc_xc desloc_yc', real=True, positive=True)\nmatriz_deslocamentos = sp.Matrix([desloc_xb, desloc_yb, desloc_xc, desloc_yc])\n\n# Matrizes de posi\u00e7\u00f5es e deslocamentos para a configura\u00e7\u00e3o deformada\nconf_deformada = conf_indeformada + sp.Matrix([\n[0,0], # A \n[desloc_xb, -desloc_yb], # B \n[desloc_xc, -desloc_yc], # C \n[0,0], # D \n])\n\nprint(\"Matrizes de posi\u00e7\u00f5es e deslocamentos para a configura\u00e7\u00e3o deformada\")\nconf_deformada, matriz_deslocamentos\n```\n\n Matrizes de posi\u00e7\u00f5es e deslocamentos para a configura\u00e7\u00e3o deformada\n\n\n\n\n\n$\\displaystyle \\left( \\left[\\begin{matrix}0 & 0\\\\a + desloc_{xb} & - desloc_{yb}\\\\a + desloc_{xc} & b - desloc_{yc}\\\\2 a & b\\end{matrix}\\right], \\ \\left[\\begin{matrix}desloc_{xb}\\\\desloc_{yb}\\\\desloc_{xc}\\\\desloc_{yc}\\end{matrix}\\right]\\right)$\n\n\n\n\n```\nidentifica_barra = [[0, 2], [0, 1], [2, 1], [2, 3], [1, 3]] #AC, AB, CB, CD, BD\n\nprint(\"Identifica\u00e7\u00e3o da barra\")\nprint(\" AC, AB, CB, CD, BD\")\nidentifica_barra\n```\n\n Identifica\u00e7\u00e3o da barra\n AC, AB, CB, CD, BD\n\n\n\n\n\n$\\displaystyle \\left[ \\left[ 0, \\ 2\\right], \\ \\left[ 0, \\ 1\\right], \\ \\left[ 2, \\ 1\\right], \\ \\left[ 2, \\ 3\\right], \\ \\left[ 1, \\ 3\\right]\\right]$\n\n\n\n#Alternativa B) \n\nDetermine as condi\u00e7\u00f5es de compatibilidade para a estrutura na configura\u00e7\u00e3o deformada. Escreva as equa\u00e7\u00f5es das deforma\u00e7\u00f5es nas barras em fun\u00e7\u00e3o\n\n* (a) dos deslocamentos nos n\u00f3s livres para se deslocar e
                                        \n* (b) as informa\u00e7\u00f5es geom\u00e9tricas da estrutura.\n\nUse uma nota\u00e7\u00e3o vetorial para as condi\u00e7\u00f5es de compatibilidade.\n\n\n```\nepsilon = []\nfor pos, nos in enumerate(identifica_barra):\n # Auxiliares para salvar o valor dos n\u00f3s barras\n primeiro_no_lista = nos[0]\n ultimo_no_lista = nos[1]\n # C\u00e1lculo das condi\u00e7\u00f5es de compatibilidade\n comp_f = conf_deformada[ultimo_no_lista,:] - conf_deformada[primeiro_no_lista,:]\n comp_i = conf_indeformada[ultimo_no_lista,:] - conf_indeformada[primeiro_no_lista,:]\n epsilon.insert(pos, ((comp_f.norm() - comp_i.norm())/comp_i.norm()))\n\n# Lista de compatibilidades\nprint(\"Condi\u00e7\u00f5es de compatibilidade para a estrutura deformada em forma de lista e matriz\")\nprint(\"\\n\\t\\t\\tAC, \\t\\t\\t\\t\\tAB, \\t\\t\\t\\t\\t\\tCB, \\t\\t\\t\\t\\tCD, \\t\\t\\t\\t\\tBD\")\ndisplay(epsilon)\nprint(\"\\n\")\n\n# Convers\u00e3o da lista de condi\u00e7\u00f5es de compatibilidade para uma matriz de compatibilidade\nmatriz_epsilon=[]\nfor item in epsilon:\n matriz_epsilon.append(sp.Matrix([item]))\nmatriz_epsilon = sp.Matrix(matriz_epsilon)\ndisplay(matriz_epsilon)\n```\n\n Condi\u00e7\u00f5es de compatibilidade para a estrutura deformada em forma de lista e matriz\n \n \t\t\tAC, \t\t\t\t\tAB, \t\t\t\t\t\tCB, \t\t\t\t\tCD, \t\t\t\t\tBD\n\n\n\n$\\displaystyle \\left[ \\frac{- \\sqrt{a^{2} + b^{2}} + \\sqrt{\\left(a + desloc_{xc}\\right)^{2} + \\left(b - desloc_{yc}\\right)^{2}}}{\\sqrt{a^{2} + b^{2}}}, \\ \\frac{- a + \\sqrt{desloc_{yb}^{2} + \\left(a + desloc_{xb}\\right)^{2}}}{a}, \\ \\frac{- b + \\sqrt{\\left(desloc_{xb} - desloc_{xc}\\right)^{2} + \\left(b + desloc_{yb} - desloc_{yc}\\right)^{2}}}{b}, \\ \\frac{- a + \\sqrt{desloc_{yc}^{2} + \\left(a - desloc_{xc}\\right)^{2}}}{a}, \\ \\frac{- \\sqrt{a^{2} + b^{2}} + \\sqrt{\\left(a - desloc_{xb}\\right)^{2} + \\left(b + desloc_{yb}\\right)^{2}}}{\\sqrt{a^{2} + b^{2}}}\\right]$\n\n\n \n \n\n\n\n$\\displaystyle \\left[\\begin{matrix}\\frac{- \\sqrt{a^{2} + b^{2}} + \\sqrt{\\left(a + desloc_{xc}\\right)^{2} + \\left(b - desloc_{yc}\\right)^{2}}}{\\sqrt{a^{2} + b^{2}}}\\\\\\frac{- a + \\sqrt{desloc_{yb}^{2} + \\left(a + desloc_{xb}\\right)^{2}}}{a}\\\\\\frac{- b + \\sqrt{\\left(desloc_{xb} - desloc_{xc}\\right)^{2} + \\left(b + desloc_{yb} - desloc_{yc}\\right)^{2}}}{b}\\\\\\frac{- a + \\sqrt{desloc_{yc}^{2} + \\left(a - desloc_{xc}\\right)^{2}}}{a}\\\\\\frac{- \\sqrt{a^{2} + b^{2}} + \\sqrt{\\left(a - desloc_{xb}\\right)^{2} + \\left(b + desloc_{yb}\\right)^{2}}}{\\sqrt{a^{2} + b^{2}}}\\end{matrix}\\right]$\n\n\n#Alternativa C)\nLinearize as condi\u00e7\u00f5es de compatibilidade em fun\u00e7\u00e3o dos deslocamentos. Reescreva-as usando a matriz jacobiana da forma vetorial das condi\u00e7\u00f5es de compatibilidade. Neste caso, a transforma\u00e7\u00e3o linear representada pela Jacobiana ${\\bf J(p)}$ \u00e9 a aproxima\u00e7\u00e3o linear da fun\u00e7\u00e3o vetorial ${\\bf f(u)}$ nas proximidades do ponto ${\\bf p}$, e pode ser representada pela express\u00e3o $$ {\\bf f(u)} - {\\bf f(p)} = {\\bf J(p)} {\\bf (u-p)} $$\nonde ${\\bf u}$ \u00e9 um ponto de interesse na vizinhan\u00e7a de ${\\bf p}$. \n\n$ {\\bf f(u)} - {\\bf f(p)} = {\\bf J(p)} {\\bf (u-p)} \\Rightarrow$
                                        \n$ {\\bf f(u)} = {\\bf J(p)} {\\bf (u-p) + {\\bf f(p)}} \\Rightarrow$
                                        \n$ {\\bf f(u)} = {\\bf J(0)} {\\bf (u-0)} + {\\bf f(0)}$\n\n\n```\nfrom sympy import latex, Eq #Lineariza\u00e7\u00e3o das condi\u00e7\u00f5es de compatibilidade\n\nJacobiano = matriz_epsilon.jacobian(matriz_deslocamentos) #Calculo do Jacobiano\nprint(\"J(p) = \")\ndisplay(Jacobiano)\n\nprint(\"\\nJ(0) = \") #Substitui\u00e7\u00e3o das posi\u00e7\u00f5es iniciais no jacobiano\nJacobiano_0 = Jacobiano.subs([(desloc_xb, 0), (desloc_xc, 0), (desloc_yb, 0), (desloc_yc, 0)])\ndisplay(Jacobiano_0)\n\nprint(\"\\nf(0) = \") #Fun\u00e7\u00e3o vetorial no ponto inicial\nfuncaoVetorial = matriz_epsilon.subs([(desloc_xb, 0), (desloc_xc, 0), (desloc_yb, 0), (desloc_yc, 0)])\ndisplay(funcaoVetorial)\n\nprint(\"\\nf(u) = \") #Transforma\u00e7\u00e3o linear\nepsilon_linearizado = Jacobiano_0 * matriz_deslocamentos + funcaoVetorial\ndisplay(epsilon_linearizado)\n```\n\n J(p) = \n\n\n\n$\\displaystyle \\left[\\begin{matrix}0 & 0 & \\frac{a + desloc_{xc}}{\\sqrt{a^{2} + b^{2}} \\sqrt{\\left(a + desloc_{xc}\\right)^{2} + \\left(b - desloc_{yc}\\right)^{2}}} & \\frac{- b + desloc_{yc}}{\\sqrt{a^{2} + b^{2}} \\sqrt{\\left(a + desloc_{xc}\\right)^{2} + \\left(b - desloc_{yc}\\right)^{2}}}\\\\\\frac{a + desloc_{xb}}{a \\sqrt{desloc_{yb}^{2} + \\left(a + desloc_{xb}\\right)^{2}}} & \\frac{desloc_{yb}}{a \\sqrt{desloc_{yb}^{2} + \\left(a + desloc_{xb}\\right)^{2}}} & 0 & 0\\\\\\frac{desloc_{xb} - desloc_{xc}}{b \\sqrt{\\left(desloc_{xb} - desloc_{xc}\\right)^{2} + \\left(b + desloc_{yb} - desloc_{yc}\\right)^{2}}} & \\frac{b + desloc_{yb} - desloc_{yc}}{b \\sqrt{\\left(desloc_{xb} - desloc_{xc}\\right)^{2} + \\left(b + desloc_{yb} - desloc_{yc}\\right)^{2}}} & \\frac{- desloc_{xb} + desloc_{xc}}{b \\sqrt{\\left(desloc_{xb} - desloc_{xc}\\right)^{2} + \\left(b + desloc_{yb} - desloc_{yc}\\right)^{2}}} & \\frac{- b - desloc_{yb} + desloc_{yc}}{b \\sqrt{\\left(desloc_{xb} - desloc_{xc}\\right)^{2} + \\left(b + desloc_{yb} - desloc_{yc}\\right)^{2}}}\\\\0 & 0 & \\frac{- a + desloc_{xc}}{a \\sqrt{desloc_{yc}^{2} + \\left(a - desloc_{xc}\\right)^{2}}} & \\frac{desloc_{yc}}{a \\sqrt{desloc_{yc}^{2} + \\left(a - desloc_{xc}\\right)^{2}}}\\\\\\frac{- a + desloc_{xb}}{\\sqrt{a^{2} + b^{2}} \\sqrt{\\left(a - desloc_{xb}\\right)^{2} + \\left(b + desloc_{yb}\\right)^{2}}} & \\frac{b + desloc_{yb}}{\\sqrt{a^{2} + b^{2}} \\sqrt{\\left(a - desloc_{xb}\\right)^{2} + \\left(b + desloc_{yb}\\right)^{2}}} & 0 & 0\\end{matrix}\\right]$\n\n\n \n J(0) = \n\n\n\n$\\displaystyle \\left[\\begin{matrix}0 & 0 & \\frac{a}{a^{2} + b^{2}} & - \\frac{b}{a^{2} + b^{2}}\\\\\\frac{1}{a} & 0 & 0 & 0\\\\0 & \\frac{1}{b} & 0 & - \\frac{1}{b}\\\\0 & 0 & - \\frac{1}{a} & 0\\\\- \\frac{a}{a^{2} + b^{2}} & \\frac{b}{a^{2} + b^{2}} & 0 & 0\\end{matrix}\\right]$\n\n\n \n f(0) = \n\n\n\n$\\displaystyle \\left[\\begin{matrix}0\\\\0\\\\0\\\\0\\\\0\\end{matrix}\\right]$\n\n\n \n f(u) = \n\n\n\n$\\displaystyle \\left[\\begin{matrix}\\frac{a desloc_{xc}}{a^{2} + b^{2}} - \\frac{b desloc_{yc}}{a^{2} + b^{2}}\\\\\\frac{desloc_{xb}}{a}\\\\\\frac{desloc_{yb}}{b} - \\frac{desloc_{yc}}{b}\\\\- \\frac{desloc_{xc}}{a}\\\\- \\frac{a desloc_{xb}}{a^{2} + b^{2}} + \\frac{b desloc_{yb}}{a^{2} + b^{2}}\\end{matrix}\\right]$\n\n\n# Alternativa D) \nResponda \u00e0 seguinte quest\u00e3o: Qual a hip\u00f3tese envolvida no processo de lineariza\u00e7\u00e3o e por que a lineariza\u00e7\u00e3o \u00e9 necess\u00e1ria?\n\nR - A hip\u00f3tese envolvida \u00e9 de que as deforma\u00e7\u00f5es s\u00e3o pequenas. Dito isso, o processo de lineariza\u00e7\u00e3o \u00e9 importante para facilitar os c\u00e1lculos, j\u00e1 que passamos a usar equa\u00e7\u00f5es lineares ao inv\u00e9s de equa\u00e7\u00f5es n\u00e3o lineares. De modo geral, o objetivo \u00e9 simplificar a resolu\u00e7\u00e3o do problema. \n\n#Alternativa E)\n\nEscreva as condi\u00e7\u00f5es de equil\u00edbrio na configura\u00e7\u00e3o indeformada. Use uma nota\u00e7\u00e3o vetorial para as equa\u00e7\u00f5es de equil\u00edbrio.\n\nConsiderando que $ \\sum Y = 0 $ e $ \\sum X = 0 $, temos:\n\n\n```\nNac, Nab, Ncb, Ncd, Nbd = sp.var('N_{ac} N_{ab} N_{cb} N_{cd} N_{bd}', real=True, positive=True)\nNormais_nos = [Nac, Nab, Ncb, Ncd, Nbd]\n\n#Condi\u00e7\u00f5es de equilibrio para o eixo X\nBX = Nbd*(a/sp.sqrt(a**2 + b**2)) - Nab\nCX = Ncd - Nac*a/(sp.sqrt(a**2 + b**2))\n\n#Condi\u00e7\u00f5es de equilibrio para o eixo Y\nBY = Ncb + Nbd*b/(sp.sqrt(a**2 + b**2))\nCY = - Ncb - Nac*b/(sp.sqrt(a**2 + b**2)) + P\n\nprint(\"Condi\u00e7\u00f5es de equilibrio para os pontos B e C das dire\u00e7\u00f5es X e Y:\\n\")\nprint(\"\\tBX \\t\\tBY \\t\\tCX \\t\\t\\tCY\")\ndisplay([BX, BY, CX, CY])\n```\n\n Condi\u00e7\u00f5es de equilibrio para os pontos B e C das dire\u00e7\u00f5es X e Y:\n \n \tBX \t\tBY \t\tCX \t\t\tCY\n\n\n\n$\\displaystyle \\left[ - N_{ab} + \\frac{N_{bd} a}{\\sqrt{a^{2} + b^{2}}}, \\ \\frac{N_{bd} b}{\\sqrt{a^{2} + b^{2}}} + N_{cb}, \\ - \\frac{N_{ac} a}{\\sqrt{a^{2} + b^{2}}} + N_{cd}, \\ - \\frac{N_{ac} b}{\\sqrt{a^{2} + b^{2}}} - N_{cb} - 40000\\right]$\n\n\n#Alternativa F)\n\nUsando as rela\u00e7\u00f5es constitutivas, reescreva as for\u00e7as internas em fun\u00e7\u00e3o dos deslocamentos e reescreva as equa\u00e7\u00f5es de equil\u00edbrio em fun\u00e7\u00e3o dos deslocamentos.\n\nEqua\u00e7\u00e3o Constitutiva:
                                        \n$\\frac{N}{A} = E \\epsilon \\Rightarrow$
                                        \n$N = E \\epsilon A$\n\n\n```\nNormais_nos = [Nac, Nab, Ncb, Ncd, Nbd]\nNormais_nos\n```\n\n\n\n\n$\\displaystyle \\left[ N_{ac}, \\ N_{ab}, \\ N_{cb}, \\ N_{cd}, \\ N_{bd}\\right]$\n\n\n\n\n```\nA, E = sp.var('A E', real=True, positive=True)\n\nequacoesEquilibrio = []\nfor item in range(len(Normais_nos)):\n equacoesEquilibrio.append(E * epsilon[item] * A) #epsilon\n\nprint(\"\\t\\t\\tAC, \\t\\t\\t\\t\\t\\tAB, \\t\\t\\t\\t\\t\\t\\tCB, \\t\\t\\t\\t\\t\\t\\tCD, \\t\\t\\t\\t\\tBD\")\ndisplay(equacoesEquilibrio)\n```\n\n \t\t\tAC, \t\t\t\t\t\tAB, \t\t\t\t\t\t\tCB, \t\t\t\t\t\t\tCD, \t\t\t\t\tBD\n\n\n\n$\\displaystyle \\left[ \\frac{A E \\left(- \\sqrt{a^{2} + b^{2}} + \\sqrt{\\left(a + desloc_{xc}\\right)^{2} + \\left(b - desloc_{yc}\\right)^{2}}\\right)}{\\sqrt{a^{2} + b^{2}}}, \\ \\frac{A E \\left(- a + \\sqrt{desloc_{yb}^{2} + \\left(a + desloc_{xb}\\right)^{2}}\\right)}{a}, \\ \\frac{A E \\left(- b + \\sqrt{\\left(desloc_{xb} - desloc_{xc}\\right)^{2} + \\left(b + desloc_{yb} - desloc_{yc}\\right)^{2}}\\right)}{b}, \\ \\frac{A E \\left(- a + \\sqrt{desloc_{yc}^{2} + \\left(a - desloc_{xc}\\right)^{2}}\\right)}{a}, \\ \\frac{A E \\left(- \\sqrt{a^{2} + b^{2}} + \\sqrt{\\left(a - desloc_{xb}\\right)^{2} + \\left(b + desloc_{yb}\\right)^{2}}\\right)}{\\sqrt{a^{2} + b^{2}}}\\right]$\n\n\n\n```\nnormal = []\nfor pos, nos in enumerate(identifica_barra):\n # Auxiliares para salvar o valor dos n\u00f3s das barras\n primeiro_no_lista = nos[0]\n ultimo_no_lista = nos[1]\n\n # C\u00e1lculo das condi\u00e7\u00f5es de compatibilidade\n comp_f = conf_deformada[ultimo_no_lista,:] - conf_deformada[primeiro_no_lista,:]\n comp_f /= comp_f.norm()\n normal.append(comp_f)\n\n# Calculo da matriz normal\nnormal_mat = sp.Matrix(normal)\ndisplay(normal_mat)\ndisplay(normal_mat.subs([(desloc_xb, 0), (desloc_xc, 0), (desloc_yb, 0), (desloc_yc, 0)]))\n```\n\n\n$\\displaystyle \\left[\\begin{matrix}\\frac{a + desloc_{xc}}{\\sqrt{\\left(a + desloc_{xc}\\right)^{2} + \\left(b - desloc_{yc}\\right)^{2}}} & \\frac{b - desloc_{yc}}{\\sqrt{\\left(a + desloc_{xc}\\right)^{2} + \\left(b - desloc_{yc}\\right)^{2}}}\\\\\\frac{a + desloc_{xb}}{\\sqrt{desloc_{yb}^{2} + \\left(a + desloc_{xb}\\right)^{2}}} & - \\frac{desloc_{yb}}{\\sqrt{desloc_{yb}^{2} + \\left(a + desloc_{xb}\\right)^{2}}}\\\\\\frac{desloc_{xb} - desloc_{xc}}{\\sqrt{\\left(desloc_{xb} - desloc_{xc}\\right)^{2} + \\left(b + desloc_{yb} - desloc_{yc}\\right)^{2}}} & \\frac{- b - desloc_{yb} + desloc_{yc}}{\\sqrt{\\left(desloc_{xb} - desloc_{xc}\\right)^{2} + \\left(b + desloc_{yb} - desloc_{yc}\\right)^{2}}}\\\\\\frac{a - desloc_{xc}}{\\sqrt{desloc_{yc}^{2} + \\left(a - desloc_{xc}\\right)^{2}}} & \\frac{desloc_{yc}}{\\sqrt{desloc_{yc}^{2} + \\left(a - desloc_{xc}\\right)^{2}}}\\\\\\frac{a - desloc_{xb}}{\\sqrt{\\left(a - desloc_{xb}\\right)^{2} + \\left(b + desloc_{yb}\\right)^{2}}} & \\frac{b + desloc_{yb}}{\\sqrt{\\left(a - desloc_{xb}\\right)^{2} + \\left(b + desloc_{yb}\\right)^{2}}}\\end{matrix}\\right]$\n\n\n\n$\\displaystyle \\left[\\begin{matrix}\\frac{a}{\\sqrt{a^{2} + b^{2}}} & \\frac{b}{\\sqrt{a^{2} + b^{2}}}\\\\1 & 0\\\\0 & -1\\\\1 & 0\\\\\\frac{a}{\\sqrt{a^{2} + b^{2}}} & \\frac{b}{\\sqrt{a^{2} + b^{2}}}\\end{matrix}\\right]$\n\n\n#Alternativa G)\n\nResolva o sistema de equa\u00e7\u00f5es de equil\u00edbrio para os deslocamentos. Assuma valores unit\u00e1rios para as \u00e1reas das se\u00e7\u00f5es transversais (A). O m\u00f3dulo de elasticidade (E) tem valor igual 150000 N/m 2 .\n\n\n```\nsistema = []\nfor eq in equacoesEquilibrio:\n sistema.append(eq.subs([(a, 4),(b, 3),(A, 1),(E, 150000)]))\ndisplay(sistema)\n#display(sp.linsolve(sistema, [desloc_xc, desloc_xb, desloc_yc, desloc_yb]))\n```\n\n\n$\\displaystyle \\left[ 30000 \\sqrt{\\left(3 - desloc_{yc}\\right)^{2} + \\left(desloc_{xc} + 4\\right)^{2}} - 150000, \\ 37500 \\sqrt{desloc_{yb}^{2} + \\left(desloc_{xb} + 4\\right)^{2}} - 150000, \\ 50000 \\sqrt{\\left(desloc_{xb} - desloc_{xc}\\right)^{2} + \\left(desloc_{yb} - desloc_{yc} + 3\\right)^{2}} - 150000, \\ 37500 \\sqrt{desloc_{yc}^{2} + \\left(4 - desloc_{xc}\\right)^{2}} - 150000, \\ 30000 \\sqrt{\\left(4 - desloc_{xb}\\right)^{2} + \\left(desloc_{yb} + 3\\right)^{2}} - 150000\\right]$\n\n\nAo usar a fun\u00e7\u00e3o *linsolve* na c\u00e9lula acima, n\u00f3s recebemos a seguinte mensagem de erro: \"*During handling of the above exception, another exception occurred:*\". Aparentemente, esse problema tem rela\u00e7\u00e3o com as ra\u00edzes quadradas das equa\u00e7\u00f5es de equil\u00edbrio que encontramos. Mesmo pesquisando, n\u00e3o conseguimos encontrar uma solu\u00e7\u00e3o para esse problema. Dessa forma, deixamos a linha comentada.\n\n#Alternativa H)\nCompare o resultado encontrado com o pacote anastruct.\n\n\n```\n! pip install anastruct\n```\n\n Collecting anastruct\n Downloading anastruct-1.2.0-py3-none-any.whl (69 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 69 kB 3.7 MB/s \n \u001b[?25hRequirement already satisfied: numpy>=1.15.4 in /usr/local/lib/python3.7/dist-packages (from anastruct) (1.19.5)\n Requirement already satisfied: matplotlib>=3.0 in /usr/local/lib/python3.7/dist-packages (from anastruct) (3.2.2)\n Requirement already satisfied: scipy>=1.1.0 in /usr/local/lib/python3.7/dist-packages (from anastruct) (1.4.1)\n Requirement already satisfied: kiwisolver>=1.0.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib>=3.0->anastruct) (1.3.2)\n Requirement already satisfied: python-dateutil>=2.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib>=3.0->anastruct) (2.8.2)\n Requirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.7/dist-packages (from matplotlib>=3.0->anastruct) (0.11.0)\n Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib>=3.0->anastruct) (3.0.6)\n Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.7/dist-packages (from python-dateutil>=2.1->matplotlib>=3.0->anastruct) (1.15.0)\n Installing collected packages: anastruct\n Successfully installed anastruct-1.2.0\n\n\n\n```\n# importando os pacotes\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom anastruct import SystemElements\n\nE, A = 150000,1\nss = SystemElements(EA=E*A)\n\n# Define e posiciona os n\u00f3s da estrutura\nnode={\"A\":(0,0), \"B\":(4,0), \"C\":(4,3), \"D\":(8,3)}\n\n# # Define as conectividades/membros entre os n\u00f3s\nss.add_element(location=[node['A'], node['B']], spring={2:0})\nss.add_element(location=[node['A'], node['C']], spring={2:0})\nss.add_element(location=[node['D'], node['B']], spring={2:0})\nss.add_element(location=[node['D'], node['C']], spring={2:0})\nss.add_element(location=[node['B'], node['C']], spring={1:0, 2:0})\n\n# Adiciona os suportes fixos\nnode_id = ss.find_node_id(node['A'])\nss.add_support_hinged(node_id=node_id)\nnode_id = ss.find_node_id(node['D'])\nss.add_support_hinged(node_id=node_id)\n\n#Adiciona a carga pontual\nF=-40000 #=-40kn\nnode_C = ss.find_node_id(node['C'])\nss.point_load(node_id=node_C, Fy=F, rotation=0) # Add um carregamento ao n\u00f3 \"C\" da estrutura\n\nss.solve()\n\nss.show_structure(scale=0.7, figsize=(7, 4.5), offset=(0, 0))\nss.show_displacement(scale=0.7, figsize=(7,4.5), offset=(0,0))\nprint(\"N\u00f3 A: \\nDeslocamento em x= \" + str(ss.get_node_displacements()[0][1]) + \"\\nDeslocamento em y= \" + str(ss.get_node_displacements()[0][2]), \"\\nphi_y = \" + str(ss.get_node_displacements()[0][3]) + \"\\n\")\nprint(\"N\u00f3 B: \\nDeslocamento em x= \" + str(ss.get_node_displacements()[1][1]) + \"\\nDeslocamento em y= \" + str(ss.get_node_displacements()[1][2]), \"\\nphi_y = \" + str(ss.get_node_displacements()[1][3]) + \"\\n\")\nprint(\"N\u00f3 C: \\nDeslocamento em x= \" + str(ss.get_node_displacements()[2][1]) + \"\\nDeslocamento em y= \" + str(ss.get_node_displacements()[2][2]), \"\\nphi_y = \" + str(ss.get_node_displacements()[2][3]) + \"\\n\")\nprint(\"N\u00f3 D: \\nDeslocamento em x= \" + str(ss.get_node_displacements()[3][1]) + \"\\nDeslocamento em y= \" + str(ss.get_node_displacements()[3][2]), \"\\nphi_y = \" + str(ss.get_node_displacements()[3][3]) + \"\\n\")\n```\n\n\n```\n\n```\n", "meta": {"hexsha": "de0e04ce8bcdd60f5bbb7d2e91f961cdb753ceac", "size": 165752, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "[MAC023]_Trabalho_01.ipynb", "max_stars_repo_name": "MathewsJosh/mecanica-estruturas-ufjf", "max_stars_repo_head_hexsha": "7f94b9a7cdacdb5a22b8fc491959f309be65acfb", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "[MAC023]_Trabalho_01.ipynb", "max_issues_repo_name": "MathewsJosh/mecanica-estruturas-ufjf", "max_issues_repo_head_hexsha": "7f94b9a7cdacdb5a22b8fc491959f309be65acfb", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "[MAC023]_Trabalho_01.ipynb", "max_forks_repo_name": "MathewsJosh/mecanica-estruturas-ufjf", "max_forks_repo_head_hexsha": "7f94b9a7cdacdb5a22b8fc491959f309be65acfb", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 121.7869213813, "max_line_length": 56810, "alphanum_fraction": 0.7091980791, "converted": true, "num_tokens": 7967, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.7826624789529376, "lm_q1q2_score": 0.42791143737042586}} {"text": "\n\n\n# Javascript Interface Code for Head-On Black Hole Collision with Psi4\n\n## Author: Karinne Summers\n### Based on the Start-to-Finish Example: Head-On Black Hole Collision by Zach Etienne with formatting improvements courtesy Brandon Clark\n\n## This module implements a basic numerical relativity code to merge two black holes, as well as the gravitational wave analysis provided by the $\\psi_4$ NRPy+ tutorial notebooks ([$\\psi_4$](Tutorial-Psi4.ipynb) & [$\\psi_4$ tetrad](Tutorial-Psi4_tetrads.ipynb)).\n\n### Here we place the black holes initially on the $z$-axis, so the entire simulation is axisymmetric about the $\\phi$-axis. Minimal sampling in the $\\phi$ direction greatly speeds up the simulation.\n\n## Introduction:\nHere we use NRPy+ to generate the C source code necessary to set up initial data for two black holes (Brill-Lindquist, [Brill & Lindquist, Phys. Rev. 131, 471, 1963](https://journals.aps.org/pr/abstract/10.1103/PhysRev.131.471); see also Eq. 1 of [Brandt & Br\u00fcgmann, arXiv:gr-qc/9711015v1](https://arxiv.org/pdf/gr-qc/9711015v1.pdf)). Then we use it to generate the RHS expressions for [Method of Lines](https://reference.wolfram.com/language/tutorial/NDSolveMethodOfLines.html) time integration based on the [explicit Runge-Kutta fourth-order scheme](https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods) (RK4).\n\nThe entire algorithm is outlined as follows, with links to the relevant NRPy+ tutorial notebooks listed at each step:\n\n1. Allocate memory for gridfunctions, including temporary storage for the Method of Lines time integration\n * [**NRPy+ tutorial on Method of Lines algorithm**](Tutorial-Method_of_Lines-C_Code_Generation.ipynb).\n1. Set gridfunction values to initial data \n * [**NRPy+ tutorial on Brill-Lindquist initial data**](Tutorial-ADM_Initial_Data-Brill-Lindquist.ipynb)\n * [**NRPy+ tutorial on validating Brill-Lindquist initial data**](Tutorial-Start_to_Finish-BSSNCurvilinear-Setting_up_Exact_Initial_Data.ipynb).\n1. Next, integrate the initial data forward in time using the Method of Lines coupled to a Runge-Kutta explicit timestepping algorithm:\n 1. At the start of each iteration in time, output the Hamiltonian constraint violation \n * [**NRPy+ tutorial on BSSN constraints**](Tutorial-BSSN_constraints.ipynb).\n 1. At each RK time substep, do the following:\n 1. Evaluate BSSN RHS expressions \n * [**NRPy+ tutorial on BSSN right-hand sides**](Tutorial-BSSN_time_evolution-BSSN_RHSs.ipynb)\n * [**NRPy+ tutorial on BSSN gauge condition right-hand sides**](Tutorial-BSSN_time_evolution-BSSN_gauge_RHSs.ipynb) \n 1. Apply singular, curvilinear coordinate boundary conditions [*a la* the SENR/NRPy+ paper](https://arxiv.org/abs/1712.07658)\n * [**NRPy+ tutorial on setting up singular, curvilinear boundary conditions**](Tutorial-Start_to_Finish-Curvilinear_BCs.ipynb)\n 1. Enforce constraint on conformal 3-metric: $\\det{\\bar{\\gamma}_{ij}}=\\det{\\hat{\\gamma}_{ij}}$ \n * [**NRPy+ tutorial on enforcing $\\det{\\bar{\\gamma}_{ij}}=\\det{\\hat{\\gamma}_{ij}}$ constraint**](Tutorial-BSSN-Enforcing_Determinant_gammabar_equals_gammahat_Constraint.ipynb)\n\n\n\n# Table of Contents\n$$\\label{toc}$$\n\nThis notebook is organized as follows\n\n1. [Step 1](#initializenrpy): Set core NRPy+ parameters for numerical grids and reference metric\n1. [Step 2](#adm_id): Import Brill-Lindquist ADM initial data C function from the [`BSSN.BrillLindquist`](../edit/BSSN/BrillLindquist.py) NRPy+ module\n1. [Step 3](#bssn): Output C code for BSSN spacetime solve\n 1. [Step 3.a](#bssnrhs): Output C code for BSSN RHS expressions\n 1. [Step 3.b](#hamconstraint): Output C code for Hamiltonian constraint\n 1. [Step 3.c](#enforce3metric): Enforce conformal 3-metric $\\det{\\bar{\\gamma}_{ij}}=\\det{\\hat{\\gamma}_{ij}}$ constraint\n 1. [Step 3.d](#psi4): Compute $\\psi_4$, which encodes gravitational wave information in our numerical relativity calculations\n 1. [Step 3.e](#decomposepsi4): Decompose $\\psi_4$ into spin-weight -2 spherical harmonics\n 1. [Step 3.e.i](#spinweight): Output ${}^{-2}Y_{\\ell,m}$, up to and including $\\ell=\\ell_{\\rm max}$=`l_max` (set to 2 here)\n 1. [Step 3.e.ii](#full_diag): Decomposition of $\\psi_4$ into spin-weight -2 spherical harmonics: Full C-code diagnostic implementation \n 1. [Step 3.f](#coutput): Output all NRPy+ C-code kernels, in parallel if possible \n 1. [Step 3.g](#cparams_rfm_and_domainsize): Output C codes needed for declaring and setting Cparameters; also set `free_parameters.h`\n1. [Step 4](#bc_functs): Set up boundary condition functions for chosen singular, curvilinear coordinate system\n1. [Step 5](#mainc): `BrillLindquist_Playground.c`: The Main C Code\n1. [Step 6](#compileexec): Compile generated C codes & perform the black hole collision calculation\n\n\n```python\n# nrpytutorial should be in a subdirectory, so, do the following.\nimport os,sys\nnrpy_dir_path = os.path.join(\"nrpytutorial\")\nif nrpy_dir_path not in sys.path:\n sys.path.append(nrpy_dir_path)\n```\n\n\n\n# Step 1: Set core NRPy+ parameters for numerical grids and reference metric \\[Back to [top](#toc)\\]\n$$\\label{initializenrpy}$$\n\n\n```python\n# Step P1: Import needed NRPy+ core modules:\nfrom outputC import lhrh,outCfunction,outC_function_dict # NRPy+: Core C code output module\nimport finite_difference as fin # NRPy+: Finite difference C code generation module\nimport NRPy_param_funcs as par # NRPy+: Parameter interface\nimport grid as gri # NRPy+: Functions having to do with numerical grids\nimport indexedexp as ixp # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support\nimport reference_metric as rfm # NRPy+: Reference metric support\nimport cmdline_helper as cmd # NRPy+: Multi-platform Python command-line interface\nimport shutil, os, sys, time # Standard Python modules for multiplatform OS-level functions, benchmarking\n\n# Step P2: Create C code output directory:\nCcodesdir = os.path.join(\"EMCC_BSSN_Two_BHs_Collide_Psi4/\")\n# First remove C code output directory if it exists\n# Courtesy https://stackoverflow.com/questions/303200/how-do-i-remove-delete-a-folder-that-is-not-empty\nshutil.rmtree(Ccodesdir, ignore_errors=True)\n# Then create a fresh directory\ncmd.mkdir(Ccodesdir)\n\n# Step P3: Create executable output directory:\noutdir = os.path.join(Ccodesdir,\"output/\")\ncmd.mkdir(outdir)\n\n# Step 1: Set the spatial dimension parameter\n# to three (BSSN is a 3+1 decomposition\n# of Einstein's equations), and then read\n# the parameter as DIM.\nDIM = 3\npar.set_parval_from_str(\"grid::DIM\",DIM)\n\n# Step 1.a: Enable SIMD-optimized code?\n# I.e., generate BSSN and Ricci C code kernels using SIMD-vectorized\n# compiler intrinsics, which *greatly improve the code's performance*,\n# though at the expense of making the C-code kernels less\n# human-readable.\n# * Important note in case you wish to modify the BSSN/Ricci kernels\n# here by adding expressions containing transcendental functions\n# (e.g., certain scalar fields):\n# Note that SIMD-based transcendental function intrinsics are not\n# supported by the default installation of gcc or clang (you will\n# need to use e.g., the SLEEF library from sleef.org, for this\n# purpose). The Intel compiler suite does support these intrinsics\n# however without the need for external libraries.\nenable_SIMD = True\n\n# Step 2: Set some core parameters, including CoordSystem MoL timestepping algorithm,\n# FD order, floating point precision, and CFL factor:\n# Choices are: Spherical, SinhSpherical, SinhSphericalv2, Cylindrical, SinhCylindrical,\n# SymTP, SinhSymTP\nCoordSystem = \"SinhSpherical\"\n\n# Decompose psi_4 (second time derivative of gravitational\n# wave strain) into all spin-weight=-2\n# l,m spherical harmonics, starting at l=2\n# going up to and including l_max, set here:\nl_max = 2\n\n# domain_size sets the default value for:\n# * Spherical's params.RMAX\n# * SinhSpherical*'s params.AMAX\n# * Cartesians*'s -params.{x,y,z}min & .{x,y,z}max\n# * Cylindrical's -params.ZMIN & .{Z,RHO}MAX\n# * SinhCylindrical's params.AMPL{RHO,Z}\n# * *SymTP's params.AMAX\ndomain_size = 10 # Length scale of computational domain\nFD_order = 8 # Finite difference order: even numbers only, starting with 2. 12 is generally unstable\n\n# sinh_width sets the default value for:\n# * SinhSpherical's params.SINHW\n# * SinhCylindrical's params.SINHW{RHO,Z}\n# * SinhSymTP's params.SINHWAA\nsinh_width = 0.2 # If Sinh* coordinates chosen\n\n# sinhv2_const_dr sets the default value for:\n# * SinhSphericalv2's params.const_dr\n# * SinhCylindricalv2's params.const_d{rho,z}\nsinhv2_const_dr = 0.05 # If Sinh*v2 coordinates chosen\n\n# SymTP_bScale sets the default value for:\n# * SinhSymTP's params.bScale\nSymTP_bScale = 0.5 # If SymTP chosen\n\n# Step 2.b: Set the timestepping order,\n# the core data type, and the CFL factor.\n# Step 2.b: Set the order of spatial and temporal derivatives;\n# the core data type, and the CFL factor.\n# RK_method choices include: Euler, \"RK2 Heun\", \"RK2 MP\", \"RK2 Ralston\", RK3, \"RK3 Heun\", \"RK3 Ralston\",\n# SSPRK3, RK4, DP5, DP5alt, CK5, DP6, L6, DP8\nRK_method = \"RK4\"\nREAL = \"double\" # Best to use double here.\nCFL_FACTOR = 0.1 # (GETS OVERWRITTEN WHEN EXECUTED.) In pure axisymmetry (symmetry_axes = 2 below) 1.0 works fine. Otherwise 0.5 or lower.\n\n# Step 3: Generate Runge-Kutta-based (RK-based) timestepping code.\n# As described above the Table of Contents, this is a 3-step process:\n# 3.A: Evaluate RHSs (RHS_string)\n# 3.B: Apply boundary conditions (post_RHS_string, pt 1)\n# 3.C: Enforce det(gammahat) = det(gammahat) constraint (post_RHS_string, pt 2)\nimport MoLtimestepping.C_Code_Generation as MoL\nfrom MoLtimestepping.RK_Butcher_Table_Dictionary import Butcher_dict\nRK_order = Butcher_dict[RK_method][1]\ncmd.mkdir(os.path.join(Ccodesdir,\"MoLtimestepping/\"))\nMoL.MoL_C_Code_Generation(RK_method,\n RHS_string = \"\"\"\nRicci_eval(&rfmstruct, ¶ms, RK_INPUT_GFS, auxevol_gfs);\nrhs_eval(&rfmstruct, ¶ms, auxevol_gfs, RK_INPUT_GFS, RK_OUTPUT_GFS);\"\"\",\n post_RHS_string = \"\"\"\napply_bcs_curvilinear(¶ms, &bcstruct, NUM_EVOL_GFS, evol_gf_parity, RK_OUTPUT_GFS);\nenforce_detgammahat_constraint(&rfmstruct, ¶ms, RK_OUTPUT_GFS);\\n\"\"\",\n outdir = os.path.join(Ccodesdir,\"MoLtimestepping/\"))\n\n# Step 4: Set the coordinate system for the numerical grid\npar.set_parval_from_str(\"reference_metric::CoordSystem\",CoordSystem)\nrfm.reference_metric() # Create ReU, ReDD needed for rescaling B-L initial data, generating BSSN RHSs, etc.\n\n# Step 5: Set the finite differencing order to FD_order (set above).\npar.set_parval_from_str(\"finite_difference::FD_CENTDERIVS_ORDER\", FD_order)\n\n# Step 6: Copy SIMD/SIMD_intrinsics.h to $Ccodesdir/SIMD/SIMD_intrinsics.h\ncmd.mkdir(os.path.join(Ccodesdir,\"SIMD\"))\nshutil.copy(os.path.join(\"nrpytutorial/SIMD/\")+\"SIMD_intrinsics.h\",os.path.join(Ccodesdir,\"SIMD/\"))\n\n# Step 7: Set the direction=2 (phi) axis to be the symmetry axis; i.e.,\n# axis \"2\", corresponding to the i2 direction.\n# This sets all spatial derivatives in the phi direction to zero.\npar.set_parval_from_str(\"indexedexp::symmetry_axes\",\"2\")\n```\n\n\n\n## Step 1.c: Output needed C code for finding the minimum proper distance between grid points, needed for [CFL](https://en.wikipedia.org/w/index.php?title=Courant%E2%80%93Friedrichs%E2%80%93Lewy_condition&oldid=806430673)-limited timestep \\[Back to [top](#toc)\\]\n$$\\label{cfl}$$\n\nIn order for our explicit-timestepping numerical solution to the scalar wave equation to be stable, it must satisfy the [CFL](https://en.wikipedia.org/w/index.php?title=Courant%E2%80%93Friedrichs%E2%80%93Lewy_condition&oldid=806430673) condition:\n$$\n\\Delta t \\le \\frac{\\min(ds_i)}{c},\n$$\nwhere $c$ is the wavespeed, and\n$$ds_i = h_i \\Delta x^i$$ \nis the proper distance between neighboring gridpoints in the $i$th direction (in 3D, there are 3 directions), $h_i$ is the $i$th reference metric scale factor, and $\\Delta x^i$ is the uniform grid spacing in the $i$th direction:\n\n\n```python\n# Output the find_timestep() function to a C file.\nrfm.out_timestep_func_to_file(os.path.join(Ccodesdir,\"find_timestep.h\"))\n```\n\n\n\n# Step 2: Import Brill-Lindquist ADM initial data C function from the [`BSSN.BrillLindquist`](../edit/BSSN/BrillLindquist.py) NRPy+ module \\[Back to [top](#toc)\\]\n$$\\label{adm_id}$$\n\nThe [`BSSN.BrillLindquist`](../edit/BSSN/BrillLindquist.py) NRPy+ module does the following:\n\n1. Set up Brill-Lindquist initial data [ADM](https://en.wikipedia.org/wiki/ADM_formalism) quantities in the **Cartesian basis**, as [documented here](Tutorial-ADM_Initial_Data-Brill-Lindquist.ipynb). \n1. Convert the ADM **Cartesian quantities** to **BSSN quantities in the desired Curvilinear basis** (set by reference_metric::CoordSystem), as [documented here](Tutorial-ADM_Initial_Data-Converting_ADMCartesian_to_BSSNCurvilinear.ipynb).\n1. Sets up the standardized C function for setting all BSSN Curvilinear gridfunctions in a pointwise fashion, as [written here](../edit/BSSN/BSSN_ID_function_string.py), and returns the C function as a Python string.\n\n\n```python\nimport BSSN.BrillLindquist as bl\ndef BrillLindquistID():\n print(\"Generating optimized C code for Brill-Lindquist initial data. May take a while, depending on CoordSystem.\")\n start = time.time()\n\n bl.BrillLindquist() # Registers ID C function in dictionary, used below to output to file.\n with open(os.path.join(Ccodesdir,\"initial_data.h\"),\"w\") as file:\n file.write(outC_function_dict[\"initial_data\"])\n end = time.time()\n print(\"(BENCH) Finished BL initial data codegen in \"+str(end-start)+\" seconds.\")\n```\n\n\n\n# Step 3: Output C code for BSSN spacetime solve \\[Back to [top](#toc)\\]\n$$\\label{bssn}$$\n\n\n\n## Step 3.a: Output C code for BSSN RHS expressions \\[Back to [top](#toc)\\]\n$$\\label{bssnrhs}$$\n\n\n```python\nimport BSSN.BSSN_RHSs as rhs\nimport BSSN.BSSN_gauge_RHSs as gaugerhs\n# Set the *covariant*, second-order Gamma-driving shift condition\npar.set_parval_from_str(\"BSSN.BSSN_gauge_RHSs::ShiftEvolutionOption\", \"GammaDriving2ndOrder_Covariant\")\n\nprint(\"Generating symbolic expressions for BSSN RHSs...\")\nstart = time.time()\n# Enable rfm_precompute infrastructure, which results in\n# BSSN RHSs that are free of transcendental functions,\n# even in curvilinear coordinates, so long as\n# ConformalFactor is set to \"W\" (default).\ncmd.mkdir(os.path.join(Ccodesdir,\"rfm_files/\"))\npar.set_parval_from_str(\"reference_metric::enable_rfm_precompute\",\"True\")\npar.set_parval_from_str(\"reference_metric::rfm_precompute_Ccode_outdir\",os.path.join(Ccodesdir,\"rfm_files/\"))\n\n# Evaluate BSSN + BSSN gauge RHSs with rfm_precompute enabled:\nimport BSSN.BSSN_quantities as Bq\npar.set_parval_from_str(\"BSSN.BSSN_quantities::LeaveRicciSymbolic\",\"True\")\n\nrhs.BSSN_RHSs()\ngaugerhs.BSSN_gauge_RHSs()\n\n# We use betaU as our upwinding control vector:\nBq.BSSN_basic_tensors()\nbetaU = Bq.betaU\n\nimport BSSN.Enforce_Detgammahat_Constraint as EGC\nenforce_detg_constraint_symb_expressions = EGC.Enforce_Detgammahat_Constraint_symb_expressions()\n\n# Next compute Ricci tensor\npar.set_parval_from_str(\"BSSN.BSSN_quantities::LeaveRicciSymbolic\",\"False\")\nBq.RicciBar__gammabarDD_dHatD__DGammaUDD__DGammaU()\n\n# Now register the Hamiltonian as a gridfunction.\nH = gri.register_gridfunctions(\"AUX\",\"H\")\n# Then define the Hamiltonian constraint and output the optimized C code.\nimport BSSN.BSSN_constraints as bssncon\nbssncon.BSSN_constraints(add_T4UUmunu_source_terms=False)\n\n# Now that we are finished with all the rfm hatted\n# quantities in generic precomputed functional\n# form, let's restore them to their closed-\n# form expressions.\npar.set_parval_from_str(\"reference_metric::enable_rfm_precompute\",\"False\") # Reset to False to disable rfm_precompute.\nrfm.ref_metric__hatted_quantities()\nend = time.time()\nprint(\"(BENCH) Finished BSSN symbolic expressions in \"+str(end-start)+\" seconds.\")\n\ndef BSSN_RHSs():\n print(\"Generating C code for BSSN RHSs in \"+par.parval_from_str(\"reference_metric::CoordSystem\")+\" coordinates.\")\n start = time.time()\n\n # Construct the left-hand sides and right-hand-side expressions for all BSSN RHSs\n lhs_names = [ \"alpha\", \"cf\", \"trK\"]\n rhs_exprs = [gaugerhs.alpha_rhs, rhs.cf_rhs, rhs.trK_rhs]\n for i in range(3):\n lhs_names.append( \"betU\"+str(i))\n rhs_exprs.append(gaugerhs.bet_rhsU[i])\n lhs_names.append( \"lambdaU\"+str(i))\n rhs_exprs.append(rhs.lambda_rhsU[i])\n lhs_names.append( \"vetU\"+str(i))\n rhs_exprs.append(gaugerhs.vet_rhsU[i])\n for j in range(i,3):\n lhs_names.append( \"aDD\"+str(i)+str(j))\n rhs_exprs.append(rhs.a_rhsDD[i][j])\n lhs_names.append( \"hDD\"+str(i)+str(j))\n rhs_exprs.append(rhs.h_rhsDD[i][j])\n\n # Sort the lhss list alphabetically, and rhss to match.\n # This ensures the RHSs are evaluated in the same order\n # they're allocated in memory:\n lhs_names,rhs_exprs = [list(x) for x in zip(*sorted(zip(lhs_names,rhs_exprs), key=lambda pair: pair[0]))]\n\n # Declare the list of lhrh's\n BSSN_evol_rhss = []\n for var in range(len(lhs_names)):\n BSSN_evol_rhss.append(lhrh(lhs=gri.gfaccess(\"rhs_gfs\",lhs_names[var]),rhs=rhs_exprs[var]))\n\n # Set up the C function for the BSSN RHSs\n desc=\"Evaluate the BSSN RHSs\"\n name=\"rhs_eval\"\n outCfunction(\n outfile = os.path.join(Ccodesdir,name+\".h\"), desc=desc, name=name,\n params = \"\"\"rfm_struct *restrict rfmstruct,const paramstruct *restrict params,\n const REAL *restrict auxevol_gfs,const REAL *restrict in_gfs,REAL *restrict rhs_gfs\"\"\",\n body = fin.FD_outputC(\"returnstring\",BSSN_evol_rhss, params=\"outCverbose=False,enable_SIMD=True\",\n upwindcontrolvec=betaU),\n loopopts = \"InteriorPoints,enable_SIMD,enable_rfm_precompute\")\n end = time.time()\n print(\"(BENCH) Finished BSSN_RHS C codegen in \" + str(end - start) + \" seconds.\")\n\ndef Ricci():\n print(\"Generating C code for Ricci tensor in \"+par.parval_from_str(\"reference_metric::CoordSystem\")+\" coordinates.\")\n start = time.time()\n desc=\"Evaluate the Ricci tensor\"\n name=\"Ricci_eval\"\n outCfunction(\n outfile = os.path.join(Ccodesdir,name+\".h\"), desc=desc, name=name,\n params = \"\"\"rfm_struct *restrict rfmstruct,const paramstruct *restrict params,\n const REAL *restrict in_gfs,REAL *restrict auxevol_gfs\"\"\",\n body = fin.FD_outputC(\"returnstring\",\n [lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"RbarDD00\"),rhs=Bq.RbarDD[0][0]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"RbarDD01\"),rhs=Bq.RbarDD[0][1]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"RbarDD02\"),rhs=Bq.RbarDD[0][2]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"RbarDD11\"),rhs=Bq.RbarDD[1][1]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"RbarDD12\"),rhs=Bq.RbarDD[1][2]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"RbarDD22\"),rhs=Bq.RbarDD[2][2])],\n params=\"outCverbose=False,enable_SIMD=True\"),\n loopopts = \"InteriorPoints,enable_SIMD,enable_rfm_precompute\")\n end = time.time()\n print(\"(BENCH) Finished Ricci C codegen in \" + str(end - start) + \" seconds.\")\n```\n\n Generating symbolic expressions for BSSN RHSs...\n (BENCH) Finished BSSN symbolic expressions in 10.577486991882324 seconds.\n\n\n\n\n## Step 3.b: Output C code for Hamiltonian constraint \\[Back to [top](#toc)\\]\n$$\\label{hamconstraint}$$\n\nNext output the C code for evaluating the Hamiltonian constraint [(**Tutorial**)](Tutorial-BSSN_constraints.ipynb). In the absence of numerical error, this constraint should evaluate to zero. However it does not due to numerical (typically truncation and roundoff) error. We will therefore measure the Hamiltonian constraint violation to gauge the accuracy of our simulation, and, ultimately determine whether errors are dominated by numerical finite differencing (truncation) error as expected.\n\n\n```python\ndef Hamiltonian():\n start = time.time()\n print(\"Generating optimized C code for Hamiltonian constraint. May take a while, depending on CoordSystem.\")\n # Set up the C function for the Hamiltonian RHS\n desc=\"Evaluate the Hamiltonian constraint\"\n name=\"Hamiltonian_constraint\"\n outCfunction(\n outfile = os.path.join(Ccodesdir,name+\".h\"), desc=desc, name=name,\n params = \"\"\"rfm_struct *restrict rfmstruct,const paramstruct *restrict params,\n REAL *restrict in_gfs, REAL *restrict aux_gfs\"\"\",\n body = fin.FD_outputC(\"returnstring\",lhrh(lhs=gri.gfaccess(\"aux_gfs\", \"H\"), rhs=bssncon.H),\n params=\"outCverbose=False\"),\n loopopts = \"InteriorPoints,enable_rfm_precompute\")\n\n end = time.time()\n print(\"(BENCH) Finished Hamiltonian C codegen in \" + str(end - start) + \" seconds.\")\n```\n\n\n\n## Step 3.c: Enforce conformal 3-metric $\\det{\\bar{\\gamma}_{ij}}=\\det{\\hat{\\gamma}_{ij}}$ constraint \\[Back to [top](#toc)\\]\n$$\\label{enforce3metric}$$\n\nThen enforce conformal 3-metric $\\det{\\bar{\\gamma}_{ij}}=\\det{\\hat{\\gamma}_{ij}}$ constraint (Eq. 53 of [Ruchlin, Etienne, and Baumgarte (2018)](https://arxiv.org/abs/1712.07658)), as [documented in the corresponding NRPy+ tutorial notebook](Tutorial-BSSN-Enforcing_Determinant_gammabar_equals_gammahat_Constraint.ipynb)\n\nApplying curvilinear boundary conditions should affect the initial data at the outer boundary, and will in general cause the $\\det{\\bar{\\gamma}_{ij}}=\\det{\\hat{\\gamma}_{ij}}$ constraint to be violated there. Thus after we apply these boundary conditions, we must always call the routine for enforcing the $\\det{\\bar{\\gamma}_{ij}}=\\det{\\hat{\\gamma}_{ij}}$ constraint:\n\n\n```python\ndef gammadet():\n start = time.time()\n print(\"Generating optimized C code for gamma constraint. May take a while, depending on CoordSystem.\")\n\n # Set up the C function for the det(gammahat) = det(gammahat)\n EGC.output_Enforce_Detgammahat_Constraint_Ccode(Ccodesdir,exprs=enforce_detg_constraint_symb_expressions)\n end = time.time()\n print(\"(BENCH) Finished gamma constraint C codegen in \" + str(end - start) + \" seconds.\")\n```\n\n\n\n## Step 3.d: Compute $\\psi_4$, which encodes gravitational wave information in our numerical relativity calculations \\[Back to [top](#toc)\\]\n$$\\label{psi4}$$\n\nThe [Weyl scalar](https://en.wikipedia.org/wiki/Weyl_scalar) $\\psi_4$ encodes gravitational wave information in our numerical relativity calculations. For more details on how it is computed, see [this NRPy+ tutorial notebook for information on $\\psi_4$](Tutorial-Psi4.ipynb) and [this one on the Quasi-Kinnersley tetrad](Tutorial-Psi4_tetrads.ipynb) (as implemented in [Baker, Campanelli, Lousto (2001)](https://arxiv.org/pdf/gr-qc/0104063.pdf)).\n\n$\\psi_4$ is related to the gravitational wave strain via\n$$\n\\psi_4 = \\ddot{h}_+ - i \\ddot{h}_\\times,\n$$\nwhere $\\ddot{h}_+$ is the second time derivative of the $+$ polarization of the gravitational wave strain $h$, and $\\ddot{h}_\\times$ is the second time derivative of the $\\times$ polarization of the gravitational wave strain $h$.\n\n\n```python\nimport BSSN.Psi4_tetrads as BP4t\npar.set_parval_from_str(\"BSSN.Psi4_tetrads::TetradChoice\",\"QuasiKinnersley\")\n#par.set_parval_from_str(\"BSSN.Psi4_tetrads::UseCorrectUnitNormal\",\"True\")\nimport BSSN.Psi4 as BP4\nprint(\"Generating symbolic expressions for psi4...\")\nstart = time.time()\nBP4.Psi4()\nend = time.time()\nprint(\"(BENCH) Finished psi4 symbolic expressions in \"+str(end-start)+\" seconds.\")\n\npsi4r_0pt = gri.register_gridfunctions(\"AUX\",\"psi4r_0pt\")\npsi4r_1pt = gri.register_gridfunctions(\"AUX\",\"psi4r_1pt\")\npsi4r_2pt = gri.register_gridfunctions(\"AUX\",\"psi4r_2pt\")\npsi4i_0pt = gri.register_gridfunctions(\"AUX\",\"psi4i_0pt\")\npsi4i_1pt = gri.register_gridfunctions(\"AUX\",\"psi4i_1pt\")\npsi4i_2pt = gri.register_gridfunctions(\"AUX\",\"psi4i_2pt\")\n\n\ndesc=\"\"\"Since it's so expensive to compute, instead of evaluating\npsi_4 at all interior points, this functions evaluates it on a\npoint-by-point basis.\"\"\"\nname=\"psi4\"\noutCfunction(\n outfile = os.path.join(Ccodesdir,name+\".h\"), desc=desc, name=name,\n params = \"\"\"const paramstruct *restrict params,\n const int i0,const int i1,const int i2,\n REAL *restrict xx[3], const REAL *restrict in_gfs, REAL *restrict aux_gfs\"\"\",\n body = \"\"\"\n const int idx = IDX3S(i0,i1,i2);\n const REAL xx0 = xx[0][i0];const REAL xx1 = xx[1][i1];const REAL xx2 = xx[2][i2];\n// Real part of psi_4, divided into 3 terms\n {\n#include \"Psi4re_pt0_lowlevel.h\"\n }\n {\n#include \"Psi4re_pt1_lowlevel.h\"\n }\n {\n#include \"Psi4re_pt2_lowlevel.h\"\n }\n// Imaginary part of psi_4, divided into 3 terms\n {\n#include \"Psi4im_pt0_lowlevel.h\"\n }\n {\n#include \"Psi4im_pt1_lowlevel.h\"\n }\n {\n#include \"Psi4im_pt2_lowlevel.h\"\n }\"\"\")\n\ndef Psi4re(part):\n print(\"Generating C code for psi4_re_pt\"+str(part)+\" in \"+par.parval_from_str(\"reference_metric::CoordSystem\")+\" coordinates.\")\n start = time.time()\n Psi4re_pt = fin.FD_outputC(\"returnstring\",\n [lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"psi4r_\"+str(part)+\"pt\"),rhs=BP4.psi4_re_pt[part])],\n params=\"outCverbose=False,CSE_sorting=none\") # Generating the CSE for psi4 is the slowest\n # operation in this notebook, and much of the CSE\n # time is spent sorting CSE expressions. Disabling\n # this sorting makes the C codegen 3-4x faster,\n # but the tradeoff is that every time this is\n # run, the CSE patterns will be different\n # (though they should result in mathematically\n # *identical* expressions). You can expect\n # roundoff-level differences as a result.\n with open(os.path.join(Ccodesdir,\"Psi4re_pt\"+str(part)+\"_lowlevel.h\"), \"w\") as file:\n file.write(Psi4re_pt)\n end = time.time()\n print(\"(BENCH) Finished generating psi4_re_pt\"+str(part)+\" in \"+str(end-start)+\" seconds.\")\n\ndef Psi4im(part):\n print(\"Generating C code for psi4_im_pt\"+str(part)+\" in \"+par.parval_from_str(\"reference_metric::CoordSystem\")+\" coordinates.\")\n start = time.time()\n Psi4im_pt = fin.FD_outputC(\"returnstring\",\n [lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"psi4i_\"+str(part)+\"pt\"),rhs=BP4.psi4_im_pt[part])],\n params=\"outCverbose=False,CSE_sorting=none\") # Generating the CSE for psi4 is the slowest\n # operation in this notebook, and much of the CSE\n # time is spent sorting CSE expressions. Disabling\n # this sorting makes the C codegen 3-4x faster,\n # but the tradeoff is that every time this is\n # run, the CSE patterns will be different\n # (though they should result in mathematically\n # *identical* expressions). You can expect\n # roundoff-level differences as a result.\n with open(os.path.join(Ccodesdir,\"Psi4im_pt\"+str(part)+\"_lowlevel.h\"), \"w\") as file:\n file.write(Psi4im_pt)\n end = time.time()\n print(\"(BENCH) Finished generating psi4_im_pt\"+str(part)+\" in \"+str(end-start)+\" seconds.\")\n```\n\n Generating symbolic expressions for psi4...\n (BENCH) Finished psi4 symbolic expressions in 44.386873722076416 seconds.\n Output C function psi4() to file EMCC_BSSN_Two_BHs_Collide_Psi4/psi4.h\n\n\n\n\n## Step 3.e: Decompose $\\psi_4$ into spin-weight -2 spherical harmonics \\[Back to [top](#toc)\\]\n$$\\label{decomposepsi4}$$ \n\nInstead of measuring $\\psi_4$ for all possible (gravitational wave) observers in our simulation domain, we instead decompose it into a natural basis set, which by convention is the spin-weight -2 spherical harmonics.\n\nHere we implement the algorithm for decomposing $\\psi_4$ into spin-weight -2 spherical harmonic modes. The decomposition is defined as follows:\n\n$$\n{}^{-2}\\left[\\psi_4\\right]_{\\ell,m}(t,R) = \\int \\int \\psi_4(t,R,\\theta,\\phi)\\ \\left[{}^{-2}Y^*_{\\ell,m}(\\theta,\\phi)\\right] \\sin \\theta d\\theta d\\phi,\n$$\n\nwhere\n\n* ${}^{-2}Y^*_{\\ell,m}(\\theta,\\phi)$ is the complex conjugate of the spin-weight $-2$ spherical harmonic $\\ell,m$ mode\n* $R$ is the (fixed) radius at which we extract $\\psi_4$ information\n* $t$ is the time coordinate\n* $\\theta,\\phi$ are the polar and azimuthal angles, respectively (we use [the physics notation for spherical coordinates](https://en.wikipedia.org/wiki/Spherical_coordinate_system) here)\n\n\n\n### Step 3.e.i Output ${}^{-2}Y_{\\ell,m}$, up to and including $\\ell=\\ell_{\\rm max}$=`l_max` (set to 2 here) \\[Back to [top](#toc)\\]\n$$\\label{spinweight}$$ \n\nHere we output all spin-weight $-2$ spherical harmonics [**Tutorial Module**](Tutorial-SpinWeighted_Spherical_Harmonics.ipynb) for $\\ell=0$ up to and including $\\ell=\\ell_{\\rm max}$=`l_max` (set to 2 here).\n\n\n```python\nimport SpinWeight_minus2_SphHarmonics.SpinWeight_minus2_SphHarmonics as swm2\ncmd.mkdir(os.path.join(Ccodesdir,\"SpinWeight_minus2_SphHarmonics\"))\nswm2.SpinWeight_minus2_SphHarmonics(maximum_l=l_max,\n filename=os.path.join(Ccodesdir,\"SpinWeight_minus2_SphHarmonics/SpinWeight_minus2_SphHarmonics.h\"))\n```\n\n\n\n### Step 3.e.ii Decomposition of $\\psi_4$ into spin-weight -2 spherical harmonics: Full C-code diagnostic implementation \\[Back to [top](#toc)\\]\n$$\\label{full_diag}$$ \n\nNote that this diagnostic implementation assumes that `Spherical`-like coordinates are used (e.g., `SinhSpherical` or `Spherical`), which are the most natural coordinate system for decomposing $\\psi_4$ into spin-weight -2 modes.\n\nFirst we process the inputs needed to compute $\\psi_4$ at all needed $\\theta,\\phi$ points \n\n\n```python\n## Code for this moved to step 5\n```\n\nNext we implement the integral:\n\n$$\n{}^{-2}\\left[\\psi_4\\right]_{\\ell,m}(t,R) = \\int \\int \\psi_4(t,R,\\theta,\\phi)\\ \\left[{}^{-2}Y^*_{\\ell,m}(\\theta,\\phi)\\right] \\sin \\theta d\\theta d\\phi.\n$$\n\nSince $\\psi_4(t,R,\\theta,\\phi)$ and $\\left[{}^{-2}Y^*_{\\ell,m}(\\theta,\\phi)\\right]$ are generally complex, for simplicity let's define\n\\begin{align}\n\\psi_4(t,R,\\theta,\\phi)&=a+i b \\\\\n\\left[{}^{-2}Y_{\\ell,m}(\\theta,\\phi)\\right] &= c + id\\\\\n\\implies \\left[{}^{-2}Y^*_{\\ell,m}(\\theta,\\phi)\\right] = \\left[{}^{-2}Y_{\\ell,m}(\\theta,\\phi)\\right]^* &=c-i d\n\\end{align}\n\nThen the product (appearing within the integral) will be given by\n\\begin{align}\n(a + i b) (c-i d) &= (ac + bd) + i(bc - ad),\n\\end{align}\nwhich cleanly splits the real and complex parts. For better modularity, we output this algorithm to a function `decompose_psi4_into_swm2_modes()` in file `decompose_psi4_into_swm2_modes.h`. Here, we will call this function from within `output_psi4_spinweight_m2_decomposition()`, but in general it could be called from codes that do not use spherical coordinates, and the `psi4r_at_R_ext[]` and `psi4i_at_R_ext[]` arrays are filled using interpolations.\n\nFinally, we complete the function `output_psi4_spinweight_m2_decomposition()`, now calling the above routine and freeing all allocated memory.\n\n\n```python\n## Code for this moved to step 5\n```\n\n\n\n## Step 3.f: Output all NRPy+ C-code kernels, in parallel if possible \\[Back to [top](#toc)\\]\n$$\\label{coutput}$$ \n\n\n```python\n# Step 0: Import the multiprocessing module.\nimport multiprocessing\n\n# Step 1: Create a list of functions we wish to evaluate in parallel\nfuncs = [Psi4re,Psi4re,Psi4re,Psi4im,Psi4im,Psi4im, BrillLindquistID,BSSN_RHSs,Ricci,Hamiltonian,gammadet]\n\n# Step 1.a: Define master function for calling all above functions.\n# Note that lambdifying this doesn't work in Python 3\ndef master_func(idx):\n if idx < 3: # Call Psi4re(arg)\n funcs[idx](idx)\n elif idx < 6: # Call Psi4im(arg-3)\n funcs[idx](idx-3)\n else: # All non-Psi4 functions:\n funcs[idx]()\n\ntry:\n if os.name == 'nt':\n # It's a mess to get working in Windows, so we don't bother. :/\n # https://medium.com/@grvsinghal/speed-up-your-python-code-using-multiprocessing-on-windows-and-jupyter-or-ipython-2714b49d6fac\n raise Exception(\"Parallel codegen currently not available in Windows\")\n # Step 1.b: Import the multiprocessing module.\n import multiprocessing\n\n # Step 1.c: Evaluate list of functions in parallel if possible;\n # otherwise fallback to serial evaluation:\n pool = multiprocessing.Pool()\n pool.map(master_func,range(len(funcs)))\nexcept:\n # Steps 1.b-1.c, alternate: As fallback, evaluate functions in serial.\n # This will happen on Android and Windows systems\n for idx in range(len(funcs)):\n master_func(idx)\n```\n\n Generating C code for psi4_re_pt0 in SinhSpherical coordinates.Generating optimized C code for Brill-Lindquist initial data. May take a while, depending on CoordSystem.Generating C code for psi4_re_pt1 in SinhSpherical coordinates.Generating C code for psi4_im_pt0 in SinhSpherical coordinates.Generating C code for psi4_re_pt2 in SinhSpherical coordinates.Generating C code for BSSN RHSs in SinhSpherical coordinates.Generating C code for Ricci tensor in SinhSpherical coordinates.Generating optimized C code for Hamiltonian constraint. May take a while, depending on CoordSystem.Generating optimized C code for gamma constraint. May take a while, depending on CoordSystem.\n \n \n \n \n \n \n \n \n Generating C code for psi4_im_pt2 in SinhSpherical coordinates.\n Generating C code for psi4_im_pt1 in SinhSpherical coordinates.\n Output C function enforce_detgammahat_constraint() to file EMCC_BSSN_Two_BHs_Collide_Psi4/enforce_detgammahat_constraint.h\n (BENCH) Finished gamma constraint C codegen in 0.3458900451660156 seconds.\n (BENCH) Finished generating psi4_im_pt1 in 34.87725281715393 seconds.\n (BENCH) Finished generating psi4_im_pt2 in 48.07617902755737 seconds.\n (BENCH) Finished generating psi4_re_pt2 in 62.37357544898987 seconds.\n (BENCH) Finished generating psi4_re_pt1 in 64.43620777130127 seconds.\n Output C function rhs_eval() to file EMCC_BSSN_Two_BHs_Collide_Psi4/rhs_eval.h\n (BENCH) Finished BSSN_RHS C codegen in 64.70630764961243 seconds.\n Output C function Ricci_eval() to file EMCC_BSSN_Two_BHs_Collide_Psi4/Ricci_eval.h\n (BENCH) Finished Ricci C codegen in 69.96846079826355 seconds.\n (BENCH) Finished generating psi4_im_pt0 in 113.97330212593079 seconds.\n Output C function Hamiltonian_constraint() to file EMCC_BSSN_Two_BHs_Collide_Psi4/Hamiltonian_constraint.h\n (BENCH) Finished Hamiltonian C codegen in 127.71309232711792 seconds.\n (BENCH) Finished generating psi4_re_pt0 in 196.5384418964386 seconds.\n (BENCH) Finished BL initial data codegen in 286.0503706932068 seconds.\n\n\n\n\n## Step 3.g: Output C codes needed for declaring and setting Cparameters; also set `free_parameters.h` \\[Back to [top](#toc)\\]\n$$\\label{cparams_rfm_and_domainsize}$$\n\nBased on declared NRPy+ Cparameters, first we generate `declare_Cparameters_struct.h`, `set_Cparameters_default.h`, and `set_Cparameters[-SIMD].h`.\n\nThen we output `free_parameters.h`, which sets initial data parameters, as well as grid domain & reference metric parameters, applying `domain_size` and `sinh_width`/`SymTP_bScale` (if applicable) as set above\n\n\n```python\n# Step 3.f.i: Generate declare_Cparameters_struct.h, set_Cparameters_default.h, and set_Cparameters[-SIMD].h\npar.generate_Cparameters_Ccodes(os.path.join(Ccodesdir))\n\n# Step 3.f.ii: Set free_parameters.h\nwith open(os.path.join(Ccodesdir,\"free_parameters.h\"),\"w\") as file:\n file.write(\"\"\"\n// Set free-parameter values.\n\n// Set free-parameter values for BSSN evolution:\nparams.eta = 2.0;\n\n// Set free parameters for the (Brill-Lindquist) initial data\nparams.BH1_posn_x = 0.0; params.BH1_posn_y = 0.0; params.BH1_posn_z =+0.25;\nparams.BH2_posn_x = 0.0; params.BH2_posn_y = 0.0; params.BH2_posn_z =-0.25;\nparams.BH1_mass = 0.5; params.BH2_mass = 0.5;\\n\"\"\")\n\n# Append to $Ccodesdir/free_parameters.h reference metric parameters based on generic\n# domain_size,sinh_width,sinhv2_const_dr,SymTP_bScale,\n# parameters set above.\nrfm.out_default_free_parameters_for_rfm(os.path.join(Ccodesdir,\"free_parameters.h\"),\n domain_size,sinh_width,sinhv2_const_dr,SymTP_bScale)\n\n# Step 3.f.iii: Generate set_Nxx_dxx_invdx_params__and__xx.h:\nrfm.set_Nxx_dxx_invdx_params__and__xx_h(Ccodesdir)\n\n# Step 3.f.iv: Generate xx_to_Cart.h, which contains xx_to_Cart() for\n# (the mapping from xx->Cartesian) for the chosen\n# CoordSystem:\nrfm.xx_to_Cart_h(\"xx_to_Cart\",\"./set_Cparameters.h\",os.path.join(Ccodesdir,\"xx_to_Cart.h\"))\n\n# Step 3.f.v: Generate declare_Cparameters_struct.h, set_Cparameters_default.h, and set_Cparameters[-SIMD].h\npar.generate_Cparameters_Ccodes(os.path.join(Ccodesdir))\n```\n\n\n\n# Step 4: Set up boundary condition functions for chosen singular, curvilinear coordinate system \\[Back to [top](#toc)\\]\n$$\\label{bc_functs}$$\n\nNext apply singular, curvilinear coordinate boundary conditions [as documented in the corresponding NRPy+ tutorial notebook](Tutorial-Start_to_Finish-Curvilinear_BCs.ipynb)\n\n\n```python\nimport CurviBoundaryConditions.CurviBoundaryConditions as cbcs\ncbcs.Set_up_CurviBoundaryConditions(os.path.join(Ccodesdir,\"boundary_conditions/\"),\n Cparamspath=os.path.join(\"../\"),path_prefix=\"../nrpytutorial/\")\n```\n\n Wrote to file \"EMCC_BSSN_Two_BHs_Collide_Psi4/boundary_conditions/parity_conditions_symbolic_dot_products.h\"\n Evolved parity: ( aDD00:4, aDD01:5, aDD02:6, aDD11:7, aDD12:8, aDD22:9,\n alpha:0, betU0:1, betU1:2, betU2:3, cf:0, hDD00:4, hDD01:5, hDD02:6,\n hDD11:7, hDD12:8, hDD22:9, lambdaU0:1, lambdaU1:2, lambdaU2:3, trK:0,\n vetU0:1, vetU1:2, vetU2:3 )\n Auxiliary parity: ( H:0, psi4i_0pt:0, psi4i_1pt:0, psi4i_2pt:0,\n psi4r_0pt:0, psi4r_1pt:0, psi4r_2pt:0 )\n AuxEvol parity: ( RbarDD00:4, RbarDD01:5, RbarDD02:6, RbarDD11:7,\n RbarDD12:8, RbarDD22:9 )\n Wrote to file \"EMCC_BSSN_Two_BHs_Collide_Psi4/boundary_conditions/EigenCoord_Cart_to_xx.h\"\n\n\n\n\n# Step 5: `BrillLindquist_Playground.c`: The Main C Code \\[Back to [top](#toc)\\]\n\n$$\\label{mainc}$$\n\n## Psi4 Header Files\n\n*(Code moved here from 3.e.ii)*\n\n\n```python\n%%writefile $Ccodesdir/lowlevel_decompose_psi4_into_swm2_modes.h\n\nvoid lowlevel_decompose_psi4_into_swm2_modes(const paramstruct *restrict params,\n const REAL curr_time, const REAL R_ext,\n const REAL *restrict th_array,const REAL *restrict sinth_array,const REAL *restrict ph_array,\n const REAL *restrict psi4r_at_R_ext,const REAL *restrict psi4i_at_R_ext) {\n #include \"set_Cparameters.h\"\n for(int l=2;l<=L_MAX;l++) { // L_MAX is a global variable, since it must be set in Python (so that SpinWeight_minus2_SphHarmonics() computes enough modes)\n for(int m=-l;m<=l;m++) {\n // Parallelize the integration loop:\n REAL psi4r_l_m = 0.0;\n REAL psi4i_l_m = 0.0;\n #pragma omp parallel for reduction(+:psi4r_l_m,psi4i_l_m)\n for(int i1=0;i1=0) sprintf(filename,\"outpsi4_l%d_m+%d-%d-r%.2f.txt\",l,m, Nxx0,(double)R_ext);\n FILE *outpsi4_l_m;\n // 0 = n*dt when n=0 is exactly represented in double/long double precision,\n // so no worries about the result being ~1e-16 in double/ld precision\n if(curr_time==0) outpsi4_l_m = fopen(filename, \"w\");\n else outpsi4_l_m = fopen(filename, \"a\");\n fprintf(outpsi4_l_m,\"%e %.15e %.15e\\n\", (double)(curr_time),\n (double)psi4r_l_m,(double)psi4i_l_m);\n fclose(outpsi4_l_m);\n }\n }\n}\n```\n\n Writing EMCC_BSSN_Two_BHs_Collide_Psi4//lowlevel_decompose_psi4_into_swm2_modes.h\n\n\n\n```python\n%%writefile $Ccodesdir/driver_psi4_spinweightm2_decomposition.h\n\n// Define global variable to be accessed in main C code\nREAL *restrict diagnostic_output_gfs_p;\n\nvoid driver_psi4_spinweightm2_decomposition(const paramstruct *restrict params,\n const REAL curr_time, const int R_ext_idx,\n REAL *restrict xx[3],\n const REAL *restrict y_n_gfs,\n REAL *restrict diagnostic_output_gfs,\n int i0, int i1, int i2, int lastRun) {\n\n // Step 0: Set global variable to local value\n diagnostic_output_gfs_p = diagnostic_output_gfs;\n\n #include \"set_Cparameters.h\"\n // Step 1: Set the extraction radius R_ext based on the radial index R_ext_idx\n REAL R_ext;\n {\n REAL xx0 = xx[0][R_ext_idx];\n REAL xx1 = xx[1][1];\n REAL xx2 = xx[2][1];\n REAL xCart[3];\n xx_to_Cart(params,xx,R_ext_idx,1,1,xCart);\n R_ext = sqrt(xCart[0]*xCart[0] + xCart[1]*xCart[1] + xCart[2]*xCart[2]);\n }\n\n // Step 2: Compute psi_4 at a specific point on this extraction radius and store to a local 2D array.\n const int sizeof_2Darray = sizeof(REAL)*(Nxx_plus_2NGHOSTS1-2*NGHOSTS)*(Nxx_plus_2NGHOSTS2-2*NGHOSTS);\n REAL *restrict psi4r_at_R_ext = (REAL *)malloc(sizeof_2Darray);\n REAL *restrict psi4i_at_R_ext = (REAL *)malloc(sizeof_2Darray);\n\n // ... also store theta, sin(theta), and phi to corresponding 1D arrays.\n REAL *restrict sinth_array = (REAL *restrict)malloc(sizeof(REAL)*(Nxx_plus_2NGHOSTS1-2*NGHOSTS));\n REAL *restrict th_array = (REAL *restrict)malloc(sizeof(REAL)*(Nxx_plus_2NGHOSTS1-2*NGHOSTS));\n REAL *restrict ph_array = (REAL *restrict)malloc(sizeof(REAL)*(Nxx_plus_2NGHOSTS2-2*NGHOSTS));\n\n#pragma omp parallel for\n th_array[i1-NGHOSTS] = xx[1][i1];\n sinth_array[i1-NGHOSTS] = sin(xx[1][i1]);\n ph_array[i2-NGHOSTS] = xx[2][i2];\n\n psi4(params, i0,i1,i2, xx, y_n_gfs, diagnostic_output_gfs);\n\n if(lastRun==1){\n // Step 3: Once all points at this extraction radius have been compute perform integrations across all\n // l,m modes from l=2 up to and including L_MAX (global variable):\n lowlevel_decompose_psi4_into_swm2_modes(params, curr_time,R_ext, th_array,sinth_array, ph_array,\n psi4r_at_R_ext,psi4i_at_R_ext);\n }\n}\n```\n\n Writing EMCC_BSSN_Two_BHs_Collide_Psi4//driver_psi4_spinweightm2_decomposition.h\n\n\n## Main c files\n\n\n```python\n# Part P0: Define REAL, set the number of ghost cells NGHOSTS (from NRPy+'s FD_CENTDERIVS_ORDER),\n# and set the CFL_FACTOR (which can be overwritten at the command line)\n\nwith open(os.path.join(Ccodesdir,\"BSSN_Playground_REAL__NGHOSTS__CFL_FACTOR.h\"), \"w\") as file:\n file.write(\"\"\"\n// Part P0.a: Set the number of ghost cells, from NRPy+'s FD_CENTDERIVS_ORDER\n#define NGHOSTS \"\"\"+str(int(FD_order/2)+1)+\"\"\"\n// Part P0.b: Set the numerical precision (REAL) to double, ensuring all floating point\n// numbers are stored to at least ~16 significant digits\n#define REAL \"\"\"+REAL+\"\"\"\n// Part P0.c: Set the number of ghost cells, from NRPy+'s FD_CENTDERIVS_ORDER\nREAL CFL_FACTOR = \"\"\"+str(CFL_FACTOR)+\"\"\"; // Set the CFL Factor. Can be overwritten at command line.\n// Part P0.d: We decompose psi_4 into all spin-weight=-2\n// l,m spherical harmonics, starting at l=2,\n// going up to and including l_max, set here:\n#define L_MAX \"\"\"+str(l_max)+\"\"\"\n\"\"\")\n```\n\n\n```python\n%%writefile $Ccodesdir/BrillLindquist_Playground.c\n\n// Step P0: Define REAL and NGHOSTS; and declare CFL_FACTOR. This header is generated in NRPy+.\n#include \"BSSN_Playground_REAL__NGHOSTS__CFL_FACTOR.h\"\n\n#include \"rfm_files/rfm_struct__declare.h\"\n\n#include \"declare_Cparameters_struct.h\"\n\n#include \"emscripten.h\"\n\n// All SIMD intrinsics used in SIMD-enabled C code loops are defined here:\n#include \"SIMD/SIMD_intrinsics.h\"\n\n// Step P1: Import needed header files\n#include \"stdio.h\"\n#include \"stdlib.h\"\n#include \"math.h\"\n#include \"time.h\"\n#include \"stdint.h\" // Needed for Windows GCC 6.x compatibility\n#ifndef M_PI\n#define M_PI 3.141592653589793238462643383279502884L\n#endif\n#ifndef M_SQRT1_2\n#define M_SQRT1_2 0.707106781186547524400844362104849039L\n#endif\n#define wavespeed 1.0 // Set CFL-based \"wavespeed\" to 1.0.\n\n// Step P2: Declare the IDX4S(gf,i,j,k) macro, which enables us to store 4-dimensions of\n// data in a 1D array. In this case, consecutive values of \"i\"\n// (all other indices held to a fixed value) are consecutive in memory, where\n// consecutive values of \"j\" (fixing all other indices) are separated by\n// Nxx_plus_2NGHOSTS0 elements in memory. Similarly, consecutive values of\n// \"k\" are separated by Nxx_plus_2NGHOSTS0*Nxx_plus_2NGHOSTS1 in memory, etc.\n// #define IDX4SS(g,i,j,k) IDX4S(g,i,j,k)\n#define IDX4S(g,i,j,k) \\\n( (i) + Nxx_plus_2NGHOSTS0 * ( (j) + Nxx_plus_2NGHOSTS1 * ( (k) + Nxx_plus_2NGHOSTS2 * (g) ) ) )\n#define IDX4ptS(g,idx) ( (idx) + (Nxx_plus_2NGHOSTS0*Nxx_plus_2NGHOSTS1*Nxx_plus_2NGHOSTS2) * (g) )\n#define IDX3S(i,j,k) ( (i) + Nxx_plus_2NGHOSTS0 * ( (j) + Nxx_plus_2NGHOSTS1 * ( (k) ) ) )\n#define LOOP_REGION(i0min,i0max, i1min,i1max, i2min,i2max) \\\n for(int i2=i2min;i2Cartesian via\n// {xx[0][i0],xx[1][i1],xx[2][i2]}->{xCart[0],xCart[1],xCart[2]}\n#include \"xx_to_Cart.h\"\n\n// Step P5: Defines set_Nxx_dxx_invdx_params__and__xx(const int EigenCoord, const int Nxx[3],\n// paramstruct *restrict params, REAL *restrict xx[3]),\n// which sets params Nxx,Nxx_plus_2NGHOSTS,dxx,invdx, and xx[] for\n// the chosen Eigen-CoordSystem if EigenCoord==1, or\n// CoordSystem if EigenCoord==0.\n#include \"set_Nxx_dxx_invdx_params__and__xx.h\"\n\n// Step P6: Include basic functions needed to impose curvilinear\n// parity and boundary conditions.\n#include \"boundary_conditions/CurviBC_include_Cfunctions.h\"\n\n// Step P7: Implement the algorithm for upwinding.\n// *NOTE*: This upwinding is backwards from\n// usual upwinding algorithms, because the\n// upwinding control vector in BSSN (the shift)\n// acts like a *negative* velocity.\n//#define UPWIND_ALG(UpwindVecU) UpwindVecU > 0.0 ? 1.0 : 0.0\n\n// Step P8: Include function for enforcing detgammahat constraint.\n#include \"enforce_detgammahat_constraint.h\"\n\n// Step P9: Find the CFL-constrained timestep\n#include \"find_timestep.h\"\n\n// Step P10: Declare function necessary for setting up the initial data.\n// Step P10.a: Define BSSN_ID() for BrillLindquist initial data\n// Step P10.b: Set the generic driver function for setting up BSSN initial data\n#include \"initial_data.h\"\n\n// Step P11: Declare function for evaluating Hamiltonian constraint (diagnostic)\n#include \"Hamiltonian_constraint.h\"\n\n// Step P12: Declare rhs_eval function, which evaluates BSSN RHSs\n#include \"rhs_eval.h\"\n\n// Step P13: Declare Ricci_eval function, which evaluates Ricci tensor\n#include \"Ricci_eval.h\"\n\n// Step P14: Declare function for evaluating real and imaginary parts of psi4 (diagnostic)\n#include \"psi4.h\"\n#include \"SpinWeight_minus2_SphHarmonics/SpinWeight_minus2_SphHarmonics.h\"\n#include \"lowlevel_decompose_psi4_into_swm2_modes.h\"\n#include \"driver_psi4_spinweightm2_decomposition.h\"\n\n// Step P15: Declare global pointers and variables to be referenced in getter function.\n// This must be done in order to access variables created in the initialize\n// function in the stepfoward function.\nint dimensions[4] = {28, 8, 2, 7};\nint arrNGHOSTS[4];\nparamstruct params_p;\nrfm_struct rfmstruct_p;\nbc_struct bcstruct_p;\nREAL N_final_p, output_every_N_p, dt_p;\nREAL *y_n_gfs_p, *auxevol_gfs_p, *k_odd_gfs_p, *k_even_gfs_p, *y_nplus1_running_total_gfs_p, *xx_p[3];\n\n// initialize() function:\n// Step 0: Set up grid structure, allocate memory for gridfunctions, set up coordinates\n// Step 1: Set up initial data to an exact solutiondef\n// Step 2: Initialize global pointers and variables\n\nint EMSCRIPTEN_KEEPALIVE initialize(REAL CFL_FACTOR){\n\n paramstruct params;\n#include \"set_Cparameters_default.h\"\n\n // Step 0a: Set up numerical grid structure, first in space...\n int n1 = dimensions[0];\n int n2 = dimensions[1];\n int n3 = dimensions[2];\n const int Nxx[3] = {n1,n2,n3};\n\n // Step 0b: Set free parameters, overwriting Cparameters defaults\n // by hand or with command-line input, as desired.\n#include \"free_parameters.h\"\n\n // Step 0c: Uniform coordinate grids are stored to *xx[3]\n REAL *xx[3];\n // Step 0c.i: Set bcstruct\n bc_struct bcstruct;\n {\n int EigenCoord = 1;\n // Step 0c.ii: Call set_Nxx_dxx_invdx_params__and__xx(), which sets\n // params Nxx,Nxx_plus_2NGHOSTS,dxx,invdx, and xx[] for the\n // chosen Eigen-CoordSystem.\n set_Nxx_dxx_invdx_params__and__xx(EigenCoord, Nxx, ¶ms, xx);\n // Step 0c.iii: Set Nxx_plus_2NGHOSTS_tot\n#include \"set_Cparameters-nopointer.h\"\n const int Nxx_plus_2NGHOSTS_tot = Nxx_plus_2NGHOSTS0*Nxx_plus_2NGHOSTS1*Nxx_plus_2NGHOSTS2;\n // Step 0d: Find ghostzone mappings; set up bcstruct\n#include \"boundary_conditions/driver_bcstruct.h\"\n // Step 0d.i: Free allocated space for xx[][] array\n for(int i=0;i<3;i++) free(xx[i]);\n }\n\n // Step 0e: Call set_Nxx_dxx_invdx_params__and__xx(), which sets\n // params Nxx,Nxx_plus_2NGHOSTS,dxx,invdx, and xx[] for the\n // chosen (non-Eigen) CoordSystem.\n int EigenCoord = 0;\n set_Nxx_dxx_invdx_params__and__xx(EigenCoord, Nxx, ¶ms, xx);\n\n // Step 0f: Set all C parameters \"blah\" for params.blah, including\n // Nxx_plus_2NGHOSTS0 = params.Nxx_plus_2NGHOSTS0, etc.\n #include \"set_Cparameters-nopointer.h\"\n const int Nxx_plus_2NGHOSTS_tot = Nxx_plus_2NGHOSTS0*Nxx_plus_2NGHOSTS1*Nxx_plus_2NGHOSTS2;\n\n // Step 0g: Time coordinate parameters\n const REAL t_final = 1.5*domain_size;\n\n // Step 0h: Set timestep based on smallest proper distance between gridpoints and CFL factor\n REAL dt = find_timestep(¶ms, xx);\n\n N_final_p = (int)(t_final / dt + 0.5); // The number of points in time.\n // Add 0.5 to account for C rounding down\n // typecasts to integers.\n REAL out_approx_every_t = 0.2;\n output_every_N_p = (int)(out_approx_every_t*((REAL)N_final_p)/t_final);\n\n // Step 0i: Error out if the number of auxiliary gridfunctions outnumber evolved gridfunctions.\n // This is a limitation of the RK method. You are always welcome to declare & allocate\n // additional gridfunctions by hand.\n if(NUM_AUX_GFS > NUM_EVOL_GFS) {\n fprintf(stderr,\"Error: NUM_AUX_GFS > NUM_EVOL_GFS. Either reduce the number of auxiliary gridfunctions,\\n\");\n fprintf(stderr,\" or allocate (malloc) by hand storage for *diagnostic_output_gfs. \\n\");\n exit(1);\n }\n\n // Step 0j: Allocate memory for gridfunctions\n#include \"MoLtimestepping/RK_Allocate_Memory.h\"\n REAL *restrict auxevol_gfs = (REAL *)malloc(sizeof(REAL) * NUM_AUXEVOL_GFS * Nxx_plus_2NGHOSTS_tot);\n\n // Step 0k: Set up precomputed reference metric arrays\n // Step 0k.i: Allocate space for precomputed reference metric arrays.\n#include \"rfm_files/rfm_struct__malloc.h\"\n\n // Step 0k.ii: Define precomputed reference metric arrays.\n {\n #include \"set_Cparameters-nopointer.h\"\n #include \"rfm_files/rfm_struct__define.h\"\n }\n\n // Step 1a: Set up initial data to an exact solution\n initial_data(¶ms, xx, y_n_gfs);\n\n // Step 1b: Apply boundary conditions, as initial data\n // are sometimes ill-defined in ghost zones.\n // E.g., spherical initial data might not be\n // properly defined at points where r=-1.\n apply_bcs_curvilinear(¶ms, &bcstruct, NUM_EVOL_GFS,evol_gf_parity, y_n_gfs);\n enforce_detgammahat_constraint(&rfmstruct, ¶ms, y_n_gfs);\n\n //Step 2: Assign pointers/Initialize global variables\n arrNGHOSTS[0] = NGHOSTS;\n arrNGHOSTS[1] = Nxx_plus_2NGHOSTS0;\n arrNGHOSTS[2] = Nxx_plus_2NGHOSTS1;\n arrNGHOSTS[3] = Nxx_plus_2NGHOSTS2;\n dt_p = dt;\n params_p = params;\n rfmstruct_p = rfmstruct;\n bcstruct_p = bcstruct;\n y_n_gfs_p = y_n_gfs;\n auxevol_gfs_p = auxevol_gfs;\n k_odd_gfs_p = k_odd_gfs;\n k_even_gfs_p = k_even_gfs;\n y_nplus1_running_total_gfs_p = y_nplus1_running_total_gfs;\n xx_p[0]=xx[0];\n xx_p[1]=xx[1];\n xx_p[2]=xx[2];\n\n return 0;\n}\n\n// stepForward() function:\n// Step 1: Define and initialize variables from initialize() function so they can be used in the RK-like Method\n// of Lines timestepping algorithm\n// Step 2: Step forward one timestep (t -> t+dt) in time using chosen RK-like MoL timestepping algorithm\n\nvoid EMSCRIPTEN_KEEPALIVE stepForward(){\n\n // Step 1: Redefine and initialize variables. In order to call each time-step one by one, we need to redefine\n // some variables used in the MoL timestepping algorithm with the saved values from the initialization\n // step\n int Nxx_plus_2NGHOSTS0 = arrNGHOSTS[1];\n int Nxx_plus_2NGHOSTS1 = arrNGHOSTS[2];\n int Nxx_plus_2NGHOSTS2 = arrNGHOSTS[3];\n int Nxx_plus_2NGHOSTS_tot = Nxx_plus_2NGHOSTS0*Nxx_plus_2NGHOSTS1*Nxx_plus_2NGHOSTS2;\n int N_final = N_final_p;\n int output_every_N = output_every_N_p;\n REAL *restrict diagnostic_output_gfs = diagnostic_output_gfs_p;\n REAL dt = dt_p;\n paramstruct params = params_p;\n rfm_struct rfmstruct = rfmstruct_p;\n bc_struct bcstruct = bcstruct_p;\n REAL *y_n_gfs = y_n_gfs_p;\n REAL *restrict auxevol_gfs = auxevol_gfs_p;\n REAL *k_odd_gfs = k_odd_gfs_p;\n REAL *k_even_gfs = k_even_gfs_p;\n REAL *restrict y_nplus1_running_total_gfs = y_nplus1_running_total_gfs_p;\n REAL *xx[3];\n xx[0]=xx_p[0];\n xx[1]=xx_p[1];\n xx[2]=xx_p[2];\n#include \"boundary_conditions/driver_bcstruct.h\"\n\n // Step 2: Step forward one timestep (t -> t+dt) in time using\n // chosen RK-like MoL timestepping algorithm\n\n#include \"MoLtimestepping/RK_MoL.h\"\n}\n\n// runSpinWeightFx() function:\n// Step 1: Runs Psi4 spinweight function given a specific point on a specific extraction radius\n\nvoid EMSCRIPTEN_KEEPALIVE runSpinWeightFx(int n, const int R_ext_idx, int i0, int i1, int i2, int lastRun) {\n // Step 1: Call spinweight decomposition function\n driver_psi4_spinweightm2_decomposition(¶ms_p, ((REAL)n)*dt_p, R_ext_idx, xx_p, y_n_gfs_p,\n diagnostic_output_gfs_p, i0, i1, i2, lastRun);\n}\n\n// Getter functions used to access the data in javascript after compliling.\n\n// getNFinal(): returns final time-step value\nREAL EMSCRIPTEN_KEEPALIVE getNFinal(){\n return N_final_p;\n}\n\n// getNGHOSTS(): returns desired dimensions including ghosts shells\nREAL EMSCRIPTEN_KEEPALIVE getNGHOSTS(int i){\n return arrNGHOSTS[i];\n}\n\n// getFxVal(): returns desired function value at specfic point in space, index i is determined from the\n// IDX3S and IDX4ptS arrays\nREAL EMSCRIPTEN_KEEPALIVE getFxVal(int i){\n return y_n_gfs_p[i];\n}\n\n// getIDX3S(): returns IDX3S index for a given point\nREAL EMSCRIPTEN_KEEPALIVE getIDX3S(int i0, int i1, int i2){\n int Nxx_plus_2NGHOSTS0 = arrNGHOSTS[1];\n int Nxx_plus_2NGHOSTS1 = arrNGHOSTS[2];\n int Nxx_plus_2NGHOSTS2 = arrNGHOSTS[3];\n return IDX3S(i0,i1,i2);\n}\n\n// getIDX4ptS(): returns IDX4ptS index to be used in getFxVal() using information from getIDX3S()\nREAL EMSCRIPTEN_KEEPALIVE getIDX4ptS(int fx, REAL idx){\n int Nxx_plus_2NGHOSTS0 = arrNGHOSTS[1];\n int Nxx_plus_2NGHOSTS1 = arrNGHOSTS[2];\n int Nxx_plus_2NGHOSTS2 = arrNGHOSTS[3];\n return IDX4ptS(fx,idx);\n}\n\n// getCart(): returns the cartesian version of a given spherical coordinate\nREAL EMSCRIPTEN_KEEPALIVE getCart(int i0, int i1, int i2, int index){\n REAL xCart[3];\n xx_to_Cart(¶ms_p,xx_p,i0,i1,i2,xCart);\n return xCart[index];\n}\n\n// setDim(): setter function allowing the user to change the resolution of the simulation\nvoid EMSCRIPTEN_KEEPALIVE setDim(int dim1, int dim2, int dim3){\n dimensions[0] = dim1;\n dimensions[1] = dim2;\n dimensions[2] = dim3;\n dimensions[3]= dim1/4;\n}\n\n// getDim(): getter function for the dimensions/resolution of the simulation\nint EMSCRIPTEN_KEEPALIVE getDim(int i){\n return dimensions[i];\n}\n\n// getPsi4R(): getter function for the real part of psi 4 at a given point\nREAL EMSCRIPTEN_KEEPALIVE getPsi4R(int idx3d){\n int Nxx_plus_2NGHOSTS0 = arrNGHOSTS[1];\n int Nxx_plus_2NGHOSTS1 = arrNGHOSTS[2];\n int Nxx_plus_2NGHOSTS2 = arrNGHOSTS[3];\n REAL *restrict diagnostic_output_gfs = diagnostic_output_gfs_p;\n\n const REAL psi4r = (+diagnostic_output_gfs[IDX4ptS(PSI4R_0PTGF,idx3d)]\n +diagnostic_output_gfs[IDX4ptS(PSI4R_1PTGF,idx3d)]\n +diagnostic_output_gfs[IDX4ptS(PSI4R_2PTGF,idx3d)]);\n return psi4r;\n}\n\n// getPsi4R(): getter function for the imaginary part of psi 4 at a given point\nREAL EMSCRIPTEN_KEEPALIVE getPsi4I(int idx3d){\n int Nxx_plus_2NGHOSTS0 = arrNGHOSTS[1];\n int Nxx_plus_2NGHOSTS1 = arrNGHOSTS[2];\n int Nxx_plus_2NGHOSTS2 = arrNGHOSTS[3];\n REAL *restrict diagnostic_output_gfs = diagnostic_output_gfs_p;\n\n const REAL psi4i = (+diagnostic_output_gfs[IDX4ptS(PSI4I_0PTGF,idx3d)]\n +diagnostic_output_gfs[IDX4ptS(PSI4I_1PTGF,idx3d)]\n +diagnostic_output_gfs[IDX4ptS(PSI4I_2PTGF,idx3d)]);\n return psi4i;\n}\n\n// runsim(): function that runs the initialize function with a specific resolution and CFL factor\nvoid EMSCRIPTEN_KEEPALIVE runsim(){\n CFL_FACTOR = 1.0;\n \n initialize(CFL_FACTOR);\n}\n```\n\n Writing EMCC_BSSN_Two_BHs_Collide_Psi4//BrillLindquist_Playground.c\n\n\n\n\n# Step 6: Compile\n\n\\[Back to [top](#toc)\\]\n$$\\label{compileexec}$$\n\nUse emscripten to generate compilied html, wasm and js files. Copy the wasm and js files to the main javascript directory.\n\n\n```python\nmain_c_file = os.path.join(Ccodesdir,\"BrillLindquist_Playground.c\")\nmain_output_file = os.path.join(outdir,\"BHCollidePsi4.html\")\nprint(\"Attempting to compile\\n\", main_c_file, \"\\nto\\n\", main_output_file,\"\\n\")\ncmd.C_compile(main_c_file, main_output_file, compile_mode=\"emscripten\")\n\nprint(\"\\nFiles in output directory are:\\n\", os.listdir(outdir))\n```\n\n Attempting to compile\n EMCC_BSSN_Two_BHs_Collide_Psi4/BrillLindquist_Playground.c \n to\n EMCC_BSSN_Two_BHs_Collide_Psi4/output/BHCollidePsi4.html \n \n Compiling executable...\n (EXEC): Executing `emcc -std=gnu99 -s -O3 -march=native -funroll-loops -s ALLOW_MEMORY_GROWTH=1 EMCC_BSSN_Two_BHs_Collide_Psi4/BrillLindquist_Playground.c -o EMCC_BSSN_Two_BHs_Collide_Psi4/output/BHCollidePsi4.html -lm `...\n (BENCH): Finished executing in 44.89996314048767 seconds.\n Finished compilation.\n \n Files in output directory are:\n ['BHCollidePsi4.html', 'BHCollidePsi4.js', 'BHCollidePsi4.wasm']\n\n\n\n```bash\n%%bash\ncp EMCC_BSSN_Two_BHs_Collide_Psi4/output/BHCollidePsi4.wasm .\ncp EMCC_BSSN_Two_BHs_Collide_Psi4/output/BHCollidePsi4.js .\n```\n", "meta": {"hexsha": "f7a93868138762f11ddc24325e3a5add9ac8d9bf", "size": 78687, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "EMCC-BSSNCurvilinear-Two_BHs_Collide-Psi4.ipynb", "max_stars_repo_name": "wucap/NRPyJS", "max_stars_repo_head_hexsha": "2029b169f05011b686cc7d270849ff18af4115d4", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-09-23T02:03:49.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-23T02:03:49.000Z", "max_issues_repo_path": "EMCC-BSSNCurvilinear-Two_BHs_Collide-Psi4.ipynb", "max_issues_repo_name": "karinnes/NRPyJS", "max_issues_repo_head_hexsha": "9a9e73050de8fc41d0d0c0971971d0712ac1ec2c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "EMCC-BSSNCurvilinear-Two_BHs_Collide-Psi4.ipynb", "max_forks_repo_name": "karinnes/NRPyJS", "max_forks_repo_head_hexsha": "9a9e73050de8fc41d0d0c0971971d0712ac1ec2c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 50.6351351351, "max_line_length": 685, "alphanum_fraction": 0.5941388031, "converted": true, "num_tokens": 17861, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7826624789529375, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.42791143737042575}} {"text": "```python\n%matplotlib inline\n\nimport matplotlib\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nplt.rcParams[\"figure.figsize\"] = (12, 9)\nplt.rcParams[\"font.size\"] = 18\n```\n\n# Reactor Physics\n\nThis is not a reactor physics class. However, to understand and analyze fuel management, advanced reactor designs, and spent fuel recycling, one must first have a background in reactor physics.\n\n## Learning Objectives\n\n- Describe criticality in terms of the life cycle of a fission neutron. \n- Differentiate concepts such as neutron flux, reaction rates, cross sections, reactivity, multiplication factor.\n- Calculate the multiplication factor using the 6 factor formula. \n- Calculate the dynamic impact of reactivity feedback on criticality.\n- Recognize the impact of moderator material choice on reactor behavior.\n- Analyze reactor behavior.\n- Calculate the impact of geometry on criticality.\n\n\n## Neutron Multiplication Fundamentals\n\n### Neutron interactions with matter.\n\n\\begin{align}\n^1_0n + {^a_z}X \\longrightarrow \\left({^{a+1}_z}X\\right)^* \\longrightarrow \n\\begin{cases}\n^1_0n + {^a_z}X , & \\mbox{Resonance Elastic Scattering}\\\\\n^1_0n + \\left({^a_z}X\\right)^* , & \\mbox{Elastic Scattering}\\\\\n{^{a+1}_z}X +\\gamma , & \\mbox{Radiative Capture}\\\\\n{^{a_1}_{z_2}}X + {^{a_2}_{z_2}}X + \\nu^1_0n, & \\mbox{Fission}\n\\end{cases}\n\\end{align}\n\n\n\n\n### Cross sections\n\nThe likelihood of each of these events is captured by cross sections. \n\n- $\\sigma_x = $ microscopic cross section $[cm^2]$\n- $\\Sigma_x = $ macroscopic cross section $[1/length]$\n- $\\Sigma_x = N\\sigma_x $\n- $N = $ number density of target atoms $[\\#/volume]$\n\n\n### Cross sections are in units of area. Explain this to your neighbor.\n\n\n### Reaction Rates\n\n- The microscopic cross section is just the likelihood of the event reported as an area (typically in units of $10^{-24}cm^2$ or \"barns\" which, to a neutron is like hitting the \"broad side of a barn\") \n- The macroscopic cross section is just the likelihood of the event per unit area of a certain density of target isotopes.\n- The reaction rate is the macroscopic cross section times the flux of incident neutrons.\n\n\\begin{align}\nR_{i,j}(\\vec{r}) &= N_j(\\vec{r})\\int dE \\phi(\\vec{r},E)\\sigma_{i,j}(E)\\\\\nR_{i,j}(\\vec{r}) &= \\mbox{reactions of type i involving isotope j } [reactions/cm^3s]\\\\\nN_j(\\vec{r}) &= \\mbox{number of nuclei participating in the reactions } [\\#/cm^3]\\\\\nE &= \\mbox{energy} [MeV]\\\\\n\\phi(\\vec{r},E)&= \\mbox{flux of neutrons with energy E at position i } [\\#/cm^2s]\\\\\n\\sigma_{i,j}(E)&= \\mbox{cross section } [cm^2]\\\\\n\\end{align}\n\n\nThis can be written more simply as $R_x = \\Sigma_x I N$, where I is intensity of the neutron flux.\n\n\n### Neutron attenuation\n\n\n$$\n\\begin{align}\nI(x) &= I_0e^{-\\Sigma_t x}\\\\\n\\end{align}\n$$\n\nwhere\n\n$$\n\\begin{align}\n I(x) &= \\mbox{intensity at distance x}\\\\\n I_0 &= \\mbox{initial intensity}\\\\\n \\Sigma_t &= \\mbox{macroscopic total cross section} \\\\\n x &= \\mbox{distance into material [m]}\\\\\n\\end{align}\n$$\n\n\n```python\nimport math\ndef attenuation(distance, initial=100, sig_t=1):\n \"\"\"This function describes neutron attenuation into the slab\"\"\"\n return initial*math.exp(-sig_t*distance)\n\n```\n\nRather than intensity, one can find the probability density:\n\nIn the case of decay:\n\n\\begin{align}\nP(t)dt &= \\lambda e^{-\\lambda t}dt\n\\end{align}\n\n\nFrom this, one can find the mean lifetime of a neutron before absorption:\n\n\\begin{align}\n\\bar{t} &= \\int_0^\\infty t'P(t')dt'\\\\\n &= \\int_0^\\infty t'\\lambda e^{-\\lambda t'}dt'\\\\ \n &= \\frac{1}{\\lambda}\n\\end{align}\n\nIn the case of attenuation:\n\\begin{align}\nP(x)dx &= \\Sigma_te^{-\\Sigma_tx}dx\n\\end{align}\n\nSo, the mean free path is:\n\n\\begin{align}\n\\bar{l} &= \\int_0^\\infty x'P(x')dx'\\\\\n &= \\int_0^\\infty x'\\Sigma_te^{-\\Sigma_t x'}dx'\\\\ \n &= \\frac{1}{\\Sigma_t}\n\\end{align}\n\n\n```python\ndef prob_dens(distance, initial=100, sig_t=1):\n return sig_t*attenuation(distance, initial=100, sig_t=1)\n\n```\n\n\n```python\nsig_t = 0.05\ni_0 = 100\n\n# This code plots attenuation\nimport numpy as np\nz = np.arange(24)\ny = np.arange(24)\nx = np.arange(24)\nfor h in range(0,24):\n x[h] = h\n y[h] = attenuation(h, initial=i_0, sig_t=sig_t)\n z[h] = prob_dens(h, initial=i_0, sig_t=sig_t)\n\n# creates a figure and axes with matplotlib\nfig, ax = plt.subplots()\nscatter = plt.scatter(x, y, color='blue', s=y*20, alpha=0.4) \nax.plot(x, y, color='red') \nax.plot(x, z, color='green') \n\n\n# adds labels to the plot\nax.set_ylabel('Percent of Neutrons')\nax.set_xlabel('Distance into slab')\nax.set_title('Attenuation')\n\n# adds tooltips\nimport mpld3\nlabels = ['{0}% intensity'.format(i) for i in y]\ntooltip = mpld3.plugins.PointLabelTooltip(scatter, labels=labels)\nmpld3.plugins.connect(fig, tooltip)\n\nmpld3.display()\n```\n\n\n\n\n\n\n\n\n
                                        \n\n\n\n\n### Source term\n\nThe source of neutrons in a reactor are the neutrons from fission. \n\n\\begin{align}\ns &=\\nu \\Sigma_f \\phi\n\\end{align}\n\nwhere\n\n\\begin{align}\ns &= \\mbox{neutrons available for next generation of fissions}\\\\\n\\nu &= \\mbox{the number of neutrons born per fission}\\\\\n\\Sigma_f &= \\mbox{the cross section for fissions in the material}\\\\\n\\phi &= \\mbox{initial neutron flux}\n\\end{align}\n\nThis can also be written as:\n\n\\begin{align}\ns &= \\nu\\Sigma_f\\phi\\\\\n &= \\nu\\frac{\\Sigma_f}{\\Sigma_{a,fuel}}\\frac{\\Sigma_{a,fuel}}{\\Sigma_a}{\\Sigma_a} \\phi\\\\\n &= \\eta f {\\Sigma_a} \\phi\\\\\n\\eta &= \\frac{\\nu\\Sigma_f}{\\Sigma_{a,fuel}} \\\\\n &= \\mbox{number of neutrons produced per neutron absorbed by the fuel, \"neutron reproduction factor\"}\\\\\nf &= \\frac{\\Sigma_{a,fuel}}{\\Sigma_a} \\\\\n &= \\mbox{number of neutrons absorbed in the fuel per neutron absorbed anywhere, \"fuel utilization factor\"}\\\\\n\\end{align}\n\nThis absorption and flux term at the end seeks to capture the fact that some of the neutrons escape. However, if we assume an infinite reactor, we know that all the neutrons are eventually absorbed in either the fuel or the coolant, so we can normalize by $\\Sigma_a\\phi$ and therefore:\n\n\n\\begin{align}\nk_\\infty &= \\frac{\\eta f \\Sigma_a\\phi}{\\Sigma_a \\phi}\\\\\n&= \\eta f\n\\end{align}\n\n### Fission Spectrum\n\n$\\chi(E)$ is an empirical probability density function describing the energies of prompt fission neutrons. \n\n\\begin{align}\n\\chi (E) &= 0.453e^{-1.036E}\\sinh\\left(\\sqrt{2.29E}\\right)\\\\\n\\end{align}\n\n\n```python\nimport numpy as np\nimport math\ndef chi(energy):\n return 0.453*np.exp(-1.036*energy)*np.sinh(np.sqrt(2.29*energy))\n\nenergies = np.arange(0.0,10.0, 0.1)\n\nplt.plot(energies, chi(energies))\nplt.title(r'Prompt Neutron Energy Distribution $\\chi(E)$')\nplt.xlabel(\"Prompt Neutron Energy [MeV]\")\nplt.ylabel(\"probability\")\n```\n\n#### Questions about this plot:\n\n- What is the most likely prompt neutron energy?\n- Can you write an equation for the average neutron energy?\n\n\n\n\n\n```python\n### Max function\nprint(max([chi(e) for e in energies]), chi(0.7))\n```\n\n 0.3581027022872076 0.35810270228720753\n\n\n#### Expectation Value\n\nRecall that the average energy will be the expectation value of the probability density function.\n\n\n\n\\begin{align}\n<\\chi(E)> &= \\int\\chi(E)dE\\\\\n&= E \\chi(E)\n\\end{align}\n\n\n```python\nplt.plot(energies, [chi(e)*e for e in energies])\n```\n\n### What energy neutron do we prefer for fission in U235?\n\n\n\n\n### Four Factor Formula\n\nSo, then, if we consider the impact of neutron energy on the likelihood of absorption, we can break the neutrons up into a thermal and a fast group. \n\n\nSo, we can say more explicitly:\n\n\n\\begin{align}\nk_\\infty =& \\epsilon p \\eta f\\\\\n\\epsilon =& \\mbox{ fast fission factor}\\\\\n =& \\frac{\\mbox{neutrons produced by all fissions}}{\\mbox{neutrons produced by thermal fissions}}\\\\\np =& \\mbox{ resonance escape probability}\\\\\n =& \\frac{\\mbox{neutrons that reach thermal energies}}{\\mbox{number of fast neutrons that start to slow down}}\n\\end{align}\n\nIf we consider the non-infinite nature of most reactors, we have to add two more factors to the formula:\n\n\\begin{align}\nk_\\infty =& P_{fnl} P_{tnl}\\epsilon p \\eta f\\\\\nP_{fnl} =& \\mbox{ fast neutron non leakage probability}\\\\\n =& \\frac{\\mbox{fast neutrons that do not leak from the reacotr}}{\\mbox{fast neutrons produced by thermal fissions}}\\\\\nP_{tnl} =& \\mbox{ thermal neutron non leakage probability}\\\\\n =& \\frac{\\mbox{thermal neutrons that do not leak from the reacotr}}{\\mbox{neutrons that reach thermal energies}}\\\\\n\\end{align}\n\n\n## Neutron Transport Fundamentals\n\n\n### Fast Homogenous Reactor\n\nConsider a reactor that is:\n\n- fast\n- critical\n- the fuel and coolant are a homogeneous mixture\n- the reactor has only one region, no reflector (\"bare\" reactor)\n\nThis reactor can be described by the one group diffusion equation:\n\n\\begin{align}\nD\\nabla^2\\phi-\\Sigma_a\\phi + s &= \\frac{1}{v}\\frac{\\partial \\phi}{\\partial t}\\\\\nD &= \\mbox{ the diffusion coefficient}\\\\\n\\phi &= \\mbox{ flux}\\\\\nv &= \\mbox{ neutron speed}\n\\end{align}\n\nIf the fission source, $s$ does not balance neutron absorption and leakage, then the right hand side of the one-group diffusion equation is nonzero and the power may increase or decrease with time.\n\n### Reactivity is deviation from k\n\n\n\\begin{align}\n\\rho &= \\frac{k-1}{k}\n\\end{align}\n\n\n\n```python\ndef rho(k):\n \"\"\"reactivity, rho\n :param k: multiplication factor\n \"\"\"\n return (k-1)/k\n\n# Plot reactivity as a function of k\nimport numpy as np\nk_vals = np.arange(0.7, 2.0, 0.01)\nr = [rho(k) for k in k_vals]\nplt.plot(k_vals, r)\nplt.ylabel(r'reactivity $\\rho$ $[\\frac{\\delta k}{k}]$',fontsize=20,)\nplt.xlabel(r'multiplication factor $k$ $[-]$',fontsize=20,)\nplt.title(r'$\\rho = \\frac{k-1}{k}$', fontsize=20, y=1.02)\n```\n\n#### Steady state\n\n \n\nThe multiplication factor, k, can be used to adjust the source strength and reach a steady state diffusion equation:\n\n\n\\begin{align}\nD\\nabla^2\\phi-\\Sigma_a\\phi + \\frac{1}{k}\\nu\\Sigma_f\\phi &= 0\\\\\n\\end{align}\n\n\n\n\n## Diffusion Solution\n\n\n```python\n## This code example was adapted from\n## https://github.com/marort91/AlphaEigenvalue/blob/master/RadiationTransportCoding/NeutronDiffusion_Python/NDE_CriticalityEigenvalue.ipynb\n\ndef diff(sig_tr):\n return 1.0/(3.0*sig_tr)\n\ndef sig_tr(e):\n sig_t(e) - mu*sig_s(e)\n\nD = diff(0.1)\nnusigf = 0.70\nsiga = 0.066 \n\nLx = np.pi*((nusigf-siga)/D)**(-0.5)\n\nN = 50;\nh = Lx/(N-1)\n\nx = np.zeros(N)\n\nfor i in range(N-1):\n x[i+1] = x[i] + h\n \n \nL = np.zeros((N,N))\nA = np.zeros((N,N))\nM = np.zeros((N,N))\n\nfor i in range(N):\n L[i][i] = L[i][i] + (-2*(-D/(h**2)))\n \nfor i in range(1,N):\n L[i][i-1] = L[i][i-1] + (-D/h**2)\n \nfor i in range(N-1):\n L[i][i+1] = L[i][i+1] + (-D/h**2)\n \nfor i in range(N):\n A[i][i] = A[i][i] + siga\n \nM = L + A\n\nM[0][0] = 1\nM[0][1] = 0\nM[N-1][N-1] = 1\nM[N-1][N-2] = 0\n\nphi0 = np.ones((N,1))\nphi0[0] = 0\nphi0[N-1] = 0\n\ntol = 1e-15\nk = 1.00\n\n\ndef is_converged(k_old, k, tol):\n return np.abs(k - k_old) <= tol\n\nfor i in range(100):\n # update k\n k_old = k\n # solve for psi\n psi = np.linalg.solve(M, nusigf*phi0)\n \n # solve for k\n k = sum(nusigf*psi)/sum(nusigf*phi0)\n \n # solve for phi\n phi0 = (1/k)*psi\n phi0[0] = 0\n phi0[N-1] = 0\n \n # determine convergence\n if is_converged(k_old, k, tol):\n break\n \nplt.plot(x, phi0)\nplt.xlabel('Slab (cm)')\nplt.ylabel('Neutron Flux')\nplt.grid()\n\nprint(\"k-effective = \", k)\n\nprint(\" approx alpha = \", rho(k) * sum(nusigf*phi0)/sum(phi0))\n```\n\n### One Group Reactor Equation\n\nWe can define a quantity, geometric bucking, as:\n\n\\begin{align}\nB^2 &= \\frac{1}{D}\\left(\\frac{\\nu}{k}\\Sigma_f - \\Sigma_a\\right)\\\\\n\\end{align}\n\nNext, we can simplify the previous equation using this definition, so that the one-group reactor equation becomes:\n\n\n\\begin{align}\n\\nabla^2\\phi + B^2\\phi &= 0\\\\\n\\end{align}\n\nTo find the criticality of a reactor with a specific geometry, then, we can solve the geometric buckling equation for k:\n\n\\begin{align}\nk &= \\frac{\\nu\\Sigma_f}{DB^2 + \\Sigma_a}\\\\\n\\end{align}\n\n\nThe buckling, B, is used to help describe the geometry of the system. The solutions of the one group reactor equation for boundary conditions corresponding to canonical shapes provide both flux and buckling formulae for each canonical shape:\n\n#### Slab\nThickness a:\n\\begin{align}\n\\phi &= cos\\left(\\frac{\\pi x}{a}\\right)\\\\\nB^2 &= \\frac{\\pi^2}{a^2} \n\\end{align}\n\n#### Cylinder\nHeight H, Radius R:\n\\begin{align}\n\\phi &= J_0\\left(\\frac{\\nu_0r}{R}\\right)cos\\left(\\frac{\\pi z}{H}\\right)\\\\\nB^2 &= \\frac{\\nu_0^2}{R^2} + \\frac{\\pi^2}{H^2}\n\\end{align}\n\n#### Sphere\nRadius R:\n\\begin{align}\n\\phi &= \\left(\\frac{1}{r}\\right)sin\\left(\\frac{\\pi r}{R}\\right)\\\\\nB^2 &= \\frac{\\pi^2}{R^2} \n\\end{align}\n\n\n```python\ndef k(nusigf, diff, bsq, siga):\n return nusigf/(diff*bsq + siga)\n\n\ndef bsq_sphere(r):\n return (np.pi**2)/(r**2)\n\n\n\nnusigf = 0.3\ndiff = 1.1\nsiga =0.01\n\nfig, ax1 = plt.subplots()\nradii = np.arange(0.1, 5.0, 0.01)\nax1.plot(radii, k(nusigf, diff, bsq_sphere(radii), siga), 'b-', label='k')\nax1.set_xlabel('radius (r)')\nax1.set_ylabel('k', color='b')\nax1.set_title('Criticality and Geometric Buckling of a Sphere')\nfor tl in ax1.get_yticklabels():\n tl.set_color('b')\n\n \nax2 = ax1.twinx()\n\nax2.plot(radii, bsq_sphere(radii), 'darkorange', label=r'$B^2$')\nax2.set_ylabel(r'$B^2$', color='darkorange')\nfor tl in ax2.get_yticklabels():\n tl.set_color('darkorange')\n\nax1.legend()\nax2.legend(loc=2)\n```\n\n### Multigroup and beyond\n\nTo capture the variation of neutron energies in the diffusion equation, we can discretize the flux into energy groups.\n\n\\begin{align}\n\\phi &= \\sum_{g=1}^{g=G}\\phi_g\\\\\n\\phi_g &= \\int_{E_g}^{E_{g-1}}\\phi(E)dE\\rvert_{g = 1,G}\n\\end{align}\n\nThe diffusion coefficient also needs to be individually evaluated for each energy group such that we have a $D_g$ for each group g. Also, it is important to consider possible paths of demise for potential fission neutrons. \n\nThe derivation, based on integration, can be found in your book, pages 128-130. It is based on the following integration:\n\n\\begin{align}\n\\int_{E_g}^{E_{g-1}}dE\\left[-\\nabla D(\\vec{r},E)\\nabla\\phi(\\vec{r},E)+ \\Sigma_t(\\vec{r},E)\\phi(\\vec{r},E)\\right] &= \\int_{E_g}^{E_{g-1}}dE\\left[\\int_{E'}\\Sigma_{s}(\\vec{r},E'\\rightarrow E)\\phi(\\vec{r},E') + S(\\vec{r},E)\\right]\\\\\n\\end{align}\n\nOnce this integration is completed for a set of group boundaries, $g\\in[1,G] \\rightarrow E_g\\in[E_1,E_G]$, one can generically write:\n\n\\begin{align}\n-\\nabla D_g\\nabla\\phi_g+ \\Sigma_{t,g}\\phi_g &= \\sum_{g'=1}^{g'=G}\\Sigma_{s}^{g'\\rightarrow g}\\phi_{g'} + S\\int_{E_g}^{E_{g-1}}\\chi(E)dE\\\\\n\\end{align}\n\nRecall, however, that the fission spectrum is a probability density function, so one can also write the following identities:\n\n\\begin{align}\n\\int_0^\\infty\\chi(E) &= 1\\\\ \n\\chi_g &= \\int_{E_g}^{E_{g-1}}\\chi(E)dE\\\\ \n\\sum_{g=1}^{g=G}\\chi_g &= 1\\\\\n\\end{align}\n\nThis simplifies the diffusion equation:\n\n\\begin{align}\n-\\nabla D_g\\nabla^2\\phi_g+ \\Sigma_{t,g}\\phi_g &= \\sum_{g'=1}^{g'=G}\\Sigma_{s}^{g'\\rightarrow g}\\phi_{g'} + \\chi_gS\\\\\n\\end{align}\n\nwhere\n\n\\begin{align}\nS_g &= \\sum_{g'=1}^{g'=G}(\\nu\\Sigma_f)_{g'}\\phi_{g'}\n\\end{align}\n\n#### Group Removal Cross Section\n\nThe right hand side summation of the scattering cross section is confusing though. Most of the scattering is from other groups into this one, but some of the scattering is from this group into itself. Keeping that term, $\\Sigma_s^{g'\\rightarrow g}$ on the right hand side is misleading. It's not a source of new neutrons. So, we have a change of variables that can clean this up, the group removal cross section.\n\n\n\\begin{align}\n\\Sigma_{R,g} &= \\Sigma_{t,g} - \\Sigma_{s}^{g'\\rightarrow g}\n\\end{align}\n\nIf we use the group removal cross section, then we arrive at the following form of the multigroup equation.\n\n\n\\begin{align}\n-\\nabla D_g\\nabla^2\\phi_g+ \\Sigma_{R,g}\\phi_g &= \\sum_{g'\\ne G}^{g'=G}\\Sigma_{s}^{g'\\rightarrow g}\\phi_{g'} + \\chi_g\\sum_{g'=1}^{g'=G}(\\nu\\Sigma_f)_{g'}\\phi_{g'}\\\\\n\\end{align}\n\n### Two Group Diffusion\n\nLet's define just two groups, fast and thermal. Let's also state that all prompt neutrons are born fast and that the diffusion coefficient and cross sections do not vary in space. With these assumptions, we arrive at the following two equations:\n\n\n\\begin{align}\n-D_1\\nabla^2\\phi_1 + \\Sigma_{R,1}\\phi_1 &= \\Sigma_{s}^{2\\rightarrow 1}\\phi_{2} + \\left[(\\nu\\Sigma_f)_{1}\\phi_{1} + (\\nu\\Sigma_f)_{2}\\phi_{2}\\right]\\\\\n\\end{align}\n\n\\begin{align}\n- D_2\\nabla^2\\phi_2+ \\Sigma_{R,2}\\phi_2 &= \\Sigma_{s}^{1\\rightarrow 2}\\phi_{1}\\\\\n\\end{align}\n\n**Discussion: What happened to the prompt fission spectrum, $\\chi_g$ in the above equations?**\n\n\n\n**Discussion: If we neglect upscattering, which term or terms in the above two equations will disappear?**\n\n\n\n\n### Criticality calculation\n\n\nFor criticality calculations, one might normalize the prompt fission spectrum with k as we have done before. For simplicity, one can move the scattering term to the left hand side, as if to say \"diffusion, group removal, and scattering are balanced at criticality.\"\n\n\n\\begin{align}\n-\\nabla D_g\\nabla\\phi_g+ \\Sigma_{R,g}\\phi_g - \\sum_{g'\\ne G}^{g'=G}\\Sigma_{s}^{g'\\rightarrow g}\\phi_{g'} &= \\frac{1}{k}\\chi_g\\sum_{g'=1}^{g'=G}(\\nu\\Sigma_f)_{g'}\\phi_{g'}\\\\\n\\end{align}\n\nThis change propagates to the explicit equations for two groups thus:\n\n\n\\begin{align}\n- D_1\\nabla^2\\phi_1 + \\Sigma_{R,1}\\phi_1 - \\Sigma_{s}^{2\\rightarrow 1}\\phi_{2} &= \\frac{1}{k}\\chi_1\\left[(\\nu\\Sigma_f)_{1}\\phi_{1} + (\\nu\\Sigma_f)_{2}\\phi_{2}\\right]\\\\\n\\end{align}\n\n\\begin{align}\n- D_2\\nabla^2\\phi_2+ \\Sigma_{R,2}\\phi_2 - \\Sigma_{s}^{1\\rightarrow 2}\\phi_{1} &= 0\\\\\n\\end{align}\n\n### Addition of chemical shim\n\nTogether let us consider the impact of a chemical shim, some absorber introduced to the coolant.\n\n**Discussion: In the two group equations, which parameter will change?**\n\n\n\n\n\nIn the thermal group, group 2, the removal cross section will change, because it involves absorption. The amount of chemical shim will impact the absorption cross section thus:\n\n\\begin{align}\n\\Sigma_{a',2} &= \\Sigma_{a,2} + \\Sigma_{shim,2} \\\\\n\\Sigma_{shim, 2} &= \\Sigma_{a',2} - \\Sigma_{a,2}\\\\\n&= \\rho_{shim}\\frac{N_{avo}}{A_{shim}}\\sigma_{a,shim,2}\\\\\n\\end{align}\n\n## Wrap-up\n\n- Describe criticality in terms of the life cycle of a fission neutron. \n- Differentiate concepts such as neutron flux, reaction rates, cross sections, reactivity, multiplication factor.\n- Calculate the multiplication factor using the 6 factor formula. \n- Calculate the dynamic impact of reactivity feedback on criticality.\n- Recognize the impact of moderator material choice on reactor behavior.\n- Analyze reactor behavior.\n- Calculate the impact of geometry on criticality.\n\n## References\n\nThis section was developed to complement pages YY-ZZ of [1]. \n\n[1] N. Tsoulfanidis, The Nuclear Fuel Cycle. La Grange Park, Illinois, USA: American Nuclear Society, 2013.\n\n", "meta": {"hexsha": "be0d5e9a1179c9de84244a4616a92ed5f6c9b189", "size": 241874, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "reactor-physics/reactor-physics.ipynb", "max_stars_repo_name": "abachma2/npre412", "max_stars_repo_head_hexsha": "3f105a15edc07745f1dd65cd791777a01136ec23", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "reactor-physics/reactor-physics.ipynb", "max_issues_repo_name": "abachma2/npre412", "max_issues_repo_head_hexsha": "3f105a15edc07745f1dd65cd791777a01136ec23", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "reactor-physics/reactor-physics.ipynb", "max_forks_repo_name": "abachma2/npre412", "max_forks_repo_head_hexsha": "3f105a15edc07745f1dd65cd791777a01136ec23", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 235.9746341463, "max_line_length": 46034, "alphanum_fraction": 0.8736242837, "converted": true, "num_tokens": 5973, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.4277530921163461}} {"text": "```python\nimport sys\nsys.path.append('..')\nimport torch\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nfrom sklearn.preprocessing import LabelEncoder, MinMaxScaler\nfrom sympy import simplify_logic\n\nfrom lens.utils.relu_nn import get_reduced_model, prune_features\nfrom lens import logic\nfrom lens.utils.base import collect_parameters\n\ntorch.manual_seed(0)\nnp.random.seed(0)\n```\n\n\n```python\ngene_expression_matrix = pd.read_csv('reduced_w_1/data.csv', index_col=None, header=None)\nlabels = pd.read_csv('reduced_w_1/tempLabels_W-1.csv', index_col=None, header=None)\ngenes = pd.read_csv('reduced_w_1/features.csv', index_col=None, header=None)\n```\n\n\n```python\ngene_expression_matrix\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
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                                        03.3200003.3200003.320006.9415366.590419
                                        14.2329783.3200003.320007.2795486.476784
                                        23.3200004.2006093.320007.7416004.643134
                                        33.3200003.3200003.320007.2766005.953452
                                        43.3200003.3200003.320007.2246286.555227
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                                        563.3200003.3200003.320007.6601826.128603
                                        573.3200003.7004303.451317.8098266.153968
                                        583.3200003.3200003.320007.5805886.134398
                                        594.1743193.3200003.320007.0160047.124143
                                        603.6992513.3200003.320007.5680446.236546
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                                        61 rows \u00d7 5 columns

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                                        \n\n\n\n\n```python\nlabels\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        0
                                        0diagnosis: healthy control
                                        1diagnosis: healthy control
                                        2diagnosis: healthy control
                                        3diagnosis: healthy control
                                        4diagnosis: healthy control
                                        ......
                                        56omalizumab responder status: Responder
                                        57omalizumab responder status: Responder
                                        58omalizumab responder status: Responder
                                        59omalizumab responder status: Responder
                                        60omalizumab responder status: Responder
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                                        61 rows \u00d7 1 columns

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                                        \n\n\n\n\n```python\nencoder = LabelEncoder()\nlabels_encoded = encoder.fit_transform(labels.values)\nlabels_encoded_noncontrols = labels_encoded[labels_encoded!=0] - 1\n\ndata_controls = gene_expression_matrix[labels_encoded==0]\ndata = gene_expression_matrix[labels_encoded!=0]\n\ngene_signature = data_controls.mean(axis=0)\ndata_scaled = data - gene_signature\n\nscaler = MinMaxScaler((0, 1))\nscaler.fit(data_scaled)\ndata_normalized = scaler.transform(data_scaled)\n\nx_train = torch.FloatTensor(data_normalized)\ny_train = torch.FloatTensor(labels_encoded_noncontrols).unsqueeze(1)\nprint(x_train.shape)\nprint(y_train.shape)\n```\n\n torch.Size([40, 5])\n torch.Size([40, 1])\n\n\n c:\\users\\pietr\\anaconda3\\envs\\deep-logic\\lib\\site-packages\\sklearn\\utils\\validation.py:63: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().\n return f(*args, **kwargs)\n\n\n\n```python\ntorch.manual_seed(0)\nnp.random.seed(0)\ndevice = torch.device('cuda:0' if torch.cuda.is_available() else 'cpu')\nx_train = x_train.to(device)\ny_train = y_train.to(device)\n\nlayers = [\n torch.nn.Linear(x_train.size(1), 10, bias=True),\n torch.nn.LeakyReLU(),\n torch.nn.Linear(10, 5, bias=True),\n torch.nn.LeakyReLU(),\n torch.nn.Linear(5, 1, bias=True),\n torch.nn.Sigmoid(),\n]\nmodel = torch.nn.Sequential(*layers).to(device)\n\noptimizer = torch.optim.Adam(model.parameters(), lr=0.01)\nmodel.train()\nneed_pruning = True\nfor epoch in range(1, 3001):\n # forward pass\n optimizer.zero_grad()\n y_pred = model(x_train)\n # Compute Loss\n loss = torch.nn.functional.binary_cross_entropy(y_pred, y_train)\n\n for module in model.children():\n if isinstance(module, torch.nn.Linear):\n loss += 0.0 * torch.norm(module.weight, 1)\n\n # backward pass\n loss.backward()\n optimizer.step()\n\n # compute accuracy\n if epoch % 1000 == 0:\n y_pred_d = (y_pred > 0.5)\n accuracy = (y_pred_d.eq(y_train).sum(dim=1) == y_train.size(1)).sum().item() / y_train.size(0)\n print(f'Epoch {epoch}: train accuracy: {accuracy:.4f}')\n \n if epoch > 8000 and need_pruning and epoch % 3000 == 0:\n prune_features(model, 1, device)\n need_pruning = True\n```\n\n Epoch 1000: train accuracy: 1.0000\n Epoch 2000: train accuracy: 1.0000\n Epoch 3000: train accuracy: 1.0000\n\n\n## Local explanations\n\n\n```python\nnp.set_printoptions(precision=2, suppress=True)\noutputs = []\nfor i, (xin, yin) in enumerate(zip(x_train, y_train)):\n model_reduced = get_reduced_model(model, xin.to(device), bias=False).to(device)\n for module in model_reduced.children():\n if isinstance(module, torch.nn.Linear):\n wa = module.weight.cpu().detach().numpy()\n break\n output = model_reduced(xin)\n \n pred_class = torch.argmax(output)\n true_class = torch.argmax(y_train[i])\n\n # generate local explanation only if the prediction is correct\n if pred_class.eq(true_class):\n local_explanation = logic.relu_nn.explain_local(model.to(device), x_train, y_train, xin, yin, device=device)\n print(f'Input {(i+1)}')\n print(f'\\tx={xin.cpu().detach().numpy()}')\n print(f'\\ty={y_train[i].cpu().detach().numpy()}')\n print(f'\\ty={output.cpu().detach().numpy()}')\n #print(f'\\tw={wa}')\n print(f'\\tExplanation: {local_explanation}')\n print()\n outputs.append(output)\n if i > 1:\n break\n```\n\n Input 1\n \tx=[1. 0. 0.06 0.55 0.07]\n \ty=[0.]\n \ty=[0.]\n \tExplanation: feature0000000000 & ~feature0000000001 & ~feature0000000002 & feature0000000003 & ~feature0000000004\n \n Input 2\n \tx=[0.13 0.89 0.35 0.49 0.47]\n \ty=[0.]\n \ty=[0.]\n \tExplanation: ~feature0000000000 & feature0000000001 & ~feature0000000002 & ~feature0000000003 & ~feature0000000004\n \n Input 3\n \tx=[0.72 0.38 0. 0.69 0. ]\n \ty=[0.]\n \ty=[0.]\n \tExplanation: feature0000000000 & ~feature0000000001 & ~feature0000000002 & feature0000000003 & ~feature0000000004\n \n\n\n# Combine local explanations\n\n\n```python\nglobal_explanation, predictions, counter = logic.combine_local_explanations(model, x=x_train, y=y_train, \n target_class=0,\n topk_explanations=10, device=device)\n\nynp = y_train.cpu().detach().numpy()[:, 0]\naccuracy = np.sum(predictions == ynp) / len(ynp)\nprint(f'Accuracy of when using the formula \"{global_explanation}\": {accuracy:.4f}')\n```\n\n Accuracy of when using the formula \"(~feature0000000000 & ~feature0000000001 & ~feature0000000002) | (feature0000000000 & feature0000000004 & ~feature0000000002 & ~feature0000000003) | (feature0000000003 & feature0000000004 & ~feature0000000000 & ~feature0000000001) | (feature0000000003 & feature0000000004 & ~feature0000000000 & ~feature0000000002) | (feature0000000003 & ~feature0000000001 & ~feature0000000002 & ~feature0000000004) | (~feature0000000000 & ~feature0000000002 & ~feature0000000003 & ~feature0000000004)\": 0.7750\n\n\n\n```python\nglobal_explanation = logic.relu_nn.explain_global(model, n_classes=1, target_class=0, device=device)\nexplanation = logic.relu_nn.explain_global(model, n_classes=1, target_class=0, device=device)\nif explanation not in ['False', 'True', 'The formula is too complex!']:\n accuracy, _ = logic.relu_nn.test_explanation(explanation, target_class=0, x=x_train.cpu(), y=y_train.cpu())\n print(f'Class {0} - Global explanation: \"{global_explanation}\" - Accuracy: {accuracy:.4f}')\n```\n\n Class 0 - Global explanation: \"(feature0000000004 & ~feature0000000000 & ~feature0000000002) | (~feature0000000000 & ~feature0000000001 & ~feature0000000002) | (~feature0000000000 & ~feature0000000002 & ~feature0000000003) | (feature0000000004 & ~feature0000000001 & ~feature0000000002 & ~feature0000000003)\" - Accuracy: 0.9250\n\n\n\n```python\nw, b = collect_parameters(model, device)\nfeature_weights = w[0]\nfeature_used_bool = np.sum(np.abs(feature_weights), axis=0) > 0\nfeature_used = np.nonzero(feature_used_bool)[0]\ngenes.iloc[feature_used]\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        0
                                        0ILMN_3286286
                                        1ILMN_1775520
                                        2ILMN_1656849
                                        3ILMN_1781198
                                        4ILMN_1665457
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                                        \n\n\n\nILMN_3286286, ILMN_1775520, ILMN_1656849, ILMN_1781198, ILMN_1665457\n\n\n```python\nsum(y_train == 0).item() / len(y_train)\n```\n\n\n\n\n 0.25\n\n\n", "meta": {"hexsha": "b01ecfbd23fa97918ab93cfbb8c892a442fd4261", "size": 20441, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/example_03_omalizumab_nobias_reduced.ipynb", "max_stars_repo_name": "pietrobarbiero/logic_explained_networks", "max_stars_repo_head_hexsha": "238f2a220ae8fc4f31ab0cf12649603aba0285d5", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 18, "max_stars_repo_stars_event_min_datetime": "2021-05-24T07:47:57.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-05T14:48:39.000Z", "max_issues_repo_path": "examples/example_03_omalizumab_nobias_reduced.ipynb", "max_issues_repo_name": "pietrobarbiero/logic_explained_networks", "max_issues_repo_head_hexsha": "238f2a220ae8fc4f31ab0cf12649603aba0285d5", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-08-25T16:33:10.000Z", "max_issues_repo_issues_event_max_datetime": "2021-08-25T16:33:10.000Z", "max_forks_repo_path": "examples/example_03_omalizumab_nobias_reduced.ipynb", "max_forks_repo_name": "pietrobarbiero/deep-logic", "max_forks_repo_head_hexsha": "238f2a220ae8fc4f31ab0cf12649603aba0285d5", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2021-05-26T08:15:14.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-23T18:58:16.000Z", "avg_line_length": 30.6461769115, "max_line_length": 542, "alphanum_fraction": 0.4571204931, "converted": true, "num_tokens": 3609, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.6791786926816161, "lm_q1q2_score": 0.4277294945410075}} {"text": "```python\nfrom IPython.display import Image\nImage('../../Python_probability_statistics_machine_learning_2E.png',width=200)\n```\n\n\n\n\n\n\n\n## Boosting Trees\n\nTo understand additive modeling using trees, recall the\nGram-Schmidt orthogonalization procedure for vectors. The\npurpose of this orthogonalization procedure is to create an\northogonal set of vectors starting with a given vector\n$\\mathbf{u}_1$. We have already discussed the projection\noperator in the section [ch:prob:sec:projection](#ch:prob:sec:projection). The\nGram-Schmidt orthogonalization procedure starts with a\nvector $\\mathbf{v}_1$, which we define as the following:\n\n$$\n\\mathbf{u}_1 = \\mathbf{v}_1\n$$\n\n with the corresponding projection operator $proj_{\\mathbf{u}_1}$. The\nnext step in the procedure is to remove the residual of $\\mathbf{u}_1$\nfrom $\\mathbf{v}_2$, as in the following:\n\n$$\n\\mathbf{u}_2 = \\mathbf{v}_2 - proj_{\\mathbf{u}_1}(\\mathbf{v}_2)\n$$\n\n This procedure continues for $\\mathbf{v}_3$ as in the following:\n\n$$\n\\mathbf{u}_3 = \\mathbf{v}_3 - proj_{\\mathbf{u}_1}(\\mathbf{v}_3) - proj_{\\mathbf{u}_2}(\\mathbf{v}_3)\n$$\n\n and so on. The important aspect of this procedure is that new\nincoming vectors (i.e., $\\mathbf{v}_k$) are stripped of any pre-existing\ncomponents already present in the set of $\\lbrace\\mathbf{u}_1,\\mathbf{u}_2,\\ldots,\\mathbf{u}_M \\rbrace$.\n\nNote that this procedure is sequential. That is, the *order*\nof the incoming $\\mathbf{v}_i$ matters [^rotation]. Thus,\nany new vector can be expressed using the so-constructed\n$\\lbrace\\mathbf{u}_1,\\mathbf{u}_2,\\ldots,\\mathbf{u}_M\n\\rbrace$ basis set, as in the following:\n\n[^rotation]: At least up to a rotation of the resulting\northonormal basis.\n\n$$\n\\mathbf{x} = \\sum \\alpha_i \\mathbf{u}_i\n$$\n\n The idea behind additive trees is to reproduce this procedure for\ntrees instead of vectors. There are many natural topological and algebraic\nproperties that we lack for the general problem, however. For example, we already\nhave well established methods for measuring distances between vectors for\nthe Gram-Schmidt procedure outlined above (namely, the $L_2$\ndistance), which we lack here. Thus, we need the concept of *loss function*,\nwhich is a way of measuring how well the process is working out at each\nsequential step. This loss function is parameterized by the training data and\nby the classification function under consideration: $ L_{\\mathbf{y}}(f(x)) $.\nFor example, if we want a classifier ($f$) that selects the label $y_i$\nbased upon the input data $\\mathbf{x}_i$ ($f :\\mathbf{x}_i \\rightarrow y_i)$,\nthen the squared error loss function would be the following:\n\n$$\nL_{\\mathbf{y}}(f(x)) = \\sum_i (y_i - f(x_i))^2\n$$\n\n We represent the classifier in terms of a set of basis trees:\n\n$$\nf(x) = \\sum_k \\alpha_k u_{\\mathbf{x}}(\\theta_k)\n$$\n\n The general algorithm for forward stagewise additive modeling is \nthe following:\n\n * Initialize $f(x)=0$\n\n * For $m=1$ to $m=M$, compute the following:\n\n$$\n(\\beta_m,\\gamma_m) = \\argmin_{\\beta,\\gamma} \\sum_i L(y_i,f_{m-1}(x_i)+\\beta b(x_i;\\gamma))\n$$\n\n* Set $f_m(x) = f_{m-1}(x) +\\beta_m b(x;\\gamma_m) $\n\nThe key point is that the residuals from the prior step are used to fit the\nbasis function for the subsequent iteration. That is, the following \nequation is being sequentially approximated.\n\n$$\nf_m(x) - f_{m-1}(x) =\\beta_m b(x_i;\\gamma_m)\n$$\n\n Let's see how this works for decision trees and the exponential loss\nfunction.\n\n$$\nL(x,f(x)) = \\exp(-y f(x))\n$$\n\n Recall that for the classification problem, $y\\in \\lbrace -1,1 \\rbrace$. For \nAdaBoost, the basis functions are the individual classifiers, $G_m(x)\\mapsto \\lbrace -1,1 \\rbrace$\nThe key step in the algorithm is the minimization step for\nthe objective function\n\n$$\nJ(\\beta,G) = \\sum_i \\exp(y_i(f_{m-1}(x_i)+\\beta G(x_i)))\n$$\n\n$$\n(\\beta_m,G_m) = \\argmin_{\\beta,G} \\sum_i \\exp(y_i(f_{m-1}(x_i)+\\beta G(x_i)))\n$$\n\n Now, because of the exponential, we can factor out the following:\n\n$$\nw_i^{(m)} = \\exp(y_i f_{m-1}(x_i))\n$$\n\n as a weight on each data element and re-write the objective function as the following:\n\n$$\nJ(\\beta,G) = \\sum_i w_i^{(m)} \\exp(y_i \\beta G(x_i))\n$$\n\n The important observation here is that $y_i G(x_i)\\mapsto 1$ if\nthe tree classifies $x_i$ correctly and $y_i G(x_i)\\mapsto -1$ otherwise. \nThus, the above sum has terms like the following:\n\n$$\nJ(\\beta,G) = \\sum_{y_i\\ne G(x_i)} w_i^{(m)} \\exp(-\\beta) + \\sum_{y_i= G(x_i)} w_i^{(m)}\\exp(\\beta)\n$$\n\n For $\\beta>0$, this means that the best $G(x)$ is\nthe one that incorrectly classifies for the largest weights. Thus, \nthe minimizer is the following:\n\n$$\nG_m = \\argmin_G \\sum_i w_i^{(m)} I(y_i \\ne G(x_i))\n$$\n\n where $I$ is the indicator function (i.e.,\n$I(\\texttt{True})=1,I(\\texttt{False})=0$). \n\n For $\\beta>0$, we can re-write the objective function\nas the following:\n\n$$\nJ= (\\exp(\\beta)-\\exp(-\\beta))\\sum_i w_i^{(m)} I(y_i \\ne G(x_i)) + \\exp(-\\beta) \\sum_i w_i^{(m)}\n$$\n\n and substitute $\\theta=\\exp(-\\beta)$ so that\n\n\n
                                        \n\n$$\n\\begin{equation}\n\\frac{J}{\\sum_i w_i^{(m)}} = \\left(\\frac{1}{\\theta} - \\theta\\right)\\epsilon_m+\\theta\n\\label{eq:bt001} \\tag{1}\n\\end{equation}\n$$\n\n where\n\n$$\n\\epsilon_m = \\frac{\\sum_i w_i^{(m)}I(y_i\\ne G(x_i))}{\\sum_i w_i^{(m)}}\n$$\n\n is the error rate of the classifier with $0\\le \\epsilon_m\\le 1$. Now,\nfinding $\\beta$ is a straight-forward calculus minimization exercise on the\nright side of Equation ([1](#eq:bt001)), which gives the following:\n\n$$\n\\beta_m = \\frac{1}{2}\\log \\frac{1-\\epsilon_m}{\\epsilon_m}\n$$\n\n Importantly, $\\beta_m$ can become negative if\n$\\epsilon_m<\\frac{1}{2}$, which would would violate our assumptions on $\\beta$.\nThis is captured in the requirement that the base learner be better than just\nrandom guessing, which would correspond to $\\epsilon_m>\\frac{1}{2}$.\nPractically speaking, this means that boosting cannot fix a base learner that\nis no better than a random guess. Formally speaking, this is known as the\n*empirical weak learning assumption* [[schapire2012boosting]](#schapire2012boosting). \n\nNow we can move to the iterative weight update. Recall that\n\n$$\nw_i^{(m+1)} = \\exp(y_i f_{m}(x_i)) = w_i^{(m)}\\exp(y_i \\beta_m G_m(x_i))\n$$\n\n which we can rewrite as the following:\n\n$$\nw_i^{(m+1)} = w_i^{(m)}\\exp(\\beta_m)\\exp(-2\\beta_m I(G_m(x_i)=y_i))\n$$\n\n This means that the data elements that are incorrectly classified have their\ncorresponding weights increased by $\\exp(\\beta_m)$ and those that are correctly\nclassified have their corresponding weights reduced by $\\exp(-\\beta_m)$.\nThe reason for the choice of the exponential loss function comes from the\nfollowing:\n\n$$\nf^{*}(x) = \\argmin_{f(x)} \\mathbb{E}_{Y\\vert x}(\\exp(-Y f(x))) = \\frac{1}{2}\\log \\frac{\\mathbb{P}(Y=1\\vert x)}{\\mathbb{P}(Y=\\minus 1\\vert x)}\n$$\n\n This means that boosting is approximating a $f(x)$ that is actually\nhalf the log-odds of the conditional class probabilities. This can be\nrearranged as the following\n\n$$\n\\mathbb{P}(Y=1\\vert x) = \\frac{1}{1+\\exp(\\minus 2f^{*}(x))}\n$$\n\n The important benefit of this general formulation for boosting, as a sequence\nof additive approximations, is that it opens the door to other choices of loss\nfunction, especially loss functions that are based on robust statistics that\ncan account for errors in the training data (c.f. Hastie).\n\n### Gradient Boosting\n\nGiven a differentiable loss function, the optimization process can be\nformulated using numerical gradients. The fundamental idea is to treat the\n$f(x_i)$ as a scalar parameter to be optimized over. Generally speaking,\nwe can think of the following loss function,\n\n$$\nL(f) = \\sum_{i=1}^N L(y_i,f(x_i))\n$$\n\n as a vectorized quantity\n\n$$\n\\mathbf{f} = \\lbrace f(x_1),f(x_2),\\ldots, f(x_N) \\rbrace\n$$\n\n so that the optimization is over this vector\n\n$$\n\\hat{\\mathbf{f}} = \\argmin_{\\mathbf{f}} L(\\mathbf{f})\n$$\n\n With this general formulation we can use numerical optimization methods \nto solve for the optimal $\\mathbf{f}$ as a sum of component vectors\nas in the following:\n\n$$\n\\mathbf{f}_M = \\sum_{m=0}^M \\mathbf{h}_m\n$$\n\n Note that this leaves aside the prior assumption that $f$ \nis parameterized as a sum of individual decision trees.\n\n$$\ng_{i,m} = \\left[ \\frac{\\partial L(y_i,f(x_i))}{\\partial f(x_i)} \\right]_{f(x_i)=f_{m-1}(x_i)}\n$$\n", "meta": {"hexsha": "27831c5cab6794944e211ef4f3c339c4789ef124", "size": 192061, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "chapter/machine_learning/boosting_trees.ipynb", "max_stars_repo_name": "derakding/Python-for-Probability-Statistics-and-Machine-Learning-2E", "max_stars_repo_head_hexsha": "9d12a298d43ae285d9549a79bb5544cf0a9b7516", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 224, "max_stars_repo_stars_event_min_datetime": "2019-05-07T08:56:01.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-25T15:50:41.000Z", "max_issues_repo_path": "chapter/machine_learning/boosting_trees.ipynb", "max_issues_repo_name": "derakding/Python-for-Probability-Statistics-and-Machine-Learning-2E", "max_issues_repo_head_hexsha": "9d12a298d43ae285d9549a79bb5544cf0a9b7516", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 9, "max_issues_repo_issues_event_min_datetime": "2019-08-27T12:57:17.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-21T15:45:13.000Z", "max_forks_repo_path": "chapter/machine_learning/boosting_trees.ipynb", "max_forks_repo_name": "derakding/Python-for-Probability-Statistics-and-Machine-Learning-2E", "max_forks_repo_head_hexsha": "9d12a298d43ae285d9549a79bb5544cf0a9b7516", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 73, "max_forks_repo_forks_event_min_datetime": "2019-05-25T07:15:47.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-07T00:22:37.000Z", "avg_line_length": 326.6343537415, "max_line_length": 176652, "alphanum_fraction": 0.9325995387, "converted": true, "num_tokens": 2618, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.7931059609645723, "lm_q1q2_score": 0.42747080518728586}} {"text": "\n\n# Deep Learning, uncertainty and probabilistic models.\n\n\n> **Certainty** is perfect knowledge that has total security from error, or the mental state of being without doubt. (*Wikipedia*)\n\n> **Uncertainty** has been called *an unintelligible expression without a straightforward description*. It describes a situation involving insecurity and/or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. (*Wikipedia*)\n\nA common understanding of the term **uncertainty** in scientific works is that of a measure that reflects the amount of dispersion of a random variable. In other words, it is a scalar measure of how \"random\" a random variable is. But this a reduction of the term. \n\n> There is **no single formula for uncertainty** because there are many different ways to measure dispersion: standard deviation, variance, and entropy are all appropriate measures. However, it's important to keep in mind that a single scalar number cannot paint a full picture of \"randomness'', as that would require communicating the entire random variable itself! *Eric Jang*.\n\nWhen working with predictive systems, measuring uncertainty is **important for dealing with the risk associated to decisions**. If we can measure risk, we can define policies related to making use of predictions.\n\nRegarding uncertainty, we can be in several **states**:\n\n+ Complete certainty. This is the case of macroscopic mechanics, where given an input there is no uncertainty about the output.\n+ Uncertainty with risk. This is the case of a game with an uncertain output, but where we exactly know the probability distribution over outcomes.\n+ Fully reducible uncertainty. This is the case of a game where by gathering sufficient data and/or using the right model we can get complete certainty.\n+ Partially reducible uncertainty. This is the case of games where we have no full knowledge about the probability distribution over outcomes. By gathering sufficient data we can know the probability distribution over outcomes.\n+ Etc.\n\nUncertainty in predictive systems $y = f_w(x)$ can be **caused** by several factors (see next). Different types of uncertainty must be measured in different ways. \n\n## Epistemic uncertainty\n\nEpistemic uncertainty corresponds to the uncertainty originated by the lack of knowledge about the model. It can be explained as to which extent is our model able to describe the distribution that generated the data. \n\nIn this case we can talk of two different types of uncertainties caused by whether the model has been **trained with enough data**, so it\u2019s been able to learn the full distributionof the data, or whether the **expressiveness of the model is able to capture the complexity of the distribution**. \n\nWhen using a expressive enough model, this type of uncertainty *can be reduced* by providing more samples during the training phase.\n\n## Aleatoric uncertainty\n\n In this case the uncertainty measured belongs to the data. This uncertainty is inherent to the data and *can\u2019t be reduced* by adding more data to the training process. Its cause can be the lack of information in $x$ to determine $y$ or the lack of precision when measuring $x$ or $y$.\n\n**Aleatoric uncertainty propagates from the inputs to the model predictions**. Consider a simple model $y=5x$, which takes in normally-distributed input $x \\sim N(0,1)$. In this case, $y \\sim N(0,5)$, so the aleatoric uncertainty of the predictive distribution can be described by $\\sigma=5$. \n\nThere are two types of aleatoric uncertainty: \n\n+ **Homoscedastic**: Uncertainty remains constant for all the data. For example: $y \\sim N(0,5)$.\n+ **Heteroscedastic**: Uncertainty depends on the input. For example: $y \\sim N(0,f(x))$. \n \n## Out of Distribution (OoD) \n\nDetermining whether inputs are \"valid'' is a serious problem for deploying ML in the wild, and is known as the Out of Distribution (OoD) problem. \n\nThere are two ways to handle OoD inputs for a machine learning model: 1) Model the bad inputs and detect them before we even put them through the model 2) Let the \"weirdness'' of model predictions imply to us that the input was probably malformed.\n\nModeling bad inputs is a difficult task. Training a discriminator is not completely robust because it can give arbitrary predictions for an input that lies in neither distribution. Instead of a discriminator, we could build a density model of the in-distribution data, such as a kernel density estimator.\n\nThe second approach to OoD detection involves using the predictive (epistemic) uncertainty of the task model to tell us when inputs are OoD. Ideally, malformed inputs to a model ought to generate \"weird'' predictive distribution $p(y|x)$. For instance, it has been showed that the maximum softmax probability (the predicted class) for OoD inputs tends to be lower than that of in-distribution inputs. \n\n## Example\n\nUncertainty can be understood from a **simple machine learning model** that attempts to predict daily rainfall from a sequence of barometer readings. Aleatoric uncertainty is irreducible randomness that arises from the data collection process. Epistemic uncertainty reflects confidence that our model is making the correct predictions. Finally, out-of-distribution errors arise when the model sees an input that differs from its training data (e.g. temperature of the sun, other anomalies). *Eric Jang*.\n\n\n\n## Calibration\n\nJust because a model is able to output a \"confidence interval\" for a prediction doesn't mean that the confidence interval actually reflects the actual probabilities of outcomes in reality! \n\nSuppose our rainfall model tells us that there will be $N(4,1)$ inches of rain today. If our model is **calibrated**, then if we were to repeat this experiment over and over again under identical conditions (possibly re-training the model each time), we really would observe empirical rainfall to be distributed exactly $N(4,1)$.\n\nMachine Learning models mostly optimize for test accuracy or some fitness function. Researchers are not performing model selection by deploying the model in repeated identical experiments and measuring calibration error, so unsurprisingly, this **models tend to be poorly calibrated**.\n\n\n\n## Uncertainty and Neural Networks\n\nA trained neural network $f$ can be viewed as the instantiation of a specific probabilistic model $p(y|x,D)$. \n\nFor classification, $y$ is a set of classes and $p(y|x,D)$ is a **categorical distribution**. \n\nFor regression, $y$ is a continuous variable and $p(y|x,D)$ can be modelled with a **Gaussian (or Laplacian) distribution**. \n\nOur training goal can be to **find the most probable network instance** (represented by parameters $w$) that generated the outputs:\n\n$$\np(y | x, D) = \\int_w p(y | x, w)p(w|D)dw\n$$\n\nWe can observe that the distribution of the output depends on two terms: one that depends on the application of the **model to the input data**, and a second one that is measuring **how the model may vary depending on the training data**. \n\nFrom this definition we can say that the first term is modeling the **aleatoric uncertainty**, as it is measuring how the output is distributed when the input is $x$, and the second term is modeling the **epistemic uncertainty** as it is measuring the uncertainty induced by the parameters of the model.\n\n## Epistemic Uncertainty \n\nIn the case of the epistemic uncertainty, let\u2019s assume that we want to train a model with parameters $w$ that produces an output $y = f_w(x)$.\n\nTo capture epistemic uncertainty in a neural network we will adopt a Bayesian point of view and we will put a prior distribution over its weights, for example a Gaussian prior distribution: $w \u223c \\mathcal{N} (0, I)$. Such a model is referred to as a **Bayesian neural network**. Bayesian neural networks replace the deterministic network\u2019s weight parameters with distributions over these parameters.\n\nIn the case of regression, we suppose the model likelihood is $p(y | x,w) \u223c \\mathcal{N} (f_w(x), \\sigma^2)$. \n\nFor classification tasks, we assume a softmax likelihood:\n\n$$\np(y = d | x, w) = \\frac{\\exp(f^d_w(x))}{\\sum_{d'}\\exp(f^{d'}_w(x))}\n$$\n\nGiven a dataset $D = (X,Y)$ we then look for the posterior distribution over the space of parameters by invoking Bayes\u2019 theorem:\n\n$$\np(w|X,Y) = \\frac{p(Y|X,w) p(w)}{p(Y|X)}\n$$\n\nThe true posterior $p(w|X, Y)$ cannot usually be evaluated analytically, but we can use indirect methods such as variational inference or MC-dropout. \n\nThis distribution captures the most probable function parameters given our observed data. With it we can predict an output for a new input point $x$ by integrating:\n\n$$\np(y|x,X,Y) = \\int p(y|x,w)p(w|X,Y)dw\n$$\n\nThis integral can be computed by Monte Carlos method:\n\n$$\n\\mathop{\\mathbb{E}}(y|x, X,Y) \\approx \\frac{1}{M}\\sum_{j=1}^{M} f_{{w_{j}}}(x)\n$$\n\nFinaly we can calculate the corresponding variance:\n$$\n\\mathop{\\mathbb{Var}} (y) \\approx \\frac{1}{M}\\sum_{j=1}^{M} f_{w_{j}}(x)^2 - \\mathop{\\mathbb{E}}(y|x, X,Y)^2\n$$\n\n\nWhich can be used as a measure of the **epistemic uncertainty**.\n\n## Aleatoric Uncertainty\n\nFor obtaining the heteroscedastic uncertainty, we follow the same approach as in the epistemic case: we consider that we have a **fixed model** $f_w(x)$ and we want to observe the variability of the term $p(y|f_w(x))$. \n\nIn **regression tasks**, , we can assume that $y \\sim \\mathcal{N}(f_{{w}}(x), \\sigma(x)^2)$, where $f_{w}(x)$ is the predicted value of $x$ and $\\sigma_(x)$ is the unknown variance of this value. How do we estimate this value? \n\nIn the same way we use a deep learning model to estimate $f_w(x)$, we can use a deep learning model to estimate variance: $\\sigma_w(x)$, and then $y \\sim \\mathcal{N}(f_{{w}}(x), \\sigma_w(x)^2)$.\n\nApplying this approximation to the log-likelihood adds an additional term to the loss that depends on $\\sigma_w(x)$:\n\n$$\n\\mathcal L = - \\frac{1}{N} \\sum_{i=1}^N \\log p(y|f_w(x_i))\n$$\n\n$$\n= \\frac{1}{2 \\sigma(x)^2} ( y - f_w(x))^2 + \\frac{1}{2} \\log \\sigma_w(x)^2\n$$\n\nWe can easily optimiza a model that outputs $(f_{{w}}(x), \\sigma_w(x))$ by using the reparametrization trick in the last layer of the network.\n\nIn **classification tasks**, this approximation is not as straightforward as in regression, as for classification there is already an uncertainty component due to the distribution of the probabilities when applying the softmax to the logits. \n\nIn this scenario, we can apply the same assumption but to the logits space instead of the output directly:\n\n$$\\begin{split}\nlogits \\sim \\mathcal{N}(f_{w}(x), diag(\\sigma_w(x)^2)) \\\\\np = softmax(logits) \\\\\ny \\sim Categorical(p)\n\\end{split}\n$$\n\nHere we can apply the reparametrization trick for computing the logits, $u$:\n\n$$\n\\begin{split}\nu \\sim \\mathcal{N}(f_{w}(x), diag(\\sigma_w(x)^2)) \\\\\nu = f_{w}(x) + \\sqrt{diag(\\sigma_w(x)^2)}* \\epsilon\\\\\nu = f_{w}(x) + diag(\\sigma_w(x))* \\epsilon, \\epsilon\\sim\\mathcal{N}(0,1)\n\\end{split}\n$$\n\nAnd then apply Monte Carlo sampling to obtain the expectation of the probability:\n\n$$\n\\mathop{\\mathbb{E}}[p] = \\frac{1}{M}\\sum^{M}_{m=1}softmax(u^{m})\n$$\n\nWhen applied to a crossentropy loss we have that:\n\n$$\n\\begin{align}\n\\ell(W) & = -\\frac{1} {C}\\sum^{C}_{c=1}y_{i,c}\\log{(p_{i,c})} \\\\\n& = -\\frac{1}{C}\\sum^{C}_{c=1}y_{i,c}\\log{\\frac{1}{M}\\sum^{M}_{m=1}softmax(u^{m})}\\\\\n& = -\\frac{1}{C}\\sum^{C}_{c=1}y_{i,c}\\log{\\frac{1}{M}\\sum^{M}_{m=1}\\frac{\\exp{u^{m}_{i,c,m}}}{\\sum^{C}_{c{}'=1}\\exp{u^{m}_{i,c{}',m}}}} \\\\\n& = -\\frac{1}{C}\\sum^{C}_{c=1}y_{i,c}\\log{\\sum^{M}_{m=1}\\exp({u^{m}_{i,c,m}}} - \\log{\\sum^{C}_{c'=1}\\exp{u^{m}_{i,c',m}})}-\\log{M} \\\\\n\\end{align} \n$$\n\nWhere $C$ is the number of classes and $M$ is the number of Monte Carlo samples taken.\n\nOnce trained, the sigmas can be used to obtain a measure of the aleatoric uncertainty associated to the input by calculating the mean variance of the logits:\n\n$$\n\\begin{align}\n\\mathop{\\mathbb{U}} & = \\frac{1}{C}\\sum^{C}_{c=1}var[p_{c}] \\\\\n&= \\frac{1}{C}\\sum^{C}_{c=1}var(softmax(u^{m}_{c})), m \\in \\left \\{1,M\\right \\}\\end{align} \n$$\n\n\n## Total uncertainty\n\nEpistemic and aleatoric uncertinty can be combined in a model. Then, the predictive uncertainty for $y$ can be approaximated by:\n\n$$\nVar(y) \\approx \\frac{1}{T} \\sum_{t=1}^{T} y_t^2 - (\\frac{1}{T} \\sum_{t=1}^{T} y)^2 + \\frac{1}{T} \\sum_{t=1}^{T} \\sigma_t(x)^2\n$$\n\nwith ${y_t,\\sigma_t(x)}$ a set of $T$ sampled outputs: $y_t, \\sigma_t (x) = f_{w_t} (x)$ for random weights $w_t \\sim p(w)$. \n\n\n# Predicting probability distributions\n\n#### Regression with a pre-defined homoscedastic noise model\n\nA network can now be trained with a Gaussian negative log likelihood function (`neg_log_likelihood`) as loss function assuming a **fixed standard deviation** (`noise`). \n\n\nThis is equivalent to consider the following loss function:\n\n$$\nLogLoss = \\sum_i \\frac{(y_i - f(x_i))^2}{2 \\sigma^2}+\\frac{1}{2} \\log \\sigma^2\n$$\n\nwhere the model predicts a mean $f(x_i)$. \n\n\n\n\n```\nimport tensorflow\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n%matplotlib inline\n\ndef f(x, sigma):\n epsilon = np.random.randn(*x.shape) * sigma\n return 10 * np.sin(2 * np.pi * (x)) + epsilon\n\ntrain_size = 320\nnoise = 1.0\n\nplt.figure(figsize=(8,4))\nX = np.linspace(-0.8, 0.8, train_size).reshape(-1, 1)\ny = f(X, sigma=noise)\ny_true = f(X, sigma=0.0)\n\nplt.scatter(X, y, marker='+', label='Training data')\nplt.plot(X, y_true, label='Truth', color='r')\nplt.title('Noisy training data and ground truth')\nplt.legend();\n\nprint(X[0],y[0],y_true[0])\n```\n\n\n```\nfrom tensorflow.keras import backend as K\nfrom tensorflow.keras import activations, initializers\nfrom tensorflow.keras.layers import Input, Dense\nfrom tensorflow.keras.models import Model\nfrom tensorflow.keras import callbacks, optimizers\nimport tensorflow as tf\nimport tensorflow_probability as tfp\n\nx_in = Input(shape=(1,))\nx = Dense(20, activation='relu')(x_in)\nx = Dense(20, activation='relu')(x)\nx = Dense(1)(x)\n\nmodel = Model(x_in, x)\nmodel.summary()\n```\n\n \n WARNING: The TensorFlow contrib module will not be included in TensorFlow 2.0.\n For more information, please see:\n * https://github.com/tensorflow/community/blob/master/rfcs/20180907-contrib-sunset.md\n * https://github.com/tensorflow/addons\n If you depend on functionality not listed there, please file an issue.\n \n WARNING:tensorflow:From /usr/local/lib/python2.7/dist-packages/tensorflow/python/ops/resource_variable_ops.py:435: colocate_with (from tensorflow.python.framework.ops) is deprecated and will be removed in a future version.\n Instructions for updating:\n Colocations handled automatically by placer.\n _________________________________________________________________\n Layer (type) Output Shape Param # \n =================================================================\n input_1 (InputLayer) (None, 1) 0 \n _________________________________________________________________\n dense (Dense) (None, 20) 40 \n _________________________________________________________________\n dense_1 (Dense) (None, 20) 420 \n _________________________________________________________________\n dense_2 (Dense) (None, 1) 21 \n =================================================================\n Total params: 481\n Trainable params: 481\n Non-trainable params: 0\n _________________________________________________________________\n\n\n\n```\ndef neg_log_likelihood(y_true, y_pred, sigma=1.0):\n dist = tfp.distributions.Normal(loc=y_pred, scale=sigma)\n return K.sum(-dist.log_prob(y_true))\n\nmodel.compile(loss=neg_log_likelihood, optimizer=optimizers.Adam(lr=0.01), metrics=['mse'])\nmodel.fit(X, y, batch_size=10, epochs=150, verbose=0);\n```\n\n\n```\nfrom tqdm import tqdm_notebook\n\nX_test = np.linspace(-0.8, 0.8, 1000).reshape(-1, 1)\ny_pred_list = []\n\ny_pred = model.predict(X_test)\n \ny_mean = np.mean(y_pred, axis=1)\ny_sigma = noise\n\nplt.figure(figsize=(8,4))\nplt.plot(X_test, y_mean, 'r-', label='Predictive mean');\nplt.scatter(X, y, marker='+', label='Training data')\nplt.fill_between(X_test.ravel(), \n y_mean + 2 * y_sigma, \n y_mean - 2 * y_sigma, \n alpha=0.2, label='Uncertainty (Confidence Interval)')\nplt.title('Prediction')\nplt.legend();\n```\n\n#### Regression with an heteroscedastic noise model\n\nThis is equivalent to consider the following loss function:\n\n$$\nLogLoss = \\sum_i \\frac{(y_i - f(x_i))^2}{2 \\sigma^2(x_i)}+\\frac{1}{2} \\log \\sigma^2(x_i)\n$$\n\n\n```\n# https://engineering.taboola.com/predicting-probability-distributions/\n\nimport numpy as np\nimport pandas as pd\nimport seaborn as sns\nimport tensorflow as tf\nimport matplotlib.pyplot as plt\nfrom sklearn.utils import shuffle\ndef f(x):\n return x**2-6*x+9\n\ndef data_generator(x,sigma_0,samples):\n return np.random.normal(f(x),sigma_0*x,samples)\n\nsigma_0 = 0.1\nx_vals = np.arange(1,5.2,0.2)\nx_arr = np.array([])\ny_arr = np.array([])\nsamples = 50\nfor x in x_vals:\n x_arr = np.append(x_arr, np.full(samples,x))\n y_arr = np.append(y_arr, data_generator(x,sigma_0,samples))\nx_arr, y_arr = shuffle(x_arr, y_arr)\nx_test = np.arange(1.1,5.1,0.2)\n\nfig, ax = plt.subplots(figsize=(10,10))\nplt.grid(True)\nplt.xlabel('x')\nplt.ylabel('g(x)')\nax.scatter(x_arr,y_arr,label='sampled data')\nax.plot(x_vals,list(map(f,x_vals)),c='m',label='f(x)')\nax.legend(loc='upper center',fontsize='large',shadow=True)\nplt.show()\n```\n\n\n```\nepochs = 500\nbatch_size = 50\nlearning_rate = 0.0003\ndisplay_step = 50\nbatch_num = int(len(x_arr) / batch_size)\n\ntf.reset_default_graph()\nx = tf.placeholder(name='x',shape=(None,1),dtype=tf.float32)\ny = tf.placeholder(name='y',shape=(None,1),dtype=tf.float32)\n\nlayer = x\nfor _ in range(3):\n layer = tf.layers.dense(inputs=layer, units=12, activation=tf.nn.tanh)\n\noutput = tf.layers.dense(inputs=layer, units=1)\n\n# cot function\ncost = tf.reduce_mean(tf.losses.mean_squared_error(labels=y,predictions=output))\n\noptimizer = tf.train.AdamOptimizer(learning_rate).minimize(cost)\nx_batches = np.array_split(x_arr, batch_num)\ny_batches = np.array_split(y_arr, batch_num)\n\nwith tf.Session() as sess:\n sess.run(tf.global_variables_initializer())\n for epoch in range(epochs):\n avg_cost = 0.0\n x_batches, y_batches = shuffle(x_batches, y_batches)\n for i in range(batch_num):\n x_batch = np.expand_dims(x_batches[i],axis=1)\n y_batch = np.expand_dims(y_batches[i],axis=1)\n _, c = sess.run([optimizer,cost], feed_dict={x:x_batch, y:y_batch})\n avg_cost += c/batch_num\n if epoch % display_step == 0:\n print('Epoch {0} | cost = {1:.4f}'.format(epoch,avg_cost))\n y_pred = sess.run(output,feed_dict={x:np.expand_dims(x_test,axis=1)})\n print('Final cost: {0:.4f}'.format(avg_cost))\n\nfig, ax = plt.subplots(figsize=(10,10))\nplt.grid(True)\nplt.xlabel('x')\nplt.ylabel('y')\nax.scatter(x_arr,y_arr,c='b',label='sampled data')\nax.scatter(x_test,y_pred,c='r',label='predicted values')\nax.plot(x_vals,list(map(f,x_vals)),c='m',label='f(x)')\nax.legend(loc='upper center',fontsize='large',shadow=True)\nplt.show()\n```\n\n\n```\n\n#new cost function\ndef mdn_cost(mu, sigma, y):\n dist = tf.distributions.Normal(loc=mu, scale=sigma)\n return tf.reduce_mean(-dist.log_prob(y))\n\nepochs = 500\nbatch_size = 50\nlearning_rate = 0.0003\ndisplay_step = 50\nbatch_num = int(len(x_arr) / batch_size)\n\ntf.reset_default_graph()\nx = tf.placeholder(name='x',shape=(None,1),dtype=tf.float32)\ny = tf.placeholder(name='y',shape=(None,1),dtype=tf.float32)\n\nlayer = x\nfor _ in range(3):\n layer = tf.layers.dense(inputs=layer, units=12, activation=tf.nn.tanh)\n \n# now we have two different outputs\nmu = tf.layers.dense(inputs=layer, units=1)\nsigma = tf.layers.dense(inputs=layer, units=1, activation=lambda x: tf.nn.elu(x) + 1)\n\ncost = mdn_cost(mu, sigma, y)\n\noptimizer = tf.train.AdamOptimizer(learning_rate).minimize(cost)\nx_batches = np.array_split(x_arr, batch_num)\ny_batches = np.array_split(y_arr, batch_num)\nwith tf.Session() as sess:\n sess.run(tf.global_variables_initializer())\n for epoch in range(epochs):\n avg_cost = 0.0\n x_batches, y_batches = shuffle(x_batches, y_batches)\n for i in range(batch_num):\n x_batch = np.expand_dims(x_batches[i],axis=1)\n y_batch = np.expand_dims(y_batches[i],axis=1)\n _, c = sess.run([optimizer,cost], feed_dict={x:x_batch, y:y_batch})\n avg_cost += c/batch_num\n if epoch % display_step == 0:\n print('Epoch {0} | cost = {1:.4f}'.format(epoch,avg_cost))\n mu_pred, sigma_pred = sess.run([mu,sigma],feed_dict={x:np.expand_dims(x_test,axis=1)})\n print('Final cost: {0:.4f}'.format(avg_cost))\n\nfig, ax = plt.subplots(figsize=(10,10))\nplt.grid(True)\nplt.xlabel('x')\nplt.ylabel('y')\nax.errorbar(x_test,mu_pred,yerr=np.absolute(sigma_pred),c='r',ls='None',marker='.',ms=10,label='predicted distributions')\nax.scatter(x_arr,y_arr,c='b',alpha=0.05,label='sampled data')\nax.errorbar(x_vals,list(map(f,x_vals)),yerr=list(map(lambda x: sigma_0*x,x_vals)),c='b',lw=2,ls='None',marker='.',ms=10,label='true distributions')\nax.plot(x_vals,list(map(f,x_vals)),c='m',label='f(x)')\nax.legend(loc='upper center',fontsize='large',shadow=True)\nplt.show()\n\n```\n\n## Epistemic and Total Uncertainty\n\n##### Copyright 2019 The TensorFlow Authors.\n\nLicensed under the Apache License, Version 2.0 (the \"License\");\n\n\n```\n#@title Licensed under the Apache License, Version 2.0 (the \"License\"); { display-mode: \"form\" }\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n# https://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\n```\n\n# TFP Probabilistic Layers: Regression\n\n\n \n \n
                                        \n Run in Google Colab\n \n View source on GitHub\n
                                        \n\nIn this example we show how to fit regression models using TFP's \"probabilistic layers.\"\n\n### Dependencies & Prerequisites\n\n\n\n```\n#@title Install { display-mode: \"form\" }\nTF_Installation = 'TF2 Nightly (GPU)' #@param ['TF2 Nightly (GPU)', 'TF2 Stable (GPU)', 'TF1 Nightly (GPU)', 'TF1 Stable (GPU)','System']\n\nif TF_Installation == 'TF2 Nightly (GPU)':\n !pip install -q --upgrade tf-nightly-gpu-2.0-preview\n print('Installation of `tf-nightly-gpu-2.0-preview` complete.')\nelif TF_Installation == 'TF2 Stable (GPU)':\n !pip install -q --upgrade tensorflow-gpu==2.0.0-alpha0\n print('Installation of `tensorflow-gpu==2.0.0-alpha0` complete.')\nelif TF_Installation == 'TF1 Nightly (GPU)':\n !pip install -q --upgrade tf-nightly-gpu\n print('Installation of `tf-nightly-gpu` complete.')\nelif TF_Installation == 'TF1 Stable (GPU)':\n !pip install -q --upgrade tensorflow-gpu\n print('Installation of `tensorflow-gpu` complete.')\nelif TF_Installation == 'System':\n pass\nelse:\n raise ValueError('Selection Error: Please select a valid '\n 'installation option.')\n```\n\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 349.2MB 52kB/s \n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 3.1MB 38.7MB/s \n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 61kB 26.3MB/s \n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 430kB 59.4MB/s \n \u001b[?25h Building wheel for wrapt (setup.py) ... \u001b[?25l\u001b[?25hdone\n \u001b[31mERROR: thinc 6.12.1 has requirement wrapt<1.11.0,>=1.10.0, but you'll have wrapt 1.11.1 which is incompatible.\u001b[0m\n Installation of `tf-nightly-gpu-2.0-preview` complete.\n\n\n\n```\n#@title Install { display-mode: \"form\" }\nTFP_Installation = \"Nightly\" #@param [\"Nightly\", \"Stable\", \"System\"]\n\nif TFP_Installation == \"Nightly\":\n !pip install -q tfp-nightly\n print(\"Installation of `tfp-nightly` complete.\")\nelif TFP_Installation == \"Stable\":\n !pip install -q --upgrade tensorflow-probability\n print(\"Installation of `tensorflow-probability` complete.\")\nelif TFP_Installation == \"System\":\n pass\nelse:\n raise ValueError(\"Selection Error: Please select a valid \"\n \"installation option.\")\n```\n\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 983kB 3.4MB/s \n \u001b[?25hInstallation of `tfp-nightly` complete.\n\n\n\n```\n#@title Import { display-mode: \"form\" }\n\nfrom __future__ import absolute_import\nfrom __future__ import division\nfrom __future__ import print_function\n\nfrom pprint import pprint\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport seaborn as sns\n\nimport tensorflow as tf\nfrom tensorflow.python import tf2\nif not tf2.enabled():\n import tensorflow.compat.v2 as tf\n tf.enable_v2_behavior()\n assert tf2.enabled()\n\nimport tensorflow_probability as tfp\n\nsns.reset_defaults()\n#sns.set_style('whitegrid')\n#sns.set_context('talk')\nsns.set_context(context='talk',font_scale=0.7)\n\n%matplotlib inline\n\ntfd = tfp.distributions\n```\n\n### Make things Fast!\n\nBefore we dive in, let's make sure we're using a GPU for this demo. \n\nTo do this, select \"Runtime\" -> \"Change runtime type\" -> \"Hardware accelerator\" -> \"GPU\".\n\nThe following snippet will verify that we have access to a GPU.\n\n\n```\nif tf.test.gpu_device_name() != '/device:GPU:0':\n print('WARNING: GPU device not found.')\nelse:\n print('SUCCESS: Found GPU: {}'.format(tf.test.gpu_device_name()))\n```\n\n SUCCESS: Found GPU: /device:GPU:0\n\n\nNote: if for some reason you cannot access a GPU, this colab will still work. (Training will just take longer.)\n\n## Motivation\n\nWouldn't it be great if we could use TFP to specify a probabilistic model then simply minimize the negative log-likelihood, i.e.,\n\n\n```\nnegloglik = lambda y, rv_y: -rv_y.log_prob(y)\n```\n\nWell not only is it possible, but this colab shows how! (In context of linear regression problems.)\n\n\n```\n#@title Synthesize dataset.\nw0 = 0.125\nb0 = 5.\nx_range = [-20, 60]\n\ndef load_dataset(n=150, n_tst=150):\n np.random.seed(43)\n def s(x):\n g = (x - x_range[0]) / (x_range[1] - x_range[0])\n return 3 * (0.25 + g**2.)\n x = (x_range[1] - x_range[0]) * np.random.rand(n) + x_range[0]\n eps = np.random.randn(n) * s(x)\n y = (w0 * x * (1. + np.sin(x)) + b0) + eps\n x = x[..., np.newaxis]\n x_tst = np.linspace(*x_range, num=n_tst).astype(np.float32)\n x_tst = x_tst[..., np.newaxis]\n return y, x, x_tst\n\ny, x, x_tst = load_dataset()\n```\n\n### Case 1: No Uncertainty\n\n\n```\n# Build model.\nmodel = tf.keras.Sequential([\n tf.keras.layers.Dense(1),\n tfp.layers.DistributionLambda(lambda t: tfd.Normal(loc=t, scale=1)),\n])\n\n# Do inference.\nmodel.compile(optimizer=tf.optimizers.Adam(learning_rate=0.01), loss=negloglik)\nmodel.fit(x, y, epochs=1000, verbose=False);\n\n# Profit.\n[print(np.squeeze(w.numpy())) for w in model.weights];\nyhat = model(x_tst)\nassert isinstance(yhat, tfd.Distribution)\n```\n\n 0.13275796\n 5.1289654\n\n\n\n```\n#@title Figure 1: No uncertainty.\nw = np.squeeze(model.layers[-2].kernel.numpy())\nb = np.squeeze(model.layers[-2].bias.numpy())\n\nplt.figure(figsize=[6, 1.5]) # inches\n#plt.figure(figsize=[8, 5]) # inches\nplt.plot(x, y, 'b.', label='observed');\nplt.plot(x_tst, yhat.mean(),'r', label='mean', linewidth=4);\nplt.ylim(-0.,17);\nplt.yticks(np.linspace(0, 15, 4)[1:]);\nplt.xticks(np.linspace(*x_range, num=9));\n\nax=plt.gca();\nax.xaxis.set_ticks_position('bottom')\nax.yaxis.set_ticks_position('left')\nax.spines['left'].set_position(('data', 0))\nax.spines['top'].set_visible(False)\nax.spines['right'].set_visible(False)\n#ax.spines['left'].set_smart_bounds(True)\n#ax.spines['bottom'].set_smart_bounds(True)\nplt.legend(loc='center left', fancybox=True, framealpha=0., bbox_to_anchor=(1.05, 0.5))\n\nplt.savefig('/tmp/fig1.png', bbox_inches='tight', dpi=300)\n```\n\n### Case 2: Aleatoric Uncertainty\n\n\n```\n# Build model.\nmodel = tf.keras.Sequential([\n tf.keras.layers.Dense(1 + 1),\n tfp.layers.DistributionLambda(\n lambda t: tfd.Normal(loc=t[..., :1],\n scale=1e-3 + tf.math.softplus(0.05 * t[...,1:]))),\n])\n\n# Do inference.\nmodel.compile(optimizer=tf.optimizers.Adam(learning_rate=0.01), loss=negloglik)\nmodel.fit(x, y, epochs=1000, verbose=False);\n\n# Profit.\n[print(np.squeeze(w.numpy())) for w in model.weights];\nyhat = model(x_tst)\nassert isinstance(yhat, tfd.Distribution)\n```\n\n [0.13226233 0.41329 ]\n [5.1153 1.2915019]\n\n\n\n```\n#@title Figure 2: Aleatoric Uncertainty\nplt.figure(figsize=[6, 1.5]) # inches\nplt.plot(x, y, 'b.', label='observed');\n\nm = yhat.mean()\ns = yhat.stddev()\n\nplt.plot(x_tst, m, 'r', linewidth=4, label='mean');\nplt.plot(x_tst, m + 2 * s, 'g', linewidth=2, label=r'mean + 2 stddev');\nplt.plot(x_tst, m - 2 * s, 'g', linewidth=2, label=r'mean - 2 stddev');\n\nplt.ylim(-0.,17);\nplt.yticks(np.linspace(0, 15, 4)[1:]);\nplt.xticks(np.linspace(*x_range, num=9));\n\nax=plt.gca();\nax.xaxis.set_ticks_position('bottom')\nax.yaxis.set_ticks_position('left')\nax.spines['left'].set_position(('data', 0))\nax.spines['top'].set_visible(False)\nax.spines['right'].set_visible(False)\n#ax.spines['left'].set_smart_bounds(True)\n#ax.spines['bottom'].set_smart_bounds(True)\nplt.legend(loc='center left', fancybox=True, framealpha=0., bbox_to_anchor=(1.05, 0.5))\n\nplt.savefig('/tmp/fig2.png', bbox_inches='tight', dpi=300)\n```\n\n### Case 3: Epistemic Uncertainty\n\n\n```\n# Specify the surrogate posterior over `keras.layers.Dense` `kernel` and `bias`.\ndef posterior_mean_field(kernel_size, bias_size=0, dtype=None):\n n = kernel_size + bias_size\n c = np.log(np.expm1(1.))\n return tf.keras.Sequential([\n tfp.layers.VariableLayer(2 * n, dtype=dtype),\n tfp.layers.DistributionLambda(lambda t: tfd.Independent(\n tfd.Normal(loc=t[..., :n],\n scale=1e-5 + tf.nn.softplus(c + t[..., n:])),\n reinterpreted_batch_ndims=1)),\n ])\n```\n\n\n```\n# Specify the surrogate posterior over `keras.layers.Dense` `kernel` and `bias`.\ndef prior_trainable(kernel_size, bias_size=0, dtype=None):\n n = kernel_size + bias_size\n return tf.keras.Sequential([\n tfp.layers.VariableLayer(n, dtype=dtype),\n tfp.layers.DistributionLambda(lambda t: tfd.Independent(\n tfd.Normal(loc=t, scale=1),\n reinterpreted_batch_ndims=1)),\n ])\n```\n\n\n```\n# Build model.\nmodel = tf.keras.Sequential([\n tfp.layers.DenseVariational(1, posterior_mean_field, prior_trainable),\n tfp.layers.DistributionLambda(lambda t: tfd.Normal(loc=t, scale=1)),\n])\n\n# Do inference.\nmodel.compile(optimizer=tf.optimizers.Adam(learning_rate=0.01), loss=negloglik)\nmodel.fit(x, y, epochs=1000, verbose=False);\n\n# Profit.\n[print(np.squeeze(w.numpy())) for w in model.weights];\nyhat = model(x_tst)\nassert isinstance(yhat, tfd.Distribution)\n```\n\n [ 0.1374202 5.1056857 -3.7132006 -0.5256554]\n [0.1592448 5.12206 ]\n\n\n\n```\n#@title Figure 3: Epistemic Uncertainty\nplt.figure(figsize=[6, 1.5]) # inches\nplt.clf();\nplt.plot(x, y, 'b.', label='observed');\n\nyhats = [model(x_tst) for _ in range(100)]\navgm = np.zeros_like(x_tst[..., 0])\nfor i, yhat in enumerate(yhats):\n m = np.squeeze(yhat.mean())\n s = np.squeeze(yhat.stddev())\n if i < 25:\n plt.plot(x_tst, m, 'r', label='ensemble means' if i == 0 else None, linewidth=0.5)\n avgm += m\nplt.plot(x_tst, avgm/len(yhats), 'r', label='overall mean', linewidth=4)\n\nplt.ylim(-0.,17);\nplt.yticks(np.linspace(0, 15, 4)[1:]);\nplt.xticks(np.linspace(*x_range, num=9));\n\nax=plt.gca();\nax.xaxis.set_ticks_position('bottom')\nax.yaxis.set_ticks_position('left')\nax.spines['left'].set_position(('data', 0))\nax.spines['top'].set_visible(False)\nax.spines['right'].set_visible(False)\n#ax.spines['left'].set_smart_bounds(True)\n#ax.spines['bottom'].set_smart_bounds(True)\nplt.legend(loc='center left', fancybox=True, framealpha=0., bbox_to_anchor=(1.05, 0.5))\n\nplt.savefig('/tmp/fig3.png', bbox_inches='tight', dpi=300)\n```\n\n### Case 4: Aleatoric & Epistemic Uncertainty\n\n\n```\n# Build model.\nmodel = tf.keras.Sequential([\n tfp.layers.DenseVariational(1 + 1, posterior_mean_field, prior_trainable),\n tfp.layers.DistributionLambda(\n lambda t: tfd.Normal(loc=t[..., :1],\n scale=1e-3 + tf.math.softplus(0.01 * t[...,1:]))),\n])\n\n# Do inference.\nmodel.compile(optimizer=tf.optimizers.Adam(learning_rate=0.01), loss=negloglik)\nmodel.fit(x, y, epochs=1000, verbose=False);\n\n# Profit.\n[print(np.squeeze(w.numpy())) for w in model.weights];\nyhat = model(x_tst)\nassert isinstance(yhat, tfd.Distribution)\n```\n\n [ 0.13091448 2.8183267 5.1185093 2.954348 -3.344546 -0.684905\n -0.5135757 0.05770317]\n [0.13267924 2.7312424 5.191225 2.9794762 ]\n\n\n\n```\n#@title Figure 4: Both Aleatoric & Epistemic Uncertainty\nplt.figure(figsize=[6, 1.5]) # inches\nplt.plot(x, y, 'b.', label='observed');\n\nyhats = [model(x_tst) for _ in range(100)]\navgm = np.zeros_like(x_tst[..., 0])\nfor i, yhat in enumerate(yhats):\n m = np.squeeze(yhat.mean())\n s = np.squeeze(yhat.stddev())\n if i < 15:\n plt.plot(x_tst, m, 'r', label='ensemble means' if i == 0 else None, linewidth=1.)\n plt.plot(x_tst, m + 2 * s, 'g', linewidth=0.5, label='ensemble means + 2 ensemble stdev' if i == 0 else None);\n plt.plot(x_tst, m - 2 * s, 'g', linewidth=0.5, label='ensemble means - 2 ensemble stdev' if i == 0 else None);\n avgm += m\nplt.plot(x_tst, avgm/len(yhats), 'r', label='overall mean', linewidth=4)\n\nplt.ylim(-0.,17);\nplt.yticks(np.linspace(0, 15, 4)[1:]);\nplt.xticks(np.linspace(*x_range, num=9));\n\nax=plt.gca();\nax.xaxis.set_ticks_position('bottom')\nax.yaxis.set_ticks_position('left')\nax.spines['left'].set_position(('data', 0))\nax.spines['top'].set_visible(False)\nax.spines['right'].set_visible(False)\n#ax.spines['left'].set_smart_bounds(True)\n#ax.spines['bottom'].set_smart_bounds(True)\nplt.legend(loc='center left', fancybox=True, framealpha=0., bbox_to_anchor=(1.05, 0.5))\n\nplt.savefig('/tmp/fig4.png', bbox_inches='tight', dpi=300)\n```\n", "meta": {"hexsha": "a9c9a845f92a510b54bcc5eb707d5bc2d671f41d", "size": 408579, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "14. Uncertainty_and_Probabilistic_Layers.ipynb", "max_stars_repo_name": "DataScienceUB/DeepLearningMaster2019-2020", "max_stars_repo_head_hexsha": "01a973c2ba2daf855701e87048378789f6e44546", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 26, "max_stars_repo_stars_event_min_datetime": "2019-01-29T19:24:53.000Z", "max_stars_repo_stars_event_max_datetime": "2021-03-31T05:59:51.000Z", "max_issues_repo_path": "14. Uncertainty_and_Probabilistic_Layers.ipynb", "max_issues_repo_name": "bernatesquirol/DeepLearningMaster20192020", "max_issues_repo_head_hexsha": "8501e1dff74af0ef582800d9a53d1b99b9d53870", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 15, "max_issues_repo_issues_event_min_datetime": "2019-12-16T22:03:46.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-11T23:54:18.000Z", "max_forks_repo_path": "14. 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                                        \nAssignment: Teach a Deep Neural Network to Paint\n

                                        \n\n

                                        \nParag K. Mital
                                        \nCreative Applications of Deep Learning w/ Tensorflow
                                        \nKadenze Academy
                                        \n#CADL\n

                                        \n\nThis work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.\n\n# Learning Goals\n\n* Learn how to create a Neural Network\n* Learn to use a neural network to paint an image\n* Apply creative thinking to the inputs, outputs, and definition of a network\n\n# Outline\n\n\n\n- [Assignment Synopsis](#assignment-synopsis)\n- [Part One - Fully Connected Network](#part-one---fully-connected-network)\n - [Instructions](#instructions)\n - [Code](#code)\n - [Variable Scopes](#variable-scopes)\n- [Part Two - Image Painting Network](#part-two---image-painting-network)\n - [Instructions](#instructions-1)\n - [Preparing the Data](#preparing-the-data)\n - [Cost Function](#cost-function)\n - [Explore](#explore)\n - [A Note on Crossvalidation](#a-note-on-crossvalidation)\n- [Part Three - Learning More than One Image](#part-three---learning-more-than-one-image)\n - [Instructions](#instructions-2)\n - [Code](#code-1)\n- [Part Four - Open Exploration \\(Extra Credit\\)](#part-four---open-exploration-extra-credit)\n- [Assignment Submission](#assignment-submission)\n\n\n\nThis next section will just make sure you have the right version of python and the libraries that we'll be using. Don't change the code here but make sure you \"run\" it (use \"shift+enter\")!\n\n\n```python\n# First check the Python version\nimport sys\nif sys.version_info < (3,4):\n print('You are running an older version of Python!\\n\\n' \\\n 'You should consider updating to Python 3.4.0 or ' \\\n 'higher as the libraries built for this course ' \\\n 'have only been tested in Python 3.4 and higher.\\n')\n print('Try installing the Python 3.5 version of anaconda '\n 'and then restart `jupyter notebook`:\\n' \\\n 'https://www.continuum.io/downloads\\n\\n')\n\n# Now get necessary libraries\ntry:\n import os\n import numpy as np\n import matplotlib.pyplot as plt\n from skimage.transform import resize\n from skimage import data\n from scipy.misc import imresize\nexcept ImportError:\n print('You are missing some packages! ' \\\n 'We will try installing them before continuing!')\n !pip install \"numpy>=1.11.0\" \"matplotlib>=1.5.1\" \"scikit-image>=0.11.3\" \"scikit-learn>=0.17\" \"scipy>=0.17.0\"\n import os\n import numpy as np\n import matplotlib.pyplot as plt\n from skimage.transform import resize\n from skimage import data\n from scipy.misc import imresize\n print('Done!')\n\n# Import Tensorflow\ntry:\n import tensorflow as tf\nexcept ImportError:\n print(\"You do not have tensorflow installed!\")\n print(\"Follow the instructions on the following link\")\n print(\"to install tensorflow before continuing:\")\n print(\"\")\n print(\"https://github.com/pkmital/CADL#installation-preliminaries\")\n\n# This cell includes the provided libraries from the zip file\n# and a library for displaying images from ipython, which\n# we will use to display the gif\ntry:\n from libs import utils, gif\n import IPython.display as ipyd\nexcept ImportError:\n print(\"Make sure you have started notebook in the same directory\" +\n \" as the provided zip file which includes the 'libs' folder\" +\n \" and the file 'utils.py' inside of it. You will NOT be able\"\n \" to complete this assignment unless you restart jupyter\"\n \" notebook inside the directory created by extracting\"\n \" the zip file or cloning the github repo.\")\n\n# We'll tell matplotlib to inline any drawn figures like so:\n%matplotlib inline\nplt.style.use('ggplot')\n```\n\n\n```python\n# Bit of formatting because I don't like the default inline code style:\nfrom IPython.core.display import HTML\nHTML(\"\"\"\"\"\")\n```\n\n\n\n\n\n\n\n\n\n# Assignment Synopsis\n\nIn this assignment, we're going to create our first neural network capable of taking any two continuous values as inputs. Those two values will go through a series of multiplications, additions, and nonlinearities, coming out of the network as 3 outputs. Remember from the last homework, we used convolution to filter an image so that the representations in the image were accentuated. We're not going to be using convolution w/ Neural Networks until the next session, but we're effectively doing the same thing here: using multiplications to accentuate the representations in our data, in order to minimize whatever our cost function is. To find out what those multiplications need to be, we're going to use Gradient Descent and Backpropagation, which will take our cost, and find the appropriate updates to all the parameters in our network to best optimize the cost. In the next session, we'll explore much bigger networks and convolution. This \"toy\" network is really to help us get up and running with neural networks, and aid our exploration of the different components that make up a neural network. You will be expected to explore manipulations of the neural networks in this notebook as much as possible to help aid your understanding of how they effect the final result.\n\nWe're going to build our first neural network to understand what color \"to paint\" given a location in an image, or the row, col of the image. So in goes a row/col, and out goes a R/G/B. In the next lesson, we'll learn what this network is really doing is performing regression. For now, we'll focus on the creative applications of such a network to help us get a better understanding of the different components that make up the neural network. You'll be asked to explore many of the different components of a neural network, including changing the inputs/outputs (i.e. the dataset), the number of layers, their activation functions, the cost functions, learning rate, and batch size. You'll also explore a modification to this same network which takes a 3rd input: an index for an image. This will let us try to learn multiple images at once, though with limited success.\n\nWe'll now dive right into creating deep neural networks, and I'm going to show you the math along the way. Don't worry if a lot of it doesn't make sense, and it really takes a bit of practice before it starts to come together.\n\n\n# Part One - Fully Connected Network\n\n\n## Instructions\nCreate the operations necessary for connecting an input to a network, defined by a `tf.Placeholder`, to a series of fully connected, or linear, layers, using the formula: \n\n$$\\textbf{H} = \\phi(\\textbf{X}\\textbf{W} + \\textbf{b})$$\n\nwhere $\\textbf{H}$ is an output layer representing the \"hidden\" activations of a network, $\\phi$ represents some nonlinearity, $\\textbf{X}$ represents an input to that layer, $\\textbf{W}$ is that layer's weight matrix, and $\\textbf{b}$ is that layer's bias. \n\nIf you're thinking, what is going on? Where did all that math come from? Don't be afraid of it. Once you learn how to \"speak\" the symbolic representation of the equation, it starts to get easier. And once we put it into practice with some code, it should start to feel like there is some association with what is written in the equation, and what we've written in code. Practice trying to say the equation in a meaningful way: \"The output of a hidden layer is equal to some input multiplied by another matrix, adding some bias, and applying a non-linearity\". Or perhaps: \"The hidden layer is equal to a nonlinearity applied to an input multiplied by a matrix and adding some bias\". Explore your own interpretations of the equation, or ways of describing it, and it starts to become much, much easier to apply the equation.\n\nThe first thing that happens in this equation is the input matrix $\\textbf{X}$ is multiplied by another matrix, $\\textbf{W}$. This is the most complicated part of the equation. It's performing matrix multiplication, as we've seen from last session, and is effectively scaling and rotating our input. The bias $\\textbf{b}$ allows for a global shift in the resulting values. Finally, the nonlinearity of $\\phi$ allows the input space to be nonlinearly warped, allowing it to express a lot more interesting distributions of data. Have a look below at some common nonlinearities. If you're unfamiliar with looking at graphs like this, it is common to read the horizontal axis as X, as the input, and the vertical axis as Y, as the output.\n\n\n```python\nxs = np.linspace(-6, 6, 100)\nplt.plot(xs, np.maximum(xs, 0), label='relu')\nplt.plot(xs, 1 / (1 + np.exp(-xs)), label='sigmoid')\nplt.plot(xs, np.tanh(xs), label='tanh')\nplt.xlabel('Input')\nplt.xlim([-6, 6])\nplt.ylabel('Output')\nplt.ylim([-1.5, 1.5])\nplt.title('Common Activation Functions/Nonlinearities')\nplt.legend(loc='lower right')\n```\n\nRemember, having series of linear followed by nonlinear operations is what makes neural networks expressive. By stacking a lot of \"linear\" + \"nonlinear\" operations in a series, we can create a deep neural network! Have a look at the output ranges of the above nonlinearity when considering which nonlinearity seems most appropriate. For instance, the `relu` is always above 0, but does not saturate at any value above 0, meaning it can be anything above 0. That's unlike the `sigmoid` which does saturate at both 0 and 1, meaning its values for a single output neuron will always be between 0 and 1. Similarly, the `tanh` saturates at -1 and 1.\n\nChoosing between these is often a matter of trial and error. Though you can make some insights depending on your normalization scheme. For instance, if your output is expected to be in the range of 0 to 1, you may not want to use a `tanh` function, which ranges from -1 to 1, but likely would want to use a `sigmoid`. Keep the ranges of these activation functions in mind when designing your network, especially the final output layer of your network.\n\n\n## Code\n\nIn this section, we're going to work out how to represent a fully connected neural network with code. First, create a 2D `tf.placeholder` called $\\textbf{X}$ with `None` for the batch size and 2 features. Make its `dtype` `tf.float32`. Recall that we use the dimension of `None` for the batch size dimension to say that this dimension can be any number. Here is the docstring for the `tf.placeholder` function, have a look at what args it takes:\n\nHelp on function placeholder in module `tensorflow.python.ops.array_ops`:\n\n```python\nplaceholder(dtype, shape=None, name=None)\n```\n\n Inserts a placeholder for a tensor that will be always fed.\n\n **Important**: This tensor will produce an error if evaluated. Its value must\n be fed using the `feed_dict` optional argument to `Session.run()`,\n `Tensor.eval()`, or `Operation.run()`.\n\n For example:\n\n```python\nx = tf.placeholder(tf.float32, shape=(1024, 1024))\ny = tf.matmul(x, x)\n\nwith tf.Session() as sess:\n print(sess.run(y)) # ERROR: will fail because x was not fed.\n\n rand_array = np.random.rand(1024, 1024)\n print(sess.run(y, feed_dict={x: rand_array})) # Will succeed.\n```\n\n Args:\n dtype: The type of elements in the tensor to be fed.\n shape: The shape of the tensor to be fed (optional). If the shape is not\n specified, you can feed a tensor of any shape.\n name: A name for the operation (optional).\n\n Returns:\n A `Tensor` that may be used as a handle for feeding a value, but not\n evaluated directly.\n\n

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                                        \n\n\n```python\n# Create a placeholder with None x 2 dimensions of dtype tf.float32, and name it \"X\":\nX = ...\n```\n\nNow multiply the tensor using a new variable, $\\textbf{W}$, which has 2 rows and 20 columns, so that when it is left mutiplied by $\\textbf{X}$, the output of the multiplication is None x 20, giving you 20 output neurons. Recall that the `tf.matmul` function takes two arguments, the left hand ($\\textbf{X}$) and right hand side ($\\textbf{W}$) of a matrix multiplication.\n\nTo create $\\textbf{W}$, you will use `tf.get_variable` to create a matrix which is `2 x 20` in dimension. Look up the docstrings of functions `tf.get_variable` and `tf.random_normal_initializer` to get familiar with these functions. There are many options we will ignore for now. Just be sure to set the `name`, `shape` (this is the one that has to be [2, 20]), `dtype` (i.e. tf.float32), and `initializer` (the `tf.random_normal_intializer` you should create) when creating your $\\textbf{W}$ variable with `tf.get_variable(...)`.\n\nFor the random normal initializer, often the mean is set to 0, and the standard deviation is set based on the number of neurons. But that really depends on the input and outputs of your network, how you've \"normalized\" your dataset, what your nonlinearity/activation function is, and what your expected range of inputs/outputs are. Don't worry about the values for the initializer for now, as this part will take a bit more experimentation to understand better!\n\nThis part is to encourage you to learn how to look up the documentation on Tensorflow, ideally using `tf.get_variable?` in the notebook. If you are really stuck, just scroll down a bit and I've shown you how to use it. \n\n

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                                        \n\n\n```python\nW = tf.get_variable(...\nh = tf.matmul(...\n```\n\nAnd add to this result another new variable, $\\textbf{b}$, which has [20] dimensions. These values will be added to every output neuron after the multiplication above. Instead of the `tf.random_normal_initializer` that you used for creating $\\textbf{W}$, now use the `tf.constant_initializer`. Often for bias, you'll set the constant bias initialization to 0 or 1.\n\n

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                                        \n\n\n```python\nb = tf.get_variable(...\nh = tf.nn.bias_add(...\n```\n\nSo far we have done:\n$$\\textbf{X}\\textbf{W} + \\textbf{b}$$\n\nFinally, apply a nonlinear activation to this output, such as `tf.nn.relu`, to complete the equation:\n\n$$\\textbf{H} = \\phi(\\textbf{X}\\textbf{W} + \\textbf{b})$$\n\n

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                                        \n\n\n```python\nh = ...\n```\n\nNow that we've done all of this work, let's stick it inside a function. I've already done this for you and placed it inside the `utils` module under the function name `linear`. We've already imported the `utils` module so we can call it like so, `utils.linear(...)`. The docstring is copied below, and the code itself. Note that this function is slightly different to the one in the lecture. It does not require you to specify `n_input`, and the input `scope` is called `name`. It also has a few more extras in there including automatically converting a 4-d input tensor to a 2-d tensor so that you can fully connect the layer with a matrix multiply (don't worry about what this means if it doesn't make sense!).\n\n```python\nutils.linear??\n```\n\n```python\ndef linear(x, n_output, name=None, activation=None, reuse=None):\n \"\"\"Fully connected layer\n\n Parameters\n ----------\n x : tf.Tensor\n Input tensor to connect\n n_output : int\n Number of output neurons\n name : None, optional\n Scope to apply\n\n Returns\n -------\n op : tf.Tensor\n Output of fully connected layer.\n \"\"\"\n if len(x.get_shape()) != 2:\n x = flatten(x, reuse=reuse)\n\n n_input = x.get_shape().as_list()[1]\n\n with tf.variable_scope(name or \"fc\", reuse=reuse):\n W = tf.get_variable(\n name='W',\n shape=[n_input, n_output],\n dtype=tf.float32,\n initializer=tf.contrib.layers.xavier_initializer())\n\n b = tf.get_variable(\n name='b',\n shape=[n_output],\n dtype=tf.float32,\n initializer=tf.constant_initializer(0.0))\n\n h = tf.nn.bias_add(\n name='h',\n value=tf.matmul(x, W),\n bias=b)\n\n if activation:\n h = activation(h)\n\n return h, W\n```\n\n\n## Variable Scopes\n\nNote that since we are using `variable_scope` and explicitly telling the scope which name we would like, if there is *already* a variable created with the same name, then Tensorflow will raise an exception! If this happens, you should consider one of three possible solutions:\n\n1. If this happens while you are interactively editing a graph, you may need to reset the current graph:\n```python\n tf.reset_default_graph()\n```\nYou should really only have to use this if you are in an interactive console! If you are creating Python scripts to run via command line, you should really be using solution 3 listed below, and be explicit with your graph contexts! \n2. If this happens and you were not expecting any name conflicts, then perhaps you had a typo and created another layer with the same name! That's a good reason to keep useful names for everything in your graph!\n3. More likely, you should be using context managers when creating your graphs and running sessions. This works like so:\n\n ```python\n g = tf.Graph()\n with tf.Session(graph=g) as sess:\n Y_pred, W = linear(X, 2, 3, activation=tf.nn.relu)\n ```\n\n or:\n\n ```python\n g = tf.Graph()\n with tf.Session(graph=g) as sess, g.as_default():\n Y_pred, W = linear(X, 2, 3, activation=tf.nn.relu)\n ```\n\nYou can now write the same process as the above steps by simply calling:\n\n\n```python\nh, W = utils.linear(\n x=X, n_output=20, name='linear', activation=tf.nn.relu)\n```\n\n\n# Part Two - Image Painting Network\n\n\n## Instructions\n\nFollow along the steps below, first setting up input and output data of the network, $\\textbf{X}$ and $\\textbf{Y}$. Then work through building the neural network which will try to compress the information in $\\textbf{X}$ through a series of linear and non-linear functions so that whatever it is given as input, it minimized the error of its prediction, $\\hat{\\textbf{Y}}$, and the true output $\\textbf{Y}$ through its training process. You'll also create an animated GIF of the training which you'll need to submit for the homework!\n\nThrough this, we'll explore our first creative application: painting an image. This network is just meant to demonstrate how easily networks can be scaled to more complicated tasks without much modification. It is also meant to get you thinking about neural networks as building blocks that can be reconfigured, replaced, reorganized, and get you thinking about how the inputs and outputs can be anything you can imagine.\n\n\n## Preparing the Data\n\nWe'll follow an example that Andrej Karpathy has done in his online demonstration of \"image inpainting\". What we're going to do is teach the network to go from the location on an image frame to a particular color. So given any position in an image, the network will need to learn what color to paint. Let's first get an image that we'll try to teach a neural network to paint.\n\n

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                                        \n\n\n```python\n# First load an image\nimg = ...\n\n# Be careful with the size of your image.\n# Try a fairly small image to begin with,\n# then come back here and try larger sizes.\nimg = imresize(img, (100, 100))\nplt.figure(figsize=(5, 5))\nplt.imshow(img)\n\n# Make sure you save this image as \"reference.png\"\n# and include it in your zipped submission file\n# so we can tell what image you are trying to paint!\nplt.imsave(fname='reference.png', arr=img)\n```\n\nIn the lecture, I showed how to aggregate the pixel locations and their colors using a loop over every pixel position. I put that code into a function `split_image` below. Feel free to experiment with other features for `xs` or `ys`.\n\n\n```python\ndef split_image(img):\n # We'll first collect all the positions in the image in our list, xs\n xs = []\n\n # And the corresponding colors for each of these positions\n ys = []\n\n # Now loop over the image\n for row_i in range(img.shape[0]):\n for col_i in range(img.shape[1]):\n # And store the inputs\n xs.append([row_i, col_i])\n # And outputs that the network needs to learn to predict\n ys.append(img[row_i, col_i])\n\n # we'll convert our lists to arrays\n xs = np.array(xs)\n ys = np.array(ys)\n return xs, ys\n```\n\nLet's use this function to create the inputs (xs) and outputs (ys) to our network as the pixel locations (xs) and their colors (ys):\n\n\n```python\nxs, ys = split_image(img)\n\n# and print the shapes\nxs.shape, ys.shape\n```\n\nAlso remember, we should normalize our input values!\n\n

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                                        \n\n\n```python\n# Normalize the input (xs) using its mean and standard deviation\nxs = ...\n\n# Just to make sure you have normalized it correctly:\nprint(np.min(xs), np.max(xs))\nassert(np.min(xs) > -3.0 and np.max(xs) < 3.0)\n```\n\nSimilarly for the output:\n\n\n```python\nprint(np.min(ys), np.max(ys))\n```\n\nWe'll normalize the output using a simpler normalization method, since we know the values range from 0-255:\n\n\n```python\nys = ys / 255.0\nprint(np.min(ys), np.max(ys))\n```\n\nScaling the image values like this has the advantage that it is still interpretable as an image, unlike if we have negative values.\n\nWhat we're going to do is use regression to predict the value of a pixel given its (row, col) position. So the input to our network is `X = (row, col)` value. And the output of the network is `Y = (r, g, b)`.\n\nWe can get our original image back by reshaping the colors back into the original image shape. This works because the `ys` are still in order:\n\n\n```python\nplt.imshow(ys.reshape(img.shape))\n```\n\nBut when we give inputs of (row, col) to our network, it won't know what order they are, because we will randomize them. So it will have to *learn* what color value should be output for any given (row, col).\n\nCreate 2 placeholders of `dtype` `tf.float32`: one for the input of the network, a `None x 2` dimension placeholder called $\\textbf{X}$, and another for the true output of the network, a `None x 3` dimension placeholder called $\\textbf{Y}$.\n\n

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                                        \n\n\n```python\n# Let's reset the graph:\ntf.reset_default_graph()\n\n# Create a placeholder of None x 2 dimensions and dtype tf.float32\n# This will be the input to the network which takes the row/col\nX = tf.placeholder(...\n\n# Create the placeholder, Y, with 3 output dimensions instead of 2.\n# This will be the output of the network, the R, G, B values.\nY = tf.placeholder(...\n```\n\nNow create a deep neural network that takes your network input $\\textbf{X}$ of 2 neurons, multiplies it by a linear and non-linear transformation which makes its shape [None, 20], meaning it will have 20 output neurons. Then repeat the same process again to give you 20 neurons again, and then again and again until you've done 6 layers of 20 neurons. Then finally one last layer which will output 3 neurons, your predicted output, which I've been denoting mathematically as $\\hat{\\textbf{Y}}$, for a total of 6 hidden layers, or 8 layers total including the input and output layers. Mathematically, we'll be creating a deep neural network that looks just like the previous fully connected layer we've created, but with a few more connections. So recall the first layer's connection is:\n\n\\begin{align}\n\\textbf{H}_1=\\phi(\\textbf{X}\\textbf{W}_1 + \\textbf{b}_1) \\\\\n\\end{align}\n\nSo the next layer will take that output, and connect it up again:\n\n\\begin{align}\n\\textbf{H}_2=\\phi(\\textbf{H}_1\\textbf{W}_2 + \\textbf{b}_2) \\\\\n\\end{align}\n\nAnd same for every other layer:\n\n\\begin{align}\n\\textbf{H}_3=\\phi(\\textbf{H}_2\\textbf{W}_3 + \\textbf{b}_3) \\\\\n\\textbf{H}_4=\\phi(\\textbf{H}_3\\textbf{W}_4 + \\textbf{b}_4) \\\\\n\\textbf{H}_5=\\phi(\\textbf{H}_4\\textbf{W}_5 + \\textbf{b}_5) \\\\\n\\textbf{H}_6=\\phi(\\textbf{H}_5\\textbf{W}_6 + \\textbf{b}_6) \\\\\n\\end{align}\n\nIncluding the very last layer, which will be the prediction of the network:\n\n\\begin{align}\n\\hat{\\textbf{Y}}=\\phi(\\textbf{H}_6\\textbf{W}_7 + \\textbf{b}_7)\n\\end{align}\n\nRemember if you run into issues with variable scopes/names, that you cannot recreate a variable with the same name! Revisit the section on Variable Scopes if you get stuck with name issues.\n\n

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                                        \n\n\n```python\n# We'll create 6 hidden layers. Let's create a variable\n# to say how many neurons we want for each of the layers\n# (try 20 to begin with, then explore other values)\nn_neurons = ...\n\n# Create the first linear + nonlinear layer which will\n# take the 2 input neurons and fully connects it to 20 neurons.\n# Use the `utils.linear` function to do this just like before,\n# but also remember to give names for each layer, such as\n# \"1\", \"2\", ... \"5\", or \"layer1\", \"layer2\", ... \"layer6\".\nh1, W1 = ...\n\n# Create another one:\nh2, W2 = ...\n\n# and four more (or replace all of this with a loop if you can!):\nh3, W3 = ...\nh4, W4 = ...\nh5, W5 = ...\nh6, W6 = ...\n\n# Now, make one last layer to make sure your network has 3 outputs:\nY_pred, W7 = utils.linear(h6, 3, activation=None, name='pred')\n```\n\n\n```python\nassert(X.get_shape().as_list() == [None, 2])\nassert(Y_pred.get_shape().as_list() == [None, 3])\nassert(Y.get_shape().as_list() == [None, 3])\n```\n\n\n## Cost Function\n\nNow we're going to work on creating a `cost` function. The cost should represent how much `error` there is in the network, and provide the optimizer this value to help it train the network's parameters using gradient descent and backpropagation.\n\nLet's say our error is `E`, then the cost will be:\n\n$$cost(\\textbf{Y}, \\hat{\\textbf{Y}}) = \\frac{1}{\\text{B}} \\displaystyle\\sum\\limits_{b=0}^{\\text{B}} \\textbf{E}_b\n$$\n\nwhere the error is measured as, e.g.:\n\n$$\\textbf{E} = \\displaystyle\\sum\\limits_{c=0}^{\\text{C}} (\\textbf{Y}_{c} - \\hat{\\textbf{Y}}_{c})^2$$\n\nDon't worry if this scares you. This is mathematically expressing the same concept as: \"the cost of an actual $\\textbf{Y}$, and a predicted $\\hat{\\textbf{Y}}$ is equal to the mean across batches, of which there are $\\text{B}$ total batches, of the sum of distances across $\\text{C}$ color channels of every predicted output and true output\". Basically, we're trying to see on average, or at least within a single minibatches average, how wrong was our prediction? We create a measure of error for every output feature by squaring the predicted output and the actual output it should have, i.e. the actual color value it should have output for a given input pixel position. By squaring it, we penalize large distances, but not so much small distances.\n\nConsider how the square function (i.e., $f(x) = x^2$) changes for a given error. If our color values range between 0-255, then a typical amount of error would be between $0$ and $128^2$. For example if my prediction was (120, 50, 167), and the color should have been (0, 100, 120), then the error for the Red channel is (120 - 0) or 120. And the Green channel is (50 - 100) or -50, and for the Blue channel, (167 - 120) = 47. When I square this result, I get: (120)^2, (-50)^2, and (47)^2. I then add all of these and that is my error, $\\textbf{E}$, for this one observation. But I will have a few observations per minibatch. So I add all the error in my batch together, then divide by the number of observations in the batch, essentially finding the mean error of my batch. \n\nLet's try to see what the square in our measure of error is doing graphically.\n\n\n```python\nerror = np.linspace(0.0, 128.0**2, 100)\nloss = error**2.0\nplt.plot(error, loss)\nplt.xlabel('error')\nplt.ylabel('loss')\n```\n\nThis is known as the $l_2$ (pronounced el-two) loss. It doesn't penalize small errors as much as it does large errors. This is easier to see when we compare it with another common loss, the $l_1$ (el-one) loss. It is linear in error, by taking the absolute value of the error. We'll compare the $l_1$ loss with normalized values from $0$ to $1$. So instead of having $0$ to $255$ for our RGB values, we'd have $0$ to $1$, simply by dividing our color values by $255.0$.\n\n\n```python\nerror = np.linspace(0.0, 1.0, 100)\nplt.plot(error, error**2, label='l_2 loss')\nplt.plot(error, np.abs(error), label='l_1 loss')\nplt.xlabel('error')\nplt.ylabel('loss')\nplt.legend(loc='lower right')\n```\n\nSo unlike the $l_2$ loss, the $l_1$ loss is really quickly upset if there is *any* error at all: as soon as error moves away from $0.0$, to $0.1$, the $l_1$ loss is $0.1$. But the $l_2$ loss is $0.1^2 = 0.01$. Having a stronger penalty on smaller errors often leads to what the literature calls \"sparse\" solutions, since it favors activations that try to explain as much of the data as possible, rather than a lot of activations that do a sort of good job, but when put together, do a great job of explaining the data. Don't worry about what this means if you are more unfamiliar with Machine Learning. There is a lot of literature surrounding each of these loss functions that we won't have time to get into, but look them up if they interest you.\n\nDuring the lecture, we've seen how to create a cost function using Tensorflow. To create a $l_2$ loss function, you can for instance use tensorflow's `tf.squared_difference` or for an $l_1$ loss function, `tf.abs`. You'll need to refer to the `Y` and `Y_pred` variables only, and your resulting cost should be a single value. Try creating the $l_1$ loss to begin with, and come back here after you have trained your network, to compare the performance with a $l_2$ loss.\n\nThe equation for computing cost I mentioned above is more succintly written as, for $l_2$ norm:\n\n$$cost(\\textbf{Y}, \\hat{\\textbf{Y}}) = \\frac{1}{\\text{B}} \\displaystyle\\sum\\limits_{b=0}^{\\text{B}} \\displaystyle\\sum\\limits_{c=0}^{\\text{C}} (\\textbf{Y}_{c} - \\hat{\\textbf{Y}}_{c})^2$$\n\nFor $l_1$ norm, we'd have:\n\n$$cost(\\textbf{Y}, \\hat{\\textbf{Y}}) = \\frac{1}{\\text{B}} \\displaystyle\\sum\\limits_{b=0}^{\\text{B}} \\displaystyle\\sum\\limits_{c=0}^{\\text{C}} \\text{abs}(\\textbf{Y}_{c} - \\hat{\\textbf{Y}}_{c})$$\n\nRemember, to understand this equation, try to say it out loud: the $cost$ given two variables, $\\textbf{Y}$, the actual output we want the network to have, and $\\hat{\\textbf{Y}}$ the predicted output from the network, is equal to the mean across $\\text{B}$ batches, of the sum of $\\textbf{C}$ color channels distance between the actual and predicted outputs. If you're still unsure, refer to the lecture where I've computed this, or scroll down a bit to where I've included the answer.\n\n

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                                        \n\n\n```python\n# first compute the error, the inner part of the summation.\n# This should be the l1-norm or l2-norm of the distance\n# between each color channel.\nerror = ...\nassert(error.get_shape().as_list() == [None, 3])\n```\n\n

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                                        \n\n\n```python\n# Now sum the error for each feature in Y. \n# If Y is [Batch, Features], the sum should be [Batch]:\nsum_error = ...\nassert(sum_error.get_shape().as_list() == [None])\n```\n\n

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                                        \n\n\n```python\n# Finally, compute the cost, as the mean error of the batch.\n# This should be a single value.\ncost = ...\nassert(cost.get_shape().as_list() == [])\n```\n\nWe now need an `optimizer` which will take our `cost` and a `learning_rate`, which says how far along the gradient to move. This optimizer calculates all the gradients in our network with respect to the `cost` variable and updates all of the weights in our network using backpropagation. We'll then create mini-batches of our training data and run the `optimizer` using a `session`.\n\n

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                                        \n\n\n```python\n# Refer to the help for the function\noptimizer = tf.train....minimize(cost)\n\n# Create parameters for the number of iterations to run for (< 100)\nn_iterations = ...\n\n# And how much data is in each minibatch (< 500)\nbatch_size = ...\n\n# Then create a session\nsess = tf.Session()\n```\n\nWe'll now train our network! The code below should do this for you if you've setup everything else properly. Please read through this and make sure you understand each step! Note that this can take a VERY LONG time depending on the size of your image (make it < 100 x 100 pixels), the number of neurons per layer (e.g. < 30), the number of layers (e.g. < 8), and number of iterations (< 1000). Welcome to Deep Learning :)\n\n\n```python\n# Initialize all your variables and run the operation with your session\nsess.run(tf.global_variables_initializer())\n\n# Optimize over a few iterations, each time following the gradient\n# a little at a time\nimgs = []\ncosts = []\ngif_step = n_iterations // 10\nstep_i = 0\n\nfor it_i in range(n_iterations):\n \n # Get a random sampling of the dataset\n idxs = np.random.permutation(range(len(xs)))\n \n # The number of batches we have to iterate over\n n_batches = len(idxs) // batch_size\n \n # Now iterate over our stochastic minibatches:\n for batch_i in range(n_batches):\n \n # Get just minibatch amount of data\n idxs_i = idxs[batch_i * batch_size: (batch_i + 1) * batch_size]\n\n # And optimize, also returning the cost so we can monitor\n # how our optimization is doing.\n training_cost = sess.run(\n [cost, optimizer],\n feed_dict={X: xs[idxs_i], Y: ys[idxs_i]})[0]\n\n # Also, every 20 iterations, we'll draw the prediction of our\n # input xs, which should try to recreate our image!\n if (it_i + 1) % gif_step == 0:\n costs.append(training_cost / n_batches)\n ys_pred = Y_pred.eval(feed_dict={X: xs}, session=sess)\n img = np.clip(ys_pred.reshape(img.shape), 0, 1)\n imgs.append(img)\n # Plot the cost over time\n fig, ax = plt.subplots(1, 2)\n ax[0].plot(costs)\n ax[0].set_xlabel('Iteration')\n ax[0].set_ylabel('Cost')\n ax[1].imshow(img)\n fig.suptitle('Iteration {}'.format(it_i))\n plt.show()\n```\n\n\n```python\n# Save the images as a GIF\n_ = gif.build_gif(imgs, saveto='single.gif', show_gif=False)\n```\n\nLet's now display the GIF we've just created:\n\n\n```python\nipyd.Image(url='single.gif?{}'.format(np.random.rand()),\n height=500, width=500)\n```\n\n\n## Explore\n\nGo back over the previous cells and exploring changing different parameters of the network. I would suggest first trying to change the `learning_rate` parameter to different values and see how the cost curve changes. What do you notice? Try exponents of $10$, e.g. $10^1$, $10^2$, $10^3$... and so on. Also try changing the `batch_size`: $50, 100, 200, 500, ...$ How does it effect how the cost changes over time?\n\nBe sure to explore other manipulations of the network, such as changing the loss function to $l_2$ or $l_1$. How does it change the resulting learning? Also try changing the activation functions, the number of layers/neurons, different optimizers, and anything else that you may think of, and try to get a basic understanding on this toy problem of how it effects the network's training. Also try comparing creating a fairly shallow/wide net (e.g. 1-2 layers with many neurons, e.g. > 100), versus a deep/narrow net (e.g. 6-20 layers with fewer neurons, e.g. < 20). What do you notice?\n\n\n## A Note on Crossvalidation\n\nThe cost curve plotted above is only showing the cost for our \"training\" dataset. Ideally, we should split our dataset into what are called \"train\", \"validation\", and \"test\" sets. This is done by taking random subsets of the entire dataset. For instance, we partition our dataset by saying we'll only use 80% of it for training, 10% for validation, and the last 10% for testing. Then when training as above, you would only use the 80% of the data you had partitioned, and then monitor accuracy on both the data you have used to train, but also that new 10% of unseen validation data. This gives you a sense of how \"general\" your network is. If it is performing just as well on that 10% of data, then you know it is doing a good job. Finally, once you are done training, you would test one last time on your \"test\" dataset. Ideally, you'd do this a number of times, so that every part of the dataset had a chance to be the test set. This would also give you a measure of the variance of the accuracy on the final test. If it changes a lot, you know something is wrong. If it remains fairly stable, then you know that it is a good representation of the model's accuracy on unseen data.\n\nWe didn't get a chance to cover this in class, as it is less useful for exploring creative applications, though it is very useful to know and to use in practice, as it avoids overfitting/overgeneralizing your network to all of the data. Feel free to explore how to do this on the application above!\n\n\n# Part Three - Learning More than One Image\n\n\n## Instructions\n\nWe're now going to make use of our Dataset from Session 1 and apply what we've just learned to try and paint every single image in our dataset. How would you guess is the best way to approach this? We could for instance feed in every possible image by having multiple row, col -> r, g, b values. So for any given row, col, we'd have 100 possible r, g, b values. This likely won't work very well as there are many possible values a pixel could take, not just one. What if we also tell the network *which* image's row and column we wanted painted? We're going to try and see how that does.\n\nYou can execute all of the cells below unchanged to see how this works with the first 100 images of the celeb dataset. But you should replace the images with your own dataset, and vary the parameters of the network to get the best results!\n\nI've placed the same code for running the previous algorithm into two functions, `build_model` and `train`. You can directly call the function `train` with a 4-d image shaped as N x H x W x C, and it will collect all of the points of every image and try to predict the output colors of those pixels, just like before. The only difference now is that you are able to try this with a few images at a time. There are a few ways we could have tried to handle multiple images. The way I've shown in the `train` function is to include an additional input neuron for *which* image it is. So as well as receiving the row and column, the network will also receive as input which image it is as a number. This should help the network to better distinguish the patterns it uses, as it has knowledge that helps it separates its process based on which image is fed as input.\n\n\n```python\ndef build_model(xs, ys, n_neurons, n_layers, activation_fn,\n final_activation_fn, cost_type):\n \n xs = np.asarray(xs)\n ys = np.asarray(ys)\n \n if xs.ndim != 2:\n raise ValueError(\n 'xs should be a n_observates x n_features, ' +\n 'or a 2-dimensional array.')\n if ys.ndim != 2:\n raise ValueError(\n 'ys should be a n_observates x n_features, ' +\n 'or a 2-dimensional array.')\n \n n_xs = xs.shape[1]\n n_ys = ys.shape[1]\n \n X = tf.placeholder(name='X', shape=[None, n_xs],\n dtype=tf.float32)\n Y = tf.placeholder(name='Y', shape=[None, n_ys],\n dtype=tf.float32)\n\n current_input = X\n for layer_i in range(n_layers):\n current_input = utils.linear(\n current_input, n_neurons,\n activation=activation_fn,\n name='layer{}'.format(layer_i))[0]\n\n Y_pred = utils.linear(\n current_input, n_ys,\n activation=final_activation_fn,\n name='pred')[0]\n \n if cost_type == 'l1_norm':\n cost = tf.reduce_mean(tf.reduce_sum(\n tf.abs(Y - Y_pred), 1))\n elif cost_type == 'l2_norm':\n cost = tf.reduce_mean(tf.reduce_sum(\n tf.squared_difference(Y, Y_pred), 1))\n else:\n raise ValueError(\n 'Unknown cost_type: {}. '.format(\n cost_type) + 'Use only \"l1_norm\" or \"l2_norm\"')\n \n return {'X': X, 'Y': Y, 'Y_pred': Y_pred, 'cost': cost}\n```\n\n\n```python\ndef train(imgs,\n learning_rate=0.0001,\n batch_size=200,\n n_iterations=10,\n gif_step=2,\n n_neurons=30,\n n_layers=10,\n activation_fn=tf.nn.relu,\n final_activation_fn=tf.nn.tanh,\n cost_type='l2_norm'):\n\n N, H, W, C = imgs.shape\n all_xs, all_ys = [], []\n for img_i, img in enumerate(imgs):\n xs, ys = split_image(img)\n all_xs.append(np.c_[xs, np.repeat(img_i, [xs.shape[0]])])\n all_ys.append(ys)\n xs = np.array(all_xs).reshape(-1, 3)\n xs = (xs - np.mean(xs, 0)) / np.std(xs, 0)\n ys = np.array(all_ys).reshape(-1, 3)\n ys = ys / 127.5 - 1\n\n g = tf.Graph()\n with tf.Session(graph=g) as sess:\n model = build_model(xs, ys, n_neurons, n_layers,\n activation_fn, final_activation_fn,\n cost_type)\n optimizer = tf.train.AdamOptimizer(\n learning_rate=learning_rate).minimize(model['cost'])\n sess.run(tf.global_variables_initializer())\n gifs = []\n costs = []\n step_i = 0\n for it_i in range(n_iterations):\n # Get a random sampling of the dataset\n idxs = np.random.permutation(range(len(xs)))\n\n # The number of batches we have to iterate over\n n_batches = len(idxs) // batch_size\n training_cost = 0\n\n # Now iterate over our stochastic minibatches:\n for batch_i in range(n_batches):\n\n # Get just minibatch amount of data\n idxs_i = idxs[batch_i * batch_size:\n (batch_i + 1) * batch_size]\n\n # And optimize, also returning the cost so we can monitor\n # how our optimization is doing.\n cost = sess.run(\n [model['cost'], optimizer],\n feed_dict={model['X']: xs[idxs_i],\n model['Y']: ys[idxs_i]})[0]\n training_cost += cost\n\n print('iteration {}/{}: cost {}'.format(\n it_i + 1, n_iterations, training_cost / n_batches))\n\n # Also, every 20 iterations, we'll draw the prediction of our\n # input xs, which should try to recreate our image!\n if (it_i + 1) % gif_step == 0:\n costs.append(training_cost / n_batches)\n ys_pred = model['Y_pred'].eval(\n feed_dict={model['X']: xs}, session=sess)\n img = ys_pred.reshape(imgs.shape)\n gifs.append(img)\n return gifs\n```\n\n\n## Code\n\nBelow, I've shown code for loading the first 100 celeb files. Run through the next few cells to see how this works with the celeb dataset, and then come back here and replace the `imgs` variable with your own set of images. For instance, you can try your entire sorted dataset from Session 1 as an N x H x W x C array. Explore!\n\n

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                                        \n\n\n```python\nceleb_imgs = utils.get_celeb_imgs()\nplt.figure(figsize=(10, 10))\nplt.imshow(utils.montage(celeb_imgs).astype(np.uint8))\n# It doesn't have to be 100 images, explore!\nimgs = np.array(celeb_imgs).copy()\n```\n\nExplore changing the parameters of the `train` function and your own dataset of images. Note, you do not have to use the dataset from the last assignment! Explore different numbers of images, whatever you prefer.\n\n

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                                        \n\n\n```python\n# Change the parameters of the train function and\n# explore changing the dataset\ngifs = train(imgs=imgs)\n```\n\nNow we'll create a gif out of the training process. Be sure to call this 'multiple.gif' for your homework submission:\n\n\n```python\nmontage_gifs = [np.clip(utils.montage(\n (m * 127.5) + 127.5), 0, 255).astype(np.uint8)\n for m in gifs]\n_ = gif.build_gif(montage_gifs, saveto='multiple.gif')\n```\n\nAnd show it in the notebook\n\n\n```python\nipyd.Image(url='multiple.gif?{}'.format(np.random.rand()),\n height=500, width=500)\n```\n\nWhat we're seeing is the training process over time. We feed in our `xs`, which consist of the pixel values of each of our 100 images, it goes through the neural network, and out come predicted color values for every possible input value. We visualize it above as a gif by seeing how at each iteration the network has predicted the entire space of the inputs. We can visualize just the last iteration as a \"latent\" space, going from the first image (the top left image in the montage), to the last image, (the bottom right image).\n\n\n```python\nfinal = gifs[-1]\nfinal_gif = [np.clip(((m * 127.5) + 127.5), 0, 255).astype(np.uint8) for m in final]\ngif.build_gif(final_gif, saveto='final.gif')\n```\n\n\n```python\nipyd.Image(url='final.gif?{}'.format(np.random.rand()),\n height=200, width=200)\n```\n\n\n# Part Four - Open Exploration (Extra Credit)\n\nI now what you to explore what other possible manipulations of the network and/or dataset you could imagine. Perhaps a process that does the reverse, tries to guess where a given color should be painted? What if it was only taught a certain palette, and had to reason about other colors, how it would interpret those colors? Or what if you fed it pixel locations that weren't part of the training set, or outside the frame of what it was trained on? Or what happens with different activation functions, number of layers, increasing number of neurons or lesser number of neurons? I leave any of these as an open exploration for you.\n\nTry exploring this process with your own ideas, materials, and networks, and submit something you've created as a gif! To aid exploration, be sure to scale the image down quite a bit or it will require a much larger machine, and much more time to train. Then whenever you think you may be happy with the process you've created, try scaling up the resolution and leave the training to happen over a few hours/overnight to produce something truly stunning!\n\nMake sure to name the result of your gif: \"explore.gif\", and be sure to include it in your zip file.\n\n

                                        TODO! COMPLETE THIS SECTION!

                                        \n\n\n```python\n# Train a network to produce something, storing every few\n# iterations in the variable gifs, then export the training\n# over time as a gif.\n...\n\n\ngif.build_gif(montage_gifs, saveto='explore.gif')\n```\n\n\n```python\nipyd.Image(url='explore.gif?{}'.format(np.random.rand()),\n height=500, width=500)\n```\n\n\n# Assignment Submission\n\nAfter you've completed the notebook, create a zip file of the current directory using the code below. This code will make sure you have included this completed ipython notebook and the following files named exactly as:\n\n
                                        \n    session-2/\n      session-2.ipynb\n      single.gif\n      multiple.gif\n      final.gif\n      explore.gif*\n      libs/\n        utils.py\n        \n    * = optional/extra-credit\n
                                        \n\nYou'll then submit this zip file for your second assignment on Kadenze for \"Assignment 2: Teach a Deep Neural Network to Paint\"! If you have any questions, remember to reach out on the forums and connect with your peers or with me.\n\nTo get assessed, you'll need to be a premium student! This will allow you to build an online portfolio of all of your work and receive grades. If you aren't already enrolled as a student, register now at http://www.kadenze.com/ and join the [#CADL](https://twitter.com/hashtag/CADL) community to see what your peers are doing! https://www.kadenze.com/courses/creative-applications-of-deep-learning-with-tensorflow/info\n\nAlso, if you share any of the GIFs on Facebook/Twitter/Instagram/etc..., be sure to use the #CADL hashtag so that other students can find your work!\n\n\n```python\nutils.build_submission('session-2.zip',\n ('reference.png',\n 'single.gif',\n 'multiple.gif',\n 'final.gif',\n 'session-2.ipynb'),\n ('explore.gif'))\n```\n", "meta": {"hexsha": "72933a125123ba8b7e2634a625e4984be84f2cea", "size": 65222, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "session-2/session-2.ipynb", "max_stars_repo_name": "itamaro/CADL", "max_stars_repo_head_hexsha": "b5de17485962577fc51156cd12da1b16d66dbb26", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 1628, "max_stars_repo_stars_event_min_datetime": "2016-07-19T22:21:56.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-27T15:19:45.000Z", "max_issues_repo_path": "session-2/session-2.ipynb", "max_issues_repo_name": "itamaro/CADL", "max_issues_repo_head_hexsha": "b5de17485962577fc51156cd12da1b16d66dbb26", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 75, "max_issues_repo_issues_event_min_datetime": "2016-07-22T02:05:05.000Z", "max_issues_repo_issues_event_max_datetime": "2019-05-20T21:00:34.000Z", "max_forks_repo_path": "session-2/session-2.ipynb", "max_forks_repo_name": "itamaro/CADL", "max_forks_repo_head_hexsha": "b5de17485962577fc51156cd12da1b16d66dbb26", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 955, "max_forks_repo_forks_event_min_datetime": "2016-07-22T00:10:52.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-24T15:06:09.000Z", "avg_line_length": 44.8569463549, "max_line_length": 1298, "alphanum_fraction": 0.6022201098, "converted": true, "num_tokens": 12494, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.6825737408694988, "lm_q1q2_score": 0.42737592189956014}} {"text": "\n \n\n\n# **Quantum Computing 101: it's easier than it sounds!**\n\nBy the end of this notebook we will write 2 simple quantum programs that will demonstrate the basic components of quantum mechanics which may make quantum computers faster than traditional computers:\n\n* Superposition (Quantum systems can be in multiple states at once)\n* Entanglement (Quantum computers can do multiple operations at once)\n\n
                                        \n\nSo let's jump right in.\n# **The code**\n\nLet's start by importing the shor library, and necessary components.\n\n\n```\n# Install shor library from pypi\n!pip install shor\n\n# Import the Circuit object for constructing Quantum Programs\nfrom shor.quantum import Circuit\n\n# The quantum gates necessary for these examples\nfrom shor.gates import CNOT, Hadamard\n\n# The input layer for the program\nfrom shor.layers import Qbits\n\n# The measure operation to get a binary output (classical) for the program\nfrom shor.operations import Measure\n```\n\n Requirement already satisfied: shor in /usr/local/lib/python3.6/dist-packages (0.0.6)\n Requirement already satisfied: matplotlib<4.0.0,>=3.3.3 in /usr/local/lib/python3.6/dist-packages (from shor) (3.3.4)\n Requirement already satisfied: qiskit<0.24.0,>=0.23.1 in /usr/local/lib/python3.6/dist-packages (from shor) (0.23.5)\n Requirement already satisfied: numpy<2.0.0,>=1.19.4 in /usr/local/lib/python3.6/dist-packages (from shor) (1.19.5)\n Requirement already satisfied: kiwisolver>=1.0.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib<4.0.0,>=3.3.3->shor) (1.3.1)\n Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.3 in /usr/local/lib/python3.6/dist-packages (from matplotlib<4.0.0,>=3.3.3->shor) (2.4.7)\n Requirement already satisfied: pillow>=6.2.0 in /usr/local/lib/python3.6/dist-packages (from matplotlib<4.0.0,>=3.3.3->shor) (7.0.0)\n Requirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.6/dist-packages (from matplotlib<4.0.0,>=3.3.3->shor) (0.10.0)\n Requirement already satisfied: python-dateutil>=2.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib<4.0.0,>=3.3.3->shor) (2.8.1)\n Requirement already satisfied: qiskit-aqua==0.8.2 in /usr/local/lib/python3.6/dist-packages (from qiskit<0.24.0,>=0.23.1->shor) (0.8.2)\n Requirement already satisfied: qiskit-ibmq-provider==0.11.1 in /usr/local/lib/python3.6/dist-packages (from qiskit<0.24.0,>=0.23.1->shor) (0.11.1)\n Requirement already satisfied: qiskit-ignis==0.5.2 in /usr/local/lib/python3.6/dist-packages (from qiskit<0.24.0,>=0.23.1->shor) (0.5.2)\n Requirement already satisfied: qiskit-aer==0.7.4 in /usr/local/lib/python3.6/dist-packages (from qiskit<0.24.0,>=0.23.1->shor) (0.7.4)\n Requirement already satisfied: qiskit-terra==0.16.4 in /usr/local/lib/python3.6/dist-packages (from qiskit<0.24.0,>=0.23.1->shor) (0.16.4)\n Requirement already satisfied: six in /usr/local/lib/python3.6/dist-packages (from cycler>=0.10->matplotlib<4.0.0,>=3.3.3->shor) (1.15.0)\n Requirement already satisfied: fastdtw in /usr/local/lib/python3.6/dist-packages (from qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (0.3.4)\n Requirement already satisfied: yfinance in /usr/local/lib/python3.6/dist-packages (from qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (0.1.55)\n Requirement already satisfied: quandl in /usr/local/lib/python3.6/dist-packages (from qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (3.6.0)\n Requirement already satisfied: retworkx>=0.5.0 in /usr/local/lib/python3.6/dist-packages (from qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (0.7.2)\n Requirement already satisfied: docplex==2.15.194 in /usr/local/lib/python3.6/dist-packages (from qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (2.15.194)\n Requirement already satisfied: h5py in /usr/local/lib/python3.6/dist-packages (from qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (2.10.0)\n Requirement already satisfied: pandas in /usr/local/lib/python3.6/dist-packages (from qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (1.1.5)\n Requirement already satisfied: scipy>=1.4 in /usr/local/lib/python3.6/dist-packages (from qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (1.4.1)\n Requirement already satisfied: setuptools>=40.1.0 in /usr/local/lib/python3.6/dist-packages (from qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (53.0.0)\n Requirement already satisfied: scikit-learn>=0.20.0 in /usr/local/lib/python3.6/dist-packages (from qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (0.22.2.post1)\n Requirement already satisfied: dataclasses; python_version < \"3.7\" in /usr/local/lib/python3.6/dist-packages (from qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (0.8)\n Requirement already satisfied: dlx in /usr/local/lib/python3.6/dist-packages (from qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (1.0.4)\n Requirement already satisfied: sympy>=1.3 in /usr/local/lib/python3.6/dist-packages (from qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (1.7.1)\n Requirement already satisfied: psutil>=5 in /usr/local/lib/python3.6/dist-packages (from qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (5.4.8)\n Requirement already satisfied: requests>=2.19 in /usr/local/lib/python3.6/dist-packages (from qiskit-ibmq-provider==0.11.1->qiskit<0.24.0,>=0.23.1->shor) (2.23.0)\n Requirement already satisfied: websockets>=8 in /usr/local/lib/python3.6/dist-packages (from qiskit-ibmq-provider==0.11.1->qiskit<0.24.0,>=0.23.1->shor) (8.1)\n Requirement already satisfied: nest-asyncio!=1.1.0,>=1.0.0 in /usr/local/lib/python3.6/dist-packages (from qiskit-ibmq-provider==0.11.1->qiskit<0.24.0,>=0.23.1->shor) (1.5.1)\n Requirement already satisfied: requests-ntlm>=1.1.0 in /usr/local/lib/python3.6/dist-packages (from qiskit-ibmq-provider==0.11.1->qiskit<0.24.0,>=0.23.1->shor) (1.1.0)\n Requirement already satisfied: urllib3>=1.21.1 in /usr/local/lib/python3.6/dist-packages (from qiskit-ibmq-provider==0.11.1->qiskit<0.24.0,>=0.23.1->shor) (1.24.3)\n Requirement already satisfied: networkx>=2.2 in /usr/local/lib/python3.6/dist-packages (from qiskit-ignis==0.5.2->qiskit<0.24.0,>=0.23.1->shor) (2.5)\n Requirement already satisfied: cython>=0.27.1 in /usr/local/lib/python3.6/dist-packages (from qiskit-aer==0.7.4->qiskit<0.24.0,>=0.23.1->shor) (0.29.21)\n Requirement already satisfied: pybind11>=2.4 in /usr/local/lib/python3.6/dist-packages (from qiskit-aer==0.7.4->qiskit<0.24.0,>=0.23.1->shor) (2.6.2)\n Requirement already satisfied: contextvars>=2.4; python_version < \"3.7\" in /usr/local/lib/python3.6/dist-packages (from qiskit-terra==0.16.4->qiskit<0.24.0,>=0.23.1->shor) (2.4)\n Requirement already satisfied: dill>=0.3 in /usr/local/lib/python3.6/dist-packages (from qiskit-terra==0.16.4->qiskit<0.24.0,>=0.23.1->shor) (0.3.3)\n Requirement already satisfied: fastjsonschema>=2.10 in /usr/local/lib/python3.6/dist-packages (from qiskit-terra==0.16.4->qiskit<0.24.0,>=0.23.1->shor) (2.15.0)\n Requirement already satisfied: ply>=3.10 in /usr/local/lib/python3.6/dist-packages (from qiskit-terra==0.16.4->qiskit<0.24.0,>=0.23.1->shor) (3.11)\n Requirement already satisfied: python-constraint>=1.4 in /usr/local/lib/python3.6/dist-packages (from qiskit-terra==0.16.4->qiskit<0.24.0,>=0.23.1->shor) (1.4.0)\n Requirement already satisfied: jsonschema>=2.6 in /usr/local/lib/python3.6/dist-packages (from qiskit-terra==0.16.4->qiskit<0.24.0,>=0.23.1->shor) (2.6.0)\n Requirement already satisfied: multitasking>=0.0.7 in /usr/local/lib/python3.6/dist-packages (from yfinance->qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (0.0.9)\n Requirement already satisfied: lxml>=4.5.1 in /usr/local/lib/python3.6/dist-packages (from yfinance->qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (4.6.2)\n Requirement already satisfied: inflection>=0.3.1 in /usr/local/lib/python3.6/dist-packages (from quandl->qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (0.5.1)\n Requirement already satisfied: more-itertools in /usr/local/lib/python3.6/dist-packages (from quandl->qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (8.7.0)\n Requirement already satisfied: pytz>=2017.2 in /usr/local/lib/python3.6/dist-packages (from pandas->qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (2018.9)\n Requirement already satisfied: joblib>=0.11 in /usr/local/lib/python3.6/dist-packages (from scikit-learn>=0.20.0->qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (1.0.0)\n Requirement already satisfied: mpmath>=0.19 in /usr/local/lib/python3.6/dist-packages (from sympy>=1.3->qiskit-aqua==0.8.2->qiskit<0.24.0,>=0.23.1->shor) (1.1.0)\n Requirement already satisfied: certifi>=2017.4.17 in /usr/local/lib/python3.6/dist-packages (from requests>=2.19->qiskit-ibmq-provider==0.11.1->qiskit<0.24.0,>=0.23.1->shor) (2020.12.5)\n Requirement already satisfied: idna<3,>=2.5 in /usr/local/lib/python3.6/dist-packages (from requests>=2.19->qiskit-ibmq-provider==0.11.1->qiskit<0.24.0,>=0.23.1->shor) (2.10)\n Requirement already satisfied: chardet<4,>=3.0.2 in /usr/local/lib/python3.6/dist-packages (from requests>=2.19->qiskit-ibmq-provider==0.11.1->qiskit<0.24.0,>=0.23.1->shor) (3.0.4)\n Requirement already satisfied: cryptography>=1.3 in /usr/local/lib/python3.6/dist-packages (from requests-ntlm>=1.1.0->qiskit-ibmq-provider==0.11.1->qiskit<0.24.0,>=0.23.1->shor) (3.4.5)\n Requirement already satisfied: ntlm-auth>=1.0.2 in /usr/local/lib/python3.6/dist-packages (from requests-ntlm>=1.1.0->qiskit-ibmq-provider==0.11.1->qiskit<0.24.0,>=0.23.1->shor) (1.5.0)\n Requirement already satisfied: decorator>=4.3.0 in /usr/local/lib/python3.6/dist-packages (from networkx>=2.2->qiskit-ignis==0.5.2->qiskit<0.24.0,>=0.23.1->shor) (4.4.2)\n Requirement already satisfied: immutables>=0.9 in /usr/local/lib/python3.6/dist-packages (from contextvars>=2.4; python_version < \"3.7\"->qiskit-terra==0.16.4->qiskit<0.24.0,>=0.23.1->shor) (0.15)\n Requirement already satisfied: cffi>=1.12 in /usr/local/lib/python3.6/dist-packages (from cryptography>=1.3->requests-ntlm>=1.1.0->qiskit-ibmq-provider==0.11.1->qiskit<0.24.0,>=0.23.1->shor) (1.14.4)\n Requirement already satisfied: pycparser in /usr/local/lib/python3.6/dist-packages (from cffi>=1.12->cryptography>=1.3->requests-ntlm>=1.1.0->qiskit-ibmq-provider==0.11.1->qiskit<0.24.0,>=0.23.1->shor) (2.20)\n\n\n### **Program 1: A Quantum Coin - Superposition on a quantum computer**\n\nThis program can be written with just a single qbit, and demonstrates quantum superposition.\n\n\n\n```\n# Program 1: A Quantum Coin - Superposition on a quantum computer\n\n# Build the circuit.\nqc = Circuit()\nqc.add(Qbits(1))\nqc.add(Hadamard(0))\nqc.add(Measure(0))\n\n# Draw the circuit diagram.\nprint(qc.draw())\n\n# Run the progam using a quantum simulator\njob = qc.run(100)\nprint(job.result.counts)\nprint(f'With our quantum coin, we flipped {job.result[0]} heads and {job.result[1]} tails')\n\n```\n\n \u250c\u2500\u2500\u2500\u2510 \u2591 \u250c\u2500\u2510\n q_0: \u2524 H \u251c\u2500\u2591\u2500\u2524M\u251c\n \u2514\u2500\u2500\u2500\u2518 \u2591 \u2514\u2565\u2518\n c: 1/\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256c\u2550\n \u2551 \n measure: 1/\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2569\u2550\n 0 \n {0: 47, 1: 53}\n With our quantum coin, we flipped 47 heads and 53 tails\n\n\n\n
                                        \n\n# **Understanding the code**\nQuantum computing can be a bit daunting at first, but the rest of this notebook should help give you a basic understanding of the math behind quantum computers.\n\nTo understand these programs, and quantum computers in general, it is simpler to start with something familiar, everyday classical computers, and work our way up from there.\n\n## **What is a computer?**\nA device which stores information (bits), and changes that information by following a set of rules (computation) to do something clever or useful.\n\n
                                        \n\n### ***Bit: A tiny switch, represents information***\nWhen we say information, we mean a thing or a concept. To define a thing or a concept, you must define what makes that thing or concept different.\n\nBits represent differences at the simplest level by enhabiting one of two possibilites: this or that, left or right, on or off, etc. A bit is a \"two state system\" and can represent any two things, states, or classes.\n\nTypically we will see bits representing states like:\n\n_0, off, or false_\n\n_1, on , or true_\n\n
                                        \n\nIt can be helpful to think of bits like tiny light switches.\n\n
                                        \n\n### ***Multiple Bits: Represent more information***\nIn order to represent more complicated things, like numbers, you need multiple bits.\n\nFor the numbers 0 through 7 (an 8 state system) we need 3 bits, since 3 bits have 8 possible combinations:\n\n$$Zero: 000\t\\quad\tOne: 001\t\\quad\tTwo: 010\t\\quad\tThree: 011\t\\quad\tFour: 100\t\\quad\tFive: 101\t\\quad\tSix: 110 \\quad Seven: 111$$\n\nIn general, $n$ bits can represent one of $2 ^ n$ possible states.\n\n
                                        \n\n### ***Computation / Gates: Flipping bits***\nA computation, or gate, changes information (a list of bits) from one state to another. It flips bits.\n\nA simple computational gate, NOT, flips a single bit:\n$$0 \\rightarrow 1$$\n$$1 \\rightarrow 0$$\n\nMore complicated gates use multiple input bits, and can operate on seperate output bits.\n\nThe \"and\" gate uses 2 input bits and 1 output bit. It outputs 1 if and only if both the input bits are 1:\n\n$$00 \\rightarrow 0$$\n$$01 \\rightarrow 0$$\n$$10 \\rightarrow 0$$\n$$11 \\rightarrow 1$$\n\n## **State Vector: Another way to view bits**\nA useful mathematical way to view bits is to use a list of numbers called the \"state vector\" (also called a \"one hot\" encoding.)\n\nThe state vector is a list of numbers, all of which are either zero or one.\nIt has two properties:\n- Only a single number is set to 1, everything else is 0\n- The length of the list is equal to the number of possible states it represents\n\n
                                        \n\nFor example, a single bit is 2 state system. The state vector representation would look like this:\n\n$\\vec{0} = |0\\rangle=\\begin{bmatrix}1 \\\\ 0 \\end{bmatrix}\\qquad$$\\vec{1} = |1\\rangle = \\begin{bmatrix}0 \\\\1 \\end{bmatrix}$\n\n\n
                                        \n\nFor some numbers between 0 - 7 (8 state system) their state vector representation looks like this:\n\n$Six=\\vec{6}=|110\\rangle=\\begin{bmatrix}0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\1 \\\\0\\end{bmatrix}\\quad$\n$Two=\\vec{2}=|010\\rangle=\\begin{bmatrix}0 \\\\0 \\\\1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0\\end{bmatrix}\\quad$\n$Zero=\\vec{0}=|010\\rangle=\\begin{bmatrix}1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0\\end{bmatrix}\\quad$\n\n
                                        \n\n### ***Computing with state vectors***\nComputations can be represented using matrix multiplication when using the state vector representation.\n\nThe NOT gate can be represented as the following 2 x 2 matrix:\n$$NOT =\\bigoplus=\\begin{pmatrix}0 & 1 \\\\ 1 & 0\\end{pmatrix}$$\n\nAs we can see, :\n$$NOT\\; |0\\rangle = \\begin{pmatrix}0 & 1 \\\\ 1 & 0\\end{pmatrix} \\begin{bmatrix}1 \\\\ 0\\end{bmatrix} = \\begin{bmatrix}0 \\\\ 1\\end{bmatrix} = |1\\rangle$$\n\n
                                        \n\n$$NOT \\, |1\\rangle = \\begin{pmatrix}0 & 1 \\\\ 1 & 0\\end{pmatrix} \\begin{bmatrix}0 \\\\ 1\\end{bmatrix}= \\begin{bmatrix}1 \\\\ 0\\end{bmatrix} = |0\\rangle$$\n\n\n## **What is a quantum computer?**\nA device which stores information (qbits), and changes that information by following a set of rules (quantum computation) to do something clever or useful.\n
                                        \n\n### ***Qbit / qubit: A tiny wheel, represent information***\nThe biggest difference between qbits and bits are the two properties:\n- Superposition\n- Entanglement\n\n### ***A Single Qbit: Superposition***\nSuperposition refers to the ability of the qbit to be in multiple positions at the same time. But, this is an overly complicated way of viewing it.\n\n
                                        \n\nMore simply put:\n\nBits are like traditional light switches:\n\n\n\nQbits are like circular dimmers (before they are measured):\n\n\n\n
                                        \n\nWith a bit, you have two possible values: on or off, 1 or 0.\n\n
                                        \n\nWith a qbit, you can spin it around, you have infinitely many rotational states. Straight up means the qbit is all the way on ($|1\\rangle$ state), and straight down means it is all the way off ($|0\\rangle$ state), but every other position on that dimmer is partially on and partially off.\n\n
                                        \n\nIn a sense, the qbit can be both on and off at the same time. This is known as superposition.\n\n
                                        \n\nWe are going from digital to analog, from discrete to continuous, and because of that, we can store a LOT more (infinitely more) information in a single qbit than we could hope to store in a regular bit. However, there is a catch:\n\n### ***Measurement: Changes the qbit***\n\nWhen we actually measure a qbit, it snaps to either $|1\\rangle$ or $|0\\rangle$ and nothing in between.\n\nIt is as if we rotate the dimmer switch, and then open our eyes to find the lights either all the way on or all the off, no matter how we rotate the dimmer switch.\n\n
                                        \n\nThat was a lot, so let's use our familiar state vector to make this more concrete.\n\n### ***State Vector: Another way to view qbits***\n\nWe can represent qbits using the state vector representation, in fact, we were doing quantum comuting all along! The very same representation we used for bits applies to qbits:\n\n$\\vec{0} = |0\\rangle=\\begin{bmatrix}1 \\\\ 0 \\end{bmatrix}\\qquad$$\\vec{1} = |1\\rangle = \\begin{bmatrix}0 \\\\1 \\end{bmatrix}$\n\n
                                        \n\nThese are the 0 and 1 states for the qbit, but more generally, since the qbit is like a wheel, it can take on many more states:\n\n$\\begin{bmatrix}\\alpha \\\\ \\beta \\end{bmatrix}\\qquad$ where $\\qquad \\alpha ^2 + \\beta ^2 = 1$\n\nThe second term is the pythagorean theorem. It ensures that the qbit's components have a fixed radius of 1, or in other words, a qbit's state is a rotation on a circle.\n\nAll \"superposition\" is referring to, is that we have a continuous set of values ($\\alpha$ and $\\beta$) instead of 2 states: just 0 or just 1.\n\n
                                        \n\n### ***Multiple Qbits: Entanglement***\n\nIn a similar way to bits, we can store WAY more information with multiple qbits than with a single qbit.\n\nIf we look at the state vector representation for 3 qbits:\n$$\\begin{bmatrix}a \\\\b \\\\c \\\\d \\\\e \\\\f \\\\g \\\\h\\end{bmatrix}$$\nwhere:\n$$a^2 + b^2 + c ^2 + d ^2 + e ^2 + f ^2 + g^2 + h^2 = 1$$\n\n
                                        \n\nAmazingly, if we have $n$ qbits, we have $2^n$ continuous angles to work with!! This scaling factor is the second reason that quantum computers have an advantage.\n\nEven with just 30 qbits, that gives:\n\n$2^{30} = 1,073,741,824$ continuous variables.\n\n\n\n\n# **Making the connection**\n\nArmed with our mathematical representation, we can re-visit our first program.\n\n## Program 1: A Quantum Coin - Superposition\n\n#### ***Building the circuit***\n* We will create a `Circuit()` with a single qbit, which by convention starts in the $|0\\rangle$ state.\n\n* Then we will add a `Hadamard` gate, which rotates the qbit precisely halfway between $|0\\rangle$ and $|1\\rangle$.\n\n* Then we add a `Measure` operation our qbit, which will \"collapse\" the qbit into either $|0\\rangle$ or $|1\\rangle$.\n\n* Finally, we will run the circuit 100 times`\n\n
                                        \n\n#### ***What should we expect?***\nSince the qbit is rotated halfway between $|0\\rangle$ and $|1\\rangle$, we expect the measured qbit to \"collapse\" to those states with equal probability (about 50 times each).\n\n\n#### ***Why is this \"superposition\"?***\nBecause our circuit is moving the qbit into a state that is an approximately equal linear combination of the basis states $|0\\rangle$ or $|1\\rangle$\n\nRather than just being 0, or just being 1 there is an equal chance of measuring either.\n\n\n```\n# Program 1: A Quantum Coin - Superposition on a quantum computer\n\nqc = Circuit()\nqc.add(Qbits(1)) # Add 1 Qbit, which by convention starts in the 0 / down / off state.\nqc.add(Hadamard(0)) # Apply the Hadamard gate to the 1st qbit (index 0), rotating it to a 50 - 50 superposition\nqc.add(Measure(0)) # Measure the 1st qbit (index 0)\n\nprint(qc.draw())\n\njob = qc.run(100) # Run the circuit 100 times (using a simulator)\nprint(job.result.counts)\n```\n\n \u250c\u2500\u2500\u2500\u2510 \u2591 \u250c\u2500\u2510\n q_0: \u2524 H \u251c\u2500\u2591\u2500\u2524M\u251c\n \u2514\u2500\u2500\u2500\u2518 \u2591 \u2514\u2565\u2518\n c: 1/\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256c\u2550\n \u2551 \n measure: 1/\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2569\u2550\n 0 \n {0: 46, 1: 54}\n\n\nAs expected, we see about half of the 100 results are in the $|0\\rangle$ state and half are in the $|1\\rangle$!\n\nThat is it! We just demonstrated quantum superposition.\n\n\n### **Program 2: Quantum Entanglement**\n\nNow we will introduce a second qbit into our circuit, and a new gate `CNOT`\n\n#### ***CNOT gate***\n`CNOT` or the \"controlled not\" gate, takes 2 qbits as input. It flips the second (target) qbit if and only if the first (control) qbit is 1.\n\nThe `CNOT` transitions the input states to the following outputs:\n$$|00\\rangle \\rightarrow |00\\rangle$$\n$$|01\\rangle \\rightarrow |01\\rangle$$\n$$|10\\rangle \\rightarrow |11\\rangle$$\n$$|11\\rangle \\rightarrow |10\\rangle$$\n\n#### ***Building the circuit***\n* In our circuit, we will add 2 qbits which start in the $|0\\rangle$ state.\n* Then, we apply the `Hadamard` gate to move the first (control) qbit into a 50 - 50 superposition state\n* Then, we will apply the entanglement gate `CNOT` to the first (control) qbit, and the second (target) qbit\n* Then we add a `Measurement` operation on both of our qbits\n* Finally, we run the circuit 100 times\n\n
                                        \n\n#### ***What should we expect?***\nAs before, we expect the first qbit to be $|0\\rangle$ about half of the time, and $|1\\rangle$ the other half of the time.\n\nAs for the second qbit, it should become \"entangled\" with the first, through the `CNOT` gate. Referring to the state transitions above, since we expect the inputs at that point in the circuit to be $|00\\rangle$ or $|10\\rangle$ we expect the output states to be $|00\\rangle$ and $|11\\rangle$.\n\n#### ***Why is this \"entanglement\"?***\nEntanglement means the state of the system only makes sense with multiple qbits, or in other words, we can't factor the state back into individual qbits.\n\nIn our case, the second qbit's state depends strongly on the first qbits. We can't replicate the result if we try to seperate the circuit into two sub-circutis with 1 qbit each.\n\n\n\n```\n# Program 2: Entanglement Quantum Circuit with 2 Qbits\n\nqc = Circuit()\nqc.add(Qbits(2))\nqc.add(Hadamard(0))\nqc.add(CNOT(0, 1)) # Add a control not gate with the 0th qbit as the control bit\nqc.add(Measure(0, 1))\n\n# Draw the circuit diagramt\nprint(qc.draw())\n\n# Run the program 100 times\nprint(qc.run(100).result)\n```\n\n \u250c\u2500\u2500\u2500\u2510 \u2591 \u250c\u2500\u2510 \n q_0: \u2524 H \u251c\u2500\u2500\u25a0\u2500\u2500\u2500\u2591\u2500\u2524M\u251c\u2500\u2500\u2500\n \u2514\u2500\u2500\u2500\u2518\u250c\u2500\u2534\u2500\u2510 \u2591 \u2514\u2565\u2518\u250c\u2500\u2510\n q_1: \u2500\u2500\u2500\u2500\u2500\u2524 X \u251c\u2500\u2591\u2500\u2500\u256b\u2500\u2524M\u251c\n \u2514\u2500\u2500\u2500\u2518 \u2591 \u2551 \u2514\u2565\u2518\n c: 2/\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256c\u2550\u2550\u256c\u2550\n \u2551 \u2551 \n measure: 2/\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2569\u2550\u2550\u2569\u2550\n 0 1 \n {'00000': 57, '00011': 43}\n\n\n## ***We just demonstrated quantum entanglement***\n\nAs you can see, the results are as expected, and are tied together (entangled) such that only $|00\\rangle$ and $|11\\rangle$ are observed.\n\n", "meta": {"hexsha": "b8e1d36a4c2cdcc674c98cab008aa741e657605d", "size": 29363, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/intro/01_superposition_and_entanglement.ipynb", "max_stars_repo_name": "Shor-dev/shor", "max_stars_repo_head_hexsha": "8814030a1a7890084362327a14dd8838c246dd8a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 8, "max_stars_repo_stars_event_min_datetime": "2020-05-14T23:31:43.000Z", "max_stars_repo_stars_event_max_datetime": "2021-02-24T20:38:32.000Z", "max_issues_repo_path": "tutorials/intro/01_superposition_and_entanglement.ipynb", "max_issues_repo_name": "Shor-dev/shor", "max_issues_repo_head_hexsha": "8814030a1a7890084362327a14dd8838c246dd8a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 41, "max_issues_repo_issues_event_min_datetime": "2020-06-08T22:42:17.000Z", "max_issues_repo_issues_event_max_datetime": "2021-04-21T17:34:42.000Z", "max_forks_repo_path": "tutorials/intro/01_superposition_and_entanglement.ipynb", "max_forks_repo_name": "Shor-dev/shor", "max_forks_repo_head_hexsha": "8814030a1a7890084362327a14dd8838c246dd8a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2020-06-30T04:09:55.000Z", "max_forks_repo_forks_event_max_datetime": "2020-09-27T20:58:32.000Z", "avg_line_length": 52.0620567376, "max_line_length": 306, "alphanum_fraction": 0.603787079, "converted": true, "num_tokens": 7553, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737214979746, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.42737590977058043}} {"text": "# Batch Normalization\nOne way to make deep networks easier to train is to **use more sophisticated optimization procedures such as SGD+momentum, RMSProp, or Adam**. Another strategy is to **change the architecture of the network to make it easier to train**. \nOne idea along these lines is batch normalization which was proposed by [1] in 2015.\n\nThe idea is relatively straightforward. **Machine learning methods tend to work better when their input data consists of uncorrelated features with zero mean and unit variance**. When training a neural network, we can preprocess the data before feeding it to the network to explicitly decorrelate its features; this will ensure that the first layer of the network sees data that follows a nice distribution. However, even if we preprocess the input data, the activations at deeper layers of the network will likely no longer be decorrelated and will no longer have zero mean or unit variance since they are output from earlier layers in the network. Even worse, during the training process the distribution of features at each layer of the network will shift as the weights of each layer are updated.\n\nThe authors of [1] hypothesize that the shifting distribution of features inside deep neural networks may make training deep networks more difficult. To overcome this problem, [1] proposes to insert **batch normalization layers** into the network. At training time, a **batch normalization layer uses a minibatch of data to estimate the mean and standard deviation of each feature. These estimated means and standard deviations are then used to center and normalize the features of the minibatch. A running average of these means and standard deviations is kept during training, and at test time these running averages are used to center and normalize features**.\n\nIt is possible that this normalization strategy could reduce the representational power of the network, since it may sometimes be optimal for certain layers to have features that are not zero-mean or unit variance. To this end, the batch normalization layer includes learnable shift and scale parameters for each feature dimension.\n\n[1] [Sergey Ioffe and Christian Szegedy, \"Batch Normalization: Accelerating Deep Network Training by Reducing\nInternal Covariate Shift\", ICML 2015.](https://arxiv.org/abs/1502.03167)\n\n\n```python\n# As usual, a bit of setup\nimport time\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom cs231n.classifiers.fc_net import *\nfrom cs231n.data_utils import get_CIFAR10_data\nfrom cs231n.gradient_check import eval_numerical_gradient, eval_numerical_gradient_array\nfrom cs231n.solver import Solver\n\n%matplotlib inline\nplt.rcParams['figure.figsize'] = (10.0, 8.0) # set default size of plots\nplt.rcParams['image.interpolation'] = 'nearest'\nplt.rcParams['image.cmap'] = 'gray'\n\n# for auto-reloading external modules\n# see http://stackoverflow.com/questions/1907993/autoreload-of-modules-in-ipython\n%load_ext autoreload\n%autoreload 2\n\ndef rel_error(x, y):\n \"\"\" returns relative error \"\"\"\n return np.max(np.abs(x - y) / (np.maximum(1e-8, np.abs(x) + np.abs(y))))\n\ndef print_mean_std(x,axis=0):\n print(' means: ', x.mean(axis=axis))\n print(' stds: ', x.std(axis=axis))\n print() \n```\n\n\n```python\n# Load the (preprocessed) CIFAR10 data.\ndata = get_CIFAR10_data()\nfor k, v in data.items():\n print('%s: ' % k, v.shape)\n```\n\n X_train: (49000, 3, 32, 32)\n y_train: (49000,)\n X_val: (1000, 3, 32, 32)\n y_val: (1000,)\n X_test: (1000, 3, 32, 32)\n y_test: (1000,)\n\n\n## Batch normalization: forward\nIn the file `cs231n/layers.py`, implement the batch normalization forward pass in the function `batchnorm_forward`. Once you have done so, run the following to test your implementation.\n\nReferencing the paper linked to above in [1] may be helpful!\n\n\n```python\n# Check the training-time forward pass by checking means and variances\n# of features both before and after batch normalization \n\n# Simulate the forward pass for a two-layer network\nnp.random.seed(231)\nN, D1, D2, D3 = 200, 50, 60, 3\nX = np.random.randn(N, D1)\nW1 = np.random.randn(D1, D2)\nW2 = np.random.randn(D2, D3)\na = np.maximum(0, X.dot(W1)).dot(W2)\n\nprint('Before batch normalization:')\nprint_mean_std(a,axis=0)\n\ngamma = np.ones((D3,))\nbeta = np.zeros((D3,))\n# Means should be close to zero and stds close to one\nprint('After batch normalization (gamma=1, beta=0)')\na_norm, _ = batchnorm_forward(a, gamma, beta, {'mode': 'train'})\nprint_mean_std(a_norm,axis=0)\n\ngamma = np.asarray([1.0, 2.0, 3.0])\nbeta = np.asarray([11.0, 12.0, 13.0])\n# Now means should be close to beta and stds close to gamma\nprint('After batch normalization (gamma=', gamma, ', beta=', beta, ')')\na_norm, _ = batchnorm_forward(a, gamma, beta, {'mode': 'train'})\nprint_mean_std(a_norm,axis=0)\n```\n\n Before batch normalization:\n means: [ -2.3814598 -13.18038246 1.91780462]\n stds: [27.18502186 34.21455511 37.68611762]\n \n After batch normalization (gamma=1, beta=0)\n means: [5.99520433e-17 6.93889390e-17 8.32667268e-19]\n stds: [0.99999999 1. 1. ]\n \n After batch normalization (gamma= [1. 2. 3.] , beta= [11. 12. 13.] )\n means: [11. 12. 13.]\n stds: [0.99999999 1.99999999 2.99999999]\n \n\n\n\n```python\n# Check the test-time forward pass by running the training-time\n# forward pass many times to warm up the running averages, and then\n# checking the means and variances of activations after a test-time\n# forward pass.\n\nnp.random.seed(231)\nN, D1, D2, D3 = 200, 50, 60, 3\nW1 = np.random.randn(D1, D2)\nW2 = np.random.randn(D2, D3)\n\nbn_param = {'mode': 'train'}\ngamma = np.ones(D3)\nbeta = np.zeros(D3)\n\nfor t in range(50):\n X = np.random.randn(N, D1)\n a = np.maximum(0, X.dot(W1)).dot(W2)\n batchnorm_forward(a, gamma, beta, bn_param)\n\nbn_param['mode'] = 'test'\nX = np.random.randn(N, D1)\na = np.maximum(0, X.dot(W1)).dot(W2)\na_norm, _ = batchnorm_forward(a, gamma, beta, bn_param)\n\n# Means should be close to zero and stds close to one, but will be\n# noisier than training-time forward passes.\nprint('After batch normalization (test-time):')\nprint_mean_std(a_norm,axis=0)\n```\n\n After batch normalization (test-time):\n means: [-0.03927354 -0.04349152 -0.10452688]\n stds: [1.01531428 1.01238373 0.97819988]\n \n\n\n## Batch normalization: backward\nNow implement the backward pass for batch normalization in the function `batchnorm_backward`.\n\nTo derive the backward pass you should write out the computation graph for batch normalization and backprop through each of the intermediate nodes. Some intermediates may have multiple outgoing branches; make sure to sum gradients across these branches in the backward pass.\n\nOnce you have finished, run the following to numerically check your backward pass.\n\n\n```python\n# Gradient check batchnorm backward pass\nnp.random.seed(231)\nN, D = 4, 5\nx = 5 * np.random.randn(N, D) + 12\ngamma = np.random.randn(D)\nbeta = np.random.randn(D)\ndout = np.random.randn(N, D)\n\nbn_param = {'mode': 'train'}\nfx = lambda x: batchnorm_forward(x, gamma, beta, bn_param)[0]\nfg = lambda a: batchnorm_forward(x, a, beta, bn_param)[0]\nfb = lambda b: batchnorm_forward(x, gamma, b, bn_param)[0]\n\ndx_num = eval_numerical_gradient_array(fx, x, dout)\nda_num = eval_numerical_gradient_array(fg, gamma.copy(), dout)\ndb_num = eval_numerical_gradient_array(fb, beta.copy(), dout)\n\n_, cache = batchnorm_forward(x, gamma, beta, bn_param)\ndx, dgamma, dbeta = batchnorm_backward(dout, cache)\n#You should expect to see relative errors between 1e-13 and 1e-8\nprint('dx error: ', rel_error(dx_num, dx))\nprint('dgamma error: ', rel_error(da_num, dgamma))\nprint('dbeta error: ', rel_error(db_num, dbeta))\n```\n\n dx error: 1.6674604875341426e-09\n dgamma error: 7.417225040694815e-13\n dbeta error: 2.379446949959628e-12\n\n\n## Batch normalization: alternative backward\nIn class we talked about two different implementations for the sigmoid backward pass. One strategy is to write out a computation graph composed of simple operations and backprop through all intermediate values. Another strategy is to work out the derivatives on paper. For example, you can derive a very simple formula for the sigmoid function's backward pass by simplifying gradients on paper.\n\nSurprisingly, it turns out that you can do a similar simplification for the batch normalization backward pass too! \n\nIn the forward pass, given a set of inputs $X=\\begin{bmatrix}x_1\\\\x_2\\\\...\\\\x_N\\end{bmatrix}$, \n\nwe first calculate the mean $\\mu$ and variance $v$.\nWith $\\mu$ and $v$ calculated, we can calculate the standard deviation $\\sigma$ and normalized data $Y$.\nThe equations and graph illustration below describe the computation ($y_i$ is the i-th element of the vector $Y$).\n\n\\begin{align}\n& \\mu=\\frac{1}{N}\\sum_{k=1}^N x_k & v=\\frac{1}{N}\\sum_{k=1}^N (x_k-\\mu)^2 \\\\\n& \\sigma=\\sqrt{v+\\epsilon} & y_i=\\frac{x_i-\\mu}{\\sigma}\n\\end{align}\n\n\n\nThe meat of our problem during backpropagation is to compute $\\frac{\\partial L}{\\partial X}$, given the upstream gradient we receive, $\\frac{\\partial L}{\\partial Y}.$ To do this, recall the chain rule in calculus gives us $\\frac{\\partial L}{\\partial X} = \\frac{\\partial L}{\\partial Y} \\cdot \\frac{\\partial Y}{\\partial X}$.\n\nThe unknown/hart part is $\\frac{\\partial Y}{\\partial X}$. We can find this by first deriving step-by-step our local gradients at \n$\\frac{\\partial v}{\\partial X}$, $\\frac{\\partial \\mu}{\\partial X}$,\n$\\frac{\\partial \\sigma}{\\partial v}$, \n$\\frac{\\partial Y}{\\partial \\sigma}$, and $\\frac{\\partial Y}{\\partial \\mu}$,\nand then use the chain rule to compose these gradients (which appear in the form of vectors!) appropriately to compute $\\frac{\\partial Y}{\\partial X}$.\n\nIf it's challenging to directly reason about the gradients over $X$ and $Y$ which require matrix multiplication, try reasoning about the gradients in terms of individual elements $x_i$ and $y_i$ first: in that case, you will need to come up with the derivations for $\\frac{\\partial L}{\\partial x_i}$, by relying on the Chain Rule to first calculate the intermediate $\\frac{\\partial \\mu}{\\partial x_i}, \\frac{\\partial v}{\\partial x_i}, \\frac{\\partial \\sigma}{\\partial x_i},$ then assemble these pieces to calculate $\\frac{\\partial y_i}{\\partial x_i}$. \n\nYou should make sure each of the intermediary gradient derivations are all as simplified as possible, for ease of implementation. \n\nAfter doing so, implement the simplified batch normalization backward pass in the function `batchnorm_backward_alt` and compare the two implementations by running the following. Your two implementations should compute nearly identical results, but the alternative implementation should be a bit faster.\n\n\n```python\nnp.random.seed(231)\nN, D = 100, 500\nx = 5 * np.random.randn(N, D) + 12\ngamma = np.random.randn(D)\nbeta = np.random.randn(D)\ndout = np.random.randn(N, D)\n\nbn_param = {'mode': 'train'}\nout, cache = batchnorm_forward(x, gamma, beta, bn_param)\n\nt1 = time.time()\ndx1, dgamma1, dbeta1 = batchnorm_backward(dout, cache)\nt2 = time.time()\ndx2, dgamma2, dbeta2 = batchnorm_backward_alt(dout, cache)\nt3 = time.time()\n\nprint('dx difference: ', rel_error(dx1, dx2))\nprint('dgamma difference: ', rel_error(dgamma1, dgamma2))\nprint('dbeta difference: ', rel_error(dbeta1, dbeta2))\nprint('speedup: %.2fx' % ((t2 - t1) / (t3 - t2)))\n```\n\n dx difference: 3.423369562270315e-13\n dgamma difference: 0.0\n dbeta difference: 0.0\n speedup: 1.96x\n\n\n## Fully Connected Nets with Batch Normalization\nNow that you have a working implementation for batch normalization, go back to your `FullyConnectedNet` in the file `cs231n/classifiers/fc_net.py`. Modify your implementation to add batch normalization.\n\nConcretely, when the `normalization` flag is set to `\"batchnorm\"` in the constructor, you should insert a batch normalization layer before each ReLU nonlinearity. The outputs from the last layer of the network should not be normalized. Once you are done, run the following to gradient-check your implementation.\n\nHINT: You might find it useful to define an additional helper layer similar to those in the file `cs231n/layer_utils.py`. If you decide to do so, do it in the file `cs231n/classifiers/fc_net.py`.\n\n\n```python\nnp.random.seed(231)\nN, D, H1, H2, C = 2, 15, 20, 30, 10\nX = np.random.randn(N, D)\ny = np.random.randint(C, size=(N,))\n\n# You should expect losses between 1e-4~1e-10 for W, \n# losses between 1e-08~1e-10 for b,\n# and losses between 1e-08~1e-09 for beta and gammas.\nfor reg in [0, 3.14]:\n print('Running check with reg = ', reg)\n model = FullyConnectedNet([H1, H2], input_dim=D, num_classes=C,\n reg=reg, weight_scale=5e-2, dtype=np.float64,\n normalization='batchnorm')\n\n loss, grads = model.loss(X, y)\n print('Initial loss: ', loss)\n\n for name in sorted(grads):\n f = lambda _: model.loss(X, y)[0]\n grad_num = eval_numerical_gradient(f, model.params[name], verbose=False, h=1e-5)\n print('%s relative error: %.2e' % (name, rel_error(grad_num, grads[name])))\n if reg == 0: print()\n```\n\n Running check with reg = 0\n Initial loss: 2.2611955101340957\n W1 relative error: 1.10e-04\n W2 relative error: 3.11e-06\n W3 relative error: 4.05e-10\n b1 relative error: 4.44e-08\n b2 relative error: 2.22e-08\n b3 relative error: 1.01e-10\n beta1 relative error: 7.33e-09\n beta2 relative error: 1.89e-09\n gamma1 relative error: 6.96e-09\n gamma2 relative error: 2.41e-09\n \n Running check with reg = 3.14\n Initial loss: 6.996533220108303\n W1 relative error: 1.98e-06\n W2 relative error: 2.29e-06\n W3 relative error: 1.11e-08\n b1 relative error: 5.55e-09\n b2 relative error: 2.22e-08\n b3 relative error: 2.10e-10\n beta1 relative error: 6.65e-09\n beta2 relative error: 3.39e-09\n gamma1 relative error: 6.27e-09\n gamma2 relative error: 5.28e-09\n\n\n# Batchnorm for deep networks\nRun the following to train a six-layer network on a subset of 1000 training examples both with and without batch normalization.\n\n\n```python\nnp.random.seed(231)\n# Try training a very deep net with batchnorm\nhidden_dims = [100, 100, 100, 100, 100]\n\nnum_train = 1000\nsmall_data = {\n 'X_train': data['X_train'][:num_train],\n 'y_train': data['y_train'][:num_train],\n 'X_val': data['X_val'],\n 'y_val': data['y_val'],\n}\n\nweight_scale = 2e-2\nbn_model = FullyConnectedNet(hidden_dims, weight_scale=weight_scale, normalization='batchnorm')\nmodel = FullyConnectedNet(hidden_dims, weight_scale=weight_scale, normalization=None)\n\nprint('Solver with batch norm:')\nbn_solver = Solver(bn_model, small_data,\n num_epochs=10, batch_size=50,\n update_rule='adam',\n optim_config={\n 'learning_rate': 1e-3,\n },\n verbose=True,print_every=20)\nbn_solver.train()\n\nprint('\\nSolver without batch norm:')\nsolver = Solver(model, small_data,\n num_epochs=10, batch_size=50,\n update_rule='adam',\n optim_config={\n 'learning_rate': 1e-3,\n },\n verbose=True, print_every=20)\nsolver.train()\n```\n\n Solver with batch norm:\n (Epoch 0 / 10) train acc: 0.107000; val_acc: 0.115000\n (Iteration 20 / 200) loss: 1.940574\n (Epoch 1 / 10) train acc: 0.314000; val_acc: 0.266000\n (Iteration 40 / 200) loss: 1.733858\n (Epoch 2 / 10) train acc: 0.385000; val_acc: 0.279000\n (Iteration 60 / 200) loss: 1.588897\n (Epoch 3 / 10) train acc: 0.493000; val_acc: 0.309000\n (Iteration 80 / 200) loss: 1.532039\n (Epoch 4 / 10) train acc: 0.531000; val_acc: 0.307000\n (Iteration 100 / 200) loss: 1.309579\n (Epoch 5 / 10) train acc: 0.574000; val_acc: 0.313000\n (Iteration 120 / 200) loss: 1.090331\n (Epoch 6 / 10) train acc: 0.634000; val_acc: 0.338000\n (Iteration 140 / 200) loss: 1.044100\n (Epoch 7 / 10) train acc: 0.683000; val_acc: 0.323000\n (Iteration 160 / 200) loss: 0.726503\n (Epoch 8 / 10) train acc: 0.769000; val_acc: 0.327000\n (Iteration 180 / 200) loss: 0.856740\n (Epoch 9 / 10) train acc: 0.793000; val_acc: 0.335000\n (Iteration 200 / 200) loss: 0.810437\n (Epoch 10 / 10) train acc: 0.783000; val_acc: 0.327000\n \n Solver without batch norm:\n (Epoch 0 / 10) train acc: 0.129000; val_acc: 0.131000\n (Iteration 20 / 200) loss: 2.011618\n (Epoch 1 / 10) train acc: 0.283000; val_acc: 0.250000\n (Iteration 40 / 200) loss: 1.853587\n (Epoch 2 / 10) train acc: 0.316000; val_acc: 0.277000\n (Iteration 60 / 200) loss: 1.903742\n (Epoch 3 / 10) train acc: 0.373000; val_acc: 0.282000\n (Iteration 80 / 200) loss: 1.557937\n (Epoch 4 / 10) train acc: 0.390000; val_acc: 0.310000\n (Iteration 100 / 200) loss: 1.309605\n (Epoch 5 / 10) train acc: 0.434000; val_acc: 0.300000\n (Iteration 120 / 200) loss: 1.414160\n (Epoch 6 / 10) train acc: 0.535000; val_acc: 0.345000\n (Iteration 140 / 200) loss: 1.308890\n (Epoch 7 / 10) train acc: 0.530000; val_acc: 0.304000\n (Iteration 160 / 200) loss: 1.029080\n (Epoch 8 / 10) train acc: 0.628000; val_acc: 0.339000\n (Iteration 180 / 200) loss: 0.934465\n (Epoch 9 / 10) train acc: 0.654000; val_acc: 0.342000\n (Iteration 200 / 200) loss: 0.793320\n (Epoch 10 / 10) train acc: 0.712000; val_acc: 0.328000\n\n\nRun the following to visualize the results from two networks trained above. You should find that using batch normalization helps the network to converge much faster.\n\n\n```python\ndef plot_training_history(title, label, baseline, bn_solvers, plot_fn, bl_marker='.', bn_marker='.', labels=None):\n \"\"\"utility function for plotting training history\"\"\"\n plt.title(title)\n plt.xlabel(label)\n bn_plots = [plot_fn(bn_solver) for bn_solver in bn_solvers]\n bl_plot = plot_fn(baseline)\n num_bn = len(bn_plots)\n for i in range(num_bn):\n label='with_norm'\n if labels is not None:\n label += str(labels[i])\n plt.plot(bn_plots[i], bn_marker, label=label)\n label='baseline'\n if labels is not None:\n label += str(labels[0])\n plt.plot(bl_plot, bl_marker, label=label)\n plt.legend(loc='lower center', ncol=num_bn+1) \n\n \nplt.subplot(3, 1, 1)\nplot_training_history('Training loss','Iteration', solver, [bn_solver], \\\n lambda x: x.loss_history, bl_marker='o', bn_marker='o')\nplt.subplot(3, 1, 2)\nplot_training_history('Training accuracy','Epoch', solver, [bn_solver], \\\n lambda x: x.train_acc_history, bl_marker='-o', bn_marker='-o')\nplt.subplot(3, 1, 3)\nplot_training_history('Validation accuracy','Epoch', solver, [bn_solver], \\\n lambda x: x.val_acc_history, bl_marker='-o', bn_marker='-o')\n\nplt.gcf().set_size_inches(15, 15)\nplt.show()\n```\n\n# Batch normalization and initialization\nWe will now run a small experiment to study the interaction of batch normalization and weight initialization.\n\nThe first cell will train 8-layer networks both with and without batch normalization using different scales for weight initialization. The second layer will plot training accuracy, validation set accuracy, and training loss as a function of the weight initialization scale.\n\n\n```python\nnp.random.seed(231)\n# Try training a very deep net with batchnorm\nhidden_dims = [50, 50, 50, 50, 50, 50, 50]\nnum_train = 1000\nsmall_data = {\n 'X_train': data['X_train'][:num_train],\n 'y_train': data['y_train'][:num_train],\n 'X_val': data['X_val'],\n 'y_val': data['y_val'],\n}\n\nbn_solvers_ws = {}\nsolvers_ws = {}\nweight_scales = np.logspace(-4, 0, num=20)\nfor i, weight_scale in enumerate(weight_scales):\n print('Running weight scale %d / %d' % (i + 1, len(weight_scales)))\n bn_model = FullyConnectedNet(hidden_dims, weight_scale=weight_scale, normalization='batchnorm')\n model = FullyConnectedNet(hidden_dims, weight_scale=weight_scale, normalization=None)\n\n bn_solver = Solver(bn_model, small_data,\n num_epochs=10, batch_size=50,\n update_rule='adam',\n optim_config={\n 'learning_rate': 1e-3,\n },\n verbose=False, print_every=200)\n bn_solver.train()\n bn_solvers_ws[weight_scale] = bn_solver\n\n solver = Solver(model, small_data,\n num_epochs=10, batch_size=50,\n update_rule='adam',\n optim_config={\n 'learning_rate': 1e-3,\n },\n verbose=False, print_every=200)\n solver.train()\n solvers_ws[weight_scale] = solver\n```\n\n Running weight scale 1 / 20\n Running weight scale 2 / 20\n Running weight scale 3 / 20\n Running weight scale 4 / 20\n Running weight scale 5 / 20\n Running weight scale 6 / 20\n Running weight scale 7 / 20\n Running weight scale 8 / 20\n Running weight scale 9 / 20\n Running weight scale 10 / 20\n Running weight scale 11 / 20\n Running weight scale 12 / 20\n Running weight scale 13 / 20\n Running weight scale 14 / 20\n Running weight scale 15 / 20\n Running weight scale 16 / 20\n Running weight scale 17 / 20\n Running weight scale 18 / 20\n Running weight scale 19 / 20\n Running weight scale 20 / 20\n\n\n\n```python\n# Plot results of weight scale experiment\nbest_train_accs, bn_best_train_accs = [], []\nbest_val_accs, bn_best_val_accs = [], []\nfinal_train_loss, bn_final_train_loss = [], []\n\nfor ws in weight_scales:\n best_train_accs.append(max(solvers_ws[ws].train_acc_history))\n bn_best_train_accs.append(max(bn_solvers_ws[ws].train_acc_history))\n \n best_val_accs.append(max(solvers_ws[ws].val_acc_history))\n bn_best_val_accs.append(max(bn_solvers_ws[ws].val_acc_history))\n \n final_train_loss.append(np.mean(solvers_ws[ws].loss_history[-100:]))\n bn_final_train_loss.append(np.mean(bn_solvers_ws[ws].loss_history[-100:]))\n \nplt.subplot(3, 1, 1)\nplt.title('Best val accuracy vs weight initialization scale')\nplt.xlabel('Weight initialization scale')\nplt.ylabel('Best val accuracy')\nplt.semilogx(weight_scales, best_val_accs, '-o', label='baseline')\nplt.semilogx(weight_scales, bn_best_val_accs, '-o', label='batchnorm')\nplt.legend(ncol=2, loc='lower right')\n\nplt.subplot(3, 1, 2)\nplt.title('Best train accuracy vs weight initialization scale')\nplt.xlabel('Weight initialization scale')\nplt.ylabel('Best training accuracy')\nplt.semilogx(weight_scales, best_train_accs, '-o', label='baseline')\nplt.semilogx(weight_scales, bn_best_train_accs, '-o', label='batchnorm')\nplt.legend()\n\nplt.subplot(3, 1, 3)\nplt.title('Final training loss vs weight initialization scale')\nplt.xlabel('Weight initialization scale')\nplt.ylabel('Final training loss')\nplt.semilogx(weight_scales, final_train_loss, '-o', label='baseline')\nplt.semilogx(weight_scales, bn_final_train_loss, '-o', label='batchnorm')\nplt.legend()\nplt.gca().set_ylim(1.0, 3.5)\n\nplt.gcf().set_size_inches(15, 15)\nplt.show()\n```\n\n## Inline Question 1:\nDescribe the results of this experiment. How does the scale of weight initialization affect models with/without batch normalization differently, and why?\n\n## Answer:\n- model with batch normalization is less sensitive to the sacle of weight initialization, since the change with scale is smoother than that of model without batch normalization\n- model with batchnorm can work with small weight initialization scale ($10^{-4} - 10^{-2.5}$), but model without batchnorm performs badly when small weight initialization scale is applied\n- the best weight initialization scale for model with batchnorm is around $10^{-2}$, while it is around $10^{-1}$ for model without batchnorm\n\n\n# Batch normalization and batch size\nWe will now run a small experiment to study the interaction of batch normalization and batch size.\n\nThe first cell will train 6-layer networks both with and without batch normalization using different batch sizes. The second layer will plot training accuracy and validation set accuracy over time.\n\n\n```python\ndef run_batchsize_experiments(normalization_mode):\n np.random.seed(231)\n # Try training a very deep net with batchnorm\n hidden_dims = [100, 100, 100, 100, 100]\n num_train = 1000\n small_data = {\n 'X_train': data['X_train'][:num_train],\n 'y_train': data['y_train'][:num_train],\n 'X_val': data['X_val'],\n 'y_val': data['y_val'],\n }\n n_epochs=10\n weight_scale = 2e-2\n batch_sizes = [5,10,50]\n lr = 10**(-3.5)\n solver_bsize = batch_sizes[0]\n\n print('No normalization: batch size = ',solver_bsize)\n model = FullyConnectedNet(hidden_dims, weight_scale=weight_scale, normalization=None)\n solver = Solver(model, small_data,\n num_epochs=n_epochs, batch_size=solver_bsize,\n update_rule='adam',\n optim_config={\n 'learning_rate': lr,\n },\n verbose=False)\n solver.train()\n \n bn_solvers = []\n for i in range(len(batch_sizes)):\n b_size=batch_sizes[i]\n print('Normalization: batch size = ',b_size)\n bn_model = FullyConnectedNet(hidden_dims, weight_scale=weight_scale, normalization=normalization_mode)\n bn_solver = Solver(bn_model, small_data,\n num_epochs=n_epochs, batch_size=b_size,\n update_rule='adam',\n optim_config={\n 'learning_rate': lr,\n },\n verbose=False)\n bn_solver.train()\n bn_solvers.append(bn_solver)\n \n return bn_solvers, solver, batch_sizes\n\nbatch_sizes = [5,10,50]\nbn_solvers_bsize, solver_bsize, batch_sizes = run_batchsize_experiments('batchnorm')\n```\n\n No normalization: batch size = 5\n Normalization: batch size = 5\n Normalization: batch size = 10\n Normalization: batch size = 50\n\n\n\n```python\nplt.subplot(2, 1, 1)\nplot_training_history('Training accuracy (Batch Normalization)','Epoch', solver_bsize, bn_solvers_bsize, \\\n lambda x: x.train_acc_history, bl_marker='-^', bn_marker='-o', labels=batch_sizes)\nplt.subplot(2, 1, 2)\nplot_training_history('Validation accuracy (Batch Normalization)','Epoch', solver_bsize, bn_solvers_bsize, \\\n lambda x: x.val_acc_history, bl_marker='-^', bn_marker='-o', labels=batch_sizes)\n\nplt.gcf().set_size_inches(15, 10)\nplt.show()\n```\n\n## Inline Question 2:\nDescribe the results of this experiment. What does this imply about the relationship between batch normalization and batch size? Why is this relationship observed?\n\n## Answer:\n- generally, model with batchnorm and large batch size converges faster, and is easier to overfit the trainning data, compared with model without batchnorm\n- when the batch size is small (e.g. 5), model without batchnorm could be better, that is, model with batchnorm and small batch size (e.g. 5) converges slower than model with the same batch size but without batchnorm\n- within models with batchnorm, large batch size makes model converges faster but introduces sever overfitting\n\n\n\n# Layer Normalization\nBatch normalization has proved to be effective in making networks easier to train, but the dependency on batch size makes it less useful in complex networks which have a cap on the input batch size due to hardware limitations. \n\nSeveral alternatives to batch normalization have been proposed to mitigate this problem; one such technique is Layer Normalization [2]. Instead of normalizing over the batch, we normalize over the features. In other words, when using **Layer Normalization, each feature vector corresponding to a single datapoint is normalized based on the sum of all terms within that feature vector**.\n\n[2] [Ba, Jimmy Lei, Jamie Ryan Kiros, and Geoffrey E. Hinton. \"Layer Normalization.\" stat 1050 (2016): 21.](https://arxiv.org/pdf/1607.06450.pdf)\n\n## Inline Question 3:\nWhich of these data preprocessing steps is analogous to batch normalization, and which is analogous to layer normalization?\n\n1. Scaling each image in the dataset, so that the RGB channels for each row of pixels within an image sums up to 1.\n2. Scaling each image in the dataset, so that the RGB channels for all pixels within an image sums up to 1. \n3. Subtracting the mean image of the dataset from each image in the dataset.\n4. Setting all RGB values to either 0 or 1 depending on a given threshold.\n\n## Answer:\n- 3 is analogous to batch normalization\n- 2 is analogous to layer normalization\n\n\n# Layer Normalization: Implementation\n\nNow you'll implement layer normalization. This step should be relatively straightforward, as conceptually the implementation is almost identical to that of batch normalization. One significant difference though is that for layer normalization, we do not keep track of the moving moments, and the testing phase is identical to the training phase, where **the mean and variance are directly calculated per datapoint**.\n\nHere's what you need to do:\n\n* In `cs231n/layers.py`, implement the forward pass for layer normalization in the function `layernorm_forward`. \n\nRun the cell below to check your results.\n* In `cs231n/layers.py`, implement the backward pass for layer normalization in the function `layernorm_backward`. \n\nRun the second cell below to check your results.\n* Modify `cs231n/classifiers/fc_net.py` to add layer normalization to the `FullyConnectedNet`. When the `normalization` flag is set to `\"layernorm\"` in the constructor, you should insert a layer normalization layer before each ReLU nonlinearity. \n\nRun the third cell below to run the batch size experiment on layer normalization.\n\n\n```python\n# Check the training-time forward pass by checking means and variances\n# of features both before and after layer normalization \n\n# Simulate the forward pass for a two-layer network\nnp.random.seed(231)\nN, D1, D2, D3 =4, 50, 60, 3\nX = np.random.randn(N, D1)\nW1 = np.random.randn(D1, D2)\nW2 = np.random.randn(D2, D3)\na = np.maximum(0, X.dot(W1)).dot(W2)\n\nprint('Before layer normalization:')\nprint_mean_std(a,axis=1)\n\ngamma = np.ones(D3)\nbeta = np.zeros(D3)\n# Means should be close to zero and stds close to one\nprint('After layer normalization (gamma=1, beta=0)')\na_norm, _ = layernorm_forward(a, gamma, beta, {'mode': 'train'})\nprint_mean_std(a_norm,axis=1)\n\ngamma = np.asarray([3.0,3.0,3.0])\nbeta = np.asarray([5.0,5.0,5.0])\n# Now means should be close to beta and stds close to gamma\nprint('After layer normalization (gamma=', gamma, ', beta=', beta, ')')\na_norm, _ = layernorm_forward(a, gamma, beta, {'mode': 'train'})\nprint_mean_std(a_norm,axis=1)\n```\n\n Before layer normalization:\n means: [-59.06673243 -47.60782686 -43.31137368 -26.40991744]\n stds: [10.07429373 28.39478981 35.28360729 4.01831507]\n \n After layer normalization (gamma=1, beta=0)\n means: [ 4.81096644e-16 -7.40148683e-17 2.22044605e-16 -5.92118946e-16]\n stds: [0.99999995 0.99999999 1. 0.99999969]\n \n After layer normalization (gamma= [3. 3. 3.] , beta= [5. 5. 5.] )\n means: [5. 5. 5. 5.]\n stds: [2.99999985 2.99999998 2.99999999 2.99999907]\n \n\n\n\n```python\n# Gradient check layer normalization backward pass\nnp.random.seed(231)\nN, D = 4, 5\nx = 5 * np.random.randn(N, D) + 12\ngamma = np.random.randn(D)\nbeta = np.random.randn(D)\ndout = np.random.randn(N, D)\n\nln_param = {}\nfx = lambda x: layernorm_forward(x, gamma, beta, ln_param)[0]\nfg = lambda a: layernorm_forward(x, a, beta, ln_param)[0]\nfb = lambda b: layernorm_forward(x, gamma, b, ln_param)[0]\n\ndx_num = eval_numerical_gradient_array(fx, x, dout)\nda_num = eval_numerical_gradient_array(fg, gamma.copy(), dout)\ndb_num = eval_numerical_gradient_array(fb, beta.copy(), dout)\n\n_, cache = layernorm_forward(x, gamma, beta, ln_param)\ndx, dgamma, dbeta = layernorm_backward(dout, cache)\n\n#You should expect to see relative errors between 1e-12 and 1e-8\nprint('dx error: ', rel_error(dx_num, dx))\nprint('dgamma error: ', rel_error(da_num, dgamma))\nprint('dbeta error: ', rel_error(db_num, dbeta))\n```\n\n dx error: 2.107277492956569e-09\n dgamma error: 4.519489546032799e-12\n dbeta error: 2.5842537629899423e-12\n\n\n# Layer Normalization and batch size\n\nWe will now run the previous batch size experiment with layer normalization instead of batch normalization. Compared to the previous experiment, **you should see a markedly smaller influence of batch size on the training history!**\n\n\n```python\nln_solvers_bsize, solver_bsize, batch_sizes = run_batchsize_experiments('layernorm')\n\nplt.subplot(2, 1, 1)\nplot_training_history('Training accuracy (Layer Normalization)','Epoch', solver_bsize, ln_solvers_bsize, \\\n lambda x: x.train_acc_history, bl_marker='-^', bn_marker='-o', labels=batch_sizes)\nplt.subplot(2, 1, 2)\nplot_training_history('Validation accuracy (Layer Normalization)','Epoch', solver_bsize, ln_solvers_bsize, \\\n lambda x: x.val_acc_history, bl_marker='-^', bn_marker='-o', labels=batch_sizes)\n\nplt.gcf().set_size_inches(15, 10)\nplt.show()\n```\n\n## Inline Question 4:\nWhen is layer normalization likely to not work well, and why?\n\n1. Using it in a very deep network\n2. Having a very small dimension of features\n3. Having a high regularization term\n\n\n## Answer:\n2, a few features is not enough to estimated mean and variance\n\n", "meta": {"hexsha": "e0dee576f82c80b18f5bdfb88b979acb71eb7ad9", "size": 430794, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "assignment2/BatchNormalization.ipynb", "max_stars_repo_name": "VincentYCYao/stanford-cs231n-2019", "max_stars_repo_head_hexsha": "c293f754ef4b9a14e1ebff7a9d743e6d103072a5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-08-20T15:17:09.000Z", "max_stars_repo_stars_event_max_datetime": "2020-08-20T15:17:09.000Z", "max_issues_repo_path": "assignment2/BatchNormalization.ipynb", "max_issues_repo_name": "VincentYCYao/stanford-cs231n-2019", "max_issues_repo_head_hexsha": "c293f754ef4b9a14e1ebff7a9d743e6d103072a5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "assignment2/BatchNormalization.ipynb", "max_forks_repo_name": "VincentYCYao/stanford-cs231n-2019", "max_forks_repo_head_hexsha": "c293f754ef4b9a14e1ebff7a9d743e6d103072a5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 381.2336283186, "max_line_length": 107672, "alphanum_fraction": 0.9273132866, "converted": true, "num_tokens": 9198, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185498374789, "lm_q2_score": 0.7549149923816048, "lm_q1q2_score": 0.42729588923840733}} {"text": "
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                                        INF285/ILI285 Computaci\u00f3n Cient\u00edfica

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                                        Tarea N\u00b01, v1.00

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                                        \n\n## Instrucciones\n\nFelipe Montero Concha 201473611-8\n\n* La tarea es individual.\n* Las consultas sobre las tareas se deben realizar por medio de la plataforma Aula.\n* La tarea debe ser realizada en `Jupyter Notebook` (`Python3`).\n* Se evaluar\u00e1 la correcta utilizaci\u00f3n de librerias `NumPy`, `SciPy`, entre otras, as\u00ed como la correcta implementaci\u00f3n de algoritmos de forma vectorizada.\n* **El archivo de entrega debe denominarse ROL-tarea-numero.ipynb**. _De no respetarse este formato existir\u00e1 un descuento de **50 puntos**_\n* La fecha de entrega es el jueves 30 de Abril a las **18:00 hrs**. Se aceptar\u00e1n entregas hasta las 19:00 hrs sin descuento en caso de existir algun problema, posteriormente existir\u00e1 un descuento lineal hasta las 20:00 hrs del mismo d\u00eda.\n* Las tareas que sean entregadas antes del jueves a mediod\u00eda recibir\u00e1n una bonificaci\u00f3n de 10 puntos\n* Debe citar cualquier c\u00f3digo ajeno utilizado (incluso si proviene de los Jupyter Notebooks del curso).\n\n\n\n\n## Introducci\u00f3n \n\nEn esta primera tarea de INF/ILI-285, versi\u00f3n 2020-1, estudiaremos la importancia de los primeros temas estudiados en el curso, los cuales son: Representaci\u00f3n de Punto Flotante, P\u00e9rdida de Significancia, Errores de Cancelaci\u00f3n y B\u00fasqueda de Ceros. El desarrollo de cada uno de esos temas se presenta en una serie de preguntas, donde ustedes deben ir decidiendo, pregunta a pregunta, como cada uno de los temas se aplica. En general, los temas no se analizan por separado, sino de manera acoplada. Es muy importante que cada uno de los problemas sea analizado te\u00f3ricamente primero, para luego poner su conocimiento en acci\u00f3n. Cada problema puede ser desarrollado de diversas formas, sin embargo, es muy importante determinar al final si el camino elegido resuelve la pregunta presentada.\n\n## Problemas\n\n\n```python\nimport numpy as np\nimport bitstring\nimport scipy.special\n```\n\n### 1. Simulador (50 ptos) \n\nDada la familia de polinomios de grado 3 con la forma:\n\n\n\\begin{equation}\nf(x)=x^3 - A\\,x^2 + A\\,x - 1\n\\end{equation}\n\n\nSe pide implementar un algoritmo que reciba como par\u00e1metros el valor de $A$ (con $|A|>>1$) y la cantidad de bits que tiene en la mantisa un computador ficticio, el manejo del exponente debe considerarse de las misma forma que lo maneja _double precision_.\nEsta implementaci\u00f3n debe calcular las ra\u00edces de $f$ sin perdida de significancia con la cantidad de bits disponibles para la mantisa.\nPara obtener las ra\u00edces de $f$ usted debe encontrar de forma algebraica sus ra\u00edces y luego proponer un algoritmo basado en las _f\u00f3rmulas_ obtenidas.\n\nConsidere que en ese computador ficticio cuenta con las operaciones matem\u00e1ticas necesarias para obtener las ra\u00edces. Considere el siguiente ejemplo:\n\n```python\n# Alg. Base\na = 9819824.624837\nb = 148736.523476\nc = a+b\n\n# Alg. con Representaci\u00f3n de Punto Flotante de 'bits_mant' bits en la mantisa.\nam = f_new_rep(9819824.624837,bits_mant) # Aproximar el input en la nueva representaci\u00f3n.\nbm = f_new_rep(148736.523476,bits_mant) # Aproximar el input en la nueva representaci\u00f3n.\ncm = f_new_rep(m,exp,am+bm) # Aproximar el output de la suma y cada operaci\u00f3n en la nueva representaci\u00f3n.\n```\n\n\n\n```python\n\"\"\"\ninput\nx : (double) valor a evaluar\nbits_mant : (int) cantidad de bits de la mantisa\noutput\neval : (double) resultado obtenido\n\"\"\"\n#Esta funcion lo unico que hace es sumar el 1 en caso de que sea necesario a un numero binario, \n#acarreando el 1 cuando es necesario\ndef sumar_1_bin(num):\n a_sumar = str(num)[::-1]\n result = \"\"\n debe = True\n for bit in a_sumar:\n if bit == \"1\" and debe:\n result += \"0\"\n elif bit == \"0\" and debe:\n result += \"1\"\n debe = False\n else:\n result += bit\n return result[::-1]\n \n\ndef f_new_rep(x,bits_mant):\n # Algoritmo de representaci\u00f3n de punto flotante modificada.\n rep_IEEE = bitstring.BitArray(float= x, length = 64)\n ceros_unos = rep_IEEE.bin\n\n signo = ceros_unos[0]\n exponente = ceros_unos[1:12]\n mantisa = ceros_unos[12:]\n \n new_mantisa = mantisa[0:bits_mant]\n check_mant = mantisa[bits_mant:53]\n \n if check_mant[0] == \"0\":\n new_mantisa = new_mantisa\n elif check_mant[0] == \"1\":\n if \"1\" in check_mant[1:]:\n new_mantisa = sumar_1_bin(new_mantisa)\n elif \"1\" not in check_mant[1:]:\n if new_mantisa[-1] == \"1\":\n new_mantisa = sumar_1_bin(new_mantisa)\n dec_exponente = int(exponente,2)-1023\n total = 2**dec_exponente\n k = dec_exponente - 1 \n for i in range(0,bits_mant):\n total += int(new_mantisa[i])*(2**(k))\n k-=1\n if int(signo):\n total = total*-1\n return total\n\n\n\n\n\nprint(f_new_rep(90,32))\n\n```\n\n\n```python\n#Para todo polinomio cubico de esta caracteristica, el 1 siempre es raiz por lo tanto se procede a hacer ruffini en el codigo\n# dividiendo el polinomio en 1, obteniendo los coeficientes de la cuadratica correspondiente para luego obtener a traves de\n#la formula de soluciones cuadraticas el resto de las raices del polinomio cubico original.\ndef f_find_roots(A,bits_mant):\n \n coeficientes = [1,-A,A,-1]\n c_cuadraticos = [coeficientes[0]]\n for i in range(0,len(coeficientes)):\n if i == 0:\n a_sumar = coeficientes[0]*1\n else:\n c_cuadraticos.append(coeficientes[i]+a_sumar)\n a_sumar = coeficientes[i] + a_sumar\n print(c_cuadraticos)\n \n a = f_new_rep(c_cuadraticos[0],bits_mant) \n b = f_new_rep(c_cuadraticos[1],bits_mant) \n c = f_new_rep(c_cuadraticos[2],bits_mant)\n \n b_cuadrado = f_new_rep(b**2,bits_mant)\n dos_a = f_new_rep(2*a,bits_mant)\n ace = f_new_rep(4*a*c,bits_mant)\n \n resta_raiz = f_new_rep(b_cuadrado-ace,bits_mant)\n \n \n raiz = f_new_rep(np.sqrt(resta_raiz),bits_mant)\n \n \n sum1 = f_new_rep(-b + raiz , bits_mant)\n sum2 = f_new_rep(-b - raiz , bits_mant)\n \n x1 = f_new_rep(sum1/dos_a,bits_mant)\n x2 = f_new_rep(sum2/dos_a,bits_mant)\n \n x_roots = (1,x1,x2)\n return x_roots\n\nf_find_roots(90,32)\n```\n\n### Polinomios de Legendre (50 puntos)\nDada la funci\u00f3n compuesta $f$\n\\begin{equation}\nf_{n,m}(x) = L_n(C_m(x)),\n\\end{equation}\ndonde $L_n$ es conocido como el polinomio de Legendre de grado $n$ definido de la siguiente forma:\n\\begin{equation}\nL_{n}(x)=\\frac{1}{2^{n}} \\sum_{k=0}^{n}\\left(\\begin{array}{l}\nn \\\\\nk\n\\end{array}\\right)^{2}(x-1)^{n-k}(x+1)^{k},\n\\end{equation}\ny $C_m$ es el polinomio de Chebyshev\n\\begin{equation}\nC_m(x) = \\cos(m \\cdot \\arccos(x)).\n\\end{equation}\n\nUtilizando el m\u00e9todo de Bisecci\u00f3n y Punto fijo se pide obtener la ra\u00edz de $f$ m\u00e1s cercana a $0.5$ dado un valor de $m$ y $n$\n\n\n\n*Hint: Las ra\u00edces de Legendre son conocidas*\n\n\n\n\n```python\n#Por bisecci\u00f3n se procede conociendo que el chebyshev de grado n se comporta de forma muy similar al legendre de grado n\n# por lo tanto se le realiza una bisecci\u00f3n en base a una raiz encontrada al chebyshev de grado n , la cual actua como \n# x_0 para la funci\u00f3n utilizada a comparar con el chebyshev de grado m\n\"\"\"\ninput\nn: (int) grado del polinomio de Legendre\nm: (int) grado del polinomio de Chebyshev\ntol: (double) tolerancia para la detenci\u00f3n del algoritmo\noutput\nroot: (double) raiz obtenida\n\"\"\"\ndef f_Biseccion(n, m, tol):\n i = 0\n menor_dif = float(\"inf\")\n while i < n:\n valor_a_buscar = np.cos((np.pi*((2*i)-1))/(2*n))\n if abs(valor_a_buscar-0.5) < menor_dif:\n elegido = valor_a_buscar\n menor_dif = abs(valor_a_buscar-0.5)\n i+=1\n \n b = 1\n a = np.cos((np.pi/n)*(n-2))\n \n funcioncita = lambda x : elegido - np.cos(m*np.arccos(x))\n \n\n i = 0\n c = 0\n anterior = float(\"inf\")\n \n while (abs(a-b) > tol) and (anterior != c):\n anterior = c\n c = (b+a)/2\n if funcioncita(c) == 0:\n return c\n else:\n if funcioncita(a)*funcioncita(c) < 0:\n b = c\n else:\n a = c\n i+=1\n\n x_m = np.cos(m*np.arccos(c))\n return x_m\n\"\"\"\ndef f_FPI(n, m, tol):\n x = x_0 \n iteraciones=[]\n errores = []\n i = 0\n while abs(x_calc-g(x)) < tol:\n try:\n x_calc = g(x)\n except OverflowError:\n return None\n if x_calc == x and (x_calc not in iteraciones):\n iteraciones.append(x_calc)\n elif x_calc == x and (x_calc in iteraciones):\n i = n_iter\n else:\n iteraciones.append(x_calc)\n x = x_calc\n i+=1\n \n \n return root\n\"\"\"\nf_Biseccion(4,2,1e-8)\n```\n\n### Determinantes (20 puntos)\n\nDada una matriz de dimensiones $ n \\times n$ de la forma:\n\\begin{equation}\nA\n=\n\\begin{pmatrix}\na_{1,1} & a_{1,2} & \\dots & a_{1,n} \\\\\na_{2,1} & a_{2,2} & \\dots & a_{2,n} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\na_{n,1} & a_{n,2} & \\dots & a_{n,n}\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n\\mathbf{r}_1 \\\\\n\\mathbf{r}_2 \\\\\n\\vdots \\\\\n\\mathbf{r}_n \\\\\n\\end{pmatrix},\n\\end{equation}\ndonde $\\mathbf{r}_k$ representa la $k$-\u00e9sima fila de la matriz $A$.\nConsidere la siguiente matriz $C_{i,j}(\\alpha)$,\n\\begin{equation}\nC_{i,j}(\\alpha)\n=\n\\begin{pmatrix}\n\\mathbf{r}_1 \\\\\n\\vdots \\\\\n\\mathbf{r}_i\\,(1-\\alpha)+\\mathbf{r}_j\\,\\alpha \\\\\n\\vdots \\\\\n\\mathbf{r}_j\\,(1-\\alpha)+\\mathbf{r}_i\\,\\alpha \\\\\n\\vdots \\\\\n\\mathbf{r}_n \\\\\n\\end{pmatrix},\n\\end{equation}\nde lo cual sabemos que $C_{i,j}(0)=A$ y que $C_{i,j}(1)$ es la matriz $A$ donde se intercambiaron las filas $i$ y $j$, es decir:\n\\begin{equation}\nC_{i,j}(1)\n=\n\\begin{pmatrix}\n\\mathbf{r}_1 \\\\\n\\vdots \\\\\n\\mathbf{r}_j \\\\\n\\vdots \\\\\n\\mathbf{r}_i \\\\\n\\vdots \\\\\n\\mathbf{r}_n \\\\\n\\end{pmatrix}.\n\\end{equation}\nDe las relaciones anteriores podemos concluir que el determinante de la matriz $A$, denominado $D=\\det(A)$, es igual al determinante de $C_{i,j}(0)$, es decir $\\det(C_{i,j}(0))=\\det(A)=D$.\nPor el otro lado, el determinante de $C_{i,j}(1)$ es $-D$, dado que es el intercambio de las filas $i$ y $j$ de la matriz $A$.\nPor lo cual podemos concluir que $-D\\leq \\det(C_{i,j}(\\alpha))\\leq D$.\n\nUtilizando el m\u00e9todo de Bisecci\u00f3n debe encontrar el valor de $\\alpha$ con $p$ decimales de precisi\u00f3n que permitan que, dado los \u00edndices de las filas $i$, $j$, y $i\\neq j$, el determinante de la matriz sea igual a $d$, donde $d\\in[-D,D]$. \n\nPara esto se debe implementar una funci\u00f3n que reciba la matriz $A$, las filas $i$ y $j$, y $p$; y retorne $\\widehat{\\alpha}$ tal que $\\det(C_{i,j}(\\widehat{\\alpha}))=d$.\n\n\n\n```python\n#Aca se determino que hay que calcular la biusecci\u00f3n a parti de la funci\u00f3n d - det(M(alpha))\n# el intervalo de itaraci\u00f3n de alpha es entre 0 y 1 ya que no puede salir de esos valores, considerando que\n#los valores alpha y 1-alpha estan relacionados.\n\"\"\"\ninput\nA: (array n x n) matriz\ni: (int) \u00edndice de la fila \"i\".\nj: (int) \u00edndice de la fila \"j\".\np: (int) cantidad de decimales de precision \nd: (double) valor requerido del determinante de $C_{i,j}(\\alpha)$, i.e. $\\det(C_{i,j}(\\widehat{\\alpha}))=d$.\noutput\nalpha_hat: (double) alpha_hat tal que det(C_{i,j}(alpha_hat))=d.\n\"\"\"\ndef generate_c(A,i,j,alpha):\n a_aux = np.array(A)\n fila_i = np.array(A[i])\n fila_j = np.array(A[j])\n aux = 0\n filas = []\n for x in range(0,len(A)):\n if x == i:\n aux = fila_i*alpha + fila_j*(1.-alpha)\n filas.append(aux)\n elif x == j:\n aux = fila_i*alpha + fila_j*(1.-alpha)\n filas.append(aux)\n else:\n filas.append(a_aux[x])\n print(\"este es el alfa de la funcion\"+str(alpha))\n retorno = np.array(filas)\n print(retorno)\n return retorno\n\n\n\ndef find_alpha_hat(A,i,j,p,d):\n tolerancia =10**(-p-1)\n \n funcioncita = lambda C,i,j,alpha_i : d - np.linalg.det(generate_c(C,i,j,alpha_i))\n a = 0\n b = 1\n anterior = float(\"inf\")\n print(generate_c(A,i,j,0.75))\n\n while (abs(a-b) > tolerancia) :\n c = (b+a)/2\n if funcioncita(A,i,j,c) == 0:\n alpha_hat = c\n return alpha_hat\n else:\n if funcioncita(A,i,j,a)*funcioncita(A,i,j,c) < 0:\n b = c\n else:\n a = c\n alpha_hat = c\n print(np.linalg.det(generate_c(A,i,j,alpha_hat)))\n return alpha_hat\n\n\nA = np.array([[19,12,33,9],[3,4,6,4],[7,8,9,2],[5,9,6,1]])\nprint(np.linalg.det(A))\nprint(find_alpha_hat(A,0,2,7,4))\n```\n\n# Referencias\n\nSi corresponde\n\n\n```python\n\n```\n", "meta": {"hexsha": "0e7b5858e68790b144e1b9a42e8e97f041147a4e", "size": 18474, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "201473611-8-tarea-1.ipynb", "max_stars_repo_name": "Felipitoo/CC", "max_stars_repo_head_hexsha": "2ce7bac8c02b5ef7089e752f2143e13a4b77afc2", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "201473611-8-tarea-1.ipynb", "max_issues_repo_name": "Felipitoo/CC", "max_issues_repo_head_hexsha": "2ce7bac8c02b5ef7089e752f2143e13a4b77afc2", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "201473611-8-tarea-1.ipynb", "max_forks_repo_name": "Felipitoo/CC", "max_forks_repo_head_hexsha": "2ce7bac8c02b5ef7089e752f2143e13a4b77afc2", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 34.2111111111, "max_line_length": 792, "alphanum_fraction": 0.5220850926, "converted": true, "num_tokens": 4031, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185205547239, "lm_q2_score": 0.7549149923816048, "lm_q1q2_score": 0.4272958671324166}} {"text": "# Circuitos de 2\u00aa Ordem\nJupyter Notebook desenvolvido por [Gustavo S.S.](https://github.com/GSimas)\n\n> \"Todos os que podem cursar um mestrado em engenharia devem faz\u00ea-lo a fim de estender o sucesso de sua\ncarreira! Se voc\u00ea quer trabalhar com pesquisa, o estado da arte em engenharia, lecionar em uma universidade\nou iniciar seu pr\u00f3prio neg\u00f3cio, voc\u00ea realmente precisa cursar um doutorado!\" - **Charles K. Alexander**\n\nCircuitos contendo dois elementos de armazenamento,\nque s\u00e3o conhecidos como circuitos de segunda ordem, porque\nsuas respostas s\u00e3o descritas como equa\u00e7\u00f5es diferenciais contendo derivadas\nsegundas.\n\nExemplos comuns de circuitos de segunda ordem s\u00e3o os RLC, onde est\u00e3o\npresentes os tr\u00eas tipos de elementos passivos, como mostram Figuras 8.1a e b.\nOutros exemplos s\u00e3o circuitos RL e RC, como os indicados nas Figuras 8.1c e\nd.\n\n\n\nUm circuito com amplificadores operacionais com dois elementos de armazenamento\ntamb\u00e9m pode ser um circuito de segunda ordem. Assim como nos\ncircuitos de primeira ordem, o de segunda ordem pode conter v\u00e1rios resistores\ne fontes dependentes e independentes.\n\n**Um circuito de segunda ordem \u00e9 caracterizado por uma equa\u00e7\u00e3o diferencial\nde segunda ordem. Ele \u00e9 formado por resistores e o equivalente de dois\nelementos de armazenamento.**\n\n## Determina\u00e7\u00e3o dos valores inicial e final\n\nPara a an\u00e1lise de circuitos de 2\u00aa Ordem \u00e9 preciso obter v(0), i(0), dv(0)/dt, di(0)/dt, i(\u221e) e v(\u221e), onde v \u00e9 a tens\u00e3o no capacitor e i a corrente no indutor.\n**A tens\u00e3o no capacitor \u00e9 sempre cont\u00ednua\nde modo que**\n\n\\begin{align}\n{\\Large v(0^+) = v(0^-) = v(0)}\n\\end{align}\n\n**E a corrente no indutor \u00e9 sempre cont\u00ednua de modo que**\n\n\\begin{align}\n{\\Large i(0^+) = i(0^-) = i(0)}\n\\end{align}\n\nonde t = 0\u2013 representa o instante imediatamente anterior ao evento de comuta\u00e7\u00e3o\ne t = 0+ \u00e9 o instante imediatamente ap\u00f3s o evento de comuta\u00e7\u00e3o, supondo\nque esse evento ocorra em t = 0.\n\n**Exemplo 8.1**\n\nA chave na Figura 8.2 foi fechada h\u00e1 um bom tempo. Ela \u00e9 aberta em t = 0. Determine:\n\na) v(0+), i(0+)\n\nb) dv(0+)/dt, di(0+)/dt\n\nc) i(\u221e) e v(\u221e)\n\n\n\n\n```python\nprint(\"Exemplo 8.1\")\n\nL = 0.25\nC = 0.1\nVs = 12\n\n#para t < 0\ni0 = Vs/(4 + 2)\nv0 = i0*2\nprint(\"Corrente i(0+):\",i0,\"A\")\nprint(\"Tens\u00e3o v(0+):\",v0,\"V\")\n\n#para t = 0+\n#vl = L*di/dt\n#di/dt = vl/L\nvl = Vs - i0*4 - v0\ndi = vl/L\n#ic = C*dv/dt\n#dv/dt = ic/C\ndv = i0/C\nprint(\"Taxa di/dt:\",di,\"A/s\")\nprint(\"Taxa dv/dt:\",dv,\"V/s\")\n\n#para t = infinito\nv = Vs\ni = 0\nprint(\"Para t infinito, v:\",v,\"V\")\nprint(\"Para t infinito, i:\",i,\"A\")\n```\n\n Exemplo 8.1\n Corrente i(0+): 2.0 A\n Tens\u00e3o v(0+): 4.0 V\n Taxa di/dt: 0.0 A/s\n Taxa dv/dt: 20.0 V/s\n Para t infinito, v: 12 V\n Para t infinito, i: 0 A\n\n\n**Problema Pr\u00e1tico 8.1**\n\nA chave na Figura 8.4 foi aberta h\u00e1 um bom tempo, entretanto, foi fechada em t = 0.\n\na) v(0+), i(0+)\n\nb) dv(0+)/dt, di(0+)/dt\n\nc) i(\u221e) e v(\u221e)\n\n\n\n\n```python\nprint(\"Problema Pr\u00e1tico 8.1\")\n\nL = 0.4\nC = 1/20\nVs = 24\n\n#para t < 0\ni0 = Vs/(10 + 2)\nv0 = i0*2\nprint(\"Corrente i(0+):\",i0,\"A\")\nprint(\"Tens\u00e3o v(0+):\",v0,\"V\")\n\n#para t = 0+\n#di/dt = vl/L\nvl = Vs - v0\ndi = vl/L\n#dv/dt = ic/C\ndv = 0\nprint(\"Taxa di/dt:\",di,\"A/s\")\nprint(\"Taxa dv/dt:\",dv,\"V/s\")\n\n#para t = infinito\ni = Vs/2\nv = i*2\nprint(\"Corrente i(\u221e)\",i,\"A\")\nprint(\"Tens\u00e3o v(\u221e)\",v,\"V\")\n```\n\n Problema Pr\u00e1tico 8.1\n Corrente i(0+): 2.0 A\n Tens\u00e3o v(0+): 4.0 V\n Taxa di/dt: 50.0 A/s\n Taxa dv/dt: 0 V/s\n Corrente i(\u221e) 12.0 A\n Tens\u00e3o v(\u221e) 24.0 V\n\n\n**Exemplo 8.2**\n\nNo circuito da Figura 8.5, calcule: \n\n(a) iL(0+), vC(0+); vR(0+); \n\n\n(b) diL(0+)/dt, dvC(0+)/dt, dvR(0+)/dt; \n\n(c) iL(\u221e), vC(\u221e), vR(\u221e).\n\n\n\n\n```python\nprint(\"Exemplo 8.2\")\n\nL = 0.6\nC = 1/2\nVs = 20\nCs = 3\n\n#para t < 0\nv0 = -Vs\ni0 = 0\nvr0 = Cs*4/(2 + 4) * 2\nprint(\"Corrente i(0+):\",i0,\"A\")\nprint(\"Tens\u00e3o v(0+):\",v0,\"V\")\nprint(\"Tens\u00e3o Vr(0+):\",vr0,\"V\")\n\n#para t = 0+\n#di/dt = vl/L\nvl = 0\ndi = vl/L\n#dv/dt = ic/C\nic = Cs*2/(2 + 4)\ndv = ic/C\n#3 = vr/2 + v0/4\n#0 = 2dvr/dt + dv0/dt\n#-vr + v0 + vc + 20 = 0\n#-dvr + dv0 + 2 = 0 => dv0 = dvr - 2\ndvr = 2/3\nprint(\"Taxa di/dt:\",di,\"A/s\")\nprint(\"Taxa dv/dt:\",dv,\"V/s\")\nprint(\"Taxa dvr/dt:\",dvr,\"V/s\")\n\n#para t = \u221e\ni = Cs*2/(4 + 2)\nv = -Vs\nvr = (Cs - i)*2\nprint(\"Tens\u00e3o i(\u221e):\",i,\"A\")\nprint(\"Corrente v(\u221e):\",v,\"V\")\nprint(\"Tens\u00e3o vr(\u221e):\",vr,\"V\")\n```\n\n Exemplo 8.2\n Corrente i(0+): 0 A\n Tens\u00e3o v(0+): -20 V\n Tens\u00e3o Vr(0+): 4.0 V\n Taxa di/dt: 0.0 A/s\n Taxa dv/dt: 2.0 V/s\n Taxa dvr/dt: 0.6666666666666666 V/s\n Tens\u00e3o i(\u221e): 1.0 A\n Corrente v(\u221e): -20 V\n Tens\u00e3o vr(\u221e): 4.0 V\n\n\n**Problema Pr\u00e1tico 8.2**\n\nPara o circuito da Figura 8.7, determine: \n\n(a) iL(0+), vC(0+); vR(0+); \n\n(b) diL(0+)/dt, dvC(0+)/dt, dvR(0+)/dt; \n\n(c) iL(\u221e), vC(\u221e), vR(\u221e).\n\n\n\n\n```python\nprint(\"Problema Pr\u00e1tico 8.2\")\n\nC = 1/5\nL = 2\nCs1 = 6\nCs2 = 4\n\n#para t < 0\ni0 = -Cs1\nv0 = 0\nvr0 = 0\nprint(\"Corrente i(0+):\",i0,\"A\")\nprint(\"Tens\u00e3o v(0+):\",v0,\"V\")\nprint(\"Tens\u00e3o vr(0+):\",vr0,\"V\")\n\n#para t = 0+\n#di/dt = vl/L\nvl = 0\ndi = vl/L\n#dv/dt = ic/C\nic = 4\ndv = ic/C\n#vr = vc - vl = 0\n#dvr = 20 - dvl\n#6 = vr/5 + il\n#0 = dvr + 5di\n#0 = dvr\ndvr = 0\nprint(\"Taxa di/dt:\",di,\"A/s\")\nprint(\"Taxa dv/dt:\",dv,\"V/s\")\nprint(\"Taxa dvr/dt:\",dvr,\"V/s\")\n\n#para t = \u221e\ni = Cs2 - Cs1\nvr = Cs2*5\nvc = vr\nprint(\"Corrente i(\u221e):\",i,\"A\")\nprint(\"Tens\u00e3o v(\u221e):\",v,\"V\")\nprint(\"Tens\u00e3o vr(\u221e):\",vr,\"V\")\n```\n\n Problema Pr\u00e1tico 8.2\n Corrente i(0+): -6 A\n Tens\u00e3o v(0+): 0 V\n Tens\u00e3o vr(0+): 0 V\n Taxa di/dt: 0.0 A/s\n Taxa dv/dt: 20.0 V/s\n Taxa dvr/dt: 0 V/s\n Corrente i(\u221e): -2 A\n Tens\u00e3o v(\u221e): -20 V\n Tens\u00e3o vr(\u221e): 20 V\n\n\n# Circuito RLC S\u00e9rie sem fonte\n\nO circuito\n\u00e9 excitado pela energia inicialmente armazenada no capacitor e indutor, representada\npela tens\u00e3o inicial V0 no capacitor e pela corrente inicial I0 no indutor.\nPortanto, em t = 0.\n\n\n\n\\begin{align}\n{\\Large v(0) = \\frac{1}{C} \\int_{- \\infty}^{0} idt = V_0}\n\\end{align}\n\n\\begin{align}\n{\\Large i(0) = I_0}\n\\end{align}\n\nAplicando a LKT no circuito da Figura 8.8:\n\n\\begin{align}\n{\\Large Ri + L \\frac{di}{dt} + \\frac{1}{C} \\int_{- \\infty}^{0} i(\\tau)d \\tau = 0}\n\\end{align}\n\nPara eliminar a integral, diferenciamos em rela\u00e7\u00e3o a t e reorganizamos os termos:\n\n\\begin{align}\n{\\Large \\frac{d^2i}{dt^2} + \\frac{R}{L} \\frac{di}{dt} + \\frac{i}{LC} = 0}\n\\end{align}\n\nEsta \u00e9 uma equa\u00e7\u00e3o diferencial de segunda ordem e \u00e9 o motivo para os circuitos\nRLC neste cap\u00edtulo serem chamados circuitos de segunda ordem. Obtemos o valor inicial da\nderivada de i a partir de:\n\n\\begin{align}\n{\\Large Ri(0) + L \\frac{di(0)}{dt} + V_0 = 0}\n\\end{align}\n\nNossa experi\u00eancia no cap\u00edtulo anterior sobre circuitos de primeira ordem nos sugere que a solu\u00e7\u00e3o \u00e9 na forma exponencial.\nPortanto, fa\u00e7amos:\n\n\\begin{align}\n{\\Large i = Ae^{st}}\n\\end{align}\n\nonde A e s s\u00e3o constantes a serem determinadas. Substituindo nas equa\u00e7\u00f5es anteriores:\n\n\\begin{align}\n{\\Large As^2e^{st} + \\frac{AR}{L}se^{st} + \\frac{A}{LC}e^{st} = 0}\n\\end{align}\n\nou:\n\n\\begin{align}\n{\\Large Ae^{st} (s^2 + \\frac{R}{L}s + \\frac{1}{LC}) = 0}\n\\end{align}\n\nJ\u00e1 que i = Ae^(st) \u00e9 a solu\u00e7\u00e3o pressuposta de que estamos tentando encontrar,\napenas a express\u00e3o entre par\u00eanteses pode ser zero:\n\n\\begin{align}\n{\\Large s^2 + \\frac{R}{L}s + \\frac{1}{LC} = 0}\nend{align}\n\nEsta equa\u00e7\u00e3o quadr\u00e1tica \u00e9 conhecida como **equa\u00e7\u00e3o caracter\u00edstica da Equa\u00e7\u00e3o\ndiferencial** uma vez que as ra\u00edzes da equa\u00e7\u00e3o ditam as caracter\u00edsticas\nb\u00e1sicas de i. As duas ra\u00edzes s\u00e3o:\n\n\\begin{align}\n{\\Large s_1 = -\\frac{R}{2L} + \\sqrt{ (\\frac{R}{2L})^2 - \\frac{1}{LC}}}\n\\\\{\\Large s_2 = -\\frac{R}{2L} - \\sqrt{ (\\frac{R}{2L})^2 - \\frac{1}{LC}}}\n\\end{align}\n\n**Uma forma mais condensada de expressar as ra\u00edzes \u00e9**:\n\n\\begin{align}\n{\\Large s_1 = -\\alpha + \\sqrt{ \\alpha^2 - \\omega_0^2 }}\n\\\\{\\Large s_2 = -\\alpha - \\sqrt{ \\alpha^2 - \\omega_0^2 }}\n\\end{align}\n\n**onde**\n\n\\begin{align}\n{\\Large \\alpha = \\frac{R}{2L} }\n\\\\{\\Large \\omega_0 = \\frac{1}{ \\sqrt{LC} }}\n\\end{align}\n\nAs ra\u00edzes s1 e s2 s\u00e3o chamadas **frequ\u00eancias naturais**, medidas em nepers\npor segundo (Np/s), pois est\u00e3o associadas \u00e0 resposta natural do circuito; v0 \u00e9\nconhecida como **frequ\u00eancia ressonante** ou estritamente como a **frequ\u00eancia natural\nn\u00e3o amortecida** expressa em radianos por segundo (rad/s); e a \u00e9 a frequ\u00eancia\nde neper ou fator de amortecimento expresso em nepers por segundo.\n\nAssim, pode-se escrever:\n\n\\begin{align}\n{\\Large s^2 + 2 \\alpha s + \\omega_0^2 = 0}\n\\end{align}\n\nA raz\u00e3o \u03b1/\u03c90 \u00e9 conhecida como fator de amortecimento, \u03b6.\n\n## Resposta do circuito RLC s\u00e9rie:\n\nConsequentemente, a resposta natural do circuito RLC em s\u00e9rie \u00e9:\n\n\\begin{align}\n{\\Large i(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t}}\n\\end{align}\n\nonde as constantes A1 e A2 s\u00e3o determinadas a partir dos valores iniciais i(0) e\ndi(0)/dt.\n\nAssim, podemos inferir que existem tr\u00eas tipos de solu\u00e7\u00f5es:\n\n1. Se \u03b1 > \u03c90 temos o caso de **amortecimento supercr\u00edtico**\n2. Se \u03b1 = \u03c90 temos o caso de **amortecimento cr\u00edtico**\n3. Se \u03b1 < \u03c90 temos o caso de **subamortecimento**\n\n## Amortecimento Supercr\u00edtico (\u03b1 > \u03c90)\n\nSe \u03b1 > \u03c90 implica C > 4L/R^2. Quando isso acontece, ambas as ra\u00edzes s1 e s2 s\u00e3o negativas e reais. A resposta \u00e9:\n\n\\begin{align}\n{\\Large i(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t}}\n\\end{align}\n\nque decai e se aproxima de zero \u00e0 medida que t aumenta.\n\n\n\n## Amortecimento Cr\u00edtico (\u03b1 = \u03c90)\n\nQuando \u03b1 = \u03c90, C = 4L/R^2 e:\n\n\\begin{align}\n{\\Large s_1 = s_2 = -\\alpha = -\\frac{R}{2L}}\n\\end{align}\n\nAssim:\n\n\\begin{align}\n{\\Large i(t) = A_1 e^{-\\alpha t} + A_2 e^{-\\alpha t} = A_3 e^{-\\alpha t}}\n\\end{align}\n\nIsso n\u00e3o pode ser a solu\u00e7\u00e3o, pois as duas condi\u00e7\u00f5es iniciais n\u00e3o podem ser satisfeitas com a constante \u00fanica A3. Assim, de acordo com a resolucao de equa\u00e7\u00f5es diferenciais de segunda ordem, resposta natural de um circuito com amortecimento\ncr\u00edtico \u00e9 a soma de dois termos: exponencial negativa e exponencial\nnegativa multiplicada por um termo linear:\n\n\\begin{align}\n{\\Large i(t) = (A_2 + A_1 t)e^{-\\alpha t}}\n\\end{align}\n\n\n\nessa figura \u00e9 um esbo\u00e7o de i(t) = te^(\u2013\u03b1t), que\natinge um valor m\u00e1ximo igual a e^(\u20131)/\u03b1 em t = 1/\u03b1, uma constante de tempo, e\nent\u00e3o decresce at\u00e9 chegar a zero.\n\n## Subamortecimento\n\nPara \u03b1 < \u03c90, temos C < 4L/R^2. As ra\u00edzes podem ser escritas como:\n\n\\begin{align}\n{\\Large s_1 = -\\alpha + \\sqrt{ \\alpha^2 - \\omega_0^2 } = -\\alpha + j\\omega_d}\n\\\\{\\Large s_2 = -\\alpha - \\sqrt{ \\alpha^2 - \\omega_0^2 } = -\\alpha - j\\omega_d}\n\\end{align}\n\nonde **j = \u221a-1 e \u03c9d = \u221a(\u03c90^2 - \u03b1^2)** que \u00e9 chamada **frequ\u00eancia de amortecimento**.\nTanto v0 como vd s\u00e3o frequ\u00eancias naturais, porque ajudam a determinar\na resposta natural; enquanto \u03c90 \u00e9 muitas vezes denominada frequ\u00eancia\nnatural n\u00e3o amortecida, \u03c9d \u00e9 chamada frequ\u00eancia natural amortecida. A resposta\nnatural \u00e9:\n\n\\begin{align}\n{\\Large i(t) = e^{-\\alpha t} (B_1 cos(\\omega_d t) + B_2 sen(\\omega_d t))}\n\\end{align}\n\nonde B1 = A1 + A2 e B2 = j(A1 - A2)\n\n\n\n## Considera\u00e7\u00f5es\n\nR = 0 produz uma resposta\n**perfeitamente senoidal**. Essa\nresposta n\u00e3o pode ser concretizada\nna pr\u00e1tica com L e C em virtude\ndas perdas inerentes nesses\nelementos. Um dispositivo eletr\u00f4nico chamado\n**oscilador** \u00e9 capaz de produzir uma\nresposta perfeitamente senoidal.\n\nConclu\u00edmos esta se\u00e7\u00e3o enfatizando as seguintes propriedades interessantes\ne peculiares de um circuito RLC:\n\n1. O comportamento de um circuito destes pode ser compreendido pelo conceito de **amortecimento**, que \u00e9 a **perda gradual da energia inicial armazenada**. Se R = 0 diz-se que o circuito est\u00e1 sem perdas, pois o elemento amortecedor ou dissipador (R) n\u00e3o est\u00e1 presente. Ajustando-se o valor de R, a resposta pode ser n\u00e3o amortecida, com amortecimento supercr\u00edtico, com amortecimento cr\u00edtico ou ent\u00e3o subamortecida.\n\n2. A **resposta oscilat\u00f3ria** \u00e9 poss\u00edvel em raz\u00e3o da presen\u00e7a de dois tipos de elementos de armazenamento. Ter tanto L como C possibilita que o fluxo de energia fique indo e vindo entre eles. A oscila\u00e7\u00e3o amortecida, exibida pela resposta subamortecida, \u00e9 conhecida como **oscila\u00e7\u00e3o circular**.\n\n3. O caso com amortecimento cr\u00edtico \u00e9 a fronteira entre os casos de subamortecimento e de amortecimento supercr\u00edtico e ela cai de forma mais r\u00e1pida. Com as mesmas condi\u00e7\u00f5es iniciais, o caso de amortecimento **supercr\u00edtico tem o tempo de acomoda\u00e7\u00e3o mais longo**, pois ele leva o maior tempo para dissipar a energia inicialmente armazenada. Se desejarmos uma resposta que se aproxime do valor final mais **rapidamente sem oscila\u00e7\u00e3o ou com oscila\u00e7\u00e3o circular**, o circuito com **amortecimento cr\u00edtico** \u00e9 o mais indicado. \n\n**Exemplo 8.3**\n\nNa Figura 8.8, R = 40 \u03a9 , L = 4 H e C = 1/4 F. Calcule as ra\u00edzes caracter\u00edsticas do circuito.\nA resposta natural \u00e9 com amortecimento supercr\u00edtico, com subamortecimento ou\ncom amortecimento cr\u00edtico?\n\n\n\n\n```python\nprint(\"Exemplo 8.3\")\nR = 40\nL = 4\nC = 1/4\n\ndef sqrt(x):\n r = x**(1/2)\n return r\n\nalpha = R/(2*L)\nomega = 1/(sqrt(L*C))\ns1 = -alpha + sqrt(alpha**2 - omega**2)\ns2 = -alpha - sqrt(alpha**2 - omega**2)\n\nresposta = \"\"\n\nif alpha > omega:\n resposta = \"superamortecimento\"\nelif alpha == omega:\n resposta = \"amortecimento cr\u00edtico\"\nelse:\n resposta = \"subamortecimento\"\n\nprint(\"Raiz s1:\",s1)\nprint(\"Raiz s2:\",s2)\nprint(\"Resposta:\",resposta)\n```\n\n Exemplo 8.3\n Raiz s1: -0.10102051443364424\n Raiz s2: -9.898979485566356\n Resposta: superamortecimento\n\n\n**Problema Pr\u00e1tico 8.3**\n\nSe R = 10, L = 5 H e C = 2 mF na Figura 8.8, determine alpha, omega0, s1 e s2. Qual \u00e9 o tipo\nde resposta natural que o circuito apresentar\u00e1?\n\n\n```python\nprint(\"Problema Pr\u00e1tico 8.3\")\n\nm = 10**(-3) ##definicao de mili\n\nR = 10\nL = 5\nC = 2*m\n\nalpha = R/(2*L)\nomega = 1/(sqrt(L*C))\ns1 = -alpha + sqrt(alpha**2 - omega**2)\ns2 = -alpha - sqrt(alpha**2 - omega**2)\n\nresposta = \"\"\n\nif alpha > omega:\n resposta = \"superamortecimento\"\nelif alpha == omega:\n resposta = \"amortecimento cr\u00edtico\"\nelse:\n resposta = \"subamortecimento\"\n\nprint(\"Alpha:\",alpha)\nprint(\"Omega:\",omega)\nprint(\"Raiz s1:\",s1)\nprint(\"Raiz s2:\",s2)\nprint(\"Resposta:\",resposta)\n```\n\n Problema Pr\u00e1tico 8.3\n Alpha: 1.0\n Omega: 10.0\n Raiz s1: (-0.9999999999999994+9.9498743710662j)\n Raiz s2: (-1.0000000000000007-9.9498743710662j)\n Resposta: subamortecimento\n\n\n**Exemplo 8.4**\n\nDetermine i(t) no circuito da Figura 8.10. Suponha que o circuito tenha atingido o estado\nest\u00e1vel em t = 0\u2013.\n\n\n\n\n```python\nprint(\"Exemplo 8.4\")\n\nfrom sympy import *\n\nt = symbols('t')\nA2 = symbols('A2')\n\nC = 20*m\nL = 0.5\nVs = 10\n\n#tensao e corrente iniciais\nv0 = Vs*6/(6 + 4)\ni0 = v0/6\n\n#t > 0\nR = 3 + 6\nalpha = R/(2*L)\nomega = 1/(sqrt(L*C))\n\ns1 = -alpha + sqrt(alpha**2 - omega**2)\ns2 = -alpha - sqrt(alpha**2 - omega**2)\n\ndef resposta_rlc(alpha, omega): #funcao para verificar tipo de resposta\n resposta = \"\"\n if alpha > omega:\n resposta = \"superamortecimento\"\n elif alpha == omega:\n resposta = \"amortecimento cr\u00edtico\"\n else:\n resposta = \"subamortecimento\"\n return resposta\n\nresposta = resposta_rlc(alpha,omega)\n \nomegad = sqrt(omega**2 - alpha**2)\n\n#Tipo de resposta:\n#alpha < omega -> subamortecido -> i(t) = e^(-alpha t)*(A1*cos(wd*t) + A2*sen(wd*t))\n#i(0) = 1 -> A1 = 1\n\nprint(\"Tensao inicial capacitor:\",v0)\nprint(\"Corrente inicial indutor:\", i0)\nprint(\"Alpha:\",alpha)\nprint(\"Omega:\",omega)\nprint(\"Raiz s1:\",s1)\nprint(\"Raiz s2:\",s2)\nprint(\"Resposta:\",resposta)\n\n#resposta completa\nr = exp(-alpha*t)*(cos(omegad*t) + A2*sin(omegad*t))\n#di/dt = -1/L * [Ri(0) + v(0)] = -6 A/s\n#porem\nr2 = r.diff(t)\nprint(\"i(0):\",r2.subs(t,0))\n#assim\nA2 = (-6 + 9)/4.35\nprint(\"Constante A2:\",A2)\nr = exp(-alpha*t)*(cos(omegad*t) + A2*sin(omegad*t))\nprint(\"Resposta completa:\",r)\n```\n\n Exemplo 8.4\n Tensao inicial capacitor: 6.0\n Corrente inicial indutor: 1.0\n Alpha: 9.0\n Omega: 10.0000000000000\n Raiz s1: -9.0 + 4.35889894354067*I\n Raiz s2: -9.0 - 4.35889894354067*I\n Resposta: subamortecimento\n i(0): 4.35889894354067*A2 - 9.0\n Constante A2: 0.6896551724137931\n Resposta completa: (0.689655172413793*sin(4.35889894354067*t) + cos(4.35889894354067*t))*exp(-9.0*t)\n\n\n**Problema Pr\u00e1tico 8.4**\n\nO circuito da Figura 8.12 atingiu o estado est\u00e1vel em t = 0\u2013. Se o interruptor muda para\na posi\u00e7\u00e3o b em t = 0, calcule i(t) para t > 0.\n\n\n\n\n```python\nprint(\"Problema Pr\u00e1tico 8.4\")\n\nA2 = symbols('A2')\n\nL = 1\nC = 1/9\nVs = 100\n\n#para t < 0\ni0 = Vs/10\nv0 = 0\n\n#para t > 0\nR = 5\nalpha = R/(2*L)\nomega = 1/(sqrt(L*C))\nresposta = resposta_rlc(alpha,omega)\nomegad = sqrt(omega**2 - alpha**2)\n\n#Encontrar constantes A1 e A2\n\n#i(0) = Vs/10 = 10 A\n#i(t) = exp(-alpha*t)*(A1*cos(omegad*t) + A2*sin(omegad*t))\n#i(0) = A1\nA1 = 10\n#di/dt = -1/L * [Ri(0) + v(0)]\ndi = -1/L * (R*i0 + v0)\nprint(\"di/dt:\",di) #di = -50\nr = exp(-alpha*t)*(A1*cos(omegad*t) + A2*sin(omegad*t))\nr2 = r.diff(t)\nprint(\"i(0):\",r2.subs(t,0)) #1.65831*A2 - 25\nA2 = (di + 25)/1.65831\nprint(\"Constante A2:\",A2)\n\n#Resposta completa\nr = exp(-alpha*t)*(A1*cos(omegad*t) + A2*sin(omegad*t))\n\nprint(\"Alpha:\",alpha)\nprint(\"Omega0:\",omega)\nprint(\"Resposta:\",resposta)\nprint(\"i(t):\",r)\n```\n\n Problema Pr\u00e1tico 8.4\n di/dt: -50.0\n i(0): 1.6583123951777*A2 - 25.0\n Constante A2: -15.075589003262358\n Alpha: 2.5\n Omega0: 3.00000000000000\n Resposta: subamortecimento\n i(t): (-15.0755890032624*sin(1.6583123951777*t) + 10*cos(1.6583123951777*t))*exp(-2.5*t)\n\n", "meta": {"hexsha": "5bb03584786e8556ad02035ff2e895913004af5f", "size": 26302, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Aula 13 - Circuitos de Segunda Ordem.ipynb", "max_stars_repo_name": "ofgod2/Circuitos-electricos-Boylestad-12ed-Portugues", "max_stars_repo_head_hexsha": "60e815f6904858f3cda8b5c7ead8ea77aa09c7fd", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2019-08-13T13:33:15.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-16T16:46:06.000Z", "max_issues_repo_path": "Aula 13 - Circuitos de Segunda Ordem.ipynb", "max_issues_repo_name": "ofgod2/Circuitos-electricos-Boylestad-12ed-Portugues", "max_issues_repo_head_hexsha": "60e815f6904858f3cda8b5c7ead8ea77aa09c7fd", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2017-08-24T17:36:15.000Z", "max_issues_repo_issues_event_max_datetime": "2017-08-24T17:36:15.000Z", "max_forks_repo_path": "Aula 13 - Circuitos de Segunda Ordem.ipynb", "max_forks_repo_name": "ofgod2/Circuitos-electricos-Boylestad-12ed-Portugues", "max_forks_repo_head_hexsha": "60e815f6904858f3cda8b5c7ead8ea77aa09c7fd", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 8, "max_forks_repo_forks_event_min_datetime": "2019-03-29T14:31:49.000Z", "max_forks_repo_forks_event_max_datetime": "2021-12-30T17:59:23.000Z", "avg_line_length": 30.232183908, "max_line_length": 525, "alphanum_fraction": 0.5032697133, "converted": true, "num_tokens": 6742, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.6859494614282923, "lm_q1q2_score": 0.42697564299892715}} {"text": "```python\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\n```\n\n\n```python\ndf1 = pd.read_csv('data/data.csv')\ndf1\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        dayHumidity(%)Tmax(oC)Tmin(oc)Rain(mm)
                                        0194.713.05.60.0
                                        12100.016.42.59.9
                                        2398.422.02.30.3
                                        3498.419.51.00.0
                                        4596.820.01.40.0
                                        ..................
                                        35535695.420.52.70.0
                                        35635794.220.53.00.0
                                        35735895.620.03.70.0
                                        35835991.119.22.20.0
                                        35936096.919.11.40.0
                                        \n

                                        360 rows \u00d7 5 columns

                                        \n
                                        \n\n\n\n\n```python\ndf1 = df1.set_index('day')\ndf1\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Humidity(%)Tmax(oC)Tmin(oc)Rain(mm)
                                        day
                                        194.713.05.60.0
                                        2100.016.42.59.9
                                        398.422.02.30.3
                                        498.419.51.00.0
                                        596.820.01.40.0
                                        ...............
                                        35695.420.52.70.0
                                        35794.220.53.00.0
                                        35895.620.03.70.0
                                        35991.119.22.20.0
                                        36096.919.11.40.0
                                        \n

                                        360 rows \u00d7 4 columns

                                        \n
                                        \n\n\n\n\n```python\ndf1['temp(oC)'] = (df1['Tmax(oC)']+df1['Tmin(oc)'])/2\ndf1\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Humidity(%)Tmax(oC)Tmin(oc)Rain(mm)temp(oC)
                                        day
                                        194.713.05.60.09.30
                                        2100.016.42.59.99.45
                                        398.422.02.30.312.15
                                        498.419.51.00.010.25
                                        596.820.01.40.010.70
                                        ..................
                                        35695.420.52.70.011.60
                                        35794.220.53.00.011.75
                                        35895.620.03.70.011.85
                                        35991.119.22.20.010.70
                                        36096.919.11.40.010.25
                                        \n

                                        360 rows \u00d7 5 columns

                                        \n
                                        \n\n\n\n\n```python\ndf1.iloc[1,4]\n```\n\n\n\n\n 9.45\n\n\n\n\n```python\ndf1.describe().T\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        countmeanstdmin25%50%75%max
                                        Humidity(%)360.088.2155568.91373352.182.77589.90095.225100.00
                                        Tmax(oC)360.025.8947224.12092311.222.65027.15029.22533.10
                                        Tmin(oc)360.012.8222226.513911-1.37.50013.40019.50021.50
                                        Rain(mm)360.04.0686119.9430380.00.0000.0001.60075.00
                                        temp(oC)360.019.3584725.0675839.114.77520.72523.82526.55
                                        \n
                                        \n\n\n\n\n```python\nplt.figure(figsize = [8,6])\nplt.plot(df1.index,df1['Humidity(%)'])\nplt.xlabel('Day no.')\nplt.ylabel('Humidity(%)')\nplt.title('Daily variation of Humidity')\nplt.show()\n```\n\n\n```python\nplt.figure(figsize = [8,6])\nplt.plot(df1.index,df1['Tmax(oC)'],label='Max. temp')\nplt.plot(df1.index,df1['Tmin(oc)'],label='Min. temp')\nplt.xlabel('Day no.')\nplt.ylabel('Temperature($^o$C)')\nplt.title('Daily variation of temperature')\nplt.legend()\nplt.show()\n```\n\n\n```python\nday=[0,30,60,90,120,150,180,210,240,270,300,330,360]\nmtemp=[]\nmRH=[]\nmrain=[]\nfor i in range(12):\n mRH.append(np.sum(df1.iloc[day[i]+1:day[i+1],0])/30)\n mtemp.append(np.sum(df1.iloc[day[i]+1:day[i+1],4])/30)\n mrain.append(np.sum(df1.iloc[day[i]+1:day[i+1],3]))\n```\n\n\n```python\nm = np.arange(1,13)\nmonth=['Jan','Feb','March','April','May','June','July','Aug','Sept','Oct','Nov','Dec']\nplt.bar(m,mtemp,color='cyan',edgecolor='black') \nplt.xticks(m, month )\nplt.xlabel('Month', fontsize=14)\nplt.ylabel('Temperature($^o$C)', fontsize=14) \nplt.tick_params(axis='x', rotation= 45)\nplt.title('Monthly variation of temperature', fontsize = 18 ,color='green') \nplt.tight_layout() \nplt.show()\n```\n\n\n```python\nplt.bar(m,mRH,color='green',edgecolor='black') \nplt.xticks(m, month )\nplt.xlabel('Month', fontsize=14)\nplt.ylabel('Humidity(%)', fontsize=14) \nplt.tick_params(axis='x', rotation= 45)\nplt.title('Monthly variation of Humidity', fontsize = 18 ,color='green') \nplt.tight_layout() \nplt.show()\n```\n\n\n```python\nplt.bar(m,mrain,color='yellow',edgecolor='black') \nplt.xticks(m, month )\nplt.xlabel('Month', fontsize=14)\nplt.ylabel('Rainfall(mm)', fontsize=14) \nplt.tick_params(axis='x', rotation= 45)\nplt.title('Monthly variation of rainfall', fontsize = 18 ,color='green') \nplt.tight_layout() \nplt.show()\n```\n\n\\begin{equation}\ndew~point=\\frac{b(aT/(b+T)+ln(RH))}{a-(aT/(b+T)+ln(RH)}\n\\end{equation}\n\n\n```python\ndef DewPoin(t,rh):\n a=17.27\n b=237.7\n x=a*t/(b+t)\n nm=b*(x+np.log(rh/100))\n dm=a-(x+np.log(rh/100))\n y=nm/dm\n return y\n```\n\n\n```python\ndt=DewPoin(26,78)\ndt\n```\n\n\n\n\n 21.85732599954956\n\n\n\n\n```python\ndf1['dewpoint(oC)']=DewPoin(df1['temp(oC)'],df1['Humidity(%)'])\ndf1\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Humidity(%)Tmax(oC)Tmin(oc)Rain(mm)temp(oC)dewpoint(oC)
                                        day
                                        194.713.05.60.09.308.493324
                                        2100.016.42.59.99.459.450000
                                        398.422.02.30.312.1511.904965
                                        498.419.51.00.010.2510.008675
                                        596.820.01.40.010.7010.212111
                                        .....................
                                        35695.420.52.70.011.6010.889071
                                        35794.220.53.00.011.7510.847578
                                        35895.620.03.70.011.8511.169239
                                        35991.119.22.20.010.709.306803
                                        36096.919.11.40.010.259.779279
                                        \n

                                        360 rows \u00d7 6 columns

                                        \n
                                        \n\n\n\n\n```python\nplt.figure(figsize = [8,6])\nplt.plot(df1.index,df1['dewpoint(oC)'])\nplt.xlabel('Day no.')\nplt.ylabel('dew point($^o$C)')\nplt.title('Daily variation of Dew point')\nplt.show()\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "86a2ca2ad34e66e371d0d05db72aec224eb750c2", "size": 251095, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "PMS_data.ipynb", "max_stars_repo_name": "joshidot/NPS", "max_stars_repo_head_hexsha": "0b5b7dde9b5a9769c8a437d193b210545f9344ca", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "PMS_data.ipynb", "max_issues_repo_name": "joshidot/NPS", "max_issues_repo_head_hexsha": "0b5b7dde9b5a9769c8a437d193b210545f9344ca", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "PMS_data.ipynb", "max_forks_repo_name": "joshidot/NPS", "max_forks_repo_head_hexsha": "0b5b7dde9b5a9769c8a437d193b210545f9344ca", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 237.1057601511, "max_line_length": 62188, "alphanum_fraction": 0.8915032159, "converted": true, "num_tokens": 5264, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.7025300573952052, "lm_q1q2_score": 0.42690164157965366}} {"text": "```python\n%matplotlib inline\n\nimport matplotlib\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nplt.rcParams[\"figure.figsize\"] = (12, 9)\nplt.rcParams[\"font.size\"] = 18\n```\n\n# In Core Fuel Management\n\nIn core fuel management focuses on the study of requirements and operational considerations impacting fuel performance in the reactor core, power history, core loading patterns, and refuelling activities.\n\n\n## Learning Objectives\n\nAt the end of this lesson, you will be equipped to:\n\n- List safety constraints driving in core fuel management decisions.\n- Calculate capacity and availability factors.\n- Calculate the mass required for each reactor year of operation.\n- Calculate core and assembly discharge burnup based on power output.\n- Analyze the reactivity evolution of a core based on burnup.\n- Apply burnup calculations to multiple batch cores.\n- Recognize the relationship between the number of batches and the final burnup. \n- Understand the goals driving choices in various fuel loading patterns. \n- Apply these lessons to pebble-fuelled and liquid-fueled advanced reactor designs.\n- Recognize the impact of extended burnup on fuel utilization, SWU utilization, and fuel cycle cost.\n- Understand how isotopic activities can be used to determine fuel burnup.\n- Calculate burnup based on key activity ratios.\n\n## Safety Constraints\n\n\n- $\\frac{P_{peak}}{P_{avg}}$, peak to average power ratio.\n- $T_{max}$, maximimum core temperature.\n- Departure from Nucleate Boiling Ratio (DNBR)\n- $\\rho$, reactivity in the core.\n- $\\alpha_T$, temperature coefficient of reactivity \n\nPrimarily, there is a loss of coolant accident (LOCA) peak clad temp (PCT) limit of 1205 $^\\circ C$, which limits the maximum pellet linear power density to approx 48 kW/m at Hot Full Power(HFP).\n\n- Critical Heat Flux (CHF), which denotes departure from nuclear boiling (DNB) for a PWR and Dryout for a BWR, not being exceeded during anticipated transients, which limits the maximum average fuel pin linear power density to approximately 29 kW/m at HFP.\n- Fuel cladding strain limit not exceeded during anticipated transients\n\n### Safety Variables\n\n- Fuel enrichment\n- Re-load batch size & number of assemblies\n- Fuel loading pattern of fresh and partially spent fuel assemblies\n- Control mechanisms\n\n\n## Mass Required\n\nThe simplest possible representation of the mass of fuel that must be added into a reactor is:\n\n\\begin{align}\nM(t) &= \\frac{Q}{BU}\n\\end{align}\n\nwhere\n\\begin{align}\nM &= \\mbox{mass of heavy metal (e.g., uranium) in the core }[MTHM/yr]\\\\\nQ &= \\mbox{annual thermal energy output }[GWd/yr]\\\\\nBU &= \\mbox{burnup }[GWd/MTIHM]\n\\end{align}\n\n\nBut, Q itself typically needs to be back-calculated from energy produced.\n\n\\begin{align}\nQ &= \\frac{P_0\\cdot CF\\cdot T}{\\eta_{th}}\n\\end{align}\n\nwhere\n\\begin{align}\nP_0 &= \\mbox{installed electric capacity }[GWe]\\\\\nCF &= \\mbox{capacity factor }[-]\\\\\nT &= \\mbox{time in core } [days]\\\\\n\\eta_{th} &= \\mbox{thermal efficiency }[GWe/GWth]\\\\\n\\end{align}\n\n\n\n\n```python\ndef m(q, bu):\n return q/bu\n\ndef q(p0, cf, t, eta_th):\n return p0*cf*t/eta_th\n\np0 = 1.500 # installed electric capacity MWe\ncf = 0.9 # capacity factor\nt = 365 # days per year\neta_th = 0.33 # thermal efficiency MWe/MWth\nbu = 50 # burnup GWd/MTIHM\n\nprint(m(q(p0, cf, t, eta_th), bu))\n\n```\n\n 29.863636363636363\n\n\n## Capacity and Availability Factors\n\nThe capacity factor is representative of the plant's tendency to acheive its rated power capacity.\n\n\\begin{align}\nCF &= \\frac{\\mbox{actual power generated over time T}}{\\mbox{rated power potential over time T}}\\\\\n &=\\frac{\\int_0^T P(t)dt}{P_0T}\\\\\nP(t) &= \\mbox{ thermal power at time t during period T}\n\\end{align}\n\nThe capacity factor, integrated over time, gives Effective Full Power Days (EFPD), the equivalent number of days at full power.\n\n\\begin{align}\nEFPD &= \\int_0^TCF(t)dt\\\\\n &= \\int_0^T \\frac{\\int_0^T P(t)dt}{P_0T}\\\\\n\\end{align}\n\n\n\nThe availability factor is always greater than the capacity factor. \n\n\\begin{align}\nAF &= \\frac{\\mbox{time during which the reactor was operational during time period T}}{T}\n\\end{align}\n\n\n\n\n```python\n# The reactor shuts down:\n# for a few days during the 10th month\n# for one month during month 18\nshutdowns = {10:10.1,\n 18.5:19.5}\n\nimport numpy as np\ndef A(t, shutdowns):\n to_ret = 1.0*(t > 0)\n for start,stop in shutdowns.items():\n if start < t and t < stop:\n to_ret = 0\n return to_ret\n\ntimes = np.arange(0.0, 20.0, 0.01)\nhist = np.arange(0.0, 20.0, 0.01)\ncf = np.arange(0.0, 20.0, 0.01)\n\nfor i in range(0, times.size):\n hist[i] = A(times[i], shutdowns)\n cf[i] = A(times[i], shutdowns)*(1.-0.01*np.random.random())\n \nplt.plot(times, hist, label='Availability')\nplt.plot(times, cf, label='Capacity')\n\nplt.ylim([-0.5, 1.5])\nplt.title('Capacity and Availabilty')\nplt.xlabel('Time (months)')\nplt.ylabel('Factor [-]')\nplt.legend()\n\n```\n\nWe can do a quick numeric integral to get each factor as an integral over the 20 month cycle.\n\n\\begin{align}\nAF &= \\frac{\\int_0^{20}A(t)dt}{T}\\\\\nCF &= \\frac{\\int_0^{20}P(t)dt}{P_0T}\\\\\n\\end{align}\n\n\n\n```python\nprint(\"Availability Factor = \", hist.sum()/hist.shape[0])\nprint(\"Capacity Factor = \", cf.sum()/cf.shape[0])\n```\n\n Availability Factor = 0.9455\n Capacity Factor = 0.9408645280541623\n\n\n## Simple Reactivity Model\n- On each cycle (1/n)th of the fuel is replaced\n- Each fuel batch experiences a discharge burnup of Bd\n- Each fuel batch on each cycle experiences a burnup of Bd/n\n- $k_{reactor}$ is the uncontrolled multiplication factor (excess reactivity)\n- $k_i$ is the infinite multiplication factor of a fuel batch (excess reactivity)\n \nEach batch of fuel will have a different burn-up and $k_i(B)$ since each batch has been in the reactor a different length of time. The reactivity of the reactor is found by summing over the reactivities of all the batches of fuel, for n batches:\n\n\\begin{align}\nk_{reactor} = \\frac{1}{n}\\sum_{i=1}^{n}k_i(B)\n\\end{align}\n\n\n\\begin{align}\nk_i(B) = k_0 - \\alpha B_n\n\\end{align}\n\n- $k_0$ is the uncontrolled infinite multiplication factor of the fuel batch when it is fresh.\n- $B_n$ is the burnup of the batch in a single cycle. The n refers to the number of batches that the reload scheme includes.\n- $\\alpha$ is a constant of proportionality with units of 1/Bn. Uniform linear depletion.\n- $k_F$ is the uncontrolled infinite multiplication factor necessary to sustain a chain reaction at the end of an operating cycle\n\n\n\n\n```python\n\ndef ki(k0, alpha, b):\n return k0 - alpha*b\n\ndef k(ki, n):\n return (1/n)*np.sum(ki)\n\nn=0\nk0 =4.5\nalpha = (k0 - 1)/20000\nbu = np.arange(0, 50000., 1000.)\nplt.plot(bu, ki(k0, alpha, bu))\nplt.plot(bu, np.zeros(bu.shape), color='r')\nplt.ylabel(r'$k_i(B)$')\nplt.xlabel(r'$B$')\nplt.title('Excess Reactivity Using Linear Depletion Model')\n```\n\nThis approximation is somewhat accurate and gives an intuition for the impact of reloading on excess reactivity in the core. \n\n\n\n## Single Cycle Refuelling\n\n\n\n\n\\begin{align}\nk_{reactor} = k_1(B_1)\n\\end{align}\n\n\n\\begin{align}\nk_1(B_1) = k_0 - \\alpha B_1\n\\end{align}\n\nTherefore the fuel burnup capability is:\n\n\\begin{align}\nB_1 &= \\frac{k_0-k_F}{\\alpha}\n\\end{align}\n\n\n## Two Cycle Refuelling\n\n\n\nAt the end of each cycle one batch of fuel has been burned for one cycle and the other batch has been burned for two cycles. Thus:\n\n\\begin{align}\nk_F &= \\frac{k_0 - \\alpha B_2}{2} + \\frac{k_0 - 2\\alpha B_2}{2}\\\\\n &= k_0 - \\frac{3\\alpha B_2}{2}\\\\\nB_2 &= \\frac{2(k_0 - k_F)}{3\\alpha}\\\\\n &= \\frac{2}{3}B_1\n\\end{align}\n\n- Each batch in the two cycle reload scheme is burned for $2B_2$. \n\nSo, in terms of the single cycle reload burnup:\n\n\\begin{align}\n2B_2 &= 2\\left(\\frac{2}{3}B_1\\right)\\\\\n &= \\frac{4}{3}B_1\\\\\n\\end{align}\n\n**This means there is 1/3 more burnup in the two cycle reload, for the same initial and final multiplication factors $k_0$ and $k_F$ (exactly the same fuel.)**\n\n\n## N Cycle Reload Scheme\n\nThe relation between end-of-cycle core multiplication factor kF and the fresh fuel batch infinite multiplication factor k0 and the batch burnup in general is\n\n\n\\begin{align}\nk_F &= k_0 - \\frac{1}{n}\\sum_{i=1}^{n}i\\alpha B_n\\\\\n\\end{align}\n\nRecall from your geometric series intution:\n\\begin{align}\n\\sum_{i=1}^{n}i &= \\frac{n(n + 1)}{2}\\\\\n\\end{align}\n\nTherefore:\n\n\\begin{align}\nk_F &= k_0 - \\left(\\frac{n + 1}{2}\\right)\\alpha B_n\\\\\n\\end{align}\n\nThe batch burnup in a single cycle is then the result of solving for $B_n$:\n\n\\begin{align}\nB_n &= \\frac{2(k_0 - k_F)}{\\alpha(n + 1)}\n\\end{align}\n\n\nThe discharge burnup of batch n, is the batch burnup in a cycle times the number of cycles:\n\n\\begin{align}\nB_n^d &= nB_n\\\\\n&= \\frac{2n(k_0 - k_F)}{\\alpha(n + 1)}\\\\\n &= \\left(\\frac{2n}{n + 1}\\right)\\frac{k_0 - k_F}{\\alpha} \\\\\n &= \\left(\\frac{2n}{n + 1}\\right)B_1 \\\\\n\\end{align}\n\n\n\n```python\ndef bd(n, b1):\n num = 2*n*b1\n denom = n+1\n return num/denom\n\nb1 = 12000\nn = np.arange(1,50)\nplt.plot(n, bd(n, b1))\n```\n\n### Discussion: What is the primary drawback of many batches per core?\n \n \n\n## Fuel Loading Patterns \n\nVarious fuel loading patterns are used to acheive improved fuel utilization (higher burnup), better core control, and lower leakage to the pressure vessel. \n\n\n\n\n\n\n\n\n## Many and $\\infty$ Batch Reactor Designs\n\nInfinite batch refuelling (a.k.a. online refuelling) is possible in liquid fuelled cores with online reprocessing.\n\n\n\nWhat exactly is a pebble core, then, in terms of batches?\n\n\n
                                        Aufiero, 2016
                                        \n\n\n\n\n## Determining Burnup\n\n- Direct methods occur while the fuel is still in the core (using ion chambers and in-core flux probes)\n- Indirect methods use measurements of activity after the fuel has been removed.\n\n\\begin{align}\nBU &= a + bA(^{137}Cs)\\\\\nBU &= c(e, r) + d(e, r) \\left[A(^{134}Cs)/A(^{137}Cs)\\right]\\\\\nBU &= a\\cdot exp{\\left[b\\cdot ln\\left(\\frac{A(^{106}Ru)A(^{137}Cs)}{[A(^{134}Cs)^2}\\right)\\right]}\\\\\na, b, c, d &= \\mbox{calibration constants}\\\\\ne &= \\mbox{enrichment}\\\\\nr &= \\mbox{power rating}\n\\end{align}\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "27a9652058e42ed84228e0a4f373f4d5e1553b33", "size": 121502, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "in-core/in-core.ipynb", "max_stars_repo_name": "abachma2/npre412", "max_stars_repo_head_hexsha": "3f105a15edc07745f1dd65cd791777a01136ec23", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "in-core/in-core.ipynb", "max_issues_repo_name": "abachma2/npre412", "max_issues_repo_head_hexsha": "3f105a15edc07745f1dd65cd791777a01136ec23", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "in-core/in-core.ipynb", "max_forks_repo_name": "abachma2/npre412", "max_forks_repo_head_hexsha": "3f105a15edc07745f1dd65cd791777a01136ec23", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 211.6759581882, "max_line_length": 39904, "alphanum_fraction": 0.9041497259, "converted": true, "num_tokens": 2991, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631556226292, "lm_q2_score": 0.7025300573952054, "lm_q1q2_score": 0.4269016315965173}} {"text": "# **CS224W - Colab 3**\n\nIn Colab 2 we constructed GNN models by using PyTorch Geometric's built in GCN layer, `GCNConv`. In this Colab we will go a step deeper and implement the **GraphSAGE** ([Hamilton et al. (2017)](https://arxiv.org/abs/1706.02216)) layer directly. Then we will run our models on the CORA dataset, which is a standard citation network benchmark dataset.\n\n**Note**: Make sure to **sequentially run all the cells in each section** so that the intermediate variables / packages will carry over to the next cell\n\nHave fun and good luck on Colab 3 :)\n\n# Device\nWe recommend using a GPU for this Colab.\n\nPlease click `Runtime` and then `Change runtime type`. Then set the `hardware accelerator` to **GPU**.\n\n## Installation\n\n\n```python\n# Install torch geometric\nimport os\nif 'IS_GRADESCOPE_ENV' not in os.environ:\n !pip install torch-scatter -f https://pytorch-geometric.com/whl/torch-1.9.0+cu111.html\n !pip install torch-sparse -f https://pytorch-geometric.com/whl/torch-1.9.0+cu111.html\n !pip install torch-geometric\n !pip install -q git+https://github.com/snap-stanford/deepsnap.git\n```\n\n Looking in links: https://pytorch-geometric.com/whl/torch-1.9.0+cu111.html\n Collecting torch-scatter\n Downloading https://data.pyg.org/whl/torch-1.9.0%2Bcu111/torch_scatter-2.0.8-cp37-cp37m-linux_x86_64.whl (10.4 MB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 10.4 MB 12.5 MB/s \n \u001b[?25hInstalling collected packages: torch-scatter\n Successfully installed torch-scatter-2.0.8\n 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Installing collected packages: isodate, yacs, rdflib, torch-geometric\n Successfully installed isodate-0.6.0 rdflib-6.0.2 torch-geometric-2.0.1 yacs-0.1.8\n Building wheel for deepsnap (setup.py) ... \u001b[?25l\u001b[?25hdone\n\n\n\n```python\nimport torch_geometric\ntorch_geometric.__version__\n```\n\n\n\n\n '2.0.2'\n\n\n\n# 1) GNN Layers\n\n## Implementing Layer Modules\n\nIn Colab 2, we implemented a GCN model for node and graph classification tasks. However, for that notebook we took advantage of PyG's built in GCN module. For Colab 3, we provide a build upon a general Graph Neural Network Stack, into which we will be able to plugin our own module implementations: GraphSAGE and GAT.\n\nWe will then use our layer implemenations to complete node classification on the CORA dataset, a standard citation network benchmark. In this dataset, nodes correspond to documents and edges correspond to undirected citations. Each node or document in the graph is assigned a class label and features based on the documents binarized bag-of-words representation. Specifically, the Cora graph has 2708 nodes, 5429 edges, 7 prediction classes, and 1433 features per node. \n\n## GNN Stack Module\n\nBelow is the implementation of a general GNN stack, where we can plugin any GNN layer, such as **GraphSage**, **GAT**, etc. This module is provided for you. Your implementations of the **GraphSage** and **GAT** (Colab 4) layers will function as components in the GNNStack Module.\n\n\n```python\nimport torch\nimport torch_scatter\nimport torch.nn as nn\nimport torch.nn.functional as F\n\nimport torch_geometric.nn as pyg_nn\nimport torch_geometric.utils as pyg_utils\n\nfrom torch import Tensor\nfrom typing import Union, Tuple, Optional\nfrom torch_geometric.typing import (OptPairTensor, Adj, Size, NoneType,\n OptTensor)\n\nfrom torch.nn import Parameter, Linear\nfrom torch_sparse import SparseTensor, set_diag\nfrom torch_geometric.nn.conv import MessagePassing\nfrom torch_geometric.utils import remove_self_loops, add_self_loops, softmax\n\nclass GNNStack(torch.nn.Module):\n def __init__(self, input_dim, hidden_dim, output_dim, args, emb=False):\n super(GNNStack, self).__init__()\n conv_model = self.build_conv_model(args.model_type)\n self.convs = nn.ModuleList()\n self.convs.append(conv_model(input_dim, hidden_dim))\n assert (args.num_layers >= 1), 'Number of layers is not >=1'\n for l in range(args.num_layers-1):\n self.convs.append(conv_model(args.heads * hidden_dim, hidden_dim))\n\n # post-message-passing\n self.post_mp = nn.Sequential(\n nn.Linear(args.heads * hidden_dim, hidden_dim), nn.Dropout(args.dropout), \n nn.Linear(hidden_dim, output_dim))\n\n self.dropout = args.dropout\n self.num_layers = args.num_layers\n\n self.emb = emb\n\n def build_conv_model(self, model_type):\n if model_type == 'GraphSage':\n return GraphSage\n elif model_type == 'GAT':\n # When applying GAT with num heads > 1, you need to modify the \n # input and output dimension of the conv layers (self.convs),\n # to ensure that the input dim of the next layer is num heads\n # multiplied by the output dim of the previous layer.\n # HINT: In case you want to play with multiheads, you need to change the for-loop that builds up self.convs to be\n # self.convs.append(conv_model(hidden_dim * num_heads, hidden_dim)), \n # and also the first nn.Linear(hidden_dim * num_heads, hidden_dim) in post-message-passing.\n return GAT\n\n def forward(self, data):\n x, edge_index, batch = data.x, data.edge_index, data.batch\n \n for i in range(self.num_layers):\n x = self.convs[i](x, edge_index)\n x = F.relu(x)\n x = F.dropout(x, p=self.dropout,training=self.training)\n\n x = self.post_mp(x)\n\n if self.emb == True:\n return x\n\n return F.log_softmax(x, dim=1)\n\n def loss(self, pred, label):\n return F.nll_loss(pred, label)\n```\n\n## Creating Our Own Message Passing Layer\n\nNow let's start implementing our own message passing layers! Working through this part will help us become acutely familiar with the behind the scenes work of implementing Pytorch Message Passing Layers, allowing us to build our own GNN models. To do so, we will work with and implement 3 critcal functions needed to define a PyG Message Passing Layer: `forward`, `message`, and `aggregate`.\n\nBefore diving head first into the coding details, let us quickly review the key components of the message passing process. To do so, we will focus on a single round of messsage passing with respect to a single central node $x$. Before message passing, $x$ is associated with a feature vector $x^{l-1}$, and the goal of message passing is to update this feature vector as $x^l$. To do so, we implement the following steps: 1) each neighboring node $v$ passes its current message $v^{l-1}$ across the edge $(x, v)$ - 2) for the node $x$, we aggregate all of the messages of the neighboring nodes (for example through a sum or mean) - and 3) we transform the aggregated information by for example applying linear and non-linear transformations. Altogether, the message passing process is applied such that every node $u$ in our graph updates its embedding by acting as the central node $x$ in step 1-3 described above. \n\nNow, we extending this process to that of a single message passing layer, the job of a message passing layer is to update the current feature representation or embedding of each node in a graph by propagating and transforming information within the graph. Overall, the general paradigm of a message passing layers is: 1) pre-processing -> 2) **message passing** / propagation -> 3) post-processing. \n\nThe `forward` fuction that we will implement for our message passing layer captures this execution logic. Namely, the `forward` function handles the pre and post-processing of node features / embeddings, as well as initiates message passing by calling the `propagate` function. \n\n\nThe `propagate` function encapsulates the message passing process! It does so by calling three important functions: 1) `message`, 2) `aggregate`, and 3) `update`. Our implementation will vary slightly from this, as we will not explicitly implement `update`, but instead place the logic for updating node embeddings after message passing and within the `forward` function. To be more specific, after information is propagated (message passing), we can further transform the node embeddings outputed by `propagate`. Therefore, the output of `forward` is exactly the node embeddings after one GNN layer.\n\nLastly, before starting to implement our own layer, let us dig a bit deeper into each of the functions described above:\n\n1. \n\n```\ndef propagate(edge_index, x=(x_i, x_j), extra=(extra_i, extra_j), size=size):\n```\nCalling `propagate` initiates the message passing process. Looking at the function parameters, we highlight a couple of key parameters. \n\n - `edge_index` is passed to the forward function and captures the edge structure of the graph.\n - `x=(x_i, x_j)` represents the node features that will be used in message passing. In order to explain why we pass the tuple `(x_i, x_j)`, we first look at how our edges are represented. For every edge $(i, j) \\in \\mathcal{E}$, we can differentiate $i$ as the source or central node ($x_{central}$) and j as the neighboring node ($x_{neighbor}$). \n \n Taking the example of message passing above, for a central node $u$ we will aggregate and transform all of the messages associated with the nodes $v$ s.t. $(u, v) \\in \\mathcal{E}$ (i.e. $v \\in \\mathcal{N}_{u}$). Thus we see, the subscripts `_i` and `_j` allow us to specifcally differenciate features associated with central nodes (i.e. nodes recieving message information) and neighboring nodes (i.e. nodes passing messages). \n\n This is definitely a somewhat confusing concept; however, one key thing to remember / wrap your head around is that depending on the perspective, a node $x$ acts as a central node or a neighboring node. In fact, in undirected graphs we store both edge directions (i.e. $(i, j)$ and $(j, i)$). From the central node perspective, `x_i`, x is collecting neighboring information to update its embedding. From a neighboring node perspective, `x_j`, x is passing its message information along the edge connecting it to a different central node.\n\n - `extra=(extra_i, extra_j)` represents additional information that we can associate with each node beyond its current feature embedding. In fact, we can include as many additional parameters of the form `param=(param_i, param_j)` as we would like. Again, we highlight that indexing with `_i` and `_j` allows us to differentiate central and neighboring nodes. \n\n The output of the `propagate` function is a matrix of node embeddings after the message passing process and has shape $[N, d]$.\n\n2. \n```\ndef message(x_j, ...):\n```\nThe `message` function is called by propagate and constructs the messages from\nneighboring nodes $j$ to central nodes $i$ for each edge $(i, j)$ in *edge_index*. This function can take any argument that was initially passed to `propagate`. Furthermore, we can again differentiate central nodes and neighboring nodes by appending `_i` or `_j` to the variable name, .e.g. `x_i` and `x_j`. Looking more specifically at the variables, we have:\n\n - `x_j` represents a matrix of feature embeddings for all neighboring nodes passing their messages along their respective edge (i.e. all nodes $j$ for edges $(i, j) \\in \\mathcal{E}$). Thus, its shape is $[|\\mathcal{E}|, d]$!\n - In implementing GAT we will see how to access additional variables passed to propagate\n\n Critically, we see that the output of the `message` function is a matrix of neighboring node embeddings ready to be aggregated, having shape $[|\\mathcal{E}|, d]$.\n\n3. \n```\ndef aggregate(self, inputs, index, dim_size = None):\n```\nLastly, the `aggregate` function is used to aggregate the messages from neighboring nodes. Looking at the parameters we highlight:\n\n - `inputs` represents a matrix of the messages passed from neighboring nodes (i.e. the output of the `message` function).\n - `index` has the same shape as `inputs` and tells us the central node that corresponding to each of the rows / messages $j$ in the `inputs` matrix. Thus, `index` tells us which rows / messages to aggregate for each central node.\n\n The output of `aggregate` is of shape $[N, d]$.\n\n\nFor additional resources refer to the PyG documentation for implementing custom message passing layers: https://pytorch-geometric.readthedocs.io/en/latest/notes/create_gnn.html\n\n## GraphSage Implementation\n\nFor our first GNN layer, we will implement the well known GraphSage ([Hamilton et al. (2017)](https://arxiv.org/abs/1706.02216)) layer! \n\nFor a given *central* node $v$ with current embedding $h_v^{l-1}$, the message passing update rule to tranform $h_v^{l-1} \\rightarrow h_v^l$ is as follows: \n\n\\begin{equation}\nh_v^{(l)} = W_l\\cdot h_v^{(l-1)} + W_r \\cdot AGG(\\{h_u^{(l-1)}, \\forall u \\in N(v) \\})\n\\end{equation}\n\nwhere $W_1$ and $W_2$ are learanble weight matrices and the nodes $u$ are *neighboring* nodes. Additionally, we use mean aggregation for simplicity:\n\n\\begin{equation}\nAGG(\\{h_u^{(l-1)}, \\forall u \\in N(v) \\}) = \\frac{1}{|N(v)|} \\sum_{u\\in N(v)} h_u^{(l-1)}\n\\end{equation}\n\nOne thing to note is that we're adding a **skip connection** to our GraphSage implementation through the term $W_l\\cdot h_v^{(l-1)}$. \n\nBefore implementing this update rule, we encourage you to think about how different parts of the formulas above correspond with the functions outlined earlier: 1) `forward`, 2) `message`, and 3) `aggregate`. As a hint, we are given what the aggregation function is (i.e. mean aggregation)! Now the question remains, what are the messages passed by each neighbor nodes and when do we call the `propagate` function? \n\nNote: in this case the message function or messages are actually quite simple. Additionally, remember that the `propagate` function encapsulates the operations of / the outputs of the combined `message` and `aggregate` functions.\n\n\nLastly, $\\ell$-2 normalization of the node embeddings is applied after each iteration.\n\n\nFor the following questions, DON'T refer to any existing implementations online.\n\n\n```python\nclass GraphSage(MessagePassing):\n \n def __init__(self, in_channels, out_channels, normalize = True,\n bias = False, **kwargs): \n super(GraphSage, self).__init__(**kwargs)\n\n self.in_channels = in_channels\n self.out_channels = out_channels\n self.normalize = normalize\n\n self.lin_l = None\n self.lin_r = None\n\n ############################################################################\n # TODO: Your code here! \n # Define the layers needed for the message and update functions below.\n # self.lin_l is the linear transformation that you apply to embedding \n # for central node.\n # self.lin_r is the linear transformation that you apply to aggregated \n # message from neighbors.\n # Don't forget the bias!\n # Our implementation is ~2 lines, but don't worry if you deviate from this.\n \n self.lin_l = Linear(self.in_channels, self.out_channels, bias = bias)\n self.lin_r = Linear(self.in_channels, self.out_channels, bias = bias)\n \n ############################################################################\n \n\n self.reset_parameters()\n\n def reset_parameters(self):\n self.lin_l.reset_parameters()\n self.lin_r.reset_parameters()\n\n def forward(self, x, edge_index, size = None):\n \"\"\"\"\"\"\n\n out = None\n\n ############################################################################\n # TODO: Your code here! \n # Implement message passing, as well as any post-processing (our update rule).\n # 1. Call the propagate function to conduct the message passing.\n # 1.1 See the description of propagate above or the following link for more information: \n # https://pytorch-geometric.readthedocs.io/en/latest/notes/create_gnn.html\n # 1.2 We will only use the representation for neighbor nodes (x_j), so by default\n # we pass the same representation for central and neighbor nodes as x=(x, x). \n # 2. Update our node embedding with skip connection from the previous layer.\n # 3. If normalize is set, do L-2 normalization (defined in \n # torch.nn.functional)\n #\n # Our implementation is ~5 lines, but don't worry if you deviate from this.\n out = torch.zeros(x.shape[0],self.out_channels)\n for i in range(x.shape[0]): # loop over nodes\n index = edge_index[1][edge_index[0] == i]\n agg = x[index].mean(axis = 0)\n out_i = self.lin_l(x[i]) + self.lin_r(agg)\n if self.normalize:\n out_i = F.normalize(out_i, dim = 0)\n out[i] = out_i\n \n \n \n\n ############################################################################\n\n return out\n\n def message(self, x_j):\n\n out = None\n\n ############################################################################\n # TODO: Your code here! \n # Implement your message function here.\n # Hint: Look at the formulation of the mean aggregation function, focusing on \n # what message each neighboring node passes.\n #\n # Our implementation is ~1 lines, but don't worry if you deviate from this.\n\n ############################################################################\n\n return out\n\n def aggregate(self, inputs, index, dim_size = None):\n\n out = None\n\n # The axis along which to index number of nodes.\n node_dim = self.node_dim\n\n ############################################################################\n # TODO: Your code here! \n # Implement your aggregate function here.\n # See here as how to use torch_scatter.scatter: \n # https://pytorch-scatter.readthedocs.io/en/latest/functions/scatter.html#torch_scatter.scatter\n #\n # Our implementation is ~1 lines, but don't worry if you deviate from this.\n\n\n ############################################################################\n\n return out\n\n```\n\n## Building Optimizers\n\nThis function has been implemented for you. **For grading purposes please use the default Adam optimizer**, but feel free to play with other types of optimizers on your own.\n\n\n```python\nimport torch.optim as optim\n\ndef build_optimizer(args, params):\n weight_decay = args.weight_decay\n filter_fn = filter(lambda p : p.requires_grad, params)\n if args.opt == 'adam':\n optimizer = optim.Adam(filter_fn, lr=args.lr, weight_decay=weight_decay)\n elif args.opt == 'sgd':\n optimizer = optim.SGD(filter_fn, lr=args.lr, momentum=0.95, weight_decay=weight_decay)\n elif args.opt == 'rmsprop':\n optimizer = optim.RMSprop(filter_fn, lr=args.lr, weight_decay=weight_decay)\n elif args.opt == 'adagrad':\n optimizer = optim.Adagrad(filter_fn, lr=args.lr, weight_decay=weight_decay)\n if args.opt_scheduler == 'none':\n return None, optimizer\n elif args.opt_scheduler == 'step':\n scheduler = optim.lr_scheduler.StepLR(optimizer, step_size=args.opt_decay_step, gamma=args.opt_decay_rate)\n elif args.opt_scheduler == 'cos':\n scheduler = optim.lr_scheduler.CosineAnnealingLR(optimizer, T_max=args.opt_restart)\n return scheduler, optimizer\n```\n\n## Training and Testing\n\nHere we provide you with the functions to train and test. **Please do not modify this part for grading purposes.**\n\n\n```python\nimport time, os\n\nimport networkx as nx\nimport numpy as np\nimport torch\nimport torch.optim as optim\nfrom tqdm import trange\nimport pandas as pd\nimport copy\n\nfrom torch_geometric.datasets import TUDataset\nfrom torch_geometric.datasets import Planetoid\nfrom torch_geometric.data import DataLoader\n\nimport torch_geometric.nn as pyg_nn\n\nimport matplotlib.pyplot as plt\n\n\ndef train(dataset, args):\n \n print(\"Node task. test set size:\", np.sum(dataset[0]['test_mask'].numpy()))\n print()\n test_loader = loader = DataLoader(dataset, batch_size=args.batch_size, shuffle=False)\n\n # build model\n model = GNNStack(dataset.num_node_features, args.hidden_dim, dataset.num_classes, \n args)\n scheduler, opt = build_optimizer(args, model.parameters())\n\n # train\n losses = []\n test_accs = []\n best_acc = 0\n best_model = None\n for epoch in trange(args.epochs, desc=\"Training\", unit=\"Epochs\"):\n total_loss = 0\n model.train()\n for batch in loader:\n opt.zero_grad()\n pred = model(batch)\n label = batch.y\n pred = pred[batch.train_mask]\n label = label[batch.train_mask]\n loss = model.loss(pred, label)\n loss.backward()\n opt.step()\n total_loss += loss.item() * batch.num_graphs\n total_loss /= len(loader.dataset)\n losses.append(total_loss)\n\n if epoch % 10 == 0:\n test_acc = test(test_loader, model)\n test_accs.append(test_acc)\n if test_acc > best_acc:\n best_acc = test_acc\n best_model = copy.deepcopy(model)\n else:\n test_accs.append(test_accs[-1])\n \n return test_accs, losses, best_model, best_acc, test_loader\n\ndef test(loader, test_model, is_validation=False, save_model_preds=False, model_type=None):\n test_model.eval()\n\n correct = 0\n # Note that Cora is only one graph!\n for data in loader:\n with torch.no_grad():\n # max(dim=1) returns values, indices tuple; only need indices\n pred = test_model(data).max(dim=1)[1]\n label = data.y\n\n mask = data.val_mask if is_validation else data.test_mask\n # node classification: only evaluate on nodes in test set\n pred = pred[mask]\n label = label[mask]\n\n if save_model_preds:\n print (\"Saving Model Predictions for Model Type\", model_type)\n\n data = {}\n data['pred'] = pred.view(-1).cpu().detach().numpy()\n data['label'] = label.view(-1).cpu().detach().numpy()\n\n df = pd.DataFrame(data=data)\n # Save locally as csv\n df.to_csv('CORA-Node-' + model_type + '.csv', sep=',', index=False)\n \n correct += pred.eq(label).sum().item()\n\n total = 0\n for data in loader.dataset:\n total += torch.sum(data.val_mask if is_validation else data.test_mask).item()\n\n return correct / total\n \nclass objectview(object):\n def __init__(self, d):\n self.__dict__ = d\n\n```\n\n## Let's Start the Training!\n\nWe will be working on the CORA dataset on node-level classification.\n\nThis part is implemented for you. **For grading purposes, please do not modify the default parameters.** However, feel free to play with different configurations just for fun!\n\n**Submit your best accuracy and loss on Gradescope.**\n\n\n```python\nif 'IS_GRADESCOPE_ENV' not in os.environ:\n for args in [\n {'model_type': 'GraphSage', 'dataset': 'cora', 'num_layers': 2, 'heads': 1, 'batch_size': 32, 'hidden_dim': 32, 'dropout': 0.5, 'epochs': 500, 'opt': 'adam', 'opt_scheduler': 'none', 'opt_restart': 0, 'weight_decay': 5e-3, 'lr': 0.01},\n ]:\n args = objectview(args)\n for model in ['GraphSage']:\n args.model_type = model\n\n # Match the dimension.\n if model == 'GAT':\n args.heads = 2\n else:\n args.heads = 1\n\n if args.dataset == 'cora':\n dataset = Planetoid(root='/tmp/cora', name='Cora')\n else:\n raise NotImplementedError(\"Unknown dataset\") \n test_accs, losses, best_model, best_acc, test_loader = train(dataset, args) \n\n print(\"Maximum test set accuracy: {0}\".format(max(test_accs)))\n print(\"Minimum loss: {0}\".format(min(losses)))\n\n # Run test for our best model to save the predictions!\n test(test_loader, best_model, is_validation=False, save_model_preds=True, model_type=model)\n print()\n\n plt.title(dataset.name)\n plt.plot(losses, label=\"training loss\" + \" - \" + args.model_type)\n plt.plot(test_accs, label=\"test accuracy\" + \" - \" + args.model_type)\n plt.legend()\n plt.show()\n\n```\n\n## Question 1.1: What is the maximum accuracy obtained on the test set for GraphSage? (10 points)\n\nRunning the cell above will show the results of your best model and save your best model's predictions to a file named *CORA-Node-GraphSage.csv*. \n\nAs we have seen before you can view this file by clicking on the *Folder* icon on the left side pannel. When you sumbit your assignment, you will have to download this file and attatch it to your submission.\n", "meta": {"hexsha": "bb514264ce9db6a5db87148493ea2c39d9e58b2c", "size": 65336, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Colab 3/CS224W - Colab 3_tobias.ipynb", "max_stars_repo_name": "victorcroisfelt/aau-cs224w-ml-with-graphs", "max_stars_repo_head_hexsha": "adb38651be8da98cc574f127763c785ed16dfb5a", "max_stars_repo_licenses": ["Unlicense", "MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Colab 3/CS224W - Colab 3_tobias.ipynb", "max_issues_repo_name": "victorcroisfelt/aau-cs224w-ml-with-graphs", "max_issues_repo_head_hexsha": "adb38651be8da98cc574f127763c785ed16dfb5a", "max_issues_repo_licenses": ["Unlicense", "MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Colab 3/CS224W - Colab 3_tobias.ipynb", "max_forks_repo_name": "victorcroisfelt/aau-cs224w-ml-with-graphs", "max_forks_repo_head_hexsha": "adb38651be8da98cc574f127763c785ed16dfb5a", "max_forks_repo_licenses": ["Unlicense", "MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 82.2871536524, "max_line_length": 27244, "alphanum_fraction": 0.7448267418, "converted": true, "num_tokens": 7150, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.7490872243177518, "lm_q1q2_score": 0.42686934091794226}} {"text": "| |Pierre Proulx, ing, professeur|\n|:---|:---|\n|D\u00e9partement de g\u00e9nie chimique et de g\u00e9nie biotechnologique |** GCH200-Ph\u00e9nom\u00e8nes d'\u00e9changes I **|\n\n## Sections 0, 1 et 2: Sections 1.0, 1.1 et 1.2: la viscosit\u00e9 des fluides et comment la viscosit\u00e9 affecte le mouvement des fluides. \n>#### Pour les deux premi\u00e8res sections, lire les pages 1-23 du document suivant:\n>http://gch200-pp.espaceweb.usherbrooke.ca/Chap1.pdf \n#### elles vous pr\u00e9sentent le concept de viscosit\u00e9 et donnent des exemples de valeurs typiques des viscosit\u00e9s de certains fluides. \n\n# *\n\n## Section 1.3, calcul de la viscosit\u00e9 r\u00e9duite de fluides. La temp\u00e9rature, la pression et la viscosit\u00e9 sont normalis\u00e9es en les divisant par la propri\u00e9t\u00e9 critique. Le graphe 1.3-1 montre ces viscosit\u00e9s r\u00e9duites.\n\n> #### Les packages python \"CoolProp\" et \"thermo\" ont \u00e9t\u00e9 d\u00e9velopp\u00e9s pour calculer les propri\u00e9t\u00e9s thermodynamiques et de transport en utilisant les m\u00e9thodes les plus r\u00e9centes de calcul. Au lieu d'utiliser la figure 1.3-1 et l'annexe E pour estimer la viscosit\u00e9 en fonction de la temp\u00e9rature et de la pression, il est facile de trouver la viscosit\u00e9 en utilisant simplement, par exemple pour la viscosit\u00e9 du CO2 \u00e0 300 K et 101325 Pascals (1 atm):\n from CoolProp.CoolProp import PropsSI\n muCO2C=PropsSI('V','T',300,'P',101325,'CO2')\n import thermo as th\n CO2=th.Chemical('CO2',T=300,P=101325) # thermo utilise une approche objet\n print(muCO2C,CO2.mug) # ainsi CO2.mug est la viscosit\u00e9 \u00e0 l'\u00e9tat gazeux du CO2\n#### Dans ce qui suit, on utilise CoolProp pour reconstruire le graphe 1.3-1. Par la suite on utilisera aussi thermo\n\n#### Voyons d'abord si le calcul des propri\u00e9t\u00e9s critiques est bien fait, comparons avec l'annexe E, rappelons-nous que le SI est pour Syst\u00e8me International, donc M\u00e8tres, Kilogramme, Seconde, Joule, etc... Pas d'unit\u00e9s mixtes comme les poises, les atmosph\u00e8res, les PSI, les livres-forces, et autres unit\u00e9s d\u00e9velopp\u00e9es sans standard coh\u00e9rent. V\u00e9rifiez dans l'annexe E si les calculs de CoolProp coincident bien avec les donn\u00e9es de Transport Phenomena.\n\n\n```python\nfrom CoolProp.CoolProp import PropsSI\nmuCO2C=PropsSI('V','T',300,'P',101325,'CO2')\nprint(muCO2C)\n```\n\n 1.5021470647392787e-05\n\n\n\n```python\nimport thermo as th\nCO2=th.Chemical('CO2',T=300,P=101325)\nprint(CO2.mug)\n```\n\n 1.5021470647392787e-05\n\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom CoolProp.CoolProp import PropsSI\n%matplotlib inline\nprint(('{:<12}'*4).format('Fluide','Tc (K)','Pc (Pa)','muc (Pa*s)'))\nprint('-'*48)\nfor fluide in ['methane','ethane','propane','argon','N2','O2','CO2','air']:\n Tc=PropsSI('Tcrit',fluide)\n Pc=PropsSI('Pcrit',fluide)\n vc=PropsSI('V','T',Tc,'P',Pc,fluide)\n print('{:<12}{:<12.3f}{:<12.3e}{:<12.3e}'.format(fluide,Tc,Pc,vc))\n```\n\n Fluide Tc (K) Pc (Pa) muc (Pa*s) \n ------------------------------------------------\n methane 190.564 4.599e+06 1.593e-05 \n ethane 305.322 4.872e+06 2.187e-05 \n propane 369.890 4.251e+06 2.393e-05 \n argon 150.687 4.863e+06 2.731e-05 \n N2 126.192 3.396e+06 1.808e-05 \n O2 154.581 5.043e+06 2.632e-05 \n CO2 304.128 7.377e+06 3.404e-05 \n air 132.531 3.786e+06 2.008e-05 \n\n\n#### Maintenant construisons le graphe en nous basant sur la figure 1.3-1\n\n\n```python\nfluide='argon'\nTc=PropsSI('Tcrit',fluide)\nPc=PropsSI('Pcrit',fluide)\nvc=PropsSI('V','T',Tc,'P',Pc,fluide)\n\nT = np.logspace(-0.2, 1) \nP = np.logspace(-0.3, 1.3979, 6)\n\nvr = np.zeros((len(T),len(P)))\nfor i,Tr in enumerate(T):\n for j,Pr in enumerate(P):\n vr[i,j]=PropsSI('V','T',Tr*Tc,'P',Pr*Pc,fluide)/vc\n\nwith plt.xkcd():\n plt.rcParams['figure.figsize'] = 10,8 \n plt.loglog(T,vr)\n plt.xlabel('temperature reduite')\n plt.ylabel('viscosite reduite')\n plt.grid(True,which=\"both\")\n vr=[0.4,0.8,0.9,1,2,3,4,5,6,7,8,9,10,20]\n plt.xlim(T[0],T[len(T)-1])\n plt.ylim(np.amin(vr),np.amax(vr))\n plt.legend(['{:0.1f}'.format(p) for p in P])\n plt.title('Viscosite reduite en fonction de la temperature \\n et de la pressions reduites')\nplt.show()\n```\n\n#### Comparez avec la figure 1.3-1, vous verrez que les r\u00e9sultats sont tr\u00e8s similaires. On peut donc utiliser CoolProp avec confiance sur une grande gamme de temp\u00e9ratures et de pressions pour gaz et liquides.\n\n>>\n\n# *\n\n## Section 1.4, calcul des viscosit\u00e9s de gaz \u00e0 des pressions inf\u00e9rieures \u00e0 10% de la pression critique (CoolProp et thermo)\n\nLa viscosit\u00e9 individuelle peut \u00eatre \u00e0 partir de l'\u00e9quation 1.4-14 en utilisant l'annexe E2.\n\n>>> $\n\\begin{equation*}\n \\boxed{ \n \\mu = 2.6693 \\times 10^{-5} \\frac{\\sqrt{MT}}{\\sigma^2 \\Omega_{\\mu}}\n }\n \\end{equation*}\n$ \n\n\n> Cependant, on a vu CoolProp qui effectue ce calcul, maintenant regardons comment un autre package python (thermo) utilise lui-m\u00eame CoolProp pour calculer la viscosit\u00e9 \u00e0 partir de 1.4-14, et utilise une syntaxe qui peut sembler plus facile et ajoute les calculs de m\u00e9lange de gaz comme l'\u00e9quation 1.4-15, 1.4-16 (Wilkes). \n>Pour un m\u00e9lange de gaz on peut approximer la viscosit\u00e9 en utilisant une pond\u00e9ration propos\u00e9e par Wilkes en 1949.\n\n>>> $\n\\begin{equation*}\n \\boxed{\n \\mu_{mix} = \\sum_{i=0}^N \\frac {x_i \\mu_i} {\\sum_{j=0}^N (x_j \\phi_{ij})}\n }\n\\end{equation*}\n$\n\n\navec \n\n>>> \n$ \\begin{equation*}\n \\boxed{\n \\phi_{ij} = \\frac {1} { \\sqrt{8} }\n \\bigg ( 1+ \\frac{M_i}{M_j} \\bigg )^{-1/2}\n \\bigg [ 1+ \\bigg ( \\frac{\\mu_i}{\\mu_j} \\bigg )^{1/2}\n \\bigg ( \\frac{M_j}{M_i} \\bigg )^{1/4}\n \\bigg ]^{2}\n }\n \\end{equation*}\n $\n\n Voyons comment thermo r\u00e9soud l'exemple 1.4-1. Il calcule les viscosit\u00e9s individuelles des 3 gaz en utilisant 1.4-14, 1.4-15 et 1.4-16. Le r\u00e9sultat est l\u00e9g\u00e8rement diff\u00e9rent de celui de Transport Phenomena, pas moins pr\u00e9cis.\n\n\n```python\n# package thermo\nimport thermo as th\n#\n# regardons d'abord comment thermo calcule la viscosit\u00e9 d'un gaz et comparons avec\n# la syntaxe de Coolprop\nthermoMu = th.Chemical('CO2',300,101325).mu\ncoolPropMu = PropsSI('V','T',300,'P',101325,'CO2')\nExempleMu = th.Mixture(['CO2','O2','N2'],zs=[0.133,0.039,0.828],T=293).mu\n\nprint('Avec thermo : {:8.3e}'.format(thermoMu))\nprint('Avec Coolprop : {:8.3e}'.format(coolPropMu))\nprint('Calcul de l exemple 1.4-1 : {:8.3e}'.format(ExempleMu))\n```\n\n Avec thermo : 1.502e-05\n Avec Coolprop : 1.502e-05\n Calcul de l exemple 1.4-1 : 1.744e-05\n\n\n# *\n\n## Section 1.5, calcul des viscosit\u00e9s de liquides (CoolProp et thermo)\n\n>> Avec thermo, le calcul des propri\u00e9t\u00e9s des liquides est tr\u00e8s simple et tr\u00e8s naturel, par exemple pour l'eau liquide ou la vapeur d'eau \u00e0 50 degr\u00e9s C et pression atmosph\u00e9rique (on utilise toujours des unit\u00e9s du syst\u00e8me international dans thermo et CoolProp, donc P=101325 Pascals = 1 atmosph\u00e8re):\n\n\n```python\nprint(th.Chemical('water',273+50,101325).mug)\n# une autre fa\u00e7on de faire le m\u00eame travail, utilisant la nature objet du programme thermo\neau=th.Chemical('water',273+50,101325)\neau.mug\n```\n\n 1.0511439780362781e-05\n\n\n\n\n\n 1.0511439780362781e-05\n\n\n\n\n```python\nprint(th.Chemical('water',273+50,101325).mul)\n# ou puisqu'on a d\u00e9j\u00e0 d\u00e9fini l'objet eau de la classe Chemical, avec les T et P pr\u00e9c\u00e9dents\neau.mul # on a pas besoin de print, car la derni\u00e8re ligne de chaque cellule est imprim\u00e9e...\n```\n\n 0.0005478953637518872\n\n\n\n\n\n 0.0005478953637518872\n\n\n\n#### Comparons les valeurs obtenues avec thermo avec celles des \u00e9quations propos\u00e9es dans Transport Phenomena\n\n\n```python\n# viscosit\u00e9 du benz\u00e8ne, exemple 1.5-1\nprint(th.Chemical('benzene', T=273+20, P=101325).mul)\n```\n\n 0.0006482417530392874\n\n\n#### Le r\u00e9sultat est bien meilleur, on peut le comparer avec le tableau 1.1-3, donc on utilisera donc thermo pour le calcul des viscosit\u00e9s des liquides. Continuons \u00e0 comparer les valeurs pr\u00e9dites par thermo avec le tableau 1.1-3. Les m\u00e9thodes utilis\u00e9es par CoolProp et thermo pour calculer la viscosit\u00e9 des liquides sont beaucoup plus pr\u00e9cises que celle propos\u00e9e dans la section 1.5, alors on utilisera thermo.\n\n\n```python\nprint('***Tableau 1.1-3 Transport Phenomena, \u00e0 comparer\\n')\n#Nom, Nom Thermo, temperature (\u00b0C), pression (kPa)\nelements=[\n ['brome','bromine', 25,101.325],\n ['C2H5OH 0\u00b0C','C2H5OH', 0,101.325],\n ['C2H5OH 25\u00b0C','C2H5OH', 25,101.325],\n ['C2H5OH 50\u00b0C','C2H5OH', 50,101.325],\n ['Mercure','mercury', 20,101.325]\n]\n\nfor elem in elements:\n mul=th.Chemical(elem[1], T=273+elem[2], P=elem[3]*1000).mul\n print('{:<20}{:10.6e} Pa*s'.format(elem[0],mul))\n```\n\n ***Tableau 1.1-3 Transport Phenomena, \u00e0 comparer\n \n brome 9.755912e-04 Pa*s\n C2H5OH 0\u00b0C 1.824754e-03 Pa*s\n C2H5OH 25\u00b0C 1.085494e-03 Pa*s\n C2H5OH 50\u00b0C 6.907656e-04 Pa*s\n Mercure 1.558125e-03 Pa*s\n\n\n# *\n\n## Section 1.6- Calcul de la viscosit\u00e9 des suspensions.\n\nEn 1906, un jeune chercheur totalement inconnu de 27 ans propose une expression qui permet de calculer la viscosit\u00e9 de suspensions (dilu\u00e9es) de sph\u00e8res solides dans un liquide. Cette expression se lit:\n\n>>> $\n\\begin{equation*}\n \\boxed{ \n \\frac {\\mu_{eff}}{\\mu_0}= 1 + \\frac {5}{2}\\phi\n }\n \\end{equation*} (1.6-1)\n$ \n\nEn fait, il y avait une petite erreur de calcul, au lieu de 5/2 Einstein a donn\u00e9 2 comme coefficient devant $\\phi$. Il l'a corrig\u00e9e quelques ann\u00e9es plus tard, en 1911.\n\nCette \u00e9quation a \u00e9t\u00e9 utilis\u00e9e comme base pour plusieurs travaux par la suite , celle de Mooney par exemple:\n\n>>> $\n\\begin{equation*}\n \\boxed{ \n \\frac {\\mu_{eff}}{\\mu_0}= exp \\bigg(\n { \\frac {\\frac {5}{2} \\phi} {1-(\\phi/\\phi_0)} } \\bigg )\n }\n \\end{equation*} (1.6-2)\n$ \n\n\nLa cellule suivante effectue l'impl\u00e9mentation de la formule de Mooney et la compare avec un travail (Vand et al.) ou les auteurs ont liss\u00e9 des r\u00e9sultats exp\u00e9rimentaux. C'est l'exercice 1B-3\n\n\n```python\n# Exercice 1B3\n#\nimport sympy as sp\nfrom IPython.display import *\nfrom matplotlib import rcParams\n%matplotlib inline\n\nsp.init_printing(use_latex=True)\nphi,phi0=sp.symbols('phi,phi_0')\nmuV=sp.Function('mu_V')(phi)\nmuV=1+2.5*phi+7.17*phi**2+16.2*phi**3\nmuM=sp.Function('mu_M')(phi)\nmuM=sp.exp((5./2.*phi)/(1-phi/phi0))\nmuM=muM.subs(phi0,0.7)\nrcParams['lines.linewidth'] = 3\nplt.rcParams['figure.figsize'] = 10,8 \np1=sp.plot(muV,(phi,0,0.4),ylim=(1,5),legend=True,\n xlabel='$\\phi$',ylabel='${\\mu/\\mu_0}$',show=False)\np2=sp.plot(muM,(phi,0,0.4),ylim=(1,5),show=False)\np1.append(p2[0])\np1[0].label='Vand, fit valide environ jusqu''\u00e0 0.5'\np1[0].line_color='red'\np1[1].label='Mooney'\np1[1].line_color='black'\np1.show()\n```\n\n> #### On constate que l'approximation de Mooney n'est valide que pour des valeurs de $\\phi$ qui ne d\u00e9passent pas 25%, environ. Une suspension qui contient 25% par volume de solides est consid\u00e9r\u00e9e habituellement comme tr\u00e8s concentr\u00e9e.\n\n# *\n\n### Section 1.7: Flux de quantit\u00e9 de mouvement.\n\n>http://pierreproulx.espaceweb.usherbrooke.ca//images/GCH200_Ch1_resume.pdf\n", "meta": {"hexsha": "e54640c59e52bc3f45c74a4b1ea80360a81337d3", "size": 159841, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Chap-1.ipynb", "max_stars_repo_name": "Spationaute/GCH200", "max_stars_repo_head_hexsha": "55144f5b2a59a7240d36c985997387f5036149f7", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-02-26T16:29:58.000Z", "max_stars_repo_stars_event_max_datetime": "2018-02-26T16:29:58.000Z", "max_issues_repo_path": "Chap-1.ipynb", "max_issues_repo_name": "Spationaute/GCH200", "max_issues_repo_head_hexsha": "55144f5b2a59a7240d36c985997387f5036149f7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Chap-1.ipynb", "max_forks_repo_name": "Spationaute/GCH200", "max_forks_repo_head_hexsha": "55144f5b2a59a7240d36c985997387f5036149f7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2018-02-27T15:04:33.000Z", "max_forks_repo_forks_event_max_datetime": "2021-06-03T16:38:07.000Z", "avg_line_length": 290.62, "max_line_length": 105424, "alphanum_fraction": 0.9203208188, "converted": true, "num_tokens": 3830, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.749087201911703, "lm_q1q2_score": 0.426869328149796}} {"text": "\n\n# \u30e1\u30e2\n\nhttps://github.com/drvinceknight/Python-Mathematics-Handbook\n\n\u3068\u3044\u3046 github \u4e0a\u306e jupyter \u3067 3 \u5e74\u524d\u306b\u66f8\u304b\u308c\u3066\u3044\u308b\u672c\u306e\u7b2c2\u7ae0\u304c\n\nSymbolic mathematics with Sympy (Sympy \u306b\u3088\u308b\u6570\u5f0f\u51e6\u7406)\n\n\u3068\u8a00\u3046\u7ae0\u306b\u306a\u3063\u3066\u3044\u308b\u3002\n\n\u3053\u308c\u3092\u8aad\u3080\u3002\n\n\n# Sympy \u306b\u3088\u308b\u6570\u5f0f\u51e6\u7406 \n\nSymbolic Mathematics with Sympy\n\n\nSympy \u306b\u3088\u308b\u6570\u5f0f\u51e6\u7406\u3067\u3067\u304d\u308b\u3053\u3068\n\n* \u6570\u5f0f\u3092\u6271\u3046\n* \u6570\u5f0f\u3092\u89e3\u304f\n* \u6570\u5f0f\u3092\u5fae\u7a4d\u5206\u3059\u308b\n* \u6570\u5f0f\u3092\u30b0\u30e9\u30d5\u306b\u3059\u308b\n\n\u307b\u304b\u306b\u3082\u3044\u308d\u3044\u308d\u3067\u304d\u308b\u3053\u3068\u304c\u3042\u308b\u3002 sympy \u306e\u30db\u30fc\u30e0\u30da\u30fc\u30b8\u3092\u53c2\u7167\u3002\n\n \n\n\n\n\n```python\n# sympy \u3092\u4f7f\u3046\u305f\u3081\u306e\u6e96\u5099\nfrom sympy import *\nx = symbols ('x')\ndisplay(x)\ndisplay(x - x == 0)\ndisplay(x - x)\n```\n\n\n$\\displaystyle x$\n\n\n\n True\n\n\n\n$\\displaystyle 0$\n\n\n# sympy \u3092\u4f7f\u3046\u305f\u3081\u306e\u6e96\u5099\n\nsympy \u306f import \u304c\u5fc5\u8981\u3002\n\n\u5909\u6570\u3082\u5ba3\u8a00\u304c\u5fc5\u8981\u3002 from sympy.abc import * \u3068\u3044\u3046\u624b\u3082\u3042\u308b\u3002\n\nprint \u306e\u4ee3\u308f\u308a\u306b display \u3092\u4f7f\u3046\u3068 latex \u3067\u8868\u793a\u3057\u3066\u304f\u308c\u308b\u3002 \u6570\u5b66\u306b\u304a\u3044\u3066\u30b7\u30f3\u30dc\u30eb\u306f\u5927\u5207\u3002\n\nx \u306b\u5024\u3092\u5165\u308c\u306a\u304f\u3066\u3082 x - x == 0 \u3068\u3044\u3046\u6f14\u7b97\u304c\u53ef\u80fd\u3002\n\n \n\n\n\u6b21\u306e\u5f0f\u3092\u78ba\u304b\u3081\u3066\u307f\u3088\u3046\u3002\n\n$$(a + b) ^ 2 = a ^ 2 + 2ab + b ^2$$\n\n\n \n\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = (a + b) ** 2 \ndisplay(expr)\n```\n\n\n$\\displaystyle \\left(a + b\\right)^{2}$\n\n\n\u53f3\u8fba\u306e\u3088\u3046\u306a\u5165\u308c\u65b9\u3092\u3059\u308b\u3068\u3069\u3046\u306a\u308b\u304b\u3002\n\n\n```python\nexpr2 = a**2+2*a*b+b**2\ndisplay(expr2)\n```\n\n\n$\\displaystyle a^{2} + 2 a b + b^{2}$\n\n\n\u5c55\u958b expand \u3057\u305f\u308a\u3001\u56e0\u6570\u5206\u89e3 factor \u3057\u305f\u308a\u3067\u304d\u308b\u3002\n\n\n```python\ndisplay(expand(expr))\n```\n\n\n$\\displaystyle a^{2} + 2 a b + b^{2}$\n\n\n\n```python\ndisplay(factor(expr2))\n```\n\n\n$\\displaystyle \\left(a + b\\right)^{2}$\n\n\n\u540c\u7b49\u6027\u3092\u78ba\u304b\u3081\u308b\u3002\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = (a + b) ** 2 \nexpr2 = a**2+2*a*b+b**2\ndisplay(expr)\ndisplay(expr2)\ndisplay(expr == expr)\ndisplay(expr == expr2)\ndisplay(expr == expand(expr))\na=2;b=3\nexpr = (a + b) ** 2 \nexpr2 = a**2+2*a*b+b**2\ndisplay(expr == expr2)\ndisplay(expr == expand(expr))\n```\n\n\n$\\displaystyle \\left(a + b\\right)^{2}$\n\n\n\n$\\displaystyle a^{2} + 2 a b + b^{2}$\n\n\n\n True\n\n\n\n False\n\n\n\n False\n\n\n\n True\n\n\n\n True\n\n\n\u4e0a\u306e\u5b9f\u9a13\u304b\u3089\u308f\u304b\u308b\u3053\u3068\u306f\u6570\u5f0f\u306e\u5f62\u304c\u9055\u3046\u3068 `==` \u306f\u771f\u306b\u306f\u306a\u3089\u306a\u3044\u3002\n\n\u5024\u304c\u5165\u308c\u3070\u3001\u5f53\u7136 `==` \u304c\u771f\u306b\u306a\u308b\u3002\n\nlatex \u3068\u3044\u3046\u95a2\u6570\u3067 latex \u8868\u73fe\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = (a + b) ** 2 \nlatex(expand(expr))\n```\n\n\n\n\n 'a^{2} + 2 a b + b^{2}'\n\n\n\n`a^{2} + 2 a b + b^{2}`\n$$a^{2} + 2 a b + b^{2}$$\n\n \n\n\n\n---\n**\u7df4\u7fd2\u554f\u984c** $\\quad$ sympy \u3092\u4f7f\u3063\u3066\u4e0b\u8a18\u306e\u7b49\u5f0f\u3092\u78ba\u8a8d\u3059\u308b\u3002\n\n* $(a - b) ^ 2 = a ^ 2 - 2 a b + b^2$\n* $a ^ 2 - b ^ 2 = (a - b) (a + b)\\quad$ (`expand` \u3067\u306f\u306a\u304f `factor` \u3092\u4f7f\u3063\u3066\u307f\u308b)\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = (a-b)**2\ndisplay(expr)\ndisplay(expand(expr))\n```\n\n\n$\\displaystyle \\left(a - b\\right)^{2}$\n\n\n\n$\\displaystyle a^{2} - 2 a b + b^{2}$\n\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = a**2-b**2\ndisplay(expr)\ndisplay(factor(expr))\n```\n\n\n$\\displaystyle a^{2} - b^{2}$\n\n\n\n$\\displaystyle \\left(a - b\\right) \\left(a + b\\right)$\n\n\n# \u6570\u5f0f\u3092\u89e3\u304f\n\nsympy \u3092\u4f7f\u3063\u3066 $x$ \u306b\u3064\u3044\u3066\u3001\u6b21\u306e 2\u6b21\u65b9\u7a0b\u5f0f (quadratic equasion) \u3092\u89e3\u304f\u3002\n\n$$a x ^ 2 + b x + c = 0$$\n\n \n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = a * x ** 2 + b * x + c\ndisplay(solveset(expr,x))\n```\n\nsympy \u306e `solveset` \u306f\u5f15\u6570\u3092 2 \u3064\u53d6\u308a\u3001\u7b2c 1 \u5f15\u6570\u306f\u6570\u5f0f\u3067\u3001\u7b2c 2 \u5f15\u6570\u306f\u89e3\u304f\u5bfe\u8c61\u3068\u306a\u308b\u5909\u6570\u3067\u3042\u308b\u3002\n\n \n\n\n---\n**\u7df4\u7fd2\u554f\u984c** $\\quad$ sympy \u3092\u4f7f\u3063\u3066 3 \u6b21\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\u3092\u6c42\u3081\u308b\u3002\n\n$$a x ^ 3 + b x ^ 2 + c x + d = 0$$\n\n \n\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = a * x ** 3 + b * x**2 + c*x + d\ndisplay (expr)\ndisplay(solveset(expr,x))\n```\n\n\n$\\displaystyle a x^{3} + b x^{2} + c x + d$\n\n\n\n$\\displaystyle \\left\\{- \\frac{- \\frac{3 c}{a} + \\frac{b^{2}}{a^{2}}}{3 \\sqrt[3]{\\frac{\\sqrt{- 4 \\left(- \\frac{3 c}{a} + \\frac{b^{2}}{a^{2}}\\right)^{3} + \\left(\\frac{27 d}{a} - \\frac{9 b c}{a^{2}} + \\frac{2 b^{3}}{a^{3}}\\right)^{2}}}{2} + \\frac{27 d}{2 a} - \\frac{9 b c}{2 a^{2}} + \\frac{b^{3}}{a^{3}}}} - \\frac{\\sqrt[3]{\\frac{\\sqrt{- 4 \\left(- \\frac{3 c}{a} + \\frac{b^{2}}{a^{2}}\\right)^{3} + \\left(\\frac{27 d}{a} - \\frac{9 b c}{a^{2}} + \\frac{2 b^{3}}{a^{3}}\\right)^{2}}}{2} + \\frac{27 d}{2 a} - \\frac{9 b c}{2 a^{2}} + \\frac{b^{3}}{a^{3}}}}{3} - \\frac{b}{3 a}, - \\frac{- \\frac{3 c}{a} + \\frac{b^{2}}{a^{2}}}{3 \\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{\\sqrt{- 4 \\left(- \\frac{3 c}{a} + \\frac{b^{2}}{a^{2}}\\right)^{3} + \\left(\\frac{27 d}{a} - \\frac{9 b c}{a^{2}} + \\frac{2 b^{3}}{a^{3}}\\right)^{2}}}{2} + \\frac{27 d}{2 a} - \\frac{9 b c}{2 a^{2}} + \\frac{b^{3}}{a^{3}}}} - \\frac{\\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{\\sqrt{- 4 \\left(- \\frac{3 c}{a} + \\frac{b^{2}}{a^{2}}\\right)^{3} + \\left(\\frac{27 d}{a} - \\frac{9 b c}{a^{2}} + \\frac{2 b^{3}}{a^{3}}\\right)^{2}}}{2} + \\frac{27 d}{2 a} - \\frac{9 b c}{2 a^{2}} + \\frac{b^{3}}{a^{3}}}}{3} - \\frac{b}{3 a}, - \\frac{- \\frac{3 c}{a} + \\frac{b^{2}}{a^{2}}}{3 \\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{\\sqrt{- 4 \\left(- \\frac{3 c}{a} + \\frac{b^{2}}{a^{2}}\\right)^{3} + \\left(\\frac{27 d}{a} - \\frac{9 b c}{a^{2}} + \\frac{2 b^{3}}{a^{3}}\\right)^{2}}}{2} + \\frac{27 d}{2 a} - \\frac{9 b c}{2 a^{2}} + \\frac{b^{3}}{a^{3}}}} - \\frac{\\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{\\sqrt{- 4 \\left(- \\frac{3 c}{a} + \\frac{b^{2}}{a^{2}}\\right)^{3} + \\left(\\frac{27 d}{a} - \\frac{9 b c}{a^{2}} + \\frac{2 b^{3}}{a^{3}}\\right)^{2}}}{2} + \\frac{27 d}{2 a} - \\frac{9 b c}{2 a^{2}} + \\frac{b^{3}}{a^{3}}}}{3} - \\frac{b}{3 a}\\right\\}$\n\n\n `solveset` \u306b\u5f15\u6570\u3092\u8db3\u3057\u3066\u3001\u89e3\u306e\u7bc4\u56f2 domain \u3092\u6307\u5b9a\u3059\u308b\u3053\u3068\u304c\u53ef\u80fd\u3067\u3042\u308b\u3002\n\n \u6b21\u306e\u5f0f\u306e domain \u3092 $\\mathbb{R}$ \u3068\u3057\u3066\u89e3\u3044\u3066\u307f\u308b\u3002\n\n$$x^2=-1$$\n\n\u30e1\u30e2 $\\quad$ domain \u306f S.Reals \u3068\u304b\u3067\u6307\u5b9a\u3059\u308b\u3002 S \u306e\u66f8\u5f0f\u306f dir(S) \u3067\u8abf\u3079\u3089\u308c\u308b\u3002\u9ed2\u677f\u30dc\u30fc\u30eb\u30c9\u306f `\\mathbb` \u3067\u8868\u8a18\u3067\u304d\u308b\u3002\n\n\n```python\nsolveset(x ** 2 + 1, x, domain=S.Reals)\n```\n\n\n\n\n$\\displaystyle \\emptyset$\n\n\n\n$\\emptyset$ \u306f\u7a7a\u96c6\u5408\u306e\u610f\u5473\u3067\u3042\u308b\u3002\n\n\u7a7a\u96c6\u5408\u306f `\\emptyset` `\\varnothing` `\\phi` \u3067\u8868\u3055\u308c\u308b\u3002$\\emptyset, \\varnothing, \\phi$\n\n \n\n\n---\n**\u7df4\u7fd2\u554f\u984c** $\\quad$ sympy \u3092\u4f7f\u3063\u3066\u6b21\u306e\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\n\n* $x ^ 2 = 2$ $\\quad (x \\in \\mathbb{N})$\n* $x ^ 3 + 2 x = 0$ $\\quad (x \\in \\mathbb{R})$\n\n \n\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = x**2 - 2\ndisplay (expr)\ndisplay(solveset(expr,x,domain=S.Naturals))\n```\n\n\n$\\displaystyle x^{2} - 2$\n\n\n\n$\\displaystyle \\emptyset$\n\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = x**3 + 2*x\ndisplay (expr)\ndisplay(solveset(expr,x,domain=S.Reals))\n```\n\n\n$\\displaystyle x^{3} + 2 x$\n\n\n\n$\\displaystyle \\left\\{0\\right\\}$\n\n\n# \u5fae\u5206\u7a4d\u5206\u65b9\u7a0b\u5f0f \nsymbolic calculus\n\nsympy \u3067\u6975\u9650 limit \u3092\u6271\u3046\u3053\u3068\u304c\u3067\u304d\u308b\u3002\n\n\u6b21\u306e\u5f0f\u3092\u8003\u3048\u3088\u3046\u3002\n\n$$\\lim_{x\\to 0^+}\\frac{1}{x}$$\n\n \n\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\ndisplay(limit(1/x, x, 0, dir=\"+\"))\n```\n\n\n$\\displaystyle \\infty$\n\n\n---\n**\u7df4\u7fd2\u554f\u984c** $\\quad$ \u6b21\u306e\u6975\u9650\u3092\u8a08\u7b97\u3059\u308b\u3002\n\n1. $\\displaystyle \\lim_{x\\to 0^-}\\frac{1}{x}$\n2. $\\displaystyle \\lim_{x\\to 0}\\frac{1}{x^2}$\n\n \n\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\ndisplay(limit(1/x, x, 0, dir=\"-\"))\ndisplay(limit(1/x**2, x, 0))\n\n```\n\n\n$\\displaystyle -\\infty$\n\n\n\n$\\displaystyle \\infty$\n\n\nsympy \u3092\u4f7f\u3063\u3066\u5fae\u5206 differentiate/Derivative \u3084\u7a4d\u5206 integrate/Integral \u304c\u3067\u304d\u308b\u3002\n\n\u6b21\u306e\u5f0f\u3092\u5fae\u5206\u3059\u308b\u3002\n\n$$x ^ 2 - \\cos(x)$$\n\n \n\n\n\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f `Derivative` \u3001\u5fae\u5206\u8a08\u7b97\u306f `diff`\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = x**2 - cos(x)\ndisplay(Derivative(x ** 2 - cos(x), x))\ndisplay(Derivative(expr, x))\ndisplay(diff(x ** 2 - cos(x), x))\ndisplay(diff(expr, x))\n```\n\n\n$\\displaystyle \\frac{d}{d x} \\left(x^{2} - \\cos{\\left(x \\right)}\\right)$\n\n\n\n$\\displaystyle \\frac{d}{d x} \\left(x^{2} - \\cos{\\left(x \\right)}\\right)$\n\n\n\n$\\displaystyle 2 x + \\sin{\\left(x \\right)}$\n\n\n\n$\\displaystyle 2 x + \\sin{\\left(x \\right)}$\n\n\n\u7a4d\u5206\u306f Integral \u3068 integrate\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = x**2 - cos(x)\ndisplay(expr)\ndisplay(Integral(expr, x))\ndisplay(integrate(expr, x))\nintegrate(x ** 2 - cos(x), x)\n```\n\n\n$\\displaystyle x^{2} - \\cos{\\left(x \\right)}$\n\n\n\n$\\displaystyle \\int \\left(x^{2} - \\cos{\\left(x \\right)}\\right)\\, dx$\n\n\n\n$\\displaystyle \\frac{x^{3}}{3} - \\sin{\\left(x \\right)}$\n\n\n\n\n\n$\\displaystyle \\frac{x^{3}}{3} - \\sin{\\left(x \\right)}$\n\n\n\n\u5b9a\u7a4d\u5206\u306f\u6b21\u306e\u3088\u3046\u306b\u884c\u3046\u3002\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = x**2 - cos(x)\ndisplay(Integral(expr, (x, 0, 5)))\ndisplay(integrate(expr, (x, 0, 5)))\nintegrate(x ** 2 - cos(x), (x, 0, 5))\n```\n\n\n$\\displaystyle \\int\\limits_{0}^{5} \\left(x^{2} - \\cos{\\left(x \\right)}\\right)\\, dx$\n\n\n\n$\\displaystyle \\frac{125}{3} - \\sin{\\left(5 \\right)}$\n\n\n\n\n\n$\\displaystyle \\frac{125}{3} - \\sin{\\left(5 \\right)}$\n\n\n\n---\n\n**\u7df4\u7fd2\u554f\u984c** $\\quad$ sympy \u3092\u4f7f\u3063\u3066\u4e0b\u8a18\u306e\u5f0f\u3092\u8a08\u7b97\u3059\u308b\u3002\n\n1. $\\displaystyle \\frac{d\\sin(x ^2)}{dx}$\n2. $\\displaystyle \\frac{d(x ^2 + xy - \\ln(y))}{dy}$\n3. $\\displaystyle \\int e^x \\cos(x)\\;dx$\n4. $\\displaystyle \\int_0^5 e^{2x}\\;dx$\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = sin (x**2)\ndisplay(Derivative(expr, x))\ndisplay(diff(expr, x))\n\n```\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = x**2+x*y-log(y)\ndisplay(Derivative(expr, y))\ndisplay(diff(expr, y))\n```\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = exp(x)*cos(x)\ndisplay(expr)\ndisplay(Integral(expr, x))\ndisplay(integrate(expr, x))\n```\n\n\n$\\displaystyle e^{x} \\cos{\\left(x \\right)}$\n\n\n\n$\\displaystyle \\int e^{x} \\cos{\\left(x \\right)}\\, dx$\n\n\n\n$\\displaystyle \\frac{e^{x} \\sin{\\left(x \\right)}}{2} + \\frac{e^{x} \\cos{\\left(x \\right)}}{2}$\n\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = exp(2*x)\ndisplay(expr)\ndisplay(Integral(expr, (x,0,5)))\ndisplay(integrate(expr, (x,0,5)))\n```\n\n\n$\\displaystyle e^{2 x}$\n\n\n\n$\\displaystyle \\int\\limits_{0}^{5} e^{2 x}\\, dx$\n\n\n\n$\\displaystyle - \\frac{1}{2} + \\frac{e^{10}}{2}$\n\n\n# \u30b0\u30e9\u30d5\u63cf\u753b \n\nplotting\n\n\u30e1\u30e2 $\\quad$ \u3053\u3053\u3067\u6ce8\u610f\u3002 sympy \u3067\u7c21\u6613\u306b\u4f8b\u3048\u3070 $y=x^2$ \u306e\u30b0\u30e9\u30d5\u304c\u63cf\u3051\u308b\u304c\u3001\u30b0\u30e9\u30d5\u306f\u672c\u6765\u5ea7\u6a19\u306e\u7bc4\u56f2\u3092\u6c7a\u3081\u305f\u308a\u3001\u30bc\u30ed\u3092\u3069\u3053\u306b\u7f6e\u304f\u304b\u3001\u76ee\u76db\u308a\u3092\u3069\u3046\u3059\u308b\u304b\u306a\u3069\u9762\u5012\u306a\u8a2d\u5b9a\u304c\u5fc5\u8981\u306a\u3068\u3053\u308d\u3092\u88cf\u3067\u51e6\u7406\u3057\u3066\u3044\u308b\u3002 \n\n\u88cf\u3067\u52d5\u3044\u3066\u3044\u308b\u306e\u304c matplotlib (http://matplotlib.org/)\u3068\u3044\u3046 python \u306e\u30e9\u30a4\u30d6\u30e9\u30ea\u30fc\u3060\u304c\u3001\u3053\u308c\u3082\u7c21\u6613\u306a\u4f7f\u3044\u65b9\u3068\u7d30\u304b\u304f\u8a2d\u5b9a\u3057\u3066\u4f7f\u3046\u65b9\u6cd5\u304c\u3042\u308b\u3002\n\nsympy \u3067\u30b0\u30e9\u30d5\u63cf\u753b\u3059\u308b\u969b\u306b\u306f matplotlib \u306e\u3069\u306e\u6a5f\u80fd\u3092\u4f7f\u3063\u3066\u3044\u308b\u3068\u304b\u3001\u610f\u8b58\u3057\u306a\u304c\u3089\u4f7f\u3046\u306e\u304c\u3088\u3044\u3068\u601d\u3046\u3002\n\nnumpy \u3068 matplotlib \u3092\u660e\u793a\u7684\u306b\u4f7f\u3046\n\n \n\n\n$x^2$ \u306e\u30b0\u30e9\u30d5\u3092\u63cf\u304f\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = x ** 2\np = plot(expr);\n```\n\nmatplotlib \u3068 numpy \u3067\u306f\u6b21\u306e\u3088\u3046\u306b\u63cf\u304f\u3002\n\n\n```python\nimport matplotlib.pyplot as plt\nimport numpy as np\nx = np.linspace(-10, 10, 100) # 100\u306b\u523b\u3080\nfig, ax = plt.subplots() # figure \u3068 ax \u3092\u4f5c\u308b\nax.spines['left'].set_position(('data', 0))\nax.spines['right'].set_visible(False)\nax.spines['top'].set_visible(False)\nax.spines['bottom'].set_position(('data', 0))\nax.set_xlabel('x') # x\u8ef8\u306e\u30e9\u30d9\u30eb\nax.set_ylabel('f(x)') # y\u8ef8\u306e\u30e9\u30d9\u30eb\nax.plot(x, x**2) # \u30d7\u30ed\u30c3\u30c8\u3059\u308b\nplt.show()\n```\n\n\u30e1\u30e2 $\\quad$ xlabel \u3084 ylabel \u306e\u4f4d\u7f6e\u306f\u3069\u3046\u3084\u3063\u3066\u6307\u793a\u3059\u308b\u306e\u304b\u3002 \n\n\u5ea7\u6a19\u306e\u4f4d\u7f6e\u3092\u5909\u3048\u308b\u305f\u3081\u306b spines \u3092\u8a2d\u5b9a\u3059\u308b\u306e\u306f\u624b\u9593\u304c\u591a\u3044\u3002 \n\n\u30c6\u30ad\u30b9\u30c8\u306b\u306f pdf \u3067\u4fdd\u5b58\u3059\u308b\u65b9\u6cd5\u304c\u66f8\u3044\u3066\u3042\u308b\u3002\n\n\n```python\np.save(\"x_squared.pdf\")\n```\n\n\u30e1\u30e2 $\\quad$ `p.save(\"x_squared.pdf\")` \u3067 pdf \u3092\u4f5c\u308b\u3068\u540c\u6642\u306b\u753b\u9762\u4e0a\u306b\u63cf\u753b\u3092\u3057\u3066\u3044\u308b\u3002 \n\nnumpy+matplotlib \u306e `plt.savefig(\"temp.svg\", format=\"svg\")` \u3082\u540c\u3058\u3060\u3063\u305f!!!!\n\n\n\n\n```python\nprint(type(p) )\nprint(type(ax))\nprint(len(dir(p)))\nprint(len(dir(ax)))\n```\n\n \n \n 54\n 449\n\n\n\u4e0a\u306e p \u3068 ax \u306e dir \u3092\u53d6\u3063\u3066\u307f\u308b\u3068\u5727\u5012\u7684\u306b ax \u306e\u65b9\u304c\u6a5f\u80fd\u304c\u591a\u3044\u3002\n\n\u305d\u3046\u3044\u3046\u3053\u3068\u306a\u306e\u3060\u308d\u3046\u3002\n\n \n\n\n---\n**\u7df4\u7fd2\u554f\u984c** $\\quad$ \u6b21\u306e\u95a2\u6570\u3092\u30b0\u30e9\u30d5\u63cf\u753b\u3059\u308b\u3002\n\n* $y=x + cos(x)$\n* $y=x ^ 2 - e^x$ (you might find `ylim` helpful as an argument)\n\n\n\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = x + cos(x)\np = plot(expr)\n```\n\n\u540c\u3058\u30b0\u30e9\u30d5\u3092 numpy+matplotlib \u3067\u63cf\u304f\u3002\n\n\n```python\nimport matplotlib.pyplot as plt\nimport numpy as np\nx = np.linspace(-10, 10, 100) # 100\u306b\u523b\u3080\nfig, ax = plt.subplots() # figure \u3068 ax \u3092\u4f5c\u308b\nax.spines['left'].set_position(('data', 0))\nax.spines['right'].set_visible(False)\nax.spines['top'].set_visible(False)\nax.spines['bottom'].set_position(('data', 0))\nax.set_xlabel('x') # x\u8ef8\u306e\u30e9\u30d9\u30eb\nax.set_ylabel('f(x)') # y\u8ef8\u306e\u30e9\u30d9\u30eb\nax.plot(x, x + np.cos(x)) # cos \u306f numpy \u306e cos \u3092\u4f7f\u3046\u306e\u3067\u6ce8\u610f\nplt.show()\n```\n\n\n```python\nfrom sympy.abc import *\nfrom sympy import *\nexpr = x**2 - exp(x)\np = plot(expr, xlim=(-1,10))\n```\n\n\u540c\u3058\u30b0\u30e9\u30d5\u3092 numpy+matplotlib \u3067\u63cf\u304f\u3002\n\n\n```python\nimport matplotlib.pyplot as plt\nimport numpy as np\nx = np.linspace(-1, 10, 100) # 100\u306b\u523b\u3080\nfig, ax = plt.subplots() # figure \u3068 ax \u3092\u4f5c\u308b\nax.spines['left'].set_position(('data', 0))\nax.spines['right'].set_visible(False)\nax.spines['top'].set_visible(False)\nax.spines['bottom'].set_position(('data', 0))\nax.set_xlabel('x') # x\u8ef8\u306e\u30e9\u30d9\u30eb\nax.set_ylabel('f(x)') # y\u8ef8\u306e\u30e9\u30d9\u30eb\nax.plot(x, x**2 - np.exp(x)) # exp \u306f numpy \u306e exp \u3092\u4f7f\u3046\u306e\u3067\u6ce8\u610f\nplt.show()\n```\n", "meta": {"hexsha": "b36623ef4f8255f9a3a252bf098c54006eac8e6c", "size": 162482, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "old_math_sympy.ipynb", "max_stars_repo_name": "kalz2q/-yjupyternotebooks", "max_stars_repo_head_hexsha": "ba37ac7822543b830fe8602b3f611bb617943463", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-09-16T03:45:19.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-16T03:45:19.000Z", "max_issues_repo_path": "old_math_sympy.ipynb", "max_issues_repo_name": "kalz2q/-yjupyternotebooks", "max_issues_repo_head_hexsha": "ba37ac7822543b830fe8602b3f611bb617943463", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "old_math_sympy.ipynb", "max_forks_repo_name": "kalz2q/-yjupyternotebooks", "max_forks_repo_head_hexsha": "ba37ac7822543b830fe8602b3f611bb617943463", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 89.3249037933, "max_line_length": 18766, "alphanum_fraction": 0.7959220098, "converted": true, "num_tokens": 5308, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702761768249, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4268682268572087}} {"text": "

                                        Table of Contents

                                        \n
                                        \n\n\n```python\n# code for loading the format for the notebook\nimport os\n\n# path : store the current path to convert back to it later\npath = os.getcwd()\nos.chdir(os.path.join('..', 'notebook_format'))\nfrom formats import load_style\nload_style(css_style = 'custom2.css', plot_style = False)\n```\n\n\n\n\n\n\n\n\n\n\n\n\n```python\nos.chdir(path)\n\n# 1. magic to print version\n# 2. magic so that the notebook will reload external python modules\n%load_ext watermark\n%load_ext autoreload\n%autoreload 2\n\nimport numpy as np\nimport pandas as pd\nfrom math import ceil\nfrom tqdm import trange\nfrom subprocess import call\nfrom scipy.sparse import csr_matrix, dok_matrix\n\n%watermark -a 'Ethen' -d -t -v -p numpy,pandas,sklearn,tqdm,scipy\n```\n\n Ethen 2018-04-07 14:28:04 \n \n CPython 3.6.4\n IPython 6.2.1\n \n numpy 1.14.2\n pandas 0.22.0\n sklearn 0.19.1\n tqdm 4.19.8\n scipy 1.0.0\n\n\n# Collaborative Filtering for Implicit Feedback Datasets\n\nOne common scenario in real-world recommendation system is we only have **implicit** instead of **explicit** user-item interaction data. To elaborate on this a little bit more, a user may be searching for an item on the web, or listening to songs. Unlike a rating data, where we have direct access to the user's preference towards an item, these type of actions do not **explicitly** state or quantify any preference of the user for the item, but instead gives us **implicit confidence** about the user's opinion.\n\nEven when we do have explicit data, it might still be a good idea to incorporate implicit data into the model. Consider, for example, listening to songs. When users listen to music on a streaming service, they might rarely ever rate a song that he/she like or dislike. But more often they skip a song, or listen only halfway through it if they dislike it. If the user really liked a song, they will often come back and listen to it. So, to infer a user's musical taste profile, their listens, repeat listens, skips and fraction of tracks listened to, etc. might be far more valuable signals than explicit ratings.\n\n## Formulation\n\nRecall from the previous notebook that the loss function for training the recommendation model on explicit feedback data was:\n\n$$\n\\begin{align}\nL_{explicit} &= \\sum\\limits_{u,i \\in S}( r_{ui} - x_{u} y_{i}^{T} )^{2} + \\lambda \\big( \\sum\\limits_{u} \\left\\Vert x_{u} \\right\\Vert^{2} + \\sum\\limits_{i} \\left\\Vert y_{i} \\right\\Vert^{2} \\big)\n\\end{align}\n$$\n\nWhere:\n\n- $r_{ui}$ is the true rating given by user $u$ to the item $i$\n- $x_u$ and $y_i$ are user u's and item i's latent factors, both are $1\u00d7d$ dimensional, where $d$ the number of latent factors that the user can specify\n- $S$ was the set of all user-item ratings\n- $\\lambda$ controls the regularization strength that prevents overfitting the user and item vectors\n\nTo keep it concrete, let's assume we're working music data and the value of our $r_{ui}$ will consists of implicit ratings that counts the number of times a user has listened to a song (song listen count). Then new formulation becomes:\n\n$$\n\\begin{align}\nL_{implicit} &= \\sum\\limits_{u,i} c_{ui}( p_{ui} - x_{u} y_{i}^{T} )^2 + \\lambda \\big( \\sum\\limits_{u} \\left\\Vert x_{u} \\right\\Vert^{2} + \\sum\\limits_{i} \\left\\Vert y_{i} \\right\\Vert^{2} \\big)\n\\end{align}\n$$\n\nRecall that with implicit feedback, we do not have ratings anymore; rather, we have users' preferences for items. Therefore, in the new loss function, the ratings $r_{ui}$ has been replaced with a preference $p_{ui}$ indicating the preference of user $u$ to item $i$. $p_{ui}$ is a set of binary variables and is computed by binarizing $r_{ui}$.\n\n$$\n\\begin{align}\np_{ui} &= \\begin{cases} 1 &\\mbox{if } r_{ui} > 0 \\\\ 0 & \\mbox{otherwise} \\end{cases}\n\\end{align}\n$$\n\nWe make the assumption that if a user has interacted at all with an item ($r_{ui} > 0$), then we set $p_{ui} = 1$ to indicate that user $u$ has a liking/preference for item $i$. Otherwise, we set $p_{ui} = 0$. However, these assumptions comes with varying degrees of confidence. First of all, when $p_{ui} = 0$, we assume that it should be associated with a lower confidence, as there are many reasons beyond disliking the item as to why the user has not interacted with it. e.g. Unaware of it's existence. On the other hand, as the number of implicit feedback, $r_{ui}$, grows, we have a stronger indication that the user does indeed like the item (regardless of whether he/she if buying a gift for someone else). So to measure the level of confidence mentioned above, we introduce another set of variables $c_{ui}$ that measures our confidence in observing $p_{ui}$:\n\n$$\n\\begin{align}\nc_{ui} = 1 + \\alpha r_{ui}\n\\end{align}\n$$\n\nWhere the 1 ensures we have some minimal confidence for every user-item pair, and as we observe more and more implicit feedback (as $r_{ui}$ gets larger and larger), our confidence in $p_{ui} = 1$ increases accordingly. The term $\\alpha$ is a parameter that we have to specify to control the rate of the increase. This formulation makes intuitive sense when we look back at the $c_{ui}( p_{ui} - x_{u} y_{i}^{T} )^2$ term in the loss function. A larger $c_{ui}$ means that the prediction $x_{u} y_{i}^{T}$ has to be that much closer to $p_{ui}$ so that term will not contribute too much to the total loss.\n\nThe implementation in the later section will be based on the formula above, but note that there are many ways in which we can tune the formulation above. For example, we can derive $p_{ui}$ from $r_{ui}$ differently. So instead of setting the binary cutoff at 0, we can set it at another threshold that we feel is appropriate for the domain. Similarly, there are many ways to transform $r_{ui}$ into the confidence level $c_{ui}$. e.g. we can use:\n\n$$\n\\begin{align}\nc_{ui} = 1 + \\alpha log \\left( 1 + r_{ui} / \\epsilon \\right)\n\\end{align}\n$$\n\nRegardless of the scheme, it's important to realize that we are transforming the raw observation $r_{ui}$ into two distinct representation, the preference $p_{ui}$ and the confidence levels of the preference $c_{ui}$.\n\n## Alternating Least Squares\n\nLet's assume we have $m$ users and $n$ items. Now, to solve for the loss function above, we start by treating $y_i$ as constant and solve the loss function with respect to $x_u$. To do this, we rewrite and expand the first term in the loss function (excluding the regularization terms), $\\sum\\limits_{u,i} c_{ui}( p_{ui} - x_{u} y_{i}^{T} )^2$ part as:\n\n$$\n\\begin{align}\n\\sum\\limits_{u,i} c_{ui}( p_{ui} - x_{u} y_{i}^{T} )^2\n&= \\sum_u c_u( p_u^T - x_u Y^T )^2 \\\\\n&= \\sum_u p_u^T C^u p_u - 2 x_u Y^T C^u p_u + x_u Y^T C^u Y x_u^T\n\\end{align}\n$$\n\nWhere: \n\n- $Y \\in \\mathbb{R}^{n \\times d}$ represents all item row vectors vertically stacked on each other\n- $p_u \\in \\mathbb{R^{n \\times 1}}$ contains element all of the preferences of the user\n- The diagonal matrix $C^u \\in \\mathbb{R^{n \\times n}}$ consists of $c_{ui}$ in row/column $i$, which is the user's confidence across all the items. e.g. if $u = 0$ then the matrix for user $u_0$ will look like:\n\n$$\n\\begin{align}\n{C}^{u_0} = \\begin{bmatrix} c_{u_{01}} & 0 & 0 & 0 & ... & 0 \\\\ 0 & c_{u_{02}} & 0 & 0 & ... &0\\\\ ... \\\\ ... \\\\ 0 & 0 & 0 & 0 & ... & c_{u_{0n}}\\end{bmatrix}\n\\end{align}\n$$\n\nThe formula above can also be used to monitor the loss function at each iteration. If we set $A = Y^T C^u Y$ and $b = Y^T C^u$, the last two terms can be rewritten as $(A x_u^T - 2b p_u) x_u$. As for the first term $p_u^T C^u p_u$ we can leverage the fact that $p_u$ is 1 for all positive items, and just sum the confidence term $C^u$.\n\nNow for the derivation of the partial derivative.\n\n$$\n\\begin{align}\n\\frac{\\partial L_{implicit}}{\\partial x_u} \n&= -2 Y^T C^u p_u + 2 Y^T C^u Y x_u + 2 \\lambda x_u = 0 \\\\\n&= (Y^T C^u Y + \\lambda I)x_u = Y^T C^u p_{u} \\\\\n&= x_u = (Y^T C^u Y + \\lambda I)^{-1} Y^T C^u p_u\n\\end{align}\n$$\n\nThe main computational bottleneck in the expression above is the need to compute $Y^T C^u Y$ for every user. Speedup can be obtained by re-writing the expression as:\n\n$$\n\\begin{align}\n{Y}^T {C}^{u} {Y} &= Y^T Y + {Y}^T \\left( C^u - I \\right) Y\n\\end{align}\n$$\n\nNow the term $Y^T Y$ becomes independent of each user and can be computed independently, next notice $\\left(C^u - I \\right)$ has only $n_u$ non-zero elements, where $n_u$ is the number of items for which $r_{ui} > 0$. Similarly, $C^u p_u$ contains only $n_u$ non-zero elements since $p_u$ is a binary transformation of $r_{ui}$. Thus the final formulation becomes:\n\n$$\n\\begin{align}\n\\frac{\\partial L_{implicit}}{\\partial x_u} \n&= x_u = (Y^T Y + Y^T \\left( C^u - I \\right) Y + \\lambda I)^{-1} Y^T C^u p_u\n\\end{align}\n$$\n\nAfter solving for $x_u$ the same procedure can be carried out to solve for $y_i$ giving a similar expression:\n\n$$\n\\begin{align}\n\\frac{\\partial L_{implicit}}{\\partial y_i} \n&= y_i = (X^T X + X^T \\left( C^i - I \\right) X + \\lambda I)^{-1} X^T C^i p_i\n\\end{align}\n$$\n\n## Implementation\n\nWe'll use the same movielens dataset like the previous notebook. The movielens data is not an implicit feedback dataset as the user did provide explicit ratings, but we will use it for now to test out our implementation. The overall preprocessing procedure of loading the data and doing the train/test split is the same as the previous notebook. But here we'll do it in a sparse matrix fashion.\n\n\n```python\nfile_dir = 'ml-100k'\nfile_path = os.path.join(file_dir, 'u.data')\nif not os.path.isdir(file_dir):\n call(['curl', '-O', 'http://files.grouplens.org/datasets/movielens/' + file_dir + '.zip'])\n call(['unzip', file_dir + '.zip'])\n\nnames = ['user_id', 'item_id', 'rating', 'timestamp']\ndf = pd.read_csv(file_path, sep = '\\t', names = names)\nprint('data dimension: \\n', df.shape)\ndf.head()\n```\n\n data dimension: \n (100000, 4)\n\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        user_iditem_idratingtimestamp
                                        01962423881250949
                                        11863023891717742
                                        2223771878887116
                                        3244512880606923
                                        41663461886397596
                                        \n
                                        \n\n\n\n\n```python\ndef create_matrix(data, user_col, item_col, rating_col):\n \"\"\"\n creates the sparse user-item interaction matrix\n \n Parameters\n ----------\n data : DataFrame\n implicit rating data\n\n user_col : str\n user column name\n\n item_col : str\n item column name\n \n ratings_col : str\n implicit rating column name\n\n Returns\n -------\n ratings : scipy sparse csr_matrix [n_users, n_items]\n user/item ratings matrix\n\n data : DataFrame\n the implict rating data that retains only the positive feedback\n (if specified to do so)\n \"\"\"\n # map each item and user to a unique numeric value\n for col in (item_col, user_col):\n data[col] = data[col].astype('category')\n \n # create a sparse matrix of using the (rating, (rows, cols)) format\n rows = data[user_col].cat.codes\n cols = data[item_col].cat.codes\n rating = data[rating_col]\n ratings = csr_matrix((rating, (rows, cols)))\n ratings.eliminate_zeros()\n return ratings, data\n```\n\n\n```python\nuser_col = 'user_id'\nitem_col = 'item_id'\nrating_col = 'rating'\nX, df = create_matrix(df, user_col, item_col, rating_col)\nX\n```\n\n\n\n\n <943x1682 sparse matrix of type ''\n \twith 100000 stored elements in Compressed Sparse Row format>\n\n\n\nThe following train and test set split function is assuming that you're doing a train/test split using the current dataset. Though it's probably better to use time to perform the train/test split. For example, using the year 2016's data as training and the 1 first month of 2017's data as testing.\n\n\n```python\ndef create_train_test(ratings, test_size = 0.2, seed = 1234):\n \"\"\"\n split the user-item interactions matrix into train and test set\n by removing some of the interactions from every user and pretend\n that we never seen them\n \n Parameters\n ----------\n ratings : scipy sparse csr_matrix\n The user-item interactions matrix\n \n test_size : float between 0.0 and 1.0, default 0.2\n Proportion of the user-item interactions for each user\n in the dataset to move to the test set; e.g. if set to 0.2\n and a user has 10 interactions, then 2 will be moved to the\n test set\n \n seed : int, default 1234\n Seed for reproducible random splitting the \n data into train/test set\n \n Returns\n -------\n train : scipy sparse csr_matrix\n Training set\n \n test : scipy sparse csr_matrix\n Test set\n \"\"\"\n assert test_size < 1.0 and test_size > 0.0\n\n # Dictionary Of Keys based sparse matrix is more efficient\n # for constructing sparse matrices incrementally compared with csr_matrix\n train = ratings.copy().todok()\n test = dok_matrix(train.shape)\n \n # 1. for all the users assign randomly chosen interactions\n # to the test and assign those interactions to zero in the training;\n # when computing the interactions to go into the test set, \n # remember to round up the numbers (e.g. a user has 4 ratings, if the\n # test_size is 0.2, then 0.8 ratings will go to test, thus we need to\n # round up to ensure the test set gets at least 1 rating);\n # 2. note that we can easily the parallelize the for loop if we were to\n # aim for a more efficient implementation\n rstate = np.random.RandomState(seed)\n for u in range(ratings.shape[0]):\n split_index = ratings[u].indices\n n_splits = ceil(test_size * split_index.shape[0])\n test_index = rstate.choice(split_index, size = n_splits, replace = False)\n test[u, test_index] = ratings[u, test_index]\n train[u, test_index] = 0\n \n train, test = train.tocsr(), test.tocsr()\n return train, test\n```\n\n\n```python\nseed = 1234\ntest_size = 0.2\nX_train, X_test = create_train_test(X, test_size, seed)\nX_train\n```\n\n\n\n\n <943x1682 sparse matrix of type ''\n \twith 79619 stored elements in Compressed Sparse Row format>\n\n\n\nThe following implementation uses some tricks to speed up the procedure. First of all, when we need to solve $Ax = b$ where $A$ is an $n \\times n$ matrix, a lot of books might write the solution as $x = A^{-1} b$, however, in practice there is hardly ever a good reason to calculate that it that way as solving the equation $Ax = b$ is faster than finding $A^{-1}$.\n\nThe next one is the idea of computing matrix product $X^T X$ using a [outer product](https://docs.scipy.org/doc/numpy/reference/generated/numpy.outer.html) of each row.\n\n\n```python\n# example matrix\nX = np.array([[9, 3, 5], [4, 1, 2]]).T\nX\n```\n\n\n\n\n array([[9, 4],\n [3, 1],\n [5, 2]])\n\n\n\n\n```python\n# normal matrix product\nX.T.dot(X)\n```\n\n\n\n\n array([[115, 49],\n [ 49, 21]])\n\n\n\n\n```python\n# intialize an empty array\nend_result = np.zeros((2, 2))\n\n# loop through each row add up the outer product\nfor i in range(X.shape[0]):\n out = np.outer(X[i], X[i])\n end_result += out\n print('row:\\n', X[i])\n print('outer product of row:\\n', out)\n\nend_result\n```\n\n row:\n [9 4]\n outer product of row:\n [[81 36]\n [36 16]]\n row:\n [3 1]\n outer product of row:\n [[9 3]\n [3 1]]\n row:\n [5 2]\n outer product of row:\n [[25 10]\n [10 4]]\n\n\n\n\n\n array([[115., 49.],\n [ 49., 21.]])\n\n\n\nThe reason why this can speed things up is that the matrix product is now the sum of the outer products of the rows, where each row's computation is independent of another can be computed in the parallelized fashion then added back together!\n\nLast but not least, is exploiting the property of scipy's [Compressed Sparse Row Matrix](https://docs.scipy.org/doc/scipy-0.18.1/reference/generated/scipy.sparse.csr_matrix.html) to access the non-zero elements. For those that are unfamiliar with it, the following link has a pretty decent quick introduction. [Blog: Empty rows in sparse arrays](http://mike.place/2015/sparse/).\n\n\n```python\nclass ALSWR:\n \"\"\"\n Alternating Least Squares with Weighted Regularization\n for implicit feedback\n\n Parameters\n ----------\n n_iters : int\n number of iterations to train the algorithm\n\n n_factors : int\n number/dimension of user and item latent factors\n\n alpha : int\n scaling factor that indicates the level of confidence in preference\n\n reg : int\n regularization term for the user and item latent factors\n\n seed : int\n seed for the randomly initialized user, item latent factors\n\n Reference\n ---------\n Y. Hu, Y. Koren, C. Volinsky Collaborative Filtering for Implicit Feedback Datasets\n http://yifanhu.net/PUB/cf.pdf\n \"\"\"\n def __init__(self, n_iters, n_factors, alpha, reg, seed):\n self.reg = reg\n self.seed = seed\n self.alpha = alpha\n self.n_iters = n_iters\n self.n_factors = n_factors\n \n def fit(self, ratings):\n \"\"\"\n ratings : scipy sparse csr_matrix [n_users, n_items]\n sparse matrix of user-item interactions\n \"\"\" \n # the original confidence vectors should include a + 1,\n # but this direct addition is not allowed when using sparse matrix,\n # thus we'll have to deal with this later in the computation\n Cui = ratings.copy().tocsr()\n Cui.data *= self.alpha\n Ciu = Cui.T.tocsr()\n self.n_users, self.n_items = Cui.shape\n \n # initialize user latent factors and item latent factors\n # randomly with a specified set seed\n rstate = np.random.RandomState(self.seed)\n self.user_factors = rstate.normal(size = (self.n_users, self.n_factors))\n self.item_factors = rstate.normal(size = (self.n_items, self.n_factors))\n \n for _ in trange(self.n_iters, desc = 'training progress'):\n self._als_step(Cui, self.user_factors, self.item_factors)\n self._als_step(Ciu, self.item_factors, self.user_factors) \n \n return self\n \n def _als_step(self, Cui, X, Y):\n \"\"\"\n when solving the user latent vectors,\n the item vectors will be fixed and vice versa\n \"\"\"\n # the variable name follows the notation when holding\n # the item vector Y constant and solving for user vector X\n \n # YtY is a d * d matrix that is computed\n # independently of each user\n YtY = Y.T.dot(Y)\n data = Cui.data\n indptr, indices = Cui.indptr, Cui.indices\n\n # for every user build up A and b then solve for Ax = b,\n # this for loop is the bottleneck and can be easily parallized\n # as each users' computation is independent of one another\n for u in range(self.n_users):\n # initialize a new A and b for every user\n b = np.zeros(self.n_factors)\n A = YtY + self.reg * np.eye(self.n_factors)\n \n for index in range(indptr[u], indptr[u + 1]):\n # indices[index] stores non-zero positions for a given row\n # data[index] stores corresponding confidence,\n # we also add 1 to the confidence, since we did not \n # do it in the beginning, when we were to give every \n # user-item pair and minimal confidence\n i = indices[index]\n confidence = data[index] + 1\n factor = Y[i]\n\n # for b, Y^T C^u p_u\n # there should be a times 1 for the preference \n # Pui = 1\n # b += confidence * Y[i] * Pui\n # but the times 1 can be dropped\n b += confidence * factor\n \n # for A, Y^T (C^u - I) Y\n A += (confidence - 1) * np.outer(factor, factor)\n\n X[u] = np.linalg.solve(A, b)\n \n return self\n\n def predict(self):\n \"\"\"predict ratings for every user and item\"\"\"\n prediction = self.user_factors.dot(self.item_factors.T)\n return prediction\n \n def _predict_user(self, user):\n \"\"\"\n returns the predicted ratings for the specified user,\n this is mainly used in computing evaluation metric\n \"\"\"\n user_pred = self.user_factors[user].dot(self.item_factors.T)\n return user_pred\n```\n\n\n```python\nals = ALSWR(n_iters = 15, n_factors = 20, alpha = 15, reg = 0.01, seed = 1234)\nals.fit(X_train)\n```\n\n training progress: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 15/15 [00:28<00:00, 1.93s/it]\n\n\n\n\n\n <__main__.ALSWR at 0x1079186d8>\n\n\n\n# Ranking Metrics\n\nNow that we've built our recommendation engine, the next important thing to do is to evaluate our model's offline performance.\n\nLet's say that there are some users and some items, like movies, songs or jobs. Each user might be interested in some items. The client asks us to recommend a few items (the number is x) for each user. In other words, what we're after is the top-N recommendation for each user and after recommending these items to the user, we need a way to measure whether the recommendation is useful or not. One key thing to note is that metrics such as RMSE might not be the best at assessing the quality of recommendations because the training focused on items with the most ratings, achieving a good fit for those. The items with few ratings don't mean much in terms of their impact on the loss. As a result, predictions for them will be off. \n\nLong story short, we need metrics specifically crafted for ranking evaluation and the two most popular ranking metrics are **MAP (Mean Average Precision)** and **NDCG (Normalized Discounted Cumulative Gain)**. The main difference between the two is that MAP assumes binary relevance (an item is either of interest or not, e.g. a user clicked a link, watched a video, purchased a product), while NDCG can be used in any case where we can assign relevance score to a recommended item (binary, integer or real). The relationship is just like classification and regression.\n\n## Mean Average Precision\n\nFor this section of the content, a large portion is based on the excellent blog post at the following link. [Blog: What you wanted to know about Mean Average Precision](http://fastml.com/what-you-wanted-to-know-about-mean-average-precision/). This documentation builds on top of it by carrying out the educational implementation.\n\nLet's say that there are some users and some items, like movies, songs or jobs. Each user might be interested in some items. The client asks us to recommend a few items (the number is x) for each user. After recommending the items to the user, we need a way to measure whether the recommendation is useful or not. One way to do this is using **MAP@k (Mean Average Precision at k)** .\n\nThe intuition behind this evaluation metric is that:\n\n- We can recommend at most k items for each user (this is essentially the `@k` part), but we will be penalized for bad recommendations\n- Order matters, so it's better to submit more certain recommendations first, followed by recommendations we are less sure about\n\nDiving a bit deeper, we will first ignore `@k` and get M out of the way. MAP is just the mean of APs, or average precision, for all users. If we have 1000 users, we sum APs for each user and divide the sum by 1000. This is MAP. So now, what is AP, or average precision?\n\nOne way to understand average precision is this way: we type something in Google and it shows us 10 results. It's probably best if all of them were relevant. If only some are relevant, say five of them, then it's much better if the relevant ones are shown first. It would be bad if first five were irrelevant and good ones only started from sixth, wouldn't it? The formula for computing AP is: \n\nsum i=1:k of (precision at i * change in recall at i)\n\nWhere precision at i is a percentage of correct items among first i recommendations. Change in recall is 1/k if item at i is correct (for every correct item), otherwise zero. Note that this is base on the assumption that the number of relevant items is bigger or equal to k: r >= k. If not, change in recall is 1/r for each correct i instead of 1/k.\n\nFor example, If the actual items were [1 2 3 4 5] and we recommend [6 4 7 1 2]. In this case we get 4, 1 and 2 right, but with some incorrect guesses in between. Now let's say we were to compute AP@2, so only two first predictions matter: 6 and 4. First is wrong, so precision@1 is 0. Second is right, so precision@2 is 0.5. Change in recall is 0 and 0.5 (that's 1/k) respectively, so AP@2 = 0 * 0 + 0.5 * 0.5 = 0.25\n\n\n```python\ndef compute_apk(y_true, y_pred, k):\n \"\"\"\n average precision at k, y_pred is assumed \n to be truncated to length k prior to feeding\n it to the function\n \"\"\"\n # convert to set since membership \n # testing in a set is vastly faster\n actual = set(y_true)\n \n # precision at i is a percentage of correct \n # items among first i recommendations; the\n # correct count will be summed up by n_hit\n n_hit = 0\n precision = 0\n for i, p in enumerate(y_pred, 1):\n if p in actual:\n n_hit += 1\n precision += n_hit / i\n\n # divide by recall at the very end\n avg_precision = precision / min(len(actual), k)\n return avg_precision\n```\n\n\n```python\n# example 1\n\n# y_true, is the true interaction of a user\n# and y_pred is the recommendation we decided\n# to give to the user\nk = 2\ny_true = np.array([1, 2, 3, 4, 5])\ny_pred = np.array([6, 4, 7, 1, 2])\ncompute_apk(y_true, y_pred[:k], k) # 0.25\n```\n\n\n\n\n 0.25\n\n\n\n\n```python\n# example 2\n\nk = 5\ny_true = np.array([1, 2])\ny_pred = np.array([6, 4, 7, 1, 2])\ncompute_apk(y_true, y_pred[:k], k) # 0.325\n```\n\n\n\n\n 0.325\n\n\n\nAfter computing the average precision for this individual user, we then compute this for every single user and take the mean of these values, then that would essentially be our MAP@k, mean average precision at k. For this metric, the bigger the better.\n\n\n```python\ndef mapk_score(model, ratings, k):\n \"\"\"\n mean average precision at rank k for the ALS model\n\n Parameters\n ----------\n model : ALSWR instance\n fitted ALSWR model\n\n ratings : scipy sparse csr_matrix [n_users, n_items]\n sparse matrix of user-item interactions\n\n k : int\n mean average precision at k's k\n \n Returns\n -------\n mapk : float\n the mean average precision at k's score\n \"\"\"\n # compare the top k predictions' index to the actual index,\n # the model is assumed to have the _predict_user method\n mapk = 0\n n_users = ratings.shape[0]\n for u in range(n_users):\n y_true = ratings[u].indices\n u_pred = model._predict_user(u)\n y_pred = np.argsort(u_pred)[::-1][:k]\n mapk += compute_apk(y_true, y_pred, k)\n\n mapk /= n_users\n return mapk\n```\n\n\n```python\nk = 5\nmapk_train = mapk_score(als, X_train, k)\nmapk_test = mapk_score(als, X_test, k)\nprint('mapk training:', mapk_train)\nprint('mapk testing:', mapk_test)\n```\n\n mapk training: 0.20038882997525595\n mapk testing: 0.05916489925768833\n\n\nNote that it's normal for this metric to be low. We can compare this metric with a baseline to get a sense of how well the algorithm is performing. And a nice baseline for recommendation engine is to simply recommend every user the most popular items (items that has the most user interaction)\n\n## NDCG\n\nSuppose that on a four-point scale, we give a 0 score for an irrelevant result, 1 for a partially relevant, 2 for relevant, and 3 for perfect. Suppose also that a query is judged by one of our judges, and the first four results that the search engine returns are assessed as relevant (2), irrelevant (0), perfect (3), and relevant (2) by a judge. \n\nThe idea behind **NDCG** is: A recommender returns some items and we'd like to compute how good the list is. Each item has a relevance score, usually a non-negative number. That's **gain (G)**. For items we don't have user feedback for we usually set the gain to zero. Now we add up those scores, that's **cumulative gain (CG)**. So, the cumulative gain for the four results is the sum of the scores for each result: 2 + 0 + 3 + 2 = 7. In mathematical notations, the CG at a particular rank ${\\displaystyle k}$ is defined as:\n\n\\begin{align}\nCG_k = \\sum_{i=1}^k rel_i\n\\end{align}\n\nWhere $rel_i$ is the graded relevance of the result at position ${\\displaystyle i}$. As we can see from the formula, the value computed with this function is unaffected by changes in the ordering of search results, thus DCG is used in place of CG for a more accurate measure about the usefulness of results' ranking.\n\nWhen evaluating rankings, we\u2019d prefer to see the most relevant items at the top of the list, i.e the first result in our search results is more important than the second, the second more important than the third, and so on. Therefore before summing the scores we divide each by a growing number, which is the **discount (D)**. One simple way to make position one more important than two (and so on) is to divide each score by the rank. So, for example, if the third result is \"great\", its contribution is $3 / 3 = 1$ (since the score for \"great\" is 3 , and the rank of the result is 3). If \"great\" were the first result, its contribution would be 3 / 1 = 3. Though in practice, it's more common to discount it using a logarithm of the item position, giving us:\n\n\\begin{align}\nDCG_k = \\sum_{i=1}^k \\frac{rel_i}{\\log_2{\\left(i+1\\right)}}\n\\end{align}\n\n\n```python\ndef dcg_at_k(score, k = None):\n \"\"\"\n discounted cumulative gain (dcg)\n \n Parameters\n ----------\n score : 1d nd.array\n ranking/relevance score\n \n k : int, default None\n evaluate the measure for the top-k ranking score,\n default None evaluates all\n \n Returns\n -------\n dcg: float\n \"\"\"\n if k is not None:\n score = score[:k]\n\n discounts = np.log2(np.arange(2, score.size + 2))\n dcg = np.sum(score / discounts)\n return dcg\n\n\nscore = np.array([2, 0, 3, 2])\ndcg_at_k(score)\n```\n\n\n\n\n 4.361353116146786\n\n\n\nThere's an alternative formulation of DCG that places stronger emphasis on retrieving relevant documents:\n\n\\begin{align}\nDCG_k = \\sum_{i=1}^k \\frac{2^{rel_i} - 1}{\\log_2{\\left(i+1\\right)}}\n\\end{align}\n\nThis formula is commonly used in industry including major web search companies and data science competition platform such as Kaggle.\n\n\n```python\ndef dcg_at_k(score, k = None):\n \"\"\"\n discounted cumulative gain (dcg)\n \n Parameters\n ----------\n score : 1d nd.array\n ranking/relevance score\n \n k : int, default None\n evaluate the measure for the top-k ranking score,\n default None evaluates all\n \n Returns\n -------\n dcg: float\n \"\"\"\n if k is not None:\n score = score[:k]\n\n gain = 2 ** score - 1\n discounts = np.log2(np.arange(2, score.size + 2))\n dcg = np.sum(gain / discounts)\n return dcg\n\n\nscore = np.array([2, 0, 3, 2])\ndcg_at_k(score)\n```\n\n\n\n\n 7.79202967422018\n\n\n\nThe final touch to this metric is **Normalized (N)**. It's not fair to compare DCG values across queries because some queries are easier than others or result lists vary in length depending on the query, so we normalize them by: First, figure out what the best ranking score is for this result and compute DCG for that, then we divide the raw DCG by this ideal DCG to get NDCG@K, a number between 0 and 1. In our previous example, we had 2, then 0, 3, and a 2. The best arrangement of these same results would have been: 3, 2, 2, 0, that is, if the \"great\" result had been ranked first, followed by the two \"relevant\" ones, and then the \"irrelevant\". So we compute the DCG score for the rank 3, 2, 2, 0 to obtain our ideal DCG (IDCG) and simply perform:\n\n\\begin{align}\nNDCG_k = \\frac{DCG_k}{IDCG_k}\n\\end{align}\n\nto obtain our final ndcg.\n\n\n```python\ndef ndcg_at_k(score, k = None):\n \"\"\"\n normalized discounted cumulative gain (ndcg)\n \n Parameters\n ----------\n score : 1d nd.array\n ranking/relevance score\n \n k : int, default None\n evaluate the measure for the top-k ranking score,\n default None evaluates all\n \n Returns\n -------\n ndcg: float, 0.0 ~ 1.0\n \"\"\"\n actual_dcg = dcg_at_k(score, k)\n sorted_score = np.sort(score)[::-1]\n best_dcg = dcg_at_k(sorted_score, k)\n ndcg = actual_dcg / best_dcg\n return ndcg\n\n\nndcg_at_k(score)\n```\n\n\n\n\n 0.7497534568197889\n\n\n\nThe next section modifies the function API a little bit so it becomes more suitable for evaluating the recommendation engine.\n\n\n```python\ndef ndcg_score(model, ratings, k):\n \"\"\"\n Normalized discounted cumulative gain (NDCG) at rank k\n for the ALS model; which computes the ndcg score for\n each users' recommendation and does a simply average\n \n Parameters\n ----------\n model : ALSWR instance\n fitted ALSWR model\n\n ratings : scipy sparse csr_matrix [n_users, n_items]\n sparse matrix of user-item interactions\n\n k : int\n rank k's k\n \n Returns\n -------\n avg_ndcg : float\n ndcg at k's score averaged across all users\n \"\"\"\n ndcg = 0.0\n n_users, n_items = ratings.shape\n for u in range(n_users):\n y_true = np.zeros(n_items)\n y_true[ratings[u].indices] = 1\n u_pred = model._predict_user(u)\n ndcg += ndcg_at_k(y_true, u_pred, k)\n \n avg_ndcg = ndcg / n_users\n return avg_ndcg\n\n\ndef ndcg_at_k(y_true, y_score, k = 10):\n \"\"\"\n Normalized discounted cumulative gain (NDCG) at rank k\n \n Parameters\n ----------\n y_true : array-like, shape = [n_samples]\n Ground truth (true relevance labels)\n \n y_score : array-like, shape = [n_samples]\n Predicted scores\n \n k : int\n Rank\n\n Returns\n -------\n ndcg : float, 0.0 ~ 1.0\n \"\"\"\n actual = dcg_at_k(y_true, y_score, k)\n best = dcg_at_k(y_true, y_true, k) \n ndcg = actual / best\n return ndcg\n\n\ndef dcg_at_k(y_true, y_score, k = 10):\n \"\"\"\n Discounted cumulative gain (DCG) at rank k\n \n Parameters\n ----------\n y_true : array-like, shape = [n_samples]\n Ground truth (true relevance labels)\n \n y_score : array-like, shape = [n_samples]\n Predicted scores\n \n k : int\n Rank\n\n Returns\n -------\n dcg : float\n \"\"\"\n order = np.argsort(y_score)[::-1]\n y_true = np.take(y_true, order[:k])\n gains = 2 ** y_true - 1\n discounts = np.log2(np.arange(2, gains.size + 2))\n dcg = np.sum(gains / discounts)\n return dcg\n```\n\n\n```python\nk = 5\nndcg_train = ndcg_score(als, X_train, k)\nndcg_test = ndcg_score(als, X_test, k)\nprint('ndcg training:', ndcg_train)\nprint('ndcg testing:', ndcg_test)\n```\n\n ndcg training: 0.2959125755797325\n ndcg testing: 0.11226091289209723\n\n\n# Reference\n\n- [Blog: Don\u2019t invert that matrix](https://www.johndcook.com/blog/2010/01/19/dont-invert-that-matrix/)\n- [Blog: Empty rows in sparse arrays](http://mike.place/2015/sparse/)\n- [Blog: Implicit Feedback and Collaborative Filtering](http://datamusing.info/blog/2015/01/07/implicit-feedback-and-collaborative-filtering/)\n- [Paper: Y. Hu, Y. Koren, C. Volinsky Collaborative Filtering for Implicit Feedback Datasets](http://yifanhu.net/PUB/cf.pdf)\n- [StackExchange: Analytic solution for matrix factorization using alternating least squares](http://math.stackexchange.com/questions/1072451/analytic-solution-for-matrix-factorization-using-alternating-least-squares)\n\n---\n\n\n- [Gist: Ranking Metrics](https://gist.github.com/bwhite/3726239)\n- [Gist: Learning to rank metrics](https://gist.github.com/mblondel/7337391)\n- [Github: Average Precision](https://github.com/benhamner/Metrics/blob/master/Python/ml_metrics/average_precision.py)\n- [Github: Average Precision Unit Test](https://github.com/benhamner/Metrics/blob/master/Python/ml_metrics/test/test_average_precision.py)\n- [Wiki: Discounted cumulative gain](https://en.wikipedia.org/wiki/Discounted_cumulative_gain)\n- [Blog: Measuring Search Relevance](http://www.ebaytechblog.com/2010/11/10/measuring-search-relevance/)\n- [Blog: Evaluating recommender systems](http://fastml.com/evaluating-recommender-systems/)\n- [Blog: What you wanted to know about Mean Average Precision](http://fastml.com/what-you-wanted-to-know-about-mean-average-precision/)\n", "meta": {"hexsha": "96f5f3bc42e9ff8f478ba53114863a256ad70d9d", "size": 65073, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "recsys/2_implicit.ipynb", "max_stars_repo_name": "certara-ShengnanHuang/machine-learning", "max_stars_repo_head_hexsha": "d21dfbeabf2876ffe49fcef444ca4516c4d36df0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2104, "max_stars_repo_stars_event_min_datetime": "2016-04-15T13:35:55.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-28T10:39:51.000Z", "max_issues_repo_path": "recsys/2_implicit.ipynb", "max_issues_repo_name": "certara-ShengnanHuang/machine-learning", "max_issues_repo_head_hexsha": "d21dfbeabf2876ffe49fcef444ca4516c4d36df0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 10, "max_issues_repo_issues_event_min_datetime": "2017-04-07T14:25:23.000Z", "max_issues_repo_issues_event_max_datetime": "2021-05-18T03:16:15.000Z", "max_forks_repo_path": "recsys/2_implicit.ipynb", "max_forks_repo_name": "certara-ShengnanHuang/machine-learning", "max_forks_repo_head_hexsha": "d21dfbeabf2876ffe49fcef444ca4516c4d36df0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 539, "max_forks_repo_forks_event_min_datetime": "2015-12-10T04:23:44.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-31T07:15:28.000Z", "avg_line_length": 37.3339070568, "max_line_length": 1533, "alphanum_fraction": 0.5211531664, "converted": true, "num_tokens": 12178, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891163376236, "lm_q2_score": 0.7248702761768248, "lm_q1q2_score": 0.42686821639717953}} {"text": "# Coase and Property\n\n> Coase, R. H. 1960. \u201cThe Problem of Social Cost.\u201d *The Journal of Law and Economics* 3:1\u201344.\n\n> Coase, Ronald H. 1937. \u201cThe Nature of the Firm.\u201d *Economica* 4 (16):386\u2013405.\n\n**Slideshow mode**: this notebook can be viewed as a slideshow by pressing Alt-R if run on a server.\n\n## Coase (1960) The Problem of Social Cost\n\n### A rancher and wheat farmer.\n\nBoth are utilizing adjacent plots of land. No fence separates the lands. \n\n**The Wheat Farmer:** chooses a production method that delivers a maximum profit of $\\Pi_W =8$. \n- to keep this simple suppose this is the farmer's only production choice.\n\n**The Rancher:** chooses herd size $x$ to maximize profits $\\Pi_C(x) = P \\cdot F(x) - c \\cdot x^2$\n\n- $P$ is cattle price and $c$ is the cost of feeding each animal. \n\n- The herd size $x^*$ that maximizes profits given by:\n\n$$P \\cdot F'(x^*) = c$$\n\n**Example:** If $F(x) = x$, $c=\\frac{1}{2}$. \n\nThe FOC are $x^{*} = P_c$ \n\nWith $P_c=4$ and $c=\\frac{1}{2}$, the rancher's privately optimal herd size of $x^* = 4$\n\n#### Missing Property Rights impose external costs\n\nWith no effective barrier separating the fields cattle sometimes strays into the wheat farmer's fields, damaging crops and reducing wheat farmer's profits.\n\nAssume that if the rancher keeps a herd size $x$ net profits in wheat are reduced from $\\Pi_W$ to:\n\n$$\\Pi_W(x) = \\Pi_W - d \\cdot x^2$$\n\n**The external cost**\n\nSuppose $d=\\frac{1}{2}$\n\nAt the rancher's private optimum herd size of $x^*=4$, the farmer's profit is reduced from 8 to zero:\n\n$$\\begin{align}\n\\Pi_W(x) &= \\Pi_W - d \\cdot x^2 \\\\\n & = 8 - \\frac{1}{2} \\cdot 4^2 = 0\n \\end{align}$$\n\n\n```python\nfrom coase import *\nfrom ipywidgets import interact, fixed\n```\n\nAt private optimum Rancher earns \\$8 but imposes external costs that drive the farmer's earnings to zero.\n\n\n```python\ncoaseplot1()\n```\n\nPrivate and social marginal benefits and costs can be plotted to see deadweight loss (DWL) differently:\n\n\n```python\ncoaseplot2()\n```\n\n## The assignment of property rights (liability)\n\n**Scenario 1:** Farmer is given the right to enjoin (i.e. limit or prohibit) cattle herding.\n\nIf the farmer enforces a prohibition on all cattle herding:\n\n- Rancher now earns \\$0. \n- Farmer earns \\$8. \n\n- But this is not efficient! Total output is smaller than it could be. \n- If transactions costs are low the two parties can bargain to a more efficient outcome.\n\n**Scenario 1:** Farmer is given the right to enjoin (i.e. limit or prohibit) cattle herding.\n\nRancher reasons that if she were permitted to herd 2 cattle she'd earn $\\$6$ while imposing \\$2 in damage.\n - She could offer $\\$2$ in full compensation for damage, pocketing remaining \\$4 \n - or they could bargain over how to divide the gains to trade of \\$4 in other ways.\n\n**Scenario 2:** Rancher is granted right to graze with impunity.\n\nFarmer reasons that if herd size could be reduced from 4 to 2\n- farm profits would rise from $\\$0$ to $\\$6$\n- rancher's profits would fall from $\\$8$ to $\\$6$\n\n- So farmer could offer to fully compensate rancher for $\\$2$ loss and pocket remaining $\\$4$\n- or they could bargain to divide those gains to trade of $\\$4$ in other ways.\n\n### Who causes the externality?\n\n- The rancher, because his cows trample the crops?\n- The farmer, for placing his field too close to the rancher?\n- Ronald Coase point is that there is no clear answer to this question.\n - Hence Pigouvian tax/subsidy 'solutions' are not obvious. Should we tax the rancher, or subsidize them to keep their herd size down? \n - 'Externality' problem is due to the non-assignment of property rights.\n\n## The 'Coase Theorem'\n\n### With zero/low transactions costs\n\n- **The initial assignment of property rights does not matter for efficiency:** The parties traded to an efficient solution no matter who first got the rights. \n\n- **The 'emergence' of property rights**: Even with no initial third-party assignment of property rights, it should be in the interests of the parties to create such rights and negotiate/trade to an efficient outcome. \n\n- **The initial allocation does matter for the distribution of benefits between parties.** Legally tradable entitlements are valuable, generate income to those who can then sell.\n\n### Coase Theorem: True, False or Tautology?\n\n> \"Costless bargaining is efficient tautologically; if I assume people can agree on socially efficient bargains, then of course they will... In the absence of property rights, a bargain *establishes* a contract between parties with novel rights that needn\u2019t exist ex-ante.\"\nCooter (1990)\n\nIn the Farmer and Rancher example there was a missing market for legal entitlements. \n\nOnce the market is made complete (by an assumed third party) then the First Welfare Theorem applies: complete competitive markets will lead to efficient allocations, regardless of initial allocation of property rights. \n\nThe \"Coase Theorem\" makes legal entitlements tradable.\n\nIn this view insuring efficiency is matter or removing impediments to free exchange of legal entitlements. However, \n\n>\"The interesting case is when transaction costs make bargaining difficult. What you should take from Coase is that social efficiency can be enhanced by institutions (including the firm!) which allow socially efficient bargains to be reached by removing restrictive transaction costs, and particularly that the assignment of property rights to different parties can either help or hinder those institutions.\"\n\nGood further discussions from [D. Mcloskey](http://www.deirdremccloskey.com/docs/pdf/Article_306.pdf) and [here](https://afinetheorem.wordpress.com/2013/09/03/on-coases-two-famous-theorems/): \n \n\n## When initial rights allocations matters for efficiency\n\n- 'Coase Theorem' (Stigler) interpretation sweeps under the rug the complicated political question of who gets initial rights.\n - Parties may engage in costly conflict, expend real resources to try to establish control over initial allocation of rights.\n - The [Myerson Satterthaite theorem](https://en.wikipedia.org/wiki/Myerson%E2%80%93Satterthwaite_theorem) establishes that when parties are asymmetrically informed about each other's valuations (e.g. here about the value of damages or benefits) then efficient exchange may become difficult/impossible. Each party may try to extract rents by trying to \"hold-up\" the other. \n - Suppose we had many farmers and ranchers. It might be costly/difficult to bring all relevant ranchers and farmers together and to agree on bargain terms. \n- Coase himself thought transactions costs mattered and hence initial allocation mechanisms had to be thought through carefully (e.g. spectrum auctions). \n\n## A Coasian view of land market development\n\nSuppose there is an open field. In the absence of a land market whoever gets to the land first (possibly the more powerful in the the village) will prepare/clear land until the marginal value product of the last unit of land is equal to the clearing cost. We contrast two situations:\n\n(1) Open frontier: where land is still abundant\n\n(2) Land Scarcity.\n\nThere will be a misallocation in (2) shown by DWL in the diagram... but also an incentive for the parties to bargain to a more efficient outcome. A well functionining land market would also deliver that outcome. \n\n#### Abundant land environment\n\n$\\bar T$ units of land and $N$=2 households.\n\nLand clearing cost $c$. Frontier land not yet exhausted.\n\nMaximize profits at $P \\cdot F_T(T) = c$\n\nLand demand for each farmer is given by $P\\cdot F_T(T_i) = r$. So for this production $P \\frac{1}{\\sqrt T_i} = r$ or $P \\frac{1}{\\sqrt T_i} = cl$ so we can write\n\n$$T^*_i(r) = (P/r)^2$$\n\nIf there is an open frontier the sum or demands falls short of total land supply and the marginal cost of land is the cost of clearing $r=c_l$. \n\n'Land scarcity' results on the other hand when there is an equilibrium price of land $r>c_l$ where $r$ is found from \n\n$$\\sum T^*_i(r) = \\bar T$$\n\nNow land rent $r-c$ can be charged on the right to access and use land. Trade in these legal entitlements can raise output and efficiency. But there may be conflict and a 'scramble' to establish those rights of first access. \n\n#### 'Customary' land rights\n\n- Suppose norm is that all in the village can use as much land as they can farm\n- Higher status individuals get allocation first\n- As long as land is abundant everyone gets the land they want\n- No \"land rent\" -- cannot charge rent above $c$ since villagers are free to clear at cost $c$\n\n\n```python\nlandmarket(P=5, cl = 3, title = 'Open Frontier')\n```\n\n### The closing of the frontier\n- Rising population or improving price or technology increases demand for land.\n- Suppose price at which product can be sold increases\n - demand for land increases.\n- Suppose total demandat clearing cost $c$ exceedsavailable land supply. \n - High-status individuals (who have first-access) leave less land available than is needed to satisfy remaining villagers demand.\n- Inefficient allocation of land\n - marginal products of land not equalized across households.\n - output would increase if we establish a market for trading land\n\n\n```python\nlandmarket(P=8, cl = 3, title = 'Land Scarcity')\n```\n\nWe can solve for the equilibrium rental rate $r$ given environmental paramters including the price $P$, land endowment $\\bar T$, population size $N$ and technology parameters $A)\n\nTo do:\n(things to still do in this notebook)\n - indicate DWL on landmarket diagrams\n - create widget to see how diagram shifts with changing parameters\n\n\n```python\ninteract(landmarket, P=(4,10,0.2), cl = (0,5,0.5), \n title = fixed('Land'), A=fixed(1));\n```\n", "meta": {"hexsha": "cbd2f7f3ceebd86e456b90e7f30d78486e7f962b", "size": 250056, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/Coase.ipynb", "max_stars_repo_name": "jhconning/DevII", "max_stars_repo_head_hexsha": "fc9c9e91dae79eea3e55beb67b2de01d8c375b04", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/Coase.ipynb", "max_issues_repo_name": "jhconning/DevII", "max_issues_repo_head_hexsha": "fc9c9e91dae79eea3e55beb67b2de01d8c375b04", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/Coase.ipynb", "max_forks_repo_name": "jhconning/DevII", "max_forks_repo_head_hexsha": "fc9c9e91dae79eea3e55beb67b2de01d8c375b04", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 260.475, "max_line_length": 66452, "alphanum_fraction": 0.923241194, "converted": true, "num_tokens": 2380, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.7154239897159438, "lm_q1q2_score": 0.42670258109947135}} {"text": "```python\nclean_up=True # removes gams-related files in work-folder if true\n%run StdPackages.ipynb\n```\n\n# **Static, partial equilibrium model**\n### **Production module, example 3:**\n\n*Example 3 defines a partial equilibrium model from data, including nesting structure, and calibration targets. The model is as stylized as possible, ignoring investments, trade and taxes.*\n\n**NB: The module can run with the cobb-douglas specification ($\\sigma=1$). However, when this is the case, the sum of $\\mu$-parameters should equal 1 (to keep constant returns to scale).**\n\n### **1: Read in data, and nesting structure**\n\nRead in general equilibrium data:\n\n\n```python\ndata_folder = os.getcwd()+'\\\\PE_P\\\\Example_3'\ndata = {'Production_v': data_folder+'\\\\Production_v.xlsx', 'Production_p': data_folder+'\\\\Production_p.xlsx'}\ncomponents = ['domestic']\ndb_GE = ReadData.read_data.main(data,components=components)\n```\n\nExtract data on sector 's1' from GE data:\n\n\n```python\ns0 = 's1'\ndb_PE = ReadData.PE_from_GE(db_GE,s0)\n```\n\nRead in nesting:\n\n\n```python\nnt = nesting_tree.nesting_tree(name='PE_Example3')\n```\n\n*Input-part:*\n\n\n```python\nnt.add_tree(data_folder+'\\\\nest_s1_in.xlsx',name='s1_in')\nread_type = {'1dvars': {'sheets': ['sigma'],'names':{}}, 'vars_panel': {'sheets': {'mu': 2},'names': {}}}\nnt.trees['s1_in'].database.read_from_excel(data_folder+'\\\\nest_s1_in.xlsx',read_type)\n```\n\n*Output-split:*\n\n\n```python\nnt.add_tree(data_folder+'\\\\nest_s1_out.xlsx',name='s1_out',**{'type_io': 'output', 'type_f': 'CET'})\nnt.trees['s1_out'].database.read_from_excel(data_folder+'\\\\nest_s1_out.xlsx',read_type)\n```\n\n*NB: Note that both Y1,Y2 are used as inputs and outputs. To accomodate for this, we use the feature of temporarily applying new set-names to selected elements of the nesting tree. A dict keeps track of the original names and reverts the names after collecting the nesting trees later on. This feature is called by adding 'temp_namespace' as in the following:*\n\n\n```python\nnt.trees['s1_out'].temp_namespace = {'Y1': 'Y1_out', 'Y2': 'Y2_out'}\n```\n\n### **2: Set up baseline model**\n\n*Compile tree attributes:*\n\n*Note that once the aggregate tree has been compiled, the temporary namespace used to have Y1,Y2 as both inputs and outputs are reversed per default - hence the 'temporary' in temporary namespace. If this is not the case, it should be specified through other means than the 'self.temp_namspace' attribute.*\n\n\n```python\nnt.run_all()\n```\n\n*Initiate production module with calibration data:*\n\n\n```python\npm = PE.GPM_STA_PE.production(nt=nt,**{'calib_db': db_PE, 'work_folder': work_folder, 'data_folder': data_folder})\n```\n\n*Run calibration function. This works as follows:*\n1. Runs baseline model:\n * Initialize variables (if no levels are given, some default values are provided).\n * Write gams-code to the folder self.data_folder (specified in the options above).\n * Create a model-instance named 'baseline' per default (can be specified with kwargs 'name_base': name). \n * Create a checkpoint named 'based 'baseline', and solve model.\n2. Create and ready calibration model instance:\n * Create model-instance named 'calib' per default (can be specified with kwargs 'name_calib': name).\n * Endogenize/exogenize relevant variables for calibration: self.calib contains gams-code for endogenizing $\\mu$-parameters, and exogenizing prices/quantities. Note: Initially, the calibration data for quantities/prices are not applied; this is to check that the endogenizing/exogenizing does not break the 'squareness' of the model.\n3. Run calibration model:\n * The default mode is to calibrate the model in a 'sneaky' manner: The solve_sneakily method creates a database with grids of data between the initial solution, and the data provided in self.calib_db. We then gradually update exogenous variables in 'n_steps' towards the calibration targets.\n * Note: When calibrating the standard partial equilibrium model, sneaking up on the solution is not as straightforward: If the production sector has constant returns to scale, it has to hold that the value of inputs equals the value of outputs; calibration of $\\mu$-parameters cannot alter this fact. Thus, making linear grids between the initial solution and the calibration target can result in unbalanced data (making the model potentially infeasible to solve for). To remedy this, the solve_sneakily method allow the user to apply one of three approaches that guarantees balanced data: \n * For $type\\_=$'both': Input prices and quantities are linearly spaced. Let $(p0,pT,q0,qT)$ denote the initial and final (last step of the sneaky-solve) price/quantity vectors for outputs in the sector. Let $i$ index the outputs quantities/prices, and $j$ the intermediate step in the sneaky-solve. We then solve:\n $$\\begin{align}\n \\sum_i (x_jpT(i)+(1-x_j)p0(i))*(x_jqT(i)+(1-x_j)q0(i)) = TC_j,\n \\end{align}$$\n where $TC_j$ is the total costs in the j'th step (defined by summing over inputs). That is, $x_j$ solved for the weight on the final step $T$. \n * For $type\\_=$'price': Input prices and quantities are linearly spaced. Output prices are linearly spaced as well. Quantities are spaced by solving for $x_j$:\n $$\\begin{align}\n \\sum_i p_j(i)*(x_jqT(i)+(1-x_j)q0(i)) = TC_j.\n \\end{align}$$\n Thus output quantities are non-linearly spaced to ensure that the step is feasible.\n * For $type\\_=$'quant': The reverse case of 'price'; output quantities are linearly spaced, output prices are non-linearly spaced to ensure feasibility.\n * Also note: Besides choosing different versions of creating balanced data, the module has been updated to create different types of nonlinear grids. Thus gridtype can be specified as $\\lbrace pol,rust,linear\\rbrace$. The 'linear' option simply applies the 'np.linspace' method. 'pol' and 'rust' methods specify grids as follows:\n $$\\begin{align}\n \\text{Polynomial (pol):} && x_i &= x_0+(x_N-x_0)\\left(\\dfrac{i-1}{N}\\right)^{\\phi} \\\\ \n \\text{Rust-spaced (rust):} && x_i &= x_{i-1}+\\dfrac{x_N-x_{i-1}}{(N-i+1)^{\\phi}}.\n \\end{align}$$\n In both cases, a $\\phi=1$ implies a linear grid, $\\phi<1$ places more gridpoints close to the end of the grid $(x_N)$, and vice versa for $\\phi>1$. \n\n*Here we calibrate by taking a loooooot of steps, to make sure to get it right (this adds the polynomially spaced grid with weights in the end of the grid):*\n\n\n```python\npm.calibrate(type_='price',kwargs_shock={'n_steps': 10,'gridtype':'pol','phi': 0.5})\n```\n\n*If succesfull, store calibrated data:*\n\n\n```python\nif pm.model_instances['calib'].modelstat == 16 and pm.model_instances['calib'].solvestat==1:\n pm.model_instances['calib'].out_db.db_Gdx.export(pm.data_folder+'\\\\calibrated.gdx')\n```\n\n### **A: How to pickle a model/model-instance**\n\nExport model-instance by specifying folder+name:\n\n\n```python\npm.model_instances['calib'].export(pm.data_folder,'PE_Production_3_calib')\n```\n\nNote that the gams workspace is not stored, only settings and data.\n\nLoad model-instance from pickle by specifying the kwarg 'pickle_path':\n\n\n```python\ntest_model = DB2Gams.gams_model(pickle_path=pm.data_folder+'\\\\PE_Production_3_calib')\n```\n\n### **B: How to pickle the production module**\n\nExport production module by specifycing name (folder is default set at self.data_folder, but can be specified using a kwarg): \n\n\n```python\npm.export('test')\n```\n\nLoad again by specifying the path to the pickle (here callced test)::\n\n\n```python\ntest_pm = PE.GPM_STA_PE.production(pickle_path=pm.data_folder+'\\\\test.pkl')\n```\n\nNote: Currently, only the self.model attribute, and self.model_instances are stored, and not gams checkpoints and calib_db.\n", "meta": {"hexsha": "56e2ab1d6f5bfc6997f28e827113feca31f07433", "size": 23796, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Examples/PE_Production_3.ipynb", "max_stars_repo_name": "ChampionApe/GamsPythonModels", "max_stars_repo_head_hexsha": "aaa234b2627cda2b92e478e8e8503bf9778aebeb", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Examples/PE_Production_3.ipynb", "max_issues_repo_name": "ChampionApe/GamsPythonModels", "max_issues_repo_head_hexsha": "aaa234b2627cda2b92e478e8e8503bf9778aebeb", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Examples/PE_Production_3.ipynb", "max_forks_repo_name": "ChampionApe/GamsPythonModels", "max_forks_repo_head_hexsha": "aaa234b2627cda2b92e478e8e8503bf9778aebeb", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 62.2931937173, "max_line_length": 2488, "alphanum_fraction": 0.6659942848, "converted": true, "num_tokens": 1977, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421276, "lm_q2_score": 0.6513548782017745, "lm_q1q2_score": 0.4265688764702192}} {"text": "# \u7b2c\u4e03\u8bb2\uff1a\u6700\u4f18\u95f4\u9694\u5206\u7c7b\u5668\u3001\u62c9\u683c\u6717\u65e5\u5bf9\u5076\u3001\u652f\u6301\u5411\u91cf\u673a\n\n\u7b80\u5355\u7684\u56de\u987e\u4e0a\u4e00\u8bb2\u7684\u5185\u5bb9\uff1a\n\n* \u6211\u4eec\u5b9a\u4e49\u4e86\u65b0\u7684\u5047\u8bbe\u51fd\u6570\uff1a\n\n $$\\begin{align}h_{w,b}&=g\\left(w^Tx+b\\right)\\\\g(z)&=\\begin{cases}1 &z\\gt 0\\\\-1 &z\\lt 0\\end{cases}\\\\y&=\\{-1,1\\}\\end{align}$$\n\n* \u7ed9\u51fa\u4e86\u51fd\u6570\u95f4\u9694\u7684\u5b9a\u4e49\uff1a\n\n $$\\hat{\\gamma}^{(i)}=y^{(i)}\\left(w^Tx^{(i)}+b\\right)$$\n\n* \u4ee5\u53ca\u51e0\u4f55\u95f4\u9694\u7684\u5b9a\u4e49\uff1a\n\n $$\\gamma^{(i)}=y^{(i)}\\left(\\left(\\frac{w}{\\lVert w\\rVert}\\right)^Tx^{(i)}+\\frac{b}{\\lVert w\\rVert}\\right)$$\n\n* \u4e86\u89e3\u4e86\u51fd\u6570\u95f4\u9694\u4e0e\u51e0\u4f55\u95f4\u9694\u7684\u5173\u7cfb\uff1a\n\n $$\\gamma=\\frac{\\hat\\gamma}{\\lVert w\\rVert}$$\n\n* \u540c\u65f6\u4e5f\u4e86\u89e3\u4e86\u5176\u51e0\u4f55\u610f\u4e49\uff1a\u7531$w^Tx+b=0$\u786e\u5b9a\u7684\u5206\u7c7b\u8d85\u5e73\u9762\u5c06\u6b63\u8d1f\u6837\u672c\u5206\u9694\u5f00\uff0c\u540c\u65f6\u4f7f\u8d85\u5e73\u9762\u4e0e\u6837\u672c\u95f4\u7684\u6700\u5c0f\u95f4\u9694\u5c3d\u53ef\u80fd\u5927\u3002\u6211\u4eec\u4e5f\u53d1\u73b0\uff0c\u5982\u679c\u6309\u6bd4\u4f8b\u7f29\u653e\u53c2\u6570$(w,b)$\uff0c\u5e76\u4e0d\u4f1a\u5bf9\u5047\u8bbe\u7ed3\u679c\u9020\u6210\u4efb\u4f55\u5f71\u54cd\uff0c\u56e0\u4e3a\u7ed9\u53c2\u6570\u540c\u65f6\u4e58\u4ee5\u4e00\u4e2a\u7cfb\u6570\u5e76\u4e0d\u4f1a\u5f71\u54cd\u8d85\u5e73\u9762\u7684\u4f4d\u7f6e\u3002\n\n\u63a5\u7740\u4e0a\u4e00\u8bb2\uff0c\u6211\u4eec\u7ee7\u7eed\u4ecb\u7ecd\u652f\u6301\u5411\u91cf\u673a\u7684\u76f8\u5173\u77e5\u8bc6\u3002\n\n## 4. \u6700\u4f18\u95f4\u9694\u5206\u7c7b\u5668\uff08optimal margin classifier\uff09\n\n\u901a\u8fc7\u4e0a\u4e00\u8bb2\u7684\u4ecb\u7ecd\uff0c\u5bf9\u4e8e\u7ed9\u5b9a\u7684\u8bad\u7ec3\u96c6\uff0c\u6211\u4eec\u4f1a\u81ea\u7136\u7684\u60f3\u8981\u627e\u5230\u4e00\u4e2a\u80fd\u591f\u4f7f\uff08\u51e0\u4f55\uff09\u95f4\u9694\u8fbe\u5230\u6700\u5927\u7684\u5224\u5b9a\u8fb9\u754c\uff0c\u56e0\u4e3a\u8fd9\u6837\u7684\u5224\u5b9a\u8fb9\u754c\u80fd\u591f\u5f97\u5230\u9ad8\u5ea6\u53ef\u4fe1\u7684\u9884\u6d4b\u96c6\uff0c\u540c\u65f6\u4e5f\u662f\u5bf9\u8bad\u7ec3\u96c6\u5f88\u597d\u7684\u62df\u5408\u3002\u8fd9\u4e5f\u5c06\u5e2e\u52a9\u6211\u4eec\u5f97\u5230\u4e00\u4e2a\u5c06\u6b63\u8d1f\u6837\u672c\u5206\u9694\uff08\u51e0\u4f55\u95f4\u9694\uff09\u5f00\u7684\u5206\u7c7b\u5668\u3002\n\n\u76ee\u524d\u4e3a\u6b62\uff0c\u6211\u4eec\u90fd\u5047\u8bbe\u8bad\u7ec3\u96c6\u662f\u7ebf\u6027\u53ef\u5206\u7684\uff0c\u5373\u6211\u4eec\u4e00\u5b9a\u53ef\u4ee5\u627e\u5230\u4e00\u4e2a\u80fd\u591f\u5c06\u6b63\u8d1f\u6837\u672c\u5206\u5f00\u7684\u5206\u7c7b\u8d85\u5e73\u9762\u3002\u90a3\u4e48\uff0c\u6211\u4eec\u600e\u6837\u627e\u5230\u5c06\u51e0\u4f55\u95f4\u9694\u6700\u5927\u5316\u7684\u5206\u7c7b\u8d85\u5e73\u9762\u5462\uff1f\u8ba8\u8bba\u4e0b\u9762\u7684\u4f18\u5316\u95ee\u9898\uff1a\n\n$$\\begin{align}\\displaystyle\\operatorname*{max}_{w,b}&\\quad\\gamma\\\\\\mathrm{s.t.}&\\quad y^{(i)}\\left(w^Tx^{(i)}+b\\right)\\geq\\gamma,\\quad i=1,\\cdots,m\\\\&\\quad\\lVert w\\rVert=1\\end{align}$$\n\n\u4e0a\u5f0f\u8868\u793a\uff0c\u5728\u6ee1\u8db3\u6bcf\u4e00\u4e2a\u8bad\u7ec3\u6837\u672c\u7684\u51fd\u6570\u95f4\u9694\u90fd\u4e0d\u5c0f\u4e8e$\\gamma$\u7684\u60c5\u51b5\u4e0b\uff0c\u4f7f\u7528$(w,b)$\u6784\u9020$\\gamma$\u7684\u80fd\u591f\u53d6\u5230\u7684\u6700\u5927\u503c\uff0c\u540c\u65f6\u6709\u9650\u5236\u6761\u4ef6$\\lVert w\\rVert=1$\u4fdd\u8bc1\u51fd\u6570\u95f4\u9694\u7b49\u4e8e\u51e0\u4f55\u95f4\u9694\u3002\u901a\u8fc7\u6c42\u5f97\u7684$(w,b)$\u5c31\u53ef\u4ee5\u8ba1\u7b97\u51fa\u8be5\u8bad\u7ec3\u96c6\u7684\u6700\u5927\u51e0\u4f55\u95f4\u9694\u3002\uff08\u6ce8\uff1a$\\mathrm{s.t.}$\u662f\u201csubject to\u201d\u7684\u7f29\u5199\uff0c\u610f\u4e3a\u201c\u53d7\u9650\u4e8e\u201d\u3002\uff09\n\n\u4e0a\u9762\u7684\u4f18\u5316\u95ee\u9898\u6709\u4e2a\u68d8\u624b\u7684\u9650\u5236\u6761\u4ef6$\\lVert w\\rVert=1$\uff08\u8fd9\u662f\u4e00\u4e2a\u7cdf\u7cd5\u7684\u975e\u51f8\u6027\u7ea6\u675f\uff0c\u5373\u53c2\u6570$w$\u53ef\u80fd\u4f4d\u4e8e\u4e00\u4e2a\u5355\u4f4d\u5706\u73af/\u7403\u9762\u4e0a\uff09\uff0c\u8fd9\u663e\u7136\u4e0d\u662f\u53ef\u4ee5\u63d0\u4ea4\u7ed9\u6807\u51c6\u7684\u51f8\u4f18\u5316\u8f6f\u4ef6\uff08\u5982\u68af\u5ea6\u4e0b\u964d\u6cd5\u3001\u725b\u987f\u6cd5\u7b49\uff09\u53bb\u89e3\u51b3\u7684\u95ee\u9898\u3002\u6240\u4ee5\uff0c\u6211\u4eec\u5c06\u4e0a\u9762\u7684\u95ee\u9898\u53d8\u5f62\u4e3a\u53e6\u4e00\u79cd\u7684\u5f62\u5f0f\uff1a\n\n$$\\begin{align}\\displaystyle\\operatorname*{max}_{w,b}&\\quad\\frac{\\hat\\gamma}{\\lVert w\\rVert}\\\\\\mathrm{s.t.}&\\quad y^{(i)}\\left(w^Tx^{(i)}+b\\right)\\geq\\hat\\gamma,\\quad i=1,\\cdots,m\\end{align}$$\n\n\u4e5f\u5c31\u662f\u6211\u4eec\u9009\u62e9\u6700\u5927\u5316\u51e0\u4f55\u95f4\u9694$\\frac{\\hat\\gamma}{\\lVert w\\rVert}$\uff0c\u540c\u65f6\u6ee1\u8db3\u6240\u6709\u6837\u672c\u7684\u51fd\u6570\u95f4\u9694\u4e0d\u5c0f\u4e8e$\\hat\\gamma$\u7684\u6761\u4ef6\u3002\u56e0\u4e3a\u51fd\u6570\u95f4\u9694\u548c\u51e0\u4f55\u95f4\u9694\u7684\u5173\u7cfb\u4e3a$\\gamma=\\frac{\\hat\\gamma}{\\lVert w\\rVert}$\uff0c\u800c\u53c8\u820d\u5f03\u4e86\u9650\u5236\u6761\u4ef6$\\lVert w\\rVert=1$\uff0c\u6240\u4ee5\u8fd9\u6b21\u6211\u4eec\u89c9\u5f97\u5e94\u8be5\u53ef\u4ee5\u6c42\u51fa\u6700\u5927\u503c\u4e86\u3002\u4e0d\u8fc7\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u8fd9\u6b21\u6700\u5927\u5316\u7684\u5bf9\u8c61$\\frac{\\hat\\gamma}{\\lVert w\\rVert}$\u662f\u975e\u51f8\u7684\uff0c\u800c\u6211\u4eec\u540c\u6837\u6ca1\u6709\u73b0\u6210\u7684\u51f8\u4f18\u5316\u8f6f\u4ef6\u6765\u89e3\u51b3\u6b64\u7c7b\u95ee\u9898\u3002\n\n\u4e8e\u662f\uff0c\u6211\u4eec\u53d1\u73b0\u5728\u7b2c\u4e00\u4e2a\u4f18\u5316\u95ee\u9898\u4e2d\uff0c\u5b58\u5728\u975e\u51f8\u6027\u7684\u9650\u5236\u6761\u4ef6\uff0c\u800c\u5728\u7b2c\u4e8c\u4e2a\u4f18\u5316\u95ee\u9898\u4e2d\uff0c\u5b58\u5728\u975e\u51f8\u6027\u7684\u4f18\u5316\u76ee\u6807\uff0c\u6240\u4ee5\u6211\u4eec\u5e76\u4e0d\u80fd\u4fdd\u8bc1\u8f6f\u4ef6\u53ef\u4ee5\u627e\u5230\u5168\u5c40\u6700\u5c0f\u503c\u3002\uff08\u8981\u5e26\u4e0a\u9650\u5236\u6761\u4ef6$\\lVert w\\rVert=1$\u662f\u56e0\u4e3a\u6211\u4eec\u7684\u4f18\u5316\u60f3\u8981\u5ea6\u91cf\u7684\u662f\u51e0\u4f55\u95f4\u9694\uff09\u3002\n\n\u56de\u60f3\u524d\u4e00\u8bb2\uff0c\u6211\u4eec\u77e5\u9053\uff0c\u6309\u6bd4\u4f8b\u7f29\u653e\u53c2\u6570$(w,b)$\u5bf9\u5047\u8bbe\u7ed3\u679c\u6ca1\u6709\u4efb\u4f55\u5f71\u54cd\uff0c\u6211\u4eec\u73b0\u5728\u53ef\u4ee5\u5229\u7528\u8fd9\u4e00\u70b9\u3002\u6211\u4eec\u73b0\u5728\u6765\u5f15\u5165\u9650\u5236\u6761\u4ef6\uff1a\u5bf9\u4e8e\u7ed9\u5b9a\u7684\u8bad\u7ec3\u96c6\uff0c\u4ee5$(w,b)$\u4e3a\u53c2\u6570\u7684\u51fd\u6570\u95f4\u9694\u5fc5\u987b\u4e3a$1$\uff1a\n\n$$\\hat\\gamma=1$$\n\n\u7ed9\u53c2\u6570$w,b$\u4e58\u4ee5\u7f29\u653e\u5e38\u91cf\u4f1a\u4f7f\u5f97\u51fd\u6570\u95f4\u9694\u7f29\u653e\u540c\u6837\u7684\u500d\u6570\uff0c\u5219\u53ef\u4ee5\u901a\u8fc7\u7f29\u653e$w,b$\u6765\u6ee1\u8db3\u4e0a\u9762\u7684\u9650\u5236\u6761\u4ef6\u3002\u5c06\u8fd9\u4e00\u9650\u5236\u6761\u4ef6\u52a0\u5165\u4e0a\u9762\u7684\u5047\u8bbe\u4e2d\uff0c\u4e8e\u662f\u95ee\u9898\u53d8\u4e3a\u6700\u5927\u5316$\\frac{\\hat\\gamma}{\\lVert w\\rVert}=\\frac{1}{\\lVert w\\rVert}$\uff0c\u4e5f\u5c31\u662f\u76f8\u5f53\u4e8e\u6700\u5c0f\u5316$\\lVert w\\rVert^2$\u3002\u73b0\u5728\u7684\u4f18\u5316\u95ee\u9898\u53d8\u4e3a\uff1a\n\n$$\\begin{align}\\displaystyle\\operatorname*{min}_{w,b}&\\quad\\frac{1}{2}\\lVert w\\rVert^2\\\\\\mathrm{s.t.}&\\quad y^{(i)}\\left(w^Tx^{(i)}+b\\right)\\geq 1,\\quad i=1,\\cdots,m\\end{align}$$\n\n\u8fd9\u6837\uff0c\u6211\u4eec\u5c31\u628a\u539f\u6765\u68d8\u624b\u7684\u95ee\u9898\u53d8\u4e3a\u4e86\u53ef\u4ee5\u7528\u8f6f\u4ef6\u9ad8\u6548\u89e3\u51b3\u7684\u95ee\u9898\u3002\u4e0a\u5f0f\u662f\u4e00\u4e2a\u5e26\u6709\u7ebf\u6027\u7ea6\u675f\u7684\u51f8\u4e8c\u6b21\u578b\u95ee\u9898\uff0c\u89e3\u8be5\u5f0f\u5373\u53ef\u5f97\u5230**\u6700\u4f18\u95f4\u9694\u5206\u7c7b\u5668\uff08optimal margin classifier\uff09**\u3002\u8fd9\u4e00\u7c7b\u4f18\u5316\u95ee\u9898\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u4e00\u4e9b\u73b0\u6210\u7684\u5546\u7528\u4e8c\u6b21\u4f18\u5316\u7a0b\u5e8f\uff08QP: quadratic programming\uff09\u89e3\u51b3\u3002\n\n\u5230\u8fd9\u91cc\uff0c\u95ee\u9898\u57fa\u672c\u89e3\u51b3\uff0c\u6211\u4eec\u6682\u505c\u4e00\u4e0b\u652f\u6301\u5411\u91cf\u673a\u7684\u8bb2\u89e3\uff0c\u5148\u4e86\u89e3\u4e00\u4e0b\u62c9\u683c\u6717\u65e5\u5bf9\u5076\u3002\u5b83\u5c06\u5f15\u51fa\u6211\u4eec\u4f18\u5316\u95ee\u9898\u7684\u5bf9\u5076\u5f62\u5f0f\uff0c\u901a\u8fc7\u4f7f\u7528\u6838\u65b9\u6cd5\uff0c\u5c06\u4fdd\u8bc1\u5728\u6211\u4eec\u5c06\u6700\u4f18\u95f4\u9694\u5206\u7c7b\u5668\u95ee\u9898\u63a8\u5e7f\u5230\u6781\u9ad8\u7ef4\u5ea6\u7a7a\u95f4\u65f6\u7b97\u6cd5\u7684\u9ad8\u6548\u6c42\u89e3\u3002\u5bf9\u5076\u5f62\u5f0f\u4e5f\u5c06\u5e2e\u52a9\u6211\u4eec\u63a8\u5bfc\u51fa\u4e00\u4e2a\u7528\u4ee5\u89e3\u51b3\u4e0a\u9762\u4f18\u5316\u95ee\u9898\u7684\u9ad8\u6548\u7b97\u6cd5\uff0c\u5176\u6c42\u89e3\u901f\u5ea6\u6bd4\u5e38\u89c1\u7684QP\u8f6f\u4ef6\u5feb\u5f88\u591a\u3002\n\n## 5. \u62c9\u683c\u6717\u65e5\u5bf9\u5076\uff08Lagrange duality\uff09\n\n\u6211\u4eec\u5148\u653e\u4e00\u653e\u652f\u6301\u5411\u91cf\u673a\u548c\u6700\u5927\u95f4\u9694\u5206\u7c7b\u5668\uff0c\u6765\u804a\u4e00\u804a\u5e26\u6709\u7ea6\u675f\u6761\u4ef6\u7684\u4f18\u5316\u95ee\u9898\u3002\u8003\u8651\u4e0b\u9762\u7684\u95ee\u9898\uff1a\n\n$$\\begin{align}\\displaystyle\\operatorname*{min}_{w}&\\quad f(w)\\\\\\mathrm{s.t.}&\\quad h_i(w)=0,\\quad i=1,\\cdots,l\\end{align}$$\n\n\u56de\u5fc6\u5728\u9ad8\u7b49\u6570\u5b66\u4e2d\u5b66\u5230\u7684\u62c9\u683c\u6717\u65e5\u4e58\u6570\u6cd5\uff08[\u4e2d\u6587](https://zh.wikipedia.org/zh/%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E4%B9%98%E6%95%B0)\uff0c[\u82f1\u6587](https://en.wikipedia.org/wiki/Lagrange_multiplier)\uff09\uff0c\u6211\u4eec\u5b9a\u4e49**\u62c9\u683c\u6717\u65e5\u7b97\u5b50\uff08Lagrangian\uff09**\uff1a\n\n$$\\mathcal{L}(w,\\beta)=f(w)+\\sum_{i=1}^l\\beta_ih_i(w)$$\n\n\u6b64\u5904\u7684\u7cfb\u6570$\\beta$\u79f0\u4f5c**\u62c9\u683c\u6717\u65e5\u4e58\u6570\uff08Lagrange multiplier\uff09**\uff0c\u7136\u540e\u5c06$\\mathcal{L}$\u7684\u504f\u5bfc\u6570\u7f6e\u4e3a\u96f6\uff1a\n\n$$\\begin{align}\\frac{\\partial\\mathcal{L}}{\\partial w_i}&=0\\\\\\frac{\\partial\\mathcal{L}}{\\partial\\beta_i}&=0\\end{align}$$\n\n\u6700\u540e\u89e3\u51fa$w,\\beta$\u5373\u53ef\u3002\u76f4\u63a5\u53c2\u8003\u62c9\u683c\u6717\u65e5\u4e58\u6570\u6cd5\u7684\u4f8b\u9898\uff08[\u4e2d\u6587](https://zh.wikipedia.org/wiki/%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E4%B9%98%E6%95%B0#.E4.BE.8B.E5.AD.90)\uff0c[\u82f1\u6587](https://en.wikipedia.org/wiki/Lagrange_multiplier#Examples)\uff09\u66f4\u52a0\u76f4\u89c2\u3002\n\n\u5728\u8fd9\u4e00\u8282\u6211\u4eec\u4f1a\u4e00\u822c\u5316\u8fd9\u79cd\u5e26\u6709\u7ea6\u675f\u6761\u4ef6\u7684\u4f18\u5316\u95ee\u9898\uff0c\u4e5f\u5c31\u662f\u4e0d\u9650\u4e8e\u7b49\u5f0f\u7ea6\u675f\u6761\u4ef6\uff0c\u540e\u9762\u4f1a\u52a0\u5165\u4e0d\u7b49\u5f0f\u7ea6\u675f\u6761\u4ef6\u3002\u56e0\u4e3a\u7bc7\u5e45\u9650\u5236\uff0c\u6211\u4eec\u4e0d\u4f1a\u5728\u8fd9\u91cc\u76f4\u63a5\u63a8\u5bfc\u62c9\u683c\u6717\u65e5\u5bf9\u5076\u7406\u8bba\uff08\u611f\u5174\u8da3\u53ef\u4ee5\u53c2\u8003R.T. Rockarfeller (1970), Convex Analysis, Princeton University Press.\uff09\uff0c\u4ec5\u5728\u8fd9\u91cc\u7ed9\u51fa\u4e3b\u8981\u601d\u8def\u548c\u7ed3\u679c\uff0c\u8fdb\u800c\u5728\u5e94\u7528\u5728\u6700\u4f18\u95f4\u9694\u5206\u7c7b\u5668\u4f18\u5316\u95ee\u9898\u4e2d\u3002\n\n\u4e0b\u9762\u4ecb\u7ecd**\u539f\u59cb\u4f18\u5316\u95ee\u9898\uff08primal optimization problem\uff09**\uff1a\n\n$$\\begin{align}\\displaystyle\\operatorname*{min}_{w}&\\quad f(w)\\\\\\mathrm{s.t.}&\\quad g_i(w)\\leq 0,\\quad i=1,\\cdots ,k\\\\&\\quad h_i(w)=0,\\quad i=1,\\cdots,l\\end{align}$$\n\n\u5b9a\u4e49**\u5e7f\u4e49\u62c9\u683c\u6717\u65e5\u7b97\u5b50\uff08generalized Lagrangian\uff09**\uff1a\n\n$$\\mathcal{L}(w,\\alpha,\\beta)=f(w)+\\sum_{i=1}^k\\alpha_ig_i(w)+\\sum_{i=1}^l\\beta_ih_i(w)$$\n\n\u8fd9\u91cc\u7684$\\alpha,\\beta$\u662f\u62c9\u683c\u6717\u65e5\u4e58\u6570\uff0c\u8003\u8651\u4e0b\u9762\u7684\u91cf\uff1a\n\n$$\\theta_{\\mathcal{P}}(w)=\\operatorname*{max}_{\\alpha,\\beta:\\alpha_i\\geq 0}\\mathcal{L}(w,\\alpha,\\beta)$$\n\n\u5176\u4e2d\u4e0b\u6807$\\mathcal{P}$\u4ee3\u8868\"primal\"\u3002\u5bf9\u4e8e\u4e00\u4e9b$w$\uff0c\u5982\u679c$w$\u8fdd\u53cd\u4e86\u4efb\u4f55\u539f\u59cb\u7ea6\u675f\u6761\u4ef6\uff08\u5982$g_i(w)\\gt 0$\u6216$h_i(w)\\neq 0$\uff09\uff0c\u5219\u53ef\u4ee5\u8bc1\u660e\uff1a\n\n$$\\begin{align}\\theta_{\\mathcal{P}}(w)&=\\operatorname*{max}_{\\alpha,\\beta:\\alpha_i\\geq 0}f(w)+\\sum_{i=1}^k\\alpha_ig_i(w)+\\sum_{i=1}^l\\beta_ih_i(w)\\tag{1}\\\\&=\\infty\\tag{2}\\end{align}$$\n\n\uff08\u5982\u679c\u67d0\u4e2a$g_i(w)\\gt 0$\uff0c\u5219\u62c9\u683c\u6717\u65e5\u7b97\u5b50\u4e2d\u8be5\u9879\u76f8\u5e94\u7684\u62c9\u683c\u6717\u65e5\u4e58\u6570$\\alpha_i$\u53ea\u9700\u53d6\u65e0\u7a77\u5927\u5c31\u53ef\u4ee5\u4f7f\u6574\u4e2a\u5f0f\u5b50\u6700\u5927\u5316\uff1b\u7c7b\u4f3c\u7684\uff0c\u5982\u679c\u67d0\u4e2a$h_i(w)\\neq 0$\uff0c\u5219\u62c9\u683c\u6717\u65e5\u7b97\u5b50\u4e2d\u8be5\u9879\u76f8\u5e94\u7684\u62c9\u683c\u6717\u65e5\u4e58\u6570$\\beta_i$\u53ea\u9700\u53d6\u65e0\u7a77\u5927\u4e5f\u53ef\u4ee5\u4f7f\u6574\u4e2a\u5f0f\u5b50\u6700\u5927\u5316\u3002\uff09\n\n\u76f8\u53cd\uff0c\u5982\u679c$w$\u5728\u7ea6\u675f\u6761\u4ef6\u5185\uff0c\u5219\u6709$\\theta_{\\mathcal{P}}=f(w)$\uff0c\u4e8e\u662f\u6709\uff1a\n\n$$\\theta_{\\mathcal{P}}=\\begin{cases}f(w)&\\textrm{if }w\\textrm{ satisfies primal constraints}\\\\\\infty&\\textrm{otherwise}\\end{cases}$$\n\n\uff08\u4e5f\u5c31\u662f\u8bf4\uff0c\u5bf9\u4e8e\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684$w$\uff0c\u8981\u4f7f\u5f97\u62c9\u683c\u6717\u65e5\u7b97\u5b50\u6700\u5927\u5316\uff0c\u9700\u8981\u62c9\u683c\u6717\u65e5\u4e58\u6570\u9879\u4e4b\u548c\u4e3a\u96f6\u3002\uff09\n\n\u5219\u5bf9\u4e8e\u6ee1\u8db3\u539f\u59cb\u7ea6\u675f\u7684$w$\u6765\u8bf4\uff0c$\\theta_{\\mathcal{P}}$\u4e0e\u539f\u59cb\u4f18\u5316\u95ee\u9898\u4e2d\u7684\u76ee\u6807\u51fd\u6570\u76f8\u540c\uff1b\u5bf9\u4e8e\u8fdd\u53cd\u539f\u59cb\u7ea6\u675f\u7684$w$\u6765\u8bf4\uff0c$\\theta_{\\mathcal{P}}$\u4e3a\u6b63\u65e0\u7a77\u3002\u56e0\u6b64\uff0c\u5982\u679c\u8003\u8651\u6700\u5c0f\u5316\uff1a\n\n$$\\operatorname*{min}_{w}\\theta_{\\mathcal{P}}(w)=\\operatorname*{min}_{w}\\operatorname*{max}_{\\alpha,\\beta:\\alpha_i\\geq 0}\\mathcal{L}(w,\\alpha,\\beta)$$\n\n\u6211\u4eec\u53d1\u73b0\uff0c\u8fd9\u4e0e\u539f\u59cb\u4f18\u5316\u95ee\u9898\u662f\u4e00\u6837\u7684\uff08\u4e5f\u6709\u540c\u6837\u7684\u89e3\uff09\u3002\u4e3a\u4e86\u540e\u9762\u4f7f\u7528\u65b9\u4fbf\uff0c\u6211\u4eec\u5b9a\u4e49\u4f18\u5316\u76ee\u6807\u7684\u6700\u4f18\u503c\u4e3a$p^*=\\displaystyle\\operatorname*{min}_{w}\\theta_{\\mathcal{P}}(w)$\uff0c\u79f0\u4e3a\u539f\u59cb\u95ee\u9898\u7684**\u6700\u4f18\u503c\uff08optimal value\uff09**\u3002\n\n\u73b0\u5728\uff0c\u6211\u4eec\u6765\u770b\u4e00\u4e2a\u7565\u5fae\u4e0d\u540c\u7684\u95ee\u9898\uff0c\u5b9a\u4e49\u5173\u4e8e\u62c9\u683c\u6717\u65e5\u4e58\u6570\u7684\u51fd\u6570\uff1a\n\n$$\\theta_{\\mathcal{D}}(\\alpha,\\beta)=\\operatorname*{min}_{w}\\mathcal{L}(w,\\alpha,\\beta)$$\n\n\u5176\u4e2d\u4e0b\u6807$\\mathcal{D}$\u4ee3\u8868\u201cdual\u201d\u3002\u7559\u610f\u5230\u5728$\\theta_{\\mathcal{P}}$\u7684\u5b9a\u4e49\u4e2d\uff0c\u6211\u4eec\u4f18\u5316\uff08\u6700\u5927\u5316\uff09\u5173\u4e8e$\\alpha,\\beta$\u7684\u51fd\u6570\uff1b\u800c\u5728\u8fd9\u91cc\uff0c\u6211\u4eec\u6700\u5c0f\u5316\u5173\u4e8e$w$\u7684\u51fd\u6570\u3002\n\n\u5f15\u5165**\u5bf9\u5076\u4f18\u5316\u95ee\u9898\uff08dual optimization problem\uff09**\uff1a\n\n$$\\operatorname*{max}_{\\alpha,\\beta:\\alpha_i\\geq 0}\\theta_{\\mathcal{D}}(\\alpha,\\beta)=\\operatorname*{max}_{\\alpha,\\beta:\\alpha_i\\geq 0}\\operatorname*{min}_{w}\\mathcal{L}(w,\\alpha,\\beta)$$\n\n\u8fd9\u4e2a\u5f0f\u5b50\u9664\u4e86\u201cmax\u201d\u548c\u201cmin\u201d\u7684\u987a\u5e8f\u53d1\u751f\u6539\u53d8\u4ee5\u5916\uff0c\u5176\u4f59\u7684\u540c\u524d\u9762\u7684\u539f\u59cb\u95ee\u9898\u4e00\u6837\u3002\u540c\u6837\u7684\uff0c\u5b9a\u4e49\u5bf9\u5076\u4f18\u5316\u95ee\u9898\u7684\u6700\u4f18\u503c\u4e3a$d^*=\\displaystyle\\operatorname*{max}_{\\alpha,\\beta:\\alpha_i\\geq 0}\\theta_{\\mathcal{D}}(\\alpha,\\beta)$\u3002\n\n\u539f\u59cb\u95ee\u9898\u4e0e\u5bf9\u5076\u95ee\u9898\u7684\u5173\u7cfb\u53ef\u4ee5\u7528\u4e0b\u9762\u7684\u5f0f\u5b50\u7b80\u5355\u8868\u793a\uff1a\n\n$$d^*=\\operatorname*{max}_{\\alpha,\\beta:\\alpha_i\\geq 0}\\operatorname*{min}_{w}\\mathcal{L}(w,\\alpha,\\beta)\\leq p^*=\\operatorname*{min}_{w}\\operatorname*{max}_{\\alpha,\\beta:\\alpha_i\\geq 0}\\mathcal{L}(w,\\alpha,\\beta)$$\n\n\uff08\u4e00\u4e2a\u666e\u904d\u4e8b\u5b9e\uff1a\u201cmin max\u201d\u67d0\u51fd\u6570\u603b\u662f\u5927\u4e8e\u7b49\u4e8e\u201cmax min\u201d\u67d0\u51fd\u6570\uff0c\u6bd4\u5982$\\displaystyle\\operatorname*{max}_{y\\in\\{0,1\\}}\\underbrace{\\left(\\displaystyle\\operatorname*{min}_{x\\in\\{0,1\\}}1\\{x=y\\}\\right)}_{0}\\leq\\displaystyle\\operatorname*{min}_{x\\in\\{0,1\\}}\\underbrace{\\left(\\displaystyle\\operatorname*{max}_{y\\in\\{0,1\\}}1\\{x=y\\}\\right)}_{1}$\uff09\u5728\u67d0\u4e9b\u7279\u5b9a\u60c5\u51b5\u4e0b\uff0c\u80fd\u591f\u5f97\u5230\uff1a\n\n$$d^*=p^*$$\n\n\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u89e3\u5bf9\u5076\u4f18\u5316\u95ee\u9898\u6765\u5f97\u5230\u539f\u59cb\u4f18\u5316\u95ee\u9898\u7684\u6700\u4f18\u503c\uff08\u8fd9\u4e48\u505a\u7684\u539f\u56e0\u662f\uff0c\u5bf9\u5076\u95ee\u9898\u901a\u5e38\u66f4\u52a0\u7b80\u5355\uff0c\u800c\u4e14\u4e0e\u539f\u59cb\u95ee\u9898\u76f8\u6bd4\uff0c\u5bf9\u5076\u95ee\u9898\u5177\u6709\u66f4\u591a\u6709\u7528\u7684\u6027\u8d28\uff0c\u7a0d\u540e\u6211\u4eec\u4f1a\u5728\u6700\u4f18\u95f4\u9694\u5206\u7c7b\u5668\u5373\u652f\u6301\u5411\u91cf\u673a\u95ee\u9898\u4e2d\u89c1\u5230\uff09\uff0c\u63a5\u4e0b\u6765\u770b\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u6b64\u5f0f\u6210\u7acb\u3002\n\n\u5047\u8bbe$f$\u548c$g_i$\u662f\u51f8\u51fd\u6570\uff08\u5bf9\u4e8e$f$\u7684\u6d77\u68ee\u77e9\u9635\u6709\uff0c\u5f53\u4e14\u4ec5\u5f53\u6d77\u68ee\u77e9\u9635\u534a\u6b63\u5b9a\u65f6\uff0c$f$\u662f\u51f8\u51fd\u6570\u3002\u4e3e\u4e2a\u4f8b\u5b50\uff1a$f(w)=w^Tw$\u662f\u51f8\u7684\uff0c\u7c7b\u4f3c\u7684\uff0c\u6240\u6709\u7ebf\u6027\uff08\u548c\u4eff\u5c04\uff09\u51fd\u6570\u4e5f\u662f\u51f8\u7684\u3002\u53e6\u5916\uff0c\u5373\u4f7f$f$\u4e0d\u53ef\u5fae\uff0c\u5b83\u4e5f\u53ef\u4ee5\u662f\u51f8\u7684\uff0c\u4e0d\u8fc7\u73b0\u5728\u6211\u4eec\u4e0d\u9700\u8981\u8fd9\u79cd\u66f4\u4e00\u822c\u5316\u7684\u5173\u4e8e\u51f8\u6027\u7684\u5b9a\u4e49\uff09\uff0c$h_i$\u662f\u4eff\u5c04\u51fd\u6570\uff08[\u4e2d\u6587](https://zh.wikipedia.org/wiki/%E4%BB%BF%E5%B0%84%E5%8F%98%E6%8D%A2)\uff0c[\u82f1\u6587](https://en.wikipedia.org/wiki/Affine_transformation)\uff09\uff08\u4eff\u5c04\u51fd\u6570/\u53d8\u6362\u662f\u6307\u5b58\u5728$a_i,b_i$\u4f7f\u5f97$h_i(w)=a_i^Tw+b_i$\uff0c\u800c\u201c\u4eff\u5c04\u53d8\u6362\u201d\u662f\u6307\u7ebf\u6027\u53d8\u6362\u540e\u52a0\u4e0a\u622a\u8ddd\u9879$b_i$\u4f7f\u6574\u4f53\u5e73\u79fb\uff0c\u5373\u7ebf\u6027\u53d8\u6362\u662f\u56fa\u5b9a\u539f\u70b9\u7684\uff0c\u800c\u4eff\u5c04\u53d8\u6362\u662f\u53ef\u4ee5\u5e73\u79fb\u7684\uff09\u3002\u8fdb\u4e00\u6b65\u5047\u8bbe$g_i$\u662f\u4e25\u683c\u53ef\u7528\u7684\uff0c\u5373\u5bf9\u4e8e\u6240\u6709$i$\u5b58\u5728$w$\u80fd\u591f\u4f7f$g_i(w)\\lt 0$\u3002\n\n\u5728\u4e0a\u8ff0\u5047\u8bbe\u6761\u4ef6\u4e0b\uff0c\u9644\u52a0\u6761\u4ef6$w^*,\\alpha^*,\\beta^*$\u4e00\u5b9a\u5b58\u5728\uff08$w^*$\u662f\u539f\u59cb\u95ee\u9898\u7684\u89e3\uff0c$\\alpha^*,\\beta^*$\u662f\u5bf9\u5076\u95ee\u9898\u7684\u89e3\uff09\uff0c\u518d\u9644\u52a0\u6761\u4ef6$p^*=d^*=\\mathcal{L}(w^*,\\alpha^*,\\beta^*)$\uff0c\u518d\u9644\u52a0\u6761\u4ef6$w^*,\\alpha^*,\\beta^*$\u6ee1\u8db3**KKT\u6761\u4ef6\uff08Karush-Kuhn-Tucker conditions\uff09**\uff1a\n\n$$\\begin{align}\\frac{\\partial}{\\partial w_i}\\mathcal{L}(w^*,\\alpha^*,\\beta^*)&=0,\\quad i=1,\\cdots,n\\tag{3}\\\\\\frac{\\partial}{\\partial \\beta_i}\\mathcal{L}(w^*,\\alpha^*,\\beta^*)&=0,\\quad i=1,\\cdots,l\\tag{4}\\\\\\alpha_i^*g_i(w^*)&=0,\\quad i=1,\\cdots,k\\tag{5}\\\\g_i(w^*)&\\leq0,\\quad i=1,\\cdots,k\\tag{6}\\\\\\alpha_i^*&\\geq0,\\quad i=1,\\cdots,k\\tag{7}\\end{align}$$\n\n\u5982\u679c\u5b58\u5728\u6ee1\u8db3KKT\u6761\u4ef6\u7684$w^*,\\alpha^*,\\beta^*$\uff0c\u5219\u539f\u59cb\u95ee\u9898\u4e0e\u5bf9\u5076\u95ee\u9898\u4e00\u5b9a\u6709\u89e3\u3002$(5)$\u5f0f\u53c8\u79f0\u4e3a**KKT\u5bf9\u5076\u4e92\u8865\u6761\u4ef6\uff08KKT dual complementarity condition\uff09**\uff0c\u8fd9\u4e2a\u6761\u4ef6\u8868\u660e\u5982\u679c$a_i^*\\gt0$\u5219$g_i(w^*)=0$\uff08\u5373\u7ea6\u675f\u6761\u4ef6$g_i(w^*)\\leq0$\u201c\u6fc0\u6d3b\u201d\uff0c\u6210\u4e3a\u4e00\u4e2a**\u6d3b\u52a8\u7ea6\u675f\uff08active constraint\uff09**\u5e76\u5904\u4e8e\u53d6\u7b49\u53f7\u7684\u72b6\u6001\uff09\u3002\u5728\u540e\u9762\u7684\u8bfe\u7a0b\u4e2d\uff0c\u6211\u4eec\u5c06\u901a\u8fc7\u8fd9\u4e2a\u6761\u4ef6\u77e5\u9053\u652f\u6301\u5411\u91cf\u673a\u53ea\u6709\u4e00\u5c0f\u90e8\u5206\u201c\u652f\u6301\u5411\u91cf\u201d\u3002\u5f53\u8bb2\u5230\u5e8f\u5217\u6700\u5c0f\u4f18\u5316\u7b97\u6cd5\uff08SMO\uff09\u65f6\uff0cKKT\u5bf9\u5076\u4e92\u8865\u6761\u4ef6\u4e5f\u4f1a\u7ed9\u6211\u4eec\u4e00\u4e2a\u9a8c\u8bc1\u5176\u6536\u655b\u7279\u5f81\u7684\u65b9\u6cd5\u3002\n\n## 6. \u6700\u4f18\u95f4\u9694\u5206\u7c7b\u5668\uff08\u7eed\uff09\n\n\u5728\u4e0a\u4e00\u8282\uff0c\u6211\u4eec\u5f15\u5165\u4e86\u539f\u59cb\u4f18\u5316\u95ee\u9898\uff0c\u7528\u4ee5\u6c42\u89e3\u6700\u4f18\u95f4\u9694\u5206\u7c7b\u5668\uff1a\n\n$$\\begin{align}\\displaystyle\\operatorname*{min}_{w,b}&\\quad\\frac{1}{2}\\lVert w\\rVert^2\\\\\\mathrm{s.t.}&\\quad y^{(i)}\\left(w^Tx^{(i)}+b\\right)\\geq 1,\\quad i=1,\\cdots,m\\end{align}$$\n\n\u6211\u4eec\u53ef\u4ee5\u5c06\u7ea6\u675f\u6761\u4ef6\u5199\u4e3a\uff1a\n\n$$g_i(w,b)=-y^{(i)}\\left(w^Tx^{(i)}+b\\right)+1\\leq0$$\n\n\u5bf9\u4e8e\u6bcf\u4e2a\u8bad\u7ec3\u6837\u672c\u90fd\u6709\u8fd9\u6837\u4e00\u4e2a\u7ea6\u675f\u6761\u4ef6\u3002\u4ece\u5bf9\u5076\u4e92\u8865\u7ea6\u675f\u6761\u4ef6\u53ef\u77e5\uff0c\u5f53$\\alpha_i\\gt0\\implies g_i(w,b)=0$\uff08\u6b64\u65f6\u662f\u4e00\u4e2a\u6d3b\u52a8\u7ea6\u675f\uff09$\\iff$\u6837\u672c$(x^{(i)},y^{(i)})$\u7684\u51fd\u6570\u95f4\u9694\u4e3a$1$\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5982\u679c\u8be5\u7ea6\u675f\u662f\u4e00\u4e2a\u6d3b\u52a8\u7ea6\u675f\uff0c\u90a3\u4e48\u5b83\u5b9e\u9645\u4e0a\u662f\u5c06\u4e00\u4e2a\u4e0d\u7b49\u5f0f\u6761\u4ef6\u53d8\u4e3a\u7b49\u5f0f\u6761\u4ef6\uff0c\u8fd9\u610f\u5473\u7740\u7b2c$i$\u4e2a\u8bad\u7ec3\u6837\u672c\u7684\u51fd\u6570\u95f4\u9694\u4e00\u5b9a\u7b49\u4e8e$1$\u3002\u8003\u8651\u4e0b\u56fe\uff0c\u5b9e\u7ebf\u8868\u793a\u5177\u6709\u6700\u5927\u95f4\u9694\u7684\u5206\u7c7b\u8d85\u5e73\u9762\uff1a\n\n\n\n\u56fe\u4e2d\u6700\u9760\u8fd1\u5224\u5b9a\u8fb9\u754c\uff08\u5373$w^Tx+b=0$\uff09\u7684\u70b9\u5c31\u662f\u5177\u6709\u6700\u5c0f\u95f4\u9694\u7684\u70b9\uff0c\u5728\u4e0a\u56fe\u4e2d\u5171\u6709\u4e09\u4e2a\u8fd9\u6837\u7684\u70b9\uff08\u4e00\u8d1f\u4e24\u6b63\uff09\uff0c\u5728\u5e73\u884c\u4e8e\u5224\u522b\u8fb9\u754c\u7684\u865a\u7ebf\u4e0a\u3002\u4e8e\u662f\uff0c\u4e00\u5171\u6709\u4e09\u4e2a$\\alpha_i$\u5728\u6211\u4eec\u89e3\u4f18\u5316\u95ee\u9898\u7684\u8fc7\u7a0b\u4e2d\u4e0d\u4e3a\u96f6\uff08\u4e5f\u5c31\u662f\u865a\u7ebf\u4e0a\u7684\u4e09\u4e2a\u70b9\uff0c\u53ea\u6709\u8fd9\u4e09\u4e2a\u70b9\u7684\u62c9\u683c\u6717\u65e5\u4e58\u6570\u4e0d\u4e3a\u96f6\uff0c\u4e5f\u53ea\u6709\u8fd9\u4e09\u4e2a\u6837\u672c\u7684\u51fd\u6570\u95f4\u9694\u7b49\u4e8e$1$\uff0c\u5176\u4f59\u6837\u672c\u7684\u51fd\u6570\u95f4\u9694\u90fd\u4e25\u683c\u5927\u4e8e$1$\u3002\u518d\u591a\u8bf4\u4e00\u70b9\uff0c\u6709\u65f6\u4f1a\u6709$g_i,\\alpha_i$\u90fd\u7b49\u4e8e\u96f6\u7684\u60c5\u51b5\uff0c\u4f46\u901a\u5e38$g_i=0$\u65f6$\\alpha_i$\u662f\u975e\u96f6\u7684\uff0c\u6240\u4ee5\u90a3\u4e9b\u51fd\u6570\u95f4\u9694\u4e3a$1$\u7684\u6837\u672c\u5c31\u662f\u90a3\u4e9b$\\alpha_i$\u4e0d\u7b49\u4e8e\u96f6\u7684\u6837\u672c\uff09\uff0c\u8fd9\u4e09\u4e2a\u6837\u672c\u4e5f\u79f0\u4e3a\u8fd9\u4e2a\u95ee\u9898\u7684**\u652f\u6301\u5411\u91cf\uff08support vectors\uff09**\u3002\u652f\u6301\u5411\u91cf\u7684\u6570\u91cf\u901a\u5e38\u90fd\u6bd4\u8bad\u7ec3\u96c6\u6837\u672c\u603b\u91cf\u5c11\u5f88\u591a\u3002\n\n\u968f\u7740\u6211\u4eec\u5bf9\u5bf9\u5076\u95ee\u9898\u7684\u6df1\u5165\u7406\u89e3\uff0c\u5176\u4e2d\u7684\u5173\u952e\u601d\u60f3\u5c31\u662f\u5c1d\u8bd5\u5c06\u7b97\u6cd5\u7528\u8f93\u5165\u7279\u5f81\u7a7a\u95f4\u4e2d\u70b9\u7684\u5411\u91cf\u5185\u79ef$\\left\\langle x^{(i)},x^{(j)}\\right\\rangle$\u7684\u5f62\u5f0f\u8868\u8fbe\u51fa\u6765\uff08\u53ef\u4ee5\u770b\u505a$\\left(x^{(i)}\\right)^Tx^{(j)}$\uff09\u3002\u5b9e\u9645\u4e0a\u5f53\u6211\u4eec\u4f7f\u7528\u6838\u65b9\u6cd5\u65f6\uff0c\u8fd9\u79cd\u8868\u8fbe\u6cd5\u5c06\u6210\u4e3a\u7b97\u6cd5\u7684\u5173\u952e\u3002\n\n\u4e3a\u4f18\u5316\u95ee\u9898\u6784\u9020\u62c9\u683c\u6717\u65e5\u7b97\u5b50\uff1a\n\n$$\\mathcal{L}(w,b,\\alpha)=\\frac{1}{2}\\lVert w\\rVert^2-\\sum_{i=1}^m\\alpha_i\\left[y^{(i)}\\left(w^Tx^{(i)}+b\\right)-1\\right]\\tag{8}$$\n\n\u5f0f\u4e2d\u53ea\u6709\u62c9\u683c\u6717\u65e5\u4e58\u6570$\\alpha_i$\u800c\u6ca1\u6709$\\beta_i$\uff0c\u56e0\u4e3a\u6b64\u95ee\u9898\u4e2d\u53ea\u542b\u6709\u4e0d\u7b49\u5f0f\u7ea6\u675f\u6761\u4ef6\u3002\n\n\u6211\u4eec\u9700\u8981\u627e\u51fa\u6b64\u95ee\u9898\u7684\u5bf9\u5076\u5f62\u5f0f\u3002\u8981\u5f97\u5230\u5bf9\u5076\u95ee\u9898\u7684$\\theta_{\\mathcal{D}}$\uff0c\u9996\u5148\u9700\u8981\u6700\u5c0f\u5316\u5173\u4e8e$w,b$\uff08\u5c06$\\alpha$\u4f5c\u4e3a\u5e38\u91cf\uff09\u7684\u51fd\u6570$\\mathcal{L}(w,b,\\alpha)$\uff0c\u4e5f\u5c31\u662f\u5bf9$\\mathcal{L}$\u5206\u522b\u6c42\u5173\u4e8e$w,b$\u7684\u504f\u5bfc\uff0c\u5e76\u5c06\u504f\u5bfc\u6570\u7f6e\u4e3a\u96f6\uff1a\n\n$$\\nabla_w\\mathcal{L}(w,b,\\alpha)=w-\\sum_{i=1}^m\\alpha_iy^{(i)}x^{(i)}=0$$\n\n\u4e8e\u662f\u5f97\u5230$w$\uff1a\n\n$$w=\\sum_{i=1}^m\\alpha_iy^{(i)}x^{(i)}\\tag{9}$$\n\n\u53ef\u4ee5\u770b\u51fa$w$\u5b9e\u9645\u4e0a\u662f\u7531$\\alpha_i$\u8bbe\u7f6e\u6743\u91cd\u540e\u8f93\u5165\u7279\u5f81\u503c\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\u3002\u7ee7\u7eed\u5bf9$b$\u6c42\u504f\u5bfc\uff1a\n\n$$\\frac{\\partial}{\\partial b}\\mathcal{L}(w,b,\\alpha)=\\sum_{i=1}^m\\alpha_iy^{(i)}=0\\tag{10}$$\n\n\u73b0\u5728\u5c06$(9)$\u5f0f\u4ee3\u5165$(8)$\u5f0f\u5e76\u5316\u7b80\uff0c\u6709\uff1a\n\n$$\\mathcal{L}(w,b,\\alpha)=\\sum_{i=1}^m\\alpha_i-\\frac{1}{2}\\sum_{i,j=1}^m\\alpha_i\\alpha_jy^{(i)}y^{(j)}\\left(x^{(i)}\\right)^Tx^{(j)}-b\\sum_{i=1}^m\\alpha_iy^{(i)}$$\n\n\u6839\u636e$(10)$\u5f0f\uff0c\u6700\u540e\u4e00\u9879\u5e94\u4e3a$0$\uff0c\u4e8e\u662f\u5f97\u5230\uff1a\n\n$$\\mathcal{L}(w,b,\\alpha)=\\sum_{i=1}^m\\alpha_i-\\frac{1}{2}\\sum_{i,j=1}^m\\alpha_i\\alpha_jy^{(i)}y^{(j)}\\left(x^{(i)}\\right)^Tx^{(j)}$$\n\n\u8fd9\u4e2a\u5f0f\u5b50\u662f\u6211\u4eec\u901a\u8fc7\u6c42\u51fd\u6570$\\mathcal{L}$\u5173\u4e8e$w,b$\u7684\u51fd\u6570\u6700\u5c0f\u503c\u5f97\u5230\u7684\uff0c\u518d\u52a0\u4e0a\u7ea6\u675f\u6761\u4ef6$\\alpha_i\\geq0$\uff08\u59cb\u7ec8\u5b58\u5728\u7684\u7ea6\u675f\uff09\u548c$(10)$\u5f0f\uff0c\u6700\u7ec8\u5f97\u5230\u4e0b\u9762\u7684\u5bf9\u5076\u4f18\u5316\u95ee\u9898\uff08\u6b64\u65f6\u5c06$\\mathcal{L}$\u770b\u505a\u5173\u4e8e$\\alpha$\u7684\u51fd\u6570\uff09\uff1a\n\n$$\\begin{align}\\displaystyle\\operatorname*{max}_{\\alpha}&\\quad W(\\alpha)=\\sum_{i=1}^m\\alpha_i-\\frac{1}{2}\\sum_{i,j=1}^m\\alpha_i\\alpha_jy^{(i)}y^{(j)}\\left\\langle x^{(i)},x^{(j)}\\right\\rangle\\\\\\mathrm{s.t.}&\\quad \\alpha_i\\geq 0,\\quad i=1,\\cdots,m\\\\&\\quad\\sum_{i=1}^m\\alpha_iy^{(i)}=0\\end{align}$$\n\n\uff08\u5728\u8fd9\u91cc\u7b80\u5355\u7684\u89e3\u91ca\u4e00\u4e0b\u7b2c\u4e8c\u4e2a\u7ea6\u675f\u6761\u4ef6\uff0c\u5373\u62c9\u683c\u6717\u65e5\u7b97\u5b50\u5bf9$b$\u6c42\u504f\u5bfc\u7684\u7ed3\u679c\u3002\u5982\u679c$\\displaystyle\\sum_{i=1}^m\\alpha_iy^{(i)}\\neq0$\uff0c\u5219$\\theta_{\\mathcal{D}}(\\alpha)=-\\infty$\u3002\u6362\u53e5\u8bdd\u8bf4\u62c9\u683c\u6717\u65e5\u7b97\u5b50\u662f\u53c2\u6570$b$\u7684\u7ebf\u6027\u51fd\u6570\u3002\u6240\u4ee5\u5982\u679c\u6211\u4eec\u7684\u76ee\u6807\u662f$\\displaystyle\\operatorname*{max}_{\\alpha\\geq0}\\theta_{\\mathcal{D}}(\\alpha)$\uff0c\u90a3\u4e48\u5c31\u5e94\u8be5\u9009\u62e9\u4f7f\u5f97$\\displaystyle\\sum_{i=1}^m\\alpha_iy^{(i)}=0$\u7684$\\alpha$\uff0c\u56e0\u4e3a\u5f53$\\displaystyle\\sum_{i=1}^m\\alpha_iy^{(i)}=0$\u65f6$\\theta_{\\mathcal{D}}(\\alpha)=W(\\alpha)$\u3002\uff09\n\n\u6613\u8bc1$p^*=d^*$\u53caKKT\u6761\u4ef6\uff08(3)-(7)\u5f0f\uff09\u5728\u6b64\u4f18\u5316\u95ee\u9898\u4e2d\u6210\u7acb\u3002\u90a3\u4e48\u6211\u4eec\u5c31\u53ef\u4ee5\u901a\u8fc7\u8ba1\u7b97\u5bf9\u5076\u95ee\u9898\u6765\u5f97\u5230\u539f\u59cb\u95ee\u9898\u7684\u89e3\u3002\u4e0a\u9762\u7684\u5bf9\u5076\u95ee\u9898\u662f\u4e00\u4e2a\u6c42\u5173\u4e8e$\\alpha_i$\u7684\u51fd\u6570\u7684\u6700\u5927\u503c\u7684\u95ee\u9898\u3002\u6211\u4eec\u7a0d\u540e\u518d\u8ba8\u8bba\u5bf9\u5076\u95ee\u9898\u7684\u7279\u5b9a\u89e3\u6cd5\uff0c\u5148\u5047\u8bbe\u6211\u4eec\u80fd\u591f\u89e3\u51fa\u6700\u4f18\u503c\uff08\u5373\u627e\u5230\u5728\u7ea6\u675f\u6761\u4ef6\u4e0b\u4f7f$W(\\alpha)$\u6700\u5927\u5316\u7684$\\alpha$\u7684\u53d6\u503c\uff09\uff0c\u90a3\u4e48\u5c31\u53ef\u4ee5\u4f7f\u7528$(9)$\u5f0f\u627e\u5230\u6700\u4f18\u503c$w$\uff08$w$\u662f\u4e00\u4e2a\u5173\u4e8e$\\alpha$\u7684\u51fd\u6570\uff09\u3002\u8003\u8651\u539f\u59cb\u95ee\u9898\uff0c\u6709\u4e86$w^*$\u5219\u53ef\u4ee5\u76f4\u63a5\u6c42\u51fa\u622a\u8ddd\u9879$b$\uff0c\u56e0\u4e3a\u6b64\u65f6\u8d85\u5e73\u9762\u7684\u6cd5\u5411\u91cf\u5df2\u7ecf\u786e\u5b9a\uff0c\u6211\u4eec\u53ea\u9700\u8981\u5728\u6ee1\u8db3\u8be5\u6cd5\u5411\u91cf\u7684\u8d85\u5e73\u9762\u4e2d\u201c\u56fa\u5b9a\u201d\u9002\u5f53\u7684\u622a\u8ddd\uff0c\u4f7f\u5f97\u8d85\u5e73\u9762\u5230\u6b63\u8d1f\u6837\u672c\u7684\u8ddd\u79bb\u76f8\u7b49\u5373\u53ef\uff08\u5373\u627e\u5230\u56fe\u4e2d\u4e24\u865a\u7ebf\u4e4b\u95f4\u7684\u5b9e\u73b0\uff09\u3002\u6211\u4eec\u5c06$a,w$\u5e26\u5165\u539f\u59cb\u95ee\u9898\u6c42\u4e2d\u89e3$b$\uff1a\n\n$$b^*=\\frac{\\displaystyle\\operatorname*{max}_{i:y^{(i)}=-1}w^{*T}x^{(i)}+\\displaystyle\\operatorname*{min}_{i:y^{(i)}=1}w^{*T}x^{(i)}}{2}\\tag{11}$$\n\n\u518d\u7ee7\u7eed\u4e4b\u524d\uff0c\u6211\u4eec\u5148\u4ed4\u7ec6\u89c2\u5bdf\u4e00\u4e0b$(9)$\u5f0f\uff0c\u4e5f\u5c31\u662f\u6700\u4f18\u503c$w$\u5173\u4e8e\u6700\u4f18\u503c$\\alpha$\u7684\u51fd\u6570\u3002\u5047\u8bbe\u6211\u4eec\u5df2\u7ecf\u901a\u8fc7\u62df\u5408\u8bad\u7ec3\u96c6\u5f97\u5230\u6a21\u578b\u7684\u53c2\u6570\uff0c\u73b0\u5728\u60f3\u8981\u9884\u6d4b\u4e00\u4e2a\u65b0\u7684\u8f93\u5165\u70b9$x$\uff0c\u90a3\u4e48\u6211\u4eec\u4f1a\u8ba1\u7b97$w^Tx+b$\uff0c\u5f53\u4e14\u4ec5\u5f53\u8fd9\u4e2a\u503c\u5927\u4e8e$1$\u65f6\uff0c\u6a21\u578b\u624d\u4f1a\u7ed9\u51fa\u7ed3\u8bba$y=1$\u3002\u4f46\u662f\u901a\u8fc7$(9)$\u5f0f\uff0c\u8fd9\u4e2a\u503c\u4e5f\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a\n\n$$\\begin{align}w^Tx+b&=\\left(\\sum_{i=1}^m\\alpha_iy^{(i)}x^{(i)}\\right)^Tx+b\\tag{12}\\\\&=\\sum_{i=1}^m\\alpha_iy^{(i)}\\left\\langle x^{(i)},x\\right\\rangle+b\\tag{13}\\end{align}$$\n\n\u5982\u679c\u6211\u4eec\u5df2\u7ecf\u6c42\u51fa$\\alpha_i$\uff0c\u4e3a\u4e86\u505a\u51fa\u9884\u6d4b\uff0c\u5219\u53ea\u9700\u8981\u6309\u7167\u4e0a\u5f0f\u6c42\u51fa$x$\u4e0e\u8bad\u7ec3\u96c6\u4e2d\u6837\u672c\u7684\u5185\u79ef\u3002\u800c\u4e14\u5728\u524d\u9762\u7684\u6c42\u89e3\u4e2d\uff0c\u6211\u4eec\u77e5\u9053\uff0c\u9664\u4e86\u652f\u6301\u5411\u91cf\u4ee5\u4e3a\uff0c\u5176\u4f59\u8bad\u7ec3\u6837\u672c\u5bf9\u5e94\u7684$\\alpha_i$\u90fd\u662f\u96f6\uff0c\u56e0\u6b64\u4e0a\u9762\u7684\u6c42\u548c\u4e2d\u5f88\u591a\u9879\u90fd\u662f\u96f6\u3002\u6240\u4ee5\u6211\u4eec\u53ea\u9700\u8981\u6c42\u51fa$x$\u4e0e\u652f\u6301\u5411\u91cf\u7684\u5185\u79ef\uff0c\u7136\u540e\u518d\u6309\u7167$(13)$\u5f0f\u8ba1\u7b97\u5e76\u4f5c\u51fa\u9884\u6d4b\u5373\u53ef\u3002\n\n\u4e3a\u4e86\u8ba1\u7b97\u5bf9\u5076\u4f18\u5316\u95ee\u9898\uff0c\u6211\u4eec\u6df1\u5165\u4e86\u89e3\u4e86\u95ee\u9898\u7684\u7ed3\u6784\uff0c\u4e8e\u662f\u4ec5\u4f9d\u9760\u652f\u6301\u5411\u91cf\u7684\u5185\u79ef\u5c31\u8868\u793a\u51fa\u6574\u4e2a\u7b97\u6cd5\u3002\u5728\u4e0b\u4e00\u8282\u4e2d\uff0c\u6211\u4eec\u5c06\u7ee7\u7eed\u5206\u6790\u8fd9\u4e2a\u7279\u6027\uff0c\u8fdb\u800c\u80fd\u591f\u5c06\u6838\u65b9\u6cd5\u5e94\u7528\u5728\u5206\u7c7b\u95ee\u9898\u3002\u800c\u4f5c\u4e3a\u7ed3\u679c\u5f97\u5230\u7684**\u652f\u6301\u5411\u91cf\u673a\uff08support vector machines\uff09**\u5c06\u4f1a\u662f\u4e00\u4e2a\u80fd\u591f\u5728\u6781\u9ad8\u7ef4\u5ea6\u7a7a\u95f4\u4e2d\u9ad8\u6548\u5b66\u4e60\u7684\u7b97\u6cd5\u3002\n", "meta": {"hexsha": "e0c6c094aaab44f30d4af46dea1a56da7ffa6a52", "size": 14717, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "4-SVM/note/LSJU-SVM-02.ipynb", "max_stars_repo_name": "PeterChenYijie/MachineLearningZeroToALL", "max_stars_repo_head_hexsha": "b14005c3e0b5a39a0ba82db5c9791f682b5effd5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 8, "max_stars_repo_stars_event_min_datetime": "2018-04-20T09:10:20.000Z", "max_stars_repo_stars_event_max_datetime": "2019-02-16T07:50:32.000Z", "max_issues_repo_path": "4-SVM/note/LSJU-SVM-02.ipynb", "max_issues_repo_name": "DeepInDeeper/MachineLearningZeroToALL", "max_issues_repo_head_hexsha": "b14005c3e0b5a39a0ba82db5c9791f682b5effd5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, 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YES\n2. YES", "lm_q1_score": 0.6513548646660542, "lm_q2_score": 0.6548947223065755, "lm_q1q2_score": 0.4265688632185126}} {"text": "\n\n# Full configuration interaction theory\n\n \n**Morten Hjorth-Jensen**, [National Superconducting Cyclotron Laboratory](http://www.nscl.msu.edu/) and [Department of Physics and Astronomy](https://www.pa.msu.edu/), [Michigan State University](http://www.msu.edu/), East Lansing, MI 48824, USA\n\nDate: **Jul 19, 2018**\n\nCopyright 2013-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n\n\n\n\n\n## Slater determinants as basis states, Repetition\nThe simplest possible choice for many-body wavefunctions are **product** wavefunctions.\nThat is\n\n$$\n\\Psi(x_1, x_2, x_3, \\ldots, x_A) \\approx \\phi_1(x_1) \\phi_2(x_2) \\phi_3(x_3) \\ldots\n$$\n\nbecause we are really only good at thinking about one particle at a time. Such \nproduct wavefunctions, without correlations, are easy to \nwork with; for example, if the single-particle states $\\phi_i(x)$ are orthonormal, then \nthe product wavefunctions are easy to orthonormalize. \n\nSimilarly, computing matrix elements of operators are relatively easy, because the \nintegrals factorize.\n\n\nThe price we pay is the lack of correlations, which we must build up by using many, many product \nwavefunctions. (Thus we have a trade-off: compact representation of correlations but \ndifficult integrals versus easy integrals but many states required.)\n\n\n\n## Slater determinants as basis states, repetition\nBecause we have fermions, we are required to have antisymmetric wavefunctions, e.g.\n\n$$\n\\Psi(x_1, x_2, x_3, \\ldots, x_A) = - \\Psi(x_2, x_1, x_3, \\ldots, x_A)\n$$\n\netc. This is accomplished formally by using the determinantal formalism\n\n$$\n\\Psi(x_1, x_2, \\ldots, x_A) \n= \\frac{1}{\\sqrt{A!}} \n\\det \\left | \n\\begin{array}{cccc}\n\\phi_1(x_1) & \\phi_1(x_2) & \\ldots & \\phi_1(x_A) \\\\\n\\phi_2(x_1) & \\phi_2(x_2) & \\ldots & \\phi_2(x_A) \\\\\n \\vdots & & & \\\\\n\\phi_A(x_1) & \\phi_A(x_2) & \\ldots & \\phi_A(x_A) \n\\end{array}\n\\right |\n$$\n\nProduct wavefunction + antisymmetry = Slater determinant.\n\n\n\n## Slater determinants as basis states\n\n$$\n\\Psi(x_1, x_2, \\ldots, x_A) \n= \\frac{1}{\\sqrt{A!}} \n\\det \\left | \n\\begin{array}{cccc}\n\\phi_1(x_1) & \\phi_1(x_2) & \\ldots & \\phi_1(x_A) \\\\\n\\phi_2(x_1) & \\phi_2(x_2) & \\ldots & \\phi_2(x_A) \\\\\n \\vdots & & & \\\\\n\\phi_A(x_1) & \\phi_A(x_2) & \\ldots & \\phi_A(x_A) \n\\end{array}\n\\right |\n$$\n\nProperties of the determinant (interchange of any two rows or \nany two columns yields a change in sign; thus no two rows and no \ntwo columns can be the same) lead to the Pauli principle:\n\n* No two particles can be at the same place (two columns the same); and\n\n* No two particles can be in the same state (two rows the same).\n\n\n\n\n## Slater determinants as basis states\nAs a practical matter, however, Slater determinants beyond $N=4$ quickly become \nunwieldy. Thus we turn to the **occupation representation** or **second quantization** to simplify calculations. \n\nThe occupation representation or number representation, using fermion **creation** and **annihilation** \noperators, is compact and efficient. It is also abstract and, at first encounter, not easy to \ninternalize. It is inspired by other operator formalism, such as the ladder operators for \nthe harmonic oscillator or for angular momentum, but unlike those cases, the operators **do not have coordinate space representations**.\n\nInstead, one can think of fermion creation/annihilation operators as a game of symbols that \ncompactly reproduces what one would do, albeit clumsily, with full coordinate-space Slater \ndeterminants.\n\n\n\n## Quick repetition of the occupation representation\nWe start with a set of orthonormal single-particle states $\\{ \\phi_i(x) \\}$. \n(Note: this requirement, and others, can be relaxed, but leads to a \nmore involved formalism.) **Any** orthonormal set will do. \n\nTo each single-particle state $\\phi_i(x)$ we associate a creation operator \n$\\hat{a}^\\dagger_i$ and an annihilation operator $\\hat{a}_i$. \n\nWhen acting on the vacuum state $| 0 \\rangle$, the creation operator $\\hat{a}^\\dagger_i$ causes \na particle to occupy the single-particle state $\\phi_i(x)$:\n\n$$\n\\phi_i(x) \\rightarrow \\hat{a}^\\dagger_i |0 \\rangle\n$$\n\n## Quick repetition of the occupation representation\nBut with multiple creation operators we can occupy multiple states:\n\n$$\n\\phi_i(x) \\phi_j(x^\\prime) \\phi_k(x^{\\prime \\prime}) \n\\rightarrow \\hat{a}^\\dagger_i \\hat{a}^\\dagger_j \\hat{a}^\\dagger_k |0 \\rangle.\n$$\n\nNow we impose antisymmetry, by having the fermion operators satisfy **anticommutation relations**:\n\n$$\n\\hat{a}^\\dagger_i \\hat{a}^\\dagger_j + \\hat{a}^\\dagger_j \\hat{a}^\\dagger_i\n= [ \\hat{a}^\\dagger_i ,\\hat{a}^\\dagger_j ]_+ \n= \\{ \\hat{a}^\\dagger_i ,\\hat{a}^\\dagger_j \\} = 0\n$$\n\nso that\n\n$$\n\\hat{a}^\\dagger_i \\hat{a}^\\dagger_j = - \\hat{a}^\\dagger_j \\hat{a}^\\dagger_i\n$$\n\n## Quick repetition of the occupation representation\nBecause of this property, automatically $\\hat{a}^\\dagger_i \\hat{a}^\\dagger_i = 0$, \nenforcing the Pauli exclusion principle. Thus when writing a Slater determinant \nusing creation operators,\n\n$$\n\\hat{a}^\\dagger_i \\hat{a}^\\dagger_j \\hat{a}^\\dagger_k \\ldots |0 \\rangle\n$$\n\neach index $i,j,k, \\ldots$ must be unique.\n\nFor some relevant exercises with solutions see chapter 8 of [Lecture Notes in Physics, volume 936](http://www.springer.com/us/book/9783319533353).\n\n\n\n\n## Full Configuration Interaction Theory\nWe have defined the ansatz for the ground state as\n\n$$\n|\\Phi_0\\rangle = \\left(\\prod_{i\\le F}\\hat{a}_{i}^{\\dagger}\\right)|0\\rangle,\n$$\n\nwhere the index $i$ defines different single-particle states up to the Fermi level. We have assumed that we have $N$ fermions. \nA given one-particle-one-hole ($1p1h$) state can be written as\n\n$$\n|\\Phi_i^a\\rangle = \\hat{a}_{a}^{\\dagger}\\hat{a}_i|\\Phi_0\\rangle,\n$$\n\nwhile a $2p2h$ state can be written as\n\n$$\n|\\Phi_{ij}^{ab}\\rangle = \\hat{a}_{a}^{\\dagger}\\hat{a}_{b}^{\\dagger}\\hat{a}_j\\hat{a}_i|\\Phi_0\\rangle,\n$$\n\nand a general $NpNh$ state as\n\n$$\n|\\Phi_{ijk\\dots}^{abc\\dots}\\rangle = \\hat{a}_{a}^{\\dagger}\\hat{a}_{b}^{\\dagger}\\hat{a}_{c}^{\\dagger}\\dots\\hat{a}_k\\hat{a}_j\\hat{a}_i|\\Phi_0\\rangle.\n$$\n\n## Full Configuration Interaction Theory\nWe can then expand our exact state function for the ground state \nas\n\n$$\n|\\Psi_0\\rangle=C_0|\\Phi_0\\rangle+\\sum_{ai}C_i^a|\\Phi_i^a\\rangle+\\sum_{abij}C_{ij}^{ab}|\\Phi_{ij}^{ab}\\rangle+\\dots\n=(C_0+\\hat{C})|\\Phi_0\\rangle,\n$$\n\nwhere we have introduced the so-called correlation operator\n\n$$\n\\hat{C}=\\sum_{ai}C_i^a\\hat{a}_{a}^{\\dagger}\\hat{a}_i +\\sum_{abij}C_{ij}^{ab}\\hat{a}_{a}^{\\dagger}\\hat{a}_{b}^{\\dagger}\\hat{a}_j\\hat{a}_i+\\dots\n$$\n\nSince the normalization of $\\Psi_0$ is at our disposal and since $C_0$ is by hypothesis non-zero, we may arbitrarily set $C_0=1$ with \ncorresponding proportional changes in all other coefficients. Using this so-called intermediate normalization we have\n\n$$\n\\langle \\Psi_0 | \\Phi_0 \\rangle = \\langle \\Phi_0 | \\Phi_0 \\rangle = 1,\n$$\n\nresulting in\n\n$$\n|\\Psi_0\\rangle=(1+\\hat{C})|\\Phi_0\\rangle.\n$$\n\n## Full Configuration Interaction Theory\nWe rewrite\n\n$$\n|\\Psi_0\\rangle=C_0|\\Phi_0\\rangle+\\sum_{ai}C_i^a|\\Phi_i^a\\rangle+\\sum_{abij}C_{ij}^{ab}|\\Phi_{ij}^{ab}\\rangle+\\dots,\n$$\n\nin a more compact form as\n\n$$\n|\\Psi_0\\rangle=\\sum_{PH}C_H^P\\Phi_H^P=\\left(\\sum_{PH}C_H^P\\hat{A}_H^P\\right)|\\Phi_0\\rangle,\n$$\n\nwhere $H$ stands for $0,1,\\dots,n$ hole states and $P$ for $0,1,\\dots,n$ particle states. \nOur requirement of unit normalization gives\n\n$$\n\\langle \\Psi_0 | \\Phi_0 \\rangle = \\sum_{PH}|C_H^P|^2= 1,\n$$\n\nand the energy can be written as\n\n$$\nE= \\langle \\Psi_0 | \\hat{H} |\\Phi_0 \\rangle= \\sum_{PP'HH'}C_H^{*P}\\langle \\Phi_H^P | \\hat{H} |\\Phi_{H'}^{P'} \\rangle C_{H'}^{P'}.\n$$\n\n## Full Configuration Interaction Theory\nNormally\n\n$$\nE= \\langle \\Psi_0 | \\hat{H} |\\Phi_0 \\rangle= \\sum_{PP'HH'}C_H^{*P}\\langle \\Phi_H^P | \\hat{H} |\\Phi_{H'}^{P'} \\rangle C_{H'}^{P'},\n$$\n\nis solved by diagonalization setting up the Hamiltonian matrix defined by the basis of all possible Slater determinants. A diagonalization\n\nis equivalent to finding the variational minimum of\n\n$$\n\\langle \\Psi_0 | \\hat{H} |\\Phi_0 \\rangle-\\lambda \\langle \\Psi_0 |\\Phi_0 \\rangle,\n$$\n\nwhere $\\lambda$ is a variational multiplier to be identified with the energy of the system.\nThe minimization process results in\n\n2\n3\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\sum_{P'H'}\\left\\{\\delta[C_H^{*P}]\\langle \\Phi_H^P | \\hat{H} |\\Phi_{H'}^{P'} \\rangle C_{H'}^{P'}+\nC_H^{*P}\\langle \\Phi_H^P | \\hat{H} |\\Phi_{H'}^{P'} \\rangle \\delta[C_{H'}^{P'}]-\n\\lambda( \\delta[C_H^{*P}]C_{H'}^{P'}+C_H^{*P}\\delta[C_{H'}^{P'}]\\right\\} = 0.\n$$\n\nSince the coefficients $\\delta[C_H^{*P}]$ and $\\delta[C_{H'}^{P'}]$ are complex conjugates it is necessary and sufficient to require the quantities that multiply with $\\delta[C_H^{*P}]$ to vanish.\n\n\n\n\n\n## Full Configuration Interaction Theory\n\nThis leads to\n\n$$\n\\sum_{P'H'}\\langle \\Phi_H^P | \\hat{H} |\\Phi_{H'}^{P'} \\rangle C_{H'}^{P'}-\\lambda C_H^{P}=0,\n$$\n\nfor all sets of $P$ and $H$.\n\nIf we then multiply by the corresponding $C_H^{*P}$ and sum over $PH$ we obtain\n\n$$\n\\sum_{PP'HH'}C_H^{*P}\\langle \\Phi_H^P | \\hat{H} |\\Phi_{H'}^{P'} \\rangle C_{H'}^{P'}-\\lambda\\sum_{PH}|C_H^P|^2=0,\n$$\n\nleading to the identification $\\lambda = E$. This means that we have for all $PH$ sets\n\n\n
                                        \n\n$$\n\\begin{equation}\n\\sum_{P'H'}\\langle \\Phi_H^P | \\hat{H} -E|\\Phi_{H'}^{P'} \\rangle = 0. \\label{eq:fullci} \\tag{1}\n\\end{equation}\n$$\n\n## Full Configuration Interaction Theory\nAn alternative way to derive the last equation is to start from\n\n$$\n(\\hat{H} -E)|\\Psi_0\\rangle = (\\hat{H} -E)\\sum_{P'H'}C_{H'}^{P'}|\\Phi_{H'}^{P'} \\rangle=0,\n$$\n\nand if this equation is successively projected against all $\\Phi_H^P$ in the expansion of $\\Psi$, then the last equation on the previous slide\nresults. As stated previously, one solves this equation normally by diagonalization. If we are able to solve this equation exactly (that is\nnumerically exactly) in a large Hilbert space (it will be truncated in terms of the number of single-particle states included in the definition\nof Slater determinants), it can then serve as a benchmark for other many-body methods which approximate the correlation operator\n$\\hat{C}$.\n\n\n\n\n## Example of a Hamiltonian matrix\nSuppose, as an example, that we have six fermions below the Fermi level.\nThis means that we can make at most $6p-6h$ excitations. If we have an infinity of single particle states above the Fermi level, we will obviously have an infinity of say $2p-2h$ excitations. Each such way to configure the particles is called a **configuration**. We will always have to truncate in the basis of single-particle states.\nThis gives us a finite number of possible Slater determinants. Our Hamiltonian matrix would then look like (where each block can have a large dimensionalities):\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                        $0p-0h$ $1p-1h$ $2p-2h$ $3p-3h$ $4p-4h$ $5p-5h$ $6p-6h$
                                        $0p-0h$ x x x 0 0 0 0
                                        $1p-1h$ x x x x 0 0 0
                                        $2p-2h$ x x x x x 0 0
                                        $3p-3h$ 0 x x x x x 0
                                        $4p-4h$ 0 0 x x x x x
                                        $5p-5h$ 0 0 0 x x x x
                                        $6p-6h$ 0 0 0 0 x x x
                                        \nwith a two-body force. Why are there non-zero blocks of elements?\n\n\n\n## Example of a Hamiltonian matrix with a Hartree-Fock basis\nIf we use a Hartree-Fock basis, this corresponds to a particular unitary transformation where matrix elements of the type $\\langle 0p-0h \\vert \\hat{H} \\vert 1p-1h\\rangle =\\langle \\Phi_0 | \\hat{H}|\\Phi_{i}^{a}\\rangle=0$ and our Hamiltonian matrix becomes \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                        $0p-0h$ $1p-1h$ $2p-2h$ $3p-3h$ $4p-4h$ $5p-5h$ $6p-6h$
                                        $0p-0h$ $\\tilde{x}$ 0 $\\tilde{x}$ 0 0 0 0
                                        $1p-1h$ 0 $\\tilde{x}$ $\\tilde{x}$ $\\tilde{x}$ 0 0 0
                                        $2p-2h$ $\\tilde{x}$ $\\tilde{x}$ $\\tilde{x}$ $\\tilde{x}$ $\\tilde{x}$ 0 0
                                        $3p-3h$ 0 $\\tilde{x}$ $\\tilde{x}$ $\\tilde{x}$ $\\tilde{x}$ $\\tilde{x}$ 0
                                        $4p-4h$ 0 0 $\\tilde{x}$ $\\tilde{x}$ $\\tilde{x}$ $\\tilde{x}$ $\\tilde{x}$
                                        $5p-5h$ 0 0 0 $\\tilde{x}$ $\\tilde{x}$ $\\tilde{x}$ $\\tilde{x}$
                                        $6p-6h$ 0 0 0 0 $\\tilde{x}$ $\\tilde{x}$ $\\tilde{x}$
                                        \n\n\n\n\n## Shell-model jargon\nIf we do not make any truncations in the possible sets of Slater determinants (many-body states) we can make by distributing $A$ nucleons among $n$ single-particle states, we call such a calculation for **Full configuration interaction theory**\n\nIf we make truncations, we have different possibilities\n\n* The standard nuclear shell-model. Here we define an effective Hilbert space with respect to a given core. The calculations are normally then performed for all many-body states that can be constructed from the effective Hilbert spaces. This approach requires a properly defined effective Hamiltonian\n\n* We can truncate in the number of excitations. For example, we can limit the possible Slater determinants to only $1p-1h$ and $2p-2h$ excitations. This is called a configuration interaction calculation at the level of singles and doubles excitations, or just CISD. \n\n* We can limit the number of excitations in terms of the excitation energies. If we do not define a core, this defines normally what is called the no-core shell-model approach. \n\nWhat happens if we have a three-body interaction and a Hartree-Fock basis?\n\n\n\n## FCI and the exponential growth\nFull configuration interaction theory calculations provide in principle, if we can diagonalize numerically, all states of interest. The dimensionality of the problem explodes however quickly.\n\nThe total number of Slater determinants which can be built with say $N$ neutrons distributed among $n$ single particle states is\n\n$$\n\\left (\\begin{array}{c} n \\\\ N\\end{array} \\right) =\\frac{n!}{(n-N)!N!}.\n$$\n\nFor a model space which comprises the first for major shells only $0s$, $0p$, $1s0d$ and $1p0f$ we have $40$ single particle states for neutrons and protons. For the eight neutrons of oxygen-16 we would then have\n\n$$\n\\left (\\begin{array}{c} 40 \\\\ 8\\end{array} \\right) =\\frac{40!}{(32)!8!}\\sim 10^{9},\n$$\n\nand multiplying this with the number of proton Slater determinants we end up with approximately with a dimensionality $d$ of $d\\sim 10^{18}$.\n\n\n\n\n## Exponential wall\nThis number can be reduced if we look at specific symmetries only. However, the dimensionality explodes quickly!\n\n* For Hamiltonian matrices of dimensionalities which are smaller than $d\\sim 10^5$, we would use so-called direct methods for diagonalizing the Hamiltonian matrix\n\n* For larger dimensionalities iterative eigenvalue solvers like Lanczos' method are used. The most efficient codes at present can handle matrices of $d\\sim 10^{10}$.\n\n\n\n\n\n## A non-practical way of solving the eigenvalue problem\nTo see this, we look at the contributions arising from\n\n$$\n\\langle \\Phi_H^P | = \\langle \\Phi_0|\n$$\n\nin Eq. ([eq:fullci](#eq:fullci)), that is we multiply with $\\langle \\Phi_0 |$\nfrom the left in\n\n$$\n(\\hat{H} -E)\\sum_{P'H'}C_{H'}^{P'}|\\Phi_{H'}^{P'} \\rangle=0.\n$$\n\nIf we assume that we have a two-body operator at most, Slater's rule gives then an equation for the \ncorrelation energy in terms of $C_i^a$ and $C_{ij}^{ab}$ only. We get then\n\n$$\n\\langle \\Phi_0 | \\hat{H} -E| \\Phi_0\\rangle + \\sum_{ai}\\langle \\Phi_0 | \\hat{H} -E|\\Phi_{i}^{a} \\rangle C_{i}^{a}+\n\\sum_{abij}\\langle \\Phi_0 | \\hat{H} -E|\\Phi_{ij}^{ab} \\rangle C_{ij}^{ab}=0,\n$$\n\nor\n\n$$\nE-E_0 =\\Delta E=\\sum_{ai}\\langle \\Phi_0 | \\hat{H}|\\Phi_{i}^{a} \\rangle C_{i}^{a}+\n\\sum_{abij}\\langle \\Phi_0 | \\hat{H}|\\Phi_{ij}^{ab} \\rangle C_{ij}^{ab},\n$$\n\nwhere the energy $E_0$ is the reference energy and $\\Delta E$ defines the so-called correlation energy.\nThe single-particle basis functions could be the results of a Hartree-Fock calculation or just the eigenstates of the non-interacting part of the Hamiltonian.\n\n\n\n\n\n## A non-practical way of solving the eigenvalue problem\nTo see this, we look at the contributions arising from\n\n$$\n\\langle \\Phi_H^P | = \\langle \\Phi_0|\n$$\n\nin Eq. ([eq:fullci](#eq:fullci)), that is we multiply with $\\langle \\Phi_0 |$\nfrom the left in\n\n$$\n(\\hat{H} -E)\\sum_{P'H'}C_{H'}^{P'}|\\Phi_{H'}^{P'} \\rangle=0.\n$$\n\n\n## A non-practical way of solving the eigenvalue problem\nIf we assume that we have a two-body operator at most, Slater's rule gives then an equation for the \ncorrelation energy in terms of $C_i^a$ and $C_{ij}^{ab}$ only. We get then\n\n$$\n\\langle \\Phi_0 | \\hat{H} -E| \\Phi_0\\rangle + \\sum_{ai}\\langle \\Phi_0 | \\hat{H} -E|\\Phi_{i}^{a} \\rangle C_{i}^{a}+\n\\sum_{abij}\\langle \\Phi_0 | \\hat{H} -E|\\Phi_{ij}^{ab} \\rangle C_{ij}^{ab}=0,\n$$\n\nor\n\n$$\nE-E_0 =\\Delta E=\\sum_{ai}\\langle \\Phi_0 | \\hat{H}|\\Phi_{i}^{a} \\rangle C_{i}^{a}+\n\\sum_{abij}\\langle \\Phi_0 | \\hat{H}|\\Phi_{ij}^{ab} \\rangle C_{ij}^{ab},\n$$\n\nwhere the energy $E_0$ is the reference energy and $\\Delta E$ defines the so-called correlation energy.\nThe single-particle basis functions could be the results of a Hartree-Fock calculation or just the eigenstates of the non-interacting part of the Hamiltonian.\n\n\n\n\n\n## Rewriting the FCI equation\nIn our notes on Hartree-Fock calculations, \nwe have already computed the matrix $\\langle \\Phi_0 | \\hat{H}|\\Phi_{i}^{a}\\rangle $ and $\\langle \\Phi_0 | \\hat{H}|\\Phi_{ij}^{ab}\\rangle$. If we are using a Hartree-Fock basis, then the matrix elements\n$\\langle \\Phi_0 | \\hat{H}|\\Phi_{i}^{a}\\rangle=0$ and we are left with a *correlation energy* given by\n\n$$\nE-E_0 =\\Delta E^{HF}=\\sum_{abij}\\langle \\Phi_0 | \\hat{H}|\\Phi_{ij}^{ab} \\rangle C_{ij}^{ab}.\n$$\n\n## Rewriting the FCI equation\nInserting the various matrix elements we can rewrite the previous equation as\n\n$$\n\\Delta E=\\sum_{ai}\\langle i| \\hat{f}|a \\rangle C_{i}^{a}+\n\\sum_{abij}\\langle ij | \\hat{v}| ab \\rangle C_{ij}^{ab}.\n$$\n\nThis equation determines the correlation energy but not the coefficients $C$.\n\n\n\n## Rewriting the FCI equation, does not stop here\nWe need more equations. Our next step is to set up\n\n$$\n\\langle \\Phi_i^a | \\hat{H} -E| \\Phi_0\\rangle + \\sum_{bj}\\langle \\Phi_i^a | \\hat{H} -E|\\Phi_{j}^{b} \\rangle C_{j}^{b}+\n\\sum_{bcjk}\\langle \\Phi_i^a | \\hat{H} -E|\\Phi_{jk}^{bc} \\rangle C_{jk}^{bc}+\n\\sum_{bcdjkl}\\langle \\Phi_i^a | \\hat{H} -E|\\Phi_{jkl}^{bcd} \\rangle C_{jkl}^{bcd}=0,\n$$\n\nas this equation will allow us to find an expression for the coefficents $C_i^a$ since we can rewrite this equation as\n\n$$\n\\langle i | \\hat{f}| a\\rangle +\\langle \\Phi_i^a | \\hat{H}|\\Phi_{i}^{a} \\rangle C_{i}^{a}+ \\sum_{bj\\ne ai}\\langle \\Phi_i^a | \\hat{H}|\\Phi_{j}^{b} \\rangle C_{j}^{b}+\n\\sum_{bcjk}\\langle \\Phi_i^a | \\hat{H}|\\Phi_{jk}^{bc} \\rangle C_{jk}^{bc}+\n\\sum_{bcdjkl}\\langle \\Phi_i^a | \\hat{H}|\\Phi_{jkl}^{bcd} \\rangle C_{jkl}^{bcd}=EC_i^a.\n$$\n\n## Rewriting the FCI equation, please stop here\nWe see that on the right-hand side we have the energy $E$. This leads to a non-linear equation in the unknown coefficients. \nThese equations are normally solved iteratively ( that is we can start with a guess for the coefficients $C_i^a$). A common choice is to use perturbation theory for the first guess, setting thereby\n\n$$\nC_{i}^{a}=\\frac{\\langle i | \\hat{f}| a\\rangle}{\\epsilon_i-\\epsilon_a}.\n$$\n\n## Rewriting the FCI equation, more to add\nThe observant reader will however see that we need an equation for $C_{jk}^{bc}$ and $C_{jkl}^{bcd}$ as well.\nTo find equations for these coefficients we need then to continue our multiplications from the left with the various\n$\\Phi_{H}^P$ terms. \n\n\nFor $C_{jk}^{bc}$ we need then\n\n4\n4\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\sum_{cdkl}\\langle \\Phi_{ij}^{ab} | \\hat{H} -E|\\Phi_{kl}^{cd} \\rangle C_{kl}^{cd}+\\sum_{cdeklm}\\langle \\Phi_{ij}^{ab} | \\hat{H} -E|\\Phi_{klm}^{cde} \\rangle C_{klm}^{cde}+\\sum_{cdefklmn}\\langle \\Phi_{ij}^{ab} | \\hat{H} -E|\\Phi_{klmn}^{cdef} \\rangle C_{klmn}^{cdef}=0,\n$$\n\nand we can isolate the coefficients $C_{kl}^{cd}$ in a similar way as we did for the coefficients $C_{i}^{a}$.\n\n\n\n\n\n## Rewriting the FCI equation, more to add\nA standard choice for the first iteration is to set\n\n$$\nC_{ij}^{ab} =\\frac{\\langle ij \\vert \\hat{v} \\vert ab \\rangle}{\\epsilon_i+\\epsilon_j-\\epsilon_a-\\epsilon_b}.\n$$\n\nAt the end we can rewrite our solution of the Schroedinger equation in terms of $n$ coupled equations for the coefficients $C_H^P$.\nThis is a very cumbersome way of solving the equation. However, by using this iterative scheme we can illustrate how we can compute the\nvarious terms in the wave operator or correlation operator $\\hat{C}$. We will later identify the calculation of the various terms $C_H^P$\nas parts of different many-body approximations to full CI. In particular, we can relate this non-linear scheme with Coupled Cluster theory and\nmany-body perturbation theory.\n\n\n\n\n## Summarizing FCI and bringing in approximative methods\n\nIf we can diagonalize large matrices, FCI is the method of choice since:\n* It gives all eigenvalues, ground state and excited states\n\n* The eigenvectors are obtained directly from the coefficients $C_H^P$ which result from the diagonalization\n\n* We can compute easily expectation values of other operators, as well as transition probabilities\n\n* Correlations are easy to understand in terms of contributions to a given operator beyond the Hartree-Fock contribution. This is the standard approach in many-body theory.\n\n\n\n## Definition of the correlation energy\nThe correlation energy is defined as, with a two-body Hamiltonian,\n\n$$\n\\Delta E=\\sum_{ai}\\langle i| \\hat{f}|a \\rangle C_{i}^{a}+\n\\sum_{abij}\\langle ij | \\hat{v}| ab \\rangle C_{ij}^{ab}.\n$$\n\nThe coefficients $C$ result from the solution of the eigenvalue problem. \nThe energy of say the ground state is then\n\n$$\nE=E_{ref}+\\Delta E,\n$$\n\nwhere the so-called reference energy is the energy we obtain from a Hartree-Fock calculation, that is\n\n$$\nE_{ref}=\\langle \\Phi_0 \\vert \\hat{H} \\vert \\Phi_0 \\rangle.\n$$\n\n## FCI equation and the coefficients\n\nHowever, as we have seen, even for a small case like the four first major shells and a nucleus like oxygen-16, the dimensionality becomes quickly intractable. If we wish to include single-particle states that reflect weakly bound systems, we need a much larger single-particle basis. We need thus approximative methods that sum specific correlations to infinite order. \n\nPopular methods are\n* [Many-body perturbation theory (in essence a Taylor expansion)](http://www.sciencedirect.com/science/article/pii/0370157395000126)\n\n* [Coupled cluster theory (coupled non-linear equations)](http://iopscience.iop.org/article/10.1088/0034-4885/77/9/096302/meta)\n\n* Green's function approaches (matrix inversion)\n\n* [Similarity group transformation methods (coupled ordinary differential equations)](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.106.222502)\n\nAll these methods start normally with a Hartree-Fock basis as the calculational basis.\n\n\n\n## Important ingredients to have in codes\n\n* Be able to validate and verify the algorithms. \n\n* Include concepts like unit testing. Gives the possibility to test and validate several or all parts of the code.\n\n* Validation and verification are then included *naturally* and one can develop a better attitude to what is meant with an ethically sound scientific approach.\n\n\n\n\n\n## A structured approach to solving problems\nIn the steps that lead to the development of clean code you should think of \n 1. How to structure a code in terms of functions (use IDEs or advanced text editors like sublime or atom)\n\n 2. How to make a module\n\n 3. How to read input data flexibly from the command line or files\n\n 4. How to create graphical/web user interfaces\n\n 5. How to write unit tests \n\n 6. How to refactor code in terms of classes (instead of functions only)\n\n 7. How to conduct and automate large-scale numerical experiments\n\n 8. How to write scientific reports in various formats (LaTeX, HTML, doconce)\n\n\n\n\n## Additional benefits\nMany of the above aspetcs will save you a lot of time when you incrementally extend software over time from simpler to more complicated problems. In particular, you will benefit from many good habits:\n1. New code is added in a modular fashion to a library (modules)\n\n2. Programs are run through convenient user interfaces\n\n3. It takes one quick command to let all your code undergo heavy testing \n\n4. Tedious manual work with running programs is automated,\n\n5. Your scientific investigations are reproducible, scientific reports with top quality typesetting are produced both for paper and electronic devices. Use version control software like [git](https://git-scm.com/) and repositories like [github](https://github.com/)\n\n\n\n\n\n## Unit Testing\nUnit Testing is the practice of testing the smallest testable parts,\ncalled units, of an application individually and independently to\ndetermine if they behave exactly as expected. \n\nUnit tests (short code\nfragments) are usually written such that they can be preformed at any\ntime during the development to continually verify the behavior of the\ncode. \n\nIn this way, possible bugs will be identified early in the\ndevelopment cycle, making the debugging at later stages much\neasier.\n\n\n\n## Unit Testing, benefits\nThere are many benefits associated with Unit Testing, such as\n * It increases confidence in changing and maintaining code. Big changes can be made to the code quickly, since the tests will ensure that everything still is working properly.\n\n * Since the code needs to be modular to make Unit Testing possible, the code will be easier to reuse. This improves the code design.\n\n * Debugging is easier, since when a test fails, only the latest changes need to be debugged.\n\n * Different parts of a project can be tested without the need to wait for the other parts to be available.\n\n\n * A unit test can serve as a documentation on the functionality of a unit of the code.\n\n\n\n\n## Simple example of unit test\nLook up the guide on how to install unit tests for c++ at course webpage. This is the version with classes.\n\n #include \n \n class MyMultiplyClass{\n public:\n double multiply(double x, double y) {\n return x * y;\n }\n };\n \n TEST(MyMath) {\n MyMultiplyClass my;\n CHECK_EQUAL(56, my.multiply(7,8));\n }\n \n int main()\n {\n return UnitTest::RunAllTests();\n }\n\n\n## Simple example of unit test\nAnd without classes\n\n #include \n \n \n double multiply(double x, double y) {\n return x * y;\n }\n \n TEST(MyMath) {\n CHECK_EQUAL(56, multiply(7,8));\n }\n \n int main()\n {\n return UnitTest::RunAllTests();\n } \n\n\nFor Fortran users, the link at contains a similar\nsoftware for unit testing. For Python go to .\n\n\n\n\n\n\n\n## [Unit tests](https://github.com/philsquared/Catch/blob/master/docs/tutorial.md)\nThere are many types of **unit test** libraries. One which is very popular with C++ programmers is [Catch](https://github.com/philsquared/Catch/blob/master/docs/tutorial.md)\n\nCatch is header only. All you need to do is drop the file(s) somewhere reachable from your project - either in some central location you can set your header search path to find, or directly into your project tree itself! \n\nThis is a particularly good option for other Open-Source projects that want to use Catch for their test suite.\n\n\n\n## Examples\n\nComputing factorials\n\n inline unsigned int Factorial( unsigned int number ) {\n return number > 1 ? Factorial(number-1)*number : 1;\n }\n\n\n## Factorial Example\n\nSimple test where we put everything in a single file\n\n #define CATCH_CONFIG_MAIN // This tells Catch to provide a main()\n #include \"catch.hpp\"\n inline unsigned int Factorial( unsigned int number ) {\n return number > 1 ? Factorial(number-1)*number : 1;\n }\n \n TEST_CASE( \"Factorials are computed\", \"[factorial]\" ) {\n REQUIRE( Factorial(0) == 1 );\n REQUIRE( Factorial(1) == 1 );\n REQUIRE( Factorial(2) == 2 );\n REQUIRE( Factorial(3) == 6 );\n REQUIRE( Factorial(10) == 3628800 );\n }\n \n\n\nThis will compile to a complete executable which responds to command line arguments. If you just run it with no arguments it will execute all test cases (in this case there is just one), report any failures, report a summary of how many tests passed and failed and return the number of failed tests.\n\n## What did we do (1)?\nAll we did was\n\n #define \n\n\none identifier and\n\n #include \n\n\none header and we got everything - even an implementation of main() that will respond to command line arguments. \nOnce you have more than one file with unit tests in you'll just need to\n\n #include \"catch.hpp\" \n\n\nand go. Usually it's a good idea to have a dedicated implementation file that just has\n\n #define CATCH_CONFIG_MAIN \n #include \"catch.hpp\". \n\n\nYou can also provide your own implementation of main and drive Catch yourself.\n\n\n## What did we do (2)?\nWe introduce test cases with the\n\n TEST_CASE \n\n\nmacro.\n\nThe test name must be unique. You can run sets of tests by specifying a wildcarded test name or a tag expression. \nAll we did was **define** one identifier and **include** one header and we got everything.\n\nWe write our individual test assertions using the\n\n REQUIRE \n\n\nmacro.\n\n\n## Unit test summary and testing approach\nThree levels of tests\n1. Microscopic level: testing small parts of code, use often unit test libraries\n\n2. Mesoscopic level: testing the integration of various parts of your code\n\n3. Macroscopic level: testing that the final result is ok\n\n \n\n\n\n## Coding Recommendations\nWriting clean and clear code is an art and reflects \nyour understanding of \n\n1. derivation, verification, and implementation of algorithms\n\n2. what can go wrong with algorithms\n\n3. overview of important, known algorithms\n\n4. how algorithms are used to solve mathematical problems\n\n5. reproducible science and ethics\n\n6. algorithmic thinking for gaining deeper insights about scientific problems\n\nComputing is understanding and your understanding is reflected in your abilities to\nwrite clear and clean code.\n\n\n## Summary and recommendations\nSome simple hints and tips in order to write clean and clear code\n1. Spell out the algorithm and have a top-down approach to the flow of data\n\n2. Start with coding as close as possible to eventual mathematical expressions\n\n3. Use meaningful names for variables\n\n4. Split tasks in simple functions and modules/classes\n\n5. Functions should return as few as possible variables\n\n6. Use unit tests and make sure your codes are producing the correct results\n\n7. Where possible use symbolic coding to autogenerate code and check results\n\n8. Make a proper timing of your algorithms\n\n9. Use version control and make your science reproducible\n\n10. Use IDEs or smart editors with debugging and analysis tools.\n\n11. Automatize your computations interfacing high-level and compiled languages like C++ and Fortran.\n\n12. .....\n\n## Building a many-body basis\nHere we will discuss how we can set up a single-particle basis which we can use in the various parts of our projects, from the simple pairing model to infinite nuclear matter. We will use here the simple pairing model to illustrate in particular how to set up a single-particle basis. We will also use this do discuss standard FCI approaches like:\n 1. Standard shell-model basis in one or two major shells\n\n 2. Full CI in a given basis and no truncations\n\n 3. CISD and CISDT approximations\n\n 4. No-core shell model and truncation in excitation energy\n\n\n\n## Building a many-body basis\nAn important step in an FCI code is to construct the many-body basis. \n\nWhile the formalism is independent of the choice of basis, the **effectiveness** of a calculation \nwill certainly be basis dependent. \n\nFurthermore there are common conventions useful to know.\n\nFirst, the single-particle basis has angular momentum as a good quantum number. You can \nimagine the single-particle wavefunctions being generated by a one-body Hamiltonian, \nfor example a harmonic oscillator. Modifications include harmonic oscillator plus \nspin-orbit splitting, or self-consistent mean-field potentials, or the Woods-Saxon potential which mocks \nup the self-consistent mean-field. \nFor nuclei, the harmonic oscillator, modified by spin-orbit splitting, provides a useful language \nfor describing single-particle states.\n\n\n\n## Building a many-body basis\nEach single-particle state is labeled by the following quantum numbers: \n\n* Orbital angular momentum $l$\n\n* Intrinsic spin $s$ = 1/2 for protons and neutrons\n\n* Angular momentum $j = l \\pm 1/2$\n\n* $z$-component $j_z$ (or $m$)\n\n* Some labeling of the radial wavefunction, typically $n$ the number of nodes in the radial wavefunction, but in the case of harmonic oscillator one can also use the principal quantum number $N$, where the harmonic oscillator energy is $(N+3/2)\\hbar \\omega$. \n\nIn this format one labels states by $n(l)_j$, with $(l)$ replaced by a letter:\n$s$ for $l=0$, $p$ for $l=1$, $d$ for $l=2$, $f$ for $l=3$, and thenceforth alphabetical.\n\n\n\n## Building a many-body basis\n In practice the single-particle space has to be severely truncated. This truncation is \ntypically based upon the single-particle energies, which is the effective energy \nfrom a mean-field potential. \n\nSometimes we freeze the core and only consider a valence space. For example, one \nmay assume a frozen ${}^{4}\\mbox{He}$ core, with two protons and two neutrons in the $0s_{1/2}$ \nshell, and then only allow active particles in the $0p_{1/2}$ and $0p_{3/2}$ orbits. \n\n\nAnother example is a frozen ${}^{16}\\mbox{O}$ core, with eight protons and eight neutrons filling the \n$0s_{1/2}$, $0p_{1/2}$ and $0p_{3/2}$ orbits, with valence particles in the \n$0d_{5/2}, 1s_{1/2}$ and $0d_{3/2}$ orbits.\n\n\nSometimes we refer to nuclei by the valence space where their last nucleons go. \nSo, for example, we call ${}^{12}\\mbox{C}$ a $p$-shell nucleus, while ${}^{26}\\mbox{Al}$ is an \n$sd$-shell nucleus and ${}^{56}\\mbox{Fe}$ is a $pf$-shell nucleus.\n\n\n\n\n## Building a many-body basis\nThere are different kinds of truncations.\n\n* For example, one can start with `filled' orbits (almost always the lowest), and then allow one, two, three... particles excited out of those filled orbits. These are called 1p-1h, 2p-2h, 3p-3h excitations. \n\n* Alternately, one can state a maximal orbit and allow all possible configurations with particles occupying states up to that maximum. This is called *full configuration*.\n\n* Finally, for particular use in nuclear physics, there is the *energy* truncation, also called the $N\\hbar\\Omega$ or $N_{max}$ truncation.\n\n\n\n## Building a many-body basis\nHere one works in a harmonic oscillator basis, with each major oscillator shell assigned a principal quantum number $N=0,1,2,3,...$. \nThe $N\\hbar\\Omega$ or $N_{max}$ truncation: Any configuration is given an noninteracting energy, which is the sum \nof the single-particle harmonic oscillator energies. (Thus this ignores \nspin-orbit splitting.)\n\nExcited state are labeled relative to the lowest configuration by the \nnumber of harmonic oscillator quanta.\n\nThis truncation is useful because if one includes *all* configuration up to \nsome $N_{max}$, and has a translationally invariant interaction, then the intrinsic \nmotion and the center-of-mass motion factor. In other words, we can know exactly \nthe center-of-mass wavefunction. \n\nIn almost all cases, the many-body Hamiltonian is rotationally invariant. This means \nit commutes with the operators $\\hat{J}^2, \\hat{J}_z$ and so eigenstates will have \ngood $J,M$. Furthermore, the eigenenergies do not depend upon the orientation $M$. \n\n\nTherefore we can choose to construct a many-body basis which has fixed $M$; this is \ncalled an $M$-scheme basis. \n\n\nAlternately, one can construct a many-body basis which has fixed $J$, or a $J$-scheme \nbasis.\n\n\n\n## Building a many-body basis\nThe Hamiltonian matrix will have smaller dimensions (a factor of 10 or more) in the $J$-scheme than in the $M$-scheme. \nOn the other hand, as we'll show in the next slide, the $M$-scheme is very easy to \nconstruct with Slater determinants, while the $J$-scheme basis states, and thus the \nmatrix elements, are more complicated, almost always being linear combinations of \n$M$-scheme states. $J$-scheme bases are important and useful, but we'll focus on the \nsimpler $M$-scheme.\n\nThe quantum number $m$ is additive (because the underlying group is Abelian): \nif a Slater determinant $\\hat{a}_i^\\dagger \\hat{a}^\\dagger_j \\hat{a}^\\dagger_k \\ldots | 0 \\rangle$ \nis built from single-particle states all with good $m$, then the total\n\n$$\nM = m_i + m_j + m_k + \\ldots\n$$\n\nThis is *not* true of $J$, because the angular momentum group SU(2) is not Abelian.\n\n\n\n\n## Building a many-body basis\n\nThe upshot is that \n* It is easy to construct a Slater determinant with good total $M$;\n\n* It is trivial to calculate $M$ for each Slater determinant;\n\n* So it is easy to construct an $M$-scheme basis with fixed total $M$.\n\nNote that the individual $M$-scheme basis states will *not*, in general, \nhave good total $J$. \nBecause the Hamiltonian is rotationally invariant, however, the eigenstates will \nhave good $J$. (The situation is muddied when one has states of different $J$ that are \nnonetheless degenerate.)\n\n\n\n\n\n\n## Building a many-body basis\nExample: two $j=1/2$ orbits\n\n\n\n\n\n\n\n\n\n\n\n
                                        Index $n$ $l$ $j$ $m_j$
                                        1 0 0 1/2 -1/2
                                        2 0 0 1/2 1/2
                                        3 1 0 1/2 -1/2
                                        4 1 0 1/2 1/2
                                        \nNote that the order is arbitrary.\n\n\n\n## Building a many-body basis\nThere are $\\left ( \\begin{array}{c} 4 \\\\ 2 \\end{array} \\right) = 6$ two-particle states, \nwhich we list with the total $M$:\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                        Occupied $M$
                                        1,2 0
                                        1,3 -1
                                        1,4 0
                                        2,3 0
                                        2,4 1
                                        3,4 0
                                        \nThere are 4 states with $M= 0$, \nand 1 each with $M = \\pm 1$.\n\n\n\n\n## Building a many-body basis\nAs another example, consider using only single particle states from the $0d_{5/2}$ space. \nThey have the following quantum numbers\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                        Index $n$ $l$ $j$ $m_j$
                                        1 0 2 5/2 -5/2
                                        2 0 2 5/2 -3/2
                                        3 0 2 5/2 -1/2
                                        4 0 2 5/2 1/2
                                        5 0 2 5/2 3/2
                                        6 0 2 5/2 5/2
                                        \n\n\n\n## Building a many-body basis\nThere are $\\left ( \\begin{array}{c} 6 \\\\ 2 \\end{array} \\right) = 15$ two-particle states, \nwhich we list with the total $M$:\n\n\n\n\n\n\n\n\n\n\n\n\n
                                        Occupied $M$ Occupied $M$ Occupied $M$
                                        1,2 -4 2,3 -2 3,5 1
                                        1,3 -3 2,4 -1 3,6 2
                                        1,4 -2 2,5 0 4,5 2
                                        1,5 -1 2,6 1 4,6 3
                                        1,6 0 3,4 0 5,6 4
                                        \nThere are 3 states with $M= 0$, 2 with $M = 1$, and so on.\n\n\n\n\n\n\n\n\n\n\n\n\n## Shell-model project\n\nThe first step is to construct the $M$-scheme basis of Slater determinants.\nHere $M$-scheme means the total $J_z$ of the many-body states is fixed.\n\nThe steps could be:\n\n* Read in a user-supplied file of single-particle states (examples can be given) or just code these internally;\n\n* Ask for the total $M$ of the system and the number of particles $N$;\n\n* Construct all the $N$-particle states with given $M$. You will validate the code by comparing both the number of states and specific states.\n\n\n\n## Shell-model project\nThe format of a possible input file could be\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                        Index $n$ $l$ $2j$ $2m_j$
                                        1 1 0 1 -1
                                        2 1 0 1 1
                                        3 0 2 3 -3
                                        4 0 2 3 -1
                                        5 0 2 3 1
                                        6 0 2 3 3
                                        7 0 2 5 -5
                                        8 0 2 5 -3
                                        9 0 2 5 -1
                                        10 0 2 5 1
                                        11 0 2 5 3
                                        12 0 2 5 5
                                        \nThis represents the $1s_{1/2}0d_{3/2}0d_{5/2}$ valence space, or just the $sd$-space. There are \ntwelve single-particle states, labeled by an overall index, and which have associated quantum \nnumbers the number of radial nodes, the orbital angular momentum $l$, and the \nangular momentum $j$ and third component $j_z$. To keep everything as integers, we could store $2 \\times j$ and \n$2 \\times j_z$.\n\n\n\n## Shell-model project\nTo read in the single-particle states you need to:\n* Open the file \n\n * Read the number of single-particle states (in the above example, 12); allocate memory; all you need is a single array storing $2\\times j_z$ for each state, labeled by the index.\n\n\n* Read in the quantum numbers and store $2 \\times j_z$ (and anything else you happen to want).\n\n\n\n## Shell-model project\n\nThe next step is to read in the number of particles $N$ and the fixed total $M$ (or, actually, $2 \\times M$). \nFor this project we assume only a single species of particles, say neutrons, although this can be \nrelaxed. **Note**: Although it is often a good idea to try to write a more general code, given the \nshort time alloted we would suggest you keep your ambition in check, at least in the initial phases of the \nproject. \n\n\nYou should probably write an error trap to make sure $N$ and $M$ are congruent; if $N$ is even, then \n$2 \\times M$ should be even, and if $N$ is odd then $2\\times M$ should be odd.\n\n\n\n## Shell-model project\nThe final step is to generate the set of $N$-particle Slater determinants with fixed $M$. \nThe Slater determinants will be stored in occupation representation. Although in many codes\nthis representation is done compactly in bit notation with ones and zeros, but for \ngreater transparency and simplicity we will list the occupied single particle states.\n\n Hence we can \nstore the Slater determinant basis states as $sd(i,j)$, that is an \narray of dimension $N_{SD}$, the number of Slater determinants, by $N$, the number of occupied \nstate. So if for the 7th Slater determinant the 2nd, 3rd, and 9th single-particle states are occupied, \nthen $sd(7,1) = 2$, $sd(7,2) = 3$, and $sd(7,3) = 9$.\n\n\n\n## Shell-model project\n\nWe can construct an occupation representation of Slater determinants by the *odometer*\nmethod. Consider $N_{sp} = 12$ and $N=4$. \nStart with the first 4 states occupied, that is:\n\n* $sd(1,:)= 1,2,3,4$ (also written as $|1,2,3,4 \\rangle$)\n\nNow increase the last occupancy recursively:\n* $sd(2,:)= 1,2,3,5$\n\n* $sd(3,:)= 1,2,3,6$\n\n* $sd(4,:)= 1,2,3,7$\n\n* $\\ldots$\n\n* $sd(9,:)= 1,2,3,12$\n\nThen start over with \n* $sd(10,:)= 1,2,4,5$\n\nand again increase the rightmost digit\n\n* $sd(11,:)= 1,2,4,6$\n\n* $sd(12,:)= 1,2,4,7$\n\n* $\\ldots$\n\n* $sd(17,:)= 1,2,4,12$\n\n\n\n## Shell-model project\nWhen we restrict ourselves to an $M$-scheme basis, we could choose two paths. \nThe first is simplest (and simplest is often best, at \nleast in the first draft of a code): generate all possible Slater determinants, \nand then extract from this initial list a list of those Slater determinants with a given \n$M$. (You will need to write a short function or routine that computes $M$ for any \ngiven occupation.) \n\n\nAlternately, and not too difficult, is to run the odometer routine twice: each time, as \nas a Slater determinant is calculated, compute $M$, but do not store the Slater determinants \nexcept the current one. You can then count up the number of Slater determinants with a \nchosen $M$. Then allocated storage for the Slater determinants, and run the odometer \nalgorithm again, this time storing Slater determinants with the desired $M$ (this can be \ndone with a simple logical flag).\n\n\n\n\n## Shell-model project\n\n*Some example solutions*: Let's begin with a simple case, the $0d_{5/2}$ space containing six single-particle states\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                        Index $n$ $l$ $j$ $m_j$
                                        1 0 2 5/2 -5/2
                                        2 0 2 5/2 -3/2
                                        3 0 2 5/2 -1/2
                                        4 0 2 5/2 1/2
                                        5 0 2 5/2 3/2
                                        6 0 2 5/2 5/2
                                        \nFor two particles, there are a total of 15 states, which we list here with the total $M$:\n* $\\vert 1,2 \\rangle$, $M= -4$, $\\vert 1,3 \\rangle$, $M= -3$\n\n* $\\vert 1,4 \\rangle$, $M= -2$, $\\vert 1,5 \\rangle$, $M= -1$\n\n* $\\vert 1,5 \\rangle$, $M= 0$, $vert 2,3 \\rangle$, $M= -2$\n\n* $\\vert 2,4 \\rangle$, $M= -1$, $\\vert 2,5 \\rangle$, $M= 0$\n\n* $\\vert 2,6 \\rangle$, $M= 1$, $\\vert 3,4 \\rangle$, $M= 0$\n\n* $\\vert 3,5 \\rangle$, $M= 1$, $\\vert 3,6 \\rangle$, $M= 2$\n\n* $\\vert 4,5 \\rangle$, $M= 2$, $\\vert 4,6 \\rangle$, $M= 3$\n\n* $\\vert 5,6 \\rangle$, $M= 4$\n\nOf these, there are only 3 states with $M=0$.\n\n\n\n## Shell-model project\n*You should try* by hand to show that in this same single-particle space, that for \n$N=3$ there are 3 states with $M=1/2$ and for $N= 4$ there are also only 3 states with $M=0$. \n\n*To test your code*, confirm the above. \n\nAlso, \nfor the $sd$-space given above, for $N=2$ there are 14 states with $M=0$, for $N=3$ there are 37 \nstates with $M=1/2$, for $N=4$ there are 81 states with $M=0$.\n\n\n\n## Shell-model project\nFor our project, we will only consider the pairing model.\nA simple space is the $(1/2)^2$ space with four single-particle states\n\n\n\n\n\n\n\n\n\n\n\n
                                        Index $n$ $l$ $s$ $m_s$
                                        1 0 0 1/2 -1/2
                                        2 0 0 1/2 1/2
                                        3 1 0 1/2 -1/2
                                        4 1 0 1/2 1/2
                                        \nFor $N=2$ there are 4 states with $M=0$; show this by hand and confirm your code reproduces it.\n\n\n\n## Shell-model project\nAnother, slightly more challenging space is the $(1/2)^4$ space, that is, \nwith eight single-particle states we have\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                        Index $n$ $l$ $s$ $m_s$
                                        1 0 0 1/2 -1/2
                                        2 0 0 1/2 1/2
                                        3 1 0 1/2 -1/2
                                        4 1 0 1/2 1/2
                                        5 2 0 1/2 -1/2
                                        6 2 0 1/2 1/2
                                        7 3 0 1/2 -1/2
                                        8 3 0 1/2 1/2
                                        \nFor $N=2$ there are 16 states with $M=0$; for $N=3$ there are 24 states with $M=1/2$, and for \n$N=4$ there are 36 states with $M=0$.\n\n\n\n## Shell-model project\nIn the shell-model context we can interpret this as 4 $s_{1/2}$ levels, with $m = \\pm 1/2$, we can also think of these are simple four pairs, $\\pm k, k = 1,2,3,4$. Later on we will \nassign single-particle energies, depending on the radial quantum number $n$, that is, \n$\\epsilon_k = |k| \\delta$ so that they are equally spaced.\n\n\n\n## Shell-model project\n\nFor application in the pairing model we can go further and consider only states with \nno \"broken pairs,\" that is, if $+k$ is filled (or $m = +1/2$, so is $-k$ ($m=-1/2$). \nIf you want, you can write your code to accept only these, and obtain the following \nsix states:\n\n* $| 1, 2 , 3 , 4 \\rangle , $\n\n* $| 1 , 2 , 5 , 6 \\rangle , $\n\n* $| 1 , 2 , 7 , 8 \\rangle , $\n\n* $| 3 , 4 , 5 , 6 \\rangle , $\n\n* $| 3 , 4 , 7 , 8 \\rangle , $\n\n* $| 5 , 6 , 7 , 8 \\rangle $\n\n\n\n\n\n## Shell-model project\n**Hints for coding.**\n\n\n\n* Write small modules (routines/functions) ; avoid big functions that do everything. (But not too small.)\n\n* Use Unit tests! Write lots of error traps, even for things that are `obvious.'\n\n* Document as you go along. The Unit tests serve as documentation. For each function write a header that includes: \n\na. Main purpose of function and/or unit test\n\nb. names and brief explanation of input variables, if any \n\nc. names and brief explanation of output variables, if any\n\nd. functions called by this function\n\ne. called by which functions\n\n\n\n## Shell-model project\n\nHints for coding\n\n* Unit tests will save time. Use also IDEs for debugging. If you insist on brute force debugging, print out intermediate values. It's almost impossible to debug a code by looking at it - the code will almost always win a `staring contest.'\n\n* Validate code with SIMPLE CASES. Validate early and often. Unit tests!! \n\nThe number one mistake is using a too complex a system to test. For example ,\nif you are computing particles in a potential in a box, try removing the potential - you should get \nparticles in a box. And start with one particle, then two, then three... Don't start with \neight particles.\n\n\n\n## Shell-model project\n\nOur recommended occupation representation, e.g. $| 1,2,4,8 \\rangle$, is \neasy to code, but numerically inefficient when one has hundreds of \nmillions of Slater determinants.\n\n\nIn state-of-the-art shell-model codes, one generally uses bit \nrepresentation, i.e. $|1101000100... \\rangle$ where one stores \nthe Slater determinant as a single (or a small number of) integer.\n\n\nThis is much more compact, but more intricate to code with considerable \nmore overhead. There exist \nbit-manipulation functions. We will discuss these in more detail at the beginning of the third week.\n\n\n\n## Example case: pairing Hamiltonian\n\nWe consider a space with $2\\Omega$ single-particle states, with each \nstate labeled by \n$k = 1, 2, 3, \\Omega$ and $m = \\pm 1/2$. The convention is that \nthe state with $k>0$ has $m = + 1/2$ while $-k$ has $m = -1/2$.\n\n\nThe Hamiltonian we consider is\n\n$$\n\\hat{H} = -G \\hat{P}_+ \\hat{P}_-,\n$$\n\nwhere\n\n$$\n\\hat{P}_+ = \\sum_{k > 0} \\hat{a}^\\dagger_k \\hat{a}^\\dagger_{-{k}}.\n$$\n\nand $\\hat{P}_- = ( \\hat{P}_+)^\\dagger$.\n\nThis problem can be solved using what is called the quasi-spin formalism to obtain the \nexact results. Thereafter we will try again using the explicit Slater determinant formalism.\n\n\n\n\n\n## Example case: pairing Hamiltonian\n\nOne can show (and this is part of the project) that\n\n$$\n\\left [ \\hat{P}_+, \\hat{P}_- \\right ] = \\sum_{k> 0} \\left( \\hat{a}^\\dagger_k \\hat{a}_k \n+ \\hat{a}^\\dagger_{-{k}} \\hat{a}_{-{k}} - 1 \\right) = \\hat{N} - \\Omega.\n$$\n\nNow define\n\n$$\n\\hat{P}_z = \\frac{1}{2} ( \\hat{N} -\\Omega).\n$$\n\nFinally you can show\n\n$$\n\\left [ \\hat{P}_z , \\hat{P}_\\pm \\right ] = \\pm \\hat{P}_\\pm.\n$$\n\nThis means the operators $\\hat{P}_\\pm, \\hat{P}_z$ form a so-called $SU(2)$ algebra, and we can \nuse all our insights about angular momentum, even though there is no actual \nangular momentum involved.\n\nSo we rewrite the Hamiltonian to make this explicit:\n\n$$\n\\hat{H} = -G \\hat{P}_+ \\hat{P}_- \n= -G \\left( \\hat{P}^2 - \\hat{P}_z^2 + \\hat{P}_z\\right)\n$$\n\n## Example case: pairing Hamiltonian\n\nBecause of the SU(2) algebra, we know that the eigenvalues of \n$\\hat{P}^2$ must be of the form $p(p+1)$, with $p$ either integer or half-integer, and the eigenvalues of $\\hat{P}_z$ \nare $m_p$ with $p \\geq | m_p|$, with $m_p$ also integer or half-integer. \n\n\nBut because $\\hat{P}_z = (1/2)(\\hat{N}-\\Omega)$, we know that for $N$ particles \nthe value $m_p = (N-\\Omega)/2$. Furthermore, the values of $m_p$ range from \n$-\\Omega/2$ (for $N=0$) to $+\\Omega/2$ (for $N=2\\Omega$, with all states filled). \n\nWe deduce the maximal $p = \\Omega/2$ and for a given $n$ the \nvalues range of $p$ range from $|N-\\Omega|/2$ to $\\Omega/2$ in steps of 1 \n(for an even number of particles) \n\n\nFollowing Racah we introduce the notation\n$p = (\\Omega - v)/2$\nwhere $v = 0, 2, 4,..., \\Omega - |N-\\Omega|$ \nWith this it is easy to deduce that the eigenvalues of the pairing Hamiltonian are\n\n$$\n-G(N-v)(2\\Omega +2-N-v)/4\n$$\n\nThis also works for $N$ odd, with $v= 1,3,5, \\dots$.\n\n\n\n\n## Example case: pairing Hamiltonian\n\nLet's take a specific example: $\\Omega = 3$ so there are 6 single-particle states, \nand $N = 3$, with $v= 1,3$. Therefore there are two distinct eigenvalues,\n\n$$\nE = -2G, 0\n$$\n\nNow let's work this out explicitly. The single particle degrees of freedom are defined as\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                        Index $k$ $m$
                                        1 1 -1/2
                                        2 -1 1/2
                                        3 2 -1/2
                                        4 -2 1/2
                                        5 3 -1/2
                                        6 -3 1/2
                                        \n There are $\\left( \\begin{array}{c}6 \\\\ 3 \\end{array} \\right) = 20$ three-particle states, but there \nare 9 states with $M = +1/2$, namely\n$| 1,2,3 \\rangle, |1,2,5\\rangle, | 1,4,6 \\rangle, | 2,3,4 \\rangle, |2,3,6 \\rangle, | 2,4,5 \\rangle, | 2, 5, 6 \\rangle, |3,4,6 \\rangle, | 4,5,6 \\rangle$.\n\n\n\n\n\n\n\n## Example case: pairing Hamiltonian\n\nIn this basis, the operator\n\n$$\n\\hat{P}_+\n= \\hat{a}^\\dagger_1 \\hat{a}^\\dagger_2 + \\hat{a}^\\dagger_3 \\hat{a}^\\dagger_4 +\n\\hat{a}^\\dagger_5 \\hat{a}^\\dagger_6\n$$\n\nFrom this we can determine that\n\n$$\n\\hat{P}_- | 1, 4, 6 \\rangle = \\hat{P}_- | 2, 3, 6 \\rangle\n= \\hat{P}_- | 2, 4, 5 \\rangle = 0\n$$\n\nso those states all have eigenvalue 0.\n\n\n\n\n## Example case: pairing Hamiltonian\nNow for further example,\n\n$$\n\\hat{P}_- | 1,2,3 \\rangle = | 3 \\rangle\n$$\n\nso\n\n$$\n\\hat{P}_+ \\hat{P}_- | 1,2,3\\rangle = | 1,2,3\\rangle+ | 3,4,3\\rangle + | 5,6,3\\rangle\n$$\n\nThe second term vanishes because state 3 is occupied twice, and reordering the last \nterm we\nget\n\n$$\n\\hat{P}_+ \\hat{P}_- | 1,2,3\\rangle = | 1,2,3\\rangle+ |3, 5,6\\rangle\n$$\n\nwithout picking up a phase.\n\n\n\n## Example case: pairing Hamiltonian\n\nContinuing in this fashion, with the previous ordering of the many-body states\n( $| 1,2,3 \\rangle, |1,2,5\\rangle, | 1,4,6 \\rangle, | 2,3,4 \\rangle, |2,3,6 \\rangle, | 2,4,5 \\rangle, | 2, 5, 6 \\rangle, |3,4,6 \\rangle, | 4,5,6 \\rangle$) the \nHamiltonian matrix of this system is\n\n$$\nH = -G\\left( \n\\begin{array}{ccccccccc}\n1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \n\\end{array} \\right )\n$$\n\nThis is useful for our project. One can by hand confirm \nthat there are 3 eigenvalues $-2G$ and 6 with value zero.\n\n\n\n## Example case: pairing Hamiltonian\n\nAnother example\nUsing the $(1/2)^4$ single-particle space, resulting in eight single-particle states\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                        Index $n$ $l$ $s$ $m_s$
                                        1 0 0 1/2 -1/2
                                        2 0 0 1/2 1/2
                                        3 1 0 1/2 -1/2
                                        4 1 0 1/2 1/2
                                        5 2 0 1/2 -1/2
                                        6 2 0 1/2 1/2
                                        7 3 0 1/2 -1/2
                                        8 3 0 1/2 1/2
                                        \nand then taking only 4-particle, $M=0$ states that have no `broken pairs', there are six basis Slater \ndeterminants:\n\n* $| 1, 2 , 3 , 4 \\rangle , $\n\n* $| 1 , 2 , 5 , 6 \\rangle , $\n\n* $| 1 , 2 , 7 , 8 \\rangle , $\n\n* $| 3 , 4 , 5 , 6 \\rangle , $\n\n* $| 3 , 4 , 7 , 8 \\rangle , $\n\n* $| 5 , 6 , 7 , 8 \\rangle $\n\n\n\n## Example case: pairing Hamiltonian\n\nNow we take the following Hamiltonian\n\n$$\n\\hat{H} = \\sum_n n \\delta \\hat{N}_n - G \\hat{P}^\\dagger \\hat{P}\n$$\n\nwhere\n\n$$\n\\hat{N}_n = \\hat{a}^\\dagger_{n, m=+1/2} \\hat{a}_{n, m=+1/2} +\n\\hat{a}^\\dagger_{n, m=-1/2} \\hat{a}_{n, m=-1/2}\n$$\n\nand\n\n$$\n\\hat{P}^\\dagger = \\sum_{n} \\hat{a}^\\dagger_{n, m=+1/2} \\hat{a}^\\dagger_{n, m=-1/2}\n$$\n\nWe can write down the $ 6 \\times 6$ Hamiltonian in the basis from the prior slide:\n\n$$\nH = \\left ( \n\\begin{array}{cccccc}\n2\\delta -2G & -G & -G & -G & -G & 0 \\\\\n -G & 4\\delta -2G & -G & -G & -0 & -G \\\\\n-G & -G & 6\\delta -2G & 0 & -G & -G \\\\\n -G & -G & 0 & 6\\delta-2G & -G & -G \\\\\n -G & 0 & -G & -G & 8\\delta-2G & -G \\\\\n0 & -G & -G & -G & -G & 10\\delta -2G \n\\end{array} \\right )\n$$\n\n(You should check by hand that this is correct.) \n\nFor $\\delta = 0$ we have the closed form solution of the g.s. energy given by $-6G$.\n\n\n\n\n## Building a Hamiltonian matrix\nThe goal is to compute the matrix elements of the Hamiltonian, specifically\nmatrix elements between many-body states (Slater determinants) of two-body\noperators\n\n$$\n\\sum_{p < q, r < s}V_{pqr} \\hat{a}^\\dagger_p \\hat{a}^\\dagger_q\\hat{a}_s \\hat{a}_r\n$$\n\nIn particular we will need to compute\n\n$$\n\\langle \\beta | \\hat{a}^\\dagger_p \\hat{a}^\\dagger_q\\hat{a}_s \\hat{a}_r |\\alpha \\rangle\n$$\n\nwhere $\\alpha, \\beta$ are indices labeling Slater determinants and $p,q,r,s$ label\nsingle-particle states.\n\n\n\n\n## Building a Hamiltonian matrix\nNote: there are other, more efficient ways to do this than the method we describe, \nbut you will\nbe able to produce a working code quickly.\n\nAs we coded in the first step,\na Slater determinant $| \\alpha \\rangle$ with index $\\alpha$ is a\nlist of $N$ occupied single-particle states $i_1 < i_2 < i_3 \\ldots i_N$.\n\n\nFurthermore, for the two-body matrix elements $V_{pqrs}$ we normally assume\n$p < q$ and $r < s$. For our specific project, the interaction is much simpler and you can use this to simplify considerably the setup of a shell-model code for project 2.\n\nWhat follows here is a more general, but still brute force, approach.\n\n\n\n\n## Building a Hamiltonian matrix\nWrite a function that:\n1. Has as input the single-particle indices $p,q,r,s$ for the two-body operator and the index $\\alpha$ for the ket Slater determinant;\n\n2. Returns the index $\\beta$ of the unique (if any) Slater determinant such that\n\n$$\n| \\beta \\rangle = \\pm \\hat{a}^\\dagger_p \\hat{a}^\\dagger_q\\hat{a}_s \\hat{a}_r |\\alpha \\rangle\n$$\n\nas well as the phase\n\nThis is equivalent to computing\n\n$$\n\\langle \\beta | \\hat{a}^\\dagger_p \\hat{a}^\\dagger_q\\hat{a}_s \\hat{a}_r |\\alpha \\rangle\n$$\n\n## Building a Hamiltonian matrix, first step\nThe first step can take as input an initial Slater determinant\n(whose position in the list of basis Slater determinants is $\\alpha$) written as an\nordered listed of occupied single-particle states, e.g. $1,2,5,8$, and the\nindices $p,q,r,s$ from the two-body operator. \n\nIt will return another final Slater determinant if the single-particle states $r$ and $s$ are occupied, else it will return an \nempty Slater determinant\n(all zeroes). \n\nIf $r$ and $s$ are in the list of occupied single particle states, then\nreplace the initial single-particle states $ij$ as $i \\rightarrow r$ and $j \\rightarrow r$.\n\n\n\n\n## Building a Hamiltonian matrix, second step\nThe second step will take the final Slater determinant \nfrom the first step (if not empty),\nand then order by pairwise permutations (i.e., if the Slater determinant is\n$i_1, i_2, i_3, \\ldots$, then if $i_n > i_{n+1}$, interchange \n$i_n \\leftrightarrow i_{n+1}$.\n\n\n\n\n\n## Building a Hamiltonian matrix\n\nIt will also output a phase. If any two single-particle occupancies are repeated, \nthe phase is\n0. Otherwise it is +1 for an even permutation and -1 for an odd permutation to \nbring the final\nSlater determinant into ascending order, $j_1 < j_2 < j_3 \\ldots$.\n\n\n\n\n## Building a Hamiltonian matrix\n**Example**: Suppose in the $sd$ single-particle space that the initial \nSlater determinant\nis $1,3,9,12$. If $p,q,r,s = 2,8,1,12$, then after the first step the final Slater determinant\nis $2,3,9,8$. The second step will return $2,3,8,9$ and a phase of -1, \nbecause an odd number of interchanges is required.\n\n\n\n\n## Building a Hamiltonian matrix\n\n**Example**: Suppose in the $sd$ single-particle space that the initial \nSlater determinant\nis $1,3,9,12$. If $p,q,r,s = 3,8,1,12$, then after the first step the \nfinal Slater determinant\nis $3,3,9,8$, but after the second step the phase is 0 \nbecause the single-particle state 3 is\noccupied twice.\n\nLastly, the final step takes the ordered final Slater determinant and \nwe search through the basis list to\ndetermine its index in the many-body basis, that is, $\\beta$.\n\n\n\n\n## Building a Hamiltonian matrix\n\nThe Hamiltonian is then stored as an $N_{SD} \\times N_{SD}$ array of real numbers, which\ncan be allocated once you have created the many-body basis and know $N_{SD}$.\n\n\n\n\n## Building a Hamiltonian matrix\n\n1. Initialize $H(\\alpha,\\beta)=0.0$\n\n2. Set up an outer loop over $\\beta$\n\n3. Loop over $\\alpha = 1, NSD$\n\n4. For each $\\alpha$, loop over $a=1,ntbme$ and fetch $V(a)$ and the single-particle indices $p,q,r,s$ \n\n5. If $V(a) = 0$ skip. Otherwise, apply $\\hat{a}^\\dagger_p\\hat{a}^\\dagger_q \\hat{a}_s \\hat{a}_r$ to the Slater determinant labeled by $\\alpha$.\n\n6. Find, if any, the label $\\beta$ of the resulting Slater determinant and the phase (which is 0, +1, -1).\n\n7. If phase $\\neq 0$, then update $H(\\alpha,\\beta)$ as $H(\\alpha,\\beta) + phase*V(a)$. The sum is important because multiple operators might contribute to the same matrix element.\n\n8. Continue loop over $a$\n\n9. Continue loop over $\\alpha$.\n\n10. End the outer loop over $\\beta$.\n\nYou should force the resulting matrix $H$ to be symmetric. To do this, when\nupdating $H(\\alpha,\\beta)$, if $\\alpha \\neq \\beta$, also update $H(\\beta,\\alpha)$.\n\n\n\n\n## Building a Hamiltonian matrix\n\nYou will also need to include the single-particle energies. This is easy: they only\ncontribute to diagonal matrix elements, that is, $H(\\alpha,\\alpha)$. \nSimply find the occupied single-particle states $i$ and add the corresponding $\\epsilon(i)$.\n\n\n\n\n## Hamiltonian matrix without the bit representation\n\nConsider the many-body state $\\Psi_{\\lambda}$ expressed as linear combinations of\nSlater determinants ($SD$) of orthonormal single-particle states $\\phi({\\bf r})$:\n\n\n
                                        \n\n$$\n\\begin{equation}\n\\Psi_{\\lambda} = \\sum_i C_{\\lambda i} SD_i\n\\label{_auto1} \\tag{2}\n\\end{equation}\n$$\n\nUsing the Slater-Condon rules the matrix elements of any one-body\n(${\\cal O}_1$) or two-body (${\\cal O}_2$) operator expressed in the\ndeterminant space have simple expressions involving one- and two-fermion\nintegrals in our given single-particle basis.\nThe diagonal elements are given by:\n\n$$\n\\begin{eqnarray}\n \\langle SD | {\\cal O}_1 | SD \\rangle & = & \\sum_{i \\in SD} \\langle \\phi_i | {\\cal O}_1 | \\phi_i \\rangle \\\\\n \\langle SD | {\\cal O}_2 | SD \\rangle & = & \\frac{1}{2} \\sum_{(i,j) \\in SD} \n \\langle \\phi_i \\phi_j | {\\cal O}_2 | \\phi_i \\phi_j \\rangle - \\nonumber \\\\\n & & \n \\langle \\phi_i \\phi_j | {\\cal O}_2 | \\phi_j \\phi_i \\rangle \\nonumber \n\\end{eqnarray}\n$$\n\n## Hamiltonian matrix without the bit representation, one and two-body operators\n\nFor two determinants which differ only by the substitution of single-particle states $i$ with\na single-particle state $j$:\n\n$$\n\\begin{eqnarray}\n \\langle SD | {\\cal O}_1 | SD_i^j \\rangle & = & \\langle \\phi_i | {\\cal O}_1 | \\phi_j \\rangle \\\\\n \\langle SD | {\\cal O}_2 | SD_i^j \\rangle & = & \\sum_{k \\in SD} \n \\langle \\phi_i \\phi_k | {\\cal O}_2 | \\phi_j \\phi_k \\rangle - \n \\langle \\phi_i \\phi_k | {\\cal O}_2 | \\phi_k \\phi_j \\rangle \\nonumber\n\\end{eqnarray}\n$$\n\nFor two determinants which differ by two single-particle states\n\n$$\n\\begin{eqnarray}\n \\langle SD | {\\cal O}_1 | SD_{ik}^{jl} \\rangle & = & 0 \\\\\n \\langle SD | {\\cal O}_2 | SD_{ik}^{jl} \\rangle & = & \n \\langle \\phi_i \\phi_k | {\\cal O}_2 | \\phi_j \\phi_l \\rangle -\n \\langle \\phi_i \\phi_k | {\\cal O}_2 | \\phi_l \\phi_j \\rangle \\nonumber \n\\end{eqnarray}\n$$\n\nAll other matrix elements involving determinants with more than two\nsubstitutions are zero.\n\n\n\n## Strategies for setting up an algorithm\n\n\nAn efficient implementation of these rules requires\n\n* to find the number of single-particle state substitutions between two determinants\n\n* to find which single-particle states are involved in the substitution\n\n* to compute the phase factor if a reordering of the single-particle states has occured\n\nWe can solve this problem using our odometric approach or alternatively using a bit representation as discussed below and in more detail in \n\n* [Scemama and Gimer's article (Fortran codes)](https://github.com/scemama/slater_condon)\n\n* [Simen Kvaal's article on how to build an FCI code (C++ code)](https://arxiv.org/abs/0810.2644)\n\nWe recommend in particular the article by Simen Kvaal. It contains nice general classes for creation and annihilation operators as well as the calculation of the phase (see below).\n\n\n\n\n## Computing expectation values and transitions in the shell-model\nWhen we diagonalize the Hamiltonian matrix, the eigenvectors are the coefficients $C_{\\lambda i}$ used \nto express the many-body state $\\Psi_{\\lambda}$ in terms of a linear combinations of\nSlater determinants ($SD$) of orthonormal single-particle states $\\phi({\\bf r})$.\n\nWith these eigenvectors we can compute say the transition likelyhood of a one-body operator as\n\n$$\n\\langle \\Psi_{\\lambda} \\vert {\\cal O}_1 \\vert \\Psi_{\\sigma} \\rangle = \n\\sum_{ij}C_{\\lambda i}^*C_{\\sigma j} \\langle SD_i | {\\cal O}_1 | SD_j \\rangle .\n$$\n\nWriting the one-body operator in second quantization as\n\n$$\n{\\cal O}_1 = \\sum_{pq} \\langle p \\vert {\\cal o}_1 \\vert q\\rangle a_p^{\\dagger} a_q,\n$$\n\nwe have\n\n$$\n\\langle \\Psi_{\\lambda} \\vert {\\cal O}_1 \\vert \\Psi_{\\sigma} \\rangle = \n\\sum_{pq}\\langle p \\vert {\\cal o}_1 \\vert q\\rangle \\sum_{ij}C_{\\lambda i}^*C_{\\sigma j} \\langle SD_i |a_p^{\\dagger} a_q | SD_j \\rangle .\n$$\n\n## Computing expectation values and transitions in the shell-model and spectroscopic factors\nThe terms we need to evalute then are just the elements\n\n$$\n\\langle SD_i |a_p^{\\dagger} a_q | SD_j \\rangle,\n$$\n\nwhich can be rewritten in terms of spectroscopic factors by inserting a complete set of Slater determinats as\n\n$$\n\\langle SD_i |a_p^{\\dagger} a_q | SD_j \\rangle = \\sum_{l}\\langle SD_i \\vert a_p^{\\dagger}\\vert SD_l\\rangle \\langle SD_l \\vert a_q \\vert SD_j \\rangle,\n$$\n\nwhere $\\langle SD_l\\vert a_q(a_p^{\\dagger})\\vert SD_j\\rangle$ are the spectroscopic factors. These can be easily evaluated in $m$-scheme. Using the Wigner-Eckart theorem we can transform these to a $J$-coupled scheme through so-called reduced matrix elements.\n\n\n\n\n\n\n## Operators in second quantization\nIn the build-up of a shell-model or FCI code that is meant to tackle large dimensionalities\nwe need to deal with the action of the Hamiltonian $\\hat{H}$ on a\nSlater determinant represented in second quantization as\n\n$$\n|\\alpha_1\\dots \\alpha_n\\rangle = a_{\\alpha_1}^{\\dagger} a_{\\alpha_2}^{\\dagger} \\dots a_{\\alpha_n}^{\\dagger} |0\\rangle.\n$$\n\nThe time consuming part stems from the action of the Hamiltonian\non the above determinant,\n\n$$\n\\left(\\sum_{\\alpha\\beta} \\langle \\alpha|t+u|\\beta\\rangle a_\\alpha^{\\dagger} a_\\beta + \\frac{1}{4} \\sum_{\\alpha\\beta\\gamma\\delta}\n \\langle \\alpha \\beta|\\hat{v}|\\gamma \\delta\\rangle a_\\alpha^{\\dagger} a_\\beta^{\\dagger} a_\\delta a_\\gamma\\right)a_{\\alpha_1}^{\\dagger} a_{\\alpha_2}^{\\dagger} \\dots a_{\\alpha_n}^{\\dagger} |0\\rangle.\n$$\n\nA practically useful way to implement this action is to encode a Slater determinant as a bit pattern.\n\n\n\n\n## Operators in second quantization\nAssume that we have at our disposal $n$ different single-particle states\n$\\alpha_0,\\alpha_2,\\dots,\\alpha_{n-1}$ and that we can distribute among these states $N\\le n$ particles.\n\nA Slater determinant can then be coded as an integer of $n$ bits. As an example, if we have $n=16$ single-particle states\n$\\alpha_0,\\alpha_1,\\dots,\\alpha_{15}$ and $N=4$ fermions occupying the states $\\alpha_3$, $\\alpha_6$, $\\alpha_{10}$ and $\\alpha_{13}$\nwe could write this Slater determinant as\n\n$$\n\\Phi_{\\Lambda} = a_{\\alpha_3}^{\\dagger} a_{\\alpha_6}^{\\dagger} a_{\\alpha_{10}}^{\\dagger} a_{\\alpha_{13}}^{\\dagger} |0\\rangle.\n$$\n\nThe unoccupied single-particle states have bit value $0$ while the occupied ones are represented by bit state $1$. \nIn the binary notation we would write this 16 bits long integer as\n\n$$\n\\begin{array}{cccccccccccccccc}\n{\\alpha_0}&{\\alpha_1}&{\\alpha_2}&{\\alpha_3}&{\\alpha_4}&{\\alpha_5}&{\\alpha_6}&{\\alpha_7} & {\\alpha_8} &{\\alpha_9} & {\\alpha_{10}} &{\\alpha_{11}} &{\\alpha_{12}} &{\\alpha_{13}} &{\\alpha_{14}} & {\\alpha_{15}} \\\\\n{0} & {0} &{0} &{1} &{0} &{0} &{1} &{0} &{0} &{0} &{1} &{0} &{0} &{1} &{0} & {0} \\\\\n\\end{array}\n$$\n\nwhich translates into the decimal number\n\n$$\n2^3+2^6+2^{10}+2^{13}=9288.\n$$\n\nWe can thus encode a Slater determinant as a bit pattern.\n\n\n\n\n## Operators in second quantization\nWith $N$ particles that can be distributed over $n$ single-particle states, the total number of Slater determinats (and defining thereby the dimensionality of the system) is\n\n$$\n\\mathrm{dim}(\\mathcal{H}) = \\left(\\begin{array}{c} n \\\\N\\end{array}\\right).\n$$\n\nThe total number of bit patterns is $2^n$.\n\n\n\n## Operators in second quantization\nWe assume again that we have at our disposal $n$ different single-particle orbits\n$\\alpha_0,\\alpha_2,\\dots,\\alpha_{n-1}$ and that we can distribute among these orbits $N\\le n$ particles.\nThe ordering among these states is important as it defines the order of the creation operators.\nWe will write the determinant\n\n$$\n\\Phi_{\\Lambda} = a_{\\alpha_3}^{\\dagger} a_{\\alpha_6}^{\\dagger} a_{\\alpha_{10}}^{\\dagger} a_{\\alpha_{13}}^{\\dagger} |0\\rangle,\n$$\n\nin a more compact way as\n\n$$\n\\Phi_{3,6,10,13} = |0001001000100100\\rangle.\n$$\n\nThe action of a creation operator is thus\n\n$$\na^{\\dagger}_{\\alpha_4}\\Phi_{3,6,10,13} = a^{\\dagger}_{\\alpha_4}|0001001000100100\\rangle=a^{\\dagger}_{\\alpha_4}a_{\\alpha_3}^{\\dagger} a_{\\alpha_6}^{\\dagger} a_{\\alpha_{10}}^{\\dagger} a_{\\alpha_{13}}^{\\dagger} |0\\rangle,\n$$\n\nwhich becomes\n\n$$\n-a_{\\alpha_3}^{\\dagger} a^{\\dagger}_{\\alpha_4} a_{\\alpha_6}^{\\dagger} a_{\\alpha_{10}}^{\\dagger} a_{\\alpha_{13}}^{\\dagger} |0\\rangle=-|0001101000100100\\rangle.\n$$\n\n## Operators in second quantization\nSimilarly\n\n$$\na^{\\dagger}_{\\alpha_6}\\Phi_{3,6,10,13} = a^{\\dagger}_{\\alpha_6}|0001001000100100\\rangle=a^{\\dagger}_{\\alpha_6}a_{\\alpha_3}^{\\dagger} a_{\\alpha_6}^{\\dagger} a_{\\alpha_{10}}^{\\dagger} a_{\\alpha_{13}}^{\\dagger} |0\\rangle,\n$$\n\nwhich becomes\n\n$$\n-a^{\\dagger}_{\\alpha_4} (a_{\\alpha_6}^{\\dagger})^ 2 a_{\\alpha_{10}}^{\\dagger} a_{\\alpha_{13}}^{\\dagger} |0\\rangle=0!\n$$\n\nThis gives a simple recipe: \n* If one of the bits $b_j$ is $1$ and we act with a creation operator on this bit, we return a null vector\n\n* If $b_j=0$, we set it to $1$ and return a sign factor $(-1)^l$, where $l$ is the number of bits set before bit $j$.\n\n\n\n\n## Operators in second quantization\nConsider the action of $a^{\\dagger}_{\\alpha_2}$ on various slater determinants:\n\n$$\n\\begin{array}{ccc}\na^{\\dagger}_{\\alpha_2}\\Phi_{00111}& = a^{\\dagger}_{\\alpha_2}|00111\\rangle&=0\\times |00111\\rangle\\\\\na^{\\dagger}_{\\alpha_2}\\Phi_{01011}& = a^{\\dagger}_{\\alpha_2}|01011\\rangle&=(-1)\\times |01111\\rangle\\\\\na^{\\dagger}_{\\alpha_2}\\Phi_{01101}& = a^{\\dagger}_{\\alpha_2}|01101\\rangle&=0\\times |01101\\rangle\\\\\na^{\\dagger}_{\\alpha_2}\\Phi_{01110}& = a^{\\dagger}_{\\alpha_2}|01110\\rangle&=0\\times |01110\\rangle\\\\\na^{\\dagger}_{\\alpha_2}\\Phi_{10011}& = a^{\\dagger}_{\\alpha_2}|10011\\rangle&=(-1)\\times |10111\\rangle\\\\\na^{\\dagger}_{\\alpha_2}\\Phi_{10101}& = a^{\\dagger}_{\\alpha_2}|10101\\rangle&=0\\times |10101\\rangle\\\\\na^{\\dagger}_{\\alpha_2}\\Phi_{10110}& = a^{\\dagger}_{\\alpha_2}|10110\\rangle&=0\\times |10110\\rangle\\\\\na^{\\dagger}_{\\alpha_2}\\Phi_{11001}& = a^{\\dagger}_{\\alpha_2}|11001\\rangle&=(+1)\\times |11101\\rangle\\\\\na^{\\dagger}_{\\alpha_2}\\Phi_{11010}& = a^{\\dagger}_{\\alpha_2}|11010\\rangle&=(+1)\\times |11110\\rangle\\\\\n\\end{array}\n$$\n\nWhat is the simplest way to obtain the phase when we act with one annihilation(creation) operator\non the given Slater determinant representation?\n\n\n\n\n## Operators in second quantization\nWe have an SD representation\n\n$$\n\\Phi_{\\Lambda} = a_{\\alpha_0}^{\\dagger} a_{\\alpha_3}^{\\dagger} a_{\\alpha_6}^{\\dagger} a_{\\alpha_{10}}^{\\dagger} a_{\\alpha_{13}}^{\\dagger} |0\\rangle,\n$$\n\nin a more compact way as\n\n$$\n\\Phi_{0,3,6,10,13} = |1001001000100100\\rangle.\n$$\n\nThe action of\n\n$$\na^{\\dagger}_{\\alpha_4}a_{\\alpha_0}\\Phi_{0,3,6,10,13} = a^{\\dagger}_{\\alpha_4}|0001001000100100\\rangle=a^{\\dagger}_{\\alpha_4}a_{\\alpha_3}^{\\dagger} a_{\\alpha_6}^{\\dagger} a_{\\alpha_{10}}^{\\dagger} a_{\\alpha_{13}}^{\\dagger} |0\\rangle,\n$$\n\nwhich becomes\n\n$$\n-a_{\\alpha_3}^{\\dagger} a^{\\dagger}_{\\alpha_4} a_{\\alpha_6}^{\\dagger} a_{\\alpha_{10}}^{\\dagger} a_{\\alpha_{13}}^{\\dagger} |0\\rangle=-|0001101000100100\\rangle.\n$$\n\n## Operators in second quantization\nThe action\n\n$$\na_{\\alpha_0}\\Phi_{0,3,6,10,13} = |0001001000100100\\rangle,\n$$\n\ncan be obtained by subtracting the logical sum (AND operation) of $\\Phi_{0,3,6,10,13}$ and \na word which represents only $\\alpha_0$, that is\n\n$$\n|1000000000000000\\rangle,\n$$\n\nfrom $\\Phi_{0,3,6,10,13}= |1001001000100100\\rangle$.\n\nThis operation gives $|0001001000100100\\rangle$. \n\nSimilarly, we can form $a^{\\dagger}_{\\alpha_4}a_{\\alpha_0}\\Phi_{0,3,6,10,13}$, say, by adding \n$|0000100000000000\\rangle$ to $a_{\\alpha_0}\\Phi_{0,3,6,10,13}$, first checking that their logical sum\nis zero in order to make sure that the state $\\alpha_4$ is not already occupied.\n\n\n\n\n\n\n## Operators in second quantization\nIt is trickier however to get the phase $(-1)^l$. \nOne possibility is as follows\n* Let $S_1$ be a word that represents the 1-bit to be removed and all others set to zero.\n\nIn the previous example $S_1=|1000000000000000\\rangle$\n* Define $S_2$ as the similar word that represents the bit to be added, that is in our case\n\n$S_2=|0000100000000000\\rangle$.\n* Compute then $S=S_1-S_2$, which here becomes\n\n$$\nS=|0111000000000000\\rangle\n$$\n\n* Perform then the logical AND operation of $S$ with the word containing\n\n$$\n\\Phi_{0,3,6,10,13} = |1001001000100100\\rangle,\n$$\n\nwhich results in $|0001000000000000\\rangle$. Counting the number of 1-bits gives the phase. Here you need however an algorithm for bitcounting.\n\n\n\n\n\n\n\n\n## Bit counting\n\n\nWe include here a python program which may aid in this direction. It uses bit manipulation functions from .\n\n\n```\nimport math\n\n\"\"\"\nA simple Python class for Slater determinant manipulation\nBit-manipulation stolen from:\n\nhttp://wiki.python.org/moin/BitManipulation\n\"\"\"\n\n# bitCount() counts the number of bits set (not an optimal function)\n\ndef bitCount(int_type):\n \"\"\" Count bits set in integer \"\"\"\n count = 0\n while(int_type):\n int_type &= int_type - 1\n count += 1\n return(count)\n\n\n# testBit() returns a nonzero result, 2**offset, if the bit at 'offset' is one.\n\ndef testBit(int_type, offset):\n mask = 1 << offset\n return(int_type & mask) >> offset\n\n# setBit() returns an integer with the bit at 'offset' set to 1.\n\ndef setBit(int_type, offset):\n mask = 1 << offset\n return(int_type | mask)\n\n# clearBit() returns an integer with the bit at 'offset' cleared.\n\ndef clearBit(int_type, offset):\n mask = ~(1 << offset)\n return(int_type & mask)\n\n# toggleBit() returns an integer with the bit at 'offset' inverted, 0 -> 1 and 1 -> 0.\n\ndef toggleBit(int_type, offset):\n mask = 1 << offset\n return(int_type ^ mask)\n\n# binary string made from number\n\ndef bin0(s):\n return str(s) if s<=1 else bin0(s>>1) + str(s&1)\n\ndef bin(s, L = 0):\n ss = bin0(s)\n if L > 0:\n return '0'*(L-len(ss)) + ss\n else:\n return ss\n \n \n\nclass Slater:\n \"\"\" Class for Slater determinants \"\"\"\n def __init__(self):\n self.word = int(0)\n\n def create(self, j):\n print \"c^+_\" + str(j) + \" |\" + bin(self.word) + \"> = \",\n # Assume bit j is set, then we return zero.\n s = 0\n # Check if bit j is set.\n isset = testBit(self.word, j)\n if isset == 0:\n bits = bitCount(self.word & ((1<\"\n return s\n \n def annihilate(self, j):\n print \"c_\" + str(j) + \" |\" + bin(self.word) + \"> = \",\n # Assume bit j is not set, then we return zero.\n s = 0\n # Check if bit j is set.\n isset = testBit(self.word, j)\n if isset == 1:\n bits = bitCount(self.word & ((1<\"\n return s\n\n\n\n# Do some testing:\n\nphi = Slater()\nphi.create(0)\nphi.create(1)\nphi.create(2)\nphi.create(3)\n\nprint\n\n\ns = phi.annihilate(2)\ns = phi.create(7)\ns = phi.annihilate(0)\ns = phi.create(200)\n```\n\n## Eigenvalue problems, basic definitions\nLet us consider the matrix $\\mathbf{A}$ of dimension $n$. The eigenvalues of\n$\\mathbf{A}$ are defined through the matrix equation\n\n$$\n\\mathbf{A}\\mathbf{x}^{(\\nu)} = \\lambda^{(\\nu)}\\mathbf{x}^{(\\nu)},\n$$\n\nwhere $\\lambda^{(\\nu)}$ are the eigenvalues and $\\mathbf{x}^{(\\nu)}$ the\ncorresponding eigenvectors.\nUnless otherwise stated, when we use the wording eigenvector we mean the\nright eigenvector. The left eigenvalue problem is defined as\n\n$$\n\\mathbf{x}^{(\\nu)}_L\\mathbf{A} = \\lambda^{(\\nu)}\\mathbf{x}^{(\\nu)}_L\n$$\n\nThe above right eigenvector problem is equivalent to a set of $n$ equations with $n$ unknowns\n$x_i$.\n\n\n\n\n## Eigenvalue problems, basic definitions\nThe eigenvalue problem can be rewritten as\n\n$$\n\\left( \\mathbf{A}-\\lambda^{(\\nu)} \\mathbf{I} \\right) \\mathbf{x}^{(\\nu)} = 0,\n$$\n\nwith $\\mathbf{I}$ being the unity matrix. This equation provides\na solution to the problem if and only if the determinant\nis zero, namely\n\n$$\n\\left| \\mathbf{A}-\\lambda^{(\\nu)}\\mathbf{I}\\right| = 0,\n$$\n\nwhich in turn means that the determinant is a polynomial\nof degree $n$ in $\\lambda$ and in general we will have \n$n$ distinct zeros.\n\n\n\n\n## Eigenvalue problems, basic definitions\nThe eigenvalues of a matrix \n$\\mathbf{A}\\in {\\mathbb{C}}^{n\\times n}$\nare thus the $n$ roots of its characteristic polynomial\n\n$$\nP(\\lambda) = det(\\lambda\\mathbf{I}-\\mathbf{A}),\n$$\n\nor\n\n$$\nP(\\lambda)= \\prod_{i=1}^{n}\\left(\\lambda_i-\\lambda\\right).\n$$\n\nThe set of these roots is called the spectrum and is denoted as\n$\\lambda(\\mathbf{A})$.\nIf $\\lambda(\\mathbf{A})=\\left\\{\\lambda_1,\\lambda_2,\\dots ,\\lambda_n\\right\\}$ then we have\n\n$$\ndet(\\mathbf{A})= \\lambda_1\\lambda_2\\dots\\lambda_n,\n$$\n\nand if we define the trace of $\\mathbf{A}$ as\n\n$$\nTr(\\mathbf{A})=\\sum_{i=1}^n a_{ii}\n$$\n\nthen\n\n$$\nTr(\\mathbf{A})=\\lambda_1+\\lambda_2+\\dots+\\lambda_n.\n$$\n\n## Abel-Ruffini Impossibility Theorem\nThe *Abel-Ruffini* theorem (also known as Abel's impossibility theorem) \nstates that there is no general solution in radicals to polynomial equations of degree five or higher.\n\nThe content of this theorem is frequently misunderstood. It does not assert that higher-degree polynomial equations are unsolvable. \nIn fact, if the polynomial has real or complex coefficients, and we allow complex solutions, then every polynomial equation has solutions; this is the fundamental theorem of algebra. Although these solutions cannot always be computed exactly with radicals, they can be computed to any desired degree of accuracy using numerical methods such as the Newton-Raphson method or Laguerre method, and in this way they are no different from solutions to polynomial equations of the second, third, or fourth degrees.\n\nThe theorem only concerns the form that such a solution must take. The content of the theorem is \nthat the solution of a higher-degree equation cannot in all cases be expressed in terms of the polynomial coefficients with a finite number of operations of addition, subtraction, multiplication, division and root extraction. Some polynomials of arbitrary degree, of which the simplest nontrivial example is the monomial equation $ax^n = b$, are always solvable with a radical.\n\n\n\n\n## Abel-Ruffini Impossibility Theorem\n\nThe *Abel-Ruffini* theorem says that there are some fifth-degree equations whose solution cannot be so expressed. \nThe equation $x^5 - x + 1 = 0$ is an example. Some other fifth degree equations can be solved by radicals, \nfor example $x^5 - x^4 - x + 1 = 0$. The precise criterion that distinguishes between those equations that can be solved \nby radicals and those that cannot was given by Galois and is now part of Galois theory: \na polynomial equation can be solved by radicals if and only if its Galois group is a solvable group.\n\nToday, in the modern algebraic context, we say that second, third and fourth degree polynomial \nequations can always be solved by radicals because the symmetric groups $S_2, S_3$ and $S_4$ are solvable groups, \nwhereas $S_n$ is not solvable for $n \\ge 5$.\n\n\n\n\n## Eigenvalue problems, basic definitions\nIn the present discussion we assume that our matrix is real and symmetric, that is \n$\\mathbf{A}\\in {\\mathbb{R}}^{n\\times n}$.\nThe matrix $\\mathbf{A}$ has $n$ eigenvalues\n$\\lambda_1\\dots \\lambda_n$ (distinct or not). Let $\\mathbf{D}$ be the\ndiagonal matrix with the eigenvalues on the diagonal\n\n$$\n\\mathbf{D}= \\left( \\begin{array}{ccccccc} \\lambda_1 & 0 & 0 & 0 & \\dots &0 & 0 \\\\\n 0 & \\lambda_2 & 0 & 0 & \\dots &0 &0 \\\\\n 0 & 0 & \\lambda_3 & 0 &0 &\\dots & 0\\\\\n \\dots & \\dots & \\dots & \\dots &\\dots &\\dots & \\dots\\\\\n 0 & \\dots & \\dots & \\dots &\\dots &\\lambda_{n-1} & \\\\\n 0 & \\dots & \\dots & \\dots &\\dots &0 & \\lambda_n\n \\end{array} \\right).\n$$\n\nIf $\\mathbf{A}$ is real and symmetric then there exists a real orthogonal matrix $\\mathbf{S}$ such that\n\n$$\n\\mathbf{S}^T \\mathbf{A}\\mathbf{S}= \\mathrm{diag}(\\lambda_1,\\lambda_2,\\dots ,\\lambda_n),\n$$\n\nand for $j=1:n$ we have $\\mathbf{A}\\mathbf{S}(:,j) = \\lambda_j \\mathbf{S}(:,j)$.\n\n\n\n\n## Eigenvalue problems, basic definitions\nTo obtain the eigenvalues of $\\mathbf{A}\\in {\\mathbb{R}}^{n\\times n}$,\nthe strategy is to\nperform a series of similarity transformations on the original\nmatrix $\\mathbf{A}$, in order to reduce it either into a diagonal form as above\nor into a tridiagonal form. \n\nWe say that a matrix $\\mathbf{B}$ is a similarity\ntransform of $\\mathbf{A}$ if\n\n$$\n\\mathbf{B}= \\mathbf{S}^T \\mathbf{A}\\mathbf{S}, \\hspace{1cm} \\mathrm{where} \\hspace{1cm} \\mathbf{S}^T\\mathbf{S}=\\mathbf{S}^{-1}\\mathbf{S} =\\mathbf{I}.\n$$\n\nThe importance of a similarity transformation lies in the fact that\nthe resulting matrix has the same\neigenvalues, but the eigenvectors are in general different.\n\n\n\n\n## Eigenvalue problems, basic definitions\nTo prove this we\nstart with the eigenvalue problem and a similarity transformed matrix $\\mathbf{B}$.\n\n$$\n\\mathbf{A}\\mathbf{x}=\\lambda\\mathbf{x} \\hspace{1cm} \\mathrm{and}\\hspace{1cm} \n \\mathbf{B}= \\mathbf{S}^T \\mathbf{A}\\mathbf{S}.\n$$\n\nWe multiply the first equation on the left by $\\mathbf{S}^T$ and insert\n$\\mathbf{S}^{T}\\mathbf{S} = \\mathbf{I}$ between $\\mathbf{A}$ and $\\mathbf{x}$. Then we get\n\n\n
                                        \n\n$$\n\\begin{equation}\n (\\mathbf{S}^T\\mathbf{A}\\mathbf{S})(\\mathbf{S}^T\\mathbf{x})=\\lambda\\mathbf{S}^T\\mathbf{x} ,\n\\label{_auto2} \\tag{3}\n\\end{equation}\n$$\n\nwhich is the same as\n\n$$\n\\mathbf{B} \\left ( \\mathbf{S}^T\\mathbf{x} \\right ) = \\lambda \\left (\\mathbf{S}^T\\mathbf{x}\\right ).\n$$\n\nThe variable $\\lambda$ is an eigenvalue of $\\mathbf{B}$ as well, but with\neigenvector $\\mathbf{S}^T\\mathbf{x}$.\n\n\n\n\n## Eigenvalue problems, basic definitions\nThe basic philosophy is to\n * Either apply subsequent similarity transformations (direct method) so that\n\n\n
                                        \n\n$$\n\\begin{equation}\n \\mathbf{S}_N^T\\dots \\mathbf{S}_1^T\\mathbf{A}\\mathbf{S}_1\\dots \\mathbf{S}_N=\\mathbf{D} ,\n\\label{_auto3} \\tag{4}\n\\end{equation}\n$$\n\n* Or apply subsequent similarity transformations so that $\\mathbf{A}$ becomes tridiagonal (Householder) or upper/lower triangular (the *QR* method to be discussed later). \n\n * Thereafter, techniques for obtaining eigenvalues from tridiagonal matrices can be used.\n\n * Or use so-called power methods\n\n * Or use iterative methods (Krylov, Lanczos, Arnoldi). These methods are popular for huge matrix problems.\n\n\n\n\n\n## Discussion of methods for eigenvalues\n**The general overview.**\n\n\nOne speaks normally of two main approaches to solving the eigenvalue problem.\n * The first is the formal method, involving determinants and the characteristic polynomial. This proves how many eigenvalues there are, and is the way most of you learned about how to solve the eigenvalue problem, but for matrices of dimensions greater than 2 or 3, it is rather impractical.\n\n * The other general approach is to use similarity or unitary tranformations to reduce a matrix to diagonal form. This is normally done in two steps: first reduce to for example a *tridiagonal* form, and then to diagonal form. The main algorithms we will discuss in detail, Jacobi's and Householder's (so-called direct method) and Lanczos algorithms (an iterative method), follow this methodology.\n\n\n\n\n## Eigenvalues methods\nDirect or non-iterative methods require for matrices of dimensionality $n\\times n$ typically $O(n^3)$ operations. These methods are normally called standard methods and are used for dimensionalities\n$n \\sim 10^5$ or smaller. A brief historical overview \n\n\n\n\n\n\n\n\n\n\n\n\n
                                        Year $n$
                                        1950 $n=20$ (Wilkinson)
                                        1965 $n=200$ (Forsythe et al.)
                                        1980 $n=2000$ Linpack
                                        1995 $n=20000$ Lapack
                                        This decade $n\\sim 10^5$ Lapack
                                        \nshows that in the course of 60 years the dimension that direct diagonalization methods can handle has increased by almost a factor of\n$10^4$ (note this is for serial versions). However, it pales beside the progress achieved by computer hardware, from flops to petaflops, a factor of almost $10^{15}$. We see clearly played out in history the $O(n^3)$ bottleneck of direct matrix algorithms.\n\nSloppily speaking, when $n\\sim 10^4$ is cubed we have $O(10^{12})$ operations, which is smaller than the $10^{15}$ increase in flops.\n\n\n\n\n## Discussion of methods for eigenvalues\nIf the matrix to diagonalize is large and sparse, direct methods simply become impractical, \nalso because\nmany of the direct methods tend to destroy sparsity. As a result large dense matrices may arise during the diagonalization procedure. The idea behind iterative methods is to project the \n$n-$dimensional problem in smaller spaces, so-called Krylov subspaces. \nGiven a matrix $\\mathbf{A}$ and a vector $\\mathbf{v}$, the associated Krylov sequences of vectors\n(and thereby subspaces) \n$\\mathbf{v}$, $\\mathbf{A}\\mathbf{v}$, $\\mathbf{A}^2\\mathbf{v}$, $\\mathbf{A}^3\\mathbf{v},\\dots$, represent\nsuccessively larger Krylov subspaces. \n\n\n\n\n\n\n\n\n\n
                                        Matrix $\\mathbf{A}\\mathbf{x}=\\mathbf{b}$ $\\mathbf{A}\\mathbf{x}=\\lambda\\mathbf{x}$
                                        $\\mathbf{A}=\\mathbf{A}^*$ Conjugate gradient Lanczos
                                        $\\mathbf{A}\\ne \\mathbf{A}^*$ GMRES etc Arnoldi
                                        \n\n\n\n\n## Eigenvalues and Lanczos' method\nBasic features with a real symmetric matrix (and normally huge $n> 10^6$ and sparse) \n$\\hat{A}$ of dimension $n\\times n$:\n\n* Lanczos' algorithm generates a sequence of real tridiagonal matrices $T_k$ of dimension $k\\times k$ with $k\\le n$, with the property that the extremal eigenvalues of $T_k$ are progressively better estimates of $\\hat{A}$' extremal eigenvalues.* The method converges to the extremal eigenvalues.\n\n* The similarity transformation is\n\n$$\n\\hat{T}= \\hat{Q}^{T}\\hat{A}\\hat{Q},\n$$\n\nwith the first vector $\\hat{Q}\\hat{e}_1=\\hat{q}_1$.\n\nWe are going to solve iteratively\n\n$$\n\\hat{T}= \\hat{Q}^{T}\\hat{A}\\hat{Q},\n$$\n\nwith the first vector $\\hat{Q}\\hat{e}_1=\\hat{q}_1$.\nWe can write out the matrix $\\hat{Q}$ in terms of its column vectors\n\n$$\n\\hat{Q}=\\left[\\hat{q}_1\\hat{q}_2\\dots\\hat{q}_n\\right].\n$$\n\n## Eigenvalues and Lanczos' method, tridiagonal matrix\nThe matrix\n\n$$\n\\hat{T}= \\hat{Q}^{T}\\hat{A}\\hat{Q},\n$$\n\ncan be written as\n\n$$\n\\hat{T} = \\left(\\begin{array}{cccccc}\n \\alpha_1& \\beta_1 & 0 &\\dots & \\dots &0 \\\\\n \\beta_1 & \\alpha_2 & \\beta_2 &0 &\\dots &0 \\\\\n 0& \\beta_2 & \\alpha_3 & \\beta_3 & \\dots &0 \\\\\n \\dots& \\dots & \\dots &\\dots &\\dots & 0 \\\\\n \\dots& & &\\beta_{n-2} &\\alpha_{n-1}& \\beta_{n-1} \\\\\n 0& \\dots &\\dots &0 &\\beta_{n-1} & \\alpha_{n} \\\\\n \\end{array} \\right)\n$$\n\n## Eigenvalues and Lanczos' method, tridiagonal and orthogonal matrices\nUsing the fact that\n\n$$\n\\hat{Q}\\hat{Q}^T=\\hat{I},\n$$\n\nwe can rewrite\n\n$$\n\\hat{T}= \\hat{Q}^{T}\\hat{A}\\hat{Q},\n$$\n\nas\n\n$$\n\\hat{Q}\\hat{T}= \\hat{A}\\hat{Q}.\n$$\n\n## Eigenvalues and Lanczos' method\nIf we equate columns\n\n$$\n\\hat{T} = \\left(\\begin{array}{cccccc}\n \\alpha_1& \\beta_1 & 0 &\\dots & \\dots &0 \\\\\n \\beta_1 & \\alpha_2 & \\beta_2 &0 &\\dots &0 \\\\\n 0& \\beta_2 & \\alpha_3 & \\beta_3 & \\dots &0 \\\\\n \\dots& \\dots & \\dots &\\dots &\\dots & 0 \\\\\n \\dots& & &\\beta_{n-2} &\\alpha_{n-1}& \\beta_{n-1} \\\\\n 0& \\dots &\\dots &0 &\\beta_{n-1} & \\alpha_{n} \\\\\n \\end{array} \\right)\n$$\n\nwe obtain\n\n$$\n\\hat{A}\\hat{q}_k=\\beta_{k-1}\\hat{q}_{k-1}+\\alpha_k\\hat{q}_k+\\beta_k\\hat{q}_{k+1}.\n$$\n\n## Eigenvalues and Lanczos' method, defining the Lanczos' vectors\nWe have thus\n\n$$\n\\hat{A}\\hat{q}_k=\\beta_{k-1}\\hat{q}_{k-1}+\\alpha_k\\hat{q}_k+\\beta_k\\hat{q}_{k+1},\n$$\n\nwith $\\beta_0\\hat{q}_0=0$ for $k=1:n-1$. Remember that the vectors $\\hat{q}_k$ are orthornormal and this implies\n\n$$\n\\alpha_k=\\hat{q}_k^T\\hat{A}\\hat{q}_k,\n$$\n\nand these vectors are called Lanczos vectors.\n\n\n\n## Eigenvalues and Lanczos' method, basic steps\nWe have thus\n\n$$\n\\hat{A}\\hat{q}_k=\\beta_{k-1}\\hat{q}_{k-1}+\\alpha_k\\hat{q}_k+\\beta_k\\hat{q}_{k+1},\n$$\n\nwith $\\beta_0\\hat{q}_0=0$ for $k=1:n-1$ and\n\n$$\n\\alpha_k=\\hat{q}_k^T\\hat{A}\\hat{q}_k.\n$$\n\nIf\n\n$$\n\\hat{r}_k=(\\hat{A}-\\alpha_k\\hat{I})\\hat{q}_k-\\beta_{k-1}\\hat{q}_{k-1},\n$$\n\nis non-zero, then\n\n$$\n\\hat{q}_{k+1}=\\hat{r}_{k}/\\beta_k,\n$$\n\nwith $\\beta_k=\\pm ||\\hat{r}_{k}||_2$.\n", "meta": {"hexsha": "6d1581af9a1a59ef85a230315ae9729058a5b92d", "size": 153131, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "doc/pub/fci/ipynb/fci.ipynb", "max_stars_repo_name": "NuclearTalent/ManyBody2018", "max_stars_repo_head_hexsha": "2339ed834777fa10f6156344f17494b9a7c0bf91", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2018-07-17T01:09:17.000Z", "max_stars_repo_stars_event_max_datetime": "2021-04-08T02:34:02.000Z", "max_issues_repo_path": "doc/pub/fci/ipynb/fci.ipynb", "max_issues_repo_name": "NuclearTalent/ManyBody2018", "max_issues_repo_head_hexsha": "2339ed834777fa10f6156344f17494b9a7c0bf91", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/pub/fci/ipynb/fci.ipynb", "max_forks_repo_name": "NuclearTalent/ManyBody2018", "max_forks_repo_head_hexsha": "2339ed834777fa10f6156344f17494b9a7c0bf91", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 8, "max_forks_repo_forks_event_min_datetime": "2018-07-16T06:31:54.000Z", "max_forks_repo_forks_event_max_datetime": "2021-05-01T07:53:38.000Z", "avg_line_length": 36.073262662, "max_line_length": 516, "alphanum_fraction": 0.5240023248, "converted": true, "num_tokens": 33527, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.6893056231680122, "lm_q1q2_score": 0.4265288324859775}} {"text": "# Problema # 1 -Equilibrio de la carga de trabajo-\n\n# Parseador de funciones \n\n### Katherine Yohanna Mazariegos Guerra\n### Abner Xocop Chacach\n\n\n\n```python\n# En archivo exercise01.txt x1, x2 y x3 es el nombre de las variables, se usaron esas para seguir con el est\u00e1ndar de nombramiento en Programaci\u00f3n Lineal.\n# Es necesario escribir si es min/max para que el programa identifique si es de maximizaci\u00f3n o minimizaci\u00f3n, st para indicar que esas son las restricciones y end para indicar al programa que el problema termina ah\u00ed\n\ndef parse_coefficients(coefficient_list, monomial):\n \"\"\"\n Este es un parseador de coeficientes. Consiste en comprobar si una cadena tiene una expresi\u00f3n regular en donde pueda extraer caracteres espec\u00edficos, en este caso se busca extraer los coeficientes.\n\n Args:\n :rtype: None\n :param coefficient_list: Lista en la que se almacenar\u00e1n los coeficientes\n :param monomial: Una cadena (por ejemplo, -3x1) que ser\u00e1 analizada hasta su coeficiente (por ejemplo, -3)\n\n Verifica qu\u00e9 patr\u00f3n coincide. V\u00e1lidos son: (s)(n)lv\n Los par\u00e9ntesis indican la existencia opcional\n s es + o - (la ausencia significa +)\n n es un n\u00famero (coeficiente, la ausencia significa 1)\n l es una letra latina min\u00fascula (letra variable)\n v es un n\u00famero, probablemente incremental (n\u00famero variable)\n\n Import re:\n Una expresi\u00f3n regular (o RE) especifica un conjunto de cadenas que se corresponde con ella; las funciones de este m\u00f3dulo le permiten comprobar si una cadena particular se corresponde con una expresi\u00f3n regular dada (o si una expresi\u00f3n regular dada se corresponde con una cadena particular, que se reduce a lo mismo)\n Source: https://docs.python.org/3/library/re.html\n \"\"\"\n import re\n\n if re.match('[ ]*[\\+ ]?[\\d]+[\\.]?[\\d]*', monomial):\n float_cast = float(re.match('[ ]*[\\+ ]?[\\d]+[\\.]?[\\d]*', monomial).group(0))\n coefficient_list.append(float_cast)\n elif re.match('[ ]*[\\-][\\d]+[\\.]?[\\d]*', monomial):\n float_cast = float(re.match('[ ]*[\\-][\\d]+[\\.]?[\\d]*', monomial).group(0))\n coefficient_list.append(float_cast)\n elif re.match('[ ]*[\\+]*[a-z][\\d]+', monomial):\n coefficient_list.append(1)\n elif re.match('[ ]*[\\-][a-z][\\d]+', monomial):\n coefficient_list.append(-1)\n```\n\n\n```python\nlines = []\ndef parse_lp1(input_filename):\n \"\"\"\n Esta funci\u00f3n es la encargada de leer el archivo, har\u00e1 uso del parser para mandar l\u00ednea a l\u00ednea el contenido del .txt y seg\u00fan los coeficientes que obtenga devolver\u00e1 las matrices y arrays correspondientes. \n\n Tiene tareas como verificar que el archivo haya sido encontrado, leer si es un problema de maximizaci\u00f3n o minimizaci\u00f3n, llenar las matrices/ arrays.\n\n :rtype : tuple\n :param input_filename: Nombre de archivo de la entrada del problema lineal\n :return: Retorna A-matrix, b-vector, c-vector, MinMax\n \"\"\"\n import re\n\n error = 0 # Inicializar la variable de error. Si el error!=0 entonces hubo un problema de entrada de archivo \n\n\n try:\n infile = open('exercise01.txt')\n except FileNotFoundError:\n error = 1\n print('\\nInput file error: Archivo no encontrado.') # Archivo no encontrado\n\n #lines = []\n if error != 1:\n for line in infile:\n lines.append(line)\n infile.close()\n for line in lines:\n print(line, end='') \n\n minmax_line = '' # Verficar si el problema es de maximizaci\u00f3n o de minimizaci\u00f3n\n for line in lines:\n if re.match('^[ ]*max|min', line):\n minmax_line = line\n\n minmax = 0\n objective_function = ''\n\n if re.match('^[ ]*max', minmax_line): #Si en el archivo se encuentra la palabra 'max' entonces el problema es de maximizaci\u00f3n\n minmax = -1\n objective_function = minmax_line\n objective_function = objective_function.strip('max')\n \n elif re.match('^[ ]*min', minmax_line): # Si en el archivo se encuentra la palabra 'min' entonces el problema es de minimizaci\u00f3n\n minmax = 1\n objective_function = minmax_line\n objective_function = objective_function.strip('min')\n\n if minmax_line == '' and minmax == 0: # Si en el archivo no se encuentra ni 'max' ni 'min' entonces no hay funci\u00f3n objetivo\n error = 2\n print('\\nInput file error: Funci\u00f3n objetivo no encontrada.')\n\n c_vector = [] # Rellenar el vector c con coeficientes de funci\u00f3n objetiva\n\n regex = re.compile('^[\\+\\- ]?[\\d]*[\\.]?[\\d]*[a-z][\\d+]')\n while regex.match(objective_function):\n monomial = regex.match(objective_function).group(0)\n parse_coefficients(c_vector, monomial)\n objective_function = objective_function.replace(monomial, '', 1)\n\n a_matrix = [] # Rellenar la matriz A (coeficientes) y el vector b utilizando las restricciones del problema\n b_vector = []\n eqin = []\n\n st_line = ''\n st_index = 0\n for index, line in enumerate(lines):\n if 'st' in line:\n st_index = index\n st_line = line\n\n if re.match('^[ ]*st', st_line):\n st_line = st_line.replace('st', ' ', 1)\n\n if st_line == '':\n error = 3\n print('\\nInput file error: L\u00ednea de restricciones no encontrada. No existe la keyword \\'st\\'.')\n\n while st_index < len(lines) - 1:\n sub_a_vector = []\n a_matrix.append(sub_a_vector)\n while True:\n st_line = st_line.strip(' ')\n if re.match('^[\\+\\- ]?[\\d]*[\\.]?[\\d]*[a-z][\\d+]', st_line):\n monomial = re.match('^[\\+\\- ]?[\\d]*[\\.]?[\\d]*[a-z][\\d+]', st_line).group(0)\n parse_coefficients(sub_a_vector, monomial)\n st_line = st_line.replace(monomial, '', 1)\n elif re.match('^[<>=]+', st_line):\n monomial = re.match('^[<>=]+', st_line).group(0)\n if monomial == '<=':\n eqin.append(-1)\n elif monomial == '>=':\n eqin.append(1)\n elif monomial == '=':\n eqin.append(0)\n else:\n error = 4\n print('\\nInput file error: Caracter inesperado; esperados <=, >=, = al menos', monomial)\n st_line = st_line.replace(monomial, '', 1)\n elif re.match('^[\\d]+', st_line):\n monomial = re.match('^[\\d]+', st_line).group(0)\n int_cast = int(re.match('^[\\d]+', st_line).group(0))\n b_vector.append(int_cast)\n st_line = st_line.replace(monomial, '', 1)\n else:\n if not sub_a_vector: # Eval\u00faa true cuando las l\u00edneas est\u00e1n vac\u00edas entre las restricciones\n a_matrix.pop()\n break\n\n st_index += 1 # Incrementar el n\u00famero de l\u00ednea y obtener el siguiente\n st_line = lines[st_index]\n\n if st_line == 'end\\n' and error == 0: # B\u00fasqueda de la declaraci\u00f3n final y ausencia de errores\n print('\\nArchivo cargado exitosamente.')\n break\n\n return a_matrix, b_vector, c_vector, eqin, minmax # Devolver todas las listas y variables creadas\n```\n\n\n```python\ndef convert_to_dual(input_filename, output_filename):\n \"\"\"\n Verifica si son restricciones de >=, <= o =. Tambi\u00e9n tiene como tarea hacer un archivo de salida en el que muestre los resultados de las matrices que se llenaron.\n\n :param input_filename: Nombre de archivo de la entrada del problema lineal\n :param output_filename: Filename of the linear problem output\n :return: Returns A-matrix, b-vector, c-vector, Variable-constraints, MinMax\n \"\"\"\n\n (a_matrix, b_vector, c_vector, eqin, minmax) = parse_lp1(input_filename) # Llamar la funci\u00f3n parse_lp1\n\n variable_constraints = [] # Convertir las restricciones a equivalentes duales '*' significa libre\n if minmax == -1:\n for el in eqin:\n if el == 0:\n variable_constraints.append('*')\n elif el == 1:\n variable_constraints.append('>=0')\n elif el == -1:\n variable_constraints.append('<=0')\n\n a_matrix = list(zip(a_matrix)) # Traspuesta de A-matrix\n\n minmax = -minmax # min(max) el problema dual es max(min)\n\n outfile = open(output_filename, 'w') # Escribir el problema a un archivo de salida\n outfile.write('(Objective Function) b-vector: [' + ', '.join(map(str, b_vector)) + ']\\n')\n\n outfile.write('\\nA-matrix: [')\n thing = ''\n for index, sub_a_vector in enumerate(a_matrix):\n thing += '[ ' + ', '.join(map(str, sub_a_vector)) + ']'\n if index != (len(a_matrix) - 1):\n thing += ', '\n outfile.write(thing + ']\\n')\n\n outfile.write('\\n(Contraints) c-vector: [' + ', '.join(map(str, c_vector)) + ']\\n')\n outfile.write('\\n(Variable Contraints) variable_constraints-vector: [' + ', '.join(map(str, c_vector)) + ']\\n')\n outfile.write('\\nEqin: [' + ', '.join(map(str, eqin)) + ']\\n')\n outfile.write('\\nMinMax: [' + str(minmax) + ']\\n')\n outfile.close()\n\n return a_matrix, b_vector, c_vector, variable_constraints, eqin, minmax\n\n(a_matrix, b_vector, c_vector, variable_contraints, eqin, minmax) = convert_to_dual('input-lp1', 'output-lp2')\n```\n\n min 42x1+87x2\n st 4x1+2x2<=480\n 3x1+6x2<=480\n end\n\n# Imports \n\n\n```python\nimport numpy as np\nimport math\nfrom sympy.abc import x,y\nfrom sympy import *\n```\n\n# Lista con todas las soluciones posibles \n\n\n```python\nlista = []\n```\n\n# Soluciones(Interceptos) en x \n\n\n```python\n#Soluciones(Interceptos) en x\ndef x_intercepts():\n\n \"\"\"\n Esta funci\u00f3n nos ayuda a encontrar los interceptos en el eje x, almacenarlos en una matriz y verificar cu\u00e1l de estos interceptos pordr\u00eda ser una soluci\u00f3n \u00f3ptima para el proyecto.\n \"\"\"\n\n new_mat= np.array(a_matrix).reshape(2,2) # Pasar los valores a una matriz operable\n new_b= np.array(b_vector).reshape(2,1)\n sol = np.linalg.solve(new_mat, new_b) # Resolver para la matriz\n\n new_mat[1,1]=0 # \u00danicamente se tomar\u00e1n los valores de x\n new_mat[0,1]=0\n new_mat_2=[new_mat[0,0], 0],[0, new_mat[1,0]]\n new_mat_2 = np.array(new_mat_2)\n sol = np.linalg.solve(new_mat_2,new_b)\n #new_mat_2= [new_mat[0,1],new_mat[1,1]]\n\n x_intercepts=[[sol[0][0],0],[sol[1][0],0]]\n #print(x_intercepts)\n return x_intercepts\n \n\n#x_intercepts()\nx_i_min = math.inf\n\nfor i in x_intercepts():\n if i[0] < x_i_min:\n x_i_min = i[0]\n\nlista.append([x_i_min, 0])\nprint(lista)\n\n```\n\n [[120.0, 0]]\n\n\n# M\u00e9todo de validaci\u00f3n de Restricciones\n\n\n```python\nequations_evalued = []\ndef check(x,y):\n for i in range(1,len(lines)-5): # recorremos el array de strings para convertir cada ecuaci\u00f3n str a symbolic.\n \n string_eq = lines[i].replace(\"max \",'').replace('min ','').replace(\"st \",'').replace(\"end\",'').replace('\\n','').replace(\" \",'').replace(\"x1\",\"*x\").replace(\"x2\",\"*y\")\n if eval(string_eq):\n equations_evalued.append(string_eq) # a\u00f1adimos cada restricci\u00f3n sym a un arreglo\n else:\n pass\n \n return (equations_evalued[0]) and (equations_evalued[1]) and (x>=0) and (y>=0) # retornamos un booleano que para chequear cada punto cuando se invoque la func check()\n\n```\n\n# Soluciones(Interceptos) en y \n\n\n```python\n#Soluciones(Interceptos) en y\ndef y_intercepts():\n\n \"\"\"\n Esta funci\u00f3n nos ayuda a encontrar los interceptos en el eje y, almacenarlos en una matriz y verificar cu\u00e1l de estos interceptos pordr\u00eda ser una soluci\u00f3n \u00f3ptima para el proyecto.\n \"\"\"\n\n new_mat= np.array(a_matrix).reshape(2,2) # Pasar los valores a una matriz operable\n new_b= np.array(b_vector).reshape(2,1)\n sol = np.linalg.solve(new_mat, new_b) # Resolver para la matriz\n\n new_mat[0,0]=0 # \u00danicamente se tomar\u00e1n los valores de y\n new_mat[1,0]=0\n new_mat_2=[0,new_mat[0,1]],[new_mat[1,1],0]\n new_mat_2 = np.array(new_mat_2)\n sol = np.linalg.solve(new_mat_2,new_b)\n y_intercepts=[[0, sol[0][0]],[0,sol[1][0]]]\n #print(y_intercepts)\n return y_intercepts\n\n#y_intercepts()\n\ny_i_min = math.inf\n\nfor i in y_intercepts():\n if i[1] < y_i_min:\n y_i_min = i[1]\n\nlista.append([0, y_i_min]) # Almacena el elemento en la lista principal\nprint(lista)\n```\n\n [[120.0, 0], [0, 80.0]]\n\n\n# Restricci\u00f3n de no negatividad \n\n\n```python\n#Funcion 0,0\ndef p_origen():\n\n \"\"\"\n El objetivo de tener esta funci\u00f3n es de ingreser el origen (0, 0) dentro de la lista principal para ser verificado si es el punto \u00f3ptimo, es decir que el origen tambi\u00e9n es tomado como un candidato.\n \"\"\"\n\n origen = [0,0]\n #print(origen)\n lista.append(origen)\nprint(lista)\n```\n\n [[120.0, 0], [0, 80.0]]\n\n\n# Soluciones de las restricciones \n\n\n```python\ndef solve_equation():\n \n \"\"\"\n Aqu\u00ed se resuelve la intersecci\u00f3n entre las dos restricciones ingresadas. Los puntos (x, y) obtenidos son ingresados a la lista principal de candidatos a ser puntos \u00f3ptimos\n \"\"\"\n\n #print(a_matrix)\n new_mat= np.array(a_matrix).reshape(2,2)\n new_b= np.array(b_vector).reshape(2,1)\n sol = np.linalg.solve(new_mat, new_b)\n return sol\n\nsolve_e = [[solve_equation()[0][0], solve_equation()[1][0]]]\nprint(solve_e[0])\n\nlista.append(solve_e[0])\nprint(lista)\n\n```\n\n [106.66666666666667, 26.666666666666668]\n [[120.0, 0], [0, 80.0], [106.66666666666667, 26.666666666666668]]\n\n\n# Coeficientes funci\u00f3n objetivo \n\n\n```python\n# Se imprimen los coeficientes de la funci\u00f3n objetivo\nprint(c_vector)\n```\n\n [42.0, 87.0]\n\n\n\n```python\nsolved = False\n\n\nmaxi = 0\nx_max = 0\ny_max = 0\n```\n\n# Resultados \n\n\n```python\nsolved = False\n\n\nmaxi = 0\nx_max = 0\ny_max = 0\n\ndef resultado(): \n global maxi\n for each_point in lista:\n #print(f\"each_point {each_point}\")\n solved = check(each_point[0],each_point[1])\n #print(f\"eval_eq_arr {equations_evalued}\")\n # print(check(each_point[0],each_point[1]))\n \n if solved is False:\n lista.remove(each_point[0],each_point[1])\n solved = True\n \n if solved:\n \"\"\"\n En esta funci\u00f3n se hacen las verificaciones para saber cu\u00e1l de los puntos candidatos hacen la optimizaci\u00f3n de la funci\u00f3n objetivo. La variable maxi es la que indica la maximizaci\u00f3n, y_max y x_max son los puntos que optimizan la funci\u00f3n con las restricciones dadas\n \"\"\"\n\n \n for i in lista:\n temporal = c_vector[0]*i[0]+c_vector[1]*i[1]\n if temporal > maxi:\n maxi = temporal\n print(f\"temporal: {temporal}\")\n y_max = i[1]\n x_max = i[0]\n \n print(\"\\n\\n=== Resultados ===\")\n\n print('c-vector:', c_vector)\n print('A-matrix:', a_matrix)\n print('b-vector:', b_vector)\n print('Variable-contraints-vector:', variable_contraints)\n #print('Eqin:', eqin)\n print('MinMax:', minmax)\n print('Punto y:', y_max)\n print(\"Punto x: \", x_max)\n print('Resultado optimizaci\u00f3n:', maxi)\n\n else:\n print(\"ERROR PLS FIX ME :(\")\n\n(a_matrix, b_vector, c_vector, variable_contraints, eqin, minmax) = convert_to_dual('input-lp1', 'output-lp2')\n\n\n\nresultado()\nprint(f\"\\nmaxi {maxi}\")\n\n```\n\n min 42x1+87x2\n st 4x1+2x2<=480\n 3x1+6x2<=480\n endmin 42x1+87x2\n st 4x1+2x2<=480\n 3x1+6x2<=480\n endtemporal: 5040.0\n temporal: 6960.0\n \n \n === Resultados ===\n c-vector: [42.0, 87.0]\n A-matrix: [([4.0, 2.0],), ([3.0, 6.0],)]\n b-vector: [480, 480]\n Variable-contraints-vector: []\n MinMax: -1\n Punto y: 80.0\n Punto x: 0\n Resultado optimizaci\u00f3n: 6960.0\n \n maxi 6960.0\n\n\n# Sensibilidad \n\n\n```python\n\"\"\"\nEsto \u00fanicamente imprime las matrices importantes para la soluci\u00f3n del problema.\n c-vector es para el vector de la funci\u00f3n objetivo\n A-matrix es la matriz que contiene las restricciones\n b-vector es el vector que contiene los coeficientes del lado derecho de las restricciones\n MinMax indica si es maximizaci\u00f3n o minimizaci\u00f3n\n\"\"\"\n\nprint('\\n===Results===')\n\nprint('c-vector:', c_vector)\nprint('A-matrix:', a_matrix)\nprint('b-vector:', b_vector)\n#print('Variable-contraints-vector:', variable_contraints)\n#print('Eqin:', eqin)\nprint('MinMax:', minmax)\n```\n\n \n ===Results===\n c-vector: [42.0, 87.0]\n A-matrix: [([4.0, 2.0],), ([3.0, 6.0],)]\n b-vector: [480, 480]\n Variable-contraints-vector: []\n Eqin: [-1, -1]\n MinMax: -1\n\n\n\n```python\n\"\"\"\nLa matriz A-matrix generada en pasos anteriores se tiene lo siguiente\nA-matrix: [([4.0, 2.0],), ([3.0, 6.0],)]\n\nAqu\u00ed lo que se hace es hacer dos matrices nuevas, una llamada a_matrix_const1 para que contenga a [4.0, 2.0] y otra llamada \na_matrix_const2 = a_matrix[1][0] que contiene los valores [3.0, 6.0]\n\"\"\"\n\na_matrix_const1 = a_matrix[0][0]\nprint(a_matrix_const1)\na_matrix_const2 = a_matrix[1][0]\nprint(a_matrix_const2)\na_matrix_const1[1]\n```\n\n [4.0, 2.0]\n [3.0, 6.0]\n\n\n\n\n\n 2.0\n\n\n\n\n```python\n\"\"\"\nAqu\u00ed se calculan las pendientes de los constraints y la pendiente de la funci\u00f3n objetivo\n m_const1 es la pendiente de la constraint 1\n m_const2 es la pendiente de la constraint 2\n m_c1 es la pendiente de la funci\u00f3n objetivo, siendo c1 el primer coeficiente de la funci\u00f3n\n m_c2 es la pendiente de la funci\u00f3n objetivo, siendo c2 el segundo coeficiente de la funci\u00f3n\n \nEs MENESTER mencionar que c1 es el primer coeficiente de la funci\u00f3n objetivo, por lo que todo lo que a continuaci\u00f3n se menciona\ncomo c1, se refiere a este n\u00famero. c2 es el segundo coeficiente de la funci\u00f3n objetivo, al igual que c1, a partir de aqu\u00ed\nc2 se refiere al n\u00famero indicado.\n\"\"\"\n\nm_const1 = -1 * a_matrix_const1[0]/a_matrix_const1[1]\n\nm_const2 = -1 * a_matrix_const2[0]/a_matrix_const2[1]\n\nm_c1 = -1 / c_vector[1]\n\nm_c2 = -c_vector[0]\n\nprint(f\"m_const1: {m_const1}\")\nprint(f\"m_const2: {m_const2}\")\nprint(f\"m_c1: {m_c1}\")\nprint(f\"m_c2: {m_c2}\")\n```\n\n m_const1: -2.0\n m_const2: -0.5\n m_c1: -0.011494252873563218\n m_c2: -42.0\n\n\n\n```python\n\"\"\"\nAqu\u00ed se analizan los rangos en los que c1 y c2 son v\u00e1lidos. Lo que hace es analizar las pendientes para c1 y c2 contra\nlas pendientes de las restricciones 1 y 2. Al dividirse, se obtiene tanto el l\u00edmite superior como el l\u00edmite inferior permitido\ntanto para c1 como para c2\n\"\"\"\n\nlimit_lo_c1 = m_const1 / m_c1\n\nlimit_up_c1 = m_const2 / m_c1\n\nlimit_lo_c2 = m_c2 / m_const2\n\nlimit_up_c2 = m_c2 / m_const1\n\n#print(f\"L\u00edmite c1 permitido:\\n{limit_lo_c1}\")\n#print(f\"{limit_up_c1}\")\n#print(f\"L\u00edmite c2 permitido:\\n{limit_lo_c2}\")\n#print(f\"{limit_up_c2}\")\n```\n\n L\u00edmite c1 permitido:\n 174.0\n 43.5\n L\u00edmite c2 permitido:\n 84.0\n 21.0\n\n\n\n```python\n\"\"\"\nAqu\u00ed se calculan los decreases and increases permitidos. Lo que hace es tomar el valor de c1, eval\u00faa el l\u00edmite superior\nobtenido, la diferencia obtenida es el increase permitido. Luego toma el mismo valor de c1, eval\u00faa el l\u00edmite inferior, la\ndiferencia entre ambos es el decrease permitido. Luego eval\u00faa el valor de c2 y hace lo mismo en los l\u00edmites para calcular el\nincrease y decrease permitidos para c2.\n\"\"\"\n\n#1\nallow_increasex = limit_up_c1 - c_vector[0]\nif allow_increasex < 0:\n print(f\"Incremento en x permitido: infinito\")\nelse:\n print(f\"allow_increasex: {allow_increasex}\")\n#2\nallow_decreasex = c_vector[0] - limit_lo_c1 \n\nif allow_decreasex < 0:\n print(f\"allow_decreasex: infinito\")\nelse:\n print(f\"allow_decreasex: {allow_decreasex}\")\n\n#3\nallow_increasey = limit_up_c2 - c_vector[1]\nif allow_increasey < 0:\n print(f\"allow_increasey: infinito\")\nelse:\n print(f\"allow_increasey: {allow_increasey}\")\n\n #4\nallow_decreasey = c_vector[1] - limit_lo_c2\nif allow_decreasey < 0:\n print(f\"allow_decreasey: infinito\")\nelse:\n print(f\"allow_decreasey: {allow_decreasey}\")\n\n```\n\n allow_increasex: 1.5\n allow_decreasex: infinito\n allow_increasey: infinito\n allow_decreasey: 3.0\n\n", "meta": {"hexsha": "fa60d5686e546e2b11dfde981f66a78cfa50883c", "size": 29540, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Sensibility/Sensibility.ipynb", "max_stars_repo_name": "abnerxch/linear-programming", "max_stars_repo_head_hexsha": "9108deb4cb553e8e490927fb10caecb3751202fb", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Sensibility/Sensibility.ipynb", "max_issues_repo_name": "abnerxch/linear-programming", "max_issues_repo_head_hexsha": "9108deb4cb553e8e490927fb10caecb3751202fb", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Sensibility/Sensibility.ipynb", "max_forks_repo_name": "abnerxch/linear-programming", "max_forks_repo_head_hexsha": "9108deb4cb553e8e490927fb10caecb3751202fb", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 33.3408577878, "max_line_length": 331, "alphanum_fraction": 0.5211577522, "converted": true, "num_tokens": 5864, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.6893056167854461, "lm_q1q2_score": 0.4265288285365704}} {"text": "```python\nfrom IPython.display import HTML\ntag = HTML('''\nToggle cell visibility here.''')\ndisplay(tag)\n```\n\n\n\nToggle cell visibility here.\n\n\n\n```python\n# Erasmus+ ICCT project (2018-1-SI01-KA203-047081)\n\n%matplotlib notebook\nimport numpy as np\nimport math\nimport matplotlib.pyplot as plt\nfrom scipy import signal\nimport ipywidgets as widgets\nimport control as c\nimport sympy as sym\nfrom IPython.display import Latex, display, Markdown # For displaying Markdown and LaTeX code\nfrom fractions import Fraction\nimport matplotlib.patches as patches\n```\n\n## First-order systems with no zeros\n\n### Introduction\n\nFirst-order systems without zeros are characterized by the following transfer function:\n\n\\begin{equation}\n G(s)=\\frac{k}{s+k}.\n\\end{equation}\n\nThe $k$ value is important since it defines the following parameters:\n- $1/k$ denotes the *time constant* of the response, which defines the time needed for the step response to reach $\\approx$ 63% of its final value.\n- $t_r$ denotes the *rise time*, i.e. the time needed for the system response to go from 10\\% to 90\\% of the steady state value.\n- $t_s$ denotes the *settling time*, i.e. the time at which the system response is outside the error band (e.g. 2\\% as set in the example below) for the last time.\n\nThe step response of these systems is given by:\n\n\\begin{equation}\n c(t)=1-e^{-at},\n\\end{equation}\n\nwhere the forced response is equal to $1$ and natural response to $-e^{-at}$.\n\n---\n\n### How to use this notebook?\n\nMove the slider to define the $k$ value in the transfer function of the first-order system - $G(s)=\\frac{k}{s+k}$ and observe the unit step time response of the defined system.\n\n\n```python\n# set up plot\nfig, ax = plt.subplots(figsize=[9.8,4],num='First-order system')\nax.set_ylim([-1, 2])\nax.set_xlim([0, 5])\nax.grid(True)\nax.set_title ('Time response')\nax.set_xlabel('$t$ [s]')\nax.set_ylabel('Input, output')\nxaxis = ax.axhline(y=0,color='k',lw=1)\n\nresponse, = ax.plot([], [])\nslope, = ax.plot([], [])\nx1a, = ax.plot([], [])\ny1a, = ax.plot([], [])\ntr11, = ax.plot([], [])\ntrv1, = ax.plot([], [])\ntrv2, = ax.plot([], [])\ntrh1, = ax.plot([], [])\ntrh2, = ax.plot([], [])\nts11, = ax.plot([], [])\nts1, = ax.plot([], [])\nts2, = ax.plot([], [])\ntexttr=ax.text(0,0,'')\ntextts=ax.text(0,0,'')\n\nax.step([0,5],[0,1],color='C0',label='input')\n\n# generate x values\nt = np.linspace(0, 2 * np.pi, 10000)\n \ndef response_func(t, k):\n \"\"\"\"Return response function\"\"\"\n return 1-np.exp(-k*t)\n\n@widgets.interact(k=(1, 5, 1))\n\n\ndef update(k=1):\n \"\"\"Remove old lines from plot and plot new one\"\"\"\n global response,slope,x1a,y1a,tr11,trv1,trv2,trh1,trh2,ts11,ts1,ts2,texttr,textts\n ax.lines.remove(response)\n ax.lines.remove(slope)\n ax.lines.remove(x1a)\n ax.lines.remove(y1a)\n ax.lines.remove(tr11)\n ax.lines.remove(trv1)\n ax.lines.remove(trv2)\n ax.lines.remove(trh1)\n ax.lines.remove(trh2)\n ax.lines.remove(ts11)\n ax.lines.remove(ts1)\n ax.lines.remove(ts2)\n texttr.remove()\n textts.remove()\n response, = ax.plot(t, response_func(t,k), color='C1',lw=2)\n response.set_label('output')\n slope, = ax.plot([0,1/k], [0,1], color='C2',lw=2)\n slope.set_label('initial slope')\n x1a, = ax.plot([1/k,1/k],[0,1-np.exp(-1)],'--',color='k',lw=.8)\n y1a, = ax.plot([0,1/k],[1-np.exp(-1),1-np.exp(-1)],'--',color='k',lw=.8)\n# rise time\n tr11, = ax.plot([-np.log(0.9)/k,-np.log(0.1)/k],[-0.5,-0.5],color='k',lw=.8)\n trv1, = ax.plot([-np.log(0.9)/k,-np.log(0.9)/k],[-0.5,0.1],'--',color='k',lw=.8)\n trv2, = ax.plot([-np.log(0.1)/k,-np.log(0.1)/k],[-0.5,0.9],'--',color='k',lw=.8)\n trh1, = ax.plot([0,-np.log(0.9)/k],[0.1,0.1],'--',color='k',lw=.8)\n trh2, = ax.plot([0,-np.log(0.1)/k],[0.9,0.9],'--',color='k',lw=.8)\n# settling time\n ts11, = ax.plot([0,-np.log(0.02)/k],[-0.7,-0.7],color='k',lw=.8)\n ts1, = ax.plot([0,0],[-0.7,0],'--',color='k',lw=.8)\n ts2, = ax.plot([-np.log(0.02)/k,-np.log(0.02)/k],[-0.7,0.98],'--',color='k',lw=.8)\n ax.legend()\n texttr=ax.text((-np.log(0.1)/k-(-np.log(0.9)/k))/2,-0.45, '$t_r$',fontsize=13)\n textts=ax.text((-np.log(0.02)/k)/2-0.1,-0.65, '$t_s$',fontsize=13)\n\n plt.xticks([0,1/k,2,4], [0,'${1}/{%s}$'%k,2,4],fontsize=8)\n plt.yticks([0.1,0.5,0.63,0.9,1,1.5,2], [0.1,0.5,0.63,0.9,1,1.5,2],fontsize=8)\n \n num1=[k]\n den1=[1,k]\n display(Markdown('Transfer function of the system $G(s)$ is equal to:'))\n tf_sys1=c.TransferFunction(num1,den1)\n s=sym.Symbol('s')\n eq=(k/(s+k))\n display(eq)\n```\n\n\n \n\n\n\n\n\n\n\n interactive(children=(IntSlider(value=1, description='k', max=5, min=1), Output()), _dom_classes=('widget-inte\u2026\n\n\n\n```python\n\n```\n\n## Second-order systems\n\n### Introduction\n\nIn contrast to the first-order systems presented above, in which the parameter $k$ only affected the speed of the response, changes of the analogue parameters in the second order systems may affect the actual form of the response. The following four responses are possible in these systems:\n- *overdamped* response,\n- *underdamped* response,\n- *undapmed* response, and\n- *critically damped* response.\n\n### How to use this notebook?\n\nMove the slider to define the values of $a$ and $b$ in the transfer function of the second-order system of the form $G(s)=\\frac{b}{s^2+as+b}$ and observe the pole-zero plot and the unit step time response of the defined system.\n\n\n```python\n# set up plot\nfig1, ax1 = plt.subplots(1,2,figsize=[9.8,4],num='Second-order system')\nax1[0].set_ylim([-3.5, 3])\nax1[1].set_ylim([0, 2.5])\n# ax1.set_xlim([0, 5])\nax1[0].grid(True)\nax1[1].grid(True)\nax1[0].axhline(y=0,color='k',lw=.8)\nax1[1].axhline(y=0,color='k',lw=.8)\nax1[0].axvline(x=0,color='k',lw=.8)\nax1[1].axvline(x=0,color='k',lw=.8)\nax1[0].set_xlabel('Re')\nax1[0].set_ylabel('Im')\nax1[1].set_xlabel('$t$ [s]')\nax1[1].set_ylabel('Input, output')\nax1[0].set_title('Pole-zero plot')\nax1[1].set_title('Time response')\n\nt = np.linspace(0, 20, 10000)\n\ntextGs = ax1[0].text(0,0,'')\n\nax1[1].step([0,20],[0,1],color='C0',label='input')\n\nplotzero, = ax1[0].plot([], [])\nresponse2, = ax1[1].plot([], [])\n\ndef response_func2(t, a, b):\n num_sys=np.array([b])\n den_sys=np.array([1,a,b])\n tf_sys=c.TransferFunction(num_sys,den_sys)\n poles_sys,zeros_sys=c.pzmap(tf_sys, Plot=False)\n T, yout = c.step_response(tf_sys,t)\n return T, yout, poles_sys, tf_sys\n \n@widgets.interact(a=(0, 10, 1),b=(1,10,1))\n\ndef update(a=7,b=9):\n \"\"\" Update plots \"\"\"\n global response2, plotzero, textGs\n ax1[0].lines.remove(plotzero)\n ax1[1].lines.remove(response2)\n# textGs.remove()\n T, yout, poles_sys, tf_sys = response_func2(t, a, b)\n plotzero, = ax1[0].plot(np.real(poles_sys), np.imag(poles_sys), 'xg', markersize=10, label = 'Poles')\n# textGs = ax1[0].text(-7,1,tf_sys)\n response2, = ax1[1].plot(T,yout,color='C1',label='output')\n s=sym.Symbol('s')\n eq=b/(s**2+a*s+b)\n coeff = [1,a,b]\n rootsdenom=np.roots(coeff)\n eq2=b/((s-rootsdenom[0])*(s-rootsdenom[1]))\n display(Markdown('Transfer function of the system $G(s)$ is equal to:'))\n display(eq),display(Markdown('or')),display(eq2)\n\n if np.imag(poles_sys)[0] == 0 and np.imag(poles_sys)[1] == 0 and np.real(poles_sys)[0] < 0 and np.real(poles_sys)[1] < 0 and np.real(poles_sys)[0]!=np.real(poles_sys)[1]:\n display(Markdown('The system is **overdamped**, because both poles have only negative real parts.'))\n elif math.isclose(0, np.imag(poles_sys)[0], abs_tol=10**-6) and math.isclose(0, np.imag(poles_sys)[1], abs_tol=10**-6) and np.real(poles_sys)[1] < 0 and np.real(poles_sys)[0]==np.real(poles_sys)[1]:\n display(Markdown('The system is **critically damped** beacuse there is a double pole with negative real part only.'))\n elif np.real(poles_sys)[0] == 0 and np.real(poles_sys)[1] == 0:\n display(Markdown('The system is **undamped**, because the poles have only imaginary parts.'))\n elif np.imag(poles_sys)[0] != 0 and np.imag(poles_sys)[1] != 0 and np.real(poles_sys)[0] != 0 and np.real(poles_sys)[1] != 0:\n display(Markdown('The system is **underdamped** beacuse both poles have negative real and non-zero complex part.'))\n ax1[0].legend()\n ax1[1].legend()\n```\n\n\n \n\n\n\n\n\n\n\n interactive(children=(IntSlider(value=7, description='a', max=10), IntSlider(value=9, description='b', max=10,\u2026\n\n", "meta": {"hexsha": "483799457029575f95da80f9f6feb1b61c7b7e66", "size": 201626, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ICCT_en/examples/02/.ipynb_checkpoints/TD-10-First-and-second-order-systems-Basics-checkpoint.ipynb", "max_stars_repo_name": "ICCTerasmus/ICCT", "max_stars_repo_head_hexsha": "fcd56ab6b5fddc00f72521cc87accfdbec6068f6", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2021-05-22T18:42:14.000Z", "max_stars_repo_stars_event_max_datetime": "2021-10-03T14:10:22.000Z", "max_issues_repo_path": "ICCT/ENG/examples/02/TD-10-First-and-second-order-systems-Basics.ipynb", "max_issues_repo_name": "tuxsaurus/ICCT", "max_issues_repo_head_hexsha": "30d1aea4fb056c9736c9b4c5a0f50fff14fa6382", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ICCT/ENG/examples/02/TD-10-First-and-second-order-systems-Basics.ipynb", "max_forks_repo_name": "tuxsaurus/ICCT", "max_forks_repo_head_hexsha": "30d1aea4fb056c9736c9b4c5a0f50fff14fa6382", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2021-05-24T11:40:09.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-29T16:36:18.000Z", "avg_line_length": 104.145661157, "max_line_length": 59873, "alphanum_fraction": 0.7683731265, "converted": true, "num_tokens": 2850, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056040203135, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.42652882063775616}} {"text": "# 5 - Convolutional Sequence to Sequence Learning\n\nIn this notebook we'll be implementing the [Convolutional Sequence to Sequence Learning](https://arxiv.org/abs/1705.03122) model. \n\n\n\n## Introduction\n\nThis model is drastically different to the previous models used in these tutorials. There is are no recurrent components used at all. Instead it makes use of convolutional layers, typically used for image processing. For an introduction to convolutional layers on text for sentiment analysis, see [this](https://github.com/bentrevett/pytorch-sentiment-analysis/blob/master/4%20-%20Convolutional%20Sentiment%20Analysis.ipynb) tutorial. \n\nIn short, a convolutional layer uses *filters*. These filters have a *width* (and also a *height* in images, but usually not text). If a filter has a width of 3, then it can see 3 consecutive tokens. Each convolutional layer has many of these filters (1024 in this tutorial). Each filter will slide across the sequence, from beginning to the end, looking at all 3 consectuive tokens at a time. The idea is that each of these 1024 filters will learn to extract a different feature from the text. The result of this feature extraction will then be used by the model - potentially as input to another convolutional layer. This can then all be used to extract features from the source sentence to translate it into the target language.\n\n\n## Preparing the Data\n\nFirst, let's import all the required modules and set the random seeds for reproducability.\n\n\n```python\nimport torch\nimport torch.nn as nn\nimport torch.optim as optim\nimport torch.nn.functional as F\n\nfrom torchtext.datasets import TranslationDataset, Multi30k\nfrom torchtext.data import Field, BucketIterator\n\nimport matplotlib.pyplot as plt\nimport matplotlib.ticker as ticker\n\nimport spacy\n\nimport random\nimport math\nimport time\n```\n\n\n```python\nSEED = 1234\n\nrandom.seed(SEED)\ntorch.manual_seed(SEED)\ntorch.backends.cudnn.deterministic = True\n```\n\nNext, we'll load the spaCy models and define the tokenizers for the source and target languages.\n\n\n```python\nspacy_de = spacy.load('de')\nspacy_en = spacy.load('en')\n```\n\n\n```python\ndef tokenize_de(text):\n \"\"\"\n Tokenizes German text from a string into a list of strings\n \"\"\"\n return [tok.text for tok in spacy_de.tokenizer(text)]\n\ndef tokenize_en(text):\n \"\"\"\n Tokenizes English text from a string into a list of strings\n \"\"\"\n return [tok.text for tok in spacy_en.tokenizer(text)]\n```\n\nNext, we'll set up the `Field`s which decide how the data will be processed. By default RNN models in PyTorch require the sequence to be a tensor of shape **[sequence length, batch size]** so TorchText will, by default, return batches of tensors in the same shape. However in this notebook we are using CNNs which expect the batch dimension to be first. We tell TorchText to have batches be **[batch size, sequence length]** by setting `batch_first = True`. \n\nWe also append the start and end of sequence tokens as well as lowercasing all text.\n\n\n```python\nSRC = Field(tokenize = tokenize_de, \n init_token = '', \n eos_token = '', \n lower = True, \n batch_first = True)\n\nTRG = Field(tokenize = tokenize_en, \n init_token = '', \n eos_token = '', \n lower = True, \n batch_first = True)\n```\n\nThen, we load our dataset.\n\n\n```python\ntrain_data, valid_data, test_data = Multi30k.splits(exts=('.de', '.en'), \n fields=(SRC, TRG))\n```\n\nWe build our vocabulary as before, by converting any tokens that appear less than 2 times into `` tokens.\n\n\n```python\nSRC.build_vocab(train_data, min_freq = 2)\nTRG.build_vocab(train_data, min_freq = 2)\n```\n\nThe final bit of data preparation is defining the device and then building the iterator.\n\n\n```python\ndevice = torch.device('cuda' if torch.cuda.is_available() else 'cpu')\n```\n\n\n```python\nBATCH_SIZE = 128\n\ntrain_iterator, valid_iterator, test_iterator = BucketIterator.splits(\n (train_data, valid_data, test_data), \n batch_size = BATCH_SIZE,\n device = device)\n```\n\n## Building the Model\n\nNext up is building the model. As before, the model is made of an encoder and decoder. The encoder *encodes* the input sentence, in the source language, into a *context vector*. The decoder *decodes* the context vector to produce the output sentence in the target language.\n\n### Encoder\n\nPrevious models in these tutorials had an encoder that compresses an entire input sentence into a single context vector, $z$. The convolutional sequence-to-sequence model is a little different - it gets two context vectors for each token in the input sentence. So, if our input sentence had 6 tokens, we would get 12 context vectors, two for each token. \n\nThe two context vectors per token are a *conved* vector and a *combined* vector. The conved vector is the result of each token being passed through a few layers - which we will explain shortly. The combined vector comes from the sum of the convolved vector and the embedding of that token. Both of these are returned by the encoder to be used by the decoder.\n\nThe image below shows the result of an input sentence - *zwei menschen fechten.* - being passed through the encoder.\n\n\n\nFirst, the token is passed through a *token embedding layer* - which is standard for neural networks in natural language processing. However, as there are no recurrent connections in this model it has no idea about the order of the tokens within a sequence. To rectify this we have a second embedding layer, the *positional embedding layer*. This is a standard embedding layer where the input is not the token itself but the position of the token within the sequence - starting with the first token, the `` (start of sequence) token, in position 0.\n\nNext, the token and positional embeddings are elementwise summed together to get a vector which contains information about the token and also its position with in the sequence - which we simply call the *embedding vector*. This is followed by a linear layer which transforms the embedding vector into a vector with the required hidden dimension size. \n\nThe next step is to pass this hidden vector into $N$ *convolutional blocks*. This is where the \"magic\" happens in this model and we will detail the contents of the convolutional blocks shortly. After passing through the convolutional blocks, the vector is then fed through another linear layer to transform it back from the hidden dimension size into the embedding dimension size. This is our *conved* vector - and we have one of these per token in the input sequence. \n\nFinally, the conved vector is elementwise summed with the embedding vector via a residual connection to get a *combined* vector for each token. Again, there is a combined vector for each token in the input sequence.\n\n### Convolutional Blocks\n\nSo, how do these convolutional blocks work? The below image shows 2 convolutional blocks with a single filter (blue) that is sliding across the tokens within the sequence. In the actual implementation we will have 10 convolutional blocks with 1024 filters in each block.\n\n\n\nFirst, the input sentence is padded. This is because the convolutional layers will reduce the length of the input sentence and we want the length of the sentence coming into the convolutional blocks to equal the length of it coming into the convolutional blocks. Without padding, the length of the sequence coming out of a convolutional layer will be `filter_size - 1` shorter than the sequence entering the convolutional layer. For example, if we had a filter size of 3, the sequence will be 2 elements shorter. Thus, we pad the sentence with one padding element on each side. We can calculate the amount of padding on each side by simply doing `(filter_size - 1)/2` for odd sized filters - we will not cover even sized filters in this tutorial.\n\nThese filters are designed so the output hidden dimension of them is twice the input hidden dimension. In computer vision terminology these hidden dimensions are called *channels* - but we will stick to calling them hidden dimensions. Why do we double the size of the hidden dimension leaving the convolutional filter? This is because we are using a special activation function called *gated linear units* (GLU). GLUs have gating mechanisms (similar to LSTMs and GRUs) contained within the activation function and actually half the size of the hidden dimension - whereas usually activation functions keep the hidden dimensions the same size.\n\nAfter passing through the GLU activation the hidden dimension size for each token is the same as it was when it entered the convolutional blocks. It is now elementwise summed with its own vector before it was passed through the convolutional layer. \n\nThis concludes a single convolutional block. Subsequent blocks take the output of the previous block and perform the same steps. Each block has their own parameters, they are not shared between blocks. The output of the last block goes back to the main encoder - where it is fed through a linear layer to get the conved output and then elementwise summed with the embedding of the token to get the combined output.\n\n### Encoder Implementation\n\nTo keep the implementation simple, we only allow for odd sized kernels. This allows padding to be added equally to both sides of the source sequence.\n\nThe `scale` variable is used by the authors to \"ensure that the variance throughout the network does not change dramatically\". The performance of the model seems to vary wildly using different seeds if this is not used.\n\nThe positional embedding is initialized to have a \"vocabulary\" of 100. This means it can handle sequences up to 100 elements long, indexed from 0 to 99. This can be increased if used on a dataset with longer sequences.\n\n\n```python\nclass Encoder(nn.Module):\n def __init__(self, input_dim, emb_dim, hid_dim, n_layers, kernel_size, dropout, device):\n super().__init__()\n \n assert kernel_size % 2 == 1, \"Kernel size must be odd!\"\n \n self.device = device\n \n self.scale = torch.sqrt(torch.FloatTensor([0.5])).to(device)\n \n self.tok_embedding = nn.Embedding(input_dim, emb_dim)\n self.pos_embedding = nn.Embedding(100, emb_dim)\n \n self.emb2hid = nn.Linear(emb_dim, hid_dim)\n self.hid2emb = nn.Linear(hid_dim, emb_dim)\n \n self.convs = nn.ModuleList([nn.Conv1d(in_channels = hid_dim, \n out_channels = 2 * hid_dim, \n kernel_size = kernel_size, \n padding = (kernel_size - 1) // 2)\n for _ in range(n_layers)])\n \n self.dropout = nn.Dropout(dropout)\n \n def forward(self, src):\n \n #src = [batch size, src sent len]\n \n #create position tensor\n pos = torch.arange(0, src.shape[1]).unsqueeze(0).repeat(src.shape[0], 1).to(self.device)\n \n #pos = [0, 1, 2, 3, ..., src sent len - 1]\n \n #pos = [batch size, src sent len]\n \n #embed tokens and positions\n tok_embedded = self.tok_embedding(src)\n pos_embedded = self.pos_embedding(pos)\n \n #tok_embedded = pos_embedded = [batch size, src sent len, emb dim]\n \n #combine embeddings by elementwise summing\n embedded = self.dropout(tok_embedded + pos_embedded)\n \n #embedded = [batch size, src sent len, emb dim]\n \n #pass embedded through linear layer to convert from emb dim to hid dim\n conv_input = self.emb2hid(embedded)\n \n #conv_input = [batch size, src sent len, hid dim]\n \n #permute for convolutional layer\n conv_input = conv_input.permute(0, 2, 1) \n \n #conv_input = [batch size, hid dim, src sent len]\n \n #begin convolutional blocks...\n \n for i, conv in enumerate(self.convs):\n \n #pass through convolutional layer\n conved = conv(self.dropout(conv_input))\n\n #conved = [batch size, 2 * hid dim, src sent len]\n\n #pass through GLU activation function\n conved = F.glu(conved, dim = 1)\n\n #conved = [batch size, hid dim, src sent len]\n \n #apply residual connection\n conved = (conved + conv_input) * self.scale\n\n #conved = [batch size, hid dim, src sent len]\n \n #set conv_input to conved for next loop iteration\n conv_input = conved\n \n #...end convolutional blocks\n \n #permute and convert back to emb dim\n conved = self.hid2emb(conved.permute(0, 2, 1))\n \n #conved = [batch size, src sent len, emb dim]\n \n #elementwise sum output (conved) and input (embedded) to be used for attention\n combined = (conved + embedded) * self.scale\n \n #combined = [batch size, src sent len, emb dim]\n \n return conved, combined\n```\n\n### Decoder\n\nThe decoder takes in the actual target sentence and tries to predict it. This model differs from the recurrent neural network models previously detailed in these tutorials as it predicts all tokens within the target sentence in parallel. There is no sequential processing, i.e. no decoding loop. This will be detailed further later on in the tutorials.\n\nThe decoder is similar to the encoder, with a few changes to both the main model and the convolutional blocks inside the model.\n\n\n\nFirst, the embeddings do not have a residual connection that connects after the convolutional blocks and the transformation. Instead the embeddings are fed into the convolutional blocks to be used as residual connections there.\n\nSecond, to feed the decoder information from the encoder, the encoder conved and combined outputs are used - again, within the convolutional blocks. \n\nFinally, the output of the decoder is a linear layer from embedding dimension to output dimension. This is used make a prediction about what the next word in the translation should be.\n\n### Decoder Convolutional Blocks\n\nAgain, these are similar to the convolutional blocks within the encoder, with a few changes.\n\n\n\nFirst, the padding. Instead of padding equally on each side to ensure the length of the sentence stays the same throughout, we only pad at the beginning of the sentence. As we are processing all of the targets simultaneously in parallel, and not sequentially, we need a method of only allowing the filters translating token $i$ to only look at tokens before word $i$. If they were allowed to look at token $i+1$ (the token they should be outputting), the model will simply learn to output the next word in the sequence by directly copying it, without actually learning how to translate.\n\nLet's see what happens if we **incorrectly** padded equally on each side, like we do in the decoder.\n\n\n\nThe filter at the first position, which is trying use the first word in the sequence, `` to predict the second word, `two`, can now directly see the word `two`. This is the same for every position, the word the model trying to predict is the second element covered by the filter. Thus, the filters can learn to simply copy the second word at each position allowing for perfect translation without actually learning how to translate.\n\nSecond, after the GLU activation and before the residual connection, the block calculates and applies attention - using the encoded representations and the embedding of the current word. **Note**: we only show the connections to the rightmost token, but they are actually connected to all tokens - this was done for clarity. Each token input uses their own, and only their own, embedding for their own attention calculation.\n\nThe attention is calculated by first using a linear layer to change the hidden dimension to the same size as the embedding dimension. Then the embedding summed via a residual connection. This combination then has the standard attention calculation applied by finding how much it \"matches\" with the *encoded conved* and then this is applied by getting a weighted sum over the *encoded combined*. This is then projected back up to the hidden dimenson size and a residual connection to the initial input to the attention layer is applied.\n\nWhy do they calculate attention first with the encoded conved and then use it to calculate the weighted sum over the encoded combined? The paper argues that the encoded conved is good for getting a larger context over the encoded sequence, whereas the encoded combined has more information about the specific token and is thus therefore more useful for makng a prediction.\n\n### Decoder Impementation\n\nAs we only pad on one side the decoder is allowed to use both odd and even sized padding. Again, the `scale` is used to reduce variance throughout the model and the position embedding is initialized to have a \"vocabulary\" of 100.\n\nThis model takes in the encoder representations in its `forward` method and both are passed to the `calculate_attention` method which calculates and applies attention. It also returns the actual attention values, but we are not currently using them.\n\n\n```python\nclass Decoder(nn.Module):\n def __init__(self, output_dim, emb_dim, hid_dim, n_layers, kernel_size, dropout, pad_idx, device):\n super().__init__()\n \n self.kernel_size = kernel_size\n self.pad_idx = pad_idx\n self.device = device\n \n self.scale = torch.sqrt(torch.FloatTensor([0.5])).to(device)\n \n self.tok_embedding = nn.Embedding(output_dim, emb_dim)\n self.pos_embedding = nn.Embedding(100, emb_dim)\n \n self.emb2hid = nn.Linear(emb_dim, hid_dim)\n self.hid2emb = nn.Linear(hid_dim, emb_dim)\n \n self.attn_hid2emb = nn.Linear(hid_dim, emb_dim)\n self.attn_emb2hid = nn.Linear(emb_dim, hid_dim)\n \n self.out = nn.Linear(emb_dim, output_dim)\n \n self.convs = nn.ModuleList([nn.Conv1d(in_channels = hid_dim, \n out_channels = 2 * hid_dim, \n kernel_size = kernel_size)\n for _ in range(n_layers)])\n \n self.dropout = nn.Dropout(dropout)\n \n def calculate_attention(self, embedded, conved, encoder_conved, encoder_combined):\n \n #embedded = [batch size, trg sent len, emb dim]\n #conved = [batch size, hid dim, trg sent len]\n #encoder_conved = encoder_combined = [batch size, src sent len, emb dim]\n \n #permute and convert back to emb dim\n conved_emb = self.attn_hid2emb(conved.permute(0, 2, 1))\n \n #conved_emb = [batch size, trg sent len, emb dim]\n \n combined = (conved_emb + embedded) * self.scale\n \n #combined = [batch size, trg sent len, emb dim]\n \n energy = torch.matmul(combined, encoder_conved.permute(0, 2, 1))\n \n #energy = [batch size, trg sent len, src sent len]\n \n attention = F.softmax(energy, dim=2)\n \n #attention = [batch size, trg sent len, src sent len]\n \n attended_encoding = torch.matmul(attention, encoder_combined)\n \n #attended_encoding = [batch size, trg sent len, emd dim]\n \n #convert from emb dim -> hid dim\n attended_encoding = self.attn_emb2hid(attended_encoding)\n \n #attended_encoding = [batch size, trg sent len, hid dim]\n \n #apply residual connection\n attended_combined = (conved + attended_encoding.permute(0, 2, 1)) * self.scale\n \n #attended_combined = [batch size, hid dim, trg sent len]\n \n return attention, attended_combined\n \n def forward(self, trg, encoder_conved, encoder_combined):\n \n #trg = [batch size, trg sent len]\n #encoder_conved = encoder_combined = [batch size, src sent len, emb dim]\n \n #create position tensor\n pos = torch.arange(0, trg.shape[1]).unsqueeze(0).repeat(trg.shape[0], 1).to(self.device)\n \n #pos = [batch size, trg sent len]\n \n #embed tokens and positions\n tok_embedded = self.tok_embedding(trg)\n pos_embedded = self.pos_embedding(pos)\n \n #tok_embedded = [batch size, trg sent len, emb dim]\n #pos_embedded = [batch size, trg sent len, emb dim]\n \n #combine embeddings by elementwise summing\n embedded = self.dropout(tok_embedded + pos_embedded)\n \n #embedded = [batch size, trg sent len, emb dim]\n \n #pass embedded through linear layer to go through emb dim -> hid dim\n conv_input = self.emb2hid(embedded)\n \n #conv_input = [batch size, trg sent len, hid dim]\n \n #permute for convolutional layer\n conv_input = conv_input.permute(0, 2, 1) \n \n #conv_input = [batch size, hid dim, trg sent len]\n \n for i, conv in enumerate(self.convs):\n \n #apply dropout\n conv_input = self.dropout(conv_input)\n \n #need to pad so decoder can't \"cheat\"\n padding = torch.zeros(conv_input.shape[0], \n conv_input.shape[1], \n self.kernel_size - 1).fill_(self.pad_idx).to(self.device)\n \n padded_conv_input = torch.cat((padding, conv_input), dim = 2)\n \n #padded_conv_input = [batch size, hid dim, trg sent len + kernel size - 1]\n \n #pass through convolutional layer\n conved = conv(padded_conv_input)\n\n #conved = [batch size, 2 * hid dim, trg sent len]\n \n #pass through GLU activation function\n conved = F.glu(conved, dim = 1)\n\n #conved = [batch size, hid dim, trg sent len]\n \n #calculate attention\n attention, conved = self.calculate_attention(embedded, \n conved, \n encoder_conved, \n encoder_combined)\n \n #attention = [batch size, trg sent len, src sent len]\n \n #apply residual connection\n conved = (conved + conv_input) * self.scale\n \n #conved = [batch size, hid dim, trg sent len]\n \n #set conv_input to conved for next loop iteration\n conv_input = conved\n \n conved = self.hid2emb(conved.permute(0, 2, 1))\n \n #conved = [batch size, trg sent len, emb dim]\n \n output = self.out(self.dropout(conved))\n \n #output = [batch size, trg sent len, output dim]\n \n return output, attention\n```\n\n### Seq2Seq\n\nThe encapsulating `Seq2Seq` module is a lot different from recurrent neural network methods used in previous notebooks, especially in the decoding. \n\nOur `trg` has the `` element sliced off of the end of the sequence. This is because we do not input the `` token into the decoder.\n\nThe encoding is similar, insert the source sequence and receive a \"context vector\". However, here we have two context vectors per word in the source sequence, `encoder_conved` and `encoder_combined`. \n\nAs the decoding is done in parallel we do not need a decoding loop. All of the target sequence is input into the decoder at once and the padding is used to ensure each convolutional filter in the decoder can only see the current and previous tokens in the sequence as it slides across the sentence.\n\nThis also, however, means we cannot do teacher forcing using this model. We do not have a loop in which we can choose whether to input the predicted token or the actual token in the sequence as everything is predicted in parallel.\n\n\n```python\nclass Seq2Seq(nn.Module):\n def __init__(self, encoder, decoder):\n super().__init__()\n \n self.encoder = encoder\n self.decoder = decoder\n \n def forward(self, src, trg):\n \n #src = [batch size, src sent len]\n #trg = [batch size, trg sent len - 1] ( token sliced off the end)\n \n #calculate z^u (encoder_conved) and (z^u + e) (encoder_combined)\n #encoder_conved is output from final encoder conv. block\n #encoder_combined is encoder_conved plus (elementwise) src embedding plus positional embeddings \n encoder_conved, encoder_combined = self.encoder(src)\n \n #encoder_conved = [batch size, src sent len, emb dim]\n #encoder_combined = [batch size, src sent len, emb dim]\n \n #calculate predictions of next words\n #output is a batch of predictions for each word in the trg sentence\n #attention a batch of attention scores across the src sentence for each word in the trg sentence\n output, attention = self.decoder(trg, encoder_conved, encoder_combined)\n \n #output = [batch size, trg sent len - 1, output dim]\n #attention = [batch size, trg sent len - 1, src sent len]\n \n return output, attention\n```\n\n## Training the Seq2Seq Model\n\nThe rest of the tutorial is similar to all of the previous ones. We define all of the hyperparameters, initialize the encoder and decoder, and initialize the overall model - placing it on the GPU if we have one.\n\nIn the paper they find that it is more beneficial to use a small filter (kernel size of 3) and a high number of layers (5+).\n\n\n```python\nINPUT_DIM = len(SRC.vocab)\nOUTPUT_DIM = len(TRG.vocab)\nEMB_DIM = 256\nHID_DIM = 512 # each conv. layer has 2 * hid_dim filters\nENC_LAYERS = 10 # number of conv. blocks in encoder\nDEC_LAYERS = 10 # number of conv. blocks in decoder\nENC_KERNEL_SIZE = 3 # must be odd!\nDEC_KERNEL_SIZE = 3 # can be even or odd\nENC_DROPOUT = 0.25\nDEC_DROPOUT = 0.25\nPAD_IDX = TRG.vocab.stoi['']\n \nenc = Encoder(INPUT_DIM, EMB_DIM, HID_DIM, ENC_LAYERS, ENC_KERNEL_SIZE, ENC_DROPOUT, device)\ndec = Decoder(OUTPUT_DIM, EMB_DIM, HID_DIM, DEC_LAYERS, DEC_KERNEL_SIZE, DEC_DROPOUT, PAD_IDX, device)\n\nmodel = Seq2Seq(enc, dec).to(device)\n```\n\nWe can also see that the model has almost twice as many parameters as the attention based model (20m to 37m).\n\n\n```python\ndef count_parameters(model):\n return sum(p.numel() for p in model.parameters() if p.requires_grad)\n\nprint(f'The model has {count_parameters(model):,} trainable parameters')\n```\n\n The model has 37,351,685 trainable parameters\n\n\nNext, we define the optimizer and the loss function (criterion). As before we ignore the loss where the target sequence is a padding token.\n\n\n```python\noptimizer = optim.Adam(model.parameters())\n```\n\n\n```python\ncriterion = nn.CrossEntropyLoss(ignore_index = PAD_IDX)\n```\n\nThen, we define the training loop for the model.\n\nWe handle the sequences a little differently than previous tutorials. For all models we never put the `` into the decoder. This is handled in the RNN models by the having the decoder loop not reach having the `` as an input to the decoder. In this model, we simply slice the `` token off the end of the sequence. Thus:\n\n$$\\begin{align}\n\\text{trg} &= [sos, x_1, x_2, x_3, eos]\\\\\n\\text{trg[:-1]} &= [sos, x_1, x_2, x_3]\n\\end{align}$$\n\n$x_i$ denotes actual target sequence element. We then feed this into the model to get a predicted sequence that should hopefully predict the `` token:\n\n$$\\begin{align}\n\\text{output} &= [y_1, y_2, y_3, eos]\n\\end{align}$$\n\n$y_i$ denotes predicted target sequence element. We then calculate our loss using the original `trg` tensor with the `` token sliced off the front, leaving the `` token:\n\n$$\\begin{align}\n\\text{output} &= [y_1, y_2, y_3, eos]\\\\\n\\text{trg[1:]} &= [x_1, x_2, x_3, eos]\n\\end{align}$$\n\nWe then calculate our losses and update our parameters as is standard.\n\n\n```python\ndef train(model, iterator, optimizer, criterion, clip):\n \n model.train()\n \n epoch_loss = 0\n \n for i, batch in enumerate(iterator):\n \n src = batch.src\n trg = batch.trg\n \n optimizer.zero_grad()\n \n output, _ = model(src, trg[:,:-1])\n \n #output = [batch size, trg sent len - 1, output dim]\n #trg = [batch size, trg sent len]\n \n output = output.contiguous().view(-1, output.shape[-1])\n trg = trg[:,1:].contiguous().view(-1)\n \n #output = [batch size * trg sent len - 1, output dim]\n #trg = [batch size * trg sent len - 1]\n \n loss = criterion(output, trg)\n \n loss.backward()\n \n torch.nn.utils.clip_grad_norm_(model.parameters(), clip)\n \n optimizer.step()\n \n epoch_loss += loss.item()\n \n return epoch_loss / len(iterator)\n```\n\nThe evaluation loop is the same as the training loop, just without the gradient calculations and parameter updates.\n\n\n```python\ndef evaluate(model, iterator, criterion):\n \n model.eval()\n \n epoch_loss = 0\n \n with torch.no_grad():\n \n for i, batch in enumerate(iterator):\n\n src = batch.src\n trg = batch.trg\n\n output, _ = model(src, trg[:,:-1])\n \n #output = [batch size, trg sent len - 1, output dim]\n #trg = [batch size, trg sent len]\n\n output = output.contiguous().view(-1, output.shape[-1])\n trg = trg[:,1:].contiguous().view(-1)\n\n #output = [batch size * trg sent len - 1, output dim]\n #trg = [batch size * trg sent len - 1]\n \n loss = criterion(output, trg)\n\n epoch_loss += loss.item()\n \n return epoch_loss / len(iterator)\n```\n\nAgain, we have a function that tells us how long each epoch takes.\n\n\n```python\ndef epoch_time(start_time, end_time):\n elapsed_time = end_time - start_time\n elapsed_mins = int(elapsed_time / 60)\n elapsed_secs = int(elapsed_time - (elapsed_mins * 60))\n return elapsed_mins, elapsed_secs\n```\n\nFinally, we train our model.\n\nAlthough we have almost twice as many parameters as the attention based RNN model, it actually takes around half the time as the standard version and about the same time as the packed padded sequences version. This is due to all calculations being done in parallel using the convolutional filters instead of sequentially using RNNs. \n\n**Note**: this model always has a teacher forcing ratio of 1, i.e. it will always use the ground truth next token from the target sequence. This means we cannot compare perplexity values against the previous models when they are using a teacher forcing ratio that is not 1. See [here](https://github.com/bentrevett/pytorch-seq2seq/issues/39#issuecomment-529408483) for the results of the attention based RNN using a teacher forcing ratio of 1. \n\n\n```python\nN_EPOCHS = 10\nCLIP = 1\n\nbest_valid_loss = float('inf')\n\nfor epoch in range(N_EPOCHS):\n \n start_time = time.time()\n \n train_loss = train(model, train_iterator, optimizer, criterion, CLIP)\n valid_loss = evaluate(model, valid_iterator, criterion)\n \n end_time = time.time()\n \n epoch_mins, epoch_secs = epoch_time(start_time, end_time)\n \n if valid_loss < best_valid_loss:\n best_valid_loss = valid_loss\n torch.save(model.state_dict(), 'tut5-model.pt')\n \n print(f'Epoch: {epoch+1:02} | Time: {epoch_mins}m {epoch_secs}s')\n print(f'\\tTrain Loss: {train_loss:.3f} | Train PPL: {math.exp(train_loss):7.3f}')\n print(f'\\t Val. Loss: {valid_loss:.3f} | Val. PPL: {math.exp(valid_loss):7.3f}')\n```\n\n Epoch: 01 | Time: 0m 30s\n \tTrain Loss: 4.202 | Train PPL: 66.837\n \t Val. Loss: 2.907 | Val. PPL: 18.306\n Epoch: 02 | Time: 0m 31s\n \tTrain Loss: 3.018 | Train PPL: 20.446\n \t Val. Loss: 2.378 | Val. PPL: 10.779\n Epoch: 03 | Time: 0m 30s\n \tTrain Loss: 2.598 | Train PPL: 13.438\n \t Val. Loss: 2.121 | Val. PPL: 8.340\n Epoch: 04 | Time: 0m 31s\n \tTrain Loss: 2.360 | Train PPL: 10.591\n \t Val. Loss: 1.976 | Val. PPL: 7.210\n Epoch: 05 | Time: 0m 30s\n \tTrain Loss: 2.208 | Train PPL: 9.102\n \t Val. Loss: 1.899 | Val. PPL: 6.678\n Epoch: 06 | Time: 0m 31s\n \tTrain Loss: 2.095 | Train PPL: 8.124\n \t Val. Loss: 1.845 | Val. PPL: 6.328\n Epoch: 07 | Time: 0m 31s\n \tTrain Loss: 2.008 | Train PPL: 7.450\n \t Val. Loss: 1.801 | Val. PPL: 6.059\n Epoch: 08 | Time: 0m 31s\n \tTrain Loss: 1.933 | Train PPL: 6.909\n \t Val. Loss: 1.776 | Val. PPL: 5.908\n Epoch: 09 | Time: 0m 31s\n \tTrain Loss: 1.872 | Train PPL: 6.499\n \t Val. Loss: 1.748 | Val. PPL: 5.743\n Epoch: 10 | Time: 0m 31s\n \tTrain Loss: 1.819 | Train PPL: 6.166\n \t Val. Loss: 1.731 | Val. PPL: 5.649\n\n\nWe then load the parameters which obtained the lowest validation loss and calculate the loss over the test set. \n\n\n```python\nmodel.load_state_dict(torch.load('tut5-model.pt'))\n\ntest_loss = evaluate(model, test_iterator, criterion)\n\nprint(f'| Test Loss: {test_loss:.3f} | Test PPL: {math.exp(test_loss):7.3f} |')\n```\n\n | Test Loss: 1.800 | Test PPL: 6.047 |\n\n\n## Inference\n\n\n```python\ndef translate_sentence(sentence, src_field, trg_field, model, device, max_len = 50):\n\n model.eval()\n \n if isinstance(sentence, str):\n nlp = spacy.load('en')\n tokens = [token.text.lower() for token in nlp(sentence)]\n else:\n tokens = [token.lower() for token in sentence]\n\n tokens = [''] + tokens + ['']\n \n src_indexes = [src_field.vocab.stoi[token] for token in tokens]\n\n src_tensor = torch.LongTensor(src_indexes).unsqueeze(0).to(device)\n\n with torch.no_grad():\n encoder_conved, encoder_combined = model.encoder(src_tensor)\n\n trg_indexes = [trg_field.vocab.stoi[trg_field.init_token]]\n\n for i in range(max_len):\n\n trg_tensor = torch.LongTensor(trg_indexes).unsqueeze(0).to(device)\n\n with torch.no_grad():\n output, attention = model.decoder(trg_tensor, encoder_conved, encoder_combined)\n \n pred_token = output.argmax(2)[:,-1].item()\n \n trg_indexes.append(pred_token)\n\n if pred_token == trg_field.vocab.stoi[trg_field.eos_token]:\n break\n \n trg_tokens = [trg_field.vocab.itos[i] for i in trg_indexes]\n \n return trg_tokens[1:], attention\n```\n\n\n```python\ndef display_attention(sentence, translation, attention):\n \n fig = plt.figure(figsize=(10,10))\n ax = fig.add_subplot(111)\n \n attention = attention.squeeze(0).cpu().detach().numpy()\n \n cax = ax.matshow(attention, cmap='bone')\n \n ax.tick_params(labelsize=15)\n ax.set_xticklabels(['']+['']+[t.lower() for t in sentence]+[''], \n rotation=45)\n ax.set_yticklabels(['']+translation)\n\n ax.xaxis.set_major_locator(ticker.MultipleLocator(1))\n ax.yaxis.set_major_locator(ticker.MultipleLocator(1))\n\n plt.show()\n plt.close()\n```\n\n\n```python\nexample_idx = 10\n\nsrc = vars(train_data.examples[example_idx])['src']\ntrg = vars(train_data.examples[example_idx])['trg']\n\nprint(f'src = {src}')\nprint(f'trg = {trg}')\n```\n\n src = ['eine', 'ballettklasse', 'mit', 'f\u00fcnf', 'm\u00e4dchen', ',', 'die', 'nacheinander', 'springen', '.']\n trg = ['a', 'ballet', 'class', 'of', 'five', 'girls', 'jumping', 'in', 'sequence', '.']\n\n\n\n```python\ntranslation, attention = translate_sentence(src, SRC, TRG, model, device)\n\nprint(f'predicted trg = {translation}')\n```\n\n predicted trg = ['a', 'ballet', 'with', 'five', 'girls', 'jumping', 'in', 'sequence', '.', '']\n\n\n\n```python\ndisplay_attention(src, translation, attention)\n```\n\n\n```python\nexample_idx = 1\n\nsrc = vars(valid_data.examples[example_idx])['src']\ntrg = vars(valid_data.examples[example_idx])['trg']\n\nprint(f'src = {src}')\nprint(f'trg = {trg}')\n```\n\n src = ['ein', 'mann', 'schl\u00e4ft', 'in', 'einem', 'gr\u00fcnen', 'raum', 'auf', 'einem', 'sofa', '.']\n trg = ['a', 'man', 'sleeping', 'in', 'a', 'green', 'room', 'on', 'a', 'couch', '.']\n\n\n\n```python\ntranslation, attention = translate_sentence(src, SRC, TRG, model, device)\n\nprint(f'predicted trg = {translation}')\n```\n\n predicted trg = ['a', 'man', 'sleeping', 'in', 'a', 'green', 'room', 'on', 'a', 'couch', '.', '']\n\n\n\n```python\ndisplay_attention(src, translation, attention)\n```\n\n\n```python\nexample_idx = 20\n\nsrc = vars(test_data.examples[example_idx])['src']\ntrg = vars(test_data.examples[example_idx])['trg']\n\nprint(f'src = {src}')\nprint(f'trg = {trg}')\n```\n\n src = ['leute', ',', 'die', 'vor', 'einem', 'geb\u00e4ude', 'stehen', '.']\n trg = ['people', 'standing', 'outside', 'of', 'a', 'building', '.']\n\n\n\n```python\ntranslation, attention = translate_sentence(src, SRC, TRG, model, device)\n\nprint(f'predicted trg = {translation}')\n```\n\n predicted trg = ['people', 'standing', 'in', 'front', 'of', 'a', 'building', '.', '']\n\n\n\n```python\ndisplay_attention(src, translation, attention)\n```\n", "meta": {"hexsha": "7b32aca59d8e9729748b7463e41192b0e757d983", "size": 123501, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "5 - Convolutional Sequence to Sequence Learning.ipynb", "max_stars_repo_name": "zhao1402072392/pytorch-seq2seq", "max_stars_repo_head_hexsha": "848d0b248b855fb76e8f582a97435ae630b8662a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-10-28T23:14:17.000Z", "max_stars_repo_stars_event_max_datetime": "2019-10-28T23:14:17.000Z", "max_issues_repo_path": "5 - Convolutional Sequence to Sequence Learning.ipynb", "max_issues_repo_name": "cdxzyc/pytorch-seq2seq", "max_issues_repo_head_hexsha": "0d1dfed292b3fe0eb19ccafce36a522825a0a96d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "5 - Convolutional Sequence to Sequence Learning.ipynb", "max_forks_repo_name": "cdxzyc/pytorch-seq2seq", "max_forks_repo_head_hexsha": "0d1dfed292b3fe0eb19ccafce36a522825a0a96d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-03-24T12:56:46.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-24T12:56:46.000Z", "avg_line_length": 101.1474201474, "max_line_length": 28120, "alphanum_fraction": 0.804479316, "converted": true, "num_tokens": 8919, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175005616829, "lm_q2_score": 0.6477982247516796, "lm_q1q2_score": 0.4265216880092962}} {"text": "```python\n%matplotlib widget\n```\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pydae.ssa as ssa\n```\n\n\n```python\nfrom vsc_lcl import vsc_lcl_class \n```\n\n\n```python\n\n```\n\n## Instantiate system\n\n\n```python\nsyst = vsc_lcl_class()\nsyst.Dt = 5e-6\nsyst.decimation = 1\nsyst.N_store = 100_000\nsyst.update()\n```\n\n## Solve steady state\n\n\n```python\nsyst.initialize([{'eta_q_g01':0.8693333333333333,'G_d_g01':0.01}],xy0=100)\nssa.eval_ss(syst);\n```\n\n\n```python\nsyst.report_x()\nsyst.report_y()\nsyst.report_u()\nsyst.report_z()\n```\n\n i_tD_g01 = -0.20\n i_tQ_g01 = 1.63\n v_mD_g01 = 0.65\n v_mQ_g01 = 326.02\n i_sD_g01 = 0.20\n i_sQ_g01 = -1.63\n i_1_D = 0.20\n i_1_Q = -1.63\n v_sD_g01 = 0.00\n v_sQ_g01 = 326.00\n v_md_g01 = 0.65\n v_mq_g01 = 326.02\n v_sd_g01 = 0.00\n v_sq_g01 = 326.00\n i_td_g01 = -0.20\n i_tq_g01 = 1.63\n i_sd_g01 = 0.20\n i_sq_g01 = -1.63\n eta_D_g01 = 0.00\n eta_Q_g01 = 0.87\n v_dc_g01 = 750.00\n v_1_D = 0.00\n v_1_Q = 326.00\n phi_g01 = 0.00\n eta_d_g01 = 0.00\n eta_q_g01 = 0.87\n v_t_D = 0.00\n v_t_Q = 326.00\n damp_D = 0.01\n damp_Q = 3.26\n i_sd_g01 = 0.20\n i_sq_g01 = -1.63\n v_md_g01 = 0.65\n v_mq_g01 = 326.02\n i_td_g01 = -0.20\n i_tq_g01 = 1.63\n\n\n\n```python\nssa.damp_report(syst)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        RealImagFreq.Damp
                                        Mode 1-1265.70796320276.0360103227.0313570.062303
                                        Mode 2-1265.707963-20276.0360103227.0313570.062303
                                        Mode 3-1265.70796319647.7174793127.0313570.064287
                                        Mode 4-1265.707963-19647.7174793127.0313570.064287
                                        Mode 5-31.415927314.15926550.0000000.099504
                                        Mode 6-31.415927-314.15926550.0000000.099504
                                        \n
                                        \n\n\n\n\n```python\nssa.participation(syst).abs().round(2)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Mode 1Mode 2Mode 3Mode 4Mode 5Mode 6
                                        i_tD_g010.130.130.130.130.250.25
                                        i_tQ_g010.130.130.130.130.250.25
                                        v_mD_g010.250.250.250.250.000.00
                                        v_mQ_g010.250.250.250.250.000.00
                                        i_sD_g010.130.130.130.130.250.25
                                        i_sQ_g010.130.130.130.130.250.25
                                        \n
                                        \n\n\n\n### Open loop\n\n\n```python\n\u0394t = 50.0e-6 \ntimes = np.arange(0.0,0.2,\u0394t)\n```\n\n\n```python\nsyst.initialize([{'eta_q_g01':0.8693333333333333,'G_d_g01':0.0}],xy0=100)\neta_q_g01_0 = syst.get_value('eta_q_g01')\nit = 0\nfor t in times:\n \n eta_q_g01 = eta_q_g01_0\n if t>5e-3: \n eta_q_g01 = eta_q_g01_0*1.05\n # if t>10e-3: \n # eta_q_g01 = eta_q_g01_0\n \n events=[{'t_end':t,'eta_q_g01':eta_q_g01}]\n syst.run(events)\n\n it += 1\n \nsyst.post();\n```\n\n\n```python\nfig, axes = plt.subplots(nrows=2, ncols=1, figsize=(7, 7))\n\naxes[0].plot(syst.T,syst.get_values('i_sd_g01'),label='i_sd_g01')\naxes[0].plot(syst.T,syst.get_values('i_sq_g01'),label='i_sq_g01')\n\n\nfor ax in axes:\n ax.grid()\n ax.legend()\nax.set_xlabel('Time (s)')\n```\n\n\n Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous \u2026\n\n\n No handles with labels found to put in legend.\n\n\n\n\n\n Text(0.5, 0, 'Time (s)')\n\n\n\n\n```python\nsyst.N_store\n```\n\n\n\n\n 100000\n\n\n\n### CTRL1\n\n\n```python\nsyst = vsc_lcl_class()\nsyst.Dt = 5e-6\nsyst.decimation =1\nsyst.N_store =100_000\nsyst.update()\n\n\u0394t = 50.0e-6 \ntimes = np.arange(0.0,0.2,\u0394t)\n\nsyst.initialize([{'G_d_g01':0.005}],xy0=1000)\neta_q_g01_0 = syst.get_value('eta_q_g01')\nit = 0\n\ni_sd,i_sq,v_sd,v_sq = syst.get_mvalue(['i_sd_g01','i_sq_g01','v_sd_g01','v_sq_g01'])\nv_dc = syst.get_value('v_dc_g01')\neta_d = syst.get_value('eta_d_g01') \neta_q = syst.get_value('eta_q_g01') \n\n# control design\nR_t_g01,L_t_g01 = syst.get_value('R_t_g01'),syst.get_value('L_t_g01')\nR_s_g01,L_s_g01,C_m_g01 = syst.get_value('R_s_g01'),syst.get_value('L_s_g01'),syst.get_value('C_m_g01') \nR = R_t_g01 + R_s_g01\nL = L_t_g01 + L_s_g01\ntau_ctrl_1 = 5e-3; #Time constant of CTRL 1\nK_pi = L/tau_ctrl_1; #Proportional gain of CTRL 1\nK_ii = R/tau_ctrl_1; #Integral gain of CTRL 1\nxi = np.zeros((2,1))\n\n#u_d = K_pi*epsilon_d + K_ii*xi_d\n#u_q = K_pi*epsilon_q + K_ii*xi_q \n#u_d = eta_d*v_dc/2 - v_sd + L*i_sq*omega => eta_d = (u_d + v_sd - L*i_sq*omega)*2/v_dc\n#u_q = eta_q*v_dc/2 - v_sq - L*i_sd*omega => eta_q = (u_q + v_sq + L*i_sd*omega)*2/v_dc\nomega = 2*np.pi*50\nu_d_0 = eta_d*v_dc/2 - v_sd + L*i_sq*omega\nu_q_0 = eta_q*v_dc/2 - v_sq - L*i_sd*omega \ni_sd_ref_0 = i_sd\ni_sq_ref_0 = i_sq\n\n# simulation\nfor t in times:\n \n # measurements\n i_sd = syst.get_value('i_sd_g01')\n i_sq = syst.get_value('i_sq_g01') \n v_sd = syst.get_value('v_sd_g01')\n v_sq = syst.get_value('v_sq_g01') \n v_dc = syst.get_value('v_dc_g01')\n \n i_sd_ref = i_sd_ref_0\n i_sq_ref = i_sq_ref_0\n if t>10e-3: i_sd_ref = 20\n if t>100e-3: i_sq_ref = 30\n \n xi_d = xi[0,0]\n xi_q = xi[1,0]\n \n epsilon_d = i_sd_ref - i_sd\n epsilon_q = i_sq_ref - i_sq\n \n u_d = K_pi*epsilon_d + K_ii*xi_d + u_d_0\n u_q = K_pi*epsilon_q + K_ii*xi_q + u_q_0 \n\n eta_d = (u_d + v_sd - L*i_sq*omega)*2/v_dc\n eta_q = (u_q + v_sq + L*i_sd*omega)*2/v_dc\n \n xi[0,0] += \u0394t*epsilon_d\n xi[1,0] += \u0394t*epsilon_q\n \n events=[{'t_end':t,'eta_d_g01':eta_d,'eta_q_g01':eta_q}]\n syst.run(events)\n\n it += 1\n \nsyst.post();\n```\n\n\n```python\nplt.close('all')\nfig, axes = plt.subplots(nrows=2, ncols=1, figsize=(7, 7))\n\naxes[0].plot(syst.T,syst.get_values('i_sd_g01'),label='i_sd_g01')\naxes[0].plot(syst.T,syst.get_values('i_sq_g01'),label='i_sq_g01')\n\naxes[1].plot(syst.T,syst.get_values('eta_D_g01'),label='eta_D_g01')\naxes[1].plot(syst.T,syst.get_values('eta_Q_g01'),label='eta_Q_g01')\n\nfor ax in axes:\n ax.grid()\n ax.legend()\nax.set_xlabel('Time (s)')\n```\n\n\n Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous \u2026\n\n\n\n\n\n Text(0.5, 0, 'Time (s)')\n\n\n\n\n```python\n\n```\n\n### CTRL1 + Active damping\n\n\n```python\nsyst = vsc_lcl_class()\nsyst.Dt = 5e-6\nsyst.decimation =1\nsyst.N_store =100_000\nsyst.update()\n\n\u0394t = 50.0e-6 \ntimes = np.arange(0.0,0.2,\u0394t)\n\nsyst.initialize([{'G_d_g01':0.0}],xy0=1000)\neta_q_g01_0 = syst.get_value('eta_q_g01')\nit = 0\n\ni_sd,i_sq,i_td,i_tq,v_sd,v_sq = syst.get_mvalue(['i_sd_g01','i_sq_g01','i_td_g01','i_tq_g01','v_sd_g01','v_sq_g01'])\nv_dc = syst.get_value('v_dc_g01')\neta_d = syst.get_value('eta_d_g01') \neta_q = syst.get_value('eta_q_g01') \n\n# control design\nR_t_g01,L_t_g01 = syst.get_value('R_t_g01'),syst.get_value('L_t_g01')\nR_s_g01,L_s_g01,C_m_g01 = syst.get_value('R_s_g01'),syst.get_value('L_s_g01'),syst.get_value('C_m_g01') \nR = R_t_g01 + R_s_g01\nL = L_t_g01 + L_s_g01\ntau_ctrl_1 = 5e-3; #Time constant of CTRL 1\nK_pi = L/tau_ctrl_1; #Proportional gain of CTRL 1\nK_ii = R/tau_ctrl_1; #Integral gain of CTRL 1\nG_v = 1.0 #Active damping\n# en pu G_d = L/C*G_v\n\nxi = np.zeros((2,1))\n\n#u_d = K_pi*epsilon_d + K_ii*xi_d\n#u_q = K_pi*epsilon_q + K_ii*xi_q \n#u_d = eta_d*v_dc/2 - v_sd + L*i_sq*omega => eta_d = (u_d + v_sd - L*i_sq*omega)*2/v_dc\n#u_q = eta_q*v_dc/2 - v_sq - L*i_sd*omega => eta_q = (u_q + v_sq + L*i_sd*omega)*2/v_dc\nomega = 2*np.pi*50\nu_d_0 = eta_d*v_dc/2 - v_sd + L*i_sq*omega + G_v*(i_td - i_sd)\nu_q_0 = eta_q*v_dc/2 - v_sq - L*i_sd*omega + G_v*(i_tq - i_sq)\ni_sd_ref_0 = i_sd\ni_sq_ref_0 = i_sq\n\n# simulation\nfor t in times:\n \n # measurements\n i_sd = syst.get_value('i_sd_g01')\n i_sq = syst.get_value('i_sq_g01') \n v_sd = syst.get_value('v_sd_g01')\n v_sq = syst.get_value('v_sq_g01')\n i_td = syst.get_value('i_td_g01')\n i_tq = syst.get_value('i_tq_g01') \n v_dc = syst.get_value('v_dc_g01')\n \n i_sd_ref = i_sd_ref_0\n i_sq_ref = i_sq_ref_0\n if t>10e-3: i_sd_ref = 20\n if t>100e-3: i_sq_ref = 30\n \n xi_d = xi[0,0]\n xi_q = xi[1,0]\n \n epsilon_d = i_sd_ref - i_sd\n epsilon_q = i_sq_ref - i_sq\n \n u_d = K_pi*epsilon_d + K_ii*xi_d + u_d_0\n u_q = K_pi*epsilon_q + K_ii*xi_q + u_q_0 \n\n eta_d = (u_d + v_sd - L*i_sq*omega - G_v*(i_td - i_sd))*2/v_dc\n eta_q = (u_q + v_sq + L*i_sd*omega - G_v*(i_tq - i_sq))*2/v_dc\n \n xi[0,0] += \u0394t*epsilon_d\n xi[1,0] += \u0394t*epsilon_q\n \n events=[{'t_end':t,'eta_d_g01':eta_d,'eta_q_g01':eta_q}]\n syst.run(events)\n\n it += 1\n \nsyst.post();\n```\n\n\n```python\nplt.close('all')\nfig, axes = plt.subplots(nrows=2, ncols=1, figsize=(7, 7))\n\naxes[0].plot(syst.T,syst.get_values('i_sd_g01'),label='i_sd_g01')\naxes[0].plot(syst.T,syst.get_values('i_sq_g01'),label='i_sq_g01')\n\naxes[1].plot(syst.T,syst.get_values('eta_D_g01'),label='eta_D_g01')\naxes[1].plot(syst.T,syst.get_values('eta_Q_g01'),label='eta_Q_g01')\n\nfor ax in axes:\n ax.grid()\n ax.legend()\nax.set_xlabel('Time (s)')\n```\n\n\n Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous \u2026\n\n\n\n\n\n Text(0.5, 0, 'Time (s)')\n\n\n\n\n```python\n\n```\n\n### CTRL1 + Active damping + delay\n\n\n```python\nsyst = vsc_lcl_class()\nsyst.Dt = 5e-6\nsyst.decimation =1\nsyst.N_store =100_000\nsyst.update()\n\n\u0394t = 500.0e-6 \ntimes = np.arange(0.0,0.2,\u0394t)\n\nsyst.initialize([{'G_d_g01':0.0, 'C_m_g01':4e-6}],xy0=1000)\neta_q_g01_0 = syst.get_value('eta_q_g01')\nit = 0\n\ni_sd,i_sq,i_td,i_tq,v_sd,v_sq = syst.get_mvalue(['i_sd_g01','i_sq_g01','i_td_g01','i_tq_g01','v_sd_g01','v_sq_g01'])\nv_dc = syst.get_value('v_dc_g01')\neta_d = syst.get_value('eta_d_g01') \neta_q = syst.get_value('eta_q_g01') \n\n# control design\nR_t_g01,L_t_g01 = syst.get_value('R_t_g01'),syst.get_value('L_t_g01')\nR_s_g01,L_s_g01,C_m_g01 = syst.get_value('R_s_g01'),syst.get_value('L_s_g01'),syst.get_value('C_m_g01') \nR = R_t_g01 + R_s_g01\nL = L_t_g01 + L_s_g01\ntau_ctrl_1 = 5e-3; #Time constant of CTRL 1\nK_pi = L/tau_ctrl_1; #Proportional gain of CTRL 1\nK_ii = R/tau_ctrl_1; #Integral gain of CTRL 1\nG_v = 0.0 #Active damping\n\nxi = np.zeros((2,1))\n\n#u_d = K_pi*epsilon_d + K_ii*xi_d\n#u_q = K_pi*epsilon_q + K_ii*xi_q \n#u_d = eta_d*v_dc/2 - v_sd + L*i_sq*omega => eta_d = (u_d + v_sd - L*i_sq*omega)*2/v_dc\n#u_q = eta_q*v_dc/2 - v_sq - L*i_sd*omega => eta_q = (u_q + v_sq + L*i_sd*omega)*2/v_dc\nomega = 2*np.pi*50\nu_d_0 = eta_d*v_dc/2 - v_sd + L*i_sq*omega + G_v*(i_td - i_sd)\nu_q_0 = eta_q*v_dc/2 - v_sq - L*i_sd*omega + G_v*(i_tq - i_sq)\ni_sd_ref_0 = i_sd\ni_sq_ref_0 = i_sq\neta_d_prev = eta_d\neta_q_prev = eta_q\ndamp_d_list = []\ndamp_q_list = []\n# simulation\nfor t in times:\n \n # measurements\n i_sd = syst.get_value('i_sd_g01')\n i_sq = syst.get_value('i_sq_g01') \n v_sd = syst.get_value('v_sd_g01')\n v_sq = syst.get_value('v_sq_g01')\n i_td = syst.get_value('i_td_g01')\n i_tq = syst.get_value('i_tq_g01') \n v_dc = syst.get_value('v_dc_g01')\n \n i_sd_ref = i_sd_ref_0\n i_sq_ref = i_sq_ref_0\n if t>10e-3: i_sd_ref = 20\n if t>100e-3: i_sq_ref = 30\n \n xi_d = xi[0,0]\n xi_q = xi[1,0]\n \n epsilon_d = i_sd_ref - i_sd\n epsilon_q = i_sq_ref - i_sq\n \n u_d = K_pi*epsilon_d + K_ii*xi_d + u_d_0\n u_q = K_pi*epsilon_q + K_ii*xi_q + u_q_0 \n\n i_m_d_0 = i_td - i_sd\n i_m_q_0 = i_tq - i_sq\n i_m_d_90 = (i_tq - i_sq) \n i_m_q_90 = (i_td - i_sd) \n \n K_0 = -0.6453\n K_90 = -(1-0.6453)\n \n K_0 = -0.5286373998102673\n K_90 = -0.8488477481397001\n\n K_0 = -1.0\n K_90 = -0.0\n\n damp_d = G_v*(K_0*i_m_d_0+K_90*i_m_d_90)\n damp_q = G_v*(K_0*i_m_q_0-K_90*i_m_q_90)\n \n eta_d = (u_d + v_sd - L*i_sq*omega + damp_d)*2/v_dc\n eta_q = (u_q + v_sq + L*i_sd*omega + damp_q)*2/v_dc\n \n xi[0,0] += \u0394t*epsilon_d\n xi[1,0] += \u0394t*epsilon_q\n \n events=[{'t_end':t,'eta_d_g01':eta_d_prev,'eta_q_g01':eta_q_prev}]\n syst.run(events)\n\n eta_d_prev = eta_d\n eta_q_prev = eta_q\n\n damp_d_list += [damp_d]\n damp_q_list += [damp_q]\n\n it += 1\n \nsyst.post();\n\nDamp_d = np.array(damp_d_list)\nDamp_q = np.array(damp_q_list) \n```\n\n\n```python\nplt.close('all')\nfig, axes = plt.subplots(nrows=2, ncols=1, figsize=(7, 7),sharex=True)\n\naxes[0].plot(syst.T,syst.get_values('i_sd_g01'),label='i_sd_g01')\naxes[0].plot(syst.T,syst.get_values('i_sq_g01'),label='i_sq_g01')\n\n#axes[0].plot(syst.T,syst.get_values('damp_D')-syst.get_values('damp_D')[0],label='damp_D')\naxes[1].plot(syst.T,syst.get_values('damp_Q'),label='damp_Q')\n\n#axes[1].plot(syst.T,syst.get_values('eta_D_g01'),label='eta_D_g01')\n#axes[1].plot(syst.T,syst.get_values('eta_Q_g01'),label='eta_Q_g01')\n#axes[0].plot(times,Damp_d-Damp_d[0],label='Damp_d')\naxes[1].plot(times,Damp_q-Damp_q[0],label='Damp_q')\nfor ax in axes:\n ax.grid()\n ax.legend()\nax.set_xlabel('Time (s)')\n```\n\n\n Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous \u2026\n\n\n\n\n\n Text(0.5, 0, 'Time (s)')\n\n\n1 = K_d_0**2+K_d_90**2\nphase = \u0394t/(T_2damp*np.pi*2)\nK_d_0/K_d_90 = np.tan(phase) = tangente\nK_d_0 = tangente*K_d_90\n1 = (tangente*K_d_90)**2 + K_d_90**2\n1 = (tangente**2+1)*K_d_90**2\n\n```python\nT_2damp = 1/(3227.031357)\nphase = \u0394t/(T_2damp)*2*np.pi\ntangente = np.tan(phase)\n\nK_90 = (1/((1/tangente)**2+1))**0.5\nK_0 = (1 - K_90**2)**0.5\n\nprint(f' K_0 = {-K_0}')\nprint(f' K_90 = {-K_90}')\n```\n\n K_0 = -0.7562459294598116\n K_90 = -0.6542874705933666\n\n\n\n```python\nT_2damp = 1/(3227.031357)\nphase = \u0394t/(T_2damp)*2*np.pi\n```\n\n\n```python\nnp.rad2deg(phase)*20/3.227\n```\n\n\n\n\n 3600.034981468857\n\n\n\n\n```python\nT_2damp\n```\n\n\n\n\n 0.00030988233127354764\n\n\n\n## CTRL1 in state feedback\n\n\n```python\nimport pydae.ssa as ssa\nimport scipy.signal as sctrl\n```\n\n\n```python\nssa.eval_ss(syst);\n```\n\n\n```python\nsyst = vsc_lcl_class()\nsyst.Dt = 5e-6\nsyst.decimation =1\nsyst.N_store =100_000\nsyst.update()\n\n\u0394t = 50e-6 \n#x_d_ctrl_list = ['i'] # states to consider in the reduction\nz_ctrl_list = [ 'i_sd_g01', 'i_sq_g01'] # outputs to consider in the controller\nu_ctrl_list = ['eta_d_g01','eta_q_g01'] # intputs to consider in the controller\nz_ctrl_idxs = [syst.outputs_list.index(item) for item in z_ctrl_list]\nu_ctrl_idxs = [syst.inputs_run_list.index(item) for item in u_ctrl_list]\n\nsyst.\u0394t = \u0394t\n\n## Calculate equilibirum point\nsyst.initialize([{'G_d_g01':0.0,'eta_d_g01':0.0,'eta_q_g01':-0.8693333,'v_1_Q':-326,'v_1_D':0.0}],xy0=1000)\nssa.eval_ss(syst)\n\n# linear continous plant\nA_p = syst.A\nB_p = syst.B\nC_p = syst.C\nD_p = syst.D\n\n# plant discretization\nA_d,B_d,C_d,D_d,Dt = sctrl.cont2discrete((A_p,B_p,C_p,D_p),\u0394t,method='zoh')\n\nN_z_d,N_x_d = C_d.shape # discreticed plant dimensions\nN_x_d,N_u_d = B_d.shape\n\n# convenient matrices\nO_ux = np.zeros((N_u_d,N_x_d))\nO_xu = np.zeros((N_x_d,N_u_d))\nO_uu = np.zeros((N_u_d,N_u_d))\nI_uu = np.eye(N_u_d)\n\nsyst.A_d = A_d\nsyst.B_d = B_d\n\n\n# Controller ##################################################################################\nB_c = B_d[:,u_ctrl_idxs]\nC_c = C_d[z_ctrl_idxs,:]\nD_c = D_d[z_ctrl_idxs,:]\n\nN_x_c,N_u_d = B_c.shape\nN_z_c,N_x_c = C_c.shape\n\n\nO_ux = np.zeros((N_u_d,N_x_d))\nO_xu = np.zeros((N_x_d,N_u_d))\nO_uu = np.zeros((N_u_d,N_u_d))\nI_uu = np.eye(N_u_d)\n\n\n# discretized plant:\n# \u0394x_d = A_d*\u0394x_d + B_d*\u0394u_d\n# \u0394z_c = C_c*\u0394x_d + D_c*\u0394u_d\n\n# dinamic extension:\n# \u0394x_d = A_d*\u0394x_d + B_d*\u0394u_d\n# \u0394x_i = \u0394x_i + \u0394t*(\u0394z_c-\u0394z_c_ref) = \u0394x_i + \u0394t*C_c*\u0394x_d - Dt*\u0394z_c_ref\n# \u0394z_c = z_c - z_c_0\n# \u0394z_c_ref = z_c_ref - z_c_0\n# (\u0394z_c-\u0394z_c_ref) = z_c - z_c_ref\n\nA_e = np.block([\n [ A_d, O_xu], # \u0394x_d\n [ \u0394t*C_c, I_uu], # \u0394x_i \n ])\n\nB_e = np.block([\n [ B_c],\n [ O_uu], \n ])\n\n\n\n# weighting matrices\nQ_c = np.eye(A_e.shape[0])\nQ_c[-1,-1] = 1e4\nQ_c[-2,-2] = 1e4\n\nR_c = np.eye(B_c.shape[1])*100000\n\n\nK_c,S_c,E_c = ssa.dlqr(A_e,B_e,Q_c,R_c)\n\nE_cont = np.log(E_c)/\u0394t\n```\n\n\n```python\n-E_c.real/np.abs(E_c)\n```\n\n\n\n\n array([-0.4895141 , -0.4895141 , -0.51666282, -0.51666282, -0.99987441,\n -0.99987441, -0.99999999, -0.99999999])\n\n\n\n\n```python\nE_c\n```\n\n\n\n\n array([0.37274688+6.63992221e-01j, 0.37274688-6.63992221e-01j,\n 0.39312068+6.51460819e-01j, 0.39312068-6.51460819e-01j,\n 0.97147951+1.53979236e-02j, 0.97147951-1.53979236e-02j,\n 0.99690471+1.40510238e-04j, 0.99690471-1.40510238e-04j])\n\n\n\n\n```python\nsyst = vsc_lcl_class()\nsyst.Dt = 5e-6\nsyst.decimation =10\nsyst.N_store =1_000_000\nsyst.update()\ntimes = np.arange(0.0,0.2,\u0394t)\n\nsyst.initialize([{'G_d_g01':0.0,'eta_d_g01':0.0,'eta_q_g01':-0.8693333,'v_1_Q':-326,'v_1_D':0.0}],xy0=1000)\ni_sd = syst.get_value('i_sd_g01')\ni_sq = syst.get_value('i_sq_g01') \nv_sd = syst.get_value('v_sd_g01')\nv_sq = syst.get_value('v_sq_g01')\ni_td = syst.get_value('i_td_g01')\ni_tq = syst.get_value('i_tq_g01') \nv_md = syst.get_value('v_md_g01')\nv_mq = syst.get_value('v_mq_g01') \nv_dc = syst.get_value('v_dc_g01')\neta_d = syst.get_value('eta_d_g01')\neta_q = syst.get_value('eta_q_g01')\neta_d_prev = eta_d\neta_q_prev = eta_q \ni_sd_ref_0 = i_sd\ni_sq_ref_0 = i_sq\nx_d_0 = np.array([i_td,i_tq,v_md,v_mq,i_sd,i_sq]).reshape(6,1)\nu_d_0 = np.array([eta_d,eta_q]).reshape(2,1)\nsyst.\u0394xi = np.zeros((2,1)) \nit = 0\nfor t in times:\n \n # measurements\n i_sd = syst.get_value('i_sd_g01')\n i_sq = syst.get_value('i_sq_g01') \n v_sd = syst.get_value('v_sd_g01')\n v_sq = syst.get_value('v_sq_g01')\n i_td = syst.get_value('i_td_g01')\n i_tq = syst.get_value('i_tq_g01') \n v_md = syst.get_value('v_md_g01')\n v_mq = syst.get_value('v_mq_g01') \n v_dc = syst.get_value('v_dc_g01')\n\n x_d = np.array([i_td,i_tq,v_md,v_mq,i_sd,i_sq]).reshape(6,1)\n \u0394x_d = x_d - x_d_0 \n \n \u0394x_i = syst.\u0394xi \n \n i_sd_ref = i_sd_ref_0\n i_sq_ref = i_sq_ref_0\n if t>10e-3: i_sd_ref = 20\n if t>100e-3: i_sq_ref = 30\n \n epsilon_d = i_sd - i_sd_ref\n epsilon_q = i_sq - i_sq_ref \n \n epsilon = np.block([[epsilon_d],[epsilon_q]])\n \n\n \u0394x_e = np.block([[\u0394x_d], [\u0394x_i]])\n \n \u0394u_d = -K_c @ \u0394x_e\n \n u_d = \u0394u_d + u_d_0\n \n syst.\u0394xi += \u0394t*epsilon\n \n eta_d = u_d[0,0]\n eta_q = u_d[1,0]\n\n \n events=[{'t_end':t,'eta_d_g01':eta_d_prev,'eta_q_g01':eta_q_prev}]\n syst.run(events)\n\n eta_d_prev = eta_d\n eta_q_prev = eta_q\n it += 1\n \nsyst.post();\n```\n\n\n```python\nplt.close('all')\nfig, axes = plt.subplots(nrows=2, ncols=1, figsize=(7, 7),sharex=True)\n\naxes[0].plot(syst.T,syst.get_values('i_sd_g01'),label='i_sd_g01')\naxes[0].plot(syst.T,syst.get_values('i_sq_g01'),label='i_sq_g01')\n\n\nfor ax in axes:\n ax.grid()\n ax.legend()\nax.set_xlabel('Time (s)')\n```\n\n\n Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous \u2026\n\n\n No handles with labels found to put in legend.\n\n\n\n\n\n Text(0.5, 0, 'Time (s)')\n\n\n\n\n```python\nL_t\n```\n\n\n```python\nK_c\n```\n\n\n\n\n matrix([[ 3.05930459e-02, 2.28423291e-04, -5.37914607e-04,\n -4.28085968e-06, -2.70547919e-02, -2.11027070e-04,\n 2.11938528e-01, -1.06665661e-01],\n [-2.28423291e-04, 3.05930459e-02, 4.28085968e-06,\n -5.37914607e-04, 2.11027070e-04, -2.70547919e-02,\n 1.06665661e-01, 2.11938528e-01]])\n\n\n\n\n```python\n\n```\n\n\n```python\nsyst = vsc_lcl_class()\nsyst.Dt = 5e-6\nsyst.decimation =1\nsyst.N_store =100_000\nsyst.update()\n\n\u0394t = 100e-6 \n#x_d_ctrl_list = ['i'] # states to consider in the reduction\nz_ctrl_list = [ 'i_sd_g01', 'i_sq_g01'] # outputs to consider in the controller\nu_ctrl_list = ['eta_d_g01','eta_q_g01'] # intputs to consider in the controller\nz_ctrl_idxs = [syst.outputs_list.index(item) for item in z_ctrl_list]\nu_ctrl_idxs = [syst.inputs_run_list.index(item) for item in u_ctrl_list]\n\nsyst.\u0394t = \u0394t\n\n## Calculate equilibirum point\nsyst.initialize([{'G_d_g01':0.0,'eta_d_g01':0.0,'eta_q_g01':-0.8693333,'v_1_Q':-326,'v_1_D':0.0, 'C_m_g01':4e-6}],xy0=1000)\nssa.eval_ss(syst)\n\n# linear continous plant\nA_p = syst.A\nB_p = syst.B\nC_p = syst.C\nD_p = syst.D\n\n# plant discretization\nA_d,B_d,C_d,D_d,Dt = sctrl.cont2discrete((A_p,B_p,C_p,D_p),\u0394t,method='zoh')\n\nN_z_d,N_x_d = C_d.shape # discreticed plant dimensions\nN_x_d,N_u_d = B_d.shape\n\n# convenient matrices\nO_ux = np.zeros((N_u_d,N_x_d))\nO_xu = np.zeros((N_x_d,N_u_d))\nO_uu = np.zeros((N_u_d,N_u_d))\nI_uu = np.eye(N_u_d)\n\nsyst.A_d = A_d\nsyst.B_d = B_d\n\n\n# Controller ##################################################################################\nB_c = B_d[:,u_ctrl_idxs]\nC_c = C_d[z_ctrl_idxs,:]\nD_c = D_d[z_ctrl_idxs,:][:,u_ctrl_idxs]\n\nN_x_c,N_u_d = B_c.shape\nN_z_c,N_x_c = C_c.shape\n\n\nO_ux = np.zeros((N_u_d,N_x_d))\nO_xu = np.zeros((N_x_d,N_u_d))\nO_uu = np.zeros((N_u_d,N_u_d))\nI_uu = np.eye(N_u_d)\n\n\n# discretized plant:\n# \u0394x_d = A_d*\u0394x_d + B_d*\u0394u_d\n# \u0394z_c = C_c*\u0394x_d + D_c*\u0394u_d\n\n# dinamic extension:\n# \u0394x_d = A_d*\u0394x_d + B_d*\u0394u_d\n# \u0394x_i = \u0394x_i + \u0394t*(\u0394z_c-\u0394z_c_ref) = \u0394x_i + \u0394t*C_c*\u0394x_d - Dt*\u0394z_c_ref\n# \u0394z_c = z_c - z_c_0\n# \u0394z_c_ref = z_c_ref - z_c_0\n# (\u0394z_c-\u0394z_c_ref) = z_c - z_c_ref\nomega_b = 2*np.pi*50\n\nW = np.block([\n [ np.cos(omega_b*\u0394t), -np.sin(omega_b*\u0394t)], \n [ np.sin(omega_b*\u0394t), np.cos(omega_b*\u0394t)], \n ])\n\nA_e = np.block([\n [ A_d, B_c@W, O_xu], # \u0394x_d\n [ O_ux, O_uu, O_uu], # \u0394x_r\n [ \u0394t*C_c, \u0394t*D_c, I_uu], # \u0394x_i \n ])\n\nB_e = np.block([\n [ O_xu],\n [ I_uu],\n [ O_uu], \n ])\n\nA_ctrl = A_e[N_x_d:,N_x_d:]\nB_ctrl = B_e[N_x_d:]\n\n# weighting matrices\nQ_c = np.eye(A_e.shape[0])\nQ_c[-1,-1] = 1e7\nQ_c[-2,-2] = 1e7\n\nR_c = np.eye(B_c.shape[1])\n\nK_c,S_c,E_c = ssa.dlqr(A_e,B_e,Q_c,R_c)\n\nE_cont = np.log(E_c)/\u0394t\n\nsyst.A_ctrl = A_ctrl\nsyst.B_ctrl = B_ctrl\nsyst.K_c = K_c\nsyst.N_x_d = N_x_d # number of plant states\nsyst.N_u_d = N_u_d # number of plant inputs\nsyst.N_z_c = N_z_c # number of plant outputs considered for the controller\n```\n\n\n```python\nE_cont.imag/2/np.pi\n```\n\n\n\n\n array([ 4961.58042816, -4961.58042816, 181.10033477, -181.10033477,\n 131.13747671, -131.13747671, 4938.38239422, -4938.38239422,\n 0. , 5000. ])\n\n\n\n\n```python\nW\n```\n\n\n\n\n array([[ 0.99950656, -0.03141076],\n [ 0.03141076, 0.99950656]])\n\n\n\n\n```python\n-E_cont.real/np.abs(E_cont)\n```\n\n\n\n\n array([0.02782544, 0.02782544, 0.75202455, 0.75202455, 0.82785763,\n 0.82785763, 0.95872622, 0.95872622, 1. , 0.99213562])\n\n\n\n\n```python\nE_cont\n```\n\n\n\n\n array([ -867.78091113+31174.52924662j, -867.78091113-31174.52924662j,\n -1298.24512951 +1137.88696253j, -1298.24512951 -1137.88696253j,\n -1216.03836363 +823.96106691j, -1216.03836363 -823.96106691j,\n -104624.92913934+31028.77170057j, -104624.92913934-31028.77170057j,\n -248040.19728861 +0.j , -249017.20157367+31415.9265359j ])\n\n\n\n\n```python\nsyst = vsc_lcl_class()\nsyst.Dt = 5e-6\nsyst.decimation =1\nsyst.N_store =100_000\nsyst.update()\ntimes = np.arange(0.0,0.1,\u0394t)\n\nsyst.initialize([{'G_d_g01':0.0,'eta_d_g01':0.0,'eta_q_g01':-0.8693333,'v_1_Q':-326,'v_1_D':0.0, 'C_m_g01':4e-6}],xy0=1000)\nssa.eval_A(syst)\ni_sd = syst.get_value('i_sd_g01')\ni_sq = syst.get_value('i_sq_g01') \nv_sd = syst.get_value('v_sd_g01')\nv_sq = syst.get_value('v_sq_g01')\ni_td = syst.get_value('i_td_g01')\ni_tq = syst.get_value('i_tq_g01') \nv_md = syst.get_value('v_md_g01')\nv_mq = syst.get_value('v_mq_g01') \nv_dc = syst.get_value('v_dc_g01')\neta_d = syst.get_value('eta_d_g01')\neta_q = syst.get_value('eta_q_g01')\ni_sd_ref_0 = i_sd\ni_sq_ref_0 = i_sq\nv_sq_0 = v_sq\nv_sd_0 = v_sd\nx_d_0 = np.array([i_td,i_tq,v_md,v_mq,i_sd,i_sq]).reshape(6,1)\nu_d_0 = np.array([eta_d,eta_q]).reshape(2,1)\nx_r_0 = u_d_0\nsyst.\u0394x_e = np.zeros((10,1))\nit = 0\nfor t in times:\n \n \u0394x_e = syst.\u0394x_e\n # measurements\n i_sd = syst.get_value('i_sd_g01')\n i_sq = syst.get_value('i_sq_g01') \n v_sd = syst.get_value('v_sd_g01')\n v_sq = syst.get_value('v_sq_g01')\n i_td = syst.get_value('i_td_g01')\n i_tq = syst.get_value('i_tq_g01') \n v_md = syst.get_value('v_md_g01')\n v_mq = syst.get_value('v_mq_g01') \n v_dc = syst.get_value('v_dc_g01')\n\n x_d = np.array([i_td,i_tq,v_md,v_mq,i_sd,i_sq]).reshape(6,1)\n \n \u0394x_d = x_d - x_d_0 \n \u0394x_r = syst.\u0394x_e[N_x_c:-N_u_d,:] \n \u0394x_i = syst.\u0394x_e[(N_x_c+N_u_d):,:] \n \n i_sd_ref = i_sd_ref_0\n i_sq_ref = i_sq_ref_0\n v_sq = v_sq_0\n v_sd = v_sd_0\n if t>20e-3: i_sd_ref = 20\n if t>30e-3: i_sq_ref = 30\n if t>45e-3: v_sd = 163 \n if t>45e-3: v_sq = -163\n epsilon_d = i_sd - i_sd_ref\n epsilon_q = i_sq - i_sq_ref \n \n epsilon = np.block([[epsilon_d],[epsilon_q]])\n \n \u0394u_r = -K_c @ \u0394x_e + np.block([[ (v_sd-v_sd_0)*2/v_dc],[(v_sq-v_sq_0)*2/v_dc]])\n \n \n \u0394x_r = \u0394u_r\n \u0394x_i += \u0394t*epsilon\n \n \u0394x_e = np.block([[\u0394x_d],[\u0394x_r],[\u0394x_i]])\n \n syst.\u0394x_e = \u0394x_e\n \n x_r = \u0394x_r + x_r_0 \n \n eta_dq = W@x_r\n eta_d = eta_dq[0,0] \n eta_q = eta_dq[1,0] \n\n \n events=[{'t_end':t,'eta_d_g01':eta_d,'eta_q_g01':eta_q,'v_1_Q':v_sq,'v_1_D':v_sd}]\n syst.run(events)\n\n# eta_d_prev = eta_d\n# eta_q_prev = eta_q\n it += 1\n \nsyst.post();\n```\n\n\n```python\nplt.close('all')\nfig, axes = plt.subplots(nrows=2, ncols=1, figsize=(7, 7),sharex=True)\n\naxes[0].plot(syst.T,syst.get_values('i_sd_g01'),label='i_sd_g01')\naxes[0].plot(syst.T,syst.get_values('i_sq_g01'),label='i_sq_g01')\n\naxes[1].plot(syst.T,syst.get_values('eta_D_g01'),label='eta_D_g01')\naxes[1].plot(syst.T,syst.get_values('eta_Q_g01'),label='eta_Q_g01')\nfor ax in axes:\n ax.grid()\n ax.legend()\nax.set_xlabel('Time (s)')\n```\n\n\n Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous \u2026\n\n\n\n\n\n Text(0.5, 0, 'Time (s)')\n\n\n\n\n```python\nssa.damp_report(syst)\n\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        RealImagFreq.Damp
                                        Mode 1-15.70796320314.1530973233.097880.000773
                                        Mode 2-15.707963-20314.1530973233.097880.000773
                                        Mode 3-15.70796319685.8345663133.097880.000798
                                        Mode 4-15.707963-19685.8345663133.097880.000798
                                        Mode 5-31.415927314.15926550.000000.099504
                                        Mode 6-31.415927-314.15926550.000000.099504
                                        \n
                                        \n\n\n\n\n```python\nimport sympy as sym\n\nx_d_1,x_d_2,x_d_3,x_d_4,x_d_5,x_d_6 = sym.symbols('Dx_d_1,Dx_d_2,Dx_d_3,Dx_d_4,Dx_d_5,Dx_d_6')\nx_r_1,x_r_2 = sym.symbols('Dx_r_1,Dx_r_2')\nx_i_1,x_i_2 = sym.symbols('Dx_i_1,Dx_i_2')\n\nx_e = sym.Matrix([x_d_1,x_d_2,x_d_3,x_d_4,x_d_5,x_d_6,x_r_1,x_r_2,x_i_1,x_i_2])\nu_r = -K_c * x_e\n```\n\n\n```python\nprint(u_r[0])\n```\n\n 0.023914820787791*Dx_d_1 + 0.00124312201525765*Dx_d_2 + 0.00210868343440974*Dx_d_3 + 9.76673467759675e-5*Dx_d_4 - 0.0363072666129747*Dx_d_5 - 0.0013695946212219*Dx_d_6 - 10.022203616505*Dx_i_1 + 1.45742192739155*Dx_i_2 - 0.327372378893965*Dx_r_1 - 0.000502327486946109*Dx_r_2\n\n\n\n```python\nprint(u_r[0])\n```\n\n 0.023914820787791*Dx_d_1 + 0.00124312201525765*Dx_d_2 + 0.00210868343440974*Dx_d_3 + 9.76673467759675e-5*Dx_d_4 - 0.0363072666129747*Dx_d_5 - 0.0013695946212219*Dx_d_6 - 10.022203616505*Dx_i_1 + 1.45742192739155*Dx_i_2 - 0.327372378893965*Dx_r_1 - 0.000502327486946109*Dx_r_2\n\n\n\n```python\nprint(u_r[1])\n```\n\n -0.00124312201525856*Dx_d_1 + 0.0239148207877911*Dx_d_2 - 9.76673467759516e-5*Dx_d_3 + 0.00210868343440976*Dx_d_4 + 0.00136959462122251*Dx_d_5 - 0.0363072666129741*Dx_d_6 - 1.45742192739181*Dx_i_1 - 10.0222036165041*Dx_i_2 + 0.000502327486927267*Dx_r_1 - 0.327372378893964*Dx_r_2\n\n\n\n```python\nsyst.get_value('C_m_g01')\n```\n\n\n\n\n 4e-06\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "2f12b6c608bfe35dc74b548c497a3725e09b2ef5", "size": 53735, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/machines/vsc/vsc_lcl.ipynb", "max_stars_repo_name": "pydae/pydae", "max_stars_repo_head_hexsha": "8076bcfeb2cdc865a5fc58561ff8d246d0ed7d9d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-12-20T03:45:26.000Z", "max_stars_repo_stars_event_max_datetime": "2020-12-20T03:45:26.000Z", "max_issues_repo_path": "examples/machines/vsc/vsc_lcl.ipynb", "max_issues_repo_name": "pydae/pydae", "max_issues_repo_head_hexsha": "8076bcfeb2cdc865a5fc58561ff8d246d0ed7d9d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "examples/machines/vsc/vsc_lcl.ipynb", "max_forks_repo_name": "pydae/pydae", "max_forks_repo_head_hexsha": "8076bcfeb2cdc865a5fc58561ff8d246d0ed7d9d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 28.0161626694, "max_line_length": 289, "alphanum_fraction": 0.4717037313, "converted": true, "num_tokens": 12441, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417487156366, "lm_q2_score": 0.6477982315512489, "lm_q1q2_score": 0.4265216838023111}} {"text": "
                                        \n

                                        Tutorial 2

                                        \n

                                        Physics Informed Neural Networks Part 4

                                        \n

                                        Navier Stokes Hidden Fluid Mechanics Example

                                        \n
                                        \n\n# Overview\n\nThis notebook is based on two papers: *[Physics-Informed Neural Networks: A Deep LearningFramework for Solving Forward and Inverse ProblemsInvolving Nonlinear Partial Differential Equations](https://www.sciencedirect.com/science/article/pii/S0021999118307125)* and *[Hidden Physics Models: Machine Learning of NonlinearPartial Differential Equations](https://www.sciencedirect.com/science/article/pii/S0021999117309014)* with the help of Fergus Shone and Michael Macraild.\n\nThese tutorials will go through solving Partial Differential Equations using Physics Informed Neuaral Networks focusing on the 1D Heat Equation and a more complex example using the Navier Stokes Equation\n\n**This notebook is a breif illustrative overview of Hidden Physics Models beyond the scope of these tutorials**\n\n
                                        \n \nIf you have not already then in your repositoy directory please run the following code in your terminal (linux or Mac) or via git bash. \n \n```bash\ngit submodule init\ngit submodule update --init --recursive\n```\n \n **If this does not work please clone the [PINNs](https://github.com/maziarraissi/PINNs) repository into your Physics_Informed_Neural_Networks folder**\n
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                                        \n\n

                                        Physics Informed Neural Networks

                                        \n\nFor a typical Neural Network using algorithims like gradient descent to look for a hypothesis, data is the only guide, however if the data is noisy or sparse and we already have governing physical models we can use the knowledge we already know to optamize and inform the algoithms. This can be done via [feature enginnering]() or by adding a physicall inconsistency term to the loss function.\n\n\n \n \n \n## The very basics\n\nIf you know nothing about neural networks there is a [toy neural network python code example](https://github.com/cemac/LIFD_ENV_ML_NOTEBOOKS/tree/main/ToyNeuralNetwork) included in the [LIFD ENV ML Notebooks Repository]( https://github.com/cemac/LIFD_ENV_ML_NOTEBOOKS). Creating a 2 layer neural network to illustrate the fundamentals of how Neural Networks work and the equivlent code using the python machine learning library [tensorflow](https://keras.io/). \n\n \n## Recommended reading \n \nThe in-depth theory behind neural networks will not be covered here as this tutorial is focusing on application of machine learning methods. If you wish to learn more here are some great starting points. \n\n* [All you need to know on Neural networks](https://towardsdatascience.com/nns-aynk-c34efe37f15a) \n* [Introduction to Neural Networks](https://victorzhou.com/blog/intro-to-neural-networks/)\n* [Physics Guided Neural Networks](https://towardsdatascience.com/physics-guided-neural-networks-pgnns-8fe9dbad9414)\n* [Maziar Rassi's Physics informed GitHub web Page](https://maziarraissi.github.io/PINNs/)\n\n
                                        \n\n\n
                                        \n\n# Hidden Fluid Mechanics #\n\nIn this notebook, we will utilise a more advanced implementation of PINNs, taken from Maziar Raissi's paper Hidden Fluid Mechanics. In many fluid flow scenarios, direct measurement of variables such as velocity and pressure is not possible; However, we may have access to measurements of some passive scalar field (such as smoke concentration in wind tunnel testing). This work aims to use PINNs to uncover hidden velocity and pressure fields for flow problems, utilising data drawn only from measurements of some passive scalar, $c(t,x,y)$. This scalar is governed by the transport equation:\n\n\\begin{equation}\nc_t + u c_x + v c_y = \\text{Pec}^{-1} \\left(c_{xx} + c_{yy}\\right)\n\\end{equation}\n\nwhere $\\mathrm{Pec}$ is the Peclet number, and $u, v$ are the two velocity components. Then, the Navier-Stokes equations are given by:\n\n\\begin{equation} \nu_t + uu_x + vu_y = -p_x +\\mathrm{Re}^{-1}(u_{xx} + u_{yy}),\n\\\\\nv_t + uv_x + vv_y = -p_y +\\mathrm{Re}^{-1}(v_{xx} + v_{yy}),\n\\\\\nu_x + v_y = 0\n\\end{equation}\n\nfor pressure $p$ and Reynolds number $\\mathrm{Re}$. Note that the only constraint on the velocity and pressure predictions is that they satisfy the above equations.\n\nThe structure of the neural network is seen below:\n\n\n\nThe network has four inputs, as expected, and six outputs, namely the three velocity components, the pressure, the concentration, c, and one final variable, d. d(t,x,y,z) is an 'auxilliary variable', defined to be the complement of c (i.e. d = 1 - c), and is governed by the same transport equation as c. Its inclusion improves prediction accuracy, and helps in detecting boundary locations.\n\nThe data for this notebook was generated by Raissi et al. using a spectral element sovler called NekTar. The Navier-Stokes and transport equations are numerically approximated to a high degree of accuracy. \n\nThe fluid problem at hand is 2D channel flow over an obstacle. A crude diagram of the flow domain can be seen below:\n\n\nIt may be useful to read the original paper alongside this notebook, which can be found at https://arxiv.org/pdf/1808.04327.pdf. The source code can be accessed via this Github repository: https://github.com/maziarraissi/HFM.\n\nThis script was orignally written by Maziar Raissi et al. in their PINNs repository, before being modified by Michael MacRaild and Fergus Shone. All figures are from https://arxiv.org/pdf/1808.04327.pdf.\n
                                        \n\n
                                        \n\n

                                        Python

                                        \n\n \n## Tensorflow \n \nThere are many machine learning python libraries available, [TensorFlow](https://www.tensorflow.org/) a is one such library. If you have GPUs on the machine you are using TensorFlow will automatically use them and run the code even faster!\n\n## Further Reading\n\n* [Running Jupyter Notebooks](https://jupyter.readthedocs.io/en/latest/running.html#running)\n* [Tensorflow optimizers](https://www.tutorialspoint.com/tensorflow/tensorflow_optimizers.htm)\n\n
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                                        \n\n
                                        \n \n

                                        Requirements

                                        \n\nThese notebooks should run with the following requirements satisfied\n\n

                                        Python Packages:

                                        \n\n* Python 3\n* tensorflow > 2\n* numpy \n* matplotlib\n* scipy\n\n

                                        Data Requirements

                                        \n \nThis notebook referes to some data included in the git hub repositroy\n \n
                                        \n\n\n**Contents:**\n\n1. [1D Heat Equation Non ML Example](PINNs_1DHeatEquations_nonML.ipynb)\n2. [1D Equation PINN Example](PINNs_1DHeatEquationExample.ipynb)\n3. [Navier-Stokes PINNs discovery of PDE\u2019s](PINNs_NavierStokes_example.ipynb)\n3. **[Navier-Stokes PINNs Hidden Fluid Mechanics](PINNs_Navier_Stokes_HFM.ipynb)**\n\n
                                        \n\n
                                        \nLoad in all required modules (includig some auxillary code) and turn off warnings. Make sure Keras session is clear\n
                                        \n\n\n```python\n# For readability: disable warnings\nimport warnings\nwarnings.filterwarnings('ignore')\n```\n\n\n```python\nimport sys\nsys.path.insert(0, 'PINNs/Utilities/')\nimport tensorflow as tf\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport scipy.io\nfrom scipy.interpolate import griddata\nfrom itertools import product, combinations\nfrom mpl_toolkits.mplot3d import Axes3D\nfrom mpl_toolkits.mplot3d.art3d import Poly3DCollection\nfrom mpl_toolkits.axes_grid1 import make_axes_locatable\nimport matplotlib.gridspec as gridspec\nfrom time import time\nimport scipy.sparse as sp\nimport scipy.sparse.linalg as la\n```\n\n
                                        \n\n# Define PINN Functions\n \nIn this section we define a number of functions to be called later in the script, such as error metrics, the general neural network class etc. In the original github repository, these functions are defined in a separate script called 'utilities.py', which is called in each example script. To keep this notebook contained, we include them here instead.\n\nThe basic functionality of this code is the same as the previous examples, but some of the syntax is updated to streamline the code. \n \n
                                        \n\n\n```python\n# generate tensorflow session and initialise variables\ndef tf_session():\n # tf session\n config = tf.compat.v1.ConfigProto(allow_soft_placement=True,\n log_device_placement=True)\n config.gpu_options.force_gpu_compatible = True\n sess = tf.compat.v1.Session(config=config)\n \n # init\n init = tf.compat.v1.global_variables_initializer()\n sess.run(init)\n \n return sess\n\n# produce relative error between prediction and ground truth\ndef relative_error(pred, exact):\n if type(pred) is np.ndarray:\n return np.sqrt(np.mean(np.square(pred - exact))/np.mean(np.square(exact - np.mean(exact))))\n return tf.sqrt(tf.reduce_mean(tf.square(pred - exact))/tf.reduce_mean(tf.square(exact - tf.reduce_mean(exact))))\n\n# produce MSE of predicition and ground truth\ndef mean_squared_error(pred, exact):\n if type(pred) is np.ndarray:\n return np.mean(np.square(pred - exact))\n return tf.reduce_mean(tf.square(pred - exact))\n\n# generate gradient of Y w.r.t. x using automatic differentiation\ndef fwd_gradients(Y, x):\n dummy = tf.ones_like(Y)\n G = tf.compat.v1.gradients(Y, x, grad_ys=dummy, colocate_gradients_with_ops=True)[0]\n Y_x = tf.compat.v1.gradients(G, dummy, colocate_gradients_with_ops=True)[0]\n return Y_x\n\n# calculate 2D velocity gradients\ndef Gradient_Velocity_2D(u, v, x, y):\n \n Y = tf.concat([u, v], 1)\n \n Y_x = fwd_gradients(Y, x)\n Y_y = fwd_gradients(Y, y)\n \n u_x = Y_x[:,0:1]\n v_x = Y_x[:,1:2]\n \n u_y = Y_y[:,0:1]\n v_y = Y_y[:,1:2]\n \n return [u_x, v_x, u_y, v_y]\n\n# basic neural network class. This function constructs the network architecture, \n# normalises the inputs using the mean and standard deviation and specifies the\n# activation function, which is chosen here to be sigmoid.\nclass neural_net(object):\n def __init__(self, *inputs, layers):\n \n self.layers = layers\n self.num_layers = len(self.layers)\n \n if len(inputs) == 0:\n in_dim = self.layers[0]\n self.X_mean = np.zeros([1, in_dim])\n self.X_std = np.ones([1, in_dim])\n else:\n X = np.concatenate(inputs, 1)\n self.X_mean = X.mean(0, keepdims=True)\n self.X_std = X.std(0, keepdims=True)\n \n self.weights = []\n self.biases = []\n self.gammas = []\n \n for l in range(0,self.num_layers-1):\n in_dim = self.layers[l]\n out_dim = self.layers[l+1]\n W = np.random.normal(size=[in_dim, out_dim])\n b = np.zeros([1, out_dim])\n g = np.ones([1, out_dim])\n # tensorflow variables\n self.weights.append(tf.Variable(W, dtype=tf.float32, trainable=True))\n self.biases.append(tf.Variable(b, dtype=tf.float32, trainable=True))\n self.gammas.append(tf.Variable(g, dtype=tf.float32, trainable=True))\n \n def __call__(self, *inputs):\n \n H = (tf.concat(inputs, 1) - self.X_mean)/self.X_std\n \n for l in range(0, self.num_layers-1):\n W = self.weights[l]\n b = self.biases[l]\n g = self.gammas[l]\n # weight normalization\n V = W/tf.norm(W, axis = 0, keepdims=True)\n # matrix multiplication\n H = tf.matmul(H, V)\n # add bias\n H = g*H + b\n # activation\n if l < self.num_layers-2:\n H = H*tf.sigmoid(H)\n \n Y = tf.split(H, num_or_size_splits=H.shape[1], axis=1)\n \n return Y\n\n# produce the necessary output gradients and then form the Navier-Stokes residuals for 2D flow\ndef Navier_Stokes_2D(c, u, v, p, t, x, y, Pec, Rey):\n \n Y = tf.concat([c, u, v, p], 1)\n \n Y_t = fwd_gradients(Y, t)\n Y_x = fwd_gradients(Y, x)\n Y_y = fwd_gradients(Y, y)\n Y_xx = fwd_gradients(Y_x, x)\n Y_yy = fwd_gradients(Y_y, y)\n \n c = Y[:,0:1]\n u = Y[:,1:2]\n v = Y[:,2:3]\n p = Y[:,3:4]\n \n c_t = Y_t[:,0:1]\n u_t = Y_t[:,1:2]\n v_t = Y_t[:,2:3]\n \n c_x = Y_x[:,0:1]\n u_x = Y_x[:,1:2]\n v_x = Y_x[:,2:3]\n p_x = Y_x[:,3:4]\n \n c_y = Y_y[:,0:1]\n u_y = Y_y[:,1:2]\n v_y = Y_y[:,2:3]\n p_y = Y_y[:,3:4]\n \n c_xx = Y_xx[:,0:1]\n u_xx = Y_xx[:,1:2]\n v_xx = Y_xx[:,2:3]\n \n c_yy = Y_yy[:,0:1]\n u_yy = Y_yy[:,1:2]\n v_yy = Y_yy[:,2:3]\n \n e1 = c_t + (u*c_x + v*c_y) - (1.0/Pec)*(c_xx + c_yy)\n e2 = u_t + (u*u_x + v*u_y) + p_x - (1.0/Rey)*(u_xx + u_yy) \n e3 = v_t + (u*v_x + v*v_y) + p_y - (1.0/Rey)*(v_xx + v_yy)\n e4 = u_x + v_y\n \n return e1, e2, e3, e4\n\n# calculate the strain rate for generating results later in the script\ndef Strain_Rate_2D(u, v, x, y):\n \n [u_x, v_x, u_y, v_y] = Gradient_Velocity_2D(u, v, x, y)\n \n eps11dot = u_x\n eps12dot = 0.5*(v_x + u_y)\n eps22dot = v_y\n \n return [eps11dot, eps12dot, eps22dot]\n```\n\n
                                        \n\n# Define PINN (HFM) Class\n \nHere the main PINN class is defined. It is shorter than the PINN class in the previous notebooks as many of the functions that were previously included inside are defined externally in the previous section, and called within this class. There is also a change in the way the network is trained, with the implementation of batch training (or mini-batch training). This is useful when very large amounts of data are used, as it divides the entire training set into subsets called batches, avoiding issues with exceeding data allocation. Also, when implemented correctly, batch training can be more efficient. It is worth reading up on batch training in general, as it is a useful machine learning approach. \n \n
                                        \n\n\n```python\nclass HFM(object):\n # notational conventions\n # _tf: placeholders for input/output data and points used to regress the equations\n # _pred: output of neural network\n # _eqns: points used to regress the equations\n # _data: input-output data\n # _star: preditions\n\n def __init__(self, t_data, x_data, y_data, c_data,\n t_eqns, x_eqns, y_eqns,\n layers, batch_size):\n \n # specs\n self.layers = layers\n self.batch_size = batch_size\n \n # flow properties\n self.Pec = tf.Variable(15.0, dtype=tf.float32, trainable = True)\n self.Rey = tf.Variable(5.0, dtype=tf.float32, trainable = True)\n \n # data\n [self.t_data, self.x_data, self.y_data, self.c_data] = [t_data, x_data, y_data, c_data]\n [self.t_eqns, self.x_eqns, self.y_eqns] = [t_eqns, x_eqns, y_eqns]\n \n # placeholders\n [self.t_data_tf, self.x_data_tf, self.y_data_tf, self.c_data_tf] = [tf.compat.v1.placeholder(tf.float32, shape=[None, 1]) for _ in range(4)]\n [self.t_eqns_tf, self.x_eqns_tf, self.y_eqns_tf] = [tf.compat.v1.placeholder(tf.float32, shape=[None, 1]) for _ in range(3)]\n \n # physics \"uninformed\" neural networks\n self.net_cuvp = neural_net(self.t_data, self.x_data, self.y_data, layers = self.layers)\n \n [self.c_data_pred,\n self.u_data_pred,\n self.v_data_pred,\n self.p_data_pred] = self.net_cuvp(self.t_data_tf,\n self.x_data_tf,\n self.y_data_tf)\n \n # physics \"informed\" neural networks\n [self.c_eqns_pred,\n self.u_eqns_pred,\n self.v_eqns_pred,\n self.p_eqns_pred] = self.net_cuvp(self.t_eqns_tf,\n self.x_eqns_tf,\n self.y_eqns_tf)\n \n [self.e1_eqns_pred,\n self.e2_eqns_pred,\n self.e3_eqns_pred,\n self.e4_eqns_pred] = Navier_Stokes_2D(self.c_eqns_pred,\n self.u_eqns_pred,\n self.v_eqns_pred,\n self.p_eqns_pred,\n self.t_eqns_tf,\n self.x_eqns_tf,\n self.y_eqns_tf,\n self.Pec,\n self.Rey)\n \n [self.eps11dot_eqns_pred,\n self.eps12dot_eqns_pred,\n self.eps22dot_eqns_pred] = Strain_Rate_2D(self.u_eqns_pred,\n self.v_eqns_pred,\n self.x_eqns_tf,\n self.y_eqns_tf)\n \n # loss\n self.loss = mean_squared_error(self.c_data_pred, self.c_data_tf) + \\\n mean_squared_error(self.e1_eqns_pred, 0.0) + \\\n mean_squared_error(self.e2_eqns_pred, 0.0) + \\\n mean_squared_error(self.e3_eqns_pred, 0.0) + \\\n mean_squared_error(self.e4_eqns_pred, 0.0)\n \n # optimizers\n self.learning_rate = tf.compat.v1.placeholder(tf.float32, shape=[])\n self.optimizer = tf.compat.v1.train.AdamOptimizer(learning_rate = self.learning_rate)\n self.train_op = self.optimizer.minimize(self.loss)\n \n self.sess = tf_session()\n \n # Add ops to save and restore all the variables.\n self.netSaveDir = \"modelckpts/HFM/\"\n self.saver = tf.compat.v1.train.Saver()\n\n def train(self, total_time, learning_rate):\n \n N_data = self.t_data.shape[0]\n N_eqns = self.t_eqns.shape[0]\n \n start_time = time()\n running_time = 0\n it = 0\n while running_time < total_time:\n \n idx_data = np.random.choice(N_data, min(self.batch_size, N_data))\n idx_eqns = np.random.choice(N_eqns, self.batch_size)\n \n (t_data_batch,\n x_data_batch,\n y_data_batch,\n c_data_batch) = (self.t_data[idx_data,:],\n self.x_data[idx_data,:],\n self.y_data[idx_data,:],\n self.c_data[idx_data,:])\n\n (t_eqns_batch,\n x_eqns_batch,\n y_eqns_batch) = (self.t_eqns[idx_eqns,:],\n self.x_eqns[idx_eqns,:],\n self.y_eqns[idx_eqns,:])\n\n\n tf_dict = {self.t_data_tf: t_data_batch,\n self.x_data_tf: x_data_batch,\n self.y_data_tf: y_data_batch,\n self.c_data_tf: c_data_batch,\n self.t_eqns_tf: t_eqns_batch,\n self.x_eqns_tf: x_eqns_batch,\n self.y_eqns_tf: y_eqns_batch,\n self.learning_rate: learning_rate}\n \n self.sess.run([self.train_op], tf_dict)\n \n # Print\n if it % 10 == 0:\n elapsed = time() - start_time\n running_time += elapsed/3600.0\n [loss_value,\n Pec_value,\n Rey_value,\n learning_rate_value] = self.sess.run([self.loss,\n self.Pec,\n self.Rey,\n self.learning_rate], tf_dict)\n print('It: %d, Loss: %.3e, Pec: %.3f, Rey: %.3f, Time: %.2fs, Running Time: %.2fh, Learning Rate: %.1e'\n %(it, loss_value, Pec_value, Rey_value, elapsed, running_time, learning_rate_value))\n sys.stdout.flush()\n start_time = time()\n it += 1\n\n if it % 1000 == 0:\n save_path = self.saver.save(self.sess, self.netSaveDir + 'model_at_iter%s.ckpt'%(it))\n print('Model saved in path: %s' % save_path)\n save_path = self.saver.save(self.sess, self.netSaveDir + 'model.ckpt')\n print('Model saved in path: %s' % save_path)\n \n def predict(self, t_star, x_star, y_star):\n tf_dict = {self.t_data_tf: t_star, self.x_data_tf: x_star, self.y_data_tf: y_star}\n \n c_star = self.sess.run(self.c_data_pred, tf_dict)\n u_star = self.sess.run(self.u_data_pred, tf_dict)\n v_star = self.sess.run(self.v_data_pred, tf_dict)\n p_star = self.sess.run(self.p_data_pred, tf_dict)\n \n return c_star, u_star, v_star, p_star\n \n def predict_eps_dot(self, t_star, x_star, y_star):\n \n tf_dict = {self.t_eqns_tf: t_star, self.x_eqns_tf: x_star, self.y_eqns_tf: y_star}\n \n eps11dot_star = self.sess.run(self.eps11dot_eqns_pred, tf_dict)\n eps12dot_star = self.sess.run(self.eps12dot_eqns_pred, tf_dict)\n eps22dot_star = self.sess.run(self.eps22dot_eqns_pred, tf_dict)\n \n return eps11dot_star, eps12dot_star, eps22dot_star\n```\n\n
                                        \n\n# Main Code\n \nIn this section we run each of our previously defined functions and classes to train the network and generate results. We specify the batch size, network width and depth, process the training and test data, train the network and output error results. Note that the 'train' function is commented out, so if you would like to re-train the network feel free to uncomment that line (although be warned: it will overwrite the previously trained weights). \n\n\n
                                        \n\n\n```python\n# This line of code is required to prevent some tensorflow errors arrising from the\n# inclusion of some tensorflw v 1 code \ntf.compat.v1.disable_eager_execution()\n\nbatch_size = 10000\n\nlayers = [3] + 10*[4*50] + [4]\n\n# Load Data\ndata = scipy.io.loadmat('Data/Stenosis2D.mat')\n\nt_star = data['t_star'] # T x 1\nx_star = data['x_star'] # N x 1\ny_star = data['y_star'] # N x 1\n\nT = t_star.shape[0]\nN = x_star.shape[0]\n\nU_star = data['U_star'] # N x T\nV_star = data['V_star'] # N x T\nP_star = data['P_star'] # N x T\nC_star = data['C_star'] # N x T \n\n# Rearrange Data \nT_star = np.tile(t_star, (1,N)).T # N x T\nX_star = np.tile(x_star, (1,T)) # N x T\nY_star = np.tile(y_star, (1,T)) # N x T \n\n######################################################################\n######################## Noiseless Data ##############################\n######################################################################\n\nT_data = T # int(sys.argv[1])\nN_data = N # int(sys.argv[2])\nidx_t = np.concatenate([np.array([0]), np.random.choice(T-2, T_data-2, replace=False)+1, np.array([T-1])] )\nidx_x = np.random.choice(N, N_data, replace=False)\nt_data = T_star[:, idx_t][idx_x,:].flatten()[:,None]\nx_data = X_star[:, idx_t][idx_x,:].flatten()[:,None]\ny_data = Y_star[:, idx_t][idx_x,:].flatten()[:,None]\nc_data = C_star[:, idx_t][idx_x,:].flatten()[:,None]\n\nT_eqns = T\nN_eqns = N\nidx_t = np.concatenate([np.array([0]), np.random.choice(T-2, T_eqns-2, replace=False)+1, np.array([T-1])] )\nidx_x = np.random.choice(N, N_eqns, replace=False)\nt_eqns = T_star[:, idx_t][idx_x,:].flatten()[:,None]\nx_eqns = X_star[:, idx_t][idx_x,:].flatten()[:,None]\ny_eqns = Y_star[:, idx_t][idx_x,:].flatten()[:,None]\n```\n\n\n```python\n# Training\nmodel = HFM(t_data, x_data, y_data, c_data,\n t_eqns, x_eqns, y_eqns,\n layers, batch_size)\n```\n\n
                                        \n \n# Loading Pre trained model option \n \nIf the training time is too slow you can skip the following line and load in a pretrained model instead set `loadweights = True` in the next cell. You can play around with different number of iterations to see the effects e.g. setting `saver.restore(sess, netSaveDir + 'model_at_iter3000.ckpt')`\n\n
                                        \n\n\n```python\n# model.train(total_time = 40, learning_rate=1e-3) # UNCOMMENT THIS LINE IF YOU WOULD LIKE TO RE-TRAIN\n```\n\n\n```python\nloadweights = True\nif loadweights:\n print(\"loading pre trained model\")\n netSaveDir = \"modelckpts/HFM/\"\n saver = tf.compat.v1.train.Saver()\n saver.restore(model.sess, netSaveDir + 'model_at_iter3000.ckpt')\n```\n\n\n```python\nShear = np.zeros((300,t_star.shape[0]))\n\nfor snap in range(0,t_star.shape[0]):\n\n x1_shear = np.linspace(15,25,100)[:,None]\n x2_shear = np.linspace(25,35,100)[:,None]\n x3_shear = np.linspace(35,55,100)[:,None]\n\n x_shear = np.concatenate([x1_shear,x2_shear,x3_shear], axis=0)\n\n y1_shear = 0.0*x1_shear\n y2_shear = np.sqrt(25.0 - (x2_shear - 30.0)**2)\n y3_shear = 0.0*x3_shear\n\n y_shear = np.concatenate([y1_shear,y2_shear,y3_shear], axis=0)\n\n t_shear = T_star[0,snap] + 0.0*x_shear\n\n eps11_dot_shear, eps12_dot_shear, eps22_dot_shear = model.predict_eps_dot(t_shear, x_shear, y_shear)\n\n nx1_shear = 0.0*x1_shear\n nx2_shear = 6.0 - x2_shear/5.0\n nx3_shear = 0.0*x3_shear\n\n nx_shear = np.concatenate([nx1_shear,nx2_shear,nx3_shear], axis=0)\n\n ny1_shear = -1.0 + 0.0*y1_shear\n ny2_shear = -y2_shear/5.0\n ny3_shear = -1.0 + 0.0*y3_shear\n\n ny_shear = np.concatenate([ny1_shear,ny2_shear,ny3_shear], axis=0)\n\n shear_x = 2.0*(1.0/5.0)*(eps11_dot_shear*nx_shear + eps12_dot_shear*ny_shear)\n shear_y = 2.0*(1.0/5.0)*(eps12_dot_shear*nx_shear + eps22_dot_shear*ny_shear)\n\n shear = np.sqrt(shear_x**2 + shear_y**2)\n\n Shear[:,snap] = shear.flatten()\n```\n\n\n```python\n# Test Data\nsnap = np.array([55])\nt_test = T_star[:,snap]\nx_test = X_star[:,snap]\ny_test = Y_star[:,snap]\n\nc_test = C_star[:,snap]\nu_test = U_star[:,snap]\nv_test = V_star[:,snap]\np_test = P_star[:,snap]\n```\n\n\n```python\n# Prediction\nc_pred, u_pred, v_pred, p_pred = model.predict(t_test, x_test, y_test)\n```\n\n\n```python\n# Error\nerror_c = relative_error(c_pred, c_test)\nerror_u = relative_error(u_pred, u_test)\nerror_v = relative_error(v_pred, v_test)\nerror_p = relative_error(p_pred - np.mean(p_pred), p_test - np.mean(p_test))\n\nprint('Error c: %e' % (error_c))\nprint('Error u: %e' % (error_u))\nprint('Error v: %e' % (error_v))\nprint('Error p: %e' % (error_p))\n```\n\n\n```python\n################# Save Data ###########################\n\nC_pred = 0*C_star\nU_pred = 0*U_star\nV_pred = 0*V_star\nP_pred = 0*P_star\n```\n\n\n```python\n# calculate the relative error of each variable per time-step\nfor snap in range(0,t_star.shape[0]):\n t_test = T_star[:,snap:snap+1]\n x_test = X_star[:,snap:snap+1]\n y_test = Y_star[:,snap:snap+1]\n\n c_test = C_star[:,snap:snap+1]\n u_test = U_star[:,snap:snap+1]\n v_test = V_star[:,snap:snap+1]\n p_test = P_star[:,snap:snap+1]\n\n # Prediction\n c_pred, u_pred, v_pred, p_pred = model.predict(t_test, x_test, y_test)\n\n C_pred[:,snap:snap+1] = c_pred\n U_pred[:,snap:snap+1] = u_pred\n V_pred[:,snap:snap+1] = v_pred\n P_pred[:,snap:snap+1] = p_pred\n\n # Error\n error_c = relative_error(c_pred, c_test)\n error_u = relative_error(u_pred, u_test)\n error_v = relative_error(v_pred, v_test)\n error_p = relative_error(p_pred - np.mean(p_pred), p_test - np.mean(p_test))\n print('Time step: ', snap)\n print('Error c: %e' % (error_c))\n print('Error u: %e' % (error_u))\n print('Error v: %e' % (error_v))\n print('Error p: %e' % (error_p))\n \n \n```\n\n
                                        \n\n \n# Plots\n \nSince the original code included no Python-based plotting functionality, the plots have been taken from the original [paper](https://www.sciencedirect.com/science/article/pii/S0021999117309014). Here we see, at one time step, the concentration fields c and d. The inverted nature of these two variables can be seen. \n\n\n\nIn the plots below, we see comparisons between ground truth and prediction for each output variable at one time step. We see that the PINN is capable of replicating each variable with a good degree of accuracy, although it is worth noting that the pressure prediction can only be accurate up to a constant, as it only appears in the Navier-Stokes equation in derivative form. This discrepency can be seen in the colourbar alongside the pressure plots.\n\n\n\nFinally, plots for ground truth and predicted wall shear stress (WSS) can be found below. Here, plotted is the WSS magnitude on the lower boundary of the domain at every time step. What is most impressive here is that we have not imposed any boundary conditions on the lower wall and we still obtain very accurate predictions based only on information from within the flow.\n\n\n \n \n
                                        \n\n\n```python\n\n```\n", "meta": {"hexsha": "eb2cba0e88a060834a7e8747e69fbcee6af5ddd9", "size": 38809, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Physics_Informed_Neural_Networks/PINNs_NavierStokes_HFM.ipynb", "max_stars_repo_name": "wolfiex/LIFD_ENV_ML_NOTEBOOKS", "max_stars_repo_head_hexsha": "324949a841a7a0977b2358d1eaf3eb9b22924ce2", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2021-08-13T21:38:21.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-21T06:16:23.000Z", "max_issues_repo_path": "Physics_Informed_Neural_Networks/PINNs_NavierStokes_HFM.ipynb", "max_issues_repo_name": "wolfiex/LIFD_ENV_ML_NOTEBOOKS", "max_issues_repo_head_hexsha": "324949a841a7a0977b2358d1eaf3eb9b22924ce2", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2021-07-26T10:10:12.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-24T12:08:01.000Z", "max_forks_repo_path": "Physics_Informed_Neural_Networks/PINNs_NavierStokes_HFM.ipynb", "max_forks_repo_name": "wolfiex/LIFD_ENV_ML_NOTEBOOKS", "max_forks_repo_head_hexsha": "324949a841a7a0977b2358d1eaf3eb9b22924ce2", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-08-13T21:39:34.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-13T21:39:34.000Z", "avg_line_length": 41.0243128964, "max_line_length": 714, "alphanum_fraction": 0.5405962534, "converted": true, "num_tokens": 7637, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.658417487156366, "lm_q1q2_score": 0.4265216748484004}} {"text": "# Linear Algebraic Systems \n\n
                                        \n\n# Learning Objectives \n\n### *Setting up a linear system*\n- Be able to distinguish a linear system from a nonlinear system.\n- Learn to write a linear system in a matrix-vector format.\n- Input a linear system into a program\n- Write applicable engineering models in a linear format.\n\n### *Characterizing linear systems*\n- Determine the bandwidth(s) of the linear system.*\n- Determine if a linear system has a unique solution.\n- Will solving a linear system provide a meaningful result (is it well-posed).\n\n### *Solving a linear system*\n- Solve a linear system by Gauss-Elimination\n- Solve a linear system by LU factorizations.\n- State when numerical errors occur may occur in Gauss-Elimination and how to resolve them.\\*\n- Be able to state some conditions for the convergence of Jacobi and Gauss-Siedel methods.\\*\n\n\\*Not covered in tutorial.\n\n
                                        \n\n# Relevance \n\nIt would be only a slight overstatement to say that linear algebra underlies all modern numerical algorithms to one degree or another. Even software which specifically addresses nonlinear or complicated forms will often make use of linear algebra in a myriad of subroutines. In essence, many complicated problems can be reduced to repeated formulating and solving linear systems. \n\n
                                        \n\n# Chlorination of Benzene \n\nMonochlorobenzene is a high boiling point solvent used in many industrial applications ranging from the manufacture of [dyes](https://patentimages.storage.googleapis.com/76/55/12/31756682f755c5/US6013811.pdf), [plastic and rubber processing](https://patentimages.storage.googleapis.com/66/c8/e1/69918f557fbd35/US8563079.pdf), [paints](https://patentimages.storage.googleapis.com/28/17/a7/a40e46f75806cf/US9062206.pdf), and [waxes](https://patents.google.com/patent/US3999960A/en). Approximately, [$1 billion](https://www.prnewswire.com/news-releases/global-chlorobenzene-industry-300939916.html) worth of this solvent are used per year. It is [produced by combining chlorine with benzene](https://onlinelibrary.wiley.com/doi/abs/10.1002/14356007.o06_o03) in the presence of a Lewis acid catalyst such as Ferric Chloride FeCl3 or Anhydrous Aluminum Chloride AlCl3. When benzene and chlorine are mixed two key reactions occur: (1) the chlorination of benzene and (2) the secondary undesirable chlorination of the resulting monochlorobenzene. Product purity is ensured by performing a series of separation steps after the initial reaction and reagant usage is reduced by including a recycle loop. \n\n
                                        \nObjective: A large customer order has been placed for monochlorobenzene that needs to be fulfilled on a tight timeline. The small chemical company you work for has occasionally produced this solvent in the past and has an existing 6m3 continuously stirred tank reactor (CSTR) that will need to be repurposed to produce 25kmol/h monochlorobenzene. Moreover, there isn't sufficient development time available to attempt to improve the processing conditions for the reactor or separators. As a result, you'll need to use the documented process conditions for the reactor and separators which means supplying to the first separator an effluent consisting of 90% benzene, 7% monochlorobenzene, and 3% dichlorobenzene by mole. We need to determine the recycle flow rates and reaction rates required needed to achieve the desired rate at which monochlorobenzene need to be produced. \n
                                        \n\nDenote benzene as (A), monochlorobenzene as (B), and dichlorobenzene as (C). The mole fraction of species A in stream S1 is denote by y1,A and the molar flowrate F [kmol/h] for stream S1 is denoted F1.\n\n- The feed stream is composed entirely of A and Cl2.\n- An excess amount of Cl2 is provided and HCl removal can be assumed to be trivial. So it's reasonable exclude it from our design considerations. Moreover, this means the reactions which take place in the reactor vessel can be simplified to A → B (at rate r1 [kmol/h]) and B → C (at rate r2 [kmol/h]).\n\n\n\n# Variables and Mass Balances \n\nFor streams containing only a pure substance, we need only keep track of the total flow rate. Enumerating each quantity in the model is given by $F_1$, $F_2$, $F_3$, $F_4$, $F_5$, $F_6$, $F_7$, $y_{3,A}$, $y_{3,B}$, $y_{3,C}$, $y_{4,B}$, $y_{4,C}$, $r_1$, $r_2$, and $\\nu$. Five of these are specified (e.g. these are **parameters** with fixed values). This leaves ten degrees of freedom associated with a variable. We'll now need to write out the equations which govern the system (physical laws, equipment specifications, and performance specifications). We'll start with writing the mass balances over each unit operation. \n\nMixer (Overall Mass Balance): \n\n\\begin{align}\n F_1 + F_7 - F_2 &= 0\n\\end{align}\n\nReactor (Individual Component Mass Balances):\n\n\\begin{align}\n F_2 - r_1\\nu - y_{3,A}F_3 &= 0 \\\\\n (r_1 - r_2)\\nu + y_{3,B}F_3 &= 0 \\\\\n r_2\\nu - y_{3,C}F_3 &= 0\n\\end{align}\n\nSeparator #1 (Individual Component Mass Balances):\n\n\\begin{align}\n F_3 - F_4 - F_7 &= 0 \\\\\n y_{3,B}F_3 - y_{4,B}F_4 &= 0 \\\\\n y_{3,C}F_3 - y_{4,C}F_4 &= 0\n\\end{align}\n\nSeparator #2 (Individual Component Mass Balances):\n\n\\begin{align}\n F_4 - F_5 - F_6 &= 0 \\\\\n y_{4,B}F_4 - F_5 &= 0\n\\end{align}\n\nIn order to write these equations in a matrix-vector form: we'll first need to associate each variable (excluding **parameters**) in the model with a component in a vector of variables, $\\mathbf{x}$. \n\n\\begin{align}\n \\mathbf{x} = (F_1, F_2, F_3, F_4, F_6, F_7, y_{4,B}, y_{4,C}, r_1, r_2)\n\\end{align}\n\nSo, $x_2 = F_2$, $x_3 = F_3$ and so on until we have $x_{10} = r_2$. \n\n
                                        \nNow, take a minute to inspect the model and categorize the system of equations. \n
                                        \n\nThe 2nd and 3rd equations for Separator #1 as well as the 2nd equation in Separator #2 contain the expressions $y_{4,B}F_4$ and $y_{4,C}F_4$. These terms consist of two variables being multiplied by one another (a bilinear term) which is one of simplest commonly encountered nonlinear term. All other expressions consist of addition, subtraction, and the multiplication of a parameter by a variable. So this system of equations is nonlinear but would be linear if we ommitted the expressions: $y_{4,B}F_4$ and $y_{4,C}F_4$.\n\n
                                        \nIt's often useful to reformulate a model (algebraicly rearrange and potentially introduce new variables) to arrive at a more easily solvable form. \n
                                        \n\nWe'll start by introducing two new variables: the molar flowrates $F_{4,B}$ and $F_{4,C}$ defined as by\n\n\\begin{align}\n F_{4,B} = y_{4,B}F_4 \\\\\n F_{4,C} = y_{4,C}F_4\n\\end{align}\n\nWe can then write an additional equation for the molar flowrates of S4:\n\nMolar flowrate expressions for S4:\n\n\\begin{align}\n F_{4,B} + F_{4,C} &= F_4\n\\end{align}\n\nNext, we'll introduce the variables: $F_{3,A}$, $F_{3,B}$, and $F_{3,C}$ defined as:\n\n\\begin{align}\n F_{3,A} = y_{3,A}F_3 \\\\\n F_{3,B} = y_{3,B}F_3 \\\\\n F_{3,C} = y_{3,C}F_3\n\\end{align}\n\nWe can then write an additional equation for the molar flowrates of S3:\n\nMolar flowrate expressions for S3:\n\n\\begin{align}\nF_{3,A} + F_{3,B} + F_{3,C} = F_3\n\\end{align}\n\nWe can then rearrange the mass balances for the Reactor, Separator #1, and Separator #2.\n\nReactor (Individual Component Mass Balances):\n\n\\begin{align}\n F_2 - r_1\\nu - F_{3,A} &= 0 \\\\\n (r_1 - r_2)\\nu + F_{3,B} &= 0 \\\\\n r_2\\nu - F_{3,C} &= 0\n\\end{align}\n\nReformulated Separator #1 Mass Balances:\n\n\\begin{align}\n F_3 - F_4 - F_7 &= 0 \\\\\n F_{3,B} - F_{4,B} &= 0 \\\\\n F_{3,C} - F_{4,C} &= 0\n\\end{align}\n\nReformulated Separator #2 Mass Balances:\n\n\\begin{align}\n F_4 - F_5 - F_6 &= 0 \\\\\n F_{4,B} - F_5 &= 0\n\\end{align}\n\n\\begin{align}\n \\mathbf{x'} = (F_1, F_2, F_{3,A}, F_{3,B}, F_{3,C}, F_3, F_{4,B}, F_{4,C}, F_4, F_6, F_7, r_1, r_2)\n\\end{align}\n\n\n
                                        \nINTERACTIVE! Now we enter the above linear equation, Mx' = b in matrix-vector form.\n
                                        \n\n\n```julia\n# Define the Mx' = b linear system\n\n\n# Define storage for linear system\nM = zeros(13,13) # Makes an 2d array of size 13-by-13. This array has 13 rows and 13 columns.\nb = zeros(13) # Makes a column vector of size 10. A vector with 13 rows and 1 column.\n\n# Input parameters\ny3A = 0.90 # effluent benzene concentration of reactor\ny3B = 0.07 # effluent monochlorobenzene concentration of reactor\ny3C = 0.03 # effluent dichlorobenzene concentration of reactor\nV = 6 # reactor volume\nF_5 = 25.0 # flow rate of monochlorobenzene required\n\n### MASS BALANCE (MIXER) ###\n\n# fills in the first row\nM[1,1] = 1.0 # Set the matrix M's entry in the first row and first column to 1\nM[1,2] = -1.0 # Set the matrix M's entry in the first row and second column to -1\nM[1,11] = 1.0 # Set the matrix M's entry in the first row and eighth column to 1\n\n# b[1] and all other entries of M in row #1, so no need to change these values\n\nM[2,2] = 1.0 # reactor - equation #1 \nM[2,3] = -1.0\nM[2,12] = -V\n\nM[3,4] = 1.0 # reactor - equation #2 \nM[3,12] = -V\nM[3,13] = V\n\nM[4,5] = -1.0 # reactor - equation #3 \nM[4,13] = V\n\nM[10,8] = 1.0 # stream 4 balance \nM[10,9] = -1.0\nM[10,7] = 1.0\n\nM[11,3] = 1.0 # stream 3 balance #1 \nM[11,4] = 1.0\nM[11,5] = 1.0\nM[11,6] = -1.0\n\nM[12,3] = 1.0 # stream 3 balance #2\nM[12,6] = -y3A\n\nM[13,4] = 1.0 # stream 4 balance #3\nM[13,6] = -y3B\n\n### SEPARATOR #1 & 2 - MASS BALANCES ###\n\n# FILL IN THE REST BELOW HERE IN ROWS 5 to 9 of M!!!!!!!!\n```\n\n\n\n\n -0.07\n\n\n\n\n```julia\n# Run this to display the input matrix and vector\ndisplay(\"This Matrix (M) is \") \ndisplay(M)\ndisplay(\"The r.h.s vector (b) is \") \ndisplay(b)\nx = M\\b\ndisplay(x)\n```\n\n\n \"This Matrix (M) is \"\n\n\n\n 13\u00d713 Array{Float64,2}:\n 1.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0\n 0.0 1.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -6.0 0.0\n 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -6.0 6.0\n 0.0 0.0 0.0 0.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.0\n 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 -1.0 0.0 -1.0 0.0 0.0\n 0.0 0.0 0.0 1.0 0.0 0.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0\n 0.0 0.0 0.0 0.0 1.0 0.0 0.0 -1.0 0.0 0.0 0.0 0.0 0.0\n 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 -1.0 0.0 0.0 0.0\n 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0\n 0.0 0.0 0.0 0.0 0.0 0.0 1.0 1.0 -1.0 0.0 0.0 0.0 0.0\n 0.0 0.0 1.0 1.0 1.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n 0.0 0.0 1.0 0.0 0.0 -0.9 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n 0.0 0.0 0.0 1.0 0.0 -0.07 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n\n\n\n \"The r.h.s vector (b) is \"\n\n\n\n 13-element Array{Float64,1}:\n 0.0\n 0.0\n 0.0\n 0.0\n 0.0\n 0.0\n 0.0\n 25.0\n 25.0\n 0.0\n 0.0\n 0.0\n 0.0\n\n\n\n 13-element Array{Float64,1}:\n 83.33333333333337 \n 833.3333333333339 \n 750.0000000000006 \n 58.33333333333337 \n 25.0 \n 833.3333333333339 \n 58.33333333333337 \n 25.0 \n 83.33333333333337 \n 58.33333333333337 \n 750.0000000000006 \n 13.888888888888896\n 4.166666666666667\n\n\n# Existence and Uniqueness of the Solution \n\nNow that the linear system has been formulated, let's check to see whether the system has a solution. For a square matrix M, M system is invertible\n\n
                                        \n INTERACTIVE! Make use of the function det to check see if matrix M is invertible. Does the linear equation Mx = b have a unique solution? \n
                                        \n\n\n```julia\nusing LinearAlgebra: det\n\n# FILL IN THE REST BELOW HERE\ndet(M)\n```\n\n\n\n\n -1.0799999999999992\n\n\n\n# Well-posed problems and the Condition Number \n\nWhile a linear system, `Mx = b`, may have a unique solution, we also seek to understand whether using a particular algorithm to solve `Mx = b` will yeild an accurate solution. We often don't know `M` or `b` exactly. As a consequence, solutions which vary greatly when `M` or `b` are slightly perturbed may be suspect. This is generally assessed by evaluating the **condition number** of a linear system and is defined as: $\\text{cond}(M) \\equiv ||M|| \\times ||M^{-1}||$. A derivation of this is given in section 2.3.4 of Dorfman and Daoutidis, Numerical Methods with Chemical Engineering Applications which arrives at the following inequality:\n\n$||\\Delta x || \\leq \\text{cond}(M)\\bigg(\\frac{|| \\Delta b || || x ||}{|| b ||}\\bigg)$\n\nFor a fixed condition number, $\\text{cond}(M)$, solution $x$, and $b$ if we slightly perturb $b$ by $\\Delta b$ then $x$ may change by at most an amount proportional to the condition number. Another way of interpreting this is by applying the following rule of thumb: the condition number $\\kappa$ means that the method looses $\\log_{10}{\\kappa}$ of accuracy relative to rounding error.\n\nIf an application leads to an ill-posed problem (values of **M** and **b** may be known to very high precision), there are a multitude of ways this may be dealt with. This most common and often most robust approach is to apply a [**preconditioner**](http://www.mathcs.emory.edu/~benzi/Web_papers/survey.pdf). That is we find an appropriate matrix **Y** and multiply both sides of the original linear system to create an equivalent new linear system, `(YM)x = (Yb)`, which has a lower condition number. We can then solve the equivalent modified system. \n\n
                                        \nINTERACTIVE! Make use of the function cond to compute the condition number using the $L^2$-norm. Is the matrix A well-conditioned? \n
                                        \n\n\n```julia\nusing LinearAlgebra: cond\n\n# FILL IN THE REST BELOW HERE\n```\n\n\n\n\n 1494.7023893088221\n\n\n\n# Solving the linear equation with Gauss Elimination \n\nNow let's write a quick script to solve Mx = b. We'll do this in three steps. Defining a function to perform forward elimination, a function to back substitution, and lastly a main function to perform each of the prior functions in sequence.\n\n\n```julia\nfunction forward_elimination!(M,b,n)\n \n for k = 1:(n - 1)\n for i = (k + 1):n\n m = M[i,k]/M[k,k]\n for j = (k + 1):n\n M[i,j] = M[i,j] - m*M[k,j]\n end\n b[i] = b[i] - m*b[k]\n end\n end\n \n return nothing\nend\n```\n\n
                                        \nComment: Note that while M, b are used in prior cells these variables don't interfere in the definition of forward_elimination!. The variables listed in the argument of forward_elimination!, that is (M,x,b,n), are evaluated in local scope when forward_elimination! is called.\n
                                        \n\n
                                        \nINTERACTIVE! The overall gauss elimination function is defined below. You just need to finish coding the back_substitution! function.\n
                                        \n\n\n```julia\nfunction back_substitution!(M,x,b,n)\n \n # FILL IN THE REST BELOW HERE\n \n return nothing\nend\n```\n\n\n```julia\nfunction Gauss_Elimination!(M, b)\n n = length(b)\n x = zeros(n)\n \n forward_elimination!(M,b,n)\n back_substitution!(M,x,b,n) \n \n return x\nend\n\n# Runs a Gauss elimination routine using Mx = b has inputs\nx = Gauss_Elimination!(M, b)\n\ndisplay(\"The solution via Gauss elimination is \"); \ndisplay(x)\n```\n\n# Solving with an LU factorization \n\n
                                        \nINTERACTIVE! Run the following snippet of code to solve the linear system using LU factorization to solve the linear system.\n
                                        \n\n\n```julia\nusing LinearAlgebra: lu\n\nlu_fact = lu(M) # performs an LU factorization of M. Note that pivot = Val(false)\n # means that the LU factorization is performed without pivoting\n # (otherwise a permutation matrix describing the pivots is computed as well)\n\ny = lu_fact.L\\b # solve Ly = b for y\nx = lu_fact.U\\y # solve Ux = y for x\n\ndisplay(\"The solution via LU factorization is \"); \ndisplay(x)\n```\n\n\n \"The solution via LU factorization is \"\n\n\n\n 13-element Array{Float64,1}:\n 83.33333333333337 \n 833.3333333333339 \n 750.0000000000006 \n 58.33333333333337 \n 24.999999999999996\n 833.3333333333339 \n 58.33333333333337 \n 25.0 \n 83.33333333333337 \n 58.33333333333337 \n 750.0000000000006 \n 13.888888888888896\n 4.166666666666666\n\n\n
                                        \n\n# Questions for reflection \n\n- What varieties of problems may result in a banded matrix?\n- What does it mean that an iterative method converged? \n- When is an absolute or relative convergence criteria preferable?\n- When is solving by one method versus another preferable (e.g. Banded Gauss-Elimination versus Gauss-Siedel)?\n", "meta": {"hexsha": "4f8a1509dc3198fd9be7fb6cbca731245b5f1771", "size": 24963, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorial/julia/Tutorial 1 - Linear Algebra/J1_LAsystems.ipynb", "max_stars_repo_name": "PSORLab/Chemical_Engineering_Analysis_Notebooks", "max_stars_repo_head_hexsha": "fce093cc35e7cccbbb34efd6940c725e3af0757d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2021-06-10T17:41:10.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-24T15:01:21.000Z", "max_issues_repo_path": "tutorial/julia/Tutorial 1 - Linear Algebra/J1_LAsystems.ipynb", "max_issues_repo_name": "PSORLab/Chemical_Engineering_Analysis_Notebooks", "max_issues_repo_head_hexsha": "fce093cc35e7cccbbb34efd6940c725e3af0757d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorial/julia/Tutorial 1 - Linear Algebra/J1_LAsystems.ipynb", "max_forks_repo_name": "PSORLab/Chemical_Engineering_Analysis_Notebooks", "max_forks_repo_head_hexsha": "fce093cc35e7cccbbb34efd6940c725e3af0757d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2021-06-10T17:48:30.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-08T04:37:33.000Z", "avg_line_length": 39.3118110236, "max_line_length": 1239, "alphanum_fraction": 0.5541000681, "converted": true, "num_tokens": 6034, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.7690802317779601, "lm_q1q2_score": 0.4264322737057987}} {"text": "```python\nfrom ngames.evaluation.extensivegames import ExtensiveFormGame, plot_game\nfrom ngames.evaluation.build import build_full_game\nfrom ngames.evaluation.equilibrium import minimize_incentives, \\\n subgame_perfect_equilibrium, outcome_probability\n```\n\n# Default configuration\n\nBoth fishers start at the shore. They can go to one of two fishing spots. If fishers go to different spots, they stay there. If they go to the same spot, they can choose to stay or leave. If, even after then, both fishers end up at the same spot, they fight. The probability of any fisher $i$ winning the fight is given by:\n\n\\begin{equation}\nP\\left[\\text{$i$ wins fight}\\right]=\\frac{S_i}{\\sum_j S_j}\n\\end{equation}\n\nwhere $S_i$ denotes the strength of fisher $i$, a parameter between 1 (weakest) and 10 (strongest).\n\nTo assign the utilities at the terminal nodes of the games, we associate the following actions and outcomes with some payoff:\n\n* Travel between spots: $c<0$.\n\n* Fish alone at spot $i$ / win fight over spot $i$: $v_i$, with $v_1>v_2$.\n\n* Loose a fight: $d<0$.\n\n\n```python\nimport re\nfrom networkx.algorithms.shortest_paths.generic import shortest_path\n\ndef assign_utilities(G, spot_value={'spot1': 10, 'spot2': 5}, c=-2, d=-6):\n for z in G.game_tree.terminal_nodes:\n utility = {p: 0 for p in G.players}\n fluents = G.state_fluents[z]\n agents_location = {}\n fight_winner = None\n for f in fluents:\n parsed_f = re.split('\\(|, |\\)', f)\n if parsed_f[0] == 'at':\n agents_location[parsed_f[1]] = parsed_f[2]\n if parsed_f[0] == 'won_fight':\n fight_winner = parsed_f[1]\n # fishers keeps a spot to himself\n if len(agents_location.values()) == len(set(agents_location.values())):\n for ag in G.players:\n utility[ag] += spot_value[agents_location[ag]]\n # fisher wins a fight for a spot\n if fight_winner:\n utility[fight_winner] += spot_value[agents_location[fight_winner]]\n # everyone else loses\n for p in G.players:\n if p != fight_winner:\n utility[p] += d\n # a fisher travels between spots\n path_from_root = shortest_path(G.game_tree, G.game_tree.root, z)\n for i in range(len(path_from_root) -1):\n edge = (path_from_root[i], path_from_root[i+1])\n if G.game_tree.edges[edge]['action'] == 'leave':\n player = G.turn_function[path_from_root[i]]\n utility[player] += c\n G.set_utility(z,utility)\n```\n\n\n```python\ngame_default = build_full_game('.', 'fishers', 0)\nassign_utilities(game_default)\n```\n\n\n```python\nmy_fig_kwargs = dict(figsize=(20, 10), frameon=False)\nmy_node_kwargs = dict(font_size=10, node_size=500, edgecolors='k',\n linewidths=1.5)\nmy_edge_kwargs = dict(arrowsize=15, width=1.5)\nmy_edge_labels_kwargs = dict(font_size=10)\nmy_patch_kwargs = dict(linewidth=2)\nmy_legend_kwargs = dict(fontsize=16, loc='upper right', edgecolor='white')\nmy_info_sets_kwargs = dict(linestyle='--', linewidth=3)\nmy_utility_label_kwargs = dict(horizontalalignment='center', fontsize=10,\n weight='bold')\n\nposition_colors = {'alice': 'aquamarine', 'bob': 'greenyellow'}\n\nfig = plot_game(game_default,\n position_colors,\n fig_kwargs=my_fig_kwargs,\n node_kwargs=my_node_kwargs,\n edge_kwargs=my_edge_kwargs,\n edge_labels_kwargs=my_edge_labels_kwargs,\n patch_kwargs=my_patch_kwargs,\n legend_kwargs=my_legend_kwargs,\n utility_label_kwargs=my_utility_label_kwargs,\n utility_label_shift=0.08,\n decimals=0,\n info_sets_kwargs=my_info_sets_kwargs)\n```\n\n\n```python\nfor n, f in game_default.state_fluents.items():\n f.sort()\n print(\"{} -- {}\".format(n, ', '.join(f)))\n```\n\n 1 -- at(alice, shore), at(bob, shore)\n 4 -- at(alice, spot1), at(bob, spot1)\n 5 -- at(alice, spot1), at(bob, spot2)\n 6 -- at(alice, spot2), at(bob, spot1)\n 7 -- at(alice, spot2), at(bob, spot2)\n 14 -- at(alice, spot1), at(bob, spot1), won_fight(bob)\n 15 -- at(alice, spot1), at(bob, spot1), won_fight(alice)\n 11 -- at(alice, spot1), at(bob, spot2)\n 12 -- at(alice, spot2), at(bob, spot1)\n 16 -- at(alice, spot2), at(bob, spot2), won_fight(bob)\n 17 -- at(alice, spot2), at(bob, spot2), won_fight(alice)\n 24 -- at(alice, spot2), at(bob, spot2), won_fight(bob)\n 25 -- at(alice, spot2), at(bob, spot2), won_fight(alice)\n 21 -- at(alice, spot2), at(bob, spot1)\n 22 -- at(alice, spot1), at(bob, spot2)\n 26 -- at(alice, spot1), at(bob, spot1), won_fight(bob)\n 27 -- at(alice, spot1), at(bob, spot1), won_fight(alice)\n\n\n\n```python\nsubgame_mixed_strat, back_utilities, incentives = \\\nsubgame_perfect_equilibrium(game_default, minimize_incentives)\n\noutcome_prob = {t: outcome_probability(game_default, subgame_mixed_strat, t)\n for t in game_default.game_tree.terminal_nodes}\nexpected_utilities = {p: 0 for p in game_default.players}\n\nfor n, p in outcome_prob.items():\n for player in game_default.players:\n expected_utilities[player] += p*game_default.utility[n][player]\n print(\"{} -- {:.2f}\".format(n, p))\n \nprint(expected_utilities)\nprint(sum(outcome_prob.values()))\n```\n\n 5 -- 0.11\n 6 -- 0.25\n 11 -- 0.00\n 12 -- 0.11\n 14 -- 0.18\n 15 -- 0.31\n 16 -- 0.00\n 17 -- 0.00\n 21 -- 0.01\n 22 -- 0.01\n 24 -- 0.00\n 25 -- 0.00\n 26 -- 0.01\n 27 -- 0.01\n {'alice': 4.805554360942283, 'bob': 4.16907862629295}\n 1.0\n\n\n\n```python\nviolent_outcomes = [14, 15, 16, 17,24, 25, 26, 27]\nviolent_outcomes_probs = [outcome_prob[t] for t in violent_outcomes]\nviolence_prob = sum(violent_outcomes_probs)\nprint(violence_prob)\n```\n\n 0.5151145211131013\n\n\n# *First-in-time, first-in-right* configuration\n\nNow fishers also start from the shore, but if they go to the same fishing spot, the fisher who gets there first is commited to the spot, while the fisher who has lost the race has to leave. The probability that fisher $i$ wins the race if both fishers go to the same spot is:\n\n\\begin{equation}\nP(\\text{$i$ wins race}) = \\frac{V_i}{\\sum_j V_j}\n\\end{equation}\n\nwhere $V_i$ denotes the speed of fisher $i$, a parameter between 1 (slowest) and 10 (fastest).\n\n\n```python\ngame_race = build_full_game('.', 'fishers', 1)\nassign_utilities(game_race)\n\nfig = plot_game(game_race,\n position_colors,\n fig_kwargs=my_fig_kwargs,\n node_kwargs=my_node_kwargs,\n edge_kwargs=my_edge_kwargs,\n edge_labels_kwargs=my_edge_labels_kwargs,\n patch_kwargs=my_patch_kwargs,\n legend_kwargs=my_legend_kwargs,\n utility_label_kwargs=my_utility_label_kwargs,\n utility_label_shift=0.08,\n decimals=0,\n info_sets_kwargs=my_info_sets_kwargs)\n```\n\n\n```python\nfor n, f in game_race.state_fluents.items():\n f.sort()\n print(\"{} -- {}\".format(n, ', '.join(f)))\n```\n\n 1 -- at(alice, shore), at(bob, shore)\n 8 -- at(alice, spot1), at(bob, spot1), won_race(bob)\n 9 -- at(alice, spot1), at(bob, spot1), won_race(alice)\n 5 -- at(alice, spot1), at(bob, spot2)\n 6 -- at(alice, spot2), at(bob, spot1)\n 10 -- at(alice, spot2), at(bob, spot2), won_race(bob)\n 11 -- at(alice, spot2), at(bob, spot2), won_race(alice)\n 13 -- at(alice, spot2), at(bob, spot1), won_race(bob)\n 15 -- at(alice, spot1), at(bob, spot2), won_race(alice)\n 17 -- at(alice, spot1), at(bob, spot2), won_race(bob)\n 19 -- at(alice, spot2), at(bob, spot1), won_race(alice)\n\n\n\n```python\nsubgame_mixed_strat, back_utilities, incentives = \\\n subgame_perfect_equilibrium(game_race, minimize_incentives)\n\noutcome_prob = {t: outcome_probability(game_race, subgame_mixed_strat, t)\n for t in game_race.game_tree.terminal_nodes}\nexpected_utilities = {p: 0 for p in game_race.players}\n\nfor n, p in outcome_prob.items():\n for player in game_race.players:\n expected_utilities[player] += p*game_race.utility[n][player]\n print(\"{} -- {:.2f}\".format(n, p))\n\nprint(expected_utilities)\n```\n\n 5 -- 0.00\n 6 -- 0.00\n 13 -- 0.62\n 15 -- 0.38\n 17 -- 0.00\n 19 -- 0.00\n {'alice': 5.6923076923076925, 'bob': 7.307692307692308}\n\n\n# *First-to-announce, first-in-right* configuration\n\nIn this configuration, one of the fishers is randomly assigned to the position of announcer. When both fishers are in the shore, anyone can go to any spot before the announcer has declared to one spot. Then, both fishers can go to any of the spots. If the fishers both go the spot that the announcer has declared, the announcer is guaranted to win the race to the spot. If the fishers both go to the spot that has not been declared by the announcer, any of them might win the race to get there. The winner of the race is commited to the spot (she stays) while the looser has to leave.\n\nNote that the rules for this configuration are *in addition* to the *first-in-time, first-in-right* rules, and they have been assigned priority 2 (*first-in-time, first-in-right* rules where assigned priority 1, and the default rules were assigned priority 0, as usual).\n\n\n```python\ngame_announce = build_full_game('.', 'fishers', 2)\nassign_utilities(game_announce)\n\nmy_fig_kwargs = dict(figsize=(20, 14), frameon=False)\nfig = plot_game(game_announce,\n position_colors,\n fig_kwargs=my_fig_kwargs,\n node_kwargs=my_node_kwargs,\n edge_kwargs=my_edge_kwargs,\n edge_labels_kwargs=my_edge_labels_kwargs,\n patch_kwargs=my_patch_kwargs,\n legend_kwargs=my_legend_kwargs,\n utility_label_kwargs=my_utility_label_kwargs,\n utility_label_shift=0.07,\n decimals=0,\n info_sets_kwargs=my_info_sets_kwargs)\n```\n\n\n```python\nfor n, f in game_announce.state_fluents.items():\n f.sort()\n print(\"{} -- {}\".format(n, ', '.join(f)))\n```\n\n 1 -- at(alice, shore), at(bob, shore)\n 2 -- announced(alice, spot1), at(alice, shore), at(bob, shore)\n 3 -- announced(alice, spot2), at(alice, shore), at(bob, shore)\n 6 -- announced(alice, spot1), at(alice, spot1), at(bob, spot1), won_race(alice)\n 7 -- announced(alice, spot1), at(alice, spot1), at(bob, spot2)\n 8 -- announced(alice, spot1), at(alice, spot2), at(bob, spot1)\n 10 -- announced(alice, spot1), at(alice, spot2), at(bob, spot2), won_race(bob)\n 11 -- announced(alice, spot1), at(alice, spot2), at(bob, spot2), won_race(alice)\n 18 -- announced(alice, spot2), at(alice, spot1), at(bob, spot1), won_race(bob)\n 19 -- announced(alice, spot2), at(alice, spot1), at(bob, spot1), won_race(alice)\n 15 -- announced(alice, spot2), at(alice, spot1), at(bob, spot2)\n 16 -- announced(alice, spot2), at(alice, spot2), at(bob, spot1)\n 17 -- announced(alice, spot2), at(alice, spot2), at(bob, spot2), won_race(alice)\n 21 -- announced(alice, spot1), at(alice, spot1), at(bob, spot2), won_race(alice)\n 23 -- announced(alice, spot1), at(alice, spot1), at(bob, spot2), won_race(bob)\n 25 -- announced(alice, spot1), at(alice, spot2), at(bob, spot1), won_race(alice)\n 27 -- announced(alice, spot2), at(alice, spot2), at(bob, spot1), won_race(alice)\n 29 -- announced(alice, spot2), at(alice, spot2), at(bob, spot1), won_race(bob)\n 31 -- announced(alice, spot2), at(alice, spot1), at(bob, spot2), won_race(alice)\n\n\n\n```python\nsubgame_mixed_strat, back_utilities, incentives = \\\n subgame_perfect_equilibrium(game_announce, minimize_incentives)\n\noutcome_prob = {t: outcome_probability(game_announce, subgame_mixed_strat, t)\n for t in game_announce.game_tree.terminal_nodes}\nexpected_utilities = {p: 0 for p in game_announce.players}\n\nfor n, p in outcome_prob.items():\n for player in game_announce.players:\n expected_utilities[player] += p*game_announce.utility[n][player]\n print(\"{} -- {:.2f}\".format(n, p))\n\nprint(expected_utilities)\n```\n\n 7 -- 1.00\n 8 -- 0.00\n 15 -- 0.00\n 16 -- 0.00\n 21 -- 0.00\n 23 -- 0.00\n 25 -- 0.00\n 27 -- 0.00\n 29 -- 0.00\n 31 -- 0.00\n {'alice': 9.999999999997518, 'bob': 5.000000000001026}\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "bb8e1ebc56d8d476b3daa4503838a257dc70e86b", "size": 355161, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/fishers/.ipynb_checkpoints/fishers-checkpoint.ipynb", "max_stars_repo_name": "nmontesg/norms-games", "max_stars_repo_head_hexsha": "ee4d7ad4f3cc774020cd5617e6957e804995ef70", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-07-22T14:28:31.000Z", "max_stars_repo_stars_event_max_datetime": "2021-07-22T14:28:31.000Z", "max_issues_repo_path": "examples/fishers/fishers.ipynb", "max_issues_repo_name": "nmontesg/norms-games", "max_issues_repo_head_hexsha": "ee4d7ad4f3cc774020cd5617e6957e804995ef70", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "examples/fishers/fishers.ipynb", "max_forks_repo_name": "nmontesg/norms-games", "max_forks_repo_head_hexsha": "ee4d7ad4f3cc774020cd5617e6957e804995ef70", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 686.9651837524, "max_line_length": 136344, "alphanum_fraction": 0.942958264, "converted": true, "num_tokens": 3581, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.66192288918838, "lm_q2_score": 0.6442250996557036, "lm_q1q2_score": 0.4264273392517754}} {"text": "```python\n%load_ext autoreload\n%autoreload 2\n```\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport sympy as sy\nfrom simtk import openmm as mm\nfrom simtk.openmm import app\nimport simtk.unit as unit\n```\n\n Warning: importing 'simtk.openmm' is deprecated. Import 'openmm' instead.\n\n\n# Molecular dynamics\n\n## With OpenMM\n\n```python\nimport openmm as mm\nfrom openmm import unit\nfrom uibcdf_systems import TwoLJParticles\n\nbox=[[2.5, 0.0, 0.0], [0.0, 2.5, 0.0], [0.0, 0.0, 2.5]]*unit.nanometers\nmolecular_system = TwoLJParticles(atom_1='Ar', atom_2='Xe', box=box)\n\nintegrator = mm.LangevinIntegrator(300.0*unit.kelvin, 1.0/unit.picoseconds, 0.1*unit.picoseconds)\nplatform = Platform.getPlatformByName('CUDA')\nsimulation = Simulation(molecular_system.topology, molecular_system.system, integrator, platform)\n\ncoordinates=[[0.0, 0.0, 0.0], [1.0, 0.0, 0.0]]*unit.nanometers\nsimulation.context.setPositions(coordinates)\n\nvelocities = np.zeros([2, 3], np.float32) * unit.nanometers/unit.picoseconds\nsimulation.context.setVelocities(velocities)\n\nsimulation.step(1000)\n```\n\n## With this library\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom openmm import unit\n\nfrom uibcdf_systems import TwoLJParticles\nfrom uibcdf_systems.tools import langevin\n\nbox=[[2.5, 0.0, 0.0], [0.0, 2.5, 0.0], [0.0, 0.0, 2.5]]*unit.nanometers\n\nmolecular_system = TwoLJParticles(atom_1='Ar', atom_2='Xe', box=box)\n```\n\n### Newtonian dynamics\n\n\n```python\nd_min = molecular_system.get_distance_minimum()\ncoordinates = np.zeros([2, 3], np.float32) * unit.nanometers\ncoordinates[0,0] = 1.0 * unit.nanometers\ncoordinates[1,0] = coordinates[0,0] + d_min + 0.05 * unit.angstroms\n\nvelocities = np.zeros([2, 3], np.float32) * unit.nanometers/unit.picoseconds\n\nmolecular_system.set_coordinates(coordinates)\nmolecular_system.set_velocities(velocities)\n\ntraj_dict = langevin(molecular_system,\n friction=0.0/unit.picoseconds,\n temperature=0.0*unit.kelvin,\n time=10.0*unit.picoseconds,\n saving_timestep=0.05*unit.picoseconds,\n integration_timestep=0.05*unit.picoseconds)\n```\n\n\n 0%| | 0/200 [00:00\n\n# Table of Contents\n$$\\label{toc}$$\n\nThis notebook is organized as follows\n\n1. [Step 1](#initializenrpy): Initialize core Python/NRPy+ modules and parameters\n1. [Step 2](#adding_cfuncs_to_dict): Adding C functions to the dictionary\n 1. [Step 2.a](#cfunc__id_lorene_adm_quantities): `ID_Lorene_ADM_quantities_Cartesian`\n 1. [Step 2.b](#cfunc__set_hydro_quantities): `set_hydro_quantities`\n 1. [Step 2.c](#cfunc__initial_data): `initial_data`\n 1. [Step 2.d](#cfunc__hamiltonian_constraint_source_term): `Hamiltonian_constraint_source_term`\n 1. [Step 2.d.i](#hydro_quantities_eos): The equation of state & derived hydrodynamics quantities \n 1. [Step 2.d.ii](#grmhd_rho): The energy density $\\rho$\n 1. [Step 2.d.iii](#adding_ham_constraint_source_term_to_dict): Adding the function to the dictionary\n 1. [Step 2.e](#cfunc__hamiltonian_constraint_no_source_term): `Hamiltonian_constraint_no_source_term`\n 1. [Step 2.e.i](#hamiltonian_constraint_no_source_symb): The Hamiltonian constraint without source terms\n 1. [Step 2.e.ii](#adding_ham_constraint_no_source_term_to_dict): Adding the function to the dictionary\n 1. [Step 2.f](#cfunc__main): `main`\n1. [Step 3](#ccode_kernels_generation): C code kernels generation\n 1. [Step 3.a](#cparams_rfm_and_domainsize): Set `free_parameters.h`; also output C codes needed for declaring and setting Cparameters\n 1. [Step 3.b](#add_all): Add all NRPy+Lorene BNS initial data C codes to C function dictionary\n 1. [Step 3.c](#generate_c_code): Generating C code for setting Lorene BNS initial data in NRPy+\n1. [Step 4](#compiling_and_running): Compiling and running the code\n1. [Step 5](#constraint_violations): Visualization: convergence of Hamiltonian constraint\n1. [Step 6](#latex_pdf_output): Output this notebook to $\\LaTeX$-formatted PDF file\n\n\n\n# Step 1: Initialize core Python/NRPy+ modules and parameters \\[Back to [top](#toc)\\]\n$$\\label{initializenrpy}$$\n\nLet's start by importing all the needed modules from Python/NRPy+:\n\n\n```python\n# Step 1: Initialize core Python/NRPy+ modules and parameters\n# Step 1.a: Add the NRPy+ base directory to the path\nimport os,sys,shutil # Standard Python modules for multiplatform OS-level functions\nnrpy_dir_path = os.path.join(\"..\")\nif nrpy_dir_path not in sys.path:\n sys.path.append(nrpy_dir_path)\n\n# Step 1.b: Import core NRPy+ modules\nimport sympy as sp # SymPy: The Python computer algebra package upon which NRPy+ depends\nimport outputC as outC # NRPy+: Core C code output module\nimport finite_difference as fin # NRPy+: Finite difference C code generation module\nimport cmdline_helper as cmd # NRPy+: Multi-platform Python command-line interface\nimport reference_metric as rfm # NRPy+: Reference metric support\nimport NRPy_param_funcs as par # NRPy+: Parameter interface\nimport indexedexp as ixp # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support\nimport grid as gri # NRPy+: Functions having to do with numerical grids\n\n# Step 1.c: Check if lorene_standalone has been downloaded and compiled\nif not os.path.exists(os.path.join(\"lorene_standalone\", \"Lorene\", \"Lib\", \"liblorenef77_g.a\")):\n print(\"\"\"# Error: Lorene hasn't been compiled yet. Please run the following from nrpytutorial/:\ncd in_progress # Make sure you're in the nrpytutorial root directory.\ngit clone https://bitbucket.org/zach_etienne/lorene_standalone.git\ncd lorene_standalone/\n# For \"Lorene1\": wget http://astro.phys.wvu.edu/zetienne/resu.d\n# Lorene2 (latest):\nwget --no-check-certificate https://ccrgpages.rit.edu/~jfaber/BNSID/Data/simple_polytrope/gam2.5/gam2.5_1.4_1.4/gam2.5_1.4_1.4_hr/gam2.5_1.4_1.4_hr_50/resu_5.000000e+01_1.520000e+00_1.520000e+00.d -O resu.d\ncd Lorene/\nHOME_LORENE=`pwd` make -j20\n\"\"\")\n sys.exit(1)\n\n# Step 1.d: Check if the initial data file exists\nif not os.path.exists(os.path.join(\"lorene_standalone\", \"resu.d\")):\n print(\"\"\"# Error: resu.d not found.\n# Be sure to go into nrpytutorial\n# and run:\ncd in_progress/lorene_standalone\n# For \"Lorene1\": wget http://astro.phys.wvu.edu/zetienne/resu.d\n# Lorene2 (latest):\nwget --no-check-certificate https://ccrgpages.rit.edu/~jfaber/BNSID/Data/simple_polytrope/gam2.5/gam2.5_1.4_1.4/gam2.5_1.4_1.4_hr/gam2.5_1.4_1.4_hr_50/resu_5.000000e+01_1.520000e+00_1.520000e+00.d -O resu.d\n\"\"\")\n sys.exit(1)\n\n# Step P1: Create C code output directory:\nCcodesdir = os.path.join(\"lorene_standalone\", \"interpolator\")\n# Step P1.a: First remove C code output directory if it exists\n# Courtesy https://stackoverflow.com/questions/303200/how-do-i-remove-delete-a-folder-that-is-not-empty\nshutil.rmtree(Ccodesdir, ignore_errors=True)\n# Step P1.b: Then create a fresh directory\ncmd.mkdir(Ccodesdir)\n\n# Step P2: Set basic NRPy+ parameters\nCoordSystem = \"Cartesian\"\npar.set_parval_from_str(\"reference_metric::CoordSystem\",CoordSystem)\n\n# Step P3: Set finite difference order\nFD_order = 4\npar.set_parval_from_str(\"finite_difference::FD_CENTDERIVS_ORDER\",FD_order)\n\n# Step P4: Enable rfm_precompute\nenable_rfm_precompute = True\n\n# Step P5: Disable FD_functions\nenable_FD_functions = False\n\n# Step P6: Disable SIMD\nenable_SIMD = False\n \n# Step P7: Parameter used to compute dt_min\nthismodule = \"Lorene_ID\"\n_wavespeed = par.Cparameters(\"REAL\",thismodule,\"wavespeed\",1.0)\nCFL_FACTOR = 0.5\n\n# Step P8: Set grid parameters defaults\n# Step P8.a: domain_size sets the default value for:\n# * Spherical's params.RMAX\n# * SinhSpherical*'s params.AMAX\n# * Cartesians*'s -params.{x,y,z}min & .{x,y,z}max\n# * Cylindrical's -params.ZMIN & .{Z,RHO}MAX\n# * SinhCylindrical's params.AMPL{RHO,Z}\n# * *SymTP's params.AMAX\ndomain_size = 12.5 # Needed for all coordinate systems.\n\n# Step P8.b: sinh_width sets the default value for:\n# * SinhSpherical's params.SINHW\n# * SinhCylindrical's params.SINHW{RHO,Z}\n# * SinhSymTP's params.SINHWAA\nsinh_width = 0.2 # If Sinh* coordinates chosen\n\n# Step P8.c: sinhv2_const_dr sets the default value for:\n# * SinhSphericalv2's params.const_dr\n# * SinhCylindricalv2's params.const_d{rho,z}\nsinhv2_const_dr = 0.05 # If Sinh*v2 coordinates chosen\n\n# Step P8.d: SymTP_bScale sets the default value for:\n# * SinhSymTP's params.bScale\nSymTP_bScale = 1.0 # If SymTP chosen\n\n# Step P9: Create rfm_files directory, if rfm_precompute is enabled\nif enable_rfm_precompute:\n cmd.mkdir(os.path.join(Ccodesdir, \"rfm_files/\"))\n par.set_parval_from_str(\"reference_metric::rfm_precompute_to_Cfunctions_and_NRPy_basic_defines\",\"True\")\n par.set_parval_from_str(\"reference_metric::rfm_precompute_Ccode_outdir\", os.path.join(Ccodesdir, \"rfm_files/\"))\n\n# Step P10: Create SIMD directory and copy intrinsics, if SIMD is enabled\nif enable_SIMD:\n cmd.mkdir(os.path.join(Ccodesdir,\"SIMD\"))\n shutil.copy(os.path.join(\"SIMD/\")+\"SIMD_intrinsics.h\",os.path.join(Ccodesdir,\"SIMD/\"))\n```\n\n\n\n# Step 2: Adding C functions to the dictionary \\[Back to [top](#toc)\\]\n$$\\label{adding_cfuncs_to_dict}$$\n\nWe will now add all C functions that we will need to read in the [LORENE](https://lorene.obspm.fr/) initial data and interpolate it onto the NRPy+ grids.\n\n\n\n## Step 2.a: `ID_Lorene_ADM_quantities_Cartesian` \\[Back to [top](#toc)\\]\n$$\\label{cfunc__id_lorene_adm_quantities}$$\n\nThe first function we will write is the one that sets up the ADM quantities from the [LORENE](https://lorene.obspm.fr/) initial data file, typically named `resu.d`. NRPy+ needs us to define the following ADM quantities:\n\n$$\n\\left(\\alpha,\\beta^{i},\\gamma_{ij},K_{ij}\\right),\n$$\n\nwhere $\\alpha$ is the lapse function, $\\beta^{i}$ is the shift vector, $\\gamma_{ij}$ is the physical spatial metric, and $K_{ij}$ is the extrinsic curvature.\n\nWe note here that these quantities will all be read from the [LORENE](https://lorene.obspm.fr/) initial data file and the interpolated onto the NRPy+ grid in the `main` function [below](#cfunc__main).\n\nIn this step, we write the function `ID_Lorene_ADM_quantities_Cartesian`, which is required by the C functions generated by the [`ADM_Numerical_Spherical_or_Cartesian_to_BSSNCurvilinear.py`](ADM_Numerical_Spherical_or_Cartesian_to_BSSNCurvilinear.py) \\[[**edit**](../edit/ADM_Numerical_Spherical_or_Cartesian_to_BSSNCurvilinear.py)\\] NRPy+ module. The goal of this function is to initialize local variables for the ADM quantities above from a user-defined struct of type `ID_inputs`. In our case this is a very simple task since our struct, which is defined in [Step 3.c](#ccode_kernels__defines_prototypes) below, contains pointers to the [LORENE](https://lorene.obspm.fr/) gridfunctions arrays, and therefore all we need to do is copy over the data to our local variables at the particular gridpoint we are interested in.\n\nNote that we also initialize $B^{i}$, which is used by the second-order Gamma-driver shift condition, to zero.\n\n\n```python\n# Step 2: Adding C functions to the directionary\n# Step 2.a: Adding ID_Lorene_ADM_quantities_Cartesian to the C functions dictionary\ndef add_to_Cfunction_dict_ID_Lorene_ADM_quantities_Cartesian():\n desc = \"\"\"\n(c) 2021 Leo Werneck\nThis function sets the initial data for all ADM quantities.\n\"\"\"\n includes = [\"NRPy_basic_defines.h\"]\n prefunc = \"\"\n c_type = \"void\"\n name = \"ID_Lorene_ADM_quantities_Cartesian\"\n \n print(\"Writing \"+name+\" function\")\n \n params = \"\"\"const paramstruct *restrict params,\n const int i0i1i2[3],const REAL xyz_or_rthph[3],\n const ID_inputs other_inputs,\n REAL *gammaDD00,REAL *gammaDD01,REAL *gammaDD02,REAL *gammaDD11,REAL *gammaDD12,REAL *gammaDD22,\n REAL *KDD00,REAL *KDD01,REAL *KDD02,REAL *KDD11,REAL *KDD12,REAL *KDD22,\n REAL *alpha,\n REAL *betaU0,REAL *betaU1,REAL *betaU2,\n REAL *BU0,REAL *BU1,REAL *BU2\"\"\"\n body = \"\"\"\n // Set useful indices\n const int i0 = i0i1i2[0];\n const int i1 = i0i1i2[1];\n const int i2 = i0i1i2[2];\n const int idx = IDX3S(i0,i1,i2);\n \n // Lapse function alpha\n *alpha = other_inputs.alp[idx];\n \n // Shift vector beta^{i}\n *betaU0 = other_inputs.betax[idx];\n *betaU1 = other_inputs.betay[idx];\n *betaU2 = other_inputs.betaz[idx];\n \n // B^{i}\n *BU0 = 0.0;\n *BU1 = 0.0;\n *BU2 = 0.0;\n\n // Spatial metric gamma_{ij}\n *gammaDD00 = other_inputs.gxx[idx];\n *gammaDD01 = other_inputs.gxy[idx];\n *gammaDD02 = other_inputs.gxz[idx];\n *gammaDD11 = other_inputs.gyy[idx];\n *gammaDD12 = other_inputs.gyz[idx];\n *gammaDD22 = other_inputs.gzz[idx];\n\n // Extrinsic curvature K_{ij}\n *KDD00 = other_inputs.kxx[idx];\n *KDD01 = other_inputs.kxy[idx];\n *KDD02 = other_inputs.kxz[idx];\n *KDD11 = other_inputs.kyy[idx];\n *KDD12 = other_inputs.kyz[idx];\n *KDD22 = other_inputs.kzz[idx]; \n\"\"\"\n loopopts = \"InteriorPoints\"\n outC.add_to_Cfunction_dict(\n includes=includes,\n prefunc=prefunc,\n desc=desc,\n c_type=c_type, name=name, params=params,\n body=body,enableCparameters=True)\n \n print(\"Finished writing \"+name+\" function\")\n```\n\n\n\n## Step 2.b: `set_hydro_quantities` \\[Back to [top](#toc)\\]\n$$\\label{cfunc__set_hydro_quantities}$$\n\nWe now focus on the task of setting up the hydro quantities needed by our code, namely\n\n$$\n\\left(\\rho_{\\rm b},u^{\\mu},b^{\\mu}\\right),\n$$\n\nwhere $\\rho_{\\rm b}$ is the baryonic density, $u^{\\mu}$ is the fluid four-velocity, and $b^{\\mu}$ is related to the magnetic field $B^{i}$ via (see e.g. Eqs. 31, 23, and 24 of [Duez *et al*. (2005)](https://arxiv.org/pdf/astro-ph/0503420.pdf))\n\n$$\n\\begin{align}\nb^{\\mu} &= \\frac{B^{\\mu}_{(u)}}{\\sqrt{4\\pi}},\\\\\nB^{0}_{(u)} &= \\frac{u_{i}B^{i}}{\\alpha},\\\\\nB^{i}_{(u)} &= \\frac{B^{i}/\\alpha + B^{0}_{(u)}u^{i}}{u^{0}}.\n\\end{align}\n$$\n\nWe will assume that our initial data is unmagnetized and therefore will set $b^{\\mu} = 0$. Note also that other hydro quantities, such as the pressure $P$ and the specific internal energy $\\epsilon$, can be computed from the base quantities above and therefore we will not store them in gridfunctions.\n\nThe [LORENE](https://lorene.obspm.fr/) initial data file contains the base hydro quantities that we need in our code, namely the baryonic density $\\rho_{\\rm b}$ and the fluid 3-velocity in the Eulerian reference frame $v^{i}_{(n)}$. We are then left with the task of computing $u^{\\mu}$ from the input data, a task which we now explain in detail.\n\nAfter reading in the local value of $v^{i}_{(n)}$ must determine $u^{0}$. This is done by first remembering that the [Lorentz factor](https://en.wikipedia.org/wiki/Lorentz_factor), $W$, is given by\n\n$$\nW = \\frac{1}{\\sqrt{1 - \\gamma_{ij}v^{i}_{(n)}v^{j}_{(n)}}}.\n$$\n\nFurthermore, remember that (see e.g. Eq. 15 in [Noble *et al*. (2006)](https://arxiv.org/pdf/astro-ph/0512420.pdf), noticing that they use the notation $W\\to\\gamma$)\n\n$$\nW = -n_{\\mu}u^{\\mu} = \\alpha u^{0},\n$$\n\nwhere $n_{\\mu}=\\left(\\alpha,0,0,0\\right)$ is the unit vector normal to spatial hypersurfaces. We thus find the identity\n\n$$\n\\frac{1}{W^{2}} = \\frac{1}{\\left(\\alpha u^{0}\\right)^{2}} = 1 - \\gamma_{ij}v^{i}_{(n)}v^{j}_{(n)}\n\\implies\nA \\equiv 1 - \\frac{1}{\\left(\\alpha u^{0}\\right)^{2}} = \\gamma_{ij}v^{i}_{(n)}v^{j}_{(n)}.\n$$\n\nAfter $A$ is computed we can determine $u^{0}$ trivially using\n\n$$\n\\frac{1}{\\left(\\alpha u^{0}\\right)^{2}} = 1 - A \\implies u^{0} = \\frac{1}{\\alpha\\sqrt{1-A}}.\n$$\n\nAt this point we can compute the fluid 3-velocity $u^{i}$ using (see e.g. Eq. 10 [Etienne *et al.* (2015)](https://arxiv.org/pdf/1501.07276.pdf))\n\n$$\nu^{i} = u^{0}\\left(\\alpha v^{i}_{(n)} - \\beta^{i}\\right).\n$$\n\n\n```python\n# Step 2.b: Adding set_hydro_quantities to the C functions dictionary\ndef add_to_Cfunction_dict_set_hydro_quantities():\n desc = \"\"\"\n(c) 2021 Leo Werneck\nThis function sets the initial data for all hydro quantities.\n\"\"\"\n includes = [\"NRPy_basic_defines.h\",\"bin_ns.h\"]\n prefunc = \"\"\n c_type = \"void\"\n name = \"set_hydro_quantities\"\n params = \"\"\"const paramstruct *restrict params, const ID_inputs other_inputs, REAL *restrict aux_gfs\"\"\"\n body = r\"\"\"\n // Set the index\n const int idx = IDX3S(i0,i1,i2);\n \n // Read in needed metric quantities\n const REAL alpL = other_inputs.alp[idx];\n const REAL gxxL = other_inputs.gxx[idx];\n const REAL gxyL = other_inputs.gxy[idx];\n const REAL gxzL = other_inputs.gxz[idx];\n const REAL gyyL = other_inputs.gyy[idx];\n const REAL gyzL = other_inputs.gyz[idx];\n const REAL gzzL = other_inputs.gzz[idx];\n const REAL betaxL = 0.0;//other_inputs.betax[idx];\n const REAL betayL = 0.0;//other_inputs.betay[idx];\n const REAL betazL = 0.0;//other_inputs.betaz[idx];\n \n // rho_b (don't forget that we need to floor it!)\n const REAL rho_b = std::max(other_inputs.nbar[idx] / other_inputs.rho_b_unit,1e-12);\n \n // Velocities (remember that Lorene gives the Valencia velocity)\n REAL velx = other_inputs.u_euler_x[idx];\n REAL vely = other_inputs.u_euler_y[idx];\n REAL velz = other_inputs.u_euler_z[idx];\n \n // Adapted from IllinoisGRMHD\n REAL vsqrd = gxxL * velx * velx +\n 2.0*gxyL * velx * vely +\n 2.0*gxzL * velx * velz +\n gyyL * vely * vely +\n 2.0*gyzL * vely * velz +\n gzzL * velz * velz;\n\n // Apply velocity limit (W is the Lorentz factor)\n REAL W = 1.0/sqrt(1.0 - vsqrd);\n REAL W_max = 10.0;\n if( W > W_max ) {\n REAL correction_fac = W_max/W;\n velx *= correction_fac;\n vely *= correction_fac;\n velz *= correction_fac;\n W = W_max;\n fprintf(stderr,\"BAAAD: Initial data with very high velocities!\\n\");\n }\n \n // Now compute u^{mu}\n // Remember that: W = alpha u^{0} => u^{0} = W/alpha\n const REAL u4U0 = W/alpL;\n const REAL u4U1 = u4U0 * ( velx * alpL - betaxL );\n const REAL u4U2 = u4U0 * ( vely * alpL - betayL );\n const REAL u4U3 = u4U0 * ( velz * alpL - betazL );\n \n // Set the gridfunctions\n aux_gfs[IDX4ptS(RHOBGF,idx)] = rho_b;\n aux_gfs[IDX4ptS(U4U0GF,idx)] = u4U0;\n aux_gfs[IDX4ptS(U4U1GF,idx)] = u4U1;\n aux_gfs[IDX4ptS(U4U2GF,idx)] = u4U2;\n aux_gfs[IDX4ptS(U4U3GF,idx)] = u4U3;\n \n // TODO: extend to nonzero magnetic fields\n aux_gfs[IDX4ptS(SMALLB4U0GF,idx)] = 0.0;\n aux_gfs[IDX4ptS(SMALLB4U1GF,idx)] = 0.0;\n aux_gfs[IDX4ptS(SMALLB4U2GF,idx)] = 0.0;\n aux_gfs[IDX4ptS(SMALLB4U3GF,idx)] = 0.0;\n\"\"\"\n loopopts = \"AllPoints\"\n outC.add_to_Cfunction_dict(\n includes=includes,\n prefunc=prefunc,\n desc=desc,\n c_type=c_type, name=name, params=params,\n body=body,loopopts=loopopts)\n \n print(\"Finished writing \"+name+\" function\")\n```\n\n\n\n## Step 2.c: `initial_data` \\[Back to [top](#toc)\\]\n$$\\label{cfunc__initial_data}$$\n\nWe now write the core initial data driver function, which is the only one that the user has to directly use to set up initial data. This function performs the following tasks:\n\n1. Initializes all BSSN curvilinear quantities from the input ADM quantities. This is a two-step process, where the functions `ID_BSSN__ALL_BUT_LAMBDAs` and `ID_BSSN_lambdas`, defined by the [`ADM_Numerical_Spherical_or_Cartesian_to_BSSNCurvilinear.py`](ADM_Numerical_Spherical_or_Cartesian_to_BSSNCurvilinear.py) \\[[**edit**] NRPy+ module, are called.\n1. Initializes all hydro quantities. This is done by calling the `set_hydro_quantities` function that we have written in [Step 2.b](cfunc__set_hydro_quantities).\n\n\n```python\n# Step 2.c: initial_data\n# Step 2.c.i: First add the core NRPy+ ADM_Cartesian to\n# BSSN_Curvilinear C functions to the dictionary\nimport ADM_Numerical_Spherical_or_Cartesian_to_BSSNCurvilinear as atob\nrfm.reference_metric()\natob.Convert_Spherical_or_Cartesian_ADM_to_BSSN_curvilinear(\"Cartesian\",\"ID_Lorene_ADM_quantities_Cartesian\",\n Ccodesdir=Ccodesdir,loopopts=\"\")\n\n# Step 2.c.ii: Adding initial_data to the C functions dictionary\ndef add_to_Cfunction_dict_initial_data():\n desc = \"\"\"\n(c) 2021 Leo Werneck\nThis is the initial data driver and is responsible for setting\nall metric and matter field on the initial hypersurface.\n\"\"\"\n includes = [\"NRPy_basic_defines.h\",\"NRPy_function_prototypes.h\",\n \"ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h\",\n \"ID_BSSN__ALL_BUT_LAMBDAs.h\",\n \"ID_BSSN_lambdas.h\"]\n prefunc = \"\"\n c_type = \"void\"\n name = \"initial_data\"\n \n print(\"Writing \"+name+\" function\")\n \n params = \"\"\"const paramstruct *restrict params,REAL *restrict xx[3],ID_inputs other_inputs,\n REAL *restrict in_gfs,REAL *restrict aux_gfs\"\"\"\n body = r\"\"\"\n \n // Set up hydro quantities\n set_hydro_quantities(params,other_inputs,aux_gfs);\n\n // Set up BSSN quantities\n ID_BSSN__ALL_BUT_LAMBDAs(params,xx,other_inputs,in_gfs);\n ID_BSSN_lambdas( params,xx, in_gfs);\n\"\"\"\n outC.add_to_Cfunction_dict(\n includes=includes,\n prefunc=prefunc,\n desc=desc,\n c_type=c_type, name=name, params=params,\n body=body,enableCparameters=False)\n \n print(\"Finished writing \"+name+\" function\")\n```\n\n Output C function ID_BSSN_lambdas() to file lorene_standalone/interpolator/ID_BSSN_lambdas.h\n Output C function ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs() to file lorene_standalone/interpolator/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h\n Output C function ID_BSSN__ALL_BUT_LAMBDAs() to file lorene_standalone/interpolator/ID_BSSN__ALL_BUT_LAMBDAs.h\n\n\n\n\n## Step 2.d: `Hamiltonian_constraint_source_term` \\[Back to [top](#toc)\\]\n$$\\label{cfunc__hamiltonian_constraint_source_term}$$\n\nWe now focus on the computation of the energy density $\\rho$ which appears on the right-hand side of the [Hamiltonian constriant](../Tutorial-BSSN_stress_energy_source_terms.ipynb). The definition of the energy density is (cf. Eq. 10a in [Baumgarte *et al.* (2013)](https://arxiv.org/pdf/1211.6632.pdf))\n\n$$\n\\rho \\equiv n_{\\mu}n_{\\nu}T^{\\mu\\nu}.\n$$\n\nHere, $T^{\\mu\\nu} = T^{\\mu\\nu}_{\\rm GRMHD}$ is the energy-momentum tensor of general relativistic magnetohydrodynamics (GRMHD) and is given by (cf. Eq. 33 in [Duez *et al*. (2005)](https://arxiv.org/pdf/astro-ph/0503420.pdf))\n\n$$\nT^{\\mu\\nu}_{\\rm GRMHD} = \\left(\\rho_{b}h + b^{2}\\right)u^{\\mu}u^{\\nu} + \\left(P^{2} + \\frac{b^{2}}{2}\\right)g^{\\mu\\nu} - b^{\\mu}b^{\\nu},\n$$\n\nwhere $h = 1 + \\epsilon + P/\\rho$ is the specific enthalpy, $P$ is the pressure, $b^{2}\\equiv b_{\\mu}b^{\\mu}$, and $g^{\\mu\\nu}$ is the inverse of the spacetime metric $g_{\\mu\\nu}$.\n\n\n\n### Step 2.d.i: The equation of state & derived hydrodynamics quantities \\[Back to [top](#toc)\\]\n$$\\label{hydro_quantities_eos}$$\n\nWe must now compute some of the derived hydrodynamics quantities defined above, such as the pressure and specific internal energy. We compute these quantities by employing an [equation of state (EOS)](https://en.wikipedia.org/wiki/Equation_of_state). We will assume here a very simple EOS which is known as a [simple polytrope](https://en.wikipedia.org/wiki/Polytrope). This EOS relates the pressure with the baryonic density via\n\n$$\nP(\\rho_{\\rm b}) = K \\rho_{\\rm b}^{\\Gamma},\n$$\n\nwhere $K$ is a constant of proportinality and $\\Gamma$ is the adiabatic index. From this we can determine the specific internal energy using\n\n$$\n\\epsilon = \\int d\\rho_{\\rm b} \\frac{P}{\\rho_{\\rm b}^{2}} = K\\int d\\rho_{\\rm b} \\rho_{\\rm b}^{\\Gamma-2} = \\frac{K\\rho_{\\rm b}^{\\Gamma-1}}{\\Gamma-1} = \\frac{P}{\\rho_{\\rm b}\\left(\\Gamma-1\\right)},\n$$\n\nwhere we have fixed the integration constant to zero by demanding that $\\lim_{\\rho_{\\rm b}\\to0}\\epsilon=0$.\n\n\n```python\n# Step 2.d.i: Derived hydro quantities\n# Step 2.d.i.A: Register rho_b as an auxiliary gridfunction\nrho_b = gri.register_gridfunctions(\"AUX\",\"rhob\")\n\n# Step 2.d.i.B: Define K and Gamma as symbols\nK,Gamma = sp.symbols(\"K Gamma\",real=True)\n\n# Step 2.d.i.C: Compute the pressure: P = K rho_{b}^Gamma\nP = K * rho_b**Gamma\n\n# Step 2.d.i.D: Compute the specific internal energy: epsilon = P/( rho_{b}(Gamma-1) )\nepsilon = P / ( rho_b*(Gamma-1) )\n```\n\n\n\n### Step 2.d.ii: The energy density $\\rho$ \\[Back to [top](#toc)\\]\n$$\\label{grmhd_rho}$$\n\nWe now compute the symbolic expressions for the GRMHD energy density $\\rho$. This requires a few steps, which we outline below:\n\n1. Define symbolic expressions for the ADM variables used by the C code\n1. Define symbolic expressions for $u^{\\mu}$ and $b^{\\mu}$ used by the C code\n1. Use the function `compute_smallbsquared` from the [GRFFE/equations.py](../GRFFE/equations.py) \\[[**edit**](../edit/GRFFE/equations.py), [**tutorial**](../Tutorial-GRFFE_Equations-Cartesian.ipynb)\\] NRPy+ module to compute $b^{2}$.\n1. Use the function `compute_GRMHD_T4UU` from the [GRMHD/equations.py](../GRMHD/equations.py) \\[[**edit**](../edit/GRMHD/equations.py), [**tutorial**](../Tutorial-GRMHD_Equations-Cartesian.ipynb)\\] NRPy+ module to compute $T^{\\mu\\nu}_{\\rm GRMHD}$.\n1. Declare symbolic expressions for $n_{\\mu} = \\left(-\\alpha,0,0,0\\right)$.\n1. Compute $\\rho = n_{\\mu}n_{\\nu}T^{\\mu\\nu}_{\\rm GRMHD}$.\n\n\n```python\n# Step 2.d.ii: The energy density rho\n# Step 2.d.ii.A: Import GRFFE/equations.py and GRMHD/equations.py NRPy+ modules\nimport GRFFE.equations as GRFFE\nimport GRMHD.equations as GRMHD\nimport BSSN.ADMBSSN_tofrom_4metric as AB4m\n\n# Step 2.d.ii.B: Define symbolic expressions for metric quantities\nDIM = 3\nalpha = sp.Symbol(\"other_inputs.alp[IDX3S(i0,i1,i2)]\",real=True)\nbetaU = ixp.zerorank1()\ngammaDD = ixp.zerorank2()\nfor i in range(DIM):\n betaU[i] = sp.Symbol(\"other_inputs.beta\"+chr(ord('x')+i)+\"[IDX3S(i0,i1,i2)]\",real=True)\n for j in range(i,DIM):\n gammaDD[i][j] = gammaDD[j][i] = sp.Symbol(\"other_inputs.g\"+chr(ord('x')+i)+chr(ord('x')+j)+\"[IDX3S(i0,i1,i2)]\",real=True)\n\ngammaUU,_ = ixp.symm_matrix_inverter3x3(gammaDD)\n \n# Step 2.d.ii.C: Define symbolic expressions for hydro quantities\nu4U = ixp.register_gridfunctions_for_single_rank1(\"AUX\",\"u4U\",DIM=4)\nsmallb4U = ixp.register_gridfunctions_for_single_rank1(\"AUX\",\"smallb4U\",DIM=4)\n\n# Step 2.d.ii.D: b^{2} = b_{\\mu}b^{\\mu}\nGRFFE.compute_smallbsquared(gammaDD, betaU, alpha, smallb4U)\n\n# Step 2.d.ii.E: Compute the GRMHD energy-momentum tensor\nGRMHD.compute_GRMHD_T4UU(gammaDD,betaU,alpha, rho_b,P,epsilon,u4U, smallb4U, GRFFE.smallbsquared)\n\n# Step 2.d.ii.F: ADM 4-metric\nAB4m.g4DD_ito_BSSN_or_ADM(\"ADM\",gammaDD=gammaDD,betaU=betaU,alpha=alpha)\n\n# Step 2.d.ii.G: Unit 4-vector\nn4D = ixp.zerorank1(DIM=4)\nn4D[0] = -alpha\n\n# Step 2.d.ii.H: Induced metric\ngamma4DD = ixp.zerorank2(DIM=4)\nfor mu in range(4):\n for nu in range(4):\n gamma4DD[mu][nu] = AB4m.g4DD[mu][nu] + n4D[mu] * n4D[nu]\n\n# Step 2.d.ii.I: Energy density\nrhoADM = sp.sympify(0)\nfor mu in range(4):\n for nu in range(4):\n rhoADM += n4D[mu] * n4D[nu] * GRMHD.T4UU[mu][nu]\n\n# Step 2.d.ii.J: Momentum density\nSD = ixp.zerorank1()\nfor i in range(3):\n for mu in range(4):\n for nu in range(4):\n SD[i] += - gamma4DD[i+1][mu] * n4D[nu] * GRMHD.T4UU[mu][nu]\n \n# Step 2.d.ii.L: S^{i} = gamma^{ij}S_{j}\nSU = ixp.zerorank1()\nfor i in range(3):\n for j in range(3):\n SU[i] += gammaUU[i][j] * SD[j]\n\n# Step 2.d.ii.K: Sources\nM_PI = par.Cparameters(\"REAL\", thismodule, [\"M_PI\"], \"3.14159265358979323846264338327950288\")\nsourceH = -16 * M_PI * rhoADM\nsourceMU = ixp.zerorank1()\nfor i in range(3):\n sourceMU[i] = -8 * M_PI * SU[i] / rfm.ReU[i]\n```\n\n\n\n### Step 2.d.iii: Adding the function to the dictionary \\[Back to [top](#toc)\\]\n$$\\label{adding_ham_constraint_source_term_to_dict}$$\n\nHaving defined everything we need, we now add the function `Hamiltonian_constraint_source_term` to our C function dictionary.\n\n\n```python\n# Step 2.e.i: The Hamiltonian constraint without source terms\n# Step 2.e.i.A: Import the BSSN/BSSN_constraints.py NRPy+ module\nimport BSSN.BSSN_constraints as bssncon\nimport BSSN.BSSN_stress_energy_source_terms as Bsest\n\n# Step 2.e.i.B: Adjust reference metric environment if rfm_precompute is enabled\nif enable_rfm_precompute:\n par.set_parval_from_str(\"reference_metric::enable_rfm_precompute\",\"True\")\n rfm.reference_metric()\n\n# Step 2.e.i.C: Now register the Hamiltonian constraint as an auxiliary gridfunction\nH = gri.register_gridfunctions(\"AUX\",\"H\")\nMU = ixp.register_gridfunctions_for_single_rank1(\"AUX\",\"MU\")\n_ = gri.register_gridfunctions(\"AUX\",\"sourceH\")\n_ = ixp.register_gridfunctions_for_single_rank1(\"AUX\",\"sourceMU\")\n\n# Step 2.e.i.D: Set symbolic expressions for the constraints\nbssncon.BSSN_constraints(add_T4UUmunu_source_terms=False)\n\n# Step 2.e.i.E: Reset the reference metric environment if rfm_precompute is enabled\nif enable_rfm_precompute:\n par.set_parval_from_str(\"reference_metric::enable_rfm_precompute\",\"False\")\n rfm.ref_metric__hatted_quantities()\n```\n\n\n\n## Step 2.e: `Hamiltonian_constraint_no_source_term` \\[Back to [top](#toc)\\]\n$$\\label{cfunc__hamiltonian_constraint_no_source_term}$$\n\n\n```python\n# Step 2.d.iii: Adding Hamiltonian_constraint_source_term\n# to the C functions dictionary\ndef add_to_Cfunction_dict_Hamiltonian_and_momentum_constraints_source_terms():\n desc = \"\"\"\n(c) 2021 Leo Werneck\nThis function computes the energy density rho, which appears\nin the source term of the Hamiltonian constraint.\n\"\"\"\n includes = [\"NRPy_basic_defines.h\"]\n prefunc = \"\"\n c_type = \"void\"\n name = \"Hamiltonian_and_momentum_constraints_source_terms\"\n \n print(\"Writing \"+name+\" function\")\n \n params = \"\"\"rfm_struct *restrict rfmstruct,const paramstruct *restrict params,\n const REAL K, const REAL Gamma, ID_inputs other_inputs,\n REAL *restrict aux_gfs\"\"\"\n body = \"\"\"\n const int idx = IDX3S(i0,i1,i2);\n const REAL rhob = aux_gfs[IDX4ptS(RHOBGF,idx)];\n const REAL u4U0 = aux_gfs[IDX4ptS(U4U0GF,idx)];\n const REAL u4U1 = aux_gfs[IDX4ptS(U4U1GF,idx)];\n const REAL u4U2 = aux_gfs[IDX4ptS(U4U2GF,idx)];\n const REAL u4U3 = aux_gfs[IDX4ptS(U4U3GF,idx)];\n const REAL smallb4U0 = aux_gfs[IDX4ptS(SMALLB4U0GF,idx)];\n const REAL smallb4U1 = aux_gfs[IDX4ptS(SMALLB4U1GF,idx)];\n const REAL smallb4U2 = aux_gfs[IDX4ptS(SMALLB4U2GF,idx)];\n const REAL smallb4U3 = aux_gfs[IDX4ptS(SMALLB4U3GF,idx)];\n\n\"\"\"+outC.outputC([sourceH,\n sourceMU[0],\n sourceMU[1],\n sourceMU[2]],\n [gri.gfaccess(\"aux_gfs\", \"sourceH\"),\n gri.gfaccess(\"aux_gfs\", \"sourceMU0\"),\n gri.gfaccess(\"aux_gfs\", \"sourceMU1\"),\n gri.gfaccess(\"aux_gfs\", \"sourceMU2\")], \"returnstring\",\n params=\"outCverbose=False,includebraces=False\")\n loopopts = \"InteriorPoints\"\n if enable_SIMD:\n loopopts +=\",enable_SIMD\"\n if enable_rfm_precompute:\n loopopts +=\",enable_rfm_precompute\"\n outC.add_to_Cfunction_dict(\n includes=includes,\n prefunc=prefunc,\n desc=desc,\n c_type=c_type, name=name, params=params,\n body=body,loopopts=loopopts)\n \n print(\"Finished writing \"+name+\" function\")\n```\n\n\n\n### Step 2.e.i: The Hamiltonian constraint without source terms \\[Back to [top](#toc)\\]\n$$\\label{hamiltonian_constraint_no_source_symb}$$\n\nWe now declare the symbolic expression for the Hamiltonian constraint by invoking the [`BSSN/BSSN_constraints.py`](../BSSN/BSSN_constraints.py) \\[[**edit**](../edit/BSSN/BSSN_constraints.py), [**tutorial**](../Tutorial-BSSN_constraints.ipynb)\\] NRPy+ module.\n\n\n```python\n\n```\n\n\n\n### Step 2.e.ii: Adding the function to the dictionary \\[Back to [top](#toc)\\]\n$$\\label{adding_ham_constraint_no_source_term_to_dict}$$\n\nHaving defined everything we need, we now add the function `Hamiltonian_constraint_no_source_term` to our C function dictionary.\n\n\n```python\n# Step 2.e.ii: Adding Hamiltonian_constraint_no_source_term\n# to the C functions dictionary\ndef add_to_Cfunction_dict_Hamiltonian_and_momentum_constraints_no_source_terms():\n desc = \"\"\"\n(c) 2021 Leo Werneck\nThis function computes the metric terms of the Hamiltonian constraint.\n\"\"\"\n includes = [\"NRPy_basic_defines.h\"]\n prefunc = \"\"\n c_type = \"void\"\n name = \"Hamiltonian_and_momentum_constraints_no_source_terms\"\n \n print(\"Writing \"+name+\" function\")\n \n params = \"\"\"rfm_struct *restrict rfmstruct,const paramstruct *restrict params,\n REAL *restrict in_gfs, REAL *restrict aux_gfs\"\"\"\n body = \"\"\"\n const int idx = IDX3S(i0,i1,i2);\n const REAL rhob = aux_gfs[IDX4ptS(RHOBGF,idx)];\n const REAL u4U0 = aux_gfs[IDX4ptS(U4U0GF,idx)];\n const REAL u4U1 = aux_gfs[IDX4ptS(U4U1GF,idx)];\n const REAL u4U2 = aux_gfs[IDX4ptS(U4U2GF,idx)];\n const REAL u4U3 = aux_gfs[IDX4ptS(U4U3GF,idx)];\n const REAL smallb4U0 = aux_gfs[IDX4ptS(SMALLB4U0GF,idx)];\n const REAL smallb4U1 = aux_gfs[IDX4ptS(SMALLB4U1GF,idx)];\n const REAL smallb4U2 = aux_gfs[IDX4ptS(SMALLB4U2GF,idx)];\n const REAL smallb4U3 = aux_gfs[IDX4ptS(SMALLB4U3GF,idx)];\n\\n\"\"\"+fin.FD_outputC(\"returnstring\",\n [outC.lhrh(lhs=gri.gfaccess(\"aux_gfs\", \"H\") , rhs=bssncon.H),\n outC.lhrh(lhs=gri.gfaccess(\"aux_gfs\", \"MU0\"), rhs=bssncon.MU[0]),\n outC.lhrh(lhs=gri.gfaccess(\"aux_gfs\", \"MU1\"), rhs=bssncon.MU[1]),\n outC.lhrh(lhs=gri.gfaccess(\"aux_gfs\", \"MU2\"), rhs=bssncon.MU[2])],\n params=\"outCverbose=False\")\n loopopts = \"InteriorPoints,enable_rfm_precompute\"\n outC.add_to_Cfunction_dict(\n includes=includes,\n prefunc=prefunc,\n desc=desc,\n c_type=c_type, name=name, params=params,\n body=body,loopopts=loopopts)\n \n print(\"Finished writing \"+name+\" function\")\n```\n\n\n\n## Step 2.f: `main` \\[Back to [top](#toc)\\]\n$$\\label{cfunc__main}$$\n\nWe now add the `main` function to the C functions dictionary. This function combines all the other functions that we have defined this far into a C program that is able to:\n\n1. Set a NRPy+ grid.\n1. Convert the NRPy+ grid to Cartesian coordinates, which is the coordinate system expected by [LORENE](https://lorene.obspm.fr/).\n1. Read the initial data from the input `resu.d` file and interpolate the solution onto the NRPy+ grid.\n1. Set up BSSN and GRMHD quantities from the input.\n1. Compute the Hamiltonian constraint violations on the NRPy+ grid.\n1. Output the results to file.\n\n\n```python\n# Step 2.f: Adding main to the C functions dictionary\ndef add_to_Cfunction_dict_main():\n print(\"Writing main function\")\n desc = \"\"\"\n(c) 2009 Erik Schnetter\n(c) 2010 Frank Loeffler\nEdits by Z Etienne & L Werneck 2021\n\"\"\"\n\n includes = [\"\", \"\", \"\", \"\", \"\",\n \"assert.h\", \"stdlib.h\", \"bin_ns.h\", \"unites.h\",\n \"NRPy_basic_defines.h\",\"NRPy_function_prototypes.h\"]\n\n prefunc = r\"\"\"\nusing namespace std;\n\n// define namespace here for old versions of Lorene that don't do so\nnamespace Lorene {}\nusing namespace Lorene;\n\"\"\"\n\n c_type = \"int\"\n name = \"main\"\n \n print(\"Writing \"+name+\" function\")\n \n params = \"int argc, const char *argv[]\"\n body = r\"\"\"\n // Step 0: Check correct usage\n if((argc < 9) || (argc > 10)) {\n fprintf(stderr,\"Error, incorrect usage. Usage: ./standalone_interpolator [Nx0] [Nx1] [Nx2] [offset_axis] [offset_star_1] [offset_star_2] [Gamma] [filename (resu.d)] [(optional) initial shift (zero or lorene)]\\n\");\n exit(1);\n }\n\n // Step 0.a: Set up physical constants for converting quantities\n // from SI units (Lorene) to Geometrized units (NRPy)\n // Be aware: these are the constants Lorene uses. They do differ from other\n // conventions, but they gave the best results in some tests.\n double const c_light = Unites::c_si; // speed of light [m/s]\n double const nuc_dens = Unites::rhonuc_si; // Nuclear density as used in Lorene units [kg/m^3]\n double const G_grav = Unites::g_si; // gravitational constant [m^3/kg/s^2]\n double const M_sun = Unites::msol_si; // solar mass [kg]\n\n // Step 0.b: Geometrized units in terms of SI units:\n // (These are derived from M = M_sun, c = G = 1,\n // and using 1/M_sun for the magnetic field)\n double const geomM = M_sun;\n double const geomL = geomM * G_grav / pow(c_light,2);\n double const geomT = geomL / c_light;\n\n // Step 0.c: Other quantities\n double const coord_unit = geomL / 1.0e+3; // from km (~1.477)\n double const rho_b_unit = geomM / pow(geomL,3); // from kg/m^3\n \n printf(\"coord_unit = %.15e\\n\",coord_unit);\n\n // Step 0.d: Set initial shift from user input or default to zero\n char initial_shift[256];\n if( argc == 10 ) {\n sprintf(initial_shift,\"%s\",argv[9]);\n }\n else {\n sprintf(initial_shift,\"%s\",\"zero\");\n }\n\n // Step 0.f: Set up numerical grid structure, first in space...\n const int Nxx[3] = { atoi(argv[1]), atoi(argv[2]), atoi(argv[3]) };\n if(Nxx[0]%2 != 0 || Nxx[1]%2 != 0 || Nxx[2]%2 != 0) {\n printf(\"Error: Cannot guarantee a proper cell-centered grid if number of grid cells not set to even number.\\n\");\n printf(\" For example, in case of angular directions, proper symmetry zones will not exist.\\n\");\n exit(1);\n }\n\n int offset_index;\n REAL offset_star1_xyz[3] = {0,0,0};\n REAL offset_star2_xyz[3] = {0,0,0};\n char offset_axis[10];\n sprintf(offset_axis,\"%s\",argv[4]);\n \n if( !strcmp(offset_axis,\"x\") ) {\n offset_index = 0;\n }\n else if( !strcmp(offset_axis,\"y\") ) {\n offset_index = 1;\n }\n else if( !strcmp(offset_axis,\"z\") ) {\n offset_index = 2;\n }\n else {\n fprintf(stderr,\"Error: unsupported offset axis: %s. Supported options are: x, y, and z\\n\",offset_axis);\n exit(1);\n }\n offset_star1_xyz[offset_index] = strtod(argv[5],NULL) / coord_unit;\n offset_star2_xyz[offset_index] = strtod(argv[6],NULL) / coord_unit;\n\n printf(\"Beginning analysis of Lorene initial data.\\n\");\n printf(\"Grid #1 will be centered at (x,y,z) = (%g,%g,%g)\\n\",offset_star1_xyz[0],offset_star1_xyz[1],offset_star1_xyz[2]);\n printf(\"Grid #2 will be centered at (x,y,z) = (%g,%g,%g)\\n\",offset_star2_xyz[0],offset_star2_xyz[1],offset_star2_xyz[2]);\n printf(\"Grid #3 will be centered at (x,y,z) = (0,0,0)\\n\");\n\n const int ngrids = 1;\n for(int n_grid=1;n_grid<=ngrids;n_grid++) {\n\n printf(\"Beginning analysis of Grid #%d\\n\",n_grid);\n\n // Step 0.e: Set up NRPy parameter struct\n paramstruct params;\n set_Cparameters_to_default(¶ms);\n\n // Step 0.g: Set free parameters, overwriting Cparameters defaults\n // by hand or with command-line input, as desired.\n#include \"free_parameters.h\"\n\n if(n_grid == 1) {\n params.Cart_originx = offset_star1_xyz[0];\n params.Cart_originy = offset_star1_xyz[1];\n params.Cart_originz = offset_star1_xyz[2];\n //params.RMAX = 24;\n }\n else if(n_grid == 2) {\n params.Cart_originx = offset_star2_xyz[0];\n params.Cart_originy = offset_star2_xyz[1];\n params.Cart_originz = offset_star2_xyz[2];\n //params.RMAX = 24;\n }\n \n // Step 0.h: Uniform coordinate grids are stored to *xx[3]\n REAL *xx[3];\n\n // Step 0.i: Call set_Nxx_dxx_invdx_params__and__xx(), which sets\n // params Nxx,Nxx_plus_2NGHOSTS,dxx,invdx, and xx[] for the\n // chosen (non-Eigen) CoordSystem.\n int EigenCoord = 1;\n set_Nxx_dxx_invdx_params__and__xx(EigenCoord, Nxx, ¶ms, xx);\n\n // Step 0.j: Set all C parameters \"blah\" for params.blah, including\n // Nxx_plus_2NGHOSTS0 = params.Nxx_plus_2NGHOSTS0, etc.\n#include \"set_Cparameters-nopointer.h\"\n const int Nxx_plus_2NGHOSTS_tot = Nxx_plus_2NGHOSTS0*Nxx_plus_2NGHOSTS1*Nxx_plus_2NGHOSTS2;\n\n // Step 0.l: Allocate memory for initial data gridfunctions on NRPy grid\n REAL *id_gfs = (REAL *)malloc(NUM_EVOL_GFS*Nxx_plus_2NGHOSTS_tot*sizeof(REAL));\n REAL *aux_gfs = (REAL *)malloc(NUM_AUX_GFS*Nxx_plus_2NGHOSTS_tot*sizeof(REAL));\n\n // Step 0.m: Set up precomputed reference metric arrays\n // Step 0.m.i: Allocate space for precomputed reference metric arrays.\n rfm_struct rfmstruct;\n rfm_precompute_rfmstruct_malloc(¶ms, &rfmstruct);\n\n // Step 0.m.ii: Define precomputed reference metric arrays.\n {\n#include \"set_Cparameters-nopointer.h\"\n rfm_precompute_rfmstruct_define(¶ms, xx, &rfmstruct);\n }\n\n // LORENE COORDINATES, != NRPy COORDINATES\n vector x_Lorene(Nxx_plus_2NGHOSTS_tot);\n vector y_Lorene(Nxx_plus_2NGHOSTS_tot);\n vector z_Lorene(Nxx_plus_2NGHOSTS_tot);\n\n#pragma omp parallel for\n for(int i2=0;i2\n\n# Step 3: C code kernels generation \\[Back to [top](#toc)\\]\n$$\\label{ccode_kernels_generation}$$\n\nWe now generate the C code kernels that are needed by our program. These include:\n\n1. `free_parameters.h`: a file containing initialization values for all the free parameters in our program.\n1. All the C functions that we have added to the C functions dictionary.\n1. `NRPy_basic_defines.h`: a C header file that contains all the data structures and definitions which are used by NRPy+ programs.\n1. `NRPy_function_prototypes.h`: a C header file that contains the prototypes of all the functions that have been added to the C functions dictionary.\n\n\n\n## Step 3.a: Set `free_parameters.h`; also output C codes needed for declaring and setting Cparameters \\[Back to [top](#toc)\\]\n$$\\label{cparams_rfm_and_domainsize}$$\n\nFirst we output `free_parameters.h`, which sets initial data parameters, as well as grid domain & reference metric parameters, applying `domain_size` and `sinh_width`/`SymTP_bScale` (if applicable) as set above.\n\n\n```python\n# Step 3.a.i: Set free_parameters.h\nwith open(os.path.join(Ccodesdir,\"free_parameters.h\"),\"w\") as file:\n file.write(\"\"\"\n// Free parameters related to physical system:\nparams.wavespeed = 1.0;\n\n// Free parameters related to numerical timestep:\nREAL CFL_FACTOR = \"\"\"+str(CFL_FACTOR)+\";\\n\")\n\n# Step 3.a.2: Append to $Ccodesrootdir/free_parameters.h reference metric parameters\n# based on generic domain_size,sinh_width,sinhv2_const_dr,SymTP_bScale,\n# parameters set above.\nrfm.out_default_free_parameters_for_rfm(os.path.join(Ccodesdir,\"free_parameters.h\"),\n domain_size,sinh_width,sinhv2_const_dr,SymTP_bScale)\n```\n\n\n\n## Step 3.b: Add all NRPy+Lorene BNS initial data C codes to C function dictionary \\[Back to [top](#toc)\\]\n$$\\label{add_all}$$\n\n\n```python\n# Step 3.b: Add all NRPy+Lorene BNS C codes to C function to dictionary\ndef BNS_Initial_Data_NRPyLorene_register_C_functions():\n add_to_Cfunction_dict_ID_Lorene_ADM_quantities_Cartesian()\n add_to_Cfunction_dict_set_hydro_quantities()\n add_to_Cfunction_dict_initial_data()\n add_to_Cfunction_dict_Hamiltonian_and_momentum_constraints_source_terms()\n add_to_Cfunction_dict_Hamiltonian_and_momentum_constraints_no_source_terms()\n add_to_Cfunction_dict_main()\n```\n\n\n\n## Step 3.c: Generating C code for setting Lorene BNS initial data in NRPy+ \\[Back to [top](#toc)\\]\n$$\\label{generate_c_code}$$\n\n\n```python\n# Step 3.c: Generating C code for setting Lorene BNS initial data in NRPy+\n# Step 3.c.i: Generate all the C codes for the C functions\nBNS_Initial_Data_NRPyLorene_register_C_functions()\n\n# Step 3.c.ii: Define a custom dictionary entry for NRPy_basic_defines.h\ndef generate_supplementary_dict():\n supplementary_dict = {}\n supplementary_dict[\"Lorene_ID\"] = r\"\"\"\ntypedef struct of_ID_inputs {\n // This struct is used to store Lorene\n // initial data in Cartesian basis\n REAL *alp,*betax,*betay,*betaz;\n REAL *gxx,*gxy,*gxz,*gyy,*gyz,*gzz;\n REAL *kxx,*kxy,*kxz,*kyy,*kyz,*kzz;\n REAL *nbar, *u_euler_x,*u_euler_y,*u_euler_z;\n REAL rho_b_unit;\n} ID_inputs;\n\"\"\"\n return supplementary_dict\n\n# Step 3.c.iii: Register all NRPy+ related C functions and\n# add entries to NRPy_basic_defines.h\noutC.outputC_register_C_functions_and_NRPy_basic_defines() # #define M_PI, etc.\noutC.NRPy_param_funcs_register_C_functions_and_NRPy_basic_defines(directory=Ccodesdir)\ngri.register_C_functions_and_NRPy_basic_defines() # #define IDX3S(), etc.\nrfm.register_C_functions_and_NRPy_basic_defines(enable_rfm_precompute=enable_rfm_precompute)\nfin.register_C_functions_and_NRPy_basic_defines(NGHOSTS_account_for_onezone_upwind=False)\n\n# Step 3.c.iv: Output functions for computing all finite-difference stencils.\n# Must be called after defining all functions depending on FD stencils.\nif enable_FD_functions:\n fin.output_finite_difference_functions_h(path=Ccodesdir)\n\n# Step 3.c.v: Call this last: Set up NRPy_basic_defines.h and NRPy_function_prototypes.h.\noutC.construct_NRPy_basic_defines_h(Ccodesdir, enable_SIMD=enable_SIMD,\n supplemental_dict=generate_supplementary_dict())\noutC.construct_NRPy_function_prototypes_h(Ccodesdir)\n```\n\n Writing ID_Lorene_ADM_quantities_Cartesian function\n Finished writing ID_Lorene_ADM_quantities_Cartesian function\n Finished writing set_hydro_quantities function\n Writing initial_data function\n Finished writing initial_data function\n Writing Hamiltonian_and_momentum_constraints_source_terms function\n Finished writing Hamiltonian_and_momentum_constraints_source_terms function\n Writing Hamiltonian_and_momentum_constraints_no_source_terms function\n Finished writing Hamiltonian_and_momentum_constraints_no_source_terms function\n Writing main function\n Writing main function\n Finished writing main function\n\n\n\n\n# Step 4: Compiling and running the code \\[Back to [top](#toc)\\]\n$$\\label{compiling_and_running}$$\n\n\n```python\ndata_dir = os.path.join(\"lorene_standalone\",\"interpolator\")\n\noutfile_1 = os.path.join(data_dir,\"initial_data_grid_1.x.asc\")\noutfile_2 = os.path.join(data_dir,\"initial_data_grid_2.x.asc\")\noutfile_3 = os.path.join(data_dir,\"initial_data_grid_3.x.asc\")\n\n# outfile_1_lr = os.path.join(data_dir,\"initial_data_grid_1_lr.x.asc\")\n# outfile_2_lr = os.path.join(data_dir,\"initial_data_grid_2_lr.x.asc\")\n# outfile_3_lr = os.path.join(data_dir,\"initial_data_grid_3_lr.x.asc\")\n\n# outfile_1_hr = os.path.join(data_dir,\"initial_data_grid_1_hr.x.asc\")\n# outfile_2_hr = os.path.join(data_dir,\"initial_data_grid_2_hr.x.asc\")\n# outfile_3_hr = os.path.join(data_dir,\"initial_data_grid_3_hr.x.asc\")\n\ncmd.new_C_compile(Ccodesdir, \"interpolator\",\n addl_CFLAGS=[\"-I../Lorene/Export/C++/Include\",\n \"-I../Lorene/C++/Include\"],\n addl_libraries=[\"-L../Lorene/Lib/\",\n \"-llorene_export\", \"-llorene\",\n \"-llorenef77\", \"-lgfortran\", \"-lfftw3\", \"-lgsl\",\n \"-lgslcblas\", \"-llapack\", \"-lblas\"],\n CC=\"g++\")\n\n# shutil.copy(outfile_1,outfile_1_lr)\n# shutil.copy(outfile_2,outfile_2_lr)\n# shutil.copy(outfile_3,outfile_3_lr)\n\n# os.chdir(Ccodesdir)\n# cmd.Execute(\"interpolator\", \"128 32 32 x 25 -25 2.5 ../resu.d\")\n# os.chdir(os.path.join(\"..\",\"..\"))\n\n# shutil.copy(outfile_1,outfile_1_hr)\n# shutil.copy(outfile_2,outfile_2_hr)\n# shutil.copy(outfile_3,outfile_3_hr)\n```\n\n (EXEC): Executing `make -j10`...\n ld: warning: ld: warning: dylib (/usr/local/opt/gsl/lib/libgsl.dylib) was built for newer macOS version (11.0) than being linked (10.16)dylib (/usr/local/opt/fftw/lib/libfftw3.dylib) was built for newer macOS version (11.2) than being linked (10.16)\n \n ld: warning: dylib (/usr/local/opt/gsl/lib/libgslcblas.dylib) was built for newer macOS version (11.0) than being linked (10.16)\n (BENCH): Finished executing in 4.877245187759399 seconds.\n Finished compilation.\n\n\n\n```python\nos.chdir(Ccodesdir)\ncmd.Execute(\"interpolator\", \"40 40 40 x 25 -25 2.5 ../resu_lowres.d\",\"out.txt\")\nos.chdir(os.path.join(\"..\",\"..\"))\n```\n\n (EXEC): Executing `./interpolator 40 40 40 x 25 -25 2.5 ../resu_lowres.d`...\n (BENCH): Finished executing in 1.0230820178985596 seconds.\n\n\n\n\n# Step 5: Visualization: convergence of Hamiltonian constraint \\[Back to [top](#toc)\\]\n$$\\label{constraint_violations}$$\n\n\n```python\n# import numpy as np\n# import matplotlib.pyplot as plt\n# from IPython.display import Image\n\n# outfile_1_etk = os.path.join(\"ETK_data\",\"ETK_data_lr_40.asc\")\n# outfile_2_etk = os.path.join(\"ETK_data\",\"ETK_data_mr_40.asc\")\n# outfile_3_etk = os.path.join(\"ETK_data\",\"ETK_data_hr_40.asc\")\n# outfile_1_nrpy = os.path.join(data_dir,\"initial_data_grid_lr_40.x.asc\")\n# outfile_2_nrpy = os.path.join(data_dir,\"initial_data_grid_mr_40.x.asc\")\n# outfile_3_nrpy = os.path.join(data_dir,\"initial_data_grid_hr_40.x.asc\")\n\n# data_1_etk = np.loadtxt(outfile_1_etk).T\n# data_2_etk = np.loadtxt(outfile_2_etk).T\n# data_3_etk = np.loadtxt(outfile_3_etk).T\n# data_1_nrpy = np.loadtxt(outfile_1_nrpy).T\n# data_2_nrpy = np.loadtxt(outfile_2_nrpy).T\n# data_3_nrpy = np.loadtxt(outfile_3_nrpy).T\n\n# fig = plt.figure()\n\n# plt.grid()\n# plt.ylabel(r\"$\\log_{10}\\left|\\mathcal{H}\\right|$\")\n# plt.xlabel(r\"$x$ [km]\")\n# plt.plot(data_1_etk[0],np.log10(np.maximum(np.abs(data_1_etk[4][:]),1e-15)),'blue',label=r\"ETK, low spectral resolution\")\n# plt.plot(data_1_nrpy[0],np.log10(np.maximum(np.abs(data_1_nrpy[3][:]),1e-15)),'orange',ls='--',label=\"NRPy+, low spectral resolution\")\n# plt.plot(data_2_etk[0],np.log10(np.maximum(np.abs(data_2_etk[4][:]),1e-15)),'purple',label=r\"ETK, med spectral resolution\")\n# plt.plot(data_2_nrpy[0],np.log10(np.maximum(np.abs(data_2_nrpy[3][:]),1e-15)),'cyan',ls='--',label=\"NRPy+, med spectral resolution\")\n# plt.plot(data_3_etk[0],np.log10(np.maximum(np.abs(data_3_etk[4][:]),1e-15)),'green',ls=\":\",label=r\"ETK, high spectral resolution\")\n# plt.plot(data_3_nrpy[0],np.log10(np.maximum(np.abs(data_3_nrpy[3][:]),1e-15)),'red',ls='-.',label=\"NRPy+, high spectral resolution\")\n# plt.legend()\n\n# outfig = \"constraint_violations.png\"\n# plt.savefig(outfig,dpi=150,facecolor='white')\n# plt.close(fig)\n# Image(outfig)\n```\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom IPython.display import Image\n\noutfile_1_etk = os.path.join(\"ETK_data\",\"ETK_data_lr_40.asc\")\n# outfile_2_etk = os.path.join(\"ETK_data\",\"ETK_data_hr_80.asc\")\noutfile_1_nrpy = os.path.join(data_dir,\"initial_data_grid_1.x.asc\")\n# outfile_2_nrpy = os.path.join(data_dir,\"initial_data_grid_hr_80.x.asc\")\n\ndata_1_etk = np.loadtxt(outfile_1_etk).T\n# data_2_etk = np.loadtxt(outfile_2_etk).T\ndata_1_nrpy = np.loadtxt(outfile_1_nrpy).T\n# data_2_nrpy = np.loadtxt(outfile_2_nrpy).T\n\nfig = plt.figure()\n\nplt.title(\"BNS Initial Data - High spectral resolution\")\nplt.grid()\nplt.ylabel(r\"$\\log_{10}\\left|\\mathcal{H}\\right|$\")\nplt.xlabel(r\"$x$ [km]\")\nplt.plot(data_1_etk[0],np.log10(np.maximum(np.abs(data_1_etk[7][:]),1e-15)),'blue',label=r\"ETK, $N=40^{3}$\")\nplt.plot(data_1_nrpy[0],np.log10(np.maximum(np.abs(data_1_nrpy[6][:]),1e-15)),'orange',ls='--',label=\"NRPy+, $N=40^{3}$\")\n# plt.plot(data_2_etk[0],np.log10(np.maximum(np.abs(data_2_etk[4][:]),1e-15)),'green',ls=\":\",label=r\"ETK, $N=80^{3}$\")\n# plt.plot(data_2_nrpy[0],np.log10(np.maximum(np.abs(data_2_nrpy[3][:]),1e-15)),'red',ls='-.',label=\"NRPy+, $N=80^{3}$\")\nplt.legend()\n\noutfig = \"constraint_violations.png\"\nplt.savefig(outfig,dpi=150,facecolor='white')\nplt.close(fig)\nImage(outfig)\n```\n\n\n```python\n# fig,axs = plt.subplots(figsize=(9,3),ncols=3,nrows=1,sharex=True)\n# for i in range(len(axs)):\n# axs[i].plot(grid_etk_v_data_lr[0],grid_etk_v_data_lr[idx],c=\"red\",ls=\":\",label=\"vel ETK\")\n# axs[i].plot(grid_etk_v_data_lr[0],grid_etk_v_data_lr[idx+3],c=\"magenta\",ls=\":\",label=\"v ETK\")\n# axs[i].plot(vels_nrpy[0],vels_nrpy[idx],c='black',ls=\"-.\",label=\"vel NRPy\")\n# axs[i].grid()\n# axs[i].legend()\n \n# plt.tight_layout()\n# plt.savefig(\"velocities.png\",dpi=150,facecolor='white')\n# plt.close(fig)\n# Image(\"velocities.png\")\n```\n\n\n```python\n# outfile_1_etk = os.path.join(\"ETK_data\",\"all_data.asc\")\n# # outfile_1_etk_vel = os.path.join(\"ETK_data\",\"vx_vy_vz.x.asc\")\n# grid_etk_x_data_lr = np.loadtxt(outfile_1_etk).T\n# grid_1_x_data_lr = np.loadtxt(outfile_1).T\n# # grid_etk_v_data_lr = np.loadtxt(outfile_1_etk_vel).T\n# # vels_nrpy = np.loadtxt(os.path.join(data_dir,\"vx_vy_vz_NRPy.x.asc\")).T\n\n# X = 0\n# H = 4\n# ALP = 9\n# GXX = 13\n# GYY = 16\n# GZZ = 18\n# RHOB = 25\n# VELX = 26\n# VELY = VELX+1\n# VELZ = VELY+1\n# VX = 29\n# VY = VX+1\n# VZ = VY+1\n\n# fig,axs = plt.subplots(figsize=(9,4.5),ncols=3,nrows=3,sharex=True)\n# axs = axs.flatten()\n# ylabels = [r\"$\\log_{10}\\left|\\mathcal{H}\\right|$\",\n# r\"$\\alpha$\",\n# r\"$\\rho_{b}$\",\n# r\"$\\gamma_{xx}$\",\n# r\"$\\gamma_{yy}$\",\n# r\"$\\gamma_{zz}$\",\n# r\"$v^{x} = u^{x}/u^{0}$\",\n# r\"$v^{y} = u^{y}/u^{0}$\",\n# r\"$v^{z} = u^{z}/u^{0}$\"]\n# qETK = [np.log10(np.maximum(np.abs(grid_etk_x_data_lr[H][:]),1e-15)),\n# grid_etk_x_data_lr[ALP],grid_etk_x_data_lr[RHOB],\n# grid_etk_x_data_lr[GXX],grid_etk_x_data_lr[GYY],grid_etk_x_data_lr[GZZ],\n# grid_etk_x_data_lr[VX],grid_etk_x_data_lr[VY],grid_etk_x_data_lr[VZ]]\n\n# # xCart[0],xCart[1],xCart[2],H,rho_b,alp,gxx,gyy,gzz,vx,vy,vz\n# qNRPy =[np.log10(np.maximum(np.abs(grid_1_x_data_lr[3][:]),1e-15)),\n# grid_1_x_data_lr[5],grid_1_x_data_lr[4],\n# grid_1_x_data_lr[6],grid_1_x_data_lr[7],grid_1_x_data_lr[8],\n# grid_1_x_data_lr[9],grid_1_x_data_lr[10],grid_1_x_data_lr[11]]\n\n# for i in range(len(axs)):\n# axs[i].grid()\n# axs[i].set_ylabel(ylabels[i])\n# axs[i].plot(grid_etk_x_data_lr[0],qETK[i],c='blue',ls='-',label=\"ETK\")\n# axs[i].plot(grid_1_x_data_lr[0],qNRPy[i],c='orange',ls='--',label=\"NRPy+\")\n# axs[i].legend()\n\n# plt.tight_layout()\n# outfig = \"constraint_violations.png\"\n# plt.savefig(outfig,dpi=150,facecolor='white')\n# plt.close(fig)\n# Image(outfig)\n```\n\n\n```python\n# import glob\n\n# file_list = sorted(glob.glob(os.path.join(\"ETK_data\",\"*.x.asc\")))\n\n# master_file = os.path.join(\"ETK_data\",\"ETK_data_mr_40.asc\")\n# all_data = []\n# local_data = np.loadtxt(file_list[0]).T\n# all_data.append(local_data[9])\n# all_data.append(local_data[12])\n# for i in range(1,len(file_list)):\n# local_data = np.loadtxt(file_list[i]).T\n# all_data.append(local_data[12])\n\n# np.savetxt(master_file,list(zip(*all_data)))\n```\n\n\n```python\n# string = \"# Column 1: x\\n\"\n# counter = 2\n# for i in file_list:\n# string += \"# Column \"+str(counter)+\": \"+i.split(\".\")[0].split(\"/\")[1]+\"\\n\"\n# counter += 1\n# print(string)\n```\n\n\n\n# Step 6: Output this notebook to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n$$\\label{latex_pdf_output}$$\n\nThe following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n[Tutorial-ADM_Initial_Data-Lorene_BNS.pdf](Tutorial-ADM_Initial_Data-Lorene_BNS.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means.)\n\n\n```python\ncmd.output_Jupyter_notebook_to_LaTeXed_PDF(\"Tutorial-ADM_Initial_Data-Lorene_BNS\")\n```\n\n Created Tutorial-ADM_Initial_Data-Lorene_BNS.tex, and compiled LaTeX file\n to PDF file Tutorial-ADM_Initial_Data-Lorene_BNS.pdf\n\n", "meta": {"hexsha": "ae3ae934b33bea809bb3cdfc041296f132e3f744", "size": 163519, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "in_progress/Tutorial-ADM_Initial_Data-Lorene_BNS.ipynb", "max_stars_repo_name": "Harmohit-Singh/nrpytutorial", "max_stars_repo_head_hexsha": "81e6fe09c6882a2d95e1d0ea57f465fc7eda41e1", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "in_progress/Tutorial-ADM_Initial_Data-Lorene_BNS.ipynb", "max_issues_repo_name": "Harmohit-Singh/nrpytutorial", "max_issues_repo_head_hexsha": "81e6fe09c6882a2d95e1d0ea57f465fc7eda41e1", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "in_progress/Tutorial-ADM_Initial_Data-Lorene_BNS.ipynb", "max_forks_repo_name": "Harmohit-Singh/nrpytutorial", "max_forks_repo_head_hexsha": "81e6fe09c6882a2d95e1d0ea57f465fc7eda41e1", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 89.1597600872, "max_line_length": 80948, "alphanum_fraction": 0.7628471309, "converted": true, "num_tokens": 19683, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6442250928250375, "lm_q2_score": 0.66192288918838, "lm_q1q2_score": 0.4264273347304011}} {"text": "## Tex Preamble\n$$\n\\newcommand{\\bm}[1]{{\\bf #1}}\n\\newcommand{\\eq}[1]{Eq.(\\ref{eq:#1})}\n\\newcommand{\\zu}[1]{\u56f3\\ref{fig:#1}}\n\\newcommand{\\shiki}[1]{\u5f0f(\\ref{eq:#1})}\n\\newcommand{\\fig}[1]{Fig.\\ \\ref{fig:#1}}\n\\newcommand{\\grp}[1]{Fig.\\ \\ref{grp:#1}}\n\\newcommand{\\tab}[1]{Table\\ \\ref{tab:#1}}\n\\newcommand{\\defeq}{:=}\n\\newcommand{\\tvec}[1]{\\bvec{#1}^{\\mathsf{T}}} \n\\newcommand{\\mr}[1]{\\mathrm{#1}}%rm\u4f53\n\\newcommand{\\ml}[1]{\\mathcal{#1}}%frac\u4f53\n\\newcommand{\\bmat}[1]{\\begin{bmatrix} #1 \\end{bmatrix}}%\u884c\u5217\u7c21\u7565\u5316\n\\newcommand{\\pmat}[1]{\\begin{pmatrix} #1 \\end{pmatrix}}%\u884c\u5217\u7c21\u7565\u5316\n\\newcommand{\\mat}[1]{\\left( \\begin{matrix} #1 \\end{matrix} \\right)}%\u884c\u5217\u7c21\u7565\u5316\n%\\newcommand{\\nom}[1]{ \\ml{N} \\{ 0,\\sigma_{ #1 } \\} }\n\\newcommand{\\expo}[1]{ \\mathrm{e}^{#1} }\n$$\n\nTex\u30d7\u30ea\u30a2\u30f3\u30d6\u30eb\uff0bYalmip\u306e\u521d\u671f\u8a2d\u5b9a\u8a18\u8ff0\u7b87\u6240\u3002\n\naddpath\u3067SeDumi\u3084SDpt3\u306a\u3069\u306e\u30bd\u30eb\u30d0\u3092\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u3002\n\n\n\n```matlab\naddpath(genpath('C:\\Users\\hkedw\\Documents\\MATLAB\\SeDuMi_1_3'))\naddpath(genpath('C:\\Users\\hkedw\\Documents\\MATLAB\\SDpt3-4.0'))\naddpath(genpath('C:\\Users\\hkedw\\Documents\\MATLAB\\YALMIP-master'))\n```\n\n \n\n\n\n# LMI\u3067\u306e\u6975\u914d\u7f6e\u30c6\u30b9\u30c8\n\u30b7\u30b9\u30c6\u30e0\n\n$$\n\\dot{x}=Ax\n$$\n\u306b\u5bfe\u3057\u3066\uff0c\u56fa\u6709\u5024\u3092\u3069\u3053\u306b\u304a\u304f\u304b\u3068\u3044\u3046\u554f\u984c\u3092\u8003\u3048\u308b\u3002\n\n\u30a2\u30eb\u30d5\u30a1\u9818\u57df\uff08$-\\alpha$\u3088\u308a\u5de6\u534a\u9762\uff09\uff0c\u5186\u9818\u57df\uff0c\u30bb\u30af\u30bf\u9818\u57df\u306b\u6975\u3092\u914d\u7f6e\u3059\u308bLMI\u3092\u89e3\u304f\u3002\n\n\u9069\u5f53\u306a\u884c\u5217\u306e\u6210\u5206$\\bm{\\Psi}=\\bmat{r & s\\\\s&q}$\u3092\u7528\u3044\u3066\n\n$$\n\\bmat{\\lambda & 1} \\bm{\\Psi} \\bmat{\\lambda \\\\1}\\prec 0\n$$\n\u3092\u6e80\u305f\u3059$\\lambda$\u306e\u9818\u57df\u3078\u3068\u6975\u914d\u7f6e\u3059\u308b\u3002\n\n\u4f8b\u3048\u3070$-\\alpha$\u306e\u5de6\u5074\u306e\u30a2\u30eb\u30d5\u30a1\u9818\u57df\u306f\u4ee5\u4e0b\u306e\u901a\u308a\uff0c\n$$\n\\bm{\\Psi}_\\alpha=\\bmat{0&1\\\\1&2\\alpha}\n$$\n\n\u4e2d\u5fc3$c$\uff0c\u534a\u5f84$r$\u306e\u5186\u9818\u57df\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b\u3002\n$$\n\\bm{\\Psi}_C=\\bmat{1&-c\\\\-c&c^2-r^2}\n$$\n\n\u5f93\u3063\u3066\u4ee5\u4e0b\u306e\u6570\u5f0f\u3092\u6e80\u305f\u3059\u884c\u5217$P \\succ0$\u304c\u3042\u308c\u3070\u8a72\u5f53\u306e\u4f4d\u7f6e\u306b\u6975\u914d\u7f6e\u304c\u306a\u3055\u308c\u3066\u3044\u308b\u3002\n$$\n\\bmat{A^\\top&I} \\bmat{rP&sP\\\\sP&qP}\\bmat{A\\\\I}\\prec 0\n$$\n\n\u5c55\u958b\u3059\u308b\u3068\uff0c\n$$\nrA^\\top P A+sPA+sA^\\top P+qP\\prec0\n$$\n\n\u307e\u305f\uff0c$X:=P^{-1}$\u306b\u3064\u3044\u3066\u66f8\u304d\u76f4\u3059\u3068\uff0c\n\n$$\n\\bmat{A&I} \\bmat{rX&sX\\\\sX&qX}\\bmat{A^\\top\\\\I}\\prec 0\n$$\n\n\n```matlab\n%{\n%% Parameters\nm2 = 0.002;\nl2 = 0.1;\ng = 9.8;\nL1 = 11.6e-2;\n% tekito\nI2 = L1/2;\nc2 = 1.1;\nb = 1.4;\na = 2.1;\n%% Equation\nA = [0 0 1 0;\n 0 0 0 1;\n 0 0 -a 0;\n 0 m2*g*l2/I2 m2*L1*l2*a/I2 -c2/I2];\n\nB = [0;0;b;-m2*L1*l2*b/I2];\n%}\n\n% ----------\nJ = 0.0712; M = 0.390; mu = 0.695;\nl = 0.204; g = 9.81;\n\nA = [ 0 1\n M*g*l/J -mu/J ];\nB = [ 0\n 1/J ];\n\nrank(ctrb(A,B))\n```\n\n \n ans =\n \n 2\n \n \n\n\n## \u30a2\u30eb\u30d5\u30a1\u9818\u57df\uff08\u53ce\u675f\u901f\u5ea6\u5236\u7d04\uff09\n\n\u5148\u7a0b\u306e\u5f0f\u3092\u5c55\u958b\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5236\u7d04\u3092\u3068\u3051\u3070\u826f\u3044\u3002\n\n$$\nPA+A^\\top P+2\\alpha P \\prec 0, P\\succ0\n$$\n\n\u3053\u308c\u306f$X:=P^{-1}$\u3092\u4e21\u8fba\u304b\u3089\u304b\u3051\u308b\u3053\u3068\u3067\u7b49\u4fa1\u7684\u306b\n\n$$\nAX+XA^\\top +2\\alpha X \\prec 0, X\\succ0\n$$\n\n\u3068\u8868\u305b\u308b\u3002\n\n\n### BMI\u304b\u3089LMI\u3078\u306e\u5909\u63db\u3092\u7528\u3044\u305f \u8a2d\u8a08\u554f\u984c\n\u3053\u3053\u3067\u4ee5\u4e0b\u306e\u30b2\u30a4\u30f3K\u306e\u72b6\u614bFB\u30b7\u30b9\u30c6\u30e0\u3092\u5b89\u5b9a\u5316\u3055\u305b\u308b\u3053\u3068\u3092\u8003\u3048\u308b\u3002\n$$\n\\dot{x}=Ax+Bu=(A+BK)x\n$$\n\n\u5148\u307b\u3069\u306e\u5f0f\u306b\u4ee3\u5165\u3057\u3066\uff0c\uff08\u3053\u306e\u307e\u307e\u3060\u3068BMI\uff1a\u53cc\u7dda\u5f62\u884c\u5217\u4e0d\u7b49\u5f0f\u306b\u306a\u308b\u306e\u3067\uff09\n$$\n(A+BK)X+X(A+BK)^\\top +2\\alpha X \\prec 0, X\\succ0\n$$\n\n\u3053\u3053\u3067\uff0c$KX=Y$\u3068\u5909\u63db\u3059\u308b\u3053\u3068\u3067\uff0cLMI\u306b\u518d\u3073\u5909\u63db\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\n\n$$\n(AX+BY)+(AX+BY)^\\top +2\\alpha X \\prec 0, X\\succ0\n$$\n\n\n\n```matlab\nalpha = 5;\n\nrow=size(A,1);col=size(A,2);\n\n\nX = sdpvar(row,col,'symmetric')\nY = sdpvar(1,col,'full')\n\nLMI = [A*X+B*Y+(A*X+B*Y)'+ 2*alpha*X<0 ,X>0];\n\nsol = solvesdp(LMI,Y*Y.');\n\nX = double(X);\nY = double(Y);\nK = Y/X;\n\n% LMI1 result\npole1 = eig(A+B*K);\n\ndisplay(pole1);\n```\n\n Linear matrix variable 2x2 (symmetric, real, 3 variables)\n Linear matrix variable 1x2 (full, real, 2 variables)\n \u8b66\u544a: Strict inequalities are not supported. A non-strict has been added instead')\n > In < (line 12)\n \u8b66\u544a: Strict inequalities are not supported. A non-strict has been added instead')\n > In > (line 12)\n SeDuMi 1.3 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.\n Alg = 2: xz-corrector, theta = 0.250, beta = 0.500\n eqs m = 6, order n = 7, dim = 13, blocks = 4\n nnz(A) = 16 + 0, nnz(ADA) = 30, nnz(L) = 18\n it : b*y gap delta rate t/tP* t/tD* feas cg cg prec\n 0 : 1.51E+02 0.000\n 1 : -2.79E-01 1.40E+01 0.000 0.0928 0.9900 0.9900 1.54 1 1 4.9E+00\n 2 : -1.70E-02 3.10E+00 0.000 0.2213 0.9000 0.9000 1.36 1 1 1.6E+00\n 3 : 1.07E-03 8.88E-01 0.000 0.2862 0.9000 0.9000 0.66 1 1 1.2E+00\n 4 : -2.91E-03 2.51E-01 0.000 0.2831 0.9000 0.9000 0.82 1 1 1.3E+00\n 5 : -5.27E-04 5.03E-02 0.000 0.2002 0.9000 0.9000 1.19 1 1 9.4E-01\n 6 : -1.26E-04 1.30E-02 0.000 0.2588 0.9000 0.9000 1.38 1 1 1.7E-01\n 7 : -2.82E-05 3.73E-03 0.000 0.2865 0.9000 0.9000 1.25 1 1 3.6E-02\n 8 : -7.36E-06 1.10E-03 0.000 0.2949 0.9000 0.9000 1.13 1 1 9.1E-03\n 9 : -1.94E-06 3.26E-04 0.000 0.2968 0.9000 0.9000 1.07 1 1 2.4E-03\n 10 : -6.07E-07 1.01E-04 0.000 0.3105 0.9000 0.9000 1.04 1 1 7.1E-04\n 11 : -1.51E-07 3.19E-05 0.000 0.3149 0.9000 0.9000 1.02 1 1 2.2E-04\n 12 : -7.32E-08 1.08E-05 0.000 0.3383 0.9000 0.9000 1.01 1 1 7.1E-05\n 13 : -1.34E-08 3.53E-06 0.000 0.3272 0.9000 0.9000 1.01 1 1 2.3E-05\n 14 : -7.19E-09 1.10E-06 0.000 0.3123 0.9000 0.9000 1.00 2 2 7.0E-06\n 15 : -1.41E-09 3.25E-07 0.000 0.2948 0.9000 0.9000 1.00 2 2 2.1E-06\n 16 : -5.66E-10 9.62E-08 0.000 0.2958 0.9000 0.9000 1.00 2 2 6.0E-07\n 17 : -1.28E-10 2.87E-08 0.000 0.2987 0.9000 0.9000 1.00 2 2 1.8E-07\n 18 : -5.44E-11 8.90E-09 0.000 0.3099 0.9000 0.9000 1.00 2 2 5.5E-08\n 19 : -1.14E-11 2.77E-09 0.000 0.3115 0.9000 0.9000 1.00 2 2 1.7E-08\n 20 : -5.45E-12 8.66E-10 0.000 0.3122 0.9000 0.9000 1.00 3 3 5.3E-09\n 21 : -1.09E-12 2.64E-10 0.000 0.3053 0.9000 0.9000 1.00 3 3 1.6E-09\n 22 : -4.86E-13 8.00E-11 0.000 0.3028 0.9000 0.9000 1.00 3 3 4.9E-10\n \n iter seconds digits c*x b*y\n 22 0.3 7.6 6.5901333299e-13 -4.8648236002e-13\n |Ax-b| = 1.7e-11, [Ay-c]_+ = 2.3E-13, |x|= 7.1e-01, |y|= 1.3e-01\n \n Detailed timing (sec)\n Pre IPM Post\n 7.001E-03 1.280E-01 4.003E-03 \n Max-norms: ||b||=1, ||c|| = 1,\n Cholesky |add|=1, |skip| = 0, ||L.L|| = 3.03988.\n \n pole1 =\n \n -10.9597 + 0.2407i\n -10.9597 - 0.2407i\n \n \n\n\n\u6975\u914d\u7f6e\u306f\u305f\u3060\u306e\u6761\u4ef6\u3067\u3042\u308a\uff0c\u5b9f\u969b\u306e\u8a08\u7b97\u3067\u306f\u3053\u308c\u306b\u66f4\u306b\u5236\u7d04\u6761\u4ef6\u3092\u4e0e\u3048\u308b\u3002\n\u30b3\u30de\u30f3\u30c9\u3067\u306f\u6700\u5c0f\u5316\u3059\u308b\u6761\u4ef6\u3092\u5165\u308c\u3066\u3044\u308b\u3002\n\n\n\n \n\n\n### \u30b3\u30c4\u3068memo\n\n- \u30b2\u30a4\u30f3K\u306fY/X\u306a\u306e\u3067Y\u3092\u6700\u5c0f\u5316\u3059\u308b\u3053\u3068\u3067\u5c0f\u3055\u3081\u306e\u30b2\u30a4\u30f3\u3092\u671f\u5f85\u3067\u304d\u308b\u3002\uff08X\u306e\u5024\u306b\u3088\u3063\u3066\u306f\u5176\u306e\u9650\u308a\u3067\u306f\u306a\u3044\u304c\uff09\n- \u4fdd\u967a\u3068\u3057\u3066X>0\u3067\u306f\u306a\u304fX-I>0\u3068\u5236\u7d04\u3092\u304a\u3044\u3066\u30b2\u30a4\u30f3\u306e\u767a\u6563\u3092\u6291\u3048\u308b\n- \u5927\u304d\u3044\u884c\u5217\u3067\u89e3\u3044\u305f\u65b9\u304c\u697d\u306a\u306e\u304b\n\n\n#### X>0\u3068X>eye\u306e\u9055\u3044\n\nX>0\u3068\u3057\u305f\u6642\u3002\n\n\n```\npole1 =\n\n -10.9597 + 0.2407i\n -10.9597 - 0.2407i\n```\n\nX>eye\u3068\u3057\u305f\u6642\u3002\n```\npole1 =\n\n -5.0001\n -10.7783\n```\n\u3068\u306a\u308b\u3002X\u3092\u7e1b\u308b\u306e\u306f\u306a\u304b\u306a\u304b\u610f\u5473\u304c\u3042\u308a\u305d\u3046\u3002\n\n\n```matlab\n% Solve LMI 2\n\nX = sdpvar(row,col,'symmetric')\nY = sdpvar(1,col,'full')\ngamma = sdpvar(1);\n\n% LMI \nLMI = [A*X+B*Y+(A*X+B*Y).'+ 2*alpha*X<0,[gamma Y;Y.' eye(size(A))]>0,X>eye(size(A))];\n\nsol = solvesdp(LMI,gamma);\n\nX = double(X);\nY = double(Y);\nK = Y/X;\n\n% LMI1 result\npole2 = eig(A+B*K);\n\ndisplay(pole2);\n```\n\n Linear matrix variable 2x2 (symmetric, real, 3 variables)\n Linear matrix variable 1x2 (full, real, 2 variables)\n \u8b66\u544a: Strict inequalities are not supported. A non-strict has been added instead')\n > In < (line 12)\n \u8b66\u544a: Strict inequalities are not supported. A non-strict has been added instead')\n > In > (line 12)\n \u8b66\u544a: Strict inequalities are not supported. A non-strict has been added instead')\n > In > (line 12)\n SeDuMi 1.3 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.\n Alg = 2: xz-corrector, theta = 0.250, beta = 0.500\n eqs m = 6, order n = 8, dim = 18, blocks = 4\n nnz(A) = 15 + 0, nnz(ADA) = 30, nnz(L) = 18\n it : b*y gap delta rate t/tP* t/tD* feas cg cg prec\n 0 : 1.25E+02 0.000\n 1 : -1.15E+00 4.02E+01 0.000 0.3220 0.9000 0.9000 0.84 1 1 2.8E+01\n 2 : -1.66E+01 7.76E+00 0.000 0.1932 0.9000 0.9000 -0.76 1 1 4.2E+01\n 3 : -8.40E+01 1.42E+00 0.000 0.1828 0.9000 0.9000 -0.77 1 1 3.4E+01\n 4 : -1.73E+02 4.14E-01 0.000 0.2921 0.9000 0.9000 -0.67 1 1 2.5E+01\n 5 : -1.28E+02 1.15E-01 0.000 0.2768 0.9000 0.9000 0.08 1 1 8.4E+00\n 6 : -7.55E+01 2.78E-02 0.000 0.2419 0.9000 0.9000 1.15 1 1 1.8E+00\n 7 : -4.79E+01 6.90E-03 0.000 0.2485 0.9000 0.9000 1.08 1 1 4.5E-01\n 8 : -3.54E+01 1.80E-03 0.000 0.2617 0.9000 0.9000 0.71 1 1 1.5E-01\n 9 : -2.89E+01 4.42E-04 0.000 0.2449 0.9000 0.9000 0.45 1 1 5.5E-02\n 10 : -2.58E+01 1.20E-04 0.000 0.2715 0.9000 0.9000 0.49 1 1 2.0E-02\n 11 : -2.43E+01 3.87E-05 0.000 0.3223 0.9000 0.9000 0.55 1 1 8.3E-03\n 12 : -2.34E+01 1.42E-05 0.000 0.3683 0.9000 0.9000 0.48 1 1 4.3E-03\n 13 : -2.28E+01 5.10E-06 0.000 0.3578 0.9000 0.9000 0.58 1 1 2.0E-03\n 14 : -2.25E+01 2.13E-06 0.000 0.4180 0.9000 0.9000 0.42 1 1 1.2E-03\n 15 : -2.22E+01 7.43E-07 0.000 0.3485 0.9000 0.9000 0.59 1 1 5.3E-04\n 16 : -2.20E+01 2.87E-07 0.000 0.3861 0.9000 0.9000 0.43 2 2 3.1E-04\n 17 : -2.18E+01 1.01E-07 0.000 0.3518 0.9000 0.9000 0.56 2 2 1.4E-04\n 18 : -2.17E+01 3.97E-08 0.000 0.3938 0.9000 0.9000 0.40 2 2 8.3E-05\n 19 : -2.17E+01 1.37E-08 0.000 0.3446 0.9000 0.9000 0.56 2 2 3.6E-05\n 20 : -2.16E+01 5.21E-09 0.000 0.3802 0.9000 0.9000 0.41 2 2 2.1E-05\n 21 : -2.16E+01 1.81E-09 0.000 0.3481 0.9000 0.9000 0.54 2 2 9.4E-06\n 22 : -2.16E+01 7.04E-10 0.000 0.3885 0.9000 0.9000 0.40 2 2 5.6E-06\n 23 : -2.15E+01 2.43E-10 0.000 0.3451 0.9000 0.9000 0.55 3 2 2.5E-06\n 24 : -2.15E+01 9.28E-11 0.000 0.3819 0.9000 0.9000 0.40 4 5 1.4E-06\n 25 : -2.15E+01 3.22E-11 0.000 0.3473 0.9000 0.9000 0.54 5 3 6.4E-07\n 26 : -2.15E+01 1.25E-11 0.000 0.3867 0.9000 0.9000 0.40 5 5 3.8E-07\n 27 : -2.15E+01 4.31E-12 0.000 0.3457 0.9000 0.9000 0.55 5 5 1.7E-07\n 28 : -2.15E+01 1.65E-12 0.000 0.3830 0.9000 0.9000 0.40 5 5 9.8E-08\n 29 : -2.15E+01 5.73E-13 0.000 0.3469 0.9000 0.9000 0.54 5 5 4.4E-08\n 30 : -2.15E+01 2.21E-13 0.000 0.3857 0.9000 0.9000 0.40 5 5 2.6E-08\n 31 : -2.15E+01 7.64E-14 0.000 0.3460 0.9000 0.9000 0.55 5 5 1.1E-08\n 32 : -2.15E+01 2.93E-14 0.000 0.3837 0.9000 0.9000 0.40 5 5 6.7E-09\n 33 : -2.15E+01 1.02E-14 0.000 0.3467 0.9000 0.9000 0.55 5 5 3.0E-09\n 34 : -2.15E+01 3.91E-15 0.000 0.3852 0.9000 0.9000 0.40 5 5 1.7E-09\n 35 : -2.15E+01 1.36E-15 0.000 0.3462 0.9000 0.9000 0.55 6 6 7.7E-10\n \n iter seconds digits c*x b*y\n 35 0.6 5.5 -2.1506133405e+01 -2.1506059306e+01\n |Ax-b| = 1.0e-09, [Ay-c]_+ = 0.0E+00, |x|= 4.9e+01, |y|= 1.5e+06\n \n Detailed timing (sec)\n Pre IPM Post\n 1.200E-02 2.710E-01 2.002E-03 \n Max-norms: ||b||=1, ||c|| = 1,\n Cholesky |add|=0, |skip| = 1, ||L.L|| = 9.2353.\n \n pole2 =\n \n -5.0001\n -10.7783\n \n \n\n\n## \u5186\u9818\u57df\u3078\u306e\u6975\u914d\u7f6e\n\n\u89e3\u304f\u3079\u304dLMI\u306f\n\n$$\ncA^\\top P A-cPA-cA^\\top P+(c^2-r^2)P\\prec0\\\\\ncA X A^\\top -cXA^\\top-cA X+(c^2-r^2)X\\prec0\n$$\n\u3067\u3042\u308b\u3002\n\n\u3053\u306e\u307e\u307e\u3060\u3068LMI\u3068\u3057\u3066\u89e3\u3051\u306a\u3044\u3002\u305d\u3053\u3067Schur\u306e\u88dc\u984c\u3092\u7528\u3044\u3066\u3053\u308c\u3092LMI\u306e\u5f62\u5f0f\u3078\u3068\u5909\u63db\u3059\u308b\u3053\u3068\u3092\u8003\u3048\u308b\u3002\n\n\n### Schur\u306e\u88dc\u984c\u306e\u9069\u7528\nSchur\uff08\u30b7\u30e5\u30fc\u30eb\uff09\u306e\u88dc\u984c\u3068\u306f\u4ee5\u4e0b\u306e\u88dc\u984c\u3067\u3042\u308b\u3002\n$$\n\\bmat{A&B\\\\C&D} \\prec 0 \\Leftrightarrow A-BD^{-1}C \\prec 0\n$$\n\u305f\u3060\u3057\uff0c$D$\u306f\u5f53\u7136\u9006\u884c\u5217\u304c\u7121\u304f\u3066\u306f\u306a\u3089\u306a\u3044\u3002\n\n\u3053\u306e\u88dc\u984c\u3092\u7528\u3044\u308b\u3053\u3068\u3067\u6b21\u306e\u3088\u3046\u306a\u95a2\u4fc2\u304c\u5c0e\u3051\u308b\u3002\n\n$$\n\\begin{align}\nc(XA^\\top+A X)+(r^2-c^2)X - cA X A^\\top \\succ 0 &\\Leftrightarrow \\\\\n&\\bmat{c(XA^\\top+A X)+(r^2-c^2)X & AX \\\\ XA^\\top & X}\\succ 0\n\\end{align}\n$$\n\n\u3057\u305f\u304c\u3063\u3066LMI\u306b\u5909\u63db\u3067\u304d\u305f\u306e\u3067\u3053\u308c\u3092\u7528\u3044\u3066\u5909\u63db\u3092\u884c\u3046\u3002\n\n### FB\u7cfb\u306e\u8a2d\u8a08\n\n\u5148\u7a0b\u3084\u3063\u305f\u3068\u304a\u308a\uff0c$A+BK$\u3092\u4ee3\u5165\u3057\uff0c$Y=KX$\u306e\u5909\u63db\u3092\u884c\u3046\u3002\n\u7d50\u8ad6\u304b\u3089\u8a00\u3046\u3068\uff0c\u4ee5\u4e0b\u306e\u901a\u308a\u306b\u306a\u308b\u3002\n\n\n$$\nM=AX+BY\\\\\n\\bmat{c(M^\\top+M)+(r^2-c^2)X & M \\\\ M^\\top & X}\\succ 0\n$$\n\n\n\n\n```matlab\nrow=size(A,1);col=size(A,2);\n\n\nX = sdpvar(row,col,'symmetric')\nY = sdpvar(1,col,'full')\n\n\nM = A*X + B*Y;\n\nqc = -5; rc = 2;\nM_D3 = [qc*(M+M')+(rc^2-qc^2)*X M\n M' X ];\n\nLMI = [ M_D3 > 0 , X>0];\n\noptions = sdpsettings('solver','sedumi');\nsolvesdp(LMI,[],options) ;\n\nX_opt = double(X);\nY_opt = double(Y);\nK_opt = Y_opt/X_opt;\n\n% LMI3 result\npole3 = eig(A+B*K_opt);\n\ndisplay(pole3);\n```\n\n Linear matrix variable 2x2 (symmetric, real, 3 variables)\n Linear matrix variable 1x2 (full, real, 2 variables)\n \u8b66\u544a: Strict inequalities are not supported. A non-strict has been added instead')\n > In > (line 12)\n \u8b66\u544a: Strict inequalities are not supported. A non-strict has been added instead')\n > In > (line 12)\n SeDuMi 1.3 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.\n Alg = 2: xz-corrector, theta = 0.250, beta = 0.500\n eqs m = 5, order n = 7, dim = 21, blocks = 3\n nnz(A) = 23 + 0, nnz(ADA) = 25, nnz(L) = 15\n it : b*y gap delta rate t/tP* t/tD* feas cg cg prec\n 0 : 1.15E+04 0.000\n 1 : 0.00E+00 9.45E+02 0.000 0.0821 0.9900 0.9900 1.00 1 0 2.3E+01\n 2 : 0.00E+00 1.97E+02 0.000 0.2080 0.9000 0.9000 1.00 1 1 4.8E+00\n 3 : 0.00E+00 1.61E+01 0.000 0.0817 0.9900 0.9900 1.00 1 1 3.9E-01\n 4 : 0.00E+00 3.48E+00 0.000 0.2168 0.9000 0.9000 1.00 1 1 8.6E-02\n 5 : 0.00E+00 1.99E-01 0.000 0.0570 0.9900 0.9900 1.00 1 1 4.9E-03\n 6 : 0.00E+00 8.23E-05 0.000 0.0004 0.9999 0.9999 1.00 1 1 2.0E-06\n 7 : 0.00E+00 3.15E-11 0.000 0.0000 1.0000 1.0000 1.00 1 1 2.8E-12\n \n iter seconds digits c*x b*y\n 7 0.0 Inf 0.0000000000e+00 0.0000000000e+00\n |Ax-b| = 5.3e-13, [Ay-c]_+ = 0.0E+00, |x|= 1.0e-11, |y|= 4.3e-01\n \n Detailed timing (sec)\n Pre IPM Post\n 8.006E-03 3.900E-02 4.003E-03 \n Max-norms: ||b||=0, ||c|| = 0,\n Cholesky |add|=0, |skip| = 0, ||L.L|| = 3.77576.\n \n pole3 =\n \n -3.9254 + 0.5682i\n -3.9254 - 0.5682i\n \n \n\n\n\u6761\u4ef6\u3092\u7c21\u5358\u306b\u3057\u305f\u3089\u304d\u3061\u3093\u3068\u5186\u306e\u4e2d\u306b\u5165\u3063\u3066\u3044\u308b\u3002\n\n## \u30bb\u30af\u30bf\u6761\u4ef6\n\n\u6975\u3068\u539f\u70b9\u306e\u7dda\u5206\u304c\u5b9f\u8ef8\u3068\u4ea4\u308f\u308b\u89d2\u5ea6\u304c$\\theta$\u4ee5\u5185\u3067\u3042\u308b\u3068\u3044\u3046\u5236\u7d04\u3092$k=\\tan\\theta$\u3067\u8868\u3059\u3002\n\nLMI\u6761\u4ef6\u306f\n$$\n\\bmat{k(PA+A^\\top P) & PA-A^\\top P\\\\A^\\top P-PA &k(PA+A^\\top P)} \\prec 0\n$$\n\u3042\u308b\u3044\u306f\n$$\n\\bmat{k(AX+XA^\\top ) & AX-XA^\\top \\\\ XA^\\top-AX &k(AX+XA^\\top)} \\prec0\n$$\n\u3068\u306a\u308b\u3002\n\n\n\u4eca\u307e\u3067\u3068\u540c\u69d8\u306b\u3057\u3066FB\u306e\u305f\u3081\u306e\u6761\u4ef6\u306f\uff0c\n$$\nM=AX+BY\\\\\n\\bmat{k(M+M^\\top ) & M-M^\\top \\\\ M^\\top-M &k(M+M^\\top)} \\prec0\n$$\n\n\u3067\u3042\u308b\u3002\n\n\n```matlab\nk=tand(15)\nX = sdpvar(row,col,'symmetric')\nY = sdpvar(1,col,'full')\n\nM = A*X + B*Y;\n\nqc = -5; rc = 2;\ncircle = [qc*(M+M')+(rc^2-qc^2)*X M\n M' X ];\nsector = [k*(M+M') M-M.'; -M+M.' k*(M+M')];\n\nLMI = [ circle > 0 , sector<0, X>0];\n\noptions = sdpsettings('solver','sedumi');\nsol=solvesdp(LMI,[],options);\n\nsol\n\nX_opt = double(X);\nY_opt = double(Y);\nK_opt = Y_opt/X_opt;\n\n% LMI3 result\npole4 = eig(A+B*K_opt);\n\ndisplay(pole4);\n```\n\n \n k =\n \n 0.2679\n \n Linear matrix variable 2x2 (symmetric, real, 3 variables)\n Linear matrix variable 1x2 (full, real, 2 variables)\n \u8b66\u544a: Strict inequalities are not supported. A non-strict has been added instead')\n > In > (line 12)\n \u8b66\u544a: Strict inequalities are not supported. A non-strict has been added instead')\n > In < (line 12)\n \u8b66\u544a: Strict inequalities are not supported. A non-strict has been added instead')\n > In > (line 12)\n SeDuMi 1.3 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.\n Alg = 2: xz-corrector, theta = 0.250, beta = 0.500\n eqs m = 5, order n = 11, dim = 37, blocks = 4\n nnz(A) = 47 + 0, nnz(ADA) = 25, nnz(L) = 15\n it : b*y gap delta rate t/tP* t/tD* feas cg cg prec\n 0 : 8.96E+03 0.000\n 1 : 0.00E+00 2.24E+03 0.000 0.2501 0.9000 0.9000 1.00 1 0 7.8E+01\n 2 : 0.00E+00 4.68E+02 0.000 0.2088 0.9000 0.9000 1.00 1 1 1.6E+01\n 3 : 0.00E+00 4.18E+01 0.000 0.0892 0.9900 0.9900 1.00 1 1 1.5E+00\n 4 : 0.00E+00 1.17E+01 0.000 0.2811 0.9000 0.9000 1.00 1 1 4.1E-01\n 5 : 0.00E+00 8.16E-01 0.000 0.0694 0.9900 0.9900 1.00 1 1 2.8E-02\n 6 : 0.00E+00 3.48E-03 0.000 0.0043 0.9990 0.9990 1.00 1 1 1.2E-04\n 7 : 0.00E+00 6.71E-10 0.000 0.0000 1.0000 1.0000 1.00 1 1 2.3E-11\n \n iter seconds digits c*x b*y\n 7 0.0 Inf 0.0000000000e+00 0.0000000000e+00\n |Ax-b| = 1.7e-11, [Ay-c]_+ = 0.0E+00, |x|= 5.9e-10, |y|= 5.4e-01\n \n Detailed timing (sec)\n Pre IPM Post\n 7.001E-03 2.900E-02 2.997E-03 \n Max-norms: ||b||=0, ||c|| = 0,\n Cholesky |add|=0, |skip| = 0, ||L.L|| = 4.75102.\n \n sol = \n \n \u30d5\u30a3\u30fc\u30eb\u30c9\u3092\u3082\u3064 struct:\n \n yalmiptime: 0.0527\n solvertime: 0.0423\n info: 'Successfully solved (SeDuMi-1.3)'\n problem: 0\n \n \n pole4 =\n \n -3.3251\n -5.3127\n \n \n\n\n\n```matlab\nsol\n```\n\n \n sol = \n \n \u30d5\u30a3\u30fc\u30eb\u30c9\u3092\u3082\u3064 struct:\n \n yalmiptime: 0.0527\n solvertime: 0.0423\n info: 'Successfully solved (SeDuMi-1.3)'\n problem: 0\n \n \n\n\n\u5148\u7a0b\u306e\u5186\u306e\u7bc4\u56f2\u3068\u6bd4\u8f03\u3059\u308b\u3068\uff0c\u8907\u7d20\u6210\u5206\u3092\u524a\u3063\u3066\u3053\u306e\u3088\u3046\u306b\u89e3\u306e\u7bc4\u56f2\u3092\u7d5e\u308c\u308b\u3002\n\n\u9006\u3082\u3067\u304d\u308b\u306e\u3060\u308d\u3046\u304b\uff1f\u3068\u601d\u3063\u305f\u304c\u53f3\u534a\u5e73\u9762\u306b\u884c\u3063\u3061\u3083\u3046\u306e\u3067\u7121\u3057\u3067\u3002\n\n\n```matlab\n\n```\n", "meta": {"hexsha": "bab1ab0ef193bdbfc67aae327bfaee3203edff82", "size": 23412, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/poleplacement/LMI study Test.ipynb", "max_stars_repo_name": "YoshiRi/LMI_study", "max_stars_repo_head_hexsha": "ba48e75e1ea05f5a7d1ea84faebc2c500ce786f2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-03-24T07:20:02.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-24T07:20:02.000Z", "max_issues_repo_path": "notebooks/poleplacement/LMI study Test.ipynb", "max_issues_repo_name": "YoshiRi/LMI_study", "max_issues_repo_head_hexsha": "ba48e75e1ea05f5a7d1ea84faebc2c500ce786f2", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/poleplacement/LMI study Test.ipynb", "max_forks_repo_name": "YoshiRi/LMI_study", "max_forks_repo_head_hexsha": "ba48e75e1ea05f5a7d1ea84faebc2c500ce786f2", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 31.9399727149, "max_line_length": 94, "alphanum_fraction": 0.4516914403, "converted": true, "num_tokens": 9352, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124811, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.4263394396504612}} {"text": "```python\nfrom IPython.display import HTML\ntag = HTML('''\nToggle cell visibility here.''')\ndisplay(tag)\n```\n\n\n\nToggle cell visibility here.\n\n\n\n```python\n# Erasmus+ ICCT project (2018-1-SI01-KA203-047081)\n\n%matplotlib notebook\nimport numpy as np\nimport math\nimport matplotlib.pyplot as plt\nfrom scipy import signal\nimport ipywidgets as widgets\nimport control as c\nimport sympy as sym\nfrom IPython.display import Latex, display, Markdown # For displaying Markdown and LaTeX code\nfrom fractions import Fraction\nimport matplotlib.patches as patches\n```\n\n## Aproksimacija dominantnega pola\n\nPri analizi sistemov sisteme pogosto aproksimiramo z dominantnim polom ali parom kompleksnih dominantnih polov. V tem interaktivnem primeru je prikazan ta postopek.\n\nSisteme drugega reda lahko zapi\u0161emo z naslednjo prenosno funkcijo:\n\n\\begin{equation}\n G(s)=\\frac{\\alpha\\beta}{(s+\\alpha)(s+\\beta)}=\\frac{1}{(\\frac{1}{\\alpha}s+1)(\\frac{1}{\\beta}s+1)},\n\\end{equation}\n\nkjer je $\\beta=1$, $\\alpha$ pa je iterabilen parameter.\n\nSisteme tretjega reda lahko zapi\u0161emo z naslednjo prenosno funkcijo:\n\n\\begin{equation}\n G(s)=\\frac{\\alpha{\\omega_0}^2}{\\big(s+\\alpha\\big)\\big(s^2+2\\zeta\\omega_0s+\\omega_0^2\\big)}=\\frac{1}{(\\frac{1}{\\alpha}s+1)(\\frac{1}{\\omega_0^2}s^2+\\frac{2\\zeta\\alpha}{\\omega_0}s+1)},\n\\end{equation}\n\nkjer so $\\beta=1$, $\\omega_0=4.1$ in $\\zeta=0.24$, $\\alpha$ pa je iterabilen parameter.\n\n---\n\n### Kako upravljati s tem interaktivnim primerom?\n\nPreklapljaj med sistemom drugega in tretjega reda ter z uporabo drsnika spreminjaj lego premakljivega pola $\\alpha$.\n\nTa interaktivni primer temelji na [primeru](https://lpsa.swarthmore.edu/PZXferStepBode/DomPole.html \"The Dominant Pole Approximation\") Prof. Erika Cheeverja.\n\n\n\n\n```python\n# System selector buttons\nstyle = {'description_width': 'initial'}\ntypeSelect = widgets.ToggleButtons(\n options=[('sistem drugega reda', 0), ('sistem tretjega reda', 1),],\n description='Izberi: ',style=style)\n```\n\n\n```python\ndisplay(typeSelect)\ncontinuous_update=False\n\n# set up plot \n\nfig, ax = plt.subplots(2,1,figsize=[9.8,7],num='Aproksmiacija dominantnih polov')\nplt.subplots_adjust(hspace=0.35)\nax[0].grid(True)\nax[1].grid(True)\n# ax[2].grid(which='both', axis='both', color='lightgray')\nax[0].axhline(y=0,color='k',lw=.8)\nax[1].axhline(y=0,color='k',lw=.8)\nax[0].axvline(x=0,color='k',lw=.8)\nax[1].axvline(x=0,color='k',lw=.8)\nax[0].set_xlabel('Re')\nax[0].set_ylabel('Im')\nax[0].set_xlim([-10,0.5])\nax[1].set_xlim([-0.5,20])\nax[1].set_xlabel('$t$ [s]')\nax[1].set_ylabel('vhod, izhod')\nax[0].set_title('Diagram lege ni\u010del in polov')\nax[1].set_title('\u010casovni odziv sistema')\n\nplotzero, = ax[0].plot([], [])\nresponse, = ax[1].plot([], [])\nresponseAdom, = ax[1].plot([], [])\nresponseBdom, = ax[1].plot([], [])\n\nax[1].step([0,50],[0,1],color='C0',label='vhod')\n\n# generate x values\n \ndef response_func(a,index):\n \n global plotzero, response, responseAdom, responseBdom \n# global bodePlot, bodePlotAdom, bodePlotBdom\n\n t = np.linspace(0, 50, 1000)\n \n if index==0:\n b=1\n num=a*b\n den=([1,a+b,a*b])\n tf_sys=c.TransferFunction(num,den)\n poles_sys,zeros_sys=c.pzmap(tf_sys, Plot=False)\n tout, yout = c.step_response(tf_sys,t)\n den1=([1,a])\n tf_sys1=c.TransferFunction(a,den1)\n toutA, youtA = c.step_response(tf_sys1,t)\n den2=([1,b])\n tf_sys2=c.TransferFunction(b,den2)\n toutB, youtB = c.step_response(tf_sys2,t)\n mag, phase, omega = c.bode_plot(tf_sys, Plot=False) # Bode-plot\n magA, phase, omegaA = c.bode_plot(tf_sys1, Plot=False) # Bode-plot\n magB, phase, omegaB = c.bode_plot(tf_sys2, Plot=False) # Bode-plot\n s=sym.Symbol('s')\n eq=(a*b/((s+a)*(s+b)))\n eq1=1/(((1/a)*s+1)*((1/b)*s+1))\n display(Markdown('Premakljivi pol (vijoli\u010dna krivulja) $\\\\alpha$ je enak %.1f, fiksni pol (rde\u010da krivulja) $b$ je enak %i; Prenosna funkcija sistema je'%(a,1)))\n display(eq),display(Markdown('oz.')),display(eq1)\n\n elif index==1:\n omega0=4.1\n zeta=0.24\n num=a*omega0**2\n den=([1,2*zeta*omega0+a,omega0**2+2*zeta*omega0*a,a*omega0**2])\n tf_sys=c.TransferFunction(num,den)\n poles_sys,zeros_sys=c.pzmap(tf_sys, Plot=False)\n tout, yout = c.step_response(tf_sys,t)\n den1=([1,a])\n tf_sys1=c.TransferFunction(a,den1)\n toutA, youtA = c.step_response(tf_sys1,t)\n den2=([1,2*zeta*omega0,omega0**2])\n tf_sys2=c.TransferFunction(omega0**2,den2)\n toutB, youtB = c.step_response(tf_sys2,t)\n mag, phase, omega = c.bode_plot(tf_sys, Plot=False) # Bode-plot\n magA, phase, omegaA = c.bode_plot(tf_sys1, Plot=False) # Bode-plot\n magB, phase, omegaB = c.bode_plot(tf_sys2, Plot=False) # Bode-plot\n s=sym.Symbol('s')\n eq=(a*omega0**2/((s+a)*(s**2+2*zeta*omega0*s+omega0*omega0)))\n eq1=1/(((1/a)*s+1)*((1/(omega0*omega0))*s*s+(2*zeta*a/omega0)*s+1))\n \n display(Markdown('Premakljivi pol (vijoli\u010dna krivulja) $\\\\alpha$ je enak %.1f, fiksni pol (rde\u010da krivulja) $\\\\beta$ je enak $1\\pm4j$ (vrednost $\\omega_0$ je nastavljena na 4.1, vrednost $\\zeta$ pa na 0.24). Prenosna funkcija sistema je enaka:'%(a)))\n display(eq),display(Markdown('oz.')),display(eq1)\n \n ax[0].lines.remove(plotzero)\n ax[1].lines.remove(response)\n ax[1].lines.remove(responseAdom)\n ax[1].lines.remove(responseBdom)\n \n plotzero, = ax[0].plot(np.real(poles_sys), np.imag(poles_sys), 'xg', markersize=10, label = 'pol')\n response, = ax[1].plot(tout,yout,color='C1',label='odziv sistema',lw=3)\n responseAdom, = ax[1].plot(toutA,youtA,color='C4',label='odziv zaradi premakljivega pola/para polov')\n responseBdom, = ax[1].plot(toutB,youtB,color='C3',label='odziv zaradi fiksnega pola')\n\n ax[0].legend()\n ax[1].legend()\n \na_slider=widgets.FloatSlider(value=0.1, min=0.1, max=10, step=.1,\n description='$\\\\alpha$:',disabled=False,continuous_update=False,\n orientation='horizontal',readout=True,readout_format='.2f',)\n\ninput_data=widgets.interactive_output(response_func,{'a':a_slider,'index':typeSelect})\n\ndef update_slider(index):\n global a_slider\n \n aval=[0.1,0.1]\n a_slider.value=aval[index] \n\ninput_data2=widgets.interactive_output(update_slider,{'index':typeSelect})\n\ndisplay(a_slider,input_data)\n\n```\n\n\n ToggleButtons(description='Izberi: ', options=(('sistem drugega reda', 0), ('sistem tretjega reda', 1)), style\u2026\n\n\n\n \n\n\n\n\n\n\n\n FloatSlider(value=0.1, continuous_update=False, description='$\\\\alpha$:', max=10.0, min=0.1)\n\n\n\n Output()\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "e06ad2f28a6474f34458b9cbd94e9a7df5354701", "size": 188277, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ICCT_si/examples/02/.ipynb_checkpoints/TD-12-Aproksimacija-dominantnega-pola-checkpoint.ipynb", "max_stars_repo_name": "ICCTerasmus/ICCT", "max_stars_repo_head_hexsha": "fcd56ab6b5fddc00f72521cc87accfdbec6068f6", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2021-05-22T18:42:14.000Z", "max_stars_repo_stars_event_max_datetime": "2021-10-03T14:10:22.000Z", "max_issues_repo_path": "ICCT_si/examples/02/TD-12-Aproksimacija-dominantnega-pola.ipynb", "max_issues_repo_name": "ICCTerasmus/ICCT", "max_issues_repo_head_hexsha": "fcd56ab6b5fddc00f72521cc87accfdbec6068f6", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ICCT_si/examples/02/TD-12-Aproksimacija-dominantnega-pola.ipynb", "max_forks_repo_name": "ICCTerasmus/ICCT", "max_forks_repo_head_hexsha": "fcd56ab6b5fddc00f72521cc87accfdbec6068f6", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2021-05-24T11:40:09.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-29T16:36:18.000Z", "avg_line_length": 166.1756398941, "max_line_length": 140863, "alphanum_fraction": 0.8488662981, "converted": true, "num_tokens": 2648, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124812, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.42633943965046117}} {"text": "```python\n# import numpy as np\n\n# # !/usr/bin/env python3\n# # -*- coding: utf-8 -*-\n# \"\"\"\n# Created on 20181219\n\n# @author: zhangji\n\n# Trajection of a ellipse, Jeffery equation. \n# \"\"\"\n\n# %pylab inline\n# pylab.rcParams['figure.figsize'] = (25, 11)\n# fontsize = 40\n\n# import numpy as np\n# import scipy as sp\n# from scipy.optimize import leastsq, curve_fit\n# from scipy import interpolate\n# from scipy.interpolate import interp1d\n# from scipy.io import loadmat, savemat\n# # import scipy.misc\n\n# import matplotlib\n# from matplotlib import pyplot as plt\n# from matplotlib import animation, rc\n# import matplotlib.ticker as mtick\n# from mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\n# from mpl_toolkits.mplot3d import Axes3D, axes3d\n\n# from sympy import symbols, simplify, series, exp\n# from sympy.matrices import Matrix\n# from sympy.solvers import solve\n\n# from IPython.display import display, HTML\n# from tqdm import tqdm_notebook as tqdm\n# import pandas as pd\n# import re\n# from scanf import scanf\n# import os\n# import glob\n\n# from codeStore import support_fun as spf\n# from src.support_class import *\n# from src import stokes_flow as sf\n\n# rc('animation', html='html5')\n# PWD = os.getcwd()\n# font = {'size': 20}\n# matplotlib.rc('font', **font)\n# np.set_printoptions(linewidth=90, precision=5)\n\n%load_ext autoreload\n%autoreload 2\n\nfrom tqdm.notebook import tqdm as tqdm_notebook\nimport os\nimport glob\nimport natsort \nimport numpy as np\nimport scipy as sp\nfrom scipy.optimize import leastsq, curve_fit\nfrom scipy import interpolate, integrate\nfrom scipy import spatial\n# from scipy.interpolate import interp1d\nfrom scipy.io import loadmat, savemat\n# import scipy.misc\n# import importlib\nfrom IPython.display import display, HTML, Math, Latex\nimport pandas as pd\nimport pickle\n\nimport matplotlib\nfrom matplotlib import pyplot as plt\nimport matplotlib.colors as colors\nfrom matplotlib import animation, rc\nimport matplotlib.ticker as mtick\nfrom mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\nfrom mpl_toolkits.mplot3d import Axes3D, axes3d\nfrom mpl_toolkits.axes_grid1.axes_divider import make_axes_locatable\nfrom mpl_toolkits.mplot3d.art3d import Line3DCollection\nfrom matplotlib import cm\n\nfrom time import time\nfrom src.support_class import *\nfrom src import jeffery_model as jm\nfrom codeStore import support_fun as spf\nfrom codeStore import support_fun_table as spf_tb\nfrom codeStore import support_fun_baseflow as spf_bf\n# %matplotlib notebook\n\nfrom sympy.parsing import mathematica\nimport sympy\nfrom sympy.printing.latex import LatexPrinter, print_latex\nfrom sympy.utilities.lambdify import lambdify, lambdastr\nimport inspect\n\n%matplotlib inline\nparams = {'animation.html': 'html5',\n 'font.family': 'sans-serif'}\nparams['text.latex.preamble'] = [r'\\usepackage{bm}',\n r'\\usepackage{amsmath}',\n r'\\usepackage{amssymb}',\n r'\\usepackage{mathrsfs}',\n r'\\DeclareMathOperator{\\Tr}{Tr}', ]\nparams['text.usetex'] = True\nplt.rcParams.update(params)\n\nPWD = os.getcwd()\nnp.set_printoptions(linewidth=120, precision=5)\n\nfig = plt.figure(figsize=(2, 2))\nfig.patch.set_facecolor('white')\nax0 = fig.add_subplot(1, 1, 1)\n```\n\n\n```python\n0.5236 / np.pi, 1.0472 / np.pi\n```\n\n\n\n\n (0.1666670564058328, 0.3333341128116656)\n\n\n\n\n```python\n3 * 29\n```\n\n\n\n\n 87\n\n\n\n## mdf .pickle\n\n\n```python\n# load_pickle = '/home/zhangji/stokes_flow_master/src/ellipsoidB05_baseFlow_theo.pickle'\n# save_pickle = '/home/zhangji/stokes_flow_master/src/dbg_baseFlow.pickle'\n\nload_pickle = '/home/zhangji/stokes_flow_master/src/ecoB01B05_4tail_baseFlow.pickle'\nsave_pickle = '/home/zhangji/stokes_flow_master/src/dbg_baseFlow.pickle'\n\nwith open(load_pickle, 'rb') as handle:\n tpick = pickle.load(handle)\n\nuw_Base_list = tpick['uw_Base_list']\nfor ui in uw_Base_list:\n print(ui)\n\ntw0 = uw_Base_list[2][5]\ntw1 = (uw_Base_list[4][3] + uw_Base_list[5][4]) / 2\ntw2 = (uw_Base_list[4][4] - uw_Base_list[5][3]) / 2\ntw3 = uw_Base_list[9][5]\ntu0 = uw_Base_list[2][2]\ntu1 = (uw_Base_list[4][0] + uw_Base_list[5][1]) / 2\ntu2 = (uw_Base_list[4][1] - uw_Base_list[5][0]) / 2\ntu3 = uw_Base_list[9][2]\n\nuw_Base_list2 = [np.zeros(6) for _ in range(10)]\nuw_Base_list2[2][5] = tw0\nuw_Base_list2[4][3] = tw1\nuw_Base_list2[5][4] = tw1\nuw_Base_list2[4][4] = tw2\nuw_Base_list2[5][3] = -tw2\nuw_Base_list2[9][5] = tw3\n\nuw_Base_list2[2][2] = tu0\nuw_Base_list2[4][0] = tu1\nuw_Base_list2[5][1] = tu1\nuw_Base_list2[4][1] = tu2\nuw_Base_list2[5][0] = -tu2\nuw_Base_list2[9][2] = tu3\n\nprint()\nfor ui in uw_Base_list2:\n print(ui)\n\ntpick['uw_Base_list'] = uw_Base_list2\nwith open(save_pickle, 'wb') as handle:\n pickle.dump(tpick, handle, protocol=4)\nprint('save to %s' % save_pickle)\n```\n\n [ -0.00032 -0.00148 0.00005 -0.00087 0.99083 0.00000]\n [ -0.00000 0.00000 -0.00000 0.00000 -0.00000 0.00000]\n [ -0.00000 -0.00000 -0.02014 0.00000 -0.00000 0.04047]\n [ -0.00000 -0.00000 -0.00000 0.00000 -0.00000 -0.00000]\n [ -0.00064 -0.00295 -0.00000 -0.00174 0.98166 -0.00000]\n [ 0.00295 -0.00064 0.00000 -0.98166 -0.00174 -0.00000]\n [ 0.00000 0.00000 -0.00000 1.00000 0.00000 0.00000]\n [ -0.00000 -0.00000 -0.00000 -0.00000 1.00000 -0.00000]\n [ -0.00000 0.00000 -0.00000 -0.00000 0.00000 1.00000]\n [ -0.00000 -0.00000 0.00103 0.00000 -0.00000 -0.60277]\n \n [ 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000]\n [ 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000]\n [ 0.00000 0.00000 -0.02014 0.00000 0.00000 0.04047]\n [ 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000]\n [ -0.00064 -0.00295 0.00000 -0.00174 0.98166 0.00000]\n [ 0.00295 -0.00064 0.00000 -0.98166 -0.00174 0.00000]\n [ 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000]\n [ 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000]\n [ 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000]\n [ 0.00000 0.00000 0.00103 0.00000 0.00000 -0.60277]\n save to /home/zhangji/stokes_flow_master/src/dbg_baseFlow.pickle\n\n\n\n```python\n\n```\n\n\n```python\nt_theta, t_phi, t_psi = 0, 0, 0\n# t_theta, t_phi, t_psi = np.pi / 2, 0, np.pi / 2\n# max_t, eval_dt = 0.1, 0.01\n# update_fun, rtol, atol = '1fe', 1e-9, 1e-12\nmax_t, eval_dt = 10, 0.0001\nupdate_fun, rtol, atol = '5bs', 1e-9, 1e-12\nsave_every = 1\nomega_tail = 0\n# table_name2 = 'ecoB01B05_baseFlow'\ntable_name2 = 'dbg_baseFlow'\n# table_name2 = 'ellipsoidB05_baseFlow_theo'\n# table_name2 = 'ellipsoidB05_act_baseFlow'\nABCFlowkwargs = {'ABC_A': 1, \n 'ABC_B': 1, \n 'ABC_C': 1, \n 'name': 'ABCFlowProblem'}\nini_center = np.array((0, 0, 0))\nproblemHandle=jm.ABCFlowProblem\n\n# check rotational symmetric\nrotM0 = Rloc2glb(t_theta, t_phi, t_psi)\nrotM1 = np.array((rotM0[:, 1], rotM0[:, 2], rotM0[:, 0])).T\nrotM2 = np.array((rotM0[:, 2], rotM0[:, 0], rotM0[:, 1])).T\nq0 = Quaternion()\nq0.from_matrix(rotM0)\nq1 = Quaternion()\nq1.from_matrix(rotM1)\nq2 = Quaternion()\nq2.from_matrix(rotM2)\ntheta0, phi0, psi0 = q0.get_thphps()\ntheta1, phi1, psi1 = q1.get_thphps()\ntheta2, phi2, psi2 = q2.get_thphps()\n# display(Math('\\\\theta_{ini} = %.4f, \\\\phi_{ini} = %.4f, \\\\psi_{ini} = %.4f, ' % (theta0, phi0, psi0)))\n# display(Math('\\\\theta_{ini} = %.4f, \\\\phi_{ini} = %.4f, \\\\psi_{ini} = %.4f, ' % (theta1, phi1, psi1)))\n# display(Math('\\\\theta_{ini} = %.4f, \\\\phi_{ini} = %.4f, \\\\psi_{ini} = %.4f, ' % (theta2, phi2, psi2)))\ndisplay(Math('\\\\boldsymbol p_1 = (%.2f, %.2f, %.2f), \\\\boldsymbol p_2 = (%.2f, %.2f, %.2f)' % \\\n (rotM0[0, 2], rotM0[1, 2], rotM0[2, 2], rotM0[0, 0], rotM0[1, 0], rotM0[2, 0], )))\ndisplay(Math('\\\\boldsymbol p_1 = (%.2f, %.2f, %.2f), \\\\boldsymbol p_2 = (%.2f, %.2f, %.2f)' % \\\n (rotM1[0, 2], rotM1[1, 2], rotM1[2, 2], rotM1[0, 0], rotM1[1, 0], rotM1[2, 0], )))\ndisplay(Math('\\\\boldsymbol p_1 = (%.2f, %.2f, %.2f), \\\\boldsymbol p_2 = (%.2f, %.2f, %.2f)' % \\\n (rotM2[0, 2], rotM2[1, 2], rotM2[2, 2], rotM2[0, 0], rotM2[1, 0], rotM2[2, 0], )))\n\ntdata0 = spf_bf.do_GivenFlowObj(theta0, phi0, psi0, max_t, table_name=table_name2, \n update_fun=update_fun, rtol=rtol, atol=atol, eval_dt=eval_dt, \n save_every=save_every, tqdm_fun=tqdm_notebook, \n omega_tail=omega_tail, ini_center=ini_center, \n problemHandle=problemHandle, **ABCFlowkwargs)\n\ntdata1 = spf_bf.do_GivenFlowObj(theta1, phi1, psi1, max_t, table_name=table_name2, \n update_fun=update_fun, rtol=rtol, atol=atol, eval_dt=eval_dt, \n save_every=save_every, tqdm_fun=tqdm_notebook, \n omega_tail=omega_tail, ini_center=ini_center, \n problemHandle=problemHandle, **ABCFlowkwargs)\n\ntdata2 = spf_bf.do_GivenFlowObj(theta2, phi2, psi2, max_t, table_name=table_name2, \n update_fun=update_fun, rtol=rtol, atol=atol, eval_dt=eval_dt, \n save_every=save_every, tqdm_fun=tqdm_notebook, \n omega_tail=omega_tail, ini_center=ini_center, \n problemHandle=problemHandle, **ABCFlowkwargs)\n```\n\n\n$\\displaystyle \\boldsymbol p_1 = (0.00, 0.00, 1.00), \\boldsymbol p_2 = (1.00, 0.00, -0.00)$\n\n\n\n$\\displaystyle \\boldsymbol p_1 = (1.00, 0.00, -0.00), \\boldsymbol p_2 = (-0.00, 1.00, 0.00)$\n\n\n\n$\\displaystyle \\boldsymbol p_1 = (-0.00, 1.00, 0.00), \\boldsymbol p_2 = (0.00, 0.00, 1.00)$\n\n\n\n HBox(children=(HTML(value=''), FloatProgress(value=0.0), HTML(value='')))\n\n\n \n\n\n\n HBox(children=(HTML(value=''), FloatProgress(value=0.0), HTML(value='')))\n\n\n \n\n\n\n HBox(children=(HTML(value=''), FloatProgress(value=0.0), HTML(value='')))\n\n\n \n\n\n\n```python\n%matplotlib inline\nfigsize, dpi = np.array((16, 9)) * 0.5, 100 \nbase_t_min, base_t_max = 0, np.inf\nshow_handle = spf_bf.core_show_thphps_X_t\n\nfor use_data in (tdata0, tdata1, tdata2):\n base_t, base_dt, base_X, base_thphps, base_U, base_W, base_psi_t = use_data\n tidx = (base_t >= base_t_min) * (base_t <= base_t_max)\n spf_bf.show_fun(show_handle, base_t[tidx], base_thphps[tidx], base_psi_t[tidx], base_X[tidx], \n figsize=figsize, dpi=dpi)\n```\n\n\n```python\n%matplotlib inline\nfigsize, dpi = np.array((16, 9)) * 0.5, 100 \nbase_t_min, base_t_max = 0, np.inf\nshow_handle = spf_bf.core_show_P1P2_t\n\nfor use_data in (tdata0, tdata1, tdata2):\n base_t, base_dt, base_X, base_thphps, base_U, base_W, base_psi_t = use_data\n tidx = (base_t >= base_t_min) * (base_t <= base_t_max)\n spf_bf.show_fun(show_handle, base_t[tidx], base_thphps[tidx], base_psi_t[tidx], \n figsize=figsize, dpi=dpi)\n```\n\n\n```python\n\n```\n\n## varify the method of base flow for the ABC flow\ncurrent version fix_x=True, fix_y=True, fix_z=True\n\n\n```python\nini_theta, ini_phi, ini_psi = 0, 0, 0\n# ini_theta, ini_phi, ini_psi = np.random.sample(3) * (1, 2, 2) * np.pi\nini_center = np.array((0, 0, 0))\nini_t, max_t = 0, 0.01\nupdate_fun = '1fe'\nrtol, atol = 1e-9, 1e-12\neval_dt = 0.01\nsave_every = 1\nomega_tail = 0\ntable_name = 'ABC_dbg_baseFlow'\ndbg_DEF, dbg_G, dbg_H, dbg_I = (np.random.sample(4) - 0.5) * 2 * np.pi * (0.01, 1, 1, 1)\nproblem_kwargs = {'ABC_A': 1, \n 'ABC_B': 1, \n 'ABC_C': 1, \n 'ABC_D': dbg_DEF, \n 'ABC_E': dbg_DEF, \n 'ABC_F': dbg_DEF, \n 'ABC_G': dbg_G, \n 'ABC_H': dbg_H, \n 'ABC_I': dbg_I, \n 'name': 'ABC_dbg'}\nproblemHandle = jm.ABCFlowProblem_DEFHIJ\n\nproblem = problemHandle(**problem_kwargs)\nobj_kwargs = spf_bf.do_GivenFlowObj_kwargs(ini_center, ini_theta, ini_phi, ini_psi,\n omega_tail=omega_tail, table_name=table_name,\n name='GivenFlowObj')\nobj = jm.GivenFlowObj(**obj_kwargs)\nobj.set_update_para(fix_x=True, fix_y=True, fix_z=True, update_fun=update_fun,\n rtol=rtol, atol=atol, save_every=save_every, tqdm_fun=tqdm_notebook)\nproblem.add_obj(obj)\n# base_t, base_dt, base_X, base_thphps, base_U, base_W, base_psi_t \\\n# = obj.update_self(t0=ini_t, t1=max_t, eval_dt=eval_dt)\n\nRlog2glb = Rloc2glb(ini_theta, ini_phi, ini_psi)\nUp, Wp = obj.calc_Up_fun(ini_theta, ini_phi, ini_psi, ini_center, Rlog2glb)\nprint('Up', Up)\nprint('Wp', Wp)\n\nts = '\\n' \nts = ts + 'mpirun -n 4 python ../ecoli_ABC_Flow.py -main_fun_ABC 1 -sm lg_rs -legendre_m 3 -legendre_k 2 -epsilon 3.000000 -rh11 0.100000 -rh12 0.100000 -rh2 0.030000 -ch 1.000000 -nth 12 -eh 0 -ph 0.666667 -hfct 1.000000 -n_tail 1 -with_cover 2 -left_hand 0 -rs1 0.5 -rs2 0.5 -ds 0.05 -es 0 -with_T_geo 0 -dist_hs 0.500000 -ksp_max_it 100 -plot_geo 0 -ffweight 2.000000 -f ABC_dbg '\nts = ts + ' -ini_theta %.4f -ini_phi %.4f -ini_psi %.4f ' % (ini_theta, ini_phi, ini_psi)\nts = ts + ''.join([' -%s %.6f' % (i0, problem_kwargs[i0]) for i0 in problem_kwargs.keys() if 'ABC' in i0])\nprint(ts)\n\n```\n\n Up [ -0.95170 1.64168 -0.29683]\n Wp [ 0.00075 -0.00313 0.00052]\n \n mpirun -n 4 python ../ecoli_ABC_Flow.py -main_fun_ABC 1 -sm lg_rs -legendre_m 3 -legendre_k 2 -epsilon 3.000000 -rh11 0.100000 -rh12 0.100000 -rh2 0.030000 -ch 1.000000 -nth 12 -eh 0 -ph 0.666667 -hfct 1.000000 -n_tail 1 -with_cover 2 -left_hand 0 -rs1 0.5 -rs2 0.5 -ds 0.05 -es 0 -with_T_geo 0 -dist_hs 0.500000 -ksp_max_it 100 -plot_geo 0 -ffweight 2.000000 -f ABC_dbg -ini_theta 0.0000 -ini_phi 0.0000 -ini_psi 0.0000 -ABC_C 1.000000 -ABC_I 2.654199 -ABC_D -0.003372 -ABC_E -0.003372 -ABC_B 1.000000 -ABC_G -0.068131 -ABC_H 2.442069 -ABC_F -0.003372 -ABC_A 1.000000\n\n\n\n```python\nts = '\\n' \nts = ts + 'mpirun -n 4 python ../ecoli_ABC_Flow.py -main_fun_ABC 1 -sm lg_rs -legendre_m 3 -legendre_k 2 -epsilon 3.000000 -rh11 0.100000 -rh12 0.100000 -rh2 0.030000 -ch 1.000000 -nth 12 -eh 0 -ph 0.666667 -hfct 1.000000 -n_tail 1 -with_cover 2 -left_hand 0 -rs1 0.5 -rs2 0.5 -ds 0.05 -es 0 -with_T_geo 0 -dist_hs 0.500000 -ksp_max_it 100 -plot_geo 0 -ffweight 2.000000 -f ABC_dbg '\nts = ts + ' -ini_theta %.4f -ini_phi %.4f -ini_psi %.4f ' % (ini_theta, ini_phi, ini_psi)\nts = ts + ''.join([' -%s %.6f' % (i0, problem_kwargs[i0]) for i0 in problem_kwargs.keys() if 'ABC' in i0])\nprint(ts)\n\n```\n\n \n mpirun -n 4 python ../ecoli_ABC_Flow.py -main_fun_ABC 1 -sm lg_rs -legendre_m 3 -legendre_k 2 -epsilon 3.000000 -rh11 0.100000 -rh12 0.100000 -rh2 0.030000 -ch 1.000000 -nth 12 -eh 0 -ph 0.666667 -hfct 1.000000 -n_tail 1 -with_cover 2 -left_hand 0 -rs1 0.5 -rs2 0.5 -ds 0.05 -es 0 -with_T_geo 0 -dist_hs 0.500000 -ksp_max_it 100 -plot_geo 0 -ffweight 2.000000 -f ABC_dbg -ini_theta 0.0000 -ini_phi 0.0000 -ini_psi 0.0000 -ABC_C 1.000000 -ABC_I 1.000000 -ABC_D 1.000000 -ABC_E 1.000000 -ABC_B 1.000000 -ABC_G 1.000000 -ABC_H 1.000000 -ABC_F 1.000000 -ABC_A 1.000000\n\nUp [ 1.05890 1.05029 0.99960]\nWp [ -0.81586 0.80493 0.50229]\n\n```python\n# simple shear case, for dbg\n\nini_theta, ini_phi, ini_psi = 0, 0, 0\nini_center = np.array((0, 0, 0))\nini_t, max_t = 0, 0.01\nupdate_fun = '1fe'\nrtol, atol = 1e-9, 1e-12\neval_dt = 0.01\nsave_every = 1\nomega_tail = 0\ntable_name = 'ABC_dbg_baseFlow'\nproblem_kwargs = {'planeShearRate': np.array((1, 0, 0)), }\nproblemHandle = jm.ShearJefferyProblem\n\nproblem = problemHandle(**problem_kwargs)\nobj_kwargs = spf_bf.do_GivenFlowObj_kwargs(ini_center, ini_theta, ini_phi, ini_psi,\n omega_tail=omega_tail, table_name=table_name,\n name='GivenFlowObj')\nobj = jm.GivenFlowObj(**obj_kwargs)\nobj.set_update_para(fix_x=True, fix_y=True, fix_z=True, update_fun=update_fun,\n rtol=rtol, atol=atol, save_every=save_every, tqdm_fun=tqdm_notebook)\nproblem.add_obj(obj)\n# base_t, base_dt, base_X, base_thphps, base_U, base_W, base_psi_t \\\n# = obj.update_self(t0=ini_t, t1=max_t, eval_dt=eval_dt)\n\nRlog2glb = Rloc2glb(ini_theta, ini_phi, ini_psi)\nUp, Wp = obj.calc_Up_fun(ini_theta, ini_phi, ini_psi, ini_center, Rlog2glb)\nprint('Up', Up)\nprint('Wp', Wp)\n\n```\n\n Up [ 0.05767 -0.00362 0.00021]\n Wp [ -0.00508 0.80559 -0.00217]\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "4c54f55051980a95dc42e5e66897470ac1daff2d", "size": 316884, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ABC_Flow/database/Try_sim.ipynb", "max_stars_repo_name": "pcmagic/stokes_flow", "max_stars_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-11-11T05:00:53.000Z", "max_stars_repo_stars_event_max_datetime": "2018-11-11T05:00:53.000Z", "max_issues_repo_path": "ABC_Flow/database/Try_sim.ipynb", "max_issues_repo_name": "pcmagic/stokes_flow", "max_issues_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ABC_Flow/database/Try_sim.ipynb", "max_forks_repo_name": "pcmagic/stokes_flow", "max_forks_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 434.0876712329, "max_line_length": 50340, "alphanum_fraction": 0.9347269032, "converted": true, "num_tokens": 6203, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6654105587468141, "lm_q2_score": 0.640635854839898, "lm_q1q2_score": 0.42628586212225944}} {"text": "# Opencast Mining\n\n## Objective and Prerequisites\n\nHow can a mining company use mathematical optimization to identify which excavation locations to choose in order to maximize the gross margins of extracting ore? Try this modeling example to find out!\n\nThis model is example 14 from the fifth edition of Model Building in Mathematical Programming by H. Paul Williams on pages 269-270 and 324-325.\n\nThis example is at the intermediate level where we assume that you know Python and the Gurobi Python API and that you have some knowledge of building mathematical optimization models.\n\n**Download the Repository**
                                        \nYou can download the repository containing this and other examples by clicking [here](https://github.com/Gurobi/modeling-examples/archive/master.zip). \n\n**Gurobi License**
                                        \nIn order to run this Jupyter Notebook properly, you must have a Gurobi license. If you do not have one, you can request an [evaluation license](https://www.gurobi.com/downloads/request-an-evaluation-license/?utm_source=3PW&utm_medium=OT&utm_campaign=WW-MU-MUI-OR-O_LEA-PR_NO-Q3_FY20_WW_JPME_OPENCAST_MINING_COM_EVAL_GitHub&utm_term=Opencast_Mining&utm_content=C_JPM) as a *commercial user*, or download a [free license](https://www.gurobi.com/academia/academic-program-and-licenses/?utm_source=3PW&utm_medium=OT&utm_campaign=WW-MU-EDU-OR-O_LEA-PR_NO-Q3_FY20_WW_JPME_OPENCAST_MINING_ACADEMIC_EVAL_GitHub&utm_term=Opencast_Mining&utm_content=C_JPM) as an *academic user*.\n\n## Problem Description\n\nA company has obtained permission to conduct opencast mining within a square plot 200 ft $\\times$ 200 ft. The angle of slip of the soil is such that it is not possible for the sides of the excavation to be steeper than 45 degrees. The company has obtained estimates for the value of the ore in various places at various depths. Bearing in mind the restrictions imposed by the angle of slip, the company decides to consider the problem as one of the extracting of rectangular blocks. Each block has horizontal dimensions 50 ft $\\times$ 50 ft and a vertical dimension of 25 ft. If the blocks are chosen to lie above one another then it is only possible to excavate blocks forming an upturned pyramid. \nThe three dimensional representation below shows four levels of excavation. We have numbered, in black, each block at each level, and the number in red represents the block underneath the four blocks of the level. For example, block 17 of level 2 lies underneath the blocks 1,2,5,and 6 of level 1.\n\n\nThe profit for the extraction of ore at each block has been estimated. The goal is to find an ore extraction plan that maximizes total profit.\n\n## Model Formulation\n\n### Sets and Indices\n\n$b,b2 \\in \\text{Blocks}=\\{1,...,30 \\}$.\n\n### Parameters\n\n$\\text{profit}_{b} \\in \\mathbb{R}^+$: Profit from extracting ore from block $b$.\n\n$(b,b2) \\in Arcs = Blocks \\times Blocks$: This parameter represent the arcs in the series-parallel graph describing the rules of extraction. The arc $(b,b2)$ in the adjacency matrix of this series-parallel graph has a value of 1 if block b2 is one of the four blocks above block b, and 0 otherwise. For example, arc $(29,24)$ represents that block 24 is one of the four blocks above block 29.\n\n### Decision Variables\n\n$\\text{extract}_{b} \\in \\{0,1\\}$: This binary variable is equal 1, if block $b$ is selected, and 0 otherwise.\n\n### Constraints\n\n**Extraction**: If a block is extracted, then the four blocks above it must also be extracted..\n\n\\begin{equation}\n\\text{extract}_{b2} \\geq \\text{extract}_{b} \\quad \\forall (b,b2) \\in \\text{Arcs}\n\\end{equation}\n\n### Objective Function\n\n**Profits**: Maximize profits from the extraction of ore.\n\n\\begin{equation}\n\\text{Maximize} \\quad \\sum_{b \\in Blocks} \\text{profit}_{b}*\\text{extract}_{b}\n\\end{equation}\n\n## Python Implementation\n\nWe import the Gurobi Python Module.\n\n\n```python\nimport numpy as np\nimport pandas as pd\n\nimport gurobipy as gp\nfrom gurobipy import GRB\n\n# tested with Python 3.7.0 & Gurobi 9.0\n```\n\n## Input data\n\nWe define all the input data for the model and other Python libraries.\n\n\n```python\n# Create a dictionary to capture the profit of the extracting of ore at each block.\n\nblocks, profit = gp.multidict({\n ('1'): 0,\n ('2'): 0,\n ('3'): 0,\n ('4'): -1500,\n ('5'): 0,\n ('6'): 1000,\n ('7'): 0,\n ('8'): -1500,\n ('9'): -1000,\n ('10'): -1000,\n ('11'): -1500,\n ('12'): -2000,\n ('13'): -1500,\n ('14'): -1500,\n ('15'): -2000,\n ('16'): -2500,\n ('17'): 2000,\n ('18'): 2000,\n ('19'): -2000,\n ('20'): 0,\n ('21'): 0,\n ('22'): -4000,\n ('23'): -2000,\n ('24'): -2000,\n ('25'): -5000,\n ('26'): 16000,\n ('27'): 4000,\n ('28'): 2000,\n ('29'): 0,\n ('30'): 2000\n})\n\n# Create a dictionary for the adjacency matrix of the series-parallel graph.\n\narcs, value = gp.multidict({\n ('30','26'): 1,\n ('30','27'): 1,\n ('30','28'): 1,\n ('30','29'): 1,\n ('29','21'): 1,\n ('29','22'): 1,\n ('29','24'): 1,\n ('29','25'): 1,\n ('28','20'): 1,\n ('28','21'): 1,\n ('28','23'): 1,\n ('28','24'): 1,\n ('27','18'): 1,\n ('27','19'): 1,\n ('27','21'): 1,\n ('27','22'): 1,\n ('26','17'): 1,\n ('26','18'): 1,\n ('26','20'): 1,\n ('26','21'): 1,\n ('25','11'): 1,\n ('25','12'): 1,\n ('25','15'): 1,\n ('25','16'): 1,\n ('24','10'): 1,\n ('24','11'): 1,\n ('24','14'): 1,\n ('24','15'): 1,\n ('23','9'): 1,\n ('23','10'): 1,\n ('23','13'): 1,\n ('23','14'): 1,\n ('22','7'): 1,\n ('22','8'): 1,\n ('22','11'): 1,\n ('22','12'): 1,\n ('21','6'): 1,\n ('21','7'): 1,\n ('21','10'): 1,\n ('21','11'): 1,\n ('20','5'): 1,\n ('20','6'): 1,\n ('20','9'): 1,\n ('20','10'): 1,\n ('19','3'): 1,\n ('19','4'): 1,\n ('19','7'): 1,\n ('19','8'): 1,\n ('18','2'): 1,\n ('18','3'): 1,\n ('18','6'): 1,\n ('18','7'): 1,\n ('17','1'): 1,\n ('17','2'): 1,\n ('17','5'): 1,\n ('17','6'): 1\n})\n```\n\n## Model Deployment\n\nWe create a model and the variables. These binary decision variables define which block to extract ore from.\n\nNotice that the matrix of coefficients of the constraints is totally unimodular, therefore the decision variables can be defined in the interval $[0,1]$ and the problem can be solved as a linear programming problem.\n\n\n```python\nmodel = gp.Model('opencastMining')\n\n# Decision variable to extract ore from block\nextract = model.addVars(blocks, ub=1, vtype=GRB.CONTINUOUS, name=\"extract\" )\n```\n\n Using license file c:\\gurobi\\gurobi.lic\n\n\nThe following constraints ensure that if a block is extracted, then the four blocks above it must also be extracted.\n\n\n```python\n# Extraction constraints\n\nextractionConstrs = model.addConstrs( (extract[b] <= extract[b2] for b,b2 in arcs), name='extractionConstrs')\n```\n\nWe want to maximize the profits from the extraction of ore.\n\n\n```python\n# Objective function\n\nextractionProfit = gp.quicksum(profit[b]*extract[b] for b in blocks )\n\nmodel.setObjective(extractionProfit, GRB.MAXIMIZE)\n```\n\n\n```python\n# Verify model formulation\n\nmodel.write('opencastMining.lp')\n\n# Run optimization engine\n\nmodel.optimize()\n```\n\n Gurobi Optimizer version 9.1.0 build v9.1.0rc0 (win64)\n Thread count: 4 physical cores, 8 logical processors, using up to 8 threads\n Optimize a model with 56 rows, 30 columns and 112 nonzeros\n Model fingerprint: 0x83427493\n Coefficient statistics:\n Matrix range [1e+00, 1e+00]\n Objective range [1e+03, 2e+04]\n Bounds range [1e+00, 1e+00]\n RHS range [0e+00, 0e+00]\n Presolve removed 22 rows and 12 columns\n Presolve time: 0.01s\n Presolved: 34 rows, 18 columns, 68 nonzeros\n \n Iteration Objective Primal Inf. Dual Inf. Time\n 0 2.3014500e+04 1.100300e+01 0.000000e+00 0s\n 9 1.7500000e+04 0.000000e+00 0.000000e+00 0s\n \n Solved in 9 iterations and 0.01 seconds\n Optimal objective 1.750000000e+04\n\n\n## Analysis\n\nThe total profit generated from the optimal ore extraction plan is $\\$17,500.00$. \nThe block to extract ore from and its associated profit or loss are shown in the following table.\n\n\n```python\n# Output reports\n\ncount = 0\nextraction_plan = pd.DataFrame(columns=[\"Block\", \"Profit/Loss\"])\nfor b in blocks:\n if(extract[b].x > 0.5):\n count += 1\n extraction_plan = extraction_plan.append({\"Block\": b, \"Profit/Loss\": '${:,.2f}'.format(profit[b]*round(extract[b].x)) }, ignore_index=True) \nextraction_plan.index=[''] * count\nextraction_plan\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        BlockProfit/Loss
                                        1$0.00
                                        2$0.00
                                        3$0.00
                                        5$0.00
                                        6$1,000.00
                                        7$0.00
                                        9$-1,000.00
                                        10$-1,000.00
                                        11$-1,500.00
                                        17$2,000.00
                                        18$2,000.00
                                        20$0.00
                                        21$0.00
                                        26$16,000.00
                                        \n
                                        \n\n\n\n## References\n\nH. Paul Williams, Model Building in Mathematical Programming, fifth edition.\n\nCopyright \u00a9 2020 Gurobi Optimization, LLC\n\n\n```python\n\n```\n", "meta": {"hexsha": "7cdfeeed2a4aae56dc7e17d321073f8916362ff3", "size": 16151, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "opencast_mining/opencast_mining.ipynb", "max_stars_repo_name": "anupamsharmaberkeley/Gurobi_Optimization", "max_stars_repo_head_hexsha": "701200b5bfd9bf46036675f5b157b3d8e3728ff9", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 153, "max_stars_repo_stars_event_min_datetime": "2019-07-11T15:08:37.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-25T10:12:54.000Z", "max_issues_repo_path": "opencast_mining/opencast_mining.ipynb", "max_issues_repo_name": "anupamsharmaberkeley/Gurobi_Optimization", "max_issues_repo_head_hexsha": "701200b5bfd9bf46036675f5b157b3d8e3728ff9", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 7, "max_issues_repo_issues_event_min_datetime": "2020-10-29T12:34:13.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-28T14:16:43.000Z", "max_forks_repo_path": "opencast_mining/opencast_mining.ipynb", "max_forks_repo_name": "anupamsharmaberkeley/Gurobi_Optimization", "max_forks_repo_head_hexsha": "701200b5bfd9bf46036675f5b157b3d8e3728ff9", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 91, "max_forks_repo_forks_event_min_datetime": "2019-11-11T17:04:54.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-30T21:34:20.000Z", "avg_line_length": 31.544921875, "max_line_length": 710, "alphanum_fraction": 0.4752646895, "converted": true, "num_tokens": 3182, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6654105720171531, "lm_q2_score": 0.6406358411176238, "lm_q1q2_score": 0.4262858614927681}} {"text": "```python\n%pylab inline\nimport pandas as pd\n```\n\n Populating the interactive namespace from numpy and matplotlib\n\n\n\n```python\ndfy = pd.read_csv('GPU_logYOLO.txt')\ndft = pd.read_csv('GPU_logYOLO-tiny.txt')\n```\n\n\n```python\n# figure( figsize= (8,4))\ndates = list(df['Date'][::40])\nplot(df['Date'],dfy['GPU Chip Power Draw [W]'],'b',label='YOLOv3 608')\nplot(dft['GPU Chip Power Draw [W]'],'r',label='YOLOv3 tiny')\nxticks(arange(0,len(df['Date']),40),arange(0,len(df['Date']),40) ,rotation=45)\n# grid()\nylabel('$p_i [W]$')\nxlabel('$t [s]$')\ntitle('GPU power consumption')\nlegend()\n```\n\n\n```python\ndef w2kwh(df):\n d = df['GPU Chip Power Draw [W]']\n w=d.sum()/len(d)\n kwh = w/1e3\n return kwh\n\nw2kwh(dfy), w2kwh(dft)\n```\n\n\n\n\n (0.01351986301369863, 0.007358333333333334)\n\n\n\n\n```python\ndef w2kwy(df):\n d = df['GPU Chip Power Draw [W]']\n w=d.sum()/len(d)\n kwy = w/1e3*(24*365)\n return kwy\nw2kwy(dft)\n```\n\n\n\n\n 64.459\n\n\n\n\\begin{equation}\nkWh = \\frac{W*t}{1000}\\\\\nW = \\frac{1}{1000*3600}=kWh\n\\end{equation}\n\n\n```python\nfrom PIL import Image\n```\n\n\n```python\nimg = Image.open('augmentation/IMG_20190702_165555.jpg')\n\nangles = arange(0,181,30)\nfigure(figsize(20,20))\n\nfor i, angle in enumerate(angles):\n subplot(1,len(angles),i+1)\n rot = img.rotate(angle)\n imshow(rot)\n rot.save(f'augmentation/rotated/img{angle}.jpg')\n xticks([]), yticks([])\n title(f'{angle}$\\deg$')\n\n```\n\n\n```python\nimgs = os.listdir('augmentation/rotated/')\nimgs = [x for x in imgs if x.startswith('box')]\nfor i,v in enumerate(imgs):\n subplot(2,len(imgs),i+1)\n img = Image.open(f'augmentation/rotated/{v}')\n imshow(img)\n xticks([]), yticks([])\n angle = v.split('_')[1].split('.')[0][3:]\n title(f'{angle}$\\deg$')\n```\n\n\n```python\nimg = cv2.imread('augmentation/IMG_20190702_165555.jpg')\n\n\nangles = arange(0,181,30)\nfigure(figsize(20,20))\n\nfor i, angle in enumerate(angles):\n subplot(1,len(angles),i+1)\n imshow(loc)\n # loc.save(f'augmentation/rotated/img{angle}.jpg')\n xticks([]), yticks([])\n title(f'{angle}$\\deg$')\n\n```\n\n\n\n\n array(['box_img0.jpg', 'box_img120.jpg', 'box_img150.jpg',\n 'box_img180.jpg', 'box_img30.jpg', 'box_img60.jpg',\n 'box_img90.jpg'], dtype=' a, y => b) diff(g, y)(x => a, y => b)] #gradient of g\n(\"\u2207f at max:\", dg(max[1], max[2]), \"\u2207g at max:\", df), (\"\u2207f at min:\", dg(min[1], min[2]), \"\u2207g at min:\", df)\n```\n\n\n\n\n ((\"\u2207f at max:\", Sym[6/5 8/5], \"\u2207g at max:\", Sym[3 4]), (\"\u2207f at min:\", Sym[-6/5 -8/5], \"\u2207g at min:\", Sym[3 4]))\n\n\n\n### Answer\nHere we have computed the gradient of f and g and evaluated at our max and min from above. We can see that in the case of the the max the multiplier is 5/2 and in the case of the min the multiplier is -5/2. \u03bb is the ratio \u2207f / \u2207g. However we could also interpret it in another way. \u03bb tells us how a slight change in our constraint level set will effect f(x). \n\n\n```julia\nusing Plots\nplotlyjs()\nf(x) = 3x[1] + 4x[2]\ng(x) = x[1]^2 + x[2]^2 - 1\nsurface(-2:0.1:2, -2:0.1:2, (x\u2081,x\u2082) -> f([x\u2081,x\u2082]))\nsurface!(-2:0.1:2, -2:0.1:2, (x\u2081,x\u2082) -> g([x\u2081,x\u2082]))\n\u03b8s = LinRange(0, 2\u03c0, 100) \npath3d!(cos.(\u03b8s), sin.(\u03b8s), fill(0.1,length(\u03b8s)), linewidth = 8)\n```\n\n\n\n\n```julia\nusing Pkg\nPkg.instantiate()\n```\n\n \u001b[32m\u001b[1m Installed\u001b[22m\u001b[39m LaTeXStrings \u2500 v1.2.0\n\n\n## Problem 2\n\nWe would like to examine the details of matrix differentiation in this problem. Let's say that $\\mathbf{x} = [x_1, x_2, x_3]$ and \n$$A = \\left[\\begin{matrix} 1 & 2 & 0 \\\\ 2 & 1 & 0 \\\\ 0 & 1 & 1 \\end{matrix}\\right]$$\n\n(a) Given an $\\mathbb{R}^3$-valued function $\\mathbf{a}$ of a vector $\\mathbf{x}$ in $\\mathbb{R}^3$, recall that the derivative of $\\mathbf{a}$ with respect to $\\mathbf{x}$ is \n$$\n\\frac{\\partial \\mathbf{a}}{\\partial \\mathbf{x}} = \n\\left(\\begin{matrix}\n\\frac{\\partial a_1}{\\partial x_1} & \\frac{\\partial a_1}{\\partial x_2} & \\frac{\\partial a_1}{\\partial x_3} \\\\\n\\frac{\\partial a_2}{\\partial x_1} & \\frac{\\partial a_2}{\\partial x_2} & \\frac{\\partial a_2}{\\partial x_3} \\\\\n\\frac{\\partial a_3}{\\partial x_1} & \\frac{\\partial a_3}{\\partial x_2} & \\frac{\\partial a_3}{\\partial x_3}\n\\end{matrix}\\right)\n$$\nwhere $\\mathbf{a} = [a_1, a_2, a_3]$, $\\mathbf{x} = [x_1, x_2, x_3]$.\n\nUsing this definition, find $\\frac{\\partial (A\\mathbf{x})}{\\partial \\mathbf{x}}$ and show that it is equal to the matrix $A$.\n\n### Answer:\nWe can see that : $A x=\\left[\\begin{array}{l}1 x_{1}+2 x_{2}+0 x_{3} \\\\ 2 x_{1}+1 x_{2}+0 x_{3} \\\\ 0 x_{1}+1 x_{2}+1 x_{3}\\end{array}\\right]$ so by the above definition we can see that: $$\n\\frac{\\partial(A x)}{\\partial x}=\\left[\\begin{array}{lll}\n1 & 2 & 0 \\\\\n2 & 1 & 0 \\\\\n0 & 1 & 1\n\\end{array}\\right] = A\n$$\n\n\n(b) Similarly, the derivative of a real-valued function with respect to a vector in $\\mathbb{R}^3$ is\n$$\\frac{\\partial a}{\\partial \\mathbf{x}} = \\left[\\begin{array}{ccc} \\frac{\\partial a}{\\partial x_1} & \\frac{\\partial a}{\\partial x_2} & \\frac{\\partial a}{\\partial x_3}\\end{array}\\right],$$ where $\\mathbf{x} = (x_1, x_2, x_3)$\n\nUsing this definition, find $\\mathbf{x}'A\\mathbf{x}$, then find $\\frac{\\partial \\mathbf{x}'A\\mathbf{x}}{\\partial \\mathbf{x}}$, and show that it is equal to $\\mathbf{x}'(A + A')$.\n\n### Answer:\n\nUsing $Ax$ from above we have $x'Ax=x'\\left[\\begin{array}{l}1 x_{1}+2 x_{2}+0 x_{3} \\\\ 2 x_{1}+1 x_{2}+0 x_{3} \\\\ 0 x_{1}+1 x_{2}+1 x_{3}\\end{array}\\right] = \\left[x_{1}^{2}+4 x_{1} x_{2}+x_{2}^{2}+x_{2} x_{3}+x_{3}^{2}\\right]$\nFrom the fact given above we now have $$\n\\frac{\\partial x^{\\prime} A x}{\\partial x}=[\\ 2 x_{1}+4 x_{2} \\quad 4 x_{1}+2 x_{2}+x_{3} \\quad x_{2}+2 x_{3}] =\n\\left[\\begin{array}{lll}\nx_{1} & x_{2} & x_{3}\n\\end{array}\\right]\\left(\\left[\\begin{array}{lll}\n1 & 2 & 0 \\\\\n2 & 1 & 0 \\\\\n0 & 1 & 1\n\\end{array}\\right]+\\left[\\begin{array}{lll}\n1 & 2 & 0 \\\\\n2 & 1 & 1 \\\\\n0 & 0 & 1\n\\end{array}\\right]\\right)=x^{\\prime}\\left(A+A^{\\prime}\\right)\n$$\n\n## Problem 3\n\nIn class, we differentiated the function $x \\mapsto \\exp(\\cos^2(x))$ at the point $x = \\pi/4$ \"autodiff-style\", meaning that we substituted as soon as possible in our function evaluations and derivative calculations, so that we could have nothing but functions and numbers the whole way through. In other words, we avoided having to represent any symbolic expressions in the computation. This is what it looks like drawn out in a diagram, with derivative values in purple and function values in green:\n\n\n\nIn this problem, we're going to do the same thing but with matrix derivatives in place of the single-variable derivatives. \n\nConsider the function $f(\\mathbf{x}) = [\\sigma(x_1), \\sigma(x_2), \\sigma(x_3)]$, where $\\sigma(x) = \\frac{1}{1 + \\exp(-x)}$. Let $A = \\left[\\begin{matrix} 1 & 2 & 0 \\\\ 3 & -2 & 1 \\\\ 0 & 3 & 2 \\end{matrix}\\right]$ and $B = \\left[\\begin{matrix} 4 & -1 & 2 \\\\ 0 & 0 & 2 \\\\ 3 & 0 & 0 \\end{matrix}\\right]$. Differentiate the function $\\mathbf{x} \\mapsto Bf(A\\mathbf{x})$ with respect to $\\mathbf{x}$ at the point $\\mathbf{x} = [-1, 0, 2]$, using a diagram as similar as possible to the one above. Actually, it should be exactly the same, [mutatis mutandis](https://en.wikipedia.org/wiki/Mutatis_mutandis). \n\nNotes: \n1. The function we're differentiating here is a composition of the functions \"multiply by $B$\", f, and \"multiply by $A$\".\n2. To make life easier, feel free to take the equation $\\frac{d\\sigma}{dx} = \\frac{e^{- x}}{\\left(1 + e^{- x}\\right)^{2}}$ as given.\n3. This function is not only related to neural networks, it **is** a neural network. Differentiating (specifically using this autodifferentiation technique) is how we train neural networks.\n4. Also, feel free to evaluate each expression numerically or exactly, as you prefer.\n5. Here's some code to help get you started:\n\n### Please note that the 3x3 matrices next to the 3x1 columns vectors on top of the image are not meant to be multiplied. In each case the one on the left is the derivative and the one on the right is each function evaluated at Ax, f(Ax) and Bf(Ax) respectively. \n\n\n\n\n```julia\nusing LinearAlgebra\nA = [1 2 0; 3 -2 1; 0 3 2]\nB = [4 -1 2; 0 0 2; 3 0 0]\nx = [-1, 0, 2] \n#\u03c3(x) = 1/(1 + sympy.exp(-x)) # exact approach\n\u03c3(x) = 1/(1 + exp(-x)) # numerical approach\nd\u03c3(x) = exp(-x) / (1 + exp(-x))^2\nAx = A*x\nB*diagm([d\u03c3(Ax[1]), d\u03c3(Ax[2]), d\u03c3(Ax[3])])*A\n```\n\n\n\n\n 3\u00d73 Array{Float64,2}:\n 0.196612 2.0721 -0.125961\n 0.0 0.105976 0.0706508\n 0.589836 1.17967 0.0\n\n\n\n### Answer\nIn this problem we compute the derivative \"auto-diff\" style by, at each step computing the the value of the function and the derivative at the output from the previous step. At the end we multiply each of the three derivatives to get the final answer as shown above. \n\n## Problem 4\n\nGiven a square matrix $A$, its matrix exponential $\\exp(A)$ is defined to be $I + A + \\frac12A^2 + \\frac16A^3 + \\cdots$. In other languages, the function `exp` exponentiates a matrix entry-by-entry, while the matrix exponential is a different function, like `expm`. In Julia, `exp` computes the matrix exponential, while `exp.` is the component-wise version: \n\n\n```julia\nA = [1 2; 3 4]\n```\n\n\n\n\n 2\u00d72 Array{Int64,2}:\n 1 2\n 3 4\n\n\n\n(a) Numerically investigate the claim that the derivative of $\\exp(tA)$ with respect to $t \\in \\mathbb{R}$ is $A\\exp(tA)$. Or should it be $\\exp(tA)A$?\n\nNote 1: the derivative of a matix-valued function of a real variable is defined entry-by-entry. \n\nNote 2: \"Numerically investigate\" just means \"make up a few small examples and see if the results you get for those examples make sense\". You can use difference quotients to approximate derivatives. Even though that method loses a lot of precision, it's fine because you're just doing an approximate check anyway. \n\n\n```julia\nfunction is_this_correct(matrix, t, \u03f5)\n println(\"diff quotient:\", (exp((t + \u03f5)*matrix) - exp(t*matrix)) / \u03f5), \n println(\"A*exp(tA): \", matrix*exp(t*matrix)) \n println(\"exp(tA)*A: \", exp(t*matrix) * matrix)\nend\nprintln(is_this_correct(A, 1, 10^-6))\nprintln(is_this_correct(A, 0, 10^-6))\nprintln(is_this_correct(A, -1, 10^-6))\n```\n\n diff quotient:[276.17939234403354 402.88525265452773; 604.3278789604753 880.5072713187197]\n A*exp(tA): [276.17864989971486 402.8841706654232; 604.3262559981348 880.5049058978498]\n exp(tA)*A: [276.1786498997149 402.8841706654232; 604.3262559981348 880.5049058978498]\n nothing\n diff quotient:[1.000003500228885 2.000005000009; 3.0000075000135 4.000010999982704]\n A*exp(tA): [1.0 2.0; 3.0 4.0]\n exp(tA)*A: [1.0 2.0; 3.0 4.0]\n nothing\n diff quotient:[-0.40519235122715697 0.19675712203959242; 0.2951356834479667 -0.11005666772367913]\n A*exp(tA): [-0.4051924439546264 0.1967571339831402; 0.29513570097471 -0.11005674297991597]\n exp(tA)*A: [-0.4051924439546264 0.19675713398314; 0.2951357009747102 -0.11005674297991597]\n nothing\n\n\n### Answer\nFor this question we approximate the derivative of exp(tA) by using difference quotients for various values of t and epsilon of 10^-6. We compare this value to A*exp(tA) and exp(tA)*A. A*exp(tA) and exp(tA)*A are the same up to a very high degree of precision and our difference quotient is also very close. It seems that both A*exp(tA) and exp(tA)*A are the correct derivative. \n\n(b) Numerically investigate the claim that if $A$ is diagonalizable as $V D V^{-1}$, then the matrix exponential can be calcluated as $V \\exp.(D) V^{-1}$. In other words, the idea is that you can matrix exponentiate by diagonalizing and applying the exponential function entry-wise to the diagonal matrix.\n\n\n```julia\nB = [1 2; 2 1]\nfunction is_this_correct2(matrix)\n D, V = eigen(matrix)\n println(V*exp.(diagm(D))*inv(V))\n println(exp(matrix))\nend\nis_this_correct2(A)\nis_this_correct2(B)\n```\n\n [52.82644912441753 75.25106032243072; 112.61934260593233 163.21631012349727]\n [51.968956198705044 74.73656456700328; 112.10484685050491 164.07380304920997]\n [9.226708182179554 9.85882874100811; 9.858828741008113 11.226708182179555]\n [10.226708182179554 9.85882874100811; 9.85882874100811 10.226708182179554]\n\n\n### Answer\nFor this question we have a function that prints two matrices. One evaluated as V*exp.(D)*V inverse and the other as exp() of our matrix. We run this function on two diagnolizable matrices, one that is symmetric and one that is not. For both we see that our resulting matrices are close but off by enough to raise concern. It is possible that this is do to machine error being amplified during computation. However we observe that if we replace exp.(D) with exp(D) our results are within an extremely small margin of error. \n\n\n```julia\nB = [1 2; 2 1]\nfunction is_this_correct2(matrix)\n D, V = eigen(matrix)\n println(V*exp(diagm(D))*inv(V))\n println(exp(matrix))\nend\nis_this_correct2(A)\nis_this_correct2(B)\n```\n\n [51.96895619870499 74.73656456700319; 112.10484685050479 164.0738030492098]\n [51.968956198705044 74.73656456700328; 112.10484685050491 164.07380304920997]\n [10.226708182179555 9.858828741008113; 9.858828741008113 10.226708182179555]\n [10.226708182179554 9.85882874100811; 9.85882874100811 10.226708182179554]\n\n\n*Solution.* \n\n## Problem 5\n\nIn this problem, we're going to look at a couple virtues of having the `Int8` and `Int64` data types \"wrap around\" from $2^n-1$ to $-2^n$ (for $n = 8$ and $n = 64$, respectively).\n\n(a) Use ordinary, by-hand arithmetic to add the binary numbers `00011011` and `00000101`. \n\n*Hint: this is exactly like decimal addition, with place-value and carrying and all that, but with 2 in place of 10.*\n\n00011011 + 00000101 = 00100000\n\n(b) Now apply the same, by-hand algorithm to add the Int8 numbers `00000011` and `11111001`. Does the algorithm yield the correct result, despite the second number being negative? \n\n00000011 + 11111001 = 11111100 which is 3 - 7 = -4 so it works\n\n(c) Julia, like many languages, includes *bitshift* operators. As the name suggests, these operators shift a value's underlying bits over a specified number of positions:\n\n\n```julia\ntwentyone = parse(Int8, \"21\")\nbitstring(twentyone)\n```\n\n\n\n\n \"00010101\"\n\n\n\n\n```julia\nbitstring(twentyone >>> 1)\n```\n\n\n\n\n \"00001010\"\n\n\n\n\n```julia\nbitstring(twentyone >>> -1)\n```\n\n\n\n\n \"00101010\"\n\n\n\nWe can use the bitshift operator to calculate the midpoint between two integers `lo` and `hi`: \n\n\n\n```julia\nlo = 2\nhi = 14\n(lo + hi) >>> 1\n```\n\n\n\n\n 8\n\n\n\n(d) Explain why the formula `(lo + hi) >>> 1` gives the midpoint between `lo` and `hi`.\n\nThis works because the formula for mid point is (hi + lo) / 2 and moving the bits over 1 to the right is the same as division by 2. This is because for any given bit n, n-1 is half its value\n\n(e) Show that if `n` is equal to `2^62`, this formula gives the correct midpoint between `n` and `2n`, even though the intermediate value `2n` is actually *negative* in the Int64 system. Show also that the more straightforward formula `(n + 2n) \u00f7 2` gives a mathematically incorrect result in this overflow context.\n\n## Problem 6\n\nConsider the following PRNG (which was actually widely used in the early 1970s): we begin with an odd positive integer $a_1$ less than $2^{31}$ and for all $n \\geq 2$, we define $a_n$ to be the remainder when dividing $65539a_{n-1}$ by $2^{31}$.\n\nUse Julia to calculate $9a_{3n+1} - 6 a_{3n+2} + a_{3n+3}$ for the first $10^6$ values of $n$, and show that there are only $\\textit{15}$ unique values in the resulting list (!). Based on this observation, describe what you would see if you plotted many points of the form $(a_{3n+1},a_{3n+2},a_{3n+3})$ in three-dimensional space.\n\n\n```julia\nfunction PRNG1970(oddint, n)\n a = [oddint]\n for i in 1:n\n push!(a, mod(65539*a[end], 2^31))\n end\n a\nend\nfunction f(oddint, n)\n seq = PRNG1970(oddint, n)\n length(Set([9*seq[3*n+1] - 6*seq[3*n+2] + seq[3*n+3] for n in 1:10^3]))\nend\nplot([( \n```\n\n\n\n\n 15\n\n\n\n#### This problem is purely optional:\n---\n\n## Bonus Problem\n\nWe have learned the representation of numbers with the `Int64` and `Float64` format. We will look a bit more into how one can implement calculations at the machine level.\n\n(a) Calculate the sum of the Int8 value with bitstring $00110101$ and the Int8 value with bitstring $b = 00011011$. \n\n(b) Descibe an algorithm for multiplying two `Int8` numbers. *Hint: You will want to multiply digit by digit, similar to hand-multiplication in base 10.*\n\n(c) Now we want to add the two `Int8` numbers $a=01110010$ and $b=00101011$, first, please convert these numbers to decimal numbers and calculate the correct sum.\n\n(d) Using `Julia` and the builtin `Int8()` function, calculate the sum described in (c). What do you find? Can you provide an explaination for this behavior?\n\nNow we will consider two `Float64` numbers. We would like to look at one way of implementing addition of `Float64` numbers, described below. For the sake of simplicity, we shall assume that we are only adding positive numbers and they do not exceed $2^{1024}$. Remember that for every `Float64` number, we have an exponent value $e$ that ranges from $0$ to $2^{11} - 1$, and a mantissa value $f$ that ranges from $0$ to $2^{52} - 1$.\n\nSay we have now a new data type `intinf`, the rules are:\n- Every digit is either 0 or 1, there is a \"decimal point\" and an unlimited number of digits on both sides of the \"decimal point\".\n- If the $n^{th}$ digit above the \"decimal point\" is a $1$, it represents $2^{n-1}$\n- If the $n^{th}$ digit below the \"decimal point\" is a $1$, it represents $2^{-n}$ \nIf you are familiar with radix points in binary numbers, this is exactly that. For example, $110.01_{[intinf]}$ = $1 * 2^2 + 1 * 2^1 + 1 * 2^{-2} = 6.25$. \n\n(e) What is $100.101_{[intinf]}$? What about $10.001_{[intinf]}$? What is $100.101_{[intinf]} + 10.001_{[intinf]}$?\n\n(f) Given the $e$ and $f$ of a `Float64` number, please represent that number in `intinf` format. You will need to use the symbols `>>` and `<<`. `a >> x` means to move the digits of `a` right by `x` spaces while holding the \"decimal point\" still. E.g. `1.0 >> 2 = 0.01`, `0.101 << 2 = 10.1`\n\n(g) To add two `Float64` numbers together, we will first convert them into `intinf` numbers, then add them together, and finally convert the sum back into `Float64`. With this process in mind, please write down the specific steps of adding two `Float64` numbers represented by $a = (e_a, f_a)$ and $b = (e_b, f_b)$. Your final answer should be another `Float64` number. Please give explaination to all your procedures. \nYou are not required to use mathematical representations from start to finish, feel free to explain in words where necessary.\n\nBonus: How would you implement multiplication of two `Float64` numbers? Again, please give sufficient reasoning and/or steps of calculation where necessary. (This is not required and will not affect your grade.)\n\n\n```julia\n2+2\n```\n\n\n\n\n 4\n\n\n\n\n```julia\n\n```\n", "meta": {"hexsha": "3f10227fda1c7c963a4ea1124b1a352948d2519d", "size": 758135, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": ".ipynb_checkpoints/data1010-hw-03-Copy1-checkpoint.ipynb", "max_stars_repo_name": "ncl11/spotify-project", "max_stars_repo_head_hexsha": "e6387554f05feaef865b97f38620226a9c77d3f7", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": ".ipynb_checkpoints/data1010-hw-03-Copy1-checkpoint.ipynb", "max_issues_repo_name": "ncl11/spotify-project", "max_issues_repo_head_hexsha": "e6387554f05feaef865b97f38620226a9c77d3f7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": ".ipynb_checkpoints/data1010-hw-03-Copy1-checkpoint.ipynb", "max_forks_repo_name": "ncl11/spotify-project", "max_forks_repo_head_hexsha": "e6387554f05feaef865b97f38620226a9c77d3f7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 960.8808618504, "max_line_length": 727728, "alphanum_fraction": 0.9449570327, "converted": true, "num_tokens": 6658, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353744, "lm_q2_score": 0.7634837689358857, "lm_q1q2_score": 0.4262735979338165}} {"text": "# Markov Logic Networks to ProbLog\n\nJune 2015 \n[Wannes Meert](mailto:wannes.meert@cs.kuleuven.be), [Guy Van den Broeck](mailto:guy.vandenbroeck@cs.kuleuven.be), [Anton Dries](mailto:anton.dries@cs.kuleuven.be), DTAI Research Group, KU Leuven\n\nAlthough the semantics of ProbLog and MLNs are different, you can transform any first-order MLN into a ProbLog program. This is discussed in:\n\n> Daan Fierens, Guy Van den Broeck, Maurice Bruynooghe, and Luc De Raedt.\n> \"Constraints for probabilistic logic programming.\"\n> In Proceedings of the NIPS probabilistic programming workshop, pp. 1-4. 2012.\n\nThe converse is false. For example, a ProbLog program with inductive definitions such as path or reachability cannot be expressed in first-order logic (although you can always reduce to SAT, propositionally).\n\nFor an overview of the MLN semantics, we refer to:\n\n> Stanley Kok, Parag Singla, Matthew Richardson and Pedro Domingos (2005). \n> \"The Alchemy System for Statistical Relational AI\", Technical Report, \n> Department of Computer Science and Engineering, \n> University of Washington, Seattle, WA. \n> http://www.cs.washington.edu/ai/alchemy.\n\n\n## Using the WFOMC Implementation\n\nThe conversion from MLN to ProbLog is built into the Weighted First-Order Model Counting (WFOMC) application available from https://dtai.cs.kuleuven.be/wfomc.\n\nThe complete chain looks as follows:\n\n $ java -jar wfomc.jar --problog-out friendsmoker.pl friendsmoker.mln\n $ echo -e \"query(friends(guy,wannes)).\\nquery(smokes(guy)).\" >> friendsmoker.pl\n $ ./problog-cli.py friendsmoker.pl\n \tfriends(guy,wannes) : 0.45804886\n\t smokes(guy) : 0.5\n\nThe first step creates a ProbLog theory that is equivalent to the given MLN file. The second stap adds the query statements to the ProbLog file. Finally, ProbLog is called to compute the probabilities of the queries.\n\n## Manual Conversion\n\nIt is also possible to perform the computation manually. This is based on the method mentioned in the cProbLog paper and the transformation to weighted first-order model counting explained in:\n\n> Van den Broeck, Guy, Nima Taghipour, Wannes Meert, Jesse Davis, and Luc De Raedt.\n> \"Lifted probabilistic inference by first-order knowledge compilation.\"\n> In Proceedings of the Twenty-Second international joint conference on Artificial Intelligence-Volume Volume Three, pp. 2178-2185. AAAI Press, 2011.\n\n### MLN to WFOMC\n\nAn MLN\n\n people = {A,B}\n Smokes(people)\n Friends(people,people)\n w Smokes(x) ^ Friends(x,y) => Smokes(y)\n\nis translated to an MLN where only literal formulas have a weight (ignoring the type definitions):\n\n w R(x,y)\n R(x,y) <=> (Smokes(x) ^ Friends(x,y) => Smokes(y)).\n\n\n### WFOMC to ProbLog\n\nThe WFOMC program above can be written to cProbLog:\n\n people(a). people(b).\n 0.5 :: smokes(X) :- people(X).\n 0.5 :: friends(X) :- people(X), people(Y).\n w' :: r(X,Y).\n evidence(r(X,Y) <=> smokes(X) ^ friends(X,Y) => smokes(Y), true) :- people(X), people(Y).\n\nwith $w' = \\frac{e^w}{e^w+1}$, upercase symbols for variables and lowercase symbols for constants and predicates. The types of the predicates are introduced by adding a `people` predicate that lists the domain elements and linking this to the `smokes` and `friends` predicates. The 0.5 probability is a default for MLNs if a predicate is not introduced explicitely ($\\frac{e^0}{e^0+1} = 0.5$). If a sentence `w_s Smokes(x)` is present in the MLN, the 0.5 would be replaced by $w_s' = \\frac{e^{w_s}}{e^{w_s}+1}$ for `smokes`.\n\nThe current ProbLog implementation does not support first-order logic constraints, so this has to be rewritten to ProbLog rules. This can be achieved by turning the constraint into a CNF and adding a ProbLog clauses `ev :- not_c_i` for every clause `c_i` in the CNF. The `not_c_i` is the negation of clause `c_i`. Finally, you set the evidence on `ev` to false.\n\n people(a). people(b).\n 0.5 :: smokes(X) :- people(X).\n 0.5 :: friends(X) :- people(X), people(Y).\n w' :: r(X,Y).\n ev :- people(X), people(Y), \\+f_1(X,Y), \\+friends(X,Y).\n ev :- people(X), people(Y), \\+f_1(X,Y), \\+smokes(X).\n ev :- people(Y), smokes(X), \\+f_1(Y,X).\n ev :- smokes(X), friends(X,Y), f_1(X,Y), \\+smokes(Y).\n evidence(ev,false).\n\n\n## Partition Function\n\nIf you want to compute the partition function of an MLN you can query the probability of the evidence `ev` (do not forget to remove the `evidence(ev,false)` rule. Since the ProbLog probabilities are a normalization of the original weights, you have to denormalize them. This can be done by multiplying the probability of evidence with $1+e^w$ for all grounded sentences that appear in the MLN (or equivalently all groundings of all `r` that appear in ProbLog).\n\n\\begin{equation}\n \\textit{partitionfunction(program)} = Pr(ev)\\cdot \\prod_{w = \\textit{weight(c)} \\land c \\in \\textit{clauses(grounding)}} (1+e^w)\n\\end{equation}\n\n\n## More information\n\nhttps://dtai.cs.kuleuven.be/problog\n", "meta": {"hexsha": "62ec33fa334159bd024dc0c3e1ada2933ac64e97", "size": 6718, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "usecases/mln/MLN.ipynb", "max_stars_repo_name": "HEmile/problog", "max_stars_repo_head_hexsha": "576b6fd305f72b12125111c8d4d62cf8a7bbda0f", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 189, "max_stars_repo_stars_event_min_datetime": "2019-05-27T08:20:10.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-28T09:29:22.000Z", "max_issues_repo_path": "usecases/mln/MLN.ipynb", "max_issues_repo_name": "HEmile/problog", "max_issues_repo_head_hexsha": "576b6fd305f72b12125111c8d4d62cf8a7bbda0f", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 60, "max_issues_repo_issues_event_min_datetime": "2019-06-11T15:07:48.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-25T02:31:23.000Z", "max_forks_repo_path": "usecases/mln/MLN.ipynb", "max_forks_repo_name": "HEmile/problog", "max_forks_repo_head_hexsha": "576b6fd305f72b12125111c8d4d62cf8a7bbda0f", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 33, "max_forks_repo_forks_event_min_datetime": "2019-07-03T13:14:24.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-20T01:07:15.000Z", "avg_line_length": 43.3419354839, "max_line_length": 536, "alphanum_fraction": 0.6137243227, "converted": true, "num_tokens": 1381, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802735722129, "lm_q2_score": 0.6370308082623216, "lm_q1q2_score": 0.4260973413044296}} {"text": "# LSTM Structure and Hidden State\n\nWe know that RNNs are used to maintain a kind of memory by linking the output of one node to the input of the next. In the case of an LSTM, for each piece of data in a sequence (say, for a word in a given sentence),\nthere is a corresponding *hidden state* $h_t$. This hidden state is a function of the pieces of data that an LSTM has seen over time; it contains some weights and, represents both the short term and long term memory components for the data that the LSTM has already seen. \n\nSo, for an LSTM that is looking at words in a sentence, **the hidden state of the LSTM will change based on each new word it sees. And, we can use the hidden state to predict the next word in a sequence** or help identify the type of word in a language model, and lots of other things!\n\n\n## LSTMs in Pytorch\n\nTo create and train an LSTM, you have to know how to structure the inputs, and hidden state of an LSTM. In PyTorch an LSTM can be defined as: `lstm = nn.LSTM(input_size=input_dim, hidden_size=hidden_dim, num_layers=n_layers)`.\n\nIn PyTorch, an LSTM expects all of its inputs to be 3D tensors, with dimensions defined as follows:\n>* `input_dim` = the number of inputs (a dimension of 20 could represent 20 inputs)\n>* `hidden_dim` = the size of the hidden state; this will be the number of outputs that each LSTM cell produces at each time step.\n>* `n_layers ` = the number of hidden LSTM layers to use; this is typically a value between 1 and 3; a value of 1 means that each LSTM cell has one hidden state. This has a default value of 1.\n\n\n \n### Hidden State\n\nOnce an LSTM has been defined with input and hidden dimensions, we can call it and retrieve the output and hidden state at every time step.\n `out, hidden = lstm(input.view(1, 1, -1), (h0, c0))` \n\nThe inputs to an LSTM are **`(input, (h0, c0))`**.\n>* `input` = a Tensor containing the values in an input sequence; this has values: (seq_len, batch, input_size)\n>* `h0` = a Tensor containing the initial hidden state for each element in a batch\n>* `c0` = a Tensor containing the initial cell memory for each element in the batch\n\n`h0` nd `c0` will default to 0, if they are not specified. Their dimensions are: (n_layers, batch, hidden_dim).\n\nThese will become clearer in the example in this notebook. This and the following notebook are modified versions of [this PyTorch LSTM tutorial](https://pytorch.org/tutorials/beginner/nlp/sequence_models_tutorial.html#lstm-s-in-pytorch).\n\nLet's take a simple example and say we want to process a single sentence through an LSTM. If we want to run the sequence model over one sentence \"Giraffes in a field\", our input should look like this `1x4` row vector of individual words:\n\n\\begin{align}\\begin{bmatrix}\n \\text{Giraffes } \n \\text{in } \n \\text{a } \n \\text{field} \n \\end{bmatrix}\\end{align}\n\nIn this case, we know that we have **4 inputs words** and we decide how many outputs to generate at each time step, say we want each LSTM cell to generate **3 hidden state values**. We'll keep the number of layers in our LSTM at the default size of 1.\n\nThe hidden state and cell memory will have dimensions (n_layers, batch, hidden_dim), and in this case that will be (1, 1, 3) for a 1 layer model with one batch/sequence of words to process (this one sentence) and 3 genereated, hidden state values.\n\n\n### Example Code\n\nNext, let's see an example of one LSTM that is designed to look at a sequence of 4 values (numerical values since those are easiest to create and track) and generate 3 values as output. This is what the sentence processing network from above will look like, and you are encouraged to change these input/hidden-state sizes to see the effect on the structure of the LSTM!\n\n\n```python\nimport torch\nimport torch.nn as nn\nimport matplotlib.pyplot as plt\n\n%matplotlib inline\n\ntorch.manual_seed(2) # so that random variables will be consistent and repeatable for testing\n```\n\n\n\n\n \n\n\n\n### Define a simple LSTM\n\n\n**A note on hidden and output dimensions**\n\nThe `hidden_dim` and size of the output will be the same unless you define your own LSTM and change the number of outputs by adding a linear layer at the end of the network, ex. fc = nn.Linear(hidden_dim, output_dim).\n\n\n```python\nfrom torch.autograd import Variable\n\n# define an LSTM with an input dim of 4 and hidden dim of 3\n# this expects to see 4 values as input and generates 3 values as output\ninput_dim = 4\nhidden_dim = 3\nlstm = nn.LSTM(input_size=input_dim, hidden_size=hidden_dim) \n\n# make 5 input sequences of 4 random values each\ninputs_list = [torch.randn(1, input_dim) for _ in range(5)]\nprint('inputs: \\n', inputs_list)\nprint('\\n')\n\n# initialize the hidden state\n# (1 layer, 1 batch_size, 3 outputs)\n# first tensor is the hidden state, h0\n# second tensor initializes the cell memory, c0\nh0 = torch.randn(1, 1, hidden_dim)\nc0 = torch.randn(1, 1, hidden_dim)\n\n\nh0 = Variable(h0)\nc0 = Variable(c0)\n# step through the sequence one element at a time.\nfor i in inputs_list:\n # wrap in Variable \n i = Variable(i)\n \n # after each step, hidden contains the hidden state\n out, hidden = lstm(i.view(1, 1, -1), (h0, c0))\n print('out: \\n', out)\n print('hidden: \\n', hidden)\n\n```\n\n inputs: \n [tensor([[ 0.6048, 0.1659, -0.1222, -0.4764]]), tensor([[-0.3905, 0.7649, 0.0500, 2.2558]]), tensor([[-0.6688, 0.3069, -1.1182, 0.0612]]), tensor([[ 0.7757, 0.9996, -0.2380, -1.7623]]), tensor([[0.4873, 1.4592, 1.4165, 1.0032]])]\n \n \n out: \n tensor([[[ 0.2856, 0.1553, -0.2874]]], grad_fn=)\n hidden: \n (tensor([[[ 0.2856, 0.1553, -0.2874]]], grad_fn=), tensor([[[ 0.4711, 0.6539, -0.5247]]], grad_fn=))\n out: \n tensor([[[ 0.1673, 0.3217, -0.2189]]], grad_fn=)\n hidden: \n (tensor([[[ 0.1673, 0.3217, -0.2189]]], grad_fn=), tensor([[[ 0.3727, 0.5381, -0.2726]]], grad_fn=))\n out: \n tensor([[[ 0.1575, 0.2889, -0.3375]]], grad_fn=)\n hidden: \n (tensor([[[ 0.1575, 0.2889, -0.3375]]], grad_fn=), tensor([[[ 0.2843, 0.5862, -0.5245]]], grad_fn=))\n out: \n tensor([[[ 0.3591, 0.0996, -0.0849]]], grad_fn=)\n hidden: \n (tensor([[[ 0.3591, 0.0996, -0.0849]]], grad_fn=), tensor([[[ 0.4852, 0.5292, -0.2144]]], grad_fn=))\n out: \n tensor([[[ 0.2744, 0.1383, -0.0651]]], grad_fn=)\n hidden: \n (tensor([[[ 0.2744, 0.1383, -0.0651]]], grad_fn=), tensor([[[ 0.3959, 0.4187, -0.0939]]], grad_fn=))\n\n\nYou should see that the output and hidden Tensors are always of length 3, which we specified when we defined the LSTM with `hidden_dim`. \n\n### All at once\n\nA for loop is not very efficient for large sequences of data, so we can also, **process all of these inputs at once.**\n\n1. concatenate all our input sequences into one big tensor, with a defined batch_size\n2. define the shape of our hidden state\n3. get the outputs and the *most recent* hidden state (created after the last word in the sequence has been seen)\n\n\nThe outputs may look slightly different due to our differently initialized hidden state.\n\n\n```python\n# turn inputs into a tensor with 5 rows of data\n# add the extra 2nd dimension (1) for batch_size\ninputs = torch.cat(inputs_list).view(len(inputs_list), 1, -1)\n\n# print out our inputs and their shape\n# you should see (number of sequences, batch size, input_dim)\nprint('inputs size: \\n', inputs.size())\nprint('\\n')\n\nprint('inputs: \\n', inputs)\nprint('\\n')\n\n# initialize the hidden state\nh0 = torch.randn(1, 1, hidden_dim)\nc0 = torch.randn(1, 1, hidden_dim)\n\n# wrap everything in Variable\ninputs = Variable(inputs)\nh0 = Variable(h0)\nc0 = Variable(c0)\n# get the outputs and hidden state\nout, hidden = lstm(inputs, (h0, c0))\n\nprint('out: \\n', out)\nprint('hidden: \\n', hidden)\n```\n\n inputs size: \n torch.Size([5, 1, 4])\n \n \n inputs: \n tensor([[[ 0.6048, 0.1659, -0.1222, -0.4764]],\n \n [[-0.3905, 0.7649, 0.0500, 2.2558]],\n \n [[-0.6688, 0.3069, -1.1182, 0.0612]],\n \n [[ 0.7757, 0.9996, -0.2380, -1.7623]],\n \n [[ 0.4873, 1.4592, 1.4165, 1.0032]]])\n \n \n out: \n tensor([[[-0.3738, -0.3674, 0.1176]],\n \n [[-0.0939, -0.3720, -0.0641]],\n \n [[-0.1766, -0.3323, -0.1559]],\n \n [[-0.2950, -0.1399, 0.0756]],\n \n [[-0.0830, -0.1071, 0.1872]]], grad_fn=)\n hidden: \n (tensor([[[-0.0830, -0.1071, 0.1872]]], grad_fn=), tensor([[[-0.1329, -0.2052, 0.3845]]], grad_fn=))\n\n\n### Next: Hidden State and Gates\n\nThis notebooks shows you the structure of the input and output of an LSTM in PyTorch. Next, you'll learn more about how exactly an LSTM represents long-term and short-term memory in it's hidden state, and you'll reach the next notebook exercise.\n\n#### Part of Speech\n\nIn the notebook that comes later in this lesson, you'll see how to define a model to tag parts of speech (nouns, verbs, determinants), include an LSTM and a Linear layer to define a desired output size, *and* finally train our model to create a distribution of class scores that associates each input word with a part of speech.\n\n\n```python\n\n```\n", "meta": {"hexsha": "07b52360a83a80e21dd350a6388e33aa33942498", "size": 12596, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "2_4_LSTMs/.ipynb_checkpoints/1. 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LSTM Structure-checkpoint.ipynb", "max_forks_repo_name": "adl-aleb/ComputerVision_ND-", "max_forks_repo_head_hexsha": "ae02f79af5ddef683a05fc01ca5983087b1a3197", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 41.7086092715, "max_line_length": 375, "alphanum_fraction": 0.5857415052, "converted": true, "num_tokens": 2852, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370308082623217, "lm_q2_score": 0.6688802603710086, "lm_q1q2_score": 0.4260973328948558}} {"text": "# Development: Ikeda for many ships\nThere are some interesting section data in the manoeuvring database, here it is investigated if this can be used as input to *ScoresII*.\n\n\n```python\n# %load ../../imports.py\n\"\"\"\nThese is the standard setup for the notebooks.\n\"\"\"\n\n%matplotlib inline\n%load_ext autoreload\n%autoreload 2\n\nfrom jupyterthemes import jtplot\njtplot.style(theme='onedork', context='notebook', ticks=True, grid=False)\n\nimport pandas as pd\npd.options.display.max_rows = 999\npd.options.display.max_columns = 999\npd.set_option(\"display.max_columns\", None)\nimport numpy as np\nimport os\nimport matplotlib.pyplot as plt\n#plt.style.use('paper')\n\n#import data\nimport copy\nfrom rolldecay.bis_system import BisSystem\nfrom rolldecay import database\nfrom mdldb.tables import Run\n\nfrom sklearn.pipeline import Pipeline\nfrom rolldecayestimators.transformers import CutTransformer, LowpassFilterDerivatorTransformer, ScaleFactorTransformer, OffsetTransformer\nfrom rolldecayestimators.direct_estimator_cubic import EstimatorQuadraticB, EstimatorCubic\nfrom rolldecayestimators.ikeda_estimator import IkedaQuadraticEstimator\nimport rolldecayestimators.equations as equations\nimport rolldecayestimators.lambdas as lambdas\nfrom rolldecayestimators.substitute_dynamic_symbols import lambdify\nimport rolldecayestimators.symbols as symbols\nimport sympy as sp\n\nfrom sklearn.metrics import r2_score\n\n\n```\n\n\n```python\ndf_sections = pd.read_csv('sections.csv', sep=';', index_col=0)\nmask=df_sections['b20'].notnull()\ndf_sections=df_sections.loc[mask].copy()\ndf_sections.drop(index=['DAE18A','DAE18B'], inplace=True) # Bad project number\ndf_sections['Project No']=df_sections['Project No'].astype(int)\n```\n\n\n```python\ndf_data = pd.read_csv('ship_data.csv', sep=';', index_col=0)\n```\n\n\n```python\ndf_data.head()\n```\n\n\n```python\ndf_sections_data=pd.merge(left=df_sections, right=df_data, how='left', left_on='SHIP', right_on='SHIP', \n suffixes=('','_data'))\ndf_sections_data.set_index('SHIP', inplace=True)\nassert df_sections_data.index.is_unique\n\n```\n\n\n```python\ndf_sections_data.describe()\n```\n\n\n```python\ndf_rolldecay = database.load(rolldecay_table_name='rolldecay_quadratic_b', limit_score=0.99, \n exclude_table_name='rolldecay_exclude')\n```\n\n\n```python\ninteresting_projects = set(df_rolldecay['project_number']) & set(df_sections_data['Project No'])\ninteresting_projects\n```\n\n\n```python\ndf_=df_sections_data.reset_index()\ndf = pd.merge(left=df_, right=df_rolldecay, how='inner', left_on=('Project No','DISP'), \n right_on=('project_number','Volume'), suffixes=('','_model'))\n```\n\n\n```python\ndf.describe()\n```\n\n\n```python\nmask=df['id'].notnull()\ndf = df.loc[mask].copy()\n```\n\n\n```python\ndef assemble_sections(row, lpp):\n \n b=np.zeros(21)\n area=np.zeros(21)\n t=np.zeros(21)\n for i in range(21):\n b_key='b%i' % i\n area_key='area%i' % i\n t_key = 't%i' % i\n b[i]=row[b_key]\n area[i]=row[area_key]\n t[i]=row[t_key]\n \n df_sections = pd.DataFrame()\n df_sections['b']=b\n df_sections['area']=area\n df_sections['t']=t\n df_sections['x']=np.linspace(0,lpp,21)\n df_sections['SHIP']=int(row['SHIP'])\n df_sections['Project No']=row['Project No']\n \n df_sections.dropna(inplace=True)\n return df_sections\n```\n\n\n```python\ndf_sections_data\n```\n\n\n```python\ndf_all_sections = pd.DataFrame()\n\nfor index, row in df_sections.iterrows():\n data = df_sections_data.loc[row.SHIP]\n lpp=data.LPP\n df_ = assemble_sections(row=row, lpp=lpp)\n df_all_sections=df_all_sections.append(df_)\n\n```\n\n\n```python\ndf_all_sections.head()\n```\n\n\n```python\ndef count_variations(df_group):\n df_group=df_group.copy()\n \n for i in range(21):\n b_key='b%i' % i\n area_key='area%i' % i\n t_key = 't%i' % i\n \n keys=[b_key, area_key, t_key]\n \n for key in keys:\n variations = len(df_group[key].unique())\n if not variations==1:\n break\n \n if not variations==1:\n break\n \n return variations\n \n```\n\n\n```python\ndf.groupby(by='Project No').apply(func=count_variations) \n```\n\n\n```python\nmask = ~df.duplicated(['SHIP','loading_condition_id'], keep='first')\ndf_ids = df.loc[mask,['SHIP','loading_condition_id']].copy()\ndf_ids.set_index('SHIP', inplace=True)\nassert df_ids.index.is_unique\n```\n\n\n```python\nloading_condition_ids = df_ids['loading_condition_id']\n```\n\n\n```python\ndf_all_sections_id=pd.merge(left=df_all_sections, right=loading_condition_ids, how='outer', left_on='SHIP', right_index=True)\nif 'Project No' in df_all_sections:\n df_all_sections_id.drop(columns=['Project No'], inplace=True)\ndf_all_sections_id.to_csv('all_sections.csv', sep=';', index=False)\n```\n\n## Run ScoresII for one of the loading conditions\n\n\n```python\nfrom scipy.integrate import simps\ndef calculate_lcb(x, area):\n \"\"\"\n Calculate lcb from AP\n \"\"\"\n return simps(y=area*x,x=x)/np.trapz(y=area,x=x)\n```\n\n\n```python\nrow=df.iloc[0]\nsections = df_all_sections_id.groupby('SHIP').get_group(row.SHIP)\n\nfig,axes=plt.subplots(nrows=3)\nax=axes[0]\nax.set_title('Project: %s loading_condition: %i' % (row.project_number, row.loading_condition_id))\nsections.plot(x='x', y='b', ax=ax);\nax.plot([0,row.LPP], [row.B,row.B], 'r--')\n\nax=axes[1]\nsections.plot(x='x', y='t', ax=ax);\nax.plot([0,row.LPP], [row.TF,row.TF], 'r--')\n\nlcb=calculate_lcb(x=sections['x'], area=sections['area'])\nax=axes[2]\nsections.plot(x='x', y='area', ax=ax);\nax.plot([lcb,lcb],[0,0],'go')\nplt.tight_layout()\n```\n\n\n```python\nlcb-row.lpp/2\n```\n\n\n```python\nrow\n```\n\n\n```python\nfrom pyscores2.indata import Indata\nfrom pyscores2.runScores2 import Calculation\nfrom pyscores2.output import OutputFile\nfrom pyscores2 import TDPError\nfrom rolldecayestimators.ikeda import Ikeda, IkedaR\n```\n\n\n```python\nsections['cScores']=sections['area']/(sections['b']*sections['t'])\nmask=sections['cScores']>1\nsections.loc[mask,'cScores']=1\nsections\n```\n\n\n```python\ndef calculate_dispacement(x, area, **kwargs):\n \"\"\"\n Calculate displacement\n \"\"\"\n return np.trapz(y=area,x=x)\n```\n\n\n```python\nindata = Indata()\n\nindata.cScores=sections['cScores']\nindata.ts=sections['t']\nindata.bs=sections['b']\nindata.zbars=np.zeros_like(sections['b']) # Guessing...\n\nbeam=sections['b'].max()\nindata.lpp=row.lpp\n#indata.displacement=row.Volume\nindata.displacement=calculate_dispacement(**sections)\n\ndraught=(row.TA+row.TF)/2\nindata.draught=draught\ng=9.81\nindata.g=g\nindata.kxx=row.KXX\nindata.kyy=row.KYY\n#indata.lcb=row.LCG\nlcb=calculate_lcb(x=sections['x'], area=sections['area'])\nindata.lcb=lcb-row.lpp/2\n\nindata.lpp=row.lpp\nindata.projectName=str(row.loading_condition_id)\nrho=1000\nindata.rho=rho\nindata.zcg=row.kg-draught\n#indata.waveFrequenciesMin=0.2\n#indata.waveFrequenciesMax=0.5\n#indata.waveFrequenciesIncrement=0.006\nw=row.omega0/np.sqrt(row.scale_factor)\nindata.waveFrequenciesMin=w*0.5\nindata.waveFrequenciesMax=w*2.0\nN=40\nindata.waveFrequenciesIncrement=(indata.waveFrequenciesMax-indata.waveFrequenciesMin)/N\nindata.runOptions[\"IE\"].set_value(1)\n```\n\n\n```python\nindata.lcb\n```\n\n\n```python\nindata.runOptions[\"IG\"].set_value(0)\n```\n\n\n```python\nindata.speedMax\n```\n\n\n```python\nindata.save('test.in')\n```\n\n\n```python\ncalculation = Calculation(outDataDirectory='scores2/result')\ntry:\n calculation.run(indata=indata)\nexcept TDPError:\n print('Dissregarding the TDPError')\n```\n\n\n```python\nindata.lcb\n```\n\n\n```python\noutput_file = OutputFile(filePath=calculation.outDataPath)\n#df = output_file.get_result()\n#\n#fig,ax=plt.subplots()\n#for index, group in df.groupby(by=['speed','wave direction'], sort=False):\n# group.plot(x='frequencies', y='rollAmplitude', style='o-', label=index, ax=ax)\n# \n#ax.grid(True)\n#ax.legend();\n#ax.set_ylabel('Roll');\n```\n\n\n```python\nw,B_W0=output_file.calculate_B_W0()\n```\n\n\n```python\nfi_a = np.deg2rad(10)\nw = row.omega0\nscale_factor=row.scale_factor\nV = row.ship_speed*1.852/3.6/np.sqrt(scale_factor)\nR = 0.01*row.beam/scale_factor\nlBK=row.BKL/scale_factor\nbBK=row.BKB/scale_factor\nikeda = Ikeda.load_scoresII(V=V, w=w, fi_a=fi_a, indata=indata, output_file=output_file, \n scale_factor=scale_factor, lBK=lBK, bBK=bBK)\n\nikeda.R = R\nikeda.calculate_B44()\n```\n\n\n```python\nikeda_r = IkedaR.load_scoresII(V=V, w=w, fi_a=fi_a, indata=indata, output_file=output_file, \n scale_factor=scale_factor, lBK=lBK, bBK=bBK)\n\nikeda_r.calculate_B44()\n```\n\n\n```python\nB_e = lambdas.B_e_lambda(B_1=row['B_1'], B_2=row['B_2'], phi_a=fi_a, \n omega0=row['omega0'])\n\nvolume=row.Volume/(scale_factor**3)\nbeam=row.beam/scale_factor\n\nB_e_hat = lambdas.B_e_hat_lambda(B_e=B_e, Disp=volume, beam=beam, \n g=g, rho=rho)\nB_e_hat\n```\n\n\n```python\ndef calculate(inputs, ikeda):\n\n output = inputs.copy()\n output['B_44_hat'] = ikeda.calculate_B44()\n output['B_W0'] =ikeda.calculate_B_W0()\n output['B_W'] =ikeda.calculate_B_W()\n output['B_F'] =ikeda.calculate_B_F()\n output['B_E'] =ikeda.calculate_B_E()\n output['B_BK'] =ikeda.calculate_B_BK()\n output['B_L'] =ikeda.calculate_B_L()\n output['Bw_div_Bw0'] =ikeda.calculate_Bw_div_Bw0()\n \n return output\n```\n\n\n```python\nscale_factor=row.scale_factor\ninputs = pd.DataFrame()\ninputs['Fn']=np.linspace(0,0.2,100)\ninputs['lpp']=indata.lpp/scale_factor\ninputs['V']=inputs['Fn']*np.sqrt(inputs['lpp']*g)\n\nfi_a = np.deg2rad(10)\nw = row.omega0\n\nR = 0.01*row.beam/scale_factor\nlBK=row.BKL/scale_factor\nbBK=row.BKB/scale_factor\nikeda = Ikeda.load_scoresII(V=inputs['V'], w=w, fi_a=fi_a, indata=indata, output_file=output_file, \n scale_factor=scale_factor, lBK=lBK, bBK=bBK)\nikeda.R=R\n```\n\n\n```python\noutput = calculate(inputs=inputs, ikeda=ikeda)\n\nfig,ax=plt.subplots()\noutput.plot.area(x='Fn', y = ['B_BK','B_F','B_E','B_L','B_W',], ax=ax)\nax.legend()\n#ax.set_ylim(0,0.014)\n#ax.set_title('Original Ikeda compared to model tests');\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "66ec864229b9fda6b7f1ce077770143e926f635b", "size": 17415, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/rolldecay/06_ikeda/01.03_ikeda_many_dev.ipynb", "max_stars_repo_name": "martinlarsalbert/rolldecay", "max_stars_repo_head_hexsha": "bee335c27e6519bd32ed9488751bc13e7f6fa9b3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": 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YES\n2. YES", "lm_q1_score": 0.6688802735722128, "lm_q2_score": 0.6370307875894139, "lm_q1q2_score": 0.42609732747672935}} {"text": "# Introduction to zfit\n\nIn this notebook, we will have a walk through the main components of zfit and their features. Especially the extensive model building part will be discussed separately.\nzfit consists of 5 mostly independent parts. Other libraries can rely on this parts to do plotting or statistical inference, such as hepstats does. Therefore we will discuss two libraries in this tutorial: zfit to build models, data and a loss, minimize it and get a fit result and hepstats, to use the loss we built here and do inference.\n\n\n\n\n## Data\n\nThis component in general plays a minor role in zfit: it is mostly to provide a unified interface for data.\n\nPreprocessing is therefore not part of zfit and should be done beforehand. Python offers many great possibilities to do so (e.g. Pandas).\n\nzfit `Data` can load data from various sources, most notably from Numpy, Pandas DataFrame, TensorFlow Tensor and ROOT (using uproot). It is also possible, for convenience, to convert it directly `to_pandas`. The constructors are named `from_numpy`, `from_root` etc.\n\n\n```python\nimport zfit\nfrom zfit import z\nimport tensorflow as tf\nimport numpy as np\nimport matplotlib.pyplot as plt\n```\n\nA `Data` needs not only the data itself but also the observables: the human readable string identifiers of the axes (corresponding to \"columns\" of a Pandas DataFrame). It is convenient to define the `Space` not only with the observable but also with a limit: this can directly be re-used as the normalization range in the PDF.\n\nFirst, let's define our observables\n\n\n```python\nobs = zfit.Space('obs1', (-5, 10))\n```\n\nThis `Space` has limits. Next to the effect of handling the observables, we can also play with the limits: multiple `Spaces` can be added to provide disconnected ranges. More importantly, `Space` offers functionality:\n- limit1d: return the lower and upper limit in the 1 dimensional case (raises an error otherwise)\n- rect_limits: return the n dimensional limits\n- area(): calculate the area (e.g. distance between upper and lower)\n- inside(): return a boolean Tensor corresponding to whether the value is _inside_ the `Space`\n- filter(): filter the input values to only return the one inside\n\n\n```python\nsize_normal = 10000\ndata_normal_np = np.random.normal(size=size_normal, scale=2)\n\ndata_normal = zfit.Data.from_numpy(obs=obs, array=data_normal_np)\n```\n\nThe main functionality is\n- nevents: attribute that returns the number of events in the object\n- data_range: a `Space` that defines the limits of the data; if outside, the data will be cut\n- n_obs: defines the number of dimensions in the dataset\n- with_obs: returns a subset of the dataset with only the given obs\n- weights: event based weights\n\nFurthermore, `value` returns a Tensor with shape `(nevents, n_obs)`.\n\nTo retrieve values, in general `z.unstack_x(data)` should be used; this returns a single Tensor with shape (nevents) or a list of tensors if `n_obs` is larger then 1.\n\n\n```python\nprint(f\"We have {data_normal.nevents} events in our dataset with the minimum of {np.min(data_normal.unstack_x())}\") # remember! The obs cut out some of the data\n```\n\n We have 9946 events in our dataset with the minimum of -4.978826801974078\n\n\n\n```python\ndata_normal.n_obs\n```\n\n\n\n\n 1\n\n\n\n## Model\n\nBuilding models is by far the largest part of zfit. We will therefore cover an essential part, the possibility to build custom models, in an extra chapter. Let's start out with the idea that you define your parameters and your observable space; the latter is the expected input data.\n\nThere are two types of models in zfit:\n- functions, which are rather simple and \"underdeveloped\"; their usage is often not required.\n- PDF that are function which are normalized (over a specified range); this is the main model and is what we gonna use throughout the tutorials.\n\nA PDF is defined by\n\n\\begin{align}\n\\mathrm{PDF}_{f(x)}(x; \\theta) = \\frac{f(x; \\theta)}{\\int_{a}^{b} f(x; \\theta)}\n\\end{align}\n\nwhere a and b define the normalization range (`norm_range`), over which (by inserting into the above definition) the integral of the PDF is unity.\n\nzfit has a modular approach to things and this is also true for models. While the normalization itself (e.g. what are parameters, what is normalized data) will already be pre-defined in the model, models are composed of functions that are transparently called inside. For example, a Gaussian would usually be implemented by writing a Python function `def gauss(x, mu, sigma)`, which does not care about the normalization and then be wrapped in a PDF, where the normalization and what is a parameter is defined.\n\nIn principle, we can go far by using simply functions (e.g. [TensorFlowAnalysis/AmpliTF](https://github.com/apoluekt/AmpliTF) by Anton Poluektov uses this approach quite successfully for Amplitude Analysis), but this design has limitations for a more general fitting library such as zfit (or even [TensorWaves](https://github.com/ComPWA/tensorwaves), being built on top of AmpliTF).\nThe main thing is to keep track of the different ordering of the data and parameters, especially the dependencies. \n\n\nLet's create a simple Gaussian PDF. We already defined the `Space` for the data before, now we only need the parameters. This are a different object than a `Space`.\n\n### Parameter\nA `Parameter` (there are different kinds actually, more on that later) takes the following arguments as input:\n`Parameter(human readable name, initial value[, lower limit, upper limit])` where the limits are recommended but not mandatory. Furthermore, `step_size` can be given (which is useful to be around the given uncertainty, e.g. for large yields or small values it can help a lot to set this). Also, a `floating` argument is supported, indicating whether the parameter is allowed to float in the fit or not (just omitting the limits does _not_ make a parameter constant).\n\nParameters have a unique name. This is served as the identifier for e.g. fit results. However, a parameter _cannot_ be retrieved by its string identifier (its name) but the object itself should be used. In places where a parameter maps to something, the object itself is needed, not its name.\n\n\n```python\nmu = zfit.Parameter('mu', 1, -3, 3, step_size=0.2)\nsigma_num = zfit.Parameter('sigma42', 1, 0.1, 10, floating=False)\n```\n\nThese attributes can be changed:\n\n\n```python\nprint(f\"sigma is float: {sigma_num.floating}\")\nsigma_num.floating = True\nprint(f\"sigma is float: {sigma_num.floating}\")\n```\n\n sigma is float: False\n sigma is float: True\n\n\n*PITFALL NOTEBOOKS: since the parameters have a unique name, a second parameter with the same name cannot be created; the behavior is undefined and therefore it raises an error.\nWhile this does not pose a problem in a normal Python script, it does in a Jupyter-like notebook, since it is an often practice to \"rerun\" a cell as an attempt to \"reset\" things. Bear in mind that this does not make sense, from a logic point of view. The parameter already exists. Best practice: write a small wrapper, do not rerun the parameter creation cell or simply rerun the notebook (restart kernel & run all). For further details, have a look at the discussion and arguments [here](https://github.com/zfit/zfit/issues/186)*\n\nNow we have everything to create a Gaussian PDF:\n\n\n```python\ngauss = zfit.pdf.Gauss(obs=obs, mu=mu, sigma=sigma_num)\n```\n\nSince this holds all the parameters and the observables are well defined, we can retrieve them\n\n\n```python\ngauss.n_obs # dimensions\n```\n\n\n\n\n 1\n\n\n\n\n```python\ngauss.obs\n```\n\n\n\n\n ('obs1',)\n\n\n\n\n```python\ngauss.space\n```\n\n\n\n\n \n\n\n\n\n```python\ngauss.norm_range\n```\n\n\n\n\n \n\n\n\nAs we've seen, the `obs` we defined is the `space` of Gauss: this acts as the default limits whenever needed (e.g. for sampling). `gauss` also has a `norm_range`, which equals by default as well to the `obs` given, however, we can explicitly change that with `set_norm_range`.\n\nWe can also access the parameters of the PDF in two ways, depending on our intention: \neither by _name_ (the parameterization name, e.g. `mu` and `sigma`, as defined in the `Gauss`), which is useful if we are interested in the parameter that _describes_ the shape\n\n\n```python\ngauss.params\n```\n\n\n\n\n OrderedDict([('mu', ),\n ('sigma', )])\n\n\n\nor to retrieve all the parameters that the PDF depends on. As this now may sounds trivial, we will see later that models can depend on other models (e.g. sums) and parameters on other parameters. There is one function that automatically retrieves _all_ dependencies, `get_params`. It takes three arguments to filter:\n- floating: whether to filter only floating parameters, only non-floating or don't discriminate\n- is_yield: if it is a yield, or not a yield, or both\n- extract_independent: whether to recursively collect all parameters. This, and the explanation for why independent, can be found later on in the `Simultaneous` tutorial.\n\nUsually, the default is exactly what we want if we look for _all free parameters that this PDF depends on_.\n\n\n```python\ngauss.get_params()\n```\n\n\n\n\n OrderedSet([, ])\n\n\n\nThe difference will also be clear if we e.g. use the same parameter twice:\n\n\n```python\ngauss_only_mu = zfit.pdf.Gauss(obs=obs, mu=mu, sigma=mu)\nprint(f\"params={gauss_only_mu.params}\")\nprint(f\"get_params={gauss_only_mu.get_params()}\")\n```\n\n params=OrderedDict([('mu', ), ('sigma', )])\n get_params=OrderedSet([])\n\n\n## Functionality\n\nPDFs provide a few useful methods. The main features of a zfit PDF are:\n\n- `pdf`: the normalized value of the PDF. It takes an argument `norm_range` that can be set to `False`, in which case we retrieve the unnormalized value\n- `integrate`: given a certain range, the PDF is integrated. As `pdf`, it takes a `norm_range` argument that integrates over the unnormalized `pdf` if set to `False`\n- `sample`: samples from the pdf and returns a `Data` object\n\n\n```python\nintegral = gauss.integrate(limits=(-1, 3)) # corresponds to 2 sigma integral\nintegral\n```\n\n\n\n\n \n\n\n\n### Tensors\n\nAs we see, many zfit functions return Tensors. This is however no magical thing! If we're outside of models, than we can always safely convert them to a numpy array by calling `zfit.run(...)` on it (or any structure containing potentially multiple Tensors). However, this may not even be required often! They can be added just like numpy arrays and interact well with Python and Numpy:\n\n\n```python\nnp.sqrt(integral)\n```\n\n\n\n\n array([0.97698502])\n\n\n\nThey also have shapes, dtypes, can be slices etc. So do not convert them except you need it. More on this can be seen in the talk later on about zfit and TensorFlow 2.0.\n\n\n```python\nsample = gauss.sample(n=1000) # default space taken as limits\nsample\n```\n\n\n\n\n \n\n\n\n\n```python\nsample.unstack_x()[:10]\n```\n\n\n\n\n \n\n\n\n\n```python\nsample.n_obs\n```\n\n\n\n\n 1\n\n\n\n\n```python\nsample.obs\n```\n\n\n\n\n ('obs1',)\n\n\n\nWe see that sample returns also a zfit `Data` object with the same space as it was sampled in. This can directly be used e.g.\n\n\n```python\nprobs = gauss.pdf(sample)\nprobs[:10]\n```\n\n\n\n\n \n\n\n\n**NOTE**: In case you want to do this repeatedly (e.g. for toy studies), there is a way more efficient way (see later on)\n\n## Plotting\n\nso far, we have a dataset and a PDF. Before we go for fitting, we can make a plot. This functionality is not _directly_ provided in zfit (but can be added to [zfit-physics](https://github.com/zfit/zfit-physics)). It is however simple enough to do it:\n\n\n```python\ndef plot_model(model, data, scale=1, plot_data=True): # we will use scale later on\n\n nbins = 50\n\n lower, upper = data.data_range.limit1d\n x = tf.linspace(lower, upper, num=1000) # np.linspace also works\n y = model.pdf(x) * size_normal / nbins * data.data_range.area()\n y *= scale\n plt.plot(x, y)\n data_plot = zfit.run(z.unstack_x(data)) # we could also use the `to_pandas` method\n if plot_data:\n plt.hist(data_plot, bins=nbins)\n```\n\n\n```python\nplot_model(gauss, data_normal)\n```\n\nWe can of course do better (and will see that later on, continuously improve the plots), but this is quite simple and gives us the full power of matplotlib.\n\n### Different models\n\nzfit offers a selection of predefined models (and extends with models from zfit-physics that contain physics specific models such as ARGUS shaped models).\n\n\n```python\nprint(zfit.pdf.__all__)\n```\n\n ['BasePDF', 'BaseFunctor', 'Exponential', 'CrystalBall', 'DoubleCB', 'Gauss', 'Uniform', 'TruncatedGauss', 'WrapDistribution', 'Cauchy', 'Chebyshev', 'Legendre', 'Chebyshev2', 'Hermite', 'Laguerre', 'RecursivePolynomial', 'ProductPDF', 'SumPDF', 'GaussianKDE1DimV1', 'ZPDF', 'SimplePDF', 'SimpleFunctorPDF']\n\n\nTo create a more realistic model, we can build some components for a mass fit with a\n- signal component: CrystalBall\n- combinatorial background: Exponential\n- partial reconstructed background on the left: Kernel Density Estimation\n\n\n```python\nmass_obs = zfit.Space('mass', (0, 1000))\n```\n\n\n```python\n# Signal component\n\nmu_sig = zfit.Parameter('mu_sig', 400, 100, 600)\nsigma_sig = zfit.Parameter('sigma_sig', 50, 1, 100)\nalpha_sig = zfit.Parameter('alpha_sig', 300, 100, 400)\nn_sig = zfit.Parameter('n sig', 4, 0.1, 30)\nsignal = zfit.pdf.CrystalBall(obs=mass_obs, mu=mu_sig, sigma=sigma_sig, alpha=alpha_sig, n=n_sig)\n```\n\n\n```python\n# combinatorial background\n\nlam = zfit.Parameter('lambda', -0.01, -0.05, -0.001)\ncomb_bkg = zfit.pdf.Exponential(lam, obs=mass_obs)\n```\n\n\n```python\npart_reco_data = np.random.normal(loc=200, scale=150, size=700)\npart_reco_data = zfit.Data.from_numpy(obs=mass_obs, array=part_reco_data) # we don't need to do this but now we're sure it's inside the limits\n\npart_reco = zfit.pdf.GaussianKDE1DimV1(obs=mass_obs, data=part_reco_data, bandwidth='adaptive')\n```\n\n## Composing models\n\nWe can also compose multiple models together. Here we'll stick to one dimensional models, the extension to multiple dimensions is explained in the \"custom models tutorial\".\n\nHere we will use a `SumPDF`. This takes pdfs and fractions. If we provide n pdfs and:\n- n - 1 fracs: the nth fraction will be 1 - sum(fracs)\n- n fracs: no normalization attempt is done by `SumPDf`. If the fracs are not implicitly normalized, this can lead to bad fitting\n behavior if there is a degree of freedom too much\n \n\n\n\n```python\nsig_frac = zfit.Parameter('sig_frac', 0.3, 0, 1)\ncomb_bkg_frac = zfit.Parameter('comb_bkg_frac', 0.25, 0, 1)\nmodel = zfit.pdf.SumPDF([signal, comb_bkg, part_reco], [sig_frac, comb_bkg_frac])\n```\n\nIn order to have a corresponding data sample, we can just create one. Since we want to fit to this dataset later on, we will create it with slightly different values. Therefore, we can use the ability of a parameter to be set temporarily to a certain value with\n\n\n```python\nprint(f\"before: {sig_frac}\")\nwith sig_frac.set_value(0.25):\n print(f\"new value: {sig_frac}\")\nprint(f\"after 'with': {sig_frac}\")\n```\n\n before: \n new value: \n after 'with': \n\n\nWhile this is useful, it does not fully scale up. We can use the `zfit.param.set_values` helper therefore.\n(_Sidenote: instead of a list of values, we can also use a `FitResult`, the given parameters then take the value from the result_)\n\n\n```python\nwith zfit.param.set_values([mu_sig, sigma_sig, sig_frac, comb_bkg_frac, lam], [370, 34, 0.18, 0.15, -0.006]):\n data = model.sample(n=10000)\n```\n\n\n```python\nplot_model(model, data);\n```\n\nPlotting the components is not difficult now: we can either just plot the pdfs separately (as we still can access them) or in a generalized manner by accessing the `pdfs` attribute:\n\n\n```python\ndef plot_comp_model(model, data):\n for mod, frac in zip(model.pdfs, model.params.values()):\n plot_model(mod, data, scale=frac, plot_data=False)\n plot_model(model, data)\n```\n\n\n```python\nplot_comp_model(model, data)\n```\n\nNow we can add legends etc. Btw, did you notice that actually, the `frac` params are zfit `Parameters`? But we just used them as if they were Python scalars and it works.\n\n\n```python\nprint(model.params)\n```\n\n OrderedDict([('frac_0', ), ('frac_1', ), ('frac_2', ), ('param_1', )]) value=0.45>)])\n\n\n### Extended PDFs\n\nSo far, we have only looked at normalized PDFs that do contain information about the shape but not about the _absolute_ scale. We can make a PDF extended by adding a yield to it.\n\nThe behavior of the new, extended PDF does **NOT change**, any methods we called before will act the same. Only exception, some may require an argument _less_ now. All the methods we used so far will return the same values. What changes is that the flag `model.is_extended` now returns `True`. Furthermore, we have now a few more methods that we can use which would have raised an error before:\n- `get_yield`: return the yield parameter (notice that the yield is _not_ added to the shape parameters `params`)\n- `ext_{pdf,integrate}`: these methods return the same as the versions used before, however, multiplied by the yield\n- `sample` is still the same, but does not _require_ the argument `n` anymore. By default, this will now equal to a _poissonian sampled_ n around the yield.\n\nThe `SumPDF` now does not strictly need `fracs` anymore: if _all_ input PDFs are extended, the sum will be as well and use the (normalized) yields as fracs\n\nThe preferred way to create an extended PDf is to use `PDF.create_extended(yield)`. However, since this relies on copying the PDF (which may does not work for different reasons), there is also a `set_yield(yield)` method that sets the yield in-place. This won't lead to ambiguities, as everything is supposed to work the same.\n\n\n```python\nyield_model = zfit.Parameter('yield_model', 10000, 0, 20000, step_size=10)\nmodel_ext = model.create_extended(yield_model)\n```\n\nalternatively, we can create the models as extended and sum them up\n\n\n```python\nsig_yield = zfit.Parameter('sig_yield', 2000, 0, 10000, step_size=1)\nsig_ext = signal.create_extended(sig_yield)\n\ncomb_bkg_yield = zfit.Parameter('comb_bkg_yield', 6000, 0, 10000, step_size=1)\ncomb_bkg_ext = comb_bkg.create_extended(comb_bkg_yield)\n\npart_reco_yield = zfit.Parameter('part_reco_yield', 2000, 0, 10000, step_size=1)\npart_reco.set_yield(part_reco_yield) # unfortunately, `create_extended` does not work here. But no problem, it won't change anyting.\npart_reco_ext = part_reco\n```\n\n\n```python\nmodel_ext_sum = zfit.pdf.SumPDF([sig_ext, comb_bkg_ext, part_reco_ext])\n```\n\n# Loss\n\nA loss combines the model and the data, for example to build a likelihood. Furthermore, it can contain constraints, additions to the likelihood. Currently, if the `Data` has weights, these are automatically taken into account.\n\n\n```python\nnll_gauss = zfit.loss.UnbinnedNLL(gauss, data_normal)\n```\n\nThe loss has several attributes to be transparent to higher level libraries. We can calculate the value of it using `value`.\n\n\n```python\nnll_gauss.value()\n```\n\n\n\n\n \n\n\n\nNotice that due to graph building, this will take significantly longer on the first run. Rerun the cell above and it will be way faster.\n\n\n\nFurthermore, the loss also provides a possibility to calculate the gradients or, often used, the value and the gradients.\n\nWe can access the data and models (and possible constraints)\n\n\n```python\nnll_gauss.model\n```\n\n\n\n\n [0]\n\n\n\n\n```python\nnll_gauss.data\n```\n\n\n\n\n []\n\n\n\n\n```python\nnll_gauss.constraints\n```\n\n\n\n\n []\n\n\n\nSimilar to the models, we can also get the parameters via `get_params`.\n\n\n```python\nnll_gauss.get_params()\n```\n\n\n\n\n OrderedSet([, ])\n\n\n\n### Extended loss\n\nMore interestingly, we can now build a loss for our composite sum model using the sampled data. Since we created an extended model, we can now also create an extended likelihood, taking into account a Poisson term to match the yield to the number of events.\n\n\n```python\nnll = zfit.loss.ExtendedUnbinnedNLL(model_ext_sum, data)\n```\n\n\n```python\nnll.get_params()\n```\n\n\n\n\n OrderedSet([, , , , , , , ])\n\n\n\n# Minimization\n\nWhile a loss is interesting, we usually want to minimize it. Therefore we can use the minimizers in zfit, most notably `Minuit`, a wrapper around the [iminuit minimizer](https://github.com/scikit-hep/iminuit).\n\nThe philosophy is to create a minimizer instance that is mostly _stateless_, e.g. does not remember the position (there are considerations to make it possible to have a state, in case you feel interested, [contact us](https://github.com/zfit/zfit#contact))\n\nGiven that iminuit provides us with a very reliable and stable minimizer, it is usually recommended to use this. Others are implemented as well and could easily be wrapped, however, the convergence is usually not as stable.\n\nMinuit has a few options:\n- `tolerance`: the Estimated Distance to Minimum (EDM) criteria for convergence (default 1e-3)\n- `verbosity`: between 0 and 10, 5 is normal, 7 is verbose, 10 is maximum\n- `use_minuit_grad`: if True, uses the Minuit numerical gradient instead of the TensorFlow gradient. This is usually more stable for smaller fits; furthermore the TensorFlow gradient _can_ (experience based) sometimes be wrong.\n\n\n```python\nminimizer = zfit.minimize.Minuit(use_minuit_grad=True)\n```\n\nFor the minimization, we can call `minimize`, which takes a\n- loss as we created above\n- optionally: the parameters to minimize\n\nBy default, `minimize` uses all the free floating parameters (obtained with `get_params`). We can also explicitly specify which ones to use by giving them (or better, objects that depend on them) to `minimize`; note however that non-floating parameters, even if given explicitly to `minimize` won 't be minimized.\n\n## Pre-fit parts of the PDF\n\nBefore we want to fit the whole PDF however, it can be useful to pre-fit it. A way can be to fix the combinatorial background by fitting the exponential to the right tail.\n\nTherefore we create a new data object with an additional cut and furthermore, set the normalization range of the background pdf to the range we are interested in.\n\n\n```python\nvalues = z.unstack_x(data)\nobs_right_tail = zfit.Space('mass', (700, 1000))\ndata_tail = zfit.Data.from_tensor(obs=obs_right_tail, tensor=values)\nwith comb_bkg.set_norm_range(obs_right_tail):\n nll_tail = zfit.loss.UnbinnedNLL(comb_bkg, data_tail)\n minimizer.minimize(nll_tail)\n```\n\n ------------------------------------------------------------------\n | FCN = 313 | Ncalls=17 (17 total) |\n | EDM = 1.65e-08 (Goal: 0.001) | up = 0.5 |\n ------------------------------------------------------------------\n | Valid Min. | Valid Param. | Above EDM | Reached call limit |\n ------------------------------------------------------------------\n | True | True | False | False |\n ------------------------------------------------------------------\n | Hesse failed | Has cov. | Accurate | Pos. def. | Forced |\n ------------------------------------------------------------------\n | False | True | True | True | False |\n ------------------------------------------------------------------\n\n\nSince we now fit the lambda parameter of the exponential, we can fix it.\n\n\n```python\nlam.floating = False\nlam\n```\n\n\n\n\n \n\n\n\n\n```python\nresult = minimizer.minimize(nll)\n```\n\n ------------------------------------------------------------------\n | FCN = -1.954e+04 | Ncalls=167 (167 total) |\n | EDM = 0.000128 (Goal: 0.001) | up = 0.5 |\n ------------------------------------------------------------------\n | Valid Min. | Valid Param. | Above EDM | Reached call limit |\n ------------------------------------------------------------------\n | True | True | False | False |\n ------------------------------------------------------------------\n | Hesse failed | Has cov. | Accurate | Pos. def. | Forced |\n ------------------------------------------------------------------\n | False | True | True | True | False |\n ------------------------------------------------------------------\n\n\n\n```python\nplot_comp_model(model_ext_sum, data)\n```\n\n# Fit result\n\nThe result of every minimization is stored in a `FitResult`. This is the last stage of the zfit workflow and serves as the interface to other libraries. Its main purpose is to store the values of the fit, to reference to the objects that have been used and to perform (simple) uncertainty estimation.\n\n\n```python\nprint(result)\n```\n\n FitResult of\n 0] data=[] constraints=[]> \n with\n \n \n \u2552\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2555\n \u2502 valid \u2502 converged \u2502 param at limit \u2502 edm \u2502 min value \u2502\n \u255e\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2561\n \u2502 True \u2502 True \u2502 False \u2502 0.00013 \u2502 -1.954e+04 \u2502\n \u2558\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255b\n \n Parameters\n name value at limit\n --------------- ------- ----------\n sig_yield 1851 False\n comb_bkg_yield 1131 False\n part_reco_yield 7018 False\n alpha_sig 300 False\n mu_sig 369.7 False\n n sig 4 False\n sigma_sig 34.09 False\n\n\nThis gives an overview over the whole result. Often we're mostly interested in the parameters and their values, which we can access with a `params` attribute.\n\n\n```python\nprint(result.params)\n```\n\n name value at limit\n --------------- ------- ----------\n sig_yield 1851 False\n comb_bkg_yield 1131 False\n part_reco_yield 7018 False\n alpha_sig 300 False\n mu_sig 369.7 False\n n sig 4 False\n sigma_sig 34.09 False\n\n\nThis is a `dict` which stores any knowledge about the parameters and can be accessed by the parameter (object) itself:\n\n\n```python\nresult.params[mu_sig]\n```\n\n\n\n\n {'value': 369.7337120757637}\n\n\n\n'value' is the value at the minimum. To obtain other information about the minimization process, `result` contains more attributes:\n- fmin: the function minimum\n- edm: estimated distance to minimum\n- info: contains a lot of information, especially the original information returned by a specific minimizer\n- converged: if the fit converged\n\n\n```python\nresult.fmin\n```\n\n\n\n\n -19539.388894488584\n\n\n\n## Estimating uncertainties\n\nThe `FitResult` has mainly two methods to estimate the uncertainty:\n- a profile likelihood method (like MINOS)\n- Hessian approximation of the likelihood (like HESSE)\n\nWhen using `Minuit`, this uses (currently) it's own implementation. However, zfit has its own implementation, which are likely to become the standard and can be invoked by changing the method name.\n\nHesse is also [on the way to implement](https://github.com/zfit/zfit/pull/244) the [corrections for weights](https://inspirehep.net/literature/1762842).\n\nWe can explicitly specify which parameters to calculate, by default it does for all.\n\n\n```python\nresult.hesse()\n```\n\n\n\n\n OrderedDict([(,\n {'error': 73.52924652265973}),\n (,\n {'error': 79.49051306516832}),\n (,\n {'error': 136.98379193128127}),\n (,\n {'error': 141.4213562373095}),\n (,\n {'error': 1.3913704142629884}),\n (,\n {'error': 10.069756698215553}),\n (,\n {'error': 1.3407261987832484})])\n\n\n\n\n```python\n# result.hesse(method='hesse_np')\n```\n\nWe get the result directly returned. This is also added to `result.params` for each parameter and is nicely displayed with an added column\n\n\n```python\nprint(result.params)\n```\n\n name value minuit_hesse at limit\n --------------- ------- -------------- ----------\n sig_yield 1851 +/- 69 False\n comb_bkg_yield 1131 +/- 75 False\n part_reco_yield 7018 +/- 1.2e+02 False\n alpha_sig 300 +/- 1.4e+02 False\n mu_sig 369.7 +/- 1.4 False\n n sig 4 +/- 10 False\n sigma_sig 34.09 +/- 1.3 False\n\n\n\n```python\nerrors, new_result = result.errors(params=[sig_yield, part_reco_yield, mu_sig]) # just using three for speed reasons\n```\n\n /home/user/.local/lib/python3.6/site-packages/zfit/minimizers/fitresult.py:359: FutureWarning: 'minuit_minos' will be changed as the default errors method to a custom implementationwith the same functionality. If you want to make sure that 'minuit_minos' will be used in the future, add it explicitly as in `errors(method='minuit_minos')`\n \"in the future, add it explicitly as in `errors(method='minuit_minos')`\", FutureWarning)\n\n\n\n```python\n# errors, new_result = result.errors(params=[yield_model, sig_frac, mu_sig], method='zfit_error')\n```\n\n\n```python\nprint(errors)\n```\n\n OrderedDict([(, MError(name='sig_yield', is_valid=True, lower=-73.19610573010954, upper=73.2494131480733, lower_valid=True, upper_valid=True, at_lower_limit=False, at_upper_limit=False, at_lower_max_fcn=False, at_upper_max_fcn=False, lower_new_min=False, upper_new_min=False, nfcn=208, min=1850.6452612293656)), (, MError(name='part_reco_yield', is_valid=True, lower=-134.46090913541406, upper=138.70099174669127, lower_valid=True, upper_valid=True, at_lower_limit=False, at_upper_limit=False, at_lower_max_fcn=False, at_upper_max_fcn=False, lower_new_min=False, upper_new_min=False, nfcn=204, min=7017.782731457284)), (, MError(name='mu_sig', is_valid=True, lower=-1.3846589189505978, upper=1.391857347531778, lower_valid=True, upper_valid=True, at_lower_limit=False, at_upper_limit=False, at_lower_max_fcn=False, at_upper_max_fcn=False, lower_new_min=False, upper_new_min=False, nfcn=131, min=369.7337120757637))])\n\n\n\n```python\nprint(result.params)\n```\n\n name value minuit_hesse minuit_minos at limit\n --------------- ------- -------------- ------------------- ----------\n sig_yield 1851 +/- 69 - 73 + 73 False\n comb_bkg_yield 1131 +/- 75 False\n part_reco_yield 7018 +/- 1.2e+02 -1.3e+02 +1.4e+02 False\n alpha_sig 300 +/- 1.4e+02 False\n mu_sig 369.7 +/- 1.4 - 1.4 + 1.4 False\n n sig 4 +/- 10 False\n sigma_sig 34.09 +/- 1.3 False\n\n\n#### What is 'new_result'?\n\nWhen profiling a likelihood, such as done in the algorithm used in `errors`, a new minimum can be found. If this is the case, this new minimum will be returned, otherwise `new_result` is `None`. Furthermore, the current `result` would be rendered invalid by setting the flag `valid` to `False`. _Note_: this behavior only applies to the zfit internal error estimator.\n\n### A simple profile\n\nThere is no default function (yet) for simple profiling plot. However, again, we're in Python and it's simple enough to do that for a parameter. Let's do it for `sig_yield`\n\n\n```python\nx = np.linspace(1600, 2000, num=50)\ny = []\nsig_yield.floating = False\nfor val in x:\n sig_yield.set_value(val)\n y.append(nll.value())\n\nsig_yield.floating = True\nzfit.param.set_values(nll.get_params(), result)\n```\n\n\n```python\nplt.plot(x, y)\n```\n\nWe can also access the covariance matrix of the parameters\n\n\n```python\nresult.covariance()\n```\n\n\n\n\n array([[ 4.80313887e+03, 1.02095082e+03, -3.43531769e+03,\n 0.00000000e+00, -2.08964215e+01, 0.00000000e+00,\n 4.67706435e+01],\n [ 1.02095082e+03, 5.63159948e+03, -5.91552419e+03,\n 0.00000000e+00, -7.87129146e+00, 0.00000000e+00,\n 1.77558437e+01],\n [-3.43531769e+03, -5.91552419e+03, 1.46369813e+04,\n 0.00000000e+00, 3.07633305e+01, 0.00000000e+00,\n -6.37418378e+01],\n [ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,\n 2.00000000e+04, 0.00000000e+00, 0.00000000e+00,\n 0.00000000e+00],\n [-2.08964215e+01, -7.87129146e+00, 3.07633305e+01,\n 0.00000000e+00, 1.93158341e+00, 0.00000000e+00,\n -4.28578345e-01],\n [ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,\n 0.00000000e+00, 0.00000000e+00, 1.01400000e+02,\n 0.00000000e+00],\n [ 4.67706435e+01, 1.77558437e+01, -6.37418378e+01,\n 0.00000000e+00, -4.28578345e-01, 0.00000000e+00,\n 1.74205276e+00]])\n\n\n\n# End of zfit\n\nThis is where zfit finishes and other libraries take over.\n\n# Beginning of hepstats\n\n`hepstats` is a library containing statistical tools and utilities for high energy physics. In particular you do statistical inferences using the models and likelhoods function constructed in `zfit`.\n\nShort example: let's compute for instance a confidence interval at 68 % confidence level on the mean of the gaussian defined above.\n\n\n```python\nfrom hepstats.hypotests.parameters import POIarray\nfrom hepstats.hypotests.calculators import AsymptoticCalculator\nfrom hepstats.hypotests import ConfidenceInterval\n```\n\n\n```python\ncalculator = AsymptoticCalculator(input=result, minimizer=minimizer)\n```\n\n\n```python\nvalue = result.params[mu_sig][\"value\"]\nerror = result.params[mu_sig][\"minuit_hesse\"][\"error\"]\n\nmean_scan = POIarray(mu_sig, np.linspace(value - 1.5*error, value + 1.5*error, 10))\n```\n\n\n```python\nci = ConfidenceInterval(calculator, mean_scan)\n```\n\n\n```python\nci.interval()\n```\n\n \n Confidence interval on mu_sig:\n \t368.3536149718213 < mu_sig < 371.12167557864825 at 68.0% C.L.\n\n\n\n\n\n {'observed': 369.7420306479183,\n 'upper': 371.12167557864825,\n 'lower': 368.3536149718213}\n\n\n\n\n```python\nfrom utils import one_minus_cl_plot\n\nax = one_minus_cl_plot(ci)\nax.set_xlabel(\"mean\")\n```\n\nThere will be more of `hepstats` later.\n", "meta": {"hexsha": "18a790f05dfcf1d7e36683b20f1caf9fbe69c27d", "size": 877127, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Introduction.ipynb", "max_stars_repo_name": "zfit/PyHEP2020", "max_stars_repo_head_hexsha": "8eb962c720ce48694f1c0031614639ead45db195", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2020-07-13T09:01:18.000Z", "max_stars_repo_stars_event_max_datetime": "2020-11-02T14:57:30.000Z", "max_issues_repo_path": "Introduction.ipynb", "max_issues_repo_name": "zfit/PyHEP2020", "max_issues_repo_head_hexsha": "8eb962c720ce48694f1c0031614639ead45db195", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Introduction.ipynb", "max_forks_repo_name": "zfit/PyHEP2020", "max_forks_repo_head_hexsha": "8eb962c720ce48694f1c0031614639ead45db195", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2020-07-16T17:05:18.000Z", "max_forks_repo_forks_event_max_datetime": "2020-11-02T14:58:04.000Z", "avg_line_length": 334.7812977099, "max_line_length": 252392, "alphanum_fraction": 0.9320805311, "converted": true, "num_tokens": 9764, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.6688802603710086, "lm_q1q2_score": 0.42609732367638914}} {"text": "```python\n# Tested on python 3.6.4 \n%matplotlib inline\n\nimport numpy as np # 1.13.3\nfrom scipy.integrate import odeint # 1.0.0\nimport scipy.optimize as op\nimport scipy.stats as stats\nimport matplotlib.pyplot as plt # 2.1.1\nfrom matplotlib.ticker import AutoMinorLocator\nfrom matplotlib.ticker import MaxNLocator\n\nimport pandas as pd # 0.22.0\nimport emcee # 2.2.1\nimport corner # 2.0.1\nimport progressbar # 3.34.3\nimport seaborn as sns # 0.8.1\nfrom cycler import cycler # 0.10.0\n\nprint('emcee version', emcee.__version__)\n\n# Directories defined here \nDIR_DATA = './data/'\nDIR_PLOTS = './plots/'\nDIR_OUT = './output/'\n```\n\n emcee version 2.2.1\n\n\nThis jupyter notebook analyses TXTL dynamics from the 2018 Swank et al. paper. The code in this notebook requires the following data files, which are located in `DIR_DATA`:\n\n dynamics_chip.csv\n dynamics_PR.csv\n \nThe results are used to generate Supplementary Figure S3. Plots are written into `DIR_PLOTS`.\n\n\n```python\ndef plotlin(data,name,DIR_PLOTS):\n\n plt.close(\"all\")\n\n my_dpi=150\n\n figure_options={'figsize':(8.27,5.83)} #figure size in inches. A4=11.7x8.3. A5=8.27,5.83\n font_options={'size':'28','family':'sans-serif','sans-serif':'Arial'}\n plt.rc('figure', **figure_options)\n plt.rc('font', **font_options)\n\n current_palette=sns.color_palette(\"deep\", 4)\n plt.rc('axes',prop_cycle=(cycler('color',current_palette)))\n f, axarr=plt.subplots()\n plt.subplots_adjust(left=0.25,bottom=0.2,right=0.95,top=0.95)\n \n # Plot data\n sns.regplot(x='init',y='fin',data=data,ci=95); \n formatplot(axarr,'Initial rate (RFU/h)','Final level (RFU)', xlim=False,ylim=False)\n plt.savefig(DIR_PLOTS+name+'.pdf',dpi=my_dpi,transparent=True)\n\ndef plotlinPR(data,name,DIR_PLOTS):\n\n plt.close(\"all\")\n\n my_dpi=150\n\n figure_options={'figsize':(8.27,5.83)} #figure size in inches. A4=11.7x8.3. A5=8.27,5.83\n font_options={'size':'28','family':'sans-serif','sans-serif':'Arial'}\n plt.rc('figure', **figure_options)\n plt.rc('font', **font_options)\n\n current_palette=sns.color_palette(\"deep\", 4)\n plt.rc('axes',prop_cycle=(cycler('color',current_palette)))\n f, axarr=plt.subplots()\n plt.subplots_adjust(left=0.25,bottom=0.2,right=0.95,top=0.95)\n \n # Plot data\n sns.regplot(x='Initial Rate',y='Final Level',data=df2,ci=95); \n formatplot(axarr,'Initial rate (RFU/h)','Final level (RFU)', xlim=False,ylim=False)\n plt.savefig(DIR_PLOTS+name+'.pdf',dpi=my_dpi,transparent=True)\n \ndef formatplot(ax,xlabel,ylabel,xlim,ylim,logx=False,logy=False,logxy=False,symlogx=False):\n\n ######### SET TITLES AND LABLES #########\n\n #ax.set_title('Plot title')\n if xlabel!=False:\n ax.set_xlabel(xlabel, labelpad=12)\n if ylabel!=False: \n ax.set_ylabel(ylabel, labelpad=12)\n\n ######### SET AXES LIMITS #########\n\n if xlim!=False:\n ax.set_xlim(xlim)\n if ylim!=False:\n ax.set_ylim(ylim)\n\n ######### SET TICK VALUES #########\n\n #ax.set_xticks([0,0.5,1])\n #\tax.set_yticks([0,2,4,6,8])\n\n ######### SET LINE THICKNESSES #########\n\n #ax.xaxis.set_major_formatter(mpl.ticker.FormatStrFormatter(\"%1.e\"))\n #ax.axhline(linewidth=2, color='k') \n #ax.axvline(linewidth=2, color='k')\n ax.spines['bottom'].set_linewidth(2)\n ax.spines['top'].set_linewidth(2)\n ax.spines['left'].set_linewidth(2)\n ax.spines['right'].set_linewidth(2) \n\n ######### SET TICKS #########\n\n if logx==True:\n\n ax.set_xscale(\"log\")\n\n elif logy==True:\n\n ax.set_yscale(\"log\")\n\n elif logxy==True:\n\n ax.set_xscale(\"log\")\n ax.set_yscale(\"log\")\n \n elif symlogx==True:\n\n ax.set_xscale(\"symlog\",linthreshx=1e-4)\n ax.set_yscale(\"log\")\n\n else:\n minorLocatorx=AutoMinorLocator(2) # Number of minor intervals per major interval\n minorLocatory=AutoMinorLocator(2)\n ax.xaxis.set_minor_locator(minorLocatorx)\n ax.yaxis.set_minor_locator(minorLocatory)\n\n ax.tick_params(which='major', width=2, length=8, pad=9,direction='in',top='on',right='on')\n ax.tick_params(which='minor', width=2, length=4, pad=9,direction='in',top='on',right='on')\n```\n\nOur thermodynamic models calculate RNAP occupancies, and the validity of their comparisons with experiments relies on the proportionality between occupancy, transcription rate, and translation rate:\n\n\\begin{equation}\ny=Ap_{bound}.\n\\end{equation}\n\nIn our experiments we measure dynamic GFP production, but generally report only the final level. We can check to ensure that the final level (whose value is determined by cell-free exhaustion) is related to the initial GFP production rate (which is proportional to translation rate). A linear relationship between the two quantities would validate our use of final levels (and fold repressions calculated from those quantities) as a proxy to measure translation rate. \n\n\n```python\n# Load timecourse data\ndf=pd.read_csv('data/dynamics_chip.csv',delimiter=',')\n\n# Plot timecourse data\ninit=np.zeros(df.shape[0]*(df.shape[1]-1))\nfin=np.zeros(df.shape[0]*(df.shape[1]-1))\n\nd={'init': init, 'fin': fin}\ndataF=pd.DataFrame(data=d)\ni=0\nfor k in range((df.shape[1]-1)): \n dataF['init'].iloc[i]=df.iloc[1,k+1]/df.iloc[1,0]*60\n dataF['fin'].iloc[i]=df.iloc[-1,k+1]\n i+=1\n\n# Uncomment this to look at time courses\n#f,ax=plt.subplots()\n#i=0\n#for k in range((df.shape[1]-1)): \n# ax.plot(df.iloc[:,0],df.iloc[:,k+1],'o-');\n# dataF['init'].iloc[i]=df.iloc[1,k+1]/df.iloc[1,0]*60\n# dataF['fin'].iloc[i]=df.iloc[-1,k+1]\n# i+=1\n#plt.show()\n\nplotlin(dataF,'rates_timecoursechip',DIR_PLOTS)\n```\n\n\n```python\n# Load and plot plate reader ZF data (orthogonality matrix)\ndf2=pd.read_csv(DIR_DATA+'dynamics_PR.csv',delimiter=',')\nplotlinPR(df2,'rates_PR',DIR_PLOTS)\n```\n\nWe observe a linear relationship between initial rates and final levels both on the plate reader as well as on chip, thus validating our use of the final level as a proxy for translation rate.\n\n\n```python\n\n```\n", "meta": {"hexsha": "8bc0d1661e9fe2d7e432d534612b3b19b1f54aa4", "size": 90615, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "NB_TXTLdynamics.ipynb", "max_stars_repo_name": "lbnc-epfl/2019_Swank_analysis", "max_stars_repo_head_hexsha": "9cb1f427b81fb3837cd1d37f9862c5a8cf3ba58f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "NB_TXTLdynamics.ipynb", "max_issues_repo_name": "lbnc-epfl/2019_Swank_analysis", "max_issues_repo_head_hexsha": "9cb1f427b81fb3837cd1d37f9862c5a8cf3ba58f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "NB_TXTLdynamics.ipynb", "max_forks_repo_name": "lbnc-epfl/2019_Swank_analysis", "max_forks_repo_head_hexsha": "9cb1f427b81fb3837cd1d37f9862c5a8cf3ba58f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2018-12-14T14:19:08.000Z", "max_forks_repo_forks_event_max_datetime": "2018-12-14T14:19:08.000Z", "avg_line_length": 300.0496688742, "max_line_length": 41268, "alphanum_fraction": 0.922363847, "converted": true, "num_tokens": 1757, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419958239132, "lm_q2_score": 0.6150878555160666, "lm_q1q2_score": 0.4260356798516991}} {"text": "\n\n# Tutorial 3: Dimensionality Reduction & Reconstruction\n**Week 1, Day 5: Dimensionality Reduction**\n\n**By Neuromatch Academy**\n\n__Content creators:__ Alex Cayco Gajic, John Murray\n\n__Content reviewers:__ Roozbeh Farhoudi, Matt Krause, Spiros Chavlis, Richard Gao, Michael Waskom, Siddharth Suresh, Natalie Schaworonkow, Ella Batty\n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n

                                        \n\n---\n# Tutorial Objectives\n\n*Estimated timing of tutorial: 50 minutes*\n\nIn this notebook we'll learn to apply PCA for dimensionality reduction, using a classic dataset that is often used to benchmark machine learning algorithms: MNIST. We'll also learn how to use PCA for reconstruction and denoising.\n\nOverview:\n- Perform PCA on MNIST\n- Calculate the variance explained\n- Reconstruct data with different numbers of PCs\n- (Bonus) Examine denoising using PCA\n\nYou can learn more about MNIST dataset [here](https://en.wikipedia.org/wiki/MNIST_database).\n\n\n```python\n# @title Tutorial slides\n\n# @markdown These are the slides for the videos in all tutorials today\nfrom IPython.display import IFrame\nIFrame(src=f\"https://mfr.ca-1.osf.io/render?url=https://osf.io/kaq2x/?direct%26mode=render%26action=download%26mode=render\", width=854, height=480)\n```\n\n\n```python\n# @title Video 1: PCA for dimensionality reduction\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = 'https://player.bilibili.com/player.html?bvid={0}&page={1}'.format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=\"BV1up4y1S7xs\", width=854, height=480, fs=1)\n print('Video available at https://www.bilibili.com/video/{0}'.format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=\"oO0bbInoO_0\", width=854, height=480, fs=1, rel=0)\n print('Video available at https://youtube.com/watch?v=' + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n---\n# Setup\n\n\n```python\n# Imports\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n```\n\n\n```python\n# @title Figure Settings\nimport ipywidgets as widgets # interactive display\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/course-content/master/nma.mplstyle\")\n```\n\n\n```python\n# @title Plotting Functions\n\ndef plot_variance_explained(variance_explained):\n \"\"\"\n Plots eigenvalues.\n\n Args:\n variance_explained (numpy array of floats) : Vector of variance explained\n for each PC\n\n Returns:\n Nothing.\n\n \"\"\"\n\n plt.figure()\n plt.plot(np.arange(1, len(variance_explained) + 1), variance_explained,\n '--k')\n plt.xlabel('Number of components')\n plt.ylabel('Variance explained')\n plt.show()\n\n\ndef plot_MNIST_reconstruction(X, X_reconstructed):\n \"\"\"\n Plots 9 images in the MNIST dataset side-by-side with the reconstructed\n images.\n\n Args:\n X (numpy array of floats) : Data matrix each column\n corresponds to a different\n random variable\n X_reconstructed (numpy array of floats) : Data matrix each column\n corresponds to a different\n random variable\n\n Returns:\n Nothing.\n \"\"\"\n\n plt.figure()\n ax = plt.subplot(121)\n k = 0\n for k1 in range(3):\n for k2 in range(3):\n k = k + 1\n plt.imshow(np.reshape(X[k, :], (28, 28)),\n extent=[(k1 + 1) * 28, k1 * 28, (k2 + 1) * 28, k2 * 28],\n vmin=0, vmax=255)\n plt.xlim((3 * 28, 0))\n plt.ylim((3 * 28, 0))\n plt.tick_params(axis='both', which='both', bottom=False, top=False,\n labelbottom=False)\n ax.set_xticks([])\n ax.set_yticks([])\n plt.title('Data')\n plt.clim([0, 250])\n ax = plt.subplot(122)\n k = 0\n for k1 in range(3):\n for k2 in range(3):\n k = k + 1\n plt.imshow(np.reshape(np.real(X_reconstructed[k, :]), (28, 28)),\n extent=[(k1 + 1) * 28, k1 * 28, (k2 + 1) * 28, k2 * 28],\n vmin=0, vmax=255)\n plt.xlim((3 * 28, 0))\n plt.ylim((3 * 28, 0))\n plt.tick_params(axis='both', which='both', bottom=False, top=False,\n labelbottom=False)\n ax.set_xticks([])\n ax.set_yticks([])\n plt.clim([0, 250])\n plt.title('Reconstructed')\n plt.tight_layout()\n\n\ndef plot_MNIST_sample(X):\n \"\"\"\n Plots 9 images in the MNIST dataset.\n\n Args:\n X (numpy array of floats) : Data matrix each column corresponds to a\n different random variable\n\n Returns:\n Nothing.\n\n \"\"\"\n\n fig, ax = plt.subplots()\n k = 0\n for k1 in range(3):\n for k2 in range(3):\n k = k + 1\n plt.imshow(np.reshape(X[k, :], (28, 28)),\n extent=[(k1 + 1) * 28, k1 * 28, (k2+1) * 28, k2 * 28],\n vmin=0, vmax=255)\n plt.xlim((3 * 28, 0))\n plt.ylim((3 * 28, 0))\n plt.tick_params(axis='both', which='both', bottom=False, top=False,\n labelbottom=False)\n plt.clim([0, 250])\n ax.set_xticks([])\n ax.set_yticks([])\n plt.show()\n\n\ndef plot_MNIST_weights(weights):\n \"\"\"\n Visualize PCA basis vector weights for MNIST. Red = positive weights,\n blue = negative weights, white = zero weight.\n\n Args:\n weights (numpy array of floats) : PCA basis vector\n\n Returns:\n Nothing.\n \"\"\"\n\n fig, ax = plt.subplots()\n cmap = plt.cm.get_cmap('seismic')\n plt.imshow(np.real(np.reshape(weights, (28, 28))), cmap=cmap)\n plt.tick_params(axis='both', which='both', bottom=False, top=False,\n labelbottom=False)\n plt.clim(-.15, .15)\n plt.colorbar(ticks=[-.15, -.1, -.05, 0, .05, .1, .15])\n ax.set_xticks([])\n ax.set_yticks([])\n plt.show()\n\n\ndef plot_eigenvalues(evals, limit=True):\n \"\"\"\n Plots eigenvalues.\n\n Args:\n (numpy array of floats) : Vector of eigenvalues\n\n Returns:\n Nothing.\n\n \"\"\"\n\n plt.figure()\n plt.plot(np.arange(1, len(evals) + 1), evals, 'o-k')\n plt.xlabel('Component')\n plt.ylabel('Eigenvalue')\n plt.title('Scree plot')\n if limit:\n plt.show()\n```\n\n\n```python\n# @title Helper Functions\n\ndef add_noise(X, frac_noisy_pixels):\n \"\"\"\n Randomly corrupts a fraction of the pixels by setting them to random values.\n\n Args:\n X (numpy array of floats) : Data matrix\n frac_noisy_pixels (scalar) : Fraction of noisy pixels\n\n Returns:\n (numpy array of floats) : Data matrix + noise\n\n \"\"\"\n\n X_noisy = np.reshape(X, (X.shape[0] * X.shape[1]))\n N_noise_ixs = int(X_noisy.shape[0] * frac_noisy_pixels)\n noise_ixs = np.random.choice(X_noisy.shape[0], size=N_noise_ixs,\n replace=False)\n X_noisy[noise_ixs] = np.random.uniform(0, 255, noise_ixs.shape)\n X_noisy = np.reshape(X_noisy, (X.shape[0], X.shape[1]))\n\n return X_noisy\n\n\ndef change_of_basis(X, W):\n \"\"\"\n Projects data onto a new basis.\n\n Args:\n X (numpy array of floats) : Data matrix each column corresponding to a\n different random variable\n W (numpy array of floats) : new orthonormal basis columns correspond to\n basis vectors\n\n Returns:\n (numpy array of floats) : Data matrix expressed in new basis\n \"\"\"\n\n Y = np.matmul(X, W)\n\n return Y\n\n\ndef get_sample_cov_matrix(X):\n \"\"\"\n Returns the sample covariance matrix of data X.\n\n Args:\n X (numpy array of floats) : Data matrix each column corresponds to a\n different random variable\n\n Returns:\n (numpy array of floats) : Covariance matrix\n\"\"\"\n\n X = X - np.mean(X, 0)\n cov_matrix = 1 / X.shape[0] * np.matmul(X.T, X)\n return cov_matrix\n\n\ndef sort_evals_descending(evals, evectors):\n \"\"\"\n Sorts eigenvalues and eigenvectors in decreasing order. Also aligns first two\n eigenvectors to be in first two quadrants (if 2D).\n\n Args:\n evals (numpy array of floats) : Vector of eigenvalues\n evectors (numpy array of floats) : Corresponding matrix of eigenvectors\n each column corresponds to a different\n eigenvalue\n\n Returns:\n (numpy array of floats) : Vector of eigenvalues after sorting\n (numpy array of floats) : Matrix of eigenvectors after sorting\n \"\"\"\n\n index = np.flip(np.argsort(evals))\n evals = evals[index]\n evectors = evectors[:, index]\n if evals.shape[0] == 2:\n if np.arccos(np.matmul(evectors[:, 0],\n 1 / np.sqrt(2) * np.array([1, 1]))) > np.pi / 2:\n evectors[:, 0] = -evectors[:, 0]\n if np.arccos(np.matmul(evectors[:, 1],\n 1 / np.sqrt(2)*np.array([-1, 1]))) > np.pi / 2:\n evectors[:, 1] = -evectors[:, 1]\n\n return evals, evectors\n\n\ndef pca(X):\n \"\"\"\n Performs PCA on multivariate data. Eigenvalues are sorted in decreasing order\n\n Args:\n X (numpy array of floats) : Data matrix each column corresponds to a\n different random variable\n\n Returns:\n (numpy array of floats) : Data projected onto the new basis\n (numpy array of floats) : Vector of eigenvalues\n (numpy array of floats) : Corresponding matrix of eigenvectors\n\n \"\"\"\n\n X = X - np.mean(X, 0)\n cov_matrix = get_sample_cov_matrix(X)\n evals, evectors = np.linalg.eigh(cov_matrix)\n evals, evectors = sort_evals_descending(evals, evectors)\n score = change_of_basis(X, evectors)\n\n return score, evectors, evals\n```\n\n---\n# Section 1: Perform PCA on MNIST\n\nThe MNIST dataset consists of a 70,000 images of individual handwritten digits. Each image is a 28x28 pixel grayscale image. For convenience, each 28x28 pixel image is often unravelled into a single 784 (=28*28) element vector, so that the whole dataset is represented as a 70,000 x 784 matrix. Each row represents a different image, and each column represents a different pixel.\n \nEnter the following cell to load the MNIST dataset and plot the first nine images. It may take a few minutes to load.\n\n\n```python\nfrom sklearn.datasets import fetch_openml\n\n# GET mnist data\nmnist = fetch_openml(name='mnist_784', as_frame = False)\nX = mnist.data\n\n# Visualize\nplot_MNIST_sample(X)\n```\n\nThe MNIST dataset has an extrinsic dimensionality of 784, much higher than the 2-dimensional examples used in the previous tutorials! To make sense of this data, we'll use dimensionality reduction. But first, we need to determine the intrinsic dimensionality $K$ of the data. One way to do this is to look for an \"elbow\" in the scree plot, to determine which eigenvalues are signficant.\n\n## Coding Exercise 1: Scree plot of MNIST\n\nIn this exercise you will examine the scree plot in the MNIST dataset.\n\n**Steps:**\n- Perform PCA on the dataset using our function `pca` from tutorial 2 (already loaded in) and examine the scree plot. \n- When do the eigenvalues appear (by eye) to reach zero? (**Hint:** use `plt.xlim` to zoom into a section of the plot).\n\n\n\n```python\nhelp(pca)\n```\n\n Help on function pca in module __main__:\n \n pca(X)\n Performs PCA on multivariate data. Eigenvalues are sorted in decreasing order\n \n Args:\n X (numpy array of floats) : Data matrix each column corresponds to a\n different random variable\n \n Returns:\n (numpy array of floats) : Data projected onto the new basis\n (numpy array of floats) : Vector of eigenvalues\n (numpy array of floats) : Corresponding matrix of eigenvectors\n \n\n\n\n```python\n#################################################\n## TODO for students\n# Fill out function and remove\n# raise NotImplementedError(\"Student excercise: perform PCA and visualize scree plot\")\n#################################################\n\n# Perform PCA\nscore, evectors, evals = pca(X)\n\n# Plot the eigenvalues\nplot_eigenvalues(evals, limit=False)\nplt.xlim([0, 100]) # limit x-axis up to 100 for zooming\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W1D5_DimensionalityReduction/solutions/W1D5_Tutorial3_Solution_d8411e54.py)\n\n*Example output:*\n\n\n\n\n\n---\n# Section 2: Calculate the variance explained\n\n*Estimated timing to here from start of tutorial: 15 min*\n\nThe scree plot suggests that most of the eigenvalues are near zero, with fewer than 100 having large values. Another common way to determine the intrinsic dimensionality is by considering the variance explained. This can be examined with a cumulative plot of the fraction of the total variance explained by the top $K$ components, i.e.,\n\n\\begin{align}\n\\text{var explained} = \\frac{\\sum_{i=1}^K \\lambda_i}{\\sum_{i=1}^N \\lambda_i}\n\\end{align}\n\nwhere $\\lambda_i$ is the $i^{th}$ eigenvalue and $N$ is the total number of components (the original number of dimensions in the data).\n\nThe intrinsic dimensionality is often quantified by the $K$ necessary to explain a large proportion of the total variance of the data (often a defined threshold, e.g., 90%).\n\n## Coding Exercise 2: Plot the explained variance\n\nIn this exercise you will plot the explained variance.\n\n**Steps:**\n- Fill in the function below to calculate the fraction variance explained as a function of the number of principal componenets. **Hint:** use `np.cumsum`.\n- Plot the variance explained using `plot_variance_explained`.\n\n**Questions:**\n- How many principal components are required to explain 90% of the variance?\n- How does the intrinsic dimensionality of this dataset compare to its extrinsic dimensionality?\n\n\n\n```python\ndef get_variance_explained(evals):\n \"\"\"\n Calculates variance explained from the eigenvalues.\n\n Args:\n evals (numpy array of floats) : Vector of eigenvalues\n\n Returns:\n (numpy array of floats) : Vector of variance explained\n\n \"\"\"\n\n #################################################\n ## TO DO for students: calculate the explained variance using the equation\n ## from Section 2.\n # Comment once you've filled in the function\n # raise NotImplementedError(\"Student excercise: calculate explaine variance!\")\n #################################################\n\n # Cumulatively sum the eigenvalues\n csum = np.cumsum(evals)\n\n # Normalize by the sum of eigenvalues\n variance_explained = csum / np.sum(evals)\n\n return variance_explained\n\n\n# Calculate the variance explained\nvariance_explained = get_variance_explained(evals)\n\n# Visualize\nplot_variance_explained(variance_explained)\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W1D5_DimensionalityReduction/solutions/W1D5_Tutorial3_Solution_a4ac4c9c.py)\n\n*Example output:*\n\n\n\n\n\n---\n# Section 3: Reconstruct data with different numbers of PCs\n\n*Estimated timing to here from start of tutorial: 25 min*\n\n\n\n\n```python\n# @title Video 2: Data Reconstruction\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = 'https://player.bilibili.com/player.html?bvid={0}&page={1}'.format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=\"BV1XK4y1s7KF\", width=854, height=480, fs=1)\n print('Video available at https://www.bilibili.com/video/{0}'.format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=\"ZCUhW26AdBQ\", width=854, height=480, fs=1, rel=0)\n print('Video available at https://youtube.com/watch?v=' + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nNow we have seen that the top 100 or so principal components of the data can explain most of the variance. We can use this fact to perform *dimensionality reduction*, i.e., by storing the data using only 100 components rather than the samples of all 784 pixels. Remarkably, we will be able to reconstruct much of the structure of the data using only the top 100 components. To see this, recall that to perform PCA we projected the data $\\bf X$ onto the eigenvectors of the covariance matrix:\n\\begin{align}\n\\bf S = X W\n\\end{align}\nSince $\\bf W$ is an orthogonal matrix, ${\\bf W}^{-1} = {\\bf W}^T$. So by multiplying by ${\\bf W}^T$ on each side we can rewrite this equation as \n\\begin{align}\n{\\bf X = S W}^T.\n\\end{align}\nThis now gives us a way to reconstruct the data matrix from the scores and loadings. To reconstruct the data from a low-dimensional approximation, we just have to truncate these matrices. Let's denote ${\\bf S}_{1:K}$ and ${\\bf W}_{1:K}$ the matrices with only the first $K$ columns of $\\bf S$ and $\\bf W$, respectively. Then our reconstruction is:\n\\begin{align}\n{\\bf \\hat X = S}_{1:K} ({\\bf W}_{1:K})^T.\n\\end{align}\n\n\n## Coding Exercise 3: Data reconstruction\n\nFill in the function below to reconstruct the data using different numbers of principal components. \n\n**Steps:**\n\n* Fill in the following function to reconstruct the data based on the weights and scores. Don't forget to add the mean!\n* Make sure your function works by reconstructing the data with all $K=784$ components. The two images should look identical.\n\n\n```python\nevectors[0:100,0:100].shape\n```\n\n\n\n\n (100, 100)\n\n\n\n\n```python\nX_mean.shape\n```\n\n\n\n\n (784,)\n\n\n\n\n```python\nscore.shape\n```\n\n\n\n\n (70000, 784)\n\n\n\n\n```python\n784*784\n```\n\n\n\n\n 614656\n\n\n\n\n```python\ndef reconstruct_data(score, evectors, X_mean, K):\n \"\"\"\n Reconstruct the data based on the top K components.\n\n Args:\n score (numpy array of floats) : Score matrix\n evectors (numpy array of floats) : Matrix of eigenvectors\n X_mean (numpy array of floats) : Vector corresponding to data mean\n K (scalar) : Number of components to include\n\n Returns:\n (numpy array of floats) : Matrix of reconstructed data\n\n \"\"\"\n\n #################################################\n ## TO DO for students: Reconstruct the original data in X_reconstructed\n # Comment once you've filled in the function\n # raise NotImplementedError(\"Student excercise: reconstructing data function!\")\n #################################################\n\n # Reconstruct the data from the score and eigenvectors\n # Don't forget to add the mean!!\n # Grab all rows and 0th-Kth columns in both score and w.T\n # and multiply those together, then add back the mean we sutracted out earlier\n X_reconstructed = np.matmul(score[:, :K], evectors[:, :K].T) + X_mean\n\n return X_reconstructed\n\n\n# K = 100\nK = 784\n\n# Reconstruct the data based on all components\nX_mean = np.mean(X, 0)\nX_reconstructed = reconstruct_data(score, evectors, X_mean, K)\n\n# Plot the data and reconstruction\nplot_MNIST_reconstruction(X, X_reconstructed)\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W1D5_DimensionalityReduction/solutions/W1D5_Tutorial3_Solution_e3395916.py)\n\n*Example output:*\n\n\n\n\n\n## Interactive Demo 3: Reconstruct the data matrix using different numbers of PCs\n\nNow run the code below and experiment with the slider to reconstruct the data matrix using different numbers of principal components.\n\n1. How many principal components are necessary to reconstruct the numbers (by eye)? How does this relate to the intrinsic dimensionality of the data?\n2. Do you see any information in the data with only a single principal component?\n\n\n```python\n# @title\n\n# @markdown Make sure you execute this cell to enable the widget!\n\n\ndef refresh(K=100):\n X_reconstructed = reconstruct_data(score, evectors, X_mean, K)\n plot_MNIST_reconstruction(X, X_reconstructed)\n plt.title('Reconstructed, K={}'.format(K))\n\n\n_ = widgets.interact(refresh, K=(1, 784, 10))\n```\n\n\n interactive(children=(IntSlider(value=100, description='K', max=784, min=1, step=10), Output()), _dom_classes=\u2026\n\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W1D5_DimensionalityReduction/solutions/W1D5_Tutorial3_Solution_f659a785.py)\n\n\n\n---\n# Section 4: Visualize PCA components\n\n*Estimated timing to here from start of tutorial: 40 min*\n\n## Coding Exercise 4: Visualization of the weights\n\nNext, let's take a closer look at the first principal component by visualizing its corresponding weights. \n\n**Steps:**\n\n* Enter `plot_MNIST_weights` to visualize the weights of the first basis vector.\n* What structure do you see? Which pixels have a strong positive weighting? Which have a strong negative weighting? What kinds of images would this basis vector differentiate (hint: think about the last demo with 1 component)?\n* Try visualizing the second and third basis vectors. Do you see any structure? What about the 100th basis vector? 500th? 700th?\n\n\n```python\nhelp(plot_MNIST_weights)\n```\n\n Help on function plot_MNIST_weights in module __main__:\n \n plot_MNIST_weights(weights)\n Visualize PCA basis vector weights for MNIST. Red = positive weights,\n blue = negative weights, white = zero weight.\n \n Args:\n weights (numpy array of floats) : PCA basis vector\n \n Returns:\n Nothing.\n \n\n\n\n```python\n################################################################\n# Comment once you've filled in the function\n# raise NotImplementedError(\"Student excercise: visualize PCA components\")\n################################################################\n\n# Plot the weights of the first principal component\n\nplot_MNIST_weights(evectors[:,0])\n\n\n# pixels with strong weighting are differentiating features for that number/shape\n# pixels with strong neg weighting are negative spaces\n\n```\n\n\n```python\nplot_MNIST_weights(np.sum(evectors[:,0:4],1))\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W1D5_DimensionalityReduction/solutions/W1D5_Tutorial3_Solution_d27990ad.py)\n\n*Example output:*\n\n\n\n\n\n---\n# Summary\n\n*Estimated timing of tutorial: 50 minutes*\n\n* In this tutorial, we learned how to use PCA for dimensionality reduction by selecting the top principal components. This can be useful as the intrinsic dimensionality ($K$) is often less than the extrinsic dimensionality ($N$) in neural data. $K$ can be inferred by choosing the number of eigenvalues necessary to capture some fraction of the variance.\n* We also learned how to reconstruct an approximation of the original data using the top $K$ principal components. In fact, an alternate formulation of PCA is to find the $K$ dimensional space that minimizes the reconstruction error.\n* Noise tends to inflate the apparent intrinsic dimensionality, however the higher components reflect noise rather than new structure in the data. PCA can be used for denoising data by removing noisy higher components.\n* In MNIST, the weights corresponding to the first principal component appear to discriminate between a 0 and 1. We will discuss the implications of this for data visualization in the following tutorial.\n\n---\n# Notation\n\n\\begin{align}\nK &\\quad \\text{selected number of principal components}\\\\\nN &\\quad \\text{total number of principal components}\\\\\n\\bf W &\\quad \\text{weights, loadings matrix}\\\\\n{\\bf X} &\\quad \\text{original data matrix}\\\\\n\\bf S &\\quad \\text{projected matrix, scores}\\\\\n{\\bf S}_{1:K} &\\quad \\text{first K columns of score matrix } \\bf S\\\\\n{\\bf W}_{1:K} &\\quad \\text{first K columns of weight matrix } \\bf W\\\\\n\\end{align}\n\n---\n# Bonus\n\n---\n## Bonus Section 1: Examine denoising using PCA\n\nIn this lecture, we saw that PCA finds an optimal low-dimensional basis to minimize the reconstruction error. Because of this property, PCA can be useful for denoising corrupted samples of the data.\n\n### Bonus Coding Exercise 1: Add noise to the data\nIn this exercise you will add salt-and-pepper noise to the original data and see how that affects the eigenvalues. \n\n**Steps:**\n- Use the function `add_noise` to add noise to 20% of the pixels.\n- Then, perform PCA and plot the variance explained. How many principal components are required to explain 90% of the variance? How does this compare to the original data? \n\n\n\n```python\nhelp(add_noise)\n```\n\n Help on function add_noise in module __main__:\n \n add_noise(X, frac_noisy_pixels)\n Randomly corrupts a fraction of the pixels by setting them to random values.\n \n Args:\n X (numpy array of floats) : Data matrix\n frac_noisy_pixels (scalar) : Fraction of noisy pixels\n \n Returns:\n (numpy array of floats) : Data matrix + noise\n \n\n\n\n```python\n#################################################\n## TO DO for students\n# Comment once you've filled in the function\n# raise NotImplementedError(\"Student excercise: make MNIST noisy and compute PCA!\")\n#################################################\n\nnp.random.seed(2020) # set random seed\n\n# Add noise to data\nX_noisy = ...\n\n# Perform PCA on noisy data\nscore_noisy, evectors_noisy, evals_noisy = ...\n\n# Compute variance explained\nvariance_explained_noisy = ...\n\n# Visualize\nplot_MNIST_sample(X_noisy)\nplot_variance_explained(variance_explained_noisy)\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W1D5_DimensionalityReduction/solutions/W1D5_Tutorial3_Solution_9615a6cd.py)\n\n*Example output:*\n\n\n\n\n\n\n\n### Bonus Coding Exercise 2: Denoising\n\nNext, use PCA to perform denoising by projecting the noise-corrupted data onto the basis vectors found from the original dataset. By taking the top K components of this projection, we can reduce noise in dimensions orthogonal to the K-dimensional latent space. \n\n**Steps:**\n- Subtract the mean of the noise-corrupted data.\n- Project the data onto the basis found with the original dataset (`evectors`, not `evectors_noisy`) and take the top $K$ components. \n- Reconstruct the data as normal, using the top 50 components. \n- Play around with the amount of noise and K to build intuition.\n\n\n\n```python\n#################################################\n## TO DO for students\n# Comment once you've filled in the function\nraise NotImplementedError(\"Student excercise: reconstruct noisy data from PCA\")\n#################################################\n\n# Compute mean of noise-corrupted data\nX_noisy_mean = ...\n\n# Project onto the original basis vectors\nprojX_noisy = ...\n\n# Reconstruct the data using the top 50 components\nX_reconstructed = ...\n\n# Visualize\nplot_MNIST_reconstruction(X_noisy, X_reconstructed)\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W1D5_DimensionalityReduction/solutions/W1D5_Tutorial3_Solution_4c4115fa.py)\n\n*Example output:*\n\n\n\n\n", "meta": {"hexsha": "8497be5b93d5fa997c9c759995f530fb211750f8", "size": 373879, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "W1D5_Tutorial3.ipynb", "max_stars_repo_name": "wasita/course-content", "max_stars_repo_head_hexsha": "7d349723c2524333f5af40d617770aefb008aaa3", "max_stars_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "W1D5_Tutorial3.ipynb", "max_issues_repo_name": "wasita/course-content", "max_issues_repo_head_hexsha": "7d349723c2524333f5af40d617770aefb008aaa3", "max_issues_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "W1D5_Tutorial3.ipynb", "max_forks_repo_name": "wasita/course-content", "max_forks_repo_head_hexsha": "7d349723c2524333f5af40d617770aefb008aaa3", "max_forks_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 158.4233050847, "max_line_length": 52786, "alphanum_fraction": 0.8572827038, "converted": true, "num_tokens": 6864, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419831347361, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.4260356720467403}} {"text": "\n\n\n# Welcome to Session 01\n\nThis is a [Google Colaboratory](https://colab.research.google.com/notebooks/welcome.ipynb) notebook file. Python programs are run directly in the browser\u2014a great way to learn and use TensorFlow. To follow this tutorial, run the notebook in Google Colab by clicking the button at the top of this page.\n\n1. In Colab, connect to a Python runtime: At the top-right of the menu bar, select *CONNECT*.\n2. Run all the notebook code cells: Select *Runtime* > *Run all*.\n\n\n```python\n#@title ## Mounting Gdrive\n\nUSE_G_COLAB = True #@param {type:\"boolean\"}\n\nif USE_G_COLAB:\n from google.colab import drive\n\n \n drive.mount('/content/drive', force_remount=True)\n```\n\n Mounted at /content/drive\n\n\n\n```python\n#@title ## Project Root\n\nroot_dir = ''\n\nif USE_G_COLAB:\n root_dir = '/content/drive/My Drive/dl_app/' #@param {type:\"string\"}\n```\n\n\n```python\n!ls $root_dir\n```\n\n ls: cannot access '/content/drive/My': No such file or directory\n ls: cannot access 'Drive/workshops/2019_07_21/sessions_01/': No such file or directory\n\n\n\n```python\n#@title ## Installing requried packages\n\n#@markdown ---\n#@markdown - [TensorFlow 2.0-beta](https://www.tensorflow.org/install/gpu)\n#@markdown - [Watermark](https://github.com/rasbt/watermark)\n!pip install -q tensorflow-gpu==2.0.0-beta1\n!pip install -qU watermark\n```\n\n\n```python\n#@title ## Custom Matplotlib Style\nmpl_style = \"https://gist.githubusercontent.com/m3hrdadfi/af8aca01094afb7d3e5b46de9ad8d509/raw/871ec5d721a3b438c3c896718ea4aafc91ea9744/gadfly.mplstyle\" #@param {type:\"string\"}\n\n!wget -q $mpl_style -O /root/.config/matplotlib/matplotlibrc\n```\n\n\n```python\n#@title ## General Paramas\n\n#@markdown > A random seed is a number used to initialize a pseudorandom number generator. For a seed to be used in a pseudorandom number generator, it does not need to be random\nRANDOM_SEED = 141 #@param {type:\"integer\"}\n```\n\n# Overview\n\n\n\n$z_2 = XW_1$\n\n$a_2 = f(z_2)$\n\n$z_3 = a_2W_2$\n\n$\\hat{y} = softmax(z_3)$ # classification\n\n*where*:\n* $X$ = inputs | $\\in \\mathbb{R}^{NXD}$ ($D$ is the number of features)\n* $W_1$ = 1st layer weights | $\\in \\mathbb{R}^{DXH}$ ($H$ is the number of hidden units in layer 1)\n* $z_2$ = outputs from first layer's weights $\\in \\mathbb{R}^{NXH}$\n* $f$ = non-linear activation function\n* $a_2$ = activation applied first layer's outputs | $\\in \\mathbb{R}^{NXH}$\n* $W_2$ = 2nd layer weights | $\\in \\mathbb{R}^{HXC}$ ($C$ is the number of classes)\n* $\\hat{y}$ = prediction | $\\in \\mathbb{R}^{NXC}$ ($N$ is the number of samples)\n\nThis is a simple two-layer MLP. \n\n* **Objective:** Predict the probability of class $y$ given the inputs $X$. Non-linearity is introduced to model the complex, non-linear data.\n* **Advantages:**\n * Can model non-linear patterns in the data really well.\n* **Disadvantages:**\n * Overfits easily.\n * Computationally intensive as network increases in size.\n * Not easily interpretable.\n* **Miscellaneous:** Future neural network architectures that we'll see use the MLP as a modular unit for feed forward operations (affine transformation (XW) followed by a non-linear operation).\n\n## 7 Common Nonlinear Activation Functions and How to Choose an Activation Function\n\n#### SIGMOID / LOGISTIC\n\n*ADVANTAGES*\n- Smooth gradient, preventing \u201cjumps\u201d in output values.\n- Output values bound between 0 and 1, normalizing the output of each neuron.\n- Clear predictions\u2014For X above 2 or below -2, tends to bring the Y value (the prediction) to the edge of the curve, very close to 1 or 0. This enables clear predictions.\n\n*DISADVANTAGES*\n- Vanishing gradient\u2014for very high or very low values of X, there is almost no change to the prediction, causing a vanishing gradient problem. This can result in the network refusing to learn further, or being too slow to reach an accurate prediction.\n- Outputs not zero centered.\n- Computationally expensive\n

                                        \n\n

                                        \n\n#### TANH / HYPERBOLIC TANGENT\n\n*ADVANTAGES*\n- Zero centered\u2014making it easier to model inputs that have strongly negative, neutral, and strongly positive values.\n- Otherwise like the Sigmoid function.\n\n*DISADVANTAGES*\n- Like the Sigmoid function\n

                                        \n\n

                                        \n\n#### RELU (RECTIFIED LINEAR UNIT)\n\n*ADVANTAGES*\n- Computationally efficient\u2014allows the network to converge very quickly\n- Non-linear\u2014although it looks like a linear function, ReLU has a derivative function and allows for backpropagation.\n\n*DISADVANTAGES*\n- The Dying ReLU problem\u2014when inputs approach zero, or are negative, the gradient of the function becomes zero, the network cannot perform backpropagation and cannot learn.\n- Like the Sigmoid function\n

                                        \n\n

                                        \n \n#### LEAKY RELU\n\n*ADVANTAGES*\n- Prevents dying ReLU problem\u2014this variation of ReLU has a small positive slope in the negative area, so it does enable backpropagation, even for negative input values\nOtherwise like ReLU\n\n*DISADVANTAGES*\nResults not consistent\u2014leaky ReLU does not provide consistent predictions for negative input values.\n

                                        \n\n

                                        \n\n#### PARAMETRIC RELU\n\n*ADVANTAGES*\n- Allows the negative slope to be learned\u2014unlike leaky ReLU, this function provides the slope of the negative part of the function as an argument. It is, therefore, possible to perform backpropagation and learn the most appropriate value of \u03b1.\nOtherwise like ReLU\n\n*DISADVANTAGES*\nMay perform differently for different problems.\n\n

                                        \n$ f(x) = max( \\alpha x,x)$\n

                                        \n \n#### SOFTMAX\n\n*ADVANTAGES*\n- Able to handle multiple classes only one class in other activation functions\u2014normalizes the outputs for each class between 0 and 1, and divides by their sum, giving the probability of the input value being in a specific class.\n- Useful for output neurons\u2014typically Softmax is used only for the output layer, for neural networks that need to classify inputs into multiple categories.\n\n

                                        \n$ \\frac{ e^{ z_{j} } }{ \\sum_{k=1}^c e^{ z_{k} } } $\n

                                        \n \n#### SWISH\n\nSwish is a new, self-gated activation function discovered by researchers at Google. According to their paper, it performs better than ReLU with a similar level of computational efficiency. In experiments on ImageNet with identical models running ReLU and Swish, the new function achieved top -1 classification accuracy 0.6-0.9% higher.\n

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                                        \n\n# Training\n\n*Steps*:\n\n1. Randomly initialize the model's weights $W$ (we'll cover more effective initalization strategies in future lessons).\n2. Feed inputs $X$ into the model to do the forward pass and receive the probabilities.\n3. Compare the predictions $\\hat{y}$ (ex. [0.3, 0.3, 0.4]]) with the actual target values $y$ (ex. class 2 would look like [0, 0, 1]) with the objective (cost) function to determine loss $J$. A common objective function for classification tasks is cross-entropy loss. \n * $z_2 = XW_1$\n * $a_2 = max(0, z_2)$ # ReLU activation\n * $z_3 = a_2W_2$\n * $\\hat{y} = softmax(z_3)$\n * $J(\\theta) = - \\sum_i y_i ln (\\hat{y_i}) $\n4. Calculate the gradient of loss $J(\\theta)$ w.r.t to the model weights. \n * $ \\frac{\\partial{J}}{\\partial{W_{2j}}} = a_2\\hat{y}, \\frac{\\partial{J}}{\\partial{W_{2y}}} = a_2(\\hat{y}-1) $\n * $ \\frac{\\partial{J}}{\\partial{W_1}} = \\frac{\\partial{J}}{\\partial{\\hat{y}}} \\frac{\\partial{\\hat{y}}}{\\partial{a_2}} \\frac{\\partial{a_2}}{\\partial{z_2}} \\frac{\\partial{z_2}}{\\partial{W_1}} = W_2(\\partial{scores})(\\partial{ReLU})X $\n \n5. Apply backpropagation to update the weights $W$ using gradient descent. The updates will penalize the probabiltiy for the incorrect classes (j) and encourage a higher probability for the correct class (y).\n * $W_i = W_i - \\alpha\\frac{\\partial{J}}{\\partial{W_i}}$\n6. Repeat steps 2 - 4 until model performs well.\n\n# Terminologies\n\n## Neural Network Learning as Optimization\nA deep learning neural network learns to map a set of inputs to a set of outputs from training data.\n\nWe cannot calculate the perfect weights for a neural network; there are too many unknowns. Instead, the problem of learning is cast as a search or optimization problem and an algorithm is used to navigate the space of possible sets of weights the model may use in order to make good or good enough predictions.\n\nTypically, a neural network model is trained using the stochastic gradient descent optimization algorithm and weights are updated using the backpropagation of error algorithm.\n\nThe \u201cgradient\u201d in gradient descent refers to an error gradient. The model with a given set of weights is used to make predictions and the error for those predictions is calculated.\n\nThe gradient descent algorithm seeks to change the weights so that the next evaluation reduces the error, meaning the optimization algorithm is navigating down the gradient (or slope) of error.\n\nNow that we know that training neural nets solves an optimization problem, we can look at how the error of a given set of weights is calculated.\n\n## What Is a Loss Function and Loss?\n\nIn the context of an optimization algorithm, the function used to evaluate a candidate solution (i.e. a set of weights) is referred to as the objective function.\n\nWe may seek to maximize or minimize the objective function, meaning that we are searching for a candidate solution that has the highest or lowest score respectively.\n\nTypically, with neural networks, we seek to minimize the error. As such, the objective function is often referred to as a cost function or a loss function and the value calculated by the loss function is referred to as simply \u201closs.\u201d\nThe cost or loss function has an important job in that it must faithfully distill all aspects of the model down into a single number in such a way that improvements in that number are a sign of a better model.\nWe will review best practice or default values for each problem type with regard to the output layer and loss function.\n\n## Regression Problem\n\nA problem where you predict a real-value quantity.\n\nOutput Layer Configuration: One node with a linear activation unit.\nLoss Function: Mean Squared Error (MSE).\n\n## Binary Classification Problem\n\nA problem where you classify an example as belonging to one of two classes.\n\nThe problem is framed as predicting the likelihood of an example belonging to class one, e.g. the class that you assign the integer value 1, whereas the other class is assigned the value 0.\n\nOutput Layer Configuration: One node with a sigmoid activation unit.\nLoss Function: Cross-Entropy, also referred to as Logarithmic loss.\n\n## Multi-Class Classification Problem\n\nA problem where you classify an example as belonging to one of more than two classes.\n\nThe problem is framed as predicting the likelihood of an example belonging to each class.\n\nOutput Layer Configuration: One node for each class using the softmax activation function.\nLoss Function: Cross-Entropy, also referred to as Logarithmic loss.\n\n## How to Implement Loss Functions\n\nIn order to make the loss functions concrete, this section explains how each of the main types of loss function works.\n\n## Mean Squared Error Loss\n\nMean Squared Error loss, or MSE for short, is calculated as the average of the squared differences between the predicted and actual values.\n\nThe result is always positive regardless of the sign of the predicted and actual values and a perfect value is 0.0. The loss value is minimized, although it can be used in a maximization optimization process by making the score negative.\n\n## Cross-Entropy Loss (or Log Loss)\n\nCross-entropy loss is often simply referred to as \u201ccross-entropy,\u201d \u201clogarithmic loss,\u201d \u201clogistic loss,\u201d or \u201clog loss\u201d for short.\n\nEach predicted probability is compared to the actual class output value (0 or 1) and a score is calculated that penalizes the probability based on the distance from the expected value. The penalty is logarithmic, offering a small score for small differences (0.1 or 0.2) and enormous score for a large difference (0.9 or 1.0).\n\nCross-entropy loss is minimized, where smaller values represent a better model than larger values. A model that predicts perfect probabilities has a cross entropy or log loss of 0.0.\n\nCross-entropy for a binary or two class prediction problem is actually calculated as the average cross entropy across all examples.\n\n## An overview of gradient descent optimization algorithms\n\nGradient descent is one of the most popular algorithms to perform optimization and by far the most common way to optimize neural networks. At the same time, every state-of-the-art Deep Learning library contains implementations of various algorithms to optimize gradient descent. These algorithms, however, are often used as black-box optimizers, as practical explanations of their strengths and weaknesses are hard to come by.\n\n## Gradient descent variants\n\nThere are three variants of gradient descent, which differ in how much data we use to compute the gradient of the objective function. Depending on the amount of data, we make a trade-off between the accuracy of the parameter update and the time it takes to perform an update.\n\n## Batch gradient descent\n\nVanilla gradient descent, aka batch gradient descent, computes the gradient of the cost function w.r.t. to the parameters \u03b8 for the entire training dataset:\n

                                        \n$ \\theta = \\theta - \\eta \\nabla _{ \\theta } J( \\theta )$\n

                                        \n We then update our parameters in the opposite direction of the gradients with the learning rate determining how big of an update we perform. Batch gradient descent is guaranteed to converge to the global minimum for convex error surfaces and to a local minimum for non-convex surfaces.\n As we need to calculate the gradients for the whole dataset to perform just one update, batch gradient descent can be very slow and is intractable for datasets that don't fit in memory. Batch gradient descent also doesn't allow us to update our model online, i.e. with new examples on-the-fly.\n \n## Stochastic gradient descent\n\nStochastic gradient descent (SGD) in contrast performs a parameter update for each training example x(i) and label y(i):\n

                                        \n$\\theta = \\theta - \\eta \\nabla _{ \\theta } J( \\theta; x^{i}; y^{i})$\n

                                        \n Batch gradient descent performs redundant computations for large datasets, as it recomputes gradients for similar examples before each parameter update. SGD does away with this redundancy by performing one update at a time. It is therefore usually much faster and can also be used to learn online.\n While batch gradient descent converges to the minimum of the basin the parameters are placed in, SGD's fluctuation, on the one hand, enables it to jump to new and potentially better local minima. On the other hand, this ultimately complicates convergence to the exact minimum, as SGD will keep overshooting. However, it has been shown that when we slowly decrease the learning rate, SGD shows the same convergence behaviour as batch gradient descent, almost certainly converging to a local or the global minimum for non-convex and convex optimization respectively.\n \n## Mini-batch gradient descent\n\nMini-batch gradient descent finally takes the best of both worlds and performs an update for every mini-batch of n training examples:\n

                                        \n$\\theta = \\theta - \\eta \\nabla _{ \\theta } J( \\theta; x^{i:i+n}; y^{i:i+n})$\n

                                        \n \nThis way, it a) reduces the variance of the parameter updates, which can lead to more stable convergence; and b) can make use of highly optimized matrix optimizations common to state-of-the-art deep learning libraries that make computing the gradient w.r.t. a mini-batch very efficient. Common mini-batch sizes range between 50 and 256, but can vary for different applications. Mini-batch gradient descent is typically the algorithm of choice when training a neural network and the term SGD usually is employed also when mini-batches are used. \n\n## Gradient descent optimization algorithms\n- Momentum\n- Nesterov accelerated gradient (NAG)\n- Adagrad\n- Adadelta\n- RMSprop\n- Adam\n- AdaMax\n- Nadam\n- AMSGrad\n\n

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                                        \n \n\n

                                        \n\n## Which optimizer to use?\n\nSo, which optimizer should you now use? If your input data is sparse, then you likely achieve the best results using one of the adaptive learning-rate methods. An additional benefit is that you won't need to tune the learning rate but likely achieve the best results with the default value.\n\nIn summary, RMSprop is an extension of Adagrad that deals with its radically diminishing learning rates. It is identical to Adadelta, except that Adadelta uses the RMS of parameter updates in the numinator update rule. Adam, finally, adds bias-correction and momentum to RMSprop. Insofar, RMSprop, Adadelta, and Adam are very similar algorithms that do well in similar circumstances. It is shown that its bias-correction helps Adam slightly outperform RMSprop towards the end of optimization as gradients become sparser. Insofar, Adam might be the best overall choice.\n\n## Hyperparameters\n\nModel optimization is one of the toughest challenges in the implementation of machine learning solutions. Entire branches of machine learning and deep learning theory have been dedicated to the optimization of models. Typically, we think about model optimization as a process of regularly modifying the code of the model in order to minimize the testing error. However, deep learning optimization often entails fine tuning elements that live outside the model but that can heavily influence its behavior. Deep learning often refers to those hidden elements as hyperparameters as they are one of the most critical components of any machine learning application.\n\n#### Some Examples of Hyperparameters\n\nThe number and diversity of hyperparameters in machine learning algorithms is very specific to each model. However, there some classic hyperparameters that we should always keep our eyes on and that should help you think about this aspect of machine learning solutions:\n\u00b7 Learning Rate: The mother of all hyperparameters, the learning rate quantifies the learning progress of a model in a way that can be used to optimize its capacity.\n\u00b7 Number of Hidden Units: A classic hyperparameter in deep learning algorithms, the number of hidden units is key to regulate the representational capacity of a model.\n\u00b7 Convolution Kernel Width: In convolutional Neural Networks(CNNs), the Kernel Width influences the number of parameters in a model which, in turns, influences its capacity.\n\n## Metrics to Evaluate your Machine Learning Algorithm\n\nEvaluating your machine learning algorithm is an essential part of any project. Your model may give you satisfying results when evaluated using a metric say accuracy_score but may give poor results when evaluated against other metrics such as logarithmic_loss or any other such metric. Most of the times we use classification accuracy to measure the performance of our model, however it is not enough to truly judge our model. In this post, we will cover different types of evaluation metrics available.\n\n## Classification Accuracy\n\nClassification Accuracy is what we usually mean, when we use the term accuracy. It is the ratio of number of correct predictions to the total number of input samples.\n\n

                                        \n$Accuracy = \\frac{Number of Correct predictions}{Total Number of Predictions made}$\n

                                        \n\nIt works well only if there are equal number of samples belonging to each class.\nFor example, consider that there are 98% samples of class A and 2% samples of class B in our training set. Then our model can easily get 98% training accuracy by simply predicting every training sample belonging to class A.\nWhen the same model is tested on a test set with 60% samples of class A and 40% samples of class B, then the test accuracy would drop down to 60%. Classification Accuracy is great, but gives us the false sense of achieving high accuracy.\nThe real problem arises, when the cost of misclassification of the minor class samples are very high. If we deal with a rare but fatal disease, the cost of failing to diagnose the disease of a sick person is much higher than the cost of sending a healthy person to more tests.\n\n## Confusion Matrix\n\nConfusion Matrix as the name suggests gives us a matrix as output and describes the complete performance of the model.\nLets assume we have a binary classification problem. We have some samples belonging to two classes : YES or NO. Also, we have our own classifier which predicts a class for a given input sample. On testing our model on 165 samples ,we get the following result.\n\n\n

                                        \n $LogLoss = - \\frac{1}{n} \\sum\\limits_{i=1}^n [y_i \\cdot log_e(\\hat{y_i}) + (1-y_i) \\cdot log_e(1-\\hat{y_i}) ]$\n

                                        \n\nwhere,\ny_ij, indicates whether sample i belongs to class j or not\np_ij, indicates the probability of sample i belonging to class j\nLog Loss has no upper bound and it exists on the range Log Loss nearer to 0 indicates higher accuracy, whereas if the Log Loss is away from 0 then it indicates lower accuracy.\nIn general, minimising Log Loss gives greater accuracy for the classifier.\n\n| n=165 | Predicted: NO | Predicted: YES |\n|---|---|---|\n| Actual: NO | 50 | 10 |\n| Actual: YES | 5 | 100 |\n\nThere are 4 important terms :\n- True Positives : The cases in which we predicted YES and the actual output was also YES.\n- True Negatives : The cases in which we predicted NO and the actual output was NO.\n- False Positives : The cases in which we predicted YES and the actual output was NO.\n- False Negatives : The cases in which we predicted NO and the actual output was YES.\nAccuracy for the matrix can be calculated by taking average of the values lying across the \u201cmain diagonal\u201d i.e\n

                                        \n$Accuracy = \\frac{TruePositive+ TrueNegative}{Total Number of Samples}$\n
                                        \n $Accuracy = \\frac{100+50}{165} = 0.91$\n

                                        \n\nConfusion Matrix forms the basis for the other types of metrics.\n\n## Area Under Curve\n\nArea Under Curve(AUC) is one of the most widely used metrics for evaluation. It is used for binary classification problem. AUC of a classifier is equal to the probability that the classifier will rank a randomly chosen positive example higher than a randomly chosen negative example. Before defining AUC, let us understand two basic terms :\n- True Positive Rate (Sensitivity) : True Positive Rate is defined as TP/ (FN+TP). True Positive Rate corresponds to the proportion of positive data points that are correctly considered as positive, with respect to all positive data points.\n

                                        \n$TruePositiveRate = \\frac{TruePositive}{FalseNegative+TruePositive}$\n

                                        \n- False Positive Rate (Specificity) : False Positive Rate is defined as FP / (FP+TN). False Positive Rate corresponds to the proportion of negative data points that are mistakenly considered as positive, with respect to all negative data points.\n

                                        \n$FalsePositiveRate = \\frac{FalsePositive}{TrueNegative+FalsePositive}$\n

                                        \n False Positive Rate and True Positive Rate both have values in the range [0, 1]. FPR and TPR bot hare computed at threshold values such as (0.00, 0.02, 0.04, \u2026., 1.00) and a graph is drawn. AUC is the area under the curve of plot False Positive Rate vs True Positive Rate at different points in [0, 1].\n

                                        \n\n

                                        \nAs evident, AUC has a range of [0, 1]. The greater the value, the better is the performance of our model.\n\n## F1 Score\n\nF1 Score is used to measure a test\u2019s accuracy\nF1 Score is the Harmonic Mean between precision and recall. The range for F1 Score is [0, 1]. It tells you how precise your classifier is (how many instances it classifies correctly), as well as how robust it is (it does not miss a significant number of instances).\nHigh precision but lower recall, gives you an extremely accurate, but it then misses a large number of instances that are difficult to classify. The greater the F1 Score, the better is the performance of our model. Mathematically, it can be expressed as :\n\n

                                        \n$F1 = 2*\\frac{1}{\\frac{1}{precision}+\\frac{1}{recall}}$\n

                                        \n- Precision : It is the number of correct positive results divided by the number of positive results predicted by the classifier.\n\n

                                        \n$Precision = 2*\\frac{TruePositive}{TruePositive+FalsePositive}$\n

                                        \n\nRecall : It is the number of correct positive results divided by the number of all relevant samples (all samples that should have been identified as positive).\n\n

                                        \n$Recall = 2*\\frac{TruePositive}{TruePositive+FalsePositive}$\n

                                        \n\n\n\n# Preparation\n\n\n```python\nfrom __future__ import absolute_import, division, print_function, unicode_literals\n\nimport tensorflow as tf\n\nimport numpy as np\n\nfrom sklearn.model_selection import train_test_split\nfrom sklearn.metrics import classification_report, confusion_matrix\n\nimport matplotlib as mpl\nimport matplotlib.pyplot as plt\n\nimport itertools\nfrom datetime import datetime\nfrom tqdm import tqdm\nimport os\n\n%matplotlib inline\nmpl.rc_file(mpl.matplotlib_fname())\n```\n\n\n```python\n%reload_ext watermark\n%watermark -m -n -p tensorflow,numpy,matplotlib,sklearn -g\n```\n\n Sun Jul 21 2019 \n \n tensorflow 2.0.0-beta1\n numpy 1.16.4\n matplotlib 3.0.3\n sklearn 0.21.2\n \n compiler : GCC 8.0.1 20180414 (experimental) [trunk revision 259383\n system : Linux\n release : 4.14.79+\n machine : x86_64\n processor : x86_64\n CPU cores : 2\n interpreter: 64bit\n Git hash :\n\n\n\n```python\n#@title ## Random seed\n\nnp.random.seed(RANDOM_SEED)\n```\n\n# Data\n\n\n\nWe're going to first generate some non-linear data for a classification task.\n\n**A spiral**: is a curve which emanates from a point, moving farther away as it revolves around the point.\n\n\\begin{align}\n Spiral =\\begin{cases} X_{1\\theta} = r_{\\theta} \\cos(\\theta) \\\\X_{2\\theta} = r_{\\theta} \\sin(\\theta)\\end{cases}\n\\end{align}\n\n\n\n\n```python\ndef generate_data(num_samples, dimensions, num_classes):\n \"\"\" Generate non-linear dataset.\n \n Args:\n num_samples (int): number of samples which we want to produce.\n dimensions (int): the dimension of the data which we plan to produce ex. 2d.\n num_classes (int): number of classes which the samples are going to generate ex. 2#.\n \n Examples:\n >>> x, y = generate_data(2, 2, 2)\n (array([[ 0. , 0. ],\n [-0.712528 , -0.70164368],\n [-0. , -0. ],\n [ 0.90006129, -0.43576333]]), array([0, 0, 1, 1], dtype=uint8))\n \n Returns:\n x (ndarray): a numpy array in a shape like this (num_samples, dimensions).\n y (ndarray): a numpy array in a shape like this (num_samples, num_classes).\n \"\"\"\n \n x_origin = np.zeros((num_samples * num_classes, dimensions))\n y = np.zeros(num_samples * num_classes, dtype='uint8')\n \n for i in tqdm(range(num_classes), position=0):\n idx = range(num_samples * i, num_samples * (i + 1))\n radius = np.linspace(0.0, 1, num_samples)\n theta = np.linspace(i * 4, (i + 1) * 4, num_samples) + np.random.randn(num_samples) * 0.2\n \n x_origin[idx] = np.c_[radius * np.sin(theta), radius * np.cos(theta)]\n y[idx] = i\n \n x = np.hstack([x_origin])\n \n return x, y\n```\n\n\n```python\nnum_samples = 500\ndimensions = 2\nnum_classes = 3\n\nx, y = generate_data(num_samples, dimensions, num_classes)\n\nprint()\nprint('x: %s' % str(x.shape))\nprint('y: %s' % str(y.shape))\n```\n\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 3/3 [00:00<00:00, 1237.62it/s]\n\n \n x: (1500, 2)\n y: (1500,)\n\n\n \n\n\n\n```python\nxfig = 5.0\nyfig = 5.0\n\nplt.figure(figsize=(xfig, yfig))\nplt.title('Generated non-linear data')\nplt.scatter(x[:, 0], x[:, 1], c=y)\nplt.show()\n```\n\n## One-hot Encoding\n\nConsider an array of 5 labels out of a set of 3 classes {0, 1, 2}:\n\n```python\n> labels\narray([0, 2, 1, 2, 0])\n```\n\n**`to_categorical`** converts this into a matrix with as many\ncolumns as there are classes. The number of rows\nstays the same.\n\n```python\n> to_categorical(labels)\narray([[ 1., 0., 0.],\n [ 0., 0., 1.],\n [ 0., 1., 0.],\n [ 0., 0., 1.],\n [ 1., 0., 0.]], dtype=float32)\n```\n\n\n\n---\n\n\n\n**to_categorical**\n\n```\ntf.keras.utils.to_categorical(y, num_classes=None, dtype='float32')\n```\nConverts a class vector (integers) to binary class matrix.\n\n**Arguments:**\n\n- **`y`**: class vector to be converted into a matrix (integers from 0 to num_classes).\n- **`num_classes`**: total number of classes.\n- **`dtype`**: The data type expected by the input, as a string (float32, float64, int32...)\n\n\n\n```python\nx_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.3, random_state=RANDOM_SEED)\n\nprint(x_train.shape, y_train.shape)\n\ny_train_c = tf.keras.utils.to_categorical(y_train, num_classes)\ny_test_c = tf.keras.utils.to_categorical(y_test, num_classes)\n\nprint(y_train_c.shape)\n```\n\n (1050, 2) (1050,)\n (1050, 3)\n\n\n# Linear Model\n\nBefore we get to our neural network, we're going to implement a linear model (logistic regression). We want to see why linear models won't suffice for our dataset.\n\n\n```python\ndef build_linear(n_units, n_features): \n model = tf.keras.Sequential([\n tf.keras.layers.Input(shape=[n_features]),\n tf.keras.layers.Dense(n_units, activation='softmax')\n ])\n \n opt = tf.keras.optimizers.Adam(lr=0.0001)\n model.compile(optimizer=opt, \n loss='categorical_crossentropy', \n metrics=['accuracy'])\n \n return model\n```\n\n\n```python\nlinear_model = build_linear(n_units=3, n_features=2)\nlinear_model.summary()\n```\n\n Model: \"sequential_3\"\n _________________________________________________________________\n Layer (type) Output Shape Param # \n =================================================================\n dense_3 (Dense) (None, 3) 9 \n =================================================================\n Total params: 9\n Trainable params: 9\n Non-trainable params: 0\n _________________________________________________________________\n\n\n\n```python\nr = linear_model.fit(x_train, y_train_c, \n validation_split=0.1, \n epochs=100, \n verbose=1)\n\n\nhistory_dict = r.history\nhistory_list = list(history_dict.keys())\nprint(history_list)\n```\n\n Train on 945 samples, validate on 105 samples\n Epoch 1/100\n 945/945 [==============================] - 0s 227us/sample - loss: 0.9801 - accuracy: 0.4381 - val_loss: 0.9817 - val_accuracy: 0.4381\n Epoch 2/100\n 945/945 [==============================] - 0s 81us/sample - loss: 0.9793 - accuracy: 0.4392 - val_loss: 0.9810 - val_accuracy: 0.4381\n Epoch 3/100\n 945/945 [==============================] - 0s 75us/sample - loss: 0.9785 - accuracy: 0.4402 - val_loss: 0.9803 - val_accuracy: 0.4381\n Epoch 4/100\n 945/945 [==============================] - 0s 76us/sample - loss: 0.9776 - accuracy: 0.4413 - val_loss: 0.9796 - val_accuracy: 0.4381\n Epoch 5/100\n 945/945 [==============================] - 0s 76us/sample - loss: 0.9768 - accuracy: 0.4413 - val_loss: 0.9790 - val_accuracy: 0.4381\n Epoch 6/100\n 945/945 [==============================] - 0s 75us/sample - loss: 0.9760 - accuracy: 0.4423 - val_loss: 0.9783 - val_accuracy: 0.4381\n Epoch 7/100\n 945/945 [==============================] - 0s 91us/sample - loss: 0.9752 - accuracy: 0.4444 - val_loss: 0.9777 - val_accuracy: 0.4381\n Epoch 8/100\n 945/945 [==============================] - 0s 78us/sample - loss: 0.9744 - accuracy: 0.4476 - val_loss: 0.9770 - val_accuracy: 0.4381\n Epoch 9/100\n 945/945 [==============================] - 0s 74us/sample - loss: 0.9735 - accuracy: 0.4476 - val_loss: 0.9764 - val_accuracy: 0.4381\n Epoch 10/100\n 945/945 [==============================] - 0s 72us/sample - loss: 0.9728 - accuracy: 0.4508 - val_loss: 0.9757 - val_accuracy: 0.4381\n Epoch 11/100\n 945/945 [==============================] - 0s 75us/sample - loss: 0.9720 - accuracy: 0.4519 - val_loss: 0.9751 - val_accuracy: 0.4381\n Epoch 12/100\n 945/945 [==============================] - 0s 71us/sample - loss: 0.9711 - accuracy: 0.4561 - val_loss: 0.9744 - val_accuracy: 0.4381\n Epoch 13/100\n 945/945 [==============================] - 0s 75us/sample - loss: 0.9703 - accuracy: 0.4571 - val_loss: 0.9738 - val_accuracy: 0.4476\n Epoch 14/100\n 945/945 [==============================] - 0s 73us/sample - loss: 0.9696 - accuracy: 0.4593 - val_loss: 0.9731 - val_accuracy: 0.4476\n Epoch 15/100\n 945/945 [==============================] - 0s 75us/sample - loss: 0.9688 - accuracy: 0.4603 - val_loss: 0.9725 - val_accuracy: 0.4476\n Epoch 16/100\n 945/945 [==============================] - 0s 71us/sample - loss: 0.9680 - accuracy: 0.4614 - val_loss: 0.9719 - val_accuracy: 0.4476\n Epoch 17/100\n 945/945 [==============================] - 0s 72us/sample - loss: 0.9672 - accuracy: 0.4603 - val_loss: 0.9712 - val_accuracy: 0.4476\n Epoch 18/100\n 945/945 [==============================] - 0s 77us/sample - loss: 0.9664 - accuracy: 0.4603 - val_loss: 0.9706 - val_accuracy: 0.4476\n Epoch 19/100\n 945/945 [==============================] - 0s 80us/sample - loss: 0.9656 - accuracy: 0.4603 - val_loss: 0.9700 - val_accuracy: 0.4476\n Epoch 20/100\n 945/945 [==============================] - 0s 77us/sample - loss: 0.9649 - accuracy: 0.4614 - val_loss: 0.9693 - val_accuracy: 0.4476\n Epoch 21/100\n 945/945 [==============================] - 0s 88us/sample - loss: 0.9641 - accuracy: 0.4614 - val_loss: 0.9687 - val_accuracy: 0.4571\n Epoch 22/100\n 945/945 [==============================] - 0s 71us/sample - loss: 0.9633 - accuracy: 0.4635 - val_loss: 0.9681 - val_accuracy: 0.4571\n Epoch 23/100\n 945/945 [==============================] - 0s 71us/sample - loss: 0.9626 - accuracy: 0.4667 - val_loss: 0.9675 - val_accuracy: 0.4571\n Epoch 24/100\n 945/945 [==============================] - 0s 77us/sample - loss: 0.9618 - accuracy: 0.4667 - val_loss: 0.9669 - val_accuracy: 0.4571\n Epoch 25/100\n 945/945 [==============================] - 0s 78us/sample - loss: 0.9610 - accuracy: 0.4677 - val_loss: 0.9663 - val_accuracy: 0.4571\n Epoch 26/100\n 945/945 [==============================] - 0s 76us/sample - loss: 0.9603 - accuracy: 0.4698 - val_loss: 0.9657 - val_accuracy: 0.4571\n Epoch 27/100\n 945/945 [==============================] - 0s 75us/sample - loss: 0.9595 - accuracy: 0.4709 - val_loss: 0.9650 - val_accuracy: 0.4571\n Epoch 28/100\n 945/945 [==============================] - 0s 76us/sample - loss: 0.9588 - accuracy: 0.4709 - val_loss: 0.9645 - val_accuracy: 0.4667\n Epoch 29/100\n 945/945 [==============================] - 0s 71us/sample - loss: 0.9580 - accuracy: 0.4698 - val_loss: 0.9639 - val_accuracy: 0.4667\n Epoch 30/100\n 945/945 [==============================] - 0s 64us/sample - loss: 0.9573 - accuracy: 0.4709 - val_loss: 0.9632 - val_accuracy: 0.4571\n Epoch 31/100\n 945/945 [==============================] - 0s 72us/sample - loss: 0.9565 - accuracy: 0.4720 - val_loss: 0.9626 - val_accuracy: 0.4571\n Epoch 32/100\n 945/945 [==============================] - 0s 63us/sample - loss: 0.9558 - accuracy: 0.4741 - val_loss: 0.9620 - val_accuracy: 0.4571\n Epoch 33/100\n 945/945 [==============================] - 0s 69us/sample - loss: 0.9550 - accuracy: 0.4741 - val_loss: 0.9614 - val_accuracy: 0.4571\n Epoch 34/100\n 945/945 [==============================] - 0s 64us/sample - loss: 0.9543 - accuracy: 0.4772 - val_loss: 0.9608 - val_accuracy: 0.4571\n Epoch 35/100\n 945/945 [==============================] - 0s 69us/sample - loss: 0.9535 - accuracy: 0.4783 - val_loss: 0.9602 - val_accuracy: 0.4571\n Epoch 36/100\n 945/945 [==============================] - 0s 85us/sample - loss: 0.9528 - accuracy: 0.4783 - val_loss: 0.9596 - val_accuracy: 0.4571\n Epoch 37/100\n 945/945 [==============================] - 0s 68us/sample - loss: 0.9521 - accuracy: 0.4783 - val_loss: 0.9591 - val_accuracy: 0.4667\n Epoch 38/100\n 945/945 [==============================] - 0s 67us/sample - loss: 0.9513 - accuracy: 0.4783 - val_loss: 0.9585 - val_accuracy: 0.4667\n Epoch 39/100\n 945/945 [==============================] - 0s 62us/sample - loss: 0.9506 - accuracy: 0.4804 - val_loss: 0.9579 - val_accuracy: 0.4667\n Epoch 40/100\n 945/945 [==============================] - 0s 70us/sample - loss: 0.9499 - accuracy: 0.4804 - val_loss: 0.9573 - val_accuracy: 0.4667\n Epoch 41/100\n 945/945 [==============================] - 0s 67us/sample - loss: 0.9492 - accuracy: 0.4804 - val_loss: 0.9567 - val_accuracy: 0.4667\n Epoch 42/100\n 945/945 [==============================] - 0s 72us/sample - loss: 0.9484 - accuracy: 0.4794 - val_loss: 0.9561 - val_accuracy: 0.4762\n Epoch 43/100\n 945/945 [==============================] - 0s 68us/sample - loss: 0.9477 - accuracy: 0.4794 - val_loss: 0.9556 - val_accuracy: 0.4667\n Epoch 44/100\n 945/945 [==============================] - 0s 65us/sample - loss: 0.9470 - accuracy: 0.4794 - val_loss: 0.9550 - val_accuracy: 0.4667\n Epoch 45/100\n 945/945 [==============================] - 0s 70us/sample - loss: 0.9463 - accuracy: 0.4794 - val_loss: 0.9544 - val_accuracy: 0.4762\n Epoch 46/100\n 945/945 [==============================] - 0s 66us/sample - loss: 0.9456 - accuracy: 0.4794 - val_loss: 0.9538 - val_accuracy: 0.4667\n Epoch 47/100\n 945/945 [==============================] - 0s 65us/sample - loss: 0.9449 - accuracy: 0.4804 - val_loss: 0.9533 - val_accuracy: 0.4667\n Epoch 48/100\n 945/945 [==============================] - 0s 70us/sample - loss: 0.9442 - accuracy: 0.4836 - val_loss: 0.9527 - val_accuracy: 0.4667\n Epoch 49/100\n 945/945 [==============================] - 0s 68us/sample - loss: 0.9435 - accuracy: 0.4847 - val_loss: 0.9521 - val_accuracy: 0.4667\n Epoch 50/100\n 945/945 [==============================] - 0s 70us/sample - loss: 0.9427 - accuracy: 0.4857 - val_loss: 0.9516 - val_accuracy: 0.4667\n Epoch 51/100\n 945/945 [==============================] - 0s 86us/sample - loss: 0.9420 - accuracy: 0.4857 - val_loss: 0.9510 - val_accuracy: 0.4762\n Epoch 52/100\n 945/945 [==============================] - 0s 68us/sample - loss: 0.9413 - accuracy: 0.4857 - val_loss: 0.9504 - val_accuracy: 0.4762\n Epoch 53/100\n 945/945 [==============================] - 0s 63us/sample - loss: 0.9406 - accuracy: 0.4899 - val_loss: 0.9499 - val_accuracy: 0.4762\n Epoch 54/100\n 945/945 [==============================] - 0s 70us/sample - loss: 0.9399 - accuracy: 0.4921 - val_loss: 0.9493 - val_accuracy: 0.4857\n Epoch 55/100\n 945/945 [==============================] - 0s 70us/sample - loss: 0.9392 - accuracy: 0.4931 - val_loss: 0.9487 - val_accuracy: 0.4857\n Epoch 56/100\n 945/945 [==============================] - 0s 68us/sample - loss: 0.9386 - accuracy: 0.4931 - val_loss: 0.9482 - val_accuracy: 0.4857\n Epoch 57/100\n 945/945 [==============================] - 0s 73us/sample - loss: 0.9379 - accuracy: 0.4963 - val_loss: 0.9476 - val_accuracy: 0.4857\n Epoch 58/100\n 945/945 [==============================] - 0s 67us/sample - loss: 0.9372 - accuracy: 0.4952 - val_loss: 0.9471 - val_accuracy: 0.4857\n Epoch 59/100\n 945/945 [==============================] - 0s 63us/sample - loss: 0.9365 - accuracy: 0.4974 - val_loss: 0.9465 - val_accuracy: 0.4857\n Epoch 60/100\n 945/945 [==============================] - 0s 69us/sample - loss: 0.9358 - accuracy: 0.4995 - val_loss: 0.9460 - val_accuracy: 0.4857\n Epoch 61/100\n 945/945 [==============================] - 0s 72us/sample - loss: 0.9351 - accuracy: 0.4995 - val_loss: 0.9454 - val_accuracy: 0.4857\n Epoch 62/100\n 945/945 [==============================] - 0s 64us/sample - loss: 0.9344 - accuracy: 0.4995 - val_loss: 0.9448 - val_accuracy: 0.4952\n Epoch 63/100\n 945/945 [==============================] - 0s 71us/sample - loss: 0.9338 - accuracy: 0.5026 - val_loss: 0.9443 - val_accuracy: 0.4952\n Epoch 64/100\n 945/945 [==============================] - 0s 63us/sample - loss: 0.9331 - accuracy: 0.5037 - val_loss: 0.9437 - val_accuracy: 0.4952\n Epoch 65/100\n 945/945 [==============================] - 0s 68us/sample - loss: 0.9324 - accuracy: 0.5037 - val_loss: 0.9432 - val_accuracy: 0.4952\n Epoch 66/100\n 945/945 [==============================] - 0s 82us/sample - loss: 0.9318 - accuracy: 0.5058 - val_loss: 0.9427 - val_accuracy: 0.4952\n Epoch 67/100\n 945/945 [==============================] - 0s 70us/sample - loss: 0.9311 - accuracy: 0.5037 - val_loss: 0.9421 - val_accuracy: 0.4952\n Epoch 68/100\n 945/945 [==============================] - 0s 75us/sample - loss: 0.9304 - accuracy: 0.5058 - val_loss: 0.9416 - val_accuracy: 0.4952\n Epoch 69/100\n 945/945 [==============================] - 0s 68us/sample - loss: 0.9298 - accuracy: 0.5058 - val_loss: 0.9411 - val_accuracy: 0.4952\n Epoch 70/100\n 945/945 [==============================] - 0s 63us/sample - loss: 0.9291 - accuracy: 0.5058 - val_loss: 0.9405 - val_accuracy: 0.4952\n Epoch 71/100\n 945/945 [==============================] - 0s 68us/sample - loss: 0.9284 - accuracy: 0.5048 - val_loss: 0.9400 - val_accuracy: 0.4952\n Epoch 72/100\n 945/945 [==============================] - 0s 64us/sample - loss: 0.9278 - accuracy: 0.5058 - val_loss: 0.9394 - val_accuracy: 0.4952\n Epoch 73/100\n 945/945 [==============================] - 0s 73us/sample - loss: 0.9271 - accuracy: 0.5048 - val_loss: 0.9389 - val_accuracy: 0.4952\n Epoch 74/100\n 945/945 [==============================] - 0s 70us/sample - loss: 0.9264 - accuracy: 0.5058 - val_loss: 0.9384 - val_accuracy: 0.4952\n Epoch 75/100\n 945/945 [==============================] - 0s 68us/sample - loss: 0.9258 - accuracy: 0.5058 - val_loss: 0.9379 - val_accuracy: 0.4952\n Epoch 76/100\n 945/945 [==============================] - 0s 70us/sample - loss: 0.9251 - accuracy: 0.5069 - val_loss: 0.9373 - val_accuracy: 0.4952\n Epoch 77/100\n 945/945 [==============================] - 0s 68us/sample - loss: 0.9245 - accuracy: 0.5079 - val_loss: 0.9368 - val_accuracy: 0.4952\n Epoch 78/100\n 945/945 [==============================] - 0s 63us/sample - loss: 0.9239 - accuracy: 0.5101 - val_loss: 0.9363 - val_accuracy: 0.4952\n Epoch 79/100\n 945/945 [==============================] - 0s 73us/sample - loss: 0.9232 - accuracy: 0.5111 - val_loss: 0.9358 - val_accuracy: 0.4952\n Epoch 80/100\n 945/945 [==============================] - 0s 62us/sample - loss: 0.9225 - accuracy: 0.5111 - val_loss: 0.9352 - val_accuracy: 0.4952\n Epoch 81/100\n 945/945 [==============================] - 0s 86us/sample - loss: 0.9219 - accuracy: 0.5090 - val_loss: 0.9347 - val_accuracy: 0.4952\n Epoch 82/100\n 945/945 [==============================] - 0s 72us/sample - loss: 0.9213 - accuracy: 0.5101 - val_loss: 0.9342 - val_accuracy: 0.4952\n Epoch 83/100\n 945/945 [==============================] - 0s 69us/sample - loss: 0.9206 - accuracy: 0.5101 - val_loss: 0.9337 - val_accuracy: 0.4857\n Epoch 84/100\n 945/945 [==============================] - 0s 64us/sample - loss: 0.9200 - accuracy: 0.5111 - val_loss: 0.9332 - val_accuracy: 0.4857\n Epoch 85/100\n 945/945 [==============================] - 0s 69us/sample - loss: 0.9193 - accuracy: 0.5111 - val_loss: 0.9327 - val_accuracy: 0.4857\n Epoch 86/100\n 945/945 [==============================] - 0s 69us/sample - loss: 0.9187 - accuracy: 0.5122 - val_loss: 0.9321 - val_accuracy: 0.4857\n Epoch 87/100\n 945/945 [==============================] - 0s 69us/sample - loss: 0.9181 - accuracy: 0.5122 - val_loss: 0.9316 - val_accuracy: 0.4857\n Epoch 88/100\n 945/945 [==============================] - 0s 66us/sample - loss: 0.9174 - accuracy: 0.5122 - val_loss: 0.9312 - val_accuracy: 0.4857\n Epoch 89/100\n 945/945 [==============================] - 0s 70us/sample - loss: 0.9168 - accuracy: 0.5143 - val_loss: 0.9306 - val_accuracy: 0.4857\n Epoch 90/100\n 945/945 [==============================] - 0s 68us/sample - loss: 0.9162 - accuracy: 0.5143 - val_loss: 0.9302 - val_accuracy: 0.4857\n Epoch 91/100\n 945/945 [==============================] - 0s 63us/sample - loss: 0.9156 - accuracy: 0.5143 - val_loss: 0.9297 - val_accuracy: 0.4857\n Epoch 92/100\n 945/945 [==============================] - 0s 67us/sample - loss: 0.9150 - accuracy: 0.5153 - val_loss: 0.9292 - val_accuracy: 0.4857\n Epoch 93/100\n 945/945 [==============================] - 0s 72us/sample - loss: 0.9143 - accuracy: 0.5153 - val_loss: 0.9287 - val_accuracy: 0.4857\n Epoch 94/100\n 945/945 [==============================] - 0s 62us/sample - loss: 0.9137 - accuracy: 0.5153 - val_loss: 0.9282 - val_accuracy: 0.4857\n Epoch 95/100\n 945/945 [==============================] - 0s 76us/sample - loss: 0.9131 - accuracy: 0.5175 - val_loss: 0.9277 - val_accuracy: 0.4762\n Epoch 96/100\n 945/945 [==============================] - 0s 89us/sample - loss: 0.9125 - accuracy: 0.5164 - val_loss: 0.9272 - val_accuracy: 0.4762\n Epoch 97/100\n 945/945 [==============================] - 0s 68us/sample - loss: 0.9119 - accuracy: 0.5175 - val_loss: 0.9267 - val_accuracy: 0.4762\n Epoch 98/100\n 945/945 [==============================] - 0s 69us/sample - loss: 0.9113 - accuracy: 0.5175 - val_loss: 0.9262 - val_accuracy: 0.4762\n Epoch 99/100\n 945/945 [==============================] - 0s 65us/sample - loss: 0.9106 - accuracy: 0.5153 - val_loss: 0.9257 - val_accuracy: 0.4762\n Epoch 100/100\n 945/945 [==============================] - 0s 68us/sample - loss: 0.9100 - accuracy: 0.5175 - val_loss: 0.9252 - val_accuracy: 0.4762\n ['loss', 'accuracy', 'val_loss', 'val_accuracy']\n\n\n\n```python\nevaluation = linear_model.evaluate(x_test, y_test_c, verbose=0)\nprint(evaluation)\n```\n\n [0.9198821216159396, 0.5311111]\n\n\n\n```python\nloss = history_dict['loss']\nval_loss = history_dict['val_loss']\nepochs = range(1, len(loss) + 1)\n\nplt.plot(epochs, loss, label='Train Loss')\nplt.plot(epochs, val_loss, label='Valid Loss')\nplt.legend()\nplt.show()\n```\n\n\n```python\naccuracy = history_dict['accuracy']\nval_accuracy = history_dict['val_accuracy']\nepochs = range(1, len(accuracy) + 1)\n\nplt.plot(epochs, accuracy, label='Train Accuracy')\nplt.plot(epochs, val_accuracy, label='Valid Accuracy')\nplt.legend()\nplt.show()\n```\n\n\n```python\n#@title ## Multiclass decision boundary\n\ndef plot_mc_decision_boundary(model, x, y, steps=1000):\n \"\"\" Plot multiclass decision boundary,\n \n Args:\n model (keras.Model): a keras model.\n x (ndarray): a numpy array.\n y (ndarray): a numpy array.\n steps (integer): number of linear spaces.\n \n Returns:\n None\n \"\"\"\n x_min, x_max = x[:, 0].min() - 0.1, x[:, 0].max() + 0.1\n y_min, y_max = x[:, 1].min() - 0.1, x[:, 1].max() + 0.1\n \n x_span = np.linspace(x_min, x_max, steps)\n y_span = np.linspace(y_min, y_max, steps)\n xx, yy = np.meshgrid(x_span, y_span)\n \n y_pred = model.predict(np.c_[xx.ravel(), yy.ravel()])\n y_pred = np.argmax(y_pred, axis=1)\n y_pred = y_pred.reshape(xx.shape)\n \n plt.contourf(xx, yy, y_pred, alpha=0.5)\n plt.scatter(x[:, 0], x[:, 1], c=y)\n plt.xlim(xx.min(), xx.max())\n plt.ylim(yy.min(), yy.max())\n \n return None\n```\n\n\n```python\n#@title ## Visualize the decision boundary\n```\n\n\n```python\n#@title ## Plot confusion matrix\n\ndef plot_confusion_matrix(cm, classes):\n \"\"\" Plot confusion matrix\n \n Args:\n cm (ndarray): the input of confusion matrix.\n classes (integer): number of classes.\n \n Returns:\n None\n \"\"\"\n cmap=plt.cm.Blues\n plt.imshow(cm, interpolation='nearest', cmap=cmap)\n plt.title(\"Confusion Matrix\")\n plt.colorbar()\n tick_marks = np.arange(len(classes))\n plt.xticks(tick_marks, classes, rotation=45)\n plt.yticks(tick_marks, classes)\n plt.grid(False)\n\n fmt = 'd'\n thresh = cm.max() / 2.\n for i, j in itertools.product(range(cm.shape[0]), range(cm.shape[1])):\n plt.text(j, i, format(cm[i, j], fmt), horizontalalignment=\"center\", \n color=\"white\" if cm[i, j] > thresh else \"black\")\n\n plt.ylabel('True label')\n plt.xlabel('Predicted label')\n plt.tight_layout()\n \n return None\n```\n\n## Classification Report\n\n`sklearn.metrics.classification_report`\n\nBuild a text report showing the main classification metrics\n\n**Params:**\n- `y_true` : 1d array-like, or label indicator array / sparse matrix\n- `y_pred` : 1d array-like, or label indicator array / sparse matrix\n\n\n**Output:**\n```bash\n precision recall f1-score support\n\n 1 1.00 0.67 0.80 3\n 2 0.00 0.00 0.00 0\n 3 0.00 0.00 0.00 0\n\n micro avg 1.00 0.67 0.80 3\n macro avg 0.33 0.22 0.27 3\nweighted avg 1.00 0.67 0.80 3\n```\n\n\n```python\n#@title ## Confusion matrix\n```\n\n# Non-linear Model\n\nNow let's see how the MLP performs on the data. Note that the only difference is the addition of the non-linear activation function (we use $ReLU$ which is just $max(0, z))$. \n\n\n```python\n#@title ## Build non-linear model [MLP Model](https://en.wikipedia.org/wiki/Multilayer_perceptron)\n\ndef build_non_linear(n_classes, n_units, n_features):\n model = None\n \n return model\n```\n\n\n```python\n#@title ## Create non-linear model\n```\n\n\n```python\n#@title ## Fit non-linear model.\n```\n\n\n```python\n#@title ## Evaluating\n```\n\n\n```python\n#@title ## Plotting\n```\n\n\n```python\n#@title ## Visualize the decision boundary\n```\n\n\n```python\n#@title ## Confusion matrix\n```\n\n# Overfitting\n\nThough neural networks are great at capturing non-linear relationships they are highly susceptible to overfitting to the training data and failing to generalize on test data. Just take a look at the example below where we generate completely random data and are able to fit a model with [$2*N*C + D$](https://arxiv.org/abs/1611.03530) hidden units. The training performance is great but the overfitting leads to very poor test performance. We'll be covering strategies to tackle overfitting in future lessons.\n\nLet me define overfitting more formally. Overfitting refers to a model that models the \u201ctraining data\u201d too well. Overfitting happens when a model learns the detail and noise in the training data to the extent that it negatively impacts the performance of the model on new data.\n\n# How do you know your NN is overfitting?\nIn practice, detecting that our model is overfitting is difficult. It\u2019s not uncommon that our trained model is already in production and then we start to realize that something is wrong. In fact, it is only by confronting new data that you can make sure that everything is working properly. However, during the training we should try to reproduce the real conditions as much as possible. For this reason, it is good practice to divide our dataset into three parts - training set, dev set (also known as cross-validation or hold-out) and test set. Our model learns by seeing only the first of these parts. Hold-out is used to track our progress and draw conclusions to optimise the model. While, we use a test set at the end of the training process to evaluate the performance of our model. Using completely new data allows us to get an unbiased opinion on how well our algorithm works.\n\nIt is very important to make sure that your cross-validation and test set come from the same distribution as well as that they accurately reflect data that we expect to receive in the future. Only then we can be sure that the decisions we make during the learning process bring us closer to a better solution. I know what you are thinking about\u2026 \u201cHow should I divide my dataset?\u201d Until recently, one of the most frequently recommended splits was 60/20/20, but in the era of big data, when our dataset can count millions of entries, those fixed proportions are no longer appropriate. In short, everything depends on the size of the dataset we work with. If we have millions of entries at our disposal, perhaps it would be better idea to divide them in 98/1/1 ratio. Our dev and test sets should be simply large enough to give us high confidence in the performance of our model. \n\n

                                        \n\n

                                        \n\n# Bias and Variance\nTo give us a better understanding of this some how complex issue, we will use a simple example, that hopefully allow us to develop a valuable intuition. Our dataset consisting of two classes of points, located in a two-dimensional space.\n\n

                                        \n\n

                                        \n\nThe first model in the top right corner is very simple and therefore has a high bias, i.e. it is not able to find all significant links between features and result. This is understandable - our dataset has a lot of noise in it, and therefore simple linear regression, is not able to deal with it effectively. Neural networks performed much better, but the first one (shown in the lower left corner) fitted into the data too closely, which made it work significantly worse on the hold-out set. This means that it has a high variance - it fits into the noise and not into the intended output. This undesirable effect was mitigated, in the last model, by the use of regularisation.\n\n

                                        \n\n

                                        \n\n# Ways to prevent overfitting\n\n### L1 and L2 Regularizations\n\nOne of the first methods we should try when we need to reduce overfitting is regularisation. It involves adding an extra element to the loss function, which punishes our model for being too complex or, in simple words, for using too high values in the weight matrix. This way we try to limit its flexibility, but also encourage it to build solutions based on multiple features. Two popular versions of this method are L1 - Least Absolute Deviations (LAD) and L2 - Least Square Errors (LS). Equations describing these regularisations are given below.\nIn most cases the use of L1 is preferable, because it reduces the weight values of less important features to zero, very often eliminating them completely from the calculations. In a way, it is a built-in mechanism for automatic featur selection. Moreover, L2 does not perform very well on datasets with a large number of outliers. The use of value squares results in the model minimizing the impact of outliers at the expense of more popular examples.\n\n

                                        \n$$\n\\begin{array}{rlrl}{J_{L 1}(W, b)} & {=\\frac{1}{m} \\sum_{i=1}^{m} L\\left(\\hat{y}^{(i)}, y^{(i)}\\right)+\\lambda\\|w\\|_{1}} & {\\|w\\|_{1}} & {=\\sum_{j=1}^{n_{x}}\\left|w_{j}\\right|} \\\\ {J_{L 2}(W, b)} & {=\\frac{1}{m} \\sum_{i=1}^{m} L\\left(\\hat{y}^{(i)}, y^{(i)}\\right)+\\lambda\\|w\\|_{2}} & {\\|w\\|_{2}} & {=\\sum_{j=1}^{n_{x}} w_{j}^{2}}\\end{array}\n$$\n

                                        \n\nIncreasing the \u03bb value also increases the regularisation effect. Models with a very low \u03bb coefficient value are very \u201cturbulent\u201d.\n\n# Dropout\nAnother very popular method of regularization of neural networks is dropout. This idea is actually very simple - every unit of our neural network (except those belonging to the output layer) is given the probability p of being temporarily ignored in calculations. Hyper parameter p is called dropout rate and very often its default value is set to 0.5. Then, in each iteration, we randomly select the neurons that we drop according to the assigned probability. As a result, each time we work with a smaller neural network. The visualization below shows an example of a neural network subjected to a dropout. We can see how in each iteration random neurons from second and fourth layer are deactivated.\n\n

                                        \n\n

                                        \n\n# Early Stopping\nThe graph below shows the change in accuracy values calculated on the test and cross-validation sets during subsequent iterations of learning process. We see right away that the model we get at the end is not the best we could have possibly create. To be honest, it is much worse than what we have had after 150 epochs. Why not interrupt the learning process before the model starts overfitting? This observation inspired one of the popular overfitting reduction method, namely early stopping.\n

                                        \n\n

                                        \n\n\n```python\n#@title ## Generate random x, y\n\nov_num_samples = 40 #@param {type:\"integer\"}\nov_dimensions = 2 #@param {type:\"integer\"}\nov_num_classes = 3 #@param {type:\"integer\"}\n\nov_x = np.random.randn(ov_num_samples * ov_num_classes, ov_dimensions)\nov_y = np.array([[i] * ov_num_samples for i in range(ov_num_classes)])\nov_y = ov_y.flatten()\n\n\nprint()\nprint('x: %s' % str(ov_x.shape))\nprint('y: %s' % str(ov_y.shape))\n```\n\n \n x: (120, 2)\n y: (120,)\n\n\n\n```python\n#@title ## Preprocessing\n```\n\n\n```python\n#@title ## Create non-linear model\n```\n\n\n```python\n#@title ## Fit overfie model.\n```\n\n\n```python\n#@title ## Evaluating\n```\n\n\n```python\n#@title ## Plotting\n```\n\n\n```python\n#@title ## Visualize the decision boundary\n```\n\n\n```python\n#@title ## Confusion matrix\n```\n\n# Regularization\n\n\n```python\n#@title ## Build non-linear model [MLP Model](https://en.wikipedia.org/wiki/Multilayer_perceptron)\n\ndef build_non_linear_with_regularization(n_classes, n_units, n_features):\n model = None\n \n return model\n```\n\n\n```python\n#@title ## Create non-linear model\n```\n\n\n```python\n#@title ## Fit non-linear model with regularization.\n```\n\n\n```python\n#@title ## Evaluating\n```\n\n\n```python\n#@title ## Plotting\n```\n\n\n```python\n#@title ## Visualize the decision boundary\n```\n\n\n```python\n#@title ## Confusion matrix\n```\n\n# Dropout\n\nA great technique to overcome overfitting is to increase the size of your data but this isn't always an option. Fortuntely, there are methods like regularization and dropout that can help create a more robust model. We've already seen regularization and we can easily add it in our optimizer to use it in PyTorch. \n\nDropout is a technique (used only during training) that allows us to zero the outputs of neurons. We do this for p% of the total neurons in each layer and it changes every batch. Dropout prevents units from co-adapting too much to the data and acts as a sampling strategy since we drop a different set of neurons each time.\n\n\n\n\n```python\n#@title ## Build non-linear model [MLP Model](https://en.wikipedia.org/wiki/Multilayer_perceptron)\n\ndef build_non_linear_with_dropout(n_classes, n_units, n_features):\n model = None\n \n return model\n```\n\n\n```python\n#@title ## Create non-linear model\n```\n\n\n```python\n#@title ## Fit non-linear model with dropout.\n```\n\n\n```python\n#@title ## Evaluating\n```\n\n\n```python\n#@title ## Plotting\n```\n\n\n```python\n#@title ## Visualize the decision boundary\n```\n\n\n```python\n#@title ## Confusion matrix\n```\n\n# Visualization\n\n\n```python\n#@title ## Build non-linear model [MLP Model](https://en.wikipedia.org/wiki/Multilayer_perceptron)\n\ndef build_non_linear_v2(n_classes, n_units, n_features):\n model = None\n \n return model\n```\n\n\n```python\n#@title ## Create non-linear model\n```\n\n\n```python\n#@title # Summary of the model\n```\n\n\n```python\n#@title ## Model Arch\n```\n\n\n```python\n#@title ## Tensorboard callbacks\n```\n\n\n```python\n#@title ## Fit non-linear model.\n```\n\n\n```python\n#@title ## Plotting\n```\n\n\n```python\n#@title ## Visualize the decision boundary\n```\n\n\n```python\n#@title ## Confusion matrix\n```\n\n# Tensorboard\n\n\n```python\n# %load_ext tensorboard\n```\n\n\n```python\n# %tensorboard --logdir logs/scalars\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "1a6c12f9fc71a8ad209b3102707ec4af7f83f4c6", "size": 289462, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ws-dl-tf2-s1/sessions/01/Notebook 003 Part 1.ipynb", "max_stars_repo_name": "i-ml/tf2-workshop", "max_stars_repo_head_hexsha": "2bced4cfa528ac20f12098a7c5feb52ecf6b81e0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "ws-dl-tf2-s1/sessions/01/Notebook 003 Part 1.ipynb", "max_issues_repo_name": "i-ml/tf2-workshop", "max_issues_repo_head_hexsha": "2bced4cfa528ac20f12098a7c5feb52ecf6b81e0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ws-dl-tf2-s1/sessions/01/Notebook 003 Part 1.ipynb", "max_forks_repo_name": "i-ml/tf2-workshop", "max_forks_repo_head_hexsha": "2bced4cfa528ac20f12098a7c5feb52ecf6b81e0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2019-07-11T18:38:29.000Z", "max_forks_repo_forks_event_max_datetime": "2019-11-01T09:58:27.000Z", "avg_line_length": 130.1537769784, "max_line_length": 91862, "alphanum_fraction": 0.8381687406, "converted": true, "num_tokens": 16675, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.7577943767446202, "lm_q1q2_score": 0.4260141911092248}} {"text": "```python\n# import numpy as np\n\n# # !/usr/bin/env python3\n# # -*- coding: utf-8 -*-\n# \"\"\"\n# Created on 20181219\n\n# @author: zhangji\n\n# Trajection of a ellipse, Jeffery equation. \n# \"\"\"\n\n# %pylab inline\n# pylab.rcParams['figure.figsize'] = (25, 11)\n# fontsize = 40\n\n# import numpy as np\n# import scipy as sp\n# from scipy.optimize import leastsq, curve_fit\n# from scipy import interpolate\n# from scipy.interpolate import interp1d\n# from scipy.io import loadmat, savemat\n# # import scipy.misc\n\n# import matplotlib\n# from matplotlib import pyplot as plt\n# from matplotlib import animation, rc\n# import matplotlib.ticker as mtick\n# from mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\n# from mpl_toolkits.mplot3d import Axes3D, axes3d\n\n# from sympy import symbols, simplify, series, exp\n# from sympy.matrices import Matrix\n# from sympy.solvers import solve\n\n# from IPython.display import display, HTML\n# from tqdm import tqdm_notebook as tqdm\n# import pandas as pd\n# import re\n# from scanf import scanf\n# import os\n# import glob\n\n# from codeStore import support_fun as spf\n# from src.support_class import *\n# from src import stokes_flow as sf\n\n# rc('animation', html='html5')\n# PWD = os.getcwd()\n# font = {'size': 20}\n# matplotlib.rc('font', **font)\n# np.set_printoptions(linewidth=90, precision=5)\n\n%load_ext autoreload\n%autoreload 2\n\nfrom tqdm import tqdm_notebook\nimport os\nimport glob\nimport natsort \nimport numpy as np\nimport scipy as sp\nfrom scipy.optimize import leastsq, curve_fit\nfrom scipy import interpolate, integrate\nfrom scipy import spatial\n# from scipy.interpolate import interp1d\nfrom scipy.io import loadmat, savemat\n# import scipy.misc\nimport importlib\nfrom IPython.display import display, HTML\nimport pandas as pd\nimport pickle\nimport re\nfrom scanf import scanf\n\nimport matplotlib\nfrom matplotlib import pyplot as plt\nimport matplotlib.colors as colors\nfrom matplotlib import animation, rc\nimport matplotlib.ticker as mtick\nfrom mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\nfrom mpl_toolkits.mplot3d import Axes3D, axes3d\nfrom mpl_toolkits.axes_grid1.axes_divider import make_axes_locatable\nfrom mpl_toolkits.mplot3d.art3d import Line3DCollection\nfrom matplotlib import cm\n\nfrom time import time\nfrom datetime import datetime\nfrom src.support_class import *\nfrom src.objComposite import *\nfrom src import jeffery_model as jm\nfrom codeStore import support_fun as spf\nfrom codeStore import support_fun_table as spf_tb\nfrom codeStore import support_fun_resistance as spf_re\n# %matplotlib notebook\n\nPWD = os.getcwd()\nnp.set_printoptions(linewidth=110, precision=5)\n\nparams = {'animation.html': 'html5',\n 'font.family': 'sans-serif',\n 'font.size': 20, }\npreamble = r' '\npreamble = preamble + '\\\\usepackage{bm} '\npreamble = preamble + '\\\\usepackage{amsmath} '\npreamble = preamble + '\\\\usepackage{amssymb} '\npreamble = preamble + '\\\\usepackage{mathrsfs} '\npreamble = preamble + '\\\\DeclareMathOperator{\\\\Tr}{Tr} '\nparams['text.latex.preamble'] = preamble\nparams['text.usetex'] = True\nplt.rcParams.update(params)\n\n```\n\n\n```python\n\n```\n\n## single helix\n\n\n```python\njob_dir = 'hlxPart_th0_b2'\n\ntdir = os.path.join(os.getcwd(), job_dir)\nproblem_kwarg_list, A_list, B1_list, B2_list, C_list = spf_re.load_ABC_list(tdir)\nTrA_list = np.array([np.trace(i0) for i0 in A_list])\nTrB1_list = np.array([np.trace(i0) for i0 in B1_list])\nTrB2_list = np.array([np.trace(i0) for i0 in B2_list])\nTrC_list = np.array([np.trace(i0) for i0 in C_list])\n\nA_00_list = np.array([i0[0, 0] for i0 in A_list])\nB1_00_list = np.array([i0[0, 0] for i0 in B1_list])\nB2_00_list = np.array([i0[0, 0] for i0 in B2_list])\nC_00_list = np.array([i0[0, 0] for i0 in C_list])\nA_11_list = np.array([i0[1, 1] for i0 in A_list])\nB1_11_list = np.array([i0[1, 1] for i0 in B1_list])\nB2_11_list = np.array([i0[1, 1] for i0 in B2_list])\nC_11_list = np.array([i0[1, 1] for i0 in C_list])\nA_22_list = np.array([i0[2, 2] for i0 in A_list])\nB1_22_list = np.array([i0[2, 2] for i0 in B1_list])\nB2_22_list = np.array([i0[2, 2] for i0 in B2_list])\nC_22_list = np.array([i0[2, 2] for i0 in C_list])\n\nph_list = np.array([i0['ph'] for i0 in problem_kwarg_list])\nch_list = np.array([i0['ch'] for i0 in problem_kwarg_list])\n\ndata_hlx = pd.DataFrame({'TrA': TrA_list, \n 'TrB1': TrB1_list, \n 'TrB2': TrB2_list, \n 'TrC': TrC_list, \n 'A_00': A_00_list, \n 'B1_00': B1_00_list, \n 'B2_00': B2_00_list, \n 'C_00': C_00_list, \n 'A_11': A_11_list, \n 'B1_11': B1_11_list, \n 'B2_11': B2_11_list, \n 'C_11': C_11_list, \n 'A_22': A_22_list, \n 'B1_22': B1_22_list, \n 'B2_22': B2_22_list, \n 'C_22': C_22_list, \n 'ph': ph_list, \n 'ch': ch_list, \n }).pivot_table(index=['ph'], columns=['ch'])\n\n# %matplotlib notebook\n%matplotlib inline\nfrom matplotlib.colors import Normalize\n\nplt.rcParams.update({'font.size': 30})\nfigsize=np.array((16, 9)) * 1\ndpi = 500 if 'inline' in matplotlib.get_backend() else 100\nvmin, vmax, midpoint = -0.03, 0.005, 0\n# vmin, vmax = -0.02, 0.02\ncmap = plt.get_cmap('seismic')\n\nfig, axs = plt.subplots(1, 1, figsize=figsize, dpi=dpi)\nfig.patch.set_facecolor('white')\nnorm = spf.midLinearNorm(midpoint=midpoint, vmin=vmin, vmax=vmax)\n\nt1 = data_hlx.TrB1 / data_hlx.TrA\ntph = t1.index.values # ph\ntch = t1.columns.values # ch\ntvalue = t1.values.T\naxi = axs\nim = axi.pcolormesh(tph, tch, tvalue, cmap=cmap, norm=norm, shading='auto')\ncs = axi.contour(tph, tch, tvalue, [0, ], colors='k')\naxi.clabel(cs, inline=True)\nfig.colorbar(im, ax=axi, orientation='vertical')\naxi.set_xlabel('$\\\\lambda$')\naxi.set_ylabel('$n$')\naxi.set_xlim((0, axi.get_xlim()[1]))\naxi.set_ylim((0, axi.get_ylim()[1]))\naxi.patch.set_facecolor('grey')\naxi.patch.set_alpha(0.2)\naxi.set_title('single helix, $\\\\Tr \\\\mathbb{B}^v / \\\\Tr \\\\mathbb{A}^v$')\n\nplt.tight_layout()\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "e31ee0c567b9ad5fa522ea3eb7a42dc5ba1b7e64", "size": 442772, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "HelicodsParticles/helicoid_hlx/main_show_v2.ipynb", "max_stars_repo_name": "pcmagic/stokes_flow", "max_stars_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-11-11T05:00:53.000Z", "max_stars_repo_stars_event_max_datetime": "2018-11-11T05:00:53.000Z", "max_issues_repo_path": "HelicodsParticles/helicoid_hlx/main_show_v2.ipynb", "max_issues_repo_name": "pcmagic/stokes_flow", "max_issues_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, 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YES\n2. YES", "lm_q1_score": 0.727975460709318, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.4259392822926509}} {"text": "```python\nfrom pycalphad import Database\nfrom pycalphad.tests.datasets import DIFFUSION_TDB, ALFE_TDB, ALNIPT_TDB\n```\n\n\n```python\ndbf = Database(ALNIPT_TDB)\n```\n\n\n```python\nother = Database.from_string(dbf.to_string(fmt='tdb'), fmt='tdb')\n\nprint(dbf.phases.keys() == other.phases.keys())\nfor phase in dbf.phases.keys():\n if dbf.phases[phase] != other.phases[phase]:\n print(phase + ' differs')\n else:\n print(phase + ' eq')\n```\n\n True\n PT2AL3 eq\n AL3NI5 eq\n ALPT eq\n AL3NI2 eq\n ALPT2 eq\n PT5AL21 eq\n FCC_L12 eq\n FCC_A1 eq\n BCC_B2 eq\n AL3NI1 eq\n PT2AL eq\n LIQUID eq\n PT8AL21 eq\n PT5AL3 eq\n\n\n\n```python\ndbf == other\n```\n\n\n\n\n True\n\n\n\n\n```python\ndef param_sort_key(x):\n return x['phase_name'], x['parameter_type'], x['constituent_array'], x['parameter_order'], x['diffusing_species']\n\nself_params = sorted(dbf._parameters.all(), key=param_sort_key)\nother_params = sorted(other._parameters.all(), key=param_sort_key)\nself_params == other_params\nfor s, o in zip(self_params, other_params):\n if s != o:\n if s['parameter'] != o['parameter']:\n print('params differ')\n print('self', s['parameter'])\n print('other', o['parameter'])\n else:\n print('eq')\n```\n\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n eq\n\n\n\n```python\none = sorted(self_params[0].items(), key=operator.itemgetter(0))\n```\n\n\n```python\ntwo = sorted(other_params[0].items(), key=operator.itemgetter(0))\n```\n\n\n```python\none == two\n```\n\n\n```python\none[2][1]\n```\n\n\n```python\ntwo[2][1]\n```\n\n\n```python\none[2][1].args[1].args[1] == two[2][1].args[1].args[1]\n```\n\n\n```python\nimport numpy as np\nnp.array(one[2][1].args[1].args[0], dtype=np.longdouble)\n```\n\n\n```python\nnp.array(two[2][1].args[1].args[0], dtype=np.longdouble)\n```\n\n\n```python\nfrom sympy import sympify\nsympy.evaluate.global_evaluate = False\nsympify('ln(-5.01E-4, evaluate=False)', evaluate=False).__class__\n```\n\n\n```python\nsympify('ln(x)', evaluate=False).xreplace({'x': 5.01E-4})\n```\n\n\n```python\nsympy.S(5.01e-4).__class__\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "f140725775074aa27de6ed958365dc9408f6b633", "size": 10708, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Diffusion.ipynb", "max_stars_repo_name": "richardotis/pycalphad-sandbox", "max_stars_repo_head_hexsha": "43d8786eee8f279266497e9c5f4630d19c893092", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2017-03-08T18:21:30.000Z", "max_stars_repo_stars_event_max_datetime": "2017-03-08T18:21:30.000Z", "max_issues_repo_path": "Diffusion.ipynb", "max_issues_repo_name": "richardotis/pycalphad-sandbox", "max_issues_repo_head_hexsha": "43d8786eee8f279266497e9c5f4630d19c893092", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Diffusion.ipynb", "max_forks_repo_name": "richardotis/pycalphad-sandbox", "max_forks_repo_head_hexsha": "43d8786eee8f279266497e9c5f4630d19c893092", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2018-11-03T01:31:57.000Z", "max_forks_repo_forks_event_max_datetime": "2018-11-03T01:31:57.000Z", "avg_line_length": 18.2730375427, "max_line_length": 687, "alphanum_fraction": 0.4011019798, "converted": true, "num_tokens": 1533, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.7279754548076477, "lm_q1q2_score": 0.42593927883957683}} {"text": "```python\nimport numpy as np\nimport matplotlib\nimport matplotlib.pyplot as plt\nfrom matplotlib.transforms import Affine2D\nfrom matplotlib.projections import PolarAxes\nimport mpl_toolkits.axisartist.floating_axes as floating_axes\nimport mpl_toolkits.axisartist.angle_helper as angle_helper\nfrom mpl_toolkits.axisartist.grid_finder import FixedLocator, MaxNLocator, DictFormatter\n\nfrom astropy.table import Table\nfrom astropy.io import fits\nfrom astropy.cosmology import FlatLambdaCDM, z_at_value\n\nfrom scipy.spatial import cKDTree, ConvexHull\n\nfrom sympy import solve_poly_system, im\nfrom sympy.abc import x,y\n\nmatplotlib.rcParams.update({'font.size': 38})\n```\n\n\n```python\n#galaxy data file\ngdata = fits.open(\"ALL.fits\")\n#VoidFinder maximal sphere output\nvfdata = Table.read(\"DR7_comoving_maximal.txt\",format='ascii.commented_header')\n#VoidFinder hole output\nvfdata2 = Table.read(\"DR7_comoving_holes.txt\",format='ascii.commented_header')\n#Vsquared triangle output\ntridata = Table.read(\"DR7_triangles.dat\",format='ascii.commented_header')\ngzdata = Table.read(\"DR7_galzones.dat\",format='ascii.commented_header')\nzvdata = Table.read(\"DR7_zonevoids.dat\",format='ascii.commented_header')\n```\n\n\n```python\nD2R = np.pi/180.\n\ndef toSky(cs):\n c1 = cs.T[0]\n c2 = cs.T[1]\n c3 = cs.T[2]\n r = np.sqrt(c1**2.+c2**2.+c3**2.)\n dec = np.arcsin(c3/r)/D2R\n ra = (np.arccos(c1/np.sqrt(c1**2.+c2**2.))*np.sign(c2)/D2R)%360\n return r,ra,dec\n\ndef toCoord(r,ra,dec):\n c1 = r*np.cos(ra*D2R)*np.cos(dec*D2R)\n c2 = r*np.sin(ra*D2R)*np.cos(dec*D2R)\n c3 = r*np.sin(dec*D2R)\n return c1,c2,c3\n```\n\n\n```python\ngr = gdata[1].data['Rgal']\ngra = gdata[1].data['ra']\ngdec = gdata[1].data['dec']\n\ngx,gy,gz = toCoord(gr,gra,gdec)\nkdt = cKDTree(np.array([gx,gy,gz]).T)\n\ng_z = gzdata['zone']\nz_v = zvdata['void0']\n\nvfx = vfdata['x']\nvfy = vfdata['y']\nvfz = vfdata['z']\nvfr,vfra,vfdec = toSky(np.array([vfx,vfy,vfz]).T)\nvfrad = vfdata['radius']\nvfc = matplotlib.cm.nipy_spectral(np.linspace(0,1,len(vfr)))\nvfcc = np.random.choice(range(len(vfc)),len(vfc),replace=False)\n\nvflag = vfdata2['flag']\nvfx2 = vfdata2['x']\nvfy2 = vfdata2['y']\nvfz2 = vfdata2['z']\nvfr1,vfra1,vfdec1 = toSky(np.array([vfx2,vfy2,vfz2]).T)\nvfrad1 = vfdata2['radius']\nvfx4 = [vfx2[vflag==vfl] for vfl in np.unique(vflag)]\nvfy4 = [vfy2[vflag==vfl] for vfl in np.unique(vflag)]\nvfz4 = [vfz2[vflag==vfl] for vfl in np.unique(vflag)]\nvfr2 = [vfr1[vflag==vfl] for vfl in np.unique(vflag)]\nvfra2 = [vfra1[vflag==vfl] for vfl in np.unique(vflag)]\nvfdec2 = [vfdec1[vflag==vfl] for vfl in np.unique(vflag)]\nvfrad2 = [vfrad1[vflag==vfl] for vfl in np.unique(vflag)]\n\ngflag_vf = np.zeros(len(gx),dtype=bool)\ngflag_v2 = np.zeros(len(gx),dtype=bool)\n\nfor vfl in np.unique(vflag):\n vfx3 = vfx2[vflag==vfl]\n vfy3 = vfy2[vflag==vfl]\n vfz3 = vfz2[vflag==vfl]\n vfrad3 = vfrad1[vflag==vfl]\n for i in range(len(vfx3)):\n galinds = kdt.query_ball_point([vfx3[i],vfy3[i],vfz3[i]],vfrad3[i])\n gflag_vf[galinds] = True\n\nfor z in range(len(z_v)):\n if z_v[z] > -1:\n gflag_v2[g_z==z] = True\n\nwflag_vf = (1-gflag_vf).astype(bool)\nwflag_v2 = (1-gflag_v2).astype(bool)\n\np1_r,p1_ra,p1_dec = toSky(np.array([tridata['p1_x'],tridata['p1_y'],tridata['p1_z']]).T)\np2_r,p2_ra,p2_dec = toSky(np.array([tridata['p2_x'],tridata['p2_y'],tridata['p2_z']]).T)\np3_r,p3_ra,p3_dec = toSky(np.array([tridata['p3_x'],tridata['p3_y'],tridata['p3_z']]).T)\np1_x = tridata['p1_x']\np1_y = tridata['p1_y']\np1_z = tridata['p1_z']\np2_x = tridata['p2_x']\np2_y = tridata['p2_y']\np2_z = tridata['p2_z']\np3_x = tridata['p3_x']\np3_y = tridata['p3_y']\np3_z = tridata['p3_z']\ntrivids = np.array(tridata['void_id'])\nv2c = matplotlib.cm.nipy_spectral(np.linspace(0,1,np.amax(trivids)+1))\nv2cc = np.random.choice(range(len(v2c)),len(v2c),replace=False)\n```\n\n\n```python\n#calculate radii of maximal sphere-slice intersections\ndef cint(dec):\n cr = []\n for i in range(len(vfr)):\n dtd = np.abs(vfr[i]*np.sin((vfdec[i]-dec)*D2R))\n if dtd>vfrad[i]:\n cr.append(0.)\n else:\n cr.append(np.sqrt(vfrad[i]**2.-dtd**2.))\n return cr\n\n#calculate radii of hole-slice intersections\ndef cint2(dec):\n cr = []\n for i in range(len(vfr2)):\n cr.append([])\n for j in range(len(vfr2[i])):\n dtd = np.abs(vfr2[i][j]*np.sin((vfdec2[i][j]-dec)*D2R))\n if dtd>vfrad2[i][j]:\n cr[i].append(0.)\n else:\n cr[i].append(np.sqrt(vfrad2[i][j]**2.-dtd**2.))\n return cr\n\ndef isin(p,ps,ch,chavg,chrad):\n if np.sum((p-chavg)**2.)0:\n continue\n elif p1[0]>p[0] and p2[0]>p[0]:\n nc = nc+1\n elif ((p2[1]-p1[1])/(p2[0]-p1[0]))*((p1[1]-p[1])-((p2[1]-p1[1])/(p2[0]-p1[0]))*(p1[0]-p[0]))<1:\n nc = nc+1\n return nc%2==0\n\n#calculate coordinates of triangle-slice intersections\ndef trint(dec):\n decsum = np.array([(p1_dec>dec).astype(int),(p2_dec>dec).astype(int),(p3_dec>dec).astype(int)]).T\n intr = [[] for _ in range(np.amax(trivids)+1)]\n intra = [[] for _ in range(np.amax(trivids)+1)]\n for i in range(len(trivids)):\n if np.sum(decsum[i])==0:\n continue\n if np.sum(decsum[i])==3:\n continue\n cv = trivids[i]\n if np.sum(decsum[i])==1:\n if decsum[i][0]==1:\n intr[cv].append((p1_r[i]+p2_r[i])/2.)\n intr[cv].append((p1_r[i]+p3_r[i])/2.)\n intra[cv].append((p1_ra[i]+p2_ra[i])/2.)\n intra[cv].append((p1_ra[i]+p3_ra[i])/2.)\n elif decsum[i][1]==1:\n intr[cv].append((p2_r[i]+p1_r[i])/2.)\n intr[cv].append((p2_r[i]+p3_r[i])/2.)\n intra[cv].append((p2_ra[i]+p1_ra[i])/2.)\n intra[cv].append((p2_ra[i]+p3_ra[i])/2.)\n elif decsum[i][2]==1:\n intr[cv].append((p3_r[i]+p1_r[i])/2.)\n intr[cv].append((p3_r[i]+p2_r[i])/2.)\n intra[cv].append((p3_ra[i]+p1_ra[i])/2.)\n intra[cv].append((p3_ra[i]+p2_ra[i])/2.)\n elif np.sum(decsum[i])==2:\n if decsum[i][0]==0:\n intr[cv].append((p1_r[i]+p2_r[i])/2.)\n intr[cv].append((p1_r[i]+p3_r[i])/2.)\n intra[cv].append((p1_ra[i]+p2_ra[i])/2.)\n intra[cv].append((p1_ra[i]+p3_ra[i])/2.)\n elif decsum[i][1]==0:\n intr[cv].append((p2_r[i]+p1_r[i])/2.)\n intr[cv].append((p2_r[i]+p3_r[i])/2.)\n intra[cv].append((p2_ra[i]+p1_ra[i])/2.)\n intra[cv].append((p2_ra[i]+p3_ra[i])/2.)\n elif decsum[i][2]==0:\n intr[cv].append((p3_r[i]+p1_r[i])/2.)\n intr[cv].append((p3_r[i]+p2_r[i])/2.)\n intra[cv].append((p3_ra[i]+p1_ra[i])/2.)\n intra[cv].append((p3_ra[i]+p2_ra[i])/2.)\n return intr,intra\n\ndef getinx(xx,aa,yy,bb,zz,cc,dd):\n negb = -1.*aa*xx-bb*yy+cc*dd*dd*zz\n sqto = 0.5*np.sqrt((2.*aa*xx+2.*bb*yy-2.*cc*dd*dd*zz)**2.-4.*(aa**2.+bb**2.-cc*cc*dd*dd)*(xx**2.+yy**2.-zz*zz*dd*dd))\n twa = aa**2.+bb**2.-cc*cc*dd*dd\n tt = (negb+sqto)/twa\n if tt>0 and tt<1:\n tt = tt\n else:\n tt = (negb-sqto)/twa\n return xx+aa*tt,yy+bb*tt,zz+cc*tt\n\ndef trint2(dec):\n decsum = np.array([(p1_dec>dec).astype(int),(p2_dec>dec).astype(int),(p3_dec>dec).astype(int)]).T\n intr = [[] for _ in range(np.amax(trivids)+1)]\n intra = [[] for _ in range(np.amax(trivids)+1)]\n for i in range(len(trivids)):\n if np.sum(decsum[i])==0:\n continue\n if np.sum(decsum[i])==3:\n continue\n cv = trivids[i]\n if np.sum(decsum[i])==1:\n if decsum[i][0]==1:\n sss = getinx(p1_x[i],p2_x[i]-p1_x[i],p1_y[i],p2_y[i]-p1_y[i],p1_z[i],p2_z[i]-p1_z[i],1./np.tan(dec*D2R))\n sst = getinx(p1_x[i],p3_x[i]-p1_x[i],p1_y[i],p3_y[i]-p1_y[i],p1_z[i],p3_z[i]-p1_z[i],1./np.tan(dec*D2R))\n elif decsum[i][1]==1:\n sss = getinx(p1_x[i],p2_x[i]-p1_x[i],p1_y[i],p2_y[i]-p1_y[i],p1_z[i],p2_z[i]-p1_z[i],1./np.tan(dec*D2R))\n sst = getinx(p3_x[i],p2_x[i]-p3_x[i],p3_y[i],p2_y[i]-p3_y[i],p3_z[i],p2_z[i]-p3_z[i],1./np.tan(dec*D2R))\n elif decsum[i][2]==1:\n sss = getinx(p1_x[i],p3_x[i]-p1_x[i],p1_y[i],p3_y[i]-p1_y[i],p1_z[i],p3_z[i]-p1_z[i],1./np.tan(dec*D2R))\n sst = getinx(p3_x[i],p2_x[i]-p3_x[i],p3_y[i],p2_y[i]-p3_y[i],p3_z[i],p2_z[i]-p3_z[i],1./np.tan(dec*D2R))\n elif np.sum(decsum[i])==2:\n if decsum[i][0]==0:\n sss = getinx(p1_x[i],p2_x[i]-p1_x[i],p1_y[i],p2_y[i]-p1_y[i],p1_z[i],p2_z[i]-p1_z[i],1./np.tan(dec*D2R))\n sst = getinx(p1_x[i],p3_x[i]-p1_x[i],p1_y[i],p3_y[i]-p1_y[i],p1_z[i],p3_z[i]-p1_z[i],1./np.tan(dec*D2R))\n elif decsum[i][1]==0:\n sss = getinx(p1_x[i],p2_x[i]-p1_x[i],p1_y[i],p2_y[i]-p1_y[i],p1_z[i],p2_z[i]-p1_z[i],1./np.tan(dec*D2R))\n sst = getinx(p3_x[i],p2_x[i]-p3_x[i],p3_y[i],p2_y[i]-p3_y[i],p3_z[i],p2_z[i]-p3_z[i],1./np.tan(dec*D2R))\n elif decsum[i][2]==0:\n sss = getinx(p1_x[i],p3_x[i]-p1_x[i],p1_y[i],p3_y[i]-p1_y[i],p1_z[i],p3_z[i]-p1_z[i],1./np.tan(dec*D2R))\n sst = getinx(p3_x[i],p2_x[i]-p3_x[i],p3_y[i],p2_y[i]-p3_y[i],p3_z[i],p2_z[i]-p3_z[i],1./np.tan(dec*D2R))\n intr[cv].append(np.sqrt(np.sum(np.array(sss)**2.)))\n intr[cv].append(np.sqrt(np.sum(np.array(sst)**2.)))\n intra[cv].append((np.arccos(sss[0]/np.sqrt(sss[0]**2.+sss[1]**2.))*np.sign(sss[1])/D2R)%360)\n intra[cv].append((np.arccos(sst[0]/np.sqrt(sst[0]**2.+sst[1]**2.))*np.sign(sst[1])/D2R)%360)\n return intr,intra\n```\n\n\n```python\n#convert a circle's coordinates to ordered boundary\ndef gcp(cc1,cc2,crad,npt):\n ccx = cc1*np.cos(cc2*D2R)\n ccy = cc1*np.sin(cc2*D2R)\n Cx = np.linspace(0.,2*np.pi,npt)\n Cy = np.linspace(0.,2*np.pi,npt)\n Cx = np.cos(Cx)*crad+ccx\n Cy = np.sin(Cy)*crad+ccy\n C1 = np.sqrt(Cx**2.+Cy**2.)\n C2 = (np.sign(Cy)*np.arccos(Cx/C1)+np.pi*(1.-np.sign(Cy)))/D2R\n return C1,C2\n\n#convert circles' coordinates to ordered boundary\ndef gcp2(cc1,cc2,crad,npt,chkdpth):\n ccx = cc1*np.cos(cc2*D2R)\n ccy = cc1*np.sin(cc2*D2R)\n Cx = [np.linspace(0.,2*np.pi,int(npt*crad[k]/10)) for k in range(len(ccx))]\n Cy = [np.linspace(0.,2*np.pi,int(npt*crad[k]/10)) for k in range(len(ccx))]\n Cx = [np.cos(Cx[k])*crad[k]+ccx[k] for k in range(len(ccx))]\n Cy = [np.sin(Cy[k])*crad[k]+ccy[k] for k in range(len(ccx))]\n for i in range(len(ccx)):\n for j in range(len(ccx)):\n if i==j:\n continue\n cut = (Cx[j]-ccx[i])**2.+(Cy[j]-ccy[i])**2.>crad[i]**2.\n Cx[j] = Cx[j][cut]\n Cy[j] = Cy[j][cut]\n Cp = []\n for i in range(len(ccx)):\n Cp.extend(np.array([Cx[i],Cy[i]]).T.tolist())\n Cp = np.array(Cp)\n kdt = cKDTree(Cp)\n Cpi = [0]\n while len(Cpi)srad[i]:\n continue\n else:\n frad = np.sqrt(srad[i]**2.-dtd**2.)\n Cx.append([])\n Cy.append([])\n A = srad[i]**2.\n B = (1.-np.sin(dec*D2R))*2.*sz[i]*srad[i]\n C = -1.*np.sin(dec*D2R)*2.*srad[i]*(sx[i]*np.cos(np.arange(int(npt*frad/20.))*20.*np.pi/(npt*frad))+sy[i]*np.sin(np.arange(int(npt*frad/20.))*20.*np.pi/(npt*frad)))\n D = (sz[i]**2.)-np.sin(dec*D2R)*(sx[i]**2+sy[i]**2.+sz[i]**2.+srad[i]**2.)\n adjstr = 0\n for j in range(len(C)):\n if C[j] != 0:\n try:\n print(A)\n print(B)\n print(C[j])\n print(D)\n sps = solve_poly_system([A*(x**2)+B*x+C[j]*y+D,x**2+y**2-1],x,y)\n print(\".\",end='',flush=True)\n except:\n #print(A)\n #print(B)\n #print(C[j])\n #print(D)\n #print(sdec[i])\n #print(dec)\n #print(srad[i])\n #print(np.sqrt(sx[i]**2.+sy[i]**2.+sz[i]**2.))\n aaaa = 1\n try:\n if im(sps[0][0])==0:\n Cx[-1].insert(j-adjstr,sps[0][0])\n Cx[-1].append(sps[1][0])\n Cy[-1].insert(j-adjstr,sps[0][1])\n Cy[-1].append(sps[1][1])\n else:\n adjstr = adjstr + 1\n except:\n adjstr = adjstr + 1\n else:\n sps = solve([A*(x**2)+B*x+D],x)\n Cx[-1].insert(j,sps[0][0])\n Cx[-1].append(sps[1][0])\n Cy[-1].insert(j,np.sqrt(1.-sps[0][0]**2.))\n Cy[-1].append(-1.*np.sqrt(1.-sps[1][1]**2.))\n if len(Cx[-1])==0:\n del Cx[-1]\n del Cy[-1]\n continue\n Cx[-1] = np.array(Cx[-1])\n Cy[-1] = np.array(Cy[-1])\n Cz.append(sz[i] + srad[i]*Cx[-1])\n Cx[-1] = sx[i] + srad[i]*Cy[-1]*np.cos(np.arange(len(Cz[-1]))*2.*np.pi/len(Cz[-1]))\n Cy[-1] = sy[i] + srad[i]*Cy[-1]*np.sin(np.arange(len(Cz[-1]))*2.*np.pi/len(Cz[-1]))\n try:\n Cr.append(np.sqrt(Cx[-1]**2.+Cy[-1]**2.+Cz[-1]**2.))\n except:\n #print(Cx[-1])\n #print(Cy[-1])\n #print(Cz[-1])\n aaaa = 1\n Cra.append((np.arccos(Cx[-1]/np.sqrt(Cx[-1]**2.+Cy[-1]**2.))*np.sign(Cy[-1])/D2R)%360)\n Cs.append(np.array([Cr[-1]*np.cos(Cra[-1]),Cr[-1]*np.sin(Cra[-1])]).T)\n Cs2.append(Cs[-1])\n Ch.append(ConvexHull(Cs[-1]))\n Chavg.append(np.array([np.sum(Cs[-1].T[0]),np.sum(Cs[-1].T[1])])/len(Cs[-1]))\n Chrad.append(np.amin(np.sum((Cs[-1]-Chavg[-1])**2.,axis=1)))\n if len(Cx)==0:\n return np.array([]),np.array([])\n for i in range(len(Cs2)):\n for j in range(len(Cs)):\n if i==j:\n continue\n cut = np.ones(len(Cs2[i]))\n for k in range(len(Cs2[i])):\n if isin(Cs2[i][k],Cs[j],Ch[j],Chavg[j],Chrad[j]):\n cut[k] = False\n Cs2[i] = Cs2[i][cut]\n Cp = []\n for i in range(len(Cx)):\n print(Cs2[i].tolist())\n Cp.extend(Cs2[i].tolist())\n Cp = np.array(Cp)\n kdt = cKDTree(Cp)\n Cpi = [0]\n while len(Cpi)2:\n print(\"0\",end='',flush=True)\n dists = []\n pairs = []\n for i in range(len(xs)):\n if scut[i]:\n for j in range(i+1,len(xs)):\n if scut[j]:\n dists.append((xs[i]-xs[j])**2.+(ys[i]-ys[j])**2.)\n pairs.append([i,j])\n pairs = np.array(pairs)[np.argsort(dists)]\n paird = scut\n xs2 = xs.tolist()\n ys2 = ys.tolist()\n cmp = np.arange(len(xs)).tolist()\n for i in range(len(pairs)):\n if paird[pairs[i][0]] and paird[pairs[i][1]]:\n paird[pairs[i][0]] = False\n paird[pairs[i][1]] = False\n xs2.extend([xs[pairs[i][0]],xs[pairs[i][1]]])\n ys2.extend([ys[pairs[i][0]],ys[pairs[i][1]]])\n cmp.extend([pairs[i][0],pairs[i][1]])\n xs2 = np.array(xs2)\n ys2 = np.array(ys2)\n lcut = np.ones(len(xs2),dtype=bool)\n for i in range(len(xs2)):\n if lcut[i]:\n chains.append([])\n chains[-1].append(cmp[i])\n lcut[i] = False\n j = i + 1 - 2*(i%2)\n while xs2[j] != xs2[i]:\n lcut[j] = False\n k = np.where(xs2==xs2[j])[0]\n k = k[k != j][0]\n chains[-1].append(cmp[k])\n lcut[k] = False\n j = k + 1 - 2*(k%2)\n if chains[-1][0] != chains[-1][-1]:\n chains[-1].append(chains[-1][0])\n return chains\n\ndef convint3(intr,intra):\n intx = np.array(intr)*np.cos(np.array(intra)*D2R)\n inty = np.array(intr)*np.sin(np.array(intra)*D2R)\n chkl = []\n ccut = np.ones(len(intr),dtype=bool)\n for i in range(int(len(intr)/2)):\n chkl.append(intx[2*i]+intx[2*i+1])\n chkl = np.array(chkl)\n for i in range(len(chkl)):\n if len(chkl[chkl==chkl[i]])>1:\n ccut[2*i] = False\n ccut[2*i+1] = False\n intx = intx[ccut]\n inty = inty[ccut]\n ocut = getorder(intx,inty)\n icut = np.zeros(len(ocut),dtype=bool)\n lens = np.zeros(len(ocut))\n for i in range(len(ocut)):\n for j in range(len(ocut[i])-1):\n lens[i] = lens[i] + np.sqrt((intx[ocut[i][j+1]]-intx[ocut[i][j]])**2.+(inty[ocut[i][j+1]]-inty[ocut[i][j]])**2.)\n mlh = np.amax(lens)\n for i in range(len(ocut)):\n if lens[i]==mlh:\n continue\n o = ocut[i]\n P = np.array([intx[o][0],inty[o][0]])\n for j in range(len(ocut)):\n if j==i:\n continue\n o1 = ocut[j]\n Ps = np.array([intx[o1],inty[o1]]).T\n if isin2(P,Ps):\n icut[i] = True\n break\n return [[np.array(intr)[ccut][o].tolist(),np.array(intra)[ccut][o].tolist()] for o in ocut],icut \n```\n\n\n```python\ndef setup_axes3(fig, rect):\n \"\"\"\n Sometimes, things like axis_direction need to be adjusted.\n \"\"\"\n\n # rotate a bit for better orientation\n tr_rotate = Affine2D().translate(-95, 0)\n\n # scale degree to radians\n tr_scale = Affine2D().scale(np.pi/180., 1.)\n\n tr = tr_rotate + tr_scale + PolarAxes.PolarTransform()\n\n grid_locator1 = angle_helper.LocatorDMS(4)\n tick_formatter1 = angle_helper.FormatterDMS()\n \n grid_locator2 = MaxNLocator(3)\n\n ra0, ra1 = 108, 263\n cz0, cz1 = 0., 306.\n grid_helper = floating_axes.GridHelperCurveLinear(tr,\n extremes=(ra0, ra1, cz0, cz1),\n grid_locator1=grid_locator1,\n grid_locator2=grid_locator2,\n tick_formatter1=tick_formatter1,\n tick_formatter2=None,\n )\n\n ax1 = floating_axes.FloatingSubplot(fig, rect, grid_helper=grid_helper)\n fig.add_subplot(ax1)\n\n # adjust axis\n ax1.axis[\"left\"].set_axis_direction(\"bottom\")\n ax1.axis[\"right\"].set_axis_direction(\"top\")\n\n ax1.axis[\"bottom\"].set_visible(False)\n ax1.axis[\"top\"].set_axis_direction(\"bottom\")\n ax1.axis[\"top\"].toggle(ticklabels=True, label=True)\n ax1.axis[\"top\"].major_ticklabels.set_axis_direction(\"top\")\n ax1.axis[\"top\"].label.set_axis_direction(\"top\")\n\n ax1.axis[\"left\"].label.set_text(r\"r [Mpc h$^{-1}$]\")\n ax1.axis[\"top\"].label.set_text(r\"$\\alpha$\")\n\n\n # create a parasite axes whose transData in RA, cz\n aux_ax = ax1.get_aux_axes(tr)\n\n aux_ax.patch = ax1.patch # for aux_ax to have a clip path as in ax\n ax1.patch.zorder=0.8 # but this has a side effect that the patch is\n # drawn twice, and possibly over some other\n # artists. So, we decrease the zorder a bit to\n # prevent this.\n aux_ax.set_facecolor(\"white\")\n\n return ax1, aux_ax\n```\n\n\n```python\n#plot VoidFinder maximal spheres\ndef pvf(dec,wdth,npc):\n #fig = plt.figure(1, figsize=(12,6))\n fig = plt.figure(1, figsize=(1600/96,800/96))\n ax3, aux_ax3 = setup_axes3(fig, 111)\n Cr = cint(dec)\n for i in range(len(vfr)):\n if Cr[i]>0:\n Cr2,Cra2 = gcp(vfr[i],vfra[i],Cr[i],npc)\n aux_ax3.plot(Cra2,Cr2,color='blue')\n aux_ax3.fill(Cra2,Cr2,alpha=0.2,color='blue')\n gdcut = (gr[wflag_vf]*np.sin((gdec[wflag_vf]-dec)*D2R))**2.0:\n Cr2,Cra2 = gcp2(vfr2[i],vfra2[i],Cr[i],npc,chkdpth)\n aux_ax3.plot(Cra2,Cr2,color='blue')\n aux_ax3.fill(Cra2,Cr2,alpha=0.2,color='blue')\n #Cr2,Cra2 = gcp3(vfx4[i],vfy4[i],vfz4[i],vfr2[i],vfrad2[i],vfdec2[i],dec,npc,chkdpth)\n #if len(Cr2)>0:\n # aux_ax3.plot(Cra2,Cr2,color='blue')\n # aux_ax3.fill(Cra2,Cr2,alpha=0.5,color='blue')\n gdcut = (gr[wflag_vf]*np.sin((gdec[wflag_vf]-dec)*D2R))**2.0:\n #Intr2 = convint(Intr[i])\n #Intra2 = convint(Intra[i])\n #Intr2,Intra2 = convint2(Intr[i],Intra[i])\n Intc2,Icut = convint3(Intr[i],Intra[i])\n Intr2 = [Intc[0] for Intc in Intc2]\n Intra2 = [Intc[1] for Intc in Intc2]\n for j in range(len(Intr2)):\n #if Icut[j]:\n # continue\n aux_ax3.plot(Intra2[j],Intr2[j],color='blue')\n aux_ax3.fill(Intra2[j],Intr2[j],alpha=0.1,color='blue')\n #for j in range(len(Intr2)):\n # if Icut[j]:\n # aux_ax3.plot(Intra2[j],Intr2[j],color='blue')\n # aux_ax3.fill(Intra2[j],Intr2[j],color='white')\n gdcut = (gr[wflag_v2]*np.sin((gdec[wflag_v2]-dec)*D2R))**2.here.''')\ndisplay(tag)\n```\n\n\n\nToggle cell visibility here.\n\n\n\n```python\n%matplotlib notebook\nimport numpy as np\nimport control as control\nimport matplotlib.pyplot as plt\nimport matplotlib.image as mpimg\nfrom ipywidgets import widgets\nfrom ipywidgets import interact\nimport scipy.signal as signal\nimport sympy as sym\n\n```\n\n## Mechanikai rendszerek\n\n#### \u00c1ltal\u00e1nos t\u00f6meg-rug\u00f3-csillap\u00edt\u00e1s modell\n> A t\u00f6meg-rugr\u00f3-csillap\u00edt\u00e1s modell egy pontszer\u0171 t\u00f6megekb\u0151l \u00e1ll, amiket rug\u00f3k \u00e9s csillap\u00edt\u00e1sok kapcsolnak \u00f6ssze. A modell alkalmas \u00f6sszetett anyagtulajdons\u00e1gok, mint p\u00e9ld\u00e1ul nemlinearit\u00e1s \u00e9s viszkoelaszticit\u00e1s (forr\u00e1s: [Wikipedia](https://en.wikipedia.org/wiki/Mass-spring-damper_model \"Mass-spring-model\"))\n#### Negyedaut\u00f3 modell\n> A negyed aut\u00f3modellt a leng\u00e9scsillap\u00edt\u00f3 rendszerek min\u0151s\u00edt\u00e9s\u00e9re alkalmazz\u00e1k. A t\u00f6meg $m_1$ a \"rug\u00f3z\u00f3 t\u00f6meg\", ami negyede a j\u00e1rm\u00fa teljes t\u00f6meg\u00e9nek, \u00e9s a leng\u00e9scsillap\u00edt\u00f3 rendszer t\u00e1masztja al\u00e1. Ezzel szemben $m_2$ a \"nem-rug\u00f3z\u00f3 t\u00f6meg\", ami egy ker\u00e9k, \u00e9s a f\u00e9l tengely t\u00f6meg\u00e9nek \u00f6sszege, valamint a leng\u00e9scsillap\u00edt\u00f3 rendszer\u00e9 \u00e9s felf\u00fcggeszt\u00e9s\u00e9. A rendszer merevs\u00e9g\u00e9t \u00e9s csillap\u00edt\u00e1s\u00e1t egy ide\u00e1lis rug\u00f3\u00e1lland\u00f3val ($k_1$) \u00e9s egy surl\u00f3d\u00e1si egy\u00fctthat\u00f3val ($B$) modellezz\u00fck. A gumiker\u00e9k rugalmass\u00e1g\u00e1t egy tov\u00e1bbi, $k_2$ rug\u00f3\u00e1lland\u00f3val vessz\u00fck figyelembe. (forr\u00e1s: [Chegg Study](https://www.chegg.com/homework-help/questions-and-answers/figure-p230-shows-1-4-car-model-used-analyze-ride-quality-automotive-suspension-systems-ma-q26244005 \"1/4 car model\"))\n\n---\n\n### Hogyan kezelhet\u0151 a p\u00e9lda?\n1. A *t\u00f6meg-rug\u00f3-csillap\u00edt\u00e1s* vallamint a *negyed aut\u00f3modell* k\u00f6z\u00f6tt a megfelel\u0151 gombokkal lehet v\u00e1ltani.\n2. Az er\u0151hat\u00e1s f\u00fcggv\u00e9nye ($F$) lehet *egys\u00e9gugr\u00e1s*, *impulzus*, *r\u00e1mpa* \u00e9s *szinusz* alak\u00fa.\n3. A megfelel\u0151 cs\u00faszk\u00e1kkal m\u00f3dos\u00edthat\u00f3ak a t\u00f6megek ($m$, $m_1$ \u00e9s $m_2$), rug\u00f3\u00e1lland\u00f3k ($k$, $k_1$ \u00e9s $k_2$) \u00e9s a csillap\u00edt\u00e1s ($B$) \u00e9rt\u00e9ke, valamint a bemen\u0151 jel er\u0151s\u00edt\u00e9se \u00e9s a kezdeti felt\u00e9telek ($x_0$, $\\dot{x}_0$, $y_0$, $\\dot{y}_0$).\n\n\n \n \n \n \n \n \n \n \n \n \n
                                        Mass-spring-damper1/4 car model
                                        \n\n\n```python\n# create figure\nfig = plt.figure(figsize=(9.8, 4),num='Mechanikai rendszerek')\n\n# add sublot\nax = fig.add_subplot(111)\nax.set_title('Id\u0151 v\u00e1lasz')\nax.set_ylabel('bemenet, kimenet')\nax.set_xlabel('$t$ [s]')\n\nax.grid(which='both', axis='both', color='lightgray')\n\ninputf, = ax.plot([], [])\nresponsef, = ax.plot([], [])\nresponsef2, = ax.plot([], [])\narrowf, = ax.plot([],[])\n\nstyle = {'description_width': 'initial'}\n\nselectSystem=widgets.ToggleButtons(\n options=[('t\u00f6meg-rug\u00f3-csillap\u00edt\u00e1s',0),('negyedaut\u00f3 modell',1)],\n description='Rendszer: ', style=style) # define toggle buttons\nselectForce = widgets.ToggleButtons(\n options=[('egys\u00e9gugr\u00e1s f\u00fcggv\u00e9ny', 0), ('impulzus f\u00fcggv\u00e9ny', 1), ('r\u00e1mpa f\u00fcggv\u00e9ny', 2), ('szinusz f\u00fcggv\u00e9ny', 3)],\n description='V\u00e1lasszon $F$ f\u00fcggv\u00e9nyt: ', style=style)\ndisplay(selectSystem)\ndisplay(selectForce)\n\ndef build_model(M,K,B,M1,M2,B1,K1,K2,amp,x0,xpika0,y0,ypika0,select_System,index):\n \n num_of_samples = 1000\n total_time = 25\n t = np.linspace(0, total_time, num_of_samples) # time for which response is calculated (start, stop, step)\n \n global inputf, responsef, responsef2, arrowf\n \n if select_System==0:\n \n system0 = control.TransferFunction([1], [M, B, K])\n \n if index==0:\n inputfunc = np.ones(len(t))*amp\n inputfunc[0]=0\n time, response, xx = control.forced_response(system0, t, inputfunc, X0=[xpika0,x0]) \n \n elif index==1:\n inputfunc=signal.unit_impulse(1000, 0)*amp\n time, response, xx = control.forced_response(system0, t, inputfunc, X0=[xpika0,x0]) \n \n elif index==2:\n inputfunc=t;\n time, response, xx = control.forced_response(system0, t, inputfunc, X0=[xpika0,x0]) \n \n elif index==3:\n inputfunc=np.sin(t)*amp\n time, response, xx = control.forced_response(system0, t, inputfunc, X0=[xpika0,x0]) \n \n elif select_System==1:\n \n system1=control.TransferFunction([M2, B1, K1+K2], [M1*M2, M1*B1+M2*B1, M2*K1+M1*(K1+K2), K2*B1, K1*K2])\n system2=control.TransferFunction([B1*K1*M2**2, B1**2*K1*M2, B1*K1**2*M2 + 2*B1*K1*K2*M2,\n B1**2*K1*K2, B1*K1**2*K2 + B1*K1*K2**2],\n [M1**2*M2**2, B1*M1**2*M2 + 2*B1*M1*M2**2, \n B1**2*M1*M2 + B1**2*M2**2 + K1*M1**2*M2 + 2*K1*M1*M2**2 + 2*K2*M1**2*M2 + K2*M1*M2**2,\n 2*B1*K1*M1*M2 + 2*B1*K1*M2**2 + B1*K2*M1**2 + 5*B1*K2*M1*M2 + B1*K2*M2**2,\n B1**2*K2*M1 + 2*B1**2*K2*M2 + K1**2*M1*M2 + K1**2*M2**2 + K1*K2*M1**2 + 5*K1*K2*M1*M2 + K1*K2*M2**2 + K2**2*M1**2 + 2*K2**2*M1*M2,\n 2*B1*K1*K2*M1 + 4*B1*K1*K2*M2 + 3*B1*K2**2*M1 + 2*B1*K2**2*M2,\n B1**2*K2**2 + K1**2*K2*M1 + 2*K1**2*K2*M2 + 3*K1*K2**2*M1 + 2*K1*K2**2*M2 + K2**3*M1,\n 2*B1*K1*K2**2 + B1*K2**3,\n K1**2*K2**2 + K1*K2**3])\n if index==0:\n inputfunc = np.ones(len(t))*amp\n inputfunc[0]=0 \n time, response, xx = control.forced_response(system1, t, inputfunc, X0=[0,0,xpika0,x0])\n time2, response2, xx2 = control.forced_response(system2, t, inputfunc, X0=[0,0,0,0,0,0,ypika0,y0])\n \n elif index==1:\n inputfunc=signal.unit_impulse(1000, 0)*amp\n time, response, xx = control.forced_response(system1, t, inputfunc, X0=[0,0,xpika0,x0])\n time2, response2, xx2 = control.forced_response(system2, t, inputfunc, X0=[0,0,0,0,0,0,ypika0,y0])\n \n elif index==2:\n inputfunc=t;\n time, response, xx = control.forced_response(system1, t, inputfunc, X0=[0,0,xpika0,x0])\n time2, response2, xx2 = control.forced_response(system2, t, inputfunc, X0=[0,0,0,0,0,0,ypika0,y0])\n \n elif index==3:\n inputfunc=np.sin(t)*amp\n time, response, xx = control.forced_response(system1, t, inputfunc, X0=[0,0,xpika0,x0])\n time2, response2, xx2 = control.forced_response(system2, t, inputfunc, X0=[0,0,0,0,0,0,ypika0,y0])\n\n \n ax.lines.remove(responsef)\n ax.lines.remove(inputf)\n ax.lines.remove(responsef2)\n ax.lines.remove(arrowf)\n \n inputf, = ax.plot(t,inputfunc,label='$F$',color='C0')\n responsef, = ax.plot(time, response,label='$x$',color='C3')\n \n if select_System==1:\n responsef2, = ax.plot(time, response2,label='$y$',color='C2')\n elif select_System==0:\n responsef2, = ax.plot([],[])\n \n if index==1:\n if amp>0:\n arrowf, = ax.plot([-0.1,0,0.1],[amp-((amp*0.05)/2),amp,amp-((amp*0.05)/2)],color='C0',linewidth=4)\n elif amp==0:\n arrowf, = ax.plot([],[])\n elif amp<0:\n arrowf, = ax.plot([-0.1,0,0.1],[amp-((amp*0.05)/2),amp,amp-(amp*(0.05)/2)],color='C0',linewidth=4)\n else:\n arrowf, = ax.plot([],[])\n \n ax.relim()\n ax.autoscale_view()\n \n ax.legend() \n \ndef update_sliders(index):\n global m1_slider, b1_slider, k1_slider, m21_slider, m22_slider, b2_slider, k21_slider, k22_slider\n global x0_slider, xpika0_slider, y0_slider, ypika0_slider\n\n m1val = [0.1,0.1,0.1,0.1]\n k1val = [1,1,1,1]\n b1val = [0.1,0.1,0.1,0.1]\n m21val = [0.1,0.1,0.1,0.1]\n m22val = [0.1,0.1,0.1,0.1]\n b2val = [0.1,0.1,0.1,0.1]\n k21val = [1,1,1,1]\n k22val = [1,1,1,1]\n x0val = [0,0,0,0]\n xpika0val = [0,0,0,0]\n y0val = [0,0,0,0]\n ypika0val = [0,0,0,0]\n \n m1_slider.value = m1val[index]\n k1_slider.value = k1val[index]\n b1_slider.value = b1val[index]\n m21_slider.value = m21val[index]\n m22_slider.value = m22val[index]\n b2_slider.value = b2val[index]\n k21_slider.value = k21val[index]\n k22_slider.value = k22val[index]\n x0_slider.value = x0val[index]\n xpika0_slider.value = xpika0val[index]\n y0_slider.value = y0val[index]\n ypika0_slider.value = ypika0val[index] \n \ndef draw_controllers(type_select,index):\n \n global m1_slider, b1_slider, k1_slider, m21_slider, m22_slider, b2_slider, k21_slider, k22_slider\n global x0_slider, xpika0_slider, y0_slider, ypika0_slider\n \n \n x0_slider=widgets.FloatSlider(value=0, min=-1, max=1., step=.1,\n description='$x_0$ [dm]:',disabled=False,continuous_update=False,\n orientation='horizontal',readout=True,readout_format='.2f',)\n xpika0_slider=widgets.FloatSlider(value=0, min=-1, max=1., step=.1,\n description='${\\dot{x}}_0$ [dm/s]:',disabled=False,continuous_update=False,\n orientation='horizontal',readout=True,readout_format='.2f',)\n \n if type_select==0:\n \n amp_slider = widgets.FloatSlider(value=1.,min=-2.,max=2.,step=0.1,\n description='Bemen\u0151 jel er\u0151s\u00edt\u00e9se:',disabled=False,continuous_update=False,orientation='horizontal',readout=True,readout_format='.1f',style=style)\n \n m1_slider = widgets.FloatSlider(value=.1, min=.01, max=1., step=.01,\n description='$m$ [kg]:',disabled=False,continuous_update=False,\n orientation='horizontal',readout=True,readout_format='.2f',)\n k1_slider = widgets.FloatSlider(value=1.,min=0.,max=20.,step=.1,\n description='$k$ [N/m]:',disabled=False,continuous_update=False,\n orientation='horizontal',readout=True,readout_format='.1f',)\n b1_slider = widgets.FloatSlider(value=.1,min=0.0,max=0.5,step=.01,\n description='$B$ [Ns/m]:',disabled=False,continuous_update=False,\n rientation='horizontal',readout=True,readout_format='.2f',)\n m21_slider = widgets.FloatSlider(value=.1,min=.01,max=1.,step=.01,\n description='$m_1$ [kg]:',disabled=True,continuous_update=False,orientation='horizontal',readout=True,readout_format='.2f',\n )\n m22_slider = widgets.FloatSlider(value=.1,min=.0,max=1.,step=.01,\n description='$m_2$ [kg]:',disabled=True,continuous_update=False,orientation='horizontal',readout=True,readout_format='.2f',\n )\n b2_slider = widgets.FloatSlider(value=.1,min=0.0,max=2,step=.01,\n description='$B$ [Ns/m]:',disabled=True,continuous_update=False,orientation='horizontal',readout=True,readout_format='.2f',\n )\n k21_slider = widgets.FloatSlider(value=1.,min=0.,max=20.,step=.1,\n description='$k_1$ [N/m]:',disabled=True,continuous_update=False,orientation='horizontal',readout=True,readout_format='.1f',\n )\n k22_slider = widgets.FloatSlider(value=1.,min=0.,max=20.,step=.1,\n description='$k_2$ [N/m]:',disabled=True,continuous_update=False,orientation='horizontal',readout=True,readout_format='.1f',\n )\n \n y0_slider=widgets.FloatSlider(value=0, min=-1, max=1., step=.1,\n description='$y_0$ [dm]:',disabled=True,continuous_update=False,\n orientation='horizontal',readout=True,readout_format='.2f',)\n ypika0_slider=widgets.FloatSlider(value=0, min=-1, max=1., step=.1,\n description='${\\dot{y}}_0$ [dm/s]:',disabled=True,continuous_update=False,\n orientation='horizontal',readout=True,readout_format='.2f',)\n elif type_select==1:\n \n amp_slider = widgets.FloatSlider(value=1.,min=-2.,max=2.,step=0.1,\n description='Bemen\u0151 jel er\u0151s\u00edt\u00e9se:',disabled=False,continuous_update=False,orientation='horizontal',readout=True,readout_format='.1f',style=style)\n \n m1_slider = widgets.FloatSlider(value=.1, min=.01, max=1., step=.01,\n description='$m$ [kg]:',disabled=True,continuous_update=False,\n orientation='horizontal',readout=True,readout_format='.2f',)\n k1_slider = widgets.FloatSlider(value=1.,min=0.,max=20.,step=.1,\n description='$k$ [N/m]:',disabled=True,continuous_update=False,\n orientation='horizontal',readout=True,readout_format='.1f',)\n b1_slider = widgets.FloatSlider(value=.1,min=0.0,max=0.5,step=.01,\n description='$B$ [Ns/m]:',disabled=True,continuous_update=False,\n rientation='horizontal',readout=True,readout_format='.2f',)\n m21_slider = widgets.FloatSlider(value=.1,min=.01,max=1.,step=.01,\n description='$m_1$ [kg]:',disabled=False,continuous_update=False,orientation='horizontal',readout=True,readout_format='.2f',\n )\n m22_slider = widgets.FloatSlider(value=.1,min=.0,max=1.,step=.01,\n description='$m_2$ [kg]:',disabled=False,continuous_update=False,orientation='horizontal',readout=True,readout_format='.2f',\n )\n b2_slider = widgets.FloatSlider(value=.1,min=0.0,max=2,step=.01,\n description='$B$ [Ns/m]:',disabled=False,continuous_update=False,orientation='horizontal',readout=True,readout_format='.2f',\n )\n k21_slider = widgets.FloatSlider(value=1.,min=0.,max=20.,step=.1,\n description='$k_1$ [N/m]:',disabled=False,continuous_update=False,orientation='horizontal',readout=True,readout_format='.1f',\n )\n k22_slider = widgets.FloatSlider(value=1.,min=0.,max=20.,step=.1,\n description='$k_2$ [N/m]:',disabled=False,continuous_update=False,orientation='horizontal',readout=True,readout_format='.1f',\n )\n \n y0_slider=widgets.FloatSlider(value=0, min=-1, max=1., step=.1,\n description='$y_0$ [dm]:',disabled=False,continuous_update=False,\n orientation='horizontal',readout=True,readout_format='.2f',)\n ypika0_slider=widgets.FloatSlider(value=0, min=-1, max=1., step=.1,\n description='${\\dot{y}}_0$ [dm/s]:',disabled=False,continuous_update=False,\n orientation='horizontal',readout=True,readout_format='.2f',) \n input_data = widgets.interactive_output(build_model, {'M':m1_slider, 'K':k1_slider, 'B':b1_slider, 'M1':m21_slider,\n 'M2':m22_slider, 'B1':b2_slider, 'K1':k21_slider, 'K2':k22_slider, 'amp':amp_slider,\n 'x0':x0_slider,'xpika0':xpika0_slider,'y0':y0_slider,'ypika0':ypika0_slider, \n 'select_System':selectSystem,'index':selectForce}) \n \n input_data2 = widgets.interactive_output(update_sliders, {'index':selectForce})\n \n box_layout = widgets.Layout(border='1px solid black',\n width='auto',\n height='',\n flex_flow='row',\n display='flex')\n\n buttons1=widgets.HBox([widgets.VBox([amp_slider],layout=widgets.Layout(width='auto')),\n widgets.VBox([x0_slider,xpika0_slider]),\n widgets.VBox([y0_slider,ypika0_slider])],layout=box_layout)\n display(widgets.VBox([widgets.Label('V\u00e1lassza meg a bemeneti er\u0151s\u00edt\u00e9st \u00e9s a kezdeti felt\u00e9teleket:'), buttons1]))\n display(widgets.HBox([widgets.VBox([m1_slider,k1_slider,b1_slider], layout=widgets.Layout(width='45%')),\n widgets.VBox([m21_slider,m22_slider,k21_slider,k22_slider,b2_slider], layout=widgets.Layout(width='45%'))]), input_data)\n \nwidgets.interactive_output(draw_controllers, {'type_select':selectSystem,'index':selectForce})\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "c995c06655d0ec436a6a4e6b31bde6afa87750c8", "size": 21192, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ICCT_hu/examples/02/TD-03-Mechanikai_rendszerek.ipynb", "max_stars_repo_name": "ICCTerasmus/ICCT", "max_stars_repo_head_hexsha": "fcd56ab6b5fddc00f72521cc87accfdbec6068f6", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2021-05-22T18:42:14.000Z", "max_stars_repo_stars_event_max_datetime": "2021-10-03T14:10:22.000Z", "max_issues_repo_path": "ICCT_hu/examples/02/TD-03-Mechanikai_rendszerek.ipynb", "max_issues_repo_name": "ICCTerasmus/ICCT", "max_issues_repo_head_hexsha": "fcd56ab6b5fddc00f72521cc87accfdbec6068f6", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ICCT_hu/examples/02/TD-03-Mechanikai_rendszerek.ipynb", "max_forks_repo_name": "ICCTerasmus/ICCT", "max_forks_repo_head_hexsha": "fcd56ab6b5fddc00f72521cc87accfdbec6068f6", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2021-05-24T11:40:09.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-29T16:36:18.000Z", "avg_line_length": 52.7164179104, "max_line_length": 763, "alphanum_fraction": 0.5229331823, "converted": true, "num_tokens": 4976, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943925708562, "lm_q2_score": 0.5926665999540697, "lm_q1q2_score": 0.42588689539102936}} {"text": "#
                                        Using Statistical Models in Accelerator Physics
                                        \n\n
                                        by Kelly Anderson
                                        \n\nStatistical methods are used a lot in accelerator physics. Even though my research focuses mostly on single-particle dynamics, accelerators are always used with many particles at once. Particles in an accelerator are sent in bunches or tight groups of particles. This is done for a few reasons, one is that when the most efficient method for accelerating particles is using radio frequency cavities. Even though they provide a large amount of acceleration they can only provide it for a finite window. The particles are sent in bunches that match these windows so they are accelerated.\n\nAnother reason is that a larger number of particles means a larger number, and more frequent, collisions, this gives more data to researchers. It is also easier to get two bunches of particles to collide than two individual particles. Bunches will also provide a number of particles from different states to collide which provides more diverse data, but that can be negative in some cases.\n\nBecause of the high number of particles, it is best to describe then bunches as a whole. This is where statistical methods come into play. One concept used often is emittance. This emittance refers to the area that a bunch of particles occupies. This could be the physical profile of the bunch or the area that the bunch covers in phase space. The emittance of a bunch has to fit in the aperture of each element which is the area in phase space in which a particle can enter the element and not be lost due to unwanted collision with the surroundings. Even though a bunch\u2019s area in phase space is conserved in many cases, that does not mean its emittance is constant. This is because the area can take up many shapes, like galaxy looking spirals, but the general ellipse that encompasses all particles is generally the emittance that is being referred to and it can grow in size.\n\n
                                        \n\nWhen we describe this emittance we refer to it by the statistical average of the momentum and position of all the particles. One can also look at the dynamic aperture of an entire accelerator to know how particles must enter the machine in order to remain stable in cases like storage rings. To do this we look at how each element in the machine maps particles from beginning to the end. Many common elements, like a FODO cell, will rotate the ellipse in phase space which also needs to be taken into account.\n\n\\begin{equation}\n \\epsilon = \\sqrt{\\langle{x^2}\\rangle \\langle{x'^2}\\rangle - \\langle{x x'}\\rangle^2}\n\\end{equation}\n\nCloser to my research, there are statistical methods used in analyzing particle motion. Starting from a mathematical transformation that maps the initial state of a particle to the final state one can compare that to the data from beam position monitors in the accelerator to test the accuracy of the mapping used.\n\n---\n# References\n\nhttps://drive.google.com/file/d/1k5gs2YPnN-AwNJ3PK7JNBFejN91W15ln/view\n", "meta": {"hexsha": "228b6276159b7c58d4a597b0ffe8b5015474985b", "size": 4329, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Reports/1002-REPORT-Stats.ipynb", "max_stars_repo_name": "anderske-msu/Accelerator-Programs", "max_stars_repo_head_hexsha": "7d0e82611be864138db8fcc66372c13bd1896264", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Reports/1002-REPORT-Stats.ipynb", "max_issues_repo_name": "anderske-msu/Accelerator-Programs", "max_issues_repo_head_hexsha": "7d0e82611be864138db8fcc66372c13bd1896264", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Reports/1002-REPORT-Stats.ipynb", "max_forks_repo_name": "anderske-msu/Accelerator-Programs", "max_forks_repo_head_hexsha": "7d0e82611be864138db8fcc66372c13bd1896264", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-10-10T12:35:54.000Z", "max_forks_repo_forks_event_max_datetime": "2020-10-10T12:35:54.000Z", "avg_line_length": 44.6288659794, "max_line_length": 885, "alphanum_fraction": 0.6895356895, "converted": true, "num_tokens": 630, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.63341024983754, "lm_q1q2_score": 0.42586178955084936}} {"text": "# \u7b80\u5355\u7406\u89e3 LT-DF SOS RMP2\n\n> \u521b\u5efa\u65f6\u95f4\uff1a2019-10-17\uff1b\u6700\u540e\u4fee\u6539\uff1a2021-06-12\n\n\u8fd9\u4e00\u8282\u627f\u63a5\u4e0a\u4e00\u8282\u5bf9 DF-MP2 \u7684\u8ba8\u8bba\u3002\u6211\u4eec\u77e5\u9053 DF-MP2 \u7684\u8ba1\u7b97\u8017\u65f6\u5c3d\u7ba1\u6bd4\u4f20\u7edf MP2 \u4f4e\u4f46\u4ecd\u7136\u662f $O(N^5)$\uff1b\u8fd9\u4e00\u8282\u4f5c\u7684 LT-DF SOS \u8fd1\u4f3c\u5219\u53ef\u4ee5\u786e\u5b9e\u5730\u5c06\u8ba1\u7b97\u590d\u6742\u5ea6\u964d\u4f4e\u4e3a $O(N^4)$\u3002\n\n:::{note}\n\nLT-DF SOS MP2 \u7684\u5199\u6cd5\u5e76\u4e0d\u662f\u76ee\u524d\u901a\u7528\u7684\u5199\u6cd5\uff1b\u53ef\u80fd\u76ee\u524d\u4e5f\u6ca1\u6709\u901a\u7528\u7684\u5199\u6cd5\u3002\n\n\u8be5\u540d\u79f0\u7684\u542b\u4e49\u662f Laplace-Transformation Density-Fitting Scaled Opposite-Spin Second Order Moller-Plesset\u3002\n\n\u4e0d\u8981\u88ab\u8fd9\u4e48\u957f\u7684\u540d\u79f0\u5413\u5230\u3002\u9664\u4e86 LT \u4e4b\u5916\uff0c\u5176\u5b83\u7684\u8fc7\u7a0b\u6216\u8005\u975e\u5e38\u5bb9\u6613\u5b9e\u73b0 (SOS)\uff0c\u6216\u8005\u662f\u6211\u4eec\u5df2\u7ecf\u7406\u89e3\u7684\u5185\u5bb9 (DF, MP2)\u3002\n\n:::\n\n\n```python\n%matplotlib notebook\n\nimport numpy as np\nimport scipy\nimport quadpy\nfrom functools import partial\nimport matplotlib.pyplot as plt\n\nfrom pyscf import gto, scf, mp, df\n\nnp.einsum = partial(np.einsum, optimize=[\"greedy\", 1024 ** 3 * 2 / 8])\nnp.set_printoptions(8, linewidth=150, suppress=True)\n```\n\n\u5728\u8fd9\u7bc7\u6587\u6863\u4e2d\uff0c\u6211\u4eec\u5c06\u4f1a\u5c06\u6d4b\u8bc4\u4f53\u7cfb\u66f4\u6362\u4e3a\u4e59\u70ef\u5206\u5b50\u3002\u8fd9\u4e3b\u8981\u662f\u4e3a\u4e86\u4e0e Jung et al. [^Jung-Head-Gordon.JCP.2004.121] \u6587\u7ae0\u4e2d Table I \u7684\u6570\u636e\u4f5c\u6bd4\u8f83\u3002\n\n:::{admonition} \u51bb\u6838\u8fd1\u4f3c\u8bf4\u660e\n\n\u8fd9\u7bc7\u6587\u7ae0\u91c7\u7528\u4e86 MP2 \u7684\u51bb\u6838\u8fd1\u4f3c (Frozen Core)\uff1b\u56e0\u6b64\u6211\u4eec\u4e4b\u540e\u7684 MP2 \u8ba1\u7b97\u4e5f\u9700\u8981\u76f8\u5e94\u5730\u91c7\u7528\u51bb\u6838\u8fd1\u4f3c\u3002\u636e\u6211\u6240\u77e5 PySCF \u6ca1\u6709\u660e\u786e\u5730\u5199\u51fa\u51bb\u6838\u8fd1\u4f3c\u4ee3\u7801\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u624b\u52a8\u6307\u5b9a\u51bb\u6838\u3002\u5bf9\u4e8e\u4e59\u70ef\u5206\u5b50\uff0c\u4e24\u4e2a\u78b3\u539f\u5b50\u6709\u9700\u8981\u88ab\u51bb\u7ed3\u7684 1s \u8f68\u9053\uff0c\u56e0\u6b64\u88ab\u51bb\u7ed3\u7684\u5360\u636e\u8f68\u9053\u6570\u4e3a 2\uff1b\u800c\u975e\u5360\u8f68\u9053\u7684\u51bb\u7ed3\u6570\u4e3a\u96f6\u3002\n\n\u7a0b\u5e8f\u4e2d\u4f7f\u7528 `sf` \u8868\u793a\u51bb\u7ed3\u540e\u7684\u5360\u636e\u8f68\u9053\u5206\u5272\uff0c`Cf` \u4e0e `ef` \u8868\u793a\u51bb\u7ed3\u540e\u7684\u5206\u5b50\u8f68\u9053\u7cfb\u6570\u4e0e\u8f68\u9053\u80fd\u91cf\u3002\u4e3a\u4e86\u7b26\u53f7\u7b80\u4fbf\u8d77\u89c1\uff0c\u5176\u5b83\u7684\u7a0b\u5e8f\u7b26\u53f7\u57fa\u672c\u6cbf\u7528\u4e0a\u4e00\u7bc7\u6587\u6863\u3002\n\n:::\n\n:::{admonition} \u5206\u5b50\u7ed3\u6784\u8bf4\u660e\n\n\u76ee\u524d\u4f7f\u7528\u7684\u5206\u5b50\u7ed3\u6784\u662f\u5728 MP2/6-31G* \u4e0b\u4f18\u5316\u7684\u4e59\u70ef\u5206\u5b50\u3002\u4f18\u5316\u4f7f\u7528 Gaussian 09 rev B01 \u5b9e\u73b0\u3002\u5c3d\u7ba1\u80fd\u91cd\u590d Jung et al. [^Jung-Head-Gordon.JCP.2004.121] \u6587\u7ae0\u4e2d\u4e59\u70ef\u5206\u5b50\u7684\u7ed3\u679c\uff0c\u4f46\u4f1a\u5728\u76f8\u5173\u80fd\u7684\u7b2c 4 \u4f4d\u4e0a\u6709\u4e9b\u5fae\u5dee\u8ddd\u3002\u6211\u8ba4\u4e3a\u8fd9\u5e76\u4e0d\u5f71\u54cd\u8ba8\u8bba\u3002\n\nMP2 \u7ed3\u6784\u4f18\u5316\u8f93\u5165\u5361 {download}`assets/C2H4-opt.gjf` QCISD(T) \u80fd\u91cf\u8f93\u5165\u5361 {download}`assets/C2H4-eng.gjf`\n\n:::\n\n\n```python\nmol = gto.Mole()\nmol.atom = \"\"\"\nC -0.668188207 0.000000075 -0.0000028526\nC 0.668188207 0.0000000727 0.0000028526\nH -1.2382848349 -0.0000000262 -0.9232789581\nH -1.2382926466 -0.0000000483 0.9232684342\nH 1.2382848349 -0.0000000255 0.9232789581\nH 1.2382926466 -0.0000000477 -0.9232684342\n\"\"\"\nmol.basis = \"cc-pVTZ\"\nmol.verbose = 0\nmol.build()\n\nnatm = mol.natm\nnao = nmo = mol.nao\nnocc = mol.nelec[0]\nnvir = nmo - nocc\nso, sv, sa = slice(0, nocc), slice(nocc, nmo), slice(0, nmo)\nsf = slice(2, nocc)\n```\n\n## \u53c2\u8003\u503c\n\n### \u53c2\u8003\u503c\u4e0e SOS-MP2\n\n\u6839\u636e Jung et al. [^Jung-Head-Gordon.JCP.2004.121] \u6587\u7ae0\u7684\u7ed3\u679c\uff0c\u6211\u4eec\u7ed9\u51fa\u4e0b\u8ff0\u53c2\u8003\u503c\u5b57\u5178 `ref`\uff1b\u6bcf\u4e2a\u952e\u503c\u5bf9\u5e94\u7684\u6570\u503c\u5355\u4f4d\u662f mH (milli-Hartree)\u3002\n\n\n```python\nref = {\n \"QCISDT\": -375.4,\n \"SS\": -35.5,\n \"OS\": -264.7,\n \"MP2\": -375.4 * 0.894,\n \"SCS\": -375.4 * 0.909,\n \"SOS\": -375.4 * 0.917,\n}\n```\n\n\u5176\u4e2d\uff0c\u952e\u503c `MP2` (Second Order Moller-Plesset), `SCS` (Spin-Component Scaled) \u4e0e `SOS` (Scaled Opposite-Spin) \u539f\u5219\u4e0a\u90fd\u53ef\u4ee5\u901a\u8fc7 `SS` (Same-Spin) \u4e0e `OS` (Opposite-Spin) \u5bf9\u76f8\u5173\u80fd\u7684\u8d21\u732e\u7ed9\u51fa\uff1a\n\n$$\nE_\\mathrm{MP2, corr} = 2 E_\\mathrm{SS} + E_\\mathrm{OS}\n$$\n\n\n```python\nprint(ref[\"MP2\"])\nprint(ref[\"SS\"] * 2 + ref[\"OS\"] * 1)\n```\n\n -335.6076\n -335.7\n\n\n$$\nE_\\text{SCS-MP2, corr} = 2 \\times \\frac{1}{3} E_\\mathrm{SS} + \\frac{6}{5} E_\\mathrm{OS}\n$$\n\n\n```python\nprint(ref[\"SCS\"])\nprint(ref[\"SS\"] * 2 * 1/3 + ref[\"OS\"] * 6/5)\n```\n\n -341.23859999999996\n -341.3066666666667\n\n\n$$\nE_\\text{SOS-MP2, corr} = 1.3 E_\\mathrm{OS}\n$$\n\n\n```python\nprint(ref[\"SOS\"])\nprint(ref[\"OS\"] * 1.3)\n```\n\n -344.2418\n -344.11\n\n\n\u5173\u4e8e\u8fd9\u4e9b\u65b9\u6cd5\u7684\u7ec6\u8282\uff0c\u6211\u4eec\u4e0d\u5728\u8fd9\u91cc\u7ec6\u7a76\u3002\u5728\u8fd9\u7bc7\u6587\u6863\u4e2d\uff0c\u6211\u4eec\u53ea\u5173\u5fc3\u5230 SOS \u65b9\u6cd5\u53ea\u4f7f\u7528\u4e86 MP2 \u76f8\u5173\u80fd\u4e2d OS \u90e8\u5206\u7684\u8d21\u732e\u3002\u8fd9\u5bf9\u6211\u4eec\u540e\u9762\u7684\u8ba8\u8bba\u81f3\u5173\u91cd\u8981\u3002\u6211\u4eec\u5728\u8fd9\u7bc7\u6587\u6863\u5c06\u8981\u91cd\u590d\u7684\u5c31\u662f SOS-MP2\u3002\n\n### SCF \u8ba1\u7b97\n\n\u5728\u7ee7\u7eed\u6587\u6863\u4ee5\u524d\uff0c\u6211\u4eec\u9700\u8981\u7ed9\u51fa\u4e59\u70ef\u5206\u5b50\u7684\u5404\u79cd\u8ba1\u7b97\u7ed3\u679c\u3002\u5927\u591a\u6570\u4ee3\u7801\u8bb0\u53f7\u4e0e\u516c\u5f0f\u90fd\u53ef\u4ee5\u4e0e\u524d\u4e24\u7bc7\u6587\u6863\u6bd4\u5bf9\u3002\u8865\u5145\u51fa\u6765\u7684\u8bb0\u53f7\u9664\u4e86\u51bb\u7ed3\u8f68\u9053\u8fd1\u4f3c\u7684\u90e8\u5206\u4e4b\u5916\uff0c\u8fd8\u6709\n\n* $D_i^a$ `D_ia`\uff1a$\\varepsilon_i - \\varepsilon_a$\n\n\u7531\u4e8e\u6211\u4eec\u5173\u5fc3\u7684\u662f MP2 \u90e8\u5206\u7684 DF\uff0c\u56e0\u6b64\u81ea\u6d3d\u573a\u5373\u4f7f\u4e0d\u4f7f\u7528 DF \u4e5f\u4e0d\u5f71\u54cd\u6211\u4eec\u7684\u8ba8\u8bba\u3002\n\n\n```python\nscf_normal = scf.RHF(mol).run()\nmp2_normal = mp.MP2(scf_normal).run()\n```\n\n\n```python\nC, e = scf_normal.mo_coeff, scf_normal.mo_energy\nCf, Cv = C[:, sf], C[:, sv]\nef, ev = e[sf], e[sv]\nD_iajb = ef[:, None, None, None] - ev[None, :, None, None] + ef[None, None, :, None] - ev[None, None, None, :]\nD_ia = ef[:, None] - ev[None, :]\n```\n\n### SS/OS \u8ba1\u7b97\n\nMP2 \u80fd\u91cf\u9664\u4e86\u4f7f\u7528\u4e0a\u4e00\u4efd\u6587\u6863\u7684 $E_\\mathrm{MP2, corr} = T_{ij}^{ab} t_{ij}^{ab} D_{ij}^{ab}$ \u4e4b\u5916\uff0c\u8fd8\u53ef\u4ee5\u4f7f\u7528\u4e0b\u8ff0\u7684\u516c\u5f0f\u8ba1\u7b97\uff1a\n\n$$\nE_\\mathrm{MP2, corr} = \\frac{1}{D_{ij}^{ab}} (ia|jb) \\big[ 2 (ia|jb) - (ib|ja) \\big]\n$$\n\n\u4e0a\u5f0f\u53ef\u4ee5\u62c6\u5206\u4e3a\n\n$$\n\\begin{align}\nE_\\mathrm{SS} &= \\frac{1}{2} \\frac{1}{D_{ij}^{ab}} (ia|jb) \\big[ (ia|jb) - (ib|ja) \\big] \\\\\nE_\\mathrm{OS} &= \\frac{1}{D_{ij}^{ab}} (ia|jb)^2 \\\\\nE_\\mathrm{MP2, corr} &= 2 E_\\mathrm{SS} + E_\\mathrm{OS}\n\\end{align}\n$$\n\n\u5176\u62c6\u5206\u4f9d\u636e\u662f\u901a\u8fc7\u81ea\u65cb\u8f68\u9053\u4e0b\u7684 MP2 \u8868\u8fbe\u5f0f\u4ea7\u751f\u800c\u6765\uff0c\u8fd9\u91cc\u4e0d\u4f5c\u5c55\u5f00\u3002\u4e3a\u4e86\u8ba1\u7b97 $E_\\mathrm{OS}$\uff0c\u9664\u4e86 $D_{ij}^{ab}$ `(D_iajb)` \u4e4b\u5916\uff0c\u6211\u4eec\u8fd8\u9700\u8981 $(ia|jb)$\u3002\u8be5\u53d8\u91cf\u50a8\u5b58\u4e8e `eri_iajb` \u4e2d\uff1a\n\n\n```python\neri_iajb = np.einsum(\"ui, va, uvkl, kj, lb -> iajb\", Cf, Cv, mol.intor(\"int2e\"), Cf, Cv)\n```\n\n\u968f\u540e\u6211\u4eec\u5c31\u53ef\u4ee5\u8ba1\u7b97 $E_\\mathrm{SS}$ `E_SS_normal` \u4e0e $E_\\mathrm{OS}$ `E_OS_normal` \u4e86\uff1a\n\n\n```python\nE_SS_normal = 0.5 * (eri_iajb * (eri_iajb - eri_iajb.swapaxes(-1, -3)) / D_iajb).sum()\nE_OS_normal = (eri_iajb ** 2 / D_iajb).sum()\nprint(\"E_SS in mH: \", E_SS_normal * 1000)\nprint(\"E_OS in mH: \", E_OS_normal * 1000)\nprint(\"E_MP2_corr in mH: \", (E_OS_normal + 2 * E_SS_normal) * 1000)\n```\n\n E_SS in mH: -35.46668203417094\n E_OS in mH: -264.8091129392841\n E_MP2_corr in mH: -335.742477007626\n\n\n\u8fd9\u7bc7\u6587\u6863\u7684\u4e3b\u8981\u76ee\u6807\u5e76\u975e\u91cd\u590d $E_\\mathrm{MP2, corr}$\uff0c\u800c\u662f\u91cd\u590d $E_\\text{SOS-MP2, corr}$ `E_SOS_normal`\uff1a\n\n\n```python\nE_SOS_normal = 1.3 * E_OS_normal\nprint(\"E_SOS_corr in mH: \", E_SOS_normal * 1000)\n```\n\n E_SOS_corr in mH: -344.25184682106936\n\n\n## Laplace \u53d8\u6362 (LT) \u539f\u7406\n\n### $x^{-1}$ LT \u6570\u503c\u79ef\u5206\n\n\u5728\u8fd9\u91cc\uff0c\u6211\u4eec\u4e0d\u8ba8\u8bba\u66f4\u4e3a\u5e7f\u6cdb\u7684 LT \u7684\u539f\u7406\uff0c\u53ea\u8ba8\u8bba\u4e0e MP2 \u8ba1\u7b97\u76f8\u5173\u7684 $x^{-1} = (D_{ij}^{ab})^{-1}$ \u7684\u8ba1\u7b97\u95ee\u9898\u3002\u5bf9\u4e8e LT \u4e2d\u6d89\u53ca\u5230 MP2 \u7684\u95ee\u9898\u4f1a\u653e\u5728\u4e0b\u4e00\u5c0f\u8282\u4e2d\u8ba8\u8bba\uff1b\u8fd9\u91cc\u6211\u4eec\u53ea\u9700\u8981\u77e5\u9053\u4e59\u70ef\u5206\u5b50 $x$ \u7684\u53d6\u503c\u8303\u56f4\u5927\u7ea6\u4f1a\u662f\n\n\n```python\nprint(\"|D_iajb| max in Hartree: \", np.abs(D_iajb).max())\nprint(\"|D_iajb| min in Hartree: \", np.abs(D_iajb).min())\n```\n\n |D_iajb| max in Hartree: 31.40203636576501\n |D_iajb| min in Hartree: 1.0534258481856735\n\n\n$x^{-1}$ \u53ef\u4ee5\u901a\u8fc7\u4e0b\u8ff0\u65b9\u5f0f\u83b7\u5f97\uff1a\n\n$$\nx^{-1} = \\int_0^{+ \\infty} e^{- x t} \\, \\mathrm{d} t\n$$\n\n\u6839\u636e\u683c\u70b9\u79ef\u5206\u7684\u539f\u7406\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4e0a\u5f0f\u5728\u683c\u70b9 $\\{g\\}$ \u4e0b\u91cd\u65b0\u5199\u4e3a\n\n$$\nx^{-1} \\simeq w_g e^{- x t_g}\n$$\n\n\u73b0\u5728\u6211\u4eec\u4ece\u7a0b\u5e8f\u7684\u89d2\u5ea6\u5b9e\u73b0\u4e0a\u8ff0\u683c\u70b9\u79ef\u5206\u3002\u6211\u4eec\u77e5\u9053\uff0c$e^{-xt}$ \u51fd\u6570\u5728\u96f6\u70b9\u5904\u7684\u503c\u8f83\u5927\uff0c\u4f46\u5728\u8fdc\u79bb\u96f6\u70b9\u5904\u7684\u503c\u8f83\u5c0f\u3002\u540c\u65f6\uff0c\u7531\u4e8e $e^{-xt}$ \u672c\u8eab\u5c31\u662f\u4e00\u79cd\u6307\u6570\u5f62\u5f0f\uff0c\u56e0\u6b64\u6211\u4eec\u5bb9\u6613\u60f3\u5230\u4f7f\u7528\u6307\u6570\u7684\u65b9\u5f0f\u6784\u9020\u683c\u70b9\u3002\u4ee4\u5750\u6807\u53d8\u91cf $t_g$ `grid_points`\n\n$$\nt_g = a^g\n$$\n\n\u90a3\u4e48\u5bf9\u5e94\u7684\u683c\u70b9\u6743\u91cd $w_g$ `grid_weights` \u5e94\u5f53\u63a5\u8fd1\n\n$$\nw_g \\simeq \\frac{\\partial}{\\partial g} t_g = \\log a \\cdot a^g\n$$\n\n\u5728\u8fd9\u91cc\uff0c\u6211\u4eec\u53d6 $a = 2.5$\uff0c\u800c $g \\in \\left[ -10, 5 \\right)$\u3002\u8fd9\u79cd\u53d6\u6cd5\u4e00\u5b9a\u7a0b\u5ea6\u4e0a\u662f\u4efb\u610f\u7684\uff0c\u4f46\u8003\u8651\u5230\u6211\u4eec\u5e0c\u671b\u5bf9 $x \\in (0.1, 100)$ \u7684\u533a\u95f4\uff0c\u7279\u522b\u662f\u9760\u8fd1 $x \\in (0.1, 2)$ \u7684\u533a\u57df\u6709\u6bd4\u8f83\u826f\u597d\u7684\u8fd1\u4f3c (MP2 \u4e2d\u4e3b\u8981\u4ea7\u751f\u8d21\u732e\u7684\u90e8\u5206\u5728\u5206\u6bcd $x$ \u503c\u504f\u5c0f\u7684\u533a\u57df)\uff0c\u56e0\u6b64\u8fd9\u4e9b\u53c2\u6570\u5e76\u4e0d\u662f\u5f7b\u5e95\u4efb\u610f\u5730\u9009\u53d6\u7684\u3002\n\n\n```python\ngrid_points = 2.5 ** np.arange(-12, 6)\ngrid_weights = np.log(2.5) * grid_points\ngrid_points\n```\n\n\n\n\n array([ 0.00001678, 0.00004194, 0.00010486, 0.00026214, 0.00065536, 0.0016384 , 0.004096 , 0.01024 , 0.0256 , 0.064 , 0.16 ,\n 0.4 , 1. , 2.5 , 6.25 , 15.625 , 39.0625 , 97.65625 ])\n\n\n\n\u4e3e\u4f8b\u6765\u8bf4\uff0c\u5982\u679c\u6211\u4eec\u8981\u7528\u683c\u70b9\u8fd1\u4f3c $2^{-1} = 0.5 \\simeq w_g e^{- 2 t_g}$\uff0c\u90a3\u4e48\u7a0b\u5e8f\u53ef\u4ee5\u7f16\u5199\u5982\u4e0b\uff1a\n\n\n```python\n(grid_weights * np.exp(- 1 * grid_points)).sum()\n```\n\n\n\n\n 1.0001747291147214\n\n\n\n\u8fd9\u5e94\u8be5\u662f\u4e00\u4e2a\u6bd4\u8f83\u6709\u6548\u7684\u8fd1\u4f3c\u4e86\u3002\n\n### LT \u6570\u503c\u79ef\u5206\u7cbe\u5ea6\n\n\u5bf9\u4e8e\u66f4\u5e7f\u6cdb\u7684\u6570\u503c\u7cbe\u5ea6\u5206\u6790\uff0c\u6211\u4eec\u5728\u4e0a\u8ff0 LT \u683c\u70b9\u7684\u6761\u4ef6\u4e0b\uff0c\u7ed9\u51fa\u5982\u4e0b\u7684\u56fe\u50cf\uff1a\n\n\n```python\nfig, ax = plt.subplots()\n\nx_axis = np.logspace(-2, 3, 100)\ny_axis = (grid_weights * np.exp(- x_axis[:, None] * grid_points)).sum(axis=-1) * x_axis\nax.plot(x_axis, y_axis)\nax.plot([1e-2, 1e3], [1.001, 1.001], color=\"C1\", linestyle=\":\")\nax.plot([1e-2, 1e3], [1, 1], color=\"black\", linestyle=\":\")\nax.plot([1e-2, 1e3], [0.999, 0.999], color=\"C1\", linestyle=\":\")\nax.plot([1e-1, 1e-1], [0, 2], color=\"C2\", linestyle=\":\")\nax.plot([1e2, 1e2], [0, 2], color=\"C2\", linestyle=\":\")\nax.set_ylim(0.99,1.002)\nax.set_xscale(\"log\")\n\nax.set_xlabel(\"$x$\")\nax.set_ylabel(\"Relative error\")\nax.set_title(\"Ratio of $\\sum_g w_g e^{-x t_g}$ to $1/x$\")\n```\n\n\n \n\n\n\n\n\n\n\n\n\n Text(0.5, 1.0, 'Ratio of $\\\\sum_g w_g e^{-x t_g}$ to $1/x$')\n\n\n\n\u4e0a\u56fe\u8868\u793a\u7684\u662f\u5728\u4e0d\u540c $x$ \u53d6\u503c\u4e0b\uff0c\u6211\u4eec\u6240\u7ed9\u51fa\u7684\u683c\u70b9\u79ef\u5206 $w_g e^{-x t_g}$ \u7684\u7cbe\u5ea6\uff1b\u5728 $x \\in (0.1, 100)$ \u7684\u533a\u95f4\u5185\u6211\u4eec\u4ecd\u7136\u80fd\u4fdd\u8bc1 $0.1\\%$ \u7684\u7cbe\u5ea6\uff0c\u4f46\u8d85\u8fc7\u8fd9\u4e9b\u533a\u57df\u4e4b\u540e\uff0c\u7ed3\u679c\u5c31\u4f1a\u8fc5\u901f\u5730\u53d8\u5dee\u3002\n\n\u6211\u4eec\u77e5\u9053\uff0c\u82e5\u5206\u5b50\u5448\u73b0\u8fd1\u7b80\u5e76\u7684\u60c5\u5f62\uff0c\u90a3\u4e48 $x = D_{ij}^{ab}$ \u4f1a\u975e\u5e38\u5c0f\uff1b\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0cLT \u6240\u63d0\u4f9b\u7684\u683c\u70b9\u79ef\u5206\u5c31\u5f88\u53ef\u80fd\u65e0\u6cd5\u6b63\u786e\u5730\u63cf\u8ff0\u5206\u6bcd $D_{ij}^{ab}$ \u7684\u884c\u4e3a\uff0c\u4f7f\u5f97\u672c\u6765\u5728\u8fd1\u7b80\u5e76\u5c31\u65e0\u6cd5\u6b63\u5e38\u8ba1\u7b97\u7684 MP2 \u7684\u60c5\u51b5\u96ea\u4e0a\u52a0\u971c\u3002\u4f46\u82e5\u5206\u5b50\u7684 HOMO/LUMO \u7684\u80fd\u7ea7\u5dee\u8db3\u591f\u5927\u5e76\u4e14\u5927\u4e8e $0.1$ \u4e2a Hartree\uff0c\u90a3\u4e48\u6211\u4eec\u5e94\u5f53\u9884\u671f\u5f53\u524d\u683c\u70b9\u4e0b\u7684 LT \u8fd1\u4f3c\u53ef\u4ee5\u51e0\u4e4e\u7cbe\u786e\u5730\u7ed9\u51fa\u76f8\u5173\u80fd\u3002\n\n### LT SOS MP2\n\n\u73b0\u5728\u6211\u4eec\u5c31\u53ef\u4ee5\u7528\u4e0a\u521a\u624d\u7684\u6280\u672f\uff0c\u5bf9 LT SOS MP2 \u8fdb\u884c\u8ba1\u7b97\u3002\u8fd9\u5e76\u4e0d\u4f1a\u63d0\u5347\u4efb\u4f55\u8ba1\u7b97\u6548\u7387\uff1b\u8fd9\u91cc\u53ea\u662f\u7528\u6765\u9a8c\u8bc1\u6211\u4eec\u65b9\u624d\u4f5c\u7684 LT \u8fd1\u4f3c\u7684\u6709\u6548\u6027\u3002\n\n\u5728 LT \u8fd1\u4f3c\u4e0b\uff0cSOS \u80fd\u91cf\u53ef\u4ee5\u8868\u8fbe\u4e3a\n\n$$\n\\begin{align}\nE_\\text{SOS-MP2, corr} &= 1.3 \\frac{1}{D_{ij}^{ab}} (ia|jb)^2 \\\\\n&= - 1.3 (ia|jb)^2 \\int_0^{+\\infty} e^{D_{ij}^{ab} t} \\, \\mathrm{d} t \\\\\n&\\simeq -1.3 w_g (ia|jb)^2e^{D_{ij}^{ab} t_g}\n\\end{align}\n$$\n\n\u8fd9\u91cc\u7684\u516c\u5f0f\u7684\u6b63\u8d1f\u53f7\u53ef\u80fd\u4e0d\u592a\u76f4\u89c2\u3002\u8fd9\u662f\u56e0\u4e3a\u5728\u8fd9\u7bc7\u6587\u6863\u7684\u8bb0\u53f7\u4e0b\uff0c$D_{ij}^{ab} < 0$\u3002\u6211\u4eec\u901a\u8fc7\u5982\u4e0b\u4ee3\u7801\uff0c\u7ed9\u51fa LT SOS MP2 \u80fd\u91cf\uff1a\n\n\n```python\nE_SOS_LT = - 1.3 * np.einsum(\"g, iajb, giajb ->\", grid_weights, eri_iajb**2, np.exp(D_iajb * grid_points[:, None, None, None, None]))\nprint(\"LT-SOS-MP2 corr in mH: \", E_SOS_LT * 1000)\nprint(\"Deviation to SOS-MP2 in mH: \", (E_SOS_LT - E_SOS_normal) * 1000)\n```\n\n LT-SOS-MP2 corr in mH: -344.2429384155852\n Deviation to SOS-MP2 in mH: 0.00890840548417593\n\n\n\u6211\u4eec\u53ef\u4ee5\u53d1\u73b0\u8fd9\u662f\u4e00\u4e2a\u6709\u6548\u7684\u8fd1\u4f3c\u3002\n\n## LT-DF SOS MP2\n\n### DF \u73af\u5883\u8bbe\u7f6e\n\n\u5728\u7ee7\u7eed\u6587\u6863\u4e4b\u524d\uff0c\u6211\u4eec\u9700\u8981\u5bf9 DF \u7684\u73af\u5883\u4f5c\u4e00\u4e9b\u7b80\u5355\u7684\u8bbe\u7f6e\u3002\n\n\n```python\nauxmol = mol.copy()\nauxmol.basis = \"cc-pvtz-ri\"\nauxmol.build()\n\nnao_df = auxmol.nao\n```\n\n\n```python\nint2c2e = auxmol.intor(\"int2c2e\")\nint2c2e.shape\nint3c2e = df.incore.aux_e2(mol, auxmol)\n\nint2c2e_half = scipy.linalg.cholesky(int2c2e, lower=True)\nV_df_mp2 = scipy.linalg.solve_triangular(int2c2e_half, int3c2e.reshape(-1, nao_df).T, lower=True)\\\n .reshape(nao_df, nao, nao).transpose((1, 2, 0))\n```\n\n\u5c3d\u7ba1\u8fd9\u91cc\u7684 $V_{ia, P}$ `V_df_ia` \u4e0e\u524d\u4e00\u7bc7\u6587\u6863\u4e2d\u7684 $V_{ia, P}^\\mathrm{MP2}$ \u662f\u76f8\u540c\u7684\uff0c\u4f46\u9700\u8981\u6ce8\u610f\u5230\u8fd9\u91cc\u7684\u8f68\u9053 $i$ \u662f\u4ef7\u5c42\u7684\u5360\u636e\u8f68\u9053\uff0c\u800c\u4e0d\u80fd\u662f\u51bb\u7ed3\u7684\u5360\u636e\u8f68\u9053\u3002\n\n\n```python\nV_df_ia = np.einsum(\"uvP, ui, va -> iaP\", V_df_mp2, Cf, Cv)\n```\n\n### LT-DF SOS MP2 \u8868\u8fbe\u5f0f\n\n\u6211\u4eec\u5bf9\u4e0a\u9762\u63d0\u5230\u7684 LT SOS MP2 \u7684\u8fd1\u4f3c\u8868\u8fbe\u5f0f\u518d\u4f5c DF \u8fd1\u4f3c\uff1a\n\n$$\n\\begin{align}\nE_\\text{SOS-MP2, corr} &\\simeq -1.3 w_g (ia|jb)^2 e^{D_{ij}^{ab} t_g} \\\\\n&\\simeq -1.3 w_g V_{ia, P} V_{jb, P} V_{ia, Q} V_{jb, Q} e^{D_{ij}^{ab} t_g} \\\\\n&= -1.3 w_g V_{ia, P} V_{jb, P} V_{ia, Q} V_{jb, Q} e^{D_i^a t_g} e^{D_j^b t_g}\n\\end{align}\n$$\n\n\u6ce8\u610f\u5230\u6700\u540e\u4e00\u4e2a\u7b49\u53f7\u5229\u7528\u5230 $D_{ij}^{ab} = D_i^a + D_j^b$\u3002\n\n\n```python\nE_SOS_LTDF = - 1.3 * np.einsum(\"g, iaP, jbP, iaQ, jbQ, gia, gjb ->\",\n grid_weights, V_df_ia, V_df_ia, V_df_ia, V_df_ia,\n np.exp(D_ia * grid_points[:, None, None]),\n np.exp(D_ia * grid_points[:, None, None]))\nprint(\"LT-DF-SOS-MP2 corr in mH: \", E_SOS_LTDF * 1000)\nprint(\"Deviation to SOS-MP2 in mH: \", (E_SOS_LTDF - E_SOS_normal) * 1000)\n```\n\n LT-DF-SOS-MP2 corr in mH: -344.1316103156453\n Deviation to SOS-MP2 in mH: 0.12023650542408726\n\n\n\u6211\u4eec\u8ba4\u4e3a\u8fd9\u57fa\u672c\u91cd\u590d\u51fa\u4e86\u539f\u59cb\u7684 SOS MP2 \u80fd\u91cf\u4e86\u3002\n\n### \u7b80\u5316\u516c\u5f0f\u8868\u8fbe\n\n\u663e\u7136\u4e0a\u9762\u7684\u516c\u5f0f\u8868\u8fbe\u592a\u8fc7\u4e8e\u7e41\u6742\u3002\u5728\u5b9e\u9645\u5b9e\u73b0\u65f6\uff0c\u4e00\u822c\u5148\u5c06\u5176\u4e2d\u7684\u4e09\u4e2a\u5f20\u91cf\u7f29\u5e76\u4e3a `X_gPQ`\uff1a\n\n$$\nX_{gPQ} = V_{ia, P} V_{ia, Q} e^{D_i^a t_g}\n$$\n\n\n```python\nX_gPQ = np.einsum(\"iaP, iaQ, gia -> gPQ\", V_df_ia, V_df_ia, np.exp(D_ia * grid_points[:, None, None]))\n```\n\n\u90a3\u4e48 LT-DF \u8fd1\u4f3c\u540e\u7684 SOS MP2 \u8868\u8fbe\u5f0f\u53ef\u4ee5\u7b80\u5199\u4e3a\n\n$$\nE_\\text{SOS-MP2, corr} \\simeq -1.3 w_g X_{gPQ}^2\n$$\n\n\n```python\nE_SOS_LTDF_simp = - 1.3 * np.einsum(\"g, gPQ ->\", grid_weights, X_gPQ**2)\nprint(\"LT-DF-SOS-MP2 corr in mH: \", E_SOS_LTDF_simp * 1000)\nprint(\"Deviation to SOS-MP2 in mH: \", (E_SOS_LTDF_simp - E_SOS_normal) * 1000)\n```\n\n LT-DF-SOS-MP2 corr in mH: -344.13161031563544\n Deviation to SOS-MP2 in mH: 0.12023650543391273\n\n\n### \u8ba1\u7b97\u6548\u7387\u6bd4\u8f83\n\n\u8fd9\u91cc\u6211\u4eec\u5bf9 SOS MP2\uff0cLT SOS MP2 \u4e0e LT-DF SOS MP2 \u7684\u8ba1\u7b97\u6548\u7387\u8fdb\u884c\u6bd4\u8f83\uff0c\u5de5\u5177\u662f `np.einsum_path`\u3002\u6211\u4eec\u5047\u5b9a\u5185\u5b58\u7684\u7a7a\u95f4\u662f\u8db3\u591f\u591a\u7684\uff0c\u5e76\u4e14\u4e0d\u8003\u8651\u4e3a\u751f\u6210\u5404\u79cd\u79ef\u5206\u3001\u6216\u5404\u79cd DF \u4e09\u4e2d\u5fc3\u79ef\u5206\u6240\u9700\u8981\u8017\u8d39\u7684\u65f6\u95f4\u3002\n\n**\u666e\u901a MP2**\n\n\u6548\u7387\u8017\u65f6\u6700\u5927\u7684\u90e8\u5206\u5728\u53cc\u7535\u5b50\u79ef\u5206\u8f6c\u6362\u8fc7\u7a0b\u3002\n\n\n```python\nprint(np.einsum_path(\"ui, va, uvkl, kj, lb -> iajb\",\n Cf, Cv, mol.intor(\"int2e\"), Cf, Cv,\n optimize=[\"greedy\", 1024 ** 3 * 2 / 8])[1]) # Given 2GB memory space\n```\n\n Complete contraction: ui,va,uvkl,kj,lb->iajb\n Naive scaling: 8\n Optimized scaling: 5\n Naive FLOP count: 3.801e+14\n Optimized FLOP count: 2.487e+09\n Theoretical speedup: 152841.286\n Largest intermediate: 9.365e+06 elements\n --------------------------------------------------------------------------\n scaling current remaining\n --------------------------------------------------------------------------\n 5 uvkl,ui->iklv va,kj,lb,iklv->iajb\n 5 iklv,kj->ijlv va,lb,ijlv->iajb\n 5 ijlv,va->ijal lb,ijal->iajb\n 5 ijal,lb->iajb iajb->iajb\n\n\n**DF SOS MP2**\n\n\u6548\u7387\u8017\u65f6\u6700\u5927\u7684\u90e8\u5206\u5728\u4ece 3c2e \u79ef\u5206\u751f\u6210\u53cc\u7535\u5b50\u79ef\u5206\u8fc7\u7a0b\u3002\n\n\n```python\nprint(np.einsum_path(\"iaP, jbP -> iajb\",\n V_df_ia, V_df_ia, \n optimize=[\"greedy\", 1024 ** 3 * 2 / 8])[1]) # Given 2GB memory space\n```\n\n Complete contraction: iaP,jbP->iajb\n Naive scaling: 5\n Optimized scaling: 5\n Naive FLOP count: 2.368e+08\n Optimized FLOP count: 2.368e+08\n Theoretical speedup: 1.000\n Largest intermediate: 4.199e+05 elements\n --------------------------------------------------------------------------\n scaling current remaining\n --------------------------------------------------------------------------\n 5 jbP,iaP->iajb iajb->iajb\n\n\n**LT-DF SOS MP2**\n\n\u6548\u7387\u8017\u65f6\u6700\u5927\u7684\u90e8\u5206\u5728\u751f\u6210\u4e2d\u95f4\u5f20\u91cf $X_{gPQ}$ \u7684\u8fc7\u7a0b\u3002\n\n\n```python\nprint(np.einsum_path(\"iaP, iaQ, gia -> gPQ\",\n V_df_ia, V_df_ia, np.exp(D_ia * grid_points[:, None, None]),\n optimize=[\"greedy\", 1024 ** 3 * 2 / 8])[1]) # Given 2GB memory space\n```\n\n Complete contraction: iaP,iaQ,gia->gPQ\n Naive scaling: 5\n Optimized scaling: 5\n Naive FLOP count: 2.783e+09\n Optimized FLOP count: 1.858e+09\n Theoretical speedup: 1.497\n Largest intermediate: 3.289e+06 elements\n --------------------------------------------------------------------------\n scaling current remaining\n --------------------------------------------------------------------------\n 4 gia,iaP->igaP iaQ,igaP->gPQ\n 5 igaP,iaQ->gPQ gPQ->gPQ\n\n\n\u5bf9\u4e8e\u8fd9\u4e2a\u5206\u5b50\u800c\u8a00\uff0c\u653e\u5f00\u4f18\u5316\u5f20\u91cf\u7f29\u5e76\u8fc7\u7a0b\uff0c\u7ed3\u679c\u53cd\u800c\u4f1a\u56de\u5230\u7c7b\u4f3c\u4e8e DF MP2 \u7684\u7f29\u5e76\u8fc7\u7a0b\uff1a\n\n\n```python\nprint(np.einsum_path(\"g, iaP, jbP, iaQ, jbQ, gia, gjb ->\",\n grid_weights, V_df_ia, V_df_ia, V_df_ia, V_df_ia,\n np.exp(D_ia * grid_points[:, None, None]),\n np.exp(D_ia * grid_points[:, None, None]),\n optimize=[\"greedy\", 1024 ** 3 * 2 / 8])[1]) # Given 2GB memory space\n```\n\n Complete contraction: g,iaP,jbP,iaQ,jbQ,gia,gjb->\n Naive scaling: 7\n Optimized scaling: 5\n Naive FLOP count: 4.207e+12\n Optimized FLOP count: 4.892e+08\n Theoretical speedup: 8600.264\n Largest intermediate: 4.199e+05 elements\n --------------------------------------------------------------------------\n scaling current remaining\n --------------------------------------------------------------------------\n 3 gia,g->iga iaP,jbP,iaQ,jbQ,gjb,iga->\n 5 jbP,iaP->ijab iaQ,jbQ,gjb,iga,ijab->\n 5 jbQ,iaQ->ijab gjb,iga,ijab,ijab->\n 4 ijab,ijab->ijab gjb,iga,ijab->\n 5 ijab,gjb->iga iga,iga->\n 3 iga,iga-> ->\n\n\n**\u7b80\u5355\u603b\u7ed3**\n\n\u4e8b\u5b9e\u4e0a\uff0cLT-DF SOS MP2 \u7684\u4f18\u52bf\u5e76\u4e0d\u4f53\u73b0\u5728\u5c0f\u5206\u5b50\u7684\u8ba1\u7b97\u4e0a\uff0c\u8fd9\u4ece\u521a\u624d\u7684\u6548\u7387\u5206\u6790\u4e0a\u5c31\u80fd\u770b\u51fa\u6765\u3002\u4f46\u662f\uff0c\u5bf9\u4e8e\u5927\u5206\u5b50 (\u5360\u636e\u7535\u5b50\u6570\u8f83\u591a\u7684\u60c5\u51b5\uff0c\u800c\u975e\u57fa\u7ec4\u975e\u5e38\u5e9e\u5927\u7684\u60c5\u51b5)\uff0cLT-DF MP2 \u5c31\u6709\u4e0d\u5c0f\u7684\u4f18\u52bf\u3002\u8fd9\u662f\u56e0\u4e3a\u8017\u65f6\u6700\u5927\u7684\u90e8\u5206\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u522b\u662f\n\n* LT-DF SOS MP2: $O(n_\\mathrm{grid} n_\\mathrm{nocc} n_\\mathrm{nvir} n_\\mathrm{aux}^2)$\n\n* DF SOS MP2: $O(n_\\mathrm{occ}^2 n_\\mathrm{vir}^2 n_\\mathrm{aux})$\n\n\u6211\u4eec\u53ef\u80fd\u4f1a\u8ba4\u4e3a $n_\\mathrm{occ} < n_\\mathrm{vir}$\uff0c\u800c $n_\\mathrm{aux}$ \u5728\u57fa\u7ec4\u8f83\u5927\u65f6\u662f $n_\\mathrm{vir}$ \u7684 3 \u500d\u5de6\u53f3\u3002\u7531\u4e8e $n_\\mathrm{grid}$ \u4e0d\u968f\u4f53\u7cfb\u589e\u5927\u800c\u53d8\u5316\uff0c\u56e0\u6b64\u5728 $n_\\mathrm{grid} \\ll n_\\mathrm{occ}$ \u7684\u60c5\u51b5\u4e0b\uff0cLT \u7684\u4f18\u52bf\u4f1a\u51f8\u663e\u3002\n\n## \u5907\u6ce8\n\n\u8fd9\u4efd\u6587\u6863\u989c\u6587\u6770\u63d0\u4f9b\u4e86 QChem \u4ee3\u7801\u4e2d\u5173\u4e8e 7 \u70b9 LT \u683c\u70b9\u7684\u652f\u6301\uff1b\u5c3d\u7ba1\u6700\u7ec8\u6587\u6863\u4f7f\u7528\u4e86\u81ea\u5b9a\u4e49\u7684\u7b49\u6bd4\u7ea7\u6570\u683c\u70b9\uff0c\u4f46 7 \u70b9 LT \u683c\u70b9\u5e2e\u52a9\u4e86\u6587\u6863\u7684\u5b8c\u6210\u7684\u8fc7\u7a0b\u3002\n\n[^Jung-Head-Gordon.JCP.2004.121]: Jung, Y.; Lochan, R. C.; Dutoi, A. D.; Head-Gordon, M. Scaled Opposite-Spin Second Order M\u00f8ller\u2013Plesset Correlation Energy: An Economical Electronic Structure Method. *J. Chem. Phys.* **2004**, *121* (20), 9793\u20139802. doi: [10.1063/1.1809602](https://doi.org/10.1063/1.1809602).\n", "meta": {"hexsha": "985db2dd0ab016d2a2de89d9c010bc6239cf8aec", "size": 146215, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "source/QC_Notes/DF_Series/LT_MP2.ipynb", "max_stars_repo_name": "ajz34/ajz34.readthedocs.io", "max_stars_repo_head_hexsha": "73be05a73241c18b98fd0d4dbdc48c643278c3da", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-07-30T12:31:14.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-14T03:56:56.000Z", "max_issues_repo_path": "source/QC_Notes/DF_Series/LT_MP2.ipynb", "max_issues_repo_name": "ajz34/ajz34.readthedocs.io", "max_issues_repo_head_hexsha": "73be05a73241c18b98fd0d4dbdc48c643278c3da", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "source/QC_Notes/DF_Series/LT_MP2.ipynb", "max_forks_repo_name": "ajz34/ajz34.readthedocs.io", "max_forks_repo_head_hexsha": "73be05a73241c18b98fd0d4dbdc48c643278c3da", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-07-30T12:32:09.000Z", "max_forks_repo_forks_event_max_datetime": "2020-07-30T12:32:09.000Z", "avg_line_length": 77.7325890484, "max_line_length": 83051, "alphanum_fraction": 0.7533495195, "converted": true, "num_tokens": 7579, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7956581000631541, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.42575539906782894}} {"text": "\n\n\n# Tutorial-IllinoisGRMHD: mhdflux.C\n\n## Authors: Leo Werneck & Zach Etienne\n\n**This module is currently under development**\n\n## In this tutorial module we explain how the MHD fluxes are evaluated within IllinoisGRMHD.\n\n### Required and recommended citations:\n\n* **(Required)** Etienne, Z. B., Paschalidis, V., Haas R., M\u00f6sta P., and Shapiro, S. L. IllinoisGRMHD: an open-source, user-friendly GRMHD code for dynamical spacetimes. Class. Quantum Grav. 32 (2015) 175009. ([arxiv:1501.07276](http://arxiv.org/abs/1501.07276)).\n* **(Required)** Noble, S. C., Gammie, C. F., McKinney, J. C., Del Zanna, L. Primitive Variable Solvers for Conservative General Relativistic Magnetohydrodynamics. Astrophysical Journal, 641, 626 (2006) ([astro-ph/0512420](https://arxiv.org/abs/astro-ph/0512420)).\n* **(Recommended)** Del Zanna, L., Bucciantini N., Londrillo, P. An efficient shock-capturing central-type scheme for multidimensional relativistic flows - II. Magnetohydrodynamics. A&A 400 (2) 397-413 (2003). DOI: 10.1051/0004-6361:20021641 ([astro-ph/0210618](https://arxiv.org/abs/astro-ph/0210618)).\n\n\n\n# Table of Contents\n$$\\label{toc}$$\n\nThis module is organized as follows\n\n0. [Step 0](#src_dir): **Source directory creation**\n1. [Step 1](#introduction): **Introduction**\n1. [Step 2](#basic_quantities): **Basic quantities**\n1. [Step 3](#eos_quantities): **Equation of State (EOS) quantities**\n1. [Step 4](#impose_speed_limit_output_u0): **Impose a speed limit and determine $u^{0}$**\n1. [Step 5](#magnetic_field_comoving_frame): **The magnetic field measured in the comoving fluid frame, $b^{\\mu}$**\n1. [Step 6](#min_max_characteristic_speeds): **The minimum and maximum characteristic speeds**\n1. [Step 7](#rho_star_flux): **MHD flux: $\\rho_{\\star}$**\n1. [Step 8](#energy_flux): **MHD flux: $\\tilde{\\tau}$**\n1. [Step 9](#momentum_flux): **MHD flux: $\\tilde{S}_{i}$**\n1. [Step 10](#code_validation): **Code validation**\n1. [Step 11](#latex_pdf_output): **Output this module to $\\LaTeX$-formatted PDF file**\n\n\n\n# Step 0: Source directory creation \\[Back to [top](#toc)\\]\n$$\\label{src_dir}$$\n\nWe will now use the [cmdline_helper.py NRPy+ module](Tutorial-Tutorial-cmdline_helper.ipynb) to create the source directory within the `IllinoisGRMHD` NRPy+ directory, if it does not exist yet.\n\n\n```python\n# Step 0: Creation of the IllinoisGRMHD source directory\n# Step 0a: Add NRPy's directory to the path\n# https://stackoverflow.com/questions/16780014/import-file-from-parent-directory\nimport os,sys\nnrpy_dir_path = os.path.join(\"..\",\"..\")\nif nrpy_dir_path not in sys.path:\n sys.path.append(nrpy_dir_path)\n\n# Step 0b: Load up cmdline_helper and create the directory\nimport cmdline_helper as cmd\nIGM_src_dir_path = os.path.join(\"..\",\"src\")\ncmd.mkdir(IGM_src_dir_path)\n\n# Step 0c: Create the output file path \noutfile_path__mhdflux__C = os.path.join(IGM_src_dir_path,\"mhdflux.C\")\n```\n\n\n\n# Step 1: Introduction \\[Back to [top](#toc)\\]\n$$\\label{introduction}$$\n\nIn this tutorial module we will compute the flux terms for the conservative variables $U=\\left\\{\\rho_{\\star},\\tilde{\\tau},\\tilde{S}_{i}\\right\\}$, which are defined in terms of the primitive variables as\n\n$$\n\\left(\n\\begin{matrix}\n\\rho_{\\star}\\\\\n\\tilde{\\tau}\\\\\n\\tilde{S}_{i}\n\\end{matrix}\n\\right)\n=\n\\left(\n\\begin{matrix}\n\\alpha\\sqrt{\\gamma}\\rho_{b}u^{0}\\\\\n\\alpha^{2}\\sqrt{\\gamma}T^{00}-\\rho_{\\star}\\\\\n\\left(\\rho_{\\star}h + \\alpha u^{0}\\sqrt{\\gamma}b^{2}\\right)u_{i} - \\alpha\\sqrt{\\gamma}b^{0}b_{i}\n\\end{matrix}\n\\right)\\ .\n$$\n\nThe flux terms for these conservative variables are\n\n$$\n\\boldsymbol{F} \n= \n\\left(\n\\begin{matrix}\nF^{i}_{\\rho_{\\star}}\\\\\nF^{i}_{\\tilde{\\tau}}\\\\\n\\left(F_{\\tilde{S}}\\right)^{j}_{\\ i}\n\\end{matrix}\n\\right)\n=\n\\left(\n\\begin{matrix}\n\\rho_{\\star}v^{i}\\\\\n\\alpha^{2}\\sqrt{\\gamma}T^{0j} - \\rho_{\\star}v^{j}\\\\\n\\alpha\\sqrt{\\gamma}T^{j}_{\\ i}\n\\end{matrix}\n\\right)\\ .\n$$\n\nThe MHD flux algorithm computes, for each of the fluxes above, the standard Harten-Lax-van Leer (HLL) flux\n\n$$\n\\boxed{F^{\\rm HLL} = \\frac{c^{-}F_{r} + c^{+}F_{l} - c^{+}c^{-}\\left(U_{r} - U_{l}\\right)}{c^{+} + c^{-}}}\\ .\n$$\n\nAs we go through the implementation, we will notice that will need a lot of the functions defined within the `inlined_functions.C` code file of `IllinoisGRMHD`. We will therefore explain the algorithm of each function in quite some detail, so that the reader can go through this tutorial without having to stop and read the [`inlined_functions.C` tutorial module](Tutorial-IllinoisGRMHD_inlined_functions.ipynb). We do encourage the reader to go through the module as well, though, since we will be presenting the math behind each function, but not the code itself.\n\n\n\n# Step 2: Basic quantities \\[Back to [top](#toc)\\]\n$$\\label{basic_quantities}$$\n\nWe start by defining useful quantities, such as $\\psi^{\\pm4}$, $\\psi^{6}$, $\\alpha\\sqrt{\\gamma}$, $\\alpha^{-1}$, and $\\alpha^{-2}$.\n\n\n```python\n%%writefile $outfile_path__mhdflux__C\n//-----------------------------------------------------------------------------\n// Compute the flux for advecting rho_star, tau (Font's energy variable), \n// and S_i .\n//-----------------------------------------------------------------------------\nstatic inline void mhdflux(int i,int j,int k,const int flux_dirn,CCTK_REAL *Ul,CCTK_REAL *Ur, CCTK_REAL *FACEVAL,CCTK_REAL *FACEVAL_LAPSE_PSI4,eos_struct &eos, CCTK_REAL Gamma_th,\n CCTK_REAL &cmax,CCTK_REAL &cmin,\n CCTK_REAL &rho_star_flux,CCTK_REAL &tau_flux,CCTK_REAL &st_x_flux,CCTK_REAL &st_y_flux,CCTK_REAL &st_z_flux) {\n\n CCTK_REAL psi4 = FACEVAL_LAPSE_PSI4[PSI4];\n CCTK_REAL psi6 = FACEVAL_LAPSE_PSI4[PSI4]*FACEVAL_LAPSE_PSI4[PSI2];\n CCTK_REAL psim4 = 1.0/(psi4);\n\n CCTK_REAL alpha_sqrt_gamma = FACEVAL_LAPSE_PSI4[LAPSE]*psi6;\n CCTK_REAL ONE_OVER_LAPSE = 1.0/FACEVAL_LAPSE_PSI4[LAPSE];\n CCTK_REAL ONE_OVER_LAPSE_SQUARED=SQR(ONE_OVER_LAPSE);\n```\n\n Overwriting ../src/mhdflux.C\n\n\n\n\n# Step 3: Equation of State (EOS) quantities \\[Back to [top](#toc)\\]\n$$\\label{eos_quantities}$$\n\nNext we compute the quantities $P_{\\rm cold}$, $\\epsilon_{\\rm cold}$, $\\frac{dP_{\\rm cold}}{d\\rho}$, $\\epsilon_{\\rm th}$, $h$, and $\\Gamma_{\\rm cold}$. We have written $\\rho_{b}\\equiv\\rho$ so that the discussion constains a notation that is slightly less cluttered with indices.\n\nThe function `compute_P_cold__eps_cold__dPcold_drho__eps_th__h__Gamma_cold()` is defined within the `inlined_functions.C` code file of `IllinoisGRMHD`. It currently implementes 3 different cases:\n\n1. **Case 1: $\\boldsymbol{\\rho_{b} = 0}$**\n\n In this case, we set $\\Gamma_{\\rm cold} = \\Gamma_{\\rm tab}$, i.e. it's tabulated input value, and all other quantities to zero: $P_{\\rm cold}=\\epsilon_{\\rm cold}=\\frac{dP_{\\rm cold}}{d\\rho}=h=\\epsilon_{\\rm th}=0$\n \n2. **Case 2: Single Polytrope EOS**\n\n In this case we have have the relations:\n \n $$\n \\boxed{\n \\begin{align}\n P_{\\rm cold} &= \\kappa \\rho^{\\Gamma_{\\rm cold}}\\ ,\\\\\n \\epsilon_{\\rm cold} &= \\left.\\left(\\frac{P_{\\rm cold}}{\\rho}\\right)\\middle/\\left(\\Gamma_{\\rm cold}-1\\right)\\right.\\ ,\\\\\n \\frac{dP_{\\rm cold}}{d\\rho} &= \\frac{\\Gamma_{\\rm cold}P_{\\rm cold}}{\\rho}\\ ,\\\\\n \\epsilon_{\\rm th} &= \\left.\\left(\\frac{P-P_{\\rm cold}}{\\rho}\\right)\\middle/\\left(\\Gamma_{\\rm cold}-1\\right)\\right.\\ ,\\\\\n h &= 1+\\epsilon_{\\rm cold}+\\epsilon_{\\rm th} + \\frac{P}{\\rho}\\ ,\\\\\n \\Gamma_{\\rm cold} &= \\Gamma_{\\rm tab}\\ .\n \\end{align}}\n $$\n \n2. **Case 3: Piecewise Polytrope EOS**\n\n We now consider a set of dividing densities $\\rho_{\\min} < \\rho_{1} < \\cdots < \\rho_{n-1} < \\rho_{\\max}$ such that the pressure and energy density are everywhere continuous. In each interval, the EOS is a single polytrope, with its own $K_{i}$ and $\\left(\\Gamma_{cold}\\right)_{i}\\equiv\\Gamma_{i}$, so that\n \n $$\n \\boxed{\n P_{\\rm cold} =\n \\left\\{\n \\begin{matrix}\n K_{0}\\rho^{\\Gamma_{0}} & , & \\rho \\leq \\rho_{0}\\\\\n K_{1}\\rho^{\\Gamma_{1}} & , & \\rho_{0} \\leq \\rho \\leq \\rho_{1}\\\\\n \\vdots & & \\vdots\\\\\n K_{j}\\rho^{\\Gamma_{j}} & , & \\rho_{j-1} \\leq \\rho \\leq \\rho_{j}\\\\\n \\vdots & & \\vdots\\\\\n K_{N-1}\\rho^{\\Gamma_{N-1}} & , & \\rho_{N-2} \\leq \\rho \\leq \\rho_{N-1}\\\\\n K_{N}\\rho^{\\Gamma_{N}} & , & \\rho \\geq \\rho_{N-1}\n \\end{matrix}\n \\right.\n }\\ .\n $$\n \n Then, for each set $\\left\\{P_{i},K_{i},\\Gamma_{i}\\right\\}$ the desired quantities are computed using the same equations used in Case 2.\n\n\n```python\n%%writefile -a $outfile_path__mhdflux__C\n\n\n // First compute P_{cold}, \\epsilon_{cold}, dP_{cold}/drho, \\epsilon_{th}, h, and \\Gamma_{cold},\n // for right and left faces:\n CCTK_REAL P_coldr,eps_coldr,dPcold_drhor=0,eps_thr=0,h_r=0,Gamma_coldr;\n compute_P_cold__eps_cold__dPcold_drho__eps_th__h__Gamma_cold(Ur,eos,Gamma_th,P_coldr,eps_coldr,dPcold_drhor,eps_thr,h_r,Gamma_coldr);\n CCTK_REAL P_coldl,eps_coldl,dPcold_drhol=0,eps_thl=0,h_l=0,Gamma_coldl;\n compute_P_cold__eps_cold__dPcold_drho__eps_th__h__Gamma_cold(Ul,eos,Gamma_th,P_coldl,eps_coldl,dPcold_drhol,eps_thl,h_l,Gamma_coldl);\n```\n\n Appending to ../src/mhdflux.C\n\n\n\n\n# Step 4: Impose a speed limit and determine $u^{0}$ \\[Back to [top](#toc)\\]\n$$\\label{impose_speed_limit_output_u0}$$\n\nWe now call upon the `impose_speed_limit_output_u0()` function inside the `inlined_functions.C` code file of `IllinoisGRMHD`. The basic algorithm performed by this function is summarized here. We start by evaluating the quantity\n\n$$\n\\begin{align}\n{\\rm one\\_minus\\_one\\_over\\_alpha\\_u0\\_squared} \\equiv A \n&= \\gamma_{ij}\\left(\\frac{v^{i}+\\beta^{i}}{\\alpha}\\right)\\left(\\frac{v^{j}+\\beta^{j}}{\\alpha}\\right)\\\\\n&= \\frac{\\gamma_{ij}}{\\alpha^{2}}\\left[\\frac{\\gamma^{ik}u_{k}}{u^{0}} - \\beta^{i} + \\beta^{i}\\right]\\left[\\frac{\\gamma^{j\\ell}u_{\\ell}}{u^{0}} - \\beta^{j} + \\beta^{j}\\right]\\\\\n&=\\frac{\\gamma_{ij}u^{i}u^{j}}{\\left(\\alpha u^{0}\\right)^{2}}\\\\\n&=\\frac{\\left(\\alpha u^{0}\\right)^{2}-1}{\\left(\\alpha u^{0}\\right)^{2}}\\\\\n&=1 - \\frac{1}{\\left(\\alpha u^{0}\\right)^{2}}\\ \\\\\n\\implies \\boxed{A = 1 - \\frac{1}{\\left(\\alpha u^{0}\\right)^{2}}}\\ ,\n\\end{align}\n$$\n\nwhere when going from line 1 to 2 and from line 3 to 4 we have used eqs. (53) and (56) from [Duez *et al.*](https://arxiv.org/pdf/astro-ph/0503420.pdf), respectively. Then we construct the \"speed limit quantity\"\n\n$$\n{\\rm ONE\\_MINUS\\_ONE\\_OVER\\_GAMMA\\_SPEED\\_LIMIT\\_SQUARED} \\equiv B = 1-\\frac{1}{\\gamma^{2}_{\\rm speed\\ limit}}\\ .\n$$\n\nIf $A > B$, then we construct the correction factor $C\\equiv A / B$, and adjust the velocities using\n\n$$\n\\boxed{v^{i} \\to \\left(v^{i}+\\beta^{i}\\right)C - \\beta^{i}}\\ .\n$$\n\nFinally, since $A$ is evaluated using the first line above, namely\n\n$$\nA = \\gamma_{ij}\\left(\\frac{v^{i}+\\beta^{i}}{\\alpha}\\right)\\left(\\frac{v^{j}+\\beta^{j}}{\\alpha}\\right)\\ ,\n$$\n\nbut we know, from the last line how $A$ and $u^{0}$ are related, we compute\n\n$$\n\\boxed{u^{0} = \\frac{1}{\\alpha\\sqrt{1-A}}}\\ .\n$$\n\n\n```python\n%%writefile -a $outfile_path__mhdflux__C\n\n\n //Compute face velocities\n // Begin by computing u0\n output_stats stats; stats.failure_checker=0;\n CCTK_REAL u0_r,u0_l;\n impose_speed_limit_output_u0(FACEVAL,Ur,psi4,ONE_OVER_LAPSE,stats,u0_r);\n impose_speed_limit_output_u0(FACEVAL,Ul,psi4,ONE_OVER_LAPSE,stats,u0_l);\n```\n\n Appending to ../src/mhdflux.C\n\n\n\n\n# Step 5: The magnetic field measured in the comoving fluid frame, $b^{\\mu}$ \\[Back to [top](#toc)\\]\n$$\\label{magnetic_field_comoving_frame}$$\n\nNow we use th function `compute_smallba_b2_and_u_i_over_u0_psi4()` from the `inlined_functions.C` code file of `IllinoisGRMHD`.\n\nWe will need the following identities\n\n$$\n\\begin{align}\nv^{i} &= \\frac{u^{i}}{u^{0}}\\ ,\\\\\nB^{0}_{(u)} &= \\frac{u_{i}B^{i}}{\\alpha}\\ ,\\\\\nB^{i}_{(u)} &= \\frac{1}{u^{0}}\\left(\\frac{B^{i}}{\\alpha} + u^{i}B^{0}_{(u)}\\right)\\ ,\\\\\nb^{\\mu} &= \\frac{B^{\\mu}_{(u)}}{\\sqrt{4\\pi}}\\ .\n\\end{align}\n$$\n\nWe start by setting the relation\n\n$$\nb^{0} = \\frac{u_{i}B^{i}}{\\alpha\\sqrt{4\\pi}} \\implies \\boxed{\\alpha\\sqrt{4\\pi}b^{0} = u_{i}B^{i}}\\ .\n$$\n\nThen we compute\n\n$$\n\\begin{align}\nb^{i} &= \\frac{B^{i}_{(u)}}{\\sqrt{4\\pi}}\\\\\n &= \\frac{1}{u^{0}\\sqrt{4\\pi}}\\left(\\frac{B^{i}}{\\alpha} + B^{0}_{(u)}u^{i}\\right)\\\\\n &= \\frac{1}{u^{0}\\sqrt{4\\pi}}\\left(\\frac{B^{i}}{\\alpha} + \\sqrt{4\\pi}b^{0}u^{i}\\right)\\\\\n &= \\frac{1}{\\alpha\\sqrt{4\\pi}}\\left(\\frac{B^{i}}{u^{0}} + \\alpha\\sqrt{4\\pi}b^{0}\\frac{u^{i}}{u^{0}}\\right)\\\\\n\\implies &\\boxed{b^{i} = \\frac{1}{\\alpha\\sqrt{4\\pi}}\\left(\\frac{B^{i}}{u^{0}} + \\alpha\\sqrt{4\\pi}b^{0}v^{i}\\right)}\\ .\n\\end{align}\n$$\n\nFinally, we compute\n\n$$\n\\begin{align}\nb^{2} &= g_{\\mu\\nu}b^{\\mu}b^{\\nu}\\\\\n &= g_{00}\\left(b^{0}\\right)^{2} + g_{ij}b^{i}b^{j} + 2g_{0i}b^{0}b^{i}\\\\\n &= \\left(-\\alpha^{2} + \\gamma_{ij}\\beta^{i}\\beta^{j}\\right)\\left(b^{0}\\right)^{2} + \\gamma_{ij}b^{i}b^{j} + 2b^{0}\\gamma_{ij}\\beta^{j}b^{i}\\\\\n &= -\\left(\\alpha b^{0}\\right)^{2} + \\gamma_{ij}\\left[b^{i}b^{j} + 2b^{0}b^{i}\\beta^{j} + \\left(b^{0}\\right)^{2}\\beta^{i}\\beta^{j}\\right]\\\\\n\\implies &\\boxed{b^{2} = -\\left(\\alpha b^{0}\\right)^{2} + \\gamma_{ij}\\left(b^{i} + b^{0}\\beta^{i}\\right)\\left(b^{j} + b^{0}\\beta^{j}\\right)}\n\\end{align}\n$$\n\n\n```python\n%%writefile -a $outfile_path__mhdflux__C\n\n\n //Next compute b^{\\mu}, the magnetic field measured in the comoving fluid frame:\n CCTK_REAL ONE_OVER_LAPSE_SQRT_4PI = ONE_OVER_LAPSE*ONE_OVER_SQRT_4PI;\n /***********************************************************/\n /********** RIGHT FACE ************/\n // Note that smallbr[4] = b^a defined in Gammie's paper, on the right face.\n CCTK_REAL u_x_over_u0_psi4r,u_y_over_u0_psi4r,u_z_over_u0_psi4r;\n CCTK_REAL smallbr[NUMVARS_SMALLB];\n // Compute b^{a}, b^2, and u_i over u^0\n compute_smallba_b2_and_u_i_over_u0_psi4(FACEVAL,FACEVAL_LAPSE_PSI4,Ur,u0_r,ONE_OVER_LAPSE_SQRT_4PI, \n u_x_over_u0_psi4r,u_y_over_u0_psi4r,u_z_over_u0_psi4r,smallbr);\n // Then compute u_xr,u_yr, and u_zr. We need to set the zeroth component so we can specify U_LOWER{r,l}[{UX,UY,UZ}] (UX=1,UY=2,UZ=3).\n CCTK_REAL U_LOWERr[4] = { 0.0, u_x_over_u0_psi4r*u0_r*FACEVAL_LAPSE_PSI4[PSI4], u_y_over_u0_psi4r*u0_r*FACEVAL_LAPSE_PSI4[PSI4], \n u_z_over_u0_psi4r*u0_r*FACEVAL_LAPSE_PSI4[PSI4] };\n /********** LEFT FACE ************/\n // Note that smallbl[4] = b^a defined in Gammie's paper, on the left face.\n CCTK_REAL u_x_over_u0_psi4l,u_y_over_u0_psi4l,u_z_over_u0_psi4l;\n CCTK_REAL smallbl[NUMVARS_SMALLB];\n // Compute b^{a}, b^2, and u_i over u^0\n compute_smallba_b2_and_u_i_over_u0_psi4(FACEVAL,FACEVAL_LAPSE_PSI4,Ul,u0_l,ONE_OVER_LAPSE_SQRT_4PI, \n u_x_over_u0_psi4l,u_y_over_u0_psi4l,u_z_over_u0_psi4l,smallbl);\n // Then compute u_xr,u_yr, and u_zr. We need to set the zeroth component so we can specify U_LOWER{r,l}[{UX,UY,UZ}]\n CCTK_REAL U_LOWERl[4] = { 0.0, u_x_over_u0_psi4l*u0_l*FACEVAL_LAPSE_PSI4[PSI4], u_y_over_u0_psi4l*u0_l*FACEVAL_LAPSE_PSI4[PSI4], \n u_z_over_u0_psi4l*u0_l*FACEVAL_LAPSE_PSI4[PSI4] };\n /***********************************************************/\n```\n\n Appending to ../src/mhdflux.C\n\n\n\n\n# Step 6: The minimum and maximum characteristic speeds \\[Back to [top](#toc)\\]\n$$\\label{min_max_characteristic_speeds}$$\n\nWe will now explain two different functions contained in the `inlined_functions.C` code file of `IllinoisGRMHD`. These functions are `compute_v02()` and `find_cp_cm`. These functions are called with the objective of computing the minimum ($-$) and maximum ($+$) characteristic speeds at each cell interface, $c_{\\pm}^{r,l}$.\n\nWe approximate the general GRMHD dispersion relation (eq. 27 of [Gammie & McKinney (2003)](https://arxiv.org/pdf/astro-ph/0301509.pdf)) by the simpler expression\n\n$$\n\\omega_{\\rm cm}^{2} = \\left[v_{\\rm A}^{2} + c_{\\rm s}^{2}\\left(1-v_{\\rm A}^{2}\\right)\\right]k_{\\rm cm}^{2}\\ ,\n$$\n\nwhere $\\omega_{\\rm cm}=-k_{\\mu}u^{\\mu}$ is the frequency and $k_{\\rm cm}^{2} = K_{\\mu}K^{\\mu}$ the wavenumber of an MHD wave mode in the frame comoving with the fluid, where $K_{\\mu}$ is defined as the projection of the wave vector $k^{\\nu}$ onto the direction normal to $u^{\\nu}$: $K_{\\mu} = \\left(g_{\\mu\\nu}+u_{\\mu}u_{\\nu}\\right)k^{\\nu}$. $c_{\\rm s}$ is the sound speed, and $v_{\\rm A}$ is the Alfv\u00e9n speed, given by\n\n$$\nv_{\\rm A} = \\sqrt{\\frac{b^{2}}{\\rho_{b}h + b^{2}}}\\ .\n$$\n\nWith these definitions, we may then solve the approximate dispersion relation above along direction $i$, noting that in the comoving frame $k_{\\mu} = \\left(-\\omega,k_{j}\\delta^{j}_{\\ i}\\right)$ and the wave (phase) velocity is $c_{\\pm} = \\left.\\omega\\middle/\\left(k_{j}\\delta^{j}_{\\ i}\\right)\\right.$. The dispersion can then be written as a quadratic equation for $c_{\\pm}$:\n\n$$\nac_{\\pm}^{2} + bc_{\\pm} + c = 0\\ ,\n$$\n\nwith\n\n$$\n\\boxed{\n\\begin{align}\na &= \\left(1-v_{0}^{2}\\right)\\left(u^{0}\\right)^{2} - v_{0}^{2}g^{00}\\ ,\\\\\nb &= 2v_{0}^{2}g^{i0} - 2u^{i}u^{0}\\left(1-v^{2}_{0}\\right)\\ ,\\\\\nc &= \\left(1-v_{0}^{2}\\right)\\left(u^{i}\\right)^{2} - v_{0}^{2}g^{ii}\\ ,\\\\\nv_{0}^{2} &= v_{\\rm A}^{2} + c_{\\rm s}^{2}\\left(1-v_{\\rm A}^{2}\\right)\\ ,\\\\\nc_{\\rm s} &= \\left.\\left[\\frac{dP_{\\rm cold}}{d\\rho_{b}} + \\Gamma_{\\rm th}\\left(\\Gamma_{\\rm th}-1\\right)\\epsilon_{\\rm th}\\right]\\middle/h\\right.\\ ,\\\\\nc_{+} &= \\max\\left(\\frac{-b \\pm \\sqrt{b^{2}-4ac}}{2a}\\right)\\ ,\\\\\nc_{-} &= \\min\\left(\\frac{-b \\pm \\sqrt{b^{2}-4ac}}{2a}\\right)\\ .\n\\end{align}\n}\n$$\n\nFinally, after the maximum and minimum speeds $c_{\\pm}$ have been computed at left and right facs, the standard Harten-Lax-van Leer (HLL), approximate Riemann solve is applied to compute fluxes for the three conservative variables $U = \\left\\{\\rho_{\\star},\\tilde{\\tau},\\tilde{S}_{i}\\right\\}$:\n\n$$\nF^{\\rm HLL} = \\frac{c^{-}F_{r} + c^{+}F_{l} - c^{+}c^{-}\\left(U_{r} - U_{l}\\right)}{c^{+} + c^{-}}\\ ,\n$$\n\nwhere\n\n$$\n\\boxed{c^{\\pm} = \\pm\\max\\left(0,c^{r}_{\\pm},c^{l}_{\\pm}\\right)}\\ .\n$$\n\n\n```python\n%%writefile -a $outfile_path__mhdflux__C\n\n\n // Compute v02 = v_A^2 + c_s^2*(1.0-v_A^2), where c_s = sound speed, and v_A = Alfven velocity\n CCTK_REAL v02r,v02l;\n // First right face\n compute_v02(dPcold_drhor,Gamma_th,eps_thr,h_r,smallbr,Ur,v02r);\n // Then left face.\n compute_v02(dPcold_drhol,Gamma_th,eps_thl,h_l,smallbl,Ul,v02l);\n\n int offset=flux_dirn-1;\n\n // Now that we have computed v02 = v_A^2 + c_s^2*(1.0-v_A^2), we can\n // next compute c_+ and c_- using a simplified dispersion relation. \n // Note that, in solving the simplified disp. relation, we overestimate \n // c_+ and c_- by around a factor of 2, making the MHD evolution more\n // diffusive (and potentially more *stable*) than it could be.\n CCTK_REAL cplusr,cminusr,cplusl,cminusl;\n find_cp_cm(cplusr,cminusr,v02r,u0_r,\n Ur[VX+offset],ONE_OVER_LAPSE_SQUARED,FACEVAL[SHIFTX+offset],psim4,FACEVAL[GUPXX+offset]);\n find_cp_cm(cplusl,cminusl,v02l,u0_l,\n Ul[VX+offset],ONE_OVER_LAPSE_SQUARED,FACEVAL[SHIFTX+offset],psim4,FACEVAL[GUPXX+offset]);\n\n // Then compute cmax, cmin. This is required for the HLL flux.\n CCTK_REAL cmaxL = MAX(0.0,MAX(cplusl,cplusr));\n CCTK_REAL cminL = -MIN(0.0,MIN(cminusl,cminusr));\n```\n\n Appending to ../src/mhdflux.C\n\n\n\n\n# Step 7: MHD flux: $\\rho_{\\star}$ \\[Back to [top](#toc)\\]\n$$\\label{rho_star_flux}$$\n\nWe now evaluate the flux for $U=\\rho_{\\star}$. First, remember that\n\n$$\n\\rho_{\\star} = \\alpha\\sqrt{\\gamma}\\rho_{b}u^{0}\\ .\n$$\n\n$$\nF^{i}_{\\rho_{\\star}} = \\rho_{\\star} v^{i}\\ ,\n$$\n\nwhere $i$ is the current flux direction. We can then evaluate the HLL flux in the $i$-direction,\n\n$$\n\\boxed{F^{\\rm HLL} = \\frac{c^{-}F_{r} + c^{+}F_{l} - c^{+}c^{-}\\left(U_{r} - U_{l}\\right)}{c^{+} + c^{-}}}\\ .\n$$\n\n\n```python\n%%writefile -a $outfile_path__mhdflux__C\n\n\n //*********************************************************************\n // density flux = \\rho_* v^m, where m is the current flux direction (the m index)\n //*********************************************************************\n CCTK_REAL rho_star_r = alpha_sqrt_gamma*Ur[RHOB]*u0_r;\n CCTK_REAL rho_star_l = alpha_sqrt_gamma*Ul[RHOB]*u0_l;\n CCTK_REAL Fr = rho_star_r*Ur[VX+offset]; // flux_dirn = 2, so offset = 1, implies Ur[VX] -> Ur[VY]\n CCTK_REAL Fl = rho_star_l*Ul[VX+offset]; // flux_dirn = 2, so offset = 1, implies Ul[VX] -> Ul[VY]\n\n // HLL step for rho_star:\n rho_star_flux = (cminL*Fr + cmaxL*Fl - cminL*cmaxL*(rho_star_r-rho_star_l) )/(cmaxL + cminL);\n```\n\n Appending to ../src/mhdflux.C\n\n\n\n\n# Step 8: MHD flux: $\\tilde{\\tau}$ \\[Back to [top](#toc)\\]\n$$\\label{energy_flux}$$\n\nWe then evaluate the flux for $U=\\tilde{\\tau}$. First let us remember that\n\n$$\n\\tilde{\\tau} = \\alpha^{2}\\sqrt{\\gamma}T^{00} - \\rho_{\\star}\\ .\n$$\n\nWe then have the flux term\n\n$$\nF^{i}_{\\tilde{\\tau}} = \\alpha^{2}\\sqrt{\\gamma}T^{0j} - \\rho_{\\star}v^{j}\\ ,\n$$\n\nwhere $i$ is the current flux direction. We can then evaluate the HLL flux in the $i$-direction,\n\n$$\n\\boxed{F^{\\rm HLL} = \\frac{c^{-}F_{r} + c^{+}F_{l} - c^{+}c^{-}\\left(U_{r} - U_{l}\\right)}{c^{+} + c^{-}}}\\ .\n$$\n\n\n```python\n%%writefile -a $outfile_path__mhdflux__C\n\n\n //*********************************************************************\n // energy flux = \\alpha^2 \\sqrt{\\gamma} T^{0m} - \\rho_* v^m, where m is the current flux direction (the m index)\n //*********************************************************************\n // First compute some useful metric quantities:\n CCTK_REAL alpha_squared_sqrt_gamma = FACEVAL_LAPSE_PSI4[LAPSE]*alpha_sqrt_gamma;\n CCTK_REAL g4uptm = ONE_OVER_LAPSE_SQUARED*FACEVAL[SHIFTX+offset];\n CCTK_REAL g4uptt = -ONE_OVER_LAPSE_SQUARED;\n /********** RIGHT FACE ************/\n // Compute a couple useful hydro quantities:\n CCTK_REAL rho0_h_plus_b2_r = (Ur[RHOB]*h_r + smallbr[SMALLB2]);\n CCTK_REAL P_plus_half_b2_r = (Ur[PRESSURE]+0.5*smallbr[SMALLB2]);\n // Then compute T^{0m} and the flux:\n CCTK_REAL TUP0m_r = rho0_h_plus_b2_r*SQR(u0_r)*Ur[VX+offset] + P_plus_half_b2_r*g4uptm - smallbr[SMALLBT]*smallbr[SMALLBX+offset];\n Fr = alpha_squared_sqrt_gamma * TUP0m_r - rho_star_r * Ur[VX+offset];\n // Finally compute tau\n CCTK_REAL TUP00_r = rho0_h_plus_b2_r*u0_r*u0_r + P_plus_half_b2_r*g4uptt - smallbr[SMALLBT]*smallbr[SMALLBT];\n CCTK_REAL tau_r = alpha_squared_sqrt_gamma * TUP00_r - rho_star_r;\n /********** LEFT FACE *************/\n // Compute a couple useful hydro quantities:\n CCTK_REAL rho0_h_plus_b2_l = (Ul[RHOB]*h_l + smallbl[SMALLB2]);\n CCTK_REAL P_plus_half_b2_l = (Ul[PRESSURE]+0.5*smallbl[SMALLB2]);\n // Then compute T^{0m} and the flux:\n CCTK_REAL TUP0m_l = rho0_h_plus_b2_l*SQR(u0_l)*Ul[VX+offset] + P_plus_half_b2_l*g4uptm - smallbl[SMALLBT]*smallbl[SMALLBX+offset];\n Fl = alpha_squared_sqrt_gamma * TUP0m_l - rho_star_l * Ul[VX+offset];\n // Finally compute tau\n CCTK_REAL TUP00_l = rho0_h_plus_b2_l*u0_l*u0_l + P_plus_half_b2_l*g4uptt - smallbl[SMALLBT]*smallbl[SMALLBT];\n CCTK_REAL tau_l = alpha_squared_sqrt_gamma * TUP00_l - rho_star_l;\n\n // HLL step for tau:\n tau_flux = (cminL*Fr + cmaxL*Fl - cminL*cmaxL*(tau_r-tau_l) )/(cmaxL + cminL);\n```\n\n Appending to ../src/mhdflux.C\n\n\n\n\n# Step 9: MHD flux: $\\tilde{S}_{i}$ \\[Back to [top](#toc)\\]\n$$\\label{momentum_flux}$$\n\nWe then evaluate the flux for $U=\\tilde{S}_{i}$. First let us remember that\n\n$$\n\\tilde{S}_{i} = \\left(\\rho_{\\star} h + \\alpha u^{0} \\sqrt{\\gamma}b^{2}\\right)u_{i} - \\alpha\\sqrt{\\gamma}b^{0}b_{i}\\ .\n$$\n\nWe then have the flux term\n\n$$\n\\left(F_{\\tilde{S}}\\right)^{j}_{\\ i} = \\alpha\\sqrt{\\gamma}T^{j}_{\\ i}\\ ,\n$$\n\nwhere $j$ is the current flux direction and $i$ correspond to the component of $\\tilde{S}_{i}$ we are interested in. We can then evaluate the HLL flux in the $i$-direction,\n\n$$\n\\boxed{F^{\\rm HLL} = \\frac{c^{-}F_{r} + c^{+}F_{l} - c^{+}c^{-}\\left(U_{r} - U_{l}\\right)}{c^{+} + c^{-}}}\\ .\n$$\n\n\n```python\n%%writefile -a $outfile_path__mhdflux__C\n\n\n //*********************************************************************\n // momentum flux = \\alpha \\sqrt{\\gamma} T^m_j, where m is the current flux direction (the m index)\n //*********************************************************************\n // b_j = g_{ij} (b^i + b^t shift^i), g_{ij} = physical metric\n //CCTK_REAL sbtr=0,sbtl=0;\n CCTK_REAL smallb_lowerr[NUMVARS_SMALLB],smallb_lowerl[NUMVARS_SMALLB];\n lower_4vector_output_spatial_part(psi4,FACEVAL,smallbr,smallb_lowerr);\n lower_4vector_output_spatial_part(psi4,FACEVAL,smallbl,smallb_lowerl);\n\n /********** Flux for S_x **********/\n // [S_x flux] = \\alpha \\sqrt{\\gamma} T^m_x, where m is the current flux direction (the m index)\n // Again, offset = 0 for reconstruction in x direction, 1 for y, and 2 for z\n // Note that kronecker_delta[flux_dirn][0] = { 1 if flux_dirn==1, 0 otherwise }.\n Fr = alpha_sqrt_gamma*( rho0_h_plus_b2_r*(u0_r*Ur[VX+offset])*U_LOWERr[UX] \n + P_plus_half_b2_r*kronecker_delta[flux_dirn][0] - smallbr[SMALLBX+offset]*smallb_lowerr[SMALLBX] );\n Fl = alpha_sqrt_gamma*( rho0_h_plus_b2_l*(u0_l*Ul[VX+offset])*U_LOWERl[UX] \n + P_plus_half_b2_l*kronecker_delta[flux_dirn][0] - smallbl[SMALLBX+offset]*smallb_lowerl[SMALLBX] );\n\n // S_x =\\alpha\\sqrt{\\gamma}( T^0_x )\n CCTK_REAL st_x_r = alpha_sqrt_gamma*( rho0_h_plus_b2_r*u0_r*U_LOWERr[UX] - smallbr[SMALLBT]*smallb_lowerr[SMALLBX] );\n CCTK_REAL st_x_l = alpha_sqrt_gamma*( rho0_h_plus_b2_l*u0_l*U_LOWERl[UX] - smallbl[SMALLBT]*smallb_lowerl[SMALLBX] );\n\n // HLL step for Sx:\n st_x_flux = (cminL*Fr + cmaxL*Fl - cminL*cmaxL*(st_x_r-st_x_l) )/(cmaxL + cminL);\n\n /********** Flux for S_y **********/\n // [S_y flux] = \\alpha \\sqrt{\\gamma} T^m_y, where m is the current flux direction (the m index)\n // Again, offset = 1 for reconstruction in x direction, 2 for y, and 3 for z\n // Note that kronecker_delta[flux_dirn][1] = { 1 if flux_dirn==2, 0 otherwise }.\n Fr = alpha_sqrt_gamma*( rho0_h_plus_b2_r*(u0_r*Ur[VX+offset])*U_LOWERr[UY] + P_plus_half_b2_r*kronecker_delta[flux_dirn][1] \n - smallbr[SMALLBX+offset]*smallb_lowerr[SMALLBY] );\n Fl = alpha_sqrt_gamma*( rho0_h_plus_b2_l*(u0_l*Ul[VX+offset])*U_LOWERl[UY] + P_plus_half_b2_l*kronecker_delta[flux_dirn][1] \n - smallbl[SMALLBX+offset]*smallb_lowerl[SMALLBY] );\n\n // S_y =\\alpha\\sqrt{\\gamma}( T^0_y )\n CCTK_REAL st_y_r = alpha_sqrt_gamma*( rho0_h_plus_b2_r*u0_r*U_LOWERr[UY] - smallbr[SMALLBT]*smallb_lowerr[SMALLBY] );\n CCTK_REAL st_y_l = alpha_sqrt_gamma*( rho0_h_plus_b2_l*u0_l*U_LOWERl[UY] - smallbl[SMALLBT]*smallb_lowerl[SMALLBY] );\n\n // HLL step for Sy:\n st_y_flux = (cminL*Fr + cmaxL*Fl - cminL*cmaxL*(st_y_r-st_y_l) )/(cmaxL + cminL);\n\n /********** Flux for S_z **********/\n // [S_z flux] = \\alpha \\sqrt{\\gamma} T^m_z, where m is the current flux direction (the m index)\n // Again, offset = 1 for reconstruction in x direction, 2 for y, and 3 for z\n // Note that kronecker_delta[flux_dirn][2] = { 1 if flux_dirn==3, 0 otherwise }.\n Fr = alpha_sqrt_gamma*( rho0_h_plus_b2_r*(u0_r*Ur[VX+offset])*U_LOWERr[UZ] + P_plus_half_b2_r*kronecker_delta[flux_dirn][2] \n - smallbr[SMALLBX+offset]*smallb_lowerr[SMALLBZ] );\n Fl = alpha_sqrt_gamma*( rho0_h_plus_b2_l*(u0_l*Ul[VX+offset])*U_LOWERl[UZ] + P_plus_half_b2_l*kronecker_delta[flux_dirn][2] \n - smallbl[SMALLBX+offset]*smallb_lowerl[SMALLBZ] );\n\n // S_z =\\alpha\\sqrt{\\gamma}( T^0_z )\n CCTK_REAL st_z_r = alpha_sqrt_gamma*( rho0_h_plus_b2_r*u0_r*U_LOWERr[UZ] - smallbr[SMALLBT]*smallb_lowerr[SMALLBZ] );\n CCTK_REAL st_z_l = alpha_sqrt_gamma*( rho0_h_plus_b2_l*u0_l*U_LOWERl[UZ] - smallbl[SMALLBT]*smallb_lowerl[SMALLBZ] );\n\n // HLL step for Sz:\n st_z_flux = (cminL*Fr + cmaxL*Fl - cminL*cmaxL*(st_z_r-st_z_l) )/(cmaxL + cminL);\n\n cmax = cmaxL;\n cmin = cminL;\n}\n\n\n```\n\n Appending to ../src/mhdflux.C\n\n\n\n\n# Step 10: Code validation \\[Back to [top](#toc)\\]\n$$\\label{code_validation}$$\n\nFirst we download the original `IllinoisGRMHD` source code and then compare it to the source code generated by this tutorial notebook.\n\n\n```python\n# Verify if the code generated by this tutorial module\n# matches the original IllinoisGRMHD source code\n\n# First download the original IllinoisGRMHD source code\nimport urllib\nfrom os import path\n\noriginal_IGM_file_url = \"https://bitbucket.org/zach_etienne/wvuthorns/raw/5611b2f0b17135538c9d9d17c7da062abe0401b6/IllinoisGRMHD/src/mhdflux.C\"\noriginal_IGM_file_name = \"mhdflux-original.C\"\noriginal_IGM_file_path = os.path.join(IGM_src_dir_path,original_IGM_file_name)\n\n# Then download the original IllinoisGRMHD source code\n# We try it here in a couple of ways in an attempt to keep\n# the code more portable\ntry:\n original_IGM_file_code = urllib.request.urlopen(original_IGM_file_url).read()\n # Write down the file the original IllinoisGRMHD source code\n with open(original_IGM_file_path,\"w\") as file:\n file.write(original_IGM_file_code)\nexcept:\n try:\n original_IGM_file_code = urllib.urlopen(original_IGM_file_url).read()\n # Write down the file the original IllinoisGRMHD source code\n with open(original_IGM_file_path,\"w\") as file:\n file.write(original_IGM_file_code)\n except:\n # If all else fails, hope wget does the job\n !wget -O $original_IGM_file_path $original_IGM_file_url\n\n# Perform validation\nValidation__mhdflux__C = !diff $original_IGM_file_path $outfile_path__mhdflux__C\n\nif Validation__mhdflux__C == []:\n # If the validation passes, we do not need to store the original IGM source code file\n !rm $original_IGM_file_path\n print(\"Validation test for mhdflux.C: PASSED!\")\nelse:\n # If the validation fails, we keep the original IGM source code file\n print(\"Validation test for mhdflux.C: FAILED!\")\n # We also print out the difference between the code generated\n # in this tutorial module and the original IGM source code\n print(\"Diff:\")\n for diff_line in Validation__mhdflux__C:\n print(diff_line)\n```\n\n Validation test for mhdflux.C: FAILED!\n Diff:\n 5c5\n < static inline void mhdflux(int i,int j,int k,const int flux_dirn,CCTK_REAL *Ul,CCTK_REAL *Ur, CCTK_REAL *FACEVAL,CCTK_REAL *FACEVAL_LAPSE_PSI4,eos_struct &eos,\n ---\n > static inline void mhdflux(int i,int j,int k,const int flux_dirn,CCTK_REAL *Ul,CCTK_REAL *Ur, CCTK_REAL *FACEVAL,CCTK_REAL *FACEVAL_LAPSE_PSI4,eos_struct &eos, CCTK_REAL Gamma_th,\n 9d8\n < \n 20,23c19,22\n < CCTK_REAL P_coldr,eps_coldr,dPcold_drhor=0,eps_thr=0,h_r=0,gamma_coldr;\n < compute_P_cold__eps_cold__dPcold_drho__eps_th__h__gamma_cold(Ur,eos,P_coldr,eps_coldr,dPcold_drhor,eps_thr,h_r,gamma_coldr);\n < CCTK_REAL P_coldl,eps_coldl,dPcold_drhol=0,eps_thl=0,h_l=0,gamma_coldl;\n < compute_P_cold__eps_cold__dPcold_drho__eps_th__h__gamma_cold(Ul,eos,P_coldl,eps_coldl,dPcold_drhol,eps_thl,h_l,gamma_coldl);\n ---\n > CCTK_REAL P_coldr,eps_coldr,dPcold_drhor=0,eps_thr=0,h_r=0,Gamma_coldr;\n > compute_P_cold__eps_cold__dPcold_drho__eps_th__h__Gamma_cold(Ur,eos,Gamma_th,P_coldr,eps_coldr,dPcold_drhor,eps_thr,h_r,Gamma_coldr);\n > CCTK_REAL P_coldl,eps_coldl,dPcold_drhol=0,eps_thl=0,h_l=0,Gamma_coldl;\n > compute_P_cold__eps_cold__dPcold_drho__eps_th__h__Gamma_cold(Ul,eos,Gamma_th,P_coldl,eps_coldl,dPcold_drhol,eps_thl,h_l,Gamma_coldl);\n 60c59\n < compute_v02(dPcold_drhor,eos.gamma_th,eps_thr,h_r,smallbr,Ur,v02r);\n ---\n > compute_v02(dPcold_drhor,Gamma_th,eps_thr,h_r,smallbr,Ur,v02r);\n 62c61\n < compute_v02(dPcold_drhol,eos.gamma_th,eps_thl,h_l,smallbl,Ul,v02l);\n ---\n > compute_v02(dPcold_drhol,Gamma_th,eps_thl,h_l,smallbl,Ul,v02l);\n\n\n\n\n# Step 11: Output this module to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n$$\\label{latex_pdf_output}$$\n\nThe following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n[Tutorial-IllinoisGRMHD__mhdflux.pdf](Tutorial-IllinoisGRMHD__mhdflux.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means).\n\n\n```python\nlatex_nrpy_style_path = os.path.join(nrpy_dir_path,\"latex_nrpy_style.tplx\")\n#!jupyter nbconvert --to latex --template $latex_nrpy_style_path Tutorial-IllinoisGRMHD__mhdflux.ipynb\n#!pdflatex -interaction=batchmode Tutorial-IllinoisGRMHD__mhdflux.tex\n#!pdflatex -interaction=batchmode Tutorial-IllinoisGRMHD__mhdflux.tex\n#!pdflatex -interaction=batchmode Tutorial-IllinoisGRMHD__mhdflux.tex\n!rm -f Tut*.out Tut*.aux Tut*.log\n```\n", "meta": {"hexsha": "e3785a6a952b9335ff3781e1a445761e1f49fa4b", "size": 43131, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "IllinoisGRMHD/doc/Tutorial-IllinoisGRMHD__mhdflux.ipynb", "max_stars_repo_name": "dinatraykova/nrpytutorial", "max_stars_repo_head_hexsha": "74d1bab0c45380727975568ba956b69c082e2293", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "IllinoisGRMHD/doc/Tutorial-IllinoisGRMHD__mhdflux.ipynb", "max_issues_repo_name": "dinatraykova/nrpytutorial", "max_issues_repo_head_hexsha": "74d1bab0c45380727975568ba956b69c082e2293", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "IllinoisGRMHD/doc/Tutorial-IllinoisGRMHD__mhdflux.ipynb", "max_forks_repo_name": "dinatraykova/nrpytutorial", "max_forks_repo_head_hexsha": "74d1bab0c45380727975568ba956b69c082e2293", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2019-11-14T03:31:18.000Z", "max_forks_repo_forks_event_max_datetime": "2019-12-12T13:42:52.000Z", "avg_line_length": 46.2282958199, "max_line_length": 571, "alphanum_fraction": 0.5557487654, "converted": true, "num_tokens": 11556, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.737158174177441, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.4257054289963075}} {"text": "# Part II. Approximate Riemann Solvers\n\nIn Part II of this book we present a number of *approximate Riemann solvers*. We have already seen that for many important hyperbolic systems it is possible to work out the exact Riemann solution for arbitrary left and right states. However, for complicated nonlinear systems, such as the Euler equations (see [Euler](Euler.ipynb)), this exact solution can only be determined by solving a nonlinear system of algebraic equations for the intermediate states and the waves that connect them. This can be done to arbitrary precision, but only at some computational expense. The cost of exactly solving a single Riemann problem may seem insignificant, but it can become prohibitively expensive when the Riemann solver is used as a building block in a finite volume method. In this case a Riemann problem must be solved at every cell edge at every time step.\n\nFor example, if we consider a very coarse grid in one space dimension with only 100 cells and take 100 time steps, then 10,000 Riemann problems must be solved. In solving practical problems in two or three space dimensions it is not unusual to require the solution of billions or trillions of Riemann problems. In this context it can be very important to develop efficient approximate Riemann solvers that quickly produce a sufficiently good approximation to the true Riemann solution.\n\nThe following points have helped to guide the development of approximate Riemann solvers:\n\n - If the solution is smooth over much of the domain, then the jump in states between neighboring cells will be very small (on the order of $\\Delta x$, the cell size) for most of the Riemann problems encountered in the numerical solution. Even if the hyperbolic system being studied is nonlinear, for such data the equations can be approximated by a linearization and we have seen that linear Riemann problems can be solved more easily than nonlinear ones. Rather than solving a nonlinear system of equations by some iterative method, one need only solve a linear system (provided the eigenvalues and eigenvectors of the Jacobian matrix are known analytically, as they often are for practical problems). In many cases the solution of this linear system can also be worked out analytically and is easy to implement, so numerical linear algebra is not required. \n \n - In spite of smoothness over much of the domain, in interesting problems there are often isolated discontinuities such as shock waves that are important to model accurately. So some Riemann problems arising in a finite volume method may have large jumps between the left and right states. Hence a robust approximate Riemann solver must also handle these cases without introducing too much error.\n \n - But even in the case of large jumps in the data, it may not be necessary or worthwhile to solve the Riemann problem exactly. The information produced by the Riemann solver goes into a numerical method that updates the approximate solution in each grid cell and the exact structure of the Riemann solution is lost in the process.\n\nEach chapter in this part of the book illustrates some common approximate Riemann solvers in the context of one of the nonlinear systems studied in part 1. We focus on two popular approaches to devising approximate Riemann solvers, though these are certainly not the only approaches: linearized solvers and two-wave solvers.\n\n## Finite volume methods\n\nWe give a short review of Riemann-based finite volume methods to illustrate what is typically needed from a Riemann solver in order to implement such methods. In one space dimension, a finite volume approximation to the solution $q(x,t_n)$ at the $n$th time step consists of discrete values $Q_j^n$, each of which can be viewed as approximating the cell average of the solution over a grid cell $x_{j-1/2} < x < x_{j+1/2}$ for some discrete grid. The cell length $\\Delta x_j = x_{j+1/2} - x_{j-1/2}$ is often uniform, but this is not required. Many methods for hyperbolic conservation laws are written in *conservation form*, in which the numerical solution is advanced from time $t_n$ to $t_{n+1} = t_n + \\Delta t_n$ by the explicit formula\n\n\\begin{align}\\label{FVupdate}\nQ_j^{n+1} = Q_j^n - \\frac{\\Delta t_n}{\\Delta x_j} (F_{j+1/2}^n - F_{j-1/2}^n),\n\\end{align}\n\nfor some definition of the *numerical flux* $F_{j-1/2}^n$, typically based on $Q_{j-1}^n$ and $Q_j^n$ (and possibly other nearby cell values at time $t_n$).\n\nDividing (\\ref{FVupdate}) by $\\Delta t_n$ and rearranging, this form can be viewed as a discretization of $q_t + f(q)_x = 0$, provided the numerical flux is *consistent* with the true flux $f(q)$ in a suitable manner. In particular if the $Q_i$ used in defining $F_{j-1/2}^n$ are all equal to the same value $\\bar q$, then $F_{j-1/2}^n$ should reduce to $f(\\bar q)$.\n\nA big advantage of using conservation form is that the numerical method is conservative. The sum $\\sum \\Delta x_j Q_j^n$ approximates the integral $\\int q(x,t_n)\\,dx$. Multiplying (\\ref{FVupdate}) by $\\Delta x_j$ and summing shows that at time $t_{n+1}$ this sum only changes due to fluxes at the boundaries of the region in question (due to cancellation of the flux differences when summing), a property shared with the true solution. For problems with shock waves, using methods in conservation form is particularly important since nonconservative formulations can lead to methods that converge to discontinuous solutions that look fine but are not correct, e.g. the shock wave might propagate at entirely the wrong speed. \n\n### Godunov's method\n\nTrying to compute numerical approximations to hyperbolic problems with strong shock waves is challenging because of the discontinuities in the solution --- classical interpretations of $(F_{j+1/2}^n - F_{j-1/2}^n)/\\Delta x_j \\approx \\partial f/ \\partial x$ break down, oscillations near discontinuities often appear, and methods can easily go catastrophically unstable.\n\nThe landmark paper of Godunov (Godunov 1959) was the first to suggest using the solution to Riemann problems in defining the numerical flux: $F_{j-1/2}^n$ is obtained by evaluating $f(Q_{j-1/2}^*)$, where $Q_{j-1/2}^*$ is the Riemann solution evaluated along the ray $x/t = 0$ after the standard Riemann problem is solved between states $q_\\ell=Q_{j-1}^n$ and $q_r=Q_j^n$ (with the discontinuity placed at $x=t=0$ as usual in the definition of the Riemann problem). If the numerical solution is now defined as a piecewise constant function with value $Q_j^n$ in the $j$th cell at time $t_n$, then the exact solution takes this value along the cell interface $x_{j-1/2}$ for sufficiently small later times $t > t_n$ (until waves from other cell interfaces begin to interact).\n\nThe classic Godunov method was developed for gas dynamics and the exact Riemann solution was used, but since only one value is used from this solution and the rest of the structure is thrown away, it is natural to use some *approximate Riemann solver* that more cheaply estimates $Q_{j-1/2}^*$. The approximations discussed in the next few chapters are often suitable.\n\nGodunov's method turns out to be very robust -- because the shock structure of the solution is used in defining the interface flux, the method generally remains stable provided that the *Courant-Friedrichs-Lewy (CFL) Condition* is satisfied, which restricts the allowable time step relative to the cell sizes and wave speeds by requiring that no wave can pass through more than one grid cell in a single time step. This is clearly a necessary condition for convergence of the method, based on domain of dependence arguments, and for Godunov's method (with the exact solver) this is generally sufficient as well, as verified in countless simulations (though seemingly impossible to prove in complete generality for nonlinear systems of equations). When the exact solution is replaced by an approximate solution, the method may not work as well, and so some care has to be used in defining a suitable approximation.\n\n\n### High-resolution methods\n\nIn spite of its robustness, Godunov's method is seldom used as just described because the built-in *numerical viscosity* that gives it robustness also leads to very smeared out solutions, particularly around discontinuities, unless a very fine computational grid is used. For the advection equation, Godunov's method reduces to the standard \"first-order upwind\" method and in general it is only first order accurate even on smooth solutions.\n\nA wide variety of higher order Godunov-type (i.e., Riemann solver based) methods have been developed. One approach first reconstructs better approximations to the solution at each time from the states $Q_j^n$, e.g. a piecewise polynomial that is linear or quadratic in each grid cell rather than constant, and then uses the states from these polynomials evaluated at the cell interfaces to define the Riemann problems. Another approach, used in the \"wave propagation methods\" developed in (LeVeque, 2002), for example, is to take the waves that come out of the Riemann solution based on the original data to also define second order correction terms. In either case some *limiters* must generally be applied in order to avoid nonphysical oscillations in solutions, particularly when the true solution has discontinuities. There is a vast literature on such methods; see for example many of the books cited in the [Preface](Preface.ipynb). For our present purposes the main point is that an approximate Riemann solver is a necessary ingredient in many methods that are commonly used to obtain high-resolution approximations to hyperbolic PDEs.\n\n## Notation and structure of approximate solutions\n\nWe consider a single Riemann problem with left state $q_\\ell$ and right state $q_r$. These states might be $Q_{j-1}^n$ and $Q_j^n$ for a typical interace when using Godunov's method, or other states defined from them, e.g. after doing a polynomial reconstruction. At any rate, from now on we will not discuss the numerical methods or the grid in its totality, but simply focus on how to define an approximate Riemann solution based an an arbitrary pair of states. The resulting \"interface solution\" and \"interface flux\" will be denoted simply by $Q^*$ and $F^*$, respectively, as approximations to the Riemann solution and flux along $x/t =0$ in the similarity solution.\n\nThe Riemann solution gives a resolution of the jump $\\Delta q = (q_r - q_\\ell)$ into a set of propagating waves. In both of the approaches described below, the approximate Riemann solution consists entirely of traveling discontinuities, i.e., there are no rarefaction waves in the approximate solution, although there may be a discontinuity that approximates such a wave. One should rightly worry about whether the approximate solution generated with such a method will satisfy the required entropy condition and end up with rarefaction waves where needed rather than entropy-violating shocks, and we address this to some extent in the examples in the following chapters. It is important to remember that we are discussing the approximate solver that will be used *at every grid interface* in every time step, and the numerical viscosity inherent in the numerical method can lead to rarefactions in the overall numerical approximation even if each approximate Riemann solution lacks rarefactions. Nonetheless some care is needed, particularly in the case of *transonic* rarefactions, as we will see.\n\nFollowing (LeVeque, 2002), we refer to these traveling discontinuities as *waves* and denote them by ${\\cal W}_p \\in {\\mathbb R}^m$, where the index $p$ denotes the characteristic family and typically ranges from $1$ to $m$ for a system of $m$ equations, although in an approximate solver the number of waves may be smaller (or possibly larger). At any rate, they always have the property that\n\\begin{align}\\label{Wsum}\nq_r - q_\\ell = \\sum_{p} {\\cal W}_p.\n\\end{align}\nFor each wave, the approximate solver must also give a wave speed $s_p \\in{\\mathbb R}$. For a linear system such as acoustics, the true solution has this form with the $s_p$ being eigenvalues of the coefficient matrix and each wave is a corresponding eigenvector, as described in [Acoustics](Acoustics.ipynb). One class of approximate Riemann solvers descussed below is based on approximating a nonlinear problem by a linearization locally at each interface.\n\nOnce a set of waves and speeds have been defined, we can define an interface flux as follows: the waves for which $s_p < 0$ are traveling to the left while those with $s_p>0$ are traveling to the right, and so as an interface flux we could use $f(Q^*)$, where $Q^*$ is defined by either\n$$\nQ^* = q_\\ell + \\sum_{p: s_p < 0} {\\cal W}_p.\n$$\nor\n$$\nQ^* = q_r - \\sum_{p: s_p > 0} {\\cal W}_p.\n$$\nThese two expressions give the same value for $Q^*$ unless there is a wave with $s_p=0$, in which case they could be different if the corresponding ${\\cal W}_p$ is nonzero. However, if we are using the *exact* Riemann solution (e.g. for a linear system or a nonlinear problem in which the solution consists only of shock waves), then a stationary discontinuity with $s_p=0$ must have no jump in flux across it (by the Rankine-Hugoniot condition) and so even if the two values of $Q^*$ differ, the flux $f(Q^*)$ is uniquely defined. For an approximate Riemann solution this might not be true.\n\nAnother way to define the interface flux $F^*$ would be as\n\\begin{align}\\label{Fstar}\nF^* = f(q_\\ell) + \\sum_{p: s_p \\leq 0} s_p{\\cal W}_p\n= f(q_r) - \\sum_{p: s_p \\geq 0} s_p{\\cal W}_p\n\\end{align}\nsuggested by the fact that this is the correct flux along $x/t = 0$ for a linear system $f(q)=Aq$ or for a nonlinear system with only shocks in the solution; in these cases the Rankine-Hugoniot condition implies that $s_p{\\cal W}_p$ is equal to the jump in flux across each wave. Note that in this expression the terms in the sum for $s_p=0$ drop out so the two expressions always agree. \n\nThe wave propagation algorithms described in (LeVeque, 2002) and implemented in Clawpack use a form of Godunov's method based on the sums appearing in (\\ref{Fstar}), called \"fluctuations\", to update the neighboring cell averages, rather than the flux difference form. In these methods the waves and speeds which are further used (after applying a limiter to the waves) to obtain the high-resolution corrections. An advantage of working with fluctuations, waves, and speeds rather than interface fluxes is that these quantities often make sense also for *non-conservative hyperbolic systems,* such as the variable coefficient linear problem $q_t + A(x)q_x = 0$, for which there is no \"flux function\". A Riemann problem is defined by prescribing matrices $A_\\ell$ and $A_r$ along with the initial data $q_\\ell$ and $q_r$, for example by using the material properties in the grid cells to the left and right of the interface for acoustics through a heterogeneous material. The waves are then naturally defined using the eigenvectors of $A_\\ell$ corresponding to negative eigenvalues for the left-going waves, and using eigenvectors of $A_r$ corresponding to positive eigenvalues for the right-going waves. See (LeVeque, 2002) for more details.\n\nUpdating cell averages by fluctuations rather than flux differencing will give a conservative method (when applied to a hyperbolic problem in conservation form) only if the waves and speeds in the approximate solver satisfy\n\\begin{align}\n\\label{adqdf}\n\\sum_{p=1}^m s_p {\\cal W}_p = f(q_r) - f(q_\\ell).\n\\end{align}\nThis is a natural condition to require of our approximate Riemann solvers in general, even though the flux-differencing form (\\ref{FVupdate}) always leads to a conservative method, since this is satisfied by the exact solution in cases where it consists only of discontinuities and each wave satisfies the Rankine-Hugoniot condition. When (\\ref{adqdf}) is satisfied we say the approximate solver is conservative.\n\n## Linearized Riemann solvers \nConsider a nonlinear system $q_t + f(q)_x = 0$. If $q_\\ell$ and $q_r$ are close to each other, as is often the case over smooth regions of a more general solution, then the nonlinear system can be approximated by a linear problem of the form $q_t + \\hat A q_x = 0$. The coefficient matrix $\\hat A$ should be some approximation to $f'(q_\\ell) \\approx f'(q_r)$ in the case where $\\|q_\\ell-q_r\\|$ is small. The idea of a general linearized Riemann solver is to define a matrix $\\hat A(q_\\ell, q_r)$ that has this property but also makes sense as an approximation in the case when $\\|q_\\ell-q_r\\|$ is not small. For many nonlinear systems there is a *Roe linearization*, a particular function that works works very well based on ideas introduced originally by Roe (Roe, 1981). For systems such as the shallow water equations or the Euler equations, there are closed-form expressions for the eigenvalues and eigenvectors of $\\hat A$ and the solution of the linearized Riemann problem, leading to efficient solvers. These will be presented in the next few chapters.\n\n## Two-wave solvers \nSince the Riemann solution impacts the overall numerical solution only based on how it modifies the two neighboring solution values, it seems reasonable to consider approximations in which only a single wave propagates in each direction. The solution will have a single intermediate state $q_m$ such that ${\\cal W}_1 = q_m - q_\\ell$ and ${\\cal W}_2 = q_r-q_m$. There are apparently $m+2$ values to be determined: the middle state $q_m \\in {\\mathbb R}^m$ and the speeds $s_1, s_2$. In order for the approximate solver to be conservative, it must satisfy (\\ref{adqdf}), and hence the $m$ conditions \n\\begin{align}\nf(q_r) - f(q_\\ell) = s_1 {\\cal W}_1 + s_2 {\\cal W}_2.\n\\end{align} \nThis can be solved for the middle state to find \n\\begin{align} \\label{AS:middle_state}\nq_m = \\frac{f(q_r) - f(q_\\ell) - s_2 q_r + s_1 q_\\ell}{s_1 - s_2}.\n\\end{align} \nIt remains only to specify the wave speeds, and it is in this specification that the various two-wave solvers differ. In the following sections we briefly discuss the choice of wave speed for a scalar problem; the choice for systems will be elaborated in subsequent chapters.\n\nTypically $s_1 < 0 < s_2$ and so the intermediate state we need is $Q^* = q_m$ and $F^* = f(Q^*)$. However, in some cases both $s_1$ and $s_2$ could have the same sign, in which case $F^*$ is either $f(q_\\ell)$ or $f(q_r)$.\n\nIn addition to the references provided below, this class of solvers is also an ingredient in the so-called *central schemes*. Due to the extreme simplicity of two-wave solvers, the resulting central schemes are often even referred to as being \"Riemann-solver-free\".\n\n### Lax-Friedrichs (LF) and local-Lax-Friedrichs (LLF)\n\nThe simplest such solver is the *Lax-Friedrichs method*, in which it is assumed that both waves have the same speed, in opposite directions:\n$$-s_1 = s_2 = a,$$\nwhere $a\\ge 0$. Then (\\ref{AS:middle_state}) becomes\n$$q_m = -\\frac{f(q_r) - f(q_\\ell)}{2a} + \\frac{q_r + q_\\ell}{2}.$$\nIn the original Lax-Friedrichs method, the wave speed $a$ is taken to be the same in every Riemann problem over the entire grid; in the *local Lax Friedrichs (LLF) method*, a different speed $a$ may be chosen for each Riemann problem.\n\nFor stability reasons, the wave speed should be chosen at least as large as the fastest wave speed appearing in the true Riemann solution. However, choosing a wave speed that is too large leads to excess diffusion. For the LLF method (originally due to Rusanov), the wave speed is chosen as\n$$a(q_r, q_\\ell) = \\max(|f'(q)|)$$ \nwhere the maximum is taken over all values of $q$ between $q_r$ and $q_\\ell$. This ensures stability, but may still introduce substantial damping of slower waves.\n\n### Harten-Lax-van Leer (HLL)\n\nA less dissipative solver can be obtained by allowing the left- and right-going waves to have different speeds.\nThis approach was developed in (Harten, Lax, and van Leer). The solution is then determined by (\\ref{AS:middle_state}). In the original HLL solver, it was suggested to again to use speeds that bound the possible speeds occurring in the true solution. For a scalar problem, this translates to \n\\begin{align*}\ns_1 & = \\min(f'(q)) \\\\\ns_2 & = \\max(f'(q)),\n\\end{align*} \nwhere again the minimum and maximum are taken over all values between $q_r$ and $q_\\ell$. Many refinements of this choice have been proposed in the context of systems of equations, some of which will be discussed in later chapters.\n\n\n```python\n\n```\n", "meta": {"hexsha": "ddf9d41817073918e39a88e942909f45fe4f2302", "size": 24120, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "src/approx/Approximate_solvers.ipynb", "max_stars_repo_name": "alsam/Claw.jl", "max_stars_repo_head_hexsha": "81479366c2be728101d6499c27b95ebf5fcdf7a3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2018-12-24T01:58:45.000Z", "max_stars_repo_stars_event_max_datetime": "2020-01-14T06:22:20.000Z", "max_issues_repo_path": "src/approx/Approximate_solvers.ipynb", "max_issues_repo_name": "alsam/Claw.jl", "max_issues_repo_head_hexsha": "81479366c2be728101d6499c27b95ebf5fcdf7a3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/approx/Approximate_solvers.ipynb", "max_forks_repo_name": "alsam/Claw.jl", "max_forks_repo_head_hexsha": "81479366c2be728101d6499c27b95ebf5fcdf7a3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 96.48, "max_line_length": 1399, "alphanum_fraction": 0.714800995, "converted": true, "num_tokens": 5245, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.7520125737597972, "lm_q1q2_score": 0.4256530554485706}} {"text": "# $\\text{interp_sphgrid_MO_ETK}$: An Einstein Toolkit Module for Interpolation to Spherical Grids\n\n## Author: Zach Etienne\n### Formatting improvements courtesy Brandon Clark\n\n## This module is designed to interpolate arbitrary quantities on [Einstein Toolkit](https://einsteintoolkit.org/) Adaptive-Mesh Refinement (AMR) grids (using the [Carpet](https://carpetcode.org/) AMR infrastructure) to numerical grids with spherical sampling.\n\n**This module has not yet undergone validation testing.**\n\n## Introduction:\nGiven some set of $N$ quantities $\\mathbf{Q}=\\{Q_0,Q_1,Q_2,...,Q_{N-2},Q_{N-1}\\}$, this module performs the following for each $Q_i$:\n\n1. Evaluate $Q_i$ at all gridpoints that are not ghost zones. Sometimes $Q_i$ is computed using finite difference derivatives, so this is necessary.\n1. Call upon Carpet's interpolation and interprocessor synchronization functions to fill in $Q_i$ at all ghost zones, *except* at the outer boundary. We do not generally trust $Q_i$ at the outer boundary due to errors associated with the approximate outer boundary conditions. \n1. At this point, $Q_i$ is set at all gridpoints except ghost zones at the outer boundary. Interpolate $Q_i$ to the spherical grids, **maintaining the Cartesian basis for all vectors and tensors**, and append the result to a file.\n\nThis tutorial module takes a three-part structure. First, all the needed core Einstein Toolkit (ETK) C routines for interpolation are presented. Second, NRPy+ is used to output gridfunctions needed on the spherical grids. Third, the needed files for interfacing this module with the rest of the Einstein Toolkit (ccl files) are specified.\n\n\n\n# Table of Contents: \n$$\\label{toc}$$ \n\nThis module is organized as follows\n\n1. [Step 1](#etkmodule): Setting up the Core C Code for the Einstein Toolkit Module\n 1. [Setp 1.a](#etk_interp): Low-Level Einstein Toolkit Interpolation Function\n 1. [Step 1.b](#sphericalgridnotes): Setting up the Spherical Grids\n 1. [Step 1.c](#fileformat): Outputting to File\n 1. [Step 1.d](#maininterpolator): The Main Interpolation Driver Function\n1. [Step 2](#nrpy): Use NRPy+ C Output to Set All Output Gridfunctions\n 1. [Step 2.a](#nrpy_list_of_funcs_interp): Set up NRPy-based \"list_of_functions_to_interpolate.h\"\n 1. [Step 2.a.i](#nrpygrmhd): GRMHD quantities (IN PROGRESS)\n 1. [Step 2.a.ii](#nrpy4metric): Compute all 10 components of the 4-metric $g_{\\mu\\nu}$\n 1. [Step 2.a.iii](#nrpy4christoffels): Compute all 40 4-Christoffels $\\Gamma^{\\mu}_{\\nu\\delta}$\n 1. [Step 2.b](#nrpy_c_callingfunction): C code calling function for the NRPy+ C output\n 1. [Step 2.c](#nrpygetgfname): The `get_gf_name()` function\n 1. [Step 2.d](#nrpy_interp_counter): C Code for Initializing and incrementing \"InterpCounter\"\n1. [Step 3](#cclfiles): Interfacing with the rest of the Einstein Toolkit; Setting up CCL files\n 1. [Step 3.a](#makecodedefn): $\\text{make.code.defn}$\n 1. [Step 3.b](#interfaceccl): $\\text{interface.ccl}$\n 1. [Step 3.c](#paramccl): $\\text{param.ccl}$\n 1. [Step 3.d](#scheduleccl): $\\text{schedule.ccl}$\n1. [Step 4](#readingoutputfile): Python Script for Reading the Output File\n1. [Step 5](#latex_pdf_output): Output this module to $\\LaTeX$-formatted PDF\n\n\n\n# Step 1: Setting up the Core C Code for the Einstein Toolkit Module \\[Back to [top](#toc)\\]\n$$\\label{etkmodule}$$\n\nFirst we set up the output directories for the ETK module:\n\n\n```python\n!mkdir interp_sphgrid_MO_ETK 2>/dev/null # 2>/dev/null: Don't throw an error or warning if the directory already exists.\n!mkdir interp_sphgrid_MO_ETK/src 2>/dev/null # 2>/dev/null: Don't throw an error or warning if the directory already exists.\n```\n\n\n\n## Step 1.a: Low-Level ETK Interpolation Function \\[Back to [top](#toc)\\]\n$$\\label{etk_interp}$$\n\nWe start by writing the low-level interpolation function **Interpolate_to_sph_grid()**, which to file. \n\n**Interpolate_to_sph_grid()** takes as input\n* **cctkGH**: Information about the underlying Cactus/Carpet grid hierarchy.\n* **interp_num_points**: Number of destination interpolation points\n* **point_x_temp, point_y_temp, point_z_temp**: Cartesian $(x,y,z)$ location for each of the **interp_num_points** interpolation points.\n* **input_array_names[1]**: List of input gridfunction names to interpolate. We will do this only one gridfunction at a time, for gridfunction $Q_i$, as described above.\n\n**Interpolate_to_sph_grid()** outputs:\n* **output_f[1]**: The gridfunction **input_array_names[1]** interpolated to the set of **interp_num_points** specified in the input.\n\n\n```python\n%%writefile interp_sphgrid_MO_ETK/src/Interpolate_to_sph_grid.h\n\nvoid Interpolate_to_sph_grid(cGH *cctkGH,CCTK_INT interp_num_points, CCTK_INT interp_order,\n CCTK_REAL *point_x_temp,CCTK_REAL *point_y_temp,CCTK_REAL *point_z_temp, \n const CCTK_STRING input_array_names[1], CCTK_REAL *output_f[1]) {\n DECLARE_CCTK_PARAMETERS; \n CCTK_INT ierr;\n\n const CCTK_INT NUM_INPUT_ARRAYS=1;\n const CCTK_INT NUM_OUTPUT_ARRAYS=1;\n\n CCTK_STRING coord_system = \"cart3d\";\n\n // Set up handles\n const CCTK_INT coord_system_handle = CCTK_CoordSystemHandle(coord_system);\n if (coord_system_handle < 0) {\n CCTK_VWarn(0, __LINE__, __FILE__, CCTK_THORNSTRING,\n \"can't get coordinate system handle for coordinate system \\\"%s\\\"!\",\n coord_system);\n }\n\n const CCTK_INT operator_handle = CCTK_InterpHandle(interpolator_name);\n if (operator_handle < 0)\n CCTK_VWarn(0, __LINE__, __FILE__, CCTK_THORNSTRING,\n \"couldn't find interpolator \\\"%s\\\"!\",\n interpolator_name);\n\n char interp_order_string[10];\n snprintf(interp_order_string, 10, \"order=%d\", interp_order);\n CCTK_STRING interpolator_pars = interp_order_string;\n CCTK_INT param_table_handle = Util_TableCreateFromString(interpolator_pars);\n if (param_table_handle < 0) {\n CCTK_VWarn(0, __LINE__, __FILE__, CCTK_THORNSTRING,\n \"bad interpolator parameter(s) \\\"%s\\\"!\",\n interpolator_pars);\n }\n \n CCTK_INT operand_indices[NUM_INPUT_ARRAYS]; //NUM_OUTPUT_ARRAYS + MAX_NUMBER_EXTRAS];\n for(int i = 0 ; i < NUM_INPUT_ARRAYS ; i++) {\n operand_indices[i] = i;\n }\n Util_TableSetIntArray(param_table_handle, NUM_OUTPUT_ARRAYS,\n operand_indices, \"operand_indices\");\n \n\n CCTK_INT opcodes[NUM_INPUT_ARRAYS];\n for(int i = 0 ; i < NUM_INPUT_ARRAYS ; i++) {\n opcodes[i] = 0;\n }\n Util_TableSetIntArray(param_table_handle, NUM_OUTPUT_ARRAYS, \n opcodes, \"opcodes\");\n\n const void* interp_coords[3] \n = { (const void *) point_x_temp,\n (const void *) point_y_temp,\n (const void *) point_z_temp };\n\n CCTK_INT input_array_indices[NUM_INPUT_ARRAYS];\n for(int i = 0 ; i < NUM_INPUT_ARRAYS ; i++) {\n input_array_indices[i] = CCTK_VarIndex(input_array_names[i]);\n if(input_array_indices[i] < 0) {\n CCTK_VWarn(0, __LINE__, __FILE__, CCTK_THORNSTRING,\n \"COULD NOT FIND VARIABLE '%s'.\",\n input_array_names[i]);\n exit(1);\n }\n }\n\n CCTK_INT output_array_types[NUM_OUTPUT_ARRAYS];\n for(int i = 0 ; i < NUM_OUTPUT_ARRAYS ; i++) {\n output_array_types[i] = CCTK_VARIABLE_REAL;\n }\n\n void * output_arrays[NUM_OUTPUT_ARRAYS]\n = { (void *) output_f[0] };\n\n // actual interpolation call\n ierr = CCTK_InterpGridArrays(cctkGH,\n 3, // number of dimensions \n operator_handle,\n param_table_handle,\n coord_system_handle,\n interp_num_points,\n CCTK_VARIABLE_REAL,\n interp_coords,\n NUM_INPUT_ARRAYS, // Number of input arrays\n input_array_indices,\n NUM_OUTPUT_ARRAYS, // Number of output arrays\n output_array_types,\n output_arrays);\n if (ierr<0) {\n CCTK_WARN(1,\"interpolation screwed up\");\n Util_TableDestroy(param_table_handle);\n exit(1);\n }\n\n ierr = Util_TableDestroy(param_table_handle);\n if (ierr != 0) {\n CCTK_WARN(1,\"Could not destroy table\");\n exit(1);\n }\n}\n```\n\n Overwriting interp_sphgrid_MO_ETK/src/Interpolate_to_sph_grid.h\n\n\n\n\n## Step 1.b: Setting up the Spherical Grids \\[Back to [top](#toc)\\]\n$$\\label{sphericalgridnotes}$$\n\n* By default, we set logarithmic radial coordinates: $r(x_{0,i}) = R_0 + e^{x_{0,i}}$, where\n + $x_{0,i} = x_{0, \\mathrm{beg}} + \\left(i+\\frac{1}{2}\\right) \\Delta x_0$\n + $x_{0, {\\mathrm{beg}}} = \\log\\left( R_{\\mathrm{in}} - R_0 \\right)$\n + $\\Delta x_0 = \\frac{1}{N_0}\\log\\left(\\frac{R_\\mathrm{out} - R_0}{R_\\mathrm{in} - R_0}\\right)$\n* As for the polar angle $\\theta$, there are two options:\n + **Option 1**: \n $$\n \\theta(x_{1,j}) \\, = \\, \\theta_c \\, + \\, \\left( \\pi - 2 \\theta_c \\right) x_{1,j} \\, \n+ \\, \\xi \\, \\sin\\left(2 \\pi x_{1,j} \\right) \\text{, where}\n$$\n + $x_{1,j} = x_{1, \\mathrm{beg}} + \\left(j+\\frac{1}{2}\\right) \\Delta x_1$\n + $\\Delta x_1 = \\frac{1}{N_1}$\n + **Option 2**: \n $$\n \\theta(x_{1,j}) = \\frac{\\pi}{2} \\left[ 1 + \\left(1-\\xi \\right) \\left(2 x_{1,j} - 1 \\right) + \\left( \\xi - \\frac{2 \\theta_c}{\\pi} \\right) \\left( 2 x_{1,j} - 1 \\right)^n \\right] \\text{, where}\n $$\n + $n$ is odd\n + $x_{1,j} = x_{1, \\mathrm{beg}} + \\left(j+\\frac{1}{2}\\right) \\Delta x_1$\n + $\\Delta x_1 = \\frac{1}{N_1}$\n* The azimuthal angle $\\phi$ is uniform, so that $\\phi(x_{2,k}) = x_{2,k}$:\n + $x_{2,k} \\in [0,2\\pi]$\n + $x_{2,k} = x_{2, \\mathrm{beg}} + \\left(k+\\frac{1}{2}\\right)\\Delta x_{2}$\n + $\\Delta x_{2} = \\frac{ 2 \\pi }{N_2}$\n\n\n```python\n%%writefile interp_sphgrid_MO_ETK/src/Set_up_interp_points_on_sph_grid.h\n\nvoid sph_grid_Interpolate_many_pts__set_interp_pts(CCTK_ARGUMENTS) {\n DECLARE_CCTK_ARGUMENTS;\n DECLARE_CCTK_PARAMETERS;\n\n CCTK_REAL dx0 = log( (Rout - R0) / (Rin - R0) ) / ((CCTK_REAL)N0);\n CCTK_REAL dx1 = 1.0 / ((CCTK_REAL)N1);\n CCTK_REAL dx2 = 2.0*M_PI / ((CCTK_REAL)N2);\n CCTK_REAL x0_beg = log( Rin - R0 );\n CCTK_INT which_pt = 0;\n for(CCTK_INT k=0;k\n\n## Step 1.c: Outputting to File (File format notes) \\[Back to [top](#toc)\\]\n$$\\label{fileformat}$$\n\nSince they take almost no space relative to the data chunks, we attach the entire metadata to each interpolated function that is output:\n\n\n```python\n%%writefile interp_sphgrid_MO_ETK/src/output_to_file.h\n\n#include \"define_NumInterpFunctions.h\"\n\n// output_to_file() starts order and InterpCounter both with the value 1\nvoid output_to_file(CCTK_ARGUMENTS,char gf_name[100],int *order,CCTK_REAL *output_f[1]) {\n DECLARE_CCTK_ARGUMENTS;\n DECLARE_CCTK_PARAMETERS;\n\n char filename[100];\n sprintf (filename, \"%s/interp_sph_grids_MO.dat\", out_dir);\n FILE *file;\n if(*InterpCounter == 1 && *order==1) {\n file = fopen (filename,\"w\");\n } else {\n file = fopen (filename,\"a+\");\n }\n if (! file) {\n CCTK_VWarn (1, __LINE__, __FILE__, CCTK_THORNSTRING,\n \"interp_sph_grid__ET_thorn: Cannot open output file '%s'\", filename);\n exit(1);\n }\n\n fwrite(gf_name, 100*sizeof(char), 1, file);\n fwrite(order, sizeof(CCTK_INT), 1, file);\n \n fwrite(&N0, sizeof(CCTK_INT), 1, file);\n fwrite(&R0, sizeof(CCTK_REAL), 1, file);\n fwrite(&Rin, sizeof(CCTK_REAL), 1, file);\n fwrite(&Rout, sizeof(CCTK_REAL), 1, file);\n\n fwrite(&N1, sizeof(CCTK_INT), 1, file);\n fwrite(&x1_beg, sizeof(CCTK_REAL), 1, file);\n fwrite(&theta_option, sizeof(CCTK_INT), 1, file);\n fwrite(&th_c, sizeof(CCTK_REAL), 1, file);\n fwrite(&xi, sizeof(CCTK_REAL), 1, file);\n fwrite(&th_n, sizeof(CCTK_INT), 1, file);\n\n fwrite(&N2, sizeof(CCTK_INT), 1, file);\n fwrite(&x2_beg, sizeof(CCTK_REAL), 1, file);\n\n CCTK_REAL magic_number = 1.130814081305130e-21;\n fwrite(&magic_number, sizeof(CCTK_REAL), 1, file);\n fwrite(&cctk_iteration, sizeof(CCTK_INT), 1, file);\n fwrite(&cctk_time, sizeof(CCTK_REAL), 1, file);\n for(CCTK_INT i=0;i<1;i++) {\n fwrite(output_f[i], sizeof(CCTK_REAL)*N0*N1*N2, 1, file);\n }\n\n fclose(file);\n}\n```\n\n Overwriting interp_sphgrid_MO_ETK/src/output_to_file.h\n\n\n\n\n## Step 1.d: The Main Interpolation Driver Function \\[Back to [top](#toc)\\]\n$$\\label{maininterpolator}$$\n\nThe **Interpolate_to_sph_grid_main_function()** function calls the above functions as follows:\n1. **sph_grid_Interpolate_many_pts__set_interp_pts()**: First set up the spherical grids\n1. **Interpolate_to_sph_grid()**: Output\n\n\n```python\n%%writefile interp_sphgrid_MO_ETK/src/main_function.cc\n\n// Include needed ETK & C library header files:\n#include \n#include \n#include \n#include \n// Needed for dealing with Cactus/ETK infrastructure\n#include \"cctk.h\"\n#include \"cctk_Arguments.h\"\n#include \"cctk_Parameters.h\"\n// Needed for low-level interpolation functions\n#include \"util_Table.h\"\n#include \"util_String.h\"\n\n// Include locally-defined C++ functions:\n#include \"Set_up_interp_points_on_sph_grid.h\"\n#include \"Interpolate_to_sph_grid.h\"\n#include \"output_to_file.h\"\n#include \"get_gf_name.h\"\n\nvoid Interpolate_to_sph_grid_main_function(CCTK_ARGUMENTS) {\n DECLARE_CCTK_ARGUMENTS;\n DECLARE_CCTK_PARAMETERS;\n\n // Perform interpolation only at iteration == interp_out_iteration:\n if(cctk_iteration != interp_out_iteration) return;\n\n // Set up spherically sampled interpolation grid arrays points_x,points_y,points_z: \n sph_grid_Interpolate_many_pts__set_interp_pts(CCTK_PASS_CTOC);\n\n // Set up output array:\n CCTK_REAL *output_f[1];\n output_f[0] = output_interped;\n // The name of the input gridfunction is always \"interp_sphgrid_MO_ETK::interped_gf\":\n const CCTK_STRING input_array_names[1] = { \"interp_sphgrid_MO_ETK::interped_gf\" };\n\n // Perform interpolation!\n for(int order=1; order <= 4; order *=2) {\n char gf_name[100];\n get_gf_name(*InterpCounter,gf_name);\n printf(\"Interpolating\\033[1m %s \\033[0m... using interpolation order = %d\\n\",gf_name,order);\n Interpolate_to_sph_grid(cctkGH, N0*N1*N2, order,\n points_x,points_y,points_z, input_array_names, output_f);\n\n if(CCTK_MyProc(cctkGH)==0) {\n for(int i=0;i 1e20) {\n printf(\"BAD POINT: %s %d %e %e %e %e\\n\",gf_name,i,points_x[i],points_y[i],points_z[i], output_f[0][i]);\n }\n }\n output_to_file(CCTK_PASS_CTOC,gf_name,&order,output_f);\n printf(\"Interpolate_to_sph_grid_main_function(): Just output to file at iteration %d\\n\",cctk_iteration);\n } else {\n printf(\"Interpolate_to_sph_grid_main_function(): Process !=0 waiting for file output at iteration %d\\n\",cctk_iteration);\n }\n }\n}\n```\n\n Overwriting interp_sphgrid_MO_ETK/src/main_function.cc\n\n\n\n\n# Step 2: Use NRPy+ C Output to Set All Output Gridfunctions \\[Back to [top](#toc)\\]\n$$ \\label{nrpy}$$\n\n\n```python\n# Step 2: Import needed NRPy+ parameters\nimport indexedexp as ixp\nimport grid as gri\nimport finite_difference as fin\nfrom outputC import *\nimport sympy as sp\nimport NRPy_param_funcs as par\nimport loop\n\npar.set_parval_from_str(\"grid::GridFuncMemAccess\",\"ETK\")\n\nfrom collections import namedtuple\ngf_interp = namedtuple('gf_interp', 'gf_description')\ngf_interp_list = []\ngf_interp_list.append(gf_interp(\"dummy -- used because this is a 1-offset array\"))\n\ninterped_gf = gri.register_gridfunctions(\"AUX\",\"interped_gf\")\n\ndef interp_fileout(which_InterpCounter, expression, filename):\n kernel = fin.FD_outputC(\"returnstring\",lhrh(lhs=gri.gfaccess(\"out_gfs\",\"interped_gf\"),rhs=expression),\"outCverbose=False\")\n output_type=\"a\"\n if which_InterpCounter == 1:\n output_type=\"w\"\n \n with open(filename, output_type) as file:\n file.write(\"if(*InterpCounter == \"+str(which_InterpCounter)+\") {\\n\")\n file.write(loop.loop([\"i2\",\"i1\",\"i0\"],\n [\"cctk_nghostzones[2]\",\"cctk_nghostzones[1]\",\"cctk_nghostzones[0]\"],\\\n [\"cctk_lsh[2]-cctk_nghostzones[2]\",\n \"cctk_lsh[1]-cctk_nghostzones[1]\",\n \"cctk_lsh[0]-cctk_nghostzones[0]\"],\\\n [\"1\",\"1\",\"1\"],\\\n [\"#pragma omp parallel for\",\"\",\"\"],\" \",kernel))\n file.write(\"}\\n\")\n # If successful, return incremented which_InterpCounter:\n return which_InterpCounter+1\n```\n\n\n\n## Step 2.a: Set up NRPy-based \"list_of_functions_to_interpolate.h\" \\[Back to [top](#top)\\]\n$$\\label{nrpy_list_of_funcs_interp}$$\n\n**First specify NRPy+ output file and initialize which_InterpCounter, which keeps track of the number of interpolated functions on the grid**\n\n\n```python\nNRPyoutfilename = \"interp_sphgrid_MO_ETK/src/list_of_functions_to_interpolate.h\"\n\nwhich_InterpCounter = 1\n```\n\n\n\n### Step 2.a.i: GRMHD quantities (*IN PROGRESS; still working on adding vector potential*) \\[Back to [top](#toc)\\]\n$$\\label{nrpygrmhd}$$\n\nThese include\n* $\\rho_b$, the baryonic density (i.e., the HydroBase variable $\\verb|rho|$)\n* $P$, the total gas pressure (i.e., the HydroBase variable $\\verb|press|$)\n* $\\Gamma v_{(n)}^i$, the Valencia 3-velocity times the Lorentz factor (i.e., the HydroBase 3-gridfuntion $\\verb|vel|$, multiplied by the Lorentz factor). This definition of velocity has the advantage that after interpolation, it will not violate $u^\\mu u_\\mu = -1$. In terms of the IllinoisGRMHD 3-velocity $v^i = u^i / u^0$, the Valencia 3-velocity is given by (Eq. 11 of [Etienne *et al*](https://arxiv.org/pdf/1501.07276.pdf)):\n$$\nv_{(n)}^i = \\frac{1}{\\alpha} \\left(v^i + \\beta^i\\right).\n$$\nFurther, $\\Gamma = \\alpha u^0$ is given by (as shown [here](Tutorial-u0_smallb_Poynting-Cartesian.ipynb)):\n$$\n\\Gamma = \\alpha u^0 = \\sqrt{\\frac{1}{1 - \\gamma_{ij}v^i_{(n)}v^j_{(n)}}}.\n$$\nTherefore, $\\Gamma v_{(n)}^i$ is given by\n$$\n\\Gamma v_{(n)}^i = \\frac{1}{\\alpha} \\left(v^i + \\beta^i\\right) \\sqrt{\\frac{1}{1 - \\gamma_{ij}v^i_{(n)}v^j_{(n)}}}.\n$$\n* $A_i$, the *unstaggered* magnetic vector potential.\n* $B^i$, the *unstaggered* magnetic field vector (output only for validation purposes).\n\n\n```python\n# INPUT GRIDFUNCTIONS: The AUX or EVOL designation is *not* used in diagnostic modules.\ngammaDD = ixp.register_gridfunctions_for_single_rank2(\"AUX\",\"gammaDD\", \"sym01\")\nbetaU = ixp.register_gridfunctions_for_single_rank1(\"AUX\",\"betaU\")\nalpha = gri.register_gridfunctions(\"AUX\",\"alpha\")\n\nDIM=3\n\ngf_interp_list.append(gf_interp(\"IGM density primitive\"))\nrho_b = gri.register_gridfunctions(\"AUX\",\"rho_b\")\ninterp_expr = rho_b\nwhich_InterpCounter = interp_fileout(which_InterpCounter,interp_expr,NRPyoutfilename)\n\ngf_interp_list.append(gf_interp(\"IGM pressure primitive\"))\nP = gri.register_gridfunctions(\"AUX\",\"P\")\ninterp_expr = P\nwhich_InterpCounter = interp_fileout(which_InterpCounter,interp_expr,NRPyoutfilename)\n```\n\nNext we implement:\n$$\nv_{(n)}^i = \\frac{1}{\\alpha} \\left(v^i + \\beta^i\\right),\n$$\nand\n$$\n\\Gamma v_{(n)}^i = \\sqrt{\\frac{1}{1 - \\gamma_{ij}v^i_{(n)}v^j_{(n)}}} v_{(n)}^i.\n$$\n\n\n```python\nIGMvU = ixp.register_gridfunctions_for_single_rank1(\"AUX\",\"IGMvU\")\nValenciav = ixp.zerorank1()\nfor i in range(DIM):\n Valenciav[i] = 1/alpha * (IGMvU[i] + betaU[i])\nv_dot_v = sp.sympify(0)\nfor i in range(DIM):\n for j in range(DIM):\n v_dot_v += gammaDD[i][j]*Valenciav[i]*Valenciav[j]\n\nGamma_times_ValenciavU = ixp.zerorank1()\nfor i in range(DIM):\n Gamma_times_ValenciavU[i] = sp.sqrt(1/(1 - v_dot_v))*Valenciav[i]\n gf_interp_list.append(gf_interp(\"Lorentz factor, times Valencia vU\"+str(i)))\n interp_expr = Gamma_times_ValenciavU[i]\n which_InterpCounter = interp_fileout(which_InterpCounter,interp_expr,NRPyoutfilename)\n\n# For testing:\n# gf_interp_list.append(gf_interp(\"Lorentz factor\"))\n# interp_expr = v_dot_v\n# which_InterpCounter = interp_fileout(which_InterpCounter,interp_expr,NRPyoutfilename)\n\n# for i in range(DIM):\n# gf_interp_list.append(gf_interp(\"Valencia vU\"+str(i)))\n# interp_expr = Valenciav[i]\n# which_InterpCounter = interp_fileout(which_InterpCounter,interp_expr,NRPyoutfilename)\n\nBU = ixp.register_gridfunctions_for_single_rank1(\"AUX\",\"BU\")\nfor i in range(DIM):\n gf_interp_list.append(gf_interp(\"IGM magnetic field component B\"+str(i)))\n interp_expr = BU[i]\n which_InterpCounter = interp_fileout(which_InterpCounter,interp_expr,NRPyoutfilename)\n```\n\n\n\n### Step 2.a.ii: Compute all 10 components of the 4-metric $g_{\\mu\\nu}$ \\[Back to [top](#toc)\\]\n$$\\label{nrpy4metric}$$\n\nWe are given $\\gamma_{ij}$, $\\alpha$, and $\\beta^i$ from ADMBase, and the 4-metric is given in terms of these quantities as\n$$\ng_{\\mu\\nu} = \\begin{pmatrix} \n-\\alpha^2 + \\beta^k \\beta_k & \\beta_i \\\\\n\\beta_j & \\gamma_{ij}\n\\end{pmatrix}.\n$$\n\n\n```python\n# Eq. 2.121 in B&S\nbetaD = ixp.zerorank1()\nfor i in range(DIM):\n for j in range(DIM):\n betaD[i] += gammaDD[i][j]*betaU[j]\n\n# Now compute the beta contraction.\nbeta2 = sp.sympify(0)\nfor i in range(DIM):\n beta2 += betaU[i]*betaD[i]\n\n# Eq. 2.122 in B&S\ng4DD = ixp.zerorank2(DIM=4)\ng4DD[0][0] = -alpha**2 + beta2\nfor i in range(DIM):\n g4DD[i+1][0] = g4DD[0][i+1] = betaD[i]\nfor i in range(DIM):\n for j in range(DIM):\n g4DD[i+1][j+1] = gammaDD[i][j]\n\nfor mu in range(4):\n for nu in range(mu,4):\n gf_interp_list.append(gf_interp(\"4-metric component g4DD\"+str(mu)+str(nu)))\n interp_expr = g4DD[mu][nu]\n which_InterpCounter = interp_fileout(which_InterpCounter,interp_expr,NRPyoutfilename)\n```\n\n\n\n### Step 2.a.iii: Compute all 40 4-Christoffels $\\Gamma^{\\mu}_{\\nu\\delta}$ \\[Back to [top](#toc)\\]\n$$\\label{nrpy4christoffels}$$\n\nBy definition,\n$$\n\\Gamma^{\\mu}_{\\nu\\delta} = \\frac{1}{2} g^{\\mu\\eta} \\left(g_{\\eta\\nu,\\delta} + g_{\\eta\\delta,\\nu} - g_{\\nu\\delta,\\eta} \\right)\n$$\n\nRecall that $g_{\\mu\\nu}$ is given from $\\gamma_{ij}$, $\\alpha$, and $\\beta^i$ via\n$$\ng_{\\mu\\nu} = \\begin{pmatrix} \n-\\alpha^2 + \\beta^k \\beta_k & \\beta_i \\\\\n\\beta_j & \\gamma_{ij}\n\\end{pmatrix}.\n$$\n\nThe derivatives $g_{\\mu\\nu,\\eta}$ are then computed in terms of finite-difference derivatives of the input ADM gridfunctions $\\gamma_{ij}$, $\\alpha$, and $\\beta^i$, **assuming that the 4-metric is static, so that $\\partial_t g_{\\mu\\nu}=0$ for all $\\mu$ and $\\nu$**.\n\nTo compute $g^{\\mu\\nu}$, we use the standard formula (Eq. 4.49 in [Gourgoulhon](https://arxiv.org/pdf/gr-qc/0703035.pdf)):\n$$\ng^{\\mu\\nu} = \\begin{pmatrix} \n-\\frac{1}{\\alpha^2} & \\frac{\\beta^i}{\\alpha^2} \\\\\n\\frac{\\beta^i}{\\alpha^2} & \\gamma^{ij} - \\frac{\\beta^i\\beta^j}{\\alpha^2}\n\\end{pmatrix},\n$$\nwhere $\\gamma^{ij}$ is given by the inverse of $\\gamma_{ij}$.\n\n\n```python\nbetaDdD = ixp.zerorank2()\ngammaDD_dD = ixp.declarerank3(\"gammaDD_dD\",\"sym01\")\nbetaU_dD = ixp.declarerank2(\"betaU_dD\",\"nosym\")\nfor i in range(DIM):\n for j in range(DIM):\n for k in range(DIM):\n # Recall that betaD[i] = gammaDD[i][j]*betaU[j] (Eq. 2.121 in B&S)\n betaDdD[i][k] += gammaDD_dD[i][j][k]*betaU[j] + gammaDD[i][j]*betaU_dD[j][k]\n\n# Eq. 2.122 in B&S\ng4DDdD = ixp.zerorank3(DIM=4)\nalpha_dD = ixp.declarerank1(\"alpha_dD\")\nfor i in range(DIM):\n # Recall that g4DD[0][0] = -alpha^2 + betaU[i]*betaD[i]\n g4DDdD[0][0][i+1] += -2*alpha*alpha_dD[i] \n for j in range(DIM):\n g4DDdD[0][0][i+1] += betaU_dD[j][i]*betaD[j] + betaU[j]*betaDdD[j][i]\n\nfor i in range(DIM):\n for j in range(DIM):\n # Recall that g4DD[i][0] = g4DD[0][i] = betaD[i]\n g4DDdD[i+1][0][j+1] = g4DDdD[0][i+1][j+1] = betaDdD[i][j]\nfor i in range(DIM):\n for j in range(DIM):\n for k in range(DIM):\n # Recall that g4DD[i][j] = gammaDD[i][j]\n g4DDdD[i+1][j+1][k+1] = gammaDD_dD[i][j][k]\n\ngammaUU, dummyDET = ixp.symm_matrix_inverter3x3(gammaDD)\n\ng4UU = ixp.zerorank2(DIM=4)\ng4UU[0][0] = -1 / alpha**2\nfor i in range(DIM):\n g4UU[0][i+1] = g4UU[i+1][0] = betaU[i]/alpha**2\nfor i in range(DIM):\n for j in range(DIM):\n g4UU[i+1][j+1] = gammaUU[i][j] - betaU[i]*betaU[j]/alpha**2\n```\n\nAgain, we are to compute:\n$$\n\\Gamma^{\\mu}_{\\nu\\delta} = \\frac{1}{2} g^{\\mu\\eta} \\left(g_{\\eta\\nu,\\delta} + g_{\\eta\\delta,\\nu} - g_{\\nu\\delta,\\eta} \\right)\n$$\n\n\n```python\nGamma4UDD = ixp.zerorank3(DIM=4)\nfor mu in range(4):\n for nu in range(4):\n for delta in range(4):\n for eta in range(4):\n Gamma4UDD[mu][nu][delta] += sp.Rational(1,2)*g4UU[mu][eta]*\\\n (g4DDdD[eta][nu][delta] + g4DDdD[eta][delta][nu] - g4DDdD[nu][delta][eta])\n\n# Now output the 4-Christoffels to file:\nfor mu in range(4):\n for nu in range(4):\n for delta in range(nu,4):\n gf_interp_list.append(gf_interp(\"4-Christoffel GammaUDD\"+str(mu)+str(nu)+str(delta)))\n interp_expr = Gamma4UDD[mu][nu][delta]\n which_InterpCounter = interp_fileout(which_InterpCounter,interp_expr,NRPyoutfilename)\n```\n\n\n\n## Step 2.b: C code calling function for the NRPy+ C output \\[Back to [top](#toc)\\]\n$$\\label{nrpy_c_callingfunction}$$\n\nIn the above blocks, we wrote and appended to a file \"list_of_functions_to_interpolate.h\". Here we write the calling function for this C code.\n\n\n```python\n%%writefile interp_sphgrid_MO_ETK/src/construct_function_to_interpolate__store_to_interped_gf.cc\n#include \n#include \n#include \"cctk.h\"\n#include \"cctk_Arguments.h\"\n#include \"cctk_Parameters.h\"\n\n// Set the gridfunction interped_gf, according to the interpolation counter variable interp_counter.\n// For example, we might interpolate \"IllinoisGRMHD::rho_b\" if interp_counter==0. The following\n// function takes care of these\nvoid list_of_functions_to_interpolate(cGH *cctkGH,const CCTK_INT *cctk_lsh,const CCTK_INT *cctk_nghostzones,\n const CCTK_REAL invdx0,const CCTK_REAL invdx1,const CCTK_REAL invdx2,\n const CCTK_INT *InterpCounter,\n const CCTK_REAL *rho_bGF,const CCTK_REAL *PGF,\n const CCTK_REAL *IGMvU0GF,const CCTK_REAL *IGMvU1GF,const CCTK_REAL *IGMvU2GF,\n const CCTK_REAL *BU0GF,const CCTK_REAL *BU1GF,const CCTK_REAL *BU2GF,\n const CCTK_REAL *gammaDD00GF,const CCTK_REAL *gammaDD01GF,const CCTK_REAL *gammaDD02GF,\n const CCTK_REAL *gammaDD11GF,const CCTK_REAL *gammaDD12GF,const CCTK_REAL *gammaDD22GF,\n const CCTK_REAL *betaU0GF,const CCTK_REAL *betaU1GF,const CCTK_REAL *betaU2GF,\n const CCTK_REAL *alphaGF, CCTK_REAL *interped_gfGF) {\n#include \"list_of_functions_to_interpolate.h\"\n}\n\nvoid construct_function_to_interpolate__store_to_interped_gf(CCTK_ARGUMENTS) {\n DECLARE_CCTK_ARGUMENTS;\n DECLARE_CCTK_PARAMETERS;\n const CCTK_REAL invdx0 = 1.0 / CCTK_DELTA_SPACE(0);\n const CCTK_REAL invdx1 = 1.0 / CCTK_DELTA_SPACE(1);\n const CCTK_REAL invdx2 = 1.0 / CCTK_DELTA_SPACE(2);\n list_of_functions_to_interpolate(cctkGH,cctk_lsh,cctk_nghostzones,invdx0,invdx1,invdx2,\n InterpCounter,\n rho_b,P,\n vx,vy,vz,\n Bx,By,Bz,\n gxx,gxy,gxz,gyy,gyz,gzz,\n betax,betay,betaz,alp, interped_gf);\n// interped_gf will be interpolated across AMR boundaries, meaning that\n// it must be prolongated. Only gridfunctions with 3 timelevels stored\n// may be prolongated (provided time_interpolation_order is set to the\n// usual value of 2). We should only call this interpolation routine\n// at iterations in which all gridfunctions are on the same timelevel\n// (usually a power of 2), which will ensure that the following \n// \"filling of the timelevels\" is completely correct.\n#pragma omp parallel for\n for(int i=0;i\n\n## Step 2.c: The `get_gf_name()` function \\[Back to [top](#toc)\\]\n$$\\label{nrpygetgfname}$$\n\n\n```python\nwith open(\"interp_sphgrid_MO_ETK/src/get_gf_name.h\", \"w\") as file:\n file.write(\"void get_gf_name(const int InterpCounter,char gf_name[100]) {\\n\")\n for i in range(1,which_InterpCounter):\n file.write(\" if(InterpCounter==\"+str(i)+\") { snprintf(gf_name,100,\\\"\"+gf_interp_list[i].gf_description+\"\\\"); return; }\\n\")\n file.write(\" printf(\\\"Error. InterpCounter = %d unsupported. I should not be here.\\\\n\\\",InterpCounter); exit(1);\\n\")\n file.write(\"}\\n\")\n```\n\n\n\n## Step 2.d: C Code for Initializing and incrementing \"InterpCounter\" \\[Back to [top](#toc)\\]\n$$\\label{nrpy_interp_counter}$$\nThe gridfunctions are interpolated one at a time based on the current value of the index quantity \"InterpCounter\". Here we write the C code needed for initializing and incrementing this variable.\n\n\n```python\nwith open(\"interp_sphgrid_MO_ETK/src/define_NumInterpFunctions.h\", \"w\") as file:\n file.write(\"#define NumInterpFunctions \"+str(which_InterpCounter)+\"\\n\") \n```\n\n\n```python\n%%writefile interp_sphgrid_MO_ETK/src/interp_counter.cc\n#include \n#include \n#include \n#include \n#include \n#include \n#include \"cctk.h\"\n#include \"cctk_Arguments.h\"\n#include \"cctk_Parameters.h\"\n\n#include \"define_NumInterpFunctions.h\"\n\nvoid SphGrid_InitializeInterpCounterToZero(CCTK_ARGUMENTS)\n{\n DECLARE_CCTK_ARGUMENTS;\n DECLARE_CCTK_PARAMETERS;\n *InterpCounter = 0;\n\n if(verbose==2) printf(\"interp_sphgrid_MO_ETK: Just set InterpCounter to %d\\n\",*InterpCounter);\n}\n\nvoid SphGrid_InitializeInterpCounter(CCTK_ARGUMENTS)\n{\n DECLARE_CCTK_ARGUMENTS;\n DECLARE_CCTK_PARAMETERS;\n\n if(cctk_iteration == interp_out_iteration) {\n *InterpCounter = 1;\n if(verbose==2) printf(\"interp_sphgrid_MO_ETK: Just set InterpCounter to %d ; ready to start looping over interpolated gridfunctions!\\n\",\n *InterpCounter);\n }\n}\n\n// This function increments InterpCounter if we are at the interp_out_iteration until\n// it hits NumInterpFunctions. At this iteration, InterpCounter is set to zero, which\n// exits the loop.\nvoid SphGrid_IncrementInterpCounter(CCTK_ARGUMENTS)\n{\n DECLARE_CCTK_ARGUMENTS;\n DECLARE_CCTK_PARAMETERS;\n\n if(*InterpCounter == NumInterpFunctions-1) {\n *InterpCounter = 0;\n if(verbose==2) printf(\"interp_sphgrid_MO_ETK: Finished! Just zeroed InterpCounter.\\n\");\n } else {\n (*InterpCounter)++;\n if(verbose==2) printf(\"interp_sphgrid_MO_ETK: Just incremented InterpCounter to %d of %d\\n\",*InterpCounter,NumInterpFunctions-1);\n }\n}\n```\n\n Overwriting interp_sphgrid_MO_ETK/src/interp_counter.cc\n\n\n\n\n# Step 3: Define how this module interacts and interfaces with the larger Einstein Toolkit infrastructure \\[Back to [top](#toc)\\]\n$$\\label{cclfiles}$$\n\nWriting a module (\"thorn\") within the Einstein Toolkit requires that three \"ccl\" files be constructed, all in the root directory of the thorn:\n\n1. $\\text{interface.ccl}$: defines the gridfunction groups needed, and provides keywords denoting what this thorn provides and what it should inherit from other thorns.\n1. $\\text{param.ccl}$: specifies free parameters within the thorn.\n1. $\\text{schedule.ccl}$: allocates storage for gridfunctions, defines how the thorn's functions should be scheduled in a broader simulation, and specifies the regions of memory written to or read from gridfunctions.\n\n\n\n## Step 3.a: $\\text{make.code.defn}$ \\[Back to [top](#toc)\\]\n$$\\label{makecodedefn}$$\n\nBefore writing the \"ccl\" files, we first add Einstein Toolkit's equivalent of a Makefile, the make.code.defn file:\n\n\n```python\n%%writefile interp_sphgrid_MO_ETK/src/make.code.defn\n# Main make.code.defn file for thorn interp_sphgrid_MO_ETK\n\n# Source files in this directory\nSRCS = main_function.cc interp_counter.cc construct_function_to_interpolate__store_to_interped_gf.cc\n```\n\n Overwriting interp_sphgrid_MO_ETK/src/make.code.defn\n\n\n\n\n## Step 3.b: $\\text{interface.ccl}$ \\[Back to [top](#toc)\\]\n$$\\label{interfaceccl}$$\n\nLet's now write $\\text{interface.ccl}$. The [official Einstein Toolkit (Cactus) documentation](http://cactuscode.org/documentation/referencemanual/ReferenceManual.html) defines what must/should be included in an interface.ccl file [**here**](http://cactuscode.org/documentation/referencemanual/ReferenceManualch8.html#x12-260000C2.2). \n\n\n```python\n%%writefile interp_sphgrid_MO_ETK/interface.ccl\n\n# With \"implements\", we give our thorn its unique name.\nimplements: interp_sphgrid_MO_ETK\n\n# By \"inheriting\" other thorns, we tell the Toolkit that we \n# will rely on variables/function that exist within those\n# functions. \ninherits: admbase IllinoisGRMHD Grid\n\n# Tell the Toolkit that we want \"interped_gf\" and \"InterpCounter\"\n# and invariants to NOT be visible to other thorns, by using \n# the keyword \"private\". Note that declaring these \n# gridfunctions here *does not* allocate memory for them;\n# that is done by the schedule.ccl file.\nprivate:\nCCTK_REAL interpolation_gf type=GF timelevels=3 tags='Checkpoint=\"no\"'\n{\n interped_gf\n} \"Gridfunction containing output from interpolation.\"\n\nint InterpCounterVar type = SCALAR tags='checkpoint=\"no\"'\n{\n InterpCounter\n} \"Counter that keeps track of which function we are interpolating.\"\n\nCCTK_REAL interp_pointcoords_and_output_arrays TYPE=ARRAY DISTRIB=CONSTANT DIM=1 SIZE=N0*N1*N2 tags='checkpoint=\"no\"'\n{\n points_x,points_y,points_z,\n output_interped\n}\n```\n\n Overwriting interp_sphgrid_MO_ETK/interface.ccl\n\n\n\n\n## Step 3.c: $\\text{param.ccl}$ \\[Back to [top](#toc)\\]\n$$\\label{paramccl}$$\n\nWe will now write the file $\\text{param.ccl}$. This file allows the listed parameters to be set at runtime. We also give allowed ranges and default values for each parameter. More information on this file's syntax can be found in the [official Einstein Toolkit documentation](http://cactuscode.org/documentation/referencemanual/ReferenceManualch8.html#x12-265000C2.3). \n\n\n```python\n%%writefile interp_sphgrid_MO_ETK/param.ccl\n\n# Output the interpolated data to the IO::out_dir directory:\nshares: IO\nUSES STRING out_dir\n\nrestricted:\n \n########################################\n# BASIC THORN STEERING PARAMETERS\nCCTK_INT interp_out_iteration \"Which iteration to interpolate to spherical grids?\" STEERABLE=ALWAYS\n{\n 0:* :: \"\"\n} 960000\n\n## Interpolator information\nCCTK_STRING interpolator_name \"Which interpolator to use?\" STEERABLE=ALWAYS\n{\n \".+\" :: \"Any nonempty string; an unsupported value will throw an error.\"\n} \"Lagrange polynomial interpolation\"\n\nCCTK_INT verbose \"Set verbosity level: 1=useful info; 2=moderately annoying (though useful for debugging)\" STEERABLE=ALWAYS\n{\n 0:2 :: \"0 = no output; 1=useful info; 2=moderately annoying (though useful for debugging)\"\n} 2\n########################################\n# SPHERICAL COORDINATE SYSTEM PARAMETERS\nCCTK_INT N0 \"Number of points in r direction\" STEERABLE=ALWAYS\n{\n 0:* :: \"\"\n} 96\n\nCCTK_INT N1 \"Number of points in theta direction\" STEERABLE=ALWAYS\n{\n 0:* :: \"\"\n} 96\n\nCCTK_INT N2 \"Number of points in phi direction\" STEERABLE=ALWAYS\n{\n 0:* :: \"\"\n} 96\n\n##########\n# Cartesian position of center of spherical grid (usually center of BH) -- CURRENTLY UNSUPPORTED!\nCCTK_REAL x_center \"x-position of center.\" STEERABLE=ALWAYS\n{\n 0:* :: \"\"\n} 0.0\n\nCCTK_REAL y_center \"y-position of center.\" STEERABLE=ALWAYS\n{\n 0:* :: \"\"\n} 0.0\n\nCCTK_REAL z_center \"z-position of center.\" STEERABLE=ALWAYS\n{\n 0:* :: \"\"\n} 0.0\n\n##########\n# Radial parameters:\nCCTK_REAL R0 \"Radial offset: r(x0) = R_0 + exp(x0). Probably should keep it set to zero.\" STEERABLE=ALWAYS\n{\n 0:* :: \"\"\n} 0.0\n\nCCTK_REAL Rin \"x0 offset: x0 = log(Rin-R0) + (i + 0.5)Dx0.\" STEERABLE=ALWAYS\n{\n 0:* :: \"\"\n} 1.08986052555408\n\nCCTK_REAL Rout \"Dx0 = log( (Rout-R0) / (Rin-R0) )/N0\" STEERABLE=ALWAYS\n{\n 0:* :: \"\"\n} 80.0\n\n##########\n# Theta parameters:\nCCTK_REAL x1_beg \"x1 offset: x1 = x1_beg + (j + 0.5)Dx1. Probably should keep it set to zero.\" STEERABLE=ALWAYS\n{\n 0:* :: \"\"\n} 0.0\n\nCCTK_INT theta_option \"Which prescription for theta should be used? 1 or 2?\" STEERABLE=ALWAYS\n{\n 1:2 :: \"\"\n} 1\n\nCCTK_REAL th_c \"theta_c: Angular cutout size for theta = 0 and pi\" STEERABLE=ALWAYS\n{\n 0:* :: \"\"\n} 0.053407075111026485 # 0.017*pi\n\nCCTK_REAL xi \"Amplitude of nonlinear part of the theta distribution.\" STEERABLE=ALWAYS\n{\n 0:* :: \"\"\n} 0.25\n\nCCTK_INT th_n \"Power of nonlinear part of theta distribution. Only for theta_option=2\" STEERABLE=ALWAYS\n{\n 0:* :: \"\"\n} 9\n\n##########\n# Phi parameters:\nCCTK_REAL x2_beg \"x2 offset: x2 = x2_beg + (k + 0.5)Dx2. Probably should keep it set to zero.\" STEERABLE=ALWAYS\n{\n 0:* :: \"\"\n} 0.0\n########################################\n```\n\n Overwriting interp_sphgrid_MO_ETK/param.ccl\n\n\n\n\n## Step 3.d: $\\text{schedule.ccl}$ \\[Back to [top](#toc)\\]\n$$\\label{scheduleccl}$$\n\nFinally, we will write the file $\\text{schedule.ccl}$; its official documentation is found [here](http://cactuscode.org/documentation/referencemanual/ReferenceManualch8.html#x12-268000C2.4). \n\nThis file declares storage for variables declared in the $\\text{interface.ccl}$ file and specifies when the various parts of the thorn will be run:\n\n\n```python\n%%writefile interp_sphgrid_MO_ETK/schedule.ccl\n\nSTORAGE: interpolation_gf[3]\nSTORAGE: InterpCounterVar\nSTORAGE: interp_pointcoords_and_output_arrays\n \n#############################\nSCHEDULE SphGrid_InitializeInterpCounterToZero AT CCTK_INITIAL\n{\n LANG: C\n OPTIONS: GLOBAL\n} \"Initialize InterpCounter variable to zero\"\n\nSCHEDULE SphGrid_InitializeInterpCounterToZero AT CCTK_POST_RECOVER_VARIABLES\n{\n LANG: C\n OPTIONS: GLOBAL\n} \"Initialize InterpCounter variable to zero\"\n\nSCHEDULE SphGrid_InitializeInterpCounter before SphGrid_InterpGroup AT CCTK_ANALYSIS\n{\n LANG: C\n OPTIONS: GLOBAL\n} \"Initialize InterpCounter variable\"\n##################\n\nSCHEDULE GROUP SphGrid_InterpGroup AT CCTK_ANALYSIS BEFORE CarpetLib_printtimestats BEFORE CarpetLib_printmemstats AFTER Convert_to_HydroBase WHILE interp_sphgrid_MO_ETK::InterpCounter\n{\n} \"Perform all spherical interpolations. This group is only actually scheduled at cctk_iteration==interp_out_iteration.\"\n\nSCHEDULE construct_function_to_interpolate__store_to_interped_gf in SphGrid_InterpGroup before DoSum\n{\n STORAGE: interpolation_gf[3],InterpCounterVar,interp_pointcoords_and_output_arrays\n OPTIONS: GLOBAL,LOOP-LOCAL\n SYNC: interpolation_gf\n LANG: C\n} \"Construct the function to interpolate\"\n\nSCHEDULE Interpolate_to_sph_grid_main_function in SphGrid_InterpGroup after construct_function_to_interpolate__store_to_interped_gf\n{\n OPTIONS: GLOBAL\n LANG: C\n} \"Perform interpolation and output result to file.\"\n#######\nSCHEDULE SphGrid_IncrementInterpCounter in SphGrid_InterpGroup after Interpolate_to_sph_grid_main_function\n{\n LANG: C\n OPTIONS: GLOBAL\n} \"Increment InterpCounter variable, or set to zero once loop is complete.\"\n##################\n```\n\n Overwriting interp_sphgrid_MO_ETK/schedule.ccl\n\n\n\n\n# Step 4: Python Script for Reading the Output File \\[Back to [top](#toc)\\]\n$$\\label{readingoutputfile}$$\n\nHere is a Python code for reading the output file generated by this thorn. It is based on a collection of Python scripts written by Bernard Kelly, available [here](https://bitbucket.org/zach_etienne/nrpy/src/master/mhd_diagnostics/). \n\nAfter generating the output file \"interp_sphgrid_MO_ETK.dat\" using the Einstein Toolkit thorn above, this script will read in all the data. Processing can then be done by straightforward modification of this script. Save the script as Interp_Sph_ReadIn.py, and run it using the command\n\n**python Interp_Sph_ReadIn.py interp_sphgrid_MO_ETK.dat 58 outfile**\n\nCurrently the last parameter \"outfile\" is required but not used.\n\n```python\n\"\"\"\ninterp_sphgrid_MO_ETK.dat File Reader. Compatible with Python 2.7+ and 3.6+ at least.\n\nZachariah B. Etienne\n\nBased on Python scripts written by Bernard Kelly:\nhttps://bitbucket.org/zach_etienne/nrpy/src/master/mhd_diagnostics/\n\nFind the latest version of this reader at the bottom of this Jupyter notebook:\nhttps://github.com/zachetienne/nrpytutorial/blob/master/Tutorial-ETK_thorn-Interpolation_to_Spherical_Grids.ipynb\n\nUsage instructions:\n\nFrom the command-line, run via:\npython Interp_Sph_ReadIn.py interp_sphgrid_MO_ETK.dat [number of gridfunctions (58 or so)] [outfile]\n\nCurrently the last parameter \"outfile\" is required but not actually used.\n\"\"\"\nimport numpy as np\nimport struct\nimport sys\nimport argparse\n\nparser = argparse.ArgumentParser(description='Read file.')\nparser.add_argument(\"datafile\", help=\"main data file\")\nparser.add_argument(\"number_of_gridfunctions\", help=\"number of gridfunctions\")\n\nparser.add_argument(\"outfileroot\", help=\"root of output file names\")\n\nargs = parser.parse_args()\n\ndatafile = args.datafile\noutfileroot = args.outfileroot\nnumber_of_gridfunctions = int(args.number_of_gridfunctions)\n\nprint(\"reading from \"+str(datafile))\n\n\"\"\"\nread_char_array():\nReads a character array of size=\"size\"\nfrom a file (with file handle = \"filehandle\")\nand returns the character array as a proper \nPython string.\n\"\"\"\ndef read_char_array(filehandle,size):\n reached_end_of_string = False\n chartmp = struct.unpack(str(size)+'s', filehandle.read(size))[0]\n\n #https://docs.python.org/3/library/codecs.html#codecs.decode\n char_array_orig = chartmp.decode('utf-8',errors='ignore')\n \n char_array = \"\"\n for i in range(len(char_array_orig)):\n char = char_array_orig[i]\n # C strings end in '\\0', which in Python-ese is '\\x00'.\n # As characters read after the end of the string will\n # generally be gibberish, we no longer append \n # to the output string after '\\0' is reached.\n if sys.version_info[0]==3 and bytes(char.encode('utf-8')) == b'\\x00':\n reached_end_of_string = True\n elif sys.version_info[0]==2 and char == '\\x00':\n reached_end_of_string = True\n\n if reached_end_of_string == False:\n char_array += char\n else:\n pass # Continue until we've read 'size' bytes\n return char_array\n\n\"\"\"\nread_header()\nReads the header from a file.\n\"\"\"\ndef read_header(filehandle):\n # This function makes extensive use of Python's struct.unpack\n # https://docs.python.org/3/library/struct.html\n # First store gridfunction name and interpolation order used:\n # fwrite(gf_name, 100*sizeof(char), 1, file);\n gf_name = read_char_array(filehandle,100)\n # fwrite(order, sizeof(CCTK_INT), 1, file);\n order = struct.unpack('i',filehandle.read(4))[0]\n\n # Then the radial grid parameters:\n # fwrite( & N0, sizeof(CCTK_INT), 1, file);\n N0 = struct.unpack('i',filehandle.read(4))[0]\n # fwrite( & R0, sizeof(CCTK_REAL), 1, file);\n R0 = struct.unpack('d',filehandle.read(8))[0]\n # fwrite( & Rin, sizeof(CCTK_REAL), 1, file);\n Rin = struct.unpack('d',filehandle.read(8))[0]\n # fwrite( & Rout, sizeof(CCTK_REAL), 1, file);\n Rout = struct.unpack('d',filehandle.read(8))[0]\n\n # Then the grid parameters related to the theta coordinate:\n # fwrite( & N1, sizeof(CCTK_INT), 1, file);\n N1 = struct.unpack('i', filehandle.read(4))[0]\n # fwrite( & x1_beg, sizeof(CCTK_REAL), 1, file);\n x1_beg = struct.unpack('d', filehandle.read(8))[0]\n # fwrite( & theta_option, sizeof(CCTK_INT), 1, file);\n theta_option = struct.unpack('i', filehandle.read(4))[0]\n # fwrite( & th_c, sizeof(CCTK_REAL), 1, file);\n th_c = struct.unpack('d', filehandle.read(8))[0]\n # fwrite( & xi, sizeof(CCTK_REAL), 1, file);\n xi = struct.unpack('d', filehandle.read(8))[0]\n # fwrite( & th_n, sizeof(CCTK_INT), 1, file);\n th_n = struct.unpack('i', filehandle.read(4))[0]\n\n # Then the grid parameters related to the phi coordinate:\n # fwrite( & N2, sizeof(CCTK_INT), 1, file);\n N2 = struct.unpack('i', filehandle.read(4))[0]\n # fwrite( & x2_beg, sizeof(CCTK_REAL), 1, file);\n x2_beg = struct.unpack('d', filehandle.read(8))[0]\n\n magic_number_check = 1.130814081305130e-21\n # fwrite( & magic_number, sizeof(CCTK_REAL), 1, file);\n magic_number = struct.unpack('d', filehandle.read(8))[0]\n if magic_number != magic_number_check:\n print(\"Error: Possible file corruption: Magic number mismatch. Found magic number = \"+str(magic_number)+\" . Expected \"+str(magic_number_check))\n exit(1)\n # fwrite( & cctk_iteration, sizeof(CCTK_INT), 1, file);\n cctk_iteration = struct.unpack('i', filehandle.read(4))[0]\n # fwrite( & cctk_time, sizeof(CCTK_REAL), 1, file);\n cctk_time = struct.unpack('d', filehandle.read(8))[0]\n\n return gf_name,order,N0,R0,Rin,Rout,N1,x1_beg,theta_option,th_c,xi,th_n,N2,x2_beg,cctk_iteration,cctk_time\n\n# Now open the file and read all the data\nwith open(datafile,\"rb\") as f:\n # Main loop over all gridfunctions\n for i in range(number_of_gridfunctions):\n # Data are output in chunks, one gridfunction at a time, with metadata\n # for each gridfunction stored at the top of each chunk\n # First read in the metadata:\n gf_name, order, N0, R0, Rin, Rout, N1, x1_beg, theta_option, th_c, xi, th_n, N2, x2_beg, cctk_iteration, cctk_time = read_header(f)\n print(\"\\nReading gridfunction \"+gf_name+\", stored at interp order = \"+str(order))\n data_chunk_size = N0*N1*N2*8 # 8 bytes per double-precision number\n # Next read in the full gridfunction data\n bytechunk = f.read(data_chunk_size)\n # Process the data using NumPy's frombuffer() function:\n # https://docs.scipy.org/doc/numpy/reference/generated/numpy.frombuffer.html\n buffer_res = np.frombuffer(bytechunk)\n # Reshape the data into a 3D NumPy array:\n # https://docs.scipy.org/doc/numpy/reference/generated/numpy.reshape.html\n this_data = buffer_res.reshape(N0,N1,N2)\n \n # Sanity check: Make sure the output in the \"middle\" of the grid looks reasonable.\n ii = int(N0/2)\n jj = int(N1/2)\n kk = int(N2/2)\n with open(\"output-gf\"+str(i)+\".txt\",\"w\") as file:\n for ii in range(N0):\n for kk in range(N2):\n r = ii*1.0/N0\n th = (jj*1.0)*np.pi/N1\n ph = (kk*1.0)*2.0*np.pi/N2\n xx = r*np.sin(th)*np.cos(ph)\n yy = r*np.sin(th)*np.sin(ph)\n zz = r*np.cos(th)\n file.write(str(xx)+\" \"+str(yy)+\" \"+str(zz)+\" \"+str(this_data[kk,jj,ii])+\"\\n\")\n\n```\n\n\n\n# Step 5: Output this module to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n$$\\label{latex_pdf_output}$$\n\nThe following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n[Tutorial-ETK_thorn-Interpolation_to_Spherical_Grids_multi_order.pdf](Tutorial-ETK_thorn-Interpolation_to_Spherical_Grids_multi_order.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means.)\n\n\n```python\n!jupyter nbconvert --to latex --template latex_nrpy_style.tplx Tutorial-ETK_thorn-Interpolation_to_Spherical_Grids_multi_order.ipynb\n!pdflatex -interaction=batchmode Tutorial-ETK_thorn-Interpolation_to_Spherical_Grids_multi_order.tex\n!pdflatex -interaction=batchmode Tutorial-ETK_thorn-Interpolation_to_Spherical_Grids_multi_order.tex\n!pdflatex -interaction=batchmode Tutorial-ETK_thorn-Interpolation_to_Spherical_Grids_multi_order.tex\n!rm -f Tut*.out Tut*.aux Tut*.log\n```\n\n [NbConvertApp] Converting notebook Tutorial-ETK_thorn-Interpolation_to_Spherical_Grids_multi_order.ipynb to latex\n [NbConvertApp] Writing 142601 bytes to Tutorial-ETK_thorn-Interpolation_to_Spherical_Grids_multi_order.tex\n This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex)\n restricted \\write18 enabled.\n entering extended mode\n This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex)\n restricted \\write18 enabled.\n entering extended mode\n This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex)\n restricted \\write18 enabled.\n entering extended mode\n\n", "meta": {"hexsha": "aacb53a668d392dd276803286f6539890e8aea80", "size": 66310, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Tutorial-ETK_thorn-Interpolation_to_Spherical_Grids_multi_order.ipynb", "max_stars_repo_name": "Lituchy/nrpyunittesting", "max_stars_repo_head_hexsha": "c611891a2f6f07afd762121e59d2c5ab857d925b", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Tutorial-ETK_thorn-Interpolation_to_Spherical_Grids_multi_order.ipynb", "max_issues_repo_name": "Lituchy/nrpyunittesting", "max_issues_repo_head_hexsha": "c611891a2f6f07afd762121e59d2c5ab857d925b", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Tutorial-ETK_thorn-Interpolation_to_Spherical_Grids_multi_order.ipynb", "max_forks_repo_name": "Lituchy/nrpyunittesting", "max_forks_repo_head_hexsha": "c611891a2f6f07afd762121e59d2c5ab857d925b", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 41.9418089817, "max_line_length": 444, "alphanum_fraction": 0.5669129845, "converted": true, "num_tokens": 14920, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191460821871, "lm_q2_score": 0.6001883592602049, "lm_q1q2_score": 0.42554503797113935}} {"text": "These lecture notes are provided under Creative Commons Attribution license, CC-BY. \nAll code is subject to the terms of the FSF-approved BSD-3 license.\n(c) 2018 Rachel B. Getman, Sapna Sarupria, Ulf D. Schiller\n\n```python\nfrom IPython.display import YouTubeVideo\n```\n\n# MSE 8900 / CHE 8450 - Multiscale Modeling\n\nRachel B. Getman, Sapna Sarupria, Ulf D. Schiller\n\nClemson University\n\n## Lecture 3.5: Lattice Boltzmann Methods\n\n1. The lattice Boltzmann equation\n2. Fluctuating lattice Boltzmann\n3. Chapman-Enskog expansion\n\n### References\n1. Sauro Succi, The Lattice Boltzmann Equation for Complex States of Flowing Matter, Oxford University Press, 2018. \n2. B. Duenweg and A. J. C. Ladd, Lattice Boltzmann Simualtions of Soft Matter, Adv. Polym. Sci. 221, 89, 2009.\n\n### From Newton to Navier-Stokes\n\n\n\n### Kinetic Theory: Boltzmann Equation\n\n* evolution equation for the (one-)particle distribution function\n$$\n\\left( \\frac{\\partial}{\\partial t} + \\vec{v}\\cdot\\frac{\\partial}{\\partial\\vec{r}} + \\frac{\\vec{F}}{m}\\cdot\\frac{\\partial}{\\partial\\vec{v}} \\right) f(\\vec{r},\\vec{v},t) = \\mathcal{C}[f]\n$$\n\n* Boltzmann collision operator\n$$\n\\mathcal{C}[f] = \\int d\\vec{v}_1 \\int d\\Omega \\, \\sigma(v_\\text{rel},\\Omega) \\, v_\\text{rel} \\left[ f(\\vec{r},\\vec{V}',t) f(\\vec{r},\\vec{v}_1',t) - f(\\vec{r},\\vec{v},t) f(\\vec{r},\\vec{v}_1,t) \\right] \n$$\n\n* Detailed balance\n$$\nf(\\vec{r},\\vec{v}_1',t) f(\\vec{r},\\vec{v}_2',t) = f(\\vec{r},\\vec{v}_1,t) f(\\vec{r},\\vec{v}_2,t)\n$$\n\n* Equilibrium distribution (Maxwell-Boltzmann) $f=f^\\text{eq} + f^\\text{neq}$\n\\begin{equation*}\n\\ln f^\\text{eq} = \\gamma_0 + \\gamma\\,\\vec{v} + \\gamma_4 \\vec{v}^2\n\\end{equation*}\n\n\n\n### Hydrodynamic Fields\n\n* \"average\" of polynomials $\\psi(\\vec{v})$ in components of $\\vec{v}$\n$$\\begin{aligned}\nm_\\psi(\\vec{r},t) = \\int \\psi(\\vec{v}) \\, f(\\vec{r},\\vec{v},t) d\\vec{v}\n\\end{aligned}$$\n\n* density, momentum density, stress tensor\n$$\\begin{aligned}\n\\rho(\\vec{r}, t) &= m \\int f d\\vec{v} \\\\\n\\vec{j}(\\vec{r}, t) &= m \\int \\vec{v} f d\\vec{v} \\\\\n\\Pi(\\vec{r} ,t) &= m \\int \\vec{v} \\otimes \\vec{v} f d\\vec{v}\n\\end{aligned}$$\n\n\n### Separation of scales\n\n* Observation: not all $m_\\psi$ show up in the macroscopic equations of motion\n\n* $\\rho$, $\\vec{j}$ (and $e$) are collisional invariants\n\\begin{equation*}\n\\int d\\vec{r} d\\vec{v} \\frac{\\delta m_{\\rho,\\vec{j},e}(f)}{\\delta f} \\mathcal{C}[f] = 0\n\\end{equation*}\n\n* local equilibrium (Maxwell-Boltzmann) $f^\\text{eq}(\\rho,\\vec{j},e)$\n\n* Hydrodynamics describes variation of $\\rho$ and $\\vec{j}$ (and $e$) through \\emph{transport} (over a macroscopic distance $\\sim L$)\n\n* all other variables relax rapidly through \\emph{collisions} ($\\sim \\lambda$ mean free path)\n\n* scale separation: $\\epsilon \\sim Kn = \\frac{\\lambda}{L} \\ll 1$ \\qquad Knudsen number $Kn=\\frac{\\lambda}{L}$\n\n#### How can we exploit the scale separation?\n\n* we are only interested in the dynamics of the slow variables up to a certain order\n* the dynamics of the fast variables beyond that order is unimportant\n* any set of fast variables that leaves the slow dynamics unchanged will do\n* the number of degrees of freedom can be greatly reduced!\n* Caveat: imperfect scale separation $\\rightarrow$ fast variables can couple to slow dynamics\n\n\n\n### Discretization a la Grad\n\n* Truncated Hermite expansion\n$$f^N(\\vec{r},\\vec{v},t)=\\omega(\\vec{v}) \\sum_{n=0}^N \\frac{1}{n!}\\mathsf{a}^{(n)}(\\vec{r},t) \\mathcal{H}^{(n)}(\\vec{v})$$\n\n($a^{(0)}=\\rho$, $a^{(1)}=\\vec{j}$, $a^{(2)} = \\Pi - \\rho\\mathsf{I}$, $\\dots$)\n\n* Gauss-Hermite quadrature\n$$\\begin{aligned}\n\\mathsf{a}^{(n)} &= \\int \\mathcal{H}^{(n)}(\\vec{v}) f^N(\\vec{r},\\vec{v},t) \\, d\\vec{v} \n= \\sum_{i} w_i \\, \\frac{\\mathcal{H}^{(n)}(\\vec{c}_i) f^N(\\vec{r},\\vec{c}_i,t)}{\\omega(\\vec{c}_i)}\\\\\n&= \\sum \\mathcal{H}^{(n)}(\\vec{c}_i) f_i(\\vec{r},t)\n\\end{aligned}$$\n\n$$\\int \\omega(\\vec{v}) \\, p(\\vec{r},\\vec{v},t) \\, d\\vec{v} = \\sum_{i} w_i \\, p(\\vec{r},\\vec{c}_i,t)$$\n\n* Discrete velocity model (DVM)\n\n$$\\left( \\frac{\\partial}{\\partial t} + \\vec{c}_i \\cdot \\frac{\\partial}{\\partial\\vec{r}} \\right) f_i = - \\sum_j \\Omega_{ij} (f_j - f_j^\\text{eq})$$\n\n### Quadratures\n\n| Quadrature | LB model | $q$ | $b_q$ | $w_q$ | $\\vec{c}_q$ |\n|---|---|---|---|---|---|\n| $E_{1,5}^{3}$ | D1Q3 | 0 | 1 | $\\frac{2}{3}$ | 0 |\n| | | 1 | 2 | $\\frac{1}{6}$ | $\\pm\\sqrt{3}$ |\n| $E_{2,5}^{9}$ | D2Q9 | 0 | 1 | $\\frac{4}{9}$ | $(0,0)$ |\n| | | 1 | 4 | $\\frac{1}{9}$ | $(\\pm\\sqrt{3},0)$, $(0,\\pm\\sqrt{3})$ |\n| | | 2 | 4 | $\\frac{1}{36}$ | $(\\pm\\sqrt{3},\\pm\\sqrt{3})$ |\n| $E_{3,5}^{15}$ | D3Q15 | 0 | 1 | $\\frac{2}{9}$ | $(0,0,0)$ |\n| | | 1 | 6 | $\\frac{1}{9}$ | $(\\pm\\sqrt{3},0,0)$, $(0,\\pm\\sqrt{3},0)$, $(0,0,\\sqrt{3})$ |\n| | | 3 | 8 | $\\frac{1}{72}$ | $(\\pm\\sqrt{3},\\pm\\sqrt{3},\\pm\\sqrt{3})$ |\n| $E_{3,5}^{19}$ | D3Q19 | 0 | 1 | $\\frac{1}{3}$ | $(0,0,0)$ |\n| | | 1 | 6 | $\\frac{1}{18}$ | $(\\pm\\sqrt{3},0,0)$, $(0,\\pm\\sqrt{3},0)$, $(0,0,\\sqrt{3})$ |\n| | | 2 | 12 | $\\frac{1}{36}$ | $(\\pm\\sqrt{3},\\pm\\sqrt{3},0)$, $(\\pm\\sqrt{3},0,\\pm\\sqrt{3})$, $(0,\\pm\\sqrt{3},\\pm\\sqrt{3})$ |\n| $E_{3,5}^{27}$ | D3Q27 | 0 | 1 | $\\frac{8}{27}$ | $(0,0,0)$ |\n| | | 1 | 6 | $\\frac{2}{27}$ | $(\\pm\\sqrt{3},0,0)$, $(0,\\pm\\sqrt{3},0)$, $(0,0,\\sqrt{3})$ |\n| | | 2 | 12 | $\\frac{1}{54}$ | $(\\pm\\sqrt{3},\\pm\\sqrt{3},0)$, $(\\pm\\sqrt{3},0,\\pm\\sqrt{3})$, $(0,\\pm\\sqrt{3},\\pm\\sqrt{3})$ |\n| | | 3 | 8 | $\\frac{1}{216}$ | $(\\pm\\sqrt{3},\\pm\\sqrt{3},\\pm\\sqrt{3})$\n\nNotation $E^n_{D,d}$: $D$ dimensions, $d$ degree, $n$ abscissae\n$q$: neighbor shell, $b_q$: number of neighbors, $w_q$ weights, $\\vec{c}_q$ velocities\n\n\\begin{equation*}\n\\mathsf{T}^{(n)} = \\sum_i w_i \\vec{c}_i \\dots \\vec{c}_i = \n\\begin{cases}\n0 & n \\text{ odd} \\\\\n{\\delta}^{(n)} & n \\text{ even}\n\\end{cases}, \\quad \\forall n \\leq d.\n\\end{equation*}\n\n### Models with polynomial equilibrium\n\n* Ansatz: expansion in the velocities $\\vec{u}$ (Euler stress is quadratic in $\\vec{u}$)\n\\begin{equation*}\nf_i^\\text{eq}(\\rho,\\vec{u}) = w_i \\rho \\left[ 1 + A \\vec{u}\\cdot\\vec{c}_i + B (\\vec{u}\\cdot\\vec{c}_i)^2 + C\\vec{u}^2 \\right]\n\\end{equation*}\n\n* cubic symmetry of lattice tensors $\\mathsf{T}^{(n)}$\n \\begin{alignat*}{2}\n &\\sum_i w_i = 1 &\n &\\sum_i w_i c_{i \\alpha} = 0 \\\\\n &\\sum_i w_i c_{i \\alpha} c_{i \\beta}\n = \\sigma_2 \\, \\delta_{\\alpha \\beta} &\n &\\sum_i w_i c_{i \\alpha} c_{i \\beta} c_{i \\gamma} = 0 \\\\\n &\\sum_i w_i c_{i \\alpha} c_{i \\beta} c_{i \\gamma} c_{i \\delta}\n = \\kappa_4 \\, \\delta_{\\alpha \\beta \\gamma \\delta} &&\n + \\sigma_4 \\left(\n \\delta_{\\alpha \\beta} \\delta_{\\gamma \\delta} \n + \\delta_{\\alpha \\gamma} \\delta_{\\beta \\delta}\n + \\delta_{\\alpha \\delta} \\delta_{\\beta \\gamma}\n \\right)\n\\end{alignat*}\n\n* at least three shells required to satisfy the conditions\n\\begin{alignat*}{4}\n\\sum_i w_i &= 1 & \\qquad\n\\kappa_4 &= 0 & \\qquad\n\\sigma_4 &= \\sigma_2^2 & \\qquad\nc_s^2 &= \\sigma_2 \n\\end{alignat*}\n\n### The D3Q19 model\n\n\n\n* Equilibrium distribution:\n\\begin{equation*}\nf_i^\\text{eq}(\\rho,\\vec{u}) = w_i \\rho \\left[ 1 + \\frac{\\vec{u}\\cdot\\vec{c}_i}{c_s^2} + \\frac{\\vec{u}\\vec{u} \\colon (\\vec{c}_i\\vec{c}_i - c_s^2 \\mathsf{I})}{2c_s^4} \\right]\n\\end{equation*}\n\n* Moments:\n\\begin{align*}\n\\sum_i f_i^\\text{eq} &= \\rho \\\\\n\\sum_i f_i^\\text{eq}\\vec{c}_i &= \\rho\\vec{u} \\\\\n\\sum_i f_i^\\text{eq}\\vec{c}_i\\vec{c}_i &= \\rho c_s^2 \\mathsf{I} + \\rho\\vec{u}\\vec{u}\n\\end{align*}\n\n* Weight coefficients:\n\\begin{equation*}\nw_i =1/3 \\quad \\text{for } |\\vec{c}_i|=0, \\qquad\nw_i = 1/18 \\quad \\text{for } |\\vec{c}_i|=1, \\qquad\nw_i = 1/36 \\quad \\text{for } |\\vec{c}_i|=\\sqrt{2}\n\\end{equation*}\n* Speed of sound: $c_s = \\frac{1}{\\sqrt3}\\left(\\frac{a}{h}\\right)$\n\n### The lattice Boltzmann equation\n\n* recall the Boltzmann equation with linearised collisions\n\n\\begin{equation*}\n\\left( \\frac{\\partial}{\\partial t} + \\vec{v} \\cdot \\frac{\\partial}{\\partial\\vec{r}} \\right) f(\\vec{r},\\vec{v},t) = - \\Omega \\left[ f(\\vec{r},\\vec{v},t) - f^{\\text{eq}}(\\vec{v}) \\right]\n\\end{equation*}\n\n* systematic discretization $\\rightarrow$ lattice Boltzmann equation\n\n\\begin{equation*}\n\\bar f_i(\\vec{r}+h\\vec{c}_i,t+h) = \\bar f_i^*(\\vec{r},t) = \\bar f_i(\\vec{r},t) + \\sum_j \\Lambda_{ij} \\left[ \\bar f_j(\\vec{r},t) - f_j^{\\text{eq}}(\\rho,\\vec{u}) \\right]\n\\end{equation*}\n\n* But this is only first-order in time!?\n\n### Splitting of the discrete Boltzmann equation\n\n* Discrete velocity model (now with force term)\n\n\\begin{equation*}\n%\\frac{df_i}{dt} + {\\lambda} f_i = {\\lambda} f_i^\\text{eq}.\n\\frac{\\partial}{\\partial t} f_i = - \\vec{c}_i \\cdot \\frac{\\partial}{\\partial\\vec{r}} f_i - \\sum_j \\Omega_{ij} (f_j - f_j^\\text{eq}) + G_i = \\left( \\mathcal{S} + \\mathcal{C} + \\mathcal{F} \\right) f_i\n\\end{equation*}\n\n* Formal solution\n\n$$f_i(t+h) = \\exp\\left[ {h(\\mathcal{S} + \\mathcal{C} + \\mathcal{F})} \\right] \\, f_i(t)$$\n\n* Baker-Campbell-Hausdorff (for sufficiently differentiable operators)\n\n\\begin{equation*}\n\\exp \\left[ h(\\mathcal{S} + \\mathcal{C} + \\mathcal{F}) \\right] = e^{ \\frac{h}{2} \\left( \\mathcal{C} + \\mathcal{F} \\right) } e^{ h \\mathcal{S} } e^{ \\frac{h}{2} \\left( \\mathcal{C} + \\mathcal{F} \\right) } + \\mathcal{O}(h^3) \\approx \\mathsf{C}^\\frac12 \\mathsf{S} \\mathsf{C}^\\frac12\n\\end{equation*}\n\n* Second-order in time lattice Boltzmannn update\n\n\\begin{equation*}\n\\mathbf{f}(t+h) = \\mathsf{C}^\\frac12 \\mathsf{S} \\mathsf{C}^\\frac12 \\, \\mathbf{f}(t)\n\\end{equation*}\n\nP. Dellar, Comp. Math. App. 65, 129-141 (2013) \nUDS, Comp. Phys. Comm. 185, 2586-2597 (2014)\n\n### Splitting of the discrete Boltzmann equation\n\n* Streaming (Integration along characteristic)\n\n\\begin{equation*}\nf_i(\\vec{r}+\\vec{c}_i,t+h) = f_i(\\vec{r},t)\n\\end{equation*}\n\n* Collisions (Crank-Nicolson rule)\n\n\\begin{equation*}\n \\mathbf{f}^\\text{nq}(t+h) = \\left( \\mathsf{I} + \\frac{h}{2} \\mathsf{\\Omega} \\right)^{-1} \n \\left( \\mathsf{I} - \\frac{h}{2} \\mathsf{\\Omega} \\right) \\mathbf{f}^\\text{nq}(t) + \\mathcal{O}((h/\\tau)^3)\n\\end{equation*}\n\n* The error $\\mathcal{O}((h/\\tau)^3)$ can cause non-linear instabilities!\n\n* Why not exact solution?\n\n\\begin{equation*}\n \\mathbf{f}^\\text{nq}(t+h) = \\exp [ - h \\mathsf{\\Omega} ] \\, \\mathbf{f}^\\text{nq}(t)\n\\end{equation*}\n\n* in practice less accurate!\n\nP. Dellar, Comp. Math. App. 65, 129-141 (2013) \nBrownlee et al., Phys. Rev. E 75, 036711 (2007)\n\n### The classical LB algorithm\n\n1. streaming step: move $f_i^*(\\vec{r},t)$ along $\\vec{c}_i$ to the next lattice site, increment $t$ by $h$\n\n$$\nf_i(\\vec{r}+h\\vec{c}_i,t+h) = f_i^*(\\vec{r},t)\n$$\n\n2. collision step: apply $\\Lambda_{ij}$ and compute the post-collisional $f_i^*(\\vec{r},t)$ on every lattice site\n\n$$\nf_i^*(\\vec{r},t) = f(\\vec{r},t) + \\sum_j \\Lambda_{ij} \\left[ f_j(\\vec{r},t) - f_j^\\text{eq}(\\rho,\\vec{u}) \\right]\n$$\n\n### Macroscopic moments in lattice Boltzmann\n\n* macroscopic fields are velocity moments of the populations\n\n\\begin{align*}\n\\rho &= \\sum_i f_i & \\rho\\vec{u} &= \\sum_i f_i\\vec{c}_i+\\frac{h}{2}\\mathbf{f} & \\mathsf{\\Pi} &= \\sum_i \\frac{f_i+f_i^*}{2} \\vec{c}_i\\otimes\\vec{c}_i\n\\end{align*}\n\n* construct orthogonal basis $e_{ki}$ for moments (recall $\\psi(\\vec{v})$ and $m_\\psi$)\n\\begin{equation*}\nm_k = \\sum_i e_{ki} f_i\n\\end{equation*}\n\n$0 \\leq k \\leq 9$: hydrodynamic modes (slow), $k\\geq10$: kinetic modes (fast)\n\n* collision matrix is diagonal in mode space\n\\begin{equation*}\n\\Lambda (\\mathbf{f} - \\mathbf{f}^\\text{eq}) = \\mathsf{M}^{-1} {\\left( \\mathsf{M}\\Lambda\\mathsf{M}^{-1} \\right)}{ \\mathsf{M} (\\mathbf{f} - \\mathbf{f}^\\text{eq})} = \\mathsf{M}^{-1} {\\hat\\Lambda}{ (\\mathbf{m}-\\mathbf{m}^\\text{eq}) }\n\\end{equation*}\n\n* MRT model\n\\begin{align*}\n(m_k - m_k^\\text{eq})^* = \\gamma_k (m_k - m_k^\\text{eq})\n\\end{align*}\n\n### Choice of the moment basis\n\n\\begin{align*}\n& m_0 = \\rho = \\sum_i f_i && \\text{mass}\\\\\n& m_1 = j_x = \\sum_i f_i c_{ix} && \\text{momentum $x$}\\\\\n& m_2 = j_y = \\sum_i f_i c_{iy} && \\text{momentum $y$}\\\\\n& m_3 = j_z = \\sum_i f_i c_{iz} && \\text{momentum $z$}\\\\\n& m_4 = \\text{tr}(\\mathsf{\\Pi}) && \\text{bulk stress}\\\\\n& m_5, \\ldots, m_9 \\simeq \\overline{\\mathsf{\\Pi}} && \\text{shear stresses}\\\\\n& m_{10}, \\ldots, m_{18} && \\text{\"kinetic modes\", \"ghost modes\"}\\\\\n\\end{align*}\n\n### Multiple relaxation time model (MRT)\n\n\\begin{align*}\n&\\gamma_0 = \\gamma_1 = \\gamma_2 = \\gamma_3 = 0 \\quad \\text{mass and momentum conservation} \\\\\n&\\gamma_4 = \\gamma_b \\quad \\text{bulk stress} \\\\\n&\\gamma_5 = \\ldots = \\gamma_9 = \\gamma_s \\quad \\text{shear stress} \\\\\n&\\gamma_{10} = \\ldots = \\gamma_{18} = 0 \\quad \\text{simplest choice, careful with boundaries!}\n\\end{align*}\n\n* Remark: we could also relax the populations directly:\n\\begin{equation*}\nf_i^{\\text{nq}*} = \\sum_j \\Lambda_{ij} f_j^\\text{nq}\n\\end{equation*}\n* simplest choice $\\Lambda_{ij}=\\lambda^{-1}\\delta_{ij}$ $\\rightarrow$ lattice BGK \n* not a particularly good choice (less stable, less accurate)\n\n### Lattice moments vs. hydrodynamic moments\n\n* Concatenation of lattice Boltzmann updates\n\n\\begin{equation*}\n\\mathbf{f}(t+N h) = \\mathsf{C}^\\frac12 \\left( \\mathsf{S} \\mathsf{C} \\right)^N \\mathsf{C}^{-\\frac{1}{2}} \\, \\mathbf{f}(t)\n\\end{equation*}\n\n* density\n$$\n\\rho = \\sum_i ( \\mathsf{C}^\\frac12 \\mathbf{f} )_i = \\sum_i f_i\n$$\n\n* momentum density\n$$\n\\vec{j} = \\sum_i ( \\mathsf{C}^\\frac12 \\mathbf{f} )_i \\vec{c}_i \n= \\sum_i f_i \\vec{c}_i + \\frac{h}{2} \\mathbf{f}\n$$\n\n* viscous stress\n\n\\begin{equation*}\n\\begin{split}\n\\sigma &= - \\sum_i ( \\mathsf{C}^\\frac12 \\mathbf{f}^\\text{nq} )_i \\vec{c}_i\\vec{c}_i\n%= \\mathsf{C}^\\frac12 \\Pi^\\text{nq} \\\\\n%&\\approx \\frac{1}{2} \\left( \\1 + \\mathsf{C} \\right) \\Pi^\\text{nq} \n= - \\left( \\mathsf{I}+ \\frac{h}{2} \\Omega \\right)^{-1} \\sum_i f^\\text{nq}_i \\vec{c}_i\\vec{c}_i %\\Pi^\\text{nq}\n\\end{split}\n\\end{equation*}\n\n* {Note: if $\\mathsf{C}^\\frac12$ includes fluctuations, it may not be invertible!}\n\n### Viscous stress relaxation\n\n$$ {\\Pi} = \\overline{{\\Pi}} + \\frac{1}{3} \\text{tr}({\\Pi}) \\mathsf{I}$$\n\n* recall: collision step applies linear relaxation to the moments\n\n\\begin{eqnarray*}\n\\overline{{\\Pi}}^{*\\text{neq}} &=& \\gamma_s \\overline{{\\Pi}}^{\\text{neq}} \\\\\n\\text{tr}({\\Pi}^{*\\text{neq}}) &=& \\gamma_b \\text{tr}({\\Pi}^{\\text{neq}})\n\\end{eqnarray*}\n\n* Chapman-Enskog expansion\n\n\\begin{equation*}\n-\\frac{1}{2} \\left ( \\mathsf{\\Pi}^{*\\text{neq}} + \\mathsf{\\Pi}^\\text{neq} \\right) = \\sigma = \\eta \\left( \\nabla\\vec{u} + (\\nabla\\vec{u})^t \\right) + \\left( \\zeta - \\frac{2}{3} \\eta \\right) (\\nabla\\cdot\\vec{u}) \\mathsf{I}\n\\end{equation*}\n\n* shear and bulk viscosities are determined by the relaxation parameters\n\n\\begin{align*}\n\\eta &= \\frac{\\rho c_s^2 h}{2} \\frac{1+\\gamma_s}{1-\\gamma_s}& \\eta_b &= \\frac{\\rho c_s^2 h}{3} \\frac{1+\\gamma_b}{1-\\gamma_b}\n\\end{align*}\n\n### Viscosity of the lattice Boltzmann fluid\n\n* incompressible Navier-Stokes equation is recovered\n\n* $-1 \\leq \\gamma_s \\leq 1$ $\\Leftrightarrow$ positive viscosities\n\n* any viscosity value is accessible\n\n\n\n### The force term in LB\n\n* force term $G_i$ in the discrete velocity model has the same\n first moments as the acceleration term $-\\frac{1}{\\rho} \\vec{F} \\cdot\n \\nabla_\\vec{c} f$\n\n\\begin{align*}\\label{eq:force-moments}\n\\sum_i G_i &= 0, & \\sum_i \\vec{c}_i G_i &= \\vec{F}, & \\sum_i \\vec{c}_i \\vec{c}_i G_i &= \\vec{F} \\vec{u} + \\vec{u} \\vec{F},\n\\end{align*}\n\n* second-order accuracy requires to transform the force term\n\n\\begin{equation*}\n\\bar{\\vec{G}} = \\left( \\mathsf{I} + \\frac{h}{2} \\Omega \\right)^{-1} h \\Omega \\, \\vec{G} = \\left( \\mathsf{I} - \\frac12 \\Lambda \\right) \\vec{G}\n\\end{equation*}\n\n* leads to a self-consistency problem when forces are velocity dependent\n\n* collide-stream-collide scheme\n\n### Units in LB\n\n* grid spacing $a$, time step $h$, particle mass $m_p$\n\n* these merely control the _accuracy_ and _stability_ of LB!\n\n* physically relevant: mass density $\\rho$, viscosity $\\eta$, temperature $k_BT$\n\n* recall:\n\n\\begin{align*}\nc_s^2 &= \\frac{1}{3} \\frac{a^2}{h^2} = \\hat{c}_s^2 \\frac{{a^2}}{{h^2}}\\\\\n\\eta &= \\frac{\\rho c_s^2 h}{2} \\frac{1+\\gamma_s}{1-\\gamma_s} = \\hat{\\rho} \\hat{c}_s^2 \\hat{\\eta} \\frac{m_p}{{a}{h}} \\\\\nk_BT &= m_p c_s^2 = m_p \\hat{c}_s^2 \\frac{{a^2}}{{h^2}}\n\\end{align*}\n\n* always make sure you are simulating the right _physics_!\n\n* for comparison with experiments: match dimensionless numbers!\n($Ma$, $Re$, $Pe$, $Sc$, $Kn$, $Pr$, $Wi$, $De$, ...)\n\n### Dimensionless numbers and LB parameters\n\n* connection between dimensionless numbers and simulation parameters\n\n\\begin{equation*}\n\\frac{1+\\gamma}{1-\\gamma} = \\frac{2\\nu}{c_s^2 h} = 2 \\sqrt{3} \\hat{L} \\frac{Ma}{Re} > 0\n\\end{equation*}\n\n* Knudsen number\n\n\\begin{equation*}\nKn\\propto\\frac{Ma}{Re} = \\frac{\\eta}{\\rho c_s L} = \\frac{c_s h}{2 L} \\frac{1 + \\gamma}{1 - \\gamma} = \\frac{1}{\\sqrt{3} \\hat L} \\left( \\frac{1}{\\lambda} - \\frac{1}{2} \\right)\n\\end{equation*}\n\n* diffusive scaling: $a \\sim \\epsilon L$ \\quad $h \\sim \\epsilon^2 T$\n\n\\begin{align*}\n\\frac{{a^2}}{{h}} &= \\text{const.} &\n\\frac{{a}}{{h}} \\sim \\frac{1}{\\epsilon}\\frac{L}{T} \\underset{\\epsilon\\rightarrow 0}{\\rightarrow} \\infty\n\\end{align*}\n\n* $Ma \\rightarrow 0$ at fixed Reynolds number\n\n* Mach number annealing: $h \\sim \\epsilon T$\n\n* run simulation with higher Mach number until transients have decayed\n\n### Thermal fluctuations\n\n* so far the LB model is athermal and entirely deterministic\n\n* for Brownian motion, we need fluctuations!\n\n\n\n\n#### Do we need fluctuations?\n\n* Ideal gas, temp. $T$, particle mass $m_p$, sound speed $c_s$:\n\n$$ k_B T = m_p c_s^2$$\n\n* $c_s \\sim a / h$ ($a$ lattice spacing, $h$ time step)\n* $c_s$ as small as possible\n\n* Example (water)\n\n|||\n|---|---|\n| sound speed realistic: | $1.5 \\times 10^3 m / s$ |\n| sound speed artificial: | $c_s = 10^2 m / s$ |\n| temperature | $T \\approx 300 K$, $k_B T = 4 \\times 10^{-21}$ |\n| particle mass: | $m_p = 4 \\times 10^{-25} kg$ |\n\n| | macroscopic scale | molecular scale |\n|---|---|---|\n| lattice spacing | $a = 1 mm$ | $a = 1 nm$ |\n| time step | $h = 10^{-5} s$ | $h = 10^{-11} s$ |\n| mass of a site | $m_a = 10^{-6} kg$ | $m_a = 10^{-24} kg$ |\n| \"Boltzmann number\" | $Bo = (m_p/m_a)^{1/2}$ | $Bo = (m_p/m_a)^{1/2}$ |\n| | $ = 6 \\times 10^{-10}$ | $ = 0.6$ |\n\n### Low Mach number physics\n\n* LB requires $u \\ll c_i$ hence $u \\ll c_s$\n* low Mach number $Ma=u/c_s \\ll 1$\n* compressibility does not matter\n* equation of state does not matter $\\rightarrow$ choose ideal gas!\n\n$m_p$ particle mass\n\n\\begin{equation*}\n p = \\frac{\\rho}{m_p} k_B T\n\\end{equation*}\n\n\\begin{equation*}\n c_s^2 = \\frac{\\partial p}{\\partial \\rho} = \\frac{1}{m_p} k_B T\n\\end{equation*}\n\n\\begin{equation*}\n p = \\rho c_s^2\n\\end{equation*}\n\n\\begin{equation*}\n k_B T = m_p c_s^2\n\\end{equation*}\n\n\n### Generalized lattice gas model (GLG)\n\n* consider integer population numbers ($m_p$ mass of an LB particle)\n\n\\begin{equation*}\n\\nu_i = \\frac{f_i}{\\mu} \\qquad \\mu=\\frac{m_p}{a^3} \\qquad \\mu \\nu_i = {w_i}\\rho\n\\end{equation*}\n\n* each lattice site in contact with a heat bath\n\n* Poisson + constraints\n\n\\begin{equation*}\n P \\left( \\left\\{ \\nu_i \\right\\} \\right)\n \\propto\n \\prod_i\n \\frac{ {\\bar \\nu}_i^{\\nu_i} }{\\nu_i !} e^{-{\\bar \\nu}_i}\n \\delta \\left( \\mu \\sum_i \\nu_i - \\rho \\right)\n \\delta \\left( \\mu \\sum_i \\nu_i \\vec c_i - \\vec j \\right)\n\\end{equation*}\n\n[B. Duenweg, UDS, A. J. C. Ladd, Phys. Rev. E 76, 036704 (2007)]\n\n### Entropy\n\n* associated entropy\n\\begin{equation*}\n P % \\left( \\left\\{ \\nu_i \\right\\} \\right) \n \\propto\n \\exp \\left[S \\left( \\left\\{ \\nu_i \\right\\} \\right)\\right]\n \\delta \\left( \\mu \\sum_i \\nu_i - \\rho \\right)\n \\delta \\left( \\mu \\sum_i \\nu_i \\vec c_i - \\vec j \\right)\n\\end{equation*}\n\n* Stirling: $\\nu_i! = \\exp \\left( \\nu_i \\ln \\nu_i - \\nu_i \\right)$\n\\begin{align*}\n S \\left( \\left\\{ \\nu_i \\right\\} \\right) \n &= - \\sum_i \\left( \\nu_i \\ln \\nu_i -\\nu_i - \\nu_i \\ln {\\bar \\nu}_i\n + {\\bar \\nu}_i \\right) \\\\\n &= \\frac{1}{\\mu} \\sum_i \\rho w_i \\left( \\frac{f_i}{\\rho w_i}-\\frac{f_i}{\\rho w_i} \\ln\\frac{f_i}{\\rho w_i} - 1 \\right)\n\\end{align*}\n\n* $\\mu$ controls the mean square fluctuations (degree of coarse-graining)\n\n### Maximum entropy principle\n\n* maximize entropy $S$ subject to constraints for mass and momentum conservation\n\n\\begin{equation*}\n\\frac{\\partial S}{\\partial \\nu_i} + \\chi\n+ \\lambda \\cdot \\vec c_i = 0 \\qquad\n\\mu \\sum_i \\nu_i - \\rho = 0 \\qquad\n\\mu \\sum_i \\nu_i \\vec c_i - \\vec{j} = 0\n\\end{equation*}\n\n* formal solution\n\n\\begin{equation*}\nf_i^{\\text{eq}} = w_i \\, \\rho \\exp \\left( \\chi + \\lambda \\cdot \\vec{c}_i \\right)\n\\end{equation*}\n\n* expansion up to $\\mathcal{O}(u^2)$\n\n\\begin{equation*}\nf_i^\\text{eq}(\\rho,\\vec{u}) = w_i \\rho \\left[ 1 + \\frac{\\vec{u}\\cdot\\vec{c}_i}{c_s^2} + \\frac{\\vec{u}\\vec{u} \\colon (\\vec{c}_i\\vec{c}_i - c_s^2 \\mathsf{I})}{2c_s^4} \\right]\n\\end{equation*}\n\n\n\n### Fluctuations around equilibrium\n\n* Gauss distribution for non-equilibrium part\n\\begin{equation*}\n P % \\left( \\left\\{ f_i^{neq} \\right\\} \\right)\n \\propto\n \\exp \\left( - \\sum_i \\frac{ \\left( f_i^{\\text{nq}} \\right)^2 }\n { 2 \\mu \\rho w_i } \\right)\n \\delta \\left( \\sum_i f_i^{\\text{nq}} \\right)\n \\delta \\left( \\sum_i \\vec c_i \\, f_i^{\\text{nq}} \\right)\n\\end{equation*}\n\n* transform to the modes ($b_k = \\sum_i w_i e_{ki}^2$, Basis $e_{ki}$)\n\\begin{equation*}\n P\\left( \\left\\{ {m}_k^{\\text{nq}} \\right\\} \\right)\n \\propto \\exp \\left( - \\sum_{k \\ge 4}\n \\frac{\\left( {m}_k^{\\text{nq}} \\right)^2}{2 \\mu \\rho b_k} \\right)\n\\end{equation*}\n\n* more convenient: ortho-_normal_ modes\n\\begin{equation*}\n \\hat{m}_k = \\sum_i \\hat{e}_{ki} \n \\frac{f_i}{\\sqrt{w_i \\mu \\rho}}\n\\end{equation*}\n\n### Implementation of the fluctuations\n\n* introduce stochastic term into the collision step\n\\begin{equation*}\n {m}_k^{* \\text{nq}} = \\gamma_k {m}_k^{\\text{nq}} + \\varphi_k r_k\n\\end{equation*}\n$r_k$ random number from normal distribution\n\n* ensure detailed balance (like in Monte-Carlo)\n\\begin{equation*}\n\\frac{p(m \\rightarrow m^*)}{p(m^* \\rightarrow m)} = \\frac{\\exp(-m^{*2}/2)}{\\exp(-m^2/2)} \\qquad \\Rightarrow \\qquad {\\varphi_k = \\sqrt{\\mu\\rho b_k \\left( 1 - \\gamma_k^2 \\right)}}\n\\end{equation*}\n\n* $\\varphi_k \\neq 0$ for \\emph{all} non-conserved modes\n* all modes have to be thermalized\n\n[A. J. C. Ladd, J. Fluid Mech. 271, 285-309 (1994)] \n[Adhikari et al., Europhys. Lett. 71, 473-479 (2005)] \n[B. Duenweg, UDS, A. J. C. Ladd, Phys. Rev. E 76, 036704 (2007)]\n\n### Eliminating fast variables: Chapman-Enskog\n\n* introduce length and time scales\n\\begin{alignat*}{3}\n&\\text{coarse-grained length:} &\\quad& \\vec{r}_1 = \\epsilon \\vec{r} &\\quad\\rightarrow\\quad& \\frac{\\partial}{\\partial\\vec{r}}=\\epsilon\\frac{\\partial}{\\partial\\vec{r}_1} \\\\\n&\\text{convective time scale:} &\\quad& t_1 = \\epsilon t \\\\\n&\\text{diffusive time scale:} &\\quad& t_2 = \\epsilon^2 t &\\quad\\rightarrow\\quad& \\frac{\\partial}{\\partial t} = \\epsilon \\frac{\\partial}{\\partial t_1} + \\epsilon^2 \\frac{\\partial}{\\partial t_2}\n\\end{alignat*}\n\n* use $\\epsilon$ as a perturbation parameter and expand $f$\n\\begin{eqnarray*}\nf &=& f^{(0)} + \\epsilon f^{(1)} + \\epsilon^2 f^{(2)} + \\dots \\\\\n\\mathcal{C}[f] &=& \\mathcal{C}[f^{(0)}] + \\epsilon \\int d\\vec{r} d\\vec{v} \\frac{\\delta C[f]}{\\delta f} f^{(1)}(\\vec{r},\\vec{v}) + \\dots\n\\end{eqnarray*}\n\n* solve for each order in $\\epsilon$\n\n### Chapman-Enskog expansion\n\n* $\\epsilon^0$: yields the collisional invariants, and the equilibrium distribution $f^{(0)}=f^\\text{eq}$\n\n* $\\epsilon^1$: yields the Euler equations, and the first order correction $f^{(1)}$\n\n\\begin{equation}\n\\tag{*}\n\\left( \\frac{\\partial}{\\partial t_1} + \\vec{c}\\frac{\\partial}{\\partial\\vec{r}_1} \\right) f^\\text{(0)} = \\int d\\vec{r}' d\\vec{c}' \\frac{\\delta C[f^{(0)}]}{\\delta f^{(0)}} f^{(1)}(\\vec{r}',\\vec{c}')\n\\end{equation}\n\n* $\\epsilon^2$: adds viscous terms to the Euler equation\n\n* Navier-Stokes!\n\n* the \"only\" difficulty is: no explicit solution of (*) is known...\\\\ (except for Maxwell molecules)\n\n### Chapman-Enskog expansion for LB\n\n* original LBE\n\\begin{equation*}\nf_i(\\vec{r}+h\\vec{c}_i,t+h) - f_i(\\vec{r},t) = \\delta_i\n\\end{equation*}\n\n* recall: coarse-grained length $\\vec{r}_1$, convective time scale $t_1$, diffusive time scale~$t_2$\n\n\\begin{equation*}\nf_i(\\vec{r}_1+\\epsilon h\\vec{c}_i,t_1+\\epsilon h,t_2+\\epsilon^2h) - f_i(\\vec{r}_1,t_1,t_2) = \\delta_i\n\\end{equation*}\n\n* Taylor expansion:\n\n\\begin{equation*}\n\\epsilon h \\left( \\frac{\\partial}{\\partial t_1} + \\vec{c}_i\\cdot\\frac{\\partial}{\\partial\\vec{r}_1} \\right) f_i + \\epsilon^2h \\left[ \\frac{\\partial}{\\partial t_2} + \\frac{h}{2} \\left( \\frac{\\partial}{\\partial t_1} + \\vec{c}_i\\cdot\\frac{\\partial}{\\partial\\vec{r}_1}\\right) \\right] f_i = \\delta_i\n\\end{equation*}\n\n### Chapman-Enskog expansion for LB\n\n* expand $f_i$ and $\\Delta_i$ in powers of $\\epsilon$\n\\begin{eqnarray*}\nf_i &=& f_i^{(0)} + \\epsilon f_i^{(1)} + \\epsilon^2 f_i^{(2)} + \\dots \\\\\n\\Delta_i &=& \\Delta_i^{(0)} + \\epsilon \\Delta_i^{(1)} + \\dots\n%&=& \\Delta_i^{(0)} + \\epsilon \\sum_j \\textcolor{blue}{\\left. \\frac{\\partial \\Delta_i}{\\partial f_j}\\right|_{f_j=f_j^{(0)}}} f_j^{(1)} + \\dots \\\\\n%&=:& \\epsilon \\sum_j \\textcolor{blue}{\\Lambda_{ij}} f_j^{(1)}\n\\end{eqnarray*}\n\n* hierarchy of equations at different powers of $\\epsilon$\n\\begin{align*}\n\\mathcal{O}(\\epsilon^0): \\quad & \\Delta_i^{(0)} = 0 \\\\\n\\mathcal{O}(\\epsilon^1): \\quad & \\left( \\frac{\\partial}{\\partial t_1} + \\vec{c}_i\\cdot\\frac{\\partial}{\\partial\\vec{r}_1} \\right) f_i^{(0)} = \\frac{1}{h} \\Delta_i^{(1)} \\\\\n\\mathcal{O}(\\epsilon^2): \\quad & \\left[ \\frac{\\partial}{\\partial t_2} + \\frac{h}{2} \\left( \\frac{\\partial}{\\partial t_1} + \\vec{c}_i\\cdot\\frac{\\partial}{\\partial\\vec{r}_1} \\right)^2 \\right] f_i^{(0)} + \\left( \\frac{\\partial}{\\partial t_1} + \\vec{c}_i\\cdot\\frac{\\partial}{\\partial\\vec{r}_1} \\right) f_i^{(1)} = \\frac{1}{h} \\Delta_i^{(2)}\n\\end{align*}\n\n### Zeroth order $\\epsilon^0$\n\n* no expansion for conserved quantities!\n\\begin{eqnarray*}\nf_i^{(0)} &=& f_i^\\text{eq} \\\\\n\\rho^{(0)} &=& \\rho = \\sum_i f_i^\\text{eq} \\\\\n\\vec{j}^{(0)} &=& \\vec{j} = \\sum_i f_i^\\text{eq} \\vec{c}_i\n\\end{eqnarray*}\n\n* linear part of collision operator\n\\begin{eqnarray*}\n\\Delta_i &=& \\epsilon \\Delta_i^{(1)} = \\epsilon \\sum_j {\\left. \\frac{\\partial \\Delta_i}{\\partial f_j} \\right|_{f^{(0)}}} f_j^{(1)} = \\sum_j {\\Lambda_{ij}} f_j^{(1)}\n\\end{eqnarray*}\n\n### Equations for the mass density\n\n* \n\\begin{align*}\n&\\frac{\\partial}{\\partial t_1} \\rho + \\frac{\\partial}{\\partial\\vec{r}_1} \\cdot \\vec{j} = 0 \\\\\n&\\frac{\\partial}{\\partial t_2} \\rho = 0\n\\end{align*}\n\n* continuity equation holds!\n\n### Equations for the momentum density\n\n* \n\\begin{align*}\n&\\frac{\\partial}{\\partial t_1} \\vec{j} + \\frac{\\partial}{\\partial\\vec{r}_1} \\cdot {\\mathsf{\\Pi}^{(0)}} = 0 \\\\\n&\\frac{\\partial}{\\partial t_2} \\vec{j} + \\frac{1}{2} \\frac{\\partial}{\\partial\\vec{r}_1} \\cdot \\left( {\\mathsf{\\Pi}^{*(1)} + \\mathsf{\\Pi}^{(1)}} \\right) = 0 \n\\end{align*}\n\n* Euler stress\n\\begin{equation*}\n{\\Pi}^{(0)} = \\rho c_s^2 \\mathsf{I} + \\rho \\vec{u}\\vec{u} = \\mathsf{\\Pi}^\\text{eq}\n\\end{equation*}\n\n* Newtonian viscous stress\n\\begin{equation*}\n\\frac{\\epsilon}{2} \\left( \\mathsf{\\Pi}^{*(1)} + \\mathsf{\\Pi}^{(1)} \\right) = - \\mathsf{\\Pi}^\\text{visc}\n\\end{equation*}\n\n* incompressible Navier-Stokes equation holds!\n\n### Diggin' deeper: Third moment\n\n* the third moment ${\\Phi}^{(0)} = \\sum_i f_i^{(0)} \\vec{c}_i\\vec{c}_i\\vec{c}_i$ enters through its equilibrium part!\n\\begin{equation*}\n\\frac{\\partial}{\\partial t_1} \\mathsf{\\Pi}^{(0)} + \\frac{\\partial}{\\partial\\vec{r}_1} \\cdot {{\\Phi}^{(0)}} = \\frac{1}{h} \\sum_i \\Delta_i^{(1)} \\vec{c}_i \\vec{c}_i = \\frac{1}{h} \\left( \\mathsf{\\Pi}^{*(1)} - \\mathsf{\\Pi}^{(1)} \\right)\n\\end{equation*}\n\n* explicit form\n\\begin{equation*}\n{\\Phi}^{(0)}_{\\alpha\\beta\\gamma} = \\rho c_s^2 \\left( u_\\alpha \\delta_{\\beta\\gamma} + u_\\beta\\delta_{\\alpha\\gamma} + u_\\gamma\\delta_{\\alpha\\beta} \\right)\n\\end{equation*}\n\n* from continuity and Euler equation\n\\begin{equation*}\n\\frac{\\partial}{\\partial t_1} \\mathsf{\\Pi}^{(0)} = \\frac{\\partial}{\\partial t_1} \\left( \\rho c_s^2 \\mathsf{I} + \\rho \\vec{u}\\vec{u} \\right) = \\dots\n\\end{equation*}\n\n* neglecting terms of $\\mathcal{O}(u^3)$\n\\begin{equation*}\n\\mathsf{\\Pi}^{*(1)} - \\mathsf{\\Pi}^{(1)} = \\rho c_s^2 h \\left( \\nabla\\vec{u} + (\\nabla\\vec{u})^t \\right)\n\\end{equation*}\n\n### Suitable LB models\n\n* equilibrium values of the moments up to ${\\Phi}^\\text{eq}$\n\\begin{eqnarray*}\n\\rho^\\text{eq}&=&\\rho \\\\\n\\vec{j}^\\text{eq}&=&\\vec{j} \\\\\n{\\Pi}^\\text{eq}&=&\\rho c_s^2 \\mathsf{I} + \\rho \\vec{u} \\vec{u} \\\\\n{\\Phi}^\\text{eq}_{\\alpha\\beta\\gamma} &=& \\rho c_s^2 \\left( u_\\alpha \\delta_{\\beta\\gamma} + u_\\beta\\delta_{\\alpha\\gamma} + u_\\gamma\\delta_{\\alpha\\beta} \\right)\n\\end{eqnarray*}\n\n* collision operator\n\\begin{align*}\n\\sum_i \\Delta_i &= 0 & \n\\sum_i \\Delta_i \\vec{c}_i &= 0 \\\\\n\\overline{{\\Pi}}^{*\\text{neq}} &= \\gamma_s \\overline{{\\Pi}}^{\\text{neq}} &\n\\text{tr}({\\Pi}^{*\\text{neq}}) &= \\gamma_b \\text{tr}({\\Pi}^{\\text{neq}})\n\\end{align*}\n\n### Hands-On Activity\n\n1. Play around with the LB code available at https://gist.github.com/uschille/8f65dd40572b2d943409.\n\n### References\n\n1. Schiller, U. D., Kr\u00fcger, T. & Henrich, O. Mesoscopic modelling and simulation of soft matter. Soft Matter 14, 9\u201326 (2017).\n4. Schiller, U. D. A unified operator splitting approach for multi-scale fluid\u2013particle coupling in the lattice Boltzmann method. Comput. Phys. Commun. 185, 2586\u20132597 (2014).\n2. D\u00fcnweg, B., Schiller, U. D. & Ladd, A. J. C. Progress in the understanding of the fluctuating lattice Boltzmann equation. Comput. Phys. Commun. 180, 605\u2013608 (2009).\n5. D\u00fcnweg, B., Schiller, U. D. & Ladd, A. J. C. Statistical mechanics of the fluctuating lattice Boltzmann equation. Phys. Rev. E 76, 036704 (2007).\n3. Schiller, U. D. Thermal fluctuations and boundary conditions in the lattice Boltzmann method. (Johannes Gutenberg University Mainz, 2008).\n", "meta": {"hexsha": "031f1a43516c714f12e5821ffce43506c7dc979b", "size": 44044, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Lecture-3-5_Lattice-Boltzmann-Methods.ipynb", "max_stars_repo_name": "Clemson-MSE/MSE8900", "max_stars_repo_head_hexsha": "705aa98a26f2421ac7b8c92729e1d3ffb2eace0b", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2018-11-06T00:09:51.000Z", "max_stars_repo_stars_event_max_datetime": "2020-12-11T03:57:32.000Z", "max_issues_repo_path": "Lecture-3-5_Lattice-Boltzmann-Methods.ipynb", "max_issues_repo_name": "Clemson-MSE/MSE8900", "max_issues_repo_head_hexsha": "705aa98a26f2421ac7b8c92729e1d3ffb2eace0b", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Lecture-3-5_Lattice-Boltzmann-Methods.ipynb", "max_forks_repo_name": "Clemson-MSE/MSE8900", "max_forks_repo_head_hexsha": "705aa98a26f2421ac7b8c92729e1d3ffb2eace0b", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2018-11-20T01:43:52.000Z", "max_forks_repo_forks_event_max_datetime": "2018-11-20T01:43:52.000Z", "avg_line_length": 37.0428931876, "max_line_length": 383, "alphanum_fraction": 0.4878984652, "converted": true, "num_tokens": 11781, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.611381973294151, "lm_q2_score": 0.695958331339634, "lm_q1q2_score": 0.42549637794493}} {"text": "# Dynamic Simulation: The Basic Procedure\nOnce a set of dynamic mass balance equations has been formulated, they can be numerically solved, and thus the behavior of a network can be simulated in response to environmental and genetic changes. Simulation results can be obtained using a number of different software packages. Dynamic simulation generates the time dependent behavior of the concentrations, i.e., $\\textbf{x}$(t). This solution can be obtained in response to several different types of perturbations and the results graphically displayed. The basic principles and procedures associated with dynamic simulation are covered in this chapter. The following three chapters then apply the simulation process to a set of simple but progressively more complex and relevant examples. \n\n## Numerical Solutions\nNetwork dynamics are described by a set of ordinary differential equations (ODEs): the dynamic mass balance equations; see Eq. (1.1). To obtain the dynamic solutions, we need three things: first, the equations themselves; second, the numerical values for the kinetic constants that are in the equations; and third, the initial conditions and parameters that are being perturbed. We describe each briefly. \n\n**1.** To formulate the mass balances we have to specify the system boundary, the fluxes in and out of the system, and the reactions that take place in the network. From the set of reactions that are taking place, a stoichiometric matrix is formed. This matrix is then put into Eq. (1.1) . One can multiply out the individual dynamic mass balances, as was done in Eq. (2.13) for the adenosine phosphate network, to prevent a large number of numerical operations that involve multiplication of reaction rates by the zero elements in $\\textbf{S}$. The reaction rate laws for the reactions are then identified and substituted into the equations. Typically, one would use elementary kinetics as shown in Eq. (2.6), or apply more complex rate laws if they are appropriate and available. This process leads to the definition of the dynamic mass balances. \n\n**2.** The numerical values for the kinetic parameters in the rate laws have to be specified, as do any imposed fluxes across the system boundary. Obtaining numerical values for the kinetic constants is typically difficult. They are put into a parameter vector designated by $\\textbf{k}$. In select cases, detailed kinetic characterization has been carried out. More often, though, one only knows these values approximately. It is important to make sure that all units are consistent throughout the equations and that the numerical values used are in the appropriate units. \n\n**3.** With the equations and numerical values for the kinetic constants specified $(\\textbf{k})$, we can simulate the responses of the network that they represent. To do so, we have to set initial conditions $(x_0)$. This leads to the numerical solution of\n\n$$\\begin{equation} \\frac{d\\textbf{x}}{dt} = \\textbf{Sv(x;k)},\\ \\textbf{x}(t = 0) = \\textbf{x}_0 \\tag{3.1} \\end{equation}$$\n\nThere are three conditions that are typically considered. \n\n1. First, the initial conditions for the concentrations are set, and the motion of the network into a steady state (open system) or equilibrium (closed system) is simulated. This scenario is typically physiologically unrealistic since individual concentrations cannot simply change individually in a living cell.\n2. Second, a change in an input flux is imposed on a network that is in a steady state. This scenario can be used to simulate the response of a cell to a change in its environment. \n3. Third, a change in a kinetic parameter is implemented at the initial time. The initial concentrations are typically set at the steady state values with the nominal value of the parameter. The equations are then simulated to a long time to obtain the steady state values that correspond to the altered kinetic parameters. These are set as the initial conditions when examining the responses of the system with the altered kinetic properties. \n\n**4.** Once the solution has been obtained it can be graphically displayed and the results analyzed. There are several ways to accomplish this step, as detailed in the next two sections. The analysis of the results can lead to post-processing of the output to form an alternative set of dynamic variables. \n\nThe simulation is implemented using a numerical solver. Currently, such implementation is carried out using standard and readily available software, such as Mathematica or MATLAB. Specialized simulation packages are also available (Table\u00a03.1). After the simulation is set up and the conditions specified, the software computes the concentrations as a function of time. The output is a file that contains the numerical values of the concentrations at a series of time points (Figure 3.1). This set of numbers is typically graphically displayed, and/or used for subsequent computations. \n\n**Table 3.1:** Available software for dynamic simulation. Assembled by Neema Jamshidi.\n\n\n\nWe will be using iPython notebooks using the python software package **MASSpy** as our simulation software.\n\n## Graphically Displaying the Solution\nThe simulation procedure described in the previous section results in a file that contains the concentrations as a function of time (Figure 3.1). These results are graphically displayed, typically in two ways: by plotting the concentrations as a function of time, or by plotting two concentrations against one another with time as a parameter along the trajectory. \n\n\n\n**Figure 3.1:** The fundamental structure of the file $\\textbf{x}$(t) that results from a numerical simulation. The two vertical bars show the list of values that would be used to compute $\\sigma_{12}$(2) (see Eq. 3.8); that is, the correlation between $x_1$ and $x_2$ with a time lag of 2.\n\nBefore describing these methods, we observe certain fundamental aspects of the equations that we are solving. The dynamic mass balances can be expanded as: \n\n$$\\begin{equation} \\frac{d\\textbf{x}}{dt} = \\textbf{Sv(x)} = \\sum\\textbf{s}_i v_i(\\textbf{x}) \\tag{3.2} \\end{equation}$$\n\nIn other words, the time derivatives are linear combinations of the reaction vectors $(\\textbf{s}_i)$ weighted by the reaction rates, that in turn change with the concentrations that are time varying. Thus, the motions are linear combinations of the directions specified by $\\textbf{s}_i$. This characteristic is important because if the $v_i$. have different time constants, the motion can be decomposed in time along these reaction vectors. \n\n### Time Profiles\nThe simulation results in a file that contains the vector $\\textbf{x}$(t) and the time points at which the numerical values for the concentrations are given. These time points can be specified by the user or are automatically generated by the solver used. Typically, the user specifies the initial time, the final time, and sometimes the time increment between the time points where the simulator stores the computed concentration values in the file. The results can then be graphically displayed depending on a few features of the solution. Some of these are shown in Figure 3.2 and are now briefly described: \n\n* Panel A: The most common way to display a dynamic solution is to plot the concentration as a function of time. \n\n* Panel B: If there are many concentration variables they are often displayed on the same graph. \n\n* Panel C: In many cases there are different response times and one plots multiple time profiles where the x-axis on each plot is scaled to a particular response time. Alternatively, one can use a logarithmic scale for time. \n\n* Panel D: If a variable moves on many time scales changing over many orders of magnitude, the y-axis is often displayed on a logarithmic scale. \n\n\n\n**Figure 3.2:** Graphing concentrations over time. (a) A single concentration shown as a function of time. (b) Many concentrations shown as a function of time. (c) A single concentration shown as a function of time separately on different time scales. (d) The logarithm of a single concentration shown as a function of time to distinguish the decay on different time scales.\n\nThe solution can thus be displayed in different ways depending on the characteristics of the time profiles. One normally plays with these representations to get an understanding of the responses of the network that they have formulated and to represent the features in which one is interested. \n\n### Dynamic phase portraits\nDynamic phase portraits represent trajectories formed when two concentrations plotted against each other, parameterized with respect to time (Figure 3.3). The dynamic trajectories in the diagram move from an initial state to a final state. Analysis of these trajectories can point to key dynamic relationships between compounds in a biochemical reaction network. For example, if a system is dynamically stable, the dynamic trajectories will converge to a single point in the plane, known as an attracting fixed pointattracting fixed point. A stable steady-state point would represent a homeostatic state. Conversely, if the system is unstable, the trajectories will not approach a fixed point but diverge away from it. The former is essentially always the case for biochemical reaction networks representing real cells. The way the trajectories converge on the steady state is highly informative as different dynamic characteristics are evident from the trajectory. \n\n\n\n**Figure 3.3:** A dynamic phase portrait.\n\n### Characteristic features of phase portraits\nA trajectory in the phase portraitphase portrait may indicate the presence of one or more general dynamic features. Namely, the shapes of the trajectories contain significant information about the dynamic characteristics of a network. Some important features of trajectories in a phase portrait are shown in Figure 3.4\n\n\n\n**Figure 3.4:** General features of dynamic phase portraits. Dynamic phase portraits are formed by graphing the time dependent concentrations of two concentrations $(x_1$ and $x_2)$ against one another. Phase portraits have certain characteristic features. (a) Conservation relationship. (b) A pair of concentrations that could be in quasi-equilibrium with one another. (c) Motion of the two concentrations dynamically independent of one another. (d) Closed loop traces representing either a periodic motion or a return to the original steady state. Modified from Kauffman 2002 [64].\n\n1. When the trajectory has a negative slope, it indicates that one concentration is increasing while the other is decreasing. The concentrations are moving on the same time scales but in opposite directions; that is, one is consumed while the other is produced. This feature might represent the substrate concentration versus the product concentration of a given reaction. Such behavior helps define aggregate concentration variablesaggregate concentration variables. \n\n2. When a trajectory in the phase portrait between two concentrations is a straight line with a positive slope, it means that the two concentrations are moving in tandem; i.e., as one increases so does the other. This feature is observed when two or more concentrations move on the same time scales and are in quasi-equilibrium with one another. Such behavior helps define aggregate concentration variables. \n\n3. When a trajectory is vertical or horizontal, it indicates that one of the concentrations is changing while the other remains constant. This feature implies either that the motions of the concentrations during the trajectory are independent of one another or that the dynamic motions of the concentrations progress on different characteristic time scales. Such behavior helps define time scale decomposition.\n\n4. When a trajectory forms a closed loop, it implies one of two possibilities. The system never converges to a steady state over time but oscillates forming a closed loop trajectory. On the other hand, if the orbit begins at one point, moves away from it, then returns to the same point after a sufficiently long time interval, then it implies that a change in another variable in the system forced it away from its steady state temporarily, but it returned to the original steady state. Such behavior helps define disturbance rejection characteristics. \n\n\n\n**Figure 3.5:** A schematic of a tiled phase portrait.The matrix is symmetric, making it possible to display statistical information about a phase portrait in the mirror position.The diagonal elements are meaningless.Originally developed in Kauffman 2002 [64].\n\nThe qualitative characteristics of dynamic phase portraitsphase portrait can provide insight into the dynamic features of a network. A trajectory may have more than one of these basic features. For instance, there can be a fast independent motion (i.e., a horizontal phase portrait trajectory) followed by a line with a positive slope after an equilibrium state has been reached.\n\n### Tiling dynamic phase portraits\nPhase portraits show the dynamic relationships between two variables on multiple time scales, see Figure 3.5. If a system has $n$ variables, then there are $n^2$ dynamic phase portraits. All pair-wise phase portraits can be tiled in a matrix form where the $\\textit{i}$, $\\textit{j}$ entry represents the dynamic phase portrait between variables $x_i$ and $x_j$. Note that such an array is symmetric and that the diagonal elements are un-informative. Thus, there are $(n^2-n)/2$ phase portraits of interest. This feature of this graphical representation opens the possibility of putting the phase portrait in the $\\textit{i}$, $\\textit{j}$ position in the array and showing other information (such as a regression coefficient or a slope) in the corresponding $\\textit{j}$, $\\textit{i}$ position. \n\nSince the time scales in biochemical reaction networks typically vary over many orders of magnitude, it often makes sense to make a series of tiled phase portraits, each of which represents a key time scale. For instance, rapid equilibration leads to straight lines with positive slopes in the phase portrait (Figure 3.4b) where the slope is the equilibrium constant of the reaction. This may be one of many dynamic events taking place. If a phase portrait is graphed separately on this time scale alone, the positive line will show up with a high regression coefficient and a slope that corresponds to the equilibrium constant.\n\n## Post-Processing the Solution \n\nThe initial suggestions obtained from graphing and visualizing the concentration vector $\\textbf{x}$(t) can lead to a more formal analysis of the results. We describe three post-processing procedures of $\\textbf{x}$(t). \n\n### Computing the fluxes from the concentration variables:\nThe solution for the concentrations $\\textbf{x}$(t)can be used to compute the fluxes from\n\n$$\\begin{equation} \\textbf{v}(t)= \\textbf{v}(\\textbf{x}(t)) \\tag{3.3} \\end{equation}$$\n\nand subsequently we can plot the fluxes in the same way as the concentrations. Graphical information about both the $\\textbf{x}$(t) and $\\textbf{v}$(t) is useful. \n\n### Combining concentrations to form aggregate variables:\nThe graphical and statistical multi-time scale analysis discussed above may lead to the identification of aggregate variables. Pooled variables, p, are computed from\n\n$$\\begin{equation} \\textbf{p}(t)= \\textbf{Px}(t)) \\tag{3.4} \\end{equation}$$\n\nwhere the pool transformation matrix, $\\textbf{P}$, defines the linear combination of the concentration variables that forms the aggregate variables. For instance, if we find that a logical way to pool two variables, $x_1$ and $x_2$, into new aggregate variables is $p_1 = x_1 + x_2$ and $p_2 = x_1 - x_2$, then we form the following matrix equation describing these relationships as: \n\n$$\\begin{equation} \\textbf{p}(t) = \\textbf{Px}(t) = \\begin{pmatrix} {p_1(t)} \\\\ {p_2(t)} \\end{pmatrix} = \\begin{pmatrix} {1} & {1} \\\\ {1} & {-1} \\end{pmatrix} \\begin{pmatrix} {x_1(t)} \\\\ {x_2(t)} \\end{pmatrix} = \\begin{pmatrix} {x_1(t) + x_2(t)} \\\\ {x_1(t) - x_2(t)} \\end{pmatrix} \\end{equation}$$\n\nThe dynamic variables, $\\textbf{p}$(t), can be graphically studied as described in the previous section. \n\n#### Example: The Phosphorylated Adenosines\nThe pool formation discussed in Chapter\u00a02 can be described by the pool transformation matrix: \n\n$$\\begin{equation} \\textbf{P} = \\begin{pmatrix} {1} & {1} & {0} \\\\ {2} & {1} & {0} \\\\ {1} & {1} & {1} \\end{pmatrix} \\end{equation} \\tag{3.5}$$\n\nand thus \n\n$$\\begin{equation} \\textbf{p} = \\textbf{Px} = \\textbf{P}\\begin{pmatrix} {\\text{ATP}} \\\\ {\\text{ADP}} \\\\ {\\text{AMP}} \\end{pmatrix} = \\begin{pmatrix} {\\text{ATP} + \\text{ADP}} \\\\ {2 \\text{ATP} + \\text{ADP}} \\\\ {\\text{ATP} + \\text{ADP} + \\text{AMP}} \\end{pmatrix} \\end{equation} \\tag{3.6}$$\n\nThe pool sizes $p_i$(t) can then be graphed as a function of time. \n\n### Correlating concentrations over time:\nOne can construct the time-separated correlation matrix, $\\textbf{R}$, based on a time scale structure of a system. In this matrix, we compute the correlation between two concentrations on a time scale as: \n\n$$\\begin{equation} \\textbf{R}(\\tau) = (r_{ij}) = \\frac{\\sigma_{ij}(\\tau)}{\\sqrt{\\sigma_{ii}\\sigma_{jj}}} \\tag{3.7} \\end{equation}$$\n\nin which $\\sigma_{ii}$ is the variance of the dataset $x_i(k)$ and $\\sigma_{ij}(\\tau)$ is the time-lagged covariance between the discrete, uniformly sampled datasets $x_i(k)$ and $x_j(k + \\tau)$, determined as, \n\n$$\\begin{equation} \\sigma_{ij}(\\tau) = \\frac{1}{n}\\sum\\limits_{k=1}^{n-\\tau} (x_i(k) - \\bar{x_i})(x_j(k + \\tau) - \\bar{x_j}) \\tag{3.8} \\end{equation}$$\n\nin which $n$ is the number of data points in the series, and $\\bar{x_i}$ indicates the average value of the series $x_i$. The values in $\\textbf{R}$ range from -1 to 1, indicating perfect anti-correlation or correlation, respectively, between two datasets with a delay of time steps. Elements in $\\textbf{R}$ equal to zero indicate that the two corresponding datasets are completely uncorrelated. If such correlation computations were done for the cases shown in Figure 3.4, one would expect to find a strong negative correlation for the data shown in Figure 3.4a, a strong positive correlation for Figure 3.4b, and no correlation for Figure 3.4c, \n\nThe correlation computations can be performed with an increment, $\\tau$, offset in time between two concentrations. An example of a time offset is shown in Figure 3.2 showing the values used from the output file to compute the correlation between $x_1$ and $x_2$ with a time lag of 2. \n\nThe matrix of phase portraits is symmetric with uninformative diagonal elements. One can therefore enter a correlation coefficient corresponding to a particular phase portrait in the transpose position to the phase portrait in the matrix. A correlation coefficient provides a quantitative description of the phase portrait's linearity between the two variables over the time scale displayed. In addition to the correlation coefficient, the slope can be computed and displayed, giving the equilibrium constant between the two compounds displayed. \n\n## Demonstration of the Simulation Procedure in MASSpy\n### Setting up the model\nThe following builds the model of three reactions in series that is described on pages 51-56 in the book. We show how the model is built, simulated, solutions graphically displayed, solutions post processed and analyzed mathematically.\n\nTo construct a model in **MASSpy**, the `MassModel`, `MassReaction`, and `MassMetabolite` objects need to be imported into the environment. \n\n\n```python\nfrom mass import MassModel, MassMetabolite, MassReaction\n```\n\n#### Defining metabolites and reactions\nOne method for creating the model is to objects that represent the metabolites and reactions. \nMetabolite are represented by `MassMetabolite` objects, and can be created by providing a unique identifier for that object. Therefore we can define the four metabolites, $x_1, x_2, x_3$, and $x_4$ by the following;\n\n\n```python\nx1 = MassMetabolite('x1')\nx2 = MassMetabolite('x2')\nx3 = MassMetabolite('x3')\nx4 = MassMetabolite('x4')\n```\n\nReactions are represented by `MassReaction` objects, and like metabolites, they can be also created by providing a unique identifier for that object.\n\n\n```python\nv1 = MassReaction('v1')\nv2 = MassReaction('v2')\n```\n\nBy default, a reaction is considered reversible. However, if we wish to make an irreversible reaction, we set the `reversible` argument to `False`. \n\n\n```python\nv3 = MassReaction('v3', reversible=False)\n```\n\nOnce the `MassReaction` objects have been created, metabolites can be added to the reaction using the `MassReaction.add_metabolites` method. \nTo quickly see how this method is used, we can use the `help()` function. Alternatively, we can go to the [API documentation](https://masspy.readthedocs.io/en/latest/autoapi/index.html) and read about how the [MassReaction.add_metabolites](https://masspy.readthedocs.io/en/latest/autoapi/mass/core/mass_reaction/index.html#mass.core.mass_reaction.MassReaction.add_metabolites) method works.\n\n\n```python\nhelp(MassReaction.add_metabolites)\n```\n\n Help on function add_metabolites in module mass.core.mass_reaction:\n \n add_metabolites(self, metabolites_to_add, combine=True, reversibly=True)\n Add metabolites and their coefficients to the reaction.\n \n If the final coefficient for a metabolite is 0 then it is removed from\n the reaction.\n \n The change is reverted upon exit when using the :class:`~.MassModel`\n as a context.\n \n Notes\n -----\n * A final coefficient of < 0 implies a reactant and a final\n coefficient of > 0 implies a product.\n \n * Extends :meth:`~cobra.core.reaction.Reaction.add_metabolites` of the\n :class:`cobra.Reaction ` by first\n ensuring that the metabolites to be added are\n :class:`.MassMetabolite`\\ s and not\n :class:`cobra.Metabolites `.\n and error message raised reflects the :mod:`mass` object.\n \n * If a :class:`cobra.Metabolite ` is\n provided. a warning is raised and a :class:`.MassMetabolite`\n will be instantiated using the\n :class:`cobra.Metabolite `.\n \n Parameters\n ----------\n metabolites_to_add : dict\n A ``dict`` with :class:`.MassMetabolite`\\ s or metabolite\n identifiers as keys and stoichiometric coefficients as values. If\n keys are strings (id of a metabolite), the reaction must already\n be part of a :class:`~.MassModel` and a metabolite with the given\n id must already exist in the :class:`~.MassModel`.\n combine : bool\n Describes the behavior of existing metabolites.\n If ``True``, the metabolite coefficients are combined together.\n If ``False`` the coefficients are replaced.\n reversibly : bool\n Whether to add the change to the context to make the change\n reversible (primarily intended for internal use).\n \n See Also\n --------\n :meth:`subtract_metabolites`\n \n\n\nTo use `MassReaction.add_metabolites`, a dictionary input is required, where the `MassMetabolite` objects are keys and the value is their stoichiometric coefficient. Reactants are defined with negative coefficients, while products are defined with positive coefficients. \n\n\n```python\nv1.add_metabolites({x1 : -1, x2 : 1})\nv2.add_metabolites({x2 : -1, x3 : 1})\nv3.add_metabolites({x3 : -1, x4 : 1})\n```\n\nReactions, e.g., $v_1$ can be used to define any kind of chemical transformation, association, activation etc. A series of methods are provided for inspection of the reaction.\n\n\n```python\nv1.id\n```\n\n\n\n\n 'v1'\n\n\n\n\n```python\nv1.reactants\n```\n\n\n\n\n []\n\n\n\n\n```python\nv1.products\n```\n\n\n\n\n []\n\n\n\n\n```python\nv1.stoichiometry\n```\n\n\n\n\n [-1, 1]\n\n\n\n\n```python\nv1.reversible\n```\n\n\n\n\n True\n\n\n\nCheck the documentation for the `MassReaction` class for further details.\n\n#### Model Setup\nTo construct a model capable of dynamic simulation, a `MassModel` object must be created. The minimal input for creating a `MassModel` object is a unique identifier. \n\n\n```python\nmodel = MassModel('Model')\n```\n\nTo add reactions and their corresponding metabolites to the model, the `MassModel.add_reactions` method can be used by providing a list of reactions to add to the model. \n\n\n```python\nmodel.add_reactions([v1, v2, v3])\n```\n\n#### Model Inspection\nSimilar to the `MassReaction` object, the `MassModel` object also has various methods that can be used to inspect the model. For example, to obtain the list of reactions and species in the system:\n\n\n```python\nmodel.reactions\n```\n\n\n\n\n [,\n ,\n ]\n\n\n\n\n```python\nmodel.metabolites\n```\n\n\n\n\n [,\n ,\n ,\n ]\n\n\n\nIn some circumstances, it is helpful to iterate through a reaction and its associated metabolites using a loop:\n\n\n```python\nprint(\"Model ID: %s\" % model.id)\nfor rxn in model.reactions:\n print(\"\\nReaction: %s\\n------------\" % rxn.id)\n for metab, stoichiometry in rxn.metabolites.items():\n print(\"%s: %s \" % (metab.id, stoichiometry))\n```\n\n Model ID: Model\n \n Reaction: v1\n ------------\n x1: -1 \n x2: 1 \n \n Reaction: v2\n ------------\n x3: 1 \n x2: -1 \n \n Reaction: v3\n ------------\n x4: 1 \n x3: -1 \n\n\nTo examine the stoichiometric matrix:\n\n\n```python\nmodel.S\n```\n\n\n\n\n array([[-1., 0., 0.],\n [ 1., -1., 0.],\n [ 0., 1., -1.],\n [ 0., 0., 1.]])\n\n\n\nThe stoichiometric matrix can also be viewed as a `pandas.DataFrame` with annotated information about the metabolites and reactions.\n\nNote: The `update_model` argument can be used to store matrix as the specified `array_type` for the next time the stoichiometric matrix is viewed. \n\n\n```python\nmodel.update_S(array_type=\"DataFrame\", update_model=True)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        v1v2v3
                                        x1-1.00.00.0
                                        x21.0-1.00.0
                                        x30.01.0-1.0
                                        x40.00.01.0
                                        \n
                                        \n\n\n\nThe rate equations can be examined,\n\n\n```python\nfor rxn, rate in model.rates.items():\n print(\"%s: %s\" % (rxn.id, rate))\n```\n\n v1: kf_v1*(x1(t) - x2(t)/Keq_v1)\n v2: kf_v2*(x2(t) - x3(t)/Keq_v2)\n v3: kf_v3*x3(t)\n\n\nor just one rate equation can be called out:\n\n\n```python\nprint(model.rates[v2])\n```\n\n kf_v2*(x2(t) - x3(t)/Keq_v2)\n\n\nThe ordinary differential equations can be also be listed in full,\n\n\n```python\nfor metab, ode in model.odes.items():\n print(\"%s: %s\" % (metab.id, ode))\n```\n\n x1: -kf_v1*(x1(t) - x2(t)/Keq_v1)\n x2: kf_v1*(x1(t) - x2(t)/Keq_v1) - kf_v2*(x2(t) - x3(t)/Keq_v2)\n x3: kf_v2*(x2(t) - x3(t)/Keq_v2) - kf_v3*x3(t)\n x4: kf_v3*x3(t)\n\n\nor just one ordiniary differential equation can be called out:\n\n\n```python\nprint(model.odes[x3])\n```\n\n kf_v2*(x2(t) - x3(t)/Keq_v2) - kf_v3*x3(t)\n\n\nNote that none of these expressions have been provided during the model construction process. Instead the expresions have been generated automatically from the provided list of reactions and their metabolites. \n\n#### Set parameters and initial condtions\nWhen using Jupyter notebooks, an overview of the model is rendered as a table when only the model object is called. Note that this also applies to metabolites and reactions.\n\n\n```python\nmodel\n```\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        NameModel
                                        Memory address0x07fe452ab6450
                                        Stoichiometric Matrix4x3
                                        Matrix Rank3
                                        Number of metabolites4
                                        Initial conditions defined0/4
                                        Number of reactions3
                                        Number of genes0
                                        Number of enzyme modules0
                                        Number of groups0
                                        Objective expression0
                                        Compartments
                                        \n\n\n\n\nFrom the model overview it can be seen that no parameters or initial conditions have been defined. Parameters can be defined directly for a specific reaction:\n\n\n```python\nv1.forward_rate_constant = 1\nv2.kf = 0.01 # Shorthand method\nv3.kf = 0.0001\n\nv1.equilibrium_constant = 1\nv2.Keq = 1 # Shorthand method\n\nfor param_type, param_dict in model.parameters.items():\n print(\"%s: %s\" %(param_type, param_dict))\n```\n\n kf: {'kf_v1': 1, 'kf_v2': 0.01, 'kf_v3': 0.0001}\n Keq: {'Keq_v1': 1, 'Keq_v2': 1, 'Keq_v3': inf}\n kr: {}\n v: {}\n Custom: {}\n Boundary: {}\n\n\nInitial conditions for metabolites can be defined directly for a specific metabolite,\n\n\n```python\nx1.initial_condition = 1\nx2.ic = 0 # Shorthand method\nmodel.initial_conditions\n```\n\n\n\n\n {: 1,\n : 0}\n\n\n\nor a dictionary can be used to define them in a model directly. The `update_metabolites` argument will subsequently update the initial condition in the metabolite object as well. \n\n\n```python\nmodel.update_initial_conditions({x3: 0, x4:0})\nmodel.initial_conditions\n```\n\n\n\n\n {: 1,\n : 0,\n : 0,\n : 0}\n\n\n\nCheck the documentation for further details on the `MassModel` class. \n\n### Simulating Dynamic Responses\n#### Simulate\nSimulating the model once it is set up properly is very simple. To set up the simulation, we use a `Simulation` object. The simulation object requires a `MassModel` for initialization.\n\n\n```python\nfrom mass import Simulation\n```\n\n\n```python\nsim = Simulation(model, verbose=True)\n```\n\n \u001b[93mWARNING:\u001b[0m \u001b[93mNo compartments found in model. Therefore creating compartment 'compartment' for entire model.\u001b[0m\n\n\n Successfully loaded MassModel 'Model' into RoadRunner.\n\n\nThe `Simulation.simulate` method from the will integrate the ordinary differential equations of the system in the provided time interval and return the dynamic responses of concentrations and fluxes.\n\n\n```python\nt0 = 0\ntf = 1e6\nnumpoints = 1e3 + 1\n\nconc_sol, flux_sol = sim.simulate(\n model, time=(t0, tf, numpoints), interpolate=True, verbose=True)\n```\n\n Getting time points\n Setting output selections\n Setting simulation values for 'Model'\n Simulating 'Model'\n Simulation for 'Model' successful\n Adding 'Model' simulation solutions to output\n Updating stored solutions\n\n\nNote: If a model is unable to be simulated, a warning will be raised. By setting the `verbose` argument to `True`, a QC/QA report outlining inconsistencies, missing values, and other issues will also be generated and displayed to assist in diagnosing the reason why a model could not be simulated. \n\n#### Inspect the solution\nAs the default setting, the Simulation object utilizes scipy interpolating functions to capture the concentration and flux responses (see documentation for scipy.interpolate for additional information). The `Simulation.simulate_model` method returns two `cobra.DictLists` containing specialized dictionaries known as `MassSolution` objects.\n\nThe first `MassSolution` object contains the `MassMetabolite` identifiers as keys, and their corresponding concentration solutions as values.\n\n\n```python\nfor metabolite, solution in conc_sol.items():\n print(metabolite, solution)\n```\n\n x1 \n x2 \n x3 \n x4 \n\n\nSimilarly, the second `MassSolution` object contains the `MassReaction` identifiers as keys, and their corresponding flux solutions as values.\n\n\n```python\nfor reaction, solution in flux_sol.items():\n print(reaction, solution)\n```\n\n v1 \n v2 \n v3 \n\n\n#### Query time responses\nThe interpolating functions are functions of time. Therefore, we can evaluate the interpolating function at a specific time point using the following: \n\n\n```python\ntime_points = 100;\nfor metabolite, interpolating_function in conc_sol.items():\n print(\"%s: %s\" % (metabolite, interpolating_function(time_points)))\nprint()\nfor reaction, interpolating_function in flux_sol.items():\n print(\"%s: %s\" % (reaction, interpolating_function(time_points)))\n```\n\n x1: 0.3710242389082219\n x2: 0.3704524448547136\n x3: 0.2569363810507253\n x4: 0.0015869284328310024\n \n v1: 0.0005717940535082393\n v2: 0.0011351606380398832\n v3: 2.5693638105072534e-05\n\n\nIt is also possible to get values for multiple time points at once:\n\n\n```python\ntime_points = [0.01, 0.1, 1, 10, 100, 1000];\nfor metabolite, interpolating_function in conc_sol.items():\n print(\"%s: %s\" % (metabolite, interpolating_function(time_points)))\nprint()\nfor reaction, interpolating_function in flux_sol.items():\n print(\"%s: %s\" % (reaction, interpolating_function(time_points)))\n```\n\n x1: [0.99009934 0.90936384 0.56699581 0.4790072 0.37102424 0.32389592]\n x2: [0.00990017 0.09058937 0.43018534 0.47682748 0.37045244 0.32388517]\n x3: [4.96651050e-07 4.67952432e-05 2.81873928e-03 4.41437817e-02\n 2.56936381e-01 3.21735095e-01]\n x4: [1.65778860e-13 1.58583691e-10 1.07541682e-07 2.15291648e-05\n 1.58692843e-03 3.04838051e-02]\n \n v1: [9.80199167e-01 8.18774471e-01 1.36810476e-01 2.17971609e-03\n 5.71794054e-04 1.07505761e-05]\n v2: [9.89967148e-05 9.05425714e-04 4.27366598e-03 4.32683701e-03\n 1.13516064e-03 2.15007648e-05]\n v3: [4.96651050e-11 4.67952432e-09 2.81873928e-07 4.41437817e-06\n 2.56936381e-05 3.21735095e-05]\n\n\nFor example, a `pandas.Dataframe` of concentration values at different time points could be generated using this method: \n\n\n```python\nimport pandas as pd\n```\n\n\n```python\ndata = [interpolating_function(time_points) \n for interpolating_function in conc_sol.values()]\nindex_col = [metabolite for metabolite in conc_sol.keys()]\npd.DataFrame(data, index=index_col, columns=time_points)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        0.010.101.0010.00100.001000.00
                                        x19.900993e-019.093638e-015.669958e-010.4790070.3710240.323896
                                        x29.900168e-039.058937e-024.301853e-010.4768270.3704520.323885
                                        x34.966510e-074.679524e-052.818739e-030.0441440.2569360.321735
                                        x41.657789e-131.585837e-101.075417e-070.0000220.0015870.030484
                                        \n
                                        \n\n\n\nThe same can be done for the fluxes: \n\n\n```python\ndata = [interpolating_function(time_points) \n for interpolating_function in flux_sol.values()]\nindex_col = [reaction for reaction in flux_sol.keys()]\npd.DataFrame(data, index=index_col, columns=time_points)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        0.010.101.0010.00100.001000.00
                                        v19.801992e-018.187745e-011.368105e-010.0021800.0005720.000011
                                        v29.899671e-059.054257e-044.273666e-030.0043270.0011350.000022
                                        v34.966510e-114.679524e-092.818739e-070.0000040.0000260.000032
                                        \n
                                        \n\n\n\n#### Filtering for specific species and fluxes\nBecause concentration and flux `MassSolution` objects are specialized dictionaries, they can be handled like any other dictionary. Therefore, obtaining the solution for individual species and fluxes can be done easily by using the `MassMetabolite` or `MassReaction` identifiers as keys.\n\n\n```python\nprint(x1.id, conc_sol[x1.id])\n```\n\n x1 \n\n\n\n```python\nfor flux in [v1, v2]:\n print(flux.id, flux_sol[flux.id])\n```\n\n v1 \n v2 \n\n\n#### Switching between numerical arrays and interpolating functions\nSuppose that instead of working with interpolating functions, we would rather work with the original time points and the corresponding solutions utilized by the ODE solver. One way this can be done would be to access the original time point values stored in the Solution object, and use those in the interpolating function:\n\n\n```python\ntime_points = conc_sol.t\n# Get a slice of the first 50 points\nprint(conc_sol[\"x1\"](time_points)[:50])\n```\n\n [1. 1. 0.99999943 0.99999734 0.99999525 0.99999316\n 0.99998821 0.99997787 0.99995537 0.99990413 0.99978077 0.99942553\n 0.99907055 0.99871581 0.99806461 0.99741426 0.99676475 0.9961161\n 0.9948862 0.99290131 0.98987466 0.9868666 0.983877 0.98090577\n 0.97795277 0.97334374 0.96877916 0.96425858 0.95727034 0.94370569\n 0.92186543 0.90109967 0.88135539 0.86258222 0.84473223 0.82775987\n 0.81162185 0.78756746 0.76536568 0.7448733 0.72595816 0.70849829\n 0.69238106 0.67750262 0.66376717 0.64105858 0.62146897 0.6045664\n 0.58997873 0.57738534]\n\n\nTo quickly convert an entire `MassSolution` object from interpolating functions to numerical arrays or vice-versa, we use the `MassSolution.interpolate` setter method:\n\n\n```python\nconc_sol.interpolate = False\n# Get a slice of the first 50 points\nconc_sol[\"x1\"][:50]\n```\n\n\n\n\n array([1. , 1. , 0.99999943, 0.99999734, 0.99999525,\n 0.99999316, 0.99998821, 0.99997787, 0.99995537, 0.99990413,\n 0.99978077, 0.99942553, 0.99907055, 0.99871581, 0.99806461,\n 0.99741426, 0.99676475, 0.9961161 , 0.9948862 , 0.99290131,\n 0.98987466, 0.9868666 , 0.983877 , 0.98090577, 0.97795277,\n 0.97334374, 0.96877916, 0.96425858, 0.95727034, 0.94370569,\n 0.92186543, 0.90109967, 0.88135539, 0.86258222, 0.84473223,\n 0.82775987, 0.81162185, 0.78756746, 0.76536568, 0.7448733 ,\n 0.72595816, 0.70849829, 0.69238106, 0.67750262, 0.66376717,\n 0.64105858, 0.62146897, 0.6045664 , 0.58997873, 0.57738534])\n\n\n\n\n```python\nconc_sol.interpolate = True\nconc_sol[\"x1\"]\n```\n\n\n\n\n \n\n\n\n\n```python\nfor key, value in conc_sol.items():\n print(key, value)\n```\n\n x1 \n x2 \n x3 \n x4 \n\n\n\n```python\nconc_sol[\"x1\"]\n```\n\n\n\n\n \n\n\n\n\n```python\nconc_sol.x1\n```\n\n\n\n\n \n\n\n\n### Visualizing the Solution Graphically\nOnce the model has been simulated, the solutions can be visualized using the visualization tools in **MASSpy**. \n\n\n```python\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfrom mass.visualization import (\n plot_phase_portrait, plot_time_profile, plot_tiled_phase_portraits)\n```\n\nAll visualization tools utilize the matplotlib python package. See documentation for the visualization class for more details on the available plotting kwargs. \n\n#### Draw time course\nPlotting the dynamic responses is straightforward using the `plot_time_profile` function:\n\n\n```python\nplot_time_profile(conc_sol);\n```\n\nFor this model and simulation, plotting on a linear scale does not provide us information about the dyanmics at various time scales. Therefore, we can use the `plot_function` kwarg to change the scale. Let us keep a linear scale on the y-axis, but change the x-axis to a logarithmic scale. \n\n\n```python\nplot_time_profile(conc_sol, plot_function=\"semilogx\");\n```\n\nThe `observable` argument allows one to specify particular solutions from the solution profile to observe while filtering out all other solutions. For example, only the solutions for $x_1$ and $x_2$ can be observed by setting observable to an list of these two keys in the solution profile. \n\n\n```python\nplot_time_profile(conc_sol, observable=[\"x1\", \"x2\"], \n plot_function=\"semilogx\");\n```\n\nThough the dynamic behavior is clear, the above plots do not provide any other information. Let us add axes labels, a title, and a legend to the plot.\n\n\n```python\nplot_time_profile(\n conc_sol, legend=\"right outside\", plot_function=\"semilogx\",\n xlabel=\"Time\", ylabel=\"Concentration\", \n title=(\"Concentration Solutions\", {\"size\": \"large\"}));\n```\n\n#### Draw phase portraits\nPlotting the dynamic responses against one another is also straightforward by using the `plot_phase_portrait` function:\n\n\n```python\nplot_phase_portrait(conc_sol, x=\"x1\", y=\"x2\",\n xlabel=\"x1\", ylabel=\"x2\");\n```\n\n$x_1$ vs $x_2$: note that you can use the `annotate_time_points` argument to highlight particular time points of interest. This argument can be utilized either by providing iterable of time points of interest. The `annotate_time_points_color` can be used to set the color of the time points. To use color to distinguish time points, the number of colors should equal the number of time points specified.\n\n\n```python\nplot_phase_portrait(\n conc_sol, x=\"x1\", y=\"x2\", xlabel=\"x1\", ylabel=\"x2\",\n annotate_time_points=[t0, 1e-1, 1e0, 1e1, 1e3, tf],\n annotate_time_points_color= [\n \"red\", \"green\", \"purple\", \"yellow\", \"cyan\", \"blue\"],\n annotate_time_points_legend=\"lower outside\");\n```\n\nAll pairwise phase portraits can be generated and viewed at once in a tiled format using the `plot_tiled_phase_portrait` function:\n\n\n```python\nplot_tiled_phase_portraits(conc_sol,\n annotate_time_points_legend=\"right outside\");\n```\n\nThis method is particularly useful for looking at correlations at various time scales. For example, looking at the overall behavior, a fast time timescale of (0, 1), an intermediate timescale of (3, 100), and a slow timescale of (300, 10000), we can generate the following:\n\n\n```python\ncorrelations = [\n np.empty((3, 3)).astype(str),\n np.empty((3, 3)).astype(str),\n np.empty((3, 3)).astype(str),\n np.empty((3, 3)).astype(str)]\nfor entry in correlations:\n entry.fill(\"\")\n\nfmt_str = \"{0:.2f}\\n{1:.2f}\"\ncorrelations[1][0, 1] = fmt_str.format(*[1, 1])\ncorrelations[2][0, 1] = fmt_str.format(*[1, 1.02])\ncorrelations[2][0, 2] = fmt_str.format(*[1, 0.51])\ncorrelations[2][1, 2] = fmt_str.format(*[1, 0.5])\ncorrelations[3][0, 1] = fmt_str.format(*[1, 1.02])\ncorrelations[3][0, 2] = fmt_str.format(*[1, 1.02])\ncorrelations[3][1, 2] = fmt_str.format(*[1, 1.02])\n\nfig, axes = plt.subplots(2, 2, figsize=(10, 10))\naxes = axes.flatten()\n\ntimes = [(0, 400000), (0, 1), (3, 100), (300, 10000)]\ntitles = [\"{0}\\nt0={1}; tf={2}\".format(label, *time)\n for label, time in zip([\"(a)\", \"(b)\", \"(c)\", \"(d)\"], times)]\n\nfor i, ax in enumerate(axes.flatten()):\n plot_tiled_phase_portraits(\n conc_sol, observable=[\"x1\", \"x2\", \"x3\"], ax=ax,\n plot_tile_placement=\"lower\", additional_data=correlations[i],\n time_vector=np.linspace(*times[i], int(1e6)),\n tile_xlabel_fontdict={\"size\": \"large\"},\n tile_ylabel_fontdict={\"size\": \"large\"},\n title=titles[i])\n```\n\n### Post process the solution\n#### Analyze pool behavior\nIn order to analyze the behavior of pools, pools can be created using the `MassSolution.make_aggregate_solution` method using the string representation of the pooling formulas. Additional parameters can also be incorporated into the pool formulation using a dictionary input for the `parameters` argument. \n\n\n```python\npools = [\"x1 - x2\", \"x1 + x2 - 2*x3\", \"x1 + x2 + x3\"]\n\nfor i, equation_str in enumerate(pools):\n pool_id = \"p\" + str(i + 1)\n conc_sol.make_aggregate_solution(\n pool_id, equation=equation_str, update=True)\n print(pool_id, conc_sol[pool_id])\n```\n\n p1 \n p2 \n p3 \n\n\nThis method utilizes the solutions for the individual metabolites over the time range input, and then creates new solutions to represent the behavior of those pools. \n\n\n```python\nplot_time_profile(\n conc_sol, observable=[\"p1\", \"p2\", \"p3\"], legend=\"best\",\n plot_function=\"semilogx\",\n xlabel=\"time\", ylabel=\"Concentrations\",\n title=(\"Pool profile\", {\"size\": \"large\"}));\n```\n\n#### Compute and plot the fluxes\nA similar process as above can be utilized to obtain behavior of the net flux through a group of reactions. Note that the `MassSolution.make_aggregate_solution` method relies on the `sympy.sympify` function and can therefore utilize specific methods, such as the absolute value function, in the string as well. \n\n\n```python\nflux_sol.make_aggregate_solution(\n \"v_net\", equation='Abs(v1) + Abs(v2) + Abs(v3)', update=True)\n```\n\n\n\n\n {'v_net': }\n\n\n\nAgain, this method obtains the solutions for the individual fluxes over the time range input, and then creates new solutions to represent the behavior of various flux combinations. \n\n\n```python\nplot_time_profile(\n flux_sol, observable=[\"v_net\"], legend=\"best\",\n plot_function=\"semilogx\", xlabel=\"time\", ylabel=\"Fluxes\", \n title=(\"Net Flux\", {\"size\": \"large\"}));\n```\n\n#### Plot phase portraits of pools\n\n\n```python\nfig, ax = plt.subplots(1, 1, figsize=(5, 5))\nplot_tiled_phase_portraits(\n conc_sol, observable=[\"p1\", \"p2\", \"p3\"], ax=ax,\n plot_tile_placement=\"lower\",\n annotate_time_points_legend=\"right outside\");\n```\n\nHere, we can see that all of the defined pools are dynamically independent of one another.\n\n## Summary\n\n* Network dynamics are described by dynamic mass balances $(d\\textbf{x}/dt = \\textbf{Sv}(\\textbf{x}; \\textbf{k}))$ that are formulated after applying a series of simplifying assumptions \n\n* To simulate the dynamic mass balances we have to specify the numerical values of the kinetic constants $(\\textbf{k})$, the initial conditions $(\\textbf{x}_0)$, and any fixed boundary fluxes.\n\n* The equations with the initial conditions can be integrated numerically. \n\n* The solution contains numerical values for the concentration variables at discrete time points. The solution is graphically displayed as concentrations over time, or in a phase portrait.\n\n* The solution can be post-processed following its initial analysis to bring out special dynamic features of the network. Such features will be described in more detail in the subsequent notebooks.\n\n$\\tiny{\\text{\u00a9 B. \u00d8. Palsson 2011;}\\ \n\\text{This publication is in copyright.}\\\\ \n\\text{Subject to statutory exception and to the provisions of relevant collective licensing agreements,}\\\\ \\text{no reproduction of any part may take place without the written permission of Cambridge University Press.}}$\n", "meta": {"hexsha": "5611f3e8c29b0e3929d26647b3246215f55899d1", "size": 303414, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/education/sb2/chapters/sb2_chapter3.ipynb", "max_stars_repo_name": "SBRG/MASSpy", "max_stars_repo_head_hexsha": "1315c1d40be8feb8731c8143dbc9ba43bf8c78ff", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 13, "max_stars_repo_stars_event_min_datetime": "2020-07-13T00:48:13.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-27T15:42:15.000Z", "max_issues_repo_path": "docs/education/sb2/chapters/sb2_chapter3.ipynb", "max_issues_repo_name": "SBRG/MASSpy", "max_issues_repo_head_hexsha": "1315c1d40be8feb8731c8143dbc9ba43bf8c78ff", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 10, "max_issues_repo_issues_event_min_datetime": "2021-02-17T18:07:43.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-23T16:22:14.000Z", "max_forks_repo_path": "docs/education/sb2/chapters/sb2_chapter3.ipynb", "max_forks_repo_name": "SBRG/MASSpy", "max_forks_repo_head_hexsha": "1315c1d40be8feb8731c8143dbc9ba43bf8c78ff", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2020-01-15T00:48:49.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-04T07:01:17.000Z", "avg_line_length": 116.2060513213, "max_line_length": 44360, "alphanum_fraction": 0.8504518579, "converted": true, "num_tokens": 13483, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583124210896, "lm_q2_score": 0.611381973294151, "lm_q1q2_score": 0.42549636637847305}} {"text": "```python\nfrom IPython.core.display import HTML\nfrom IPython.display import Image\nHTML(\"\"\"\n\n\"\"\")\n```\n\n\n\n\n\n\n\n\n\n\n# *Circuitos El\u00e9tricos I - Semana 5*\n\n### Problema 1\n \nDetermine o circuito equivalente de Th\u00e9venin ($v_{th}$, $R_{th}$) do ponto de vista dos terminais $(a,b)$ do circuito abaixo.\n\na) Determine a $v_{th}$.\\\nb) Determine a corrente de curto-circuito $i_{cc}$.\\\nc) Determine a $R_{th}$ pelo m\u00e9todo da fonte auxiliar.\n\n\n```python\nImage(\"./figures/J7C1.png\", width=600)\n```\n\n\n```python\nimport sympy as sp\nimport numpy as np\n```\n\n\n```python\n# define as N vari\u00e1veis desconhecidas\nv1, v2, v3 = sp.symbols('v1, v2, v3')\n\n# define os sistema de N equa\u00e7\u00f5es\neq1 = sp.Eq(-5.5*v1+10.5*v2-8*v3,-120) \neq2 = sp.Eq(v1-3*v2+2*v3,25) \neq3 = sp.Eq(v1+3*v2-10*v3,0)\n\n# resolve o sistema\nsoluc = sp.solve((eq1, eq2, eq3), dict=True)\n\nv1 = np.array([sol[v1] for sol in soluc])\nv2 = np.array([sol[v2] for sol in soluc]) \nv3 = np.array([sol[v3] for sol in soluc]) \n\n\nprint('Solu\u00e7\u00e3o do sistema:\\n\\n v1 = %.2f V,\\n v2 = %.2f V,\\n v3 = %.2f V.' %(v1, v2, v3))\n```\n\n Solu\u00e7\u00e3o do sistema:\n \n v1 = 15.83 V,\n v2 = -2.50 V,\n v3 = 0.83 V.\n\n", "meta": {"hexsha": "9ffa0b3d47ce22bf5b376c6db86054a0ffbe05bc", "size": 174998, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Jupyter notebooks/Circuitos Eletricos I - Semana 5.ipynb", "max_stars_repo_name": "Jefferson-Lopes/ElectricCircuits", "max_stars_repo_head_hexsha": "bf2075dc0731cacece75f7b0b378c180630bdf85", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 9, "max_stars_repo_stars_event_min_datetime": "2021-05-19T18:36:53.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-18T16:30:17.000Z", "max_issues_repo_path": "Jupyter notebooks/Circuitos Eletricos I - Semana 5.ipynb", "max_issues_repo_name": "Jefferson-Lopes/ElectricCircuits", "max_issues_repo_head_hexsha": "bf2075dc0731cacece75f7b0b378c180630bdf85", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Jupyter notebooks/Circuitos Eletricos I - Semana 5.ipynb", "max_forks_repo_name": "Jefferson-Lopes/ElectricCircuits", "max_forks_repo_head_hexsha": "bf2075dc0731cacece75f7b0b378c180630bdf85", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 10, "max_forks_repo_forks_event_min_datetime": "2021-06-25T12:52:40.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-11T14:25:48.000Z", "avg_line_length": 977.6424581006, "max_line_length": 171132, "alphanum_fraction": 0.9532737517, "converted": true, "num_tokens": 487, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.8128673110375457, "lm_q1q2_score": 0.42547129158220365}} {"text": "# Probabilistic gradient pruning\n

                                        \n\n

                                        \n\nTutorial Author: Zirui Li, Hanrui Wang\n\n\nbugs:\n - Directly run all. In the first epoch, the process will get stuck in the inference phase.\n - modify `torchquantum/plugins/qiskit_processor.py:228 parallel=True` to `parallel=False` to fix the bug.\n\n\n## Outline\n - Introduction to probabilistic gradient pruning.\n - Train a model with gradient pruning.\n\n## Introduction to probabilistic gradient pruning\n\nBy carefully investigating the on-chip training process, we observe that small gradients tend to have large relative variations\nor even wrong directions under quantum noises. Also, not all gradient computations are necessary for the\ntraining process, especially for small-magnitude gradients. \n\nThe observations provide great opportunities for us to boost the robustness and efficiency of QNN on-chip learning. Inspired by that, we\npropose a **probabilistic gradient pruning** method to predict and only\ncompute gradients of high reliability. Hence we can reduce noise impact and also save the required number of circuit runs on real\nquantum machines.\n\n### Accumulation Window and Pruning Window\n\nWe separate all the training epochs into a repeat of an accumulation window followed by a pruning window. There are three important hyper-parameters in our probabilistic gradient pruning method:\n - accumulation window width \ud835\udc64\ud835\udc4e,\n - pruning ratio \ud835\udc5f,\n - pruning window width \ud835\udc64\ud835\udc5d .\n\nIn the accumulation window, we collect the information of gradients in each training step. In each step of the pruning window, we probabilistically exempt the calculations of some gradients based on the information collected from the accumulation window and pruning ratio.\n\n
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                                        \n\nThe accumulation window width\nand pruning window width decide the reliability of the gradient\ntrend evaluation and our confidence in it, respectively. The pruning\nratio can be tuned to balance the gradient variances caused by noise\nperturbation and pruning. Thus, the percentage of the time saved\nby our probabilistic gradient pruning method is $\ud835\udc5f\\frac{\ud835\udc64\ud835\udc5d}{\ud835\udc64\ud835\udc4e+\ud835\udc64\ud835\udc5d}\u00d7 100%$.\nIn\nour experiments, we find that the setting (\ud835\udc64\ud835\udc4e=1, \ud835\udc64\ud835\udc5d=2\u223c3, \ud835\udc5f=0.3\u223c0.5)\nusually works well in all cases\n\n##Train a model with probabilistic gradient pruning\n\n###Installation\n\n\n\n\n```python\n!pip install qiskit==0.32.1\n```\n\n Collecting qiskit==0.32.1\n Downloading qiskit-0.32.1.tar.gz (13 kB)\n Collecting qiskit-terra==0.18.3\n Downloading qiskit_terra-0.18.3-cp37-cp37m-manylinux2010_x86_64.whl (6.1 MB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 6.1 MB 5.0 MB/s \n \u001b[?25hCollecting qiskit-aer==0.9.1\n Downloading 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filename=qiskit-0.32.1-py3-none-any.whl size=11777 sha256=0da44b2b1281b0ae4c6e2615fcd1db002c7b2a0f27dfff24fc5292884896d11f\n Stored in directory: /root/.cache/pip/wheels/0f/62/0a/c53eda1ead41c137c47c9730bc2771a8367b1ce00fb64e8cc6\n Building wheel for dlx (setup.py) ... \u001b[?25l\u001b[?25hdone\n Created wheel for dlx: filename=dlx-1.0.4-py3-none-any.whl size=5718 sha256=f4189a2cdf623b2017d93227cacb73900731558a0f4c543a681229a95f7a2dda\n Stored in directory: /root/.cache/pip/wheels/78/55/c8/dc61e772445a566b7608a476d151e9dcaf4e092b01b0c4bc3c\n Building wheel for docplex (setup.py) ... \u001b[?25l\u001b[?25hdone\n Created wheel for docplex: filename=docplex-2.23.222-py3-none-any.whl size=662847 sha256=06dfed71b2318068e88b361dbfd3d66d11303eb17ed9f9c8988e0d606f30fcfe\n Stored in directory: /root/.cache/pip/wheels/a7/c9/fb/cee5a89f304e77a39c466e625ac2830434b76eb8384999d116\n Building wheel for python-constraint (setup.py) ... \u001b[?25l\u001b[?25hdone\n Created wheel for python-constraint: filename=python_constraint-1.4.0-py2.py3-none-any.whl size=24081 sha256=fffdc1553687d0239ea846797253de00732d5239d91b239e0fe3423ffec55084\n Stored in directory: /root/.cache/pip/wheels/07/27/db/1222c80eb1e431f3d2199c12569cb1cac60f562a451fe30479\n Successfully built qiskit dlx docplex python-constraint\n Installing collected packages: tweedledum, symengine, retworkx, python-constraint, ply, fastjsonschema, requests, qiskit-terra, ntlm-auth, lxml, inflection, cryptography, yfinance, websocket-client, requests-ntlm, quandl, qiskit-ignis, docplex, dlx, qiskit-ibmq-provider, qiskit-aqua, qiskit-aer, qiskit\n Attempting uninstall: requests\n Found existing installation: requests 2.23.0\n Uninstalling requests-2.23.0:\n Successfully uninstalled requests-2.23.0\n Attempting uninstall: lxml\n Found existing installation: lxml 4.2.6\n Uninstalling lxml-4.2.6:\n Successfully uninstalled lxml-4.2.6\n \u001b[31mERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.\n google-colab 1.0.0 requires requests~=2.23.0, but you have requests 2.27.1 which is incompatible.\n datascience 0.10.6 requires folium==0.2.1, but you have folium 0.8.3 which is incompatible.\u001b[0m\n Successfully installed cryptography-36.0.2 dlx-1.0.4 docplex-2.23.222 fastjsonschema-2.15.3 inflection-0.5.1 lxml-4.8.0 ntlm-auth-1.5.0 ply-3.11 python-constraint-1.4.0 qiskit-0.32.1 qiskit-aer-0.9.1 qiskit-aqua-0.9.5 qiskit-ibmq-provider-0.18.1 qiskit-ignis-0.6.0 qiskit-terra-0.18.3 quandl-3.7.0 requests-2.27.1 requests-ntlm-1.1.0 retworkx-0.11.0 symengine-0.9.2 tweedledum-1.1.1 websocket-client-1.3.1 yfinance-0.1.70\n\n\n\n```python\n!git clone https://github.com/zhijian-liu/torchpack.git\n```\n\n Cloning into 'torchpack'...\n remote: Enumerating objects: 3924, done.\u001b[K\n remote: Counting objects: 100% (329/329), done.\u001b[K\n remote: Compressing objects: 100% (217/217), done.\u001b[K\n remote: Total 3924 (delta 174), reused 210 (delta 94), pack-reused 3595\u001b[K\n Receiving objects: 100% (3924/3924), 1012.96 KiB | 16.08 MiB/s, done.\n Resolving deltas: 100% (2521/2521), done.\n\n\n\n```python\n%cd torchpack\n```\n\n /content/torchpack\n\n\n\n```python\n!pip install .\n```\n\n Processing /content/torchpack\n \u001b[33m DEPRECATION: A future pip version will change local packages to be built in-place without first copying to a temporary directory. We recommend you use --use-feature=in-tree-build to test your packages with this new behavior before it becomes the default.\n pip 21.3 will remove support for this functionality. You can find discussion regarding this at https://github.com/pypa/pip/issues/7555.\u001b[0m\n Collecting loguru\n Downloading loguru-0.6.0-py3-none-any.whl (58 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 58 kB 3.3 MB/s \n \u001b[?25hCollecting multimethod\n Downloading multimethod-1.7-py3-none-any.whl (9.5 kB)\n Requirement already satisfied: numpy in /usr/local/lib/python3.7/dist-packages (from torchpack==0.3.1) (1.21.5)\n Requirement already satisfied: pyyaml in /usr/local/lib/python3.7/dist-packages (from torchpack==0.3.1) (3.13)\n Requirement already satisfied: torch>=1.5.0 in /usr/local/lib/python3.7/dist-packages (from torchpack==0.3.1) (1.10.0+cu111)\n Requirement already satisfied: torchvision in /usr/local/lib/python3.7/dist-packages (from torchpack==0.3.1) (0.11.1+cu111)\n Requirement already satisfied: tqdm in /usr/local/lib/python3.7/dist-packages (from torchpack==0.3.1) (4.63.0)\n Requirement already satisfied: typing-extensions in /usr/local/lib/python3.7/dist-packages (from torch>=1.5.0->torchpack==0.3.1) (3.10.0.2)\n Requirement already satisfied: pillow!=8.3.0,>=5.3.0 in /usr/local/lib/python3.7/dist-packages (from torchvision->torchpack==0.3.1) (7.1.2)\n Building wheels for collected packages: torchpack\n Building wheel for torchpack (setup.py) ... \u001b[?25l\u001b[?25hdone\n Created wheel for torchpack: filename=torchpack-0.3.1-py3-none-any.whl size=34632 sha256=6e05974cb0c985fbc1e541043be80ee7da98c12608ff883943ee70843656c42a\n Stored in directory: /tmp/pip-ephem-wheel-cache-njdff8xi/wheels/4f/c9/1a/b8f5a127071f807be5ee14d3d364b428a891ad020a962af415\n Successfully built torchpack\n Installing collected packages: multimethod, loguru, torchpack\n Successfully installed loguru-0.6.0 multimethod-1.7 torchpack-0.3.1\n\n\n\n```python\n%cd ..\n```\n\n /content\n\n\nDownload and cd to the repo.\n\n\n```python\n!git clone https://github.com/mit-han-lab/torchquantum.git\n```\n\n Cloning into 'torchquantum'...\n remote: Enumerating objects: 10990, done.\u001b[K\n remote: Counting objects: 100% (7782/7782), done.\u001b[K\n remote: Compressing objects: 100% (3963/3963), done.\u001b[K\n remote: Total 10990 (delta 3911), reused 7243 (delta 3410), pack-reused 3208\u001b[K\n Receiving objects: 100% (10990/10990), 6.24 MiB | 18.09 MiB/s, done.\n Resolving deltas: 100% (5878/5878), done.\n\n\n\n```python\n%cd torchquantum\n```\n\n /content/torchquantum\n\n\nInstall torch-quantum.\n\n\n```python\n!pip install --editable .\n```\n\n Obtaining file:///content/torchquantum\n Requirement already satisfied: numpy>=1.19.2 in /usr/local/lib/python3.7/dist-packages (from torchquantum==0.1.0) (1.21.5)\n Requirement already satisfied: torchvision>=0.9.0.dev20210130 in /usr/local/lib/python3.7/dist-packages (from torchquantum==0.1.0) (0.11.1+cu111)\n Requirement already satisfied: tqdm>=4.56.0 in /usr/local/lib/python3.7/dist-packages (from torchquantum==0.1.0) (4.63.0)\n Requirement already satisfied: setuptools>=52.0.0 in /usr/local/lib/python3.7/dist-packages (from torchquantum==0.1.0) (57.4.0)\n Requirement already satisfied: torch>=1.8.0 in /usr/local/lib/python3.7/dist-packages (from torchquantum==0.1.0) (1.10.0+cu111)\n Requirement already satisfied: torchpack>=0.3.0 in /usr/local/lib/python3.7/dist-packages (from torchquantum==0.1.0) (0.3.1)\n Requirement already satisfied: qiskit>=0.32.0 in /usr/local/lib/python3.7/dist-packages (from torchquantum==0.1.0) (0.32.1)\n Collecting matplotlib>=3.3.2\n Downloading matplotlib-3.5.1-cp37-cp37m-manylinux_2_5_x86_64.manylinux1_x86_64.whl (11.2 MB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 11.2 MB 5.1 MB/s \n \u001b[?25hCollecting pathos>=0.2.7\n Downloading pathos-0.2.8-py2.py3-none-any.whl (81 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 81 kB 11.2 MB/s \n \u001b[?25hRequirement already satisfied: python-dateutil>=2.7 in /usr/local/lib/python3.7/dist-packages (from matplotlib>=3.3.2->torchquantum==0.1.0) (2.8.2)\n Requirement already satisfied: pillow>=6.2.0 in /usr/local/lib/python3.7/dist-packages (from matplotlib>=3.3.2->torchquantum==0.1.0) (7.1.2)\n Collecting fonttools>=4.22.0\n Downloading fonttools-4.31.2-py3-none-any.whl (899 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 899 kB 54.9 MB/s \n \u001b[?25hRequirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.7/dist-packages (from matplotlib>=3.3.2->torchquantum==0.1.0) (0.11.0)\n Requirement already satisfied: pyparsing>=2.2.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib>=3.3.2->torchquantum==0.1.0) (3.0.7)\n Requirement already satisfied: packaging>=20.0 in /usr/local/lib/python3.7/dist-packages (from matplotlib>=3.3.2->torchquantum==0.1.0) (21.3)\n Requirement already satisfied: kiwisolver>=1.0.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib>=3.3.2->torchquantum==0.1.0) (1.4.0)\n Requirement already satisfied: typing-extensions in /usr/local/lib/python3.7/dist-packages (from kiwisolver>=1.0.1->matplotlib>=3.3.2->torchquantum==0.1.0) (3.10.0.2)\n Collecting ppft>=1.6.6.4\n Downloading ppft-1.6.6.4-py3-none-any.whl (65 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 65 kB 4.5 MB/s \n \u001b[?25hRequirement already satisfied: multiprocess>=0.70.12 in /usr/local/lib/python3.7/dist-packages (from pathos>=0.2.7->torchquantum==0.1.0) (0.70.12.2)\n Requirement already satisfied: dill>=0.3.4 in /usr/local/lib/python3.7/dist-packages (from pathos>=0.2.7->torchquantum==0.1.0) (0.3.4)\n Collecting pox>=0.3.0\n Downloading pox-0.3.0-py2.py3-none-any.whl (30 kB)\n Requirement already satisfied: six>=1.7.3 in /usr/local/lib/python3.7/dist-packages (from ppft>=1.6.6.4->pathos>=0.2.7->torchquantum==0.1.0) (1.15.0)\n Requirement already satisfied: qiskit-aqua==0.9.5 in /usr/local/lib/python3.7/dist-packages (from qiskit>=0.32.0->torchquantum==0.1.0) (0.9.5)\n Requirement already satisfied: qiskit-ignis==0.6.0 in /usr/local/lib/python3.7/dist-packages (from qiskit>=0.32.0->torchquantum==0.1.0) (0.6.0)\n Requirement already satisfied: qiskit-terra==0.18.3 in /usr/local/lib/python3.7/dist-packages (from qiskit>=0.32.0->torchquantum==0.1.0) (0.18.3)\n Requirement already satisfied: qiskit-aer==0.9.1 in /usr/local/lib/python3.7/dist-packages (from qiskit>=0.32.0->torchquantum==0.1.0) (0.9.1)\n Requirement already satisfied: qiskit-ibmq-provider==0.18.1 in /usr/local/lib/python3.7/dist-packages (from qiskit>=0.32.0->torchquantum==0.1.0) (0.18.1)\n Requirement already satisfied: scipy>=1.0 in /usr/local/lib/python3.7/dist-packages (from qiskit-aer==0.9.1->qiskit>=0.32.0->torchquantum==0.1.0) (1.4.1)\n Requirement already satisfied: h5py<3.3.0 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (3.1.0)\n Requirement already satisfied: sympy>=1.3 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (1.7.1)\n Requirement already satisfied: pandas in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (1.3.5)\n Requirement already satisfied: fastdtw<=0.3.4 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (0.3.4)\n Requirement already satisfied: dlx<=1.0.4 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (1.0.4)\n Requirement already satisfied: psutil>=5 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (5.4.8)\n Requirement already satisfied: quandl in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (3.7.0)\n Requirement already satisfied: yfinance>=0.1.62 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (0.1.70)\n Requirement already satisfied: docplex>=2.21.207 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (2.23.222)\n Requirement already satisfied: retworkx>=0.8.0 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (0.11.0)\n Requirement already satisfied: scikit-learn>=0.20.0 in /usr/local/lib/python3.7/dist-packages (from qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (1.0.2)\n Requirement already satisfied: requests>=2.19 in /usr/local/lib/python3.7/dist-packages (from qiskit-ibmq-provider==0.18.1->qiskit>=0.32.0->torchquantum==0.1.0) (2.27.1)\n Requirement already satisfied: websocket-client>=1.0.1 in /usr/local/lib/python3.7/dist-packages (from qiskit-ibmq-provider==0.18.1->qiskit>=0.32.0->torchquantum==0.1.0) (1.3.1)\n Requirement already satisfied: urllib3>=1.21.1 in /usr/local/lib/python3.7/dist-packages (from qiskit-ibmq-provider==0.18.1->qiskit>=0.32.0->torchquantum==0.1.0) (1.24.3)\n Requirement already satisfied: requests-ntlm>=1.1.0 in /usr/local/lib/python3.7/dist-packages (from qiskit-ibmq-provider==0.18.1->qiskit>=0.32.0->torchquantum==0.1.0) (1.1.0)\n Requirement already satisfied: jsonschema>=2.6 in /usr/local/lib/python3.7/dist-packages (from qiskit-terra==0.18.3->qiskit>=0.32.0->torchquantum==0.1.0) (4.3.3)\n Requirement already satisfied: symengine>0.7 in /usr/local/lib/python3.7/dist-packages (from qiskit-terra==0.18.3->qiskit>=0.32.0->torchquantum==0.1.0) (0.9.2)\n Requirement already satisfied: tweedledum<2.0,>=1.1 in /usr/local/lib/python3.7/dist-packages (from qiskit-terra==0.18.3->qiskit>=0.32.0->torchquantum==0.1.0) (1.1.1)\n Requirement already satisfied: ply>=3.10 in /usr/local/lib/python3.7/dist-packages (from qiskit-terra==0.18.3->qiskit>=0.32.0->torchquantum==0.1.0) (3.11)\n Requirement already satisfied: python-constraint>=1.4 in /usr/local/lib/python3.7/dist-packages (from qiskit-terra==0.18.3->qiskit>=0.32.0->torchquantum==0.1.0) (1.4.0)\n Requirement already satisfied: fastjsonschema>=2.10 in /usr/local/lib/python3.7/dist-packages (from qiskit-terra==0.18.3->qiskit>=0.32.0->torchquantum==0.1.0) (2.15.3)\n Requirement already satisfied: cached-property in /usr/local/lib/python3.7/dist-packages (from h5py<3.3.0->qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (1.5.2)\n Requirement already satisfied: pyrsistent!=0.17.0,!=0.17.1,!=0.17.2,>=0.14.0 in /usr/local/lib/python3.7/dist-packages (from jsonschema>=2.6->qiskit-terra==0.18.3->qiskit>=0.32.0->torchquantum==0.1.0) (0.18.1)\n Requirement already satisfied: attrs>=17.4.0 in /usr/local/lib/python3.7/dist-packages (from jsonschema>=2.6->qiskit-terra==0.18.3->qiskit>=0.32.0->torchquantum==0.1.0) (21.4.0)\n Requirement already satisfied: importlib-metadata in /usr/local/lib/python3.7/dist-packages (from jsonschema>=2.6->qiskit-terra==0.18.3->qiskit>=0.32.0->torchquantum==0.1.0) (4.11.3)\n Requirement already satisfied: importlib-resources>=1.4.0 in /usr/local/lib/python3.7/dist-packages (from jsonschema>=2.6->qiskit-terra==0.18.3->qiskit>=0.32.0->torchquantum==0.1.0) (5.4.0)\n Requirement already satisfied: zipp>=3.1.0 in /usr/local/lib/python3.7/dist-packages (from importlib-resources>=1.4.0->jsonschema>=2.6->qiskit-terra==0.18.3->qiskit>=0.32.0->torchquantum==0.1.0) (3.7.0)\n Requirement already satisfied: charset-normalizer~=2.0.0 in /usr/local/lib/python3.7/dist-packages (from requests>=2.19->qiskit-ibmq-provider==0.18.1->qiskit>=0.32.0->torchquantum==0.1.0) (2.0.12)\n Requirement already satisfied: certifi>=2017.4.17 in /usr/local/lib/python3.7/dist-packages (from requests>=2.19->qiskit-ibmq-provider==0.18.1->qiskit>=0.32.0->torchquantum==0.1.0) (2021.10.8)\n Requirement already satisfied: idna<4,>=2.5 in /usr/local/lib/python3.7/dist-packages (from requests>=2.19->qiskit-ibmq-provider==0.18.1->qiskit>=0.32.0->torchquantum==0.1.0) (2.10)\n Requirement already satisfied: ntlm-auth>=1.0.2 in /usr/local/lib/python3.7/dist-packages (from requests-ntlm>=1.1.0->qiskit-ibmq-provider==0.18.1->qiskit>=0.32.0->torchquantum==0.1.0) (1.5.0)\n Requirement already satisfied: cryptography>=1.3 in /usr/local/lib/python3.7/dist-packages (from requests-ntlm>=1.1.0->qiskit-ibmq-provider==0.18.1->qiskit>=0.32.0->torchquantum==0.1.0) (36.0.2)\n Requirement already satisfied: cffi>=1.12 in /usr/local/lib/python3.7/dist-packages (from cryptography>=1.3->requests-ntlm>=1.1.0->qiskit-ibmq-provider==0.18.1->qiskit>=0.32.0->torchquantum==0.1.0) (1.15.0)\n Requirement already satisfied: pycparser in /usr/local/lib/python3.7/dist-packages (from cffi>=1.12->cryptography>=1.3->requests-ntlm>=1.1.0->qiskit-ibmq-provider==0.18.1->qiskit>=0.32.0->torchquantum==0.1.0) (2.21)\n Requirement already satisfied: joblib>=0.11 in /usr/local/lib/python3.7/dist-packages (from scikit-learn>=0.20.0->qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (1.1.0)\n Requirement already satisfied: threadpoolctl>=2.0.0 in /usr/local/lib/python3.7/dist-packages (from scikit-learn>=0.20.0->qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (3.1.0)\n Requirement already satisfied: mpmath>=0.19 in /usr/local/lib/python3.7/dist-packages (from sympy>=1.3->qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (1.2.1)\n Requirement already satisfied: multimethod in /usr/local/lib/python3.7/dist-packages (from torchpack>=0.3.0->torchquantum==0.1.0) (1.7)\n Requirement already satisfied: loguru in /usr/local/lib/python3.7/dist-packages (from torchpack>=0.3.0->torchquantum==0.1.0) (0.6.0)\n Requirement already satisfied: pyyaml in /usr/local/lib/python3.7/dist-packages (from torchpack>=0.3.0->torchquantum==0.1.0) (3.13)\n Requirement already satisfied: multitasking>=0.0.7 in /usr/local/lib/python3.7/dist-packages (from yfinance>=0.1.62->qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (0.0.10)\n Requirement already satisfied: lxml>=4.5.1 in /usr/local/lib/python3.7/dist-packages (from yfinance>=0.1.62->qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (4.8.0)\n Requirement already satisfied: pytz>=2017.3 in /usr/local/lib/python3.7/dist-packages (from pandas->qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (2018.9)\n Requirement already satisfied: inflection>=0.3.1 in /usr/local/lib/python3.7/dist-packages (from quandl->qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (0.5.1)\n Requirement already satisfied: more-itertools in /usr/local/lib/python3.7/dist-packages (from quandl->qiskit-aqua==0.9.5->qiskit>=0.32.0->torchquantum==0.1.0) (8.12.0)\n Installing collected packages: ppft, pox, fonttools, pathos, matplotlib, torchquantum\n Attempting uninstall: matplotlib\n Found existing installation: matplotlib 3.2.2\n Uninstalling matplotlib-3.2.2:\n Successfully uninstalled matplotlib-3.2.2\n Running setup.py develop for torchquantum\n \u001b[31mERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.\n albumentations 0.1.12 requires imgaug<0.2.7,>=0.2.5, but you have imgaug 0.2.9 which is incompatible.\u001b[0m\n Successfully installed fonttools-4.31.2 matplotlib-3.5.1 pathos-0.2.8 pox-0.3.0 ppft-1.6.6.4 torchquantum-0.1.0\n\n\n\n\nChange PYTHONPATH and install other packages.\n\n\n```python\n%env PYTHONPATH=.\n```\n\n env: PYTHONPATH=.\n\n\nRun the following code to store a qiskit token. You can replace it with your own token from your IBMQ account if you like.\n\n\n\n\n```python\nfrom qiskit import IBMQ\nIBMQ.save_account('0238b0afc0dc515fe7987b02706791d1719cb89b68befedc125eded0607e6e9e9f26d3eed482f66fdc45fdfceca3aab2edb9519d96b39e9c78040194b86e7858', overwrite=True)\n```\n\n\n```python\n!pip install matplotlib==3.1.3\n```\n\n Collecting matplotlib==3.1.3\n Downloading matplotlib-3.1.3-cp37-cp37m-manylinux1_x86_64.whl (13.1 MB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 13.1 MB 5.1 MB/s \n \u001b[?25hRequirement already satisfied: python-dateutil>=2.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib==3.1.3) (2.8.2)\n Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib==3.1.3) (3.0.7)\n Requirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.7/dist-packages (from matplotlib==3.1.3) (0.11.0)\n Requirement already satisfied: kiwisolver>=1.0.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib==3.1.3) (1.4.0)\n Requirement already satisfied: numpy>=1.11 in /usr/local/lib/python3.7/dist-packages (from matplotlib==3.1.3) (1.21.5)\n Requirement already satisfied: typing-extensions in /usr/local/lib/python3.7/dist-packages (from kiwisolver>=1.0.1->matplotlib==3.1.3) (3.10.0.2)\n Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.7/dist-packages (from python-dateutil>=2.1->matplotlib==3.1.3) (1.15.0)\n Installing collected packages: matplotlib\n Attempting uninstall: matplotlib\n Found existing installation: matplotlib 3.5.1\n Uninstalling matplotlib-3.5.1:\n Successfully uninstalled matplotlib-3.5.1\n \u001b[31mERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.\n torchquantum 0.1.0 requires matplotlib>=3.3.2, but you have matplotlib 3.1.3 which is incompatible.\n albumentations 0.1.12 requires imgaug<0.2.7,>=0.2.5, but you have imgaug 0.2.9 which is incompatible.\u001b[0m\n Successfully installed matplotlib-3.1.3\n\n\n\n\n\n```python\n!ls artifact\n```\n\n aerbackend.py example2 example4 example6 README.md\n example1 example3 example5 example7\n\n\n\n```python\n!cp artifact/aerbackend.py ../../usr/local/lib/python3.7/dist-packages/qiskit/providers/aer/backends/ -r\n```\n\n### Import modules\n\n\n```python\nimport argparse\nimport os\nimport sys\nimport pdb\nimport json\nimport numpy as np\nimport torch\nimport torch.backends.cudnn\nimport torch.cuda\nimport torch.nn\nimport torch.utils.data\nimport torchquantum as tq\n\nfrom torch.optim.lr_scheduler import CosineAnnealingLR\n\nfrom torchquantum.datasets import MNIST\nfrom examples.gradient_pruning.q_models import *\nfrom torchpack.callbacks import (InferenceRunner, MeanAbsoluteError,\n MaxSaver, MinSaver,\n Saver, SaverRestore, CategoricalAccuracy)\nfrom examples.gradient_pruning.callbacks import LegalInferenceRunner, SubnetInferenceRunner, \\\n NLLError, TrainerRestore, AddNoiseInferenceRunner, GradRestore\n\n# from torchpack import distributed as dist\nfrom torchpack.environ import set_run_dir\nfrom torchpack.utils.config import configs\nfrom torchpack.utils.logging import logger\n\n```\n\n \n \n WARNING: The qiskit parameterization bug is not fixed!\n \n run python fix_qiskit_parameterization.py to fix it!\n\n\n###Function to build the callbacks\n\n\n```python\ndef get_subcallbacks(config):\n subcallbacks = []\n for subcallback in config:\n if subcallback['metrics'] == 'CategoricalAccuracy':\n subcallbacks.append(\n CategoricalAccuracy(name=subcallback['name'])\n )\n elif subcallback['metrics'] == 'MeanAbsoluteError':\n subcallbacks.append(\n MeanAbsoluteError(name=subcallback['name'])\n )\n elif subcallback['metrics'] == 'NLLError':\n subcallbacks.append(\n NLLError(name=subcallback['name'])\n )\n else:\n raise NotImplementedError(subcallback['metrics'])\n return subcallbacks\n\n\ndef make_callbacks(dataflow):\n callbacks = []\n for config in configs['callbacks']:\n if config['callback'] == 'InferenceRunner':\n callback = InferenceRunner(\n dataflow=dataflow[config['split']],\n callbacks=get_subcallbacks(config['subcallbacks'])\n )\n elif config['callback'] == 'LegalInferenceRunner':\n callback = LegalInferenceRunner(\n dataflow=dataflow[config['split']],\n callbacks=get_subcallbacks(config['subcallbacks'])\n )\n elif config['callback'] == 'SubnetInferenceRunner':\n callback = SubnetInferenceRunner(\n dataflow=dataflow[config['split']],\n callbacks=get_subcallbacks(config['subcallbacks']),\n subnet=config['subnet']\n )\n elif config['callback'] == 'AddNoiseInferenceRunner':\n callback = AddNoiseInferenceRunner(\n dataflow=dataflow[config['split']],\n callbacks=get_subcallbacks(config['subcallbacks']),\n noise_total_prob=config['noise_total_prob']\n )\n elif config['callback'] == 'SaverRestore':\n callback = SaverRestore()\n elif config['callback'] == 'Saver':\n callback = Saver(max_to_keep=config['max_to_keep'])\n elif config['callback'] == 'MaxSaver':\n callback = MaxSaver(config['name'])\n elif config['callback'] == 'MinSaver':\n callback = MinSaver(config['name'])\n elif config['callback'] == 'GradRestore':\n callback = GradRestore()\n else:\n raise NotImplementedError(config['callback'])\n callbacks.append(callback)\n\n return callbacks\n\n```\n\n###Load configs\nThe config file describes everything about the model structure and the hyper-parameters of the training process, including batch size, learning rate, number of epochs, and whether to use qiskit's processor(we use qiskit's noise processor to train and test our quantum circuit in the following example).\n\n\n```python\nconfigs.load('examples/gradient_pruning/configs.yml')\nif configs.debug.set_seed:\n torch.manual_seed(configs.debug.seed)\n np.random.seed(configs.debug.seed)\n```\n\n###Function to train\nIn this function, we create the dataset and the model according to configs. And we train our quantum model using qiskit's noise processor. The function will return a list of model accuracy after each epoch with respect to the number of inferences.\n\n\n```python\ndef train_with_configs(configs):\n if os.path.exists(\"runs/probabilistic_gradient_pruning/summary/scalars.jsonl\"):\n os.remove(\"runs/probabilistic_gradient_pruning/summary/scalars.jsonl\")\n else:\n print(\"runs/probabilistic_gradient_pruning/summary/scalars.jsonl does not exist\")\n\n device = torch.device('cuda')\n if isinstance(configs.optimizer.lr, str):\n configs.optimizer.lr = eval(configs.optimizer.lr)\n dataset = MNIST(\n root='./mnist_data',\n train_valid_split_ratio=[0.9, 0.1],\n digits_of_interest=[0, 1, 2, 3],\n n_test_samples=30,\n n_train_samples=50,\n n_valid_samples=30,\n )\n dataflow = dict()\n for split in dataset:\n sampler = torch.utils.data.RandomSampler(dataset[split])\n dataflow[split] = torch.utils.data.DataLoader(\n dataset[split],\n batch_size=configs.run.bsz,\n sampler=sampler,\n num_workers=configs.run.workers_per_gpu,\n pin_memory=True)\n\n model = QMultiFCModel0(configs.model.arch)\n\n if configs.qiskit.use_qiskit_train or configs.qiskit.use_qiskit_valid:\n from torchquantum.plugins import QiskitProcessor\n processor = QiskitProcessor(use_real_qc=configs.qiskit.use_real_qc, n_shots=configs.qiskit.n_shots, backend_name=configs.qiskit.backend_name)\n model.set_qiskit_processor(processor)\n\n model.to(device)\n\n total_params = sum(p.numel() for p in model.parameters())\n logger.info(f'Model Size: {total_params}')\n\n criterion = torch.nn.NLLLoss()\n optimizer = torch.optim.Adam(\n model.parameters(),\n lr=configs.optimizer.lr,\n weight_decay=configs.optimizer.weight_decay)\n scheduler = CosineAnnealingLR(optimizer, T_max=configs.run.n_epochs)\n\n from examples.gradient_pruning.trainers import ParamsShiftTrainer\n trainer = ParamsShiftTrainer(model=model,\n criterion=criterion,\n optimizer=optimizer,\n scheduler=scheduler)\n\n trainer.set_use_qiskit(configs)\n run_dir = 'runs/probabilistic_gradient_pruning/'\n set_run_dir(run_dir)\n\n logger.info(' '.join([sys.executable] + sys.argv))\n\n logger.info(f'Training started: \"{run_dir}\".' + '\\n' +\n f'{configs}')\n\n callbacks = make_callbacks(dataflow)\n\n trainer.train_with_defaults(\n dataflow['train'],\n num_epochs=configs.run.n_epochs,\n callbacks=callbacks)\n \n num_forward = []\n accu = []\n with open('runs/probabilistic_gradient_pruning/summary/scalars.jsonl', 'r') as json_file:\n json_list = list(json_file)\n\n for json_str in json_list:\n result = json.loads(json_str)\n if 'acc/test' in result.keys():\n num_forward.append(result['global_step'])\n accu.append(result['acc/test'])\n \n\n return num_forward, accu\n```\n\n\n```python\nnum_forward1, accu1 = train_with_configs(configs)\n```\n\n runs/probabilistic_gradient_pruning/summary/scalars.jsonl does not exist\n Downloading http://yann.lecun.com/exdb/mnist/train-images-idx3-ubyte.gz\n Downloading http://yann.lecun.com/exdb/mnist/train-images-idx3-ubyte.gz to ./mnist_data/MNIST/raw/train-images-idx3-ubyte.gz\n\n\n\n 0%| | 0/9912422 [00:00I                                         x HI x DI]. The required hyperparameters are the filter extent F and the stride S.\nThe pooling layers output is of dimension [WP x HP x DP]\n\n\\begin{equation}\n W_P = \\frac{W_I - F}{S + 1},\n \\quad\\quad\n H_P = \\frac{H_I - F}{S + 1},\n \\quad\\quad\n D_P = D_I\n\\end{equation}\n\nAccording to Karpathy the most common form is a [2x2] filter with a stride of 2.\nIf we stick to our example this is what would happen if we apply max pooling:\n\n\\begin{equation}\n \\begin{matrix} \n 3 & 7 & 11 & 6\\\\ \n 5 & 5 & 4 & 1\\\\ \n 4 & 2 & 3 & 1\\\\\n -2 & 0 & 3 & 1\n \\end{matrix}\n \\quad\n = \n \\quad\n \\begin{matrix} \n 7 & 11\\\\ \n 4 & 3\n \\end{matrix}\n\\end{equation}\n\n## CNN architectures\n\nDeciding on which CNN architecture to use is quite difficult as there is no concrete answer to that question. According to Andrey Karpathys Lecture notes[15] one should use a neural network as large as the accessible computational power allows it because the larger the network the larger the space of representable functions.\nKarpathy mentions that CNNs usually follow the following architecture pattern:\n\n`INPUT -> [[CONV -> RELU]*N -> POOL?]*M -> [FC -> RELU]*K -> FC`\n - 0 <= N <= 3\n - M >= 0\n - 0 <= K < 3 (usually)\n - * : repetition\n - ? : optional\nHe also says that there are very few situations where a CNN need to be trained from scratch and recommends using a pretrained model, with the currently best working architecture, from ImageNet.\n\nRegarding layer sizing patterns there are some recommendations:\n - **Input Layer:** Should be divisible by 2\n \n - **Conv Layer:**\n - use small kernels K: 3x3 and at most 5x5\n - use stride S = 1\n - Zero-Padding so that output dimension corresponds to input dimension. Which is the case when the following equation is applied.\n \n \\begin{equation}\n P = \\frac{K-1}{2}\n \\end{equation}\n \n \n - **Pooling Layer:** Here the most common way is to use K=2, S=2 and MAX-Pooling\n \nOver the years different types of CNN-Architectures haven been reducing the top-5 loss on imagenet.[22] The first one to significantly drop that loss, from 26% to 15.3%, was AlexNet in 2012. In 2014 the VGGNet made it's appearance in the same challenge, together with GoogLeNet/Inception which won the competition dropping the loss to 6.67%. ResNet appeared in 2015 and took the loss down to 3.57%. \nToday state of the art might be ResNet or googles Inception. New architectures keep appearing, one of them is DenseNet which was used to detect infected malaria cells in the [densenet_implementation_kernel](https://github.com/lai-la/MalariaProject/blob/master/malaria_project_densenet_implementation.ipynb) belonging to this notebook.\n\n\n\n### Dense Convolutional Network (DenseNet)\n\n\n\nOur main focus is on Tensorflow and therefore Keras. Keras provides a DenseNet implementation which we are using in our malaria_project_densenet_implementation.ipynb Kernel. The Keras documentation references a paper called \u201cDensely Connected Convolutional Networks (CVPR 2017 Best Paper Award)\u201d[9]. \n\nA DenseNet is a CNN architecture published in 2016. The main difference between traditional CNNs and the DenseNet is the density of the connected layers. Every layer is connected to every other layer. \n\nSince all layers are connected each one gets additional inputs from all previous layers. Those features maps get concatenated. Layer x has x inputs. The concatenated feature maps are the global state of the network. The global state is available to all layers in the network and therefore does not have to be recalculated or replicated over and over again. \n\nAnother advantage is that the DenseNet needs fewer parameters because the feature maps are passed directly to all the following layers and do not have to be calculated again.Narrow layers with a low number of filters create small feature maps to pass on and to add to the global state. It is efficient to train short connections from input to output.\nThe computational power needed to perform the training is lower due to the parameter efficiency. In consequence the efficiency to train the model increases and the time to do so decreases.\n\nFurthermore is the DenseNet a solution to the vanishing gradient problem. The vanishing gradient problem describes the washing out process of the input which decreases to insignificance as it passes through the layers. Activation functions map inputs into a small range. Even a large input change won\u2019t make a significant difference in the output. The gradient is small. After some activation mappings into a continuously smaller range, the output can\u2019t be changed even by big changes in the weights. The DenseNet solves this issue by shorting the way of the gradient through the network. Conquering the vanishing gradient problem allows the construction of deeper CNNs.\n\nOptimizing the feature maps reuse and thereby the information flow through the network results in a regularizing effect also on small datasets and prevents overfitting.\n\nFinally the new architecture results in higher accuracy (see Table 2)[z]. \n\n\n## CNNs for cell image classification\n\nClassifying cells is a challenging task. It is often crucial for medical diagnosis, the prevention of diseases and also for personalized treatment. Correct classification with high precision is extremely difficult, not only for computer vision but also for specialists. [10]\nUsing CNNs for cell classification tasks has proven to be very efficient in some cases. Different research groups have obtained Results where the predictions made by the ConvNet have outperformed those obtained by specialists.[10]\nThe high amount of papers written on the subject of using CNNs for different kinds of cell image classification underlines the importance of the topic. Whether it is for classifying Human Epithelial-2 (HEp-2) Cells, which are important for the diagnosis of several autoimmun diseases [11] or to differenciate between normal breast epithelial, an agressive and a less agressive form of breast cancer,[10] or again to detect malaria in blood cells[12].\nMalaria, as mentioned before, is a global health threat.[13] It caused 438.000 deaths in 2014 and has an economic inpact of about 12 billion dollar per year. Malaria is usually diagnosed visually by technicians who analyze smears of blood with a microscope. The accuracy of the diagnosis therefore depends on the experience of the technician.\nBeing a curable disease Malaria could be cured and also prevented and controlled in a more effective way if it's diagnosis would be more accurate.\nIn the paper by Dong et al. a 17 layer CNN has been used for the task of detecting malaria in blood smears. For the training 27.578 images, with an equal number of uninfected and infected cells, were used. Those images had been normalized and preprocessed to have the dimensions 44x44x3. 90% of the data was then used for training and the remaining 10 % were used for backpropagation validation. They compared the model to a transger learning model and obtained much better results with the new model. The accuracy was 97,37% and the F1 score 97,36%.\nAll this is to show that CNNs have an enormous potential to be used to help saving lifes and if used in the right places maybe also to make good diagnosis accessible in an economic way to a bigger part of the earth population.\n\n\n# Training CNNS\n\n## Activation Functions\n\nActivation functions are nodes of the CNN that are usually placed after convolutional layers or at the end. They are non-linearities and responsable for deciding whether a neuron will fire or not.[14] It's output is passed to the next layer as an input.\nThere are different types of activation functions:\n - **ReLU** (Rectified Linear Unit) has been very popular lately. When using ReLU the activations are thresholded at 0.[2]\n \n \\begin{equation}\n f(x) = max(0,x)\n \\end{equation}\n\n Its advantages are that the convergence of statistical gradient decent is being accelerated and that it doesn't use any expensive operations.\n Its biggest disadvantag is that a lot the nodes can \"die\" during training\n \n \n - **Leaky ReLU:** is trying to fix the problem of dying neurons faced by ReLU by having a negative slope of about 0.01. According to Karpathy some people have obtained good results using it. But those results are inconsistent.\n \n \\begin{equation}\n f(x) = 1(x < 0)(\\alpha x)+1(x>=0)(x)\n \\end{equation}\n \n - **Maxout:** generalizes ReLU and Leaky ReLU. Neurons where Maxout is used benefit from ReLU advantages and are not faced with the \"dying-ReLU\"-Problem.\n \n \\begin{equation}\n max(w^T_1 + b_1, w^T_2x + b_2)\n \\end{equation}\n \nAccording to Karpathy's lecture nodes ReLU should be used when working with CNNs.[15] If it is causing problems Leaky ReLU or Softmax should be used and Tanh can be tryed. Sigmoid shouldn't be used.\n\n\n## Weight Initialization\nBefore training a Convolutional Neural Network and its weights, the weights have to be initialized. It is important to avoid initializing all weights with 0. Initializing weights with zero would prevent the Network from training[16]. The derivative with respect to the loss function would be the same for every weight and would not changed in the next iteration and therefore not in any iteration. \nAlso random weights are problematic since they facilitate vanishing gradients and exploding gradients. Vanishing gradients are explained in the DenseNet section. Exploding gradients result from positive, large weights in combination with small activations. The formula weight - activation * cost shows, that big weights can produce big steps towards the minima. So big that they can be to big to pinpoint the optimum. The best practice for initializing weights depends on the activation function as we have seen with the vanishing and exploding gradient problems. Relu activation function is robust and allows bigger weights. With a normal distribution and a variance the weights can be calculated as follows:\n\n\\begin{equation}\n \\sqrt{2/size^{[l-1]}}\n\\end{equation}\n\n\n## Regularization\nRegularization generalizes the learning algorithm with small modifications to the network to reduce overfitting. Overfitting is a term to describe a poor training process of a network, which adapts too much to the training data and underperforms on the test data. Commonly used regularization techniques are L2 & L1 regularization, Dropout and Data Augmentation. \n\n\n## Data Augmentation\n\nA bigger data set can improve the accuracy of an image classifier simply by providing more training data[17]. Lack of data can be a problem especially in the medical sector because of the highly delicate nature of the data. The privacy concerns towards health related data do have the side effect of insufficient data gathering. Data augmentation could be a helpful way to close that gap. Small data sets are risking to train an overfitting CNN. With increasing size the data set will minimize the risk of overfitting.\n\nThere are two options to increase the size of the malaria cell image data set by data augmentation. One is to make the set as big as possible as has been shown by Google while training their Google Speech with a trillion words corpus[18]. The second possibility is to thoughtfully decide on specific data augmentation techniques to use and enhance the data set carefully by proven and filtered features. The first option is tempting unfortunately we don\u2019t have the training resources for that amount of data. So we will choose the second option and we will decide on specific data augmentation techniques to carefully increase the amount of reliable data and hence improve the accuracy of our CNN.\n\nThe choice of data augmentation features depends on the data we are dealing with. Our dataset consists of low quality images of roughly the same structure and size. The images are quite homogenous without many angles or colour differences, nor do they have blurry parts or weather differences. Still they vary in a small range and we will try to optimize the data set accordingly. \n\nThe malaria cell images do not have a top or bottom, left or right, which means the cells images are provided in any random rotation. Therefore it is beneficial to teach the CNN to identify cell images from every rotation and we will rotate the original cell images in various angels. The same goes for flipping the images. There is no right way in taking a cell image, so we will adapt the data set and add a flipped version of all images. The color of the cell images depends on the camera and its configuration. Some images are brighter, some are darker. By changing the color palette of the duplicated images the data set will improve further.\n\n\n## Optimization\n\nFrameworks like Keras allow the user to use optimizers easily. This section will not explain the maths behind the different algorithms but just introduce the most famous ones quickly to get an idea of what to use.\nOptimizers are used to minimize the cost or loss. The most famous one used to optimize Neural Networks is Gradient Decent.[19] However there are three variants of Gradient Decent (GD):\n - Batch-GD: used on the entire dataset, this means that all gradients have to be computed in order to perform one update. It can be very slow and might not fit into memory. It also risks to stay trapped in local minima or saddle points. It is not commonly used in practice as an optimizer for CNNs.\n - Stochastic-GD: Performs one update per parameter. It is much faster and can be used for online learning. Eventually local minima and saddle points can be crossed.\n - Mini-batch-GD: performs an update on a batch of usually <256 training examples. It combines the best of Gd and SGD and is often used in Neural Networks.\n \nThere are also a lot of GD optimization algorithms. Some of the most prominent ones are:\n - Momentum: accelerates SGD and reduce the oscillations usually observed within SGD. This is achieved by adding a friction. It is often explained with a ball rolling down a hill accumulating velocity unless there is some resistence slowing it down. This is basically what is used for the parameter update when using momentum: updates are reduced for gradients that change direction and get bigger for those who don't. The advantages are that convergence is obtained quicker and that there are less risks to get trapped in saddle points or local minima.\n - Adaptive Learning Rate Methods: Each gradient gets adjusted in an adaptive way. The idea behind those algorithms, like Adagrad, is to find a more or less straight path leading to the minimum.\n - RMSProp: Also an adaptive learning rate method, that has not been published but introduced in a lecture by Geoff Hinton. RMS stands for Root-Mean-Square. It prevents us from having an early-dying lerning rate. RMSProp and AdaDelta are pretty similar.\n - Adam (Adaptive Moment Estimator): Adam is the most commonly used optimizer when doing image classification. The Algorithm combines the adaptive learning rate and the momentum principle.\n \\begin{equation}\n m_t = \\beta _1m_{t-1} + ( 1 - \\beta_1) g_t\n \\end{equation}\n \n \\begin{equation}\n v_t = \\beta _2v_{t-1} + ( 1 - \\beta_2) g^2_t\n \\end{equation}\n \n m_t and v_t are initialized as vectors of zeros who bias towards zero if there decy rates, \\beta_1 and \\beta_2, are close to one during the initial time steps. To avoid this the two following correction terms are being used for the first initializations:\n \n \\begin{equation}\n \\hat{m}_t = \n \\frac{m_t}{(1 - \\beta^i_1)},\n \\quad\n \\hat{v}_t =\n \\frac{v_t}{(1 - \\beta^i_2)}\n \\end{equation}\n \nThe update of the parameters is performed according to the following equation:\n\n\\begin{equation}\n \\theta = \\theta_{t-1} - \n \\frac{\\alpha \\hat{m}_t}{\\sqrt{\\hat{v}_t} + \\epsilon}\n\\end{equation}\n\n\n# Evaluation of Results\n\nTo evaluate the performance of the CNN a confusion matrix can be used. It is especially useful when more then 2 classes are present within the dataset or then there is an unequal number of data belonging to the different classes.[20] With the confusion matrix, measures such as recall, precision, specifity, accuracy, and F1-Score can be measured.[2]]\nAccording to this, calculating the accuracy should be enough to adequately represent the quality of the binary classification model trained to differentiate malaria-infected and uninfected blood cells. Especially since an equal number of data is being used for both categories.\nBUT reconsidering the case it appears that when dealing with a classification task judging about someones health accuracy isn't exactly what we are looking for. The measue we would consider is called recall. \n\n\\begin{equation}\nrecall =\n\\frac{True Positives}{True Positives + False Negatives}\n\\end{equation}\n \nGetting a high Recall value would mean that most of the True Positives (TP) have been classified correctly but there are also a lot of False Positives. This is what we want because it is better to tell someone that he is sick and then find out that he isn't, then to tell someone he is healthy and risk that the disease goes unseen. Which would be a threat to his life.\n\n# Conclusion\nCNNs are the perfect choice when working on image classification tasks. This project tried to give a brief overview of what to consider when starting such a project. In practice it is extremly rare to build a CNN from scratch and frameworks such as TensorFlow and Keras make the task much easier. In the kernels accomanying this notebook a lot of the functions have been implemented by us to gain a better understanding of what actually happens.\nIn summary there is to say that when dealing with CNNs for the first time it is hardly possible to grasp the number and meaning of all involved parameters. Working with neural networks has a very steep learning curve, which is only visible after understanding the basic principles.\nWorking on this project was a fun yet sometimes frustrating task. As every time you understand something, something else appears. Wanting to understand everything doesn't help on completing a task in time, as it just takes to long. We tried our best, learned a lot and know that there is even more to be learned.\nThis project motivated us to pursue using CNNs for other projects in the future, which we will be able to begin with a higher level of understanding.\nAs for the meaning of CNNs for cell classification there seems to be an enormous potential which hopefully will be used to help people. IT in general can be used for so many things. Using it to save human lives is definitaly one of the most motivating ideas behind working in this field.\n\n\n# References\n[1] WHO (2019, March 27). Malaria. Retrieved from https://www.who.int/news-room/fact-sheets/detail/malaria\n\n[2] WHO (2018, November). World Malaria Report 2018. Retrieved from https://www.who.int/malaria/publications/world-malaria-report-2018/report/en/\n\n[3] Tangpukdee, N., Duangdee, C., Wilairatana, P., & Krudsood, S. (2009). Malaria diagnosis: a brief review. The Korean journal of parasitology.\n\n[4] Hirimutugoda, Y. M., & Wijayarathna, G. (2010). Image analysis system for detection of red cell disorders using artificial neural networks. Sri Lanka Journal of Bio-Medical Informatics.\n\n[5] Dong, Y., Jiang, Z., Shen, H., Pan, W. D., Williams, L. A., Reddy, V. V., & Bryan, A. W. (2017). Evaluations of deep convolutional neural networks for automatic identification of malaria infected cells. 2017 IEEE EMBS International Conference on Biomedical & Health Informatics.\n\n[6] Saurabh Yadav (2018, October 16). Brief Intro to Medical Image Analysis and Deep Learning. Retrieved from \nhttps://medium.com/@saurabh.yadav919/brief-intro-of-medical-image-analysis-and-deep-learning-810df940d2f7\n\n[7] Andrey Karpathy. Convolutional Neural Networks (CNNs / ConvNets), part of the lecture notes of course: CS231n Convolutional Neural Networks for Visual Recognition. Retrieved from http://cs231n.github.io/convolutional-networks/\n\n[8] Luo, W., Li, Y., Urtasun, R., & Zemel, R. (2016). Understanding the effective receptive field in deep convolutional neural networks. In Advances in neural information processing systems (pp. 4898-4906).\n\n[9] Huag, G, Liu, Z., van der Maaten, L., Weinberger, K. Q. (2016) Densely Connected Convolutional Networks. Retrieved from https://arxiv.org/abs/1608.06993\n\n[10] Oei, R. W., Hou, G., Liu, F., Zhong, J., Zhang, J., An, Z., ... & Yang, Y. (2019). Convolutional neural network for cell classification using microscope images of intracellular actin networks. PloS one, 14(3), e0213626.\n\n[11] Phan, H. T. H., Kumar, A., Kim, J., & Feng, D. (2016, April). Transfer learning of a convolutional neural network for HEp-2 cell image classification. In 2016 IEEE 13th International Symposium on Biomedical Imaging (ISBI) (pp. 1208-1211). IEEE.\n\n[12] Liang, Z., Powell, A., Ersoy, I., Poostchi, M., Silamut, K., Palaniappan, K., ... & Huang, J. X. (2016, December). CNN-based image analysis for malaria diagnosis. In 2016 IEEE International Conference on Bioinformatics and Biomedicine (BIBM) (pp. 493-496). IEEE.\n\n[13] Dong, Y., Jiang, Z., Shen, H., Pan, W. D., Williams, L. A., Reddy, V. V., ... & Bryan, A. W. (2017, February). Evaluations of deep convolutional neural networks for automatic identification of malaria infected cells. In 2017 IEEE EMBS International Conference on Biomedical & Health Informatics (BHI) (pp. 101-104). IEEE.\n\n[14] Udeme Udofia (2018, February 13). Basic Overview of Convolutional Neural Network (CNN). Retrieved from \nhttps://medium.com/@udemeudofia01/basic-overview-of-convolutional-neural-network-cnn-4fcc7dbb4f17\n\n[15] Andrey Karpathy. Convolutional Neural Networks (CNNs / ConvNets), part of the lecture notes of course: CS231n Convolutional Neural Networks for Visual Recognition Retrieved from https://cs231n.github.io/neural-networks-1/\n\n[16] Doshi, N. (26.03.2018) Deep Learning Best Practice (1) - Weight Initialization. Retrieved from https://medium.com/usf-msds/deep-learning-best-practices-1-weight-initialization-14e5c0295b94\n\n[17] Wang J. & Perez, L. (2017) The Effectiveness of Data Augmentation in Image Classi\ufb01cation using Deep Learning. Retrieved from http://cs231n.stanford.edu/reports/2017/pdfs/300.pdf\n\n[18] Halevy, A., Norvig, P. & Pereira, F. (2009) The Unreasonable Effectiveness of Data. Retrieved from https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/35179.pdf\n\n[19] Ruder, S. (2016). An overview of gradient descent optimization algorithms. arXiv preprint arXiv:1609.04747..\n\n[20] Jason Brownlee (2016, November 18). What is a Confusion Matrix in Machine Learning. Retrieved from \nhttps://machinelearningmastery.com/confusion-matrix-machine-learning/\n\n[21] Sarang Narkhede (2018, May 9). Understanding Confusion Matrix. Retrieved from\nhttps://towardsdatascience.com/understanding-confusion-matrix-a9ad42dcfd62\n\n[22] Faisal Shahbaz (2018, Noveber 11). Five Powerful CNN Architectures. Retrieved from https://medium.com/datadriveninvestor/five-powerful-cnn-architectures-b939c9ddd57b\n\n\n# License\nNotebook License (CC-BY-SA 4.0)\n\nThe following license applies to the complete notebook. It does however not apply to any referenced external media (e.g., images).\n\nMalaria Project - Theoretical Background\nDetecting Malaria cells using CNN and TF 2.0\nby Lukas Wagner, Laila Westphal\nis licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.\n\n\n```python\n\n```\n", "meta": {"hexsha": "79c4528fd57e8270c1f5e3a0ad746282ea134a67", "size": 37914, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "malaria_project_theoretical_background.ipynb", "max_stars_repo_name": "lai-la/MalariaProject", "max_stars_repo_head_hexsha": "7abd87f06672799c1e1c2edccfd9def9de401681", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "malaria_project_theoretical_background.ipynb", "max_issues_repo_name": "lai-la/MalariaProject", "max_issues_repo_head_hexsha": "7abd87f06672799c1e1c2edccfd9def9de401681", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "malaria_project_theoretical_background.ipynb", "max_forks_repo_name": "lai-la/MalariaProject", "max_forks_repo_head_hexsha": "7abd87f06672799c1e1c2edccfd9def9de401681", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 71.6710775047, "max_line_length": 963, "alphanum_fraction": 0.6908266076, "converted": true, "num_tokens": 7772, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.626124191181315, "lm_q2_score": 0.6791786861878392, "lm_q1q2_score": 0.425250205556949}} {"text": "

                                        Labor Economics

                                        \n\n### Navigation\nClick on a task in the content table in order to jump to it.
                                        \nClick on the hyperlinked parts of the task text to get back to the table of content.\n\n## Content\n- **[Introduction](#intro)**\n

                                        \n- **[Homework 1](#homework1): Get the data**

                                        \n- **[Homework 2](#homework2): Linkage validation**\n
                                        \n
                                        \n- **[Task 1: Insight & descriptive statistics](#task1)**

                                        \n - **[1.1 How many unique respondents are introduced into the panel each year?](#task1.1)**

                                        \n - **[1.2 What is the average yearly share of female respondents overall? How does the average annual share evolve from 2000-2020?](#task1.2)**

                                        \n - **[1.3 What is the average yearly share of respondents holding at least a college degree overall? How does the average annual share evolve from 2000-2020?](#task1.3)**

                                        \n - **[1.4 Plot the distribution of hourly wages for the years 2001, 2009 and 2018.](#task1.4)**

                                        \n - **[1.5 Is the data representative of the distribution of the population across all 50 U.S. states in 2017?](#task1.5)**

                                        \n - **[1.6 You have the task to create statistics of employment status, occupation, wage and education.\nYou are also asked to do this analysis for the subpopulations defined by age, race, sex and citizenship.\nHow would you proceed?](#task1.6)**\n
                                        \n
                                        \n- **[Task 2: Effects of wage on hours worked (intensive margin)](#task2)**

                                        \n - **[2.1 Examine the effect of wage on hours worked (intensive margin of labour supply) using an OLS.](#task2.1)**

                                        \n - **[2.2 Determine the correlation between wages and hours worked. What is the difference between OLS and correlation?](#task2.2)**

                                        \n - **[2.3 Add age, sex and citizenship to your model. Include the log of wage and hours. Interpret the results of a multivariate regression.](#task2.3)**

                                        \n - **[2.4 Run the following multivariate regression:

                                        \n\\\\( \\log hours = \\beta_0 + \\beta_1 \\log wage_i + \\beta_2 age_i + \\beta_3 educ2_i + \\beta_4 sex_i + \\beta_5 (educ2_i \\times sex_i)\\\\);

                                        with educ2 being a dummy variable for a respondent holding at least a college degree.\nInterpret your results.](#task2.4)**

                                        \n
                                        \n
                                        \n- **[Task 3: Visualisation of employment flows](#task3)**

                                        \n - **[3.1 Derive information on monthly employment flows. Disregard marginal in- and outflows. Do so for every month in your data.](#task3.1)**

                                        \n - **[3.2 Find out the monthly flows for the U.S. for a period of years covered by your cps sample aswell. Try to visualise the monthly flows](#task3.2)**

                                        \n - **[3.3 Calculate the monthly unemployment rate for for the months from 2018 onwards. Plot the time series.](#task3.3)**

                                        \n - **[3.4 Create a time series for the mean age of people being unemployed. What effect on the mean age did the Covid crisis show?](#task3.4)**\n
                                        \n
                                        \n- **[Task 4: Minimum wages and labor supply - a DiD-approach](#task4)**

                                        \n - **[4.1 Massachusetts increased its state level minimum wage from 10 \\\\$ to 11 \\\\$ per hour in 2017 (effective from 01.01.2017). The neighbouring New Hampshire kept its minimum wage at the federal level, i.e. 7.25 \\\\$ per hour. Using the CPS data set, run a DiD analysis to scrutinize the effect of the minimum wage increase on labour supply in Massachusetts (STATECENSUS == 14) relative to New Hampshire (STATECENSUS == 12).\n](#task4.1)**

                                        \n -**[Task 4.2 Repeat the analysis above for two or more neighbouring state of your choice for a suitable date.](#task4.2)**

                                        \n-**[Task 5 The effect of maximum duration of unemployment benefits on labor supply - Regression Discontinuity Design](#task5)**

                                        \n - **[5.1 In 2017, the maximum duration of unemployment benefits was
                                        - 12 weeks for Gerogia & Nevada,
                                        - 13 weeks for Missouri,
                                        - 14 weeks for Hawaii,
                                        - 16 weeks for Kentucky,
                                        - 20 weeks for Arizona, Minnesota & South Dakota,
                                        - 21 weeks for Indiana,
                                        - 28 weeks for Nebraska,
                                        - 30 weeks for Maryland
                                        - and 26 weeks in all other states.

                                        Use the complete dataset of IPUMS CPS for 2017.
                                        Create a dummy variable indicating an individual being unemployed longer than the maximum duration of benefits.](#task5.1)**

                                        \n - **[5.2 Have a look at the sanity of the data at hand.
                                        For every individual check the distance in weeks between two consecutive interviews as well as the difference between $durunemp_{t}$ and $durunemp_{t+1}$. What does strike your eye?](#task5.2)**

                                        \n - **[5.3 Create variables indicating whether an individual was employed or unemployed at the next interview.
                                        \nCreate another variable indicating whether a person might not have been unemployed between the interviews.](#task5.3)**

                                        \n - **[5.4 How many observations of unemployed individuals are still eligible for unemployment benefits?
                                        Find a way to detect the number of unemployment spells for an individual throughout her survey lifetime.](#task5.4)**

                                        \n - **[5.5 Create a running variable indicating the time in weeks an individual is away from her unemployment benefits to expire.
                                        Considering maximum duration of continuous unemployment, how many individuals were still eligible for benefits?](#task5.5)**

                                        \n - **[5.6 Perform an analysis of the effect of the expiration of unemployment benefits on the employment. Use a regression discontinuity design for linear fit and polynomial fit (2nd and 3rd degree).](#task5.6)**\n\n\n### [Exam](#content)\n- project work:\n - by yourself or in a group\n - presentation at the end of the term\n - brief written summary of your work, 3-5 pages\n\n### Exercise class\n- biweekly\n- working on data\n- small exercises\n- working on your own project\n\n### Tools\n- Stata\n
                                        Feel free to use any code you like for your own analysis (R, SAS, Stata, Python, ...). However, since we had to decide\non a tool to be used in this class, I am going to work with Stata.\n\n### Data\n- Current Population Survey (U.S.)\n- CPS-Supplements\n\n## Stata\n\ngetting started\n- https://ell.stackexchange.com/questions/91623/difference-between-by-and-via
                                        List of helpful links for beggining to work with stata
                                        \n\n- https://www.stata.com/features/
                                        List of features in Stata
                                        \n\n- vast amount of literature and tutorials available online
                                        \n\n\n\n\n## Current Population Survey\n\n- cps homepage\n- started in 1940, open data, free access\n- monthly survey of ~ 60.000 U.S. households\n- participants: over 15 years old, not part of the armed forces, not working in institutions such as prisons, long-term care hospitals and nursing homes\n- conducted by United States Census Bureau, commissioned by Bureau of Labor Statistics\n\n- questionnaire covers demographic and economic (especially labor force) topics\n- expanded by regular (at least yearly) supplements, such as\n - Social and Economic Supplement\n - Computer and Internet Use\n - Tobacco Use\n - Civic Engagement\n - ...\n \n (complete list of supplements: LINK)\n\n- process of data collection (monthly data):\n - households are interviewed in the week of the 19th of a month\n - revisited for three following months\n - revisited again in the next year and the very same four months\n \n
                                        4-8-4 pattern
                                        \n\n## IPUMS\n\n- problem: CPS solely assigns unique identifiers to households, not to individuals\n - i.e. the same household is revisited seven times - regardless of which family or individual answers the door\n- longitudinal research on an individual level complicated, but not impossible\n\n- linking items via matching making use of variables such as household identifier, line number within household and demographic characteristics (e.g. sex, age, race)
                                        \n - Drew, J.A., Flood, S. and Warren, J.R. (2014) \"Making full use of the longitudinal design of the Current Population Survey:\n Methods for linking records across 16 months\", Journal of Economic and Social Measurement, 39, pp. 121-144.
                                        DOI 10.3233/JEM-140388                                        \n- accordingly prepared microdata available in the Integrated Public Use Microdata Series (IPUMS-CPS) LINK\n\n\n## [Homework 1](#content)\n\nVisit https://cps.ipums.org/cps/ and create an account. Register with any valid e-mail adress, preferably your uni-mail adress. Select all the variables and samples potentially relevant to your project.\nUse the desciprtion of the variables to decide which to include in your extract. Try to minimise the number of variables in your extract.\nSubmit your request. It may take some time for your extract to be prepared and ready for download. The download link will be provided to you via e-mail once the data is processed.\n\n\nHint #1: If you already stumbled upon research papers dealing with questions similar to your project and working with CPS-data, you may include the author's choice of variables in your set as well.\n\nHint #2: take your time. This task is not trivial. Your work depends on the data at hand. Carefully chosen variables suited for answering a particular question facilitate your research.\n\n- We confine our analysis to the years from 2000 onwards\n- Choose your variables wisely: The number of variables inflates the size of the data set. E.g. 80 variables across monthly data between 2000 and 2019 constitute more than 7 gb of data\n- adding supplements you receive data sets as large as 50 gb and more\n\n- the bigger your extract,\n - the more time it takes the IPUMS server to process your request (from several hours to possibly several days)\n - the more time it takes you to download it\n - the more time and processing power is required to modify and analyse the data\n- Good practice: Select cases to reduce the size, drop redundant variables (e.g. recodes), do not include every supplement (since for most individuals in the monthly survey the values are blank anyway)\n\n\n\n## [Homework](#content)\n\nValidate the CPSID linkages following the hints in the description of the variable. (LINK to description)\n\nIt is recommended to verify the linkages by checking the consistency of the variables AGE, RACE and SEX.\nFor a linkage to be valid, a CPSID should have a unique gender and race. Though age is not a static variable, it should be in a certain range for the survey lifetime of an individual, i.e. between $AGE_{i, t=1} = AGE_{i, t=0} + d$ with $d \\in [0, 2]$.\n\nHint: This is not an easy task to do. You might come up with an own approach outside of Stata, you might be able to write proper code in Stata.\nFurthermore, you have to consider whether you want to check the linkages for two consecutive months only or if you want to check the whole survey lifetime of a participant.\n\nHere is my approach for validating the linkages between two consecutive months:\n\n\n```stata\n* loading data\ncd \"C:\\Users\\Hannes\\Documents\"\nuse \"session01.dta\", clear\n\n* Creating lead variables\nbysort cpsidp (mish): gen lead_mish = mish[_n+1]\nbysort cpsidp (mish): gen lead_age = age[_n+1] if lead_mish - mish == 1\nbysort cpsidp (mish): gen lead_race = race[_n+1] if lead_mish - mish == 1\nbysort cpsidp (mish): gen lead_sex = sex[_n+1] if lead_mish - mish == 1\n```\n\n \n C:\\Users\\Hannes\\Documents\n \n \n (1,773,756 missing values generated)\n \n (1,993,760 missing values generated)\n \n (1,993,760 missing values generated)\n \n (1,993,760 missing values generated)\n\n\n\n```stata\n* Creating lead variables\nbysort cpsidp (mish): gen lag_mish = mish[_n-1]\nbysort cpsidp (mish): gen lag_age = age[_n-1] if mish - lag_mish == 1\nbysort cpsidp (mish): gen lag_race = race[_n-1] if mish - lag_mish == 1\nbysort cpsidp (mish): gen lag_sex = sex[_n-1] if mish - lag_mish == 1\n```\n\n \n (1,773,756 missing values generated)\n \n (1,993,760 missing values generated)\n \n (1,993,760 missing values generated)\n \n (1,993,760 missing values generated)\n\n\n\n```stata\n* validation\ngen linked_f = .m\nreplace linked_f = 1 if lead_mish - mish == 1 & lead_race == race & ///\nlead_sex == sex & lead_age - age >= 0 & lead_age - age <= 1\n```\n\n \n \n lead_mish not found\n\n\n r(111);\n r(111);\n\n\n \n \n\n\n\n```stata\n* flag for linked backwards and/or forwards\ngen linked = linked_f == 1 | linked_b == 1\n```\n\n\n```stata\ntab linked\n```\n\n \n linked | Freq. Percent Cum.\n ------------+-----------------------------------\n 0 | 197,314 1.74 1.74\n 1 | 11,157,023 98.26 100.00\n ------------+-----------------------------------\n Total | 11,354,337 100.00\n\n\nWhat makes the linking invalid for those 197,5 k observations?\n\n\n```stata\ngen age_diff = lead_age - age\ntab age_diff if linked == 0 & !inrange(age_diff, 0, 2)\n```\n\n \n (1,993,760 missing values generated)\n \n \n age_diff | Freq. Percent Cum.\n ------------+-----------------------------------\n -52 | 1 0.00 0.00\n -51 | 2 0.01 0.01\n -50 | 4 0.01 0.02\n -49 | 2 0.01 0.03\n -48 | 3 0.01 0.03\n -47 | 4 0.01 0.05\n -46 | 6 0.02 0.06\n -45 | 9 0.03 0.09\n -44 | 17 0.05 0.14\n -43 | 12 0.03 0.17\n -42 | 17 0.05 0.22\n -41 | 20 0.06 0.27\n -40 | 22 0.06 0.34\n -39 | 30 0.09 0.42\n -38 | 38 0.11 0.53\n -37 | 28 0.08 0.61\n -36 | 43 0.12 0.73\n -35 | 49 0.14 0.87\n -34 | 42 0.12 0.99\n -33 | 53 0.15 1.14\n -32 | 58 0.16 1.30\n -31 | 63 0.18 1.48\n -30 | 75 0.21 1.70\n -29 | 78 0.22 1.92\n -28 | 95 0.27 2.19\n -27 | 93 0.26 2.45\n -26 | 105 0.30 2.75\n -25 | 103 0.29 3.04\n -24 | 105 0.30 3.34\n -23 | 132 0.37 3.71\n -22 | 147 0.42 4.13\n -21 | 160 0.45 4.58\n -20 | 204 0.58 5.16\n -19 | 189 0.54 5.69\n -18 | 192 0.54 6.24\n -17 | 203 0.58 6.81\n -16 | 216 0.61 7.43\n -15 | 241 0.68 8.11\n -14 | 268 0.76 8.87\n -13 | 308 0.87 9.74\n -12 | 330 0.94 10.68\n -11 | 354 1.00 11.68\n -10 | 632 1.79 13.47\n -9 | 518 1.47 14.94\n -8 | 607 1.72 16.66\n -7 | 665 1.89 18.55\n -6 | 729 2.07 20.61\n -5 | 1,339 3.80 24.41\n -4 | 1,283 3.64 28.05\n -3 | 1,812 5.14 33.18\n -2 | 2,689 7.62 40.80\n -1 | 6,466 18.33 59.13\n 3 | 2,292 6.50 65.63\n 4 | 1,598 4.53 70.16\n 5 | 1,762 4.99 75.15\n 6 | 1,012 2.87 78.02\n 7 | 876 2.48 80.51\n 8 | 688 1.95 82.46\n 9 | 625 1.77 84.23\n 10 | 788 2.23 86.46\n 11 | 482 1.37 87.83\n 12 | 340 0.96 88.79\n 13 | 321 0.91 89.70\n 14 | 316 0.90 90.60\n 15 | 297 0.84 91.44\n 16 | 227 0.64 92.08\n 17 | 208 0.59 92.67\n 18 | 224 0.63 93.31\n 19 | 217 0.62 93.92\n 20 | 221 0.63 94.55\n 21 | 171 0.48 95.03\n 22 | 175 0.50 95.53\n 23 | 139 0.39 95.92\n 24 | 144 0.41 96.33\n 25 | 114 0.32 96.66\n 26 | 127 0.36 97.02\n 27 | 117 0.33 97.35\n 28 | 106 0.30 97.65\n 29 | 99 0.28 97.93\n 30 | 102 0.29 98.22\n 31 | 89 0.25 98.47\n 32 | 72 0.20 98.67\n 33 | 66 0.19 98.86\n 34 | 60 0.17 99.03\n 35 | 49 0.14 99.17\n 36 | 46 0.13 99.30\n 37 | 49 0.14 99.44\n 38 | 29 0.08 99.52\n 39 | 22 0.06 99.58\n 40 | 29 0.08 99.67\n 41 | 28 0.08 99.74\n 42 | 33 0.09 99.84\n 43 | 13 0.04 99.88\n 44 | 15 0.04 99.92\n 45 | 8 0.02 99.94\n 46 | 8 0.02 99.96\n 47 | 4 0.01 99.97\n 48 | 3 0.01 99.98\n 49 | 1 0.00 99.99\n 50 | 4 0.01 100.00\n 51 | 1 0.00 100.00\n ------------+-----------------------------------\n Total | 35,278 100.00\n\n\n\n```stata\nnumlabel, add\ntab sex lead_sex if linked == 0 & sex ~= lead_sex\n```\n\n \n \n \n | lead_sex\n sex | 1 2 | Total\n -----------+----------------------+----------\n 1. male | 0 1,059 | 1,059 \n 2. female | 1,328 0 | 1,328 \n -----------+----------------------+----------\n Total | 1,328 1,059 | 2,387 \n\n\n\n```stata\ntab race lead_race if linked == 0 & race ~= lead_race\n```\n\n \n | lead_race\n race | 100 200 300 650 | Total\n ----------------------+--------------------------------------------+----------\n 100. white | 0 688 132 47 | 1,573 \n 200. black/negro | 643 0 17 8 | 789 \n 300. american indian/ | 151 22 0 0 | 226 \n 650. asian or pacific | 82 9 1 0 | 476 \n 651. asian only | 355 63 8 0 | 471 \n 652. hawaiian/pacific | 43 11 2 0 | 91 \n 801. white-black | 45 22 2 0 | 79 \n 802. white-american i | 59 4 4 0 | 74 \n 803. white-asian | 16 0 0 0 | 24 \n 804. white-hawaiian/p | 2 1 0 0 | 7 \n 805. black-american i | 4 1 0 0 | 8 \n 806. black-asian | 0 1 0 0 | 9 \n 808. american indian- | 0 0 0 0 | 1 \n 809. asian-hawaiian/p | 2 0 0 0 | 8 \n 810. white-black-amer | 1 0 0 0 | 3 \n 811. white-black-asia | 0 0 0 0 | 1 \n 812. white-american i | 1 0 0 0 | 1 \n 813. white-asian-hawa | 1 0 0 0 | 3 \n 816. white-black--haw | 0 0 0 0 | 1 \n 820. two or three rac | 0 0 0 0 | 9 \n ----------------------+--------------------------------------------+----------\n Total | 1,405 822 166 55 | 3,854 \n \n \n | lead_race\n race | 651 652 801 802 | Total\n ----------------------+--------------------------------------------+----------\n 100. white | 371 52 84 136 | 1,573 \n 200. black/negro | 64 4 25 3 | 789 \n 300. american indian/ | 13 0 1 35 | 226 \n 650. asian or pacific | 336 27 0 1 | 476 \n 651. asian only | 0 16 5 0 | 471 \n 652. hawaiian/pacific | 31 0 0 0 | 91 \n 801. white-black | 2 0 0 4 | 79 \n 802. white-american i | 5 0 2 0 | 74 \n 803. white-asian | 5 1 0 1 | 24 \n 804. white-hawaiian/p | 2 0 0 0 | 7 \n 805. black-american i | 1 0 1 0 | 8 \n 806. black-asian | 8 0 0 0 | 9 \n 808. american indian- | 1 0 0 0 | 1 \n 809. asian-hawaiian/p | 4 0 0 0 | 8 \n 810. white-black-amer | 0 0 2 0 | 3 \n 811. white-black-asia | 0 0 1 0 | 1 \n 812. white-american i | 0 0 0 0 | 1 \n 813. white-asian-hawa | 0 1 0 0 | 3 \n 816. white-black--haw | 0 0 0 0 | 1 \n 820. two or three rac | 0 0 0 0 | 9 \n ----------------------+--------------------------------------------+----------\n Total | 843 101 121 180 | 3,854 \n \n \n | lead_race\n race | 803 804 805 806 | Total\n ----------------------+--------------------------------------------+----------\n 100. white | 37 8 5 1 | 1,573 \n 200. black/negro | 4 0 14 0 | 789 \n 300. american indian/ | 0 0 2 0 | 226 \n 650. asian or pacific | 7 5 0 0 | 476 \n 651. asian only | 16 0 1 1 | 471 \n 652. hawaiian/pacific | 0 3 0 0 | 91 \n 801. white-black | 2 0 0 0 | 79 \n 802. white-american i | 0 0 0 0 | 74 \n 803. white-asian | 0 0 0 0 | 24 \n 804. white-hawaiian/p | 0 0 0 0 | 7 \n 805. black-american i | 0 0 0 0 | 8 \n 806. black-asian | 0 0 0 0 | 9 \n 808. american indian- | 0 0 0 0 | 1 \n 809. asian-hawaiian/p | 0 0 0 0 | 8 \n 810. white-black-amer | 0 0 0 0 | 3 \n 811. white-black-asia | 0 0 0 0 | 1 \n 812. white-american i | 0 0 0 0 | 1 \n 813. white-asian-hawa | 0 1 0 0 | 3 \n 816. white-black--haw | 0 0 0 0 | 1 \n 820. two or three rac | 0 0 0 0 | 9 \n ----------------------+--------------------------------------------+----------\n Total | 66 17 22 2 | 3,854 \n \n \n | lead_race\n race | 807 808 809 810 | Total\n ----------------------+--------------------------------------------+----------\n 100. white | 0 2 3 1 | 1,573 \n 200. black/negro | 2 0 0 2 | 789 \n 300. american indian/ | 0 1 0 0 | 226 \n 650. asian or pacific | 0 0 4 0 | 476 \n 651. asian only | 0 0 4 1 | 471 \n 652. hawaiian/pacific | 0 0 1 0 | 91 \n 801. white-black | 0 0 0 1 | 79 \n 802. white-american i | 0 0 0 0 | 74 \n 803. white-asian | 0 0 0 0 | 24 \n 804. white-hawaiian/p | 0 0 0 0 | 7 \n 805. black-american i | 0 0 0 1 | 8 \n 806. black-asian | 0 0 0 0 | 9 \n 808. american indian- | 0 0 0 0 | 1 \n 809. asian-hawaiian/p | 0 0 0 0 | 8 \n 810. white-black-amer | 0 0 0 0 | 3 \n 811. white-black-asia | 0 0 0 0 | 1 \n 812. white-american i | 0 0 0 0 | 1 \n 813. white-asian-hawa | 0 0 0 0 | 3 \n 816. white-black--haw | 1 0 0 0 | 1 \n 820. two or three rac | 0 0 1 0 | 9 \n ----------------------+--------------------------------------------+----------\n Total | 3 3 13 6 | 3,854 \n \n \n | lead_race\n race | 811 812 813 815 | Total\n ----------------------+--------------------------------------------+----------\n 100. white | 3 0 2 0 | 1,573 \n 200. black/negro | 1 1 0 0 | 789 \n 300. american indian/ | 0 0 0 1 | 226 \n 650. asian or pacific | 0 1 1 0 | 476 \n 651. asian only | 0 0 1 0 | 471 \n 652. hawaiian/pacific | 0 0 0 0 | 91 \n 801. white-black | 0 0 1 0 | 79 \n 802. white-american i | 0 0 0 0 | 74 \n 803. white-asian | 0 1 0 0 | 24 \n 804. white-hawaiian/p | 0 0 2 0 | 7 \n 805. black-american i | 0 0 0 0 | 8 \n 806. black-asian | 0 0 0 0 | 9 \n 808. american indian- | 0 0 0 0 | 1 \n 809. asian-hawaiian/p | 0 0 2 0 | 8 \n 810. white-black-amer | 0 0 0 0 | 3 \n 811. white-black-asia | 0 0 0 0 | 1 \n 812. white-american i | 0 0 0 0 | 1 \n 813. white-asian-hawa | 0 0 0 0 | 3 \n 816. white-black--haw | 0 0 0 0 | 1 \n 820. two or three rac | 0 0 8 0 | 9 \n ----------------------+--------------------------------------------+----------\n Total | 4 3 17 1 | 3,854 \n \n \n | lead_race\n race | 820 830 | Total\n ----------------------+----------------------+----------\n 100. white | 1 0 | 1,573 \n 200. black/negro | 0 1 | 789 \n 300. american indian/ | 0 0 | 226 \n 650. asian or pacific | 2 0 | 476 \n 651. asian only | 0 0 | 471 \n 652. hawaiian/pacific | 0 0 | 91 \n 801. white-black | 0 0 | 79 \n 802. white-american i | 0 0 | 74 \n 803. white-asian | 0 0 | 24 \n 804. white-hawaiian/p | 0 0 | 7 \n 805. black-american i | 0 0 | 8 \n 806. black-asian | 0 0 | 9 \n 808. american indian- | 0 0 | 1 \n 809. asian-hawaiian/p | 0 0 | 8 \n 810. white-black-amer | 0 0 | 3 \n 811. white-black-asia | 0 0 | 1 \n 812. white-american i | 0 0 | 1 \n 813. white-asian-hawa | 0 0 | 3 \n 816. white-black--haw | 0 0 | 1 \n 820. two or three rac | 0 0 | 9 \n ----------------------+----------------------+----------\n Total | 3 1 | 3,854 \n\n\n\n```stata\nby cpsidp: egen maxmish = max(mish)\ntab maxmish\n```\n\n \n \n \n maxmish | Freq. Percent Cum.\n ------------+-----------------------------------\n 1 | 8,880 0.08 0.08\n 2 | 28,653 0.25 0.33\n 3 | 62,278 0.55 0.88\n 4 | 700,541 6.17 7.05\n 5 | 123,755 1.09 8.14\n 6 | 170,636 1.50 9.64\n 7 | 302,570 2.66 12.31\n 8 | 9,957,024 87.69 100.00\n ------------+-----------------------------------\n Total | 11,354,337 100.00\n\n\n\n```stata\negen tag_help = tag(cpsidp month year)\negen obs = total(tag_help), by(cpsidp)\ndrop tag_help\n\ntab obs if obs == 1 & obs ~= maxmish\n\n```\n\n \n \n \n \n \n obs | Freq. Percent Cum.\n ------------+-----------------------------------\n 1 | 19,057 100.00 100.00\n ------------+-----------------------------------\n Total | 19,057 100.00\n\n\n\n```stata\ngen mish_delta = lead_mish - mish\ntab mish_delta if mish_delta ~= 1 & linked == 0\n```\n\n \n (1,773,756 missing values generated)\n \n \n mish_delta | Freq. Percent Cum.\n ------------+-----------------------------------\n 2 | 44,397 71.52 71.52\n 3 | 9,370 15.09 86.61\n 4 | 6,290 10.13 96.74\n 5 | 1,237 1.99 98.74\n 6 | 517 0.83 99.57\n 7 | 267 0.43 100.00\n ------------+-----------------------------------\n Total | 62,078 100.00\n\n\nSo why do we make such an effort looking at invalid linkages?\n\nBecause you ought to know the data you are about to dismiss as well as the data you would like to workt with.\nChecking you data for sanity might reduce the risk of measurement errors (wrong observations).\nApparently the concept of CPSIDP is not as trustworthy as one would have thought beforehand.\n\n
                                        Now that we have our data set, we can start working with it.
                                        \n\n\n# [Insight & descriptive statistics (Task 1)](#content)\n\n\n### 1.1 [How many unique respondents are introduced into the panel each year?](#content)\n\n\n```stata\n* count unique new individuals by year\nsort year month\negen tag_help = tag(cpsidp)\negen num_entries = total(tag_help), by(year)\ntabdisp year, c(num_entries)\n```\n\n \n \n \n \n \n -----------------------\n survey |\n year | num_entries\n ----------+------------\n 2000 | 118318\n 2001 | 90452\n 2002 | 88235\n 2003 | 85354\n 2004 | 85452\n 2005 | 88015\n 2006 | 85214\n 2007 | 84712\n 2008 | 83922\n 2009 | 85156\n 2010 | 84026\n 2011 | 82859\n 2012 | 82899\n 2013 | 80171\n 2014 | 82359\n 2015 | 83782\n 2016 | 80282\n 2017 | 77176\n 2018 | 74552\n 2019 | 70910\n 2020 | 53685\n 2021 | 26225\n -----------------------\n\n\n\n### [1.2 What is the average yearly share of female respondents overall? How does the average annual share evolve from 2000-2020?](#content)\n\n\n```stata\n* share of female responses\n\ntab sex\nreplace sex = sex - 1\nlabel define sex 0 \"male\" 1 \"female\"\nlabel value sex sex\n\n* share overall\nmean sex\n\n* or simply \ntab sex\n\n* share per year\nbysort year: egen fem = mean(sex)\ntable year, content(max fem)\n* or \ntab year sex, row\n```\n\n \n \n Variable | Obs Mean Std. Dev. Min Max\n -------------+---------------------------------------------------------\n sex | 11,354,337 1.514263 .4997965 1 2\n \n (11,354,337 real changes made)\n \n \n \n \n Mean estimation Number of obs = 11,354,337\n \n --------------------------------------------------------------\n | Mean Std. Err. [95% Conf. Interval]\n -------------+------------------------------------------------\n sex | .5142631 .0001483 .5139724 .5145539\n --------------------------------------------------------------\n \n \n \n ----------------------\n survey |\n year | max(fem)\n ----------+-----------\n 2000 | .5150207\n 2001 | .515788\n 2002 | .5150558\n 2003 | .515885\n 2004 | .5148885\n 2005 | .5155335\n 2006 | .5166667\n 2007 | .5157396\n 2008 | .5139523\n 2009 | .5114723\n 2010 | .5131696\n 2011 | .515455\n 2012 | .5148475\n 2013 | .5144394\n 2014 | .5130008\n 2015 | .5133336\n 2016 | .5131161\n 2017 | .5143098\n 2018 | .5136473\n 2019 | .5123831\n 2020 | .5126786\n 2021 | .5099174\n ----------------------\n \n \n +----------------+\n | Key |\n |----------------|\n | frequency |\n | row percentage |\n +----------------+\n \n survey | sex\n year | male female | Total\n -----------+----------------------+----------\n 2000 | 195,532 207,644 | 403,176 \n | 48.50 51.50 | 100.00 \n -----------+----------------------+----------\n 2001 | 257,579 274,376 | 531,955 \n | 48.42 51.58 | 100.00 \n -----------+----------------------+----------\n 2002 | 281,561 299,044 | 580,605 \n | 48.49 51.51 | 100.00 \n -----------+----------------------+----------\n 2003 | 281,372 299,837 | 581,209 \n | 48.41 51.59 | 100.00 \n -----------+----------------------+----------\n 2004 | 273,908 290,721 | 564,629 \n | 48.51 51.49 | 100.00 \n -----------+----------------------+----------\n 2005 | 271,854 289,287 | 561,141 \n | 48.45 51.55 | 100.00 \n -----------+----------------------+----------\n 2006 | 272,846 291,663 | 564,509 \n | 48.33 51.67 | 100.00 \n -----------+----------------------+----------\n 2007 | 271,411 289,054 | 560,465 \n | 48.43 51.57 | 100.00 \n -----------+----------------------+----------\n 2008 | 272,943 288,613 | 561,556 \n | 48.60 51.40 | 100.00 \n -----------+----------------------+----------\n 2009 | 276,748 289,746 | 566,494 \n | 48.85 51.15 | 100.00 \n -----------+----------------------+----------\n 2010 | 275,194 290,083 | 565,277 \n | 48.68 51.32 | 100.00 \n -----------+----------------------+----------\n 2011 | 269,893 287,110 | 557,003 \n | 48.45 51.55 | 100.00 \n -----------+----------------------+----------\n 2012 | 268,463 284,895 | 553,358 \n | 48.52 51.48 | 100.00 \n -----------+----------------------+----------\n 2013 | 264,413 280,139 | 544,552 \n | 48.56 51.44 | 100.00 \n -----------+----------------------+----------\n 2014 | 258,617 272,425 | 531,042 \n | 48.70 51.30 | 100.00 \n -----------+----------------------+----------\n 2015 | 253,397 267,282 | 520,679 \n | 48.67 51.33 | 100.00 \n -----------+----------------------+----------\n 2016 | 254,819 268,548 | 523,367 \n | 48.69 51.31 | 100.00 \n -----------+----------------------+----------\n 2017 | 249,060 263,736 | 512,796 \n | 48.57 51.43 | 100.00 \n -----------+----------------------+----------\n 2018 | 239,464 252,903 | 492,367 \n | 48.64 51.36 | 100.00 \n -----------+----------------------+----------\n 2019 | 229,355 241,004 | 470,359 \n | 48.76 51.24 | 100.00 \n -----------+----------------------+----------\n 2020 | 190,645 200,565 | 391,210 \n | 48.73 51.27 | 100.00 \n -----------+----------------------+----------\n 2021 | 106,146 110,442 | 216,588 \n | 49.01 50.99 | 100.00 \n -----------+----------------------+----------\n Total | 5,515,220 5,839,117 |11,354,337 \n | 48.57 51.43 | 100.00 \n\n\nIf you trying to calculate the share of a variable, yet it is not binary coded, you can try different approaches.\n\ntab gives you pretty much everything you want:\n\n>tab year sex, row\n\nIf you want to store the share as a ne variable, you have to come up with something different.\n\nLet us generate a variable for the share of unique female respondents:\n\n\n```stata\negen fem_counter = tag(year cpsidp) if sex == 1\negen fem_total = total(fem_counter), by(year)\negen both_counter = tag(year cpsidp)\negen respondents_total = total(both_counter), by(year)\ngen fem_share = fem_total / respondents_total\n```\n\n\n```stata\nsave \"workfile01.dta\", replace\n```\n\n file workfile01.dta saved\n\n\n\n```stata\ncollapse fem_share, by(year)\ntwoway (line fem_share year, sort), ytitle(share) xtitle(year) title(Share of female respondents) yscale(range(0.00 1.00)) ylabel(0(0.1)1)\n```\n\n\n \n\n\n\n\n```stata\nuse \"workfile01.dta\", clear\n```\n\nI strongly recommend to drop the auxiliary variables after that, so that the table stays clear and tidy.\n\n\n```stata\ndrop fem_* both_* respondents_*\n```\n\n\n### [1.3 What is the average yearly share of respondents holding at least a college degree overall? How does the average annual share evolve from 2000-2020?](#content)\n\nFor the average share of respondents holding a college degree you have to calculated the share of respondents instead the share of responses. You also have to keep in mind that individuals are paticipating up to four times a year in the survey.\nThe following code considers this as well as the fact that a person can obtain college degree between the waves of interviews. The highes degree obtained within a year will be counted.\n\n\n```stata\ntab educ\ngen college = .\nreplace college = 1 if inlist(educ, 125, 124, 092, 123, 111)\n\negen tag_college = tag(year cpsidp college)\negen tag_respondents = tag(year cpsidp)\n\negen college_total = total(tag_college), by(year)\negen respondents_total = total(tag_respondents), by(year)\n\ngen college_share = college_total / respondents_total\n```\n\n \n \n educational attainment recode | Freq. Percent Cum.\n ----------------------------------------+-----------------------------------\n 1. niu or blank | 9 0.00 0.00\n 2. none or preschool | 29,001 0.26 0.26\n 10. grades 1, 2, 3, or 4 | 55,654 0.49 0.75\n 20. grades 5 or 6 | 124,978 1.10 1.85\n 30. grades 7 or 8 | 221,665 1.95 3.80\n 40. grade 9 | 335,882 2.96 6.76\n 50. grade 10 | 414,008 3.65 10.40\n 60. grade 11 | 447,258 3.94 14.34\n 71. 12th grade, no diploma | 165,357 1.46 15.80\n 73. high school diploma or equivalent | 3,219,208 28.35 44.15\n 81. some college but no degree | 2,086,324 18.37 62.53\n 91. associate's degree, occupational/vo | 506,645 4.46 66.99\n 92. associate's degree, academic progra | 545,433 4.80 71.79\n 111. bachelor's degree | 2,106,549 18.55 90.34\n 123. master's degree | 798,942 7.04 97.38\n 124. professional school degree | 150,658 1.33 98.71\n 125. doctorate degree | 146,766 1.29 100.00\n ----------------------------------------+-----------------------------------\n Total | 11,354,337 100.00\n \n (11,354,337 missing values generated)\n \n (3,748,348 real changes made)\n \n \n \n \n \n\n\n\n```stata\nsave \"workfile01.dta\", replace\n```\n\n file workfile01.dta saved\n\n\n\n```stata\ncollapse college_share, by(year)\ntwoway (line college_share year, sort), ytitle(share) xtitle(year) title(Share of respondents holding at least a college degree) yscale(range(0.00 1.00)) ylabel(0(0.1)1)\n```\n\n\n \n\n\n\n\n```stata\nuse \"workfile01.dta\", clear\n```\n\n\n```stata\ndrop college* respondents*\n```\n\n\n### [1.4 Plot the distribution of hourly wages for the years 2001, 2009 and 2018.](#content)\n\n\n```stata\nmvdecode(hourwage), mv(999.99)\n\nqui histogram hourwage if year==2001, name(h2001) yscale(range(0 .1)) ylabel(0(0.02)0.1)\nqui histogram hourwage if year==2009, name(h2009) yscale(range(0 .1)) ylabel(0(0.02)0.1)\nqui histogram hourwage if year==2018, name(h2018) yscale(range(0 .1)) ylabel(0(0.02)0.1)\ngraph combine h2001 h2009 h2018, col(3)\n```\n\n\n \n\n\n\n\n### [1.5 Is the data representative of the distribution of the population across all 50 U.S. states in 2017?](#content)\n\n\n```stata\ntab statecensus\n```\n\n \n state (census code) | Freq. Percent Cum.\n ----------------------------------------+-----------------------------------\n 11. maine | 171,775 1.51 1.51\n 12. new hampshire | 208,615 1.84 3.35\n 13. vermont | 161,983 1.43 4.78\n 14. massachusetts | 206,391 1.82 6.59\n 15. rhode island | 163,331 1.44 8.03\n 16. connecticut | 200,132 1.76 9.80\n 21. new york | 508,971 4.48 14.28\n 22. new jersey | 249,338 2.20 16.47\n 23. pennsylvania | 358,675 3.16 19.63\n 31. ohio | 323,555 2.85 22.48\n 32. indiana | 185,484 1.63 24.12\n 33. illinois | 362,981 3.20 27.31\n 34. michigan | 289,975 2.55 29.87\n 35. wisconsin | 202,315 1.78 31.65\n 41. minnesota | 229,674 2.02 33.67\n 42. iowa | 183,425 1.62 35.29\n 43. missouri | 174,191 1.53 36.82\n 44. north dakota | 153,968 1.36 38.18\n 45. south dakota | 158,552 1.40 39.57\n 46. nebraska | 164,934 1.45 41.03\n 47. kansas | 159,595 1.41 42.43\n 51. delaware | 152,790 1.35 43.78\n 52. maryland | 210,504 1.85 45.63\n 53. district of columbia | 145,550 1.28 46.91\n 54. virginia | 225,874 1.99 48.90\n 55. west virginia | 170,770 1.50 50.41\n 56. north carolina | 241,747 2.13 52.54\n 57. south carolina | 154,789 1.36 53.90\n 58. georgia | 228,556 2.01 55.91\n 59. florida | 463,151 4.08 59.99\n 61. kentucky | 155,967 1.37 61.36\n 62. tennessee | 169,203 1.49 62.85\n 63. alabama | 166,799 1.47 64.32\n 64. mississippi | 138,288 1.22 65.54\n 71. arkansas | 145,579 1.28 66.82\n 72. louisiana | 156,549 1.38 68.20\n 73. oklahoma | 145,215 1.28 69.48\n 74. texas | 574,448 5.06 74.54\n 81. montana | 143,027 1.26 75.80\n 82. idaho | 147,604 1.30 77.10\n 83. wyoming | 152,347 1.34 78.44\n 84. colorado | 209,008 1.84 80.28\n 85. new mexico | 133,566 1.18 81.46\n 86. arizona | 153,674 1.35 82.81\n 87. utah | 155,571 1.37 84.18\n 88. nevada | 165,476 1.46 85.64\n 91. washington | 201,853 1.78 87.42\n 92. oregon | 165,215 1.46 88.87\n 93. california | 952,149 8.39 97.26\n 94. alaska | 148,309 1.31 98.57\n 95. hawaii | 162,899 1.43 100.00\n ----------------------------------------+-----------------------------------\n Total | 11,354,337 100.00\n\n\n\n```stata\negen distinct2017 = tag(cpsidp year) if year == 2017\ntab statecensus if distinct2017 == 1\n```\n\n \n \n \n state (census code) | Freq. Percent Cum.\n ----------------------------------------+-----------------------------------\n 11. maine | 1,399 0.89 0.89\n 12. new hampshire | 2,136 1.36 2.26\n 13. vermont | 2,058 1.31 3.57\n 14. massachusetts | 3,368 2.15 5.72\n 15. rhode island | 1,436 0.92 6.64\n 16. connecticut | 1,602 1.02 7.66\n 21. new york | 6,411 4.09 11.75\n 22. new jersey | 3,165 2.02 13.78\n 23. pennsylvania | 4,433 2.83 16.61\n 31. ohio | 4,024 2.57 19.17\n 32. indiana | 2,568 1.64 20.81\n 33. illinois | 4,561 2.91 23.73\n 34. michigan | 3,556 2.27 26.00\n 35. wisconsin | 2,151 1.37 27.37\n 41. minnesota | 2,077 1.33 28.70\n 42. iowa | 1,758 1.12 29.82\n 43. missouri | 2,238 1.43 31.25\n 44. north dakota | 2,239 1.43 32.68\n 45. south dakota | 1,710 1.09 33.77\n 46. nebraska | 1,934 1.23 35.00\n 47. kansas | 1,965 1.25 36.26\n 51. delaware | 1,712 1.09 37.35\n 52. maryland | 2,031 1.30 38.65\n 53. district of columbia | 2,646 1.69 40.34\n 54. virginia | 2,914 1.86 42.20\n 55. west virginia | 3,453 2.20 44.40\n 56. north carolina | 3,531 2.25 46.66\n 57. south carolina | 2,411 1.54 48.20\n 58. georgia | 3,560 2.27 50.47\n 59. florida | 6,820 4.35 54.82\n 61. kentucky | 2,024 1.29 56.12\n 62. tennessee | 3,014 1.92 58.04\n 63. alabama | 3,209 2.05 60.09\n 64. mississippi | 2,881 1.84 61.93\n 71. arkansas | 2,767 1.77 63.70\n 72. louisiana | 3,525 2.25 65.95\n 73. oklahoma | 2,258 1.44 67.39\n 74. texas | 9,006 5.75 73.14\n 81. montana | 2,903 1.85 74.99\n 82. idaho | 2,394 1.53 76.52\n 83. wyoming | 2,245 1.43 77.95\n 84. colorado | 1,882 1.20 79.15\n 85. new mexico | 2,710 1.73 80.88\n 86. arizona | 2,342 1.50 82.38\n 87. utah | 2,340 1.49 83.87\n 88. nevada | 1,945 1.24 85.12\n 91. washington | 2,846 1.82 86.93\n 92. oregon | 2,321 1.48 88.41\n 93. california | 13,892 8.87 97.28\n 94. alaska | 1,957 1.25 98.53\n 95. hawaii | 2,296 1.47 100.00\n ----------------------------------------+-----------------------------------\n Total | 156,624 100.00\n\n\nCompare the frequencies with official figures, which you can finde here:\ncensus.gov\n\n\n### [1.6 You have the task to create statistics of employment status, occupation, wage and education.
                                        You are also asked to do this analysis for the subpopulations defined by age, race, sex and citizenship.
                                        How would you proceed?](#content)\n\nThe idea of this task is to rise awareness of the different scale of measurements present in this data set.\nE.g., there is no point in calculating 5 point statistics for a nominal variable. Frequency tables are a helpful start for a data analysis.\n\nThere are different ways in stata to create meaningful statistics. When doing so for various subpopulations, you should avoid to render the statistics to granular, i.e. analsing too many groups. The more result tables you have, the harder it gets to compare them.\n\n\n\n```stata\n/* There exist a variety of commands that yield a quick overview of the data as well as some related basic statistics\nand further information.*/\ndescribe\n```\n\n\n```stata\ncodebook\n```\n\n\n```stata\ntab year\n```\n\n\n```stata\nsummarize age, detail\n```\n\n\n```stata\n// Create frequency tables using the tabulate command\ntabulate educ sex, row\n```\n\n\n```stata\nnumlabel, add\ntab educ\n```\n\nValue 92 represents associate's degrees. Those degrees academic degrees in the US and in other countries who introduced it. However, in most countries of the EU those degrees are not considered academic.\nNow we have to decide whether to label those degrees as academic college degrees or not.\nI choose not to do so, since the results of my analysis would be hard to compare with the situation in other countries where such a degree is unknown.\n\nIf you happen to work with this variable in your project, you may decide to label associate's degrees differently. However you proceed, please elaborate on your decision.\n\n\n```stata\n/* create new variable college that is 0 for every degree below college, 1 for every degree equivalent\nor above college and .m for missing values */\n\ngen college = 0\nreplace college = 1 if educ > 92\nreplace college = .m if educ == 1\n```\n\n\n```stata\ntab sex college, row\n```\n\n\n```stata\nbysort sex college: tab occ empstat, row\n```\n\n\n```stata\nnumlabel, add\ntabulate faminc\n```\n\n\n```stata\nmvdecode(faminc), mv(996 997 999)\n```\n\n\n```stata\ntabstat faminc, by(year) statistics(p25, median, p75)\n```\n\n\n```stata\ngraph bar (count), over(faminc) over(year, label(angle(v))) asyvars stack percent\n```\n\n\n# [2. Effects of wage on hours worked (intensive margin)](#content)\n\n\n```stata\n*load data\ncd \"C:\\Users\\Hannes\\Documents\"\nuse \"session01.dta\", clear\n```\n\n \n C:\\Users\\Hannes\\Documents\n \n\n\n\n```stata\ngen hwage_r = round(hourwage, 1)\n```\n\n\n```stata\ntab hwage_r\n```\n\n IPUM's description of hourwage:\n>HOURWAGE reports how much the respondent earned per hour in the current job, for those workers paid an hourly wage (and coded as \"2\" in PAIDHOUR). Amounts are expressed as they were reported to the interviewer; users must adjust for inflation using Consumer Price Index adjustment factors. Researchers should use the EARNWT weight with this variable.

                                        Users should note that HOURWAGE originally had two implied decimal places, but was revised so that the command files provided by IPUMS divide HOURWAGE by 100.

                                        HOURWAGE is one of the Outgoing Rotation/Earner Study questions.\n\n\n```stata\nsummarize hourwage, detail\n```\n\n\nThe 10th percentile is 999.99, which means \"not in universe\". Thus we are lacking information on hourly wage for at least 90% of the respondents.\nThe variable for wages in topcoded for the sake of discretion: Every wage that would result in a yearly income of more than \\$ 100,000 (1985-2002) and \\\\$ 150,000 (since 2003) is coded as 99.99 in hourwage.\n\n\n```stata\nmvdecode(hourwage), mv(999.99)\n```\n\n hourwage: 10334665 missing values generated\n\n\n\n```stata\nsummarize hourwage, detail\n```\n\nSo how many topcoded values do we have in hourwage?\n\n\n```stata\ntab hourwage if hourwage == 99.99\n```\n\n \n hourly wage | Freq. Percent Cum.\n ------------+-----------------------------------\n 99.99 | 147 100.00 100.00\n ------------+-----------------------------------\n Total | 147 100.00\n\n\n\n```stata\nsum hourwage if hourwage < 99.99\n```\n\n\n```stata\nmvdecode(hourwage), mv(99.99)\n```\n\n hourwage: 147 missing values generated\n\n\n\n```stata\ntab month if !missing(hourwage)\n```\n\n\n### [2.1 Examine the effect of wage on hours worked (intensive margin of labour supply) using an OLS.](#content)\n\n\n```stata\nnumlabel , add\ntabulate uhrswork1\n```\n\nuhrswork1 = 999 denotes missing values. But how do we handle varying hours (i.e. uhrswork1=997)? In a way we are lacking information here as well.\n\n\n```stata\ntab empstat\n```\n\nFrom our understanding individuals with $empstat \\in [1, 12]$ should work at least some hours a week.\n\n\n```stata\ntab uhrswork1 if inrange(empstat, 1, 12)\n```\n\n \n hours usually |\n worked per week |\n at main job | Freq. Percent Cum.\n -----------------+-----------------------------------\n 0. 0 hours | 6,588 0.09 0.09\n 1 | 3,188 0.04 0.14\n 2 | 7,959 0.11 0.25\n 3 | 7,698 0.11 0.36\n 4 | 13,467 0.19 0.54\n 5 | 17,100 0.24 0.78\n 6 | 15,871 0.22 1.00\n 7 | 6,084 0.09 1.09\n 8 | 32,912 0.46 1.55\n 9 | 5,634 0.08 1.63\n 10 | 65,531 0.92 2.54\n 11 | 2,631 0.04 2.58\n 12 | 38,263 0.53 3.12\n 13 | 4,294 0.06 3.18\n 14 | 7,845 0.11 3.29\n 15 | 82,197 1.15 4.43\n 16 | 41,570 0.58 5.02\n 17 | 6,080 0.08 5.10\n 18 | 18,750 0.26 5.36\n 19 | 3,857 0.05 5.42\n 20 | 269,134 3.76 9.18\n 21 | 6,728 0.09 9.27\n 22 | 10,534 0.15 9.42\n 23 | 6,799 0.10 9.51\n 24 | 68,599 0.96 10.47\n 25 | 125,220 1.75 12.22\n 26 | 7,552 0.11 12.33\n 27 | 8,527 0.12 12.45\n 28 | 21,295 0.30 12.75\n 29 | 3,968 0.06 12.80\n 30 | 223,764 3.13 15.93\n 31 | 2,277 0.03 15.96\n 32 | 99,667 1.39 17.35\n 33 | 9,441 0.13 17.49\n 34 | 8,833 0.12 17.61\n 35 | 227,885 3.19 20.79\n 36 | 83,390 1.17 21.96\n 37 | 39,541 0.55 22.51\n 38 | 87,401 1.22 23.73\n 39 | 9,936 0.14 23.87\n 40 | 3,957,809 55.32 79.19\n 41 | 4,113 0.06 79.25\n 42 | 35,985 0.50 79.75\n 43 | 18,161 0.25 80.01\n 44 | 20,602 0.29 80.30\n 45 | 315,324 4.41 84.70\n 46 | 10,018 0.14 84.84\n 47 | 7,996 0.11 84.95\n 48 | 56,433 0.79 85.74\n 49 | 3,654 0.05 85.79\n 50 | 506,182 7.08 92.87\n 51 | 952 0.01 92.88\n 52 | 7,930 0.11 92.99\n 53 | 3,187 0.04 93.04\n 54 | 4,298 0.06 93.10\n 55 | 107,358 1.50 94.60\n 56 | 9,850 0.14 94.74\n 57 | 1,450 0.02 94.76\n 58 | 2,914 0.04 94.80\n 59 | 440 0.01 94.80\n 60 | 231,627 3.24 98.04\n 61 | 207 0.00 98.04\n 62 | 1,082 0.02 98.06\n 63 | 1,036 0.01 98.07\n 64 | 916 0.01 98.09\n 65 | 26,040 0.36 98.45\n 66 | 1,588 0.02 98.47\n 67 | 393 0.01 98.48\n 68 | 828 0.01 98.49\n 69 | 165 0.00 98.49\n 70 | 45,307 0.63 99.13\n 71 | 80 0.00 99.13\n 72 | 7,385 0.10 99.23\n 73 | 139 0.00 99.23\n 74 | 239 0.00 99.24\n 75 | 6,575 0.09 99.33\n 76 | 287 0.00 99.33\n 77 | 516 0.01 99.34\n 78 | 447 0.01 99.34\n 79 | 66 0.00 99.35\n 80 | 23,988 0.34 99.68\n 81 | 66 0.00 99.68\n 82 | 210 0.00 99.68\n 83 | 78 0.00 99.69\n 84 | 6,603 0.09 99.78\n 85 | 1,331 0.02 99.80\n 86 | 176 0.00 99.80\n 87 | 124 0.00 99.80\n 88 | 205 0.00 99.80\n 89 | 43 0.00 99.80\n 90 | 4,008 0.06 99.86\n 91 | 205 0.00 99.86\n 92 | 65 0.00 99.86\n 93 | 43 0.00 99.86\n 94 | 101 0.00 99.87\n 95 | 271 0.00 99.87\n 96 | 624 0.01 99.88\n 97 | 30 0.00 99.88\n 98 | 573 0.01 99.89\n 99 | 8,066 0.11 100.00\n -----------------+-----------------------------------\n Total | 7,154,399 100.00\n\n\nuhrswork1 = 997 affects approximately 7.6 % of all respondents having a job. We are going to exclude 997 an 999 from the analysis, since we are interested in effects on the intensive margin.\nKeep in mind that the missing of data (or in our case a varying in hours) could yield some important implications as well.\n\nTo elaborate on that, let us have a brief look at missing values.\n\n#### Missing Completely At Random (MCAR)\nThe occurence of a missing value in Y does not depend on realisations of any other variable $X_i$ nor does it depend on other realisations of Y itself.
                                        \ne.g.\n- transmission errors\n\n#### Missing At Random (MAR)\nThe occurence of a missing value can be fully explained by the other variables $X_i$.
                                        \ne.g.\n- males are less likely to state existing mental problems - regardless of whether they have mentral problems or not\n\n#### Missing Not At Random (MNAR)\nThe occurence of a missing value in Y does depend on realisations of Y itself and can also depend on realisations of other variables $X_i$.
                                        \ne.g.\n- People with high income are less likely to state their income.\n\n\n\nKeep in mind that the missing of data might hold some information as well.\n\n\n```stata\nmvdecode(uhrswork1), mv(999 997)\n```\n\n\n```stata\nreg uhrswork1 hourwage [pweight=earnwt]\n```\n\n (sum of wgt is 8,740,213,304.43)\n \n Linear regression Number of obs = 952,574\n F(1, 952572) = 22347.36\n Prob > F = 0.0000\n R-squared = 0.0501\n Root MSE = 9.6503\n \n ------------------------------------------------------------------------------\n | Robust\n uhrswork1 | Coef. Std. Err. t P>|t| [95% Conf. Interval]\n -------------+----------------------------------------------------------------\n hourwage | .2388114 .0015975 149.49 0.000 .2356803 .2419425\n _cons | 32.61111 .0278884 1169.34 0.000 32.55645 32.66577\n ------------------------------------------------------------------------------\n\n\nOLS minimizes\n$$\\sum_{i=1}^n (y_i - (\\alpha1 + \\beta x_i))^2$$\nIntroducing a weight $w_i$ (probability weight) for every observation:\n$$\\sum_{i=1}^n w_i^2 (y_i - (\\alpha1 + \\beta x_i))^2$$\n$$\\sum_{i=1}^n (w_iy_i - (\\alpha w_i + \\beta w_i x_i))^2$$\n\n\n### [2.2 Determine the correlation between wages and hours worked. What is the difference between OLS and correlation?](#content)\n\n\n```stata\ncorr uhrswork1 hourwage\n```\n\n (obs=955,096)\n \n | uhrswo~1 hourwage\n -------------+------------------\n uhrswork1 | 1.0000\n hourwage | 0.2343 1.0000\n \n\n\nThe default correlation command does not feature pweights. However, for point estimates of the correlation it is recommended to use aweight instead - though it delivers bogus p-values.\nhttps://www.stata.com/support\n\n\n```stata\ncorr uhrswork1 hourwage [aweight=earnwt]\ndisplay round(r(rho)^2, .0001)\n```\n\n \n (sum of wgt is 8,740,213,304.43)\n (obs=952,574)\n \n | uhrswo~1 hourwage\n -------------+------------------\n uhrswork1 | 1.0000\n hourwage | 0.2239 1.0000\n \n \n .0501\n\n\n\n### [2.3 Add age, sex and citizenship to your model. Include the log of wage and hours. Interpret the results of a multivariate regression.](#content)\n\n\n```stata\nnumlabel, add\ntab citizen\n```\n\n \n \n \n citizenship status | Freq. Percent Cum.\n -----------------------------------+-----------------------------------\n 1. born in u.s | 9,742,527 85.80 85.80\n 2. born in u.s. outlying | 58,904 0.52 86.32\n 3. born abroad of american parents | 100,987 0.89 87.21\n 4. naturalized citizen | 636,648 5.61 92.82\n 5. not a citizen | 815,271 7.18 100.00\n -----------------------------------+-----------------------------------\n Total | 11,354,337 100.00\n\n\n\n```stata\nrecode citizen (1/4 = 1) (5 = 0), gen (ctz)\nlabel define citizen 0 \"not a citizen\" 1 \"citizen\"\nlabel value ctz citizen\n```\n\n \n (1611810 differences between citizen and ctz)\n \n \n\n\n\n```stata\ntab ctz, missing\n```\n\n \n RECODE of |\n citizen |\n (citizenship |\n status) | Freq. Percent Cum.\n --------------+-----------------------------------\n not a citizen | 815,271 7.18 7.18\n citizen | 10,539,066 92.82 100.00\n --------------+-----------------------------------\n Total | 11,354,337 100.00\n\n\n\n```stata\nhist age\n```\n\n (bin=70, start=15, width=.74285714)\n\n\n\n \n\n\n\n \n\n\nIn your own data extract, you might notice the accumulation at ages 80 and 85?

                                        \nThe description of the variable age on the ipums website holds the following:
                                        \n>Top codes:
                                        \n1962-1987: 99+ coded as 99
                                        \n1988-2001: 90+ coded as 90
                                        \n2002-2003: 80+ coded as 80
                                        \n2004-onward: 80-84 coded as 80, 85+ coded as 85
                                        \n\nDoes this affect our analysis? How many people aged 80 and above still participate in the job market?
                                        \n(First guess: Not that much.)\n\n\n```stata\n* creating logs\ngen log_uhrswork1 = log(uhrswork1)\ngen log_hourwage = log(hourwage)\n```\n\n \n (4,206,526 missing values generated)\n \n (10,334,812 missing values generated)\n\n\n\n```stata\n* linear regression\nreg log_uhrswork1 log_hourwage age i.sex i.ctz [pweight=earnwt]\n```\n\n (sum of wgt is 8,735,618,218.398)\n \n Linear regression Number of obs = 952,041\n F(4, 952036) = 19380.34\n Prob > F = 0.0000\n R-squared = 0.1133\n Root MSE = .3694\n \n ------------------------------------------------------------------------------\n | Robust\n log_uhrswo~1 | Coef. Std. Err. t P>|t| [95% Conf. Interval]\n -------------+----------------------------------------------------------------\n log_hourwage | .1672318 .0010219 163.65 0.000 .1652289 .1692347\n age | .003772 .0000397 95.09 0.000 .0036942 .0038497\n |\n sex |\n 2. female | -.1185369 .0008945 -132.51 0.000 -.1202902 -.1167837\n |\n ctz |\n 1. citizen | -.0798118 .001299 -61.44 0.000 -.0823578 -.0772658\n _cons | 3.08505 .0029665 1039.98 0.000 3.079236 3.090864\n ------------------------------------------------------------------------------\n\n\nNote: Stata created a appropriate cutoff for the dummies i.sex by itself. No need to transform the variable into a binary 0-1 format.\n\nInstead of creating a new dummy variable for citizen you can also tell stata where to set the cutoff for a dummy in the regression by typing **i5.citizen**.\nStata will construct a dummy variable that is one for citizen greater or equal 5 and zero otherwise.\nNote that this is the exact opposite definition of the dummy variable we have created.\n\n\n```stata\nreg log_uhrswork1 log_hourwage age i.sex i5.citizen [pweight=earnwt]\n```\n\n (sum of wgt is 8,735,618,218.398)\n \n Linear regression Number of obs = 952,041\n F(4, 952036) = 19380.34\n Prob > F = 0.0000\n R-squared = 0.1133\n Root MSE = .3694\n \n -------------------------------------------------------------------------------\n | Robust\n log_uhrswork1 | Coef. Std. Err. t P>|t| [95% Conf. Interval]\n --------------+----------------------------------------------------------------\n log_hourwage | .1672318 .0010219 163.65 0.000 .1652289 .1692347\n age | .003772 .0000397 95.09 0.000 .0036942 .0038497\n |\n sex |\n 2. female | -.1185369 .0008945 -132.51 0.000 -.1202902 -.1167837\n |\n citizen |\n 5. not a c.. | .0798118 .001299 61.44 0.000 .0772658 .0823578\n _cons | 3.005238 .0030812 975.36 0.000 2.999199 3.011277\n -------------------------------------------------------------------------------\n\n\nThe absolute value of the citizen dummy is the same as for the former regression, only the sign of the coefficient, the t-value and the confidence interval changes.\nBecause of the changing reference level of both models (1 denotes two different things in citizen2 and i5.citizen) the intercept is slightly different, too.\n\n\n**[Run the following multivariate regression:](#content)

                                        \n\\\\( \\log hours = \\beta_0 + \\beta_1 \\log wage_i + \\beta_2 age_i + \\beta_3 educ2_i + \\beta_4 sex_i + \\beta_5 (educ2_i \\times sex_i)\\\\);

                                        with educ2 being a dummy variable for a respondent holding at least a college degree.\nInterpret your results.**\n\n\n```stata\nnumlabel, add\ntab educ\n```\n\n \n \n \n educational attainment recode | Freq. Percent Cum.\n ----------------------------------------+-----------------------------------\n 1. niu or blank | 9 0.00 0.00\n 2. none or preschool | 29,001 0.26 0.26\n 10. grades 1, 2, 3, or 4 | 55,654 0.49 0.75\n 20. grades 5 or 6 | 124,978 1.10 1.85\n 30. grades 7 or 8 | 221,665 1.95 3.80\n 40. grade 9 | 335,882 2.96 6.76\n 50. grade 10 | 414,008 3.65 10.40\n 60. grade 11 | 447,258 3.94 14.34\n 71. 12th grade, no diploma | 165,357 1.46 15.80\n 73. high school diploma or equivalent | 3,219,208 28.35 44.15\n 81. some college but no degree | 2,086,324 18.37 62.53\n 91. associate's degree, occupational/vo | 506,645 4.46 66.99\n 92. associate's degree, academic progra | 545,433 4.80 71.79\n 111. bachelor's degree | 2,106,549 18.55 90.34\n 123. master's degree | 798,942 7.04 97.38\n 124. professional school degree | 150,658 1.33 98.71\n 125. doctorate degree | 146,766 1.29 100.00\n ----------------------------------------+-----------------------------------\n Total | 11,354,337 100.00\n\n\n\n```stata\nrecode educ (1 = .m) (092 111 123 124 125 = 1) (else = 0), gen(educ2)\n```\n\n (11354337 differences between educ and educ2)\n\n\n\n```stata\nlabel define educ2 .m \"missing answer\" 0 \"no college\" 1 \"college\"\nlabel value educ2 educ2\n```\n\n\n```stata\ntab educ2, missing\n```\n\n \n RECODE of educ |\n (educational |\n attainment |\n recode) | Freq. Percent Cum.\n ---------------+-----------------------------------\n no college | 7,605,980 66.99 66.99\n college | 3,748,348 33.01 100.00\n missing answer | 9 0.00 100.00\n ---------------+-----------------------------------\n Total | 11,354,337 100.00\n\n\n\n```stata\nreg log_uhrswork1 log_hourwage age i.sex##i.educ2 [pweight=earnwt]\n```\n\n (sum of wgt is 8,735,618,218.398)\n \n Linear regression Number of obs = 952,041\n F(5, 952035) = 15926.79\n Prob > F = 0.0000\n R-squared = 0.1128\n Root MSE = .36952\n \n ------------------------------------------------------------------------------\n | Robust\n log_uhrswo~1 | Coef. Std. Err. t P>|t| [95% Conf. Interval]\n -------------+----------------------------------------------------------------\n log_hourwage | .1770907 .0010586 167.29 0.000 .1750159 .1791655\n age | .0037393 .0000399 93.65 0.000 .0036611 .0038176\n |\n sex |\n 2. female | -.1105777 .0010262 -107.75 0.000 -.1125891 -.1085663\n |\n educ2 |\n 1. college | -.0389158 .0013397 -29.05 0.000 -.0415415 -.0362901\n |\n sex#educ2 |\n 2. female #|\n 1. college | -.0246875 .0020473 -12.06 0.000 -.0287002 -.0206748\n |\n _cons | 2.996977 .0030552 980.95 0.000 2.990989 3.002965\n ------------------------------------------------------------------------------\n\n\nceteris paribus:\n\n\\\\( \\beta_3 educ2_i + \\beta_4 sex_i + \\beta_5 (educ2_i \\times sex_i) \\\\)

                                        \n\nPlugging in the estimates for the coefficients and calculating the total coefficient for all four subpopulations defined by the two binary dummy variables *sex* and *college* yields:\n
                                        \n\nmale and no college ==> \\\\( \\beta_3*0 + \\beta_4*0 + \\beta_5*(0 \\times 0) \\\\)
                                        \n0 * (-.0389158) + 0 * (-.1105777) -.0246875 * (0x0) = 0 (reference group)

                                        \nmale and college ==> \\\\( \\beta_3*1 + \\beta_4*0 + \\beta_5*(1 \\times 0) \\\\)
                                        \n1 * (-.0389158) + 0 * (-.1105777) -.0246875 * (1x0) = -.0389158

                                        \nfemale and no college ==> \\\\( \\beta_3*0 + \\beta_4*1 + \\beta_5*(0 \\times 1) \\\\)
                                        \n0 * (-.0389158) + 1 * (-.1105777) -.0246875 * (0x1) = -.1105777

                                        \nfemale and collge ==> \\\\( \\beta_3*1 + \\beta_4*1 + \\beta_5*(1 \\times 1) \\\\)
                                        \n1 * (-.0389158) + 1 * (-.1105777) -.0246875 * (1x1) = -.174181\n\nThe coefficients of the dummy variable depict the effect of being a male holding a college degree as well as of being a female holding or not holding a college degree relative to being a male not holding a college degree.
                                        \nPay attention to the reference group when interpreting your results \n\n\n```stata\ntwoway (lfit uhrswork1 hourwage if sex == 1 & educ2 == 0) (lfit uhrswork1 hourwage if sex == 1 & educ2 == 1) (lfit uhrswork1 hourwage if sex == 2 & educ2 == 0) (lfit uhrswork1 hourwage if sex == 2 & educ2 == 1), legend(cols(2) order(1 \"male, no college\" 2 \"male, college\" 3 \"female, no college\" 4 \"female, no college\")) title(\"Regression function fit\", color(gs0)) \n```\n\n\n \n\n\n\n\n```stata\nreg uhrswork1 log_hourwage age i.sex##i.educ2 [pweight=earnwt]\n```\n\n (sum of wgt is 8,740,213,304.43)\n \n Linear regression Number of obs = 952,574\n F(5, 952568) = 21161.61\n Prob > F = 0.0000\n R-squared = 0.1380\n Root MSE = 9.193\n \n ------------------------------------------------------------------------------\n | Robust\n uhrswork1 | Coef. Std. Err. t P>|t| [95% Conf. Interval]\n -------------+----------------------------------------------------------------\n log_hourwage | 4.994455 .0263433 189.59 0.000 4.942823 5.046087\n age | .0913921 .0009462 96.58 0.000 .0895375 .0932467\n |\n sex |\n 2. female | -3.386369 .0252816 -133.95 0.000 -3.43592 -3.336818\n |\n educ2 |\n 1. college | -1.15064 .0375782 -30.62 0.000 -1.224292 -1.076988\n |\n sex#educ2 |\n 2. female #|\n 1. college | -.5551601 .0519744 -10.68 0.000 -.6570283 -.453292\n |\n _cons | 21.80574 .0731417 298.13 0.000 21.66239 21.9491\n ------------------------------------------------------------------------------\n\n\n\n```stata\nreg log_uhrswork1 hourwage age i.sex##i.educ2 [pweight=earnwt]\n```\n\n (sum of wgt is 8,735,618,218.398)\n \n Linear regression Number of obs = 952,041\n F(5, 952035) = 11881.97\n Prob > F = 0.0000\n R-squared = 0.0889\n Root MSE = .37445\n \n ------------------------------------------------------------------------------\n | Robust\n log_uhrswo~1 | Coef. Std. Err. t P>|t| [95% Conf. Interval]\n -------------+----------------------------------------------------------------\n hourwage | .0063358 .0000632 100.31 0.000 .006212 .0064596\n age | .0046301 .0000404 114.56 0.000 .0045509 .0047094\n |\n sex |\n 2. female | -.127375 .0010303 -123.62 0.000 -.1293944 -.1253555\n |\n educ2 |\n 1. college | -.0280367 .0013698 -20.47 0.000 -.0307215 -.0253519\n |\n sex#educ2 |\n 2. female #|\n 1. college | -.0098348 .0020711 -4.75 0.000 -.013894 -.0057755\n |\n _cons | 3.330031 .001883 1768.44 0.000 3.32634 3.333721\n ------------------------------------------------------------------------------\n\n\n\n```stata\nreg uhrswork1 hourwage age i.sex##i.educ2 [pweight=earnwt]\n```\n\n (sum of wgt is 8,740,213,304.43)\n \n Linear regression Number of obs = 952,574\n F(5, 952568) = 15812.80\n Prob > F = 0.0000\n R-squared = 0.1101\n Root MSE = 9.3406\n \n ------------------------------------------------------------------------------\n | Robust\n uhrswork1 | Coef. Std. Err. t P>|t| [95% Conf. Interval]\n -------------+----------------------------------------------------------------\n hourwage | .1860219 .001619 114.90 0.000 .1828486 .1891952\n age | .1152636 .0009575 120.38 0.000 .1133869 .1171403\n |\n sex |\n 2. female | -3.837657 .0255038 -150.47 0.000 -3.887644 -3.787671\n |\n educ2 |\n 1. college | -.8898273 .038427 -23.16 0.000 -.965143 -.8145116\n |\n sex#educ2 |\n 2. female #|\n 1. college | -.1451342 .0527644 -2.75 0.006 -.2485507 -.0417178\n |\n _cons | 31.13315 .0448581 694.04 0.000 31.04523 31.22107\n ------------------------------------------------------------------------------\n\n\n\\\\( y = \\beta_0 + \\beta_1 x \\\\)\n\n|
                                        **y / x**
                                        |
                                        **level**
                                        |
                                        **log**
                                        |\n| :--- | :----: | ---: |\n| **level** | change in x by 1 unit changes y by \\\\( \\beta_1 \\\\) | change in x by 1 % changes y by
                                        \\\\( \\beta_1 / 100\\\\) units |\n| **log** | change in x by 1 unit changes y
                                        **approximately** by \\\\( \\beta_1 \\times 100 \\% \\\\) | change in x by 1 % changes y by \\\\( \\beta_1 \\% \\\\)|\n\n\n## [3. Employment Flows](#content)\n\n### Group Work (15-20 min)\n\n**The statistics of the Bureau of Labor Statistics base on data from the CPS.**\n\n**[Derive information on monthly employment flows. Disregard marginal in- and outflows.](#content)**\n**Do so for every month in your data.**\n\n\n\n\n```stata\ncd \"C:\\Users\\Hannes\\Documents\"\nuse \"session01.dta\", clear\n```\n\n \n C:\\Users\\Hannes\\Documents\n \n\n\n\n```stata\nnumlabel, add\ntab empstat, missing\n```\n\n \n \n \n employment status | Freq. Percent Cum.\n -----------------------------------+-----------------------------------\n 1. armed forces | 50,414 0.44 0.44\n 10. at work | 7,451,380 65.63 66.07\n 12. has job, not at work last week | 299,849 2.64 68.71\n 21. unemployed, experienced worker | 406,801 3.58 72.29\n 22. unemployed, new worker | 39,493 0.35 72.64\n 32. nilf, unable to work | 588,222 5.18 77.82\n 34. nilf, other | 1,778,746 15.67 93.49\n 36. nilf, retired | 739,432 6.51 100.00\n -----------------------------------+-----------------------------------\n Total | 11,354,337 100.00\n\n\n\n```stata\nrecode empstat (1/12 = 1) (21 22 = 2) (32/36 = 3), gen(emp2)\nlabel define emp2 1 \"employed\" 2 \"unemployed\" 3 \"not in labor force\"\nlabel value emp2 emp2\n```\n\n \n (11303923 differences between empstat and emp2)\n \n \n\n\n\n```stata\ntab emp2\n```\n\n \n RECODE of empstat |\n (employment |\n status) | Freq. Percent Cum.\n -------------------+-----------------------------------\n employed | 7,801,643 68.71 68.71\n unemployed | 446,294 3.93 72.64\n not in labor force | 3,106,400 27.36 100.00\n -------------------+-----------------------------------\n Total | 11,354,337 100.00\n\n\n\n```stata\nbysort cpsidp (mish): gen lead_mish = mish[_n + 1]\nbysort cpsidp (mish): gen lead_emp = emp2[_n + 1] if lead_mish - mish == 1\n```\n\n \n (1,773,756 missing values generated)\n \n (1,993,760 missing values generated)\n\n\n\n```stata\ngen empflow = emp2 * 10 + lead_emp\nreplace empflow = emp2 * 1000 + 999 if missing(lead_emp)\nlabel define empflow 11 \"EE\" 12 \"EU\" 13 \"EN\" 21 \"UE\" 22 \"UU\" 23 \"UN\" 31 \"NE\" 32 \"NU\" 33 \"NN\" 1999 \"E.\" 2999 \"U.\" 3999 \"N.\"\nlabel value empflow empflow\ntab empflow\n```\n\n \n (1,993,760 missing values generated)\n \n (1,993,760 real changes made)\n \n \n \n \n empflow | Freq. Percent Cum.\n ------------+-----------------------------------\n EE | 6,162,616 54.28 54.28\n EU | 88,037 0.78 55.05\n EN | 197,102 1.74 56.79\n UE | 95,966 0.85 57.63\n UU | 184,199 1.62 59.25\n UN | 84,685 0.75 60.00\n NE | 179,394 1.58 61.58\n NU | 83,560 0.74 62.32\n NN | 2,285,018 20.12 82.44\n E. | 1,353,888 11.92 94.36\n U. | 81,444 0.72 95.08\n N. | 558,428 4.92 100.00\n ------------+-----------------------------------\n Total | 11,354,337 100.00\n\n\n\n```stata\ntab month empflow if year == 2020\n```\n\n \n | empflow\n month | EE EU EN UE | Total\n -------------+--------------------------------------------+----------\n 1. january | 20,794 234 601 225 | 38,391 \n 2. february | 19,648 353 697 194 | 38,412 \n 3. march | 16,075 1,746 1,061 184 | 34,763 \n 4. april | 15,014 490 483 934 | 33,541 \n 5. may | 14,684 351 437 837 | 32,119 \n 6. june | 14,229 303 457 563 | 30,236 \n 7. july | 14,744 198 509 508 | 29,965 \n 8. august | 15,163 220 554 414 | 30,231 \n 9. september | 15,967 221 471 329 | 31,793 \n 10. october | 16,036 183 507 286 | 31,641 \n 11. november | 15,248 209 475 211 | 30,659 \n 12. december | 13,360 181 387 164 | 29,459 \n -------------+--------------------------------------------+----------\n Total | 190,962 4,689 6,639 4,849 | 391,210 \n \n \n | empflow\n month | UU UN NE NU | Total\n -------------+--------------------------------------------+----------\n 1. january | 334 205 544 182 | 38,391 \n 2. february | 355 176 509 160 | 38,412 \n 3. march | 333 297 396 187 | 34,763 \n 4. april | 1,102 363 615 421 | 33,541 \n 5. may | 885 302 620 390 | 32,119 \n 6. june | 744 292 513 310 | 30,236 \n 7. july | 719 264 518 234 | 29,965 \n 8. august | 588 261 472 223 | 30,231 \n 9. september | 550 268 571 232 | 31,793 \n 10. october | 523 214 469 194 | 31,641 \n 11. november | 476 204 440 184 | 30,659 \n 12. december | 424 180 320 147 | 29,459 \n -------------+--------------------------------------------+----------\n Total | 7,033 3,026 5,987 2,864 | 391,210 \n \n \n | empflow\n month | NN E. U. N. | Total\n -------------+--------------------------------------------+----------\n 1. january | 8,208 4,793 218 2,053 | 38,391 \n 2. february | 7,863 5,832 208 2,417 | 38,412 \n 3. march | 7,279 4,820 201 2,184 | 34,763 \n 4. april | 7,441 3,918 628 2,132 | 33,541 \n 5. may | 6,736 4,162 561 2,154 | 32,119 \n 6. june | 6,080 4,189 485 2,071 | 30,236 \n 7. july | 6,181 3,891 428 1,771 | 29,965 \n 8. august | 6,408 3,870 309 1,749 | 30,231 \n 9. september | 6,754 4,183 303 1,944 | 31,793 \n 10. october | 6,714 4,297 275 1,943 | 31,641 \n 11. november | 6,327 4,484 264 2,137 | 30,659 \n 12. december | 5,693 5,679 348 2,576 | 29,459 \n -------------+--------------------------------------------+----------\n Total | 81,684 54,118 4,228 25,131 | 391,210 \n\n\n**[Find out the monthly flows for the U.S. for a period of years covered by your cps sample aswell.](#content)
                                        \nAdequate data is provided by the Bureau of Labor Statistics. LINK\n

                                        \nTry to visualise the monthly flows.**\n\n**You can also continue to work on your CPS data instead.**\n\n*Note: Not all tasks have to be solved using Stata. Especially when working with smaller data sets, it is much more convenient to explore and modify data using a spreadsheet program.*\n\nSee my example here: Link to Google Slides

                                        \nRelated Spreadsheet\n\n\n```stata\nsave \"session02.dta\", replace\n```\n\n file session02.dta saved\n\n\n\n```stata\ncollapse (count) countflow = cpsidp, by(year month empflow)\n```\n\n\n```stata\ngen date = ym(year,month)\nformat date %tmMCY\n```\n\n\n```stata\ntwoway (line countflow date if empflow == 11, lpattern(solid) lcolor(green)) ///\n(line countflow date if empflow == 12, lpattern(solid) lcolor(red)) ///\n(line countflow date if empflow == 13, lpattern(solid) lcolor(black)) ///\n(line countflow date if empflow == 21, lpattern(dash) lcolor(green)) ///\n(line countflow date if empflow == 22, lpattern(dash) lcolor(red)) ///\n(line countflow date if empflow == 23, lpattern(dash) lcolor(black)) ///\n(line countflow date if empflow == 31, lpattern(dot) lcolor(green)) ///\n(line countflow date if empflow == 32, lpattern(dot) lcolor(red)) ///\n(line countflow date if empflow == 33, lpattern(dot) lcolor(black)), ///\nlegend(cols(2) order(1 \"EE\" 2 \"EU\" 3 \"EN\" 4 \"UE\" 5 \"UU\" 6 \"UN\" 7 \"NE\" 8 \"NU\" 9 \"NN\")) ///\ntitle(\"labor force flows\", color(gs0))\n```\n\n### Job Separation Rate\nThe Job Separation Rate is the ratio of the number of individuals formerly being employed that become unemployed and the number of employed individuals in the reporting month.\n\n\\\\(seprate_{t+1} = \\frac{n_{EU, t}}{n_{E, t}}\\\\)

                                        \n\nThe number of employed individuals within a survey month equals the number of inividuals experiencing \"EE\", \"EU\", \"EN\" and \"E.\", i.e. inlist(empflow, 11, 12, 13, 1999)\n\n\\\\(seprate_{t+1} = \\frac{n_{EU,t}}{n_{EE,t}+n_{EU,t}+n_{EN,t}+n_{E.,t}}\\\\)\n\n\n```stata\n* # total employed\nbysort date: egen tot_empl = total(countflow) if inlist(empflow, 11, 12, 13, 1999)\n\n* # total separations\nbysort date: egen tot_sep = total(countflow) if empflow == 12\n\n* ratio\ngen seprate = tot_sep / tot_empl\n```\n\n \n variable tot_empl already defined\n\n\n r(110);\n r(110);\n\n\n \n \n\n\n\n```stata\nsum seprate\n```\n\n \n Variable | Obs Mean Std. Dev. Min Max\n -------------+---------------------------------------------------------\n seprate | 259 .0112869 .0046523 .0056074 .0736647\n\n\n\n```stata\ntwoway (line seprate date, lpattern(solid) lcolor(green)), title(\"job separation rate\", color(gs0))\n```\n\n### Job Finding Rate\nThe Job Finding Rate is the ratio of the number of individuals formerly being unemployed that become employed and the number of unemployed individuals in the reporting month.\n\n\\\\(findrate_{t+1} = \\frac{n_{UE, t}}{n_{U, t}}\\\\)

                                        \n\nThe number of unemployed individuals within a survey month equals the number of inividuals experiencing \"UE\", \"UU\", \"UN\" and \"U.\", i.e. inlist(empflow, 21, 22, 23, 2999)\n\n\\\\(findrate_{t+1} = \\frac{n_{UE,t}}{n_{UE,t}+n_{UU,t}+n_{UN,t}+n_{U.,t}}\\\\)\n\n\n```stata\n* # total unemployed\nbysort date: egen tot_unemp = total(countflow) if inlist(empflow, 21, 22, 23, 2999)\n\n* # total job matches\nbysort date: egen tot_find = total(countflow) if empflow == 21\n\n* ratio\ngen findrate = tot_find / tot_unemp\n```\n\n \n (2,074 missing values generated)\n \n (2,852 missing values generated)\n \n (2,852 missing values generated)\n\n\n\n```stata\ntwoway (line seprate date, lpattern(solid) lcolor(green)) ///\n(line findrate date, lpattern(dot) lcolor(red)), ///\ntitle(\"job separation rate & job finding rate\", color(gs0))\n```\n\n\n```stata\nmisstable summarize\n```\n\n Obs<.\n +------------------------------\n | | Unique\n Variable | Obs=. Obs>. Obs<. | values Min Max\n -------------+--------------------------------+------------------------------\n tot_empl | 2,074 1,037 | 257 14116 35182\n tot_sep | 2,852 259 | 184 105 1746\n seprate | 2,852 259 | 259 .0056074 .0736647\n tot_unemp | 2,074 1,037 | 244 699 3243\n tot_find | 2,852 259 | 190 145 934\n findrate | 2,852 259 | 258 .1269231 .3397762\n -----------------------------------------------------------------------------\n\n\nUsing the Hodrick-Prescott filter to decompose the flows into a cylclical and a trend component with a smoothing parameter of 14400 for monhtly data.\n\n\n```stata\ncollapse (mean) seprate = seprate findrate = findrate, by(date)\n```\n\n\n```stata\ntsset date, monthly\n```\n\n time variable: date, January2000 to August2021\n delta: 1 month\n\n\n\n```stata\ntsfilter hp ct_sep = seprate, trend(trend_sep) smooth(14400)\ntsfilter hp ct_find = findrate, trend(trend_find) smooth(14400)\n```\n\n\n```stata\ntwoway (line ct_sep date, lpattern(solid) lcolor(green)) ///\n(line ct_find date, lpattern(dot) lcolor(red)), ///\nlegend(rows(2)) title(\"job separation rate & job finding rate, cyclical component\", color(gs0))\n```\n\n\n```stata\ntwoway (line trend_sep date, lpattern(solid) lcolor(green)) ///\n(line trend_find date, lpattern(dot) lcolor(red)), ///\nlegend(rows(2)) title(\"job separation rate & job finding rate, trend component\", color(gs0))\n```\n\n\n```stata\nbysort date sex: egen total_empflow = total(countflow)\ngen share = countflow / total_empflow\n```\n\n\n```stata\ntsfilter hp ct=share, trend(trendvar) smooth(14400)\n```\n\n**[Calculate the monthly unemployment rate for for the months from 2018 onwards.](#content)**
                                        \n**Plot the time series.**\n\n\n```stata\nuse \"session02.dta\", clear\n```\n\n\n```stata\ngen date = ym(year,month)\nformat date %tmMCY\n```\n\n\n```stata\ncollapse (count) frequency = cpsidp, by(date year month emp2)\n```\n\n\n```stata\nbysort date: egen total_frequency = total(frequency)\n```\n\n\n```stata\nbysort date: egen total_laborforce = total(frequency) if emp2 < 3\n```\n\n (260 missing values generated)\n\n\n\n```stata\ngen ushare = frequency / total_laborforce if emp2 == 2\n```\n\n (520 missing values generated)\n\n\n\n```stata\ntwoway (line ushare date if year >= 2018)\n```\n\n**[Add a line for the share of individuas not being part of the labor force in the same plot.](#content)**\n\n\n```stata\ngen nshare = frequency / total_frequency if emp2 == 3\n```\n\n (520 missing values generated)\n\n\n\n```stata\ntwoway (line ushare date if year >= 2018) (line nshare date if year >= 2018)\n```\n\n**[Create a time series for the mean age of people being unemployed. What effect on the mean age did the Covid crisis show?](#content)**\n\n\n```stata\nuse \"session02.dta\", clear\n```\n\n\n```stata\ngen date = ym(year,month)\nformat date %tmMCY\n```\n\n\n```stata\ncollapse (mean) mean_age = age (count) count_in = cpsidp, by(year month date emp2)\n```\n\n\n```stata\ntwoway (line mean_age date if emp2 == 2)\n```\n\n\n```stata\ntwoway (line mean_age date if emp2 == 1) (line mean_age date if emp2 == 2) (line mean_age date if emp2 == 3)\n```\n\n\n```stata\ntsset emp2 date, monthly\n```\n\n panel variable: emp2 (strongly balanced)\n time variable: date, January2000 to August2021\n delta: 1 month\n\n\n\n```stata\ntsfilter hp ct=mean_age, trend(trendvar) smooth(14400)\n```\n\n\n```stata\ntwoway (line trendvar date if emp2 == 2)\n```\n\n\n```stata\ntwoway (line ct date if emp2 == 2)\n```\n\n\n```stata\ngen tmp_var = count_in * mean_age\nbysort date: egen total_in = total(count_in)\nbysort date: egen total_age = total(tmp_var)\ngen mean_age_t = total_age / total_in\n\ndrop tmp_var total_in total_age\n```\n\n\n```stata\ntwoway (line mean_age_t date)\n```\n\n\n```stata\ngen age_diff = mean_age - mean_age_t\n```\n\n\n```stata\ntwoway (line age_diff date)\n```\n\n\n```stata\ntwoway (line age_diff date if year >= 2018)\n```\n\n\n## [Task 4](#content)\n## Minimum wages and labour supply\n### A Difference-in-difference-approach (DiD)\n\n\n## [Task 4.1](#content)\n\nMassachusetts increased its state level minimum wage from 10 \\\\$ to 11 \\\\$ per hour in 2017 (effective from 01.01.2017).\n\nThe neighbouring New Hampshire kept its minimum wage at the federal level, i.e. 7.25 \\\\$ per hour.\n\nUsing the CPS data set, run a DiD analysis to scrutinize the effect of the minimum wage increase on labour supply in Massachusetts (STATECENSUS == 14) relative to New Hampshire (STATECENSUS == 12).\n\nData on minimum wages in the US: Historical table minimum wage per state\n\nIntroduction to DiD: https://mixtape.scunning.com/difference-in-differences.html\n\n## Difference-in-Difference (DiD)\n\n \n| | Treatment | Control |\n|-|:-:|:-:|\n| t=0 | **Y**T0 | **Y**C0 |\n| t=1 | **Y**T1 | **Y**C1 |\n\n

                                        \nDifference before the treatment:
                                        **Y**T0 - **Y**C0
                                        \nDifference after the treatment:
                                        **Y**T1 - **Y**C1
                                        \nDifference-in-differences:
                                        (**Y**T1 - **Y**C1) - (**Y**T0 - **Y**C0)


                                        \n\nThe idea behind this: The difference before the treatment captures the *normal* difference, the difference after the treatment captures the normal difference and the causal effect.\n

                                        \nThe naive evaluation of the DiD would simply be the equation given above. Note that you could also subtract the treatment effect for the control group from the treatment effect of the treatment group to obtain the same estimator, i.e.
                                        \n(**Y**T1 - **Y**C1) - (**Y**T0 - **Y**C0) = (**Y**T1 - **Y**T0) - (**Y**C1 - **Y**C0)\n\n\n\n\nLet us consider the plain graphic example given above: Two groups of patients receive different medicamentation: The one group is treated with a new drug (treatment group), the other one is treated conventionally. The mortality of both groups differ before the treatment. This could be due to sample selection, e.g. because the researchers and physicians were afraid of severe side effects they chose younger and more robust patients to be treated.\n
                                        \nFor the DiD to be applicable, the commmon trend assumptions have to be met: In absence of any policy difference or treatment, bot groups would have evolved in parrallel. This means that in absence of the treatment we would expect the development of the mortality for both groups to be parallel.
                                        \nThis assumptions is easiest to be checked graphically by visualising trends in the variable of interest.\n

                                        \n\nThe treatment effect is measured by analysing the development of the differences in mortality between both groups.
                                        \nThe mortality rates beforehand (T0 - C0) give us information about how the difference in mortality between the two groups would have been without treatment (T1 - B = T0 - C0). This scenario is represented by the dashed orange line.
                                        \nBecause of the treatment, the difference in mortality actually became T1 - C1. Hence, the treatment created a difference in differences of (T1 - C1) - (T0 - C0), which can be imagined as the vertical distance between the imaginary point B and C1 (B-C1). \n\nMost basic example:
                                        \n\\\\[ Y_{it} = \\beta_0 + \\beta_1 \\text{TREAT}_i + \\beta_2 \\text{POST}_t + \\beta_{3rDD} ( \\text{TREAT} \\times \\text{POST}) +\\epsilon_{st} \\\\]
                                        \n\\\\( \\text{TREAT} \\dots\\\\) dummy for treatment group
                                        \n\\\\( \\text{POST} \\dots\\\\) dummy for treatment time
                                        \n\\\\( \\text{TREAT} \\times \\text{POST} \\dots\\\\) interaction term, indicates observations of treatment group after the treatment\n\n\n```stata\ncd \"C:\\Users\\Hannes\\Documents\"\nuse \"cps_00028.dta\", clear\nnumlabel, add\ntab statecensus\n```\n\n \n C:\\Users\\Hannes\\Documents\n \n \n \n \n state (census code) | Freq. Percent Cum.\n ----------------------------------------+-----------------------------------\n 11. maine | 221,336 1.24 1.24\n 12. new hampshire | 295,050 1.65 2.88\n 13. vermont | 245,069 1.37 4.25\n 14. massachusetts | 324,094 1.81 6.06\n 15. rhode island | 219,778 1.23 7.29\n 16. connecticut | 274,086 1.53 8.82\n 21. new york | 727,864 4.07 12.89\n 22. new jersey | 354,383 1.98 14.87\n 23. pennsylvania | 510,357 2.85 17.72\n 31. ohio | 471,823 2.64 20.35\n 32. indiana | 288,189 1.61 21.96\n 33. illinois | 519,371 2.90 24.86\n 34. michigan | 412,841 2.31 27.17\n 35. wisconsin | 294,209 1.64 28.81\n 41. minnesota | 322,879 1.80 30.61\n 42. iowa | 263,761 1.47 32.09\n 43. missouri | 274,788 1.53 33.62\n 44. north dakota | 253,042 1.41 35.04\n 45. south dakota | 239,054 1.34 36.37\n 46. nebraska | 251,507 1.40 37.78\n 47. kansas | 242,364 1.35 39.13\n 51. delaware | 233,194 1.30 40.43\n 52. maryland | 311,361 1.74 42.17\n 53. district of columbia | 254,535 1.42 43.59\n 54. virginia | 356,334 1.99 45.58\n 55. west virginia | 301,241 1.68 47.26\n 56. north carolina | 386,950 2.16 49.43\n 57. south carolina | 261,404 1.46 50.89\n 58. georgia | 391,527 2.19 53.07\n 59. florida | 743,912 4.15 57.23\n 61. kentucky | 241,905 1.35 58.58\n 62. tennessee | 298,938 1.67 60.25\n 63. alabama | 291,533 1.63 61.88\n 64. mississippi | 263,605 1.47 63.35\n 71. arkansas | 268,818 1.50 64.85\n 72. louisiana | 312,417 1.74 66.60\n 73. oklahoma | 248,097 1.39 67.98\n 74. texas | 998,022 5.57 73.56\n 81. montana | 268,730 1.50 75.06\n 82. idaho | 259,316 1.45 76.50\n 83. wyoming | 252,640 1.41 77.92\n 84. colorado | 296,576 1.66 79.57\n 85. new mexico | 245,611 1.37 80.94\n 86. arizona | 259,501 1.45 82.39\n 87. utah | 271,656 1.52 83.91\n 88. nevada | 246,105 1.37 85.28\n 91. washington | 325,311 1.82 87.10\n 92. oregon | 267,669 1.49 88.60\n 93. california | 1,549,450 8.65 97.25\n 94. alaska | 220,721 1.23 98.48\n 95. hawaii | 271,538 1.52 100.00\n ----------------------------------------+-----------------------------------\n Total | 17,904,462 100.00\n\n\n\n```stata\nkeep if inlist(statecensus, 12, 14)\n```\n\n (17,285,318 observations deleted)\n\n\n\n```stata\nsum year\n```\n\n\n```stata\ngen qdate = qofd(dofm(ym(year, month)))\nformat qdate %tq\n```\n\n\n```stata\ntab qdate\n```\n\n \n qdate | Freq. Percent Cum.\n ------------+-----------------------------------\n 2010q1 | 14,373 2.32 2.32\n 2010q2 | 15,083 2.44 4.76\n 2010q3 | 14,874 2.40 7.16\n 2010q4 | 14,440 2.33 9.49\n 2011q1 | 14,245 2.30 11.79\n 2011q2 | 14,593 2.36 14.15\n 2011q3 | 14,800 2.39 16.54\n 2011q4 | 14,658 2.37 18.91\n 2012q1 | 14,140 2.28 21.19\n 2012q2 | 14,200 2.29 23.49\n 2012q3 | 14,517 2.34 25.83\n 2012q4 | 14,512 2.34 28.17\n 2013q1 | 14,252 2.30 30.48\n 2013q2 | 13,747 2.22 32.70\n 2013q3 | 13,888 2.24 34.94\n 2013q4 | 13,751 2.22 37.16\n 2014q1 | 13,608 2.20 39.36\n 2014q2 | 13,521 2.18 41.54\n 2014q3 | 13,497 2.18 43.72\n 2014q4 | 13,463 2.17 45.90\n 2015q1 | 13,181 2.13 48.02\n 2015q2 | 12,585 2.03 50.06\n 2015q3 | 12,402 2.00 52.06\n 2015q4 | 13,048 2.11 54.17\n 2016q1 | 13,049 2.11 56.28\n 2016q2 | 12,604 2.04 58.31\n 2016q3 | 12,931 2.09 60.40\n 2016q4 | 13,229 2.14 62.54\n 2017q1 | 13,111 2.12 64.65\n 2017q2 | 12,993 2.10 66.75\n 2017q3 | 12,713 2.05 68.81\n 2017q4 | 12,952 2.09 70.90\n 2018q1 | 12,549 2.03 72.92\n 2018q2 | 12,735 2.06 74.98\n 2018q3 | 12,600 2.04 77.02\n 2018q4 | 12,526 2.02 79.04\n 2019q1 | 12,295 1.99 81.03\n 2019q2 | 12,116 1.96 82.98\n 2019q3 | 12,175 1.97 84.95\n 2019q4 | 12,194 1.97 86.92\n 2020q1 | 11,638 1.88 88.80\n 2020q2 | 10,744 1.74 90.53\n 2020q3 | 10,924 1.76 92.30\n 2020q4 | 11,076 1.79 94.09\n 2021q1 | 10,942 1.77 95.85\n 2021q2 | 11,202 1.81 97.66\n 2021q3 | 10,937 1.77 99.43\n 2021q4 | 3,531 0.57 100.00\n ------------+-----------------------------------\n Total | 619,144 100.00\n\n\nVariable HOURWAGE - hourly wage\nhttps://cps.ipums.org/cps-action/variables/HOURWAGE#codes_section\n\nThe variable is topcoded.\n\n> For the years 2003 forward, for usual hours worked <29, the topcode is \\\\$99.99, and for usual hours worked 29+, the topcode is \\\\$2885.07/(usual hours worked).\n\n\n```stata\nmvdecode(uhrsworkorg), mv(999)\n```\n\n uhrsworkorg: 590003 missing values generated\n\n\n\n```stata\ngen wage_topcoded = 0 if !missing(hourwage)\nreplace wage_topcoded = 1 if uhrsworkorg < 29 & hourwage == 99.99\nreplace wage_topcoded = 1 if uhrsworkorg >= 29 & hourwage == round(2885.07/uhrsworkorg, 0.01)\n```\n\n \n \n (5 real changes made)\n \n (11 real changes made)\n\n\n\n```stata\ntab hourwage uhrsworkorg if wage_topcoded == 1\n```\n\n \n hourly | usual hours worked per week, outgoing rotation groups\n wage | 8 14 15 20 25 | Total\n -----------+-------------------------------------------------------+----------\n 30.69 | 0 0 0 0 0 | 1 \n 36.06 | 0 0 0 0 0 | 2 \n 48.08 | 0 0 0 0 0 | 8 \n 99.99 | 1 1 1 1 1 | 5 \n -----------+-------------------------------------------------------+----------\n Total | 1 1 1 1 1 | 16 \n \n \n | usual hours worked per week,\n hourly | outgoing rotation groups\n wage | 60 80 94 | Total\n -----------+---------------------------------+----------\n 30.69 | 0 0 1 | 1 \n 36.06 | 0 2 0 | 2 \n 48.08 | 8 0 0 | 8 \n 99.99 | 0 0 0 | 5 \n -----------+---------------------------------+----------\n Total | 8 2 1 | 16 \n\n\nSo we have 16 topcoded entries for hourwage in our data.\n\n\n```stata\nsum hourwage\nmvdecode(hourwage), mv(999.99)\nsum hourwage\n```\n\n \n \n Variable | Obs Mean Std. Dev. Min Max\n -------------+---------------------------------------------------------\n hourwage | 619,144 937.0221 240.5184 1 999.99\n \n hourwage: 579425 missing values generated\n \n \n Variable | Obs Mean Std. Dev. Min Max\n -------------+---------------------------------------------------------\n hourwage | 39,719 18.43993 11.06415 1 99.99\n\n\nLet's have a look at the usual hours worked at the main job at the wage rate of HOURWAGE.\n\nUHRSWORKORG\nhttps://cps.ipums.org/cps-action/variables/UHRSWORKORG#codes_section\n\nEdit: We are disregarding UHRSWORKORG for this task. If you want to relate the labor supply to a specific wage for an individual, you should use UHRSWORKORG. But we are merely interested in the overall effect of a minimum wage increase, so we are going to use UHRSWORKT - the usual total hours worked in a week.\n\n\n```stata\nmvdecode(uhrsworkt), mv(997 999)\nsum uhrsworkt\n```\n\n \n uhrsworkt: 322625 missing values generated\n \n \n Variable | Obs Mean Std. Dev. Min Max\n -------------+---------------------------------------------------------\n uhrsworkt | 296,519 38.97833 12.06778 0 159\n\n\n\n```stata\n*gen topcode_value = round(2885.07 / uhrsworkorg, 0.01) if uhrsworkorg >= 29\n*replace topcode_value = 99.99 if uhrsworkorg < 29\n\n*** uhrsworkorg is not part of our analysis\n```\n\n\n```stata\ntabstat hourwage if inlist(statecensus, 12, 14) & inrange(year, 2016, 2017) & wage_topcoded == 0, ///\nby(statecensus) statistics(count, mean, median, min, max)\n```\n\n \n Summary for variables: hourwage\n by categories of: statecensus (state (census code))\n \n statecensus | N mean p50 min max\n -----------------+--------------------------------------------------\n 12. new hampshir | 2946 18.1349 15.115 1.01 99.99\n 14. massachusett | 3728 19.22948 15.27 1.2 99\n -----------------+--------------------------------------------------\n Total | 6674 18.74632 15.25 1.01 99.99\n --------------------------------------------------------------------\n\n\n\n```stata\negen hwmeant = mean(hourwage) if wage_topcoded == 0, by(qdate statecensus)\n\ngen hwmean12 = .m\nreplace hwmean12 = hwmeant if statecensus == 12\n\ngen hwmean14 = .m\nreplace hwmean14 = hwmeant if statecensus == 14\n```\n\n \n (16 missing values generated)\n \n (619,144 missing values generated)\n \n (295,050 real changes made, 7 to missing)\n \n (619,144 missing values generated)\n \n (324,094 real changes made, 9 to missing)\n\n\n\n```stata\nqui graph twoway line hwmean12 hwmean14 qdate, legend(size(small)) xlabel(, angle(vertical)) sort\nqui graph export \"G:\\Meine Ablage\\Uni Rostock\\Lehrveranstaltungen\\WiSe 2020 21\\Labor Economics (Master)\\Exercise\\img\\hwmean12.png\"\n```\n\n (file G:\\Meine Ablage\\Uni Rostock\\Lehrveranstaltungen\\WiSe 2020 21\\Labor Economi\n > cs (Master)\\Exercise\\img\\hwmean12.png written in PNG format)\n\n\n\n\n\n```stata\negen hwmedian = median(hourwage) if wage_topcoded == 0, by(qdate statecensus)\n\ngen hwmedian12 = .m\nreplace hwmedian12 = hwmedian if statecensus == 12\n\ngen hwmedian14 = .m\nreplace hwmedian14 = hwmedian if statecensus == 14\n```\n\n \n (16 missing values generated)\n \n (619,144 missing values generated)\n \n (295,050 real changes made, 7 to missing)\n \n (619,144 missing values generated)\n \n (324,094 real changes made, 9 to missing)\n\n\n\n```stata\nqui graph twoway line hwmedian12 hwmedian14 qdate, legend(size(small)) xlabel(, angle(vertical)) sort\nqui graph export \"G:\\Meine Ablage\\Uni Rostock\\Lehrveranstaltungen\\WiSe 2020 21\\Labor Economics (Master)\\Exercise\\img\\hwmedian.png\"\n```\n\n \n \n (file G:\\Meine Ablage\\Uni Rostock\\Lehrveranstaltungen\\WiSe 2020 21\\Labor Economi\n > cs (Master)\\Exercise\\img\\hwmedian.png written in PNG format)\n\n\n\n\n\n```stata\negen hworkp50 = median(uhrsworkt), by(qdate statecensus)\n\ngen hworkp5012 = .m\nreplace hworkp5012 = hworkp50 if statecensus == 12\n\ngen hworkp5014 = .m\nreplace hworkp5014 = hworkp50 if statecensus == 14\n```\n\n \n \n (619,144 missing values generated)\n \n (295,050 real changes made)\n \n (619,144 missing values generated)\n \n (324,094 real changes made)\n\n\n\n```stata\nqui graph twoway line hworkp5012 hworkp5014 qdate, legend(size(small)) xlabel(, angle(vertical)) sort\nqui graph export \"G:\\Meine Ablage\\Uni Rostock\\Lehrveranstaltungen\\WiSe 2020 21\\Labor Economics (Master)\\Exercise\\img\\hworkp50.png\"\n```\n\n \n \n (file G:\\Meine Ablage\\Uni Rostock\\Lehrveranstaltungen\\WiSe 2020 21\\Labor Economi\n > cs (Master)\\Exercise\\img\\hworkp50.png written in PNG format)\n\n\n\n\n\n```stata\negen hworkmean = mean(uhrsworkt), by(qdate statecensus)\n\ngen hworkmean12 = .m\nreplace hworkmean12 = hworkmean if statecensus == 12\n\ngen hworkmean14 = .m\nreplace hworkmean14 = hworkmean if statecensus == 14\n```\n\n \n \n (619,144 missing values generated)\n \n (295,050 real changes made)\n \n (619,144 missing values generated)\n \n (324,094 real changes made)\n\n\n\n```stata\nqui graph twoway line hworkmean12 hworkmean14 qdate, legend(size(small)) xlabel(, angle(vertical)) sort\nqui graph export \"G:\\Meine Ablage\\Uni Rostock\\Lehrveranstaltungen\\WiSe 2020 21\\Labor Economics (Master)\\Exercise\\img\\hworkmean.png\"\n```\n\n\n\n\n```stata\ntab statecensus\n```\n\n \n state (census code) | Freq. Percent Cum.\n ----------------------------------------+-----------------------------------\n 12. new hampshire | 295,050 47.65 47.65\n 14. massachusetts | 324,094 52.35 100.00\n ----------------------------------------+-----------------------------------\n Total | 619,144 100.00\n\n\n\n```stata\negen hwp25 = pctile(hourwage), by(qdate statecensus) p(25)\n\ngen hwp2512 = .m\nreplace hwp2512 = hwp25 if statecensus == 12\n\ngen hwp2514 = .m\nreplace hwp2514 = hwp25 if statecensus == 14\n```\n\n \n \n (619,144 missing values generated)\n \n (295,050 real changes made)\n \n (619,144 missing values generated)\n \n (324,094 real changes made)\n\n\n\n```stata\nqui graph twoway line hwp2512 hwp2514 qdate, legend(size(small)) xlabel(, angle(vertical)) sort\nqui graph export \"G:\\Meine Ablage\\Uni Rostock\\Lehrveranstaltungen\\WiSe 2020 21\\Labor Economics (Master)\\Exercise\\img\\hwp25.png\"\n```\n\n\n\n\n```stata\negen hwp10 = pctile(hourwage), by(qdate statecensus) p(10)\n\ngen hwp1012 = .m\nreplace hwp1012 = hwp10 if statecensus == 12\n\ngen hwp1014 = .m\nreplace hwp1014 = hwp10 if statecensus == 14\n```\n\n \n \n (619,144 missing values generated)\n \n (295,050 real changes made)\n \n (619,144 missing values generated)\n \n (324,094 real changes made)\n\n\n\n```stata\nqui graph twoway line hwp1012 hwp1014 qdate, legend(size(small)) xlabel(, angle(vertical)) sort\nqui graph export \"G:\\Meine Ablage\\Uni Rostock\\Lehrveranstaltungen\\WiSe 2020 21\\Labor Economics (Master)\\Exercise\\img\\hwp10.png\"\n```\n\n\n\nDefine the treatment date as well as the treatment group.\n\n\n```stata\ngen post14 = 0\nreplace post14 = 1 if year == 2017\n\ngen treat14 = 0\nreplace treat14 = 1 if statecensus == 14\n```\n\n \n \n (51,769 real changes made)\n \n \n (324,094 real changes made)\n\n\n\n```stata\nreg uhrsworkt i.treat14 i.post14 i.treat14#i.post14 if inlist(statecensus, 12, 14) & inrange(year, 2016, 2017)\n```\n\n \n Source | SS df MS Number of obs = 51,076\n -------------+---------------------------------- F(3, 51072) = 0.81\n Model | 352.798159 3 117.599386 Prob > F = 0.4891\n Residual | 7432254.49 51,072 145.525033 R-squared = 0.0000\n -------------+---------------------------------- Adj R-squared = -0.0000\n Total | 7432607.28 51,075 145.523393 Root MSE = 12.063\n \n ------------------------------------------------------------------------------\n uhrsworkt | Coef. Std. Err. t P>|t| [95% Conf. Interval]\n -------------+----------------------------------------------------------------\n 1.treat14 | -.180106 .1543337 -1.17 0.243 -.4826017 .1223897\n 1.post14 | .0173384 .1689218 0.10 0.918 -.3137501 .3484268\n |\n treat14#|\n post14 |\n 1 1 | .0290606 .2179739 0.13 0.894 -.3981705 .4562917\n |\n _cons | 39.25552 .1192699 329.13 0.000 39.02175 39.48929\n ------------------------------------------------------------------------------\n\n\n\n```stata\nmargins i.post14#i.treat14\n```\n\n \n Adjusted predictions Number of obs = 51,076\n Model VCE : OLS\n \n Expression : Linear prediction, predict()\n \n ------------------------------------------------------------------------------\n | Delta-method\n | Margin Std. Err. t P>|t| [95% Conf. Interval]\n -------------+----------------------------------------------------------------\n post14#|\n treat14 |\n 0 0 | 39.25552 .1192699 329.13 0.000 39.02175 39.48929\n 0 1 | 39.07542 .0979468 398.95 0.000 38.88344 39.26739\n 1 0 | 39.27286 .1196213 328.31 0.000 39.0384 39.50732\n 1 1 | 39.12182 .0968735 403.84 0.000 38.93194 39.31169\n ------------------------------------------------------------------------------\n\n\n\n```stata\nqui marginsplot\nqui graph export \"G:\\Meine Ablage\\Uni Rostock\\Lehrveranstaltungen\\WiSe 2020 21\\Labor Economics (Master)\\Exercise\\img\\margins1.png\"\n```\n\n\n\n### For wages <= 10th percentile\n\n\n```stata\nreg uhrsworkt i.treat14 i.post14 i.treat14#i.post14 if inlist(statecensus, 12, 14) & inrange(year, 2016, 2017) & (hourwage <= hwp1012 | hourwage <= hwp1014) & !missing(hourwage)\n```\n\n \n Source | SS df MS Number of obs = 6,264\n -------------+---------------------------------- F(3, 6260) = 4.61\n Model | 1809.87695 3 603.292316 Prob > F = 0.0032\n Residual | 819443.966 6,260 130.901592 R-squared = 0.0022\n -------------+---------------------------------- Adj R-squared = 0.0017\n Total | 821253.843 6,263 131.127869 Root MSE = 11.441\n \n ------------------------------------------------------------------------------\n uhrsworkt | Coef. Std. Err. t P>|t| [95% Conf. Interval]\n -------------+----------------------------------------------------------------\n 1.treat14 | -.6650665 .4065078 -1.64 0.102 -1.461961 .1318283\n 1.post14 | .6568202 .4373933 1.50 0.133 -.2006207 1.514261\n |\n treat14#|\n post14 |\n 1 1 | -.6746551 .5830591 -1.16 0.247 -1.817651 .4683407\n |\n _cons | 35.87641 .3040471 118.00 0.000 35.28038 36.47245\n ------------------------------------------------------------------------------\n\n\n\n```stata\nmargins i.post14#i.treat14\n```\n\n \n Adjusted predictions Number of obs = 6,264\n Model VCE : OLS\n \n Expression : Linear prediction, predict()\n \n ------------------------------------------------------------------------------\n | Delta-method\n | Margin Std. Err. t P>|t| [95% Conf. Interval]\n -------------+----------------------------------------------------------------\n post14#|\n treat14 |\n 0 0 | 35.87641 .3040471 118.00 0.000 35.28038 36.47245\n 0 1 | 35.21135 .2698222 130.50 0.000 34.6824 35.74029\n 1 0 | 36.53323 .3144333 116.19 0.000 35.91684 37.14963\n 1 1 | 35.19351 .2753925 127.79 0.000 34.65365 35.73337\n ------------------------------------------------------------------------------\n\n\n\n```stata\nqui marginsplot\nqui graph export \"G:\\Meine Ablage\\Uni Rostock\\Lehrveranstaltungen\\WiSe 2020 21\\Labor Economics (Master)\\Exercise\\img\\margins2.png\"\n```\n\n\n\n\n```stata\ntab statecensus\n```\n\n \n state (census code) | Freq. Percent Cum.\n ----------------------------------------+-----------------------------------\n 12. new hampshire | 295,050 47.65 47.65\n 14. massachusetts | 324,094 52.35 100.00\n ----------------------------------------+-----------------------------------\n Total | 619,144 100.00\n\n\nLet us have a look at the extensive margin of labour supply.
                                        \nIn a first analysis, let us look at the share of the labour force being employed.\n\n\n```stata\ngen employed = 1 if inrange(empstat, 10, 12)\nreplace employed = 0 if empstat > 12\n```\n\n \n (298,625 missing values generated)\n \n (194,502 real changes made)\n\n\n\n```stata\nreg employed i.treat14 i.post14 i.treat14#i.post14 if inlist(statecensus, 12, 14) & inrange(year, 2016, 2017)\n```\n\n \n Source | SS df MS Number of obs = 86,933\n -------------+---------------------------------- F(3, 86929) = 25.85\n Model | 18.1152273 3 6.03840908 Prob > F = 0.0000\n Residual | 20302.3018 86,929 .23355039 R-squared = 0.0009\n -------------+---------------------------------- Adj R-squared = 0.0009\n Total | 20320.4171 86,932 .233750714 Root MSE = .48327\n \n ------------------------------------------------------------------------------\n employed | Coef. Std. Err. t P>|t| [95% Conf. Interval]\n -------------+----------------------------------------------------------------\n 1.treat14 | -.0373972 .0047552 -7.86 0.000 -.0467173 -.0280772\n 1.post14 | -.0090599 .0052331 -1.73 0.083 -.0193167 .001197\n |\n treat14#|\n post14 |\n 1 1 | .0189626 .0067136 2.82 0.005 .0058041 .0321212\n |\n _cons | .6489781 .0037088 174.98 0.000 .6417089 .6562474\n ------------------------------------------------------------------------------\n\n\n\n```stata\nmargins i.post14#i.treat14\n```\n\n \n Adjusted predictions Number of obs = 86,933\n Model VCE : OLS\n \n Expression : Linear prediction, predict()\n \n ------------------------------------------------------------------------------\n | Delta-method\n | Margin Std. Err. t P>|t| [95% Conf. Interval]\n -------------+----------------------------------------------------------------\n post14#|\n treat14 |\n 0 0 | .6489781 .0037088 174.98 0.000 .6417089 .6562474\n 0 1 | .6115809 .002976 205.51 0.000 .605748 .6174138\n 1 0 | .6399183 .0036919 173.33 0.000 .6326822 .6471544\n 1 1 | .6214837 .0029716 209.14 0.000 .6156593 .627308\n ------------------------------------------------------------------------------\n\n\n\n```stata\nqui marginsplot\nqui graph export \"G:\\Meine Ablage\\Uni Rostock\\Lehrveranstaltungen\\WiSe 2020 21\\Labor Economics (Master)\\Exercise\\img\\margins3.png\"\n```\n\n\n\n\n```stata\negen mean_empl = mean(employed), by(statecensus qdate)\ngen mean_empl12 = .m\nreplace mean_empl12 = mean_empl if statecensus == 12\n\ngen mean_empl14 = .m\nreplace mean_empl14 = mean_empl if statecensus == 14\n```\n\n \n \n (619,144 missing values generated)\n \n (295,050 real changes made)\n \n (619,144 missing values generated)\n \n (324,094 real changes made)\n\n\n\n```stata\nqui graph twoway line mean_empl14 mean_empl12 qdate, legend(size(small)) xlabel(, angle(vertical)) sort\nqui graph export \"G:\\Meine Ablage\\Uni Rostock\\Lehrveranstaltungen\\WiSe 2020 21\\Labor Economics (Master)\\Exercise\\img\\mean_emp.png\n```\n\n\n\nAccording to our plain analysis, there seems to be a positive effect on employment associated with the increase of the minimum wage in Massachusetts.\nThere are some ways to test the robustness of our analysis results.\nFirst, we are going to replace the date of the treatment by a different date.\n\n\n```stata\nssc install diff, replace\n```\n\n checking diff consistency and verifying not already installed...\n all files already exist and are up to date.\n\n\n\n```stata\ndiff employed, t(treat14) p(post)\n\n```\n\n \n DIFFERENCE-IN-DIFFERENCES ESTIMATION RESULTS\n Number of observations in the DIFF-IN-DIFF: 515021\n Before After \n Control: 228908 17135 246043\n Treated: 242530 26448 268978\n 471438 43583\n --------------------------------------------------------\n Outcome var. | emplo~d | S. Err. | |t| | P>|t|\n ----------------+---------+---------+---------+---------\n Before | | | | \n Control | 0.641 | | | \n Treated | 0.604 | | | \n Diff (T-C) | -0.037 | 0.001 | -26.28 | 0.000***\n After | | | | \n Control | 0.640 | | | \n Treated | 0.621 | | | \n Diff (T-C) | -0.018 | 0.005 | 3.88 | 0.000***\n | | | | \n Diff-in-Diff | 0.019 | 0.005 | 3.77 | 0.000***\n --------------------------------------------------------\n R-square: 0.00\n * Means and Standard Errors are estimated by linear regression\n **Inference: *** p<0.01; ** p<0.05; * p<0.1\n\n\n\n```stata\ngen post2 = 0\nreplace post2 = 1 if year == 2016\nreg employed i.treat14 i.post2 i.treat14#i.post2 if inlist(statecensus, 12, 14) & inrange(year, 2015, 2016)\n```\n\n \n \n (51,813 real changes made)\n \n \n Source | SS df MS Number of obs = 86,175\n -------------+---------------------------------- F(3, 86171) = 45.39\n Model | 31.9430405 3 10.6476802 Prob > F = 0.0000\n Residual | 20214.4597 86,171 .234585414 R-squared = 0.0016\n -------------+---------------------------------- Adj R-squared = 0.0015\n Total | 20246.4027 86,174 .234947928 Root MSE = .48434\n \n -------------------------------------------------------------------------------\n employed | Coef. Std. Err. t P>|t| [95% Conf. Interval]\n --------------+----------------------------------------------------------------\n 1.treat14 | -.0394845 .0047208 -8.36 0.000 -.0487372 -.0302317\n 1.post2 | .0075507 .0051385 1.47 0.142 -.0025208 .0176222\n |\n treat14#post2 |\n 1 1 | .0020872 .006708 0.31 0.756 -.0110605 .0152349\n |\n _cons | .6414274 .003548 180.78 0.000 .6344733 .6483815\n -------------------------------------------------------------------------------\n\n\n\n```stata\nmargins i.post2#i.treat14\n```\n\n \n Adjusted predictions Number of obs = 86,175\n Model VCE : OLS\n \n Expression : Linear prediction, predict()\n \n -------------------------------------------------------------------------------\n | Delta-method\n | Margin Std. Err. t P>|t| [95% Conf. Interval]\n --------------+----------------------------------------------------------------\n post2#treat14 |\n 0 0 | .6414274 .003548 180.78 0.000 .6344733 .6483815\n 0 1 | .601943 .0031141 193.30 0.000 .5958393 .6080466\n 1 0 | .6489781 .003717 174.60 0.000 .6416928 .6562635\n 1 1 | .6115809 .0029825 205.05 0.000 .6057351 .6174267\n -------------------------------------------------------------------------------\n\n\n\n```stata\nqui marginsplot\nqui graph export \"G:\\Meine Ablage\\Uni Rostock\\Lehrveranstaltungen\\WiSe 2020 21\\Labor Economics (Master)\\Exercise\\img\\margins4.png\"\n```\n\n\n\n\n```stata\ngen post3 = 0\nreplace post3 = 1 if year == 2018\nreg employed i.treat14 i.post3 i.treat14#i.post3 if inlist(statecensus, 12, 14) & inrange(year, 2017, 2018)\n```\n\n \n \n (50,410 real changes made)\n \n \n Source | SS df MS Number of obs = 85,740\n -------------+---------------------------------- F(3, 85736) = 12.13\n Model | 8.42934549 3 2.80978183 Prob > F = 0.0000\n Residual | 19857.9865 85,736 .231617832 R-squared = 0.0004\n -------------+---------------------------------- Adj R-squared = 0.0004\n Total | 19866.4158 85,739 .231708042 Root MSE = .48127\n \n -------------------------------------------------------------------------------\n employed | Coef. Std. Err. t P>|t| [95% Conf. Interval]\n --------------+----------------------------------------------------------------\n 1.treat14 | -.0184346 .0047196 -3.91 0.000 -.027685 -.0091842\n 1.post3 | .0084229 .0052262 1.61 0.107 -.0018204 .0186662\n |\n treat14#post3 |\n 1 1 | .0079049 .0067232 1.18 0.240 -.0052725 .0210822\n |\n _cons | .6399183 .0036766 174.05 0.000 .6327122 .6471244\n -------------------------------------------------------------------------------\n\n\n\n```stata\nmargins i.post3#i.treat14\n```\n\n \n Adjusted predictions Number of obs = 85,740\n Model VCE : OLS\n \n Expression : Linear prediction, predict()\n \n -------------------------------------------------------------------------------\n | Delta-method\n | Margin Std. Err. t P>|t| [95% Conf. Interval]\n --------------+----------------------------------------------------------------\n post3#treat14 |\n 0 0 | .6399183 .0036766 174.05 0.000 .6327122 .6471244\n 0 1 | .6214837 .0029593 210.01 0.000 .6156835 .6272839\n 1 0 | .6483412 .0037143 174.55 0.000 .6410612 .6556211\n 1 1 | .6378114 .0030216 211.08 0.000 .631889 .6437338\n -------------------------------------------------------------------------------\n\n\n\n```stata\nqui marginsplot\nqui graph export \"G:\\Meine Ablage\\Uni Rostock\\Lehrveranstaltungen\\WiSe 2020 21\\Labor Economics (Master)\\Exercise\\img\\margins5.png\"\n```\n\n\n\nIn 2016 Massachusetts increased its state minimum wage level by 1 \\\\$ (9 to 10), whereas New Hampshire kept its minimum wage at 7.25 \\\\$.
                                        \nOur analysis shows no statistical significant effect of the treatment. C.p., we would expect a similiar effect as in our analysis above.
                                        \nA pseudo treatment for 2018, when in fact the minimum wage was not increased in neither of both states, shows no statistical effect of the treatment on the treatment group.\nAs for the increase in minimum wages from 2016 to 2017, our model rather supports the notion of a positive effect of an increase in minimum wages on the extensive labour supply.

                                        \n\n\n```stata\nclear\n```\n\n\n## [Task 4.2](#content)\n\nRepeat the analysis above for two or more neighbouring state of your choice for a suitable date.
                                        \n\n\n## [Task 5](#content)\n\n### The effect of maximum duration of unemployment benefits on labor supply
                                        Regression Discontinuity Design\n\n\nFor this task we are going to employ a Regression Discontinuity approach.
                                        \nA short introduction can be found here: https://mixtape.scunning.com/regression-discontinuity.html

                                        \n\nA more sophisticated introduction into different aspects of RnD can be found here:\nCattaneo, Idrobo, Titiunik (2019) A Practical Introduction to Regression Discontinuity Designs: Foundations; \nhttps://arxiv.org/pdf/1911.09511.pdf\n\nThe first 3 tasks are designated to scrutinizing preparing the data set for the analysis. These are exhausting, yet important tasks.\n\nWe are going to have a brief look at a simple sharp regression discontinuity design. \"Sharp\" means that every individual in the sample is affected by the measure of interest. \"Fuzzy\" on the other hand means that the probability of being treated significantly increases around a cutoff, but that there still exist untreated individuals on the right of the cutoff as well as treated individuals on the left of the cutoff. See literature above for more details.\n\nAs the name suggests, regression discontinuity analyses possinle discountinuities around a threshold. This threshold could be any treatment, e.g. policy measures. The design enables us to conduct counterfactual analyis when a control group is missing.
                                        In many scenarios we are lackking a control group, because the measure affects everybody. E.g. an minimum wage increase (should) apply to every worker.

                                        \n\nIn absence of a control group we assume that the two groups of individuals near noth sides of the cutoff are on average similar to each other and that we do not have discontinuities in covariates around the cutoff. To give just one example: Let us imagine that from 01.01.2021 on every German citizen is entitled to receive unconditional basic income. You are interested in the effect of this policy measure on labor supply. Since the unconditional basic income applies to everybody, you are missing a control group. So you decide to employ a regression discontinuity design. In order for possible effects on the labor supply to be attributed to the policy measure, you need to make sure that nothing else happens around the cutoff 01.01.2021. For example, the share of male individuals in a sample could also experience a signficant change around the cutoff or the wage distribution could change crossing the cutoff.
                                        You can easily verify that covariates affecting the outcome are balanced around the cutoff by running regression discontinuity analysis around the cutoff for them as well.\n\n\n\n\nA basic model might look like that:\n\n\\\\[ y_i = \\beta_0 + \\beta_1 (x_i - c) + \\beta_2 D_i + \\beta_3 \\left(D_i \\times (x_i - c)\\right) \\\\]\n\nThe running variable \\\\( x_i \\\\) represents the distance to the treatment and eventually crosses a threshold c, which indicates wheter the treatment took place. \\\\( D_i \\\\) is an indicator variable for the treatment.\n\\\\[\n \\begin{equation}\n D=\n \\begin{cases}\n 0 & \\text{if}\\ x_i < 01.01.2021, \\\\\n 1 & \\text{if} x_i >= 01.01.2021\n \\end{cases}\n \\end{equation}\n\\\\]\n\n\\\\( \\beta_2 D_i \\\\) represents the difference in levels of the fitted lines on the left and on the right of the cutoff. The coefficient \\\\( \\beta_2 \\\\) could be interpreted as an additional intercept introduced by the treatment.\n\n\\\\( \\beta_3 \\left(D_i \\times (x_i - c)\\right) \\\\) introduces the possibility of different slopes of the fitted line on the left and on the right of the cutoff.\n\nThe bandwidth (\\\\( h \\\\)) of the running variable \\\\(x_i\\\\) you choose for the analysis affects the results of the simple regression. A high bandwidth (more data points, farther from cutoff) is associated wth a high bias and low variance. A low bandwidth (fewer data points, closer to cutoff) on the other hand is associated with a low bias and a high variance.\nThe proximity to the cutoff can affect the size of the treatment effect as well.\n\n\nThe three lovely hand-drawn examples below show you how the bandwidth affects the size of the treatment effect.\n\n\n\n\n\n\n\nIf we employ a polynomial fit the treatment effect might disappear completely.\n\n\n\nThere exists different appraches to choose the optimal bandwith.\nI am using a command of the package rdrobust.\nThe command *rdbwselect* with the specification *bwselect(mserd)* yields an estimate for a mean squared error optimal bandwidth for a linear fit *p(1)*, i.e. the bandwidth that minimizes the MSE.\n\n\n## [Task 5.1](#content)\n\nIn 2017, the maximum duration of unemployment benefits was\n- 12 weeks for Gerogia & Nevada,\n- 13 weeks for Missouri,\n- 14 weeks for Hawaii,\n- 16 weeks for Kentucky,\n- 20 weeks for Arizona, Minnesota & South Dakota,\n- 21 weeks for Indiana,\n- 28 weeks for Nebraska,\n- 30 weeks for Maryland\n\nand 26 weeks in all other states.

                                        \nUse the complete dataset of IPUMS CPS for 2017.
                                        \nCreate a dummy variable indicating an individual being unemployed longer than the maximum duration of benefits.\n\n\n```stata\n* Loading the data\ncd \"C:\\Users\\Hannes\\Documents\"\nuse \"cps_00028.dta\", clear\nkeep if year == 2017\nnumlabel, add\ntab statecensus\n```\n\n\n```stata\n* maximum duration of unemployment benefits in weeks\ngen maxui = 26\nreplace maxui = 12 if inlist(statecensus, 58, 88)\nreplace maxui = 13 if statecensus == 43\nreplace maxui = 14 if statecensus == 95\nreplace maxui = 16 if statecensus == 61\nreplace maxui = 20 if inlist(statecensus, 86, 41, 45)\nreplace maxui = 21 if statecensus == 21\nreplace maxui = 28 if statecensus == 46\nreplace maxui = 30 if statecensus == 52\n\n```\n\n\n```stata\ntab maxui\n```\n\n\n```stata\n** unemployment benefits expired?\nmvdecode(durunemp), mv(999)\ngen expui = .m\nreplace expui = 0 if durunemp <= maxui & inlist(empstat, 21, 22)\nreplace expui = 1 if durunemp > maxui & inlist(empstat, 21, 22)\ntab expui, missing\n```\n\n\n## [Task 5.2](#content)\n\nHave a look at the sanity of the data at hand.
                                        \nFor every individual check the distance in weeks between two consecutive interviews as well as the difference between $durunemp_{t}$ and $durunemp_{t+1}$. What does strike your eye?\n\n\n```stata\ntab durunemp\n```\n\n\n```stata\nnumlabel, add\ntab empstat if !missing(durunemp)\n```\n\nSo far, so good: Almost only unemployed workers report durations of unemployment. The 13 missing employment status are most likely unemployed, too. Since we only have observations where the employment status is not defined, we are going to drop these information.\n\n\n```stata\nmvdecode(empstat), mv(0)\n```\n\n\n```stata\nhist durunemp\n```\n\nProblem: The majority of individuals are reported several times in the CPS within one year.\n\n\n```stata\n** cpsidp occurence per year\negen tag = tag(cpsid mish)\negen occurence = total(tag), by(cpsid)\ndrop tag\ntab occurence\n```\n\nThe majority of survey participants are interviewed 4 times in 2017.\n\nLet us have a look at how stable the employment status for unemployed individuals is throughout the year 2017.\n\n\n```stata\n** give next mish for individual\nbysort cpsidp (mish): gen lead_mish = mish[_n+1]\n\n** Employment status of unemployed individuals in the next interview\nbysort cpsidp (mish): gen empstat_t = empstat[_n+1]\nreplace empstat_t = . if missing(lead_mish)\n\n** Generate a variable for calendar week\ngen cw = week(mdy(month, 19, year))\nbysort cpsidp (mish): gen cw_t = cw[_n+1]\n\n** duration of unemployment in next interview\nbysort cpsidp (mish): gen durunemp_t = durunemp[_n+1]\n```\n\n\n```stata\ntab empstat empstat_t if empstat != empstat_t\n```\n\n\n```stata\ntab empstat empstat_t if empstat == empstat_t\n```\n\nAt leas 92k times we have changes in the employment status.\n\nLet us now try to detect whether an individual had been continously unemployed or not.\n\n\n```stata\ngen diff_cw = cw_t - cw\ngen diff_du = durunemp_t - durunemp\ngen diff_cw_du = diff_cw - diff_du\ngen con_unemp = .\n\n** Does the difference between the differences in durunemp and cw indicate continous unemployment?\nreplace con_unemp = 1 if inrange(diff_du, diff_cw - 1, diff_cw + 1)\nreplace con_unemp = 0 if !inrange(diff_du, diff_cw - 1, diff_cw + 1) & !missing(durunemp)\n```\n\nThe idea of the snippet above is to check whether individuals who happen to have multiple appearances in the sample in the year 2017 were continuously unemployed during the time in between the interviews.
                                        \nIf so, then
                                        \n\\\\[ durunemp_{mish=t+1} - durunemp_{mish=t} \\in [CW_{mish=t+1} - CW_{mish=t} - 1, CW_{mish=t+1} - CW_{mish=t} + 1] \\\\]\n\n\n```stata\ntab con_unemp\n```\n\n\n```stata\ntab diff_cw_du\n```\n\nInteresting finding: While we might argue that a difference of -2 weeks could be due to individuals incorrectly recalling their duration of unemployment in weeks, it is harder to justify a difference of -4 weeks and more similarly.
                                        \nWhat could be other reasons for that?
                                        \nDid we stumbled upon a bug in the dataset?
                                        \nDoes the linkage by IPUMS yield false ties?
                                        \nMost likely individuals simply recall the time being unemployed wrongly. This would explain some range of the differences, but still the gap should be near 0 - not 100 weeks.
                                        \nEspecially the negative differences do not make sense, since this would mean that the duration of the unemployment increased faster than time passes. We are going to exclude such observations from our analysis, but we will do so with some tolerance.\n\n\n```stata\nkeep if diff_cw_du >= -1\n```\n\nBut how about positive values of *diff_cw_du*? They could imply that an individual was employed in between the interviews. However, this difference would be limited by the maximum interview break length of 8 months.\n\n\n```stata\ntab diff_du diff_cw if diff_cw_du > 0\n```\n\n\\\\[\\text{diff_du} < 0 \\Rightarrow du_{t+1} < du_{t} \\\\]\nThis could mean that a person was not unemployed in between both periods.\n

                                        \n\\\\[\\text{diff_du} > 0 \\Rightarrow du_{t+1} > du_{t} \\\\]\nThis is what we would expect to see for continuing unemployment. However, we cannot reject the possibility of breaks from unemployment.

                                        \n\\\\[\\text{diff_du} = 0 \\Rightarrow du_{t+1} = du_{t+1} \\\\]\nThis would make sense if \\\\( du_{t+1}, du_{t+1} \\leq \\text{diff_cw} \\\\), because otherwise this would indicate an error in reporting.
                                        \ne.g.
                                        \n$du_{t+1}, du_{t+1} = 3 \\Rightarrow \\text{diff_du} = 0 $ would be fine,
                                        \n$du_{t+1}, du_{t+1} = 33 \\Rightarrow \\text{diff_du} = 0 $ would not.

                                        \n\nWe are going to keep the osbervations of unemployed individuals for which\n\\\\[ durunemp_{mish=t+1} - durunemp_{mish=t} \\leq (cw_{mish=t+1} - cw_{mish=t}) + 1\\\\]\n\\\\[\\Leftrightarrow \\text{diff_cu} \\leq \\text{diff_cw}+1^* \\\\] \nand\n\\\\[ durunemp_{mish=t+1}, durunemp_{mish=t} \\leq \\text{diff_cw} \\quad \\forall \\quad \\text{diff_du} = 0. \\\\]\n

                                        \n\\* *Note that we already excluded negative cases, so we can concentrate on the positive gaps here.*\n\n\n```stata\nkeep if (diff_du ~= 0 & diff_du <= diff_cw + 1) | missing(diff_du) | (diff_du == 0 & durunemp < diff_cw & durunemp_t < diff_cw)\n```\n\nIf a person managed to switch its employment status away from unemployment in between interview periods after a long time of unemployment, but ends up in unemployment at the time of the next interview then this could lead to\n\\\\[durunemp_{t+1} \\leq durunemp_{t}. \\\\]\nThis implies\n\\\\[durunemp_{t+1} - durunemp_{t} \\leq 0, \\\\]\nso that\n\\\\[cw_{t+1} - cw_{t} - durunemp_{t+1} + durunemp_{t} \\geq 0. \\\\]\ndiff_cw_du can even exceed diff_cw.\n\n\n```stata\ntab diff_cw_du if diff_cw < diff_cw_du\n```\n\nLet us have a look at the reasons for being unemployed in the next interview month (since we are dealing with leading variables).\n\n\n```stata\nbysort cpsidp (mish): gen whyunemp_t = whyunemp[_n+1]\ntab whyunemp\ntab whyunemp_t con_unemp\n```\n\nWe cannot give an answer to the question of why unemployment seems to be discontinuous for most unemployed units with the data at hand.
                                        \nBear this findings in mind while continuing to work on our problem.\n\n\n## [Task 5.3](#content)\nCreate variables indicating whether an individual was employed or unemployed at the next interview.
                                        \nCreate another variable indicating whether a person might not have been unemployed between the interviews.\n\n\n```stata\n** unemployed in next period?\ngen unemployed_t = 1 if inlist(empstat_t, 21, 22)\nreplace unemployed_t = 0 if !inlist(empstat_t, 21, 22)\n\n** employed in next period?\ngen employed_t = 0 if !inlist(empstat_t, 10, 12)\nreplace employed_t = 1 if inlist(empstat_t, 10, 12)\n\n** unemployed between interviews?\ngen unemp_bw = .\nreplace unemp_bw = 1 if unemployed_t == 1\nreplace unemp_bw = 0 if diff_du < diff_cw - 1\n ** adding some tolerance, so that the durationunemployment\n ** can deviate up to 2 weeks from the difference in calender weeks\n```\n\n\n```stata\ntable unemployed_t unemp_bw employed_t, content(freq)\n```\n\n\n```stata\n** distribution of unemployed wrt max duration of unemployment in 2017\nhist durunemp if unemployed_t == 0 | (inlist(empstat, 21, 22) & missing(lead_mish))\n```\n\n\n## [Task 5.4](#content)\nHow many observations of unemployed individuals are still eligible for unemployment benefits?
                                        \nFind a way to detect the number of unemployment spells for an individual throughout her survey lifetime.\n\n\n```stata\ntab expui\n```\n\n75.26 % of all observations for unemployed individuals lie within the range of weeks for which unemployment benefits were granted.\n\n**continuous unemployment**
                                        \nWe have been looking into this earlier. A respondent could experience multiple spells of unemployment throughout the survey. We have to find a crude measure for the approximate number of such spells.\n\n\n```stata\n** break in unemployment\nbysort cpsidp (mish): gen empstat_y = empstat[_n-1]\nbysort cpsidp (mish): gen durunemp_y = durunemp[_n-1]\nbysort cpsidp (mish): gen cw_y = cw[_n-1]\n```\n\n\n```stata\n** first occurence in 2017?\negen min_mish = min(mish), by(cpsidp)\n\n** dummy for new spell of unemployment\ngen new_u_spell = .m\n\n* unemployed yestrday and unemployed today, no strong indicators of break from unenmployment, i.e. diff_du == diff_cw\nreplace new_u_spell = 0 if inlist(empstat_y, 21, 22) & inlist(empstat, 21, 22) & durunemp - durunemp_y == (cw - cw_y)\n\n* unemployed today, indicator of break from unemployment, i.e. diff_du < diff_cw\nreplace new_u_spell = 1 if inlist(empstat, 21, 22) & durunemp - durunemp_y < (cw - cw_y) & durunemp < (cw - cw_y)\n\n* very first observation of an individual, that happens to be unemployed at this point of time\nreplace new_u_spell = 1 if mish == min_mish & inlist(empstat, 21, 22)\n\n* unemployed today, but not unemployed yesterday\nreplace new_u_spell = 1 if mish > min_mish & !inlist(empstat_y, 21, 22) & inlist(empstat, 21, 22)\n\ntab new_u_spell\n```\n\n\n```stata\n* total of spells\negen num_u_spells = total(new_u_spell), by(cpsidp)\n* cumulative sum of spells over months\nbysort cpsidp (mish): gen sum_u_spells = sum(new_u_spell)\ntab num_u_spells\n```\n\n\n```stata\ntab sum_u_spells\n```\n\nThe table above gives the number of rows of oberservations per number of unemployment spells. However, remember that indiviuals appear more than once in the data. If we are interested in the number of distinct individuals per maximum number of unemployment spell, we have to do the following:\n\n\n```stata\negen tag_n_spells = tag(cpsidp sum_u_spells)\ntab sum_u_spells if tag_n_spells == 1\n```\n\nThe table above yields the number of individuals per number of unemployment spells in 2017.
                                        \nAccording to our identification of breaks in unemployment there were 25 individuals that at least reentered the state of unemployment twice. Does this seem to be plausible to you?\n\n\n```stata\ntab sum_u_spell num_u_spell if expui == 0\n```\n\n\n```stata\ntab sum_u_spell num_u_spell if expui == 1\n```\n\n\n## [Task 5.5](#content)\nCreate a running variable indicating the time in weeks an individual is away from her unemployment benefits to expire.\nConsidering maximum duration of continuous unemployment, how many individuals were still eligible for benefits? \n\n\n```stata\n** weeks until unemployment benefits expire\ngen dis_maxui = durunemp - maxui\n```\n\n\n```stata\nhist dis_maxui\n```\n\n\n```stata\negen max_mish = max(mish), by(cpsidp)\n```\n\n\n```stata\ntab dis_maxui if unemployed_t == 0 | (mish == max_mish & inrange(unemployed, 21, 22))\n```\n\n~ 73% of unemployed individuals are still eligble for unemployment benefits in their last appearance as unemployed in the survey in 2017 (with regard to the duration of their unemployment).\n\n\n```stata\ntab lead_mish if expui == 1\n```\n\n\n```stata\ntable lead_mish expui con_unemp if unemployed_t == 1, content(freq)\n```\n\n\n```stata\ngen dis_mish = lead_mish - mish\ntab dis_mish \n```\n\nSince we conition our estimation of continuity on calendar week, we do not care about the acutal distance between two consecutive interviews.\n\nHow about jobs that were taken up and were lost again between interviews?

                                        \n$durunemp_{mish+1}< cw_{mish+1} - cw_{mish}$ could indicate such an event.\n\n\n```stata\n** employed between interviews if duration of unemployment is shorter than the time passed between the interviews\nreplace employed_t = 1 if durunemp_t < diff_cw & inlist(empstat, 21, 22)\n```\n\n\n```stata\ntab employed_t\n```\n\n\n```stata\ntable expui unemployed_t employed_t if inlist(empstat, 21, 22), content(freq)\n```\n\nThe last column indicates cases where an unemployment spell might have been not continuous.m\n\n\n## [Task 5.6](#content)\nPerform an analysis of the effect of the expiration of unemployment benefits on the employment.\nUse a regression discontinuity design for linear fit and polynomial fit (2nd and 3rd degree)\n\nNow we have to install a package for the RDD.
                                        \nWe are going to install the latest version from the default repository.\n\n\n```stata\nsearch rdrobust\n```\n\nWhat is the running variable?
                                        \nWhat determines the threshold?
                                        \nWhich bandwidth are we going to choose?
                                        \nDo we have a sharp or a fuzzy RD-Design?\n\n\n```stata\n** running variable: weeks to expiration of unemployment benefits\ngen rv = durunemp - maxui\n```\n\n\n```stata\n** treatment variable: unemloyment benefits expired\ngen ubex = 0 if rv < 0\nreplace ubex = 1 if rv >= 0\n```\n\n\n```stata\nscatter ubex rv, xline(0)\n```\n\nSince the treatment is merely a dummy for exceding the limit of the duration of unemployment benefits, we are dealing with a sharp regression discontinuity design.
                                        \nCrossing the threshold perfectly assigns each observation to the treatment group. We cannot observe two different treatment states for one x value.\n\n## Linear fit\n\nThe command *rdbwselect* with the specification *bwselect(mserd)* yields an estimate for a mean squared error optimal bandwidth for a linear fit *p(1)*, i.e. the bandwidth that minimizes the MSE.\n\n$$MSE(h) = \ud835\udd3c \\left[ ( \\hat{ \\tau_{SRD}}-\\tau_{SRD})^2 \\right] = \ud835\udd3c\\left[ \\left( (\\hat{\\mu}_+ - \\mu_+) - (\\hat{\\mu}_- - \\mu_-) \\right)^2 \\right]$$\n\n$$h^* = arg \\min_h MSE(h)$$\n\nA vivid discussion on the advantages and disadvantages of different bandwidth selecting methods has emerged. It is always good practice to inform yourself on the latest developments with regard to an econometric or other analytical approach before starting to employ a standard model.
                                        \nFor the purpose of demonstration, we are going to stick to the textbook approach.\n\n\n```stata\nrdbwselect employed_t rv, bwselect(mserd) p(1) q(2)\n```\n\n\n```stata\ngen tmp_y = employed_t if inrange(rv, -7, 7)\ngen tmp_x = rv if inrange(rv, -7, 7)\n\nsu tmp_x\n\nrdrobust tmp_y tmp_x, bwselect(mserd) p(1) h(7) c(0)\n```\n\n\n```stata\nrdplot tmp_y tmp_x, genvars ci(95) p(1) binselect(es) vce(robust)\n```\n\nThe following snippets are more complex than they have to be. This is due to Jupyter not displaying the output of\n\n> rdplot y x\n\nproperly. So I had to use the source code for rdplot.

                                        \n\nYou can find information on the package here: https://rdpackages.github.io/rdrobust/.\n\n\n```stata\nreg tmp_y tmp_x ubex, robust\n```\n\n\n```stata\ngen rdplot_hat_y_0 = _b[_cons] + (tmp_x * _b[tmp_x])\ngen rdplot_hat_y_1 = _b[_cons] + (tmp_x * _b[tmp_x]) + (ubex * _b[ubex])\n```\n\n\n```stata\ntwoway (rcap rdplot_ci_l rdplot_ci_r rdplot_mean_bin, color(gs11)) ///\n(scatter rdplot_mean_y rdplot_mean_bin, sort msize(small) mcolor(gs10)) ///\n(line rdplot_hat_y_0 tmp_x if tmp_x<=0, lcolor(black) sort lwidth(medthin) lpattern(solid)) ///\n(line rdplot_hat_y_1 tmp_x if tmp_x>=0, lcolor(black) sort lwidth(medthin) lpattern(solid)), ///\nxline(0, lcolor(black) lwidth(medthin)) xscale(r(-7 7) noextend) xlabel(-7(1)7) /// \nlegend(cols(2) order(1 \"Sample average within bin\" 2 \"linear fit\" )) title(\"Regression function fit\", color(gs0)) \n```\n\n### Polynomial fit of order 2\n\n\n```stata\ngen tmp_x2 = rv * rv if inrange(rv, -7, 7)\nreg tmp_y tmp_x tmp_x2 ubex, robust \n```\n\n\n```stata\ngen rdplot_hat_y_2 = _b[_cons] + (tmp_x * _b[tmp_x]) + (tmp_x2 * _b[tmp_x2])\ngen rdplot_hat_y_3 = _b[_cons] + (tmp_x * _b[tmp_x]) + (tmp_x2 * _b[tmp_x2]) + (ubex * _b[ubex])\n```\n\n\n```stata\ntwoway (rcap rdplot_ci_l rdplot_ci_r rdplot_mean_bin, color(gs11)) ///\n(scatter rdplot_mean_y rdplot_mean_bin, sort msize(small) mcolor(gs10)) ///\n(line rdplot_hat_y_2 tmp_x if tmp_x<=0, lcolor(red) sort lwidth(medthin) lpattern(solid)) ///\n(line rdplot_hat_y_3 tmp_x if tmp_x>=0, lcolor(red) sort lwidth(medthin) lpattern(solid)) ///\n(line rdplot_hat_y_0 tmp_x if tmp_x<=0, lcolor(black) sort lwidth(medthin) lpattern(solid)) ///\n(line rdplot_hat_y_1 tmp_x if tmp_x>=0, lcolor(black) sort lwidth(medthin) lpattern(solid)), ///\nxline(0, lcolor(black) lwidth(medthin)) xscale(r(-7 7) noextend) xlabel(-7(1)7) /// \nlegend(cols(2) order(1 \"Sample average within bin\" 2 \"linear fit\" )) title(\"Regression function fit\", color(gs0)) \n```\n\n### Polynomial fit of order 3\n\n\n```stata\ngen tmp_x3 = tmp_x2 * rv if inrange(rv, -7, 7)\nreg tmp_y tmp_x tmp_x2 tmp_x3 ubex, robust\ngen rdplot_hat_y_4 = _b[_cons] + (tmp_x * _b[tmp_x]) + (tmp_x2 * _b[tmp_x2])\ngen rdplot_hat_y_5 = _b[_cons] + (tmp_x * _b[tmp_x]) + (tmp_x2 * _b[tmp_x2]) + (ubex * _b[ubex])\n```\n\n\n```stata\ntwoway (rcap rdplot_ci_l rdplot_ci_r rdplot_mean_bin, color(gs11)) ///\n(scatter rdplot_mean_y rdplot_mean_bin, sort msize(small) mcolor(gs10)) ///\n(line rdplot_hat_y_4 tmp_x if tmp_x<=0, lcolor(green) sort lwidth(medthin) lpattern(solid)) ///\n(line rdplot_hat_y_5 tmp_x if tmp_x>=0, lcolor(green) sort lwidth(medthin) lpattern(solid)) ///\n(line rdplot_hat_y_2 tmp_x if tmp_x<=0, lcolor(red) sort lwidth(medthin) lpattern(solid)) ///\n(line rdplot_hat_y_3 tmp_x if tmp_x>=0, lcolor(red) sort lwidth(medthin) lpattern(solid)) ///\n(line rdplot_hat_y_0 tmp_x if tmp_x<=0, lcolor(black) sort lwidth(medthin) lpattern(solid)) ///\n(line rdplot_hat_y_1 tmp_x if tmp_x>=0, lcolor(black) sort lwidth(medthin) lpattern(solid)), ///\nxline(0, lcolor(black) lwidth(medthin)) xscale(r(-7 7) noextend) xlabel(-7(1)7) /// \nlegend(cols(2) order(1 \"Sample average within bin\" 2 \"linear fit\" )) title(\"Regression function fit\", color(gs0)) \n```\n\n**Why did I use the predictions of y from a regression to plot the lines?**
                                        \nAs I said, I am not able to render the output of *rdplot* inside this notebook properly.
                                        \nThe actual fitted lines of *rdplot* are slightly different to my predictions of y due to differences in the variance-covariance estimation employed in both the regression in *rdplot* and my *reg y x, robust*.
                                        \n", "meta": {"hexsha": "37f28cce75525ed8bcd5c4f34b99b5486fb52ba1", "size": 818469, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebook/labor_economics-run.ipynb", "max_stars_repo_name": "makhro/LaborEconomics", "max_stars_repo_head_hexsha": "cd1f3abf8d1511e932017db09fe43f5ce81de090", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebook/labor_economics-run.ipynb", "max_issues_repo_name": "makhro/LaborEconomics", "max_issues_repo_head_hexsha": "cd1f3abf8d1511e932017db09fe43f5ce81de090", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebook/labor_economics-run.ipynb", "max_forks_repo_name": "makhro/LaborEconomics", "max_forks_repo_head_hexsha": "cd1f3abf8d1511e932017db09fe43f5ce81de090", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 101.6226719642, "max_line_length": 76316, "alphanum_fraction": 0.7270733528, "converted": true, "num_tokens": 47089, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.7217432122827968, "lm_q1q2_score": 0.42502673291879906}} {"text": "# Testing Asteroseismic Radii of Dwarf Stars with Gaia Parallaxes\n\n## Supporting material for the paper \"Asteroseismic Radii of Dwarfs: New Accuracy Constraints From Gaia DR2 Parallaxes\"\n#### Christian L. Sahlholdt, Victor Silva Aguirre\n\nIn this notebook we derive the radii of a sample of almost a hundred dwarf stars based on their parallaxes measured by the [Gaia spacecraft](http://www.esa.int/Our_Activities/Space_Science/Gaia_overview) and their angular diameters determined from the infrared flux method in a previous study ([Sahlholdt et al. 2018](https://academic.oup.com/mnras/article/476/2/1931/4848283)).\nBy comparing these radii to the ones based on [asteroseismology](https://en.wikipedia.org/wiki/Asteroseismology), we test the accuracy of the asteroseismic radii.\n\nSee README.md for help on running this notebook.\n\n## Contents\n* [Introduction](#intro)\n\n\n* [Computing Stellar Radii](#stel_rad)\n * [Stellar Data](#stel_data)\n * [Scaling Relation Parameters](#scal_par)\n * [Calculation of Gaia Radii](#rad_calc)\n\n\n* [Comparison Between Seismic and Gaia Radii](#gaia_comp)\n * [Error Distributions](#err_dist)\n * [Absolute Differences](#abs_diff)\n * [Relative Differences](#rel_diff)\n\n\n* [Correcting the Scaling Relations](#scal_corr)\n\n\n* [References](#ref)\n\n## Introduction \n\nPrecise and accurate stellar parameters like masses, radii, and ages are crucial in many different fields of astronomy.\nIn recent years, the field of asteroseismology, which is the study of intrinsic stellar oscillations, has seen great progress as a means of determining stellar parameters with high precision.\nJust like seismic measurements of earthquakes reveal the inner structure of Earth, measurements of stellar pulsations provide information about the star's interior.\nFor a star like the Sun, the outer layers undergo powerful convective motion which excites a rich spectrum of oscillation modes, and each individual pulsation carries information about the stellar structure.\nThis enables a determination of the stellar properties by matching the observed oscillations to those of a stellar model.\n\nIt is not always possible to determine the individual oscillation frequencies due to their very low amplitudes. However, usually two global parameters can be determined from the oscillations, namely the\nmean frequency separation and the frequency of maximum oscillation power.\nThese parameters follow a set of approximate *scaling relations*, tying them to the mass, radius, and temperature of the star, which makes it straightforward to estimate properties of any star with detected solar-like oscillations.\n\nSince asteroseismology holds such great potential for determining precise stellar parameters, it is important to verify that the results are also accurate.\nUnfortunately, direct measurements of radii and masses are very difficult to obtain -- especially in the large quantities possible by asteroseismology.\nIn order to test the results of asteroseismology for large stellar samples, less direct comparisons must be employed.\nOne approach is to use the trigonometric parallax with an independent measurement of the angular diameter.\nWhen combined, this gives a radius which can be compared with the seismic value.\n\nWith the release of parallaxes from the Gaia mission, a new opportunity to test asteroseismic radii has arisen.\nThe Gaia data has significantly increased the number of solar-like oscillators with precise parallax measurements and puts new observational constraints on the asteroseismic results.\n\nIn this notebook we focus on a little less than a hundred dwarf stars which have some of the most precisely determined oscillation frequencies.\nThese stars have had their individual frequencies measured, so we will test the seismic radii from both a detailed model fit of the individual frequencies and from the approximate scaling relations.\n\n## Computing Stellar Radii \n\nThe first step in the analysis is to derive the radii of the stars based on their parallaxes.\n\nIn this section the available data is presented, and we derive radii based on the parallaxes and angular diameters.\n\n### Stellar Data \n\nThe data is stored in a number of files. First of all we have the \"observed\" stellar parameters in the file `star_obs.txt`. The first column holds an ID number of the star, and the following columns hold the value and uncertainty of a number of \"observables\" (all of which are in fact derived quantities).\n\n\n```python\n!head -n15 data/star_obs.txt\n```\n\n # Observational parameters (all of which are in fact derived parameters)\r\n #\r\n # Columns (the ones left out contain the uncertainties on the previous parameter):\r\n # 0 - KIC ID\r\n # 1 - Effective temperature (K)\r\n # 3 - Metallicity ([Fe/H])\r\n # 5 - Large frequency separation (dnu) (microHz)\r\n # 7 - Frequency of maximum power (nu_max) (microHz)\r\n # 9 - Angular diameter (mas)\r\n ############################################################################################\r\n 1435467\t6382\t 80\t 0.01\t0.10\t 70.331\t0.219\t1405.99\t 8.07\t1.15e-01\t3.63e-03\r\n 2837475\t6675\t 81\t 0.01\t0.10\t 75.759\t0.197\t1560.83\t 9.69\t1.27e-01\t5.46e-03\r\n 3425851\t6469\t 80\t-0.04\t0.10\t 92.600\t1.500\t2038.00\t 60.00\t5.21e-02\t1.30e-03\r\n 3427720\t6037\t 79\t-0.06\t0.10\t119.866\t0.183\t2729.47\t 14.99\t1.14e-01\t5.45e-03\r\n 3456181\t6626\t134\t-0.15\t0.10\t 52.046\t0.234\t 971.39\t 7.72\t8.00e-02\t2.23e-03\r\n\n\nIn the file `star_mass_rad.txt` we have stellar masses, radii, surface gravities, and large frequency separations derived by fitting stellar models to the individual stellar oscillation frequencies using the BAyesian STellar Algorithm (BASTA, see [Silva Aguirre et al. (2015)](https://academic.oup.com/mnras/article-abstract/452/2/2127/1064904)).\n\n\n```python\n!head -n14 data/star_mass_rad.txt\n```\n\n # Stellar radii, masses, surface gravities and \\Delta\\nu from fitting to individual frequencies\r\n #\r\n # Columns (the ones left out contain the uncertainties on the previous parameter):\r\n # 0 - KIC ID\r\n # 1 - Radius (solar units)\r\n # 3 - Mass (solar units)\r\n # 5 - Surface gravity (log10(g/cm^3))\r\n # 7 - dnufit (muHz)\r\n #################################################################################\r\n 1435467\t1.699e+00\t2.379e-02\t1.350e+00\t4.038e-02\t4.109e+00\t8.965e-03\t7.089e+01\t5.955e-01\r\n 2837475\t1.623e+00\t2.283e-02\t1.382e+00\t4.500e-02\t4.159e+00\t1.184e-02\t7.592e+01\t1.038e+00\r\n 3425851\t1.365e+00\t1.961e-02\t1.212e+00\t4.870e-02\t4.251e+00\t1.015e-02\t9.305e+01\t6.578e-01\r\n 3427720\t1.122e+00\t1.064e-02\t1.121e+00\t1.984e-02\t4.388e+00\t4.159e-03\t1.211e+02\t7.881e-01\r\n 3456181\t2.105e+00\t3.298e-02\t1.493e+00\t3.554e-02\t3.966e+00\t1.168e-02\t5.335e+01\t7.021e-01\r\n\n\nThe files `star_par_gdr1.txt` and `star_par_gdr2.txt` contain the Gaia parallaxes from the first and second Gaia data releases. Here we focus on the data from the second release.\n\n\n```python\n!head -n11 data/star_par_gdr2.txt\n```\n\n # Stellar Parallaxes from Gaia mission (data release 2)\r\n #\r\n # Columns (the ones left out contain the uncertainties on the previous parameter):\r\n # 0 - KIC ID\r\n # 1 - Gaia parallax (mas)\r\n ########################\r\n 1435467\t7.2876\t0.0302\r\n 2837475\t8.2478\t0.0290\r\n 3425851\t3.4031\t0.0982\r\n 3427720\t10.6579\t0.0263\r\n 3456181\t4.0414\t0.0252\r\n\n\nFinally, the file `star_fdnu.txt` contains theoretical correction factors to $\\Delta\\nu$.\n\n\n```python\n!head -n13 data/star_fdnu.txt\n```\n\n # Correction factors to dnu defined as fdnu=dnufit/dnuscal, where dnufit\r\n # is calucated based on model frequencies\r\n #\r\n # Columns:\r\n # 0 - KIC ID\r\n # 1 - dnu correction factor, fdnu\r\n # 2 - dnu correction factor after correcting dnufit for the surface effect\r\n ########################\r\n 1435467 1.000 0.992\r\n 2837475 0.986 0.975\r\n 3425851 0.998 0.986\r\n 3427720 1.006 0.995\r\n 3456181 0.999 0.986\r\n\n\nBefore we can load the data we import `numpy` and `pandas` for handling the data, and `matplotlib` for plotting.\n\n\n```python\n# Import packages to be used for data analysis and plotting\nimport numpy as np\nnp.random.seed(1)\nimport pandas as pd\nimport matplotlib as mpl\nimport matplotlib.pyplot as plt\nfrom matplotlib.ticker import ScalarFormatter\n```\n\n\n```python\n# The default matplotlib settings are restored (just to be sure) and the default figure size is updated\nplt.rcParams.update(mpl.rcParamsDefault)\nplt.rcParams.update({'figure.figsize': (5, 5)})\n```\n\nUsing `pandas` we load the data from each file into a separate `DataFrame` with the stellar ID as the index.\n\n\n```python\n# Names to use for the columns in the different files\nobs_names = ['ID', 'teff', 'teff_unc', 'feh', 'feh_unc', 'dnu', 'dnu_unc',\n 'numax', 'numax_unc', 'angdiam', 'angdiam_unc']\nmass_rad_names = ['ID', 'rad', 'rad_unc', 'mass', 'mass_unc', 'logg', 'logg_unc', 'dnufit', 'dnufit_unc']\nplx_names = ['ID', 'plx', 'plx_unc']\n\n# Read data into DataFrames\nstar_obs = pd.read_table('data/star_obs.txt', header=None, skiprows=10,\n names=obs_names, index_col=0, delim_whitespace=True)\nstar_mr = pd.read_table('data/star_mass_rad.txt', header=None, skiprows=9,\n names=mass_rad_names, index_col=0, delim_whitespace=True)\nstar_plx = pd.read_table('data/star_par_gdr2.txt', header=None, skiprows=6,\n names=plx_names, index_col=0, delim_whitespace=True)\nstar_fdnu = pd.read_table('data/star_fdnu.txt', header=None, skiprows=8,\n names=['ID', 'fdnu', 'fdnu_corr'], index_col=0, delim_whitespace=True)\n\nprint('Number of stars with asteroseismic parameters: ', len(star_mr))\nprint('Number of stars with Gaia parallaxes: ', len(star_plx))\n```\n\n Number of stars with asteroseismic parameters: 95\n Number of stars with Gaia parallaxes: 93\n\n\nNot all of the stars with asteroseismic parameters are included in the Gaia data.\nWe restrict the analysis to the stars with Gaia parallaxes by joining the data sets on the indices of the parallax data.\n\n\n```python\n# Combine the DataFrames while excluding the stars without Gaia parallaxes\nsa = star_plx.join([star_obs, star_mr], how='inner')\nsa = sa.join(star_fdnu)\nsa = sa[~np.isnan(sa.numax)]\nn_star = len(sa)\n\n# Add columns for every parameter which will be derived later\nfor new_name in ['rad_scal', 'rad_scal_unc', 'mass_scal', 'mass_scal_unc',\n 'rad_scal_dnufit', 'rad_scal_dnufit_unc', 'mass_scal_dnufit', 'mass_scal_dnufit_unc',\n 'rad_scal_dnufit_corr', 'rad_scal_dnufit_corr_unc', 'mass_scal_dnufit_corr', 'mass_scal_dnufit_corr_unc',\n 'rad_gaia', 'rad_gaia_unc']:\n sa[new_name] = np.zeros(n_star)\n\n# The DataFrame is converted into a NumPy \"Records array\" since we don't need\n# the indices of the DataFrame. This makes it easier to work with the data values.\nsa = sa.to_records();\n```\n\n### Scaling Relation Parameters \n\nAs mentioned above, the masses and radii of the stars, which are included in the data we have just loaded, come from a detailed asteroseismic analysis using all individual oscillation frequencies.\nWhen individual frequencies are not available, it is possible to derive masses and radii using the average parameters of the frequency spectrum; the large frequency separation, $\\Delta\\nu$, and the frequency of maximum power, $\\nu_{\\mathrm{max}}$. The figure below shows a power spectrum of the Sun and the two average parameters.\n\n\n\nThe average parameters follow a set of approximate scaling relations given by ([Chaplin and Miglio (2013)](http://www.annualreviews.org/doi/full/10.1146/annurev-astro-082812-140938)):\n\n\\begin{align}\n \\frac{\\Delta\\nu}{\\Delta\\nu_{\\odot}} &\\simeq \\left(\\frac{M}{M_{\\odot}}\\right)^{1/2}\n \\left(\\frac{R}{R_{\\odot}}\\right)^{-3/2}\n \\label{eq:dnu_scal} \\; ,\n \\\\\n \\frac{\\nu_{\\mathrm{max}}}{\\nu_{\\mathrm{max},\\odot}} &\\simeq\n \\left(\\frac{M}{M_{\\odot}}\\right)\n \\left(\\frac{R}{R_{\\odot}}\\right)^{-2}\n \\left(\\frac{T_{\\mathrm{eff}}}{T_{\\mathrm{eff},\\odot}}\\right)^{-1/2}\n \\label{eq:numax_scal} \\; ,\n\\end{align}\n\nwhich can be rearranged to give the scaling relations for the radius and mass:\n\n\\begin{align}\n \\frac{R}{R_{\\odot}} &\\simeq \\left(\\frac{\\nu_{\\mathrm{max}}}{\\nu_{\\mathrm{max},\\odot}}\\right)\n \\left(\\frac{\\Delta\\nu}{\\Delta\\nu_{\\odot}}\\right)^{-2}\n \\left(\\frac{T_{\\mathrm{eff}}}{T_{\\mathrm{eff},\\odot}}\\right)^{1/2}\n \\; , \\label{eq:R_scal} \\\\\n \\frac{M}{M_{\\odot}} &\\simeq \\left(\\frac{\\nu_{\\mathrm{max}}}{\\nu_{\\mathrm{max},\\odot}}\\right)^{3}\n \\left(\\frac{\\Delta\\nu}{\\Delta\\nu_{\\odot}}\\right)^{-4}\n \\left(\\frac{T_{\\mathrm{eff}}}{T_{\\mathrm{eff},\\odot}}\\right)^{3/2}\n \\; . \\label{eq:M_scal}\n\\end{align}\n\nIn the code below, the scaling relations are implemented and applied to the data.\nWe calculate scaling relation masses and radii based on the observed $\\Delta\\nu$ and $\\nu_{\\mathrm{max}}$. We also repeat the calculation after correcting $\\Delta\\nu$ using the correction factors.\n\n\n```python\ndef scal_mr(dnu, numax, teff):\n '''\n Calculate mass and radius using the seismic scaling relations.\n '''\n # Constants\n dnusun = 135.1\n numaxsun = 3090.\n teffsun = 5777.\n\n # Scaling relations\n mass = np.power(numax/numaxsun, 3) * np.power(dnu/dnusun, -4) * np.power(teff/teffsun, 3/2)\n rad = (numax/numaxsun) * np.power(dnu/dnusun, -2) * np.power(teff/teffsun, 1/2)\n\n return mass, rad\n\ndef scal_mr_unc(dnu, dnu_unc, numax, numax_unc, teff, teff_unc, n_MC=100000):\n '''\n Calculate uncertainties on mass and radius by MC sampling.\n '''\n n = len(dnu)\n mass_unc, rad_unc = np.zeros(n), np.zeros(n)\n \n for i in range(n):\n teff_synth = np.random.normal(teff[i], teff_unc[i], n_MC)\n dnu_synth = np.random.normal(dnu[i], dnu_unc[i], n_MC)\n numax_synth = np.random.normal(numax[i], numax_unc[i], n_MC)\n\n mass_synth, rad_synth = scal_mr(dnu_synth, numax_synth, teff_synth)\n mass_unc[i], rad_unc[i] = np.std(mass_synth), np.std(rad_synth)\n\n return mass_unc, rad_unc\n```\n\n\n```python\n# Scaling relation masses and radii as well as uncertainties\n# are calculated and stored in the data array\nsa.mass_scal, sa.rad_scal = scal_mr(sa.dnu, sa.numax, sa.teff)\nsa.mass_scal_unc, sa.rad_scal_unc = scal_mr_unc(sa.dnu, sa.dnu_unc,\n sa.numax, sa.numax_unc,\n sa.teff, sa.teff_unc)\n\n# Scaling relation masses and radii corrected using dnufit values\nsa.mass_scal_dnufit, sa.rad_scal_dnufit = scal_mr(sa.dnu/sa.fdnu, sa.numax, sa.teff)\nsa.mass_scal_dnufit_unc, sa.rad_scal_dnufit_unc = scal_mr_unc(sa.dnu/sa.fdnu, sa.dnu_unc/sa.fdnu,\n sa.numax, sa.numax_unc,\n sa.teff, sa.teff_unc)\n\n# Scaling relation masses and radii corrected using dnufit values\n# with additional corrections for the surface effect\nsa.mass_scal_dnufit_corr, sa.rad_scal_dnufit_corr = scal_mr(sa.dnu/sa.fdnu_corr, sa.numax, sa.teff)\nsa.mass_scal_dnufit_corr_unc, sa.rad_scal_dnufit_corr_unc = scal_mr_unc(sa.dnu/sa.fdnu_corr,\n sa.dnu_unc/sa.fdnu_corr,\n sa.numax, sa.numax_unc,\n sa.teff, sa.teff_unc)\n```\n\n### Calculation of Gaia Radii \nWe now have two sets of seismic radii of the stars with Gaia parallaxes. One set are the ones we loaded from the detailed analysis based on individual frequencies. The other are the ones we just derived based on the approximate scaling relations. We now use the observed angular diameters to derive Gaia radii.\n\nBased on the radius of a star, $R$, and its angular diameter, $\\theta$, the distance and parallax can be calculated:\n\n\\begin{equation}\n d = C\\frac{2R}{\\theta} \\;\\;\\; \\Rightarrow \\;\\;\\;\n \\varpi = \\frac{1}{d} = \\frac{\\theta}{C\\times 2R} \\; .\n\\end{equation}\n\n$C$ is the factor which converts the distance into parsec which means that the parallax is in arcseconds. To get a Gaia radius based on the parallax, we isolate the radius:\n\n\\begin{equation}\n R = \\frac{\\theta}{C\\times 2\\varpi} \\; .\n\\end{equation}\n\nIn the code below the functions used to calculate Gaia radii and uncertainties are defined. We divide the radii by the solar value to get it in solar units.\n\n\n```python\ndef rad_gaia(plx, angdiam):\n '''\n Calculate the radius based on a parallax and angular diameter.\n '''\n # Constants\n radsun = 6.95508e5\n pc_conv = 6.685e-6\n\n # Radius (plx divided by 1000 to get arcseconds)\n rad = angdiam / (pc_conv * 2 * (plx/1000))\n rad /= radsun\n\n return rad\n\ndef rad_gaia_unc(plx, plx_unc, angdiam, angdiam_unc, n_MC=100000):\n '''\n Calculate uncertainties on radii using MC sampling.\n '''\n n = len(plx)\n rad_unc = np.zeros(n)\n \n for i in range(n):\n plx_synth = np.random.normal(plx[i], plx_unc[i], n_MC)\n angdiam_synth = np.random.normal(angdiam[i], angdiam_unc[i], n_MC)\n\n rad_synth = rad_gaia(plx_synth, angdiam_synth)\n rad_unc[i] = np.std(rad_synth)\n\n return rad_unc\n```\n\n\n```python\n# Gaia radii based on the parallaxes are calculated and stored\nsa.rad_gaia = rad_gaia(sa.plx, sa.angdiam)\nsa.rad_gaia_unc = rad_gaia_unc(sa.plx, sa.plx_unc, sa.angdiam, sa.angdiam_unc)\n```\n\n## Comparison Between Seismic and Gaia Radii \n\nWe have now derived both seismic and Gaia radii and want to compare them in order to test the accuracy of the seismic radii.\n\nFirst we compare the radius uncertainties and then investigate differences.\n\n### Error Distributions \n\nThe seismic and Gaia radii have different error distributions due to the different ways they are measured/derived.\nHere we plot the absolute and relative error distributions for both, taking seismic radii from the individual frequency set.\n\n\n```python\nfig, (ax1, ax2) = plt.subplots(1, 2, sharey=True, figsize=(5.5, 3.5))\nax1.hist(sa.rad_gaia_unc, alpha=1, label='Gaia data', zorder=3)\nax1.hist(sa.rad_unc, alpha=1, label='Seismic data', zorder=4)\n#ax1.hist(sa.rad_scal_unc, alpha=1, zorder=2)\nax2.hist(sa.rad_gaia_unc/sa.rad_gaia, alpha=1, label='Gaia data', zorder=3)\nax2.hist(sa.rad_unc/sa.rad, alpha=1, label='Seismic data', zorder=4)\n#ax2.hist(sa.rad_scal_unc/sa.rad_scal, alpha=1, zorder=2)\n\nax1.set_ylabel('N')\nax1.set_xlabel(r'$\\sigma_{R}$ [$R_{\\odot}$]')\nax2.set_xlabel(r'$\\sigma_{R} / R$')\nax1.grid(axis='y', zorder=0)\nax2.grid(axis='y', zorder=0)\nax1.set_xlim(left=0)\nax2.set_xlim(left=0)\nax1.legend()\n\nfig.tight_layout()\nfig.savefig('figures/rad_hist.pdf')\nplt.show()\n```\n\nThe uncertainties on the Gaia radii are dominated by uncertainties in the angular diameters as shown in the following figure.\n\n\n```python\nfig, ax = plt.subplots(figsize=(2.8, 3.5))\nax.hist(sa.plx_unc/sa.plx, label='Parallax', zorder=4)\nax.hist(sa.angdiam_unc/sa.angdiam, label='Angular diam.', zorder=3)\n\nax.set_ylabel('N')\nax.set_xlabel(r'$\\sigma_{x} / x$')\nax.grid(axis='y', zorder=0)\nax.legend()\n\nplt.show()\n```\n\n### Absolute Differences \n\nWe now look at the absolute radius differences.\n\nThe following code contains functions for calculating radius differences and ratios, and for plotting the differences.\n\n\n```python\ndef xy_diff(x, x_unc, y, y_unc):\n '''\n Calculate differences and propagated uncertainties between x and y.\n '''\n diff = x - y\n diff_unc = np.sqrt(x_unc**2 + y_unc**2)\n\n return diff, diff_unc\n\ndef xy_ratio(x, x_unc, y, y_unc):\n '''\n Calculate ratios and propagated uncertainties between x and y.\n '''\n r = x / y\n r_unc = r * np.sqrt((x_unc / x)**2 + (y_unc / y)**2)\n\n return r, r_unc\n```\n\n\n```python\nfrom scipy.stats import gaussian_kde\nfrom scipy import odr as sodr\n\n\ndef odr_poly_fit(x, y, sx, sy, order=1):\n '''\n Fit a polynomial of a given order to the data x,y with uncertainties sx, sy\n using orthogonal distance regression as implemented in scipy.\n '''\n poly_model = sodr.polynomial(order)\n odr_data = sodr.RealData(x, y, sx, sy)\n\n odr_fit = sodr.ODR(odr_data, poly_model, beta0=[1.]*(order+1))\n odr_out = odr_fit.run()\n \n return odr_out\n\n\ndef plot_radcomp(rad, rad_unc, rad_gaia, rad_gaia_unc, ax, outliers=None, polyfit=None):\n '''\n Plot radius comparison. Optionally mark outliers in red.\n '''\n # Define outliers if given\n if outliers is not None:\n bf = np.isin(sa.ID, outliers)\n else:\n bf = np.zeros(len(sa.ID), dtype=bool)\n\n # Plot points and errorbars\n ax.scatter(rad_gaia[~bf], rad[~bf], s=18, edgecolors='k', linewidth=0.8, zorder=3)\n ax.scatter(rad_gaia[bf], rad[bf], marker='^', c='r', s=18, edgecolors='k', linewidth=0.8, zorder=3)\n ax.errorbar(rad_gaia[~bf], rad[~bf], xerr=rad_gaia_unc[~bf], yerr=rad_unc[~bf],\n ls='none', linewidth=1, zorder=2)\n ax.errorbar(rad_gaia[bf], rad[bf], xerr=rad_gaia_unc[bf], yerr=rad_unc[bf],\n ls='none', c='r', linewidth=1, zorder=2)\n\n # Optionally fit polynomial and plot it\n if polyfit is not None:\n fit = odr_poly_fit(rad_gaia[~bf], rad[~bf], rad_gaia_unc[~bf], rad_unc[~bf], order=polyfit)\n #fit.pprint()\n poly = np.poly1d(fit.beta[::-1])\n\n poly_x = np.linspace(np.min(rad_gaia), np.max(rad_gaia))\n poly_y = poly(poly_x)\n\n ax.plot(poly_x, poly_y, c='k', zorder=4)\n\n ax.text(0.75, 2.25, 'Slope of fit: ' + \"{0:.3f}\".format(round(fit.beta[1], 3)) + r' $\\pm$ ' + \"{0:.3f}\".format(round(fit.sd_beta[1], 3)))\n\n ax.set_xlim([0.65, 2.5])\n ax.set_ylim([0.65, 2.5])\n ax.plot([0.65, 2.5], [0.65, 2.5], c='k', ls=':', zorder=0)\n\n ax.set_xlabel(r'$R_{\\mathrm{Gaia}}$ [$R_{\\odot}$]')\n ax.set_ylabel(r'$R_{\\mathrm{seismic}}$ [$R_{\\odot}$]')\n\n\ndef plot_radcomp2(rad, rad_unc, rad_gaia, rad_gaia_unc, axes, outliers=None, polyfit=None):\n '''\n Same as plot_radcomp but with a second subplot showing the\n differences between the two sets of radii.\n \n For best results, axes should be a tuple of two axes (ax1, ax2) where ax1 is\n on top of ax2.\n '''\n ax1, ax2 = axes\n # Plot radius comparison on first axis\n plot_radcomp(rad, rad_unc, rad_gaia, rad_gaia_unc, ax1, outliers, polyfit)\n\n rdiff, rdiff_unc = xy_diff(rad, rad_unc, rad_gaia, rad_gaia_unc)\n\n # Define outliers if given\n if outliers is not None:\n bf = np.isin(sa.ID, outliers)\n else:\n bf = np.zeros(len(sa.ID), dtype=bool)\n\n # Plot radius differences and errorbars\n ax2.scatter(rad_gaia[~bf], rdiff[~bf], s=18, edgecolors='k', linewidth=0.8, zorder=3)\n ax2.errorbar(rad_gaia[~bf], rdiff[~bf], yerr=rdiff_unc[~bf],\n ls='none', linewidth=1, zorder=2)\n ax2.axhline(c='k', ls=':', zorder=0)\n\n # Calculate and write mean difference\n mean_diff = np.sum(rdiff[~bf]/rdiff_unc[~bf]**2) / np.sum(1/rdiff_unc[~bf]**2)\n mean_diff_unc = np.sqrt(1/np.sum(1/rdiff_unc[~bf]**2))\n ax2.text(0.6, 0.2, 'Mean difference: ' + \"{0:.3f}\".format(round(mean_diff, 3)) + r' $\\pm$ ' + \"{0:.3f}\".format(round(mean_diff_unc, 3)))\n\n ax1.set_xlabel('')\n ax2.set_xlabel(r'$R_{\\mathrm{Gaia}}$ [$R_{\\odot}$]')\n ax2.set_ylabel(r'$R_{\\mathrm{seismic}}-R_{\\mathrm{Gaia}}$ [$R_{\\odot}$]')\n \n \ndef plot_raddiff(rad, rad_unc, rad_gaia, rad_gaia_unc, ax, outliers=None, relative=False,\n label=''):\n '''\n Plot radius differences as KDEs. Optionally leave out a number of outliers.\n '''\n rad_d, rad_d_unc = xy_diff(rad, rad_unc, rad_gaia, rad_gaia_unc)\n # Remove outliers if given, otherwise plot all\n if outliers is not None:\n bf = np.isin(sa.ID, outliers)\n rad_d = rad_d[~bf]\n rad_d_unc = rad_d_unc[~bf]\n\n rad_plot = rad_d/rad_d_unc if relative else rad_d\n\n rad_kde = gaussian_kde(rad_plot)\n if relative:\n x_plot = np.linspace(-3, 3, 100)\n else:\n x_plot = np.linspace(-0.2, 0.2, 100)\n \n ax.plot(x_plot, rad_kde(x_plot)/np.sum(rad_kde(x_plot)*np.diff(x_plot)[0]),\n label=label, zorder=2)\n ax.axvline(c='k', ls='--', zorder=0)\n ax.set_ylim(bottom=0)\n ax.set_xlim([x_plot[0], x_plot[-1]])\n\n if relative:\n ax.set_xlabel(r'$\\Delta R / \\sigma$')\n else:\n ax.set_xlabel(r'$\\Delta R$ [$R_{\\odot}$]')\n\n ax.set_ylabel('KDE')\n```\n\nBefore plotting the data we note that a number of the stars may have Gaia radii which differ by several standard deviations from the seismic values. By setting the variable `outlier_cut` in the cell below it is possible to remove those stars for which the radius difference deviates from zero by more than `outlier_cut` standard deviations.\n\n\n```python\noutlier_cut = 3.0\n\nrdiff, rdiff_unc = xy_diff(sa.rad, sa.rad_unc, sa.rad_gaia, sa.rad_gaia_unc)\nbad_ID = sa.ID[np.absolute(rdiff) - outlier_cut*rdiff_unc > 0]\nprint(bad_ID)\n```\n\n [ 3425851 6278762 7510397 7871531 8554498 8866102 9025370 9965715\n 10454113]\n\n\nWe also set new parameters for the plots.\n\n\n```python\nplt.rcParams.update({'xtick.top': True,\n 'ytick.right': True,\n 'xtick.direction': 'in',\n 'ytick.direction': 'in',\n 'xtick.minor.visible': True,\n 'ytick.minor.visible': True})\n```\n\nHere we plot the radius comparisons and differences for both sets of seismic parallaxes, and we mark the stars that are outliers in the individual frequency analysis (they may have ill-determined photometry).\n\n\n```python\nfig = plt.figure(figsize=(9,6))\nax1 = plt.subplot2grid((3, 2), (0, 0), rowspan=2)\nax2 = plt.subplot2grid((3, 2), (2, 0), sharex=ax1)\nax3 = plt.subplot2grid((3, 2), (0, 1), rowspan=2, sharex=ax1, sharey=ax1)\nax4 = plt.subplot2grid((3, 2), (2, 1), sharex=ax1, sharey=ax2)\n\nplot_radcomp2(sa.rad, sa.rad_unc, sa.rad_gaia, sa.rad_gaia_unc, (ax1, ax2), outliers=bad_ID, polyfit=1)\nplot_radcomp2(sa.rad_scal, sa.rad_scal_unc, sa.rad_gaia, sa.rad_gaia_unc, (ax3, ax4), outliers=bad_ID, polyfit=1)\n\nax1.set_aspect('equal')\nax3.set_aspect('equal')\nax1.set_title('Individual frequencies')\nax3.set_title('Scaling relations')\nax3.set_ylabel('')\nax4.set_ylabel('')\n\nfig.tight_layout()\nfig.savefig('figures/rad_compare.pdf')\nplt.show()\n```\n\nHere we plot the radius differences for both sets of seismic parallaxes as KDEs.\n\n\n```python\nfig, (ax1, ax2) = plt.subplots(1, 2, figsize=(8,4))\nplot_raddiff(sa.rad, sa.rad_unc, sa.rad_gaia, sa.rad_gaia_unc, ax1, outliers=bad_ID, label='Frequencies')\nplot_raddiff(sa.rad, sa.rad_unc, sa.rad_gaia, sa.rad_gaia_unc, ax2, outliers=bad_ID, relative=True)\nplot_raddiff(sa.rad_scal, sa.rad_scal_unc, sa.rad_gaia, sa.rad_gaia_unc, ax1, outliers=bad_ID, label='Scaling relations')\nplot_raddiff(sa.rad_scal, sa.rad_scal_unc, sa.rad_gaia, sa.rad_gaia_unc, ax2, outliers=bad_ID, relative=True)\nax1.legend(fontsize=9)\nfig.tight_layout()\nfig.savefig('figures/rad_diff.pdf')\nplt.show()\n```\n\nWe remove the outliers and output the mean radius offset\n\n\n```python\n# Pick out stars which are not in the bad_ID list\nsaf = sa[~np.isin(sa.ID, bad_ID)]\nprint('Stars left after outlier removal:', len(saf))\nprint()\n\n# Print weighted mean offset and uncertainty on the mean\nrdiff2, rdiff2_unc = xy_diff(saf.rad, saf.rad_unc, saf.rad_gaia, saf.rad_gaia_unc)\nprint('Weighted mean offset for individual frequency data before outlier removal:')\nprint(np.sum(rdiff/rdiff_unc**2) / np.sum(1/rdiff_unc**2), np.sqrt(1/np.sum(1/rdiff_unc**2)))\nprint('\\nWeighted mean offset for individual frequency data after outlier removeal:')\nprint(np.sum(rdiff2/rdiff2_unc**2) / np.sum(1/rdiff2_unc**2), np.sqrt(1/np.sum(1/rdiff2_unc**2)))\n```\n\n Stars left after outlier removal: 84\n \n Weighted mean offset for individual frequency data before outlier removal:\n -0.02722006277007764 0.004406069996126776\n \n Weighted mean offset for individual frequency data after outlier removeal:\n -0.015376488115215992 0.004621385381335025\n\n\n### Relative Differences \n\nIt is also interesting whether the observed offset depends on the stellar parameters like the effective temperature or the metallicity.\nIn the following we compare the ratios of seismic and Gaia radii as a function of these parameters. The ratios are binned in order to reduce the clutter of the plots.\n\n\n```python\nrrat, rrat_unc = xy_ratio(saf.rad, saf.rad_unc, saf.rad_gaia, saf.rad_gaia_unc)\nprint('Weighted mean relative offset after outlier removal:')\nprint(np.sum(rrat/rrat_unc**2) / np.sum(1/rrat_unc**2), np.sqrt(1/np.sum(1/rrat_unc**2)))\n```\n\n Weighted mean relative offset after outlier removal:\n 0.9875244337924886 0.0035312429125952806\n\n\n\n```python\ndef binned_ratios(bin_array, bins, r, r_unc):\n '''\n Bin the data in `r` according to the values in bin_array\n and the given bins.\n The weighted mean and standard deviation of the ratios in each of\n the bins are returned as arrays.\n '''\n # Find out which bin each of the values belong in\n nbins = len(bins)-1\n inds = np.digitize(bin_array, bins, right=True)\n\n # Initialize arrays\n binx = np.zeros(nbins)\n bin_mean = np.zeros(nbins)\n bin_std = np.zeros(nbins)\n\n # For each bin the weighted mean and standard deviation of the ratios\n # are calculated and stored\n for i in range(nbins):\n where = np.where(inds == i+1)\n bin_r = r[where]\n bin_r_unc = r_unc[where]\n\n bin_mean[i] = np.sum(bin_r/bin_r_unc**2) / np.sum(1/bin_r_unc**2)\n bin_std[i] = np.sqrt(1/np.sum(1/bin_r_unc**2))\n\n return bin_mean, bin_std\n```\n\n\n```python\ndef plot_rad_ratio(rad, rad_unc, rad_gaia, rad_gaia_unc, ax, xax='teff', open_marker=False,\n plabel=None):\n '''\n Plot the ratios of seismic and Gaia radii as a\n function of either effective temperature or metallicity.\n Optionally add errobars and labels.\n '''\n # Calculate parallax ratios\n rad_r, rad_r_unc = xy_ratio(rad, rad_unc, rad_gaia, rad_gaia_unc)\n\n # Define xaxis bins depending on the xaxis variable\n if xax == 'teff':\n bin_array = saf.teff\n bins = np.arange(5300, 6800, 200, dtype=int)\n bin_centers = np.arange(5400, 6700, 200, dtype=int)\n elif xax == 'feh':\n bin_array = saf.feh\n bins = np.arange(-0.6, 0.45, 0.1)\n bin_centers = np.arange(-0.55, 0.4, 0.1)\n\n # Calculate and plot binned means and standard deviations\n bin_mean, bin_std = binned_ratios(saf[xax], bins, rad_r, rad_r_unc)\n if open_marker:\n ax.errorbar(bin_centers, bin_mean, yerr=bin_std, marker='o',\n c='k', mfc='w', linestyle=':', capsize=0, label=plabel)\n else:\n ax.errorbar(bin_centers, bin_mean, yerr=bin_std, marker='o',\n c='k', linestyle='--', capsize=0, label=plabel)\n\ndef plot_rad_ratios(rad1, rad2, rad_gaia, ax, xax='teff',\n ast_label1=None, ast_label2=None):\n '''\n Plot ratios of seismic and Gaia radii for both sets of seismic\n radii with errorbars only on one of the sets.\n '''\n plot_rad_ratio(*rad1, *rad_gaia, ax, xax, open_marker=False, plabel=ast_label1)\n plot_rad_ratio(*rad2, *rad_gaia, ax, xax, open_marker=True, plabel=ast_label2)\n ax.axhline(y=1, c='k', ls='--', lw=1)\n\n if xax == 'teff':\n ax.set_xlim([5300, 6800])\n ax.set_xlabel('Effective temperature [K]')\n elif xax == 'feh':\n ax.set_xlim([-0.6, 0.4])\n ax.set_xlabel('Metallicity [Fe/H]')\n\n ax.set_ylim([0.94, 1.085])\n ax.set_ylabel(r'$R_{\\mathrm{seismic}}$ / $R_{\\mathrm{Gaia}}$')\n if ast_label1 is not None or ast_label2 is not None:\n ax.legend(ncol=2)\n```\n\n\n```python\nfig, (ax1, ax2) = plt.subplots(2, 1)\nplot_rad_ratios((saf.rad, saf.rad_unc), (saf.rad_scal, saf.rad_scal_unc),\n (saf.rad_gaia, saf.rad_gaia_unc), ax1, xax='teff')\nplot_rad_ratios((saf.rad, saf.rad_unc), (saf.rad_scal, saf.rad_scal_unc),\n (saf.rad_gaia, saf.rad_gaia_unc), ax2, xax='feh',\n ast_label1='Ind. frequencies', ast_label2='Scaling relations')\n\nfig.tight_layout()\nfig.savefig('figures/rad_ratio.pdf')\nplt.show()\n```\n\nIt is interesting that the two sets of seismic radii agree at around the solar temperature $T_{\\mathrm{eff},\\odot} = 5777$ K since this is where the scaling relations are most accurate due to the way they scale relative to the solar values.\nIn the interactive plot below, it is possible to explore changes in the plot when all angular diameters are multiplied by a common factor. It is also possible to make the factor which is multiplied on the radii temperature dependent.\n\n\n```python\ndef plot_rad_ratio_int(gaia_offset=0.0, rad_factor=1.00,\n rad_scal_factor=1.00, angdiam_factor=1.00,\n xax='teff', rfactor='const'):\n '''\n Function for interactive plotting of radius ratios.\n '''\n plx = saf.plx + gaia_offset\n # Choose between a constant factor on the radius or one that\n # increases with temperature to reach the input value at 6600 K.\n if rfactor == 'const':\n rad = saf.rad * rad_factor\n rad_scal = saf.rad_scal * rad_scal_factor\n elif rfactor == 'lin_teff':\n rad = saf.rad * (1 + (rad_factor - 1)*(saf.teff - 5400)/1200)\n rad_scal = saf.rad_scal * (1 + (rad_scal_factor - 1)*(saf.teff - 5400)/1200)\n angdiam = saf.angdiam * angdiam_factor\n \n # Calculate new Gaia radii after scaling\n radg = rad_gaia(plx, angdiam)\n \n # Plot ratios\n fig, ax = plt.subplots(figsize=(5.5,3.))\n plot_rad_ratios((rad, saf.rad_unc), (rad_scal, saf.rad_scal_unc),\n (radg, saf.rad_gaia_unc), ax, xax,\n ast_label1='Individual frequencies', ast_label2='Scaling relations')\n fig.tight_layout()\n plt.show()\n```\n\n\n```python\nfrom ipywidgets import interactive\nint_plot = interactive(plot_rad_ratio_int, gaia_offset=(-0.3, 0.3, 0.02),\n rad_factor=(0.9, 1.1, 0.01),\n rad_scal_factor=(0.9, 1.1, 0.01),\n angdiam_factor=(0.9, 1.1, 0.01),\n xax=['teff', 'feh'],\n rfactor=['const', 'lin_teff'])\nint_plot\n```\n\n\n interactive(children=(FloatSlider(value=0.0, description='gaia_offset', max=0.3, min=-0.3, step=0.02), FloatSl\u2026\n\n\n## Correcting the Scaling Relations \n\nFinally, we test the radii from the scaling relations after applying the correction factor to $\\Delta\\nu$.\nFor a few stars we were unable to correct the frequencies for the surface effect. We start by excluding these stars.\n\n\n```python\nprint('Excluded:', saf.ID[np.isnan(saf.fdnu)])\nsaf = saf[~np.isnan(saf.fdnu)]\nprint('Number of stars left:', len(saf))\n```\n\n Excluded: [4141376 5094751 6196457 8478994]\n Number of stars left: 80\n\n\n\n```python\nfrom scipy import stats\n\ndef poly_fit(x, y, y_unc, xp_lim, order=1, n_boot=10000):\n '''\n Fit a polynomial of order `order` to the data x,y with\n uncertainties in the y-values. Bootstrapping is applied\n to return lower and upper 1-\\sigma uncertainty regions.\n '''\n # Fit a polynomial to the data in the given x-range.\n px = np.linspace(xp_lim[0], xp_lim[1])\n pfit_coeff, pfit_cov = np.polyfit(x, y, order, w=1/y_unc**2, cov=True)\n if order == 1:\n slope, slope_var = pfit_coeff[0], pfit_cov[0,0]\n\n tt = slope / np.sqrt(slope_var)\n pval = stats.t.sf(np.abs(tt), len(x)-1)*2\n\n poly = np.poly1d(pfit_coeff)\n py = poly(px)\n\n # Repeat the fit for n_boot bootstrap samples.\n boot_results = np.zeros((n_boot, len(px)))\n boot_slope = np.zeros(n_boot)\n for i in range(n_boot):\n rand_per = np.random.randint(len(x), size=len(x))\n synth_x = x[rand_per]\n synth_y = y[rand_per]\n synth_y_unc = y_unc[rand_per]\n\n pfit = np.polyfit(synth_x, synth_y, order, w=1/synth_y_unc**2)\n poly = np.poly1d(pfit)\n py_temp = poly(px)\n\n boot_results[i, :] = py_temp\n boot_slope[i] = pfit[0]\n\n # Calculate the 16th and 84th percentile at each x-value\n # and return as lower and upper uncertainty regions.\n boot_lower = np.percentile(boot_results, 16, axis=0)\n boot_upper = np.percentile(boot_results, 84, axis=0)\n\n if order == 1:\n return px, py, boot_lower, boot_upper, slope, slope_var, pval\n else:\n return px, py, boot_lower, boot_upper\n```\n\n\n```python\nfig, (ax1, ax2, ax3) = plt.subplots(3, 1, figsize=(5,7), sharex=True, sharey=True)\n\naxes = (ax1, ax2, ax3)\ntitles = ('No correction',\n r'$\\Delta\\nu_{\\mathrm{fit}}$ correction',\n r'$\\Delta\\nu_{\\mathrm{fit}}$ correction with surface term')\npanels = ('a)', 'b)', 'c)')\n\nfor i, (rad, rad_unc) in enumerate(((saf.rad_scal, saf.rad_scal_unc),\n (saf.rad_scal_dnufit, saf.rad_scal_dnufit_unc),\n (saf.rad_scal_dnufit_corr, saf.rad_scal_dnufit_corr_unc))):\n ax = axes[i]\n rad_r, rad_r_unc = xy_ratio(rad, rad_unc, saf.rad_gaia, saf.rad_gaia_unc)\n rad_r_mean = np.sum(rad_r/rad_r_unc**2) / np.sum(1/rad_r_unc**2)\n rad_r_mean_unc = np.sqrt(1/np.sum(1/rad_r_unc**2))\n\n ax.scatter(saf['teff'], rad_r, edgecolors='k', s=22, linewidth=0.8,\n zorder=1)\n ax.errorbar(saf['teff'], rad_r, yerr=rad_r_unc, linestyle='none',\n c='k', capsize=0, lw=1, zorder=0)\n\n fit_filt = (saf['teff'] > 5600) & (saf['teff'] < 6400)\n #px, py, py_low, py_high, slope, slope_var, slope_pval = poly_fit(saf['teff'][fit_filt], rad_r[fit_filt], rad_r_unc[fit_filt], [5600., 6400.], 1)\n px, py, py_low, py_high, slope, slope_var, slope_pval = poly_fit(saf['teff'], rad_r, rad_r_unc, [5340., 6780.], 1)\n slope_std = np.sqrt(slope_var)\n\n ax.plot(px, py, c='k', ls='--', linewidth=2, zorder=2)\n ax.axhline(y=1, c='k', ls=':', zorder=0)\n ax.axhline(y=1.03, c='grey', ls='--', zorder=0)\n ax.axhline(y=0.97, c='grey', ls='--', zorder=0)\n\n ax.text(5300, 0.87, 'Mean ratio: ' + \"{0:.3f}\".format(round(rad_r_mean, 3)) + r' $\\pm$ ' + \"{0:.3f}\".format(round(rad_r_mean_unc, 3)))\n ax.text(5300, 1.12, 'No slope p-value: ' + \"{0:.2e}\".format(slope_pval))\n ax.text(6750, 1.11, panels[i], weight='bold')\n\n ax.set_ylim([0.85, 1.15])\n ax.set_xlim([5280, 6850])\n\n ax.set_ylabel(r'$R_{\\mathrm{seismic}}$ / $R_{\\mathrm{Gaia}}$')\n ax.set_title(titles[i])\n\nax3.set_xlabel('Effective temperature [K]')\n\nfig.tight_layout()\nfig.savefig('figures/rad_ratio_fit.pdf')\nplt.show()\n```\n\nAlternative version of figure above\n\n\n```python\nfig, (ax1, ax2, ax3) = plt.subplots(3, 1, figsize=(5,7), sharex=True, sharey=True)\n\naxes = (ax1, ax2, ax3)\ntitles = ('No correction',\n r'$\\Delta\\nu_{\\mathrm{fit}}$ correction',\n r'$\\Delta\\nu_{\\mathrm{fit}}$ correction with surface term')\npanels = ('a)', 'b)', 'c)')\n\nfor i, (rad, rad_unc) in enumerate(((saf.rad_scal, saf.rad_scal_unc),\n (saf.rad_scal_dnufit, saf.rad_scal_dnufit_unc),\n (saf.rad_scal_dnufit_corr, saf.rad_scal_dnufit_corr_unc))):\n ax = axes[i]\n rad_r, rad_r_unc = xy_ratio(rad, rad_unc, saf.rad_gaia, saf.rad_gaia_unc)\n rad_r_mean = np.sum(rad_r/rad_r_unc**2) / np.sum(1/rad_r_unc**2)\n rad_r_mean_unc = np.sqrt(1/np.sum(1/rad_r_unc**2))\n\n ax.scatter(saf['teff'], rad_r, s=18, linewidth=0.8, c='grey',\n zorder=1)\n ax.errorbar(saf['teff'], rad_r, yerr=rad_r_unc, linestyle='none',\n c='grey', capsize=0, lw=0.8, zorder=0)\n\n bin_array = saf.teff\n bins = np.arange(5300, 6800, 200, dtype=int)\n bin_centers = np.arange(5400, 6700, 200, dtype=int)\n bin_mean, bin_std = binned_ratios(bin_array, bins, rad_r, rad_r_unc)\n ax.plot(bin_centers, bin_mean, marker='o', c='k', mfc='w', linestyle=':', zorder=3)\n ax.errorbar(bin_centers, bin_mean, yerr=bin_std, c='k', capsize=0, linestyle='none', zorder=3)\n\n ax.axhline(y=1, c='k', ls='--', zorder=0)\n ax.axhline(y=1.03, c='grey', ls='--', zorder=0)\n ax.axhline(y=0.97, c='grey', ls='--', zorder=0)\n\n ax.text(5300, 0.87, 'Mean ratio: ' + \"{0:.3f}\".format(round(rad_r_mean, 3)) + r' $\\pm$ ' + \"{0:.3f}\".format(round(rad_r_mean_unc, 3)))\n ax.text(6750, 1.11, panels[i], weight='bold')\n\n ax.set_ylim([0.85, 1.15])\n ax.set_xlim([5280, 6850])\n\n ax.set_ylabel(r'$R_{\\mathrm{seismic}}$ / $R_{\\mathrm{Gaia}}$')\n ax.set_title(titles[i])\n\nax3.set_xlabel('Effective temperature [K]')\n\nfig.tight_layout()\nfig.savefig('figures/rad_ratio_binned.pdf')\nplt.show()\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "2c209c07cb25ca3e83cab6dfd22078c2aa844ffc", "size": 573042, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "testing_ast_radii_rad.ipynb", "max_stars_repo_name": "csahlholdt/testing_ast_radii", "max_stars_repo_head_hexsha": "c28c3f066c719c8c907aeea88ac3c9ed366ec090", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "testing_ast_radii_rad.ipynb", "max_issues_repo_name": "csahlholdt/testing_ast_radii", "max_issues_repo_head_hexsha": "c28c3f066c719c8c907aeea88ac3c9ed366ec090", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, 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YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4250267293941805}} {"text": "---\nlayout: global\ntitle: \u534f\u540c\u8fc7\u6ee4\ndisplayTitle: \u534f\u540c\u8fc7\u6ee4\n---\n\n# \u534f\u540c\u8fc7\u6ee4\n\n\u534f\u540c\u8fc7\u6ee4[Collaborative filtering](http://en.wikipedia.org/wiki/Recommender_system#Collaborative_filtering)\u5e38\u88ab\u7528\u4e8e\u63a8\u8350\u7cfb\u7edf\u3002\n\u8fd9\u7c7b\u6280\u672f\u76ee\u6807\u5728\u4e8e\u586b\u5145\u201c\u7528\u6237\uff0d\u5546\u54c1\u201d\u8054\u7cfb\u77e9\u9635\u4e2d\u7684\u7f3a\u5931\u9879\u3002\n`spark.ml`\u76ee\u524d\u652f\u6301\u57fa\u4e8e\u6a21\u578b\u7684\u534f\u540c\u8fc7\u6ee4\uff0c\u5176\u4e2d\u7528\u6237\u548c\u5546\u54c1\u4ee5\u5c11\u91cf\u7684\u9690\u56e0\u5b50\u6765\u63cf\u8ff0\uff0c\u7528\u4ee5\u9884\u6d4b\u7f3a\u5931\u9879\u3002\n`spark.ml`\u4f7f\u7528\u4ea4\u66ff\u6700\u5c0f\u4e8c\u4e58[alternating least squares(ALS)](http://dl.acm.org/citation.cfm?id=1608614)\u7b97\u6cd5\u6765\u5b66\u4e60\u8fd9\u4e9b\u9690\u56e0\u5b50\u3002\n`spark.ml`\u5b9e\u73b0\u7684\u63a5\u53e3\u6709\u8fd9\u4e9b\u53c2\u6570\uff1a\n\n* *numBlocks*\u662f\u7528\u6237\u548c\u5546\u54c1\u5c06\u88ab\u5212\u5206\u4e3a\u51e0\u5757\u4ee5\u5e76\u884c\u8ba1\u7b97\uff08\u9ed8\u8ba410\uff09\u3002\n* *rank*\u662f\u6a21\u578b\u4e2d\u7684\u9690\u56e0\u5b50\u6570\u91cf\uff08\u9ed8\u8ba410\uff09\u3002\n* *maxIter*\u6700\u5927\u8fed\u4ee3\u8f6e\u6570\uff08\u9ed8\u8ba410\uff09\u3002\n* *regParam*\u6307\u5b9aALS\u4e2d\u7684\u6b63\u5219\u5316\u53c2\u6570\uff08\u9ed8\u8ba41.0\uff09\u3002\n* *implicitPrefs*\u6307\u5b9a\u4f7f\u7528*\u663e\u5f0f\u53cd\u9988*\u8fd8\u662f*\u9690\u5f0f\u53cd\u9988*\u6570\u636e\u7684ALS\u53d8\u4f53\uff08\u9ed8\u8ba4\u4e3a`false`\u8868\u793a\u4f7f\u7528*\u663e\u5f0f\u53cd\u9988*\uff09\u3002\n* *alpha*\u7528\u4e8e\u9690\u5f0f\u53cd\u9988\u7684ALS\uff0c\u5f71\u54cd\u89c2\u6d4b\u5230\u7684\u504f\u597d\u4e2d\u7684*baseline*\u7f6e\u4fe1\u5ea6\uff08\u9ed8\u8ba41.0\uff09\u3002\n* *nonnegative*\u6307\u5b9a\u662f\u5426\u5bf9\u6700\u5c0f\u4e8c\u4e58\u5e94\u7528\u975e\u8d1f\u7ea6\u675f\uff08\u9ed8\u8ba4\u4e3a`false`\uff09\u3002\n\n\u53ef\u4ee5\u8c03\u6574\u8fd9\u4e9b\u53c2\u6570\uff0c\u4e0d\u65ad\u4f18\u5316\u7ed3\u679c\uff0c\u4f7f\u5747\u65b9\u5dee\u53d8\u5c0f\u3002\u6bd4\u5982\uff1aimaxIter\u8d8a\u5927\uff0cregParam\u8d8a\u5c0f\uff0c\u5747\u65b9\u5dee\u4f1a\u8d8a\u5c0f\uff0c\u63a8\u8350\u7ed3\u679c\u8f83\u4f18\u3002\n\n**\u6ce8\u610f**\uff1a\u57fa\u4e8eDataFrame\u7684ALS\u63a5\u53e3\u76ee\u524d\u4ec5\u652f\u6301\u6574\u6570\u578b\u7684\u7528\u6237\u548c\u5546\u54c1\u7f16\u53f7\u3002\n\u4e5f\u652f\u6301\u7528\u6237\u548c\u5546\u54c1\u7f16\u53f7\u5217\u7684\u5176\u4ed6\u6570\u503c\u7c7b\u578b\uff0c\u4f46\u7f16\u53f7\u5fc5\u987b\u5728\u6574\u6570\u578b\u8303\u56f4\u5185\u3002\n\n## \u4ec0\u4e48\u662fALS\n\nALS\u901a\u8fc7\u89c2\u5bdf\u5230\u7684\u6240\u6709\u7528\u6237\u7ed9\u5546\u54c1\u7684\u6253\u5206\uff0c\u6765\u63a8\u65ad\u6bcf\u4e2a\u7528\u6237\u7684\u559c\u597d\u5e76\u5411\u7528\u6237\u63a8\u8350\u9002\u5408\u7684\u5546\u54c1\u3002\n\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u6211\u4eec\u770b\u4e0b\u9762\u4e00\u4e2a`8*8`\u7684\u7528\u6237\u6253\u5206\u77e9\u9635\uff1a\n\n
                                        \n\n\u8fd9\u4e2a\u77e9\u9635$A$\u7684\u6bcf\u4e00\u884c\u4ee3\u8868\u4e00\u4e2a\u7528\u6237$(u_1,u_2,...,u_8)$\u3001\u6bcf\u4e00\u5217\u4ee3\u8868\u4e00\u4e2a\u5546\u54c1$(v_1,v_2,...,v_8)$\u3001\u7528\u6237\u7684\u6253\u5206\u4e3a`1-9`\u5206\u3002\n\u8fd9\u4e2a\u77e9\u9635\u53ea\u663e\u793a\u4e86\u89c2\u5bdf\u5230\u7684\u6253\u5206\uff0c\u6211\u4eec\u9700\u8981\u63a8\u6d4b\u6ca1\u6709\u89c2\u5bdf\u5230\u7684\u6253\u5206\u3002\u6bd4\u5982$(u_6,v_5)$\u6253\u5206\u591a\u5c11\uff1f\n\u5982\u679c\u4ee5\u6570\u72ec\u7684\u65b9\u5f0f\u6765\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u53ef\u4ee5\u5f97\u5230\u552f\u4e00\u7684\u7ed3\u679c\u3002\n\u56e0\u4e3a\u6570\u72ec\u7684\u89c4\u5219\u5f88\u5f3a\uff0c\u6bcf\u6dfb\u52a0\u4e00\u6761\u89c4\u5219\uff0c\u5c31\u8ba9\u6574\u4e2a\u7cfb\u7edf\u7684\u81ea\u7531\u5ea6\u4e0b\u964d\u4e00\u4e2a\u91cf\u7ea7\u3002\n\u5f53\u6211\u4eec\u6ee1\u8db3\u6240\u6709\u7684\u89c4\u5219\u65f6\uff0c\u6574\u4e2a\u7cfb\u7edf\u7684\u81ea\u7531\u5ea6\u5c31\u964d\u4e3a$1$\u4e86\uff0c\u4e5f\u5c31\u5f97\u51fa\u4e86\u552f\u4e00\u7684\u7ed3\u679c\u3002\n\n\u5bf9\u4e8e\u4e0a\u9762\u7684\u6253\u5206\u77e9\u9635\uff0c\u5982\u679c\u6211\u4eec\u4e0d\u6dfb\u52a0\u4efb\u4f55\u6761\u4ef6\u7684\u8bdd\uff0c\u4e5f\u5373\u6253\u5206\u4e4b\u95f4\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u6211\u4eec\u5c31\u6ca1\u6cd5\u5f97\u5230$(u_6,v_5)$\u7684\u6253\u5206\u3002\n\u6240\u4ee5\u5728\u8fd9\u4e2a\u7528\u6237\u6253\u5206\u77e9\u9635\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u9700\u8981\u63d0\u51fa\u4e00\u4e2a\u9650\u5236\u5176\u81ea\u7531\u5ea6\u7684\u5408\u7406\u5047\u8bbe\uff0c\u4f7f\u5f97\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u89c2\u5bdf\u5df2\u6709\u6253\u5206\u6765\u731c\u6d4b\u672a\u77e5\u6253\u5206\u3002\n\nALS\u7684\u6838\u5fc3\u5c31\u662f\u8fd9\u6837\u4e00\u4e2a\u5047\u8bbe\uff1a\u6253\u5206\u77e9\u9635\u662f\u8fd1\u4f3c\u4f4e\u79e9\u7684\u3002\n\u6362\u53e5\u8bdd\u8bf4\uff0c\u5c31\u662f\u4e00\u4e2a$A_{m*n}$\u7684\u6253\u5206\u77e9\u9635\u53ef\u4ee5\u7531\u5206\u89e3\u7684\u4e24\u4e2a\u5c0f\u77e9\u9635$U_{m*k}$\u548c$V_{k*n}$\u7684\u4e58\u79ef\u6765\u8fd1\u4f3c\uff0c\u5373$A=U{V}^{T},k <= m,n$\u3002\n\u8fd9\u5c31\u662fALS\u7684\u77e9\u9635\u5206\u89e3\u65b9\u6cd5\u3002\u8fd9\u6837\u6211\u4eec\u628a\u7cfb\u7edf\u7684\u81ea\u7531\u5ea6\u4ece$O(mn)$\u964d\u5230\u4e86$O((m+n)k)$\u3002\n\n\u90a3\u4e48ALS\u7684\u4f4e\u79e9\u5047\u8bbe\u4e3a\u4ec0\u4e48\u662f\u5408\u7406\u7684\u5462\uff1f\n\u6211\u4eec\u63cf\u8ff0\u4e00\u4e2a\u4eba\u7684\u559c\u597d\u7ecf\u5e38\u662f\u5728\u4e00\u4e2a\u62bd\u8c61\u7684\u4f4e\u7ef4\u7a7a\u95f4\u4e0a\u8fdb\u884c\u7684\uff0c\u5e76\u4e0d\u9700\u8981\u4e00\u4e00\u5217\u51fa\u4ed6\u559c\u597d\u7684\u4e8b\u7269\u3002\n\u4f8b\u5982\uff0c\u6211\u559c\u597d\u770b\u4fa6\u63a2\u5f71\u7247\uff0c\u53ef\u80fd\u4ee3\u8868\u6211\u559c\u6b22\u300a\u795e\u63a2\u590f\u6d1b\u7279\u300b\u3001\u300a\u795e\u63a2\u72c4\u4ec1\u6770\u300b\u7b49\u3002\n\u8fd9\u4e9b\u5f71\u7247\u90fd\u7b26\u5408\u6211\u5bf9\u81ea\u5df1\u559c\u597d\u7684\u63cf\u8ff0\uff0c\u4e5f\u5c31\u662f\u8bf4\u4ed6\u4eec\u5728\u8fd9\u4e2a\u62bd\u8c61\u7684\u4f4e\u7ef4\u7a7a\u95f4\u7684\u6295\u5f71\u548c\u6211\u7684\u559c\u597d\u76f8\u4f3c\u3002\n\n\u518d\u62bd\u8c61\u4e00\u4e9b\u6765\u63cf\u8ff0\u8fd9\u4e2a\u95ee\u9898\uff0c\u6211\u4eec\u628a\u67d0\u4e2a\u4eba\u7684\u559c\u597d\u6620\u5c04\u5230\u4e86\u4f4e\u7ef4\u5411\u91cf$u_i$\u4e0a\uff0c\u540c\u65f6\u5c06\u67d0\u4e2a\u5f71\u7247\u7684\u7279\u5f81\u6620\u5c04\u5230\u4e86\u7ef4\u5ea6\u76f8\u540c\u7684\u5411\u91cf$v_j$\u4e0a\uff0c\u90a3\u4e48\u8fd9\u4e2a\u4eba\u548c\u8fd9\u4e2a\u5f71\u7247\u7684\u76f8\u4f3c\u5ea6\u5c31\u53ef\u4ee5\u8868\u8ff0\u6210\u8fd9\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u5185\u79ef$u_{i}^{T}v_{j}$ \u3002\n\u6211\u4eec\u628a\u6253\u5206\u7406\u89e3\u6210\u76f8\u4f3c\u5ea6\uff0c\u90a3\u4e48\u6253\u5206\u77e9\u9635$A$\u5c31\u53ef\u4ee5\u7531\u7528\u6237\u559c\u597d\u77e9\u9635\u548c\u4ea7\u54c1\u7279\u5f81\u77e9\u9635\u7684\u4e58\u79ef$U{V}^{T}$\u6765\u8fd1\u4f3c\u4e86\u3002\n\n\u4f4e\u7ef4\u7a7a\u95f4\u7684\u9009\u53d6\u662f\u4e00\u4e2a\u95ee\u9898\u3002\n\u8fd9\u4e2a\u4f4e\u7ef4\u7a7a\u95f4\u8981\u80fd\u591f\u5f88\u597d\u7684\u533a\u5206\u4e8b\u7269\uff0c\u90a3\u4e48\u5c31\u9700\u8981\u4e00\u4e2a\u660e\u786e\u7684\u53ef\u91cf\u5316\u76ee\u6807\uff0c\u8fd9\u5c31\u662f\u91cd\u6784\u8bef\u5dee\u3002\n\u5728ALS\u4e2d\u6211\u4eec\u4f7f\u7528F\u8303\u6570\u6765\u91cf\u5316\u91cd\u6784\u8bef\u5dee\uff0c\u5c31\u662f\u6bcf\u4e2a\u5143\u7d20\u91cd\u6784\u8bef\u5dee\u7684\u5e73\u65b9\u548c\u3002\n\u8fd9\u91cc\u5b58\u5728\u4e00\u4e2a\u95ee\u9898\uff0c\u6211\u4eec\u53ea\u89c2\u5bdf\u5230\u90e8\u5206\u6253\u5206\uff0c$A$\u4e2d\u7684\u5927\u91cf\u672a\u77e5\u5143\u662f\u6211\u4eec\u60f3\u63a8\u65ad\u7684\uff0c\u6240\u4ee5\u8fd9\u4e2a\u91cd\u6784\u8bef\u5dee\u662f\u5305\u542b\u672a\u77e5\u6570\u7684\u3002\n\u89e3\u51b3\u65b9\u6848\u5f88\u7b80\u5355\uff1a\u53ea\u8ba1\u7b97\u5df2\u77e5\u6253\u5206\u7684\u91cd\u6784\u8bef\u5dee\u3002\n\n$$\\sum\\limits_{(i,j)\\in R}(a_{ij} - u_iv_j^T)^2$$\n\n\u4e0b\u4e00\u8282\u6211\u4eec\u5c06\u4ece\u539f\u7406\u4e0a\u8bb2\u89e3`spark.ml`\u4e2d\u5b9e\u73b0\u7684ALS\u6a21\u578b\u3002\n\n## Spark\u4e2dALS\u7684\u5b9e\u73b0\u539f\u7406\n\n`spark.ml`\u5229\u7528\u4ea4\u6362\u6700\u5c0f\u4e8c\u4e58\u89e3\u51b3\u77e9\u9635\u5206\u89e3\u95ee\u9898\uff0c\u5206\u4e24\u79cd\u60c5\u51b5\uff1a\u6570\u636e\u96c6\u662f\u663e\u5f0f\u53cd\u9988\u548c\u6570\u636e\u96c6\u662f\u9690\u5f0f\u53cd\u9988\u3002\n\u7531\u4e8e\u9690\u5f0f\u53cd\u9988\u7b97\u6cd5\u7684\u539f\u7406\u662f\u5728\u663e\u793a\u53cd\u9988\u7b97\u6cd5\u539f\u7406\u7684\u57fa\u7840\u4e0a\u4f5c\u7684\u4fee\u6539\uff0c\u6240\u4ee5\u6211\u4eec\u5728\u6b64\u53ea\u4f1a\u5177\u4f53\u8bb2\u89e3\u6570\u636e\u96c6\u4e3a\u9690\u5f0f\u53cd\u9988\u7684\u7b97\u6cd5\u3002\n\n### \u4ecb\u7ecd\n\n\u4ece\u5e7f\u4e49\u4e0a\u8bb2\uff0c\u63a8\u8350\u7cfb\u7edf\u57fa\u4e8e\u4e24\u79cd\u4e0d\u540c\u7684\u7b56\u7565\uff1a\u57fa\u4e8e\u5185\u5bb9\u7684\u65b9\u6cd5\u548c\u57fa\u4e8e\u534f\u540c\u8fc7\u6ee4\u7684\u65b9\u6cd5\u3002\n`spark.ml`\u4e2d\u4f7f\u7528\u534f\u540c\u8fc7\u6ee4\u7684\u65b9\u5f0f\u3002\n\u534f\u540c\u8fc7\u6ee4\u5206\u6790\u7528\u6237\u4ee5\u53ca\u7528\u6237\u76f8\u5173\u7684\u4ea7\u54c1\u7684\u76f8\u5173\u6027\uff0c\u7528\u4ee5\u8bc6\u522b\u65b0\u7684\u7528\u6237-\u4ea7\u54c1\u76f8\u5173\u6027\u3002\n\u534f\u540c\u8fc7\u6ee4\u7cfb\u7edf\u9700\u8981\u7684\u552f\u4e00\u4fe1\u606f\u662f\u7528\u6237\u8fc7\u53bb\u7684\u884c\u4e3a\u4fe1\u606f\uff0c\u6bd4\u5982\u5bf9\u4ea7\u54c1\u7684\u8bc4\u4ef7\u4fe1\u606f\u3002\n\u534f\u540c\u8fc7\u6ee4\u662f\u9886\u57df\u65e0\u5173\u7684\uff0c\u6240\u4ee5\u5b83\u53ef\u4ee5\u65b9\u4fbf\u89e3\u51b3\u57fa\u4e8e\u5185\u5bb9\u65b9\u6cd5\u96be\u4ee5\u89e3\u51b3\u7684\u8bb8\u591a\u95ee\u9898\u3002\n\n\u63a8\u8350\u7cfb\u7edf\u4f9d\u8d56\u4e0d\u540c\u7c7b\u578b\u7684\u8f93\u5165\u6570\u636e\uff0c\u6700\u65b9\u4fbf\u7684\u662f\u9ad8\u8d28\u91cf\u7684\u663e\u5f0f\u53cd\u9988\u6570\u636e\uff0c\u5b83\u4eec\u5305\u542b\u7528\u6237\u5bf9\u611f\u5174\u8da3\u5546\u54c1\u660e\u786e\u7684\u8bc4\u4ef7\u3002\n\u4f8b\u5982\uff0cNetflix\u6536\u96c6\u7684\u7528\u6237\u5bf9\u7535\u5f71\u8bc4\u4ef7\u7684\u661f\u661f\u7b49\u7ea7\u6570\u636e\u3002\n\u4f46\u662f\u663e\u5f0f\u53cd\u9988\u6570\u636e\u4e0d\u4e00\u5b9a\u603b\u662f\u627e\u5f97\u5230\uff0c\u56e0\u6b64\u63a8\u8350\u7cfb\u7edf\u53ef\u4ee5\u4ece\u66f4\u4e30\u5bcc\u7684\u9690\u5f0f\u53cd\u9988\u4fe1\u606f\u4e2d\u63a8\u6d4b\u7528\u6237\u7684\u504f\u597d\u3002\n\u9690\u5f0f\u53cd\u9988\u7c7b\u578b\u5305\u62ec\u8d2d\u4e70\u5386\u53f2\u3001\u6d4f\u89c8\u5386\u53f2\u3001\u641c\u7d22\u6a21\u5f0f\u751a\u81f3\u9f20\u6807\u52a8\u4f5c\u3002\n\u4f8b\u5982\uff0c\u8d2d\u4e70\u540c\u4e00\u4e2a\u4f5c\u8005\u8bb8\u591a\u4e66\u7684\u7528\u6237\u53ef\u80fd\u559c\u6b22\u8fd9\u4e2a\u4f5c\u8005\u3002\n\n\u8bb8\u591a\u7814\u7a76\u90fd\u96c6\u4e2d\u5728\u5904\u7406\u663e\u5f0f\u53cd\u9988\uff0c\u7136\u800c\u5728\u5f88\u591a\u5e94\u7528\u573a\u666f\u4e0b\uff0c\u5e94\u7528\u7a0b\u5e8f\u91cd\u70b9\u5173\u6ce8\u9690\u5f0f\u53cd\u9988\u6570\u636e\u3002\u56e0\u4e3a\u53ef\u80fd\u7528\u6237\u4e0d\u613f\u610f\u8bc4\u4ef7\u5546\u54c1\u6216\u8005\u7531\u4e8e\u7cfb\u7edf\u9650\u5236\u6211\u4eec\u4e0d\u80fd\u6536\u96c6\u663e\u5f0f\u53cd\u9988\u6570\u636e\u3002\u5728\u9690\u5f0f\u6a21\u578b\u4e2d\uff0c\u4e00\u65e6\u7528\u6237\u5141\u8bb8\u6536\u96c6\u53ef\u7528\u7684\u6570\u636e\uff0c\u5728\u5ba2\u6237\u7aef\u5e76\u4e0d\u9700\u8981\u989d\u5916\u7684\u663e\u5f0f\u6570\u636e\u3002\u6587\u732e\u4e2d\u7684\u7cfb\u7edf\u907f\u514d\u4e3b\u52a8\u5730\u5411\u7528\u6237\u6536\u96c6\u663e\u5f0f\u53cd\u9988\u4fe1\u606f\uff0c\u6240\u4ee5\u7cfb\u7edf\u4ec5\u4ec5\u4f9d\u9760\u9690\u5f0f\u4fe1\u606f\u3002\n\n\u4e86\u89e3\u9690\u5f0f\u53cd\u9988\u7684\u7279\u70b9\u975e\u5e38\u91cd\u8981\uff0c\u56e0\u4e3a\u8fd9\u4e9b\u7279\u8d28\u4f7f\u6211\u4eec\u907f\u514d\u4e86\u76f4\u63a5\u8c03\u7528\u57fa\u4e8e\u663e\u5f0f\u53cd\u9988\u7684\u7b97\u6cd5\u3002\u6700\u4e3b\u8981\u7684\u7279\u70b9\u6709\u5982\u4e0b\u51e0\u79cd\uff1a\n\n1. \u6ca1\u6709\u8d1f\u53cd\u9988\u3002\u901a\u8fc7\u89c2\u5bdf\u7528\u6237\u884c\u4e3a\uff0c\u6211\u4eec\u53ef\u4ee5\u63a8\u6d4b\u90a3\u4e2a\u5546\u54c1\u4ed6\u53ef\u80fd\u559c\u6b22\uff0c\u7136\u540e\u8d2d\u4e70\uff0c\u4f46\u662f\u6211\u4eec\u5f88\u96be\u63a8\u6d4b\u54ea\u4e2a\u5546\u54c1\u7528\u6237\u4e0d\u559c\u6b22\u3002\u8fd9\u5728\u663e\u5f0f\u53cd\u9988\u7b97\u6cd5\u4e2d\u5e76\u4e0d\u5b58\u5728\uff0c\u56e0\u4e3a\u7528\u6237\u660e\u786e\u544a\u8bc9\u4e86\u6211\u4eec\u54ea\u4e9b\u4ed6\u559c\u6b22\u54ea\u4e9b\u4ed6\u4e0d\u559c\u6b22\u3002\n\n1. \u9690\u5f0f\u53cd\u9988\u662f\u5185\u5728\u7684\u566a\u97f3\u3002\u867d\u7136\u6211\u4eec\u62fc\u547d\u7684\u8ffd\u8e2a\u7528\u6237\u884c\u4e3a\uff0c\u4f46\u662f\u6211\u4eec\u4ec5\u4ec5\u53ea\u662f\u731c\u6d4b\u4ed6\u4eec\u7684\u504f\u597d\u548c\u771f\u5b9e\u52a8\u673a\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u53ef\u80fd\u77e5\u9053\u4e00\u4e2a\u4eba\u7684\u8d2d\u4e70\u884c\u4e3a\uff0c\u4f46\u662f\u8fd9\u5e76\u4e0d\u80fd\u5b8c\u5168\u8bf4\u660e\u504f\u597d\u548c\u52a8\u673a\uff0c\u56e0\u4e3a\u8fd9\u4e2a\u5546\u54c1\u53ef\u80fd\u4f5c\u4e3a\u793c\u7269\u88ab\u8d2d\u4e70\u800c\u7528\u6237\u5e76\u4e0d\u559c\u6b22\u5b83\u3002\n\n1. \u663e\u793a\u53cd\u9988\u7684\u6570\u503c\u503c\u8868\u793a\u504f\u597d\uff08preference\uff09\uff0c\u9690\u5f0f\u56de\u9988\u7684\u6570\u503c\u503c\u8868\u793a\u4fe1\u5ea6\uff08confidence\uff09\u3002\u57fa\u4e8e\u663e\u793a\u53cd\u9988\u7684\u7cfb\u7edf\u7528\u661f\u661f\u7b49\u7ea7\u8ba9\u7528\u6237\u8868\u8fbe\u4ed6\u4eec\u7684\u559c\u597d\u7a0b\u5ea6\uff0c\u4f8b\u5982\u4e00\u9897\u661f\u8868\u793a\u5f88\u4e0d\u559c\u6b22\uff0c\u4e94\u9897\u661f\u8868\u793a\u975e\u5e38\u559c\u6b22\u3002\u57fa\u4e8e\u9690\u5f0f\u53cd\u9988\u7684\u6570\u503c\u503c\u63cf\u8ff0\u7684\u662f\u52a8\u4f5c\u7684\u9891\u7387\uff0c\u4f8b\u5982\u7528\u6237\u8d2d\u4e70\u7279\u5b9a\u5546\u54c1\u7684\u6b21\u6570\u3002\u4e00\u4e2a\u8f83\u5927\u7684\u503c\u5e76\u4e0d\u80fd\u8868\u660e\u66f4\u591a\u7684\u504f\u7231\u3002\u4f46\u662f\u8fd9\u4e2a\u503c\u662f\u6709\u7528\u7684\uff0c\u5b83\u63cf\u8ff0\u4e86\u5728\u4e00\u4e2a\u7279\u5b9a\u89c2\u5bdf\u4e2d\u7684\u4fe1\u4efb\u5ea6\u3002\u4e00\u4e2a\u53d1\u751f\u4e00\u6b21\u7684\u4e8b\u4ef6\u53ef\u80fd\u5bf9\u7528\u6237\u504f\u7231\u6ca1\u6709\u7528\uff0c\u4f46\u662f\u4e00\u4e2a\u5468\u671f\u6027\u4e8b\u4ef6\u66f4\u53ef\u80fd\u53cd\u6620\u4e00\u4e2a\u7528\u6237\u7684\u9009\u62e9\u3002\n\n1. \u8bc4\u4ef7\u9690\u5f0f\u53cd\u9988\u63a8\u8350\u7cfb\u7edf\u9700\u8981\u5408\u9002\u7684\u624b\u6bb5\u3002\n\n### \u663e\u5f0f\u53cd\u9988\u6a21\u578b\n\n\u9690\u56e0\u5b50\u6a21\u578b\u7531\u4e00\u4e2a\u9488\u5bf9\u534f\u540c\u8fc7\u6ee4\u7684\u4ea4\u66ff\u65b9\u6cd5\u7ec4\u6210\uff0c\u5b83\u4ee5\u4e00\u4e2a\u66f4\u52a0\u5168\u9762\u7684\u65b9\u5f0f\u53d1\u73b0\u6f5c\u5728\u7279\u5f81\u6765\u89e3\u91ca\u89c2\u5bdf\u7684`ratings`\u6570\u636e\u3002\n\u6211\u4eec\u5173\u6ce8\u7684\u6a21\u578b\u7531\u5947\u5f02\u503c\u5206\u89e3\uff08SVD\uff09\u63a8\u6f14\u800c\u6765\u3002\n\u4e00\u4e2a\u5178\u578b\u7684\u6a21\u578b\u5c06\u6bcf\u4e2a\u7528\u6237$u$\uff08\u5305\u542b\u4e00\u4e2a\u7528\u6237-\u7279\u5f81\u5411\u91cf$u_i$\uff09\u548c\u6bcf\u4e2a\u5546\u54c1$v$\uff08\u5305\u542b\u4e00\u4e2a\u5546\u54c1-\u7279\u5f81\u5411\u91cf$v_j$\uff09\u8054\u7cfb\u8d77\u6765\u3002\n\u9884\u6d4b\u901a\u8fc7\u5185\u79ef$r_{ij}=u_{i}^{T}v_{j}$\u6765\u5b9e\u73b0\u3002\n\u53e6\u4e00\u4e2a\u9700\u8981\u5173\u6ce8\u7684\u5730\u65b9\u662f\u53c2\u6570\u4f30\u8ba1\u3002\n\u8bb8\u591a\u5f53\u524d\u7684\u5de5\u4f5c\u90fd\u5e94\u7528\u5230\u4e86\u663e\u5f0f\u53cd\u9988\u6570\u636e\u96c6\u4e2d\uff0c\u8fd9\u4e9b\u6a21\u578b\u4ec5\u4ec5\u57fa\u4e8e\u89c2\u5bdf\u5230\u7684`rating`\u6570\u636e\u76f4\u63a5\u5efa\u6a21\uff0c\u540c\u65f6\u901a\u8fc7\u4e00\u4e2a\u9002\u5f53\u7684\u6b63\u5219\u5316\u6765\u907f\u514d\u8fc7\u62df\u5408\u3002\n\u516c\u5f0f\u5982\u4e0b\uff1a\n\n$$\\min_{u,v}\\sum\\limits_{(i,j)\\in R}[(a_{ij} - u_iv_j^T)^2+\\lambda(u_i^2+v_j^2)]$$\n\n\u5728\u516c\u5f0f(2.1)\u4e2d\uff0c$\\lambda$\u662f\u6b63\u5219\u5316\u7684\u53c2\u6570\uff0c\u6b63\u89c4\u5316\u662f\u4e3a\u4e86\u9632\u6b62\u8fc7\u62df\u5408\u7684\u60c5\u51b5\u53d1\u751f\u3002\n\u8fd9\u6837\uff0c\u6211\u4eec\u7528\u6700\u5c0f\u5316\u91cd\u6784\u8bef\u5dee\u6765\u89e3\u51b3\u534f\u540c\u63a8\u8350\u95ee\u9898\u3002\u6211\u4eec\u4e5f\u6210\u529f\u5c06\u63a8\u8350\u95ee\u9898\u8f6c\u6362\u4e3a\u4e86\u6700\u4f18\u5316\u95ee\u9898\u3002\n\n\n### \u9690\u5f0f\u53cd\u9988\u6a21\u578b\n\n\u5728\u663e\u5f0f\u53cd\u9988\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u9700\u8981\u505a\u4e00\u4e9b\u6539\u52a8\u5f97\u5230\u6211\u4eec\u7684\u9690\u5f0f\u53cd\u9988\u6a21\u578b\u3002\n`spark.ml`\u4f7f\u7528[Collaborative Filtering for Implicit Feedback Datasets](http://dx.doi.org/10.1109/ICDM.2008.22)\u63d0\u51fa\u7684\u65b9\u6cd5\u6765\u5904\u7406\u9690\u5f0f\u53cd\u9988\u6570\u636e\u3002\n\u9996\u5148\uff0c\u6211\u4eec\u9700\u8981\u5f62\u5f0f\u5316\u7531$r_{ij}$\u53d8\u91cf\u8861\u91cf\u7684\u4fe1\u4efb\u5ea6\u7684\u6982\u5ff5\u3002\n\u6211\u4eec\u5f15\u5165\u4e86\u4e00\u7ec4\u4e8c\u5143\u53d8\u91cf$p_{ij}$ \uff0c\u5b83\u8868\u793a\u7528\u6237$u$\u5bf9\u5546\u54c1$v$\u7684\u504f\u597d\u3002\n$p_{ij}$\u7684\u516c\u5f0f\u5982\u4e0b\uff1a\n\n$$\\large p_{ij}= \\Bigg\\{ ^{1, r_{ij}>0}_{0,r_{ij}=0}$$\n\n\u6362\u53e5\u8bdd\u8bf4\uff0c\u5982\u679c\u7528\u6237\u8d2d\u4e70\u4e86\u5546\u54c1\uff0c\u6211\u4eec\u8ba4\u4e3a\u7528\u6237\u559c\u6b22\u8be5\u5546\u54c1\uff0c\u5426\u5219\u6211\u4eec\u8ba4\u4e3a\u7528\u6237\u4e0d\u559c\u6b22\u8be5\u5546\u54c1\u3002\n\u7136\u800c\u6211\u4eec\u7684\u4fe1\u5ff5\uff08`beliefs`\uff09\u4e0e\u53d8\u5316\u7684\u4fe1\u4efb\uff08`confidence`\uff09\u7b49\u7ea7\u606f\u606f\u76f8\u5173\u3002\n\u9996\u5148\uff0c\u5f88\u81ea\u7136\u7684\uff0c$p_{ij}$\u7684\u503c\u4e3a0\u548c\u4f4e\u4fe1\u4efb\u6709\u5173\u3002\n\u7528\u6237\u5bf9\u4e00\u4e2a\u5546\u54c1\u6ca1\u6709\u5f97\u5230\u4e00\u4e2a\u6b63\u7684\u504f\u597d\u53ef\u80fd\u6e90\u4e8e\u591a\u65b9\u9762\u7684\u539f\u56e0\uff0c\u5e76\u4e0d\u4e00\u5b9a\u662f\u4e0d\u559c\u6b22\u8be5\u5546\u54c1\u3002\n\u4f8b\u5982\uff0c\u7528\u6237\u53ef\u80fd\u5e76\u4e0d\u77e5\u9053\u8be5\u5546\u54c1\u7684\u5b58\u5728\u3002\n\u53e6\u5916\uff0c\u7528\u6237\u8d2d\u4e70\u4e00\u4e2a\u5546\u54c1\u4e5f\u5e76\u4e0d\u4e00\u5b9a\u662f\u7528\u6237\u559c\u6b22\u5b83\u3002\n\u56e0\u6b64\u6211\u4eec\u9700\u8981\u4e00\u4e2a\u65b0\u7684\u4fe1\u4efb\u7b49\u7ea7\u6765\u663e\u793a\u7528\u6237\u504f\u7231\u67d0\u4e2a\u5546\u54c1\u3002\n\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c$r_{ij}$\u8d8a\u5927\uff0c\u8d8a\u80fd\u6697\u793a\u7528\u6237\u559c\u6b22\u67d0\u4e2a\u5546\u54c1\u3002\n\u56e0\u6b64\uff0c\u6211\u4eec\u5f15\u5165\u4e86\u4e00\u7ec4\u53d8\u91cf$c_{ij}$\uff0c\u5b83\u8861\u91cf\u4e86\u6211\u4eec\u89c2\u5bdf\u5230$p_{ij}$\u7684\u4fe1\u4efb\u5ea6\u3002\n$c_{ij}$\u4e00\u4e2a\u5408\u7406\u7684\u9009\u62e9\u5982\u4e0b\u6240\u793a\uff1a\n\n$$\\large c_{ij} = 1 + \\alpha r_{ij}$$\n\n\u6309\u7167\u8fd9\u79cd\u65b9\u5f0f\uff0c\u6211\u4eec\u5b58\u5728\u6700\u5c0f\u9650\u5ea6\u7684\u4fe1\u4efb\u5ea6\uff0c\u5e76\u4e14\u968f\u7740\u6211\u4eec\u89c2\u5bdf\u5230\u7684\u6b63\u504f\u5411\u7684\u8bc1\u636e\u8d8a\u6765\u8d8a\u591a\uff0c\u4fe1\u4efb\u5ea6\u4e5f\u4f1a\u8d8a\u6765\u8d8a\u5927\u3002\n\n\u6211\u4eec\u7684\u76ee\u7684\u662f\u627e\u5230\u7528\u6237\u5411\u91cf$u_i$\u4ee5\u53ca\u5546\u54c1\u5411\u91cf$v_j$\u6765\u8868\u660e\u7528\u6237\u504f\u597d\u3002\u8fd9\u4e9b\u5411\u91cf\u5206\u522b\u662f\u7528\u6237\u56e0\u7d20\uff08\u7279\u5f81\uff09\u5411\u91cf\u548c\u5546\u54c1\u56e0\u7d20\uff08\u7279\u5f81\uff09\u5411\u91cf\u3002\n\u672c\u8d28\u4e0a\uff0c\u8fd9\u4e9b\u5411\u91cf\u5c06\u7528\u6237\u548c\u5546\u54c1\u6620\u5c04\u5230\u4e00\u4e2a\u516c\u7528\u7684\u9690\u5f0f\u56e0\u7d20\u7a7a\u95f4\uff0c\u4ece\u800c\u4f7f\u5b83\u4eec\u53ef\u4ee5\u76f4\u63a5\u6bd4\u8f83\u3002\n\u8fd9\u548c\u7528\u4e8e\u663e\u5f0f\u6570\u636e\u96c6\u7684\u77e9\u9635\u5206\u89e3\u6280\u672f\u7c7b\u4f3c\uff0c\u4f46\u662f\u5305\u542b\u4e24\u70b9\u4e0d\u4e00\u6837\u7684\u5730\u65b9\uff1a\n\n1. \u6211\u4eec\u9700\u8981\u8003\u8651\u4e0d\u540c\u7684\u4fe1\u4efb\u5ea6\uff0c\n1. \u6700\u4f18\u5316\u9700\u8981\u8003\u8651\u6240\u6709\u53ef\u80fd\u7684$u, v$\u5bf9\uff0c\u800c\u4e0d\u4ec5\u4ec5\u662f\u548c\u89c2\u5bdf\u6570\u636e\u76f8\u5173\u7684$u, v$\u5bf9\u3002\n\n\u663e\u6027\u53cd\u9988\u7684\u77e9\u9635\u5206\u89e3\u4f18\u5316\u65f6\uff0c\u5bf9\u4e8e`missing data`(\u6ca1\u6709\u8bc4\u5206)\uff0c\u662f\u4e0d\u4f1a\u5f53\u505a\u8bad\u7ec3\u6570\u636e\u8f93\u5165\u5230\u6a21\u578b\u7684\uff0c\u4f18\u5316\u65f6\u9488\u5bf9\u5df2\u77e5\u8bc4\u5206\u6570\u636e\u4f18\u5316\u3002\n\u800c\u8fd9\u91cc\u9690\u6027\u53cd\u9988\uff0c\u662f\u5229\u7528\u6240\u6709\u53ef\u80fd\u7684$u, v$\u952e\u503c\u5bf9\uff0c\u6240\u4ee5\u603b\u7684\u6570\u636e\u662f$m*n$\uff0c\u5176\u4e2d$m$\u662f\u7528\u6237\u6570\u91cf\uff0c$n$\u662f\u7269\u54c1\u6570\u91cf\u3002\n\u8fd9\u91cc\u6ca1\u6709\u6240\u8c13\u7684`missing data`\uff0c\u56e0\u4e3a\u5047\u5982$u$\u5bf9$v$\u6ca1\u6709\u4efb\u4f55\u52a8\u4f5c\uff0c\u6211\u4eec\u5c31\u8ba4\u4e3a\u504f\u597d\u503c\u4e3a0\uff0c\u53ea\u4e0d\u8fc7\u7f6e\u4fe1\u5ea6\u8f83\u4f4e\u800c\u5df2\u3002\n\u56e0\u6b64\uff0c\u901a\u8fc7\u6700\u5c0f\u5316\u4e0b\u9762\u7684\u635f\u5931\u51fd\u6570\u6765\u8ba1\u7b97\u76f8\u5173\u56e0\u5b50\u3002\n\n$$min_{u,v}\\sum _{i,j}c_{ij}(p_{ij}-u_{i}^{T}v_{j})^{2} + \\lambda (\\sum_{i}\\left \\| u_{i} \\right \\|^{2} + \\sum_{j}\\left \\|v_{j} \\right \\|^{2})$$\n\n### \u6c42\u89e3\u6700\u5c0f\u5316\u635f\u5931\u51fd\u6570\n\n\u8003\u8651\u5230\u635f\u5931\u51fd\u6570\u5305\u542b$m*n$\u4e2a\u5143\u7d20\uff0c$m$\u662f\u7528\u6237\u7684\u6570\u91cf\uff0c$n$\u662f\u5546\u54c1\u7684\u6570\u91cf\u3002\n\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c$m*n$\u53ef\u4ee5\u5230\u8fbe\u51e0\u767e\u4ebf\u3002\n\u8fd9\u4e48\u591a\u7684\u5143\u7d20\u5e94\u8be5\u907f\u514d\u4f7f\u7528\u968f\u673a\u68af\u5ea6\u4e0b\u964d\u6cd5\u6765\u6c42\u89e3\uff0c\u56e0\u6b64\uff0cspark\u9009\u62e9\u4f7f\u7528\u4ea4\u66ff\u6700\u4f18\u5316\u65b9\u5f0f\u6c42\u89e3\u3002\n\n\u4e0a\u8ff0\u635f\u5931\u51fd\u6570\u662f\u975e\u51f8\u51fd\u6570\uff0c\u65e0\u6cd5\u6c42\u89e3\u6700\u4f18\u89e3\u3002\n\u4f46\u662f\uff0c\u56fa\u5b9a\u516c\u5f0f\u4e2d\u7684\u7528\u6237-\u7279\u5f81\u5411\u91cf\u6216\u8005\u5546\u54c1-\u7279\u5f81\u5411\u91cf\uff0c\u516c\u5f0f\u5c31\u4f1a\u53d8\u6210\u4e8c\u6b21\u65b9\u7a0b\uff0c\u53ef\u4ee5\u6c42\u51fa\u5168\u5c40\u7684\u6781\u5c0f\u503c\u3002\n\u4ea4\u66ff\u6700\u5c0f\u4e8c\u4e58\u7684\u8ba1\u7b97\u8fc7\u7a0b\u662f\uff1a\u4ea4\u66ff\u7684\u91cd\u65b0\u8ba1\u7b97\u7528\u6237-\u7279\u5f81\u5411\u91cf\u548c\u5546\u54c1-\u7279\u5f81\u5411\u91cf\uff0c\u6bcf\u4e00\u6b65\u90fd\u4fdd\u8bc1\u964d\u4f4e\u635f\u5931\u51fd\u6570\u7684\u503c\uff0c\u76f4\u5230\u627e\u5230\u6781\u5c0f\u503c\u3002\n\u4ea4\u66ff\u6700\u5c0f\u4e8c\u4e58\u6cd5\u7684\u5904\u7406\u8fc7\u7a0b\u5982\u4e0b\u6240\u793a\uff1a\n\n* \u5148\u968f\u673a\u751f\u6210\u4e00\u4e2a$U_{m*k}^{(0)}$\u3002\u4e00\u822c\u53ef\u4ee5\u53d60\u503c\u6216\u8005\u5168\u5c40\u5747\u503c\u3002\n* \u56fa\u5b9a$U^{(0)}$\uff0c\u5373\u8ba4\u4e3a\u662f\u5df2\u77e5\u7684\u5e38\u91cf\uff0c\u6765\u6c42\u89e3\uff1a\n$$\\large C = \\sum\\limits_{(i,j)\\in R}[(a_{ij} - u_i^{(0)}v_j^T)^2+\\lambda((u_i^2)^{(0)}+v_j^2)]$$\n\u7531\u4e8e\u4e0a\u5f0f\u4e2d\u53ea\u6709$v_j$\u4e00\u4e2a\u672a\u77e5\u53d8\u91cf\uff0c\u56e0\u6b64$C$\u7684\u6700\u4f18\u5316\u95ee\u9898\u8f6c\u5316\u4e3a\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\uff0c\u7528\u6700\u5c0f\u4e8c\u4e58\u6cd5\u6c42\u89e3$v_j$\u7684\u6700\u4f18\u89e3\uff1a \n\u56fa\u5b9a$j,j\\in(1,2,...,n)$\uff0c\u7b49\u5f0f\u4e24\u8fb9\u5173\u4e8e$v_j$\u6c42\u5bfc\u5f97\uff1a\n$$\n\\begin{align}\n&\\large \\frac{d(c)}{d(v_j)}\\\\\n&\\large= \\frac{d}{d(v_j)}(\\sum\\limits_{i=1}^{m}[(a_{ij} - u_i^{(0)}v_j^T)^2+\\lambda((u_i^2)^{(0)}+v_j^2)])\\\\\n&\\large= \\sum\\limits_{i=1}^m[2(a_{ij} - u_i^{(0)}v_j^T)(- (u_i^T)^{(0)})+2\\lambda v_j]\\\\\n&\\large= 2\\sum\\limits_{i=1}^m[( u_i^{(0)}(u_i^T)^{(0)}+\\lambda)v_j-a_{ij}(u_i^T)^{(0)}]\n\\end{align}\n$$\n\u4ee4$\\large \\frac{d(c)}{d(v_j)} =0$\uff0c\u53ef\u5f97\uff1a\n$$\n\\begin{align}\n&\\large\\sum\\limits_{i=1}^m[( u_i^{(0)}(u_i^T)^{(0)}+\\lambda)v_j]\\\\\n&\\large=\\sum\\limits_{i=1}^m a_{ij}(u_i^T)^{(0)}\\\\\n&\\large=> (U^{(0)}(U^T)^{(0)} + \\lambda E)v_j\\large= a_j^TU^{(0)}\n\\end{align}\n$$\n\u4ee4$M_1 = U^{(0)}(U^T)^{(0)} + \\lambda E , M_2 = a_j^TU^{(0)}$\uff0c\u5219$v_j = M_1^{-1}M_2$\n\n* \u6309\u7167\u4e0a\u5f0f\u4f9d\u6b21\u8ba1\u7b97$v_1\uff0cv_2\uff0c...\uff0cv_n$\uff0c\u4ece\u800c\u5f97\u5230$V^{(0)}$\n* \u540c\u7406\uff0c\u7528\u6b65\u9aa42\u4e2d\u7c7b\u4f3c\u7684\u65b9\u6cd5: \n$$\\large C = \\sum\\limits_{(i,j)\\in R}[(a_{ij} - u_i(v_j^T)^{(0)})^2+\\lambda(u_i^2+(v_j^2)^{(0)})]$$\n\u56fa\u5b9a$i , i\\in (1,2,...,m)$\uff0c\u7b49\u5f0f\u4e24\u8fb9\u5173\u4e8e$u_i$\u6c42\u5bfc\u5f97\uff1a\n$$\n\\begin{align}\n&\\large \\frac{d(c)}{d(u_i)}\\\\\n&\\large= \\frac{d}{d(u_i)}(\\sum\\limits_{j=1}^{n}[(a_{ij} - u_i(v_j^T)^{(0)})^2+\\lambda((u_i^2)+(v_j^2)^{(0)})])\\\\\n&\\large= \\sum\\limits_{j=1}^n[2(a_{ij} - u_i(v_j^T)^{(0)})(- (v_j^T)^{(0)})+2\\lambda u_i]\\\\\n&\\large= 2\\sum\\limits_{j=1}^n[( v_j^{(0)}(v_j^T)^{(0)}+\\lambda)u_i-a_{ij}(v_j^T)^{(0)}]\n\\end{align}\n$$\n\u4ee4$\\large \\frac{d(c)}{d(u_i)} =0$\uff0c\u53ef\u5f97\uff1a\n$$\n\\begin{align}\n&\\large\\sum\\limits_{j=1}^n[( v_j^{(0)}(v_j^T)^{(0)}+\\lambda)u_i]\\\\\n&\\large=\\sum\\limits_{j=1}^n a_{ij}(v_j^T)^{(0)}\\\\\n&\\large=>( (V^{(0)}(V^T)^{(0)} + \\lambda E)u_i = a_i^TV^{(0)}\n\\end{align}\n$$\n\u4ee4$M_1 = V^{(0)}(V^T)^{(0)} + \\lambda E , M_2 =a_i^TV^{(0)}$\uff0c\u5219$u_i = M_1^{-1}M_2$\n\n* \u6309\u7167\u4e0a\u5f0f\u4f9d\u6b21\u8ba1\u7b97$u_1\uff0cu_2\uff0c...\uff0cu_n$\uff0c\u4ece\u800c\u5f97\u5230$U^{(1)}$\n* \u5faa\u73af\u6267\u884c\u6b65\u9aa42\u30013\uff0c\u76f4\u5230\u635f\u5931\u51fd\u6570$C$\u7684\u503c\u6536\u655b\uff08\u6216\u8005\u8bbe\u7f6e\u4e00\u4e2a\u8fed\u4ee3\u6b21\u6570N\uff0c\u8fed\u4ee3\u6267\u884c\u6b65\u9aa42\u30013\uff0cN\u6b21\u540e\u505c\u6b62\uff09\u3002\u8fd9\u6837\uff0c\u5c31\u5f97\u5230\u4e86$C$\u6700\u4f18\u89e3\u5bf9\u5e94\u7684\u77e9\u9635$U$\u3001$V$\u3002\n\n\n## \u6b63\u5219\u5316\u53c2\u6570\n\n\u6211\u4eec\u8c03\u6574\u6b63\u5219\u5316\u53c2\u6570`regParam`\u6765\u89e3\u51b3\u7528\u6237\u5728\u66f4\u65b0\u7528\u6237\u56e0\u5b50\u65f6\u4ea7\u751f\u65b0\u8bc4\u5206\u6216\u8005\u5546\u54c1\u66f4\u65b0\u5546\u54c1\u56e0\u5b50\u65f6\u6536\u5230\u7684\u65b0\u8bc4\u5206\u5e26\u6765\u7684\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\u3002\n\u8fd9\u4e2a\u65b9\u6cd5\u53eb\u505a\u201cALS-WR\u201d\u6765\u81ea\u4e8e\u201c[Large-Scale Parallel Collaborative Filtering for the Netflix Prize](http://dx.doi.org/10.1007/978-3-540-68880-8_32)\u201d\u3002\n\u5b83\u964d\u4f4e`regParam`\u5bf9\u6570\u636e\u96c6\u89c4\u6a21\u7684\u4f9d\u8d56\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ee5\u5c06\u4ece\u90e8\u5206\u5b50\u96c6\u4e2d\u5b66\u4e60\u5230\u7684\u6700\u4f73\u53c2\u6570\u5e94\u7528\u5230\u6574\u4e2a\u6570\u636e\u96c6\u4e2d\u65f6\u83b7\u5f97\u540c\u6837\u7684\u6027\u80fd\u3002\n\n## \u51b7\u542f\u52a8\u7b56\u7565\n\n\u5f53\u4f7f\u7528`ALSModel`\u9884\u6d4b\u65f6\u4f1a\u7ecf\u5e38\u9047\u5230\u6d4b\u8bd5\u6570\u636e\u96c6\u4e2d\u7684\u7528\u6237\u548c/\u6216\u5546\u54c1\u6ca1\u6709\u5728\u8bad\u7ec3\u65f6\u51fa\u73b0\u3002\u8fd9\u79cd\u60c5\u51b5\u901a\u5e38\u51fa\u73b0\u5728\u4e24\u79cd\u573a\u666f\u91cc\uff1a\n\n1. \u751f\u4ea7\u7cfb\u7edf\u4e2d\uff0c\u65b0\u7528\u6237\u548c\u5546\u54c1\u6ca1\u6709\u6253\u5206\u5386\u53f2\u6240\u4ee5\u6a21\u578b\u6ca1\u6709\u4e3a\u6b64\u8bad\u7ec3\u8fc7\uff08\u5373\u201c\u51b7\u542f\u52a8\u95ee\u9898\u201d\uff09\u3002\n2. \u5728\u4ea4\u53c9\u9a8c\u8bc1\u65f6\uff0c\u6570\u636e\u5206\u5272\u6210\u8bad\u7ec3\u548c\u8bc4\u4f30\u4e24\u90e8\u5206\u3002\u82e5\u4f7f\u7528`CrossValidator`\u6216`TrainValidationSplit`\u4e4b\u7c7b\u7684\u7b80\u5355\u968f\u673a\u5206\u5272\uff0c\u7ecf\u5e38\u9047\u5230\u8bc4\u4f30\u6570\u636e\u96c6\u4e2d\u7684\u7528\u6237\u548c/\u6216\u5546\u54c1\u6ca1\u6709\u5728\u8bad\u7ec3\u6570\u636e\u4e2d\u51fa\u73b0\u3002\n\n\u5f53\u7528\u6237\u548c/\u6216\u5546\u54c1\u56e0\u5b50\u6ca1\u6709\u88ab\u6a21\u578b\u8868\u793a\u65f6\uff0cSpark`ALSModel.transform`\u9ed8\u8ba4\u9884\u6d4b\u4e3a`NaN`\u3002\u751f\u4ea7\u7cfb\u7edf\u4e2d\u8fd9\u53ef\u80fd\u6709\u7528\uff0c\u56e0\u4e3a\u8fd9\u6307\u793a\u4e86\u65b0\u7528\u6237\u548c\u5546\u54c1\uff0c\u8fd9\u6837\u7cfb\u7edf\u53ef\u4ee5\u63d0\u51fa\u4e9b\u5907\u9009\u9879\u4f5c\u4e3a\u9884\u6d4b\u3002\n\n\u5f53\u7136\u4ea4\u53c9\u9a8c\u8bc1\u65f6\u8fd9\u6837\u4e0d\u5408\u9002\uff0c\u4efb\u4f55`NaN`\u9884\u6d4b\u503c\u90fd\u4f1a\u5bfc\u81f4\u8bc4\u4f30\u6307\u6807\u53d8\u6210`NaN`\uff08\u6bd4\u5982\u4f7f\u7528`RegressionEvaluator`\uff09\u3002\n\u8fd9\u6837\u5c31\u4e0d\u80fd\u505a\u6a21\u578b\u9009\u62e9\u4e86\u3002\n\nSpark\u5141\u8bb8\u7528\u6237\u8bbe\u7f6e`coldStartStrategy`\u53c2\u6570\u6765\u629b\u5f03`DataFrame`\u4e2d\u4efb\u4f55\u5305\u542b`NaN`\u9884\u6d4b\u503c\u7684\u884c\u3002\n\u8bc4\u4f30\u6307\u6807\u5c31\u53ea\u8003\u8651\u975e`NaN`\u6570\u636e\u6240\u4ee5\u4e0d\u4f1a\u5931\u6548\u3002\n\u4e0b\u9762\u7684\u4f8b\u5b50\u5c55\u793a\u4e86\u5982\u4f55\u4f7f\u7528\u8be5\u53c2\u6570\u3002\n\n**\u6ce8\u610f**\uff1a\u76ee\u524d\u652f\u6301\u7684\u51b7\u542f\u52a8\u7b56\u7565\u6709`\"nan\"`\uff08\u4e0a\u9762\u63cf\u8ff0\u7684\u9ed8\u8ba4\u884c\u4e3a\uff09\u548c`\"drop\"`\u3002\u5c06\u6765\u53ef\u80fd\u4f1a\u652f\u6301\u66f4\u591a\u7b56\u7565\u3002\n\n**\u6837\u4f8b**\n\n\u4e0b\u9762\u7684\u4f8b\u5b50\u4e2d\u6211\u4eec\u4ece[MovieLens dataset](http://grouplens.org/datasets/movielens/)\u8f7d\u5165\u8bc4\u5206\u6570\u636e\uff0c\u6bcf\u4e00\u884c\u5305\u62ec\u4e00\u4e2a\u7528\u6237\u3001\u4e00\u4e2a\u7535\u5f71\u3001\u4e00\u4e2a\u8bc4\u5206\u548c\u4e00\u4e2a\u65f6\u95f4\u6233\u3002\n\u6211\u4eec\u9ed8\u8ba4\u5176\u8bc4\u5206\u662f\u663e\u5f0f\u7684\u6765\u8bad\u7ec3ALS\u6a21\u578b\uff08`implicitPrefs`\u8bbe\u4e3a`False`\uff09\u3002\n\u6211\u4eec\u901a\u8fc7\u9884\u6d4b\u8bc4\u5206\u7684\u5747\u65b9\u6839\u8bef\u5dee\u6765\u8bc4\u4ef7\u63a8\u8350\u6a21\u578b\u3002\n\n\u8bf7\u53c2\u8003[`ALS` Python\u6587\u6863](api/python/pyspark.ml.html#pyspark.ml.recommendation.ALS)\u4e86\u89e3\u66f4\u591a\u7ec6\u8282\u3002\n\n\u5b8c\u6574\u6837\u4f8b\u4ee3\u7801\u53ef\u4ee5\u5728[Spark\u4ed3\u5e93](https://github.com/apache/spark)\u4e2d\u7684\"examples/src/main/python/ml/als_example.py\"\u627e\u5230\n\n\n```python\nfrom pyspark.ml.evaluation import RegressionEvaluator\nfrom pyspark.ml.recommendation import ALS\nfrom pyspark.sql import Row\n\nlines = spark.read.text(\"data/mllib/als/sample_movielens_ratings.txt\").rdd\nparts = lines.map(lambda row: row.value.split(\"::\"))\nratingsRDD = parts.map(lambda p: Row(userId=int(p[0]), movieId=int(p[1]),\n rating=float(p[2]), timestamp=long(p[3])))\nratings = spark.createDataFrame(ratingsRDD)\n(training, test) = ratings.randomSplit([0.8, 0.2])\n\n# Build the recommendation model using ALS on the training data\n# Note we set cold start strategy to 'drop' to ensure we don't get NaN evaluation metrics\nals = ALS(maxIter=5, regParam=0.01, userCol=\"userId\", itemCol=\"movieId\", ratingCol=\"rating\",\n coldStartStrategy=\"drop\")\nmodel = als.fit(training)\n\n# Evaluate the model by computing the RMSE on the test data\npredictions = model.transform(test)\nevaluator = RegressionEvaluator(metricName=\"rmse\", labelCol=\"rating\",\n predictionCol=\"prediction\")\nrmse = evaluator.evaluate(predictions)\nprint(\"Root-mean-square error = \" + str(rmse))\n\n# Generate top 10 movie recommendations for each user\nuserRecs = model.recommendForAllUsers(10)\n# Generate top 10 user recommendations for each movie\nmovieRecs = model.recommendForAllItems(10)\n\n# Generate top 10 movie recommendations for a specified set of users\nusers = ratings.select(als.getUserCol()).distinct().limit(3)\nuserSubsetRecs = model.recommendForUserSubset(users, 10)\n# Generate top 10 user recommendations for a specified set of movies\nmovies = ratings.select(als.getItemCol()).distinct().limit(3)\nmovieSubSetRecs = model.recommendForItemSubset(movies, 10)\n```\n\n\u5982\u679c\u8bc4\u5206\u77e9\u9635\u6765\u81ea\u5176\u4ed6\u4fe1\u606f\u6765\u6e90\uff0c\u4e5f\u53ef\u5c06`implicitPrefs`\u8bbe\u7f6e\u4e3a`true`\u6765\u83b7\u5f97\u66f4\u597d\u7684\u7ed3\u679c\u3002\n\n\n```python\nals = ALS(maxIter=5, regParam=0.01, implicitPrefs=True,\n userCol=\"userId\", itemCol=\"movieId\", ratingCol=\"rating\")\n```\n", "meta": {"hexsha": "a392e8958734bab2b6f85addc8d50ae95e43ab1c", "size": 13890, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ml-collaborative-filtering.ipynb", "max_stars_repo_name": "lotress/sparkml-doc", "max_stars_repo_head_hexsha": "bd06c67b17d69026551b9044df258ad30ede0fa1", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "ml-collaborative-filtering.ipynb", "max_issues_repo_name": "lotress/sparkml-doc", "max_issues_repo_head_hexsha": "bd06c67b17d69026551b9044df258ad30ede0fa1", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ml-collaborative-filtering.ipynb", "max_forks_repo_name": "lotress/sparkml-doc", "max_forks_repo_head_hexsha": "bd06c67b17d69026551b9044df258ad30ede0fa1", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 40.7331378299, "max_line_length": 231, "alphanum_fraction": 0.6005039597, "converted": true, "num_tokens": 6942, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4250267293941805}} {"text": "# Training with Skewed Dataset\n\n**CS5483 Data Warehousing and Data Mining**\n___\n\n\n```python\n%reset -f\nfrom IPython import display\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\n# produce vector inline graphics\nfrom IPython.display import set_matplotlib_formats\nset_matplotlib_formats('svg')\nfrom weka.core import dataset\nimport weka.core.jvm as jvm\nfrom weka.core.converters import Loader\nfrom weka.classifiers import SingleClassifierEnhancer, Classifier, Evaluation, FilteredClassifier\nfrom weka.core.classes import Random\nimport weka.core.packages as packages\nfrom weka.core.classes import complete_classname\nfrom weka.filters import Filter\n```\n\n## Setup\n\nIn this notebook, we will train classifiers properly on the skewed dataset for detecting microcalcifications in mammograms.\n\nIn particular, we will use the meta classifier `ThresholdSelector` and the filter `SMOTE` [Synthetic Minority Over-sampling Technique](https://doi.org/10.1613/jair.953), which needs to be installed as [additional packages in WEKA](https://weka.sourceforge.io/packageMetaData/):\n\n\n```python\njvm.start(packages=True) # support custom packages\n\nimport weka.core.packages as packages\n\npkgs = [\"thresholdSelector\", \"SMOTE\"]\n\nfor item in packages.all_packages():\n if item.name in pkgs:\n print(item.name + \" \" + item.url)\n```\n\n You may install the packages using Weka package manager instead of downloading the zip files. To install them in `python-weka-wrapper`, run the following code:\n\n\n```python\nfor pkg in pkgs:\n if not packages.is_installed(pkg):\n print(f\"Installing {pkg}...\")\n packages.install_package(pkg)\n else:\n print(f\"Skipping {pkg}, already installed. \")\nelse:\n print(\"Done.\")\n```\n\nBy default, these packages are installed under your home directory `~/wekafiles/packages/`:\n\n\n```python\n!ls ~/wekafiles/packages\n```\n\nFor the packages to take effect, you must restart the kernel (`Kernel` -> `Restart`). Note that running `jvm.stop()` followed by `jvm.start(packages=True)` will not work because [`javabridge` currently does not support restarting a virtual machine](https://stackoverflow.com/questions/51795945/after-stopping-jvm-unable-to-start-it-again).\n\nAfter restarting the kernel, check that the packages have been successfully installed by running the following code:\n\n\n```python\nfrom weka.core.classes import complete_classname\n\nprint(complete_classname(\".ThresholdSelector\"))\nprint(complete_classname(\".SMOTE\"))\nprint(packages.installed_packages())\n```\n\nWe will use the same mammography dataset from\n[OpenML](https://www.openml.org/d/310) and J48 as the base classifier. The following loads the dataset into the notebook:\n\n\n```python\nloader = Loader(classname=\"weka.core.converters.ArffLoader\")\ndata = loader.load_url('https://www.openml.org/data/download/52214/phpn1jVwe')\ndata.class_is_last()\npos_class = 1\nclf = Classifier(classname=\"weka.classifiers.trees.J48\")\n```\n\n## Threshold Selector\n\nThe meta classifier `ThresholdSelector` uses the threshold-moving technique to optimize a performance measure you specify, which can be the precision, recall, $F$-score, etc. See an explanation of threshold moving technique [here](https://machinelearningmastery.com/threshold-moving-for-imbalanced-classification/).\n\nThe following shows how to maximize recall:\n\n\n```python\ntsc = SingleClassifierEnhancer(\n classname=\"weka.classifiers.meta.ThresholdSelector\",\n options=['-M', 'RECALL'])\ntsc.options=['-M', 'RECALL']\ntsc.classifier = clf\n\nevl = Evaluation(data)\nevl.crossvalidate_model(tsc, data, 10, Random(1))\n\nprint(f\"maximum recall: {evl.recall(pos_class):.3g}\")\n```\n\nThe maximum recall is 100% as expected by setting the threshold to 1.\n\n**Exercise** Using J48 as the base classifier and 10-fold cross-validation, obtain the highest precision and $F$ score. Assign the values to `max_precision` and `max_f` respectively. \n\nIf you use `python-weka-wrapper`, be careful that reseting `tsc.options` may also reset the base classifier to the default one, which is not J48. To ensure that you are using J48, set the base classifier again after the options:\n```Python\ntsc.options=['-M', ___]\ntsc.classifier = clf\n```\n\n\n```python\n# YOUR CODE HERE\nraise NotImplementedError()\nmax_precision, max_f\n```\n\n\n```python\n# tests\n```\n\n## Cost-sensitive Classifier\n\nWe can build a classifier to maximize certain cost. Weka provides a convenient interface for the [cost/benefit analysis](https://wiki.pentaho.com/display/DATAMINING/Cost+Benefit+Analysis):\n\n
                                          \n
                                        1. In the explorer interface, train J48 on the mammography dataset with 10-fold cross-validation.\n
                                        2. Right click on the result in the result list.\n
                                        3. Choose Cost/Benefit analysis and 1 as the positive class value.\n
                                        4. Specify the cost matrix:
                                          \n\\begin{align}\n\\begin{bmatrix} \\text{cost}_\\text{TP} & \\text{cost}_\\text{FN}\\\\ \\text{cost}_\\text{FP} & \\text{cost}_\\text{TN}\\end{bmatrix}.\n\\end{align}\n
                                        5. Click `Minimize Cost/Benefit` to minimize the cost:
                                          \n\\begin{align}\n\\text{cost}_\\text{TP} \\text{TP}\n+ \\text{cost}_\\text{FN} \\text{FN}\n+ \\text{cost}_\\text{FP} \\text{FP} \n+ \\text{cost}_\\text{TN} \\text{TN}.\n\\end{align}\n
                                        \n\n**Exercise** Assign to `cost_matrix` the cost matrix that achieves the maximum precision. You can define the cost matrix as follows:\n```Python\ncost_matrix = np.array([[__, __],\n [__, __]])\n```\n \n[*Hint: Pay attention to the row and column labels of the confusion matrix. It changes after you specify $1$ as the positive class value.*]\n\n\n```python\n# YOUR CODE HERE\nraise NotImplementedError()\ncost_matrix\n```\n\nThe following test cell demonstrate how to train a meta classifier to minimize the cost defined using the cost matrix you provided.\n\n\n```python\n# tests\nfrom weka.classifiers import SingleClassifierEnhancer, Classifier, Evaluation\n\ncsc = SingleClassifierEnhancer(\n classname=\"weka.classifiers.meta.CostSensitiveClassifier\",\n options=[\n \"-cost-matrix\",\n '[' + ' ; '.join(' '.join(str(entry) for entry in cost_matrix[:, i])\n for i in range(2)) + ']', \"-S\", \"1\"\n ])\ncsc.classifier = clf\n\nevl = Evaluation(data)\nevl.crossvalidate_model(csc, data, 10, Random(1))\n\nprecision = evl.precision(pos_class)\nprint(f\"maximum precision: {precision:.3g}\")\n```\n\n## SMOTE\n\nSynthetic Minority Over-sampling TEchnique (SMOTE) is a filter that up-samples the minority class. Instead of creating duplicates of the same instance, it creates new samples as convex combinations of existing ones. See a more detailed explanation of SMOTE [here](http://rikunert.com/SMOTE_explained).\n\n**Exercise** Using the FilteredClassifier with J48 as the classifer and SMOTE as the filter, try to tweek the setting of SMOTE to give the highest possilbe value of $F$ score larger than the maximum one achieved by `ThresholdSelector`. Assign to `smote.options` your choice of the filter. E.g., you can change the percentage of SMOTE instances to 150% as follows:\n```Python\nsmote.options = ['-P', '150']\n```\n\n\n```python\nsmote = Filter(classname=\"weka.filters.supervised.instance.SMOTE\")\nprint('Default smote.options:', smote.options)\n# YOUR CODE HERE\nraise NotImplementedError()\nprint('Your smote.options:', smote.options)\n```\n\n\n```python\n# tests\nfc = FilteredClassifier()\nfc.filter = smote\nfc.classifier = clf\n\nevl = Evaluation(data)\nevl.crossvalidate_model(fc, data, 10, Random(1))\n\nf_score = evl.f_measure(pos_class)\nprint(f\"F-score by SMOTE: {f_score:.3g}\")\n```\n", "meta": {"hexsha": "00e575971af595f656517a4e52876ba7a4bf657e", "size": 16224, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Tutorial5/Training with Skewed Dataset.ipynb", "max_stars_repo_name": "ccha23/cs5483", "max_stars_repo_head_hexsha": "e8fa9d9b8a0545696958ca87c2c9a8a133109191", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Tutorial5/Training with Skewed Dataset.ipynb", "max_issues_repo_name": "ccha23/cs5483", "max_issues_repo_head_hexsha": "e8fa9d9b8a0545696958ca87c2c9a8a133109191", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-04-19T09:21:04.000Z", "max_issues_repo_issues_event_max_datetime": "2021-04-19T09:21:06.000Z", "max_forks_repo_path": "Tutorial5/Training with Skewed Dataset.ipynb", "max_forks_repo_name": "ccha23/cs5483", "max_forks_repo_head_hexsha": "e8fa9d9b8a0545696958ca87c2c9a8a133109191", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-03-20T05:25:45.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-20T05:25:45.000Z", "avg_line_length": 27.9242685026, "max_line_length": 372, "alphanum_fraction": 0.5826553254, "converted": true, "num_tokens": 1839, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.7217432062975979, "lm_q1q2_score": 0.4250267293941805}} {"text": " \n#### Procesamiento Digital de Se\u00f1ales\n\n# Trabajo Pr\u00e1ctico N\u00ba0\n#### Nicol\u00e1s Ferragamo \n\n\n# Introducci\u00f3n\nJupyter Notebook es una herramienta para la confecci\u00f3n de reportes t\u00e9cnicos, dado que permite la interacci\u00f3n en el mismo ambiente de: \n1. un procesador de texto elemental (formato Markdown) que permite resaltar texto, en forma de *it\u00e1lica* o **negrita** de manera muy legible (haciendo doble click en este texto podr\u00e1s ver el c\u00f3digo fuente estilo Markdown). Cuenta con estilos predefinidos:\n\n# T\u00edtulo 1\n## T\u00edtulo 2\n### T\u00edtulo 3\n\ny tambi\u00e9n la capacidad de incluir enlaces a otras p\u00e1ginas, como por ejemplo [esta p\u00e1gina](https://medium.com/ibm-data-science-experience/markdown-for-jupyter-notebooks-cheatsheet-386c05aeebed) donde encontrar\u00e1s m\u00e1s funcionalidades del lenguaje **Markdown**\n\n2. capacidad para incluir lenguaje matem\u00e1tico estilo LaTex, tanto de forma presentada\n\n\\begin{equation}\nT(z) = \\frac{Y(z)}{X(z)} = \\frac{ b_2 \\, z^{-2} + b_1 \\, z^{-1} + b_0 }\n{a_2 \\, z^{-2} + a_1 \\, z^{-1} + a_0}\n\\end{equation}\n\ncomo *inline* en el propio p\u00e1rrafo $y[k] = \\frac{1}{a_0} \\left( \\sum_{m=0}^{M} b_m \\; x[k-m] - \\sum_{n=1}^{N} a_n \\; y[k-n] \\right) $\n\n3. La posibilidad de incluir scripts en Python, como los que usaremos para las simulaciones en los TPs de la materia. En este caso usaremos el *testbench0.py* como ejemplo. Una vez que lo probamos y estamos seguros que funciona de forma esperada en *Spyder*, podemos incluir los resultados de la simulaci\u00f3n de manera casi transparente. Solo tenemos que agregar una celda de c\u00f3digo donde incluimos el c\u00f3digo, y los resultados directamente quedan incluidos en este documento.\n\n\n```python\n# M\u00f3dulos para Jupyter\nimport warnings\nwarnings.filterwarnings('ignore')\n\nimport numpy as np\nimport matplotlib as mpl\n#%% Inicializaci\u00f3n de librer\u00edas\n# Setup inline graphics: Esto lo hacemos para que el tama\u00f1o de la salida, \n# sea un poco m\u00e1s adecuada al tama\u00f1o del documento\nmpl.rcParams['figure.figsize'] = (10,10)\n\nimport matplotlib.pyplot as plt\nimport pdsmodulos as pds\n\n#%% Esto tiene que ver con cuestiones de presentaci\u00f3n de los gr\u00e1ficos,\n# NO ES IMPORTANTE\nfig_sz_x = 14\nfig_sz_y = 13\nfig_dpi = 80 # dpi\n\nfig_font_family = 'Ubuntu'\nfig_font_size = 16\n\nplt.rcParams.update({'font.size':fig_font_size})\nplt.rcParams.update({'font.family':fig_font_family})\n\n##############################################\n#%% A partir de aqu\u00ed comienza lo IMPORTANTE #\n#############################################\n\ndef my_testbench( sig_type ):\n \n # Datos generales de la simulaci\u00f3n\n fs = 1000.0 # frecuencia de muestreo (Hz)\n N = 1000 # cantidad de muestras\n \n ts = 1/fs # tiempo de muestreo\n df = fs/N # resoluci\u00f3n espectral\n \n # grilla de sampleo temporal\n tt = np.linspace(0, (N-1)*ts, N).flatten()\n \n # grilla de sampleo frecuencial\n ff = np.linspace(0, (N-1)*df, N).flatten()\n\n # Concatenaci\u00f3n de matrices:\n # guardaremos las se\u00f1ales creadas al ir poblando la siguiente matriz vac\u00eda\n x = np.array([], dtype=np.float).reshape(N,0)\n ii = 0\n \n # estructuras de control de flujo\n if sig_type['tipo'] == 'senoidal':\n \n \n # calculo cada senoidal de acuerdo a sus par\u00e1metros\n for this_freq in sig_type['frecuencia']:\n # prestar atenci\u00f3n que las tuplas dentro de los diccionarios tambi\u00e9n pueden direccionarse mediante \"ii\"\n aux = sig_type['amplitud'][ii] * np.sin( 2*np.pi*this_freq*tt + sig_type['fase'][ii] )\n # para concatenar horizontalmente es necesario cuidar que tengan iguales FILAS\n x = np.hstack([x, aux.reshape(N,1)] )\n ii += 1\n \n elif sig_type['tipo'] == 'ruido':\n \n # calculo cada se\u00f1al de ruido incorrelado (blanco), Gausiano de acuerdo a sus par\u00e1metros\n # de varianza\n for this_var in sig_type['varianza']:\n aux = np.sqrt(this_var) * np.random.randn(N,1)\n # para concatenar horizontalmente es necesario cuidar que tengan iguales FILAS\n x = np.hstack([x, aux] )\n \n # Podemos agregar alg\u00fan dato extra a la descripci\u00f3n de forma program\u00e1tica\n # {0:.3f} significa 0: primer argunmento de format\n # .3f formato flotante, con 3 decimales\n # $ ... $ indicamos que incluiremos sintaxis LaTex: $\\hat{{\\sigma}}^2$\n sig_props['descripcion'] = [ sig_props['descripcion'][ii] + ' - $\\hat{{\\sigma}}^2$ :{0:.3f}'.format( np.var(x[:,ii])) for ii in range(0,len(sig_props['descripcion'])) ]\n \n else:\n \n print(\"Tipo de se\u00f1al no implementado.\") \n return\n \n #%% Presentaci\u00f3n gr\u00e1fica de los resultados\n \n plt.figure(1)\n line_hdls = plt.plot(tt, x)\n plt.title('Se\u00f1al: ' + sig_type['tipo'] )\n plt.xlabel('tiempo [segundos]')\n plt.ylabel('Amplitud [V]')\n # plt.grid(which='both', axis='both')\n \n # presentar una leyenda para cada tipo de se\u00f1al\n axes_hdl = plt.gca()\n \n # este tipo de sintaxis es *MUY* de Python\n axes_hdl.legend(line_hdls, sig_type['descripcion'], loc='upper right' )\n \n plt.show()\n\n```\n\nDado que nuestro *testbench* ha sido desarrollado de manera funcional, llamando a la funci\u00f3n *my_testbench()* con diferentes par\u00e1metros, podemos lograr funcionalidades diferentes, como mostramos a continuaci\u00f3n primero con una senoidal:\n\n\n```python\nsig_props = { 'tipo': 'senoidal', \n 'frecuencia': (3, 10, 20), # Uso de tuplas para las frecuencias \n 'amplitud': (1, 1, 1),\n 'fase': (0, 0, 0)\n } \n# Como tambi\u00e9n puedo agregar un campo descripci\u00f3n de manera program\u00e1tica\n# este tipo de sintaxis es *MUY* de Python\nsig_props['descripcion'] = [ str(a_freq) + ' Hz' for a_freq in sig_props['frecuencia'] ]\n \n# Invocamos a nuestro testbench exclusivamente: \nmy_testbench( sig_props )\n```\n\nY ahora con una se\u00f1al aleatoria, en este caso ruido blanco Gaussiano incorrelado de varianza $\\sigma^2$:\n\n\n```python\n# Usar CTRL+1 para comentar o descomentar el bloque de abajo.\nsig_props = { 'tipo': 'ruido', \n 'varianza': (1, 1, 1) # Uso de tuplas para las frecuencias \n } \nsig_props['descripcion'] = [ '$\\sigma^2$ = ' + str(a_var) for a_var in sig_props['varianza'] ]\n \n# Invocamos a nuestro testbench exclusivamente: \nmy_testbench( sig_props )\n\n```\n\nComo puede verse en la figura anterior, al samplear una distribuci\u00f3n estad\u00edstica de media nula y varianza $\\sigma^2=1$, obtenemos realizaciones cuyo par\u00e1metro $\\sigma^2$ estimado, es decir $\\hat\\sigma^2$, tienen una desviaci\u00f3n respecto al verdadero valor (sesgo). Nos ocuparemos de estudiar el sesgo y la varianza de algunos estimadores cuando veamos **Estimaci\u00f3n Espectral**.\n\n# Una vez terminado ...\nUna vez que hayas termiando con la confecci\u00f3n del documento, podemos utilizar una ventaja muy importante de este tipo de documentos que es la posibilidad de compartirlos *online* mediante la [p\u00e1gina de nbviewer](http://nbviewer.jupyter.org/). Para ello es necesario que tu notebook y todos los recursos asociados est\u00e9n alojados en un repositorio de [Github](https://github.com/). Como ejemplo, pod\u00e9s ver este mismo documento disponible [online](http://nbviewer.jupyter.org/github/marianux/pdstestbench/blob/master/notebook0.ipynb).\n", "meta": {"hexsha": "79e911413d2980ed0180b2030affc6115cd9d28c", "size": 217277, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebook0.ipynb", "max_stars_repo_name": "NicolasFerragamo/pdstestbench", "max_stars_repo_head_hexsha": "e87f1a0000f7a0b9593dccd4b0f6fa256ca67438", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebook0.ipynb", "max_issues_repo_name": "NicolasFerragamo/pdstestbench", "max_issues_repo_head_hexsha": "e87f1a0000f7a0b9593dccd4b0f6fa256ca67438", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebook0.ipynb", "max_forks_repo_name": "NicolasFerragamo/pdstestbench", "max_forks_repo_head_hexsha": "e87f1a0000f7a0b9593dccd4b0f6fa256ca67438", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 845.4357976654, "max_line_length": 162616, "alphanum_fraction": 0.9513478187, "converted": true, "num_tokens": 2018, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891163376236, "lm_q2_score": 0.7217432062975979, "lm_q1q2_score": 0.42502671897927563}} {"text": "\n\n# Lambda School Data Science Module 132\n## Sampling, Confidence Intervals, and Hypothesis Testing\n\n## Prepare - examine other available hypothesis tests\n\nIf you had to pick a single hypothesis test in your toolbox, t-test would probably be the best choice - but the good news is you don't have to pick just one! Here's some of the others to be aware of:\n\n\n```\nimport numpy as np\nfrom scipy.stats import chisquare # One-way chi square test\n\n# Chi square can take any crosstab/table and test the independence of rows/cols\n# The null hypothesis is that the rows/cols are independent -> low chi square\n# The alternative is that there is a dependence -> high chi square\n# Be aware! Chi square does *not* tell you direction/causation\n\nind_obs = np.array([[1, 1], [2, 2]]).T\nprint(ind_obs)\nprint(chisquare(ind_obs, axis=None))\n\ndep_obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T\nprint(dep_obs)\nprint(chisquare(dep_obs, axis=None))\n```\n\n [[1 2]\n [1 2]]\n Power_divergenceResult(statistic=0.6666666666666666, pvalue=0.8810148425137847)\n [[16 32]\n [18 24]\n [16 16]\n [14 28]\n [12 20]\n [12 24]]\n Power_divergenceResult(statistic=23.31034482758621, pvalue=0.015975692534127565)\n\n\n\n```\n# Distribution tests:\n# We often assume that something is normal, but it can be important to *check*\n\n# For example, later on with predictive modeling, a typical assumption is that\n# residuals (prediction errors) are normal - checking is a good diagnostic\n\nfrom scipy.stats import normaltest\n# Poisson models arrival times and is related to the binomial (coinflip)\nsample = np.random.poisson(5, 1000)\nprint(normaltest(sample)) # Pretty clearly not normal\n```\n\n NormaltestResult(statistic=46.24179655519528, pvalue=9.093298788095795e-11)\n\n\n\n```\n# Kruskal-Wallis H-test - compare the median rank between 2+ groups\n# Can be applied to ranking decisions/outcomes/recommendations\n# The underlying math comes from chi-square distribution, and is best for n>5\nfrom scipy.stats import kruskal\n\nx1 = [1, 3, 5, 7, 9]\ny1 = [2, 4, 6, 8, 10]\nprint(kruskal(x1, y1)) # x1 is a little better, but not \"significantly\" so\n\nx2 = [1, 1, 1]\ny2 = [2, 2, 2]\nz = [2, 2] # Hey, a third group, and of different size!\nprint(kruskal(x2, y2, z)) # x clearly dominates\n```\n\n KruskalResult(statistic=0.2727272727272734, pvalue=0.6015081344405895)\n KruskalResult(statistic=7.0, pvalue=0.0301973834223185)\n\n\nAnd there's many more! `scipy.stats` is fairly comprehensive, though there are even more available if you delve into the extended world of statistics packages. As tests get increasingly obscure and specialized, the importance of knowing them by heart becomes small - but being able to look them up and figure them out when they *are* relevant is still important.\n\n## T-test Assumptions\n\n\n\n- Independence of means\n\nAre the means of our voting data independent (do not affect the outcome of one another)?\n \nThe best way to increase thel likelihood of our means being independent is to randomly sample (which we did not do).\n\n\n\n```\nfrom scipy.stats import ttest_ind\n\n?ttest_ind\n```\n\n- \"Homogeneity\" of Variance? \n\nIs the magnitude of the variance between the two roughly the same?\n\nI think we're OK on this one for the voting data, although it probably could be better, one party was larger than the other.\n\nIf we suspect this to be a problem then we can use Welch's T-test\n\n\n```\n?ttest_ind\n```\n\n- \"Dependent Variable\" (sample means) are Distributed Normally\n\n\n\nLots of statistical tests depend on normal distributions. We can test for normality using Scipy as was shown above.\n\nThis assumption is often assumed even if the assumption is a weak one. If you strongly suspect that things are not normally distributed, you can transform your data to get it looking more normal and then run your test. This problem typically goes away for large sample sizes (yay Central Limit Theorem) and is often why you don't hear it brought up. People declare the assumption to be satisfied either way. \n\n\n\n## Central Limit Theorem\n\n\n\n\n```\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\n\nsample_means = []\nfor x in range(0,3000):\n coinflips = np.random.binomial(n=1, p=.5, size=12)\n one_sample = coinflips\n sample_means.append(coinflips.mean())\n\nprint(len(sample_means))\nprint(sample_means)\n```\n\n 3000\n [0.5, 0.6666666666666666, 0.25, 0.6666666666666666, 0.3333333333333333, 0.4166666666666667, 0.5, 0.4166666666666667, 0.4166666666666667, 0.5833333333333334, 0.16666666666666666, 0.3333333333333333, 0.4166666666666667, 0.5833333333333334, 0.5, 0.5, 0.5833333333333334, 0.3333333333333333, 0.5833333333333334, 0.4166666666666667, 0.5, 0.75, 0.4166666666666667, 0.4166666666666667, 0.4166666666666667, 0.5, 0.5, 0.75, 0.5, 0.3333333333333333, 0.5, 0.75, 0.3333333333333333, 0.75, 0.4166666666666667, 0.3333333333333333, 0.4166666666666667, 0.4166666666666667, 0.4166666666666667, 0.75, 0.4166666666666667, 0.4166666666666667, 0.4166666666666667, 0.4166666666666667, 0.75, 0.5833333333333334, 0.5833333333333334, 0.25, 0.5833333333333334, 0.5, 0.5833333333333334, 0.75, 0.5, 0.5833333333333334, 0.3333333333333333, 0.5833333333333334, 0.5833333333333334, 0.3333333333333333, 0.5833333333333334, 0.25, 0.6666666666666666, 0.5, 0.25, 0.25, 0.4166666666666667, 0.4166666666666667, 0.8333333333333334, 0.5, 0.5, 0.5833333333333334, 0.6666666666666666, 0.3333333333333333, 0.5833333333333334, 0.5, 0.5833333333333334, 0.5, 0.5833333333333334, 0.5833333333333334, 0.6666666666666666, 0.5833333333333334, 0.75, 0.6666666666666666, 0.4166666666666667, 0.5, 0.5833333333333334, 0.6666666666666666, 0.5833333333333334, 0.5833333333333334, 0.5833333333333334, 0.08333333333333333, 0.5, 0.5833333333333334, 0.5, 0.75, 0.6666666666666666, 0.6666666666666666, 0.5833333333333334, 0.4166666666666667, 0.5, 0.4166666666666667, 0.3333333333333333, 0.5833333333333334, 0.4166666666666667, 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0.75, 0.5833333333333334, 0.5833333333333334, 0.75, 0.3333333333333333, 0.8333333333333334, 0.5, 0.4166666666666667, 0.3333333333333333, 0.6666666666666666, 0.4166666666666667, 0.4166666666666667, 0.5, 0.3333333333333333, 0.4166666666666667, 0.4166666666666667, 0.3333333333333333, 0.5, 0.5, 0.75, 0.25, 0.5, 0.5, 0.5833333333333334, 0.75, 0.5, 0.5, 0.6666666666666666, 0.75, 0.5, 0.5, 0.5833333333333334, 0.4166666666666667, 0.3333333333333333, 0.75, 0.75, 0.5, 0.4166666666666667, 0.5833333333333334, 0.4166666666666667, 0.3333333333333333, 0.75, 0.4166666666666667, 0.3333333333333333, 0.25, 0.75, 0.6666666666666666, 0.4166666666666667, 0.5833333333333334, 0.6666666666666666, 0.4166666666666667, 0.5, 0.25, 0.5, 0.5, 0.4166666666666667, 0.5, 0.5833333333333334, 0.5, 0.5833333333333334, 0.6666666666666666, 0.5, 0.5833333333333334, 0.6666666666666666, 0.75, 0.5, 0.5, 0.3333333333333333, 0.5, 0.5833333333333334, 0.4166666666666667, 0.6666666666666666, 0.5833333333333334, 0.3333333333333333, 0.5, 0.5, 0.4166666666666667, 0.6666666666666666, 0.75, 0.3333333333333333, 0.4166666666666667, 0.5, 0.4166666666666667, 0.6666666666666666, 0.3333333333333333, 0.4166666666666667, 0.25, 0.3333333333333333, 0.4166666666666667, 0.5833333333333334, 0.5, 0.5, 0.8333333333333334, 0.75, 0.4166666666666667, 0.4166666666666667, 0.6666666666666666, 0.5, 0.4166666666666667, 0.4166666666666667, 0.5833333333333334, 0.6666666666666666, 0.6666666666666666, 0.75, 0.25, 0.4166666666666667, 0.4166666666666667, 0.5833333333333334, 0.5, 0.25, 0.5, 0.5833333333333334, 0.5833333333333334, 0.75, 0.5, 0.4166666666666667, 0.4166666666666667, 0.6666666666666666, 0.6666666666666666, 0.5, 0.4166666666666667, 0.5, 0.5833333333333334, 0.4166666666666667, 0.5833333333333334, 0.5833333333333334, 0.75, 0.5833333333333334, 0.6666666666666666, 0.5, 0.4166666666666667, 0.6666666666666666, 0.5833333333333334, 0.5, 0.5833333333333334, 0.5, 0.4166666666666667, 0.4166666666666667, 0.5, 0.16666666666666666, 0.5, 0.5833333333333334, 0.4166666666666667, 0.6666666666666666, 0.6666666666666666, 0.6666666666666666, 0.8333333333333334, 0.3333333333333333, 0.5, 0.5833333333333334, 0.6666666666666666, 0.5, 0.08333333333333333, 0.25, 0.8333333333333334, 0.5, 0.5833333333333334, 0.6666666666666666, 0.5833333333333334, 0.4166666666666667, 0.3333333333333333, 0.5833333333333334, 0.5, 0.5833333333333334, 0.5, 0.4166666666666667, 0.5, 0.3333333333333333, 0.8333333333333334, 0.8333333333333334, 0.4166666666666667, 0.5, 0.4166666666666667, 0.5, 0.3333333333333333, 0.6666666666666666, 0.5, 0.5833333333333334, 0.5, 0.4166666666666667, 0.4166666666666667, 0.5, 0.5, 0.25, 0.5833333333333334, 0.3333333333333333, 0.5833333333333334, 0.5, 0.5833333333333334, 0.5, 0.4166666666666667, 0.4166666666666667, 0.5833333333333334, 0.25, 0.4166666666666667, 0.6666666666666666, 0.8333333333333334, 0.5833333333333334, 0.25, 0.4166666666666667, 0.6666666666666666, 0.25, 0.4166666666666667, 0.5833333333333334, 0.5833333333333334, 0.5, 0.25, 0.75, 0.5833333333333334, 0.5, 0.25, 0.3333333333333333, 0.4166666666666667, 0.16666666666666666, 0.5, 0.6666666666666666, 0.6666666666666666, 0.5833333333333334, 0.5833333333333334, 0.4166666666666667, 0.25, 0.6666666666666666, 0.75, 0.5833333333333334, 0.5, 0.4166666666666667, 0.3333333333333333, 0.3333333333333333, 0.5, 0.4166666666666667, 0.4166666666666667, 0.4166666666666667, 0.6666666666666666, 0.25, 0.6666666666666666, 0.5833333333333334, 0.4166666666666667, 0.3333333333333333, 0.5833333333333334, 0.6666666666666666, 0.3333333333333333, 0.4166666666666667, 0.4166666666666667, 0.5833333333333334, 0.75, 0.5, 0.3333333333333333, 0.5833333333333334, 0.5833333333333334, 0.25, 0.6666666666666666, 0.5, 0.5833333333333334, 0.75, 0.5, 0.5833333333333334, 0.75, 0.6666666666666666, 0.75, 0.5, 0.5, 0.4166666666666667, 0.5, 0.4166666666666667, 0.4166666666666667, 0.6666666666666666, 0.5833333333333334, 0.5, 0.5833333333333334, 0.4166666666666667, 0.5, 0.5, 0.3333333333333333, 0.5833333333333334, 0.6666666666666666, 0.5, 0.5833333333333334, 0.4166666666666667, 0.3333333333333333, 0.4166666666666667, 0.3333333333333333, 0.3333333333333333, 0.6666666666666666, 0.5833333333333334, 0.5, 0.25, 0.4166666666666667, 0.4166666666666667, 0.5, 0.75, 0.3333333333333333, 0.5, 0.75, 0.5, 0.4166666666666667, 0.4166666666666667, 0.5, 0.3333333333333333, 0.4166666666666667, 0.6666666666666666, 0.4166666666666667, 0.4166666666666667, 0.4166666666666667, 0.5833333333333334, 0.5833333333333334, 0.75, 0.3333333333333333, 0.5, 0.5833333333333334, 0.6666666666666666, 0.4166666666666667, 0.5, 0.5833333333333334, 0.5, 0.5833333333333334, 0.5833333333333334, 0.5833333333333334, 0.5, 0.5, 0.5833333333333334, 0.4166666666666667, 0.3333333333333333, 0.75, 0.5833333333333334, 0.4166666666666667, 0.5833333333333334, 0.5, 0.5833333333333334, 0.4166666666666667, 0.6666666666666666, 0.3333333333333333, 0.5833333333333334, 0.6666666666666666, 0.08333333333333333, 0.5, 0.5833333333333334, 0.5833333333333334, 0.4166666666666667, 0.5, 0.75, 0.5, 0.6666666666666666, 0.6666666666666666, 0.5, 0.5, 0.16666666666666666, 0.4166666666666667, 1.0, 0.6666666666666666, 0.5833333333333334, 0.5833333333333334, 0.3333333333333333, 0.75, 0.6666666666666666, 0.4166666666666667, 0.6666666666666666, 0.5833333333333334, 0.75, 0.5, 0.5, 0.5, 0.5833333333333334, 0.5, 0.5833333333333334, 0.25, 0.4166666666666667, 0.75, 0.4166666666666667, 0.3333333333333333, 0.5833333333333334, 0.5, 0.6666666666666666, 0.4166666666666667, 0.5, 0.75, 0.4166666666666667, 0.25, 0.25, 0.5, 0.5, 0.3333333333333333, 0.5, 0.3333333333333333, 0.16666666666666666, 0.6666666666666666, 0.4166666666666667, 0.4166666666666667, 0.4166666666666667, 0.5, 0.4166666666666667, 0.6666666666666666, 0.4166666666666667, 0.25, 0.4166666666666667, 0.3333333333333333, 0.3333333333333333, 0.3333333333333333, 0.5833333333333334, 0.5, 0.5833333333333334, 0.75, 0.5, 0.5833333333333334, 0.75, 0.6666666666666666, 0.6666666666666666, 0.5, 0.3333333333333333, 0.5833333333333334, 0.3333333333333333, 0.4166666666666667, 0.4166666666666667, 0.5, 0.75, 0.6666666666666666, 0.4166666666666667]\n\n\n\n```\ndf = pd.DataFrame({'a': one_sample})\ndf.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        a
                                        00
                                        11
                                        21
                                        31
                                        40
                                        \n
                                        \n\n\n\n\n```\ndf.a.hist()\n```\n\n\n```\nax = plt.hist(sample_means, bins=24)\nplt.title('Distribution of 3000 sample means \\n (of 12 coinflips each)');\n```\n\nWhat does the Central Limit Theorem State? That no matter the initial distribution of the population, the distribution of sample means taken will approximate a normal distribution as $n \\rightarrow \\infty$.\n\nThis has very important implications for hypothesis testing and is precisely the reason why the t-distribution begins to approximate the normal distribution as our sample size increases. \n\n## Standard Error of the Mean\n\nWhat does it mean to \"estimate\"? the Population mean?\n\n\n```\n\n```\n\n## Build and Interpret a Confidence Interval\n\n\n\n\n```\ncoinflips_100 = np.random.binomial(n=1, p=.5, size=100)\n \nsample_std = np.std(coinflips_100, ddof=1)\nprint(\"sample standard dev\", sample_std)\n```\n\n sample standard dev 0.5024183937956914\n\n\n\n```\nsample_size = len(coinflips_100)\nprint(sample_size)\n```\n\n 100\n\n\n\n```\nstandard_error = sample_std / np.sqrt(sample_size)\n\nprint(\"Standard Error\", standard_error)\n```\n\n Standard Error 0.05024183937956914\n\n\n\n```\nfrom scipy import stats\n\nstderr = stats.sem(coinflips_100, ddof=1)\nprint(\"Scipy standard error\", stderr)\n```\n\n Scipy standard error 0.05024183937956914\n\n\n##Look at stats.t.ppf()\n\n\n```\nt=stats.t.ppf(.975, 99) \nt\n```\n\n\n\n\n 1.9842169515086827\n\n\n\n\n```\nt = stats.t.ppf(.025, 99)\nt\n```\n\n\n\n\n -1.9842169515086832\n\n\n\n\n```\n(1+.99)/2\n```\n\n\n\n\n 0.995\n\n\n\n\n```\nstats.t.ppf(.995, 99)\n```\n\n\n\n\n 2.6264054563851857\n\n\n\n\n```\ndef confidence_interval(data, confidence=0.95):\n \"\"\"\n Calculate a confidence interval around a sample mean for given data.\n Using t-distribution and two-tailed test, default 95% confidence. \n \n Arguments:\n data - iterable (list or numpy array) of sample observations\n confidence - level of confidence for the interval\n \n Returns:\n tuple of (mean, lower bound, upper bound)\n \"\"\"\n data = np.array(data)\n mean = np.mean(data)\n sample_size = len(data)\n sample_std_dev = np.std(data, ddof=1)\n stderr = stats.sem(data)\n interval = stderr * stats.t.ppf((1 + confidence) / 2.0, sample_size - 1)\n return (mean, mean - interval, mean + interval)\n```\n\n## Graphically Represent a Confidence Interval\n\nhttps://matplotlib.org/3.1.1/api/_as_gen/matplotlib.pyplot.errorbar.html\n\nhttps://jakevdp.github.io/PythonDataScienceHandbook/04.03-errorbars.html\n\nhttps://seaborn.pydata.org/generated/seaborn.barplot.html\n\n\n```\nimport seaborn as sns\n\n\ncoinflips_100 = np.random.binomial(n=1, p=.5, size=100)\nsns.kdeplot(coinflips_100)\nCI = confidence_interval(coinflips_100)\nplt.axvline(x=CI[1], color='r')\nplt.axvline(x=CI[2], color='r')\nplt.axvline(x=CI[0], color='k')\n```\n\n\n```\n\n```\n\n## Relationship between Confidence Intervals and T-tests\n\nConfidence Interval == Bounds of statistical significance for our t-test\n\nA sample mean that falls inside of our confidence interval will \"FAIL TO REJECT\" our null hypothesis\n\nA sample mean that falls outside of our confidence interval will \"REJECT\" our null hypothesis\n\n\n```\nfrom scipy.stats import t, ttest_1samp\n```\n\n\n```\nimport numpy as np\n\ncoinflip_means = []\nfor x in range(0,100):\n coinflips = np.random.binomial(n=1, p=.5, size=30)\n coinflip_means.append(coinflips.mean())\n\nprint(coinflip_means)\n```\n\n [0.6, 0.5, 0.6333333333333333, 0.4, 0.4, 0.4666666666666667, 0.5333333333333333, 0.5333333333333333, 0.4, 0.5333333333333333, 0.3, 0.5333333333333333, 0.5333333333333333, 0.6, 0.5333333333333333, 0.6333333333333333, 0.4666666666666667, 0.5333333333333333, 0.5333333333333333, 0.43333333333333335, 0.4, 0.6, 0.36666666666666664, 0.6, 0.6, 0.5666666666666667, 0.4, 0.43333333333333335, 0.6333333333333333, 0.6333333333333333, 0.5333333333333333, 0.43333333333333335, 0.5333333333333333, 0.3333333333333333, 0.4666666666666667, 0.3333333333333333, 0.4, 0.4666666666666667, 0.6333333333333333, 0.43333333333333335, 0.36666666666666664, 0.5666666666666667, 0.5666666666666667, 0.6333333333333333, 0.4666666666666667, 0.6666666666666666, 0.36666666666666664, 0.36666666666666664, 0.36666666666666664, 0.5, 0.6666666666666666, 0.5333333333333333, 0.5, 0.43333333333333335, 0.5, 0.4666666666666667, 0.36666666666666664, 0.6333333333333333, 0.43333333333333335, 0.4666666666666667, 0.5666666666666667, 0.4666666666666667, 0.5, 0.6333333333333333, 0.43333333333333335, 0.3333333333333333, 0.43333333333333335, 0.36666666666666664, 0.4, 0.43333333333333335, 0.5666666666666667, 0.5, 0.4666666666666667, 0.3333333333333333, 0.5666666666666667, 0.43333333333333335, 0.43333333333333335, 0.43333333333333335, 0.5666666666666667, 0.5333333333333333, 0.5333333333333333, 0.7, 0.43333333333333335, 0.4, 0.5, 0.43333333333333335, 0.5, 0.5333333333333333, 0.36666666666666664, 0.6, 0.43333333333333335, 0.5, 0.5, 0.6666666666666666, 0.3, 0.5666666666666667, 0.5333333333333333, 0.5333333333333333, 0.36666666666666664, 0.5333333333333333]\n\n\n\n```\n# Sample Size\nn = len(coinflip_means)\n# Degrees of Freedom\ndof = n-1\n# The Mean of Means:\nmean = np.mean(coinflip_means)\n# Sample Standard Deviation\nsample_std = np.std(coinflip_means, ddof=1)\n# Standard Error\nstd_err = sample_std/n**.5\n\nCI = t.interval(.95, dof, loc=mean, scale=std_err)\nprint(\"95% Confidence Interval: \", CI)\n```\n\n 95% Confidence Interval: (0.47229604845181783, 0.5097039515481822)\n\n\n\n```\n'''You can roll your own CI calculation pretty easily. \nThe only thing that's a little bit challenging \nis understanding the t stat lookup'''\n\n# 95% confidence interval\nt_stat = t.ppf(.975, dof)\nprint(\"t Statistic:\", t_stat)\n\nCI = (mean-(t_stat*std_err), mean+(t_stat*std_err))\nprint(\"Confidence Interval\", CI)\n```\n\n t Statistic: 1.9842169515086827\n Confidence Interval (0.47229604845181783, 0.5097039515481822)\n\n\nA null hypothesis that's just inside of our confidence interval == fail to reject\n\n\n\n\n```\nttest_1samp(coinflip_means, .49)\n```\n\n\n\n\n Ttest_1sampResult(statistic=0.10608544116451862, pvalue=0.9157292253188959)\n\n\n\nA null hypothesis that's just outside of our confidence interval == reject\n\n\n\n\n```\nttest_1samp(coinflip_means, .4818927)\n```\n\n\n\n\n Ttest_1sampResult(statistic=0.966151938317618, pvalue=0.33632241031590215)\n\n\n\n## Run a $\\chi^{2}$ Test \"by hand\" (Using Numpy)\n\n\n```\ndf = pd.read_csv('https://raw.githubusercontent.com/ryanleeallred/datasets/master/adult.csv', na_values=\" ?\")\nprint(df.shape)\ndf.head()\n```\n\n (32561, 15)\n\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        ageworkclassfnlwgteducationeducation-nummarital-statusoccupationrelationshipracesexcapital-gaincapital-losshours-per-weekcountrysalary
                                        039State-gov77516Bachelors13Never-marriedAdm-clericalNot-in-familyWhiteMale2174040United-States<=50K
                                        150Self-emp-not-inc83311Bachelors13Married-civ-spouseExec-managerialHusbandWhiteMale0013United-States<=50K
                                        238Private215646HS-grad9DivorcedHandlers-cleanersNot-in-familyWhiteMale0040United-States<=50K
                                        353Private23472111th7Married-civ-spouseHandlers-cleanersHusbandBlackMale0040United-States<=50K
                                        428Private338409Bachelors13Married-civ-spouseProf-specialtyWifeBlackFemale0040Cuba<=50K
                                        \n
                                        \n\n\n\n\n```\ndf.describe()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        agefnlwgteducation-numcapital-gaincapital-losshours-per-week
                                        count32561.0000003.256100e+0432561.00000032561.00000032561.00000032561.000000
                                        mean38.5816471.897784e+0510.0806791077.64884487.30383040.437456
                                        std13.6404331.055500e+052.5727207385.292085402.96021912.347429
                                        min17.0000001.228500e+041.0000000.0000000.0000001.000000
                                        25%28.0000001.178270e+059.0000000.0000000.00000040.000000
                                        50%37.0000001.783560e+0510.0000000.0000000.00000040.000000
                                        75%48.0000002.370510e+0512.0000000.0000000.00000045.000000
                                        max90.0000001.484705e+0616.00000099999.0000004356.00000099.000000
                                        \n
                                        \n\n\n\n\n```\ndf.describe(exclude='number')\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        workclasseducationmarital-statusoccupationrelationshipracesexcountrysalary
                                        count307253256132561307183256132561325613197832561
                                        unique816714652412
                                        topPrivateHS-gradMarried-civ-spouseProf-specialtyHusbandWhiteMaleUnited-States<=50K
                                        freq22696105011497641401319327816217902917024720
                                        \n
                                        \n\n\n\n\n```\ncut_points = [0,9,19,29,39,49,1000]\nlabel_names = ['0-9', '10-19', '20-29', '30-39', '40-49', '50+']\ndf['hours_per_week_categories'] = pd.cut(df['hours-per-week'], cut_points, labels=label_names)\n\ndf.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        ageworkclassfnlwgteducationeducation-nummarital-statusoccupationrelationshipracesexcapital-gaincapital-losshours-per-weekcountrysalaryhours_per_week_categories
                                        039State-gov77516Bachelors13Never-marriedAdm-clericalNot-in-familyWhiteMale2174040United-States<=50K40-49
                                        150Self-emp-not-inc83311Bachelors13Married-civ-spouseExec-managerialHusbandWhiteMale0013United-States<=50K10-19
                                        238Private215646HS-grad9DivorcedHandlers-cleanersNot-in-familyWhiteMale0040United-States<=50K40-49
                                        353Private23472111th7Married-civ-spouseHandlers-cleanersHusbandBlackMale0040United-States<=50K40-49
                                        428Private338409Bachelors13Married-civ-spouseProf-specialtyWifeBlackFemale0040Cuba<=50K40-49
                                        \n
                                        \n\n\n\n\n```\ndf['sex'].value_counts()\n```\n\n\n\n\n Male 21790\n Female 10771\n Name: sex, dtype: int64\n\n\n\n\n```\ndf['hours_per_week_categories'].value_counts()\n```\n\n\n\n\n 40-49 18336\n 50+ 6462\n 30-39 3667\n 20-29 2392\n 10-19 1246\n 0-9 458\n Name: hours_per_week_categories, dtype: int64\n\n\n\n\n```\ndf = df.sort_values(by='hours_per_week_categories')\n\ndf.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        ageworkclassfnlwgteducationeducation-nummarital-statusoccupationrelationshipracesexcapital-gaincapital-losshours-per-weekcountrysalaryhours_per_week_categories
                                        3129055Self-emp-not-inc41938Bachelors13Married-civ-spouseProf-specialtyWifeWhiteFemale008United-States<=50K0-9
                                        517232NaN134886HS-grad9Married-civ-spouseNaNWifeWhiteFemale002United-States>50K0-9
                                        2292817NaN33266610th6Never-marriedNaNOwn-childWhiteFemale004United-States<=50K0-9
                                        790235Private359131Bachelors13Married-civ-spouseProf-specialtyWifeWhiteFemale729808NaN>50K0-9
                                        660441Private406603HS-grad9Never-marriedOther-serviceNot-in-familyWhiteMale006Iran<=50K0-9
                                        \n
                                        \n\n\n\n\n```\ncontingency_table = pd.crosstab(df['sex'], df['hours_per_week_categories'], margins=True)\n\ncontingency_table\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        hours_per_week_categories0-910-1920-2930-3940-4950+All
                                        sex
                                        Female235671128719145636102810771
                                        Male2235751105175312700543421790
                                        All45812462392366718336646232561
                                        \n
                                        \n\n\n\n\n```\nfemalecount = contingency_table.iloc[0][0:6].values\nfemalecount\n```\n\n\n\n\n array([ 235, 671, 1287, 1914, 5636, 1028])\n\n\n\n\n```\nmalecount = contingency_table.iloc[1][0:6].values\nmalecount\n```\n\n\n\n\n array([ 223, 575, 1105, 1753, 12700, 5434])\n\n\n\n\n```\nimport matplotlib.pyplot as plt\nimport seaborn as sns\n\n#plots the bar chart\nfig = plt.figure(figsize=(10,5))\nsns.set(font_scale=1.8)\ncategories = label_names\np1 = plt.bar(categories, malecount, .55, color='#d62728')\np2 = plt.bar(categories, femalecount, .55, bottom=malecount)\nplt.legend((p2[0], p1[0]), ('Female', 'Male'))\nplt.xlabel('Count')\nplt.show()\n```\n\n##Expected Value Calculation\n\n\\begin{align}\nexpected_{i,j} =\\frac{(row_{i} \\text{total})(column_{j} \\text{total}) }{(\\text{total observations})} \n\\end{align}\n\n\n\n\n```\n#Get Row sums\nrow_sums = contingency_table.iloc[0:2, 6].values\ncol_sums = contingency_table.iloc[2, 0:6].values\nprint (row_sums)\nprint (col_sums)\n```\n\n [10771 21790]\n [ 458 1246 2392 3667 18336 6462]\n\n\n\n```\ntotal = contingency_table.loc['All', 'All']\ntotal\n```\n\n\n\n\n 32561\n\n\n\n\n```\nlen(df)\n```\n\n\n\n\n 32561\n\n\n\n\n```\n df.shape[0]\n```\n\n\n\n\n 32561\n\n\n\n\n```\nexpected = []\nfor i in range(len(row_sums)):\n expected_row = []\n for column in col_sums:\n expected_val = column*row_sums[i]/total\n expected_row.append(expected_val)\n expected.append(expected_row)\n \nexpected = np.array(expected)\nprint (expected.shape)\nprint (expected)\n```\n\n (2, 6)\n [[ 151.50388502 412.16995793 791.26046497 1213.02346365\n 6065.44811277 2137.59411566]\n [ 306.49611498 833.83004207 1600.73953503 2453.97653635\n 12270.55188723 4324.40588434]]\n\n\n\n```\nobserved = pd.crosstab(df['sex'], df['hours_per_week_categories']).values\nprint(observed.shape)\nobserved\n```\n\n (2, 6)\n\n\n\n\n\n array([[ 235, 671, 1287, 1914, 5636, 1028],\n [ 223, 575, 1105, 1753, 12700, 5434]])\n\n\n\n## Chi-Squared Statistic with Numpy\n\n\\begin{align}\n\\chi^2 = \\sum \\frac{(observed_{i}-expected_{i})^2}{(expected_{i})}\n\\end{align}\n\nFor the $observed$ values we will just use a version of our contingency table without the margins as a numpy array. In this way, if our observed values array and our expected values array are the same shape, then we can subtract them and divide them directly which makes the calculations a lot cleaner. No for loops!\n\n\n```\nchi_squared = ((observed - expected)**2/expected).sum()\nprint(f\"Chi-Squared: {chi_squared}\")\n```\n\n Chi-Squared: 2287.190943926107\n\n\n\n```\n#Calculate Degrees of Freedom\ndof = (len(row_sums) - 1) * (len(col_sums) - 1)\nprint(f\"Degrees of Freedom: {dof}\")\n```\n\n Degrees of Freedom: 5\n\n\n## Run a $\\chi^{2}$ Test using Scipy\n\n\n```\nchi_squared, p_value, dof, expected = stats.chi2_contingency(observed)\n\nprint(f\"Chi-Squared: {chi_squared}\")\nprint(f\"P-value: {p_value}\")\nprint(f\"Degrees of Freedom: {dof}\") \nprint(\"Expected: \\n\", np.array(expected))\n```\n\n Chi-Squared: 2287.190943926107\n P-value: 0.0\n Degrees of Freedom: 5\n Expected: \n [[ 151.50388502 412.16995793 791.26046497 1213.02346365\n 6065.44811277 2137.59411566]\n [ 306.49611498 833.83004207 1600.73953503 2453.97653635\n 12270.55188723 4324.40588434]]\n\n\n### Conclusion\nBased on a chi-squared statistic of 2287 and a p-value of 0. I reject the null hypothesis that hours_worked_per_week, and sex are independent, and suggest the alternative that there is an association between hours_worked_per_week and sex.\n", "meta": {"hexsha": "1e8757b4e154977d8c3ec8d729f477bbeecbb971", "size": 207952, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "module2-sampling-confidence-intervals-and-hypothesis-testing/LS_DS_132_Sampling_Confidence_Intervals_and_Hypothesis_Testing.ipynb", "max_stars_repo_name": "corbittcoder/DS-Unit-1-Sprint-3-Statistical-Tests-and-Experiments", "max_stars_repo_head_hexsha": "51861525c58093e1d6e3f3e12de96b5d993e4477", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "module2-sampling-confidence-intervals-and-hypothesis-testing/LS_DS_132_Sampling_Confidence_Intervals_and_Hypothesis_Testing.ipynb", "max_issues_repo_name": "corbittcoder/DS-Unit-1-Sprint-3-Statistical-Tests-and-Experiments", "max_issues_repo_head_hexsha": "51861525c58093e1d6e3f3e12de96b5d993e4477", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "module2-sampling-confidence-intervals-and-hypothesis-testing/LS_DS_132_Sampling_Confidence_Intervals_and_Hypothesis_Testing.ipynb", "max_forks_repo_name": "corbittcoder/DS-Unit-1-Sprint-3-Statistical-Tests-and-Experiments", "max_forks_repo_head_hexsha": "51861525c58093e1d6e3f3e12de96b5d993e4477", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 82.7175815434, "max_line_length": 45442, "alphanum_fraction": 0.7032536355, "converted": true, "num_tokens": 35251, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073802837477, "lm_q2_score": 0.7718434925908525, "lm_q1q2_score": 0.4249827234445075}} {"text": "## You don't know what to choose between Matlab and Python :\n\n# Try \n\n
                                        \n\n\n# Play with the slides \n\nhttps://mybinder.org/v2/gh/raphbacher/julia-intro/master?filepath=presentation_julia.ipynb\n\nSources:\n\nhttps://github.com/raphbacher/julia-intro\n\n# Outline\n\n- State of the art : the two language problem\n- What is Julia\n- Why Julia is fast\n- Julia Ecosystem & interop \n\n## Matlab\n\nPros :\n- Polished product with support\n- Simulink\n- High level syntax\n\nCons :\n- Closed : not everyone has access to it, impossible to put a demonstrator online.\n- Slow loops (better since 2015) : not everything is pretty once vectorized\n- Not fast per se (Fortran bindings), memory management is difficult\n- Not a generalist language (cumbersome to put a demo on the web e.g.)\n- \u20ac\u20ac each year for a lab \n\n\n\n\n## Python\n \nPros :\n- Free and open-source, generalist, widly used outside scientific community\n- Lot of scientific communities are embracing it\n- Lots of efforts to make it fast (numba, ...)\n\nCons :\n- Scientific computing is not native : \n - all fallback to C/Fortran black-boxes -> limit flexibility \n- Object Oriented paradigm can be cumbersome for scientific code\n\n## Science and the two languages problem\n\nScientists need to easily explore new ideas :\n\n- Need for mathematical abstractions\n- Need for customizations (avoid black boxes, try variations)\n\n- But also need for performance (intrinsic goal or to allow re-use)\n\nWhat is done now (when you need to go further than using existing packages, for e.g. data analysis)\n\n1) Prototyping in R/Python/Matlab\n\n2) Rewriting whole or parts (if money/man power) to a performant low-level language as C/Fortran\n\n\n# Yet another language\n\n
                                        \n\n# Here comes Julia :\n\n* innovative open-source generalist programming language with scientific computing at its core\n* **easy as Matlab, fast as Fortran, flexible as Python, deep as LISP**\n* leverages (for now...) major C/Fortran science packages, e.g. LAPACK, MPI, FFTW... \n* 5th language to run at 1 petaflops (Celeste project), after assembly, Fortran, C, C++ \n* State of the art packages in Optimization (JUMP), Differential Equations (DifEq), ... Well positioned for ML (Flux, Knet, autodiff...)\n* solves the \"two-languages problem\"\n\n\n# Background\n\n* origins at MIT, UCSB applied math & computer science, 2010 \n* founders Viral Shah, Jeff Bezanson, Stefan Karpinsky (now at Julia Computing LLC), Alan Edelman (MIT)\n* ~10-person core language team, ~870 contributors, ~2500 registered packages \n* support from Intel, Microsoft, Wall St., Moore Foundation, NumFocus\n* julia-0.1 released in 2012, julia-1.0 in August 2018, now julia-1.2\n\n\n**Its goal : Solve the two-languages problem by having a dynamic, high level language with good mathematical expressivity able to be as fast as C/Fortran.**\n\n# Disclaimer\n\n- I am not a Julia expert : I've been following the language for ~ two years and I am currently porting python research codes to Julia.\n- Julia (like Python...) is not a drop-in replacement for Matlab, it is an alternative.\n- Don't fix that ain't broken : \n - if Matlab/R/Python/C is good for you, no need to change\n - but if you are sometimes faced with some limitations of these languages, give Julia a try\n\n# Julia: expressive as Matlab\n\nJulia can as expressive as Matlab for interactive calculations with built-in numerical libraries and plotting. A few examples...\n\n### Arrays\n\n\n```julia\nA = randn(4,4) # 4 x 4 matrix of normally distributed random numbers\n```\n\n\n\n\n 4\u00d74 Array{Float64,2}:\n -1.7897 -1.27394 -0.796802 -1.12804 \n -0.843538 -0.607065 0.187264 0.0447423\n -0.83152 -0.483635 0.0594034 -0.607388 \n 0.194026 1.4971 0.533445 -0.379708 \n\n\n\n\n```julia\nA[:,1] # familiar colon syntax --extract first column of A\n```\n\n\n\n\n 4-element Array{Float64,1}:\n -1.7896952667660175 \n -0.843538211646361 \n -0.8315200775522081 \n 0.19402554880895634\n\n\n\n## Linear Algebra\n\n\n```julia\nusing LinearAlgebra\nb = randn(4) \nx = A\\b # solve Ax=b for x using backslash operator\nnorm(A*x-b)\n```\n\n\n\n\n 5.551115123125783e-16\n\n\n\n\n```julia\nU, \u03c3, V = svd(A); # singular value decomposition --note unicode variable name \u03c3\n```\n\n\n```julia\n\u03a3 = diagm(0 => \u03c3)\nnorm(A - U*\u03a3*V') # compute error of SVD factorization A = U \u03a3 V'\n```\n\n\n\n\n 8.164634113057559e-16\n\n\n\n# Fast as Fortran\n\n\n\n```julia\nfunction loop()\n a=0 \n for i in 1:1_000_000\n a+=1;\n end\nend\n@time loop() # compile time\n@time loop() # now fast\n```\n\n 0.004979 seconds (20.83 k allocations: 1.173 MiB)\n 0.000001 seconds (4 allocations: 160 bytes)\n\n\n## Compare with Matlab (R2018) \n\n*I know this is not Matlab-friendly, that's the point !*\n\n function [] = loop()\n a=0;\n for i=1:1000000\n a=i+1;\n end\n end\n \n f = @() loop();\n \n timeit(f) => 0.0018s\n\n**1000x slower !**\n\n### Benchmarks on common code pattterns\n\nTakeaway: Julia timings are clustered around C timing, Matlab/Python/R orders of magnitude slower.\n\n
                                        \n\n### KS-CNAB2 benchmark: CPU time versus lines of code \n\nTakeaway: The Julia PDE code is almost as compact as Matlab/Python, almost as fast as C/Fortran.\n
                                        \n\n\n## Julia: easy, dynamic, and fast. How? \n\n\n * **Just-in-time compilation** (JIT)\n - user-level code is compiled to machine code on-the-fly \n \n \n * **Meticulous type system**\n - designed to maximize impact of JIT\n - type inference: compiler determines types of variables\n \n \n * **Multiple dispatch**\n - functions are compiled for each set of argument types\n - function dispatch determined at compile time when possible, run time when not\n \n\n**Just-in-time ahead-of-time compilation\n\nFunctions are compiled to machine code when first run. Subsequent runs are as fast as compiled C, C++, Fortran.\nNot a tracing JIT (like pypy)\n\n# Caveats\n\n- It's a marathon not a sprint : Julia can be slow at first\n- It's an open-source language with lots of good willing contributors, it's not a product (see JuliaPro...)\n- If you directly translate from Matlab to Julia you will not always see an improvement, you have exploit Julia strengths (loops to avoid allocations, type stability, ...)\n\n# A julia type\n\nLike a C struct, or a Python object... but without methods\n\n```julia\nstruct Spaceship\n speed::Float64\n Position::Array{Float64,2}\n end\n```\n\n## Multiple dispatch\n\n```julia\ncollide(a::Asteroid, s::Spaceship) # a first method of the function collide\ncollide(s1::Spaceship, s2::Spaceship) # another method of the function collide\n```\n\n## The power of Multiple dispatch\n\nRef https://medium.com/@Jernfrost/defining-custom-units-in-julia-and-python-513c34a4c971\n\nSee also (very interesting!) : https://www.youtube.com/watch?v=kc9HwsxE1OY\n\n\n```julia\nabstract type Temperature end\n\nstruct Celsius <: Temperature\n value\nend\n\nstruct Kelvin <: Temperature\n value::Float64 \nend\n\nstruct Fahrenheit <: Temperature\n value::Float64\nend\n\nimport Base: promote_rule, convert\npromote_rule(::Type{Kelvin}, ::Type{Celsius}) = Kelvin\npromote_rule(::Type{Fahrenheit}, ::Type{Celsius}) = Celsius\npromote_rule(::Type{Fahrenheit}, ::Type{Kelvin}) = Kelvin\n\nconvert(::Type{Kelvin}, t::Celsius) = Kelvin(t.value + 273.15)\nconvert(::Type{Kelvin}, t::Fahrenheit) = Kelvin(Celsius(t))\nconvert(::Type{Celsius}, t::Kelvin) = Celsius(t.value - 273.15)\nconvert(::Type{Celsius}, t::Fahrenheit) = Celsius((t.value - 32)*5/9)\nconvert(::Type{Fahrenheit}, t::Celsius) = Fahrenheit(t.value*9/5 + 32)\nconvert(::Type{Fahrenheit}, t::Kelvin) = Fahrenheit(Celsius(t))\n\nimport Base: +,-,*\n+(x::T, y::T) where {T <: Temperature} = T(x.value + y.value)\n-(x::T, y::T) where {T <: Temperature} = T(x.value - y.value)\n\n+(x::Temperature, y::Temperature) = +(promote(x,y)...)\n-(x::Temperature, y::Temperature) = -(promote(x,y)...)\n\n*(x::Number, y::T) where {T <: Temperature} = T(x * y.value)\n\nconst \u00b0C = Celsius(1); const \u00b0F = Fahrenheit(1); const K = Kelvin(1);\n```\n\n\n```julia\n1K+1\u00b0C\n```\n\n\n\n\n Kelvin(275.15)\n\n\n\n## Introspection \n$$f(x) = 4\\, x\\, (1-x)$$\n\n\n```julia\nf(x) = 4x*(1-x) # define logistic map\n@time f(0.3); # first run is slow\n@time f(0.4); # second run is a thousand times faster\n```\n\n 0.008778 seconds (18.00 k allocations: 1.003 MiB)\n 0.000003 seconds (5 allocations: 176 bytes)\n\n\n#### Observe the compilation of $f(x)$ by stages\n\nuser Julia -> generic Julia expression -> typed Julia expression -> intermediate compiler language -> machine code\n\n\n```julia\ninclude(\"macros.jl\");\n```\n\n\n```julia\n@code_lowered f(0.3) # show f(x) as generic Julia expression\n```\n\n\n\n\n CodeInfo(\n \u001b[90m1 \u2500\u001b[39m %1 = 4 * x\n \u001b[90m\u2502 \u001b[39m %2 = 1 - x\n \u001b[90m\u2502 \u001b[39m %3 = %1 * %2\n \u001b[90m\u2514\u2500\u2500\u001b[39m return %3\n )\n\n\n\n\n```julia\n@code_typed f(0.3) # show f(x) as Julia expr with inferred types, based on the arg types\n```\n\n\n\n\n CodeInfo(\n \u001b[90m1 \u2500\u001b[39m %1 = (Base.mul_float)(4.0, x)\u001b[36m::Float64\u001b[39m\n \u001b[90m\u2502 \u001b[39m %2 = (Base.sub_float)(1.0, x)\u001b[36m::Float64\u001b[39m\n \u001b[90m\u2502 \u001b[39m %3 = (Base.mul_float)(%1, %2)\u001b[36m::Float64\u001b[39m\n \u001b[90m\u2514\u2500\u2500\u001b[39m return %3\n ) => Float64\n\n\n\n\n```julia\n@code_llvm f(0.3) # show f(x) in intermediate compiler language (LLVM)\n```\n\n \n ; @ In[9]:1 within `f'\n define double @julia_f_12939(double) {\n top:\n ; \u250c @ promotion.jl:314 within `*' @ float.jl:399\n %1 = fmul double %0, 4.000000e+00\n ; \u2514\n ; \u250c @ promotion.jl:315 within `-' @ float.jl:397\n %2 = fsub double 1.000000e+00, %0\n ; \u2514\n ; \u250c @ float.jl:399 within `*'\n %3 = fmul double %1, %2\n ; \u2514\n ret double %3\n }\n\n\n## Type inference and dispatch\n\nWe can see that if we call f on an Integer we get a code specialised for integer (i64)\n\n\n```julia\n@code_native f(0.3) # show f(x) in IA-64 assembly language\n```\n\n \t.text\n ; \u250c @ In[9]:1 within `f'\n \tmovabsq\t$140409732451304, %rax # imm = 0x7FB3B039CBE8\n ; \u2502\u250c @ promotion.jl:314 within `*' @ float.jl:399\n \tvmulsd\t(%rax), %xmm0, %xmm1\n \tmovabsq\t$140409732451312, %rax # imm = 0x7FB3B039CBF0\n ; \u2502\u2514\n ; \u2502\u250c @ promotion.jl:315 within `-' @ float.jl:397\n \tvmovsd\t(%rax), %xmm2 # xmm2 = mem[0],zero\n \tvsubsd\t%xmm0, %xmm2, %xmm0\n ; \u2502\u2514\n ; \u2502\u250c @ float.jl:399 within `*'\n \tvmulsd\t%xmm0, %xmm1, %xmm0\n ; \u2502\u2514\n \tretq\n \tnopw\t%cs:(%rax,%rax)\n ; \u2514\n\n\n\n```julia\n@code_native f(3) # show f(x) in IA-64 assembly language\n```\n\n \t.text\n ; \u250c @ In[9]:1 within `f'\n ; \u2502\u250c @ In[9]:1 within `-'\n \tmovl\t$1, %eax\n \tsubq\t%rdi, %rax\n ; \u2502\u2514\n ; \u2502\u250c @ int.jl:54 within `*'\n \timulq\t%rdi, %rax\n \tshlq\t$2, %rax\n ; \u2502\u2514\n \tretq\n \tnopw\t%cs:(%rax,%rax)\n ; \u2514\n\n\n## Genericity\n(from https://www.youtube.com/watch?v=kc9HwsxE1OY)\n\n\n```julia\nusing LinearAlgebra\n\n# First we write a generic function (that takes an array A and a list of vectors \n# and sums the inner products of A and each vector)\nfunction inner_sum(A,vs)\n t = zero(eltype(A))\n for v in vs\n t += inner(v,A,v)\n end\n return t\nend\n\n# we define the inner product used in our inner_sum function\ninner(v,A,w) = dot(v,A*w)\n\n## Now we want a new type, the famous One Hot Vector\n\nimport Base: *\n\nstruct OneHotVector <: AbstractVector{Bool}\n idx::UInt32\n len::UInt32\nend\n\nBase.size(xs::OneHotVector) = (Int64(xs.len),)\nBase.getindex(xs::OneHotVector, i::Integer) = i == xs.idx\nBase.getindex(xs::OneHotVector, ::Colon) = OneHotVector(xs.idx, xs.len)\n```\n\n\n```julia\n# It works (generic) ! but it's slow\nv = [OneHotVector(i,1000) for i in rand(1:1000,100)]\nA=rand(1000,1000)\n\ninner_sum(A,v)\n@time inner_sum(A,v)\n\n# And now specify to get speed !\nA::AbstractMatrix * b::OneHotVector = A[:, b.idx]\ninner(v::OneHotVector,A,w::OneHotVector) = A[v.idx,w.idx] # How to do that in single dispatch ??\n\ninner_sum(A,v)\n@time inner_sum(A,v)\n```\n\n 0.037617 seconds (105 allocations: 793.922 KiB)\n 0.000003 seconds (5 allocations: 176 bytes)\n\n\n\n\n\n 51.52316795511521\n\n\n\n## From prototype to performance\n\n\n```julia\na=rand(100_000_000)\n\nfunction simple_sum(a)\n res = 0\n for i in eachindex(a)\n res += a[i]\n end\n return res\nend\n\nsimple_sum(a)\n@time simple_sum(a) # naive implementation\n\nfunction simd_sum(a)\n res = zero(eltype(a)) # note the type instability of res=0\n @simd for i in eachindex(a)\n @inbounds res += a[i]\n end\n return res\nend\n\nsimd_sum(a)\n@time simd_sum(a)\nsum(a)\n@time sum(a)\n\nusing PyCall\nnp = pyimport(\"numpy\")\nnp.sum(a)\n@time np.sum(a)\n\nusing Test\n@test np.sum(a)\u2248simd_sum(a)\n```\n\n 0.215574 seconds (5 allocations: 176 bytes)\n 0.080315 seconds (5 allocations: 176 bytes)\n 0.069825 seconds (5 allocations: 176 bytes)\n 0.046125 seconds (15 allocations: 688 bytes)\n\n\n\n\n\n \u001b[32m\u001b[1mTest Passed\u001b[22m\u001b[39m\n\n\n\n## An ecosystem of packages\n\n- Most of packages are available on Github (easy collaboration)\n- Main fields are grouped in Github Organizations (see https://julialang.org/community/)\n- Julia comes with a powerful package/environment manager\n\n\n\n```julia\nusing Pkg #load the Pkg stdlib\nPkg.activate(\".\") #activate the local environment\n```\n\n\n\n\n \"/home/raphael/Projets/Perso/tests-julia/julia-intro/Project.toml\"\n\n\n\n## Solving Differential Equations\n\n Solve the Lorenz equations:\n \n\n$$\n\\begin{align}\n\\frac{dx}{dt} &= \u03c3(y-x) \\\\\n\\frac{dy}{dt} &= x(\u03c1-z) - y \\\\\n\\frac{dz}{dt} &= xy - \u03b2z \\\\\n\\end{align}\n$$\n\n
                                        \nFair Warning \u26a0 : the next cell is gonna take some time. Like really. You can go grab a coffee.\n\nThis compilation time only occurs the first time you load this version of this package on your machine. \nIf you are trying this online on Mybinder, remember that each time you connect, you are on a fresh julia install \njust built for you, so this precompilation time occurs. And Plots and DifferentialEquations are pretty big libraries (DifEq regroups a lot of solvers).\n
                                        \n\n\n```julia\n#Pkg.add(\"DifferentialEquations\") # add the Differential equation suite\nusing DifferentialEquations # first time is very slow (precompilation)\nusing Plots\ngr()\n```\n\n\n\n\n Plots.GRBackend()\n\n\n\n\n```julia\nfunction lorenz(du,u,p,t)\n du[1] = 10.0*(u[2]-u[1])\n du[2] = u[1]*(28.0-u[3]) - u[2]\n du[3] = u[1]*u[2] - (8/3)*u[3]\nend\n\nu0 = [1.0;0.0;0.0] ; tspan = (0.0,100.0);\nprob = ODEProblem(lorenz,u0,tspan)\nsol = DifferentialEquations.solve(prob)\nPlots.plot(sol,vars=(1,2,3))\n```\n\n\n\n\n \n\n \n\n\n\n## Optimization\n\n\n```julia\nusing JuMP, GLPK, Test\n\n\"\"\"\nFormulate and solve a simple LP:\n max 5x + 3y\n st 1x + 5y <= 3\n 0 <= x <= 2\n 0 <= y <= 30\n\"\"\"\nfunction example_basic()\n model = Model(with_optimizer(GLPK.Optimizer))\n\n @variable(model, 0 <= x <= 2)\n @variable(model, 0 <= y <= 30)\n\n @objective(model, Max, 5x + 3y)\n @constraint(model, 1x + 5y <= 3.0)\n\n println(model)\n JuMP.optimize!(model)\n\n obj_value = JuMP.objective_value(model)\n x_value = JuMP.value(x)\n y_value = JuMP.value(y)\n\n println(\"Objective value: \", obj_value)\n println(\"x = \", x_value)\n println(\"y = \", y_value)\n @test obj_value \u2248 10.6\n @test x_value \u2248 2\n @test y_value \u2248 0.2\n\nend\n\nexample_basic()\n\n```\n\n \u250c Info: Precompiling GLPK [60bf3e95-4087-53dc-ae20-288a0d20c6a6]\n \u2514 @ Base loading.jl:1186\n\n\n Max 5 x + 3 y\n Subject to\n x + 5 y \u2264 3.0\n x \u2265 0.0\n y \u2265 0.0\n x \u2264 2.0\n y \u2264 30.0\n \n Objective value: 10.6\n x = 2.0\n y = 0.2\n\n\n\n\n\n \u001b[32m\u001b[1mTest Passed\u001b[22m\u001b[39m\n\n\n\n## Composability\n\nSee https://tutorials.juliadiffeq.org/html/type_handling/02-uncertainties.html\n\n\n```julia\nusing DifferentialEquations, Measurements, Plots\n\ng = 9.79 \u00b1 0.02; # Gravitational constants\nL = 1.00 \u00b1 0.01; # Length of the pendulum\n\n#Initial Conditions\nu\u2080 = [0 \u00b1 0, \u03c0 / 60 \u00b1 0.01] # Initial speed and initial angle\ntspan = (0.0, 6.3)\n\n#Define the problem\nfunction simplependulum(du,u,p,t)\n \u03b8 = u[1]\n d\u03b8 = u[2]\n du[1] = d\u03b8\n du[2] = -(g/L)*\u03b8\nend\n\n#Pass to solvers\nprob = ODEProblem(simplependulum, u\u2080, tspan)\nsol = DifferentialEquations.solve(prob, Tsit5(), reltol = 1e-6)\n\n# Analytic solution\nu = u\u2080[2] .* cos.(sqrt(g / L) .* sol.t)\n\nplot(sol.t, getindex.(sol.u, 2), label = \"Numerical\")\nplot!(sol.t, u, label = \"Analytic\")\n```\n\n\n\n\n \n\n \n\n\n\n## Automatic differentiation\n\n\n\n\n```julia\n#Pkg.add(\"ForwardDiff\")\nusing ForwardDiff\n\n# Define a function f\nf(x::Vector) = sum(sin, x) + prod(tan, x) * sum(sqrt, x);\n\nx = rand(5) # small size for example's sake\n\n# Get the gradient of f\ng = x -> ForwardDiff.gradient(f, x); # g = \u2207f\n\n@show g(x)\n\n# Get the Hessian\nForwardDiff.hessian(f, x)\n\n```\n\n g(x) = [2.57287, 2.51714, 2.3224, 1.87393, 1.8107]\n\n\n\n\n\n 5\u00d75 Array{Float64,2}:\n 1.39246 4.80673 4.31075 3.52545 3.74384\n 4.80673 1.37213 4.18168 3.41991 3.63178\n 4.31075 4.18168 1.32308 3.06703 3.25711\n 3.52545 3.41991 3.06703 1.78856 2.66366\n 3.74384 3.63178 3.25711 2.66366 2.8778 \n\n\n\n# Deep learning\n\nClassic MNIST number classification with Flux.jl\n\n\n```julia\nusing Flux, Flux.Data.MNIST, Statistics\nusing Flux: onehotbatch, onecold, crossentropy, throttle\nusing Base.Iterators: repeated\n# using CuArrays\n\n# Classify MNIST digits with a simple multi-layer-perceptron\n\n# Load images\nimgs = MNIST.images()\n# Stack images into one large batch\nX = hcat(float.(reshape.(imgs, :))...) |> gpu\n\nlabels = MNIST.labels()\n# One-hot-encode the labels\nY = onehotbatch(labels, 0:9) |> gpu\n\n# Define the neural network (two Dense layers)\nm = Chain(\n Dense(28^2, 32, relu),\n Dense(32, 10),\n softmax) |> gpu\n\n# define loss function and metric\nloss(x, y) = crossentropy(m(x), y)\naccuracy(x, y) = mean(onecold(m(x)) .== onecold(y))\n\n# create batches of 100\ndataset = repeated((X, Y), 100)\n# Define callback that show the current loss\nevalcb = () -> @show(loss(X, Y))\n\nopt = ADAM()\n\n# Train\nFlux.train!(loss, params(m), dataset, opt, cb = throttle(evalcb, 10))\n@show accuracy(X, Y)\n# Test set accuracy\ntX = hcat(float.(reshape.(MNIST.images(:test), :))...) |> gpu\ntY = onehotbatch(MNIST.labels(:test), 0:9) |> gpu\n@show accuracy(tX, tY)\n```\n\n loss(X, Y) = 2.2864406f0 (tracked)\n loss(X, Y) = 1.495227f0 (tracked)\n loss(X, Y) = 1.018613f0 (tracked)\n loss(X, Y) = 0.7433482f0 (tracked)\n loss(X, Y) = 0.60261893f0 (tracked)\n loss(X, Y) = 0.51817137f0 (tracked)\n loss(X, Y) = 0.46358913f0 (tracked)\n loss(X, Y) = 0.41939938f0 (tracked)\n loss(X, Y) = 0.39135918f0 (tracked)\n accuracy(X, Y) = 0.8994333333333333\n accuracy(tX, tY) = 0.9029\n\n\n\n\n\n 0.9029\n\n\n\n### Prediction\n\nPredict the number for a given image\n\n\n```julia\nn = rand(1:length(MNIST.images(:test)))\nprint(\"Prediction:\",argmax(tY[:,n])-1)\nheatmap(MNIST.images(:test)[n])\n\n```\n\n Prediction:5\n\n\n\n\n \n\n \n\n\n\n## Deep learning + Autodiff\n\n=> Scientific Machine learning\n\nhttp://www.stochasticlifestyle.com/the-essential-tools-of-scientific-machine-learning-scientific-ml/\n\nhttps://fluxml.ai/2019/03/05/dp-vs-rl.html\n\nhttps://fluxml.ai/2019/02/07/what-is-differentiable-programming.html\n\nhttps://www.youtube.com/watch?v=FGfx8CQHdQA\n\n\n## Ecosystem\n\nAnd much more :\n\n- JuliaRobotics (real-time !)\n- Stats/Machine Learning\n- Web\n- GPU\n- ...\n\n\n\n# Interop\n\nInterop possible avec Python, Matlab, R, Java, C/Fortran,...\n\n\n```julia\n#Pkg.add(\"PyCall\") # add python binding package\n\nusing PyCall \n@pyimport math # import python math library\n@show math.sin(math.pi / 4) \n@show sin(pi / 4) ; #Julia native sinus function\n\n```\n\n math.sin(math.pi / 4) = 0.7071067811865475\n sin(pi / 4) = 0.7071067811865475\n\n\n\n```julia\n# call C\nt = ccall((:clock, \"libc.so.6\"), Int32, ())\n```\n\n\n\n\n 854653866\n\n\n\n### Macros: code that transforms code\n\n\n```julia\n# @time macro inserts timing and memory profiling into expression, then evaluates, and prints\n@time f(2//3)\n```\n\n 0.076290 seconds (114.51 k allocations: 6.258 MiB, 24.08% gc time)\n\n\n\n\n\n 8//9\n\n\n\n\n```julia\n# @which macro determines which function is called, provides link to source code on GitHub\n@which exp(\u03c0)\n```\n\n\n\n\nexp(x::Real) in Base.Math at special/exp.jl:73\n\n\n\n\n```julia\n@which exp(\u03c0*im)\n```\n\n\n\n\nexp(z::Complex) in Base at complex.jl:604\n\n\n\nMacros enable run-time code generation and transformation. \n\nApplications :\n\n * generation and execution of boilerplate code\n * run-time generation and optimization of algorithms, e.g. FFTW, ATLAS\n * symbolic mathematics, automatic differentiation\n * *all written like high-level Python, running like compiled C !!!*\n\n## Parallelism in Julia: just change the array type\n\nSome very trivial examples of Julia's built-in parallelism\n\n\n```julia\nusing Distributed\n#Pkg.add(\"DistributedArrays\")\n# add 4 process\naddprocs(4);\n```\n\n\n```julia\n; cat codes/count_heads.jl\n```\n\n function count_heads(n)\n c::Int = 0\n for i=1:n\n c += rand(Bool)\n end\n c\n end\n\n\n\n```julia\n# define function on all process\n@everywhere include(\"codes/count_heads.jl\")\n\n# dispatch two tasks\na = @spawn count_heads(10000000)\nb = @spawn count_heads(10000000)\n(fetch(a)+fetch(b))/20000000 # Get proba of drawing a head\n```\n\n\n\n\n 0.49999545\n\n\n\n### Parallel loops with reduction\n\n\n```julia\n# distribute the simulation on all process and sum results\nnheads = @distributed (+) for i=1:200000000\n Int(rand(Bool))\nend \nnheads/200000000\n```\n\n\n\n\n 0.499957515\n\n\n\nAnd more :\n- Distributed arrays\n- GPUArrays\n- TPUArrays ...\n- Nested Threading (PARTR)\n\nJust changing the behavior of the underlying structure can bring new hardware support for all packages\n\n# Helpful materials\n\n \n- Main site https://julialang.org/\n- Docs https://docs.julialang.org/en/v1/\n- Courses : https://juliaacademy.com/\n- Forum https://discourse.julialang.org/\n- Book https://benlauwens.github.io/ThinkJulia.jl/latest/book.html\n- Blog http://www.stochasticlifestyle.com/\n- All-in-one package : https://juliacomputing.com/products/juliapro.html\n- Try online : https://juliabox.com/\n\n\n# Code as a first-class citizen product of research\n\n- The (new version) Julia package manager has reproductibility at its core (each code project is linked to a deterministic set of dependencies)\n- Creating a Julia package comes with tools for documentation, unit testing, continuous integration\n- New scientific collaborations can be based on software (see the Github organizations such as JuliaDiff, JuliaStats, etc...)\n\n\n\n# Food for thoughts\n\n- Cost and open source\n - Matlab licences cost several 10K\u20ac each year to some labs\n - Julia (and Python and R) come for free but development is not free\n - A part of Matlab costs could go to financing open source software that is critical for science (see e.g. https://bitbucket.org/paugier/etude-asso-pynumfr/src/default/etude_asso_python_sciences_fr.rst?fileviewer=file-view-default)\n- The combo C(++) low-level libraries and high level bindings (as Tensorflow, Keras...) lead to black box workflows...\n\n# Conclusions\n\nJulia\n* **Easy as Matlab, fast as Fortran, flexible as Python, deep as Lisp**\n* Scientific computing, from interactive exploration to HPC\n* Paradigms (multiple dispatch) that enforce collaboration\n\nNot covered\n* large-scale programming, development ecosystem, environments, debuggers, etc.\n* Abstract Types, compositions, ...\n* rough edges: plotting, static compilation (not quite there), package loading times, young 1.x\n\n*Thanks* : the Julia community for most of the examples, xkcd\n\n \nJulia website: http://www.julialang.org, this talk: https://github.com/raphbacher/julia-intro\n\n", "meta": {"hexsha": "b75a0d3558f8cb50e51393a280b408d0a327434a", "size": 411641, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "presentation_julia.ipynb", "max_stars_repo_name": "raphbacher/julia-intro", "max_stars_repo_head_hexsha": "ecf7d683b46a75fa2214edf55e9873ee57de3834", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "presentation_julia.ipynb", "max_issues_repo_name": "raphbacher/julia-intro", "max_issues_repo_head_hexsha": "ecf7d683b46a75fa2214edf55e9873ee57de3834", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "presentation_julia.ipynb", "max_forks_repo_name": "raphbacher/julia-intro", "max_forks_repo_head_hexsha": "ecf7d683b46a75fa2214edf55e9873ee57de3834", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 85.8837888588, "max_line_length": 244, "alphanum_fraction": 0.6488517908, "converted": true, "num_tokens": 7434, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.546738151984614, "lm_q2_score": 0.7772998611746912, "lm_q1q2_score": 0.42497948963654764}} {"text": "# Analysis tools\n\nBefore we start, we copy the atmospheric setup case from the [\"Getting Started\"](getting_started.html) example:\n\nFirst setting up the Radtrans object:\n\n\n```python\nimport numpy as np\nfrom petitRADTRANS import Radtrans\n\natmosphere = Radtrans(line_species = ['H2O', 'CO_all_iso', 'CH4', 'CO2', 'Na', 'K'], \\\n rayleigh_species = ['H2', 'He'], \\\n continuum_opacities = ['H2-H2', 'H2-He'], \\\n wlen_bords_micron = [0.3, 15])\n\npressures = np.logspace(-6, 2, 100)\natmosphere.setup_opa_structure(pressures)\n```\n\n \n Read CIA opacities for H2-H2...\n Read CIA opacities for H2-He...\n Done.\n \n\n\n
                                        \n\n**Units in petitRADTRANS:** remember that all units in *petitRADTRANS* are in cgs, **except for pressure**, which is in bars, **and the mean molecular weight (MMW)**, which is in units of atomic mass units.\n
                                        \n\nAnd the atmospheric parameters:\n\n\n```python\nfrom petitRADTRANS import nat_cst as nc\nR_pl = 1.838*nc.r_jup_mean\ngravity = 1e1**2.45\nP0 = 0.01\n\nkappa_IR = 0.01\ngamma = 0.4\nT_int = 200.\nT_equ = 1500.\ntemperature = nc.guillot_global(pressures, kappa_IR, gamma, gravity, T_int, T_equ)\n\nmass_fractions = {}\nmass_fractions['H2'] = 0.74 * np.ones_like(temperature)\nmass_fractions['He'] = 0.24 * np.ones_like(temperature)\nmass_fractions['H2O'] = 0.001 * np.ones_like(temperature)\nmass_fractions['CO_all_iso'] = 0.01 * np.ones_like(temperature)\nmass_fractions['CO2'] = 0.00001 * np.ones_like(temperature)\nmass_fractions['CH4'] = 0.000001 * np.ones_like(temperature)\nmass_fractions['Na'] = 0.00001 * np.ones_like(temperature)\nmass_fractions['K'] = 0.000001 * np.ones_like(temperature)\n\nMMW = 2.33 * np.ones_like(temperature)\n```\n\n
                                        \n\n**Abundances in petitRADTRANS:** remember that abundances in petitCODE are in units of **mass fractions**, not number fractions (aka volume mixing ratio, VMR). You can convert between mass fractions and VMRs by using\n\\begin{equation}\nX_i = \\frac{\\mu_i}{\\mu}n_i,\n\\end{equation}\nwhere $X_i$ is the mass fraction of species $i$, $\\mu_i$ the mass of a single molecule/atom/ion/... of species $i$, $\\mu$ is the atmospheric mean molecular weight, and $n_i$ is the VMR of species $i$.\n\n
                                        \n\n## Transmission contribution functions\n\nWe calculate the transmission spectrum in the usual way, this time setting the ``contribution = True`` keyword argument, however. This will additionally measure the contribution of the different layers, by calculating as many transmission spectra as there are layers, iteratively turning off the opacity in one layer only. The difference to the nominal transmission spectrum then measures the influence of the respective layers. Note that calculating the contribution function will increase the computation time considerably. We plan to improve the calculation speed of the contribution function soon. The formal definition of the contribution function is (also see Molli\u00e8re et al., in prep.):\n\n\\begin{equation}\nC_{\\rm tr}^{i} = \\frac{R_{\\rm nom}^2-R^2(\\kappa_i=0)}{\\sum_{j=1}^{N_{\\rm L}}\\left[R_{\\rm nom}^2-R^2(\\kappa_j=0)\\right]},\n\\end{equation}\n\nwhere $R_{\\rm nom}$ is the nominal transmission radius of the planet and $R(\\kappa_i=0)$ is the transmission radius obtained from setting the opacity in the $i$th layer to zero. $N_{\\rm L}$ is the number of atmospheric layers.\n\nNow, to the contribution function calculation:\n\n\n```python\natmosphere.calc_transm(temperature, mass_fractions, \\\n gravity, MMW, R_pl=R_pl, P0_bar=P0, \\\n contribution = True)\n```\n\nThe transmission contribution function is plotted below, one can see that pressures above 0.5 bar cannot be probed in the wavelength range studied here.\n\n\n```python\nimport pylab as plt\nplt.rcParams['figure.figsize'] = (10, 6)\n\nwlen_mu = nc.c/atmosphere.freq/1e-4\nX, Y = np.meshgrid(wlen_mu, pressures)\nplt.contourf(X,Y,atmosphere.contr_tr,30,cmap=plt.cm.bone_r)\n\nplt.yscale('log')\nplt.xscale('log')\nplt.ylim([1e2,1e-6])\nplt.xlim([np.min(wlen_mu),np.max(wlen_mu)])\n\nplt.xlabel('Wavelength (microns)')\nplt.ylabel('P (bar)')\nplt.title('Transmission contribution function')\nplt.show()\nplt.clf()\n```\n\n## Emission contribution functions\n\nBelow, we show the same for the emission contribution functions, which are defined in the usual way, that is measuring the fraction of flux a layer contributes to the total flux, at a given wavelength. The computational time is comparable to a normal emission spectrum.\n\n\n```python\natmosphere.calc_flux(temperature, mass_fractions, \\\n gravity, MMW, \\\n contribution = True)\n```\n\nThe emission contribution function is plotted below, one can see that pressures that pressures larger than 1 bar can now be probed.\n\n\n```python\nimport pylab as plt\nplt.rcParams['figure.figsize'] = (10, 6)\n\nwlen_mu = nc.c/atmosphere.freq/1e-4\nX, Y = np.meshgrid(wlen_mu, pressures)\nplt.contourf(X,Y,atmosphere.contr_em,30,cmap=plt.cm.bone_r)\n\nplt.yscale('log')\nplt.xscale('log')\nplt.ylim([1e2,1e-6])\nplt.xlim([np.min(wlen_mu),np.max(wlen_mu)])\n\nplt.xlabel('Wavelength (microns)')\nplt.ylabel('P (bar)')\nplt.title('Emission contribution function')\nplt.show()\nplt.clf()\n```\n\nOne also sees that scattering is not included in the pRT emission spectrum here, blueward of the strong alkali lines in the optical, quite large pressures can be probed. Conversely, in the transmission contribution plot above, the Rayleigh scattering is clearly visible. Hence, we will turn on scattering in the calculation below to show its impact on the spectra.\n\n
                                        \n\n**Scattering and petitRADTRANS:** remember that scattering is included for emission spectra in petitRADTRANS only if requested specifically when generating the Radtrans object, as it increases the runtime (see [\"Scattering for Emission Spectra\"](emis_scat.html) for an example how to do this).\n
                                        \n\nFirst we reload the pRT object with scattering turned on:\n\n\n```python\natmosphere = Radtrans(line_species = ['H2O', 'CO_all_iso', 'CH4', 'CO2', 'Na', 'K'], \\\n rayleigh_species = ['H2', 'He'], \\\n continuum_opacities = ['H2-H2', 'H2-He'], \\\n wlen_bords_micron = [0.3, 15], \\\n do_scat_emis = True)\n\npressures = np.logspace(-6, 2, 100)\natmosphere.setup_opa_structure(pressures)\n```\n\n \n Read CIA opacities for H2-H2...\n Read CIA opacities for H2-He...\n Done.\n \n\n\nNow we recalculate and plot the emission contribution function:\n\n\n```python\natmosphere.calc_flux(temperature, mass_fractions, \\\n gravity, MMW, \\\n contribution = True)\n\nimport pylab as plt\nplt.rcParams['figure.figsize'] = (10, 6)\n\nwlen_mu = nc.c/atmosphere.freq/1e-4\nX, Y = np.meshgrid(wlen_mu, pressures)\nplt.contourf(X,Y,atmosphere.contr_em,30,cmap=plt.cm.bone_r)\n\nplt.yscale('log')\nplt.xscale('log')\nplt.ylim([1e2,1e-6])\nplt.xlim([np.min(wlen_mu),np.max(wlen_mu)])\n\nplt.xlabel('Wavelength (microns)')\nplt.ylabel('P (bar)')\nplt.title('Emission contribution function, *with* scattering')\nplt.show()\nplt.clf()\n```\n\nAs can be seen, the Rayleigh scattering contribution to the emitted flux leaving the atmosphere is clearly visible now.\n", "meta": {"hexsha": "71a20297cbc23f77c1bdf01e18f2ecb6c0582466", "size": 149752, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/content/notebooks/analysis.ipynb", "max_stars_repo_name": "nborsato/petitRADTRANS", "max_stars_repo_head_hexsha": "2df983bc46b892486b1b035d7c6933ab46f0d36c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/content/notebooks/analysis.ipynb", "max_issues_repo_name": "nborsato/petitRADTRANS", "max_issues_repo_head_hexsha": "2df983bc46b892486b1b035d7c6933ab46f0d36c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/content/notebooks/analysis.ipynb", "max_forks_repo_name": "nborsato/petitRADTRANS", "max_forks_repo_head_hexsha": "2df983bc46b892486b1b035d7c6933ab46f0d36c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 339.5736961451, "max_line_length": 47260, "alphanum_fraction": 0.9351661414, "converted": true, "num_tokens": 1953, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587586, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.4249111580202287}} {"text": " Trusted Notebook\" width=\"500 px\" align=\"left\">\n\n# Getting Started with Qiskit\n\nHere, we provide an overview of working with Qiskit. Qiskit provides the basic building blocks necessary to program quantum computers. The foundation of Qiskit is the Terra element. The basic concept of Qiskit Terra is an array of quantum circuits. A workflow using Terra consists of two stages: **Build** and **Execute**. **Build** allows you to make different quantum circuits that represent the problem you are solving, and **Execute** allows you to run them on different backends. After the jobs have been run, the data is collected. There are methods for putting this data together, depending on the program. This either gives you the answer you wanted, or allows you to make a better program for the next instance.\n\n\n**Contents**\n\n[Circuit basics](#circuit_basics)\n\n[Simulating circuits with Qiskit Aer](#aer_simulation)\n\n[Running circuits using the IBMQ provider](#ibmq_provider)\n\n**Code imports**\n\n\n```python\nimport numpy as np\nfrom qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister\nfrom qiskit import execute\n```\n\n## Circuit Basics \n\n\n### Building the circuit\n\nThe basic elements needed for your first program are the QuantumCircuit, and QuantumRegister.\n\n\n```python\n# Create a Quantum Register with 3 qubits.\nq = QuantumRegister(3, 'q')\n\n# Create a Quantum Circuit acting on the q register\ncirc = QuantumCircuit(q)\n```\n\n
                                        \nNote: Naming the QuantumRegister is optional and not required.\n
                                        \n\nAfter you create the circuit with its registers, you can add gates (\"operations\") to manipulate the registers. As you proceed through the documentation you will find more gates and circuits; the below is an example of a quantum circuit that makes a three-qubit GHZ state\n\n$$|\\psi\\rangle = \\left(|000\\rangle+|111\\rangle\\right)/\\sqrt{2}.$$\n\nTo create such a state, we start with a 3-qubit quantum register. By default, each qubit in the register is initialized to $|0\\rangle$. To make the GHZ state, we apply the following gates:\n* A Hadamard gate $H$ on qubit 0, which puts it into a superposition state.\n* A controlled-Not operation ($C_{X}$) between qubit 0 and qubit 1.\n* A controlled-Not operation between qubit 0 and qubit 2.\n\nOn an ideal quantum computer, the state produced by running this circuit would be the GHZ state above.\n\nIn Qiskit Terra, operations can be added to the circuit one-by-one, as shown below.\n\n\n```python\n# Add a H gate on qubit 0, putting this qubit in superposition.\ncirc.h(q[0])\n# Add a CX (CNOT) gate on control qubit 0 and target qubit 1, putting\n# the qubits in a Bell state.\ncirc.cx(q[0], q[1])\n# Add a CX (CNOT) gate on control qubit 0 and target qubit 2, putting\n# the qubits in a GHZ state.\ncirc.cx(q[0], q[2])\n```\n\n\n\n\n \n\n\n\n## Visualize Circuit\n\nYou can visualize your circuit using Qiskit Terra `QuantumCircuit.draw()`, which plots circuit in the form found in many textbooks.\n\n\n```python\ncirc.draw()\n```\n\n\n\n\n
                                                \u250c\u2500\u2500\u2500\u2510          \nq_0: |0>\u2524 H \u251c\u2500\u2500\u25a0\u2500\u2500\u2500\u2500\u25a0\u2500\u2500\n        \u2514\u2500\u2500\u2500\u2518\u250c\u2500\u2534\u2500\u2510  \u2502  \nq_1: |0>\u2500\u2500\u2500\u2500\u2500\u2524 X \u251c\u2500\u2500\u253c\u2500\u2500\n             \u2514\u2500\u2500\u2500\u2518\u250c\u2500\u2534\u2500\u2510\nq_2: |0>\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2524 X \u251c\n                  \u2514\u2500\u2500\u2500\u2518
                                        \n\n\n\nIn this circuit, the qubits are put in order with qubit zero at the top and qubit two at the bottom. The circuit is read left-to-right (meaning that gates which are applied earlier in the circuit show up further to the left).\n\n## Simulating circuits using Qiskit Aer \n\nQiskit Aer is our package for simulating quantum circuits. It provides many different backends for doing a simulation. Here we use the basic python version.\n\n### Statevector backend\n\nThe most common backend in Qiskit Aer is the `statevector_simulator`. This simulator returns the quantum \nstate which is a complex vector of dimensions $2^n$ where $n$ is the number of qubits \n(so be careful using this as it will quickly get too large to run on your machine).\n\n
                                        \n\n\nWhen representing the state of a multi-qubit system, the tensor order used in qiskit is different than that use in most physics textbooks. Suppose there are $n$ qubits, and qubit $j$ is labeled as $Q_{j}$. In most textbooks (such as Nielsen and Chuang's \"Quantum Computation and Information\"), the basis vectors for the $n$-qubit state space would be labeled as $Q_{0}\\otimes Q_{1} \\otimes \\cdots \\otimes Q_{n}$. **This is not the ordering used by qiskit!** Instead, qiskit uses an ordering in which the $n^{\\mathrm{th}}$ qubit is on the _left_ side of the tesnsor product, so that the basis vectors are labeled as $Q_n\\otimes \\cdots \\otimes Q_1\\otimes Q_0$.\n\nFor example, if qubit zero is in state 0, qubit 1 is in state 0, and qubit 2 is in state 1, qiskit would represent this state as $|100\\rangle$, whereas most physics textbooks would represent it as $|001\\rangle$.\n\nThis difference in labeling affects the way multi-qubit operations are represented as matrices. For example, qiskit represents a controlled-X ($C_{X}$) operation with qubit 0 being the control and qubit 1 being the target as\n\n$$C_X = \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & 0 \\\\\\end{pmatrix}.$$\n\n
                                        \n\nTo run the above circuit using the statevector simulator, first you need to import Aer and then set the backend to `statevector_simulator`.\n\n\n```python\n# Import Aer\nfrom qiskit import BasicAer\n\n# Run the quantum circuit on a statevector simulator backend\nbackend = BasicAer.get_backend('statevector_simulator')\n```\n\nNow we have chosen the backend it's time to compile and run the quantum circuit. In Qiskit Terra we provide the `execute` function for this. ``execute`` returns a ``job`` object that encapsulates information about the job submitted to the backend.\n\n\n
                                        \nTip: You can obtain the above parameters in Jupyter. Simply place the text cursor on a function and press Shift+Tab.\n
                                        \n\n\n```python\n# Create a Quantum Program for execution \njob = execute(circ, backend)\n```\n\nWhen you run a program, a job object is made that has the following two useful methods: \n`job.status()` and `job.result()` which return the status of the job and a result object respectively.\n\n
                                        \nNote: Jobs run asynchronously but when the result method is called it switches to synchronous and waits for it to finish before moving on to another task.\n
                                        \n\n\n```python\nresult = job.result()\n```\n\nThe results object contains the data and Qiskit Terra provides the method \n`result.get_statevector(circ)` to return the state vector for the quantum circuit.\n\n\n```python\noutputstate = result.get_statevector(circ, decimals=3)\nprint(outputstate)\n```\n\n [0.707+0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j\n 0.707+0.j]\n\n\nQiskit Terra also provides a visualization toolbox to allow you to view these results.\n\nBelow, we use the visualization function to plot the real and imaginary components of the state vector.\n\n\n```python\nfrom qiskit.tools.visualization import plot_state_city\nplot_state_city(outputstate)\n```\n\n### Unitary backend\n\nQiskit Aer also includes a `unitary_simulator` that works _provided all the elements in the circuit are unitary operations_. This backend calculates the $2^n \\times 2^n$ matrix representing the gates in the quantum circuit. \n\n\n```python\n# Run the quantum circuit on a unitary simulator backend\nbackend = BasicAer.get_backend('unitary_simulator')\njob = execute(circ, backend)\nresult = job.result()\n\n# Show the results\nprint(result.get_unitary(circ, decimals=3))\n```\n\n [[ 0.707+0.j 0.707+0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j\n 0. +0.j 0. +0.j]\n [ 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j\n 0.707+0.j -0.707+0.j]\n [ 0. +0.j 0. +0.j 0.707+0.j 0.707+0.j 0. +0.j 0. +0.j\n 0. +0.j 0. +0.j]\n [ 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0.707+0.j -0.707+0.j\n 0. +0.j 0. +0.j]\n [ 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0.707+0.j 0.707+0.j\n 0. +0.j 0. +0.j]\n [ 0. +0.j 0. +0.j 0.707+0.j -0.707+0.j 0. +0.j 0. +0.j\n 0. +0.j 0. +0.j]\n [ 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j\n 0.707+0.j 0.707+0.j]\n [ 0.707+0.j -0.707+0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j\n 0. +0.j 0. +0.j]]\n\n\n### OpenQASM backend\n\nThe simulators above are useful because they provide information about the state output by the ideal circuit and the matrix representation of the circuit. However, a real experiment terminates by _measuring_ each qubit (usually in the computational $|0\\rangle, |1\\rangle$ basis). Without measurement, we cannot gain information about the state. Measurements cause the quantum system to collapse into classical bits. \n\nFor example, suppose we make independent measurements on each qubit of the three-qubit GHZ state\n$$|\\psi\\rangle = |000\\rangle +|111\\rangle)/\\sqrt{2},$$\nand let $xyz$ denote the bitstring that results. Recall that, under the qubit labeling used by Qiskit, $x$ would correspond to the outcome on qubit 2, $y$ to the outcome on qubit 1, and $z$ to the outcome on qubit 0. This representation of the bitstring puts the most significant bit (MSB) on the left, and the least significant bit (LSB) on the right. This is the standard ordering of binary bitstrings. We order the qubits in the same way, which is why Qiskit uses a non-standard tensor product order.\n\nThe probability of obtaining outcome $xyz$ is given by\n$$\\mathrm{Pr}(xyz) = |\\langle xyz | \\psi \\rangle |^{2}.$$\nBy explicit computation, we see there are only two bitstrings that will occur: $000$ and $111$. If the bitstring $000$ is obtained, the state of the qubits is $|000\\rangle$, and if the bitstring is $111$, the qubits are left in the state $|111\\rangle$. The probability of obtaining 000 or 111 is the same; namely, 1/2:\n$$\\begin{align}\n\\mathrm{Pr}(000) &= |\\langle 000 | \\psi \\rangle |^{2} = \\frac{1}{2}\\\\\n\\mathrm{Pr}(111) &= |\\langle 111 | \\psi \\rangle |^{2} = \\frac{1}{2}.\n\\end{align}$$\n\nTo simulate a circuit that includes measurement, we need to add measurements to the original circuit above, and use a different Aer backend.\n\n\n```python\n# Create a Classical Register with 3 bits.\nc = ClassicalRegister(3, 'c')\n# Create a Quantum Circuit\nmeas = QuantumCircuit(q, c)\nmeas.barrier(q)\n# map the quantum measurement to the classical bits\nmeas.measure(q,c)\n\n# The Qiskit circuit object supports composition using\n# the addition operator.\nqc = circ+meas\n\n#drawing the circuit\nqc.draw()\n```\n\n\n\n\n
                                                \u250c\u2500\u2500\u2500\u2510           \u2591       \u250c\u2500\u2510\nq_0: |0>\u2524 H \u251c\u2500\u2500\u25a0\u2500\u2500\u2500\u2500\u25a0\u2500\u2500\u2500\u2591\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2524M\u251c\n        \u2514\u2500\u2500\u2500\u2518\u250c\u2500\u2534\u2500\u2510  \u2502   \u2591    \u250c\u2500\u2510\u2514\u2565\u2518\nq_1: |0>\u2500\u2500\u2500\u2500\u2500\u2524 X \u251c\u2500\u2500\u253c\u2500\u2500\u2500\u2591\u2500\u2500\u2500\u2500\u2524M\u251c\u2500\u256b\u2500\n             \u2514\u2500\u2500\u2500\u2518\u250c\u2500\u2534\u2500\u2510 \u2591 \u250c\u2500\u2510\u2514\u2565\u2518 \u2551 \nq_2: |0>\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2524 X \u251c\u2500\u2591\u2500\u2524M\u251c\u2500\u256b\u2500\u2500\u256b\u2500\n                  \u2514\u2500\u2500\u2500\u2518 \u2591 \u2514\u2565\u2518 \u2551  \u2551 \n c_0: 0 \u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256c\u2550\u2550\u256c\u2550\u2550\u2569\u2550\n                           \u2551  \u2551    \n c_1: 0 \u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256c\u2550\u2550\u2569\u2550\u2550\u2550\u2550\n                           \u2551       \n c_2: 0 \u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2569\u2550\u2550\u2550\u2550\u2550\u2550\u2550\n
                                        \n\n\n\nThis circuit adds a classical register, and three measurements that are used to map the outcome of qubits to the classical bits. \n\nTo simulate this circuit, we use the ``qasm_simulator`` in Qiskit Aer. Each run of this circuit will yield either the bitstring 000 or 111. To build up statistics about the distribution of the bitstrings (to, e.g., estimate $\\mathrm{Pr}(000)$), we need to repeat the circuit many times. The number of times the circuit is repeated can be specified in the ``execute`` function, via the ``shots`` keyword.\n\n\n```python\n# Use Aer's qasm_simulator\nbackend_sim = BasicAer.get_backend('qasm_simulator')\n\n# Execute the circuit on the qasm simulator.\n# We've set the number of repeats of the circuit\n# to be 1024, which is the default.\njob_sim = execute(qc, backend_sim, shots=1024)\n\n# Grab the results from the job.\nresult_sim = job_sim.result()\n```\n\nOnce you have a result object, you can access the counts via the function `get_counts(circuit)`. This gives you the _aggregated_ binary outcomes of the circuit you submitted.\n\n\n```python\ncounts = result_sim.get_counts(qc)\nprint(counts)\n```\n\n {'111': 524, '000': 500}\n\n\nApproximately 50 percent of the time the output bitstring is 000. Qiskit Terra also provides a function `plot_histogram` which allows you to view the outcomes. \n\n\n```python\nfrom qiskit.tools.visualization import plot_histogram\nplot_histogram(counts)\n```\n\nThe estimated outcome probabilities $\\mathrm{Pr}(000)$ and $\\mathrm{Pr}(111)$ are computed by taking the aggregate counts and dividing by the number of shots (times the circuit was repeated). Try changing the ``shots`` keyword in the ``execute`` function and see how the estimated probabilities change.\n\n## Running circuits using the IBMQ provider \n\nTo faciliate access to real quantum computing hardware, we have provided a simple API interface.\nTo access IBMQ devices, you'll need an API token. For the public IBM Q devices, you can generate an API token [here](https://quantumexperience.ng.bluemix.net/qx/account/advanced) (create an account if you don't already have one). For Q Network devices, login to the q-console, click your hub, group, and project, and expand \"Get Access\" to generate your API token and access url.\n\nOur IBMQ provider lets you run your circuit on real devices or on our HPC simulator. Currently, this provider exists within Qiskit, and can be imported as shown below. For details on the provider, see [The IBMQ Provider](the_ibmq_provider.ipynb).\n\n\n```python\nfrom qiskit import IBMQ\n```\n\nAfter generating your API token, call, `IBMQ.save_account('MY_TOKEN')`. For Q Network users, you'll also need to include your access url: `IBMQ.save_account('MY_TOKEN', 'URL')`\n\nThis will store your IBMQ credentials in a local file. Unless your registration information has changed, you only need to do this once. You may now load your accounts by calling,\n\n\n```python\nIBMQ.load_accounts()\n```\n\nOnce your account has been loaded, you can view the list of backends available to you.\n\n\n```python\nprint(\"Available backends:\")\nIBMQ.backends()\n```\n\n Available backends:\n\n\n\n\n\n [,\n ,\n ,\n ,\n ]\n\n\n\n### Running circuits on real devices\n\nToday's quantum information processors are small and noisy, but are advancing at a fast pace. They provide a great opportunity to explore what [noisy, intermediate-scale quantum (NISQ)](https://arxiv.org/abs/1801.00862) computers can do.\n\nThe IBMQ provider uses a queue to allocate the devices to users. We now choose a device with the least busy queue which can support our program (has at least 3 qubits).\n\n\n```python\nfrom qiskit.providers.ibmq import least_busy\n\nlarge_enough_devices = IBMQ.backends(filters=lambda x: x.configuration().n_qubits > 4 and\n not x.configuration().simulator)\nbackend = least_busy(large_enough_devices)\nprint(\"The best backend is \" + backend.name())\n```\n\n The best backend is ibmq_20_tokyo\n\n\nTo run the circuit on the backend, we need to specify the number of shots and the number of credits we are willing to spend to run the circuit. Then, we execute the circuit on the backend using the ``execute`` function.\n\n\n```python\nfrom qiskit.tools.monitor import job_monitor\nshots = 1024 # Number of shots to run the program (experiment); maximum is 8192 shots.\nmax_credits = 3 # Maximum number of credits to spend on executions. \n\njob_exp = execute(qc, backend=backend, shots=shots, max_credits=max_credits)\njob_monitor(job_exp)\n```\n\n\n HTML(value=\"

                                        Job Status: job is being initialized

                                        \")\n\n\n``job_exp`` has a ``.result()`` method that lets us get the results from running our circuit.\n\n
                                        \nNote: When the .result() method is called, the code block will wait until the job has finished before releasing the cell.\n
                                        \n\n\n```python\nresult_exp = job_exp.result()\n```\n\nLike before, the counts from the execution can be obtained using ```get_counts(qc)``` \n\n\n```python\ncounts_exp = result_exp.get_counts(qc)\nplot_histogram([counts_exp,counts])\n```\n\n### Simulating circuits using a HPC simulator\n\nThe IBMQ provider also comes with a remote optimized simulator called ``ibmq_qasm_simulator``. This remote simulator is capable of simulating up to 32 qubits. It can be used the \nsame way as the remote real backends. \n\n\n```python\nbackend = IBMQ.get_backend('ibmq_qasm_simulator', hub=None)\n```\n\n\n```python\nshots = 1024 # Number of shots to run the program (experiment); maximum is 8192 shots.\nmax_credits = 3 # Maximum number of credits to spend on executions. \n\njob_hpc = execute(qc, backend=backend, shots=shots, max_credits=max_credits)\n```\n\n\n```python\nresult_hpc = job_hpc.result()\n```\n\n\n```python\ncounts_hpc = result_hpc.get_counts(qc)\nplot_histogram(counts_hpc)\n```\n\n### Retrieving a previously ran job\n\nIf your experiment takes longer to run then you have time to wait around, or if you simply want to retrieve old jobs back, the IBMQ backends allow you to do that.\nFirst you would need to note your job's ID:\n\n\n```python\njobID = job_exp.job_id()\n\nprint('JOB ID: {}'.format(jobID)) \n```\n\n JOB ID: 5c1a7275199f9700512a2d0f\n\n\nGiven a job ID, that job object can be later reconstructed from the backend using retrieve_job:\n\n\n```python\njob_get=backend.retrieve_job(jobID)\n```\n\nand then the results can be obtained from the new job object. \n\n\n```python\njob_get.result().get_counts(qc)\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "6facd8b1686f724ceb4d12daaa87bb582b51a751", "size": 230925, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "qiskit/basics/getting_started_with_qiskit_terra.ipynb", "max_stars_repo_name": "liupibm/qiskit-tutorials", "max_stars_repo_head_hexsha": "33c26f1bffefec7bd41d460e4c6d4266e73641be", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-12-30T11:30:52.000Z", "max_stars_repo_stars_event_max_datetime": "2018-12-30T11:30:52.000Z", "max_issues_repo_path": "qiskit/basics/getting_started_with_qiskit_terra.ipynb", "max_issues_repo_name": "liupibm/qiskit-tutorials", "max_issues_repo_head_hexsha": "33c26f1bffefec7bd41d460e4c6d4266e73641be", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2019-04-12T07:43:25.000Z", "max_issues_repo_issues_event_max_datetime": "2020-02-07T13:32:18.000Z", "max_forks_repo_path": "qiskit/basics/getting_started_with_qiskit_terra.ipynb", "max_forks_repo_name": "liupibm/qiskit-tutorials", "max_forks_repo_head_hexsha": "33c26f1bffefec7bd41d460e4c6d4266e73641be", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 236.6034836066, "max_line_length": 152936, "alphanum_fraction": 0.9123135217, "converted": true, "num_tokens": 5091, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.4249111580202286}} {"text": "

                                        Table of Contents

                                        \n
                                        \n\n\n\n# Approximate Methods for Solving One-Particle Schrödinger Equations\nUp to this point, we've focused on systems for which we can solve the Schrödinger equation. Unfortunately, there are very few such systems, and their relevance for real chemical systems is very limited. This motivates the approximate methods for solving the Schrödinger equation. One must be careful, however, if one makes poor assumptions, the results of approximate methods can be very poor. Conversely, with appropriate insight, approximation techniques can be extremely useful. \n\n## Expansion in a Basis\nWe have seen the eigenvectors of a Hermitian operator are a complete basis, and can be chosen to be orthonormal. We have also seen how a wavefunction can be expanded in a basis,\n$$\n\\Psi(x) = \\sum_{k=0}^{\\infty} c_k \\phi_k(x)\n$$\nNote that there is no requirement that the basis set, $\\{\\phi_k(x) \\}$ be eigenvectors of a Hermitian operator: all that matters is that the basis set is complete. For real problems, of course, one can choose only a finite number of basis functions, \n$$\n\\Psi(x) \\approx \\sum_{k=0}^{N_{\\text{basis}}} c_k \\phi_k(x)\n$$\nbut as the number of basis functions, $N_{\\text{basis}}$, increases, results should become increasingly accurate. \n\nSubstituting this expression for the wavefunction into the time-independent Schrödinger equation, \n$$\n\\hat{H} \\Psi(x) = \\hat{H} \\sum_{k=0}^{\\infty} c_k \\phi_k(x) = E \\sum_{k=0}^{\\infty} c_k \\phi_k(x)\n$$\nMultiplying on the left by $\\left(\\phi_j(x) \\right)^*$ and integrating over all space, \n$$\n \\sum_{k=0}^{\\infty} \\left[\\int \\left(\\phi_j(x) \\right)^* \\hat{H} \\phi_k(x) dx \\right] c_k \n = E \\sum_{k=0}^{\\infty}\\left[ \\int \\left(\\phi_j(x) \\right)^* \\phi_k(x) dx\\right] c_k\n$$\nAt this stage we usually define the Hamiltonian matrix, $\\mathbf{H}$, as the matrix with elements\n$$\nh_{jk} = \\int \\left(\\phi_j(x) \\right)^* \\hat{H} \\phi_k(x) dx \n$$\nand the overlap matrix, $\\mathbf{S}$ as the matrix with elements\n$$\ns_{jk} = \\int \\left(\\phi_j(x) \\right)^* \\phi_k(x) dx\n$$\nIf the basis is orthonormal, then the overlap matrix is equal to the identity matrix, $\\mathbf{S} = \\mathbf{I}$ and its elements are therefore given by the Kronecker delta, $s_{jk} = \\delta_{jk}$.\n\nThe Schrödinger equation therefore can be written as a [generalized matrix eigenvalue problem](https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix#Generalized_eigenvalue_problem):\n$$\n\\mathbf{Hc}=E\\mathbf{Sc}\n$$\nor, in element-wise notation, as:\n$$\n \\sum_{k=0}^{\\infty} h_{jk} c_k \n = E \\sum_{k=0}^{\\infty} s_{jk} c_k\n$$\nIn the special case where the basis functions are orthogonormal, $\\mathbf{S} = \\mathbf{I}$ and this is an ordinary [matrix eigenvalue problem](https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#Eigenvalues_and_eigenvectors_of_matrices),\n$$\n\\mathbf{Hc}=E\\mathbf{c}\n$$\nor, in element-wise notation, as:\n$$\n \\sum_{k=0}^{\\infty} h_{jk} c_k \n = E c_j\n$$\n\n### Solving the Secular Equation\nIn the context of quantum chemistry, the generalized eigenvalue problem \n$$\n\\mathbf{Hc}=E\\mathbf{Sc}\n$$\nis called the *secular equation*. To solve the secular equation:\n1. Choose a basis, $\\{|\\phi_k\\rangle \\}$ and a basis-set size, $N_{\\text{basis}}$\n1. Evaluate the matrix elements of the Hamiltonian and the overlap matrix\n\\begin{align}\nh_{jk} &= \\int \\left(\\phi_j(x) \\right)^* \\hat{H} \\phi_k(x) dx \\qquad \\qquad 0 \\le j,k \\le N_{\\text{basis}} \\\\\ns_{jk} &= \\int \\left(\\phi_j(x) \\right)^* \\phi_k(x) dx \n\\end{align}\n1. Solve the generalized eigenvalue problem\n$$\n \\sum_{k=0}^{\\infty} h_{jk} c_k \n = E \\sum_{k=0}^{\\infty} s_{jk} c_k\n$$\n\nBecause of the variational principle, the lowest eigenvalue will always be greater than or equal to the true ground-state energy. \n\n### Example for the Particle-in-a-Box\nAs an example, consider an electron confined to a box with length 2 Bohr, stretching from $x=-1$ to $x=1$. We know that the exact energy of this system is \n$$E=\\tfrac{(\\pi n)^2}{8}$$ \nThe exact wavefunctions are easily seen to be \n$$\\psi_n(x) = \n\\begin{cases}\n\\cos\\left(\\tfrac{n \\pi x}{2}\\right) & n=1,3,5,\\ldots \\\\\n\\sin\\left(\\tfrac{n \\pi x}{2}\\right) & n=2,4,6,\\ldots \n\\end{cases}\n$$ \n\nHowever, for pedagogical purposes, suppose we did not know these answers. We know that the wavefunction will be zero at $x= \\pm1$, so we might hypothesize a basis like:\n$$\n\\phi_k(x) = (x-1)(x+1)x^k = x^{k+2} - x^{k} \\qquad \\qquad k=0,1,2,\\ldots\n$$\nThe overlap matrix elements are\n\n\\begin{align}\ns_{jk} &= \\int_{-1}^{1} \\left(\\phi_j(x) \\right)^* \\phi_k(x) dx \\\\\n&= \\int_{-1}^{1} \\left(x^{j+2}-x^{j}\\right) \\left(x^{k+2} - x^{k}\\right) dx \\\\\n&= \\int_{-1}^{1} \\left(x^{j+k+4}+x^{j+k} - 2 x^{j+k+2}\\right) dx \\\\\n&= \\left[\\frac{x^{k+j+5}}{k+j+5} + \\frac{x^{k+j+1}}{k+j+1} \n- 2\\frac{x^{k+j+3}}{k+j+3} \\right]_{-1}^{+1}\n\\end{align}\n\nThis integral is zero when $k+j$ is odd. Specifically,\n$$\ns_{jk} = \n\\begin{cases}\n 0 & j+k \\text{ is odd}\\\\\n 2\\left(\\frac{1}{k+j+5} - \\frac{2}{k+j+3} + \\frac{1}{k+j+1} \\right) & j+k \\text{ is even}\n\\end{cases}\n$$\nand the Hamiltonian matrix elements are\n\n\\begin{align}\nh_{jk} &= \\int_{-1}^{1} \\left(\\phi_j(x) \\right)^* \\hat{H} \\phi_k(x) dx \\\\\n&= \\int_{-1}^{1} \\left(x^{j+2}-x^{j}\\right) \\left(-\\tfrac{1}{2}\\tfrac{d^2}{dx^2}\\right) \\left(x^{k+2} - x^{k}\\right) dx \\\\\n&= -\\tfrac{1}{2}\\int_{-1}^{1} \\left(x^{j+2}-x^{j}\\right) \\left((k+2)(k+1)x^{k} - (k)(k-1)x^{k-2}\\right) dx \\\\\n&= -\\tfrac{1}{2}\\int_{-1}^{1} \\left((k+2)(k+1)x^{k+j+2} + (k)(k-1)x^{k+j-2} -\\left[(k+2)(k+1) + k(k-1) \\right]x^{k+j} \\right) dx \\\\\n&= -\\tfrac{1}{2}\\left[\\left(\\frac{(k+2)(k+1)}{k+j+3}x^{k+j+3} + \\frac{(k)(k-1)}{k+j-1}x^{k+j-1} \n- \\frac{(k+2)(k+1) + k(k-1)}{k+j+1}x^{k+j+1} \\right) \\right]_{-1}^{+1}\n\\end{align}\n\nThis integral is also zero when $k+j$ is odd. Specifically,\n$$\nh_{jk} = \n\\begin{cases}\n 0 & j+k \\text{ is odd}\\\\\n 2\\left(\\frac{(k+2)(k+1)}{k+j+3} - \\frac{(k+2)(k+1) + k(k-1)}{k+j+1} + \\frac{(k)(k-1)}{k+j-1} \\right) & j+k \\text{ is even}\n\\end{cases}\n$$\n\n\n```python\nimport numpy as np\nfrom scipy.linalg import eigh\nimport matplotlib.pyplot as plt\n\n\ndef compute_energy_ground_state(n_basis):\n \"\"\"Compute ground state energy by solving the Secular equations.\"\"\"\n # assign S & H to zero matrices\n s = np.zeros((n_basis, n_basis))\n h = np.zeros((n_basis, n_basis))\n\n # loop over upper-triangular elements & compute S & H elements\n for j in range(0, n_basis):\n for k in range(j, n_basis):\n if (j + k) % 2 == 0:\n s[j, k] = s[k, j] = 2 * (1 / (k + j + 5) - 2 / (k + j + 3) + 1 / (k + j + 1))\n h[j, k] = h[k, j] = -1 * (((k + 2) * (k + 1)) / (k + j + 3) - ((k + 2) * (k + 1) + k * (k - 1)) / (k + j + 1) + (k**2 - k) / (k + j - 1))\n \n # solve Hc = ESc to get eigenvalues E\n e_vals = eigh(h, s, eigvals_only=True)\n return e_vals[0]\n\n\n# plot basis set convergence of Secular equations\n# -----------------------------------------------\n\n# evaluate energy for a range of basis functions\nn_values = np.arange(2, 11, 1)\ne_values = np.array([compute_energy_ground_state(n) for n in n_values])\nexpected_energy = (1 * np.pi)**2 / 8.\n\nplt.rcParams['figure.figsize'] = [15, 8]\nfig, axes = plt.subplots(1, 2)\nfig.suptitle(\"Basis Set Convergence of Secular Equations\", fontsize=24, fontweight='bold')\n\nfor index, axis in enumerate(axes.ravel()):\n if index == 0:\n # plot approximate & exact energy\n axis.plot(n_values, e_values, marker='o', linestyle='--', label='Approximate')\n axis.plot(n_values, np.repeat(expected_energy, len(n_values)), marker='', linestyle='-', label='Exact')\n # set axes labels\n axis.set_xlabel(\"Number of Basis Functions\", fontsize=12, fontweight='bold')\n axis.set_ylabel(\"Ground-State Energy [a.u.]\", fontsize=12, fontweight='bold')\n axis.legend(frameon=False, fontsize=14)\n else:\n # plot log of approximate energy error (skip the last two values because they are zero)\n axis.plot(n_values[:-2], np.log10(e_values[:-2] - expected_energy), marker='o', linestyle='--')\n # set axes labels\n axis.set_xlabel(\"Number of Basis Functions\", fontsize=12, fontweight='bold')\n axis.set_ylabel(\"Log10 (Ground-State Energy Error [a.u.])\", fontsize=12, fontweight='bold')\n\nplt.show()\n```\n\n### Particle-in-a-Box with Jacobi polynomials\nSimilar results can be obtained with different basis functions. It is often convenient to use an orthonormal basis, where $s_{jk} = \\delta_{jk}$. For the particle-in-a-box with $-1 \\le x \\le 1$, one such set of basis functions can be constructed from the (normalized) [Jacobi polynomials](https://en.wikipedia.org/wiki/Jacobi_polynomials),\n$$\n\\phi_j(x) = N_j(1-x)(1+x)P_j^{(2,2)}(x)\n$$\nwhere $N_j$ is the normalization constant\n$$\nN_j = \\sqrt{\\frac{(2j+5)(j+4)(j+3)}{32(j+2)(j+1)}}\n$$\nTo evaluate the Hamiltonian it is useful to know that:\n\\begin{align}\n\\frac{d^2\\phi_j(x)}{dx^2} &= N_j\n\\left(-2 P_j^{(2,2)}(x) - 4x \\frac{d P_j^{(2,2)}(x)}{dx} + (1-x)(1+x)\\frac{d^2 P_j^{(2,2)}(x)}{dx^2} \\right) \\\\\n&= N_j\n\\left(-2 P_j^{(2,2)}(x) - 4x \\frac{j+5}{2} P_{j-1}^{(3,3)}(x) + (1-x^2)\\frac{(j+5)(j+6)}{4}P_{j-2}^{(4,4)}(x) \\right) \n\\end{align}\nThe Hamiltonian matrix elements could be evaluated analytically, but the expression is pretty complicated. It's easier to merely evaluate them numerically as:\n$$\nh_{jk} = -\\frac{1}{2}N_j N_k \\int_{-1}^1 (1-x)(1+x) P_k^{(2, 2)}(x) \\left(-2 P_j^{(2,2)}(x) - 4x \\frac{j+5}{2} P_{j-1}^{(3,3)}(x) + (1-x^2)\\frac{(j+5)(j+6)}{4}P_{j-2}^{(4,4)}(x) \\right) dx \n$$\n\n\n\n\n```python\nimport numpy as np\nfrom scipy.linalg import eigh\nfrom scipy.special import eval_jacobi\nfrom scipy.integrate import quad\nimport matplotlib.pyplot as plt\n\n\ndef compute_energy_ground_state(n_basis):\n \"\"\"Compute ground state energy for a particle-in-a-Box with Jacobi basis.\"\"\"\n \n def normalization(i):\n return np.sqrt((2 * i + 5) * (i + 4) * (i + 3) / (32 * (i + 2) * (i + 1)))\n \n def phi_squared(x, j):\n return (normalization(j) * (1 - x) * (1 + x) * eval_jacobi(j, 2, 2, x))**2\n \n def integrand(x, j, k):\n term = -2 * eval_jacobi(j, 2, 2, x)\n if j - 1 >= 0:\n term -= 2 * x * (j + 5) * eval_jacobi(j - 1, 3, 3, x)\n if j - 2 >= 0:\n term += 0.25 * (1 - x**2) * (j + 5) * (j + 6) * eval_jacobi(j - 2, 4, 4, x)\n return (1 - x) * (1 + x) * eval_jacobi(k, 2, 2, x) * term\n \n # assign H to a zero matrix\n h = np.zeros((n_basis, n_basis))\n\n # compute H elements\n for j in range(n_basis):\n for k in range(n_basis):\n integral = quad(integrand, -1.0, 1.0, args=(j, k))[0]\n h[j, k] = -0.5 * normalization(j) * normalization(k) * integral\n\n # solve Hc = Ec to get eigenvalues E\n e_vals = eigh(h, None, eigvals_only=True)\n return e_vals[0]\n\n\n# plot basis set convergence of particle-in-a-Box with Jacobi basis\n# -----------------------------------------------------------------\n\n# evaluate energy for a range of basis functions\nn_values = np.arange(2, 11, 1)\ne_values = np.array([compute_energy_ground_state(n) for n in n_values])\nexpected_energy = (1 * np.pi)**2 / 8.\n\nplt.rcParams['figure.figsize'] = [15, 8]\nfig, axes = plt.subplots(1, 2)\nfig.suptitle(\"Basis Set Convergence of Particle-in-a-Box with Jacobi Basis\", fontsize=24, fontweight='bold')\n\nfor index, axis in enumerate(axes.ravel()):\n if index == 0:\n # plot approximate & exact energy\n axis.plot(n_values, e_values, marker='o', linestyle='--', label='Approximate')\n axis.plot(n_values, np.repeat(expected_energy, len(n_values)), marker='', linestyle='-', label='Exact')\n # set axes labels\n axis.set_xlabel(\"Number of Basis Functions\", fontsize=12, fontweight='bold')\n axis.set_ylabel(\"Ground-State Energy [a.u.]\", fontsize=12, fontweight='bold')\n axis.legend(frameon=False, fontsize=14)\n else:\n # plot log of approximate energy error (skip the last two values because they are zero)\n axis.plot(n_values[:-2], np.log10(e_values[:-2] - expected_energy), marker='o', linestyle='--')\n # set axes labels\n axis.set_xlabel(\"Number of Basis Functions\", fontsize=12, fontweight='bold')\n axis.set_ylabel(\"Log10 (Ground-State Energy Error [a.u.])\", fontsize=12, fontweight='bold')\n\nplt.show()\n```\n\n#### 🤔 Thought-Provoking Question: Why does adding odd-order polynomials to the basis set not increase the accuracy for the ground state wavefunction. \nHint: The ground state wavefunction is an even function.\nA function is said to be even if it is symmetric about the origin, $f(x) = f(-x)$. A function is said to be odd if it is antisymmetric around the origin, $f(x) = - f(-x)$. Even-degree polynomials (e.g., $1, x^2, x^4, \\ldots$) are even functions; odd-degree polynomials (e.g.; $x, x^3, x^5, \\ldots$) are odd functions. $\\cos(ax)$ is an even function and $\\sin(ax)$ is an odd function. $\\cosh(ax)$ is an even function and $\\sinh(ax)$ is an odd function. In addition,\n- A linear combination of odd functions is also odd.\n- A linear combination of even functions is also even.\n- The product of two odd functions is even.\n- The product of two even functions is even.\n- The product of an odd and an even function is odd.\n- The integral of an odd function from $-a$ to $a$ is always zero.\n- The integral of an even function from $-a$ to $a$ is always twice the value of its integral from $0$ to $a$; it is also twice its integral from $-a$ to $0$.\n- The first derivative of an even function is odd.\n- The first derivative of an odd function is even.\n- The k-th derivative of an even function is odd if k is odd, and even if k is even.\n- The k-th derivative of an odd function is even if k is odd, and odd if k is even.\n\nThese properties of odd and even functions are often very useful. In particular, the first and second properties indicate that if you know that the exact wavefunction you are looking for is odd (or even), it will be a linear combination of basis functions that are odd (or even). E.g., odd basis functions are useless for approximating even eigenfunctions.\n\n#### 🤔 Thought-Provoking Question: Why does one get exactly the same results for the Jacobi polynomials and the simpler $(1-x)(1+x)x^k$ polynomials? \nHint: Can you rewrite one set of polynomials as a linear combination of the others?\n\n\n\n## Perturbation Theory\nIt is not uncommon that a Hamiltonian for which the Schrödinger equation is difficult to solve is \"close\" to another Hamiltonian that is easier to solve. In such cases, one can attempt to solve the easier problem, then *perturb* the system towards the actual, more difficult to solve, system of interest. The idea of leveraging easy problems to solve difficult problems is the essence of [perturbation theory](https://en.wikipedia.org/wiki/Perturbation_theory).\n\n\n### The Perturbed Hamiltonian\nSuppose that for some Hamiltonian, $\\hat{H}$, we know the eigenfunctions and eigenvalues,\n$$\n\\hat{H} |\\psi_k \\rangle = E_k |\\psi_k \\rangle\n$$\nHowever, we are not interested in this Hamiltonian, but a different Hamiltonian, $\\tilde{H}$, which we can write as:\n$$\n\\tilde{H} = \\hat{H} + \\hat{V}\n$$\nwhere obviously\n$$\n\\hat{V} = \\tilde{H} - \\hat{H}\n$$\n\nLet us now define a family of perturbed Hamiltonians, \n$$\n\\hat{H}(\\lambda) = \\hat{H} + \\lambda \\hat{V}\n$$\nwhere obviously:\n$$\n\\hat{H}(\\lambda) = \n\\begin{cases}\n \\hat{H} & \\lambda = 0\\\\\n \\tilde{H} & \\lambda = 1\n\\end{cases}\n$$\nWriting the Schrödinger equation for $\\hat{H}_\\lambda$, we have:\n$$\n\\hat{H}(\\lambda) |\\psi_k(\\lambda) \\rangle = E_k(\\lambda) |\\psi_k(\\lambda) \\rangle \n$$\nThis equation holds true for all values of $\\lambda$. Since we know the answer for $\\lambda = 0$, and we *assume* that the perturbed system described by $\\tilde{H}$ is close enough to $\\hat{H}$ for the solution at $\\lambda =0$ to be useful, we will write the expand the energy and wavefunction as [Taylor-MacLaurin series](https://en.wikipedia.org/wiki/Taylor_series)\n\n\\begin{align}\nE_k(\\lambda) &= E_k(\\lambda=0) + \\lambda \\left[\\frac{dE_k}{d \\lambda} \\right]_{\\lambda=0} \n+ \\frac{\\lambda^2}{2!} \\left[\\frac{d^2E_k}{d \\lambda^2} \\right]_{\\lambda=0} \n+ \\frac{\\lambda^3}{3!} \\left[\\frac{d^3E_k}{d \\lambda^3} \\right]_{\\lambda=0} + \\cdots \\\\\n|\\psi_k(\\lambda) \\rangle &= |\\psi_k(\\lambda=0) \\rangle + \\lambda \\left[\\frac{d|\\psi_k \\rangle}{d \\lambda} \\right]_{\\lambda=0} \n+ \\frac{\\lambda^2}{2!} \\left[\\frac{d^2|\\psi_k \\rangle}{d \\lambda^2} \\right]_{\\lambda=0} \n+ \\frac{\\lambda^3}{3!} \\left[\\frac{d^3|\\psi_k \\rangle}{d \\lambda^3} \\right]_{\\lambda=0} + \\cdots \n\\end{align}\n\nWhen we write this, we are implicitly assuming that the derivatives all exist, which is not true if the zeroth-order state is degenerate (unless the perturbation does not break the degeneracy).\n\nIf we insert these expressions into the Schrödinger equation for $\\hat{H}(\\lambda)$, we obtain a polynomial of the form:\n$$\n0=p(\\lambda)= a_0 + a_1 \\lambda + a_2 \\lambda^2 + a_3 \\lambda^3 + \\cdots\n$$\nThis equation can only be satisfied for *all* $\\lambda$ if all its terms are zero, so \n$$\n0 = a_0 = a_1 = a_2 = \\cdots\n$$ \nThe key equations that need to be solved are listed below. First there is the zeroth-order equation, which is automatically satisfied:\n$$\n0 = a_0 = \\left( \\hat{H}(0) - E_k(0) \\right) | \\psi_k(0) \\rangle \n$$\nThe first-order equation is:\n$$\n0 = a_1 = \\left( \\hat{H}(0) - E_k(0) \\right) \\left[\\frac{d|\\psi_k \\rangle}{d \\lambda} \\right]_{\\lambda=0} +\\left(\\hat{V} - \\left[\\frac{dE_k}{d \\lambda} \\right]_{\\lambda=0}\\right)|\\psi_k(\\lambda=0) \\rangle \n$$\nThe second-order equation is:\n$$\n0 = a_2 = \\tfrac{1}{2} \\left( \\hat{H}(0) - E_k(0) \\right) \\left[\\frac{d^2|\\psi_k \\rangle}{d \\lambda^2} \\right]_{\\lambda=0} +\\left(\\hat{V} - \\left[\\frac{dE_k}{d \\lambda} \\right]_{\\lambda=0}\\right)\\left[\\frac{d|\\psi_k \\rangle}{d \\lambda} \\right]_{\\lambda=0}\n-\\tfrac{1}{2} \\left[\\frac{d^2E_k}{d \\lambda^2} \\right]_{\\lambda=0} |\\psi_k(\\lambda=0) \\rangle \n$$\nHigher-order equations are increasingly complicated, but still tractable in some cases. One usually applies perturbation theory only when the perturbation is relatively small, which usually suffices to ensure that the Taylor series expansion converges rapidly and higher-order terms are relatively insignificant.\n\n### Hellmann-Feynman Theorem\nThe [Hellmann-Feynman theorem](https://en.wikipedia.org/wiki/Hellmann%E2%80%93Feynman_theorem) has been discovered many times, most impressively by [Richard Feynman](https://en.wikipedia.org/wiki/Richard_Feynman), who included it in his undergraduate senior thesis. In simple terms:\n> **Hellmann-Feynman Theorem**: Suppose that the Hamiltonian, $\\hat{H}(\\lambda)$ depends on a parameter. Then the first-order change in the energy with respect to the parameter is given by the equation,\n$$\n\\left[\\frac{dE}{d\\lambda}\\right]_{\\lambda = \\lambda_0} = \\int \\left( \\psi(\\lambda_0;x)\\right)^* \\left[\\frac{d\\hat{H}}{d \\lambda} \\right]_{\\lambda = \\lambda_0}\\psi(\\lambda_0;x) \\; dx\n$$\n\n#### Derivation of the Hellmann-Feynman Theorem by Differentiation Under the Integral Sign\nThe usual way to derive the Hellmann-Feynman theorem uses the technique of [differentiation under the integral sign](https://en.wikipedia.org/wiki/Leibniz_integral_rule). Therefore, \n$$\n\\frac{dE}{d\\lambda} = \\frac{d}{d\\lambda}\\int \\left( \\psi(\\lambda;x)\\right)^* \\hat{H}\\psi(\\lambda;x) \\; dx = \\int \\frac{d\\left( \\psi(\\lambda;x)\\right)^* \\hat{H}\\psi(\\lambda;x) }{d\\lambda}\\; dx\n$$\nWhile such an operation is not always mathematically permissible, it is usually permissible, as should be clear from the definition of the derivative as a limit of a difference,\n$$\n\\left[\\frac{dE}{d\\lambda}\\right]_{\\lambda = \\lambda_0} = \\lim_{h\\rightarrow0} \\frac{E(\\lambda_0 + h) - E(\\lambda_0)}{h}\n$$\nand the fact that the integral of a sum is the sum of the integrals. Using the product rule for derivatives, one obtains:\n\n\\begin{align}\n\\frac{dE}{d\\lambda} &= \\int \\frac{d\\left( \\psi(\\lambda;x)\\right)^* \\hat{H}\\psi(\\lambda;x) }{d\\lambda}\\; dx \\\\\n&=\\int \n \\frac{\\left(\\psi(\\lambda;x)\\right)^*}{d\\lambda} \\hat{H} \\psi(\\lambda;x) \n+ \\left( \\psi(\\lambda;x)\\right)^* \\frac{d\\hat{H}}{d \\lambda}\\psi(\\lambda;x) \n+ \\left( \\psi(\\lambda;x)\\right)^* \\hat{H} \\frac{d\\psi(\\lambda;x)}{d\\lambda}\n\\; dx \\\\\n&=\\int \n \\frac{\\left(\\psi(\\lambda;x)\\right)^*}{d\\lambda} E(\\lambda) \\psi(\\lambda;x) \n+ \\left( \\psi(\\lambda;x)\\right)^* \\frac{d\\hat{H}}{d \\lambda}\\psi(\\lambda;x) \n+ \\left( \\psi(\\lambda;x)\\right)^* E(\\lambda) \\frac{d\\psi(\\lambda;x)}{d\\lambda}\n\\; dx \\\\\n&=E(\\lambda) \\int \n+ \\frac{\\left(\\psi(\\lambda;x)\\right)^*}{d\\lambda} \\psi(\\lambda;x) \n+ \\left( \\psi(\\lambda;x)\\right)^* \\frac{d\\psi(\\lambda;x)}{d\\lambda}\n\\; dx\n+\\int \n\\left( \\psi(\\lambda;x)\\right)^* \\frac{d\\hat{H}}{d \\lambda}\\psi(\\lambda;x) \n\\; dx \\\\\n&=\\int \n\\left( \\psi(\\lambda;x)\\right)^* \\frac{d\\hat{H}}{d \\lambda}\\psi(\\lambda;x) \n\\; dx \n\\end{align}\n\nIn the third-from-last line we used the eigenvalue relation and the Hermitian property of the Hamiltonian; in the last step we have used the fact that the wavefunctions are normalized and the fact that the derivative of a constant is zero to infer that the terms involving the wavefunction derivatives vanish. Specifically, we used:\n\\begin{align}\n\\int &\\left(\\left[\\frac{d\\left( \\psi(\\lambda_0;x)\\right)^*}{d \\lambda}\\right]_{\\lambda = \\lambda_0} \\psi(\\lambda_0;x) \n+ \\left( \\psi(\\lambda_0;x)\\right)^* \\left[\\frac{d \\psi(\\lambda_0;x)}{d \\lambda}\\right]_{\\lambda = \\lambda_0}\\right) \\; dx \\\\\n&= \\left[\\frac{d}{d \\lambda} \\int \\left( \\psi(\\lambda_0;x)\\right)^* \\psi(\\lambda_0;x) \\; dx \\right]_{\\lambda = \\lambda_0}\\\\\n&= \\frac{d 1}{d\\lambda} \\\\\n&= 0\n\\end{align}\n\n#### Derivation of the Hellmann-Feynman Theorem from First-Order Perturbation Theory\n\nStarting with the equation from first-order perturbation theory,\n$$\n0 = a_1 = \\left( \\hat{H}(0) - E_k(0) \\right) \\left[\\frac{d|\\psi_k \\rangle}{d \\lambda} \\right]_{\\lambda=0} +\\left(\\hat{V} - \\left[\\frac{dE_k}{d \\lambda} \\right]_{\\lambda=0}\\right)|\\psi_k(0) \\rangle \n$$\nmultiply on the left-hand-side by $\\langle \\psi_k(0) |$. (I.e., multiply by $\\psi_k(0;x)^*$ and integrate.) Then: \n$$\n0 = \\langle \\psi_k(0) |\\left( \\hat{H}(0) - E_k(0) \\right) \\left[\\frac{d|\\psi_k \\rangle}{d \\lambda} \\right]_{\\lambda=0} +\\langle \\psi_k(0) |\\left(\\hat{V} - \\left[\\frac{dE_k}{d \\lambda} \\right]_{\\lambda=0}\\right)|\\psi_k(0) \\rangle \n$$\nBecause the Hamiltonian is Hermitian, the first term is zero. The second term can be rearranged to give the Hellmann-Feynman theorem, \n$$\n\\left[\\frac{dE_k}{d \\lambda} \\right]_{\\lambda=0} \\langle \\psi_k(0) |\\psi_k(0) \\rangle = \\langle \\psi_k(0) |\\hat{V}|\\psi_k(0) \\rangle = \\langle \\psi_k(0) |\\left[\\frac{d\\hat{H}}{d\\lambda} \\right]_{\\lambda=0}|\\psi_k(0) \\rangle\n$$\n\n### Perturbed Wavefunctions\nTo determine the change in the wavefunction, \n$$\\psi_k'(\\lambda) = \\frac{d |\\psi_k\\rangle}{d\\lambda}$$\nit is helpful to adopt the convention of intermediate normalization, whereby\n$$\n\\langle \\psi_k(0) | \\psi_k(\\lambda) \\rangle = 1\n$$\nfor all $\\lambda$. Inserting the series expansion for $|\\psi(\\lambda) \\rangle$ one finds that \n\\begin{align}\n1 &= \\langle \\psi_k(0) | \\psi_k(0) \\rangle + \\lambda \\langle \\psi_k(0) | \\psi_k'(0) \\rangle + \\tfrac{\\lambda^2}{2!} \\langle \\psi_k(0) | \\psi_k''(0) \\rangle + \\cdots \\\\\n1 &= 1 + \\lambda \\langle \\psi_k(0) | \\psi_k'(0) \\rangle + \\tfrac{\\lambda^2}{2!} \\langle \\psi_k(0) | \\psi_k''(0) \\rangle + \\cdots \n\\end{align} \nwhere in the second line we have used the normalization of the zeroth-order wavefunction, $\\langle \\psi_k(0) | \\psi_k(0) \\rangle = 1$. Since this equation holds for all $\\lambda$, it must be that\n$$\n0=\\langle \\psi_k(0) | \\psi_k'(0) \\rangle\\\\\n0=\\langle \\psi_k(0) | \\psi_k''(0) \\rangle\\\\\n\\vdots\n$$\nBecause the eigenfunctions of $\\hat{H}(0)$ are a complete basis, we can expand $ | \\psi_k'(0) \\rangle$ as:\n$$\n | \\psi_k'(0) \\rangle = \\sum_{j=0}^{\\infty} c_j | \\psi_j(0) \\rangle\n$$\nbut because $\\langle \\psi_k(0) | \\psi_k'(0) \\rangle=0$, it must be that $c_k = 0$. So:\n$$\n | \\psi_k'(0) \\rangle = \\sum_{j=0\\\\\n j \\ne k}^{\\infty} c_j | \\psi_j(0) \\rangle\n$$\nWe insert this expansion into the expression from first-order perturbation theory:\n$$\n0 = \\left( \\hat{H}(0) - E_k(0) \\right) \\sum_{j=0\\\\\n j \\ne k}^{\\infty} c_j | \\psi_j(0) \\rangle +\\left(\\hat{V} - \\left[\\frac{dE_k}{d \\lambda} \\right]_{\\lambda=0}\\right)|\\psi_k(0) \\rangle \n$$\nand multiply on the left by $\\langle \\psi_l(0) |$, with $l \\ne k$. \n\\begin{align}\n0 &= \\langle \\psi_l(0) |\\left( \\hat{H}(0) - E_k(0) \\right) \\sum_{j=0\\\\\n j \\ne k}^{\\infty} c_j | \\psi_j(0) \\rangle +\\langle \\psi_l(0) |\\left(\\hat{V} - \\left[\\frac{dE_k}{d \\lambda} \\right]_{\\lambda=0}\\right)|\\psi_k(0) \\rangle \\\\\n&= \\sum_{j=0\\\\ j \\ne k}^{\\infty} c_j\\langle \\psi_l(0) |\\left( E_l(0) - E_k(0) \\right) | \\psi_j(0) \\rangle +\\langle \\psi_l(0) |\\hat{V} |\\psi_k(0) \\rangle - \\left[\\frac{dE_k}{d \\lambda} \\right]_{\\lambda=0}\\langle \\psi_l(0) |\\psi_k(0) \\rangle \\\\\n&= \\sum_{j=0\\\\ j \\ne k}^{\\infty} c_j \\left( E_l(0) - E_k(0) \\right) \\delta_{lj} +\\langle \\psi_l(0) |\\hat{V} |\\psi_k(0) \\rangle \\\\\n&=c_l \\left( E_l(0) - E_k(0) \\right)+\\langle \\psi_l(0) |\\hat{V} |\\psi_k(0) \\rangle \n\\end{align}\n\nAssuming that the k-th state is nondegenerate (so that we can safely divide by $ E_l(0) - E_k(0)$), \n$$\nc_l = \\frac{\\langle \\psi_l(0) |\\hat{V} |\\psi_k(0) \\rangle }{E_k(0) - E_l(0)}\n$$\nand so:\n$$\n | \\psi_k'(0) \\rangle = \\sum_{j=0\\\\\n j \\ne k}^{\\infty} \\frac{\\langle \\psi_j(0) |\\hat{V} |\\psi_k(0) \\rangle }{E_k(0) - E_j(0)} | \\psi_j(0) \\rangle\n$$\n\nHigher-order terms can be determined in a similar way, but we will only deduce the expression for the second-order energy change. Using the second-order terms from the perturbation expansion,\n$$\n0 = a_2 = \\tfrac{1}{2} \\left( \\hat{H}(0) - E_k(0) \\right) |\\psi_k''(0) \\rangle +\\left(\\hat{V} - E_k'(0)\\right)|\\psi_k'(0) \\rangle-\\tfrac{1}{2} E_k''(0) |\\psi_k(0) \\rangle \n$$\nProjecting this expression against $\\langle \\psi_k(0) |$, one has:\n\\begin{align}\n0 &= \\tfrac{1}{2} \\langle \\psi_k(0) |\\left( \\hat{H}(0) - E_k(0) \\right) |\\psi_k''(0) \\rangle +\\langle \\psi_k(0) |\\left(\\hat{V} - E_k'(0)\\right)|\\psi_k'(0) \\rangle-\\tfrac{1}{2} \\langle \\psi_k(0) |E_k''(0) |\\psi_k(0) \\rangle \\\\\n &= \\tfrac{1}{2} \\langle \\psi_k(0) |\\left(E_k(0) - E_k(0) \\right) |\\psi_k''(0) \\rangle \n+\\langle \\psi_k(0) |\\hat{V} |\\psi_k'(0) \\rangle\n-E_k'(0)\\langle \\psi_k(0) |\\psi_k'(0) \\rangle\n-\\tfrac{1}{2} E_k''(0) \\\\\n &= \\langle \\psi_k(0) |\\hat{V} |\\psi_k'(0) \\rangle\n-\\tfrac{1}{2} E_k''(0) \n\\end{align}\nTo obtain the last line we used the intermediate normalization of the perturbed wavefunction, $\\langle \\psi_k(0) | \\psi_k'(0) \\rangle = 0$. Rewriting the expression for the second-order change in the energy, and then inserting the expression for the first-order wavefunction, gives\n\\begin{align}\n E_k''(0) &= 2\\langle \\psi_k(0) |\\hat{V} |\\psi_k'(0) \\rangle \\\\\n &= 2\\langle \\psi_k(0) |\\hat{V} \\sum_{j=0\\\\\n j \\ne k}^{\\infty} \\frac{\\langle \\psi_j(0) |\\hat{V} |\\psi_k(0) \\rangle }{E_k(0) - E_j(0)} | \\psi_j(0) \\rangle \\\\\n &= 2 \\sum_{j=0\\\\j \\ne k}^{\\infty}\\frac{\\langle \\psi_j(0) |\\hat{V} |\\psi_k(0) \\rangle \\langle \\psi_k(0) |\\hat{V} | \\psi_j(0) \\rangle}{E_k(0) - E_j(0)} \\\\\n &= 2 \\sum_{j=0\\\\j \\ne k}^{\\infty}\\frac{ \\left|\\langle \\psi_j(0) |\\hat{V} |\\psi_k(0) \\rangle \\right|^2}{E_k(0) - E_j(0)} \n\\end{align}\nNotice that for the ground state ($k=0$), where $E_0 - E_{j>0} < 0$, the second-order energy change is never positive, $ E_0''(0) \\le 0$.\n\n### The Law of Diminishing Returns and Accelerating Losses\nSuppose one is given a Hamiltonian that is parameterized in the general form used in perturbation theory, \n$$\n\\hat{H}(\\lambda) = \\hat{H}(0) + \\lambda \\hat{V}\n$$\nAccording to the Hellmann-Feynman theorem, I have:\n$$\n\\frac{dE_0}{d\\lambda} = E_0'(\\lambda) = \\langle \\psi(\\lambda) | \\hat{V} |\\psi(\\lambda) \\rangle\n$$\nConsider two distinct values for the perturbation parameter, $\\lambda_1 < \\lambda_2$. According to the variational principle, if one evaluates the expectation value of $\\hat{H}(\\lambda_1)$ with $\\psi(\\lambda_2)$ one will obtain an energy above the true ground-state energy. I.e., \n$$\nE_0(\\lambda_1) = \\langle \\psi(\\lambda_1) | \\hat{H}(\\lambda_1) |\\psi(\\lambda_1) \\rangle < \\langle \\psi(\\lambda_2) | \\hat{H}(\\lambda_1) |\\psi(\\lambda_2) \\rangle\n$$\nOr, more explicitly, \n$$\n\\langle \\psi(\\lambda_1) | \\hat{H}(0) +\\lambda_1\\hat{V} |\\psi(\\lambda_1) \\rangle < \\langle \\psi(\\lambda_2) | \\hat{H}(0) +\\lambda_1\\hat{V} |\\psi(\\lambda_2) \\rangle\n$$\nSimilarly, the energy expectation value $\\hat{H}(\\lambda_2)$ evaluated with $\\psi(\\lambda_1)$ is above the true ground-state energy, so\n$$\n\\langle \\psi(\\lambda_2) | \\hat{H}(0) +\\lambda_2\\hat{V} |\\psi(\\lambda_2) \\rangle < \\langle \\psi(\\lambda_1) | \\hat{H}(0) +\\lambda_2\\hat{V} |\\psi(\\lambda_1) \\rangle\n$$\nAdding these two inequalities and cancelling out the factors of $\\langle \\psi(\\lambda_2) | \\hat{H}(0) |\\psi(\\lambda_2) \\rangle $ that appear on both sides of the inequality, one finds that:\n$$\n\\left(\\lambda_2 - \\lambda_1 \\right) \\left(\\langle \\psi(\\lambda_2) | \\hat{V} |\\psi(\\lambda_2) \\rangle - \\langle \\psi(\\lambda_1) | \\hat{V} |\\psi(\\lambda_1) \\rangle \\right) < 0\n$$\nor, using the Hellmann-Feynman theorem (in reverse),\n$$\n\\left(\\lambda_2 - \\lambda_1 \\right) \\left( E_0'(\\lambda_2) - E_0'(\\lambda_1)\\right) < 0\n$$\n\nRecall that $\\lambda_2 > \\lambda_1$. Thus $E_0'(\\lambda_2) < E_0'(\\lambda_1)$. If the system is losing energy at $\\lambda_1$ (i.e., $E'(\\lambda_1) < 0$), then at $\\lambda_2$ the system is losing energy even faster ($E_0'(\\lambda_2)$ is more negative than $E_0'(\\lambda_1)$. This is the law of accelerating losses. If the system is gaining energy a $\\lambda_1$ (i.e., $E_0'(\\lambda_1) > 0$), then at $\\lambda_2$ the system is gaining energy more slowly (or even losing energy) ($E_0'(\\lambda_2)$ is smaller than $E_0'(\\lambda_1)$). This is the law of diminishing returns. \n\nIf the energy is a twice-differentiable function of $\\lambda$, then one can infer that the second derivative of the energy is always negative\n$$\n\\lim_{\\lambda_2 \\rightarrow \\lambda_1} \\frac{E_0'(\\lambda_2) - E_0'(\\lambda_1)}{\\lambda_2 - \\lambda_1} = \\left[\\frac{d^2E_0}{d\\lambda^2}\\right]_{\\lambda = \\lambda_1}= E_0''(\\lambda_1)< 0 \n$$\n\n### Example: Particle in a Box with a Sloped Bottom\n#### The Hamiltonian for an Applied Uniform Electric Field\nWhen a system cannot be solved exactly, one can solve it approximately using\n- perturbation theory. \n- variational methods using either an explicit wavefunction form or basis-set expansion.\n\nTo exemplify these approaches, we will use the particle-in-a-box with a sloped bottom. This is obtained when an external electric field is applied to a charged particle in the box. The force on the charged particle due to the field is\n$$\n\\text{force} = \\text{charge} \\cdot \\text{electric field} \n$$\nso for an electron in a box on which an electric field of magnitude $F$ is applied in the $+x$ direction, the force is\n$$\n\\text{force} = -e F\n$$\nwhere $e$ is the magnitude of the charge on the electron. The potential is\n$$\n\\text{potential} = - \\nabla \\text{force}\n$$\nAssuming that the potential is zero at the origin for convenience, $V(0) = 0$, the potential is thus:\n$$\nV(x) = eFx\n$$\n\nThe particle in a box with an applied field has the Hamiltonian\n$$\n\\hat{H} = -\\frac{\\hbar^2}{2m} \\frac{d^2}{dx^2} + V(x) + eFx\n$$\nor, in atomic units,\n$$\n\\hat{H} = -\\tfrac{1}{2} \\tfrac{d^2}{dx^2} + V(x) + Fx\n$$\nFor simplicity, we assume the case where the box has length 2 and is centered at the origin,\n$$\nV(x) =\n\\begin{cases}\n\\infty & x \\le -1 \\\\\n0 & -1 < x < 1 \\\\\n\\infty & 1 \\le x \n\\end{cases}\n$$\n\nFor small electric fields, we can envision solving this system by perturbation theory. We also expect that variational approaches can work well. We'll explore how these strategies can work. It turns out, however, that this system can be [solved exactly](https://www.utupub.fi/bitstream/handle/10024/117904/progradu-tuomas-riihim%C3%A4ki.pdf), though the treatment is far beyond the scope of this course. There are a few useful equations, however: for a field strength of $F=\\tfrac{1}{16}$ the ground-state energy is 1.23356 a.u. and for a field strength of $F=\\tfrac{25}{8}$ the ground-state energy is 0.9063 a.u.; these can be compared to the unperturbed result of 1.23370 a.u.. Some approximate formulas for the higher eigenvalues are available:\n\n\\begin{align}\nE\\left(F=\\tfrac{1}{16};n\\right) \n&= \\frac{10.3685}{8} \n\\left( 0.048 + 5.758 \\cdot 10^{-5} n + 0.952 n^2 + 3.054 \\cdot 10^{-7} n^3\\right) - \\frac{1}{16} \\\\\nE\\left(F=\\tfrac{25}{8};n\\right)\n&= \\frac{32.2505}{8} \n\\left( 0.688 + 0.045 n + 0.300 n^2 + 2.365 \\cdot 10^{-4} n^3\\right) - \\frac{25}{8}\n\\end{align}\n\n> Note, to obtain these numbers from the reference data in [solved exactly](https://www.utupub.fi/bitstream/handle/10024/117904/progradu-tuomas-riihim%C3%A4ki.pdf), you need to keep in mind that the reference data assumes the mass is $1/2$ instead of $1$, and that the reference data is for a box from $0 \\le x \\le 1$ instead of $-1 \\le x \\le 1$. This requires dividing the reference field by 16, and shifting the energies by the field, and dividing the energy by 8 (because both the length of the box and the mass of the particle has doubled). In the end, $F = \\tfrac{1}{16} F_{\\text{ref}}$ and $E = \\tfrac{1}{8}E_{\\text{ref}}-\\tfrac{1}{16}F_{\\text{ref}}= \\tfrac{1}{8}E_{\\text{ref}}-F$.\n\n\n#### Perturbation Theory for the Particle-in-a-Box in a Uniform Electric Field\n##### The First-Order Energy Correction is Always Zero\nThe corrections due to the perturbation are all zero to first order. To see this, consider that, from the Hellmann-Feynman theorem,\n\n\\begin{align}\n\\left[\\frac{dE_n}{dF}\\right]_{F=0} &= \\int_{-1}^{1} \\psi_n(x) \\left[ \\frac{d \\hat{H}}{dF} \\right]_{F=0} \\psi_n(x) dx \\\\\n&= \\int_{-1}^{1} x|\\psi_n(x)|^2 dx \\\\\n&= \\int_{-1}^{1} \\text{(even function)} \\text{(odd function) } dx \\\\\n&= \\int_{-1}^{1}\\text{(odd function) } dx \\\\\n&= 0\n\\end{align}\n\nThis reflects the fact that this system has a vanishing dipole moment. \n\n##### The First-Order Correction to the Wavefunction\nTo determine the first-order correction to the wavefunction, one needs to evaluate integrals that look like:\n$$\nV_{mn} = \\int_{-1}^{1} \\psi_m(x) (x) \\psi_n(x) dx $$\nFrom the properties of odd and even functions, and the fact that $\\psi_n(x)$ is odd if $n$ is even, and *vice versa*, it's clear that $V_mn = 0$ unless $m+n$ is odd. (That is, either $m$ or $n$, but not both, must be odd.) The integrals we need to evaluate all have the form\n$$\nV_{mn} = \\int_{-1}^{1} x \\sin \\left(\\frac{m \\pi x}{2} \\right)\\cos \\left(\\frac{n \\pi x}{2} \\right) dx \n$$\nwhere $m$ is even and $n$ is odd. Using the trigonometric identity\n$$\n\\sin(ax) \\cos(bx) = \\tfrac{1}{2} \\sin((a+b)x) + \\tfrac{1}{2} \\sin((a-b)x)\n$$\nwe can deduce that the integral is\n$$\nV_{mn} = \\left[2\\frac{\\sin\\left( \\frac{(m-n)\\pi x}{2} \\right)}{(m-n)^2 \\pi^2} \n+ 2\\frac{\\sin\\left( \\frac{(m+n)\\pi x}{2} \\right)}{(m+n)^2 \\pi^2} \n-\\frac{x\\cos\\left( \\frac{(m-n)\\pi x}{2} \\right)}{(m-n) \\pi} \n-\\frac{x\\cos\\left( \\frac{(m+n)\\pi x}{2} \\right)}{(m+n)^2 \\pi} \\right]_{=1}^{1}\n$$\nAs mentioned before, this integral is zero unless $m+n$ is odd. The cosine terms therefore vanish. For odd $p$, $\\sin \\tfrac{p \\pi}{2} = -1^{(p-1)/2}$, we have\n$$\nV_{mn} = \n\\begin{cases}\n0 & m+n \\text{ is even} \\\\\n\\dfrac{4}{\\pi^2} \\left( \\dfrac{-1^{(m-n-1)/2} }{(m-n)^2} + \\dfrac{-1^{(m+n-1)/2}}{(m+n)^2} \\right) & m+n \\text{ is odd}\n\\end{cases} \n$$\nThe first-order corrections to the ground-state wavefunction is then: \n$$\n \\left[\\frac{d\\psi_n(x)}{dF} \\right]_{F=0} = | \\psi_m'(0) \\rangle = \\sum_{m=1\\\\\n m \\ne n}^{\\infty} \\frac{V_{mn}}{E_n(0) - E_m(0)} | \\psi_m(0) \\rangle\n$$\n\n##### The Second-Order Correction to the Energy\nThe second-order correction to the energy is \n$$\n E_n''(0) = 2 \\sum_{m=1\\\\m \\ne n}^{\\infty}\\frac{ V_{mn}^2}{E_m(0) - E_n(0)} \n$$\nThis infinite sum is not trivial to evaluate, but we can investigate the first non-vanishing term for the ground state. (This is the so-called Unsold approximation.) Thus:\n$$\nE_0''(0) = 2 \\frac{V_{21}^2}{E_1(0) - E_2(0)} = 2 \\frac{\\left(\\tfrac{4}{\\pi^2}(1-\\tfrac{1}{9})\\right)^2}{\\tfrac{\\pi^2}{8} - \\tfrac{4\\pi^2}{8}} \n= -\\frac{16384 }{243 \\pi^6} = -0.0701\n$$\nUsing this, we can estimate the ground-state energy for different field strengths as \n$$\nE(F) \\approx E(0) - \\frac{1}{2!}\\frac{16384 }{243 \\pi^6} F^2\n$$\nFor the field strengths for which we have exact results readily available, this gives \n$$\nE(\\tfrac{1}{16}) \\approx 1.23356 \\text{ a.u.} \\\\\nE(\\tfrac{25}{8}) \\approx 0.8913 \\text{ a.u.} \\\\\n$$\nThese results are impressively accurate, especially considering all the effects we have neglected.\n\n#### Variational Approach to the Particle-in-a-Box in a Uniform Electric Field\nWhen the field is applied, it becomes more favorable for the electron to drift to the $x<0$ side of the box. To accomodate this, we can propose a wavefunction ansatz for the ground state,\n$$\n\\psi_c(x) = (1 - cx)\\cos\\left(\\frac{\\pi x}{2} \\right)\n$$\nClearly $c = 0$ in the absence of a field, but $c > 0$ is to be expected when the field is applied. We can determine the optimal value of $c$ using the variational principle. First we need to determine the energy as a function of $c$:\n$$\nE(c) = \\frac{\\langle \\psi_c | \\hat{H} | \\psi_c \\rangle}{\\langle \\psi_c | \\psi_c \\rangle}\n$$\nThe denominator of this expression is easily evaluated\n$$\n\\langle \\psi_c | \\psi_c \\rangle = 1 + \\gamma c^2 \n$$\nwhere we have defined the constant:\n$$\n\\gamma = \\int_{-1}^1 x^2 \\cos^2\\left(\\frac{\\pi x}{2}\\right) dx = \\tfrac{1}{3} - \\tfrac{2}{\\pi^2}\n$$\nThe numerator is \n\n$$\n\\langle \\psi_c | \\hat{H} | \\psi_c \\rangle = \\frac{\\pi^2}{8} \n+ c^2 \\frac{\\gamma \\pi^2}{8} - 2cF\\gamma + \\tfrac{1}{2}c^2\n$$\nwhere we have used the integral:\n\n$$\n\\int_{-1}^1 x \\cos\\left(\\frac{\\pi x}{2}\\right) \\sin\\left(\\frac{\\pi x}{2} \\right) dx = \\frac{1}{\\pi}\n$$\n\nThis equation can be solved analytically because it is a cubic equation, but it is more convenient to solve it numerically.\n\n#### Basis-Set Expansion for the Particle-in-a-Box in a Uniform Electric Field\nAs a final approach to this problem, we can expand the wavefunction in a basis set. The eigenfunctions of the unperturbed particle-in-a-box are a sensible choice here, though we could use polynomials (as we did earlier in this worksheet) without issue if one wished to do so. The eigenfunctions of the unperturbed problem are orthonormal, so the overlap matrix is the identity matrix\n$$\ns_{mn} = \\delta_{mn}\n$$\nThe Hamiltonian matrix elements are:\nand the Hamiltonian matrix elements are\n\n\\begin{align}\nh_{mn} &= \\int_{-1}^{1} \\cos\\left(\\frac{m \\pi x}{2} \\right) \\hat{H} \\cos\\left(\\frac{n \\pi x}{2} \\right) dx \\\\\n&= \\int_{-1}^{1} \\cos\\left(\\frac{m \\pi x}{2} \\right)\\left[-\\frac{1}{2}\\frac{d^2}{dx^2} + Fx \\right] \\cos\\left(\\frac{n \\pi x}{2} \\right) dx \\\\\n&= \\frac{\\pi^2 n^2}{2} \\delta_{mn} + F V_{mn}\n\\end{align}\nUsing the results we have already determined for the matrix elements, then, \n\n$$\nh_{mn} = \n\\begin{cases}\n 0 & m\\ne n \\text{ and }m+n \\text{ is even}\\\\\n \\dfrac{\\pi^2n^2}{8} & m = n \\\\\n \\dfrac{4F}{\\pi^2} \\left( \\dfrac{-1^{(m-n-1)/2} }{(m-n)^2} + \\dfrac{-1^{(m+n-1)/2}}{(m+n)^2} \\right) & m+n \\text{ is odd}\n\\end{cases}\n$$\n\n#### Demonstration\nIn the following code block, we'll demonstrate how the energy converges as we increase the number of terms in our calculation. For the excited states, it seems the reference data is likely erroneous.\n\n\n```python\nimport numpy as np\nfrom scipy.linalg import eigh\nfrom scipy.optimize import minimize_scalar\nfrom scipy.integrate import quad\nimport matplotlib.pyplot as plt\n\n\ndef compute_V(n_basis):\n \"\"\"Compute the matrix for an electron in a box from -1 to 1 in a unit external field, in a.u.\"\"\"\n \n # initialize V to a zero matrix\n V = np.zeros((n_basis,n_basis))\n \n # Because Python is zero-indexed, our V matrix will be shifted by 1. I'll\n # make this explicit by making the counters km1 and lm1 (k minus 1 and l minus 1)\n for km1 in range(n_basis):\n for lm1 in range(n_basis):\n if (km1 + lm1) % 2 == 1:\n # The matrix element is zero unless (km1 + lm1) is odd, which means that (km1 + lm1) mod 2 = 1.\n # Either km1 is even or km1 is odd. If km1 is odd, then the km1 corresponds to a sine and\n # lm1 is even, and corresponds to a cosine. If km1 is even and lm1 is odd, then the roles of the\n # sine and cosine are reversed, and one needs to multiply the first term below by -1. The\n # factor -1**lm1 achieves this switching.\n V[km1,lm1] = 4. / np.pi**2 * (-1**((km1 - lm1 - 1)/2) / (km1-lm1)**2 * -1**(lm1)\n + -1**((km1 + lm1 - 1)/2) / (km1+lm1+2)**2)\n return V\n\ndef energy_pt2(k,F,n_basis):\n \"\"\"kth excited state energy in a.u. for an electron in a box of length 2 in the field F estimated with 2nd-order PT.\n\n Parameters\n ----------\n k : scalar, int \n k = 0 is the ground state and k = 1 is the first excite state.\n F : scalar\n the external field strength\n n_basis : scalar, int\n the number of terms to include in the sum over states in the second order pert. th. correction.\n\n Returns\n -------\n energy_pt2 : scalar\n The estimated energy of the kth-excited state of the particle in a box of length 2 in field F.\n \"\"\"\n # It makes no sense for n_basis to be less than k.\n assert(k < n_basis), \"The excitation level of interest should be smaller than n_basis\" \n \n # Energy of the kth-excited state in a.u.\n energy = (np.pi**2 / 8.) * np.array([(k + 1)**2 for k in range(n_basis)])\n V = compute_V(n_basis)\n \n der2 = 0\n for j in range(n_basis):\n if j != k:\n der2 += 2*V[j,k]**2/(energy[k]-energy[j]) \n \n return energy[k] + der2 * F**2 / 2\n\n\ndef energy_variational(F):\n \"\"\"ground state energy for a electron0in-a-Box in a box of length 2 in the field F estimated with the var. principle.\n The variational wavefunction ansatz is psi(x) = (1+cx)cos(pi*x/2) where c is a variational parameter.\n \"\"\"\n gamma = 1/3 - 2/np.pi**2\n def func(c):\n return (np.pi**2/8*(1+gamma*c**2)+c**2/2 - 2*c*gamma*F) / ( 1 + gamma * c**2)\n\n res = minimize_scalar(func,(0,1))\n \n return res.fun\n\n\ndef energy_basis(F,n_basis):\n \"\"\"Eigenenergies in a.u. of an electron in a box of length 2 in the field F estimated by basis-set expansion.\n n_basis basis functions from the F=0 case are used.\n \n Parameters\n ----------\n F : scalar\n the external field strength\n n_basis : scalar, int\n the number of terms to include in the sum over states in the second order pert. th. correction.\n\n Returns\n -------\n energy_basis_exp : array_like\n list of n_basis eigenenergies\n \"\"\"\n energy = (np.pi**2 / 8.) * np.array([(k + 1)**2 for k in range(n_basis)])\n V = compute_V(n_basis)\n \n # assign Hamiltonian to the potential matrix, times the field strength:\n h = F*V \n np.fill_diagonal(h,energy)\n # solve Hc = Ec to get eigenvalues E\n e_vals = eigh(h, None, eigvals_only=True)\n return e_vals\n\nprint(\"Energy of these models vs. reference values:\")\nprint(\"Energy of the unperturbed ground state (field = 0):\", np.pi**2 / 8.) \nprint(\" \")\nprint(\"Field value: \", 1./16)\nprint(\"Exact Energy of the ground state:\", 10.3685/8 - 1./16)\nprint(\"Energy of the ground state estimated with 2nd-order perturbation theory:\", energy_pt2(0,1./16,50)) \nprint(\"Energy of the ground state estimated with the variational principle:\", energy_variational(1./16)) \nprint(\"Energy of the ground state estimated with basis set expansion:\", energy_basis(1./16,50)[0]) \nprint(\" \")\nprint(\"Field value: \", 25./4)\nprint(\"Exact Energy of the ground state:\", 32.2505/8 - 25./8)\nprint(\"Energy of the ground state estimated with 2nd-order perturbation theory:\", energy_pt2(0,25./8,50)) \nprint(\"Energy of the ground state estimated with the variational principle:\", energy_variational(25./8)) \nprint(\"Energy of the ground state estimated with basis set expansion:\", energy_basis(25./8,50)[0]) \nprint(\" \")\nprint(\" \")\nprint(\"Energy of the unperturbed first excited state (field = 0):\", np.pi**2 * 2**2 / 8.) \nprint(\" \")\nprint(\"Field value: \", 1./16)\nprint(\"Exact Energy of the first excited state:\", 39.9787/8 - 1./16)\nprint(\"Energy of the first excited state estimated with 2nd-order perturbation theory:\", energy_pt2(1,1./16,50)) \nprint(\"Energy of the first excited state estimated with basis set expansion:\", energy_basis(1./16,50)[1]) \nprint(\" \")\nprint(\"Field value: \", 25./4)\nprint(\"Exact Energy of the first excited state:\", 65.177/8 - 25./8)\nprint(\"Energy of the first excited state estimated with 2nd-order perturbation theory:\", energy_pt2(1,25./4,50)) \nprint(\"Energy of the first excited state estimated with basis set expansion:\", energy_basis(25./4,50)[1]) \nprint(\" \")\nprint(\" \")\nprint(\"Energy of the unperturbed second excited state (field = 0):\", np.pi**2 * 3**2 / 8.) \nprint(\" \")\nprint(\"Field value: \", 1./16)\nprint(\"Exact Energy of the second excited state:\", 89.3266/8 - 1./16)\nprint(\"Energy of the second excited state estimated with 2nd-order perturbation theory:\", energy_pt2(2,1./16,50)) \nprint(\"Energy of the first excited state estimated with basis set expansion:\", energy_basis(1./16,50)[2]) \nprint(\" \")\nprint(\"Field value: \", 25./4)\nprint(\"Exact Energy of the ground state:\", 114.309/8 - 25./8)\nprint(\"Energy of the second excited state estimated with 2nd-order perturbation theory:\", energy_pt2(2,25./4,50)) \nprint(\"Energy of the second excited state estimated with basis set expansion:\", energy_basis(25./4,50)[2])\n```\n\n Energy of these models vs. reference values:\n Energy of the unperturbed ground state (field = 0): 1.2337005501361697\n \n Field value: 0.0625\n Exact Energy of the ground state: 1.2335625\n Energy of the ground state estimated with 2nd-order perturbation theory: 1.2335633924534153\n Energy of the ground state estimated with the variational principle: 1.2335671162852282\n Energy of the ground state estimated with basis set expansion: 1.2335633952295426\n \n Field value: 6.25\n Exact Energy of the ground state: 0.9063125000000003\n Energy of the ground state estimated with 2nd-order perturbation theory: 0.8908063432502749\n Energy of the ground state estimated with the variational principle: 0.9250111494073084\n Energy of the ground state estimated with basis set expansion: 0.9063121281470087\n \n \n Energy of the unperturbed first excited state (field = 0): 4.934802200544679\n \n Field value: 0.0625\n Exact Energy of the first excited state: 4.9348375\n Energy of the first excited state estimated with 2nd-order perturbation theory: 4.93484310142371\n Energy of the first excited state estimated with basis set expansion: 4.934843098735417\n \n Field value: 6.25\n Exact Energy of the first excited state: 5.022125000000001\n Energy of the first excited state estimated with 2nd-order perturbation theory: 5.343810990856903\n Energy of the first excited state estimated with basis set expansion: 5.161850795200653\n \n \n Energy of the unperturbed second excited state (field = 0): 11.103304951225528\n \n Field value: 0.0625\n Exact Energy of the second excited state: 11.103325\n Energy of the second excited state estimated with 2nd-order perturbation theory: 11.103329317896025\n Energy of the first excited state estimated with basis set expansion: 11.103329317809841\n \n Field value: 6.25\n Exact Energy of the ground state: 11.163625\n Energy of the second excite state estimated with 2nd-order perturbation theory: 11.34697165619853\n Energy of the second excited state estimated with basis set expansion: 11.336913933805935\n\n\n\n```python\n# user-specified parameters\nF = 25.0 / 8\nnbasis = 20\n\n# plot basis set convergence of energy estimates at a given field\n# ---------------------------------------------------------------\n\n# evaluate energy for a range of basis functions at a given field\nn_values = np.arange(2, 11, 1)\ne_pt2_basis = np.array([energy_pt2(0, F, n) for n in n_values])\ne_var_basis = np.repeat(energy_variational(F), len(n_values))\ne_exp_basis = np.array([energy_basis(F, n)[0] for n in n_values])\n\n# evaluate energy for a range of fields at a given basis\nf_values = np.arange(0.0, 100., 5.)\ne_pt2_field = np.array([energy_pt2(0, f, nbasis) for f in f_values])\ne_var_field = np.array([energy_variational(f) for f in f_values])\ne_exp_field = np.array([energy_basis(f, nbasis)[0] for f in f_values])\n\n\nplt.rcParams['figure.figsize'] = [15, 8]\nfig, axes = plt.subplots(1, 2)\n# fig.suptitle(\"Basis Set Convergence of Particle-in-a-Box with Jacobi Basis\", fontsize=24, fontweight='bold')\n\nfor index, axis in enumerate(axes.ravel()):\n if index == 0:\n # plot approximate energy at a fixed field\n axis.plot(n_values, e_pt2_basis, marker='o', linestyle='--', label='PT2')\n axis.plot(n_values, e_var_basis, marker='', linestyle='-', label='Variational')\n axis.plot(n_values, e_exp_basis, marker='x', linestyle='-', label='Basis Expansion')\n # set axes labels\n axis.set_xlabel(\"Number of Basis Functions\", fontsize=12, fontweight='bold')\n axis.set_ylabel(\"Ground-State Energy [a.u.]\", fontsize=12, fontweight='bold')\n axis.set_title(f\"Field Strength = {F}\", fontsize=24, fontweight='bold')\n axis.legend(frameon=False, fontsize=14)\n else:\n # plot approximate energy at a fixed basis\n axis.plot(f_values, e_pt2_field, marker='o', linestyle='--', label='PT2')\n axis.plot(f_values, e_var_field, marker='', linestyle='-', label='Variational')\n axis.plot(f_values, e_exp_field, marker='x', linestyle='-', label='Basis Expansion')\n # set axes labels\n axis.set_xlabel(\"Field Strength [a.u.]\", fontsize=12, fontweight='bold')\n axis.set_ylabel(\"Ground-State Energy [a.u.]\", fontsize=12, fontweight='bold')\n axis.set_title(f\"Number of Basis = {nbasis}\", fontsize=24, fontweight='bold')\n axis.legend(frameon=False, fontsize=14)\n\nplt.show()\n```\n\n\n```python\n# user-specified parameters\nF = 25.0 / 8\nnbasis = 20\n\n# plot basis set convergence of 1st excited state energy at a given field\n# ---------------------------------------------------------------\n\n# evaluate energy for a range of basis functions at a given field\nn_values = np.arange(2, 11, 1)\ne_pt2_basis = np.array([energy_pt2(1, F, n) for n in n_values])\ne_exp_basis = np.array([energy_basis(F, n)[1] for n in n_values])\n\n# evaluate energy for a range of fields at a given basis\nf_values = np.arange(0.0, 100., 5.)\ne_pt2_field = np.array([energy_pt2(1, f, nbasis) for f in f_values])\ne_exp_field = np.array([energy_basis(f, nbasis)[1] for f in f_values])\n\n\nplt.rcParams['figure.figsize'] = [15, 8]\nfig, axes = plt.subplots(1, 2)\n# fig.suptitle(\"Basis Set Convergence of Particle-in-a-Box with Jacobi Basis\", fontsize=24, fontweight='bold')\n\nfor index, axis in enumerate(axes.ravel()):\n if index == 0:\n # plot approximate energy at a fixed field\n axis.plot(n_values, e_pt2_basis, marker='o', linestyle='--', label='PT2')\n axis.plot(n_values, e_exp_basis, marker='x', linestyle='-', label='Basis Expansion')\n # set axes labels\n axis.set_xlabel(\"Number of Basis Functions\", fontsize=12, fontweight='bold')\n axis.set_ylabel(\"1st Excited-State Energy [a.u.]\", fontsize=12, fontweight='bold')\n axis.set_title(f\"Field Strength = {F}\", fontsize=24, fontweight='bold')\n axis.legend(frameon=False, fontsize=14)\n else:\n # plot approximate energy at a fixed basis\n axis.plot(f_values, e_pt2_field, marker='o', linestyle='--', label='PT2')\n axis.plot(f_values, e_exp_field, marker='x', linestyle='-', label='Basis Expansion')\n # set axes labels\n axis.set_xlabel(\"Field Strength [a.u.]\", fontsize=12, fontweight='bold')\n axis.set_ylabel(\"1st Excited-State Energy [a.u.]\", fontsize=12, fontweight='bold')\n axis.set_title(f\"Number of Basis = {nbasis}\", fontsize=24, fontweight='bold')\n axis.legend(frameon=False, fontsize=14)\n\nplt.show()\n```\n\n## 🪞 Self-Reflection\n- When is a basis set appropriate? When is perturbation theory more appropriate?\n- Consider the hydrogen molecule ion, $\\text{H}_2^+$. Is it more sensible to use the secular equation (basis-set-expansion) or perturbation theory? What if the bond length is very small? What if the bond length is very large?\n\n## 🤔 Thought-Provoking Questions\n- Show that if you minimize the energy as a function of the basis-set coefficients using the variational principle, then you obtain the secular equation.\n- If a uniform external electric field of magnitude $F$ in the $\\hat{\\mathbf{u}} = [\\hat{u}_x,\\hat{u}_y,\\hat{u}_z]^T$ direction is applied to a particle with charge $q$, the potential $V(x,y,z) = -qF(u_x x + u_y y + u_z z)$ is added to the Hamiltonian. (This follows from the fact that the force applied to the particles is proportional to the electric field, $\\text{force} = q \\vec{E} = q F \\hat{\\mathbf{u}}$ and the force is $\\text{force} = - \\nabla V(x,y,z)$. If the field is weak, then perturbation theory can be used, and the energy can be written as a Taylor series. The coefficients of the Taylor series give the dipole moment ($\\mu$), dipole polarizability ($\\alpha$), first dipole hyperpolarizability ($\\beta$), second dipole hyperpolarizability ($\\gamma$) in the $\\hat{\\mathbf{u}}$ direction.\n - The dipole moment, $\\mu$, of any spherical system is zero. Explain why.\n - The polarizability, $\\alpha$, of any system is always positive. Explain why.\n\n\\begin{align}\nE_k(F) &= E_k(0) + F \\left[\\frac{dE_k}{d F} \\right]_{F=0} \n+ \\frac{F^2}{2!} \\left[\\frac{d^2E_k}{d F^2} \\right]_{F=0} \n+ \\frac{F^3}{3!} \\left[\\frac{d^3E_k}{d F^3} \\right]_{F=0} \n+ \\frac{F^4}{4!} \\left[\\frac{d^4E_k}{d F^4} \\right]_{F=0} + \\cdots \\\\\n&= E_k(0) - F \\mu_{F=0} \n- \\frac{F^2}{2!} \\alpha_{F=0} \n- \\frac{F^3}{3!} \\beta_{F=0} \n- \\frac{F^4}{4!} \\gamma_{F=0} + \\cdots \\\\\n\\end{align}\n\n- The Hellmann-Feynman theorem indicates that given the ground-state wavefunction for a molecule, the force on the nuclei can be obtained. Explain how.\n- What does it mean that perturbation theory is inaccurate when the perturbation is large?\n- Can you explain why the energy goes down when the electron-in-a-box is placed in an external field?\n- For a sufficiently-highly excited state, the effect of an external electric field is negligible. Why is this true intuitively? Can you show it graphically? Can you explain it mathematically?\n\n## 🔁 Recapitulation\n- What is the secular equation?\n- What is the Hellmann-Feynman theorem?\n- How is the Hellmann-Feynman theorem related to perturbation theory? \n- What is perturbation theory? What is the expression for the first-order perturbed wavefunction?\n\n## 🔮 Next Up...\n- Multielectron systems\n- Approximate methods for multielectron systems.\n\n## 📚 References\nMy favorite sources for this material are:\n- [Randy's book](https://github.com/PaulWAyers/IntroQChem/blob/main/documents/DumontBook.pdf?raw=true)\n- D. A. MacQuarrie, Quantum Chemistry (University Science Books, Mill Valley California, 1983)\n\nThere are also some excellent wikipedia articles:\n- [Perturbation theory](https://en.wikipedia.org/wiki/Perturbation_theory)\n- [Variational method](https://en.wikipedia.org/wiki/Variational_method_(quantum_mechanics))\n", "meta": {"hexsha": "1600294937a92a9ed2bbb46aba56a1be28299b64", "size": 313110, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ipynb/ApproximateMethods.ipynb", "max_stars_repo_name": "PaulWAyers/IntroQChemProblems", "max_stars_repo_head_hexsha": "0491272e0b8e71059e601537b196181ab6b367cd", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2021-01-16T15:35:43.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-17T17:43:32.000Z", "max_issues_repo_path": "ipynb/ApproximateMethods.ipynb", "max_issues_repo_name": "PaulWAyers/IntroQChemProblems", "max_issues_repo_head_hexsha": "0491272e0b8e71059e601537b196181ab6b367cd", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ipynb/ApproximateMethods.ipynb", "max_forks_repo_name": "PaulWAyers/IntroQChemProblems", "max_forks_repo_head_hexsha": "0491272e0b8e71059e601537b196181ab6b367cd", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 6, "max_forks_repo_forks_event_min_datetime": "2021-01-29T16:30:34.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-03T08:41:47.000Z", "avg_line_length": 237.564491654, "max_line_length": 67472, "alphanum_fraction": 0.870016927, "converted": true, "num_tokens": 20459, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631556226291, "lm_q2_score": 0.6992544273261175, "lm_q1q2_score": 0.42491115189208295}} {"text": "# The Mathematics of Drilling Intercepts\r\n\r\n- toc: true\r\n- branch: master\r\n- badges: true\r\n- comments: true\n\n## Introduction\n\n\nDrilling intercepts are a prominent feature of junior mining news releases. They are closely monitored by the mining investment community, and a particularly good intercept can raise the prospects for a project. \n\nAs an example, consider this November 10 2020 release from Freegold Ventures:\n\n> *Freegold Intercepts 3.78 g/t Au Over 119 Metres Including 131.5 g/t Over 3 Metres Within 573 Metres of 1.21 g/t Au at Golden Summit*\n\nThe market responded with a 3% boost in the share price the next trading day, so clearly this was regarded as a positive signal for the company's prospects. [(This is typical: capital markets tend to treat any news of this sort as good news.)](https://doi.org/10.1177%2F0312896212473401)\n\nThe implications for the economic, geological, and engineering variables surrounding the project are much less clear. *Is this a good geological result? Is it a good engineering result?* Intercepts are *highlights*: incomplete data, collected and released selectively, so is it even possible to make an informed judgement using these numbers?\n\nTo complicate things even further, the selectively reported drilling intercepts are usually released in a rather complex manner, which can make it difficult to distinguish between truly good numbers and deceptively good results. Drilling intercepts are discussed at great length in other sources (here and here and here) but we'll take a mathematical perspective and develop a model that describes nested intercept configurations of arbitrary complexity.\n \nWe'll take Great Bear Resources for an extended example. Great Bear Resources is a Canadian junior mining company whose stock gained substantially on announcement of very high grade intercepts at their Dixie project in Ontario. At time of writing, GBR is trading at a $886 million CAD market cap (which is not very bearish at all!) \n\n\n
                                        \n
                                        \n
                                        GBR Price Today by TradingView
                                        \n \n
                                        \n\n\n\n\n```\n#hide\n\nfrom google.colab import drive\ndrive.mount('/content/drive')\n\n\n!pip install anytree\n\nfrom anytree import Node, RenderTree\n\nfrom anytree import Node, RenderTree, AsciiStyle, LevelOrderIter\n\n\n```\n\n Mounted at /content/drive\n\n\n\nHere we open up the spreadsheet of drilling results (available on their website), and then filter on `Drill Hole` to consider a single hole:\n\n\n```\nimport pandas as pd\n\nintercepts = pd.read_excel('drive/My Drive/Projects/posts/data/Great_Bear/lp_drill_hole_composites_all.xlsx')\nintercepts['Record'] = intercepts.index\n\ndh = intercepts[intercepts['Drill Hole'] == 'BR-022']\n\ndh\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Drill HoleUnnamed: 1From (m)To (m)Width (m)Gold (g/t)Record
                                        95BR-022including110.0116.106.102.6295
                                        96BR-022and including111.4113.101.707.9296
                                        97BR-022and274.0299.0025.000.1997
                                        98BR-022and432.9439.006.104.0598
                                        99BR-022including432.9435.702.808.1899
                                        100BR-022and445.0452.007.000.41100
                                        101BR-022and461.6512.0050.401.78101
                                        102BR-022including471.0512.0041.002.09102
                                        103BR-022and including471.0478.007.002.37103
                                        104BR-022and including490.0491.001.008.15104
                                        105BR-022and including505.2508.753.5514.90105
                                        106BR-022and including506.2506.700.50100.48106
                                        107BR-022and600.0620.0020.000.52107
                                        \n
                                        \n\n\n\nThis is how intercepts are typically presented: a table with a `From` field describing where they started measuring, a `To` field describing where they stopped, and a `Grade` field (called `Gold` here) that tells us how enriched that interval is with the valuable stuff. `From` and `To` are typically measured downhole from the drill collar. \n\nIt's easy to establish a basic understanding of how these tables are read, and many experienced mining investors immediately recognize these grades as very high. The rest of us might need to rely on statistics, since we don't have the benefit of many years' experience with drilling results. \r\n\r\nOf course it is first necessary to determine the true assay values for each separate interval from top to bottom. Unfortunately, each row is not independent - some of the intercepts are contained in others, and the subinterval gold is INCLUDED in the parent interval calculation! So we can't just use the `Gold (g/t)` field directly, since intercepts are reported with these \"highlights\", or higher grade sections within the longer interval. \r\n\r\nSometimes this convention is used unethically to suggest larger intervals of enrichment than truly exist. This is called \"grade smearing\" and the method of residual grade calculation applied here will detect any such attempt to disguise poor results. \n\nAt first it may seem like the correct interpretation of these intervals is to imagine them intervals stacked on top of one another, but this is very misleading. We can easily visualize this to see the error:\n\n\n\n```\n#hide_input\n\nimport altair as alt\n\ny_axis = alt.Axis(\n title='Intercept ID',\n offset=5,\n ticks=False,\n domain=False\n)\n\nalt.Chart(dh).mark_bar().encode(\n alt.X('From (m):Q',\n scale=alt.Scale(zero=False)),\n x2='To (m):Q',\n y=alt.Y('Drill Hole:N', axis=y_axis),\n color=alt.Color('Gold (g/t):Q', scale=alt.Scale(scheme=\"inferno\")),\n tooltip=[\n alt.Tooltip('Width (m):Q', title='Width'),\n alt.Tooltip('Gold (g/t):Q', title='Gold Grade')\n ]\n).properties(width=800, height=100).configure(background='#D9E9F0')\n```\n\n\n\n\n\n
                                        \n\n\n\n\n\r\n- We only have the total grade, INCLUDING the high-grade, child subintervals. Considering it in that way ignores the fact that the high-grade intervals are included in the wider, lower-grade intervals, inflating the grade measured over that length. This has enormous implications for the *continuity* of the mineralization, which determines the feasibility of the project. \r\n\r\nIn order to eliminate this effect we'll need to do some math with the intercepts. This visualization attempts to show this hierarchical, branching structure:\n\n\n```\n#hide_input\n\nimport altair as alt\n\ny_axis = alt.Axis(\n title='Interval ID',\n offset=5,\n ticks=False,\n domain=False\n)\n\nalt.Chart(dh).mark_bar().encode(\n alt.X('From (m):Q',\n scale=alt.Scale(zero=False)),\n x2='To (m):Q',\n y=alt.Y('Record:N', axis=y_axis),\n color=alt.Color('Gold (g/t):Q', scale=alt.Scale(scheme=\"inferno\")),\n tooltip=[\n alt.Tooltip('Width (m):Q', title='Width'),\n alt.Tooltip('Gold (g/t):Q', title='Gold Grade')\n ]\n).properties(width=800, height=400).configure(background='#D9E9F0')\n```\n\n\n\n\n\n
                                        \n\n\n\n\n\r\n\r\nPlotted side by side, the intercepts show the parent-child overlapping relationship and capture the complexity of the problem. \r\n\r\nParent intervals can have no child intervals, a single child interval, or several child intervals. Child intervals themselves can have no child intervals, a single child interval, or several child intervals. Clearly there is a whole class of related problems we could solve with a general solution to this problem.\r\n\n\nSo far we have treated the `From` and `To` fields in isolation, and we can use a cool feature of Pandas to convert them to intervals:\n\n\n```\ndh['Interval'] = dh.apply(lambda x: pd.Interval(x['From (m)'], x['To (m)']), axis=1)\r\n\r\ndh\n```\n\n /usr/local/lib/python3.6/dist-packages/ipykernel_launcher.py:1: SettingWithCopyWarning: \n A value is trying to be set on a copy of a slice from a DataFrame.\n Try using .loc[row_indexer,col_indexer] = value instead\n \n See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n \"\"\"Entry point for launching an IPython kernel.\n\n\nSo the motivation here was to create `Interval` objects to use them with the `pd.Overlaps` function and then model the overlap relationship among the different intervals:\n\n\n\n```\nimport itertools\nimport numpy as np\n\ncross_interval = itertools.product(dh.Interval,dh.Interval)\n\n\n\noverlap_matrix = np.array([interval[0].overlaps(interval[1]) for interval in cross_interval])\n\nintersect_matrix = np.array([interval[0].intersect(interval[1]) for interval in cross_interval])\n\nns = int(np.sqrt(overlap_matrix.shape[0]))\n\n```\n\n\n```\n#hide_input\n\nimport matplotlib.pyplot as plt\n\nplt.rcParams.update({'font.size': 16})\n\nfig, ax = plt.subplots(1,1, figsize=(10,11))\n\nimg = ax.imshow(overlap_matrix.reshape(ns,ns), cmap='hot', interpolation='nearest')\n\nx_label_list = dh.Record.values\n\nax.set_xticks(np.arange(0, ns, 1))\nax.set_yticks(np.arange(0, ns, 1))\n\ntext = ax.set_xticklabels(x_label_list)\ntext = ax.set_yticklabels(x_label_list)\n\n```\n\nHere we see the overlaps: if a pixel is white, it means that the interval on the x-axis and the interval on the y-axis overlap. \n\n\nOverlap is symmetric: so each 'child' overlaps with its parent and vice versa. It should become clear that we are actually interested in the \"contains\" relationship, which is not symmetric and will help us identify parent intervals and child intervals and start reducing the intervals. \n\nFortunately this is also supported in Python:\n\n\n```\n#hide\r\n\r\nfrom sympy import Interval\r\n\r\ndh['Interval_obj'] = dh.apply(lambda x: Interval(x['From (m)'], x['To (m)']), axis=1)\r\n\r\n\r\ncross_interval = itertools.product(dh.Interval_obj,dh.Interval_obj)\r\ncontain_matrix = np.array([interval[0].is_proper_superset(interval[1]) for interval in cross_interval])\r\n\n```\n\n /usr/local/lib/python3.6/dist-packages/ipykernel_launcher.py:5: SettingWithCopyWarning:\n \n \n A value is trying to be set on a copy of a slice from a DataFrame.\n Try using .loc[row_indexer,col_indexer] = value instead\n \n See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n \n\n\n\n```\n#hide_input\n\nimport matplotlib.pyplot as plt\n\nplt.rcParams.update({'font.size': 16})\n\nfig, ax = plt.subplots(1,1, figsize=(10,11))\n\nimg = ax.imshow(contain_matrix.reshape(ns,ns), cmap='hot', interpolation='nearest')\n\nx_label_list = dh.Record.values\n\nax.set_xticks(np.arange(0, ns, 1))\nax.set_yticks(np.arange(0, ns, 1))\n\ntext = ax.set_xticklabels(x_label_list)\ntext = ax.set_yticklabels(x_label_list)\n```\n\nNow we can pull out a tree \n\nOf the machine-intelligible formats, a **tree data structure** is clearly the most suited to representing the intervals. \n\n\n```\n#hide\nfrom anytree import Node, RenderTree, AsciiStyle, PreOrderIter\n\ninterval_pairs = list(itertools.product(dh.Record.values, dh.Record.values))\nparent_child = list(contain_matrix)\n\nroot = Node(\"DH\")\n\nall_str_to_node = {str(dh['Record'].values[i]) : Node(str(dh['Record'].values[i]), parent = root) for i, column in enumerate(contain_matrix.reshape(ns,ns).T)}\n\n#[all_str_to_node[str(dh_101['Record'].values[i])].parent := root for i, column in enumerate(contain_matrix.reshape(14,14).T) if ~np.any(column)]\n\ncontain_matrix_sq = contain_matrix.reshape(ns,ns)\n\n#for i, col in enumerate(contain_matrix_sq.T):\n# print(col)\n# if ~np.any(col):\n# #all_str_to_node[str(dh_101['Record'].values[i])].parent = root\n# print(\"Root: {}\".format(dh_101['Record'].values[i]))\n# else:\n# print(\"Parent: {}\".format(dh_101['Record'].values[::-1][np.argmax(col[::-1])]))\n# print(\"Child: {}\".format(dh_101['Record'].values[i]))\n\n\n```\n\n\n```\nfor i, col in enumerate(contain_matrix_sq.T):\n\n if ~np.any(col):\n\n all_str_to_node[str(dh['Record'].values[i])].parent = root\n\n else:\n\n all_str_to_node[str(dh['Record'].values[i])].parent = all_str_to_node[str(dh['Record'].values[::-1][np.argmax(col[::-1])])]\n\nprint(RenderTree(root, style=AsciiStyle()).by_attr())\n```\n\n DH\n |-- 95\n | +-- 96\n |-- 97\n |-- 98\n | +-- 99\n |-- 100\n |-- 101\n | +-- 102\n | |-- 103\n | |-- 104\n | +-- 105\n | +-- 106\n +-- 107\n\n\nNow we are really getting somewhere- we can actually start looking at the global picture (since we now know which intervals are not \"child\" intervals)\n\nThese are the direct children. We can go ahead and plot them and have a totally accurate picture of the log: \n\n\n```\n#hide_input\n\ndh_prime = dh[dh.Record.isin((child.name for child in root.children))]\n\n\ny_axis = alt.Axis(\n title='Intercept ID',\n offset=5,\n ticks=False,\n domain=False\n)\n\n\nreqd_cols = ['From (m)', 'To (m)', 'Gold (g/t)', 'Width (m)']\n\n\nalt.Chart(dh_prime[reqd_cols]).mark_bar().encode(\n alt.X('From (m):Q',\n scale=alt.Scale(zero=False)),\n x2='To (m):Q',\n y=alt.Y('Drill Hole:N', axis=y_axis),\n color=alt.Color('Gold (g/t):Q', scale=alt.Scale(scheme=\"inferno\")),\n tooltip=[\n alt.Tooltip('Width (m):Q', title='Width'),\n alt.Tooltip('Gold (g/t):Q', title='Gold Grade')\n ]\n).properties(width=800, height=100).configure(background='#D9E9F0')\n```\n\n\n\n\n\n
                                        \n\n\n\n\n\n```\ndh_prime.dtypes\n```\n\n\n\n\n Drill Hole object\n Unnamed: 1 object\n From (m) float64\n To (m) float64\n Width (m) float64\n Gold (g/t) float64\n Record int64\n Interval interval[float64]\n Interval_obj object\n dtype: object\n\n\n\nWhile that is correct, it is not complete: we have left out all of the additional information provided by the smaller sub-intervals! \n\nIn order to incorporate that we will have to remove them from the parent intervals and determine the residual grade (whatever is left once we pull out the gold contained in the subinterval) \n\n\n```\n((119) * (3.78) - (3) * (131.5)) / (119 - 3)\n```\n\n\n\n\n 0.47689655172413786\n\n\n\nAs an example of this kind of calculation, a simpler set of intervals from a Freegold Ventures press release: \n\n> *Freegold Intercepts 3.78 g/t Au Over 119 Metres Including 131.5 g/t Over 3 Metres Within 573 Metres of 1.21 g/t Au at Golden Summit*\n\nWe know the gold grade over the whole 119 meters, and the gold grade over 3 meters, but what is the gold grade over the $119 - 3 = 116 m$?\n\nThe solution is a simple weighted average calculation, like compositing over a drillhole: \n\n$\\frac{119 \\times 3.78-3 \\times 131.5}{119-3} = 0.477 g/t$\n\n\nCredit to https://twitter.com/BrentCo77759016/status/1326183861722599424 and \n\n\nSo now we have to do this, but with every single subinterval until we get the residual grade at every point along the drillhole\n\nFortunately, the tree data structure we selected has specialized methods that make a traversal very simple. \n\n\n\n```\nlevelord_nodes = [(node.name, node.children) for node in LevelOrderIter(root)]\n\nlevelord_nodes\n```\n\n\n\n\n [('DH',\n (Node('/DH/95'),\n Node('/DH/97'),\n Node('/DH/98'),\n Node('/DH/100'),\n Node('/DH/101'),\n Node('/DH/107'))),\n ('95', (Node('/DH/95/96'),)),\n ('97', ()),\n ('98', (Node('/DH/98/99'),)),\n ('100', ()),\n ('101', (Node('/DH/101/102'),)),\n ('107', ()),\n ('96', ()),\n ('99', ()),\n ('102',\n (Node('/DH/101/102/103'), Node('/DH/101/102/104'), Node('/DH/101/102/105'))),\n ('103', ()),\n ('104', ()),\n ('105', (Node('/DH/101/102/105/106'),)),\n ('106', ())]\n\n\n\n\n```\nnn_np_loi = [(node.name, node.parent) for node in LevelOrderIter(root)]\n```\n\n\n```\nall_str_to_node\n```\n\n\n\n\n {'100': Node('/DH/100'),\n '101': Node('/DH/101'),\n '102': Node('/DH/101/102'),\n '103': Node('/DH/101/102/103'),\n '104': Node('/DH/101/102/104'),\n '105': Node('/DH/101/102/105'),\n '106': Node('/DH/101/102/105/106'),\n '107': Node('/DH/107'),\n '95': Node('/DH/95'),\n '96': Node('/DH/95/96'),\n '97': Node('/DH/97'),\n '98': Node('/DH/98'),\n '99': Node('/DH/98/99')}\n\n\n\n\n```\nfor node, parent in nn_np_loi[::-1][:-1]:\r\n print(node)\r\n for child in all_str_to_node[node].children:\r\n print(child)\n```\n\n 106\n 105\n Node('/DH/101/102/105/106')\n 104\n 103\n 102\n Node('/DH/101/102/103')\n Node('/DH/101/102/104')\n Node('/DH/101/102/105')\n 99\n 96\n 107\n 101\n Node('/DH/101/102')\n 100\n 98\n Node('/DH/98/99')\n 97\n 95\n Node('/DH/95/96')\n\n\n\n```\ndh\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Drill HoleUnnamed: 1From (m)To (m)Width (m)Gold (g/t)RecordIntervalInterval_obj
                                        95BR-022including110.0116.106.102.6295(110.0, 116.1]Interval(110.000000000000, 116.100000000000)
                                        96BR-022and including111.4113.101.707.9296(111.4, 113.1]Interval(111.400000000000, 113.100000000000)
                                        97BR-022and274.0299.0025.000.1997(274.0, 299.0]Interval(274.000000000000, 299.000000000000)
                                        98BR-022and432.9439.006.104.0598(432.9, 439.0]Interval(432.900000000000, 439.000000000000)
                                        99BR-022including432.9435.702.808.1899(432.9, 435.7]Interval(432.900000000000, 435.700000000000)
                                        100BR-022and445.0452.007.000.41100(445.0, 452.0]Interval(445.000000000000, 452.000000000000)
                                        101BR-022and461.6512.0050.401.78101(461.6, 512.0]Interval(461.600000000000, 512.000000000000)
                                        102BR-022including471.0512.0041.002.09102(471.0, 512.0]Interval(471.000000000000, 512.000000000000)
                                        103BR-022and including471.0478.007.002.37103(471.0, 478.0]Interval(471.000000000000, 478.000000000000)
                                        104BR-022and including490.0491.001.008.15104(490.0, 491.0]Interval(490.000000000000, 491.000000000000)
                                        105BR-022and including505.2508.753.5514.90105(505.2, 508.75]Interval(505.200000000000, 508.750000000000)
                                        106BR-022and including506.2506.700.50100.48106(506.2, 506.7]Interval(506.200000000000, 506.700000000000)
                                        107BR-022and600.0620.0020.000.52107(600.0, 620.0]Interval(600.000000000000, 620.000000000000)
                                        \n
                                        \n\n\n\n\n```\ncross_interval = itertools.product(dh.Interval_obj,dh.Interval_obj)\r\n\r\nintersect_matrix = np.array([interval[0].intersect(interval[1]) for interval in cross_interval])\r\n\r\nintersect_matrix\n```\n\n\n\n\n array([Interval(110.000000000000, 116.100000000000),\n Interval(111.400000000000, 113.100000000000), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n Interval(111.400000000000, 113.100000000000),\n Interval(111.400000000000, 113.100000000000), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n EmptySet(), EmptySet(),\n Interval(274.000000000000, 299.000000000000), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n EmptySet(), EmptySet(),\n Interval(432.900000000000, 439.000000000000),\n Interval(432.900000000000, 435.700000000000), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n Interval(432.900000000000, 435.700000000000),\n Interval(432.900000000000, 435.700000000000), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n EmptySet(), EmptySet(),\n Interval(445.000000000000, 452.000000000000), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n EmptySet(), EmptySet(),\n Interval(461.600000000000, 512.000000000000),\n Interval(471.000000000000, 512.000000000000),\n Interval(471.000000000000, 478.000000000000),\n Interval(490.000000000000, 491.000000000000),\n Interval(505.200000000000, 508.750000000000),\n Interval(506.200000000000, 506.700000000000), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n EmptySet(), Interval(471.000000000000, 512.000000000000),\n Interval(471.000000000000, 512.000000000000),\n Interval(471.000000000000, 478.000000000000),\n Interval(490.000000000000, 491.000000000000),\n Interval(505.200000000000, 508.750000000000),\n Interval(506.200000000000, 506.700000000000), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n EmptySet(), Interval(471.000000000000, 478.000000000000),\n Interval(471.000000000000, 478.000000000000),\n Interval(471.000000000000, 478.000000000000), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n Interval(490.000000000000, 491.000000000000),\n Interval(490.000000000000, 491.000000000000), EmptySet(),\n Interval(490.000000000000, 491.000000000000), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(),\n Interval(505.200000000000, 508.750000000000),\n Interval(505.200000000000, 508.750000000000), EmptySet(),\n EmptySet(), Interval(505.200000000000, 508.750000000000),\n Interval(506.200000000000, 506.700000000000), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n EmptySet(), Interval(506.200000000000, 506.700000000000),\n Interval(506.200000000000, 506.700000000000), EmptySet(),\n EmptySet(), Interval(506.200000000000, 506.700000000000),\n Interval(506.200000000000, 506.700000000000), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n EmptySet(), EmptySet(), EmptySet(), EmptySet(), EmptySet(),\n EmptySet(), EmptySet(),\n Interval(600.000000000000, 620.000000000000)], dtype=object)\n\n\n\n\n```\ndh['grade_len'] = dh['Gold (g/t)'] * dh['Width (m)']\n```\n\n /usr/local/lib/python3.6/dist-packages/ipykernel_launcher.py:1: SettingWithCopyWarning: \n A value is trying to be set on a copy of a slice from a DataFrame.\n Try using .loc[row_indexer,col_indexer] = value instead\n \n See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n \"\"\"Entry point for launching an IPython kernel.\n\n\n\n```\ndh\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Drill HoleUnnamed: 1From (m)To (m)Width (m)Gold (g/t)RecordIntervalInterval_objgrade_len
                                        95BR-022including110.0116.106.102.6295(110.0, 116.1]Interval(110.000000000000, 116.100000000000)15.982
                                        96BR-022and including111.4113.101.707.9296(111.4, 113.1]Interval(111.400000000000, 113.100000000000)13.464
                                        97BR-022and274.0299.0025.000.1997(274.0, 299.0]Interval(274.000000000000, 299.000000000000)4.750
                                        98BR-022and432.9439.006.104.0598(432.9, 439.0]Interval(432.900000000000, 439.000000000000)24.705
                                        99BR-022including432.9435.702.808.1899(432.9, 435.7]Interval(432.900000000000, 435.700000000000)22.904
                                        100BR-022and445.0452.007.000.41100(445.0, 452.0]Interval(445.000000000000, 452.000000000000)2.870
                                        101BR-022and461.6512.0050.401.78101(461.6, 512.0]Interval(461.600000000000, 512.000000000000)89.712
                                        102BR-022including471.0512.0041.002.09102(471.0, 512.0]Interval(471.000000000000, 512.000000000000)85.690
                                        103BR-022and including471.0478.007.002.37103(471.0, 478.0]Interval(471.000000000000, 478.000000000000)16.590
                                        104BR-022and including490.0491.001.008.15104(490.0, 491.0]Interval(490.000000000000, 491.000000000000)8.150
                                        105BR-022and including505.2508.753.5514.90105(505.2, 508.75]Interval(505.200000000000, 508.750000000000)52.895
                                        106BR-022and including506.2506.700.50100.48106(506.2, 506.7]Interval(506.200000000000, 506.700000000000)50.240
                                        107BR-022and600.0620.0020.000.52107(600.0, 620.0]Interval(600.000000000000, 620.000000000000)10.400
                                        \n
                                        \n\n\n\n\n```\nfrom sympy import Union\nimport functools\n\nresid_grades, resid_len = {}, {} \n\nfor node in levelord_nodes[1:]:\n\n parent_interval = dh[dh['Record'] == float(node[0])]\n child_names = [child.name for child in node[1]]\n child_intervals = [dh[dh['Record'] == float(child)] for child in child_names]\n\n new_interval_obj = parent_interval.Interval_obj.values[0] - Union([child.Interval_obj.values[0] for child in child_intervals])\n\n l_child_int = Union([intv['Interval_obj'].values[0] for intv in child_intervals])._measure\n\n lg_child_int = [dh.loc[int(child_name)]['grade_len'] for child_name in child_names]\n\n lg_total_int = parent_interval.grade_len.values[0]\n\n residual_grade = (lg_total_int - sum(lg_child_int)) / (new_interval_obj._measure)\n\n resid_grades[node[0]] = residual_grade\n resid_len[node[0]] = new_interval_obj\n\n print(\"Interval:\")\n print(node[0])\n\n print(\"Length x Grade:\")\n print(lg_total_int - sum(lg_child_int))\n\n print(\"Residual Grade:\")\n print(residual_grade)\n\n```\n\n Interval:\n 95\n Length x Grade:\n 2.517999999999999\n Residual Grade:\n 0.572272727272726\n Interval:\n 97\n Length x Grade:\n 4.75\n Residual Grade:\n 0.190000000000000\n Interval:\n 98\n Length x Grade:\n 1.801000000000002\n Residual Grade:\n 0.545757575757574\n Interval:\n 100\n Length x Grade:\n 2.8699999999999997\n Residual Grade:\n 0.410000000000000\n Interval:\n 101\n Length x Grade:\n 4.022000000000006\n Residual Grade:\n 0.427872340425534\n Interval:\n 107\n Length x Grade:\n 10.4\n Residual Grade:\n 0.520000000000000\n Interval:\n 96\n Length x Grade:\n 13.464\n Residual Grade:\n 7.92000000000005\n Interval:\n 99\n Length x Grade:\n 22.903999999999996\n Residual Grade:\n 8.17999999999997\n Interval:\n 102\n Length x Grade:\n 8.055000000000007\n Residual Grade:\n 0.273514431239389\n Interval:\n 103\n Length x Grade:\n 16.59\n Residual Grade:\n 2.37000000000000\n Interval:\n 104\n Length x Grade:\n 8.15\n Residual Grade:\n 8.15000000000000\n Interval:\n 105\n Length x Grade:\n 2.654999999999994\n Residual Grade:\n 0.870491803278683\n Interval:\n 106\n Length x Grade:\n 50.24\n Residual Grade:\n 100.480000000000\n\n\nCheck these solutions: 95 should be easy to validate\r\n\r\nincludes 96\r\n\r\n95 extends from 110.00 m to 116.10 m (l = 6.10 m)\r\n\r\n96 is the interval from 111.40 to 113.10 m (l = 1.70 m)\r\n\r\nLarger interval \r\n\r\n\r\n(6.10 m)(2.62 g/t) = (1.7 m)(7.92 g/t) + (4.4 m)(x g/t)\r\n\r\nRearranging terms, we get:\r\n\r\n$x = \\frac{(6.10 m)(2.62 g/t) - (1.7 m)(7.92 g/t)}{ 4.4 m }$\r\n\r\nSo the residual grade is 0.5723 g/t, which matches the value found above! \n\n\n```\n((6.10 * 2.62) - (1.7)*(7.92)) / (4.4)\n```\n\n\n\n\n 0.5722727272727269\n\n\n\n\n```\nresid_len\n```\n\n\n\n\n {'100': Interval(445.000000000000, 452.000000000000),\n '101': Interval.Ropen(461.600000000000, 471.000000000000),\n '102': Union(Interval.open(478.000000000000, 490.000000000000), Interval.open(491.000000000000, 505.200000000000), Interval.Lopen(508.750000000000, 512.000000000000)),\n '103': Interval(471.000000000000, 478.000000000000),\n '104': Interval(490.000000000000, 491.000000000000),\n '105': Union(Interval.Ropen(505.200000000000, 506.200000000000), Interval.Lopen(506.700000000000, 508.750000000000)),\n '106': Interval(506.200000000000, 506.700000000000),\n '107': Interval(600.000000000000, 620.000000000000),\n '95': Union(Interval.Ropen(110.000000000000, 111.400000000000), Interval.Lopen(113.100000000000, 116.100000000000)),\n '96': Interval(111.400000000000, 113.100000000000),\n '97': Interval(274.000000000000, 299.000000000000),\n '98': Interval.Lopen(435.700000000000, 439.000000000000),\n '99': Interval(432.900000000000, 435.700000000000)}\n\n\n\n\n```\nInterval(445.000000000000, 452.000000000000).end\n```\n\n\n\n\n 452.000000000000\n\n\n\n\n```\ndh['Record'].astype(str).map(resid_len)\r\ndh['Record'].astype(str).map(resid_grades)\n```\n\n\n\n\n 95 0.572272727272726\n 96 7.92000000000005\n 97 0.190000000000000\n 98 0.545757575757574\n 99 8.17999999999997\n 100 0.410000000000000\n 101 0.427872340425534\n 102 0.273514431239389\n 103 2.37000000000000\n 104 8.15000000000000\n 105 0.870491803278683\n 106 100.480000000000\n 107 0.520000000000000\n Name: Record, dtype: object\n\n\n\nTODO: Need to split up the non-contiguous segments so that you can actually plot them \r\n\r\n\r\nIdea: use .args attribute\r\nProblem: This is defined for Interval objects as well and it gets us something we don't want\n\n\n```\ndh['Record'].astype(str).map(resid_len)\n```\n\n\n\n\n 95 Union(Interval.Ropen(110.000000000000, 111.400...\n 96 Interval(111.400000000000, 113.100000000000)\n 97 Interval(274.000000000000, 299.000000000000)\n 98 Interval.Lopen(435.700000000000, 439.000000000...\n 99 Interval(432.900000000000, 435.700000000000)\n 100 Interval(445.000000000000, 452.000000000000)\n 101 Interval.Ropen(461.600000000000, 471.000000000...\n 102 Union(Interval.open(478.000000000000, 490.0000...\n 103 Interval(471.000000000000, 478.000000000000)\n 104 Interval(490.000000000000, 491.000000000000)\n 105 Union(Interval.Ropen(505.200000000000, 506.200...\n 106 Interval(506.200000000000, 506.700000000000)\n 107 Interval(600.000000000000, 620.000000000000)\n Name: Record, dtype: object\n\n\n\n\n```\n\r\n\r\ndef args_extended(interval):\r\n if type(interval) == Union:\r\n return interval.args\r\n else:\r\n return interval\r\n\r\nremaining_interval_grade = pd.DataFrame({'interval' : dh['Record'].astype(str).map(resid_len), 'grade' : dh['Record'].astype(str).map(resid_grades)})\r\n\r\nremaining_interval_grade['split_intervals'] = remaining_interval_grade.interval.apply(args_extended)\r\n\r\nrig_exploded = remaining_interval_grade.explode('split_intervals')\n```\n\n\n```\nrig_exploded\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        intervalgradesplit_intervals
                                        95Union(Interval.Ropen(110.000000000000, 111.400...0.572272727272726Interval.Ropen(110.000000000000, 111.400000000...
                                        95Union(Interval.Ropen(110.000000000000, 111.400...0.572272727272726Interval.Lopen(113.100000000000, 116.100000000...
                                        96Interval(111.400000000000, 113.100000000000)7.92000000000005Interval(111.400000000000, 113.100000000000)
                                        97Interval(274.000000000000, 299.000000000000)0.190000000000000Interval(274.000000000000, 299.000000000000)
                                        98Interval.Lopen(435.700000000000, 439.000000000...0.545757575757574Interval.Lopen(435.700000000000, 439.000000000...
                                        99Interval(432.900000000000, 435.700000000000)8.17999999999997Interval(432.900000000000, 435.700000000000)
                                        100Interval(445.000000000000, 452.000000000000)0.410000000000000Interval(445.000000000000, 452.000000000000)
                                        101Interval.Ropen(461.600000000000, 471.000000000...0.427872340425534Interval.Ropen(461.600000000000, 471.000000000...
                                        102Union(Interval.open(478.000000000000, 490.0000...0.273514431239389Interval.open(478.000000000000, 490.000000000000)
                                        102Union(Interval.open(478.000000000000, 490.0000...0.273514431239389Interval.open(491.000000000000, 505.200000000000)
                                        102Union(Interval.open(478.000000000000, 490.0000...0.273514431239389Interval.Lopen(508.750000000000, 512.000000000...
                                        103Interval(471.000000000000, 478.000000000000)2.37000000000000Interval(471.000000000000, 478.000000000000)
                                        104Interval(490.000000000000, 491.000000000000)8.15000000000000Interval(490.000000000000, 491.000000000000)
                                        105Union(Interval.Ropen(505.200000000000, 506.200...0.870491803278683Interval.Ropen(505.200000000000, 506.200000000...
                                        105Union(Interval.Ropen(505.200000000000, 506.200...0.870491803278683Interval.Lopen(506.700000000000, 508.750000000...
                                        106Interval(506.200000000000, 506.700000000000)100.480000000000Interval(506.200000000000, 506.700000000000)
                                        107Interval(600.000000000000, 620.000000000000)0.520000000000000Interval(600.000000000000, 620.000000000000)
                                        \n
                                        \n\n\n\n\n```\nrig_exploded['From'] = rig_exploded.split_intervals.apply(lambda x: x.start).astype(float)\r\nrig_exploded['To'] = rig_exploded.split_intervals.apply(lambda x: x.end).astype(float)\r\nrig_exploded['Width'] = (rig_exploded.To - rig_exploded.From).astype(float)\r\n\r\nrig_exploded['grade'] = rig_exploded.grade.astype(float)\r\n\r\nrig_exploded['drillhole'] = 'BR-022'\t\n```\n\n\n```\nrig_exploded[['From', 'To', 'grade', 'Width']]\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        FromTogradeWidth
                                        95110.00111.400.5722731.40
                                        95113.10116.100.5722733.00
                                        96111.40113.107.9200001.70
                                        97274.00299.000.19000025.00
                                        98435.70439.000.5457583.30
                                        99432.90435.708.1800002.80
                                        100445.00452.000.4100007.00
                                        101461.60471.000.4278729.40
                                        102478.00490.000.27351412.00
                                        102491.00505.200.27351414.20
                                        102508.75512.000.2735143.25
                                        103471.00478.002.3700007.00
                                        104490.00491.008.1500001.00
                                        105505.20506.200.8704921.00
                                        105506.70508.750.8704922.05
                                        106506.20506.70100.4800000.50
                                        107600.00620.000.52000020.00
                                        \n
                                        \n\n\n\n\n```\n\r\ny_axis = alt.Axis(\r\n title='Intercept ID',\r\n offset=5,\r\n ticks=False,\r\n domain=False\r\n)\r\n\r\n\r\nreqd_cols = ['From', 'To', 'grade', 'Width', 'drillhole']\r\n\r\n\r\nalt.Chart(rig_exploded[reqd_cols]).mark_bar().encode(\r\n alt.X('From:Q',\r\n scale=alt.Scale(zero=False)),\r\n x2='To:Q',\r\n y=alt.Y('drillhole:N', axis=y_axis),\r\n color=alt.Color('grade:Q', scale=alt.Scale(scheme=\"inferno\")),\r\n tooltip=[\r\n alt.Tooltip('Width:Q', title='Width'),\r\n alt.Tooltip('grade:Q', title='Gold Grade')\r\n ]\r\n).properties(width=800, height=100).configure(background='#D9E9F0').interactive()\n```\n\n\n\n\n\n
                                        \n\n\n\n", "meta": {"hexsha": "cc56332e7e1f4143ce64e48c6d264348f5419885", "size": 152751, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "_notebooks/2020-12-02-Mathematics-of-Drilling-Intercepts.ipynb", "max_stars_repo_name": "nshea3/blog", "max_stars_repo_head_hexsha": "782287d448e7a7100820b93c0e81b12b846255f7", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "_notebooks/2020-12-02-Mathematics-of-Drilling-Intercepts.ipynb", "max_issues_repo_name": "nshea3/blog", "max_issues_repo_head_hexsha": "782287d448e7a7100820b93c0e81b12b846255f7", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2021-09-28T05:39:21.000Z", "max_issues_repo_issues_event_max_datetime": "2021-12-07T04:13:51.000Z", "max_forks_repo_path": "_notebooks/2020-12-02-Mathematics-of-Drilling-Intercepts.ipynb", "max_forks_repo_name": "nshea3/blog", "max_forks_repo_head_hexsha": "782287d448e7a7100820b93c0e81b12b846255f7", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 55.6875683558, "max_line_length": 17266, "alphanum_fraction": 0.5391846862, "converted": true, "num_tokens": 15286, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.596433160611502, "lm_q2_score": 0.7122321903471563, "lm_q1q2_score": 0.42479889637800733}} {"text": "## Thi\u1ebft K\u1ebf H\u1ed3i Quy Gi\u00e1n \u0110o\u1ea1n\n\nD\u00f9 kh\u00f4ng ng\u1eabm ngh\u0129 nhi\u1ec1u v\u1ec1 n\u00f3 nh\u01b0ng ch\u00fang ta kh\u00f4ng kh\u1ecfi \u1ea5n t\u01b0\u1ee3ng b\u1edfi t\u00ednh li\u00ean t\u1ee5c c\u1ee7a t\u1ef1 nhi\u00ean. B\u1ea1n kh\u00f4ng th\u1ec3 tr\u1ed3ng c\u00e2y m\u00e0 n\u1ee5 kh\u00f4ng n\u1edf tr\u01b0\u1edbc, b\u1ea1n kh\u00f4ng th\u1ec3 d\u1ecbch chuy\u1ec3n t\u1ee9c th\u1eddi t\u1eeb n\u01a1i n\u00e0y sang n\u01a1i kh\u00e1c, v\u1ebft th\u01b0\u01a1ng c\u1ea7n th\u1eddi gian \u0111\u1ec3 ch\u1eefa l\u00e0nh. Ngay c\u1ea3 trong l\u0129nh v\u1ef1c x\u00e3 h\u1ed9i, s\u1ef1 li\u00ean t\u1ee5c d\u01b0\u1eddng nh\u01b0 l\u00e0 m\u1ed9t chu\u1ea9n m\u1ef1c. B\u1ea1n kh\u00f4ng th\u1ec3 ph\u00e1t tri\u1ec3n doanh nghi\u1ec7p ch\u1ec9 trong m\u1ed9t ng\u00e0y, c\u1ea7n ph\u1ea3i c\u00f3 s\u1ef1 ki\u00ean \u0111\u1ecbnh v\u00e0 ch\u0103m ch\u1ec9 \u0111\u1ec3 t\u00edch lu\u1ef9 c\u1ee7a c\u1ea3i v\u00e0 ph\u1ea3i m\u1ea5t \u0111\u1ebfn nhi\u1ec1u n\u0103m \u0111\u1ec3 b\u1ea1n n\u1eafm \u0111\u01b0\u1ee3c c\u00e1ch h\u1ed3i quy tuy\u1ebfn t\u00ednh ho\u1ea1t \u0111\u1ed9ng. Trong nh\u1eefng tr\u01b0\u1eddng h\u1ee3p th\u01b0\u1eddng th\u1ea5y, t\u1ef1 nhi\u00ean r\u1ea5t c\u00f3 t\u00ednh li\u00ean k\u1ebft v\u00e0 kh\u00f4ng thay \u0111\u1ed5i nhi\u1ec1u.\n\n```\nL\u00e0m sao \u0111em h\u1ebft x\u00e1c h\u1ed3n,\nH\u00f2a m\u00ecnh v\u1edbi \u0110\u1ea1o ch\u1eb3ng c\u00f2n l\u00eca xa.\n```\n\\- \u0110\u1ea1o \u0110\u1ee9c Kinh, L\u00e3o T\u1eed.\n\nC\u00f3 ngh\u0129a l\u00e0 **khi ch\u00fang ta nh\u00ecn th\u1ea5y nh\u1eefng thay \u0111\u1ed5i \u0111\u1ed9t ng\u1ed9t, c\u00f3 th\u1ec3 ch\u00fang l\u00e0 nh\u00e2n t\u1ea1o**, hay n\u00f3i c\u00e1ch kh\u00e1c l\u00e0 nh\u1eefng t\u00ecnh hu\u1ed1ng do con ng\u01b0\u1eddi t\u1ea1o ra. Nh\u1eefng s\u1ef1 ki\u1ec7n n\u00e0y th\u01b0\u1eddng \u0111i k\u00e8m v\u1edbi nh\u1eefng gi\u1ea3 t\u01b0\u1edfng \u0111\u1ed1i v\u1edbi nh\u1eefng chuy\u1ec7n th\u00f4ng th\u01b0\u1eddng: n\u1ebfu m\u1ed9t \u0111i\u1ec1u k\u1ef3 l\u1ea1 x\u1ea3y ra, n\u00f3 cho ch\u00fang ta m\u1ed9t s\u1ed1 hi\u1ec3u bi\u1ebft v\u1ec1 nh\u1eefng g\u00ec c\u00f3 th\u1ec3 \u0111\u00e3 x\u1ea3y ra n\u1ebfu t\u1ef1 nhi\u00ean v\u1eadn h\u00e0nh theo m\u1ed9t h\u01b0\u1edbng kh\u00e1c. Kh\u00e1m ph\u00e1 nh\u1eefng bi\u1ebfn \u0111\u1ed5i nh\u00e2n t\u1ea1o n\u00e0y l\u00e0 c\u1ed1t l\u00f5i c\u1ee7a m\u00f4 h\u00ecnh Thi\u1ebft K\u1ebf H\u1ed3i Quy Gi\u00e1n \u0110o\u1ea1n.\n\n\n\nThi\u1ebft l\u1eadp c\u01a1 b\u1ea3n c\u00f3 d\u1ea1ng nh\u01b0 sau. H\u00e3y t\u01b0\u1edfng t\u01b0\u1ee3ng b\u1ea1n c\u00f3 m\u1ed9t bi\u1ebfn can thi\u1ec7p \\\\(T\\\\) v\u00e0 c\u00e1c k\u1ebft qu\u1ea3 ti\u1ec1m n\u0103ng \\\\(Y_0\\\\) v\u00e0 \\\\(Y_1\\\\). Can thi\u1ec7p T l\u00e0 m\u1ed9t h\u00e0m gi\u00e1n \u0111o\u1ea1n c\u1ee7a bi\u1ebfn ch\u1ec9 \u0111\u1ecbnh quan s\u00e1t \u0111\u01b0\u1ee3c \\\\(R\\\\) sao cho\n\n\n$\nD_i = \\mathcal{1}\\{R_i>c\\}\n$\n\nN\u00f3i c\u00e1ch kh\u00e1c, can thi\u1ec7p c\u00f3 gi\u00e1 tr\u1ecb 0 khi \\\\(R\\\\) d\u01b0\u1edbi ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh \\\\(c\\\\) v\u00e0 c\u00f3 gi\u00e1 tr\u1ecb 1 \u0111\u1ed1i v\u1edbi tr\u01b0\u1eddng h\u1ee3p ng\u01b0\u1ee3c l\u1ea1i. \u0110i\u1ec1u n\u00e0y c\u00f3 ngh\u0129a l\u00e0 ch\u00fang ta ph\u1ea3i quan s\u00e1t \\\\(Y_1\\\\) khi \\\\(R>c\\\\) v\u00e0 \\\\(Y_0\\\\) khi \\\\(R\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        agecellallmvasuicide
                                        019.06849392.82540035.82932711.203714
                                        119.15068495.10074035.63925612.193368
                                        219.23287692.14429534.20565011.715812
                                        319.31507088.42776032.27895711.275010
                                        419.39726088.70494032.65096710.984314
                                        \n
                                        \n\n\n\n\u0110\u1ec3 d\u1ec5 h\u00ecnh dung, c\u0169ng nh\u01b0 v\u00ec m\u1ed9t l\u00ed do quan tr\u1ecdng kh\u00f4ng k\u00e9m \u1edf ph\u1ea7n sau, ch\u00fang ta s\u1ebd t\u1eadp trung bi\u1ebfn ch\u1ec9 \u0111\u1ecbnh `agecell` t\u1ea1i ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh \u1edf tu\u1ed5i 21.\n\n\n```python\ndrinking[\"agecell\"] -= 21\n```\n\nN\u1ebfu ch\u00fang ta v\u1ebd bi\u1ec3u \u0111\u1ed3 c\u1ee7a nhi\u1ec1u bi\u1ebfn k\u1ebft qu\u1ea3 (`all`,`mva`, `suicide`) v\u1edbi bi\u1ebfn ch\u1ec9 \u0111\u1ecbnh tr\u00ean tr\u1ee5c x, ch\u00fang ta th\u1ea5y \u0111\u01b0\u1ee3c r\u1eb1ng t\u1ef7 l\u1ec7 t\u1eed vong c\u0169ng t\u0103ng l\u00ean khi ta v\u01b0\u1ee3t qua \u0111\u1ed9 tu\u1ed5i u\u1ed1ng r\u01b0\u1ee3u h\u1ee3p ph\u00e1p.\n\n\n```python\nplt.figure(figsize=(8,8))\nax = plt.subplot(3,1,1)\ndrinking.plot.scatter(x=\"agecell\", y=\"all\", ax=ax)\nplt.title(\"Death Cause by Age (Centered at 0)\")\n\nax = plt.subplot(3,1,2, sharex=ax)\ndrinking.plot.scatter(x=\"agecell\", y=\"mva\", ax=ax)\n\nax = plt.subplot(3,1,3, sharex=ax)\ndrinking.plot.scatter(x=\"agecell\", y=\"suicide\", ax=ax);\n\n```\n\nC\u00f3 m\u1ed9t s\u1ed1 d\u1ea5u hi\u1ec7u, nh\u01b0ng ch\u00fang ta c\u1ea7n nhi\u1ec1u h\u01a1n th\u1ebf. Ch\u00ednh x\u00e1c th\u00ec vi\u1ec7c u\u1ed1ng r\u01b0\u1ee3u bia t\u00e1c \u0111\u1ed9ng l\u00ean t\u1ef7 l\u1ec7 t\u1eed vong t\u1ea1i ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh n\u00e0o? V\u00e0 sai s\u1ed1 chu\u1ea9n cho \u01b0\u1edbc l\u01b0\u1ee3ng \u0111\u00f3 l\u00e0 bao nhi\u00eau?\n\n## \u01af\u1edbc l\u01b0\u1ee3ng RDD \n\nGi\u1ea3 thi\u1ebft ch\u1ee7 \u0111\u1ea1o m\u00e0 RDD d\u1ef1a v\u00e0o l\u00e0 s\u1ef1 li\u00ean t\u1ee5c c\u1ee7a k\u1ebft qu\u1ea3 ti\u1ec1m n\u0103ng t\u1ea1i ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh. N\u00f3i m\u1ed9t c\u00e1ch ch\u00ednh x\u00e1c th\u00ec gi\u1edbi h\u1ea1n c\u1ee7a c\u00e1c k\u1ebft qu\u1ea3 ti\u1ec1m n\u0103ng khi bi\u1ebfn ch\u1ec9 \u0111\u1ecbnh ti\u1ec7m c\u1eadn ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh t\u1eeb b\u00ean ph\u1ea3i v\u00e0 b\u00ean tr\u00e1i ph\u1ea3i nh\u01b0 nhau.\n\n$$\n\\lim_{r \\to c^-} E[Y_{ti}|R_i=r] = \\lim_{r \\to c^+} E[Y_{ti}|R_i=r]\n$$\n\nN\u1ebfu \u0111i\u1ec1u ki\u1ec7n n\u00e0y tho\u1ea3 m\u00e3n, ch\u00fang ta c\u00f3 th\u1ec3 t\u00ecm \u0111\u01b0\u1ee3c t\u00e1c \u0111\u1ed9ng nh\u00e2n qu\u1ea3 t\u1ea1i ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh\n\n\\begin{align}\n\\lim_{r \\to c^+} E[Y_{ti}|R_i=r] - \\lim_{r \\to c^-} E[Y_{ti}|R_i=r]=&\\lim_{r \\to c^+} E[Y_{1i}|R_i=r] - \\lim_{r \\to c^-} E[Y_{0i}|R_i=r] \\\\\n=& E[Y_{1i}|R_i=r] - E[Y_{0i}|R_i=r] \\\\\n=& E[Y_{1i} - Y_{0i}|R_i=r]\n\\end{align}\n\nTheo m\u1ed9t c\u00e1ch n\u00e0o \u0111\u00f3 th\u00ec \u0111\u00e2y ch\u00ednh l\u00e0 m\u1ed9t d\u1ea1ng c\u1ee7a T\u00e1c \u0110\u1ed9ng Can Thi\u1ec7p B\u00ecnh Qu\u00e2n C\u1ee5c B\u1ed9 (LATE), v\u00ec ch\u00fang ta ch\u1ec9 c\u00f3 th\u1ec3 bi\u1ebft n\u00f3 t\u1ea1i ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh. Trong thi\u1ebft l\u1eadp n\u00e0y, ch\u00fang ta c\u00f3 th\u1ec3 coi RDD nh\u01b0 m\u1ed9t th\u1eed nghi\u1ec7m ng\u1eabu nhi\u00ean c\u1ee5c b\u1ed9. \u0110\u1ed1i v\u1edbi nh\u1eefng ng\u01b0\u1eddi t\u1ea1i ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh, can thi\u1ec7p c\u00f3 th\u1ec3 di\u1ec5n ra m\u1ed9t c\u00e1ch t\u00ecnh c\u1edd theo c\u1ea3 hai h\u01b0\u1edbng, m\u1ed9t s\u1ed1 ng\u01b0\u1eddi n\u1eb1m d\u01b0\u1edbi ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh, v\u00e0 m\u1ed9t s\u1ed1 ng\u01b0\u1eddi l\u1ea1i n\u1eb1m tr\u00ean. Trong v\u00ed d\u1ee5 c\u1ee7a ch\u00fang ta, t\u1ea1i c\u00f9ng m\u1ed9t th\u1eddi \u0111i\u1ec3m, m\u1ed9t s\u1ed1 ng\u01b0\u1eddi ch\u1ec9 v\u1eeba qua \u0111\u1ed9 tu\u1ed5i 21 v\u00e0 m\u1ed9t s\u1ed1 ng\u01b0\u1eddi s\u1eafp tr\u00f2n 21 tu\u1ed5i. Nh\u1eefng th\u1ee9 quy\u1ebft \u0111\u1ecbnh \u0111i\u1ec1u n\u00e0y l\u00e0 li\u1ec7u c\u00f3 ai \u0111\u00f3 \u0111\u01b0\u1ee3c sinh ra sau \u0111\u00f3 v\u00e0i ng\u00e0y hay kh\u00f4ng, \u0111i\u1ec1u n\u00e0y kh\u00e1 ng\u1eabu nhi\u00ean. V\u00ec l\u00fd do n\u00e0y, RDD cho th\u1ea5y m\u1ed9t c\u00e2u chuy\u1ec7n v\u1ec1 nh\u00e2n qu\u1ea3 r\u1ea5t h\u1ea5p d\u1eabn. N\u00f3 kh\u00f4ng ph\u1ea3i l\u00e0 ti\u00eau chu\u1ea9n v\u00e0ng nh\u01b0 RCT, nh\u01b0ng n\u00f3 c\u0169ng g\u1ea7n nh\u01b0 th\u1ebf.\n\nB\u00e2y gi\u1edd, \u0111\u1ec3 \u01b0\u1edbc l\u01b0\u1ee3ng t\u00e1c \u0111\u1ed9ng can thi\u1ec7p t\u1ea1i ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh, t\u1ea5t c\u1ea3 nh\u1eefng g\u00ec ch\u00fang ta c\u1ea7n l\u00e0m l\u00e0 \u01b0\u1edbc l\u01b0\u1ee3ng c\u1ea3 hai gi\u1edbi h\u1ea1n trong c\u00f4ng th\u1ee9c tr\u00ean v\u00e0 so s\u00e1nh ch\u00fang. C\u00e1ch \u0111\u01a1n gi\u1ea3n nh\u1ea5t \u0111\u1ec3 l\u00e0m \u0111i\u1ec1u \u0111\u00f3 l\u00e0 ch\u1ea1y m\u1ed9t m\u00f4 h\u00ecnh h\u1ed3i quy tuy\u1ebfn t\u00ednh\n\n\n\n\u0110\u1ec3 cho n\u00f3 ho\u1ea1t \u0111\u1ed9ng, ch\u00fang ta cho bi\u1ebfn gi\u1ea3 tr\u00ean ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh t\u00e1c \u0111\u1ed9ng l\u00ean bi\u1ebfn ch\u1ec9 \u0111\u1ecbnh\n\n$\ny_i = \\beta_0 + \\beta_1 r_i + \\beta_2 \\mathcal{1}\\{r_i>c\\} + \\beta_3 \\mathcal{1}\\{r_i>c\\} r_i\n$\n\n\u0110i\u1ec1u m\u00e0 ta \u0111ang l\u00e0m l\u00e0 \u00e1p m\u1ed9t h\u1ed3i quy tuy\u1ebfn t\u00ednh tr\u00ean ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh v\u00e0 m\u1ed9t h\u1ed3i quy kh\u00e1c \u1edf d\u01b0\u1edbi ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh \u0111\u00f3. Tham s\u1ed1 \\\\(\\beta_0\\\\) l\u00e0 h\u1ec7 s\u1ed1 ch\u1eb7n c\u1ee7a h\u1ed3i quy d\u01b0\u1edbi ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh v\u00e0 \\\\(\\beta_0+\\beta_2\\\\) l\u00e0 h\u1ec7 s\u1ed1 ch\u1eb7n c\u1ee7a h\u1ed3i quy tr\u00ean ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh.\n\nV\u00e0 \u0111\u00e2y l\u00e0 l\u00fac th\u1ee7 thu\u1eadt \u0111\u1eb7t g\u1ed1c t\u1ecda \u0111\u1ed9 0 t\u1ea1i ng\u01b0\u1ee1ng ch\u1ec9 \u0111\u1ecbnh can thi\u1ec7p ph\u00e1t huy t\u00e1c d\u1ee5ng. Sau b\u01b0\u1edbc ti\u1ec1n x\u1eed l\u00fd n\u00e0y, ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh s\u1ebd tr\u1edf th\u00e0nh 0. \u0110i\u1ec1u n\u00e0y khi\u1ebfn h\u1ec7 s\u1ed1 ch\u1eb7n \\\\(\\beta_0\\\\) l\u00e0 gi\u00e1 tr\u1ecb \u0111\u01b0\u1ee3c d\u1ef1 \u0111o\u00e1n \u1edf ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh, cho h\u1ed3i quy b\u00ean d\u01b0\u1edbi n\u00f3. N\u00f3i c\u00e1ch kh\u00e1c, \\\\(\\beta_0=\\lim_{r \\to c^-} E[Y_{ti}|R_i=r]\\\\). V\u1edbi l\u1eadp lu\u1eadn t\u01b0\u01a1ng t\u1ef1, \\\\(\\beta_0+\\beta_2\\\\) l\u00e0 gi\u1edbi h\u1ea1n c\u1ee7a k\u1ebft qu\u1ea3 \u1edf tr\u00ean. Ngh\u0129a l\u00e0\n\n$\n\\lim_{r \\to c^+} E[Y_{ti}|R_i=r] - \\lim_{r \\to c^-} E[Y_{ti}|R_i=r]=\\beta_2=E[ATE|R=c]\n$\n\n\u0110i\u1ec1u n\u00e0y \u0111\u01b0\u1ee3c th\u1ec3 hi\u1ec7n qua \u0111o\u1ea1n code d\u01b0\u1edbi \u0111\u00e2y trong tr\u01b0\u1eddng h\u1ee3p ch\u00fang ta \u01b0\u1edbc l\u01b0\u1ee3ng t\u00e1c \u0111\u1ed9ng c\u1ee7a vi\u1ec7c u\u1ed1ng r\u01b0\u1ee3u \u0111\u1ebfn t\u1ef7 l\u1ec7 t\u1eed vong g\u00e2y ra b\u1edfi m\u1ecdi nguy\u00ean nh\u00e2n \u1edf \u0111\u1ed9 tu\u1ed5i 21.\n\n\n```python\nrdd_df = drinking.assign(threshold=(drinking[\"agecell\"] > 0).astype(int))\n\nmodel = smf.wls(\"all~agecell*threshold\", rdd_df).fit()\n\nmodel.summary().tables[1]\n```\n\n\n\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        coef std err t P>|t| [0.025 0.975]
                                        Intercept 93.6184 0.932 100.399 0.000 91.739 95.498
                                        agecell 0.8270 0.819 1.010 0.318 -0.823 2.477
                                        threshold 7.6627 1.319 5.811 0.000 5.005 10.320
                                        agecell:threshold -3.6034 1.158 -3.111 0.003 -5.937 -1.269
                                        \n\n\n\nM\u00f4 h\u00ecnh n\u00e0y cho ch\u00fang ta bi\u1ebft r\u1eb1ng t\u1ef7 l\u1ec7 t\u1eed vong t\u0103ng 7.6627 \u0111i\u1ec3m khi u\u1ed1ng r\u01b0\u1ee3u. M\u1ed9t c\u00e1ch n\u00f3i kh\u00e1c l\u00e0 r\u01b0\u1ee3u l\u00e0m t\u0103ng 8% nguy c\u01a1 t\u1eed vong g\u00e2y ra b\u1edfi m\u1ecdi nguy\u00ean nh\u00e2n ((7.6627+93.6184)/93.6184). L\u01b0u \u00fd r\u1eb1ng \u0111i\u1ec1u n\u00e0y c\u0169ng cung c\u1ea5p cho ch\u00fang ta c\u00e1c sai s\u1ed1 chu\u1ea9n cho \u01b0\u1edbc l\u01b0\u1ee3ng c\u1ee7a t\u00e1c \u0111\u1ed9ng nh\u00e2n qu\u1ea3. Trong tr\u01b0\u1eddng h\u1ee3p n\u00e0y, t\u00e1c \u0111\u1ed9ng c\u00f3 \u00fd ngh\u0129a th\u1ed1ng k\u00ea, v\u00ec tr\u1ecb s\u1ed1 p c\u00f3 gi\u00e1 tr\u1ecb d\u01b0\u1edbi 0.01.\n\nN\u1ebfu ch\u00fang ta mu\u1ed1n x\u00e1c minh m\u00f4 h\u00ecnh n\u00e0y b\u1eb1ng \u0111\u1ed3 th\u1ecb, ta c\u00f3 th\u1ec3 bi\u1ec3u th\u1ecb c\u00e1c gi\u00e1 tr\u1ecb d\u1ef1 \u0111o\u00e1n tr\u00ean d\u1eef li\u1ec7u m\u00e0 ch\u00fang ta c\u00f3. B\u1ea1n c\u00f3 th\u1ec3 th\u1ea5y r\u1eb1ng ch\u00fang ta c\u00f3 2 m\u00f4 h\u00ecnh h\u1ed3i quy: m\u1ed9t cho nh\u1eefng ng\u01b0\u1eddi c\u00f3 \u0111\u1ed9 tu\u1ed5i tr\u00ean 21 v\u00e0 m\u1ed9t cho \u0111\u1ed9 tu\u1ed5i d\u01b0\u1edbi 21.\n\n\n```python\nax = drinking.plot.scatter(x=\"agecell\", y=\"all\", color=\"C0\")\ndrinking.assign(predictions=model.fittedvalues).plot(x=\"agecell\", y=\"predictions\", ax=ax, color=\"C1\")\nplt.title(\"Regression Discontinuity\");\n```\n\nN\u1ebfu ta l\u00e0m t\u01b0\u01a1ng t\u1ef1 cho nh\u1eefng t\u00e1c nh\u00e2n kh\u00e1c, ta s\u1ebd c\u00f3 nh\u01b0 sau\n\n\n```python\nplt.figure(figsize=(8,8))\n\nfor p, cause in enumerate([\"all\", \"mva\", \"suicide\"], 1):\n ax = plt.subplot(3,1,p)\n drinking.plot.scatter(x=\"agecell\", y=cause, ax=ax)\n m = smf.wls(f\"{cause}~agecell*threshold\", rdd_df).fit()\n ate_pct = 100*((m.params[\"threshold\"] + m.params[\"Intercept\"])/m.params[\"Intercept\"] - 1)\n drinking.assign(predictions=m.fittedvalues).plot(x=\"agecell\", y=\"predictions\", ax=ax, color=\"C1\")\n plt.title(f\"Impact of Alcohol on Death: {np.round(ate_pct, 2)}%\")\n\nplt.tight_layout()\n```\n\nRDD cho th\u1ea5y u\u1ed1ng r\u01b0\u1ee3u l\u00e0m t\u0103ng 15% nguy c\u01a1 t\u1eed vong do t\u1ef1 t\u1eed v\u00e0 tai n\u1ea1n xe h\u01a1i, \u0111\u00e2y l\u00e0 m\u1ed9t con s\u1ed1 kh\u00e1 l\u1edbn. Nh\u1eefng k\u1ebft qu\u1ea3 n\u00e0y l\u00e0 nh\u1eefng l\u00fd l\u1ebd thuy\u1ebft ph\u1ee5c \u0111\u1ec3 kh\u00f4ng h\u1ea1 \u0111\u1ed9 tu\u1ed5i u\u1ed1ng r\u01b0\u1ee3u t\u1ed1i thi\u1ec3u n\u1ebfu ch\u00fang ta mu\u1ed1n gi\u1ea3m thi\u1ec3u t\u1ef7 l\u1ec7 t\u1eed vong.\n\n### T\u1ef7 Tr\u1ecdng Kernel\n\nH\u1ed3i Quy Gi\u00e1n \u0110o\u1ea1n ph\u1ea7n l\u1edbn d\u1ef1a v\u00e0o c\u00e1c \u0111\u1eb7c \u0111i\u1ec3m ngo\u1ea1i suy c\u1ee7a h\u1ed3i quy tuy\u1ebfn t\u00ednh. V\u00ec ch\u00fang ta \u0111ang xem x\u00e9t c\u00e1c gi\u00e1 tr\u1ecb t\u1ea1i \u0111i\u1ec3m \u0111\u1ea7u v\u00e0 cu\u1ed1i c\u1ee7a 2 \u0111\u01b0\u1eddng h\u1ed3i quy, t\u1ed1t h\u01a1n h\u1ebft ch\u00fang ta n\u00ean x\u00e1c \u0111\u1ecbnh \u0111\u00fang c\u00e1c gi\u1edbi h\u1ea1n \u0111\u00f3. Nh\u1eefng g\u00ec c\u00f3 th\u1ec3 x\u1ea3y ra l\u00e0 h\u1ed3i quy c\u00f3 th\u1ec3 \u0111\u00e3 t\u1eadp trung qu\u00e1 nhi\u1ec1u v\u00e0o vi\u1ec7c kh\u1edbp c\u00e1c \u0111i\u1ec3m d\u1eef li\u1ec7u kh\u00e1c \u0111\u1ec3 \u0111\u1ed5i l\u1ea1i t\u1ef7 l\u1ec7 kh\u1edbp \u0111\u00e1ng th\u1ea5t v\u1ecdng t\u1ea1i ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh. N\u1ebfu \u0111i\u1ec1u n\u00e0y x\u1ea3y ra, r\u1ea5t c\u00f3 th\u1ec3 ch\u00fang ta \u0111\u00e3 \u0111o l\u01b0\u1eddng sai t\u00e1c \u0111\u1ed9ng can thi\u1ec7p.\n\nM\u1ed9t c\u00e1ch \u0111\u1ec3 gi\u1ea3i quy\u1ebft v\u1ea5n \u0111\u1ec1 tr\u00ean l\u00e0 g\u00e1n tr\u1ecdng s\u1ed1 cao h\u01a1n cho c\u00e1c \u0111i\u1ec3m n\u1eb1m g\u1ea7n ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh h\u01a1n. C\u00f3 nhi\u1ec1u c\u00e1ch \u0111\u1ec3 l\u00e0m \u0111\u01b0\u1ee3c \u0111i\u1ec1u n\u00e0y, nh\u01b0ng m\u1ed9t c\u00e1ch ph\u1ed5 bi\u1ebfn l\u00e0 \u0111i\u1ec1u ch\u1ec9nh l\u1ea1i tr\u1ecdng s\u1ed1 cho c\u00e1c m\u1eabu th\u00f4ng qua **h\u00e0m tam gi\u00e1c kernel**\n\n$\nK(R, c, h) = \\mathcal{1}\\{|R-c| \\leq h\\} * \\bigg(1-\\frac{|R-c|}{h}\\bigg)\n$\n\nPh\u1ea7n \u0111\u1ea7u ti\u00ean c\u1ee7a kernel l\u00e0 m\u1ed9t h\u00e0m ch\u1ec9 s\u1ed1 cho bi\u1ebft ch\u00fang ta \u0111\u00e3 g\u1ea7n ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh hay ch\u01b0a. G\u1ea7n nh\u01b0 th\u1ebf n\u00e0o? \u0110i\u1ec1u n\u00e0y \u0111\u01b0\u1ee3c x\u00e1c \u0111\u1ecbnh b\u1edfi tham s\u1ed1 d\u1ea3i t\u1ea7n su\u1ea5t \\\\(h\\\\). Ph\u1ea7n th\u1ee9 hai c\u1ee7a kernel l\u00e0 m\u1ed9t h\u00e0m t\u1ef7 tr\u1ecdng. Khi ch\u00fang ta r\u1eddi xa ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh, tr\u1ecdng s\u1ed1 ng\u00e0y c\u00e0ng nh\u1ecf h\u01a1n. C\u00e1c tr\u1ecdng s\u1ed1 \u0111\u01b0\u1ee3c chia cho d\u1ea3i t\u1ea7n su\u1ea5t. N\u1ebfu d\u1ea3i t\u1ea7n su\u1ea5t l\u1edbn, tr\u1ecdng s\u1ed1 s\u1ebd gi\u1ea3m d\u1ea7n v\u1edbi t\u1ed1c \u0111\u1ed9 ch\u1eadm. N\u1ebfu d\u1ea3i t\u1ea7n su\u1ea5t nh\u1ecf, tr\u1ecdng s\u1ed1 s\u1ebd nhanh ch\u00f3ng ti\u1ebfn v\u1ec1 0.\n\n\u0110\u1ec3 d\u1ec5 hi\u1ec3u h\u01a1n, ta s\u1ebd bi\u1ec3u di\u1ec5n c\u00e1c t\u1ef7 tr\u1ecdng \u0111\u1ed1i v\u1edbi kernel \u0111\u01b0\u1ee3c \u00e1p d\u1ee5ng trong b\u00e0i to\u00e1n n\u00e0y. T\u00f4i \u0111\u00e3 thi\u1ebft l\u1eadp d\u1ea3i t\u1ea7n su\u1ea5t t\u1ea1i \u0111\u00e2y l\u00e0 1, c\u00f3 ngh\u0129a l\u00e0 ch\u00fang ta s\u1ebd ch\u1ec9 xem x\u00e9t d\u1eef li\u1ec7u t\u1eeb nh\u1eefng ng\u01b0\u1eddi kh\u00f4ng qu\u00e1 22 tu\u1ed5i v\u00e0 kh\u00f4ng d\u01b0\u1edbi 20 tu\u1ed5i.\n\n\n```python\ndef kernel(R, c, h):\n indicator = (np.abs(R-c) <= h).astype(float)\n return indicator * (1 - np.abs(R-c)/h)\n```\n\n\n```python\nplt.plot(drinking[\"agecell\"], kernel(drinking[\"agecell\"], c=0, h=1))\nplt.xlabel(\"agecell\")\nplt.ylabel(\"Weight\")\nplt.title(\"Kernel Weight by Age\");\n```\n\nN\u1ebfu ch\u00fang ta \u00e1p d\u1ee5ng nh\u1eefng tr\u1ecdng s\u1ed1 n\u00e0y cho b\u00e0i to\u00e1n \u0111\u1ea7u ti\u00ean, t\u00e1c \u0111\u1ed9ng c\u1ee7a r\u01b0\u1ee3u s\u1ebd l\u1edbn h\u01a1n, \u00edt nh\u1ea5t l\u00e0 \u0111\u1ed1i v\u1edbi tr\u01b0\u1eddng h\u1ee3p t\u1eed vong g\u00e2y ra b\u1edfi m\u1ecdi nguy\u00ean nh\u00e2n. K\u1ebft qu\u1ea3 t\u0103ng v\u1ecdt t\u1eeb 7.6627 l\u00ean 9.7004. K\u1ebft qu\u1ea3 v\u1eabn c\u00f3 \u00fd ngh\u0129a th\u1ed1ng k\u00ea. Ngo\u00e0i ra, h\u00e3y l\u01b0u \u00fd r\u1eb1ng t\u00f4i \u0111ang s\u1eed d\u1ee5ng `wls` thay v\u00ec `ols`\n\n\n```python\nmodel = smf.wls(\"all~agecell*threshold\", rdd_df,\n weights=kernel(drinking[\"agecell\"], c=0, h=1)).fit()\n\nmodel.summary().tables[1]\n```\n\n\n\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        coef std err t P>|t| [0.025 0.975]
                                        Intercept 93.2002 0.731 127.429 0.000 91.726 94.674
                                        agecell 0.4109 1.789 0.230 0.819 -3.196 4.017
                                        threshold 9.7004 1.034 9.378 0.000 7.616 11.785
                                        agecell:threshold -7.1759 2.531 -2.835 0.007 -12.276 -2.075
                                        \n\n\n\n\n```python\nax = drinking.plot.scatter(x=\"agecell\", y=\"all\", color=\"C0\")\ndrinking.assign(predictions=model.fittedvalues).plot(x=\"agecell\", y=\"predictions\", ax=ax, color=\"C1\")\nplt.title(\"Regression Discontinuity (Local Regression)\");\n```\n\nV\u00e0 \u0111\u00e2y l\u00e0 nh\u1eefng g\u00ec ta c\u00f3 \u0111\u01b0\u1ee3c \u0111\u1ed1i v\u1edbi t\u1ef7 l\u1ec7 t\u1eed vong g\u00e2y ra b\u1edfi c\u00e1c t\u00e1c nh\u00e2n kh\u00e1c. L\u01b0u \u00fd r\u1eb1ng h\u1ed3i quy t\u1ea1i nh\u1eefng \u0111i\u1ec3m tr\u00ean ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh c\u00f3 xu h\u01b0\u1edbng gi\u1ea3m l\u1edbn h\u01a1n v\u00ec n\u00f3 kh\u00f4ng xem x\u00e9t c\u00e1c \u0111i\u1ec3m n\u1eb1m xa ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh.\n\n\n```python\nplt.figure(figsize=(8,8))\nweights = kernel(drinking[\"agecell\"], c=0, h=1)\n\nfor p, cause in enumerate([\"all\", \"mva\", \"suicide\"], 1):\n ax = plt.subplot(3,1,p)\n drinking.plot.scatter(x=\"agecell\", y=cause, ax=ax)\n m = smf.wls(f\"{cause}~agecell*threshold\", rdd_df, weights=weights).fit()\n ate_pct = 100*((m.params[\"threshold\"] + m.params[\"Intercept\"])/m.params[\"Intercept\"] - 1)\n drinking.assign(predictions=m.fittedvalues).plot(x=\"agecell\", y=\"predictions\", ax=ax, color=\"C1\")\n plt.title(f\"Impact of Alcohol on Death: {np.round(ate_pct, 2)}%\")\n\nplt.tight_layout()\n```\n\nNgo\u1ea1i tr\u1eeb t\u00e1c nh\u00e2n t\u1ef1 s\u00e1t, c\u00f3 v\u1ebb nh\u01b0 vi\u1ec7c th\u00eam t\u1ef7 tr\u1ecdng kernel \u0111\u00e3 l\u00e0m cho t\u00e1c \u0111\u1ed9ng ti\u00eau c\u1ef1c c\u1ee7a r\u01b0\u1ee3u l\u1edbn h\u01a1n. M\u1ed9t l\u1ea7n n\u1eefa, n\u1ebfu ch\u00fang ta mu\u1ed1n gi\u1ea3m thi\u1ec3u t\u1ef7 l\u1ec7 t\u1eed vong, ch\u00fang ta KH\u00d4NG n\u00ean khuy\u1ebfn ngh\u1ecb h\u1ea1 \u0111\u1ed9 tu\u1ed5i u\u1ed1ng r\u01b0\u1ee3u h\u1ee3p ph\u00e1p, b\u1edfi v\u00ec r\u00f5 r\u00e0ng r\u01b0\u1ee3u c\u00f3 t\u00e1c \u0111\u1ed9ng \u0111\u1ebfn t\u1ef7 l\u1ec7 t\u1eed vong.\n\nTr\u01b0\u1eddng h\u1ee3p \u0111\u01a1n gi\u1ea3n n\u00e0y bao qu\u00e1t nh\u1eefng g\u00ec x\u1ea3y ra khi thi\u1ebft k\u1ebf h\u1ed3i quy gi\u00e1n \u0111o\u1ea1n v\u1eadn h\u00e0nh m\u1ed9t c\u00e1ch ho\u00e0n h\u1ea3o. Ti\u1ebfp theo, ch\u00fang ta s\u1ebd t\u00ecm hi\u1ec3u m\u1ed9t s\u1ed1 ch\u1ea9n \u0111o\u00e1n m\u00e0 ta n\u00ean ch\u1ea1y \u0111\u1ec3 ki\u1ec3m tra xem ch\u00fang ta c\u00f3 th\u1ec3 tin t\u01b0\u1edfng RDD \u0111\u1ebfn m\u1ee9c n\u00e0o v\u00e0 c\u00f9ng n\u00f3i v\u1ec1 m\u1ed9t ch\u1ee7 \u0111\u1ec1 \u0111\u00e3 r\u1ea5t \u0111\u1ed7i quen thu\u1ed9c: t\u00e1c \u0111\u1ed9ng c\u1ee7a gi\u00e1o d\u1ee5c \u0111\u1ed1i v\u1edbi thu nh\u1eadp.\n\n## Hi\u1ec7u \u1ee8ng Da C\u1eebu v\u00e0 RDD M\u1edd \n\nKhi n\u00f3i \u0111\u1ebfn t\u00e1c \u0111\u1ed9ng c\u1ee7a gi\u00e1o d\u1ee5c \u0111\u1ed1i v\u1edbi thu nh\u1eadp, c\u00f3 hai quan \u0111i\u1ec3m ch\u00ednh trong kinh t\u1ebf h\u1ecdc. L\u1eadp lu\u1eadn \u0111\u1ea7u ti\u00ean \u0111\u01b0\u1ee3c bi\u1ebft \u0111\u1ebfn r\u1ed9ng r\u00e3i r\u1eb1ng gi\u00e1o d\u1ee5c l\u00e0m t\u0103ng v\u1ed1n nh\u00e2n l\u01b0c, t\u0103ng n\u0103ng su\u1ea5t v\u00e0 do \u0111\u00f3, t\u0103ng thu nh\u1eadp. Theo quan \u0111i\u1ec3m n\u00e0y, gi\u00e1o d\u1ee5c th\u1ef1c s\u1ef1 thay \u0111\u1ed5i b\u1ea1n theo h\u01b0\u1edbng t\u1ed1t h\u01a1n. M\u1ed9t quan \u0111i\u1ec3m kh\u00e1c cho r\u1eb1ng gi\u00e1o d\u1ee5c ch\u1ec9 \u0111\u01a1n gi\u1ea3n l\u00e0 m\u1ed9t c\u01a1 ch\u1ebf ph\u00e1t t\u00edn hi\u1ec7u. N\u00f3 ch\u1ec9 \u0111\u1ea9y b\u1ea1n \u0111\u1ebfn v\u1edbi c\u00e1c b\u00e0i ki\u1ec3m tra v\u00e0 b\u00e0i t\u1eadp kh\u00f3 nh\u1eb1n, v\u00e0 n\u1ebfu b\u1ea1n c\u00f3 th\u1ec3 l\u00e0m \u0111\u01b0\u1ee3c ch\u00fang, n\u00f3 ph\u00e1t t\u00edn hi\u1ec7u cho th\u1ecb tr\u01b0\u1eddng r\u1eb1ng b\u1ea1n l\u00e0 m\u1ed9t nh\u00e2n vi\u00ean gi\u1ecfi. Theo quan \u0111i\u1ec3m n\u00e0y, gi\u00e1o d\u1ee5c kh\u00f4ng l\u00e0m cho b\u1ea1n n\u0103ng su\u1ea5t h\u01a1n. N\u00f3 ch\u1ec9 cho th\u1ecb tr\u01b0\u1eddng bi\u1ebft b\u1ea1n lu\u00f4n l\u00e0m vi\u1ec7c hi\u1ec7u qu\u1ea3 nh\u01b0 th\u1ebf n\u00e0o. \u0110i\u1ec1u quan tr\u1ecdng \u1edf \u0111\u00e2y l\u00e0 b\u1eb1ng t\u1ed1t nghi\u1ec7p. N\u1ebfu b\u1ea1n c\u00f3 n\u00f3, b\u1ea1n s\u1ebd \u0111\u01b0\u1ee3c tr\u1ea3 l\u01b0\u01a1ng cao h\u01a1n. Ch\u00fang ta g\u1ecdi \u0111\u00e2y l\u00e0 **hi\u1ec7u \u1ee9ng da c\u1eebu**, b\u1edfi v\u00ec tr\u01b0\u1edbc \u0111\u00e2y b\u1eb1ng c\u1ea5p \u0111\u01b0\u1ee3c in b\u1eb1ng da c\u1eebu.\n\n\u0110\u1ec3 ki\u1ec3m \u0111\u1ecbnh gi\u1ea3 thuy\u1ebft n\u00e0y, [Clark v\u00e0 Martorell](https://faculty.smu.edu/millimet/classes/eco7321/papers/clark%20martorell%202014.pdf) \u0111\u00e3 s\u1eed d\u1ee5ng h\u1ed3i quy gi\u00e1n \u0111o\u1ea1n \u0111\u1ec3 \u0111o l\u01b0\u1eddng t\u00e1c \u0111\u1ed9ng c\u1ee7a vi\u1ec7c t\u1ed1t nghi\u1ec7p l\u1edbp 12 \u0111\u1ed1i v\u1edbi thu nh\u1eadp. \u0110\u1ec3 l\u00e0m \u0111\u01b0\u1ee3c \u0111i\u1ec1u \u0111\u00f3, h\u1ecd \u0111\u00e3 ph\u1ea3i c\u00e2n nh\u1eafc m\u1ed9t s\u1ed1 bi\u1ebfn ch\u1ec9 \u0111\u1ecbnh, theo \u0111\u00f3 n\u1ebfu h\u1ecdc sinh v\u01b0\u1ee3t qua ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh \u0111\u00f3 th\u00ec t\u1ed1t nghi\u1ec7p v\u00e0 ng\u01b0\u1ee3c l\u1ea1i. H\u1ecd \u0111\u00e3 t\u00ecm th\u1ea5y d\u1eef li\u1ec7u nh\u01b0 v\u1eady trong h\u1ec7 th\u1ed1ng gi\u00e1o d\u1ee5c Texas.\n\n\u0110\u1ec3 t\u1ed1t nghi\u1ec7p \u1edf Texas, h\u1ecdc sinh ph\u1ea3i v\u01b0\u1ee3t qua m\u1ed9t k\u1ef3 thi. C\u00e1c k\u1ef3 ki\u1ec3m tra b\u1eaft \u0111\u1ea7u t\u1eeb n\u0103m l\u1edbp 10 v\u00e0 h\u1ecdc sinh c\u00f3 th\u1ec3 l\u00e0m nhi\u1ec1u l\u1ea7n, nh\u01b0ng cu\u1ed1i c\u00f9ng, c\u00e1c b\u1ea1n tr\u1ebb ph\u1ea3i \u0111\u1ed1i m\u1eb7t v\u1edbi k\u1ef3 thi cu\u1ed1i v\u00e0o cu\u1ed1i n\u0103m l\u1edbp 12. \u00dd t\u01b0\u1edfng l\u00e0 l\u1ea5y d\u1eef li\u1ec7u t\u1eeb nh\u1eefng h\u1ecdc sinh \u0111\u00e3 tham gia c\u00e1c k\u1ef3 thi cu\u1ed1i \u0111\u00f3 v\u00e0 so s\u00e1nh nh\u1eefng h\u1ecdc sinh thi tr\u01b0\u1ee3t c\u1eadn \u0111i\u1ec3m \u0111\u1ed7 v\u1edbi nh\u1eefng b\u1ea1n thi \u0111\u1eadu c\u1eadn \u0111i\u1ec3m \u0111\u1ed7. Nh\u1eefng h\u1ecdc sinh n\u00e0y s\u1ebd c\u00f3 v\u1ed1n nh\u00e2n l\u1ef1c r\u1ea5t t\u01b0\u01a1ng \u0111\u1ed3ng, nh\u01b0ng kh\u00e1c bi\u1ec7t v\u1ec1 t\u00edn hi\u1ec7u ch\u1ee9ng ch\u1ec9. C\u1ee5 th\u1ec3, nh\u1eefng h\u1ecdc sinh v\u1eeba \u0111\u1ee7 \u0111i\u1ec3m \u0111\u1ed7, s\u1ebd nh\u1eadn \u0111\u01b0\u1ee3c b\u1eb1ng t\u1ed1t nghi\u1ec7p.\n\n\n```python\nsheepskin = pd.read_csv(\"./data/sheepskin.csv\")[[\"avgearnings\", \"minscore\", \"receivehsd\", \"n\"]]\nsheepskin.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        avgearningsminscorereceivehsdn
                                        011845.086-30.00.41666712
                                        19205.679-29.00.38709731
                                        28407.745-28.00.31818244
                                        311114.087-27.00.37777845
                                        410814.624-26.00.30666775
                                        \n
                                        \n\n\n\nM\u1ed9t l\u1ea7n n\u1eefa, d\u1eef li\u1ec7u \u0111\u01b0\u1ee3c nh\u00f3m theo bi\u1ebfn ch\u1ec9 \u0111\u1ecbnh. N\u00f3 kh\u00f4ng ch\u1ec9 ch\u1ee9a bi\u1ebfn ch\u1ec9 \u0111\u1ecbnh (`minscore`, v\u1edbi 0 l\u00e0 gi\u00e1 tr\u1ecb trung t\u00e2m) v\u00e0 k\u1ebft qu\u1ea3 (`avgearnings`), m\u00e0 n\u00f3 c\u00f2n c\u00f3 x\u00e1c su\u1ea5t nh\u1eadn b\u1eb1ng t\u1ed1t nghi\u1ec7p theo \u00f4 \u0111i\u1ec3m \u0111\u00f3 v\u00e0 k\u00edch th\u01b0\u1edbc c\u1ee7a \u00f4 \u0111i\u1ec3m (`n`). V\u00ec v\u1eady, v\u00ed d\u1ee5, trong s\u1ed1 12 h\u1ecdc sinh c\u00f3 k\u1ebft qu\u1ea3 -30 d\u01b0\u1edbi ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh \u0111i\u1ec3m, ch\u1ec9 c\u00f3 5 ng\u01b0\u1eddi c\u00f3 th\u1ec3 nh\u1eadn \u0111\u01b0\u1ee3c b\u1eb1ng t\u1ed1t nghi\u1ec7p.\n\n\u0110i\u1ec1u n\u00e0y c\u00f3 ngh\u0129a l\u00e0 c\u00f3 s\u1ef1 gi\u1ea3m s\u00fat trong ch\u1ec9 \u0111\u1ecbnh can thi\u1ec7p. M\u1ed9t s\u1ed1 h\u1ecdc sinh c\u00f3 s\u1ed1 \u0111i\u1ec3m d\u01b0\u1edbi ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh \u0111\u1ed7 v\u1eabn c\u00f3 th\u1ec3 xoay x\u1edf \u0111\u1ec3 l\u1ea5y \u0111\u01b0\u1ee3c b\u1eb1ng t\u1ed1t nghi\u1ec7p b\u1eb1ng m\u1ecdi c\u00e1ch. T\u1ea1i \u0111\u00e2y, h\u1ed3i quy gi\u00e1n \u0111o\u1ea1n c\u00f3 d\u1ea1ng **m\u1edd**, thay v\u00ec s\u1eafc . L\u01b0u \u00fd r\u1eb1ng x\u00e1c su\u1ea5t nh\u1eadn \u0111\u01b0\u1ee3c b\u1eb1ng t\u1ed1t nghi\u1ec7p kh\u00f4ng t\u0103ng v\u1ecdt t\u1eeb 0 l\u00ean 1 t\u1ea1i ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh m\u00e0 n\u00f3 nh\u1ea3y t\u1eeb kho\u1ea3ng 50% l\u00ean 90%.\n\n\n```python\nsheepskin.plot.scatter(x=\"minscore\", y=\"receivehsd\", figsize=(10,5))\nplt.xlabel(\"\u0110i\u1ec3m thi so v\u1edbi Ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh\")\nplt.ylabel(\"T\u1ec9 l\u1ec7 nh\u1eadn b\u1eb1ng\")\nplt.title(\"B\u00e0i thi cu\u1ed1i c\u00f9ng\");\n```\n\nCh\u00fang ta c\u00f3 th\u1ec3 ngh\u0129 v\u1ec1 RD m\u1edd nh\u01b0 m\u1ed9t lo\u1ea1i kh\u00f4ng tu\u00e2n th\u1ee7. Vi\u1ec7c v\u01b0\u1ee3t qua ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh n\u00e0y s\u1ebd khi\u1ebfn m\u1ecdi h\u1ecdc sinh nh\u1eadn \u0111\u01b0\u1ee3c b\u1eb1ng t\u1ed1t nghi\u1ec7p, nh\u01b0ng m\u1ed9t s\u1ed1 h\u1ecdc sinh, nh\u1eefng ng\u01b0\u1eddi lu\u00f4n kh\u00f4ng nh\u1eadn, kh\u00f4ng nh\u1eadn \u0111\u01b0\u1ee3c b\u1eb1ng. T\u01b0\u01a1ng t\u1ef1 nh\u01b0 v\u1eady, nh\u1eefng b\u1ea1n c\u00f3 \u0111i\u1ec3m d\u01b0\u1edbi ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh s\u1ebd kh\u00f4ng nh\u1eadn \u0111\u01b0\u1ee3c b\u1eb1ng t\u1ed1t nghi\u1ec7p, nh\u01b0ng m\u1ed9t s\u1ed1 h\u1ecdc sinh, nh\u1eefng ngu\u1eddi lu\u00f4n nh\u1eadn, v\u1eabn xoay x\u1edf \u0111\u1ec3 l\u1ea5y \u0111\u01b0\u1ee3c b\u1eb1ng.\n\nC\u0169ng nh\u01b0 k\u1ebft qu\u1ea3 ti\u1ec1m n\u0103ng, ch\u00fang ta c\u00f3 tr\u1ea1ng th\u00e1i can thi\u1ec7p ti\u1ec1m n\u0103ng trong t\u00ecnh hu\u1ed1ng n\u00e0y. \\\\(T_1\\\\) l\u00e0 can thi\u1ec7p m\u00e0 m\u1ecdi ng\u01b0\u1eddi nh\u1eadn \u0111\u01b0\u1ee3c n\u1ebfu h\u1ecd v\u01b0\u1ee3t qu\u00e1 ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh. \\\\(T_0\\\\) l\u00e0 can thi\u1ec7p m\u00e0 m\u1ecdi ng\u01b0\u1eddi nh\u1eadn \u0111\u01b0\u1ee3c n\u1ebfu h\u1ecd \u1edf d\u01b0\u1edbi ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh. Nh\u01b0 b\u1ea1n \u0111\u00e3 th\u1ea5y, ch\u00fang ta c\u00f3 th\u1ec3 xem **ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh nh\u01b0 m\u1ed9t Bi\u1ebfn C\u00f4ng C\u1ee5**. C\u0169ng gi\u1ed1ng nh\u01b0 trong IV, y\u1ebfu t\u1ed1 kh\u00f4ng tu\u00e2n th\u1ee7 l\u00e0m cho t\u00e1c \u0111\u1ed9ng can thi\u1ec7p m\u00e0 ch\u00fang ta \u01b0\u1edbc l\u01b0\u1ee3ng b\u1ecb thi\u00ean l\u1ec7ch v\u1ec1 0.\n\n\n\nX\u00e1c su\u1ea5t \u0111\u1ec3 can thi\u1ec7p nh\u1ecf h\u01a1n 1, th\u1eadm ch\u00ed \u1edf tr\u00ean ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh, l\u00e0m cho k\u1ebft qu\u1ea3 ch\u00fang ta quan s\u00e1t \u0111\u01b0\u1ee3c th\u1ea5p h\u01a1n k\u1ebft qu\u1ea3 ti\u1ec1m n\u0103ng th\u1ef1c \\\\(Y_1\\\\). T\u01b0\u01a1ng t\u1ef1, k\u1ebft qu\u1ea3 m\u00e0 ch\u00fang ta quan s\u00e1t d\u01b0\u1edbi ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh cao h\u01a1n k\u1ebft qu\u1ea3 ti\u1ec1m n\u0103ng th\u1ef1c \\\\(Y_0\\\\). \u0110i\u1ec1u n\u00e0y l\u00e0m cho t\u00e1c \u0111\u1ed9ng can thi\u1ec7p t\u1ea1i ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh c\u00f3 v\u1ebb nh\u01b0 nh\u1ecf h\u01a1n so v\u1edbi th\u1ef1c t\u1ebf v\u00e0 ch\u00fang t\u00f4i s\u1ebd ph\u1ea3i s\u1eed d\u1ee5ng k\u1ef9 thu\u1eadt IV \u0111\u1ec3 \u0111i\u1ec1u ch\u1ec9nh.\n\nC\u0169ng nh\u01b0 tr\u01b0\u1edbc \u0111\u00e2y ch\u00fang ta \u0111\u1eb7t gi\u1ea3 thi\u1ebft r\u1eb1ng k\u1ebft qu\u1ea3 ti\u1ec1m n\u0103ng s\u1ebd li\u00ean t\u1ee5c, b\u00e2y gi\u1edd ch\u00fang ta c\u0169ng l\u00e0m t\u01b0\u01a1ng t\u1ef1 cho can thi\u1ec7p ti\u1ec1m n\u0103ng. Ngo\u00e0i ra, ch\u00fang ta c\u1ea7n \u0111\u1eb7t gi\u1ea3 thi\u1ebft cho \u0111\u01a1n \u0111i\u1ec7u, gi\u1ed1ng nh\u01b0 trong IV. N\u00f3 n\u00f3i r\u1eb1ng \\\\(T_{i1}>T_{i0} \\ \\forall i\\\\). \u0110i\u1ec1u n\u00e0y c\u00f3 ngh\u0129a l\u00e0 vi\u1ec7c v\u01b0\u1ee3t qua ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh t\u1eeb tr\u00e1i sang ph\u1ea3i ch\u1ec9 l\u00e0m t\u0103ng kh\u1ea3 n\u0103ng nh\u1eadn \u0111\u01b0\u1ee3c b\u1eb1ng t\u1ed1t nghi\u1ec7p (ho\u1eb7c kh\u00f4ng c\u00f3 ng\u01b0\u1eddi lu\u00f4n kh\u00f4ng nh\u1eadn). V\u1edbi 2 gi\u1ea3 thi\u1ebft n\u00e0y, ch\u00fang ta c\u00f3 M\u00f4 H\u00ecnh \u01af\u1edbc L\u01b0\u1ee3ng Wald cho LATE.\n\n$$\n\\dfrac{\\lim_{r \\to c^+} E[Y_i|R_i=r] - \\lim_{r \\to c^-} E[Y_i|R_i=r]}{\\lim_{r \\to c^+} E[T_i|R_i=r] - \\lim_{r \\to c^-} E[T_i|R_i=r]} = E[Y_{1i} - Y_{0i} | T_{1i} > T_{0i}, R_i=c]\n$$\n\nL\u01b0u \u00fd r\u1eb1ng \u0111\u00e2y l\u00e0 m\u1ed9t \u01b0\u1edbc l\u01b0\u1ee3ng c\u1ee5c b\u1ed9 theo hai ngh\u0129a. \u0110\u1ea7u ti\u00ean, n\u00f3 l\u00e0 c\u1ee5c b\u1ed9 v\u00ec n\u00f3 ch\u1ec9 cho th\u1ea5y t\u00e1c \u0111\u1ed9ng can thi\u1ec7p t\u1ea1i ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh \\\\(c\\\\). \u0110\u00e2y l\u00e0 RD c\u1ee5c b\u1ed9. Th\u1ee9 hai, n\u00f3 l\u00e0 c\u1ee5c b\u1ed9 v\u00ec n\u00f3 ch\u1ec9 \u01b0\u1edbc l\u01b0\u1ee3ng t\u00e1c \u0111\u1ed9ng can thi\u1ec7p cho ng\u01b0\u1eddi tu\u00e2n th\u1ee7. \u0110\u00e2y l\u00e0 IV c\u1ee5c b\u1ed9.\n\n\u0110\u1ec3 \u01b0\u1edbc l\u01b0\u1ee3ng, ch\u00fang ta s\u1ebd s\u1eed d\u1ee5ng 2 h\u1ed3i quy tuy\u1ebfn t\u00ednh. T\u1eed s\u1ed1 c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c \u01b0\u1edbc l\u01b0\u1ee3ng theo c\u00e1ch ch\u00fang ta \u0111\u00e3 l\u00e0m tr\u01b0\u1edbc \u0111\u00f3. \u0110\u1ec3 c\u00f3 \u0111\u01b0\u1ee3c m\u1eabu s\u1ed1, ch\u00fang ta ch\u1ec9 c\u1ea7n thay k\u1ebft qu\u1ea3 b\u1eb1ng can thi\u1ec7p. Nh\u01b0ng tr\u01b0\u1edbc ti\u00ean, h\u00e3y b\u00e0n v\u1ec1 ki\u1ec3m \u0111\u1ecbnh gi\u1ea3 thi\u1ebft m\u00e0 ch\u00fang ta c\u1ea7n ch\u1ea1y \u0111\u1ec3 \u0111\u1ea3m b\u1ea3o r\u1eb1ng \u01b0\u1edbc l\u01b0\u1ee3ng RDD c\u00f3 th\u1ec3 tin t\u01b0\u1edfng \u0111\u01b0\u1ee3c.\n\n### McCrary Test\n\nM\u1ed9t th\u1ee9 c\u00f3 th\u1ec3 ph\u00e1 v\u1ee1 l\u1eadp lu\u1eadn RDD c\u1ee7a ch\u00fang ta l\u00e0 n\u1ebfu m\u1ecdi ng\u01b0\u1eddi chi ph\u1ed1i v\u1ecb tr\u00ed c\u1ee7a h\u1ecd t\u1ea1i ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh. N\u00f3 c\u00f3 th\u1ec3 x\u1ea3y ra trong v\u00ed d\u1ee5 v\u1ec1 da c\u1eebu n\u1ebfu h\u1ecdc sinh c\u00f3 \u0111i\u1ec3m s\u1ed1 ngay d\u01b0\u1edbi ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh t\u00ecm ra c\u00e1ch \u0111\u1ec3 n\u00e2ng \u0111i\u1ec3m b\u00e0i ki\u1ec3m tra l\u00ean m\u1ed9t ch\u00fat. M\u1ed9t v\u00ed d\u1ee5 kh\u00e1c l\u00e0 khi b\u1ea1n c\u1ea7n c\u00f3 thu nh\u1eadp d\u01b0\u1edbi m\u1ed9t m\u1ee9c nh\u1ea5t \u0111\u1ecbnh \u0111\u1ec3 nh\u1eadn \u0111\u01b0\u1ee3c tr\u1ee3 c\u1ea5p ch\u00ednh ph\u1ee7. M\u1ed9t s\u1ed1 gia \u0111\u00ecnh c\u00f3 th\u1ec3 c\u1ed1 t\u00ecnh gi\u1ea3m thu nh\u1eadp ch\u1ec9 \u0111\u1ec3 \u0111\u1ee7 \u0111i\u1ec1u ki\u1ec7n tham gia g\u00f3i tr\u1ee3 c\u1ea5p.\n\nTrong c\u00e1c t\u00ecnh hu\u1ed1ng n\u00e0y, ch\u00fang ta c\u00f3 xu h\u01b0\u1edbng th\u1ea5y m\u1ed9t hi\u1ec7n t\u01b0\u1ee3ng g\u1ecdi l\u00e0 t\u1eadp h\u1ee3p m\u1eadt \u0111\u1ed9 c\u1ee7a bi\u1ebfn ch\u1ec9 \u0111\u1ecbnh. Ngh\u0129a l\u00e0 ch\u00fang ta s\u1ebd c\u00f3 r\u1ea5t nhi\u1ec1u \u0111\u1ed1i t\u01b0\u1ee3ng ngay tr\u00ean ho\u1eb7c ngay d\u01b0\u1edbi ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh. \u0110\u1ec3 ki\u1ec3m tra \u0111i\u1ec1u n\u00e0y, ch\u00fang ta c\u00f3 th\u1ec3 v\u1ebd bi\u1ec3u \u0111\u1ed3 h\u00e0m m\u1eadt \u0111\u1ed9 c\u1ee7a bi\u1ebfn ch\u1ec9 \u0111\u1ecbnh v\u00e0 xem li\u1ec7u c\u00f3 b\u1ea5t k\u1ef3 \u0111\u1ed9t bi\u1ebfn n\u00e0o xung quanh ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh hay kh\u00f4ng. \u0110\u1ed1i v\u1edbi tr\u01b0\u1eddng h\u1ee3p c\u1ee7a ch\u00fang ta, m\u1eadt \u0111\u1ed9 \u0111\u01b0\u1ee3c bi\u1ec3u th\u1ecb b\u1edfi c\u1ed9t `n` trong d\u1eef li\u1ec7u.\n\n\n```python\nplt.figure(figsize=(8,8))\n\nax = plt.subplot(2,1,1)\nsheepskin.plot.bar(x=\"minscore\", y=\"n\", ax=ax)\nplt.title(\"Ki\u1ec3m \u0111\u1ecbnh McCrary\")\nplt.ylabel(\"\u0110\u1ed9 tr\u01a1n t\u1ea1i ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh\")\n\nax = plt.subplot(2,1,2, sharex=ax)\nsheepskin.replace({1877:1977, 1874:2277}).plot.bar(x=\"minscore\", y=\"n\", ax=ax)\nplt.xlabel(\"\u0110i\u1ec3m thi so v\u1edbi ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh\")\nplt.ylabel(\"\u0110\u1ed9t bi\u1ebfn t\u1ea1i ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh\");\n```\n\nBi\u1ec3u \u0111\u1ed3 \u0111\u1ea7u ti\u00ean cho th\u1ea5y m\u1eadt \u0111\u1ed9 d\u1eef li\u1ec7u tr\u00f4ng nh\u01b0 th\u1ebf n\u00e0o. Nh\u01b0 ch\u00fang ta c\u00f3 th\u1ec3 th\u1ea5y, kh\u00f4ng c\u00f3 \u0111\u1ed9t bi\u1ebfn n\u00e0o xung quanh ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh, c\u00f3 ngh\u0129a l\u00e0 kh\u00f4ng c\u00f3 h\u1ed9i t\u1ee5. H\u1ecdc sinh kh\u00f4ng thao t\u00fang \u0111i\u1ec3m s\u1ed1 c\u1ee7a ch\u00fang t\u1ea1i ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh. Ch\u1ec9 nh\u1eb1m m\u1ee5c \u0111\u00edch minh h\u1ecda, bi\u1ec3u \u0111\u1ed3 th\u1ee9 hai cho th\u1ea5y t\u1eadp h\u1ee3p s\u1ebd nh\u01b0 th\u1ebf n\u00e0o n\u1ebfu h\u1ecdc sinh c\u00f3 th\u1ec3 thao t\u00fang \u0111i\u1ec3m s\u1ed1 t\u1ea1i ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh. Ch\u00fang ta s\u1ebd th\u1ea5y m\u1eadt \u0111\u1ed9 c\u1ee7a c\u00e1c \u00f4 ngay tr\u00ean ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh t\u0103ng \u0111\u1ed9t bi\u1ebfn, v\u00ec nhi\u1ec1u h\u1ecdc sinh c\u00f3 s\u1ed1 \u0111i\u1ec3m t\u1ea1i \u0111\u00f3. Nh\u1eb1m gi\u1ea3i quy\u1ebft v\u1ea5n \u0111\u1ec1 n\u00e0y, ch\u00fang ta c\u00f3 th\u1ec3 quay l\u1ea1i \u0111\u1ec3 \u01b0\u1edbc l\u01b0\u1ee3ng hi\u1ec7u \u1ee9ng da c\u1eebu.\n\nNh\u01b0 t\u00f4i \u0111\u00e3 n\u00f3i tr\u01b0\u1edbc \u0111\u00e2y, t\u1eed s\u1ed1 c\u1ee7a M\u00f4 h\u00ecnh \u01b0\u1edbc l\u01b0\u1ee3ng wald c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c \u01b0\u1edbc l\u01b0\u1ee3ng gi\u1ed1ng v\u1edbi c\u00e1ch ta \u0111\u00e3 l\u00e0m trong RDD s\u1eafc . T\u1ea1i \u0111\u00e2y, ch\u00fang ta s\u1ebd s\u1eed d\u1ee5ng tr\u1ecdng s\u1ed1 c\u1ee7a kernel v\u1edbi d\u1ea3i t\u1ea7n su\u1ea5t l\u00e0 15. V\u00ec ch\u00fang ta c\u0169ng c\u00f3 k\u00edch th\u01b0\u1edbc c\u1ee7a \u00f4 \u0111i\u1ec3m, ch\u00fang ta s\u1ebd nh\u00e2n kernel v\u1edbi k\u00edch th\u01b0\u1edbc m\u1eabu \u0111\u1ec3 c\u00f3 \u0111\u01b0\u1ee3c tr\u1ecdng s\u1ed1 cu\u1ed1i c\u00f9ng cho \u00f4 \u0111i\u1ec3m.\n\n\n```python\nsheepsking_rdd = sheepskin.assign(threshold=(sheepskin[\"minscore\"]>0).astype(int))\nmodel = smf.wls(\"avgearnings~minscore*threshold\",\n sheepsking_rdd,\n weights=kernel(sheepsking_rdd[\"minscore\"], c=0, h=15)*sheepsking_rdd[\"n\"]).fit()\n\nmodel.summary().tables[1]\n```\n\n\n\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        coef std err t P>|t| [0.025 0.975]
                                        Intercept 1.399e+04 83.678 167.181 0.000 1.38e+04 1.42e+04
                                        minscore 181.6636 16.389 11.084 0.000 148.588 214.739
                                        threshold -97.7571 145.723 -0.671 0.506 -391.839 196.325
                                        minscore:threshold 18.1955 30.311 0.600 0.552 -42.975 79.366
                                        \n\n\n\nN\u00f3 cho ch\u00fang ta bi\u1ebft r\u1eb1ng t\u00e1c \u0111\u1ed9ng c\u1ee7a b\u1eb1ng t\u1ed1t nghi\u1ec7p l\u00e0 -97,7571, nh\u01b0ng con s\u1ed1 n\u00e0y kh\u00f4ng mang \u00fd ngh\u0129a th\u1ed1ng k\u00ea (tr\u1ecb s\u1ed1 p l\u00e0 0,5). N\u1ebfu ch\u00fang ta bi\u1ec3u di\u1ec5n c\u00e1c k\u1ebft qu\u1ea3, ta c\u00f3 m\u1ed9t \u0111\u01b0\u1eddng ch\u1ea1y li\u00ean t\u1ee5c qua ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh. Nh\u1eefng ng\u01b0\u1eddi c\u00f3 nhi\u1ec1u n\u0103m h\u1ecdc v\u1ea5n h\u01a1n th\u1ef1c s\u1ef1 ki\u1ebfm \u0111\u01b0\u1ee3c nhi\u1ec1u ti\u1ec1n h\u01a1n, nh\u01b0ng kh\u00f4ng c\u00f3 \u0111\u1ed9t bi\u1ebfn n\u00e0o \u1edf th\u1eddi \u0111i\u1ec3m h\u1ecd nh\u1eadn \u0111\u01b0\u1ee3c b\u1eb1ng t\u1ed1t nghi\u1ec7p l\u1edbp 12. \u0110\u00e2y l\u00e0 m\u1ed9t l\u1eadp lu\u1eadn \u1ee7ng h\u1ed9 quan \u0111i\u1ec3m r\u1eb1ng gi\u00e1o d\u1ee5c t\u0103ng thu nh\u1eadp b\u1eb1ng c\u00e1ch gi\u00fap cho m\u1ecdi ng\u01b0\u1eddi l\u00e0m vi\u1ec7c hi\u1ec7u qu\u1ea3 h\u01a1n, b\u1edfi v\u00ec kh\u00f4ng c\u00f3 s\u1ef1 xu\u1ea5t hi\u1ec7n c\u1ee7a hi\u1ec7u \u1ee9ng da c\u1eebu.\n\n\n```python\nax = sheepskin.plot.scatter(x=\"minscore\", y=\"avgearnings\", color=\"C0\")\nsheepskin.assign(predictions=model.fittedvalues).plot(x=\"minscore\", y=\"predictions\", ax=ax, color=\"C1\", figsize=(8,5))\nplt.xlabel(\"Test Scores Relative to Cutoff\")\nplt.ylabel(\"Average Earnings\")\nplt.title(\"Last-chance Exams\");\n```\n\nTuy nhi\u00ean, nh\u01b0 ch\u00fang ta \u0111\u00e3 bi\u1ebft t\u1eeb c\u00e1ch ho\u1ea1t \u0111\u1ed9ng c\u1ee7a thi\u00ean l\u1ec7ch b\u1ea5t tu\u00e2n, k\u1ebft qu\u1ea3 n\u00e0y thi\u00ean l\u1ec7ch v\u1ec1 0. \u0110\u1ec3 ch\u1ec9nh s\u1eeda, ch\u00fang ta c\u1ea7n chia t\u1ef7 tr\u1ecdng \u1edf giai \u0111o\u1ea1n \u0111\u1ea7u v\u00e0 d\u00f9ng M\u00f4 h\u00ecnh \u01b0\u1edbc l\u01b0\u1ee3ng wald. Th\u1eadt kh\u00f4ng may, Python kh\u00f4ng c\u00f3 s\u1eb5n g\u00f3i c\u00f4ng c\u1ee5 n\u00e0o \u0111\u1ee7 t\u1ed1t \u0111\u1ec3 gi\u1ea3i quy\u1ebft v\u1ea5n \u0111\u1ec1 n\u00e0y, v\u00ec v\u1eady ch\u00fang ta s\u1ebd ph\u1ea3i th\u1ef1c hi\u1ec7n ch\u00fang th\u1ee7 c\u00f4ng v\u00e0 s\u1eed d\u1ee5ng bootstrap \u0111\u1ec3 \u0111o \u0111\u01b0\u1ee3c sai s\u1ed1 chu\u1ea9n.\n\n\u0110o\u1ea1n code d\u01b0\u1edbi \u0111\u00e2y \u01b0\u1edbc l\u01b0\u1ee3ng t\u1eed s\u1ed1 c\u1ee7a M\u00f4 h\u00ecnh \u01b0\u1edbc l\u01b0\u1ee3ng wald gi\u1ed1ng nh\u01b0 ch\u00fang ta \u0111\u00e3 l\u00e0m tr\u01b0\u1edbc \u0111\u00e2y v\u00e0 n\u00f3 c\u0169ng x\u00e2y d\u1ef1ng m\u1eabu s\u1ed1 b\u1eb1ng c\u00e1ch thay th\u1ebf bi\u1ebfn m\u1ee5c ti\u00eau b\u1eb1ng bi\u1ebfn can thi\u1ec7p `receivehsd`. B\u01b0\u1edbc cu\u1ed1i c\u00f9ng ch\u1ec9 c\u1ea7n chia t\u1eed s\u1ed1 cho m\u1eabu s\u1ed1.\n\n\n```python\ndef wald_rdd(data):\n weights=kernel(data[\"minscore\"], c=0, h=15)*data[\"n\"]\n denominator = smf.wls(\"receivehsd~minscore*threshold\", data, weights=weights).fit()\n numerator = smf.wls(\"avgearnings~minscore*threshold\", data, weights=weights).fit()\n return numerator.params[\"threshold\"]/denominator.params[\"threshold\"]\n```\n\n\n```python\nfrom joblib import Parallel, delayed \n\nnp.random.seed(45)\nbootstrap_sample = 1000\nates = Parallel(n_jobs=4)(delayed(wald_rdd)(sheepsking_rdd.sample(frac=1, replace=True))\n for _ in range(bootstrap_sample))\nates = np.array(ates)\n```\n\nV\u1edbi c\u00e1c m\u1eabu bootstrap, ch\u00fang ta c\u00f3 th\u1ec3 bi\u1ec3u di\u1ec5n ph\u00e2n ph\u1ed1i c\u1ee7a c\u00e1c ATE v\u00e0 xem th\u1eed kho\u1ea3ng tin c\u1eady 95% n\u1eb1m \u1edf \u0111\u00e2u.\n\n\n```python\nsns.distplot(ates, kde=False)\nplt.vlines(ates.mean()-1.96*ates.std(), 0, 100, linestyles=\"dotted\")\nplt.vlines(ates.mean()+1.96*ates.std(), 0, 100, linestyles=\"dotted\", label=\"95% CI\")\nplt.title(\"ATE Bootstrap Distribution\")\nplt.xlim([-10000, 10000])\nplt.legend();\n```\n\nB\u1ea1n th\u1ea5y \u0111\u1ea5y, ngay c\u1ea3 khi ch\u00fang ta chia t\u1ef7 tr\u1ecdng t\u00e1c \u0111\u1ed9ng ngay giai \u0111o\u1ea1n \u0111\u1ea7u, n\u00f3 v\u1eabn kh\u00f4ng kh\u00e1c 0 v\u1ec1 m\u1eb7t th\u1ed1ng k\u00ea. \u0110i\u1ec1u n\u00e0y c\u00f3 ngh\u0129a l\u00e0 gi\u00e1o d\u1ee5c kh\u00f4ng l\u00e0m t\u0103ng thu nh\u1eadp \u0111\u01a1n gi\u1ea3n ch\u1ec9 b\u1edfi m\u1ed9t hi\u1ec7u \u1ee9ng da c\u1eebu \u0111\u01a1n thu\u1ea7n, m\u00e0 do n\u00f3 gi\u00fap gia t\u0103ng n\u0103ng su\u1ea5t c\u1ee7a con ng\u01b0\u1eddi.\n\n## \u00dd t\u01b0\u1edfng ch\u1ee7 \u0111\u1ea1o\n\nCh\u00fang ta \u0111\u00e3 h\u1ecdc c\u00e1ch t\u1eadn d\u1ee5ng s\u1ef1 gi\u00e1n \u0111o\u1ea1n nh\u00e2n t\u1ea1o \u0111\u1ec3 \u01b0\u1edbc l\u01b0\u1ee3ng t\u00e1c \u0111\u1ed9ng nh\u00e2n qu\u1ea3. \u00dd t\u01b0\u1edfng l\u00e0 ch\u00fang ta s\u1ebd c\u00f3 m\u1ed9t s\u1ed1 ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh nh\u00e2n t\u1ea1o khi\u1ebfn x\u00e1c su\u1ea5t c\u1ee7a s\u1ef1 can thi\u1ec7p nh\u1ea3y v\u1ecdt. M\u1ed9t v\u00ed d\u1ee5 m\u00e0 ch\u00fang ta \u0111\u00e3 th\u1ea5y l\u00e0 \u0111\u1ed9 tu\u1ed5i l\u00e0m cho x\u00e1c su\u1ea5t u\u1ed1ng r\u01b0\u1ee3u t\u0103ng v\u1ecdt \u1edf \u0111\u1ed9 tu\u1ed5i 21, ch\u00fang ta c\u00f3 th\u1ec3 s\u1eed d\u1ee5ng \u0111i\u1ec1u n\u00e0y \u0111\u1ec3 \u01b0\u1edbc l\u01b0\u1ee3ng t\u00e1c \u0111\u1ed9ng c\u1ee7a vi\u1ec7c u\u1ed1ng r\u01b0\u1ee3u. Th\u1ef1c t\u1ebf ch\u1ec9 ra r\u1eb1ng khi \u1edf g\u1ea7n v\u1edbi ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh, ch\u00fang ta c\u00f3 m\u1ed9t ph\u01b0\u01a1ng ph\u00e1p g\u1ea7n v\u1edbi th\u1eed nghi\u1ec7m ng\u1eabu nhi\u00ean. C\u00e1c \u0111\u1ed1i t\u01b0\u1ee3ng \u1edf r\u1ea5t g\u1ea7n ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh c\u00f3 th\u1ec3 \u0111\u00e3 \u0111i theo m\u1ed9t trong hai h\u01b0\u1edbng v\u00e0 nh\u1eefng g\u00ec c\u00f3 th\u1ec3 x\u00e1c \u0111\u1ecbnh n\u01a1i ch\u00fang thu\u1ed9c v\u1ec1 l\u00e0 ng\u1eabu nhi\u00ean. V\u1edbi \u0111i\u1ec1u n\u00e0y, ch\u00fang ta c\u00f3 th\u1ec3 so s\u00e1nh nh\u1eefng \u0111\u1ed1i t\u01b0\u1ee3ng c\u1eadn tr\u00ean v\u00e0 c\u1eadn d\u01b0\u1edbi ng\u01b0\u1ee1ng quy\u1ebft \u0111\u1ecbnh \u0111\u1ec3 c\u00f3 \u0111\u01b0\u1ee3c t\u00e1c \u0111\u1ed9ng can thi\u1ec7p. Ch\u00fang ta \u0111\u00e3 bi\u1ebft c\u00e1ch l\u00e0m \u0111i\u1ec1u \u0111\u00f3 v\u1edbi h\u1ed3i quy tuy\u1ebfn t\u00ednh theo t\u1ef7 tr\u1ecdng b\u1eb1ng kernel v\u00e0 n\u00f3 th\u1eadm ch\u00ed c\u00f2n cung c\u1ea5p sai s\u1ed1 chu\u1ea9n cho ATE m\u00e0 ch\u00fang ta kh\u00f4ng c\u1ea7n ph\u1ea3i m\u1ea5t t\u00ed c\u00f4ng s\u1ee9c n\u00e0o.\n\nSau \u0111\u00f3, ch\u00fang ta xem x\u00e9t \u0111i\u1ec1u g\u00ec s\u1ebd x\u1ea3y ra trong thi\u1ebft k\u1ebf RD m\u1edd, khi c\u00f3 s\u1ef1 xu\u1ea5t hi\u1ec7n c\u1ee7a vi\u1ec7c kh\u00f4ng tu\u00e2n th\u1ee7. Ch\u00fang ta th\u1ea5y r\u1eb1ng c\u00e1ch ta c\u00f3 th\u1ec3 ti\u1ebfp c\u1eadn t\u00ecnh hu\u1ed1ng c\u0169ng gi\u1ed1ng nh\u01b0 c\u00e1ch ch\u00fang ta \u0111\u00e3 l\u00e0m v\u1edbi IV.\n\n## T\u00e0i li\u1ec7u tham kh\u1ea3o\n\nT\u00f4i mu\u1ed1n d\u00e0nh lo\u1ea1t b\u00e0i vi\u1ebft n\u00e0y \u0111\u1ec3 vinh danh Joshua Angrist, Alberto Abadie and Christopher Walters v\u00ec kh\u00f3a h\u1ecdc Kinh t\u1ebf l\u01b0\u1ee3ng tuy\u1ec7t c\u00fa m\u00e8o c\u1ee7a h\u1ecd. Ph\u1ea7n l\u1edbn \u00fd t\u01b0\u1edfng trong lo\u1ea1t b\u00e0i n\u00e0y \u0111\u01b0\u1ee3c l\u1ea5y t\u1eeb c\u00e1c b\u00e0i gi\u1ea3ng c\u1ee7a h\u1ecd \u0111\u01b0\u1ee3c t\u1ed5 ch\u1ee9c b\u1edfi Hi\u1ec7p h\u1ed9i Kinh t\u1ebf M\u0129. Theo d\u00f5i c\u00e1c b\u00e0i gi\u1ea3ng n\u00e0y l\u00e0 nh\u1eefng g\u00ec t\u00f4i l\u00e0m trong su\u1ed1t n\u0103m 2020 kh\u00f3 nh\u1eb1n.\n* [Kinh t\u1ebf l\u01b0\u1ee3ng v\u1edbi d\u1eef li\u1ec7u ch\u00e9o](https://www.aeaweb.org/conference/cont-ed/2017-webcasts)\n* [Luy\u1ec7n ch\u01b0\u1edfng Kinh t\u1ebf l\u01b0\u1ee3ng G\u1ea7n nh\u01b0 V\u00f4 h\u1ea1i](https://www.aeaweb.org/conference/cont-ed/2020-webcasts)\n\nT\u00f4i c\u0169ng mu\u1ed1n gi\u1edbi thi\u1ec7u cu\u1ed1n s\u00e1ch l\u00fd th\u00fa c\u1ee7a Angrist. Ch\u00fang cho t\u00f4i th\u1ea5y Kinh t\u1ebf l\u01b0\u1ee3ng, ho\u1eb7c 'L\u01b0\u1ee3ng theo c\u00e1ch h\u1ecd g\u1ecdi kh\u00f4ng ch\u1ec9 v\u00f4 c\u00f9ng h\u1eefu \u00edch m\u00e0 c\u00f2n r\u1ea5t vui.\n\n* [Kinh t\u1ebf l\u01b0\u1ee3ng G\u1ea7n nh\u01b0 V\u00f4 h\u1ea1i](https://www.mostlyharmlesseconometrics.com/)\n* [Luy\u1ec7n ch\u01b0\u1edfng 'L\u01b0\u1ee3ng](https://www.masteringmetrics.com/)\n\nT\u00e0i li\u1ec7u tham kh\u1ea3o cu\u1ed1i c\u00f9ng c\u1ee7a t\u00f4i l\u00e0 cu\u1ed1n s\u00e1ch c\u1ee7a Miguel Hernan and Jamie Robins. N\u00f3 l\u00e0 ng\u01b0\u1eddi b\u1ea1n \u0111\u1ed3ng h\u00e0nh tin c\u1eady v\u1edbi t\u00f4i khi tr\u1ea3 l\u1eddi nh\u1eefng c\u00e2u h\u1ecfi nh\u00e2n qu\u1ea3 kh\u00f3 nh\u1eb1n.\n\n* [S\u00e1ch Suy Lu\u1eadn Nh\u00e2n Qu\u1ea3](https://www.hsph.harvard.edu/miguel-hernan/causal-inference-book/)\n\n# B\u1ea3ng T\u1eeb Vi\u1ebft t\u1eaft \n|Vi\u1ebft t\u1eaft| Ti\u1ebfng Anh | Ti\u1ebfng Vi\u1ec7t |\n| --- | --- | --- | \n|IV|Instrumental Variable|Bi\u1ebfn C\u00f4ng c\u1ee5| \n|RD|Regression Discontinuity|H\u1ed3i quy Gi\u00e1n \u0111o\u1ea1n| \n|RDD|Regression Discontinuity Design|Thi\u1ebft k\u1ebf H\u1ed3i quy Gi\u00e1n \u0111o\u1ea1n| \n\n\n# B\u1ea3ng Thu\u1eadt ng\u1eef \n| Thu\u1eadt ng\u1eef | Ti\u1ebfng Anh |\n| --- | --- | \n|bi\u1ebfn can thi\u1ec7p|treatment variable| \n|bi\u1ebfn c\u00f4ng c\u1ee5|instrumental-variable, instrumental variable, instrument, instrument variable| \n|bi\u1ebfn gi\u1ea3|dummy, dummy variable| \n|bi\u1ebfn k\u1ebft qu\u1ea3|outcome variable| \n|bootstrap|bootstrap| \n|can thi\u1ec7p ti\u1ec1m n\u0103ng|potential treatment| \n|ch\u1ec9 \u0111\u1ecbnh can thi\u1ec7p|treatment assignment| \n|code|code| \n|d\u1ea3i t\u1ea7n su\u1ea5t|bandwidth| \n|gi\u00e1 tr\u1ecb d\u1ef1 \u0111o\u00e1n|predicted value| \n|gi\u1ea3 thuy\u1ebft|hypothesis| \n|hi\u1ec7u \u1ee9ng da c\u1eebu|sheepskin effect| \n|h\u00e0m gi\u00e1n \u0111o\u1ea1n|discontinuous function| \n|h\u00e0m m\u1eadt \u0111\u1ed9|density function| \n|h\u00e0m tam gi\u00e1c kernel|triangular kernel| \n|h\u00e0m t\u1ef7 tr\u1ecdng|weighting function| \n|h\u1ec7 s\u1ed1 ch\u1eb7n|intercept| \n|h\u1ed3i quy|regression, regress| \n|h\u1ed3i quy gi\u00e1n \u0111o\u1ea1n|regression discontinuity| \n|h\u1ed3i quy tuy\u1ebfn t\u00ednh|linear regression| \n|ki\u1ec3m \u0111\u1ecbnh gi\u1ea3 thi\u1ebft|sanity check| \n|k\u00edch th\u01b0\u1edbc m\u1eabu|sample size| \n|k\u1ebft qu\u1ea3 ti\u1ec1m n\u0103ng|potential outcome| \n|m\u00f4 h\u00ecnh h\u1ed3i quy|regression model| \n|m\u00f4 h\u00ecnh \u01b0\u1edbc l\u01b0\u1ee3ng wald|wald estimator| \n|m\u1eabu|sample| \n|m\u1eadt \u0111\u1ed9|density| \n|m\u1eadt \u0111\u1ed9 d\u1eef li\u1ec7u|data density| \n|ng\u01b0\u1ee1ng|threshold| \n|ph\u00e2n ph\u1ed1i|distribution| \n|sai s\u1ed1 chu\u1ea9n|standard error| \n|tham s\u1ed1|parameter| \n|tham s\u1ed1 d\u1ea3i t\u1ea7n su\u1ea5t|bandwidth parameter| \n|thi\u00ean l\u1ec7ch|bias| \n|thi\u00ean l\u1ec7ch b\u1ea5t tu\u00e2n|non compliance bias| \n|thi\u1ebft k\u1ebf h\u1ed3i quy gi\u00e1n \u0111o\u1ea1n|regression discontinuity design| \n|th\u1eed nghi\u1ec7m ng\u1eabu nhi\u00ean|randomised trial| \n|th\u1eed nghi\u1ec7m ng\u1eabu nhi\u00ean c\u1ee5c b\u1ed9|local randomized trial| \n|tr\u1ecb s\u1ed1 p|p-value| \n|t\u00e1c \u0111\u1ed9ng can thi\u1ec7p|treatment effect, treatment impact| \n|t\u00e1c \u0111\u1ed9ng can thi\u1ec7p b\u00ecnh qu\u00e2n c\u1ee5c b\u1ed9|local average treatment effect| \n|x\u00e1c su\u1ea5t|probability| \n|\u0111i\u1ec3m d\u1eef li\u1ec7u|data point| \n|\u01b0\u1edbc l\u01b0\u1ee3ng c\u1ee5c b\u1ed9|local estimate| \n\n", "meta": {"hexsha": "9d709f26499a22e80d076acb60aff751ed1da075", "size": 489580, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ipynb/16-Thi\u1ebft-k\u1ebf-h\u1ed3i-quy-gi\u00e1n-\u0111o\u1ea1n-QA.ipynb", "max_stars_repo_name": "vietecon/NhanQuaPython", "max_stars_repo_head_hexsha": "0cf0e78faa9eeef23d8ed2c41125b58cd5a5330b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2020-11-19T11:59:09.000Z", "max_stars_repo_stars_event_max_datetime": "2021-01-20T02:06:07.000Z", "max_issues_repo_path": "ipynb/16-Thi\u1ebft-k\u1ebf-h\u1ed3i-quy-gi\u00e1n-\u0111o\u1ea1n-QA.ipynb", "max_issues_repo_name": "vietecon/NhanQuaPython", "max_issues_repo_head_hexsha": "0cf0e78faa9eeef23d8ed2c41125b58cd5a5330b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ipynb/16-Thi\u1ebft-k\u1ebf-h\u1ed3i-quy-gi\u00e1n-\u0111o\u1ea1n-QA.ipynb", "max_forks_repo_name": "vietecon/NhanQuaPython", "max_forks_repo_head_hexsha": "0cf0e78faa9eeef23d8ed2c41125b58cd5a5330b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2020-11-21T09:09:36.000Z", "max_forks_repo_forks_event_max_datetime": "2021-12-14T12:25:37.000Z", "avg_line_length": 489580.0, "max_line_length": 489580, "alphanum_fraction": 0.9311654888, "converted": true, "num_tokens": 14902, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.7879312006227324, "lm_q1q2_score": 0.42468169619195406}} {"text": "# Molecular Dynamics\n\n##### Syma Khalid\n\n\n```\nfrom IPython.core.display import HTML\ncss_file = '../ipython_notebook_styles/ngcmstyle.css'\nHTML(open(css_file, \"r\").read())\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n## Velocity Verlet algorithm\n\nConsiders both the atomic positions and velocities at the same time.\n\nDivide the procedure into three steps:\n\n$$\n\\begin{align}\n v(t + \\delta t / 2) & = v(t) = \\frac{a(t) \\, \\delta t}{2}, \\\\\n r(t + \\delta t) & = r(t) + v(t + \\delta t / 2) \\, \\delta t \\\\\n v(t + \\delta t) & = v(t + \\delta t / 2) + \\frac{a(t + \\delta t) \\, \\delta t}{2}.\n\\end{align}\n$$\n\nOnly set of positions, forces and velocities need to be carried at any one time.\n\n##### Advantages\n\n* Simple to code\n* Relatively modest memory requirements\n* Velocities explictly calculated, thus temperature is reliably calculated\n* Energy well conserved.\n\nThe algorithm of choice for most popular MD codes.\n\n## Temperature\n\nTemperature specifies thermodynamic state - important concept in MD:\n\n$$\n\\begin{equation}\n T(t) = \\sum_{i=1}^N \\frac{m_i v_i^2(t)}{K_B (3 N -3)}\n\\end{equation}\n$$\n\n\n\n$N$ is the number of particles and $K_B$ the Boltzmann constant.\n\n## Timesteps\n\nHow to choose $\\delta t$? Must strike balance between conserving energy and efficient simulation.\n\nHigh frequency motions can be problematic; get around this by \"freezing out\" - constraining bonds to their equilibrium values.\n\nFor biomolecules $\\delta t = 2$fs is often used.\n\n## System setup\n\nNeed coordinates for the molecules to start.\n\nUsual to get structure from x-ray crystallography or NMR (for biological systems).\n\nEmbed structure in an environment that mimics experimental conditions.\n\nSet system up carefully, taking care to remove any \"bad contacts\" - eg, use energy miinimisation.\n\n## Periodic Boundaries\n\nOnly small numbers of molecules in a cell (max 150k as of 2014). Molecules on the surface experience different forces to bulk molecules, so to avoid this use periodic boundaries.\n\nIf a particle leaves the central box one of its images will enter the box through the opposite face.\n\nThe number of particles in the central box (which is actually simulated) remains constant.\n\n## System Equilibriation\n\nThe equivalent of a \"well-stirred\" solution in synthetic chemistry.\n\nWhen doing a simulation in the **NPT** (Number of particles, Pressure, Temperature: these are *forced* to remain constant) ensemble, then we monitor the system volume (ie, the volume of the box).\n\nSimilarly, when using an **NVT** (Volume...) ensemble, we monitor the system pressure.\n\nOften start by equilibriating in the NVT ensemble, then switching to the NPT ensemble.\n\nCan study the systematic drift in the structure by monitoring the root-mean-square deviation\n\n## Timescale limitations\n\nOne of the major limitations: timescale that can be simulated.\n\nState-of-the-art: simulations max 1-10 $\\mu$s of \"real time\".\n\nProblem as conformation changes can take ms, s, to hours or longer.\n\nAdditional problem: encounters between two molecules can lead to changes on diffusion timescale of ms or longer.\n\n## Enhanced Sampling Methods\n\nMany different methods to tackle the timescale limitations. Focus on Metadynamics.\n\nAssume system can be described by a few selected degrees of freedom, called *collective variables* (CV).\n\nSampling facilitated by addition of a history-dependent bias potential that acts on the CVs.\n\nCalculate the location of the system in the space determined by the CVs, add a positive Gaussian to discourage the system from revisiting that location (as it has already been sampled).\n\nHow to choose CVs?\n\nThey are a function of the microscopic coordinates; not unique.\n\n* Should distinguish between initial and final state and also describe key intermediates (but none of these are necessarilly known!).\n* Should include all slow modes of the systesm.\n* Should be limited in number.\n\nOnce full energy landscape sampled, the *free-energy* is a constant as a function of the CVs. Can construct the true free-energy by substracting the Gaussians added (big additional benefit).\n\nHow many Gaussians to add, and the width, are tuneable parameters (plenty of trial and error).\n\n## Energy minimisation\n\nMethod employed to bring system to its (local) energy minimum. To find the minimum points\n\n* map the complete *hypersurface* (potential energy wrt all possible parameters) and thereby find all possible values\n* want approach that keeps computational cost realistic\n* tyipcally a $3N-5$ or $3N-6$ dimensional problem.\n\nVarious algorithms that work.\n\n* No single algorithm always works\n* Algorithms \"go downhill\", locating the nearest minimum (not necessarily the *global* minimum).\n\nTwo general categories:\n\n* Those that use derivative information\n* Those that don't.\n\nFocus on one example of a derivative method.\n\n### First Derivative Method - Steepest Descent.\n\nFunction must be continuous and differentiable. First order methods use the gradients. Clearly related to the force $\\vec{F} = -\\nabla V$.\n\nImagine standing on top of a hill and looking for the steepest slope that takes you down.\n\n* At each interation the search direction is taken as the negative gradient of the function.\n* Negative goes downhill.\n* How far to go? Use *line search*.\n\n* Locate minimum along the specified direction.\n* Bracket the minimum. Then use Bisection or similar to find the minimum.\n* From this point, recalculate the search direction; must be perpendicular to the first.\n\n**Advantages**:\n\n* Quickly removes worst steric clashes\n* Rapidly brings bond lengths and angles to ideal values \n\n**Disadvantages**\n\n* Inefficient in valleys and near the minimum\n\n\n```\n\n```\n", "meta": {"hexsha": "de7abfef36fe871ee353dbcaf49f90b96e82fd1e", "size": 16066, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "FEEG6016 Simulation and Modelling/2014/Molecular Dynamics Lecture 2.ipynb", "max_stars_repo_name": "ngcm/training-public", "max_stars_repo_head_hexsha": "e5a0d8830df4292315c8879c4b571eef722fdefb", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2015-06-23T05:50:49.000Z", "max_stars_repo_stars_event_max_datetime": "2016-06-22T10:29:53.000Z", "max_issues_repo_path": "FEEG6016 Simulation and Modelling/2014/Molecular Dynamics Lecture 2.ipynb", "max_issues_repo_name": "Jhongesell/training-public", "max_issues_repo_head_hexsha": "e5a0d8830df4292315c8879c4b571eef722fdefb", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2017-11-28T08:29:55.000Z", "max_issues_repo_issues_event_max_datetime": "2017-11-28T08:29:55.000Z", "max_forks_repo_path": "FEEG6016 Simulation and Modelling/2014/Molecular Dynamics Lecture 2.ipynb", "max_forks_repo_name": "Jhongesell/training-public", "max_forks_repo_head_hexsha": "e5a0d8830df4292315c8879c4b571eef722fdefb", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 24, "max_forks_repo_forks_event_min_datetime": "2015-04-18T21:44:48.000Z", "max_forks_repo_forks_event_max_datetime": "2019-01-09T17:35:58.000Z", "avg_line_length": 31.7509881423, "max_line_length": 206, "alphanum_fraction": 0.5022407569, "converted": true, "num_tokens": 2210, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736783928749127, "lm_q2_score": 0.7401743620390162, "lm_q1q2_score": 0.4246220384617566}} {"text": "# Robinson Crusoe or 10 minutes to respy\n\nThis is a short introduction to respy for new users. As economists love Robinsonades [1](#fn1), we will showcase the implementation of a Robinson Crusoe economy as a discrete choice dynamic programming model. Throughout the notebook you find italic text which make you familiar with Robinson's story. We will first set the scene with a broad introduction and then turn to the precise model specification. We continue by simulating the model and analyze its comparative statics. We then extend the model and showcase the estimation of the model parameters. \n\nJust to be clear, don't misinterpret the fact that we explain **respy** using such a simplistic model. **respy** is not a toy and can just as well solve state-of-the-art structural models. It's just easier to explain **respy** in a situation where we don't have to explain a complicated model at the same time. \n\n\n```python\n%matplotlib inline\n\nimport io\nimport matplotlib.pyplot as plt\nimport pandas as pd\nimport respy as rp\nimport yaml\nimport seaborn as sns\nimport numpy as np\n\nfrom pathlib import Path\nfrom time import time\n\nplt.style.use(\"../_static/respy.mplstyle\")\n```\n\n## Introduction \n\n---\n*After setting sail against his parents' wishes, being captured by pirates, escaping from them, building a plantation, and setting sail again to capture slaves in Africa, [Robinson Crusoe](https://en.wikipedia.org/wiki/Robinson_Crusoe) stranded on a small island. He is alone with one dog, two cats, and only some supplies. He goes fishing to make ends meet and if he is too tired he will relax in his hammock. But, he cannot relax to often as storing food is a difficult task on a tropical island.*\n\n---\n\nIn the discrete choice dynamic programming model, Robinson chooses every period $t = 0, \\dots, T - 1$ to either go fishing, $a = 0$, or spend the day in the hammock, $a = 1$, to maximize his expected sum of discounted lifetime utility. The utility of a choice, $U(s_t, a_t)$, depends on the state $s_t$, which contains information on the individual's characteristics, and the chosen alternative $a_t$. For working alternatives like fishing utility consists of two components, a wage and a non-pecuniary component.\n\n$$\n U(s_t, a_t) = W(s_t, a_t) + N(s_t, a_t)\n$$\n\nFor non-working alternatives like the hammock, $W(s_t, a_t) = 0$. The wage is defined as\n\n$$\\begin{align}\n W(s_t, a_t) &= r_a \\exp\\{x^w_{at} \\beta^w_a + \\epsilon_{at}\\}\\\\\n \\ln(W(s_t, a_t)) &= \\ln(r_a) + x^w_{at} \\beta^w_a + \\epsilon_{at}\n\\end{align}$$\n\n\nwhere $r_a$ is normally a market rental price for the skill units generated in the exponential expression. Another interpretation is that $ln(r_a)$ is simply the constant in the skill units. The skill units are generated by two components. $x^w_{at}$ and $\\beta^w_a$ are the choice- and time-dependent covariates and parameters related to the wage signaled by superscript $w$. $\\epsilon_{at}$ is a choice-specific random shock from the shock vector $\\epsilon_t \\sim \\mathcal{N}(0, \\Sigma)$ for all choices. Shocks are usually correlated across choices in one period, but are independent across periods.\n\nThe non-pecuniary rewards for working alternatives are a vector dot product of covariates $x_t^w$ and parameters $\\beta^w$. The superscript $w$ signals that the components belong to working alternatives.\n\n$$\n N^w(s_t, a_t) = x_t^w\\beta^w\n$$\n\nThe non-pecuniary reward for non-working alternatives is very similar except that the shocks enter the equation additively. Superscript $n$ stands for non-pecuniary.\n\n$$\n N^n(s_t, a_t) = x_t^n\\beta^n + \\epsilon_{at}\n$$\n\nAlong with the lower triangular elements of the shock variance-covariance matrix of $\\epsilon_t$, the utility parameters $\\beta_a^w$, $\\beta_a^n$ and $r_a$ form the main parameters of the model.\n\nIf Robinson chooses to go fishing, he gains one additional unit of experience in the next period. Experience starts at zero and goes over 1, 2, 3 up to $T - 1$.\n\nThe general assumption imposed on Robinson is that he is forward-looking and maximizes the expected present value of utility over the remaining lifetime. W.l.o.g. $t = 0$ and let $V(s_0)$ define the value of the maximization which is achieved by a sequence of choices, $\\{a_t\\}^T_{t = 0}$, such that every action is in the choice set, $a_t \\in C(s_t)$ and $s_{t + 1}$ adheres to the law of motion. Then, the expected present value of lifetime utility in state $s_0$ is\n\n$$\n V(s_0) = \\text{E} \\max_{\\{a_t\\}^T_{t = 0}} \\left[\n \\sum^T_{t = 0} \\delta^t U(s_t, a_t) \\, \\Big|\n \\, a_t \\in C(s_t), s_{t+1} = m(s_t, a_t)\n \\right]\n$$\n\nNote that the shocks in period $t = 0$ are not stochastic. Thus, one can extract the utility in the current period $U(s_0, a_0)$, also called the flow utility, and the discount factor $\\delta$ from the expectation. Then, the formula can be rewritten so that the second term becomes the maximum over alternative-specific value functions at time $t = 1$ which are also called continuation values.\n\n$$\\begin{align}\n V(s_0) &= \\max_{a_0} \\, U(s_0, a_0) + \\delta \\text{E} \\max_{\\{a_t\\}^T_{t = 1}}\n \\left[\n \\sum^T_{t = 1} \\delta^{t - 1} U(s_t, a_t) \\, \\Big|\n \\, a_t \\in C(s_t), s_{t + 1} = m(s_t, a_t)\n \\right] \\\\\n &= \\max_{a_0} \\, U(s_0, a_0)\n + \\delta \\text{E} \\max_{a_1} V_{a_1}(s_1)\n\\end{align}$$\n\nThe maximum over alternative-specific value functions can be rewritten as the value function of state $s_1$ or $V(s_1) = \\max_{a_1} V_{a_1}(s_1)$ which yields the famous Bellman equation. Due to the recursive nature of the problem, the alternative-specific value functions are defined as\n\n$$\\begin{equation}\n V_a(s_t) = \\begin{cases}\n U(s_t, a_t) + \\delta \\text{E} V(s_{t+1}) & \\text{if } t < T \\\\\n U(s_t, a_t) & \\text{if } t = T\n \\end{cases}\n\\end{equation}$$\n\nThe former equation shows that the shocks in period $t + 1$ are unknown to the individual in period $t$. Thus, utility must be maximized given the joint distribution of shocks in period $t + 1$ which is a maximization problem over a two-dimensional integral. Denote the non-stochastic part of a state as $s^-$. Then, Robinson maximizes\n\n$$\\begin{equation}\n V(s_t) = \\max_{a_t}\\{\n U(s_t, a_t) + \\delta \\int_{\\epsilon_{0, t + 1}} \\dots \\int_{\\epsilon_{2, t + 1}}\n \\max_{a_{t + 1}} V_{a_{t + 1}}(s^-_{t + 1}, \\epsilon_{t + 1})\n f_\\epsilon(\\epsilon_{t + 1})\n d_{\\epsilon_{0, t + 1}} \\dots d_{\\epsilon_{2, t + 1}}\n \\}\n\\end{equation}$$\n\n## Specification\n\nHow can we express the equations and parameters with **respy**? The following cell contains the code to write a `.csv` file which is the cornerstone of a model as it contains all parameters and some other specifications. With `io.StringIO` we can pretend it is an actual file on the filesystem and easily loaded with `pandas`.\n\n\n```python\nparams = \"\"\"category,name,value\ndelta,delta,0.95\nwage_fishing,exp_fishing,0.1\nnonpec_fishing,constant,-1\nnonpec_hammock,constant,2.5\nnonpec_hammock,not_fishing_last_period,-1\nshocks_sdcorr,sd_fishing,1\nshocks_sdcorr,sd_hammock,1\nshocks_sdcorr,corr_hammock_fishing,-0.2\nlagged_choice_1_hammock,constant,1\n\"\"\"\n```\n\n\n```python\nparams = pd.read_csv(io.StringIO(params), index_col=[\"category\", \"name\"])\nparams\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        value
                                        categoryname
                                        deltadelta0.95
                                        wage_fishingexp_fishing0.10
                                        nonpec_fishingconstant-1.00
                                        nonpec_hammockconstant2.50
                                        not_fishing_last_period-1.00
                                        shocks_sdcorrsd_fishing1.00
                                        sd_hammock1.00
                                        corr_hammock_fishing-0.20
                                        lagged_choice_1_hammockconstant1.00
                                        \n
                                        \n\n\n\nThe DataFrame for the parameters contains a two-level MultiIndex to group parameters in categories. `name` should be uniquely assigned in each category or otherwise only the sum of identically named parameters is identified. `value` contains the value of the parameter. Note that we named Robinson's alternatives `\"fishing\"` and `\"hammock\"` and we have to use the names consistently. As long as you stick to lowercase letters separated by underscores, you can choose any name you want.\n\nThe parameter specification contains following entries:\n\n- The first entry contains the discount factor of individuals.\n- The second category `\"wage_fishing\"` contains the parameters of the log wage equation for fishing. The group contains only one name called `\"exp_fishing\"` where `\"exp_*\"` is an identifier in the model for experience accumulated in a certain alternative. **respy** requires that you respect those identifiers of which there are not many and reference your alternatives consistently with the same name. If you stick to lowercase letters possibly separated by underscores, you are fine.\n- The third and fourth categories concern the non-pecuniary reward of fishing and relaxing in the hammock.\n- `\"shocks_sdcorr\"` groups the lower triangular of the variance-covariance matrix of shocks.\n- `\"lagged_choice_1_hammock\"` governs the distribution of previous choices at the begin of the model horizon.\n\n`params` is complemented with `options` which contains additional information. Here is short description:\n\n- `\"n_periods\"` defines the number of periods for which decision rules are computed.\n- `\"_seed\"`: Seeds are used in every model component to ensure reproducibility. You can use any seed you would like or even repeat the same seed number. Internally, we ensure that randomness is completely uncorrelated.\n- `\"estimation_draws\"` defines the number of draws used to simulate the choice probabilities with Monte Carlo simulation in the maximum likelihood estimation.\n- `\"estimation_tau\"` controls the temperature of the softmax function to avoid zero-valued choice probabilities.\n- `\"interpolation_points\"` controls how many states are used to approximate the value functions of others states in each period. `-1` turns the approximation off. The approximation is detailed in Keane and Wolpin (1994).\n- ``\"simulation_agents\"`` defines how many individuals are simulated.\n- ``\"solution_draws\"`` defines the number of draws used to simulate the expected value functions in the solution.\n- `\"covariates\"` is another dictionary where the key determines the covariate's name and the value is its definition. Here, we have to define what `\"constant\"` means.\n\n\n```python\noptions = \"\"\"n_periods: 10\nestimation_draws: 200\nestimation_seed: 500\nestimation_tau: 0.001\ninterpolation_points: -1\nsimulation_agents: 1_000\nsimulation_seed: 132\nsolution_draws: 500\nsolution_seed: 456\ncovariates:\n constant: \"1\"\n not_fishing_last_period: \"lagged_choice_1 != 'fishing'\"\n\"\"\"\n```\n\n\n```python\noptions = yaml.safe_load(options)\noptions\n```\n\n\n\n\n {'n_periods': 10,\n 'estimation_draws': 200,\n 'estimation_seed': 500,\n 'estimation_tau': 0.001,\n 'interpolation_points': -1,\n 'simulation_agents': 1000,\n 'simulation_seed': 132,\n 'solution_draws': 500,\n 'solution_seed': 456,\n 'covariates': {'constant': '1',\n 'not_fishing_last_period': \"lagged_choice_1 != 'fishing'\"}}\n\n\n\n## Simulation\n\nWe are now ready to simulate the model.\n\n\n```python\nsimulate = rp.get_simulate_func(params, options)\ndf = simulate(params)\n```\n\n\n```python\ndf.head(15)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Experience_FishingLagged_Choice_1Shock_Reward_FishingMeas_Error_Wage_FishingShock_Reward_HammockMeas_Error_Wage_HammockDense_KeyCore_IndexChoiceWage...Nonpecuniary_Reward_FishingWage_FishingFlow_Utility_FishingValue_Function_FishingContinuation_Value_FishingNonpecuniary_Reward_HammockWage_HammockFlow_Utility_HammockValue_Function_HammockContinuation_Value_Hammock
                                        IdentifierPeriod
                                        000hammock-0.03503510.040965101fishing0.965571...-10.965571-0.03442918.43014519.4363931.5NaN1.54714518.41463817.755256
                                        11fishing0.07425411.506491110hammockNaN...-11.1903580.19035817.89191218.6332152.5NaN3.96120320.07221616.958962
                                        21hammock-0.35456011.185316120hammockNaN...-10.775258-0.22474215.43311516.4819551.5NaN2.73228016.93671714.952039
                                        31hammock-0.1093971-0.785877131fishing0.990647...-10.990647-0.00935313.64500114.3730051.5NaN0.75188013.07314912.969757
                                        42fishing-1.06370511.245234146hammockNaN...-10.421597-0.57840311.72337012.9492352.5NaN3.93281614.96762911.615592
                                        52hammock-0.1062351-1.854834151fishing1.098301...-11.0983010.09830110.09015510.5177411.5NaN-0.2961118.5608569.323124
                                        63fishing-0.6926031-1.2385331610hammockNaN...-10.675297-0.3247037.8064398.5590982.5NaN1.4250118.5191457.467509
                                        73hammock-0.5552171-0.5420481713hammockNaN...-10.774749-0.2252515.4081865.9299341.5NaN1.0799475.8326615.002857
                                        83hammock-0.9434241-0.581919181hammockNaN...-10.525490-0.4745102.6642553.3039631.5NaN1.1185233.4273032.430295
                                        93hammock-1.54173810.487632193hammockNaN...-10.288882-0.711118-0.7111180.0000001.5NaN2.2861282.2861280.000000
                                        100hammock-0.71313711.418950101hammockNaN...-10.490104-0.50989617.95467819.4363931.5NaN3.03290919.90040217.755256
                                        10hammock-0.4641341-0.384774113hammockNaN...-10.628679-0.37132116.26839917.5154941.5NaN1.21582716.37756415.959723
                                        20hammock0.0667551-0.844060121fishing1.069034...-11.0690340.06903414.82809415.5358541.5NaN0.65964314.05338414.098675
                                        31fishing-0.7485651-1.348821135hammockNaN...-10.522795-0.47720513.17715014.3730052.5NaN1.32814413.64941212.969757
                                        41hammock-0.1305741-0.539693149fishing0.969889...-10.969889-0.03011111.56296112.2032341.5NaN0.99732611.38607910.935529
                                        \n

                                        15 rows \u00d7 22 columns

                                        \n
                                        \n\n\n\nWe can inspect Robinson's decisions period by period.\n\n\n```python\nfig, ax = plt.subplots()\n\ndf.groupby(\"Period\").Choice.value_counts(normalize=True).unstack().plot.bar(\n stacked=True, ax=ax\n)\n\nplt.xticks(rotation=\"horizontal\")\n\nplt.legend(loc=\"lower center\", bbox_to_anchor=(0.5,-0.275), ncol=2)\n\nplt.show()\nplt.close()\n```\n\nWe can also analyze the persistence in decisions.\n\n\n```python\ndata = pd.crosstab(df.Lagged_Choice_1, df.Choice, normalize=True)\nsns.heatmap(data, cmap=\"Blues\", annot=True)\n```\n\n## Analysis\n\nWe now study how Robinson's behavior changes as we increase the returns to experience. We do so by plotting the average level of final experience in the sample under the different parameterizations.\n\nThis analysis of the comparative statics of the model is straightforward to implement. In models of educational choice, this type of analysis is often applied to evaluate the effect of alternative tuition policies on average educational attainment. See Keane & Wolpin (1997, 2001) for example. The basic structure of the analysis remains the same.\n\n\n```python\n# Specification of grid for evaluation\nnum_points = 15 \ngrid_start = 0.0\ngrid_stop = 0.3\n\ngrid_points = np.linspace(grid_start, grid_stop, num_points)\n\nrslts = list()\nfor value in grid_points:\n \n params.loc[\"wage_fishing\", \"exp_fishing\"] = value\n\n df = simulate(params)\n\n stat = df.groupby(\"Identifier\")[\"Experience_Fishing\"].max().mean()\n rslts.append(stat)\n```\n\nWe collected all results in `rslts` and are ready to create a basic visualization.\n\n\n```python\nfig, ax = plt.subplots()\n\nax.plot(grid_points, rslts)\n\nax.set_ylim([0, 10])\nax.set_xlabel(\"Return to experience\")\nax.set_ylabel(\"Average final level of exerience\")\n\nplt.show()\nplt.close()\n```\n\nIn the absence of any returns to experience, Robinson still spends more than two periods fishing. This share then increases with the return. Starting at around 0.2, Robinson spends all his time fishing.\n\n## Extension\n\nLet us make the model more interesting!\n\n---\n*At some point Crusoe notices that a group of cannibals occasionally visits the island and celebrate one of their dark rituals. But then, a prisoner can escape and becomes Crusoe's new friend Friday whom he teaches English. In return Friday can share his knowledge once to help Robinson improve his fishing skills, but that is only possible after Robinson tried at least once to go fishing.*\n\n---\n\nA common extension to structural models is to increase the choice set. Here, we want to add another choice called `\"friday\"` which affects the utility of fishing. The choice should be available once, starting with the third period, and only after Robinson has been fishing before.\n\nNote that, we load the example models with the function, `rp.get_example_model`. The command for the former model is `params, options, df = rp.get_example_model(\"robinson_crusoe_basic\")`. You can use `with_data=False` to suppress the automatic simulation of a sample with this parameterization.\n\n\n```python\nparams, options = rp.get_example_model(\"robinson_crusoe_extended\", with_data=False)\n```\n\nAt first, take a look at the parameterization. There is a new positive parameter called `\"contemplation_with_friday\"` which enters the wage equation of fishing. The choice `\"friday\"` itself has a negative constant utility term which models the effort costs of learning and the food penalty. The variance-covariance matrix is also adjusted.\n\n\n```python\nparams\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        value
                                        categoryname
                                        deltadelta0.95
                                        wage_fishingexp_fishing0.10
                                        contemplation_with_friday0.40
                                        nonpec_fishingconstant-1.00
                                        nonpec_fridayconstant-1.00
                                        not_fishing_last_period-1.00
                                        nonpec_hammockconstant2.50
                                        not_fishing_last_period-1.00
                                        shocks_sdcorrsd_fishing1.00
                                        sd_friday1.00
                                        sd_hammock1.00
                                        corr_friday_fishing0.00
                                        corr_hammock_fishing0.00
                                        corr_hammock_friday0.00
                                        lagged_choice_1_hammockconstant1.00
                                        \n
                                        \n\n\n\nTurning to the `options`, we can see that the new covariate `\"contemplation_with_friday\"` is only affecting utility if Robinson is experienced in fishing and only for one interaction with friday. This naturally limits the interaction with Friday. The key `\"negative_choice_set\"` can be used to restrict the choice Friday to the third and following periods. The first key matches a choice. The value of the key can be a list of strings. If the string evaluates to `True`, a utility penalty ensures that individuals will never choose the corresponding states. There exist some states in the state space which will never be reached because choices are mutually exclusive or are affected by other restrictions. Filters under `\"core_state_space_filters\"` can be used to purge those states from the state space, reducing runtime and memory consumption.\n\n\n```python\noptions\n```\n\n\n\n\n {'n_periods': 10,\n 'estimation_draws': 200,\n 'estimation_seed': 500,\n 'estimation_tau': 0.001,\n 'interpolation_points': -1,\n 'simulation_agents': 1000,\n 'simulation_seed': 132,\n 'solution_draws': 500,\n 'solution_seed': 456,\n 'covariates': {'constant': '1',\n 'contemplation_with_friday': 'exp_friday == 1 and exp_fishing >= 1',\n 'not_fishing_last_period': \"lagged_choice_1 != 'fishing'\"},\n 'negative_choice_set': {'friday': ['period < 2', 'exp_fishing == 0']},\n 'core_state_space_filters': [\"period > 0 and exp_fishing + exp_friday == period and lagged_choice_1 == 'hammock'\",\n 'period <= 2 and exp_friday != 0',\n 'period >= 3 and period - exp_friday < 2',\n 'exp_friday > 0 and exp_fishing == 0',\n \"exp_friday > 0 and exp_fishing == 1 and lagged_choice_1 == 'fishing'\",\n \"period - exp_friday == 2 and lagged_choice_1 != 'friday' and period > 2\",\n \"exp_{choices_w_exp} == 0 and lagged_choice_1 == '{choices_w_exp}'\"]}\n\n\n\nNow, let us simulate a sample of the new model.\n\n\n```python\nsimulate = rp.get_simulate_func(params, options)\n```\n\n\n```python\ndf = simulate(params)\n```\n\n\n```python\nfig, ax = plt.subplots()\n\ndf.groupby(\"Period\").Choice.value_counts(normalize=True).unstack().plot.bar(\n stacked=True, ax=ax, color=[\"C0\", \"C2\", \"C1\"],\n)\n\nplt.xticks(rotation=\"horizontal\")\n\nplt.legend(loc=\"lower center\", bbox_to_anchor=(0.5,-0.275), ncol=3)\n\nplt.show()\nplt.close()\n```\n\n## Estimation\n\nFor model calibration, **respy** supports estimation via maximum likelihood and the method of simulated moments. An appropriate criterion function function for both methods can be constructed in just a few steps.\n\nFor optimization of the criterion, the use of external optimization libraries is requried. We recommend [estimagic](https://github.com/OpenSourceEconomics/estimagic), an open-source tool to estimate structural models and more. **estimagic** can be used for the optimization and standard error calculation of a criterion function produced by **respy**. \n\nUnlike other optimization libraries, ``estimagic`` does not optimize over a simple vector of parameters, but instead stores parameters in a ``pd.DataFrame``, which makes it easier to parse them into the quantities we need, store lower and upper bounds together with parameters and express constraints on the parameters. \n\nFor ``estimagic``, we need to pass constraints on the parameters in a list containing dictionaries. Each dictionary is a constraint. A constraint includes two components: First, we need to tell ``estimagic`` which parameters we want to constrain. This is achieved by specifying an index location which will be passed to `df.loc`. Then, define the type of the constraint. Here, we only impose the constraint that the shock parameters have to be valid variances and correlations.\n\n*Note*: It is possible to utilize other optimization libraries but we recommend **estimagic** due to the reasons stated above.\n\n\n```python\nfrom estimagic import maximize\n```\n\n\n```python\ncrit_func = rp.get_log_like_func(params, options, df)\ncrit_func(params)\n\nconstr = rp.get_parameter_constraints(\"robinson_crusoe\")\n```\n\n\n```python\nresults = maximize(\n criterion=crit_func, \n params=params, \n algorithm=\"scipy_lbfgsb\", \n algo_options={\"stopping_max_criterion_evaluations\": 3}, \n constraints=constr,\n)\n```\n\nRunning the minimal working example shown above will start an optimization that is limited to three criterion evaluations (for the sake of this tutorial). **estimagic** will also produce a logging database called `logging.db` that stores information about the optimization. The package offers many useful options to set up a tractable estimation for your model.\n\nMore information can be found in the **estimagic** documentation: https://estimagic.readthedocs.io/.\n\n## Footnotes\n\n1\n One of the earliest references of Robinsonades in economics can be found in Marx (1867). In the 37th footnote, he mentions that even Ricardo used the theme before him.\n\n\n## References\n\n> Bellman, R. (1957). Dynamic Programming. *Princeton University Press*, Princeton, NJ.\n\n> Keane, M. P. and Wolpin, K. I. (1997). [The Career Decisions of Young Men](https://doi.org/10.1086/262080). *Journal of Political Economy*, 105(3): 473-522.\n\n> Marx, K. (1867). Das Kapital, Bd. 1. *MEW*, Bd, 23, 405\n", "meta": {"hexsha": "817529bcd9cbdb5a279c442cb6eb4095af9c1ea3", "size": 246269, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/tutorials/robinson_crusoe.ipynb", "max_stars_repo_name": "restudToolbox/respy", "max_stars_repo_head_hexsha": "19b9602c6f34f39034b00a88f36219ed3c4cfe5a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 58, "max_stars_repo_stars_event_min_datetime": "2018-04-10T19:52:43.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-28T19:45:06.000Z", "max_issues_repo_path": "docs/tutorials/robinson_crusoe.ipynb", "max_issues_repo_name": "restudToolbox/respy", "max_issues_repo_head_hexsha": "19b9602c6f34f39034b00a88f36219ed3c4cfe5a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 319, "max_issues_repo_issues_event_min_datetime": "2018-03-29T07:06:47.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-27T18:03:10.000Z", "max_forks_repo_path": "docs/tutorials/robinson_crusoe.ipynb", "max_forks_repo_name": "restudToolbox/respy", "max_forks_repo_head_hexsha": "19b9602c6f34f39034b00a88f36219ed3c4cfe5a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 39, "max_forks_repo_forks_event_min_datetime": "2018-04-02T09:38:02.000Z", "max_forks_repo_forks_event_max_datetime": "2021-12-30T01:13:05.000Z", "avg_line_length": 269.4409190372, "max_line_length": 40318, "alphanum_fraction": 0.6632462876, "converted": true, "num_tokens": 9890, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494678483918, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.42445210924159105}} {"text": "\n\n# Research question: When to Accept or Reject an Offer? \r\n\r\nReferences: J J McCall. Economics of Information and Job Search. The Quarterly Journal of Economics, 84(1):113\u2013126, 1970.\n\n### install the QuantEcon Library \n\n\n```\n!pip install quantecon ###if you use anaconda, type in the command instead: !conda install -y quantecon\n```\n\n Requirement already satisfied: quantecon in /usr/local/lib/python3.6/dist-packages (0.4.8)\n Requirement already satisfied: requests in /usr/local/lib/python3.6/dist-packages (from quantecon) (2.23.0)\n Requirement already satisfied: numpy in /usr/local/lib/python3.6/dist-packages (from quantecon) (1.19.5)\n Requirement already satisfied: sympy in /usr/local/lib/python3.6/dist-packages (from quantecon) (1.1.1)\n Requirement already satisfied: scipy>=1.0.0 in /usr/local/lib/python3.6/dist-packages (from quantecon) (1.4.1)\n Requirement already satisfied: numba>=0.38 in /usr/local/lib/python3.6/dist-packages (from quantecon) (0.48.0)\n Requirement already satisfied: certifi>=2017.4.17 in /usr/local/lib/python3.6/dist-packages (from requests->quantecon) (2020.12.5)\n Requirement already satisfied: chardet<4,>=3.0.2 in /usr/local/lib/python3.6/dist-packages (from requests->quantecon) (3.0.4)\n Requirement already satisfied: urllib3!=1.25.0,!=1.25.1,<1.26,>=1.21.1 in /usr/local/lib/python3.6/dist-packages (from requests->quantecon) (1.24.3)\n Requirement already satisfied: idna<3,>=2.5 in /usr/local/lib/python3.6/dist-packages (from requests->quantecon) (2.10)\n Requirement already satisfied: mpmath>=0.19 in /usr/local/lib/python3.6/dist-packages (from sympy->quantecon) (1.1.0)\n Requirement already satisfied: setuptools in /usr/local/lib/python3.6/dist-packages (from numba>=0.38->quantecon) (51.3.3)\n Requirement already satisfied: llvmlite<0.32.0,>=0.31.0dev0 in /usr/local/lib/python3.6/dist-packages (from numba>=0.38->quantecon) (0.31.0)\n\n\n\n```\n!pip install numba \n```\n\n Requirement already satisfied: numba in /usr/local/lib/python3.6/dist-packages (0.48.0)\n Requirement already satisfied: llvmlite<0.32.0,>=0.31.0dev0 in /usr/local/lib/python3.6/dist-packages (from numba) (0.31.0)\n Requirement already satisfied: setuptools in /usr/local/lib/python3.6/dist-packages (from numba) (51.3.3)\n Requirement already satisfied: numpy>=1.15 in /usr/local/lib/python3.6/dist-packages (from numba) (1.19.5)\n\n\n\n```\nimport numpy as np\r\nfrom numba import jit, float64, jitclass\r\nimport matplotlib.pyplot as plt\r\n%matplotlib inline\r\nimport quantecon as qe\r\nfrom quantecon.distributions import BetaBinomial\n```\n\n# The Model\n\n### The Objective function\r\n\r\nThe agent is infinitely lived and aims to maximize the expected discounted sum of earnings $E\\sum_{t=0}^{\\infty}\\beta^{t}y_t$\r\n\r\n\r\n* $\\beta$ is the discount factor\r\n* $y_{t}$ is income, equal to the wage $w_{t}$ when employed, and compensation $c$ when unemployed\r\n\r\n\n\n### The State Value Function\r\n\r\n$s_t$ is an i.i.d. random variable called \"the State\", it is distributed over a set $S$ with probability $p(s)$ for $s\\in S$\r\n\r\n* $w_t =w(s_t)$\r\n* **State value function**: $v^{*}(s)$ denotes the value of the objective function given state s when an agent in this situation makes an optimal decision\r\n\r\n* **The bellman equation**: the state value function must satisfy the following condition\r\n$v^{*}=max[w(s)/(1-\\beta),c+\\beta\\sum_{s'\\in S}v*(s')q(s') ]$\r\n\r\n\r\n\r\n\r\n\r\n\n\n### The optimal policy\r\nChoose the strategy that (accept or reject an offer) gives a higher value\r\n\r\n$\\sigma(s) =1[w(s)/(1-\\beta)>c+\\beta\\sum_{s'\\in S}v*(s')q(s') ]$\r\n\r\nhere $1[P]$ =1 if statement P is true and 0 otherwise\r\n\r\nwe can also write this as $\\sigma(s) =1[w(s)>\\overline{w}]$, where $\\overline{w}$ is the reservation wage which is a constant depending on \u03b2,c,and the wage distribution. From the above optimal strategy, we can see that actually $\\overline{w}=(1-\\beta)(c+\\beta\\sum_{s'\\in S}v*(s')q(s') )$.\r\n\r\n\r\nThe agent should accept if and only if the current wage offer exceeds the reservation wage.\r\n\n\n# Simulation\n\n## Simulate wage distributions\r\n\r\n###BetaBinominal distribution: https://en.wikipedia.org/wiki/Beta-binomial_distribution\r\n\r\nBetaBinominal(n,a,b) is a distribution on finite support {0, 1, 2, ..,n} with parameter a and b. \r\n\r\nSpecial case: \r\n\r\n\r\n* It reduces to the Bernoulli distribution as a special case when n = 1.\r\n* For a = b = 1, it is the discrete uniform distribution from 0 to n.\r\n\r\n\r\nChange following two boxes only to simulate the model for your own parameter of choices. \r\n\n\n\n```\nn, a, b = 50, 1, 1 # default parameters\r\nq_default = BetaBinomial(n, a, b).pdf() # default choice of q\n```\n\n\n```\nw_min, w_max = 50, 100\r\nw_default = np.linspace(w_min, w_max, n+1)\n```\n\n\n```\nfig, ax = plt.subplots()\r\nax.plot(w_default, q_default, '-o', label='$q(w(i))$')\r\nax.set_xlabel('wages')\r\nax.set_ylabel('probabilities')\r\n\r\nplt.show()\n```\n\n## Solve for the optimal policy and compute reservation wage\r\n\r\nreferences: compiling python code with @jit https://numba.pydata.org/numba-doc/latest/user/jit.html\n\n\n```\nmccall_data = [\r\n ('c', float64), # unemployment compensation\r\n ('\u03b2', float64), # discount factor\r\n ('w', float64[:]), # array of wage values, w[i] = wage at state i\r\n ('q', float64[:]) # array of probabilities\r\n]\n```\n\n\n```\n@jitclass(mccall_data)\r\nclass McCallModel:\r\n\r\n def __init__(self, c=25, \u03b2=0.99, w=w_default, q=q_default):\r\n\r\n self.c, self.\u03b2 = c, \u03b2\r\n self.w, self.q = w_default, q_default\r\n\r\n def state_action_values(self, i, v):\r\n \"\"\"\r\n The values of state-action pairs.\r\n \"\"\"\r\n # Simplify names\r\n c, \u03b2, w, q = self.c, self.\u03b2, self.w, self.q\r\n # Evaluate value for each state-action pair\r\n # Consider action = accept or reject the current offer\r\n accept = w[i] / (1 - \u03b2)\r\n reject = c + \u03b2 * np.sum(v * q)\r\n\r\n return np.array([accept, reject])\n```\n\n\n```\ndef plot_value_function_seq(mcm, ax, num_plots=6):\r\n \"\"\"\r\n Plot a sequence of value functions.\r\n\r\n * mcm is an instance of McCallModel\r\n * ax is an axes object that implements a plot method.\r\n\r\n \"\"\"\r\n\r\n n = len(mcm.w)\r\n v = mcm.w / (1 - mcm.\u03b2)\r\n v_next = np.empty_like(v)\r\n for i in range(num_plots):\r\n ax.plot(mcm.w, v, '-', alpha=0.4, label=f\"iterate {i}\")\r\n # Update guess\r\n for i in range(n):\r\n v_next[i] = np.max(mcm.state_action_values(i, v))\r\n v[:] = v_next # copy contents into v\r\n\r\n ax.legend(loc='lower right')\n```\n\n\n```\nmcm = McCallModel()\r\n\r\nfig, ax = plt.subplots()\r\nax.set_xlabel('wage')\r\nax.set_ylabel('value')\r\nplot_value_function_seq(mcm, ax)\r\nplt.show()\n```\n\n\n```\n@jit(nopython=True)\r\ndef compute_reservation_wage(mcm,\r\n max_iter=500,\r\n tol=1e-6):\r\n\r\n # Simplify names\r\n c, \u03b2, w, q = mcm.c, mcm.\u03b2, mcm.w, mcm.q\r\n\r\n # == First compute the value function == #\r\n\r\n n = len(w)\r\n v = w / (1 - \u03b2) # initial guess\r\n v_next = np.empty_like(v)\r\n i = 0\r\n error = tol + 1\r\n while i < max_iter and error > tol:\r\n\r\n for i in range(n):\r\n v_next[i] = np.max(mcm.state_action_values(i, v))\r\n\r\n error = np.max(np.abs(v_next - v))\r\n i += 1\r\n\r\n v[:] = v_next # copy contents into v\r\n\r\n # == Now compute the reservation wage == #\r\n\r\n return (1 - \u03b2) * (c + \u03b2 * np.sum(v * q))\n```\n\n\n```\ncompute_reservation_wage(mcm) \n```\n\n\n\n\n 92.17437718608396\n\n\n\n### change the parameter in the following box to calculate new reservation wage for parameters of your choice. \n\n\n```\nmcm = McCallModel(c=20, \u03b2=0.999)\r\ncompute_reservation_wage(mcm)\n```\n\n\n\n\n 97.67814959177011\n\n\n\n## Comparative Statics \n\n### the resevation wage\n\n\n```\ngrid_size = 25\r\nR = np.empty((grid_size, grid_size))\r\n\r\nc_vals = np.linspace(10.0, 30.0, grid_size) ### change parameter of the unemployment conpensation\r\n\u03b2_vals = np.linspace(0.9, 0.99, grid_size) ### change parameter of the discount factors\r\n\r\nfor i, c in enumerate(c_vals):\r\n for j, \u03b2 in enumerate(\u03b2_vals):\r\n mcm = McCallModel(c=c, \u03b2=\u03b2)\r\n R[i, j] = compute_reservation_wage(mcm)\n```\n\ncontour plot : \r\n\r\nhttps://problemsolvingwithpython.com/06-Plotting-with-Matplotlib/06.14-Contour-Plots/\n\n\n```\nfig, ax = plt.subplots()\r\n\r\ncs1 = ax.contourf(c_vals, \u03b2_vals, R.T, alpha=0.75)\r\nctr1 = ax.contour(c_vals, \u03b2_vals, R.T)\r\n\r\nplt.clabel(ctr1, inline=1, fontsize=13)\r\nplt.colorbar(cs1, ax=ax)\r\n\r\n\r\nax.set_title(\"reservation wage\")\r\nax.set_xlabel(\"$c$\", fontsize=16)\r\nax.set_ylabel(\"$\u03b2$\", fontsize=16)\r\n\r\nax.ticklabel_format(useOffset=False)\r\n\r\nplt.show()\n```\n\n### in the following box, we keep unemployment compensation c fixed, and we discuss how the reservation wage change when the discount factor $\\beta$ changes\n\n\n```\nline_lens =100 ### change to value of your choice\r\nc=60 # please change c to value of your choice\r\n\u03b2_vals = np.linspace(0, 0.9999, line_lens) # please change the range of beta to value of your choice\r\nR = np.empty((line_lens,1))\r\nfor j, \u03b2 in enumerate(\u03b2_vals):\r\n mcm = McCallModel(c=c, \u03b2=\u03b2)\r\n R[j] = compute_reservation_wage(mcm)\n```\n\n\n```\nfig, ax = plt.subplots()\r\n\r\ncs2 = ax.scatter(\u03b2_vals, R)\r\n\r\n\r\nax.set_xlabel(\"$\u03b2$\", fontsize=16)\r\nax.set_ylabel(\"$reservation wage$\", fontsize=16)\r\n\r\nax.ticklabel_format(useOffset=False)\r\n\r\nplt.show()\n```\n\n### in the following box, we keep the discount factor $\\beta$ constant, we discuss how the reservation wage change when the unemployment compensation changes\n\n\n```\nline_lens =25 ## change line_lens to value of your choice\r\n\u03b2=0.5 ### change beta to value of your choice\r\nc_vals = np.linspace(10, 80, line_lens) ### change c to the range of your choice\r\nR = np.empty((line_lens,1))\r\nfor i, c in enumerate(c_vals):\r\n mcm = McCallModel(c=c, \u03b2=\u03b2)\r\n R[i] = compute_reservation_wage(mcm)\n```\n\n\n```\nfig, ax = plt.subplots()\r\n\r\ncs3 = ax.scatter(c_vals, R)\r\n\r\n\r\nax.set_xlabel(\"$c$\", fontsize=16)\r\nax.set_ylabel(\"$reservation wage$\", fontsize=16)\r\n\r\nax.ticklabel_format(useOffset=False)\r\n\r\nplt.show()\n```\n\n### the duration of unemploymnet \n\ncomparative statics for the unemployment compensation\n\n\n```\ncdf = np.cumsum(q_default)\r\n\r\n@jit(nopython=True)\r\ndef compute_stopping_time(w_bar, seed=1234):\r\n\r\n np.random.seed(seed)\r\n t = 1\r\n while True:\r\n # Generate a wage draw\r\n w = w_default[qe.random.draw(cdf)]\r\n # Stop when the draw is above the reservation wage\r\n if w >= w_bar:\r\n stopping_time = t\r\n break\r\n else:\r\n t += 1\r\n return stopping_time\r\n\r\n@jit(nopython=True)\r\ndef compute_mean_stopping_time(w_bar, num_reps=100000):\r\n obs = np.empty(num_reps)\r\n for i in range(num_reps):\r\n obs[i] = compute_stopping_time(w_bar, seed=i)\r\n return obs.mean()\r\n\r\n\u03b2 =0.6 ### change the discount factor beta to value of your choice\r\nc_vals = np.linspace(10, 80, 25) ### change the unemployment compensation to value of your choice\r\n\r\n\r\nstop_times = np.empty_like(c_vals)\r\nfor i, c in enumerate(c_vals):\r\n mcm = McCallModel(c=c, \u03b2=\u03b2)\r\n w_bar = compute_reservation_wage(mcm)\r\n stop_times[i] = compute_mean_stopping_time(w_bar)\r\n\r\nfig, ax = plt.subplots()\r\n\r\nax.plot(c_vals, stop_times, label=\"mean unemployment duration\")\r\nax.set(xlabel=\"unemployment compensation\", ylabel=\"months\")\r\nax.legend()\r\n\r\nplt.show()\n```\n\ncomparative statics for the discount factor\n\n\n```\ncdf = np.cumsum(q_default)\r\n\r\n@jit(nopython=True)\r\ndef compute_stopping_time(w_bar, seed=1234):\r\n\r\n np.random.seed(seed)\r\n t = 1\r\n while True:\r\n # Generate a wage draw\r\n w = w_default[qe.random.draw(cdf)]\r\n # Stop when the draw is above the reservation wage\r\n if w >= w_bar:\r\n stopping_time = t\r\n break\r\n else:\r\n t += 1\r\n return stopping_time\r\n\r\n@jit(nopython=True)\r\ndef compute_mean_stopping_time(w_bar, num_reps=100000):\r\n obs = np.empty(num_reps)\r\n for i in range(num_reps):\r\n obs[i] = compute_stopping_time(w_bar, seed=i)\r\n return obs.mean()\r\n\r\nc=60 ### change the unemployment compensation c to values of your choice\r\n\u03b2_vals = np.linspace(0, 0.99, 100) ### change discount factor beta to range of your choice.\r\nstop_times = np.empty_like(\u03b2_vals)\r\nfor j, \u03b2 in enumerate(\u03b2_vals):\r\n mcm = McCallModel(c=c, \u03b2=\u03b2)\r\n w_bar = compute_reservation_wage(mcm)\r\n stop_times[j] = compute_mean_stopping_time(w_bar)\r\n\r\nfig, ax = plt.subplots()\r\n\r\nax.plot(\u03b2_vals, stop_times, label=\"discount factor\")\r\nax.set(xlabel=\"discount factor\", ylabel=\"months\")\r\nax.legend()\r\n\r\nplt.show()\n```\n\n\n```\n\n```\n", "meta": {"hexsha": "7db600b9f21e9f5726374ff93e1d033391dd9791", "size": 143919, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Simulation/Job_Searching_Model.ipynb", "max_stars_repo_name": "sunshineluyao/Econ-211", "max_stars_repo_head_hexsha": "c088b164a4670e2cd03faede7a97b7a2f0af8077", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Simulation/Job_Searching_Model.ipynb", "max_issues_repo_name": "sunshineluyao/Econ-211", "max_issues_repo_head_hexsha": "c088b164a4670e2cd03faede7a97b7a2f0af8077", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Simulation/Job_Searching_Model.ipynb", "max_forks_repo_name": "sunshineluyao/Econ-211", "max_forks_repo_head_hexsha": "c088b164a4670e2cd03faede7a97b7a2f0af8077", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 8, "max_forks_repo_forks_event_min_datetime": "2021-01-13T04:29:25.000Z", "max_forks_repo_forks_event_max_datetime": "2021-01-25T04:12:23.000Z", "avg_line_length": 157.8059210526, "max_line_length": 34622, "alphanum_fraction": 0.8702117163, "converted": true, "num_tokens": 3574, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494421679929, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.42445209335106265}} {"text": "```python\nimport os\nimport sys\njupyter_module_path = os.path.join( os.getcwd(), '..', '..' )\nsys.path.append( jupyter_module_path )\njupyter_module_path = os.path.join( os.getcwd(), '..', '..', 'EfJupyter' )\nsys.path.append( jupyter_module_path )\n\nfrom EfJupyter import EfConf\n```\n\nIn this example, exact analytical trajectory of a charged particle moving in a uniform electric field is compared with results of numerical simulation. \n\nIn uniform electric field charged particle moves with uniform acceleration, directed along the field.\n\n\\begin{align}\n& \\mathbf{v} = \\mathbf{v}_0 + \\frac{q}{m} \\mathbf{E}_0 t\n\\\\\n& \\mathbf{r} = \\mathbf{r}_0 + \\mathbf{v}_0 t + \\frac{1}{2} \\frac{q}{m} \\mathbf{E}_0 t^2\n\\end{align}\n\nPart of a config file, regarding single particle, is similar to \n['singe particle in free space'](https://github.com/epicf/ef/wiki/Ex1:-Single-Particle-In-Free-Space). \n\n_\u043f\u043e\u043c\u0435\u043d\u044f\u0442\u044c \u043d\u0430 \u0440\u0430\u0432\u043d\u043e\u043c\u0435\u0440\u043d\u043e\u0435 \u043f\u043e\u043b\u0435_\n\nAs of uniform electric field, currently ef can not use any analytical expressions for fields, nor load precomputed numerical fields. However, a good approximation for uniform field is field between two parallel charged plates. \nThe simplest way to model this in ef is to specify appropriate boundary conditions. We need nonzero potential difference on two opposing sides of the computational volume. \n\n [Boundary conditions]\n boundary_phi_left = 0.0\n boundary_phi_right = 0.0\n boundary_phi_bottom = 0.0\n boundary_phi_top = 0.0\n boundary_phi_near = 3.0\n boundary_phi_far = -3.0\n \nTo ensure a good degree of field uniformity, we need distance between two \"plates\" (i.e. sides of the computational volume with potential difference) to be much less than size of the plates\n\n [Spatial mesh]\n grid_x_size = 10.0\n grid_x_step = 0.25\n grid_y_size = 10.0\n grid_y_step = 0.25\n grid_z_size = 1.0\n grid_z_step = 0.01\n \n\nInitial position of the particle had to be adjusted:\n\n [Particle_source_box.emit_single_particle]\n ...\n box_x_left = 5.00\n box_x_right = 5.01\n box_y_bottom = 5.00\n box_y_top = 5.01\n box_z_near = 0.10\n box_z_far = 0.11\n ...\n\n\n\nAfter the simulation, it is useful to glance over potential distribution in the volume.\n3d figure is not very informative. \nUse 2d colormaps at XY and XZ planes at nodes near particle initial position.\nRelevant plotting script is [link].\n\nThe results are\n\n\n\nElectric field is directed perpendicular to equipotential surfaces. It can be seen, that it \nthe middle of computational region electric field is directed along Z. \n\n\n```python\n\n```\n\nTrajectories are plotted similarly to previous examples.\nMagnitude of electric field equals potential difference divided by distance between plates:\n[E_0 = - delta phi / d ][eqn]\n\nFigures of trajectories: (todo: something wrong with kin. energy.)\n\n\n\n\n\nDegree of field uniformity can be estimated by evaluating difference\nof field components near initial and final points of particle trajectory:\n[code]\n\nFor present choice of parameters: \n\n ( Exf - Exi ) / Ez0 = , \n ( Eyf - Eyi ) / Ez0 = , \n ( Ezf - Ezi ) / Ez0 = \n\nIt should not be very significant.\n\n\n```python\n\n```\n", "meta": {"hexsha": "8eef8fa08d74bd1943e73346bc0049038fce3044", "size": 5682, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/single_particle_in_electric_field/single_particle_in_uniform_electric_field.ipynb", "max_stars_repo_name": "JacobMSD/ef_python", "max_stars_repo_head_hexsha": "13d785c10dd293c60ab90065c518e5afb14e5a02", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "examples/single_particle_in_electric_field/single_particle_in_uniform_electric_field.ipynb", "max_issues_repo_name": "JacobMSD/ef_python", "max_issues_repo_head_hexsha": "13d785c10dd293c60ab90065c518e5afb14e5a02", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "examples/single_particle_in_electric_field/single_particle_in_uniform_electric_field.ipynb", "max_forks_repo_name": "JacobMSD/ef_python", "max_forks_repo_head_hexsha": "13d785c10dd293c60ab90065c518e5afb14e5a02", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.1016949153, "max_line_length": 236, "alphanum_fraction": 0.5902851109, "converted": true, "num_tokens": 825, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.7549149813536518, "lm_q1q2_score": 0.42439546268366346}} {"text": "# Description:\n\n* calculations for modeling fragments in a CsCl gradient under non-equilibrium conditions\n\n# Notes\n\n* Good chapter on determining G+C content from CsCl gradient analysis\n * http://www.academia.edu/428160/Using_Analytical_Ultracentrifugation_of_DNA_in_CsCl_Gradients_to_Explore_Large-Scale_Properties_of_Genomes\n\n* http://www.analyticalultracentrifugation.com/dynamic_density_gradients.htm\n\n* Meselson et al. - 1957 - Equilibrium Sedimentation of Macromolecules in Den\n\n* Vinograd et al. - 1963 - Band-Centrifugation of Macromolecules and Viruses \n\n* http://onlinelibrary.wiley.com.proxy.library.cornell.edu/doi/10.1002/bip.360101011/pdf\n\n\n## Ultracentrigation book\nhttp://books.google.com/books?hl=en&lr=&id=vxcSBQAAQBAJ&oi=fnd&pg=PA143&dq=Measurement+of+Density+Heterogeneity+by+Sedimentation+in&ots=l8ObYN-zVv&sig=Vcldf9_aqrJ-u7nQ1lBRKbknHps#v=onepage&q&f=false\n\n\n## Forum info\n* http://stackoverflow.com/questions/18624005/how-do-i-perform-a-convolution-in-python-with-a-variable-width-gaussian\n* http://timstaley.co.uk/posts/convolving-pdfs-in-python/\n\n\n## Possible workflows:\n\n### KDE convolution\n* KDE of fragment GC values\n* bandwidth cross validation: https://jakevdp.github.io/blog/2013/12/01/kernel-density-estimation/\n* convolution of KDE with diffusion function:\n\t* gaussian w/ mean of 0 and scale param = 44.5 (kb) / (mean fragment length)\n\t\t* http://www.academia.edu/428160/Using_Analytical_Ultracentrifugation_of_DNA_in_CsCl_Gradients_to_Explore_Large-Scale_Properties_of_Genomes\n\t* http://nbviewer.ipython.org/github/timstaley/ipython-notebooks/blob/compiled/probabilistic_programming/convolving_distributions_illustration.ipynb\n\n## variable KDE \n* variable KDE of fragment GC values where kernel sigma is determined by mean fragment length\n\t* gaussian w/ scale param = 44.5 (kb) / fragment length\n\n\n\n# Standard deviation of homogeneous DNA fragments\n\nVinograd et al., 1963; (band-centrifugation): \n\n\\begin{align}\n\\sigma^2 = \\frac{r_0}{r_0^0} \\left\\{ \\frac{r_0}{r_0^0} + 2D \\left( t - t^0 \\right) \\right\\}\n\\end{align}\n\n## Standard deviation of Gaussian band (assuming equilibrium), Meselson et al., 1957:\n\n\\begin{align}\n\\sigma^2 = -\\sqrt{w} \\\\\nw = \\textrm{molecular weight}\n\\end{align}\n\n## Standard deviation of Gaussian band at a given time, Meselson et al., 1957:\n\n\\begin{equation}\nt^* = \\frac{\\sigma^2}{D} \\left(ln \\frac{L}{\\sigma} + 1.26 \\right), \\quad L\\gg\\sigma \\\\\n\\sigma^2 = \\textrm{stdev at equilibrium} \\\\\nL = \\textrm{length of column}\n\\end{equation}\n\n\n\n\n* Gaussian within 1% of equillibrium value from center.\n* ! assumes density gradient established at t = 0\n\n\n### Alternative form (from Birne and Rickwood 1978; eq 6.22):\n\n\\begin{align}\nt = \\frac{\\beta^{\\circ}(p_p - p_m)}{w^4 r_p^2 s} \\left(1.26 + ln \\frac{r_b - r_t}{\\sigma}\\right)\n\\end{align}\n\n\n\\begin{equation}\nt = \\textrm{time in seconds} \\\\\n\\beta^{\\circ} = \\beta^{\\circ} \\textrm{ of salt forming the density gradient (CsCl = ?)} \\\\\np_p = \\textrm{buoyant density of the the particle at equilibrium} \\\\\np_m = \\textrm{average density of the medium} \\\\\nw = \\textrm{angular velocity} \\\\\nr_p = \\textrm{distance (cm) of particle from from the axis of rotation (at equilibrium)} \\\\\ns = \\textrm{sedimentation rate} (S_{20,w} * 10^{-13}) \\\\\nr_b = \\textrm{distance to top of gradient (cm)} \\\\\nr_t = \\textrm{distance to bottom of gradient (cm)} \\\\\nr_b - r_t = \\textrm{length of gradient (L)}\n\\end{equation}\n\n\n### Solving for sigma:\n\n\\begin{align}\n\\sigma = \\frac{L}{e^{\\left(\\frac{t w^4 r_p^2 s}{\\beta^{\\circ}(p_p - p_m)} - 1.26\\right)}}\n\\end{align}\n\n### sigma (alternative; but assuming sedimentation equilibrium reached; no time component)\n\n\\begin{align}\n{\\sigma} = \\frac{\\theta}{M_{app}} \\frac{RT}{ \\frac{w^2r_c}{\\beta} * w^2r_o }\n\\end{align}\n\n\n\\begin{equation}\n{\\theta} = \\textrm{buoyant dnesity of the macromolecules} \\\\\nM_{app} = \\textrm{apparent molecular weight oif the solvated macromolecules} \\\\\nR = \\textrm{universal gas constant} \\\\\nT = \\textrm{Temperate in K} \\\\\nw = \\textrm{angular velocity} \\\\\n\\beta^{\\circ} = \\beta^{\\circ} \\textrm{ coef. of salt forming the density gradient} \\\\\nr_c = \\textrm{isoconcentration point} \\\\\nr_o = \\textrm{distance (cm) of particle from from the axis of rotation (at equilibrium)} \\\\\n\\end{equation}\n\n## Clay et al., 2003 method (assumes sedimentation equilibrium)\n\n\\begin{align}\n\\sigma = \\sqrt{\\frac{\\rho R T}{B^2 G M_C l}} \n\\end{align}\n\n\\begin{equation}\n{\\rho} = \\textrm{buoyant dnesity of the macromolecules} \\\\\nR = \\textrm{universal gas constant} \\\\\nT = \\textrm{Temperate in K} \\\\\n\\beta = \\beta^{\\circ} \\textrm{ coef. of salt forming the density gradient} \\\\\nM_C = \\textrm{molecular weight per base pair of dry cesium DNA} \\\\\nG = \\textrm{Constant from Clay et al., 2003 (7.87x10^-10) } \\\\\nl = \\textrm{fragment length (bp)} \\\\\n\\end{equation}\n\n\n# Variables specific to the Buckley lab setup\n\n\\begin{equation}\n\\omega = (2\\pi \\times \\textrm{RPM}) /60, \\quad \\textrm{RPM} = 55000 \\\\\n\\beta^{\\circ} = 1.14 \\times 10^9 \\\\\nr_b = 4.85 \\\\\nr_t = 2.6 \\\\\nL = r_b - r_t \\\\\ns = S_{20,w} * 10^{-13} \\\\\nS_{20,w} = 2.8 + 0.00834 * (l*666)^{0.479}, \\quad \\textrm{where l = length of fragment; S in Svedberg units} \\\\\np_m = 1.7 \\\\\np_p = \\textrm{buoyant density of the particle in CsCl} \\\\\nr_p = ? \\\\\nt = \\textrm{independent variable}\n\\end{equation}\n\n\n__isoconcentration point__\n\n\\begin{equation}\nr_c = \\sqrt{(r_t^2 + r_t * r_b + r_b^2)/3}\n\\end{equation}\n\n__rp in relation to the particle's buoyant density__\n\n\\begin{equation}\nr_p = \\sqrt{ ((p_p-p_m)\\frac{2\\beta^{\\circ}}{w}) + r_c^2 } \\\\\np_p = \\textrm{buoyant density}\n\\end{equation}\n\n__buoyant density of a DNA fragment in CsCl__\n\n\\begin{equation}\np_p = 0.098F + 1.66, \\quad \\textrm{where F = G+C molar fraction}\n\\end{equation}\n\n\n__info needed on a DNA fragment to determine it's sigma of the Guassian distribution__\n\n* fragment length\n* fragment G+C\n\n# Coding equations\n\n\n```python\n%load_ext rpy2.ipython\n```\n\n The rpy2.ipython extension is already loaded. To reload it, use:\n %reload_ext rpy2.ipython\n\n\n\n```r\n%%R\nlibrary(dplyr)\nlibrary(tidyr)\nlibrary(ggplot2)\nlibrary(gridExtra)\n```\n\n\n```r\n%%R\n\n\nGC2MW = function(x){ \n A = 313.2\n T = 304.2\n C = 289.2\n G = 329.2\n GC = G + C\n AT = A + T\n x = x / 100\n x*GC + (1-x)*AT \n}\n\n\nGC2BD = function(GC){\n # GC = percentage\n BD = GC / 100 * 0.098 + 1.66\n return(BD)\n}\n\ncalc_BD_macro = function(p_i, w, B, r)\n\nrpm2w2 = function(rpm){\n x = 2 * pi * rpm / 60\n return(x**2)\n}\n\ncalc_R_c = function(r_t, r_b){\n x = r_t**2 + r_t * r_b + r_b**2\n return(sqrt(x/3))\n}\n\n\ncalc_R_p = function(p_p, p_m, B, w, r_c){\n # distance of the particle from the axis of rotation (at equilibrium)\n x = ((p_p - p_m) * (2 * B / w)) + r_c**2\n return(sqrt(x))\n}\n\ncalc_S = function(l, GC){\n # l = dsDNA length (bp)\n MW = GC2MW(GC)\n S = 0.00834 * (l * MW)**0.479 + 2.8\n S = S * 1e-13\n return(S)\n}\n\ncalc_dif_sigma_OLD = function(L, w, r_p, S, t, B, p_p, p_m){\n nom = w**2 * r_p**2 * S \n denom = B * (p_p - p_m)\n x = nom / denom * t - 1.26\n sigma = L / exp(x)\n return(sigma)\n}\n \ncalc_dif_sigma = function(L, w, r_c, S, t, B, p_p, p_m){\n nom = w**2 * r_c**2 * S \n denom = B * (p_p - p_m)\n x = nom / denom * t - 1.26\n sigma = L / exp(x)\n return(sigma)\n}\n\nR_p2BD = function(r_p, p_m, B, w, r_c){\n # converting a distance from center of rotation of a particle to buoyant density\n ## inverse of `calc_R_p`\n nom = (r_p**2 - r_c**2) * w\n return(nom / (2 * B) + p_m)\n}\n\nsigma2BD = function(r_p, sigma, p_m, B, w, r_c){\n BD_low = R_p2BD(r_p - sigma, p_m, B, w, r_c)\n BD_high = R_p2BD(r_p + sigma, p_m, B, w, r_c)\n return(BD_high - BD_low)\n}\n \ntime2eq = function(B, p_p, p_m, w, r_c, s, L, sigma){\n x = (B * (p_p - p_m)) / (w**2 * r_c**2 * s) \n y = 1.26 + log(L / sigma)\n return(x * y)\n}\n \n```\n\n# Time to equilibrium\n\n\n```r\n%%R -w 450 -h 300\n# time to eq \n\ncalc_time2eq = function(x, B, L, rpm, r_t, r_b, sigma, p_m){\n l = x[1]\n GC = x[2]\n s = calc_S(l, GC)\n w = rpm2w2(rpm)\n p_p = GC2BD(GC)\n r_c = calc_R_c(r_t, r_b)\n #r_p = calc_R_p(p_p, p_m, B, w, r_c)\n t = time2eq(B, p_p, p_m, w, r_c, s, L, sigma)\n t = t / 360\n return(t)\n} \n\nrpm = 55000 \nB = 1.14e9\nr_b = 4.85\nr_t = 2.6\nL = r_b - r_t\np_m = 1.7\n\nl = seq(100,20000,100) # bp\nGC = 1:100 # percent\nsigma = 0.01\n\ndf = expand.grid(l, GC)\ndf$t = apply(df, 1, calc_time2eq, B=B, L=L, rpm=rpm, r_t=r_t, r_b=r_b, sigma=sigma, p_m=p_m)\ncolnames(df) = c('length', 'GC', 'time')\ndf %>% head\n\ncols = rev(rainbow(12))\np1 = ggplot(df, aes(GC, length, fill=time)) +\n geom_tile() +\n scale_x_continuous(expand=c(0,0)) +\n scale_y_continuous(expand=c(0,0)) +\n scale_fill_gradientn(colors=cols) +\n geom_hline(yintercept=4000, linetype='dashed', color='black') +\n #geom_vline(xintercept=60*60*66, linetype='dashed', color='black') +\n labs(x='GC (%)', y='dsDNA length (bp)') +\n theme_bw() +\n theme(\n text = element_text(size=16)\n )\n\np1\n```\n\n# sigma as a function of time & fragment length\n\n\n```r\n%%R\nrpm = 55000\nB = 1.14e9\nr_b = 4.85\nr_t = 2.6\nL = r_b - r_t\np_m = 1.7\n\nl = 500 # bp\nGC = 50 # pebrcent\nt = 60 * 60 * 66 # sec\n\nS = calc_S(l, GC)\nw2 = rpm2w2(rpm)\np_p = GC2BD(GC)\nr_c = calc_R_c(r_t, r_b)\nr_p = calc_R_p(p_p, p_m, B, w2, r_c)\nsigma = calc_dif_sigma(L, w2, r_p, S, t, B, p_p, p_m)\nprint(sigma)\n#sigma_BD = sigma2BD(r_p, sigma, p_m, B, w2, r_c)\n#print(sigma_BD)\n```\n\n\n [1] 9.800228e-105\n\n\n\n\n```r\n%%R\n\n#-- alternative calculation\np_p = 1.7\n\nM = l * 882\nR = 8.3144598 #J mol^-1 K^-1\nT = 293.15\n\ncalc_stdev(p_p, M, R, T, w2, r_c, B, r_p)\n```\n\n\n [1] 6.665251e-10\n\n\n\n# Graphing sigma as a function of time & fragment length\n\n\n```r\n%%R -h 300 -w 850\ncalc_sigma_BD = function(x, rpm, GC, r_t, r_b, p_m, B, L){\n l = x[1]\n t = x[2]\n S = calc_S(l, GC)\n w2 = rpm2w2(rpm)\n p_p = GC2BD(GC)\n r_c = calc_R_c(r_t, r_b)\n r_p = calc_R_p(p_p, p_m, B, w2, r_c)\n sigma = calc_dif_sigma(L, w2, r_p, S, t, B, p_p, p_m)\n if (sigma > L){\n return(NA)\n } else {\n return(sigma)\n }\n}\n\n\n# params\nGC = 50\nrpm = 55000\nB = 1.14e9\nr_b = 4.85\nr_t = 2.6\nL = r_b - r_t\np_m = 1.66\n\n# pairwise calculations of all parameters\nl = 50**seq(1,3, by=0.05)\nt = 6**seq(3,8, by=0.05)\ndf = expand.grid(l, t)\ndf$sigma = apply(df, 1, calc_sigma_BD, rpm=rpm, GC=GC, r_t=r_t, r_b=r_b, p_m=p_m, B=B, L=L)\ncolnames(df) = c('length', 'time', 'sigma')\ndf= df %>%\n mutate(sigma = ifelse((sigma < 1e-20 | sigma > 1e20), NA, sigma))\n\n# plotting\ncols = rev(rainbow(12))\np1 = ggplot(df, aes(time, length, fill=sigma)) +\n geom_tile() +\n scale_x_log10(expand=c(0,0)) +\n scale_y_log10(expand=c(0,0)) +\n scale_fill_gradientn(colors=cols) +\n #geom_hline(yintercept=4000, linetype='dashed', color='black') +\n geom_vline(xintercept=60*60*66, linetype='dashed', color='black') +\n labs(x='Time', y='Length') +\n theme_bw() +\n theme(\n text = element_text(size=16)\n )\n\np2 = p1 + scale_fill_gradientn(colors=cols, trans='log10')\ngrid.arrange(p1, p2, ncol=2)\n```\n\n### Low GC\n\n\n```r\n%%R -h 300 -w 850\n\n# params\nGC = 20\nrpm = 55000\nB = 1.14e9\nr_b = 4.85\nr_t = 2.6\nL = r_b - r_t\np_m = 1.66\n\n# pairwise calculations of all parameters\nl = 50**seq(1,3, by=0.05)\nt = 6**seq(3,8, by=0.05)\ndf = expand.grid(l, t)\ndf$sigma = apply(df, 1, calc_sigma_BD, rpm=rpm, GC=GC, r_t=r_t, r_b=r_b, p_m=p_m, B=B, L=L)\ncolnames(df) = c('length', 'time', 'sigma')\ndf= df %>%\n mutate(sigma = ifelse((sigma < 1e-20 | sigma > 1e20), NA, sigma))\n\n# plotting\ncols = rev(rainbow(12))\np1 = ggplot(df, aes(time, length, fill=sigma)) +\n geom_tile() +\n scale_x_log10(expand=c(0,0)) +\n scale_y_log10(expand=c(0,0)) +\n scale_fill_gradientn(colors=cols) +\n #geom_hline(yintercept=4000, linetype='dashed', color='black') +\n geom_vline(xintercept=60*60*66, linetype='dashed', color='black') +\n labs(x='Time', y='Length') +\n theme_bw() +\n theme(\n text = element_text(size=16)\n )\n\np2 = p1 + scale_fill_gradientn(colors=cols, trans='log10')\ngrid.arrange(p1, p2, ncol=2)\n```\n\n### High GC\n\n\n```r\n%%R -h 300 -w 850\n\n# params\nGC = 80\nrpm = 55000\nB = 1.14e9\nr_b = 4.85\nr_t = 2.6\nL = r_b - r_t\np_m = 1.66\n\n# pairwise calculations of all parameters\nl = 50**seq(1,3, by=0.05)\nt = 6**seq(3,8, by=0.05)\ndf = expand.grid(l, t)\ndf$sigma = apply(df, 1, calc_sigma_BD, rpm=rpm, GC=GC, r_t=r_t, r_b=r_b, p_m=p_m, B=B, L=L)\ncolnames(df) = c('length', 'time', 'sigma')\ndf= df %>%\n mutate(sigma = ifelse((sigma < 1e-20 | sigma > 1e20), NA, sigma))\n\n# plotting\ncols = rev(rainbow(12))\np1 = ggplot(df, aes(time, length, fill=sigma)) +\n geom_tile() +\n scale_x_log10(expand=c(0,0)) +\n scale_y_log10(expand=c(0,0)) +\n scale_fill_gradientn(colors=cols) +\n #geom_hline(yintercept=4000, linetype='dashed', color='black') +\n geom_vline(xintercept=60*60*66, linetype='dashed', color='black') +\n labs(x='Time', y='Length') +\n theme_bw() +\n theme(\n text = element_text(size=16)\n )\n\np2 = p1 + scale_fill_gradientn(colors=cols, trans='log10')\ngrid.arrange(p1, p2, ncol=2)\n```\n\n## Plotting Clay et al,. method\n\n\n```r\n%%R\ncalc_dif_sigma_Clay = function(rho, R, T, B, G, M, l){\n sigma = sqrt((rho*R*T)/(B**2*G*M*l))\n return(sigma)\n}\n```\n\n\n```r\n%%R -w 850 -h 300\n\nwrap_calc_sigma_Clay = function(x, R, T, B, G, m){\n l= x[1]\n GC = x[2]\n rho = GC2BD(GC)\n sigma = calc_dif_sigma_Clay(rho, R, T, B, G, m, l)\n return(sigma)\n}\n\n# params\nR = 8.3145e7\nT = 293.15\nG = 7.87e-10\nM = 882\nB = 1.14e9\n\nl = 50**seq(1,3, by=0.05)\nGC = 1:100\n\n# pairwise calculations of all parameters\n\ndf = expand.grid(l, GC)\n\ndf$sigma = apply(df, 1, wrap_calc_sigma_Clay, R=R, T=T, B=B, G=G, m=M)\ncolnames(df) = c('length', 'GC', 'sigma')\n\n\n# plotting\ncols = rev(rainbow(12))\np1 = ggplot(df, aes(GC, length, fill=sigma)) +\n geom_tile() +\n scale_y_log10(expand=c(0,0)) +\n scale_x_continuous(expand=c(0,0)) +\n scale_fill_gradientn(colors=cols) +\n labs(y='length (bp)', x='G+C') +\n theme_bw() +\n theme(\n text = element_text(size=16)\n )\n\np2 = p1 + scale_fill_gradientn(colors=cols, trans='log10')\ngrid.arrange(p1, p2, ncol=2)\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n# --Sandbox--\n\n# Graphing the equations above\n\n\n```python\n%pylab inline\n```\n\n\n```python\nimport scipy as sp\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nimport mixture\n#import sklearn.mixture as mixture\n```\n\n## Generating fragments\n\n\n```python\nn_frags = 10000\nfrag_GC = np.random.normal(0.5,0.1,n_frags)\nfrag_GC[frag_GC < 0] = 0\nfrag_GC[frag_GC > 1] = 1\nfrag_len = np.random.normal(10000,1000,n_frags)\n```\n\n\n```python\nret = plt.hist2d(frag_GC, frag_len, bins=100)\n```\n\n## Setting variables\n\n\n```python\nRPM = 55000\nomega = (2 * np.pi * RPM) / 60\n\nbeta_o = 1.14 * 10**9\n\nradius_bottom = 4.85 \nradius_top = 2.6 \ncol_len = radius_bottom - radius_top\n\ndensity_medium = 1.7\n```\n\n## Calculation functions\n\n\n```python\n# BD from GC\nfrag_BD = 0.098 * frag_GC + 1.66\n\nret = plt.hist(frag_BD, bins=100)\n```\n\n\n```python\nsedimentation = (frag_len*666)**0.479 * 0.00834 + 2.8 # l = length of fragment\n\nret = plt.hist(sedimentation, bins=100)\n```\n\n\n```python\n# sedimentation as a function of fragment length \nlen_range = np.arange(1,10000, 100)\n\nret = plt.scatter(len_range, 2.8 + 0.00834 * (len_range*666)**0.479 )\n```\n\n\n```python\n# isoconcentration point\niso_point = sqrt((radius_top**2 + radius_top * radius_bottom + radius_bottom**2)/3)\niso_point\n```\n\n\n```python\n# radius of particle\n\n#radius_particle = np.sqrt( (frag_BD - density_medium)*2*(beta_o/omega) + iso_point**2 )\n\n\n#ret = plt.hist(radius_particle)\n```\n\n\n```python\n\n```\n\n# Testing out speed of mixture models\n\n\n```python\nn_dists = 10\nn_samp = 10000\n```\n\n\n```python\ndef make_mm(n_dists):\n dist_loc = np.random.uniform(0,1,n_dists)\n dist_scale = np.random.uniform(0,0.1, n_dists)\n dists = [mixture.NormalDistribution(x,y) for x,y in zip(dist_loc, dist_scale)]\n eq_weights = np.array([1.0 / n_dists] * n_dists)\n eq_weights[0] += 1.0 - np.sum(eq_weights)\n return mixture.MixtureModel(n_dists, eq_weights, dists)\n```\n\n\n```python\nmm = make_mm(n_dists)\n```\n\n\n```python\n%%timeit\nsmp = mm.sampleDataSet(n_samp).getInternalFeature(0).flatten()\n```\n\n\n```python\n%%timeit\nsmp = np.array([mm.sample() for i in arange(n_samp)])\n```\n\n\n```python\nn_dists = 1000\nmm = make_mm(n_dists)\n```\n\n\n```python\n%%timeit\nsmp = mm.sampleDataSet(n_samp).getInternalFeature(0).flatten()\n```\n\n\n```python\n%%timeit\nsmp = np.array([mm.sample() for i in arange(n_samp)])\n```\n\n\n```python\nn_dists = 10000\nmm = make_mm(n_dists)\n```\n\n\n```python\n%%timeit\nsmp = mm.sampleDataSet(n_samp).getInternalFeature(0).flatten()\n```\n\n\n```python\n%%timeit\nsmp = np.array([mm.sample() for i in arange(n_samp)])\n```\n\n\n```python\nn_samp = 100000\n```\n\n\n```python\n%%timeit\nsmp = mm.sampleDataSet(n_samp).getInternalFeature(0).flatten()\n```\n\n\n```python\n%%timeit\nsmp = np.array([mm.sample() for i in arange(n_samp)])\n```\n\n__Notes:__\n\n* a mixture model with many distributions (>1000) is very slow for sampling\n\n\n```python\nx = np.random.normal(3, 1, 100)\ny = np.random.normal(1, 1, 100)\nH, xedges, yedges = np.histogram2d(y, x, bins=100)\n```\n\n\n```python\nH\n```\n\n***\n***\n\n# Workflow for modeling DNA fragment locations in a gradient\n\nFor each genome in mock community, simulate N fragments and calculate their Guassian distributions in the gradient.\nCreate a mixture model of those Guassian distributions to sample Aa fragments,\nwhere Aa = the absolute abundance of the taxon in the mock community.\nOne mixture model per genome.\n\n## User defined:\n\n* Rotor specs\n* cfg parameters (RPM, time)\n\n## Generate fragment density distributions\n\n* For each genome in the mock community:\n * Simulate fragments\n * Calculate sigma of Gaussian density distribution\n * Create mixture model from all Gaussians of the fragments\n \n## Simulate fraction communities\n\n* For each genome in mock community:\n * sample fragments from mixture model based on total abundance of taxon in mock community\n * bin fragments into gradient fractions\n\n\n```python\n\n```\n", "meta": {"hexsha": "e4f29fe11e4958688182f64dc62ca1a89742a873", "size": 175257, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ipynb/theory/non-equilibrium_calcs.ipynb", "max_stars_repo_name": "arischwartz/test", "max_stars_repo_head_hexsha": "87a8306a294f59b0eef992529ce900cea876c605", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2019-03-15T09:46:48.000Z", "max_stars_repo_stars_event_max_datetime": "2019-06-05T18:16:39.000Z", "max_issues_repo_path": "ipynb/theory/non-equilibrium_calcs.ipynb", "max_issues_repo_name": "arischwartz/test", "max_issues_repo_head_hexsha": "87a8306a294f59b0eef992529ce900cea876c605", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-11-01T23:18:10.000Z", "max_issues_repo_issues_event_max_datetime": "2020-11-01T23:18:10.000Z", "max_forks_repo_path": "ipynb/theory/non-equilibrium_calcs.ipynb", "max_forks_repo_name": "arischwartz/test", "max_forks_repo_head_hexsha": "87a8306a294f59b0eef992529ce900cea876c605", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 136.386770428, "max_line_length": 35503, "alphanum_fraction": 0.868142214, "converted": true, "num_tokens": 6497, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421276, "lm_q2_score": 0.6477982315512489, "lm_q1q2_score": 0.4242396473257246}} {"text": "# Model Interpretation Methods\n\nWelcome to the final assignment of course 3! In this assignment we will focus on the interpretation of machine learning and deep learning models. Using the techniques we've learned this week we'll revisit some of the models we've built throughout the course and try to understand a little more about what they're doing.\n\nIn this assignment you'll use various methods to interpret different types of machine learning models. In particular, you'll learn about the following topics:\n\n- Interpreting Deep Learning Models\n - Understanding output using GradCAMs\n- Feature Importance in Machine Learning\n - Permutation Method\n - SHAP Values\n\nLet's get started.\n\n### This assignment covers the folowing topics:\n\n- [1. Interpreting Deep Learning Models](#1)\n - [1.1 GradCAM](#1-1)\n - [1.1.1 Getting Intermediate Layers](#1-1-1)\n - [1.1.2 Getting Gradients](#1-1-2)\n - [1.1.3 Implementing GradCAM](#1-1-3)\n - [Exercise 1](#ex-01)\n - [1.1.4 Using GradCAM to Visualize Multiple Labels](#1-1-4)\n - [Exercise 2](#ex-02)\n- [2. Feature Importance in Machine Learning](#2)\n - [2.1 Permuation Method for Feature Importance](#2-1)\n - [2.1.1 Implementing Permutation](#2-1-1)\n - [Exercise 3](#ex-03)\n - [2.1.2 Implementing Importance](#2-1-2)\n - [Exercise 4](#ex-04)\n - [2.1.3 Computing our Feature Importance](#2-1-3)\n - [2.2 Shapley Values for Random Forests](#2-2)\n - [2.2.1 Visualizing Feature Importance on Specific Individuals](#2-2-1)\n - [2.2.2 Visualizing Feature Importance on Aggregate](#2-2-2)\n - [2.2.3 Visualizing Interactions between Features](#2-2-3)\n\n## Packages\n\nWe'll first import the necessary packages for this assignment.\n\n- `keras`: we'll use this framework to interact with our deep learning model\n- `matplotlib`: standard plotting library\n- `pandas`: we'll use this to manipulate data\n- `numpy`: standard python library for numerical operations\n- `cv2`: library that contains convenience functions for image processing\n- `sklearn`: standard machine learning library\n- `lifelines`: we'll use their implementation of the c-index\n- `shap`: library for interpreting and visualizing machine learning models using shapley values\n\n\n\n```python\nimport keras\nfrom keras import backend as K\nimport matplotlib.pyplot as plt\nimport pandas as pd\nimport numpy as np\nimport cv2\nimport sklearn\nimport lifelines\nimport shap\n\n\nfrom util import *\n\n# This sets a common size for all the figures we will draw.\nplt.rcParams['figure.figsize'] = [10, 7]\n```\n\n Using TensorFlow backend.\n\n\n\n## 1 Interpreting Deep Learning Models\n\nTo start, let's try understanding our X-ray diagnostic model from Course 1 Week 1. Run the next cell to load in the model (it should take a few seconds to complete).\n\n\n```python\nmodel = load_C3M3_model()\n```\n\n Got loss weights\n Loaded DenseNet\n Added layers\n Compiled Model\n Loaded Weights\n\n\nLet's load in an X-ray image to develop on. Run the next cell to load and show the image.\n\n\n```python\nIMAGE_DIR = 'nih_new/images-small/'\ndf = pd.read_csv(\"nih_new/train-small.csv\")\nim_path = IMAGE_DIR + '00016650_000.png'\nx = load_image(im_path, df, preprocess=False)\nplt.imshow(x, cmap = 'gray')\nplt.show()\n```\n\nNext, let's get our predictions. Before we plug the image into our model, we have to normalize it. Run the next cell to compute the mean and standard deviation of the images in our training set. \n\n\n```python\nmean, std = get_mean_std_per_batch(df)\n```\n\nNow we are ready to normalize and run the image through our model to get predictions.\n\n\n```python\nlabels = ['Cardiomegaly', 'Emphysema', 'Effusion', 'Hernia', 'Infiltration', 'Mass', 'Nodule', 'Atelectasis',\n 'Pneumothorax', 'Pleural_Thickening', 'Pneumonia', 'Fibrosis', 'Edema', 'Consolidation']\n\nprocessed_image = load_image_normalize(im_path, mean, std)\npreds = model.predict(processed_image)\npred_df = pd.DataFrame(preds, columns = labels)\npred_df.loc[0, :].plot.bar()\nplt.title(\"Predictions\")\nplt.show()\n```\n\nWe see, for example, that the model predicts Cardiomegaly (enlarged heart) with high probability. Indeed, this patient was diagnosed with cardiomegaly. However, we don't know where the model is looking when it's making its own diagnosis. To gain more insight into what the model is looking at, we can use GradCAMs.\n\n\n### 1.1 GradCAM\n\nGradCAM is a technique to visualize the impact of each region of an image on a specific output for a Convolutional Neural Network model. Through GradCAM, we can generate a heatmap by computing gradients of the specific class scores we are interested in visualizing.\n\n\n#### 1.1.1 Getting Intermediate Layers\n\nPerhaps the most complicated part of computing GradCAM is accessing intermediate activations in our deep learning model and computing gradients with respect to the class output. Now we'll go over one pattern to accomplish this, which you can use when implementing GradCAM.\n\nIn order to understand how to access intermediate layers in a computation, first let's see the layers that our model is composed of. This can be done by calling Keras convenience function `model.summary()`. Do this in the cell below.\n\n\n```python\nmodel.summary()\n```\n\n __________________________________________________________________________________________________\n Layer (type) Output Shape Param # Connected to \n ==================================================================================================\n input_1 (InputLayer) (None, None, None, 3 0 \n __________________________________________________________________________________________________\n zero_padding2d_1 (ZeroPadding2D (None, None, None, 3 0 input_1[0][0] \n __________________________________________________________________________________________________\n conv1/conv (Conv2D) (None, None, None, 6 9408 zero_padding2d_1[0][0] \n __________________________________________________________________________________________________\n conv1/bn (BatchNormalization) (None, None, None, 6 256 conv1/conv[0][0] \n __________________________________________________________________________________________________\n conv1/relu (Activation) (None, None, None, 6 0 conv1/bn[0][0] \n __________________________________________________________________________________________________\n zero_padding2d_2 (ZeroPadding2D (None, None, None, 6 0 conv1/relu[0][0] \n __________________________________________________________________________________________________\n pool1 (MaxPooling2D) (None, None, None, 6 0 zero_padding2d_2[0][0] \n __________________________________________________________________________________________________\n conv2_block1_0_bn (BatchNormali (None, None, None, 6 256 pool1[0][0] \n __________________________________________________________________________________________________\n conv2_block1_0_relu (Activation (None, None, None, 6 0 conv2_block1_0_bn[0][0] \n __________________________________________________________________________________________________\n conv2_block1_1_conv (Conv2D) (None, None, None, 1 8192 conv2_block1_0_relu[0][0] \n __________________________________________________________________________________________________\n conv2_block1_1_bn (BatchNormali (None, None, None, 1 512 conv2_block1_1_conv[0][0] \n __________________________________________________________________________________________________\n conv2_block1_1_relu (Activation (None, None, None, 1 0 conv2_block1_1_bn[0][0] \n __________________________________________________________________________________________________\n conv2_block1_2_conv (Conv2D) (None, None, None, 3 36864 conv2_block1_1_relu[0][0] \n __________________________________________________________________________________________________\n conv2_block1_concat (Concatenat (None, None, None, 9 0 pool1[0][0] \n conv2_block1_2_conv[0][0] \n __________________________________________________________________________________________________\n conv2_block2_0_bn (BatchNormali (None, None, None, 9 384 conv2_block1_concat[0][0] \n __________________________________________________________________________________________________\n conv2_block2_0_relu (Activation (None, None, None, 9 0 conv2_block2_0_bn[0][0] \n __________________________________________________________________________________________________\n conv2_block2_1_conv (Conv2D) (None, None, None, 1 12288 conv2_block2_0_relu[0][0] \n __________________________________________________________________________________________________\n conv2_block2_1_bn (BatchNormali (None, None, None, 1 512 conv2_block2_1_conv[0][0] \n __________________________________________________________________________________________________\n conv2_block2_1_relu (Activation (None, None, None, 1 0 conv2_block2_1_bn[0][0] \n __________________________________________________________________________________________________\n conv2_block2_2_conv (Conv2D) (None, None, None, 3 36864 conv2_block2_1_relu[0][0] \n __________________________________________________________________________________________________\n conv2_block2_concat (Concatenat (None, None, None, 1 0 conv2_block1_concat[0][0] \n conv2_block2_2_conv[0][0] \n __________________________________________________________________________________________________\n conv2_block3_0_bn (BatchNormali (None, None, None, 1 512 conv2_block2_concat[0][0] \n __________________________________________________________________________________________________\n conv2_block3_0_relu (Activation (None, None, None, 1 0 conv2_block3_0_bn[0][0] \n __________________________________________________________________________________________________\n conv2_block3_1_conv (Conv2D) (None, None, None, 1 16384 conv2_block3_0_relu[0][0] \n __________________________________________________________________________________________________\n conv2_block3_1_bn (BatchNormali (None, None, None, 1 512 conv2_block3_1_conv[0][0] \n __________________________________________________________________________________________________\n conv2_block3_1_relu (Activation (None, None, None, 1 0 conv2_block3_1_bn[0][0] \n __________________________________________________________________________________________________\n conv2_block3_2_conv (Conv2D) (None, None, None, 3 36864 conv2_block3_1_relu[0][0] \n __________________________________________________________________________________________________\n conv2_block3_concat (Concatenat (None, None, None, 1 0 conv2_block2_concat[0][0] \n conv2_block3_2_conv[0][0] \n __________________________________________________________________________________________________\n conv2_block4_0_bn (BatchNormali (None, None, None, 1 640 conv2_block3_concat[0][0] \n __________________________________________________________________________________________________\n conv2_block4_0_relu (Activation (None, None, None, 1 0 conv2_block4_0_bn[0][0] \n __________________________________________________________________________________________________\n conv2_block4_1_conv (Conv2D) (None, None, None, 1 20480 conv2_block4_0_relu[0][0] \n __________________________________________________________________________________________________\n conv2_block4_1_bn (BatchNormali (None, None, None, 1 512 conv2_block4_1_conv[0][0] \n __________________________________________________________________________________________________\n conv2_block4_1_relu (Activation (None, None, None, 1 0 conv2_block4_1_bn[0][0] \n __________________________________________________________________________________________________\n conv2_block4_2_conv (Conv2D) (None, None, None, 3 36864 conv2_block4_1_relu[0][0] \n __________________________________________________________________________________________________\n conv2_block4_concat (Concatenat (None, None, None, 1 0 conv2_block3_concat[0][0] \n conv2_block4_2_conv[0][0] \n __________________________________________________________________________________________________\n conv2_block5_0_bn (BatchNormali (None, None, None, 1 768 conv2_block4_concat[0][0] \n __________________________________________________________________________________________________\n conv2_block5_0_relu (Activation (None, None, None, 1 0 conv2_block5_0_bn[0][0] \n __________________________________________________________________________________________________\n conv2_block5_1_conv (Conv2D) (None, None, None, 1 24576 conv2_block5_0_relu[0][0] \n __________________________________________________________________________________________________\n conv2_block5_1_bn (BatchNormali (None, None, None, 1 512 conv2_block5_1_conv[0][0] \n __________________________________________________________________________________________________\n conv2_block5_1_relu (Activation (None, None, None, 1 0 conv2_block5_1_bn[0][0] \n __________________________________________________________________________________________________\n conv2_block5_2_conv (Conv2D) (None, None, None, 3 36864 conv2_block5_1_relu[0][0] \n __________________________________________________________________________________________________\n conv2_block5_concat (Concatenat (None, None, None, 2 0 conv2_block4_concat[0][0] \n conv2_block5_2_conv[0][0] \n __________________________________________________________________________________________________\n conv2_block6_0_bn (BatchNormali (None, None, None, 2 896 conv2_block5_concat[0][0] \n __________________________________________________________________________________________________\n conv2_block6_0_relu (Activation (None, None, None, 2 0 conv2_block6_0_bn[0][0] \n __________________________________________________________________________________________________\n conv2_block6_1_conv (Conv2D) (None, None, None, 1 28672 conv2_block6_0_relu[0][0] \n __________________________________________________________________________________________________\n conv2_block6_1_bn (BatchNormali (None, None, None, 1 512 conv2_block6_1_conv[0][0] \n __________________________________________________________________________________________________\n conv2_block6_1_relu (Activation (None, None, None, 1 0 conv2_block6_1_bn[0][0] \n __________________________________________________________________________________________________\n conv2_block6_2_conv (Conv2D) (None, None, None, 3 36864 conv2_block6_1_relu[0][0] \n __________________________________________________________________________________________________\n conv2_block6_concat (Concatenat (None, None, None, 2 0 conv2_block5_concat[0][0] \n conv2_block6_2_conv[0][0] \n __________________________________________________________________________________________________\n pool2_bn (BatchNormalization) (None, None, None, 2 1024 conv2_block6_concat[0][0] \n __________________________________________________________________________________________________\n pool2_relu (Activation) (None, None, None, 2 0 pool2_bn[0][0] \n __________________________________________________________________________________________________\n pool2_conv (Conv2D) (None, None, None, 1 32768 pool2_relu[0][0] \n __________________________________________________________________________________________________\n pool2_pool (AveragePooling2D) (None, None, None, 1 0 pool2_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block1_0_bn (BatchNormali (None, None, None, 1 512 pool2_pool[0][0] \n __________________________________________________________________________________________________\n conv3_block1_0_relu (Activation (None, None, None, 1 0 conv3_block1_0_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block1_1_conv (Conv2D) (None, None, None, 1 16384 conv3_block1_0_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block1_1_bn (BatchNormali (None, None, None, 1 512 conv3_block1_1_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block1_1_relu (Activation (None, None, None, 1 0 conv3_block1_1_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block1_2_conv (Conv2D) (None, None, None, 3 36864 conv3_block1_1_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block1_concat (Concatenat (None, None, None, 1 0 pool2_pool[0][0] \n conv3_block1_2_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block2_0_bn (BatchNormali (None, None, None, 1 640 conv3_block1_concat[0][0] \n __________________________________________________________________________________________________\n conv3_block2_0_relu (Activation (None, None, None, 1 0 conv3_block2_0_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block2_1_conv (Conv2D) (None, None, None, 1 20480 conv3_block2_0_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block2_1_bn (BatchNormali (None, None, None, 1 512 conv3_block2_1_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block2_1_relu (Activation (None, None, None, 1 0 conv3_block2_1_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block2_2_conv (Conv2D) (None, None, None, 3 36864 conv3_block2_1_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block2_concat (Concatenat (None, None, None, 1 0 conv3_block1_concat[0][0] \n conv3_block2_2_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block3_0_bn (BatchNormali (None, None, None, 1 768 conv3_block2_concat[0][0] \n __________________________________________________________________________________________________\n conv3_block3_0_relu (Activation (None, None, None, 1 0 conv3_block3_0_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block3_1_conv (Conv2D) (None, None, None, 1 24576 conv3_block3_0_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block3_1_bn (BatchNormali (None, None, None, 1 512 conv3_block3_1_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block3_1_relu (Activation (None, None, None, 1 0 conv3_block3_1_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block3_2_conv (Conv2D) (None, None, None, 3 36864 conv3_block3_1_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block3_concat (Concatenat (None, None, None, 2 0 conv3_block2_concat[0][0] \n conv3_block3_2_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block4_0_bn (BatchNormali (None, None, None, 2 896 conv3_block3_concat[0][0] \n __________________________________________________________________________________________________\n conv3_block4_0_relu (Activation (None, None, None, 2 0 conv3_block4_0_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block4_1_conv (Conv2D) (None, None, None, 1 28672 conv3_block4_0_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block4_1_bn (BatchNormali (None, None, None, 1 512 conv3_block4_1_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block4_1_relu (Activation (None, None, None, 1 0 conv3_block4_1_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block4_2_conv (Conv2D) (None, None, None, 3 36864 conv3_block4_1_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block4_concat (Concatenat (None, None, None, 2 0 conv3_block3_concat[0][0] \n conv3_block4_2_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block5_0_bn (BatchNormali (None, None, None, 2 1024 conv3_block4_concat[0][0] \n __________________________________________________________________________________________________\n conv3_block5_0_relu (Activation (None, None, None, 2 0 conv3_block5_0_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block5_1_conv (Conv2D) (None, None, None, 1 32768 conv3_block5_0_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block5_1_bn (BatchNormali (None, None, None, 1 512 conv3_block5_1_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block5_1_relu (Activation (None, None, None, 1 0 conv3_block5_1_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block5_2_conv (Conv2D) (None, None, None, 3 36864 conv3_block5_1_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block5_concat (Concatenat (None, None, None, 2 0 conv3_block4_concat[0][0] \n conv3_block5_2_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block6_0_bn (BatchNormali (None, None, None, 2 1152 conv3_block5_concat[0][0] \n __________________________________________________________________________________________________\n conv3_block6_0_relu (Activation (None, None, None, 2 0 conv3_block6_0_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block6_1_conv (Conv2D) (None, None, None, 1 36864 conv3_block6_0_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block6_1_bn (BatchNormali (None, None, None, 1 512 conv3_block6_1_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block6_1_relu (Activation (None, None, None, 1 0 conv3_block6_1_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block6_2_conv (Conv2D) (None, None, None, 3 36864 conv3_block6_1_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block6_concat (Concatenat (None, None, None, 3 0 conv3_block5_concat[0][0] \n conv3_block6_2_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block7_0_bn (BatchNormali (None, None, None, 3 1280 conv3_block6_concat[0][0] \n __________________________________________________________________________________________________\n conv3_block7_0_relu (Activation (None, None, None, 3 0 conv3_block7_0_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block7_1_conv (Conv2D) (None, None, None, 1 40960 conv3_block7_0_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block7_1_bn (BatchNormali (None, None, None, 1 512 conv3_block7_1_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block7_1_relu (Activation (None, None, None, 1 0 conv3_block7_1_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block7_2_conv (Conv2D) (None, None, None, 3 36864 conv3_block7_1_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block7_concat (Concatenat (None, None, None, 3 0 conv3_block6_concat[0][0] \n conv3_block7_2_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block8_0_bn (BatchNormali (None, None, None, 3 1408 conv3_block7_concat[0][0] \n __________________________________________________________________________________________________\n conv3_block8_0_relu (Activation (None, None, None, 3 0 conv3_block8_0_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block8_1_conv (Conv2D) (None, None, None, 1 45056 conv3_block8_0_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block8_1_bn (BatchNormali (None, None, None, 1 512 conv3_block8_1_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block8_1_relu (Activation (None, None, None, 1 0 conv3_block8_1_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block8_2_conv (Conv2D) (None, None, None, 3 36864 conv3_block8_1_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block8_concat (Concatenat (None, None, None, 3 0 conv3_block7_concat[0][0] \n conv3_block8_2_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block9_0_bn (BatchNormali (None, None, None, 3 1536 conv3_block8_concat[0][0] \n __________________________________________________________________________________________________\n conv3_block9_0_relu (Activation (None, None, None, 3 0 conv3_block9_0_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block9_1_conv (Conv2D) (None, None, None, 1 49152 conv3_block9_0_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block9_1_bn (BatchNormali (None, None, None, 1 512 conv3_block9_1_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block9_1_relu (Activation (None, None, None, 1 0 conv3_block9_1_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block9_2_conv (Conv2D) (None, None, None, 3 36864 conv3_block9_1_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block9_concat (Concatenat (None, None, None, 4 0 conv3_block8_concat[0][0] \n conv3_block9_2_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block10_0_bn (BatchNormal (None, None, None, 4 1664 conv3_block9_concat[0][0] \n __________________________________________________________________________________________________\n conv3_block10_0_relu (Activatio (None, None, None, 4 0 conv3_block10_0_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block10_1_conv (Conv2D) (None, None, None, 1 53248 conv3_block10_0_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block10_1_bn (BatchNormal (None, None, None, 1 512 conv3_block10_1_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block10_1_relu (Activatio (None, None, None, 1 0 conv3_block10_1_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block10_2_conv (Conv2D) (None, None, None, 3 36864 conv3_block10_1_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block10_concat (Concatena (None, None, None, 4 0 conv3_block9_concat[0][0] \n conv3_block10_2_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block11_0_bn (BatchNormal (None, None, None, 4 1792 conv3_block10_concat[0][0] \n __________________________________________________________________________________________________\n conv3_block11_0_relu (Activatio (None, None, None, 4 0 conv3_block11_0_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block11_1_conv (Conv2D) (None, None, None, 1 57344 conv3_block11_0_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block11_1_bn (BatchNormal (None, None, None, 1 512 conv3_block11_1_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block11_1_relu (Activatio (None, None, None, 1 0 conv3_block11_1_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block11_2_conv (Conv2D) (None, None, None, 3 36864 conv3_block11_1_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block11_concat (Concatena (None, None, None, 4 0 conv3_block10_concat[0][0] \n conv3_block11_2_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block12_0_bn (BatchNormal (None, None, None, 4 1920 conv3_block11_concat[0][0] \n __________________________________________________________________________________________________\n conv3_block12_0_relu (Activatio (None, None, None, 4 0 conv3_block12_0_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block12_1_conv (Conv2D) (None, None, None, 1 61440 conv3_block12_0_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block12_1_bn (BatchNormal (None, None, None, 1 512 conv3_block12_1_conv[0][0] \n __________________________________________________________________________________________________\n conv3_block12_1_relu (Activatio (None, None, None, 1 0 conv3_block12_1_bn[0][0] \n __________________________________________________________________________________________________\n conv3_block12_2_conv (Conv2D) (None, None, None, 3 36864 conv3_block12_1_relu[0][0] \n __________________________________________________________________________________________________\n conv3_block12_concat (Concatena (None, None, None, 5 0 conv3_block11_concat[0][0] \n conv3_block12_2_conv[0][0] \n __________________________________________________________________________________________________\n pool3_bn (BatchNormalization) (None, None, None, 5 2048 conv3_block12_concat[0][0] \n __________________________________________________________________________________________________\n pool3_relu (Activation) (None, None, None, 5 0 pool3_bn[0][0] \n __________________________________________________________________________________________________\n pool3_conv (Conv2D) (None, None, None, 2 131072 pool3_relu[0][0] \n __________________________________________________________________________________________________\n pool3_pool (AveragePooling2D) (None, None, None, 2 0 pool3_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block1_0_bn (BatchNormali (None, None, None, 2 1024 pool3_pool[0][0] \n __________________________________________________________________________________________________\n conv4_block1_0_relu (Activation (None, None, None, 2 0 conv4_block1_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block1_1_conv (Conv2D) (None, None, None, 1 32768 conv4_block1_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block1_1_bn (BatchNormali (None, None, None, 1 512 conv4_block1_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block1_1_relu (Activation (None, None, None, 1 0 conv4_block1_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block1_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block1_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block1_concat (Concatenat (None, None, None, 2 0 pool3_pool[0][0] \n conv4_block1_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block2_0_bn (BatchNormali (None, None, None, 2 1152 conv4_block1_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block2_0_relu (Activation (None, None, None, 2 0 conv4_block2_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block2_1_conv (Conv2D) (None, None, None, 1 36864 conv4_block2_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block2_1_bn (BatchNormali (None, None, None, 1 512 conv4_block2_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block2_1_relu (Activation (None, None, None, 1 0 conv4_block2_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block2_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block2_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block2_concat (Concatenat (None, None, None, 3 0 conv4_block1_concat[0][0] \n conv4_block2_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block3_0_bn (BatchNormali (None, None, None, 3 1280 conv4_block2_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block3_0_relu (Activation (None, None, None, 3 0 conv4_block3_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block3_1_conv (Conv2D) (None, None, None, 1 40960 conv4_block3_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block3_1_bn (BatchNormali (None, None, None, 1 512 conv4_block3_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block3_1_relu (Activation (None, None, None, 1 0 conv4_block3_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block3_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block3_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block3_concat (Concatenat (None, None, None, 3 0 conv4_block2_concat[0][0] \n conv4_block3_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block4_0_bn (BatchNormali (None, None, None, 3 1408 conv4_block3_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block4_0_relu (Activation (None, None, None, 3 0 conv4_block4_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block4_1_conv (Conv2D) (None, None, None, 1 45056 conv4_block4_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block4_1_bn (BatchNormali (None, None, None, 1 512 conv4_block4_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block4_1_relu (Activation (None, None, None, 1 0 conv4_block4_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block4_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block4_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block4_concat (Concatenat (None, None, None, 3 0 conv4_block3_concat[0][0] \n conv4_block4_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block5_0_bn (BatchNormali (None, None, None, 3 1536 conv4_block4_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block5_0_relu (Activation (None, None, None, 3 0 conv4_block5_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block5_1_conv (Conv2D) (None, None, None, 1 49152 conv4_block5_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block5_1_bn (BatchNormali (None, None, None, 1 512 conv4_block5_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block5_1_relu (Activation (None, None, None, 1 0 conv4_block5_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block5_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block5_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block5_concat (Concatenat (None, None, None, 4 0 conv4_block4_concat[0][0] \n conv4_block5_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block6_0_bn (BatchNormali (None, None, None, 4 1664 conv4_block5_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block6_0_relu (Activation (None, None, None, 4 0 conv4_block6_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block6_1_conv (Conv2D) (None, None, None, 1 53248 conv4_block6_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block6_1_bn (BatchNormali (None, None, None, 1 512 conv4_block6_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block6_1_relu (Activation (None, None, None, 1 0 conv4_block6_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block6_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block6_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block6_concat (Concatenat (None, None, None, 4 0 conv4_block5_concat[0][0] \n conv4_block6_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block7_0_bn (BatchNormali (None, None, None, 4 1792 conv4_block6_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block7_0_relu (Activation (None, None, None, 4 0 conv4_block7_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block7_1_conv (Conv2D) (None, None, None, 1 57344 conv4_block7_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block7_1_bn (BatchNormali (None, None, None, 1 512 conv4_block7_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block7_1_relu (Activation (None, None, None, 1 0 conv4_block7_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block7_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block7_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block7_concat (Concatenat (None, None, None, 4 0 conv4_block6_concat[0][0] \n conv4_block7_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block8_0_bn (BatchNormali (None, None, None, 4 1920 conv4_block7_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block8_0_relu (Activation (None, None, None, 4 0 conv4_block8_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block8_1_conv (Conv2D) (None, None, None, 1 61440 conv4_block8_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block8_1_bn (BatchNormali (None, None, None, 1 512 conv4_block8_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block8_1_relu (Activation (None, None, None, 1 0 conv4_block8_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block8_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block8_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block8_concat (Concatenat (None, None, None, 5 0 conv4_block7_concat[0][0] \n conv4_block8_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block9_0_bn (BatchNormali (None, None, None, 5 2048 conv4_block8_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block9_0_relu (Activation (None, None, None, 5 0 conv4_block9_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block9_1_conv (Conv2D) (None, None, None, 1 65536 conv4_block9_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block9_1_bn (BatchNormali (None, None, None, 1 512 conv4_block9_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block9_1_relu (Activation (None, None, None, 1 0 conv4_block9_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block9_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block9_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block9_concat (Concatenat (None, None, None, 5 0 conv4_block8_concat[0][0] \n conv4_block9_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block10_0_bn (BatchNormal (None, None, None, 5 2176 conv4_block9_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block10_0_relu (Activatio (None, None, None, 5 0 conv4_block10_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block10_1_conv (Conv2D) (None, None, None, 1 69632 conv4_block10_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block10_1_bn (BatchNormal (None, None, None, 1 512 conv4_block10_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block10_1_relu (Activatio (None, None, None, 1 0 conv4_block10_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block10_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block10_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block10_concat (Concatena (None, None, None, 5 0 conv4_block9_concat[0][0] \n conv4_block10_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block11_0_bn (BatchNormal (None, None, None, 5 2304 conv4_block10_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block11_0_relu (Activatio (None, None, None, 5 0 conv4_block11_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block11_1_conv (Conv2D) (None, None, None, 1 73728 conv4_block11_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block11_1_bn (BatchNormal (None, None, None, 1 512 conv4_block11_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block11_1_relu (Activatio (None, None, None, 1 0 conv4_block11_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block11_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block11_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block11_concat (Concatena (None, None, None, 6 0 conv4_block10_concat[0][0] \n conv4_block11_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block12_0_bn (BatchNormal (None, None, None, 6 2432 conv4_block11_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block12_0_relu (Activatio (None, None, None, 6 0 conv4_block12_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block12_1_conv (Conv2D) (None, None, None, 1 77824 conv4_block12_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block12_1_bn (BatchNormal (None, None, None, 1 512 conv4_block12_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block12_1_relu (Activatio (None, None, None, 1 0 conv4_block12_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block12_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block12_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block12_concat (Concatena (None, None, None, 6 0 conv4_block11_concat[0][0] \n conv4_block12_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block13_0_bn (BatchNormal (None, None, None, 6 2560 conv4_block12_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block13_0_relu (Activatio (None, None, None, 6 0 conv4_block13_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block13_1_conv (Conv2D) (None, None, None, 1 81920 conv4_block13_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block13_1_bn (BatchNormal (None, None, None, 1 512 conv4_block13_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block13_1_relu (Activatio (None, None, None, 1 0 conv4_block13_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block13_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block13_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block13_concat (Concatena (None, None, None, 6 0 conv4_block12_concat[0][0] \n conv4_block13_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block14_0_bn (BatchNormal (None, None, None, 6 2688 conv4_block13_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block14_0_relu (Activatio (None, None, None, 6 0 conv4_block14_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block14_1_conv (Conv2D) (None, None, None, 1 86016 conv4_block14_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block14_1_bn (BatchNormal (None, None, None, 1 512 conv4_block14_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block14_1_relu (Activatio (None, None, None, 1 0 conv4_block14_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block14_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block14_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block14_concat (Concatena (None, None, None, 7 0 conv4_block13_concat[0][0] \n conv4_block14_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block15_0_bn (BatchNormal (None, None, None, 7 2816 conv4_block14_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block15_0_relu (Activatio (None, None, None, 7 0 conv4_block15_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block15_1_conv (Conv2D) (None, None, None, 1 90112 conv4_block15_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block15_1_bn (BatchNormal (None, None, None, 1 512 conv4_block15_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block15_1_relu (Activatio (None, None, None, 1 0 conv4_block15_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block15_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block15_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block15_concat (Concatena (None, None, None, 7 0 conv4_block14_concat[0][0] \n conv4_block15_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block16_0_bn (BatchNormal (None, None, None, 7 2944 conv4_block15_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block16_0_relu (Activatio (None, None, None, 7 0 conv4_block16_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block16_1_conv (Conv2D) (None, None, None, 1 94208 conv4_block16_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block16_1_bn (BatchNormal (None, None, None, 1 512 conv4_block16_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block16_1_relu (Activatio (None, None, None, 1 0 conv4_block16_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block16_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block16_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block16_concat (Concatena (None, None, None, 7 0 conv4_block15_concat[0][0] \n conv4_block16_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block17_0_bn (BatchNormal (None, None, None, 7 3072 conv4_block16_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block17_0_relu (Activatio (None, None, None, 7 0 conv4_block17_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block17_1_conv (Conv2D) (None, None, None, 1 98304 conv4_block17_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block17_1_bn (BatchNormal (None, None, None, 1 512 conv4_block17_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block17_1_relu (Activatio (None, None, None, 1 0 conv4_block17_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block17_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block17_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block17_concat (Concatena (None, None, None, 8 0 conv4_block16_concat[0][0] \n conv4_block17_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block18_0_bn (BatchNormal (None, None, None, 8 3200 conv4_block17_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block18_0_relu (Activatio (None, None, None, 8 0 conv4_block18_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block18_1_conv (Conv2D) (None, None, None, 1 102400 conv4_block18_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block18_1_bn (BatchNormal (None, None, None, 1 512 conv4_block18_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block18_1_relu (Activatio (None, None, None, 1 0 conv4_block18_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block18_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block18_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block18_concat (Concatena (None, None, None, 8 0 conv4_block17_concat[0][0] \n conv4_block18_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block19_0_bn (BatchNormal (None, None, None, 8 3328 conv4_block18_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block19_0_relu (Activatio (None, None, None, 8 0 conv4_block19_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block19_1_conv (Conv2D) (None, None, None, 1 106496 conv4_block19_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block19_1_bn (BatchNormal (None, None, None, 1 512 conv4_block19_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block19_1_relu (Activatio (None, None, None, 1 0 conv4_block19_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block19_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block19_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block19_concat (Concatena (None, None, None, 8 0 conv4_block18_concat[0][0] \n conv4_block19_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block20_0_bn (BatchNormal (None, None, None, 8 3456 conv4_block19_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block20_0_relu (Activatio (None, None, None, 8 0 conv4_block20_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block20_1_conv (Conv2D) (None, None, None, 1 110592 conv4_block20_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block20_1_bn (BatchNormal (None, None, None, 1 512 conv4_block20_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block20_1_relu (Activatio (None, None, None, 1 0 conv4_block20_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block20_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block20_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block20_concat (Concatena (None, None, None, 8 0 conv4_block19_concat[0][0] \n conv4_block20_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block21_0_bn (BatchNormal (None, None, None, 8 3584 conv4_block20_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block21_0_relu (Activatio (None, None, None, 8 0 conv4_block21_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block21_1_conv (Conv2D) (None, None, None, 1 114688 conv4_block21_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block21_1_bn (BatchNormal (None, None, None, 1 512 conv4_block21_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block21_1_relu (Activatio (None, None, None, 1 0 conv4_block21_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block21_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block21_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block21_concat (Concatena (None, None, None, 9 0 conv4_block20_concat[0][0] \n conv4_block21_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block22_0_bn (BatchNormal (None, None, None, 9 3712 conv4_block21_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block22_0_relu (Activatio (None, None, None, 9 0 conv4_block22_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block22_1_conv (Conv2D) (None, None, None, 1 118784 conv4_block22_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block22_1_bn (BatchNormal (None, None, None, 1 512 conv4_block22_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block22_1_relu (Activatio (None, None, None, 1 0 conv4_block22_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block22_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block22_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block22_concat (Concatena (None, None, None, 9 0 conv4_block21_concat[0][0] \n conv4_block22_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block23_0_bn (BatchNormal (None, None, None, 9 3840 conv4_block22_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block23_0_relu (Activatio (None, None, None, 9 0 conv4_block23_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block23_1_conv (Conv2D) (None, None, None, 1 122880 conv4_block23_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block23_1_bn (BatchNormal (None, None, None, 1 512 conv4_block23_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block23_1_relu (Activatio (None, None, None, 1 0 conv4_block23_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block23_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block23_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block23_concat (Concatena (None, None, None, 9 0 conv4_block22_concat[0][0] \n conv4_block23_2_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block24_0_bn (BatchNormal (None, None, None, 9 3968 conv4_block23_concat[0][0] \n __________________________________________________________________________________________________\n conv4_block24_0_relu (Activatio (None, None, None, 9 0 conv4_block24_0_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block24_1_conv (Conv2D) (None, None, None, 1 126976 conv4_block24_0_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block24_1_bn (BatchNormal (None, None, None, 1 512 conv4_block24_1_conv[0][0] \n __________________________________________________________________________________________________\n conv4_block24_1_relu (Activatio (None, None, None, 1 0 conv4_block24_1_bn[0][0] \n __________________________________________________________________________________________________\n conv4_block24_2_conv (Conv2D) (None, None, None, 3 36864 conv4_block24_1_relu[0][0] \n __________________________________________________________________________________________________\n conv4_block24_concat (Concatena (None, None, None, 1 0 conv4_block23_concat[0][0] \n conv4_block24_2_conv[0][0] \n __________________________________________________________________________________________________\n pool4_bn (BatchNormalization) (None, None, None, 1 4096 conv4_block24_concat[0][0] \n __________________________________________________________________________________________________\n pool4_relu (Activation) (None, None, None, 1 0 pool4_bn[0][0] \n __________________________________________________________________________________________________\n pool4_conv (Conv2D) (None, None, None, 5 524288 pool4_relu[0][0] \n __________________________________________________________________________________________________\n pool4_pool (AveragePooling2D) (None, None, None, 5 0 pool4_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block1_0_bn (BatchNormali (None, None, None, 5 2048 pool4_pool[0][0] \n __________________________________________________________________________________________________\n conv5_block1_0_relu (Activation (None, None, None, 5 0 conv5_block1_0_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block1_1_conv (Conv2D) (None, None, None, 1 65536 conv5_block1_0_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block1_1_bn (BatchNormali (None, None, None, 1 512 conv5_block1_1_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block1_1_relu (Activation (None, None, None, 1 0 conv5_block1_1_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block1_2_conv (Conv2D) (None, None, None, 3 36864 conv5_block1_1_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block1_concat (Concatenat (None, None, None, 5 0 pool4_pool[0][0] \n conv5_block1_2_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block2_0_bn (BatchNormali (None, None, None, 5 2176 conv5_block1_concat[0][0] \n __________________________________________________________________________________________________\n conv5_block2_0_relu (Activation (None, None, None, 5 0 conv5_block2_0_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block2_1_conv (Conv2D) (None, None, None, 1 69632 conv5_block2_0_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block2_1_bn (BatchNormali (None, None, None, 1 512 conv5_block2_1_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block2_1_relu (Activation (None, None, None, 1 0 conv5_block2_1_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block2_2_conv (Conv2D) (None, None, None, 3 36864 conv5_block2_1_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block2_concat (Concatenat (None, None, None, 5 0 conv5_block1_concat[0][0] \n conv5_block2_2_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block3_0_bn (BatchNormali (None, None, None, 5 2304 conv5_block2_concat[0][0] \n __________________________________________________________________________________________________\n conv5_block3_0_relu (Activation (None, None, None, 5 0 conv5_block3_0_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block3_1_conv (Conv2D) (None, None, None, 1 73728 conv5_block3_0_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block3_1_bn (BatchNormali (None, None, None, 1 512 conv5_block3_1_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block3_1_relu (Activation (None, None, None, 1 0 conv5_block3_1_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block3_2_conv (Conv2D) (None, None, None, 3 36864 conv5_block3_1_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block3_concat (Concatenat (None, None, None, 6 0 conv5_block2_concat[0][0] \n conv5_block3_2_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block4_0_bn (BatchNormali (None, None, None, 6 2432 conv5_block3_concat[0][0] \n __________________________________________________________________________________________________\n conv5_block4_0_relu (Activation (None, None, None, 6 0 conv5_block4_0_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block4_1_conv (Conv2D) (None, None, None, 1 77824 conv5_block4_0_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block4_1_bn (BatchNormali (None, None, None, 1 512 conv5_block4_1_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block4_1_relu (Activation (None, None, None, 1 0 conv5_block4_1_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block4_2_conv (Conv2D) (None, None, None, 3 36864 conv5_block4_1_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block4_concat (Concatenat (None, None, None, 6 0 conv5_block3_concat[0][0] \n conv5_block4_2_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block5_0_bn (BatchNormali (None, None, None, 6 2560 conv5_block4_concat[0][0] \n __________________________________________________________________________________________________\n conv5_block5_0_relu (Activation (None, None, None, 6 0 conv5_block5_0_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block5_1_conv (Conv2D) (None, None, None, 1 81920 conv5_block5_0_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block5_1_bn (BatchNormali (None, None, None, 1 512 conv5_block5_1_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block5_1_relu (Activation (None, None, None, 1 0 conv5_block5_1_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block5_2_conv (Conv2D) (None, None, None, 3 36864 conv5_block5_1_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block5_concat (Concatenat (None, None, None, 6 0 conv5_block4_concat[0][0] \n conv5_block5_2_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block6_0_bn (BatchNormali (None, None, None, 6 2688 conv5_block5_concat[0][0] \n __________________________________________________________________________________________________\n conv5_block6_0_relu (Activation (None, None, None, 6 0 conv5_block6_0_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block6_1_conv (Conv2D) (None, None, None, 1 86016 conv5_block6_0_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block6_1_bn (BatchNormali (None, None, None, 1 512 conv5_block6_1_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block6_1_relu (Activation (None, None, None, 1 0 conv5_block6_1_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block6_2_conv (Conv2D) (None, None, None, 3 36864 conv5_block6_1_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block6_concat (Concatenat (None, None, None, 7 0 conv5_block5_concat[0][0] \n conv5_block6_2_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block7_0_bn (BatchNormali (None, None, None, 7 2816 conv5_block6_concat[0][0] \n __________________________________________________________________________________________________\n conv5_block7_0_relu (Activation (None, None, None, 7 0 conv5_block7_0_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block7_1_conv (Conv2D) (None, None, None, 1 90112 conv5_block7_0_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block7_1_bn (BatchNormali (None, None, None, 1 512 conv5_block7_1_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block7_1_relu (Activation (None, None, None, 1 0 conv5_block7_1_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block7_2_conv (Conv2D) (None, None, None, 3 36864 conv5_block7_1_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block7_concat (Concatenat (None, None, None, 7 0 conv5_block6_concat[0][0] \n conv5_block7_2_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block8_0_bn (BatchNormali (None, None, None, 7 2944 conv5_block7_concat[0][0] \n __________________________________________________________________________________________________\n conv5_block8_0_relu (Activation (None, None, None, 7 0 conv5_block8_0_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block8_1_conv (Conv2D) (None, None, None, 1 94208 conv5_block8_0_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block8_1_bn (BatchNormali (None, None, None, 1 512 conv5_block8_1_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block8_1_relu (Activation (None, None, None, 1 0 conv5_block8_1_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block8_2_conv (Conv2D) (None, None, None, 3 36864 conv5_block8_1_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block8_concat (Concatenat (None, None, None, 7 0 conv5_block7_concat[0][0] \n conv5_block8_2_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block9_0_bn (BatchNormali (None, None, None, 7 3072 conv5_block8_concat[0][0] \n __________________________________________________________________________________________________\n conv5_block9_0_relu (Activation (None, None, None, 7 0 conv5_block9_0_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block9_1_conv (Conv2D) (None, None, None, 1 98304 conv5_block9_0_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block9_1_bn (BatchNormali (None, None, None, 1 512 conv5_block9_1_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block9_1_relu (Activation (None, None, None, 1 0 conv5_block9_1_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block9_2_conv (Conv2D) (None, None, None, 3 36864 conv5_block9_1_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block9_concat (Concatenat (None, None, None, 8 0 conv5_block8_concat[0][0] \n conv5_block9_2_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block10_0_bn (BatchNormal (None, None, None, 8 3200 conv5_block9_concat[0][0] \n __________________________________________________________________________________________________\n conv5_block10_0_relu (Activatio (None, None, None, 8 0 conv5_block10_0_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block10_1_conv (Conv2D) (None, None, None, 1 102400 conv5_block10_0_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block10_1_bn (BatchNormal (None, None, None, 1 512 conv5_block10_1_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block10_1_relu (Activatio (None, None, None, 1 0 conv5_block10_1_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block10_2_conv (Conv2D) (None, None, None, 3 36864 conv5_block10_1_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block10_concat (Concatena (None, None, None, 8 0 conv5_block9_concat[0][0] \n conv5_block10_2_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block11_0_bn (BatchNormal (None, None, None, 8 3328 conv5_block10_concat[0][0] \n __________________________________________________________________________________________________\n conv5_block11_0_relu (Activatio (None, None, None, 8 0 conv5_block11_0_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block11_1_conv (Conv2D) (None, None, None, 1 106496 conv5_block11_0_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block11_1_bn (BatchNormal (None, None, None, 1 512 conv5_block11_1_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block11_1_relu (Activatio (None, None, None, 1 0 conv5_block11_1_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block11_2_conv (Conv2D) (None, None, None, 3 36864 conv5_block11_1_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block11_concat (Concatena (None, None, None, 8 0 conv5_block10_concat[0][0] \n conv5_block11_2_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block12_0_bn (BatchNormal (None, None, None, 8 3456 conv5_block11_concat[0][0] \n __________________________________________________________________________________________________\n conv5_block12_0_relu (Activatio (None, None, None, 8 0 conv5_block12_0_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block12_1_conv (Conv2D) (None, None, None, 1 110592 conv5_block12_0_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block12_1_bn (BatchNormal (None, None, None, 1 512 conv5_block12_1_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block12_1_relu (Activatio (None, None, None, 1 0 conv5_block12_1_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block12_2_conv (Conv2D) (None, None, None, 3 36864 conv5_block12_1_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block12_concat (Concatena (None, None, None, 8 0 conv5_block11_concat[0][0] \n conv5_block12_2_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block13_0_bn (BatchNormal (None, None, None, 8 3584 conv5_block12_concat[0][0] \n __________________________________________________________________________________________________\n conv5_block13_0_relu (Activatio (None, None, None, 8 0 conv5_block13_0_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block13_1_conv (Conv2D) (None, None, None, 1 114688 conv5_block13_0_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block13_1_bn (BatchNormal (None, None, None, 1 512 conv5_block13_1_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block13_1_relu (Activatio (None, None, None, 1 0 conv5_block13_1_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block13_2_conv (Conv2D) (None, None, None, 3 36864 conv5_block13_1_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block13_concat (Concatena (None, None, None, 9 0 conv5_block12_concat[0][0] \n conv5_block13_2_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block14_0_bn (BatchNormal (None, None, None, 9 3712 conv5_block13_concat[0][0] \n __________________________________________________________________________________________________\n conv5_block14_0_relu (Activatio (None, None, None, 9 0 conv5_block14_0_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block14_1_conv (Conv2D) (None, None, None, 1 118784 conv5_block14_0_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block14_1_bn (BatchNormal (None, None, None, 1 512 conv5_block14_1_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block14_1_relu (Activatio (None, None, None, 1 0 conv5_block14_1_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block14_2_conv (Conv2D) (None, None, None, 3 36864 conv5_block14_1_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block14_concat (Concatena (None, None, None, 9 0 conv5_block13_concat[0][0] \n conv5_block14_2_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block15_0_bn (BatchNormal (None, None, None, 9 3840 conv5_block14_concat[0][0] \n __________________________________________________________________________________________________\n conv5_block15_0_relu (Activatio (None, None, None, 9 0 conv5_block15_0_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block15_1_conv (Conv2D) (None, None, None, 1 122880 conv5_block15_0_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block15_1_bn (BatchNormal (None, None, None, 1 512 conv5_block15_1_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block15_1_relu (Activatio (None, None, None, 1 0 conv5_block15_1_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block15_2_conv (Conv2D) (None, None, None, 3 36864 conv5_block15_1_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block15_concat (Concatena (None, None, None, 9 0 conv5_block14_concat[0][0] \n conv5_block15_2_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block16_0_bn (BatchNormal (None, None, None, 9 3968 conv5_block15_concat[0][0] \n __________________________________________________________________________________________________\n conv5_block16_0_relu (Activatio (None, None, None, 9 0 conv5_block16_0_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block16_1_conv (Conv2D) (None, None, None, 1 126976 conv5_block16_0_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block16_1_bn (BatchNormal (None, None, None, 1 512 conv5_block16_1_conv[0][0] \n __________________________________________________________________________________________________\n conv5_block16_1_relu (Activatio (None, None, None, 1 0 conv5_block16_1_bn[0][0] \n __________________________________________________________________________________________________\n conv5_block16_2_conv (Conv2D) (None, None, None, 3 36864 conv5_block16_1_relu[0][0] \n __________________________________________________________________________________________________\n conv5_block16_concat (Concatena (None, None, None, 1 0 conv5_block15_concat[0][0] \n conv5_block16_2_conv[0][0] \n __________________________________________________________________________________________________\n bn (BatchNormalization) (None, None, None, 1 4096 conv5_block16_concat[0][0] \n __________________________________________________________________________________________________\n global_average_pooling2d_1 (Glo (None, 1024) 0 bn[0][0] \n __________________________________________________________________________________________________\n dense_1 (Dense) (None, 14) 14350 global_average_pooling2d_1[0][0] \n ==================================================================================================\n Total params: 7,051,854\n Trainable params: 6,968,206\n Non-trainable params: 83,648\n __________________________________________________________________________________________________\n\n\nThere are a lot of layers, but typically we'll only be extracting one of the last few. Remember that the last few layers usually have more abstract information. To access a layer, we can use `model.get_layer(layer).output`, which takes in the name of the layer in question. Let's try getting the `conv5_block16_concat` layer, the raw output of the last convolutional layer.\n\n\n```python\nspatial_maps = model.get_layer('conv5_block16_concat').output\nprint(spatial_maps)\n```\n\n Tensor(\"conv5_block16_concat/concat:0\", shape=(?, ?, ?, 1024), dtype=float32)\n\n\nNow, this tensor is just a placeholder, it doesn't contain the actual activations for a particular image. To get this we will use [Keras.backend.function](https://www.tensorflow.org/api_docs/python/tf/keras/backend/function) to return intermediate computations while the model is processing a particular input. This method takes in an input and output placeholders and returns a function. This function will compute the intermediate output (until it reaches the given placeholder) evaluated given the input. For example, if you want the layer that you just retrieved (conv5_block16_concat), you could write the following:\n\n\n```python\nget_spatial_maps = K.function([model.input], [spatial_maps])\nprint(get_spatial_maps)\n```\n\n \n\n\nWe see that we now have a `Function` object. Now, to get the actual intermediate output evaluated with a particular input, we just plug in an image to this function:\n\n\n```python\n# get an image\nx = load_image_normalize(im_path, mean, std)\nprint(f\"x is of type {type(x)}\")\nprint(f\"x is of shape {x.shape}\")\n```\n\n x is of type \n x is of shape (1, 320, 320, 3)\n\n\n\n```python\n# get the spatial maps layer activations (a list of numpy arrays)\nspatial_maps_x_l = get_spatial_maps([x])\n\nprint(f\"spatial_maps_x_l is of type {type(spatial_maps_x_l)}\")\nprint(f\"spatial_maps_x_l is has length {len(spatial_maps_x_l)}\")\n```\n\n spatial_maps_x_l is of type \n spatial_maps_x_l is has length 1\n\n\n\n```python\n# get the 0th item in the list\nspatial_maps_x = spatial_maps_x_l[0]\nprint(f\"spatial_maps_x is of type {type(spatial_maps_x)}\")\nprint(f\"spatial_maps_x is of shape {spatial_maps_x.shape}\")\n```\n\n spatial_maps_x is of type \n spatial_maps_x is of shape (1, 10, 10, 1024)\n\n\nNotice that the shape is (1, 10, 10, 1024). The 0th dimension of size 1 is the batch dimension. Remove the batch dimension for later calculations by taking the 0th index of spatial_maps_x.\n\n\n```python\n# Get rid of the batch dimension\nspatial_maps_x = spatial_maps_x[0] # equivalent to spatial_maps_x[0,:]\nprint(f\"spatial_maps_x without the batch dimension has shape {spatial_maps_x.shape}\")\nprint(\"Output some of the content:\")\nprint(spatial_maps_x[0])\n```\n\n spatial_maps_x without the batch dimension has shape (10, 10, 1024)\n Output some of the content:\n [[-0.64217424 -0.22323257 0.05218337 ... 0.10186759 -0.04801503\n 0.11582824]\n [-0.5476699 -0.16134709 -0.55162615 ... 0.16851762 -0.07845385\n 0.20111586]\n [-0.8194008 -0.23154774 -0.43870574 ... 0.18792555 -0.09083493\n 0.26589885]\n ...\n [-0.4833299 0.3569351 0.17129314 ... 0.05083632 -0.03575559\n 0.02226869]\n [-0.8109025 -0.52058625 -0.0482365 ... 0.15647241 -0.06991401\n 0.00422742]\n [-1.6978302 -1.2835201 0.60596764 ... 0.10288006 0.01210845\n -0.00818749]]\n\n\nWe now have the activations for that particular image, and we can use it for interpretation. The function that is returned by calling `K.function([model.input], [spatial_maps])` (saved here in the variable `get_spatial_maps`) is sometimes referred to as a \"hook\", letting you peek into the intermediate computations in the model. \n\n\n#### 1.1.2 Getting Gradients\n\nThe other major step in computing GradCAMs is getting gradients with respect to the output for a particular class. Luckily, Keras makes getting gradients simple. We can use the [Keras.backend.gradients](https://www.tensorflow.org/api_docs/python/tf/keras/backend/gradients) function. The first parameter is the value you are taking the gradient of, and the second is the parameter you are taking that gradient with respect to. We illustrate below: \n\n\n```python\n# get the output of the model\noutput_with_batch_dim = model.output\nprint(f\"Model output includes batch dimension, has shape {output_with_batch_dim.shape}\")\n```\n\n Model output includes batch dimension, has shape (?, 14)\n\n\nTo get the output without the batch dimension, you can take the 0th index of the tensor. Note that because the batch dimension is 'None', you could actually enter any integer index, but let's just use 0.\n\n\n```python\n# Get the output without the batch dimension\noutput_all_categories = output_with_batch_dim[0]\nprint(f\"The output for all 14 categories of disease has shape {output_all_categories.shape}\")\n```\n\n The output for all 14 categories of disease has shape (14,)\n\n\nThe output has 14 categories, one for each disease category, indexed from 0 to 13. Cardiomegaly is the disease category at index 0.\n\n\n```python\n# Get the first category's output (Cardiomegaly) at index 0\ny_category_0 = output_all_categories[0]\nprint(f\"The Cardiomegaly output is at index 0, and has shape {y_category_0.shape}\")\n```\n\n The Cardiomegaly output is at index 0, and has shape ()\n\n\n\n```python\n# Get gradient of y_category_0 with respect to spatial_maps\n\ngradient_l = K.gradients(y_category_0, spatial_maps)\nprint(f\"gradient_l is of type {type(gradient_l)} and has length {len(gradient_l)}\")\n\n# gradient_l is a list of size 1. Get the gradient at index 0\ngradient = gradient_l[0]\nprint(gradient)\n```\n\n gradient_l is of type and has length 1\n Tensor(\"gradients/AddN:0\", shape=(?, ?, ?, 1024), dtype=float32)\n\n\nAgain, this is just a placeholder. Just like for intermediate layers, we can use `K.function` to compute the value of the gradient for a particular input. \n\nThe K.function() takes in\n- a list of inputs: in this case, one input, 'model.input'\n- a list of tensors: in this case, one output tensor 'gradient'\n\nIt returns a function that calculates the activations of the list of tensors.\n- This returned function returns a list of the activations, one for each tensor that was passed into K.function().\n\n\n```python\n# Create the function that gets the gradient\nget_gradient = K.function([model.input], [gradient])\ntype(get_gradient)\n```\n\n\n\n\n keras.backend.tensorflow_backend.Function\n\n\n\n\n```python\n# get an input x-ray image\nx = load_image_normalize(im_path, mean, std)\nprint(f\"X-ray image has shape {x.shape}\")\n```\n\n X-ray image has shape (1, 320, 320, 3)\n\n\nThe `get_gradient` function takes in a list of inputs, and returns a list of the gradients, one for each image.\n\n\n```python\n# use the get_gradient function to get the gradient (pass in the input image inside a list)\ngrad_x_l = get_gradient([x])\nprint(f\"grad_x_l is of type {type(grad_x_l)} and length {len(grad_x_l)}\")\n\n# get the gradient at index 0 of the list.\ngrad_x_with_batch_dim = grad_x_l[0]\nprint(f\"grad_x_with_batch_dim is type {type(grad_x_with_batch_dim)} and shape {grad_x_with_batch_dim.shape}\")\n\n# To remove the batch dimension, take the value at index 0 of the batch dimension\ngrad_x = grad_x_with_batch_dim[0]\nprint(f\"grad_x is type {type(grad_x)} and shape {grad_x.shape}\")\n\nprint(\"Gradient grad_x (show some of its content:\")\nprint(grad_x[0])\n```\n\n grad_x_l is of type and length 1\n grad_x_with_batch_dim is type and shape (1, 10, 10, 1024)\n grad_x is type and shape (10, 10, 1024)\n Gradient grad_x (show some of its content:\n [[-1.4299641e-10 2.8810271e-10 3.3761882e-08 ... 9.4272409e-06\n -6.3098059e-06 6.5744803e-06]\n [-1.4299641e-10 2.8810271e-10 3.3761882e-08 ... 9.4272409e-06\n -6.3098059e-06 6.5744803e-06]\n [-1.4299641e-10 2.8810271e-10 3.3761882e-08 ... 9.4272409e-06\n -6.3098059e-06 6.5744803e-06]\n ...\n [-1.4299641e-10 2.8810271e-10 3.3761882e-08 ... 9.4272409e-06\n -6.3098059e-06 6.5744803e-06]\n [-1.4299641e-10 2.8810271e-10 3.3761882e-08 ... 9.4272409e-06\n -6.3098059e-06 6.5744803e-06]\n [-1.4299641e-10 2.8810271e-10 3.3761882e-08 ... 9.4272409e-06\n -6.3098059e-06 6.5744803e-06]]\n\n\nJust like we had a hook into the penultimate layer, we now have a hook into the gradient! This allows us to easily compute pretty much anything relevant to our model output. \n\nWe can also combine the two to have one function call which gives us both the gradient and the last layer (this might come in handy when implementing GradCAM in the next section).\n\n\n```python\n# Use K.function to generate a single function\n# Notice that a list of two tensors, is passed in as the second argument of K.function()\nget_spatial_maps_and_gradient = K.function([model.input], [spatial_maps, gradient])\nprint(type(get_spatial_maps_and_gradient))\n```\n\n \n\n\n\n```python\n# The returned function returns a list of the evaluated tensors\ntensor_eval_l = get_spatial_maps_and_gradient([x])\nprint(f\"tensor_eval_l is type {type(tensor_eval_l)} and length {len(tensor_eval_l)}\")\n```\n\n tensor_eval_l is type and length 2\n\n\n\n```python\n# store the two numpy arrays from index 0 and 1 into their own variables\nspatial_maps_x_with_batch_dim, grad_x_with_batch_dim = tensor_eval_l\nprint(f\"spatial_maps_x_with_batch_dim has shape {spatial_maps_x_with_batch_dim.shape}\")\nprint(f\"grad_x_with_batch_dim has shape {grad_x_with_batch_dim.shape}\")\n```\n\n spatial_maps_x_with_batch_dim has shape (1, 10, 10, 1024)\n grad_x_with_batch_dim has shape (1, 10, 10, 1024)\n\n\n\n```python\n# Note: you could also do this directly from the function call:\nspatial_maps_x_with_batch_dim, grad_x_with_batch_dim = get_spatial_maps_and_gradient([x])\nprint(f\"spatial_maps_x_with_batch_dim has shape {spatial_maps_x_with_batch_dim.shape}\")\nprint(f\"grad_x_with_batch_dim has shape {grad_x_with_batch_dim.shape}\")\n```\n\n spatial_maps_x_with_batch_dim has shape (1, 10, 10, 1024)\n grad_x_with_batch_dim has shape (1, 10, 10, 1024)\n\n\n\n```python\n# Remove the batch dimension by taking the 0th index at the batch dimension\nspatial_maps_x = spatial_maps_x_with_batch_dim[0]\ngrad_x = grad_x_with_batch_dim[0]\nprint(f\"spatial_maps_x shape {spatial_maps_x.shape}\")\nprint(f\"grad_x shape {grad_x.shape}\")\n\nprint(\"\\nSpatial maps (print some content):\")\nprint(spatial_maps_x[0])\nprint(\"\\nGradient (print some content:\")\nprint(grad_x[0])\n```\n\n spatial_maps_x shape (10, 10, 1024)\n grad_x shape (10, 10, 1024)\n \n Spatial maps (print some content):\n [[-0.64217424 -0.22323257 0.05218337 ... 0.10186759 -0.04801503\n 0.11582824]\n [-0.5476699 -0.16134709 -0.55162615 ... 0.16851762 -0.07845385\n 0.20111586]\n [-0.8194008 -0.23154774 -0.43870574 ... 0.18792555 -0.09083493\n 0.26589885]\n ...\n [-0.4833299 0.3569351 0.17129314 ... 0.05083632 -0.03575559\n 0.02226869]\n [-0.8109025 -0.52058625 -0.0482365 ... 0.15647241 -0.06991401\n 0.00422742]\n [-1.6978302 -1.2835201 0.60596764 ... 0.10288006 0.01210845\n -0.00818749]]\n \n Gradient (print some content:\n [[-1.4299641e-10 2.8810271e-10 3.3761882e-08 ... 9.4272409e-06\n -6.3098059e-06 6.5744803e-06]\n [-1.4299641e-10 2.8810271e-10 3.3761882e-08 ... 9.4272409e-06\n -6.3098059e-06 6.5744803e-06]\n [-1.4299641e-10 2.8810271e-10 3.3761882e-08 ... 9.4272409e-06\n -6.3098059e-06 6.5744803e-06]\n ...\n [-1.4299641e-10 2.8810271e-10 3.3761882e-08 ... 9.4272409e-06\n -6.3098059e-06 6.5744803e-06]\n [-1.4299641e-10 2.8810271e-10 3.3761882e-08 ... 9.4272409e-06\n -6.3098059e-06 6.5744803e-06]\n [-1.4299641e-10 2.8810271e-10 3.3761882e-08 ... 9.4272409e-06\n -6.3098059e-06 6.5744803e-06]]\n\n\n\n#### 1.1.3 Implementing GradCAM\n\n\n### Exercise 1\n\nIn the next cell, fill in the `grad_cam` method to produce GradCAM visualizations for an input model and image. This is fairly complicated, so it might help to break it down into these steps:\n\n1. Hook into model output and last layer activations.\n2. Get gradients of last layer activations with respect to output.\n3. Compute value of last layer and gradients for input image.\n4. Compute weights from gradients by global average pooling.\n5. Compute the dot product between the last layer and weights to get the score for each pixel.\n6. Resize, take ReLU, and return cam. \n\n
                                        \n \n Hints\n\n \n The following hints follow the order of the sections described above.\n 1. Remember that the output shape of our model will be [1, class_amount]. \n 1. The input in this case will always have batch_size = 1\n 2. See [K.gradients](https://www.tensorflow.org/api_docs/python/tf/keras/backend/gradients)\n 3. Follow the procedure we used in the previous two sections.\n 4. Check the axis; make sure weights have shape (C)!\n 5. See [np.dot](https://docs.scipy.org/doc/numpy/reference/generated/numpy.dot.html)\n
                                        \n\nTo test, you will compare your output on an image to the output from a correct implementation of GradCAM. You will receive full credit if the pixel-wise mean squared error is less than 0.05.\n\n\n```python\n# UNQ_C1 (UNIQUE CELL IDENTIFIER, DO NOT EDIT)\ndef grad_cam(input_model, image, category_index, layer_name):\n \"\"\"\n GradCAM method for visualizing input saliency.\n \n Args:\n input_model (Keras.model): model to compute cam for\n image (tensor): input to model, shape (1, H, W, 3)\n cls (int): class to compute cam with respect to\n layer_name (str): relevant layer in model\n H (int): input height\n W (int): input width\n Return:\n cam ()\n \"\"\"\n cam = None\n \n ### START CODE HERE (REPLACE INSTANCES OF 'None' with your code) ###\n\n # 1. Get placeholders for class output and last layer\n # Get the model's output\n output_with_batch_dim = input_model.output\n \n # Remove the batch dimension\n output_all_categories = output_with_batch_dim[0]\n \n # Retrieve only the disease category at the given category index\n y_c = output_all_categories[category_index]\n \n # Get the input model's layer specified by layer_name, and retrive the layer's output tensor\n spatial_map_layer = input_model.get_layer(layer_name).output\n \n # 2. Get gradients of last layer with respect to output\n\n # get the gradients of y_c with respect to the spatial map layer (it's a list of length 1)\n grads_l = K.gradients(y_c,spatial_map_layer)\n \n # Get the gradient at index 0 of the list\n grads = grads_l[0]\n \n # 3. Get hook for the selected layer and its gradient, based on given model's input\n # Hint: Use the variables produced by the previous two lines of code\n spatial_map_and_gradient_function = K.function([input_model.input],[spatial_map_layer, grads])\n \n # Put in the image to calculate the values of the spatial_maps (selected layer) and values of the gradients\n spatial_map_all_dims, grads_val_all_dims = spatial_map_and_gradient_function([image])\n \n # Reshape activations and gradient to remove the batch dimension\n # Shape goes from (B, H, W, C) to (H, W, C)\n # B: Batch. H: Height. W: Width. C: Channel \n # Reshape spatial map output to remove the batch dimension\n spatial_map_val = spatial_map_all_dims[0]\n \n # Reshape gradients to remove the batch dimension\n grads_val = grads_val_all_dims[0]\n # print(spatial_map_val.shape,type(spatial_map_val),\"---\",grads_val.shape,type(grads_val))\n # 4. Compute weights using global average pooling on gradient \n # grads_val has shape (Height, Width, Channels) (H,W,C)\n # Take the mean across the height and also width, for each channel\n # Make sure weights have shape (C)\n weights = np.mean(grads_val,axis=(0,1))\n \n # 5. Compute dot product of spatial map values with the weights\n cam = np.dot(spatial_map_val, weights)\n\n ### END CODE HERE ###\n \n # We'll take care of the postprocessing.\n H, W = image.shape[1], image.shape[2]\n cam = np.maximum(cam, 0) # ReLU so we only get positive importance\n cam = cv2.resize(cam, (W, H), cv2.INTER_NEAREST)\n cam = cam / cam.max()\n\n return cam\n```\n\nBelow we generate the CAM for the image and compute the error (pixel-wise mean squared difference) from the expected values according to our reference. \n\n\n```python\nim = load_image_normalize(im_path, mean, std)\ncam = grad_cam(model, im, 5, 'conv5_block16_concat') # Mass is class 5\n\n# Loads reference CAM to compare our implementation with.\nreference = np.load(\"reference_cam.npy\")\nerror = np.mean((cam-reference)**2)\n\nprint(f\"Error from reference: {error:.4f}, should be less than 0.05\")\n```\n\n Error from reference: 0.0014, should be less than 0.05\n\n\nRun the next cell to visualize the CAM and the original image. \n\n\n```python\nplt.imshow(load_image(im_path, df, preprocess=False), cmap='gray')\nplt.title(\"Original\")\nplt.axis('off')\n\nplt.show()\n\nplt.imshow(load_image(im_path, df, preprocess=False), cmap='gray')\nplt.imshow(cam, cmap='magma', alpha=0.5)\nplt.title(\"GradCAM\")\nplt.axis('off')\nplt.show()\n```\n\nWe can see that it focuses on the large white area in the middle of the chest cavity. Indeed this is a clear case of cardiomegaly, that is, an enlarged heart. \n\n\n#### 1.1.4 Using GradCAM to Visualize Multiple Labels\n\n\n### Exercise 2\n\nWe can use GradCAMs for multiple labels on the same image. Let's do it for the labels with best AUC for our model, Cardiomegaly, Mass, and Edema. \n\n\n```python\n# UNQ_C2 (UNIQUE CELL IDENTIFIER, DO NOT EDIT)\ndef compute_gradcam(model, img, mean, std, data_dir, df, \n labels, selected_labels, layer_name='conv5_block16_concat'):\n \"\"\"\n Compute GradCAM for many specified labels for an image. \n This method will use the `grad_cam` function.\n \n Args:\n model (Keras.model): Model to compute GradCAM for\n img (string): Image name we want to compute GradCAM for.\n mean (float): Mean to normalize to image.\n std (float): Standard deviation to normalize the image.\n data_dir (str): Path of the directory to load the images from.\n df(pd.Dataframe): Dataframe with the image features.\n labels ([str]): All output labels for the model.\n selected_labels ([str]): All output labels we want to compute the GradCAM for.\n layer_name: Intermediate layer from the model we want to compute the GradCAM for.\n \"\"\"\n img_path = data_dir + img\n preprocessed_input = load_image_normalize(img_path, mean, std)\n predictions = model.predict(preprocessed_input)\n print(\"Ground Truth: \", \", \".join(np.take(labels, np.nonzero(df[df[\"Image\"] == img][labels].values[0]))[0]))\n\n plt.figure(figsize=(15, 10))\n plt.subplot(151)\n plt.title(\"Original\")\n plt.axis('off')\n plt.imshow(load_image(img_path, df, preprocess=False), cmap='gray')\n \n j = 1\n \n ### START CODE HERE (REPLACE INSTANCES OF 'None' with your code) ### \n # Loop through all labels\n for i in range(len(labels)): # complete this line\n # Compute CAM and show plots for each selected label.\n \n # Check if the label is one of the selected labels\n if labels[i] in selected_labels: # complete this line\n \n # Use the grad_cam function to calculate gradcam\n gradcam = grad_cam(model,preprocessed_input,i,layer_name)\n \n ### END CODE HERE ###\n \n print(\"Generating gradcam for class %s (p=%2.2f)\" % (labels[i], round(predictions[0][i], 3)))\n plt.subplot(151 + j)\n plt.title(labels[i] + \": \" + str(round(predictions[0][i], 3)))\n plt.axis('off')\n plt.imshow(load_image(img_path, df, preprocess=False), cmap='gray')\n plt.imshow(gradcam, cmap='magma', alpha=min(0.5, predictions[0][i]))\n j +=1\n```\n\nRun the following cells to print the ground truth diagnosis for a given case and show the original x-ray as well as GradCAMs for Cardiomegaly, Mass, and Edema.\n\n\n```python\ndf = pd.read_csv(\"nih_new/train-small.csv\")\n\nimage_filename = '00016650_000.png'\nlabels_to_show = ['Cardiomegaly', 'Mass', 'Edema']\ncompute_gradcam(model, image_filename, mean, std, IMAGE_DIR, df, labels, labels_to_show)\n```\n\nThe model correctly predicts absence of mass or edema. The probability for mass is higher, and we can see that it may be influenced by the shapes in the middle of the chest cavity, as well as around the shoulder. We'll run it for two more images. \n\n\n```python\nimage_filename = '00005410_000.png'\ncompute_gradcam(model, image_filename, mean, std, IMAGE_DIR, df, labels, labels_to_show)\n```\n\nIn the example above, the model correctly focuses on the mass near the center of the chest cavity. \n\n\n```python\nimage_name = '00004090_002.png'\ncompute_gradcam(model, image_name, mean, std, IMAGE_DIR, df, labels, labels_to_show)\n```\n\nHere the model correctly picks up the signs of edema near the bottom of the chest cavity. We can also notice that Cardiomegaly has a high score for this image, though the ground truth doesn't include it. This visualization might be helpful for error analysis; for example, we can notice that the model is indeed looking at the expected area to make the prediction.\n\nThis concludes the section on GradCAMs. We hope you've gained an appreciation for the importance of interpretation when it comes to deep learning models in medicine. Interpretation tools like this one can be helpful for discovery of markers, error analysis, and even in deployment. \n\n\n## 2 Feature Importance in Machine Learning\n\nWhen developing predictive models and risk measures, it's often helpful to know which features are making the most difference. This is easy to determine in simpler models such as linear models and decision trees. However as we move to more complex models to achieve high performance, we usually sacrifice some interpretability. In this assignment we'll try to regain some of that interpretability using Shapley values, a technique which has gained popularity in recent years, but which is based on classic results in cooperative game theory. \n\nWe'll revisit our random forest model from course 2 module 2 and try to analyze it more closely using Shapley values. Run the next cell to load in the data and model from that assignment and recalculate the test set c-index.\n\n\n```python\nrf = pickle.load(open('nhanes_rf.sav', 'rb')) # Loading the model\ntest_df = pd.read_csv('nhanest_test.csv')\ntest_df = test_df.drop(test_df.columns[0], axis=1)\nX_test = test_df.drop('y', axis=1)\ny_test = test_df.loc[:, 'y']\ncindex_test = cindex(y_test, rf.predict_proba(X_test)[:, 1])\n\nprint(\"Model C-index on test: {}\".format(cindex_test))\n```\n\n Model C-index on test: 0.7776169781865744\n\n\nRun the next cell to print out the riskiest individuals according to our model. \n\n\n```python\nX_test_risky = X_test.copy(deep=True)\nX_test_risky.loc[:, 'risk'] = rf.predict_proba(X_test)[:, 1] # Predicting our risk.\nX_test_risky = X_test_risky.sort_values(by='risk', ascending=False) # Sorting by risk value.\nX_test_risky.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        AgeDiastolic BPPoverty indexRaceRed blood cellsSedimentation rateSerum AlbuminSerum CholesterolSerum IronSerum MagnesiumSerum ProteinSexSystolic BPTIBCTSWhite blood cellsBMIPulse pressurerisk
                                        57270.080.0312.01.054.87.04.4222.052.01.577.21.0180.0417.012.57.545.770473100.00.77
                                        19069.0100.0316.01.077.726.04.2197.065.01.497.51.0165.0298.021.88.822.12901865.00.69
                                        130073.080.0999.01.052.635.03.9258.061.01.666.81.0150.0314.019.49.426.46685070.00.69
                                        63466.0100.069.02.042.947.03.8233.0170.01.428.61.0180.0411.041.47.222.12949880.00.68
                                        122174.080.067.01.040.324.03.7139.028.01.916.42.0140.0495.05.74.122.06638960.00.68
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                                        \n\n\n\n\n### 2.1 Permuation Method for Feature Importance\n\nFirst we'll try to determine feature importance using the permutation method. In the permutation method, the importance of feature $i$ would be the regular performance of the model minus the performance with the values for feature $i$ permuted in the dataset. This way we can assess how well a model without that feature would do without having to train a new model for each feature. \n\n\n#### 2.1.1 Implementing Permutation\n\n\n### Exercise 3\n\nComplete the implementation of the function below, which given a feature name returns a dataset with those feature values randomly permuted. \n\n
                                        \n \n Hints\n\n \n
                                        \n\n\n```python\n# UNQ_C3 (UNIQUE CELL IDENTIFIER, DO NOT EDIT)\ndef permute_feature(df, feature):\n \"\"\"\n Given dataset, returns version with the values of\n the given feature randomly permuted. \n\n Args:\n df (dataframe): The dataset, shape (num subjects, num features)\n feature (string): Name of feature to permute\n Returns:\n permuted_df (dataframe): Exactly the same as df except the values\n of the given feature are randomly permuted.\n \"\"\"\n permuted_df = df.copy(deep=True) # Make copy so we don't change original df\n\n ### START CODE HERE (REPLACE INSTANCES OF 'None' with your code) ###\n\n # Permute the values of the column 'feature'\n permuted_features = np.random.permutation(permuted_df[feature])\n \n # Set the column 'feature' to its permuted values.\n permuted_df[feature] = permuted_features\n \n ### END CODE HERE ###\n\n return permuted_df\n```\n\n\n```python\nprint(\"Test Case\")\n\nexample_df = pd.DataFrame({'col1': [0, 1, 2], 'col2':['A', 'B', 'C']})\nprint(\"Original dataframe:\")\nprint(example_df)\nprint(\"\\n\")\n\nprint(\"col1 permuted:\")\nprint(permute_feature(example_df, 'col1'))\n\nprint(\"\\n\")\nprint(\"Compute average values over 1000 runs to get expected values:\")\ncol1_values = np.zeros((3, 1000))\nnp.random.seed(0) # Adding a constant seed so we can always expect the same values and evaluate correctly. \nfor i in range(1000):\n col1_values[:, i] = permute_feature(example_df, 'col1')['col1'].values\n\nprint(\"Average of col1: {}, expected value: [0.976, 1.03, 0.994]\".format(np.mean(col1_values, axis=1)))\n```\n\n Test Case\n Original dataframe:\n col1 col2\n 0 0 A\n 1 1 B\n 2 2 C\n \n \n col1 permuted:\n col1 col2\n 0 2 A\n 1 0 B\n 2 1 C\n \n \n Compute average values over 1000 runs to get expected values:\n Average of col1: [0.976 1.03 0.994], expected value: [0.976, 1.03, 0.994]\n\n\n\n#### 2.1.2 Implementing Importance\n\n\n### Exercise 4\n\nNow we will use the function we just created to compute feature importances (according to the permutation method) in the function below.\n\n
                                        \n \n Hints\n\n\\begin{align}\nI_x = \\left\\lvert perf - perf_x \\right\\rvert\n\\end{align}\n\nwhere $I_x$ is the importance of feature $x$ and\n\\begin{align}\nperf_x = \\frac{1}{n}\\cdot \\sum_{i=1}^{n} perf_i^{sx}\n\\end{align}\n\nwhere $perf_i^{sx}$ is the performance with the feature $x$ shuffled in the $i$th permutation.\n
                                        \n\n\n```python\n# UNQ_C4 (UNIQUE CELL IDENTIFIER, DO NOT EDIT)\ndef permutation_importance(X, y, model, metric, num_samples = 100):\n \"\"\"\n Compute permutation importance for each feature.\n\n Args:\n X (dataframe): Dataframe for test data, shape (num subject, num features)\n y (np.array): Labels for each row of X, shape (num subjects,)\n model (object): Model to compute importances for, guaranteed to have\n a 'predict_proba' method to compute probabilistic \n predictions given input\n metric (function): Metric to be used for feature importance. Takes in ground\n truth and predictions as the only two arguments\n num_samples (int): Number of samples to average over when computing change in\n performance for each feature\n Returns:\n importances (dataframe): Dataframe containing feature importance for each\n column of df with shape (1, num_features)\n \"\"\"\n\n importances = pd.DataFrame(index = ['importance'], columns = X.columns)\n \n # Get baseline performance (note, you'll use this metric function again later)\n baseline_performance = metric(y, model.predict_proba(X)[:, 1])\n\n ### START CODE HERE (REPLACE INSTANCES OF 'None' with your code) ###\n\n # Iterate over features (the columns in the importances dataframe)\n for feature in importances.columns: # complete this line\n \n # Compute 'num_sample' performances by permutating that feature\n \n # You'll see how the model performs when the feature is permuted\n # You'll do this num_samples number of times, and save the performance each time\n # To store the feature performance,\n # create a numpy array of size num_samples, initialized to all zeros\n feature_performance_arr = np.zeros(num_samples)\n \n # Loop through each sample\n for i in range(num_samples): # complete this line\n \n # permute the column of dataframe X\n perm_X = permute_feature(X,feature)\n \n # calculate the performance with the permuted data\n # Use the same metric function that was used earlier\n feature_performance_arr[i] = metric(y, model.predict_proba(perm_X)[:, 1])\n \n \n # Compute importance: absolute difference between \n # the baseline performance and the average across the \n importances[feature]['importance'] = np.abs(baseline_performance - np.mean(feature_performance_arr))\n \n ### END CODE HERE ###\n\n return importances\n```\n\n**Test Case**\n\n\n```python\nprint(\"Test Case\")\nprint(\"\\n\")\nprint(\"We check our answers on a Logistic Regression on a dataset\")\nprint(\"where y is given by a sigmoid applied to the important feature.\") \nprint(\"The unimportant feature is random noise.\")\nprint(\"\\n\")\nexample_df = pd.DataFrame({'important': np.random.normal(size=(1000)), 'unimportant':np.random.normal(size=(1000))})\nexample_y = np.round(1 / (1 + np.exp(-example_df.important)))\nexample_model = sklearn.linear_model.LogisticRegression(fit_intercept=False).fit(example_df, example_y)\n\nexample_importances = permutation_importance(example_df, example_y, example_model, cindex, num_samples=100)\nprint(\"Computed importances:\")\nprint(example_importances)\nprint(\"\\n\")\nprint(\"Expected importances (approximate values):\")\nprint(pd.DataFrame({\"important\": 0.50, \"unimportant\": 0.00}, index=['importance']))\nprint(\"If you round the actual values, they will be similar to the expected values\")\n```\n\n Test Case\n \n \n We check our answers on a Logistic Regression on a dataset\n where y is given by a sigmoid applied to the important feature.\n The unimportant feature is random noise.\n \n \n Computed importances:\n important unimportant\n importance 0.496674 2.89012e-05\n \n \n Expected importances (approximate values):\n important unimportant\n importance 0.5 0.0\n If you round the actual values, they will be similar to the expected values\n\n\n\n#### 2.1.3 Computing our Feature Importance\n\nNext, we compute importances on our dataset. Since we are computing the permutation importance for all the features, it might take a few minutes to run.\n\n\n```python\nimportances = permutation_importance(X_test, y_test, rf, cindex, num_samples=100)\nimportances\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        AgeDiastolic BPPoverty indexRaceRed blood cellsSedimentation rateSerum AlbuminSerum CholesterolSerum IronSerum MagnesiumSerum ProteinSexSystolic BPTIBCTSWhite blood cellsBMIPulse pressure
                                        importance0.1477720.01130340.01111480.0004491580.0008056940.0062850.005271720.0008481180.0002037890.002740190.001548670.02723370.006189490.002259220.0004252880.002561280.003048840.00379624
                                        \n
                                        \n\n\n\nLet's plot these in a bar chart for easier comparison.\n\n\n```python\nimportances.T.plot.bar()\nplt.ylabel(\"Importance\")\nl = plt.legend()\nl.remove()\nplt.show()\n```\n\nYou should see age as by far the best prediction of near term mortality, as one might expect. Next is sex, followed by diastolic blood pressure. Interestingly, the poverty index also has a large impact, despite the fact that it is not directly related to an individual's health. This alludes to the importance of social determinants of health in our model. \n\n\n### 2.2 Shapley Values for Random Forests\n\nWe'll contrast the permutation method with a more recent technique known as Shapley values (actually, Shapley values date back to the mid 20th century, but have only been applied to machine learning very recently). \n\n\n#### 2.2.1 Visualizing Feature Importance on Specific Individuals\n\nWe can use Shapley values to try and understand the model output on specific individuals. In general Shapley values take exponential time to compute, but luckily there are faster approximations for forests in particular that run in polynomial time. Run the next cell to display a 'force plot' showing how each feature influences the output for the first person in our dataset. If you want more information about 'force plots' and other decision plots, please take a look at [this notebook](https://github.com/slundberg/shap/blob/master/notebooks/plots/decision_plot.ipynb) by the `shap` library creators.\n\n\n```python\nexplainer = shap.TreeExplainer(rf)\ni = 0 # Picking an individual\nshap_value = explainer.shap_values(X_test.loc[X_test_risky.index[i], :])[1]\nshap.force_plot(explainer.expected_value[1], shap_value, feature_names=X_test.columns, matplotlib=True)\n```\n\nFor this individual, their age, pulse pressure, and sex were the biggest contributors to their high risk prediction. Note how shapley values give us greater granularity in our interpretations. \n\nFeel free to change the `i` value above to explore the feature influences for different individuals.\n\n\n#### 2.2.2 Visualizing Feature Importance on Aggregate\n\nJust like with the permutation method, we might also want to understand model output in aggregate. Shapley values allow us to do this as well. Run the next cell to initialize the shapley values for each example in the test set (this may also take a few minutes). \n\n\n```python\nshap_values = shap.TreeExplainer(rf).shap_values(X_test)[1]\n```\n\nYou can ignore the `setting feature_perturbation` message.\n\nRun the next cell to see a summary plot of the shapley values for each feature on each of the test examples. The colors indicate the value of the feature. The features are listed in terms of decreasing absolute average shapley value over all the individuals in the dataset.\n\n\n```python\nshap.summary_plot(shap_values, X_test)\n```\n\nIn the above plot, you might be able to notice a high concentration of points on specific SHAP value ranges. This means that a high proportion of our test set lies on those ranges.\n\nAs with the permutation method, age, sex, poverty index, and diastolic BP seem to be the most important features. Being older has a negative impact on mortality, and being a woman (sex=2.0) has a positive effect. \n\n\n#### 2.2.3 Visualizing Interactions between Features\n\nThe `shap` library also lets you visualize interactions between features using dependence plots. These plot the Shapley value for a given feature for each data point, and color the points in using the value for another feature. This lets us begin to explain the variation in shapley value for a single value of the main feature.\n\nRun the next cell to see the interaction between Age and Sex. \n\n\n```python\nshap.dependence_plot('Age', shap_values, X_test, interaction_index = 'Sex')\n```\n\nWe see that while Age > 50 is generally bad (positive Shapley value), being a woman (red points) generally reduces the impact of age. This makes sense since we know that women generally live longer than men. \n\nRun the next cell to see the interaction between Poverty index and Age \n\n\n```python\nshap.dependence_plot('Poverty index', shap_values, X_test, interaction_index='Age')\n```\n\nWe see that the impact of poverty index drops off quickly, and for higher income individuals age begins to explain much of variation in the impact of poverty index. We encourage you to try some other pairs and see what other interesting relationships you can find!\n\nCongratulations! You've completed the final assignment of course 3, well done! \n", "meta": {"hexsha": "dfd92b3ac38601f51c631bf231cf0d0e5adda5de", "size": 834001, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "AI for Medical Treatement/Week 3/utf-8''C3M3_Assignment.ipynb", "max_stars_repo_name": "Rishit-dagli/AI-for-Medicine", "max_stars_repo_head_hexsha": "b6690f836993ef60af69ba182dee0f820f82db25", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2020-09-19T07:50:21.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-05T10:41:23.000Z", "max_issues_repo_path": "AI for Medical Treatement/Week 3/utf-8''C3M3_Assignment.ipynb", "max_issues_repo_name": "Rishit-dagli/AI-for-Medicine", "max_issues_repo_head_hexsha": "b6690f836993ef60af69ba182dee0f820f82db25", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-04-01T00:39:21.000Z", "max_issues_repo_issues_event_max_datetime": "2021-04-01T00:39:21.000Z", "max_forks_repo_path": "AI for Medical Treatement/Week 3/utf-8''C3M3_Assignment.ipynb", "max_forks_repo_name": "Rishit-dagli/AI-for-Medicine", "max_forks_repo_head_hexsha": "b6690f836993ef60af69ba182dee0f820f82db25", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2020-06-24T21:11:12.000Z", "max_forks_repo_forks_event_max_datetime": "2020-12-01T03:40:51.000Z", "avg_line_length": 244.7186032864, "max_line_length": 148640, "alphanum_fraction": 0.8946224285, "converted": true, "num_tokens": 31266, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.6548947357776795, "lm_q1q2_score": 0.42423964278299897}} {"text": "```python\n%matplotlib widget\n%matplotlib \n# check your backend is ipympl, which needs to be installed by yourself manually pip install ipympl\nimport matplotlib.pyplot as plt\nimport numpy as np\nplt.ion()\n```\n\n Using matplotlib backend: module://ipympl.backend_nbagg\n\n\n\n\n\n \n\n\n\n\n```python\nfrom matplotlib import pyplot as plt\n\nclass LineBuilder:\n def __init__(self, line):\n self.line = line\n self.xs = list(line.get_xdata())\n self.ys = list(line.get_ydata())\n self.cid_mouse_press = line.figure.canvas.mpl_connect('button_press_event', self)\n self.cid_key_press = line.figure.canvas.mpl_connect('key_press_event', self.finish)\n\n def __call__(self, event):\n# print('click', event)\n if event.inaxes!=self.line.axes: return\n self.xs.append(event.xdata)\n self.ys.append(event.ydata)\n self.line.set_data(self.xs, self.ys)\n self.line.figure.canvas.draw()\n \n def finish(self, event):\n if event.inaxes!=self.line.axes: return\n if event.key == 'enter':\n self.xs.append(self.xs[0])\n self.ys.append(self.ys[0])\n self.line.set_data(self.xs, self.ys)\n self.line.figure.canvas.draw()\n line.figure.canvas.mpl_disconnect(self.cid_mouse_press)\n line.figure.canvas.mpl_disconnect(self.cid_key_press)\n \n \n\nfig, ax = plt.subplots()\nax.set_title('click to build line segments')\nline, = ax.plot([], []) # empty line\nlinebuilder = LineBuilder(line)\nax.set_xlim(-9, 9)\nax.set_ylim(-9, 9)\n\nplt.show()\n```\n\n\n```python\ndef uniformly_scattered_points_in_polygon(polygon, ny=40):\n if polygon.xs[-1] != polygon.xs[0]:\n raise ValueError(\"The polygon is not closed in x coordinate.\")\n if polygon.ys[-1] != polygon.ys[0]:\n raise ValueError(\"The polygon is not closed in y coordinate.\")\n poly_edge_num = len(polygon.xs) - 1\n \n ymin, ymax = min(polygon.ys), max(polygon.ys)\n ygap = ymax - ymin\n dy = ygap / ny\n dx = dy\n yarr = np.linspace(ymin+dy, ymax, num=ny, endpoint=False)\n \n scattered_points_x = []\n scattered_points_y = []\n def line_of_two_points(x1, y1, x2, y2):\n '''\n return a, b, c of line ax + by + c = 0\n '''\n if abs(y2-y1)> abs(x2-x1): # the line is steeo, let x = k2 * y + b\n k2 = (x2 - x1) / (y2 - y1)\n c = x1 - k2*y1\n return 1, -k2, -c\n else: # let y = k1 * x + b\n k1 = (y2 - y1) / (x2 - x1)\n c = y1 - k1*x1\n return -k1, 1, -c\n for y in yarr:\n intersected_points_x = list()\n for i in range(poly_edge_num):\n if (polygon.ys[i] - y)*(polygon.ys[i+1] - y) < 0: # the segment intersects with y = {y} horizontal line.\n seg_a, seg_b, seg_c = line_of_two_points(\n polygon.xs[i], polygon.ys[i],\n polygon.xs[i+1], polygon.ys[i+1])\n intersected_points_x.append( -(seg_b*y+seg_c) / seg_a )\n intersected_points_x.sort()\n if len(intersected_points_x)%2 != 0:\n print(intersected_points_x)\n raise RuntimeError(f\"The polygon is strange that we don't have even number of intersected points with y={y} line.\")\n \n for i in range(int(len(intersected_points_x)/2)):\n horizon_seg_xbeg = intersected_points_x[2*i]\n horizon_seg_xend = intersected_points_x[2*i+1]\n scat_x = horizon_seg_xbeg\n while True:\n scattered_points_x.append(scat_x)\n scattered_points_y.append(y)\n scat_x+= dx\n if scat_x > horizon_seg_xend:\n break\n \n return scattered_points_x, scattered_points_y\n \n \n```\n\n\n```python\n\n```\n\n\n```python\nfrom sympy import Function, Symbol, symbols, lambdify\nfrom sympy import sin, pi\nepsilon, rho_i, theta_i = symbols(\"\\\\epsilon, r_i, \\\\theta_i \", real=True)\nrho_ip1, theta_ip1 = symbols(\"r_{i+1}, \\\\theta_{i+1}\", real=True)\n\nrho_ip1 = rho_ip1.subs(rho_ip1, rho_i + epsilon*sin(theta_i))\ntheta_ip1 = theta_ip1.subs(theta_ip1, theta_i + rho_ip1)\n\n\nfixed_points_rho = np.asarray([2*np.pi, 2*np.pi])\nfixed_points_theta = np.asarray([0, np.pi])\nfixed_points_x = fixed_points_rho * np.cos(fixed_points_theta)\nfixed_points_y = fixed_points_rho * np.sin(fixed_points_theta)\nfixed_points_scat = ax.scatter(fixed_points_x, fixed_points_y)\n\n\ninit_scat_x, init_scat_y = uniformly_scattered_points_in_polygon(linebuilder)\ninit_scat_x, init_scat_y = np.asarray(init_scat_x), np.asarray(init_scat_y)\ninit_scat_rho = np.sqrt( init_scat_x ** 2 + init_scat_y ** 2 )\ninit_scat_theta = np.arctan2( init_scat_y, init_scat_x )\n# scat = ax.scatter(init_scat_x, init_scat_y)\n\nparameter_dict = {epsilon: 0.3}\n\nlamb_rho_ip1 = lambdify([rho_i, theta_i], rho_ip1.subs(parameter_dict))\nlamb_theta_ip1 = lambdify([rho_i, theta_i], theta_ip1.subs(parameter_dict))\n```\n\n\n```python\nfrom sympy import Function, Symbol, symbols, lambdify\nfrom sympy import sin, pi\nepsilon, rho_i, theta_i = symbols(\"\\\\epsilon, r_i, \\\\theta_i \", real=True)\nrho_ip1, theta_ip1 = symbols(\"r_{i+1}, \\\\theta_{i+1}\", real=True)\n\nrho_ip1 = rho_ip1.subs(rho_ip1, rho_i + epsilon*sin(theta_i))\ntheta_ip1 = theta_ip1.subs(theta_ip1, theta_i + rho_ip1)\n\n\nfixed_points_rho = np.asarray([2*np.pi, 2*np.pi])\nfixed_points_theta = np.asarray([0, np.pi])\nfixed_points_x = fixed_points_rho * np.cos(fixed_points_theta)\nfixed_points_y = fixed_points_rho * np.sin(fixed_points_theta)\nfixed_points_scat = ax.scatter(fixed_points_x, fixed_points_y)\n\n\ninit_scat_x, init_scat_y = uniformly_scattered_points_in_polygon(linebuilder)\ninit_scat_x, init_scat_y = np.asarray(init_scat_x), np.asarray(init_scat_y)\ninit_scat_rho = np.sqrt( init_scat_x ** 2 + init_scat_y ** 2 )\ninit_scat_theta = np.arctan2( init_scat_y, init_scat_x )\n# scat = ax.scatter(init_scat_x, init_scat_y)\n\nparameter_dict = {epsilon: 0.3}\n\nlamb_rho_ip1 = lambdify([rho_i, theta_i], rho_ip1.subs(parameter_dict))\nlamb_theta_ip1 = lambdify([rho_i, theta_i], theta_ip1.subs(parameter_dict))\n```\n\n\n```python\n\n```\n\n\n```python\n# mapped_scat_rho = lamb_rho_ip1(init_scat_rho, init_scat_theta)\n# mapped_scat_theta = lamb_theta_ip1(init_scat_rho, init_scat_theta)\n# mapped_scat_x = mapped_scat_rho * np.cos(mapped_scat_theta)\n# mapped_scat_y = mapped_scat_rho * np.sin(mapped_scat_theta)\n# scat.remove()\n# scat = ax.scatter(mapped_scat_x, mapped_scat_y)\n# init_scat_rho = mapped_scat_rho\n# init_scat_theta = mapped_scat_theta\n```\n\n\n```python\nniter = 50\nmapped_scat_rho = np.empty((len(init_scat_rho), niter))\nmapped_scat_theta = np.empty((len(init_scat_rho), niter))\nmapped_scat_x = np.empty((len(init_scat_rho), niter))\nmapped_scat_y = np.empty((len(init_scat_rho), niter))\nmapped_scat_rho[:,0] = init_scat_rho\nmapped_scat_theta[:,0] = init_scat_theta\nmapped_scat_x[:,0] = init_scat_x\nmapped_scat_y[:,0] = init_scat_y\n\nscat_list = []\nfor i in range(niter-1):\n mapped_scat_rho[:,i+1] = lamb_rho_ip1(mapped_scat_rho[:,i], mapped_scat_theta[:,i])\n mapped_scat_theta[:,i+1] = lamb_theta_ip1(mapped_scat_rho[:,i], mapped_scat_theta[:,i])\n mapped_scat_x[:,i+1] = mapped_scat_rho[:,i] * np.cos(mapped_scat_theta[:,i])\n mapped_scat_y[:,i+1] = mapped_scat_rho[:,i] * np.sin(mapped_scat_theta[:,i])\nfor i in range(len(init_scat_rho)):\n scat_list.append( ax.scatter(mapped_scat_x[i,:], mapped_scat_y[i,:], s=0.02) )\n \n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\"\"\"\nIllustrate the figure and axes enter and leave events by changing the\nframe colors on enter and leave\n\"\"\"\nimport matplotlib.pyplot as plt\n\ndef enter_axes(event):\n print('enter_axes', event.inaxes)\n event.inaxes.patch.set_facecolor('yellow')\n event.canvas.draw()\n\ndef leave_axes(event):\n print('leave_axes', event.inaxes)\n event.inaxes.patch.set_facecolor('white')\n event.canvas.draw()\n\ndef enter_figure(event):\n print('enter_figure', event.canvas.figure)\n event.canvas.figure.patch.set_facecolor('red')\n event.canvas.draw()\n\ndef leave_figure(event):\n print('leave_figure', event.canvas.figure)\n event.canvas.figure.patch.set_facecolor('grey')\n event.canvas.draw()\n\nfig1, axs = plt.subplots(2)\nfig1.suptitle('mouse hover over figure or axes to trigger events')\n\nfig1.canvas.mpl_connect('figure_enter_event', enter_figure)\nfig1.canvas.mpl_connect('figure_leave_event', leave_figure)\nfig1.canvas.mpl_connect('axes_enter_event', enter_axes)\nfig1.canvas.mpl_connect('axes_leave_event', leave_axes)\n\nfig2, axs = plt.subplots(2)\nfig2.suptitle('mouse hover over figure or axes to trigger events')\n\nfig2.canvas.mpl_connect('figure_enter_event', enter_figure)\nfig2.canvas.mpl_connect('figure_leave_event', leave_figure)\nfig2.canvas.mpl_connect('axes_enter_event', enter_axes)\nfig2.canvas.mpl_connect('axes_leave_event', leave_axes)\n\nplt.show()\n```\n", "meta": {"hexsha": "3bce3e3a682ab2a5e06eca894fc6b148b61e3fe8", "size": 83785, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "nb/pick_points_on_poloidal_cross_section.ipynb", "max_stars_repo_name": "WenyinWei/MHDpy", "max_stars_repo_head_hexsha": "a1ba3cd4b1ca8287e32a97170c685dff1d7cd276", "max_stars_repo_licenses": ["MIT"], 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2487, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175139669997, "lm_q2_score": 0.6442250996557035, "lm_q1q2_score": 0.42416908855045093}} {"text": "
                                        \n

                                        Universidad Nacional de C\u00f3rdoba - Facultad de Matem\u00e1tica, Astronom\u00eda, F\u00edsica y Computaci\u00f3n

                                        \n

                                        Diplomatura en Ciencia de Datos, Aprendizaje Autom\u00e1tico y sus Aplicaciones

                                        \n
                                        \n\n

                                        Regresi\u00f3n Linear - Ejemplo

                                        \n

                                        An\u00e1lisis y Visualizaci\u00f3n de Datos - 2019

                                        \n\nEn este ejemplo veremos c\u00f3mo implementar una regresi\u00f3n log\u00edstica para predecir una variable num\u00e9rica. Volveremos a utilizar el dataset [Human Freedom Index 2018](https://www.cato.org/human-freedom-index-new) de el instituto Cato.\n\nUsaremos la misma versi\u00f3n del conjunto de datos que en el pr\u00e1ctico.\n\nEn esta notebook vamos a tratar de estimar una funci\u00f3n lineal que modele el cambio a trav\u00e9s del tiempo de la libertad personal y la econ\u00f3mica.\n\n\n```python\n%matplotlib inline\nimport matplotlib.pyplot as plt\nimport numpy\nimport pandas\nimport seaborn\n```\n\n /usr/local/lib/python3.6/dist-packages/statsmodels/tools/_testing.py:19: FutureWarning: pandas.util.testing is deprecated. Use the functions in the public API at pandas.testing instead.\n import pandas.util.testing as tm\n\n\n\n```python\nseaborn.set_context(context='talk', font_scale=1.2)\n```\n\n\n```python\nBLUE = '#35A7FF'\nRED = '#FF5964'\nGREEN = '#6BF178'\nYELLOW = '#FFE74C'\n```\n\n\n```python\ndataset = pandas.read_csv(\n 'https://object.cato.org/sites/cato.org/files/human-freedom-index-files/human-freedom-index-2019.csv')\ndataset.shape\n```\n\n\n\n\n (1620, 120)\n\n\n\n\n```python\nscore_cols = [col for col in dataset.columns if 'pf_identity' in col] + [\n 'pf_score', # Personal Freedom (score)\n 'pf_rank', # Personal Freedom (rank)\n 'ef_score', # Economic Freedom (score)\n 'ef_rank', # Economic Freedom (rank)\n 'hf_score', # Human Freedom (score)\n 'hf_rank', # Human Freedom (rank)\n]\n\nimportant_cols = ['year', 'ISO_code', 'countries', 'region'] + score_cols\n```\n\n\n```python\ndataset = dataset[important_cols].replace('-', numpy.nan)\nfor score_col in score_cols:\n dataset[score_col] = pandas.to_numeric(dataset[score_col])\ndataset\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        yearISO_codecountriesregionpf_identity_legalpf_identity_sex_malepf_identity_sex_femalepf_identity_sexpf_identity_divorcepf_identitypf_scorepf_rankef_scoreef_rankhf_scorehf_rank
                                        02017ALBAlbaniaEastern Europe0.010.010.010.07.55.88.0146.07.6730.07.8438.0
                                        12017DZAAlgeriaMiddle East & North AfricaNaN0.00.00.00.00.05.20146.04.77159.04.99155.0
                                        22017AGOAngolaSub-Saharan Africa10.00.00.00.05.05.05.98121.04.83158.05.40151.0
                                        32017ARGArgentinaLatin America & the Caribbean10.010.010.010.010.010.08.0441.05.67147.06.8677.0
                                        42017ARMArmeniaCaucasus & Central Asia7.010.010.010.07.58.27.1572.07.7027.07.4254.0
                                        ...................................................
                                        16152008AUSAustraliaOceaniaNaN10.010.010.010.010.09.297.08.186.08.734.0
                                        16162008DNKDenmarkWestern EuropeNaN10.010.010.010.010.09.493.07.989.08.734.0
                                        16172008CHESwitzerlandWestern EuropeNaN10.010.010.010.010.09.316.08.354.08.833.0
                                        16182008NZLNew ZealandOceaniaNaN10.010.010.010.010.09.424.08.463.08.942.0
                                        16192008HKGHong KongEast AsiaNaN10.010.010.010.010.09.1312.09.111.09.121.0
                                        \n

                                        1620 rows \u00d7 16 columns

                                        \n
                                        \n\n\n\nEn el pr\u00e1ctico hab\u00edamos trabajado sobre las variables `ef_score` y `pf_score`, que hacen referencia a los \u00edndices de libertad personal y libertad econ\u00f3mica de cada p\u00e1is. Adem\u00e1s, sabemos que el dataset incluye una medici\u00f3n del \u00edndice anual por pa\u00eds desde 2008 hasta 2016, aunque hay datos faltantes de algunos indicadores.\n\nLa motivaci\u00f3n de este an\u00e1lisis comienza con este gr\u00e1fico, que muestra una tendencia decreciente de la libertad personal y una tendencia ascendiente de la libertad econ\u00f3mica. La libertad humana o `pf_score` es el promedio de ambos indicadores\n\n\n```python\nplt.figure(figsize=(10,6))\nseaborn.lineplot(data=dataset,\n x='year', y='ef_score',\n label='Economic Freedom')\nseaborn.lineplot(data=dataset,\n x='year', y='pf_score',\n label='Personal Freedom')\n\nseaborn.lineplot(data=dataset,\n x='year', y='hf_score',\n label='Human Freedom')\nplt.legend()\nplt.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.)\nplt.ylabel('Index score')\nseaborn.despine()\n```\n\nEste fen\u00f3meno podr\u00eda estar dado por varios factores:\n * Hay pocos pa\u00edses en los que la libertad personal est\u00e1 decreciendo, pero su libertad econ\u00f3mica se mantiene constante.\n * Los pa\u00edses para los cuales sube la libertad econ\u00f3mica decrecen en libertad personal.\n * **\u00bfOtras?**\n\nVeamos qu\u00e9 sucede en Argentina. Si graficamos ambas variables, vemos que \"van bajando\". Formalmente, esto significa que hay la recta que las modela tiene una pendiente negativa.\n\n**\u00bfY esto, es grave?**\n\n\n```python\nplt.figure(figsize=(15, 8))\nseaborn.regplot(data=dataset[(dataset.ISO_code == 'ARG')],\n x='year', y='ef_score', label='Economic Freedom')\nseaborn.regplot(data=dataset[(dataset.ISO_code == 'ARG')],\n x='year', y='pf_score', label='Personal Freedom')\nseaborn.regplot(data=dataset[(dataset.ISO_code == 'ARG')],\n x='year', y='hf_score', label='Human Freedom')\nplt.legend()\nplt.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.)\nseaborn.despine()\nplt.xlim(2008, 2016)\n```\n\nPodemos graficar varios pa\u00edses, pero es dif\u00edcil comparar visualmente entre tantas variables, qu\u00e9 pa\u00edses \"decrecen\" m\u00e1s r\u00e1pido que otros.\n\n\n```python\ncountries = ['Argentina', 'Brazil', 'Mexico', 'Bolivia',\n 'Uruguay', 'Peru', 'Colombia', 'Venezuela']\ng = seaborn.FacetGrid(dataset, col=\"countries\",\n margin_titles=True, aspect=2, col_wrap=2,\n col_order=countries)\ng.map(seaborn.lineplot, \"year\", \"ef_score\", color=BLUE)\ng.map(seaborn.lineplot, \"year\", \"pf_score\", color=RED)\ng.set(xlim=(2008, 2016))\n\nprint('Puntajes de libertad para Am\u00e9rica Latina');\n```\n\nPara poder comparar la situaci\u00f3n de Argentina con otros pa\u00edses, podemos comparar la pendiente de la recta de la regresi\u00f3n lineal. A partir del gr\u00e1fico anterior pudimos ver que la mayor\u00eda de los pa\u00edses tiene tendencias similares y que se pueden estimar con una recta sin perder generalidad. Esto es posible tambi\u00e9n, en cierta medida, porque tenemos pocos puntos para estimar.\n\n## Regresi\u00f3n lineal\n\nQueremos ver cu\u00e1l es el coeficiente que relaciona ambas variables.\n\n\nPrimero: **\u00bfCu\u00e1l es la variable dependiente? \u00bfCu\u00e1l la independiente?**\n\nUna vez que hemos determinado eso, lo que queremos encontrar es la funci\u00f3n de la siguiente forma:\n\n$$ef = a * year + b$$\n\nReescribiremos esto como una funci\u00f3n $e$ (por economic), cuyo par\u00e1metro es el valor $y$ (por year):\n\n$$e(y) = a * y + b$$\n\nVamos a describir los ejemplos como pares $(x_y, x_e)$, donde $x_y$ denota el `year` y $x_e$ denota `ef_score`. \n\nPara encontrar la recta $e$ que mejor describe los datos, queremos minimizar el error cuadr\u00e1tico medio, definido como:\n\n$$mse = \\frac{1}{|X|} \\sum_{x \\in X} (e(x_y) - x_e)^2 $$\n\nRecordemos que para minimizar una funci\u00f3n, una buena opci\u00f3n es comenzar por buscar los puntos estacionarios, donde la derivada se anula. Por suerte, la funci\u00f3n $mse$ es convexa, y por lo tanto tiene todos sus puntos estacionarios son minimizadores. El minimizador es el valor de los par\u00e1metros $a$ y $b$ que minimizan la funci\u00f3n. Ahora, en hemos cambiado nuestras \"variables\", lo que buscamos es encontrar la funci\u00f3n adecuada, por lo tanto lo que cambia son los valores de los par\u00e1metros que definen la funci\u00f3n. \n\nPrimero, notemos que:\n\n$$\\frac{\\partial}{\\partial a}e(y) = x_p$$\n\n$$\\frac{\\partial}{\\partial b}e(y) = 1$$\n\nCon eso, calculamos las derivadas parciales para cada par\u00e1metro de la funci\u00f3n $mse$.\n\n$$\\frac{\\partial}{\\partial a}mse = \\frac{2}{|X|} \\sum_{x \\in X} (e(x_p) - x_e) \\frac{\\partial}{\\partial a} (e(x_p) - x_e) = \\frac{2}{|X|} \\sum_{x \\in X} (e(x_p) - x_e) e_p $$\n\n$$\\frac{\\partial}{\\partial b}mse = \\frac{2}{|X|} \\sum_{x \\in X} \\frac{\\partial}{\\partial b} e(x_p) - x_e = \\frac{2}{|X|} \\sum_{x \\in X} e(x_p) - x_e $$\n\n\nA pesar del formuler\u00edo, es bastante simple. S\u00f3lo reemplazamos $mse$ por su definici\u00f3n, y luego aplicamos un par de reglas como \"la derivada de la suma es la suma de las derivadas\", la regla de la cadena, o la definici\u00f3n de la derivada de la funci\u00f3n cuadr\u00e1tica.\n\nUna vez que tenemos esos valores, tenemos que igualarlos a cero para encontrar los puntos estacionarios.\n\n\\begin{align}\n \\frac{\\partial}{\\partial a}mse &= \\frac{2}{|X|} \\sum_{x \\in X} (e(x_y) - x_e) x_y = 0 \\\\\n &\\Rightarrow a = \\frac{\\bar{x_y} \\bar{x_e} - \\overline{x_yx_e}}{(\\bar{x_y})^2 - \\overline{x_y^2}} \n\\end{align}\n\n\\begin{align}\n \\frac{\\partial}{\\partial b}mse &= \\frac{2}{|X|} \\sum_{x \\in X} e(x_y) - x_e = 0 \\\\\n &\\Rightarrow b = \\bar{x_e} - a \\bar{x_y}\n\\end{align}\n\nDonde $\\bar{x}$ es la media del valor para todos los ejemplos. Vamos a confiar en estas f\u00f3rmulas, pero una demostraci\u00f3n de las mismas est\u00e1 en:\n\nhttps://medium.freecodecamp.org/machine-learning-mean-squared-error-regression-line-c7dde9a26b93\n\n\n```python\ndef estimate_params(X_y, X_e):\n \"\"\"Caculates the value of a using all the examples.\"\"\"\n num = numpy.mean(X_y)*numpy.mean(X_e) - numpy.mean(numpy.multiply(X_y, X_e))\n denom = numpy.mean(X_y)**2 - numpy.mean(numpy.multiply(X_y, X_y))\n a = num / denom\n b = numpy.mean(X_e) - a * numpy.mean(X_y)\n return a, b\n```\n\n\n```python\n# Asumimos que todos los registros que tienen pf_score tienen el a\u00f1o.\na, b = estimate_params(\n dataset[(dataset.ISO_code == 'ARG') &\n (dataset.pf_score.notnull())].year.dropna(),\n dataset[dataset.ISO_code == 'ARG'].pf_score)\na, b\n```\n\n\n\n\n (-0.02048484848483261, 49.32575757572563)\n\n\n\n\n```python\ndef base_linear_regression(x_y, a):\n return a * x_y\n```\n\n\n```python\ndef regplot2(data, x, y, reg_func, **reg_func_args):\n \"\"\"Plots the x, y columns from data and builds a\n line with the regression reg_func.\"\"\"\n seaborn.scatterplot(data=data, x=x, y=y, color=BLUE)\n minimum = data[x].min()\n maximum = data[x].max()\n plt.plot([minimum, maximum],\n [reg_func(minimum, **reg_func_args),\n reg_func(maximum, **reg_func_args)],\n color=GREEN)\n seaborn.despine()\n plt.show()\n```\n\n\n```python\nregplot2(dataset[dataset.ISO_code == 'ARG'],\n x='year', y='pf_score', reg_func=base_linear_regression,\n a=a)\n```\n\nVemos que la recta va en el sentido correcto, pero est\u00e1 demasiado abajo. Esto ocurre porque no hemos usado el t\u00e9rmino de bias.\n\nRedefinamos entonces la regresi\u00f3n log\u00edstica\n\n\n```python\ndef linear_regression(x_y, a, b):\n return a * x_y + b\n```\n\n\n```python\nregplot2(dataset[dataset.ISO_code == 'ARG'],\n x='year', y='pf_score', reg_func=linear_regression,\n a=a, b=b)\n```\n\n## Continuamos el an\u00e1lisis\n\nPerfecto! Ahora podemos calcular las pendientes y los biases para todos los a\u00f1os, para regresiones que estimen el `pf_score`.\n\n\n```python\ndef build_regressions(data, x_var='year', y_var='pf_score'):\n records = []\n for code in data.ISO_code.unique():\n record = [code, data[data.ISO_code == code].region.values[0],\n data[data.ISO_code == code].countries.values[0]]\n y_data = data[data.ISO_code == code][y_var].dropna()\n # Comprobamos que hay datos en el intervalo\n if len(y_data) <= 1:\n continue\n x_data = data[(data.ISO_code == code) &\n (data[y_var].notnull())][x_var].dropna()\n # Estimamos los par\u00e1metros\n a, b = estimate_params(x_data, y_data)\n # Calculamos el error cuadr\u00e1tico medio de la regresi\u00f3n lineal estimada\n predictions = numpy.apply_along_axis(\n lambda x: linear_regression(x, a, b), 0, x_data)\n mse = numpy.mean(numpy.power(predictions - y_data, 2))\n record.extend([a, b, mse])\n # Agregamos el registro\n records.append(record)\n return pandas.DataFrame.from_records(\n records, columns=['ISO_code', 'region', 'country',\n 'slope', 'bias', 'mse']\n )\n```\n\n\n```python\npf_regressions = build_regressions(dataset).set_index('ISO_code')\npf_regressions[:10]\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        regioncountryslopebiasmse
                                        ISO_code
                                        ALBEastern EuropeAlbania-0.03690982.0845450.044086
                                        DZAMiddle East & North AfricaAlgeria-0.00612117.5879390.009140
                                        AGOSub-Saharan AfricaAngola0.087636-170.6781820.039059
                                        ARGLatin America & the CaribbeanArgentina-0.02048549.3257580.003178
                                        ARMCaucasus & Central AsiaArmenia-0.01666740.7566670.008773
                                        AUSOceaniaAustralia-0.01157632.5132120.002516
                                        AUTWestern EuropeAustria0.037394-66.0483030.007005
                                        AZECaucasus & Central AsiaAzerbaijan-0.105758218.9581210.060136
                                        BHSLatin America & the CaribbeanBahamas-0.03290974.1705450.009934
                                        BHRMiddle East & North AfricaBahrain-0.089515186.3882420.048762
                                        \n
                                        \n\n\n\n\n```python\nplt.figure(figsize=(10,6))\nseaborn.distplot(\n pf_regressions[pf_regressions.region == 'Latin America & the Caribbean'].slope,\n color=BLUE, label='Latam y Caribe')\nseaborn.distplot(pf_regressions.slope, color=RED, label='Global')\nplt.xlabel('Slope of linear regression that fits the pf_score of each country')\nplt.legend()\nseaborn.despine()\n```\n\n\n```python\ndef plot_regressions(regressions):\n plt.figure(figsize=(10,6))\n colors = seaborn.color_palette(\"cubehelix\", len(regressions))\n for color, (year, row) in zip(colors, regressions.iterrows()):\n minimum, maximum = 2008, 2016\n plt.plot([minimum, maximum],\n [linear_regression(minimum, row['slope'], row['bias']),\n linear_regression(maximum, row['slope'], row['bias'])],\n color=color, label=str(year), linestyle='--')\n plt.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.)\n seaborn.despine()\n```\n\n\n```python\nplot_regressions(pf_regressions.loc[['ARG', 'BRA', 'VEN', 'CAN']])\nplt.xlabel('year')\nplt.ylabel('pf_score')\nplt.ylim(4, 10)\n```\n\n### Libertad Econ\u00f3mica\n\n\n```python\nef_regressions = build_regressions(dataset, y_var='ef_score').set_index('ISO_code')\nef_regressions[:10]\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        regioncountryslopebiasmse
                                        ISO_code
                                        ALBEastern EuropeAlbania0.047333-87.8013330.003397
                                        DZAMiddle East & North AfricaAlgeria-0.04260690.8056970.009553
                                        AGOSub-Saharan AfricaAngola0.015394-25.7863030.084069
                                        ARGLatin America & the CaribbeanArgentina-0.083273173.0183640.132188
                                        ARMCaucasus & Central AsiaArmenia0.013333-19.1653330.003609
                                        AUSOceaniaAustralia-0.01090930.0265450.002294
                                        AUTWestern EuropeAustria-0.00848524.8197580.000690
                                        AZECaucasus & Central AsiaAzerbaijan0.042242-78.7878790.007204
                                        BHSLatin America & the CaribbeanBahamas-0.02666761.0386670.003569
                                        BHRMiddle East & North AfricaBahrain-0.01036428.0978180.009243
                                        \n
                                        \n\n\n\n\n```python\nplot_regressions(ef_regressions.loc[['ARG', 'BRA', 'VEN', 'CAN']])\nplt.xlabel('year')\nplt.ylabel('ef_score')\n```\n\n\n```python\nplt.figure(figsize=(10,6))\nseaborn.distplot(\n ef_regressions[ef_regressions.region == 'Latin America & the Caribbean'].slope,\n color=BLUE, label='Latam y Caribe')\nseaborn.distplot(ef_regressions.slope, color=RED, label='Global')\nplt.xlabel('Slope of linear regression that fits the ef_score of each country')\nplt.legend()\nseaborn.despine()\n```\n\n## An\u00e1lisis conjunto\n\n**\u00bfCu\u00e1les es el 10% de pa\u00edses en los que la libertad humana disminuye m\u00e1s r\u00e1pidamente?**\n\n\n```python\nquantil = pf_regressions.slope.quantile(0.1)\npf_regressions[pf_regressions.slope < quantil].country\n```\n\n\n\n\n ISO_code\n BTN Bhutan\n BRA Brazil\n BRN Brunei Darussalam\n BDI Burundi\n EGY Egypt\n GMB Gambia, The\n MUS Mauritius\n NPL Nepal\n NER Niger\n RWA Rwanda\n SYR Syria\n TJK Tajikistan\n THA Thailand\n TUR Turkey\n UKR Ukraine\n VEN Venezuela\n YEM Yemen, Rep.\n Name: country, dtype: object\n\n\n\n**\u00bfCu\u00e1les es el 10% de pa\u00edses en los que la libertad econ\u00f3mica disminuye m\u00e1s r\u00e1pidamente?**\n\n\n```python\nquantil = ef_regressions.slope.quantile(0.1)\nef_regressions[ef_regressions.slope < quantil].country\n```\n\n\n\n\n ISO_code\n ARG Argentina\n BRA Brazil\n BRN Brunei Darussalam\n EGY Egypt\n FJI Fiji\n GHA Ghana\n IRQ Iraq\n KWT Kuwait\n LBR Liberia\n LBY Libya\n PNG Pap. New Guinea\n SLE Sierra Leone\n SDN Sudan\n SYR Syria\n TUN Tunisia\n VEN Venezuela\n ZMB Zambia\n Name: country, dtype: object\n\n\n\n**\u00bfCu\u00e1les son los paises en los que la libertad econ\u00f3mica aumenta pero la libertad personal disminuye (r\u00e1pidamente)?**\n\n\n```python\nall_countries = dataset.ISO_code.unique()\ncodes = []\nfor code in all_countries:\n if (code in ef_regressions.index and code in pf_regressions.index and\n ef_regressions.loc[code].slope > 0.02 and\n pf_regressions.loc[code].slope < -0.02):\n codes.append(code)\nef_regressions.loc[codes].country\n```\n\n\n\n\n ISO_code\n ALB Albania\n AZE Azerbaijan\n BGR Bulgaria\n BDI Burundi\n KHM Cambodia\n CPV Cape Verde\n CHN China\n GMB Gambia, The\n GTM Guatemala\n GIN Guinea\n ISL Iceland\n IDN Indonesia\n KAZ Kazakhstan\n LAO Laos\n MYS Malaysia\n MLT Malta\n MEX Mexico\n MAR Morocco\n MMR Myanmar\n NPL Nepal\n NER Niger\n PRY Paraguay\n PHL Philippines\n RUS Russia\n RWA Rwanda\n ESP Spain\n TZA Tanzania\n VNM Vietnam\n Name: country, dtype: object\n\n\n\n# Errores\n\nCalculamos el mse pero nunca lo usamos. Veamos c\u00f3mo son los pa\u00edses para los que la regresi\u00f3n linear no produce una buena aproximaci\u00f3n\n\n\n```python\npf_regressions.mse.sort_values()[-10:]\n```\n\n\n\n\n ISO_code\n TLS 0.106220\n SYC 0.107328\n TUR 0.126490\n CAF 0.142989\n VEN 0.145224\n COD 0.165123\n LBY 0.182368\n GNB 0.194737\n YEM 0.195064\n BDI 0.206774\n Name: mse, dtype: float64\n\n\n\n\n```python\nplt.figure(figsize=(10,6))\ncountries = ['BDI', 'YEM', 'GNB', 'LBY']\nseaborn.lineplot(data=dataset[dataset.ISO_code.isin(countries)], x='year', y='hf_score',\n hue='countries')\nplt.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.)\nseaborn.despine()\n```\n\nClaramente se ve que estas funciones no pod\u00edan ser estimadas satisfactoriamente con una recta, pero a\u00fan as\u00ed, la tendencia general (descendiente o ascendiente) habr\u00eda sido aproximada\n\n\n```python\n\n```\n", "meta": {"hexsha": "e0ba3357b210548893d8cd839cef3925336f7277", "size": 510630, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "09_regresion_lineal.ipynb", "max_stars_repo_name": "danilo-91/practico-diplo-ayvd", "max_stars_repo_head_hexsha": "2b5f146c632769d47b828c14da50a456c9045325", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "09_regresion_lineal.ipynb", "max_issues_repo_name": "danilo-91/practico-diplo-ayvd", "max_issues_repo_head_hexsha": "2b5f146c632769d47b828c14da50a456c9045325", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "09_regresion_lineal.ipynb", "max_forks_repo_name": "danilo-91/practico-diplo-ayvd", "max_forks_repo_head_hexsha": "2b5f146c632769d47b828c14da50a456c9045325", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 273.9431330472, "max_line_length": 72594, "alphanum_fraction": 0.8935961459, "converted": true, "num_tokens": 8496, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.7248702761768248, "lm_q1q2_score": 0.4241224352386751}} {"text": "   \n\n# Tutorial 3: Deep linear neural networks\n**Week 1, Day 2: Linear Deep Learning**\n\n**By Neuromatch Academy**\n\n__Content creators:__ Saeed Salehi, Spiros Chavlis, Andrew Saxe\n\n__Content reviewers:__ Polina Turishcheva, Antoine De Comite\n\n__Content editors:__ Anoop Kulkarni\n\n__Production editors:__ Khalid Almubarak, Spiros Chavlis\n\n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n

                                        \n\n---\n# Tutorial Objectives\n\n* Deep linear neural networks\n* Learning dynamics and singular value decomposition\n* Representational Similarity Analysis\n* Illusory correlations & ethics\n\n\n```python\n# @title Tutorial slides\n\n# @markdown These are the slides for the videos in this tutorial\n\n# @markdown If you want to locally download the slides, click [here](https://osf.io/bncr8/download)\nfrom IPython.display import IFrame\nIFrame(src=f\"https://mfr.ca-1.osf.io/render?url=https://osf.io/bncr8/?direct%26mode=render%26action=download%26mode=render\", width=854, height=480)\n```\n\n---\n# Setup\n\nThis a GPU-Free tutorial!\n\n\n```python\n# @title Install dependencies\n!pip install git+https://github.com/NeuromatchAcademy/evaltools --quiet\n\nfrom evaltools.airtable import AirtableForm\n```\n\n\n```python\n# Imports\nimport math\nimport torch\nimport matplotlib\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nimport torch.nn as nn\nimport torch.optim as optim\n```\n\n\n```python\n# @title Figure settings\n\nfrom matplotlib import gridspec\nfrom ipywidgets import interact, IntSlider, FloatSlider, fixed\nfrom ipywidgets import FloatLogSlider, Layout, VBox\nfrom ipywidgets import interactive_output\nfrom mpl_toolkits.axes_grid1 import make_axes_locatable\n\nimport warnings\nwarnings.filterwarnings(\"ignore\")\n\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/content-creation/main/nma.mplstyle\")\n```\n\n\n```python\n# @title Plotting functions\n\ndef plot_x_y_hier_data(im1, im2, subplot_ratio=[1, 2]):\n fig = plt.figure(figsize=(12, 5))\n gs = gridspec.GridSpec(1, 2, width_ratios=subplot_ratio)\n ax0 = plt.subplot(gs[0])\n ax1 = plt.subplot(gs[1])\n ax0.imshow(im1, cmap=\"cool\")\n ax1.imshow(im2, cmap=\"cool\")\n # plt.suptitle(\"The whole dataset as imshow plot\", y=1.02)\n ax0.set_title(\"Labels of all samples\")\n ax1.set_title(\"Features of all samples\")\n ax0.set_axis_off()\n ax1.set_axis_off()\n plt.tight_layout()\n plt.show()\n\n\ndef plot_x_y_hier_one(im1, im2, subplot_ratio=[1, 2]):\n fig = plt.figure(figsize=(12, 1))\n gs = gridspec.GridSpec(1, 2, width_ratios=subplot_ratio)\n ax0 = plt.subplot(gs[0])\n ax1 = plt.subplot(gs[1])\n ax0.imshow(im1, cmap=\"cool\")\n ax1.imshow(im2, cmap=\"cool\")\n ax0.set_title(\"Labels of a single sample\")\n ax1.set_title(\"Features of a single sample\")\n ax0.set_axis_off()\n ax1.set_axis_off()\n plt.tight_layout()\n plt.show()\n\n\ndef plot_tree_data(label_list = None, feature_array = None, new_feature = None):\n cmap = matplotlib.colors.ListedColormap(['cyan', 'magenta'])\n n_features = 10\n n_labels = 8\n im1 = np.eye(n_labels)\n if feature_array is None:\n im2 = np.array([[1, 1, 1, 1, 1, 1, 1, 1],\n [0, 0, 0, 0, 0, 0, 0, 0],\n [0, 0, 0, 0, 1, 1, 1, 1],\n [1, 1, 1, 1, 0, 0, 0, 0],\n [0, 0, 0, 0, 0, 0, 1, 1],\n [0, 0, 1, 1, 0, 0, 0, 0],\n [1, 1, 0, 0, 0, 0, 0, 0],\n [0, 0, 0, 0, 1, 1, 0, 0],\n [0, 1, 1, 1, 0, 0, 0, 0],\n [0, 0, 0, 0, 1, 1, 0, 1]]).T\n im2[im2 == 0] = -1\n feature_list = ['can_grow',\n 'is_mammal',\n 'has_leaves',\n 'can_move',\n 'has_trunk',\n 'can_fly',\n 'can_swim',\n 'has_stem',\n 'is_warmblooded',\n 'can_flower']\n else:\n im2 = feature_array\n if label_list is None:\n label_list = ['Goldfish', 'Tuna', 'Robin', 'Canary',\n 'Rose', 'Daisy', 'Pine', 'Oak']\n fig = plt.figure(figsize=(12, 7))\n gs = gridspec.GridSpec(1, 2, width_ratios=[1, 1.35])\n ax1 = plt.subplot(gs[0])\n ax2 = plt.subplot(gs[1])\n ax1.imshow(im1, cmap=cmap)\n if feature_array is None:\n implt = ax2.imshow(im2, cmap=cmap, vmin=-1.0, vmax=1.0)\n else:\n implt = ax2.imshow(im2[:, -n_features:], cmap=cmap, vmin=-1.0, vmax=1.0)\n divider = make_axes_locatable(ax2)\n cax = divider.append_axes(\"right\", size=\"5%\", pad=0.1)\n cbar = plt.colorbar(implt, cax=cax, ticks=[-0.5, 0.5])\n cbar.ax.set_yticklabels(['no', 'yes'])\n ax1.set_title(\"Labels\")\n ax1.set_yticks(ticks=np.arange(n_labels))\n ax1.set_yticklabels(labels=label_list)\n ax1.set_xticks(ticks=np.arange(n_labels))\n ax1.set_xticklabels(labels=label_list, rotation='vertical')\n ax2.set_title(\"{} random Features\".format(n_features))\n ax2.set_yticks(ticks=np.arange(n_labels))\n ax2.set_yticklabels(labels=label_list)\n if feature_array is None:\n ax2.set_xticks(ticks=np.arange(n_features))\n ax2.set_xticklabels(labels=feature_list, rotation='vertical')\n else:\n ax2.set_xticks(ticks=[n_features-1])\n ax2.set_xticklabels(labels=[new_feature], rotation='vertical')\n plt.tight_layout()\n plt.show()\n\n\ndef plot_loss(loss_array, title=\"Training loss (Mean Squared Error)\", c=\"r\"):\n plt.figure(figsize=(10, 5))\n plt.plot(loss_array, color=c)\n plt.xlabel(\"Epoch\")\n plt.ylabel(\"MSE\")\n plt.title(title)\n plt.show()\n\n\ndef plot_loss_sv(loss_array, sv_array):\n n_sing_values = sv_array.shape[1]\n sv_array = sv_array / np.max(sv_array)\n cmap = plt.cm.get_cmap(\"Set1\", n_sing_values)\n\n _, (plot1, plot2) = plt.subplots(2, 1, sharex=True, figsize=(10, 10))\n plot1.set_title(\"Training loss (Mean Squared Error)\")\n plot1.plot(loss_array, color='r')\n\n plot2.set_title(\"Evolution of singular values (modes)\")\n for i in range(n_sing_values):\n plot2.plot(sv_array[:, i], c=cmap(i))\n plot2.set_xlabel(\"Epoch\")\n plt.show()\n\n\ndef plot_loss_sv_twin(loss_array, sv_array):\n n_sing_values = sv_array.shape[1]\n sv_array = sv_array / np.max(sv_array)\n cmap = plt.cm.get_cmap(\"winter\", n_sing_values)\n\n fig = plt.figure(figsize=(10, 5))\n ax1 = plt.gca()\n ax1.set_title(\"Learning Dynamics\")\n ax1.set_xlabel(\"Epoch\")\n ax1.set_ylabel(\"Mean Squared Error\", c='r')\n ax1.tick_params(axis='y', labelcolor='r')\n ax1.plot(loss_array, color='r')\n\n ax2 = ax1.twinx()\n ax2.set_ylabel(\"Singular values (modes)\", c='b')\n ax2.tick_params(axis='y', labelcolor='b')\n for i in range(n_sing_values):\n ax2.plot(sv_array[:, i], c=cmap(i))\n\n fig.tight_layout()\n plt.show()\n\n\ndef plot_ills_sv_twin(ill_array, sv_array, ill_label):\n n_sing_values = sv_array.shape[1]\n sv_array = sv_array / np.max(sv_array)\n cmap = plt.cm.get_cmap(\"winter\", n_sing_values)\n\n fig = plt.figure(figsize=(10, 5))\n ax1 = plt.gca()\n ax1.set_title(\"Network training and the Illusory Correlations\")\n ax1.set_xlabel(\"Epoch\")\n ax1.set_ylabel(ill_label, c='r')\n ax1.tick_params(axis='y', labelcolor='r')\n ax1.plot(ill_array, color='r', linewidth=3)\n ax1.set_ylim(-1.05, 1.05)\n # ax1.set_yticks([-1, 0, 1])\n # ax1.set_yticklabels(['False', 'Not sure', 'True'])\n\n ax2 = ax1.twinx()\n ax2.set_ylabel(\"Singular values (modes)\", c='b')\n ax2.tick_params(axis='y', labelcolor='b')\n for i in range(n_sing_values):\n ax2.plot(sv_array[:, i], c=cmap(i))\n\n fig.tight_layout()\n plt.show()\n\n\ndef plot_loss_sv_rsm(loss_array, sv_array, rsm_array, i_ep):\n n_ep = loss_array.shape[0]\n rsm_array = rsm_array / np.max(rsm_array)\n sv_array = sv_array / np.max(sv_array)\n\n n_sing_values = sv_array.shape[1]\n cmap = plt.cm.get_cmap(\"winter\", n_sing_values)\n\n fig = plt.figure(figsize=(14, 5))\n gs = gridspec.GridSpec(1, 2, width_ratios=[5, 3])\n\n ax0 = plt.subplot(gs[1])\n ax0.yaxis.tick_right()\n implot = ax0.imshow(rsm_array[i_ep], cmap=\"Purples\", vmin=0.0, vmax=1.0)\n divider = make_axes_locatable(ax0)\n cax = divider.append_axes(\"right\", size=\"5%\", pad=0.9)\n cbar = plt.colorbar(implot, cax=cax, ticks=[])\n cbar.ax.set_ylabel('Similarity', fontsize=12)\n ax0.set_title(\"RSM at epoch {}\".format(i_ep), fontsize=16)\n # ax0.set_axis_off()\n ax0.set_yticks(ticks=np.arange(n_sing_values))\n ax0.set_yticklabels(labels=item_names)\n # ax0.set_xticks([])\n ax0.set_xticks(ticks=np.arange(n_sing_values))\n ax0.set_xticklabels(labels=item_names, rotation='vertical')\n\n ax1 = plt.subplot(gs[0])\n ax1.set_title(\"Learning Dynamics\", fontsize=16)\n ax1.set_xlabel(\"Epoch\")\n ax1.set_ylabel(\"Mean Squared Error\", c='r')\n ax1.tick_params(axis='y', labelcolor='r', direction=\"in\")\n ax1.plot(np.arange(n_ep), loss_array, color='r')\n ax1.axvspan(i_ep-2, i_ep+2, alpha=0.2, color='m')\n\n ax2 = ax1.twinx()\n ax2.set_ylabel(\"Singular values\", c='b')\n ax2.tick_params(axis='y', labelcolor='b', direction=\"in\")\n for i in range(n_sing_values):\n ax2.plot(np.arange(n_ep), sv_array[:, i], c=cmap(i))\n ax1.set_xlim(-1, n_ep+1)\n ax2.set_xlim(-1, n_ep+1)\n\n plt.show()\n```\n\n\n```python\n#@title Helper functions\n\natform = AirtableForm('appn7VdPRseSoMXEG',\\\n 'W1D2_T3','https://portal.neuromatchacademy.org/api/redirect/to/f60119ed-1c22-4dae-9e18-b6a767f477e1')\n\ndef build_tree(n_levels, n_branches, probability, to_np_array=True):\n \"\"\"Builds a tree\n \"\"\"\n assert 0.0 <= probability <= 1.0\n\n tree = {}\n\n tree[\"level\"] = [0]\n for i in range(1, n_levels+1):\n tree[\"level\"].extend([i]*(n_branches**i))\n\n tree[\"pflip\"] = [probability]*len(tree[\"level\"])\n\n tree[\"parent\"] = [None]\n k = len(tree[\"level\"])-1\n for j in range(k//n_branches):\n tree[\"parent\"].extend([j]*n_branches)\n\n if to_np_array:\n tree[\"level\"] = np.array(tree[\"level\"])\n tree[\"pflip\"] = np.array(tree[\"pflip\"])\n tree[\"parent\"] = np.array(tree[\"parent\"])\n\n return tree\n\n\ndef sample_from_tree(tree, n):\n \"\"\" Generates n samples from a tree\n \"\"\"\n items = [i for i, v in enumerate(tree[\"level\"]) if v == max(tree[\"level\"])]\n n_items = len(items)\n x = np.zeros(shape=(n, n_items))\n rand_temp = np.random.rand(n, len(tree[\"pflip\"]))\n flip_temp = np.repeat(tree[\"pflip\"].reshape(1, -1), n, 0)\n samp = (rand_temp > flip_temp) * 2 - 1\n\n for i in range(n_items):\n j = items[i]\n prop = samp[:, j]\n while tree[\"parent\"][j] is not None:\n j = tree[\"parent\"][j]\n prop = prop * samp[:, j]\n x[:, i] = prop.T\n return x\n\n\ndef generate_hsd():\n # building the tree\n n_branches = 2 # 2 branches at each node\n probability = .15 # flipping probability\n n_levels = 3 # number of levels (depth of tree)\n tree = build_tree(n_levels, n_branches, probability, to_np_array=True)\n tree[\"pflip\"][0] = 0.5\n n_samples = 10000 # Sample this many features\n\n tree_labels = np.eye(n_branches**n_levels)\n tree_features = sample_from_tree(tree, n_samples).T\n return tree_labels, tree_features\n\n\ndef linear_regression(X, Y):\n \"\"\"Analytical Linear regression\n\n \"\"\"\n assert isinstance(X, np.ndarray)\n assert isinstance(Y, np.ndarray)\n M, Dx = X.shape\n N, Dy = Y.shape\n assert Dx == Dy\n W = Y @ X.T @ np.linalg.inv(X @ X.T)\n return W\n\n\ndef add_feature(existing_features, new_feature):\n assert isinstance(existing_features, np.ndarray)\n assert isinstance(new_feature, list)\n new_feature = np.array([new_feature]).T\n # return np.hstack((tree_features, new_feature*2-1))\n return np.hstack((tree_features, new_feature))\n\n\ndef net_svd(model, in_dim):\n \"\"\"Performs a Singular Value Decomposition on a given model weights\n\n Args:\n model (torch.nn.Module): neural network model\n in_dim (int): the input dimension of the model\n\n Returns:\n U, \u03a3, V (Tensors): Orthogonal, diagonal, and orthogonal matrices\n \"\"\"\n W_tot = torch.eye(in_dim)\n for weight in model.parameters():\n W_tot = weight.detach() @ W_tot\n U, SIGMA, V = torch.svd(W_tot)\n return U, SIGMA, V\n\n\ndef net_rsm(h):\n \"\"\"Calculates the Representational Similarity Matrix\n\n Arg:\n h (torch.Tensor): activity of a hidden layer\n\n Returns:\n (torch.Tensor): Representational Similarity Matrix\n \"\"\"\n rsm = h @ h.T\n return rsm\n\n\ndef initializer_(model, gamma=1e-12):\n \"\"\"(in-place) Re-initialization of weights\n\n Args:\n model (torch.nn.Module): PyTorch neural net model\n gamma (float): initialization scale\n \"\"\"\n for weight in model.parameters():\n n_out, n_in = weight.shape\n sigma = gamma / math.sqrt(n_in + n_out)\n nn.init.normal_(weight, mean=0.0, std=sigma)\n\n\ndef test_initializer_ex(seed):\n torch.manual_seed(seed)\n model = LNNet(5000, 5000, 1)\n try:\n ex_initializer_(model, gamma=1)\n std = torch.std(next(iter(model.parameters())).detach()).item()\n if -1e-5 <= (std - 0.01) <= 1e-5:\n print(\"Well done! Seems to be correct!\")\n else:\n print(\"Please double check your implementation!\")\n except:\n print(\"Faulty Implementation!\")\n\n\ndef test_net_svd_ex(seed):\n torch.manual_seed(seed)\n model = LNNet(8, 30, 100)\n try:\n U_ex, \u03a3_ex, V_ex = ex_net_svd(model, 8)\n U, \u03a3, V = net_svd(model, 8)\n if (torch.all(torch.isclose(U_ex.detach(), U.detach(), atol=1e-6)) and\n torch.all(torch.isclose(\u03a3_ex.detach(), \u03a3.detach(), atol=1e-6)) and\n torch.all(torch.isclose(V_ex.detach(), V.detach(), atol=1e-6))):\n print(\"Well done! Seems to be correct!\")\n else:\n print(\"Please double check your implementation!\")\n except:\n print(\"Faulty Implementation!\")\n\n\ndef test_net_rsm_ex(seed):\n torch.manual_seed(seed)\n x = torch.rand(7, 17)\n try:\n y_ex = ex_net_rsm(x)\n y = x @ x.T\n if (torch.all(torch.isclose(y_ex, y, atol=1e-6))):\n print(\"Well done! Seems to be correct!\")\n else:\n print(\"Please double check your implementation!\")\n except:\n print(\"Faulty Implementation!\")\n```\n\n\n```python\n#@title Set random seed\n\n#@markdown Executing `set_seed(seed=seed)` you are setting the seed\n\n# for DL its critical to set the random seed so that students can have a\n# baseline to compare their results to expected results.\n# Read more here: https://pytorch.org/docs/stable/notes/randomness.html\n\n# Call `set_seed` function in the exercises to ensure reproducibility.\nimport random\nimport torch\n\ndef set_seed(seed=None, seed_torch=True):\n if seed is None:\n seed = np.random.choice(2 ** 32)\n random.seed(seed)\n np.random.seed(seed)\n if seed_torch:\n torch.manual_seed(seed)\n torch.cuda.manual_seed_all(seed)\n torch.cuda.manual_seed(seed)\n torch.backends.cudnn.benchmark = False\n torch.backends.cudnn.deterministic = True\n\n print(f'Random seed {seed} has been set.')\n\n\n# In case that `DataLoader` is used\ndef seed_worker(worker_id):\n worker_seed = torch.initial_seed() % 2**32\n np.random.seed(worker_seed)\n random.seed(worker_seed)\n```\n\n\n```python\n#@title Set device (GPU or CPU). Execute `set_device()`\n# especially if torch modules used.\n\n# inform the user if the notebook uses GPU or CPU.\n\ndef set_device():\n device = \"cuda\" if torch.cuda.is_available() else \"cpu\"\n if device != \"cuda\":\n print(\"GPU is not enabled in this notebook. \\n\"\n \"If you want to enable it, in the menu under `Runtime` -> \\n\"\n \"`Hardware accelerator.` and select `GPU` from the dropdown menu\")\n else:\n print(\"GPU is enabled in this notebook. \\n\"\n \"If you want to disable it, in the menu under `Runtime` -> \\n\"\n \"`Hardware accelerator.` and select `None` from the dropdown menu\")\n\n return device\n```\n\n\n```python\nSEED = 2021\nset_seed(seed=SEED)\nDEVICE = set_device()\n```\n\nThis colab notebook is GPU free!\n\n---\n# Section 0: Prelude\n*Time estimate: ~10 mins*\n\n\n\n**A note on the exercises**: Most of the exercises are marked Optional (Bonus) and should only read through them if you are in a tight timeline. Therefore we would not rely on the implementation of the exercises. If necessary, you can look at the *Helper Functions* cell above to find the functions and classes used in this tutorial.\n\nThroughout this tutorial, we will use a linear neural net with a single hidden layer. We have also excluded `bias` from the layers. Please note that the forward loop returns the hidden activation, besides the network output (prediction). we will need it in section 3.\n\n\n```python\nclass LNNet(nn.Module):\n \"\"\"A Linear Neural Net with one hidden layer\n \"\"\"\n\n def __init__(self, in_dim, hid_dim, out_dim):\n \"\"\"\n Args:\n in_dim (int): input dimension\n out_dim (int): ouput dimension\n hid_dim (int): hidden dimension\n \"\"\"\n super().__init__()\n self.in_hid = nn.Linear(in_dim, hid_dim, bias=False)\n self.hid_out = nn.Linear(hid_dim, out_dim, bias=False)\n\n def forward(self, x):\n \"\"\"\n Args:\n x (torch.Tensor): input tensor\n \"\"\"\n hid = self.in_hid(x) # hidden activity\n out = self.hid_out(hid) # output (prediction)\n return out, hid\n```\n\nOther than `net_svd` and `net_rsm` functions, the training loop should be mostly familiar to you. We will define these functions in the coming sections.\n\n**important**: Please note that the two functions are part of inner training loop and are therefore executed and recorded at every iteration.\n\n\n```python\ndef train(model, inputs, targets, n_epochs, lr, illusory_i=0):\n \"\"\"Training function\n\n Args:\n model (torch nn.Module): the neural network\n inputs (torch.Tensor): features (input) with shape `[batch_size, input_dim]`\n targets (torch.Tensor): targets (labels) with shape `[batch_size, output_dim]`\n n_epochs (int): number of training epochs (iterations)\n lr (float): learning rate\n illusory_i (int): index of illusory feature\n\n Returns:\n np.ndarray: record (evolution) of training loss\n np.ndarray: record (evolution) of singular values (dynamic modes)\n np.ndarray: record (evolution) of representational similarity matrices\n np.ndarray: record of network prediction for the last feature\n \"\"\"\n in_dim = inputs.size(1)\n\n losses = np.zeros(n_epochs) # loss records\n modes = np.zeros((n_epochs, in_dim)) # singular values (modes) records\n rs_mats = [] # representational similarity matrices\n illusions = np.zeros(n_epochs) # prediction for the given feature\n\n optimizer = optim.SGD(model.parameters(), lr=lr)\n criterion = nn.MSELoss()\n\n for i in range(n_epochs):\n optimizer.zero_grad()\n predictions, hiddens = model(inputs)\n loss = criterion(predictions, targets)\n loss.backward()\n optimizer.step()\n\n # Section 2 Singular value decomposition\n U, \u03a3, V = net_svd(model, in_dim)\n\n # Section 3 calculating representational similarity matrix\n RSM = net_rsm(hiddens.detach())\n\n # Section 4 network prediction of illusory_i inputs for the last feature\n pred_ij = predictions.detach()[illusory_i, -1]\n\n # logging (recordings)\n losses[i] = loss.item()\n modes[i] = \u03a3.detach().numpy()\n rs_mats.append(RSM.numpy())\n illusions[i] = pred_ij.numpy()\n\n return losses, modes, np.array(rs_mats), illusions\n```\n\nWe also need take over the initialization of the weights. In PyTorch, [`nn.init`](https://pytorch.org/docs/stable/nn.init.html) provides us with the functions to initialize tensors from a given distribution.\n\n## Coding Exercise 0: Re-initialization (Optional)\n\nComplete the function `ex_initializer_`, such that the weights are sampled from the following distribution:\n\n\\begin{equation}\n\\mathcal{N}\\left(\\mu=0, ~~\\sigma=\\gamma \\sqrt{\\dfrac{1}{n_{in} + n_{out}}} \\right)\n\\end{equation}\n\nwhere $\\gamma$ is the initialization scale, $n_{in}$ and $n_{out}$ are respectively input and output dimensions of the layer. the Underscore (\"_\") in `ex_initializer_` and other functions, denotes \"[in-place](https://discuss.pytorch.org/t/what-is-in-place-operation/16244/2)\" operation.\n\n**important note**: since we did not include bias in the layers, the `model.parameters()` would only return the weights in each layer.\n\n\n```python\n#add event to airtable\natform.add_event('Coding Exercise 0: Re-initialization')\n\ndef ex_initializer_(model, gamma=1e-12):\n \"\"\"(in-place) Re-initialization of weights\n\n Args:\n model (torch.nn.Module): PyTorch neural net model\n gamma (float): initialization scale\n \"\"\"\n for weight in model.parameters():\n n_out, n_in = weight.shape\n #################################################\n ## Define the standard deviation (sigma) for the normal distribution\n # as given in the equation above\n # Complete the function and remove or comment the line below\n raise NotImplementedError(\"Function `ex_initializer_`\")\n #################################################\n sigma = ...\n nn.init.normal_(weight, mean=0.0, std=sigma)\n\n\n\n## uncomment and run\n# test_initializer_ex(SEED)\n```\n\n\n```python\n# to_remove solution\n\n#add event to airtable\natform.add_event('Coding Exercise 0: Re-initialization')\n\n\ndef ex_initializer_(model, gamma=1e-12):\n \"\"\"(in-place) Re-initialization of weights\n\n Args:\n model (torch.nn.Module): PyTorch neural net model\n gamma (float): initialization scale\n \"\"\"\n for weight in model.parameters():\n n_out, n_in = weight.shape\n sigma = gamma / math.sqrt(n_in + n_out)\n nn.init.normal_(weight, mean=0.0, std=sigma)\n\n\n## uncomment and run\ntest_initializer_ex(SEED)\n```\n\n---\n# Section 1: Deep Linear Neural Nets\n\n*Time estimate: ~20 mins*\n\n\n```python\n# @title Video 1: Intro to Representation Learning\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1iM4y1T7eJ\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"DqMSU4Bikt0\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 1: Intro to Representation Learning')\n\ndisplay(out)\n```\n\nSo far, depth just seems to slow down the learning. And we know that a single nonlinear hidden layer (given enough number of neurons and infinite training samples) has the potential to approximate any function. So it seems fair to ask: **What is depth good for**?\n\nOne reason can be that shallow nonlinear neural networks hardly meet their true potential in practice. In the contrast, deep neural nets are often surprisingly powerful in learning complex functions without sacrificing generalization. A core intuition behind deep learning is that deep nets derive their power through learning internal representations. How does this work? To address representation learning, we have to go beyond the 1D chain.\n\nFor this and the next couple of exercises, we use syntactically generated hierarchically structured data through a *branching diffusion process* (see [this reference](https://www.pnas.org/content/pnas/suppl/2019/05/16/1820226116.DCSupplemental/pnas.1820226116.sapp.pdf) for more details).\n\n
                                        \n\n
                                        hierarchically structured data (a tree)
                                        \n\nThe inputs to the network are labels (i.e. names), while the outputs are the features (i.e. attributes). For example, for the label \"Goldfish\", the network has to learn all the (artificially created) features, such as \"*can swim*\", \"*is cold-blooded*\", \"*has fins*\", and more. Given that we are training on hierarchically structured data, network could also learn the tree structure, that Goldfish and Tuna have rather similar features, and Robin has more in common with Tuna, compared to Rose.\n\n\n```python\n# @markdown #### Run to generate and visualize training samples from tree\n\ntree_labels, tree_features = generate_hsd()\n\n# convert (cast) data from np.ndarray to torch.Tensor\nlabel_tensor = torch.tensor(tree_labels).float()\nfeature_tensor = torch.tensor(tree_features).float()\n\nitem_names = ['Goldfish', 'Tuna', 'Robin', 'Canary',\n 'Rose', 'Daisy', 'Pine', 'Oak']\nplot_tree_data()\n\n# dimensions\nprint(\"---------------------------------------------------------------\")\nprint(\"Input Dimension: {}\".format(tree_labels.shape[1]))\nprint(\"Output Dimension: {}\".format(tree_features.shape[1]))\nprint(\"Number of samples: {}\".format(tree_features.shape[0]))\n```\n\nTo continue this tutorial, it is vital to understand the premise of our training data and what the task is. Therefore, please take your time to discuss them with your pod.\n\n
                                        \n\n
                                        The neural network used for this tutorial
                                        \n\n## Interactive Demo 1: Training the deep LNN\n\nTraining a neural net on our data is straight forward. But before executing the next cell, remember the training loss curve from previous tutorial.\n\n\n```python\n# @markdown #### Make sure you execute this cell to train the network and plot\n\nlr = 100.0 # learning rate\ngamma = 1e-12 # initialization scale\nn_epochs = 250 # number of epochs\ndim_input = 8 # input dimension = `label_tensor.size(1)`\ndim_hidden = 30 # hidden neurons\ndim_output = 10000 # output dimension = `feature_tensor.size(1)`\n\n# model instantiation\ndlnn_model = LNNet(dim_input, dim_hidden, dim_output)\n\n# weights re-initialization\ninitializer_(dlnn_model, gamma)\n\n# training\nlosses, *_ = train(dlnn_model,\n label_tensor,\n feature_tensor,\n n_epochs=n_epochs,\n lr=lr)\n\n# plotting\nplot_loss(losses)\n```\n\n**Think!**\n\nWhy haven't we seen these \"bumps\" in training before? And should we look for them in the future? What do these bumps mean?\n\nRecall from previous tutorial, that we are always interested in learning rate ($\\eta$) and initialization ($\\gamma$) that would give us the fastest but yet stable (reliable) convergence. Try finding the optimal $\\eta$ and $\\gamma$ using the following widgets. More specifically, try large $\\gamma$ and see if we can recover the bumps by tuning the $\\eta$.\n\n\n```python\n# @markdown #### Make sure you execute this cell to enable the widget!\n\ndef loss_lr_init(lr, gamma):\n \"\"\"Trains and plots the loss evolution given lr and initialization\n Args:\n lr (float): learning rate\n gamma (float): initialization scale\n \"\"\"\n n_epochs = 250 # number of epochs\n dim_input = 8 # input dimension = `label_tensor.size(1)`\n dim_hidden = 30 # hidden neurons\n dim_output = 10000 # output dimension = `feature_tensor.size(1)`\n\n # model instantiation\n dlnn_model = LNNet(dim_input, dim_hidden, dim_output)\n\n # weights re-initialization\n initializer_(dlnn_model, gamma)\n\n losses, *_ = train(dlnn_model,\n label_tensor,\n feature_tensor,\n n_epochs=n_epochs,\n lr=lr)\n\n plot_loss(losses)\n\n_ = interact(loss_lr_init,\n lr = FloatSlider(min=1.0, max=200.0,\n step=1.0, value=100.0,\n continuous_update=False,\n readout_format='.1f',\n description='eta'),\n epochs = fixed(250),\n gamma = FloatLogSlider(min=-15, max=1,\n step=1, value=1e-12, base=10,\n continuous_update=False,\n description='gamma')\n )\n```\n\n---\n# Section 2: Singular Value Decomposition (SVD)\n\n*Time estimate: ~20 mins*\n\n\n```python\n# @title Video 2: SVD\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1bw411R7DJ\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"18oNWRziskM\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 2: SVD')\n\ndisplay(out)\n```\n\nIn this section, we intend to study the learning (training) dynamics we just saw. First, we should know that a linear neural network is performing sequential matrix multiplications, which can be simplified to:\n\n\\begin{align}\n\\mathbf{y} &= \\mathbf{W}_{L}~\\mathbf{W}_{L-1}~\\dots~\\mathbf{W}_{1} ~ \\mathbf{x} \\\\\n &= (\\prod_{i=1}^{L}{\\mathbf{W}_{i}}) ~ \\mathbf{x} \\\\\n &= \\mathbf{W}_{tot} ~ \\mathbf{x}\n\\end{align}\n\nwhere $L$ denotes the number of layers in our network.\n\n[Saxe et al. (2013)](https://arxiv.org/abs/1312.6120) showed that to analyze and to understanding the nonlinear learning dynamics of a deep LNN, we can use [Singular Value Decomposition (SVD)](https://en.wikipedia.org/wiki/Singular_value_decomposition) to decompose the $\\mathbf{W}_{tot}$ into orthogonal vectors, where orthogonality of the vectors would ensure their \"individuality (independence)\". This means we can break a deep wide LNN into multiple deep narrow LNN, so their activity is untangled from each other.\n\n
                                        \n\n__A Quick intro to SVD__\n\nAny real-valued matix $A$ (yes, ANY) can be decomposed (factorized) to 3 matrices:\n\n\\begin{equation}\n\\mathbf{A} = \\mathbf{U} \\mathbf{\u03a3} \\mathbf{V}^{\\top}\n\\end{equation}\n\nwhere $U$ is an orthogonal matrix, $\\Sigma$ is a diagonal matrix, and $V$ is again an orthogonal matrix. The diagonal elements of $\\Sigma$ are called **singular values**.\n\nThe main difference between SVD and EigenValue Decomposition (EVD), is that EVD requires $A$ to be squared and does not guarantee the eigenvectors to be orthogonal. \n\nWe strongly recommend the [Singular Value Decomposition (the SVD)](https://www.youtube.com/watch?v=mBcLRGuAFUk) by the amazing [Gilbert Strang](http://www-math.mit.edu/~gs/) if you would like to learn more.\n\n\n## Coding Exercise 2: SVD (Optional)\n\nThe goal is to perform the SVD on $\\mathbf{W}_{tot}$ in every epoch, and record the singular values (modes) during the training.\n\nComplete the function `ex_net_svd`, by first calculating the $\\mathbf{W}_{tot} = \\prod_{i=1}^{L}{\\mathbf{W}_{i}}$ and finally performing SVD on the $\\mathbf{W}_{tot}$. Please use the PyTorch [`torch.svd`](https://pytorch.org/docs/stable/generated/torch.svd.html) instead of NumPy [`np.linalg.svd`](https://numpy.org/doc/stable/reference/generated/numpy.linalg.svd.html).\n\n\n```python\ndef ex_net_svd(model, in_dim):\n \"\"\"Performs a Singular Value Decomposition on a given model weights\n\n Args:\n model (torch.nn.Module): neural network model\n in_dim (int): the input dimension of the model\n\n Returns:\n U, \u03a3, V (Tensors): Orthogonal, diagonal, and orthogonal matrices\n \"\"\"\n W_tot = torch.eye(in_dim)\n for weight in model.parameters():\n #################################################\n ## Calculate the W_tot by multiplication of all weights\n # and then perform SVD on the W_tot using pytorch's `torch.svd`\n # Remember that weights need to be `.detach()` from the graph\n # Complete the function and remove or comment the line below\n raise NotImplementedError(\"Function `ex_net_svd`\")\n #################################################\n W_tot = ...\n U, \u03a3, V = ...\n return U, \u03a3, V\n\n#add event to airtable\natform.add_event('Coding Exercise 2: SVD')\n\n## Uncomment and run\n# test_net_svd_ex(SEED)\n```\n\n\n```python\n# to_remove solution\ndef ex_net_svd(model, in_dim):\n \"\"\"Performs a Singular Value Decomposition on a given model weights\n\n Args:\n model (torch.nn.Module): neural network model\n in_dim (int): the input dimension of the model\n\n Returns:\n U, \u03a3, V (Tensors): Orthogonal, diagonal, and orthogonal matrices\n \"\"\"\n W_tot = torch.eye(in_dim)\n for weight in model.parameters():\n W_tot = weight @ W_tot\n U, \u03a3, V = torch.svd(W_tot)\n return U, \u03a3, V\n\n#add event to airtable\natform.add_event('Coding Exercise 2: SVD')\n\n\n## Uncomment and run\ntest_net_svd_ex(SEED)\n```\n\n\n```python\n# @markdown #### Make sure you execute this cell to train the network and plot\n\nlr = 100.0 # learning rate\ngamma = 1e-12 # initialization scale\nn_epochs = 250 # number of epochs\ndim_input = 8 # input dimension = `label_tensor.size(1)`\ndim_hidden = 30 # hidden neurons\ndim_output = 10000 # output dimension = `feature_tensor.size(1)`\n\n# model instantiation\ndlnn_model = LNNet(dim_input, dim_hidden, dim_output)\n\n# weights re-initialization\ninitializer_(dlnn_model, gamma)\n\n# training\nlosses, modes, *_ = train(dlnn_model,\n label_tensor,\n feature_tensor,\n n_epochs=n_epochs,\n lr=lr)\n\nplot_loss_sv_twin(losses, modes)\n```\n\n**Think!**\n\nIn EigenValue decomposition, the amount of variance explained by eigenvectors is proportional to the corresponding eigenvalues. What about the SVD? We see that the gradient descent guides the network to first learn the features that carry more information (have higher singular value)!\n\n\n```python\n# @title Video 3: SVD - Discussion\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1t54y1J7Tb\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"JEbRPPG2kUI\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 3: SVD - Discussion')\n\ndisplay(out)\n```\n\n---\n# Section 3: Representational Similarity Analysis (RSA)\n\n*Time estimate: ~20 mins*\n\n\n```python\n# @title Video 4: RSA\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV19f4y157zD\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"YOs1yffysX8\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 4: RSA')\n\ndisplay(out)\n```\n\nThe previous section ended with an interesting remark. SVD helped to break our deep \"wide\" linear neural net into 8 deep \"narrow\" linear neural nets.\n\nThe first narrow net (highest singular value) converges fastest, while the last four narrow nets, converge almost simultaneously and have the smallest singular values. Can it be that the narrow net with larger mode is learning the difference between \"living things\" and \"objects\", while another narrow net with smaller mode is learning the difference between Fish and Birds? how could we check this hypothesis?\n\nRepresentational Similarity Analysis (RSA) is an approach that could help us understand the internal representation of our network. The main idea is that the activity of hidden units (neurons) in the network must be similar when the network is presented with similar input. For our dataset (hierarchically structured data), we expect the activity of neurons in the hidden layer to be more similar for Tuna and Canary, and less similar for Tuna and Oak.\n\nFor similarity measure, we can use the good old dot (scalar) product, which is also called cosine similarity. For calculating the dot product between multiple vectors (which would be our case), we can simply use matrix multiplication. Therefore the Representational Similarity Matrix for multiple-input (batch) activity could be calculated as follow:\n\n\\begin{equation}\nRSM = \\mathbf{H} \\mathbf{H}^{\\top}\n\\end{equation}\n\nwhere $\\mathbf{H} = \\mathbf{X} \\mathbf{W_1}$ is the activity of hidden neurons for a given batch $\\mathbf{X}$.\n\n\n## Coding Exercise 3: RSA (Optional)\n\nThe task is simple. We would need to measure the similarity between hidden layer activities $~\\mathbf{h} = \\mathbf{x} ~\\mathbf{W_1}$) for every input $\\mathbf{x}$.\n\nIf we perform RSA in every iteration, we could also see the evolution of representation learning.\n\n\n```python\ndef ex_net_rsm(h):\n \"\"\"Calculates the Representational Similarity Matrix\n\n Arg:\n h (torch.Tensor): activity of a hidden layer\n\n Returns:\n (torch.Tensor): Representational Similarity Matrix\n \"\"\"\n #################################################\n ## Calculate the Representational Similarity Matrix\n # Complete the function and remove or comment the line below\n raise NotImplementedError(\"Function `ex_net_rsm`\")\n #################################################\n rsm = ...\n return rsm\n\n#add event to airtable\natform.add_event(' Coding Exercise 3: RSA')\n\n## Uncomment and run\n# test_net_rsm_ex(SEED)\n```\n\n\n```python\n# to_remove solution\ndef ex_net_rsm(h):\n \"\"\"Calculates the Representational Similarity Matrix\n\n Arg:\n h (torch.Tensor): activity of a hidden layer\n\n Returns:\n (torch.Tensor): Representational Similarity Matrix\n \"\"\"\n rsm = h @ h.T\n return rsm\n\n#add event to airtable\natform.add_event(' Coding Exercise 3: RSA')\n\n## Uncomment and run\ntest_net_rsm_ex(SEED)\n```\n\nNow we can train the model while recording the losses, modes, and RSMs at every iteration. First, use the epoch slider to explore the evolution of RSM without changing default lr ($\\eta$) and initialization ($\\gamma$). Then, as we did before, set $\\eta$ and $\\gamma$ to larger values to see whether you can retrieve the sequential structured learning of representations.\n\n\n```python\n#@markdown #### Make sure you execute this cell to enable widgets\n\ndef loss_svd_rsm_lr_gamma(lr, gamma, i_ep):\n \"\"\"\n Args:\n lr (float): learning rate\n gamma (float): initialization scale\n i_ep (int): which epoch to show\n\n \"\"\"\n n_epochs = 250 # number of epochs\n dim_input = 8 # input dimension = `label_tensor.size(1)`\n dim_hidden = 30 # hidden neurons\n dim_output = 10000 # output dimension = `feature_tensor.size(1)`\n\n # model instantiation\n dlnn_model = LNNet(dim_input, dim_hidden, dim_output)\n\n # weights re-initialization\n initializer_(dlnn_model, gamma)\n\n # training\n losses, modes, rsms, _ = train(dlnn_model,\n label_tensor,\n feature_tensor,\n n_epochs=n_epochs,\n lr=lr)\n plot_loss_sv_rsm(losses, modes, rsms, i_ep)\n\ni_ep_slider = IntSlider(min=10, max=241, step=1, value=61,\n continuous_update=False,\n description='Epoch',\n layout=Layout(width='630px'))\n\nlr_slider = FloatSlider(min=20.0, max=200.0, step=1.0, value=100.0,\n continuous_update=False,\n readout_format='.1f',\n description='eta')\n\ngamma_slider = FloatLogSlider(min=-15, max=1, step=1,\n value=1e-12, base=10,\n continuous_update=False,\n description='gamma')\n\nwidgets_ui = VBox([lr_slider, gamma_slider, i_ep_slider])\n\nwidgets_out = interactive_output(loss_svd_rsm_lr_gamma,\n {'lr': lr_slider,\n 'gamma': gamma_slider,\n 'i_ep': i_ep_slider})\n\ndisplay(widgets_ui, widgets_out)\n```\n\nLet's take a moment to analyze this more. A deep neural net is learning the representations, rather than a naive mapping (look-up table). This is thought to be the reason for deep neural nets supreme generalization and transfer learning ability. Unsurprisingly, neural nets with no hidden layer are incapable of representation learning, even with extremely small initialization.\n\n\n```python\n# @title Video 5: RSA - Discussion\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV18y4y1j7Xr\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"vprldATyq1o\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 5: RSA - Discussion')\ndisplay(out)\n```\n\n---\n# Section 4: Illusory Correlations\n\n*Time estimate: ~20-30 mins*\n\n\n```python\n# @title Video 6: Illusory Correlations\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1vv411E7Sq\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"RxsAvyIoqEo\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 6: Illusory Correlations')\n\ndisplay(out)\n```\n\nLet's recall the training loss curves. There was often a long plateau (where the weights are stuck at a saddle point), followed by a sudden drop. For very deep complex neural nets, such plateaus can take hours of training, and we are often tempted to stop the training, because we believe it is \"as good as it gets\"! Another side effect of \"immature interruption\" of training is the network finding (learning) illusory correlations.\n\nTo better understand this, let's do the next demonstration and exercise.\n\n## Demonstration: Illusory Correlations\n\nOur original dataset has 4 animals: Canary, Robin, Goldfish, and Tuna. These animals all have bones. Therefore if we include a \"has bone\" feature, the network would learn it at the second level (i.e. second bump, second mode convergence), when it learns the animal-plants distinction.\n\nWhat if the dataset has Shark instead of Goldfish. Sharks don't have bones (their skeletons are made of cartilaginous, which is much lighter than true bone and more flexible). Then we will have a feature which is *True* (i.e. +1) for Tuna, Robin, and Canary, but *False* (i.e. -1) for all the plants and the shark! Let's see what the network does.\n\nFirst, we add the new feature to the targets. We then start training our LNN and in every epoch, record the network prediction for \"sharks having bones\".\n\n
                                        \n\n\n```python\n# sampling new data from the tree\ntree_labels, tree_features = generate_hsd()\n\n# replacing Goldfish with Shark\nitem_names = ['Shark', 'Tuna', 'Robin', 'Canary',\n 'Rose', 'Daisy', 'Pine', 'Oak']\n\n# index of label to record\nillusion_idx = 0 # Shark is the first element\n\n# the new feature (has bones) vector\nnew_feature = [-1, 1, 1, 1, -1, -1, -1, -1]\nits_label = 'has_bones'\n\n# adding feature has_bones to the feature array\ntree_features = add_feature(tree_features, new_feature)\n\n# plotting\nplot_tree_data(item_names, tree_features, its_label)\n```\n\nYou can see the new feature shown in the last column of the plot above.\n\nNow we can train the network on the new data, and record the network prediction (output) for Shark (indexed 0) label and \"has bone\" feature (last feature, indexed -1), during the training.\n\nHere is the snippet from the training loop that keeps track of network prediction for `illusory_i`th label and last (`-1`) feature:\n\n```python\npred_ij = predictions.detach()[illusory_i, -1]\n```\n\n\n```python\n#@markdown #### Make sure you execute this cell to train the network and plot\n\n# convert (cast) data from np.ndarray to torch.Tensor\nlabel_tensor = torch.tensor(tree_labels).float()\nfeature_tensor = torch.tensor(tree_features).float()\n\nlr = 100.0 # learning rate\ngamma = 1e-12 # initialization scale\nn_epochs = 250 # number of epochs\ndim_input = 8 # input dimension = `label_tensor.size(1)`\ndim_hidden = 30 # hidden neurons\ndim_output = feature_tensor.size(1)\n\n# model instantiation\ndlnn_model = LNNet(dim_input, dim_hidden, dim_output)\n\n# weights re-initialization\ninitializer_(dlnn_model, gamma)\n\n# training\n_, modes, _, ill_predictions = train(dlnn_model,\n label_tensor,\n feature_tensor,\n n_epochs=n_epochs,\n lr=lr,\n illusory_i=illusion_idx)\n\n# a label for the plot\nill_label = f\"Prediction for {item_names[illusion_idx]} {its_label}\"\n\n# plotting\nplot_ills_sv_twin(ill_predictions, modes, ill_label)\n```\n\nIt seems that the network starts by learning an \"illusory correlation\" that sharks have bones, and in later epochs, as it learns deeper representations, it can see (learn) beyond the illusory correlation. This is important to remember that we never presented the network with any data saying that sharks have bones.\n\n## Exercise 4: Illusory Correlations\n\nThis exercise is just for you to explore the idea of illusory correlations. Think of medical, natural, or possibly social illusory correlations which can test the learning power of deep linear neural nets.\n\n**important notes**: the generated data is independent of tree labels, therefore the names are just for convenience.\n\nHere is our example for **Non-human Living things do not speak**. The lines marked by `{edit}` are for you to change in your example.\n\n\n\n```python\n# sampling new data from the tree\ntree_labels, tree_features = generate_hsd()\n\n# {edit} replacing Canary with Parrot\nitem_names = ['Goldfish', 'Tuna', 'Robin', 'Parrot',\n 'Rose', 'Daisy', 'Pine', 'Oak']\n\n# {edit} index of label to record\nillusion_idx = 3 # Parrot is the fourth element\n\n# {edit} the new feature (cannot speak) vector\nnew_feature = [1, 1, 1, -1, 1, 1, 1, 1]\nits_label = 'cannot_speak'\n\n# adding feature has_bones to the feature array\ntree_features = add_feature(tree_features, new_feature)\n\n# plotting\nplot_tree_data(item_names, tree_features, its_label)\n```\n\n\n```python\n# @markdown #### Make sure you execute this cell to train the network and plot\n\n# convert (cast) data from np.ndarray to torch.Tensor\nlabel_tensor = torch.tensor(tree_labels).float()\nfeature_tensor = torch.tensor(tree_features).float()\n\nlr = 100.0 # learning rate\ngamma = 1e-12 # initialization scale\nn_epochs = 250 # number of epochs\ndim_input = 8 # input dimension = `label_tensor.size(1)`\ndim_hidden = 30 # hidden neurons\ndim_output = feature_tensor.size(1)\n\n# model instantiation\ndlnn_model = LNNet(dim_input, dim_hidden, dim_output)\n\n# weights re-initialization\ninitializer_(dlnn_model, gamma)\n\n# training\n_, modes, _, ill_predictions = train(dlnn_model,\n label_tensor,\n feature_tensor,\n n_epochs=n_epochs,\n lr=lr,\n illusory_i=illusion_idx)\n\n# a label for the plot\nill_label = f\"Prediction for {item_names[illusion_idx]} {its_label}\"\n\n# plotting\nplot_ills_sv_twin(ill_predictions, modes, ill_label)\n```\n\n\n```python\n# @title Video 7: Illusory Correlations - Discussion\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1vv411E7rg\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"6VLHKQjQJmI\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 7: Illusory Correlations')\n\ndisplay(out)\n```\n\n---\n# Summary\n\nThe second day of the course has ended. So, in the third tutorial of the linear deep learning day we have learned more advanced topics. In the beginning we implemented a deep linear neural network and then we studied its learning dynamics using the linear algebra tool called singular value decomposition. Then, we learned about the representational similarity analysis and the illusory correlation.\n\n\n```python\n# @title Video 8: Outro\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1AL411n7ns\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"N2szOIsKyXE\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n#add event to airtable\natform.add_event('Video 8: Outro')\n\ndisplay(out)\n```\n\n\n```python\n# @title Airtable Submission Link\nfrom IPython import display as IPydisplay\nIPydisplay.HTML(\n f\"\"\"\n
                                        \n \n \n
                                        \"\"\" )\n```\n\n---\n# Bonus\n\n*Time estimate: ~20-30 mins*\n\n\n```python\n# @title Video 9: Linear Regression\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1Pf4y1L71L\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"uULOAbhYaaE\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n\n## Section 5.1: Linear Regression\n\nGenerally, *regression* refers to a set of methods for modeling the mapping (relationship) between one (or more) independent variable(s) (i.e., features) and one (or more) dependent variable(s) (i.e., labels). For example, if we want to examine the relative impacts of calendar date, GPS coordinates, and time of the say (the independent variables) on air temperature (the dependent variable). On the other hand, regression can be used for predictive analysis. Thus the independent variables are also called predictors. When the model contains more than one predictor, then the method is called *multiple regression*, and if it contains more than one dependent variable called *multivariate regression*. Regression problems pop up whenever we want to predict a numerical (usually continuous) value.\n\nThe independent variables are collected in vector $\\mathbf{x} \\in \\mathbb{R}^M$, where $M$ denotes the number of independent variables, while the dependent variables are collected in vector $\\mathbf{y} \\in \\mathbb{R}^N$, where $N$ denotes the number of dependent variables. And the mapping between them is represented by the weight matrix $\\mathbf{W} \\in \\mathbb{R}^{N \\times M}$ and a bias vector $\\mathbf{b} \\in \\mathbb{R}^{N}$ (generalizing to affine mappings).\n\nThe multivariate regression model can be written as:\n\n\\begin{equation}\n\\mathbf{y} = \\mathbf{W} ~ \\mathbf{x} + \\mathbf{b}\n\\end{equation}\n\nor it can be written in matrix format as:\n\n\\begin{equation}\n\\begin{bmatrix} y_{1} \\\\ y_{2} \\\\ \\vdots \\\\ y_{N} \\\\ \\end{bmatrix} = \\begin{bmatrix} w_{1,1} & w_{1,2} & \\dots & w_{1,M} \\\\ w_{2,1} & w_{2,2} & \\dots & w_{2,M} \\\\ \\vdots & \\ddots & \\ddots & \\vdots \\\\ w_{N,1} & w_{N,2} & \\dots & w_{N,M} \\end{bmatrix} \\begin{bmatrix} x_{1} \\\\ x_{2} \\\\ \\vdots \\\\ x_{M} \\\\ \\end{bmatrix} + \\begin{bmatrix} b_{1} \\\\ b_{2} \\\\ \\vdots \\\\b_{N} \\\\ \\end{bmatrix}\n\\end{equation}\n\n\n## Section 5.2: Vectorized regression\n\nLinear regression can be simply extended to multi-samples ($D$) input-output mapping, which we can collect in a matrix $\\mathbf{X} \\in \\mathbb{R}^{M \\times D}$, sometimes called the design matrix. The sample dimension also shows up in the output matrix $\\mathbf{Y} \\in \\mathbb{R}^{N \\times D}$. Thus, linear regression takes the following form:\n\n\\begin{equation}\n\\mathbf{Y} = \\mathbf{W} ~ \\mathbf{X} + \\mathbf{b}\n\\end{equation}\n\nwhere matrix $\\mathbf{W} \\in \\mathbb{R}^{N \\times M}$ and the vector $\\mathbf{b} \\in \\mathbb{R}^{N}$ (broadcasted over sample dimension) are the desired parameters to find.\n\n## Section 5.3: Analytical Linear Regression\n\nLinear regression is a relatively simple optimization problem. Unlike most other models that we will see in this course, linear regression for mean squared loss can be solved analytically.\n\nFor $D$ samples (batch size), $\\mathbf{X} \\in \\mathbb{R}^{M \\times D}$, and $\\mathbf{Y} \\in \\mathbb{R}^{N \\times D}$, the goal of linear regression is to find $\\mathbf{W} \\in \\mathbb{R}^{N \\times M}$ such that:\n\n\\begin{equation}\n\\mathbf{Y} = \\mathbf{W} ~ \\mathbf{X}\n\\end{equation}\n\nGiven the Squared Error loss function, we have:\n\n\\begin{equation}\nLoss(\\mathbf{W}) = ||\\mathbf{Y} - \\mathbf{W} ~ \\mathbf{X}||^2\n\\end{equation}\n\nSo, using matrix notation, the optimization problem is given by:\n\n\\begin{align}\n\\mathbf{W^{*}} &= \\underset{\\mathbf{W}}{\\mathrm{argmin}} \\left( Loss (\\mathbf{W}) \\right) \\\\\n &= \\underset{\\mathbf{W}}{\\mathrm{argmin}} \\left( ||\\mathbf{Y} - \\mathbf{W} ~ \\mathbf{X}||^2 \\right) \\\\\n&= \\underset{\\mathbf{W}}{\\mathrm{argmin}} \\left( \\left( \\mathbf{Y} - \\mathbf{W} ~ \\mathbf{X}\\right)^{\\top} \\left( \\mathbf{Y} - \\mathbf{W} ~ \\mathbf{X}\\right) \\right)\n\\end{align}\n\nTo solve the minimization problem, we can simply set the derivative of the loss with respect to $\\mathbf{W}$ to zero.\n\n\\begin{equation}\n\\dfrac{\\partial Loss}{\\partial \\mathbf{W}} = 0\n\\end{equation}\n\nAssuming that $\\mathbf{X}\\mathbf{X}^{\\top}$ is full-rank, and thus it is invertible we can write:\n\n\\begin{equation}\n\\mathbf{W}^{\\mathbf{*}} = \\mathbf{Y} \\mathbf{X}^{\\top} \\left( \\mathbf{X} \\mathbf{X}^{\\top} \\right) ^{-1}\n\\end{equation}\n\n\n\n### Coding Exercise 5.3.1: Analytical solution to LR\n\nComplete the function `linear_regression` for finding the analytical solution to linear regression.\n\n\n\n```python\ndef linear_regression(X, Y):\n \"\"\"Analytical Linear regression\n\n Args:\n X (np.ndarray): design matrix\n Y (np.ndarray): target ouputs\n\n return:\n np.ndarray: estimated weights (mapping)\n \"\"\"\n assert isinstance(X, np.ndarray)\n assert isinstance(Y, np.ndarray)\n M, Dx = X.shape\n N, Dy = Y.shape\n assert Dx == Dy\n #################################################\n ## Complete the linear_regression_exercise function\n # Complete the function and remove or comment the line below\n raise NotImplementedError(\"Linear Regression `linear_regression`\")\n #################################################\n W = ...\n\n return W\n\n\nW_true = np.random.randint(low=0, high=10, size=(3, 3)).astype(float)\n\nX_train = np.random.rand(3, 37) # 37 samples\nnoise = np.random.normal(scale=0.01, size=(3, 37))\nY_train = W_true @ X_train + noise\n\n## Uncomment and run\n# W_estimate = linear_regression(X_train, Y_train)\n# print(f\"True weights:\\n {W_true}\")\n# print(f\"\\nEstimated weights:\\n {np.round(W_estimate, 1)}\")\n```\n\n\n```python\n# to_remove solution\ndef linear_regression(X, Y):\n \"\"\"Analytical Linear regression\n\n Args:\n X (np.ndarray): design matrix\n Y (np.ndarray): target ouputs\n\n return:\n np.ndarray: estimated weights (mapping)\n \"\"\"\n assert isinstance(X, np.ndarray)\n assert isinstance(Y, np.ndarray)\n M, Dx = X.shape\n N, Dy = Y.shape\n assert Dx == Dy\n\n W = Y @ X.T @ np.linalg.inv(X @ X.T)\n\n return W\n\n\nW_true = np.random.randint(low=0, high=10, size=(3, 3)).astype(float)\n\nX_train = np.random.rand(3, 37) # 37 samples\nnoise = np.random.normal(scale=0.01, size=(3, 37))\nY_train = W_true @ X_train + noise\n\n## Uncomment and run\nW_estimate = linear_regression(X_train, Y_train)\nprint(f\"True weights:\\n {W_true}\")\nprint(f\"\\nEstimated weights:\\n {np.round(W_estimate, 1)}\")\n```\n\n## Demonstration: Linear Regression vs. DLNN\n\nA linear neural network with NO hidden layer is very similar to linear regression in its core. We also know that no matter how many hidden layers a linear network has, it can be compressed to linear regression (no hidden layers).\n\nIn this demonstration, we use the hierarchically structured data to:\n\n* analytically find the mapping between features and labels\n* train a zero-depth LNN to find the mapping \n* compare them to the $W_{tot}$ from the already trained deep LNN\n\n\n```python\n# sampling new data from the tree\ntree_labels, tree_features = generate_hsd()\n\n# convert (cast) data from np.ndarray to torch.Tensor\nlabel_tensor = torch.tensor(tree_labels).float()\nfeature_tensor = torch.tensor(tree_features).float()\n```\n\n\n```python\n# calculating the W_tot for deep network (already trained model)\n\nlr = 100.0 # learning rate\ngamma = 1e-12 # initialization scale\nn_epochs = 250 # number of epochs\ndim_input = 8 # input dimension = `label_tensor.size(1)`\ndim_hidden = 30 # hidden neurons\ndim_output = 10000 # output dimension = `feature_tensor.size(1)`\n\n# model instantiation\ndlnn_model = LNNet(dim_input, dim_hidden, dim_output)\n\n# weights re-initialization\ninitializer_(dlnn_model, gamma)\n\n# training\nlosses, modes, rsms, ills = train(dlnn_model,\n label_tensor,\n feature_tensor,\n n_epochs=n_epochs,\n lr=lr)\n\ndeep_W_tot = torch.eye(dim_input)\nfor weight in dlnn_model.parameters():\n deep_W_tot = weight @ deep_W_tot\ndeep_W_tot = deep_W_tot.detach().numpy()\n```\n\n\n```python\n# analytically estimation of weights\n# our data is batch first dimension, so we need to transpose our data\nanalytical_weights = linear_regression(tree_labels.T, tree_features.T)\n```\n\n\n```python\nclass LRNet(nn.Module):\n \"\"\"A Linear Neural Net with ZERO hidden layer (LR net)\n \"\"\"\n\n def __init__(self, in_dim, out_dim):\n \"\"\"\n Args:\n in_dim (int): input dimension\n hid_dim (int): hidden dimension\n \"\"\"\n super().__init__()\n self.in_out = nn.Linear(in_dim, out_dim, bias=False)\n\n def forward(self, x):\n \"\"\"\n Args:\n x (torch.Tensor): input tensor\n \"\"\"\n out = self.in_out(x) # output (prediction)\n return out\n```\n\n\n```python\nlr = 1000.0 # learning rate\ngamma = 1e-12 # initialization scale\nn_epochs = 250 # number of epochs\ndim_input = 8 # input dimension = `label_tensor.size(1)`\ndim_output = 10000 # output dimension = `feature_tensor.size(1)`\n\n# model instantiation\nLR_model = LRNet(dim_input, dim_output)\noptimizer = optim.SGD(LR_model.parameters(), lr=lr)\ncriterion = nn.MSELoss()\n\nlosses = np.zeros(n_epochs) # loss records\nfor i in range(n_epochs): # training loop\n optimizer.zero_grad()\n predictions = LR_model(label_tensor)\n loss = criterion(predictions, feature_tensor)\n loss.backward()\n optimizer.step()\n losses[i] = loss.item()\n\n# trained weights from zero_depth_model\nLR_model_weights = next(iter(LR_model.parameters())).detach().numpy()\n\nplot_loss(losses, \"Training loss for zero depth LNN\", c=\"r\")\n```\n\n\n```python\nprint(\"The final weights from all methods are approximately equal?! \"\n\"{}!\".format(\n (np.allclose(analytical_weights, LR_model_weights, atol=1e-02) and \\\n np.allclose(analytical_weights, deep_W_tot, atol=1e-02))\n )\n)\n```\n\nAs you may have guessed, they all arrive at the same results but through very different paths.\n\n\n```python\n# @title Video 10: Linear Regression - Discussion\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV18v411E7Wg\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"gG15_J0i05Y\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n", "meta": {"hexsha": "80c4b056589d1924f124a43177503977fd5c99ea", "size": 90866, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W1D2_LinearDeepLearning/W1D2_Tutorial3.ipynb", "max_stars_repo_name": "justynaekert/course-content-dl", "max_stars_repo_head_hexsha": "aa64d9feb1ae92ad4b7afaf13b13616b3a020c20", "max_stars_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-11-30T08:42:05.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-30T08:42:05.000Z", "max_issues_repo_path": "tutorials/W1D2_LinearDeepLearning/W1D2_Tutorial3.ipynb", "max_issues_repo_name": "justynaekert/course-content-dl", "max_issues_repo_head_hexsha": "aa64d9feb1ae92ad4b7afaf13b13616b3a020c20", "max_issues_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/W1D2_LinearDeepLearning/W1D2_Tutorial3.ipynb", "max_forks_repo_name": "justynaekert/course-content-dl", "max_forks_repo_head_hexsha": "aa64d9feb1ae92ad4b7afaf13b13616b3a020c20", "max_forks_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.4777197912, "max_line_length": 807, "alphanum_fraction": 0.5605396958, "converted": true, "num_tokens": 17222, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499941, "lm_q2_score": 0.640635868562172, "lm_q1q2_score": 0.4240515364913317}} {"text": "\n\n# Deep Reinforcement Learning\n\nIn this session, we will use Open AI Gym to implement Q-learning, a classic algorthim in RL, and Deep Q-Networks (DQN), its deep learning counterpart. Hopefully this will give us some intuition for what we gain by moving from tabular methods to using neural networks for function approximation.\n\n## Setup\n\n\n\nFirst, let's install the necessary dependencies. This does not need to be run again if you restart the runtime, but it does if you factory reset the runtime. \n\n\n```\n!sudo apt-get install -y xvfb ffmpeg\n!apt-get install x11-utils\n!pip install 'gym==0.17.1'\n!pip install 'pyglet==1.4.0'\n!pip install pyvirtualdisplay\n!pip install --upgrade tensorflow-probability\n!pip install imageio-ffmpeg\n```\n\n Reading package lists... Done\n Building dependency tree \n Reading state information... Done\n ffmpeg is already the newest version (7:3.4.6-0ubuntu0.18.04.1).\n The following NEW packages will be installed:\n xvfb\n 0 upgraded, 1 newly installed, 0 to remove and 25 not upgraded.\n Need to get 784 kB of archives.\n After this operation, 2,266 kB of additional disk space will be used.\n Get:1 http://archive.ubuntu.com/ubuntu bionic-updates/universe amd64 xvfb amd64 2:1.19.6-1ubuntu4.4 [784 kB]\n Fetched 784 kB in 2s (355 kB/s)\n debconf: unable to initialize frontend: Dialog\n debconf: (No usable dialog-like program is installed, so the dialog based frontend cannot be used. at /usr/share/perl5/Debconf/FrontEnd/Dialog.pm line 76, <> line 1.)\n debconf: falling back to frontend: Readline\n debconf: unable to initialize frontend: Readline\n debconf: (This frontend requires a controlling tty.)\n debconf: falling back to frontend: Teletype\n dpkg-preconfigure: unable to re-open stdin: \n Selecting previously unselected package xvfb.\n (Reading database ... 144568 files and directories currently installed.)\n Preparing to unpack .../xvfb_2%3a1.19.6-1ubuntu4.4_amd64.deb ...\n Unpacking xvfb (2:1.19.6-1ubuntu4.4) ...\n Setting up xvfb (2:1.19.6-1ubuntu4.4) ...\n Processing triggers for man-db (2.8.3-2ubuntu0.1) ...\n Reading package lists... Done\n Building dependency tree \n Reading state information... Done\n The following additional packages will be installed:\n libxxf86dga1\n Suggested packages:\n mesa-utils\n The following NEW packages will be installed:\n libxxf86dga1 x11-utils\n 0 upgraded, 2 newly installed, 0 to remove and 25 not upgraded.\n Need to get 209 kB of archives.\n After this operation, 711 kB of additional disk space will be used.\n Get:1 http://archive.ubuntu.com/ubuntu bionic/main amd64 libxxf86dga1 amd64 2:1.1.4-1 [13.7 kB]\n Get:2 http://archive.ubuntu.com/ubuntu bionic/main amd64 x11-utils amd64 7.7+3build1 [196 kB]\n Fetched 209 kB in 2s (112 kB/s)\n Selecting previously unselected package libxxf86dga1:amd64.\n (Reading database ... 144575 files and directories currently installed.)\n Preparing to unpack .../libxxf86dga1_2%3a1.1.4-1_amd64.deb ...\n Unpacking libxxf86dga1:amd64 (2:1.1.4-1) ...\n Selecting previously unselected package x11-utils.\n Preparing to unpack .../x11-utils_7.7+3build1_amd64.deb ...\n Unpacking x11-utils (7.7+3build1) ...\n Setting up libxxf86dga1:amd64 (2:1.1.4-1) ...\n Setting up x11-utils (7.7+3build1) ...\n Processing triggers for man-db (2.8.3-2ubuntu0.1) ...\n Processing triggers for libc-bin (2.27-3ubuntu1) ...\n /sbin/ldconfig.real: /usr/local/lib/python3.6/dist-packages/ideep4py/lib/libmkldnn.so.0 is not a symbolic link\n \n Requirement already satisfied: gym==0.17.1 in /usr/local/lib/python3.6/dist-packages (0.17.1)\n Requirement already satisfied: scipy in /usr/local/lib/python3.6/dist-packages (from gym==0.17.1) (1.4.1)\n Requirement already satisfied: six in /usr/local/lib/python3.6/dist-packages (from gym==0.17.1) (1.12.0)\n Requirement already satisfied: pyglet<=1.5.0,>=1.4.0 in /usr/local/lib/python3.6/dist-packages (from gym==0.17.1) (1.4.0)\n Requirement already satisfied: numpy>=1.10.4 in /usr/local/lib/python3.6/dist-packages (from gym==0.17.1) (1.18.3)\n Requirement already satisfied: cloudpickle<1.4.0,>=1.2.0 in /usr/local/lib/python3.6/dist-packages (from gym==0.17.1) (1.3.0)\n Requirement already satisfied: future in /usr/local/lib/python3.6/dist-packages (from pyglet<=1.5.0,>=1.4.0->gym==0.17.1) (0.16.0)\n Requirement already satisfied: pyglet==1.4.0 in /usr/local/lib/python3.6/dist-packages (1.4.0)\n Requirement already satisfied: future in /usr/local/lib/python3.6/dist-packages (from pyglet==1.4.0) (0.16.0)\n Requirement already satisfied: pyvirtualdisplay in /usr/local/lib/python3.6/dist-packages (0.2.5)\n Requirement already satisfied: EasyProcess in /usr/local/lib/python3.6/dist-packages (from pyvirtualdisplay) (0.2.10)\n Requirement already up-to-date: tensorflow-probability in /usr/local/lib/python3.6/dist-packages (0.10.0rc0)\n Requirement already satisfied, skipping upgrade: six>=1.10.0 in /usr/local/lib/python3.6/dist-packages (from tensorflow-probability) (1.12.0)\n Requirement already satisfied, skipping upgrade: gast>=0.3.2 in /usr/local/lib/python3.6/dist-packages (from tensorflow-probability) (0.3.3)\n Requirement already satisfied, skipping upgrade: cloudpickle>=1.2.2 in /usr/local/lib/python3.6/dist-packages (from tensorflow-probability) (1.3.0)\n Requirement already satisfied, skipping upgrade: decorator in /usr/local/lib/python3.6/dist-packages (from tensorflow-probability) (4.4.2)\n Requirement already satisfied, skipping upgrade: numpy>=1.13.3 in /usr/local/lib/python3.6/dist-packages (from tensorflow-probability) (1.18.3)\n Requirement already satisfied: imageio-ffmpeg in /usr/local/lib/python3.6/dist-packages (0.4.1)\n\n\nNow let's import the packages we will be using.\n\n\n```\nfrom __future__ import absolute_import, division, print_function\n\nimport base64\nimport imageio\nimport IPython\nimport matplotlib\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport PIL.Image\nimport pyvirtualdisplay\nimport random\nimport tensorflow as tf\nimport gym\n\nfrom gym.spaces import Discrete\nfrom gym.spaces import Box\nfrom tensorflow.keras import Sequential\nfrom tensorflow.keras.layers import Dense\nfrom tensorflow.keras.optimizers import Adam\nfrom collections import defaultdict\nfrom collections import deque\nfrom pyvirtualdisplay import Display\n```\n\n\n```\nprint(tf.version.VERSION)\nprint(gym.version.VERSION)\n```\n\n 2.2.0-rc3\n 0.17.1\n\n\n## Introduction to Open AI Gym\n\nGym provides you with a set of [environments](https://gym.openai.com/envs/#classic_control). If we think of the classic RL framework schematic, Gym takes care of the environment, and you take care of the agent. \n\n\n\nFrom this diagram, we can expect that we will interact with a Gym environment by giving it an action as input, and receiving a next state and reward as output. \n\nIt is then our job to implement some algorithm to learn the policy $\\pi(a|s)$ for our agent to use to act within the environment. \n\n### Setting up visualization\n\nFirst, let's get plotting for Gym working in Colab. This will help give us an intuitive feel for how to work with the Gym environments.\n\n\n```\ndef embed_mp4(filename):\n \"\"\"Embeds an mp4 file in the notebook.\"\"\"\n video = open(filename,'rb').read()\n b64 = base64.b64encode(video)\n tag = '''\n '''.format(b64.decode())\n\n return IPython.display.HTML(tag)\n```\n\nBefore we get to implementing algorithms for our agents to learn a good policy, let's visualize an agent acting according to a random policy. At this point the visualizatation is just to give us an idea of what what an environment looks like, and later on we'll come back to see how we generate this video. \n\n\n```\ndef create_random_policy_video(env, filename, num_episodes=5, fps=30):\n \"\"\"Generates a visualization of an agent acting according to a random\n policy in the given environment.\"\"\"\n display = Display(visible=0, size=(400, 300))\n display.start()\n filename = filename + \".mp4\"\n with imageio.get_writer(filename, fps=fps) as video:\n for _ in range(num_episodes):\n done = False\n observation = env.reset()\n video.append_data(env.render(mode='rgb_array'))\n while not done:\n action = env.action_space.sample()\n observation, reward, done, info = env.step(action)\n video.append_data(env.render(mode='rgb_array'))\n display.stop()\n return embed_mp4(filename)\n```\n\n\n```\nenv = gym.make(\"MsPacman-v0\")\ncreate_random_policy_video(env, \"video\", num_episodes=1)\n```\n\n WARNING:root:IMAGEIO FFMPEG_WRITER WARNING: input image is not divisible by macro_block_size=16, resizing from (210, 160) to (224, 160) to ensure video compatibility with most codecs and players. To prevent resizing, make your input image divisible by the macro_block_size or set the macro_block_size to None (risking incompatibility). You may also see a FFMPEG warning concerning speedloss due to data not being aligned.\n\n\n\n\n\n\n\n\n\n\nThis is pretty cool! In this example Gym gives us a great visualization of our agent playing PacMan (though not very well). \n\n### Gym basics\n\n#### Environments\n\nAs we would expect, an environment is defined by its state and action spaces.\n\nFirst, of all there are two types of Gym [spaces](https://gym.openai.com/docs/#spaces):\n- **`observation_space`**: defines the state space of the environment\n- **`action_space`**: defines the action space of the environment\n\nSpaces can be: \n- `gym.spaces.Discrete`: fixed range of n values \n- `gym.spaces.Box`: n-dimensional box\n\nYou can inspect the valid range for a `Box` by calling `Box.low()` and `Box.high()`.\n\n\n\n\n```\nenv = gym.make(\"MsPacman-v0\")\nprint(f'Action space: {env.action_space}')\nprint(f'Observation space: {env.observation_space}')\n```\n\n Action space: Discrete(9)\n Observation space: Box(210, 160, 3)\n\n\nWe can see here that for `MsPacman-v0`, the `action_space` consits 9 possible actions, and the `observation_space` is a 210 x 160 x 3 box (rgb-image). We can also extract these dimensions using `Discrete.n` and `Box.shape`. \n\n\n\n\n```\nprint(env.action_space.n)\nprint(env.observation_space.shape)\n```\n\n 9\n (210, 160, 3)\n\n\nIf we're curious, we can find the action meanings by calling,\n\n\n```\nprint(env.unwrapped.get_action_meanings())\n```\n\n ['NOOP', 'UP', 'RIGHT', 'LEFT', 'DOWN', 'UPRIGHT', 'UPLEFT', 'DOWNRIGHT', 'DOWNLEFT']\n\n\nI would've guessed that the `action_space` would be just up, down, left, right, but apparently this implementation includes combination actions as well. Theoretically you don't \"need\" to know these details about the environment you're using because your algorithm should learn a good policy given whatever the available action space is, but I think it's still nice to get a sense of.\n\n#### Key functions for interacting with the environment\n\n\nWe will mainly use three functions for interacting with the environment.\n\n**`observation = env.reset()`**\n\n- This function returns the starting state of an environment. We will call this function any time we want to start a new episode. \n\n\n**`observation, reward, done, info = env.step(action)`**\n\n- This function is how your agent takes actions in the environment; it defines the transition and reward function. It takes in an `action` as an argument, and returns the next `observation`(next_state), the `reward` (**float**), if the episode is `Done` (**bool**), and `info`, which we won't be using here but can contain helpful information for debugging. \n\n**`action = env.action_space.sample()`**\n- This is a helpful function for sampling a random action from the `action_space`. We will be using the $\\epsilon$-greedy exploration strategy, so we will use this function when we want to select a random action.\n\n\nIf we look back at the code for `create_random_policy_video()`, we can see how we used these three functions to get the data for the video. Stripping away all code for plotting, the main loop is:\n\n\n```\nnum_episodes = 1\nenv = gym.make(\"MsPacman-v0\")\n\nfor _ in range(num_episodes):\n observation = env.reset()\n done = False\n while not done:\n action = env.action_space.sample()\n observation, reward, done, info = env.step(action)\n```\n\nIn this notebook, we will generally replace the term `observation` with `state` becuase this is the wording we're more familiar with.\n\n## Implementing RL algorithms\n\nNow that we have all of the setup done, let's get to the fun part of actually implementing the algorithms!\n\n### CartPole environment\n\nFor our implementations we are going to use the `CartPole-v1` environment. This is a simple environment where both of our algorithms (Q-learning and DQN) will be able to learn a good policy within a reasonably short amount of time. \n\n\n\n\n```\nenv = gym.make(\"CartPole-v1\")\n```\n\nThe goal in this environment is to move the cart left or right in order to balance the pole so that it remains upright. \n\n\n```\ncreate_random_policy_video(env, \"video\", num_episodes= 10)\n```\n\n WARNING:root:IMAGEIO FFMPEG_WRITER WARNING: input image is not divisible by macro_block_size=16, resizing from (400, 600) to (400, 608) to ensure video compatibility with most codecs and players. To prevent resizing, make your input image divisible by the macro_block_size or set the macro_block_size to None (risking incompatibility). You may also see a FFMPEG warning concerning speedloss due to data not being aligned.\n\n\n\n\n\n\n\n\n\n\n\n```\nprint(f'Action space: {env.action_space}')\nprint(f'State space: {env.observation_space}')\n```\n\n Action space: Discrete(2)\n State space: Box(4,)\n\n\nThis is a dramatically simpler environment than the MsPacman environment. The `observation_space` is a 4 dimensional array, and there are 2 possible actions. The CartPole [documentation](https://github.com/openai/gym/wiki/CartPole-v0) tells us the meanings of the observation and action spaces.\n\n Observation: \n Type: Box(4)\n Num\tObservation Min Max\n 0\tCart Position -4.8 4.8\n 1\tCart Velocity -Inf Inf\n 2\tPole Angle -24 deg 24 deg\n 3\tPole Velocity At Tip -Inf Inf\n \n Actions:\n Type: Discrete(2)\n Num\tAction\n 0\tPush cart to the left\n 1\tPush cart to the right\n\nAn episode terminates when the pole falls more than 12 degrees from vertical, the cart position moves off-screen, or the number of steps within the episode exceeds 500.\n\n### Required functions for learning a policy\n\n\n```\nclass Agent:\n\n def create_model(self):\n \"\"\"This model will be used by the act() method to select actions, \n and will be updated during training \"\"\"\n pass\n\n def act(self, test=False):\n \"\"\"This function implements your policy and choose actions based on the \n current model. If test=True, actions chosen without exploration.\"\"\"\n pass\n\n def update_model(self):\n \"\"\"This function specifies how to update the model based on experience \n in the environment\"\"\"\n pass\n\n def train(self):\n \"\"\"The main loop for training the model by selecting actions, \n interacting with the environment, and updating the model\"\"\"\n pass\n```\n\nOnce we have a trained model, we can evaluate it's performance using a similar loop to the one above used to visualize the random policy. The only difference is that we replace the `env.action_space.sample()` function call with `agent.act()`.\n\n agent = Agent()\n agent = Agent.train()\n \n # run trained agent\n for _ in range(num_episodes):\n state = agent.env.reset()\n done = False\n while not done:\n action = agent.env.act(state, test=True)\n next_state, reward, done, info = env.step(action)\n\n### Evaluation functions\n\nNow we can use this loop to generate a video visualizing the learned policy.\n\n\n```\ndef create_learned_policy_video(agent, filename, num_episodes=5, fps=30):\n \"\"\"Generates a video of the given agent acting accoring to its learned \n policy for the specified number of episodes.\"\"\"\n display = Display(visible=0, size=(400, 300))\n display.start()\n filename = filename + \".mp4\"\n with imageio.get_writer(filename, fps=fps) as video:\n for _ in range(num_episodes):\n done = False\n state = agent.env.reset()\n video.append_data(agent.env.render(mode='rgb_array'))\n while not done:\n action = agent.act(state, test=True)\n state, reward, done, info = agent.env.step(action)\n video.append_data(agent.env.render(mode='rgb_array'))\n display.stop()\n return embed_mp4(filename)\n```\n\nWe will also want to evaluate the performance of the learned model. For evaluation trials we will not use $\\epsilon$-greedy exploration, but instead always choose the best action according to our learned policy.\n\n\n```\ndef evaluate_policy(agent, num_episodes=10):\n \"\"\"Runs the agent through the specified number of episodes and prints the \n average return. \"\"\"\n reward_history = []\n for _ in range(num_episodes):\n state = agent.env.reset()\n total_reward = 0\n done = False\n while not done:\n action = agent.act(state, test=True)\n next_state, reward, done, _ = agent.env.step(action)\n total_reward += reward\n state = next_state\n reward_history.append(total_reward) \n\n print(\"Exploit reward average: {}\".format(np.mean(reward_history).round(2)))\n```\n\n## Q-Learning\n\n\n\nTabular Q-learning stores and updates a Q-value estimate for each state-action pair, $Q(s,a)$. Each of these Q-values is stored in a look-up table.\n\n**Discretize environment observations**\n\nSince the cartpole environment has an `observation_space` with continuous values, the number of Q-values we need to store and update would quickly explode, and also not be very useful. To avoid this we are going to use a wrapper for the environment that transforms the `obervation_space` from a continuous-valued `Box` to a discrete-valued `Discrete`. I got this wrapper from Lillian Weng's Q-learning [implementation](https://github.com/lilianweng/deep-reinforcement-learning-gym/blob/master/playground/utils/wrappers.py) (understanding the details of how she implements this isn't important for what we're focusing on).\n\n\n```\nclass DiscretizedObservationWrapper(gym.ObservationWrapper):\n \"\"\"This wrapper converts a Box observation into a single integer.\n \"\"\"\n def __init__(self, env, n_bins=10, low=None, high=None):\n super().__init__(env)\n assert isinstance(env.observation_space, Box)\n\n low = self.observation_space.low if low is None else low\n high = self.observation_space.high if high is None else high\n\n self.n_bins = n_bins\n self.val_bins = [np.linspace(l, h, n_bins + 1) for l, h in\n zip(low.flatten(), high.flatten())]\n self.observation_space = Discrete(n_bins ** low.flatten().shape[0])\n\n def _convert_to_one_number(self, digits):\n return sum([d * ((self.n_bins + 1) ** i) for i, d in enumerate(digits)])\n\n def observation(self, observation):\n digits = [np.digitize([x], bins)[0]\n for x, bins in zip(observation.flatten(), self.val_bins)]\n return self._convert_to_one_number(digits)\n```\n\n### Algorithm\n\nThe next step is to implement the q-learning algorithm. The method names give you the skeleton of the implementation but the content is left for you to fill in. We've inserted detailed comments to guide your implementation. We've also left some code in that is not essential to the algorithm (e.g decaying the epsilon parameter each step, keeping track of reward history).\n\nYou need to fill in the content for three methods:\n\n- `create_model()` - filled in already\n- `act()`\n- `update_model()`\n- `train()`\n\n\n\n\n#### create_model()\n\nWe've left the code for `create_model()` filled in because it is only creating a dictionary for storing the Q values. We are using `defaultdict (float)` rather than `dict` for a more efficient implementation. This automatically initializes any key entry to 0.0, rather than returning `KeyError`. \n\n\n```\n# define environement\nenv = gym.make(\"CartPole-v1\")\nenv_discrete = DiscretizedObservationWrapper(\n env,\n n_bins=8,\n low=np.array([-2.4, -2.0, -0.42, -3.5]),\n high=np.array([2.4, 2.0, 0.42, 3.5])\n)\n\n# get example state-action pair\nstate = env_discrete.reset()\naction = env_discrete.action_space.sample()\n\n# define defaultdict and query the state-action pair\nexample = defaultdict(float)\nexample[state, action] # *no KeyError*\n```\n\n\n\n\n 0.0\n\n\n\n#### act() \n\nFor our implementation, we will be using the $\\epsilon$-greedy exploration policy.\n\n\\begin{equation}\n a =\n \\begin{cases}\n \\text{random} & \\text{with probability $\\epsilon$}\\\\\n \\arg\\max_a Q(s,a) & \\text{otherwise}\\\\\n \\end{cases} \n\\end{equation}\n\n#### update_model()\n\nThis function should update the Q-value estimate using the Q-learning update rule based on the $(s,a,r,s'\\text{done})$.\n\n$$ Q(s,a) \\leftarrow Q(s,a) + \\alpha \\left[r_t + \\gamma \\max_{a'} Q(s',a') - Q(s,a) \\right]$$\n\nIf the state is terminal (`done=True`) the update will be \n$$ Q(s,a) \\leftarrow Q(s,a) + \\alpha \\left[r_t - Q(s,a) \\right]$$\n\n#### train()\n\nThis function will run the main training loop. Here is the pseudocode for the Q-learning algorithm.\n\n create model (initialize q_values)\n for n_episodes\n initialize state\n while not done\n select action according to policy\n execute action; observe reward and next_state\n update model\n\nThis function will be used to train the agent as follows:\n\n agent = Agent(env)\n agent = Agent.train()\n\n\n\nRemember, these are the environment api calls you will need to use in your implementation.\n- `observation = env.reset()`\n- `observation, reward, done, info = env.step(action)`\n- `action = env.action_space.sample()`\n\n#### Implementation\n\n\n```\nclass QLearning:\n\n def __init__(self, env, gamma=0.9, alpha=0.5, epsilon=0.99, \n epsilon_decay=0.9999, epsilon_min=0.1):\n self.env = env\n self.gamma = gamma\n self.alpha = alpha\n self.epsilon = epsilon\n self.epsilon_decay = epsilon_decay\n self.epsilon_min = epsilon_min\n self.actions = range(self.env.action_space.n)\n\n def create_model(self):\n \"\"\"\"For Q-learning the model is simply a dictionary for storing the\n tabular Q values.\"\"\"\n self.Q = defaultdict(float)\n\n def act(self, state, test=False):\n \"\"\"Choose action based on your current model using epsilon-greedy\n exploration\"\"\"\n # update epsilon\n self.epsilon *= self.epsilon_decay\n self.epsilon = max(self.epsilon_min, self.epsilon)\n epsilon = 0 if test else self.epsilon\n # take a random action with probability epsilon\n if np.random.random() < epsilon:\n return self.env.action_space.sample()\n # Pick the action with highest q value.\n qvals = {action: self.Q[state, action] for action in self.actions}\n max_q = max(qvals.values())\n # In case multiple actions have the same maximum q value.\n actions_with_max_q = [action for action, q in qvals.items() if q == max_q]\n action = np.random.choice(actions_with_max_q)\n return action\n\n def update_model(self, state, action, reward, next_state, done):\n # get max q value in next state\n max_q_next = max([self.Q[next_state, action] for a in self.actions]) \n # Do not include the next state's value if currently at the terminal state.\n max_q_next = max_q_next * (1.0 - done)\n # Update q value of current state-action pair\n self.Q[state, action] += self.alpha * \\\n (reward + self.gamma * max_q_next - self.Q[state, action])\n\n def train(self, num_episodes=20):\n \"\"\"This is the main training loop for interacting with the environment \n and updating your model. We've left in code for storing training history.\"\"\"\n self.reward_history = []\n # create model\n self.create_model()\n for episode in range(num_episodes):\n total_reward = 0.0\n done = False\n # initialize state\n state = self.env.reset()\n while not done:\n # select action according to policy\n action = self.act(state)\n # execute action\n next_state, reward, done, _ = self.env.step(action)\n # update model\n self.update_model(state, action, reward, next_state, done)\n # keep track of total reward in episode\n total_reward += reward\n # update state\n state = next_state\n \n # save total reward from epsisode and print training progress \n self.reward_history.append(total_reward) \n if episode % 500 == 0:\n print(\"episode {}: {} average reward\".format(\n episode, \n np.mean(self.reward_history[max(0,episode-500):episode+1]).round(2)))\n\n```\n\n### Training an agent\n\nOnce you have your algorithm implemented, let's train an agent!\n\n\n```\nenv = gym.make('CartPole-v1')\nenv_discrete = DiscretizedObservationWrapper(\n env,\n n_bins=8,\n low=np.array([-2.4, -2.0, -0.42, -3.5]),\n high=np.array([2.4, 2.0, 0.42, 3.5])\n)\n\nseed = 0\nenv.seed(seed)\nenv.action_space.seed(seed)\nnp.random.seed(seed)\nrandom.seed(seed)\n\nqlearning_agent = QLearning(env_discrete)\nqlearning_agent.train(num_episodes=5000)\n```\n\n episode 0: 52.0 average reward\n episode 500: 42.61 average reward\n episode 1000: 79.64 average reward\n episode 1500: 94.46 average reward\n episode 2000: 87.87 average reward\n episode 2500: 96.78 average reward\n episode 3000: 101.91 average reward\n episode 3500: 118.41 average reward\n episode 4000: 117.37 average reward\n episode 4500: 113.55 average reward\n\n\n\n```\n# visualize total reward per episode across training\nplt.figure(figsize=(7,4))\nplt.plot(qlearning_agent.reward_history, alpha=.3, color='teal', label='raw')\nplt.plot(np.convolve(qlearning_agent.reward_history, np.ones((50,))/50, mode='valid'), color='purple', label='smoothed')\nplt.xlabel('episode #', fontsize=15)\nplt.ylabel('total reward per episode', fontsize=15)\nplt.legend()\nplt.show()\n```\n\n###Evaluating the agent\n\nFirst, let's see what the average reward is across 100 trials when the agent is exploiting its learned policy (not using $\\epsilon$-greedy exploration).\n\nNow, let's visualize the agent acting according to its learned policy.\n\n\n```\nevaluate_policy(qlearning_agent, num_episodes=100)\n```\n\n Exploit reward average: 121.67\n\n\n\n```\ncreate_learned_policy_video(qlearning_agent, \"video\", num_episodes=1)\n```\n\n WARNING:root:IMAGEIO FFMPEG_WRITER WARNING: input image is not divisible by macro_block_size=16, resizing from (400, 600) to (400, 608) to ensure video compatibility with most codecs and players. To prevent resizing, make your input image divisible by the macro_block_size or set the macro_block_size to None (risking incompatibility). You may also see a FFMPEG warning concerning speedloss due to data not being aligned.\n\n\n\n\n\n\n\n\n\n\nWoo, it learned something! This is definitely an improvement from random, but it's certainly not optimal (a reward of 500 is optimal). This agent could get better with more training but it would probably take a long time for it to reach optimal performance.\n\n### What is the model learning?\n\nWe can tell that the model has learned something, but it would be nice to get some idea of what it's learned. In order to get a sense of this, we can visualize the learned Q-values across a set of states. For this example, we are going to plot Q-values as a function of pole velocity, while the cart position, cart velocity, and pole angle are all 0 (the pole is in the center, not moving, and upright). \n\nIntuitively, the agent should have learned to push the cart right if the pole velocity is to the right (>0), and to push the cart left if the pole velocity is to the left (<0). \n\n\n\n```\nn_obs = 40\nobs_array = np.zeros((n_obs, env.observation_space.shape[0]))\nobs_array[:,0] = 0\nobs_array[:,1] = 0\nobs_array[:,2] = 0\nobs_array[:,3] = np.linspace(-5, 5, n_obs)\n# run model\nq_values = np.zeros((n_obs, env.action_space.n))\nfor i, obs in enumerate(obs_array):\n obs_disctete = env_discrete.observation(obs)\n q_values[i] = [qlearning_agent.Q[obs_disctete, action] for action in qlearning_agent.actions]\n# visualize results\nplt.figure(figsize=(8,5))\nplt.plot(obs_array[:,3], q_values[:,0], color='purple', label='push cart left', linewidth=2)\nplt.plot(obs_array[:,3], q_values[:,1], color='green', label='push cart right', linewidth=2)\nplt.vlines(0, q_values.min(), q_values.max(), linestyle='--', color='dimgray')\nplt.xlabel('Pole Velocity', fontsize=15)\nplt.ylabel('Q value', fontsize=15)\nplt.title('Cart Position=0, Cart Velocity=0, Pole Angle=0', fontsize=15)\nplt.legend(fontsize=13)\nplt.show()\n```\n\nIt does what we expect! The Q-values for a=right are larger than the Q-values for a=left when the pole velocity is greater than 0, and vice versa for when the pole velocity is less than 0. \n\n## Deep Q-Networks (DQN)\n\n\nNow that we've implemented Q-learning, let's move on to implementing DQNs!\n\n### Algorithm\n\nSimilar to Q-Learning, the method names for the DQN class give you the skeleton of the implementation, while the content is left for you to fill in. For DQNs you will need to write a few more functions than you needed for Q-Learning.\n\nYou need to fill in the content for six functions:\n\n- `create_model` - filled in already\n- `act()` \n- `remember()`\n- `update_model()`\n- `update_target()` - fillded in already\n- `train()`\n\n\n\n\n#### create_model()\n\nFor this implementation, we're going to use a two-layer densely connected network. This network will take in a state as input and output a Q-value estimate for each action within this state. Consequently, the input dim is the same as the `observation_space` shape (4 for the CartPole environment), and the output dim is the same as the `action_space` shape (2 for the CartPole environment). \n\nWe've left the code for `create_model()` filled in because it is largely determined by learning TensorFlow syntax, which isn't our focus here. We will use ReLu activation function and the Adam optimizer.\n\nWe will use mean squared error loss, as specified by the DQN loss function.\n\n def create_model(self):\n model = Sequential()\n model.add(Dense(24, input_dim=self.state_shape[0], activation=\"relu\"))\n model.add(Dense(16, activation=\"relu\"))\n model.add(Dense(self.env.action_space.n))\n model.compile(loss=\"mean_squared_error\", optimizer=Adam(lr=self.learning_rate))\n return model\n\n\n\n\n#### act() \n\nWe will again use the $\\epsilon$-greedy exploration policy. \n\n\\begin{equation}\n a =\n \\begin{cases}\n \\text{random} & \\text{with probability $\\epsilon$}\\\\\n \\arg\\max_a Q(s,a;\\theta) & \\text{otherwise}\\\\\n \\end{cases} \n\\end{equation}\n\nTo get the Q-values from your Q-network, you can run\n\n q_values = self.model.predict(state)\n\n#### remember()\n\nAfter each step, we need to store the $(s, a,r,s', \\text{done})$ experience in the replay memory. In this implementation we will store memories in a `deque` with a specified maximum legnth (memory capacity).\n\n\n```\nreplay_memory = deque(maxlen=5)\nprint(replay_memory)\n\nfor i in range(7):\n replay_memory.append(i)\n print(replay_memory)\n\n# in your implementation you will append the experience\n# [state, action, reward, next_state, done] instead of i\n```\n\n deque([], maxlen=5)\n deque([0], maxlen=5)\n deque([0, 1], maxlen=5)\n deque([0, 1, 2], maxlen=5)\n deque([0, 1, 2, 3], maxlen=5)\n deque([0, 1, 2, 3, 4], maxlen=5)\n deque([1, 2, 3, 4, 5], maxlen=5)\n deque([2, 3, 4, 5, 6], maxlen=5)\n\n\n#### update_model()\n\nTo update the model, you'll need to:\n\n1. sample a batch of experiences $(s_j, a_j, r_j, s_j', \\text{done}_j)$ from memory\n2. calculate the target output (`done` indicates if $s$ is terminal)\n\n\\begin{equation}\n y_j =\n \\begin{cases}\n r_j + \\gamma \\max_{a'} Q(s_j',a'; \\theta^-) & \\text{if $s$ is not terminal}\\\\\n r_j & \\text{if $s$ is terminal}\\\\\n \\end{cases} \n\\end{equation}\n\n3. Perform gradient descent step according to\n\n$$ L(\\theta) = \\left\\langle \\big( y_j - Q(s_j,a_j; \\theta) \\big)^2\\right\\rangle_{(s,a,r,s') \\sim Uniform(Memory)}$$\n\nFor the third step, the TensorFlow code you will need is\n\n model.fit(batch_states, batch_target, epochs=1, verbose=0)\n\n\n**NOTE**\n\nThe `batch_target` must be the same dimensions as the model output. This means you must have a target for every action for each input state in your batch of experiences $(s_j,a_j,r_j,s_j')$. For each action $a$ that is not in the experience batch, use the current output of the target model as the target value, $Q(s,a;\\theta^-)$. Therefore, for each state $s_j$, \n\n\\begin{equation}\n \\text{target} =\n \\begin{cases}\n y_j & \\text{if $a$ is in experience batch $(s_j, a_j, r_j, s_j')$}\\\\\n Q(s,a;\\theta^-) & \\text{if $a$ is NOT in experience batch $(s_j, a_j, r_j, s_j')$}\\\\\n \\end{cases} \n\\end{equation}\n\n**NOTE 2**\n\nHere is a helpful line of code for reformating samples from memory, each of which will be a list `[state, action, reward, next_state, done]`, to a set of `np.array`s with dim (n_batch x __).\n\n batch_states, batch_actions, batch_rewards, batch_next_states, batch_done = map(np.asarray, zip(*memory_samples))\n\n\n#### update_tartget()\n\nThis function is used to set the target network weights equal to the model network weights. This is only done periodically thoughout training, which reduces variance in the gradient across steps and stabilizes training. \n\nWe've left the code for `update_target()` filled in because, again, it is largely just Tensorflow syntax. You'll have to use this function appropriately within the main training loop.\n\n def update_target(self):\n weights = self.model.get_weights()\n self.target_model.set_weights(weights)\n\n#### train()\n\nThis function will run the main training loop. Here is the pseudocode for the DQN algorithm.\n\n initialize Q-network (create model)\n initialize target network (create model and set weights equal to Q-network)\n for n_episodes\n initialize state\n while not done\n select action according to policy\n execute action; observe reward and next_state\n add experience (state, action, reward, next_state, done) to memory\n sample batch of (state, action, reward, next_state, done) experiences from memory and update model\n every C steps, update target model\n\n\n\nSame as for Q-learning, the Gym api calls you will need are:\n- `observation = env.reset()`\n- `observation, reward, done, info = env.step(action)`\n- `action = env.action_space.sample()`\n\nThe Tensorflow api calls you will need are: \n- `model_output = model.predict(model_input)`\n- `model.fit(model_input, model_target, epochs=1, verbose=0)`\n\n#### Implementation\n\n\n```\nclass DQN:\n def __init__(self, env, memory_cap=1000, gamma=0.9, epsilon=0.99, \n epsilon_decay=0.995, epsilon_min=0.01, learning_rate=0.005, \n batch_size=32, C=20):\n\n self.env = env\n self.memory = deque(maxlen=memory_cap)\n self.state_shape = env.observation_space.shape\n\n self.gamma = gamma\n self.epsilon = epsilon \n self.epsilon_min = epsilon_min\n self.epsilon_decay = epsilon_decay \n self.learning_rate = learning_rate\n self.batch_size = batch_size\n self.C = C\n\n def create_model(self):\n \"\"\"We will use a two-layer perceptron. The input dim must equal the\n state space dim and the output dim must equal the action space dim, \n but you can play around with the size of the hidden layers. For DQNs,\n we need mean squared error loss.\"\"\"\n model = Sequential()\n model.add(Dense(24, input_dim=self.state_shape[0], activation=\"relu\"))\n model.add(Dense(16, activation=\"relu\"))\n model.add(Dense(self.env.action_space.n))\n model.compile(loss=\"mean_squared_error\", optimizer=Adam(lr=self.learning_rate))\n return model\n \n def act(self, state, test=False):\n \"\"\"Choose action based on your current model using epsilon-greedy\n exploration\"\"\"\n # update epsilon\n self.epsilon *= self.epsilon_decay\n self.epsilon = max(self.epsilon_min, self.epsilon)\n epsilon = 0.01 if test else self.epsilon \n # take random action with probability epsilon\n if np.random.random() < epsilon:\n return self.env.action_space.sample()\n # reshape state to feed into model (tensorflow thing, shape must be\n # (1, input_dim), not (input_dim,).)\n state = state.reshape((1, self.state_shape[0]))\n # get q_values from model\n q_values = self.model.predict(state)[0]\n # get action (argmax of Q-values)\n action = np.argmax(q_values)\n return action\n\n def remember(self, state, action, reward, new_state, done):\n \"\"\"Append experience to memory\"\"\"\n self.memory.append([state, action, reward, new_state, done])\n\n def update_model(self):\n \"\"\"This function updates the q-network model. You must 1) sample a\n batch of experiences from the replay memory 2) calculate the target for\n each expereince, 3) update the model by calling model.fit()\"\"\"\n # only update model once have sufficient number of experiences in memory\n if len(self.memory) < self.batch_size:\n return\n # sample a batch of experiences from memory\n memory_samples = random.sample(self.memory, self.batch_size)\n # reformat samples\n batch_states, batch_actions, batch_rewards, batch_next_states, batch_done = map(np.asarray, zip(*memory_samples))\n # get target model predictions for current state\n batch_target = self.target_model.predict(batch_states)\n # get target model predictions for future state\n q_future = self.target_model.predict(batch_next_states).max(axis=1)\n # for actions within experience batch, replace target model predictions \n # with the target value (r + gamma * max Q(s,a) if s not terminal, r is s terminal)\n batch_target[range(self.batch_size), batch_actions] = batch_rewards + (1 - batch_done) * q_future * self.gamma\n self.model.fit(batch_states, batch_target, epochs=1, verbose=0)\n\n def update_target(self):\n \"\"\"\"Sets target weights equal to model weights.\"\"\"\n weights = self.model.get_weights()\n self.target_model.set_weights(weights)\n\n def train(self, num_episodes=50):\n \"\"\"This function implements the main training loop.\"\"\"\n # keep track of total reward per episode\n self.reward_history = []\n # initialize model and target model\n self.model = self.create_model()\n self.target_model = self.create_model()\n self.target_model.set_weights(self.model.get_weights())\n # we need to keep track of steps now so we can update the model every \n # C steps\n step = 0\n # interact with environment and update model\n for episode in range(num_episodes):\n total_reward = 0\n done = False\n state = self.env.reset()\n while not done:\n # select action according to policy\n action = self.act(state)\n # execute action\n next_state, reward, done, _ = self.env.step(action)\n # add experience to memory\n self.remember(state, action, reward, next_state, done)\n # update model (sample batch of experiences for update)\n self.update_model()\n \n total_reward += reward\n step += 1\n state = next_state\n if step % self.C:\n self.update_target()\n \n # save total reward from epsisode and print training progress \n self.reward_history.append(total_reward) \n print(\"episode {}: {} reward\".format(episode, total_reward))\n```\n\n### Training an agent\n\n\n```\nenv = gym.make('CartPole-v1') \n\nseed = 2\n\nenv.seed(seed)\nenv.action_space.seed(seed)\nnp.random.seed(seed)\nrandom.seed(seed)\ntf.random.set_seed(seed)\n\ndqn_agent = DQN(env, batch_size=32)\ndqn_agent.train(num_episodes=35)\n```\n\n episode 0: 12.0 reward\n episode 1: 14.0 reward\n episode 2: 19.0 reward\n episode 3: 19.0 reward\n episode 4: 10.0 reward\n episode 5: 10.0 reward\n episode 6: 21.0 reward\n episode 7: 16.0 reward\n episode 8: 30.0 reward\n episode 9: 11.0 reward\n episode 10: 11.0 reward\n episode 11: 50.0 reward\n episode 12: 26.0 reward\n episode 13: 23.0 reward\n episode 14: 40.0 reward\n episode 15: 113.0 reward\n episode 16: 24.0 reward\n episode 17: 43.0 reward\n episode 18: 38.0 reward\n episode 19: 8.0 reward\n episode 20: 42.0 reward\n episode 21: 64.0 reward\n episode 22: 54.0 reward\n episode 23: 70.0 reward\n episode 24: 91.0 reward\n episode 25: 30.0 reward\n episode 26: 121.0 reward\n episode 27: 213.0 reward\n episode 28: 500.0 reward\n episode 29: 500.0 reward\n episode 30: 364.0 reward\n episode 31: 401.0 reward\n episode 32: 342.0 reward\n episode 33: 309.0 reward\n episode 34: 290.0 reward\n\n\n\n```\n# visualize total reward per episode across training\nplt.figure(figsize=(7,4))\nplt.plot(dqn_agent.reward_history, alpha=1, color='purple', label='raw')\nplt.xlabel('episode #', fontsize=15)\nplt.ylabel('total reward per episode', fontsize=15)\nplt.title('DQN Agent')\nplt.show()\n```\n\n### Evaluating the agent\n\n\n```\nevaluate_policy(dqn_agent, num_episodes=1)\n```\n\n Exploit reward average: 227.0\n\n\n\n```\ncreate_learned_policy_video(dqn_agent, \"video\", num_episodes=1)\n```\n\n xdpyinfo was not found, X start can not be checked! Please install xdpyinfo!\n WARNING:root:IMAGEIO FFMPEG_WRITER WARNING: input image is not divisible by macro_block_size=16, resizing from (400, 600) to (400, 608) to ensure video compatibility with most codecs and players. To prevent resizing, make your input image divisible by the macro_block_size or set the macro_block_size to None (risking incompatibility). You may also see a FFMPEG warning concerning speedloss due to data not being aligned.\n\n\n\n\n\n\n\n\n\n\nYay, it learned the task! This is also not quite optimal, but its only been 35 sessions! This is a huge change from our Q-learning agent, which took 5000 episode to reach less than the perfomance of our DQN agents. This tells us how useful it is to keep the continous-valued inputs and use a neural network to appromimate the Q-value function. \n\n### What is the model learning?\n\n\n```\nn_obs = 40\nobs_array = np.zeros((n_obs, env.observation_space.shape[0]))\nobs_array[:,0] = 0\nobs_array[:,1] = 0\nobs_array[:,2] = 0\nobs_array[:,3] = np.linspace(-5, 5, n_obs)\n# run model\nq_values = dqn_agent.model.predict(obs_array)\n# visualize results\nplt.figure(figsize=(8,5))\nplt.plot(obs_array[:,3], q_values[:,0], color='purple', label='push cart left', linewidth=2)\nplt.plot(obs_array[:,3], q_values[:,1], color='green', label='push cart right', linewidth=2)\nplt.vlines(0, q_values.min(), q_values.max(), linestyle='--', color='dimgray')\nplt.xlabel('Pole Velocity', fontsize=15)\nplt.ylabel('Q value', fontsize=15)\nplt.title('Cart Position=0, Cart Velocity=0, Pole Angle=0', fontsize=15)\nplt.legend(fontsize=13)\nplt.show()\n```\n\nWe can see here that our agent learned the switch between pushing the car right and left right at 0. It was able to learn this behavior from few examples because even if it didn't encounter every state along this continuum, it can generalize from nearby experiences. Pretty cool!\n\n## Summary\n\nTo summarize, in this notebook we implemented the Q-Learning and DQN algorithms and trained an agent on each. Moving from Q-learning to DQNs gave us similar performance (slightly better for the DQN) after 5000 episodes for Q-learning and only 35 episodes for DQNs. This is a pretty amazing improvement, and gives us a hands-on feel for how much you gain by using neural networks for function approximation. \n\n\nWe focused on DQNs here, but Policy Gradient algorithms are a very effective class of algorithms for RL tasks. Our \"What is the model learning?\" plot may give us a hint for why this is. DQNs need to approximate the Q-value for each action, and must get the highest Q-value right in order for the policy to be correct. If the Q-values are very simialar this might by tricky and lead to noise in selecting the best action. Perhaps it is more simple to directly learn the action space, as Policy Gradients do. \n\nThere is a lot to learn in DeepRL, and this at least gives us a start!\n\n## Citations\n\nA lot of the code in the Colab notebook was inpired by some really helpful posts from other people:\n\n- Lillian Weng's [deep-reinforcement-learning-gym](https://github.com/lilianweng/deep-reinforcement-learning-gym) repo (adapted her Q-learning implementation)\n- Anita Hu's [TF2-RL](https://github.com/anita-hu/TF2-RL) repo (adapted her DQN implementation)\n- TF [tutorial](https://www.tensorflow.org/agents/tutorials/1_dqn_tutorial) for training DQN with TF-Agents (adapted TF-Agents visualization functions for Gym)\n", "meta": {"hexsha": "2d3f136d948faa9e16678ad5fe5a288b917e6a35", "size": 512918, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "DeepRL/DeepRL-solutions.ipynb", "max_stars_repo_name": "john-vastola/ML-from-scratch-seminar", "max_stars_repo_head_hexsha": "5df9db96ab5012929403fa9a90545b142a721612", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 8, "max_stars_repo_stars_event_min_datetime": "2019-01-24T16:43:18.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-04T22:56:02.000Z", "max_issues_repo_path": "DeepRL/DeepRL-solutions.ipynb", "max_issues_repo_name": "john-vastola/ML-from-scratch-seminar", "max_issues_repo_head_hexsha": "5df9db96ab5012929403fa9a90545b142a721612", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "DeepRL/DeepRL-solutions.ipynb", "max_forks_repo_name": "john-vastola/ML-from-scratch-seminar", "max_forks_repo_head_hexsha": "5df9db96ab5012929403fa9a90545b142a721612", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 5, "max_forks_repo_forks_event_min_datetime": "2019-02-12T23:11:13.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-06T23:23:29.000Z", "avg_line_length": 228.573083779, "max_line_length": 149769, "alphanum_fraction": 0.8926280614, "converted": true, "num_tokens": 11392, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.6619228625116081, "lm_q1q2_score": 0.42405151886319636}} {"text": "\n\n# Tutorial 3: Learning to Act: Q-Learning\n**Week 3, Day 4: Reinforcement Learning**\n\n**By Neuromatch Academy**\n\n__Content creators:__ Marcelo Mattar and Eric DeWitt with help from Byron Galbraith\n\n__Content reviewers:__ Ella Batty, Matt Krause and Michael Waskom\n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n

                                        \n\n---\n\n# Tutorial Objectives\n \n*Estimated timing of tutorial: 40 min*\n\nIn this tutorial you will learn how to act in the more realistic setting of sequential decisions, formalized by Markov Decision Processes (MDPs). In a sequential decision problem, the actions executed in one state not only may lead to immediate rewards (as in a bandit problem), but may also affect the states experienced next (unlike a bandit problem). Each individual action may therefore affect affect all future rewards. Thus, making decisions in this setting requires considering each action in terms of their expected **cumulative** future reward.\n\nWe will consider here the example of spatial navigation, where actions (movements) in one state (location) affect the states experienced next, and an agent might need to execute a whole sequence of actions before a reward is obtained.\n\nBy the end of this tutorial, you will learn\n* what grid worlds are and how they help in evaluating simple reinforcement learning agents\n* the basics of the Q-learning algorithm for estimating action values\n* how the concept of exploration and exploitation, reviewed in the bandit case, also applies to the sequential decision setting\n\n\n```python\n# @title Tutorial slides\n\n# @markdown These are the slides for all videos in this tutorial.\nfrom IPython.display import IFrame\nIFrame(src=f\"https://mfr.ca-1.osf.io/render?url=https://osf.io/2jzdu/?direct%26mode=render%26action=download%26mode=render\", width=854, height=480)\n```\n\n---\n# Setup\n\n\n```python\n# Imports\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.signal import convolve as conv\n```\n\n\n```python\n#@title Figure Settings\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/course-content/master/nma.mplstyle\")\n```\n\n\n```python\n#@title Plotting Functions\n\ndef plot_state_action_values(env, value, ax=None):\n \"\"\"\n Generate plot showing value of each action at each state.\n \"\"\"\n if ax is None:\n fig, ax = plt.subplots()\n\n for a in range(env.n_actions):\n ax.plot(range(env.n_states), value[:, a], marker='o', linestyle='--')\n ax.set(xlabel='States', ylabel='Values')\n ax.legend(['R','U','L','D'], loc='lower right')\n\n\ndef plot_quiver_max_action(env, value, ax=None):\n \"\"\"\n Generate plot showing action of maximum value or maximum probability at\n each state (not for n-armed bandit or cheese_world).\n \"\"\"\n if ax is None:\n fig, ax = plt.subplots()\n\n X = np.tile(np.arange(env.dim_x), [env.dim_y,1]) + 0.5\n Y = np.tile(np.arange(env.dim_y)[::-1][:,np.newaxis], [1,env.dim_x]) + 0.5\n which_max = np.reshape(value.argmax(axis=1), (env.dim_y,env.dim_x))\n which_max = which_max[::-1,:]\n U = np.zeros(X.shape)\n V = np.zeros(X.shape)\n U[which_max == 0] = 1\n V[which_max == 1] = 1\n U[which_max == 2] = -1\n V[which_max == 3] = -1\n\n ax.quiver(X, Y, U, V)\n ax.set(\n title='Maximum value/probability actions',\n xlim=[-0.5, env.dim_x+0.5],\n ylim=[-0.5, env.dim_y+0.5],\n )\n ax.set_xticks(np.linspace(0.5, env.dim_x-0.5, num=env.dim_x))\n ax.set_xticklabels([\"%d\" % x for x in np.arange(env.dim_x)])\n ax.set_xticks(np.arange(env.dim_x+1), minor=True)\n ax.set_yticks(np.linspace(0.5, env.dim_y-0.5, num=env.dim_y))\n ax.set_yticklabels([\"%d\" % y for y in np.arange(0, env.dim_y*env.dim_x,\n env.dim_x)])\n ax.set_yticks(np.arange(env.dim_y+1), minor=True)\n ax.grid(which='minor',linestyle='-')\n\n\ndef plot_heatmap_max_val(env, value, ax=None):\n \"\"\"\n Generate heatmap showing maximum value at each state\n \"\"\"\n if ax is None:\n fig, ax = plt.subplots()\n\n if value.ndim == 1:\n value_max = np.reshape(value, (env.dim_y,env.dim_x))\n else:\n value_max = np.reshape(value.max(axis=1), (env.dim_y,env.dim_x))\n value_max = value_max[::-1,:]\n\n im = ax.imshow(value_max, aspect='auto', interpolation='none', cmap='afmhot')\n ax.set(title='Maximum value per state')\n ax.set_xticks(np.linspace(0, env.dim_x-1, num=env.dim_x))\n ax.set_xticklabels([\"%d\" % x for x in np.arange(env.dim_x)])\n ax.set_yticks(np.linspace(0, env.dim_y-1, num=env.dim_y))\n if env.name != 'windy_cliff_grid':\n ax.set_yticklabels(\n [\"%d\" % y for y in np.arange(\n 0, env.dim_y*env.dim_x, env.dim_x)][::-1])\n return im\n\n\ndef plot_rewards(n_episodes, rewards, average_range=10, ax=None):\n \"\"\"\n Generate plot showing total reward accumulated in each episode.\n \"\"\"\n if ax is None:\n fig, ax = plt.subplots()\n\n smoothed_rewards = (conv(rewards, np.ones(average_range), mode='same')\n / average_range)\n\n ax.plot(range(0, n_episodes, average_range),\n smoothed_rewards[0:n_episodes:average_range],\n marker='o', linestyle='--')\n ax.set(xlabel='Episodes', ylabel='Total reward')\n\n\ndef plot_performance(env, value, reward_sums):\n fig, axes = plt.subplots(nrows=2, ncols=2, figsize=(16, 12))\n plot_state_action_values(env, value, ax=axes[0,0])\n plot_quiver_max_action(env, value, ax=axes[0,1])\n plot_rewards(n_episodes, reward_sums, ax=axes[1,0])\n im = plot_heatmap_max_val(env, value, ax=axes[1,1])\n fig.colorbar(im)\n```\n\n---\n# Section 1: Markov Decision Processes\n\n\n```python\n# @title Video 1: MDPs and Q-learning\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = 'https://player.bilibili.com/player.html?bvid={0}&page={1}'.format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=\"BV1ft4y1Q7bX\", width=854, height=480, fs=1)\n print('Video available at https://www.bilibili.com/video/{0}'.format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=\"8yvwMrUQJOU\", width=854, height=480, fs=1, rel=0)\n print('Video available at https://youtube.com/watch?v=' + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n**Grid Worlds**\n\nAs pointed out, bandits only have a single state and immediate rewards for our actions. Many problems we are interested in have multiple states and delayed rewards, i.e. we won't know if the choices we made will pay off over time, or which actions we took contributed to the outcomes we observed.\n\nIn order to explore these ideas, we turn the a common problem setting: the grid world. Grid worlds are simple environments where each state corresponds to a tile on a 2D grid, and the only actions the agent can take are to move up, down, left, or right across the grid tiles. The agent's job is almost always to find a way to a goal tile in the most direct way possible while overcoming some maze or other obstacles, either static or dynamic.\n\nFor our discussion we will be looking at the classic Cliff World, or Cliff Walker, environment. This is a 4x10 grid with a starting position in the lower-left and the goal position in the lower-right. Every tile between these two is the \"cliff\", and should the agent enter the cliff, they will receive a -100 reward and be sent back to the starting position. Every tile other than the cliff produces a -1 reward when entered. The goal tile ends the episode after taking any action from it.\n\n\n\nGiven these conditions, the maximum achievable reward is -11 (1 up, 9 right, 1 down). Using negative rewards is a common technique to encourage the agent to move and seek out the goal state as fast as possible.\n\n---\n# Section 2: Q-Learning\n\n*Estimated timing to here from start of tutorial: 20 min*\n\nNow that we have our environment, how can we solve it? \n\nOne of the most famous algorithms for estimating action values (aka Q-values) is the Temporal Differences (TD) **control** algorithm known as *Q-learning* (Watkins, 1989). \n\n\\begin{align}\nQ(s_t,a_t) \\leftarrow Q(s_t,a_t) + \\alpha \\big(r_t + \\gamma\\max_\\limits{a} Q(s_{t+1},a_{t+1}) - Q(s_t,a_t)\\big)\n\\end{align}\n\nwhere $Q(s,a)$ is the value function for action $a$ at state $s$, $\\alpha$ is the learning rate, $r$ is the reward, and $\\gamma$ is the temporal discount rate.\n\nThe expression $r_t + \\gamma\\max_\\limits{a} Q(s_{t+1},a_{t+1})$ is referred to as the TD target while the full expression \n\\begin{align}\nr_t + \\gamma\\max_\\limits{a} Q(s_{t+1},a_{t+1}) - Q(s_t,a_t),\n\\end{align}\ni.e. the difference between the TD target and the current Q-value, is referred to as the TD error, or reward prediction error.\n\nBecause of the max operator used to select the optimal Q-value in the TD target, Q-learning directly estimates the optimal action value, i.e. the cumulative future reward that would be obtained if the agent behaved optimally, regardless of the policy currently followed by the agent. For this reason, Q-learning is referred to as an **off-policy** method.\n\n## Coding Exercise 2: Implement the Q-learning algorithm\n\nIn this exercise you will implement the Q-learning update rule described above. It takes in as arguments the previous state $s_t$, the action $a_t$ taken, the reward received $r_t$, the current state $s_{t+1}$, the Q-value table, and a dictionary of parameters that contain the learning rate $\\alpha$ and discount factor $\\gamma$. The method returns the updated Q-value table. For the parameter dictionary, $\\alpha$: `params['alpha']` and $\\gamma$: `params['gamma']`. \n\nOnce we have our Q-learning algorithm, we will see how it handles learning to solve the Cliff World environment. \n\nYou will recall from the previous tutorial that a major part of reinforcement learning algorithms are their ability to balance exploitation and exploration. For our Q-learning agent, we again turn to the epsilon-greedy strategy. At each step, the agent will decide with probability $1 - \\epsilon$ to use the best action for the state it is currently in by looking at the value function, otherwise just make a random choice.\n\nThe process by which our the agent will interact with and learn about the environment is handled for you in the helper function `learn_environment`. This implements the entire learning episode lifecycle of stepping through the state observation, action selection (epsilon-greedy) and execution, reward, and state transition. Feel free to review that code later to see how it all fits together, but for now let's test out our agent.\n\n\n```python\n# @markdown Execute to get helper functions `epsilon_greedy`, `CliffWorld`, and `learn_environment`\n\ndef epsilon_greedy(q, epsilon):\n \"\"\"Epsilon-greedy policy: selects the maximum value action with probabilty\n (1-epsilon) and selects randomly with epsilon probability.\n\n Args:\n q (ndarray): an array of action values\n epsilon (float): probability of selecting an action randomly\n\n Returns:\n int: the chosen action\n \"\"\"\n if np.random.random() > epsilon:\n action = np.argmax(q)\n else:\n action = np.random.choice(len(q))\n\n return action\n\n\nclass CliffWorld:\n \"\"\"\n World: Cliff world.\n 40 states (4-by-10 grid world).\n The mapping from state to the grids are as follows:\n 30 31 32 ... 39\n 20 21 22 ... 29\n 10 11 12 ... 19\n 0 1 2 ... 9\n 0 is the starting state (S) and 9 is the goal state (G).\n Actions 0, 1, 2, 3 correspond to right, up, left, down.\n Moving anywhere from state 9 (goal state) will end the session.\n Taking action down at state 11-18 will go back to state 0 and incur a\n reward of -100.\n Landing in any states other than the goal state will incur a reward of -1.\n Going towards the border when already at the border will stay in the same\n place.\n \"\"\"\n def __init__(self):\n self.name = \"cliff_world\"\n self.n_states = 40\n self.n_actions = 4\n self.dim_x = 10\n self.dim_y = 4\n self.init_state = 0\n\n def get_outcome(self, state, action):\n if state == 9: # goal state\n reward = 0\n next_state = None\n return next_state, reward\n reward = -1 # default reward value\n if action == 0: # move right\n next_state = state + 1\n if state % 10 == 9: # right border\n next_state = state\n elif state == 0: # start state (next state is cliff)\n next_state = None\n reward = -100\n elif action == 1: # move up\n next_state = state + 10\n if state >= 30: # top border\n next_state = state\n elif action == 2: # move left\n next_state = state - 1\n if state % 10 == 0: # left border\n next_state = state\n elif action == 3: # move down\n next_state = state - 10\n if state >= 11 and state <= 18: # next is cliff\n next_state = None\n reward = -100\n elif state <= 9: # bottom border\n next_state = state\n else:\n print(\"Action must be between 0 and 3.\")\n next_state = None\n reward = None\n return int(next_state) if next_state is not None else None, reward\n\n def get_all_outcomes(self):\n outcomes = {}\n for state in range(self.n_states):\n for action in range(self.n_actions):\n next_state, reward = self.get_outcome(state, action)\n outcomes[state, action] = [(1, next_state, reward)]\n return outcomes\n\n\ndef learn_environment(env, learning_rule, params, max_steps, n_episodes):\n # Start with a uniform value function\n value = np.ones((env.n_states, env.n_actions))\n\n # Run learning\n reward_sums = np.zeros(n_episodes)\n\n # Loop over episodes\n for episode in range(n_episodes):\n state = env.init_state # initialize state\n reward_sum = 0\n\n for t in range(max_steps):\n # choose next action\n action = epsilon_greedy(value[state], params['epsilon'])\n\n # observe outcome of action on environment\n next_state, reward = env.get_outcome(state, action)\n\n # update value function\n value = learning_rule(state, action, reward, next_state, value, params)\n\n # sum rewards obtained\n reward_sum += reward\n\n if next_state is None:\n break # episode ends\n state = next_state\n\n reward_sums[episode] = reward_sum\n\n return value, reward_sums\n```\n\n\n```python\ndef q_learning(state, action, reward, next_state, value, params):\n \"\"\"Q-learning: updates the value function and returns it.\n\n Args:\n state (int): the current state identifier\n action (int): the action taken\n reward (float): the reward received\n next_state (int): the transitioned to state identifier\n value (ndarray): current value function of shape (n_states, n_actions)\n params (dict): a dictionary containing the default parameters\n\n Returns:\n ndarray: the updated value function of shape (n_states, n_actions)\n \"\"\"\n # Q-value of current state-action pair\n q = value[state, action]\n\n ##########################################################\n ## TODO for students: implement the Q-learning update rule\n # Fill out function and remove\n #raise NotImplementedError(\"Student excercise: implement the Q-learning update rule\")\n ##########################################################\n\n # write an expression for finding the maximum Q-value at the current state\n if next_state is None:\n max_next_q = 0\n else:\n max_next_q = np.max(value[next_state])\n\n # write the expression to compute the TD error\n td_error = reward + params['gamma'] * max_next_q - q\n # write the expression that updates the Q-value for the state-action pair\n value[state, action] = q + params['alpha'] *td_error\n\n return value\n\n\n# set for reproducibility, comment out / change seed value for different results\nnp.random.seed(1)\n\n# parameters needed by our policy and learning rule\nparams = {\n 'epsilon': 0.1, # epsilon-greedy policy\n 'alpha': 0.1, # learning rate\n 'gamma': 1.0, # discount factor\n}\n\n# episodes/trials\nn_episodes = 500\nmax_steps = 1000\n\n# environment initialization\nenv = CliffWorld()\n\n# solve Cliff World using Q-learning\nresults = learn_environment(env, q_learning, params, max_steps, n_episodes)\nvalue_qlearning, reward_sums_qlearning = results\n\n# Plot results\nplot_performance(env, value_qlearning, reward_sums_qlearning)\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W3D4_ReinforcementLearning/solutions/W3D4_Tutorial3_Solution_7a5ca920.py)\n\n*Example output:*\n\n\n\n\n\nIf all went well, we should see four plots that show different aspects on our agent's learning and progress.\n\n* The top left is a representation of the Q-table itself, showing the values for different actions in different states. Notably, going right from the starting state or down when above the cliff is clearly very bad.\n* The top right figure shows the greedy policy based on the Q-table, i.e. what action would the agent take if it only took its best guess in that state.\n* The bottom right is the same as the top, only instead of showing the action, it's showing a representation of the maximum Q-value at a particular state.\n* The bottom left is the actual proof of learning, as we see the total reward steadily increasing after each episode until asymptoting at the maximum possible reward of -11.\n\nFeel free to try changing the parameters or random seed and see how the agent's behavior changes.\n\n---\n# Summary\n\n*Estimated timing of tutorial: 40 min*\n\nIn this tutorial you implemented a reinforcement learning agent based on Q-learning to solve the Cliff World environment. Q-learning combined the epsilon-greedy approach to exploration-expoitation with a table-based value function to learn the expected future rewards for each state.\n\n---\n# Bonus\n\n---\n## Bonus Section 1: SARSA\n\nAn alternative to Q-learning, the SARSA algorithm also estimates action values. However, rather than estimating the optimal (off-policy) values, SARSA estimates the **on-policy** action value, i.e. the cumulative future reward that would be obtained if the agent behaved according to its current beliefs.\n\n\\begin{align}\nQ(s_t,a_t) \\leftarrow Q(s_t,a_t) + \\alpha \\big(r_t + \\gamma Q(s_{t+1},a_{t+1}) - Q(s_t,a_t)\\big)\n\\end{align}\n\nwhere, once again, $Q(s,a)$ is the value function for action $a$ at state $s$, $\\alpha$ is the learning rate, $r$ is the reward, and $\\gamma$ is the temporal discount rate.\n\nIn fact, you will notices that the *only* difference between Q-learning and SARSA is the TD target calculation uses the policy to select the next action (in our case epsilon-greedy) rather than using the action that maximizes the Q-value.\n\n### Bonus Coding Exercise 1: Implement the SARSA algorithm\n\nIn this exercise you will implement the SARSA update rule described above. Just like Q-learning, it takes in as arguments the previous state $s_t$, the action $a_t$ taken, the reward received $r_t$, the current state $s_{t+1}$, the Q-value table, and a dictionary of parameters that contain the learning rate $\\alpha$ and discount factor $\\gamma$. The method returns the updated Q-value table. You may use the `epsilon_greedy` function to acquire the next action. For the parameter dictionary, $\\alpha$: `params['alpha']`, $\\gamma$: `params['gamma']`, and $\\epsilon$: `params['epsilon']`. \n\nOnce we have an implementation for SARSA, we will see how it tackles Cliff World. We will again use the same setup we tried with Q-learning.\n\n\n```python\ndef sarsa(state, action, reward, next_state, value, params):\n \"\"\"SARSA: updates the value function and returns it.\n\n Args:\n state (int): the current state identifier\n action (int): the action taken\n reward (float): the reward received\n next_state (int): the transitioned to state identifier\n value (ndarray): current value function of shape (n_states, n_actions)\n params (dict): a dictionary containing the default parameters\n\n Returns:\n ndarray: the updated value function of shape (n_states, n_actions)\n \"\"\"\n # value of previous state-action pair\n q = value[state, action]\n\n ##########################################################\n ## TODO for students: implement the SARSA update rule\n # Fill out function and remove\n raise NotImplementedError(\"Student excercise: implement the SARSA update rule\")\n ##########################################################\n\n # select the expected value at current state based on our policy by sampling\n # from it\n if next_state is None:\n policy_next_q = 0\n else:\n # write an expression for selecting an action using epsilon-greedy\n policy_action = ...\n # write an expression for obtaining the value of the policy action at the\n # current state\n policy_next_q = ...\n\n # write the expression to compute the TD error\n td_error = ...\n # write the expression that updates the Q-value for the state-action pair\n value[state, action] = ...\n\n return value\n\n\n# set for reproducibility, comment out / change seed value for different results\nnp.random.seed(1)\n\n# parameters needed by our policy and learning rule\nparams = {\n 'epsilon': 0.1, # epsilon-greedy policy\n 'alpha': 0.1, # learning rate\n 'gamma': 1.0, # discount factor\n}\n\n# episodes/trials\nn_episodes = 500\nmax_steps = 1000\n\n# environment initialization\nenv = CliffWorld()\n\n# learn Cliff World using Sarsa\nresults = learn_environment(env, sarsa, params, max_steps, n_episodes)\nvalue_sarsa, reward_sums_sarsa = results\n\n# Plot results\nplot_performance(env, value_sarsa, reward_sums_sarsa)\n```\n\n\n```python\ndef sarsa(state, action, reward, next_state, value, params):\n \"\"\"SARSA: updates the value function and returns it.\n\n Args:\n state (int): the current state identifier\n action (int): the action taken\n reward (float): the reward received\n next_state (int): the transitioned to state identifier\n value (ndarray): current value function of shape (n_states, n_actions)\n params (dict): a dictionary containing the default parameters\n\n Returns:\n ndarray: the updated value function of shape (n_states, n_actions)\n \"\"\"\n # value of previous state-action pair\n q = value[state, action]\n\n # select the expected value at current state based on our policy by sampling\n # from it\n if next_state is None:\n policy_next_q = 0\n else:\n # write an expression for selecting an action using epsilon-greedy\n policy_action = epsilon_greedy(value[next_state], params['epsilon'])\n # write an expression for obtaining the value of the policy action at the\n # current state\n policy_next_q = value[next_state, policy_action]\n\n # write the expression to compute the TD error\n td_error = reward + params['gamma'] * policy_next_q - q\n # write the expression that updates the Q-value for the state-action pair\n value[state, action] = q + params['alpha'] * td_error\n\n return value\n\n\n# set for reproducibility, comment out / change seed value for different results\nnp.random.seed(1)\n\n# parameters needed by our policy and learning rule\nparams = {\n 'epsilon': 0.1, # epsilon-greedy policy\n 'alpha': 0.1, # learning rate\n 'gamma': 1.0, # discount factor\n}\n\n# episodes/trials\nn_episodes = 500\nmax_steps = 1000\n\n# environment initialization\nenv = CliffWorld()\n\n# learn Cliff World using Sarsa\nresults = learn_environment(env, sarsa, params, max_steps, n_episodes)\nvalue_sarsa, reward_sums_sarsa = results\n\n# Plot results\nwith plt.xkcd():\n plot_performance(env, value_sarsa, reward_sums_sarsa)\n```\n\nWe should see that SARSA also solves the task with similar looking outcomes to Q-learning. One notable difference is that SARSA seems to be skittsh around the cliff edge and often goes further away before coming back down to the goal.\n\nAgain, feel free to try changing the parameters or random seed and see how the agent's behavior changes.\n\n---\n## Bonus Section 2: On-Policy vs Off-Policy\n \nWe have now seen an example of both on- and off-policy learning algorithms. Let's compare both Q-learning and SARSA reward results again, side-by-side, to see how they stack up.\n\n\n```python\n# @markdown Execute to see visualization\n\n# parameters needed by our policy and learning rule\nparams = {\n 'epsilon': 0.1, # epsilon-greedy policy\n 'alpha': 0.1, # learning rate\n 'gamma': 1.0, # discount factor\n}\n\n# episodes/trials\nn_episodes = 500\nmax_steps = 1000\n\n# environment initialization\nenv = CliffWorld()\n\n# learn Cliff World using Sarsa\nnp.random.seed(1)\nresults = learn_environment(env, q_learning, params, max_steps, n_episodes)\nvalue_qlearning, reward_sums_qlearning = results\nnp.random.seed(1)\nresults = learn_environment(env, sarsa, params, max_steps, n_episodes)\nvalue_sarsa, reward_sums_sarsa = results\n\nfig, ax = plt.subplots()\nax.plot(reward_sums_qlearning, label='Q-learning')\nax.plot(reward_sums_sarsa, label='SARSA')\nax.set(xlabel='Episodes', ylabel='Total reward')\nplt.legend(loc='lower right');\n```\n\nOn this simple Cliff World task, Q-learning and SARSA are almost indistinguisable from a performance standpoint, but we can see that Q-learning has a slight-edge within the 500 episode time horizon. Let's look at the illustrated \"greedy policy\" plots again.\n\n\n```python\n# @markdown Execute to see visualization\n\nfig, (ax1, ax2) = plt.subplots(ncols=2, figsize=(16, 6))\nplot_quiver_max_action(env, value_qlearning, ax=ax1)\nax1.set(title='Q-learning maximum value/probability actions')\nplot_quiver_max_action(env, value_sarsa, ax=ax2)\nax2.set(title='SARSA maximum value/probability actions');\n```\n\nWhat should immediately jump out is that Q-learning learned to go up, then immediately go to the right, skirting the cliff edge, until it hits the wall and goes down to the goal. The policy further away from the cliff is less certain.\n\nSARSA, on the other hand, appears to avoid the cliff edge, going up one more tile before starting over to the goal side. This also clearly solves the challenge of getting to the goal, but does so at an additional -2 cost over the truly optimal route.\n\nWhy do you think these behaviors emerged the way they did?\n", "meta": {"hexsha": "416fedc12b416da0752053ae18c9fa684b18976d", "size": 347895, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W3D4_ReinforcementLearning/student/W3D4_Tutorial3.ipynb", "max_stars_repo_name": "luisarai/NMA2021", "max_stars_repo_head_hexsha": "d6cd66bf32d929f3030d0d66c2c92de55bd2d886", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tutorials/W3D4_ReinforcementLearning/student/W3D4_Tutorial3.ipynb", "max_issues_repo_name": "luisarai/NMA2021", "max_issues_repo_head_hexsha": "d6cd66bf32d929f3030d0d66c2c92de55bd2d886", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/W3D4_ReinforcementLearning/student/W3D4_Tutorial3.ipynb", "max_forks_repo_name": "luisarai/NMA2021", "max_forks_repo_head_hexsha": "d6cd66bf32d929f3030d0d66c2c92de55bd2d886", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 278.9855653569, "max_line_length": 261822, "alphanum_fraction": 0.9014932666, "converted": true, "num_tokens": 6684, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872131147276, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.42399724710132797}} {"text": "\n\n# Tutorial 2: Regularization techniques part 2\n**Week 1, Day 5: Regularization**\n\n**By Neuromatch Academy**\n\n\n__Content creators:__ Ravi Teja Konkimalla, Mohitrajhu Lingan Kumaraian, Kevin Machado Gamboa, Kelson Shilling-Scrivo, Lyle Ungar\n\n__Content reviewers:__ Piyush Chauhan, Siwei Bai, Kelson Shilling-Scrivo\n\n__Content editors:__ Roberto Guidotti, Spiros Chavlis\n\n__Production editors:__ Saeed Salehi, Spiros Chavlis\n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n

                                        \n\n---\n# Tutorial Objectives\n\n1. Regularization as shrinkage of overparameterized models: L1, L2\n2. Regularization by Dropout\n3. Regularization by Data Augmentation\n4. Perils of Hyper-Parameter Tuning\n5. Rethinking generalization\n\n\n```python\n\n# @title Tutorial slides\n\n# @markdown These are the slides for the videos in this tutorial\nfrom IPython.display import IFrame\nIFrame(src=f\"https://mfr.ca-1.osf.io/render?url=https://osf.io/7um6p/?direct%26mode=render%26action=download%26mode=render\", width=854, height=480)\n```\n\n\n\n\n\n\n\n\n\n\n---\n# Setup\nNote that some of the code for today can take up to an hour to run. We have therefore \"hidden\" that code and shown the resulting outputs.\n\n\n\n```python\n# @title Install dependencies\n#!apt-get install -y ffmpeg --quiet\n!pip install imageio-ffmpeg --quiet\n\n!pip install git+https://github.com/NeuromatchAcademy/evaltools --quiet\nfrom evaltools.airtable import AirtableForm\n\natform = AirtableForm('appn7VdPRseSoMXEG','W1D5_T2', 'https://portal.neuromatchacademy.org/api/redirect/to/a76f99c1-9005-4566-8bcd-bed4e53d21f1')\n```\n\n\n```python\n# Imports\nfrom __future__ import print_function\n\nimport copy\nimport torch\nimport random\nimport pathlib\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport matplotlib.animation as animation\n\nimport torch.nn as nn\nimport torch.optim as optim\nimport torch.nn.functional as F\n\nfrom torchvision import datasets, transforms\nfrom torchvision.datasets import ImageFolder\nfrom torch.optim.lr_scheduler import StepLR\n\nfrom tqdm.auto import tqdm\nfrom IPython.display import HTML, display\n```\n\n\n```python\n# @title Figure Settings\nimport ipywidgets as widgets\n%matplotlib inline\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/content-creation/main/nma.mplstyle\")\n```\n\n\n```python\n# @title Loading Animal Faces data\nimport requests, os\nfrom zipfile import ZipFile\n\nprint(\"Start downloading and unzipping `AnimalFaces` dataset...\")\nname = 'afhq'\nfname = f\"{name}.zip\"\nurl = f\"https://osf.io/kgfvj/download\"\n\nif not os.path.exists(fname):\n r = requests.get(url, allow_redirects=True)\n with open(fname, 'wb') as fh:\n fh.write(r.content)\n\n if os.path.exists(fname):\n with ZipFile(fname, 'r') as zfile:\n zfile.extractall(f\".\")\n os.remove(fname)\n\nprint(\"Download completed.\")\n```\n\n Start downloading and unzipping `AnimalFaces` dataset...\n Download completed.\n\n\n\n```python\n# @title Loading Animal Faces Randomized data\nfrom IPython.display import clear_output\n\nprint(\"Start downloading and unzipping `Randomized AnimalFaces` dataset...\")\n\nnames = ['afhq_random_32x32', 'afhq_10_32x32']\nurls = [\"https://osf.io/9sj7p/download\",\n \"https://osf.io/wvgkq/download\"]\n\n\nfor i, name in enumerate(names):\n url = urls[i]\n fname = f\"{name}.zip\"\n\n if not os.path.exists(fname):\n r = requests.get(url, allow_redirects=True)\n with open(fname, 'wb') as fh:\n fh.write(r.content)\n\n if os.path.exists(fname):\n with ZipFile(fname, 'r') as zfile:\n zfile.extractall(f\".\")\n os.remove(fname)\n\nprint(\"Download completed.\")\n```\n\n Start downloading and unzipping `Randomized AnimalFaces` dataset...\n Download completed.\n\n\n\n```python\n# @title Plotting functions\n\n\ndef imshow(img):\n img = img / 2 + 0.5 # unnormalize\n npimg = img.numpy()\n plt.imshow(np.transpose(npimg, (1, 2, 0)))\n plt.axis(False)\n plt.show()\n\n\ndef plot_weights(norm, labels, ws, title='Weight Size Measurement'):\n plt.figure(figsize=[8, 6])\n plt.title(title)\n plt.ylabel('Frobenius Norm Value')\n plt.xlabel('Model Layers')\n plt.bar(labels, ws)\n plt.axhline(y=norm,\n linewidth=1,\n color='r',\n ls='--',\n label='Total Model F-Norm')\n plt.legend()\n plt.show()\n\n\ndef visualize_data(dataloader):\n\n for idx, (data,label) in enumerate(dataloader):\n plt.figure(idx)\n # Choose the datapoint you would like to visualize\n index = 22\n\n # choose that datapoint using index and permute the dimensions\n # and bring the pixel values between [0,1]\n data = data[index].permute(1, 2, 0) * \\\n torch.tensor([0.5, 0.5, 0.5]) + \\\n torch.tensor([0.5, 0.5, 0.5])\n\n # Convert the torch tensor into numpy\n data = data.numpy()\n\n plt.imshow(data)\n plt.axis(False)\n image_class = classes[label[index].item()]\n print(f'The image belongs to : {image_class}')\n\n plt.show()\n```\n\n\n```python\n# @title Helper functions\n\n## Network Class - Animal Faces\nclass AnimalNet(nn.Module):\n def __init__(self):\n super(AnimalNet, self).__init__()\n self.fc1 = nn.Linear(3 * 32 * 32, 128)\n self.fc2 = nn.Linear(128, 32)\n self.fc3 = nn.Linear(32, 3)\n\n def forward(self, x):\n x = x.view(x.shape[0], -1)\n x = F.relu(self.fc1(x))\n x = F.relu(self.fc2(x))\n x = self.fc3(x)\n output = F.log_softmax(x, dim=1)\n return output\n\n\n# Simple Net\nclass Net(nn.Module):\n def __init__(self):\n super(Net, self).__init__()\n\n self.fc1 = nn.Linear(1, 300)\n self.fc2 = nn.Linear(300, 500)\n self.fc3 = nn.Linear(500, 1)\n\n def forward(self, x):\n x = F.leaky_relu(self.fc1(x))\n x = F.leaky_relu(self.fc2(x))\n output = self.fc3(x)\n return output\n\n\n# Network Class - Animal Faces\nclass BigAnimalNet(nn.Module):\n def __init__(self):\n super(BigAnimalNet, self).__init__()\n self.fc1 = nn.Linear(3*32*32, 124)\n self.fc2 = nn.Linear(124, 64)\n self.fc3 = nn.Linear(64, 3)\n\n def forward(self, x):\n x = x.view(x.shape[0],-1)\n x = F.leaky_relu(self.fc1(x))\n x = F.leaky_relu(self.fc2(x))\n x = self.fc3(x)\n output = F.log_softmax(x, dim=1)\n return output\n\n\ndef train(args, model, train_loader, optimizer, epoch,\n reg_function1=None, reg_function2=None, criterion=F.nll_loss):\n \"\"\"\n Trains the current inpur model using the data\n from Train_loader and Updates parameters for a single pass\n \"\"\"\n\n device = args['device']\n model.train()\n for batch_idx, (data, target) in enumerate(train_loader):\n data, target = data.to(device), target.to(device)\n optimizer.zero_grad()\n output = model(data)\n # L1 regularization\n if reg_function2 is None and reg_function1 is not None:\n loss = criterion(output, target) + args['lambda1']*reg_function1(model)\n # L2 regularization\n elif reg_function1 is None and reg_function2 is not None:\n loss = criterion(output, target) + args['lambda2']*reg_function2(model)\n # No regularization\n elif reg_function1 is None and reg_function2 is None:\n loss = criterion(output, target)\n # Both L1 and L2 regularizations\n else:\n loss = criterion(output, target) + args['lambda1']*reg_function1(model) + args['lambda2']*reg_function2(model)\n loss.backward()\n optimizer.step()\n\n return model\n\n\ndef test(model, test_loader, loader='Test', criterion=F.nll_loss,\n device='cpu'):\n \"\"\"\n Tests the current Model\n \"\"\"\n model.eval()\n test_loss = 0\n correct = 0\n with torch.no_grad():\n for data, target in test_loader:\n data, target = data.to(device), target.to(device)\n output = model(data)\n test_loss += criterion(output, target, reduction='sum').item() # sum up batch loss\n pred = output.argmax(dim=1, keepdim=True) # get the index of the max log-probability\n correct += pred.eq(target.view_as(pred)).sum().item()\n\n test_loss /= len(test_loader.dataset)\n def __init__(self):\n super(BigAnimalNet, self).__init__()\n self.fc1 = nn.Linear(3*32*32, 124)\n self.fc2 = nn.Linear(124, 64)\n self.fc3 = nn.Linear(64, 3)\n\n def forward(self, x):\n x = x.view(x.shape[0],-1)\n x = F.leaky_relu(self.fc1(x))\n x = F.leaky_relu(self.fc2(x))\n x = self.fc3(x)\n output = F.log_softmax(x, dim=1)\n return output\n return 100. * correct / len(test_loader.dataset)\n\n\ndef main(args, model, train_loader, val_loader, test_data,\n reg_function1=None, reg_function2=None, criterion=F.nll_loss):\n \"\"\"\n Trains the model with train_loader and tests the learned model using val_loader\n \"\"\"\n\n device = args['device']\n\n model = model.to(device)\n optimizer = optim.SGD(model.parameters(), lr=args['lr'], momentum=args['momentum'])\n\n val_acc_list, train_acc_list,param_norm_list = [], [], []\n for epoch in tqdm(range(args['epochs'])):\n trained_model = train(args, model, train_loader, optimizer, epoch,\n reg_function1=reg_function1,\n reg_function2=reg_function2)\n train_acc = test(trained_model, train_loader, loader='Train', device=device)\n val_acc = test(trained_model, val_loader, loader='Val', device=device)\n param_norm = calculate_frobenius_norm(trained_model)\n train_acc_list.append(train_acc)\n val_acc_list.append(val_acc)\n param_norm_list.append(param_norm)\n\n return val_acc_list, train_acc_list, param_norm_list, model\n\n\ndef calculate_frobenius_norm(model):\n norm = 0.0\n # Sum the square of all parameters\n for name,param in model.named_parameters():\n norm += torch.norm(param).data**2\n # Return a square root of the sum of squares of all the parameters\n return norm**0.5\n\n\ndef early_stopping_main(args, model, train_loader, val_loader, test_data):\n\n device = args['device']\n\n model = model.to(device)\n optimizer = optim.SGD(model.parameters(), lr=args['lr'], momentum=args['momentum'])\n\n best_acc = 0.0\n best_epoch = 0\n\n # Number of successive epochs that you want to wait before stopping training process\n patience = 20\n\n # Keps track of number of epochs during which the val_acc was less than best_acc\n wait = 0\n\n val_acc_list, train_acc_list = [], []\n for epoch in tqdm(range(args['epochs'])):\n trained_model = train(args, model, device, train_loader, optimizer, epoch)\n train_acc = test(trained_model, train_loader, loader='Train', device=device)\n val_acc = test(trained_model, val_loader, loader='Val', device=device)\n if (val_acc > best_acc):\n best_acc = val_acc\n best_epoch = epoch\n best_model = copy.deepcopy(trained_model)\n wait = 0\n else:\n wait += 1\n if (wait > patience):\n print(f'early stopped on epoch: {epoch}')\n break\n train_acc_list.append(train_acc)\n val_acc_list.append(val_acc)\n\n return val_acc_list, train_acc_list, best_model, best_epoch\n```\n\n\n```python\n# @title Set random seed\n# @markdown Executing `set_seed(seed=seed)` you are setting the seed\n\n# for DL its critical to set the random seed so that students can have a\n# baseline to compare their results to expected results.\n# Read more here: https://pytorch.org/docs/stable/notes/randomness.html\n\n# Call `set_seed` function in the exercises to ensure reproducibility.\nimport random\nimport torch\n\ndef set_seed(seed=None, seed_torch=True):\n if seed is None:\n seed = np.random.choice(2 ** 32)\n random.seed(seed)\n np.random.seed(seed)\n if seed_torch:\n torch.manual_seed(seed)\n torch.cuda.manual_seed_all(seed)\n torch.cuda.manual_seed(seed)\n torch.backends.cudnn.benchmark = False\n torch.backends.cudnn.deterministic = True\n\n print(f'Random seed {seed} has been set.')\n\n\n# In case that `DataLoader` is used\ndef seed_worker(worker_id):\n worker_seed = torch.initial_seed() % 2**32\n np.random.seed(worker_seed)\n random.seed(worker_seed)\n```\n\n\n```python\n# @title Set device (GPU or CPU). Execute `set_device()`\n# especially if torch modules used.\n\n# inform the user if the notebook uses GPU or CPU.\n\ndef set_device():\n device = \"cuda\" if torch.cuda.is_available() else \"cpu\"\n if device != \"cuda\":\n print(\"WARNING: For this notebook to perform best, \"\n \"if possible, in the menu under `Runtime` -> \"\n \"`Change runtime type.` select `GPU` \")\n else:\n print(\"GPU is enabled in this notebook.\")\n\n return device\n```\n\n\n```python\nSEED = 2021\nset_seed(seed=SEED)\nDEVICE = set_device()\n```\n\n Random seed 2021 has been set.\n GPU is enabled in this notebook.\n\n\n\n```python\n# @title Dataloaders for the Dataset\n## Dataloaders for the Dataset\nbatch_size = 128\nclasses = ('cat', 'dog', 'wild')\n\ntrain_transform = transforms.Compose([\n transforms.ToTensor(),\n transforms.Normalize((0.5, 0.5, 0.5), (0.5, 0.5, 0.5))\n ])\ndata_path = pathlib.Path('.')/'afhq' # using pathlib to be compatible with all OS's\nimg_dataset = ImageFolder(data_path/'train', transform=train_transform)\n\n\n####################################################\ng_seed = torch.Generator()\ng_seed.manual_seed(SEED)\n\n\n##Dataloaders for the Original Dataset\nimg_train_data, img_val_data,_ = torch.utils.data.random_split(img_dataset,\n [100, 100, 14430])\n\n#Creating train_loader and Val_loader\ntrain_loader = torch.utils.data.DataLoader(img_train_data,\n batch_size=batch_size,\n worker_init_fn=seed_worker,\n num_workers=2,\n generator=g_seed)\nval_loader = torch.utils.data.DataLoader(img_val_data,\n batch_size=1000,\n num_workers=2,\n worker_init_fn=seed_worker,\n generator=g_seed)\n\n#creating test dataset\ntest_transform = transforms.Compose([\n transforms.ToTensor(),\n transforms.Normalize((0.5, 0.5, 0.5), (0.5, 0.5, 0.5))\n ])\nimg_test_dataset = ImageFolder(data_path/'val', transform=test_transform)\n\n\n####################################################\n\n## Dataloaders for the Random Dataset\n\n# splitting randomized data into training and validation data\ndata_path = pathlib.Path('.')/'afhq_random_32x32/afhq_random' # using pathlib to be compatible with all OS's\nimg_dataset = ImageFolder(data_path/'train', transform=train_transform)\nrandom_img_train_data, random_img_val_data,_ = torch.utils.data.random_split(img_dataset, [100,100,14430])\n\n#Randomized train and validation dataloader\nrand_train_loader = torch.utils.data.DataLoader(random_img_train_data,\n batch_size=batch_size,\n num_workers=2,\n worker_init_fn=seed_worker,\n generator=g_seed)\nrand_val_loader = torch.utils.data.DataLoader(random_img_val_data,\n batch_size=1000,\n num_workers=2,\n worker_init_fn=seed_worker,\n generator=g_seed)\n\n####################################################\n\n## Dataloaders for the Partially Random Dataset\n\n# Splitting data between training and validation dataset for partially randomized data\ndata_path = pathlib.Path('.')/'afhq_10_32x32/afhq_10' # using pathlib to be compatible with all OS's\nimg_dataset = ImageFolder(data_path/'train', transform=train_transform)\npartially_random_train_data, partially_random_val_data, _ = torch.utils.data.random_split(img_dataset, [100,100,14430])\n\n#Training and Validation loader for partially randomized data\npartial_rand_train_loader = torch.utils.data.DataLoader(partially_random_train_data,\n batch_size=batch_size,\n num_workers=2,\n worker_init_fn=seed_worker,\n generator=g_seed)\npartial_rand_val_loader = torch.utils.data.DataLoader(partially_random_val_data,\n batch_size=1000,\n num_workers=2,\n worker_init_fn=seed_worker,\n generator=g_seed)\n```\n\n---\n# Section 1: L1 and L2 Regularization\n\n*Time estimate: ~30 mins*\n\n\n\n```python\n# @title Video 1: L1 and L2 regression\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV19h41167H7\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"oQNdloKdysM\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 1: L1 and L2 regression')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nSome of you might have already come across L1 and L2 regularization before in other courses. L1 and L2 are the most common types of regularization. These update the general cost function by adding another term known as the regularization term.\n\n***Cost function = Loss (say, binary cross entropy) + Regularization term***\n\nThis regularization term makes the parameters smaller, giving simpler models that will overfit less.\n\nDiscuss among your teammates whether the above assumption is good or bad?\n\n## Section 1.1: Unregularized Model\n\n\n```python\n# @markdown #### Dataloaders for Regularization\ndata_path = pathlib.Path('.')/'afhq' # using pathlib to be compatible with all OS's\nimg_dataset = ImageFolder(data_path/'train', transform=train_transform)\n\n# Splitting dataset\nreg_train_data, reg_val_data,_ = torch.utils.data.random_split(img_dataset,\n [30, 100, 14500])\ng_seed = torch.Generator()\ng_seed.manual_seed(SEED)\n\n# Creating train_loader and Val_loader\nreg_train_loader = torch.utils.data.DataLoader(reg_train_data,\n batch_size=batch_size,\n worker_init_fn=seed_worker,\n num_workers=2,\n generator=g_seed)\nreg_val_loader = torch.utils.data.DataLoader(reg_val_data,\n batch_size=1000,\n worker_init_fn=seed_worker,\n num_workers=2,\n generator=g_seed)\n```\n\nNow let's train a model without any regularization and keep it aside as our benchmark for this section.\n\n\n```python\n# Set the arguments\nargs = {\n 'epochs': 150,\n 'lr': 5e-3,\n 'momentum': 0.99,\n 'device': DEVICE,\n}\n\n# intialize the model\nset_seed(seed=SEED)\nmodel = AnimalNet()\n\n# Train the model\nval_acc_unreg, train_acc_unreg, param_norm_unreg, _ = main(args,\n model,\n reg_train_loader,\n reg_val_loader,\n img_test_dataset)\n\n# Train and Test accuracy plot\nplt.figure()\nplt.plot(val_acc_unreg, label='Val Accuracy', c='red', ls='dashed')\nplt.plot(train_acc_unreg, label='Train Accuracy', c='red', ls='solid')\nplt.axhline(y=max(val_acc_unreg), c='green', ls='dashed')\nplt.title('Unregularized Model')\nplt.ylabel('Accuracy (%)')\nplt.xlabel('Epoch')\nplt.legend()\nplt.show()\nprint(f\"maximum Validation Accuracy reached: {max(val_acc_unreg)}\")\n```\n\n## Section 1.2: L1 Regularization\n\nL1 (or \"LASSO\") Regularization uses a penalty which is the sum of the absolute value of all the weights in the DLN, resulting in the following loss function ($L$ is the usual Cross Entropy loss):\n\n\\begin{equation}\nL_R=L + \\lambda \\sum \\left| w^{(r)}_{ij} \\right|\n\\end{equation}\n\nwhere $r$ denotes the layer, and $ij$ the specific weight in that layer.\n\nAt a high level, L1 Regularization is similar to L2 Regularization since it leads to smaller weights (you will see the analogy in the next subsection). It results in the following weight update equation when using Stochastic Gradient Descent:\n\n\\begin{equation}\nw^{(r)}_{ij}\u2190w^{(r)}_{ij}\u2212\\eta \\lambda \\text{sgn}\\left(w^{(r)}_{ij}\\right)\u2212\\eta \\frac{\\partial L}{\\partial w_{ij}^{(r)}} \n\\end{equation}\n\nwhere $\\text{sgn}(\\cdot)$ is the sign function, such that\n\n\\begin{equation}\n\\text{sgn}(w) = \n\\left\\{\n \\begin{array}{ll}\n +1 & \\mbox{if } w > 0 \\\\\n -1 & \\mbox{if } w < 0 \\\\\n 0 & \\mbox{if } w = 0\n \\end{array}\n\\right.\n\\end{equation}\n\n### Coding Exercise 1.1: L1 Regularization\n\nWrite a function which calculates the L1 norm of all the tensors of a Pytorch model.\n\n\n```python\ndef l1_reg(model):\n \"\"\"\n Inputs: Pytorch model\n This function calculates the l1 norm of the all the tensors in the model\n \"\"\"\n l1 = 0.0\n ####################################################################\n # Fill in all missing code below (...),\n # then remove or comment the line below to test your function\n #raise NotImplementedError(\"Complete the l1_reg function\")\n ####################################################################\n for param in model.parameters():\n l1 += torch.sum(torch.abs(param.data))\n\n return l1\n\n# add event to airtable\natform.add_event('Coding Exercise 1.1: L1 Regularization')\n\nset_seed(seed=SEED)\n## uncomment to test\nnet = nn.Linear(20, 20)\nprint(f\"L1 norm of the model: {l1_reg(net)}\")\n```\n\n Random seed 2021 has been set.\n L1 norm of the model: 48.445133209228516\n\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content-dl/tree/main//tutorials/W1D5_Regularization/solutions/W1D5_Tutorial2_Solution_f9f318de.py)\n\n\n\n```\nRandom seed 2021 has been set.\nL1 norm of the model: 48.445133209228516\n```\n\nNow, let's train a classifier which uses L1 regularization. Tune the hyperparameter `lambda1` such that the validation accuracy is higher than that of the unregularized model.\n\n\n```python\n# Set the arguments\nargs1 = {\n 'test_batch_size': 1000,\n 'epochs': 150,\n 'lr': 5e-3,\n 'momentum': 0.99,\n 'device': DEVICE,\n 'lambda1': 0.001 # <<<<<<<< Tune the hyperparameter lambda\n}\n\n# intialize the model\nset_seed(seed=SEED)\nmodel = AnimalNet()\n\n# Train the model\nval_acc_l1reg, train_acc_l1reg, param_norm_l1reg, _ = main(args1,\n model,\n reg_train_loader,\n reg_val_loader,\n img_test_dataset,\n reg_function1=l1_reg)\n\n# Train and Test accuracy plot\nplt.figure()\nplt.plot(val_acc_l1reg, label='Val Accuracy L1 Regularized',\n c='red', ls='dashed')\nplt.plot(train_acc_l1reg, label='Train Accuracy L1 regularized',\n c='red', ls='solid')\nplt.axhline(y=max(val_acc_l1reg), c='green', ls='dashed')\nplt.title('L1 regularized model')\nplt.ylabel('Accuracy (%)')\nplt.xlabel('Epoch')\nplt.legend()\nplt.show()\nprint(f\"maximum Validation Accuracy reached: {max(val_acc_l1reg)}\")\n```\n\nWhat value of `lambda1` hyperparameter, worked for L1 Regularization? (Note that the $\\lambda$ in the equations is the `lambda1` in the code for clarity)\n\n## Section 1.3: L2 / Ridge Regularization\n\nL2 Regularization, sometimes referred to as \u201cWeight Decay\u201d, is widely used. It works by adding a quadratic penalty term to the Cross Entropy Loss Function $L$, which results in a new Loss Function $L_R$ given by:\n\n\\begin{equation}\nL_R = L + \\lambda \\sum \\left( w^{(r)}_{ij} \\right)^2\n\\end{equation}\n\nwhere, again, $r$ denotes the layer, and $ij$ the specific weight in that layer.\n\nIn order to get further insight into L2 Regularization, we investigate its effect on the Gradient Descent based update equations for the weight and bias parameters. Taking the derivative on both sides of the above equation, we obtain\n\n\\begin{equation}\n\\frac{\\partial L_R}{\\partial w^{(r)}_{ij}}=\\frac{\\partial L}{\\partial w^{(r)}_{ij}} + 2\\lambda w^{(r)}_{ij}\n\\end{equation}\n\nThus the weight update rule becomes:\n\n\\begin{equation}\nw^{(r)}_{ij}\u2190w^{(r)}_{ij}\u2212\u03b7\\frac{\\partial L}{\\partial w^{(r)}_{ij}}\u22122 \\eta \\lambda w^{(r)}_{ij}=(1\u22122 \\eta \\lambda)w^{(r)}_{ij} \u2212 \\eta \\frac{\\partial L}{\\partial w^{(r)}_{ij}}\n\\end{equation}\n\nwhere, $\\eta$ is learning rate.\n\n### Coding Exercise 1.2: L2 Regularization\n\nWrite a function which calculates the L2 norm of all the tensors of a Pytorch model. (What did we call this before?)\n\n\n```python\ndef l2_reg(model):\n\n \"\"\"\n Inputs: Pytorch model\n This function calculates the l2 norm of the all the tensors in the model\n \"\"\"\n\n l2 = 0.0\n ####################################################################\n # Fill in all missing code below (...),\n # then remove or comment the line below to test your function\n #raise NotImplementedError(\"Complete the l2_reg function\")\n ####################################################################\n for param in model.parameters(): \n l2 += torch.sum(torch.abs(param.data)**2)\n\n return l2\n\n# add event to airtable\natform.add_event('Coding Exercise 1.2: L2 Regularization')\n\nset_seed(SEED)\n## uncomment to test\nnet = nn.Linear(20, 20)\nprint(f\"L2 norm of the model: {l2_reg(net)}\")\n```\n\n Random seed 2021 has been set.\n L2 norm of the model: 7.328375816345215\n\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content-dl/tree/main//tutorials/W1D5_Regularization/solutions/W1D5_Tutorial2_Solution_8505e24d.py)\n\n\n\n```\nRandom seed 2021 has been set.\nL2 norm of the model: 7.328375816345215\n```\n\nNow we'll train a classifier which uses L2 regularization. Tune the hyperparameter `lambda` such that the val accuracy is higher than that of the unregularized model.\n\n\n```python\n# Set the arguments\nargs2 = {\n 'test_batch_size': 1000,\n 'epochs': 150,\n 'lr': 5e-3,\n 'momentum': 0.99,\n 'device': DEVICE,\n 'lambda2': 0.001 # <<<<<<<< Tune the hyperparameter lambda\n}\n\n# intialize the model\nset_seed(seed=SEED)\nmodel = AnimalNet()\n\n# Train the model\nval_acc_l2reg, train_acc_l2reg, param_norm_l2reg, model = main(args2,\n model,\n train_loader,\n val_loader,\n img_test_dataset,\n reg_function2=l2_reg)\n\n## Train and Test accuracy plot\nplt.figure()\nplt.plot(val_acc_l2reg, label='Val Accuracy L2 regularized',\n c='red', ls='dashed')\nplt.plot(train_acc_l2reg, label='Train Accuracy L2 regularized',\n c='red', ls='solid')\nplt.axhline(y=max(val_acc_l2reg), c='green', ls='dashed')\nplt.title('L2 Regularized Model')\nplt.ylabel('Accuracy (%)')\nplt.xlabel('Epoch')\nplt.legend()\nplt.show()\nprint(f\"maximum Validation Accuracy reached: {max(val_acc_l2reg)}\")\n```\n\nWhat value `lambda2` worked for L2 Regularization? (Note that the $\\lambda$ in the equations is the `lambda2` in the code for clarity)\n\nNow, let's run a model with both L1 and L2 regularization terms.\n\n\n```python\n# @markdown Visualize all of them together (Run Me!)\n\n# @markdown `lambda1=0.001` and `lambda2=0.001`\n\nargs3 = {\n 'test_batch_size': 1000,\n 'epochs': 150,\n 'lr': 5e-3,\n 'momentum': 0.99,\n 'device': DEVICE,\n 'lambda1': 0.001,\n 'lambda2': 0.001\n}\n\n# Intialize the model\nset_seed(seed=SEED)\nmodel = AnimalNet()\nval_acc_l1l2reg, train_acc_l1l2reg, param_norm_l1l2reg, _ = main(args3,\n model,\n train_loader,\n val_loader,\n img_test_dataset,\n reg_function1=l1_reg,\n reg_function2=l2_reg)\n\nplt.figure()\n\nplt.plot(val_acc_l2reg, c='red', ls='dashed')\nplt.plot(train_acc_l2reg,\n label=f\"L2 regularized, $\\lambda_2$={args2['lambda2']}\",\n c='red', ls='solid')\nplt.axhline(y=max(val_acc_l2reg), c='red', ls='dashed')\n\nplt.plot(val_acc_l1reg, c='green', ls = 'dashed')\nplt.plot(train_acc_l1reg,\n label=f\"L1 regularized, $\\lambda_1$={args1['lambda1']}\",\n c='green', ls='solid')\nplt.axhline(y=max(val_acc_l1reg), c='green', ls='dashed')\n\nplt.plot(val_acc_unreg, c='blue', ls = 'dashed')\nplt.plot(train_acc_unreg,\n label='Unregularized', c='blue', ls='solid')\nplt.axhline(y=max(val_acc_unreg), c='blue', ls='dashed')\n\nplt.plot(val_acc_l1l2reg, c='orange', ls='dashed')\nplt.plot(train_acc_l1l2reg,\n label=f\"L1+L2 regularized, $\\lambda_1$={args3['lambda1']}, $\\lambda_2$={args3['lambda2']}\",\n c='orange', ls='solid')\nplt.axhline(y=max(val_acc_l1l2reg), c='orange', ls = 'dashed')\n\nplt.xlabel('epoch')\nplt.ylabel('Accuracy (%)')\nplt.legend()\nplt.show()\n```\n\nNow, let's visualize what these different regularization does to the parameters of the model. We observe the effect by computing the size (technically, the Frobenius norm) of the model parameters\n\n\n```python\n# @markdown #### Visualize Norm of the Models (Train Me!)\nplt.figure()\nplt.plot(param_norm_unreg, label='Unregularized', c='blue')\nplt.plot(param_norm_l1reg, label='L1 Regularized', c='green')\nplt.plot(param_norm_l2reg, label='L2 Regularized', c='red')\nplt.plot(param_norm_l1l2reg, label='L1+L2 Regularized', c='orange')\nplt.xlabel('epoch')\nplt.ylabel('Parameter Norms')\nplt.legend()\nplt.show()\n```\n\nIn the above plots, you should have seen that even after the model achieves 100% train accuracy the val accuracies are fluctuating. This suggests that the model is still trying to learn something. Why would this be the case?\n\n---\n# Section 2: Dropout\n\n*Time estimate: ~25 mins*\n\n\n\n```python\n# @title Video 2: Dropout\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1gU4y1G7V2\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"UZfUzawej3A\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 2: Dropout')\n\ndisplay(out)\n```\n\nIn dropout, we literally drop out (zero out) some neurons during training. Throughout training, on each iteration, standard dropout zeros out some fraction (usually 1/2) of the nodes in each layer before calculating the subsequent layer. Randomly selecting different subsets to dropout introduces noise into the process and reduces overfitting.\n\n
                                        \n\n\nNow let's revisit the toy dataset that we generated above to visualize how the dropout stabilizes training on a noisy dataset. We will slightly modify the architecture we used above to add dropout layers.\n\n\n```python\n# Network Class - 2D\nclass NetDropout(nn.Module):\n def __init__(self):\n super(NetDropout, self).__init__()\n\n self.fc1 = nn.Linear(1, 300)\n self.fc2 = nn.Linear(300, 500)\n self.fc3 = nn.Linear(500, 1)\n # We add two dropout layers\n self.dropout1 = nn.Dropout(0.4)\n self.dropout2 = nn.Dropout(0.2)\n\n def forward(self, x):\n x = F.leaky_relu(self.dropout1(self.fc1(x)))\n x = F.leaky_relu(self.dropout2(self.fc2(x)))\n output = self.fc3(x)\n return output\n```\n\n\n```python\n# @markdown #### Run to train the default network\nset_seed(seed=SEED)\n# creating train data\nX = torch.rand((10, 1))\nX.sort(dim = 0)\nY = 2*X + 2*torch.empty((X.shape[0], 1)).normal_(mean=0, std=1) # adding small error in the data\n\nX = X.unsqueeze_(1)\nY = Y.unsqueeze_(1)\n\n# creating test dataset\nX_test = torch.linspace(0, 1, 40)\nX_test = X_test.reshape((40, 1, 1))\n\n# train the network on toy dataset\nmodel = Net()\ncriterion = nn.MSELoss()\noptimizer = optim.Adam(model.parameters(), lr=1e-4)\nmax_epochs = 10000\niters = 0\n\nrunning_predictions = np.empty((40, (int)(max_epochs/500 + 1)))\n\ntrain_loss = []\ntest_loss = []\nmodel_norm = []\n\nfor epoch in tqdm(range(max_epochs)):\n\n #training\n model_norm.append(calculate_frobenius_norm(model))\n model.train()\n optimizer.zero_grad()\n predictions = model(X)\n loss = criterion(predictions,Y)\n loss.backward()\n optimizer.step()\n\n train_loss.append(loss.data)\n model.eval()\n Y_test = model(X_test)\n loss = criterion(Y_test, 2*X_test)\n test_loss.append(loss.data)\n\n if (epoch % 500 == 0 or epoch == max_epochs - 1):\n running_predictions[:, iters] = Y_test[:, 0, 0].detach().numpy()\n iters += 1\n```\n\n\n```python\n# train the network on toy dataset\n\n# Intialize the model\nset_seed(seed=SEED)\nmodel = NetDropout()\ncriterion = nn.MSELoss()\noptimizer = optim.Adam(model.parameters(), lr=1e-4)\nmax_epochs = 10000\niters = 0\n\nrunning_predictions_dp = np.empty((40, (int)(max_epochs / 500)))\n\ntrain_loss_dp = []\ntest_loss_dp = []\nmodel_norm_dp = []\n\nfor epoch in tqdm(range(max_epochs)):\n\n # training\n model_norm_dp.append(calculate_frobenius_norm(model))\n model.train()\n optimizer.zero_grad()\n predictions = model(X)\n loss = criterion(predictions, Y)\n loss.backward()\n optimizer.step()\n\n train_loss_dp.append(loss.data)\n model.eval()\n Y_test = model(X_test)\n loss = criterion(Y_test, 2*X_test)\n test_loss_dp.append(loss.data)\n\n if (epoch % 500 == 0 or epoch == max_epochs):\n running_predictions_dp[:, iters] = Y_test[:, 0, 0].detach().numpy()\n iters += 1\n```\n\nNow that we have finished training, let's see how the model has evolved over the training process.\n\n\n```python\n# @markdown Animation! (Run Me!)\nset_seed(seed=SEED)\n\nfig = plt.figure(figsize=(8, 6))\nax = plt.axes()\ndef frame(i):\n ax.clear()\n ax.scatter(X[:, 0, :].numpy(), Y[:, 0, :].numpy())\n plot = ax.plot(X_test[:, 0, :].detach().numpy(),\n running_predictions_dp[:, i])\n title = f\"Epoch: {i*500}\"\n plt.title(title)\n ax.set_xlabel(\"X axis\")\n ax.set_ylabel(\"Y axis\")\n return plot\n\n\nanim = animation.FuncAnimation(fig, frame, frames=range(20),\n blit=False, repeat=False,\n repeat_delay=10000)\nhtml_anim = HTML(anim.to_html5_video());\nplt.close()\ndisplay(html_anim)\n```\n\n\n```python\n# @markdown Plot the train and test losses\nplt.figure()\nplt.plot(test_loss_dp, label='test loss dropout', c='blue', ls='dashed')\nplt.plot(test_loss, label='test loss', c='red', ls='dashed')\nplt.ylabel('loss')\nplt.xlabel('epochs')\nplt.title('dropout vs without dropout')\nplt.legend()\nplt.show()\n```\n\n\n```python\n# @markdown Plot the train and test losses\nplt.figure()\nplt.plot(train_loss_dp, label='train loss dropout', c='blue', ls='dashed')\nplt.plot(train_loss, label='train loss', c='red', ls='dashed')\nplt.ylabel('loss')\nplt.xlabel('epochs')\nplt.title('dropout vs without dropout')\nplt.legend()\nplt.show()\n```\n\n\n```python\n# @markdown Plot model weights with epoch\nplt.figure()\nplt.plot(model_norm_dp, label='dropout')\nplt.plot(model_norm, label='no dropout')\nplt.ylabel('norm of the model')\nplt.xlabel('epochs')\nplt.legend()\nplt.title('Size of the model vs Epochs')\nplt.show()\n```\n\nDo you think this performed better than the initial model?\n\n## Section 2.1: Dropout Implementation Caveats\n\n\n* Dropout is used only during training, during testing the complete model weights are used and hence it is important to use model.eval() before testing the model. \n\n* Dropout reduces the capacity of the model during training and hence as a general practice wider networks are used when using dropout. If you are using a dropout with a random probability of 0.5 then you might want to double the number of hidden neurons in that layer.\n\nNow, let's see how dropout fares on the Animal Faces Dataset. We first modify the existing model to include dropout and then train the model.\n\n\n```python\n# Network Class - Animal Faces\nclass AnimalNetDropout(nn.Module):\n def __init__(self):\n super(AnimalNetDropout, self).__init__()\n self.fc1 = nn.Linear(3*32*32, 248)\n self.fc2 = nn.Linear(248, 210)\n self.fc3 = nn.Linear(210, 3)\n self.dropout1 = nn.Dropout(p=0.5)\n self.dropout2 = nn.Dropout(p=0.3)\n\n def forward(self, x):\n x = x.view(x.shape[0], -1)\n x = F.leaky_relu(self.dropout1(self.fc1(x)))\n x = F.leaky_relu(self.dropout2(self.fc2(x)))\n x = self.fc3(x)\n output = F.log_softmax(x, dim=1)\n return output\n```\n\n\n```python\n# Set the arguments\nargs = {\n 'test_batch_size': 1000,\n 'epochs': 200,\n 'lr': 5e-3,\n 'batch_size': 32,\n 'momentum': 0.9,\n 'device': DEVICE,\n 'log_interval': 100\n}\n\n# intialize the model\nset_seed(seed=SEED)\nmodel = AnimalNetDropout()\n\n# Train the model with Dropout\nval_acc_dropout, train_acc_dropout, _, model_dp = main(args,\n model,\n train_loader,\n val_loader,\n img_test_dataset)\n\n# intialize the BigAnimalNet model\nset_seed(seed=SEED)\nmodel = BigAnimalNet()\n\n# Train the model\nval_acc_big, train_acc_big, _, model_big = main(args,\n model,\n train_loader,\n val_loader,\n img_test_dataset)\n\n\n# Train and Test accuracy plot\nplt.figure()\nplt.plot(val_acc_big, label='Val - Big', c='blue', ls='dashed')\nplt.plot(train_acc_big, label='Train - Big', c='blue', ls='solid')\nplt.plot(val_acc_dropout, label='Val - DP', c='magenta', ls='dashed')\nplt.plot(train_acc_dropout, label='Train - DP', c='magenta', ls='solid')\nplt.title('Dropout')\nplt.ylabel('Accuracy (%)')\nplt.xlabel('Epoch')\nplt.legend()\nplt.show()\n```\n\nWhen do you think dropouts can perform bad and do you think their placement within a model matters?\n\n---\n# Section 3: Data Augmentation\n\n*Time estimate: ~15 mins*\n\n\n\n```python\n# @title Video 3: Data Augmentation\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1Xw411d7Pz\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"nm44FhjL3xc\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 3: Data Augmentation')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nData augmentation is often used to increase the number of training samples. Now we will explore the effects of data augmentation on regularization. Here regularization is achieved by adding noise into training data after every epoch.\n\nPytorch's torchvision module provides a few built-in data augmentation techniques, which we can use on image datasets. Some of the techniques we most frequently use are:\n\n\n* Random Crop\n* Random Rotate\n* Vertical Flip\n* Horizontal Flip\n\n\n\n\n```python\n# @markdown #### Data Loader without Data Augmentation\n\n# For reproducibility\ng_seed = torch.Generator()\ng_seed.manual_seed(SEED)\n\n\ntrain_transform = transforms.Compose([\n transforms.ToTensor(),\n transforms.Normalize((0.5, 0.5, 0.5), (0.5, 0.5, 0.5))\n ])\ndata_path = pathlib.Path('.')/'afhq' # using pathlib to be compatible with all OS's\nimg_dataset = ImageFolder(data_path/'train', transform=train_transform)\n\n# Splitting dataset\nimg_train_data, img_val_data,_ = torch.utils.data.random_split(img_dataset, [250,100,14280])\n\n# Creating train_loader and Val_loader\ntrain_loader = torch.utils.data.DataLoader(img_train_data,\n batch_size=batch_size,\n num_workers=2,\n worker_init_fn=seed_worker,\n generator=g_seed)\nval_loader = torch.utils.data.DataLoader(img_val_data,\n batch_size=1000,\n num_workers=2,\n worker_init_fn=seed_worker,\n generator=g_seed)\n```\n\nDefine a DataLoader using [torchvision.transforms](https://pytorch.org/docs/stable/torchvision/transforms.html) which randomly augments the data for us. \n\n\n```python\n# Data Augmentation using transforms\nnew_transforms = transforms.Compose([\n transforms.RandomHorizontalFlip(p=0.1),\n transforms.RandomVerticalFlip(p=0.1),\n transforms.ToTensor(),\n transforms.Normalize((0.5, 0.5, 0.5),\n (0.5, 0.5, 0.5))\n ])\n\ndata_path = pathlib.Path('.')/'afhq' # using pathlib to be compatible with all OS's\nimg_dataset = ImageFolder(data_path/'train', transform=new_transforms)\n# Splitting dataset\nnew_train_data, _,_ = torch.utils.data.random_split(img_dataset,\n [250, 100, 14280])\n\n# For reproducibility\ng_seed = torch.Generator()\ng_seed.manual_seed(SEED)\n\n# Creating train_loader and Val_loader\nnew_train_loader = torch.utils.data.DataLoader(new_train_data,\n batch_size=batch_size,\n worker_init_fn=seed_worker,\n generator=g_seed)\n```\n\n\n```python\n# Set the arguments\nargs = {\n 'epochs': 250,\n 'lr': 1e-3,\n 'momentum': 0.99,\n 'device': DEVICE,\n}\n\n# Intialize the model\nset_seed(seed=SEED)\nmodel_aug = AnimalNet()\n\n# train the model\nval_acc_dataaug, train_acc_dataaug, param_norm_datadug, _ = main(args,\n model_aug,\n new_train_loader,\n val_loader,\n img_test_dataset)\n# Intialize the model\nset_seed(seed=SEED)\nmodel_pure = AnimalNet()\n\nval_acc_pure, train_acc_pure, param_norm_pure, _, = main(args,\n model_pure,\n train_loader,\n val_loader,\n img_test_dataset)\n\n\n# Train and Test accuracy plot\nplt.figure()\nplt.plot(val_acc_pure, label='Val Accuracy Pure',\n c='red', ls='dashed')\nplt.plot(train_acc_pure, label='Train Accuracy Pure',\n c='red', ls='solid')\nplt.plot(val_acc_dataaug, label='Val Accuracy data augment',\n c='blue', ls='dashed')\nplt.plot(train_acc_dataaug, label='Train Accuracy data augment',\n c='blue', ls='solid')\nplt.axhline(y=max(val_acc_pure), c='red', ls='dashed')\nplt.axhline(y=max(val_acc_dataaug), c='blue', ls='dashed')\nplt.title('Data Augmentation')\nplt.ylabel('Accuracy (%)')\nplt.xlabel('Epoch')\nplt.legend()\nplt.show()\n```\n\n\n```python\n# Plot together: without and with augmenetation\nplt.figure()\nplt.plot(param_norm_pure, c='red', label='Without Augmentation')\nplt.plot(param_norm_datadug, c='blue', label='With Augmentation')\nplt.title('Norm of parameters as a function of training epoch')\nplt.xlabel('epoch')\nplt.ylabel('Norm of model parameters')\nplt.legend()\nplt.show()\n```\n\nCan you think of more ways of augmenting training data? (Think of other problems beyond object recogition.)\n\n### Think! 3.1: Thought Question\nWhy does it work better to regularize an overparameterized ANN than to start with a smaller one? Think about the regularization methods you know.\nEach group has a 10 min discussion.\n\n\n```python\n# @title Student Response\nfrom ipywidgets import widgets\n\n\ntext=widgets.Textarea(\n value='Type your answer here and click on `Submit!`',\n placeholder='Type something',\n description='',\n disabled=False\n)\n\nbutton = widgets.Button(description=\"Submit!\")\n\ndisplay(text,button)\n\ndef on_button_clicked(b):\n atform.add_answer('q1', text.value)\n print(\"Submission successful!\")\n\n\nbutton.on_click(on_button_clicked)\n```\n\n\n Textarea(value='Type your answer here and click on `Submit!`', placeholder='Type something')\n\n\n\n Button(description='Submit!', style=ButtonStyle())\n\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content-dl/tree/main//tutorials/W1D5_Regularization/solutions/W1D5_Tutorial2_Solution_519e352b.py)\n\n\n\n---\n# Section 4: Stochastic Gradient Descent\n\n*Time estimate: ~20 mins*\n\n\n```python\n# @title Video 4: SGD\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1nM4y1K7wP\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"rjzlFvJhNqE\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 4: SGD')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n## Section 4.1: Learning Rate\nIn this section, we will see how learning rate can act as regularizer while training a neural network. In summary:\n\n\n* Smaller learning rates regularize less. They slowly converge to deep minima. \n* Larger learning rates regularizes more by missing local minima and converging to broader, flatter minima, which often generalize better.\n\nBut beware, a very large learning rate may result in overshooting or finding a really bad local minimum.\n\n\n\nIn the block below, we will train the Animal Net model with different learning rates and see how that affects the regularization.\n\n\n```python\n# @markdown #### Generating Data Loaders\n\n# For reproducibility\ng_seed = torch.Generator()\ng_seed.manual_seed(SEED)\n\nbatch_size = 128\ntrain_transform = transforms.Compose([\n transforms.ToTensor(),\n transforms.Normalize((0.5, 0.5, 0.5), (0.5, 0.5, 0.5))\n ])\n\ndata_path = pathlib.Path('.')/'afhq' # using pathlib to be compatible with all OS's\nimg_dataset = ImageFolder(data_path/'train', transform=train_transform)\nimg_train_data, img_val_data, = torch.utils.data.random_split(img_dataset, [11700,2930])\n\nfull_train_loader = torch.utils.data.DataLoader(img_train_data,\n batch_size=batch_size,\n num_workers=2,\n worker_init_fn=seed_worker,\n generator=g_seed)\nfull_val_loader = torch.utils.data.DataLoader(img_val_data,\n batch_size=1000,\n num_workers=2,\n worker_init_fn=seed_worker,\n generator=g_seed)\n\ntest_transform = transforms.Compose([\n transforms.ToTensor(),\n transforms.Normalize((0.5, 0.5, 0.5), (0.5, 0.5, 0.5)) # [TO-DO]\n ])\nimg_test_dataset = ImageFolder(data_path/'val', transform=test_transform)\n# img_test_loader = DataLoader(img_test_dataset, batch_size=batch_size,shuffle=False, num_workers=1)\nclasses = ('cat', 'dog', 'wild')\n```\n\n\n```python\n# Set the arguments\nargs = {\n 'test_batch_size': 1000,\n 'epochs': 350,\n 'batch_size': 32,\n 'momentum': 0.99,\n 'device': DEVICE\n}\n\nlearning_rates = [5e-4, 1e-3, 5e-3]\nacc_dict = {}\n\nfor i, lr in enumerate(learning_rates):\n # Initialize the model\n set_seed(seed=SEED)\n model = AnimalNet()\n # Learning rate\n args['lr'] = lr\n # Train the model\n val_acc, train_acc, param_norm, _ = main(args,\n model,\n train_loader,\n val_loader,\n img_test_dataset)\n # store the outputs\n acc_dict[f'val_{i}'] = val_acc\n acc_dict[f'train_{i}'] = train_acc\n acc_dict[f'param_norm_{i}'] = param_norm\n```\n\n\n```python\n# @markdown Plot Train and Validation accuracy (Run me)\n\nplt.figure()\nfor i, lr in enumerate(learning_rates):\n plt.plot(acc_dict[f'val_{i}'], linestyle='dashed',\n label=f'lr={lr:0.1e} - validation')\n plt.plot(acc_dict[f'train_{i}'], label=f'{lr:0.1e} - train')\n\n print(f\"Maximum Test Accuracy obtained with lr={lr:0.1e}: {max(acc_dict[f'val_{i}'])}\")\n\nplt.title('Optimal Learning Rate')\nplt.ylabel('Accuracy (%)')\nplt.xlabel('Epoch')\nplt.legend()\nplt.show()\n```\n\n\n```python\n# @markdown Plot parametric norms (Run me)\nplt.figure()\n\nfor i, lr in enumerate(learning_rates):\n plt.plot(acc_dict[f'param_norm_{i}'],label=f'lr={lr:0.2e}')\n\nplt.legend()\nplt.xlabel('epoch')\nplt.ylabel('parameter norms')\nplt.show()\n```\n\nIn the model above, we observe something different from what we expected. Why do you think this is happening?\n\n---\n# Section 5: Hyperparameter Tuning\n\n*Time estimate: ~5 mins*\n\n\n\n```python\n# @title Video 5: Hyperparameter tuning\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1E44y127Sn\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"HgkiKRYc-3A\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 5: Hyperparameter tuning')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n\n\nHyper-Parameter tuning is often difficult and time consuming. It is a key part of training any Deep Learning model to give good generalization. There are a few techniques that we can use to guide us during the search. \n\n\n\n* Grid Search: Try all possible combinations of hyperparameters\n* Random Search: Randomly try different combinations of hyperparameters\n* Coordinate-wise Gradient Descent: Start at one set of hyperparameters and try changing one at a time, accept any changes that reduce your validation error\n* Bayesian Optimization/ Auto ML: Start from a set of hyperparameters that have worked well on a similar problem, and then do some sort of local exploration (e.g., gradient descent) from there.\n\nThere are lots of choices, like what range to explore over, which parameter to optimize first, etc. Some hyperparameters don\u2019t matter much (people use a dropout of either 0.5 or 0, but not much else). Others can matter a lot more (e.g., size and depth of the neural net). The key is to see what worked on similar problems.\n\nOne can automate the process of tuning the network Architecture using \"Neural Architecture Search\", which designs new architectures using a few building blocks (Linear, Convolutional, Convolution Layers, etc.) and optimizes the design based on performance using a wide range of techniques such as Grid Search, Reinforcement Learning, GD, Evolutionary Algorithms, etc. This obviously requires very high computer power. Read this [article](https://lilianweng.github.io/lil-log/2020/08/06/neural-architecture-search.html) to learn more about NAS. \n\n\n## Think! 5: Overview of regularization techniques\n\nWhich regularization technique today do you think had the biggest effect on the network? Why might do you think so? Can you apply all of the regularization methods on the same network?\n\n\n```python\n# @title Student Response\nfrom ipywidgets import widgets\n\n\ntext=widgets.Textarea(\n value='Type your answer here and click on `Submit!`',\n placeholder='Type something',\n description='',\n disabled=False\n)\n\nbutton = widgets.Button(description=\"Submit!\")\n\ndisplay(text,button)\n\ndef on_button_clicked(b):\n atform.add_answer('q2', text.value)\n print(\"Submission successful!\")\n\n\nbutton.on_click(on_button_clicked)\n```\n\n\n Textarea(value='Type your answer here and click on `Submit!`', placeholder='Type something')\n\n\n\n Button(description='Submit!', style=ButtonStyle())\n\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content-dl/tree/main//tutorials/W1D5_Regularization/solutions/W1D5_Tutorial2_Solution_a308b739.py)\n\n\n\n---\n# Summary\n\nCongratulations! The first week of NMA-DL has ended! In this tutorial, you learned more techniques of regulariation, i.e., L1 and L2 regularization, Dropout, and Data Augmenetation. Finally, you have seen the learning rate of SGD can act as a reularizer. An iteresting paper can be found [here](https://arxiv.org/abs/1611.03530). \n\nIf you have time left, see the bonus material on *Adversarial Attacks*!\n\n\n```python\n# @title Airtable Submission Link\nfrom IPython import display as IPydisplay\nIPydisplay.HTML(\n f\"\"\"\n
                                        \n \n \n
                                        \"\"\" )\n```\n\n\n\n\n\n
                                        \n \n \n
                                        \n\n\n\n---\n# Bonus: Adversarial Attacks\n\n*Time estimate: ~15 mins*\n\n\n```python\n# @title Video 6: Adversarial Attacks\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV19o4y1X74u\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"LzPPoiKi5jE\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 6: Adversarial Attacks')\n\ndisplay(out)\n```\n\nDesigning perturbations to the input data to trick a machine learning model is called an \"adversarial attack\". These attacks are an inevitable consequence of learning in high dimensional space with complex decision boundaries. Depending on the application, these attacks can be very dangerous.\n\n\n\n\nHence, it is important for us to build models which can defend against such attacks. One possible way to do it is by regularizing the networks, which smooths the decision boundaries. A few ways of building models robust to such attacks are:\n\n\n\n* [Defensive Distillation](https://deepai.org/machine-learning-glossary-and-terms/defensive-distillation) : Models trained via distillation are less prone to such attacks as they are trained on soft labels as there is an element of randomness in the training process.\n* [Feature Squeezing](https://evademl.org/squeezing/): Identifies adversarial attacks for on-line classifiers whose model is being used by comparing model's perdiction before and after squeezing the input. \n* [SGD](https://arxiv.org/abs/1706.06083) You can also pick weight to minimize what the adversary is trying to maximize via SGD.\n\n\nRead more about adversarial attacks [here](https://openai.com/blog/adversarial-example-research/)\n\n", "meta": {"hexsha": "e193ba8ad6a408f79f95484f3c1ceec412827e08", "size": 572384, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W1D5_Regularization/student/ED_W1D5_Tutorial2.ipynb", "max_stars_repo_name": "eduardojdiniz/course-content-dl", "max_stars_repo_head_hexsha": "8d66641683651bce7b0179b6d890aef5a048a8b9", "max_stars_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tutorials/W1D5_Regularization/student/ED_W1D5_Tutorial2.ipynb", "max_issues_repo_name": "eduardojdiniz/course-content-dl", "max_issues_repo_head_hexsha": "8d66641683651bce7b0179b6d890aef5a048a8b9", "max_issues_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/W1D5_Regularization/student/ED_W1D5_Tutorial2.ipynb", "max_forks_repo_name": "eduardojdiniz/course-content-dl", "max_forks_repo_head_hexsha": "8d66641683651bce7b0179b6d890aef5a048a8b9", "max_forks_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 191.7534338358, "max_line_length": 136216, "alphanum_fraction": 0.8851679991, "converted": true, "num_tokens": 15902, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878414043816, "lm_q2_score": 0.6893056295505783, "lm_q1q2_score": 0.42398351174815346}} {"text": "\n\n# DataRecord Tutorial\n\n
                                        \nFor more Landlab tutorials, click here: https://landlab.readthedocs.io/en/latest/user_guide/tutorials.html\n
                                        \n\n\nThis tutorial illustrates how to record variables of a Landlab model using DataRecord.\n\n## What is DataRecord?\nDataRecord is a data structure that can hold data variables relating to a Landlab model or to items living on the [Landlab grid](../grid_object_demo/grid_object_demo.ipynb).\n\nDataRecord is built on [xarray](http://xarray.pydata.org/en/stable/index.html)'s Dataset structure: a multi-dimensional, in memory, array database. Dataset implements the mapping interface with keys given by variable names and values given by DataArray objects for each variable name. DataRecord inherits all the methods and attributes from xarray.Dataset.\n\nA DataRecord can have one or both (or none) of the following dimensions:\n- `time`: The simulated time in the model.\n- `item_id`: An identifier of a generic item in the model.\n\nCoordinates are one dimensional arrays used for label-based indexing. \n\nThe examples below illustrate different use cases for DataRecord. \n\nWe start by importing the necessary libraries:\n\n\n```python\nimport numpy as np\nfrom landlab import RasterModelGrid\nfrom landlab.data_record import DataRecord\n\nfrom landlab import imshow_grid\nimport matplotlib.pyplot as plt\nfrom matplotlib.pyplot import plot, subplot, xlabel, ylabel, title, legend, figure\n%matplotlib inline\n```\n\n## Case 1. DataRecord with 1 dimension: time\nLet's start with an example where we set DataRecord to have only `time` as a dimension.\nAn example variable that varies over time and relates to the Landlab grid could be the mean elevation of the topographic surface. We will store this example variable in DataRecord.\n\nWe create a Raster grid, create a field (at nodes) called `topographic__elevation` and populate it with random values.\n\n\n```python\ngrid_1 = RasterModelGrid((10, 10), (1., 1.))\nz = np.random.rand(100)\n_ = grid_1.add_field('topographic__elevation', z, at='node')\n```\n\nPrint the current mean elevation.\n\n\n```python\ncurrent_mean = np.mean(grid_1.at_node['topographic__elevation'])\nprint(current_mean)\n```\n\nNow we will create a DataRecord that will hold the data variable `mean_elevation` relating to `grid_1`. The first value, at time=0 is the current mean elevation on the grid.\n\n\n```python\ndr_1 = DataRecord(grid_1,\n time=[0.],\n items=None,\n data_vars={'mean_elevation': (['time'], ([current_mean]))},\n attrs={'mean_elevation': 'y'})\n```\n\nThe input arguments passed in this case are: the grid, time (as a 1-element list), a data variable dictionary and an attributes dictionary. Note that `items` is not filled, we will see its use in other cases below.\n\nNote the format of the `data_vars` dictionary: \n```python\n {'variable_name_1' : (['dimensions'], variable_data_1),\n 'variable_name_2' : (['dimensions'], variable_data_2),\n ...}\n```\n\nThe attributes dictionary `attrs` can be used to store metadata about the variables: in this example, we use it to store the variable units.\n\nSo far, our DataRecord `dr_1` holds one variable `mean_elevation` with one record at time=0.\n\n\n\n```python\ndr_1\n```\n\nWe can visualise this data structure as a [pandas dataframe](https://pandas.pydata.org/pandas-docs/stable/generated/pandas.DataFrame.html):\n\n\n```python\ndr_1.dataset.to_dataframe()\n```\n\nNow we will run a simple model where the grid surface is uplifted several times and the mean elevation is recorded at every time step. We use the method `add_record` to put the new value in the DataRecord `dr_1`:\n\n\n```python\ntotal_time = 100\ndt = 20\n\nuplift_rate = 0.01 # m/y\n\nfor t in range(20, total_time, dt):\n grid_1.at_node['topographic__elevation'] += uplift_rate * dt\n dr_1.add_record(time=[t],\n new_record={\n 'mean_elevation':\n (['time'],\n ([np.mean(grid_1.at_node['topographic__elevation'])]))\n })\n```\n\nLet's see what was recorded:\n\n\n```python\ndr_1.dataset['mean_elevation'].values\n```\n\nThe corresponding time coordinates are:\n\n\n```python\ndr_1.dataset.time.values\n```\n\nNotice the different syntax used here: \n- `time` is a **dimension** and can be called by `dr_1.time` (or `dr_1['time']`)\n- whereas `mean_elevation` is a **variable** and must be called by `dr_1['mean_elevation']`\n\nDataRecord also has the handy property `time_coordinates` that returns these values as a list:\n\n\n\n\n\n```python\ndr_1.time_coordinates\n```\n\nYou can use the methods `get_data` and `set_data` to access and change the data:\n\n\n```python\ndr_1.get_data(time=[20.], data_variable='mean_elevation')\n```\n\n\n```python\ndr_1.set_data(time=[80.], data_variable='mean_elevation', new_value=1.5)\n\ndr_1.dataset['mean_elevation']\n```\n\n## Case 2. DataRecord with 1 dimension: item_id\nAn important feature of DataRecord is that it allows to create **items** that live on grid elements, and variables describing them. For instance, we can create *boulders* and store information about their *size* and *lithology*.\n\nTo create items, we need to instantiate a DataRecord and pass it a dictionary describing where each item lives on the Landlab grid. The format of this dictionary is: \n```python\n {'grid_element' : [grid_element],\n 'element_id' : [element_id]}\n```\n \nwhere:\n- `grid_element` is a str or number-of-items-long array containing strings of the grid element(s) on which the items live (e.g.: node, link). Valid locations depend on the grid type (`my_grid.groups` gives the valid locations for your grid). If `grid_element` is provided as a string, it is assumed that all items live on the same type of grid element.\n- `element_id` is an array of integers identifying the grid element IDs on which each item resides. For each item, `element_id` must be less than the number of this item's `grid_element` that exist on the grid. For example, if the grid has 10 links, no item can live at link 10 or link -3 because only links 0 to 9 exist in this example.\n\n\n\n```python\ngrid_2 = RasterModelGrid((5, 5), (2, 2))\n\nboulders = {\n 'grid_element': 'node',\n 'element_id': np.array([6, 11, 12, 17, 12])\n}\n\ninitial_boulder_sizes = np.array([1, 1.5, 3, 1, 2])\nboulder_lithologies = np.array(\n ['sandstone', 'granite', 'sandstone', 'sandstone', 'limestone'])\n\ndr_2 = DataRecord(grid_2,\n time=None,\n items=boulders,\n data_vars={\n 'boulder_size': (['item_id'], initial_boulder_sizes),\n 'boulder_litho': (['item_id'], boulder_lithologies)\n },\n attrs={'boulder_size': 'm'})\ndr_2.dataset.to_dataframe()\n```\n\nEach *item* (in this case, each boulder) is designated by an `item_id`, its position on the grid is described by a `grid_element` and an `element_id`.\n\nWe can use the method `add_item` to add new boulders to the record:\n\n\n```python\ndr_2.add_item(\n new_item={\n 'grid_element': np.array(['link', 'node']),\n 'element_id': np.array([24, 8])\n },\n new_item_spec={'boulder_size': (['item_id'], np.array([1.2, 2.]))})\n\ndr_2.dataset.to_dataframe()\n```\n\nNotice that we did not specify the lithologies of the new boulders, their recorded values are thus set as `NaN`. We can use the `set_data` method to report the boulder lithologies: \n\n\n```python\ndr_2.set_data(data_variable='boulder_litho',\n item_id=[5, 6],\n new_value=['sandstone', 'granite'])\ndr_2.dataset.to_dataframe()\n```\n\nWe can use the method `calc_aggregate_value` to apply a function to a variable aggregated at grid elements. For example, we can calculate the mean size of boulders on each node:\n\n\n```python\nmean_size = dr_2.calc_aggregate_value(func=np.mean,\n data_variable='boulder_size')\nmean_size\n```\n\nNotice that boulder #5 is on a link so it is not taken into account in this calculation.\n\n\n```python\n# replace nans with 0:\nmean_size[np.isnan(mean_size)] = 0\n\n# show unfiltered mean sizes on the grid:\nimshow_grid(grid_2, mean_size)\n```\n\nBefore doing this calculation we could filter by lithology and only use the 'sandstone' boulders in the calculation:\n\n\n```python\n# define a filter array:\nfilter_litho = (dr_2.dataset['boulder_litho'] == 'sandstone')\n\n# aggregate by node and apply function numpy.mean on boulder_size\nfiltered_mean = dr_2.calc_aggregate_value(func=np.mean,\n data_variable='boulder_size',\n at='node',\n filter_array=filter_litho)\n\nfiltered_mean\n```\n\n## Case 3. DataRecord with 2 dimensions: item_id and time\n\nWe may want to record variables that have both dimensions `time` *and* `item_id`.\n\nIn the previous example, some variables that characterize the items (boulders) may not vary with time, such as `boulder_lithology`. Although it can be interesting to keep track of the change in size through time. We will redefine the DataRecord such that the variable `boulder_size` varies among the items/boulders (identified by `item_id`) and through `time`. The variable `boulder_litho` varies only among the items/boulders and this lithogy variable does not vary through time.\n\n\n```python\ngrid_3 = RasterModelGrid((5, 5), (2, 2))\n\ninitial_boulder_sizes_3 = np.array([[10], [4], [8], [3], [5]])\n# boulder_lithologies = np.array(['sandstone', 'granite', 'sandstone', 'sandstone', 'limestone']) #same as above, already run\n\nboulders_3 = {\n 'grid_element': 'node',\n 'element_id': np.array([[6], [11], [12], [17], [12]])\n}\n\ndr_3 = DataRecord(grid_3,\n time=[0.],\n items=boulders_3,\n data_vars={\n 'boulder_size': (['item_id',\n 'time'], initial_boulder_sizes_3),\n 'boulder_litho': (['item_id'], boulder_lithologies)\n },\n attrs={'boulder_size': 'm'})\ndr_3\n```\n\nNote that the syntax to define the `initial_boulder_sizes_3` (as well as `element_id`) has changed: they are number-of-items-by-1 arrays because they vary along both `time` and `item_id` (compared to `boulder_lithologies` which is just number-of-items long as it only varies along `item_id`).\n\n\n```python\nboulder_lithologies.shape, initial_boulder_sizes.shape, initial_boulder_sizes_3.shape\n```\n\nLet's define a very simple erosion law for the boulders:\n\n$$\n\\begin{equation}\n\\frac{dD}{dt} = -k_{b} . D\n\\end{equation}\n$$\n\nwhere $D$ is the boulder diameter $[L]$ (this value represents the `boulder_size` variable), $t$ is time, and $k_{b}$ is the block erodibility $[L.T^{-1}]$.\n\nWe will now model boulder erosion and use DataRecord to store their size through time.\n\n\n```python\ndt = 100\ntotal_time = 100000\n\ntime_index = 1\n\nfor t in range(dt, total_time, dt):\n\n # create a new time coordinate:\n dr_3.add_record(time=np.array([t]))\n\n # this propagates grid_element and element_id values forward in time (instead of the 'nan' default filling):\n dr_3.ffill_grid_element_and_id()\n\n for i in range(0, dr_3.number_of_items):\n # value of block erodibility:\n if dr_3.dataset['boulder_litho'].values[i] == 'limestone':\n k_b = 10**-5\n elif dr_3.dataset['boulder_litho'].values[i] == 'sandstone':\n k_b = 3 * 10**-6\n elif dr_3.dataset['boulder_litho'].values[i] == 'granite':\n k_b = 3 * 10**-7\n else:\n print('Unknown boulder lithology')\n\n dr_3.dataset['boulder_size'].values[i, time_index] = dr_3.dataset[\n 'boulder_size'].values[i, time_index - 1] - k_b * dr_3.dataset[\n 'boulder_size'].values[i, time_index - 1] * dt\n\n time_index += 1\n\nprint('Done')\n```\n\n\n```python\nfigure(figsize=(15, 8))\n\ntime = range(0, total_time, dt)\nboulder_size = dr_3.dataset['boulder_size'].values\n\nsubplot(121)\nplot(time, boulder_size[1], label='granite')\nplot(time, boulder_size[3], label='sandstone')\nplot(time, boulder_size[-1], label='limestone')\nxlabel('Time (yr)')\nylabel('Boulder size (m)')\nlegend(loc='lower left')\ntitle('Boulder erosion by lithology')\n\n# normalized plots\nsubplot(122)\nplot(time, boulder_size[1] / boulder_size[1, 0], label='granite')\nplot(time, boulder_size[2] / boulder_size[2, 0], label='sandstone')\nplot(time, boulder_size[-1] / boulder_size[-1, 0], label='limestone')\nxlabel('Time (yr)')\nylabel('Boulder size normalized to size at t=0 (m)')\nlegend(loc='lower left')\ntitle('Normalized boulder erosion by lithology')\nplt.show()\n```\n\n## Other properties provided by DataRecord\n\n\n```python\ndr_3.variable_names\n```\n\n\n```python\ndr_3.number_of_items\n```\n\n\n```python\ndr_3.item_coordinates\n```\n\n\n```python\ndr_3.number_of_timesteps\n```\n\n\n```python\ndr_1.time_coordinates\n```\n\n\n```python\ndr_1.earliest_time\n```\n\n\n```python\ndr_1.latest_time\n```\n\n\n```python\ndr_1.prior_time\n```\n\n# More on DataRecord\n\nDataRecord is the data structure on which the following Landlab components are based:\n- ClastTracker (coming soon)\n- SpeciesEvolver (coming soon)\n\n\n```python\n\n```\n", "meta": {"hexsha": "0262c56cec086b4fdd91bce6ad37a36f84129451", "size": 21042, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/tutorials/data_record/DataRecord_tutorial.ipynb", "max_stars_repo_name": "amanaster2/landlab", "max_stars_repo_head_hexsha": "ea17f8314eb12e3fc76df66c9b6ff32078caa75c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2019-04-05T16:41:40.000Z", "max_stars_repo_stars_event_max_datetime": "2021-06-11T20:33:14.000Z", "max_issues_repo_path": "notebooks/tutorials/data_record/DataRecord_tutorial.ipynb", "max_issues_repo_name": "amanaster2/landlab", "max_issues_repo_head_hexsha": "ea17f8314eb12e3fc76df66c9b6ff32078caa75c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 7, "max_issues_repo_issues_event_min_datetime": "2020-04-03T21:40:07.000Z", "max_issues_repo_issues_event_max_datetime": "2020-12-02T22:57:12.000Z", "max_forks_repo_path": "notebooks/tutorials/data_record/DataRecord_tutorial.ipynb", "max_forks_repo_name": "amanaster2/landlab", "max_forks_repo_head_hexsha": "ea17f8314eb12e3fc76df66c9b6ff32078caa75c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2018-05-21T17:40:58.000Z", "max_forks_repo_forks_event_max_datetime": "2020-08-21T05:44:41.000Z", "avg_line_length": 30.4515195369, "max_line_length": 486, "alphanum_fraction": 0.5632069195, "converted": true, "num_tokens": 3419, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858117, "lm_q2_score": 0.734119521083126, "lm_q1q2_score": 0.423950620917933}} {"text": "# Data analysis in the Amazon audit\nMost of the materials in this notebook were kindly provided by Piotr Sapiezynski (www.sapiezynski.com)\n\n### The motivation behind the analysis\nRead it here\nhttps://themarkup.org/amazons-advantage/2021/10/14/amazon-puts-its-own-brands-first-above-better-rated-products\n\n\n```python\n# installing the packages for today (if needed)\n!pip install -U pandas\n!pip install -U scikit-learn\n!pip install -U statsmodels\n```\n\n Requirement already satisfied: pandas in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (1.4.1)\n Collecting pandas\n Downloading pandas-1.4.2-cp39-cp39-macosx_10_9_x86_64.whl (11.1 MB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 11.1 MB 2.7 MB/s eta 0:00:01\n \u001b[?25hRequirement already satisfied: python-dateutil>=2.8.1 in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (from pandas) (2.8.2)\n Requirement already satisfied: pytz>=2020.1 in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (from pandas) (2021.3)\n Requirement already satisfied: numpy>=1.18.5 in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (from pandas) (1.20.3)\n Requirement already satisfied: six>=1.5 in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (from python-dateutil>=2.8.1->pandas) (1.16.0)\n Installing collected packages: pandas\n Attempting uninstall: pandas\n Found existing installation: pandas 1.4.1\n Uninstalling pandas-1.4.1:\n Successfully uninstalled pandas-1.4.1\n Successfully installed pandas-1.4.2\n Requirement already satisfied: scikit-learn in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (1.0.2)\n Requirement already satisfied: numpy>=1.14.6 in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (from scikit-learn) (1.20.3)\n Requirement already satisfied: threadpoolctl>=2.0.0 in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (from scikit-learn) (2.2.0)\n Requirement already satisfied: joblib>=0.11 in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (from scikit-learn) (1.1.0)\n Requirement already satisfied: scipy>=1.1.0 in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (from scikit-learn) (1.7.1)\n Requirement already satisfied: statsmodels in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (0.13.2)\n Requirement already satisfied: packaging>=21.3 in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (from statsmodels) (21.3)\n Requirement already satisfied: patsy>=0.5.2 in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (from statsmodels) (0.5.2)\n Requirement already satisfied: numpy>=1.17 in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (from statsmodels) (1.20.3)\n Requirement already satisfied: scipy>=1.3 in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (from statsmodels) (1.7.1)\n Requirement already satisfied: pandas>=0.25 in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (from statsmodels) (1.4.2)\n Requirement already satisfied: pyparsing!=3.0.5,>=2.0.2 in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (from packaging>=21.3->statsmodels) (3.0.4)\n Requirement already satisfied: pytz>=2020.1 in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (from pandas>=0.25->statsmodels) (2021.3)\n Requirement already satisfied: python-dateutil>=2.8.1 in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (from pandas>=0.25->statsmodels) (2.8.2)\n Requirement already satisfied: six in /Users/roberta/opt/anaconda3/lib/python3.9/site-packages (from patsy>=0.5.2->statsmodels) (1.16.0)\n\n\n\n```python\nimport pandas as pd\nfrom sklearn.model_selection import train_test_split\n```\n\n\n```python\n# this is the data from the markup audit, downloaded from their github: \n# https://github.com/the-markup/investigation-amazon-brands \n\ndata = pd.read_csv('https://www.sapiezynski.com/cs4910/markup/data.csv')\n```\n\n\n```python\ndata\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        search_termplaced_higherstars_deltareviews_deltais_shipped_by_amazonis_sold_by_amazonis_amazonis_top_clickedrandom_noiseasin_1asin_2
                                        0#10 envelopeTrue0.01654.00220-2.569736B06VVLD2GLB01D0OANU4
                                        1#6 envelopeTrue0.17844.002220.951532B06X15WSLLB07JNXMBSX
                                        21 inch binderFalse0.0-9383.000-200.618294B00A45VF2SB01BRGTWOA
                                        31% milkFalse0.0-183.0000-20.645435B07W5Z8SJ8B07WC9MMPD
                                        410 dollar giftsTrue0.875410.00200-1.075435B00F4CEHNKB07FCNYND8
                                        ....................................
                                        1410zwave hubTrue0.8848.000020.047647B07D19VVTXB077Y939JQ
                                        1411zxzy womens hoodiesTrue-0.16.000000.009533B08F7RFZGPB08KT3BD2X
                                        1412zyliss knivesTrue0.03010.00000-0.148278B01DKEB5OCB0753SKVZX
                                        1413zyrtec for kidsTrue-0.22184.022001.474693B004E0QGOQB088X7151W
                                        1414zz plantTrue0.1379.0222-21.686695B07QD5FXRZB07RJYZ7W3
                                        \n

                                        1415 rows \u00d7 11 columns

                                        \n
                                        \n\n\n\n\n\n\n```python\ncolumns = ['stars_delta', 'reviews_delta', 'is_amazon',\n 'is_shipped_by_amazon', 'is_sold_by_amazon',\n 'is_top_clicked', 'placed_higher']\n```\n\n\n```python\ndataset = data[columns]\n```\n\n\n```python\ntrain, test = train_test_split(dataset, test_size=0.2, stratify=dataset['placed_higher'])\n```\n\n\n```python\nfeatures = ['is_amazon',\n 'is_shipped_by_amazon', 'is_sold_by_amazon',\n 'is_top_clicked', 'stars_delta', 'reviews_delta', ]\n```\n\n\n```python\nfrom sklearn.ensemble import RandomForestClassifier\nclf = RandomForestClassifier(n_estimators=500, max_depth=3, max_features = 6)\nclf.fit(train[features], train['placed_higher'])\n\n```\n\n\n\n\n RandomForestClassifier(max_depth=3, max_features=6, n_estimators=500)\n\n\n\n\n```python\n# let's see how well it does in predicting the test set\n# 1. predict the test data\ny_pred = clf.predict(test[features])\n\n# 2. calculate accuracy, i.e. the fraction of time that the prediction was correct out of all predictions\n(y_pred == test['placed_higher']).sum()/len(y_pred)\n```\n\n\n\n\n 0.6855123674911661\n\n\n\n\n```python\nfor feature, importance in zip(features, clf.feature_importances_):\n print(feature, importance)\n```\n\n is_amazon 0.9032928487280287\n is_shipped_by_amazon 0.007487632851518247\n is_sold_by_amazon 0.002829947430288105\n is_top_clicked 0.002445399880340501\n stars_delta 0.019225107462222012\n reviews_delta 0.06471906364760226\n\n\n\n\nSo clearly, `is_amazon` is the most important, but what exactly does it mean, how does it translate the the probability of being the top result? RandomForests won't tell us... But logistic regression can.\n\nFirst, quick reminder on regression. \n\n**Linear** regression has the following formula:\n\n$$ y = \\beta_0 + \\beta_1x_1 + \\beta_2x_2 + ... $$\n\nThis means that:\n* when all variables are equal to 0, the outcome variable $y$ is equal to $\\beta_0$, the intercept.\n* a **unit** change in $x_1$ corresponds to $y$ changing by $\\beta_1$ \n\nIt might not be a powerful model, but it offers a clear interpretation/explanation in regression problems.\n\n**Logistic** regression is mostly used for binary classification and it looks similar to the linear regression, but now the outcome variable is the logarithm of odds of success.\n\n$$ ln(\\frac{\\pi}{1-\\pi}) = \\beta_0 + \\beta_1x_1 + \\beta_2x_2 + ... $$\n\nwhere $pi$ is the probability of success, so $1-\\pi$ is the probability of failure.\n\nThen, a unit change in $x_1$ corresponds to $\\beta_1$ change in log odd ratios... but that is not intuitive at all. Let's rewrite it to calculate the probability directly:\n\n$$\\begin{align}\nln(\\frac{\\pi}{1-\\pi}) &= \\beta_0 + \\beta_1x_1 + \\beta_2x_2 + ... \\\\\n\\frac{\\pi}{1-\\pi} &= e^{\\beta_0 + \\beta_1x_1 + \\beta_2x_2 + ...} \\\\\n{\\pi} &= ({1-\\pi})e^{\\beta_0 + \\beta_1x_1 + \\beta_2x_2 + ...} \\\\\n{\\pi} &= e^{\\beta_0 + \\beta_1x_1 + \\beta_2x_2 + ...}-\\pi e^{\\beta_0 + \\beta_1x_1 + \\beta_2x_2 + ...}\\\\\n{\\pi}(1+e^{\\beta_0 + \\beta_1x_1 + \\beta_2x_2 + ...}) &= e^{\\beta_0 + \\beta_1x_1 + \\beta_2x_2 + ...} \\\\\n{\\pi} &= \\frac{e^{\\beta_0 + \\beta_1x_1 + \\beta_2x_2 + ...}}{1+e^{\\beta_0 + \\beta_1x_1 + \\beta_2x_2 + ...}}\n\\end{align}$$\n\nSo, while we cannot directly say from that a unit change of $x_1$ corresponds to a certain change in the probability of the outcome, we can just plug zeros and ones into the equation above and compute the changes.\n\nLet's load the data from scratch and do some cleaning (I'm dividing by two, because The Markup left the values of -2, 0, 2 so they never have a \"unit\" change. If we divide them by two, the unit change actually is meaningful).\n\n\n```python\ndata = pd.read_csv('https://www.sapiezynski.com/cs4910/markup/data.csv')\ndata['is_shipped_by_amazon'] /= 2\ndata['is_sold_by_amazon'] /= 2\ndata['is_amazon'] /= 2\ndata['is_top_clicked'] /= 2\ndataset = data[columns]\n\ntrain, test = train_test_split(dataset, test_size=0.2, stratify=dataset['placed_higher'])\n\n```\n\nLet's start with a model that only has the intercept (constant) and `is_amazon` as features.\n\n\n```python\nimport statsmodels.api as sm\nfrom statsmodels.tools.tools import add_constant\n\n# adding the intercept as a variable\ntrain = add_constant(train)\n\n# selecting which features to train on\nfeatures = ['const', 'is_amazon']\n\n# training a logistic regression model. First the dependent (outcome variable), then the features\nlog_reg = sm.Logit(train['placed_higher'], train[features]).fit()\n```\n\n Optimization terminated successfully.\n Current function value: 0.494906\n Iterations 6\n\n\n\n```python\nlog_reg.summary()\n```\n\n\n\n\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        Logit Regression Results
                                        Dep. Variable: placed_higher No. Observations: 1132
                                        Model: Logit Df Residuals: 1130
                                        Method: MLE Df Model: 1
                                        Date: Mon, 07 Feb 2022 Pseudo R-squ.: 0.2859
                                        Time: 11:20:43 Log-Likelihood: -560.23
                                        converged: True LL-Null: -784.58
                                        Covariance Type: nonrobust LLR p-value: 1.391e-99
                                        \n\n\n \n\n\n \n\n\n \n\n
                                        coef std err z P>|z| [0.025 0.975]
                                        const 0.0883 0.073 1.208 0.227 -0.055 0.232
                                        is_amazon 2.4883 0.163 15.278 0.000 2.169 2.807
                                        \n\n\n\nWe will mostly be looking at the bottom table.\n\n`const` (the intercept) translates to the probability if all other variables are equal to 0\n\n$$ p = \\frac{e^{\\beta_{const}}}{1+e^{\\beta_{const}}} $$\n\n\n```python\nimport numpy as np\n\n# the beta coefficients are stored in the .params list, with the constant being first\n# get the constant out of the fit model:\nconst = log_reg.params[0]\n\nnp.e**const/(1+np.e**const) \n\n```\n\n\n\n\n 0.5220557281748975\n\n\n\n\n```python\n\n```\n\n\n\n\n 0.5220557281748975\n\n\n\nThis should be very close to 50%. In our case if all variables are zero, it means that either:\n1. The products are both from amazon\n2. The products are both not from amazon\n\nWe don't have a way to differentiate them, so it's a coin toss (50%)\n\nThe other coefficients describe the change in odds that corresponds to a unit change in the variable.\n* if the coefficient is positive, it means that an **increase** of variable corresponds to an **increase** in the odds of the positive outcome\n* if the coefficient is positive, it means that an **increase** of variable corresponds to an **decrease** in the odds of the positive outcome\n\n\nSo what happens when `is_amazon` is equal to 1, i.e. one product is an amazon product and the other isn't?\nBy looking at the `coef` we already know it's going to go up (because the coefficient is positive). But by How much? Let's calculate:\n\n\n$$ p = \\frac{e^{\\beta_{const} + \\beta_{is\\_amazon}}}{1+e^{\\beta_{const} + \\beta_{is\\_amazon}}} $$\n\n\n```python\nnumerator = np.e**(log_reg.params[0] + log_reg.params[1])\nprint(numerator/(1+numerator))\n```\n\n 0.9293369710808036\n\n\nThat means that in this dataset 93\\% of the time we have two products where one is from amazon and the other isn't, it's the amazon product that will be on the first place.\n\nSolved? Well no, because maybe this is just because amazon products are better. Let's include the star difference in the analysis to **control** for this:\n\n\n```python\nfeatures = ['const', 'is_amazon','stars_delta', ]\nlog_reg = sm.Logit(train['placed_higher'], train[features]).fit()\nlog_reg.summary()\n```\n\n Optimization terminated successfully.\n Current function value: 0.494329\n Iterations 6\n\n\n\n\n\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        Logit Regression Results
                                        Dep. Variable: placed_higher No. Observations: 1132
                                        Model: Logit Df Residuals: 1129
                                        Method: MLE Df Model: 2
                                        Date: Mon, 07 Feb 2022 Pseudo R-squ.: 0.2868
                                        Time: 11:21:33 Log-Likelihood: -559.58
                                        converged: True LL-Null: -784.58
                                        Covariance Type: nonrobust LLR p-value: 1.925e-98
                                        \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        coef std err z P>|z| [0.025 0.975]
                                        const 0.0867 0.073 1.185 0.236 -0.057 0.230
                                        is_amazon 2.4786 0.163 15.202 0.000 2.159 2.798
                                        stars_delta -0.2737 0.241 -1.138 0.255 -0.745 0.198
                                        \n\n\n\nNow notice the last two columns - that's the 95% confidence interval. If one value is positive, and the other negative, the interval includes 0, which means we can't really tell if the effect is positive, or negative - it's not significant. Turns out, controling for the star rating doesn't change anything.\n\nLet's add the review delta:\n\n\n```python\nfeatures = ['const', 'is_amazon','stars_delta', 'reviews_delta']\nlog_reg = sm.Logit(train['placed_higher'], train[features]).fit()\nlog_reg.summary()\n```\n\n Optimization terminated successfully.\n Current function value: 0.493737\n Iterations 6\n\n\n\n\n\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        Logit Regression Results
                                        Dep. Variable: placed_higher No. Observations: 1132
                                        Model: Logit Df Residuals: 1128
                                        Method: MLE Df Model: 3
                                        Date: Mon, 07 Feb 2022 Pseudo R-squ.: 0.2876
                                        Time: 11:21:57 Log-Likelihood: -558.91
                                        converged: True LL-Null: -784.58
                                        Covariance Type: nonrobust LLR p-value: 1.673e-97
                                        \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        coef std err z P>|z| [0.025 0.975]
                                        const 0.0854 0.073 1.167 0.243 -0.058 0.229
                                        is_amazon 2.4875 0.163 15.216 0.000 2.167 2.808
                                        stars_delta -0.2588 0.241 -1.075 0.282 -0.730 0.213
                                        reviews_delta -1.353e-06 1.17e-06 -1.157 0.247 -3.64e-06 9.39e-07
                                        \n\n\n\nSame story, so let's include the rest of our variables.\n\n\n```python\nfeatures = ['const', 'is_amazon','stars_delta', 'reviews_delta',\n 'is_shipped_by_amazon', 'is_sold_by_amazon',\n 'is_top_clicked']\nlog_reg = sm.Logit(train['placed_higher'], train[features]).fit()\nlog_reg.summary()\n```\n\n Optimization terminated successfully.\n Current function value: 0.485827\n Iterations 6\n\n\n\n\n\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        Logit Regression Results
                                        Dep. Variable: placed_higher No. Observations: 1132
                                        Model: Logit Df Residuals: 1125
                                        Method: MLE Df Model: 6
                                        Date: Mon, 07 Feb 2022 Pseudo R-squ.: 0.2990
                                        Time: 11:22:17 Log-Likelihood: -549.96
                                        converged: True LL-Null: -784.58
                                        Covariance Type: nonrobust LLR p-value: 3.532e-98
                                        \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        coef std err z P>|z| [0.025 0.975]
                                        const 0.1059 0.074 1.428 0.153 -0.039 0.251
                                        is_amazon 2.2388 0.174 12.889 0.000 1.898 2.579
                                        stars_delta -0.3405 0.246 -1.384 0.166 -0.823 0.142
                                        reviews_delta -1.501e-06 1.17e-06 -1.283 0.200 -3.79e-06 7.93e-07
                                        is_shipped_by_amazon -0.2945 0.240 -1.225 0.221 -0.766 0.177
                                        is_sold_by_amazon 0.7100 0.172 4.124 0.000 0.373 1.047
                                        is_top_clicked 0.0046 0.144 0.032 0.974 -0.277 0.286
                                        \n\n\n\nThe only other significant variable is `is_sold_by_amazon`. How do we interpret its value?\n\nFor exaple, both products are from amazon or are not from amazon, but one is sold by amazon, so `is_amazon = 0` and `is_sold_by_amazon = 1`\n\n\n\n\n```python\nnumerator = np.e**(log_reg.params[0] + log_reg.params[1] * 0 + log_reg.params[5] * 1)\nprint(numerator/(1+numerator))\n```\n\n 0.6933580136415483\n\n\nThat's the probability the product sold by amazon will be first.\n\nFor another example, first product is from amazon and sold by amazon, the other is neither:\n\n\n\n```python\nnumerator = np.e**(log_reg.params[0] + log_reg.params[1] * 1 + log_reg.params[5] * 1)\nprint(numerator/(1+numerator))\n```\n\n 0.9549840944621973\n\n\nThat's the probability the product sold by amazon will be first - it's slightly higher than when the first product is from amazon and the other isn't but we don't know who sells it.\n\n\nWhat about the predictive performance? The Markup uses RandomForests with hyperparameter tuning to get the best prediction and ends up with 73.2% accuracy\n\n\n```python\ntest = add_constant(test)\ny_pred = log_reg.predict(test[features])\n```\n\n\n```python\n((y_pred > 0.5) == test['placed_higher']).mean()\n```\n\n\n\n\n 0.696113074204947\n\n\n\n\n```python\ntrain.shape\n```\n\n\n\n\n (1132, 8)\n\n\n\n\n```python\ntrain\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        conststars_deltareviews_deltais_amazonis_shipped_by_amazonis_sold_by_amazonis_top_clickedplaced_higher
                                        661.00.211367.01.00.01.00.0True
                                        7061.00.1-17591.0-1.00.00.00.0False
                                        4281.00.01159.00.01.00.01.0False
                                        2761.0-0.320532.00.00.00.0-1.0False
                                        6751.00.00.00.00.00.00.0False
                                        ...........................
                                        7161.0-0.8-310.00.0-1.0-1.00.0True
                                        11351.0-0.2-7298.00.00.00.0-1.0True
                                        5341.00.13515.00.00.00.00.0True
                                        1931.0-0.1-484236.0-1.00.0-1.0-1.0True
                                        11691.0-0.2-2208.00.00.00.00.0False
                                        \n

                                        1132 rows \u00d7 8 columns

                                        \n
                                        \n\n\n\n\n```python\ntest\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        conststars_deltareviews_deltais_amazonis_shipped_by_amazonis_sold_by_amazonis_top_clickedplaced_higher
                                        2831.00.10.026419-1.00.00.00.0False
                                        9251.0-0.10.0470670.00.00.00.0True
                                        3941.0-0.2-0.0112490.00.00.00.0True
                                        4421.00.0-0.0481700.00.00.00.0True
                                        4671.00.40.2369890.01.00.00.0True
                                        ...........................
                                        5611.00.1-0.6102180.00.00.01.0False
                                        13351.0-0.20.152138-1.00.0-1.00.0False
                                        11541.00.10.4000560.00.01.00.0True
                                        851.00.10.0065530.00.00.00.0False
                                        1031.0-0.7-0.2122690.00.00.00.0True
                                        \n

                                        283 rows \u00d7 8 columns

                                        \n
                                        \n\n\n\nThat's not any worse than random forest - we can an equally good fit but we can actually interpret just how important the different features are!\n\nThe problem that we looked at so far is defined as follows: given the top two items, which one is going to be be placed in the first position?\n\nNow, let's solve a slightly different problem formulation (not covered in the Markup write up): given characteristics of the product, how likely is it the top product on its search page?\n\n\n```python\n# let's get the data first:\ndataset = pd.read_csv('https://www.sapiezynski.com/cs4910/markup/dataset_full.csv')\n```\n\n\n```python\ndataset\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        starsreviewsis_amazonis_sold_by_amazonis_shipped_by_amazontop_clickedis_top
                                        04.82120.00.01.01.00.01.0
                                        14.73747.00.01.01.01.00.0
                                        24.86031.00.01.01.01.00.0
                                        34.87017.00.01.01.01.00.0
                                        44.8455.00.01.01.01.00.0
                                        ........................
                                        838984.7248.00.00.01.00.00.0
                                        838994.8557.00.00.01.00.00.0
                                        839004.7214.00.00.00.01.00.0
                                        839014.8451.00.00.01.01.00.0
                                        839024.6347.00.00.01.00.00.0
                                        \n

                                        83903 rows \u00d7 7 columns

                                        \n
                                        \n\n\n\nThis dataset is an aggregation of all first search result pages from the Markup audit that had any amazon products on them. The sponsored results are removed.\n\nIn this dataset each row is a product with its star rating, the number of reviews, the binary indicators of whether it's an amazon product, it's sold or shipped by amazon, whether it was among the most clicked products, and the outcome variable - whether it was the top product in its search results page.\n\nWe're looking at at most 24 first results, so on average only one in 24 rows is the top placed product:\n\n\n\n```python\ndataset['is_top'].mean()\n```\n\n\n\n\n 0.04551684683503569\n\n\n\n\n```python\n# the lowest star rating is 1, not 0, let's adjust the rating such that it starts at 0:\ndataset['stars'] -= 1\n```\n\n
                                        \n
                                        Excercise 1: Interpreting the meaning of logistic regression coefficients
                                        \nIn this exercise you will interpret the values of the coefficients of a logistic regression model that tries to predict whether a product was the top product on its search page.\n\nIn particular, please answer these questions:\n1. What is the probability that a hypothetical product with the lowest rating, no reviews, not sold/shipped/produced by amazon, and not among the top clicked products will be placed at the top?\n1. Increase in which variables corresponds to a higher probability of being placed at the top?\n1. How do you interpret the `stars` coefficient and its statistical significance?\n1. If a product receives one more review without changing any other of its characteristics, how does it affect its chances to placed at the top? \n * How about 1000000 more reviews?\n1. What is the probability that a five-star product with 1000 reviews, produced, sold, and shipped by amazon that was among the top clicked results will be placed on the top of the result list? \n * Is this number higher, or lower than you expected? Does it change how you interpret the results of the model for only the top two positions? Why?\n\n
                                        \n\n\n```python\ndataset = add_constant(dataset)\ncols =['const', 'stars', 'reviews', 'is_amazon', 'is_sold_by_amazon', 'is_shipped_by_amazon', 'top_clicked']\nlog_reg = sm.Logit(dataset['is_top'],\n dataset[cols]).fit()\n```\n\n Optimization terminated successfully.\n Current function value: 0.180478\n Iterations 7\n\n\n\n```python\nlog_reg.summary()\n```\n\n\n\n\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        Logit Regression Results
                                        Dep. Variable: is_top No. Observations: 83903
                                        Model: Logit Df Residuals: 83896
                                        Method: MLE Df Model: 6
                                        Date: Mon, 07 Feb 2022 Pseudo R-squ.: 0.02496
                                        Time: 12:22:50 Log-Likelihood: -15143.
                                        converged: True LL-Null: -15530.
                                        Covariance Type: nonrobust LLR p-value: 3.631e-164
                                        \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
                                        coef std err z P>|z| [0.025 0.975]
                                        const -3.4286 0.221 -15.494 0.000 -3.862 -2.995
                                        stars -0.0472 0.061 -0.770 0.441 -0.167 0.073
                                        reviews 1.284e-06 4.15e-07 3.092 0.002 4.7e-07 2.1e-06
                                        is_amazon 1.1472 0.047 24.576 0.000 1.056 1.239
                                        is_sold_by_amazon -0.1224 0.039 -3.171 0.002 -0.198 -0.047
                                        is_shipped_by_amazon 0.2063 0.068 3.040 0.002 0.073 0.339
                                        top_clicked 0.4318 0.035 12.174 0.000 0.362 0.501
                                        \n\n\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n# What is the probability that a hypothetical product with the lowest rating, no reviews, \n# not sold/shipped/produced by amazon, and not among the top clicked products will be placed at the top?\nnp.e**(log_reg.params[0])/(1+ np.e**(log_reg.params[0]))\n```\n\n\n\n\n 0.03141344729908379\n\n\n\n- Increase in which variables corresponds to a higher probability of being placed at the top?\n>reviews, is_amazon, is_shipped_by_amazon, top_clicked (all ones that have positive coefs and are significant)\n\n- How do you interpret the stars coefficient and its statistical significance?\n> It is not statistically significant, so it appears not to have an effect on the placement.\n> its value is negative, so if it was significant it would indicate that higher rated items are less likely to be on top\n\n- If a product with no reviews receives its first one without changing any other of its characteristics, how does it affect its chances to placed at the top? How about when it receives 1000000 reviews instead?\n\n\n```python\np0 = np.e**(log_reg.params[0])/(1+ np.e**(log_reg.params[0]))\np1 = np.e**(log_reg.params[0] + log_reg.params[2])/(1+ np.e**(log_reg.params[0]+log_reg.params[2]))\nprint(p1-p0)\n```\n\n 3.907903880356889e-08\n\n\n\n```python\np2 = np.e**(log_reg.params[0] + log_reg.params[2] * 1000000)/(1+ np.e**(log_reg.params[0]+log_reg.params[2]*1000000))\n```\n\n\n```python\n\n```\n\n\n```python\nprint(p2-p0)\n```\n\n 0.07345786543985264\n\n\n\n```python\np0\n```\n\n\n\n\n 0.03141344729908379\n\n\n\n\n```python\np2\n```\n\n\n\n\n 0.10487131273893643\n\n\n\n> The probability of being placed at the top grows by 7 percentage points from 3% to 10%\n\n- What is the probability that a five-star product with 100000 reviews, produced, sold, and shipped by amazon that was among the top clicked results will be placed on the top of the result list? \n\n\n```python\nnumerator = np.e**(log_reg.params[0]\\\n + log_reg.params[1]*4\\\n + log_reg.params[2]*100000\\\n + log_reg.params[3]\\\n + log_reg.params[4]\\\n + log_reg.params[5]\\\n + log_reg.params[6])\n```\n\n\n```python\nnumerator/(1+ numerator)\n```\n\n\n\n\n 0.13871729855646864\n\n\n\n- Is this number higher, or lower than you expected? Does it change how you interpret the results of the model for only the top two positions? Why? \n\n> Any answer beyond \"it's what I expected, it doesn't change anything\" goes. \n>\n> To me personally it's surprisingly low - given the results from the markup I would naively expect that being an amazon product nearly guarantees the first spot but of course it doesn't - there are likely multiple amazon products on the front page and only one will make it to the top, and it's also not always that one is at the top.\n\n\n```python\n\n```\n", "meta": {"hexsha": "4e10fb6a37a7045ef3a5927ca4f38df347b65fac", "size": 82365, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Week 10/solution.ipynb", "max_stars_repo_name": "carlomarxdk/algorithmic-fairness", "max_stars_repo_head_hexsha": "34db31915b9d2e13f86a68ac1f69e558467b88c1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Week 10/solution.ipynb", "max_issues_repo_name": "carlomarxdk/algorithmic-fairness", "max_issues_repo_head_hexsha": "34db31915b9d2e13f86a68ac1f69e558467b88c1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Week 10/solution.ipynb", "max_forks_repo_name": "carlomarxdk/algorithmic-fairness", "max_forks_repo_head_hexsha": "34db31915b9d2e13f86a68ac1f69e558467b88c1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.0933391762, "max_line_length": 341, "alphanum_fraction": 0.4055848965, "converted": true, "num_tokens": 14383, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.6654105720171531, "lm_q1q2_score": 0.42388702534773326}} {"text": "Install py-pde library\n\n\n```python\n%pip install py-pde\n```\n\n Requirement already satisfied: py-pde in /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages (0.17.1)\n Requirement already satisfied: numba>=0.50.0 in /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages (from py-pde) (0.55.1)\n Requirement already satisfied: numpy>=1.18.0 in /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages (from py-pde) (1.21.5)\n Requirement already satisfied: scipy>=1.4.0 in /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages (from py-pde) (1.7.3)\n Requirement already satisfied: sympy>=1.5.0 in /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages (from py-pde) (1.9)\n Requirement already satisfied: matplotlib>=3.1.0 in /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages (from py-pde) (3.5.1)\n Requirement already satisfied: python-dateutil>=2.7 in /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages (from matplotlib>=3.1.0->py-pde) (2.8.2)\n Requirement already satisfied: fonttools>=4.22.0 in /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages (from matplotlib>=3.1.0->py-pde) (4.29.1)\n Requirement already satisfied: packaging>=20.0 in /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages (from matplotlib>=3.1.0->py-pde) (21.3)\n Requirement already satisfied: kiwisolver>=1.0.1 in /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages (from matplotlib>=3.1.0->py-pde) (1.3.2)\n Requirement already satisfied: pyparsing>=2.2.1 in /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages (from matplotlib>=3.1.0->py-pde) (3.0.7)\n Requirement already satisfied: cycler>=0.10 in /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages (from matplotlib>=3.1.0->py-pde) (0.11.0)\n Requirement already satisfied: pillow>=6.2.0 in /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages (from matplotlib>=3.1.0->py-pde) (9.0.1)\n Requirement already satisfied: llvmlite<0.39,>=0.38.0rc1 in /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages (from numba>=0.50.0->py-pde) (0.38.0)\n Requirement already satisfied: setuptools in /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages (from numba>=0.50.0->py-pde) (58.0.4)\n Requirement already satisfied: six>=1.5 in /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages (from python-dateutil>=2.7->matplotlib>=3.1.0->py-pde) (1.16.0)\n Requirement already satisfied: mpmath>=0.19 in /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages (from sympy>=1.5.0->py-pde) (1.2.1)\n Note: you may need to restart the kernel to use updated packages.\n\n\nGenerate dataset\n\n## Basic Parameter Set\n\n\n```python\nimport numpy as np\nimport torch\n\nf = 4410 # sampling frequency \nT = 1/f # sampling time \ndur = 0.1 # simulation duration\n\nt = np.arange(0, dur+T/2, T) # 0:T:dur; # time vector \n\nl = 5 # length of the pipe \ndx = 1e-2 # spatial stepsize \nxs = np.arange(0, l+dx/2, dx) # 0:dx:l; # space vector \nnumXs = np.size(xs)\n\n#c0 = 340\nc0 = 30 # propagation speed\n```\n\n## FTM Stuff\n\n\n```python\nMu = 250 # number of eigenvalues \nmu = np.arange(1, Mu+1) # 1:Mu; \n\ntest = 1j*c0*mu*np.pi/l\n\ngmu = np.concatenate((mu*np.pi/l, mu*np.pi/l))\nsmu = np.concatenate((1j*c0*mu*np.pi/l, -1j*c0*mu*np.pi/l))\n\nK1 = lambda x: 1j*np.sin(gmu*x) # @(x) 1j*sin(gmu*x); \nK2 = lambda x: 1j*smu*np.sin(gmu*x)\nKa1 = lambda x: 1j/c0**2*np.conj(smu)*np.sin(gmu*x)\nKa2 = lambda x: 1j*np.sin(gmu*x)\n\nnmu = 1./(l/2*(c0**2*smu + np.conj(smu)))\n\nA = np.diag(np.exp(smu*T)); \n\n```\n\n\n```python\nxeVec = np.array([0.1*l, 0.2*l, 0.3*l]) # vector of excitaion positions (can be extended) \n\nnum_param_steps = 1323\nfield_values = np.linspace(0,10,num_param_steps)\ngrid_size = numXs\n\ntraining_input = torch.zeros(num_param_steps, grid_size,2)\ntraining_output = torch.zeros(num_param_steps, grid_size,1)\n\n# grid = CartesianGrid([[0, 1]], grid_size, periodic=False)\n\nindex = 0\n\nfor xe, xeVal in enumerate(xeVec): #for xe = 1:length(xeVec)\n # Excitation for the wave equation is a simple delta-impulse at\n # position xe\n # Possible extensions: \n # - exciation by a hamming window to have a more smooth excitation \n # - combination with a temporal exciation shape \n yi = Ka2(xeVal)*T; # set initial values for states\n \n # vectors \n ybar = np.zeros((2*Mu, np.size(t)),dtype=complex); \n \n # set initial states\n ybar[:,0] = yi; \n \n test = range(1,np.size(t))\n \n # processing to create time progression of individual states\n for k in range(1,np.size(t)) :\n ybar[:,k] = A@ybar[:,k-1]\n \n \n # create output signal over time at a single observation position\n # (maybe this part is not necessary, therefore it is commented)\n xo = 0.7*l; \n c1 = K1(xo); \n y = c1@ybar; # recover deflection from states (inverse transformation)\n y = np.real(y)\n \n \n # create spatial vectors. \n # Result y_x: spatial distribution of the deflection y on the pipe at all\n # temportal sampling points\n \n K1_x = np.zeros((np.size(xs), 2*Mu)); \n y_x = np.zeros((np.size(xs), np.size(t))); \n\n for xi in range(np.size(xs)) : #1:length(xs) \n K1_x[xi,:] = K1(xs[xi])/nmu; \n y_x[xi,:] = K1_x[xi,:]@ybar; \n \n # take the real part because there might be a small imaginary part \n y_x = np.real(y_x) \n y_x = y_x / 10**6 # scale the output to less than 1\n\n for k in range(1,np.size(t)) :\n #field = ScalarField(grid, val)\n #result = solve_poisson_equation(field, bc=[{\"value\": 0}, {\"derivative\": 1}])\n training_input[index,:,0] = torch.tensor(y_x[:,k-1])\n training_input[index,:,1] = torch.linspace(0,1, grid_size)\n training_output[index,:,0] = torch.tensor(y_x[:,k])\n index += 1\n```\n\n /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages/ipykernel_launcher.py:51: ComplexWarning: Casting complex values to real discards the imaginary part\n /Users/sebastian/opt/anaconda3/envs/neuralOperator37/lib/python3.7/site-packages/ipykernel_launcher.py:52: ComplexWarning: Casting complex values to real discards the imaginary part\n\n\nModel definitions copied from https://github.com/zongyi-li/fourier_neural_operator\n\n\n```python\nimport torch.nn as nn\nimport torch.nn.functional as F\n\n################################################################\n# 1d fourier layer\n################################################################\nclass SpectralConv1d(nn.Module):\n def __init__(self, in_channels, out_channels, modes1):\n super(SpectralConv1d, self).__init__()\n\n \"\"\"\n 1D Fourier layer. It does FFT, linear transform, and Inverse FFT. \n \"\"\"\n\n self.in_channels = in_channels\n self.out_channels = out_channels\n self.modes1 = modes1 #Number of Fourier modes to multiply, at most floor(N/2) + 1\n\n self.scale = (1 / (in_channels*out_channels))\n self.weights1 = nn.Parameter(self.scale * torch.rand(in_channels, out_channels, self.modes1, dtype=torch.cfloat))\n\n # Complex multiplication\n def compl_mul1d(self, input, weights):\n # (batch, in_channel, x ), (in_channel, out_channel, x) -> (batch, out_channel, x)\n return torch.einsum(\"bix,iox->box\", input, weights)\n\n def forward(self, x):\n batchsize = x.shape[0]\n #Compute Fourier coeffcients up to factor of e^(- something constant)\n x_ft = torch.fft.rfft(x)\n\n # Multiply relevant Fourier modes\n out_ft = torch.zeros(batchsize, self.out_channels, x.size(-1)//2 + 1, device=x.device, dtype=torch.cfloat)\n out_ft[:, :, :self.modes1] = self.compl_mul1d(x_ft[:, :, :self.modes1], self.weights1)\n\n #Return to physical space\n x = torch.fft.irfft(out_ft, n=x.size(-1))\n return x\n\nclass FNO1d(nn.Module):\n def __init__(self, modes, width):\n super(FNO1d, self).__init__()\n\n \"\"\"\n The overall network. It contains 4 layers of the Fourier layer.\n 1. Lift the input to the desire channel dimension by self.fc0 .\n 2. 4 layers of the integral operators u' = (W + K)(u).\n W defined by self.w; K defined by self.conv .\n 3. Project from the channel space to the output space by self.fc1 and self.fc2 .\n \n input: the solution of the initial condition and location (a(x), x)\n input shape: (batchsize, x=s, c=2)\n output: the solution of a later timestep\n output shape: (batchsize, x=s, c=1)\n \"\"\"\n\n self.modes1 = modes\n self.width = width\n self.padding = 2 # pad the domain if input is non-periodic\n self.fc0 = nn.Linear(2, self.width) # input channel is 2: (a(x), x)\n\n self.conv0 = SpectralConv1d(self.width, self.width, self.modes1)\n self.conv1 = SpectralConv1d(self.width, self.width, self.modes1)\n self.conv2 = SpectralConv1d(self.width, self.width, self.modes1)\n self.conv3 = SpectralConv1d(self.width, self.width, self.modes1)\n self.w0 = nn.Conv1d(self.width, self.width, 1)\n self.w1 = nn.Conv1d(self.width, self.width, 1)\n self.w2 = nn.Conv1d(self.width, self.width, 1)\n self.w3 = nn.Conv1d(self.width, self.width, 1)\n\n self.fc1 = nn.Linear(self.width, 128)\n self.fc2 = nn.Linear(128, 1)\n\n def forward(self, x):\n x = self.fc0(x)\n x = x.permute(0, 2, 1)\n # x = F.pad(x, [0,self.padding]) # pad the domain if input is non-periodic\n\n x1 = self.conv0(x)\n x2 = self.w0(x)\n x = F.gelu(x1) + x2\n\n x1 = self.conv1(x)\n x2 = self.w1(x)\n x = F.gelu(x1) + x2\n\n x1 = self.conv2(x)\n x2 = self.w2(x)\n x = F.gelu(x1) + x2\n\n x1 = self.conv3(x)\n x2 = self.w3(x)\n x = F.gelu(x1) + x2\n\n # x = x[..., :-self.padding] # pad the domain if input is non-periodic\n x = x.permute(0, 2, 1)\n x = self.fc1(x)\n x = F.gelu(x)\n x = self.fc2(x)\n return x\n\n```\n\n\n```python\nmodes = 32 # 32\nwidth = 16 # 16\n\nepochs = 30\nlearning_rate = 1e-4\nbatch_size = 64\n\n\nmodel = FNO1d(modes, width) #.to('cuda')\n\noptimizer = torch.optim.AdamW(model.parameters(), lr=learning_rate, weight_decay=1e-4)\nscheduler = torch.optim.lr_scheduler.OneCycleLR(optimizer, max_lr = 1e-3, epochs=epochs, steps_per_epoch= 32)\n\ndataloader = torch.utils.data.DataLoader(torch.utils.data.TensorDataset(training_input, training_output), batch_size=batch_size, shuffle=True)\n\n\nfor ep in range(epochs):\n for input, output in dataloader:\n #input, output = input.cuda(), output.cuda()\n optimizer.zero_grad()\n pred_output = model(input)\n loss = torch.nn.functional.mse_loss(pred_output, output)\n loss.backward()\n optimizer.step()\n scheduler.step()\n print(\"\\r\",'loss:' + str(loss.detach().cpu().numpy()), end = \"\")\n \n\n```\n\nCheck output\n\n\n```python\nimport matplotlib.pyplot as plt\n\ngrid_start = 0\ngrid_end = 1\ntest_grid_size = 64\nfield_val = 2\n\nmodel_input = torch.zeros(1,grid_size,2)\nmodel_output = torch.zeros(1,grid_size,1)\n\ntestNum = 30\nmodel_input[:,:,:] = training_input[testNum,:,:]\nmodel_output[:,:,:] = training_output[testNum,:,:]\n#model_input = model_input.to('cuda')\ninput_field = model_input[0,:,0] # training_input[10,:,0]\n\nmodel_result = model(model_input)\n\nloss = torch.nn.functional.mse_loss(model_result, model_output)\nprint(\"\\r\",'loss:' + str(loss.detach().cpu().numpy()), end = \"\")\n\nplt.figure()\nplt.plot(input_field.data)\nplt.plot(model_output[0,:,0].data)\nplt.plot(model_result.detach().cpu().flatten().numpy())\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "50407e0f335d45d2d0462bf91caf6e43ee287b50", "size": 45948, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "legacy/Notebooks/NeuralOperators-WaveMarkov.ipynb", "max_stars_repo_name": "julian-parker/DAFX22_FNO", "max_stars_repo_head_hexsha": "72f30144317a3f8ba8ea23ecf9a0333c81fc87db", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "legacy/Notebooks/NeuralOperators-WaveMarkov.ipynb", "max_issues_repo_name": "julian-parker/DAFX22_FNO", "max_issues_repo_head_hexsha": "72f30144317a3f8ba8ea23ecf9a0333c81fc87db", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "legacy/Notebooks/NeuralOperators-WaveMarkov.ipynb", "max_forks_repo_name": "julian-parker/DAFX22_FNO", "max_forks_repo_head_hexsha": "72f30144317a3f8ba8ea23ecf9a0333c81fc87db", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 83.0886075949, "max_line_length": 16870, "alphanum_fraction": 0.7409462871, "converted": true, "num_tokens": 3550, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.665410558746814, "lm_q1q2_score": 0.42388701689411856}} {"text": "```julia\nusing NPZ\nusing PyPlot\nusing SpecialFunctions\n```\n\n Unable to init server: Could not connect: Connection refused\n Unable to init server: Could not connect: Connection refused\n \n (.:14921): Gdk-CRITICAL **: 18:21:09.991: gdk_cursor_new_for_display: assertion 'GDK_IS_DISPLAY (display)' failed\n \n (.:14921): Gdk-CRITICAL **: 18:21:10.006: gdk_cursor_new_for_display: assertion 'GDK_IS_DISPLAY (display)' failed\n\n\n\n# Table of Contents\n1. [Loading crystal data](#loading_crystal_data)\n2. [Sorting $i,j,k$](#sorting_ijk)\n3. [Manybody potential at single $z$-slice](#manybody_potential_at_single_z-slice)\n4. [3D manybody potential](#3d_manybody_potential)\n\n\n## Loading crystal data\n[Table of Contents](#table_of_contents)\n\nFirst, we load the data generated from [this notebook](effective_parameters_to_cylindrical_potential_MCM-41_PYTHON.ipynb).\n\n\n```julia\n#xyz atom positions withn the unit cell\nmcm41_xyz_shifted = NPZ.npzread(\"data/mcm41_xyz_shifted_CVFF.npz\");\nmcm41_x = mcm41_xyz_shifted[\"x\"];\nmcm41_y = mcm41_xyz_shifted[\"y\"];\nmcm41_z = mcm41_xyz_shifted[\"z\"];\n\n#mixed sigma and epsilon parameters for the Lennard-Jones potential (helium and MCM-41 atoms)\nsigma = mcm41_xyz_shifted[\"sigma\"];\nepsilon = mcm41_xyz_shifted[\"epsilon\"];\n```\n\n\n```julia\n#lattice vectors for MCM-41 unit cell\nmcm41_lattice_vectors = NPZ.npzread(\"data/mcm41_lattice_vectors.npz\");\nA1 = mcm41_lattice_vectors[\"A1\"];\nA2 = mcm41_lattice_vectors[\"A2\"];\nA3 = mcm41_lattice_vectors[\"A3\"];\n```\n\nRecall, we wish to generate the manybody potential within a single pore by summing a 12-6 Lennard-Jones potential over the atoms within a semi-infinite MCM-41 supercell\n\\begin{equation}\n U_\\mathrm{MCM-41}^\\mathrm{He}(\\vec{r}) = \\sum_{i,j,k = -\\infty}^\\infty \\sum_\\alpha^N 4\\varepsilon_\\alpha \\biggl( \\frac{\\sigma_\\alpha^{12}}{\\lvert \\vec{r} - \\vec{r}_{ijk\\alpha}\\rvert^{12}} - \\frac{\\sigma_\\alpha^6}{\\lvert \\vec{r} - \\vec{r}_{ijk\\alpha}\\rvert^6} \\biggr)\n\\end{equation}\nwhere $\\alpha$ indexes over each atom within the unit cell, $\\varepsilon_\\alpha$ and $\\sigma_\\alpha$ are the appropriately mixed Lennard-Jones parameters, and\n\\begin{equation}\n\\vec{r}_{ijk\\alpha} = \\vec{r}_\\alpha + i\\vec{A}_a + j\\vec{A}_b + k\\vec{A}_c\n\\end{equation}\nis the position of each atom within the semi-infinite supercell with unit cell vectors $\\vec{A}_{\\{a,b,c\\}}$.\n\nThe procedure will be to expand the unit cell in each unit cell direction and perform the summation until we reach the single precision floating point limit. Here we show a birds-eye view example of expanding once in each lattice vector direction.\n\n\n```julia\nfig,ax = subplots(figsize=(10,10))\nk = 0\nfor i in -1:1:1\n for j in -1:1:1\n _A = ((i .* A1) .+ (j .* A2) .+ (k .* A3))\n ax.scatter(mcm41_x .+ _A[1], mcm41_y .+ _A[2])\n end\nend\nax.set_xlabel(L\"$x\\ \\mathrm{[\\AA]}$\")\nax.set_ylabel(L\"$y\\ \\mathrm{[\\AA]}$\")\n\narrowprops = PyPlot.PyDict()\narrowprops[\"arrowstyle\"]=\"->\"\narrowprops[\"shrinkA\"]=0.0\narrowprops[\"shrinkB\"]=0.0\nax.annotate(\"\", xy=(A1[1], A1[2]), xytext=(0, 0), arrowprops=arrowprops)\nax.annotate(L\"$\\vec{A}_a$\", xy=(A1[1], A1[2]))\nax.annotate(\"\", xy=(A2[1], A2[2]), xytext=(0, 0), arrowprops=arrowprops)\nax.annotate(L\"$\\vec{A}_b$\", xy=(A2[1], A2[2]))\nax.set_aspect(\"equal\")\n```\n\n\n## Sorting $i,j,k$\n[Table of Contents](#table_of_contents)\n\nWe created a sorted array over the summation integers $i,j,k$ since the atoms closest to the central pore will contribute the most the the manybody potential. The array is sorted by $\\lvert\\vec{r}_{ijk}\\rvert=\\lvert i\\vec{A}_a + j\\vec{A}_b + k\\vec{A}_c \\rvert$.\n\n\n```julia\n# we create a large collection of ijk indices\ni_arr = collect(-200:1:200);\nj_arr = collect(-200:1:200);\nk_arr = collect(-200:1:200);\nr_arr = Array{Float64,3}(undef,size(i_arr,1),size(j_arr,1),size(k_arr,1));\n```\n\n\n```julia\n# Here we find the magnitude to center of each replicated unit cell within the very large supercell\nfor i in i_arr\n for j in j_arr\n for k in k_arr\n _A = ((i .* A1) .+ (j .* A2) .+ (k .* A3));\n _r = sqrt(sum(_A.^2))\n r_arr[i+201,j+201,k+201] = _r\n end\n end\nend\n```\n\n\n```julia\n#initialize and fill the arrays to hold the combinations of ijk\n_i_arr = Array{Int64,3}(undef,size(i_arr,1),size(j_arr,1),size(k_arr,1));\n_j_arr = Array{Int64,3}(undef,size(i_arr,1),size(j_arr,1),size(k_arr,1));\n_k_arr = Array{Int64,3}(undef,size(i_arr,1),size(j_arr,1),size(k_arr,1));\n\nfor i in i_arr\n for j in j_arr\n for k in k_arr\n _i_arr[i+201,j+201,k+201] = i\n _j_arr[i+201,j+201,k+201] = j\n _k_arr[i+201,j+201,k+201] = k\n end\n end\nend\n```\n\n\n```julia\n#flatten each ijk array and sort by the distance to each unit cell within the supercell\nr_arr_flat = vec(r_arr);\n_i_arr_flat = vec(_i_arr);\n_j_arr_flat = vec(_j_arr);\n_k_arr_flat = vec(_k_arr);\n\np = sortperm(r_arr_flat); #get the index for sorting by distance\n\nr_arr_flat = r_arr_flat[p];\n_i_arr_flat = _i_arr_flat[p];\n_j_arr_flat = _j_arr_flat[p];\n_k_arr_flat = _k_arr_flat[p];\n```\n\n\n## Manybody potential at single $z$-slice\n[Table of Contents](#table_of_contents)\n\nNow that we have sorted the summation indeces for a suitably large supercell, we can calculate the manybody potential at a single $z$-slice within a nanopore in the MCM-41. First we define the Lennard-Jones potential functions.\n\n\n```julia\nfunction U(r::Float64,sigma::Float64,epsilon::Float64)\n pf = 4 * epsilon;\n t12 = (sigma/r)^12;\n t6 = (sigma/r)^6;\n return pf * (t12 - t6)\nend\n\nfunction U(r::Float32,sigma::Float32,epsilon::Float32)\n pf = 4 * epsilon;\n t12 = (sigma/r)^12;\n t6 = (sigma/r)^6;\n return pf * (t12 - t6)\nend\n\nfunction U(r::Float64,sigma::Float64,epsilon::Float64)\n pf = 4 * epsilon;\n t12 = (sigma/r)^12;\n t6 = (sigma/r)^6;\n return pf * (t12 - t6)\nend\n\nfunction U_mb(r::Array{Float64,1},sigma::Array{Float64,1},epsilon::Array{Float64,1})\n V = 0.0;\n @inbounds for i in 1:size(r)\n V += U(r[i],sigma[i],epsilon[i]);\n end\n return V\nend\n```\n\n\n\n\n U_mb (generic function with 1 method)\n\n\n\nNext, we generate a mesh of points in the xy-plane ecompassing the pore.\n\n\n```julia\nx_min = -20.0;\nx_max = 20.0;\ny_min = -20.0;\ny_max = 20.0;\nres = 101;\nx = zeros(res,res);\ny = zeros(res,res);\nx_range = collect(range(x_min,x_max,length=res));\ny_range = collect(range(y_min,y_max,length=res));\nfor i in 1:res\n _x = x_range[i];\n for j in 1:res\n _y = y_range[j];\n x[i,j] = _x;\n y[i,j] = _y;\n end\nend\n```\n\nNow we convert some the variables to Julia Float32 types since we are calculating the manybody potential to within the single precision floating point limit.\n\n\n```julia\nr_tmp = zeros(Float32,size(x));\n\nx32 = convert.(Float32,x);\ny32 = convert.(Float32,y);\n\nA1_32 = convert.(Float32,A1);\nA2_32 = convert.(Float32,A2);\nA3_32 = convert.(Float32,A3);\n\n# We will look at the z-slice for z=0.0 Angstroms\nZ_VALUE = 0.0;\nZ_VALUE32 = convert(Float32,Z_VALUE);\n\nV = zeros(Float32,size(x32));\nV_old = zeros(Float32,size(x32));\n```\n\nFinally, everything is set up to generate the manybody potential at a single $z$-slice.\n\n\n```julia\nii_max = 10000 #Set some maximum just in case\nfor ii = 1:ii_max\n i = _i_arr_flat[ii];\n j = _j_arr_flat[ii];\n k = _k_arr_flat[ii];\n for atom in 1:size(mcm41_x,1)\n # get distance to single atom within supercell\n r_tmp .= sqrt.(((mcm41_x[atom] + (A1_32[1]*i) + (A2_32[1]*j)) .- x32).^2 .+ ((mcm41_y[atom] + (A2_32[2]*j)) .- y32).^2 .+ ((mcm41_z[atom] + (A3_32[3]*k)) - Z_VALUE32)^2);\n \n #Calculate the Lennard-Jones potential at the separation for helium and MCM-41 atom and increment the total potential\n V .+= U.(r_tmp,sigma[atom],epsilon[atom])\n end\n if all(V .== V_old)\n # if the old potential is the same as the new potential for\n # every point within the xy-mesh, then we have reached the\n # single precision floating point limit\n println(ii) # Show progress for each slice iteration\n break\n elseif (ii == ii_max)\n println(\"WARNING: ii_max reached\")\n break\n else\n # if we haven't reached the stopping critereon, copy current potential\n copy!(V_old,V);\n end\nend\n\n```\n\n 574\n\n\nWe have calculated the manybody potential for a single $z$-slice. Here is a view of the potential.\n\n\n```julia\nextent = (-20.0,20.0,-20.0,20.0)\nfig,ax = plt.subplots()\nim = ax.imshow(transpose(V),origin=\"lower\",vmax=200,vmin=-200,extent=extent)\nax.set_xlabel(L\"$x\\ \\mathrm{[\\AA]}$\")\nax.set_ylabel(L\"$y\\ \\mathrm{[\\AA]}$\")\nfig.colorbar(im)\n```\n\n\n## 3D manybody potential\n[Table of Contents](#table_of_contents)\n\nNow that we have seen how to calculate the manybody potential for a single slice, we are ready to caculate the full 3D manybody potential. Here, we will calculate from the bottom of the single unitcell to the top.\n\n\n```julia\nz_min = -A3[3]/2; # bottom of unit cell\nz_max = A3[3]/2; # top of unit cell\nz_range = collect(range(z_min,z_max,length=res));\n\nV_3D = zeros(Float32,size(x32,1),size(x32,2),res); # initialize 3D potential\n```\n\n\n```julia\nfor jj in 1:res\n print(\"jj = $jj: \") # A more descriptive output for each slice\n Z_VALUE = z_range[jj];\n Z_VALUE32 = convert(Float32,Z_VALUE);\n V = zeros(Float32,size(x32));\n V_old = zeros(Float32,size(x32));\n ii_max = 10000\n for ii = 1:ii_max\n i = _i_arr_flat[ii];\n j = _j_arr_flat[ii];\n k = _k_arr_flat[ii];\n for atom in 1:size(mcm41_x,1)\n r_tmp .= sqrt.(((mcm41_x[atom] + (A1_32[1]*i) + (A2_32[1]*j)) .- x32).^2 .+ ((mcm41_y[atom] + (A2_32[2]*j)) .- y32).^2 .+ ((mcm41_z[atom] + (A3_32[3]*k)) - Z_VALUE32)^2);\n V .+= U.(r_tmp,sigma[atom],epsilon[atom])\n end\n if all(V .== V_old)\n println(ii)\n break\n elseif (ii == ii_max)\n break\n else\n copy!(V_old,V);\n end\n end\n V_3D[:,:,jj] .= V;\nend\n```\n\n jj = 1: 1009\n jj = 2: 383\n jj = 3: 1402\n jj = 4: 575\n jj = 5: 1010\n jj = 6: 383\n jj = 7: 557\n jj = 8: 575\n jj = 9: 1471\n jj = 10: 2078\n jj = 11: 385\n jj = 12: 587\n jj = 13: 313\n jj = 14: 1083\n jj = 15: 383\n jj = 16: 313\n jj = 17: 575\n jj = 18: 704\n jj = 19: 575\n jj = 20: 1404\n jj = 21: 575\n jj = 22: 441\n jj = 23: 739\n jj = 24: 385\n jj = 25: 575\n jj = 26: 1180\n jj = 27: 2117\n jj = 28: 739\n jj = 29: 1143\n jj = 30: 312\n jj = 31: 431\n jj = 32: 440\n jj = 33: 1143\n jj = 34: 1180\n jj = 35: 312\n jj = 36: 2877\n jj = 37: 1143\n jj = 38: 575\n jj = 39: 794\n jj = 40: 849\n jj = 41: 709\n jj = 42: 312\n jj = 43: 848\n jj = 44: 3022\n jj = 45: 689\n jj = 46: 501\n jj = 47: 440\n jj = 48: 849\n jj = 49: 836\n jj = 50: 440\n jj = 51: 574\n jj = 52: 440\n jj = 53: 849\n jj = 54: 1530\n jj = 55: 574\n jj = 56: 789\n jj = 57: 440\n jj = 58: 440\n jj = 59: 1394\n jj = 60: 574\n jj = 61: 1595\n jj = 62: 1180\n jj = 63: 312\n jj = 64: 551\n jj = 65: 2103\n jj = 66: 1494\n jj = 67: 574\n jj = 68: 574\n jj = 69: 428\n jj = 70: 312\n jj = 71: 312\n jj = 72: 2116\n jj = 73: 574\n jj = 74: 1136\n jj = 75: 574\n jj = 76: 554\n jj = 77: 428\n jj = 78: 312\n jj = 79: 5830\n jj = 80: 312\n jj = 81: 550\n jj = 82: 738\n jj = 83: 426\n jj = 84: 1470\n jj = 85: 312\n jj = 86: 2116\n jj = 87: 738\n jj = 88: 427\n jj = 89: 738\n jj = 90: 574\n jj = 91: 378\n jj = 92: 440\n jj = 93: 514\n jj = 94: 555\n jj = 95: 550\n jj = 96: 374\n jj = 97: 374\n jj = 98: 468\n jj = 99: 252\n jj = 100: 519\n jj = 101: 1005\n\n\nNow that we have the 3D manybody potential, we can save it and continue our analysis in the [effective parameters to cylindrical potential notebook](effective_parameters_to_cylindrical_potential_MCM-41_PYTHON.ipynb#determining_the_effective_parameters_to_cylindrical_potential).\n\n\n```julia\nNPZ.npzwrite(\"data/V_3D_-20.0_20.0_-20.0_20.0_-6.10_6.10_101_CVFF.npz\",\n Dict(\"x_range\" => convert.(Float32,x_range),\n \"y_range\" => convert.(Float32,y_range),\n \"z_range\" => convert.(Float32,z_range),\n \"V\" => V_3D,\n \"x_grid\" => x32,\n \"y_grid\" => y32,\n \"extent\" => convert.(Float32,[x_min,x_max,y_min,y_max])\n ))\n```\n", "meta": {"hexsha": "857d08b1586c36c2cf0509a634e8b77185257f27", "size": 143267, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Notebooks/calculating_the_effective_manybody_potential_JULIA.ipynb", "max_stars_repo_name": "DelMaestroGroup/PlatedHe4Nanopores", "max_stars_repo_head_hexsha": "a23f74cb27d12f57cb533af9a64ab170bca05a84", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Notebooks/calculating_the_effective_manybody_potential_JULIA.ipynb", "max_issues_repo_name": "DelMaestroGroup/PlatedHe4Nanopores", "max_issues_repo_head_hexsha": "a23f74cb27d12f57cb533af9a64ab170bca05a84", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Notebooks/calculating_the_effective_manybody_potential_JULIA.ipynb", "max_forks_repo_name": "DelMaestroGroup/PlatedHe4Nanopores", "max_forks_repo_head_hexsha": "a23f74cb27d12f57cb533af9a64ab170bca05a84", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 217.7310030395, "max_line_length": 90042, "alphanum_fraction": 0.9047512686, "converted": true, "num_tokens": 4436, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124812, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.42349626015400155}} {"text": "## 5-3. Quantum Approximate Optimazation Algorithm (QAOA): \u91cf\u5b50\u8fd1\u4f3c\u6700\u9069\u5316\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\n\n### \u6982\u8981\n\n\u3053\u306e\u7bc0\u3067\u306f\u3001NISQ\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u4e00\u3064\u3068\u8003\u3048\u3089\u308c\u308b Quantum Approximate Optimazation Algorithm (QAOA; \u91cf\u5b50\u8fd1\u4f3c\u6700\u9069\u5316\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0)\u3092\u5b66\u3076\u3002QAOA\u306f\u3001\u91cf\u5b50\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u3068\u540c\u69d8\u306b\u3001\u7d44\u307f\u5408\u308f\u305b\u6700\u9069\u5316\u554f\u984c\u306e\u89e3\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3067\u3042\u308b\u3002\n\n### \u554f\u984c\u8a2d\u5b9a\nQAOA\u3067\u306f\u3001$z = z_{1}z_{2}\\cdots z_{n} \\: (z_i =0,1)$ \u3068\u3044\u3046$n$\u6841\u306e\u30d3\u30c3\u30c8\u5217$z$\u306b\u95a2\u3057\u3066\u3001\u30b3\u30b9\u30c8\u95a2\u6570$C(z) = \\sum_\\alpha C_\\alpha(z)$\u304c\u6700\u5c0f\u306b\u306a\u308b\u3088\u3046\u306a$z$\u3092\u63a2\u3059\u554f\u984c\u3092\u8003\u3048\u308b\u3002$C_\\alpha(z)$\u306f\u30d3\u30c3\u30c8\u5217$z$\u3092\u5f15\u6570\u306b\u3068\u308b\u4f55\u3089\u304b\u306e\u95a2\u6570\u3067\u3001\u3053\u3053\u3067\u306f\u7279\u306b\u3001\u30a4\u30b8\u30f3\u30b0\u30e2\u30c7\u30eb\u7684\u306a$C_\\alpha(z) = z_i\\cdot z_j$\u3068\u3044\u3063\u305f\u9805\u3092\u8003\u3048\u308c\u3070\u826f\u3044\u3002\n\n\u3053\u306e\u6700\u5c0f\u5316\u554f\u984c\u3092\u89e3\u304f\u305f\u3081\u306b\u3001$n$**\u30d3\u30c3\u30c8\u306e\u91cf\u5b50\u7cfb\u3092\u7528\u3044\u308b**\u3002\u305d\u3057\u3066\u3001$\\beta = (\\beta^{(1)}, \\cdots \\beta^{(p)}), \\gamma = (\\gamma^{(1)}, \\cdots \\gamma^{(p)})$ \u3092\u30d1\u30e9\u30e1\u30fc\u30bf\u3068\u3057\u3066\u3001\u6b21\u306e\u3088\u3046\u306a\u91cf\u5b50\u72b6\u614b\u3092\u8003\u3048\u308b\u3002\n\n\\begin{align}\n&|s\\rangle = |+\\rangle^{\\otimes n} = \\frac{1}{2^{n/2}} \\sum_{z=0}^{2^{n}-1} |z\\rangle, \\\\\n&|\\beta, \\gamma \\rangle = U_X(\\beta^{(p)}) U_C(\\gamma^{(p)}) \\cdots U_X(\\beta^{(1)}) U_C(\\gamma^{(1)}) |s\\rangle. \n\\end{align}\n\n\u3053\u3053\u3067 $|+\\rangle=\\frac{1}{\\sqrt{2}}(|0\\rangle+|1\\rangle)$\u306f$X$\u6f14\u7b97\u5b50\u306e\u56fa\u6709\u72b6\u614b$X|+\\rangle=|+\\rangle$\u3067\u3042\u308a\u3001 $U_C(\\gamma), U_X(\\beta)$ \u306f\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b\u3002\n\n$$\nU_C(\\gamma^{(i)}) = e^{-i\\gamma^{(i)} C(Z)} = \\prod_{\\alpha} e^{-i\\gamma^{(i)} C_{\\alpha}(Z)}, \\\\\nU_X(\\beta^{(i)}) = e^{-i\\beta^{(i)} \\sum_{j=1}^n X_j} = \\prod_{j =1}^n e^{-i\\beta^{(i)} X_j}.\n$$\n\n\u72b6\u614b$|\\beta, \\gamma \\rangle$\u3084\u3053\u308c\u3089\u306e\u6f14\u7b97\u5b50\u306e\u610f\u5473\u3092\u8aac\u660e\u3059\u308b\u306b\u306f\u91cf\u5b50\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u306e\u77e5\u8b58\u304c\u5fc5\u8981\u306b\u306a\u308b\u3002\u3068\u308a\u3042\u3048\u305a\u3001QAOA\u3092\u4f7f\u3046\u3060\u3051\u306a\u3089\u3053\u3046\u3044\u3046\u3082\u306e\u3060\u3068\u53d7\u3051\u5165\u308c\u3066\u4f7f\u3063\u3066\u3057\u307e\u3048\u3070\u826f\u3044\u3002 \n\uff08\u306a\u304a\u3001$C(Z)$\u3068\u3044\u3046\u306e\u306f\u30d3\u30c3\u30c8\u5217\u3092\u5f15\u6570\u306b\u3068\u308b\u95a2\u6570$C(z)$\u306e\u5165\u529b\u306b\u30d1\u30a6\u30ea\u6f14\u7b97\u5b50$Z_1\\cdots Z_n$\u3068\u4ee3\u5165\u3057\u305f\u3082\u306e\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002\uff09\n\n\u305d\u3057\u3066\u3001$F(\\beta, \\gamma) = \\langle{\\bf \\gamma, \\,\\beta}|C(Z)|{\\bf \\gamma, \\,\\beta}\\rangle$ \u3092\u6700\u5c0f\u306b\u3059\u308b\u3088\u3046\u306a$\\beta,\\gamma$\u3092\u63a2\u7d22\u3059\u308b\u3053\u3068\u3067\u3001\u5143\u3005\u306e\u6700\u9069\u5316\u554f\u984c\u306e\u7b54\u3048\u3092\u63a2\u305d\u3046\u3068\u3059\u308b\u306e\u304c\u3001QAOA\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3067\u3042\u308b\u3002\n\n### QAOA\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u624b\u9806\n\u5177\u4f53\u7684\u306aQAOA\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u624b\u9806\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3067\u3042\u308b\u3002\n\n1. \u91cf\u5b50\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u4e0a\u3067\u91cd\u306d\u5408\u308f\u305b\u72b6\u614b$|s\\rangle = |+\\rangle^{\\otimes n}$\u3092\u4f5c\u308b\u3002\n2. \u30d1\u30e9\u30e1\u30fc\u30bf$\\beta, \\gamma$\u306b\u5fdc\u3058\u3066\u3001\u91cf\u5b50\u72b6\u614b\u306b$U_C(\\gamma^{(i)}),U_X(\\beta^{(i)})$\u3092\u304b\u3051\u3066\u3044\u304d\u3001\u72b6\u614b$|\\beta, \\gamma \\rangle$\u3092\u5f97\u308b\u3002\n3. \u91cf\u5b50\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u3092\u7528\u3044\u3066 $\\langle \\beta, \\gamma |C(Z)|\\beta, \\gamma \\rangle$ \u3092\u6e2c\u5b9a\u3059\u308b\u3002\n4. \u53e4\u5178\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u3067\u3001$\\langle \\beta, \\gamma |C(Z)|\\beta, \\gamma \\rangle$ \u304c\u3088\u308a\u5c0f\u3055\u304f\u306a\u308b\u3088\u3046\u306b\u30d1\u30e9\u30e1\u30fc\u30bf $\\beta, \\gamma$ \u3092\u30a2\u30c3\u30d7\u30c7\u30fc\u30c8\u3059\u308b\u3002\n5. 1\u301c4\u3092\u7e70\u308a\u8fd4\u3057\u3001\u6700\u9069\u306a $\\beta^*, \\gamma^*$ \u3092\u5f97\u308b\u3002\n6. \u72b6\u614b $|\\beta^*, \\gamma^* \\rangle$ \u306b\u5bfe\u3057\u3066\u3001$z$\u65b9\u5411\u306e\u5c04\u5f71\u6e2c\u5b9a\u3092\u8907\u6570\u56de\u5b9f\u884c\u3057\u3001**\u5f97\u3089\u308c\u305f\uff08\u826f\u3055\u305d\u3046\u306a\uff09\u6e2c\u5b9a\u7d50\u679c** $z_1\\cdots z_n$ **\u3092\u5143\u3005\u306e\u6700\u9069\u5316\u554f\u984c\u306e\u89e3\u3068\u3057\u3066\u63a1\u7528**\u3059\u308b\u3002\uff08\u6ce8\uff1a\u6e2c\u5b9a\u7d50\u679c $z_1\\cdots z_n$ \u306f\u53e4\u5178\u30d3\u30c3\u30c8\uff09\n\n\u5c11\u3005\u3084\u3084\u3053\u3057\u3044\u306e\u3067\u3001\u5177\u4f53\u4f8b\u3092\u5b9f\u88c5\u3057\u306a\u304c\u3089\u78ba\u8a8d\u3057\u3066\u3044\u3053\u3046\u3002\n\n### \u5b9f\u88c5\uff1aMaxcut\u554f\u984c\u3092QAOA\u3067\u89e3\u304f\n\n\u3053\u3053\u3067\u306f\u5177\u4f53\u4f8b\u3068\u3057\u3066Maxcut\u554f\u984c\u3068\u3044\u3046\u554f\u984c\u3092\u53d6\u308a\u4e0a\u3052\u308b\u3002 \n[Maxcut\u554f\u984c](https://ja.wikipedia.org/wiki/\u30ab\u30c3\u30c8_(\u30b0\u30e9\u30d5\u7406\u8ad6))\u306f\u3001$n$\u500b\u306e\u9802\u70b9\u3092\u6301\u3064\u30b0\u30e9\u30d5\uff08\u4f8b\u3048\u3070\u4e0b\u56f3\uff09\u3092\uff12\u3064\u306b\u5206\u5272\u3059\u308b\u6642\u306b\u3001\u5206\u5272\u3055\u308c\u308b\u8fba\u306e\u6570\u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u308b\u554f\u984c\u3067\u3042\u308b\u3002\n\n\n(\u56f3\u306e\u51fa\u5178\uff1aWikipedia [\u30ab\u30c3\u30c8_(\u30b0\u30e9\u30d5\u7406\u8ad6)](https://ja.wikipedia.org/wiki/\u30ab\u30c3\u30c8_(\u30b0\u30e9\u30d5\u7406\u8ad6)))\n\n\u3053\u306e\u554f\u984c\u3092QAOA\u3067\u6271\u3048\u308b\u3088\u3046\u306a\u6700\u9069\u5316\u554f\u984c\u306b\u5e30\u7740\u3055\u305b\u308b\u306b\u306f\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3059\u308b\u3002\n\u9802\u70b9\u3092\uff12\u3064\u306e\u30b0\u30eb\u30fc\u30d7\u306b\u5206\u3051\u305f\u6642\u3001\u7247\u65b9\u306e\u30b0\u30eb\u30fc\u30d7\u306b\u5c5e\u3059\u308b\u9802\u70b9\u306b+1\u3001\u3082\u3046\u4e00\u65b9\u306e\u30b0\u30eb\u30fc\u30d7\u306b-1\u3092\u4ed8\u4e0e\u3059\u308b\u3068\u3059\u308c\u3070\u3001\u30b3\u30b9\u30c8\u95a2\u6570\n\n$$\nC(z) = -\\frac{1}{2} \\sum_{\\text{\u8fba\u3067\u7e4b\u304c\u3063\u3066\u3044\u308b\u9802\u70b9}i,j} ( 1 - z_i z_j)\n$$\n\n\u306f (\u30b0\u30eb\u30fc\u30d7\u5206\u3051\u306b\u3088\u3063\u3066\u5206\u5272\u3055\u308c\u308b\u8fba\u306e\u6570) $\\times (-1)$ \u3092\u8868\u3059\u3002\n\u3086\u3048\u306b\u3001$C(z)$\u3092\u6700\u5c0f\u5316\u3059\u308b\u3088\u3046\u306a\u30d3\u30c3\u30c8\u5217$z=z_1\\cdots z_n$\u3092\u898b\u3064\u3051\u308c\u3070\u3001\u5206\u5272\u3059\u308b\u8fba\u306e\u6570\u3092\u6700\u5927\u5316\u3059\u308b\u3088\u3046\u306a\u9802\u70b9\u306e\u5206\u3051\u65b9\u3092\u898b\u3064\u3051\u305f\u3053\u3068\u306b\u306a\u308b\u3002\n\n\u4ee5\u4e0b\u3067\u306f\u3001\u9577\u65b9\u5f62(\u9802\u70b9\u304c\uff14\u3064\u306e\u56f3\u5f62)\u306emaxcut\u554f\u984c\u3092\u89e3\u3044\u3066\u307f\u3088\u3046\u3002\n\n\n\u3053\u306e\u5834\u5408\u3001$C(Z)$\u306f\n\n$$\n\\begin{eqnarray}\nC(Z) &=& -\\frac{1}{2}(1-Z_{0}Z_{1})-\\frac{1}{2}(1-Z_{1}Z_{2})-\\frac{1}{2}(1-Z_{2}Z_{3})-\\frac{1}{2}(1-Z_{3}Z_{1})\\\\\n&=&\\frac{1}{2}(Z_{0}Z_{1}+Z_{1}Z_{2}+Z_{2}Z_{3}+Z_{3}Z_{1}) - 2\n\\end{eqnarray}\n$$\n\n\u3068\u306a\u308b\u3002\u7b2c\u4e8c\u9805\u306f\u5b9a\u6570\u3060\u304b\u3089\u3001\u4ee5\u4e0b\u3067\u306f\n\n$$\nC(Z) = \\frac{1}{2}(Z_{0}Z_{1}+Z_{1}Z_{2}+Z_{2}Z_{3}+Z_{3}Z_{1}) \n$$\n\n\u3068\u304a\u304f\u3002\n\n#### $p=1$\u306e\u5834\u5408\n\u307e\u305a\u306f$p=1$\u306e\u5834\u5408\u306e\u5b9f\u88c5\u3092\u3084\u3063\u3066\u307f\u3088\u3046\u3002\u3053\u306e\u6642\u3001$|\\beta, \\gamma \\rangle = U_X(\\beta^{(1)}) U_C(\\gamma^{(1)}) |s\\rangle$ \u3067\u3042\u308b\u3002\n\n$U_C(\\gamma^{(1)}) = \\prod_{i=0}^3 e^{-i\\gamma^{(1)} Z_i Z_{i+1} }$ \u3092\u5b9f\u88c5\u3059\u308b\u306b\u306f\u30014-2\u7bc0\u3067\u3082\u7528\u3044\u305f\u95a2\u4fc2\n\n$$\ne^{-i \\delta Z_i Z_{i+1}}\u3000= \\operatorname{CNOT}_{i,i+1} \\cdot e^{-i\\delta Z_{i+1}} \\cdot \\operatorname{CNOT}_{i,i+1}.\n$$\n\n\u3092\u4f7f\u3048\u3070\u826f\u3044\u3002\uff08\u884c\u5217\u306b\u76f4\u3057\u3066\u8a08\u7b97\u3059\u308b\u3068\u3001\u5408\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\uff09\n\n\u4ee5\u4e0a\u3092\u8e0f\u307e\u3048\u3066\u3001$|\\beta, \\gamma \\rangle$ \u3092\u69cb\u6210\u3057\u3066 $\\langle \\beta, \\gamma | C(Z) |\\beta, \\gamma \\rangle$ \u3092\u6e2c\u5b9a\u3057\u3001\u305d\u308c\u3092\u6700\u5c0f\u5316\u3059\u308b\u5de5\u7a0b\u3092\u5b9f\u88c5\u3059\u308b\u3068\u3053\u306e\u3088\u3046\u306b\u306a\u308b\u3002\n\n\n```python\n## Google Colaboratory\u306e\u5834\u5408\u30fbQulacs\u304c\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u3055\u308c\u3066\u3044\u306a\u3044local\u74b0\u5883\u306e\u5834\u5408\u306e\u307f\u5b9f\u884c\u3057\u3066\u304f\u3060\u3055\u3044\n!pip install qulacs\n```\n\n\n```python\n#\u5fc5\u8981\u306a\u30e9\u30a4\u30d6\u30e9\u30ea\u3092\u30a4\u30f3\u30dd\u30fc\u30c8\u3059\u308b\nfrom qulacs import QuantumState, QuantumCircuit, Observable, PauliOperator\nfrom qulacs.gate import H, CNOT, RX, RZ\nfrom scipy.optimize import minimize\nimport numpy as np\n\n## \u9802\u70b9\u306e\u6570\nn = 4 \n\n## C(Z)\u3092qulacs.Observable\u3068\u3057\u3066\u5b9a\u7fa9\ncost_observable = Observable(n) \nfor i in range(n):\n cost_observable.add_operator( PauliOperator(\"Z {:} Z {:}\".format(i, (i+1)%n), 0.5) )\n\n# circuit \u306b U_C(gamma) \u3092\u52a0\u3048\u308b\u95a2\u6570\ndef add_U_C(circuit, gamma):\n for i in range(n):\n j = (i+1) % n\n circuit.add_CNOT_gate(i, j)\n circuit.add_gate(RZ(j, -2*gamma)) ## qulacs\u3067\u306f RZ(theta)=e^{i*theta/2*Z}\n circuit.add_CNOT_gate(i, j)\n return circuit\n \n# circuit \u306b U_X(beta) \u3092\u52a0\u3048\u308b\u95a2\u6570\ndef add_U_X(circuit, beta):\n for i in range(n):\n circuit.add_gate(RX(i, -2*beta))\n return circuit\n\n# p=1 \u306e |beta, gamma> \u3092\u4f5c\u3063\u3066 \u3092\u8fd4\u3059\u95a2\u6570\n# x = [beta, gamma]\ndef QAOA_output_onelayer(x): \n beta, gamma = x\n\n circuit = QuantumCircuit(n)\n ## \u91cd\u306d\u5408\u308f\u305b\u3092\u4f5c\u308b\u305f\u3081\u3001\u30a2\u30c0\u30de\u30fc\u30eb\u30b2\u30fc\u30c8\u3092\u304b\u3051\u308b\n for i in range(n):\n circuit.add_H_gate(i)\n ## U_C, U_X\u3092\u304b\u3051\u308b\n circuit = add_U_C(circuit, gamma)\n circuit = add_U_X(circuit, beta)\n\n ## |beta, gamma>\u3092\u4f5c\u308b\n state = QuantumState(n)\n state.set_zero_state() \n circuit.update_quantum_state(state)\n return cost_observable.get_expectation_value(state) \n\n## \u521d\u671f\u5024\nx0 = np.array( [0.1, 0.1 ])\n\n## scipy.minimize \u3092\u7528\u3044\u3066\u6700\u5c0f\u5316\nresult = minimize(QAOA_output_onelayer, x0, options={'maxiter':500}, method='powell')\nprint(result.fun) # \u6700\u9069\u5316\u5f8c\u306e\u5024\nprint(result.x) # \u6700\u9069\u5316\u5f8c\u306e(beta, gamma)\n```\n\n -0.999999999499185\n [1.17809152 0.39269362]\n\n\n\u6700\u5c0f\u5024-1\u3068\u3001\u305d\u306e\u6642\u306e$\\beta^{(1)}, \\gamma^{(1)}$\u306e\u5024\u304c\u5f97\u3089\u308c\u305f\u3002 \n\u3053\u306e\u6700\u9069\u306a\u72b6\u614b $|\\beta, \\gamma\\rangle$ \u3092$z$\u65b9\u5411\u306b\u5c04\u5f71\u6e2c\u5b9a\u3057\u305f\u6642\u306b\u3069\u306e\u3088\u3046\u306a\u5024\u304c\u5f97\u3089\u308c\u308b\u304b\u3001\u5177\u4f53\u7684\u306b\u898b\u3066\u307f\u3088\u3046\u3002\n\n\n```python\n# \u6700\u9069\u306abeta, gamma\u3092\u4f7f\u3063\u3066 |beta, gamma> \u3092\u3064\u304f\u308b\nbeta_opt, gamma_opt = result.x\n\ncircuit = QuantumCircuit(n)\n## \u91cd\u306d\u5408\u308f\u305b\u3092\u4f5c\u308b\u305f\u3081\u3001\u30a2\u30c0\u30de\u30fc\u30eb\u30b2\u30fc\u30c8\u3092\u304b\u3051\u308b\nfor i in range(n):\n circuit.add_H_gate(i)\n## U_C, U_X\u3092\u304b\u3051\u308b\ncircuit = add_U_C(circuit, gamma_opt)\ncircuit = add_U_X(circuit, beta_opt)\n\n## |beta, gamma>\u3092\u4f5c\u308b\nstate = QuantumState(n)\nstate.set_zero_state() \ncircuit.update_quantum_state(state)\n\n## z\u65b9\u5411\u306b\u89b3\u6e2c\u3057\u305f\u6642\u306e\u78ba\u7387\u5206\u5e03\u3092\u6c42\u3081\u308b. (\u72b6\u614b\u30d9\u30af\u30c8\u30eb\u306e\u5404\u6210\u5206\u306e\u7d76\u5bfe\u5024\u306e\u4e8c\u4e57=\u89b3\u6e2c\u78ba\u7387)\nprobs = np.abs(state.get_vector())**2\nprint(probs)\n```\n\n [0.01562503 0.01562568 0.01562568 0.0781236 0.01562568 0.26562503\n 0.0781236 0.01562568 0.01562568 0.0781236 0.26562503 0.01562568\n 0.0781236 0.01562568 0.01562568 0.01562503]\n\n\n\n```python\n# \u30d7\u30ed\u30c3\u30c8\u3059\u308b\nimport matplotlib.pyplot as plt\n%matplotlib inline\n\n## z\u65b9\u5411\u306b\u5c04\u5f71\u6e2c\u5b9a\u3057\u305f\u6642\u306b\u5f97\u3089\u308c\u308b\u53ef\u80fd\u6027\u304c\u3042\u308b\u30d3\u30c3\u30c8\u5217\nz_basis = [format(i,\"b\").zfill(n) for i in range(probs.size)]\n\nplt.figure(figsize=(10, 5))\nplt.xlabel(\"states\")\nplt.ylabel(\"probability(%)\")\nplt.bar(z_basis, probs*100)\nplt.show()\n```\n\n\u3064\u307e\u308a\u3001$z$\u65b9\u5411\u306e\u5c04\u5f71\u6e2c\u5b9a\u3092\u884c\u3046\u3068\u3001`0101` \u304b `1010` \u304c\u6e2c\u5b9a\u3055\u308c\u308b\u78ba\u7387\u304c\u9ad8\u3044\u3053\u3068\u304c\u5206\u304b\u3063\u305f\u3002\u3053\u308c\u3089\u306e\u30d3\u30c3\u30c8\u5217\u306f\u9802\u70b91\u3068\u9802\u70b93\u3001\u9802\u70b92\u3068\u9802\u70b94\u304c\u540c\u3058\u30b0\u30eb\u30fc\u30d7\u306b\u306a\u308b\u3068\u3044\u3046\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u304b\u3089\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5206\u5272\u3092\u8868\u3057\u3066\u3044\u308b\u3002\n\n\n\n\u3053\u306e\u6642\u3001\u56f3\u5f62\u3092\u5206\u5272\u3059\u308b\u66f2\u7dda\u304c\u6a2a\u5207\u308b\u8fba\u306e\u6570\u306f4\u672c\u3067\u3042\u308a\u3001\u3053\u306e\u56f3\u5f62\u3092\u5206\u5272\u3059\u308b\u6642\u306b\u901a\u308b\u8fba\u306e\u6570\u306e\u6700\u5927\u5024\u3067\u3042\u308b\u3002\n\n\u3088\u3063\u3066\u3001\u6700\u9069\u5316\u3057\u305f$|\\beta, \\gamma\\rangle$\u306b\u5c04\u5f71\u6e2c\u5b9a\u3092\u884c\u3044\u3001\u3042\u308b\u7a0b\u5ea6\u591a\u6570\u306e\u6e2c\u5b9a\u7d50\u679c\u3092\u884c\u3063\u3066\u6e2c\u5b9a\u78ba\u7387\u304c\u9ad8\u3044\u30d3\u30c3\u30c8\u5217\u3092\u63a1\u7528\u3059\u308c\u3070\u3001\u3082\u3068\u3082\u3068\u89e3\u304d\u305f\u304b\u3063\u305f\u6700\u9069\u5316\u554f\u984c$C(z)$\u306e\u89e3\u304c\u5f97\u3089\u308c\u305f\u3001\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002 \n\u4e00\u5fdc\u3001\u3053\u308c\u3067\u3081\u3067\u305f\u3057\u3081\u3067\u305f\u3057\u3001\u3068\u8a00\u3048\u308b\u306e\u3060\u304c\u3001\u6700\u9069\u5316\u3057\u305f\u30b3\u30b9\u30c8\u95a2\u6570$\\langle \\beta, \\gamma | C(Z) |\\beta, \\gamma \\rangle$ \u306e\u5024\u306f\u22121\u3060\u3063\u305f\u3053\u3068\u3092\u601d\u3044\u51fa\u3057\u3066\u307b\u3057\u3044\u3002$\\langle 0101 | C(Z) |0101 \\rangle = \\langle 1010 | C(Z) |1010 \\rangle = -2$\u306a\u306e\u3067\u3001\u30b3\u30b9\u30c8\u95a2\u6570\u306b\u3064\u3044\u3066\u306f\u6b63\u3057\u3044\u5024\u304c\u5f97\u3089\u308c\u3066\u3044\u306a\u3044\uff01 \u3053\u308c\u306fvariational\u306a\u72b6\u614b$|\\beta, \\gamma \\rangle$ \u304c\u5341\u5206\u306a\u8868\u73fe\u80fd\u529b\u3092\u6301\u305f\u305a\u3001\u771f\u306e\u89e3$|0101\\rangle, |1010\\rangle$ \u3092\u8868\u73fe\u3067\u304d\u306a\u304b\u3063\u305f\u3053\u3068\u306b\u7531\u6765\u3059\u308b\u3002\n\n\u305d\u3053\u3067\u3001\u56de\u8def\u3092\u3088\u308a\u8907\u96d1\u306b\u3057\u305f$p=2$\u306e\u5834\u5408\u306b\u7d50\u679c\u304c\u3069\u3046\u5909\u308f\u308b\u304b\u898b\u3066\u307f\u3088\u3046\u3002\n\n\u203b \u3061\u306a\u307f\u306b\u3001$|\\beta, \\gamma\\rangle$\u306b100\u56de\u306e\u6e2c\u5b9a\u3092\u884c\u3063\u3066\u30d3\u30c3\u30c8\u5217$z$\u3092100\u901a\u308a\u5f97\u3066\u3001\u305d\u308c\u305e\u308c\u306b\u3064\u3044\u3066$C(z)$\u3092\u53e4\u5178\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u3067\u8a08\u7b97\u3057\u3066\u307f\u3066\u6700\u3082\u826f\u304b\u3063\u305f\u3082\u306e\u3092\u63a1\u7528\u3059\u308b\u3001\u3068\u3044\u3063\u305f\u6226\u7565\u3092\u7528\u3044\u308c\u3070\u3001\u3053\u306e\u3088\u3046\u306a\u554f\u984c\u306f\u751f\u3058\u306a\u3044\u304b\u3082\u3057\u308c\u306a\u3044\u3002\n\n#### $p=2$\u306e\u5834\u5408\n\n\n```python\n#\u5fc5\u8981\u306a\u30e9\u30a4\u30d6\u30e9\u30ea\u3092\u30a4\u30f3\u30dd\u30fc\u30c8\u3059\u308b\nfrom qulacs import QuantumState, QuantumCircuit, Observable, PauliOperator\nfrom qulacs.gate import H, CNOT, RX, RZ\nfrom scipy.optimize import minimize\nimport numpy as np\n\n## \u9802\u70b9\u306e\u6570\nn = 4 \n\n## C(Z)\u3092qulacs.Observable\u3068\u3057\u3066\u5b9a\u7fa9\ncost_observable = Observable(n) \nfor i in range(n):\n cost_observable.add_operator( PauliOperator(\"Z {:} Z {:}\".format(i, (i+1)%n), 0.5) )\n\n# circuit \u306b U_C(gamma) \u3092\u52a0\u3048\u308b\u95a2\u6570\ndef add_U_C(circuit, gamma):\n for i in range(n):\n j = (i+1) % n\n circuit.add_CNOT_gate(i, j)\n circuit.add_gate(RZ(j, -2*gamma)) ## qulacs\u3067\u306f RZ(theta)=e^{i*theta/2*Z}\n circuit.add_CNOT_gate(i, j)\n return circuit\n \n# circuit \u306b U_X(beta) \u3092\u52a0\u3048\u308b\u95a2\u6570\ndef add_U_X(circuit, beta):\n for i in range(n):\n circuit.add_gate(RX(i, -2*beta))\n return circuit\n\n# p=2 \u306e |beta, gamma> \u3092\u4f5c\u3063\u3066 \u3092\u8fd4\u3059\u95a2\u6570\n# x = [beta0, beta1, gamma0, gamma1]\ndef QAOA_output_twolayer(x): \n beta0, beta1, gamma0, gamma1 = x\n\n circuit = QuantumCircuit(n)\n ## \u91cd\u306d\u5408\u308f\u305b\u3092\u4f5c\u308b\u305f\u3081\u3001\u30a2\u30c0\u30de\u30fc\u30eb\u30b2\u30fc\u30c8\u3092\u304b\u3051\u308b\n for i in range(n):\n circuit.add_H_gate(i)\n ## U_C, U_X\u3092\u304b\u3051\u308b\n circuit = add_U_C(circuit, gamma0)\n circuit = add_U_X(circuit, beta0)\n circuit = add_U_C(circuit, gamma1)\n circuit = add_U_X(circuit, beta1)\n\n ## |beta, gamma>\u3092\u4f5c\u308b\n state = QuantumState(n)\n state.set_zero_state() \n circuit.update_quantum_state(state)\n return cost_observable.get_expectation_value(state) \n\n## \u521d\u671f\u5024\nx0 = np.array( [0.1, 0.1, 0.2, 0.3 ])\n\n## scipy.minimize \u3092\u7528\u3044\u3066\u6700\u5c0f\u5316\nresult = minimize(QAOA_output_twolayer, x0, options={'maxiter':500}, method='powell')\nprint(result.fun) # \u6700\u9069\u5316\u5f8c\u306e\u5024\nprint(result.x) # \u6700\u9069\u5316\u5f8c\u306e[beta0, beta1, gamma0, gamma1]\n\n## \u6700\u9069\u5316\u5f8c\u306e\u72b6\u614b\u3092\u6e2c\u5b9a\u3057\u305f\u6642\u306e\u78ba\u7387\u5206\u5e03\u3092\u8abf\u3079\u308b\nbeta0, beta1, gamma0, gamma1 = result.x\n\ncircuit = QuantumCircuit(n)\n## \u91cd\u306d\u5408\u308f\u305b\u3092\u4f5c\u308b\u305f\u3081\u3001\u30a2\u30c0\u30de\u30fc\u30eb\u30b2\u30fc\u30c8\u3092\u304b\u3051\u308b\nfor i in range(n):\n circuit.add_H_gate(i)\n## U_C, U_X\u3092\u304b\u3051\u308b\ncircuit = add_U_C(circuit, gamma0)\ncircuit = add_U_X(circuit, beta0)\ncircuit = add_U_C(circuit, gamma1)\ncircuit = add_U_X(circuit, beta1)\n\n## |beta, gamma>\u3092\u4f5c\u308b\nstate = QuantumState(n)\nstate.set_zero_state() \ncircuit.update_quantum_state(state)\n\n## \u72b6\u614b\u30d9\u30af\u30c8\u30eb\u306e\u5404\u6210\u5206\u306e\u7d76\u5bfe\u5024\u306e\u4e8c\u4e57=\u89b3\u6e2c\u78ba\u7387\nprobs = np.abs(state.get_vector())**2\nprint(probs)\n\n## z\u65b9\u5411\u306b\u5c04\u5f71\u6e2c\u5b9a\u3057\u305f\u6642\u306b\u5f97\u3089\u308c\u308b\u53ef\u80fd\u6027\u304c\u3042\u308b\u30d3\u30c3\u30c8\u5217\nz_basis = [format(i,\"b\").zfill(n) for i in range(probs.size)]\n\nimport matplotlib.pyplot as plt\nplt.figure(figsize=(10, 5))\nplt.xlabel(\"states\")\nplt.ylabel(\"probability(%)\")\nplt.bar(z_basis, probs*100)\nplt.show()\n```\n\n$p=1$\u306e\u6642\u306b\u6bd4\u3079\u3001\u5727\u5012\u7684\u306b\u5927\u304d\u3044\u78ba\u7387\u3067\u771f\u306e\u89e3$|0101\\rangle, |1010\\rangle$\u304c\u5f97\u3089\u308c\u308b\u78ba\u7387\u304c\u9ad8\u3044\u3053\u3068\u304c\u5206\u304b\u308b\u3002\u307e\u305f\u3001\u30b3\u30b9\u30c8\u95a2\u6570\u306e\u5024\u3082\u6b63\u3057\u304f\u22122\u306b\u8fd1\u3065\u3044\u3066\u3044\u308b\u3002\n\n\u3053\u306e\u3088\u3046\u306b\u3001QAOA\u3092\u7528\u3044\u308b\u969b\u306b\u306f\u3001\u5909\u5206\u91cf\u5b50\u56de\u8def\u306e\u8907\u96d1\u3055$p$\u306e\u5927\u304d\u3055\u306b\u3082\u6ce8\u610f\u3057\u306a\u304c\u3089\u5b9f\u88c5\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\u3060\u308d\u3046\u3002\n\n### \u53c2\u8003\u6587\u732e\n\n[1] E. Farhi, J. Goldstone, and S. Gutmann, \u201cA Quantum Approximate Optimization Algorithm\u201d, [arXiv:1411.4028](https://arxiv.org/abs/1411.4028) (2014).\n\n[2] Eddie Farhi: A Quantum Approximate Optimization Algorithm, https://www.youtube.com/watch?v=J8y0VhnISi8\n", "meta": {"hexsha": "d733bbfa70b8884a9b6630e4ca6ea55d62b5dba1", "size": 33161, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/5.3_quantum_approximate_optimazation_algorithm.ipynb", "max_stars_repo_name": "shnchr/quantum-native-dojo", "max_stars_repo_head_hexsha": "7846dca31b4e47cec44fd21be2098a143402fc9f", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/5.3_quantum_approximate_optimazation_algorithm.ipynb", "max_issues_repo_name": "shnchr/quantum-native-dojo", "max_issues_repo_head_hexsha": "7846dca31b4e47cec44fd21be2098a143402fc9f", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/5.3_quantum_approximate_optimazation_algorithm.ipynb", "max_forks_repo_name": "shnchr/quantum-native-dojo", "max_forks_repo_head_hexsha": "7846dca31b4e47cec44fd21be2098a143402fc9f", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 65.2775590551, "max_line_length": 8396, "alphanum_fraction": 0.7507614366, "converted": true, "num_tokens": 5382, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833945721304, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.42349429370633557}} {"text": "# Episodic Lunar Lander with function appoximation and control\n\nThis Notebook is intended to solve the Episodic Lunar Lander problem using Semi-gradient Expected sarsa with neural networks for function approximation.\n\nThe description of the problem is given below:\n\n\"Landing pad is always at coordinates (0,0). Coordinates are the first two numbers in state vector. Reward for moving from the top of the screen to landing pad and zero speed is about 100..140 points. If lander moves away from landing pad it loses reward back. Episode finishes if the lander crashes or comes to rest, receiving additional -100 or +100 points. Each leg ground contact is +10. Firing main engine is -0.3 points each frame. Solved is 200 points. Landing outside landing pad is possible. Fuel is infinite, so an agent can learn to fly and then land on its first attempt. Four discrete actions available: do nothing, fire left orientation engine, fire main engine, fire right orientation engine.\" \n\n\n\nImage and Text taken from [Official documentaiton Lunar Lander](https://gym.openai.com/envs/LunarLander-v2/).\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom tqdm import tqdm\nfrom copy import deepcopy\n\nimport gym\nfrom gym.wrappers import Monitor\nfrom utils import *\n\nimport torch\nfrom torch import nn\nimport torch.nn.functional as F\nfrom torch import optim\n\n%matplotlib inline\n\ndevice = torch.device('cuda' if torch.cuda.is_available() else 'cpu')\n```\n\n## Markov Decision Process \n\nAs a quick recap, the diagram below explains the workflow of a Markov Decision Process (MDP)\n\n\n\nImage taken from [Section 3.1 Reinforment Learning an Introduction](http://www.incompleteideas.net/book/RLbook2018.pdf#page=70)\n\n## Environment and Agent specifications\n\nThe states, actions, reward and termination are given as follows for the lunar lander problem.\n\n**Observation**: \n\n Type: Box(8)\n Num Observation Min Max\n 0 X position -inf inf\n 1 Y position -inf inf\n 2 X velocity -inf inf\n 3 Y velocity -inf inf\n 4 Theta w.r.t ground -inf inf\n 5 Theta rate -inf inf\n 6 1 if first leg has contact, else 0 -inf inf\n 7 1 if second leg has contact, else 0 -inf inf\n \n**Actions**:\n\n Type: Discrete(4)\n Num Action\n 0 Do nothing\n 1 Fire left engine\n 2 Fire main engine\n 3 Fire right engine\n\n \n**Reward**:\n\n Reward of 0 is awarded if the agent reached the flag(position = 0.5) on top of the mountain\n Reward of -1 is awarded if the position of the agent is less than 0.5\n Reward of -100 for flying off the screen\n Reward of +100 for successful landing\n Reward of -0.3 for firing main thrusters\n Reward of -0.03 for firing side thrusters\n Reward of +10 for each leg touching fround\n \n**Starting State**:\n\n The starting position is above the landing target\n \n**Episode Termination**:\n\n The lander crashes\n The lander comes to rest\n Episode length is greater than 200\n \nFor further information see [Github source code](https://github.com/openai/gym/blob/master/gym/envs/box2d/lunar_lander.py).\n\nThe next cell aims to show how to iterate with the action and observation space of the agent and extract relevant information from it\n\n\n```python\nenv = gym.make(\"LunarLander-v2\")\nobservation = env.reset() \n\n# Object's type in the action Space\nprint(\"The Action Space is an object of type: {0}\\n\".format(env.action_space))\n# Shape of the action Space\nprint(\"The shape of the action space is: {0}\\n\".format(env.action_space.n))\n# Object's type in the Observation Space\nprint(\"The Environment Space is an object of type: {0}\\n\".format(env.observation_space))\n# Shape of the observation space\nprint(\"The Shape of the dimension Space are: {0}\\n\".format(env.observation_space.shape))\n# The high and low values in the observation space\nprint(\"The High values in the observation space are {0}, the low values are {1}\\n\".format(\n env.observation_space.high, env.observation_space.low))\n# Example of observation\nprint(\"The Observations at a given timestep are {0}\\n\".format(env.observation_space.sample()))\n```\n\n The Action Space is an object of type: Discrete(4)\n \n The shape of the action space is: 4\n \n The Environment Space is an object of type: Box(8,)\n \n The Shape of the dimension Space are: (8,)\n \n The High values in the observation space are [inf inf inf inf inf inf inf inf], the low values are [-inf -inf -inf -inf -inf -inf -inf -inf]\n \n The Observations at a given timestep are [-1.3609413 -0.81943125 0.00691082 0.965331 -1.3784901 1.0290705\n -1.7937465 0.85192055]\n \n\n\n## Computing action-values with neural networks\n\nTo compute action-values, a feed-forward neural network is used. This apporach allows us to compute action-values using the weights of the neural network.\n\n$$ q_\\pi(s) \\approx \\hat{q}(s, a, w) = NN(s,a,w) $$\n\nNeural networks are used to solve the control problem in RL, particularly, this networl is going to be used with an Episodic Semi-gradient Expected Sarsa agent. The inputs of the network are the states, which in this case are eight, the number of hidden layers and hidden units can vary. Finally, the number of inputs is equals to the number of actions in the problem, therefore, four output nodes are needed in the final layer. Each output node corresponds to the action value of a particular action.\n\n\n\n\nFor further information about Neural Networks for function approximation see [Section 9.7 of Reinforment Learning an Introduction](http://www.incompleteideas.net/book/RLbook2018.pdf#page=246)\n\nImage taken from [Reinforcement learning specialization, C4L5S1](https://www.coursera.org/learn/complete-reinforcement-learning-system/lecture/CVH40/meeting-with-adam-getting-the-agent-details-right)\n\n\n```python\n# Neural Netork to compute action values\nclass ActionValueNetwork(nn.Module):\n # Work Required: Yes. Fill in the layer_sizes member variable (~1 Line).\n def __init__(self, network_config):\n super().__init__()\n \n # Number of states\n self.state_dim = network_config.get(\"state_dim\")\n # Hidden units\n self.num_hidden_units = network_config.get(\"num_hidden_units\")\n # Actions or output units\n self.num_actions = network_config.get(\"num_actions\")\n \n # Initialzie hidden layer \n self.hidden = nn.Linear(self.state_dim, self.num_hidden_units)\n # Initialize output layer\n self.output = nn.Linear(self.num_hidden_units, self.num_actions)\n \n \n def forward(self, s):\n \"\"\"\n This is a feed-forward pass in the network\n Args:\n s (Numpy array): The state, a 2D array of shape (batch_size, state_dim)\n Returns:\n The action-values (Torch array) calculated using the network's weights.\n A 2D array of shape (batch_size, num_actions)\n \"\"\"\n # Transform observations into a pytorch tensor\n s = torch.Tensor(s)\n \n q_vals = F.relu(self.hidden(s))\n q_vals = self.output(q_vals)\n\n return q_vals\n```\n\n## Replay Buffer\n\nExperience replay is a technique very similar to planning in RL. Overall, this technique is used to update the action-values of the agent with a set of \"experience\" collected in a model. This experience allows the model to learn without interacting with the environment using simulated experience.\n\nExperience replay is a simple method that can get some of the advantages of planning by saving a buffer of experience and using the data stored in the buffer as a model. This view of prior data as a model works because the data represents actual transitions from the underlying MDP. The data stored in the model are the state, action, reward and next state. \n\nThe model will be filled until a queue size is reached, only then the model will drop its oldest observation and add a new one. With this buffer of information, it is possible to sample \"batches\" and update the action values of the agent.\n\nAs a quick recap, the next pseudocode shows the pseudocode for Dyna-Q algorithm where the agent performs planning steps, improving the learning process of the agent with simulated experience.\n\n\n\nThe planning process is given in the next image.\n\n\n\nFor further information about planning see [Section 8.2 of Reinforment Learning an Introduction](http://www.incompleteideas.net/book/RLbook2018.pdf#page=185). **Note**: Images taken from the last reference.\n\n\n\n```python\nclass ReplayBuffer:\n def __init__(self, size, minibatch_size):\n \"\"\"\n Args:\n size (integer): The size of the replay buffer. \n minibatch_size (integer): The sample size.\n \"\"\"\n # Create the buffer\n self.buffer = [] \n self.minibatch_size = minibatch_size\n self.max_size = size\n\n def append(self, state, action, reward, terminal, next_state):\n \"\"\"\n Args:\n state (Numpy array): The state of size (state_dim) \n action (integer): The action.\n reward (float): The reward.\n terminal (integer): 1 if the next state is a terminal state and 0 otherwise.\n next_state (Numpy array): The next state of size (state_dim) . \n \"\"\"\n if len(self.buffer) == self.max_size:\n # Delete first position of the buffer if the Queue size is equals to max size\n del self.buffer[0]\n # Append new step\n self.buffer.append([state, action, reward, terminal, next_state])\n\n def sample(self):\n \"\"\"\n Returns:\n A list of transition tuples including state, action, reward, terinal, and next_state\n The return of this function is of size (minibatch_size)\n \"\"\"\n idxs = np.random.choice(np.arange(len(self.buffer)), size=self.minibatch_size)\n return [self.buffer[idx] for idx in idxs]\n\n def size(self):\n return len(self.buffer)\n```\n\n## Softmax Policy\n\nTo compute the actions, a softmax policy is used. One advantage of a softmax policy is that it explores according to the action-values, meaning that an action with a moderate value has a higher chance of getting selected compared to an action with a lower value. This sort of policies provides a feasible alternative to do exploration.\n\nThe probability of selecting each action according to the softmax policy is shown below:\n\n$$Pr{(A_t=a | S_t=s)} \\hspace{0.1cm} \\dot{=} \\hspace{0.1cm} \\frac{e^{Q(s, a)/\\tau}}{\\sum_{b \\in A}e^{Q(s, b)/\\tau}}$$\n\nHere, $\\tau$ is the temperature parameter which controls how much the agent focuses on the highest valued actions. The smaller the temperature, the more the agent selects the greedy action. Conversely, when the temperature is high, the agent selects among actions more uniformly random.\n\nGiven that a softmax policy exponentiates action values, if those values are large, exponentiating them could get very large. To implement the softmax policy in a numerically stable way,the maximum action-value is substracted from the action-values. Doing so, the probability of selecting each action looks as follows:\n\n$$Pr{(A_t=a | S_t=s)} \\hspace{0.1cm} \\dot{=} \\hspace{0.1cm} \\frac{e^{Q(s, a)/\\tau - max_{c}Q(s, c)/\\tau}}{\\sum_{b \\in A}e^{Q(s, b)/\\tau - max_{c}Q(s, c)/\\tau}}$$\n\nRecall that changing the action preferences (action-values in this case) for a constant, would not change the final value of the softmax probability. This Softmax implementation is different than the one provided by Pytorch.\n\nFor further informartion about Softmax policies and action preferences see [Section 13.1 of Reinforment Learning an Introduction](http://www.incompleteideas.net/book/RLbook2018.pdf#page=344)\n\n\n\n```python\ndef softmax(action_values, tau=1.0):\n \"\"\"\n Args:\n action_values (Tensor array): A 2D array of shape (batch_size, num_actions). \n The action-values computed by an action-value network. \n tau (float): The temperature parameter scalar.\n Returns:\n A 2D Tensor array of shape (batch_size, num_actions). Where each column is a probability distribution \n over the actions representing the policy.\n \"\"\"\n \n # Compute the preferences\n preferences = action_values / tau\n # Compute the maximum preference across the actions (Max action per row or batch)\n max_preference = torch.max(preferences, dim = 1)[0]\n\n # Reshape max_preference to [Batch, 1] \n reshaped_max_preference = max_preference.view((-1, 1))\n \n # Computing numerator\n exp_preferences = torch.exp(preferences - reshaped_max_preference)\n # Computing the denominator suming over rows (batches).\n sum_of_exp_preferences = torch.sum(exp_preferences, dim = 1)\n \n # Reshape sum_of_exp_preferences array to [Batch, 1] \n reshaped_sum_of_exp_preferences = sum_of_exp_preferences.view((-1, 1))\n \n # Computing action probabilities\n action_probs = exp_preferences / reshaped_sum_of_exp_preferences\n action_probs = action_probs.squeeze()\n \n return action_probs\n```\n\n\n```python\n# Testing the Softmax implementation\nrand_generator = np.random.RandomState(0)\naction_values = torch.tensor(rand_generator.normal(0, 1, (2, 4)))\ntau = 0.5\n\naction_probs = softmax(action_values, tau)\nprint(\"action_probs\", action_probs)\n\nassert(np.allclose(action_probs, np.array([\n [0.25849645, 0.01689625, 0.05374514, 0.67086216],\n [0.84699852, 0.00286345, 0.13520063, 0.01493741]\n])))\n\n```\n\n action_probs tensor([[0.2585, 0.0169, 0.0537, 0.6709],\n [0.8470, 0.0029, 0.1352, 0.0149]], dtype=torch.float64)\n\n\n## Computing TD target and TD estimate\n\nThe TD target and TD estimate's computation will be done in the next lines. The main idea here is to obtain the action-value network updates with experience sampled from the experience replay buffer.\n\nAt time $t$, there is an action-value function represented as a neural network, say $Q_t$. The idea is to update the action-value function and get a new one we can use at the next timestep. We will get this $Q_{t+1}$ using multiple replay steps that each result in an intermediate action-value function $Q_{t+1}^{i}$ where $i$ indexes which replay step we are at.\n\nIn each replay step, we sample a batch of experiences from the replay buffer and compute a minibatch Expected-SARSA update. Across these N replay steps, we will use the current \"un-updated\" action-value network at time $t$, $Q_t$, for computing the action-values of the next-states. This contrasts using the most recent action-values from the last replay step $Q_{t+1}^{i}$. We make this choice to have targets that are stable across replay steps. Here is the pseudocode for performing the updates:\n\n$$\n\\begin{align}\n& Q_t \\leftarrow \\text{action-value network at timestep t (current action-value network)}\\\\\n& \\text{Initialize } Q_{t+1}^1 \\leftarrow Q_t\\\\\n& \\text{For } i \\text{ in } [1, ..., N] \\text{ (i.e. N} \\text{ replay steps)}:\\\\\n& \\hspace{1cm} s, a, r, t, s'\n\\leftarrow \\text{Sample batch of experiences from experience replay buffer} \\\\\n& \\hspace{1cm} \\text{Do Expected Sarsa update with } Q_t: Q_{t+1}^{i+1}(s, a) \\leftarrow Q_{t+1}^{i}(s, a) + \\alpha \\cdot \\left[r + \\gamma \\left(\\sum_{b} \\pi(b | s') Q_t(s', b)\\right) - Q_{t+1}^{i}(s, a)\\right]\\\\\n& \\hspace{1.5cm} \\text{ making sure to add the } \\gamma \\left(\\sum_{b} \\pi(b | s') Q_t(s', b)\\right) \\text{ for non-terminal transitions only.} \\\\\n& \\text{After N replay steps, we set } Q_{t+1}^{N} \\text{ as } Q_{t+1} \\text{ and have a new } Q_{t+1} \\text{for time step } t + 1 \\text{ that we will fix in the next set of updates. }\n\\end{align}\n$$\n\nAs you can see in the pseudocode, after sampling a batch of experiences, we do many computations. The basic idea however is that we are looking to compute a form of a TD error. \n\n$$ R_{t+1} + \\gamma \\hat{q}(S_{t+1}, A_{t+1}, w)- \\hat{q}(S_t, A_t, w) $$\n \nRecall that the for this problem, the TD Target is given by.\n\n$$ r + \\gamma \\left(\\sum_{b} \\pi(b | s') Q_t(s', b)\\right) $$\n\nSimilarly, the TD estimate is.\n\n$$ Q_{t+1}^{i}(s, a) $$\n\nThe Semi-gradient Expected Sarsa update is given below.\n\n$$w \\leftarrow w + \\alpha[R_{t+1} + \\gamma \\sum_{a'}\\pi(a' | S_{t+1}) \\hat{q}(S_{t+1}, a', w) - \\hat{q}(S_t, A_t, w)]\\nabla \\hat{q}(S_t, A_t, w)$$\n\n\nFor further explanation about Episodic semi-gradient control see [Section 10.1 of Reinforment Learning an Introduction](http://www.incompleteideas.net/book/RLbook2018.pdf#page=265z)\n\n\n\n```python\n# Method to compute the TD Target and TD estimate\ndef get_td(states, next_states, actions, rewards, discount, terminals, network, current_q, tau):\n \"\"\"\n Args:\n states (Numpy array): The batch of states with the shape (batch_size, state_dim).\n next_states (Numpy array): The batch of next states with the shape (batch_size, state_dim).\n actions (Numpy array): The batch of actions with the shape (batch_size,).\n rewards (Numpy array): The batch of rewards with the shape (batch_size,).\n discount (float): The discount factor (gamma).\n terminals (Numpy array): The batch of terminals with the shape (batch_size,).\n network (ActionValueNetwork): The latest state of the network that is getting replay updates.\n current_q (ActionValueNetwork): The fixed network used for computing the targets, \n and particularly, the action-values at the next-states.\n Returns:\n target_vec (Tensor array): The TD Target for actions taken, of shape (batch_size,)\n estimate_vec (Tensor array): The TD estimate for actions taken, of shape (batch_size,)\n \"\"\"\n \n # network is the latest state of the network that is getting replay updates. In other words, \n # network represents Q_{t+1}^{i} whereas current_q represents Q_t, the fixed network used \n # for computing the targets, and particularly, the action-values at the next-states.\n \n # q_next_mat is a 2D Tensor of shape (batch_size, num_actions)\n # used to compute the action-values of the next states\n # Detach is used to remove this graph from the main graph\n q_next_mat = current_q.forward(next_states).detach()\n \n # Compute policy at next state. \n # probs_mat is a 2D Tensor of shape (batch_size, num_actions)\n probs_mat = softmax(q_next_mat, tau)\n \n # Sum of the action-values for the next_states weighted by the policy, probs_mat.\n # (1 - terminals) to make sure v_next_vec is zero for terminal next states.\n # v_next_vec is a 1D Tensor of shape (batch_size,)\n v_next_vec = torch.zeros((q_next_mat.shape[0]), dtype=torch.float64).detach()\n # Sum over rows axis (batches)\n v_next_vec = torch.sum(probs_mat * q_next_mat, dim = 1) * (1 - torch.tensor(terminals)) \n \n # Compute Expected Sarsa target\n # target_vec is a 1D Tensor of shape (batch_size,)\n target_vec = torch.tensor(rewards) + (discount * v_next_vec)\n \n # Computing action values at the current states for all actions using network\n # q_mat is a 2D Tensor of shape (batch_size, num_actions)\n q_mat = network.forward(states)\n \n # Batch Indices is an array from 0 to the batch size - 1. \n batch_indices = torch.arange(q_mat.shape[0])\n\n # Compute q_vec by selecting q(s, a) from q_mat for taken actions\n # q_vec are the estimates\n # q_vec is a 1D Tensor of shape (batch_size)\n estimate_vec = q_mat[batch_indices, actions] \n\n return target_vec, estimate_vec\n```\n\n## Computing Network's optmizer\n\nOne important step is to optimize the network using the TD estimate and the TD target computed previously. As a quick recap, the Mean squared value error is given below.\n\n$$\\overline{VE} = \\sum_s\\mu(s)[V_\\pi(s) - \\hat{v}(s,w)]^2$$\n\nThe idea is to use $\\overline{VE}$ as the Loss function to optimize the action-value network. For this particular problem, the MSE implementation provided by Pytorch is used. Additionally, the Adam optimizer is used to optimize the weights of the neural network.\n\nSee [Section 9.2 of Reinforment Learning an Introduction](http://www.incompleteideas.net/book/RLbook2018.pdf#page=221)\n\n\n```python\n### Work Required: Yes. Fill in code in optimize_network (~2 Lines).\ndef optimize_network(experiences, discount, optimizer, network, current_q, tau, criterion):\n \"\"\"\n Args:\n experiences (Numpy array): The batch of experiences including the states, actions, \n rewards, terminals, and next_states.\n discount (float): The discount factor.\n network (ActionValueNetwork): The latest state of the network that is getting replay updates.\n current_q (ActionValueNetwork): The fixed network used for computing the targets, \n and particularly, the action-values at the next-states.\n Return:\n Loss (float): The loss value for the current batch.\n \"\"\"\n \n # Get states, action, rewards, terminals, and next_states from experiences\n states, actions, rewards, terminals, next_states = map(list, zip(*experiences))\n states = np.concatenate(states) # Batch per states\n next_states = np.concatenate(next_states) # Batch per states\n rewards = np.array(rewards) # Batch size\n terminals = np.array(terminals) # Batch size\n batch_size = states.shape[0] # Batch size\n \n # Computing TD target and estimate using get_td function\n td_target, td_estimate = get_td(states, next_states, actions, rewards, discount, terminals, \\\n network, current_q, tau)\n \n # zero the gradients buffer\n optimizer.zero_grad()\n # Compute the Mean squared value error loss\n loss = criterion(td_estimate.double().to(device), td_target.to(device))\n # Backprop the error\n loss.backward()\n # Optimize the network\n optimizer.step()\n \n return (loss / batch_size).detach().numpy()\n```\n\n## Implementing Expected-Sarsa Agent\n\nThe final step is to use all the methods implemented above in the Expected-Sarsa Agent.\n\n\n```python\n### Expected Expected-Sarsa Agent\nclass ExpectedSarsaAgent():\n def __init__(self):\n self.name = \"expected_sarsa_agent\"\n \n def agent_init(self, agent_config):\n \"\"\"Setup for the agent called when the experiment first starts.\n\n Set parameters needed to setup the agent.\n\n Assume agent_config dict contains:\n {\n network_config: dictionary,\n optimizer_config: dictionary,\n replay_buffer_size: integer,\n minibatch_sz: integer, \n num_replay_updates_per_step: float\n discount_factor: float,\n }\n \"\"\"\n self.replay_buffer = ReplayBuffer(agent_config['replay_buffer_size'], \n agent_config['minibatch_sz'])\n # Add model to CPU or GPU respectively\n self.network = ActionValueNetwork(agent_config['network_config']).to(device)\n self.optimizer = optim.Adam(self.network.parameters(), lr = agent_config['optimizer_config']['step_size'], \n betas=(agent_config['optimizer_config']['beta_m'], agent_config['optimizer_config']['beta_v']),\n eps=agent_config['optimizer_config']['epsilon']) \n self.criterion = nn.MSELoss()\n self.num_actions = agent_config['network_config']['num_actions']\n self.num_replay = agent_config['num_replay_updates_per_step']\n self.discount = agent_config['gamma']\n self.tau = agent_config['tau']\n \n self.last_state = None\n self.last_action = None\n \n self.sum_rewards = 0\n self.episode_steps = 0\n self.loss = 0\n\n def policy(self, state):\n \"\"\"\n Args:\n state (Numpy array): the state.\n Returns:\n the action. \n \"\"\"\n action_values = self.network.forward(state)\n probs_batch = softmax(action_values, self.tau).detach().numpy()\n action = np.random.choice(self.num_actions, p=probs_batch.squeeze())\n return action\n\n def agent_start(self, state):\n \"\"\"The first method called when the experiment starts, called after\n the environment starts.\n Args:\n state (Numpy array): the state from the\n environment's evn_start function.\n Returns:\n The first action the agent takes.\n \"\"\"\n self.sum_rewards = 0\n self.episode_steps = 0\n self.last_state = np.array([state])\n self.last_action = self.policy(self.last_state)\n return self.last_action\n\n def agent_step(self, reward, state):\n \"\"\"A step taken by the agent.\n Args:\n reward (float): the reward received for taking the last action taken\n state (Numpy array): the state from the\n environment's step based, where the agent ended up after the\n last step\n Returns:\n The action the agent is taking.\n \"\"\"\n \n # Add current reward to the sum of rewards\n self.sum_rewards += reward\n self.episode_steps += 1\n\n # Make state an array of shape (1, state_dim) to add a batch dimension and\n # to later match the forward() and get_td() functions\n state = np.array([state])\n\n # Select action\n action = self.policy(state) #change for state for submission, normally, it is self.last_state\n \n # Append new experience to replay buffer\n self.replay_buffer.append(self.last_state, self.last_action, reward, 0, state)\n \n # Perform replay steps:\n if self.replay_buffer.size() > self.replay_buffer.minibatch_size:\n # Make a copy of the current network to obtain stable targets\n current_q = deepcopy(self.network)\n for _ in range(self.num_replay): \n # Get sample experiences from the replay buffer\n experiences = self.replay_buffer.sample()\n \n # Call optimize_network to update the weights of the network \n self.loss +=optimize_network(experiences, self.discount, self.optimizer, self.network, current_q, self.tau,\n self.criterion)\n \n # Update the last state and last action.\n self.last_state = state\n self.last_action = action\n \n return self.last_action\n\n # update of the weights using optimize_network \n def agent_end(self, reward):\n \"\"\"Run when the agent terminates.\n Args:\n reward (float): the reward the agent received for entering the\n terminal state.\n \"\"\"\n self.sum_rewards += reward\n self.episode_steps += 1\n \n # Set terminal state to an array of zeros\n state = np.zeros_like(self.last_state)\n\n self.replay_buffer.append(self.last_state, self.last_action, reward, 1, state)\n \n # Perform replay steps:\n if self.replay_buffer.size() > self.replay_buffer.minibatch_size:\n current_q = deepcopy(self.network)\n for _ in range(self.num_replay):\n \n # Get sample experiences from the replay buffer\n experiences = self.replay_buffer.sample()\n \n # Call optimize_network to update the weights of the network\n self.loss += optimize_network(experiences, self.discount, self.optimizer, self.network, current_q, self.tau,\n self.criterion)\n \n \n def agent_message(self, message):\n if message == \"get_sum_reward\":\n return self.sum_rewards, self.episode_steps\n else:\n raise Exception(\"Unrecognized Message!\")\n\n```\n\n## Running the experiment\n\nThe following lines solves the Lunar Lander problem and plot the average reward obtained over episodes, steps taken to solve the challenge at a specific episode and average loss over episodes.\n\n\n```python\n# Test the expected Sarsa Agent \n#model = ActionValueNetwork(network_config).to(device)\nnum_runs = 1\nnum_episodes = 1000\n\n# Experiment parameters\nagent_info = {\n 'network_config': {\n 'state_dim': env.observation_space.shape[0],\n 'num_hidden_units': 256,\n 'num_actions': env.action_space.n\n },\n 'optimizer_config': {\n 'step_size': 1e-3, \n 'beta_m': 0.9, \n 'beta_v': 0.999,\n 'epsilon': 1e-8\n },\n 'replay_buffer_size': 50000,\n 'minibatch_sz': 8,\n 'num_replay_updates_per_step': 4,\n 'gamma': 0.99,\n 'tau': 0.001}\n\n# Variable to store the amount of steps taken to solve the challeng\nall_steps = []\n# Variable to save the rewards in an episode\nall_rewards = []\nall_loss = []\n\n# Agent\nagent = ExpectedSarsaAgent()\n\n# Environment\nenv = gym.make('LunarLander-v2')\nenv.reset()\n# Maximum number of possible iterations (default was 200)\nenv._max_episode_steps = 10000\n\n# Number of runs are the times the experiment will start again (a.k.a episode)\nfor n_runs in range(num_runs):\n \n # Resets environment\n observation = env.reset()\n # Reset agent\n agent.agent_init(agent_info)\n # Generate last state and action in the agent\n last_action = agent.agent_start(observation)\n # Steps, rewards and loss at each episode to solve the challenge\n steps_per_episode = []\n rewards_per_episode = []\n loss_per_episode = []\n \n # Times the environment will start again without resetting the agent\n for t in tqdm(range(num_episodes)):\n \n # Reset done flag\n done = False\n # Set rewards, steps and loss to zero\n rewards = 0\n n_steps = 0\n agent.loss = 0\n # Reset environment\n observation = env.reset()\n # Run until the experiment is over\n while not done:\n \n # Render the environment only after t > # episodes\n if t > 300:\n env.render()\n\n # Take a step with the environment\n observation, reward, done, info = env.step(last_action)\n \n rewards += reward\n n_steps += 1\n\n # If the goal has been reached stop\n if done:\n # Last step with the agent\n agent.agent_end(reward)\n else:\n # Take a step with the agent\n last_action = agent.agent_step(reward, observation)\n \n # Append steps taken to solve the episode\n steps_per_episode.append(n_steps)\n # Reward obtained during the episode\n rewards_per_episode.append(rewards)\n # Loss obtained solving the experiment\n loss_per_episode.append(agent.loss)\n\n # Steps taken to solve the experiment during all\n all_steps.append(np.array(steps_per_episode))\n # Awards obtained during all episode\n all_rewards.append(np.array(rewards_per_episode))\n # Loss obtained during all episodes\n all_loss.append(loss_per_episode)\n\nenv.close()\n```\n\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 1000/1000 [51:03<00:00, 3.06s/it] \n\n\n\n```python\nsteps_average = np.mean(np.array(all_steps), axis=0)\nplt.plot(steps_average, label = 'Steps')\nplt.xlabel(\"Episodes\")\nplt.ylabel(\"Iterations\",rotation=0, labelpad=40)\nplt.xlim(-0.2, num_episodes)\nplt.ylim(steps_average.min(), steps_average.max())\nplt.title(\"Average iterations to solve the experiment over runs\")\nplt.legend()\nplt.show()\nprint(\"The Minimum number of iterations used to solve the experiment were: {0}\\n\".format(np.array(all_steps).min()))\nprint(\"The Maximum number of iterations used to solve the experiment were: {0}\\n\".format(np.array(all_steps).max()))\n```\n\n\n```python\nrewards_average = np.mean(all_rewards, axis=0)\nplt.plot(rewards_average, label = 'Average Reward')\nplt.xlabel(\"Episodes\")\nplt.ylabel(\"Sum of\\n rewards\\n during\\n episode\" ,rotation=0, labelpad=40)\nplt.xlim(-0.2, num_episodes)\nplt.ylim(rewards_average.min(), rewards_average.max())\nplt.title(\"Average reward to solve the experiment over runs\")\nplt.legend()\nplt.show()\nprint(\"The best reward obtained solving the experiment was: {0}\\n\".format(np.array(all_rewards).max()))\nprint(\"The Worst reward obtained solving the experiment was: {0}\\n\".format(np.array(all_rewards).min()))\n```\n\n\n```python\nloss_average = np.mean(np.array(all_loss), axis=0)\nplt.plot(loss_average, label = 'Steps')\nplt.xlabel(\"Episodes\")\nplt.ylabel(\"Average loss\",rotation=0, labelpad=40)\nplt.xlim(-0.2, num_episodes)\nplt.ylim(loss_average.min(), loss_average.max())\nplt.title(\"Average loss over iterations\")\nplt.legend()\nplt.show()\nprint(\"The best loss obtained solving the experiment was: {0}\\n\".format(np.array(loss_average).min()))\nprint(\"The Worst loss obtained solving the experiment was: {0}\\n\".format(np.array(loss_average).max()))\n\n```\n\n## Using the last trained Agent \n\nThis lines shows in a video the performance of the last trained agent and save a video with the results.\n\n\n```python\n# Test Sarsa Agent \nnum_runs = 1\nnum_episodes = 1000\n\n# Environment\nenv_to_wrap = gym.make('LunarLander-v2')\n# Maximum number of possible iterations (default was 200)\nenv_to_wrap._max_episode_steps = 1500\nenv = Monitor(env_to_wrap, \"./videos/lunarLander\", video_callable=lambda episode_id: True, force=True)\n\n\n# Number of runs are the times the experiment will start again (a.k.a episode)\nfor n_runs in tqdm(range(num_runs)):\n \n # Resets environment\n observation = env.reset()\n # Generate last state and action in the agent\n last_action = agent.agent_start(observation)\n \n # Times the environment will start again without resetting the agent\n for t in tqdm(range(num_episodes)):\n\n # View environment\n env.render()\n # Take a step with the environment\n observation, reward, done, info = env.step(last_action)\n\n # If the goal has been reached stop\n if done:\n # Last step with the agent\n agent.agent_end(reward)\n break\n else:\n # Take a step with the agent\n last_action = agent.agent_step(reward, observation)\n\nenv.close()\nenv_to_wrap.close()\n\nprint(\"Episode finished after {} timesteps\".format(t+1))\n```\n\n 0%| | 0/1 [00:00 \u57fa\u672c\u67e5\u8a62\uff1a\u96a8\u5802\u7df4\u7fd2\n\n\u90ed\u8000\u4ec1\n\n\n```python\nimport sqlite3\nimport pandas as pd\nfrom sqlFrameCheck import checkAnsQuery\nfrom test_queries.test_queries_02 import extract_test_queries as etq\n\nconn_twelection = sqlite3.connect('twelection.db')\nconn_nba = sqlite3.connect('nba.db')\n```\n\n## \u96a8\u5802\u7df4\u7fd2\uff1a\u4ee5 `||` \u904b\u7b97\u7b26\u5c07 `presidential2020` \u8868\u683c\u4e2d `county`\u3001`town` \u8207 `village` \u9023\u7d50\u8aa0\u4e00\u500b\u65b0\u7684\u6b04\u4f4d\u4e26\u547d\u540d\u70ba `combined_key`\uff0c\u518d\u4ee5 `DISTINCT` \u66b8\u89e3\u53f0\u7063\u7368\u4e00\u7684\u9078\u8209\u5340\u6709\u54ea\u4e9b\n\n\n```python\nans_query = \"\"\"\n-- \u5c07\u67e5\u8a62\u8a9e\u6cd5\u5beb\u5728\u9019\u88e1\n\"\"\"\n# \u8a66\u8dd1\u770b\u770b\u7d50\u679c\npd.read_sql(ans_query, conn_twelection)\n```\n\n## \u6e2c\u8cc7\u6bd4\u5c0d\n\n\n```python\ncaq = checkAnsQuery(etq('0201'), ans_query, conn_twelection)\ncaq.run_test()\n```\n\n## \u96a8\u5802\u7df4\u7fd2\uff1a\u8a08\u7b97 NBA \u7403\u54e1\u7684\u8eab\u9ad4\u8cea\u91cf\u6307\u6578\uff08BMI\uff09\u4e26\u7531\u5927\u5230\u5c0f\u6392\u5e8f\uff0c\u9078\u64c7 `firstName`\u3001`lastName` \u8207 `bmi` \u9019\u4e09\u500b\u8b8a\u6578\n\n\n```python\nans_query = \"\"\"\n-- \u5c07\u67e5\u8a62\u8a9e\u6cd5\u5beb\u5728\u9019\u88e1\n\"\"\"\n# \u8a66\u8dd1\u770b\u770b\u7d50\u679c\npd.read_sql(ans_query, conn_nba)\n```\n\n## \u6e2c\u8cc7\u6bd4\u5c0d\n\n\n```python\ncaq = checkAnsQuery(etq('0202'), ans_query, conn_nba)\ncaq.run_test()\n```\n\n## \u96a8\u5802\u7df4\u7fd2\uff1a\u4f7f\u7528 `WHERE` \u5c07 `presidential2020` \u4e2d\u81fa\u5317\u5e02\u7684\u8cc7\u6599\u7be9\u9078\u51fa\u4f86\n\n\n```python\nans_query = \"\"\"\n-- \u5c07\u67e5\u8a62\u8a9e\u6cd5\u5beb\u5728\u9019\u88e1\n\"\"\"\n# \u8a66\u8dd1\u770b\u770b\u7d50\u679c\npd.read_sql(ans_query, conn_twelection)\n```\n\n## \u6e2c\u8cc7\u6bd4\u5c0d\n\n\n```python\ncaq = checkAnsQuery(etq('0203'), ans_query, conn_twelection)\ncaq.run_test()\n```\n\n## \u96a8\u5802\u7df4\u7fd2\uff1a\u4f7f\u7528 `WHERE` \u5c07 `presidential2020` \u8868\u683c\u4e2d\u7684\u516d\u90fd\uff08\u81fa\u5317\u5e02\u3001\u65b0\u5317\u5e02\u3001\u6843\u5712\u5e02\u3001\u81fa\u4e2d\u5e02\u3001\u81fa\u5357\u5e02\u8207\u9ad8\u96c4\u5e02\uff09\u7684\u8cc7\u6599\u7be9\u9078\u51fa\u4f86\n\n\n```python\nans_query = \"\"\"\n-- \u5c07\u67e5\u8a62\u8a9e\u6cd5\u5beb\u5728\u9019\u88e1\n\"\"\"\n# \u8a66\u8dd1\u770b\u770b\u7d50\u679c\npd.read_sql(ans_query, conn_twelection)\n```\n\n## \u6e2c\u8cc7\u6bd4\u5c0d\n\n\n```python\ncaq = checkAnsQuery(etq('0204'), ans_query, conn_twelection)\ncaq.run_test()\n```\n\n## \u96a8\u5802\u7df4\u7fd2\uff1a\u67e5\u8a62 `careerSummaries` \u751f\u6daf\u5834\u5747\u5f97\u5206 `ppg` \u8d85\u904e 20 \u5206\u7684\u7403\u54e1 ID `personId`\u3001\u5834\u5747\u5f97\u5206 `ppg`\n\n\n```python\nans_query = \"\"\"\n-- \u5c07\u67e5\u8a62\u8a9e\u6cd5\u5beb\u5728\u9019\u88e1\n\"\"\"\n# \u8a66\u8dd1\u770b\u770b\u7d50\u679c\npd.read_sql(ans_query, conn_nba)\n```\n\n## \u6e2c\u8cc7\u6bd4\u5c0d\n\n\n```python\ncaq = checkAnsQuery(etq('0205'), ans_query, conn_nba)\ncaq.run_test()\n```\n\n## \u96a8\u5802\u7df4\u7fd2\uff1a\u5f9e `careerSummaries` \u4e2d\u627e\u51fa\u751f\u6daf\u52a9\u653b\u5931\u8aa4\u6bd4\uff08assists/turnovers\uff09\u6700\u9ad8\u7684\u524d 10 \u500b\u7403\u54e1 ID `personId`\u3001\u52a9\u653b\u5931\u8aa4\u6bd4 `ast_to_ratio`\n\n\\begin{equation}\nast\\_to\\_ratio = \\frac{assists}{turnovers}\n\\end{equation}\n\n\n```python\nans_query = \"\"\"\n-- \u5c07\u67e5\u8a62\u8a9e\u6cd5\u5beb\u5728\u9019\u88e1\n\"\"\"\n# \u8a66\u8dd1\u770b\u770b\u7d50\u679c\npd.read_sql(ans_query, conn_nba)\n```\n\n## \u6e2c\u8cc7\u6bd4\u5c0d\n\n\n```python\ncaq = checkAnsQuery(etq('0206'), ans_query, conn_nba)\ncaq.run_test()\n```\n", "meta": {"hexsha": "4dd2636040be8a0ac534806e33bf0d05e16cc287", "size": 6261, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "02-exercises.ipynb", "max_stars_repo_name": "datainpoint/classroom-introduction-to-sql", "max_stars_repo_head_hexsha": "7bc46c09084bd56a2ce89f5bdecb6cf6a4e3022c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "02-exercises.ipynb", "max_issues_repo_name": "datainpoint/classroom-introduction-to-sql", "max_issues_repo_head_hexsha": "7bc46c09084bd56a2ce89f5bdecb6cf6a4e3022c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2020-08-01T04:05:06.000Z", "max_issues_repo_issues_event_max_datetime": "2020-08-01T04:05:26.000Z", "max_forks_repo_path": "02-exercises.ipynb", "max_forks_repo_name": "yaojenkuo/introduction-to-sql", "max_forks_repo_head_hexsha": "d94657463ab0685b743ec8d168086e3ab512f626", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 18.4690265487, "max_line_length": 134, "alphanum_fraction": 0.4967257627, "converted": true, "num_tokens": 1082, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819874558604, "lm_q2_score": 0.6926419767901475, "lm_q1q2_score": 0.4234688283653163}} {"text": "# Phosphofructokinase (PFK) Model Construction\n\nBased on Chapter 14 of Systems Biology: Simulation of Dynamic Network States\n\nTo construct the phosphofructokinase module, first we import **MASSpy** and other essential packages. Constants used throughout the notebook are also defined.\n\n\n```python\nfrom operator import attrgetter\nfrom os import path\n\nimport matplotlib.pyplot as plt\n\nimport numpy as np\n\nfrom scipy import optimize\n\nimport sympy as sym\n\nfrom cobra import DictList\n\nfrom mass import MassConfiguration, MassMetabolite, Simulation, UnitDefinition\nfrom mass.enzyme_modules import EnzymeModule\nfrom mass.io import json, sbml\nfrom mass.example_data import create_example_model\nfrom mass.util.expressions import Keq2k, k2Keq, strip_time\nfrom mass.util.matrix import matrix_rank\nfrom mass.util.qcqa import qcqa_model\n\nmass_config = MassConfiguration()\n\nmass_config.irreversible_Keq = float(\"inf\")\n```\n\nNote that the total enzyme concentration of PFK is $33 nM = 0.033 \\mu M = 0.000033 mM$.\n\nFor the construction of the `EnzymeModule` for PFK, the following assumptions were made:\n\n1. The enzyme is a homotetramer.\n2. The enzyme binding and catalyzation of substrates occurs in an ordered sequential mechanism.\n3. The mechanism of allosteric regulation is based on the Monod-Wyman-Changeux (MWC) model for allosteric transitions of homoproteins.\n\n## Module Construction\nThe first step of creating the PFK module is to define the `EnzymeModule`. The `EnzymeModule` is an extension of the `MassModel`, with additional enzyme-specific attributes that aid in the construction, validation, and utilization of the module.\n\n__Note:__ All `EnzymeModule` specific attributes start will start the prefix \"enzyme\" or \"enzyme_module\".\n\n\n```python\nPFK = EnzymeModule(\"PFK\", name=\"Phosphofructokinase\", \n subsystem=\"Glycolysis\")\n```\n\n Set parameter Username\n\n\n### Metabolites\n#### Ligands\nThe next step is to define all of the metabolites using the `MassMetabolite` object. For `EnzymeModule` objects, the `MassMetabolite` objects will be refered to as ligands, for these `MassMetabolite` form a complex with the enzyme to serve some biological purpose. Some considerations for this step include the following:\n\n1. It is important to use a clear and consistent format for identifiers and names when defining the `MassMetabolite` objects for various reasons, some of which include improvements to model clarity and utility, assurance of unique identifiers (required to add metabolites to the model), and consistency when collaborating and communicating with others. \n\n2. In order to ensure our model is physiologically accurate, it is important to provide the `formula` argument with a string representing the chemical formula for each metabolite, and the `charge` argument with an integer representing the metabolite's ionic charge (Note that neutrally charged metabolites are provided with 0). These attributes can always be set later if necessary using the `formula` and `charge` attribute setter methods.\n\n3. To indicate that the cytosol is the cellular compartment in which the reactions occur, the string \"c\" is provided to the `compartment` argument.\n\nThis model will be created using identifiers and names found in the [BiGG Database](http://bigg.ucsd.edu/).\n\nThe ligands correspond to the activators, inhibitors, cofactors, substrates, and products involved in the enzyme catalyzed reaction. In this model, there are 6 species which must be considered.\n\n\n```python\nf6p_c = MassMetabolite(\n \"f6p_c\",\n name=\"D-Fructose 6-phosphate\",\n formula=\"C6H11O9P\",\n charge=-2,\n compartment=\"c\")\nfdp_c = MassMetabolite(\n \"fdp_c\",\n name=\"D-Fructose 1,6-bisphosphate\",\n formula=\"C6H10O12P2\",\n charge=-4,\n compartment=\"c\")\natp_c = MassMetabolite(\n \"atp_c\",\n name=\"ATP\",\n formula=\"C10H12N5O13P3\",\n charge=-4,\n compartment=\"c\")\nadp_c = MassMetabolite(\n \"adp_c\",\n name=\"ADP\",\n formula=\"C10H12N5O10P2\",\n charge=-3,\n compartment=\"c\")\namp_c = MassMetabolite(\n \"amp_c\",\n name=\"AMP\",\n formula=\"C10H12N5O7P\",\n charge=-2,\n compartment=\"c\")\nh_c = MassMetabolite(\n \"h_c\",\n name=\"H+\",\n formula=\"H\",\n charge=1,\n compartment=\"c\") \n```\n\nAfter generating the ligands, they are added to the `EnzymeModule` through the `add_metabolites` method. The ligands of the `EnzymeModule` can be viewed as a `DictList` through the `enzyme_module_ligands` attribute.\n\n\n```python\n# Add the metabolites to the EnzymeModule\nPFK.add_metabolites([f6p_c, fdp_c, atp_c, adp_c, amp_c, h_c])\n# Access DictList of ligands and print\nprint(\"All {0} Ligands: {1}\".format(\n PFK.id, \"; \".join([m.id for m in PFK.enzyme_module_ligands])))\n```\n\n All PFK Ligands: f6p_c; fdp_c; atp_c; adp_c; amp_c; h_c\n\n\nThe `enzyme_module_ligands_categorized` attribute can be used to assign metabolites to groups of user-defined categories by providing a dictionary where keys are the categories and values are the metabolites. Note that any metabolite can be placed in more than one category.\n\n\n```python\nPFK.enzyme_module_ligands_categorized = {\n \"Substrates\": f6p_c,\n \"Cofactors\": atp_c,\n \"Activators\": amp_c,\n \"Inhibitors\": atp_c,\n \"Products\": [fdp_c, adp_c, h_c]}\n\n# Access DictList of ligands and print\nprint(\"All {0} ligands ({1} total):\\n{2}\\n\".format(\n PFK.id, len(PFK.enzyme_module_ligands),\n str([m.id for m in PFK.enzyme_module_ligands])))\n\n# Access categorized attribute for ligands and print\nfor group in PFK.enzyme_module_ligands_categorized:\n print(\"{0}: {1}\".format(\n group.id, str([m.id for m in group.members])))\n```\n\n All PFK ligands (6 total):\n ['f6p_c', 'fdp_c', 'atp_c', 'adp_c', 'amp_c', 'h_c']\n \n Substrates: ['f6p_c']\n Cofactors: ['atp_c']\n Activators: ['amp_c']\n Inhibitors: ['atp_c']\n Products: ['fdp_c', 'adp_c', 'h_c']\n\n\n#### EnzymeModuleForms\n\nThe next step is to define the various states of the enzyme and enzyme-ligand complexes. These states can be represented through an `EnzymeModuleForm` object. Just like how `EnzymeModule` objects extend `MassModels`, the `EnzymeModuleForm` objects extend `MassMetabolite` objects, giving them the same functionality as a `MassMetabolite`. However, there are two important additional attrubutes that are specific to the `EnzymeModuleForm`.\n\n* The first attribute is the `enzyme_module_id`. It is meant to hold the identifier or name of the corresponding `EnzymeModule`.\n* The second attribute is the `bound_metabolites` attribute, designed to contain metabolites bound to the enzymatic site(s).\n* Automatic generation of the `name`, `formula`, and `charge` attributes attributes utilize the `bound_metabolites` attribute, which can aid in identification of `EnzymeModuleForm` and mass and charge balancing of the reactions.\n\nThe most convenient way to make an `EnzymeModuleForm` is through the `EnzymeModule.make_enzyme_module_form` method. There are several reasons to use this method to generate the `EnzymeModuleForm` objects:\n\n1. The only requirement to creating an `EnzymeModuleForm` is the identifier.\n2. A string can optionally be provided for the `name` argument to set the corresponding `name` attribute, or it can automatically be generated and set by setting the string \"Automatic\" (case sensitve). \n3. The `enzyme_module_id`, `formula` and `charge` attributes are set based on the identifier of the EnzymeModule and the MassMetabolite objects found in `bound_metabolites`.\n4. Just like the `enzyme_module_ligands_categorized` attribute, there is the `enzyme_module_forms_categorized` attribute that behaves in a similar manner. Categories can be set at the time of construction by providing a string or a list of strings to the `categories` argument. \n5. `EnzymeModuleForm` objects are automatically added to the `EnzymeModule` once created.\n\nFor this module, there are 20 `EnzymeModuleForm` objects that must be created. Because of the assumptions made for this module, a loop can be used to help automate the construction of the `EnzymeModuleForm` objects.\n\n\n```python\n# Number of identical subunits\nn_subunits = 4\n\nfor i in range(n_subunits + 1):\n # Make enzyme module forms per number of bound activators (Up to 4 Total)\n PFK.make_enzyme_module_form(\n \"pfk_R{0:d}_c\".format(i), \n name=\"Automatic\", \n categories=(\"Relaxed\", \"Free_Catalytic\"),\n bound_metabolites={amp_c: i},\n compartment=\"c\");\n\n PFK.make_enzyme_module_form(\n \"pfk_R{0:d}_A_c\".format(i), \n name=\"Automatic\", \n categories=(\"Relaxed\", \"Complexed_ATP\"),\n bound_metabolites={atp_c: 1, amp_c: i},\n compartment=\"c\");\n \n PFK.make_enzyme_module_form(\n \"pfk_R{0:d}_AF_c\".format(i), \n name=\"Automatic\", \n categories=(\"Relaxed\", \"Complexed_ATP_F6P\"),\n bound_metabolites={atp_c: 1, f6p_c: 1, amp_c: i},\n compartment=\"c\");\n\n # Make enzyme module forms per number of bound inhibitors (Up to 4 Total)\n PFK.make_enzyme_module_form(\n \"pfk_T{0:d}_c\".format(i), \n name=\"Automatic\", \n categories=\"Tense\",\n bound_metabolites={atp_c: i},\n compartment=\"c\");\n\n# Access DictList of enzyme module forms and print\nprint(\"All {0} enzyme module forms ({1} total):\\n{2}\\n\".format(\n PFK.id, len(PFK.enzyme_module_forms),\n str([m.id for m in PFK.enzyme_module_forms])))\n\n# Access categorized attribute for enzyme module forms and print\nfor group in PFK.enzyme_module_forms_categorized:\n print(\"{0}: {1}\\n\".format(\n group.id, str(sorted([m.id for m in group.members]))))\n```\n\n All PFK enzyme module forms (20 total):\n ['pfk_R0_c', 'pfk_R0_A_c', 'pfk_R0_AF_c', 'pfk_T0_c', 'pfk_R1_c', 'pfk_R1_A_c', 'pfk_R1_AF_c', 'pfk_T1_c', 'pfk_R2_c', 'pfk_R2_A_c', 'pfk_R2_AF_c', 'pfk_T2_c', 'pfk_R3_c', 'pfk_R3_A_c', 'pfk_R3_AF_c', 'pfk_T3_c', 'pfk_R4_c', 'pfk_R4_A_c', 'pfk_R4_AF_c', 'pfk_T4_c']\n \n Relaxed: ['pfk_R0_AF_c', 'pfk_R0_A_c', 'pfk_R0_c', 'pfk_R1_AF_c', 'pfk_R1_A_c', 'pfk_R1_c', 'pfk_R2_AF_c', 'pfk_R2_A_c', 'pfk_R2_c', 'pfk_R3_AF_c', 'pfk_R3_A_c', 'pfk_R3_c', 'pfk_R4_AF_c', 'pfk_R4_A_c', 'pfk_R4_c']\n \n Free_Catalytic: ['pfk_R0_c', 'pfk_R1_c', 'pfk_R2_c', 'pfk_R3_c', 'pfk_R4_c']\n \n Complexed_ATP: ['pfk_R0_A_c', 'pfk_R1_A_c', 'pfk_R2_A_c', 'pfk_R3_A_c', 'pfk_R4_A_c']\n \n Complexed_ATP_F6P: ['pfk_R0_AF_c', 'pfk_R1_AF_c', 'pfk_R2_AF_c', 'pfk_R3_AF_c', 'pfk_R4_AF_c']\n \n Tense: ['pfk_T0_c', 'pfk_T1_c', 'pfk_T2_c', 'pfk_T3_c', 'pfk_T4_c']\n \n\n\n## Reactions\n### EnzymeModuleReactions\nOnce all of the `MassMetabolite` and `EnzymeModuleForm` objects have been created, the next step is to define all of the enzyme-ligand binding reactions and conformation trasitions that occur in its mechanism.\n\nThese reactions can be represented through an `EnzymeModuleReaction` object. As with the previous enzyme objects, `EnzymeModuleReactions` extend `MassReaction` objects to maintain the same functionality. However, as with the `EnzymeModuleForm`, the `EnzymeModuleReaction` has additional enzyme-specific attributes, such as the `enzyme_module_id`.\n\nThe most conveient way to make an `EnzymeModuleReaction` is through the `EnzymeModule.make_enzyme_module_reaction` method. There are several reasons to use this method to generate the EnzymeModuleReactions:\n\n1. The only requirement to creating an `EnzymeModuleReaction` is an identifier.\n2. A string can optionally be provided for the `name` argument to set the corresponding `name` attribute, or it can automatically be generated and set by setting the string \"Automatic\" (case sensitve). \n3. There is an `enzyme_module_reactions_categorized` attribute that behaves in a similar manner as the previous categorized attributes. Categories can be set at the time of construction by providing a string or a list of strings to the `categories` argument. \n4. `MassMetabolite` and `EnzymeModuleForm` objects that already exist in the `EnzymeModule` can be directly added to the newly created `EnzymeModuleReaction` by providing a dictionary to the optional `metabolites_to_add` argument using string identifiers (or the objects) as keys and their stoichiometric coefficients as the values.\n5. `EnzymeModuleReactions` are automatically added to the `EnzymeModule` once created.\n\nFor this module, there are 24 `EnzymeModuleReactions` that must be created. Because of the assumptions made for this module, a loop can be used to help automate the construction of the `EnzymeModuleReactions`.\n\n\n```python\nfor i in range(n_subunits + 1):\n # Make reactions for enzyme-ligand binding and catalytzation per number of bound activators (Up to 4 Total)\n PFK.make_enzyme_module_reaction(\n \"PFK_R{0:d}1\".format(i), \n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=True,\n categories=\"atp_c_binding\",\n metabolites_to_add={\n \"pfk_R{0:d}_c\".format(i): -1, \n \"atp_c\": -1, \n \"pfk_R{0:d}_A_c\".format(i): 1})\n \n PFK.make_enzyme_module_reaction(\n \"PFK_R{0:d}2\".format(i), \n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=True,\n categories=\"f6p_c_binding\",\n metabolites_to_add={\n \"pfk_R{0:d}_A_c\".format(i): -1, \n \"f6p_c\": -1, \n \"pfk_R{0:d}_AF_c\".format(i): 1})\n \n PFK.make_enzyme_module_reaction(\n \"PFK_R{0:d}3\".format(i), \n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=False,\n categories=\"catalyzation\",\n metabolites_to_add={\n \"pfk_R{0:d}_AF_c\".format(i): -1, \n \"pfk_R{0:d}_c\".format(i): 1, \n \"adp_c\": 1, \n \"fdp_c\": 1,\n \"h_c\": 1})\n \n if i < n_subunits:\n # Make enzyme reactions for enzyme-activator binding\n PFK.make_enzyme_module_reaction(\n \"PFK_R{0:d}0\".format(i + 1), \n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=True,\n categories=\"amp_c_activation\",\n metabolites_to_add={\n \"pfk_R{0:d}_c\".format(i): -1, \n \"amp_c\": -1, \n \"pfk_R{0:d}_c\".format(i + 1): 1})\n\n # Make enzyme reactions for enzyme-inhibitor binding\n PFK.make_enzyme_module_reaction(\n \"PFK_T{0:d}\".format(i + 1), \n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=True,\n categories=\"atp_c_inhibition\",\n metabolites_to_add={\n \"pfk_T{0:d}_c\".format(i): -1, \n \"atp_c\": -1, \n \"pfk_T{0:d}_c\".format(i + 1): 1})\n\n# Make reaction representing enzyme transition from R to T state\nPFK.make_enzyme_module_reaction(\n \"PFK_L\",\n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=True,\n categories=\"RT_transition\",\n metabolites_to_add={\n \"pfk_R0_c\": -1, \n \"pfk_T0_c\": 1})\n\n# Access DictList of enzyme module reactions and print\nprint(\"All {0} enzyme module reactions ({1} total):\\n{2}\\n\".format(\n PFK.id, len(PFK.enzyme_module_reactions),\n str([m.name for m in PFK.enzyme_module_reactions])))\n\n# Access categorized attribute for enzyme module reactions and print\nfor group in PFK.enzyme_module_reactions_categorized:\n print(\"{0}: {1}\\n\".format(\n group.id, str(sorted([m.id for m in group.members]))))\n```\n\n All PFK enzyme module reactions (24 total):\n ['pfk_R0-atp binding', 'pfk_R0_A-f6p binding', 'pfk_R0_AF catalyzation', 'pfk_R0-amp binding', 'pfk_T0-atp binding', 'pfk_R1-atp binding', 'pfk_R1_A-f6p binding', 'pfk_R1_AF catalyzation', 'pfk_R1-amp binding', 'pfk_T1-atp binding', 'pfk_R2-atp binding', 'pfk_R2_A-f6p binding', 'pfk_R2_AF catalyzation', 'pfk_R2-amp binding', 'pfk_T2-atp binding', 'pfk_R3-atp binding', 'pfk_R3_A-f6p binding', 'pfk_R3_AF catalyzation', 'pfk_R3-amp binding', 'pfk_T3-atp binding', 'pfk_R4-atp binding', 'pfk_R4_A-f6p binding', 'pfk_R4_AF catalyzation', 'pfk_R0-pfk_T0 transition']\n \n atp_c_binding: ['PFK_R01', 'PFK_R11', 'PFK_R21', 'PFK_R31', 'PFK_R41']\n \n f6p_c_binding: ['PFK_R02', 'PFK_R12', 'PFK_R22', 'PFK_R32', 'PFK_R42']\n \n catalyzation: ['PFK_R03', 'PFK_R13', 'PFK_R23', 'PFK_R33', 'PFK_R43']\n \n amp_c_activation: ['PFK_R10', 'PFK_R20', 'PFK_R30', 'PFK_R40']\n \n atp_c_inhibition: ['PFK_T1', 'PFK_T2', 'PFK_T3', 'PFK_T4']\n \n RT_transition: ['PFK_L']\n \n\n\n### Create and Unify Rate Parameters\nThe next step is to unify rate parameters of binding steps that are not unique, allowing for those parameter values to be defined once and stored in the same place. Therefore, custom rate laws with custom parameters are used to reduce the number of parameters that need to be defined and better represent the module.\n\nThe rate law parameters can be unified using the `EnzymeModule.unify_rate_parameters` class method. This method requires a list of reactions whose rate laws that should be identical, along with a string representation of the new identifier to use on the unified parameters. There is also the optional `prefix ` argument, which if set to True, will ensure the new parameter identifiers are prefixed with the `EnzymeModule` identifier. This can be used to help prevent custom parameters from being replaced when multiple models are merged.\n\n#### Allosteric Transitions: Symmetry Model\n\nOnce rate parameters are unified, the allosteric regulation of this enzyme must be accounted for. Because this module is to be based on the (Monod-Wyman-Changeux) MWC model for ligand binding and allosteric regulation, the rate laws of the allosteric binding reactions must be adjusted to reflect the symmetry in the module using the number of identical binding sites to help determine the scalars for the parameters. \n\nFor this module, PFK is considered a homotetramer, meaning it has four identical subunits $\\nu = 4$. Each subunit can be allosterically activated by AMP or inhibited by ATP. The helper functions `k2Keq`, `Keq2k`, and `strip_time` from the `mass.util` submodule will be used to help facilitate the rate law changes in this example so that the final rate laws are dependent on the forward rate (kf) and equilibrium (Keq) constants.\n\n\n```python\nabbreviations = [\"A\", \"F\", \"I\", \"ACT\"]\nligands = [atp_c, f6p_c, atp_c, amp_c]\n\nfor met, unified_id in zip(ligands, abbreviations):\n category = {\"A\": \"binding\",\n \"F\": \"binding\",\n \"I\": \"inhibition\",\n \"ACT\": \"activation\"}[unified_id]\n group = PFK.enzyme_module_reactions_categorized.get_by_id(\n \"_\".join((met.id, category)))\n reactions = sorted(group.members, key=attrgetter(\"id\"))\n PFK.unify_rate_parameters(reactions, unified_id,\n rate_type=2, enzyme_prefix=True)\n # Add the coefficients to make symmetry model rate laws for activation and inhibition \n if unified_id in [\"I\", \"ACT\"]:\n for i, reaction in enumerate(reactions):\n custom_rate = str(strip_time((reaction.rate)))\n custom_rate = custom_rate.replace(\n \"kf_\", \"{0:d}*kf_\".format(n_subunits - i))\n custom_rate = custom_rate.replace(\n \"kr_\", \"{0:d}*kr_\".format(i + 1))\n PFK.add_custom_rate(reaction, custom_rate)\n \nPFK.unify_rate_parameters(\n PFK.enzyme_module_reactions_categorized.get_by_id(\"catalyzation\").members,\n \"PFK\")\n# Update rate laws to be in terms of kf and Keq\nPFK.custom_rates.update(k2Keq(PFK.custom_rates))\n\n# Access categorized attribute for enzyme module reactions and print\nfor group in PFK.enzyme_module_reactions_categorized:\n header = \"Category: \" + group.id\n print(\"\\n\" + header + \"\\n\" + \"-\" * len(header))\n for reaction in sorted(group.members, key=attrgetter(\"id\")):\n print(reaction.id + \": \" + str(reaction.rate))\n```\n\n \n Category: atp_c_binding\n -----------------------\n PFK_R01: kf_PFK_A*(atp_c(t)*pfk_R0_c(t) - pfk_R0_A_c(t)/Keq_PFK_A)\n PFK_R11: kf_PFK_A*(atp_c(t)*pfk_R1_c(t) - pfk_R1_A_c(t)/Keq_PFK_A)\n PFK_R21: kf_PFK_A*(atp_c(t)*pfk_R2_c(t) - pfk_R2_A_c(t)/Keq_PFK_A)\n PFK_R31: kf_PFK_A*(atp_c(t)*pfk_R3_c(t) - pfk_R3_A_c(t)/Keq_PFK_A)\n PFK_R41: kf_PFK_A*(atp_c(t)*pfk_R4_c(t) - pfk_R4_A_c(t)/Keq_PFK_A)\n \n Category: f6p_c_binding\n -----------------------\n PFK_R02: kf_PFK_F*(f6p_c(t)*pfk_R0_A_c(t) - pfk_R0_AF_c(t)/Keq_PFK_F)\n PFK_R12: kf_PFK_F*(f6p_c(t)*pfk_R1_A_c(t) - pfk_R1_AF_c(t)/Keq_PFK_F)\n PFK_R22: kf_PFK_F*(f6p_c(t)*pfk_R2_A_c(t) - pfk_R2_AF_c(t)/Keq_PFK_F)\n PFK_R32: kf_PFK_F*(f6p_c(t)*pfk_R3_A_c(t) - pfk_R3_AF_c(t)/Keq_PFK_F)\n PFK_R42: kf_PFK_F*(f6p_c(t)*pfk_R4_A_c(t) - pfk_R4_AF_c(t)/Keq_PFK_F)\n \n Category: catalyzation\n ----------------------\n PFK_R03: kf_PFK*pfk_R0_AF_c(t)\n PFK_R13: kf_PFK*pfk_R1_AF_c(t)\n PFK_R23: kf_PFK*pfk_R2_AF_c(t)\n PFK_R33: kf_PFK*pfk_R3_AF_c(t)\n PFK_R43: kf_PFK*pfk_R4_AF_c(t)\n \n Category: amp_c_activation\n --------------------------\n PFK_R10: kf_PFK_ACT*(4*amp_c(t)*pfk_R0_c(t) - pfk_R1_c(t)/Keq_PFK_ACT)\n PFK_R20: kf_PFK_ACT*(3*amp_c(t)*pfk_R1_c(t) - 2*pfk_R2_c(t)/Keq_PFK_ACT)\n PFK_R30: kf_PFK_ACT*(2*amp_c(t)*pfk_R2_c(t) - 3*pfk_R3_c(t)/Keq_PFK_ACT)\n PFK_R40: kf_PFK_ACT*(amp_c(t)*pfk_R3_c(t) - 4*pfk_R4_c(t)/Keq_PFK_ACT)\n \n Category: atp_c_inhibition\n --------------------------\n PFK_T1: kf_PFK_I*(4*atp_c(t)*pfk_T0_c(t) - pfk_T1_c(t)/Keq_PFK_I)\n PFK_T2: kf_PFK_I*(3*atp_c(t)*pfk_T1_c(t) - 2*pfk_T2_c(t)/Keq_PFK_I)\n PFK_T3: kf_PFK_I*(2*atp_c(t)*pfk_T2_c(t) - 3*pfk_T3_c(t)/Keq_PFK_I)\n PFK_T4: kf_PFK_I*(atp_c(t)*pfk_T3_c(t) - 4*pfk_T4_c(t)/Keq_PFK_I)\n \n Category: RT_transition\n -----------------------\n PFK_L: kf_PFK_L*(pfk_R0_c(t) - pfk_T0_c(t)/Keq_PFK_L)\n\n\n## The Steady State\n### Solve steady state concentrations symbolically\nTo determine the steady state of the enzyme, a dictionary of the ordinary differential equations as symbolic expressions for each of the `EnzymeModuleForm` objects. The ligands are first removed from the equations by assuming their values are taken into account in a lumped rate constant parameter.\n\nFor handling of all symbolic expressions, the **SymPy** package is used.\n\n\n```python\n# Make a dictionary of ODEs and lump ligands into rate parameters by giving them a value of 1\node_dict = {}\nlump_ligands = {sym.Symbol(met.id): 1 for met in PFK.enzyme_module_ligands}\nfor enzyme_module_form in PFK.enzyme_module_forms:\n symbol_key = sym.Symbol(enzyme_module_form.id)\n ode = sym.Eq(strip_time(enzyme_module_form.ode), 0)\n ode_dict[symbol_key] = ode.subs(lump_ligands)\n\nrank = matrix_rank(PFK.S[6:])\nprint(\"Rank Deficiency: {0}\".format(len(ode_dict) - rank))\n```\n\n Rank Deficiency: 1\n\n\nIn order to solve the system of ODEs for the steady state concentrations, an additional equation is required due to the rank deficiency of the stoichiometric matrix. Therefore, the equation for the steady state flux through the enzyme, which will be referred to as the \"enzyme net flux equation\", must be defined. \n\nTo define the enzyme net flux equation, the `EnzymeModule.make_enzyme_netflux_equation` class method can be used. \n\n* This equation is made by providing a reaction, or a list of reactions to add together.\n* Passing a bool to `use_rates` argument determines whether a symbolic equation is a summation of the flux symbols returned by `EnzymeModuleReaction.flux_symbol_str`, or a summation of the rates laws for those reactions.\n* The `update_enzyme` argument determines whether the new rate equation is set in the `enzyme_rate_equation` attribute.\n\nThe flux through the enzyme typically corresponds to the sum of the fluxes through the catalytic reaction steps.\nBecause the catalyzation reactions were assigned to the \"catalyzation\" cateogry, they can be accessed through the `enzyme_module_reactions_categorized` attribute to create the equation for $v_{\\mathrm{PFK}}$.\n\n\n```python\nreactions = PFK.enzyme_module_reactions_categorized.get_by_id(\n \"catalyzation\").members\nPFK.make_enzyme_rate_equation(\n reactions,\n use_rates=True, update_enzyme=True)\nsym.pprint(PFK.enzyme_rate_equation)\n```\n\n kf_PFK\u22c5(pfk_R0_AF_c(t) + pfk_R1_AF_c(t) + pfk_R2_AF_c(t) + pfk_R3_AF_c(t) + pfk_R4_AF_c(t))\n\n\nThe next step is to identify equations for the unknown concentrations in each reaction. These equations will need to be solved with a dependent variable before accounting for the enzyme net flux equation. The completely free form of the enzyme with no bound species will be treated as the dependent variable. \n\nTo verify that all equations are in terms of the lumped rate parameters, and the dependent variable, the solutions can be iterated through using the atoms method to identify the equation arguments. There should be no `EnzymeModuleForm` identifiers with the exception of the dependent variable. \n\n\n```python\n# Get enzyme module forms\nenzyme_module_forms = PFK.enzyme_module_forms.copy()\n# Reverse list for increased performance (due to symmetry assumption)\n# by solving for the most activated/inhibitors bound first.\nenzyme_module_forms.reverse()\n\nenzyme_solutions = {}\nfor enzyme_module_form in enzyme_module_forms:\n # Skip dependent variable\n if \"pfk_R0_c\" == str(enzyme_module_form):\n continue\n enzyme_module_form = sym.Symbol(enzyme_module_form.id)\n # Susbtitute in previous solutions and solve for the enzyme module form, \n equation = ode_dict[enzyme_module_form]\n sol = sym.solveset(equation.subs(enzyme_solutions), enzyme_module_form)\n enzyme_solutions[enzyme_module_form] = list(sol)[0]\n # Update the dictionary of solutions with the solutions\n enzyme_solutions.update({\n enzyme_module_form: sol.subs(enzyme_solutions) \n for enzyme_module_form, sol in enzyme_solutions.items()})\nargs = set()\nfor sol in enzyme_solutions.values():\n args.update(sol.atoms(sym.Symbol))\nprint(args)\n```\n\n {Keq_PFK_I, kf_PFK_F, Keq_PFK_F, Keq_PFK_L, Keq_PFK_ACT, pfk_R0_c, kf_PFK, kf_PFK_A, Keq_PFK_A}\n\n\nThe enzyme net flux equation can then be utilized as the last equation required to solve for the final unknown concentration variable in terms of the rate and equilibrium constants, allowing for all of the concentration variables to be defined in terms of the rate and equilibrium constants. Once the unknown variable has been solved for, the solution can be substituted back into the other equations. Because `sympy.solveset` function expects the input equations to be equal to 0, the `EnzymeModule.enzyme_rate_error` method with the `use_values` argument set to `False` to get the appropriate expression.\n\n\n```python\nenzyme_rate_equation = strip_time(PFK.enzyme_rate_error(False))\nprint(\"Enzyme Net Flux Equation\\n\" + \"-\"*24)\nsym.pprint(enzyme_rate_equation)\n\n# Solve for last unknown concentration symbolically\nsol = sym.solveset(enzyme_rate_equation.subs(enzyme_solutions), \"pfk_R0_c\")\n\n# Update solution dictionary with the new solution\nenzyme_solutions[sym.Symbol(\"pfk_R0_c\")] = list(sol)[0]\n\n# Update solutions with free variable solutions\nenzyme_solutions = {\n enzyme_module_form: sym.simplify(solution.subs(enzyme_solutions))\n for enzyme_module_form, solution in enzyme_solutions.items()}\n\nargs = set()\nfor sol in enzyme_solutions.values():\n args.update(sol.atoms(sym.Symbol))\nprint(\"\\n\", args)\n```\n\n Enzyme Net Flux Equation\n ------------------------\n -kf_PFK\u22c5(pfk_R0_AF_c + pfk_R1_AF_c + pfk_R2_AF_c + pfk_R3_AF_c + pfk_R4_AF_c) + v_PFK\n \n {Keq_PFK_I, kf_PFK_F, Keq_PFK_L, kf_PFK, Keq_PFK_ACT, v_PFK, Keq_PFK_F, kf_PFK_A, Keq_PFK_A}\n\n\n#### Numerical Values\nAt this point, numerical values are defined for the dissociation constants and the concentrations of the substrates, cofactors, activators, and inhibitors. Providing these numerical values will speed up the subsequent calculations. \n\nTo do this, experimental data is used to define the dissociations constants for the different binding steps under the QEA. The concentrations of the non-enzyme species are taken from the glycolysis model. Experimental data gives the following for the dissociation constants: \n\n$$K_i=0.1 mM,\\\nK_a=0.033 mM,\\\nK_A=0.068 mM,\\\nK_F=0.1 mM$$\n\nand an allosteric constant of $K_L = 0.0011$.\n\n__Note:__ The $K_i$ binding constant for ATP as an inhibitor was increased by a factor of ten since magnesium complexing of ATP is not considered here. \n\n\n```python\nnumerical_values = {}\n\n# Get ligand IDs and parameter IDs\nligand_ids = sorted([str(ligand) for ligand in PFK.enzyme_module_ligands])\nparameter_ids = [\"_\".join((PFK.id, abbrev)) for abbrev in abbreviations + [\"L\"]]\nprint(\"Ligand IDs: \" + str(ligand_ids))\nprint(\"Parameter IDs: \" + str(parameter_ids))\n\n# Load the glycolysis model to extract steady state values\nglycolysis = create_example_model(\"SB2_Glycolysis\")\n\n# Get the steady state flux value and add to numerical values\nPFK.enzyme_rate = glycolysis.reactions.get_by_id(PFK.id).steady_state_flux\nnumerical_values.update({PFK.enzyme_flux_symbol_str: PFK.enzyme_rate})\n\n# Get the steady state concentration values and add to numerical values\ninitial_conditions = {\n str(ligand): glycolysis.initial_conditions[glycolysis.metabolites.get_by_id(ligand)]\n for ligand in ligand_ids}\n\n# Define parameter values and add to numerical values\n# Because of the QEA, invert dissociation constants for Keq\nparameter_values = {\n \"Keq_\" + parameter_id: value \n for parameter_id, value in zip(parameter_ids, [1/0.068, 1/0.1, 1/0.1, 1/0.033, 0.0011])}\n\n# Display numerical values\nprint(\"\\nNumerical Values\\n----------------\")\nfor k, v in numerical_values.items():\n print(\"{0} = {1}\".format(k, v))\n```\n\n Ligand IDs: ['adp_c', 'amp_c', 'atp_c', 'f6p_c', 'fdp_c', 'h_c']\n Parameter IDs: ['PFK_A', 'PFK_F', 'PFK_I', 'PFK_ACT', 'PFK_L']\n \n Numerical Values\n ----------------\n v_PFK = 1.12\n\n\nThe next step is to define the numerical values, $K_i=0.1/1.6$, $K_a=0.033/0.0867$, $K_A=0.068/1.6$, $K_F=0.1/0.0198$, $v_{PFK}=1.12 \\text{mM/hr}$, and $K_L=1/0.0011$ using the dissociation constant values and the steady state concentrations of the ligands and introduce them into the solution to get the steady state concentrations of the enzyme module forms in terms of the rate constants. The values of the equilirbium constants and initial conditions are also stored for later use.\n\n\n```python\n# Match abbreviations to their corresponding ligands\nabbreviation_dict = {\"PFK_A\": \"atp_c\", \"PFK_F\": \"f6p_c\", \"PFK_ACT\": \"amp_c\", \"PFK_I\": \"atp_c\", \"PFK_L\": \"\"}\n\nk2K = {sym.Symbol(\"kr_\" + p): sym.Symbol(\"kf_\" + p)*sym.Symbol(\"K_\" + p) for p in abbreviation_dict.keys()}\nenzyme_solutions = {met: sym.simplify(Keq2k(solution).subs(enzyme_solutions).subs(k2K))\n for met, solution in enzyme_solutions.items()}\nK_values = dict(zip([\"K_\" + p for p in abbreviation_dict], [0.068, 0.1, 0.033, 0.1, 0.0011]))\n\nfor abbrev, ligand_id in abbreviation_dict.items():\n K_str = \"K_\" + abbrev\n if ligand_id:\n numerical_value = K_values[K_str]/initial_conditions[ligand_id]\n else:\n numerical_value = 1/K_values[K_str]\n numerical_values[sym.Symbol(K_str)] = numerical_value\n \nenzyme_solutions = {met: sym.simplify(solution.subs(numerical_values))\n for met, solution in enzyme_solutions.items()}\n\n# Display numerical values\nprint(\"\\nNumerical Values\\n----------------\")\nfor k, v in numerical_values.items():\n print(\"{0} = {1}\".format(k, v))\n```\n\n \n Numerical Values\n ----------------\n v_PFK = 1.12\n K_PFK_A = 0.0425\n K_PFK_F = 5.05050505050505\n K_PFK_ACT = 0.3804995151513754\n K_PFK_I = 0.0625\n K_PFK_L = 909.090909090909\n\n\nThe last part of this step is to simplify the solutions for the enzyme module forms and, as a QA check, ensure that only rate constants are the only symbolic arguments in the solutions. \n\n\n```python\n# Substitute values into equations\nenzyme_solutions = {\n enzyme_module_form: sym.simplify(solution.subs(numerical_values))\n for enzyme_module_form, solution in enzyme_solutions.items()}\n\nargs = set()\nfor sol in enzyme_solutions.values():\n args.update(sol.atoms(sym.Symbol))\nprint(args)\n```\n\n {kf_PFK_F, kf_PFK, kf_PFK_A}\n\n\n### Determine rate constants\n#### Total Enzyme Concentration and $r_{T}$ \nAfter solving for the enzyme module forms, the next step is to define equations for the total enzyme concentration and for the fraction of the enzyme in the T state. These two equations can be used as constraints for determining the rate parameters. To view the equation for the total enzyme concentration, we can use the `EnzymeModule.enzyme_concentration_total_equation` property.\n\n\n```python\nsym.pprint(PFK.enzyme_concentration_total_equation)\n```\n\n pfk_R0_AF_c(t) + pfk_R0_A_c(t) + pfk_R0_c(t) + pfk_R1_AF_c(t) + pfk_R1_A_c(t) + pfk_R1_c(t) + pfk_R2_AF_\n c(t) + pfk_R2_A_c(t) + pfk_R2_c(t) + pfk_R3_AF_c(t) + pfk_R3_A_c(t) + pfk_R3_c(t) + pfk_R4_AF_c(t) + pfk\n _R4_A_c(t) + pfk_R4_c(t) + pfk_T0_c(t) + pfk_T1_c(t) + pfk_T2_c(t) + pfk_T3_c(t) + pfk_T4_c(t)\n\n\nThe total concentration of PFK is 33 nM (=0.000033 mM). The `EnzymeModule.enzyme_concentration_total` atrribute can be used to set and store this concentration.\n\n\n```python\nPFK.enzyme_concentration_total = 33e-6\nprint(PFK.enzyme_concentration_total)\n```\n\n 3.3e-05\n\n\nTo determine the rate constants, an optimization problem where the objective function is to minimize the error between the measured and calculated total enzyme concentrations. To create the objective function, the `EnzymeModule.enzyme_concentration_total_error` method with the `use_values` argument set as False to get the symbolic expression of the constraint. \n\n\n```python\nenzyme_total_constraint = abs(strip_time(PFK.enzyme_concentration_total_error(use_values=False)))\nsym.pprint(enzyme_total_constraint)\n```\n\n \u2502-PFK_Total + pfk_R0_AF_c + pfk_R0_A_c + pfk_R0_c + pfk_R1_AF_c + pfk_R1_A_c + pfk_R1_c + pfk_R2_AF_c + \n pfk_R2_A_c + pfk_R2_c + pfk_R3_AF_c + pfk_R3_A_c + pfk_R3_c + pfk_R4_AF_c + pfk_R4_A_c + pfk_R4_c + pfk_\n T0_c + pfk_T1_c + pfk_T2_c + pfk_T3_c + pfk_T4_c\u2502\n\n\nSubstitute the solutions for the enzyme forms to get an equation for the error in the enzyme total concentration in terms of the rate constants.\n\n\n```python\n# Substitute value for enzyme concentration total\nenzyme_total_constraint = enzyme_total_constraint.subs({PFK.enzyme_total_symbol_str: PFK.enzyme_concentration_total})\n# Substitute solutions into constraint and simplify\nenzyme_total_constraint = sym.simplify(enzyme_total_constraint.subs(enzyme_solutions))\nsym.pprint(enzyme_total_constraint)\n```\n\n \u2502 1.19283868483391 1.71385140785683 7.14443780219149\u2502\n \u2502-3.3e-5 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2502\n \u2502 kf_PFK_F kf_PFK_A kf_PFK \u2502\n\n\nTo create the objective function in a format suitable for the minimization method from the `scipy.optimize` submodule, the `sympy.lambdify` function can be used to convert the symbolic expression into a lambda function with the rate constants as the arguments. This lambda function can then be used to generate the objective function for the `optimize.minimize` method.\n\n\n```python\n# Create a sorted tuple of the arguments to ensure the input format does not change\nargs = tuple(sorted([str(arg) for arg in list(args)]))\n# Create the objective function as a lambda function\nobjective_function = lambda x: sym.lambdify(args, enzyme_total_constraint)(*x)\n```\n\nAnother constraint can be set on the amount of inhibited enzyme in the steady state of the system using the T fraction (denoted as $r_{T}$). This fraction is simply the amount of inhibited enzyme over the total amount of enzyme. The enzyme is inhibited between 10-15% under physiological conditions (Ponce et al. Biochimica et Biophysica Acta 1971 250(1):63-74)\n\nTo make the fraction as a symbolic expression, we can use the `EnzymeModule.make_enzyme_fraction` method. This method is designed to assist in making fractions and ratios by passing to the function:\n1. A string to the `categorized_attr` argument identifying which categorized attribute (either \"forms\" for the `EnzymeModule.enzyme_module_forms_categorized` or \"reactions\" for the `EnzymeModule.enzyme_module_reactions_categorized`).\n2. A string for the `top` argument and a string for the `bottom` argument identifying the categories to sum and use in the numerator and the denominator, respectively.\n3. A bool to the `use_values` argument indicating whether to substitute numerical values into the expression to return a float or to keep the ratio as a **SymPy** expression.\n\n__Note:__ The string \"Equation\" can be passed to either the `top` or `bottom` arguments to utilize the equation stored either in `enzyme_concentration_total_equation` (for `categorized_attr`=\"forms\"), or `enzyme_rate_equation` (for `categorized_attr`=\"reactions\").\n\n\n```python\n# Set the values for the constraint bounds\nr_T_lb, r_T_ub = (0.10, 0.15)\n# Make a symbolic expression for enzyme fraction.\nr_T_expr = PFK.make_enzyme_fraction(\n categorized_attr=\"forms\", top=\"Tense\", bottom=\"Equation\",\n use_values=False)\n# Substitute solutions into the expression to make\n# solely dependent on the rate constants\nr_T_expr = sym.simplify(strip_time(r_T_expr).subs(enzyme_solutions))\n\n# Make lambda functions for the T fraction constraint\nr_T_lb_constraint = lambda x: sym.lambdify(args, r_T_expr - r_T_lb)(*x)\nr_T_ub_constraint = lambda x: sym.lambdify(args, r_T_ub - r_T_expr)(*x)\n```\n\nLastly, we place lower and upper bounds on the rate constants to ensure that the values are non-negative and are within physiological limits, and then we solve the optmization problem. Once the optimization has finished, we check whether it was successful, and if so, what the optimality and errors are associated with this particular solution instance.\n\n\n```python\nprint(\"Ordered Args: {0}\\n\".format(str(args)))\n# Set arguments for minimization\nkf_bounds = ((1e2, 1e8), (1e2, 1e8), (1e2, 1e8))\ninitial_guess = [\n 3.07e5,\n 2e5,\n 1e6,]\n\n# Find a feasible solution\nsol = optimize.minimize(\n objective_function, x0=initial_guess,\n method=\"trust-constr\",\n bounds=kf_bounds,\n options={\"gtol\": 1e-20, \"xtol\": 1e-20, \"maxiter\": 1e4, \"disp\": True})\n\n# Check whether optimzation was successful\nprint(\"\\nOptimization Success: {0}\".format(sol.success))\nif sol.success:\n # Update the paramter values dictionary with the feasible solution\n parameter_values.update(dict(zip(args, [round(x) for x in sol.x])))\n print(\"Optimization Optimality: {0:.4e}\".format(sol.optimality))\n print(\"Parameter Solutions: {:}\".format(str({arg: parameter_values[arg] for arg in args})))\n # Plug solutions back into constraints for validation\n print(\"Optimization Error: {0:.4e}\".format(enzyme_total_constraint.subs(parameter_values)))\n```\n\n Ordered Args: ('kf_PFK', 'kf_PFK_A', 'kf_PFK_F')\n \n `xtol` termination condition is satisfied.\n Number of iterations: 104, function evaluations: 224, CG iterations: 116, optimality: 3.60e-11, constraint violation: 0.00e+00, execution time: 0.67 s.\n \n Optimization Success: True\n Optimization Optimality: 3.6029e-11\n Parameter Solutions: {'kf_PFK': 307263, 'kf_PFK_A': 200325, 'kf_PFK_F': 1000059}\n Optimization Error: 1.2079e-11\n\n\nWith a successful optimization, the module is updated with the parameter values. The inhibition and activation reactions are set to have a high forward rate constant and the allosteric transition even higher, limiting the amount of unbound enzyme and ensuring that the dynamics are determined by the dissociation and allosteric constants. \n\n__Note:__ This assumption for the rate constants can be made because none of the enzyme concentrations are dependendent on the activation, inhibition, and allosteric rate constants.\n\n\n```python\n# Add the activation, inhibition, and allosteric rate constants\nfor abbrev, value in zip([\"I\", \"ACT\", \"L\"], [1e6, 1e6, 1e6**2]):\n # Account for the enzyme prefix if used in the previous function\n to_join = (\"kf\", PFK.id, abbrev)\n param = \"_\".join(to_join)\n parameter_values.update({param: value})\n \n# Display numerical values\nfor k, v in parameter_values.items():\n print(\"{0} = {1}\".format(k, v))\n```\n\n Keq_PFK_A = 14.705882352941176\n Keq_PFK_F = 10.0\n Keq_PFK_I = 10.0\n Keq_PFK_ACT = 30.3030303030303\n Keq_PFK_L = 0.0011\n kf_PFK = 307263\n kf_PFK_A = 200325\n kf_PFK_F = 1000059\n kf_PFK_I = 1000000.0\n kf_PFK_ACT = 1000000.0\n kf_PFK_L = 1000000000000.0\n\n\n### Solve steady state concentrations numerically\n\nOnce the rate constants have been defined, the steady state concentrations of the enzyme can be determined.\n\n\n```python\n# Substitute values into equations\ninitial_conditions.update({\n str(enzyme_module_form): float(sym.simplify(solution.subs(parameter_values)))\n for enzyme_module_form, solution in enzyme_solutions.items()})\n\nfor header, dictlist in zip([\"Ligand\", \"\\nEnzyme\"], [PFK.enzyme_module_ligands, PFK.enzyme_module_forms]):\n header += \" Concentrations\"\n print(\"\\n\".join([header, \"-\" * len(header)]))\n for form in dictlist:\n ic = initial_conditions[form.id]\n print(\"{0} = {1}\".format(form.id, ic))\n```\n\n Ligand Concentrations\n ---------------------\n f6p_c = 0.0198\n fdp_c = 0.0146\n atp_c = 1.6\n adp_c = 0.29\n amp_c = 0.0867281\n h_c = 8.99757e-05\n \n Enzyme Concentrations\n ----------------------\n pfk_R0_c = 3.705684451779081e-08\n pfk_R0_A_c = 1.1270977736701491e-07\n pfk_R0_AF_c = 2.1036774576199985e-08\n pfk_T0_c = 4.0762528969569896e-11\n pfk_R1_c = 3.895599656998077e-07\n pfk_R1_A_c = 1.1848611930259641e-06\n pfk_R1_AF_c = 2.2114902898450058e-07\n pfk_T1_c = 2.6088018540524733e-09\n pfk_R2_c = 1.5357179846004314e-06\n pfk_R2_A_c = 4.670943637948869e-06\n pfk_R2_AF_c = 8.71810686394121e-07\n pfk_T2_c = 6.261124449725935e-08\n pfk_R3_c = 2.690705109903528e-06\n pfk_R3_A_c = 8.183880139927137e-06\n pfk_R3_AF_c = 1.5274845331446054e-06\n pfk_T3_c = 6.678532746374332e-07\n pfk_R4_c = 1.7678768321380616e-06\n pfk_R4_A_c = 5.377063448209202e-06\n pfk_R4_AF_c = 1.0036047828713532e-06\n pfk_T4_c = 2.6714130985497327e-06\n\n\n#### Set Initial Conditions and Parameters\nOnce the steady state concentrations have been determined, the initial conditions and parameters are added to the module. All custom parameter are added to the custom_parameter attribute. The allosteric transition uses the standard parameter identifiers (returned by `kf_str` and `Keq_str` properties of the `EnzymeModuleReaction`), so they are popped out of the custom parameters and set through their respective attribute setter methods. \n\n\n```python\n# Set initial conditions\nfor met, concentration in initial_conditions.items():\n PFK.metabolites.get_by_id(str(met)).ic = concentration\n\n# Add the custom parameters and values for kf and Keq to model\nPFK.custom_parameters.update(parameter_values)\n# PFK_L uses standard reaction parameters and not custom parameters\nPFK_L = PFK.enzyme_module_reactions.PFK_L\nPFK_L.kf = PFK.custom_parameters.pop(PFK_L.kf_str)\nPFK_L.Keq = PFK.custom_parameters.pop(PFK_L.Keq_str)\n\n# Set parameter values in reaction fields\nfor group in PFK.enzyme_module_reactions_categorized:\n if group.id == \"atp_c_binding\":\n param_id = \"PFK_A\"\n elif group.id == \"f6p_c_binding\":\n param_id = \"PFK_F\"\n elif group.id == \"catalyzation\":\n param_id = \"PFK\"\n elif group.id == \"atp_c_inhibition\":\n param_id = \"PFK_I\"\n elif group.id == \"amp_c_activation\":\n param_id = \"PFK_ACT\"\n else:\n continue\n for reaction in group.members:\n kf, Keq = (\"kf_\" + param_id, \"Keq_\" + param_id)\n if kf in PFK.custom_parameters:\n reaction.kf = PFK.custom_parameters[kf]\n if Keq in PFK.custom_parameters:\n reaction.Keq = PFK.custom_parameters[Keq]\n```\n\n#### Ordering of internal species and reactions\n\nSometimes, it is also desirable to reorder the metabolite and reaction objects inside the model to follow the physiology. To reorder the internal objects, one can use `cobra.DictList` containers and the `DictList.get_by_any` method with the list of object identifiers in the desirable order. To ensure all objects are still present and not forgotten in the model, a small QA check is also performed. \n\n\n```python\nnew_metabolite_order = ['f6p_c', 'fdp_c', 'amp_c', 'adp_c', 'atp_c', 'h_c',\n 'pfk_R0_c', 'pfk_R0_A_c', 'pfk_R0_AF_c', \n 'pfk_R1_c', 'pfk_R1_A_c', 'pfk_R1_AF_c', \n 'pfk_R2_c', 'pfk_R2_A_c', 'pfk_R2_AF_c', \n 'pfk_R3_c', 'pfk_R3_A_c', 'pfk_R3_AF_c',\n 'pfk_R4_c', 'pfk_R4_A_c', 'pfk_R4_AF_c', \n 'pfk_T0_c','pfk_T1_c', 'pfk_T2_c', 'pfk_T3_c', 'pfk_T4_c']\n\nif len(glycolysis.metabolites) == len(new_metabolite_order):\n PFK.metabolites = DictList(\n PFK.metabolites.get_by_any(new_metabolite_order))\n\nif len(PFK.metabolites) == len(new_metabolite_order):\n PFK.metabolites = DictList(PFK.metabolites.get_by_any(new_metabolite_order))\n \nnew_reaction_order = [\"PFK_R01\", 'PFK_R02', \"PFK_R03\", \"PFK_R10\", \n \"PFK_R11\", \"PFK_R12\", \"PFK_R13\", \"PFK_R20\", \n \"PFK_R21\", \"PFK_R22\", \"PFK_R23\", \"PFK_R30\", \n \"PFK_R31\", \"PFK_R32\", \"PFK_R33\", \"PFK_R40\", \n \"PFK_R41\", \"PFK_R42\", \"PFK_R43\", \"PFK_L\", \n \"PFK_T1\", \"PFK_T2\", \"PFK_T3\", \"PFK_T4\"]\n\nif len(PFK.reactions) == len(new_reaction_order):\n PFK.reactions = DictList(\n PFK.reactions.get_by_any(new_reaction_order))\n \nPFK.update_S(array_type=\"DataFrame\", dtype=int)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        PFK_R01PFK_R02PFK_R03PFK_R10PFK_R11PFK_R12PFK_R13PFK_R20PFK_R21PFK_R22...PFK_R33PFK_R40PFK_R41PFK_R42PFK_R43PFK_LPFK_T1PFK_T2PFK_T3PFK_T4
                                        f6p_c0-1000-1000-1...000-1000000
                                        fdp_c0010001000...1000100000
                                        amp_c000-1000-100...0-100000000
                                        adp_c0010001000...1000100000
                                        atp_c-1000-1000-10...00-1000-1-1-1-1
                                        h_c0010001000...1000100000
                                        pfk_R0_c-101-1000000...00000-10000
                                        pfk_R0_A_c1-100000000...0000000000
                                        pfk_R0_AF_c01-10000000...0000000000
                                        pfk_R1_c0001-101-100...0000000000
                                        pfk_R1_A_c00001-10000...0000000000
                                        pfk_R1_AF_c000001-1000...0000000000
                                        pfk_R2_c00000001-10...0000000000
                                        pfk_R2_A_c000000001-1...0000000000
                                        pfk_R2_AF_c0000000001...0000000000
                                        pfk_R3_c0000000000...1-100000000
                                        pfk_R3_A_c0000000000...0000000000
                                        pfk_R3_AF_c0000000000...-1000000000
                                        pfk_R4_c0000000000...01-10100000
                                        pfk_R4_A_c0000000000...001-1000000
                                        pfk_R4_AF_c0000000000...0001-100000
                                        pfk_T0_c0000000000...000001-1000
                                        pfk_T1_c0000000000...0000001-100
                                        pfk_T2_c0000000000...00000001-10
                                        pfk_T3_c0000000000...000000001-1
                                        pfk_T4_c0000000000...0000000001
                                        \n

                                        26 rows \u00d7 24 columns

                                        \n
                                        \n\n\n\n## Module Validation \n### QC/QA model\nBefore saving the module, it is important to ensure that the module is elementally balanced, and that the module can be integrated into a larger network for simulation. Therefore, the `qcqa_model` function from `mass.util.qcqa` is used to provide a report on the module quality and and indicate whether simulation is possible and if not, what parameters and/or initial conditions are missing. \n\n\n```python\nqcqa_model(PFK, parameters=True, concentrations=True, \n fluxes=False, superfluous=True, elemental=True)\n```\n\n \u2552\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2555\n \u2502 MODEL ID: PFK \u2502\n \u2502 SIMULATABLE: True \u2502\n \u2502 PARAMETERS NUMERICALY CONSISTENT: True \u2502\n \u255e\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2561\n \u2558\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255b\n\n\n### Constraint Satisfaction and Error Values\nAnother QA check we perform is to substitute the steady state numerical values back into the constraints used in determining the rate constants in order to ensure that the constraints remain satisified, and that errors are small. \n\n\n```python\nt_fraction = PFK.make_enzyme_fraction(\"forms\", top=\"Tense\",\n bottom=\"Equation\", use_values=True)\nprint(\"Enzyme T-fraction: {:.4f}\".format(t_fraction))\n\nprint(\"Concentration Absolute Error: {0:.4e}\".format(\n abs(PFK.enzyme_concentration_total_error(use_values=True))))\nprint(\"Flux Absolute Error: {0:.4e}\".format(\n abs(PFK.enzyme_rate_error(use_values=True))))\n```\n\n Enzyme T-fraction: 0.1032\n Concentration Absolute Error: 1.2079e-11\n Flux Absolute Error: 2.2204e-16\n\n\n### Add Enzyme to MassModel\nIn order to determine whether the module can be successfully integrated into a model, another model can be loaded, merged with the module, and simulated. To validate this module, it will be merged with a glycolysis model. \n\nTo integrate the `EnzymeModule` into the `MassModel`, the reaction that the EnzymeModule will be replacing is first removed. The `MassModel.merge` method can then be utilized to add the `EnzymeModule` to the `MassModel`. \n\nWhen merging an `EnzymeModule` and a `MassModel`, the `EnzymeModule` should always be merged into the `MassModel`.\n\n\n```python\n# Load and merge glycolysis with PFK model\nglycolysis = create_example_model(\"SB2_Glycolysis.json\")\n# Remove the PFK MassReaction, then merge the EnzymeModule into the MassModel\nglycolysis.remove_reactions([glycolysis.reactions.get_by_id(\"PFK\")])\nglycolysis_PFK = glycolysis.merge(PFK)\nglycolysis_PFK\n```\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        NameGlycolysis
                                        Memory address0x07ff5e94eb7f0
                                        Stoichiometric Matrix40x44
                                        Matrix Rank37
                                        Number of metabolites40
                                        Initial conditions defined40/40
                                        Number of reactions44
                                        Number of genes0
                                        Number of enzyme modules1
                                        Number of groups16
                                        Objective expression0
                                        CompartmentsCytosol
                                        \n\n\n\n\nUsing `MassModel.merge` class method enables the `EnzymeModule` and `MassModel` to be merged like as if they were both `MassModel` objects. However, all attributes specific to the `EnzymeModule` (e.g the categorized attributes) are condensed into a speciailzed container called an `EnzymeModuleDict`.\n\nThe `EnzymeModuleDict` behaves like an ordered dictionary, but is unique in that its contents can be accessed as if they were attributes. These attributes can be viewed using `EnzymeModuleDict.keys` method. All `EnzymeModuleDicts` associated with a `MassModel` can be accessed via `MassModel.enzyme_modules` attribute.\n\n\n```python\nprint(str(glycolysis_PFK.enzyme_modules) + \"\\n\")\nprint(\"Attribute Accessors:\\n-------------------\\n\" + \"\\n\".join(list(\n glycolysis_PFK.enzyme_modules.PFK.keys())) + \"\\n\")\nglycolysis_PFK.enzyme_modules.PFK\n```\n\n []\n \n Attribute Accessors:\n -------------------\n id\n name\n subsystem\n enzyme_module_ligands\n enzyme_module_forms\n enzyme_module_reactions\n enzyme_module_ligands_categorized\n enzyme_module_forms_categorized\n enzyme_module_reactions_categorized\n enzyme_concentration_total\n enzyme_rate\n enzyme_concentration_total_equation\n enzyme_rate_equation\n S\n model\n \n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        NamePFK
                                        Memory address0x07ff5f8aabcc0
                                        Stoichiometric Matrix26x24
                                        Matrix Rank20
                                        SubsystemGlycolysis
                                        Number of Ligands6
                                        Number of EnzymeForms20
                                        Number of EnzymeModuleReactions24
                                        Enzyme Concentration Total3.3e-05
                                        Enzyme Net Flux1.12
                                        \n\n\n\n\n### Validate Steady State\n\nTo find the steady state of the model and perform simulations, the model must first be loaded into a `Simulation`. In order to load a model into a `Simulation`, the model must be simulatable, meaning there are no missing numerical values that would prevent the integration of the ODEs that comprise the model. The `verbose` argument can be used while loading a model to produce a message indicating the successful loading of a model, or why a model could not load.\n\nOnce loaded into a `Simulation`, the `find_steady_state` method can be used with the `update_values` argument in order to update the initial conditions and fluxes of the model to a steady state. The model can then be simulated using the `simulate` method by passing the model to simulate, and a tuple containing the start time and the end time. The number of time points can also be included, but is optional.\n\nAfter a successful simulation, two `MassSolution` objects are returned. The first `MassSolution` contains the concentration results of the simulation, and the second contains the flux results of the simulation. \n\nTo visually validate the steady state of the model, concentration and flux solutions can be plotted using the `plot_time_profile` function from `mass.visualization`. Alternatively, the `MassSolution.view_time_profile` property can be used to quickly generate a time profile for the results.\n\n\n```python\n# Setup simulation object, ensure model is at steady state\nsim = Simulation(glycolysis_PFK, verbose=True)\nsim.find_steady_state(glycolysis_PFK, strategy=\"simulate\",\n update_values=True, verbose=True,\n tfinal=1e4, steps=1e6)\n\n# Simulate from 0 to 1000 with 10001 points in the output\nconc_sol, flux_sol = sim.simulate(\n glycolysis_PFK,time=(0, 1e3))\n# Quickly render and display time profiles\nconc_sol.view_time_profile()\n```\n\n### Storing information and references\n#### Compartment\nBecause the character \"c\" represents the cytosol compartment, it is recommended to define and set the compartment in the `EnzymeModule.compartments` attribute.\n\n\n```python\nPFK.compartments = {\"c\": \"Cytosol\"}\nprint(PFK.compartments)\n```\n\n {'c': 'Cytosol'}\n\n\n#### Units\nAll of the units for the numerical values used in this model are \"Millimoles\" for amount and \"Liters\" for volume (giving a concentration unit of 'Millimolar'), and \"Hours\" for time. In order to ensure that future users understand the numerical values for model, it is important to define the `MassModel.units` attribute.\n\nThe `MassModel.units` is a `cobra.DictList` that contains only `UnitDefinition` objects from the `mass.core.unit` submodule. Each `UnitDefinition` is created from `Unit` objects representing the base units that comprise the `UnitDefinition`. These `Units` are stored in the `list_of_units` attribute. Pre-built units can be viewed using the `print_defined_unit_values` function from the `mass.core.unit` submodule. Alternatively, custom units can also be created using the `UnitDefinition.create_unit` method. For more information about units, please see the module docstring for `mass.core.unit` submodule.\n\n__Note:__ It is important to note that this attribute will NOT track units, but instead acts as a reference for the user and others so that they can perform necessary unit conversions.\n\n\n```python\n# Using pre-build units to define UnitDefinitions\nconcentration = UnitDefinition(\"mM\", name=\"Millimolar\",\n list_of_units=[\"millimole\", \"per_litre\"])\ntime = UnitDefinition(\"hr\", name=\"hour\", list_of_units=[\"hour\"])\n\n# Add units to model\nPFK.add_units([concentration, time])\nprint(PFK.units)\n```\n\n [, ]\n\n\n## Export\n\nAfter validation, the model is ready to be saved. The model can either be exported as a \".json\" file or as an \".sbml\" (\".xml\") file using their repsective submodules in `mass.io`.\n\nTo export the model, only the path to the directory and the model object itself need to be specified.\n\n### Export using SBML\n\n\n```python\nsbml.write_sbml_model(mass_model=PFK, filename=\"SB2_\" + PFK.id + \".xml\")\n```\n\n### Export using JSON\n\n\n```python\njson.save_json_model(mass_model=PFK, filename=\"SB2_\" + PFK.id + \".json\")\n```\n", "meta": {"hexsha": "329e04ff58517debfc2e582390cf6fdf3e671ebc", "size": 162715, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/education/sb2/model_construction/sb2_pfk.ipynb", "max_stars_repo_name": "z-haiman/MASSpy", "max_stars_repo_head_hexsha": "aeeed1e3f9d1058e9485247a86f85cb94eeecbc9", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/education/sb2/model_construction/sb2_pfk.ipynb", "max_issues_repo_name": "z-haiman/MASSpy", "max_issues_repo_head_hexsha": "aeeed1e3f9d1058e9485247a86f85cb94eeecbc9", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/education/sb2/model_construction/sb2_pfk.ipynb", "max_forks_repo_name": "z-haiman/MASSpy", "max_forks_repo_head_hexsha": "aeeed1e3f9d1058e9485247a86f85cb94eeecbc9", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 58.9974619289, "max_line_length": 50960, "alphanum_fraction": 0.6510770365, "converted": true, "num_tokens": 20999, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419704455588, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.4234688146773548}} {"text": "# Calculating Thermodynamics Observables with a quantum computer\n\n\n```python\n# imports\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nfrom functools import partial\n\nfrom qiskit.utils import QuantumInstance\nfrom qiskit import Aer\nfrom qiskit.algorithms import NumPyMinimumEigensolver, VQE\n\nfrom qiskit_nature.drivers import UnitsType, Molecule\nfrom qiskit_nature.drivers.second_quantization import (\n ElectronicStructureDriverType,\n ElectronicStructureMoleculeDriver,\n)\nfrom qiskit_nature.problems.second_quantization import ElectronicStructureProblem\nfrom qiskit_nature.converters.second_quantization import QubitConverter\nfrom qiskit_nature.mappers.second_quantization import JordanWignerMapper\nfrom qiskit_nature.algorithms import GroundStateEigensolver\nimport qiskit_nature.constants as const\nfrom qiskit_nature.algorithms.pes_samplers import BOPESSampler, EnergySurface1DSpline\nfrom thermodynamics_utils.thermodynamics import constant_volume_heat_capacity\nfrom thermodynamics_utils.vibrational_structure_fd import VibrationalStructure1DFD\nfrom thermodynamics_utils.partition_function import DiatomicPartitionFunction\nfrom thermodynamics_utils.thermodynamics import Thermodynamics\n\nimport warnings\n\nwarnings.simplefilter(\"ignore\", np.RankWarning)\n```\n\nA preliminary draft with more information related to this tutorial can be found in preprint: Stober et al, arXiv 2003.02303 (2020)\n\n### Calculation of the Born Oppenheimer Potential Energy Surface (BOPES) \n\nTo compute thermodynamic observables we begin with single point energy calculation which calculates the wavefunction and charge density and therefore the energy of a particular arrangement of nuclei. Here we compute the Born-Oppenheimer potential energy surface of a hydrogen molecule, as an example, which is simply the electronic energy as a function of bond length. \n\n\n```python\nqubit_converter = QubitConverter(mapper=JordanWignerMapper())\nquantum_instance = QuantumInstance(backend=Aer.get_backend(\"aer_simulator_statevector\"))\nsolver = VQE(quantum_instance=quantum_instance)\n\nme_gss = GroundStateEigensolver(qubit_converter, solver)\n```\n\n\n```python\nstretch1 = partial(Molecule.absolute_distance, atom_pair=(1, 0))\nmol = Molecule(\n geometry=[(\"H\", [0.0, 0.0, 0.0]), (\"H\", [0.0, 0.0, 0.2])],\n degrees_of_freedom=[stretch1],\n masses=[1.6735328e-27, 1.6735328e-27],\n)\n\n\n# pass molecule to PSYCF driver\ndriver = ElectronicStructureMoleculeDriver(mol, driver_type=ElectronicStructureDriverType.PYSCF)\n\nes_problem = ElectronicStructureProblem(driver)\n```\n\n\n```python\n# BOPES sampler testing\nbs = BOPESSampler(me_gss, bootstrap=True)\npoints = np.linspace(0.45, 5, 50)\nres = bs.sample(es_problem, points)\n```\n\n\n```python\nenergies = []\nbs_res_full = res.raw_results\nfor point in points:\n energy = bs_res_full[point].computed_energies + bs_res_full[point].nuclear_repulsion_energy\n energies.append(energy)\n```\n\n\n```python\nfig = plt.figure()\nplt.plot(points, energies)\nplt.title(\"Dissociation profile\")\nplt.xlabel(\"Interatomic distance\")\nplt.ylabel(\"Energy\")\n```\n\n\n```python\nenergy_surface = EnergySurface1DSpline()\n\nxdata = res.points\nydata = res.energies\nenergy_surface.fit(xdata=xdata, ydata=ydata)\n```\n\n\n```python\nplt.plot(xdata, ydata, \"kx\")\nx = np.arange(min(xdata) - 0.25, max(xdata) + 0.25, 0.05)\nplt.plot(x, energy_surface.eval(x), \"r-\")\nplt.xlabel(r\"distance, $\\AA$\")\nplt.ylabel(\"energy, Hartree\")\ndist = max(ydata) - min(ydata)\nplt.ylim(min(ydata) - 0.1 * dist, max(ydata) + 0.1 * dist)\n```\n\n### Calculation of the molecular Vibrational Energy levels\n\nThe Born-Oppeheimer approximation removes internuclear vibrations from the molecular Hamiltonian and the energy computed from quantum mechanical ground-state energy calculations using this approximation contain only the electronic energy. Since even at absolute zero internuclear vibrations still occur, a correction is required to obtain the true zero-temperature energy of a molecule. This correction is called the zero-point vibrational energy (ZPE), which is computed by summing the contribution from internuclear vibrational modes. Therefore, the next step in computing thermodynamic observables is determining the vibrational energy levels. This can be done by constructing the Hessian matrix based on computed single point energies close to the equilibrium bond length. The eigenvalues of the Hessian matrix can then be used to determine the vibrational energy levels and the zero-point vibrational energy \t\n\\begin{equation}\n{\\rm ZPE} = \\frac{1}{2}\\, \\sum_i ^M \\nu_i \\, ,\n\\end{equation}\nwith $\\nu_i$ being the vibrational frequencies, $M = 3N \u2212 6$ or $M = 3N \u2212 5$ for non-linear or linear molecules, respectively, and $N$ is number of the particles. \n\nHere we fit a \"full\" energy surface using a 1D spline potential and use it to evaluate molecular vibrational energy levels.\n\n\n\n```python\nvibrational_structure = VibrationalStructure1DFD(mol, energy_surface)\n\nplt.plot(xdata, ydata, \"kx\")\nx = np.arange(min(xdata) - 0.25, max(xdata) + 0.25, 0.05)\nplt.plot(x, energy_surface.eval(x), \"r-\")\nplt.xlabel(r\"distance, $\\AA$\")\nplt.ylabel(\"energy, Hartree\")\ndist = max(ydata) - min(ydata)\nplt.ylim(min(ydata) - 0.1 * dist, max(ydata) + 0.1 * dist)\nfor N in range(15):\n on = np.ones(x.shape)\n on *= energy_surface.eval(\n energy_surface.get_equilibrium_geometry()\n ) + vibrational_structure.vibrational_energy_level(N)\n plt.plot(x, on, \"g:\")\non = np.ones(x.shape)\n\nplt.show()\n```\n\n### Create a partition function for the calculation of heat capacity\n\n\nThe partition function for a molecule is the product of contributions from translational, rotational, vibrational, electronic, and nuclear degrees of freedom. Having the vibrational frequencies, now we can obtain the vibrational partition function $q_{\\rm vibration}$ to compute the whole molecular partition function \n\\begin{equation}\nq_{\\rm vibration} = \\prod_{i=1} ^M \\frac{\\exp\\,(-\\Theta_{\\nu_i}/2T)}{1-\\exp\\,(-\\Theta_{\\nu_i}/2T)} \\, . \n\\end{equation} \nHere $\\Theta_{\\nu_i}= h\\nu_i/k_B$, $T$ is the temperature and $k_B$ is the Boltzmann constant. \n\nThe single-point energy calculations and the resulting partition function can be used to calculate the (constant volume or constant pressure) heat capacity of the molecules. The constant volume heat capacity, for example, is given by \n\n\\begin{equation}\nC_v = \\left.\\frac{\\partial U}{\\partial T}\\right|_{N,V}\\, ,\n\\qquad\n{\\rm with} \\quad\nU=k_B T^2 \\left.\\frac{\\partial {\\rm ln} Q}{\\partial T}\\right|_{N,V} .\n\\end{equation}\n\n$U$ is the internal energy, $V$ is the volume and $Q$ is the partition function. \n\n\n\nHere we illustrate the simplest usage of the partition function, namely creating a Thermodynamics object to compute properties like the constant pressure heat capacity defined above. \n\n\n```python\nQ = DiatomicPartitionFunction(mol, energy_surface, vibrational_structure)\n\nP = 101350 # Pa\ntemps = np.arange(10, 1050, 5) # K\n\nmol.spins = [1 / 2, 1 / 2]\n\ntd = Thermodynamics(Q, pressure=101350)\ntd.set_pressure(101350)\ntemps = np.arange(10, 1500, 5)\nymin = 5\nymax = 11\n\nplt.plot(temps, td.constant_pressure_heat_capacity(temps) / const.CAL_TO_J)\nplt.xlim(0, 1025)\nplt.ylim(ymin, ymax)\nplt.xlabel(\"Temperature, K\")\nplt.ylabel(\"Cp, cal mol$^{-1}$ K$^{-1}$\")\n\nplt.show()\n```\n\nHere we demonstrate how to access particular components (the rotational part) of the partition function, which in the H2 case we can further split to para-hydrogen and ortho-hydrogen components.\n\n\n```python\neq = Q.get_partition(part=\"rot\", split=\"eq\")\npara = Q.get_partition(part=\"rot\", split=\"para\")\northo = Q.get_partition(part=\"rot\", split=\"ortho\")\n```\n\nWe will now plot the constant volume heat capacity (of the rotational part) demonstrating how we can call directly the functions in the 'thermodynamics' module, providing a callable object for the partition function (or in this case its rotational component). Note that in the plot we normalize the plot dividing by the universal gas constant R (Avogadro's number times Boltzmann's constant) and we use crossed to compare with experimental data found in literature.\n\n\n```python\n# REFERENCE DATA from literature\ndf_brink_T = [80.913535, 135.240157, 176.633783, 219.808499, 246.226899]\ndf_brink_Cv = [0.118605, 0.469925, 0.711510, 0.833597, 0.895701]\n\ndf_eucken_T = [\n 25.120525,\n 30.162485,\n 36.048121,\n 41.920364,\n 56.195875,\n 62.484934,\n 72.148692,\n 73.805910,\n 73.804236,\n 92.214423,\n 180.031917,\n 230.300866,\n]\ndf_eucken_Cv = [\n 0.012287,\n 0.012354,\n 0.008448,\n 0.020478,\n 0.032620,\n 0.048640,\n 0.048768,\n 0.076678,\n 0.078670,\n 0.170548,\n 0.667731,\n 0.847681,\n]\n\ndf_gia_T = [\n 190.919338,\n 195.951254,\n 202.652107,\n 204.292585,\n 209.322828,\n 225.300754,\n 234.514217,\n 243.747768,\n]\ndf_gia_Cv = [0.711700, 0.723719, 0.749704, 0.797535, 0.811546, 0.797814, 0.833793, 0.845868]\n\ndf_parting_T = [80.101665, 86.358919, 185.914204, 239.927797]\ndf_parting_Cv = [0.084730, 0.138598, 0.667809, 0.891634]\n\ndf_ce_T = [\n 80.669344,\n 135.550569,\n 145.464190,\n 165.301153,\n 182.144856,\n 203.372528,\n 237.993108,\n 268.696642,\n 294.095771,\n 308.872014,\n]\ndf_ce_Cv = [\n 0.103048,\n 0.467344,\n 0.541364,\n 0.647315,\n 0.714078,\n 0.798258,\n 0.891147,\n 0.944848,\n 0.966618,\n 0.985486,\n]\n```\n\n\n```python\nHeatCapacity = constant_volume_heat_capacity\n\nR = const.N_A * const.KB_J_PER_K\nplt.plot(temps, HeatCapacity(eq, temps) / R, \"-k\", label=\"Cv_rot Equilibrium\")\nplt.plot(temps, HeatCapacity(para, temps) / R, \"-b\", label=\"Cv_rot Para\")\nplt.plot(temps, HeatCapacity(ortho, temps) / R, \"-r\", label=\"Cv_rot Ortho\")\nplt.plot(\n temps,\n 0.25 * HeatCapacity(para, temps) / R + 0.75 * HeatCapacity(ortho, temps) / R,\n \"-g\",\n label=\"Cv_rot 1:3 para:ortho\",\n)\nplt.plot(df_brink_T, df_brink_Cv, \"+g\")\nplt.plot(df_eucken_T, df_eucken_Cv, \"+g\")\nplt.plot(df_gia_T, df_gia_Cv, \"+g\")\nplt.plot(df_parting_T, df_parting_Cv, \"+g\")\nplt.plot(df_ce_T, df_ce_Cv, \"+g\", label=\"experimental data\")\nplt.legend(loc=\"upper right\", frameon=False)\nplt.xlim(10, 400)\nplt.ylim(-0.1, 2.8)\nplt.xlabel(\"Temperature, K\")\nplt.ylabel(\"Cv (rotational)/R\")\nplt.tight_layout()\nplt.show()\n```\n\n\n```python\nimport qiskit.tools.jupyter\n\n%qiskit_version_table\n%qiskit_copyright\n```\n\n\n

                                        Version Information

                                        Qiskit SoftwareVersion
                                        qiskit-terra0.20.1
                                        qiskit-aer0.10.4
                                        qiskit-ignis0.7.0
                                        qiskit-ibmq-provider0.19.0
                                        qiskit-nature0.4.0
                                        System information
                                        Python version3.8.12
                                        Python compilerClang 10.0.0
                                        Python builddefault, Oct 12 2021 06:23:56
                                        OSDarwin
                                        CPUs8
                                        Memory (Gb)64.0
                                        Thu Jun 09 09:01:03 2022 CEST
                                        \n\n\n\n

                                        This code is a part of Qiskit

                                        © Copyright IBM 2017, 2022.

                                        This code is licensed under the Apache License, Version 2.0. You may
                                        obtain a copy of this license in the LICENSE.txt file in the root directory
                                        of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.

                                        Any modifications or derivative works of this code must retain this
                                        copyright notice, and modified files need to carry a notice indicating
                                        that they have been altered from the originals.

                                        \n\n", "meta": {"hexsha": "567dc8a54c5958637dded13287543c629ab4ce6a", "size": 114351, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/tutorials/06_calculating_thermodynamic_observables.ipynb", "max_stars_repo_name": "kevinsung/qiskit-nature", "max_stars_repo_head_hexsha": "407533e05ca33fa53eb4e9cd7b089a0a99f9540e", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/tutorials/06_calculating_thermodynamic_observables.ipynb", "max_issues_repo_name": "kevinsung/qiskit-nature", "max_issues_repo_head_hexsha": "407533e05ca33fa53eb4e9cd7b089a0a99f9540e", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/tutorials/06_calculating_thermodynamic_observables.ipynb", "max_forks_repo_name": "kevinsung/qiskit-nature", "max_forks_repo_head_hexsha": "407533e05ca33fa53eb4e9cd7b089a0a99f9540e", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 201.6772486772, "max_line_length": 33320, "alphanum_fraction": 0.9041197716, "converted": true, "num_tokens": 3512, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929002541068, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.4234295066831888}} {"text": "```python\nfrom sympy.matrices import Matrix \nimport sympy as sp\nimport numpy as np\nfrom Exercise import Exercise, MarkdownBlock\n\nfrom process_latex import process_sympy \n\ntry:\n from config import URL, TOKEN\nexcept:\n None\n\n# TODO: replace with supplied strings\nExercise.URL = URL\nExercise.TOKEN = TOKEN\n```\n\n\n```python\n\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}4\\\\6\\end{matrix}\\right]$\n\n\n\n## Introduction\nIn this notebook, you are about to create some (linear algebra) exercises using the developed `Exercise` Python library aiming to facilitate authoring parameterized mathematics exercises at a high level of abstraction (i.e. access to a scripting language and the libraries available in there, including as SymPy, NumPy and Matplotlib).\nCreated exercises can be 'played' inline, using the web-based player developed as part of this project.\nRoughly speaking this project is new combination of existing approaches: MEGUA-like parameterized text, SymPy's CAS functionality and exercise-setup as used by Grasple and SageMath for working with mathematical objects in notebooks.\n\nThe goal is to evaluate the usability of the developed library and the authoring setup (this notebook).\nNote that by no means you or your skills are being tested, it is by no means a problem if exercises are left uncompleted.\nNotes, comments and suggestions are very welcome, please write these either as code-comments or in the Markdown cells in the notebook.\nAll feedback will be reported and reflected upon anonymously.\nCompleting the notebook should take about 30 minutes, depending on setup time, prior knowledge about this project, familiarity with linear algebra and the supplied frameworks etc.\nPlease download the notebook when done and send it by email.\nAfter completion, in a brief semi-structured interview, you can further elaborate upon your experiences.\n\nTo start creating exercises, please replace the `URL` and `TOKEN` in the block above with the strings supplied by email:\n```\nExercise.URL = \"\"\nExercise.TOKEN = \"\"\n```\n\nAssumptions:\n- Familiarity with Python, Markdown, LaTeX\n- Familiarity with Jupyter-Notebook\n- Familiarity with the very basics of linear algebra\n\nRecommendations:\n- Use Binder (www.mybinder.org) to edit this notebook, if you prefer local setup instead, see README.md.\n- Use Firefox, the iFrame exercise player embeddings do not work in Chrome or Safari due to global cross-origin policies set by these browsers.\n- Other browsers (Chrome, Safari) can still be used, however, playing exercises is only possible outside of the notebook by clicking the generated exercise links, which is rather inconvenient.\n\nNotes:\n- Documentation can for the Python library can be found in the `html` directory.\n- Within Jupyter-Notebook, function documentation can be viewed by writing a `?` after the function, like so: `Exercise(\"What is $1 + 1$?\").add_answer?`\n- Within exercises, only inline math notation is supported.\n- Preview-exercises are purged from the server from time to time, don't expect long-term, persistent availability of any played exercises.\n- Please skip an exercise in case completing it requires more than a few minutes.\n\nHappy coding ;)\n\n## Exercise Basics\nThe most basic exercise contains a Markdown string with the exercise content and a single answer rule specifying the correct answer.\nMathematics notation can be written inline in LaTeX between dollar signs.\n\n\n```python\n# Create an exercise instance\ne = Exercise(\"What is $1 + 1$?\")\n# Add 2 as a correct answer\ne.add_answer(2, True, \"Correct!\")\n# Verify that the exercise is working correctly\ne.play()\n# Note: as of now, all basic arithmatic is simplified by sp.simplify(...), there is not yet a way to control this behaviour;\n# therefore writing 1 + 1 in the answer box is accepted correct\n# Details on what is simplified: https://docs.sympy.org/latest/tutorial/simplification.html\n```\n\n\n\n\n\n\n\n Published succesfully, preview at: https://www.mscthesis.nl/preview?id=1f62594b-70fc-4cd7-ad2b-b6f50c01c85c\n\n\nLet's imagine the typical student mistake for this exercise is computing $1 - 1 = 0$ instead.\nWe add an answer rule to catch that error and provide the student with answer-specific feedback.\n\n\n```python\ne.add_answer(0, False, \"\ud83e\udd14 That's not right, did you compute $1 - 1 = 0$ instead?\")\n# Verify that the specific feedback is shown\ne.play()\n```\n\n\n\n\n\n\n\n Published succesfully, preview at: https://www.mscthesis.nl/preview?id=59116756-efb1-4e2b-a102-b4494b2b0120\n\n\n### Task 1\nCreate an exercise asking learners to compute $3/3$.\nProvide answer-specific feedback in case learners compute $3*3$ instead.\nAdd default feedback (using `e.add_default_feedback(...)`) with a link pointing to a source of preference explaining (integer) devision (hint: `[link](www.example.com)`).\nFeel free to embed your favorite meme or xkcd at a correct/incorrect answer (hint ``).\n\n\n```python\n# Task 1 user code:\n```\n\n## Templating Exercises\nExercises can be parameterized/templated (still looking for the correct terminology on this one), this allows for two things:\n1. Randomization. By making part of the content random, multiple instances can be generated, allowing for repeated practice.\n2. Abstraction. By utilizing the functionality of SymPy objects to be translated to LaTeX, authoring exercises remains efficient and effective.\n\nThe integer-exercise can be randomized as follows:\n\n\n```python\nstring = \"\"\"\n### Integer addition\n\nPlease compute $@a + @b$\n\"\"\"\n\nparams = {}\n# avoid 0 + 0 instance, since 0 + 0 == 0 - 0, answer same in case our typical mistake is made\nparams[\"a\"] = np.random.randint(0, 10)\nparams[\"b\"] = np.random.randint(1, 10)\nparams[\"ans_correct\"] = params[\"a\"] + params[\"b\"]\nparams[\"ans_incorrect\"] = params[\"a\"] - params[\"b\"]\n\ne = Exercise(MarkdownBlock(string, params))\ne.add_answer(params[\"ans_correct\"], True, \"Correct!\")\ne.add_answer(params[\"ans_incorrect\"], False, MarkdownBlock(\"Did you compute $@a - @b = @ans_incorrect$ instead?\", params))\n\ne.play()\n```\n\n\n\n\n\n\n\n Published succesfully, preview at: https://www.mscthesis.nl/preview?id=aec6d9f1-b0d1-44d8-98fe-5d095810c2b5\n\n\n\n```python\ns = \"\"\"\nWhat is $@a^\\intercal$?\n\"\"\"\n\nparams = {}\nparams[\"a\"] = sp.Matrix([[1, 2], [3, 4]])\nparams[\"ans\"] = params[\"a\"].T\ne = Exercise(MarkdownBlock(s, params))\ne.add_answer(params[\"ans\"], True, \"You are right!\")\ne.write(\"demo_transpose\")\n# e.play()\n\n\n\n\n\n\n\n```\n\n\n```python\ns = \"What is $@a^\\intercal$?\"\n\nparams = {}\nparams[\"a\"] = sp.Matrix([[1, 2], [3, 4]])\nparams[\"ans\"] = params[\"a\"].T\n\ne = Exercise(MarkdownBlock(s, params))\ne.add_answer(params[\"ans\"], True, \"You are right!\")\ne.play()\n```\n\n\n\n\n\n\n\n Published succesfully, preview at: https://www.mscthesis.nl/preview?id=aaab78fc-982f-4a11-833c-b86622e8f67d\n\n\nCurrently, only a single instance is generated played at a time. Support for multi-instance generation is planned. \n\n### Working with SymPy objects to represent mathematical objects\nWe can work with SymPy objects to represent mathematical objects, like vectors and matrices.\nAn vector addition exercise can be created as follows:\n\n\n```python\nstring = \"What is $@v_1 + @v_2$?\"\n\nparams[\"v_1\"] = sp.Matrix([1, 2, 3])\nparams[\"v_2\"] = sp.Matrix([4, 5, 6])\nparams[\"ans\"] = params[\"v_1\"] + params[\"v_2\"]\n\ne = Exercise(MarkdownBlock(string, params))\ne.add_answer(params[\"ans\"], True, \"That's right!\")\n\ne.play()\n```\n\n### Task 2 Parameterized vector addition\nCreate an exercise asking learners to compute the sum of two vectors of random length (within reasonable limits), with random integer values. \nNote: if you prefer NumPy for working with matrices, you are in luck! NumPy objects can be passed to the SymPy matrix constructor, e.g. `sp.Matrix(np.arange(4))`.\n\n\n```python\n# Task 2 user code:\n```\n\n### Task 3 - Matrix indexing \nCreate an exercise asking learners to identify a value at randomized indices (but within bounds) in a 5 by 5 matrix.\nPlease make sure all values are unique so there is only one correct answer.\n\n\n```python\n# Task 3 user code:\n```\n\n### Task 4 - Matrix multiplication \nCreate an exercise asking users to multiply two matrices. \nProvide a default answer explaining the procedure in case a wrong answer is supplied.\nYou can use the `symbolic_matrix` and `explain_multiply` functions supplied in `helpers.py` as follows:\n\n\n```python\nfrom helpers import symbolic_matrix, explain_multiply\na = symbolic_matrix(\"a\", 2, 2)\nb = symbolic_matrix(\"b\", 2, 2)\ndisplay(explain_multiply(a, b))\n\na = sp.Matrix([1,2,3])\nb = sp.Matrix(np.matrix([5,6,7]).reshape(-1))\ndisplay(explain_multiply(a, b))\n```\n\n\n$\\displaystyle \\left[\\begin{matrix}{a}_{1, 1} \\cdot {b}_{1, 1} + {a}_{1, 2} \\cdot {b}_{2, 1} & {a}_{1, 1} \\cdot {b}_{1, 2} + {a}_{1, 2} \\cdot {b}_{2, 2}\\\\{a}_{2, 1} \\cdot {b}_{1, 1} + {a}_{2, 2} \\cdot {b}_{2, 1} & {a}_{2, 1} \\cdot {b}_{1, 2} + {a}_{2, 2} \\cdot {b}_{2, 2}\\end{matrix}\\right]$\n\n\n\n$\\displaystyle \\left[\\begin{matrix}1 \\cdot 5 & 1 \\cdot 6 & 1 \\cdot 7\\\\2 \\cdot 5 & 2 \\cdot 6 & 2 \\cdot 7\\\\3 \\cdot 5 & 3 \\cdot 6 & 3 \\cdot 7\\end{matrix}\\right]$\n\n\n\n```python\n# Task 4 user code:\n```\n\nHooray!\nIf you made it this far, you completed the notebook! \nPlease add any additonal comments below.\nThank you for participating!\n\nWrite any additional comments here...\n", "meta": {"hexsha": "81c5c4b7f78c3557eac4d7e65c77d50850679e46", "size": 19917, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "usability_evaluation.ipynb", "max_stars_repo_name": "rkeulemans/exercise_public", "max_stars_repo_head_hexsha": "5f8020198b8b234169eea4d5e08c98344438de5d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "usability_evaluation.ipynb", "max_issues_repo_name": "rkeulemans/exercise_public", "max_issues_repo_head_hexsha": "5f8020198b8b234169eea4d5e08c98344438de5d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "usability_evaluation.ipynb", "max_forks_repo_name": "rkeulemans/exercise_public", "max_forks_repo_head_hexsha": "5f8020198b8b234169eea4d5e08c98344438de5d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 35.3765541741, "max_line_length": 1712, "alphanum_fraction": 0.5677059798, "converted": true, "num_tokens": 2393, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.6723316991792861, "lm_q1q2_score": 0.4234174318961771}} {"text": "\n\n---\n## Archivos `.dat` + `ReadMe` \n\n\nEduard Larra\u00f1aga (ealarranaga@unal.edu.co)\n\n---\n\n### Resumen\n\nEn este cuaderno se utiliza la librer\u00eda `astropy.io` para leer una arcivo de datos de texto plano con extensi\u00f3n .dat (formato ascii) junto con un archivo adicional ReadMe que contiene los metadatos. \n\n---\n\n## 1. Obtenci\u00f3n y Lectura de los Datos\n\nConsideraremos el conjunto de datos reportado Greene and Ho [2006], el cual contiene informaci\u00f3n de 88 galaxias cercanas. \n\nGreene, J. E. and Ho, L. C. *The MBH \u2212 \u03c3\u2217 Relation in Local Active Galaxies*. ApJ 641 L21 (2006)\nhttps://ui.adsabs.harvard.edu/abs/2006ApJ...641L..21G/abstract\n\nEl conjunto de datos est\u00e1 disponibele, en varios formatos en\n\nhttp://vizier.cfa.harvard.edu/viz-bin/VizieR?-source=J/ApJ/641/L21.\n\nAl hacer click en el link ReadMe+ftp link se puede acceder a los datos via FTP.\n\n---\n\n### Abrir los archivos .dat+ReadMe.\n\nPara acceder a la informaci\u00f3n del archivo plano .dat file (en formato ascii) y los metadatos en el archivo ReadMe se utilizar\u00e1 el comando [astropy.io.ascii](https://docs.astropy.org/en/stable/io/ascii/)\n \n\n\n```python\nimport numpy as np\nfrom astropy.io import ascii\nimport matplotlib.pyplot as plt\n\n\n\ndata = ascii.read('table1.dat', readme='ReadMe')\ndata\n```\n\n WARNING: UnitsWarning: '[10-7W]' did not parse as cds unit: Syntax error If this is meant to be a custom unit, define it with 'u.def_unit'. To have it recognized inside a file reader or other code, enable it with 'u.add_enabled_units'. For details, see https://docs.astropy.org/en/latest/units/combining_and_defining.html [astropy.units.core]\n\n\n\n\n\nTable length=88\n\n\n\n\n\n\n\n\n
                                        Namezsigma*e_sigma*n_sigma*FWHMe_FWHMlogLe_logLlogME_logMe_logM
                                        km / skm / skm / skm / s[10-7W][10-7W]dex(Msun)dex(Msun)dex(Msun)
                                        str24float64float64float64str1int64int64float64float64float64float64float64
                                        SDSS J000805.62+145023.40.0454140.027.0--761038041.130.047.7--0.1
                                        ....................................
                                        POX 520.021836.05.0----------5.2--0.3
                                        NGC 43950.00094730.05.0----------4.9--0.4
                                        \n\n\n\nLa longitud de la tabla es de 88 muestras y 12 caracter\u00edsticas, incluyendo 'Name', 'z', 'sigma*', etc.\n\n---\nEs posible acceder a cada una de las caracter\u00edsticas,\n\n\n```python\ndata['z']\n```\n\n\n\n\n<Column name='z' dtype='float64' description='Redshift' length=88>\n\n\n\n\n\n\n\n\n\n\n
                                        0.0454
                                        0.0419
                                        0.0456
                                        0.0772
                                        ...
                                        0.0172
                                        0.0163
                                        0.0218
                                        0.000947
                                        \n\n\n\n\n```python\ndata['logL']\n```\n\n\n\n\n<MaskedColumn name='logL' dtype='float64' unit='[10-7W]' description='Log of H{alpha} luminosity in erg/s' length=88>\n\n\n\n\n\n\n\n\n\n\n
                                        41.13
                                        41.58
                                        41.45
                                        41.13
                                        ...
                                        --
                                        --
                                        --
                                        --
                                        \n\n\n\nTambi\u00e9n se puede acceder a los datos completos de una muestra particular o un conjunto de muestras,\n\n\n```python\ndata[1]\n```\n\n\n\n\nRow index=1\n\n\n\n\n\n
                                        Namezsigma*e_sigma*n_sigma*FWHMe_FWHMlogLe_logLlogME_logMe_logM
                                        km / skm / skm / skm / s[10-7W][10-7W]dex(Msun)dex(Msun)dex(Msun)
                                        str24float64float64float64str1int64int64float64float64float64float64float64
                                        SDSS J004236.86-104921.80.041978.410.0--19609741.580.146.7--0.1
                                        \n\n\n\n\n```python\ndata[[0,1,3,4]]\n```\n\n\n\n\nTable length=4\n\n\n\n\n\n\n\n\n
                                        Namezsigma*e_sigma*n_sigma*FWHMe_FWHMlogLe_logLlogME_logMe_logM
                                        km / skm / skm / skm / s[10-7W][10-7W]dex(Msun)dex(Msun)dex(Msun)
                                        str24float64float64float64str1int64int64float64float64float64float64float64
                                        SDSS J000805.62+145023.40.0454140.027.0--761038041.130.047.7--0.1
                                        SDSS J004236.86-104921.80.041978.410.0--19609741.580.146.7--0.1
                                        SDSS J020459.25-080816.00.0772121.09.4a372018041.130.057.0--0.1
                                        SDSS J020615.99-001729.10.0426216.030.0--386019041.910.077.5--0.1
                                        \n\n\n\nDistribuiremos la informaci\u00f3n en un conjunto de arreglos de numpy,\n\n\n```python\nz = np.array(data[\"z\"])\nsigma_star = np.array(data[\"sigma*\"])\ne_sigma_star = np.array(data[\"e_sigma*\"])\nlogL = np.array(data[\"logL\"])\ne_logL = np.array(data[\"e_logL\"])\nlogM = np.array(data[\"logM\"])\ne_logM = np.array(data[\"e_logM\"])\n```\n\n## 2. Visualizar la informaci\u00f3n\n\nVeamos diferentes combinaciones de caracter\u00edsticas, buscando alguna clase de patr\u00f3n de inter\u00e9s,\n\n\n```python\nplt.scatter(z, logM, color='red')\nplt.xlabel(r'$z$')\nplt.ylabel(r'$\\log M$')\nplt.show()\n```\n\n\n```python\nplt.figure(figsize=(10, 10))\n\nplt.subplot(221)\nplt.plot(z, logM, 'b.')\nplt.xlabel(r'$z$')\nplt.ylabel(r'$\\log M$')\n\nplt.subplot(222)\nplt.plot(logL, logM, 'r.')\nplt.xlabel(r'$\\log L$')\nplt.ylabel(r'$\\log M$')\n\nplt.subplot(223)\nplt.plot(z, logL, 'g.')\nplt.xlabel(r'$z$')\nplt.ylabel(r'$\\log L$')\n\nplt.subplot(224)\nplt.plot(sigma_star, logM, 'k.')\nplt.xlabel(r'$\\sigma_{*}$')\nplt.ylabel(r'$\\log M$')\n\nplt.show()\n```\n\nAparecen alguna informaci\u00f3n o patrones intersantes. Por ejemplo,\n\n- Se obsrevan dos puntos separados (outcaast) en el gr\u00e1fico de $\\log M$ vs $z$.\n- Parece que existen dos c\u00famulos separados en el gr\u00e1fico de $\\log M$ vs $\\log L$\n- Parece que existen tres c\u00famulos separados (o dos cumulos y dos puntos separados) en el gr\u00e1fico de $\\log L$ vs $z$\n- Existe una tendencia de correlaci\u00f3n en el gr\u00e1fico de $\\log M$ vs $\\sigma_{*}$\n\nA pesar de que estos comportamientos parecen existir, algunos de ellos deben revisarse con cuidado. Por ejemplo, los aparentes c\u00famulos en las figuras 2 y 3 corresponden a una ausencia de datos, mal interpreata al convertir la informaci\u00f3n a arreglos. Al revisar la informaci\u00f3n en $\\log L$ se tiene\n\n\n```python\nlogL\n```\n\n\n\n\n array([41.13, 41.58, 41.45, 41.13, 41.91, 41.24, 41.58, 40.45, 41.63,\n 41.67, 40.14, 40.42, 41.17, 41.31, 41.27, 40.72, 41.14, 40.8 ,\n 41.46, 41.74, 41.51, 41.65, 41.17, 41.66, 40.1 , 41.57, 41.3 ,\n 40.92, 41.62, 41.86, 40.58, 40.62, 41.81, 41.55, 41.62, 41.23,\n 41.29, 41.1 , 41.84, 40.73, 40.46, 41.18, 42.09, 41.4 , 41.51,\n 41.41, 41.86, 41.09, 40.99, 41.92, 41.24, 41.41, 41.83, 41.21,\n 41.65, 41.87, 42.57, 41.93, 42.03, 40.47, 42.14, 42.68, 42.57,\n 42.35, 42.75, 41.87, 42.67, 42.19, 41.22, 42.96, 43.61, 0. ,\n 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n 0. , 0. , 0. , 0. , 0. , 0. , 0. ])\n\n\n\ndonde se observa una gran cantidad de valores $0.$ que en realidad correspondian a datos ausentes en el conjunto inicial. Estos valores de cero fueron incorporados al utilizar el comando `logL = np.array(data[\"logL\"])`. \n\nNOTA: Es importante manjear adecuadamente la ausencia de datos!\n\n---\nPor otra parte, podemos analizar con mayor detalle la aparente correlaci\u00f3n entre $\\log M$ y $\\sigma_{*}$\n\n\n```python\nplt.plot(sigma_star, logM, 'k.')\nplt.xlabel(r'$\\sigma_{*}$')\nplt.ylabel(r'$\\log M$')\nplt.show()\n```\n\nPuede parecer una dependencia lineal entre las dos variables. Sin embargo, para tener una mejor apreciaci\u00f3n, incluiremos las barras de error. Para ello utilizamos la funci\u00f3n [matplotlib.pyplot.errorbar](https://matplotlib.org/3.2.1/api/_as_gen/matplotlib.pyplot.errorbar.html)\n\n\n```python\nplt.errorbar(sigma_star, logM, e_logM, e_sigma_star, fmt='k.')\nplt.xlabel(r'$\\sigma_{*}$')\nplt.ylabel(r'$\\log M$')\nplt.show()\n```\n\nEsta visualizaci\u00f3n miestra una correlaci\u00f3n no-lineal. Para comprobarlo, podemos modificar la gr\u00e1fica incluyendo el logaritmo de la dispersion de velocidades, $\\log M$ vs $\\log \\sigma_{*}$,con lo cual la tendencia lineal es evidente,\n\n\n```python\nplt.plot(np.log10(sigma_star), logM, 'k.')\nplt.xlabel(r'$\\log \\sigma_{*}$')\nplt.ylabel(r'$\\log M$')\nplt.show()\n```\n\nPara confirmar esta apreciaci\u00f3n, incluimos nuevamente las incertidumbres asociadas. Para ello es necesario recordar la propagaci\u00f3n de errores,\n\n\\begin{align}\nf(x) =& \\log_{10} x = \\frac{\\ln x}{\\ln 10}\\\\\n\\Delta f = &\\frac{1}{\\ln 10} \\frac{\\Delta x}{x}\n\\end{align}\n\n\n```python\n\ne_log_sigma = e_sigma_star/(np.log(10.)*sigma_star)\n\n\nplt.errorbar(np.log10(sigma_star), logM, e_logM, e_log_sigma, fmt='k.')\nplt.xlabel(r'$\\log \\sigma_{*}$')\nplt.ylabel(r'$\\log M$')\nplt.show()\n```\n\nLuego volveremos sobre este conjunto de datos para analizar mejor esta relaci\u00f3n.\n\n\n```python\n\n```\n", "meta": {"hexsha": "ec4eb3921b5ef2b88fb6568b5a742a72c6bca9e7", "size": 100543, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "15. Astrophysics Data/02. Archivos DAT+ReadMe/DATFiles.ipynb", "max_stars_repo_name": "ashcat2005/AstrofisicaComputacional2022", "max_stars_repo_head_hexsha": "67463ec4041eb08c0f326792fed0dcf9e970e9b7", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2022-03-08T06:18:56.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-10T04:55:53.000Z", "max_issues_repo_path": "15. Astrophysics Data/02. Archivos DAT+ReadMe/DATFiles.ipynb", "max_issues_repo_name": "ashcat2005/AstrofisicaComputacional2022", "max_issues_repo_head_hexsha": "67463ec4041eb08c0f326792fed0dcf9e970e9b7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "15. Astrophysics Data/02. Archivos DAT+ReadMe/DATFiles.ipynb", "max_forks_repo_name": "ashcat2005/AstrofisicaComputacional2022", "max_forks_repo_head_hexsha": "67463ec4041eb08c0f326792fed0dcf9e970e9b7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2022-03-09T17:47:43.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-21T02:29:36.000Z", "avg_line_length": 141.0140252454, "max_line_length": 28180, "alphanum_fraction": 0.8614722059, "converted": true, "num_tokens": 4144, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.672331699179286, "lm_q1q2_score": 0.42341743189617703}} {"text": "   \n\n# Tutorial 3: Learning to Act: Q-Learning\n**Week 3, Day 4: Reinforcement Learning**\n\n**By Neuromatch Academy**\n\n__Content creators:__ Marcelo Mattar and Eric DeWitt with help from Byron Galbraith\n\n__Content reviewers:__ Ella Batty, Matt Krause and Michael Waskom\n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n

                                        \n\n---\n\n# Tutorial Objectives\n \n*Estimated timing of tutorial: 40 min*\n\nIn this tutorial you will learn how to act in the more realistic setting of sequential decisions, formalized by Markov Decision Processes (MDPs). In a sequential decision problem, the actions executed in one state not only may lead to immediate rewards (as in a bandit problem), but may also affect the states experienced next (unlike a bandit problem). Each individual action may therefore affect affect all future rewards. Thus, making decisions in this setting requires considering each action in terms of their expected **cumulative** future reward.\n\nWe will consider here the example of spatial navigation, where actions (movements) in one state (location) affect the states experienced next, and an agent might need to execute a whole sequence of actions before a reward is obtained.\n\nBy the end of this tutorial, you will learn\n* what grid worlds are and how they help in evaluating simple reinforcement learning agents\n* the basics of the Q-learning algorithm for estimating action values\n* how the concept of exploration and exploitation, reviewed in the bandit case, also applies to the sequential decision setting\n\n\n```python\n# @title Tutorial slides\n\n# @markdown These are the slides for all videos in this tutorial.\nfrom IPython.display import IFrame\nIFrame(src=f\"https://mfr.ca-1.osf.io/render?url=https://osf.io/2jzdu/?direct%26mode=render%26action=download%26mode=render\", width=854, height=480)\n```\n\n---\n# Setup\n\n\n```python\n# Imports\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.signal import convolve as conv\n```\n\n\n```python\n#@title Figure Settings\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/course-content/master/nma.mplstyle\")\n```\n\n\n```python\n#@title Plotting Functions\n\ndef plot_state_action_values(env, value, ax=None):\n \"\"\"\n Generate plot showing value of each action at each state.\n \"\"\"\n if ax is None:\n fig, ax = plt.subplots()\n\n for a in range(env.n_actions):\n ax.plot(range(env.n_states), value[:, a], marker='o', linestyle='--')\n ax.set(xlabel='States', ylabel='Values')\n ax.legend(['R','U','L','D'], loc='lower right')\n\n\ndef plot_quiver_max_action(env, value, ax=None):\n \"\"\"\n Generate plot showing action of maximum value or maximum probability at\n each state (not for n-armed bandit or cheese_world).\n \"\"\"\n if ax is None:\n fig, ax = plt.subplots()\n\n X = np.tile(np.arange(env.dim_x), [env.dim_y,1]) + 0.5\n Y = np.tile(np.arange(env.dim_y)[::-1][:,np.newaxis], [1,env.dim_x]) + 0.5\n which_max = np.reshape(value.argmax(axis=1), (env.dim_y,env.dim_x))\n which_max = which_max[::-1,:]\n U = np.zeros(X.shape)\n V = np.zeros(X.shape)\n U[which_max == 0] = 1\n V[which_max == 1] = 1\n U[which_max == 2] = -1\n V[which_max == 3] = -1\n\n ax.quiver(X, Y, U, V)\n ax.set(\n title='Maximum value/probability actions',\n xlim=[-0.5, env.dim_x+0.5],\n ylim=[-0.5, env.dim_y+0.5],\n )\n ax.set_xticks(np.linspace(0.5, env.dim_x-0.5, num=env.dim_x))\n ax.set_xticklabels([\"%d\" % x for x in np.arange(env.dim_x)])\n ax.set_xticks(np.arange(env.dim_x+1), minor=True)\n ax.set_yticks(np.linspace(0.5, env.dim_y-0.5, num=env.dim_y))\n ax.set_yticklabels([\"%d\" % y for y in np.arange(0, env.dim_y*env.dim_x,\n env.dim_x)])\n ax.set_yticks(np.arange(env.dim_y+1), minor=True)\n ax.grid(which='minor',linestyle='-')\n\n\ndef plot_heatmap_max_val(env, value, ax=None):\n \"\"\"\n Generate heatmap showing maximum value at each state\n \"\"\"\n if ax is None:\n fig, ax = plt.subplots()\n\n if value.ndim == 1:\n value_max = np.reshape(value, (env.dim_y,env.dim_x))\n else:\n value_max = np.reshape(value.max(axis=1), (env.dim_y,env.dim_x))\n value_max = value_max[::-1,:]\n\n im = ax.imshow(value_max, aspect='auto', interpolation='none', cmap='afmhot')\n ax.set(title='Maximum value per state')\n ax.set_xticks(np.linspace(0, env.dim_x-1, num=env.dim_x))\n ax.set_xticklabels([\"%d\" % x for x in np.arange(env.dim_x)])\n ax.set_yticks(np.linspace(0, env.dim_y-1, num=env.dim_y))\n if env.name != 'windy_cliff_grid':\n ax.set_yticklabels(\n [\"%d\" % y for y in np.arange(\n 0, env.dim_y*env.dim_x, env.dim_x)][::-1])\n return im\n\n\ndef plot_rewards(n_episodes, rewards, average_range=10, ax=None):\n \"\"\"\n Generate plot showing total reward accumulated in each episode.\n \"\"\"\n if ax is None:\n fig, ax = plt.subplots()\n\n smoothed_rewards = (conv(rewards, np.ones(average_range), mode='same')\n / average_range)\n\n ax.plot(range(0, n_episodes, average_range),\n smoothed_rewards[0:n_episodes:average_range],\n marker='o', linestyle='--')\n ax.set(xlabel='Episodes', ylabel='Total reward')\n\n\ndef plot_performance(env, value, reward_sums):\n fig, axes = plt.subplots(nrows=2, ncols=2, figsize=(16, 12))\n plot_state_action_values(env, value, ax=axes[0,0])\n plot_quiver_max_action(env, value, ax=axes[0,1])\n plot_rewards(n_episodes, reward_sums, ax=axes[1,0])\n im = plot_heatmap_max_val(env, value, ax=axes[1,1])\n fig.colorbar(im)\n```\n\n---\n# Section 1: Markov Decision Processes\n\n\n```python\n# @title Video 1: MDPs and Q-learning\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = 'https://player.bilibili.com/player.html?bvid={0}&page={1}'.format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=\"BV1ft4y1Q7bX\", width=854, height=480, fs=1)\n print('Video available at https://www.bilibili.com/video/{0}'.format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=\"8yvwMrUQJOU\", width=854, height=480, fs=1, rel=0)\n print('Video available at https://youtube.com/watch?v=' + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n\n**Grid Worlds**\n\nAs pointed out, bandits only have a single state and immediate rewards for our actions. Many problems we are interested in have multiple states and delayed rewards, i.e. we won't know if the choices we made will pay off over time, or which actions we took contributed to the outcomes we observed.\n\nIn order to explore these ideas, we turn the a common problem setting: the grid world. Grid worlds are simple environments where each state corresponds to a tile on a 2D grid, and the only actions the agent can take are to move up, down, left, or right across the grid tiles. The agent's job is almost always to find a way to a goal tile in the most direct way possible while overcoming some maze or other obstacles, either static or dynamic.\n\nFor our discussion we will be looking at the classic Cliff World, or Cliff Walker, environment. This is a 4x10 grid with a starting position in the lower-left and the goal position in the lower-right. Every tile between these two is the \"cliff\", and should the agent enter the cliff, they will receive a -100 reward and be sent back to the starting position. Every tile other than the cliff produces a -1 reward when entered. The goal tile ends the episode after taking any action from it.\n\n\n\nGiven these conditions, the maximum achievable reward is -11 (1 up, 9 right, 1 down). Using negative rewards is a common technique to encourage the agent to move and seek out the goal state as fast as possible.\n\n---\n# Section 2: Q-Learning\n\n*Estimated timing to here from start of tutorial: 20 min*\n\nNow that we have our environment, how can we solve it? \n\nOne of the most famous algorithms for estimating action values (aka Q-values) is the Temporal Differences (TD) **control** algorithm known as *Q-learning* (Watkins, 1989). \n\n\\begin{align}\nQ(s_t,a_t) \\leftarrow Q(s_t,a_t) + \\alpha \\big(r_t + \\gamma\\max_\\limits{a} Q(s_{t+1},a_{t+1}) - Q(s_t,a_t)\\big)\n\\end{align}\n\nwhere $Q(s,a)$ is the value function for action $a$ at state $s$, $\\alpha$ is the learning rate, $r$ is the reward, and $\\gamma$ is the temporal discount rate.\n\nThe expression $r_t + \\gamma\\max_\\limits{a} Q(s_{t+1},a_{t+1})$ is referred to as the TD target while the full expression \n\\begin{align}\nr_t + \\gamma\\max_\\limits{a} Q(s_{t+1},a_{t+1}) - Q(s_t,a_t),\n\\end{align}\ni.e. the difference between the TD target and the current Q-value, is referred to as the TD error, or reward prediction error.\n\nBecause of the max operator used to select the optimal Q-value in the TD target, Q-learning directly estimates the optimal action value, i.e. the cumulative future reward that would be obtained if the agent behaved optimally, regardless of the policy currently followed by the agent. For this reason, Q-learning is referred to as an **off-policy** method.\n\n## Coding Exercise 2: Implement the Q-learning algorithm\n\nIn this exercise you will implement the Q-learning update rule described above. It takes in as arguments the previous state $s_t$, the action $a_t$ taken, the reward received $r_t$, the current state $s_{t+1}$, the Q-value table, and a dictionary of parameters that contain the learning rate $\\alpha$ and discount factor $\\gamma$. The method returns the updated Q-value table. For the parameter dictionary, $\\alpha$: `params['alpha']` and $\\gamma$: `params['gamma']`. \n\nOnce we have our Q-learning algorithm, we will see how it handles learning to solve the Cliff World environment. \n\nYou will recall from the previous tutorial that a major part of reinforcement learning algorithms are their ability to balance exploitation and exploration. For our Q-learning agent, we again turn to the epsilon-greedy strategy. At each step, the agent will decide with probability $1 - \\epsilon$ to use the best action for the state it is currently in by looking at the value function, otherwise just make a random choice.\n\nThe process by which our the agent will interact with and learn about the environment is handled for you in the helper function `learn_environment`. This implements the entire learning episode lifecycle of stepping through the state observation, action selection (epsilon-greedy) and execution, reward, and state transition. Feel free to review that code later to see how it all fits together, but for now let's test out our agent.\n\n\n```python\n# @markdown Execute to get helper functions `epsilon_greedy`, `CliffWorld`, and `learn_environment`\n\ndef epsilon_greedy(q, epsilon):\n \"\"\"Epsilon-greedy policy: selects the maximum value action with probabilty\n (1-epsilon) and selects randomly with epsilon probability.\n\n Args:\n q (ndarray): an array of action values\n epsilon (float): probability of selecting an action randomly\n\n Returns:\n int: the chosen action\n \"\"\"\n if np.random.random() > epsilon:\n action = np.argmax(q)\n else:\n action = np.random.choice(len(q))\n\n return action\n\n\nclass CliffWorld:\n \"\"\"\n World: Cliff world.\n 40 states (4-by-10 grid world).\n The mapping from state to the grids are as follows:\n 30 31 32 ... 39\n 20 21 22 ... 29\n 10 11 12 ... 19\n 0 1 2 ... 9\n 0 is the starting state (S) and 9 is the goal state (G).\n Actions 0, 1, 2, 3 correspond to right, up, left, down.\n Moving anywhere from state 9 (goal state) will end the session.\n Taking action down at state 11-18 will go back to state 0 and incur a\n reward of -100.\n Landing in any states other than the goal state will incur a reward of -1.\n Going towards the border when already at the border will stay in the same\n place.\n \"\"\"\n def __init__(self):\n self.name = \"cliff_world\"\n self.n_states = 40\n self.n_actions = 4\n self.dim_x = 10\n self.dim_y = 4\n self.init_state = 0\n\n def get_outcome(self, state, action):\n if state == 9: # goal state\n reward = 0\n next_state = None\n return next_state, reward\n reward = -1 # default reward value\n if action == 0: # move right\n next_state = state + 1\n if state % 10 == 9: # right border\n next_state = state\n elif state == 0: # start state (next state is cliff)\n next_state = None\n reward = -100\n elif action == 1: # move up\n next_state = state + 10\n if state >= 30: # top border\n next_state = state\n elif action == 2: # move left\n next_state = state - 1\n if state % 10 == 0: # left border\n next_state = state\n elif action == 3: # move down\n next_state = state - 10\n if state >= 11 and state <= 18: # next is cliff\n next_state = None\n reward = -100\n elif state <= 9: # bottom border\n next_state = state\n else:\n print(\"Action must be between 0 and 3.\")\n next_state = None\n reward = None\n return int(next_state) if next_state is not None else None, reward\n\n def get_all_outcomes(self):\n outcomes = {}\n for state in range(self.n_states):\n for action in range(self.n_actions):\n next_state, reward = self.get_outcome(state, action)\n outcomes[state, action] = [(1, next_state, reward)]\n return outcomes\n\n\ndef learn_environment(env, learning_rule, params, max_steps, n_episodes):\n # Start with a uniform value function\n value = np.ones((env.n_states, env.n_actions))\n\n # Run learning\n reward_sums = np.zeros(n_episodes)\n\n # Loop over episodes\n for episode in range(n_episodes):\n state = env.init_state # initialize state\n reward_sum = 0\n\n for t in range(max_steps):\n # choose next action\n action = epsilon_greedy(value[state], params['epsilon'])\n\n # observe outcome of action on environment\n next_state, reward = env.get_outcome(state, action)\n\n # update value function\n value = learning_rule(state, action, reward, next_state, value, params)\n\n # sum rewards obtained\n reward_sum += reward\n\n if next_state is None:\n break # episode ends\n state = next_state\n\n reward_sums[episode] = reward_sum\n\n return value, reward_sums\n```\n\n\n```python\ndef q_learning(state, action, reward, next_state, value, params):\n \"\"\"Q-learning: updates the value function and returns it.\n\n Args:\n state (int): the current state identifier\n action (int): the action taken\n reward (float): the reward received\n next_state (int): the transitioned to state identifier\n value (ndarray): current value function of shape (n_states, n_actions)\n params (dict): a dictionary containing the default parameters\n\n Returns:\n ndarray: the updated value function of shape (n_states, n_actions)\n \"\"\"\n # Q-value of current state-action pair\n q = value[state, action]\n\n ##########################################################\n ## TODO for students: implement the Q-learning update rule\n # Fill out function and remove\n raise NotImplementedError(\"Student excercise: implement the Q-learning update rule\")\n ##########################################################\n\n # write an expression for finding the maximum Q-value at the current state\n if next_state is None:\n max_next_q = 0\n else:\n max_next_q = ...\n\n # write the expression to compute the TD error\n td_error = ...\n # write the expression that updates the Q-value for the state-action pair\n value[state, action] = ...\n\n return value\n\n\n# set for reproducibility, comment out / change seed value for different results\nnp.random.seed(1)\n\n# parameters needed by our policy and learning rule\nparams = {\n 'epsilon': 0.1, # epsilon-greedy policy\n 'alpha': 0.1, # learning rate\n 'gamma': 1.0, # discount factor\n}\n\n# episodes/trials\nn_episodes = 500\nmax_steps = 1000\n\n# environment initialization\nenv = CliffWorld()\n\n# solve Cliff World using Q-learning\nresults = learn_environment(env, q_learning, params, max_steps, n_episodes)\nvalue_qlearning, reward_sums_qlearning = results\n\n# Plot results\nplot_performance(env, value_qlearning, reward_sums_qlearning)\n```\n\n[*Click for solution*](https://github.com/NeuromatchAcademy/course-content/tree/master//tutorials/W3D4_ReinforcementLearning/solutions/W3D4_Tutorial3_Solution_7a5ca920.py)\n\n*Example output:*\n\n\n\n\n\nIf all went well, we should see four plots that show different aspects on our agent's learning and progress.\n\n* The top left is a representation of the Q-table itself, showing the values for different actions in different states. Notably, going right from the starting state or down when above the cliff is clearly very bad.\n* The top right figure shows the greedy policy based on the Q-table, i.e. what action would the agent take if it only took its best guess in that state.\n* The bottom right is the same as the top, only instead of showing the action, it's showing a representation of the maximum Q-value at a particular state.\n* The bottom left is the actual proof of learning, as we see the total reward steadily increasing after each episode until asymptoting at the maximum possible reward of -11.\n\nFeel free to try changing the parameters or random seed and see how the agent's behavior changes.\n\n---\n# Summary\n\n*Estimated timing of tutorial: 40 min*\n\nIn this tutorial you implemented a reinforcement learning agent based on Q-learning to solve the Cliff World environment. Q-learning combined the epsilon-greedy approach to exploration-expoitation with a table-based value function to learn the expected future rewards for each state.\n\n---\n# Bonus\n\n---\n## Bonus Section 1: SARSA\n\nAn alternative to Q-learning, the SARSA algorithm also estimates action values. However, rather than estimating the optimal (off-policy) values, SARSA estimates the **on-policy** action value, i.e. the cumulative future reward that would be obtained if the agent behaved according to its current beliefs.\n\n\\begin{align}\nQ(s_t,a_t) \\leftarrow Q(s_t,a_t) + \\alpha \\big(r_t + \\gamma Q(s_{t+1},a_{t+1}) - Q(s_t,a_t)\\big)\n\\end{align}\n\nwhere, once again, $Q(s,a)$ is the value function for action $a$ at state $s$, $\\alpha$ is the learning rate, $r$ is the reward, and $\\gamma$ is the temporal discount rate.\n\nIn fact, you will notices that the *only* difference between Q-learning and SARSA is the TD target calculation uses the policy to select the next action (in our case epsilon-greedy) rather than using the action that maximizes the Q-value.\n\n### Bonus Coding Exercise 1: Implement the SARSA algorithm\n\nIn this exercise you will implement the SARSA update rule described above. Just like Q-learning, it takes in as arguments the previous state $s_t$, the action $a_t$ taken, the reward received $r_t$, the current state $s_{t+1}$, the Q-value table, and a dictionary of parameters that contain the learning rate $\\alpha$ and discount factor $\\gamma$. The method returns the updated Q-value table. You may use the `epsilon_greedy` function to acquire the next action. For the parameter dictionary, $\\alpha$: `params['alpha']`, $\\gamma$: `params['gamma']`, and $\\epsilon$: `params['epsilon']`. \n\nOnce we have an implementation for SARSA, we will see how it tackles Cliff World. We will again use the same setup we tried with Q-learning.\n\n\n```python\ndef sarsa(state, action, reward, next_state, value, params):\n \"\"\"SARSA: updates the value function and returns it.\n\n Args:\n state (int): the current state identifier\n action (int): the action taken\n reward (float): the reward received\n next_state (int): the transitioned to state identifier\n value (ndarray): current value function of shape (n_states, n_actions)\n params (dict): a dictionary containing the default parameters\n\n Returns:\n ndarray: the updated value function of shape (n_states, n_actions)\n \"\"\"\n # value of previous state-action pair\n q = value[state, action]\n\n ##########################################################\n ## TODO for students: implement the SARSA update rule\n # Fill out function and remove\n raise NotImplementedError(\"Student excercise: implement the SARSA update rule\")\n ##########################################################\n\n # select the expected value at current state based on our policy by sampling\n # from it\n if next_state is None:\n policy_next_q = 0\n else:\n # write an expression for selecting an action using epsilon-greedy\n policy_action = ...\n # write an expression for obtaining the value of the policy action at the\n # current state\n policy_next_q = ...\n\n # write the expression to compute the TD error\n td_error = ...\n # write the expression that updates the Q-value for the state-action pair\n value[state, action] = ...\n\n return value\n\n\n# set for reproducibility, comment out / change seed value for different results\nnp.random.seed(1)\n\n# parameters needed by our policy and learning rule\nparams = {\n 'epsilon': 0.1, # epsilon-greedy policy\n 'alpha': 0.1, # learning rate\n 'gamma': 1.0, # discount factor\n}\n\n# episodes/trials\nn_episodes = 500\nmax_steps = 1000\n\n# environment initialization\nenv = CliffWorld()\n\n# learn Cliff World using Sarsa\nresults = learn_environment(env, sarsa, params, max_steps, n_episodes)\nvalue_sarsa, reward_sums_sarsa = results\n\n# Plot results\nplot_performance(env, value_sarsa, reward_sums_sarsa)\n```\n\n\n```python\ndef sarsa(state, action, reward, next_state, value, params):\n \"\"\"SARSA: updates the value function and returns it.\n\n Args:\n state (int): the current state identifier\n action (int): the action taken\n reward (float): the reward received\n next_state (int): the transitioned to state identifier\n value (ndarray): current value function of shape (n_states, n_actions)\n params (dict): a dictionary containing the default parameters\n\n Returns:\n ndarray: the updated value function of shape (n_states, n_actions)\n \"\"\"\n # value of previous state-action pair\n q = value[state, action]\n\n # select the expected value at current state based on our policy by sampling\n # from it\n if next_state is None:\n policy_next_q = 0\n else:\n # write an expression for selecting an action using epsilon-greedy\n policy_action = epsilon_greedy(value[next_state], params['epsilon'])\n # write an expression for obtaining the value of the policy action at the\n # current state\n policy_next_q = value[next_state, policy_action]\n\n # write the expression to compute the TD error\n td_error = reward + params['gamma'] * policy_next_q - q\n # write the expression that updates the Q-value for the state-action pair\n value[state, action] = q + params['alpha'] * td_error\n\n return value\n\n\n# set for reproducibility, comment out / change seed value for different results\nnp.random.seed(1)\n\n# parameters needed by our policy and learning rule\nparams = {\n 'epsilon': 0.1, # epsilon-greedy policy\n 'alpha': 0.1, # learning rate\n 'gamma': 1.0, # discount factor\n}\n\n# episodes/trials\nn_episodes = 500\nmax_steps = 1000\n\n# environment initialization\nenv = CliffWorld()\n\n# learn Cliff World using Sarsa\nresults = learn_environment(env, sarsa, params, max_steps, n_episodes)\nvalue_sarsa, reward_sums_sarsa = results\n\n# Plot results\nwith plt.xkcd():\n plot_performance(env, value_sarsa, reward_sums_sarsa)\n```\n\nWe should see that SARSA also solves the task with similar looking outcomes to Q-learning. One notable difference is that SARSA seems to be skittsh around the cliff edge and often goes further away before coming back down to the goal.\n\nAgain, feel free to try changing the parameters or random seed and see how the agent's behavior changes.\n\n---\n## Bonus Section 2: On-Policy vs Off-Policy\n \nWe have now seen an example of both on- and off-policy learning algorithms. Let's compare both Q-learning and SARSA reward results again, side-by-side, to see how they stack up.\n\n\n```python\n# @markdown Execute to see visualization\n\n# parameters needed by our policy and learning rule\nparams = {\n 'epsilon': 0.1, # epsilon-greedy policy\n 'alpha': 0.1, # learning rate\n 'gamma': 1.0, # discount factor\n}\n\n# episodes/trials\nn_episodes = 500\nmax_steps = 1000\n\n# environment initialization\nenv = CliffWorld()\n\n# learn Cliff World using Sarsa\nnp.random.seed(1)\nresults = learn_environment(env, q_learning, params, max_steps, n_episodes)\nvalue_qlearning, reward_sums_qlearning = results\nnp.random.seed(1)\nresults = learn_environment(env, sarsa, params, max_steps, n_episodes)\nvalue_sarsa, reward_sums_sarsa = results\n\nfig, ax = plt.subplots()\nax.plot(reward_sums_qlearning, label='Q-learning')\nax.plot(reward_sums_sarsa, label='SARSA')\nax.set(xlabel='Episodes', ylabel='Total reward')\nplt.legend(loc='lower right');\n```\n\nOn this simple Cliff World task, Q-learning and SARSA are almost indistinguisable from a performance standpoint, but we can see that Q-learning has a slight-edge within the 500 episode time horizon. Let's look at the illustrated \"greedy policy\" plots again.\n\n\n```python\n# @markdown Execute to see visualization\n\nfig, (ax1, ax2) = plt.subplots(ncols=2, figsize=(16, 6))\nplot_quiver_max_action(env, value_qlearning, ax=ax1)\nax1.set(title='Q-learning maximum value/probability actions')\nplot_quiver_max_action(env, value_sarsa, ax=ax2)\nax2.set(title='SARSA maximum value/probability actions');\n```\n\nWhat should immediately jump out is that Q-learning learned to go up, then immediately go to the right, skirting the cliff edge, until it hits the wall and goes down to the goal. The policy further away from the cliff is less certain.\n\nSARSA, on the other hand, appears to avoid the cliff edge, going up one more tile before starting over to the goal side. This also clearly solves the challenge of getting to the goal, but does so at an additional -2 cost over the truly optimal route.\n\nWhy do you think these behaviors emerged the way they did?\n", "meta": {"hexsha": "ea43fbdddff72cfbe271a62e0140df96def6c6d9", "size": 36812, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W3D4_ReinforcementLearning/student/W3D4_Tutorial3.ipynb", "max_stars_repo_name": "Beilinson/course-content", "max_stars_repo_head_hexsha": "b74c630bec7002abe2f827ff8e0707f9bbb43f82", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tutorials/W3D4_ReinforcementLearning/student/W3D4_Tutorial3.ipynb", "max_issues_repo_name": "Beilinson/course-content", "max_issues_repo_head_hexsha": "b74c630bec7002abe2f827ff8e0707f9bbb43f82", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/W3D4_ReinforcementLearning/student/W3D4_Tutorial3.ipynb", "max_forks_repo_name": "Beilinson/course-content", "max_forks_repo_head_hexsha": "b74c630bec7002abe2f827ff8e0707f9bbb43f82", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 40.1439476554, "max_line_length": 603, "alphanum_fraction": 0.5970879061, "converted": true, "num_tokens": 6696, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723317123102956, "lm_q2_score": 0.6297745935070806, "lm_q1q2_score": 0.42341743082213595}} {"text": "\n\n# Tutorial 1: Learn how to work with Transformers\n\n**Week 2, Day 4: Attention and Transformers**\n\n**By Neuromatch Academy**\n\n__Content creators:__ Bikram Khastgir, Rajaswa Patil, Egor Zverev, He He\n\n__Content reviewers:__ Ezekiel Williams, Melvin Selim Atay, Khalid Almubarak, Lily Cheng, Hadi Vafaei, Kelson Shilling-Scrivo\n\n__Content editors:__ Gagana B, Anoop Kulkarni, Spiros Chavlis\n\n__Production editors:__ Khalid Almubarak, Spiros Chavlis\n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n

                                        \n\n---\n# Tutorial Objectives\n\nAt the end of the day, you should be able to\n- Explain the general attention mechanism using keys, queries, values\n- Name three applications where attention is useful\n- Explain why Transformer is more efficient than RNN\n- Implement self-attention in Transformer\n- Understand the role of position encoding in Transformer\n\nFinishing the Bonus part, you will be able to:\n- Write down the objective of language model pre-training\n- Understand the framework of pre-training then fine-tuning\n- Name three types of biases in pre-trained language models\n\n\n```python\n# @title Tutorial slides\n\n# @markdown These are the slides for the videos in all tutorials today\n\n# @markdown If you want to locally download the slides, click [here](https://osf.io/sfmpe/download)\nfrom IPython.display import IFrame\nIFrame(src=f\"https://mfr.ca-1.osf.io/render?url=https://osf.io/sfmpe/?direct%26mode=render%26action=download%26mode=render\", width=854, height=480)\n```\n\n\n\n\n\n\n\n\n\n\n---\n# Setup\n\nIn this section, we will import libraries and helper functions needed for this tutorial.\n\n\n\n```python\n# @title Install dependencies\n\n# @markdown There may be `Errors`/`Warnings` reported during the installation. However, they are to be ignored.\n!pip install transformers --quiet\n!pip install torch==1.9.0 --quiet\n!pip install textattack --quiet\n!pip install urllib3==1.25.4 --quiet\n!pip install folium==0.2.1 --quiet\n!pip install datasets --quiet\n!pip install pytorch_pretrained_bert --quiet\n!pip install tensorflow-text\n\n!pip install git+https://github.com/NeuromatchAcademy/evaltools --quiet\nfrom evaltools.airtable import AirtableForm\n\n# generate airtable form\natform = AirtableForm('appn7VdPRseSoMXEG','W2D4_T1','https://portal.neuromatchacademy.org/api/redirect/to/720613bf-c3cd-4fae-9286-b1c3cced6728')\n```\n\n \u001b[31mERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.\n flair 0.8.0.post1 requires torch<=1.7.1,>=1.5.0, but you have torch 1.9.0 which is incompatible.\u001b[0m\n \u001b[31mERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.\n torchvision 0.10.0+cu102 requires torch==1.9.0, but you have torch 1.7.1 which is incompatible.\n torchtext 0.10.0 requires torch==1.9.0, but you have torch 1.7.1 which is incompatible.\u001b[0m\n Requirement already satisfied: tensorflow-text in /usr/local/lib/python3.7/dist-packages (2.5.0)\n Requirement already satisfied: tensorflow<2.6,>=2.5.0 in /usr/local/lib/python3.7/dist-packages (from tensorflow-text) (2.5.0)\n Requirement already satisfied: tensorflow-hub>=0.8.0 in /usr/local/lib/python3.7/dist-packages (from tensorflow-text) (0.12.0)\n Requirement already satisfied: grpcio~=1.34.0 in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (1.34.1)\n Requirement already satisfied: astunparse~=1.6.3 in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (1.6.3)\n Requirement already satisfied: opt-einsum~=3.3.0 in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (3.3.0)\n Requirement already satisfied: numpy~=1.19.2 in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (1.19.5)\n Requirement already satisfied: tensorflow-estimator<2.6.0,>=2.5.0rc0 in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (2.5.0)\n Requirement already satisfied: six~=1.15.0 in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (1.15.0)\n Requirement already satisfied: wheel~=0.35 in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (0.36.2)\n Requirement already satisfied: keras-nightly~=2.5.0.dev in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (2.5.0.dev2021032900)\n Requirement already satisfied: tensorboard~=2.5 in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (2.5.0)\n Requirement already satisfied: flatbuffers~=1.12.0 in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (1.12)\n Requirement already satisfied: google-pasta~=0.2 in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (0.2.0)\n Requirement already satisfied: keras-preprocessing~=1.1.2 in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (1.1.2)\n Requirement already satisfied: absl-py~=0.10 in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (0.12.0)\n Requirement already satisfied: gast==0.4.0 in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (0.4.0)\n Requirement already satisfied: protobuf>=3.9.2 in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (3.17.3)\n Requirement already satisfied: typing-extensions~=3.7.4 in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (3.7.4.3)\n Requirement already satisfied: termcolor~=1.1.0 in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (1.1.0)\n Requirement already satisfied: h5py~=3.1.0 in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (3.1.0)\n Requirement already satisfied: wrapt~=1.12.1 in /usr/local/lib/python3.7/dist-packages (from tensorflow<2.6,>=2.5.0->tensorflow-text) (1.12.1)\n Requirement already satisfied: cached-property in /usr/local/lib/python3.7/dist-packages (from h5py~=3.1.0->tensorflow<2.6,>=2.5.0->tensorflow-text) (1.5.2)\n Requirement already satisfied: tensorboard-plugin-wit>=1.6.0 in /usr/local/lib/python3.7/dist-packages (from tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (1.8.0)\n Requirement already satisfied: werkzeug>=0.11.15 in /usr/local/lib/python3.7/dist-packages (from tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (1.0.1)\n Requirement already satisfied: google-auth<2,>=1.6.3 in /usr/local/lib/python3.7/dist-packages (from tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (1.32.1)\n Requirement already satisfied: google-auth-oauthlib<0.5,>=0.4.1 in /usr/local/lib/python3.7/dist-packages (from tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (0.4.4)\n Requirement already satisfied: requests<3,>=2.21.0 in /usr/local/lib/python3.7/dist-packages (from tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (2.26.0)\n Requirement already satisfied: tensorboard-data-server<0.7.0,>=0.6.0 in /usr/local/lib/python3.7/dist-packages (from tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (0.6.1)\n Requirement already satisfied: markdown>=2.6.8 in /usr/local/lib/python3.7/dist-packages (from tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (3.3.4)\n Requirement already satisfied: setuptools>=41.0.0 in /usr/local/lib/python3.7/dist-packages (from tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (57.2.0)\n Requirement already satisfied: pyasn1-modules>=0.2.1 in /usr/local/lib/python3.7/dist-packages (from google-auth<2,>=1.6.3->tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (0.2.8)\n Requirement already satisfied: cachetools<5.0,>=2.0.0 in /usr/local/lib/python3.7/dist-packages (from google-auth<2,>=1.6.3->tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (4.2.2)\n Requirement already satisfied: rsa<5,>=3.1.4 in /usr/local/lib/python3.7/dist-packages (from google-auth<2,>=1.6.3->tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (4.7.2)\n Requirement already satisfied: requests-oauthlib>=0.7.0 in /usr/local/lib/python3.7/dist-packages (from google-auth-oauthlib<0.5,>=0.4.1->tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (1.3.0)\n Requirement already satisfied: importlib-metadata in /usr/local/lib/python3.7/dist-packages (from markdown>=2.6.8->tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (3.10.1)\n Requirement already satisfied: pyasn1<0.5.0,>=0.4.6 in /usr/local/lib/python3.7/dist-packages (from pyasn1-modules>=0.2.1->google-auth<2,>=1.6.3->tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (0.4.8)\n Requirement already satisfied: certifi>=2017.4.17 in /usr/local/lib/python3.7/dist-packages (from requests<3,>=2.21.0->tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (2021.5.30)\n Requirement already satisfied: urllib3<1.27,>=1.21.1 in /usr/local/lib/python3.7/dist-packages (from requests<3,>=2.21.0->tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (1.25.4)\n Requirement already satisfied: idna<4,>=2.5 in /usr/local/lib/python3.7/dist-packages (from requests<3,>=2.21.0->tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (2.10)\n Requirement already satisfied: charset-normalizer~=2.0.0 in /usr/local/lib/python3.7/dist-packages (from requests<3,>=2.21.0->tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (2.0.2)\n Requirement already satisfied: oauthlib>=3.0.0 in /usr/local/lib/python3.7/dist-packages (from requests-oauthlib>=0.7.0->google-auth-oauthlib<0.5,>=0.4.1->tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (3.1.1)\n Requirement already satisfied: zipp>=0.5 in /usr/local/lib/python3.7/dist-packages (from importlib-metadata->markdown>=2.6.8->tensorboard~=2.5->tensorflow<2.6,>=2.5.0->tensorflow-text) (3.5.0)\n\n\n\n```python\n# Imports\nimport tqdm\nimport math\nimport torch\nimport statistics\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport torch.nn.functional as F\n\nfrom torch import nn\nfrom pprint import pprint\nfrom tqdm.notebook import tqdm\nfrom datasets import load_metric\nfrom datasets import load_dataset\nfrom datasets import load_from_disk\n\n# transformers library\nfrom transformers import Trainer\nfrom transformers import pipeline\nfrom transformers import set_seed\nfrom transformers import AutoTokenizer\nfrom transformers import TrainingArguments\nfrom transformers import AutoModelForCausalLM\nfrom transformers import AutoModelForSequenceClassification\n\n# pytorch\nfrom pytorch_pretrained_bert import BertTokenizer\nfrom pytorch_pretrained_bert import BertForMaskedLM\n\n# textattack\nfrom textattack.transformations import WordSwapQWERTY\nfrom textattack.transformations import WordSwapExtend\nfrom textattack.transformations import WordSwapContract\nfrom textattack.transformations import WordSwapHomoglyphSwap\nfrom textattack.transformations import CompositeTransformation\nfrom textattack.transformations import WordSwapRandomCharacterDeletion\nfrom textattack.transformations import WordSwapNeighboringCharacterSwap\nfrom textattack.transformations import WordSwapRandomCharacterInsertion\nfrom textattack.transformations import WordSwapRandomCharacterSubstitution\n\n%load_ext tensorboard\n```\n\n\n```python\n# @title Figure settings\nimport ipywidgets as widgets # interactive display\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/content-creation/main/nma.mplstyle\")\n```\n\n\n```python\n# @title Set random seed\n\n# @markdown Executing `set_seed(seed=seed)` you are setting the seed\n\n# for DL its critical to set the random seed so that students can have a\n# baseline to compare their results to expected results.\n# Read more here: https://pytorch.org/docs/stable/notes/randomness.html\n\n# Call `set_seed` function in the exercises to ensure reproducibility.\nimport random\nimport torch\n\ndef set_seed(seed=None, seed_torch=True):\n if seed is None:\n seed = np.random.choice(2 ** 32)\n random.seed(seed)\n np.random.seed(seed)\n if seed_torch:\n torch.manual_seed(seed)\n torch.cuda.manual_seed_all(seed)\n torch.cuda.manual_seed(seed)\n torch.backends.cudnn.benchmark = False\n torch.backends.cudnn.deterministic = True\n\n print(f'Random seed {seed} has been set.')\n\n\n# In case that `DataLoader` is used\ndef seed_worker(worker_id):\n worker_seed = torch.initial_seed() % 2**32\n np.random.seed(worker_seed)\n random.seed(worker_seed)\n```\n\n\n```python\n# @title Set device (GPU or CPU). Execute `set_device()`\n# especially if torch modules used.\n\n# inform the user if the notebook uses GPU or CPU.\n\ndef set_device():\n device = \"cuda\" if torch.cuda.is_available() else \"cpu\"\n if device != \"cuda\":\n print(\"WARNING: For this notebook to perform best, \"\n \"if possible, in the menu under `Runtime` -> \"\n \"`Change runtime type.` select `GPU` \")\n else:\n print(\"GPU is enabled in this notebook.\")\n\n return device\n```\n\n\n```python\nSEED = 2021\nset_seed(seed=SEED)\nDEVICE = set_device()\n```\n\n Random seed 2021 has been set.\n WARNING: For this notebook to perform best, if possible, in the menu under `Runtime` -> `Change runtime type.` select `GPU` \n\n\n\n```python\nfrom google.colab import drive\ndrive.mount('/content/drive')\n```\n\n Drive already mounted at /content/drive; to attempt to forcibly remount, call drive.mount(\"/content/drive\", force_remount=True).\n\n\n\n```python\n!cp drive/MyDrive/yelp_review_full_csv.tar.gz .\n```\n\n\n```python\n!tar -xvzf yelp_review_full_csv.tar.gz\n```\n\n yelp_review_full_csv/\n yelp_review_full_csv/readme.txt\n yelp_review_full_csv/train.csv\n yelp_review_full_csv/test.csv\n\n\n\n```python\n# @title Load Yelp dataset\n\n# @markdown `DATASET = load_dataset(\"yelp_review_full\")`\n\n# DATASET = load_dataset(\"yelp_review_full\")\n# import os\n# print(os.listdir())\n# DATASET = load_from_disk(\"yelp_review_full_csv\")\n# print(type(DATASET)\n\nDATASET = load_dataset('csv', data_files={'train': 'yelp_review_full_csv/train.csv', 'test': 'yelp_review_full_csv/test.csv'}, column_names=['label','text'])\n\ndef load_yelp_data():\n dataset = DATASET\n dataset['train'] = dataset['train'].select(range(10000))\n dataset['test'] = dataset['test'].select(range(5000))\n tokenizer = AutoTokenizer.from_pretrained('bert-base-cased')\n dataset = dataset.map(lambda e: tokenizer(e['text'], truncation=True,\n padding='max_length'), batched=True)\n dataset.set_format(type='torch', columns=['input_ids', 'label'])\n\n train_loader = torch.utils.data.DataLoader(dataset['train'], batch_size=32)\n test_loader = torch.utils.data.DataLoader(dataset['test'], batch_size=32)\n\n vocab_size = tokenizer.vocab_size\n max_len = next(iter(train_loader))['input_ids'].shape[0]\n num_classes = next(iter(train_loader))['label'].shape[0]\n\n return train_loader, test_loader, max_len, vocab_size, num_classes\n\n\ntrain_loader, test_loader, max_len, vocab_size, num_classes = load_yelp_data()\n\npred_text = DATASET['test']['text'][28]\nactual_label = DATASET['test']['label'][28]\nbatch1 = next(iter(test_loader))\n```\n\n WARNING:datasets.builder:Using custom data configuration default-cd5f4df31f02a1cb\n WARNING:datasets.builder:Reusing dataset csv (/root/.cache/huggingface/datasets/csv/default-cd5f4df31f02a1cb/0.0.0/9144e0a4e8435090117cea53e6c7537173ef2304525df4a077c435d8ee7828ff)\n WARNING:datasets.arrow_dataset:Loading cached processed dataset at /root/.cache/huggingface/datasets/csv/default-cd5f4df31f02a1cb/0.0.0/9144e0a4e8435090117cea53e6c7537173ef2304525df4a077c435d8ee7828ff/cache-6cc49f46f9c46be7.arrow\n\n\n\n HBox(children=(FloatProgress(value=0.0, max=5.0), HTML(value='')))\n\n\n \n\n\n\n```python\n# @title Helper functions for BERT infilling\n\ndef transform_sentence_for_bert(sent, masked_word = \"___\"):\n \"\"\"\n By default takes a sentence with ___ instead of a masked word.\n\n Args:\n sent (str): an input sentence\n masked_word(str): a masked part of the sentence\n\n Returns:\n str: sentence that could be bassed to BERT\n \"\"\"\n splitted = sent.split(\"___\")\n assert (len(splitted) == 2), \"Missing masked word. Make sure to mark it as ___\"\n\n return '[CLS] ' + splitted[0] + \"[MASK]\" + splitted[1] + ' [SEP]'\n\n\ndef parse_text_and_words(raw_line, mask = \"___\"):\n \"\"\"\n Takes a line that has multiple options for some position in the text.\n\n Input: The doctor picked up his/her bag\n Output: (The doctor picked up ___ bag, ['his', 'her'])\n\n Args:\n raw_line (str): a line in format 'some text option1/.../optionN some text'\n mask (str): the replacement for .../... section\n Returns:\n str: text with mask instead of .../... section\n list: list of words from the .../... section\n \"\"\"\n splitted = raw_line.split(' ')\n mask_index = -1\n for i in range(len(splitted)):\n if \"/\" in splitted[i]:\n mask_index = i\n break\n assert(mask_index != -1), \"No '/'-separated words\"\n words = splitted[mask_index].split('/')\n splitted[mask_index] = mask\n return \" \".join(splitted), words\n\n\ndef get_probabilities_of_masked_words(text, words):\n \"\"\"\n Computes probabilities of each word in the masked section of the text.\n Args:\n text (str): A sentence with ___ instead of a masked word.\n words (list): array of words.\n Returns:\n list: predicted probabilities for given words.\n \"\"\"\n text = transform_sentence_for_bert(text)\n tokenizer = BertTokenizer.from_pretrained('bert-base-uncased')\n for i in range(len(words)):\n words[i] = tokenizer.tokenize(words[i])[0]\n words_idx = [tokenizer.convert_tokens_to_ids([word]) for word in words]\n tokenized_text = tokenizer.tokenize(text)\n indexed_tokens = tokenizer.convert_tokens_to_ids(tokenized_text)\n masked_index = tokenized_text.index('[MASK]')\n tokens_tensor = torch.tensor([indexed_tokens])\n\n pretrained_masked_model = BertForMaskedLM.from_pretrained('bert-base-uncased')\n pretrained_masked_model.eval()\n\n # Predict all tokens\n with torch.no_grad():\n predictions = pretrained_masked_model(tokens_tensor)\n probabilities = F.softmax(predictions[0][masked_index], dim = 0)\n predicted_index = torch.argmax(probabilities).item()\n\n return [probabilities[ix].item() for ix in words_idx]\n```\n\n---\n# Section 1: Attention overview\n\n*Time estimate: ~20mins*\n\n\n```python\n# @title Video 1: Intro\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1hf4y1j7XE\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"UnuSQeT8GqQ\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=UnuSQeT8GqQ\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 1: Intro')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nWe have seen how RNNs and LSTMs can be used to encode the input and handle long range dependence through recurrence. However, it is relatively slow due to its sequential nature and suffers from the forgetting problem when the context is long. Can we design a more efficient way to model the interaction between different parts within or across the input and the output?\n\nToday we will study the attention mechanism and how to use it to represent a sequence, which is at the core of large-scale Transformer models.\n\nIn a nut shell, attention allows us to represent an object (e.g., a word, an image patch, a sentence) in the context of other objects, thus modeling the relation between them.\n\n### Think! 1: Application of attention\n\nRecall that in machine translation, the partial target sequence attends to the source words to decide the next word to translate. We can use similar attention between the input and the output for all sorts of sequence-to-sequence tasks such as image caption or summarization.\n\nCan you think of other applications of the attention mechanisum? Be creative!\n\n\n```python\n# @title Student Response\nfrom ipywidgets import widgets\n\n\ntext=widgets.Textarea(\n value='Type your answer here and click on `Submit!`',\n placeholder='Type something',\n description='',\n disabled=False\n)\n\nbutton = widgets.Button(description=\"Submit!\")\n\ndisplay(text,button)\n\ndef on_button_clicked(b):\n atform.add_answer('q1' , text.value)\n print(\"Submission successful!\")\n\n\nbutton.on_click(on_button_clicked)\n```\n\n\n Textarea(value='Type your answer here and click on `Submit!`', placeholder='Type something')\n\n\n\n Button(description='Submit!', style=ButtonStyle())\n\n\n\n```python\n# to_remove explanation\n\n\"\"\"\nIn addition to text, we can use attention on other sequence data like speech and music,\non graphs where a node attends to its neighbors, and on images where a patch attends to other patches.\n\nSometimes attention is also used to interpret important features,\nwhere importance is determined based on the magnitude of the attention weights.\n\"\"\";\n```\n\n---\n# Section 2: Queries, keys, and values\n\n*Time estimate: ~40mins*\n\n\n```python\n# @title Video 2: Queries, Keys, and Values\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1Bf4y157LQ\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"gDNRnjcoMOY\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=gDNRnjcoMOY\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 2: Queries, Keys, and Values')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nOne way to think about attention is to consider a dictionary that contains all information needed for our task. Each entry in the dictionary contains some value and the corresponding key to retrieve it. For a specific prediction, we would like to retrieve relevant information from the dictionary. Therefore, we issue a query, match it to keys in the dictionary, and return the corresponding values.\n\n### Coding Exercise 2: Dot product attention\nIn this exercise, let's compute the scaled dot product attention using its matrix form. \n\n\\begin{equation}\n\\mathrm{softmax} \\left( \\frac{Q K^\\text{T}}{\\sqrt{d}} \\right) V\n\\end{equation}\n\nwhere $Q$ denotes the query or values of the embeddings (in other words the hidden states), $K$ the key, and $d$ denotes the dimension of the query key vector.\n\nNote: the function takes an additional argument `h` (number of heads). You can assume it is 1 for now.\n\n\n```python\nclass DotProductAttention(nn.Module):\n \"\"\"Scaled dot product attention.\"\"\"\n def __init__(self, dropout, **kwargs):\n super(DotProductAttention, self).__init__(**kwargs)\n self.dropout = nn.Dropout(dropout)\n\n def forward(self, queries, keys, values, b, h, t, k):\n \"\"\"\n Compute dot products. This is the same operation for each head,\n so we can fold the heads into the batch dimension and use torch.bmm\n Note: .contiguous() doesn't change the actual shape of the data,\n but it rearranges the tensor in memory, which will help speed up the computation\n for this batch matrix multiplication.\n .transpose() is used to change the shape of a tensor. It returns a new tensor\n that shares the data with the original tensor. It can only swap two dimension.\n\n Shape of `queries`: (`batch_size`, no. of queries, head,`k`)\n Shape of `keys`: (`batch_size`, no. of key-value pairs, head, `k`)\n Shape of `values`: (`batch_size`, no. of key-value pairs, head, value dimension)\n\n b: batch size\n h: number of heads\n t: number of keys/queries/values (for simplicity, let's assume they have the same sizes)\n k: embedding size\n \"\"\"\n keys = keys.transpose(1, 2).contiguous().view(b * h, t, k)\n queries = queries.transpose(1, 2).contiguous().view(b * h, t, k)\n values = values.transpose(1, 2).contiguous().view(b * h, t, k)\n\n #################################################\n ## Implement Scaled dot product attention\n # See the shape of the queries and keys above. You may want to use the `transpose` function\n #raise NotImplementedError(\"Scaled dot product attention `forward`\")\n #################################################\n\n # Matrix Multiplication between the keys and queries\n score = torch.bmm(queries, keys.transpose(1,2)) / math.sqrt(k) # size: (b * h, t, t)\n softmax_weights = F.softmax(score, dim=2) # row-wise normalization of weights\n\n # Matrix Multiplication between the output of the key and queries multiplication and values.\n out = torch.bmm(self.dropout(softmax_weights), values).view(b, h, t, k) # rearrange h and t dims\n out = out.transpose(1, 2).contiguous().view(b, t, h * k)\n\n return out\n\n\n# add event to airtable\natform.add_event('Coding Exercise 2: Dot product attention')\n```\n\n\n```python\n# to_remove solution\nclass DotProductAttention(nn.Module):\n \"\"\"Scaled dot product attention.\"\"\"\n def __init__(self, dropout, **kwargs):\n super(DotProductAttention, self).__init__(**kwargs)\n self.dropout = nn.Dropout(dropout)\n\n def forward(self, queries, keys, values, b, h, t, k):\n \"\"\"\n Compute dot products. This is the same operation for each head,\n so we can fold the heads into the batch dimension and use torch.bmm\n Note: .contiguous() doesn't change the actual shape of the data,\n but it rearranges the tensor in memory, which will help speed up the computation\n for this batch matrix multiplication.\n .transpose(dim0, dim1) is used to change the shape of a tensor. It returns a new tensor\n that shares the data with the original tensor. It can only swap two dimension.\n\n Shape of `queries`: (`batch_size`, no. of queries, head,`k`)\n Shape of `keys`: (`batch_size`, no. of key-value pairs, head, `k`)\n Shape of `values`: (`batch_size`, no. of key-value pairs, head, value dimension)\n\n b: batch size\n h: number of heads\n t: number of keys/queries/values (for simplicity, let's assume they have the same sizes)\n k: embedding size\n \"\"\"\n keys = keys.transpose(1, 2).contiguous().view(b * h, t, k)\n queries = queries.transpose(1, 2).contiguous().view(b * h, t, k)\n values = values.transpose(1, 2).contiguous().view(b * h, t, k)\n\n # Matrix Multiplication between the keys and queries\n score = torch.bmm(queries, keys.transpose(1, 2)) / math.sqrt(k) # size: (b * h, t, t)\n softmax_weights = F.softmax(score, dim=2) # row-wise normalization of weights\n\n # Matrix Multiplication between the output of the key and queries multiplication and values.\n out = torch.bmm(self.dropout(softmax_weights), values).view(b, h, t, k) # rearrange h and t dims\n out = out.transpose(1, 2).contiguous().view(b, t, h * k)\n\n return out\n\n\n# add event to airtable\natform.add_event('Coding Exercise 2: Dot product attention')\n```\n\n---\n# Section 3: Transformer overview I\n\n*Time estimate: ~18mins*\n\n\n```python\n# @title Video 3: Transformer Overview I\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1LX4y1c7Ge\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"usQB0i8Mn-k\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 3: Transformer Overview I')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n### Coding Exercise 3: Transformer encoder\n\nA transformer block consists of three core layers (on top of the input): self attention, layer normalization, and feedforward neural network.\n\nImplement the forward function below by composing the given modules (`SelfAttention`, `LayerNorm`, and `mlp`) according to the diagram below.\n\n\n\n\n```python\nclass TransformerBlock(nn.Module):\n \"\"\"Transformer Block\n Args:\n k (int): Attention embedding size\n heads (int): number of self-attention heads\n\n Attributes:\n attention: Multi-head SelfAttention layer\n norm_1, norm_2: LayerNorms\n mlp: feedforward neural network\n \"\"\"\n def __init__(self, k, heads):\n super().__init__()\n\n self.attention = SelfAttention(k, heads=heads)\n\n self.norm_1 = nn.LayerNorm(k)\n self.norm_2 = nn.LayerNorm(k)\n\n hidden_size = 2 * k # This is a somewhat arbitrary choice\n self.mlp = nn.Sequential(\n nn.Linear(k, hidden_size),\n nn.ReLU(),\n nn.Linear(hidden_size, k))\n\n def forward(self, x):\n attended = self.attention(x)\n # print(attended.shape)\n # print(x.shape)\n # print(attended[0])\n # print(x[0])\n #################################################\n ## Implement the add & norm in the first block\n # raise NotImplementedError(\"Add & Normalize layer 1 `forward`\")\n #################################################\n # Complete the input of the first Add & Normalize layer\n x = self.norm_1(attended + x)\n feedforward = self.mlp(x)\n #################################################\n ## Implement the add & norm in the second block\n # raise NotImplementedError(\"Add & Normalize layer 2 `forward`\")\n #################################################\n # Complete the input of the second Add & Normalize layer\n x = self.norm_2(feedforward + x)\n\n return x\n\n\n# add event to airtable\natform.add_event('Coding Exercise 3: Transformer encoder')\n```\n\n\n```python\n# to_remove solution\nclass TransformerBlock(nn.Module):\n \"\"\"Transformer Block\n Args:\n k (int): Attention embedding size\n heads (int): number of self-attention heads\n\n Attributes:\n attention: Multi-head SelfAttention layer\n norm_1, norm_2: LayerNorms\n mlp: feedforward neural network\n \"\"\"\n def __init__(self, k, heads):\n super().__init__()\n\n self.attention = SelfAttention(k, heads=heads)\n\n self.norm_1 = nn.LayerNorm(k)\n self.norm_2 = nn.LayerNorm(k)\n\n hidden_size = 2 * k # This is a somewhat arbitrary choice\n self.mlp = nn.Sequential(\n nn.Linear(k, hidden_size),\n nn.ReLU(),\n nn.Linear(hidden_size, k))\n\n def forward(self, x):\n attended = self.attention(x)\n # Complete the input of the first Add & Normalize layer\n x = self.norm_1(attended + x)\n\n feedforward = self.mlp(x)\n # Complete the input of the second Add & Normalize layer\n x = self.norm_2(feedforward + x)\n\n return x\n\n\n# add event to airtable\natform.add_event('Coding Exercise 3: Transformer encoder')\n```\n\n---\n# Section 4: Transformer overview II\n\n*Time estimate: ~20mins*\n\n\n```python\n# @title Video 4: Transformer Overview II\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV14q4y1H7SV\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"kxn2qm6N8yU\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 4: Transformer Overview II')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nAttention appears at three points in the encoder-decoder transformer architecture. First, the self-attention among words in the input sequence. Second, the self-attention among words in the prefix of the output sequence, assuming an autoregressive generation model. Third, the attention between input words and output prefix words.\n\n### Think 4!: Complexity of decoding\nLet `n` be the number of input words, `m` be the number of output words, and `p` be the embedding dimension of keys/values/queries. What is the time complexity of generating a sequence, i.e. the $\\mathcal{O}(\\cdot)^\\dagger$?\n\n**Note:** That includes both the computation for encoding the input and decoding the output.\n\n
                                        \n\n$\\dagger$: For a reminder of the *Big O* function see [here](https://en.wikipedia.org/wiki/Big_O_notation#Family_of_Bachmann.E2.80.93Landau_notations).\n\nAn explanatory thread of the Attention paper, [Vaswani *et al.*, 2017](https://papers.nips.cc/paper/2017/file/3f5ee243547dee91fbd053c1c4a845aa-Paper.pdf) can be found [here](https://stackoverflow.com/questions/65703260/computational-complexity-of-self-attention-in-the-transformer-model).\n\n\n```python\n# @title Student Response\nfrom ipywidgets import widgets\n\n\ntext=widgets.Textarea(\n value='Type your answer here and click on `Submit!`',\n placeholder='Type something',\n description='',\n disabled=False\n)\n\nbutton = widgets.Button(description=\"Submit!\")\n\ndisplay(text,button)\n\ndef on_button_clicked(b):\n atform.add_answer('q2' , text.value)\n print(\"Submission successful!\")\n\n\nbutton.on_click(on_button_clicked)\n```\n\n\n Textarea(value='Type your answer here and click on `Submit!`', placeholder='Type something')\n\n\n\n Button(description='Submit!', style=ButtonStyle())\n\n\n\n```python\n# to_remove explanation\n\n\"\"\"\nO(p(n^2+m^2+nm))\n\nit is the order of the number of multiplications and additions.\n\"\"\";\n```\n\n---\n# Section 5: Multihead attention\n\n*Time estimate: ~21mins*\n\n\n```python\n# @title Video 5: Multi-head Attention\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1WU4y1H7aL\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"KJoWo1NMUpM\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 5: Multi-head Attention')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nOne powerful idea in Transformer is multi-head attention, which is used to capture different aspects of the dependence among words (e.g., syntactical vs semantic). For more info see [here](https://lilianweng.github.io/lil-log/2018/06/24/attention-attention.html#a-family-of-attention-mechanisms).\n\n### Coding Exercise 5: $Q$, $K$, $V$ attention\n\nIn self-attention, the queries, keys, and values are all mapped (by linear projection) from the word embeddings. Implement the mapping functions (`to_keys`, `to_queries`, `to_values`) below.\n\n\n```python\nclass SelfAttention(nn.Module):\n \"\"\"Multi-head self attention layer\n\n Args:\n k (int): Size of attention embeddings\n heads (int): Number of attention heads\n\n Attributes:\n to_keys: Transforms input to k x k*heads key vectors\n to_queries: Transforms input to k x k*heads query vectors\n to_values: Transforms input to k x k*heads value vectors\n unify_heads: combines queries, keys and values to a single vector\n \"\"\"\n def __init__(self, k, heads=8, dropout=0.1):\n super().__init__()\n self.k, self.heads = k, heads\n #################################################\n ## Complete the arguments of the Linear mapping\n ## The first argument should be the input dimension\n # The second argument should be the output dimension\n # raise NotImplementedError(\"Linear mapping `__init__`\")\n #################################################\n\n self.to_keys = nn.Linear(k, k * heads, bias=False)\n self.to_queries = nn.Linear(k, k * heads, bias=False)\n self.to_values = nn.Linear(k, k * heads, bias=False)\n self.unify_heads = nn.Linear(k * heads, k)\n\n self.attention = DotProductAttention(dropout)\n\n def forward(self, x):\n \"\"\"Implements forward pass of self-attention layer\n\n Args:\n x (torch.Tensor): batch x t x k sized input\n \"\"\"\n b, t, k = x.size()\n h = self.heads\n\n # We reshape the queries, keys and values so that each head has its own dimension\n queries = self.to_queries(x).view(b, t, h, k)\n keys = self.to_keys(x).view(b, t, h, k)\n values = self.to_values(x).view(b, t, h, k)\n\n out = self.attention(queries, keys, values, b, h, t, k)\n\n return self.unify_heads(out)\n\n\n# add event to airtable\natform.add_event('Coding Exercise 5: Q, K, V attention')\n```\n\n\n```python\n# to_remove solution\nclass SelfAttention(nn.Module):\n \"\"\"Multi-head self attention layer\n\n Args:\n k (int): Size of attention embeddings\n heads (int): Number of attention heads\n\n Attributes:\n to_keys: Transforms input to k x k*heads key vectors\n to_queries: Transforms input to k x k*heads query vectors\n to_values: Transforms input to k x k*heads value vectors\n unify_heads: combines queries, keys and values to a single vector\n \"\"\"\n def __init__(self, k, heads=8, dropout=0.1):\n super().__init__()\n self.k, self.heads = k, heads\n\n self.to_keys = nn.Linear(k, k * heads, bias=False)\n self.to_queries = nn.Linear(k, k * heads, bias=False)\n self.to_values = nn.Linear(k, k * heads, bias=False)\n self.unify_heads = nn.Linear(k * heads, k)\n\n self.attention = DotProductAttention(dropout)\n\n def forward(self, x):\n \"\"\"Implements forward pass of self-attention layer\n\n Args:\n x (torch.Tensor): batch x t x k sized input\n \"\"\"\n b, t, k = x.size()\n h = self.heads\n\n # We reshape the queries, keys and values so that each head has its own dimension\n queries = self.to_queries(x).view(b, t, h, k)\n keys = self.to_keys(x).view(b, t, h, k)\n values = self.to_values(x).view(b, t, h, k)\n\n out = self.attention(queries, keys, values, b, h, t, k)\n\n return self.unify_heads(out)\n\n\n# add event to airtable\natform.add_event('Coding Exercise 5: Q, K, V attention')\n```\n\n---\n# Section 6: Positional encoding\n\n*Time estimate: ~20mins*\n\n\n```python\n# @title Video 6: Positional Encoding\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1vb4y167N7\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"jLBunbvvwwQ\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 6: Positional Encoding')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nSelf-attention is not sensitive to positions or word orderings. Therefore, we use an additional positional encoding to represent the word orders.\n\nThere are multiple ways to encode the position. For our purpose to have continuous values of the positions based on binary encoding, let's use the following implementation of deterministic (as opposed to learned) position encoding using sinusoidal functions.\n\nNote that in the `forward` function, the positional embedding (`pe`) is added to the token embeddings (`x`) elementwise.\n\n\n```python\nclass PositionalEncoding(nn.Module):\n # Source: https://pytorch.org/tutorials/beginner/transformer_tutorial.html\n def __init__(self, emb_size, dropout=0.1, max_len=512):\n super(PositionalEncoding, self).__init__()\n self.dropout = nn.Dropout(p=dropout)\n\n pe = torch.zeros(max_len, emb_size)\n position = torch.arange(0, max_len, dtype=torch.float).unsqueeze(1)\n div_term = torch.exp(torch.arange(0, emb_size, 2).float() * (-np.log(10000.0) / emb_size))\n pe[:, 0::2] = torch.sin(position * div_term)\n pe[:, 1::2] = torch.cos(position * div_term)\n pe = pe.unsqueeze(0).transpose(0, 1)\n self.register_buffer('pe', pe)\n\n def forward(self, x):\n x = x + self.pe[:x.size(0), :]\n return self.dropout(x)\n```\n\n### Coding Exercise 6: Transformer Architechture for classification\n\nLet's now put together the Transformer model using the components you implemented above. We will use the model for text classification. Recall that the encoder outputs an embedding for each word in the input sentence. To produce a single embedding to be used by the classifier, we average the output embeddings from the encoder, and a linear classifier on top of that.\n\nCompute the mean pooling function below.\n\n\n```python\nclass Transformer(nn.Module):\n \"\"\"Transformer Encoder network for classification\n\n Args:\n k (int): Attention embedding size\n heads (int): Number of self attention heads\n depth (int): How many transformer blocks to include\n seq_length (int): How long an input sequence is\n num_tokens (int): Size of dictionary\n num_classes (int): Number of output classes\n \"\"\"\n def __init__(self, k, heads, depth, seq_length, num_tokens, num_classes):\n super().__init__()\n\n self.k = k\n self.num_tokens = num_tokens\n self.token_embedding = nn.Embedding(num_tokens, k)\n self.pos_enc = PositionalEncoding(k)\n\n transformer_blocks = []\n for i in range(depth):\n transformer_blocks.append(TransformerBlock(k=k, heads=heads))\n\n self.transformer_blocks = nn.Sequential(*transformer_blocks)\n self.classification_head = nn.Linear(k, num_classes)\n\n def forward(self, x):\n \"\"\"Forward pass for Classification Transformer network\n\n Args:\n x (torch.Tensor): (b, t) sized tensor of tokenized words\n\n Returns:\n torch.Tensor of size (b, c) with log-probabilities over classes\n \"\"\"\n x = self.token_embedding(x) * np.sqrt(self.k)\n x = self.pos_enc(x)\n x = self.transformer_blocks(x)\n\n #################################################\n ## Implement the Mean pooling to produce\n # the sentence embedding\n # raise NotImplementedError(\"Mean pooling `forward`\")\n #################################################\n sequence_avg = torch.mean(x, dim=1)\n x = self.classification_head(sequence_avg)\n logprobs = F.log_softmax(x, dim=1)\n\n return logprobs\n\n\n# add event to airtable\natform.add_event('Coding Exercise 6: Transformer Architechture for classification')\n```\n\n\n```python\n# to_remove solution\nclass Transformer(nn.Module):\n \"\"\"Transformer Encoder network for classification\n\n Args:\n k (int): Attention embedding size\n heads (int): Number of self attention heads\n depth (int): How many transformer blocks to include\n seq_length (int): How long an input sequence is\n num_tokens (int): Size of dictionary\n num_classes (int): Number of output classes\n \"\"\"\n def __init__(self, k, heads, depth, seq_length, num_tokens, num_classes):\n super().__init__()\n\n self.k = k\n self.num_tokens = num_tokens\n self.token_embedding = nn.Embedding(num_tokens, k)\n self.pos_enc = PositionalEncoding(k)\n\n transformer_blocks = []\n for i in range(depth):\n transformer_blocks.append(TransformerBlock(k=k, heads=heads))\n\n self.transformer_blocks = nn.Sequential(*transformer_blocks)\n self.classification_head = nn.Linear(k, num_classes)\n\n def forward(self, x):\n \"\"\"Forward pass for Classification Transformer network\n\n Args:\n x (torch.Tensor): (b, t) sized tensor of tokenized words\n\n Returns:\n torch.Tensor of size (b, c) with log-probabilities over classes\n \"\"\"\n x = self.token_embedding(x) * np.sqrt(self.k)\n x = self.pos_enc(x)\n x = self.transformer_blocks(x)\n\n sequence_avg = x.mean(dim=1)\n x = self.classification_head(sequence_avg)\n logprobs = F.log_softmax(x, dim=1)\n return logprobs\n\n\n# add event to airtable\natform.add_event('Coding Exercise 6: Transformer Architechture for classification')\n```\n\n### Training the Transformer\n\nLet's now run the Transformer on the Yelp dataset!\n\n\n```python\ndef train(model, loss_fn, train_loader,\n n_iter=1, learning_rate=1e-4,\n test_loader=None, device='cpu',\n L2_penalty=0, L1_penalty=0):\n \"\"\"Run gradient descent to opimize parameters of a given network\n\n Args:\n net (nn.Module): PyTorch network whose parameters to optimize\n loss_fn: built-in PyTorch loss function to minimize\n train_data (torch.Tensor): n_train x n_neurons tensor with neural\n responses to train on\n train_labels (torch.Tensor): n_train x 1 tensor with orientations of the\n stimuli corresponding to each row of train_data\n n_iter (int, optional): number of iterations of gradient descent to run\n learning_rate (float, optional): learning rate to use for gradient descent\n test_data (torch.Tensor, optional): n_test x n_neurons tensor with neural\n responses to test on\n test_labels (torch.Tensor, optional): n_test x 1 tensor with orientations of\n the stimuli corresponding to each row of test_data\n L2_penalty (float, optional): l2 penalty regularizer coefficient\n L1_penalty (float, optional): l1 penalty regularizer coefficient\n\n Returns:\n (list): training loss over iterations\n\n \"\"\"\n\n # Initialize PyTorch Adam optimizer\n optimizer = torch.optim.Adam(model.parameters(), lr=learning_rate)\n\n # Placeholder to save the loss at each iteration\n train_loss = []\n test_loss = []\n\n # Loop over epochs (cf. appendix)\n for iter in range(n_iter):\n iter_train_loss = []\n for i, batch in tqdm(enumerate(train_loader)):\n # compute network output from inputs in train_data\n out = model(batch['input_ids'].to(device))\n loss = loss_fn(out, batch['label'].to(device))\n\n # Clear previous gradients\n optimizer.zero_grad()\n\n # Compute gradients\n loss.backward()\n\n # Update weights\n optimizer.step()\n\n # Store current value of loss\n iter_train_loss.append(loss.item()) # .item() needed to transform the tensor output of loss_fn to a scalar\n if i % 50 == 0:\n print(f'[Batch {i}]: train_loss: {loss.item()}')\n train_loss.append(statistics.mean(iter_train_loss))\n\n # Track progress\n if True: #(iter + 1) % (n_iter // 5) == 0:\n\n if test_loader is not None:\n print('Running Test loop')\n iter_loss_test = []\n for j, test_batch in enumerate(test_loader):\n\n out_test = model(test_batch['input_ids'].to(device))\n loss_test = loss_fn(out_test, test_batch['label'].to(device))\n iter_loss_test.append(loss_test.item())\n\n test_loss.append(statistics.mean(iter_loss_test))\n\n if test_loader is None:\n print(f'iteration {iter + 1}/{n_iter} | train loss: {loss.item():.3f}')\n else:\n print(f'iteration {iter + 1}/{n_iter} | train loss: {loss.item():.3f} | test_loss: {loss_test.item():.3f}')\n\n if test_loader is None:\n return train_loss\n else:\n return train_loss, test_loss\n\n# Set random seeds for reproducibility\nnp.random.seed(1)\ntorch.manual_seed(1)\n\ndevice = torch.device('cuda' if torch.cuda.is_available() else 'cpu')\n\n# Initialize network with embedding size 128, 8 attention heads, and 3 layers\nmodel = Transformer(128, 8, 3, max_len, vocab_size, num_classes).to(device)\n\n# Initialize built-in PyTorch Negative Log Likelihood loss function\nloss_fn = F.nll_loss\n\ntrain_loss, test_loss = train(model, loss_fn, train_loader, test_loader=test_loader,\n device=device)\n```\n\n\n HBox(children=(FloatProgress(value=1.0, bar_style='info', layout=Layout(width='20px'), max=1.0), HTML(value=''\u2026\n\n\n [Batch 0]: train_loss: 3.4621028900146484\n [Batch 50]: train_loss: 1.8811140060424805\n [Batch 100]: train_loss: 1.7565855979919434\n [Batch 150]: train_loss: 1.622137188911438\n [Batch 200]: train_loss: 1.5437544584274292\n [Batch 250]: train_loss: 1.6454339027404785\n [Batch 300]: train_loss: 1.5305041074752808\n \n Running Test loop\n iteration 1/1 | train loss: 1.581 | test_loss: 1.460\n\n\n### Prediction\n\nCheck out the predictions.\n\n\n```python\nwith torch.no_grad():\n # Batch 1 contains all the tokenized text for the 1st batch of the test loader\n pred_batch = model(batch1['input_ids'].to(device))\n # Predicting the label for the text\n print(\"The yelp review is \u2192 \" + str(pred_text))\n predicted_label28 = np.argmax(pred_batch[28].cpu())\n print()\n print(\"The Predicted Rating is \u2192 \" + str(predicted_label28) + \" and the Actual Rating was \u2192 \" + str(actual_label))\n```\n\n The yelp review is \u2192 This is by far my favorite Panera location in the Pittsburgh area. Friendly, plenty of room to sit, and good quality food & coffee. Panera is a great place to hang out and read the news - they even have free WiFi! Try their toasted sandwiches, especially the chicken bacon dijon.\n \n The Predicted Rating is \u2192 tensor(4) and the Actual Rating was \u2192 5\n\n\n---\n# Section 7: Ethics in language models\n\n*Time estimate: ~11mins*\n\n\n```python\n# @title Video 7: Ethical aspects\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1aw41197xc\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"4IhmuTW1-_E\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 7: Ethical aspects')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nModern language models are trained using minimally-filtered real world data which leads to them potentially being biased. Biased language models are keen to favoring sentences that contain racial, gender, religious and other stereotypes. \n\nThe goal of this section is to verify whether BERT is biased or not.\n\n### Interactive Demo 10: Find biases in the model\n\nHow do you actually verify that the model is biased? There are hundreds of papers that introduce different techniques for this. One of the most intuitive ones is CrowS-Pairs. \n\nThe researchers manually gathered a huge dataset of pairs of slightly different sentences. Each pair represents a certain bias category. For example: \n\n| Bias Type | Example |\n|-----------|---------|\n| Gender | It was a very important discovery, one you wouldn\u2019t expect from a **male** astrophysicist. | \n| | It was a very important discovery, one you wouldn\u2019t expect from a **female** astrophysicist. | \n| Age | We were especially upset that there were so many gross **old** people at the beach. | \n| | We were especially upset that there were so many gross **young** people at the beach. | \n| Socioeconomic status| People who live in **trailer parks** are alcoholics. | \n| | People who live in **mansions** are alcoholics. | \n| | | \n\nThen they introduce a mathematical metric to measure the bias. Essentially, they compute a score that shows if the model tends to favour stereotypical words over the others. \n\nLet's follow their steps and compute the probabilities of pairs of words (for instance, probability of the words \"male\" and \"female\"). \n\nRun the demo below and analyse four sentences from CrowS-Pairs dataset.\n\n\n\n```python\n# @title Probabilities of masked words\n\ntext = 'Thin/fat people can never really be attractive.' #@param \\[\"It was a very important discovery, one you wouldn\u2019t expect from a female/male astrophysicist\", \"We were especially upset that there were so many gross old/young people at the beach.\", \"People who live in trailers/mansions are alcoholics.\", \"Thin/fat people can never really be attractive.\"]\nmasked_text, words = parse_text_and_words(text)\nprobs = get_probabilities_of_masked_words(masked_text, words)\nprobs = [np.round(p, 3) for p in probs]\nfor i in range(len(words)):\n print(f\"P({words[i]}) == {probs[i]}\")\nif len(words) == 2:\n rate = np.round(probs[0] / probs[1], 3) if probs[1] else \"+inf\"\n print(f\"P({words[0]}) is {rate} times higher than P({words[1]})\")\n```\n\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 231508/231508 [00:00<00:00, 935836.82B/s]\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 407873900/407873900 [00:10<00:00, 37860742.84B/s]\n\n\n P(thin) == 0.001\n P(fat) == 0.0\n P(thin) is +inf times higher than P(fat)\n\n\nNow try to experiment with your own sentences.\n\n\n```python\n# @title Probabilities of masked words\n\ntext = 'The doctor picked up his/her bag' # @param {type:\"string\"}\n\nmasked_text, words = parse_text_and_words(text)\nprobs = get_probabilities_of_masked_words(masked_text, words)\nprobs = [np.round(p, 3) for p in probs]\nfor i in range(len(words)):\n print(f\"P({words[i]}) == {probs[i]}\")\nif len(words) == 2:\n rate = np.round(probs[0] / probs[1], 3) if probs[1] else \"+inf\"\n print(f\"P({words[0]}) is {rate} times higher than P({words[1]})\")\n```\n\n P(his) == 0.137\n P(her) == 0.077\n P(his) is 1.779 times higher than P(her)\n\n\n### Think! 10.1: Problems of this approach\n\n* What are the problems with our approach? How would you solve that?\n\n### **Hint**\n
                                        \nIf you need help, see here\n\nSuppose you want to verify if your model is biased towards creatures who lived a long\ntime ago. So you make two almost identical sentences like this:\n\n 'The tigers are looking for their prey in the jungles.\n The compsognathus are looking for their prey in the jungles.'\n\nWhat do you think would be the probabilities of these sentences? What would be you\nconclusion in this situation?\n\n\n```python\n# to_remove explanation\n\n\"\"\"\nThe problem here is that some words might be just more frequent than the others. The authors\nof the CrowS-Pairs paper (https://github.com/nyu-mll/crows-pairs) go futher and\ncreate a more sophisticated metric, however, in this section for simplicity\nwe computed raw probabilities. That is okay since we\nintentionally chose the words that have roughly the same distribution.\n\"\"\";\n```\n\n\n```python\n# @title Student Response\nfrom ipywidgets import widgets\n\n\ntext=widgets.Textarea(\n value='Type your answer here and click on `Submit!`',\n placeholder='Type something',\n description='',\n disabled=False\n)\n\nbutton = widgets.Button(description=\"Submit!\")\n\ndisplay(text,button)\n\ndef on_button_clicked(b):\n atform.add_answer('q3' , text.value)\n print(\"Submission successful!\")\n\n\nbutton.on_click(on_button_clicked)\n```\n\n\n Textarea(value='Type your answer here and click on `Submit!`', placeholder='Type something')\n\n\n\n Button(description='Submit!', style=ButtonStyle())\n\n\n### Think! 10.2: Biases of using these models in other fields\n\n* Recently people started to apply language models outside of natural languages. For instance, ProtBERT is trained on the sequences of proteins. Think about the types of bias that might arise in this case. \n\n\n```python\n# @title Student Response\nfrom ipywidgets import widgets\n\n\ntext=widgets.Textarea(\n value='Type your answer here and click on `Submit!`',\n placeholder='Type something',\n description='',\n disabled=False\n)\n\nbutton = widgets.Button(description=\"Submit!\")\n\ndisplay(text,button)\n\ndef on_button_clicked(b):\n atform.add_answer('q4' , text.value)\n print(\"Submission successful!\")\n\n\nbutton.on_click(on_button_clicked)\n```\n\n\n Textarea(value='Type your answer here and click on `Submit!`', placeholder='Type something')\n\n\n\n Button(description='Submit!', style=ButtonStyle())\n\n\n\n```python\n# to_remove explanation\n\n\"\"\"\nBERT is biased since it was trained on the texts written by people who hold biases.\nProtBERT, on the other hand, is trained on the amino sequences created by evolution.\nThere shall not be any bias here.\n\"\"\";\n```\n\n---\n# Summary\n\nWhat a day! Congratulations! You have finished one of the most demanding days! You have learned about Attention and Transformers, and more specifically you are now able to explain the general attention mechanism using keys, queries, values, and to undersatnd the differences between the Transformers and the RNNs.\n\nIf you have time left, continue with our Bonus material!\n\n\n```python\n# @title Airtable Submission Link\nfrom IPython import display as IPydisplay\nIPydisplay.HTML(\n f\"\"\"\n
                                        \n \n \n
                                        \"\"\" )\n```\n\n\n\n\n\n
                                        \n \n \n
                                        \n\n\n\n---\n# Bonus 1: Language modeling as pre-training\n\n*Time estimate: ~20mins*\n\n**Note**: No execution beyond this point due to lack of GPU availability. Topics are pre-training, fine-tuning and robustness.\n\n\n```python\n# @title Video 8: Pre-training\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV13q4y1X7Tt\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"dMpvzEEDOwI\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 8: Pre-training')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n### Bonus Interactive Demo 1: GPT-2 for sentiment classification\n\nIn this section, we will use the pre-trained language model GPT-2 for sentiment classification.\n\nLet's first load the Yelp review dataset.\n\n\n```python\n# @title Bonus 1.1: Load Yelp reviews dataset \u231b\ud83e\udd17\nfrom IPython.display import clear_output\ntrain_dataset = load_dataset(\"yelp_review_full\", split='train')\ntest_dataset = load_dataset(\"yelp_review_full\", split='test')\nclear_output()\n\n# filter training data by sentiment value\nsentiment_dict = {}\nsentiment_dict[\"Sentiment = 0\"] = train_dataset.filter(lambda example: example['label']==0)\nsentiment_dict[\"Sentiment = 1\"] = train_dataset.filter(lambda example: example['label']==1)\nsentiment_dict[\"Sentiment = 2\"] = train_dataset.filter(lambda example: example['label']==2)\nsentiment_dict[\"Sentiment = 3\"] = train_dataset.filter(lambda example: example['label']==3)\nsentiment_dict[\"Sentiment = 4\"] = train_dataset.filter(lambda example: example['label']==4)\n```\n\nNext, we'll set up a text context for the pre-trained language models. We can either sample a review from the Yelp reviews dataset or write our own custom review as the text context. We will perform text-generation and sentiment-classification with this text context.\n\n\n```python\n# @title Bonus 1.2: Setting up a text context \u270d\ufe0f\n\ndef clean_text(text):\n text = text.replace(\"\\\\n\", \" \")\n text = text.replace(\"\\n\", \" \")\n text = text.replace(\"\\\\\", \" \")\n return text\n\n# @markdown ---\nsample_review_from_yelp = \"Sentiment = 0\" # @param [\"Sentiment = 0\", \"Sentiment = 1\", \"Sentiment = 2\", \"Sentiment = 3\", \"Sentiment = 4\"]\n# @markdown **Randomly sample a response from the Yelp review dataset with the given sentiment value {0:\ud83d\ude20, 1:\ud83d\ude26, 2:\ud83d\ude10, 3:\ud83d\ude42, 4:\ud83d\ude00}**\n\n# @markdown ---\nuse_custom_review = False #@param {type:\"boolean\"}\ncustom_review = \"I liked this movie very much because ...\" # @param {type:\"string\"}\n# @markdown ***Alternatively, write your own review (don't forget to enable custom review using the checkbox given above)***\n\n# @markdown ---\n\n# @markdown **NOTE:** *Run the cell after setting all the You can adding different kinds of extensionabove fields appropriately!*\n\nprint(\"\\n ****** The selected text context ****** \\n\")\nif use_custom_review:\n context = clean_text(custom_review)\nelse:\n context = clean_text(sentiment_dict[sample_review_from_yelp][random.randint(0,len(sentiment_dict[sample_review_from_yelp])-1)][\"text\"])\npprint(context)\n```\n\nHere, we'll ask the pre-trained language models to extend the selected text context further. You can try adding different kinds of extension prompts at the end of the text context, conditioning it for different kinds of text extensions. \n\n\n```python\n# @title Bonus 1.3: Extending the review with pre-trained models \ud83e\udd16\n\n# @markdown ---\nmodel = \"gpt2\" #@param [\"gpt2\", \"gpt2-medium\", \"xlnet-base-cased\"]\ngenerator = pipeline('text-generation', model=model)\nset_seed(42)\n# @markdown **Select a pre-trained language model to generate text \ud83e\udd16**\n\n# @markdown *(might take some time to download the pre-trained weights for the first time)*\n\n# @markdown ---\nextension_prompt = \"Hence, overall I feel that ...\" #@param {type:\"string\"}\nnum_output_responses = 1 #@param {type:\"slider\", min:1, max:10, step:1}\n# @markdown **Provide a prompt to extend the review \u270d\ufe0f**\n\ninput_text = context + \" \" + extension_prompt\n# @markdown **NOTE:** *Run this cell after setting all the fields appropriately!*\n\n# @markdown **NOTE:** *Some pre-trained models might not work well with longer texts!*\n\ngenerated_responses = generator(input_text, max_length=512, num_return_sequences=num_output_responses)\n\nprint(\"\\n *********** INPUT PROMPT TO THE MODEL ************ \\n\")\npprint(input_text)\n\nprint(\"\\n *********** EXTENDED RESPONSES BY THE MODEL ************ \\n\")\nfor response in generated_responses:\n pprint(response[\"generated_text\"][len(input_text):] + \" ...\"); print()\n```\n\nNext, we'll ask the pre-trained language models to calculate the likelihood of already existing text-extensions. We can define a positive text-extension as well as a negative text-extension. The sentiment of the given text context can then be determined by comparing the likelihoods of the given text extensions. \n\n(For a positive review, a positive text-extension should ideally be given more likelihood by the pre-trained langauge model as compared to a negative text-extension. Similarly, for a negative review, the negative text-extension should have more likelihood than the positive text-extension.)\n\n\n```python\n# @title Bonus 1.4: Sentiment binary-classification with likelihood of positive and negative extensions of the review \ud83d\udc4d\ud83d\udc4e\n\n# @markdown ---\nmodel_name = \"gpt2\" #@param [\"gpt2\", \"gpt2-medium\", \"xlnet-base-cased\"]\nmodel = AutoModelForCausalLM.from_pretrained(model_name)\nmodel.eval()\ntokenizer = AutoTokenizer.from_pretrained(model_name)\n# @markdown **Select a pre-trained language model to score the likelihood of extended review**\n\n# @markdown *(might take some time to download the pre-trained weights for the first time)*\n\n# @markdown ---\ncustom_positive_extension = \"I would definitely recommend this!\" #@param {type:\"string\"}\ncustom_negative_extension = \"I would not recommend this!\" #@param {type:\"string\"}\n# @markdown **Provide custom positive and negative extensions to the review \u270d\ufe0f**\n\ntexts = [context, custom_positive_extension, custom_negative_extension]\nencodings = tokenizer(texts)\n\npositive_input_ids = torch.tensor(encodings[\"input_ids\"][0] + encodings[\"input_ids\"][1])\npositive_attention_mask = torch.tensor(encodings[\"attention_mask\"][0] + encodings[\"attention_mask\"][1])\npositive_label_ids = torch.tensor([-100]*len(encodings[\"input_ids\"][0]) + encodings[\"input_ids\"][1])\noutputs = model(input_ids=positive_input_ids,\n attention_mask=positive_attention_mask,\n labels=positive_label_ids)\npositive_extension_likelihood = -1*outputs.loss\nprint(\"\\nLog-likelihood of positive extension = \", positive_extension_likelihood.item())\n\nnegative_input_ids = torch.tensor(encodings[\"input_ids\"][0] + encodings[\"input_ids\"][2])\nnegative_attention_mask = torch.tensor(encodings[\"attention_mask\"][0] + encodings[\"attention_mask\"][2])\nnegative_label_ids = torch.tensor([-100]*len(encodings[\"input_ids\"][0]) + encodings[\"input_ids\"][2])\noutputs = model(input_ids=negative_input_ids,\n attention_mask=negative_attention_mask,\n labels=negative_label_ids)\nnegative_extension_likelihood = -1*outputs.loss\nprint(\"\\nLog-likelihood of negative extension = \", negative_extension_likelihood.item())\n\nif (positive_extension_likelihood.item() > negative_extension_likelihood.item()):\n print(\"\\nPositive text-extension has greater likelihood probabilities!\")\n print(\"The given review can be predicted to be POSITIVE \ud83d\udc4d\")\nelse:\n print(\"\\nNegative text-extension has greater likelihood probabilities!\")\n print(\"The given review can be predicted to be NEGATIVE \ud83d\udc4e\")\n# @markdown **NOTE:** *Run this cell after setting all the fields appropriately!*\n\n# @markdown **NOTE:** *Some pre-trained models might not work well with longer texts!*\n```\n\n---\n# Bonus 2: Light-weight fine-tuning\n\n*Time estimate: ~10mins*\n\n\n```python\n# @title Video 9: Fine-tuning\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1CU4y1n7bV\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"buZLOKdf7Qw\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 9: Fine-tuning')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nFine-tuning these large pre-trained models with billions of parameters tends to be very slow. In this section, we will explore the effect of fine-tuning a few layers (while fixing the others) to save training time.\n\nThe HuggingFace python library provides a simplified API for training and fine-tuning transformer language models. In this exercise we will fine-tune a pre-trained language model for sentiment classification.\n\n##Bonus 2.1: Data Processing\n\nPre-trained transformer models have a fixed vocabulary of words and sub-words. The input text to a transformer model has to be tokenized into these words and sub-words during the pre-processing stage. We'll use the HuggingFace `tokenizers` to perform the tokenization here.\n\n(By default we'll use the BERT base-cased pre-trained language model here. You can try using one of the other models available [here](https://huggingface.co/transformers/pretrained_models.html) by changing the model ID values at appropriate places in the code.)\n\nMost of the pre-trained language models have a fixed maximum sequence length. With the HuggingFace `tokenizer` library, we can either pad or truncate input text sequences to maximum length with a few lines of code:\n\n\n```python\n# Tokenize the input texts\ntokenizer = AutoTokenizer.from_pretrained(\"bert-base-cased\")\ndef tokenize_function(examples):\n return tokenizer(examples[\"text\"], padding=\"max_length\", truncation=True)\n\n# Here we use the `DATASET` as defined above.\n# Recall that DATASET = load_dataset(\"yelp_review_full\")\ntokenized_datasets = DATASET.map(tokenize_function, batched=True)\n```\n\nWe'll randomly sample a subset of the [Yelp reviews dataset](https://huggingface.co/datasets/yelp_review_full) (10k train samples, 5k samples for validation & testing each). You can include more samples here for better performance (at the cost of longer training times!)\n\n\n```python\n# Select the data splits\ntrain_dataset = tokenized_datasets[\"train\"].shuffle(seed=42).select(range(10000))\ntest_dataset = tokenized_datasets[\"test\"].select(range(0,5000))\nvalidation_dataset = tokenized_datasets[\"test\"].select(range(5000, 10000))\n```\n\n## Bonus 2.2: Model Loading\n\nNext, we'll load a pre-trained checkpoint fo the model and decide which layers are to be fine-tuned.\n\nModify the `train_layers` variable below to pick which layers you would like to fine-tune (you can uncomment the print statements for this). Fine-tuning more layers might result in better performance (at the cost of longer training times). Due to computational limitations (limited GPU memory) we cannot fine-tune the entire model.\n\n\n```python\n# Load pre-trained BERT model and freeze layers\nmodel = AutoModelForSequenceClassification.from_pretrained(\"bert-base-cased\",\n num_labels=5)\ntrain_layers = [\"classifier\", \"bert.pooler\", \"bert.encoder.layer.11\"] # add/remove layers here (use layer-name sub-strings)\n\nfor name, param in model.named_parameters():\n if any(x in name for x in train_layers):\n param.requires_grad = True\n # print(\"FINE-TUNING -->\", name)\n else:\n param.requires_grad = False\n # print(\"FROZEN -->\", name)\n```\n\n## Bonus 2.3: Fine-tuning\n\nFine-tune the model! The HuggingFace `Trainer` class supports easy fine-tuning and logging. You can play around with various hyperparameters here!\n\n\n```python\n# Setup huggingface trainer\ntraining_args = TrainingArguments(output_dir=\"yelp_bert\",\n overwrite_output_dir=True,\n evaluation_strategy=\"epoch\",\n per_device_train_batch_size=64,\n per_device_eval_batch_size=64,\n learning_rate=5e-5,\n weight_decay=0.0,\n num_train_epochs=1, # students may use 5 to see a full training!\n fp16=True,\n save_steps=50,\n logging_steps=10,\n report_to=\"tensorboard\"\n )\n```\n\nWe'll use `Accuracy` as the evaluation metric for the sentiment classification task. The HuggingFace `datasets` library supports various metrics. You can try experimenting with other classification metrics here!\n\n\n```python\n# Setup evaluation metric\nmetric = load_metric(\"accuracy\")\ndef compute_metrics(eval_pred):\n logits, labels = eval_pred\n predictions = np.argmax(logits, axis=-1)\n return metric.compute(predictions=predictions, references=labels)\n```\n\nStart the training!\n\n\n```python\n# Instantiate a trainer with training and validation datasets\ntrainer = Trainer(\n model=model,\n args=training_args,\n train_dataset=train_dataset,\n eval_dataset=validation_dataset,\n compute_metrics=compute_metrics,\n)\n```\n\n\n```python\n# Train the model\ntrainer.train()\n```\n\n\n```python\n# Evaluate the model on the test dataset\ntrainer.evaluate(test_dataset)\n```\n\nWe can now visualize the `Tensorboard` logs to analyze the training process! The HuggingFace `Trainer` class will log various loss values and evaluation metrics automatically!\n\n\n```python\n# Visualize the tensorboard logs\n%tensorboard --logdir yelp_bert/runs\n```\n\n---\n# Bonus 3: Model robustness\n\n*Time estimate: ~22mins*\n\n\n```python\n# @title Video 10: Robustness\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=f\"BV1Y54y1E77J\", width=854, height=480, fs=1)\n print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=f\"hJdV2L2t4-c\", width=854, height=480, fs=1, rel=0)\n print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 10: Robustness')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nGiven the previously trained model for sentiment classification, it is possible to decieve it using various text perturbations. The text perturbations can act as previously unseen noise to the model, which might make it give out wrong values of sentiment!\n\n## Bonus Interactive Demo 3: Break the model\n\n\n```python\n# @title Bonus 3.1: Load an original review\n\ndef clean_text(text):\n text = text.replace(\"\\\\n\", \" \")\n text = text.replace(\"\\n\", \" \")\n text = text.replace(\"\\\\\", \" \")\n return text\n\n# @markdown ---\nsample_review_from_yelp = \"Sentiment = 4\" #@param [\"Sentiment = 0\", \"Sentiment = 1\", \"Sentiment = 2\", \"Sentiment = 3\", \"Sentiment = 4\"]\n# @markdown **Randomly sample a response from the Yelp review dataset with the given sentiment value {0:\ud83d\ude20, 1:\ud83d\ude26, 2:\ud83d\ude10, 3:\ud83d\ude42, 4:\ud83d\ude00}**\n\n# @markdown ---\n\ncontext = clean_text(sentiment_dict[sample_review_from_yelp][random.randint(0,len(sentiment_dict[sample_review_from_yelp])-1)][\"text\"])\n\nprint(\"Review for \", sample_review_from_yelp, \":\\n\")\npprint(context)\n```\n\nWe can apply various text perturbations to the selected review using the `textattack` python library. This will help us augment the original text to break the model!\n\n\n```python\n\"\"\"\nAugmenter Class\n===================\n\"\"\"\nfrom textattack.constraints import PreTransformationConstraint\nfrom textattack.shared import AttackedText, utils\n\nclass Augmenter:\n \"\"\"A class for performing data augmentation using TextAttack.\n\n Returns all possible transformations for a given string. Currently only\n supports transformations which are word swaps.\n\n Args:\n transformation (textattack.Transformation): the transformation\n that suggests new texts from an input.\n constraints: (list(textattack.Constraint)): constraints\n that each transformation must meet\n pct_words_to_swap: (float): [0., 1.], percentage of words to swap per augmented example\n transformations_per_example: (int): Maximum number of augmentations\n per input\n \"\"\"\n\n def __init__(\n self,\n transformation,\n constraints=[],\n pct_words_to_swap=0.1,\n transformations_per_example=1,\n ):\n assert (\n transformations_per_example > 0\n ), \"transformations_per_example must be a positive integer\"\n assert 0.0 <= pct_words_to_swap <= 1.0, \"pct_words_to_swap must be in [0., 1.]\"\n self.transformation = transformation\n self.pct_words_to_swap = pct_words_to_swap\n self.transformations_per_example = transformations_per_example\n\n self.constraints = []\n self.pre_transformation_constraints = []\n for constraint in constraints:\n if isinstance(constraint, PreTransformationConstraint):\n self.pre_transformation_constraints.append(constraint)\n else:\n self.constraints.append(constraint)\n\n def _filter_transformations(self, transformed_texts, current_text, original_text):\n \"\"\"Filters a list of ``AttackedText`` objects to include only the ones\n that pass ``self.constraints``.\"\"\"\n for C in self.constraints:\n if len(transformed_texts) == 0:\n break\n if C.compare_against_original:\n if not original_text:\n raise ValueError(\n f\"Missing `original_text` argument when constraint {type(C)} is set to compare against \"\n f\"`original_text` \"\n )\n\n transformed_texts = C.call_many(transformed_texts, original_text)\n else:\n transformed_texts = C.call_many(transformed_texts, current_text)\n return transformed_texts\n\n\n def augment(self, text):\n \"\"\"Returns all possible augmentations of ``text`` according to\n ``self.transformation``.\"\"\"\n attacked_text = AttackedText(text)\n original_text = attacked_text\n all_transformed_texts = set()\n num_words_to_swap = max(\n int(self.pct_words_to_swap * len(attacked_text.words)), 1\n )\n for _ in range(self.transformations_per_example):\n current_text = attacked_text\n words_swapped = len(current_text.attack_attrs[\"modified_indices\"])\n\n while words_swapped < num_words_to_swap:\n transformed_texts = self.transformation(\n current_text, self.pre_transformation_constraints\n )\n\n # Get rid of transformations we already have\n transformed_texts = [\n t for t in transformed_texts if t not in all_transformed_texts\n ]\n\n # Filter out transformations that don't match the constraints.\n transformed_texts = self._filter_transformations(\n transformed_texts, current_text, original_text\n )\n\n # if there's no more transformed texts after filter, terminate\n if not len(transformed_texts):\n break\n\n current_text = random.choice(transformed_texts)\n\n # update words_swapped based on modified indices\n words_swapped = max(\n len(current_text.attack_attrs[\"modified_indices\"]),\n words_swapped + 1,\n )\n all_transformed_texts.add(current_text)\n return sorted([at.printable_text() for at in all_transformed_texts])\n\n\n def augment_many(self, text_list, show_progress=False):\n \"\"\"Returns all possible augmentations of a list of strings according to\n ``self.transformation``.\n\n Args:\n text_list (list(string)): a list of strings for data augmentation\n Returns a list(string) of augmented texts.\n \"\"\"\n if show_progress:\n text_list = tqdm.tqdm(text_list, desc=\"Augmenting data...\")\n return [self.augment(text) for text in text_list]\n\n\n def augment_text_with_ids(self, text_list, id_list, show_progress=True):\n \"\"\"Supplements a list of text with more text data.\n\n Returns the augmented text along with the corresponding IDs for\n each augmented example.\n \"\"\"\n if len(text_list) != len(id_list):\n raise ValueError(\"List of text must be same length as list of IDs\")\n if self.transformations_per_example == 0:\n return text_list, id_list\n all_text_list = []\n all_id_list = []\n if show_progress:\n text_list = tqdm.tqdm(text_list, desc=\"Augmenting data...\")\n for text, _id in zip(text_list, id_list):\n all_text_list.append(text)\n all_id_list.append(_id)\n augmented_texts = self.augment(text)\n all_text_list.extend\n all_text_list.extend([text] + augmented_texts)\n all_id_list.extend([_id] * (1 + len(augmented_texts)))\n return all_text_list, all_id_list\n\n\n def __repr__(self):\n main_str = \"Augmenter\" + \"(\"\n lines = []\n # self.transformation\n lines.append(utils.add_indent(f\"(transformation): {self.transformation}\", 2))\n # self.constraints\n constraints_lines = []\n constraints = self.constraints + self.pre_transformation_constraints\n if len(constraints):\n for i, constraint in enumerate(constraints):\n constraints_lines.append(utils.add_indent(f\"({i}): {constraint}\", 2))\n constraints_str = utils.add_indent(\"\\n\" + \"\\n\".join(constraints_lines), 2)\n else:\n constraints_str = \"None\"\n lines.append(utils.add_indent(f\"(constraints): {constraints_str}\", 2))\n main_str += \"\\n \" + \"\\n \".join(lines) + \"\\n\"\n main_str += \")\"\n return main_str\n```\n\n\n```python\n# @title Bonus 3.2: Augment the original review\n\n# @markdown ---\n# @markdown Word-level Augmentations\nword_swap_contract = True # @param {type:\"boolean\"}\nword_swap_extend = False # @param {type:\"boolean\"}\nword_swap_homoglyph_swap = False # @param {type:\"boolean\"}\n\n# @markdown ---\n# @markdown Character-level Augmentations\nword_swap_neighboring_character_swap = True # @param {type:\"boolean\"}\nword_swap_qwerty = False # @param {type:\"boolean\"}\nword_swap_random_character_deletion = False # @param {type:\"boolean\"}\nword_swap_random_character_insertion = False # @param {type:\"boolean\"}\nword_swap_random_character_substitution = False # @param {type:\"boolean\"}\n# @markdown ---\n\n# @markdown Check all the augmentations that you wish to apply!\n\n# @markdown **NOTE:** *Try applying each augmentation individually, and observe the changes.*\n\n# Apply augmentations\naugmentations = []\nif word_swap_contract:\n augmentations.append(WordSwapContract())\nif word_swap_extend:\n augmentations.append(WordSwapExtend())\nif word_swap_homoglyph_swap:\n augmentations.append(WordSwapHomoglyphSwap())\nif word_swap_neighboring_character_swap:\n augmentations.append(WordSwapNeighboringCharacterSwap())\nif word_swap_qwerty:\n augmentations.append(WordSwapQWERTY())\nif word_swap_random_character_deletion:\n augmentations.append(WordSwapRandomCharacterDeletion())\nif word_swap_random_character_insertion:\n augmentations.append(WordSwapRandomCharacterInsertion())\nif word_swap_random_character_substitution:\n augmentations.append(WordSwapRandomCharacterSubstitution())\n\ntransformation = CompositeTransformation(augmentations)\naugmenter = Augmenter(transformation=transformation,\n transformations_per_example=1)\naugmented_review = clean_text(augmenter.augment(context)[0])\nprint(\"Augmented review:\\n\")\npprint(augmented_review)\n```\n\nWe can now check the predictions for the original text and its augmented version! Try to find the perfect combination of perturbations to break the model! (i.e. model giving incorrect prediction for the augmented text)\n\n\n```python\n# @title Bonus 3.3: Check model predictions\ndef getPrediction(text):\n inputs = tokenizer(text, padding=\"max_length\", truncation=True, return_tensors=\"pt\")\n for key, value in inputs.items():\n inputs[key] = value.to(model.device)\n\n outputs = model(**inputs)\n logits = outputs.logits\n pred = torch.argmax(logits, dim=1)\n return pred.item()\n\nprint(\"original Review:\\n\")\npprint(context)\nprint(\"\\nPredicted Sentiment =\", getPrediction(context))\nprint(\"########################################\")\nprint(\"\\nAugmented Review:\\n\")\npprint(augmented_review)\nprint(\"\\nPredicted Sentiment =\", getPrediction(augmented_review))\nprint(\"########################################\")\n```\n", "meta": {"hexsha": "2820fab43579e97dba82f0d9085e735bc2eff5c7", "size": 433412, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W2D4_AttentionAndTransformers/W2D4_Tutorial1.ipynb", "max_stars_repo_name": "fabxy/course-content-dl", "max_stars_repo_head_hexsha": "d2b4bf8c6d97215184d063c4dd444a99d2767ec9", "max_stars_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-11-30T08:42:05.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-30T08:42:05.000Z", "max_issues_repo_path": "tutorials/W2D4_AttentionAndTransformers/W2D4_Tutorial1.ipynb", "max_issues_repo_name": "fabxy/course-content-dl", "max_issues_repo_head_hexsha": "d2b4bf8c6d97215184d063c4dd444a99d2767ec9", "max_issues_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/W2D4_AttentionAndTransformers/W2D4_Tutorial1.ipynb", "max_forks_repo_name": "fabxy/course-content-dl", "max_forks_repo_head_hexsha": "d2b4bf8c6d97215184d063c4dd444a99d2767ec9", "max_forks_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 56.7590361446, "max_line_length": 17522, "alphanum_fraction": 0.6529422351, "converted": true, "num_tokens": 23210, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.7279754548076477, "lm_q1q2_score": 0.42317438179288525}} {"text": "last edited by Claire Valva on April 22, 2019, with cleanup on July 24, 2019\n\n# Tests simulation that includes phase locking\nthis didn't really work/ phase locking is kind of a nonfactor\n\n\n```python\n#!jupyter nbconvert --to python phase_lock_sim.ipynb\n```\n\n [NbConvertApp] Converting notebook phase_lock_sim.ipynb to python\n [NbConvertApp] Writing 8151 bytes to phase_lock_sim.py\n\n\n\n```python\nimport numpy as np\nfrom netCDF4 import Dataset, num2date \nfrom scipy.signal import get_window, csd\nfrom scipy.fftpack import fft, ifft, fftshift, fftfreq\nimport pandas as pd\nimport datetime\nfrom math import pi\nfrom scipy import optimize\nfrom astropy.stats.circstats import circmean, circvar\nfrom os import walk\nfrom multiprocessing import Pool\nfrom sympy import Poly, Eq, Function, exp, re, im\nimport pickle\nimport time\nimport random\nimport multiprocessing as mp\nfrom joblib import Parallel, delayed\n```\n\n\n```python\n#get detrended parts\nf = []\nfor (dirpath, dirnames, filenames) in walk('detrended/'):\n f.extend(filenames)\n break\n```\n\n\n```python\n# root finders\n#use these\ndef solve_f(X, solution):\n #function to solve f coeff equation for trend analysis\n x,y = X\n f = x*np.exp(1j*y) - solution\n return [np.real(f), np.imag(f)] \n\ndef real_f(X, solution):\n #function to wrap solve_f so that it can be used with fsolve\n x,y = X\n z = [x+0j,y+0j]\n actual_f = solve_f(z, solution)\n return(actual_f)\n\n# solve for phase\ndef root_find(sol):\n return optimize.root(real_f, [np.real(sol), 0], args=sol).x\n```\n\n\n```python\n# get function to generate random coeffs\ndef entry_fft(amp, phase = random.uniform(0, 2*pi)):\n # takes amplitude and phase to give corresponding fourier coeff\n entry = amp*np.exp(1j*phase)\n return entry\n\n# write functions to make a longer ifft\ndef ext_row(row, n):\n ext_f = np.zeros(((len(row) - 1) * n + 1,), dtype=\"complex128\")\n ext_f[::n] = row * n\n \n return ext_f\n\ndef ext_ifft_new(n, input_array):\n # add the zeros onto each end\n ext_f = [ext_row(entry,n) for entry in input_array]\n \n # make up for the formulat multiplying for array length\n olddim = len(input_array[5])\n newdim = len(ext_f[0])\n mult = newdim/olddim\n \n ext_f = np.multiply(ext_f, mult)\n adjusted_tested = np.fft.ifft2(ext_f)\n \n return adjusted_tested\n\n\nseason_titles = [\"Winter\", \"Spring\", \"Summer\", \"Fall\"]\n\n# flatten season for plotting\nflatten = lambda l: [item for sublist in l for item in sublist]\n```\n\n\n```python\nfile_pickle = open(\"spectra_copy/spectra_02_45.0Narr.pickle\", \"rb\")\nd2_touse, d2_seasons, d2_averages = pickle.load(file_pickle, encoding='latin1')\n```\n\n\n```python\nfiled = [\"spectra/spectra_02_45.0Sarr.pickle\", \n \"spectra/spectra_02_45.0Narr.pickle\"]\n```\n\n\n```python\nmean_phases_lock = [[-1.20929458e-16, 1.65918271e-01, -2.17292412e-01, -2.40352609e-01,\n 8.64205449e-02, 1.07202695e-02],[-1.21105919e-16, 3.96836386e-01, -5.77513605e-01, 5.62200988e-01,\n 3.64883992e-01, -7.35447431e-02]]\n```\n\n\n```python\nstds_lock = [[0. , 0.35092645, 0.37481109, 0.36100874, 0.35869798,\n 0.36139656], [0. , 0.22681627, 0.40726111, 0.39961749, 0.37641118,\n 0.36154913]]\n```\n\n\n```python\nfor heythere in range(1):\n# get function to generate random coeffs\n def entry_fft(amp,std,wavenum, phase = random.uniform(0, 2*pi)):\n # takes amplitude and phase to give corresponding fourier coeff\n if np.abs(wavenum) <= 6:\n phase = np.random.normal(loc = mean_phases_lock[ko][wavenum], scale = stds_lock[ko][wavenum])\n \n amp_new = np.random.normal(loc = amp, scale = std)\n entry = amp_new*np.exp(1j*phase)\n return entry\n \n # write functions to make a longer ifft\n def ext_row(row, n):\n ext_f = np.zeros(((len(row) - 1) * n + 1,), dtype=\"complex128\")\n ext_f[::n] = row * n\n \n return ext_f\n\n def ext_ifft_new(n, input_array):\n # add the zeros onto each end\n ext_f = [ext_row(entry,n) for entry in input_array]\n \n # make up for the formulat multiplying for array length\n olddim = len(input_array[5])\n newdim = len(ext_f[0])\n mult = newdim/olddim\n \n # ext_f = np.multiply(mult, ext_f)\n adjusted_tested = np.fft.ifft2(ext_f)\n \n return adjusted_tested\n \n def combined(amps,stds, length):\n # combines generation of random phase with inverse transform\n newarray = [[entry_fft(amp = amps[wave][timed],\n std = stds[wave][timed], wavenum = wave, \n phase = random.uniform(0, 2*pi)) \n for timed in range(len(amps[wave]))]\n for wave in range(len(amps))]\n \n newarray = [np.array(leaf) for leaf in newarray]\n iffted = ext_ifft_new(length, newarray)\n \n return iffted\n```\n\n\n```python\nfor ko in range(2):\n file_pickle = open(filed[ko], \"rb\")\n d2_touse, d2_seasons, d2_averages = pickle.load(file_pickle, encoding='latin1')\n \n \n alled = [[[[root_find(entry) for entry in sublist] \n for sublist in year] \n for year in season] \n for season in d2_seasons]\n phases = roots[:,:,:,1]\n amps = roots[:,:,:,0]\n\n \n def padded(to_pad, index):\n length = len(to_pad)\n if index == 0:\n zeros = longl - length\n to_pad = list(to_pad)\n for i in range(zeros):\n to_pad.append(0)\n return to_pad\n else:\n return to_pad\n\n #pad rows with zeros to account for leap year\n season_amps_adj = [[[padded(row, index = i) \n for row in entry] \n for entry in amps[i]] \n for i in range(4)]\n\n #get average amplitude and phases for each season\n avg_amps = [np.average(season, axis = 0) \n for season in season_amps_adj]\n\n #get average amplitude and phases for each season\n std_amps = [np.std(season, axis = 0) \n for season in season_amps_adj]\n \n def entry_fft(amp,std,wavenum, phase = random.uniform(0, 2*pi)):\n # takes amplitude and phase to give corresponding fourier coeff\n if np.abs(wavenum) <= 6:\n phase = np.random.normal(loc = mean_phases_lock[ko][wavenum], scale = stds_lock[ko][wavenum])\n \n amp_new = np.random.normal(loc = amp, scale = std)\n entry = amp_new*np.exp(1j*phase)\n return entry\n \n # write functions to make a longer ifft\n def ext_row(row, n):\n ext_f = np.zeros(((len(row) - 1) * n + 1,), dtype=\"complex128\")\n ext_f[::n] = row * n\n \n return ext_f\n\n def ext_ifft_new(n, input_array):\n # add the zeros onto each end\n ext_f = [ext_row(entry,n) for entry in input_array]\n \n # make up for the formulat multiplying for array length\n olddim = len(input_array[5])\n newdim = len(ext_f[0])\n mult = newdim/olddim\n \n # ext_f = np.multiply(mult, ext_f)\n adjusted_tested = np.fft.ifft2(ext_f)\n \n return adjusted_tested\n \n def combined(amps,stds, length):\n # combines generation of random phase with inverse transform\n newarray = [[entry_fft(amp = amps[wave][timed],\n std = stds[wave][timed], wavenum = wave, \n phase = random.uniform(0, 2*pi)) \n for timed in range(len(amps[wave]))]\n for wave in range(len(amps))]\n \n newarray = [np.array(leaf) for leaf in newarray]\n iffted = ext_ifft_new(length, newarray)\n \n return iffted\n \n def repeater(season, stds, length, times):\n # repeats the phase creation and inverse transform\n newarray = [combined(season,stds,length) for leaf in range(times)] \n return(newarray)\n \n def repeater_2(amps,stds, length, times):\n #do procedure\n repeated_comp = [repeater(amps[i],stds[i], length, times)\n for i in range(4)]\n \n #output.put(repeated_comp)\n \n \n #listed_parts.append(repeated_comp)\n \n import pickle\n \n \n file_name2 = \"sim_samp/\"\n file_pickle = open(file_name2,'wb') \n pickle.dump(repeated_comp,file_pickle)\n file_pickle.close()\n \n return repeated_comp\n \n \n runlen = 70\n runtimes = 4\n toplot = repeater_2(avg_amps,std_amps, runlen, runtimes) \n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "f1a4b48be9fb0dd2616740dac8328a2c81b063b7", "size": 12937, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lin-assumption-2/phase_lock_sim.ipynb", "max_stars_repo_name": "clairevalva/wavy-sims", "max_stars_repo_head_hexsha": "259c81078e6069777fdef455b0d806e4f8c0c262", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "lin-assumption-2/phase_lock_sim.ipynb", "max_issues_repo_name": "clairevalva/wavy-sims", "max_issues_repo_head_hexsha": "259c81078e6069777fdef455b0d806e4f8c0c262", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "lin-assumption-2/phase_lock_sim.ipynb", "max_forks_repo_name": "clairevalva/wavy-sims", "max_forks_repo_head_hexsha": "259c81078e6069777fdef455b0d806e4f8c0c262", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 31.5536585366, "max_line_length": 118, "alphanum_fraction": 0.501893793, "converted": true, "num_tokens": 2371, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943805178139, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4231724201177841}} {"text": "# \u8d1d\u53f6\u65af\u5411\u91cf\u81ea\u56de\u5f52\n\n**\u53d1\u5e03\u65e5\u671f**: 2019\u5e7410\u670815\u65e5\n\n**\u6587\u7ae0\u4f5c\u8005**: Xinyu Chen (\u9648\u65b0\u5b87) [[GitHub\u4e3b\u9875](https://github.com/xinychen)]\n\n**\u4e0b\u8f7d**: \u672cJupyter Notebook\u53ef\u5728GitHub\u4ed3\u5e93[GraphicalML](https://github.com/mobility-computing/GrapicalML/blob/master/content/bvar.ipynb)\u4e2d\u4e0b\u8f7d\u548c\u4f7f\u7528\u3002\n\n\n## 0 \u5173\u4e8e\u672c\u6587\n\n- \u8ba8\u8bba\u8d1d\u53f6\u65af\u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578b\u7684\u5f62\u5f0f\n- \u4ecb\u7ecd\u5982\u4f55\u5b9e\u73b0\u8d1d\u53f6\u65af\u5411\u91cf\u81ea\u56de\u5f52\n- \u5206\u6790\u8d1d\u53f6\u65af\u5411\u91cf\u81ea\u56de\u5f52\u7684\u5e94\u7528\n\n## 1 \u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578b\n\n\u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578b\u7684\u82f1\u6587\u540d\u79f0\u4e3aVector Autoregressive model\uff0c\u5e38\u88ab\u7b80\u5199\u6210VAR\u3002\u5411\u91cf\u81ea\u56de\u5f52\u7684\u51fa\u73b0\u7531\u6765\u5df2\u4e45\uff0c\u53ef\u4ee5\u8ffd\u6eaf\u5230\u4e0a\u4e2a\u4e16\u7eaa80\u5e74\u4ee3\uff0c\u4eba\u4eec\u6784\u5efa\u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578b\u4e3b\u8981\u51fa\u4e8e\u4ee5\u4e0b\u8003\u8651\uff1a\n\n- \u65f6\u95f4\u5e8f\u5217\u5206\u6790\u4ece\u5355\u4e00\u65f6\u95f4\u5e8f\u5217 (time series data) \u62d3\u5c55\u5230\u4e86\u591a\u5143\u65f6\u95f4\u5e8f\u5217 (multivariate time series)\uff0c\u5728\u4efb\u610f\u7b2c$t$\u4e2a\u65f6\u95f4\u95f4\u9694 (time interval)\uff0c\u89c2\u6d4b\u6837\u672c\u4ece1\u53d8\u6210\u4e86$N$\uff0c\u5176\u4e2d\uff0c$N$\u8868\u793a\u591a\u5143\u65f6\u95f4\u5e8f\u5217\u4e2d\u65f6\u95f4\u5e8f\u5217\u7684\u6570\u91cf\u3002\n- \u6807\u51c6\u7684\u81ea\u56de\u5f52\u6a21\u578b (Autoregressive model, \u7b80\u79f0AR) \u5176\u8868\u8fbe\u5f0f\u8fc7\u4e8e\u7b80\u5355\uff0c\u65e0\u6cd5\u5f88\u597d\u5730\u5728\u591a\u5143\u65f6\u95f4\u5e8f\u5217\u5206\u6790\u4e2d\u53d1\u6325\u4f5c\u7528\u3002\n\n\n### 1.1 \u6807\u51c6\u7684\u81ea\u56de\u5f52\u6a21\u578b\n\n\u5728\u7edf\u8ba1\u5b66\u3001\u7ecf\u6d4e\u5b66\u4e43\u81f3\u4fe1\u53f7\u5904\u7406\u7b49\u9886\u57df\uff0c\u81ea\u56de\u5f52\u6a21\u578b\u88ab\u5e7f\u6cdb\u5e94\u7528\u4e8e\u63cf\u8ff0\u968f\u65f6\u95f4\u53d8\u5316\u7684\u8fc7\u7a0b (\u7b80\u79f0\u65f6\u53d8\u8fc7\u7a0b)\uff0c\u5176\u4e2d\uff0c\u6700\u4e3a\u7ecf\u5178\u7684\u5e94\u7528\u5f53\u5c5e\u65f6\u95f4\u5e8f\u5217\u5206\u6790\uff0c\u5728\u8fd9\u91cc\uff0c\u81ea\u56de\u5f52\u6a21\u578b\u5047\u8bbe\u53d8\u91cf\u4e4b\u95f4\u5b58\u5728\u4e00\u4e2a\u7ebf\u6027\u7684\u4f9d\u8d56\u5173\u7cfb\uff0c\u5373\u8f93\u51fa\u53d8\u91cf (output variables) \u5982$y_t$\u4e0e\u8f93\u5165\u7684\u5386\u53f2\u53d8\u91cf (previous variables) \u5982$y_{t-1},y_{t-2},...$\u5b58\u5728\u4e00\u4e2a\u7ebf\u6027\u8868\u8fbe\u5f0f\u3002\n\n\u4e0d\u59a8\u5148\u770b\u4e00\u4e0b\u6807\u51c6\u7684\u81ea\u56de\u5f52\u6a21\u578b\uff1a\u7ed9\u5b9a\u5355\u4e00\u65f6\u95f4\u5e8f\u5217$\\boldsymbol{y}\\in\\mathbb{R}^{T}$\uff0c\u5176\u65f6\u95f4\u95f4\u9694\u7684\u6570\u91cf\u4e3a$T$\uff0c\u5219\u5bf9\u4e8e\u4efb\u610f\u7b2c$t$\u4e2a\u65f6\u95f4\u95f4\u9694\uff0c\u5b58\u5728\u5982\u4e0b\u7684\u7ebf\u6027\u8868\u8fbe\u5f0f\uff1a\n\\begin{equation}\ny_{t}=\\sum_{k=1}^{d}a_ky_{t-k}+\\epsilon_t,~t=d+1,...,T,\n\\end{equation}\n\u5176\u4e2d\uff0c$a_k,k=1,2,...,d$\u8868\u793a\u56de\u5f52\u7cfb\u6570\uff1b\u5e38\u6570$d$\u8868\u793a\u81ea\u56de\u5f52\u6a21\u578b\u7684\u9636\u6570 (order)\uff0c\u4e5f\u53ef\u4ee5\u5c06$d$\u7b80\u5355\u5730\u7406\u89e3\u6210\u5f53\u524d\u65f6\u95f4\u70b9\u5173\u8054\u8fc7\u53bb\u65f6\u95f4\u70b9\u7684\u6570\u91cf\u3002\n\n\u5728\u81ea\u56de\u5f52\u6a21\u578b\u4e2d\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u4ece\u89c2\u6d4b\u6570\u636e\u4e2d\u5b66\u4e60\u51fa\u53c2\u6570$a_k,k=1,...,d$\u3002\u73b0\u5047\u8bbe\u89c2\u6d4b\u6570\u636e\u4e3a$\\boldsymbol{y}\\in\\mathbb{R}^{T}$\uff0c\u9996\u5148\uff0c\u6211\u4eec\u9700\u8981\u5bf9\u81ea\u56de\u5f52\u6a21\u578b\u7684\u7ebf\u6027\u8868\u8fbe\u5f0f\u8fdb\u884c\u6539\u5199\uff1a\n\\begin{equation}\n\\begin{aligned}\n&y_{t}\\approx\\boldsymbol{a}^\\top{\\boldsymbol{v}}_{t},~t=d+1,...,T, \\\\\n\\Rightarrow&\\boldsymbol{z}\\approx Q\\boldsymbol{a},\n\\end{aligned}\n\\end{equation}\n\u5176\u4e2d\uff0c${\\boldsymbol{v}}_{t}=\\left(y_{t-1},y_{t-2},...,y_{t-d}\\right)\\in\\mathbb{R}^{d}$\uff1b$\\boldsymbol{z}=\\left(y_{d+1},y_{d+2},...,y_{T}\\right)\\in\\mathbb{R}^{T-d}$\uff1b$Q=\\left[\\begin{array}{c}{\\boldsymbol{v}_{d+1}^{\\top}} \\\\ {\\vdots} \\\\ {\\boldsymbol{v}_{T}^{\\top}}\\end{array}\\right] \\in \\mathbb{R}^{(T-d) \\times d}$. \u5728\u8fd9\u91cc\uff0c\u5199\u6210\u8fd9\u79cd\u5f62\u5f0f\u5b8c\u5168\u662f\u4e3a\u4e86\u7b80\u5316\u540e\u7eed\u7684\u63a8\u5bfc\u3002\n\n\u5982\u679c\u8fdb\u4e00\u6b65\u5c06$\\epsilon_t$\u4f5c\u4e3a\u9ad8\u65af\u566a\u58f0\uff0c\u91c7\u7528\u6700\u5c0f\u4e8c\u4e58\u6cd5\uff0c\u5219\u56de\u5f52\u7cfb\u6570$\\boldsymbol{a}$\u7684\u6700\u4f18\u89e3\u4e3a\n\\begin{equation}\n\\begin{aligned}\n\\boldsymbol{a}&=\\text{arg}\\min_{\\boldsymbol{x}}~\\frac{1}{2}\\sum_{t=d+1}^{T}\\left(y_{t}-\\boldsymbol{x}^\\top{\\boldsymbol{v}}_{t}\\right)^2 \\\\\n&=\\text{arg}\\min_{\\boldsymbol{x}}~\\frac{1}{2}\\left(\\boldsymbol{z}-Q\\boldsymbol{x}\\right)^\\top\\left(\\boldsymbol{z}-Q\\boldsymbol{x}\\right) \\\\\n&=\\text{arg}\\min_{\\boldsymbol{x}}~\\frac{1}{2}\\left(\\boldsymbol{x}^\\top Q^\\top Q\\boldsymbol{x}-\\boldsymbol{z}^\\top Q\\boldsymbol{x}-\\boldsymbol{x}^\\top Q^\\top\\boldsymbol{z}\\right) \\\\\n&=\\left(Q^\\top Q\\right)^{-1}Q^\\top\\boldsymbol{z}. \\\\\n\\end{aligned}\n\\end{equation}\n\n\n\u8fd9\u91cc\u91c7\u7528\u6700\u5c0f\u4e8c\u4e58\u6cd5\u5b9e\u9645\u4e0a\u80fd\u6781\u5927\u7a0b\u5ea6\u4e0a\u7b80\u5316\u7b97\u6cd5\u7684\u5b9e\u73b0\u8fc7\u7a0b\uff0c\u65e0\u9700\u8fed\u4ee3\uff0c\u53ea\u9700\u8981\u8f93\u5165\u76f8\u5e94\u7684\u53d8\u91cf$\\boldsymbol{y}$\u548c\u9636\u6570$d$\u5c31\u53ef\u4ee5\u6839\u636e\u56de\u5f52\u7cfb\u6570$\\boldsymbol{a}$\u7684\u6700\u4f18\u89e3\u8fdb\u884c\u8ba1\u7b97\u3002\n\n\n```python\nimport numpy as np\n\ndef ar_model(vec_y, order_d):\n \"\"\"\n \u7528Numpy\u5b9e\u73b0\u81ea\u56de\u5f52\u6a21\u578bAR(d).\n \u8f93\u5165\u53d8\u91cf1\uff1a\u65f6\u95f4\u5e8f\u5217\u5411\u91cfvec_y\uff1b\n \u8f93\u5165\u53d8\u91cf2\uff1a\u81ea\u56de\u5f52\u6a21\u578b\u9636\u6570order_d\uff0c\u53d6\u6b63\u6574\u6570\uff0c\u59821, 2, 3, ..., n.\n \u8f93\u51fa\u53d8\u91cf\uff1a\u81ea\u56de\u5f52\u6a21\u578b\u7684\u7cfb\u6570vec_a.\n \"\"\"\n \n T = vec_y.shape[0]\n time_lags = np.array(list(range(1, order_d + 1)))\n vec_z = vec_y[order_d :] # \u5b9a\u4e49\u5411\u91cfz\n mat_Q = np.zeros((T - order_d, order_d)) # \u5b9a\u4e49\u77e9\u9635Q\n for t in range(T - order_d):\n mat_Q[t, :] = vec_y[t + order_d - time_lags]\n \n return np.matmul(np.matmul(np.linalg.inv(np.matmul(mat_Q.T, mat_Q)), mat_Q.T), vec_z)\n```\n\n### 1.2 \u591a\u5143\u65f6\u95f4\u5e8f\u5217\n\n\u5b9e\u9645\u4e0a\uff0c\u76f8\u6bd4\u5355\u4e00\u7684\u65f6\u95f4\u5e8f\u5217\u6570\u636e\uff0c\u591a\u5143\u65f6\u95f4\u5e8f\u5217\u6570\u636e\u53cd\u800c\u66f4\u4e3a\u5e38\u89c1\uff0c\u662f\u7531\u5355\u4e00\u7684\u65f6\u95f4\u5e8f\u5217\u6784\u6210\uff0c\u5982\u4e0b\u9762\u7684\u77e9\u9635\n\\begin{equation}\nY=\\left[\\begin{array}{ccccc}\ny_{11} & \\cdots & y_{1t} & \\cdots & y_{1T} \\\\\ny_{21} & \\cdots & y_{2t} & \\cdots & y_{2T} \\\\\n\\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\\ny_{N1} & \\cdots & y_{Nt} & \\cdots & y_{NT} \\\\\n\\end{array}\n\\right]\\in\\mathbb{R}^{N\\times T}\n\\end{equation}\n\u5c31\u662f\u4e00\u822c\u5f62\u5f0f\u7684\u591a\u5143\u65f6\u95f4\u5e8f\u5217\u6570\u636e\u3002\u5728\u77e9\u9635$Y$\u4e2d\uff0c\u4efb\u610f\u7b2c$t$\u4e2a\u65f6\u95f4\u95f4\u9694\u4e0b\uff0c\u89c2\u6d4b\u503c\u4e3a\n\\begin{equation}\n\\boldsymbol{y}_{t}=\\left(y_{1t},y_{2t},...,y_{Nt}\\right)^\\top\\in\\mathbb{R}^{N},\n\\end{equation}\n\u89c2\u6d4b\u503c\u7684\u6570\u91cf\u4e3a$N$.\n\n### 1.3 \u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578b\n\n\u9488\u5bf9\u591a\u5143\u65f6\u95f4\u5e8f\u5217\u6570\u636e\uff0c\u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578b\u91c7\u7528\u4e86\u4e00\u79cd\u66f4\u4e3a\u7075\u6d3b\u7684\u65f6\u5e8f\u5efa\u6a21\u7b56\u7565\uff1a\u7ed9\u5b9a\u591a\u5143\u65f6\u95f4\u5e8f\u5217\u6570\u636e\u4e3a$Y\\in\\mathbb{R}^{N\\times T}$\uff0c\u5219\u5bf9\u4e8e\u4efb\u610f\u7b2c$t$\u4e2a\u65f6\u95f4\u95f4\u9694\uff0c\u5b58\u5728\u5982\u4e0b\u7684\u7ebf\u6027\u8868\u8fbe\u5f0f\uff1a\n\\begin{equation}\n\\boldsymbol{y}_{t}=\\sum_{k=1}^{d}A_k\\boldsymbol{y}_{t-k}+\\boldsymbol{\\epsilon}_{t},~t=d+1,...,T,\n\\end{equation}\n\u5176\u4e2d\uff0c$A_k\\in\\mathbb{R}^{N\\times N},k=1,2,...,d$\u8868\u793a\u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578b\u7684\u7cfb\u6570\u77e9\u9635\uff1b$\\boldsymbol{\\epsilon}_t$\u53ef\u89c6\u4e3a\u9ad8\u65af\u566a\u58f0\u3002\n\n\u4e3a\u65b9\u4fbf\u540e\u7eed\u63a8\u5bfc\uff0c\u4e0e\u81ea\u56de\u5f52\u6a21\u578b\u7c7b\u4f3c\uff0c\u4ee4\n\\begin{equation}\nA=\\left[A_{1}, \\ldots, A_{d}\\right]^{\\top} \\in \\mathbb{R}^{(N d) \\times N}, \\quad \\boldsymbol{v}_{t}=\\left[\\begin{array}{c}{\\boldsymbol{y}_{t-{1}}} \\\\ {\\vdots} \\\\ {\\boldsymbol{y}_{t-{d}}}\\end{array}\\right] \\in \\mathbb{R}^{(N d)},\n\\end{equation}\n\u5c06\u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578b\u8fdb\u884c\u6539\u5199\uff1a\n\\begin{equation}\n\\begin{aligned}\n\\boldsymbol{y}_{t}&\\approx \\sum_{k=1}^{d}A_k\\boldsymbol{y}_{t-k}, \\\\\n&=A^\\top\\boldsymbol{v}_{t},~t=d+1,...,T, \\\\\n\\Rightarrow Z&\\approx QA, \\\\\n\\end{aligned}\n\\end{equation}\n\u5176\u4e2d\uff0c\u516c\u5f0f\u4e2d\u7684\u77e9\u9635$Z$\u548c$Q$\u5b9a\u4e49\u5982\u4e0b\uff1a\n\\begin{equation}\nZ=\\left[\\begin{array}{c}{\\boldsymbol{y}_{{d}+1}^{\\top}} \\\\ {\\vdots} \\\\ {\\boldsymbol{y}_{T}^{\\top}}\\end{array}\\right] \\in \\mathbb{R}^{\\left(T-{d}\\right) \\times N}, \\quad Q=\\left[\\begin{array}{c}{\\boldsymbol{v}_{{d}+1}^{\\top}} \\\\ {\\vdots} \\\\ {\\boldsymbol{v}_{T}^{\\top}}\\end{array}\\right] \\in \\mathbb{R}^{(T-d) \\times(N d)}.\n\\end{equation}\n\n\u7531\u6b64\uff0c\u91c7\u7528\u6700\u5c0f\u4e8c\u4e58\u6cd5\uff0c\u7cfb\u6570\u77e9\u9635$A$\u7684\u6700\u4f18\u89e3\u4e3a\n\\begin{equation}\n\\begin{aligned}\nA&=\\text{arg}\\min_{X}~\\frac{1}{2}\\left\\|Z-QX\\right\\|_{F}^{2} \\\\\n&=\\text{arg}\\min_{X}~\\frac{1}{2}\\text{tr}\\left(\\left(Z-QX\\right)^\\top\\left(Z-QX\\right)\\right) \\\\\n&=\\text{arg}\\min_{X}~\\frac{1}{2}\\text{tr}\\left(X^\\top Q^\\top QX-Z^\\top QX-X^\\top Q^\\top Z\\right) \\\\\n&=\\left(Q^\\top Q\\right)^{-1}Q^\\top Z. \\\\\n\\end{aligned}\n\\end{equation}\n\n> \u5728\u8fd9\u91cc\uff0c\u6211\u4eec\u7528\u5230\u4e86F-\u8303\u6570\u4e0e\u77e9\u9635\u8ff9 (trace) \u4e4b\u95f4\u7684\u7b49\u4ef7\u53d8\u6362\uff0c\u5b83\u7684\u610f\u4e49\u662f\u4e3a\u4e86\u65b9\u4fbf\u63a8\u5bfc\uff0c\u5982\u4f55\u7b80\u5355\u7406\u89e3\u8fd9\u79cd\u7b49\u4ef7\u53d8\u6362\u5462\uff1f\u4e3e\u4e00\u4e2a\u4f8b\u5b50\uff1a\u7ed9\u5b9a\u4efb\u610f\u5927\u5c0f\u4e3a$2\\times 2$\u7684\u77e9\u9635$$A=\\left[\\begin{array}{cc} a_{11} & a_{12} \\\\ a_{21} & a_{22} \\\\ \\end{array}\\right]\\in\\mathbb{R}^{2\\times 2},$$\u7531\u4e8eF-\u8303\u6570\u662f\u77e9\u9635\u6240\u6709\u5143\u7d20\u7684\u5e73\u65b9\u548c\u5f00\u6839\u53f7\uff0c\u5373$$\\|A\\|_{F}=\\left(a_{11}^{2}+a_{12}^{2}+a_{21}^{2}+a_{22}^{2}\\right)^{\\frac{1}{2}},$$\u53e6\u5916\uff0c$$A^\\top A=\\left[\\begin{array}{cc} a_{11}^2+a_{21}^2 & a_{11}a_{12}+a_{21}a_{22} \\\\ a_{12}a_{11}+a_{22}a_{21} & a_{12}^{2}+a_{22}^{2} \\\\ \\end{array}\\right],$$\u56e0\u6b64\uff0c\u6839\u636e\u77e9\u9635\u8ff9\u7684\u5b9a\u4e49\uff0c\u6709$$\\text{tr}\\left(A^\\top A\\right)=a_{11}^{2}+a_{12}^{2}+a_{21}^{2}+a_{22}^{2}=\\|A\\|_{F}^2.$$\n\n\n\n\u4e0e\u81ea\u56de\u5f52\u6a21\u578b\u7684\u6c42\u89e3\u8fc7\u7a0b\u7c7b\u4f3c\uff0c\u8fd9\u91cc\u91c7\u7528\u6700\u5c0f\u4e8c\u4e58\u6cd5\u4e5f\u80fd\u6781\u5927\u7a0b\u5ea6\u4e0a\u7b80\u5316\u7b97\u6cd5\u7684\u5b9e\u73b0\u8fc7\u7a0b\uff0c\u65e0\u9700\u8fed\u4ee3\uff0c\u53ea\u9700\u8981\u8f93\u5165\u76f8\u5e94\u7684\u53d8\u91cf$Y$\u548c\u9636\u6570$d$\u5c31\u53ef\u4ee5\u6839\u636e\u7cfb\u6570\u77e9\u9635$A$\u7684\u6700\u4f18\u89e3\u8fdb\u884c\u8ba1\u7b97\u3002\n\n\n```python\nimport numpy as np\n\ndef var_model(mat_Y, order_d, num_pred):\n \"\"\"\n \u7528Numpy\u5b9e\u73b0\u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578bVAR(d).\n \u8f93\u5165\u53d8\u91cf1\uff1a\u591a\u5143\u65f6\u95f4\u5e8f\u5217\u77e9\u9635mat_Y\uff1b\n \u8f93\u5165\u53d8\u91cf2\uff1a\u81ea\u56de\u5f52\u6a21\u578b\u9636\u6570order_d\uff0c\u53d6\u6b63\u6574\u6570\uff0c\u59821, 2, 3, ..., n\uff1b\n \u8f93\u5165\u53d8\u91cf3\uff1a\u81ea\u56de\u5f52\u6a21\u578b\u7684\u9884\u6d4b\u957f\u5ea6num_pred.\n \u8f93\u51fa\u53d8\u91cf1\uff1a\u81ea\u56de\u5f52\u6a21\u578b\u7684\u7cfb\u6570mat_A\uff1b\n \u8f93\u51fa\u53d8\u91cf2\uff1a\u81ea\u56de\u5f52\u6a21\u578b\u7684\u9884\u6d4b\u503cmat_Y_new[:, T:].\n \"\"\"\n \n N, T = mat_Y.shape\n time_lags = np.array(list(range(1, order_d + 1)))\n mat_Z = mat_Y[:, order_d :].T # \u5b9a\u4e49\u77e9\u9635Z\n mat_Q = np.zeros((T - order_d, N * order_d)) # \u5b9a\u4e49\u77e9\u9635Q\n for t in range(T - order_d):\n mat_Q[t, :] = mat_Y[:, t + order_d - time_lags].reshape([N * order_d])\n mat_A = np.matmul(np.matmul(np.linalg.inv(np.matmul(mat_Q.T, mat_Q)), mat_Q.T), mat_Z) # \u8ba1\u7b97\u7cfb\u6570\u77e9\u9635A\n mat_Y_new = np.zeros((N, T + num_pred))\n mat_Y_new[:, : T] = mat_Y\n for t in range(num_pred):\n mat_Y_new[:, t + T] = np.matmul(mat_A.T, mat_Y_new[:, t + T - time_lags].reshape([N * order_d]))\n \n return mat_A, mat_Y_new[:, T :]\n```\n\n### 1.4 \u591a\u5143\u65f6\u95f4\u5e8f\u5217\u9884\u6d4b\n\n\u5f53\u5b58\u5728\u591a\u4e2a\u65f6\u95f4\u5e8f\u5217\uff0c\u4e14\u5b83\u4eec\u4e4b\u95f4\u76f8\u4e92\u5f71\u54cd\u65f6\uff0c\u5219\u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578b\u5c31\u53ef\u4ee5\u4f5c\u4e3a\u5206\u6790\u8fd9\u7c7b\u6570\u636e\u7684\u6709\u6548\u6a21\u578b\u3002\n\n\u4e0d\u8fc7\u4f7f\u7528\u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578b\u65f6\uff0c\u6211\u4eec\u8fd8\u9700\u8981\u5bf9\u53c2\u6570\u6570\u91cf\u6709\u4e00\u4e2a\u5927\u6982\u7684\u4e86\u89e3\u3002\u5f53\u591a\u5143\u65f6\u95f4\u5e8f\u5217\u6570\u636e\u662f\u7531$N$\u4e2a\u5355\u4e00\u65f6\u95f4\u5e8f\u5217\u6784\u6210\u65f6\uff0c\u82e5\u91c7\u7528\u9636\u6570\u4e3a$d$\u7684\u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578b\uff0c\u5219\u53c2\u6570\u6570\u91cf\u4e3a$N^2d$\uff0c\u6216\u8005\u5199\u6210$N\\times (Nd)$\uff1b\u5f53\u7528\u4f5c\u8bad\u7ec3\u7684\u65f6\u95f4\u5e8f\u5217\u957f\u5ea6\u4e3a$T$\u65f6\uff0c\u6211\u4eec\u76f8\u5f53\u4e8e\u62e5\u6709$N\\times T$\u7684\u89c2\u6d4b\u6837\u672c\u7528\u4f5c\u53c2\u6570\u4f30\u8ba1 (\u5b66\u4e60)\u3002\u5728\u8fd9\u91cc\uff0c\u5982\u679c\u91c7\u7528\u6700\u5c0f\u4e8c\u4e58\u6cd5\u5bf9$N\\times (Nd)$\u7684\u53c2\u6570\u8fdb\u884c\u4f30\u8ba1\uff0c\u4e3a\u4e86\u4fdd\u8bc1\u53c2\u6570\u5b66\u4e60\u7684\u6709\u6548\u6027\uff0c\u5728\u8bbe\u7f6e\u9636\u6570$d$\u65f6\u9700\u8981\u6ee1\u8db3\uff1a\n$$Nd\\ll T,$$\n\u5373\u89c2\u6d4b\u6837\u672c\u6570\u91cf\u8981\u8fdc\u5927\u4e8e\u6a21\u578b\u53c2\u6570\u6570\u91cf\u3002\n\n\n\n> \u65f6\u95f4\u5e8f\u5217\u9884\u6d4b\u95ee\u9898\uff08\u56fe\u7247\u6765\u6e90\uff1ahttps://multithreaded.stitchfix.com/blog/2017/02/28/whats-wrong-with-my-time-series/\uff09\u3002\n\n#### 1) \u5e7f\u5dde\u57ce\u5e02\u8def\u7f51\u8f66\u901f\u6570\u636e\u96c6\n\n**\u5173\u4e8e\u6570\u636e\u96c6**\n\n- \u7531214\u6761\u8def\u6bb5\u8f66\u901f\u65f6\u95f4\u5e8f\u5217\u6784\u6210\uff1b\n- \u65f6\u95f4\u95f4\u9694\u5171\u8ba1$61\\times 144=8784$.\n\n**\u9884\u6d4b\u4efb\u52a1**\n\n- \u6eda\u52a8\u9884\u6d4b\u6700\u540e5\u5929$5\\times 144=720$\u4e2a\u65f6\u95f4\u95f4\u9694\u7684\u65f6\u95f4\u5e8f\u5217\uff1b\n- \u5355\u6b65\u6eda\u52a8\u9884\u6d4b (single-step rolling prediction)\uff0c\u6bcf\u6b21\u6eda\u52a8\u7528\u5230\u5386\u53f28\u5468\u6570\u636e\uff1b\n- \u591a\u6b65\u6eda\u52a8\u9884\u6d4b (multi-step rolling prediction)\uff0c\u6bcf\u6b21\u6eda\u52a8\u7528\u5230\u5386\u53f28\u5468\u6570\u636e.\n\n**\u53c2\u6570\u8bbe\u7f6e**\n\n- \u5bf9\u4e8e\u5355\u6b65\u6eda\u52a8\u9884\u6d4b\uff0c\u9636\u6570\u4e3a$d=1,2,3,4,5$\uff1b\n- \u5bf9\u4e8e\u591a\u6b65\u6eda\u52a8\u9884\u6d4b\uff0c\u9884\u6d4b\u65f6\u95f4\u95f4\u9694\u4e3a$2,3,4,5$\uff0c\u9636\u6570\u4e3a$d=1,2,3,4,5$.\n\n**\u6a21\u578b\u8bbe\u7f6e**\n\n- \u7cfb\u6570\u77e9\u9635\u52a8\u6001\u66f4\u65b0\uff0c\u5373\u6bcf\u6b21\u6eda\u52a8\u91cd\u65b0\u4f30\u8ba1\u7cfb\u6570\u77e9\u9635\uff0c\u5e76\u8ba1\u7b97\u76f8\u5e94\u7684\u65f6\u95f4\u5e8f\u5217\u9884\u6d4b\u503c\u3002\n\n**\u6027\u80fd\u8bc4\u4f30**\n\n- MAPE (%),\n- RMSE.\n\n\n```python\nimport scipy.io\n\ntensor = scipy.io.loadmat('../datasets/Guangzhou-data-set/tensor.mat')\ntensor = tensor['tensor']\ndense_mat = tensor.reshape([tensor.shape[0], tensor.shape[1] * tensor.shape[2]])\nX = dense_mat # \u5927\u5c0f\u4e3a214-by-8784\uff0c\u6570\u636e\u96c6\u5b58\u5728\u7f3a\u5931\u6570\u636e\uff0c\u6b64\u5904\u4e0d\u4f5c\u5904\u7406\u3002\n```\n\n\n```python\nimport matplotlib.pyplot as plt\nplt.style.use('ggplot')\n## \u7ed8\u5236\u524d\u4e09\u6761\u65f6\u95f4\u5e8f\u5217\u6700\u540e5\u5929\u7684\u65f6\u95f4\u5e8f\u5217\u6570\u636e\uff1a\nfor i in range(3):\n fig = plt.figure(figsize = (8, 1.5))\n ax = fig.add_axes([0.10, 0.22, 0.85, 0.75])\n plt.plot(X[i, 56 * 144 :], color = \"black\", linewidth = 0.5)\n plt.show()\n```\n\n\n```python\nimport time\nstart = time.time()\n\norder_d = 20\nN = X.shape[0]\npred_steps = 144 * 5\nback_steps = 144 * 7 * 8\nnum_pred = 144\nstart_step = X.shape[1] - pred_steps\nmat_hat = np.zeros((N, pred_steps))\nfor t in range(int(pred_steps / num_pred)):\n if t == 0:\n A, vec = var_model(X[:, 0 : t * num_pred + start_step], order_d, num_pred)\n else:\n A, vec = var_model(X[:, t * num_pred + start_step - back_steps \n : t * num_pred + start_step], order_d, num_pred)\n if num_pred == 1:\n mat_hat[:, t] = vec.reshape(N)\n else:\n mat_hat[:, t * num_pred : (t + 1) * num_pred] = vec\n if (t + 1) % 40 == 0:\n print('The current prediction step is {}.'.format(t + 1))\n\nend = time.time()\nprint('Running time: %d seconds'%(end - start))\n```\n\n Running time: 43 seconds\n\n\n\n```python\nmat = X[:, start_step : X.shape[1]]\nmat0 = X[:, X.shape[0] - pred_steps - 1 : X.shape[0] - 1]\npos = np.where(mat != 0)\nprint('MAPE: {}'.format(np.sum(np.abs(mat[pos] - mat_hat[pos])/mat[pos])/mat[pos].shape[0]))\nprint('RMSE: {}'.format(np.sqrt(np.sum((mat[pos] - mat_hat[pos]) ** 2)/mat[pos].shape[0])))\n```\n\n MAPE: 0.16948262957368235\n RMSE: 6.195235305226982\n\n\n\n```python\nimport matplotlib.pyplot as plt\nplt.style.use('ggplot')\nfor i in range(3):\n fig = plt.figure(figsize = (10, 2))\n ax = fig.add_axes([0.13, 0.28, 0.85, 0.68])\n plt.plot(X[i, 54 * 144 :], color = \"black\", linewidth = 0.5)\n plt.plot(list(range(X.shape[1] - pred_steps - 54 * 144, X.shape[1] - 54 * 144)), \n mat_hat[i, :], color = \"#e3120b\", linewidth = 1.0)\n ax.set_ylim([0, 65])\n```\n\n**\u5355\u6b65\u6eda\u52a8\u9884\u6d4b\u7ed3\u679c** (MAPE/RMSE)\n\n| \u9636\u6570 | $d=1$ | $d=2$ | $d=3$ | $d=4$ | $d=5$ |\n|:-----------|------------:|------------:|------------:|------------:|------------:|\n|`num_pred`=1| 7.22/3.10 | 7.30/3.14 | 7.41/3.17 | 7.52/3.21 | 7.65/3.25 |\n\n\u7ed3\u679c\u5206\u6790\uff1a\u5bf9\u4e8e\u5355\u6b65\u6eda\u52a8\u9884\u6d4b\uff0c\u5f53\u9636\u6570$d$\u589e\u5927\u65f6\uff0c\u9884\u6d4b\u6548\u679c\u4f1a\u9010\u6e10\u53d8\u5dee\uff0c\u6700\u4f73\u7684\u9636\u6570\u4e3a$d=1$\u3002\n\n**\u591a\u6b65\u6eda\u52a8\u9884\u6d4b\u7ed3\u679c** (MAPE/RMSE)\n\n| \u9636\u6570 | $d=1$ | $d=2$ | $d=3$ | $d=4$ | $d=5$ |\n|:-------------|------------:|------------:|------------:|------------:|------------:|\n|`num_pred`=2 | 8.27/3.54 | 8.36/3.61 | 8.46/3.65 | 8.57/3.67 | 8.71/3.71 |\n|`num_pred`=3 | 9.05/3.88 | 9.14/3.98 | 9.21/4.01 | 9.33/4.04 | 9.45/4.07 |\n|`num_pred`=4 | 9.51/3.96 | 9.53/3.94 | 9.60/3.98 | 9.70/4.01 | 9.84/4.06 |\n|`num_pred`=5 | 10.06/4.18 | 10.06/4.15 | 10.08/4.16 | 10.16/4.18 | 10.29/4.24 |\n|`num_pred`=144| 23.77/8.32 | 22.61/7.96 | 21.40/7.57 | 20.67/7.34 | 20.24/7.21 |\n\n\u7ed3\u679c\u5206\u6790\uff1a\u5f53\u9884\u6d4b\u7684\u65f6\u95f4\u95f4\u9694\u5f88\u5c0f\u65f6\uff0cVAR(1)\u4fbf\u80fd\u53d6\u5f97\u6700\u4f73\u7684\u9884\u6d4b\u7ed3\u679c\uff1b\u968f\u7740\u9884\u6d4b\u7684\u65f6\u95f4\u95f4\u9694\u589e\u5927\uff0c\u6700\u4f73\u7684\u9636\u6570\u4e5f\u4f1a\u968f\u4e4b\u589e\u5927\uff0c\u4f8b\u5982\uff0c\u9884\u6d4b\u672a\u67655\u4e2a\u65f6\u95f4\u95f4\u9694\u7684\u65f6\u95f4\u5e8f\u5217\u65f6\uff0cVAR(2)\u53d6\u5f97\u6700\u4f73\u7684\u9884\u6d4b\u7ed3\u679c\uff0c\u6548\u679c\u4f18\u4e8eVAR(1)\u3002\n\n#### 2) \u676d\u5dde\u5730\u94c1\u5ba2\u6d41\u91cf\u6570\u636e\u96c6\n\n**\u5173\u4e8e\u6570\u636e\u96c6**\n\n- \u753180\u4e2a\u5730\u94c1\u7ad9\u7684\u5165\u7ad9\u5ba2\u6d41\u91cf\u65f6\u95f4\u5e8f\u5217\u6784\u6210\uff1b\n- \u65f6\u95f4\u95f4\u9694\u4e3a10\u5206\u949f\uff0c\u5171\u8ba1$25\\times 108=2700$\u4e2a (24:00\u81f36:00\u4e4b\u95f4\u4e0d\u5728\u670d\u52a1\u65f6\u95f4\uff0c\u5df2\u7ecf\u5254\u966425\u5929\u4e2d\u7684\u8be5\u65f6\u6bb5\u6570\u636e\uff0c\u56e0\u6b64\uff0c\u6bcf\u5929\u7684\u65f6\u95f4\u95f4\u9694\u4e2a\u6570\u4e3a108)\u3002\n\n**\u9884\u6d4b\u4efb\u52a1**\n\n- \u6eda\u52a8\u9884\u6d4b\u6700\u540e5\u5929$5\\times 108=540$\u4e2a\u65f6\u95f4\u95f4\u9694\u7684\u65f6\u95f4\u5e8f\u5217\uff1b\n- \u5355\u6b65\u6eda\u52a8\u9884\u6d4b (single-step rolling prediction)\uff0c\u6bcf\u6b21\u6eda\u52a8\u7528\u5230\u5386\u53f22\u5468\u6570\u636e\uff1b\n- \u591a\u6b65\u6eda\u52a8\u9884\u6d4b (multi-step rolling prediction)\uff0c\u6bcf\u6b21\u6eda\u52a8\u7528\u5230\u5386\u53f22\u5468\u6570\u636e.\n\n**\u53c2\u6570\u8bbe\u7f6e**\n\n- \u5bf9\u4e8e\u5355\u6b65\u6eda\u52a8\u9884\u6d4b\uff0c\u9636\u6570\u4e3a$d=1,2,3,4,5$\uff1b\n- \u5bf9\u4e8e\u591a\u6b65\u6eda\u52a8\u9884\u6d4b\uff0c\u9884\u6d4b\u65f6\u95f4\u95f4\u9694\u4e3a$2,3,4,5$\uff0c\u9636\u6570\u4e3a$d=1,2,3,4,5$.\n\n**\u6a21\u578b\u8bbe\u7f6e**\n\n- \u7cfb\u6570\u77e9\u9635\u52a8\u6001\u66f4\u65b0\uff0c\u5373\u6bcf\u6b21\u6eda\u52a8\u91cd\u65b0\u4f30\u8ba1\u7cfb\u6570\u77e9\u9635\uff0c\u5e76\u8ba1\u7b97\u76f8\u5e94\u7684\u65f6\u95f4\u5e8f\u5217\u9884\u6d4b\u503c\u3002\n\n**\u6027\u80fd\u8bc4\u4f30**\n\n- MAPE (%),\n- RMSE.\n\n\n```python\nimport scipy.io\n\ntensor = scipy.io.loadmat('../datasets/Hangzhou-data-set/tensor.mat')\ntensor = tensor['tensor']\ndense_mat = tensor.reshape([tensor.shape[0], tensor.shape[1] * tensor.shape[2]])\nX = dense_mat # \u5927\u5c0f\u4e3a80-by-2700\uff0c\u6570\u636e\u96c6\u4e0d\u5b58\u5728\u7f3a\u5931\u6570\u636e\n```\n\n\n```python\nimport time\nstart = time.time()\n\norder_d = 3\nN = X.shape[0]\npred_steps = 108 * 5\nback_steps = 108 * 7 * 2\nnum_pred = 5\nstart_step = X.shape[1] - pred_steps\nmat_hat = np.zeros((N, pred_steps))\nfor t in range(int(pred_steps/num_pred)):\n if t == 0:\n A, vec = var_model(X[:, 0 : t * num_pred + start_step], order_d, num_pred)\n else:\n A, vec = var_model(X[:, t * num_pred + start_step - back_steps \n : t * num_pred + start_step], order_d, num_pred)\n if num_pred == 1:\n mat_hat[:, t] = vec.reshape(N)\n else:\n mat_hat[:, t * num_pred : (t + 1) * num_pred] = vec\n if (t + 1) % 40 == 0:\n print('The current prediction step is {}.'.format(t + 1))\n\nend = time.time()\nprint('Running time: %d seconds'%(end - start))\n```\n\n The current prediction step is 40.\n The current prediction step is 80.\n Running time: 1 seconds\n\n\n\n```python\nmat = X[:, start_step : X.shape[1]]\nmat0 = X[:, X.shape[1] - pred_steps - 1 : X.shape[1] - 1]\npos = np.where(mat != 0)\nprint('MAPE: {}'.format(np.sum(np.abs(mat[pos] - mat_hat[pos])/mat[pos])/mat[pos].shape[0]))\nprint('RMSE: {}'.format(np.sqrt(np.sum((mat[pos] - mat_hat[pos]) ** 2)/mat[pos].shape[0])))\n```\n\n MAPE: 0.33951430128760296\n RMSE: 39.68455364476149\n\n\n\n```python\nimport matplotlib.pyplot as plt\nplt.style.use('ggplot')\nfor i in range(5):\n fig = plt.figure(figsize = (8, 2))\n ax = fig.add_axes([0.13, 0.28, 0.85, 0.68])\n plt.plot(X[i, 18 * 108 :], color = \"black\", linewidth = 0.5)\n plt.plot(list(range(X.shape[1] - pred_steps - 18 * 108, X.shape[1] - 18 * 108)), \n mat_hat[i, :], color = \"#e3120b\", linewidth = 1.0)\n```\n\n**\u5355\u6b65\u6eda\u52a8\u9884\u6d4b\u7ed3\u679c** (MAPE/RMSE)\n\n| \u9636\u6570 | $d=1$ | $d=2$ | $d=3$ | $d=4$ | $d=5$ |\n|:-----------|------------:|------------:|------------:|------------:|------------:|\n|`num_pred`=1| 23.55/33.98 | 24.26/32.56 | 24.49/32.12 | 25.62/32.64 | 27.68/33.39 |\n\n\u7ed3\u679c\u5206\u6790\uff1a\u5bf9\u4e8e\u5355\u6b65\u6eda\u52a8\u9884\u6d4b\uff0c\u5f53\u9636\u6570$d$\u589e\u5927\u65f6\uff0c\u9884\u6d4b\u6548\u679c\u4f1a\u9010\u6e10\u53d8\u5dee\uff0c\u4eceMAPE\u6307\u6807\u6765\u770b\uff0c\u6700\u4f73\u7684\u9636\u6570\u4e3a$d=1$\uff1b\u4eceRMSE\u6307\u6807\u6765\u770b\uff0c\u6700\u4f73\u7684\u9636\u6570\u4e3a$d=3$\u3002\n\n**\u591a\u6b65\u6eda\u52a8\u9884\u6d4b\u7ed3\u679c** (MAPE/RMSE)\n\n| \u9636\u6570 | $d=1$ | $d=2$ | $d=3$ | $d=4$ | $d=5$ |\n|:-------------|------------:|------------:|------------:|------------:|------------:|\n|`num_pred`=2 | 26.26/37.09 | 26.26/35.10 | 26.47/34.27 | 27.73/34.56 | 29.80/35.14 |\n|`num_pred`=3 | 29.22/39.19 | 28.48/36.41 | 28.66/35.76 | 29.45/36.11 | 31.89/36.70 |\n|`num_pred`=4 | 34.09/42.71 | 33.01/39.65 | 31.77/38.36 | 32.15/38.40 | 35.49/38.88 |\n|`num_pred`=5 | 36.86/44.32 | 34.85/40.19 | 33.95/39.68 | 34.31/39.71 | 37.02/40.34 |\n\n\u7ed3\u679c\u5206\u6790\uff1a\u9884\u6d4b\u7684\u65f6\u95f4\u95f4\u9694\u8d8a\u957f\uff0c\u9700\u8981\u7684\u9636\u6570\u5219\u5f80\u5f80\u8d8a\u5927\uff0c\u4f46\u9636\u6570\u5e76\u975e\u8d8a\u5927\u8d8a\u597d\uff0c\u8fc7\u5927\u4f1a\u8fdb\u4e00\u6b65\u5bfc\u81f4\u9884\u6d4b\u6548\u679c\u53d8\u5dee\u3002\n\n## 2 \u8d1d\u53f6\u65af\u5411\u91cf\u81ea\u56de\u5f52\n\n\u5bf9\u4e8e\u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578b\u800c\u8a00\uff0c\u4e0d\u7ba1\u91c7\u7528\u600e\u6837\u7684\u6c42\u89e3\u65b9\u6cd5\uff0c\u5176\u6c42\u89e3\u8fc7\u7a0b\u4e2d\u90fd\u4f1a\u4f34\u968f\u7740\u4e00\u5b9a\u6570\u91cf\u7684\u5f85\u4f30\u8ba1\u53c2\u6570\uff0c\u56e0\u6b64\uff0c\u5c06\u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578b\u7528\u4e8e\u5927\u89c4\u6a21\u591a\u5143\u65f6\u95f4\u5e8f\u5217\u6570\u636e\u65f6\uff0c\u4e3a\u907f\u514d\u53c2\u6570\u4f30\u8ba1\u51fa\u73b0\u8fc7\u62df\u5408\u73b0\u8c61\uff0c\u5bf9\u53c2\u6570\u8bbe\u7f6e\u5148\u9a8c\u5206\u5e03\u4e0d\u5931\u4e3a\u4e00\u79cd\u6709\u6548\u7684\u7b56\u7565\u3002\u9664\u6b64\u4e4b\u5916\uff0c\u5305\u62ecGibbs\u91c7\u6837\u5728\u5185\u7684\u4f17\u591a\u8d1d\u53f6\u65af\u63a8\u65ad\u7b97\u6cd5\u65e2\u80fd\u63d0\u4f9b\u6709\u6548\u7684\u53c2\u6570\u4f30\u8ba1\uff0c\u540c\u65f6\u53c8\u80fd\u523b\u753b\u53c2\u6570\u4f30\u8ba1\u503c\u7684\u4e0d\u786e\u5b9a\u6027 (uncertainty)\u3002\n\n### 2.1 \u56de\u987e\u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578b\n\n\u9488\u5bf9\u591a\u5143\u65f6\u95f4\u5e8f\u5217\u6570\u636e\uff0c\u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578b\u91c7\u7528\u4e86\u4e00\u79cd\u7075\u6d3b\u7684\u65f6\u5e8f\u5efa\u6a21\u7b56\u7565\uff1a\u7ed9\u5b9a\u591a\u5143\u65f6\u95f4\u5e8f\u5217\u6570\u636e\u4e3a$Y\\in\\mathbb{R}^{N\\times T}$\uff0c\u5219\u5bf9\u4e8e\u4efb\u610f\u7b2c$t$\u4e2a\u65f6\u95f4\u95f4\u9694\uff0c\u5b58\u5728\u5982\u4e0b\u7684\u7ebf\u6027\u8868\u8fbe\u5f0f\uff1a\n\\begin{equation}\n\\boldsymbol{y}_{t}=\\sum_{k=1}^{d}A_k\\boldsymbol{y}_{t-k}+\\boldsymbol{\\epsilon}_{t},~t=d+1,...,T,\n\\end{equation}\n\u5176\u4e2d\uff0c$A_k\\in\\mathbb{R}^{N\\times N},k=1,2,...,d$\u8868\u793a\u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578b\u7684\u7cfb\u6570\u77e9\u9635\uff1b$\\boldsymbol{\\epsilon}_t$\u53ef\u89c6\u4e3a\u9ad8\u65af\u566a\u58f0\u3002\n\n\u4ee4\n\\begin{equation}\nA=\\left[A_{1}, \\ldots, A_{d}\\right]^{\\top} \\in \\mathbb{R}^{(N d) \\times N}, \\quad \\boldsymbol{v}_{t}=\\left[\\begin{array}{c}{\\boldsymbol{y}_{t-{1}}} \\\\ {\\vdots} \\\\ {\\boldsymbol{y}_{t-{d}}}\\end{array}\\right] \\in \\mathbb{R}^{(N d)},\n\\end{equation}\n\u5c06\u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578b\u8fdb\u884c\u6539\u5199\uff1a\n\\begin{equation}\n\\begin{aligned}\n\\boldsymbol{y}_{t}&\\approx \\sum_{k=1}^{d}A_k\\boldsymbol{y}_{t-k}, \\\\\n&=A^\\top\\boldsymbol{v}_{t},~t=d+1,...,T, \\\\\n\\Rightarrow Z&\\approx QA, \\\\\n\\end{aligned}\n\\end{equation}\n\u5176\u4e2d\uff0c\u516c\u5f0f\u4e2d\u7684\u77e9\u9635$Z$\u548c$Q$\u5b9a\u4e49\u5982\u4e0b\uff1a\n\\begin{equation}\nZ=\\left[\\begin{array}{c}{\\boldsymbol{y}_{{d}+1}^{\\top}} \\\\ {\\vdots} \\\\ {\\boldsymbol{y}_{T}^{\\top}}\\end{array}\\right] \\in \\mathbb{R}^{\\left(T-{d}\\right) \\times N}, \\quad Q=\\left[\\begin{array}{c}{\\boldsymbol{v}_{{d}+1}^{\\top}} \\\\ {\\vdots} \\\\ {\\boldsymbol{v}_{T}^{\\top}}\\end{array}\\right] \\in \\mathbb{R}^{(T-d) \\times(N d)}.\n\\end{equation}\n\n\n\n### 2.2 \u77e9\u9635\u6b63\u6001\u5206\u5e03\n\n\u5728\u4f17\u591a\u7edf\u8ba1\u5206\u5e03\u4e2d\uff0c\u6b63\u6001\u5206\u5e03 (\u9ad8\u65af\u5206\u5e03) \u662f\u6211\u4eec\u7684\u5f88\u65e9\u5c31\u63a5\u89e6\u5230\u7684\u6982\u7387\u5206\u5e03\uff0c\u5c06\u5176\u5f62\u5f0f\u7528\u4e8e\u4ee5\u5411\u91cf\u4e3a\u968f\u673a\u53d8\u91cf$\\boldsymbol{x} \\in \\mathbb{R}^{m}$\uff0c\u4fbf\u5f62\u6210\u4e86\u6211\u4eec\u5728\u7ebf\u6027\u4ee3\u6570\u3001\u6982\u7387\u8bba\u7b49\u76f8\u5173\u8bfe\u7a0b\u4e2d\u5b66\u5230\u7684\u591a\u5143\u6b63\u6001\u5206\u5e03 (multivariate normal distribution)\uff0c\u5176\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4e3a\n\n\\begin{equation}\n\\begin{aligned}\n&\\mathcal{N}(\\boldsymbol{x} | \\boldsymbol{\\mu}, \\Sigma)=(2 \\pi)^{-m / 2}|\\Sigma|^{-1 / 2} \\exp \\left(-\\frac{1}{2}(\\boldsymbol{x}-\\boldsymbol{\\mu})^{\\top} \\Sigma^{-1}(\\boldsymbol{x}-\\boldsymbol{\\mu})\\right) \\\\ =&(2 \\pi)^{-m / 2}|\\Sigma|^{-1 / 2} \\exp \\left(-\\frac{1}{2} \\operatorname{tr}\\left[(\\boldsymbol{x}-\\boldsymbol{\\mu})(\\boldsymbol{x}-\\boldsymbol{\\mu})^{\\top} \\Sigma^{-1}\\right]\\right) \\\\\n\\end{aligned}\n\\end{equation}\n\u5176\u4e2d\uff0c$\\boldsymbol{\\mu} \\in \\mathbb{R}^{m}$\u8868\u793a\u591a\u5143\u6b63\u6001\u5206\u5e03\u7684\u5747\u503c\u5411\u91cf\uff1b$\\Sigma$\u5219\u8868\u793a\u534f\u65b9\u5dee\u77e9\u9635\u3002\n\n\u9700\u8981\u8bf4\u660e\u7684\u662f\uff0c\u8fd9\u91cc\u5c06\u591a\u5143\u6b63\u6001\u5206\u5e03\u7684\u6307\u6570\u9879\u5199\u6210\u77e9\u9635\u8ff9 (trace) \u7684\u5f62\u5f0f\u662f\u4e3a\u4e86\u65b9\u9762\u540e\u7eed\u8ba4\u8bc6\u77e9\u9635\u6b63\u6001\u5206\u5e03\uff0c\u5176\u4e2d\uff0c\u5728\u591a\u5143\u6b63\u6001\u5206\u5e03\u7684\u5199\u6cd5\u4e2d\uff0c$(\\boldsymbol{x}-\\boldsymbol{\\mu})^{\\top} \\Sigma^{-1}(\\boldsymbol{x}-\\boldsymbol{\\mu})=\\operatorname{tr}\\left[(\\boldsymbol{x}-\\boldsymbol{\\mu})(\\boldsymbol{x}-\\boldsymbol{\\mu})^{\\top} \\Sigma^{-1}\\right]$\u662f\u6052\u6210\u7acb\u7684\u3002\n\n\u5728\u591a\u5143\u6b63\u6001\u5206\u5e03\u7684\u57fa\u7840\u4e0a\uff0c\u5b9e\u9645\u4e0a\u8fd8\u5b58\u5728\u4e00\u79cd\u6b63\u6001\u5206\u5e03\uff0c\u5b83\u662f\u4ee5\u77e9\u9635\u4e3a\u968f\u673a\u53d8\u91cf\uff0c\u82e5\u968f\u673a\u77e9\u9635$X\\in\\mathbb{R}^{m\\times n}$\u670d\u4ece\u77e9\u9635\u6b63\u6001\u5206\u5e03\uff0c\u5219\u5176\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4e3a\n\n\\begin{equation}\n\\begin{aligned}\n&\\mathcal{M} \\mathcal{N}_{m \\times n}(X | M, U, V) \\\\ =&(2 \\pi)^{-m n / 2}|V|^{-m / 2}|U|^{-n / 2} \\exp \\left(-\\frac{1}{2} \\operatorname{tr}\\left[V^{-1}(X-M)^{\\top} U^{-1}(X-M)\\right]\\right)\n\\end{aligned}\n\\end{equation}\n\u5176\u4e2d\uff0c\u7b26\u53f7$\\mathcal{M N}_{m \\times n}(\\cdot)$\u6765\u81ea\u4e8e\u77e9\u9635\u6b63\u6001\u5206\u5e03 (matrix normal distribution) \u82f1\u6587\u9996\u5b57\u6bcd\u7684\u7b80\u5199\uff0c\u4e0b\u6807\u6307\u4ee3\u968f\u673a\u77e9\u9635\u7684\u5927\u5c0f\uff1b\u77e9\u9635$M \\in \\mathbb{R}^{m \\times n}$\uff0c\u4e0e\u968f\u673a\u77e9\u9635$X$\u5927\u5c0f\u76f8\u540c\uff0c\u5bf9\u5e94\u4e8e\u5747\u503c\u9879\uff1b\u77e9\u9635$U \\in \\mathbb{R}^{m \\times m}$\u3001$V \\in \\mathbb{R}^{n \\times n}$\u5bf9\u5e94\u4e8e\u534f\u65b9\u5dee\u77e9\u9635\u3002\n\n> \u6ce8\uff1a\u5173\u4e8e\u77e9\u9635\u6b63\u6001\u5206\u5e03\u66f4\u4e3a\u8be6\u7ec6\u7684\u4ecb\u7ecd\u53ef\u53c2\u8003[\u7edf\u8ba1\u5b66\u4e60 | \u77e9\u9635\u6b63\u6001\u5206\u5e03 (matrix normal distribution)\n](https://zhuanlan.zhihu.com/p/73585133)\u3002\n\n### 2.3 \u8d1d\u53f6\u65af\u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578b\n\n### 2.4 \u53c2\u6570\u7684\u540e\u9a8c\u5206\u5e03\u4e0eGibbs\u91c7\u6837\n\n\n```python\nimport numpy as np\nfrom numpy.linalg import inv as inv\nfrom scipy.stats import invwishart\nfrom scipy.stats import wishart\nfrom numpy.random import multivariate_normal as mvnrnd\n```\n\n\n```python\ndef mnrnd(M, U, V):\n \"\"\"\n Generate matrix normal distributed random matrix.\n M is a m-by-n matrix, U is a m-by-m matrix, and V is a n-by-n matrix.\n \"\"\"\n dim1, dim2 = M.shape\n X0 = np.random.rand(dim1, dim2)\n P = np.linalg.cholesky(U)\n Q = np.linalg.cholesky(V)\n return M + np.matmul(np.matmul(P, X0), Q.T)\n```\n\n\n```python\ndef sampling_MNIW(mat_Z, mat_Q, M0, Psi0, S0, nu0):\n \n var_Psi = inv(inv(Psi0) + np.matmul(mat_Q.T, mat_Q)) # \u540e\u9a8c\u53c2\u6570Psi\n var_M = np.matmul(var_Psi, np.matmul(inv(Psi0), M0) + np.matmul(mat_Q.T, mat_Z)) # \u540e\u9a8c\u53c2\u6570M\n var_S = (S0 + np.matmul(mat_Z.T, mat_Z)# + np.matmul(np.matmul(M0.T, inv(Psi0)), M0)\n - np.matmul(np.matmul(var_M.T, inv(var_Psi)), var_M)) # \u540e\u9a8c\u53c2\u6570S\n var_nu = nu0 + mat_Z.shape[0] # \u540e\u9a8c\u53c2\u6570nu\n Sigma = invwishart(df = var_nu, scale = var_S, seed = None).rvs() # \u7528inv-Wishart\u540e\u9a8c\u5206\u5e03\u5bf9Sigma\u91c7\u6837\n mat_A = mnrnd(var_M, var_Psi, Sigma) # \u7528matrix norm distribution\u540e\u9a8c\u5206\u5e03\u5bf9\u7cfb\u6570\u77e9\u9635A\u91c7\u6837\n return Sigma, mat_A\n```\n\n\n```python\ndef bvar_model(mat_Y, mat_Y_new, order_d, num_pred, num_rolling, burn_iter, gibbs_iter):\n \"\"\"\n \u7528Numpy\u5b9e\u73b0\u8d1d\u53f6\u65af\u5411\u91cf\u81ea\u56de\u5f52\u6a21\u578bBVAR(d).\n \u8f93\u5165\u53d8\u91cf1\uff1a\u591a\u5143\u65f6\u95f4\u5e8f\u5217\u77e9\u9635mat_Y\uff1b\n \u8f93\u5165\u53d8\u91cf2\uff1a\u6eda\u52a8\u9884\u6d4b\u8f93\u5165\u77e9\u9635mat_Y_new\uff1b\n \u8f93\u5165\u53d8\u91cf3\uff1a\u81ea\u56de\u5f52\u6a21\u578b\u9636\u6570order_d\uff0c\u53d6\u6b63\u6574\u6570\uff0c\u59821, 2, 3, ..., n\uff1b\n \u8f93\u5165\u53d8\u91cf4\uff1a\u81ea\u56de\u5f52\u6a21\u578b\u7684\u9884\u6d4b\u957f\u5ea6num_pred\uff1b\n \u8f93\u5165\u53d8\u91cf5\uff1a\u81ea\u56de\u5f52\u6a21\u578b\u7684\u6eda\u52a8\u9884\u6d4b\u6b21\u6570num_rolling\uff1b\n \u8f93\u5165\u53d8\u91cf6\uff1aGibbs\u91c7\u6837\u7684\u71c3\u70e7\u671f\u8fed\u4ee3\u6b21\u6570burn_iter\uff1b\n \u8f93\u5165\u53d8\u91cf7\uff1aGibbs\u91c7\u6837\u7684\u91c7\u6837\u8fed\u4ee3\u6b21\u6570gibbs_iter.\n \u8f93\u51fa\u53d8\u91cf1\uff1a\u81ea\u56de\u5f52\u6a21\u578b\u7684\u7cfb\u6570mat_A\uff1b\n \u8f93\u51fa\u53d8\u91cf2\uff1a\u81ea\u56de\u5f52\u6a21\u578b\u7684\u9884\u6d4b\u503cmat_Y_new[:, T:].\n \"\"\"\n \n N, T = mat_Y.shape\n time_lags = np.array(list(range(1, order_d + 1)))\n mat_Z = mat_Y[:, order_d :].T # \u5b9a\u4e49\u77e9\u9635Z\n mat_Q = np.zeros((T - order_d, N * order_d)) # \u5b9a\u4e49\u77e9\u9635Q\n for t in range(T - order_d):\n mat_Q[t, :] = mat_Z[t - time_lags, :].reshape([N * order_d])\n\n M0 = np.zeros((N * order_d, N))\n Psi0 = np.eye(N * order_d)\n S0 = np.eye(N)\n nu0 = N\n result = [] # \u4fdd\u5b58\u5404\u53d8\u91cf\u5728\u5404\u4ee3\u4e2d\u7684Gibbs\u91c7\u6837\u503c\n result.append(np.zeros((N, num_rolling * num_pred, gibbs_iter))) # \u4fdd\u5b58\u591a\u5143\u65f6\u95f4\u5e8f\u5217\u7684\u9884\u6d4b\u503c\n result.append(np.zeros((N * order_d, N, gibbs_iter))) # \u4fdd\u5b58\u7cfb\u6570\u77e9\u9635A\u7684\u91c7\u6837\u503c\n \n for it in range(burn_iter + gibbs_iter):\n Sigma, mat_A = sampling_MNIW(mat_Z, mat_Q, M0, Psi0, S0, nu0)\n if it >= burn_iter:\n for t0 in range(num_rolling):\n if t0 >= 1:\n mat_Z_new = np.append(mat_Z, mat_Y_new[:, (t0 - 1) * num_pred : t0 * num_pred].T, axis = 0)\n mat_Q_new = np.append(mat_Q, np.zeros((num_pred, N * order_d)), axis = 0)\n for tt in range(num_pred):\n mat_Q_new[tt - num_pred, :] = mat_Z_new[tt - num_pred - time_lags].reshape([N * order_d])\n mat_Z = mat_Z_new.copy()\n mat_Q = mat_Q_new.copy()\n result[1][:, :, it - burn_iter] = mat_A\n for t in range(num_pred):\n if t == 0:\n mat_Q_sample = mat_Q.copy()\n else:\n mat_Q_sample = np.append(mat_Q_sample, vec.reshape([1, N * order_d]), axis = 0)\n vec0 = mvnrnd(np.matmul(mat_A.T, mat_Q_sample[t0 * num_pred + t + T - order_d - 1, :]), Sigma)\n result[0][:, t0 * num_pred + t, it - burn_iter] = vec0\n vec = np.append(vec0, mat_Q_sample[-1, N :])\n if (it + 1) % 100 == 0:\n print(it + 1)\n \n return result\n```\n\n### 2.5 \u591a\u5143\u65f6\u95f4\u5e8f\u5217\u9884\u6d4b\n\n\n```python\nimport scipy.io\n\ntensor = scipy.io.loadmat('../datasets/Hangzhou-data-set/tensor.mat')\ntensor = tensor['tensor']\ndense_mat = tensor.reshape([tensor.shape[0], tensor.shape[1] * tensor.shape[2]])\nmax_const = np.max(dense_mat)\nX = dense_mat / max_const # \u5927\u5c0f\u4e3a80-by-2700\uff0c\u6570\u636e\u96c6\u4e0d\u5b58\u5728\u7f3a\u5931\u6570\u636e\n```\n\n\n```python\nimport time\nstart = time.time()\n\norder_d = 2\npred_steps = 108 * 5\nback_steps = 108 * 7 * 2\nnum_pred = 5\nnum_rolling = int(pred_steps / num_pred)\nburn_iter = 1000\ngibbs_iter = 100\n\nstart_step = X.shape[1] - pred_steps\nmat_Y = X[:, 0 : start_step]\nmat_Y_new = X[:, start_step : start_step + pred_steps - num_pred]\nresult = bvar_model(mat_Y, mat_Y_new, order_d, num_pred, num_rolling, burn_iter, gibbs_iter)\n\nend = time.time()\nprint('Running time: %d seconds'%(end - start))\n```\n\n 100\n 200\n 300\n 400\n 500\n 600\n 700\n 800\n 900\n 1000\n 1100\n Running time: 4263 seconds\n\n\n\n```python\nmat = X[:, start_step : X.shape[1]] * max_const\npos = np.where(mat != 0)\nmat_hat = np.mean(result[0], axis = 2) * max_const\nprint('MAPE: {}'.format(np.sum(np.abs(mat[pos] - mat_hat[pos])/mat[pos])/mat[pos].shape[0]))\nprint('RMSE: {}'.format(np.sqrt(np.sum((mat[pos] - mat_hat[pos]) ** 2)/mat[pos].shape[0])))\n```\n\n MAPE: 0.371662520311232\n RMSE: 44.300409121588196\n\n\n\n```python\nmat_hat90 = np.percentile(result[0], 90, axis = 2)\n```\n\n\n```python\nimport time\nstart = time.time()\n\norder_d = 2\npred_steps = 108 * 5\nback_steps = 108 * 7 * 2\nnum_pred = 5\nnum_rolling = int(pred_steps / num_pred)\nburn_iter = 5000\ngibbs_iter = 100\n\nstart_step = X.shape[1] - pred_steps\nmat_Y = X[:, 0 : start_step]\nmat_Y_new = X[:, start_step : start_step + pred_steps - num_pred]\nresult = bvar_model(mat_Y, mat_Y_new, order_d, num_pred, num_rolling, burn_iter, gibbs_iter)\n\nend = time.time()\nprint('Running time: %d seconds'%(end - start))\n```\n\n 100\n 200\n 300\n 400\n 500\n 600\n 700\n 800\n 900\n 1000\n 1100\n 1200\n 1300\n 1400\n 1500\n 1600\n 1700\n 1800\n 1900\n 2000\n 2100\n 2200\n 2300\n 2400\n 2500\n 2600\n 2700\n 2800\n 2900\n 3000\n 3100\n 3200\n 3300\n 3400\n 3500\n 3600\n 3700\n 3800\n 3900\n 4000\n 4100\n 4200\n 4300\n 4400\n 4500\n 4600\n 4700\n 4800\n 4900\n 5000\n 5100\n Running time: 2057 seconds\n\n\n\n```python\nmat = X[:, start_step : X.shape[1]] * max_const\npos = np.where(mat != 0)\nmat_hat = np.mean(result[0], axis = 2) * max_const\nprint('MAPE: {}'.format(np.sum(np.abs(mat[pos] - mat_hat[pos])/mat[pos])/mat[pos].shape[0]))\nprint('RMSE: {}'.format(np.sqrt(np.sum((mat[pos] - mat_hat[pos]) ** 2)/mat[pos].shape[0])))\n```\n\n MAPE: 0.37405811145984663\n RMSE: 44.15142376552828\n\n\n\n```python\nmat_hat10 = np.percentile(result[0], 10, axis = 2)\nmat_hat90 = np.percentile(result[0], 90, axis = 2)\n```\n\n\n```python\nimport matplotlib.pyplot as plt\nplt.style.use('ggplot')\nfigsize = 2\nfor i in range(1):\n fig = plt.figure(figsize = (8 * figsize, 2 * figsize))\n ax = fig.add_axes([0.13, 0.28, 0.85, 0.68])\n plt.plot(X[i, 18 * 108 :] * max_const, color = \"black\", linewidth = 1)\n plt.plot(list(range(X.shape[1] - pred_steps - 18 * 108, X.shape[1] - 18 * 108)), \n mat_hat[i, :], color = \"#e3120b\", linewidth = 2.0)\n plt.plot(list(range(X.shape[1] - pred_steps - 18 * 108, X.shape[1] - 18 * 108)), \n mat_hat10[i, :] * max_const, color = \"blue\", linewidth = 2.0)\n plt.plot(list(range(X.shape[1] - pred_steps - 18 * 108, X.shape[1] - 18 * 108)), \n mat_hat90[i, :] * max_const, color = \"green\", linewidth = 2.0)\n```\n", 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YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.6757646010190476, "lm_q1q2_score": 0.42311255481306387}} {"text": "```python\n%matplotlib widget\n```\n\n\n```python\nimport numpy as np\nimport scipy.optimize as sopt\nimport matplotlib.pyplot as plt\nimport pandas as pd\nimport pydae.ssa as ssa\n```\n\n\n```python\nfrom cigre_eur_lv_res_bpu import cigre_eur_lv_res_bpu_class\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\nsyst =cigre_lv_res_vsg_class()\nsyst.Dt = 0.01\nsyst.decimation = 1\nsyst.update()\n\nevents = [{ # CTRL4-3-0\n 't_end':0.0, \n 'K_f_sec':0.001,\n 'K_q_G10':0.5, 'K_q_G14':0.5,'D_G10':1,'D_G14':1,\n 'R_v_G10':0.01,'X_v_G10':0.1,'R_v_G14':0.01,'X_v_G14':0.1,\n 'K_f_G10':5,'K_f_G14':5,\n 'K_vpoi_G10':50,'K_vpoi_G14':50,\n 'K_phi_G10':1e-3,'K_phi_G14':1e-3,\n 'p_r_G10':0.0,'q_r_G10':0.0, \n 'p_r_G14':0.0,'q_r_G14':0.0, #'R_12':0.01, 'R_23':0.01, 'L_12':1e-3, 'L_23':1e-3, 'C_12':1e-6, 'C_23':1e-6,'R_t_1':0.01,'R_t_2':0.01,'L_t_1':1e-3,'L_t_2':1e-3\n 'v_s_ref_G10':1.0,'v_s_ref_G14':1.0,'omega_ref_G10':1.0,'omega_ref_G14':1.0,\n },\n {'t_end':1.0}, \n {'t_end':6.0},\n {'t_end':15.0,'p_r_G10':0.05,'p_r_G14':0.05}\n ]\n\nloads_0 = [\n {\"bus\": \"R01\", \"kVA\": 1.0, \"pf\": 0.95, \"T_i\":0.01,\"I_max\":434.78},\n {\"bus\": \"R11\", \"kVA\": 15.0, \"pf\": 0.95, \"T_i\":0.01,\"I_max\":32.6},\n {\"bus\": \"R15\", \"kVA\": 52.0, \"pf\": 0.95, \"T_i\":0.01,\"I_max\":120},\n {\"bus\": \"R16\", \"kVA\": 55.0, \"pf\": 0.95, \"T_i\":0.01,\"I_max\":120},\n {\"bus\": \"R17\", \"kVA\": 35.0, \"pf\": 0.95, \"T_i\":0.01,\"I_max\":120},\n {\"bus\": \"R18\", \"kVA\": 47.0, \"pf\": 0.95, \"T_i\":0.01,\"I_max\":120}\n ]\n\nloads_1 = [\n {\"bus\": \"R01\", \"kVA\": 1.0, \"pf\": 0.95, \"T_i\":0.01,\"I_max\":434.78},\n {\"bus\": \"R11\", \"kVA\": 15.0, \"pf\": 0.95, \"T_i\":0.01,\"I_max\":32.6},\n {\"bus\": \"R15\", \"kVA\": 52.0, \"pf\": 0.95, \"T_i\":0.01,\"I_max\":120},\n {\"bus\": \"R16\", \"kVA\": 55.0, \"pf\": 0.95, \"T_i\":0.01,\"I_max\":120},\n {\"bus\": \"R17\", \"kVA\": 35.0, \"pf\": 0.95, \"T_i\":0.01,\"I_max\":120},\n {\"bus\": \"R18\", \"kVA\": 70.0, \"pf\": 0.95, \"T_i\":0.01,\"I_max\":120}\n ]\n\n# makes all loads zero for initilization\nloads_dict = {}\nfor load in loads_0:\n p = f\"p_{load['bus']}_ref\"\n q = f\"q_{load['bus']}_ref\"\n events[0][p] = 0.0\n events[0][q] = 0.0 \n\nsyst.sopt_root_jac = True\nsyst.initialization_tol = 1e-2\nsyst.initialize(events,xy0=1)\nsyst.sopt_root_jac = True\n\nsyst.initialization_tol = 1e-12\nsyst.xy_prev[syst.x_list.index('phi_G10')] = 0.0\nsyst.xy_prev[syst.x_list.index('phi_G14')] = 0.0\n\n# assign initial loads\nfor load in loads_0:\n p_ref_name = f\"p_{load['bus']}_ref\"\n q_ref_name = f\"q_{load['bus']}_ref\"\n s = load['kVA']*1000\n p = s*load['pf']\n q = np.sign(load['pf'])*(s**2 - p**2)**0.5\n events[0][p_ref_name] = p\n events[0][q_ref_name] = q\n events[1][p_ref_name] = p\n events[1][q_ref_name] = q\n \n# assign step 1 loads\nfor load in loads_1:\n p_ref_name = f\"p_{load['bus']}_ref\"\n q_ref_name = f\"q_{load['bus']}_ref\"\n s = load['kVA']*1000\n p = s*load['pf']\n q = np.sign(load['pf'])*(s**2 - p**2)**0.5\n events[2][p_ref_name] = p\n events[2][q_ref_name] = q\n \n \nevents[0].update(loads_dict) \nsyst.initialize(events,xy0='prev')\nssa.eval_A(syst);\n```\n\n\n```python\nsyst.simulate(events,xy0='prev');\n```\n\n\n```python\nfig, axes = plt.subplots(nrows=4, figsize=(7, 10), dpi=80)\n\n\n#axes[0].plot(syst.T, syst.get_values('i_sD_G10'), label=f'i_sD_G10')\n#axes[0].plot(syst.T, syst.get_values('i_sQ_G10'), label=f'i_sQ_G10')\n\n#axes[0].plot(syst.T, syst.get_values('i_sD_G14'), label=f'i_sD_G14')\n#axes[0].plot(syst.T, syst.get_values('i_sQ_G14'), label=f'i_sQ_G14')\n\n\naxes[0].plot(syst.T, syst.get_values('p_s_pu_G10')*200, label=f'p_s_pu_G10')\naxes[0].plot(syst.T, syst.get_values('p_s_pu_G14')*200, label=f'p_s_pu_G14')\n\naxes[1].plot(syst.T, syst.get_values('q_s_pu_G10')*200, label=f'q_s_pu_G10')\naxes[1].plot(syst.T, syst.get_values('q_s_pu_G14')*200, label=f'q_s_pu_G14')\n\naxes[2].plot(syst.T, syst.get_values('omega_v_G10'), label=f'omega_v_G10')\naxes[2].plot(syst.T, syst.get_values('omega_v_G14'), label=f'omega_v_G14')\n\nfor ax in axes:\n ax.legend(loc='upper right')\n ax.grid(True)\n```\n\n\n Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous \u2026\n\n\n No handles with labels found to put in legend.\n\n\n\n```python\nloads_kva = [item['kVA'] for item in loads] \n```\n\n\n```python\nssa.damp_report(syst).round(3)\n```\n\n\n```python\nnp.sum(np.array(loads_kva))\n```\n\n\n\n\n 205.0\n\n\n\n\n```python\nget_s(syst,'R01')\n```\n\n\n\n\n (-190000.00000000006-62449.97998398403j)\n\n\n\n\n```python\nfig, axes = plt.subplots(nrows=4, figsize=(7, 10), dpi=50)\n\n\n#for ig in range(1,6):\n#axes[0].plot(T, (Z[:,0]-, label=f'$\\Delta f_{{coi}}$')\n\nbus = 'R18'\nv_d = Y[:,syst.y_list.index(f'v_{bus}_d')]\nv_q = Y[:,syst.y_list.index(f'v_{bus}_q')]\naxes[0].plot(T, np.abs(v_d+1j*v_q)*np.sqrt(3/2), label=f'{bus}: $V$')\n\nbus = 'R00'\nv_d = Z[:,0]\nv_q = Z[:,1]\n\ni_d = Y[:,syst.y_list.index(f'i_{bus}_d')]\ni_q = Y[:,syst.y_list.index(f'i_{bus}_q')]\n\np = 3/2*(i_d*v_d + i_q*v_q)\nq = 3/2*(i_q*v_d - i_d*v_q)\n\naxes[1].plot(T, p/1000, label=f'{bus}: $p$')\naxes[1].plot(T, q/1000, label=f'{bus}: $p$')\n\nv_r18_d = Y[-1,syst.y_list.index('v_R18_d')]\nv_r18_q = Y[-1,syst.y_list.index('v_R18_q')]\nv_r18_m = np.abs(v_r18_d+1j*v_r18_q)*np.sqrt(3/2)\nprint(f'V_R18 = [{v_r18_d:0.2f},{v_r18_q:0.2f}, |V_R18| = {v_r18_m:0.3f}]' )\nprint(f'V_R18 = [{p[-1]:0.2f},{q[-1]:0.2f}, |V_R18| = {v_r18_m:0.3f}]' )\n```\n\n\n Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous \u2026\n\n\n V_R18 = [-10.66,301.58, |V_R18| = 369.586]\n V_R18 = [398270.37,144564.01, |V_R18| = 369.586]\n\n\n\n```python\nbuses = [{\"bus\":\"R00\"},{\"bus\":\"R01\"},{\"bus\":\"R02\"},{\"bus\":\"R03\"},{\"bus\":\"R04\"},{\"bus\":\"R05\"},{\"bus\":\"R06\"},\n {\"bus\":\"R07\"},{\"bus\":\"R08\"},{\"bus\":\"R09\"},{\"bus\":\"R10\"},{\"bus\":\"R11\"},{\"bus\":\"R12\"},\n {\"bus\":\"R13\"},{\"bus\":\"R14\"},{\"bus\":\"R15\"},{\"bus\":\"R16\"},{\"bus\":\"R17\"},{\"bus\":\"R18\"}]\n```\n\n\n```python\nbuses_list = [bus['bus'] for bus in buses]\n```\n\n\n```python\nds_v = [1.0,0.9808973,0.9722749,0.9636541,0.9556469,0.9498654,0.9440847,0.9406105,0.9371365,\n0.9336628,0.9316682,0.9612911,0.9455948,0.9355475,0.925505,0.9154676,0.935177,\n0.9279505,0.923964]\n```\n\n\n```python\nit = 0\np_total = 0.0\nheader = f\"{'Bus':6s}| {'v_m':5s}| {'error':7s}|{'p':7s}| {'q':7s}\"\nprint(header)\nprint(':-----|:------|--------:|------:|---------:')\nfor bus in buses:\n #if bus['bus']=='R00': continue\n\n v_m = get_v(syst,bus['bus'])\n s = get_s(syst,bus['bus'])\n \n error = v_m/400 -ds_v[it] \n \n print(f\"{bus['bus']:6s}| {v_m:5.1f} | {100*np.abs(error):6.4f}% |{s.real/1000:7.2f}| {s.imag/1000:7.2f}\")\n p_total += s.real\n it+=1\n```\n\n Bus | v_m | error |p | q \n :-----|:------|--------:|------:|---------:\n R00 | 400.0 | 0.0000% | 398.27| 144.56\n R01 | 392.4 | 0.0000% |-190.00| -62.45\n R02 | 388.9 | 0.0000% | 0.00| -0.00\n R03 | 385.5 | 0.0000% | 0.00| -0.00\n R04 | 382.3 | 0.0000% | 0.00| -0.00\n R05 | 379.9 | 0.0000% | 0.00| -0.00\n R06 | 377.6 | 0.0000% | 0.00| -0.00\n R07 | 376.2 | 0.0000% | 0.00| -0.00\n R08 | 374.9 | 0.0000% | 0.00| -0.00\n R09 | 373.5 | 0.0000% | 0.00| -0.00\n R10 | 372.7 | 0.0000% | 0.00| -0.00\n R11 | 384.5 | 0.0000% | -14.25| -4.68\n R12 | 378.2 | 0.0000% | 0.00| -0.00\n R13 | 374.2 | 0.0000% | 0.00| -0.00\n R14 | 370.2 | 0.0000% | 0.00| -0.00\n R15 | 366.2 | 0.0000% | -49.40| -16.24\n R16 | 374.1 | 0.0000% | -52.25| -17.17\n R17 | 371.2 | 0.0000% | -33.25| -10.93\n R18 | 369.6 | 0.0000% | -44.65| -14.68\n\n\n\n```python\ndata = {'bus':[bus['bus'] for bus in buses],\n 'v_m':[get_v(syst,bus['bus']) for bus in buses],\n 'p':[get_s(syst,bus['bus']).real/1000 for bus in buses],\n 'q':[get_s(syst,bus['bus']).imag/1000 for bus in buses]}\ndf = pd.DataFrame(data=data)\ndf.set_index('bus')\n\ndf['v_m'] = df['v_m'].apply(lambda x: \"{:,.1f}\".format(x))\ndf['p'] = df['p'].apply(lambda x: \"{:,.1f}\".format(x))\ndf['q'] = df['q'].apply(lambda x: \"{:,.1f}\".format(x))\ndf = df.set_index('bus')\ndf\n```\n\n\n```python\n\n```\n\n\n```python\nimport sympy as sym\nv_d,v_q,i_d,i_q = sym.symbols('v_d,v_q,i_d,i_q', real=True)\ni_p_ref,i_q_ref = sym.symbols('i_p_ref,i_q_ref ', real=True)\nv_dq = v_q +1j*v_d\ni_dq = i_q +1j*i_d\ns = 3/2*v_dq*np.conj(i_dq)\n\nv_m = sym.sqrt(v_d**2+v_q**2)\n\ng_p = -i_p_ref + 3/2*(i_d*v_d + i_q*v_q)/v_m\ng_q = -i_q_ref + 3/2*(i_q*v_d - i_d*v_q)/v_m\n```\n\n\n```python\nsol = sym.solve([g_p,g_q],[i_d,i_q])\n```\n\n\n```python\nprint(sol[i_q])\n```\n\n 0.666666666666667*(i_p_ref*v_q + i_q_ref*v_d)/sqrt(v_d**2 + v_q**2)\n\n\n\n```python\ni_d_ref = 2/3*(i_p_ref*v_d - i_q_ref*v_q)/sqrt(v_d^2 + v_q^2)\ni_q_ref = 2/3*(i_p_ref*v_q + i_q_ref*v_d)/sqrt(v_d^2 + v_q^2)\n\n```\n\n\n```python\nsyst.u_run\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "662d75b769a1643173f587355322371cc039aade", "size": 21691, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/grids/grid_uri/cigre/cigre_eur_lv_res_vsg/cigre_lv_res_vsg.ipynb", "max_stars_repo_name": "pydae/pydae", "max_stars_repo_head_hexsha": "8076bcfeb2cdc865a5fc58561ff8d246d0ed7d9d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-12-20T03:45:26.000Z", "max_stars_repo_stars_event_max_datetime": "2020-12-20T03:45:26.000Z", "max_issues_repo_path": "examples/grids/grid_uri/cigre/cigre_eur_lv_res_vsg/cigre_lv_res_vsg.ipynb", "max_issues_repo_name": "pydae/pydae", "max_issues_repo_head_hexsha": "8076bcfeb2cdc865a5fc58561ff8d246d0ed7d9d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "examples/grids/grid_uri/cigre/cigre_eur_lv_res_vsg/cigre_lv_res_vsg.ipynb", "max_forks_repo_name": "pydae/pydae", "max_forks_repo_head_hexsha": "8076bcfeb2cdc865a5fc58561ff8d246d0ed7d9d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 37.2697594502, "max_line_length": 1686, "alphanum_fraction": 0.5142224886, "converted": true, "num_tokens": 3932, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646140788307, "lm_q2_score": 0.6261241632752916, "lm_q1q2_score": 0.42311255356115823}} {"text": "```python\nimport pandas as pd\nimport numpy as np\n%matplotlib inline\nimport gc\nfrom scipy import signal\nfrom sympy import fft\nfrom scipy import stats\n\n# !pip install -Uqq fastbook kaggle waterfallcharts treeinterpreter dtreeviz\n# import fastbook\n# fastbook.setup_book()\n# !pip install wwf\n# from wwf.tabular.export import *\nfrom google.colab import drive\ndrive.mount('/content/drive')\n! [ -e /content ] && pip install -Uqq fastai # upgrade fastai on colab\n# from fastbook import *\n# from kaggle import api\nfrom pandas.api.types import is_string_dtype, is_numeric_dtype, is_categorical_dtype\nfrom fastai.tabular.all import *\nfrom sklearn.ensemble import RandomForestClassifier\n# from dtreeviz.trees import *\nfrom IPython.display import Image, display_svg, SVG\n\n# pd.options.display.max_rows = 20\n# pd.options.display.max_columns = 8\n```\n\n Mounted at /content/drive\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 204kB 3.9MB/s \n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 61kB 8.0MB/s \n \u001b[?25h\n\n\n```python\n# removed atributes\n# remove = ['arCoeff', 'correlation', 'bandsEnergy', 'angle','entropy','sma',]\ntrain_df = pd.read_csv(\"/content/drive/MyDrive/UCI HAR Dataset/train.csv\")\nvalid_df = pd.read_csv(\"/content/drive/MyDrive/UCI HAR Dataset/test.csv\")\ndata = pd.concat([train_df,valid_df])\ndata.reset_index(inplace=True)\ndep_var = data['Activity']\ntrain_y = train_df['Activity']\nvalid_y = valid_df['Activity']\ndata.drop(['subject','Activity'],axis=1,inplace=True)\ntrain_df.drop(['subject','Activity'],axis=1,inplace=True)\nvalid_df.drop(['subject','Activity'],axis=1,inplace=True)\ndic = {}\nfor c in data.columns:\n dic[c]=c.replace(\"-\",\"_\")\ndata.rename(columns=dic,inplace=True)\ntrain_df.rename(columns=dic,inplace=True)\nvalid_df.rename(columns=dic,inplace=True)\n```\n\n\n```python\ntm = ['mean()','std()','mad()','max()','min()','energy()','iqr()']\n\ntxyz=['tBodyAcc'\n,'tGravityAcc'\n,'tBodyAccJerk'\n,'tBodyGyro'\n,'tBodyGyroJerk']\n\ntmag = [\n'tBodyAccMag'\n,'tGravityAccMag'\n,'tBodyAccJerkMag'\n,'tBodyGyroMag'\n,'tBodyGyroJerkMag']\n\n```\n\n\n```python\n#Dealing with txyz\nax=['X','Y','Z']\nt_cols= []\nfor f in txyz:\n for m in tm:\n for i in ax:\n t_cols.append(f+'_'+m+'_'+i)\n\n#Dealing with tmag\nfor f in tmag:\n for m in tm:\n t_cols.append(f+'_'+m)\n\n```\n\n\n```python\nclean_cols=t_cols\ntrain_X = train_df[clean_cols]\nvalid_X = valid_df[clean_cols]\ndel(train_df)\ndel(valid_df)\ncat_names = []\ncont_names = clean_cols\nsplits = (L(np.arange(7352),use_list=True),\n L(np.arange(7352, 10299), use_list=True))\nprocs= [Normalize]\ndata=data[clean_cols]\ndata.loc[:,'Activity'] = dep_var.values\n```\n\n\n```python\nlen(clean_cols)\n```\n\n\n\n\n 140\n\n\n\n\n```python\nrow1 = data[data.Activity=='LAYING'].iloc[100]\nrow2 = data[data.Activity=='SITTING'].iloc[100]\nrow3 = data[data.Activity=='STANDING'].iloc[100]\nrow4 = data[data.Activity=='WALKING'].iloc[100]\nrow5 = data[data.Activity=='WALKING_DOWNSTAIRS'].iloc[100]\nrow6 = data[data.Activity=='WALKING_UPSTAIRS'].iloc[100]\n\nrow1.drop([\"Activity\"],inplace=True)\nrow2.drop([\"Activity\"],inplace=True)\nrow3.drop([\"Activity\"],inplace=True)\nrow4.drop([\"Activity\"],inplace=True)\nrow5.drop([\"Activity\"],inplace=True)\nrow6.drop([\"Activity\"],inplace=True)\n\n```\n\n#Fastai Neural Net\n\n\n```python\nto = TabularPandas(data, procs, cat_names=cat_names, cont_names=clean_cols, y_names=\"Activity\",splits=splits,y_block = CategoryBlock(),device=torch.device('cpu'))\ntrn_dl = TabDataLoader(to.train, bs=128,shuffle=True, drop_last=True)\nval_dl = TabDataLoader(to.valid, bs=32)\ndls = DataLoaders(trn_dl, val_dl)\ngc.collect()\ndef calcHiddenLayer(data, alpha, ip, op, numHiddenLayers):\n return [(len(data.train_ds)//(alpha*(ip+op)))//numHiddenLayers]*numHiddenLayers\nlearn = tabular_learner(dls, layers=calcHiddenLayer(dls, 3, len(data.columns), 6, 2), metrics=accuracy)\n```\n\n\n```python\nlearn.summary()\n```\n\n\n\n\n\n\n\n\n TabularModel (Input shape: 128 x torch.Size([128, 140]))\n ============================================================================\n Layer (type) Output Shape Param # Trainable \n ============================================================================\n 128 x 140 \n BatchNorm1d 280 True \n ____________________________________________________________________________\n 128 x 8 \n Linear 1120 True \n ReLU \n BatchNorm1d 16 True \n Linear 64 True \n ReLU \n BatchNorm1d 16 True \n ____________________________________________________________________________\n 128 x 6 \n Linear 54 True \n ____________________________________________________________________________\n \n Total params: 1,550\n Total trainable params: 1,550\n Total non-trainable params: 0\n \n Optimizer used: \n Loss function: FlattenedLoss of CrossEntropyLoss()\n \n Callbacks:\n - TrainEvalCallback\n - Recorder\n - ProgressCallback\n\n\n\n\n```python\nlearn.lr_find()\n```\n\n\n```python\nlearn.fit_one_cycle(12, lr_max=slice(0.06309573650360108,0.12022644281387329))\n```\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        epochtrain_lossvalid_lossaccuracytime
                                        00.9500190.6606760.73837800:01
                                        10.4636360.8076950.74923700:00
                                        20.3173240.3338400.87071600:00
                                        30.2545130.6100090.79131300:00
                                        40.2274400.3200820.88530700:01
                                        50.1893600.2693920.90227300:00
                                        60.1679130.8075760.75873800:00
                                        70.1637090.6401540.85748200:01
                                        80.1529670.2446120.89854100:00
                                        90.1383390.2238860.91856100:00
                                        100.1288770.2189670.91686500:00
                                        110.1230470.2233320.92365100:00
                                        \n\n\n\n```python\nlearn.show_results()\n```\n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        tBodyAcc_mean()_XtBodyAcc_mean()_YtBodyAcc_mean()_ZtBodyAcc_std()_XtBodyAcc_std()_YtBodyAcc_std()_ZtBodyAcc_mad()_XtBodyAcc_mad()_YtBodyAcc_mad()_ZtBodyAcc_max()_XtBodyAcc_max()_YtBodyAcc_max()_ZtBodyAcc_min()_XtBodyAcc_min()_YtBodyAcc_min()_ZtBodyAcc_energy()_XtBodyAcc_energy()_YtBodyAcc_energy()_ZtBodyAcc_iqr()_XtBodyAcc_iqr()_YtBodyAcc_iqr()_ZtGravityAcc_mean()_XtGravityAcc_mean()_YtGravityAcc_mean()_ZtGravityAcc_std()_XtGravityAcc_std()_YtGravityAcc_std()_ZtGravityAcc_mad()_XtGravityAcc_mad()_YtGravityAcc_mad()_ZtGravityAcc_max()_XtGravityAcc_max()_YtGravityAcc_max()_ZtGravityAcc_min()_XtGravityAcc_min()_YtGravityAcc_min()_ZtGravityAcc_energy()_XtGravityAcc_energy()_YtGravityAcc_energy()_ZtGravityAcc_iqr()_XtGravityAcc_iqr()_YtGravityAcc_iqr()_ZtBodyAccJerk_mean()_XtBodyAccJerk_mean()_YtBodyAccJerk_mean()_ZtBodyAccJerk_std()_XtBodyAccJerk_std()_YtBodyAccJerk_std()_ZtBodyAccJerk_mad()_XtBodyAccJerk_mad()_YtBodyAccJerk_mad()_ZtBodyAccJerk_max()_XtBodyAccJerk_max()_YtBodyAccJerk_max()_ZtBodyAccJerk_min()_XtBodyAccJerk_min()_YtBodyAccJerk_min()_ZtBodyAccJerk_energy()_XtBodyAccJerk_energy()_YtBodyAccJerk_energy()_ZtBodyAccJerk_iqr()_XtBodyAccJerk_iqr()_YtBodyAccJerk_iqr()_ZtBodyGyro_mean()_XtBodyGyro_mean()_YtBodyGyro_mean()_ZtBodyGyro_std()_XtBodyGyro_std()_YtBodyGyro_std()_ZtBodyGyro_mad()_XtBodyGyro_mad()_YtBodyGyro_mad()_ZtBodyGyro_max()_XtBodyGyro_max()_YtBodyGyro_max()_ZtBodyGyro_min()_XtBodyGyro_min()_YtBodyGyro_min()_ZtBodyGyro_energy()_XtBodyGyro_energy()_YtBodyGyro_energy()_ZtBodyGyro_iqr()_XtBodyGyro_iqr()_YtBodyGyro_iqr()_ZtBodyGyroJerk_mean()_XtBodyGyroJerk_mean()_YtBodyGyroJerk_mean()_ZtBodyGyroJerk_std()_XtBodyGyroJerk_std()_YtBodyGyroJerk_std()_ZtBodyGyroJerk_mad()_XtBodyGyroJerk_mad()_YtBodyGyroJerk_mad()_ZtBodyGyroJerk_max()_XtBodyGyroJerk_max()_YtBodyGyroJerk_max()_ZtBodyGyroJerk_min()_XtBodyGyroJerk_min()_YtBodyGyroJerk_min()_ZtBodyGyroJerk_energy()_XtBodyGyroJerk_energy()_YtBodyGyroJerk_energy()_ZtBodyGyroJerk_iqr()_XtBodyGyroJerk_iqr()_YtBodyGyroJerk_iqr()_ZtBodyAccMag_mean()tBodyAccMag_std()tBodyAccMag_mad()tBodyAccMag_max()tBodyAccMag_min()tBodyAccMag_energy()tBodyAccMag_iqr()tGravityAccMag_mean()tGravityAccMag_std()tGravityAccMag_mad()tGravityAccMag_max()tGravityAccMag_min()tGravityAccMag_energy()tGravityAccMag_iqr()tBodyAccJerkMag_mean()tBodyAccJerkMag_std()tBodyAccJerkMag_mad()tBodyAccJerkMag_max()tBodyAccJerkMag_min()tBodyAccJerkMag_energy()tBodyAccJerkMag_iqr()tBodyGyroMag_mean()tBodyGyroMag_std()tBodyGyroMag_mad()tBodyGyroMag_max()tBodyGyroMag_min()tBodyGyroMag_energy()tBodyGyroMag_iqr()tBodyGyroJerkMag_mean()tBodyGyroJerkMag_std()tBodyGyroJerkMag_mad()tBodyGyroJerkMag_max()tBodyGyroJerkMag_min()tBodyGyroJerkMag_energy()tBodyGyroJerkMag_iqr()ActivityActivity_pred
                                        0-0.004209-0.0996430.120368-0.866125-0.928452-0.883030-0.858981-0.927033-0.883874-0.857038-0.943296-0.8611430.8915960.8284770.803244-0.693032-0.774962-0.691594-0.837114-0.900763-0.8792730.407910-1.158915-0.077654-0.379044-0.426713-0.456783-0.389795-0.438173-0.4572490.386393-1.198321-0.1003990.420008-1.123895-0.0369510.3384220.097407-0.568590-0.424844-0.446107-0.455938-0.029629-0.0760610.033131-0.849567-0.861085-0.796101-0.846440-0.860856-0.793040-0.797499-0.820047-0.7516660.8518610.8258510.794870-0.681675-0.715895-0.566590-0.830619-0.847676-0.779043-0.0470780.0714570.060224-0.786757-0.819251-0.850317-0.780593-0.817700-0.842022-0.774446-0.750365-0.8054650.7217290.7835840.842923-0.672667-0.600731-0.648012-0.771042-0.804949-0.8148940.162117-0.075312-0.038625-0.818781-0.722985-0.810243-0.811675-0.730790-0.810666-0.798776-0.664634-0.7995030.8073600.6894650.760415-0.606053-0.439575-0.585812-0.800132-0.735902-0.805694-0.923047-0.906590-0.907915-0.892808-0.778916-0.783610-0.901490-0.923047-0.906590-0.907915-0.892808-0.778916-0.783610-0.901490-0.858482-0.857623-0.856874-0.851606-0.792585-0.705731-0.847909-0.890511-0.854109-0.851483-0.862415-0.801820-0.736015-0.838748-0.800336-0.756808-0.767489-0.728446-0.728835-0.541019-0.7773882.02.0
                                        10.0495150.039127-0.034366-0.872706-0.955196-0.928725-0.864003-0.953899-0.933703-0.872473-0.935785-0.9008120.8999680.8920570.848458-0.693140-0.776294-0.694233-0.839057-0.929138-0.9296540.3582010.8863900.101975-0.445268-0.442748-0.539265-0.441693-0.443064-0.5301630.3350620.8617350.0637100.3791100.9159370.1438130.248040-0.048003-0.516160-0.430141-0.445438-0.506935-0.0318530.052873-0.056920-0.854304-0.873385-0.821794-0.849796-0.871566-0.822050-0.807121-0.834615-0.7745910.8603000.8379690.821208-0.681785-0.716404-0.567847-0.833598-0.853479-0.819588-0.001140-0.003076-0.049161-0.902857-0.862756-0.894041-0.896051-0.862151-0.883910-0.856164-0.800937-0.8832600.8283830.8373530.844894-0.677771-0.601898-0.649673-0.874036-0.847243-0.847241-0.0171770.0352950.032615-0.857846-0.754516-0.821759-0.853278-0.757818-0.823952-0.841181-0.723868-0.8139810.8331050.7257080.764385-0.607162-0.440179-0.586200-0.839312-0.754923-0.820435-0.947322-0.936411-0.937498-0.930520-0.814203-0.784525-0.925698-0.947322-0.936411-0.937498-0.930520-0.814203-0.784525-0.925698-0.871517-0.872845-0.868910-0.870570-0.748540-0.706251-0.856510-0.963276-0.939531-0.945097-0.925031-0.833225-0.739344-0.937832-0.829420-0.787593-0.796171-0.775032-0.748990-0.541765-0.8025751.01.0
                                        2-0.006996-0.236656-1.6142760.9231970.5567270.7889290.9945090.5241650.7456500.7035110.3030490.465480-0.748758-0.701296-1.0497210.5714530.1354890.4376751.1492360.4137730.4035010.575710-0.414306-0.354769-0.149069-0.275453-0.188092-0.113771-0.284071-0.1843480.569600-0.434085-0.3729790.583300-0.392164-0.3530150.656372-0.547786-0.579882-0.083053-0.296640-0.187247-2.285515-0.2522151.7542551.1841080.9204501.4196121.2797191.0384921.3704630.6164420.7636901.208810-1.520256-0.870731-0.9781320.9570880.5980781.1260621.3511181.2152131.127764-0.0388440.032777-0.0925760.4912460.2641051.4408480.4740590.2957531.4522230.8724290.3393800.895780-0.418652-0.070094-1.485008-0.103219-0.1687661.1965960.4537230.3279271.491969-0.5571200.4697523.4860841.0015560.3738331.0063270.9467180.4214881.1361821.2158820.3321290.575434-0.602642-0.206439-0.6006170.603428-0.0776990.5929290.7953700.4445121.4097960.8601890.5981350.5926000.7112530.9269830.5010860.6520050.8601890.5981350.5926000.7112530.9269830.5010860.6520051.2004191.0843541.1479931.1856931.2519050.9591081.3244480.5373860.4310350.4800170.375209-0.0189160.1065660.4935450.7135110.4961460.5645550.5671011.3321910.1636540.7194863.03.0
                                        3-0.0606510.028744-0.003635-0.826054-0.923434-0.909341-0.816467-0.924542-0.920752-0.840087-0.886029-0.8567250.8314090.8198920.811493-0.691673-0.775005-0.693411-0.786376-0.901641-0.929421-2.048647-2.7102430.5920720.031374-0.462999-0.5748300.000266-0.461553-0.567791-2.046780-2.7545370.553429-2.025605-2.6528840.631995-2.0501873.620023-0.192964-0.107862-0.444185-0.545934-0.0034100.031883-0.004309-0.811007-0.818506-0.785600-0.805552-0.816643-0.783920-0.772687-0.773153-0.7224250.8076710.7818100.807044-0.680138-0.713235-0.565954-0.788407-0.792043-0.7680380.054545-0.2619880.083489-0.826240-0.704668-0.865078-0.815363-0.703078-0.854556-0.787406-0.628092-0.8347690.7581470.6691120.841763-0.674994-0.590707-0.648564-0.779382-0.689030-0.821935-0.1179480.085544-0.023674-0.792632-0.610583-0.794364-0.798496-0.603581-0.795169-0.710439-0.600146-0.7851230.7541950.5869910.727894-0.604748-0.432971-0.585131-0.795185-0.588094-0.779968-0.910955-0.902941-0.901212-0.897307-0.779742-0.783077-0.881776-0.910955-0.902941-0.901212-0.897307-0.779742-0.783077-0.881776-0.824221-0.835830-0.832566-0.834710-0.677403-0.704006-0.824118-0.849100-0.817892-0.826891-0.790912-0.759346-0.732473-0.829822-0.716671-0.669640-0.679381-0.663538-0.639593-0.535805-0.7009920.00.0
                                        40.2818710.090733-0.034956-0.824818-0.951118-0.899197-0.815349-0.952366-0.902105-0.830604-0.927061-0.8746760.8617870.8847540.812307-0.691437-0.776044-0.692810-0.788604-0.935432-0.892994-2.0293991.9421871.718533-0.125121-0.445807-0.478664-0.112600-0.439259-0.467132-2.0555831.9203541.690723-2.0072511.9627201.744699-2.0480071.8793231.549093-0.064867-0.419833-0.424523-0.0161490.0209090.013480-0.837051-0.870960-0.789068-0.834258-0.872497-0.795055-0.777092-0.829682-0.7003320.8398800.8239280.782383-0.681301-0.716314-0.566173-0.816382-0.859119-0.7861390.0044470.0257120.026892-0.874388-0.765444-0.884297-0.868184-0.757483-0.878587-0.825251-0.743461-0.8209110.7982790.7392980.864423-0.677305-0.597657-0.649479-0.849238-0.730123-0.8443330.0206450.0290650.016420-0.810942-0.701331-0.812787-0.807714-0.705765-0.820028-0.797095-0.662040-0.7873530.7896270.6708040.740799-0.605725-0.438845-0.585906-0.800168-0.710052-0.820719-0.910638-0.900996-0.899641-0.905275-0.781929-0.783044-0.884865-0.910638-0.900996-0.899641-0.905275-0.781929-0.783044-0.884865-0.856719-0.847348-0.849364-0.826288-0.766370-0.705547-0.841559-0.904206-0.877952-0.881011-0.867063-0.770644-0.737065-0.867683-0.785998-0.740069-0.747180-0.728812-0.678710-0.540426-0.7535690.00.0
                                        50.204521-0.017453-0.223168-0.838747-0.932284-0.869260-0.833666-0.933643-0.868769-0.827821-0.906710-0.8832070.8363450.8503170.792391-0.692170-0.775426-0.689607-0.818910-0.920573-0.850850-1.9862642.591013-0.593369-0.271889-0.494109-0.360884-0.277143-0.488941-0.360730-2.0126642.567552-0.626651-1.9565712.607870-0.565262-2.0420933.740136-0.522361-0.278943-0.472089-0.3431020.105630-0.0180690.014023-0.814421-0.834860-0.792536-0.816641-0.837459-0.790189-0.726709-0.801053-0.7258970.8138920.7817160.801633-0.680313-0.714424-0.566382-0.811093-0.828155-0.7810300.0131810.0105420.104770-0.853633-0.775986-0.848095-0.849289-0.800081-0.844685-0.814916-0.635623-0.7435380.7537350.6901060.839940-0.676646-0.598421-0.647614-0.834271-0.809677-0.8153330.0768250.057635-0.093386-0.811094-0.653086-0.754614-0.814308-0.683158-0.769595-0.768693-0.617348-0.6573130.7771170.5404550.683655-0.605729-0.436287-0.582684-0.811618-0.711044-0.782007-0.907239-0.888056-0.884361-0.891559-0.823800-0.782746-0.854560-0.907239-0.888056-0.884361-0.891559-0.823800-0.782746-0.854560-0.838556-0.827843-0.832777-0.807501-0.720128-0.704543-0.828753-0.911027-0.835301-0.865777-0.789925-0.831540-0.736744-0.886864-0.765034-0.664479-0.700081-0.616626-0.700242-0.538097-0.7452070.00.0
                                        61.4964420.269850-0.0317500.4947690.9365630.4398060.4827230.9283520.4647280.5197500.6076910.553132-0.584616-0.828133-0.2795750.0522430.636694-0.0009770.4354350.7182080.4501350.460298-0.9427950.157486-0.055839-0.057068-0.443645-0.049558-0.025188-0.4449660.461270-0.9518360.1317840.462846-0.9190780.1898030.435611-0.159469-0.492767-0.0799050.061561-0.431661-0.320802-0.995124-0.1368960.9495751.0855860.8352950.9994991.0203290.7014181.0331191.1901870.307425-0.614565-0.721613-1.2044340.6041900.8449790.3751790.9840770.5800810.5255100.0933310.057017-0.0580270.7080760.2906280.5503720.7832680.3020320.5550530.6773980.6200490.134708-1.160279-0.162351-0.5882440.088517-0.1484240.0690590.8510140.2676830.531235-1.2605291.393179-0.3508671.7283300.6074211.2143381.5839040.6505751.1753062.0758980.8861460.871414-2.231936-0.501446-1.1745961.7206090.0852240.8638511.3651890.7279940.9626930.6329320.3149380.3458310.3981220.9584690.1844290.6004140.6329320.3149380.3458310.3981220.9584690.1844290.6004140.9739900.9439281.1090550.5092320.0038710.6529861.1774180.4230120.3334340.2606160.5139150.369172-0.0145340.2163371.0354031.0261731.0707350.8076041.9691060.5723601.0929893.03.0
                                        7-0.1714180.1318670.201359-0.798576-0.928389-0.891097-0.815421-0.929569-0.901416-0.782824-0.878718-0.8540710.6730490.8820660.782763-0.689912-0.774931-0.691626-0.819046-0.913343-0.909695-2.041373-2.5798650.937382-0.315095-0.384920-0.431255-0.306892-0.375014-0.422303-2.078895-2.6201410.913076-2.013374-2.5267450.966821-2.0494603.2106880.193071-0.287075-0.349672-0.3938990.001614-0.0119970.007614-0.771845-0.824264-0.742446-0.781704-0.829682-0.753618-0.692220-0.765793-0.6961950.7318560.8148420.696898-0.677411-0.713677-0.562604-0.776819-0.830433-0.7642410.0588610.2313760.027930-0.833727-0.601652-0.840483-0.829807-0.660805-0.840800-0.768311-0.556142-0.7685800.7679680.4674750.804877-0.675295-0.575663-0.647591-0.814461-0.739051-0.831087-0.0992610.0167720.015200-0.761194-0.529713-0.768413-0.788326-0.569807-0.797452-0.729942-0.483385-0.7695230.6502190.4038280.607780-0.602550-0.423914-0.583656-0.809281-0.613406-0.813324-0.900247-0.833899-0.850389-0.813883-0.808636-0.781527-0.870592-0.900247-0.833899-0.850389-0.813883-0.808636-0.781527-0.870592-0.810587-0.775107-0.782545-0.778531-0.665471-0.701845-0.789806-0.830705-0.648674-0.679740-0.681006-0.779715-0.725272-0.780608-0.689026-0.536080-0.558836-0.513834-0.752876-0.528485-0.6074580.00.0
                                        8-0.1338720.217271-0.3909611.2631930.9531471.0742621.3240760.9437891.0375391.3570501.0036641.176041-0.742269-0.807479-1.2938141.0906080.6604930.7589671.3432660.6440780.8657710.534743-0.503166-0.623457-0.089821-0.030210-0.199561-0.113170-0.064256-0.2003330.531873-0.484031-0.6493380.528803-0.482669-0.6115200.576944-0.506189-0.510538-0.212115-0.133353-0.2230642.789955-2.814935-2.3517320.7489711.3536661.9004970.8080851.2673231.8323060.6451201.4833852.052876-0.314738-0.682454-2.0016050.3417051.2927371.9073890.9111580.8788521.603541-2.0835131.0556350.9860740.3463310.3692130.2186930.3806030.3582990.115132-0.0884210.4112580.582959-0.813113-0.995394-0.6621460.410260-0.069505-0.1806830.4127280.489521-0.0402721.8600760.4848110.2400380.9767380.7676320.7526960.9190880.7965600.7115790.8641701.2030661.182954-0.961768-0.773819-0.6238920.5736490.2139000.2914950.8472050.9480400.6941581.1330041.2298611.1177731.3394911.7135641.0355940.6271471.1330041.2298611.1177731.3394911.7135641.0355940.6271471.1254421.3858501.4557611.333638-0.2952501.0279541.6048500.5770440.3501510.3840900.3990570.4363290.1278740.5083180.7924830.8585600.8010340.9324290.3425490.3268040.8152124.04.0
                                        \n\n\nTo get prediction on a new dataframe, you can use the `test_dl` method of the `DataLoaders`. That dataframe does not need to have the dependent variable in its column.\n\nThen `Learner.get_preds` will give you the predictions:\n\n> Note: Since machine learning models can't magically understand categories it was never trained on, the data should reflect this. If there are different missing values in your test data you should address this before training\n\n# `Random Forest` Model Interpretation\n\nAs mentioned earlier, `TabularPandas` is a powerful and easy preprocessing tool for tabular data. Integration with libraries such as Random Forests and XGBoost requires only one extra step, that the `.dataloaders` call did for us. Let's look at our `to` again. It's values are stored in a `DataFrame` like object, where we can extract the `cats`, `conts,` `xs` and `ys` if we want to:\n\nA TabularPandas behaves a lot like a fastai Datasets object, including providing train and valid attributes\n\n\n```python\n#Now we will make use of special Fastai pd.Dataframe wrapper called TabularPandas\ntor = TabularPandas(data, procs=[Normalize], cat_names=cat_names, cont_names=cont_names, y_names='Activity', splits=splits)\n```\n\n\n```python\nxs,y = tor.train.xs,tor.train.y\nvalid_xs,valid_y = tor.valid.xs,tor.valid.y\ndef r_mse(pred,y): return round(math.sqrt(((pred-y)**2).mean()), 6)\ndef m_rmse(m, xs, y): return r_mse(m.predict(xs), y)\n```\n\n\n```python\ndef rf(xs, y, n_estimators=120, max_samples=0.8,min_samples_leaf=5, **kwargs):\n return RandomForestClassifier(n_jobs=-1, n_estimators=n_estimators,\n max_samples=max_samples,min_samples_leaf=min_samples_leaf,bootstrap=True, oob_score=True).fit(xs, y)\n```\n\n\n```python\n#Fitting the Model\nm = rf(xs, y);\n```\n\n\n```python\nprint(\"Training Error = \",m_rmse(m, xs, y))\nprint(\"Validation Error = \",m_rmse(m, valid_xs, valid_y)) \nprint(\"OOB Error = \",1-m.oob_score_)\n```\n\n Training Error = 0.124523\n Validation Error = 0.600588\n OOB Error = 0.03563656147986938\n\n\n\n```python\npreds = np.stack([t.predict(valid_xs) for t in m.estimators_])\nplt.plot([r_mse(preds[:i+1].mean(0), valid_y) for i in range(40)]);\n```\n\nFor tabular data, model interpretation is particularly important. For a given model, the things we are most likely to be interested in are:\n\n> How confident are we in our predictions using a particular row of data?\n> For predicting with a particular row of data, what were the most important factors, and how did they influence that prediction?\n> Which columns are the strongest predictors, which can we ignore?\n> Which columns are effectively redundant with each other, for purposes of prediction?\n> How do predictions vary, as we vary these columns?\n\nAs we will see, random forests are particularly well suited to answering these questions. Let's start with the first one!\n\n## Feature Importance\n\n\n```python\ndef rf_feat_importance(m, df):\n return pd.DataFrame({'cols':df.columns, 'imp':m.feature_importances_}\n ).sort_values('imp', ascending=False)\n```\n\n\n```python\nfi = rf_feat_importance(m, xs)\nfi[:10] #10 Most Important Features\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        colsimp
                                        33tGravityAcc_min()_X0.085714
                                        31tGravityAcc_max()_Y0.053245
                                        36tGravityAcc_energy()_X0.049473
                                        21tGravityAcc_mean()_X0.049341
                                        34tGravityAcc_min()_Y0.044329
                                        22tGravityAcc_mean()_Y0.043795
                                        3tBodyAcc_std()_X0.038936
                                        30tGravityAcc_max()_X0.038754
                                        9tBodyAcc_max()_X0.034952
                                        37tGravityAcc_energy()_Y0.034318
                                        \n
                                        \n\n\n\n\n```python\n#Top 30 Most Imporant Features\ndef plot_fi(fi):\n return fi.plot('cols', 'imp', 'barh', figsize=(12,7), legend=False)\n\nplot_fi(fi[:30]);\n```\n\n# Ensembling with other Approaches\n\n\n```python\n# Import all machine learning algorithms\nfrom sklearn.svm import SVC\nfrom sklearn.linear_model import LogisticRegression\nfrom sklearn.neighbors import KNeighborsClassifier\nfrom sklearn.ensemble import RandomForestClassifier\n\n# Import other useful subpackage\nfrom sklearn.metrics import accuracy_score\n```\n\n\n```python\naccuracy_scores = np.zeros(4)\n\n# Support Vector Classifier\nclf1= SVC().fit(train_X, train_y)\nprediction1 = clf1.predict(valid_X)\naccuracy_scores[0] = accuracy_score(valid_y, prediction1)*100\nprint('Support Vector Classifier accuracy: {}%'.format(accuracy_scores[0]))\n\n# Logistic Regression\nclf2 = LogisticRegression().fit(train_X, train_y)\nprediction2 = clf2.predict(valid_X)\naccuracy_scores[1] = accuracy_score(valid_y, prediction2)*100\nprint('Logistic Regression accuracy: {}%'.format(accuracy_scores[1]))\n\n# K Nearest Neighbors\nclf3 = KNeighborsClassifier().fit(train_X, train_y)\nprediction3 = clf3.predict(valid_X)\naccuracy_scores[2] = accuracy_score(valid_y, prediction3)*100\nprint('K Nearest Neighbors Classifier accuracy: {}%'.format(accuracy_scores[2]))\n\n\n# Random Forest\nclf4 = RandomForestClassifier().fit(train_X, train_y)\nprediction4 = clf4.predict(valid_X)\naccuracy_scores[3] = accuracy_score(valid_y, prediction4)*100\nprint('Random Forest Classifier accuracy: {}%'.format(accuracy_scores[3]))\n\n```\n\n Support Vector Classifier accuracy: 91.72039362063114%\n\n\n /usr/local/lib/python3.7/dist-packages/sklearn/linear_model/_logistic.py:940: ConvergenceWarning: lbfgs failed to converge (status=1):\n STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n \n Increase the number of iterations (max_iter) or scale the data as shown in:\n https://scikit-learn.org/stable/modules/preprocessing.html\n Please also refer to the documentation for alternative solver options:\n https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG)\n\n\n Logistic Regression accuracy: 92.26331862911435%\n K Nearest Neighbors Classifier accuracy: 88.73430607397353%\n Random Forest Classifier accuracy: 90.29521547336275%\n\n\n\n```python\nfrom matplotlib import cm\ncolors = cm.rainbow(np.linspace(0, 1, 4))\nlabels = ['Support Vector Classifier', 'Logsitic Regression', 'K Nearest Neighbors', 'Random Forest']\nplt.bar(labels,\n accuracy_scores,\n color = colors)\nplt.rcParams[\"figure.figsize\"] = (20, 8)\nplt.xlabel('Classifiers')\nplt.ylabel('Accuracy')\nplt.title('Accuracy of various algorithms')\n\n```\n\n#Noise Removal and Filtering \n\n\n\n```python\nsample_rate=50\n\n#Input Signal = x\ndef butter_lowpass(cutoff, nyq_freq, order=3):\n normal_cutoff = float(cutoff) / nyq_freq\n b, a = signal.butter(order, normal_cutoff, btype='lowpass')\n return b, a\n\ndef butter_lowpass_filter(data, cutoff_freq, nyq_freq, order):\n b, a = butter_lowpass(cutoff_freq, nyq_freq, order=order)\n y = signal.filtfilt(b, a, data,padlen=0)\n return y\n\n#Removing the Noise from Signal using Cutoff Freq. = 20Hz and Low Pass Butterworth Filter of Order 3\ndef removeNoise(x):\n x = butter_lowpass_filter(x, 20, sample_rate/2,order=3)\n return x\n \n#filtering the signal sperate tAccXYZ -> tBodyXYZ + tGravityXYZ\ndef sep(x):\n x = signal.medfilt(x, kernel_size=3) #Using Median filter to remove extra Noise\n tBodyAcc_ = butter_lowpass_filter(x, 0.3, sample_rate/2,order=4)\n tGravityAcc_ = np.array(x)-np.array(tBodyAcc_)\n return tBodyAcc_,tGravityAcc_\n\n# Visualize\n# plt.figure(figsize=(11, 9))\n# plt.plot(x, color='red', label=\"Original signal, {} samples\".format(signal_lenght))\n# plt.plot(tBodyAcc_, color='blue', label=\"Filtered low-pass with cutoff frequency of {} Hz\".format(cutoff_frequency))\n# plt.plot(tGravityAcc_, color='gray', label=\"What has been removed\")\n# plt.title(\"Signal and its filtering\")\n# plt.xlabel('Time (1/50th sec. per tick)')\n# plt.ylabel('Amplitude')\n# plt.legend()\n# plt.show()\n```\n\n\n```python\ndf = pd.DataFrame(np.nan, index=[0], columns=clean_cols)\n```\n\n\n```python\ndef calc_mean(total_signals):\n cols= []\n for f in clean_cols:\n if \"_mean()\" in f:\n cols.append(f)\n cols_strip = [f.replace(\"_mean()\",\"\") for f in cols]\n for i in range(len(cols)):\n if cols_strip[i] in total_signals:\n df.at[0, cols[i]]=np.mean(np.array(total_signals[cols_strip[i]]))\n \n\n\ndef calc_std(total_signals):\n cols= []\n for f in clean_cols:\n if \"_std()\" in f:\n cols.append(f)\n cols_strip = [f.replace(\"_std()\",\"\") for f in cols]\n for i in range(len(cols)):\n if cols_strip[i] in total_signals:\n df.at[0, cols[i]]=np.std(np.array(total_signals[cols_strip[i]]).astype(np.float32))\n \n\ndef calc_mad(total_signals):\n cols= []\n for f in clean_cols:\n if \"_mad()\" in f:\n cols.append(f)\n cols_strip = [f.replace(\"_mad()\",\"\") for f in cols]\n for i in range(len(cols)):\n if cols_strip[i] in total_signals:\n df.at[0, cols[i]]=stats.median_absolute_deviation(np.array(total_signals[cols_strip[i]]))\n \n\ndef calc_max(total_signals):\n cols= []\n for f in clean_cols:\n if \"_max()\" in f:\n cols.append(f)\n cols_strip = [f.replace(\"_max()\",\"\") for f in cols]\n for i in range(len(cols)):\n if cols_strip[i] in total_signals:\n df.at[0, cols[i]]=np.max(np.array(total_signals[cols_strip[i]]))\n \n\ndef calc_min(total_signals):\n cols= []\n for f in clean_cols:\n if \"_min()\" in f:\n cols.append(f)\n cols_strip = [f.replace(\"_min()\",\"\") for f in cols]\n for i in range(len(cols)):\n if cols_strip[i] in total_signals:\n df.at[0, cols[i]]=np.min(np.array(total_signals[cols_strip[i]]))\n \n\ndef calc_energy(total_signals):\n cols= []\n for f in clean_cols:\n if \"_energy()\" in f:\n cols.append(f)\n cols_strip = [f.replace(\"_energy()\",\"\") for f in cols]\n for i in range(len(cols)):\n if cols_strip[i] in total_signals:\n df.at[0, cols[i]]=np.sum(np.array(total_signals[cols_strip[i]])**2)/len(np.array(total_signals[cols_strip[i]]))\n \n\n\ndef calc_iqr(total_signals):\n cols= []\n for f in clean_cols:\n if \"_iqr()\" in f:\n cols.append(f)\n cols_strip = [f.replace(\"_iqr()\",\"\") for f in cols]\n for i in range(len(cols)):\n if cols_strip[i] in total_signals:\n df.at[0, cols[i]]=stats.iqr(np.array(total_signals[cols_strip[i]]),axis=0)\n \n \n\n# def calc_MaxInds(f_signals):\n# cols= []\n# for f in clean_cols:\n# if \"_MaxInds()\" in f:\n# cols.append(f)\n# cols_strip = [f.replace(\"_MaxInds()\",\"\") for f in cols]\n# for i in range(len(cols)):\n# if cols_strip[i] in f_signals:\n# df.at[0, cols[i]]=np.array(f_signals[cols[i]]).argmax(axis=0)\n \n\n# def calc_meanFreq(f_signals):\n# cols= []\n# for f in clean_cols:\n# if \"_meanFreq()\" in f:\n# cols.append(f)\n# cols_strip = [f.replace(\"_meanFreq()\",\"\") for f in cols]\n# for i in range(len(cols)):\n# if cols_strip[i] in f_signals:\n# weights = np.array([x for x in range(len(np.array(f_signals[cols_strip[i]])))])\n# weights += 1\n# df.at[0, cols[i]]=np.mean(np.array(f_signals[cols_strip[i]]).weights,axis=0)\n \n\n\ndef calc_skewness(f_signals):\n cols= []\n for f in clean_cols:\n if \"_skewness()\" in f:\n cols.append(f)\n cols_strip = [f.replace(\"_skewness()\",\"\") for f in cols]\n for i in range(len(cols)):\n if cols_strip[i] in f_signals:\n df.at[0, cols[i]]=stats.skew(np.array(f_signals[cols_strip[i]]))\n \n\n\ndef calc_kurtosis(f_signals):\n cols= []\n for f in clean_cols:\n if \"_kurtosis()\" in f:\n cols.append(f)\n cols_strip = [f.replace(\"-kurtosis()\",\"\") for f in cols]\n for i in range(len(cols)):\n if cols_strip[i] in f_signals:\n df.at[0, cols[i]]=stats.kurtosis(np.array(f_signals[cols_strip[i]]))\n \n```\n\n\n```python\n# the body linear acceleration and angular velocity were derived in time to obtain Jerk signals (tBodyAccJerk_XYZ and tBodyGyroJerk_XYZ). \ndef jerk_norm_fft(df,tBodyAcc_X,tBodyAcc_Y,tBodyAcc_Z,tGravityAcc_X,tGravityAcc_Y,tGravityAcc_Z,tGyro_X,tGyro_Y,tGyro_Z):\n \n #jerk\n tBodyAccJerk_X = np.append(np.array([2*tBodyAcc_X[0]-tBodyAcc_X[1]]),tBodyAcc_X[1:]- tBodyAcc_X[:-1])/0.02\n tBodyAccJerk_Y = np.append(np.array([2*tBodyAcc_Y[0]-tBodyAcc_Y[1]]),tBodyAcc_Y[1:]- tBodyAcc_Y[:-1])/0.02\n tBodyAccJerk_Z = np.append(np.array([2*tBodyAcc_Z[0]-tBodyAcc_Z[1]]),tBodyAcc_Z[1:]- tBodyAcc_Z[:-1])/0.02\n tBodyGyroJerk_X = np.append(np.array([2*tGyro_X[0]-tGyro_X[1]]),tGyro_X[1:]-tGyro_X[:-1])/0.02\n tBodyGyroJerk_Y = np.append(np.array([2*tGyro_Y[0]-tGyro_Y[1]]),tGyro_Y[1:]-tGyro_Y[:-1])/0.02\n tBodyGyroJerk_Z = np.append(np.array([2*tGyro_Z[0]-tGyro_Z[1]]),tGyro_Z[1:]-tGyro_Z[:-1])/0.02\n\n\n #norm\n tBodyAccMag =np.sqrt(tBodyAcc_X**2+tBodyAcc_Y**2+tBodyAcc_Z**2)\n tGravityAccMag =np.sqrt(tGravityAcc_X**2+tGravityAcc_Y**2+tGravityAcc_Z**2)\n tBodyAccJerkMag =np.sqrt(tBodyAccJerk_X**2+tBodyAccJerk_Y**2+tBodyAccJerk_Z**2)\n tBodyGyroMag =np.sqrt(tGyro_X**2+tGyro_Y**2+tGyro_Z**2)\n tBodyGyroJerkMag=np.sqrt(tBodyGyroJerk_X**2+tBodyGyroJerk_Y**2+tBodyGyroJerk_Z**2)\n\n t_signals = { 'tBodyAcc_X':tBodyAcc_X\n ,'tBodyAcc_Y':tBodyAcc_Y\n ,'tBodyAcc_Z':tBodyAcc_Z\n ,'tGravityAcc_X':tGravityAcc_X\n ,'tGravityAcc_Y':tGravityAcc_Y\n ,'tGravityAcc_Z':tGravityAcc_Z\n ,'tBodyGyro_X':tGyro_X\n ,'tBodyGyro_Y':tGyro_Y\n ,'tBodyGyro_Z':tGyro_Z\n ,'tBodyAccJerk_X':tBodyAccJerk_X\n ,'tBodyAccJerk_Y':tBodyAccJerk_Y\n ,'tBodyAccJerk_Z':tBodyAccJerk_Z\n ,'tBodyGyroJerk_X':tBodyGyroJerk_X\n ,'tBodyGyroJerk_Y':tBodyGyroJerk_Y\n ,'tBodyGyroJerk_Z':tBodyGyroJerk_Z\n ,'tBodyAccMag':tBodyAccMag \n ,'tGravityAccMag':tGravityAccMag\n ,'tBodyAccJerkMag':tBodyAccJerkMag\n ,'tBodyGyroMag':tBodyGyroMag\n ,'tBodyGyroJerkMag': tBodyGyroJerkMag\n }\n \n return t_signals\n\n```\n\n\n```python\ndef initiate(x):\n\n tAcc_X = x[:,0]\n tAcc_Y =x[:,1]\n tAcc_Z =x[:,2]\n tGyro_X =x[:,3]\n tGyro_Y =x[:,4]\n tGyro_Z =x[:,5]\n\n #Noise Removal\n tAcc_X = removeNoise(tAcc_X)\n tAcc_Y = removeNoise(tAcc_Y)\n tAcc_Z = removeNoise(tAcc_Z)\n tGyro_X = removeNoise(tGyro_X)\n tGyro_Y = removeNoise(tGyro_Y)\n tGyro_Z = removeNoise(tGyro_Z)\n\n #Accleration Signal Seperation into Body+Gravity \n tBodyAcc_X,tGravityAcc_X = sep(tAcc_X)\n tBodyAcc_Y,tGravityAcc_Y = sep(tAcc_Y)\n tBodyAcc_Z,tGravityAcc_Z = sep(tAcc_Z)\n\n t_signals=jerk_norm_fft(df,tBodyAcc_X,tBodyAcc_Y,tBodyAcc_Z,tGravityAcc_X,tGravityAcc_Y,tGravityAcc_Z,tGyro_X,tGyro_Y,tGyro_Z)\n \n return t_signals\n\ndef preprocess(x):\n\n total_signals = initiate(x)\n calc_mean(total_signals)\n calc_std(total_signals)\n calc_mad(total_signals)\n calc_max(total_signals)\n calc_min(total_signals)\n calc_energy(total_signals)\n calc_iqr(total_signals)\n # calc_maxInds(f_signals_names,f_signals)\n # calc_meanFreq(f_signals_names,f_signals)\n # calc_skewness(f_signals)\n # calc_kurtosis(f_signals)\n \n \n\n```\n\n\n```python\ndef prediction(x):\n x = x-x.mean(axis=0)/x.std(axis=0) #Normalization\n preprocess(x) #Preprocessing\n df.iloc[0]=row5\n row = np.expand_dims(df.iloc[0].values, axis=0)\n to = learn.dls.train_ds.new(df)\n to.process()\n dl = TabDataLoader(to)\n pds = learn.get_preds(dl=dl)\n nn_pred = learn.dls.vocab[pds[0].argmax()]\n\n #print predictions\n print(\"SVM Prediction : \", clf1.predict(row))\n print(\"Logistic Regression Prediction : \", clf2.predict(row))\n print(\"KNN Classifer Prediction: \", clf3.predict(row))\n print(\"Random Forest Prediction : \", clf4.predict(row))\n print(\"Tabular Neural Net Prediction : \", nn_pred)\n \n\n```\n\n\n```python\nx1 = np.array([\n[-1.211,1.768,8.564,-2.000,0.000,-3.000],\n[-1.213,1.706,8.576,-3.000,1.000,-2.000],\n[-1.158,1.716,8.564,0.000,2.000,-1.000],\n[-1.189,1.706,8.463,2.000,0.000,0.000],\n[-1.177,1.785,8.485,-2.000,0.000,-2.000],\n[-1.158,1.711,8.631,5.000,-2.000,-2.000],\n[-1.175,1.680,8.497,5.000,0.000,-1.000],\n[-1.213,1.687,8.542,2.000,2.000,0.000],\n[-1.259,1.716,8.492,4.000,1.000,0.000],\n[-1.235,1.771,8.509,0.000,4.000,-4.000],\n[-1.172,1.766,8.530,6.000,1.000,-4.000],\n[-1.242,1.797,8.533,7.000,-2.000,-2.000],\n[-1.158,1.718,8.511,1.000,2.000,1.000],\n[-1.244,1.718,8.473,2.000,-1.000,-1.000],\n[-1.204,1.737,8.480,0.000,-3.000,1.000],\n[-1.206,1.766,8.571,-4.000,-2.000,-4.000],\n[-1.199,1.665,8.480,-8.000,-2.000,2.000],\n[-1.125,1.701,8.530,2.000,1.000,-3.000],\n[-1.268,1.742,8.545,10.000,-3.000,-6.000],\n[-1.204,1.723,8.583,1.000,1.000,0.000],\n[-1.247,1.706,8.538,1.000,-2.000,-1.000],\n[-1.196,1.737,8.504,5.000,-4.000,1.000],\n[-1.168,1.711,8.629,-2.000,0.000,-2.000],\n[-1.189,1.773,8.396,-5.000,2.000,-1.000],\n[-1.110,1.718,8.564,-2.000,-2.000,-4.000],\n[-1.182,1.687,8.554,0.000,-4.000,-8.000],\n[-1.172,1.747,8.528,3.000,-2.000,-4.000],\n[-1.194,1.718,8.439,0.000,-2.000,1.000],\n[-1.206,1.725,8.509,2.000,1.000,-3.000],\n[-1.144,1.708,8.540,-4.000,1.000,-3.000],\n[-1.196,1.744,8.480,-3.000,1.000,1.000],\n[-1.218,1.737,8.506,-1.000,-1.000,-4.000],\n[-1.199,1.752,8.581,-4.000,1.000,-5.000],\n[-1.196,1.775,8.595,-1.000,1.000,-1.000],\n[-1.259,1.692,8.497,1.000,2.000,-6.000],\n[-1.141,1.639,8.416,2.000,4.000,-2.000],\n[-1.208,1.634,8.442,2.000,3.000,-7.000],\n[-1.184,1.711,8.559,10.000,4.000,-9.000],\n[-1.268,1.716,8.480,16.000,7.000,-5.000],\n[-1.237,1.759,8.530,15.000,2.000,-3.000],\n[-1.151,1.766,8.609,6.000,6.000,-5.000],\n[-1.175,1.790,8.550,-7.000,5.000,1.000],\n[-1.175,1.728,8.554,-7.000,6.000,2.000],\n[-1.158,1.673,8.528,-3.000,4.000,-4.000],\n[-1.244,1.675,8.585,9.000,4.000,-8.000],\n[-1.204,1.711,8.595,10.000,2.000,-12.000],\n[-1.201,1.708,8.509,12.000,2.000,-8.000],\n[-1.201,1.718,8.518,10.000,0.000,-5.000],\n[-1.244,1.811,8.502,11.000,0.000,1.000],\n[-1.208,1.783,8.454,-8.000,0.000,5.000],\n[-1.223,1.694,8.554,-15.000,0.000,7.000],\n[-1.146,1.665,8.547,-2.000,0.000,2.000],\n[-1.172,1.718,8.557,4.000,-2.000,-2.000],\n[-1.170,1.701,8.578,3.000,-2.000,-3.000],\n[-1.244,1.720,8.473,3.000,-4.000,-9.000],\n[-1.170,1.752,8.471,3.000,-4.000,-8.000],\n[-1.242,1.850,8.430,-3.000,-1.000,-5.000],\n[-1.177,1.735,8.533,-3.000,0.000,1.000],\n[-1.223,1.716,8.614,-5.000,0.000,1.000],\n[-1.302,1.718,8.511,-7.000,-3.000,1.000],\n[-1.211,1.699,8.490,-4.000,-4.000,-1.000],\n[-1.218,1.692,8.569,1.000,-2.000,-2.000],\n[-1.187,1.692,8.466,2.000,-1.000,-1.000],\n[-1.254,1.723,8.576,2.000,-2.000,-5.000],\n[-1.208,1.752,8.516,4.000,-1.000,-3.000],\n[-1.182,1.735,8.480,0.000,-1.000,-2.000],\n[-1.235,1.680,8.535,0.000,-1.000,-3.000],\n[-1.206,1.752,8.511,6.000,-3.000,-1.000],\n[-1.290,1.725,8.475,6.000,-3.000,-3.000],\n[-1.228,1.768,8.518,5.000,-2.000,1.000],\n[-1.180,1.735,8.475,2.000,-2.000,0.000],\n[-1.182,1.708,8.432,1.000,-1.000,-3.000],\n[-1.239,1.747,8.430,4.000,-1.000,-3.000],\n[-1.242,1.756,8.487,-4.000,-2.000,-2.000],\n[-1.208,1.766,8.566,0.000,-1.000,-7.000],\n[-1.247,1.704,8.550,-1.000,-6.000,-3.000],\n[-1.232,1.718,8.552,-3.000,-7.000,-3.000],\n[-1.161,1.735,8.468,4.000,-7.000,-1.000],\n[-1.196,1.732,8.502,-5.000,-2.000,-2.000],\n[-1.204,1.668,8.569,-7.000,0.000,0.000],\n[-1.263,1.711,8.487,2.000,-3.000,-3.000],\n[-1.151,1.680,8.550,7.000,-2.000,-2.000],\n[-1.208,1.716,8.511,0.000,-3.000,-1.000],\n[-1.199,1.713,8.538,1.000,1.000,2.000],\n[-1.225,1.756,8.523,1.000,-3.000,-3.000],\n[-1.230,1.742,8.461,-1.000,-2.000,-5.000],\n[-1.249,1.749,8.475,-2.000,-2.000,-3.000],\n[-1.189,1.730,8.600,3.000,0.000,-5.000],\n[-1.208,1.699,8.497,1.000,0.000,1.000],\n[-1.244,1.711,8.514,0.000,0.000,-3.000],\n[-1.244,1.754,8.569,7.000,2.000,-5.000],\n[-1.228,1.706,8.480,4.000,0.000,-2.000],\n[-1.218,1.725,8.487,0.000,-2.000,-3.000],\n[-1.168,1.687,8.538,5.000,-2.000,-1.000],\n[-1.161,1.732,8.475,2.000,1.000,4.000],\n[-1.196,1.689,8.413,-4.000,-1.000,-1.000],\n[-1.204,1.725,8.432,4.000,-1.000,-3.000],\n[-1.213,1.701,8.499,7.000,-2.000,-3.000],\n[-1.223,1.665,8.442,4.000,-2.000,-3.000],\n[-1.163,1.773,8.418,5.000,-3.000,-5.000],\n[-1.199,1.773,8.564,4.000,-3.000,3.000],\n[-1.235,1.768,8.497,-4.000,-4.000,-1.000],\n[-1.204,1.816,8.466,-9.000,-3.000,1.000],\n[-1.184,1.740,8.511,-4.000,-3.000,-2.000],\n[-1.184,1.658,8.530,-1.000,-7.000,2.000],\n[-1.149,1.725,8.542,-2.000,-4.000,-1.000],\n[-1.199,1.694,8.518,2.000,-5.000,-9.000],\n[-1.218,1.742,8.588,8.000,-3.000,-5.000],\n[-1.249,1.701,8.514,1.000,-1.000,-3.000],\n[-1.232,1.692,8.497,-3.000,-2.000,0.000],\n[-1.180,1.720,8.344,4.000,-1.000,0.000],\n[-1.182,1.766,8.545,8.000,1.000,-6.000],\n[-1.220,1.742,8.581,0.000,2.000,-5.000],\n[-1.223,1.747,8.518,4.000,4.000,-5.000],\n[-1.263,1.725,8.428,10.000,2.000,-6.000],\n[-1.151,1.761,8.456,3.000,1.000,-5.000],\n[-1.237,1.708,8.466,1.000,1.000,-5.000],\n[-1.146,1.725,8.461,4.000,3.000,-3.000],\n[-1.156,1.716,8.442,-6.000,3.000,-2.000],\n[-1.232,1.787,8.459,-1.000,2.000,-4.000],\n[-1.204,1.749,8.468,-1.000,-2.000,-3.000],\n[-1.218,1.787,8.480,6.000,-3.000,-2.000],\n[-1.208,1.737,8.528,3.000,-3.000,-3.000],\n[-1.211,1.728,8.523,1.000,-4.000,-4.000],\n[-1.196,1.802,8.550,-3.000,-3.000,3.000],\n[-1.230,1.713,8.495,-4.000,-1.000,3.000],\n[-1.225,1.730,8.502,0.000,-2.000,4.000],\n[-1.220,1.680,8.547,1.000,-1.000,-3.000],\n[-1.228,1.759,8.578,5.000,1.000,-1.000],\n[-1.242,1.780,8.530,5.000,2.000,-1.000],\n[-1.204,1.754,8.425,-1.000,0.000,-2.000],\n[-1.251,1.759,8.463,-6.000,-1.000,-2.000],\n[-1.196,1.838,8.571,-3.000,0.000,-4.000],\n[-1.208,1.732,8.471,-7.000,-1.000,0.000],\n[-1.196,1.728,8.411,-1.000,2.000,-1.000],\n[-1.170,1.811,8.380,7.000,1.000,-4.000],\n[-1.230,1.790,8.454,4.000,0.000,-4.000],\n[-1.251,1.744,8.542,-2.000,-1.000,-3.000],\n[-1.242,1.802,8.404,-6.000,1.000,0.000],\n[-1.211,1.644,8.581,-2.000,3.000,0.000],\n[-1.165,1.742,8.557,3.000,4.000,-3.000],\n[-1.192,1.723,8.437,5.000,6.000,-2.000],\n[-1.206,1.708,8.506,4.000,2.000,-7.000],\n[-1.247,1.720,8.439,2.000,3.000,-3.000],\n[-1.149,1.764,8.516,0.000,3.000,0.000],\n[-1.163,1.802,8.466,-1.000,5.000,-6.000],\n[-1.228,1.723,8.576,-6.000,4.000,-5.000],\n[-1.232,1.725,8.679,-4.000,3.000,-1.000],\n[-1.283,1.728,8.483,5.000,3.000,-5.000],\n[-1.206,1.742,8.416,0.000,-2.000,-6.000],\n[-1.204,1.708,8.506,4.000,-3.000,-1.000],\n[-1.239,1.795,8.545,-5.000,-5.000,1.000],\n[-1.237,1.718,8.571,-10.000,-1.000,3.000],\n[-1.213,1.699,8.502,-7.000,3.000,2.000],\n[-1.180,1.730,8.533,-5.000,4.000,-3.000],\n[-1.242,1.680,8.535,-2.000,4.000,-3.000],\n[-1.266,1.735,8.526,9.000,1.000,-4.000],\n[-1.216,1.735,8.530,7.000,1.000,-5.000],\n[-1.180,1.740,8.370,-4.000,1.000,-5.000],\n[-1.206,1.687,8.542,-11.000,4.000,-4.000],\n[-1.242,1.759,8.463,-11.000,5.000,-4.000],\n[-1.230,1.701,8.566,-8.000,2.000,-2.000],\n[-1.244,1.658,8.518,-5.000,3.000,-3.000],\n[-1.278,1.680,8.459,4.000,-3.000,-4.000],\n[-1.213,1.713,8.466,13.000,-1.000,-2.000],\n[-1.266,1.740,8.562,-2.000,-1.000,-2.000],\n[-1.323,1.778,8.413,-3.000,-4.000,3.000],\n[-1.211,1.711,8.566,3.000,2.000,-1.000],\n[-1.213,1.747,8.550,3.000,4.000,5.000],\n[-1.144,1.673,8.518,-7.000,5.000,4.000],\n[-1.235,1.670,8.478,7.000,7.000,-3.000],\n[-1.220,1.730,8.490,9.000,3.000,-2.000],\n[-1.151,1.792,8.509,2.000,0.000,-3.000],\n[-1.168,1.742,8.392,-1.000,0.000,-3.000],\n[-1.218,1.804,8.502,4.000,3.000,-6.000],\n[-1.333,1.725,8.459,5.000,-2.000,-4.000],\n[-1.354,1.754,8.540,-2.000,-1.000,-1.000],\n[-1.302,1.725,8.504,-11.000,-8.000,1.000],\n[-1.280,1.649,8.521,-8.000,-4.000,5.000],\n[-1.235,1.771,8.473,0.000,-3.000,3.000],\n[-1.182,1.728,8.451,-3.000,-7.000,3.000],\n[-1.172,1.716,8.600,-2.000,-9.000,6.000],\n[-1.189,1.754,8.485,-3.000,-7.000,3.000],\n[-1.208,1.668,8.523,-2.000,-5.000,2.000],\n[-1.192,1.675,8.550,-6.000,-7.000,2.000],\n[-1.206,1.689,8.538,-7.000,-3.000,-7.000],\n[-1.189,1.716,8.538,5.000,-4.000,-5.000],\n[-1.177,1.732,8.490,5.000,0.000,-3.000],\n[-1.180,1.694,8.576,4.000,-6.000,-1.000],\n[-1.261,1.742,8.425,1.000,-2.000,-2.000],\n[-1.242,1.699,8.547,3.000,-2.000,-2.000],\n[-1.237,1.756,8.502,4.000,0.000,-4.000],\n[-1.261,1.735,8.459,0.000,-3.000,0.000],\n[-1.263,1.675,8.449,1.000,-4.000,2.000],\n[-1.244,1.687,8.691,7.000,-4.000,-6.000],\n[-1.189,1.730,8.499,9.000,2.000,-4.000],\n[-1.225,1.761,8.495,6.000,1.000,0.000],\n[-1.228,1.768,8.449,6.000,-1.000,-3.000],\n[-1.208,1.842,8.478,6.000,1.000,-1.000],\n[-1.196,1.730,8.432,-4.000,-1.000,-1.000],\n[-1.170,1.752,8.502,-1.000,3.000,-1.000],\n[-1.177,1.730,8.396,-2.000,1.000,-1.000],\n[-1.230,1.704,8.518,0.000,-3.000,-5.000],\n[-1.235,1.673,8.447,7.000,-1.000,-5.000],\n[-1.271,1.744,8.504,11.000,-1.000,-4.000],\n[-1.220,1.725,8.435,3.000,-2.000,-3.000],\n[-1.189,1.723,8.461,-6.000,-2.000,-2.000],\n[-1.242,1.708,8.384,-6.000,-4.000,-2.000],\n[-1.235,1.759,8.518,-6.000,-3.000,2.000],\n[-1.151,1.708,8.497,-11.000,-2.000,-1.000],\n[-1.187,1.668,8.557,4.000,3.000,-2.000],\n[-1.211,1.661,8.595,7.000,-1.000,-4.000],\n[-1.208,1.732,8.487,4.000,2.000,0.000],\n[-1.242,1.759,8.439,5.000,0.000,-2.000],\n[-1.254,1.728,8.523,0.000,-2.000,-1.000],\n[-1.220,1.723,8.535,-4.000,-3.000,2.000],\n[-1.184,1.754,8.490,-5.000,0.000,4.000],\n[-1.175,1.732,8.468,7.000,-1.000,-2.000],\n[-1.254,1.764,8.564,4.000,-2.000,3.000],\n[-1.177,1.682,8.463,-1.000,1.000,-3.000],\n[-1.211,1.752,8.435,6.000,-2.000,-4.000],\n[-1.182,1.718,8.468,4.000,0.000,-5.000],\n[-1.218,1.761,8.454,3.000,-1.000,-4.000],\n[-1.180,1.685,8.578,4.000,-2.000,-1.000],\n[-1.187,1.764,8.526,-1.000,-5.000,-4.000],\n[-1.216,1.819,8.557,-1.000,-1.000,-4.000],\n[-1.153,1.754,8.578,-7.000,-1.000,-1.000],\n[-1.208,1.723,8.461,2.000,0.000,-2.000],\n[-1.194,1.730,8.538,2.000,-2.000,-7.000],\n[-1.287,1.689,8.473,-1.000,-5.000,-4.000],\n[-1.295,1.807,8.538,5.000,-2.000,-2.000],\n[-1.182,1.701,8.569,2.000,-2.000,-6.000],\n[-1.213,1.785,8.523,-4.000,-2.000,1.000],\n[-1.206,1.668,8.492,-1.000,-1.000,-3.000],\n[-1.218,1.713,8.564,-3.000,-3.000,-3.000],\n[-1.168,1.675,8.483,0.000,-1.000,-3.000],\n[-1.182,1.694,8.516,4.000,-1.000,-2.000],\n[-1.218,1.723,8.545,7.000,1.000,0.000],\n[-1.196,1.708,8.425,12.000,0.000,-5.000],\n[-1.187,1.720,8.473,4.000,-2.000,-3.000],\n[-1.251,1.725,8.384,-4.000,-3.000,-4.000],\n[-1.218,1.737,8.466,-6.000,-1.000,0.000] \n])\n```\n\n\n```python\nprediction(x1)\n```\n\n\n\n\n\n SVM Prediction : ['LAYING']\n Logistic Regression Prediction : ['SITTING']\n KNN Classifer Prediction: ['WALKING_DOWNSTAIRS']\n Random Forest Prediction : ['WALKING_UPSTAIRS']\n Tabular Neural Net Prediction : WALKING_UPSTAIRS\n\n", "meta": {"hexsha": "3d7d3ce9d7be869ddf9aea100bfd5beba5be289b", "size": 218897, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "HAR_Final (1) (1).ipynb", "max_stars_repo_name": "lik-It/Human-Activity-Recognition-System---Mini-project", "max_stars_repo_head_hexsha": "fbdf36315c8d5f02936b525361d30373fcf86acb", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "HAR_Final (1) (1).ipynb", "max_issues_repo_name": "lik-It/Human-Activity-Recognition-System---Mini-project", "max_issues_repo_head_hexsha": "fbdf36315c8d5f02936b525361d30373fcf86acb", 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YES\n2. YES", "lm_q1_score": 0.6757645879592642, "lm_q2_score": 0.6261241632752916, "lm_q1q2_score": 0.4231125372070665}} {"text": "# Notes on current version:\nFor TOC if missing from command line try\njupyter nbextensions_configurator enable\nthen toggle nbextensions, restart.\n\n1. 1.9.2020 Managed to convert ODE models for economic extension to transition model ready for stochastic simulation, using separate birth death list\n See section on SC2UIR model. Not done for other two economic extensions yet\n2. 1.9.2020 Implemented stochastic simulation (Tau-leap method) using PyGom inbuilt capabilities: for SCIR simulation only so far\n Neeed to use integer N>>1, not 1.0, for stochastic simulation. Calculates in a few minutes for N=10000, rescaled ICUfrac to 0.02 (x10). N=100000 didn't finish in 10m.\n\n# Model Definitions\n\n## Utilities for custom extension of PyGom\n\n\n```python\n# import required packages\nimport os \nimport csv\nfrom sympy import symbols, init_printing\nimport numpy as np\nimport matplotlib\n%matplotlib inline\nimport seaborn as sb\nfrom matplotlib import pyplot as plt\nimport sympy\nimport itertools\nimport scipy\nimport datetime\nimport matplotlib.dates as mdates\nfrom pygom import DeterministicOde, Transition, SimulateOde, TransitionType, SquareLoss\nfrom scipy.optimize import minimize\n\nimport pickle as pk\nimport jsonpickle as jpk\n\nfrom cycler import cycler\nfrom pandas.plotting import register_matplotlib_converters\nregister_matplotlib_converters()\nimport pwlf\n```\n\n /Users/n/.pyenv/versions/3.7.2/lib/python3.7/site-packages/statsmodels/tools/_testing.py:19: FutureWarning: pandas.util.testing is deprecated. Use the functions in the public API at pandas.testing instead.\n import pandas.util.testing as tm\n\n\n\n```python\n\n# This cell adds two methods to the DeterministicODE class of pygom\n# dumpparams: stores params in a file './params/Model_Name.pk'\n# loadparams: loads params from same file. returns None if any problem finding the file.\n# e.g. will be accessed by SCIR.dumparams() or SCIR.loadparams()\n# This stuff needs modules os, sys, pickle as pk.\n\ndef dumpparams(self,run_id=''): # Have to add self since this will become a method\n mname = self.modelname\n dirnm = os.getcwd()\n pfile = dirnm+'/params/'+mname+'.pk'\n try:\n params = self.params.copy()\n with open(pfile,'wb') as fp:\n pk.dump(params,fp,protocol=pk.HIGHEST_PROTOCOL)\n print('dumped params to',pfile)\n if run_id != '':\n pfile2 = dirnm+'/params/'+run_id+'.pk'\n with open(pfile2,'wb') as fp:\n pk.dump(params,fp,protocol=pk.HIGHEST_PROTOCOL)\n print('dumped params to',pfile2)\n except:\n print('problem dumping params to ',pfile)\n\n\ndef loadparams(self,run_id=''): # Have to add self since this will become a method\n mname = self.modelname\n dirnm = os.getcwd()\n if run_id == '':\n pfile = dirnm+'/params/'+mname+'.pk'\n else:\n pfile = dirnm+'/params/'+run_id+'.pk'\n try:\n with open(pfile,'rb') as fp:\n params = pk.load(fp)\n print('loaded params from ',pfile,':')\n except:\n print(\"problem loading\",pfile)\n return None\n\n\n nms = [x.name for x in self.param_list]\n try:\n self.params = params.copy()\n self.parameters = params.copy()\n except:\n print('problem loading the params; none loaded')\n return None\n return True\n\nOdeClass = DeterministicOde().__class__\nsetattr(OdeClass,'dumpparams', dumpparams)\nsetattr(OdeClass,'loadparams', loadparams)\n\n\n\n\n\n\n```\n\n\n```python\ndef Float(x):\n try:\n rtn = float(x)\n except:\n rtn = float('NaN')\n return rtn\n```\n\n\n```python\ndef print_ode2(self):\n '''\n Prints the ode in symbolic form onto the screen/console in actual\n symbols rather than the word of the symbol.\n \n Based on the PyGOM built-in but adapted for Jupyter\n Corrected by John McCaskill to avoid subscript format error\n '''\n A = self.get_ode_eqn()\n B = sympy.zeros(A.rows,2)\n for i in range(A.shape[0]):\n B[i,0] = sympy.symbols('d' + '{' + str(self._stateList[i]) + '}'+ '/dt=')\n B[i,1] = A[i]\n\n return B\n```\n\n\n```python\n# Jupyter Specifics\nfrom IPython.display import display, HTML\nfrom ipywidgets.widgets import interact, interactive, IntSlider, FloatSlider, Layout, ToggleButton, ToggleButtons, fixed\ndisplay(HTML(\"\"))\nstyle = {'description_width': '100px'}\nslider_layout = Layout(width='99%')\n```\n\n\n\n\n\n## Caution Extensions to SIR Model\n\n### SIR model\n\n#### Equations\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -\\beta I S\\\\\n\\dot{I} &= \\beta I S - \\gamma I - \\mu I\\\\\n\\dot{R} & = \\gamma I \\\\\n\\dot{D} & = \\mu I\n\\end{split}\n\\end{equation}\n\n\n#### Variables\n* $S$: Susceptible individuals\n* $I$: Infected individuals \n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+I+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta$ rate at which infected individuals contact susceptibles and infect them\n* $\\gamma$ rate at which infected individuals recover from disease and become immune\n* $\\mu$ death rate for infected individuals\n\n#### Implementation\nUsing PyGOM, we will set up my simple SCIR model ODE system\nPyGOM \u2013 A Python Package for Simplifying Modelling with Systems of Ordinary Differential Equations https://arxiv.org/pdf/1803.06934.pdf\n\n\n```python\n# set up the symbolic SIR model, actually SIRD including deaths\n\nstate = ['S', 'I', 'R', 'D']\nparam_list = ['beta', 'gamma','mu','N']\n\ntransition = [\n Transition(origin='S', destination='I', equation='beta*I*S',\n transition_type=TransitionType.T),\n Transition(origin='I', destination='R', equation='gamma*I',\n transition_type=TransitionType.T),\n Transition(origin='I', destination='D', equation='mu*I',\n transition_type=TransitionType.T) \n ]\n\nSIR_model = DeterministicOde(state, param_list, transition=transition)\nSIR_model.modelname='SIR'\nSIR_model.ei=1\nSIR_model.confirmed=slice(1,4) # cases 1-3 i.e. I, R and D\nSIR_model.recovered=slice(2,3)\nSIR_model.deaths=slice(3,4)\n```\n\n\n```python\n# display equations\nprint_ode2(SIR_model)\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}d{S}/dt= & - I S \\beta\\\\d{I}/dt= & I S \\beta - I \\gamma - I \\mu\\\\d{R}/dt= & I \\gamma\\\\d{D}/dt= & I \\mu\\end{matrix}\\right]$\n\n\n\n\n```python\n# display graphical representation of the model\nSIR_model.get_transition_graph()\n```\n\n##### Derived equations, Jacobian and gradient\n\n\n```python\nSIR_model.get_ode_eqn()\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}- I S \\beta\\\\I S \\beta - I \\gamma - I \\mu\\\\I \\gamma\\\\I \\mu\\end{matrix}\\right]$\n\n\n\n\n```python\nSIR_model.get_jacobian_eqn()\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}- I \\beta & - S \\beta & 0 & 0\\\\I \\beta & S \\beta - \\gamma - \\mu & 0 & 0\\\\0 & \\gamma & 0 & 0\\\\0 & \\mu & 0 & 0\\end{matrix}\\right]$\n\n\n\n\n```python\nSIR_model.get_grad_eqn()\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}- I S & 0 & 0 & 0\\\\I S & - I & - I & 0\\\\0 & I & 0 & 0\\\\0 & 0 & I & 0\\end{matrix}\\right]$\n\n\n\n#### R0\n\n\n```python\nfrom pygom.model.epi_analysis import R0\n```\n\n\n```python\ntransition_ode = [\n Transition(origin='S', equation='-beta*I*S'),\n Transition(origin='I', equation='beta*I*S-gamma*I-mu*I'),\n Transition(origin='R', equation='gamma*I'),\n Transition(origin='D', equation='mu*I') \n ]\node = SimulateOde(state, param_list, ode=transition_ode)\node = ode.get_unrolled_obj()\node.get_transition_graph()\n```\n\n\n```python\nR0(ode,['I'])\n```\n\n\n\n\n$\\displaystyle 0$\n\n\n\n\n```python\nimport sympy.matrices.matrices\n# from sympy import *\n```\n\n### SCIR model\n\n#### Equations\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -\\beta I S + c_1 S_c - c_2*S*I\\\\\n\\dot{S_c} &= - c_0 \\beta I S_c - c_1 S_c + c_2*S*I\\\\\n\\dot{I} &= \\beta I S - \\gamma I - \\mu I\\\\\n\\dot{R} & = \\gamma I \\\\\n\\dot{D} & = \\mu I\n\\end{split}\n\\end{equation}\n\nThe use of I as a state variable triggering susceptibles to execute caution is just one choice. In contrast with deaths, it does not accumulate over time and so retains the property of an active threat to society, rather than an historical one. Alternatively, one could use the daily death rate which is proportional to it.\n\nActually, the effect of caution may be quadratic, since both the individual doing the infection and individual potentially being infected may be executing caution. The current version assumes that infectives do not change their precautionary measures in response to I. To implement this we would need to further distinguish careful and non careful infectives. This is done in the SC2IR model.\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $S_c$: Susceptible individuals exercising pandemy precautions\n* $I$: Infected individuals \n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+S_c+I+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta$ rate at which infected individuals contact susceptibles and infect them\n* $\\gamma$ rate at which infected individuals recover from disease and become immune\n* $\\mu$ death rate for infected individuals\n\n* $c_i$ three parameters characterizing cautionary response of population via class $S_c$\n\n - c_0 : reduction factor for exposure for cautioned susceptibles\n\n - c_1 : duration of caution (exponential decay time constant in days)\n\n - c_2 : rate constant for transition from uncautioned to cautioned susceptible\n\n#### Implementation\nUsing PyGOM, we will set up my simple SCIR model ODE system\nPyGOM \u2013 A Python Package for Simplifying Modelling with Systems of Ordinary Differential Equations https://arxiv.org/pdf/1803.06934.pdf\n\n\n```python\n# set up symbolic model\n\nstate = ['S', 'I', 'R', 'D', 'S_c']\nparam_list = ['beta', 'gamma', 'mu', 'c_0', 'c_1', 'c_2', 'N']\n\ntransition = [\n Transition(origin='S', destination='I', equation='beta*I*S',\n transition_type=TransitionType.T),\n Transition(origin='S', destination='S_c', equation='c_2*I*S',\n transition_type=TransitionType.T),\n Transition(origin='S_c', destination='S', equation='c_1*S_c',\n transition_type=TransitionType.T),\n Transition(origin='S_c', destination='I', equation='c_0*beta*I*S_c',\n transition_type=TransitionType.T),\n Transition(origin='I', destination='R', equation='gamma*I',\n transition_type=TransitionType.T),\n Transition(origin='I', destination='D', equation='mu*I',\n transition_type=TransitionType.T) \n ]\n\nSCIR_model = DeterministicOde(state, param_list, transition=transition)\n\nSCIR_modelS = SimulateOde(state, param_list , transition=transition)\nSCIR_model.modelname='SCIR'\nSCIR_model.ei=1\nSCIR_model.confirmed=slice(1,4) # cases 1-3 i.e. I, R and D\nSCIR_model.recovered=slice(2,3)\nSCIR_model.deaths=slice(3,4)\n\n\n\n```\n\n\n```python\n# display equations\nprint_ode2(SCIR_model)\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}d{S}/dt= & - I S \\beta - I S c_{2} + S_{c} c_{1}\\\\d{I}/dt= & I S \\beta + I S_{c} \\beta c_{0} - I \\gamma - I \\mu\\\\d{R}/dt= & I \\gamma\\\\d{D}/dt= & I \\mu\\\\d{S_c}/dt= & I S c_{2} - I S_{c} \\beta c_{0} - S_{c} c_{1}\\end{matrix}\\right]$\n\n\n\n\n```python\n# display graphical representation of the model\nSCIR_model.get_transition_graph()\n```\n\n### SC2IR model\n\n#### Equations\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -\\beta (I + c_0 I_c) S + c_1 S_c - c_2 S (I + I_c)\\\\\n\\dot{S_c} &= - c_0 \\beta (I + c_0 I_c) S_c - c_1 S_c + c_2 S (I + I_c)\\\\\n\\dot{I} &= \\beta (I + c_0 I_c) S - \\gamma I - \\mu I + c_1 I_c - c_2 I (I + I_c)\\\\\n\\dot{I_c} &= \\beta (I + c_0 I_c) S_c - \\gamma I_c - \\mu I_c - c_1 I_c + c_2 I (I + I_c)\\\\\n\\dot{R} & = \\gamma (I + I_c)\\\\\n\\dot{D} & = \\mu (I + I_c)\n\\end{split}\n\\end{equation}\n\nThe use of I as a state variable triggering susceptibles to execute caution is just one choice. In contrast with deaths, it does not accumulate over time and so retains the property of an active threat to society, rather than an historical one. Alternatively, one could use the daily death rate which is proportional to it.\n\nThe effect of caution may be quadratic, since both the individual doing the infection and individual potentially being infected may be executing caution. To implement this we distinguish careful and non careful infectives. We ignore infectives making the transition to caution or relaxing it.\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $S_c$: Susceptible individuals exercising pandemy precautions\n* $I$: Infected individuals non exercising pandemy precautions\n* $I_c$: Infected individuals exercising pandemy precautions \n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+S_c+I+I_c+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta$ rate at which infected individuals contact susceptibles and infect them\n* $\\gamma$ rate at which infected individuals recover from disease and become immune\n* $\\mu$ death rate for infected individuals\n\n* $c_i$ three parameters characterizing cautionary response of population via class $S_c$\n\n - c_0 : reduction factor for exposure for cautioned susceptibles\n\n - c_1 : duration of caution (exponential decay time constant in days)\n\n - c_2 : rate constant for transition from uncautioned to cautioned susceptible\n\n#### Implementation\nUsing PyGOM, we will set up my simple SCIR model ODE system\nPyGOM \u2013 A Python Package for Simplifying Modelling with Systems of Ordinary Differential Equations https://arxiv.org/pdf/1803.06934.pdf\n\n\n```python\n# set up symbolic model\n\nstate = ['S', 'I', 'R', 'D', 'I_c', 'S_c']\nparam_list = ['beta', 'gamma', 'mu', 'c_0', 'c_1', 'c_2', 'N']\n\ntransition = [\n Transition(origin='S', destination='I', equation='beta*(I+c_0*I_c)*S',\n transition_type=TransitionType.T),\n Transition(origin='S', destination='S_c', equation='c_2*(I+I_c)*S',\n transition_type=TransitionType.T),\n Transition(origin='S_c', destination='S', equation='c_1*S_c',\n transition_type=TransitionType.T),\n Transition(origin='S_c', destination='I_c', equation='c_0*beta*(I+c_0*I_c)*S_c',\n transition_type=TransitionType.T),\n Transition(origin='I', destination='R', equation='gamma*I',\n transition_type=TransitionType.T),\n Transition(origin='I', destination='D', equation='mu*I',\n transition_type=TransitionType.T),\n Transition(origin='I', destination='I_c', equation='c_2*(I+I_c)*I',\n transition_type=TransitionType.T),\n Transition(origin='I_c', destination='R', equation='gamma*I_c',\n transition_type=TransitionType.T),\n Transition(origin='I_c', destination='I', equation='c_1*I_c',\n transition_type=TransitionType.T),\n Transition(origin='I_c', destination='D', equation='mu*I_c',\n transition_type=TransitionType.T) #, \n ]\n\nSC2IR_model = DeterministicOde(state, param_list, transition=transition)\nSC2IR_model.modelname='SC2IR'\n\nSC2IR_model.ei=1\nSC2IR_model.confirmed=slice(1,5) # cases 1-3 i.e. I, R and D\nSC2IR_model.recovered=slice(2,3)\nSC2IR_model.deaths=slice(3,4)\n```\n\n\n```python\n# display equations\nprint_ode2(SC2IR_model)\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}d{S}/dt= & - S \\beta \\left(I + I_{c} c_{0}\\right) - S c_{2} \\left(I + I_{c}\\right) + S_{c} c_{1}\\\\d{I}/dt= & - I c_{2} \\left(I + I_{c}\\right) - I \\gamma - I \\mu + I_{c} c_{1} + S \\beta \\left(I + I_{c} c_{0}\\right)\\\\d{R}/dt= & I \\gamma + I_{c} \\gamma\\\\d{D}/dt= & I \\mu + I_{c} \\mu\\\\d{I_c}/dt= & I c_{2} \\left(I + I_{c}\\right) - I_{c} c_{1} - I_{c} \\gamma - I_{c} \\mu + S_{c} \\beta c_{0} \\left(I + I_{c} c_{0}\\right)\\\\d{S_c}/dt= & S c_{2} \\left(I + I_{c}\\right) - S_{c} \\beta c_{0} \\left(I + I_{c} c_{0}\\right) - S_{c} c_{1}\\end{matrix}\\right]$\n\n\n\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -\\beta (I + c_0 I_c) S + c_1 S_c - c_2 S (I + I_c)\\\\\n\\dot{S_c} &= - c_0 \\beta (I + c_0 I_c) S_c - c_1 S_c + c_2 S (I + I_c)\\\\\n\\dot{I} &= \\beta (I + c_0 I_c) S - \\gamma I - \\mu I\\\\\n\\dot{I_c} &= \\beta (I + c_0 I_c) S_c - \\gamma I_c - \\mu I_c\\\\\n\\dot{R} & = \\gamma (I + I_c)\\\\\n\\dot{D} & = \\mu (I + I_c)\n\\end{split}\n\\end{equation}\n\n\n\n```python\n# display graphical representation of the model\nSC2IR_model.get_transition_graph()\n```\n\n## Caution Extensions to SEIR Model\n\n### SEIR model\n\n#### Equations\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -\\beta I S\\\\\n\\dot{E} &= \\beta I S - \\alpha E\\\\\n\\dot{I} &= \\alpha E - \\gamma I - \\mu I\\\\\n\\dot{R} & = \\gamma I \\\\\n\\dot{D} & = \\mu I\n\\end{split}\n\\end{equation}\n\n\n#### Variables\n* $S$: Susceptible individuals\n* $E$: Exposed individuals\n* $I$: Infected individuals \n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+E+I+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta$ rate at which infected individuals contact susceptibles and expose them to infection\n* $\\alpha$ rate at which exposed individuals become infected (1/(incubation time)\n* $\\gamma$ rate at which infected individuals recover from disease and become immune\n* $\\mu$ death rate for infected individuals\n\n#### Implementation\nUsing PyGOM, we will set up my simple SCIR model ODE system\nPyGOM \u2013 A Python Package for Simplifying Modelling with Systems of Ordinary Differential Equations https://arxiv.org/pdf/1803.06934.pdf\n\n\n```python\n# set up the symbolic model\nstate = ['S', 'E', 'I', 'R', 'D']\nparam_list = ['beta', 'alpha', 'gamma', 'mu', 'N']\n\ntransition = [\n Transition(origin='S', destination='E', equation='beta*I*S',\n transition_type=TransitionType.T),\n Transition(origin='E', destination='I', equation='alpha*E',\n transition_type=TransitionType.T),\n Transition(origin='I', destination='R', equation='gamma*I',\n transition_type=TransitionType.T),\n Transition(origin='I', destination='D', equation='mu*I',\n transition_type=TransitionType.T) \n ]\n\nSEIR_model = DeterministicOde(state, param_list, transition=transition)\nSEIR_model.modelname='SEIR'\nSEIR_model.ei=slice(1,3) # cases 1,2 i.e. E and I\nSEIR_model.confirmed=slice(2,5) # cases 2-4 i.e. I, R and D, not E\nSEIR_model.recovered=slice(3,4)\nSEIR_model.deaths=slice(4,5)\n\n```\n\n\n```python\n# display equations\nprint_ode2(SEIR_model)\n```\n\n\n```python\n# display graphical representation of the model\nSEIR_model.get_transition_graph()\n```\n\n### SCEIR model\n\n#### Equations\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -\\beta I S + c_1 S_c - c_2*S*I\\\\\n\\dot{S_c} &= - c_0 \\beta I S_c - c_1 S_c + c_2*S*I\\\\\n\\dot{E} &= \\beta I (S + c_0 S_c) - \\alpha E\\\\\n\\dot{I} &= \\alpha E - \\gamma I - \\mu I\\\\\n\\dot{R} & = \\gamma I \\\\\n\\dot{D} & = \\mu I\n\\end{split}\n\\end{equation}\n\nThe use of I as a state variable triggering susceptibles to execute caution is just one choice. In contrast with deaths, it does not accumulate over time and so retains the property of an active threat to society, rather than an historical one. Alternatively, one could use the daily death rate which is proportional to it.\n\nActually, the effect of caution may be quadratic, since both the individual doing the infection and individual potentially being infected may be executing caution. The current version assumes that infectives do not change their precautionary measures in response to I. To implement this we would need to further distinguish careful and non careful infectives. This is done in the SC2IR model.\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $S_c$: Susceptible individuals exercising pandemy precautions\n* $E$: Exposed individuals\n* $I$: Infected individuals \n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+S_c+E+I+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta$ rate at which infected individuals contact susceptibles and infect them\n* $\\alpha$ rate at which exposed individuals become infected (1/(incubation time)\n* $\\gamma$ rate at which infected individuals recover from disease and become immune\n* $\\mu$ death rate for infected individuals\n\n* $c_i$ three parameters characterizing cautionary response of population via class $S_c$\n\n - c_0 : reduction factor for exposure for cautioned susceptibles\n\n - c_1 : duration of caution (exponential decay time constant in days)\n\n - c_2 : rate constant for transition from uncautioned to cautioned susceptible\n\n#### Implementation\nUsing PyGOM, we will set up my simple SCIR model ODE system\nPyGOM \u2013 A Python Package for Simplifying Modelling with Systems of Ordinary Differential Equations https://arxiv.org/pdf/1803.06934.pdf\n\n\n```python\n# set up the symbolic model\n\nstate = ['S', 'E', 'I', 'R', 'D', 'S_c']\nparam_list = ['beta', 'alpha', 'gamma', 'mu', 'c_0', 'c_1', 'c_2', 'N']\n\ntransition = [\n Transition(origin='S', destination='E', equation='beta*I*S',\n transition_type=TransitionType.T),\n Transition(origin='S', destination='S_c', equation='c_2*I*S',\n transition_type=TransitionType.T),\n Transition(origin='S_c', destination='S', equation='c_1*S_c',\n transition_type=TransitionType.T),\n Transition(origin='S_c', destination='E', equation='c_0*beta*I*S_c',\n transition_type=TransitionType.T),\n Transition(origin='E', destination='I', equation='alpha*E',\n transition_type=TransitionType.T),\n Transition(origin='I', destination='R', equation='gamma*I',\n transition_type=TransitionType.T),\n Transition(origin='I', destination='D', equation='mu*I',\n transition_type=TransitionType.T) \n ]\n\nSCEIR_model = DeterministicOde(state, param_list, transition=transition)\nSCEIR_model.modelname='SCEIR'\nSCEIR_model.ei=slice(1,3) # cases 1,2 i.e. E,I\nSCEIR_model.confirmed=slice(2,5) # cases 2-4 i.e. I, R and D, not E\nSCEIR_model.recovered=slice(3,4)\nSCEIR_model.deaths=slice(4,5)\n```\n\n\n```python\n# display equations\nprint_ode2(SCEIR_model)\n```\n\n\n```python\n# display graphical representation of the model\nSCEIR_model.get_transition_graph()\n```\n\n### SC3EIR model\n\n#### Equations\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -\\beta (I + c_0 I_c) S + c_1 S_c - c_2 S (I + I_c)\\\\\n\\dot{S_c} &= - c_0 \\beta (I + c_0 I_c) S_c - c_1 S_c + c_2 S (I + I_c)\\\\\n\\dot{E} &= \\beta (I + c_0 I_c) S - \\alpha E + c_1 E_c - c_2 E (I + I_c)\\\\\n\\dot{E_c} &= c_0 \\beta (I + c_0 I_c) S_c - \\alpha E_c - c_1 E_c + c_2 E (I + I_c)\\\\\n\\dot{I} &= \\alpha E - \\gamma I - \\mu I + c_1 I_c - c_2 I (I + I_c)\\\\\n\\dot{I_c} &= \\alpha E_c - \\gamma I_c - \\mu I_c - c_1 I_c + c_2 I (I + I_c)\\\\\n\\dot{R} & = \\gamma (I + I_c) \\\\\n\\dot{D} & = \\mu (I + I_c)\n\\end{split}\n\\end{equation}\n\nThe use of I as a state variable triggering susceptibles to execute caution is just one choice. In contrast with deaths, it does not accumulate over time and so retains the property of an active threat to society, rather than an historical one. Alternatively, one could use the daily death rate which is proportional to it.\n\nActually, the effect of caution may be quadratic, since both the individual doing the infection and individual potentially being infected may be executing caution. The current version assumes that infectives do not change their precautionary measures in response to I. To implement this we would need to further distinguish careful and non careful infectives. This is done in the SC2IR model.\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $S_c$: Susceptible individuals exercising pandemy precautions\n* $E$: Exposed individuals\n* $E_c$: Exposed individuals exercising caution\n* $I$: Infected individuals\n* $I_c$: Infected individuals exercising caution\n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+S_c+E+E_c+I+I_c+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta$ rate at which infected individuals contact susceptibles and infect them\n* $\\alpha$ rate at which exposed individuals become infected (1/(incubation time)\n* $\\gamma$ rate at which infected individuals recover from disease and become immune\n* $\\mu$ death rate for infected individuals\n\n* $c_i$ three parameters characterizing cautionary response of population via class $S_c$\n\n - c_0 : reduction factor for exposure for cautioned susceptibles\n\n - c_1 : duration of caution (exponential decay time constant in days)\n\n - c_2 : rate constant for transition from uncautioned to cautioned susceptible\n\n#### Implementation\nUsing PyGOM, we will set up my simple SCIR model ODE system\nPyGOM \u2013 A Python Package for Simplifying Modelling with Systems of Ordinary Differential Equations https://arxiv.org/pdf/1803.06934.pdf\n\n\n```python\n# set up the symbolic model\n\nstate = ['S', 'E', 'I', 'R', 'D', 'I_c', 'S_c', 'E_c']\nparam_list = ['beta', 'alpha', 'gamma', 'mu', 'c_0', 'c_1', 'c_2', 'N']\n\ntransition = [\n Transition(origin='S', destination='E', equation='beta*(I+c_0*I_c)*S',\n transition_type=TransitionType.T),\n Transition(origin='S', destination='S_c', equation='c_2*(I+I_c)*S',\n transition_type=TransitionType.T),\n Transition(origin='S_c', destination='S', equation='c_1*S_c',\n transition_type=TransitionType.T),\n Transition(origin='S_c', destination='E_c', equation='c_0*beta*(I+c_0*I_c)*S_c',\n transition_type=TransitionType.T),\n Transition(origin='E', destination='I', equation='alpha*E',\n transition_type=TransitionType.T),\n Transition(origin='E', destination='E_c', equation='c_2*(I+I_c)*E',\n transition_type=TransitionType.T),\n Transition(origin='E_c', destination='I_c', equation='alpha*E_c',\n transition_type=TransitionType.T),\n Transition(origin='E_c', destination='E', equation='c_1*E_c',\n transition_type=TransitionType.T),\n Transition(origin='I', destination='R', equation='gamma*I',\n transition_type=TransitionType.T),\n Transition(origin='I', destination='I_c', equation='c_2*(I+I_c)*I',\n transition_type=TransitionType.T),\n Transition(origin='I', destination='D', equation='mu*I',\n transition_type=TransitionType.T),\n Transition(origin='I_c', destination='R', equation='gamma*I_c',\n transition_type=TransitionType.T),\n Transition(origin='I_c', destination='I', equation='c_1*I_c',\n transition_type=TransitionType.T),\n Transition(origin='I_c', destination='D', equation='mu*I_c',\n transition_type=TransitionType.T)\n ]\n\nSC3EIR_model = DeterministicOde(state, param_list, transition=transition)\nSC3EIR_model.modelname='SC3EIR'\nSC3EIR_model.ei=slice(1,3) # cases 1,2 i.e. E,I # note E_c and I_c not included\nSC3EIR_model.confirmed=slice(2,6) # cases 2-5 i.e. I, R, D, and I_c, not E, E_c\nSC3EIR_model.recovered=slice(3,4)\nSC3EIR_model.deaths=slice(4,5)\n```\n\n\n```python\n# display equations\nprint_ode2(SC3EIR_model)\n```\n\n\n```python\n# display graphical representation of the model\nSC3EIR_model.get_transition_graph()\n```\n\n## Caution Extensions to SEI3R Model\n\n### SEI3R model\n\n#### Equations\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -(\\beta_1 I_1 +\\beta_2 I_2 + \\beta_3 I_3) S\\\\\n\\dot{E} &=(\\beta_1 I_1 +\\beta_2 I_2 + \\beta_3 I_3 ) S - a E \\\\\n\\dot{I_1} &= a E - \\gamma_1 I_1 - p_1 I_1 \\\\\n\\dot{I_2} &= p_1 I_1 -\\gamma_2 I_2 - p_2 I_2 \\\\\n\\dot{I_3} & = p_2 I_2 -\\gamma_3 I_3 - \\mu I_3 \\\\\n\\dot{R} & = \\gamma_1 I_1 + \\gamma_2 I_2 + \\gamma_3 I_3 \\\\\n\\dot{D} & = \\mu I_3\n\\end{split}\n\\end{equation}\n\nThis model (by Dr. Alison for example) involves exposed but not infectious individuals and three classes of infective states with increasing severity.\nThe latter two involve hospitalization with the last in ICU.\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $E$: Exposed individuals - infected but not yet infectious or symptomatic\n* $I_i$: Infected individuals in severity class $i$. Severity increaes with $i$ and we assume individuals must pass through all previous classes\n * $I_1$: Mild infection (hospitalization not required)\n * $I_2$: Severe infection (hospitalization required)\n * $I_3$: Critical infection (ICU required)\n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+E+I_1+I_2+I_3+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta_i$ rate at which infected individuals in class $I_i$ contact susceptibles and infect them\n* $a$ rate of progression from the exposed to infected class\n* $\\gamma_i$ rate at which infected individuals in class $I_i$ recover from disease and become immune\n* $p_i$ rate at which infected individuals in class $I_i$ progress to class $I_{I+1}$\n* $\\mu$ death rate for individuals in the most severe stage of disease\n\n#### Implementation\nUsing PyGOM, we will set up the model ODE system\nPyGOM \u2013 A Python Package for Simplifying Modelling with Systems of Ordinary Differential Equations https://arxiv.org/pdf/1803.06934.pdf\n\n\n```python\n# set up the symbolic SEI3R model\n\nstate = ['S', 'E', 'I_1', 'I_2','I_3','R','D']\nparam_list = ['beta_1', 'beta_2','beta_3','alpha', 'gamma_1', 'gamma_2', 'gamma_3',\n 'p_1','p_2','mu','N']\n\ntransition = [\n Transition(origin='S', destination='E', equation='(beta_1*I_1+beta_2*I_2+beta_3*I_3)*S',\n transition_type=TransitionType.T),\n Transition(origin='E', destination='I_1', equation='alpha*E',\n transition_type=TransitionType.T),\n Transition(origin='I_1', destination='R', equation='gamma_1*I_1',\n transition_type=TransitionType.T),\n Transition(origin='I_2', destination='R', equation='gamma_2*I_2',\n transition_type=TransitionType.T),\n Transition(origin='I_3', destination='R', equation='gamma_3*I_3',\n transition_type=TransitionType.T),\n Transition(origin='I_1', destination='I_2', equation='p_1*I_1',\n transition_type=TransitionType.T),\n Transition(origin='I_2', destination='I_3', equation='p_2*I_2',\n transition_type=TransitionType.T),\n Transition(origin='I_3', destination='D', equation='mu*I_3',\n transition_type=TransitionType.T) \n ]\n\n\nSEI3R_model = DeterministicOde(state, param_list, transition=transition)\nSEI3R_model.modelname='SEI3R'\nSEI3R_model.ei=slice(1,5)\nSEI3R_model.confirmed=slice(2,7) # cases 2-6 i.e. I1, I2, I3, R and D\nSEI3R_model.recovered=slice(5,6)\nSEI3R_model.deaths=slice(6,7)\n```\n\n\n```python\n# display equations\nprint_ode2(SEI3R_model)\n```\n\n\n```python\n# display graphical representation of the model\nSEI3R_model.get_transition_graph()\n```\n\n### SCEI3R model\n\n#### Equations\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -(\\beta_1 I_1 +\\beta_2 I_2 + \\beta_3 I_3) S + c_1 S_c - c_2*S*I_3\\\\\n\\dot{S_c} &= - c_0(\\beta_1 I_1 +\\beta_2 I_2 + \\beta_3 I_3) S_c - c_1 S_c + c_2*S*I_3\\\\\n\\dot{E} &=(\\beta_1 I_1 +\\beta_2 I_2 + \\beta_3 I_3 ) (S + c_0 S_c) - a E \\\\\n\\dot{I_1} &= a E - \\gamma_1 I_1 - p_1 I_1 \\\\\n\\dot{I_2} &= p_1 I_1 -\\gamma_2 I_2 - p_2 I_2 \\\\\n\\dot{I_3} & = p_2 I_2 -\\gamma_3 I_3 - \\mu I_3 \\\\\n\\dot{R} & = \\gamma_1 I_1 + \\gamma_2 I_2 + \\gamma_3 I_3 \\\\\n\\dot{D} & = \\mu I_3\n\\end{split}\n\\end{equation}\n\nThe use of I_3 as a state variable triggering susceptibles to execute caution is just one choice. In contrast with deaths, it does not accumulate over time and so retains the property of an active threat to society, rather than an historical one.\n\nActually, the effect of caution may be quadratic, since both the individual doing the infection and individual potentially being infected may be executing caution. The current version assumes that infectives do not change their precautionary measures in response to I_3. To implement this we would need to further distinguish careful and non careful infectives at least up to the I_1 level. This is done in the SC3EI3R model.\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $S_c$: Susceptible individuals exercising pandemy precautions\n* $E$: Exposed individuals - infected but not yet infectious or symptomatic\n* $I_i$: Infected individuals in severity class $i$. Severity increaes with $i$ and we assume individuals must pass through all previous classes\n * $I_1$: Mild infection (hospitalization not required)\n * $I_2$: Severe infection (hospitalization required)\n * $I_3$: Critical infection (ICU required)\n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+S_c+E+I_1+I_2+I_3+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta_i$ rate at which infected individuals in class $I_i$ contact susceptibles and infect them\n* $a$ rate of progression from the exposed to infected class\n* $\\gamma_i$ rate at which infected individuals in class $I_i$ recover from disease and become immune\n* $p_i$ rate at which infected individuals in class $I_i$ progress to class $I_{I+1}$\n* $\\mu$ death rate for individuals in the most severe stage of disease\n\n* $c_i$ three parameters characterizing cautionary response of population via class $S_c$\n\n - c_0 : reduction factor for exposure for cautioned susceptibles\n\n - c_1 : duration of caution (exponential decay time constant in days)\n\n - c_2 : rate constant for transition from uncautioned to cautioned susceptible\n\n\n\n\n#### Implementation\nUsing PyGOM, we will set up the model ODE system\nPyGOM \u2013 A Python Package for Simplifying Modelling with Systems of Ordinary Differential Equations https://arxiv.org/pdf/1803.06934.pdf\n\n\n```python\n# set up the symbolic model\n\nstate = ['S', 'E', 'I_1', 'I_2','I_3','R','D','S_c']\nparam_list = ['beta_1', 'beta_2','beta_3','alpha', 'gamma_1', 'gamma_2', 'gamma_3',\n 'p_1','p_2','mu','c_0','c_1','c_2','N']\n\ntransition = [\n Transition(origin='S', destination='E', equation='(beta_1*I_1+beta_2*I_2+beta_3*I_3)*S',\n transition_type=TransitionType.T),\n Transition(origin='S', destination='S_c', equation='c_2*I_3*S',\n transition_type=TransitionType.T),\n Transition(origin='S_c', destination='S', equation='c_1*S_c',\n transition_type=TransitionType.T),\n Transition(origin='S_c', destination='E', equation='c_0*(beta_1*I_1+beta_2*I_2+beta_3*I_3)*S_c',\n transition_type=TransitionType.T),\n Transition(origin='E', destination='I_1', equation='alpha*E',\n transition_type=TransitionType.T),\n Transition(origin='I_1', destination='R', equation='gamma_1*I_1',\n transition_type=TransitionType.T),\n Transition(origin='I_2', destination='R', equation='gamma_2*I_2',\n transition_type=TransitionType.T),\n Transition(origin='I_3', destination='R', equation='gamma_3*I_3',\n transition_type=TransitionType.T),\n Transition(origin='I_1', destination='I_2', equation='p_1*I_1',\n transition_type=TransitionType.T),\n Transition(origin='I_2', destination='I_3', equation='p_2*I_2',\n transition_type=TransitionType.T),\n Transition(origin='I_3', destination='D', equation='mu*I_3',\n transition_type=TransitionType.T) \n ]\n\n\nSCEI3R_model = DeterministicOde(state, param_list, transition=transition)\nSCEI3R_model.modelname='SCEI3R'\nSCEI3R_model.ei=slice(1,5)\nSCEI3R_model.confirmed=slice(2,7) # cases 2-6 i.e. I1, I2, I3, R and D\nSCEI3R_model.recovered=slice(5,6)\nSCEI3R_model.deaths=slice(6,7)\n```\n\n\n```python\n# display equations\nprint_ode2(SCEI3R_model)\n```\n\n\n```python\n# display graphical representation of the model\nSCEI3R_model.get_transition_graph()\n```\n\n### SC3EI3R model with caution distinguished $E$ and \ud835\udc3c1\n\n#### Equations\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -(\\beta_1 (I_1 + c_0 I_{1c}) +\\beta_2 I_2 + \\beta_3 I_3) S + c_1 S_c - c_2*S*I_3\\\\\n\\dot{S_c} &= - c_0(\\beta_1 (I_1 + c_0 I_{1c}) +\\beta_2 I_2 + \\beta_3 I_3) S_c - c_1 S_c + c_2*S*I_3\\\\\n\\dot{E} &=(\\beta_1 (I_1 + c_0 I_{1c}) +\\beta_2 I_2 + \\beta_3 I_3 ) S - a E + c_1 E_c - c_2*E*I_3\\\\\n\\dot{E_c} &=(\\beta_1 (I_1 + c_0 I_{1c}) +\\beta_2 I_2 + \\beta_3 I_3 ) c_0 S_c - a E - c_1 E_c + c_2*E*I_3\\\\\n\\dot{I_1} &= a E - \\gamma_1 I_1 - p_1 I_1 + c_1 I_{1c} - c_2*I_{1c}*I_3\\\\\n\\dot{I_{1c}} &= a E_c - \\gamma_1 I_{1c} - p_1 I_{1c} - c_1 I_{1c} + c_2*I_{1c}*I_3\\\\\n\\dot{I_2} &= p_1 (I_1 + I_{1c}) -\\gamma_2 I_2 - p_2 I_2 \\\\\n\\dot{I_3} & = p_2 I_2 -\\gamma_3 I_3 - \\mu I_3 \\\\\n\\dot{R} & = \\gamma_1 (I_1 + I_{1c}) + \\gamma_2 I_2 + \\gamma_3 I_3 \\\\\n\\dot{D} & = \\mu I_3\n\\end{split}\n\\end{equation}\n\nThe use of I_3 as a state variable triggering susceptibles to execute caution is just one choice. In contrast with deaths, it does not accumulate over time and so retains the property of an active threat to society, rather than an historical one.\n\nHere, the effect of caution is quadratic, since both the individual doing the infection and individual potentially being infected may be executing caution. To implement this we distinguish careful and non careful exposed and infectives up to the I_1 level. Once in hospital there is no difference, since all caution is executed wrt infected patients.\nWe ignore transition in caution among infected intervals as a second order effect: could be included as in SC2IR model.\n\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $S_c$: Susceptible individuals exercising pandemy precautions\n* $E$: Exposed individuals living as normal - infected but not yet infectious or symptomatic\n* $E_c$: Exposed individuals exercising pandemy precautions\n* $I_i$: Infected individuals in severity class $i$. Severity increaes with $i$ and we assume individuals must pass through all previous classes. Split non hospital cases by caution.\n * $I_1$: Mild infection (hospitalization not required), living as normal\n * $I_{1c}$: Mild infection (hospitalization not required), exercising caution\n * $I_2$: Severe infection (hospitalization required)\n * $I_3$: Critical infection (ICU required)\n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+S_c+E+E_c+I_{1c}+I_1+I_2+I_3+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta_i$ rate at which infected individuals in class $I_i$ contact susceptibles and infect them\n* $a$ rate of progression from the exposed to infected class\n* $\\gamma_i$ rate at which infected individuals in class $I_i$ recover from disease and become immune\n* $p_i$ rate at which infected individuals in class $I_i$ progress to class $I_{I+1}$\n* $\\mu$ death rate for individuals in the most severe stage of disease\n\n* $c_i$ three parameters characterizing cautionary response of population via class $S_c$\n\n - c_0 : reduction factor for exposure for cautioned susceptibles\n\n - c_1 : duration of caution (exponential decay time constant in days)\n\n - c_2 : rate constant for transition from uncautioned to cautioned susceptible\n\n#### Implementation\n\n\n```python\n# set up the symbolic model\n\nstate = ['S', 'E', 'I_1', 'I_2','I_3', 'R', 'D', 'I_c', 'S_c', 'E_c']\nparam_list = ['beta_1', 'beta_2','beta_3','alpha', 'gamma_1', 'gamma_2', 'gamma_3',\n 'p_1','p_2','mu','c_0','c_1','c_2','N']\n\ntransition = [\n Transition(origin='S', destination='E', equation='(beta_1*I_1+beta_2*I_2+beta_3*I_3+c_0*beta_1*I_c)*S',\n transition_type=TransitionType.T),\n Transition(origin='S_c', destination='E_c', equation='c_0*(beta_1*I_1+beta_2*I_2+beta_3*I_3+c_0*beta_1*I_c)*S_c',\n transition_type=TransitionType.T),\n Transition(origin='S', destination='S_c', equation='c_2*I_3*S',\n transition_type=TransitionType.T),\n Transition(origin='S_c', destination='S', equation='c_1*S_c',\n transition_type=TransitionType.T),\n Transition(origin='E', destination='I_1', equation='alpha*E',\n transition_type=TransitionType.T),\n Transition(origin='E', destination='E_c', equation='c_2*I_3*E',\n transition_type=TransitionType.T),\n Transition(origin='E_c', destination='I_c', equation='alpha*E_c',\n transition_type=TransitionType.T),\n Transition(origin='E_c', destination='E', equation='c_1*E_c',\n transition_type=TransitionType.T),\n Transition(origin='I_1', destination='R', equation='gamma_1*I_1',\n transition_type=TransitionType.T),\n Transition(origin='I_1', destination='I_c', equation='c_2*I_3*I_1', # error corrected I_1, mistakenly was I_c \n transition_type=TransitionType.T), \n Transition(origin='I_c', destination='R', equation='gamma_1*I_c',\n transition_type=TransitionType.T),\n Transition(origin='I_c', destination='I_1', equation='c_1*I_c',\n transition_type=TransitionType.T), \n Transition(origin='I_2', destination='R', equation='gamma_2*I_2',\n transition_type=TransitionType.T),\n Transition(origin='I_3', destination='R', equation='gamma_3*I_3',\n transition_type=TransitionType.T),\n Transition(origin='I_1', destination='I_2', equation='p_1*I_1',\n transition_type=TransitionType.T),\n Transition(origin='I_c', destination='I_2', equation='p_1*I_c',\n transition_type=TransitionType.T),\n Transition(origin='I_2', destination='I_3', equation='p_2*I_2',\n transition_type=TransitionType.T),\n Transition(origin='I_3', destination='D', equation='mu*I_3',\n transition_type=TransitionType.T)\n ]\n\n\nSC3EI3R_model = DeterministicOde(state, param_list, transition=transition)\nSC3EI3R_model.modelname='SC3EI3R'\nSC3EI3R_model.ei=slice(1,5) # 1,2,3,4 i.e. E,I_1,I_2,I_3 \u2013 not E_c and I_c \nSC3EI3R_model.confirmed=slice(2,8) # cases 2-7 i.e. I1, I2, I3, R, D and I_c\nSC3EI3R_model.recovered=slice(5,6)\nSC3EI3R_model.deaths=slice(6,7)\n```\n\n\n```python\n# display equations\nprint_ode2(SC3EI3R_model)\n```\n\n\n```python\n# display graphical representation of the model\nSC3EI3R_model.get_transition_graph()\n```\n\n\n```python\n# set up the symbolic model directly as ODEs to allow confirmed cases as state for fitting\n\nstate = ['S', 'E', 'I_1', 'I_2','I_3', 'R', 'D', 'I_c', 'S_c', 'E_c', 'C_f']\nparam_list = ['beta_1', 'beta_2','beta_3','alpha', 'gamma_1', 'gamma_2', 'gamma_3',\n 'p_1','p_2','mu','c_0','c_1','c_2','N']\n\ntransition = [\n Transition('S', '-(beta_1*(I_1+c_0*I_c)+beta_2*I_2+beta_3*I_3)*S - c_2*I_3*S + c_1*S_c','ODE'),\n Transition('S_c', '-c_0*(beta_1*(I_1+c_0*I_c)+beta_2*I_2+beta_3*I_3)*S_c+c_2*I_3*S-c_1*S_c','ODE'),\n Transition('E', '(beta_1*(I_1+c_0*I_c)+beta_2*I_2+beta_3*I_3)*S-alpha*E-c_2*I_3*E+c_1*E_c','ODE'),\n Transition('E_c', 'c_0*(beta_1*(I_1+c_0*I_c)+beta_2*I_2+beta_3*I_3)*S_c-alpha*E_c+c_2*I_3*E-c_1*E_c','ODE'),\n Transition('I_1', 'alpha*E-gamma_1*I_1-p_1*I_1-c_2*I_3*I_1+c_1*I_c','ODE'),\n Transition('I_c', 'alpha*E_c-gamma_1*I_c-p_1*I_c+c_2*I_3*I_1-c_1*I_c','ODE'),\n Transition('I_2', 'p_1*(I_1+I_c)-gamma_2*I_2-p_2*I_2','ODE'),\n Transition('I_3', 'p_2*I_2-gamma_3*I_3-mu*I_3','ODE'),\n Transition('R', 'gamma_1*(I_1+I_c)+gamma_2*I_2+gamma_3*I_3','ODE'),\n Transition('D', 'mu*I_3','ODE'),\n Transition('C_f', 'alpha*(E+E_c)','ODE')\n ]\n\nSC3EI3R_model = DeterministicOde(state, param_list, ode=transition)\nSC3EI3R_model.modelname='SC3EI3R'\nSC3EI3R_model.ei=slice(1,5) # 1,2,3,4 i.e. E,I_1,I_2,I_3 \u2013 not E_c and I_c \nSC3EI3R_model.confirmed=slice(10,11)\nSC3EI3R_model.recovered=slice(5,6)\nSC3EI3R_model.deaths=slice(6,7)\n```\n\n\n```python\n# display equations\nprint_ode2(SC3EI3R_model)\n```\n\n## Caution Extensions to SEIR Model with Economic Supression\nThis model is an extension of the cautionary model to include a class of susceptibles $S_u$ who are impervious to caution. \nThe main influencer for this class is the economy, which we introduce as a new state variable W, normalized to 1 in the absence of pandemic.\nThe model assumption is that fractional depression of the economy influences some susceptibles (both cautioned and uncautioned) to become uncautionable,\nwith a rate coefficient proportional to the economic depression (1-W). The economy itself is mdoelled with logistic growth to a state 1 in the absence of pandemic\nand 1- $\\kappa S_c$ with pandemic. i.e. individuals exercising caution are the main correlate of economic depression (but the only suppressor for the pandemic).\nAs for the cautioned class, uncautionable individuals also return to normal sussceptibles with exponential decay at rate $k_1$.\n\n### SC2UIR model\n\n#### Equations\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -\\beta (I + c_0 I_c) S + c_1 S_c - c_2 S (I + I_c) - k_u (1 - W) S + k_1 S_u\\\\\n\\dot{S_c} &= - c_0 \\beta (I + c_0 I_c) S_c - c_1 S_c + c_2 S (I + I_c) - k_u (1 - W) S_c\\\\\n\\dot{I} &= \\beta (I + c_0 I_c) S - \\gamma I - \\mu I + c_1 I_c - c_2 I (I + I_c)\\\\\n\\dot{I_c} &= c_0 \\beta (I + c_0 I_c) S_c - \\gamma I_c - \\mu I_c - c_1 I_c + c_2 I (I + I_c)\\\\\n\\dot{R} & = \\gamma (I + I_c) \\\\\n\\dot{D} & = \\mu (I + I_c) \\\\\n\\dot{S_u} & = -\\beta (I + c_0 I_c) S_u + k_u (1 - W) (S + S_c) - k_1 S_u \\\\\n\\dot{W} & = k_w W (1 - \\kappa S_c - W)\n\\end{split}\n\\end{equation}\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $S_c$: Susceptible individuals exercising pandemy precautions\n* $S_u$: Susceptible individuals immune to caution because of economic downturn\n* $E$: Exposed individuals\n* $E_c$: Exposed individuals exercising caution\n* $I$: Infected individuals\n* $I_c$: Infected individuals exercising caution\n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $W$: Economic status obeying a logistic law with caution individuals downturning\n* $N=S+S_c+S_u+E+E_c+I+I_c+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta$ rate at which infected individuals contact susceptibles and infect them\n* $\\alpha$ rate at which exposed individuals become infected (1/(incubation time)\n* $\\gamma$ rate at which infected individuals recover from disease and become immune\n* $\\mu$ death rate for infected individuals\n\n* $c_i$ three parameters characterizing cautionary response of population via class $S_c$\n\n - c_0 : reduction factor for exposure for cautioned susceptibles\n\n - c_1 : inverse duration of caution (exponential decay time constant in days)\n\n - c_2 : rate constant for transition from uncautioned to cautioned susceptible\n* four parameters coupling to economy and uncautionable individuals\n - k_w : rate coefficient of economy equilibration\n - k_u : rate coefficient of transition from uncautioned to uncautionable\n - k_1 : inverse duration of uncautionable state\n - $\\kappa$ : economic downturn of caution (proportional to number cautious)\n\n#### Implementation\n\n\n```python\n# set up the symbolic model\n\nstate = ['S', 'I', 'R', 'D', 'I_c', 'S_c', 'S_u', 'W']\nparam_list = ['beta', 'gamma', 'mu', 'c_0', 'c_1', 'c_2', 'k_u', 'k_1', 'k_w','kappa', 'N']\n\ntransition = [\n Transition(origin='S', equation='-beta*(I+c_0*I_c)*S+c_1*S_c-c_2*(I+I_c)*S-k_u*(1-W)*S+k_1*S_u'),\n Transition(origin='S_c', equation='-c_0*beta*(I+c_0*I_c)*S_c-c_1*S_c+c_2*(I+I_c)*S-k_u*(1-W)*S_c'),\n Transition(origin='S_u', equation='-beta*(I+c_0*I_c)*S_u+k_u*(1-W)*(S+S_c)-k_1*S_u'),\n Transition(origin='I', equation='beta*(I+c_0*I_c)*S-gamma*I-mu*I+c_1*I_c-c_2*(I+I_c)*I'),\n Transition(origin='I_c', equation='c_0*beta*(I+c_0*I_c)*S_c-gamma*I_c-mu*I_c-c_1*I_c+c_2*(I+I_c)*I'),\n Transition(origin='R', equation='gamma*(I+I_c)'),\n Transition(origin='D', equation='mu*(I+I_c)'),\n Transition(origin='W', equation='k_w*W*(1-kappa*S_c-W)')\n ]\n\nSC2UIR_model = DeterministicOde(state, param_list, ode=transition)\nSC2UIR_model.modelname='SC2UIR'\nSC2UIR_model.ei=1 # case 1 i.e. I # note I_c not included\nSC2UIR_model.confirmed=slice(1,5) # cases 1-4 i.e. I, R, D, and I_c\nSC2UIR_model.recovered=slice(2,3)\nSC2UIR_model.deaths=slice(3,4)\n# display equations\nprint_ode2(SC2UIR_model)\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}d{S}/dt= & - S \\beta \\left(I + I_{c} c_{0}\\right) - S c_{2} \\left(I + I_{c}\\right) - S k_{u} \\left(1 - W\\right) + S_{c} c_{1} + S_{u} k_{1}\\\\d{I}/dt= & - I c_{2} \\left(I + I_{c}\\right) - I \\gamma - I \\mu + I_{c} c_{1} + S \\beta \\left(I + I_{c} c_{0}\\right)\\\\d{R}/dt= & \\gamma \\left(I + I_{c}\\right)\\\\d{D}/dt= & \\mu \\left(I + I_{c}\\right)\\\\d{I_c}/dt= & I c_{2} \\left(I + I_{c}\\right) - I_{c} c_{1} - I_{c} \\gamma - I_{c} \\mu + S_{c} \\beta c_{0} \\left(I + I_{c} c_{0}\\right)\\\\d{S_c}/dt= & S c_{2} \\left(I + I_{c}\\right) - S_{c} \\beta c_{0} \\left(I + I_{c} c_{0}\\right) - S_{c} c_{1} - S_{c} k_{u} \\left(1 - W\\right)\\\\d{S_u}/dt= & - S_{u} \\beta \\left(I + I_{c} c_{0}\\right) - S_{u} k_{1} + k_{u} \\left(1 - W\\right) \\left(S + S_{c}\\right)\\\\d{W}/dt= & W k_{w} \\left(- S_{c} \\kappa - W + 1\\right)\\end{matrix}\\right]$\n\n\n\n\n```python\n# set up the symbolic model from transitions, works using separate birth and death list\n\nstate = ['S', 'I', 'R', 'D', 'I_c', 'S_c', 'S_u', 'W']\nparam_list = ['beta', 'gamma', 'mu', 'c_0', 'c_1', 'c_2', 'k_u', 'k_1', 'k_w','kappa', 'N']\n\ntransition = [\n Transition(origin='S', destination='I', equation='beta*(I+c_0*I_c)*S', transition_type=TransitionType.T),\n Transition(origin='S', destination='S_c', equation='c_2*(I+I_c)*S', transition_type=TransitionType.T),\n Transition(origin='S', destination='S_u', equation='k_u*(1-W)*S', transition_type=TransitionType.T),\n Transition(origin='S_c', destination='S', equation='c_1*S_c', transition_type=TransitionType.T),\n Transition(origin='S_c', destination='I_c', equation='c_0*beta*(I+c_0*I_c)*S_c', transition_type=TransitionType.T),\n Transition(origin='S_c', destination='S_u', equation='k_u*(1-W)*S_c', transition_type=TransitionType.T),\n Transition(origin='S_u', destination='S', equation='k_1*S_u', transition_type=TransitionType.T), \n Transition(origin='S_u', destination='I', equation='beta*(I+c_0*I_c)*S_u', transition_type=TransitionType.T), \n Transition(origin='I', destination='I_c', equation='c_2*(I+I_c)*I', transition_type=TransitionType.T), \n Transition(origin='I', destination='R', equation='gamma*I', transition_type=TransitionType.T), \n Transition(origin='I', destination='D', equation='mu*I', transition_type=TransitionType.T), \n Transition(origin='I_c', destination='I', equation='c_1*I_c', transition_type=TransitionType.T),\n Transition(origin='I_c', destination='R', equation='gamma*I_c', transition_type=TransitionType.T), \n Transition(origin='I_c', destination='D', equation='mu*I_c', transition_type=TransitionType.T),\n Transition(origin='W', destination='D', equation='0*W', transition_type=TransitionType.T)\n ]\nbdlist = [Transition(origin='W',equation='k_w*W*(1-kappa*S_c-W)', transition_type=TransitionType.B)\n ]\nSC2UIR_model = DeterministicOde(state, param_list, transition=transition)\nSC2UIR_model.birth_death_list = bdlist\nSC2UIR_model.modelname='SC2UIR'\nSC2UIR_model.ei=1 # case 1 i.e. I # note I_c not included\nSC2UIR_model.confirmed=slice(1,5) # cases 1-4 i.e. I, R, D, and I_c\nSC2UIR_model.recovered=slice(2,3)\nSC2UIR_model.deaths=slice(3,4)\n```\n\n\n```python\n# display equations\nprint_ode2(SC2UIR_model)\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}d{S}/dt= & - S \\beta \\left(I + I_{c} c_{0}\\right) - S c_{2} \\left(I + I_{c}\\right) - S k_{u} \\left(1 - W\\right) + S_{c} c_{1} + S_{u} k_{1}\\\\d{I}/dt= & - I c_{2} \\left(I + I_{c}\\right) - I \\gamma - I \\mu + I_{c} c_{1} + S \\beta \\left(I + I_{c} c_{0}\\right) + S_{u} \\beta \\left(I + I_{c} c_{0}\\right)\\\\d{R}/dt= & I \\gamma + I_{c} \\gamma\\\\d{D}/dt= & I \\mu + I_{c} \\mu\\\\d{I_c}/dt= & I c_{2} \\left(I + I_{c}\\right) - I_{c} c_{1} - I_{c} \\gamma - I_{c} \\mu + S_{c} \\beta c_{0} \\left(I + I_{c} c_{0}\\right)\\\\d{S_c}/dt= & S c_{2} \\left(I + I_{c}\\right) - S_{c} \\beta c_{0} \\left(I + I_{c} c_{0}\\right) - S_{c} c_{1} - S_{c} k_{u} \\left(1 - W\\right)\\\\d{S_u}/dt= & S k_{u} \\left(1 - W\\right) + S_{c} k_{u} \\left(1 - W\\right) - S_{u} \\beta \\left(I + I_{c} c_{0}\\right) - S_{u} k_{1}\\\\d{W}/dt= & W k_{w} \\left(- S_{c} \\kappa - W + 1\\right)\\end{matrix}\\right]$\n\n\n\n\n```python\nSC2UIR_model.get_transition_graph()\n```\n\n\n```python\node = SimulateOde(state, param_list, ode=transition)\node = ode.get_unrolled_obj()\n# R0(ode, ['I','I_c']) # produces error, no valid subset found\n```\n\n### SC3UEIR model\n\n#### Equations\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -\\beta (I + c_0 I_c) S + c_1 S_c - c_2 S (I + I_c) - k_u (1 - W) S + k_1 S_u\\\\\n\\dot{S_c} &= - c_0 \\beta (I + c_0 I_c) S_c - c_1 S_c + c_2 S (I + I_c) - k_u (1 - W) S_c\\\\\n\\dot{E} &= \\beta (I + c_0 I_c) (S + S_u) - \\alpha E + c_1 E_c - c_2 E (I + I_c)\\\\\n\\dot{E_c} &= c_0 \\beta (I + c_0 I_c) S_c - \\alpha E_c - c_1 E_c + c_2 E (I + I_c)\\\\\n\\dot{I} &= \\alpha E - \\gamma I - \\mu I + c_1 I_c - c_2 I (I + I_c)\\\\\n\\dot{I_c} &= \\alpha E_c - \\gamma I_c - \\mu I_c - c_1 I_c + c_2 I (I + I_c)\\\\\n\\dot{R} & = \\gamma (I + I_c) \\\\\n\\dot{D} & = \\mu (I + I_c) \\\\\n\\dot{S_u} & = -\\beta (I + c_0 I_c) S_u + k_u (1 - W) (S + S_c) - k_1 S_u \\\\\n\\dot{W} & = k_w W (1 - \\kappa S_c - W)\n\\end{split}\n\\end{equation}\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $S_c$: Susceptible individuals exercising pandemy precautions\n* $S_u$: Susceptible individuals immune to caution because of economic downturn\n* $E$: Exposed individuals\n* $E_c$: Exposed individuals exercising caution\n* $I$: Infected individuals\n* $I_c$: Infected individuals exercising caution\n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $W$: Economic status obeying a logistic law with caution individuals downturning\n* $N=S+S_c+S_u+E+E_c+I+I_c+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta$ rate at which infected individuals contact susceptibles and infect them\n* $\\alpha$ rate at which exposed individuals become infected (1/(incubation time)\n* $\\gamma$ rate at which infected individuals recover from disease and become immune\n* $\\mu$ death rate for infected individuals\n\n* $c_i$ three parameters characterizing cautionary response of population via class $S_c$\n\n - c_0 : reduction factor for exposure for cautioned susceptibles\n\n - c_1 : inverse duration of caution (exponential decay time constant in days)\n\n - c_2 : rate constant for transition from uncautioned to cautioned susceptible\n* four parameters coupling to economy and uncautionable individuals\n - k_w : rate coefficient of economy equilibration\n - k_u : rate coefficient of transition from uncautioned to uncautionable\n - k_1 : inverse duration of uncautionable state\n - $\\kappa$ : economic downturn of caution (proportional to number cautious)\n\n#### Implementation\n\n\n```python\n# set up the symbolic model\n\nstate = ['S', 'E', 'I', 'R', 'D', 'I_c', 'S_c', 'E_c', 'S_u', 'W']\nparam_list = ['beta', 'alpha', 'gamma', 'mu', 'c_0', 'c_1', 'c_2', 'k_u', 'k_1', 'k_w','kappa', 'N']\n\ntransition = [\n Transition(origin='S', equation='-beta*(I+c_0*I_c)*S+c_1*S_c-c_2*(I+I_c)*S-k_u*(1-W)*S+k_1*S_u'),\n Transition(origin='S_c', equation='-c_0*beta*(I+c_0*I_c)*S_c-c_1*S_c+c_2*(I+I_c)*S-k_u*(1-W)*S_c'),\n Transition(origin='S_u', equation='-beta*(I+c_0*I_c)*S_u+k_u*(1-W)*(S+S_c)-k_1*S_u'),\n Transition(origin='E', equation='beta*(I+c_0*I_c)*(S+S_u)-alpha*E+c_1*E_c-c_2*(I+I_c)*E'),\n Transition(origin='E_c', equation='c_0*beta*(I+c_0*I_c)*S_c-alpha*E_c-c_1*E_c+c_2*(I+I_c)*E'),\n Transition(origin='I', equation='alpha*E-gamma*I-mu*I+c_1*I_c-c_2*(I+I_c)*I'),\n Transition(origin='I_c', equation='alpha*E_c-gamma*I_c-mu*I_c-c_1*I_c+c_2*(I+I_c)*I'),\n Transition(origin='R', equation='gamma*(I+I_c)'),\n Transition(origin='D', equation='mu*(I+I_c)'),\n Transition(origin='W', equation='k_w*W*(1-kappa*S_c-W)')\n ]\n\nSC3UEIR_model = DeterministicOde(state, param_list, ode=transition)\nSC3UEIR_model.modelname='SC3UEIR'\nSC3UEIR_model.ei=slice(1,3) # cases 1,2 i.e. E,I # note E_c and I_c not included\nSC3UEIR_model.confirmed=slice(2,6) # cases 2-5 i.e. I, R, D, and I_c, not E, E_c\nSC3UEIR_model.recovered=slice(3,4)\nSC3UEIR_model.deaths=slice(4,5)\n```\n\n\n```python\n# display equations\nprint_ode2(SC3UEIR_model)\n```\n\n### SC3UEI3R model\n\n#### Equations \n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &={} -(\\beta_1 (I_1 + c_0 I_c) + \\beta_2 I_2 + \\beta_3 I_3) S + c_1 S_c - c_2 S I_3 - S k_u (1-W) + k_1 S_u\\\\\n\\dot{S_c} &={} - c_0 (\\beta_1 (I_1 + c_0 I_c) + \\beta_2 I_2 + \\beta_3 I_3) S_c - c_1 S_c + c_2 S I_3 - k_u (1 - W) S_c \\\\\n\\dot{E} &= \\beta_1 (I_1 + c_0 I_c) (S + S_u) - \\alpha E + c_1 E_c - c_2 I_3 E\\\\\n\\dot{E_c} &= c_0 \\beta_1 (I_1 + c_0 I_c) S_c - \\alpha E_c - c_1 E_c + c_2 I_3 E\\\\\n\\dot{I_1} &= \\alpha E - \\gamma_1 I_1 - p_1 I_1 + c_1 I_c - c_2 I_3 I_1\\\\\n\\dot{I_c} &= \\alpha E_c - \\gamma_1 I_c - p_1 I_c - c_1 I_c + c_2 I_3 I_1\\\\\n\\dot{I_2} &= p_1 (I_1 + I_c) -\\gamma_2 I_2 - p_2 I_2 \\\\\n\\dot{I_3} & = p_2 I_2 -\\gamma_3 I_3 - \\mu I_3 \\\\ \n\\dot{R} & = \\gamma_1 (I_1 + I_c) +\\gamma_2 I_2 + \\gamma_3 I_3\\\\\n\\dot{D} & = \\mu (I_3) \\\\\n\\dot{S_u} & = -(\\beta_1 (I_1 + c_0 I_c)+\\beta_2 I_2 + \\beta_3 I_3) S_u + k_u (1 - W)(S+ S_c) - k_1 S_u \\\\\n\\dot{W} & = k_w W (1 - \\kappa S_c - W)\\\\\n\\end{split}\n\\end{equation}\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $S_c$: Susceptible individuals exercising pandemy precautions\n* $S_u$: Susceptible individuals immune to caution because of economic downturn\n* $W$: Economic status obeying a logistic law with caution individuals downturning\n* $E$: Exposed individuals\n* $E_c$: Exposed individuals exercising caution\n* $I_i$: Infected individuals in severity class $i$. Severity increaes with $i$ and we assume individuals must pass through all previous classes. Split non hospital cases by caution.\n * $I_1$: Mild infection (hospitalization not required), living as normal\n * $I_c$: Mild infection (hospitalization not required), exercising caution\n * $I_2$: Severe infection (hospitalization required)\n * $I_3$: Critical infection (ICU required)\n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+S_c+S_u+E+E_c+I_1+I_c+I_2+I_3+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta$ rate at which infected individuals contact susceptibles and infect them\n* $\\alpha$ rate at which exposed individuals become infected (1/(incubation time)\n* $\\gamma$ rate at which infected individuals recover from disease and become immune\n* $\\mu$ death rate for infected individuals\n* $\\gamma_i$ rate at which infected individuals in class $I_i$ recover from disease and become immune\n* $p_i$ rate at which infected individuals in class $I_i$ progress to class $I_{I+1}$\n* $c_i$ three parameters characterizing cautionary response of population via class $S_c$\n\n - $c_0$ : reduction factor for exposure for cautioned susceptibles\n\n - $c_1$ : inverse duration of caution (exponential decay time constant in days)\n\n - $c_2$ : rate constant for transition from uncautioned to cautioned susceptible\n* four parameters coupling to economy and uncautionable individuals\n - $k_w$ : rate coefficient of economy equilibration\n - $k_u$ : rate coefficient of transition from uncautioned to uncautionable\n - $k_1$ : inverse duration of uncautionable state\n - $\\kappa$ : economic downturn of caution (proportional to number cautious)\n\n#### Implementation\n\n\n```python\n# set up the symbolic model\n\nstate = ['S', 'E', 'I_1', 'I_2', 'I_3', 'R', 'D', 'I_c', 'S_c', 'E_c', 'S_u', 'W'] # order important to allow correct plot groupings\nparam_list = ['beta_1', 'beta_2', 'beta_3', 'p_1', 'p_2', 'alpha', \n 'gamma_1', 'gamma_2', 'gamma_3','mu', 'c_0', 'c_1', 'c_2', 'k_u', 'k_1', 'k_w', 'kappa', 'N'] # order also important\n\ntransition = [\n Transition(origin='S', equation='-(beta_1*(I_1+c_0*I_c)+beta_2*I_2+beta_3*I_3)*S+c_1*S_c-c_2*(I_3)*S-k_u*(1-W)*S+k_1*S_u'),\n Transition(origin='S_c', equation='-c_0*(beta_1*(I_1+c_0*I_c)+beta_2*I_2+beta_3*I_3)*S_c-c_1*S_c+c_2*(I_3)*S-k_u*(1-W)*S_c'),\n Transition(origin='S_u', equation='-(beta_1*(I_1+c_0*I_c)+beta_2*I_2+beta_3*I_3)*S_u+k_u*(1-W)*(S+S_c)-k_1*S_u'),\n Transition(origin='W', equation='k_w*W*(1-kappa*S_c-W)'),\n Transition(origin='E', equation='beta_1*(I_1+c_0*I_c)*(S+S_u)-alpha*E-c_2*(I_3)*E+c_1*E_c'),\n Transition(origin='E_c', equation='c_0*beta_1*(I_1+c_0*I_c)*S_c-alpha*E_c+c_2*(I_3)*E-c_1*E_c'),\n Transition(origin='I_1', equation='alpha*E-gamma_1*I_1-p_1*I_1-c_2*(I_3)*I_1+c_1*I_c'),\n Transition(origin='I_c', equation='alpha*E_c-gamma_1*I_c-p_1*I_c+c_2*(I_3)*I_1-c_1*I_c'), # changed to I_c, prints better\n Transition(origin='I_2', equation='p_1*(I_1+I_c)-gamma_2*I_2-p_2*I_2'),\n Transition(origin='I_3', equation='p_2*I_2-gamma_3*I_3-mu*I_3'), # error corrected, this is equation for I_3 not I_2\n Transition(origin='R', equation='gamma_1*(I_1+I_c)+gamma_2*I_2+gamma_3*I_3'),\n Transition(origin='D', equation='mu*I_3')\n ]\n\nSC3UEI3R_model = DeterministicOde(state, param_list, ode=transition)\nSC3UEI3R_model.modelname='SC3UEI3R' # following needs to be adjusted for new models, NB add new species at end to preserve slice subsets\nSC3UEI3R_model.ei=slice(1,5) # 1,2,3,4 i.e. E,I_1,I_2,I_3 \u2013 not E_c and I_c \nSC3UEI3R_model.confirmed=slice(2,8) # cases 2-7 i.e. I1, I2, I3, R, D and I_c\nSC3UEI3R_model.recovered=slice(5,6) # case 5 R\nSC3UEI3R_model.deaths=slice(6,7) # case 6 D\n```\n\n\n```python\n# display equations\nprint_ode2(SC3UEI3R_model) # name needs to be that of current model\n```\n\n# Extract data from Johns Hopkins data base\n\n## Definition of data extraction fuctions get_data and get_country_data\n\n\n```python\nimport numpy as np\nimport csv\nimport itertools\nimport matplotlib\n%matplotlib inline\nimport seaborn as sb\nfrom matplotlib import pyplot as plt\nfrom cycler import cycler\nimport datetime\nimport matplotlib.dates as mdates\nfrom pandas.plotting import register_matplotlib_converters\nregister_matplotlib_converters()\nimport pwlf\nimport sys\nfrom IPython.core.display import display, HTML\ndisplay(HTML(\"\"))\n```\n\n\n\n\n\n\n```python\ndef get_data(jhu_file):\n dat = []\n with open(jhu_file, newline='') as csvfile:\n myreader = csv.reader(csvfile, delimiter=',')\n popdat = []\n i = 0\n for row in myreader:\n if i != 0:\n poplist = []\n j = 0\n for elt in row:\n if j >= 4:\n poplist.append(int(elt))\n elif j == 0:\n poplist.append(elt)\n elif j == 1:\n poplist[0]=(elt,poplist[0])\n j = j+1\n popdat.append(poplist)\n else:\n popdat.append(row)\n # print(popdat[i])\n i = i + 1;\n # dates\n popdat0=['dates']\n for elt in popdat[0][4:]:\n popdat0.append(elt)\n popdat[0] = [pop for pop in popdat0]\n # print('popdat[0]',popdat[0])\n # totals over all countries\n totals = np.zeros(len(popdat[0])-1,dtype=int)\n for row in popdat[1:]:\n totals = totals + np.array(row[1:])\n totals = list(np.asarray(totals))\n # print(totals)\n popkeyed = {poplist[0]: poplist[1:] for poplist in popdat}\n popkeyed.update({'dates':popdat[0][1:]})\n popkeyed.update({('World',''):totals})\n # del popkeyed[('d','a')]\n # assemble totals for countries with multiple regions\n total = np.zeros(len(popkeyed['dates']),dtype=int)\n poptotkeyed = {}\n for country,tseries in popkeyed.items():\n if country!='dates' and country[1] != '': # it seems that UK is single exception with both '' and non '' regions, UK total is then UK overseas\n countrytotal = (country[0],'Total')\n if countrytotal in poptotkeyed:\n # print(country,popkeyed[country],poptotkeyed[countrytotal])\n total = np.array(tseries)[:]+np.array(poptotkeyed[countrytotal])[:]\n else:\n total = np.array(tseries) \n poptotkeyed.update({countrytotal:list(total)})\n for countrytotal,tseries in poptotkeyed.items():\n total = np.array(tseries)\n popkeyed.update({countrytotal:list(total)})\n return popkeyed\n```\n\n\n```python\ndef Float(x):\n try:\n rtn = float(x)\n except:\n rtn = float('NaN')\n return rtn\n```\n\n\n```python\n# from covid_data_explore-jhu-j\ndef get_country_data(country_s='World', datatype='confirmed', firstdate=None, lastdate=None):\n if isinstance(country_s,str):\n country = (country_s,'')\n else: # single ('country','reg') entry\n country = country_s\n popkeyed = covid_ts[datatype]\n \n dates = popkeyed['dates']\n fmt = '%m/%d/%y'\n xx = [datetime.datetime.strptime(dd,fmt) for dd in dates ]\n if firstdate:\n firstdate_d = datetime.datetime.strptime(firstdate,fmt)\n else:\n firstdate_d = datetime.datetime.strptime(dates[0],fmt)\n if lastdate:\n lastdate_d = datetime.datetime.strptime(lastdate,fmt)\n else:\n lastdate_d = datetime.datetime.strptime(dates[-1],fmt) \n daystart = (firstdate_d-xx[0]).days\n daystop = (lastdate_d-xx[-1]).days\n\n try:\n yy = popkeyed[country]\n # print(country)\n except:\n print('country data not found',country)\n return None,None,None\n yyf = [Float(y) for y in yy]\n \n if daystart <0:\n xx0 = [xx[0]+datetime.timedelta(days=i) for i in range(daystart,0)]\n yy0 = [0.]*(-daystart)\n else:\n xx0 = []\n yy0 = []\n if daystop > 0:\n xx1 = [xx[-1]+datetime.timedelta(days=i) for i in range(daystop)]\n yy1 = [0.]*(daystop)\n else:\n xx1 = []\n yy1 = [] \n xx = xx0 + xx + xx1\n xxf = [Float((x-firstdate_d).days) for x in xx ]\n \n yy = yy0 + yyf + yy1\n return xx,xxf,yy \n\n\n```\n\n\n```python\ndef get_country_data_nyw(country_s='World', datatype='confirmed', firstdate=None, lastdate=None):\n if isinstance(country_s,str):\n country = (country_s,'')\n else: # single ('country','reg') entry\n country = country_s\n popkeyed = covid_ts[datatype]\n \n dates = popkeyed['dates']\n fmt = '%m/%d/%y'\n xx = [datetime.datetime.strptime(dd,fmt) for dd in dates ]\n if firstdate:\n firstdate_d = datetime.datetime.strptime(firstdate,fmt)\n else:\n firstdate_d = datetime.datetime.strptime(dates[0],fmt)\n if lastdate:\n lastdate_d = datetime.datetime.strptime(lastdate,fmt)\n else:\n lastdate_d = datetime.datetime.strptime(dates[-1],fmt) \n daystart = (firstdate_d-xx[0]).days\n daystop = (lastdate_d-xx[-1]).days\n \n try:\n yy = popkeyed[country]\n # print(country)\n except:\n print('country data not found',country)\n return None,None \n yyf = [Float(y) for y in yy]\n\n yy0 = []\n yy1 = [] \n if daystart>len(yyf):\n print('Error: start date does not overlap with available data')\n return None,None\n elif daystart>0:\n yyf = yyf[daystart:]\n elif daystart <0:\n yy0 = [0.]*(-daystart)\n \n if daystop < 0:\n yyf = yyf[:daystop] \n elif daystop > 0:\n yy1 = [0.]*(daystop)\n yyf = yy0 + yyf + yy1\n xxf = [float(x) for x in range(len(yyf))]\n return xxf,yyf \n```\n\n## JHU data\n\n\n```python\nbase = '../../covid-19-JH/csse_covid_19_data/csse_covid_19_time_series/'\nconfirmed = get_data(base+'time_series_covid19_confirmed_global.csv')\ndeaths = get_data(base+'time_series_covid19_deaths_global.csv')\nrecovered = get_data(base+'time_series_covid19_recovered_global.csv')\ncovid_ts = {'confirmed':confirmed,'deaths':deaths,'recovered':recovered}\ncountries_jhu = [(row[0],row[1]) for row in confirmed][1:]\nprint(\"number of countries listed\",len(countries_jhu))\ni=0\nfor country in countries_jhu:\n# print(i,country)\n i = i + 1\n```\n\n number of countries listed 274\n\n\n## Get data for one country\n\n\n```python\n# assumed data starting on firstdate\ntest_country='Germany'\nN = 80000000\nfirstdate = '01/25/20'\nlastdate = '01/08/20'\nxx,xxf,yy0 = get_country_data(test_country,'confirmed',firstdate=firstdate,lastdate=lastdate)\nxx,xxf,yy1 = get_country_data(test_country,'recovered',firstdate=firstdate,lastdate=lastdate)\nxx,xxf,yy2 = get_country_data(test_country,'deaths',firstdate=firstdate,lastdate=lastdate)\nprint(xxf)\ny_jhu={}\ny_jhu[test_country] = np.array([[yy0[i],yy1[i],yy2[i]] for i in range(0,len(yy0))])/N\n# data = np.array([[xxf[i],yy0[i],yy1[i],yy2[i]] for i in range(len(yy))])\n# print(data)\nlastday = len(y_jhu[test_country])\nprint('days 0 to',lastday,'data stored in y_jhu')\n```\n\n [-3.0, -2.0, -1.0, 0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, 32.0, 33.0, 34.0, 35.0, 36.0, 37.0, 38.0, 39.0, 40.0, 41.0, 42.0, 43.0, 44.0, 45.0, 46.0, 47.0, 48.0, 49.0, 50.0, 51.0, 52.0, 53.0, 54.0, 55.0, 56.0, 57.0, 58.0, 59.0, 60.0, 61.0, 62.0, 63.0, 64.0, 65.0, 66.0, 67.0, 68.0, 69.0, 70.0, 71.0, 72.0, 73.0, 74.0, 75.0, 76.0, 77.0, 78.0, 79.0, 80.0, 81.0, 82.0, 83.0, 84.0, 85.0, 86.0, 87.0, 88.0, 89.0, 90.0, 91.0, 92.0, 93.0, 94.0, 95.0, 96.0, 97.0, 98.0, 99.0, 100.0, 101.0, 102.0, 103.0, 104.0, 105.0, 106.0, 107.0, 108.0, 109.0, 110.0, 111.0, 112.0, 113.0, 114.0, 115.0, 116.0, 117.0, 118.0, 119.0, 120.0, 121.0, 122.0, 123.0, 124.0, 125.0, 126.0, 127.0, 128.0, 129.0, 130.0, 131.0, 132.0, 133.0, 134.0, 135.0, 136.0, 137.0, 138.0, 139.0, 140.0, 141.0, 142.0, 143.0, 144.0, 145.0, 146.0, 147.0, 148.0, 149.0, 150.0, 151.0, 152.0, 153.0, 154.0, 155.0, 156.0, 157.0, 158.0, 159.0, 160.0, 161.0, 162.0, 163.0, 164.0, 165.0, 166.0, 167.0, 168.0, 169.0, 170.0, 171.0, 172.0, 173.0, 174.0, 175.0, 176.0, 177.0, 178.0, 179.0, 180.0, 181.0, 182.0, 183.0, 184.0, 185.0, 186.0, 187.0, 188.0, 189.0, 190.0, 191.0, 192.0, 193.0, 194.0, 195.0, 196.0, 197.0, 198.0, 199.0, 200.0, 201.0, 202.0, 203.0, 204.0, 205.0, 206.0, 207.0, 208.0, 209.0, 210.0, 211.0, 212.0, 213.0, 214.0, 215.0, 216.0, 217.0, 218.0]\n days 0 to 222 data stored in y_jhu\n\n\n## OWID data\n\n\n```python\nimport csv\nowid_file = '../../covid-19-owid/public/data/owid-covid-data.csv'\ncovid_owid = []\nwith open(owid_file,'r',newline='') as fp:\n myreader = csv.DictReader(fp,delimiter=',')\n# rows = list(itertools.islice(myreader,4))\n for row in myreader:\n covid_owid.append(row)\ncovid_owid[0].keys()\n```\n\n\n\n\n odict_keys(['iso_code', 'continent', 'location', 'date', 'total_cases', 'new_cases', 'new_cases_smoothed', 'total_deaths', 'new_deaths', 'new_deaths_smoothed', 'total_cases_per_million', 'new_cases_per_million', 'new_cases_smoothed_per_million', 'total_deaths_per_million', 'new_deaths_per_million', 'new_deaths_smoothed_per_million', 'new_tests', 'total_tests', 'total_tests_per_thousand', 'new_tests_per_thousand', 'new_tests_smoothed', 'new_tests_smoothed_per_thousand', 'tests_per_case', 'positive_rate', 'tests_units', 'stringency_index', 'population', 'population_density', 'median_age', 'aged_65_older', 'aged_70_older', 'gdp_per_capita', 'extreme_poverty', 'cardiovasc_death_rate', 'diabetes_prevalence', 'female_smokers', 'male_smokers', 'handwashing_facilities', 'hospital_beds_per_thousand', 'life_expectancy'])\n\n\n\n\n```python\ndef get_data_owid(owid_file,database='owid',datatype='confirmed',dataaccum = 'cumulative'):\n import numpy as np\n import datetime\n import matplotlib.dates as mdates\n \n global covid_owid\n if not covid_owid:\n with open(owid_file, 'r', newline='') as csvfile:\n myreader = csv.DictReader(csvfile,delimiter=',')\n for row in myreader:\n covid_owid.append(row)\n close(owid_file)\n \n # for key in covid_owid[0].keys(): # to loop through all keys\n \n if datatype == 'confirmed':\n if dataaccum == 'cumulative':\n key = 'total_cases'\n elif dataaccum == 'weekly':\n key = 'new_cases_smoothed'\n else:\n key = 'new_cases'\n elif datatype == 'recovered':\n print('data for recovered cases not available in OWID database')\n key = None\n elif datatype == 'deaths':\n if dataaccum == 'cumulative':\n key = 'total_deaths'\n elif dataaccum == 'weekly':\n key = 'new_deaths_smoothed'\n else:\n key = 'new_deaths'\n elif datatype == 'tests':\n if dataaccum == 'cumulative': # reporting intervals often sporadic so better to use smoothed weekly\n # key = 'total_tests'\n key = 'new_tests_smoothed' # will adjust to cumulative below\n elif dataaccum == 'weekly':\n key = 'new_tests_smoothed'\n else:\n key = 'new_tests' # reporting intervals often sporadic so better to use smoothed weekly\n elif datatype =='stringency':\n key = 'stringency_index'\n elif datatype == 'recovered':\n print('data for recovered cases not available in OWID database')\n key = None\n return \n else:\n print('data for ', datatype,'not available or not yet translated in OWID database')\n key = None\n return\n \n countries = np.unique(np.array([dd['location'] for dd in covid_owid]))\n dates = np.unique(np.array([dd['date'] for dd in covid_owid]))\n dates.sort()\n fmt = '%Y-%m-%d'\n dates_t = [datetime.datetime.strptime(dd,fmt) for dd in dates ]\n firstdate = dates[0]\n lastdate = dates[-1]\n firstdate_t = dates_t[0]\n lastdate_t = dates_t[-1]\n\n daystart = 0\n daystop = (lastdate_t-firstdate_t).days\n \n popkeyed = {country: np.zeros(daystop+1,dtype=float) for country in countries} \n \n for dd in covid_owid:\n country = dd['location']\n day = (datetime.datetime.strptime(dd['date'],fmt)-firstdate_t).days\n popkeyed[country][day] = float(dd[key]) if not dd[key]=='' else 0.0 \n \n # popkeyed = {country: np.transpose(np.array([[dd['date'],dd[key]] for dd in covid_owid if dd['location'] == country])) for country in countries}\n # popkeyed = {country: np.array([float(dd[key]) if not dd[key]=='' else 0.0 for dd in covid_owid if dd['location'] == country]) for country in countries} \n\n if datatype == 'tests' and dataaccum == 'cumulative': # assemble cumulative tests from smooth daily tests\n for country in countries:\n data = popkeyed[country]\n sumdata= np.zeros(len(data))\n sum = 0.0\n for i,d in enumerate(data):\n sum = sum + d\n sumdata[i] = sum\n popkeyed.update({country:sumdata})\n\n fmt_jhu = '%m/%d/%y'\n popkeyed.update({'dates': [date.strftime(fmt_jhu) for date in dates_t]}) # dates are set to strings in jhu date format for compatibility\n return popkeyed\n```\n\n\n```python\nowid_file = '../../covid-19-owid/public/data/owid-covid-data.csv'\nconfirmed_owid=get_data_owid(owid_file,database='owid',datatype='confirmed',dataaccum = 'cumulative')\nrecovered_owid = None\ndeaths_owid=get_data_owid(owid_file,database='owid',datatype='deaths',dataaccum = 'cumulative')\ntests_owid=get_data_owid(owid_file,database='owid',datatype='tests',dataaccum = 'cumulative')\nstringency_owid=get_data_owid(owid_file,database='owid',datatype='stringency',dataaccum = 'daily')\ncovid_owid_ts= {'confirmed':confirmed_owid,'deaths':deaths_owid,'recovered':recovered_owid, 'tests': tests_owid , 'stringency': stringency_owid}\n```\n\n\n```python\ndef truncx(xx,daystart,daystop):\n \"\"\"truncate array xx to run from daystart to daystop\n do this before trying to extend the arrays if required\"\"\"\n daymin = max(daystart,0)\n daymax = min(daystop,(xx[-1]-xx[0]).days)\n return xx[daymin:daymax+1]\n\ndef truncy(xx,yy,daystart,daystop):\n \"\"\"truncate arrays xx and yy to run from daystart to daystop\n do this before trying to extend the arrays if required\"\"\"\n daymin = max(daystart,0)\n daymax = min(daystop,(xx[-1]-xx[0]).days)\n return yy[daymin:daymax+1]\n\ndef plotCountry_(country_s, datatype='confirmed', dataaccum='cumulative', fittype=None, ax=None, ax2=False,\n symbol='o--', step=None, firstdate=None, lastdate=None, intdates=False, linecolor=None, maxyval=None, minconfirmed=0,nsegments=3,database='jhu'):\n \"\"\" plots selected data for a list of countries or single country\n datatypes allowed are 'confirmed','deaths','recovered'\n dataaccum specifies either 'cumulative' or 'daily' or averaged over 7 days 'cum_av_weekly' or 'daily_av_weekly'\n fittypes allowed are currently None, 'piecewise-linear'\n ax graphical axes to use for plot: default None -> new axes\n ax2 true if second axes as twin axes for overlay plotting\n symbol to use for plotting\n step whether to use step plotting instead of points: default None -> points\n firstdate to plot (maybe before first date in data - pad with 0)\n lastdate to plot (maybe after last date in data - pad with 0)\n intdates : whether to plot dates as integers for compatibility (default as dates)\n linecolor is default color to use for a single trace, instead of listed set)\n \"\"\"\n global covid_ts, covid_ts_owid\n import math\n import warnings\n # extract list of countries in [(country,region),...] format from first parameter\n countries = []\n if isinstance(country_s,list):\n for country in country_s:\n if isinstance(country,str) and database == 'jhu':\n country = (country,'')\n countries.append(country)\n elif isinstance(country_s,str):\n if database == 'jhu':\n countries = [( country_s,'')]\n else:\n countries = [country_s]\n else: # single ('country','reg') entry\n countries = [country_s]\n \n # get data with datatype and extend dates to padd desired interval specified by firstdate,lastdate\n if database == 'jhu':\n popkeyed = covid_ts[datatype]\n dates = popkeyed['dates']\n fmt = '%m/%d/%y'\n elif database == 'owid':\n popkeyed = covid_owid_ts[datatype]\n dates = popkeyed['dates']\n fmt = '%m/%d/%y'\n # fmt = '%Y-%m-%d' the owid date format was converted to the jhu date format in get_data_owid\n xxd = [datetime.datetime.strptime(dd,fmt) for dd in dates ]\n if firstdate:\n firstdate_d = datetime.datetime.strptime(firstdate,fmt)\n else:\n firstdate_d = datetime.datetime.strptime(dates[0],fmt)\n if lastdate:\n lastdate_d = datetime.datetime.strptime(lastdate,fmt)\n else:\n lastdate_d = datetime.datetime.strptime(dates[-1],fmt)\n daystart = (firstdate_d-xxd[0]).days\n daystop = (lastdate_d-xxd[0]).days\n xx = [0.]*(daystop-daystart+1)\n xx = truncx(xxd,daystart,daystop)\n # print('1 len xx',len(xx))\n \n if daystart <0:\n xx0 = [xx[0]+datetime.timedelta(days=i) for i in range(daystart,0)]\n yy0 = [0.]*(-daystart)\n else:\n xx0 = []\n yy0 = []\n\n if daystop > (xxd[-1]-xxd[0]).days:\n xx1 = [xxd[-1]+datetime.timedelta(days=i) for i in range(daystop-(xxd[-1]-xxd[0]).days)]\n yy1 = [' ']*(daystop-(xxd[-1]-xxd[0]).days)\n else:\n xx1 = []\n yy1 = [] \n xx = xx0 + xx + xx1\n # print('2 len xx',len(xx))\n #print('len xx1 yy1',len(xx1),len(yy1))\n \n # print('len xx',len(xx))\n if fittype == 'piecewise-linear':\n xxi = [Float((x-xx[0]).days) for x in xx ]\n # print(xxi)\n # print('len xxi',len(xxi)) \n # locator = mdates.MonthLocator()\n locator = mdates.AutoDateLocator(minticks=5, maxticks=13)\n formatter= mdates.ConciseDateFormatter(locator)\n \n if not ax:\n fig,ax = plt.subplots(1,1,figsize=(9,6)) \n ax2 = ax\n elif ax2:\n ax2 = ax.twinx()\n else:\n ax2 = ax\n \n colors = ['k', 'b', 'c', 'm', 'y', 'g', 'olive', 'chocolate']\n \n i = 0\n j = 0\n for country in countries:\n try:\n yyd = popkeyed[country]\n if np.max(yyd) >= minconfirmed:\n j = j+1\n else:\n i = i + 1\n continue\n except:\n print('country not found',country)\n i = i + 1\n continue\n yy = truncy(xxd,yyd,daystart,daystop)\n # print(country,'1 len yy yyd',len(yy),len(yyd))\n yyf = [Float(y) for y in yy]\n yy = yy0 + yyf + yy1\n # print(country,'2 len yy',len(yy))\n # ymax=np.max(np.array(yy))\n yyf = [Float(y) for y in yy]\n if dataaccum == 'daily':\n yy = [0.]*len(yy)\n yy[0] = yyf[0]\n for k in range(1,len(yy)):\n yy[k] = yyf[k]-yyf[k-1] \n elif dataaccum == 'cum_av_weekly':\n yy = [0.]*len(yy)\n moving_av = 0.\n for k in range(len(yy)):\n if k-7 >= 0:\n moving_av = moving_av - yyf[k-7]\n moving_av = moving_av + yyf[k]\n yy[k] = moving_av/min(7.0,float(k+1))\n elif dataaccum == 'daily_av_weekly':\n yy = [0.]*len(yyf)\n yy[0] = yyf[0]\n for k in range(1,len(yy)):\n yy[k] = yyf[k]-yyf[k-1]\n yyf = [y for y in yy]\n yy = [0.]*len(yy)\n moving_av = 0.\n for k in range(len(yy)):\n if k-7 >= 0:\n moving_av = moving_av - yyf[k-7]\n moving_av = moving_av + yyf[k]\n yy[k] = moving_av/min(7.0,float(k+1))\n if intdates:\n xx = range(len(xx))\n \n if step:\n ax2.step(xx,yy,label = country[0])\n else:\n # print(ax,ax2)\n # ax2.set_ylim(ymax,0)\n if linecolor:\n color = linecolor\n else:\n color = colors[i]\n\n ax2.plot(xx, yy, symbol, markersize=3, color = color, alpha=0.8, label = country[0])\n\n if maxyval: ax.set_ylim(0,maxyval)\n if maxyval: ax2.set_ylim(0,maxyval)\n \n plt.title(country[0]+'-'+country[1]) # +' '+datatype)\n if fittype == 'piecewise-linear': \n warnings.filterwarnings(\"ignore\", message=\"Warning: zero length interval encountered in pwlf.py calc_slopes\")\n # initialize piecewise linear fit with your x and y data\n # yyf = [Float(y) for y in yy]\n yyf = [Float(y) if not math.isnan(y) else 0.0 for y in yy]\n # print(yyf)\n my_pwlf = pwlf.PiecewiseLinFit(xxi, yyf)\n # fit the data for three line segments\n res = my_pwlf.fit(nsegments)\n \n ppp = my_pwlf.p_values(method='non-linear', step_size=1e-4)\n se = my_pwlf.se # standard errors\n parameters = np.concatenate((my_pwlf.beta,\n my_pwlf.fit_breaks[1:-1]))\n header = ['Parameter type', 'Parameter value', 'Standard error', 't ', 'P > np.abs(t) (p-value)']\n print(*header, sep=' | ')\n values = np.zeros((parameters.size, 5), dtype=np.object_)\n values[:, 1] = np.around(parameters, decimals=3)\n values[:, 2] = np.around(se, decimals=3)\n values[:, 3] = np.around(parameters / se, decimals=3)\n values[:, 4] = np.around(ppp, decimals=3)\n for iii, row in enumerate(values):\n if iii < my_pwlf.beta.size:\n row[0] = 'Slope '\n print(*row, sep=' | ')\n else:\n row[0] = 'Breakpoint'\n print(*row, sep=' | ')\n print(\"\") \n # predict for the determined points\n xHat = np.linspace(min(xxi), max(xxi), num=len(xx))\n # print(len(xHat),len(xxi))\n yHat = my_pwlf.predict(xHat)\n ax2.plot(xx, yHat, color = colors[i], alpha=0.5, label = country[0]+' fit')\n \n i = i+1\n\n if j==0:\n ax.axis(\"off\")\n else:\n if j > 1:\n plt.legend(loc=\"upper left\")\n plt.title('countries '+datatype+dataaccum)\n if not intdates:\n ax2.xaxis.set_major_formatter(formatter)\n ax2.xaxis.set_major_locator(locator)\n for tick in ax2.get_xticklabels():\n tick.set_rotation(40)\n\n\n```\n\n\n```python\n\n```\n\n## Plots of data for Cautionary Model comparison\n\nComment out line 1110 in pwlf.py (in /\u2068usr\u2069/local\u2069/lib\u2069/\u2068python3.7\u2069/site-packages\u2069/pwlf\u2069 directory)\n print(\"Warning: zero length interval encountered in pwlf.py calc_slopes\").\u00a0 \nto remove repeated warnings, which don't seem to harm final result\n Warning: zero length interval encountered in pwlf.py calc_slopes\n\n\n```python\nplotCountry_(['Italy','Spain','Germany','France','United Kingdom','Sweden','Turkey'],\n 'confirmed','cum_av_weekly',firstdate='02/15/20',lastdate='09/1/20',fittype='piecewise-linear',nsegments=4)\n#plt.savefig(\"covid-19-caution/figures/fig1a.pdf\",bbox_inches='tight')\n```\n\n\n```python\nplotCountry_(['Italy','Spain','Germany','France','United Kingdom','Sweden','Turkey'],\n 'confirmed','daily_av_weekly',firstdate='02/15/20',lastdate='08/31/20',database='owid')\nplt.title(\"\");\n# plt.plot(xx,[450000*d[1] for d in dat],linewidth=6,color='salmon',alpha=0.5,linestyle='--');\n# plt.savefig(\"covid-19-caution/figures/fig1b.pdf\",bbox_inches='tight')\n```\n\n# Simulation\n\nNote: Problem with setting parameters in model.\n\nThe DeterministicOde Class method parameters, converts a dictionary or list of tuples of parameters to a dictionary with sympy symbolic keys, not strings.\nSo attempts to modify parameter values by accessing this dictionary fail. Copying the dictionary, modifying and rewriting also fail.\nInstead we store the dictionary of parameters in addition as a dictionary with string keys, under model.params. When we modify these values, they can then be copied back to the parameters method using model.parameters = model.params.\n\n## Simulation of SCIR model\n\n\n```python\n# setup time points for simulation, initial conditions and parameters\nt = np.linspace(0, lastday -1, lastday)\n\n# initial conditions assuming there is no natural immunity\nI_0 = 0.00003\nx0_SCIR = [1.0-I_0, I_0, 0.0, 0.0, 0.0]\n\n# Define parameters based on clinical observations Dr. Alison\nExposure=0.25 # Rate coefficient for exposure per individual in contact per day\nIncubPeriod=5 #Incubation period, days\nDurInf=10 #Duration of mild infections, days\nFracCritical=0.1 #Fraction of infections that are critical\nCFR=0.05 #Case fatality rate (fraction of infections resulting in death)\nTimeDeath=DurInf+7 #Time from ICU admission to death, days\n\n# Model extension by John McCaskill to include caution \nCautionFactor= 0.4 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 21. # Duration of cautionary state of susceptibles (2 weeks)\nCautionICUFrac= 0.25 # Fraction of ICUs occupied leading to transition to caution @ 1/day\nICUFrac= 0.002 # Fraction of ICUs relative to population size N\n\n# Model fitting extension to allow for incomplete detection\nFracConfirmedDet=0.125 # Fraction of recovered individuals measured\nFracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured\nFracDeathsDet=1.0\n\n# convert above parameters to model parameters\nparams = {'beta' : Exposure/sum(x0_SCIR),\n 'gamma': (1.0/DurInf),\n 'mu' : (1.0/TimeDeath)*(CFR/FracCritical),\n 'c_0' : CautionFactor,\n 'c_1' : 1.0/CautionRetention,\n 'c_2' : 1.0/(sum(x0_SCIR)*ICUFrac*CautionICUFrac),\n 'N' : sum(x0_SCIR)}\nprint(params)\n# assign x0 and params to the model, integrate over t and plot\nSCIR_model.initial_values = (x0_SCIR, t[0])\nSCIR_model.parameters = params\nSCIR_model.params = params.copy()\nsolution = SCIR_model.integrate(t[1::])\n\nSCIR_model.plot()\n\n# calculate time point when maximum number of people are infectious\npeak_i = np.argmax(solution[:,1])\nprint('Peak infection (days)', t[peak_i])\n```\n\n### Integration and plot using scipy and matplotlib directly\n\n\n```python\nsolution1 = scipy.integrate.odeint(SCIR_model.ode, x0_SCIR, t[1::])\nys = solution1.copy()\nplt.figure(figsize=(15,7))\nplt.subplot(1,2,1)\nplt.xlabel(\"Time (days)\")\nplt.ylabel(\"Number per 1000000 People\")\nplt.ylim([0,0.001])\nplt.plot(t[1::],ys,label=(\"S\",\"I\",\"R\",\"D\",\"Sc\"))\nplt.legend((\"S\",\"I\",\"R\",\"D\",\"Sc\"))\nplt.subplot(1,2,2)\nplt.xlabel(\"Time (days)\")\nplt.ylabel(\"Number per 1000000 People\")\nplt.ylim([0.000001,1])\nplt.semilogy()\nplt.plot(t[1::],ys,label=(\"S\",\"I\",\"R\",\"D\",\"Sc\"))\nplt.legend((\"S\",\"I\",\"R\",\"D\",\"Sc\"));\n```\n\n### Compare data with SCIR simulation\n\n\n```python\n# model with generating parameters \nparams1 = SCIR_model.params.copy()\nparams1['c_0']=0.85\nparams1['beta']=0.15\nSCIR_model.parameters = params1\nprint('parameters',SCIR_model.parameters)\nx0_fit = x0_SCIR.copy()\nprint('initial conditions',x0_fit)\nt_fit = t\nSCIR_model.initial_values = (x0_fit, t_fit[0])\nsol_fit = scipy.integrate.odeint(SCIR_model.ode, x0_fit, t_fit[1::])\nplt.figure(figsize=(15,10))\nplt.plot(t,y_jhu[test_country][:,1]/FracRecoveredDet, 'bo',label='R') # recovered\nplt.plot(t,y_jhu[test_country][:,2], 'ro',label='D') # died\nplt.gca().set_prop_cycle(color=['grey','orange','green','green','green','blue','red', 'black'])\nplt.plot(t_fit[1::], sol_fit)\nplt.ylim([0,0.02])\nplt.legend()\npeak_i = np.argmax(sol_fit[:,2])\nprint('Peak infection (days)', t_fit[peak_i])\n```\n\n### Stochastic simulation\n\n\n```python\nN=10000\nI_0 = 10\nx0_SCIR_S = [N-I_0, I_0, 0, 0, 0]\n# Define parameters based on clinical observations Dr. Alison\nExposure=0.25 # Rate coefficient for exposure per individual in contact per day\nIncubPeriod=5 #Incubation period, days\nDurInf=10 #Duration of mild infections, days\nFracCritical=0.1 #Fraction of infections that are critical\nCFR=0.05 #Case fatality rate (fraction of infections resulting in death)\nTimeDeath=DurInf+7 #Time from ICU admission to death, days\n\n# Model extension by John McCaskill to include caution \nCautionFactor= 0.4 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 21. # Duration of cautionary state of susceptibles (2 weeks)\nCautionICUFrac= 0.25 # Fraction of ICUs occupied leading to transition to caution @ 1/day\nICUFrac= 0.02 # Fraction of ICUs relative to population size N # increased 10X for low pop simulation\n\n# Model fitting extension to allow for incomplete detection\nFracConfirmedDet=0.125 # Fraction of recovered individuals measured\nFracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured\nFracDeathsDet=1.0\nparams_S = {'beta' : Exposure/sum(x0_SCIR_S),\n 'gamma': (1.0/DurInf),\n 'mu' : (1.0/TimeDeath)*(CFR/FracCritical),\n 'c_0' : CautionFactor,\n 'c_1' : 1.0/CautionRetention,\n 'c_2' : 1.0/(sum(x0_SCIR_S)*ICUFrac*CautionICUFrac),\n 'N' : sum(x0_SCIR_S)}\nSCIR_modelS.initial_values = (x0_SCIR_S, t[0])\nSCIR_modelS.parameters = params_S\nSCIR_modelS.params = params_S.copy()\nt_jump = np.linspace(0,100,50)\nsimX, simT =SCIR_modelS.simulate_jump(t_jump, iteration=5, full_output=True)\n```\n\n\n```python\nplt.figure(figsize=(25,15))\n\nfor iter in range(5):\n plt.subplot(2,5,iter+1)\n plt.xlabel(\"Time (days)\")\n plt.ylabel(\"Population\")\n plt.plot(simT,simX[iter],label=(\"S\",\"I\",\"R\",\"D\",\"Sc\"))\n plt.legend((\"S\",\"I\",\"R\",\"D\",\"Sc\"))\nfor iter in range(5):\n plt.subplot(2,5,5+iter+1)\n plt.xlabel(\"Time (days)\")\n plt.ylabel(\"Population\")\n plt.semilogy()\n plt.plot(simT,simX[iter],label=(\"S\",\"I\",\"R\",\"D\",\"Sc\"))\n plt.legend((\"S\",\"I\",\"R\",\"D\",\"Sc\"));\n```\n\n## Simulation of SC2IR model\n\n\n```python\n# setup time points for simulation, initial conditions and parameters\nt = np.linspace(0, lastday -1, lastday)\n\n# initial conditions assuming there is no natural immunity\nI_0 = 0.00003\nx0 = [1.0-I_0, I_0, 0.0, 0.0, 0.0, 0.0]\n\n# Define parameters based on clinical observations Dr. Alison\nExposure=0.25 # Rate coefficient for exposure per individual in contact per day\nIncubPeriod=5 #Incubation period, days\nDurInf=10 #Duration of mild infections, days\nFracCritical=0.1 #Fraction of infections that are critical\nCFR=0.05 #Case fatality rate (fraction of infections resulting in death)\nTimeDeath=DurInf+7 #Time from ICU admission to death, days\n\n# Model extension by John McCaskill to include caution \nCautionFactor= 0.4 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 21. # Duration of cautionary state of susceptibles (2 weeks)\nCautionICUFrac= 0.25 # Fraction of ICUs occupied leading to transition to caution @ 1/day\nICUFrac= 0.002 # Fraction of ICUs relative to population size N\n\n# Model fitting extension to allow for incomplete detection\nFracConfirmedDet=0.125 # Fraction of recovered individuals measured\nFracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured\nFracDeathsDet=1.0\n\n# convert above parameters to model parameters\nparams = {'beta' : Exposure/sum(x0),\n 'gamma': (1.0/DurInf),\n 'mu' : (1.0/TimeDeath)*(CFR/FracCritical),\n 'c_0' : CautionFactor,\n 'c_1' : 1.0/CautionRetention,\n 'c_2' : 1.0/(sum(x0)*ICUFrac*CautionICUFrac),\n 'N' : sum(x0)}\nprint('parameters',params)\n# assign x0 and params to the model, integrate over t and plot\nSC2IR_model.initial_values = (x0, t[0])\nSC2IR_model.parameters = params\nSC2IR_model.params = params.copy()\nsolution = SC2IR_model.integrate(t[1::])\n\nSC2IR_model.plot()\n\n# calculate time point when maximum number of people are infectious\npeak_i = np.argmax(solution[:,1])\nprint('Peak infection (days)', t[peak_i])\n```\n\n### Integration and plot using scipy and matplotlib directly\n\n\n```python\nsolution1 = scipy.integrate.odeint(SC2IR_model.ode, x0, t[1::])\nys = solution1.copy()\nplt.figure(figsize=(15,7))\nplt.subplot(1,2,1)\nplt.xlabel(\"Time (days)\")\nplt.ylabel(\"Fraction of population\")\nplt.ylim([0,1])\nplt.gca().set_prop_cycle(color=['grey','green','blue','red','darkgreen', 'black'])\nplt.plot(t[1::],ys,label=(\"S\",\"I\",\"R\",\"D\",'Ic',\"Sc\"))\nplt.legend((\"S\",\"I\",\"R\",\"D\",\"Ic\",\"Sc\"))\nplt.title(SC2IR_model.modelname)\nplt.subplot(1,2,2)\nplt.xlabel(\"Time (days)\")\nplt.ylabel(\"Fraction of population\")\nplt.ylim([0.000001,1])\nplt.semilogy()\nplt.gca().set_prop_cycle(color=['grey','green','blue','red','darkgreen', 'black'])\nplt.plot(t[1::],ys,label=(\"S\",\"I\",\"R\",\"D\",'Ic',\"Sc\"))\nplt.legend((\"S\",\"I\",\"R\",\"D\",\"Ic\",\"Sc\"))\nplt.title(SC2IR_model.modelname + ' - semilog');\n```\n\n### Compare data with SC2IR simulation\n\n\n```python\n# model with generating parameters \ndef isolveplot(beta,gamma,mu,c_0,c_1,c_2,logI_0):\n # saveparams=SC2IR_model.parameters.copy() # backup model current parameters\n # saveICs = SC2IR_model.initial_values # back model ICs\n I_0 = 10.**logI_0\n x0 = [1.0-I_0, I_0, 0.0, 0.0, 0.0, 0.0]\n params = {'beta' : beta,\n 'gamma': gamma,\n 'mu' : mu,\n 'c_0' : c_0,\n 'c_1' : c_1,\n 'c_2' : c_2,\n 'N' : sum(x0)}\n SC2IR_model.initial_values = (x0, t[0])\n SC2IR_model.parameters = params.copy()\n SC2IR_model.params = params.copy()\n SC2IR_model.dumpparams()\n \n sol_fit = scipy.integrate.odeint(SC2IR_model.ode, x0, t[1::])\n #\n plt.figure(figsize=(15,10))\n plt.plot(t,y_jhu[test_country][:,1]/FracRecoveredDet, 'bo',label='R') # recovered\n plt.semilogy()\n plt.plot(t,y_jhu[test_country][:,2], 'ro',label='D') # died\n plt.semilogy()\n plt.gca().set_prop_cycle(color=['grey','green','blue','red','darkgreen','black'])\n plt.plot(t[1::], sol_fit)\n plt.ylim([0.000001,1])\n plt.semilogy()\n plt.legend(('R','D','S','I','R','D','I_c','S_c'))\n #plt.show(())\n #ode_fit.plot()\n\n peak_i = np.argmax(sol_fit[:,2])\n print('Peak infection (days)', t_fit[peak_i])\n # SC2IR_model.parameters=saveparams.copy()\n # SC2IR_model.initial_values=saveICs\n```\n\n\n```python\nif SC2IR_model.loadparams():\n params = SC2IR_model.params.copy()\nelse:\n params = {'beta' : 0.25,\n 'gamma': 0.1,\n 'mu' : 0.05,\n 'c_0' : 0.3,\n 'c_1' : 1./14.,\n 'c_2' : 2000.,\n 'N' : 1.}\ninteract(isolveplot,\n beta=FloatSlider(min=0,max=1,step=0.01,value=params['beta'],description='beta',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n gamma=FloatSlider(min=0,max=1,step=0.01,value=params['gamma'],description='gamma',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n mu=FloatSlider(min=0,max=0.1,step=0.001,value=params['mu'],description='mu',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), \n c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_2=FloatSlider(min=0,max=5000,step=1,value=params['c_2'],description='c_2',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'),\n logI_0=FloatSlider(min=-10,max=0,step=0.01,value=np.log10(x0[1]),description='log I_0',\n style=style,layout=slider_layout,continuous_update=False))\n```\n\n loaded params from /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC2IR.pk :\n\n\n\n interactive(children=(FloatSlider(value=0.54, continuous_update=False, description='beta', layout=Layout(width\u2026\n\n\n\n\n\n \n\n\n\n\n```python\nslide_params=SC2IR_model.parameters.copy()\nprint(slide_params)\ntheta = [0.4,0.11,0.007,0.33,0.228,275.]\n```\n\n {beta: 0.46, gamma: 0.1, mu: 0.045, c_0: 0.3, c_1: 0.07142857142857142, c_2: 153.0, N: 1.0}\n\n\n\n```python\nSC2IR_model.params\n```\n\n\n\n\n {'beta': 0.25,\n 'gamma': 0.1,\n 'mu': 0.029411764705882353,\n 'c_0': 0.4,\n 'c_1': 0.047619047619047616,\n 'c_2': 2000.0,\n 'N': 1.0}\n\n\n\n## Simulation of SCEI3R model\n\n\n```python\n# setup time points for simulation, initial conditions and parameters\nt = np.linspace(0, lastday -1, lastday)\n\n# initial conditions assuming there is no natural immunity\nE_0 = 0.00003\nx0_SCEI3R = [1.0-E_0, E_0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]\n\n# Define parameters based on clinical observations Dr. Alison\nExposure=0.25 # Rate coefficient for exposure per individual in contact per day\nIncubPeriod=5 #Incubation period, days\nDurMildInf=10 #Duration of mild infections, days\nFracMild=0.70 #Fraction of infections that are mild\nFracSevere=0.20 #Fraction of infections that are severe\nFracCritical=0.1 #Fraction of infections that are critical\nCFR=0.07 #Case fatality rate (fraction of infections resulting in death)\nTimeICUDeath=7 #Time from ICU admission to death, days\nDurHosp=11 #Duration of hospitalization, days\n\n# Model extension by John McCaskill to include caution \nCautionFactor= 0.2 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 21. # Duration of cautionary state of susceptibles (2 weeks)\nCautionICUFrac= 0.3 # Fraction of ICUs occupied leading to transition to caution @ 1/day\nICUFrac= 0.002 # Fraction of ICUs relative to population size N\n\n# Model fitting extension to allow for incomplete detection\nFracConfirmedDet=0.125 # Fraction of recovered individuals measured\nFracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured\nFracDeathsDet=1.0\n\n# convert above parameters to model parameters\nparams = {'beta_1' : Exposure/sum(x0_SCEI3R),\n 'beta_2' : 0.0,\n 'beta_3' : 0.0,\n 'alpha' : 1.0/IncubPeriod,\n 'gamma_1': (1.0/DurMildInf)*FracMild,\n 'gamma_2': (1.0/DurHosp)-(1/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n 'gamma_3': (1.0/TimeICUDeath)-(1/TimeICUDeath)*(CFR/FracCritical),\n 'p_1' : (1.0/DurMildInf)-(1.0/DurMildInf)*FracMild,\n 'p_2' : (1.0/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n 'mu' : (1.0/TimeICUDeath)*(CFR/FracCritical),\n 'c_0' : CautionFactor,\n 'c_1' : 1.0/CautionRetention,\n 'c_2' : 1.0/(sum(x0_SCEI3R)*ICUFrac*CautionICUFrac),\n 'N' : sum(x0_SCEI3R)}\nprint(params)\n# assign x0 and params to the model, integrate over t and plot\nSCEI3R_model.initial_values = (x0_SCEI3R, t[0])\nSCEI3R_model.parameters = params\nSCEI3R_model.params = params.copy()\nsolution = SCEI3R_model.integrate(t[1::])\n\nSCEI3R_model.plot()\n\n# calculate time point when maximum number of people are infectious\npeak_i = np.argmax(solution[:,2])\nprint('Peak infection (days)', t[peak_i])\n```\n\n### Compare data with SCEI3R simulation\n\n\n```python\n# model with generating parameters \nparams1 = SCEI3R_model.params.copy()\nparams1['c_0']=0.7\nSCEI3R_model.parameters = params1\nprint(SCEI3R_model.parameters)\nx0_fit = x0_SCEI3R.copy()\n# x0_fit[2] = 0.00001\n#t_fit = numpy.linspace(0, 150, 1000)\nprint(x0_fit)\nt_fit = t\nprint(len(t))\nSCEI3R_model.initial_values = (x0_fit, t_fit[0])\n# %timeit sol_fit =SCEI3R_model.integrate(t_fit[1::]) # use magic %timeit to time\n#sol_fit =SCEI3R_model.integrate(t_fit[1::])\n# sol_fit = SCEI3R_model.integrate(t_fit)\nsol_fit = scipy.integrate.odeint(SCEI3R_model.ode, x0_fit, t_fit[1::])\n# print(len(sol_fit[0]))\n#\nplt.figure(figsize=(15,10))\n#plt.plot(t,y_jhu[test_country][:,0], 'go',label='I_1') # infected observations\n#plt.plot(t,y_jhu[test_country][:,1], 'go',label='I_2') # infected observations\n#plt.plot(t,y_jhu[test_country][:,2], 'go',label='I_3') # infected observations\nplt.plot(t,y_jhu[test_country][:,1]/FracRecoveredDet, 'bo',label='R') # recovered\nplt.plot(t,y_jhu[test_country][:,2], 'ro',label='D') # died\nplt.gca().set_prop_cycle(color=['grey','orange','green','green','green','blue','red','black',])\nplt.plot(t_fit[1::], sol_fit)\nplt.ylim([0,0.06])\nplt.legend()\n#plt.show(())\n#ode_fit.plot()\n\npeak_i = np.argmax(sol_fit[:,2])\nprint('Peak infection (days)', t_fit[peak_i])\n```\n\n### Integration and plot using scipy and matplotlib directly\n\n\n```python\n# solution = scipy.integrate.odeint(SCEI3R_model.ode, x0, t)\n# print(len(t))\nsolution1 = scipy.integrate.odeint(SCEI3R_model.ode, x0_SCEI3R, t[1::])\nys = solution1.copy()\nplt.figure(figsize=(15,7))\nplt.subplot(1,2,1)\nplt.xlabel(\"Time (days)\")\nplt.ylabel(\"Number per 1000000 People\")\nplt.ylim([0,1])\nplt.legend((\"S\",\"E\",\"I1\",\"I2\",\"I3\",\"R\",\"D\",\"Sc\"))\nplt.plot(t[1::],ys)\nplt.subplot(1,2,2)\nplt.xlabel(\"Time (days)\")\nplt.ylabel(\"Number per 1000000 People\")\nplt.ylim([0.000001,1])\nplt.legend((\"S\",\"E\",\"I1\",\"I2\",\"I3\",\"R\",\"D\",\"Sc\"))\nplt.semilogy()\nplt.plot(t[1::],ys);\n```\n\n## Simulation of SC3EI3R model\n\n\n```python\nlen(params)\n```\n\n\n```python\n# setup time points for simulation, initial conditions and parameters\nt = np.linspace(0, lastday -1, lastday)\ntmax=lastday-1\n# initial conditions assuming there is no natural immunity\nE_0 = 0.00003\nx0_SC3EI3R = [1.0-E_0, E_0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]\nx0_SC3EI3R = [1.0-E_0, E_0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]\n\n\n# Define parameters based on clinical observations Dr. Alison\nExposure=0.25 # Rate coefficient for exposure per individual in contact per day\nIncubPeriod=5 #Incubation period, days\nDurMildInf=10 #Duration of mild infections, days\nFracMild=0.70 #Fraction of infections that are mild\nFracSevere=0.20 #Fraction of infections that are severe\nFracCritical=0.1 #Fraction of infections that are critical\nCFR=0.07 #Case fatality rate (fraction of infections resulting in death)\nTimeICUDeath=7 #Time from ICU admission to death, days\nDurHosp=11 #Duration of hospitalization, days\n\n# Model fitting extension to allow for incomplete detection\nFracConfirmedDet=0.125 # Fraction of recovered individuals measured\nFracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured\nFracDeathsDet=1.0\n\n# Model extension by John McCaskill to include caution \nCautionFactor= 0.2 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 21. # Duration of cautionary state of susceptibles (2 weeks)\nCautionICUFrac= 0.3 # Fraction of ICUs occupied leading to transition to caution @ 1/day\nICUFrac= 0.002 # Fraction of ICUs relative to population size N\n\n\n# convert above parameters to model parameters\nparams = {'beta_1' : Exposure/sum(x0_SC3EI3R),\n 'beta_2' : 0.0,\n 'beta_3' : 0.0,\n 'alpha' : 1.0/IncubPeriod,\n 'gamma_1': (1.0/DurMildInf)*FracMild,\n 'gamma_2': (1.0/DurHosp)-(1/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n 'gamma_3': (1.0/TimeICUDeath)-(1/TimeICUDeath)*(CFR/FracCritical),\n 'p_1' : (1.0/DurMildInf)-(1.0/DurMildInf)*FracMild,\n 'p_2' : (1.0/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n 'mu' : (1.0/TimeICUDeath)*(CFR/FracCritical),\n 'c_0' : CautionFactor,\n 'c_1' : 1.0/CautionRetention,\n 'c_2' : 1.0/(sum(x0_SC3EI3R)*ICUFrac*CautionICUFrac),\n 'N' : sum(x0_SC3EI3R)}\n\nprint(params)\n# assign x0 and params to the model, integrate over t and plot\nSC3EI3R_model.initial_values = (x0_SC3EI3R, t[0])\nSC3EI3R_model.parameters = params\nSC3EI3R_model.params = params.copy()\nsolution = SC3EI3R_model.integrate(t[1::])\n\nSC3EI3R_model.plot()\n\n# calculate time point when maximum number of people are infectious\npeak_i = np.argmax(solution[:,2])\nprint('Peak infection (days)', t[peak_i])\n```\n\n\n```python\nx0_SC3EI3R\n```\n\n### Compare data with SC3EI3R simulation\n\n\n```python\n# model with generating parameters \nparams1 = SC3EI3R_model.params.copy()\nparams1['c_0']=0.35\nSC3EI3R_model.parameters = params1\nprint(SC3EI3R_model.parameters)\nx0_fit = x0_SC3EI3R.copy()\n# x0_fit[2] = 0.00001\n#t_fit = numpy.linspace(0, 150, 1000)\nprint(x0_fit)\nt_fit = t\nprint(len(t))\nSC3EI3R_model.initial_values = (x0_fit, t_fit[0])\n# %timeit sol_fit =SC3EI3R_model.integrate(t_fit[1::]) # use magic %timeit to time\nsol_fit =SC3EI3R_model.integrate(t_fit[1::])\n# sol_fit = SC3EI3R_model.integrate(t_fit)\n# sol_fit = scipy.integrate.odeint(SC3EI3R_model(params_fit).ode, x0_fit, t_fit[1::])\n# print(len(sol_fit[0]))\n#\nplt.figure(figsize=(15,10))\nplt.plot(t,y_jhu[test_country][:,1]/FracRecoveredDet, 'bo',label='R') # recovered\nplt.plot(t,y_jhu[test_country][:,2]/FracDeathsDet, 'ro',label='D') # died\nplt.gca().set_prop_cycle(color=['grey','orange','green','green','green','blue','red','darkgreen', 'black'])\n#plt.plot(t_fit[1::], sol_fit)\nplt.plot(t_fit, sol_fit)\nplt.ylim([0,0.06])\nplt.legend()\n#plt.show(())\n#ode_fit.plot()\n\npeak_i = np.argmax(sol_fit[:,2])\nprint('Peak infection (days)', t_fit[peak_i])\n```\n\n## Simulation models for range of caution parameters\n\n### SCIR, SC2IR, SCEIR, SC3EIR, SCEI3R, SC3EI3R, SC2UIR, SC3UEIR, SC3UEI3R simulations\n\n\n```python\ndef param_copy(model):\n params = model.parameters\n newparams = {}\n pkeys1 = list(model.params.keys())\n pkeys2 = list(model.parameters.keys())\n for i in range(len(pkeys1)):\n newparams[pkeys1[i]] = params[pkeys2[i]]\n print(newparams)\n model.parameters=newparams\n \ndef param_modify(model,param,value):\n params = model.parameters\n newparams = {}\n pkeys1 = list(model.params.keys())\n pkeys2 = list(model.parameters.keys())\n for i in range(len(pkeys1)):\n newparams[pkeys1[i]] = params[pkeys2[i]]\n newparams[param]=value\n print(newparams)\n model.parameters=newparams\n \nparam_modify(SCIR_model,'beta',0.721)\n```\n\n\n```python\nSCIR_model.parameters = {'gamma':0.4}\nSCIR_model.parameters\n```\n\n\n\n\n {beta: 0.721,\n gamma: 0.4,\n mu: 0.029411764705882353,\n c_0: 0.85,\n c_1: 0.047619047619047616,\n c_2: 2000.0,\n N: 1.0}\n\n\n\n\n```python\ndef vector2params_old(b,a,g,p,u,c,k,N,modelname):\n if 'I3' in modelname: # models with hospitalization\n params = {\n 'beta_1' : b[1],\n 'beta_2' : b[2],\n 'beta_3' : b[3],\n 'alpha' : a,\n 'gamma_1': g[1],\n 'gamma_2': g[2],\n 'gamma_3': g[3],\n 'p_1' : p[1],\n 'p_2' : p[2],\n 'mu' : u}\n elif 'E' in modelname:\n params = {\n 'beta' : b[1], # see above for explanations\n 'alpha' : a, \n 'gamma': g[1]+g[2]*(p[1]/(g[2]+p[2]))+g[3]*(p[1]/(g[2]+p[2]))*(p[2]/(g[3]+u)),\n 'mu' : u*(p[1]/(g[2]+p[2])*(p[2]/(g[3]+u)))} \n else:\n params = {\n 'beta' : b[1], # see above for explanations\n 'gamma': g[1]+g[2]*(p[1]/(g[2]+p[2]))+g[3]*(p[1]/(g[2]+p[2]))*(p[2]/(g[3]+u)),\n 'mu' : u*(p[1]/(g[2]+p[2])*(p[2]/(g[3]+u)))}\n \n if 'C' in modelname: # models with caution \n params['c_0'] = c[0]\n params['c_1'] = c[1]\n if 'I3' in modelname: # models with hospitalization\n params['c_2'] = c[2]\n else:\n params['c_2'] = c[2]*FracCritical\n \n if 'U' in modelname: # models with economic correction to caution \n params['k_u'] = k[0]\n params['k_1'] = k[1]\n params['k_w'] = k[2]\n params['kappa'] = k[3]\n \n params['N'] = N\n return params\n\ndef params2vector_old(params):\n b = [None,None,None]\n g = [None,None,None]\n p = [None,None,None]\n c = [None,None,None]\n b[0]=0.0\n b[1]=params['beta_1']\n b[2]=params['beta_2']\n b[3]=params['beta_3']\n g[0]=0.0\n g[1]=params['gamma_1']\n g[2]=params['gamma_2']\n g[3]=params['gamma_3']\n p[0]=0.0\n p[1]=params['p_1']\n p[2]=params['p_2']\n c[0]=params['c_1']\n c[1]=params['c_2']\n c[2]=params['c_3']\n a=params['alpha']\n u=params['mu']\n N=params['N']\n return (b,a,g,p,u,c,N)\n```\n\n\n```python\ndef vector2params(b,a,g,p,u,c,k,N,modelname):\n global FracCritical\n if 'I3' in modelname: # models with hospitalization\n params = {\n 'beta_1' : b[1],\n 'beta_2' : b[2],\n 'beta_3' : b[3],\n 'alpha' : a,\n 'gamma_1': g[1],\n 'gamma_2': g[2],\n 'gamma_3': g[3],\n 'p_1' : p[1],\n 'p_2' : p[2],\n 'mu' : u}\n elif 'E' in modelname:\n irat = 1 + p[1]/(g[2]+p[2]) + p[2]/(g[3]+u)\n #irat = 1\n params = {\n 'beta' : b[1], # see above for explanations\n 'alpha' : a, \n 'gamma': (g[1]+g[2]*(p[1]/(g[2]+p[2]))+g[3]*(p[1]/(g[2]+p[2]))*(p[2]/(g[3]+u)))/irat,\n 'mu' : u*(p[1]/(g[2]+p[2])*(p[2]/(g[3]+u))/irat)}\n else:\n irat = 1 + p[1]/(g[2]+p[2]) + p[2]/(g[3]+u)\n #irat = 1\n params = {\n #'beta' : np.sqrt(b[1]*a), # see above for explanations\n 'beta' : b[1], # see above for explanations\n 'gamma': (g[1]+g[2]*(p[1]/(g[2]+p[2]))+g[3]*(p[1]/(g[2]+p[2]))*(p[2]/(g[3]+u)))/irat,\n 'mu' : u*(p[1]/(g[2]+p[2])*(p[2]/(g[3]+u))/irat)}\n \n if 'C' in modelname: # models with caution \n params['c_0'] = c[0]\n params['c_1'] = c[1]\n if 'I3' in modelname: # models with hospitalization\n params['c_2'] = c[2]\n else:\n params['c_2'] = c[2]*FracCritical\n \n if 'U' in modelname: # models with economic correction to caution \n params['k_u'] = k[0]\n params['k_1'] = k[1]\n params['k_w'] = k[2]\n params['kappa'] = k[3]\n \n params['N'] = N\n return params\n\ndef params2vector(params):\n b = [None,None,None]\n g = [None,None,None]\n p = [None,None,None]\n c = [None,None,None]\n b[0]=0.0\n b[1]=params['beta_1']\n b[2]=params['beta_2']\n b[3]=params['beta_3']\n g[0]=0.0\n g[1]=params['gamma_1']\n g[2]=params['gamma_2']\n g[3]=params['gamma_3']\n p[0]=0.0\n p[1]=params['p_1']\n p[2]=params['p_2']\n c[0]=params['c_1']\n c[1]=params['c_2']\n c[2]=params['c_3']\n a=params['alpha']\n u=params['mu']\n N=params['N']\n return (b,a,g,p,u,c,N)\n```\n\n\n```python\ncount = 0\n\ndef difference(datain):\n dataout = np.zeros(np.shape(datain))\n for i in range(1,len(datain)):\n dataout[i,...] = datain[i,...]-datain[i-1,...]\n return dataout\n \ndef rolling_average(datain,period):\n (tmax,n) = np.shape(datain)\n dataout = np.zeros((tmax,n),dtype=float)\n moving_av = np.zeros(n,dtype=float)\n for k in range(len(datain)):\n if k-period >= 0:\n moving_av[:] = moving_av[:] - datain[k-7,...]\n moving_av[:] = moving_av[:] + datain[k,...]\n dataout[k] = moving_av/min(float(period),float(k+1))\n return dataout\n\naxes = [None]\n \ndef solveplot(smodels=['SIR','SCIR','SC2IR','SEIR','SCEIR','SC3EIR','SEI3R','SCEI3R','SC3EI3R'],species='EI',tmax=100,summing='daily',averaging='weekly',fitdata = None,scale='linear',plottitle= '',label='',\n newplot = True, gbrcolors=False, figsize = None):\n \"\"\"\n solve ODEs and plot for set of models indicated\n params: dictionary of simulation parameters\n scale: alternative 'linear' or 'log'\n species alternatives 'all', 'EI', 'confirmed', 'deaths', 'daily confirmed', 'daily deaths'\n plottitle : title for plot\n label : label for curve when called as part of multicurve plot\n newplot : whether to open new plot True/False\n models : list of models to include, default all three of those possible\n \"\"\"\n global count\n global axes\n global FracConfirmedDet,FracRecoveredDet,FracDeathsDet\n tvec=np.arange(0,tmax,1)\n tvec1 = tvec[1:]\n\n if not fitdata is None:\n tmaxf = len(fitdata)\n if fitdata.ndim != 2:\n print(\"error in number of dimensions of array\")\n else:\n print(\"fit data \",np.shape(fitdata))\n tvecf=np.arange(0,tmaxf,1)\n tvecf1 = tvecf[1:]\n \n nmodels = len(smodels)\n nm = 0\n \n count = count+1\n \n if newplot:\n axes = [None]*nmodels \n if (figsize == None):\n figsize=(nmodels*8,6)\n plt.figure(figsize=figsize)\n # fig, axeslist = plt.subplots(1, nmodels, figsize=(nmodels*8,6))\n \n solns = [] \n for smodel in smodels:\n model = cmodels[smodel]\n nm = nm + 1\n soln = scipy.integrate.odeint(model.ode, model.initial_values[0], tvec[1::])\n #Plot\n # ax = axeslist[nm]\n if axes[nm-1] == None: \n ax = axes[nm-1] = plt.subplot(1,nmodels,nm)\n else:\n ax = axes[nm-1]\n if scale == 'log': #Plot on log scale\n ax.semilogy()\n ax.ylim([0.00000001,1.0])\n if not isinstance(species,list):\n lspecies = [species]\n else:\n lspecies = species\n \n if summing == 'daily':\n ssoln = difference(soln)\n if not fitdata is None:\n sfit = difference(fitdata)\n else:\n ssoln = soln\n if not fitdata is None:\n sfit = fitdata\n \n if averaging == 'weekly':\n srsoln = rolling_average(ssoln,7)\n if not fitdata is None:\n srfit = rolling_average(sfit,7)\n else:\n srsoln = ssoln\n if not fitdata is None:\n srfit = sfit\n \n for species in lspecies:\n if species == 'confirmed':\n suma = np.sum(srsoln[:,model.confirmed],axis=1)\n if not fitdata is None:\n ax.plot(tvec1,suma,label=label,color='green')\n fita = srfit[1::,0]/FracConfirmedDet # confirmed cases data, corrected by FracConfirmedDet\n ax.plot(tvecf1,fita,'o',label=label,color='green')\n else:\n ax.plot(tvec1,suma,label=label)\n if species == 'recovered':\n suma = np.sum(srsoln[:,model.recovered],axis=1) \n if not fitdata is None:\n ax.plot(tvec1,suma,label=label,color='blue')\n fita = srfit[1::,1]/FracRecoveredDet # recovered cases data, corrected by FracRecoveredDet\n ax.plot(tvecf1,fita,'o',label=label,color='blue')\n else:\n ax.plot(tvec1,suma,label=label)\n elif species == 'deaths':\n suma = np.sum(srsoln[:,model.deaths],axis=1)\n if not fitdata is None:\n ax.plot(tvec1,suma,label=label,color='red')\n fita = srfit[1::,2]/FracDeathsDet # deaths cases data, corrected by FracDeathsDet\n ax.plot(tvecf1,fita,'o',label=label,color='red')\n else:\n ax.plot(tvec1,suma,label=label)\n elif species == 'deaths_x10':\n suma = np.sum(srsoln[:,model.deaths],axis=1)*10\n if not fitdata is None:\n ax.plot(tvec1,suma,label=label,color='red')\n fita = srfit[1::,2]*10/FracDeathsDet # deaths cases data, corrected by FracDeathsDet\n ax.plot(tvecf1,fita,'o',label=label,color='red')\n else:\n ax.plot(tvec1,suma,label=label)\n elif species == 'EI':\n ax.plot(tvec1,soln[:,model.ei],label=label)\n # ax.plot(tvec1,soln[:,model.ei],label=\"%s\" % count)\n if 'I3' in model.modelname: \n plt.legend((\"E\",\"I1\",\"I2\",\"I3\"))\n elif 'E' in model.modelname: \n plt.legend((\"E\",\"I\"))\n else:\n plt.legend((\"I\"))\n elif species == 'all':\n ax.plot(tvec1,soln,label=label)\n\n if 'I3' in model.modelname:\n if 'C3'in model.modelname:\n pspecies=(\"S\",\"E\",\"I1\",\"I2\",\"I3\",\"R\",\"D\",\"Ic\",\"Sc\",\"Ec\")\n elif 'C' in model.modelname:\n pspecies=(\"S\",\"E\",\"I1\",\"I2\",\"I3\",\"R\",\"D\",\"Sc\")\n else:\n pspecies=(\"S\",\"E\",\"I1\",\"I2\",\"I3\",\"R\",\"D\")\n elif 'E' in model.modelname:\n if 'C3'in model.modelname:\n pspecies=(\"S\",\"E\",\"I\",\"R\",\"D\",\"Ic\",\"Sc\",\"Ec\")\n else:\n pspecies=(\"S\",\"E\",\"I\",\"R\",\"D\",\"Sc\") \n else:\n if 'C2'in model.modelname:\n pspecies=(\"S\",\"I\",\"R\",\"D\",\"Ic\",\"Sc\")\n else:\n pspecies=(\"S\",\"I\",\"R\",\"D\",\"Sc\")\n plt.legend(pspecies)\n \n plt.xlabel(\"Time (days)\")\n plt.ylabel(\"Fraction of population\")\n plt.title(model.modelname +' '+plottitle)\n solns.append(soln)\n return solns\n```\n\n\n```python\n# Set up multimodel consistent sets of parameters\nExposure=0.25 # Rate coefficient for exposure per individual in contact per day\nIncubPeriod=5 #Incubation period, days \nDurMildInf=10 #Duration of mild infections, days\nFracMild=0.8 #Fraction of infections that are mild\nFracSevere=0.15 #Fraction of infections that are severe\nFracCritical=0.05 #Fraction of infections that are critical\nCFR=0.02 #Case fatality rate (fraction of infections resulting in death)\nTimeICUDeath=7 #Time from ICU admission to death, days\nDurHosp=11 #Duration of hospitalization, days\n\n# Model extension by John McCaskill to include caution \nCautionFactor= 0.3 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 14. # Duration of cautionary state of susceptibles (4 weeks)\nCautionICUFrac= 0.25 # Fraction of ICUs occupied leading to 90% of susceptibles in caution \nICUFrac= 0.001 # Fraction of ICUs relative to population size N\n\nEconomicCostOfCaution = 0.5 # Cost to economy of individual exercising caution\n\nN=1\nb=np.zeros(4) # beta\ng=np.zeros(4) # gamma\np=np.zeros(3) # progression\nc=np.zeros(3) # caution\nk=np.zeros(4) # economic caution\n\na=1/IncubPeriod # transition rate from exposed to infected\nb=Exposure*np.array([0,1,0,0])/N # hospitalized cases don't transmit\nu=(1/TimeICUDeath)*(CFR/FracCritical) # death rate from ICU\ng[3]=(1/TimeICUDeath)-u # recovery rate\n\np[2]=(1/DurHosp)*(FracCritical/(FracCritical+FracSevere))\ng[2]=(1/DurHosp)-p[2]\n\ng[1]=(1/DurMildInf)*FracMild\np[1]=(1/DurMildInf)-g[1]\n\nc[0]=CautionFactor\nc[1]=1/CautionRetention\nc[2]=1/(N*ICUFrac*CautionICUFrac) # this is the rate coefficient giving 1/day at I3 = denominator\n\nk[0]=c[1]\nk[1]=c[1]\nk[2]=c[1]\nk[3]=EconomicCostOfCaution\n\ncmodels = {'SIR':SIR_model,'SCIR':SCIR_model,'SC2IR':SC2IR_model,\n 'SEIR':SEIR_model,'SCEIR':SCEIR_model,'SC3EIR':SC3EIR_model,\n 'SEI3R':SEI3R_model,'SCEI3R':SCEI3R_model,'SC3EI3R':SC3EI3R_model,\n 'SC2UIR':SC2UIR_model,'SC3UEIR':SC3UEIR_model,'SC3UEI3R':SC3UEI3R_model}\nsmodels = ['SIR','SCIR','SC2IR','SEIR','SCEIR','SC3EIR','SEI3R','SCEI3R','SC3EI3R','SC2UIR','SC3UEIR','SC3UEI3R']\n\nfor smodel in smodels: \n params_in=vector2params(b,a,g,p,u,c,k,N,smodel)\n # print(smodel,params_in)\n cmodels[smodel].parameters = params_in\n \nI_0 = 0.00003\n\nx0_SIR = [1.0-I_0, I_0, 0.0, 0.0]\nx0_SCIR = [1.0-I_0, I_0, 0.0, 0.0, 0.0]\nx0_SC2IR = [1.0-I_0, I_0, 0.0, 0.0, 0.0, 0.0]\nSIR_model.initial_values = (x0_SIR, t[0])\nSCIR_model.initial_values = (x0_SCIR, t[0])\nSC2IR_model.initial_values = (x0_SC2IR, t[0])\n\nx0_SEIR = [1.0-I_0, 0.0, I_0, 0.0, 0.0]\nx0_SCEIR = [1.0-I_0, 0.0, I_0, 0.0, 0.0, 0.0]\nx0_SC3EIR = [1.0-I_0, 0.0, I_0, 0.0, 0.0, 0.0, 0.0, 0.0]\nSEIR_model.initial_values = (x0_SEIR, t[0])\nSCEIR_model.initial_values = (x0_SCEIR, t[0])\nSC3EIR_model.initial_values = (x0_SC3EIR, t[0])\n\nx0_SEI3R = [1.0-I_0, 0.0, I_0, 0.0, 0.0, 0.0, 0.0]\nx0_SCEI3R = [1.0-I_0, 0.0, I_0, 0.0, 0.0, 0.0, 0.0, 0.0]\nx0_SC3EI3R = [1.0-I_0, 0.0, I_0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]\nx0_SC3EI3R = [1.0-I_0, 0.0, I_0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]\nSEI3R_model.initial_values = (x0_SEI3R, t[0])\nSCEI3R_model.initial_values = (x0_SCEI3R, t[0])\nSC3EI3R_model.initial_values = (x0_SC3EI3R, t[0])\n\n\nx0_SC2UIR = [1.0-I_0, I_0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0]\nx0_SC3UEIR = [1.0-I_0, 0.0, I_0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0]\nx0_SC3UEI3R = [1.0-I_0, 0.0, I_0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0]\nSC2UIR_model.initial_values = (x0_SC2UIR, t[0])\nSC3UEIR_model.initial_values = (x0_SC3UEIR, t[0])\nSC3UEI3R_model.initial_values = (x0_SC3UEI3R, t[0])\n```\n\n\n```python\nimport os\nos.getcwd()\n```\n\n\n```python\nsmodels1 = ['SIR','SEIR','SEI3R']\nsmodels2 = ['SC2IR','SC3EIR','SC3EI3R']\n# tmax = lastday-1\ntmax = 300\n# caution standard parameters\nCautionFactor= 0.2 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 21. # Duration of cautionary state of susceptibles (4 weeks)\nCautionICUFrac= 0.3 # Fraction of ICUs occupied leading to transition to caution @ 1/day \nICUFrac= 0.002 # Fraction of ICUs relative to population size N\n\n# Sensitivity scans\nCautionFactors= [1.0,0.9,0.8,0.7,0.6,0.5] # Fractional reduction of exposure rate for cautioned individuals\nCautionFactors2= [0.5,0.4,0.3,0.2,0.1,0.0] # Fractional reduction of exposure rate for cautioned individuals\nCautionRetentions= [7.,14.,28.,56.,112.,224.] # Duration of cautionary state of susceptibles\nCautionICUFracs= [1.0,0.75,0.5,0.25,0.125,0.0625] # Fraction of ICUs occupied leading to 90% of susceptibles in caution \n\nimport os\ncwd=os.getcwd()\n\nnewplot = True \nfor smodel in smodels1:\n model = cmodels[smodel]\nlabel_c = '' \nplottitle = 'Without Caution' \nsolns=solveplot(smodels1,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n\nirat = 1 + p[1]/(g[2]+p[2]) + p[2]/(g[3]+u)\ndrat = (p[1]/(g[2]+p[2]))*(p[2]/(g[3]+u))/irat\n\nnewplot = True\nfor s in range(6):\n for smodel in smodels2:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactors[s],'c_1':1./CautionRetention}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)}\n # print(smodel,cmodels[smodel].parameters) \n \n label_c = 'CautionFactor %s' % CautionFactors[s]\n plottitle = 'Caution Factor' \n solns=solveplot(smodels2,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False \nplt.savefig(cwd+\"/figures/fig3a.pdf\",bbox_inches='tight')\n\nnewplot = True\nfor s in range(6):\n for smodel in smodels2:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactors2[s],'c_1':1./CautionRetention}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)}\n # print(smodel,cmodels[smodel].parameters) \n \n label_c = 'CautionFactor %s' % CautionFactors2[s]\n plottitle = 'Caution Factor' \n solns=solveplot(smodels2,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False \nplt.savefig(cwd+\"/figures/fig3b.pdf\",bbox_inches='tight')\n\nnewplot = True\n#for i in reversed(range(5)):\nfor s in range(6):\n for smodel in smodels2:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetentions[s]}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)}\n # print(smodel,cmodels[smodel].parameters) \n label_c = 'CautionRetention %s'% CautionRetentions[s]\n plottitle = 'Caution Retention' \n solns=solveplot(smodels2,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False\nplt.savefig(cwd+\"/figures/fig3c.pdf\",bbox_inches='tight')\n\nnewplot = True \nfor s in range(6):\n for smodel in smodels2:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetentions[s]}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFracs[s])}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFracs[s])} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFracs[s])}\n # print(smodel,cmodels[smodel].parameters) \n label_c = 'CautionICUFrac %s'% CautionICUFracs[s]\n plottitle = 'Caution ICUFrac' \n solns=solveplot(smodels2,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False\n\nplt.savefig(cwd+\"/figures/fig3d.pdf\",bbox_inches='tight')\n\n# return parameters to standard set\nfor smodel in smodels2:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1/CautionRetention,'c_2':1./(N*ICUFrac*CautionICUFrac)}\n \n\n```\n\n\n```python\nsmodels = ['SCIR','SC2IR','SCEIR','SC3EIR','SCEI3R','SC3EI3R']\n# tmax = lastday-1\ntmax = 300\n# caution standard parameters\nCautionFactor= 0.2 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 21. # Duration of cautionary state of susceptibles (4 weeks)\nCautionICUFrac= 0.3 # Fraction of ICUs occupied leading to transition to caution @ 1/day \nICUFrac= 0.002 # Fraction of ICUs relative to population size N\n\n# Sensitivity scans\nCautionFactors= [1.0,0.9,0.8,0.7,0.6,0.5] # Fractional reduction of exposure rate for cautioned individuals\nCautionFactors2= [0.5,0.4,0.3,0.2,0.1,0.0] # Fractional reduction of exposure rate for cautioned individuals\nCautionRetentions= [7.,14.,28.,56.,112.,224.] # Duration of cautionary state of susceptibles\nCautionICUFracs= [1.0,0.75,0.5,0.25,0.125,0.0625] # Fraction of ICUs occupied leading to 90% of susceptibles in caution \n\nirat = 1 + p[1]/(g[2]+p[2]) + p[2]/(g[3]+u)\ndrat = (p[1]/(g[2]+p[2]))*(p[2]/(g[3]+u))/irat\n\nnewplot = True\nfor s in range(6):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactors[s],'c_1':1./CautionRetention}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)}\n # print(smodel,cmodels[smodel].parameters) \n \n label_c = 'CautionFactor %s' % CautionFactors[s]\n plottitle = 'Caution Factor' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False \n#plt.savefig(cwd+\"/figures/fig3sa.pdf\",bbox_inches='tight')\n\nnewplot = True\nfor s in range(6):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactors2[s],'c_1':1./CautionRetention}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)}\n # print(smodel,cmodels[smodel].parameters) \n \n label_c = 'CautionFactor %s' % CautionFactors2[s]\n plottitle = 'Caution Factor' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False \n#plt.savefig(cwd+\"/figures/fig3sb.pdf\",bbox_inches='tight')\n\nnewplot = True\nfor s in range(6):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetentions[s]}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)}\n # print(smodel,cmodels[smodel].parameters) \n label_c = 'CautionRetention %s'% CautionRetentions[s]\n plottitle = 'Caution Retention' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False\n#plt.savefig(cwd+\"/figures/fig3sc.pdf\",bbox_inches='tight')\n\nnewplot = True \nfor s in range(6):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetentions[s]}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFracs[s])}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFracs[s])} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFracs[s])}\n # print(smodel,cmodels[smodel].parameters) \n label_c = 'CautionICUFrac %s'% CautionICUFracs[s]\n plottitle = 'Caution ICUFrac' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False\n\n#plt.savefig(cwd+\"/figures/fig3sd.pdf\",bbox_inches='tight')\n\n# return parameters to standard set\nfor smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1/CautionRetention,'c_2':1./(N*ICUFrac*CautionICUFrac)}\n```\n\n\n```python\nsmodels = ['SCEIR','SC3EIR','SCEI3R','SC3EI3R']\n# tmax = lastday-1\ntmax = 300\n# caution standard parameters\nCautionFactor= 0.25 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 45. # Duration of cautionary state of susceptibles (4 weeks)\nCautionICUFrac= 0.05 # Fraction of ICUs occupied leading to transition to caution @ 1/day \nICUFrac= 0.002 # Fraction of ICUs relative to population size N\n\n# Sensitivity scans\nCautionFactors= [1.0,0.9,0.75,0.5,0.25,0.0] # Fractional reduction of exposure rate for cautioned individuals\nCautionFactors2= [0.4,0.3,0.2,0.1,0.0] # Fractional reduction of exposure rate for cautioned individuals\nCautionRetentions= [7.,14.,28.,56.,112.,224.] # Duration of cautionary state of susceptibles\nCautionICUFracs= [1.0,0.75,0.5,0.25,0.125,0.0625] # Fraction of ICUs occupied leading to 90% of susceptibles in caution \n\nnewplot = True\nfor s in range(6):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n if 'E' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactors[s],'c_1':1./CautionRetention,'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_0':CautionFactors[s],'c_1':1./CautionRetention,'c_2':FracSevere*FracCritical/(N*ICUFrac*CautionICUFrac)}\n # print(smodel,cmodels[smodel].parameters) \n else:\n print(\"ERROR\")\n label_c = 'CautionFactor %s' % CautionFactors[s]\n plottitle = 'Caution Factor' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False\n\nnewplot = True\nfor s in range(5):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n if 'E' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactors2[s],'c_1':1./CautionRetention,'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_0':CautionFactors2[s],'c_1':1./CautionRetention,'c_2':FracSevere*FracCritical/(N*ICUFrac*CautionICUFrac)}\n # print(smodel,cmodels[smodel].parameters) \n else:\n print(\"ERROR\")\n label_c = 'CautionFactor %s' % CautionFactors2[s]\n plottitle = 'Caution Factor' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False\n \nnewplot = True\n#for i in reversed(range(5)):\nfor s in range(5):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n if 'E' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetentions[s],'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetentions[s],'c_2':FracSevere*FracCritical/(N*ICUFrac*CautionICUFrac)}\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1/CautionRetentions[s],'c_2':1./(N*ICUFrac*CautionICUFrac)}\n # print(smodel,cmodels[smodel].parameters) \n label_c = 'CautionRetention %s'% CautionRetentions[s]\n plottitle = 'Caution Retention' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False\n\nnewplot = True \nfor s in range(5):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1/CautionRetention,'c_2':1./(N*ICUFrac*CautionICUFracs[s])}\n # print(smodel,cmodels[smodel].parameters) \n label_c = 'CautionICUFrac %s'% CautionICUFracs[s]\n plottitle = 'Caution ICUFrac' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False\n \n# return parameters to standard set\nfor smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1/CautionRetention,'c_2':1./(N*ICUFrac*CautionICUFrac)}\n```\n\n\n```python\nsmodels = ['SCIR','SC2IR','SCEIR','SC3EIR','SCEI3R','SC3EI3R']\n# tmax = lastday-1\ntmax = 300\n# caution standard parameters\nCautionFactor= 0.2 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 21. # Duration of cautionary state of susceptibles (4 weeks)\nCautionICUFrac= 0.3 # Fraction of ICUs occupied leading to transition to caution @ 1/day \nICUFrac= 0.002 # Fraction of ICUs relative to population size N\n\n# Sensitivity scans\nCautionFactors= [1.0,0.9,0.75,0.5,0.25,0.0] # Fractional reduction of exposure rate for cautioned individuals\nCautionRetentions= [7.,14.,28.,56.,112.,224.] # Duration of cautionary state of susceptibles\nCautionICUFracs= [1.0,0.75,0.5,0.25,0.125,0.0625] # Fraction of ICUs occupied leading to 90% of susceptibles in caution \n\n\n\nnewplot = True\nfor s in range(6):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n if 'E' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactors[s],'c_1':1./CautionRetention,'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_0':CautionFactors[s],'c_1':1./CautionRetention,'c_2':FracSevere*FracCritical/(N*ICUFrac*CautionICUFrac)}\n # print(smodel,cmodels[smodel].parameters) \n else:\n print(\"ERROR\")\n label_c = 'CautionFactor %s' % CautionFactors[s]\n plottitle = 'Caution Factor' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False\n\nnewplot = True\nfor s in range(6):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n if 'E' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetentions[s],'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetentions[s],'c_2':FracSevere*FracCritical/(N*ICUFrac*CautionICUFrac)}\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1/CautionRetentions[s],'c_2':1./(N*ICUFrac*CautionICUFrac)}\n # print(smodel,cmodels[smodel].parameters) \n label_c = 'CautionRetention %s'% CautionRetentions[s]\n plottitle = 'Caution Retention' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False\n\nnewplot = True \nfor s in range(6):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1/CautionRetention,'c_2':1./(N*ICUFrac*CautionICUFracs[s])}\n # print(smodel,cmodels[smodel].parameters) \n label_c = 'CautionICUFrac %s'% CautionICUFracs[s]\n plottitle = 'Caution ICUFrac' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False\n \n# return parameters to standard set\nfor smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1/CautionRetention,'c_2':1./(N*ICUFrac*CautionICUFrac)}\n```\n\n\n```python\n# smodels = ['SC2IR','SC2UIR','SC3EIR','SC3UEIR','SC3EI3R','SC3UEI3R']\nsmodels = ['SC2UIR','SC3UEIR','SC3UEI3R']\n# tmax = lastday-1\ntmax = 300\n# caution standard parameters\nCautionFactor= 0.4 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 21. # Duration of cautionary state of susceptibles (4 weeks)\nCautionICUFrac= 0.3 # Fraction of ICUs occupied leading to transition to caution @ 1/day \nICUFrac= 0.002 # Fraction of ICUs relative to population size N\n\n# Sensitivity scans\nCautionFactors= [1.0,0.9,0.8,0.7,0.6,0.5] # Fractional reduction of exposure rate for cautioned individuals\nCautionFactors2= [0.5,0.4,0.3,0.2,0.1,0.0] # Fractional reduction of exposure rate for cautioned individuals\nCautionRetentions= [7.,14.,28.,56.,112.,224.].reverse() # Duration of cautionary state of susceptibles\nCautionICUFracs= [1.0,0.75,0.5,0.25,0.125,0.0625] # Fraction of ICUs occupied leading to 90% of susceptibles in caution \nktime = 56.\nktimes= [1., 7.,14.,28.,56.,112.] # Duration of cautionary state of susceptibles\nkappas = [1.,0.8,0.6,0.4,0.2,0.] # Economic cost of caution\nkappa = 0.5\nirat = 1 + p[1]/(g[2]+p[2]) + p[2]/(g[3]+u)\ndrat = (p[1]/(g[2]+p[2]))*(p[2]/(g[3]+u))/irat \n\nnewplot = True\nfor s in range(-1,6):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetention}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)}\n else:\n print(\"ERROR\")\n if 'U' in smodel:\n if s == -1:\n cmodels[smodel].parameters = {'k_u':0.,'k_1':1.,'k_w':1.,'kappa':0.} \n else:\n cmodels[smodel].parameters = {'k_u':1./ktime,'k_1':1./ktime,'k_w':1./ktime,'kappa':kappas[s]} \n if s == -1:\n label_c = 'no economic influence'\n else:\n label_c = 'kappa %s' % kappas[s]\n plottitle = 'Cost of caution kappa' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False \nplt.savefig(cwd+\"/figures/fig5sa.pdf\",bbox_inches='tight')\n\nnewplot = True\nfor s in range(-1,6):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetention}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)}\n else:\n print(\"ERROR\")\n if 'U' in smodel:\n if s == -1:\n cmodels[smodel].parameters = {'k_u':0.,'k_1':1.,'k_w':1.,'kappa':0.} \n else:\n cmodels[smodel].parameters = {'k_u':1./ktimes[s],'k_1':1./ktime,'k_w':1./ktime, 'kappa':kappa} \n if s == -1:\n label_c = 'no economic influence'\n else:\n label_c = 'k_u time %s' % ktimes[s]\n plottitle = 'Uncautionable decay k_u' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False \nplt.savefig(cwd+\"/figures/fig5sb.pdf\",bbox_inches='tight')\n \nnewplot = True\nfor s in range(-1,6):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetention}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)}\n else:\n print(\"ERROR\")\n if 'U' in smodel:\n if s == -1:\n cmodels[smodel].parameters = {'k_u':0.,'k_1':1.,'k_w':1.,'kappa':0.} \n else:\n cmodels[smodel].parameters = {'k_u':1./ktime,'k_1':1./ktimes[s],'k_w':1./ktime, 'kappa':kappa} \n if s == -1:\n label_c = 'no economic influence'\n else:\n label_c = 'k_1 time %s' % ktimes[s]\n plottitle = 'Uncautionable decay k_1' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False \nplt.savefig(cwd+\"/figures/fig5sc.pdf\",bbox_inches='tight')\n \nnewplot = True\nfor s in range(-1,6):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetention}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)}\n else:\n print(\"ERROR\")\n if 'U' in smodel:\n if s == -1:\n cmodels[smodel].parameters = {'k_u':0.,'k_1':1.,'k_w':1.,'kappa':0.} \n else:\n cmodels[smodel].parameters = {'k_u':1./ktime,'k_1':1./ktime,'k_w':1./ktimes[s], 'kappa':kappa} \n if s == -1:\n label_c = 'no economic influence'\n else: \n label_c = 'k_w time %s' % ktimes[s]\n plottitle = 'Economic relaxation k_w' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False \nplt.savefig(cwd+\"/figures/fig5sd.pdf\",bbox_inches='tight')\n \n# return parameters to standard set\nfor smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetention}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)}\n if 'U' in model.modelname:\n cmodels[smodel].parameters = {'k_u':1./ktime,'k_1':1./ktime,'k_w':1./ktime, 'kappa':kappa}\n```\n\n\n```python\n# smodels = ['SC2IR','SC2UIR','SC3EIR','SC3UEIR','SC3EI3R','SC3UEI3R']\nsmodels = ['SC2UIR','SC3UEIR','SC3UEI3R']\n# tmax = lastday-1\ntmax = 600\n# caution standard parameters\nCautionFactor= 0.2 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 60. # Duration of cautionary state of susceptibles (4 weeks)\nCautionICUFrac= 0.3 # Fraction of ICUs occupied leading to transition to caution @ 1/day \nICUFrac= 0.002 # Fraction of ICUs relative to population size N\n\n# Sensitivity scans\nCautionFactors= [1.0,0.9,0.8,0.7,0.6,0.5] # Fractional reduction of exposure rate for cautioned individuals\nCautionFactors2= [0.5,0.4,0.3,0.2,0.1,0.0] # Fractional reduction of exposure rate for cautioned individuals\nCautionRetentions= [7.,14.,28.,56.,112.,224.].reverse() # Duration of cautionary state of susceptibles\nCautionICUFracs= [1.0,0.75,0.5,0.25,0.125,0.0625] # Fraction of ICUs occupied leading to 90% of susceptibles in caution \nktime = 56.\nktimes= [1., 7.,14.,28.,56.,112.] # Duration of cautionary state of susceptibles\nkappas = [1.,0.8,0.6,0.4,0.2,0.] # Economic cost of caution\nkappa = 0.5\nirat = 1 + p[1]/(g[2]+p[2]) + p[2]/(g[3]+u)\ndrat = (p[1]/(g[2]+p[2]))*(p[2]/(g[3]+u))/irat \n\nnewplot = True\nfor s in range(-1,6):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetention}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)}\n else:\n print(\"ERROR\")\n if 'U' in smodel:\n if s == -1:\n cmodels[smodel].parameters = {'k_u':0.,'k_1':1.,'k_w':1.,'kappa':0.} \n else:\n cmodels[smodel].parameters = {'k_u':1./ktime,'k_1':1./ktime,'k_w':1./ktime,'kappa':kappas[s]} \n if s == -1:\n label_c = 'no economic influence'\n else:\n label_c = 'kappa %s' % kappas[s]\n plottitle = 'Cost of caution kappa' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False \nplt.savefig(cwd+\"/figures/fig6sa.pdf\",bbox_inches='tight')\n\nnewplot = True\nfor s in range(-1,6):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetention}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)}\n else:\n print(\"ERROR\")\n if 'U' in smodel:\n if s == -1:\n cmodels[smodel].parameters = {'k_u':0.,'k_1':1.,'k_w':1.,'kappa':0.} \n else:\n cmodels[smodel].parameters = {'k_u':1./ktimes[s],'k_1':1./ktime,'k_w':1./ktime, 'kappa':kappa} \n if s == -1:\n label_c = 'no economic influence'\n else:\n label_c = 'k_u time %s' % ktimes[s]\n plottitle = 'Uncautionable decay k_u' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False \nplt.savefig(cwd+\"/figures/fig6sb.pdf\",bbox_inches='tight')\n \nnewplot = True\nfor s in range(-1,6):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetention}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)}\n else:\n print(\"ERROR\")\n if 'U' in smodel:\n if s == -1:\n cmodels[smodel].parameters = {'k_u':0.,'k_1':1.,'k_w':1.,'kappa':0.} \n else:\n cmodels[smodel].parameters = {'k_u':1./ktime,'k_1':1./ktimes[s],'k_w':1./ktime, 'kappa':kappa} \n if s == -1:\n label_c = 'no economic influence'\n else:\n label_c = 'k_1 time %s' % ktimes[s]\n plottitle = 'Uncautionable decay k_1' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False \nplt.savefig(cwd+\"/figures/fig6sc.pdf\",bbox_inches='tight')\n \nnewplot = True\nfor s in range(-1,6):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetention}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)}\n else:\n print(\"ERROR\")\n if 'U' in smodel:\n if s == -1:\n cmodels[smodel].parameters = {'k_u':0.,'k_1':1.,'k_w':1.,'kappa':0.} \n else:\n cmodels[smodel].parameters = {'k_u':1./ktime,'k_1':1./ktime,'k_w':1./ktimes[s], 'kappa':kappa} \n if s == -1:\n label_c = 'no economic influence'\n else: \n label_c = 'k_w time %s' % ktimes[s]\n plottitle = 'Economic relaxation k_w' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False \nplt.savefig(cwd+\"/figures/fig6sd.pdf\",bbox_inches='tight')\n \n# return parameters to standard set\nfor smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetention}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)}\n if 'U' in model.modelname:\n cmodels[smodel].parameters = {'k_u':1./ktime,'k_1':1./ktime,'k_w':1./ktime, 'kappa':kappa}\n```\n\n\n```python\nsmodels = ['SC3UEI3R']\n# tmax = lastday-1\ntmax = 300\n# caution standard parameters\nCautionFactor= 0.4 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 60. # Duration of cautionary state of susceptibles (4 weeks)\nCautionICUFrac= 0.3 # Fraction of ICUs occupied leading to transition to caution @ 1/day \nICUFrac= 0.002 # Fraction of ICUs relative to population size N\n\n# Sensitivity scans\nCautionFactors= [1.0,0.9,0.8,0.7,0.6,0.5] # Fractional reduction of exposure rate for cautioned individuals\nCautionFactors2= [0.5,0.4,0.3,0.2,0.1,0.0] # Fractional reduction of exposure rate for cautioned individuals\nCautionRetentions= [7.,14.,28.,56.,112.,224.].reverse() # Duration of cautionary state of susceptibles\nCautionICUFracs= [1.0,0.75,0.5,0.25,0.125,0.0625] # Fraction of ICUs occupied leading to 90% of susceptibles in caution \nktime = 56.\nktimes= [2.5, 5.,10.,20.,40.,80.] # Duration of cautionary state of susceptibles\nkappas = [1.,0.8,0.6,0.4,0.2,0.] # Economic cost of caution\nkappa = 0.5\nirat = 1 + p[1]/(g[2]+p[2]) + p[2]/(g[3]+u)\ndrat = (p[1]/(g[2]+p[2]))*(p[2]/(g[3]+u))/irat \n\nnewplot = True\nfor s in range(-1,6):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetention}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)}\n else:\n print(\"ERROR\")\n if 'U' in smodel:\n if s == -1:\n cmodels[smodel].parameters = {'k_u':0.,'k_1':1.,'k_w':1.,'kappa':0.} \n else:\n cmodels[smodel].parameters = {'k_u':1./ktimes[s],'k_1':1./90.,'k_w':1./90., 'kappa':kappa} \n if s == -1:\n label_c = 'no economic influence'\n else:\n label_c = 'k_u 1/%s' % ktimes[s]\n plottitle = 'Uncautionable decay k_u' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False \n#plt.savefig(cwd+\"/figures/fig5a.pdf\",bbox_inches='tight')\n\n \n# return parameters to standard set\nfor smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetention}\n if 'E' in model.modelname:\n if 'I3' in model.modelname:\n cmodels[smodel].parameters = {'c_2':1./(N*ICUFrac*CautionICUFrac)}\n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)} \n else:\n cmodels[smodel].parameters = {'c_2':drat/(N*ICUFrac*CautionICUFrac)}\n if 'U' in model.modelname:\n cmodels[smodel].parameters = {'k_u':1./ktime,'k_1':1./ktime,'k_w':1./ktime, 'kappa':kappa}\n```\n\n\n```python\n# more extensive parameter screen for one model\nsmodels = ['SCEI3R']\n# tmax = lastday-1\ntmax = 300\n# caution standard parameters\nCautionFactor= 0.2 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 21. # Duration of cautionary state of susceptibles (4 weeks)\nCautionICUFrac= 0.3 # Fraction of ICUs occupied leading to transition to caution @ 1/day \nICUFrac= 0.002 # Fraction of ICUs relative to population size N\n\n# Sensitivity scans\nCautionFactors= [1.0,0.5,0.1] # Fractional reduction of exposure rate for cautioned individuals\nCautionRetentions= [7.,28.,112.] # Duration of cautionary state of susceptibles (2 weeks)\nCautionICUFracs= [0.0625,0.25,0.75] # Fraction of ICUs occupied leading to 90% of susceptibles in caution \n\nfor s2 in range(3):\n newplot = True\n for s3 in range(3):\n newplot = True\n for s1 in range(3):\n for smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname: \n cmodels[smodel].parameters = {'c_0':CautionFactors[s1],'c_1':1./CautionRetentions[s2],'c_2':1./(N*ICUFrac*CautionICUFracs[s3])}\n else:\n print(\"ERROR\")\n label_c = ('CF %s' % CautionFactors[s1]) + (' CICUF %s'% CautionICUFracs[s3])\n plottitle = 'Caution Factor' \n solns=solveplot(smodels,'confirmed',tmax,'daily','daily',None,'linear',plottitle,label_c,newplot)\n plt.legend()\n newplot = False\n \n# return parameters to standard set\nfor smodel in smodels:\n model = cmodels[smodel]\n if 'C' in model.modelname:\n cmodels[smodel].parameters = {'c_0':CautionFactor,'c_1':1./CautionRetention,'c_2':1./(N*ICUFrac*CautionICUFrac)}\n\n```\n\n\n```python\n# the parameters are stored not as strings but as sympy symbols\n# but they cannot be accessed externally like that\n# from sympy import Symbol\n# cmodels['SCIR'].parameters[Symbol('c_0')] # prouces KeyError: c_0\ntypes= [type(k) for k in cmodels['SCIR'].parameters.keys()]\nprint(types)\n```\n\n# Parameter fitting\n\n## Fitting via sliders\n\n### SC3EIR Model\n\n\n```python\nlen(t)\n```\n\n\n```python\nmodel = 'SC3EIR'\nif cmodels[model].loadparams():\n params = cmodels[model].params.copy()\nelse:\n params={'beta':0.25,'alpha':1./5.,'gamma':0.1,'mu':0.05,'c_0':0.3, 'c_1':1/14., 'c_2':2000}\ndef slidefitplot(beta,alpha,gamma,mu,c_0,c_1,c_2,logI_0):\n params={ 'beta':beta, 'alpha':alpha, 'gamma':gamma, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2}\n cmodels[model].parameters = params\n cmodels[model].params = params\n cmodels[model].dumpparams()\n I0 = 10**logI_0\n x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.]\n cmodels[model].initial_values = (x0,t[0])\n solveplot(smodels=[model],species=['confirmed','recovered','deaths'],tmax=len(t),summing='daily',fitdata=y_jhu[test_country],scale='linear',plottitle= '',label='confirmed',newplot = True)\n```\n\n loaded params from /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3EIR.pk :\n\n\n\n```python\ninteract(slidefitplot,\n beta=FloatSlider(min=0,max=1,step=0.01,value=params['beta'],description='beta',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n alpha=FloatSlider(min=0,max=1,step=0.01,value=params['alpha'],description='alpha',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n gamma=FloatSlider(min=0,max=1,step=0.01,value=params['gamma'],description='gamma',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), \n c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_2=FloatSlider(min=0,max=5000,step=1,value=params['c_2'],description='c_2',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'),\n logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0',\n style=style,layout=slider_layout,continuous_update=False))\n```\n\n\n interactive(children=(FloatSlider(value=0.67, continuous_update=False, description='beta', layout=Layout(width\u2026\n\n\n\n\n\n \n\n\n\n### SC3EI3R Model\n\n#### Germany\n\n\n```python\n# assumed data starting on firstdate\ntest_country='Germany'\nN = 80000000\nfirstdate = '01/25/20'\nlastdate = '01/08/20'\nxx,xxf,yy0 = get_country_data(test_country,'confirmed',firstdate=firstdate,lastdate=lastdate)\nxx,xxf,yy1 = get_country_data(test_country,'recovered',firstdate=firstdate,lastdate=lastdate)\nxx,xxf,yy2 = get_country_data(test_country,'deaths',firstdate=firstdate,lastdate=lastdate)\nprint(xxf)\ny_jhu={}\ny_jhu[test_country] = np.array([[yy0[i],yy1[i],yy2[i]] for i in range(0,len(yy0))])/N\n# data = np.array([[xxf[i],yy0[i],yy1[i],yy2[i]] for i in range(len(yy))])\n# print(data)\nlastday = len(y_jhu[test_country])\nprint('days 0 to',lastday,'data stored in y_jhu')\n```\n\n\n```python\nlen(t)\n```\n\n\n```python\n(1.0/TimeICUDeath)*(CFR/FracCritical)\n```\n\n\n```python\nmodel = 'SC3EI3R'\n\n# Define parameters based on clinical observations Dr. Alison\nExposure=0.4 # Rate coefficient for exposure per individual in contact per day\nIncubPeriod=5 #Incubation period, days\nDurMildInf=10 #Duration of mild infections, days : includes time for reg. of recovery\nFracMild=0.7 #Fraction of infections that are mild\nFracSevere=0.20 #Fraction of infections that are severe\nFracCritical=0.1 #Fraction of infections that are critical\nCFR=0.05 #Case fatality rate (fraction of infections resulting in death)\nTimeICUDeath=5 #Time from ICU admission to death, days\nDurHosp=4 #Duration of hospitalization, days : includes 4 day reg of recovery\n\n# Model fitting extension to allow for incomplete detection\nFracConfirmedDet=1.0 # Fraction of recovered individuals measured : plots made with this parameter NYI\nFracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured\nFracDeathsDet=1.0\n\n# Model extension by John McCaskill to include caution \nCautionFactor= 0.1 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 60. # Duration of cautionary state of susceptibles (2 weeks)\nCautionICUFrac= 0.1 # Fraction of ICUs occupied leading to transition to caution @ 1/day\nICUFrac= 0.001 # Fraction of ICUs relative to population size N\n\nif cmodels[model].loadparams():\n params = cmodels[model].params.copy()\nelse:\n params = {'beta_1' : Exposure/sum(x0_SC3EI3R),\n 'beta_2' : 0.0,\n 'beta_3' : 0.0,\n 'alpha' : 1.0/IncubPeriod,\n 'gamma_1': (1.0/DurMildInf)*FracMild,\n 'gamma_2': (1.0/DurHosp)-(1/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n 'gamma_3': (1.0/TimeICUDeath)-(1/TimeICUDeath)*(CFR/FracCritical),\n 'p_1' : (1.0/DurMildInf)-(1.0/DurMildInf)*FracMild,\n 'p_2' : (1.0/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n 'mu' : (1.0/TimeICUDeath)*(CFR/FracCritical),\n 'c_0' : CautionFactor,\n 'c_1' : 1.0/CautionRetention,\n 'c_2' : 1.0/(sum(x0_SC3EI3R)*ICUFrac*CautionICUFrac),\n 'N' : sum(x0_SC3EI3R)}\n\nprint(params)\nSC3EI3R_model.parameters = params\n\ndef slidefitplot(beta_1,mu,c_0,c_1,c_2,logI_0):\n params={ 'beta_1':beta_1, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2}\n cmodels[model].parameters = params\n cmodels[model].params = params\n cmodels[model].dumpparams()\n \n I0 = 10**logI_0\n x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.,0.]\n x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.]\n\n\n cmodels[model].initial_values = (x0,t[0])\n weights=np.array([1.,1.,1.])\n solveplot(smodels=[model],species=['confirmed','recovered','deaths_x10'],tmax=len(t),summing='daily',averaging='weekly',fitdata=y_jhu[test_country]*weights,scale='linear',plottitle= '',label='confirmed',newplot = True, figsize = (15,15))\n```\n\n loaded params from /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3EI3R.pk :\n {'beta_1': 0.4, 'mu': 0.1, 'c_0': 0.1, 'c_1': 0.016666666666666666, 'c_2': 10000.0}\n\n\n\n```python\nw =interactive(slidefitplot,\n beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), \n c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_2=FloatSlider(min=0,max=20000,step=1,value=params['c_2'],description='c_2',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'),\n logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0',\n style=style,layout=slider_layout,continuous_update=False)\n )\ndisplay(w)\n```\n\n\n interactive(children=(FloatSlider(value=0.4, continuous_update=False, description='beta_1', layout=Layout(widt\u2026\n\n\n\n```python\nparams=w.kwargs\n\nprint(params)\n```\n\n\n```python\n\n```\n\n#### Spain\n\n\n```python\n# assumed data starting on firstdate\ntest_country='Spain'\nN = 80000000\nfirstdate = '01/25/20'\nlastdate = '01/08/20'\nxx,xxf,yy0 = get_country_data(test_country,'confirmed',firstdate=firstdate,lastdate=lastdate)\nxx,xxf,yy1 = get_country_data(test_country,'recovered',firstdate=firstdate,lastdate=lastdate)\nxx,xxf,yy2 = get_country_data(test_country,'deaths',firstdate=firstdate,lastdate=lastdate)\nprint(xxf)\ny_jhu={}\ny_jhu[test_country] = np.array([[yy0[i],yy1[i],yy2[i]] for i in range(0,len(yy0))])/N\n# data = np.array([[xxf[i],yy0[i],yy1[i],yy2[i]] for i in range(len(yy))])\n# print(data)\nlastday = len(y_jhu[test_country])\nprint('days 0 to',lastday,'data stored in y_jhu')\n```\n\n [-3.0, -2.0, -1.0, 0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, 32.0, 33.0, 34.0, 35.0, 36.0, 37.0, 38.0, 39.0, 40.0, 41.0, 42.0, 43.0, 44.0, 45.0, 46.0, 47.0, 48.0, 49.0, 50.0, 51.0, 52.0, 53.0, 54.0, 55.0, 56.0, 57.0, 58.0, 59.0, 60.0, 61.0, 62.0, 63.0, 64.0, 65.0, 66.0, 67.0, 68.0, 69.0, 70.0, 71.0, 72.0, 73.0, 74.0, 75.0, 76.0, 77.0, 78.0, 79.0, 80.0, 81.0, 82.0, 83.0, 84.0, 85.0, 86.0, 87.0, 88.0, 89.0, 90.0, 91.0, 92.0, 93.0, 94.0, 95.0, 96.0, 97.0, 98.0, 99.0, 100.0, 101.0, 102.0, 103.0, 104.0, 105.0, 106.0, 107.0, 108.0, 109.0, 110.0, 111.0, 112.0, 113.0, 114.0, 115.0, 116.0, 117.0, 118.0, 119.0, 120.0, 121.0, 122.0, 123.0, 124.0, 125.0, 126.0, 127.0, 128.0, 129.0, 130.0, 131.0, 132.0, 133.0, 134.0, 135.0, 136.0, 137.0, 138.0, 139.0, 140.0, 141.0, 142.0, 143.0, 144.0, 145.0, 146.0, 147.0, 148.0, 149.0, 150.0, 151.0, 152.0, 153.0, 154.0, 155.0, 156.0, 157.0, 158.0, 159.0, 160.0, 161.0, 162.0, 163.0, 164.0, 165.0, 166.0, 167.0, 168.0, 169.0, 170.0, 171.0, 172.0, 173.0, 174.0, 175.0, 176.0, 177.0, 178.0, 179.0, 180.0, 181.0, 182.0, 183.0, 184.0, 185.0, 186.0, 187.0, 188.0, 189.0, 190.0, 191.0, 192.0, 193.0, 194.0, 195.0, 196.0, 197.0, 198.0, 199.0, 200.0, 201.0, 202.0, 203.0, 204.0, 205.0, 206.0, 207.0, 208.0, 209.0, 210.0, 211.0, 212.0, 213.0, 214.0, 215.0, 216.0, 217.0, 218.0]\n days 0 to 222 data stored in y_jhu\n\n\n\n```python\nlen(t)\n```\n\n\n\n\n 222\n\n\n\n\n```python\n(1.0/TimeICUDeath)*(CFR/FracCritical)\n```\n\n\n\n\n 0.1\n\n\n\n\n```python\nmodel = 'SC3EI3R'\n\n# Define parameters based on clinical observations Dr. Alison\nExposure=0.4 # Rate coefficient for exposure per individual in contact per day\nIncubPeriod=5 #Incubation period, days\nDurMildInf=10 #Duration of mild infections, days : includes time for reg. of recovery\nFracMild=0.7 #Fraction of infections that are mild\nFracSevere=0.20 #Fraction of infections that are severe\nFracCritical=0.1 #Fraction of infections that are critical\nCFR=0.05 #Case fatality rate (fraction of infections resulting in death)\nTimeICUDeath=5 #Time from ICU admission to death, days\nDurHosp=4 #Duration of hospitalization, days : includes 4 day reg of recovery\n\n# Model fitting extension to allow for incomplete detection\nFracConfirmedDet=0.5 # Fraction of confirmed individuals measured : plots made with this parameter NYI\nFracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured\nFracDeathsDet=1.0\n\n# Model extension by John McCaskill to include caution \nCautionFactor= 0.1 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 60. # Duration of cautionary state of susceptibles (2 weeks)\nCautionICUFrac= 0.1 # Fraction of ICUs occupied leading to transition to caution @ 1/day\nICUFrac= 0.001 # Fraction of ICUs relative to population size N\n\nif cmodels[model].loadparams():\n params = cmodels[model].params.copy()\nelse:\n params = {'beta_1' : Exposure/sum(x0_SC3EI3R),\n 'beta_2' : 0.0,\n 'beta_3' : 0.0,\n 'alpha' : 1.0/IncubPeriod,\n 'gamma_1': (1.0/DurMildInf)*FracMild,\n 'gamma_2': (1.0/DurHosp)-(1/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n 'gamma_3': (1.0/TimeICUDeath)-(1/TimeICUDeath)*(CFR/FracCritical),\n 'p_1' : (1.0/DurMildInf)-(1.0/DurMildInf)*FracMild,\n 'p_2' : (1.0/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n 'mu' : (1.0/TimeICUDeath)*(CFR/FracCritical),\n 'c_0' : CautionFactor,\n 'c_1' : 1.0/CautionRetention,\n 'c_2' : 1.0/(sum(x0_SC3EI3R)*ICUFrac*CautionICUFrac),\n 'N' : sum(x0_SC3EI3R)}\n\nprint(params)\ncmodels[model].parameters = params\ncmodels[model].params = params\ncmodels[model].dumpparams()\n \ndef slidefitplot(beta_1,mu,c_0,c_1,c_2,logI_0):\n params={ 'beta_1':beta_1, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2}\n cmodels[model].parameters = params\n cmodels[model].params = params\n cmodels[model].dumpparams()\n \n I0 = 10**logI_0\n x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.,0.]\n x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.]\n\n\n cmodels[model].initial_values = (x0,t[0])\n weights=np.array([1.,1.,1.])\n solveplot(smodels=[model],species=['confirmed','recovered','deaths_x10'],tmax=len(t),summing='daily',averaging='weekly',fitdata=y_jhu[test_country]*weights,scale='linear',plottitle= '',label='confirmed',newplot = True, figsize = (15,15))\n```\n\n loaded params from /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3EI3R.pk :\n {'beta_1': 0.48, 'mu': 0.1, 'c_0': 0.1, 'c_1': 0.016, 'c_2': 3819.0}\n dumped params to /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3EI3R.pk\n\n\n\n```python\nw =interactive(slidefitplot,\n beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), \n c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_2=FloatSlider(min=0,max=20000,step=1,value=params['c_2'],description='c_2',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'),\n logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0',\n style=style,layout=slider_layout,continuous_update=False)\n )\ndisplay(w)\n```\n\n\n interactive(children=(FloatSlider(value=0.48, continuous_update=False, description='beta_1', layout=Layout(wid\u2026\n\n\n\n```python\nparams=w.kwargs\n\nprint(params)\n```\n\n#### Italy\n\n\n```python\n# assumed data starting on firstdate\ntest_country='Italy'\nN = 66650000\nfirstdate = '01/25/20'\nlastdate = '01/08/20'\nxx,xxf,yy0 = get_country_data(test_country,'confirmed',firstdate=firstdate,lastdate=lastdate)\nxx,xxf,yy1 = get_country_data(test_country,'recovered',firstdate=firstdate,lastdate=lastdate)\nxx,xxf,yy2 = get_country_data(test_country,'deaths',firstdate=firstdate,lastdate=lastdate)\nprint(xxf)\ny_jhu={}\ny_jhu[test_country] = np.array([[yy0[i],yy1[i],yy2[i]] for i in range(0,len(yy0))])/N\n# data = np.array([[xxf[i],yy0[i],yy1[i],yy2[i]] for i in range(len(yy))])\n# print(data)\nlastday = len(y_jhu[test_country])\nprint('days 0 to',lastday,'data stored in y_jhu')\n```\n\n\n```python\nmodel = 'SC3EI3R'\n\n# Define parameters based on clinical observations Dr. Alison\nExposure=0.4 # Rate coefficient for exposure per individual in contact per day\nIncubPeriod=5 #Incubation period, days\nDurMildInf=8 #Duration of mild infections, days\nFracMild=0.65 #Fraction of infections that are mild\nFracSevere=0.20 #Fraction of infections that are severe\nFracCritical=0.15 #Fraction of infections that are critical\nCFR=0.1 #Case fatality rate (fraction of infections resulting in death)\nTimeICUDeath=4 #Time from ICU admission to death, days\nDurHosp=4 #Duration of hospitalization, days\n\n# Model fitting extension to allow for incomplete detection\nFracConfirmedDet=0.5 # Fraction of infected individuals confirmed\nFracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured\nFracDeathsDet=1.0\n\n# Model extension by John McCaskill to include caution \nCautionFactor= 0.1 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 60. # Duration of cautionary state of susceptibles (2 weeks)\nCautionICUFrac= 0.1 # Fraction of ICUs occupied leading to transition to caution @ 1/day\nICUFrac= 0.001 # Fraction of ICUs relative to population size N\n\nif cmodels[model].loadparams():\n params = cmodels[model].params.copy()\nelse:\n params = {'beta_1' : Exposure/sum(x0_SC3EI3R),\n 'beta_2' : 0.0,\n 'beta_3' : 0.0,\n 'alpha' : 1.0/IncubPeriod,\n 'gamma_1': (1.0/DurMildInf)*FracMild,\n 'gamma_2': (1.0/DurHosp)-(1/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n 'gamma_3': (1.0/TimeICUDeath)-(1/TimeICUDeath)*(CFR/FracCritical),\n 'p_1' : (1.0/DurMildInf)-(1.0/DurMildInf)*FracMild,\n 'p_2' : (1.0/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n 'mu' : (1.0/TimeICUDeath)*(CFR/FracCritical),\n 'c_0' : CautionFactor,\n 'c_1' : 1.0/CautionRetention,\n 'c_2' : 1.0/(sum(x0_SC3EI3R)*ICUFrac*CautionICUFrac),\n 'N' : sum(x0_SC3EI3R)}\n\nprint(params)\ncmodels[model].parameters = params\ncmodels[model].params = params\ncmodels[model].dumpparams()\n\ndef slidefitplot(beta_1,mu,c_0,c_1,c_2,logI_0):\n params={ 'beta_1':beta_1, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2}\n cmodels[model].parameters = params\n cmodels[model].params = params\n cmodels[model].dumpparams()\n \n I0 = 10**logI_0\n x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.,0.]\n x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.]\n\n\n cmodels[model].initial_values = (x0,t[0])\n weights=np.array([1.,1.,1.])\n solveplot(smodels=[model],species=['confirmed','recovered','deaths_x10'],tmax=len(t),summing='daily',averaging='weekly',fitdata=y_jhu[test_country]*weights,scale='linear',plottitle= '',label='confirmed',newplot = True, figsize = (15,15))\n```\n\n loaded params from /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3EI3R.pk :\n {'beta_1': 0.48, 'mu': 0.1, 'c_0': 0.1, 'c_1': 0.016, 'c_2': 3819.0}\n dumped params to /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3EI3R.pk\n\n\n\n```python\nw =interactive(slidefitplot,\n beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), \n c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_2=FloatSlider(min=0,max=20000,step=1,value=params['c_2'],description='c_2',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'),\n logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0',\n style=style,layout=slider_layout,continuous_update=False)\n )\ndisplay(w)\n```\n\n\n interactive(children=(FloatSlider(value=0.48, continuous_update=False, description='beta_1', layout=Layout(wid\u2026\n\n\n\n```python\nparams=w.kwargs\n\nprint(params)\n```\n\nNote that we have used 50% detection of confirmed and recovered, 100% for deaths in manual fit. \nIt appears that Italy's registration of recovery, although the right overall magnitude is markedly delayed - check reporting delays.\nItaly also had at least two successive regional infections, as seen in the dual peak confirmed data, so not easy to fit with one model.\nSee below for simulation of second peak.\n\n\n```python\nw =interactive(slidefitplot,\n beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), \n c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_2=FloatSlider(min=0,max=20000,step=1,value=params['c_2'],description='c_2',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'),\n logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0',\n style=style,layout=slider_layout,continuous_update=False)\n )\ndisplay(w)\n```\n\n#### Brazil\n\n\n```python\n# assumed data starting on firstdate\ntest_country='Brazil'\nN = 210000000\nfirstdate = '01/25/20'\nlastdate = '01/08/20'\nxx,xxf,yy0 = get_country_data(test_country,'confirmed',firstdate=firstdate,lastdate=lastdate)\nxx,xxf,yy1 = get_country_data(test_country,'recovered',firstdate=firstdate,lastdate=lastdate)\nxx,xxf,yy2 = get_country_data(test_country,'deaths',firstdate=firstdate,lastdate=lastdate)\nprint(xxf)\ny_jhu={}\ny_jhu[test_country] = np.array([[yy0[i],yy1[i],yy2[i]] for i in range(0,len(yy0))])/N\n# data = np.array([[xxf[i],yy0[i],yy1[i],yy2[i]] for i in range(len(yy))])\n# print(data)\nlastday = len(y_jhu[test_country])\nprint('days 0 to',lastday,'data stored in y_jhu')\n```\n\n\n```python\nmodel = 'SC3EI3R'\n\n# Define parameters based on clinical observations Dr. Alison\nExposure=0.4 # Rate coefficient for exposure per individual in contact per day\nIncubPeriod=5 #Incubation period, days\nDurMildInf=8 #Duration of mild infections, days\nFracMild=0.65 #Fraction of infections that are mild\nFracSevere=0.20 #Fraction of infections that are severe\nFracCritical=0.15 #Fraction of infections that are critical\nCFR=0.1 #Case fatality rate (fraction of infections resulting in death)\nTimeICUDeath=4 #Time from ICU admission to death, days\nDurHosp=8 #Duration of hospitalization, days\n\n# Model fitting extension to allow for incomplete detection\nFracConfirmedDet=0. # Fraction of recovered individuals measured\nFracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured\nFracDeathsDet=1.0\n\n# Model extension by John McCaskill to include caution \nCautionFactor= 0.1 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 60. # Duration of cautionary state of susceptibles (2 weeks)\nCautionICUFrac= 0.1 # Fraction of ICUs occupied leading to transition to caution @ 1/day\nICUFrac= 0.001 # Fraction of ICUs relative to population size N\n\n\nif cmodels[model].loadparams():\n params = cmodels[model].params.copy()\nelse:\n params = {'beta_1' : Exposure/sum(x0_SC3EI3R),\n 'beta_2' : 0.0,\n 'beta_3' : 0.0,\n 'alpha' : 1.0/IncubPeriod,\n 'gamma_1': (1.0/DurMildInf)*FracMild,\n 'gamma_2': (1.0/DurHosp)-(1/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n 'gamma_3': (1.0/TimeICUDeath)-(1/TimeICUDeath)*(CFR/FracCritical),\n 'p_1' : (1.0/DurMildInf)-(1.0/DurMildInf)*FracMild,\n 'p_2' : (1.0/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n 'mu' : (1.0/TimeICUDeath)*(CFR/FracCritical),\n 'c_0' : CautionFactor,\n 'c_1' : 1.0/CautionRetention,\n 'c_2' : 1.0/(sum(x0_SC3EI3R)*ICUFrac*CautionICUFrac),\n 'N' : sum(x0_SC3EI3R)}\n\nprint(params)\ncmodels[model].parameters = params\ncmodels[model].params = params\ncmodels[model].dumpparams()\n\ndef slidefitplot(beta_1,mu,c_0,c_1,c_2,logI_0):\n params={ 'beta_1':beta_1, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2}\n cmodels[model].parameters = params\n cmodels[model].params = params\n cmodels[model].dumpparams() \n I0 = 10**logI_0\n x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.,0.]\n x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.]\n\n cmodels[model].initial_values = (x0,t[0])\n weights=np.array([1.,1.,1.])\n solveplot(smodels=[model],species=['confirmed','recovered','deaths_x10'],tmax=len(t),summing='cumulative',averaging='weekly',fitdata=y_jhu[test_country]*weights,scale='linear',plottitle= '',label='confirmed',newplot = True, figsize = (15,15))\n```\n\n loaded params from /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3EI3R.pk :\n {'beta_1': 0.48, 'mu': 0.1, 'c_0': 0.1, 'c_1': 0.016, 'c_2': 3819.0}\n dumped params to /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3EI3R.pk\n\n\n\n```python\nw =interactive(slidefitplot,\n beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), \n c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_2=FloatSlider(min=0,max=20000,step=1,value=params['c_2'],description='c_2',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'),\n logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0',\n style=style,layout=slider_layout,continuous_update=False)\n )\ndisplay(w)\n```\n\n\n interactive(children=(FloatSlider(value=0.48, continuous_update=False, description='beta_1', layout=Layout(wid\u2026\n\n\nThe Brazil data shows that death is not as delayed as assumed. The process of progression is perhaps less clearly documented.\n\n#### Russia\n\n\n```python\n# assumed data starting on firstdate\ntest_country='Iran'\nN = 144500000\nfirstdate = '01/25/20'\nlastdate = '01/08/20'\nxx,xxf,yy0 = get_country_data(test_country,'confirmed',firstdate=firstdate,lastdate=lastdate)\nxx,xxf,yy1 = get_country_data(test_country,'recovered',firstdate=firstdate,lastdate=lastdate)\nxx,xxf,yy2 = get_country_data(test_country,'deaths',firstdate=firstdate,lastdate=lastdate)\nprint(xxf)\ny_jhu={}\ny_jhu[test_country] = np.array([[yy0[i],yy1[i],yy2[i]] for i in range(0,len(yy0))])/N\n# data = np.array([[xxf[i],yy0[i],yy1[i],yy2[i]] for i in range(len(yy))])\n# print(data)\nlastday = len(y_jhu[test_country])\nprint('days 0 to',lastday,'data stored in y_jhu')\n```\n\n\n```python\nmodel = 'SC3EI3R'\n\n# Define parameters based on clinical observations Dr. Alison\nExposure=0.4 # Rate coefficient for exposure per individual in contact per day\nIncubPeriod=5 #Incubation period, days\nDurMildInf=8 #Duration of mild infections, days\nFracMild=0.65 #Fraction of infections that are mild\nFracSevere=0.20 #Fraction of infections that are severe\nFracCritical=0.15 #Fraction of infections that are critical\nCFR=0.1 #Case fatality rate (fraction of infections resulting in death)\nTimeICUDeath=4 #Time from ICU admission to death, days\nDurHosp=8 #Duration of hospitalization, days\n\n# Model fitting extension to allow for incomplete detection\nFracConfirmedDet=0. # Fraction of recovered individuals measured\nFracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured\nFracDeathsDet=1.0\n\n# Model extension by John McCaskill to include caution \nCautionFactor= 0.1 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 60. # Duration of cautionary state of susceptibles (2 weeks)\nCautionICUFrac= 0.1 # Fraction of ICUs occupied leading to transition to caution @ 1/day\nICUFrac= 0.002 # Fraction of ICUs relative to population size N\n\nif cmodels[model].loadparams():\n params = cmodels[model].params.copy()\nelse:\n params = {'beta_1' : Exposure/sum(x0_SC3EI3R),\n 'beta_2' : 0.0,\n 'beta_3' : 0.0,\n 'alpha' : 1.0/IncubPeriod,\n 'gamma_1': (1.0/DurMildInf)*FracMild,\n 'gamma_2': (1.0/DurHosp)-(1/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n 'gamma_3': (1.0/TimeICUDeath)-(1/TimeICUDeath)*(CFR/FracCritical),\n 'p_1' : (1.0/DurMildInf)-(1.0/DurMildInf)*FracMild,\n 'p_2' : (1.0/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n 'mu' : (1.0/TimeICUDeath)*(CFR/FracCritical),\n 'c_0' : CautionFactor,\n 'c_1' : 1.0/CautionRetention,\n 'c_2' : 1.0/(sum(x0_SC3EI3R)*ICUFrac*CautionICUFrac),\n 'N' : sum(x0_SC3EI3R)}\n\nprint(params)\ncmodels[model].parameters = params\ncmodels[model].params = params\ncmodels[model].dumpparams()\n\ndef slidefitplot(beta_1,mu,c_0,c_1,c_2,logI_0):\n params={ 'beta_1':beta_1, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2}\n cmodels[model].parameters = params\n cmodels[model].params = params\n cmodels[model].dumpparams()\n\n I0 = 10**logI_0\n x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.]\n\n cmodels[model].initial_values = (x0,t[0])\n weights=np.array([1.,1.,1.])\n solveplot(smodels=[model],species=['confirmed','recovered','deaths_x10'],tmax=len(t),summing='cumulative',averaging='weekly',fitdata=y_jhu[test_country]*weights,scale='linear',plottitle= '',label='confirmed',newplot = True, figsize = (15,15))\n```\n\n loaded params from /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3EI3R.pk :\n {'beta_1': 0.48, 'mu': 0.1, 'c_0': 0.1, 'c_1': 0.016, 'c_2': 3819.0}\n dumped params to /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3EI3R.pk\n\n\n\n```python\nw =interactive(slidefitplot,\n beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), \n c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_2=FloatSlider(min=0,max=20000,step=1,value=params['c_2'],description='c_2',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'),\n logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0',\n style=style,layout=slider_layout,continuous_update=False)\n )\ndisplay(w)\n```\n\n\n interactive(children=(FloatSlider(value=0.48, continuous_update=False, description='beta_1', layout=Layout(wid\u2026\n\n\n### SC3UEIR Model\n\n\n```python\n# assumed data starting on firstdate\ntest_country='US'\nN = 66650000\nfirstdate = '01/25/20'\nlastdate = '01/08/20'\nxx,xxf,yy0 = get_country_data(test_country,'confirmed',firstdate=firstdate,lastdate=lastdate)\nxx,xxf,yy1 = get_country_data(test_country,'recovered',firstdate=firstdate,lastdate=lastdate)\nxx,xxf,yy2 = get_country_data(test_country,'deaths',firstdate=firstdate,lastdate=lastdate)\nprint(xxf)\ny_jhu={}\ny_jhu[test_country] = np.array([[yy0[i],yy1[i],yy2[i]] for i in range(0,len(yy0))])/N\n# data = np.array([[xxf[i],yy0[i],yy1[i],yy2[i]] for i in range(len(yy))])\n# print(data)\nlastday = len(y_jhu[test_country])\nprint('days 0 to',lastday,'data stored in y_jhu')\n```\n\n\n```python\nmodel = 'SC3UEIR'\nI_0 = 0.00003\nx0_SC3UEIR = [1.0-I_0, 0.0, I_0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0]\nSC3UEIR_model.initial_values = (x0_SC3UEIR, t[0])\n\n# Define parameters based on clinical observations Dr. Alison\nExposure=0.4 # Rate coefficient for exposure per individual in contact per day\nIncubPeriod=5 #Incubation period, days\nDurMildInf=8 #Duration of mild infections, days\nFracMild=0.65 #Fraction of infections that are mild\nFracSevere=0.20 #Fraction of infections that are severe\nFracCritical=0.15 #Fraction of infections that are critical\nCFR=0.1 #Case fatality rate (fraction of infections resulting in death)\nTimeICUDeath=4 #Time from ICU admission to death, days\nDurHosp=8 #Duration of hospitalization, days\n\n# Model fitting extension to allow for incomplete detection\nFracConfirmedDet=0.5 # Fraction of recovered individuals measured\nFracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured\nFracDeathsDet=1.0\n\n# Model extension by John McCaskill to include caution \nCautionFactor= 0.1 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 60. # Duration of cautionary state of susceptibles (2 weeks)\nCautionICUFrac= 0.1 # Fraction of ICUs occupied leading to transition to caution @ 1/day\nICUFrac= 0.001 # Fraction of ICUs relative to population size N\n\n# Model extension by John McCaskill to include economic influence on caution \nEconomicCostOfCaution= 0.5 # Fractional reduction of economic contribution for cautioned individuals\n\np = [0,(1.0/DurMildInf)-(1.0/DurMildInf)*FracMild, (1.0/DurHosp)*(FracCritical/(FracCritical+FracSevere))]\ng = [0,(1.0/DurMildInf)*FracMild, (1.0/DurHosp)-(1/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n (1.0/TimeICUDeath)-(1/TimeICUDeath)*(CFR/FracCritical)]\nu = (1.0/TimeICUDeath)*(CFR/FracCritical)\n \nif cmodels[model].loadparams():\n params = cmodels[model].params.copy()\nelse:\n params = {'beta' : Exposure/sum(x0_SC3UEIR),\n 'alpha' : 1.0/IncubPeriod,\n 'gamma' : g[1]+g[2]*(p[1]/(g[2]+p[2]))+g[3]*(p[1]/(g[2]+p[2]))*(p[2]/(g[3]+u)),\n 'mu' : u*(p[1]/(g[2]+p[2])*(p[2]/(g[3]+u))),\n 'c_0' : CautionFactor,\n 'c_1' : 1.0/CautionRetention,\n 'c_2' : 1.0/(sum(x0_SC3UEIR)*ICUFrac*CautionICUFrac),\n 'N' : sum(x0_SC3UEIR),\n 'k_u' : 1.0/CautionRetention,\n 'k_1' : 1.0/CautionRetention,\n 'k_w' : 1.0/CautionRetention,\n 'kappa' : EconomicCostOfCaution}\n\nprint(params)\ncmodels[model].parameters = params\ncmodels[model].params = params\ncmodels[model].dumpparams()\n# solution = SCIR_model.integrate(t[1::])\n\ndef slidefitplot(beta,alpha,gamma,mu,c_0,c_1,c_2,logI_0,k_u,k_1,k_w,kappa):\n params={ 'beta':beta, 'alpha':alpha, 'gamma':gamma, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2, 'k_u':k_u, 'k_1':k_1, 'k_w':k_w, 'kappa':kappa}\n cmodels[model].parameters = params\n cmodels[model].params = params\n cmodels[model].dumpparams() \n I0 = 10**logI_0\n x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,1.]\n\n cmodels[model].initial_values = (x0,t[0])\n weights=np.array([1.,1.,1.])\n solveplot(smodels=[model],species=['confirmed','recovered','deaths_x10'],tmax=len(t),summing='daily',averaging='weekly',fitdata=y_jhu[test_country]*weights,scale='linear',plottitle= '',label='confirmed',newplot = True, figsize = (15,15))\n```\n\n loaded params from /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3UEIR.pk :\n {'beta': 0.48, 'alpha': 0.2, 'gamma': 0.11249999999999999, 'mu': 0.0125, 'c_0': 0.1, 'c_1': 0.016666666666666666, 'c_2': 110.0, 'k_u': 0.016666666666666666, 'k_1': 0.016666666666666666, 'k_w': 0.016666666666666666, 'kappa': 0.5}\n dumped params to /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3UEIR.pk\n\n\n\n```python\nw = interactive(slidefitplot,\n beta=FloatSlider(min=0,max=1,step=0.01,value=params['beta'],description='beta',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n alpha=FloatSlider(min=0,max=1,step=0.01,value=params['alpha'],description='alpha',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n gamma=FloatSlider(min=0,max=1,step=0.01,value=params['gamma'],description='gamma',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), \n c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_2=FloatSlider(min=0,max=5000,step=1,value=params['c_2'],description='c_2',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'),\n logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0',\n style=style,layout=slider_layout,continuous_update=False),\n k_u=FloatSlider(min=0,max=1,step=0.001,value=params['k_u'],description='k_u',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n k_1=FloatSlider(min=0,max=1,step=0.001,value=params['k_1'],description='k_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n k_w=FloatSlider(min=0,max=1,step=0.001,value=params['k_w'],description='k_w',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), \n kappa=FloatSlider(min=0,max=1,step=0.001,value=params['kappa'],description='kappa',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'))\ndisplay(w)\n```\n\n\n interactive(children=(FloatSlider(value=0.48, continuous_update=False, description='beta', layout=Layout(width\u2026\n\n\n\n```python\nparams=w.kwargs\n# not a good fit yet, did better last week\nprint(params)\n```\n\n {'beta': 0.48, 'alpha': 0.2, 'gamma': 0.11249999999999999, 'mu': 0.0125, 'c_0': 0.1, 'c_1': 0.016666666666666666, 'c_2': 110.0, 'logI_0': -6.0, 'k_u': 0.016666666666666666, 'k_1': 0.016666666666666666, 'k_w': 0.016666666666666666, 'kappa': 0.5}\n\n\n### SC3UEI3R Model\n\n#### USA\n\n\n```python\n# assumed data starting on firstdate\ntest_country='US'\nN = 66650000\nfirstdate = '01/25/20'\nlastdate = '01/08/20'\nxx,xxf,yy0 = get_country_data(test_country,'confirmed',firstdate=firstdate,lastdate=lastdate)\nxx,xxf,yy1 = get_country_data(test_country,'recovered',firstdate=firstdate,lastdate=lastdate)\nxx,xxf,yy2 = get_country_data(test_country,'deaths',firstdate=firstdate,lastdate=lastdate)\nprint(xxf)\ny_jhu={}\ny_jhu[test_country] = np.array([[yy0[i],yy1[i],yy2[i]] for i in range(0,len(yy0))])/N\n# data = np.array([[xxf[i],yy0[i],yy1[i],yy2[i]] for i in range(len(yy))])\n# print(data)\nlastday = len(y_jhu[test_country])\nprint('days 0 to',lastday,'data stored in y_jhu')\n```\n\n\n```python\nmodel = 'SC3UEI3R'\nI_0 = 0.00003\nx0_SC3UEI3R = [1.0-I_0, 0.0, I_0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0]\nSC3UEI3R_model.initial_values = (x0_SC3UEI3R, t[0])\n\n# Define parameters based on clinical observations Dr. Alison\nExposure=0.4 # Rate coefficient for exposure per individual in contact per day\nIncubPeriod=5 #Incubation period, days\nDurMildInf=8 #Duration of mild infections, days\nFracMild=0.65 #Fraction of infections that are mild\nFracSevere=0.20 #Fraction of infections that are severe\nFracCritical=0.15 #Fraction of infections that are critical\nCFR=0.1 #Case fatality rate (fraction of infections resulting in death)\nTimeICUDeath=4 #Time from ICU admission to death, days\nDurHosp=5 #Duration of hospitalization, days\n\n# Model fitting extension to allow for incomplete detection\nFracConfirmedDet=0.5 # Fraction of recovered individuals measured\nFracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured\nFracDeathsDet=1.0\n\n# Model extension by John McCaskill to include caution \nCautionFactor= 0.1 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 60. # Duration of cautionary state of susceptibles (2 weeks)\nCautionICUFrac= 0.1 # Fraction of ICUs occupied leading to transition to caution @ 1/day\nICUFrac= 0.001 # Fraction of ICUs relative to population size N\n\n# Model extension by John McCaskill to include economic influence on caution \nEconomicCostOfCaution= 0.5 # Fractional reduction of economic contribution for cautioned individuals\n\np = [0,(1.0/DurMildInf)-(1.0/DurMildInf)*FracMild, (1.0/DurHosp)*(FracCritical/(FracCritical+FracSevere))]\ng = [0,(1.0/DurMildInf)*FracMild, (1.0/DurHosp)-(1/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n (1.0/TimeICUDeath)-(1/TimeICUDeath)*(CFR/FracCritical)]\nu = (1.0/TimeICUDeath)*(CFR/FracCritical)\nif cmodels[model].loadparams():\n params = cmodels[model].params.copy()\nelse:\n params = {'beta_1' : Exposure/sum(x0_SC3UEI3R),\n 'beta_2' : 0.0,\n 'beta_3' : 0.0,\n 'alpha' : 1.0/IncubPeriod,\n 'gamma_1': (1.0/DurMildInf)*FracMild,\n 'gamma_2': (1.0/DurHosp)-(1/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n 'gamma_3': (1.0/TimeICUDeath)-(1/TimeICUDeath)*(CFR/FracCritical),\n 'p_1' : (1.0/DurMildInf)-(1.0/DurMildInf)*FracMild,\n 'p_2' : (1.0/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n 'mu' : (1.0/TimeICUDeath)*(CFR/FracCritical),\n 'c_0' : CautionFactor,\n 'c_1' : 1.0/CautionRetention,\n 'c_2' : 1.0/(ICUFrac*CautionICUFrac),\n 'k_u' : 1.0/5.,\n 'k_1' : 1.0/90,\n 'k_w' : 1.0/90,\n 'kappa' : EconomicCostOfCaution,\n 'N' : sum(x0_SC3UEI3R)}\n\nprint(params)\ncmodels[model].parameters = params\ncmodels[model].params = params\ncmodels[model].dumpparams() \n```\n\n loaded params from /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3UEI3R.pk :\n {'beta_1': 0.2, 'beta_2': 0.0, 'beta_3': 0.0, 'alpha': 0.2, 'gamma_1': 0.08125, 'gamma_2': 0.11428571428571428, 'gamma_3': 0.08333333333333331, 'p_1': 0.04375, 'p_2': 0.08571428571428573, 'mu': 0.16666666666666669, 'c_0': 0.1, 'c_1': 0.016666666666666666, 'c_2': 10000.0, 'k_u': 0.2, 'k_1': 0.011111111111111112, 'k_w': 0.011111111111111112, 'kappa': 0.5, 'N': 2.0}\n dumped params to /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3UEI3R.pk\n\n\n\n```python\ndef slidefitplot(beta_1,alpha,mu,c_0,c_1,c_2,logI_0,k_u,k_1,k_w,kappa):\n params={ 'beta_1':beta_1, 'alpha':alpha, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2, 'k_u':k_u, 'k_1':k_1, 'k_w':k_w, 'kappa':kappa}\n cmodels[model].parameters = params\n \n I0 = 10**logI_0\n x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.,0.,1.]\n\n cmodels[model].initial_values = (x0,t[0])\n cmodels[model].params = params\n cmodels[model].dumpparams() \n weights=np.array([1.,1.,1.])\n solveplot(smodels=[model],species=['confirmed','recovered','deaths_x10'],tmax=len(t),summing='daily',averaging='weekly',fitdata=y_jhu[test_country]*weights,scale='linear',plottitle= '',label='confirmed',newplot = True, figsize = (15,15))\n```\n\n\n```python\nw=interactive(slidefitplot,\n beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n alpha=FloatSlider(min=0,max=1,step=0.01,value=params['alpha'],description='alpha',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), \n c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_2=FloatSlider(min=0,max=5000,step=1,value=params['c_2'],description='c_2',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'),\n logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0',\n style=style,layout=slider_layout,continuous_update=False),\n k_u=FloatSlider(min=0,max=1,step=0.001,value=params['k_u'],description='k_u',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n k_1=FloatSlider(min=0,max=1,step=0.001,value=params['k_1'],description='k_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n k_w=FloatSlider(min=0,max=1,step=0.001,value=params['k_w'],description='k_w',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), \n kappa=FloatSlider(min=0,max=1,step=0.001,value=params['kappa'],description='kappa',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'))\ndisplay(w)\n```\n\n\n interactive(children=(FloatSlider(value=0.2, continuous_update=False, description='beta_1', layout=Layout(widt\u2026\n\n\n\n```python\nparams=w.kwargs\n\nprint(params)\n```\n\n#### Spain\n\n\n```python\n# assumed data starting on firstdate\ntest_country='Spain'\nN = 66650000\nfirstdate = '01/25/20'\nlastdate = '01/08/20'\nxx,xxf,yy0 = get_country_data(test_country,'confirmed',firstdate=firstdate,lastdate=lastdate)\nxx,xxf,yy1 = get_country_data(test_country,'recovered',firstdate=firstdate,lastdate=lastdate)\nxx,xxf,yy2 = get_country_data(test_country,'deaths',firstdate=firstdate,lastdate=lastdate)\nprint(xxf)\ny_jhu={}\ny_jhu[test_country] = np.array([[yy0[i],yy1[i],yy2[i]] for i in range(0,len(yy0))])/N\n# data = np.array([[xxf[i],yy0[i],yy1[i],yy2[i]] for i in range(len(yy))])\n# print(data)\nlastday = len(y_jhu[test_country])\nprint('days 0 to',lastday,'data stored in y_jhu')\n```\n\n [-3.0, -2.0, -1.0, 0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, 32.0, 33.0, 34.0, 35.0, 36.0, 37.0, 38.0, 39.0, 40.0, 41.0, 42.0, 43.0, 44.0, 45.0, 46.0, 47.0, 48.0, 49.0, 50.0, 51.0, 52.0, 53.0, 54.0, 55.0, 56.0, 57.0, 58.0, 59.0, 60.0, 61.0, 62.0, 63.0, 64.0, 65.0, 66.0, 67.0, 68.0, 69.0, 70.0, 71.0, 72.0, 73.0, 74.0, 75.0, 76.0, 77.0, 78.0, 79.0, 80.0, 81.0, 82.0, 83.0, 84.0, 85.0, 86.0, 87.0, 88.0, 89.0, 90.0, 91.0, 92.0, 93.0, 94.0, 95.0, 96.0, 97.0, 98.0, 99.0, 100.0, 101.0, 102.0, 103.0, 104.0, 105.0, 106.0, 107.0, 108.0, 109.0, 110.0, 111.0, 112.0, 113.0, 114.0, 115.0, 116.0, 117.0, 118.0, 119.0, 120.0, 121.0, 122.0, 123.0, 124.0, 125.0, 126.0, 127.0, 128.0, 129.0, 130.0, 131.0, 132.0, 133.0, 134.0, 135.0, 136.0, 137.0, 138.0, 139.0, 140.0, 141.0, 142.0, 143.0, 144.0, 145.0, 146.0, 147.0, 148.0, 149.0, 150.0, 151.0, 152.0, 153.0, 154.0, 155.0, 156.0, 157.0, 158.0, 159.0, 160.0, 161.0, 162.0, 163.0, 164.0, 165.0, 166.0, 167.0, 168.0, 169.0, 170.0, 171.0, 172.0, 173.0, 174.0, 175.0, 176.0, 177.0, 178.0, 179.0, 180.0, 181.0, 182.0, 183.0, 184.0, 185.0, 186.0, 187.0, 188.0, 189.0, 190.0, 191.0, 192.0, 193.0, 194.0, 195.0, 196.0, 197.0, 198.0, 199.0, 200.0, 201.0, 202.0, 203.0, 204.0, 205.0, 206.0, 207.0, 208.0, 209.0, 210.0, 211.0, 212.0, 213.0, 214.0, 215.0, 216.0, 217.0, 218.0]\n days 0 to 222 data stored in y_jhu\n\n\n\n```python\nmodel = 'SC3UEI3R'\nI_0 = 0.00003\nx0_SC3UEI3R = [1.0-I_0, 0.0, I_0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0]\nSC3UEI3R_model.initial_values = (x0_SC3UEI3R, t[0])\n\n# Define parameters based on clinical observations Dr. Alison\nExposure=0.4 # Rate coefficient for exposure per individual in contact per day\nIncubPeriod=5 #Incubation period, days\nDurMildInf=8 #Duration of mild infections, days\nFracMild=0.65 #Fraction of infections that are mild\nFracSevere=0.20 #Fraction of infections that are severe\nFracCritical=0.15 #Fraction of infections that are critical\nCFR=0.1 #Case fatality rate (fraction of infections resulting in death)\nTimeICUDeath=4 #Time from ICU admission to death, days\nDurHosp=5 #Duration of hospitalization, days\n\n# Model fitting extension to allow for incomplete detection\nFracConfirmedDet=0.5 # Fraction of recovered individuals measured\nFracRecoveredDet=FracConfirmedDet # Fraction of recovered individuals measured\nFracDeathsDet=1.0\n\n# Model extension by John McCaskill to include caution \nCautionFactor= 0.1 # Fractional reduction of exposure rate for cautioned individuals\nCautionRetention= 60. # Duration of cautionary state of susceptibles (2 weeks)\nCautionICUFrac= 0.1 # Fraction of ICUs occupied leading to transition to caution @ 1/day\nICUFrac= 0.001 # Fraction of ICUs relative to population size N\n\n# Model extension by John McCaskill to include economic influence on caution \nEconomicCostOfCaution= 0.5 # Fractional reduction of economic contribution for cautioned individuals\n\np = [0,(1.0/DurMildInf)-(1.0/DurMildInf)*FracMild, (1.0/DurHosp)*(FracCritical/(FracCritical+FracSevere))]\ng = [0,(1.0/DurMildInf)*FracMild, (1.0/DurHosp)-(1/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n (1.0/TimeICUDeath)-(1/TimeICUDeath)*(CFR/FracCritical)]\nu = (1.0/TimeICUDeath)*(CFR/FracCritical)\n \nif cmodels[model].loadparams():\n params = cmodels[model].params.copy()\nelse:\n params = {'beta_1' : Exposure/sum(x0_SC3UEI3R),\n 'beta_2' : 0.0,\n 'beta_3' : 0.0,\n 'alpha' : 1.0/IncubPeriod,\n 'gamma_1': (1.0/DurMildInf)*FracMild,\n 'gamma_2': (1.0/DurHosp)-(1/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n 'gamma_3': (1.0/TimeICUDeath)-(1/TimeICUDeath)*(CFR/FracCritical),\n 'p_1' : (1.0/DurMildInf)-(1.0/DurMildInf)*FracMild,\n 'p_2' : (1.0/DurHosp)*(FracCritical/(FracCritical+FracSevere)),\n 'mu' : (1.0/TimeICUDeath)*(CFR/FracCritical),\n 'c_0' : CautionFactor,\n 'c_1' : 1.0/CautionRetention,\n 'c_2' : 1.0/(ICUFrac*CautionICUFrac),\n 'k_u' : 1.0/5.,\n 'k_1' : 1.0/90,\n 'k_w' : 1.0/90,\n 'kappa' : EconomicCostOfCaution,\n 'N' : sum(x0_SC3UEI3R)}\n\nprint(params)\ncmodels[model].parameters = params\ncmodels[model].params = params\nrun_id = '{}_{}_logI0=-6'.format(model,test_country)\ncmodels[model].dumpparams(run_id) \n\n```\n\n loaded params from /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3UEI3R.pk :\n {'beta_1': 0.35, 'alpha': 0.2, 'mu': 0.16666666666666669, 'c_0': 0.1, 'c_1': 0.016666666666666666, 'c_2': 882.0, 'k_u': 0.084, 'k_1': 0.011111111111111112, 'k_w': 0.011111111111111112, 'kappa': 0.553}\n dumped params to /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3UEI3R.pk\n dumped params to /Users/n/Projects/covid-recovery/Notebooks/covid-19-caution/params/SC3UEI3R_Spain_logI0=-6.pk\n\n\n\n```python\ndef slidefitplot(beta_1,alpha,mu,c_0,c_1,c_2,logI_0,k_u,k_1,k_w,kappa):\n params={ 'beta_1':beta_1, 'alpha':alpha, 'mu':mu, 'c_0':c_0, 'c_1':c_1, 'c_2':c_2, 'k_u':k_u, 'k_1':k_1, 'k_w':k_w, 'kappa':kappa}\n cmodels[model].parameters = params\n cmodels[model].params = params\n run_id = '{}_{}_logI0={}'.format(model,test_country,logI_0)\n cmodels[model].dumpparams(run_id) \n I0 = 10**logI_0\n x0 = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.,0.,1.]\n\n cmodels[model].initial_values = (x0,t[0])\n weights=np.array([1.,1.,1.])\n solveplot(smodels=[model],species=['confirmed','recovered','deaths_x10'],tmax=len(t),summing='daily',averaging='weekly',fitdata=y_jhu[test_country]*weights,scale='linear',plottitle= '',label='confirmed',newplot = True, figsize = (15,15))\n```\n\n\n```python\nw=interactive(slidefitplot,\n beta_1=FloatSlider(min=0,max=1,step=0.01,value=params['beta_1'],description='beta_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n alpha=FloatSlider(min=0,max=1,step=0.01,value=params['alpha'],description='alpha',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n mu=FloatSlider(min=0,max=0.2,step=0.002,value=params['mu'],description='mu',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), \n c_0=FloatSlider(min=0,max=1,step=0.01,value=params['c_0'],description='c_0',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_1=FloatSlider(min=0,max=1,step=0.001,value=params['c_1'],description='c_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n c_2=FloatSlider(min=0,max=5000,step=1,value=params['c_2'],description='c_2',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.1f'),\n logI_0=FloatSlider(min=-10,max=0,step=0.01,value=-6,description='log I_0',\n style=style,layout=slider_layout,continuous_update=False),\n k_u=FloatSlider(min=0,max=1,step=0.001,value=params['k_u'],description='k_u',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n k_1=FloatSlider(min=0,max=1,step=0.001,value=params['k_1'],description='k_1',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'),\n k_w=FloatSlider(min=0,max=1,step=0.001,value=params['k_w'],description='k_w',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'), \n kappa=FloatSlider(min=0,max=1,step=0.001,value=params['kappa'],description='kappa',\n style=style,layout=slider_layout,continuous_update=False,readout_format='.3f'))\ndisplay(w)\n```\n\n\n interactive(children=(FloatSlider(value=0.35, continuous_update=False, description='beta_1', layout=Layout(wid\u2026\n\n\n\n```python\nrun_id = '{}_{}_logI0=-6.28'.format(model,test_country)\nrun_id\n```\n\n\n\n\n 'SC3UEI3R_Spain_logI0=-6.28'\n\n\n\n\n```python\n\n```\n\n\n```python\nparams=w.kwargs\n\nprint(params)\n```\n\n## Fit SC3EI3R parameters to jhu data based on square_loss\n\n### Fit c_0 , c_1 and c_2 as well as initial value of I_1\n\n\n```python\nSC3EI3R_model.parameters\n```\n\n\n```python\n# Initial guess of parameters and initial condition, and bounding constraints\nI0 =10**-6.43\nx0_SC3EI3R = [1.-I0,0.,I0,0.,0.,0.,0.,0.,0.,0.,0.]\nSC3EI3R_model.parameters={'beta_1': 0.41, 'mu': 0.079, 'c_0': 0.1, 'c_1': 0.030303030303030304, 'c_2': 11170.0}\ncautionparams = list(params.values())[-4:-1]\ntheta = [0.1,0.07,8000.] # cautionparams\nboxBounds = [(0.05,0.4),(0.05,0.15),(1000.,200000.)]\n# set up optimization function with cost and sensitivity (Jacobian)\nobjSC3EI3R = SquareLoss(theta=theta, ode=SC3EI3R_model, x0=x0_SC3EI3R, t0=t[0], t=t[1::], y=y_jhu[test_country][1::,:],\n state_weight=[1.,1.,10.],state_name=['C_f','R','D'],\n target_param=['c_0','c_1','c_2'],target_state=['I_1'])\n# perform optimization\nres = minimize(fun=objSC3EI3R.costIV,\n jac=objSC3EI3R.sensitivityIV,\n x0=theta+[I0],\n bounds=boxBounds+[(0.0000001,0.0001)],\n #method='BFGS',\n method='SLSQP',\n #options={'disp':True,'maxiter':1000,'eps':0.01,'gtol':0.01})\n #options={'disp':True})\n options={'disp':True,'maxiter':1000,'eps':0.001,'ftol':0.001})\nprint(res)\n```\n\n\n```python\n# model with fitted parameters \nprint(params)\nprint(x0_SC3EI3R)\nparams_fit = params.copy()\n#params_fit['c_0'] = res.x[0]\n#params_fit['c_1'] = res.x[1]\n\nSC3EI3R_model.params = params_fit\nprint(SC3EI3R_model.params)\n#ode_fit = common_models.SEI3R({'beta':res.x[0], 'gamma':res.x[1],'alpha':res.x[2]})\n#x0_fit = [1-1.27e-6, 1.27e-6, 0]\nx0_fit = x0_SC3EI3R.copy()\n#x0_fit[2] = res.x[2]\n#t_fit = numpy.linspace(0, 150, 1000)\nt_fit = t\nSC3EI3R_model.initial_values = (x0_fit, t_fit[0])\n# %timeit sol_fit =SCEI3R_model.integrate(t_fit[1::]) # use magic %timeit to time\n# sol_fit =SCEI3R_model.integrate(t_fit[0::])\nsol_fit = scipy.integrate.odeint(SC3EI3R_model.ode, x0_fit, t_fit[1::])\n#\nplt.figure(figsize=(15,10))\nplt.plot(t,ynoise[:,0], 'o',color='lightgreen') # infected observations\nplt.plot(t,ynoise[:,1], 'o',color='green') # infected observations\nplt.plot(t,ynoise[:,2], 'o',color='darkgreen') # infected observations\nplt.plot(t,ynoise[:,3], 'bo') # recoverd\nplt.plot(t,ynoise[:,4], 'ro') # died\nplt.plot(t_fit[1::], sol_fit)\nplt.ylim([0,0.004])\n#plt.show(())\n#ode_fit.plot()\n\npeak_i = np.argmax(sol_fit[:,2])\nprint('Peak infection (days)', t_fit[peak_i])\n```\n\n## Testing fitting\n\n### Generate test data based on SCEI3R simulation\n\n\n```python\n# Add noise\ny = solution[:,2:7].copy()\n#print('len(y)',len(y),'t',len(t),t[0],t[1],'...',t[-1])\nnp.random.seed(seed=6)\nnoise = np.random.normal(0,1.e-2,[len(t),5])\n# ynoise = y *(1+noise)\nynoise = y *(1.0 + noise)\nynoise[ynoise<0] = 0\nplt.figure(figsize=(15,10))\nplt.plot(t,ynoise[:,0], 'go', label='I_1') \nplt.plot(t,ynoise[:,1], 'go', label='I_2') \nplt.plot(t,ynoise[:,2], 'go', label='I_3') \nplt.plot(t,ynoise[:,3], 'bo', label='R') \nplt.plot(t,ynoise[:,4], 'ro', label='D') \nplt.plot(t,y[:,0], 'g', label='I_1') \nplt.plot(t,y[:,1], 'g', label='I_2') \nplt.plot(t,y[:,2], 'g', label='I_3') \nplt.plot(t,y[:,3], 'b', label='R') \nplt.plot(t,y[:,4], 'r', label='D') \nplt.legend()\nplt.ylim(0,0.003)\nplt.show()\n```\n\n\n```python\n# model with generating parameters \nprint(params)\nparams_fit = params.copy()\nprint(params_fit['c_0'],params_fit['c_1'])\nSCEI3R_model.params = params_fit\n\nx0_fit = x0_SCEI3R.copy()\nprint(x0_fit)\n#t_fit = numpy.linspace(0, 150, 1000)\n\nt_fit = t\nSCEI3R_model.initial_values = (x0_fit, t_fit[0])\n# %timeit sol_fit =SCEI3R_model.integrate(t_fit[1::]) # use magic %timeit to time\nsol_fit = scipy.integrate.odeint(SCEI3R_model.ode, x0_fit, t_fit[1::])\n# print(len(sol_fit[0]))\n#\nplt.figure(figsize=(15,10))\nplt.plot(t,ynoise[:,0], 'go',label='I_1') # infected observations\nplt.plot(t,ynoise[:,1], 'go',label='I_2') # infected observations\nplt.plot(t,ynoise[:,2], 'go',label='I_3') # infected observations\nplt.plot(t,ynoise[:,3], 'bo',label='R') # recoverd\nplt.plot(t,ynoise[:,4], 'ro',label='D') # died\nplt.gca().set_prop_cycle(color=['grey','orange','green','green','green','blue','red', 'black'])\nplt.plot(t_fit[1::], sol_fit)\nplt.ylim([0,0.004])\nplt.legend()\n#plt.show(())\n#ode_fit.plot()\n\npeak_i = np.argmax(sol_fit[:,2])\nprint('Peak infection (days)', t_fit[peak_i])\n```\n\n\n```python\nparams # use list(...) to convert to list\n```\n\n### Fit parameters to randomized simulation data based on square_loss\n\n#### Fit c_0 and c_1 only\n\n\n```python\n# Initial guess of parameters, and bounding constraints\ncautionparams = list(params.values())[-4:-2]\ntheta = [value for value in cautionparams]\ntheta = [0.21,0.08]\nboxBounds = [(0.2,0.4),(0.05,0.15)]\nobjSCEI3R = SquareLoss(theta=theta, ode=SCEI3R_model, x0=x0_SCEI3R, t0=t[0], t=t[1::], y=ynoise[1::,:],\n state_weight=[1.,1.,1.,1.0,1.0],state_name=['I_1','I_2','I_3','R','D'],\n target_param=['c_0','c_1'])\n# perform optimization\nres = minimize(fun=objSCEI3R.cost,\n jac=objSCEI3R.sensitivity,\n x0=theta,\n #bounds=boxBounds,\n method='BFGS',\n options={'disp':True,'maxiter':1000,'eps':0.00001})# ,'ftol':0.01}) #not BFGS\nprint(res)\n```\n\n#### Fit c_0 and c_1 as well as initial value of E\n\n##### Fit c_0 and c_1 as well as initial value of E with 'SLSQP'\ndoes not work well\nnote use of special methods IV for initial value fitting of target_state\n\n\n```python\n# Initial guess of parameters and initial condition, and bounding constraints\ncautionparams = list(params.values())[-4:-2]\ntheta = [0.25,0.08]\nboxBounds = [(0.2,0.4),(0.05,0.15)]\nobjSCEI3R = SquareLoss(theta=theta, ode=SCEI3R_model, x0=x0_SCEI3R, t0=t[0], t=t[1::], y=ynoise[1::,:],\n state_weight=[1.,1.,1.,1.,1.],state_name=['I_1','I_2','I_3','R','D'],\n target_param=['c_0','c_1'],target_state=['E'])\n# perform optimization\nres = minimize(fun=objSCEI3R.costIV,\n jac=objSCEI3R.sensitivityIV,\n x0=theta+[0.00005],\n #bounds=boxBounds+[(0.0000001,0.001)],\n method='SLSQP',\n options={'disp':True,'maxiter':1000,'eps':0.01,'ftol':0.01})\nprint(res)\n```\n\n##### Fit c_0 and c_1 as well as initial value of E with BFGS\nworks well: no constraints and gtol not ftol\n\n\n```python\n# Initial guess of parameters and initial condition, and bounding constraints\ncautionparams = list(params.values())[-4:-2]\ntheta = [0.25,0.08]\nboxBounds = [(0.2,0.4),(0.05,0.15)]\nobjSCEI3R = SquareLoss(theta=theta, ode=SCEI3R_model, x0=x0_SCEI3R, t0=t[0], t=t[1::], y=ynoise[1::,:],\n state_weight=[1.,1.,1.,1.,1.],state_name=['I_1','I_2','I_3','R','D'],\n target_param=['c_0','c_1'],target_state=['E'])\n# perform optimization\nres = minimize(fun=objSCEI3R.costIV,\n jac=objSCEI3R.sensitivityIV,\n x0=theta+[0.00005],\n #bounds=boxBounds+[(0.0000001,0.001)],\n method='BFGS',\n options={'disp':True,'maxiter':1000,'eps':0.01,'gtol':0.01})\nprint(res)\n```\n\n\n```python\n# model with fitted parameters \nprint(params)\nprint(x0_SCEI3R)\nparams_fit = params.copy()\n#params_fit['c_0'] = res.x[0]\n#params_fit['c_1'] = res.x[1]\n\nSCEI3R_model.params = params_fit\nprint(SCEI3R_model.params)\n#ode_fit = common_models.SEI3R({'beta':res.x[0], 'gamma':res.x[1],'alpha':res.x[2]})\n#x0_fit = [1-1.27e-6, 1.27e-6, 0]\nx0_fit = x0.copy()\n#x0_fit[2] = res.x[2]\n#t_fit = numpy.linspace(0, 150, 1000)\nt_fit = t\nSCEI3R_model.initial_values = (x0_fit, t_fit[0])\n# %timeit sol_fit =SCEI3R_model.integrate(t_fit[1::]) # use magic %timeit to time\n# sol_fit =SCEI3R_model.integrate(t_fit[0::])\nsol_fit = scipy.integrate.odeint(SCEI3R_model.ode, x0_fit, t_fit[1::])\n#\nplt.figure(figsize=(15,10))\nplt.plot(t,ynoise[:,0], 'go') # infected observations\nplt.plot(t,ynoise[:,1], 'go') # infected observations\nplt.plot(t,ynoise[:,2], 'go') # infected observations\nplt.plot(t,ynoise[:,3], 'bo') # recoverd\nplt.plot(t,ynoise[:,4], 'ro') # died\nplt.plot(t_fit[1::], sol_fit)\nplt.ylim([0,0.004])\n#plt.show(())\n#ode_fit.plot()\n\npeak_i = np.argmax(sol_fit[:,2])\nprint('Peak infection (days)', t_fit[peak_i])\n```\n\n##### Fit c_0 and c_1 as well as initial value of E using L-BFGS-B\nthis method does not work well\n\n\n```python\n# Initial guess of parameters and initial condition, and bounding constraints\ncautionparams = list(params.values())[-4:-2]\ntheta = [0.25,0.08]\nboxBounds = [(0.2,0.4),(0.05,0.15)]\nobjSCEI3R = SquareLoss(theta=theta, ode=SCEI3R_model, x0=x0_SCEI3R, t0=t[0], t=t[1::], y=ynoise[1::,:],\n state_weight=[1.,1.,1.,1.,1.],state_name=['I_1','I_2','I_3','R','D'],\n target_param=['c_0','c_1'],target_state=['E'])\n# perform optimization\nres = minimize(fun=objSCEI3R.costIV,\n jac=objSCEI3R.sensitivityIV,\n x0=theta+[0.00005],\n bounds=boxBounds+[(0.0000001,0.001)],\n method='L-BFGS-B',\n options={'disp':True,'maxiter':1000,'eps':0.0001,'ftol':0.001})\nprint(res)\n```\n\n\n```python\nobjSCEI3R.residual()\n```\n\n##### Fit c_0 and c_1 as well as initial value of E with Nelder-Mead\nno use of Jacobian and no constraints\n\n\n```python\n# Initial guess of parameters and initial condition, and bounding constraints\ncautionparams = list(params.values())[-4:-2]\ntheta = [0.25,0.08]\nboxBounds = [(0.2,0.4),(0.05,0.15)]\nobjSCEI3R = SquareLoss(theta=theta, ode=SCEI3R_model, x0=x0_SCEI3R, t0=t[0], t=t[1::], y=ynoise[1::,:],\n state_weight=[1.,1.,1.,1.,1.],state_name=['I_1','I_2','I_3','R','D'],\n target_param=['c_0','c_1'],target_state=['E'])\n# perform optimization\nres = minimize(fun=objSCEI3R.costIV,\n #jac=objSCEI3R.sensitivityIV,\n x0=theta+[0.00005],\n #bounds=boxBounds+[(0.0000001,0.001)],\n method='Nelder-Mead',\n options={'disp':True,'maxiter':1000}) #,'eps':0.0001,'ftol':0.01}) #not NM\nprint(res)\n```\n\n\n```python\n# model with fitted parameters \nprint(params)\nprint(x0_SCEI3R)\nparams_fit = params.copy()\n#params_fit['c_0'] = res.x[0]\n#params_fit['c_1'] = res.x[1]\n\nSCEI3R_model.params = params_fit\nprint(SCEI3R_model.params)\n#ode_fit = common_models.SEI3R({'beta':res.x[0], 'gamma':res.x[1],'alpha':res.x[2]})\n#x0_fit = [1-1.27e-6, 1.27e-6, 0]\nx0_fit = x0_SCEI3R.copy()\n#x0_fit[2] = res.x[2]\n#t_fit = numpy.linspace(0, 150, 1000)\nt_fit = t\nSCEI3R_model.initial_values = (x0_fit, t_fit[0])\n# %timeit sol_fit =SCEI3R_model.integrate(t_fit[1::]) # use magic %timeit to time\n# sol_fit =SCEI3R_model.integrate(t_fit[0::])\nsol_fit = scipy.integrate.odeint(SCEI3R_model.ode, x0_fit, t_fit[1::])\n#\nplt.figure(figsize=(15,10))\nplt.plot(t,ynoise[:,0], 'go') # infected observations\nplt.plot(t,ynoise[:,1], 'go') # infected observations\nplt.plot(t,ynoise[:,2], 'go') # infected observations\nplt.plot(t,ynoise[:,3], 'bo') # recoverd\nplt.plot(t,ynoise[:,4], 'ro') # died\nplt.plot(t_fit[1::], sol_fit)\nplt.ylim([0,0.004])\n#plt.show(())\n#ode_fit.plot()\n\npeak_i = np.argmax(sol_fit[:,2])\nprint('Peak infection (days)', t_fit[peak_i])\n```\n\n## Fit SC2IR parameters to jhu data based on square_loss\n\n\n```python\nparams=SC2IR_model.parameters\nprint(params)\n```\n\n\n```python\n# Initial guess of parameters and initial condition, and bounding constraints\ntheta = [0.4,0.11,0.007,0.33,0.228,275.]\nboxBounds = [(0.2,0.5),(0.05,0.15),(0.005,0.015),(0.25,0.55),(0.15,0.4),(5.,2000.)]\n# setup cost function and Jacobian with target parameters and initial states\nobjSC2IR = SquareLoss(theta=theta, ode=SC2IR_model, x0=x0, t0=t[0], t=t[1::], y=y_jhu[test_country][1::,1:3],\n state_weight=[0.2,1.],state_name=['R','D'],\n target_param=['beta','gamma','mu','c_0','c_1','c_2'],\n target_state=['I'])\n# perform optimization\nres = minimize(fun=objSC2IR.costIV,\n jac=objSC2IR.sensitivityIV,\n x0=theta+[0.000000001],\n bounds=boxBounds+[(0.0000000001,0.000001)],\n # method='L-BFGS-B',\n # method='Nelder-Mead',\n #options={'disp':True,'maxiter':1000,'eps':0.01,'gtol':0.01})\n options={'disp':True,'maxiter':1000,'eps':0.000001,'ftol':0.000000001})\nprint(res)\n```\n\n\n```python\n# model with fitted parameters \nstartparams = SC2IR_model.parameters.copy() # save starting parameters (not fit)\nprint(params)\nprint(x0)\nparams_fit = params.copy()\nparams_fit['beta'] = res.x[0]\nparams_fit['gamma'] = res.x[1]\nparams_fit['mu'] = res.x[2]\nparams_fit['c_0'] = res.x[3]\nparams_fit['c_1'] = res.x[4]\nparams_fit['c_2'] = res.x[5]\n\nSC2IR_model.params = params_fit\nprint(SC2IR_model.params)\nx0_fit = x0.copy()\nx0_fit[1] = res.x[6]\nt_fit = t\nSC2IR_model.initial_values = (x0_fit, t_fit[0])\nsol_fit = scipy.integrate.odeint(SC2IR_model.ode, x0_fit, t_fit[1::])\n#\nplt.figure(figsize=(15,10))\nplt.semilogy()\nplt.ylim([0.000001,1])\nplt.plot(t,y_jhu[test_country][:,1], 'bo',label='R') # recovered\nplt.semilogy()\nplt.ylim([0.000001,1])\nplt.plot(t,y_jhu[test_country][:,2], 'ro',label='D') # died\nplt.semilogy()\nplt.ylim([0.000001,1])\n\nplt.gca().set_prop_cycle(color=['grey','green','blue','red', 'black'])\nplt.plot(t_fit[1::], sol_fit)\nplt.ylim([0.000001,1])\nplt.legend(('R','D','S','I','R','D','S_c','I_c'))\nplt.semilogy()\n#plt.show(())\n#ode_fit.plot()\n\npeak_i = np.argmax(sol_fit[:,1])\nprint('Peak infection (days)', t_fit[peak_i])\nSC2IR_model.parameters = startparams\n```\n\n## Fit SC3EI3R parameters to jhu data based on square_loss\n\n### Fit c_0 and c_1 only\n\n\n```python\n# Initial guess of parameters, and bounding constraints\ncautionparams = list(params.values())[-4:-1]\ntheta = [value for value in cautionparams]\nprint(theta)\ntheta = [0.3,0.08,2500.]\nboxBounds = [(0.2,0.8),(0.05,0.15),(100.,10000.)]\nobjSC3EI3R = SquareLoss(theta=theta, ode=SC3EI3R_model, x0=x0_SC3EI3R, t0=t[0], t=t[1::], y=y_jhu[test_country][1::,1:3],\n state_weight=[1.,1.],state_name=['R','D'],\n target_param=['c_0','c_1','c_2'])\n# perform optimization\nres = minimize(fun=objSC3EI3R.cost,\n #jac=objSC3EI3R.sensitivity,\n x0=theta,\n #bounds=boxBounds,\n method='L-BFGS-B',\n # method='Nelder-Mead',\n options={'disp':True,'maxiter':1000,'eps':0.00001})# ,'ftol':0.01}) #not BFGS\nprint(res)\n```\n\n### Fit c_0 and c_1 as well as initial value of E\n\n#### Fit c_0 and c_1 as well as initial value of E with 'SLSQP'\ndoes not work well\nnote use of special methods IV for initial value fitting of target_state\n\n\n```python\n# Initial guess of parameters and initial condition, and bounding constraints\ncautionparams = list(params.values())[-4:-3]\ntheta = [value for value in cautionparams]\ntheta = [0.21,0.08,2500.]\nobjSC3EI3R = SquareLoss(theta=theta, ode=SC3EI3R_model, x0=x0_SC3EI3R, t0=t[0], t=t[1::], y=y_jhu[test_country][1::,1:3],\n state_weight=[1.,1.],state_name=['R','D'],\n target_param=['c_0','c_1','c_2'],target_state=['I_1'])\n# perform optimization\nres = minimize(fun=objSC3EI3R.costIV,\n jac=objSC3EI3R.sensitivityIV,\n x0=theta+[0.00005],\n bounds=boxBounds+[(0.0000001,0.001)],\n # method='BFGS',\n method='L-BFGS-B',\n options={'disp':True,'maxiter':1000,'eps':0.01,'gtol':0.01})\nprint(res)\n```\n\n#### Fit c_0 and c_1 as well as initial value of E with BFGS\nworks well: no constraints and gtol not ftol\n\n\n```python\n# Initial guess of parameters and initial condition, and bounding constraints\ncautionparams = list(params.values())[-4:-2]\ntheta = [0.25,0.08]\nboxBounds = [(0.2,0.4),(0.05,0.15)]\nobjSCEI3R = SquareLoss(theta=theta, ode=SCEI3R_model, x0=x0_SC3EI3R, t0=t[0], t=t[1::], y=ynoise[1::,:],\n state_weight=[1.,1.,1.,1.,1.],state_name=['I_1','I_2','I_3','R','D'],\n target_param=['c_0','c_1'],target_state=['E'])\n# perform optimization\nres = minimize(fun=objSCEI3R.costIV,\n jac=objSCEI3R.sensitivityIV,\n x0=theta+[0.00005],\n #bounds=boxBounds+[(0.0000001,0.001)],\n method='BFGS',\n options={'disp':True,'maxiter':1000,'eps':0.01,'gtol':0.01})\nprint(res)\n```\n\n\n```python\n# model with fitted parameters \nprint(params)\nprint(x0_SC3EI3R)\nparams_fit = params.copy()\n#params_fit['c_0'] = res.x[0]\n#params_fit['c_1'] = res.x[1]\n\nSC3EI3R_model.params = params_fit\nprint(SC3EI3R_model.params)\n#ode_fit = common_models.SEI3R({'beta':res.x[0], 'gamma':res.x[1],'alpha':res.x[2]})\n#x0_fit = [1-1.27e-6, 1.27e-6, 0]\nx0_fit = x0_SC3EI3R.copy()\n#x0_fit[2] = res.x[2]\n#t_fit = numpy.linspace(0, 150, 1000)\nt_fit = t\nSC3EI3R_model.initial_values = (x0_fit, t_fit[0])\n# %timeit sol_fit =SCEI3R_model.integrate(t_fit[1::]) # use magic %timeit to time\n# sol_fit =SCEI3R_model.integrate(t_fit[0::])\nsol_fit = scipy.integrate.odeint(SC3EI3R_model.ode, x0_fit, t_fit[1::])\n#\nplt.figure(figsize=(15,10))\nplt.plot(t,ynoise[:,0], 'o',color='lightgreen') # infected observations\nplt.plot(t,ynoise[:,1], 'o',color='green') # infected observations\nplt.plot(t,ynoise[:,2], 'o',color='darkgreen') # infected observations\nplt.plot(t,ynoise[:,3], 'bo') # recoverd\nplt.plot(t,ynoise[:,4], 'ro') # died\nplt.plot(t_fit[1::], sol_fit)\nplt.ylim([0,0.004])\n#plt.show(())\n#ode_fit.plot()\n\npeak_i = np.argmax(sol_fit[:,2])\nprint('Peak infection (days)', t_fit[peak_i])\n```\n\n#### Fit c_0 and c_1 as well as initial value of E using L-BFGS-B\nthis method does not work well\n\n\n```python\n# Initial guess of parameters and initial condition, and bounding constraints\ncautionparams = list(params.values())[-4:-2]\ntheta = [0.25,0.08]\nboxBounds = [(0.2,0.4),(0.05,0.15)]\nobjSCEI3R = SquareLoss(theta=theta, ode=SCEI3R_model, x0=x0_SC3EI3R, t0=t[0], t=t[1::], y=ynoise[1::,:],\n state_weight=[1.,1.,1.,1.,1.],state_name=['I_1','I_2','I_3','R','D'],\n target_param=['c_0','c_1'],target_state=['E'])\n# perform optimization\nres = minimize(fun=objSCEI3R.costIV,\n jac=objSCEI3R.sensitivityIV,\n x0=theta+[0.00005],\n bounds=boxBounds+[(0.0000001,0.001)],\n method='L-BFGS-B',\n options={'disp':True,'maxiter':1000,'eps':0.0001,'ftol':0.001})\nprint(res)\n```\n\n\n```python\nobjSCEI3R.residual()\n```\n\n#### Fit c_0 and c_1 as well as initial value of E with Nelder-Mead\nno use of Jacobian and no constraints\n\n\n```python\n# Initial guess of parameters and initial condition, and bounding constraints\ncautionparams = list(params.values())[-4:-2]\ntheta = [0.25,0.08]\nboxBounds = [(0.2,0.4),(0.05,0.15)]\nobjSCEI3R = SquareLoss(theta=theta, ode=SCEI3R_model, x0=x0_SC3EI3R, t0=t[0], t=t[1::], y=ynoise[1::,:],\n state_weight=[1.,1.,1.,1.,1.],state_name=['I_1','I_2','I_3','R','D'],\n target_param=['c_0','c_1'],target_state=['E'])\n# perform optimization\nres = minimize(fun=objSCEI3R.costIV,\n #jac=objSCEI3R.sensitivityIV,\n x0=theta+[0.00005],\n #bounds=boxBounds+[(0.0000001,0.001)],\n method='Nelder-Mead',\n options={'disp':True,'maxiter':1000}) #,'eps':0.0001,'ftol':0.01}) #not NM\nprint(res)\n```\n\n\n```python\n# model with fitted parameters \nprint(params)\nprint(x0_SC3EI3R)\nparams_fit = params.copy()\nparams_fit['c_0'] = res.x[0]\nparams_fit['c_1'] = res.x[1]\n\nSC3EI3R_model.params = params_fit\nprint(SC3EI3R_model.params)\nx0_fit = x0_SC3EI3R.copy()\n#x0_fit[2] = res.x[2]\n#t_fit = numpy.linspace(0, 150, 1000)\nt_fit = t\nSC3EI3R_model.initial_values = (x0_fit, t_fit[0])\n# %timeit sol_fit =SC3EI3R_model.integrate(t_fit[1::]) # use magic %timeit to time\n# sol_fit =SC3EI3R_model.integrate(t_fit[0::])\nsol_fit = scipy.integrate.odeint(SC3EI3R_model.ode, x0_fit, t_fit[1::])\n#\nplt.figure(figsize=(15,10))\nplt.plot(t,y_jhu[:,0], 'bo') # recoverd\nplt.plot(t,y_jhu[:,1], 'ro') # died\nplt.plot(t_fit[1::], sol_fit)\nplt.ylim([0,0.004])\n#plt.show(())\n#ode_fit.plot()\n\npeak_i = np.argmax(sol_fit[:,2])\nprint('Peak infection (days)', t_fit[peak_i])\n```\n\n### Information on method options\n\n\n```python\nscipy.optimize.show_options(solver='minimize', method='SLSQP', disp=True)\nprint(' ')\nscipy.optimize.show_options(solver='minimize', method='L-BFGS-B', disp=True)\n```\n\n## Plot using full control\n\n\n```python\ndef plotmodel(solns,t,scale='linear',species='no_susc',plottitle= '',label='',\n newplot = True,models=['SEI3R','SCEI3R','SC3EI3R']):\n \"\"\"\n plot solns over \n times t interpreted as models indicated in models parameter\n scale: alternative 'linear' or 'log'\n species alternatives 'all', 'confirmed', 'deaths', 'daily confirmed', 'daily deaths'\n plottitle : title for plot\n label : label for curve when called as part of multicurve plot\n newplot : whether to open new plot True/False\n models : list of models to include, default all three of those possible\n \"\"\"\n \n nmodels = len(models)\n if len(solns) != len(models):\n print(\"Error: number of models must match number of solutions\")\n return None\n nm = 0\n \n if newplot == True:\n plt.figure(figsize=(nmodels*8,6))\n \n for nm in range(nmodels):\n soln = solns[nm]\n if models[nm] == 'SEI3R': #SEI3R\n plt.subplot(1,nmodels,nm+1)\n if scale == 'log': #Plot on log scale\n plt.semilogy()\n plt.ylim([1,10000])\n elif species != 'daily confirmed': # Plot on normal linear scale\n #plt.ylim([0,10000])\n pass\n if species == 'no_susc':\n plt.plot(tvec,soln[:,1:5],label=label)\n plt.legend((\"E\",\"I1\",\"I2\",\"I3\"))\n elif species == 'confirmed' or species == 'daily confirmed':\n suma = np.sum(soln[:,2:7],axis=1)\n # print('length=',len(suma))\n if species == 'daily confirmed':\n sumd = np.zeros(len(suma))\n for i in range(1,len(suma)):\n sumd[i] = suma[i]-suma[i-1]\n #plt.ylim([0,1000])\n plt.plot(tvec,sumd,label=label)\n else:\n #plt.ylim([0,200000])\n plt.plot(t,suma,label=label) \n elif species == 'all':\n plt.plot(tvec,soln,label=label)\n plt.legend((\"S\",\"E\",\"I1\",\"I2\",\"I3\",\"R\",\"D\"))\n plt.xlabel(\"Time (days)\")\n plt.ylabel(\"Portion of population N\")\n plt.title('SEI3R %s' % plottitle)\n\n elif models[nm] == 'SCEI3R': #SCEI3R\n #Plot\n plt.subplot(1,nmodels,nm+1)\n if scale == 'log': #Plot on log scale\n plt.semilogy()\n plt.ylim([1,10000])\n elif species != 'daily confirmed': # Plot on normal linear scale\n #plt.ylim([0,10000])\n pass\n if species == 'no_susc':\n plt.plot(t,soln[:,1:5],label=label)\n plt.legend((\"E\",\"I1\",\"I2\",\"I3\"))\n elif species == 'confirmed' or species == 'daily confirmed':\n suma = np.sum(soln[:,2:7],axis=1)\n # print('length=',len(suma))\n if species == 'daily confirmed':\n sumd = np.zeros(len(suma))\n for i in range(1,len(suma)):\n sumd[i] = suma[i]-suma[i-1]\n #plt.ylim([0,1000])\n plt.plot(t,sumd,label=label)\n else:\n #plt.ylim([0,200000])\n plt.plot(t,suma,label=label)\n elif species == 'all':\n plt.plot(t,soln,label=label)\n plt.legend((\"S\",\"E\",\"I1\",\"I2\",\"I3\",\"R\",\"D\",\"Sc\"))\n plt.xlabel(\"Time (days)\")\n plt.ylabel(\"Portion of population N\")\n plt.title('SCEI3R %s' % plottitle)\n elif models[nm] == 'SC3EI3R': #SC3EI3R\n plt.subplot(1,nmodels,nm+1)\n if scale == 'log': #Plot on log scale\n plt.semilogy()\n plt.ylim([1,10000])\n elif species != 'daily confirmed': # Plot on normal linear scale\n #plt.ylim([0,10000])\n pass\n if species == 'no_susc':\n plt.plot(t,sol[:,1:5])\n plt.legend((\"E\",\"I1\",\"I2\",\"I3\"))\n elif species == 'confirmed' or species == 'daily confirmed':\n suma = np.sum(soln[:,2:7],axis=1) + soln[:,9]\n if species == 'daily confirmed':\n sumd = np.zeros(len(suma))\n for i in range(1,len(suma)):\n sumd[i] = suma[i]-suma[i-1]\n # plt.ylim([0,1000])\n plt.plot(t,sumd,label=label)\n else:\n # plt.ylim([0,200000])\n plt.plot(t,suma,label=label)\n elif species == 'all':\n plt.plot(t,soln,label=label)\n plt.legend((\"S\",\"E\",\"I1\",\"I2\",\"I3\",\"R\",\"D\",\"Sc\",\"Ec\",\"I1c\"))\n plt.xlabel(\"Time (days)\")\n plt.ylabel(\"Portion of population N\")\n plt.title('SC3EI3R %s' % plottitle)\n return True\n```\n\n\n```python\nplotmodel([sol_fit],t_fit[1:],scale='linear',species='no_susc',plottitle= 'test',label='',\n newplot = True,models=['SCEI3R'])\n```\n", "meta": {"hexsha": "3b9173c08ec3e0fcb7b6afe31030a300c9500622", "size": 1033012, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Notebooks/covid-19-caution/SCEIR_pygom_fitting.ipynb", "max_stars_repo_name": "ProtoLife/covid-recovery", "max_stars_repo_head_hexsha": "b3989f110c6961cc51da673fc33c6384cf7d6de6", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": 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YES\n2. YES", "lm_q1_score": 0.6261241632752915, "lm_q2_score": 0.6757645879592642, "lm_q1q2_score": 0.42311253720706643}} {"text": "**Table of contents**\n\n* [PCFG lib](#lib)\n * [Span](#span)\n* [CKY+](#cky+)\n * [Item](#item)\n * [Agenda](#agenda)\n * [Inference rules](#inference-rules)\n * [Deduction](#deduction)\n* [PCFG recap](#pcfg)\n* [Inside](#inside)\n * [Semirings](#semirings)\n \n \n \n**Table of Exercises**\n\n* Theory (9 points)\n * [Exercise 7-1](#ex7-1)\n * [Exercise 7-2](#ex7-2)\n * [Exercise 7-3](#ex7-3)\n * [Exercise 7-4](#ex7-4)\n * [Exercise 7-5](#ex7-5) \n* Practicals (26 points) \n * [Exercise 7-6](#ex7-6)\n * [Exercise 7-7](#ex7-7)\n * [Exercise 7-8](#ex7-8)\n* Bonus (see below for information about points) \n * Theory: [Exercise 7-9](#ex7-9)\n * Practical: [Exercise 7-10](#ex7-10) \n\n\n**General notes**\n\n* In this notebook you are expected to use $\\LaTeX$\n* Use python3.\n* Use NLTK to read annotated data.\n* **Document your code**: TAs are more likely to understand the steps if you document them. If you don't, it's also difficult to give you partial points for exercises that are not completely correct.\n* This document contains 2 optional exercises worth bonus points.\n\n# PCFG lib\n\nWe are going to use the basic objects defined in the last lab\n\n* Symbol, Terminal, and Nonterminal\n* Rule, and CFG\n\nCheck the file `pcfglib.py` where you will find these definitions.\n\n\n\n```python\nfrom pcfglib import Symbol, Terminal, Nonterminal, Rule, CFG\n```\n\n## Span\n\n\nFor convenience, we will define one more type of Symbol, this will be a Span. A Span is just a Nonterminal decorated with two integers which represent a half-open interval $(i, j]$, that is:\n\n* start (exclusive) of phrase\n* end (inclusive) of phrase\n\nIt is very easy to define such Span class by inheriting from Nonterminal. \n\n\n```python\nclass Span(Nonterminal):\n \n def __init__(self, nonterminal: Nonterminal, start: int, end: int):\n \"\"\"\n :param nonterminal: a Nonterminal category\n :param start: start position of the phrase (exclusive)\n :param end: end position of the phrase (inclusive)\n \"\"\"\n if not isinstance(nonterminal, Nonterminal):\n raise ValueError('Only a Nonterminal can make a span')\n super(Span, self).__init__('%s:%d-%d' % (nonterminal.category, start, end))\n self._base_nonterminal = nonterminal\n self._span = (start, end)\n \n @property\n def base_nonterminal(self) -> Nonterminal:\n \"\"\"Returns the base nonterminal: the Nonterminal without span information\"\"\"\n return self._base_nonterminal\n \n @property\n def start(self):\n \"\"\"Begin of the span (open)\"\"\"\n return self._span[0]\n \n @property\n def end(self):\n \"\"\"End of the span (closed)\"\"\"\n return self._span[1]\n \n @property\n def span(self):\n \"\"\"Returns _span\"\"\"\n return self._span\n```\n\nThe function definition below constructs our running example PCFG. Note that it returns both the CFG object and the cpds.\n\nAs in the previous lab a collection of cpds is stored in a dictionary such that ```cpds[lhs]``` is a dictionary mapping from rules that rewrite that LHS symbol to their probability values.\n\n\n```python\nfrom collections import defaultdict\n\n\ndef get_toy_pcfg():\n # Some symbols\n S = Nonterminal('S')\n \n NP = Nonterminal('NP') \n VP = Nonterminal('VP')\n PP = Nonterminal('PP')\n \n NN = Nonterminal('NN')\n Vt = Nonterminal('Vt') \n Vi = Nonterminal('Vi') \n DT = Nonterminal('DT') \n IN = Nonterminal('IN') \n CC = Nonterminal('CC')\n \n # Grammar\n G = CFG(S)\n cpds = defaultdict(lambda: defaultdict(float))\n \n # Phrasal rules\n G.add(Rule(S, [NP, VP]))\n cpds[S][Rule(S, [NP, VP])] = 1.0\n \n G.add(Rule(NP, [DT, NN])) \n G.add(Rule(NP, [NN]))\n G.add(Rule(NP, [NP, PP])) \n G.add(Rule(NP, [NP, CC, NP]))\n cpds[NP][Rule(NP, [DT, NN])] = 0.4\n cpds[NP][Rule(NP, [NN])] = 0.1\n cpds[NP][Rule(NP, [NP, PP])] = 0.3\n cpds[NP][Rule(NP, [NP, CC, NP])] = 0.2\n \n G.add(Rule(VP, [Vt, NP])) \n G.add(Rule(VP, [VP, PP])) \n G.add(Rule(VP, [Vi])) \n G.add(Rule(VP, [VP, CC, VP]))\n cpds[VP][Rule(VP, [Vt, NP])] = 0.3\n cpds[VP][Rule(VP, [VP, PP])] = 0.4\n cpds[VP][Rule(VP, [Vi])] = 0.2\n cpds[VP][Rule(VP, [VP, CC, VP])] = 0.1\n \n G.add(Rule(PP, [IN, NP])) \n cpds[PP][Rule(PP, [IN, NP])] = 1.\n \n # Preterminal rules\n G.add(Rule(NN, [Terminal('dog')])) \n G.add(Rule(NN, [Terminal('cat')]))\n G.add(Rule(NN, [Terminal('man')]))\n G.add(Rule(NN, [Terminal('telescope')]))\n \n cpds[NN][Rule(NN, [Terminal('dog')])] = 0.3\n cpds[NN][Rule(NN, [Terminal('cat')])] = 0.2\n cpds[NN][Rule(NN, [Terminal('man')])] = 0.4\n cpds[NN][Rule(NN, [Terminal('telescope')])] = 0.1\n \n G.add(Rule(DT, [Terminal('the')]))\n G.add(Rule(DT, [Terminal('a')]))\n cpds[DT][Rule(DT, [Terminal('the')])] = 0.6\n cpds[DT][Rule(DT, [Terminal('a')])] = 0.4\n \n G.add(Rule(CC, [Terminal('and')]))\n G.add(Rule(CC, [Terminal(',')]))\n cpds[CC][Rule(CC, [Terminal('and')])] = 0.8\n cpds[CC][Rule(CC, [Terminal(',')])] = 0.2\n \n G.add(Rule(IN, [Terminal('with')]))\n G.add(Rule(IN, [Terminal('through')]))\n G.add(Rule(IN, [Terminal('within')]))\n cpds[IN][Rule(IN, [Terminal('with')])] = 0.5\n cpds[IN][Rule(IN, [Terminal('through')])] = 0.3\n cpds[IN][Rule(IN, [Terminal('within')])] = 0.2\n\n G.add(Rule(Vt, [Terminal('saw')]))\n G.add(Rule(Vt, [Terminal('barked')]))\n G.add(Rule(Vt, [Terminal('meowed')]))\n G.add(Rule(Vt, [Terminal('moved')]))\n cpds[Vt][Rule(Vt, [Terminal('saw')])] = 0.4\n cpds[Vt][Rule(Vt, [Terminal('barked')])] = 0.3\n cpds[Vt][Rule(Vt, [Terminal('meowed')])] = 0.2\n cpds[Vt][Rule(Vt, [Terminal('moved')])] = 0.1\n \n G.add(Rule(Vi, [Terminal('barked')]))\n G.add(Rule(Vi, [Terminal('ran')]))\n G.add(Rule(Vi, [Terminal('meowed')]))\n cpds[Vi][Rule(Vi, [Terminal('barked')])] = 0.2\n cpds[Vi][Rule(Vi, [Terminal('ran')])] = 0.7\n cpds[Vi][Rule(Vi, [Terminal('meowed')])] = 0.1\n \n return G, cpds\n```\n\nLet's inspect our grammar\n\n\n```python\nG, cpds = get_toy_pcfg()\nprint(G)\n```\n\n [S] -> [NP] [VP]\n [NP] -> [DT] [NN]\n [NP] -> [NN]\n [NP] -> [NP] [PP]\n [NP] -> [NP] [CC] [NP]\n [PP] -> [IN] [NP]\n [VP] -> [Vt] [NP]\n [VP] -> [VP] [PP]\n [VP] -> [Vi]\n [VP] -> [VP] [CC] [VP]\n [CC] -> 'and'\n [CC] -> ','\n [DT] -> 'the'\n [DT] -> 'a'\n [IN] -> 'with'\n [IN] -> 'through'\n [IN] -> 'within'\n [NN] -> 'dog'\n [NN] -> 'cat'\n [NN] -> 'man'\n [NN] -> 'telescope'\n [Vi] -> 'barked'\n [Vi] -> 'ran'\n [Vi] -> 'meowed'\n [Vt] -> 'saw'\n [Vt] -> 'barked'\n [Vt] -> 'meowed'\n [Vt] -> 'moved'\n\n\nas well as our cpds\n\n\n```python\nfor lhs, cpd in cpds.items():\n for rule, prob in cpd.items():\n print(prob, rule)\n```\n\n 0.4 [VP] -> [VP] [PP]\n 0.2 [VP] -> [Vi]\n 0.1 [VP] -> [VP] [CC] [VP]\n 0.3 [VP] -> [Vt] [NP]\n 0.3 [NN] -> 'dog'\n 0.1 [NN] -> 'telescope'\n 0.2 [NN] -> 'cat'\n 0.4 [NN] -> 'man'\n 0.7 [Vi] -> 'ran'\n 0.1 [Vi] -> 'meowed'\n 0.2 [Vi] -> 'barked'\n 1.0 [PP] -> [IN] [NP]\n 1.0 [S] -> [NP] [VP]\n 0.2 [Vt] -> 'meowed'\n 0.4 [Vt] -> 'saw'\n 0.1 [Vt] -> 'moved'\n 0.3 [Vt] -> 'barked'\n 0.4 [NP] -> [DT] [NN]\n 0.3 [NP] -> [NP] [PP]\n 0.2 [NP] -> [NP] [CC] [NP]\n 0.1 [NP] -> [NN]\n 0.2 [IN] -> 'within'\n 0.5 [IN] -> 'with'\n 0.3 [IN] -> 'through'\n 0.6 [DT] -> 'the'\n 0.4 [DT] -> 'a'\n 0.8 [CC] -> 'and'\n 0.2 [CC] -> ','\n\n\n# CKY+ \n\n\nIn this section we will implement a generalised CKY algorithm which can deal with an arbitrary epsilon-free CFG.\n\nWe will implement the parsing strategy **for you** to guarantee that it is correct. The focus of this lab is on the **inside recursion**. An extra will involve implementing a different parsing strategy, for that some of the data structures we will develop here are indeed very useful, thus take this as a learning opportunity and try to reuse some code if you decide to implement the extra.\n\nThere will be nonetheless questions throught this lab, so stay tuned.\n\n\nAgain we will use a deductive system to describe the parsing strategy:\n\n\\begin{align}\n\\text{Item} &\\qquad [i, X \\rightarrow \\alpha_\\blacksquare \\, \\bullet \\, \\beta_\\square, j] \\\\\n\\text{Goal} &\\qquad [1, S \\rightarrow \\beta_\\blacksquare \\, \\bullet, n] \\\\\n\\text{Axioms} &\\qquad [i, X \\rightarrow \\bullet \\alpha_\\square, i] &~\\text{ for all } X \\rightarrow \\alpha \\in \\mathcal R \\\\\n\\text{Scan} &\\qquad \\frac{[i, X \\rightarrow \\alpha_\\blacksquare \\, \\bullet \\, x_{j+1} \\, \\beta_\\square, j]}{[i, X \\rightarrow \\alpha_\\blacksquare \\, x_{j+1} \\bullet \\, \\beta_\\square, j + 1]} \\\\\n\\text{Complete} &\\qquad \\frac{[i, X \\rightarrow \\alpha_\\blacksquare \\, \\bullet \\, Y \\, \\beta_\\square ,k] [k, Y \\rightarrow \\gamma_\\blacksquare \\, \\bullet , j]}{[i, X \\rightarrow \\alpha_\\blacksquare \\, Y_{k,j} \\, \\bullet \\, \\beta_\\square , j]}\n\\end{align}\n\n\n\n\n**Exercise 7-1** **[1 point]** Explain the meaning of an item (make sure to discuss all elements in it).\n\n- An item is a representation of a segment of sentence $x_1^n = \\{x_1,...,x_n\\}$ which spans from $i$ to $j$\n- $X \\rightarrow \\alpha \\, \\beta \\in \\mathcal R$ corresponds to a rule\n - But because its general CNF $\\alpha$ and $\\beta$ don't correspond to one single (Non)Terminal but to subset of the RHS\n - $\\alpha$ corresponds to the part of the RHS that has been scanned\n - $\\beta$ corresponds to the part of the RHS that hasn't been scanned\n- $\\blacksquare$ represents the spans of all the elements of $\\alpha$ and is moved when the complete rule is used\n- $\\bullet$ represents the position of the \"word scanner\" which checks whether a preterminal rule can be used that matches the next word in the sentence\n\n**Exercise 7-2** **[1 point]** Explain the goal of the program\n\n\n`Typo in Goal item: should i should be 0 not 1`\n- The goal of the program is to have scanned all words in $x_1^n$ ($\\bullet$ to the right) and know for each symbol in $\\beta$ what its span is ($\\blacksquare$ to the right).\n- Goal item spans (0,n]\n\n**Exercise 7-3** **[1 point]** Explain the axioms of the program\n\n\n`Typo in Axiom item: should be S not X a nd i and j 0`\n- The axiom is the start point of where to start.\n\n\n**Exercise 7-4** **[1 point]** Explain SCAN (make sure to discuss all elements of the rule)\n\n\n\n\n\n**Exercise 7-5** **[1 point]** Explain the COMPLETE rule including all of its elements including the side condition.\n\n\n```python\n\n```\n\nThe actual **deduction** is nothing but an exhaustive enumeration of valid items.\n* we start from axioms\n* and proceed by either scanning or completing previously derived items\n* each such operation creates additional items\n* if these items were not yet discovered, they make it to what we call an **agenda**\n* the agenda is much like a queue of items yet to be processed\n* processing an item means simply giving it the chance to participate in scan and complete\n* we should be careful to never process an item twice under the same premises \n* items that are yet to be processed are called **active items**\n* items already processed are called **passive items**\n* at the end there should be no active item and many passive items\n* parsing is possible if we derive/prove/reach the goal item\n* the complete items in the passive set can be used to derive a **parse forest**\n* a parse forest is much like a CFG but its rules have symbols which are decorated with spans indicating how they parse the input sentence\n* we can use parse forests to answer questions such as: what trees can parse the sentence? And when we introduce PCFGs, we will be able to answer quetions such as: what's the best tree that parses the sentence? what's the total probability value of the sentence (once we marginalise all possible parse trees).\n\nNow we turn to implementation, which will require a few classes and data structures, but we will discuss them one by one.\n\n## Item\n\nWe have to start by turning items into code!\n\nWe are using dotted rules to represent items in CKY+. A dotted rule is basically a container for \n* a context-free production\n* a list of positions already covered in the input sentence\n * together this represents the start and end position as well as the black squares in the item\n \n \nThis is an item formally\n\n\\begin{align}\n\\qquad [i, X \\rightarrow \\alpha_\\blacksquare \\, \\bullet \\, \\beta_\\square, j]\n\\end{align}\n \nand this is how we realise it in our implementation\n\n [LHS -> RHS, [i...j]]\n\nthe first element of the pair is the rule `LHS -> RHS` and the second is a list of positions where the dot has been.\n\n\n\n```python\nclass Item: \n \"\"\"\n A dotted rule used in CKY\n \n We store a rule and a list of positions (which we call `dots` because they indicate \n positions where the dots have been)\n \n We make an Item a hashable object so that we can use it in dictionaries.\n \n \"\"\"\n \n def __init__(self, rule: Rule, dots: list):\n if len(dots) == 0:\n raise ValueError('I do not accept an empty list of dots')\n self._rule = rule\n self._dots = tuple(dots)\n \n def __eq__(self, other: 'Item'):\n \"\"\"Two items are identical if they contain the same rule and cover the same positions\"\"\"\n return self._rule == other._rule and self._dots == other._dots\n \n def __hash__(self):\n \"\"\"We let python hash the two objects that represent an Item\"\"\"\n return hash((self._rule, self._dots))\n \n def __str__(self):\n return '[{0}, {1}]'.format(self._rule, self._dots)\n \n def __repr__(self):\n return str(self)\n \n @property\n def lhs(self):\n return self._rule.lhs\n \n @property\n def rule(self):\n return self._rule\n \n @property\n def dot(self):\n return self._dots[-1]\n \n @property\n def start(self):\n return self._dots[0]\n \n @property\n def next(self):\n \"\"\"return the symbol to the right of the dot (or None, if the item is complete)\"\"\"\n if self.is_complete():\n return None\n return self._rule.rhs[len(self._dots) - 1]\n \n def state(self, i):\n return self._dots[i]\n\n def advance(self, dot):\n \"\"\"return a new item with an extended sequence of dots\"\"\"\n return Item(self._rule, self._dots + (dot,))\n \n def is_complete(self):\n \"\"\"complete items are those whose dot reached the end of the RHS sequence\"\"\"\n return len(self._rule.rhs) + 1 == len(self._dots) \n \n```\n\nLet's play a bit with item objects to see how they work\n\n\n```python\nr = Rule(Nonterminal('S'), [Nonterminal('X')])\ni1 = Item(r, [0])\ni2 = i1.advance(1)\n```\n\n\n```python\nprint(i1)\nprint(i2)\n```\n\n [[S] -> [X], (0,)]\n [[S] -> [X], (0, 1)]\n\n\n\n```python\ni1 != i2\n```\n\n\n\n\n True\n\n\n\n\n```python\ni1.is_complete()\n\n```\n\n\n\n\n False\n\n\n\n\n```python\ni2.is_complete()\n```\n\n\n\n\n True\n\n\n\n\n```python\ni1.next\n```\n\n\n\n\n [X]\n\n\n\n\n```python\ni2.next\n```\n\n## Agenda\n\nNext we need an agenda of items. In CKY+ we have to track quite a bit of information, so we will design a more complex agenda. Because there will be a lot of functionality, we will use a class. \n\nIn an agenda, some items are active, others are passive.\n\nFunctionally, the active agenda is nothing but a stack or queue, whereas the passive agenda is simply a set (all items that have already been processed). However, to make our inferences run faster, we can further organise the passive items for easy/quick access within inference rules.\n\n\n```python\nfrom collections import deque, defaultdict\n\nclass Agenda:\n \"\"\"\n An Agenda for CKY+.\n \n The agenda will organise a queue of active items as well as a set of passive items.\n \n This agenda is such that it does not push an item twice into the queue \n that is equivalent to saying that the agenda is capable of maintaining a set of already discovered items.\n \n This agenda will also organise the passive items for quick access in the COMPLETE rule. \n This means we will store complete and incomplete items separately and hash them by some useful criterion.\n A complete item essentially contributes to further advancing incomplete items. \n Incomplete items need to be further completed.\n \"\"\"\n \n def __init__(self):\n # we are organising active items in a stack (last in first out)\n self._active = deque([])\n # an item should never queue twice, thus we will manage a set of items which we have already seen\n self._discovered = set()\n\n # Passive items may be complete\n # in which case they help us complete other items\n # and they may be incomplete\n # in which case we will be trying to complete them\n # In order to make COMPLETE inferences easier, we will separate passive items into these two groups\n # and we will also organise each group conveniently.\n \n # We organise incomplete items by the symbols they wait for at a certain position\n # that is, if the key is a pair (Y, i)\n # the value is a set of items of the form\n # [X -> alpha * Y beta, [...i]]\n # in other words \"items waiting for a Y to project a span from i\"\n self._incomplete = defaultdict(set) \n \n # We organise complete items by their LHS symbol spanning from a certain position\n # if the key is a pair (X, i)\n # then the value is a set of items of the form\n # [X -> gamma *, [i ... j]]\n self._complete = defaultdict(set)\n \n def __len__(self):\n \"\"\"return the number of active items\"\"\"\n return len(self._active)\n \n def push(self, item: Item):\n \"\"\"push an item into the queue of active items\"\"\"\n if item not in self._discovered: # if an item has been seen before, we simply ignore it\n self._active.append(item)\n self._discovered.add(item)\n return True\n return False\n \n def pop(self):\n \"\"\"pop an active item\"\"\"\n if len(self._active) == 0:\n raise ValueError('I have no items left')\n return self._active.pop()\n \n def make_passive(self, item: Item):\n if item.is_complete(): # complete items offer a way to rewrite a certain LHS from a certain position\n self._complete[(item.lhs, item.start)].add(item)\n else: # incomplete items are waiting for the completion of the symbol to the right of the dot\n self._incomplete[(item.next, item.dot)].add(item)\n \n def waiting(self, symbol: Symbol, dot: int):\n return self._incomplete.get((symbol, dot), set())\n \n def complete(self, lhs: Nonterminal, start: int):\n return self._complete.get((lhs, start), set())\n \n def itercomplete(self):\n \"\"\"an iterator over complete items in arbitrary order\"\"\"\n for items in self._complete.values():\n for item in items:\n yield item\n \n```\n\nLet's see how this works\n\n\n```python\nA = Agenda()\n```\n\n\n```python\nr1 = Rule(Nonterminal('S'), [Nonterminal('S'), Nonterminal('X')])\nr1\n```\n\n\n\n\n [S] -> [S] [X]\n\n\n\nwe can push items into the agenda\n\n\n```python\nA.push(Item(r1, [0])) # S -> S X, [0] (earley axiom)\n```\n\n\n\n\n True\n\n\n\nand the agenda will make sure there are no duplicates\n\n\n```python\nA.push(Item(r1, [0]))\n```\n\n\n\n\n False\n\n\n\n\n```python\nlen(A)\n```\n\n\n\n\n 1\n\n\n\n\n```python\ni1 = Item(r1, [0])\ni1\n```\n\n\n\n\n [[S] -> [S] [X], (0,)]\n\n\n\n\n```python\nA.make_passive(i1)\n```\n\n\n```python\nA.push(Item(Rule(Nonterminal('S'), [Nonterminal('X')]), [0]))\n```\n\n\n\n\n True\n\n\n\n\n```python\nA.make_passive(Item(Rule(Nonterminal('S'), [Nonterminal('X')]), [0]))\n```\n\n\n```python\nA.push(Item(Rule(Nonterminal('S'), [Nonterminal('X')]), [0, 1]))\n```\n\n\n\n\n True\n\n\n\n\n```python\nA.make_passive(Item(Rule(Nonterminal('S'), [Nonterminal('X')]), [0, 1]))\n```\n\n\n```python\nlist(A.itercomplete())\n```\n\n\n\n\n [[[S] -> [X], (0, 1)]]\n\n\n\n## Inference rules\n\n### Basic axioms\n\nFor every rule X -> alpha, and every input position (i) between 0 and n-1, we have an item of the kind:\n\n\\begin{equation}\n[i, X \\rightarrow \\bullet \\alpha_\\square, i]\n\\end{equation}\n\nIn our implementation an axiom looks like this\n\n [X -> alpha, [i]]\n\n\n```python\ndef axioms(cfg: CFG, sentence: list):\n \"\"\"\n :params cfg: a context-free grammar (an instance of WCFG)\n :params sentence: the input sentence (as a list or tuple)\n :returns: a list of items\n \"\"\"\n items = []\n for rule in cfg:\n for i, x in enumerate(sentence): \n # We will implement a tiny optimisation here\n \n # For rules that start with terminals we can use \"look ahead\"\n if isinstance(rule.rhs[0], Terminal): \n # this is a mechanism by which we avoid constructing items which we know cannot be scanned\n # that's the terminal that starts the rule does not occur in the sentence we are parsing\n if rule.rhs[0] == x:\n items.append(Item(rule, [i]))\n else:\n items.append(Item(rule, [i]))\n return items\n```\n\nLet's have a look what type of axioms we get, note that CKY+ is very greedy. Earley parsing is an alternative strategy that's far more conservative than CKY+, for example, Earley avoids instantiating items that are not yet required and instead uses a simpler axiom (you will seee it later).\n\n\n```python\nsentence = [Terminal(w) for w in 'the man saw the dog with a telescope'.split()]\naxioms(G, sentence)\n```\n\n\n\n\n [[[Vt] -> 'saw', (2,)],\n [[NP] -> [NP] [CC] [NP], (0,)],\n [[NP] -> [NP] [CC] [NP], (1,)],\n [[NP] -> [NP] [CC] [NP], (2,)],\n [[NP] -> [NP] [CC] [NP], (3,)],\n [[NP] -> [NP] [CC] [NP], (4,)],\n [[NP] -> [NP] [CC] [NP], (5,)],\n [[NP] -> [NP] [CC] [NP], (6,)],\n [[NP] -> [NP] [CC] [NP], (7,)],\n [[DT] -> 'the', (0,)],\n [[DT] -> 'the', (3,)],\n [[PP] -> [IN] [NP], (0,)],\n [[PP] -> [IN] [NP], (1,)],\n [[PP] -> [IN] [NP], (2,)],\n [[PP] -> [IN] [NP], (3,)],\n [[PP] -> [IN] [NP], (4,)],\n [[PP] -> [IN] [NP], (5,)],\n [[PP] -> [IN] [NP], (6,)],\n [[PP] -> [IN] [NP], (7,)],\n [[VP] -> [VP] [PP], (0,)],\n [[VP] -> [VP] [PP], (1,)],\n [[VP] -> [VP] [PP], (2,)],\n [[VP] -> [VP] [PP], (3,)],\n [[VP] -> [VP] [PP], (4,)],\n [[VP] -> [VP] [PP], (5,)],\n [[VP] -> [VP] [PP], (6,)],\n [[VP] -> [VP] [PP], (7,)],\n [[NP] -> [NN], (0,)],\n [[NP] -> [NN], (1,)],\n [[NP] -> [NN], (2,)],\n [[NP] -> [NN], (3,)],\n [[NP] -> [NN], (4,)],\n [[NP] -> [NN], (5,)],\n [[NP] -> [NN], (6,)],\n [[NP] -> [NN], (7,)],\n [[NN] -> 'telescope', (7,)],\n [[DT] -> 'a', (6,)],\n [[NP] -> [NP] [PP], (0,)],\n [[NP] -> [NP] [PP], (1,)],\n [[NP] -> [NP] [PP], (2,)],\n [[NP] -> [NP] [PP], (3,)],\n [[NP] -> [NP] [PP], (4,)],\n [[NP] -> [NP] [PP], (5,)],\n [[NP] -> [NP] [PP], (6,)],\n [[NP] -> [NP] [PP], (7,)],\n [[VP] -> [VP] [CC] [VP], (0,)],\n [[VP] -> [VP] [CC] [VP], (1,)],\n [[VP] -> [VP] [CC] [VP], (2,)],\n [[VP] -> [VP] [CC] [VP], (3,)],\n [[VP] -> [VP] [CC] [VP], (4,)],\n [[VP] -> [VP] [CC] [VP], (5,)],\n [[VP] -> [VP] [CC] [VP], (6,)],\n [[VP] -> [VP] [CC] [VP], (7,)],\n [[VP] -> [Vi], (0,)],\n [[VP] -> [Vi], (1,)],\n [[VP] -> [Vi], (2,)],\n [[VP] -> [Vi], (3,)],\n [[VP] -> [Vi], (4,)],\n [[VP] -> [Vi], (5,)],\n [[VP] -> [Vi], (6,)],\n [[VP] -> [Vi], (7,)],\n [[VP] -> [Vt] [NP], (0,)],\n [[VP] -> [Vt] [NP], (1,)],\n [[VP] -> [Vt] [NP], (2,)],\n [[VP] -> [Vt] [NP], (3,)],\n [[VP] -> [Vt] [NP], (4,)],\n [[VP] -> [Vt] [NP], (5,)],\n [[VP] -> [Vt] [NP], (6,)],\n [[VP] -> [Vt] [NP], (7,)],\n [[NN] -> 'dog', (4,)],\n [[NP] -> [DT] [NN], (0,)],\n [[NP] -> [DT] [NN], (1,)],\n [[NP] -> [DT] [NN], (2,)],\n [[NP] -> [DT] [NN], (3,)],\n [[NP] -> [DT] [NN], (4,)],\n [[NP] -> [DT] [NN], (5,)],\n [[NP] -> [DT] [NN], (6,)],\n [[NP] -> [DT] [NN], (7,)],\n [[IN] -> 'with', (5,)],\n [[S] -> [NP] [VP], (0,)],\n [[S] -> [NP] [VP], (1,)],\n [[S] -> [NP] [VP], (2,)],\n [[S] -> [NP] [VP], (3,)],\n [[S] -> [NP] [VP], (4,)],\n [[S] -> [NP] [VP], (5,)],\n [[S] -> [NP] [VP], (6,)],\n [[S] -> [NP] [VP], (7,)],\n [[NN] -> 'man', (1,)]]\n\n\n\n## Scan\n\nIf the dot is placed at a position just before a *terminal*, we can **scan** it provided that the terminal matches the corresponding input position.\n\n\n\\begin{equation}\n \\frac{[i, A \\rightarrow \\alpha_\\blacksquare \\, \\bullet \\, x_{j+1} \\, \\beta_\\square, j]}{[i, A \\rightarrow \\alpha_\\blacksquare \\, x_{j+1} \\bullet \\, \\beta_\\square, j + 1]}\n\\end{equation}\n\nIn our implementation with dot lists it looks like this\n\n [X -> alpha * x beta, [i ... j]] \n --------------------------------------------\n [X -> alpha x * beta, [i ... j] + [j + 1]]\n \n\nnote that the `*` is simply indicating where the last dot would be.\n\n\n```python\ndef scan(item: Item, sentence):\n if isinstance(item.next, Terminal):\n if item.dot < len(sentence) and sentence[item.dot] == item.next:\n return item.advance(item.dot + 1)\n else:\n return None\n```\n\n\n```python\nscanned = []\nfor item in axioms(G, sentence):\n new = scan(item, sentence)\n if new is not None:\n scanned.append(new)\nscanned\n```\n\n\n\n\n [[[Vt] -> 'saw', (2, 3)],\n [[DT] -> 'the', (0, 1)],\n [[DT] -> 'the', (3, 4)],\n [[NN] -> 'telescope', (7, 8)],\n [[DT] -> 'a', (6, 7)],\n [[NN] -> 'dog', (4, 5)],\n [[IN] -> 'with', (5, 6)],\n [[NN] -> 'man', (1, 2)]]\n\n\n\n## Complete\n\nHere we let an active item interact with passive items:\n\n* either an active item is complete, then we try to advance incomplete passive items\n\n* or an active item is incomplete, in which case we try to advance the item itself by looking back to complete passive items\n\nBoth cases are covered by the inference rule\n\n\\begin{align}\n\\qquad \\frac{[i, X \\rightarrow \\alpha_\\blacksquare \\, \\bullet \\, Y \\, \\beta_\\square ,k] [k, Y \\rightarrow \\gamma_\\blacksquare \\, \\bullet , j]}{[i, X \\rightarrow \\alpha_\\blacksquare \\, Y_{k,j} \\, \\bullet \\, \\beta_\\square , j]}\n\\end{align}\n\nIn our implementation with dot lists it looks like this\n\n [X -> alpha * Y beta, [i ... k]] [Y -> gamma *, [k ... j]]\n ----------------------------------------------------------\n [X -> alpha Y * beta, [i ... k] + [j]]\n\n\n\n```python\ndef complete(item: Item, agenda: Agenda):\n items = []\n # This has two cases\n # either the input item corresponds to the second antecedent in the COMPLETE inference rule\n # in which case the item is complete (the dot stands at the end)\n # or the input item corresponds to the first antecedent in the COMPLETE inference rule\n # in which case the item is itself incomplete\n \n # When it is *complete* we use it to advance incomplete ones.\n # When it is *incomplete* we check if we know a complete item that can advance it.\n \n # First we deal with the complete case\n if item.is_complete(): # If the item is complete, it can be used to advance incomplete items\n # We then look for incomplete items that are waiting for \n # the LHS nonterminal of our complete item \n # in particular, if it matches the start position of our complete item\n for incomplete in agenda.waiting(item.lhs, item.start):\n items.append(incomplete.advance(item.dot))\n else: # Then we deal with the incomplete case\n # look for completions of item.next spanning from item.dot\n ends = set()\n for complete in agenda.complete(item.next, item.dot):\n ends.add(complete.dot)\n # advance the dot of the input item for each position that completes a span\n for end in ends:\n items.append(item.advance(end))\n return items\n```\n\n## Forest from complete items\n\nEach **complete** item in the (passive) agenda can be mapped to a new CFG rule (with nonterminal symbols annotated with spans).\nFor example, an item such as\n\n [X -> A x B *, [0,1,2,3]]\n \nresults in the rule\n\n X:0-3 -> A:0-1 x B:2-3\n \nobserve how only nonterminal nodes get annotated: this helps us keep terminals and nonterminals clearly separate.\n\n\n```python\ndef make_span(sym: Symbol, start: int, end: int):\n \"\"\"\n Helper function that returns a Span for a certain symbol.\n \n This function will only make spans out of nonterminals, terminals are return as is.\n :param sym: Terminal or Nonterminal symbol\n :param start: open begin\n :param end: closed end\n :returns: Span(sym, start, end) or sym (if Terminal)\n \"\"\"\n if isinstance(sym, Nonterminal):\n return Span(sym, start, end)\n else:\n return sym\n```\n\nMaking a forest is indeed really simple, we just need to return a new CFG with rules derived from complete items in the passive set. The rules will have their nonterminals annotated into spans.\n\n\n```python\ndef make_forest(complete_items: list, forest_start: Nonterminal):\n \"\"\"\n Converts complete items from CKY+ into a forest, that is, a CFG whose rules have spans for nonterminals.\n \n :param complete_items: a collection of dotted items which are complete (dot at the end of the RHS)\n :param forest_start: the start nonterminal (a Span) of the forest\n \"\"\"\n if not isinstance(forest_start, Span):\n raise ValueError('The start symbol of a forest should be a span')\n forest = CFG(forest_start)\n for item in complete_items: \n lhs = make_span(item.lhs, item.start, item.dot)\n rhs = []\n for i, sym in enumerate(item.rule.rhs):\n if isinstance(sym, Terminal):\n rhs.append(sym)\n else:\n rhs.append(make_span(sym, item.state(i), item.state(i + 1)))\n forest.add(Rule(lhs, rhs)) \n return forest\n```\n\n## Deduction\n\nStart with axioms and exhaustively apply inference rules\n\n\n```python\ndef cky(cfg: CFG, sentence):\n A = Agenda()\n for item in axioms(cfg, sentence):\n A.push(item)\n while A:\n item = A.pop()\n # a complete item can be used to complete other items\n # alternatively, we may be able to advance an incomplete item\n # whose next symbol is a nonterminal by combining it with some passive complete item \n if item.is_complete() or isinstance(item.next, Nonterminal):\n for new in complete(item, A):\n A.push(new)\n else: # here we have a terminal ahead of the dot, thus only scan is possible\n new = scan(item, sentence)\n if new is not None: # if we managed to scan \n A.push(new)\n A.make_passive(item)\n forest_start = make_span(cfg.start, 0, len(sentence))\n forest = make_forest(A.itercomplete(), forest_start) \n if forest.can_rewrite(forest_start):\n return forest\n else:\n return CFG(forest_start)\n```\n\n\n```python\nforest = cky(G, sentence)\n```\n\n\n```python\nforest.start\n```\n\n\n\n\n [S:0-8]\n\n\n\n\n```python\nforest.can_rewrite(forest.start)\n```\n\n\n\n\n True\n\n\n\n\n```python\nprint(forest)\n```\n\n [S:0-8] -> [NP:0-2] [VP:2-8]\n [NP:0-2] -> [DT:0-1] [NN:1-2]\n [NP:1-2] -> [NN:1-2]\n [NP:3-5] -> [DT:3-4] [NN:4-5]\n [NP:3-8] -> [NP:3-5] [PP:5-8]\n [NP:4-5] -> [NN:4-5]\n [NP:4-8] -> [NP:4-5] [PP:5-8]\n [NP:6-8] -> [DT:6-7] [NN:7-8]\n [NP:7-8] -> [NN:7-8]\n [PP:5-8] -> [IN:5-6] [NP:6-8]\n [S:0-5] -> [NP:0-2] [VP:2-5]\n [S:1-5] -> [NP:1-2] [VP:2-5]\n [S:1-8] -> [NP:1-2] [VP:2-8]\n [VP:2-5] -> [Vt:2-3] [NP:3-5]\n [VP:2-8] -> [VP:2-5] [PP:5-8]\n [VP:2-8] -> [Vt:2-3] [NP:3-8]\n [DT:0-1] -> 'the'\n [DT:3-4] -> 'the'\n [DT:6-7] -> 'a'\n [IN:5-6] -> 'with'\n [NN:1-2] -> 'man'\n [NN:4-5] -> 'dog'\n [NN:7-8] -> 'telescope'\n [Vt:2-3] -> 'saw'\n\n\nNote that if we modify the sentence in a way that it can't be parsed by G we will get an empty forest\n\n\n```python\nempty_forest = cky(G, sentence + [Terminal('!')])\n```\n\n\n```python\nempty_forest.start\n```\n\n\n\n\n [S:0-9]\n\n\n\n\n```python\nempty_forest.can_rewrite(empty_forest.start)\n```\n\n\n\n\n False\n\n\n\n# PCFG recap\n\nA probabilistic CFG is a simple extension to CFGs where we assign a joint probability distribution over the space of context-free *derivations*. \n\nA random **derivation** $D = \\langle R_1, \\ldots, R_m \\rangle$ is a sequence of $m$ *random rule applications*.\nA random rule is a pair of a random LHS nonterminal $V$ and a random RHS sequence $\\beta$, where $V \\rightarrow \\beta$ corresponds to a valid rule in the grammar.\n\nWe assume that a derivation is generated one rule at a time and each rule is generated independently. Moreover, the probability value of a rule is given by a conditional probability distribution over RHS sequences given LHS nonterminal. \n\n\n\\begin{align}\nP_{D|M}(r_1^m|m) &= \\prod_{i=1}^m P_R(r_i) \\\\\n &= \\prod_{i=1}^m P_{\\text{RHS}|\\text{LHS}}(\\beta_i | v_i)\\\\\n &= \\prod_{i=1}^m \\text{Cat}(\\beta_i | \\boldsymbol \\theta^{v_i})\\\\\n &= \\prod_{i=1}^m \\theta_{v_i \\rightarrow \\beta_i}\\\\\n\\end{align}\n\nWe can implement PCFGs rather easily by pairing a CFG grammar with a dictionary mapping from rules to their probabilities. But we must remember that for each given LHS symbol, the probability values of all of its rewriting rules must sum to 1.\n\n\\begin{equation}\n\\sum_{\\beta} \\theta_{v \\rightarrow \\beta} = 1\n\\end{equation}\n\n\n# Inside algorithm\n\n\nThis is the core of this lab, the inside recursion. The inside recursion (also known as **value recursion**) is incredibly general, it can be used to compute a range of interesting quantities.\n\nThe formula below corresponds to the recursion:\n\n\\begin{align}\n(1)\\qquad I(v) &= \n \\begin{cases}\n \\bar{1} & \\text{if }v \\text{ is terminal and } \\text{BS}(v) = \\emptyset\\\\\n \\bar{0} & \\text{if }v \\text{ is nonterminal and } \\text{BS}(v) = \\emptyset \\\\\n \\displaystyle\\bigoplus_{\\frac{a_1 \\ldots a_n}{v: \\theta} \\in \\text{BS}(v)} \\theta \\otimes \\bigotimes_{i=1}^n I(a_i) & \\text{otherwise}\n \\end{cases}\n\\end{align}\n\nIn this formula $\\text{BS}(v)$ is the *backward-star* of the node, or the set of edges **incoming** to the node. That is, all edges (rules with spans) that have that node as an LHS symbol. There is one detail important to remember. In principle only *terminal* nodes would have an empty backward-star. But because our parsing strategy can produce some dead-end nodes (nodes that cannot be expanded) we will have some nonterminal nodes with empty backward-star. Those are special cases, which we treat specially. Essentially, we give them an inside value of $\\bar 0$.\n\n## Semirings\n\nIn this formula we use generalised sum $\\oplus$ and generalised product $\\otimes$ which we explain below.\n\nA **semiring** is algebraic structure $\\langle \\mathbb K, \\oplus, \\otimes, \\bar 0, \\bar 1\\rangle$ which corresponds to a set $\\mathbb K$ equipped with addition $\\oplus$ and multiplication $\\otimes$.\n\n### Real semiring\n\nFor example, the algebra you learnt at school is a semiring! \nThe set of interest is the real line $\\mathbb K = \\mathbb R$.\n\nThen if we have two real numbers, $a \\in \\mathbb R$ and $b \\in \\mathbb R$, we define **sum** as\n\\begin{equation}\na \\oplus b = a + b\n\\end{equation}\nwhich is simply the standard addition.\nThe additive identity is the value in the set that does not affect summation, we indicate it by $\\bar 0$. In this case, we are talking about the real number 0:\n\\begin{equation}\na \\oplus \\bar 0 = a + 0 = a\n\\end{equation}\n\nWe can also define **multiplication**\n\\begin{equation}\na \\otimes b = a \\times b\n\\end{equation}\nwhich is simply the standard multiplication.\nThe multiplicative identity is the value in the set that does not affect multiplication, we indicate it by $\\bar 1$. In this case, we are talking about the read number 1:\n\\begin{equation}\na \\otimes \\bar 1 = a \\times 1 = a\n\\end{equation}\n\n### Log-Probability semiring\n\nWhen we compute a log-marginal, we are essentially using a logarithmic semiring. \n\nThen the set of interest is the set of log-probability values. Probabilities range between $0$ and $1$ and therefore log-probabilities range from $-\\infty$ (which is $\\log 0$) to $0$ (which is $\\log 1$). We denote this set $\\mathbb K = \\mathbb R_{\\le 0} \\cup \\{-\\infty\\}$.\n\nThen if we have two log-probability values $a \\in \\mathbb K$ and $b \\in \\mathbb K$, our sum becomes\n\\begin{equation}\na \\oplus b = \\log(\\exp a + \\exp b)\n\\end{equation}\nHere we first exponentiate the values bringing them back to the real semiring (where we know how to sum), then we use the standard sum (from high school), and convert the result back to the log-probability semiring by applying $\\log$ to the result.\n\nOur product becomes\n\\begin{equation}\na \\otimes b = a + b\n\\end{equation}\nwhich exploits a basic property of logarithms.\n\n\nOur additive identity is\n\\begin{equation}\na \\oplus \\bar 0 = \\log (\\exp a + \\underbrace{\\exp(-\\infty)}_{0}) = \\log \\exp a = a\n\\end{equation}\nthis is the case because exponentiating an infinitely negative number converges to $0$.\n\nFinally, our multiplicative identity is\n\\begin{equation}\na \\otimes \\bar 1 = a \\times 1 = a\n\\end{equation}\n\n\nThe interesting thing about semirings is that they manipulate different *types of numbers* but they are coherent with the basic axioms of math that we are used to. They help us realise that several algorithms are actually all the same, but they happen to operate under different algebraic structures (read: different definitions of what sum and multiplication are).\n\nWe will define a general class for semirings and you will implement various specialisations. \nThis class will only contain **class methods** this makes the class more or less like a package that can be used to organise coherent functions without really storing any content.\n\n\n```python\nclass Semiring:\n \"\"\"\n This is the interface for semirings.\n \"\"\"\n \n @classmethod\n def from_real(cls, a: float):\n \"\"\"This method takes a number in the Real semiring and converts it to the semiring of interest\"\"\"\n raise NotImplementedError('You need to implement this in the child class')\n \n @classmethod\n def to_real(cls, a: float):\n \"\"\"This method takes a number in this semiring and converts it to the Real semiring\"\"\"\n raise NotImplementedError('You need to implement this in the child class')\n \n @classmethod\n def one(cls):\n \"\"\"This method returns the multiplicative identity of the semiring\"\"\"\n raise NotImplementedError('You need to implement this in the child class')\n \n @classmethod\n def zero(cls):\n \"\"\"This method returns the additive identity of the semiring\"\"\"\n raise NotImplementedError('You need to implement this in the child class')\n \n @classmethod\n def plus(cls, a, b):\n \"\"\"\n This method sums a and b (in the semiring sense)\n where a and b are elements already converted to the type of numbers manipulated by the semiring\n \"\"\"\n raise NotImplementedError('You need to implement this in the child class')\n \n @classmethod\n def times(cls, a, b):\n \"\"\"\n This method multiplies a and b (in the semiring sense)\n where a and b are elements already converted to the type of numbers manipulated by the semiring\n \"\"\"\n raise NotImplementedError('You need to implement this in the child class')\n```\n\nWe will implement for you the *Marginal semiring*, that is, the basic algebra from school.\n\n\n```python\nclass MarginalSemiring(Semiring):\n \n @classmethod\n def from_real(cls, a: float):\n return a\n \n @classmethod\n def to_real(cls, a: float):\n return a\n \n @classmethod\n def one(cls):\n return 1.\n \n @classmethod\n def zero(cls):\n return 0.\n \n @classmethod\n def plus(cls, a, b):\n return a + b\n \n @classmethod\n def times(cls, a, b):\n return a * b\n```\n\n\n```python\nMarginalSemiring.from_real(0.2)\n```\n\n\n\n\n 0.2\n\n\n\n\n```python\nMarginalSemiring.to_real(0.5)\n```\n\n\n\n\n 0.5\n\n\n\n\n```python\nMarginalSemiring.plus(0.1, 0.2)\n```\n\n\n\n\n 0.30000000000000004\n\n\n\n\n```python\nMarginalSemiring.times(0.2, 0.3)\n```\n\n\n\n\n 0.06\n\n\n\n\n```python\nMarginalSemiring.one()\n```\n\n\n\n\n 1.0\n\n\n\n\n```python\nMarginalSemiring.zero()\n```\n\n\n\n\n 0.0\n\n\n\nand we also implement for you the *ViterbiSemiring* used to compute maximum probabilities.\n\n\n```python\nimport numpy as np\nclass ViterbiSemiring(Semiring):\n \n @classmethod\n def from_real(cls, a: float):\n return a\n \n @classmethod\n def to_real(cls, a: float):\n return a\n \n @classmethod\n def one(cls):\n return 1.\n \n @classmethod\n def zero(cls):\n return 0.\n \n @classmethod\n def plus(cls, a, b):\n return np.maximum(a, b)\n \n @classmethod\n def times(cls, a, b):\n return a * b\n```\n\n\n```python\nViterbiSemiring.times(0.2, 0.3)\n```\n\n\n\n\n 0.06\n\n\n\nnote how the following will pick the maximum rather than accumulate the numbers\n\n\n```python\nViterbiSemiring.plus(0.1, 0.4)\n```\n\n\n\n\n 0.4\n\n\n\nNow you implement the $\\log$ variants of both semirings:\n\n\n**Exercise 7-6** **[6 points]** Implement LogMarginalSemiring below as a log-variant of the MarginalSemiring as well as LogViterbiSemiring as a log-variant of the ViterbiSemiring. Run examples of all methods and confirm that the quantities they compute correspond to the correct quantities when converted back to the Real semiring using `to_real`.\n\n* **[3 points]** LogMarginalSemiring\n* **[3 points]** LogViterbiSemiring\n\n\n```python\nclass LogMarginalSemiring(Semiring): \n @classmethod\n def from_real(cls, a: float):\n return np.log(a)\n \n @classmethod\n def to_real(cls, a: float):\n return np.exp(a)\n \n @classmethod\n def one(cls):\n return 0.\n \n @classmethod\n def zero(cls):\n return -float('Inf')\n \n @classmethod\n def plus(cls, a, b):\n return np.log(np.exp(a) + np.exp(b))\n \n \n @classmethod\n def times(cls, a, b):\n return a + b\n```\n\n\n```python\nclass LogViterbiSemiring(Semiring): \n @classmethod\n def from_real(cls, a: float):\n return np.log(a)\n \n @classmethod\n def to_real(cls, a: float):\n return np.exp(a)\n \n @classmethod\n def one(cls):\n return 0.\n \n @classmethod\n def zero(cls):\n return -float('Inf')\n \n @classmethod\n def plus(cls, a, b):\n return max(a, b)\n \n \n @classmethod\n def times(cls, a, b):\n return a + b\n```\n\n## Implementing the inside recursion\n\nFor the inside recursion you need the weight (parameter converted to the appropriate semiring) of the rule that justifies each edge. \n\nFor that we provide you with a helper function. It receives an edge (Rule with spans) and the cpds of the original grammar and returns the correct parameter.\n\n\n```python\nfrom typing import Dict\n\ndef get_parameter(edge: Rule, cpds: Dict[Nonterminal, Dict[Rule, float]]):\n base_rhs = [node.base_nonterminal if isinstance(node, Span) else node for node in edge.rhs]\n base_rule = Rule(edge.lhs.base_nonterminal, base_rhs)\n return cpds[base_rule.lhs][base_rule]\n```\n\n\n```python\n# Now if you ever need to get the parameter for a rule in the grammar you can use the function above\n# For example, \nfor edge in forest:\n print(get_parameter(edge, cpds), edge)\n```\n\n 1.0 [S:1-8] -> [NP:1-2] [VP:2-8]\n 0.5 [IN:5-6] -> 'with'\n 0.3 [NN:4-5] -> 'dog'\n 1.0 [S:0-8] -> [NP:0-2] [VP:2-8]\n 0.3 [VP:2-8] -> [Vt:2-3] [NP:3-8]\n 0.3 [NP:4-8] -> [NP:4-5] [PP:5-8]\n 0.6 [DT:0-1] -> 'the'\n 0.4 [Vt:2-3] -> 'saw'\n 0.4 [NP:6-8] -> [DT:6-7] [NN:7-8]\n 0.1 [NN:7-8] -> 'telescope'\n 0.4 [NP:0-2] -> [DT:0-1] [NN:1-2]\n 1.0 [S:1-5] -> [NP:1-2] [VP:2-5]\n 0.3 [NP:3-8] -> [NP:3-5] [PP:5-8]\n 0.1 [NP:4-5] -> [NN:4-5]\n 0.4 [NN:1-2] -> 'man'\n 0.1 [NP:7-8] -> [NN:7-8]\n 0.1 [NP:1-2] -> [NN:1-2]\n 0.4 [NP:3-5] -> [DT:3-4] [NN:4-5]\n 1.0 [S:0-5] -> [NP:0-2] [VP:2-5]\n 0.3 [VP:2-5] -> [Vt:2-3] [NP:3-5]\n 0.4 [VP:2-8] -> [VP:2-5] [PP:5-8]\n 0.6 [DT:3-4] -> 'the'\n 1.0 [PP:5-8] -> [IN:5-6] [NP:6-8]\n 0.4 [DT:6-7] -> 'a'\n\n\n**Exercise 7-7** **[15 points]** Now you should implement the inside recursion below\n\n* see below for example of inside values for a correct implementation\n\n\n```python\ndef compute_inside_table(forest: CFG, cpds: Dict[Nonterminal, Dict[Rule, float]], semiring: Semiring):\n \"\"\"\n Computes the inside table, that is, the table that assigns an inside value to each \n node in the forest, where a node is a Span.\n \n For convenience, this table may also contain inside values for nodes that are not spans, such as the leaves\n or terminals of the forest, but then that inside should be semiring.one()\n \n Our parsing strategies sometimes create useless nodes, these are nonterminal nodes that have no way\n of being expanded (there are no edges incoming to those nodes, they have an empty backward-star).\n We consider those nodes have an inside value of semiring.zero().\n This is necessary to circumvent the fact that the parsing strategy can create such useless items.\n \n :param forest: a forest as produced by CKY+\n :param cpds: the cpds of the original grammar\n :param semiring: a choice of Semiring\n :return: inside table as a dictionary from a Span to an inside value (as a number in the semiring)\n \"\"\"\n \n inside_table = dict()\n # Start at S -> Find all Span(S)\n start_set = set()\n for rule in forest:\n if rule.lhs.base_nonterminal == Nonterminal('S'):\n start_set.add(rule.lhs)\n# print(cpds[rule.lhs.base_nonterminal])\n \n for s in start_set:\n iS = inside_value(s, forest, cpds, semiring, inside_table)\n inside_table[s] = iS\n return inside_table\n# print(semiring.to_real(iS))\n\ndef get_bs(item: Span, forest: CFG):\n bs = [r for r in forest if r.lhs == item]\n return bs\n\ndef inside_value(item: Span, forest: CFG, cpds, semiring:Semiring, inside_table):\n\n if isinstance(item, Terminal):\n return semiring.one()\n \n iS = semiring.zero()\n bs = get_bs(item, forest)\n if len(bs) == 0:\n return semiring.zero()\n for edge in get_bs(item, forest):\n theta = semiring.from_real(get_parameter(edge, cpds))\n prod = semiring.one()\n# print(edge)\n for sym in edge.rhs:\n if sym not in inside_table:\n \n inside_table[sym] = inside_value(sym, forest, cpds, semiring, inside_table)\n \n prod = semiring.times(prod, inside_table[sym])\n \n iS = semiring.plus(iS, semiring.times(theta, prod))\n \n return iS\n \ns = Span(Nonterminal('DT'), 3, 4)\n# print(isinstance(get_bs(s, forest)[0].rhs[0], Terminal))\n\nsemiring = LogMarginalSemiring()\ninside_table = compute_inside_table(forest, cpds, semiring)\n```\n\nMarginal probability is the inside of the GOAL item in the LogMarginalSemiring (converted back to a real number) .\n\nHere is what your result should look like\n\n```python\ninside_table = compute_inside_table(forest, cpds, LogMarginalSemiring)\nLogMarginalSemiring.to_real(inside_table[forest.start])\n4.6448640000000001e-06\n```\n\n\n```python\nLogMarginalSemiring.to_real(inside_table[forest.start])\n```\n\n\n\n\n 4.644864e-06\n\n\n\nMaximum probability is the inside of the GOAL item in the LogViterbiSemiring (converted back to a real number) .\n\nHere is what your result should look like\n\n```python\nviterbi_table = compute_inside_table(forest, cpds, LogViterbiSemiring)\nLogViterbiSemiring.to_real(viterbi_table[forest.start])\n2.6542080000000048e-06\n\n```\n\n\n```python\nviterbi_table = compute_inside_table(forest, cpds, LogViterbiSemiring)\n```\n\n\n```python\nLogViterbiSemiring.to_real(viterbi_table[forest.start])\n```\n\n\n\n\n 2.6542079999999997e-06\n\n\n\nWe can even define a semiring to count! Imagine that a semiring maps from the real numbers by saying that if something has non-zero probability it counts as $1$ and if it has zero probability it counts as $0$.\n\n\n```python\nclass CountSemiring(Semiring):\n \n @classmethod\n def from_real(cls, a: float):\n \"\"\"Map to 1 if a bigger than 0\"\"\"\n return 1. if a > 0. else 0.\n \n @classmethod\n def to_real(cls, a: float):\n return a\n \n @classmethod\n def one(cls):\n return 1.\n \n @classmethod\n def zero(cls):\n return 0.\n \n @classmethod\n def plus(cls, a, b):\n return a + b\n \n @classmethod\n def times(cls, a, b):\n return a * b\n```\n\nThen we can use the inside algorithm to find the number of **derivations** in the parse forest! \n\nIf your inside implementation is corret, this is what your result should look like:\n\n```python\ncount_table = compute_inside_table(forest, cpds, CountSemiring)\nCountSemiring.to_real(count_table[forest.start])\n2.0\n```\n\n\n```python\ncount_table = compute_inside_table(forest, cpds, CountSemiring)\n```\n\n\n```python\nCountSemiring.to_real(count_table[forest.start])\n```\n\n\n\n\n 2.0\n\n\n\nIsn't this great? :D\n\nNow you are ready to compute the actual Viterbi derivation!\n\n## Viterbi derivation\n\nThe Viterbi path is a top-down traversal of the forest, where each time we have to choose which rule/edge to use to expand a certain nonterminal symbol (span node), we choose the one whose inside value is maximum. But recall that the inside value associated with an *edge* must take into account the weight of the edge and the inside value of its children. Of course, all of this must happen within a maximising semiring (e.g. LogViterbiSemiring or ViterbiSemiring). \n\n\n\\begin{align} \n(2) \\qquad e^\\star &= \\arg\\!\\max_{e \\in \\text{BS(v)}} \\theta \\otimes \\bigotimes_{i=1}^n I(a_i) \\\\\n &~\\text{where }e:=\\frac{a_1, \\ldots, a_n}{v}:\\theta\n\\end{align}\n\n\n\n**Exercise 7-8** **[5 points]** Implement a function that returns the Viterbi derivation (a sequence of rule applications that attains maximum probability).\n\n\n```python\ndef viterbi_derivation(forest: CFG, cpds: Dict[Nonterminal, Dict[Rule, float]], inside_table: Dict[Symbol, float], semiring: Semiring):\n \"\"\"\n Return the derivation (and its yield) that attains maximum probability.\n \n This is a top-down traversal from the root, where for each node v that we need to expand, we \n solve equation (2) above.\n \n :param forest: a forest\n :param cpds: cpds of the original grammar\n :param inside_table: inside values produced with a certain maximising semiring\n :param semiring: a maximising semiring e.g. ViterbiSemiring or LogViterbiSemiring\n :returns: a tuple\n - first element is an ordered list of rule applications\n - second element is the yield of the derivation \n \"\"\" \n pass\n```\n\nIf your implementation is correct you should get\n\n```python\nviterbi_derivation(forest, cpds, viterbi_table, LogViterbiSemiring)\n(([S:0-8] -> [NP:0-2] [VP:2-8],\n [NP:0-2] -> [DT:0-1] [NN:1-2],\n [DT:0-1] -> 'the',\n [NN:1-2] -> 'man',\n [VP:2-8] -> [VP:2-5] [PP:5-8],\n [VP:2-5] -> [Vt:2-3] [NP:3-5],\n [Vt:2-3] -> 'saw',\n [NP:3-5] -> [DT:3-4] [NN:4-5],\n [DT:3-4] -> 'the',\n [NN:4-5] -> 'dog',\n [PP:5-8] -> [IN:5-6] [NP:6-8],\n [IN:5-6] -> 'with',\n [NP:6-8] -> [DT:6-7] [NN:7-8],\n [DT:6-7] -> 'a',\n [NN:7-8] -> 'telescope'),\n ('the', 'man', 'saw', 'the', 'dog', 'with', 'a', 'telescope'))\n```\n\nYou can draw trees using NLTK, here is an example, you can adjust this to visualise trees predicted by your own Viterbi derivation algorithm.\n\n\n```python\nfrom nltk.tree import Tree\nparse_sent = '(S (NP (DT the) (NN cat)) (VP (VBD ate) (NP (DT a) (NN cookie))))'\nt = Tree.fromstring(parse_sent)\nt\n```\n\n# Earley parsing\n\n\nEarley parser is a more conservative, top-down parser, it typically enumerates far less items than CKY+. \nAnswer the questions below (not optional) and as an optional extra contribute an implementation of Earley parser (you may reuse data structures developed for CKY+).\n\nMostly it requires a change of *axioms* and one additional inference rule (i.e. PREDICT). Tip: be careful not to predict the same item multiple times ;)\n\n\\begin{align}\n\\text{Item} &\\qquad [i, X \\rightarrow \\alpha_\\blacksquare \\, \\bullet \\, \\beta_\\square, j] \\\\\n\\text{Goal} &\\qquad [1, S \\rightarrow \\beta_\\blacksquare \\, \\bullet, n] \\\\\n\\text{Axioms} &\\qquad [0, S \\rightarrow \\bullet \\alpha_\\square, 0] &~\\text{ for all } S \\rightarrow \\alpha \\in \\mathcal R \\\\\n\\text{Scan} &\\qquad \\frac{[i, X \\rightarrow \\alpha_\\blacksquare \\, \\bullet \\, x_{j+1} \\, \\beta_\\square, j]}{[i, X \\rightarrow \\alpha_\\blacksquare \\, x_{j+1} \\bullet \\, \\beta_\\square, j + 1]} \\\\\n\\text{Predict} &\\qquad \\frac{[i, X \\rightarrow \\alpha_\\blacksquare \\, \\bullet \\, Y_{j+1} \\, \\beta_\\square, j]}{[j, Y \\rightarrow \\gamma_\\square, j]} &~\\text{for all } Y \\rightarrow \\gamma \\in \\mathcal R\\\\\n\\text{Complete} &\\qquad \\frac{[i, X \\rightarrow \\alpha_\\blacksquare \\, \\bullet \\, Y \\, \\beta_\\square ,k] [k, Y \\rightarrow \\gamma_\\blacksquare \\, \\bullet , j]}{[i, X \\rightarrow \\alpha_\\blacksquare \\, Y_{k,j} \\, \\bullet \\, \\beta_\\square , j]}\n\\end{align}\n\n\n\n**Exercise 7-9** ***THIS IS EXERCISE IS OPTIONAL, ITS POINTS COUNT AS BONUS*** \n\n* **[0.25 points]** Axioms\n* **[0.25 points]** Scan\n* **[0.25 points]** Predict\n* **[0.25 points]** Complete\n\n\n* Points: this exercise is worth 1 extra absolut point added directly to your final grade for *theory*. Recall that *theory* accounts at most for 20% of your grade, thus we first compute your grade without bonuses, then we add the points you earn here. The resulting grade however can never exceed 2 points (that is, 20% of the maximum possible final grade).\n* Note: we will grade this exercise using the scale: 0, 1/4, 1/2, 3/4, 1.\n\n**Exercise 7-10** ***THIS IS EXERCISE IS OPTIONAL, ITS POINTS COUNT AS BONUS*** \n\n\nImplement Earley parser and show that it works by parsing the running example and showing the inside value of the GOAL item for LogMarginalSemiring, LogViterbiSemiring, and CountSemiring.\n\n\n* Points: this exercise is worth 1 extra absolut point added directly to your final grade for *practicals*. Recall that *practicals* account at most for 40% of your grade, thus we first compute your grade without bonuses, then we add the points you earn here. The resulting grade however can never exceed 4 points (that is, 40% of the maximum possible final grade).\n* Note: we will grade this exercise using the scale: 0, 1/4, 1/2, 3/4, 1.\n\n\n", "meta": {"hexsha": "edb53137da5cbf8f12273892b870bb56bb2002ec", "size": 83518, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Lab7/Lab7.ipynb", "max_stars_repo_name": "HarmlessHarm/ntmi", "max_stars_repo_head_hexsha": "16fcb4907efe9af5ce271f36065ab121edcdd308", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Lab7/Lab7.ipynb", "max_issues_repo_name": "HarmlessHarm/ntmi", "max_issues_repo_head_hexsha": "16fcb4907efe9af5ce271f36065ab121edcdd308", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Lab7/Lab7.ipynb", "max_forks_repo_name": "HarmlessHarm/ntmi", "max_forks_repo_head_hexsha": "16fcb4907efe9af5ce271f36065ab121edcdd308", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.2338865303, "max_line_length": 574, "alphanum_fraction": 0.5034004646, "converted": true, "num_tokens": 16473, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269796369904, "lm_q2_score": 0.7577943712746406, "lm_q1q2_score": 0.42309704249968216}} {"text": "# Fairness in AI: Removing word embeddings\n\n#### Kylian van Geijtenbeek, Thom Visser, Martine Toering, Iulia Ionescu\n\n$\\textbf{Abstract:}$ This report attempts to reproduce the word embedding debiasing algorithm and replicate experiments from Bolukbasi, Chang, Zou, Saligrama, and Kalai (2016). We adapt the publicly available implementation (https://github.com/tolga-b/debiaswe) and extend it with the soft debiasing method described in their paper. Several popular benchmarks are integrated in order to evaluate the word embeddings before and after debiasing. Besides replicating results on Word2vec, the effectiveness of the debiasing algorithms is investigated on GloVe and fastText embeddings. We show that the removal of direct bias for all the embeddings barely affects their expressiveness through a comparison of benchmark scores. However, we fail to reproduce large scale soft debiasing results as the method described by the authors is not realistically applicable.\n\n\n```python\nfrom __future__ import print_function, division\n%matplotlib inline\nfrom matplotlib import pyplot as plt\nimport json\nimport random\nimport numpy as np\nimport copy\n\nimport embetter\nimport embetter.we as we\nfrom embetter.we import WordEmbedding\nimport embetter.data as data\n\nfrom embetter.debias import hard_debias\nfrom embetter.benchmarks import Benchmark\n\nfrom compare_bias import *\n```\n\n# 1 - Gender Bias in Word2vec, Glove and FastText\n\n### Load data\n\nIn this notebook, we will use one of the three different word embeddings: $\\textbf{Word2vec}$ (Mikolov et al. 2013). $\\textbf{GloVe}$ (Pennington et al. 2014) and $\\textbf{FastText}$ (Bojanowski et al. 2016) are also available.\n\nThe Word2vec embedding we use is learned from a corpus of Google News articles (https://code.google.com/archive/p/word2vec/). The embeddings are 300-dimensional for 3 million words. For GloVe we make use of the 300-dimensional vectors trained on Common Crawl (https://nlp.stanford.edu/projects/glove/). Lastly, FastText is a word embedding from Facebook AI Research lab trained on Wikipedia corpus and Common Crawl and also consists of 300-dimensional vectors (https://fasttext.cc/docs/en/english-vectors.html).\n\nWe start by loading in the data.\n\n\n```python\n# Load google news word2vec\nE = WordEmbedding(\"word2vec_small\")\n# Load soft debiased word2vec (for later)\nE_soft = WordEmbedding(\"word2vec_small_soft_debiased\")\n```\n\n Embedding shape: (26423, 300)\n 26423 words of dimension 300 : in, for, that, is, ..., Jay, Leroy, Brad, Jermaine\n Embedding shape: (26423, 300)\n 26423 words of dimension 300 : in, for, that, is, ..., Jay, Leroy, Brad, Jermaine\n\n\nAll the other embeddings can be loaded in a similar fashion, using the embedding names in the table on the GitHub Repository *(word2vec_small_hard_debiased, glove_small, glove_small_hard_debiased, glove_small_soft_debiased, word2vec_large, fasttext_small, fasttext_small_hard_debiased, fasttext_small_soft_debiased, glove_large, fasttext_large, or custom embedding file)*.\n\n## Word2vec\n\n\n```python\n# Load professions and gender related lists from Bolukbasi et al. for Word2vec\n\ngender_seed, defs, equalize_pairs, profession_words = data.load_data(E.words)\n```\n\n### Define gender direction\n\nWe define the gender direction by either PCA or by the words \"she\" and \"he\" for Word2vec. \\\nThe PCA method is generally more robust by incorporating all definitional word pairs.\n\n\n```python\n# Define gender direction with the words \"she\" and \"he\" \n# v_gender = E.diff('she', 'he')\n\n# Define gender direction with PCA\nv_gender = we.doPCA(defs, E).components_[0]\n```\n\n### Generating analogies\n\n\nBelow, we show some of the gender analogies that we can create from the embeddings. \\\nThis method is based on \"she is to X as he is to Y\" analogies, with X looping through all the words and then finding the appropriate Y. \\\n\"she\" and \"he\" are either the embeddings of these words, or the extremes of the first principal component, depending on the method used above.\n\n\n```python\n# Analogies gender\na_gender = E.best_analogies_dist_thresh(v_gender, thresh=1)\nwe.viz(a_gender)\n```\n\n Computing neighbors...\n Mean number of neighbors per word: 10.219732808538016\n Median of number of neighbors per word: 7.0\n Index Analogy Gender score\n ---------------------------------------------------------------------\n 0 herself | himself 0.94\n 1 she | he 0.94\n 2 her | his 0.91\n 3 woman | man 0.82\n 4 daughter | son 0.74\n 5 girl | boy 0.74\n 6 actress | actor 0.72\n 7 businesswoman | businessman 0.70\n 8 sister | brother 0.69\n 9 mother | father 0.69\n 10 spokeswoman | spokesman 0.67\n 11 heroine | hero 0.67\n 12 chairwoman | chairman 0.67\n 13 sisters | brothers 0.67\n 14 girls | boys 0.65\n 15 niece | nephew 0.64\n 16 councilwoman | councilman 0.62\n 17 gal | dude 0.61\n 18 aunt | uncle 0.60\n 19 granddaughter | grandson 0.59\n 20 queen | king 0.59\n 21 motherhood | fatherhood 0.59\n 22 schoolgirl | schoolboy 0.59\n 23 Anne | John 0.58\n 24 women | men 0.58\n 25 twin_sister | twin_brother 0.58\n 26 petite | lanky 0.58\n 27 matriarch | patriarch 0.56\n 28 ovarian_cancer | prostate_cancer 0.56\n 29 mom | dad 0.55\n 30 soprano | baritone 0.55\n 31 lesbian | gay 0.53\n 32 daughters | sons 0.52\n 33 mothers | fathers 0.52\n 34 grandmother | grandfather 0.52\n 35 husband | younger_brother 0.51\n 36 sorority | fraternity 0.51\n 37 lady | gentleman 0.49\n 38 blouse | shirt 0.49\n 39 queens | kings 0.48\n 40 grandma | grandpa 0.48\n 41 diva | superstar 0.47\n 42 gals | dudes 0.46\n 43 goddess | god 0.46\n 44 granddaughters | grandsons 0.46\n 45 mommy | kid 0.45\n 46 Aisha | Jamal 0.45\n 47 waitress | waiter 0.44\n 48 Sarah | Matthew 0.44\n 49 softball | baseball 0.44\n 50 mare | gelding 0.44\n 51 filly | colt 0.44\n 52 Jill | Greg 0.43\n 53 volleyball | football 0.43\n 54 hairdresser | barber 0.43\n 55 girlfriends | buddies 0.43\n 56 aunts | uncles 0.42\n 57 actresses | actors 0.42\n 58 hair | beard 0.42\n 59 heiress | magnate 0.42\n 60 feminine | manly 0.42\n 61 princess | prince 0.41\n 62 interior_designer | architect 0.41\n 63 ladies | gentlemen 0.41\n 64 bra | pants 0.41\n 65 moms | dads 0.41\n 66 congresswoman | congressman 0.39\n 67 nun | priest 0.39\n 68 nieces | nephews 0.39\n 69 babe | fella 0.39\n 70 fabulous | terrific 0.39\n 71 vocalist | guitarist 0.39\n 72 blond | burly 0.38\n 73 housewife | shopkeeper 0.38\n 74 feminism | conservatism 0.38\n 75 starlet | youngster 0.38\n 76 sewing | carpentry 0.38\n 77 sassy | snappy 0.38\n 78 glamorous | flashy 0.38\n 79 hers | yours 0.38\n 80 stepdaughter | stepson 0.38\n 81 bitch | bastard 0.38\n 82 feisty | mild_mannered 0.38\n 83 topless | shirtless 0.37\n 84 childhood | boyhood 0.37\n 85 gorgeous | brilliant 0.37\n 86 nurse | physician 0.37\n 87 compatriot | countryman 0.37\n 88 wedding_dress | tuxedo 0.37\n 89 charming | affable 0.36\n 90 breast | prostate 0.36\n 91 fillies | colts 0.36\n 92 females | males 0.35\n 93 handbag | briefcase 0.35\n 94 lovely | magnificent 0.35\n 95 kinda | guy 0.35\n 96 whore | coward 0.35\n 97 siblings | elder_brother 0.35\n 98 salon | barbershop 0.35\n 99 female | male 0.35\n 100 headscarf | turban 0.35\n 101 hubby | pal 0.35\n 102 mums | blokes 0.34\n 103 singer | frontman 0.34\n 104 cupcakes | pizzas 0.34\n 105 vocalists | trumpeter 0.34\n 106 feminist | liberal 0.34\n 107 kids | guys 0.34\n 108 salesperson | salesman 0.33\n 109 sexy | nerdy 0.33\n 110 breast_cancer | lymphoma 0.33\n 111 estrogen | testosterone 0.33\n 112 children | youngsters 0.33\n 113 vagina | penis 0.33\n 114 adorable | goofy 0.33\n 115 cheerful | jovial 0.33\n 116 witch | demon 0.32\n 117 skirts | shorts 0.32\n 118 blonde | blond 0.32\n 119 convent | cathedral 0.32\n 120 bras | trousers 0.32\n 121 dresses | shirts 0.32\n 122 luscious | crisp 0.32\n 123 giggle | chuckle 0.32\n 124 gown | blazer 0.32\n 125 giggling | grinning 0.32\n 126 Latonya | Leroy 0.31\n 127 feminists | liberalism 0.31\n 128 Laurie | Brett 0.31\n 129 stepmother | eldest_son 0.31\n 130 cosmetics | pharmaceuticals 0.31\n 131 netball | rugby 0.31\n 132 beautiful | majestic 0.31\n 133 wig | mustache 0.31\n 134 pink | blue 0.31\n 135 them | him 0.30\n 136 uterus | intestine 0.30\n 137 violinist | virtuoso 0.30\n 138 midwife | doctor 0.30\n 139 dolls | replicas 0.30\n 140 wonderful | great 0.30\n 141 nuns | monk 0.30\n 142 registered_nurse | paramedic 0.30\n 143 breasts | genitals 0.30\n 144 boobs | ass 0.29\n 145 nanny | chauffeur 0.29\n 146 beauty | grandeur 0.29\n 147 middle_aged | bearded 0.29\n 148 delightful | superb 0.29\n 149 physical_therapist | orthopedic_surgeon 0.29\n 150 teenage_girls | youths 0.29\n 151 dress | wearing 0.29\n 152 rebounder | playmaker 0.29\n 153 lingerie | menswear 0.29\n 154 hostess | bartender 0.29\n 155 friends | buddy 0.29\n 156 mammogram | colonoscopy 0.29\n 157 boyfriend | stepfather 0.29\n 158 spokespeople | spokesmen 0.29\n 159 alluring | intriguing 0.29\n 160 swimwear | sportswear 0.29\n 161 implants | stents 0.29\n 162 schoolteacher | carpenter 0.29\n 163 sultry | mellow 0.28\n 164 sexism | racism 0.28\n 165 singers | guitarists 0.28\n 166 sensual | moody 0.28\n 167 enchanting | splendid 0.28\n 168 practicality | durability 0.28\n 169 captivating | masterful 0.28\n 170 cute | clever 0.28\n 171 gymnasts | wrestlers 0.28\n 172 bride | groom 0.28\n 173 lesbians | homosexuals 0.28\n 174 cheesecake | pizza 0.28\n 175 singer_songwriter | musician 0.28\n 176 ethereal | brooding 0.28\n 177 gymnast | wrestler 0.28\n 178 libero | midfielders 0.28\n 179 guidance_counselor | headmaster 0.28\n 180 teenage_girl | teenager 0.27\n 181 rehearse | improvise 0.27\n 182 midfielder | winger 0.27\n 183 hysterical | comical 0.27\n 184 judgmental | arrogant 0.27\n 185 giggles | chuckles 0.27\n 186 gowns | robes 0.27\n 187 servicemen | veterans 0.27\n 188 gymnastics | judo 0.27\n 189 boyfriends | friends 0.27\n 190 cigarette | cigar 0.26\n 191 baking | roasting 0.26\n 192 lovers | aficionados 0.26\n 193 choreography | footwork 0.26\n 194 burlesque | rock_n_roll 0.26\n 195 kindness | humility 0.26\n 196 pianist | maestro 0.26\n 197 sexist | racist 0.26\n 198 kittens | neutered 0.26\n 199 mares | stallion 0.26\n 200 silicone | polymer 0.26\n 201 eating_disorder | alcoholism 0.26\n 202 cupcake | donut 0.26\n 203 housekeeper | janitor 0.26\n 204 designers | architects 0.26\n 205 mama | daddy 0.26\n 206 panties | socks 0.26\n 207 hurler | starter 0.26\n 208 chairperson | chaired 0.26\n 209 novelist | philosopher 0.26\n 210 backcourt | swingman 0.25\n 211 stroller | bicycle 0.25\n 212 male_counterparts | counterparts 0.25\n 213 flute | trombone 0.25\n 214 cubs | lions 0.25\n 215 crafting | drafting 0.25\n 216 manga | comic_books 0.25\n 217 presenter | broadcaster 0.25\n 218 husbands | wives 0.25\n 219 kissing | shaking_hands 0.25\n 220 lupus | multiple_myeloma 0.25\n 221 cakes | sausages 0.25\n 222 latex | rubber 0.25\n 223 hooker | fullback 0.25\n 224 bingo | gambling 0.25\n 225 veil | cloak 0.25\n 226 dancers | drummers 0.25\n 227 gender | racial 0.25\n 228 sobbed | grinned 0.25\n 229 prostitution | drug_trafficking 0.25\n 230 antiques | memorabilia 0.25\n 231 hair_salon | pizzeria 0.25\n 232 costumes | props 0.25\n 233 vocals | drummer 0.25\n 234 fiance | cousin 0.25\n 235 intuition | instincts 0.25\n 236 eating_disorders | substance_abuse 0.24\n 237 cougar | wolves 0.24\n 238 Carrie | Laurie 0.24\n 239 freshmen | rookies 0.24\n 240 amazing | unbelievable 0.24\n 241 artistry | genius 0.24\n 242 flight_attendant | pilots 0.24\n 243 teen | youth 0.24\n 244 sweaters | jerseys 0.24\n 245 smile | grin 0.24\n 246 layups | downfield 0.24\n 247 harmonies | riffs 0.24\n 248 osteoporosis | atherosclerosis 0.24\n 249 cheery | amiable 0.24\n 250 outfits | garb 0.24\n 251 beau | pals 0.24\n 252 hairstyle | facial_hair 0.24\n 253 terrifying | fearsome 0.24\n 254 child_endangerment | aggravated_assault 0.24\n 255 figure_skating | hockey 0.24\n 256 teenager | lad 0.24\n 257 jewelry | collectibles 0.24\n 258 chic | minimalist 0.24\n 259 duets | saxophonist 0.24\n 260 worker | foreman 0.24\n 261 glam | retro 0.24\n 262 uptight | cocky 0.24\n 263 sweater | jersey 0.24\n 264 rapist | thug 0.24\n 265 cervical_cancer | colorectal_cancer 0.24\n 266 hugs | shook_hands 0.24\n 267 masculine | macho 0.24\n 268 scarf | jacket 0.24\n 269 hen | cock 0.24\n 270 glitter | confetti 0.24\n 271 grad | alumnus 0.24\n 272 romantic_comedy | flick 0.24\n 273 meter_hurdles | yard_dash 0.24\n 274 cooking | grill 0.24\n 275 maids | servants 0.24\n 276 exclaimed | quipped 0.24\n 277 unconventional | unorthodox 0.24\n 278 foal | thoroughbred 0.23\n 279 seductive | mesmerizing 0.23\n 280 assistant_professor | professor_emeritus 0.23\n 281 classmates | teammates 0.23\n 282 putback | yard_touchdown 0.23\n 283 retro | throwback 0.23\n 284 underclassmen | players 0.23\n 285 baseman | offensive_lineman 0.23\n 286 transgender | homosexual 0.23\n 287 unassisted_goal | deflected 0.23\n 288 inadequate | ineffective 0.23\n 289 elegance | style 0.23\n 290 tamoxifen | statins 0.23\n 291 anime | videogames 0.23\n 292 giant_slalom | skied 0.23\n 293 menopause | heart_disease 0.23\n 294 sophomore | pounder 0.23\n 295 lover | enthusiast 0.23\n 296 plucky | hapless 0.23\n 297 dispatcher | patrolman 0.23\n 298 classmate | teammate 0.23\n 299 cats | pigeons 0.23\n 300 midwives | doctors 0.23\n 301 cosmetic_surgery | surgery 0.23\n 302 feline | bulldog 0.23\n 303 apparel | sporting_goods 0.23\n 304 warship | destroyer 0.23\n 305 breast_milk | sperm 0.23\n 306 delightfully | brilliantly 0.23\n 307 cigarettes | cigars 0.23\n 308 choral | composer 0.23\n 309 ex_boyfriend | friend 0.23\n 310 heartbreaking | humbling 0.23\n 311 seatbelt | wearing_helmet 0.23\n 312 sprinter | speedster 0.23\n 313 regionals | playoffs 0.22\n 314 spokesperson | statement 0.22\n 315 realtor | builder 0.22\n 316 lamb | butcher 0.22\n 317 ultrasound | x_ray 0.22\n 318 dancer | entertainer 0.22\n 319 gossip | rumor 0.22\n 320 stalker | pedophile 0.22\n 321 cried | chuckled 0.22\n 322 self_esteem | morale 0.22\n 323 elegantly | superbly 0.22\n 324 housewives | middle_aged 0.22\n 325 librarian | curator 0.22\n 326 frontcourt | playmakers 0.22\n 327 free_throws | touchdown_pass 0.22\n 328 singing | rapping 0.22\n 329 bartender | bouncer 0.22\n 330 pitcher | lefthander 0.22\n 331 transformative | visionary 0.22\n 332 paralegal | accountant 0.22\n 333 clarinet | saxophone 0.22\n 334 empowering | motivating 0.22\n 335 auditions | tryout 0.22\n 336 wellness | fitness 0.22\n 337 neurotic | eccentric 0.22\n 338 internship | apprenticeship 0.22\n 339 pediatrician | cardiologist 0.22\n 340 sobbing | shouted 0.22\n 341 vampire | superhero 0.22\n 342 bun | burgers 0.22\n 343 blokes | bloke 0.22\n 344 anorexia | depression 0.22\n 345 incumbents | incumbent 0.22\n 346 raped | sexually_abusing 0.22\n 347 mentoring | mentor 0.22\n 348 provocative | incendiary 0.22\n 349 cashier | robber 0.22\n 350 gosh | heck 0.22\n 351 rapes | crimes 0.22\n 352 nurses | physicians 0.21\n 353 collagen | cartilage 0.21\n 354 hitter | southpaw 0.21\n 355 pregnancy | gestation 0.21\n 356 layup | yarder 0.21\n 357 sexuality | homosexuality 0.21\n 358 resourceful | wily 0.21\n 359 plunging | slumping 0.21\n 360 badminton | snooker 0.21\n 361 shrill | louder 0.21\n 362 romance | friendship 0.21\n 363 clique | cronies 0.21\n 364 delectable | sublime 0.21\n 365 terrified | mad 0.21\n 366 Allison | Todd 0.21\n 367 thirteen | eleven 0.21\n 368 campers | camp 0.21\n 369 curves | curved 0.21\n 370 beloved | legendary 0.21\n 371 cloned | clone 0.21\n 372 exercises | drills 0.21\n 373 steals | tackles 0.21\n 374 starvation | famine 0.21\n 375 vulgar | profane 0.21\n 376 silver_medalist | compatriot 0.21\n 377 entrepreneurs | businessmen 0.21\n 378 walker | crutches 0.21\n 379 closets | lockers 0.21\n 380 graphic_designer | cartoonist 0.21\n 381 rower | skipper 0.21\n 382 artisans | craftsmen 0.21\n 383 experimenting | tinkering 0.21\n 384 shortstop | defensive_lineman 0.21\n 385 couples | married_couples 0.21\n 386 horrified | incensed 0.21\n 387 memoir | autobiography 0.21\n 388 prettiest | finest 0.21\n 389 fairy | magical 0.21\n 390 folklore | legend 0.21\n 391 scripture | disciples 0.21\n 392 satin | leather 0.21\n 393 coordinator | manager 0.21\n 394 counselor | adviser 0.21\n 395 chicks | ducks 0.21\n 396 middle_blocker | leadoff_hitter 0.21\n 397 minivan | pickup_truck 0.21\n 398 activists | supporters 0.21\n 399 caring | selfless 0.21\n 400 thighs | biceps 0.21\n 401 designed | engineered 0.21\n 402 love_triangle | saga 0.21\n 403 insecurities | frustrations 0.21\n 404 cookbook | tome 0.21\n 405 mills | steel_mills 0.20\n 406 ice_skating | skateboarding 0.20\n 407 critical_acclaim | plaudits 0.20\n 408 skirted | evaded 0.20\n 409 resignations | sacking 0.20\n 410 animal_cruelty | dogfighting 0.20\n 411 purse | wallet 0.20\n 412 abducted | gunned_down 0.20\n 413 energetic | charismatic 0.20\n 414 rehearsal | improvisation 0.20\n 415 beautify | rehabilitate 0.20\n 416 celebrities | superstars 0.20\n 417 turquoise | navy_blue 0.20\n 418 celebs | stars 0.20\n 419 holistic_approach | approach 0.20\n 420 cocktails | beers 0.20\n 421 adored | revered 0.20\n 422 shimmering | gleaming 0.20\n 423 interns | fellows 0.20\n 424 contestants | hopefuls 0.20\n 425 smitten | enamored 0.20\n 426 ribbons | plaques 0.20\n 427 golfer | sportsman 0.20\n 428 sang | chanted 0.20\n 429 demeaning | disrespectful 0.20\n 430 clothing | merchandise 0.20\n 431 flower | ornamental 0.20\n 432 strawberry | potato 0.20\n 433 makeover | revamp 0.20\n 434 courageous | honorable 0.20\n 435 demonstrates | underlines 0.20\n 436 sophomores | offensive_linemen 0.20\n 437 unsafe | dangerous 0.20\n 438 storyteller | magician 0.20\n 439 thousands | tens 0.20\n 440 doubleheader | game 0.20\n 441 hormonal | metabolic 0.20\n 442 watchers | pundits 0.20\n 443 bubbly | champagne 0.20\n 444 thyroid | inflammation 0.20\n 445 anthropologist | historian 0.20\n 446 overwhelmed | humbled 0.20\n 447 boutiques | retail_outlets 0.20\n 448 poem | eulogy 0.20\n 449 nerve_wracking | frustrating 0.20\n 450 export | import 0.20\n 451 freshman | redshirt_freshman 0.20\n 452 creepy | menacing 0.20\n 453 bondage | shackles 0.20\n 454 foul_trouble | defensively 0.20\n 455 therapist | neurologist 0.20\n 456 horrid | mediocre 0.20\n 457 workshops | sessions 0.20\n 458 articulate | thinker 0.20\n 459 tenacious | hard_nosed 0.20\n 460 drinks | beer 0.20\n 461 violin | guitar 0.20\n 462 quilting | stained_glass 0.20\n 463 boutique | shop 0.20\n 464 mysterious | enigmatic 0.20\n 465 soldiers | platoon 0.20\n 466 catcher | lineman 0.20\n 467 luminous | dazzling 0.20\n 468 dayer | seamer 0.20\n 469 backstretch | straightaway 0.20\n 470 behaviors | tendencies 0.20\n 471 attractiveness | competitiveness 0.20\n 472 vibrant | dynamic 0.20\n 473 incredibly | obviously 0.20\n 474 scandalous | disgraceful 0.20\n 475 firings | firing 0.19\n 476 sergeants | lieutenants 0.19\n 477 complainants | accusers 0.19\n 478 homemaker | schoolteacher 0.19\n 479 gender_equality | poverty_alleviation 0.19\n 480 underpass | tunnel 0.19\n 481 characters | villain 0.19\n 482 delicately | deftly 0.19\n 483 critters | beasts 0.19\n 484 advocate | proponent 0.19\n 485 soaps | sitcoms 0.19\n 486 complicit | culpable 0.19\n 487 ok | alright 0.19\n 488 interim_dividend | dividends 0.19\n 489 garments | apparel 0.19\n 490 downsize | restructure 0.19\n 491 oncoming_traffic | swerving 0.19\n 492 ex | former 0.19\n 493 manipulative | cunning 0.19\n 494 sexual_harassment | misconduct 0.19\n 495 embryo | embryonic 0.19\n 496 endearing | likeable 0.19\n 497 rape | attempted_murder 0.19\n 498 blooms | buds 0.19\n 499 stripper | strippers 0.19\n\n\nThese analogies offer an insight in potential biases along the specified bias axis (in this case gender). This is useful for a qualitative analysis.\n\n### Analyzing occupational gender bias \n\n\nThe projection of occupations on the bias axis serves as another useful source for qualitative analysis.\n\n\n```python\n# Analysis of extreme male and extreme female professions\nsp = E.profession_stereotypes(profession_words, v_gender)\n```\n\n Female | Male \n -----------------------------------------------------------------------------\n 0.38 businesswoman | maestro -0.244\n 0.379 actress | protege -0.236\n 0.378 housewife | statesman -0.222\n 0.323 homemaker | businessman -0.219\n 0.308 nurse | sportsman -0.209\n 0.302 registered_nurse | philosopher -0.196\n 0.297 waitress | marksman -0.192\n 0.28 receptionist | skipper -0.187\n 0.278 socialite | financier -0.183\n 0.277 librarian | architect -0.177\n 0.272 maid | magician -0.172\n 0.265 nun | trumpeter -0.172\n 0.261 ballerina | major_leaguer -0.16\n 0.233 nanny | salesman -0.157\n 0.232 paralegal | captain -0.154\n 0.218 hairdresser | mechanic -0.153\n 0.211 housekeeper | warrior -0.152\n 0.203 bookkeeper | lieutenant -0.152\n 0.199 stylist | gangster -0.15\n 0.198 interior_designer | fighter_pilot -0.148\n\n\n# 2 - Comparing Bias of Word2vec and FastText\n\nWe will compare the gender bias between word embeddings Word2vec and FastText. We do this by following Bolukbasi et al.'s approach shown in Figure 4 in their paper. The profession words are projected onto the gender axis for two embeddings. Each datapoint represents a profession word.\n\nBelow we compare the bias of Word2vec and FastText.\n\n\n```python\nE_f = WordEmbedding(\"fasttext_small\")\ncompare_occupational_bias(E, E_f, [\"Word2vec\", \"FastText\"])\n```\n\n# 3 - Debiasing algorithms on Word2vec\n\n## Hard debiasing\n\nIn hard debiasing, the gender neutral words are shifted to zero in the gender subspace (i.e. neutralized) by subtracting the projection of the neutral word embedding vector onto the gender subspace and renormalizing the resulting embedding to unit length. \n\n## Soft debiasing\n\nSoft debiasing is done by solving the following optimization problem as mentioned in their paper:\n\n\\begin{equation}\n \\underset{T}{\\min} || (TW)^T(TW) - W^TW||^2_F + \\lambda ||(TN)^T (TB)||^2_F\n\\end{equation}\n\nwhere W is the matrix of all embedding vectors, N is the matrix of the embedding vectors of the gender neutral words, B is the gender subspace, and T is the debiasing transformation that minimizes the projection of the neutral words onto the gender subspace but tries to maintain the pairwise inner products between the words.\n\nFor the optimization, we adapted specifics from Manzini, Chong, Black, and Tsvetkov (2019), where our code is based on the code they provide (https://github.com/TManzini/DebiasMulticlassWordEmbedding).\n\n### Hard debiasing Word2vec\nFirstly, we show how to manually debias embeddings of choice. \\\nThis overwrites the embeddings in the WordEmbedding object, so make sure to re-load the biased embeddings in a new WordEmbedding object for comparison between the benchmarks (as is done in this notebook in the Benchmarks section).\n\n\n```python\n# Pass the WordEmbedding object which contains the embeddings that should be debiased.\nhard_debias(E, gender_seed, defs, equalize_pairs)\n```\n\nSecondly, we show the effect of hard debiasing on Word2vec.\n\n\n```python\n# Hard debiased Word2vec\n# Analysis of extreme male and extreme female professions\nsp_hard_debiased = E.profession_stereotypes(profession_words, v_gender)\n```\n\n Female | Male \n -----------------------------------------------------------------------------\n 0.433 businesswoman | businessman -0.433\n 0.379 actress | congressman -0.428\n 0.378 housewife | dad -0.365\n 0.297 waitress | councilman -0.358\n 0.272 maid | statesman -0.222\n 0.265 nun | salesman -0.157\n 0.261 ballerina | handyman -0.107\n 0.0 broadcaster | monk -0.082\n 0.0 plumber | librarian -0.0\n 0.0 valedictorian | magician -0.0\n 0.0 sportswriter | graphic_designer -0.0\n 0.0 observer | novelist -0.0\n 0.0 sergeant | assistant_professor -0.0\n 0.0 doctoral_student | officer -0.0\n 0.0 fisherman | shopkeeper -0.0\n 0.0 landlord | filmmaker -0.0\n 0.0 chaplain | homemaker -0.0\n 0.0 paralegal | dancer -0.0\n 0.0 clerk | industrialist -0.0\n 0.0 superintendent | physicist -0.0\n\n\n\n```python\n# Analogies gender\na_gender_hard_debiased = E.best_analogies_dist_thresh(v_gender)\nwe.viz(a_gender_hard_debiased)\n```\n\n Computing neighbors...\n Mean number of neighbors per word: 10.218597434053665\n Median of number of neighbors per word: 7.0\n Index Analogy Gender score\n ---------------------------------------------------------------------\n 0 ex_boyfriend | ex_girlfriend 1.00\n 1 mother | father 1.00\n 2 daughters | sons 1.00\n 3 businesswoman | businessman 1.00\n 4 daughter | son 1.00\n 5 councilwoman | councilman 1.00\n 6 herself | himself 1.00\n 7 queen | king 1.00\n 8 filly | colt 1.00\n 9 husbands | wives 1.00\n 10 niece | nephew 1.00\n 11 sisters | brothers 1.00\n 12 females | males 1.00\n 13 mothers | fathers 1.00\n 14 spokeswoman | spokesman 1.00\n 15 princess | prince 1.00\n 16 schoolgirl | schoolboy 1.00\n 17 estrogen | testosterone 1.00\n 18 twin_sister | twin_brother 1.00\n 19 queens | kings 1.00\n 20 grandma | grandpa 1.00\n 21 girl | boy 1.00\n 22 women | men 1.00\n 23 female | male 1.00\n 24 granddaughters | grandsons 1.00\n 25 grandmother | grandfather 1.00\n 26 she | he 1.00\n 27 ovarian_cancer | prostate_cancer 1.0\n 28 ladies | gentlemen 1.0\n 29 her | his 1.0\n 30 motherhood | fatherhood 1.0\n 31 aunt | uncle 1.0\n 32 sorority | fraternity 1.0\n 33 congresswoman | congressman 1.0\n 34 girls | boys 1.0\n 35 convent | monastery 1.0\n 36 woman | man 1.0\n 37 gals | dudes 1.0\n 38 chairwoman | chairman 0.99\n 39 granddaughter | grandson 0.99\n 40 mom | dad 0.99\n 41 moms | dads 0.99\n 42 mare | gelding 0.99\n 43 sister | brother 0.99\n 44 actress | actor 0.64\n 45 gal | dude 0.61\n 46 lesbian | gay 0.54\n 47 compatriot | countryman 0.52\n 48 husband | younger_brother 0.51\n 49 heroine | protagonist 0.49\n 50 actresses | actors 0.48\n 51 housewife | homemaker 0.43\n 52 waitress | waiter 0.43\n 53 aunts | uncles 0.42\n 54 feminism | feminist 0.42\n 55 mustache | beard 0.42\n 56 hers | theirs 0.41\n 57 kid | guy 0.40\n 58 fella | gentleman 0.39\n 59 nieces | nephews 0.39\n 60 teenage_girls | teenagers 0.39\n 61 nun | monk 0.38\n 62 stepdaughter | stepson 0.38\n 63 childhood | boyhood 0.38\n 64 mommy | daddy 0.37\n 65 me | him 0.37\n 66 goddess | god 0.36\n 67 viagra | cialis 0.36\n 68 diva | superstar 0.36\n 69 fillies | colts 0.36\n 70 brides | bridal 0.35\n 71 matriarch | patriarch 0.35\n 72 maid | housekeeper 0.35\n 73 hostess | bartender 0.34\n 74 vagina | penis 0.33\n 75 mama | fella 0.33\n 76 teenage_girl | teenager 0.32\n 77 stepmother | eldest_son 0.31\n 78 ballerina | dancer 0.30\n 79 maternity | midwives 0.30\n 80 grandmothers | grandparents 0.30\n 81 compatriots | countrymen 0.29\n 82 witch | witchcraft 0.29\n 83 boyfriend | stepfather 0.29\n 84 uterus | intestine 0.28\n 85 menopause | puberty 0.28\n 86 heiress | socialite 0.28\n 87 bride | wedding 0.28\n 88 lesbians | gays 0.27\n 89 eldest | elder_brother 0.27\n 90 politician | statesman 0.25\n 91 maids | servants 0.24\n 92 dictator | strongman 0.24\n 93 youngster | lad 0.24\n 94 nuns | priests 0.24\n 95 maternal | infant_mortality 0.24\n 96 hubby | pal 0.23\n 97 blokes | bloke 0.22\n 98 lady | waitress 0.21\n 99 soprano | baritone 0.21\n 100 girlfriends | buddies 0.21\n 101 boyfriends | girlfriend 0.20\n 102 facial_hair | beards 0.19\n 103 womb | fetus 0.18\n 104 businesspeople | businessmen 0.18\n 105 fiance | roommate 0.18\n 106 beau | lover 0.18\n 107 salesperson | salesman 0.18\n 108 witches | vampires 0.18\n 109 estranged_husband | estranged 0.17\n 110 counterparts | brethren 0.16\n 111 bastard | chap 0.16\n 112 widow | deceased 0.16\n 113 obstetrics | pediatrics 0.16\n 114 spokespeople | spokesmen 0.16\n 115 friendship | brotherhood 0.15\n 116 hens | chickens 0.15\n 117 hen | cock 0.15\n 118 replied | sir 0.14\n 119 colon | prostate 0.14\n 120 mistress | prostitute 0.14\n 121 stallion | stud 0.13\n 122 manly | macho 0.13\n 123 wife | cousin 0.13\n 124 ma | na 0.13\n 125 carpenter | handyman 0.12\n 126 bulls | bull 0.12\n 127 widows | families 0.11\n 128 salespeople | salesmen 0.10\n 129 girlfriend | friend 0.10\n 130 suitor | takeover_bid 0.09\n 131 gaffer | lads 0.09\n 132 semen | saliva 0.09\n 133 elephants | lions 0.09\n 134 suitors | bidders 0.08\n 135 fiancee | married 0.08\n 136 guys | fellas 0.08\n 137 hair_salon | barbershop 0.07\n 138 elephant | lion 0.05\n 139 colts | mares 0.05\n 140 pa | mo 0.05\n 141 footy | blokes 0.04\n 142 aldermen | councilmen 0.04\n 143 monks | monasteries 0.04\n 144 widower | widowed 0.03\n 145 bachelor | bachelor_degree 0.02\n 146 sperm | embryos 0.02\n 147 deer | elk 0.02\n 148 residence_halls | fraternities 0.01\n 149 wedlock | fathered 0.01\n 150 penis | genitals 0.00\n 151 princes | royals 0.00\n\n\n# 4 - Benchmarks\n\nThis package includes some basic benchmarks, which allow for easy verification of the embedding's quality before and after debiassing (RG-65, WS-353 and MSR), as well as a statistical measure of bias to quantitatively inspect the effect of debiassing (WEAT).\n\n\n```python\nbenchmark = Benchmark()\nE_before = WordEmbedding(\"word2vec_small\")\n```\n\n Embedding shape: (26423, 300)\n 26423 words of dimension 300 : in, for, that, is, ..., Jay, Leroy, Brad, Jermaine\n\n\n### Word2vec\n\nBelow, we show the benchmarks for Word2vec. \\\nWhen comparing the WEAT effect size, make sure to use a single Benchmark object per embedding, benchmarking the biased version first, as is done in this notebook. (This is because the first bias axis is saved internally to measure bias along it for subsequent embeddings.)\n\n\n```python\n# Evaluate for Word2vec\nbefore_results = benchmark.evaluate(E_before, \"Before\")\nhard_results = benchmark.evaluate(E, \"Hard\")\nsoft_results = benchmark.evaluate(E_soft, \"Soft\")\n\n```\n\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 40/40 [00:05<00:00, 7.86it/s]\n\n\n +-------------------------------------------------------+\n | Results for Before |\n +---------------+-------+-----------+-------------------+\n | Dataset | Found | Not Found | Score |\n +---------------+-------+-----------+-------------------+\n | EN-RG-65 | 53 | 12 | 77.66555804950227 |\n | EN-WS-353-ALL | 318 | 35 | 68.82719646959825 |\n | MSR-analogy | 5276 | 2724 | 46.79681576952237 |\n | WEAT | - | - | 1.4849316 |\n +---------------+-------+-----------+-------------------+\n\n\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 40/40 [00:04<00:00, 8.82it/s]\n\n\n +--------------------------------------------------------+\n | Results for Hard |\n +---------------+-------+-----------+--------------------+\n | Dataset | Found | Not Found | Score |\n +---------------+-------+-----------+--------------------+\n | EN-RG-65 | 53 | 12 | 77.49622028082247 |\n | EN-WS-353-ALL | 318 | 35 | 68.52623098234018 |\n | MSR-analogy | 5276 | 2724 | 46.967399545109934 |\n | WEAT | - | - | 0.36466902 |\n +---------------+-------+-----------+--------------------+\n\n\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 40/40 [00:04<00:00, 8.34it/s]\n\n\n +-------------------------------------------------------+\n | Results for Soft |\n +---------------+-------+-----------+-------------------+\n | Dataset | Found | Not Found | Score |\n +---------------+-------+-----------+-------------------+\n | EN-RG-65 | 53 | 12 | 77.66555804950227 |\n | EN-WS-353-ALL | 318 | 35 | 68.82719646959825 |\n | MSR-analogy | 5276 | 2724 | 46.79681576952237 |\n | WEAT | - | - | -0.22672768 |\n +---------------+-------+-----------+-------------------+\n\n\nThe results of individual benchmarks can be combined into a single list and passed to benchmark's `pprint_compare` method for easy comparison.\n\n\n```python\nw2v_results = [before_results, hard_results, soft_results]\nbenchmark.pprint_compare(w2v_results, [\"Before\", \"Hard-debiased\", \"Soft-debiased\"], \"word2vec\")\n```\n\n +------------------------------------------------------------------------------------------+\n | Results for word2vec |\n +---------------+-------------------+-------------------+--------------------+-------------+\n | Score | RG-65 | WS-353 | MSR | WEAT |\n +---------------+-------------------+-------------------+--------------------+-------------+\n | Before | 77.66555804950227 | 68.82719646959825 | 46.79681576952237 | 1.4849316 |\n | Hard-debiased | 77.49622028082247 | 68.52623098234018 | 46.967399545109934 | 0.36466902 |\n | Soft-debiased | 77.66555804950227 | 68.82719646959825 | 46.79681576952237 | -0.22672768 |\n +---------------+-------------------+-------------------+--------------------+-------------+\n\n\n# 5 - Full Experiments\n\nThe full range of experiments can be executed using the `experiments.py` file from the repository. \\\nThis is best done through a terminal, to modify the behaviour using command line arguments, but its also available here. \\\nThis line outputs comparison benchmarks for all small embeddings. \\\nGender analogies and occupational gender bias are not shown for *word2vec_small*, *glove_small*, and *fasttext_small* before debiasing, after hard-debiasing and after soft-debiasing, through the `--no_show` option.\n\n\n```python\n!python experiments.py --no_show\n```\n\n ################## EXPERIMENT DETAILS ##################\n # #\n # Analogies and occupations: False #\n # Do soft debiasing from scratch: False #\n # Perform benchmarks: True #\n # #\n # Performing experiments for the following embeddings: #\n #\t- word2vec_small #\n #\t- glove_small #\n #\t- fasttext_small #\n # #\n ########################################################\n \n ########################################################\n # Doing the word2vec_small embedding #\n ########################################################\n Embedding shape: (26423, 300)\n 26423 words of dimension 300 : in, for, that, is, ..., Jay, Leroy, Brad, Jermaine\n \n Hard debiasing...\n \n Soft debiasing...\n Embedding shape: (26423, 300)\n 26423 words of dimension 300 : in, for, that, is, ..., Jay, Leroy, Brad, Jermaine\n \n Running benchmarks...\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 40/40 [00:04<00:00, 8.90it/s]\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 40/40 [00:04<00:00, 9.51it/s]\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 40/40 [00:04<00:00, 9.65it/s]\n +------------------------------------------------------------------------------------------+\n | Results for word2vec_small |\n +---------------+-------------------+-------------------+--------------------+-------------+\n | Score | RG-65 | WS-353 | MSR | WEAT |\n +---------------+-------------------+-------------------+--------------------+-------------+\n | Before | 77.66555804950227 | 68.82719646959825 | 46.79681576952237 | 1.4785795 |\n | Hard-debiased | 77.49622028082247 | 68.52623098234018 | 46.967399545109934 | 0.37883297 |\n | Soft-debiased | 77.66555804950227 | 68.82719646959825 | 46.79681576952237 | -0.15358597 |\n +---------------+-------------------+-------------------+--------------------+-------------+\n \n ########################################################\n # Doing the glove_small embedding #\n ########################################################\n Embedding shape: (42982, 300)\n 42982 words of dimension 300 : the, and, to, of, ..., cushman, darkside, motherland, chairmen\n \n Hard debiasing...\n \n Soft debiasing...\n Embedding shape: (42982, 300)\n 42982 words of dimension 300 : the, and, to, of, ..., cushman, darkside, motherland, chairmen\n \n Running benchmarks...\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 40/40 [00:07<00:00, 5.56it/s]\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 40/40 [00:07<00:00, 5.32it/s]\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 40/40 [00:07<00:00, 5.37it/s]\n +-----------------------------------------------------------------------------------------+\n | Results for glove_small |\n +---------------+-------------------+-------------------+-------------------+-------------+\n | Score | RG-65 | WS-353 | MSR | WEAT |\n +---------------+-------------------+-------------------+-------------------+-------------+\n | Before | 83.10311410756115 | 66.43748693628271 | 37.47211895910781 | -1.6155657 |\n | Hard-debiased | 83.35243100494134 | 66.61161322854848 | 37.62081784386617 | -0.39784268 |\n | Soft-debiased | 83.0703755250769 | 66.44364789148896 | 37.43494423791822 | -0.69473237 |\n +---------------+-------------------+-------------------+-------------------+-------------+\n \n ########################################################\n # Doing the fasttext_small embedding #\n ########################################################\n Embedding shape: (27014, 300)\n 27014 words of dimension 300 : the, and, of, to, ..., circumscribed, whos, salvaging, anion\n \n Hard debiasing...\n \n Soft debiasing...\n Embedding shape: (27014, 300)\n 27014 words of dimension 300 : the, and, of, to, ..., circumscribed, whos, salvaging, anion\n \n Running benchmarks...\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 40/40 [00:03<00:00, 10.44it/s]\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 40/40 [00:03<00:00, 10.90it/s]\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 40/40 [00:03<00:00, 10.88it/s]\n +----------------------------------------------------------------------------------------+\n | Results for fasttext_small |\n +---------------+-------------------+-------------------+-------------------+------------+\n | Score | RG-65 | WS-353 | MSR | WEAT |\n +---------------+-------------------+-------------------+-------------------+------------+\n | Before | 83.86622701863348 | 74.10786418997199 | 55.87019429516329 | 1.4813895 |\n | Hard-debiased | 83.50735694621694 | 74.17924052453014 | 55.99421248449773 | 0.436271 |\n | Soft-debiased | 84.26611081361189 | 74.12157140993038 | 54.73336089293096 | 0.38817888 |\n +---------------+-------------------+-------------------+-------------------+------------+\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "e36db7f02f324e506e92c42fe5c1678f9e40d69d", "size": 103889, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Fairness_in_AI_Removing_bias.ipynb", "max_stars_repo_name": "KylianvG/Embetter", "max_stars_repo_head_hexsha": "22df636089a83ef5aeeb3b52bc7ea3ddaa538385", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Fairness_in_AI_Removing_bias.ipynb", "max_issues_repo_name": "KylianvG/Embetter", "max_issues_repo_head_hexsha": "22df636089a83ef5aeeb3b52bc7ea3ddaa538385", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Fairness_in_AI_Removing_bias.ipynb", "max_forks_repo_name": "KylianvG/Embetter", "max_forks_repo_head_hexsha": "22df636089a83ef5aeeb3b52bc7ea3ddaa538385", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-10-27T10:02:54.000Z", "max_forks_repo_forks_event_max_datetime": "2020-10-27T10:02:54.000Z", "avg_line_length": 75.0643063584, "max_line_length": 21476, "alphanum_fraction": 0.4694240969, "converted": true, "num_tokens": 14342, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.607663184043154, "lm_q2_score": 0.6959583187272712, "lm_q1q2_score": 0.42290824791913384}} {"text": "# Chapter 5: Resampling Methods\n- **Chapter 5 from the book [An Introduction to Statistical Learning](https://www.statlearning.com/).**\n- **By Gareth James, Daniela Witten, Trevor Hastie and Rob Tibshirani.**\n- **Pages from $197$ to $198$**\n- **By [Mosta Ashour](https://www.linkedin.com/in/mosta-ashour/)**\n\n\n**Exercises:**\n- **[1.](#1)**\n- **[2.](#2) [(a)](#2a) [(b)](#2b) [(c)](#2c) [(d)](#2d) [(e)](#2e) [(f)](#2f) [(g)](#2g) [(h)](#2h)**\n- **[3.](#3) [(a)](#3a) [(b)](#3b)**\n- **[4.](#4)**\n\n# 5.4 Exercises \n## Conceptual \n\n\n### $1.$ Using basic statistical properties of the variance, as well as single-variable calculus, derive $(5.6)$. In other words, prove that $\\alpha$ given by $(5.6)$ does indeed minimize $\\mathrm{Var}(\\alpha X + (1 \u2212 \\alpha)Y)$.\n\n**Using the following variance [rules](http://www.kaspercpa.com/statisticalreview.htm):**\n\n$\n\\text{Var}(X+Y) = \\text{Var}(X) + \\text{Var}(Y) + 2 Cov(X,Y)\n\\\\\n\\text{Var}(cX) = c^2 Var(X)\n\\\\\n\\text{So}\\dots\n\\\\\n\\text{Var}(cX+dY) = c^2 \\text{Var}(X) + d^2 \\text{Var}(Y) + 2cd Cov(X,Y)\n$\n\n$\n\\text{thus:}\\\\\nf(\\alpha) = \\text{Var}(\\alpha X + (1 - \\alpha)Y)\\\\\n\\\\\nf(\\alpha) = \\alpha^2 \\text{Var}(X) + (1 - \\alpha)^2 \\text{Var}(Y) + 2 \\alpha (1 - \\alpha) Cov(X, Y)\\\\\n\\\\\nf(\\alpha) = \\alpha^2 \\sigma_X^2 + (1 - \\alpha)^2 \\sigma_Y^2 + 2 (\\alpha-\\alpha^2 ) \\sigma_{XY}\\\\\n$\n\n$\n\\begin{align}\n\\text{Take the first derivative:}\\\\\n\\frac {d} {d\\alpha} f(\\alpha) &= 0\\\\\n\\\\\n2 \\alpha \\sigma_X^2 + 2 (1 - \\alpha) (-1) \\sigma_Y^2 + 2 (1 - 2 \\alpha ) \\sigma_{XY} &= 0\\\\\n\\\\\n\\alpha \\sigma_X^2 + (\\alpha - 1) \\sigma_Y^2 + (-2 \\alpha + 1) \\sigma_{XY} &= 0\\\\\n\\\\\n\\color{red}{\\alpha \\sigma_X^2} + \\color{red}{\\alpha\\sigma_Y^2} - \\sigma_Y^2 \\color{red}{-2 \\alpha \\sigma_{XY}} + \\sigma_{XY} &= 0\\\\\n\\\\\n\\alpha (\\sigma_X^2 + \\sigma_Y^2 - 2 \\sigma_{XY}) - \\sigma_Y^2 + \\sigma_{XY} &= 0\\\\\n\\\\\n\\alpha (\\sigma_X^2 + \\sigma_Y^2 - 2 \\sigma_{XY}) &= \\sigma_Y^2 - \\sigma_{XY}\\\\\n\\\\\n\\text{therefore:}\n\\\\\n\\alpha &= \\frac {\\sigma_Y^2 - \\sigma_{XY}}\n {\\sigma_X^2 + \\sigma_Y^2 - 2 \\sigma_{XY}}\n\\\\\n\\text{As required.}\\\\\n\\end{align}\n$\n\n\n### $2.$ We will now derive the probability that a given observation is part of a bootstrap sample. Suppose that we obtain a bootstrap sample from a set of $n$ observations.\n\n\n**$(a)$ What is the probability that the first bootstrap observation is not the $j$th observation from the original sample? Justify your answer.**\n\n>- **Answer:**\n> - $p(n_{1} \\neq n_j) = \\frac{n-1}{n}$\n> - Which all the others observations are possible except just the first observation \"1\".\n\n\n**$(b)$ What is the probability that the second bootstrap observation is not the $j$th observation from the original sample?**\n\n>- **Answer:**\n> - The same as the previous question:\n> - $p(n_{2} \\neq n_j) = \\frac{n-1}{n}$\n> - Which all the others observations are possible except just the second observation \"1\".\n\n\n**$(c)$ Argue that the probability that the $j$th observation is not in the bootstrap sample is $(1 - 1/n)^n$.**\n\n>- **Answer:**\n> - The probability that the $j$th observation is not in the bootstrap sample is \n> - $(1\u22121/n)_1\u22c5(1\u22121/n)_2\u22c5(1\u22121/n)_3\\dots\u22c5(1\u22121/n)_n$\n> - which is: $(1\u22121/n)^n$\n> - Since choosing any sample is independent of another but has the same probability value.\n\n\n**$(d)$ When $n = 5$, what is the probability that the $j$th observation is in the bootstrap sample?**\n\n>- **Answer:**\n> - when $n = 5$ and it is **in** the bootstrap sample:\n> - $p = 1 - (1-1/5)^5$\n> - $p = 0.67232$\n\n\n**$(e)$ When n = 100, what is the probability that the $j$th observation is in the bootstrap sample?**\n\n>- **Answer:**\n> - when $n = 100$ and it is **in** the bootstrap sample:\n> - $p = 1 - (1-1/100)^{100}$\n> - $p = 0.63397$\n\n\n**$(f)$ When $n = 10,000$, what is the probability that the $j$th observation is in the bootstrap sample?**\n\n>- **Answer:**\n> - when $n = 10,000$ and it is **in** the bootstrap sample:\n> - $p = 1 - (1-1/10000)^{10000}$\n> - $p = 0.63213$\n\n\n**$(g)$ Create a plot that displays, for each integer value of $n$ from $1$ to $100,000$, the probability that the $j$th observation is in the bootstrap sample. Comment on what you observe.**\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport seaborn as sns\n\ndef prob_j_in_sample(n):\n return 1 - (1 - 1/n)**n\n\nx = np.arange(1, 100000)\ny = np.array([prob_j_in_sample(n) for n in x])\n\nplt.figure(figsize=(12,6))\nax = sns.lineplot(x=x, y=prob_j_in_sample(x))\nplt.xlabel('$n$')\nplt.ylabel('probability')\nplt.ylim((0,1));\n```\n\n>- **Comments:**\n> - We are seeing a visual representation of $\\lim\\limits_{x \\to \\infty} 1 - (1 - \\frac{1}{n})^n = 1 - \\frac{1}{\\epsilon} \\approx 0.632$\n\n\n**$(h)$ We will now investigate numerically the probability that a bootstrap sample of size $n = 100$ contains the $j$th observation. Here $j = 4$. We repeatedly create bootstrap samples, and each time we record whether or not the fourth observation is contained in the bootstrap sample.**\n```\n> store=rep (NA , 10000)\n> for (i in 1:10000) {\nstore[i]=sum(sample (1:100 , rep =TRUE)==4) >0\n}\n> mean(store)\n```\n**Comment on the results obtained.**\n\n\n```python\nstore = [] \nfor i in np.arange(1, 10000):\n store += [np.sum((np.random.randint(low=1, high=101, size=100) == 4)) > 0]\n\nnp.mean(store)\n```\n\n\n\n\n 0.6335633563356335\n\n\n\n>- **Comments:**\n> - If we created more bootstrap samples in $h$, the result observed from a numerical approach above will approximately equal to our probabilistic estimation for a sample size of $100$ in $2.e$ which was $p = 0.63397$.\n\n\n### 3. We now review $\\text{k-fold}$ cross-validation.\n\n\n**$(a)$ Explain how $\\text{k-fold}$ cross-validation is implemented.**\n>- **Answer:**\n>- This approach involves randomly dividing the set of observations into $k$ groups, or folds, of approximately equal size. \n>- The first $k-1$ folds is treated as a validation set, and the remaining fold as test set.\n>- The model is then fitted to each of these folds, and testing on the remaining fold.\n>- This procedure is repeated $k$ times; each time, a different group of observations.\n>- This process results in k estimates of the test error,\n$\\text{MSE_1},\\text{MSE_2}, \\dots ,\\text{MSE_k}$. The k-fold CV estimate is computed by averaging these values,\n$$ CV(k) =\\frac{1}{k} \\sum_{i=1}^k MSE_i. $$\n\n\n**$(b)$ What are the advantages and disadvantages of k-fold cross- validation relative to:**\n\n**$i.$ The validation set approach?**\n\n- **Answer:**\n\n>- $\\text{Advantages:}$\n> - The validation set approach is **conceptually simple** and is **easy** to implement.\n> - We can use the validation set approach in testing which predictor's transformations provides even better results, as we randomly split the observations into $k$ approximately equal sets or folds, a training set of k-1 folds, and the remaining fold is the validation set. \n> - The validation set error rates that result from fitting various regression models on the training sample and evaluating their performance on the validation sample, using $\\text{MSE}$ as a measure of validation set error, comparing the $\\text{MSE}$ results will lead us to the better predictor's transformations to use in our model.\n\n>- $\\text{Disadvantages:}$\n> - The validation estimate of the test error rate can be **highly variable**, depending on precisely which observations are included in the training and validation sets.\n> - Only a subset of the observations those that are included in the training set rather than in the validation set are used to fit the model. Since statistical methods tend to perform worse when trained on fewer observations, this suggests that the validation set error rate may tend to **overestimate the test error rate** for the model fit on the entire data set.\n> - The approach has a computational advantage: a model is trained and tested once. In $k-fold$ CV, $k$ models will be trained, and for standard values of $k$. Means that $k-fold$ CV can be far more computationally expensive for large dataset and for large values of $k$.\n\n\n\n\n**$ii.$ LOOCV?**\n\n- **Answer:**\n\n>- $\\text{Advantages:}$\n> - It has far less bias over the validation set approach.\n> - Performing LOOCV multiple times will always yield the same results: there is no randomness in the training/validation set splits.\n> - $\\text{Leave-one-out cross-validation}$ involves splitting the set of observations into two parts. However, instead of creating two subsets of comparable size, a single observation $(x_1, y_1)$ is used for the validation set, and the remaining observations ${(x_2, y_2), \\dots , (x_n, y_n)}$ make up the training set. The statistical learning method is fit on the $n - 1$ training observations, and a prediction $\\hat{y}_1$ is made for the excluded observation, using its value $x_1$. Since $(x_1, y_1)$ was not used in the fitting process, $\\text{MSE}_1 = (y_1 - \\hat{y}_1)^2$ provides an approximately unbiased estimate for the test error.\n\n>- $\\text{Disadvantages:}$\n> - Even though $\\text{MSE}_1$ is unbiased for the test error, it is a poor estimate because it is highly variable, since it is based upon a single observation $(x_1, y_1)$.\n> - LOOCV requires fitting the statistical learning method $n$ times. This has the potential to be computationally expensive.\n\n\n### 4. Suppose that we use some statistical learning method to make a prediction for the response Y for a particular value of the predictor X. Carefully describe how we might estimate the standard deviation of our prediction.\n\n>- **Answer:**\n> - Which we need to estimate the standard deviation from a given $n$ observations of the population. We could might estimate the standard deviation of our prediction by using the [sample standard deviation](https://en.wikipedia.org/wiki/Standard_deviation#Corrected_sample_standard_deviation) formula:\n$$\n\\hat{\\sigma} = \\sqrt{\\frac{\\sum_{i=1}^{n}{(\\hat{y_i} - \\bar{y})^2}}{n - 1}}\n$$\n> - The accuracy of this estimate is limited by its variability, so... \n> - We could improve the accuracy of this estimate by using the bootstrap approach. This works by randomly select $n$ observations from the original dataset to create a $B$ different bootstrap datasets. \n> - This procedure is repeated $B$ times for some large value of $B$, in order to produce $B$ different bootstrap datasets, $Z^{*1}, Z^{*2}, \\dots , Z^{*B},$ and $B$ corresponding $\\alpha$ estimates, $\\hat{\u03b1}^{\u22171}, \\hat{\u03b1}^{\u22172}, \\dots , \\hat{\u03b1}^{\u2217B}$. We can compute the standard error of these bootstrap estimates using the formula in $(5.8)$:\n$$\nSE_B(\\hat{\\alpha}) =\n\\sqrt{\n\\frac{1}{B - 1}\n\\sum^B_{r=1}\n\\bigg(\\hat{\\alpha}^{*r} - \n\\frac{1}{B}\n\\sum^B_{r'=1}\n\\hat{\\alpha}^{*r}\n\\bigg)^2\n}.\n$$\n\n# Done!\n", "meta": {"hexsha": "ad8970687d7ee0ac107399c4dc332b4633d8641b", "size": 24048, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Notebooks/5_4_0_Resampling_Methods_Conceptual.ipynb", "max_stars_repo_name": "MostaAshour/ISL-in-python", "max_stars_repo_head_hexsha": "87255625066f88d5d4625d045bdc6427a4ad9193", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Notebooks/5_4_0_Resampling_Methods_Conceptual.ipynb", "max_issues_repo_name": "MostaAshour/ISL-in-python", "max_issues_repo_head_hexsha": "87255625066f88d5d4625d045bdc6427a4ad9193", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Notebooks/5_4_0_Resampling_Methods_Conceptual.ipynb", "max_forks_repo_name": "MostaAshour/ISL-in-python", "max_forks_repo_head_hexsha": "87255625066f88d5d4625d045bdc6427a4ad9193", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 61.8200514139, "max_line_length": 8664, "alphanum_fraction": 0.7059214904, "converted": true, "num_tokens": 3364, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784220301065, "lm_q2_score": 0.7371581684030623, "lm_q1q2_score": 0.4228917348360723}} {"text": "```python\nclean_up=True # removes gams-related files in work-folder if true\n%run StdPackages.ipynb\nos.chdir(py['main'])\nimport global_settings,ReadData,ShockFunction,Production,Household,GE,Invest,Trade,Government,diagnostics\nfrom DataBase_wheels import small_updates\nos.chdir(curr)\ndata_folder = os.getcwd()+'\\\\Data\\\\IO'\ngams_folder = os.getcwd()+'\\\\gamsmodels\\\\GE'\n```\n\n The file_gams_py_gdb0.gdx is still active and was not deleted.\n The file_gams_py_gdb3.gdx is still active and was not deleted.\n\n\n# Set up a dynamic, general equilibrium model\n\n*The current general equilibrium model is a small open economy that features exogenous long run interest-, inflation-, and growth rates. These settings are defined in the global settings:*\n\n\n```python\nname = 'GE'\ngs_v = 'gs_v1'\ntindex = range(1,4)\ngs_vals = {'t':tindex}\ngs = global_settings.gs_v1(kwargs_vals=gs_vals)\n```\n\n## **1: Setup**\n\nThe fundamental setup:\n* The model consists of three fundamental sets that most equations/variables are defined over: (1) $n$ the set of goods in the economy, (2) $s$ the set of sectors, $t$ the time index.\n* We distinguish between prices of various sorts $(PbT,Peq,PwT)$, quantities $(qS,qD)$, and values $(vD,vS)$. Beyond this, a number of other variables are additionally used in various modules.\n* Market clearing ensures that the sum of supply equals the sum of supply, for a subset of goods $n$. As goods can be demanded/supplied by more than one sector, the market clearing condition is:\n$$\\begin{align}\n \\sum_{s\\in d\\_qS[s,n]} qS[t,s,n] = \\sum_{s\\in d\\_qD[s,n]} qD[t,s,n],\n\\end{align}$$\nwhere $d\\_qS$ and $d\\_qD$ are dummies identifying which sectors are active for the relevant good.\n\n## **2: Data**\n\nThe relevant data needed to run each module may vary. However, in general, the general equilibrium module should be adjusted to be consistent with input-output data. In a single year, the IO baseline data should at least cover:\n* The equilibrium price for all traded goods.\n* The inputs/outputs in values for each domestic sector, for each type of goods. The system must be balanced, such that the sum of demand equals the sum of supply for each $s$.\n\nThe following reads in the IO data and defines a number of default subsets:\n\n\n```python\ndsheets = {'Production_v': data_folder+'\\\\IO_v.xlsx', 'Production_p': data_folder+'\\\\IO_p.xlsx'}\nGE_data = ReadData.read_data.main(dsheets,name='GE_data',components=['domstic','trade','HH','tax','invest'])\n```\n\nTo the IO data, we add inventory data on durables:\n\n\n```python\nDataBase.GPM_database.merge_dbs(GE_data,excel2py.xl2PM.pm_from_workbook(data_folder+'\\\\'+'dur.xlsx',{k: 'vars' for k in ('prod','HH','G')}),'second')\n```\n\nFurthermore, we read in data on tax rates:\n\n\n```python\nDataBase.GPM_database.merge_dbs(GE_data,excel2py.xl2PM.pm_from_workbook(data_folder+'\\\\'+'Tax.xlsx',{'tbaseyear': 'vars'}),'second')\n```\n\nNote that - while not necessary - the tax rates should be of a size such that the total tax revenue corresponds to the income from the IO data; the government module can adjust some tax rates to ensure this, however, this will be in a somwhat random fashion. To check for this assertion, the following IO function computes the tax income on sectorial level, when tax rates consists of three componenets: Input taxes $tauD$, output taxes $tauSflat$ and lump sum taxes $tauLump$:\n\n\n```python\npd.DataFrame({'Model': Government.taxRevenue(GE_data), 'IO': GE_data['vD'].rctree_pd(GE_data['n_tax']).droplevel(-1)})\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        ModelIO
                                        s
                                        F0.0909422-0.018329
                                        HH1.800991.61747
                                        I_B0.2501480.263686
                                        I_M0.2075960.202365
                                        a0.4425360.058697
                                        b1.605570.827386
                                        itory0.001556620.00228146
                                        \n
                                        \n\n\n\nSome of these discrepancies can be adjusted using the lump-sum tax rate; however, not all sectors are taxed lump-sum. The following adjusts the lump sum taxes, but leaves the two sectors 'inventory' and 'foreign' sectors unbalanced:\n\n\n```python\nGE_data['tauLump'] = Government.balanceIO_lumpsum(GE_data)\npd.DataFrame({'Model': Government.taxRevenue(GE_data), 'IO': GE_data['vD'].rctree_pd(GE_data['n_tax']).droplevel(-1)})\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        ModelIO
                                        s
                                        F0.0909422-0.018329
                                        HH1.617471.61747
                                        I_B0.2636860.263686
                                        I_M0.2023650.202365
                                        a0.0586970.058697
                                        b0.8273860.827386
                                        itory0.001556620.00228146
                                        \n
                                        \n\n\n\nTo adjust the final two sectors' tax revenues, we can move the taxes on demand upward/downward by the same amount for all goods. we make this adjustment on the demand taxes ($tauD$), as the sectors that are not taxed lump-sum (itory, F) are not supplying anything in the model:\n\n\n```python\nrates = Government.balanceIO_advalorem(GE_data)\nGE_data['tauD'] = rates[~(rates==0)]\n```\n\nAnd adjust the prices with taxes, to be defined as the equilibrium price, plus taxes:\n\n\n```python\nGE_data['PwT'].vals = GE_data.get('PwT').add(GE_data.get('tauD'),fill_value=0)\n```\n\nExport the database:\n\n\n```python\nGE_data.export(repo=gams_folder)\n```\n\nBeyond this, we require a lot of data. This is handled along the way in the specific modules.\n\n## **3: Model components**\n\nIn the following we go through the various modules, including (1) which data we require for the model to run, (2) how to initialize the model, (3) run and calibrate the model separately, and (4) store the calibrated version.\n\n## **3.1: The *Production.py* module**\n\n*The production module is handled in more detail in the notebook 'Ex1\\_production.ipynb'. Here we run briefly through the model.*\n\nThe production module is build primarily on nesting trees. The trees specify how the sectors combine inputs to produce its respective outputs. A number of different type of nesting functions are available (see *gams\\_production.py*). In the following, the two production sectors in our toy model simply use the relevant inputs in a nested CES manner.\n\n### *Required Data:*\n\nThe necessary data includes:\n1. IO data on the sectors (already included).\n2. The nesting structure of inputs including technical parameters detailing the production structure (CES parameters).\n3. An inventory of the durables owned by each sector.\n\nIn the current case, the model includes two durables for each production sector: Machines ($iM$) and buildings ($iB$). This information is read from the IO data, but can also be added manually along the way. The production module includes quite a lot of flexibility (e.g. in nesting structure and functional forms). The following illustrates an example where a lot of default values are applied: \n\n*Define settings for the module:*\n\n\n```python\nname_module = 'p'\ntrees = {'a': {'file': 'S1.xlsx', 'sheets': ['lower_nests', 'upper_nest']}, \n 'b': {'file': 'S2.xlsx', 'sheets': ['lower_nests', 'upper_nest']}}\ntemp_namespace = {'a': 'a_in', 'b':'b_in'} # used when reading in nesting trees, to distinguish input and output elements with identical names.\nkwargs_st = {'sector': True, 'ss': GE_data.get('s_prod')} # settings for initializing the module\n```\n\n*Read in data to nesting tree:*\n\n\n```python\nnts = {}\nfor s,t in trees.items():\n nts[s] = nesting_tree.nesting_tree(name=name_module) # initialize tree\n for tree in t['sheets']:\n nts[s].add_tree(data_folder+'\\\\'+t['file'],tree_name=tree,**{'sheet':tree}) # add nesting structure\n DataBase.GPM_database.merge_dbs(nts[s].trees[tree].database, excel2py.xl2PM.pm_from_workbook(data_folder+'\\\\'+t['file'],{tree:'vars'}),'second') # add data\n if tree.startswith('lower'):\n nts[s].trees[tree].temp_namespace = temp_namespace\n nts[s].run_all(s0=s) # add default attributes from nesting structure\nnesting_tree.merge_nts(list(nts.values())[0], list(nts.values())[1:]) # merge trees into one.\nnt = list(nts.values())[0]\n```\n\n### *Static model:*\n\nStart by setting up a static model from the nesting tree:\n\n\n```python\ngm_static = Production.pr_static(nt=nt,work_folder = work_folder,**{'data_folder':work_folder,'name':'p_static'})\n```\n\nAdd data from the general equilibrium data, but restrict it to the data on production sectors (not necessary, but neat):\n\n\n```python\nGE_prod = small_updates.subset_db(GE_data.copy(),GE_data.get('s_prod'))\n```\n\nCalibrate the model (See *Ex1_production* for more on these settings):\n\n*1: Calibrate to inputs that are exogenous in the baseline settings:*\n\n\n```python\ngm_static.write_and_run(name='v1',add_checkpoint='v1')\ndb_temp = gm_static.slice_exo(GE_prod,copy=True)\ngm_static.model_instances['v1'].solve_sneakily(db_star=db_temp, from_cp = True, cp_init = gm_static.checkpoints['v1'], kwargs_shock={'n_steps':10})\n```\n\n\n\n\n {'Modelstat': 16.0, 'Solvestat': 1.0}\n\n\n\n*2: read current solution back to main database:*\n\n\n```python\ndb = gm_static.model_instances['v1'].out_db\n[db.series.__delitem__(sym) for sym in db.symbols if sym not in gm_static.model.database.symbols]; # delete symbols that are created only in order to calibrate model.\ngm_static.model.database.merge_dbs(gm_static.model.database,db,'second')\n```\n\n*3: calibrate to the other moments in the data:*\n\n\n```python\ngm_static.reset_settings()\ngm_static.setstate('DC')\ndb_temp = gm_static.slice_exo(GE_prod,copy=True)\ngm_static.calibrate_sneaky(db_temp,**{'n_steps':100})\n```\n\n\n\n\n {'Modelstat': 16.0, 'Solvestat': 1.0}\n\n\n\n*inspect solution compared to IO data:*\n\n\n```python\ndb = gm_static.model_instances['baseline'].out_db\n```\n\n\n```python\nvar= 'qD'\ndiagnostics.compare_data.std_diagnostics_var(db,GE_data,var)\n```\n\n### *Dynamic model:*\n\nNext, we set up the dynamic model, building on the static solution, as well as the IO data. Specifically, we: (1) Initialize using nesting structure, (2) add specification of durables from IO data, (3) Initialize parameter values and initial guess of variables using static model, (4) run.\n\n\n```python\ngm_p = Production.pr_dynamic(nt=nt,work_folder=work_folder,kwargs_st = {'sector':True, 'ss': GE_data.get('s_prod')}, gs_v=gs,**{'data_folder':gams_folder,'name':name_module}) # initialize model.\ngm_p.add_dur(GE_data.get('dur'),dur2inv = GE_data.get('dur2inv')) # add specification of durables\ngm_p.ivfs(db,merge=False) # initialzie levels from static model\ngm_p.initialize_variables(**{'check_variables': True}) # give all variables without a initial level some (semi) random number.\ngm_p.model.database[gm_p.n('mu')].vals = db.get(gm_p.n('mu')) # update values of mu-parameters to static solution\n```\n\nFinally, update the prices $PwT$ on investment goods that are not automatically loaded from the static version (where the investment behavior was not included):\n\n\n```python\ngm_p.model.database[gm_p.n('PwT')] = DataBase_wheels.repeat_variable_windex(GE_data['PwT'].rctree_pd(GE_data['inv']),gs.get('txE')).combine_first(gm_p.get('PwT'))\n```\n\nTo calibrate to IO data, we start by adding the baseline year to the IO data:\n\n\n```python\nGE_prod_t = DataBase.GPM_database()\nfor var in GE_prod.variables_flat:\n GE_prod_t[var] = DataBase_wheels.repeat_variable_windex(GE_prod.get(var),gm_p.get('t0'))\n```\n\nWe further adjust the capital depreciation rates to ensure the model is in a steady state. The depreciation rates are then fitted to data at a later point in the proces (see Example1_cont.ipynb):\n\n\n```python\ngm_p.ss_rDepr(GE_data)\n```\n\nThe calibration method we apply form a grid of values between the database in the model, and the target database (GE\\_prod\\_t in this case), and asks GAMS to solve the model on this grid. However, this only works on exogenous variables. To make sure that this works we do the following:\n1. Set the *state* of the model to calibration.\n2. Subset the target database *GE\\_prod\\_t* to only include exogenous variables (all *gmspython* models can access the exogenous/endogenous variables by calling the *self.var_exo(symbol)* method).\n3. Run the calibration function.\n\n\n```python\ngm_p.setstate('DC')\n```\n\n*Subset variables to exogenous variables:*\n\n\n```python\nGE_prod_t = gm_p.slice_exo(GE_prod_t,copy=False)\n```\n\n*Calibrate sneakily (target the database GE\\_prod\\_t, if files w. same names exists overwrite them, sneak up on the solution in n\\_steps):*\n\n\n```python\ngm_p.calibrate_sneaky(GE_prod_t,overwrite=True,**{'n_steps': 10,'diff':True})\n```\n\n\n\n\n {'Modelstat': 16.0, 'Solvestat': 1.0}\n\n\n\n*Store as pickle to run from at a later point:*\n\n\n```python\ndb = gm_p.model_instances['baseline'].out_db \n[db.series.__delitem__(sym) for sym in db.symbols if sym not in gm_p.model.database.symbols]; # delete symbols that are created only in order to calibrate model.\ngm_p.model.database.merge_dbs(gm_p.model.database,db,'second')\ngm_p.export()\n```\n\n\n\n\n 'C:\\\\Users\\\\sxj477\\\\Documents\\\\GitHub\\\\GPM_v05\\\\examples\\\\gamsmodels\\\\GE\\\\gmspython_p'\n\n\n\n\n```python\ncompare_vars,year = 'qD',2\nci = DataBase.gpy_symbol(db[compare_vars].rctree_pd(GE_prod[compare_vars]).xs(year))\npd.DataFrame({'Baseline': db[compare_vars].rctree_pd(ci).xs(year), 'IO data': GE_prod[compare_vars].rctree_pd(ci)}).plot.bar(figsize=(8,6));\n```\n\n## **3.2: The *Household.py* module**\n\n*The household module is handled in more detail in the notebook 'Ex1\\_household.ipynb'.*\n\nSimilarly to the production module, the household module is also initialized using nesting trees. The trees specify how the consumer derives utility from aggregating goods. The nesting trees used for the household are different, however, in a number of ways. For instance, whereas the 'most upper' nest of the production module always symbolizes outputs (supply), and the 'most lower' nest inputs (demand), there is no such restriction on the household nesting tree. \n\nThe current module uses a CRRA intertemporal utility setup, that is only tied to the upper-most aggregate level of consumption in the nesting tree (this can easily be altered). Furthermore, we fix the labor supply (but can similarly easily be endogenized / included in the nesting tree). \n\n### *Required Data:*\n\nThe necessary data includes:\n1. IO data on the sectors (already included).\n2. Nesting structure of the utility function, including preference parameters (impatience, CES, CRRA).\n3. An inventory of the assets/endowments of the household sector(s), as well as relevant tax rates (see tax module for more).\n\n*Define settings for the module (simpler setup than production, as we only use one tree, i.e. no looping)*\n\n\n```python\nname_module = 'HH' \ntree_name = 'HH_agg' \nfile, sheet = 'HH.xlsx', 'nesting'\n```\n\n*Define partial equilibrium data, and general equilibrium data that are subsetted for the household sector:*\n\n\n```python\nPE_data = ReadData.PE_from_GE(GE_data,GE_data.get('s_HH')[0])\nGE_HH = small_updates.subset_db(GE_data.copy(),GE_data.get('s_HH'))\n```\n\n*read in nesting tree (similar to production, but a simpler setup):*\n\n\n```python\nnt = nesting_tree.nesting_tree_hh(name=name_module,**{'version':'v1'}) # the version specifies calibration type.\nnt.add_tree(data_folder+'\\\\'+file,tree_name=tree_name,**{'sheet':sheet}) # initialize tree\ndata = excel2py.xl2PM.pm_from_workbook(data_folder+'\\\\'+file,{sheet:'vars'})\nDataBase.GPM_database.merge_dbs(nt.trees[tree_name].database, data ,'second') # add data\nnt.run_all(PE_data,s0=GE_data.get('s_HH')[0],postfix='_'+name_module,int_temp = data['crra'].index) # add attributes\n```\n\n### *Static model:*\n\nAs in the production module, we start by setting up a static version of the model and calibrate this to the IO data. The steps are generally the same here as in the production module: (1) Initialize static model from nesting tree, (2) Add IO data to the model database, (3) solve and calibrate model.\n\n*set up data and model:*\n\n\n```python\ngm_static = Household.hh_static(nt=nt,work_folder=work_folder,**{'data_folder':work_folder,'name':'hh_static'})\nDataBase.GPM_database.merge_dbs(gm_static.model.database,GE_HH,'second')\n```\n\n*slice IO data to only use exongeous data:*\n\n\n```python\ngm_static.setstate('DC') # set state to 'dynamic calibration (DC), don't initialize settings\nGE_HH_exo = gm_static.slice_exo(GE_HH,copy=True)\n```\n\n*calibrate (sneakily):*\n\n\n```python\ngm_static.calibrate_sneaky(GE_HH_exo,kwargs_init={'check_variables':True},**{'n_steps':10})\ndb_static = gm_static.model_instances['baseline'].out_db\n```\n\n### *Dynamic model:*\n\nThe dynamic model is then based on the static one (as was the case w. the production module). Specifically we (1) initialize from the nesting tree, (2) add the savings subset, (3) initialize levels from the static model solution:\n\n\n```python\ngm_hh = Household.hh_dynamic(nt=nt,work_folder=work_folder, gs_v = gs,**{'data_folder':gams_folder,'name':name_module})\ngm_hh.add_svngs() # define subset of savings; call this element 'svngs' as default.\ngm_hh.ivfs(db_static,merge=False) # initialize levels from static model\ngm_hh.initialize_variables(**{'check_variables':True})\ngm_hh.model.database[gm_hh.n('mu')].vals = db_static.get(gm_hh.n('mu')) # update calibrated parameters\n```\n\nNext, we add the baseline year to the IO data, slice the IO data to the values that are exogenous in the model and calibrate:\n\n\n```python\nGE_HH_t = DataBase.GPM_database()\nfor var in GE_HH.variables_flat:\n GE_HH_t[var] = DataBase_wheels.repeat_variable_windex(GE_HH.get(var),gm_hh.get('t0'))\ngm_hh.setstate('DC') # set state to 'dynamic calibration (DC)'\nGE_HH_t = gm_hh.slice_exo(GE_HH_t,copy=False)\ngm_hh.calibrate_sneaky(GE_HH_t,overwrite=True,**{'n_steps': 10,'diff':True})\n```\n\n\n\n\n {'Modelstat': 16.0, 'Solvestat': 1.0}\n\n\n\n*Store as pickle to run from at a later point:*\n\n\n```python\ndb = gm_hh.model_instances['baseline'].out_db \n[db.series.__delitem__(sym) for sym in db.symbols if sym not in gm_hh.model.database.symbols]; # delete symbols that are created only in order to calibrate model.\ngm_hh.model.database.merge_dbs(gm_hh.model.database,db,'second')\ngm_hh.export()\n```\n\n\n\n\n 'C:\\\\Users\\\\sxj477\\\\Documents\\\\GitHub\\\\GPM_v05\\\\examples\\\\gamsmodels\\\\GE\\\\gmspython_HH'\n\n\n\n\n```python\ncompare_vars,year = 'qD',2\nci = DataBase.gpy_symbol(db[compare_vars].rctree_pd(GE_HH[compare_vars]).xs(year))\npd.DataFrame({'Baseline': db[compare_vars].rctree_pd(ci).xs(year), 'IO data': GE_HH[compare_vars].rctree_pd(ci)}).plot.bar(figsize=(4,3));\n```\n\n## **3.3: The *investment.py* module**\n\n*The investment module is handled in more detail in the notebook 'Ex1\\_invest.ipynb'.*\n\nThe investment module essentially works as a production module (when static). The dynamic module is slightly different.\n\n### *Required Data:*\n\n*Define settings for the module:*\n\n\n```python\nname_module = 'inv'\ntrees = {'I_B': {'file': 'inv_B.xlsx', 'sheets': ['nest']}, \n 'I_M': {'file': 'inv_M.xlsx', 'sheets': ['nest']}}\nkwargs_st = {'sector': True, 'ss': GE_data.get('s_inv')} # settings for initializing the module\nnamespace = {k: name+'_'+k for k in ('inp','out','int','wT','map_all','kno_out','kno_inp','n_out','endo_PbT','exo_mu','PwT_dom')} # as the production module uses the same nesting tree, we adjust the standard names here\n```\n\n*Define tree:*\n\n\n```python\nnts = {}\nfor s,t in trees.items():\n nts[s] = nesting_tree.nesting_tree(name=name_module) # initialize tree\n for tree in t['sheets']:\n nts[s].add_tree(data_folder+'\\\\'+t['file'],tree_name=tree,**{'sheet':tree}) # add nesting structure\n DataBase.GPM_database.merge_dbs(nts[s].trees[tree].database, excel2py.xl2PM.pm_from_workbook(data_folder+'\\\\'+t['file'],{tree:'vars'}),'second') # add data\n nts[s].run_all(s0=s,**namespace) # add default attributes from nesting structure\nnesting_tree.merge_nts(list(nts.values())[0], list(nts.values())[1:]) # merge trees into one.\nnt = list(nts.values())[0]\n```\n\n### *Static model:*\n\n*similar to production module: initialize from nt, add IO data, calibrate.*\n\n\n```python\ngm_static = Invest.pr_static(nt=nt,work_folder = work_folder,kwargs_ns=namespace,**{'data_folder':work_folder,'name':'I_static'})\nGE_inv = small_updates.subset_db(GE_data.copy(),GE_data.get('s_inv'))\nDataBase.GPM_database.merge_dbs(gm_static.model.database,GE_inv,'second')\ngm_static.setstate('DC')\ndb_temp = gm_static.slice_exo(GE_inv,copy=True)\ngm_static.calibrate_sneaky(db_temp,kwargs_init={'check_variables':True},**{'n_steps':100, 'gridtype': 'pol','phi':0.9})\ndb_static = gm_static.model_instances['baseline'].out_db\n```\n\n### *Dynamic model:*\n\nSet up model and build on static setup/solution:\n\n\n```python\ngm_inv = Invest.inv_dynamic(nt=nt,work_folder=work_folder,gs_v=gs,kwargs_st=kwargs_st,kwargs_ns=namespace,**{'data_folder':gams_folder,'name': name_module})\ngm_inv.ivfs(db_static,merge=False)\ngm_inv.initialize_variables(**{'check_variables':True})\ngm_inv.model.database[gm_inv.n('mu')].vals = db_static.get(gm_inv.n('mu'))\ngm_inv.model.database[gm_inv.n('markup')].vals = db_static.get(gm_inv.n('markup'))\n```\n\nThe investment module does not need a dynamic calibration; thus we simply solve the model in baseline mode and store it:\n\n\n```python\ngm_inv.setstate('DC')\ngm_inv.setstate('B')\ngm_inv.write_and_run(overwrite=True) # the overwrite=True option overwrites existing file with same names.\n```\n\n\n```python\ndb = gm_inv.model_instances['baseline'].out_db \n[db.series.__delitem__(sym) for sym in db.symbols if sym not in gm_inv.model.database.symbols];\ngm_inv.model.database.merge_dbs(gm_inv.model.database,db,'second')\ngm_inv.export()\n```\n\n\n\n\n 'C:\\\\Users\\\\sxj477\\\\Documents\\\\GitHub\\\\GPM_v05\\\\examples\\\\gamsmodels\\\\GE\\\\gmspython_inv'\n\n\n\n\n```python\ncompare_vars,year = 'Peq',2\nci = DataBase.gpy_symbol(db[compare_vars].rctree_pd(GE_inv[compare_vars]).xs(year))\npd.DataFrame({'Baseline': db[compare_vars].rctree_pd(ci).xs(year), 'IO data': GE_inv[compare_vars].rctree_pd(ci)}).plot.bar(figsize=(6,4.5));\n```\n\n### *Inventory investments:*\n\nThe IO data can also include 'inventory investments'. In this example, we simply include a \"demand\" for inventory that tends to zero over time (in an AR(1) manner). The inclusion of inventory investments are thus primarily to match IO data; behavior is ad hoc.\n\n\n```python\nGE_itory = small_updates.subset_db(GE_data.copy(),GE_data.get('s_itory'))\nfor var in GE_itory.variables_flat:\n GE_itory[var] = DataBase_wheels.repeat_variable_windex(GE_itory.get(var),gs.get('t0'))\n```\n\n\n```python\nname_module = 'itory'\ngm_itory = Invest.itoryD(work_folder=work_folder,databases=[GE_itory],gs_v=gs,**{'data_folder':gams_folder,'name':name_module})\ngm_itory.write_and_run(kwargs_init={'check_variables':True},overwrite=True)\n```\n\nExport as pickle:\n\n\n```python\ndb = gm_itory.model_instances['baseline'].out_db \n[db.series.__delitem__(sym) for sym in db.symbols if sym not in gm_itory.model.database.symbols];\ngm_itory.model.database.merge_dbs(gm_itory.model.database,db,'second')\ngm_itory.export()\n```\n\n\n\n\n 'C:\\\\Users\\\\sxj477\\\\Documents\\\\GitHub\\\\GPM_v05\\\\examples\\\\gamsmodels\\\\GE\\\\gmspython_itory'\n\n\n\n\n```python\ncompare_vars,year = 'qD',1\nci = DataBase.gpy_symbol(db[compare_vars].rctree_pd(GE_data[compare_vars]).xs(year))\npd.DataFrame({'Baseline': db[compare_vars].rctree_pd(ci).xs(year), 'IO data': GE_data[compare_vars].rctree_pd(ci)}).plot.bar(figsize=(6,4.5));\n```\n\n## **3.4: The *Trade.py* module**\n\n*The trade module is handled in more detail in the notebook 'Ex1\\_trade.ipynb'.*\n\nThe trade model can be specified in various ways. In this simple module, the trade modules simply specifies the foreign demand for domestically produced goods in an Armington-like manner. As there are more than one foreign/domestic goods, the prices entering the demand function are the domestic price relative to the foreign price of a similar good; this similarity is declared in a mapping denoted *dom2for[n,nn]* coupling domestic goods to a foreign counterpart.\n\n*Define settings for the module:*\n\n\n```python\nname_module = 'trade'\ndata = {'file': 'trade.xlsx', 'sheets': {'vars': 'vars', 'dom2for': 'maps'}} # read in the sheet 'vars' as variables, and 'dom2for' as mappings.}\nkwargs_st = {'sector': True, 'ss': GE_data.get('s_for')} # settings for initializing the module\n```\n\n*load data:*\n\n\n```python\ndb = excel2py.xl2PM.pm_from_workbook(data_folder+'\\\\'+data['file'],data['sheets'])\n```\n\n*initialize model:*\n\n\n```python\ngm_trade = Trade.trade_dynamic(work_folder=work_folder,kwargs_st=kwargs_st,gs_v=gs,**{'data_folder':gams_folder,'name':name_module})\n```\n\n*subset GE data to relevant sectors:*\n\n\n```python\nGE_trade = small_updates.subset_db(GE_data.copy(),GE_data.get('s_for'))\n```\n\n*initialize relevant subsets from GE data, and add data loaded from excel sheets:*\n\n\n```python\ngm_trade.add_sets_from_GE(GE_trade)\nDataBase.GPM_database.merge_dbs(gm_trade.model.database,db,'second')\n```\n\n*initialize variables from GE data:*\n\n\n```python\ngm_trade.ivfs(GE_trade,merge=False) # initialize levels from static model\n```\n\n*set to calibration mode and solve:*\n\n\n```python\nGE_trade_t = DataBase.GPM_database()\nfor var in GE_trade.variables_flat:\n GE_trade_t[var] = DataBase_wheels.repeat_variable_windex(GE_trade.get(var),gm_trade.get('t0'))\ngm_trade.setstate('DC')\nGE_trade_t = gm_trade.slice_exo(GE_trade_t,copy=False)\ngm_trade.calibrate_sneaky(GE_trade_t,overwrite=True,**{'n_steps':2,'diff':True})\n```\n\n\n\n\n {'Modelstat': 15.0, 'Solvestat': 1.0}\n\n\n\n*Export as pickle:*\n\n\n```python\ndb = gm_trade.model_instances['baseline'].out_db \n[db.series.__delitem__(sym) for sym in db.symbols if sym not in gm_trade.model.database.symbols];\ngm_trade.model.database.merge_dbs(gm_trade.model.database,db,'second')\ngm_trade.export()\n```\n\n\n\n\n 'C:\\\\Users\\\\sxj477\\\\Documents\\\\GitHub\\\\GPM_v05\\\\examples\\\\gamsmodels\\\\GE\\\\gmspython_trade'\n\n\n\n\n```python\ncompare_vars,year = 'qD',2\nci = DataBase.gpy_symbol(db[compare_vars].rctree_pd(GE_data[compare_vars]).xs(year))\npd.DataFrame({'Baseline': db[compare_vars].rctree_pd(ci).xs(year), 'IO data': GE_data[compare_vars].rctree_pd(ci)}).plot.bar(figsize=(6,4.5));\n```\n\n## **3.5: The *Government.py* module**\n\nThe government sector defines the tax rates that goes into other modules. In the simplest case, we assume that taxation are in constant rates (from the perspective of the firms/consumers). This entails that three tax rates should be supplied by the government sector module:\n* $tauD[t,s,n]$: The tax rate levied in the use of inputs such that $PwT[t,s,n] = Peq[t,n]+tauD[t,s,n]$ where $PwT$ is the price with taxes for sectors $s$, good $n$. The revenue from this tax is thus defined as $\\sum_{(s,n)\\in d\\_tauD[s,n]} qD[t,s,n]*tauD[t,s,n]$.\n* $tauS[t,s,n]$: The tax rate levied on the output of a sector. This defines the difference between prices before taxes and equilibrium taxes (some variations in how they are specifically used in modules). The revenue from this tax is defined as $\\sum_{(s,n)\\in d\\_tauS[s,n]} qS[t,s,n] * tauS[t,s,n]$.\n* $tauLump[t,s]$ for $s\\in s\\_HH$: The lump sum tax that is charged on households. For other sectors (production, investment), the lump-sum tax is computed as a part of the $tauS[t,s,n]$ tax on the sectors' outputs.\n\nNote that this restricts us to only charge constant rates for supply/demand of any type, i.e. the marginal tax rate = average tax rate. However, nothing prevents us from creating more elaborate functions for how these prices may evolve (as functions e.g. of quantity variables). \n\n### *A simple setup for tax rates:*\n\nIn the following we define a simple government sector with the following instruments:\n* $tauLump[t,s]$: A lump-sum tax is levied on all sectors except (1) trade and (2) inventory investment sectors.\n* $tauS[t,s,n]$: A flat rate charged on the supply of goods.\n* $tauD[t,s,n]$: A flat rate charged on the demand of goods.\n\n### *Data requirements:*\n\nData on the specific tax rates are per default set to 0, unless otherwise specified. In this instance, we read the tax rates in the data section (from the file 'Tax.xlsx'), and added them to the IO data stored in *GE_data*. They were, however, only for the baseline year. In this instance, we assume these are kept constant (for now). Other versions of the government sector may include more elaborate tax rules, thus endogenizing the rates. This should be straightforward to provide.\n\n*settings:*\n\n\n```python\nname_module = 'G'\n```\n\n### *Static model:*\n\n\n```python\ngm_static = Government.g_static(GE_data=GE_data.copy(),work_folder=work_folder,**{'data_folder':gams_folder,'name':'g_static'})\n```\n\nRun simple baseline model:\n\n\n```python\ngm_static.write_and_run(kwargs_init={'check_variables':True},overwrite=True)\n```\n\n*Calibrate:*\n\n\n```python\ngm_static.setstate('SC')\ngm_static.reset_settings() # this resets the RunFile and CollectFile settings.\ngm_static.write_and_run(overwrite=True) # this overwrites files in the work_folder if they already exists.\ndb_static = gm_static.model_instances['baseline'].out_db\n```\n\n### *Dynamic model:*\n\n\n```python\ngm_G = Government.g_dynamic(GE_data=GE_data.copy(),work_folder=work_folder,gs_v=gs,**{'data_folder':gams_folder,'name':name_module})\ngm_G.ivfs(db_static,merge=False)\ngm_G.initialize_variables(**{'check_variables': True})\n```\n\n\n```python\nGE_G = small_updates.subset_db(GE_data.copy(),gm_G.get('s_G'))\nGE_G_t = DataBase.GPM_database()\nfor var in GE_G.variables_flat:\n GE_G_t[var] = DataBase_wheels.repeat_variable_windex(GE_G.get(var),gm_G.get('t0'))\ngm_G.setstate('DC')\nGE_G_t = gm_G.slice_exo(GE_G_t,copy=False)\ngm_G.calibrate_sneaky(GE_G_t,overwrite=True,**{'n_steps':2,'diff':True})\n```\n\n\n\n\n {'Modelstat': 15.0, 'Solvestat': 1.0}\n\n\n\n*Store as pickle to run from at a later point:*\n\n\n```python\ndb = gm_G.model_instances['baseline'].out_db \n[db.series.__delitem__(sym) for sym in db.symbols if sym not in gm_G.model.database.symbols];\ngm_G.model.database.merge_dbs(gm_G.model.database,db,'second')\ngm_G.export()\n```\n\n\n\n\n 'C:\\\\Users\\\\sxj477\\\\Documents\\\\GitHub\\\\GPM_v05\\\\examples\\\\gamsmodels\\\\GE\\\\gmspython_G'\n\n\n\n\n```python\ncompare_vars,year = 'tauD',1\nci = DataBase.gpy_symbol(db[compare_vars].rctree_pd(GE_data[compare_vars]).xs(year))\npd.DataFrame({'Baseline': db[compare_vars].rctree_pd(ci).xs(year), 'IO data': GE_data[compare_vars].rctree_pd(ci)}).plot.bar(figsize=(8,6));\n```\n", "meta": {"hexsha": "73738b09f7dca3bd14857b525744521c29634382", "size": 138138, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/.ipynb_checkpoints/Example1-checkpoint.ipynb", "max_stars_repo_name": "ChampionApe/GPM_v05", "max_stars_repo_head_hexsha": "fa44f7db58d30002726daf014c63734091c85860", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-05-18T07:11:15.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-18T07:11:15.000Z", "max_issues_repo_path": "examples/Example1.ipynb", "max_issues_repo_name": "ChampionApe/GPM_v05", "max_issues_repo_head_hexsha": "fa44f7db58d30002726daf014c63734091c85860", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "examples/Example1.ipynb", "max_forks_repo_name": "ChampionApe/GPM_v05", "max_forks_repo_head_hexsha": "fa44f7db58d30002726daf014c63734091c85860", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 75.2795640327, "max_line_length": 20968, "alphanum_fraction": 0.8085392868, "converted": true, "num_tokens": 8639, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850933, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.4228825127252626}} {"text": "#
                                        Applied Stochastic Processes HW_06
                                        \n\n**
                                        11510691 \u7a0b\u8fdc\u661f$\\DeclareMathOperator*{\\argmin}{argmin}\n\\DeclareMathOperator*{\\argmax}{argmax}\n\\DeclareMathOperator*{\\plim}{plim}\n\\newcommand{\\using}[1]{\\stackrel{\\mathrm{#1}}{=}}\n\\newcommand{\\ffrac}{\\displaystyle \\frac}\n\\newcommand{\\asim}{\\overset{\\text{a}}{\\sim}}\n\\newcommand{\\space}{\\text{ }}\n\\newcommand{\\bspace}{\\;\\;\\;\\;}\n\\newcommand{\\QQQ}{\\boxed{?\\:}}\n\\newcommand{\\void}{\\left.\\right.}\n\\newcommand{\\Tran}[1]{{#1}^{\\mathrm{T}}}\n\\newcommand{\\d}[1]{\\displaystyle{#1}}\n\\newcommand{\\CB}[1]{\\left\\{ #1 \\right\\}}\n\\newcommand{\\SB}[1]{\\left[ #1 \\right]}\n\\newcommand{\\P}[1]{\\left( #1 \\right)}\n\\newcommand{\\abs}[1]{\\left| #1 \\right|}\n\\newcommand{\\norm}[1]{\\left\\| #1 \\right\\|}\n\\newcommand{\\dd}{\\mathrm{d}}\n\\newcommand{\\Exp}{\\mathrm{E}}\n\\newcommand{\\RR}{\\mathbb{R}}\n\\newcommand{\\EE}{\\mathbb{E}}\n\\newcommand{\\NN}{\\mathbb{N}}\n\\newcommand{\\ZZ}{\\mathbb{Z}}\n\\newcommand{\\QQ}{\\mathbb{Q}}\n\\newcommand{\\AcA}{\\mathscr{A}}\n\\newcommand{\\FcF}{\\mathscr{F}}\n\\newcommand{\\Var}[2][\\,\\!]{\\mathrm{Var}_{#1}\\left[#2\\right]}\n\\newcommand{\\Cov}[2][\\,\\!]{\\mathrm{Cov}_{#1}\\left(#2\\right)}\n\\newcommand{\\Corr}[2][\\,\\!]{\\mathrm{Corr}_{#1}\\left(#2\\right)}\n\\newcommand{\\I}[1]{\\mathrm{I}\\left( #1 \\right)}\n\\newcommand{\\N}[1]{\\mathrm{N} \\left( #1 \\right)}\n\\newcommand{\\ow}{\\text{otherwise}}\\void^\\dagger$
                                        **\n\n## Question 1\n\n$\\bspace$We consider each of his step, if we consider this as the Markov chain for the drunk man, a $r.v.$ with distribution\n\n$$P\\P{X_i} = \\begin{cases}\np, & \\text{if } X_i = 1 \\\\\n1-p, & \\text{if } X_i = -1\n\\end{cases}$$\n\n$\\bspace$then by the strong law of large numbers, we have the expected value of his final step, $S = \\sum X_i$ lands on:\n\n$$\\Exp\\SB{S} = n\\Exp\\SB{X_i} = n\\P{2p-1} \\Rightarrow P\\CB{\\lim_{n\\to\\infty} \\Exp\\SB{S} = \\infty} = 1, p\\neq 0.5$$\n\n$\\bspace$meaning that the state $0$ cannot be recurrent.\n\n## Question 2\n\n$\\bspace$Suppose the little $p$ in the problem has the same meaning with the capital $P$ in the textbook (please note that next time THANKS), then we have\n\n$$\\begin{align}\np_{ij}^{\\P{n}} &= P\\CB{X_n = j \\mid X_0 = i}\\\\\n&= \\sum_{m=1}^{\\infty}P\\CB{X_n=j \\mid X_{n-m} = j}\\cdot P\\CB{X_{n-m} = j,X_k \\neq j, 1\\leq k < m \\mid X_0 = i}\\\\\n&= \\sum_{m=1}^{\\infty}f_{ij}^{m} \\cdot p_{jj}^{\\P{n-m}}\n\\end{align}$$\n\n## Question 3\n\n$\\P{1}$\n\n$\\bspace$There are two classes, $\\CB{0,1}$ and $\\CB{2}$. State $2$ is recurrent since ever in, never out. And states $0$ and $1$ are transient since it's possible they move to state $2$ and never come back.\n\n$\\P{2}$\n\n$\\bspace$Since $T>2$, $X_2 \\neq 2$ which also concludes that $X_1 \\neq 2$. So that after calculating the $\\mathbf{P}^2$\n\n$$\\mathbf{P}^2 = \\begin{Vmatrix}\n0.7 & 0.2 & 0.1 \\\\\n0.3 & 0.5 & 0.2 \\\\\n0 & 0 & 1\n\\end{Vmatrix}^2 = \\begin{Vmatrix}\n0.55 & 0.24 & 0.21 \\\\\n0.36 & 0.31 & 0.33 \\\\\n0 & 0 & 1\n\\end{Vmatrix}$$\n\n$\\bspace$Thus the probability to find is $\\ffrac{0.55} {0.55+0.24} = \\ffrac{55} {79}$\n\n>I just met a problem to find why this method is different and I just feel something is not right but don't know how. Since $T>2$, during the first two steps the $r.v.$ will never enter state $2$ so we have the new transition matrix $\\mathbf{P}'$ and its square is:\n>\n>$$\\mathbf{P'}^2 = \\begin{Vmatrix}\n7/9 & 2/9 \\\\\n3/8 & 5/8\n\\end{Vmatrix}^2 = \\begin{Vmatrix}\n223/324 & 101/324 \\\\\n101/192 & 91/192\n\\end{Vmatrix}$$\n>\n>$\\bspace$So that the probability is\n>\n>$$P\\CB{X_2 = 0\\mid X_0 = 0,T >2} = P\\CB{X_2 = 0\\mid X_0 = 0,X_1 \\neq 2,X_2 \\neq 2} = \\ffrac{223} {324}$$\n\n## Question 4\n\n$\\bspace$By the given information we can assert that $P_{ij}^{\\P{n}}>0$ for certain $n$. If $n\\leq M$ then the proof is finished, otherwise, there're must be at least one state that has been entered for at least twice.\n\n$\\bspace$So we just ignore the transitions that happen between that \"two\" identical states so that during moving from state $i$ to $j$, there's no time we're in the same state. And since there's only $M$ possible states, we conclude that there's still one possible way with less than $M$ transitions that can move from $i$ to $j$.\n\n## Question 5\n\n$$\\begin{align}\n\\;&P\\CB{X_n = m \\mid X_0 = i, X_k \\neq r, k = 1,2,\\dots,n} \\\\[0.8em]\n=\\;& \\ffrac{P\\CB{X_n = m, X_k \\neq r, k = 1,2,\\dots,n \\mid X_0 = i}} {P\\CB{X_k \\neq r, k = 1,2,\\dots,n\\mid X_0 = i}}\\\\\n=\\;& \\ffrac{P\\CB{X_n = m, X_n \\neq r \\mid X_0 = i}\\cdot \\d{\\prod_{j=1}^{n-1}P\\CB{X_j\\neq r\\mid X_0 = i}}} {\\d{\\prod_{j=1}^{n}P\\CB{X_j\\neq r\\mid X_0 = i}}}\\\\\n=\\;& \\ffrac{P\\CB{X_n = m \\mid X_0 = i}} {P\\CB{X_n \\neq r\\mid X_0 = i}}, r\\neq m,i\\\\\n=\\;& \\ffrac{P\\CB{X_n = m \\mid X_0 = i}} {1-P\\CB{X_n = r \\mid X_0 = i}} \\\\\n=\\;& \\ffrac{P_{i,m}^n} {1-P_{i,r}^n} = Q_{i,m}^{n} \n\\end{align}$$\n\n>Can't find a counterexample...\ud83d\ude2d\ud83d\ude2d\ud83d\ude2d\n", "meta": {"hexsha": "6a3a9c5f6867fb6326a3f16f1bba28bfbb755929", "size": 6832, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Probability and Statistics/Applied Random Process/HW/HW_06.ipynb", "max_stars_repo_name": "XavierOwen/Notes", "max_stars_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2018-11-27T10:31:08.000Z", "max_stars_repo_stars_event_max_datetime": "2019-01-20T03:11:58.000Z", "max_issues_repo_path": "Probability and Statistics/Applied Random Process/HW/HW_06.ipynb", "max_issues_repo_name": "XavierOwen/Notes", "max_issues_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Probability and Statistics/Applied Random Process/HW/HW_06.ipynb", "max_forks_repo_name": "XavierOwen/Notes", "max_forks_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-07-14T19:57:23.000Z", "max_forks_repo_forks_event_max_datetime": "2020-07-14T19:57:23.000Z", "avg_line_length": 39.9532163743, "max_line_length": 339, "alphanum_fraction": 0.5077576112, "converted": true, "num_tokens": 1939, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593452091672, "lm_q2_score": 0.6791787056691698, "lm_q1q2_score": 0.4227611324108412}} {"text": "\n\n# Grid vs. Random Search Hyperparameter Optimization\n\n## Setup\n\n### Installation\n\n\n```python\n!pip install matbench\n!pip install CBFV\n!pip install torch\n!pip install torchvision\n```\n\n Requirement already satisfied: matbench in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (0.5)\n Requirement already satisfied: matminer==0.7.4 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from matbench) (0.7.4)\n Requirement already satisfied: monty==2021.8.17 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from matbench) (2021.8.17)\n Requirement already satisfied: scikit-learn==1.0 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from matbench) (1.0)\n Requirement already satisfied: six>=1.16.0 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from matminer==0.7.4->matbench) (1.16.0)\n Requirement already satisfied: pymongo>=3.12.0 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from matminer==0.7.4->matbench) (4.0.1)\n Requirement already satisfied: tqdm>=4.62.0 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from matminer==0.7.4->matbench) (4.62.3)\n Requirement already satisfied: requests>=2.26.0 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from matminer==0.7.4->matbench) (2.27.1)\n Requirement already satisfied: future>=0.18.2 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from matminer==0.7.4->matbench) (0.18.2)\n Requirement already satisfied: pymatgen>=2022.0.11 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from matminer==0.7.4->matbench) (2022.2.7)\n Requirement already satisfied: sympy>=1.8 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from matminer==0.7.4->matbench) (1.8)\n Requirement already satisfied: pint>=0.17 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from matminer==0.7.4->matbench) (0.18)\n Requirement already satisfied: pandas>=1.3.1 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from matminer==0.7.4->matbench) (1.4.0)\n Requirement already satisfied: jsonschema>=3.2.0 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from matminer==0.7.4->matbench) (3.2.0)\n Requirement already satisfied: numpy>=1.21.1 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from matminer==0.7.4->matbench) (1.22.2)\n Requirement already satisfied: joblib>=0.11 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from scikit-learn==1.0->matbench) (1.1.0)\n Requirement already satisfied: scipy>=1.1.0 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from scikit-learn==1.0->matbench) (1.7.0)\n Requirement already satisfied: threadpoolctl>=2.0.0 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from scikit-learn==1.0->matbench) (3.1.0)\n Requirement already satisfied: pyrsistent>=0.14.0 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from jsonschema>=3.2.0->matminer==0.7.4->matbench) (0.18.0)\n Requirement already satisfied: setuptools in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from jsonschema>=3.2.0->matminer==0.7.4->matbench) (52.0.0.post20210125)\n Requirement already satisfied: attrs>=17.4.0 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from jsonschema>=3.2.0->matminer==0.7.4->matbench) (21.2.0)\n Requirement already satisfied: python-dateutil>=2.8.1 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from pandas>=1.3.1->matminer==0.7.4->matbench) (2.8.1)\n Requirement already satisfied: pytz>=2020.1 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from 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Requirement already satisfied: toml in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from pytest->CBFV) (0.10.2)\n Requirement already satisfied: py>=1.8.2 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from pytest->CBFV) (1.11.0)\n Requirement already satisfied: pyparsing>=2.0.2 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from packaging->pytest->CBFV) (2.4.7)\n Requirement already satisfied: torch in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (1.10.2)\n Requirement already satisfied: typing-extensions in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from torch) (4.1.1)\n Collecting torchvision\n Downloading torchvision-0.11.3-cp39-cp39-win_amd64.whl (947 kB)\n Requirement already satisfied: torch==1.10.2 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from torchvision) (1.10.2)\n Requirement already satisfied: pillow!=8.3.0,>=5.3.0 in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from torchvision) (8.3.1)\n Requirement already satisfied: numpy in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from torchvision) (1.22.2)\n Requirement already satisfied: typing-extensions in c:\\users\\taylo\\miniconda3\\envs\\my_pymatgen\\lib\\site-packages (from torch==1.10.2->torchvision) (4.1.1)\n Installing collected packages: torchvision\n Successfully installed torchvision-0.11.3\n\n\n### Imports\n\n\n```python\nimport torch\nimport torch.nn as nn # All neural network modules, nn.Linear, nn.Conv2d, BatchNorm, Loss functions\nimport torch.optim as optim # For all Optimization algorithms, SGD, Adam, etc.\nimport torch.nn.functional as F # All functions that don't have any parameters\nfrom torch.utils.data import DataLoader # Gives easier dataset managment and creates mini batches\nimport torchvision.datasets as datasets # Has standard datasets we can import in a nice and easy way\nimport torchvision.transforms as transforms # Transformations we can perform on our dataset\n\nfrom matbench.bench import MatbenchBenchmark\nfrom CBFV.composition import generate_features\nimport pandas as pd\n```\n\n### Data\n\n\n```python\nmb = MatbenchBenchmark(subset=[\"matbench_expt_is_metal\"])\ntask = list(mb.tasks)[0]\ntask.load()\nfold0 = task.folds[0]\ntrain_inputs, train_outputs = task.get_train_and_val_data(fold0)\ntest_inputs, test_outputs = task.get_test_data(fold0, include_target=True)\nprint(train_inputs[0:2], train_outputs[0:2])\nprint(train_outputs.shape, test_outputs.shape)\n \n```\n\n 2022-02-17 14:00:08 INFO Initialized benchmark 'matbench_v0.1' with 1 tasks: \n ['matbench_expt_is_metal']\n 2022-02-17 14:00:08 INFO Loading dataset 'matbench_expt_is_metal'...\n 2022-02-17 14:00:08 INFO Dataset 'matbench_expt_is_metal loaded.\n mbid\n mb-expt-is-metal-0001 Ag(AuS)2\n mb-expt-is-metal-0002 Ag(W3Br7)2\n Name: composition, dtype: object mbid\n mb-expt-is-metal-0001 True\n mb-expt-is-metal-0002 True\n Name: is_metal, dtype: bool\n (3936,) (985,)\n\n\n\n```python\ntrain_inputs.describe()\n```\n\n\n\n\n count 3936\n unique 3936\n top Ag(AuS)2\n freq 1\n Name: composition, dtype: object\n\n\n\n\n```python\ntrain_outputs.describe()\n```\n\n\n\n\n count 3936\n unique 2\n top False\n freq 1976\n Name: is_metal, dtype: object\n\n\n\n\n```python\ntrain_df = pd.DataFrame({\"formula\": train_inputs, \"target\": train_outputs})\ntest_df = pd.DataFrame({\"formula\": test_inputs, \"target\": test_outputs})\ntrain_df\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        formulatarget
                                        mbid
                                        mb-expt-is-metal-0001Ag(AuS)2True
                                        mb-expt-is-metal-0002Ag(W3Br7)2True
                                        mb-expt-is-metal-0003Ag0.5Ge1Pb1.75S4False
                                        mb-expt-is-metal-0005Ag2BBrTrue
                                        mb-expt-is-metal-0006Ag2BiO3True
                                        .........
                                        mb-expt-is-metal-4916ZrSiTeTrue
                                        mb-expt-is-metal-4917ZrTaN3False
                                        mb-expt-is-metal-4918ZrTeTrue
                                        mb-expt-is-metal-4920ZrTiF6True
                                        mb-expt-is-metal-4921ZrW2True
                                        \n

                                        3936 rows \u00d7 2 columns

                                        \n
                                        \n\n\n\n\n```python\nX_train, y_train, _, _ = generate_features(train_df)\nX_train\n```\n\n Processing Input Data: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 3936/3936 [00:00<00:00, 10725.93it/s]\n\n\n \tFeaturizing Compositions...\n\n\n Assigning Features...: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 3936/3936 [00:00<00:00, 8724.69it/s]\n\n\n \tCreating Pandas Objects...\n\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        avg_Atomic_Numberavg_Atomic_Weightavg_Periodavg_groupavg_familiesavg_Metalavg_Nonmetalavg_Metalliodavg_Mendeleev_Numberavg_l_quantum_number...mode_polarizability(A^3)mode_Melting_point_(K)mode_Boiling_Point_(K)mode_Density_(g/mL)mode_specific_heat_(J/g_K)_mode_heat_of_fusion_(kJ/mol)_mode_heat_of_vaporization_(kJ/mol)_mode_thermal_conductivity_(W/(m_K))_mode_heat_atomization(kJ/mol)mode_Cohesive_energy
                                        047.400000113.1866564.60000013.0000005.2000000.6000000.4000000.00000074.6000001.200000...2.900385.95717.852.070000.1281.717509.80000.26900279.02.85
                                        146.714286110.9316294.61904813.5714296.6666670.3333330.6666670.00000081.0000001.238095...3.100265.95331.953.120000.4735.2860015.43800.12200112.01.22
                                        236.27586285.1597384.00000014.8965526.1724140.3103450.5517240.13793183.4827590.931034...2.900385.95717.852.070000.7101.717509.80000.26900279.02.85
                                        333.50000076.6128504.00000013.0000005.5000000.5000000.2500000.25000074.2500000.500000...7.9001235.152485.1510.500000.23511.30000250.5800429.00000284.02.95
                                        433.50000078.7858283.66666714.1666675.6666670.5000000.5000000.00000079.5000000.666667...0.79354.7590.150.001430.9200.222593.40990.02674249.02.62
                                        ..................................................................
                                        393135.33333382.3031674.33333311.3333335.3333330.3333330.6666670.00000070.6666671.333333...5.400722.651262.952.330000.20016.9000052.55002.35000197.02.19
                                        393226.80000062.8384243.40000010.8000005.8000000.4000000.6000000.00000067.6000001.400000...1.10063.2577.350.001251.0400.360402.79280.02598473.04.92
                                        393346.000000109.4120005.00000010.0000005.0000000.5000000.5000000.00000067.0000001.500000...5.500722.651262.956.240000.20016.9000052.55002.35000197.02.19
                                        393414.50000031.6368022.62500013.7500007.0000000.2500000.7500000.00000080.6250001.250000...0.63453.3585.050.001700.8200.255203.26980.0279079.00.84
                                        393562.666667152.9680005.6666675.3333334.0000001.0000000.0000000.00000048.6666672.000000...11.1003683.155933.1519.300000.13035.40000824.0000174.00000849.08.90
                                        \n

                                        3936 rows \u00d7 264 columns

                                        \n
                                        \n\n\n\n\n```python\nX_test, y_test, _, _ = generate_features(test_df)\nX_test\n```\n\n Processing Input Data: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 985/985 [00:00<00:00, 12605.81it/s]\n\n\n \tFeaturizing Compositions...\n\n\n Assigning Features...: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 985/985 [00:00<00:00, 8309.95it/s]\n\n\n \tCreating Pandas Objects...\n\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        avg_Atomic_Numberavg_Atomic_Weightavg_Periodavg_groupavg_familiesavg_Metalavg_Nonmetalavg_Metalliodavg_Mendeleev_Numberavg_l_quantum_number...mode_polarizability(A^3)mode_Melting_point_(K)mode_Boiling_Point_(K)mode_Density_(g/mL)mode_specific_heat_(J/g_K)_mode_heat_of_fusion_(kJ/mol)_mode_heat_of_vaporization_(kJ/mol)_mode_thermal_conductivity_(W/(m_K))_mode_heat_atomization(kJ/mol)mode_Cohesive_energy
                                        046.206897111.0322904.55172414.8965526.1724140.3103450.5517240.13793184.0344830.931034...3.800490.15958.154.790000.3206.6940037.70000.52000227.02.46
                                        125.75000057.8154743.37500014.0000005.7500000.3750000.5000000.12500078.3750000.625000...0.79354.7590.150.001430.9200.222593.40990.02674249.02.62
                                        246.750000107.5061505.00000010.7500004.0000001.0000000.0000000.00000064.2500000.500000...7.9001235.152485.1510.500000.23511.30000250.5800429.00000284.02.95
                                        327.12500061.0840253.50000013.1250005.5000000.5000000.5000000.00000074.8750000.750000...0.79354.7590.150.001430.9200.222593.40990.02674249.02.62
                                        442.45454597.5471224.63636413.0000005.2727270.6363640.3636360.00000074.8181820.363636...7.9001235.152485.1510.500000.23511.30000250.5800429.00000284.02.95
                                        ..................................................................
                                        98024.00000051.7853333.66666712.0000006.0000000.3333330.6666670.00000073.3333331.333333...2.900385.95717.852.070000.7101.717509.80000.26900279.02.85
                                        98145.000000104.6846675.0000009.0000004.6666670.6666670.3333330.00000061.6666671.666667...6.600904.152223.156.510000.21016.9000077.140022.70000262.02.75
                                        98237.00000085.0920004.50000010.0000005.5000000.5000000.5000000.00000066.5000001.500000...3.800490.15958.154.790000.2706.6940037.70000.52000227.02.46
                                        98322.66666749.1316673.66666710.6666675.3333330.3333330.6666670.00000066.6666671.333333...5.4001683.152628.152.330000.71050.55000384.2200148.00000452.04.63
                                        98423.00000050.7458503.7500007.0000004.7500000.7500000.2500000.00000054.2500001.750000...14.6001933.153560.154.540000.52015.45000421.000021.90000470.04.85
                                        \n

                                        985 rows \u00d7 264 columns

                                        \n
                                        \n\n\n\n## construct model\n\nLet's build our nn in pytorch\n\n\n```python\n\nbatch_size = 64\ntrain_dataset = datasets.MNIST( \n root=\"dataset/\",\n train=True,\n transform=transforms.ToTensor(),\n download=True,\n)\ntrain_loader = DataLoader(\n dataset=train_dataset, batch_size=batch_size, shuffle=True\n)\ntest_dataset = datasets.MNIST(\n root=\"dataset/\",\n train=False,\n transform=transforms.ToTensor(),\n download=True,\n)\ntest_loader = DataLoader(\n dataset=test_dataset, batch_size=batch_size, shuffle=True\n)\n```\n\n\n```python\n\n```\n\nWe can do the same grid vs random search with another model, like a decision tree classifier\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n## Test\n\n\n```python\n\n```\n\n## Code Graveyard\n\n\n```python\n# from sklearn.preprocessing import StandardScaler\n# scaler = StandardScaler()\n# scaler.fit(X_train)\n# X_train = scaler.transform(X_train)\n# X_test = scaler.transform(X_test)\n# X_train\n```\n", "meta": {"hexsha": "761e7157f14bb4aec8ac122cc8e7d4dd438ed1cd", "size": 64058, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "worked_examples/vanilla_neural_net_pytorch/NN_classification_matbench.ipynb", "max_stars_repo_name": "sp8rks/MaterialsInformatics", "max_stars_repo_head_hexsha": "ed6317595dd9a7d02fe92d9d13b7c9bdad9c56bc", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 12, "max_stars_repo_stars_event_min_datetime": "2022-01-18T21:51:19.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-22T14:35:40.000Z", "max_issues_repo_path": "worked_examples/vanilla_neural_net_pytorch/NN_classification_matbench.ipynb", "max_issues_repo_name": "sp8rks/MaterialsInformatics", "max_issues_repo_head_hexsha": "ed6317595dd9a7d02fe92d9d13b7c9bdad9c56bc", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 5, "max_issues_repo_issues_event_min_datetime": "2022-01-22T21:47:13.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-13T03:52:45.000Z", "max_forks_repo_path": "worked_examples/vanilla_neural_net_pytorch/NN_classification_matbench.ipynb", "max_forks_repo_name": "sp8rks/MaterialsInformatics", "max_forks_repo_head_hexsha": "ed6317595dd9a7d02fe92d9d13b7c9bdad9c56bc", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2022-03-20T06:02:54.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-31T12:22:21.000Z", "avg_line_length": 41.3544222079, "max_line_length": 279, "alphanum_fraction": 0.4142651972, "converted": true, "num_tokens": 12043, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178686187839, "lm_q2_score": 0.6224593241981982, "lm_q1q2_score": 0.4227611060143024}} {"text": "```python\n%matplotlib inline\n```\n\n\nSynthetic seismograms using the convolutional model\n---------------------------------------------------\n\nThe simplest way to get a seismogram (in time x offset) is through the\nconvolutional model\n\n\\begin{align}trace(t) = wavelet(t) \\ast reflectivity(t)\\end{align}\n\nModule :mod:`fatiando.seismic.conv` defines functions for doing this\nconvolution, calculating the required reflectivity, and converting from depth a\nmodel into time.\n\n\n\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom fatiando.seismic import conv\nfrom fatiando.vis import mpl\n\n# Define the parameters of our depth model\nn_samples, n_traces = [600, 100]\nvelocity = 1500*np.ones((n_samples, n_traces))\n# We'll put two interfaces in depth\nvelocity[150:, :] = 2000\nvelocity[400:, :] = 3500\ndt = 2e-3\n\n# We need to convert the depth model we made above into time\nvel_l = conv.depth_2_time(velocity, velocity, dt=dt, dz=1)\n# and we'll assume the density is homogeneous\nrho_l = 2200*np.ones(np.shape(vel_l))\n# With that, we can calculate the reflectivity model in time\nrc = conv.reflectivity(vel_l, rho_l)\n# and finally perform our convolution\nsynt = conv.convolutional_model(rc, 30, conv.rickerwave, dt=dt)\n\n# We can use the utility function in fatiando.vis.mpl to plot the seismogram\nfig, axes = plt.subplots(1, 2, figsize=(8, 5))\n\nax = axes[0]\nax.set_title(\"Velocity model (in depth)\")\ntmp = ax.imshow(velocity, extent=[0, n_traces, n_samples, 0],\n cmap=\"copper\", aspect='auto', origin='upper')\nfig.colorbar(tmp, ax=ax, pad=0, aspect=50)\nax.set_xlabel('Trace')\nax.set_ylabel('Depth (m)')\n\nax = axes[1]\nax.set_title(\"Synthetic seismogram\")\nmpl.seismic_wiggle(synt[:, ::20], dt, scale=1)\nmpl.seismic_image(synt, dt, cmap=\"RdBu_r\", aspect='auto')\nax.set_xlabel('Trace')\nax.set_ylabel('Time (s)')\nplt.tight_layout()\nplt.show()\n```\n", "meta": {"hexsha": "aad950da197018e0acc7151c67b31d5c1aafd439", "size": 2882, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "_downloads/convolutional_model.ipynb", "max_stars_repo_name": "fatiando/dev", "max_stars_repo_head_hexsha": "adbaf4ed0c87ec9b724ebb3700750e05c2c563f8", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "_downloads/convolutional_model.ipynb", "max_issues_repo_name": "fatiando/dev", "max_issues_repo_head_hexsha": "adbaf4ed0c87ec9b724ebb3700750e05c2c563f8", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "_downloads/convolutional_model.ipynb", "max_forks_repo_name": "fatiando/dev", "max_forks_repo_head_hexsha": "adbaf4ed0c87ec9b724ebb3700750e05c2c563f8", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 53.3703703704, "max_line_length": 1429, "alphanum_fraction": 0.6148507981, "converted": true, "num_tokens": 524, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.42236327144181746}} {"text": "```python\nfrom IPython.display import Image\nfrom IPython.core.display import HTML \nfrom sympy import *; x,h,y,t = symbols(\"x h y t\")\nImage(url= \"https://i.imgur.com/B6ERnuf.png\")\n```\n\n\n\n\n\n\n\n\n\n```python\nexpr = (6/x**4) + (-3/x**5) + 12\ndef F(x):\n return expr\nprint(integrate(F(x)))\n```\n\n 12*x + (3 - 8*x)/(4*x**4)\n\n\n\n```python\nImage(url= \"https://i.imgur.com/vFxYle2.png\")\n```\n\n\n\n\n\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "15ca28ba90f7e58d13b35808c74a10e6cf9be779", "size": 2004, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Calculus_Homework/WWB12.8.ipynb", "max_stars_repo_name": "NSC9/Sample_of_Work", "max_stars_repo_head_hexsha": "8f8160fbf0aa4fd514d4a5046668a194997aade6", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Calculus_Homework/WWB12.8.ipynb", "max_issues_repo_name": "NSC9/Sample_of_Work", "max_issues_repo_head_hexsha": "8f8160fbf0aa4fd514d4a5046668a194997aade6", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Calculus_Homework/WWB12.8.ipynb", "max_forks_repo_name": "NSC9/Sample_of_Work", "max_forks_repo_head_hexsha": "8f8160fbf0aa4fd514d4a5046668a194997aade6", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 19.2692307692, "max_line_length": 60, "alphanum_fraction": 0.4810379242, "converted": true, "num_tokens": 135, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.4223020047509994}} {"text": "```\n%load_ext autoreload\n%autoreload 2\n\n%matplotlib inline\n```\n\n The autoreload extension is already loaded. To reload it, use:\n %reload_ext autoreload\n\n\n\n```\n\n#from google.colab import drive\n#drive.mount('/content/gdrive')\n```\n\n\n```\n#Q1. Execute the following statement. What is displayed? What does it mean?\n!pwd\n```\n\n /content\n\n\n\n```\n#Q2: Execute the following statement. What happens? Examine the left column of your colab page to see what happens.\n!git clone https://github.com/fastai/course-v3.git\n```\n\n fatal: destination path 'course-v3' already exists and is not an empty directory.\n\n\n\n```\n#export\n#Q3a: Execute the following statement. Read the error message. Explain what it means. \n\nfrom exp.nb_02 import *\n\n```\n\n\n```\n#Q3b. You can solve the problem by executing the following statement before the above statement. \n# Explain why the following statements can solve this \"module not found\" problem. \n```\n\n\n```\nimport sys\nsys.path.append('/content/course-v3/nbs/dl2')\nsys.path\n```\n\n\n```\nimport torch.nn.functional as F\n```\n\n\n```\n\n```\n\n## Initial setup\n\n### Data\n\n[Jump_to lesson 9 video](https://course.fast.ai/videos/?lesson=9&t=1786)\n\n\n```\nfrom IPython.display import Image\nfrom six.moves import urllib\n\nopener = urllib.request.build_opener()\nopener.addheaders = [('User-agent', 'Mozilla/5.0')]\nurllib.request.install_opener(opener)\n\ndef get_data():\n import os\n import torchvision.datasets as datasets\n root = '../data'\n if not os.path.exists(root):\n os.mkdir(root)\n\n train_set = datasets.MNIST(root=root, train=True, download=True)\n test_set = datasets.MNIST(root=root, train=False, download=True) #load validation set\n x_train, x_valid = train_set.data.split([50000, 10000])\n y_train, y_valid = train_set.targets.split([50000, 10000])\n return (x_train.view(50000, -1) / 256.0), y_train.float(), (x_valid.view(10000, -1))/ 256.0, y_valid.float()\n\n\n```\n\n\n```\n#mpl.rcParams['image.cmap'] = 'gray'\n```\n\n\n```\n#Q4: when you execute the following statement, where is the downloaded data stored? Examine the left column of your colab page.\nx_train,y_train,x_valid,y_valid = get_data()\n```\n\n\n```\n#Q5a: Execute the following statement. What does the number displayed mean?\nlen(x_train)\n```\n\n\n\n\n 50000\n\n\n\n\n```\n#Q5b: Execute the following statement. What do the numbers displayed refer to?\nx_train[0]\n```\n\n\n```\n#Q5c: Execute the following statement. What do the numbers displayed refer to?\ny_train[0]\n```\n\n\n\n\n tensor(5.)\n\n\n\n\n```\n#Q5d. Execute the following statement. What does the number displayed refer to? \nx_train[0].shape\n```\n\n\n\n\n torch.Size([784])\n\n\n\n\n```\n#Q5e. Execute the following statement. What do the numbers displayed refer to?\nx_train.shape\n```\n\n\n\n\n torch.Size([50000, 784])\n\n\n\n\n```\n#Q5f. Execute the following statement. What do the numbers displayed refer to?\ny_train.shape\n```\n\n\n\n\n torch.Size([50000])\n\n\n\n\n```\n#Q6: Display the values of n,m, c, and nh. What are they? For what are they used in the following code? \nn,m = x_train.shape\nc = y_train.max()+1\nnh = 50\n```\n\n\n```\n#The following defines Model class, which will be used to create a neural net.\n#Q7a: nn.Module is the parent class of class Model. Why do you want to make Model a child class to its parent nn.Module, rather than making Model stand on its own?\n#Q7b. See the definition of self.layers field. It contains nn.Linear(n_in,nh). What is the difference between nn.Linear class nn.Linear(n_in,nh)? \n#Q7c. Would you think that the weight and bias parameters of the two linear layers are initialized when object nn.Linear(n_in,nh) is constructed? If so, guess why.\n\nclass Model(nn.Module):\n def __init__(self, n_in, nh, n_out):\n super().__init__()\n self.layers = [nn.Linear(n_in,nh), nn.ReLU(), nn.Linear(nh,n_out)]\n \n def __call__(self, x):\n for l in self.layers: x = l(x)\n return x\n```\n\n\n```\n#Q8. When the following statement is executed, what happens? Explain by referring to a method in class Model.\nmodel = Model(m, nh, 10)\n```\n\n\n```\n#Q9a. When the following statement is executed, what happens? Explain by referring to a method in class Model. \n#Q9b. Draw a diagram or a graph by using arrows and boxes, which shows what computation is performed when model(x_train) is executed.\npred = model(x_train)\n```\n\n\n```\n#Q10: Execute the following statement. What do the 10 displayed numbers refer to?\npred[0]\n```\n\n\n\n\n tensor([-0.1243, -0.1045, -0.0993, -0.1187, -0.0220, 0.0487, 0.0301, -0.0693,\n -0.2306, 0.0330], grad_fn=)\n\n\n\n\n```\n#Q11. Execute the following statement. Is the resulting value near to 1? If not, what does it imply?\npred[0].sum()\n```\n\n\n\n\n tensor(-0.6569, grad_fn=)\n\n\n\n### Softmax function\n\nFirst, we will need to compute the softmax of our activations. This is defined by:\n\n$$\\hbox{softmax(x)}_{i} = \\frac{e^{x_{i}}}{e^{x_{0}} + e^{x_{1}} + \\cdots + e^{x_{n-1}}}$$\n\nor more concisely:\n\n$$\\hbox{softmax(x)}_{i} = \\frac{e^{x_{i}}}{\\sum_{0 \\leq j \\leq n-1} e^{x_{j}}}$$ \n\nIn practice, we will need the log of the softmax when we calculate the loss.\n\n\n```\ndef softmax(x): return (x.exp()/(x.exp().sum(-1,keepdim=True)))\n```\n\n\n```\nsm_pred = softmax(pred)\n\n```\n\n\n```\n#Q12. Execute the following statement. What do the 10 displayed numbers refer to?\nsm_pred[0]\n```\n\n\n\n\n tensor([0.0940, 0.0959, 0.0964, 0.0945, 0.1041, 0.1117, 0.1097, 0.0993, 0.0845,\n 0.1100], grad_fn=)\n\n\n\n\n```\n#Q13. Execute the following statement. What does the displayed numbers mean?\nsm_pred[0].sum()\n```\n\n\n\n\n tensor(1.0000, grad_fn=)\n\n\n\n\n\n## Log softmax\n\n\n```\ndef log_softmax(x): return (x.exp()/(x.exp().sum(-1,keepdim=True))).log()\n```\n\n\n```\nlog_sm_pred = log_softmax(pred)\n```\n\n\n```\nlog_sm_pred[0]\n```\n\n\n\n\n tensor([-2.3646, -2.3449, -2.3396, -2.3591, -2.2624, -2.1917, -2.2103, -2.3097,\n -2.4710, -2.2073], grad_fn=)\n\n\n\n\n\n## Cross Entropy Loss Function.\n Read the following paragraph to understand what the cross entroy loss functioni is. \n\nThe difference between two probabilities: \n https://datascience.stackexchange.com/questions/20296/cross-entropy-loss-explanation\n\nThe cross entropy formula takes in two distributions, $p(x^{(s)})$, the true distribution (defined by the label data), and $\\hat{p}(x^{(s)})$, the estimated distribution (predicted by the neural net), defined over the discrete variable $x^{(s)}$ and is given by\n\n$H(p,\\hat{p})=\u2212\\sum_{s \\in B} p(x^{(s)}) \\cdot log(\\hat{p}(x^{(s)}))$\n\n\nIn general, $ p(x^{(s)}) = [ p_{1} (x^{(s)}), ..., p_{n}(x^{(s)})]$ is a probability distribution over a set of categories. \nBut since our $ p(x^{(s)})$ are 1-hot encoded, that is, in the form of $ p(x^{(s)}) =[0,0,..,0,1,0..]$, where the probability of only one category is one and those of the other categories are all zero, this can be rewritten as\n\n \\begin{equation}\n H(p,\\hat{p})= -\\sum_{s \\in B} [ 0*\\log(\\hat{p}_{1} (x^{(s)})) \n + 1*\\log(\\hat{p}_{i} (x^{(s)}) ) +..+\n 0* \\log( \\hat{p}_{n} (x^{(s)}) ) ] \\\\ \n = -\\sum_{s \\in B} \\log(\\hat{p}_{i(s)} ) (x^{(s)}) ) \n \\tag{crossEntroyEq}\n \\end{equation}\n \n Te softmax function plays the role of the probability distribution $\\hat{p}_{1} (x^{(s)})$.\n\n Here $i(s)$ is the index of the one-hot probability distribution $p(x^{(s)})$ where the probability is one. \n \n \n\n# integer array indexing \n\n\n```\n#Q14a. Execute the following statement. What is the role of list [0,1,2] in the statement? What do the displayed numbers refer to? \nlog_sm_pred[ [0,1,2]]\n```\n\n\n\n\n tensor([[-2.3646, -2.3449, -2.3396, -2.3591, -2.2624, -2.1917, -2.2103, -2.3097,\n -2.4710, -2.2073],\n [-2.3923, -2.3010, -2.3765, -2.4043, -2.2272, -2.2951, -2.1785, -2.2922,\n -2.5106, -2.1105],\n [-2.3806, -2.3104, -2.3593, -2.4057, -2.1497, -2.3245, -2.2133, -2.2681,\n -2.4566, -2.2012]], grad_fn=)\n\n\n\n\n```\n#Q14b. Execute the following statement. What is the role of list [2,4,6] in the statement? What do the displayed numbers mean? Explain compare the result of Q15b and the result of this statement.\nlog_sm_pred[[0,1,2], [2,4,6]]\n```\n\n\n\n\n tensor([-2.3396, -2.2272, -2.2133], grad_fn=)\n\n\n\n[Jump_to lesson 9 video](https://course.fast.ai/videos/?lesson=9&t=2081)\n\n\n```\nrange(y_train.long().shape[0])\n```\n\n\n\n\n range(0, 50000)\n\n\n\n\n```\n\n```\n\n\n```\nselectOneHot = log_sm_pred[ range(y_train.long().shape[0]), y_train.long() ]\n```\n\n\n```\nselectOneHot.shape\n```\n\n\n\n\n torch.Size([50000])\n\n\n\n\n```\n#Q15. What does selectOneHot refer to?\n```\n\n\n```\ndef nll(input, target):\n \n return -input[range(target.shape[0]), target].mean()\n```\n\n\n```\ny_train.long().shape[0]\n```\n\n\n\n\n 50000\n\n\n\n\n```\n#Q16. Read nill function and explain how this function computes the result. You need to refer to {CrossEntropyLoss}.\nloss = nll(log_sm_pred, y_train.long())\n```\n\n\n```\nloss\n```\n\n\n\n\n tensor(2.3240, grad_fn=)\n\n\n\n\n```\n#Q16. Compare function nll() and the formula CrossEntropyEq. Explain that function nll() computes the cross entropy function between\n# predicted probability distrubution of the input images and the ground truth (labeled) probability distribution of the input images. \n```\n\n## Basic training loop\n\nBasically the training loop repeats over the following steps:\n- get the output of the model on a batch of inputs\n- compare the output to the labels we have and compute a loss\n- calculate the gradients of the loss with respect to every parameter of the model\n- update said parameters with those gradients to make them a little bit better\n\n[Jump_to lesson 9 video](https://course.fast.ai/videos/?lesson=9&t=2542)\n\n\n```\nloss_func = F.cross_entropy # This built-in Pytorch function combines log_softmax and nll in a single function.\n# In the following code, we will use this function, not the function nll defined above.\n```\n\n\n```\n#export\ndef accuracy(out, yb): return (torch.argmax(out, dim=1)==yb).float().mean()\n```\n\n\n```\nbs=1000 # batch size\n\nxb = x_train[0:bs] # a mini-batch from x\npreds = model(xb) # predictions\npreds[0], preds.shape\n\n#Q17. What does preds[0] refer to?\n```\n\n\n\n\n (tensor([-0.0992, -0.1511, -0.1637, -0.0874, 0.1793, -0.0683, -0.1372, 0.0398,\n -0.1217, -0.1008], grad_fn=), torch.Size([500, 10]))\n\n\n\n\n```\nyb = y_train[0:bs]\nloss = loss_func(preds, yb.long())\nloss\n#Q18. What does the value of loss refer to?\n```\n\n\n\n\n tensor(0.1827, grad_fn=)\n\n\n\n\n```\naccuracy(preds, yb)\n#Q19. What does the value of the above statement refer to? Examine the value. Is it supposed to be equal to one approximately?\n# If it is not at the current moment, what is the reason for it?\n```\n\n\n\n\n tensor(0.9460)\n\n\n\n# The mechanism for training the network: \n\n(1) computing the graidents of Loss with respect to tensors at each layer, (2) backpropgation: applying the chain rule to compute the gradient vector of the parameters, (3) updating the parameters.\n\nhttps://github.com/pytorch/pytorch/blob/35bd2b3c8b64d594d85fc740e94c30aa67892a34/torch/tensor.py\n\nhttps://github.com/pytorch/pytorch/blob/35bd2b3c8b64d594d85fc740e94c30aa67892a34/torch/tensor.py\n\nhttps://stackoverflow.com/questions/57248777/backward-function-in-pytorch\n\n\n\n```\nlr = 0.5 # learning rate\nepochs = 2 # how many epochs to train for\n```\n\n\n```\n#Q20. What does the number (n-1)//bs + 1 refer to? \nimport pdb\nfor epoch in range(epochs):\n for i in range((n-1)//bs + 1): # for each batch in the current epoch\n# pdb.set_trace()\n start_i = i*bs\n end_i = start_i+bs\n\n xb = x_train[start_i:end_i]\n yb = y_train[start_i:end_i]\n\n\n ybHat = model(xb)\n #h = ybHat.register_hook( lambda grad: print (grad) )\n\n loss = loss_func( ybHat, yb.long()) \n\n #ybHat.grad\n loss.backward() # When you call loss.backward(), all it does is compute gradient of loss \n # w.r.t all the parameters in loss that have requires_grad = True and store them in parameter.grad attribute for every parameter\n\n #ybHat.grad \n #The following performs what Pytorch function optimizer.step() does:\n # It updates all the weight and bias parameters based on the gradients of loss with respect to the weight and bias parameter \n with torch.no_grad():\n for l in model.layers:\n if hasattr(l, 'weight'):\n l.weight -= l.weight.grad * lr\n l.bias -= l.bias.grad * lr\n\n #\n # loss.backward() computes the gradient of loss w.r.t. graph LEAVES.\n # This function accumulates gradients in the leaves - you might need to zero\n # we don't care about gradients from the previous batch.\n # Not zeroing grads would lead to gradient accumulation across batches\n l.weight.grad.zero_()\n l.bias.grad.zero_()\n\n #Q21: print the loss and the accuracy after training the net using the validation dataset.\n # Explain how the loss and accuracy change as each batch is used for training the network model.\n # Try to convey detailed and specific information about the progress of training the neural net.\n # Observe the printed data carefully\n \n #Q22. Explain why you would use validation dataset to check the progress of learning of your network.\n \n \n yHatValid = model(x_valid)\n lossValid = loss_func( yHatValid, y_valid.long() ) \n\n \n print('epoch={0}, batch ={1}:'.format(epoch, i) ) \n print(' lossValid=', lossValid )\n print(' accuracyValid = ', accuracy( yHatValid, y_valid) )\n\n #Q23. Afer each epoch, print the loss and the accuracy of the network by using the training dataset. \n # Explain the result. Be attentive to the result. \n\n yHatTrain = model(x_train)\n lossTrain = loss_func( yHatTrain, y_train.long() ) \n\n print('*******************************************************') \n print('epoch={0}:', epoch ) \n print(' lossTrain=', lossTrain)\n print(' accuracyTrain = ', accuracy( yHatTrain, y_train) )\n\n #Q24. Afer each epoch, print the loss and the accuracy of the network by using the validatio dataset. \n # Explain the result. Be attentive to the result. \n\n yHatValid = model(x_valid)\n lossValid = loss_func( yHatValid, y_valid.long() ) \n\n print('*******************************************************') \n print('epoch={0}:', epoch ) \n print(' lossValid ', lossValid)\n print(' accuracyValid = ', accuracy( yHatValid, y_valid) )\n \n```\n\n\n\n\n```\nloss_func(model(xb), yb.long()), accuracy(model(xb), yb)\n```\n\n\n\n\n (tensor(0.0979, grad_fn=), tensor(0.9375))\n\n\n\n## Using parameters and optim\n\n### Parameters\n\nUse `nn.Module.__setattr__` and move relu to functional:\n\n[Jump_to lesson 9 video](https://course.fast.ai/videos/?lesson=9&t=2818)\n\n\n```\nclass Model(nn.Module):\n def __init__(self, n_in, nh, n_out):\n super().__init__()\n self.l1 = nn.Linear(n_in,nh)\n self.l2 = nn.Linear(nh,n_out)\n \n def __call__(self, x): return self.l2(F.relu(self.l1(x)))\n```\n\n\n```\nmodel = Model(m, nh, 10)\n```\n\n\n```\nfor name,l in model.named_children(): print(f\"{name}: {l}\")\n```\n\n l1: Linear(in_features=784, out_features=50, bias=True)\n l2: Linear(in_features=50, out_features=10, bias=True)\n\n\n\n```\nmodel\n```\n\n\n\n\n Model(\n (l1): Linear(in_features=784, out_features=50, bias=True)\n (l2): Linear(in_features=50, out_features=10, bias=True)\n )\n\n\n\n\n```\nmodel.l1\n```\n\n\n\n\n Linear(in_features=784, out_features=50, bias=True)\n\n\n\n\n```\ndef fit():\n for epoch in range(epochs):\n for i in range((n-1)//bs + 1):\n start_i = i*bs\n end_i = start_i+bs\n xb = x_train[start_i:end_i]\n yb = y_train[start_i:end_i]\n loss = loss_func(model(xb), yb.long())\n\n loss.backward()\n with torch.no_grad():\n for p in model.parameters(): p -= p.grad * lr\n model.zero_grad()\n```\n\n\n```\nfit()\nloss_func(model(xb), yb.long()), accuracy(model(xb), yb)\n```\n\n\n\n\n (tensor(0.0097, grad_fn=), tensor(1.))\n\n\n\nBehind the scenes, PyTorch overrides the `__setattr__` function in `nn.Module` so that the submodules you define are properly registered as parameters of the model.\n\n\n```\nclass DummyModule():\n def __init__(self, n_in, nh, n_out):\n self._modules = {}\n self.l1 = nn.Linear(n_in,nh)\n self.l2 = nn.Linear(nh,n_out)\n \n def __setattr__(self,k,v):\n if not k.startswith(\"_\"): self._modules[k] = v\n super().__setattr__(k,v)\n \n def __repr__(self): return f'{self._modules}'\n \n def parameters(self):\n for l in self._modules.values():\n for p in l.parameters(): yield p\n```\n\n\n```\nmdl = DummyModule(m,nh,10)\nmdl\n```\n\n\n\n\n {'l1': Linear(in_features=784, out_features=50, bias=True), 'l2': Linear(in_features=50, out_features=10, bias=True)}\n\n\n\n\n```\n[o.shape for o in mdl.parameters()]\n```\n\n\n\n\n [torch.Size([50, 784]),\n torch.Size([50]),\n torch.Size([10, 50]),\n torch.Size([10])]\n\n\n\n### Registering modules\n\nWe can use the original `layers` approach, but we have to register the modules.\n\n[Jump_to lesson 9 video](https://course.fast.ai/videos/?lesson=9&t=2997)\n\n\n```\nlayers = [nn.Linear(m,nh), nn.ReLU(), nn.Linear(nh,10)]\n```\n\n\n```\nclass Model(nn.Module):\n def __init__(self, layers):\n super().__init__()\n self.layers = layers\n for i,l in enumerate(self.layers): self.add_module(f'layer_{i}', l)\n \n def __call__(self, x):\n for l in self.layers: x = l(x)\n return x\n```\n\n\n```\nmodel = Model(layers)\n```\n\n\n```\nmodel\n```\n\n\n\n\n Model(\n (layer_0): Linear(in_features=784, out_features=50, bias=True)\n (layer_1): ReLU()\n (layer_2): Linear(in_features=50, out_features=10, bias=True)\n )\n\n\n\n### nn.ModuleList\n\n`nn.ModuleList` does this for us.\n\n[Jump_to lesson 9 video](https://course.fast.ai/videos/?lesson=9&t=3173)\n\n\n```\nclass SequentialModel(nn.Module):\n def __init__(self, layers):\n super().__init__()\n self.layers = nn.ModuleList(layers)\n \n def __call__(self, x):\n for l in self.layers: x = l(x)\n return x\n```\n\n\n```\nmodel = SequentialModel(layers)\n```\n\n\n```\nmodel\n```\n\n\n\n\n SequentialModel(\n (layers): ModuleList(\n (0): Linear(in_features=784, out_features=50, bias=True)\n (1): ReLU()\n (2): Linear(in_features=50, out_features=10, bias=True)\n )\n )\n\n\n\n\n```\nfit()\nloss_func(model(xb), yb.long()), accuracy(model(xb), yb)\n```\n\n\n\n\n (tensor(0.0318, grad_fn=), tensor(1.))\n\n\n\n### nn.Sequential\n\n`nn.Sequential` is a convenient class which does the same as the above:\n\n[Jump_to lesson 9 video](https://course.fast.ai/videos/?lesson=9&t=3199)\n\n\n```\nmodel = nn.Sequential(nn.Linear(m,nh), nn.ReLU(), nn.Linear(nh,10))\n```\n\n\n```\nfit()\nloss_func(model(xb), yb.long()), accuracy(model(xb), yb)\n```\n\n\n\n\n (tensor(0.0522, grad_fn=), tensor(1.))\n\n\n\n\n```\nnn.Sequential??\n```\n\n\n```\nmodel\n```\n\n\n\n\n Sequential(\n (0): Linear(in_features=784, out_features=50, bias=True)\n (1): ReLU()\n (2): Linear(in_features=50, out_features=10, bias=True)\n )\n\n\n\n### optim\n\nLet's replace our previous manually coded optimization step:\n\n```python\nwith torch.no_grad():\n for p in model.parameters(): p -= p.grad * lr\n model.zero_grad()\n```\n\nand instead use just:\n\n```python\nopt.step()\nopt.zero_grad()\n```\n\n[Jump_to lesson 9 video](https://course.fast.ai/videos/?lesson=9&t=3278)\n\n\n```\nclass Optimizer():\n def __init__(self, params, lr=0.5): self.params,self.lr=list(params),lr\n \n def step(self):\n with torch.no_grad():\n for p in self.params: p -= p.grad * lr\n\n def zero_grad(self):\n for p in self.params: p.grad.data.zero_()\n```\n\n\n```\nmodel = nn.Sequential(nn.Linear(m,nh), nn.ReLU(), nn.Linear(nh,10))\n```\n\n\n```\nopt = Optimizer(model.parameters())\n```\n\n\n```\nfor epoch in range(epochs):\n for i in range((n-1)//bs + 1):\n start_i = i*bs\n end_i = start_i+bs\n xb = x_train[start_i:end_i]\n yb = y_train[start_i:end_i]\n pred = model(xb)\n loss = loss_func(pred, yb.long())\n\n loss.backward()\n opt.step()\n opt.zero_grad()\n```\n\n\n```\nloss,acc = loss_func(model(xb), yb.long()), accuracy(model(xb), yb)\nloss,acc\n```\n\n\n\n\n (tensor(0.0091, grad_fn=), tensor(1.))\n\n\n\nPyTorch already provides this exact functionality in `optim.SGD` (it also handles stuff like momentum, which we'll look at later - except we'll be doing it in a more flexible way!)\n\n\n```\n#export\nfrom torch import optim\n```\n\n\n```\noptim.SGD.step??\n```\n\n\n```\ndef get_model():\n model = nn.Sequential(nn.Linear(m,nh), nn.ReLU(), nn.Linear(nh,10))\n return model, optim.SGD(model.parameters(), lr=lr)\n```\n\n\n```\nmodel,opt = get_model()\nloss_func(model(xb), yb.long())\n```\n\n\n\n\n tensor(2.3490, grad_fn=)\n\n\n\n\n```\nfor epoch in range(epochs):\n for i in range((n-1)//bs + 1):\n start_i = i*bs\n end_i = start_i+bs\n xb = x_train[start_i:end_i]\n yb = y_train[start_i:end_i]\n pred = model(xb)\n loss = loss_func(pred, yb.long())\n\n loss.backward()\n opt.step()\n opt.zero_grad()\n```\n\n\n```\nloss,acc = loss_func(model(xb), yb.long()), accuracy(model(xb), yb)\nloss,acc\n```\n\n\n\n\n (tensor(0.0048, grad_fn=), tensor(1.))\n\n\n\nRandomized tests can be very useful.\n\n[Jump_to lesson 9 video](https://course.fast.ai/videos/?lesson=9&t=3442)\n\n\n```\nassert acc>0.7\n```\n\n## Dataset and DataLoader\n\n### Dataset\n\nIt's clunky to iterate through minibatches of x and y values separately:\n\n```python\n xb = x_train[start_i:end_i]\n yb = y_train[start_i:end_i]\n```\n\nInstead, let's do these two steps together, by introducing a `Dataset` class:\n\n```python\n xb,yb = train_ds[i*bs : i*bs+bs]\n```\n\n[Jump_to lesson 9 video](https://course.fast.ai/videos/?lesson=9&t=3578)\n\n\n```\n#export\nclass Dataset():\n def __init__(self, x, y): self.x,self.y = x,y\n def __len__(self): return len(self.x)\n def __getitem__(self, i): return self.x[i],self.y[i]\n```\n\n\n```\ntrain_ds,valid_ds = Dataset(x_train, y_train),Dataset(x_valid, y_valid)\nassert len(train_ds)==len(x_train)\nassert len(valid_ds)==len(x_valid)\n```\n\n\n```\nxb,yb = train_ds[0:5]\nassert xb.shape==(5,28*28)\nassert yb.shape==(5,)\nxb,yb\n```\n\n\n\n\n (tensor([[0., 0., 0., ..., 0., 0., 0.],\n [0., 0., 0., ..., 0., 0., 0.],\n [0., 0., 0., ..., 0., 0., 0.],\n [0., 0., 0., ..., 0., 0., 0.],\n [0., 0., 0., ..., 0., 0., 0.]]), tensor([5., 0., 4., 1., 9.]))\n\n\n\n\n```\nmodel,opt = get_model()\n```\n\n\n```\nfor epoch in range(epochs):\n for i in range((n-1)//bs + 1):\n xb,yb = train_ds[i*bs : i*bs+bs]\n pred = model(xb)\n loss = loss_func(pred, yb.long())\n\n loss.backward()\n opt.step()\n opt.zero_grad()\n```\n\n\n```\nloss,acc = loss_func(model(xb), yb.long()), accuracy(model(xb), yb)\nassert acc>0.7\nloss,acc\n```\n\n\n\n\n (tensor(0.0098, grad_fn=), tensor(1.))\n\n\n\n### DataLoader\n\nPreviously, our loop iterated over batches (xb, yb) like this:\n\n```python\nfor i in range((n-1)//bs + 1):\n xb,yb = train_ds[i*bs : i*bs+bs]\n ...\n```\n\nLet's make our loop much cleaner, using a data loader:\n\n```python\nfor xb,yb in train_dl:\n ...\n```\n\n[Jump_to lesson 9 video](https://course.fast.ai/videos/?lesson=9&t=3674)\n\n\n```\nclass DataLoader():\n def __init__(self, ds, bs): self.ds,self.bs = ds,bs\n def __iter__(self):\n for i in range(0, len(self.ds), self.bs): yield self.ds[i:i+self.bs]\n```\n\n\n```\ntrain_dl = DataLoader(train_ds, bs)\nvalid_dl = DataLoader(valid_ds, bs)\n```\n\n\n```\nxb,yb = next(iter(valid_dl))\nassert xb.shape==(bs,28*28)\nassert yb.shape==(bs,)\n```\n\n\n```\nplt.imshow(xb[0].view(28,28))\nyb[0]\n```\n\n\n```\nmodel,opt = get_model()\n```\n\n\n```\ndef fit():\n for epoch in range(epochs):\n for xb,yb in train_dl:\n pred = model(xb)\n loss = loss_func(pred, yb.long())\n loss.backward()\n opt.step()\n opt.zero_grad()\n```\n\n\n```\nfit()\n```\n\n\n```\nloss,acc = loss_func(model(xb), yb.long()), accuracy(model(xb), yb)\nassert acc>0.7\nloss,acc\n```\n\n\n\n\n (tensor(0.0277, grad_fn=), tensor(1.))\n\n\n\n### Random sampling\n\nWe want our training set to be in a random order, and that order should differ each iteration. But the validation set shouldn't be randomized.\n\n[Jump_to lesson 9 video](https://course.fast.ai/videos/?lesson=9&t=3942)\n\n\n```\nclass Sampler():\n def __init__(self, ds, bs, shuffle=False):\n self.n,self.bs,self.shuffle = len(ds),bs,shuffle\n \n def __iter__(self):\n self.idxs = torch.randperm(self.n) if self.shuffle else torch.arange(self.n)\n for i in range(0, self.n, self.bs): yield self.idxs[i:i+self.bs]\n```\n\n\n```\nsmall_ds = Dataset(*train_ds[:10])\n```\n\n\n```\ns = Sampler(small_ds,3,False)\n[o for o in s]\n```\n\n\n\n\n [tensor([0, 1, 2]), tensor([3, 4, 5]), tensor([6, 7, 8]), tensor([9])]\n\n\n\n\n```\ns = Sampler(small_ds,3,True)\n[o for o in s]\n```\n\n\n\n\n [tensor([2, 5, 1]), tensor([3, 0, 6]), tensor([7, 8, 4]), tensor([9])]\n\n\n\n\n```\ndef collate(b):\n xs,ys = zip(*b)\n return torch.stack(xs),torch.stack(ys)\n\nclass DataLoader():\n def __init__(self, ds, sampler, collate_fn=collate):\n self.ds,self.sampler,self.collate_fn = ds,sampler,collate_fn\n \n def __iter__(self):\n for s in self.sampler: yield self.collate_fn([self.ds[i] for i in s])\n```\n\n\n```\ntrain_samp = Sampler(train_ds, bs, shuffle=True)\nvalid_samp = Sampler(valid_ds, bs, shuffle=False)\n```\n\n\n```\ntrain_dl = DataLoader(train_ds, sampler=train_samp, collate_fn=collate)\nvalid_dl = DataLoader(valid_ds, sampler=valid_samp, collate_fn=collate)\n```\n\n\n```\nxb,yb = next(iter(valid_dl))\nplt.imshow(xb[0].view(28,28))\nyb[0]\n```\n\n\n```\nxb,yb = next(iter(train_dl))\nplt.imshow(xb[0].view(28,28))\nyb[0]\n```\n\n\n```\nxb,yb = next(iter(train_dl))\nplt.imshow(xb[0].view(28,28))\nyb[0]\n```\n\n\n```\nmodel,opt = get_model()\nfit()\n\nloss,acc = loss_func(model(xb), yb), accuracy(model(xb), yb)\nassert acc>0.7\nloss,acc\n```\n\n### PyTorch DataLoader\n\n[Jump_to lesson 9 video](https://course.fast.ai/videos/?lesson=9&t=4171)\n\n\n```\n#export\nfrom torch.utils.data import DataLoader, SequentialSampler, RandomSampler\n```\n\n\n```\ntrain_dl = DataLoader(train_ds, bs, sampler=RandomSampler(train_ds), collate_fn=collate)\nvalid_dl = DataLoader(valid_ds, bs, sampler=SequentialSampler(valid_ds), collate_fn=collate)\n```\n\n\n```\nmodel,opt = get_model()\nfit()\nloss_func(model(xb), yb), accuracy(model(xb), yb)\n```\n\nPyTorch's defaults work fine for most things however:\n\n\n```\ntrain_dl = DataLoader(train_ds, bs, shuffle=True, drop_last=True)\nvalid_dl = DataLoader(valid_ds, bs, shuffle=False)\n```\n\n\n```\nmodel,opt = get_model()\nfit()\n\nloss,acc = loss_func(model(xb), yb), accuracy(model(xb), yb)\nassert acc>0.7\nloss,acc\n```\n\nNote that PyTorch's `DataLoader`, if you pass `num_workers`, will use multiple threads to call your `Dataset`.\n\n## Validation\n\nYou **always** should also have a [validation set](http://www.fast.ai/2017/11/13/validation-sets/), in order to identify if you are overfitting.\n\nWe will calculate and print the validation loss at the end of each epoch.\n\n(Note that we always call `model.train()` before training, and `model.eval()` before inference, because these are used by layers such as `nn.BatchNorm2d` and `nn.Dropout` to ensure appropriate behaviour for these different phases.)\n\n[Jump_to lesson 9 video](https://course.fast.ai/videos/?lesson=9&t=4260)\n\n\n```\ndef fit(epochs, model, loss_func, opt, train_dl, valid_dl):\n for epoch in range(epochs):\n # Handle batchnorm / dropout\n model.train()\n# print(model.training)\n for xb,yb in train_dl:\n loss = loss_func(model(xb), yb.long())\n loss.backward()\n opt.step()\n opt.zero_grad()\n\n model.eval()\n# print(model.training)\n with torch.no_grad():\n tot_loss,tot_acc = 0.,0.\n for xb,yb in valid_dl:\n pred = model(xb)\n tot_loss += loss_func(pred, yb.long())\n tot_acc += accuracy (pred,yb)\n nv = len(valid_dl)\n print(epoch, tot_loss/nv, tot_acc/nv)\n return tot_loss/nv, tot_acc/nv\n```\n\n*Question*: Are these validation results correct if batch size varies?\n\n`get_dls` returns dataloaders for the training and validation sets:\n\n\n```\n#export\ndef get_dls(train_ds, valid_ds, bs, **kwargs):\n return (DataLoader(train_ds, batch_size=bs, shuffle=True, **kwargs),\n DataLoader(valid_ds, batch_size=bs*2, **kwargs))\n```\n\nNow, our whole process of obtaining the data loaders and fitting the model can be run in 3 lines of code:\n\n\n```\ntrain_dl,valid_dl = get_dls(train_ds, valid_ds, bs)\nmodel,opt = get_model()\nloss,acc = fit(5, model, loss_func, opt, train_dl, valid_dl)\n```\n\n\n```\nassert acc>0.9\n```\n\n## Export\n\n\n```\n!python notebook2script.py 03_minibatch_training.ipynb\n```\n\n\n```\n\n```\n", "meta": {"hexsha": "310ee6ceeffe9f664e798bed67189de7637e5454", "size": 108619, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "nbs/dl2/03_minibatch_training_midTerm2021Spring.ipynb", "max_stars_repo_name": "moonryul/course-v3", "max_stars_repo_head_hexsha": "e5b13732fcbdbc75992ceef6681d00f52a8be4c2", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "nbs/dl2/03_minibatch_training_midTerm2021Spring.ipynb", "max_issues_repo_name": "moonryul/course-v3", "max_issues_repo_head_hexsha": "e5b13732fcbdbc75992ceef6681d00f52a8be4c2", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "nbs/dl2/03_minibatch_training_midTerm2021Spring.ipynb", "max_forks_repo_name": "moonryul/course-v3", "max_forks_repo_head_hexsha": "e5b13732fcbdbc75992ceef6681d00f52a8be4c2", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2020-03-03T03:24:32.000Z", "max_forks_repo_forks_event_max_datetime": "2020-09-20T11:44:38.000Z", "avg_line_length": 34.394870171, "max_line_length": 5114, "alphanum_fraction": 0.5452176875, "converted": true, "num_tokens": 8472, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6039318337259583, "lm_q2_score": 0.6992544085240401, "lm_q1q2_score": 0.42230199718088385}} {"text": "```python\nfrom IPython.core.display import HTML\nfrom IPython.display import Image\nHTML(\"\"\"\n\n\"\"\")\n```\n\n\n\n\n\n\n\n\n\n\n# *Circuitos El\u00e9tricos I*\n\n## Semana 1 - Conven\u00e7\u00f5es para aplica\u00e7\u00e3o das Leis de Kirchhoff na an\u00e1lise de circuitos\n\n\n\n### Caso 1\n\n\n```python\nImage(\"./figures/J1C1.png\", width=500)\n```\n\n#### Lei de Kirchhoff das tens\u00f5es (LKT) \n\nEm qualquer malha frechada do circuito $\\sum_k v_k = 0$\n\n`Conven\u00e7\u00e3o arbitr\u00e1ria (1): ao percorrer a malha, escolha um sinal (+ ou -) para indicar aumentos de tens\u00e3o e o sinal oposto para indicar quedas de tens\u00e3o no somat\u00f3rio da LKT.`\n\nLogo, atribuindo o sinal (-) para aumentos de tens\u00e3o e o sinal (+) para quedas de tens\u00e3o, ao aplicar a LKT no circuito mostrado acima, temos:\n\n$$ \n\\begin{align}\n-10 + v_1 + v_2 &= 0\\\\\n-v_2 + v_3 + v_4 &= 0\n\\end{align}\n$$\n\n#### Lei de Kirchhoff das correntes (LKC)\n\nEm qualquer n\u00f3 do circuito $\\sum_k i_k = 0$\n\n`Conven\u00e7\u00e3o arbitr\u00e1ria (2): para o n\u00f3 em quest\u00e3o, escolha um sinal (+ ou -) para indicar correntes chegando ao n\u00f3 e o sinal oposto para indicar correntes deixando o n\u00f3 no somat\u00f3rio da LKT.`\n\nou, para evitar erros com troca de sinais, simplesmente fa\u00e7a\n\n`Somat\u00f3rio das correntes chegando ao n\u00f3 igual ao somat\u00f3rio das correntes deixando o n\u00f3.`\n\n$$ \n\\begin{align}\ni_1 &= i_2 + i_3\\\\\ni_3 &= -0.5~A\n\\end{align}\n$$\n\n#### Lei de Ohm (+conven\u00e7\u00e3o passiva)\n\n`Conven\u00e7\u00e3o passiva (3): qualquer express\u00e3o que relacione as grandezas de tens\u00e3o e corrente num elemento ideal de dois terminais deve ser escrita de acordo com a conven\u00e7\u00e3o passiva.`\n\nA conven\u00e7\u00e3o passiva estabelece que:\n\n1. Se o sentido de refer\u00eancia adotado para corrente coincide com a queda de tens\u00e3o na polaridade de refer\u00eancia ($+ \\rightarrow -$), *qualquer express\u00e3o envolvendo $v$ e $i$* para o elemento em quest\u00e3o deve ser escrita com **sinal positivo**.\n\n\n2. Se o sentido de refer\u00eancia adotado para corrente coincide com o aumento de tens\u00e3o na polaridade de refer\u00eancia ($+ \\leftarrow -$), *qualquer express\u00e3o envolvendo $v$ e $i$* para o elemento em quest\u00e3o deve ser escrita com **sinal negativo**.\n\nA Lei de Ohm expressa a rela\u00e7\u00e3o entre tens\u00e3o, corrente e resist\u00eancia num resistor ideal. Logo, as express\u00f5es da Lei de Ohm devem obedecer a conven\u00e7\u00e3o passiva. \n\nDesse modo, podemos escrever as seguintes equa\u00e7\u00f5es para o circuito acima. \n\n$$ \n\\begin{align}\nv_1 &= 10i_1\\\\\nv_2 &= 50i_2\\\\\nv_3 &= 20i_3\n\\end{align}\n$$\n\nLogo:\n\n$$ \n\\begin{align}\n-10 + 10i_1 + 50i_2 &= 0\\\\\n-50i_2 -10 + v_4 &= 0\\\\\ni_1 - i_2 &= -0.5\n\\end{align}\n$$\n\nRearranjando as equa\u00e7\u00f5es:\n\n$$ \n\\begin{align}\n 10i_1 + 50i_2 &= 10\\\\\n-50i_2 + v_4 &= 10\\\\\ni_1 - i_2 &= -0.5\n\\end{align}\n$$\n\n### Solu\u00e7\u00e3o das equa\u00e7\u00f5es\n\n\n```python\nimport sympy as sp\nimport numpy as np\n```\n\n\n```python\n# define as N vari\u00e1veis desconhecidas\ni1, i2, v4 = sp.symbols('i1, i2, v4')\n\n# define os sistema de N equa\u00e7\u00f5es\neq1 = sp.Eq() \neq2 = sp.Eq() \neq3 = sp.Eq()\n\n# resolve o sistema\nsoluc = sp.solve((eq1, eq2, eq3), dict=True)\n\ni1 = np.array([sol[i1] for sol in soluc])\ni2 = np.array([sol[i2] for sol in soluc]) \nv4 = np.array([sol[v4] for sol in soluc]) \ni3 = -0.5\n\nprint('Solu\u00e7\u00e3o do sistema:\\n\\n i1 = %.2f A,\\n i2 = %.2f A,\\n i3 = %.2f A,\\n v4 = %.2f V.' %(i1, i2, i3, v4))\n```\n\n#### C\u00e1lculo das pot\u00eancias\n\n\n```python\n# express\u00f5es para a Lei de Ohm (conven\u00e7\u00e3o passiva)\nv1 = \nv2 = \nv3 = \n\n# express\u00f5es para as pot\u00eancias (conven\u00e7\u00e3o passiva)\np10V = \np1 = \np2 = \np3 = \np4 = \n\nprint('Pot\u00eancias:\\n\\n p10V = %.2f W\\n p1 = %.2f W,\\n p2 = %.2f W,\\n p3 = %.2f W,\\n p4 = %.2f W\\n' %(p10V, p1, p2, p3, p4))\n```\n\n\n```python\n# calcula somat\u00f3rio das pot\u00eancias\nprint('Somat\u00f3rio das pot\u00eancias : %.2f W\\n' %(p10V+p1+p2+p3+p4))\n```\n\nSimula\u00e7\u00e3o do circuito: https://tinyurl.com/yfbwd4vz\n\n### Caso 2\n\n\n```python\nImage(\"./figures/J1C2.png\", width=500)\n```\n\n\n```python\n# define as N vari\u00e1veis desconhecidas\ni1, i2, v4 = sp.symbols('i1, i2, v4')\n\n# define os sistema de N equa\u00e7\u00f5es\neq1 = sp.Eq( ) \neq2 = sp.Eq( ) \neq3 = sp.Eq( )\n\n# resolve o sistema\nsoluc = sp.solve((eq1, eq2, eq3), dict=True)\n\ni1 = np.array([sol[i1] for sol in soluc])\ni2 = np.array([sol[i2] for sol in soluc]) \nv4 = np.array([sol[v4] for sol in soluc]) \ni3 = 0.5\n\nprint('Solu\u00e7\u00e3o do sistema:\\n\\n i1 = %.2f A,\\n i2 = %.2f A,\\n i3 = %.2f A,\\n v4 = %.2f V.' %(i1, i2, i3, v4))\n```\n\n\n```python\n# express\u00f5es para a Lei de Ohm (conven\u00e7\u00e3o passiva)\nv1 = \nv2 = \nv3 = \n\n# express\u00f5es para as pot\u00eancias (conven\u00e7\u00e3o passiva)\np10V = \np1 = \np2 = \np3 = \np4 = \n\nprint('Pot\u00eancias:\\n\\n p10V = %.2f W\\n p1 = %.2f W,\\n p2 = %.2f W,\\n p3 = %.2f W,\\n p4 = %.2f W\\n' %(p10V, p1, p2, p3, p4))\n```\n\n### Caso 3\n\n\n```python\nImage(\"./figures/J1C3.png\", width=500)\n```\n\n\n```python\n# define as N vari\u00e1veis desconhecidas\ni1, i2, v4 = sp.symbols('i1, i2, v4')\n\n# define os sistema de N equa\u00e7\u00f5es\neq1 = sp.Eq( ) \neq2 = sp.Eq( ) \neq3 = sp.Eq( )\n\n# resolve o sistema\nsoluc = sp.solve((eq1, eq2, eq3), dict=True)\n\ni1 = np.array([sol[i1] for sol in soluc])\ni2 = np.array([sol[i2] for sol in soluc]) \nv4 = np.array([sol[v4] for sol in soluc]) \ni3 = 0.5\n\nprint('Solu\u00e7\u00e3o do sistema:\\n\\n i1 = %.2f A,\\n i2 = %.2f A,\\n i3 = %.2f A,\\n v4 = %.2f V.' %(i1, i2, i3, v4))\n```\n\n\n```python\n# express\u00f5es para a Lei de Ohm (conven\u00e7\u00e3o passiva)\nv1 = \nv2 = \nv3 = \n\n# express\u00f5es para as pot\u00eancias (conven\u00e7\u00e3o passiva)\np10V = \np1 = \np2 = \np3 = \np4 = \n\nprint('Pot\u00eancias:\\n\\n p10V = %.2f W\\n p1 = %.2f W,\\n p2 = %.2f W,\\n p3 = %.2f W,\\n p4 = %.2f W\\n' %(p10V, p1, p2, p3, p4))\n```\n", "meta": {"hexsha": "cad0f81ac54a719dc9748dfc87eed27ed0241758", "size": 151454, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Jupyter notebooks/.ipynb_checkpoints/Circuitos Eletricos I - 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YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.8128673246376008, "lm_q1q2_score": 0.4223019070673875}} {"text": "```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport scipy \nfrom sklearn.model_selection import ParameterGrid\nfrom sklearn.manifold import Isomap\nimport time\nfrom tqdm import tqdm\n\nimport librosa\nfrom librosa import cqt\nfrom librosa.core import amplitude_to_db\nfrom librosa.display import specshow\n\nimport os\nimport glob\n```\n\n\n```python\ndata_dir = '/Users/sripathisridhar/Desktop/SOL'\n```\n\n\n```python\nfile_paths= sorted(glob.glob(os.path.join(data_dir, '**', '*.wav')))\n\nfile_names= []\nfor file_path in file_paths:\n file_names.append(os.path.basename(file_path))\n```\n\n\n```python\nhop_size= 512\nq= 24\n```\n\n\n```python\nimport h5py \n\nwith h5py.File(\"SOL.h5\", \"r\") as f:\n features_dict = {key:f[key][()] for key in f.keys()}\n```\n\n\n```python\ngrid = {\n 'Q': [24],\n 'k': [3],\n 'comp': ['log'],\n 'instr': ['all'],\n 'dyn': ['all']\n}\n\nsettings = list(ParameterGrid(grid))\n\nfor setting in settings:\n \n if setting[\"instr\"] == 'all':\n setting['instr'] = ''\n \n if setting['dyn'] == 'all':\n setting['dyn'] = ''\n```\n\n\n```python\nbatch_str = []\nCQT_OCTAVES = 7\n\nfeatures_keys = list(features_dict.keys())\n\nfor setting in settings:\n \n q = setting['Q']\n # Batch process and store in a folder\n batch_str = [setting['instr'], setting['dyn']]\n\n batch_features = []\n for feature_key in features_keys:\n # Get features that match setting\n \n if all(x in feature_key for x in batch_str):\n batch_features.append(features_dict[feature_key])\n \n batch_features = np.stack(batch_features, axis=1)\n \n # Isomap parameters\n hop_size = 512\n compression = 'log'\n features = amplitude_to_db(batch_features)\n n_neighbors = setting['k']\n n_dimensions = 3\n n_octaves = 3 \n\n # Prune feature matrix\n bin_low = np.where((np.std(features, axis=1) / np.std(features)) > 0.1)[0][0] + q\n bin_high = bin_low + n_octaves*q \n X = features[bin_low:bin_high, :]\n\n # Z-score Standardization- improves contrast in correlation matrix\n mus = np.mean(X, axis=1)\n sigmas = np.std(X, axis=1)\n X_std = (X - mus[:, np.newaxis]) / (1e-6 + sigmas[:, np.newaxis]) # 1e-6 to avoid runtime division by zero\n\n # Pearson correlation matrix\n rho_std = np.dot(X_std, X_std.T) / X_std.shape[1]\n \n # Isomap embedding\n isomap = Isomap(n_components= n_dimensions, n_neighbors= n_neighbors)\n coords = isomap.fit_transform(rho_std)\n \n # Get note value\n freqs= librosa.cqt_frequencies(q*CQT_OCTAVES, fmin=librosa.note_to_hz('C1'), bins_per_octave=q) #librosa CQT default fmin is C1\n chroma_list= librosa.core.hz_to_note(freqs[bin_low:bin_high])\n \n notes = []\n reps = q//12\n for chroma in chroma_list:\n for i in range(reps):\n notes.append(chroma)\n```\n\n\n```python\ncurr_fig= plt.figure(figsize=(5.5, 2.75))\nax= curr_fig.add_subplot(121)\nax.axis('off')\n\nimport colorcet as cc\nsubsampled_color_ids = np.floor(np.linspace(0, 256, q, endpoint=False)).astype('int')\ncolor_list= [cc.cyclic_mygbm_30_95_c78[i] for i in subsampled_color_ids]\n\n# Plot embedding with color\nfor i in range(coords.shape[0]):\n plt.scatter(coords[i, 0], coords[i, 1], color= color_list[i%q], s=30.0)\n\nplt.plot(coords[:, 0], coords[:, 1], color='black', linewidth=0.2)\n\n# Plot Pearson correlation matrix\nrho_frequencies = freqs[bin_low:bin_high]\n\nfreq_ticklabels = ['A2', 'A3', 'A4']\nfreq_ticks = librosa.core.note_to_hz(freq_ticklabels)\n\ntick_bins = []\ntick_labels= []\nfor i,freq_tick in enumerate(freq_ticks):\n tick_bin = np.argmin(np.abs(rho_frequencies-freq_tick))\n tick_bins.append(tick_bin)\n tick_labels.append(freq_ticklabels[i])\n\nplt.figure(figsize=(2.5,2.5))\nplt.imshow(np.abs(rho_std), cmap='magma_r')\nplt.xticks(tick_bins)\nplt.gca().set_xticklabels(freq_ticklabels)\n# plt.xlabel('Log-frequency (octaves)')\nplt.yticks(tick_bins)\nplt.gca().set_yticklabels(freq_ticklabels)\n# plt.ylabel('Log-frequency (octaves)')\nplt.gca().invert_yaxis()\n\nplt.clim(0, 1)\n\n```\n\n### Circle projection\n\n\n```python\nimport circle_fit\nimport importlib\nimportlib.reload(circle_fit)\nfrom circle_fit import circle_fit\n\nA = np.transpose(coords[:,:-1])\nx, r, circle_residual = circle_fit(A, verbose=True)\n```\n\n\n```python\nimport matplotlib\nmatplotlib.rc('font', family='serif')\n\nfig, axes = plt.subplots()\nplt.scatter(A[0,:],A[1,:])\nplt.plot(x[0],x[1],'rx')\n\ncircle = plt.Circle(x, radius=r, fill=False, linestyle='-.')\n\naxes.set_aspect(1)\naxes.add_artist(circle)\n\n# axes.set_ylim([-5,6])\n# axes.set_xlim([-2,8])\n\nplt.title('Circle fit: TinySOL all instr', pad=10.0)\nplt.show()\n\nprint(np.sqrt(circle_residual)/72)\n```\n\n\n```python\nr\n```\n\n\n\n\n 6.355528108736576\n\n\n\n\n```python\ndef d_squared(a, b):\n # Takes two n-D tuples and returns euclidean distance between them\n \n # Cast to array for computation \n # Cast first to tuple in case a or b are Sympy Point objects\n p_a = np.array(tuple(a), dtype='float')\n p_b = np.array(tuple(b), dtype='float')\n \n return np.sum(np.square(p_a - p_b))\n```\n\n\n```python\nimport sympy\n\nfrom sympy.geometry import Circle, Point, Line\n\ncenter = Point(x, evaluate=False)\nc = Circle(center, r, evaluate=False)\n\nl = Line(Point(coords[0,:-1]), center, evaluate=False)\npoints = [tuple(p) for p in l.points]\n\nxy_prime = []\n\n# TODO: Optimize to a more pythonic manner\nfor x,y in coords[:,:2]:\n \n intersections = c.intersection(Line(Point(x,y), center, evaluate=False))\n \n if d_squared((x,y),intersections[0]) < d_squared((x,y), intersections[1]):\n xy_prime.append([float(p) for p in intersections[0]])\n else:\n xy_prime.append([float(p) for p in intersections[1]])\n \n```\n\n\n```python\nfig, axes = plt.subplots()\nplt.scatter(np.array(xy_prime)[:,0],np.array(xy_prime)[:,1], s=10, \n label='projected points')\nplt.scatter(A[0,:],A[1,:], s=0.5, label='isomap embedding points (2D)')\nplt.plot(center[0],center[1],'rx')\n\ncircle = plt.Circle([float(p) for p in center], radius=r, fill=False, \n linestyle='--', label='estimated circle fit')\n\naxes.set_aspect(1)\naxes.add_artist(circle)\n\nplt.title('Projected points on circle', pad=10.0)\nplt.legend(bbox_to_anchor=(1,1))\nplt.show()\n\n\n```\n\n### Line projection\n\n\n```python\nz = np.arange(len(coords[:,2]))\nz_fit = scipy.stats.linregress(z, coords[:,2])\nprint(z_fit.stderr)\n```\n\n 0.0052234352623828605\n\n\n\n```python\nplt.figure()\nplt.title('Line fit: TinySOL all instr')\nplt.scatter(np.arange(len(coords[:,2])), coords[:,2])\n\nplt.plot(z_fit.intercept + z_fit.slope*z, 'b')\n```\n\n\n```python\n# New line coordinates\nz_prime = [i * z_fit.slope + z_fit.intercept for i,_ in enumerate(coords[:,2])]\n```\n\n\n```python\ncoords_prime = np.append(np.array(xy_prime), np.expand_dims(np.array(z_prime), axis=1), axis=1)\ncoords_length = coords_prime.shape[0]\n```\n\n### Distance matrices \n\n\n```python\n# Projected helix self-distance matrix\n\nD_proj = np.zeros((coords_length, coords_length))\nfor i in range(coords_length):\n for j in range(i,coords_length):\n \n D_proj[i][j] = d_squared(coords_prime[i,:], coords_prime[j,:])\n```\n\n\n```python\n# Isomap embedding self-distance matrix\n\nD_isomap = np.zeros((coords_length, coords_length)) # Projected points same no. as isomap\nfor i in range(coords_length):\n for j in range(i, coords_length):\n \n D_isomap[i][j] = d_squared(coords[i,:], coords[j,:])\n```\n\n\n```python\n# Geodesic self-distance matrix\n\nD_geodesic = isomap.dist_matrix_\n\n# Convert to upper triangular sparse matrix\nfor i in range(coords_length):\n for j in range(i):\n D_geodesic[i,j] = 0\n```\n\n\n```python\n## Centering matrix\n\ndef centered(A, Q=24, J=3):\n # Returns centered distance matrix\n \n '''\n Inputs\n -----\n A - squared distance matrix\n Q - quality factor, 24 by default\n J - number of octaves, 3 by default\n \n Returns\n -----\n tau - MDS style diagonalized matrix of A\n '''\n \n coords_length = A.shape[0]\n H = np.zeros((coords_length, coords_length))\n\n const = 1/(Q*J)\n for i in range(coords_length):\n for j in range(coords_length):\n if j==i:\n H[i,j] = 1 - const\n else:\n H[i,j] = -const\n \n return -0.5 * np.matmul(np.matmul(H, A), H)\n```\n\n\n```python\ndef frobenius_distance(A, B):\n # Given two nxn matrices, return their 'Frobenius distance'\n \n return np.sqrt(np.sum(np.square(A - B)))\n```\n\n\n```python\nloss_isomap = frobenius_distance(centered(D_geodesic), centered(D_isomap))/coords_length\nloss_total = frobenius_distance(centered(D_geodesic), centered(D_proj))/coords_length\nloss_proj = frobenius_distance(centered(D_isomap), centered(D_proj))/coords_length\n```\n\n\n```python\nprint(f\"Isomap loss= {loss_isomap}\")\nprint(f\"Projection loss= {loss_proj}\")\nprint(f\"Total loss= {loss_total}\")\n```\n\n Isomap loss= 20.242536163460947\n Projection loss= 2.3225511781614396\n Total loss= 20.914716613612924\n\n\n\n```python\n(loss_total) - (loss_isomap + loss_proj) < 0\n```\n\n\n\n\n True\n\n\n", "meta": {"hexsha": "20879de637de542e4174fa04d9daeae795284d3a", "size": 109977, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "test-notebooks/helicalitySOL.ipynb", "max_stars_repo_name": "sripathisridhar/sridhar2020ismir", "max_stars_repo_head_hexsha": "7e7b621fdf83a67784ab0b1fce37e483931094f8", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-01-14T10:00:28.000Z", "max_stars_repo_stars_event_max_datetime": "2021-01-14T10:00:28.000Z", "max_issues_repo_path": "test-notebooks/helicalitySOL.ipynb", "max_issues_repo_name": "sripathisridhar/sridhar2020ismir", "max_issues_repo_head_hexsha": "7e7b621fdf83a67784ab0b1fce37e483931094f8", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-10-14T20:50:20.000Z", "max_issues_repo_issues_event_max_datetime": "2020-10-14T20:50:21.000Z", "max_forks_repo_path": "test-notebooks/helicalitySOL.ipynb", "max_forks_repo_name": "sripathisridhar/sridhar2020ismir", "max_forks_repo_head_hexsha": "7e7b621fdf83a67784ab0b1fce37e483931094f8", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 166.6318181818, "max_line_length": 25348, "alphanum_fraction": 0.8983423807, "converted": true, "num_tokens": 2503, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217431943272, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.4222927760391717}} {"text": "```python\n#@markdown Configuraci\u00f3n Inicial\n\nimport os, sys\nfrom IPython.utils import io\nfrom IPython.display import display, display_svg\nfrom IPython.display import Math, Latex\nfrom IPython.display import IFrame, HTML\nimport time\nfrom datetime import datetime as dt\nfrom pytz import timezone\nimport random, string\n\nprefix = 'Parallel'\nconfigdict = {\n 'Serialpc': \"DISABLE_JIT: 1\",\n 'Parallel': \"DISABLE_JIT: 0\\n\"\n \"NUM_THREADS: 2\\n\"\n \"THREADING_LAYER: 'tbb'\"\n}\n\n################################################################################\n\nworkdir = prefix +\\\n dt.now(timezone('America/Bogota')).strftime('_%y%m%d-%H%M_') +\\\n ''.join(random.sample(string.hexdigits, 4)\n )\nos.mkdir(workdir)\nos.chdir(workdir)\nprint('Working Directory:', os.getcwd(), sep='\\n')\nwith open('.numba_config.yaml','w') as cf:\n cf.write(configdict[prefix])\n print('Numba Config:', configdict[prefix], sep='\\n', end='\\n\\n')\n\n################################################################################\n\n# https://stackoverflow.com/a/57883792\n# https://stackoverflow.com/a/57113015\nwith io.capture_output() as cap:\n !pip install tbb\n !pip install setuptools\n # https://matplotlib-axes-aligner.readthedocs.io/en/latest/\n !pip install mpl-axes-aligner\n !pip install watermark\n !pip install tqdm --upgrade\n !pip install gradio\n\nwith open('pip_installs.txt', 'w') as f:\n f.write(cap.stdout)\n!pip freeze > requirements.txt\n\n################################################################################\n\nimport pandas as pd\nimport numpy as np\nfrom numba import njit, prange, config\nfrom numba.np.extensions import cross2d\n\nif config.DISABLE_JIT==1:\n cross2d = np.cross\n prange = range\n print('JIT DISABLED!')\nelse:\n print('JIT!')\nfrom IPython.display import set_matplotlib_formats\n\nfrom numpy import linalg as LA\nfrom numpy import random as rg\nfrom scipy.interpolate import interp1d\nfrom scipy import constants as const\nfrom scipy import stats as st\n\nimport plotly.graph_objects as go\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\nset_matplotlib_formats('pdf', 'svg')\nfrom matplotlib import style\n#style.use('classic')\n\n\nfrom collections import Counter\nfrom matplotlib.lines import Line2D\nfrom matplotlib.collections import LineCollection\nfrom matplotlib import colors as pltcolors\nfrom matplotlib.colors import ListedColormap, BoundaryNorm\n# https://matplotlib-axes-aligner.readthedocs.io/en/latest/\nfrom mpl_axes_aligner import align\n\nimport gradio as gr\nfrom tqdm.auto import trange, tqdm\nfrom pprint import pprint\n\nfrom sympy import Point, Polygon\n\n#https://github.com/rasbt/watermark\n%reload_ext watermark\n%watermark -v -iv -m\n\n!lscpu\n!nvidia-smi\n```\n\n Working Directory:\n /content/Parallel_211122-2102_aE53\n Numba Config:\n DISABLE_JIT: 0\n NUM_THREADS: 2\n THREADING_LAYER: 'tbb'\n \n JIT!\n Python implementation: CPython\n Python version : 3.7.12\n IPython version : 5.5.0\n \n Compiler : GCC 7.5.0\n OS : Linux\n Release : 5.4.104+\n Machine : x86_64\n Processor : x86_64\n CPU cores : 2\n Architecture: 64bit\n \n IPython : 5.5.0\n sys : 3.7.12 (default, Sep 10 2021, 00:21:48) \n [GCC 7.5.0]\n plotly : 4.4.1\n scipy : 1.4.1\n numpy : 1.19.5\n numba : 0.51.2\n matplotlib : 3.2.2\n mpl_axes_aligner: 1.3\n gradio : 2.4.6\n pandas : 1.1.5\n \n Architecture: x86_64\n CPU op-mode(s): 32-bit, 64-bit\n Byte Order: Little Endian\n CPU(s): 2\n On-line CPU(s) list: 0,1\n Thread(s) per core: 2\n Core(s) per socket: 1\n Socket(s): 1\n NUMA node(s): 1\n Vendor ID: GenuineIntel\n CPU family: 6\n Model: 79\n Model name: Intel(R) Xeon(R) CPU @ 2.20GHz\n Stepping: 0\n CPU MHz: 2200.140\n BogoMIPS: 4400.28\n Hypervisor vendor: KVM\n Virtualization type: full\n L1d cache: 32K\n L1i cache: 32K\n L2 cache: 256K\n L3 cache: 56320K\n NUMA node0 CPU(s): 0,1\n Flags: fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush mmx fxsr sse sse2 ss ht syscall nx pdpe1gb rdtscp lm constant_tsc rep_good nopl xtopology nonstop_tsc cpuid tsc_known_freq pni pclmulqdq ssse3 fma cx16 pcid sse4_1 sse4_2 x2apic movbe popcnt aes xsave avx f16c rdrand hypervisor lahf_lm abm 3dnowprefetch invpcid_single ssbd ibrs ibpb stibp fsgsbase tsc_adjust bmi1 hle avx2 smep bmi2 erms invpcid rtm rdseed adx smap xsaveopt arat md_clear arch_capabilities\n NVIDIA-SMI has failed because it couldn't communicate with the NVIDIA driver. Make sure that the latest NVIDIA driver is installed and running.\n \n\n\n\n```python\n#@markdown Definici\u00f3n de Funciones\n\n@njit(\n fastmath=True,\n error_model='numpy',\n )\ndef inner_angle(Model, i):\n k = len(Model)\n prev = Model[(i-1)%k]\n next = Model[(i+1)%k]\n here = Model[i%k]\n vetcs = prev-here, next-here\n #https://stackoverflow.com/a/14067171\n return -np.arctan2(\n cross2d(*vetcs),\n np.dot(*vetcs)\n )\n\n@njit(\n fastmath=True,\n error_model='numpy',\n )\ndef grav(Model, XZ):\n k = len(Model)\n lenXZ = len(XZ)\n xietalist = [Model-XZ[i] for i in prange(lenXZ)]\n lenxieta = len(xietalist)\n grav = np.empty(lenxieta)\n for j in prange(lenxieta):\n xi = xietalist[j].T[0]\n eta = xietalist[j].T[1]\n sum = 0\n for i in prange(k):\n A = (xi[i-1]*eta[i] - xi[i]*eta[i-1])/\\\n ((xi[i]-xi[i-1])**2 + (eta[i]-eta[i-1])**2)\n B1 = 0.5*(eta[i] - eta[i-1])*\\\n np.log((xi[i]**2 + eta[i]**2)/\\\n (xi[i-1]**2 + eta[i-1]**2))\n B2 = (xi[i] - xi[i-1])*\\\n (np.arctan(xi[i]/eta[i])-\\\n np.arctan(xi[i-1]/eta[i-1]))\n sum += A*(B1+B2)\n grav[j] = sum\n return grav\n\nclass Model(np.ndarray):\n def __new__(cls, input_array,*args,**kargs):\n return np.asarray(input_array).astype(np.float).view(cls)\n def __str__(self):\n return ','.join(map(str,self.flatten()))\n def area(self):\n # https://stackoverflow.com/a/30408825\n (x,z) = self.T\n return -0.5*(np.dot(x,np.roll(z,1))-np.dot(np.roll(x,1),z))\n def Cgeom(self):\n return np.array(np.mean(self,axis=0))\n def Cmass(self):\n # https://en.wikipedia.org/wiki/Centroid#Of_a_polygon\n k = self.__len__()\n A = self.area()\n (x,z) = self.T\n Cx = np.array(\n [(x[i%k] + x[(i+1)%k])*\\\n (x[i%k]*z[(i+1)%k] - x[(i+1)%k]*z[i%k])\\\n for i in range(k)]).sum()/A/6\n Cz = np.array(\n [(z[i%k] + z[(i+1)%k])*\\\n (x[i%k]*z[(i+1)%k] - x[(i+1)%k]*z[i%k])\\\n for i in range(k)]).sum()/A/6\n return np.array([Cx, Cz])\n def move(self, i, r, th):\n k = self.__len__()\n new = self.copy()\n new[i%k] = self[i%k] + r*np.array([np.cos(th), np.sin(th)])\n return new\n def birth(self, i, r, th):\n k = self.__len__()\n p = (self[i%k]+self[(i+1)%k])/2\n return np.insert(self, (i+1)%k, p, axis=0).move(i+1, r, th)\n def death(self, i):\n k = self.__len__()\n return np.delete(self, i%k, axis=0)\n def dvector(self, i):\n k = self.__len__()\n return np.array(\n (self[(i+1)%k] + self[(i-1)%k])/2-\n self[i%k]\n )\n def dradius(self,i):\n return norm(self.dvector(i))\n def dcat(self,i):\n k = self.__len__()\n return norm(self[(i+1)%k] - self[(i-1)%k])/2\n def bcat(self,i):\n k = self.__len__()\n return norm(self[(i+1)%k] - self[i%k])/2\n def vectors(self, i):\n k = self.__len__()\n prev = self[(i-1)%k]\n next = self[(i+1)%k]\n here = self[i%k]\n return np.array([prev-here, next-here])\n @njit(\n fastmath=True,\n error_model='numpy',\n )\n def angle(self, i):\n return inner_angle(self, i)\n def dists(self, i):\n mapvects = map(norm, self.vectors(i))\n return np.fromiter(mapvects, dtype=np.float)\n @njit(\n fastmath=True,\n error_model='numpy',\n )\n def gravitational(self, XZ, rho=600):\n g = grav(self, XZ)\n return np.column_stack((XZ, g))\n def notintersect(self):\n # https://github.com/lycantropos/bentley_ottmann \n # Shamos-Huey algorithms \n return not contour_self_intersects(\n bContour(list(map(lambda _: bPoint(*_), self))\n ))\n \ndef RegularModel(p, r, phi=np.pi/2, n=3):\n ths = phi+np.linspace(\n 0, -2*np.pi, n,\n endpoint=False\n )\n return Model(\n [p + r*np.array([np.cos(th), np.sin(th)]) for th in ths])\n \ndef str2Model(str):\n return Model(eval(str)).reshape(-1, 2)\n\ndef Model_from_file(filename):\n with open(filename, 'r') as file:\n l = list(map(str2Model, file.readlines()))\n if len(l)==1:\n return l[0]\n else:\n return l\n file.close()\n\ndef Model_from_GeoGebra(filename):\n return Model(np.genfromtxt(filename))\n\ndef xz_iterp(XZ, xx, kind='cubic'):\n zz = interp1d(*XZ.T, kind='cubic')(xx)\n return np.c_[xx, zz]\n\n\nprint('Listo')\n```\n\n Listo\n\n\n\n```python\n%%file ../TrueModel_from_GeoGebra.tsv\n188\t-95\n193\t-76\n219\t-72\n228\t-50\n252\t-54\n265\t-71\n265\t-93\n239\t-95\n229\t-119\n204\t-116\n```\n\n Overwriting ../TrueModel_from_GeoGebra.tsv\n\n\n\n```python\n#@markdown Copia y pega la tabla de los v\u00e9rtices separados por tabulaciones para cambiar la forma del modelo de prueba\nIFrame(\"https://www.geogebra.org/classic/cumzsmxp\", 1200, 500)\n```\n\n\n\n\n\n\n\n\n\n\n\n```python\ndef Model_eye(s, q=(1,1), d=(0,0), N=19):\n m = N//2\n n = (N-m)\n q = np.array(q)/2\n gauss = st.norm(0, 1/s) \n f = lambda x: (gauss.pdf(x)-gauss.pdf(1))/\\\n (gauss.pdf(0)-gauss.pdf(1))\n ii = np.r_[-1:+1:(n+1)*1j]\n jj = np.r_[+1:-1:(m+1)*1j]\n top = np.c_[ii, +f(ii)][:-1]\n bottom = np.c_[jj, -f(jj)][:-1]\n eye = q*np.r_[top, bottom] + d\n return Model(eye)\n\neye_Model = Model_eye(2.5, (200, 200*9/16), (200, -100))\nprint('Eye Model:\\n')\ndisplay_svg(Polygon(*map(Point, eye_Model)))\n```\n\n Eye Model:\n \n\n\n\n \n\n \n\n\n\n```python\n#@markdown Settings\n%%time\n\nsigma = 0.2\ngamma = 1.6\nMaxk = 20\nang = 2*np.pi\n\nold_settings = np.geterr()\nnew_settings = {\n 'divide': 'raise',\n 'invalid': 'raise',\n 'over': 'ignore',\n 'under': 'ignore'\n }\nsettlist = ['old_settings','new_settings']\ndisplay(pd.DataFrame(\n list(map(eval, settlist)),\n index=settlist\n ))\n\"\"\"\nignore: Take no action when the exception occurs.\nwarn: Print a RuntimeWarning (via the Python warnings module).\nraise: Raise a FloatingPointError.\ncall: Call a function specified using the seterrcall function.\nprint: Print a warning directly to stdout.\nlog: Record error in a Log object specified by seterrcall.\n\"\"\"\n\nGeoGebraModel = Model_from_GeoGebra('../TrueModel_from_GeoGebra.tsv')\nTrueModel = GeoGebraModel # GeoGebraModel, eye_Model\n\nw = np.ones(Maxk)\nb = np.ones(Maxk)\nd = np.ones(Maxk)\nb[-1] = 0\nd[2] = 0\nps = np.array([list(p/sum(p)) for p in np.stack((w,b,d)).T])\nps[0:2][:] = np.zeros((2,3))\n\nprint('sigma:', sigma)\nprint('gamma:', gamma)\nprint('Maxk:', Maxk)\nprint('ang:', ang)\nprint(ps)\n\n\nXZ = (lambda a:\n np.c_[0:500:a,\n 4*(1.5+np.cos(2*np.r_[0:2*np.pi:a]))]\n )(26j)\n\nXZg = TrueModel.gravitational(XZ)\ng = XZg[:,-1]\nnp.savetxt(\n '../Gravity.tsv',\n XZg,\n delimiter='\\t',\n fmt='%.6g'\n )\n\nXZmin = XZ\nXZgrid = xz_iterp(XZ, np.r_[0:500:1001j])\n\ntry:\n print('True Model:\\n')\n display_svg(Polygon(*map(Point, TrueModel.copy())))\nexcept:\n print('Error or Polygon has intersecting sides.')\n\nXg = TrueModel.gravitational(XZgrid)[:,(0,-1)]\nXgmin = TrueModel.gravitational(XZmin)[:,(0,-1)]\n```\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        divideoverunderinvalid
                                        old_settingswarnwarnignorewarn
                                        new_settingsraiseignoreignoreraise
                                        \n
                                        \n\n\n sigma: 0.2\n gamma: 1.6\n Maxk: 20\n ang: 6.283185307179586\n [[0. 0. 0. ]\n [0. 0. 0. ]\n [0.5 0.5 0. ]\n [0.33333333 0.33333333 0.33333333]\n [0.33333333 0.33333333 0.33333333]\n [0.33333333 0.33333333 0.33333333]\n [0.33333333 0.33333333 0.33333333]\n [0.33333333 0.33333333 0.33333333]\n [0.33333333 0.33333333 0.33333333]\n [0.33333333 0.33333333 0.33333333]\n [0.33333333 0.33333333 0.33333333]\n [0.33333333 0.33333333 0.33333333]\n [0.33333333 0.33333333 0.33333333]\n [0.33333333 0.33333333 0.33333333]\n [0.33333333 0.33333333 0.33333333]\n [0.33333333 0.33333333 0.33333333]\n [0.33333333 0.33333333 0.33333333]\n [0.33333333 0.33333333 0.33333333]\n [0.33333333 0.33333333 0.33333333]\n [0.5 0. 0.5 ]]\n True Model:\n \n\n\n\n \n\n \n\n\n CPU times: user 1.85 s, sys: 38.2 ms, total: 1.88 s\n Wall time: 1.88 s\n\n\n\n```python\n#@markdown Plot in 0\n\nfig, ax_z = plt.subplots(figsize=np.array([16,9])*3/5)\nax_z.tick_params(\n top=True,\n labeltop=True\n )\nax_g = ax_z.twinx()\nax_z.axhline(c='k', lw=1, zorder=0)\n\nax_g.plot(\n *Xg.T,\n color='k'\n )\nax_g.scatter(\n *Xgmin.T,\n marker='.',\n color='k'\n )\nlineg = Line2D(\n [0],[0],\n marker='.',\n color='k',\n label='Verdadera'\n )\n\nax_z.plot(\n *XZgrid.T,\n color='tab:green',\n label='_nolegend_'\n)\nax_z.fill_between(\n *XZgrid.T,\n color='tab:green',\n alpha=3/5\n)\nax_z.scatter(\n *XZmin.T,\n marker='.',\n color='tab:green',\n label='_nolegend_'\n)\nlinez = Line2D(\n [0],[0],\n marker='.',\n color='tab:green',\n label='Terreno'\n )\n\nTruepoly = patches.Polygon(\n TrueModel,\n closed=True,\n Fill=True,\n edgecolor=None,\n label='Verdadero'\n )\nax_z.add_patch(Truepoly)\n\nax_z.scatter(\n *TrueModel.Cmass(),\n marker='x',\n color='r',\n zorder=2\n )\n\nax_z.set_xlabel(r'$x$')\nax_z.set_ylabel(r'$z$', rotation=0)\nax_g.set_ylabel(r'$\\quad\\Delta{g}$', rotation=0, y=5/6)\n\nax_z.set_xlim(0, 500)\nax_z.set_ylim(ymin=-200)\nax_g.set_yticks(np.r_[0:100:5j])\nalign.yaxes(ax_z, 0, ax_g, 0, 2/3)\n\nax_g.legend(\n handles=[lineg, linez],\n title='Gravimetr\u00eda',\n bbox_to_anchor=(0.25, -.1),\n ncol=3,\n loc='upper center'\n )\nax_z.legend(\n handles=[Truepoly],\n title='Modelo',\n bbox_to_anchor=(0.75, -.1),\n ncol=2,\n loc='upper center'\n )\nplt.savefig('InitPlot.pdf', bbox_inches='tight')\nplt.show()\n```\n\n\n \n\n \n\n\n\n```python\n!cat ../Gravity.tsv\n```\n\n 0\t10\t5.00469\n 20\t9.50523\t5.81984\n 40\t8.14331\t6.80133\n 60\t6.25116\t8.02252\n 80\t4.29688\t9.59481\n 100\t2.76393\t11.6699\n 120\t2.03154\t14.4198\n 140\t2.28089\t17.9639\n 160\t3.4503\t22.2207\n 180\t5.25047\t26.7409\n 200\t7.23607\t30.6812\n 220\t8.91587\t33.0171\n 240\t9.87433\t32.9741\n 260\t9.87433\t30.536\n 280\t8.91587\t26.4886\n 300\t7.23607\t21.8943\n 320\t5.25047\t17.5795\n 340\t3.4503\t13.9663\n 360\t2.28089\t11.1482\n 380\t2.03154\t9.03614\n 400\t2.76393\t7.47582\n 420\t4.29688\t6.31369\n 440\t6.25116\t5.42419\n 460\t8.14331\t4.71545\n 480\t9.50523\t4.12576\n 500\t10\t3.61728\n\n\n\n```python\n#@title Gradio\nexamples = [\n[\"\"\"56\t-54\n106\t-31\n154\t-34\n196\t-52\n207\t-82\n181\t-96\n163\t-120\n124\t-120\n81.\t-98\n58.\t-69\"\"\"],\n[\"\"\"108\t-108\n144\t-48\n192\t-96\n120\t-120\n96\t-192\n48\t-144\"\"\"]\n]\n\ndata = examples[0][0]\ndef my_fun(data):\n df=pd.DataFrame([x.split('\\t') \n for x in data.split('\\n')], dtype=np.float)\n TrueModel=Model(df)\n\n XZ = (lambda a:\n np.c_[0:500:a,\n 4*(1.5+np.cos(2*np.r_[0:2*np.pi:a]))]\n )(26j)\n\n XZmin = XZ\n XZgrid = xz_iterp(XZ, np.r_[0:500:1001j])\n\n Xg = TrueModel.gravitational(XZgrid)[:,(0,-1)]\n Xgmin = TrueModel.gravitational(XZmin)[:,(0,-1)]\n\n fig, ax_z = plt.subplots(figsize=np.array([16,9])*3/5)\n ax_z.tick_params(\n top=True,\n labeltop=True\n )\n ax_g = ax_z.twinx()\n ax_z.axhline(c='k', lw=1, zorder=0)\n\n ax_g.plot(\n *Xg.T,\n color='k'\n )\n ax_g.scatter(\n *Xgmin.T,\n marker='.',\n color='k'\n )\n lineg = Line2D(\n [0],[0],\n marker='.',\n color='k',\n label='Verdadera'\n )\n\n ax_z.plot(\n *XZgrid.T,\n color='tab:green',\n label='_nolegend_'\n )\n ax_z.fill_between(\n *XZgrid.T,\n color='tab:green',\n alpha=3/5\n )\n ax_z.scatter(\n *XZmin.T,\n marker='.',\n color='tab:green',\n label='_nolegend_'\n )\n linez = Line2D(\n [0],[0],\n marker='.',\n color='tab:green',\n label='Terreno'\n )\n\n Truepoly = patches.Polygon(\n TrueModel,\n closed=True,\n Fill=True,\n edgecolor=None,\n label='Verdadero'\n )\n ax_z.add_patch(Truepoly)\n\n ax_z.scatter(\n *TrueModel.Cmass(),\n marker='x',\n color='r',\n zorder=2\n )\n\n ax_z.set_xlabel(r'$x$')\n ax_z.set_ylabel(r'$z$', rotation=0)\n ax_g.set_ylabel(r'$\\quad\\Delta{g}$', rotation=0, y=5/6)\n\n ax_z.set_xlim(0, 500)\n ax_z.set_ylim(ymin=-200)\n ax_g.set_yticks(np.linspace(0, 100, 5))\n align.yaxes(ax_z, 0, ax_g, 0, 2/3)\n\n ax_g.legend(\n handles=[lineg, linez],\n title='Gravimetr\u00eda',\n bbox_to_anchor=(0.25, -.1),\n ncol=3,\n loc='upper center'\n )\n ax_z.legend(\n handles=[Truepoly],\n title='Modelo',\n bbox_to_anchor=(0.75, -.1),\n ncol=2,\n loc='upper center'\n )\n fig.tight_layout()\n plt.close()\n return fig\n\ntitle = \"Metodo Directo\"\ndescription = 'Copia y pega la tabla de los v\u00e9rtices separados por tabulaciones para cambiar la forma del modelo de prueba'\narticle = \"\"\"

                                        \n \n Gravimetro | \n Github\n

                                        \"\"\"\n\narticl0 = \"\"\"

                                        \n Gravimetro | \n Github\n

                                        \"\"\"\n\n\ngrtest = gr.Interface(\n fn=my_fun,\n inputs=gr.inputs.Textbox(\n lines=11,\n placeholder=data,\n # default=data\n ),\n # inputs=gr.inputs.Dataframe(\n # col_count=2,\n # # default=w,\n # datatype='number'\n # ),\n outputs='plot',\n live=True,\n allow_flagging=False,\n allow_screenshot=False,\n title=title,\n description=description,\n article=article,\n examples=examples,\n theme='huggingface', # \"default\", \"compact\" or \"huggingface\"\n layout='unaligned' # 'horizontal', 'unaligned', 'vertical'\n )\n# grtest.launch(inline=True, debug=True)\nwith io.capture_output() as captured:\n grtest.launch(inline=True)\nprint(grtest.share_url)\nIFrame(src=grtest.share_url, width=1500, height=700)\n# grtest.close()\n```\n\n https://47988.gradio.app\n\n\n\n\n\n\n\n\n\n\n\n\n```python\n#@markdown Archivo `.zip`\n\n!rm -f zfiles.zip\n!7z a -bso0 ../{workdir}.zip *.* -x!*.zip\n!7z l ../{workdir}.zip\n```\n\n 0M Scan \b\b\b\b\b\b\b\b\b\b \b\b\b\b\b\b\b\b\b\b 0%\b\b\b\b \b\b\b\b\n 7-Zip [64] 16.02 : Copyright (c) 1999-2016 Igor Pavlov : 2016-05-21\n p7zip Version 16.02 (locale=en_US.UTF-8,Utf16=on,HugeFiles=on,64 bits,2 CPUs Intel(R) Xeon(R) CPU @ 2.20GHz (406F0),ASM,AES-NI)\n \n Scanning the drive for archives:\n 0M Scan ../\b\b\b\b\b\b\b\b\b\b\b\b\b \b\b\b\b\b\b\b\b\b\b\b\b\b1 file, 27329 bytes (27 KiB)\n \n Listing archive: ../Parallel_211122-2102_aE53.zip\n \n --\n Path = ../Parallel_211122-2102_aE53.zip\n Type = zip\n Physical Size = 27329\n \n Date Time Attr Size Compressed Name\n ------------------- ----- ------------ ------------ ------------------------\n 2021-11-23 02:03:13 ..... 29116 22289 InitPlot.pdf\n 2021-11-23 02:03:06 ..... 7683 1092 pip_installs.txt\n 2021-11-23 02:03:07 ..... 7887 3502 requirements.txt\n ------------------- ----- ------------ ------------ ------------------------\n 2021-11-23 02:03:13 44686 26883 3 files\n\n", "meta": {"hexsha": "4f9a0fae434708dd71e6f4c8d2915750662eab01", "size": 171648, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "RJMCMC/RJMCMC_Grupo1.ipynb", "max_stars_repo_name": "edwardptera/Gravimetro", "max_stars_repo_head_hexsha": "cb39d813d07a6684de2cc29cfa97087950c0c11e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "RJMCMC/RJMCMC_Grupo1.ipynb", "max_issues_repo_name": "edwardptera/Gravimetro", "max_issues_repo_head_hexsha": "cb39d813d07a6684de2cc29cfa97087950c0c11e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "RJMCMC/RJMCMC_Grupo1.ipynb", "max_forks_repo_name": "edwardptera/Gravimetro", "max_forks_repo_head_hexsha": "cb39d813d07a6684de2cc29cfa97087950c0c11e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 171648.0, "max_line_length": 171648, "alphanum_fraction": 0.7291899702, "converted": true, "num_tokens": 6974, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665855647395, "lm_q2_score": 0.7122321903471563, "lm_q1q2_score": 0.42211622038234475}} {"text": "Sebastian Raschka 2014\n\n\n```python\n%load_ext watermark\n```\n\n\n```python\n%watermark -v -u -d -p scipy,scikit-learn,numpy,matplotlib\n```\n\n last updated: 2016-07-11 \n \n CPython 3.5.1\n IPython 4.2.0\n \n scipy 0.17.1\n scikit-learn 0.17.1\n numpy 1.11.0\n matplotlib 1.5.1\n\n\n[More information](https://github.com/rasbt/watermark) about the `watermark` magic command extension.\n\n\n
                                        \nI would be happy to hear your comments and suggestions. \nPlease feel free to drop me a note via\n[twitter](https://twitter.com/rasbt), [email](mailto:bluewoodtree@gmail.com), or [google+](https://plus.google.com/+SebastianRaschka).\n
                                        \n\n
                                        \n
                                        \n\n# Kernel tricks and nonlinear dimensionality reduction via PCA\n\n## Sections\n\n- [Principal Component Analysis](#Principal-Component-Analysis)\n- [PCA and linear dimensionality reduction](#PCA-and-linear-dimensionality-reduction)\n- [Nonlinear dimensionality reduction](#Nonlinear-dimensionality-reduction)\n- [Kernel functions and the kernel trick](Kernel-functions-and-the-kernel-trick)\n- [Gaussian RBF Kernel PCA](#Gaussian-RBF-Kernel-PCA)\n- [Implementing the RBF kernel PCA step-by-step](#Implementing-the-RBF-kernel-PCA-step-by-step)\n- [Examples of RBF Kernel PCA](#Examples-of-RBF-Kernel-PCA)\n - [Half-moon shapes](#Half-moon-shapes)\n - [Concentric circles](#Concentric-circles)\n - [Swiss roll](#Swiss-roll)\n- [Appendix A: Projecting new data](#Appendix-A:-Projecting-new-data)\n- [References](#References)\n\n
                                        \n
                                        \n\n**Most machine learning algorithms have been developed and statistically validated for linearly separable data. Popular examples are linear classifiers like Support Vector Machines (SVMs) or the (standard) Principal Component Analysis (PCA) for dimensionality reduction. However, most real world data requires nonlinear methods in order to perform tasks that involve the analysis and discovery of patterns successfully.**\n\n**The focus of this article is to briefly introduce the idea of kernel methods and to implement a Gaussian radius basis function (RBF) kernel that is used to perform nonlinear dimensionality reduction via BF kernel principal component analysis (kPCA).**\n\n
                                        \n
                                        \n\n## Principal Component Analysis\n\nThe main purpose of principal component analysis (PCA) is the analysis of data to identify patterns that represent the data \"well.\" The principal components can be understood as new axes of the dataset that maximize the variance along those axes (the eigenvectors of the covariance matrix). In other words, PCA aims to find the axes with maximum variances along which the data is most spread.\n\n\n\n\n
                                        \n
                                        \n\n## PCA and linear dimensionality reduction\n\n[[back to top](#Sections)]\n\nA common application of PCA is to reduce the dimensions of the dataset with minimal loss of information.\nHere, the entire dataset (*d* dimensions) is projected onto a new subspace (*k* dimensions where *k* < *d*). \nThis method of projection is useful in order to reduce the computational costs and the error of parameter estimation (\"curse of dimensionality\").\n\nThe standard PCA approach can be summarized in six simple steps:\n\n\n\nMore details can be found in a previous article [\"Implementing a Principal Component Analysis (PCA) in Python step by step\"](http://sebastianraschka.com/Articles/2014_pca_step_by_step.html).\n\n
                                        \n
                                        \n\n## Nonlinear dimensionality reduction\n\n[[back to top](#Sections)]\n\nThe \"classic\" PCA approach described above is a linear projection technique that works well if the data is linearly separable. However, in the case of linearly inseparable data, a nonlinear technique is required if the task is to reduce the dimensionality of a dataset.\n\n\n\n\n\n\n
                                        \n
                                        \n\n## Kernel functions and the kernel trick\n\n[[back to top](#Sections)]\n\nThe basic idea to deal with linearly inseparable data is to project it onto a higher dimensional space where it becomes linearly separable. Let us call this nonlinear mapping function $\\phi$ so that the mapping of a sample $\\mathbf{x}$ can be written as $\\mathbf{x} \\rightarrow \\phi (\\mathbf{x})$, which is called \"kernel function.\" \n\nNow, the term \"kernel\" describes a function that calculates the dot product of the images of the samples $\\mathbf{x}$ under $\\phi$.\n\n\n\\begin{equation}\\kappa(\\mathbf{x_i, x_j}) = \\phi (\\mathbf{x_i}) \\phi (\\mathbf{x_j})^T \\end{equation}\n\nMore details about the derivation of this equation are provided in this excellent review article by Quan Wang: [Kernel Principal Component Analysis and its Applications in Face Recognition and Active Shape Models](http://arxiv.org/abs/1207.3538).[[1](#References)]\n\n\n\n
                                        \n\nIn other words, the function $\\phi$ maps the original d-dimensional features into a larger, k-dimensional feature space by creating nononlinear combinations of the original features. For example, if $\\mathbf{x}$ consists of 2 features:\n\n\\begin{equation} \\mathbf{x} = \\big[x_1 \\quad x_2\\big]^T \\quad \\quad \\mathbf{x} \\in I\\!R^d\\\\\n\\Downarrow \\phi \\\\\n \\mathbf{x}' = \\big[x_1 \\quad x_2 \\quad x_1 x_2 \\quad x_{1}^2 \\quad x_1 x_{2}^3 \\quad \\dots \\big]^T \\quad \\quad \\mathbf{x} \\in I\\!R^k (k >> d) \\end{equation}\n\n\n\nOften, the mathematical definition of the RBF kernel is written and implemented as\n\n\\begin{equation} \\kappa(\\mathbf{x_i, x_j}) = exp\\bigg(- \\gamma \\; \\lVert\\mathbf{x_i - x_j }\\rVert^{2}_{2} \\bigg)\\end{equation}\n\nwhere $\\textstyle\\gamma = \\tfrac{1}{2\\sigma^2}$ is a free parameter that is to be optimized. \n\n
                                        \n
                                        \n\n## Gaussian radial basis function (RBF) Kernel PCA\n\n[[back to top](#Sections)]\n\nIn the linear PCA approach, we are interested in the principal components that maximize the variance in the dataset. This is done by extracting the eigenvectors (principle components) that correspond to the largest eigenvalues based on the covariance matrix:\n\n\\begin{equation}\\text{Cov} = \\frac{1}{N} \\sum_{i=1}^{N} \\mathbf{x_i} \\mathbf{x_i}^T \\end{equation}\n\nBernhard Scholkopf ([Kernel Principal Component Analysis](http://dl.acm.org/citation.cfm?id=299113) [[2](#References)]) generalized this approach for data that was mapped onto the higher dimensional space via a kernel function:\n\n\\begin{equation}\\text{Cov} = \\frac{1}{N} \\sum_{i=1}^{N} \\phi(\\mathbf{x_i}) \\phi(\\mathbf{x_i})^T \\end{equation}\n\nHowever, in practice the the covariance matrix in the higher dimensional space is not calculated explicitly (kernel trick). Therefore, the implementation of RBF kernel PCA does not yield the principal component axes (in contrast to the standard PCA), but the obtained eigenvectors can be understood as projections of the data onto the principal components.\n\n
                                        \n
                                        \n\n## Implementing the RBF kernel PCA step-by-step\n\n[[back to top](#Sections)]\n\nIn order to implement the RBF kernel PCA we just need to consider the following two steps.\n\n##### 1. Computation of the kernel (similarity) matrix.\n\nIn this first step, we need to calculate \n\n\\begin{equation} \\kappa(\\mathbf{x_i, x_j}) = exp\\bigg(- \\gamma \\; \\lVert\\mathbf{x_i - x_j }\\rVert^{2}_{2} \\bigg)\\end{equation}\n\nfor every pair of points. E.g., if we have a dataset of 100 samples, this step would result in a symmetric 100x100 kernel matrix.\n\n##### 2. Eigendecomposition of the kernel matrix.\n\nSince it is not guaranteed that the kernel matrix is centered, we can apply the following equation to do so:\n\n\\begin{equation} K' = K - \\mathbf{1_N} K - K \\mathbf{1_N} + \\mathbf{1_N} K \\mathbf{1_N} \\end{equation}\n\nwhere $\\mathbf{1_N}$ is (like the kernel matrix) a $N\\times N$ matrix with all values equal to $\\frac{1}{N}$. [[3](#References)]\n\nNow, we have to obtain the eigenvectors of the centered kernel matrix that correspond to the largest eigenvalues. Those eigenvectors are the data points already projected onto the respective principal components. \n\n\nBelow, we implement those steps in Python to see how those computations work.\n\n
                                        \n
                                        \n\n\n```python\nfrom scipy.spatial.distance import pdist, squareform\nfrom scipy import exp\nfrom scipy.linalg import eigh\nimport numpy as np\n\ndef stepwise_kpca(X, gamma, n_components):\n \"\"\"\n Implementation of a RBF kernel PCA.\n \n Arguments:\n X: A MxN dataset as NumPy array where the samples are stored as rows (M),\n and the attributes defined as columns (N).\n gamma: A free parameter (coefficient) for the RBF kernel.\n n_components: The number of components to be returned.\n \n \"\"\"\n # Calculating the squared Euclidean distances for every pair of points\n # in the MxN dimensional dataset.\n sq_dists = pdist(X, 'sqeuclidean')\n\n # Converting the pairwise distances into a symmetric MxM matrix.\n mat_sq_dists = squareform(sq_dists)\n\n # Computing the MxM kernel matrix.\n K = exp(-gamma * mat_sq_dists)\n\n # Centering the symmetric NxN kernel matrix.\n N = K.shape[0]\n one_n = np.ones((N,N)) / N\n K = K - one_n.dot(K) - K.dot(one_n) + one_n.dot(K).dot(one_n)\n\n # Obtaining eigenvalues in descending order with corresponding \n # eigenvectors from the symmetric matrix.\n eigvals, eigvecs = eigh(K)\n\n # Obtaining the i eigenvectors that corresponds to the i highest eigenvalues.\n X_pc = np.column_stack((eigvecs[:,-i] for i in range(1,n_components+1)))\n \n return X_pc\n```\n\n
                                        \n
                                        \n\n# Examples of RBF Kernel PCA\n\n[[back to top](#Sections)]\n\nIn this section, we will apply the RBF kernel PCA to different nonlinear sample data in order to perform dimensionality reduction.\n\n
                                        \n\n## Half-moon shapes\n\n[[back to top](#Sections)]\n\nWe will start with a simple example of 2 half-moon shapes generated by the [`make_moons`](http://scikit-learn.org/stable/modules/generated/sklearn.datasets.make_moons.html) function from [scikit-learn](http://scikit-learn.org/stable/index.html).\n\n\n```python\n%matplotlib inline\n```\n\n\n```python\nimport matplotlib.pyplot as plt\n\nfrom sklearn.datasets import make_moons\nX, y = make_moons(n_samples=100, random_state=123)\n\nplt.figure(figsize=(8,6))\n\nplt.scatter(X[y==0, 0], X[y==0, 1], color='red', alpha=0.5)\nplt.scatter(X[y==1, 0], X[y==1, 1], color='blue', alpha=0.5)\n\nplt.title('A nonlinear 2Ddataset')\nplt.ylabel('y coordinate')\nplt.xlabel('x coordinate')\n\nplt.show()\n```\n\n
                                        \n
                                        \n\n### Linear PCA\n\n[[back to top](#Sections)]\n\nSince the two half-moon shapes are linearly inseparable, we expect that the \"classic\" PCA will fail to give us a \"good\" representation of the data in 1D space. Here, we will use the [`PCA`](http://scikit-learn.org/stable/modules/generated/sklearn.decomposition.PCA.html) class that is implemented in scikit-learn to perform the dimensionality reduction.\n\n\n```python\nfrom sklearn.decomposition import PCA\n\nscikit_pca = PCA(n_components=2)\nX_spca = scikit_pca.fit_transform(X)\n\nplt.figure(figsize=(8,6))\nplt.scatter(X_spca[y==0, 0], X_spca[y==0, 1], color='red', alpha=0.5)\nplt.scatter(X_spca[y==1, 0], X_spca[y==1, 1], color='blue', alpha=0.5)\n\nplt.title('First 2 principal components after Linear PCA')\nplt.xlabel('PC1')\nplt.ylabel('PC2')\nplt.show()\n```\n\n\n```python\nimport numpy as np\nscikit_pca = PCA(n_components=1)\nX_spca = scikit_pca.fit_transform(X)\n\nplt.figure(figsize=(8,6))\nplt.scatter(X_spca[y==0, 0], np.zeros((50,1)), color='red', alpha=0.5)\nplt.scatter(X_spca[y==1, 0], np.zeros((50,1)), color='blue', alpha=0.5)\n\nplt.title('First principal component after Linear PCA')\nplt.xlabel('PC1')\n\nplt.show()\n```\n\nAs we can see, the resulting principal components do not yield a subspace where the data is linearly separated well. Note that PCA is a unsupervised method and does not \"consider\" class labels in order to maximize the variance in contrast to [Linear Discriminant Analysis](http://sebastianraschka.com/Articles/2014_python_lda.html). Here, the colors blue and red are just added for visualization purposes to indicate the degree of separation.\n\n
                                        \n
                                        \n\n### Gaussian RBF kernel PCA\n\n[[back to top](#Sections)]\n\nNext, we will perform dimensionality reduction via RBF kernel PCA on our half-moon data. The choice of $\\gamma$ depends on the dataset and can be obtained via hyperparameter tuning techniques like Grid Search. Hyperparameter tuning is a broad topic itself, and here I will just use a $\\gamma$-value that I found to produce \"good\" results. \n\n\n```python\nX_pc = stepwise_kpca(X, gamma=15, n_components=2)\n\nplt.figure(figsize=(8,6))\nplt.scatter(X_pc[y==0, 0], X_pc[y==0, 1], color='red', alpha=0.5)\nplt.scatter(X_pc[y==1, 0], X_pc[y==1, 1], color='blue', alpha=0.5)\n\nplt.title('First 2 principal components after RBF Kernel PCA')\nplt.text(-0.18, 0.18, 'gamma = 15', fontsize=12)\nplt.xlabel('PC1')\nplt.ylabel('PC2')\nplt.show()\n```\n\n\n```python\nplt.figure(figsize=(8,6))\nplt.scatter(X_pc[y==0, 0], np.zeros((50)), color='red', alpha=0.5)\nplt.scatter(X_pc[y==1, 0], np.zeros((50)), color='blue', alpha=0.5)\n\nplt.title('First principal component after RBF Kernel PCA')\nplt.text(-0.17, 0.007, 'gamma = 15', fontsize=12)\nplt.xlabel('PC1')\nplt.show()\n```\n\nWe can clearly see that the projection via RBF kernel PCA yielded a subspace where the classes are separated well. Such a subspace can then be used as input for linear classification models, such as Support Vector Machines or naive Bayes classifiers, which will be covered in future articles.\n\n
                                        \n
                                        \n\n### scikit RBF kernel PCA\n\n[[back to top](#Sections)]\n\nFor our convenience, there is already an implementation of the [`KernelPCA`](http://scikit-learn.org/stable/modules/generated/sklearn.decomposition.KernelPCA.html) in scikit-learn. \nLet us confirm that the results of our implementation is consistent with scikit-learn's approach.\n\n\n```python\nfrom sklearn.decomposition import KernelPCA\n\nscikit_kpca = KernelPCA(n_components=2, kernel='rbf', gamma=15)\nX_skernpca = scikit_kpca.fit_transform(X)\n\nplt.figure(figsize=(8,6))\nplt.scatter(X_skernpca[y==0, 0], X_skernpca[y==0, 1], color='red', alpha=0.5)\nplt.scatter(X_skernpca[y==1, 0], X_skernpca[y==1, 1], color='blue', alpha=0.5)\n\nplt.text(-0.48, 0.35, 'gamma = 15', fontsize=12)\nplt.title('First 2 principal components after RBF Kernel PCA via scikit-learn')\nplt.xlabel('PC1')\nplt.ylabel('PC2')\nplt.show()\n```\n\n\n```python\nscikit_kpca = KernelPCA(n_components=1, kernel='rbf', gamma=15)\nX_skernpca = scikit_kpca.fit_transform(X)\n\nplt.figure(figsize=(8,6))\nplt.scatter(X_skernpca[y==0, 0], np.zeros((50,1)), color='red', alpha=0.5)\nplt.scatter(X_skernpca[y==1, 0], np.zeros((50,1)), color='blue', alpha=0.5)\nplt.text(-0.48, 0.007, 'gamma = 15', fontsize=12)\nplt.title('First principal component after RBF Kernel PCA')\nplt.xlabel('PC1')\nplt.show()\n```\n\n
                                        \n
                                        \n\n## Concentric circles\n\n[[back to top](#Sections)]\n\nFor our next example, we will have a look at the classic case of 2 concentric circles with random noise produced by scikit-learn's [`make_circles`](http://scikit-learn.org/stable/modules/generated/sklearn.datasets.make_circles.html).\n\n\n```python\nfrom sklearn.datasets import make_circles\n\nX, y = make_circles(n_samples=1000, random_state=123, noise=0.1, factor=0.2)\n\nplt.figure(figsize=(8,6))\n\nplt.scatter(X[y==0, 0], X[y==0, 1], color='red', alpha=0.5)\nplt.scatter(X[y==1, 0], X[y==1, 1], color='blue', alpha=0.5)\nplt.title('Concentric circles')\nplt.ylabel('y coordinate')\nplt.xlabel('x coordinate')\nplt.savefig('/Users/Sebastian/Desktop/circles1.pdf')\n```\n\n
                                        \n
                                        \n\n### Linear PCA\n\n[[back to top](#Sections)]\n\n\n```python\nscikit_pca = PCA(n_components=2)\nX_spca = scikit_pca.fit_transform(X)\n\nplt.figure(figsize=(8,6))\nplt.scatter(X[y==0, 0], np.zeros((500,1))+0.1, color='red', alpha=0.5)\nplt.scatter(X[y==1, 0], np.zeros((500,1))-0.1, color='blue', alpha=0.5)\nplt.ylim([-15,15])\nplt.text(-0.125, 12.5, 'gamma = 15', fontsize=12)\nplt.title('First principal component after Linear PCA')\nplt.xlabel('PC1')\nplt.savefig('/Users/Sebastian/Desktop/circles2.pdf')\n```\n\nAgain, the results obtained via the linear PCA approach does not produce a subspace where the 2 classes are linearly well separated.\n\n
                                        \n
                                        \n\n### Gaussian RBF kernel PCA\n\n[[back to top](#Sections)]\n\n\n```python\nX_pc = stepwise_kpca(X, gamma=15, n_components=1)\n\nplt.figure(figsize=(8,6))\nplt.scatter(X_pc[y==0, 0], np.zeros((500,1)), color='red', alpha=0.5)\nplt.scatter(X_pc[y==1, 0], np.zeros((500,1)), color='blue', alpha=0.5)\nplt.text(-0.05, 0.007, 'gamma = 15', fontsize=12)\nplt.title('First principal component after RBF Kernel PCA')\nplt.xlabel('PC1')\n\nplt.savefig('/Users/Sebastian/Desktop/circles3.pdf')\n```\n\nAnd again, this 1-dimensional subspace obtained via Gaussian RBF kernel PCA looks much better in terms of linear class separation.\n\n
                                        \n
                                        \n\n## Swiss roll\n\n[[back to top](#Sections)]\n\nUnrolling the famous Swiss roll is a more challenging task than the examples we have seen above. We will use the [`make_swiss_roll`](http://scikit-learn.org/stable/modules/generated/sklearn.datasets.make_swiss_roll.html) to create 3-dimensional Swiss roll and start with the linear PCA to project the dataset onto a 2D and 1D feature subspace.\n\n\n```python\nfrom sklearn.datasets.samples_generator import make_swiss_roll\nfrom mpl_toolkits.mplot3d import Axes3D\n\nX, color = make_swiss_roll(n_samples=800, random_state=123)\n\nfig = plt.figure(figsize=(7,7))\nax = fig.add_subplot(111, projection='3d')\nax.scatter(X[:, 0], X[:, 1], X[:, 2], c=color, cmap=plt.cm.rainbow)\nplt.title('Swiss Roll in 3D')\nplt.show()\n```\n\n
                                        \n
                                        \n\n### Linear PCA\n\n[[back to top](#Sections)]\n\n\n```python\nfrom sklearn.decomposition import PCA\n\nscikit_pca = PCA(n_components=2)\nX_spca = scikit_pca.fit_transform(X)\n\nplt.figure(figsize=(8,6))\nplt.scatter(X_spca[:, 0], X_spca[:, 1], c=color, cmap=plt.cm.rainbow)\n\nplt.title('First 2 principal components after Linear PCA')\nplt.xlabel('PC1')\nplt.ylabel('PC2')\nplt.show()\n```\n\n\n```python\nscikit_pca = PCA(n_components=1)\nX_spca = scikit_pca.fit_transform(X)\n\nplt.figure(figsize=(8,6))\nplt.scatter(X_spca, np.zeros((800,1)), c=color, cmap=plt.cm.rainbow)\nplt.title('First principal component after Linear PCA')\nplt.xlabel('PC1')\nplt.show()\n```\n\n
                                        \n
                                        \n\n### Gaussian RBF kernel PCA\n\n[[back to top](#Sections)]\n\nI haven't found a good $\\gamma$ parameter for the Gaussian RBF kernel for good linear separation of this dataset. The best result I obtained is shown in the following figures.\n\n\n```python\nX_pc = stepwise_kpca(X, gamma=0.1, n_components=2)\n\nplt.figure(figsize=(8,6))\nplt.scatter(X_pc[:, 0], X_pc[:, 1], c=color, cmap=plt.cm.rainbow)\n\nplt.title('First 2 principal components after RBF Kernel PCA')\nplt.text(-0.14, 0.14, 'gamma = 0.1', fontsize=12)\nplt.xlabel('PC1')\nplt.ylabel('PC2')\nplt.show()\n```\n\n\n```python\nplt.figure(figsize=(8,6))\nplt.scatter(X_pc[:,0], np.zeros((800,1)), c=color, cmap=plt.cm.rainbow)\n\nplt.text(-0.125, 0.007, 'gamma = 0.1', fontsize=12)\nplt.title('First principal component after RBF Kernel PCA')\nplt.xlabel('PC1')\nplt.show()\n```\n\n
                                        \n
                                        \n\n### Locally-Linear Embedding (LLE)\n\n[[back to top](#Sections)]\n\nIn 2000, Sam T. Roweis and Lawrence K. Saul ([Nonlinear dimensionality reduction by locally linear embedding](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.111.3313) [[4](#References)]) introduced an unsupervised learning algorithm called locally linear embedding (LLE) that is better suited to identify patterns in the high-dimensional feature space and solves our problem of nonlinear dimensionality reduction for the Swiss roll. \nHere, we will use the [`locally_linear_embedding`](http://scikit-learn.org/stable/modules/generated/sklearn.manifold.locally_linear_embedding.html) class from scikit-learn to \"unroll\" the Swiss roll.\n\n\n```python\nfrom sklearn.manifold import locally_linear_embedding\n\nX_lle, err = locally_linear_embedding(X, n_neighbors=12, n_components=2)\n\nplt.figure(figsize=(8,6))\nplt.scatter(X_lle[:, 0], X_lle[:, 1], c=color, cmap=plt.cm.rainbow)\n\nplt.title('First 2 principal components after Locally Linear Embedding')\nplt.show()\n```\n\n\n```python\nfrom sklearn.manifold import locally_linear_embedding\n\nX_lle, err = locally_linear_embedding(X, n_neighbors=12, n_components=1)\n\nplt.figure(figsize=(8,6))\nplt.scatter(X_lle, np.zeros((800,1)), c=color, cmap=plt.cm.rainbow)\n\nplt.title('First principal component after Locally Linear Embedding')\nplt.show()\n```\n\n
                                        \n
                                        \n\n# Appendix A: Projecting new data\n\n[[back to top](#Sections)]\n\nSo far, so good, in the sections above, we have been projecting an dataset onto a new feature subspace. However, in a real application, we are usually interested to map new data points onto the same new feature subspace (e.g., if are working with a training and a test dataset in pattern classification tasks).\n\nRemember, when we computed the eigenvectors \\\\( \\mathbf{\\alpha} \\\\) of the centered kernel matrix, those values were actually already the projected datapoints onto the principal component axis \\\\( \\mathbf{g} \\\\). \n\nIf we want to project a new data point \\\\( \\mathbf{x} \\\\) onto this principal component axis, we'd need to compute \\\\(\\phi(\\mathbf{x})^T \\mathbf{g} \\\\).\n\nFortunately, also here, we don't have to compute \\\\(\\phi(\\mathbf{x})^T \\mathbf{g} \\\\) explicitely but use the kernel trick to calculate the RBF kernel between the new data point and every data point \\\\( j \\\\) in the training dataset:\n\n$$\\phi(\\mathbf{x})^T \\mathbf{g} = \\sum_j \\alpha_{i} \\; \\phi(\\mathbf{x}) \\; \\phi(\\mathbf{x_j})^T$$\n\n$$= \\sum_j \\alpha_{i} \\; \\kappa(\\mathbf{x}, \\mathbf{x_j})$$\n\n\nand the eigenvectors \\\\( \\alpha \\\\) and eigenvalues \\\\( \\lambda \\\\) of the Kernel matrix \\\\(\\mathbf{K}\\\\) satisfy the equation\n\\\\(\\mathbf{K} \\alpha = \\lambda \\alpha \\\\), we just need to normalize the eigenvector by the corresponding eigenvalue.\n\n\nFirst, let us modify our original implemenation by returning the corresponding to return also the eigenvalues of the kernel matrix.\n\n\n```python\nfrom scipy.spatial.distance import pdist, squareform\nfrom scipy import exp\nfrom scipy.linalg import eigh\nimport numpy as np\n\ndef stepwise_kpca(X, gamma, n_components):\n \"\"\"\n Implementation of a RBF kernel PCA.\n \n Arguments:\n X: A MxN dataset as NumPy array where the samples are stored as rows (M),\n and the attributes defined as columns (N).\n gamma: A free parameter (coefficient) for the RBF kernel.\n n_components: The number of components to be returned.\n \n Returns the k eigenvectors (alphas) that correspond to the k largest \n eigenvalues (lambdas).\n \n \"\"\"\n # Calculating the squared Euclidean distances for every pair of points\n # in the MxN dimensional dataset.\n sq_dists = pdist(X, 'sqeuclidean')\n\n # Converting the pairwise distances into a symmetric MxM matrix.\n mat_sq_dists = squareform(sq_dists)\n\n # Computing the MxM kernel matrix.\n K = exp(-gamma * mat_sq_dists)\n\n # Centering the symmetric NxN kernel matrix.\n N = K.shape[0]\n one_n = np.ones((N,N)) / N\n K_norm = K - one_n.dot(K) - K.dot(one_n) + one_n.dot(K).dot(one_n)\n \n # Obtaining eigenvalues in descending order with corresponding \n # eigenvectors from the symmetric matrix.\n eigvals, eigvecs = eigh(K_norm)\n\n # Obtaining the i eigenvectors (alphas) that corresponds to the i highest eigenvalues (lambdas).\n alphas = np.column_stack((eigvecs[:,-i] for i in range(1,n_components+1)))\n lambdas = [eigvals[-i] for i in range(1,n_components+1)]\n \n return alphas, lambdas\n```\n\nNow, let's make a new half-moon dataset and project it onto a 1-dimensonal subspace using the RBF kernel PCA:\n\n\n```python\nfrom sklearn.datasets import make_moons\nX, y = make_moons(n_samples=100, random_state=123)\nalphas, lambdas = stepwise_kpca(X, gamma=15, n_components=1)\n```\n\n /Users/sebastian/miniconda3/envs/py34/lib/python3.4/site-packages/sklearn/datasets/samples_generator.py:612: DeprecationWarning: using a non-integer number instead of an integer will result in an error in the future\n y = np.hstack([np.zeros(n_samples_in, dtype=np.intp),\n\n\n
                                        \n\nTo confirm that our approach produces the correct results, let's pretend that the 24th point from the half-moon dataset is a new data point \\\\( \\mathbf{x} \\\\), and we want to project it onto this new subspace.\n\n\n```python\nx_new = X[25]\nX_proj = alphas[25] # original projection\n```\n\n\n```python\nx_new\n```\n\n\n\n\n array([ 1.8713187 , 0.00928245])\n\n\n\n\n```python\nX_proj\n```\n\n\n\n\n array([ 0.07877284])\n\n\n\n\n```python\ndef project_x(x_new, X, gamma, alphas, lambdas):\n pair_dist = np.array([np.sum((x_new-row)**2) for row in X])\n k = np.exp(-gamma * pair_dist)\n return k.dot(alphas / lambdas)\n\n# projection of the \"new\" datapoint\nx_reproj = project_x(x_new, X, gamma=15, alphas=alphas, lambdas=lambdas) \n```\n\n\n```python\nx_reproj \n```\n\n\n\n\n array([ 0.07877284])\n\n\n\n\n```python\n%matplotlib inline\n```\n\n\n```python\nimport matplotlib.pyplot as plt\nplt.figure(figsize=(8,6))\nplt.scatter(alphas[y==0, 0], np.zeros((50)), color='red', alpha=0.5)\nplt.scatter(alphas[y==1, 0], np.zeros((50)), color='blue', alpha=0.5)\nplt.scatter(X_proj, 0, color='black', label='original projection of point X[24]', marker='^', s=100)\nplt.scatter(x_reproj, 0, color='green', label='remapped point X[24]', marker='x', s=500)\nplt.legend(scatterpoints=1)\nplt.show()\n```\n\n
                                        \n
                                        \n\n# References\n\n[[back to top](#Sections)]\n\n[1] Q. Wang. [Kernel principal component analysis and its applications in face recognition and active shape models](http://arxiv.org/abs/1207.3538). CoRR, abs/1207.3538, 2012.\n\n[2] B. Scholkopf, A. Smola, and K.-R. Muller. [Kernel principal component analysis](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.128.7613). pages 583\u2013588, 1997.\n\n[3] B. Scholkopf, A. Smola, and K.-R. Muller. [Nonlinear component analysis as a kernel eigenvalue problem](http://www.mitpressjournals.org/doi/abs/10.1162/089976698300017467#.VBh9QkuCFHg). Neural computation, 10(5):1299\u20131319, 1998.\n\n[4] S. T. Roweis and L. K. Saul. [Nonlinear dimensionality reduction by locally linear embedding](http://www.sciencemag.org/content/290/5500/2323.short). Science, 290(5500):2323\u20132326, 2000.\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "08eb676c3b841acaad54aafd3aa451c4b8db1366", "size": 741791, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "pattern-classification/dimensionality_reduction/projection/kernel_pca.ipynb", "max_stars_repo_name": "gopala-kr/ds-notebooks", "max_stars_repo_head_hexsha": "bc35430ecdd851f2ceab8f2437eec4d77cb59423", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-12-13T15:41:48.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-13T15:41:48.000Z", "max_issues_repo_path": "pattern-classification/dimensionality_reduction/projection/kernel_pca.ipynb", "max_issues_repo_name": "gopala-kr/ds-notebooks", "max_issues_repo_head_hexsha": "bc35430ecdd851f2ceab8f2437eec4d77cb59423", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 15, "max_issues_repo_issues_event_min_datetime": "2021-09-12T15:06:13.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-31T19:02:08.000Z", "max_forks_repo_path": "pattern-classification/dimensionality_reduction/projection/kernel_pca.ipynb", "max_forks_repo_name": "gopala-kr/ds-notebooks", "max_forks_repo_head_hexsha": "bc35430ecdd851f2ceab8f2437eec4d77cb59423", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-01-29T00:37:52.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-29T00:37:52.000Z", "avg_line_length": 442.3321407275, "max_line_length": 119962, "alphanum_fraction": 0.9286888086, "converted": true, "num_tokens": 7172, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632683808533, "lm_q2_score": 0.7772998611746912, "lm_q1q2_score": 0.42196754314927637}} {"text": "# Theory documentation for `pneumoinfer`\n\n\n```python\n# Add pneumoinfer to the system path \nimport sys\npath = '/Users/Rob/work/pneumoinfer'\nsys.path.append(path + '/source/') \nfrom pneumoinfer import pneumoinfer\n\nimport numpy as np\nimport pandas as pd\nimport scipy.special as spec\nimport matplotlib.pyplot as plt\nimport seaborn as sns; sns.set()\n```\n\n---\n\n## Motivating background\n\nMulti-state models - stochastic processes occupying one of a finite set of states at each moment in time - appear to describe many natural phenomena, but are probably most frequently used in the mathematical modelling of population health. The statistical inference (or selection) of these models for real-world applications frequently involves data in the form of a sequence of individual state observations, which are often coupled with some diagnostic uncertainty.\n\nThere are over 90 known capsular serotypes of _Streptococcus pneumoniae_, which persist despite their mutual competition for the same ecological niche (the nasopharynx) and a known fitness gradient. Motivated by the global pneumococcal disease burden, a specific class of multi-state models has been developed to describe the carriage dynamics which offers a neat explanation of this persistence through immunity-driven stabilisation effects (see [Cobey & Lipsitch (2012)](https://pubmed.ncbi.nlm.nih.gov/22383809/)). This class of model typically uses a counting memory of past state (or serotype) occupations (or colonisations) as a model for human immunity (see, e.g., [Flasche et al. (2013)](https://royalsocietypublishing.org/doi/10.1098/rspb.2013.1939) for an alternative formulation and [L\u00f8chen & Anderson (2020)](https://pubmed.ncbi.nlm.nih.gov/31055164/) for a general review of the carriage transmission models). Building from these mathematical models, a range of statistical approaches have also been used to infer the pneumococcal carriage through a given population from nasopharyngeal swab sample data (e.g., [Lipsitch et al. (2012)](https://pubmed.ncbi.nlm.nih.gov/22441543/) and [Numminen et al. (2013)](https://pubmed.ncbi.nlm.nih.gov/23822205/)). All of this is obviously really important, e.g., to understanding more precisely how a vaccine covering a restricted range of serotypes can impact colonisation in a given community or region. \n\nThe design of policies for gathering data will always have a direct impact on the quality and utility of information that can be learned about a model via statistical inference. Therefore, it's typically useful to know _a priori_ the fundamental constraints a given policy might impose on this procedure. The purpose of the `pneumoinfer` class is to provide researchers with a rigorous framework to investigate these limitations for the inference of multi-state models with a counting memory - which are structurally inspired by the pneumococcus carriage models of [Cobey & Lipsitch (2012)](https://pubmed.ncbi.nlm.nih.gov/22383809/) and [Lipsitch et al. (2012)](https://pubmed.ncbi.nlm.nih.gov/22441543/). The framework should also useful in model inference with real data.\n\nIn this documentation, we're going to analyse the master equation of a stochastic model which includes memory effects from individual immunity and investigate a novel (to our knowledge) approximate ODE description for the dynamics, while assessing its validity. By then exploiting the new efficient ODE description, we will be able to develop a new method of inference that is very rapid in comparison to simulated likelihoods (or even ABC/likelihood-free inference methods). This is the main inference method that is implemented in the `pneumoinfer` class.\n\n---\n\n## The fixed $\\Lambda_i$ model\n\nLet's now construct a multi-state model which incorporates a counting memory of past state occupations. This will include: an event rate of state clearance $\\tilde{\\mu}_i$ - the rate at which an individual occupying the $i$-th indexed state returns to the null state; an event rate of susceptibility $\\tilde{\\Lambda}_i$ for an individual moving to the $i$-th state from the null state; and a state-specific factor matrix $f_{ii'}$ which rescales $\\tilde{\\Lambda}_{i'}$ to create an event rate for an individual moving directly between the $i$-th and $i'$-th states. \n\nNow consider $\\tilde{\\mu}_i=\\tilde{\\mu}_i( \\dots, n_{i}, \\dots )$, i.e., a function of all previous state occupations by the individual, where $n_i$ are the state-specific counts of past occupations. The rate $\\tilde{\\mu}_i$ hence maintains a 'record' of past state occupations and updates accordingly through this memory. Additionally, we will make each rate $\\tilde{\\Lambda}_i=\\tilde{\\Lambda}_i(n_{i})$, i.e., a function _only_ of the state-specific count associated to each rate, respectively. The choice in the latter case comes from interpreting the counting memory as a model for capsular immunity - this will also turn out to be quite important for our approximation further on.\n\nNote that in [Cobey & Lipsitch (2012)](https://pubmed.ncbi.nlm.nih.gov/22383809/), the models of nonspecific and specific immunity suggest choosing the following functions\n\n$$\n\\begin{align}\n\\tilde{\\mu}_i( \\dots, n_{i}, \\dots ) &= \\mu_{\\rm max} + (\\mu_i - \\mu_{\\rm max})\\exp \\bigg( -\\epsilon \\sum_{\\forall i'}n_{i'} \\bigg) \\\\\n\\tilde{\\Lambda}_i(n_i) &= \\Lambda_{i}{\\bf 1}_{n_i=0} + \\sigma \\Lambda_{i}{\\bf 1}_{n_i>0} \\,.\n\\end{align}\n$$\n\nIn the expressions above: $\\epsilon$ governs the level of (immune system maturation) with respect to the number of past state occupations; ${\\bf 1}_A$ denotes an indicator function whose argument is unity when $A$ is satisfied, else $0$; and the susceptibility of an individual is assumed to be reduced by a constant factor of $\\sigma$ after having occupied that state once or more.\n\nThe multi-state process that we're going to consider would be normally be described as a non-Markovian phenomenon. However, the modelling approach we will take is instead a bit more similar to the formal concept of a Markov embedding (as studied, e.g., recently in [Kanazawa & Sornette (2021)](https://arxiv.org/abs/2102.00242)). By creating a binary state occupation variable $x_i$ for the $i$-th serotype, and the probability of occupying state $(\\dots , x_i , \\dots , n_i , \\dots )$ at time $t$ as $P(\\dots , x_i , \\dots , n_i , \\dots , t)$, we may write a Markovian master equation for the process. Let's now define\n\n$$\n\\begin{align}\np_i(\\dots ,n_i,\\dots ,t) &\\equiv P(\\dots, x_{i}=1, x_{i'}=0, \\dots ,n_{i},\\dots ,t)\\quad \\forall i'\\neq i \\\\ \nq(\\dots ,n_i,\\dots ,t) &\\equiv P(\\dots, x_{i}=0, \\dots ,n_{i},\\dots ,t) \\quad \\forall i\\,.\n\\end{align}\n$$\n\nUsing these definitions, it is straightforward to show that the master equation satisfies\n\n$$\n\\begin{align}\n\\frac{{\\rm d}}{{\\rm d}t}p_i(\\dots ,n_i,\\dots ,t) &= \\tilde{\\Lambda}_i(n_i-1)q(\\dots ,n_{i}-1,\\dots ,t) + \\sum_{\\forall i' \\neq i}f_{i'i}\\tilde{\\Lambda}_i (n_i-1)p_{i'}(\\dots ,n_{i'}, n_i-1,\\dots ,t) \\\\\n&\\quad - \\tilde{\\mu}_i( \\dots, n_{i}, \\dots ) p_i(\\dots ,n_i,\\dots ,t) - \\sum_{\\forall i'\\neq i}f_{ii'}\\tilde{\\Lambda}_{i'} (n_{i'}) p_i(\\dots ,n_i,\\dots ,t) \\\\\n\\frac{{\\rm d}}{{\\rm d}t}q(\\dots ,n_i,\\dots ,t) &= \\sum_{\\forall i}\\tilde{\\mu}_i( \\dots, n_{i}, \\dots )p_i(\\dots ,n_i,\\dots ,t) - \\sum_{\\forall i}\\tilde{\\Lambda}_i(n_i) q(\\dots ,n_i,\\dots ,t) \\,.\n\\end{align}\n$$\n\nBy defining the state space to encode the memory of past state occupations using the $n_i$ values themselves, the process is Markovian over the full $(\\dots , x_i,\\dots ,n_i,\\dots)$ space. Note also that one may obtain the time-dependent joint distribution over the $(\\dots ,n_i,\\dots)$ space, i.e., $P(\\dots, n_i, \\dots, t)$, through marginalisation at any time\n\n$$\n\\begin{equation}\nP(\\dots, n_i, \\dots, t) = q(\\dots, n_i, \\dots, t) + \\sum_{\\forall i} p_i(\\dots, n_i, \\dots, t) \\,. \n\\end{equation}\n$$\n\nThough we intend our analysis of this class of multi-state models to apply more generally beyond immediate applications to pneumococcus, it also is worth noting that restricting individuals to occupy a single state at a time only approximates the full pneumococcal carriage dynamics. The true process actually allows for some individuals to carry more than one serotype at at time. However, due to the relatively low and variable reported prevalence of simultaneous serotype carriers (or 'co-colonised' individuals) across different studies (see, e.g., [Gratten et al. (1989)](https://pubmed.ncbi.nlm.nih.gov/2639508/), [Huebner et al. (2000)](https://journals.lww.com/pidj/fulltext/2000/10000/lack_of_utility_of_serotyping_multiple_colonies.19.aspx) and many others...), the single-state occupation model should still a good tracer model of the underlying dynamical behaviour of the system. Note also that this additional complexity in the dynamics should be straightforward to incorporate into our framework for future analyses. \n\nLet's now try an approximation for the joint distributions of $p_i(\\dots, n_i, \\dots, t)$ and $q(\\dots, n_i, \\dots, t)$ which assumes separability, such that\n\n$$\n\\begin{align}\n\\ p_i(\\dots, n_i, \\dots, t) &\\simeq p_i(t)P(\\dots, n_i, \\dots, t) \\\\\n\\ q(\\dots, n_i, \\dots, t) &\\simeq q(t)P(\\dots, n_i, \\dots, t) \\,.\n\\end{align}\n$$\n\nWe shall evaluate the quality of this approximation later on (so don't worry) under different parametric conditions, but for now, let's just treat it as an ansatz.\n\nBy marginalising over states in the master equation, then substituting in the approximations above, and finally marginalising (each a summation from $n_{i'}=0$ to $\\infty$) over the resulting relation $\\forall n_{i'} \\,\\, i'\\neq i$, one finds that the following time evolution equation is separately satisfied by each marginal $P(n_i,t)$ distribution \n\n$$\n\\begin{align}\n\\frac{{\\rm d}}{{\\rm d}t}P(n_i,t) &= \\bigg[ \\tilde{\\Lambda}_i(n_i-1)q(t) + \\sum_{\\forall i'\\neq i} f_{i'i}\\tilde{\\Lambda}_{i} (n_{i}-1) p_{i'}(t) \\bigg] P(n_{i}-1,t) \\\\\n&\\quad - \\bigg[ \\tilde{\\Lambda}_i(n_i)q(t) + \\sum_{\\forall i'\\neq i} f_{ii'}\\tilde{\\Lambda}_{i'} (n_{i'}) p_i(t)\\bigg] P(n_i,t) \\,.\n\\end{align}\n$$\n\nIn addition to the separability assumption, the key point which allowed us to derive this one-step marginal master equation was the dependence of $\\tilde{\\Lambda}_i$ on _only_ $n_i$; in contrast to all of the past recorded states $(\\dots, n_i, \\dots)$ like $\\tilde{\\mu}_i$.\n\nFrom this point on we'll focus on the specific pneumococcus model by inserting the rate function definitions from [Cobey & Lipsitch (2012)](https://pubmed.ncbi.nlm.nih.gov/22383809/) that we introduced at the start into the marginal master equation for $P(n_i,t)$. The `pneumoinfer` class is currently written for only these models (i.e., with just these choices of function), but it's useful to see how the steps above could be performed for more general models too. The solution to the marginal master equation with these function substitutions is simply a Poisson distribution $P(n_i,t) = {\\rm Poisson}[n_i;{\\rm E}_t(n_i)]$, where\n\n$$\n\\begin{equation}\n{\\rm E}_t (n_i) = {\\rm E}_{t_{\\rm init}}(n_i) + \\int^t_{t_{\\rm init}}{\\rm d}t'\\bigg[ \\sigma \\Lambda_iq(t') +\\sum_{\\forall i'\\neq i} f_{i'i}\\sigma \\Lambda_{i} p_{i'}(t')\\bigg] \\,.\n\\end{equation}\n$$\n\nExploiting the properties of this Poisson distribution, if we now return to the full master equation and marginalise them over all $n_i$, while noting that\n\n$$\n\\begin{align}\n\\ p_i(t) &= \\sum_{\\forall n_i}\\sum_{n_{i}=0}^\\infty p_i(\\dots, n_i, \\dots, t) \\\\\n\\ q(t) &= \\sum_{\\forall n_i}\\sum_{n_{i}=0}^\\infty q(\\dots, n_i, \\dots, t) \\,,\n\\end{align}\n$$\n\none arrives at the following finite (implicitly integro-differential) system\n\n$$\n\\begin{align}\n\\frac{{\\rm d}}{{\\rm d}t}p_i(t) &= \\Lambda_iF_{it} q(t) + \\sum_{\\forall i'\\neq i} f_{i'i} \\Lambda_iF_{it} p_{i'}(t) - \\mu_iG_{it} p_i(t)-\\sum_{\\forall i'\\neq i}f_{ii'}\\Lambda_{i'}F_{i't} p_i(t) \\\\\n\\frac{{\\rm d}}{{\\rm d}t}q(t) &= \\sum_{\\forall i}\\mu_iG_{it}p_i(t) - \\sum_{\\forall i}\\Lambda_iF_{it}q(t)\\,,\n\\end{align}\n$$\n\nwhere, to avoid repetition, we have defined\n\n$$\n\\begin{align}\n\\ F_{it} &= P(n_i=0,t)+\\sigma P(n_i>0,t) = e^{-{\\rm E}_t(n_i)}+\\sigma \\big[ 1-e^{-{\\rm E}_t(n_i)}\\big] \\\\\n\\ G_{it} &= \\frac{\\mu_{\\rm max}}{\\mu_i} + \\bigg( 1-\\frac{\\mu_{\\rm max}}{\\mu_i}\\bigg) e^{\\sum_{\\forall i}{\\rm E}_t(n_i)(e^{-\\epsilon}-1)}\\,, \n\\end{align}\n$$\n\nwhere to derive $G_{it}$ we have had to assume conditional independence between $n_i$ and $n_{i'}\\,\\,\\forall i'\\neq i$. The equation for ${\\rm E}_t (n_i)$ can be differentiated to provide an equation for the time derivative of ${\\rm E}_t(n_i)$ - evolving this equation alongside the system defined above yields an explicit finite ODE system. Note also that this approximation technique should apply to other forms of memory functions used for $\\tilde{\\mu}_i(\\dots, n_i, \\dots)$ and $\\tilde{\\Lambda}_i(n_i)$ by simply marginalising over their $n_i$ values, and so this approximate approach appears to be quite generalisable to other simlar systems.\n\nIn order to analyse the system properties and check the validity of the approach above, we're now going to make some decisions about the parameter space to explore. Let's independently draw the $(\\mu_i,\\Lambda_i)$ values from Gamma distributions with shapes $(\\mu_\\alpha,\\Lambda_\\alpha)$ and rates $(\\mu_\\beta,\\Lambda_\\beta)$. Let's also constrain the matrix values $f_{ii'}=f_{i}{\\bf I}_{i'}$ (where ${\\bf I}_{i'}$ denotes the elements of a simple vector of ones) which also happens to be consistent with pneumococcus data anyway (see, e.g., [Lipsitch et al. (2012)](https://pubmed.ncbi.nlm.nih.gov/22441543/)). We'll also need a metric of comparison between the marginalised distribution outputs from the fully simulated master equation and our approximation. To this end, it probably makes sense to look at the [Kullback-Leibler divergence](https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence) between the marginal distributions for $x_i$ and $n_i$ in a full stochastic simulation and our approximation. In other words\n\n$$\n\\begin{align}\n\\ D^{(x)}_{{}_{\\rm KL}} &= \\sum_{\\forall i} p_{i, {\\rm sim}}(t) \\ln \\Bigg[ \\frac{p_{i, {\\rm sim}}(t)}{p_i(t)} \\Bigg] \\\\ \n&\\simeq - \\sum_{\\forall i} \\frac{\\ln Z_{\\rm sim}(x_i, t)}{Z_{\\rm sim}(x_i, t)} -\\sum_{\\forall i} \\frac{\\ln p_i(t)}{Z_{\\rm sim}(x_i, t)} \\\\\n\\ D^{(n_i)}_{{}_{\\rm KL}} &= \\sum_{n_i=0}^{\\infty} P_{\\rm sim}(n_i, t) \\ln \\Bigg[ \\frac{P_{\\rm sim}(n_i, t)}{P(n_i,t)} \\Bigg] \\\\ \n&\\simeq - \\sum_{n_i=0}^{\\infty}\\frac{\\ln Z_{\\rm sim}(n_i, t)}{Z_{\\rm sim}(n_i, t)} - \\sum_{n_i=0}^{\\infty} \\frac{\\ln {\\rm Poisson}[n_i;{\\rm E}_t(n_i)]}{Z_{\\rm sim}(n_i, t)} \\\\\n&\\simeq - \\sum_{n_i=0}^{\\infty}\\frac{\\ln Z_{\\rm sim}(n_i, t)}{Z_{\\rm sim}(n_i, t)} - \\sum_{n_i=0}^{\\infty} \\bigg[ \\frac{n_i\\ln {\\rm E}_t(n_i) - {\\rm E}_t(n_i) - \\ln \\Gamma (n_i+1)}{Z_{\\rm sim}(n_i, t)} \\bigg] \\,,\n\\end{align}\n$$\n\nwhere $Z_{\\rm sim}(x_i, t)$ and $Z_{\\rm sim}(n_i, t)$ denote the marginal frequency counts in each state space derived from a stochastic simulation of the master equation. Note that for the whole $(\\dots, n_i, \\dots)$ space, a better comparison would involve Monte Carlo integration of the joint counts $Z_{\\rm sim}(\\dots, n_i, \\dots, t)$. However, this is quite a lot more challenging with many dimensions (usually necessitating nested sampling) and so we'll consider it to be beyond the present scope. \n\nUsing the `run_sim` method of the `pneumoinfer` class, and the equations above, we can generate numerically-approximate plots of the Kullback-Leibler divergence on the marginal distributions over a range of numbers of states, parameters and points in time.\n\n\n```python\n# Choose the number of states\nnstat = 10\n\n# Initialise pneumoinfer\npn = pneumoinfer(nstat)\n\n# Create population members with gamma-distributed\n# rate parameters\nnumpeople = 1\nkmu = 1.0\nparam_dic = {\n 'Curr' : 0,\n 'npast' : np.zeros(nstat),\n 'Lam' : np.random.gamma(kmu,1.0/kmu,size=nstat),\n 'mu' : np.random.gamma(kmu,1.0/kmu,size=nstat),\n 'f' : np.random.gamma(kmu,1.0/kmu,size=nstat),\n 'eps' : 1.0/5.0,\n 'sig' : 1.0,\n 'mumax' : 1.0\n}\npn.create_members(numpeople,param_dic)\n\n# Run the ODE and full simulation (the latter for a\n# given number of realisations\nnreals = 10000\node_tstepsize = 0.001\nsim_trejectionstepsize = 0.01\ntotal_time = 10.0\ntimes = [0.2*float(i) + 0.1 for i in range(0,40)]\npn.run_ode(total_time,ode_tstepsize)\npn.run_sim(nreals,total_time,sim_trejectionstepsize,time_snaps=times)\n\n# Setup plots for the DKL values using the outputs\nDKLx, DKLn = [], []\nfor t in times:\n counts = pd.Series(pn.sim_output['Curr'][t].flatten()).value_counts()\n co = np.zeros(nstat + 1)\n co[counts.index.values.astype(int)] = counts.values\n co = co[1:]\n it = np.argmin((pn.ode_output['time']-t)**2)\n DKLx.append(\n - np.sum(np.log(co[co>0])/co[co>0])\n - np.sum(np.log(pn.ode_output['probCurr'][it][co>0])/co[co>0])\n )\n DKLnsum = 0.0\n for i in range(0,nstat):\n ncounts = pd.Series(pn.sim_output['npastsum'][t][i].flatten()).value_counts()\n nco = np.zeros(1000)\n ns = np.arange(0, 1000, 1)\n nco[ncounts.index.values.astype(int)] = ncounts.values\n DKLnsum += (\n - np.sum(np.log(nco[nco>0])/nco[nco>0]) \n - np.sum(ns[nco>0] * np.log(pn.ode_output['Expnpast'][it][i])/nco[nco>0])\n + np.sum(pn.ode_output['Expnpast'][it][i]/nco[nco>0])\n + np.sum(spec.loggamma(ns[nco>0]+1)/nco[nco>0])\n )\n DKLn.append(DKLnsum)\n```\n\n\n```python\nfig, ax = plt.subplots(1, 2, figsize=(15,5))\nax[0].plot(times,DKLx)\nax[1].plot(times,np.asarray(DKLn)/np.asarray(times))\nax[0].set_xlabel('Time')\nax[0].set_ylabel(r'$D^{(x)}_{\\rm KL}$')\nax[1].set_xlabel('Time')\nax[1].set_ylabel(r'$\\sum_{\\forall i}D^{(n_i)}_{\\rm KL}\\,/\\,$Time')\nplt.show()\n```\n\nThe value of $D_{{}_{\\rm KL}}^{(x)}$ generally stays small (and stable) throughout for most parameter choices. Interestingly, the same cannot be said for the $D_{{}_{\\rm KL}}^{(n_i)}$ values, which appear to tend towards a deviation which is linearly proportional in time. If we now plot the time evolution of each set of quantities explicitly in time, we can see this is consistent with the observed deviations between the simulation and the ODE approximation. \n\n\n```python\nfig, ax = plt.subplots(1,2,figsize=(15,5))\ncolours = sns.color_palette()\nprobs, ncounts = [], []\nfor t in times:\n counts = pd.Series(pn.sim_output['Curr'][t].flatten()).value_counts()\n nco = np.sum(pn.sim_output['npastsum'][t],axis=1)\n ncounts.append(nco/nreals)\n pr = np.zeros(nstat + 1)\n pr[counts.index.values.astype(int)] = counts.values/float(nreals)\n probs.append(list(pr))\nfor i in range(0,nstat):\n ax[0].plot(times,np.asarray(probs)[:,i+1],label='State '+str(i+1),color=colours[i])\n ax[0].plot(pn.ode_output['time'],pn.ode_output['probCurr'][:,i],color=colours[i])\n ax[1].plot(times,np.asarray(ncounts)[:,i],label='State ' + str(i+1),color=colours[i])\n ax[1].plot(pn.ode_output['time'],pn.ode_output['Expnpast'][:,i],'--',color=colours[i])\nax[0].set_xlabel('Time')\nax[0].set_ylabel(r'$p_i$')\nax[1].set_xlabel('Time')\nax[1].set_ylabel(r'$n_i$')\nax[1].legend(bbox_to_anchor=(1.3,1.0))\nplt.show()\n```\n\n---\n\n## A varying $\\Lambda_{iu}$ model\n\nWe're now ready to introduce an alternative model which accounts for a stochastically-varying susceptibility $\\Lambda_{iu}$ (a possible model for community exposure to infectious individuals), which is now additionally indexed by '$u$' which corresponds to each individual. In this model, we have \n\n$$\n\\begin{equation}\n\\Lambda_{iu} = \\Lambda_{\\rm min} + \\lambda\\sum_{\\forall u'\\neq u}\\beta_{uu'} \\frac{x_{iu'}}{N_{\\rm p}}\\,,\n\\end{equation}\n$$\n\nwhere: the total population number is $N_{\\rm p}$; $\\beta_{uu'}$ are elements of a 'contact matrix' which rescales the event rate according to the spreading behaviour between the $u$-th and $u'$-th individuals; $\\lambda$ is a constant normalisation for $\\beta_{uu'}$; and $x_{iu'}$ records the state of the $u'$-th individual.\n\nExtending the master equation we introduced in the previous section to include the susceptibility above and the states of $N_{\\rm p}$ individuals, one can easily adapt the argument of the previous section to arrive at the following generalisation of the ODE system we found earlier\n\n$$\n\\begin{align}\n\\frac{{\\rm d}}{{\\rm d}t}p_{iu}(t) &= {\\rm E}_t(\\Lambda_{iu})F_{it} q_u(t) + \\sum_{\\forall i'\\neq i} f_{i'i} {\\rm E}_t(\\Lambda_{iu})F_{it} p_{i'u}(t) - \\mu_iG_{it} p_{iu}(t)-\\sum_{\\forall i'\\neq i}f_{ii'}{\\rm E}_t(\\Lambda_{i'u})F_{i't} p_{iu}(t) \\\\\n\\frac{{\\rm d}}{{\\rm d}t}q_u(t) &= \\sum_{\\forall i}\\mu_iG_{it}p_{iu}(t) - \\sum_{\\forall i}{\\rm E}_t(\\Lambda_{iu})F_{it}q_u(t)\\,.\n\\end{align}\n$$\n\nIn the equations above, the state occupation probabilities of separate $u$-indexed individuals (or $u$-indexed categories of individual) are $p_{iu}(t)$ and $q_u(t)$, and we've also computed the expectation\n\n$$\n\\begin{equation}\n{\\rm E}_t(\\Lambda_{iu}) = \\Lambda_{\\rm min} + \\lambda\\sum_{\\forall u'\\neq u}\\beta_{uu'} \\frac{p_{iu'}(t)}{N_{\\rm p}}\\,.\n\\end{equation}\n$$\n\nThe `pneumoinfer` class also implements ODE and full simulations for the varying $\\Lambda_{iu}$ model, and we plot an example run of this method below. \n\n\n```python\n# Once again, setup things as before\nnstat = 10\npn = pneumoinfer(nstat)\n\n# Now we add a contact matrix as well as its referenced\n# index to each population member\npn._cont_mat = np.ones((3,3))\nkmu = 1.0\nparam_dic = {\n 'npast' : np.zeros(nstat),\n 'Lam' : np.zeros(nstat),\n 'mu' : np.random.gamma(kmu,1.0/kmu,size=nstat),\n 'f' : np.random.gamma(kmu,1.0/kmu,size=nstat),\n 'eps' : 1.0/5.0,\n 'sig' : 1.0,\n 'mumax' : 1.0,\n}\nnind = 1000\nfor i in range(0,nind):\n group_param_dic = param_dic.copy()\n group_param_dic['Curr'] = np.random.randint(0,nstat+1)\n group_param_dic['cind'] = np.random.randint(0,3)\n pn.create_members(1,group_param_dic)\n \n# Running the ODE and full sim in much the same way\n# but only 1 realisation will be used for speed\nnreals = 1\node_tstepsize = 0.001\nsim_trejectionstepsize = 0.01\ntotal_time = 20.0\ntimes = [0.2*float(i) + 0.1 for i in range(0,80)]\npn.run_ode(total_time,ode_tstepsize)\npn.run_sim(nreals,total_time,sim_trejectionstepsize,time_snaps=times)\n```\n\n\n```python\ncolours = sns.color_palette()\nprobs = []\nfor t in times:\n counts = pd.Series(pn.sim_output['Curr'][t].flatten()).value_counts()\n pr = np.zeros(nstat + 1)\n pr[counts.index.values.astype(int)] = counts.values/float(nreals)\n probs.append(list(pr))\nfor i in range(0,nstat):\n plt.plot(times,np.asarray(probs)[:,i+1]/float(nind),label='State '+str(i+1),color=colours[i])\n plt.plot(pn.ode_output['time'],pn.ode_output['probCurr'][:,i],color=colours[i])\nplt.legend(bbox_to_anchor=(1.3,1.0))\nax = plt.gca()\nax.set_xlabel('Time')\nax.set_ylabel(r'$p_i$')\nplt.show()\n```\n\n---\n\n## Computing the log-likelihood for inference\n\nWe're now ready to apply our ODE approximation to the statistical inference of the full simulation. We're going to assume that all data sets considered come in the form of a sequence of state observations (or longitudinally-monitored swab samples if you're talking about pneumococcus) for each sampled individual from the population which takes the form of counts, times and associated sample sizes ${\\cal D}\\equiv \\{d_c,d_t,d_s\\}$ (where $d_c=\\{c_{ij}\\}$, $d_t=\\{t_j \\, : \\, t_{\\rm init}\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        CurrCountTime
                                        072406.3
                                        121416.3
                                        201356.3
                                        351336.3
                                        481286.3
                                        ............
                                        1054800.9
                                        1061290.9
                                        1079100.9
                                        1081060.9
                                        109650.9
                                        \n

                                        110 rows \u00d7 3 columns

                                        \n
                                        \n\n\n\nNote the form of the `pandas` dataframe given above: the `'Curr'` column denotes the state label (or serotype index) and the `'Count'` and `'Time'` columns describe the population number and time of the observation, respectively. Using this mock data, we can now compute the log-likelihood using the method provided in `pneumoinfer` to show that the sampler correctly identifies the maximum likelihood (up to some observation noise which explains the infrequent samples with a sightly higher log-likelihood).\n\n\n```python\n# Usual setup again, but this time we\n# add in the data via the keyword\nnstat = 10\npn = pneumoinfer(nstat)\npn.create_members(1,param_dic,data=df)\n\n# Generate some random samples away\n# from the known log-likelihood maximum\nmaxLL = pn.lnlike(ode_tstepsize)\nLLs = []\nmaxLams = pn.params['ode']['Lams']\nmaxmus = pn.params['ode']['mus']\nmaxfs = pn.params['ode']['fs']\nfor i in range(0,100):\n tempLams = maxLams + np.random.normal(0.0,0.01,size=nstat).reshape(nstat,1)\n tempmus = maxmus + np.random.normal(0.0,0.01,size=nstat).reshape(nstat,1)\n tempfs = maxfs + np.random.normal(0.0,0.01,size=nstat).reshape(nstat,1)\n pn.params['ode']['Lams'] = tempLams*(tempLams>0.0)\n pn.params['ode']['mus'] = tempmus*(tempmus>0.0)\n pn.params['ode']['fs'] = tempfs*(tempfs>0.0)\n LLs.append(pn.lnlike(ode_tstepsize))\n```\n\n\n```python\nplt.plot(np.asarray(LLs)-maxLL)\nax = plt.gca()\nax.set_ylabel('LL-max(LL)')\nplt.show()\n```\n\n---\n\n## Additional notes: a method to compute the gradient of the log-likelihood\n\nThe current version of `pneumoinfer` does not support a gradient calculation for the log-likelihood (mainly because I was eager to move onto some other stuff!). However, to assist anyone wanting to implement this themselves, I thought it would be helpful to go through the calculation which computes the gradient (in principle) without resorting to numerical derivatives. This makes use of the 'multiple-adjoint' method as implemented in [Zhuang et al. (2021)](https://arxiv.org/abs/2006.02493). Consider the following 'data Lagrangian'\n\n$$\n\\begin{align}\nL &= \\sum_{\\forall j \\, : \\, t_j\\,\\in \\,d_t} L_j\\\\\nL_j &= \\ln{\\cal L}( t_j \\vert \\Theta ) + \\int^{t_{j}}_{t_{j-1}}{\\rm d}t \\,{\\sf h}(t)^{\\rm T}\\bigg[ \\frac{{\\rm d}}{{\\rm d}t}{\\sf V}(t) - {\\sf M}_\\Theta (t)\\bigg] \\\\\n&= \\ln{\\cal L}( t_j \\vert \\Theta) + {\\sf h}(t_j)^{\\rm T}{\\sf V}(t_j)-{\\sf h}(t_{j-1})^{\\rm T}{\\sf V}(t_{j-1}) - \\int^{t_{j}}_{t_{j-1}}{\\rm d}t \\bigg[\\frac{{\\rm d}}{{\\rm d}t}{\\sf h}(t)^{\\rm T} {\\sf V}(t) + {\\sf h}(t)^{\\rm T}{\\sf M}_\\Theta (t)\\bigg] \\,,\n\\end{align}\n$$\n\nwhere ${\\sf V}(t)=[\\dots, {\\rm E}_t(n_i),\\dots, p_i(t), \\dots , q(t)]^{\\rm T}$, ${\\sf h}(t)$ is a dynamical vector of Lagrange multipliers and ${\\sf M}_\\Theta (t)$ is just compact notation for the vector of ODE terms on the RHS. Varying $L_j$ with respect to the boundary condition ${\\sf V}(t_j)$ and ${\\sf V}(t)$, we obtain the constraints\n\n$$\n\\begin{align}\n\\frac{\\partial L_j}{\\partial {\\sf V}(t_j)} &= 0 \\quad \\Longleftrightarrow \\quad \\frac{\\partial}{\\partial {\\sf V}(t_j)}\\ln{\\cal L}( t_j \\vert \\Theta ) + {\\sf h}(t_j ) = 0 \\\\\n\\frac{\\delta L_j}{\\delta {\\sf V}(t)} &= 0 \\quad \\Longleftrightarrow \\quad \\frac{{\\rm d}}{{\\rm d}t}{\\sf h}(t) + \\bigg[\\frac{\\partial}{\\partial {\\sf V}(t)}{\\sf M}_\\Theta (t)\\bigg]^{\\rm T}{\\sf h}(t) = 0\\,,\n\\end{align}\n$$\n\nLet us also note that if we vary $L_j$ with respect to $\\Theta$ and optimise the likelihood, one obtains\n\n$$\n\\begin{align}\n\\frac{\\partial L_j}{\\partial \\Theta} &= \\frac{\\partial}{\\partial \\Theta}\\ln{\\cal L}( t_j \\vert \\Theta ) - \\int^{t_{j}}_{t_{j-1}}{\\rm d}t \\,{\\sf h}(t)^{\\rm T}\\frac{\\partial}{\\partial \\Theta}{\\sf M}_\\Theta (t) \\\\\n&\\underset{{\\rm opt}}{\\longrightarrow} \\int_{t_{j}}^{t_{j-1}}{\\rm d}t \\,{\\sf h}(t)^{\\rm T}\\frac{\\partial}{\\partial \\Theta}{\\sf M}_\\Theta (t)\\,.\n\\end{align}\n$$\n\nThe method proposed from here would be something like:\n- Treat initial values of ${\\rm E}_{t_0}(n_i)$ as a prior that should be varied to test the robustness of the inference.\n- From the initialised states of the set of individuals run the forward ODE in time to obtain the value of $\\frac{\\partial}{\\partial \\Theta}{\\sf M}_\\Theta (t)$ at every observed timestep.\n- For each interval edge determine ${\\sf h}(t_j)$ using the first constraint equation and the ${\\sf V}(t_j)$-gradient of the forward-time likelihood.\n- For each interval solve the second equation to get its backward-time trajectory ${\\sf h}(t)$. \n- Integrate over ${\\sf h}(t)$ and $\\frac{\\partial}{\\partial \\Theta}{\\sf M}_\\Theta (t)$ to determine the gradient in the last expression.\n\nSeems like overkill, but could be interesting to implement in future if a large number of states/parameters are varied, e.g., for HMC sampling of the model posterior from a decent data set.\n\n\n```python\n\n```\n", "meta": {"hexsha": "c55d6c1991dd673a59da3d6d5e524c1cff68e3b2", "size": 291781, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/theory-docs.ipynb", "max_stars_repo_name": "umbralcalc/statemem", "max_stars_repo_head_hexsha": "94fbb4fae8cccdb53e054e4314f93dcbb059c478", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/theory-docs.ipynb", "max_issues_repo_name": "umbralcalc/statemem", "max_issues_repo_head_hexsha": "94fbb4fae8cccdb53e054e4314f93dcbb059c478", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/theory-docs.ipynb", "max_forks_repo_name": "umbralcalc/statemem", "max_forks_repo_head_hexsha": "94fbb4fae8cccdb53e054e4314f93dcbb059c478", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 382.9146981627, "max_line_length": 124040, "alphanum_fraction": 0.9105562048, "converted": true, "num_tokens": 10097, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660543, "lm_q2_score": 0.647798211152541, "lm_q1q2_score": 0.4219465161561754}} {"text": "# Notes on current version:\nFor TOC if missing from command line try\njupyter nbextensions_configurator enable\nthen toggle nbextensions, restart.\n\n1. 1.9.2020 Managed to convert ODE models for economic extension to transition model ready for stochastic simulation, using separate birth death list\n See section on SC2UIR model. Not done for other two economic extensions yet\n2. 1.9.2020 Implemented stochastic simulation (Tau-leap method) using PyGom inbuilt capabilities: for SCIR simulation only so far\n Neeed to use integer N>>1, not 1.0, for stochastic simulation. Calculates in a few minutes for N=10000, rescaled ICUfrac to 0.02 (x10). N=100000 didn't finish in 10m.\n\n# Model Definitions\n\n## Utilities for custom extension of PyGom\n\n\n```python\n# import required packages\nimport os \nimport csv\nfrom sympy import symbols, init_printing\nimport numpy as np\nimport matplotlib\n%matplotlib inline\nimport seaborn as sb\nfrom matplotlib import pyplot as plt\nimport sympy\nimport itertools\nimport scipy\nimport datetime\nimport matplotlib.dates as mdates\nfrom pygom import DeterministicOde, Transition, SimulateOde, TransitionType, SquareLoss\nfrom scipy.optimize import minimize\n\nimport pickle as pk\nimport jsonpickle as jpk\n\nfrom cycler import cycler\nfrom pandas.plotting import register_matplotlib_converters\nregister_matplotlib_converters()\nimport pwlf\n```\n\n\n```python\nfrom models import *\nfrom model_fits_age import *\n```\n\n\n\n\n\n loading data.py...\n Getting data:\n getting JHU data...\n jhu data selected from 1/22/20 to 10/8/20\n expanding JHU data : to new (daily), 7-day rolling (smoothed), reporting glitch (corrected) and combined\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='report correction deaths', max=190.0, style=ProgressStyle\u2026\n\n\n \n\n\n\n HBox(children=(FloatProgress(value=0.0, description='report correction confirmed', max=190.0, style=ProgressSt\u2026\n\n\n \n number of countries listed in JHU database 189\n done with JHU data (covid_ts dictionary keys: confirmed, deaths, recovered).\n getting owid data...\n owid data selected from 1/23/20 to 10/9/20\n expanding OWID data : to new (daily), 7-day rolling (smoothed), reporting glitch (corrected) and combined\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='report correction deaths', max=213.0, style=ProgressStyle\u2026\n\n\n \n\n\n\n HBox(children=(FloatProgress(value=0.0, description='report correction confirmed', max=213.0, style=ProgressSt\u2026\n\n\n \n number of countries listed in OWID database 212\n done with OWID data (covid_owid_ts dictionary see .keys()) .\n getting ICU and acute care data icus_2012 and WHO ...\n WHO acute file found dictionary acute_who\n ICU file found dictionary icus_2012\n mapping country names between JHU and OWID and extracting common countries...\n getting 2017 contact matrix data from 152 countries ...\n 152 country contact files found 1 A-M and 2 M-Z\n Of 186 in countries_common 145 have contact matrices\n 4 country contact matrices set equal to that of neighbour to complete cluster country set\n Kosovo:Serbia Norway:Sweden Afghanistan:Pakistan Moldova:Romania\n getting UN all sex age group data for 2020 ...\n UN contact files found 1 and 2\n Of 186 in countries_common 180 have age structure\n Kosovo age structure digitized from CIA World Fact Book Image 2018 to complete cluster country set\n extracting data sets for common countries both databases...\n extracting testing data from OWID database\n doing piecewise linear fits to testing data ... reg_testing\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='piecewise linear fit', max=186.0, style=ProgressStyle(des\u2026\n\n\n \n completed regularization of testing by pwlf and linear adjustment to confirmed cases (linr).\n constructing nonlinear adjustment to confirmed cases based on pwlf testing (nonlin and nonlinr ...\n completed nonlinear adjustment to confirmed cases.\n Done with data.\n ---------------------------------\n done with data.py.\n making the models...\n SEI3R\n SC3EI3R\n SC3UEI3R\n SIR_A4\n SC2IR_A4\n SEIR_A4\n SC3EI3R_A4\n done with the models.\n\n\n\n```python\nsavefigs = False # whether to save specific figures for paper to .../figures directory\n```\n\n\n```python\n# Jupyter Specifics\nfrom IPython.display import display, HTML\ndisplay(HTML(\"\"))\nstyle = {'description_width': '100px'}\n```\n\n\n\n\n\n## Age Extensions to SIR Model\n\n\n```python\nfit1 = ModelFit('SIR_fit1','SIR')\n```\n\n\n```python\n I_0 = 0.00003 \n age_structure = 4\n first_infected_agegroup = int(age_structure//4) # rough first approximation : would give 20-25 year olds for 16 group 0-80 in 5 year intervals\n state0 = ['S', 'I', 'R', 'D']\n param_list0 = ['beta', 'gamma','mu','N']\n state = []\n sa = {} # state age dictionary\n for s in state0:\n state_tmp = []\n for i in range(age_structure):\n state_tmp.append(s+'_'+str(i))\n state.extend(state_tmp)\n sa.update({s:state_tmp.copy()})\n param_list = param_list0\n pa = {}\n N_list = []\n for i in range(age_structure):\n N_list.append('N'+'_'+str(i))\n param_list.extend(N_list)\n pa.update({'N':N_list})\n contact = [[None]*age_structure]*age_structure\n for i in range(age_structure):\n contact[i] = ['con'+'_'+str(i)+'_'+str(j) for j in range(age_structure)]\n param_list.extend(contact[i])\n phi = [None]*age_structure\n for i in range(age_structure):\n tmp = 'beta*(0'\n for j in range(age_structure):\n tmp= tmp+'+'+contact[i][j]+'*'+sa['I'][j]+'/'+pa['N'][j]\n tmp = tmp+')'\n phi[i]=tmp[:] # Note that strings are treated like arrays and need to be copied elementwise\n transition = []\n for i in range(age_structure):\n transition.append(Transition(origin=sa['S'][i],destination=sa['I'][i], equation = phi[i]+'*'+sa['S'][i],\n transition_type=TransitionType.T))\n for i in range(age_structure):\n transition.append(Transition(origin=sa['I'][i],destination=sa['R'][i], equation = 'gamma'+'*'+sa['I'][i],\n transition_type=TransitionType.T)) \n for i in range(age_structure):\n transition.append(Transition(origin=sa['I'][i],destination=sa['D'][i], equation = 'mu'+'*'+sa['I'][i],\n transition_type=TransitionType.T)) \n\n model = DeterministicOde(state, param_list, transition=transition)\n model.modelname='SIR'+'_'+str(age_structure)\n model.ei=slice(1*age_structure,2*age_structure)\n model.confirmed=slice(1*age_structure,4*age_structure) # cases 1-3 i.e. I, R and D\n model.recovered=slice(2*age_structure,3*age_structure)\n model.deaths=slice(3*age_structure,4*age_structure)\n model.I_1 = 1*age_structure + first_infected_agegroup\n\n #x0 = [1.0-I_0, I_0, 0.0, 0.0]\n x0 = []\n for s in state0:\n state_tmp = []\n for i in range(age_structure):\n if i == first_infected_agegroup and s == 'S':\n x0.append(1.0-I_0)\n elif i == first_infected_agegroup and s == 'I':\n x0.append(I_0)\n else:\n x0.append(0.)\n model.initial_values = (x0, 0) # 0 for t[0]\n```\n\n\n```python\nphi\n```\n\n\n```python\n#model.print_ode()\nmodel.print_ode2()\n```\n\n\n```python\nmodel.get_transition_graph()\n```\n\n## Caution Extensions to SIR Model\n\n\n```python\n\n```\n\n\n```python\nfit1 = ModelFit('SEIR_fit1','SEIR')\n```\n\n\n```python\nfit1.modelname\n```\n\n\n```python\nfit1.cbparams\n```\n\n### SIR model\n\n#### Equations\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -\\beta I S\\\\\n\\dot{I} &= \\beta I S - \\gamma I - \\mu I\\\\\n\\dot{R} & = \\gamma I \\\\\n\\dot{D} & = \\mu I\n\\end{split}\n\\end{equation}\n\n\n#### Variables\n* $S$: Susceptible individuals\n* $I$: Infected individuals \n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+I+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta$ rate at which infected individuals contact susceptibles and infect them\n* $\\gamma$ rate at which infected individuals recover from disease and become immune\n* $\\mu$ death rate for infected individuals\n\n#### Implementation\nUsing PyGOM, we will set up my simple SCIR model ODE system\nPyGOM \u2013 A Python Package for Simplifying Modelling with Systems of Ordinary Differential Equations https://arxiv.org/pdf/1803.06934.pdf\n\n\n```python\nSIR_model.print_ode()\nSIR_model.print_ode2()\n```\n\n\n```python\n# display equations\nprint_ode2(SIR_model)\n```\n\n\n```python\n# display graphical representation of the model\nSIR_model.get_transition_graph()\n```\n\n##### Derived equations, Jacobian and gradient\n\n\n```python\nSIR_model.get_ode_eqn()\n```\n\n\n```python\nSIR_model.get_jacobian_eqn()\n```\n\n\n```python\nSIR_model.get_grad_eqn()\n```\n\n#### R0\n\n\n```python\nfrom pygom.model.epi_analysis import R0\n```\n\n\n```python\nstate = ['S', 'I', 'R', 'D']\nparam_list = ['beta', 'gamma','mu','N']\ntransition_ode = [\n Transition(origin='S', equation='-beta*I*S'),\n Transition(origin='I', equation='beta*I*S-gamma*I-mu*I'),\n Transition(origin='R', equation='gamma*I'),\n Transition(origin='D', equation='mu*I') \n ]\node = SimulateOde(state, param_list, ode=transition_ode)\node = ode.get_unrolled_obj()\nR0(ode,['I'])\n```\n\n### SCIR model\n\n#### Equations\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -\\beta I S + c_1 S_c - c_2*S*I\\\\\n\\dot{S_c} &= - c_0 \\beta I S_c - c_1 S_c + c_2*S*I\\\\\n\\dot{I} &= \\beta I S - \\gamma I - \\mu I\\\\\n\\dot{R} & = \\gamma I \\\\\n\\dot{D} & = \\mu I\n\\end{split}\n\\end{equation}\n\nThe use of I as a state variable triggering susceptibles to execute caution is just one choice. In contrast with deaths, it does not accumulate over time and so retains the property of an active threat to society, rather than an historical one. Alternatively, one could use the daily death rate which is proportional to it.\n\nActually, the effect of caution may be quadratic, since both the individual doing the infection and individual potentially being infected may be executing caution. The current version assumes that infectives do not change their precautionary measures in response to I. To implement this we would need to further distinguish careful and non careful infectives. This is done in the SC2IR model.\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $S_c$: Susceptible individuals exercising pandemy precautions\n* $I$: Infected individuals \n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+S_c+I+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta$ rate at which infected individuals contact susceptibles and infect them\n* $\\gamma$ rate at which infected individuals recover from disease and become immune\n* $\\mu$ death rate for infected individuals\n\n* $c_i$ three parameters characterizing cautionary response of population via class $S_c$\n\n - c_0 : reduction factor for exposure for cautioned susceptibles\n\n - c_1 : duration of caution (exponential decay time constant in days)\n\n - c_2 : rate constant for transition from uncautioned to cautioned susceptible\n\n#### Implementation\nUsing PyGOM, we will set up my simple SCIR model ODE system\nPyGOM \u2013 A Python Package for Simplifying Modelling with Systems of Ordinary Differential Equations https://arxiv.org/pdf/1803.06934.pdf\n\n\n```python\n# display equations\nprint_ode2(SCIR_model)\n```\n\n\n```python\n# display graphical representation of the model\nSCIR_model.get_transition_graph()\n```\n\n### SC2IR model\n\n#### Equations\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -\\beta (I + c_0 I_c) S + c_1 S_c - c_2 S (I + I_c)\\\\\n\\dot{S_c} &= - c_0 \\beta (I + c_0 I_c) S_c - c_1 S_c + c_2 S (I + I_c)\\\\\n\\dot{I} &= \\beta (I + c_0 I_c) S - \\gamma I - \\mu I + c_1 I_c - c_2 I (I + I_c)\\\\\n\\dot{I_c} &= \\beta (I + c_0 I_c) S_c - \\gamma I_c - \\mu I_c - c_1 I_c + c_2 I (I + I_c)\\\\\n\\dot{R} & = \\gamma (I + I_c)\\\\\n\\dot{D} & = \\mu (I + I_c)\n\\end{split}\n\\end{equation}\n\nThe use of I as a state variable triggering susceptibles to execute caution is just one choice. In contrast with deaths, it does not accumulate over time and so retains the property of an active threat to society, rather than an historical one. Alternatively, one could use the daily death rate which is proportional to it.\n\nThe effect of caution may be quadratic, since both the individual doing the infection and individual potentially being infected may be executing caution. To implement this we distinguish careful and non careful infectives. We ignore infectives making the transition to caution or relaxing it.\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $S_c$: Susceptible individuals exercising pandemy precautions\n* $I$: Infected individuals non exercising pandemy precautions\n* $I_c$: Infected individuals exercising pandemy precautions \n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+S_c+I+I_c+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta$ rate at which infected individuals contact susceptibles and infect them\n* $\\gamma$ rate at which infected individuals recover from disease and become immune\n* $\\mu$ death rate for infected individuals\n\n* $c_i$ three parameters characterizing cautionary response of population via class $S_c$\n\n - c_0 : reduction factor for exposure for cautioned susceptibles\n\n - c_1 : duration of caution (exponential decay time constant in days)\n\n - c_2 : rate constant for transition from uncautioned to cautioned susceptible\n\n#### Implementation\nUsing PyGOM, we will set up my simple SCIR model ODE system\nPyGOM \u2013 A Python Package for Simplifying Modelling with Systems of Ordinary Differential Equations https://arxiv.org/pdf/1803.06934.pdf\n\n\n```python\n# display equations\nprint_ode2(SC2IR_model)\n```\n\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -\\beta (I + c_0 I_c) S + c_1 S_c - c_2 S (I + I_c)\\\\\n\\dot{S_c} &= - c_0 \\beta (I + c_0 I_c) S_c - c_1 S_c + c_2 S (I + I_c)\\\\\n\\dot{I} &= \\beta (I + c_0 I_c) S - \\gamma I - \\mu I\\\\\n\\dot{I_c} &= \\beta (I + c_0 I_c) S_c - \\gamma I_c - \\mu I_c\\\\\n\\dot{R} & = \\gamma (I + I_c)\\\\\n\\dot{D} & = \\mu (I + I_c)\n\\end{split}\n\\end{equation}\n\n\n\n```python\n# display graphical representation of the model\nSC2IR_model.get_transition_graph()\n```\n\n## Caution Extensions to SEIR Model\n\n### SEIR model\n\n#### Equations\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -\\beta I S\\\\\n\\dot{E} &= \\beta I S - \\alpha E\\\\\n\\dot{I} &= \\alpha E - \\gamma I - \\mu I\\\\\n\\dot{R} & = \\gamma I \\\\\n\\dot{D} & = \\mu I\n\\end{split}\n\\end{equation}\n\n\n#### Variables\n* $S$: Susceptible individuals\n* $E$: Exposed individuals\n* $I$: Infected individuals \n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+E+I+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta$ rate at which infected individuals contact susceptibles and expose them to infection\n* $\\alpha$ rate at which exposed individuals become infected (1/(incubation time)\n* $\\gamma$ rate at which infected individuals recover from disease and become immune\n* $\\mu$ death rate for infected individuals\n\n#### Implementation\nUsing PyGOM, we will set up my simple SCIR model ODE system\nPyGOM \u2013 A Python Package for Simplifying Modelling with Systems of Ordinary Differential Equations https://arxiv.org/pdf/1803.06934.pdf\n\n\n```python\n# display equations\nprint_ode2(SEIR_model)\n```\n\n\n```python\n# display graphical representation of the model\nSEIR_model.get_transition_graph()\n```\n\n### SCEIR model\n\n#### Equations\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -\\beta I S + c_1 S_c - c_2*S*I\\\\\n\\dot{S_c} &= - c_0 \\beta I S_c - c_1 S_c + c_2*S*I\\\\\n\\dot{E} &= \\beta I (S + c_0 S_c) - \\alpha E\\\\\n\\dot{I} &= \\alpha E - \\gamma I - \\mu I\\\\\n\\dot{R} & = \\gamma I \\\\\n\\dot{D} & = \\mu I\n\\end{split}\n\\end{equation}\n\nThe use of I as a state variable triggering susceptibles to execute caution is just one choice. In contrast with deaths, it does not accumulate over time and so retains the property of an active threat to society, rather than an historical one. Alternatively, one could use the daily death rate which is proportional to it.\n\nActually, the effect of caution may be quadratic, since both the individual doing the infection and individual potentially being infected may be executing caution. The current version assumes that infectives do not change their precautionary measures in response to I. To implement this we would need to further distinguish careful and non careful infectives. This is done in the SC2IR model.\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $S_c$: Susceptible individuals exercising pandemy precautions\n* $E$: Exposed individuals\n* $I$: Infected individuals \n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+S_c+E+I+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta$ rate at which infected individuals contact susceptibles and infect them\n* $\\alpha$ rate at which exposed individuals become infected (1/(incubation time)\n* $\\gamma$ rate at which infected individuals recover from disease and become immune\n* $\\mu$ death rate for infected individuals\n\n* $c_i$ three parameters characterizing cautionary response of population via class $S_c$\n\n - c_0 : reduction factor for exposure for cautioned susceptibles\n\n - c_1 : duration of caution (exponential decay time constant in days)\n\n - c_2 : rate constant for transition from uncautioned to cautioned susceptible\n\n#### Implementation\nUsing PyGOM, we will set up my simple SCIR model ODE system\nPyGOM \u2013 A Python Package for Simplifying Modelling with Systems of Ordinary Differential Equations https://arxiv.org/pdf/1803.06934.pdf\n\n\n```python\n# display equations\nprint_ode2(SCEIR_model)\n```\n\n\n```python\n# display graphical representation of the model\nSCEIR_model.get_transition_graph()\n```\n\n### SC3EIR model\n\n#### Equations\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -\\beta (I + c_0 I_c) S + c_1 S_c - c_2 S (I + I_c)\\\\\n\\dot{S_c} &= - c_0 \\beta (I + c_0 I_c) S_c - c_1 S_c + c_2 S (I + I_c)\\\\\n\\dot{E} &= \\beta (I + c_0 I_c) S - \\alpha E + c_1 E_c - c_2 E (I + I_c)\\\\\n\\dot{E_c} &= c_0 \\beta (I + c_0 I_c) S_c - \\alpha E_c - c_1 E_c + c_2 E (I + I_c)\\\\\n\\dot{I} &= \\alpha E - \\gamma I - \\mu I + c_1 I_c - c_2 I (I + I_c)\\\\\n\\dot{I_c} &= \\alpha E_c - \\gamma I_c - \\mu I_c - c_1 I_c + c_2 I (I + I_c)\\\\\n\\dot{R} & = \\gamma (I + I_c) \\\\\n\\dot{D} & = \\mu (I + I_c)\n\\end{split}\n\\end{equation}\n\nThe use of I as a state variable triggering susceptibles to execute caution is just one choice. In contrast with deaths, it does not accumulate over time and so retains the property of an active threat to society, rather than an historical one. Alternatively, one could use the daily death rate which is proportional to it.\n\nActually, the effect of caution may be quadratic, since both the individual doing the infection and individual potentially being infected may be executing caution. The current version assumes that infectives do not change their precautionary measures in response to I. To implement this we would need to further distinguish careful and non careful infectives. This is done in the SC2IR model.\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $S_c$: Susceptible individuals exercising pandemy precautions\n* $E$: Exposed individuals\n* $E_c$: Exposed individuals exercising caution\n* $I$: Infected individuals\n* $I_c$: Infected individuals exercising caution\n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+S_c+E+E_c+I+I_c+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta$ rate at which infected individuals contact susceptibles and infect them\n* $\\alpha$ rate at which exposed individuals become infected (1/(incubation time)\n* $\\gamma$ rate at which infected individuals recover from disease and become immune\n* $\\mu$ death rate for infected individuals\n\n* $c_i$ three parameters characterizing cautionary response of population via class $S_c$\n\n - c_0 : reduction factor for exposure for cautioned susceptibles\n\n - c_1 : duration of caution (exponential decay time constant in days)\n\n - c_2 : rate constant for transition from uncautioned to cautioned susceptible\n\n#### Implementation\nUsing PyGOM, we will set up my simple SCIR model ODE system\nPyGOM \u2013 A Python Package for Simplifying Modelling with Systems of Ordinary Differential Equations https://arxiv.org/pdf/1803.06934.pdf\n\n\n```python\n# display equations\nprint_ode2(SC3EIR_model)\n```\n\n\n```python\n# display graphical representation of the model\nSC3EIR_model.get_transition_graph()\n```\n\n## Caution Extensions to SEI3R Model\n\n### SEI3R model\n\n#### Equations\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -(\\beta_1 I_1 +\\beta_2 I_2 + \\beta_3 I_3) S\\\\\n\\dot{E} &=(\\beta_1 I_1 +\\beta_2 I_2 + \\beta_3 I_3 ) S - a E \\\\\n\\dot{I_1} &= a E - \\gamma_1 I_1 - p_1 I_1 \\\\\n\\dot{I_2} &= p_1 I_1 -\\gamma_2 I_2 - p_2 I_2 \\\\\n\\dot{I_3} & = p_2 I_2 -\\gamma_3 I_3 - \\mu I_3 \\\\\n\\dot{R} & = \\gamma_1 I_1 + \\gamma_2 I_2 + \\gamma_3 I_3 \\\\\n\\dot{D} & = \\mu I_3\n\\end{split}\n\\end{equation}\n\nThis model (by Dr. Alison for example) involves exposed but not infectious individuals and three classes of infective states with increasing severity.\nThe latter two involve hospitalization with the last in ICU.\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $E$: Exposed individuals - infected but not yet infectious or symptomatic\n* $I_i$: Infected individuals in severity class $i$. Severity increaes with $i$ and we assume individuals must pass through all previous classes\n * $I_1$: Mild infection (hospitalization not required)\n * $I_2$: Severe infection (hospitalization required)\n * $I_3$: Critical infection (ICU required)\n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+E+I_1+I_2+I_3+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta_i$ rate at which infected individuals in class $I_i$ contact susceptibles and infect them\n* $a$ rate of progression from the exposed to infected class\n* $\\gamma_i$ rate at which infected individuals in class $I_i$ recover from disease and become immune\n* $p_i$ rate at which infected individuals in class $I_i$ progress to class $I_{I+1}$\n* $\\mu$ death rate for individuals in the most severe stage of disease\n\n#### Implementation\nUsing PyGOM, we will set up the model ODE system\nPyGOM \u2013 A Python Package for Simplifying Modelling with Systems of Ordinary Differential Equations https://arxiv.org/pdf/1803.06934.pdf\n\n\n```python\n# display equations\nprint_ode2(SEI3R_model)\n```\n\n\n```python\n# display graphical representation of the model\nSEI3R_model.get_transition_graph()\n```\n\n### SCEI3R model\n\n#### Equations\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -(\\beta_1 I_1 +\\beta_2 I_2 + \\beta_3 I_3) S + c_1 S_c - c_2*S*I_3\\\\\n\\dot{S_c} &= - c_0(\\beta_1 I_1 +\\beta_2 I_2 + \\beta_3 I_3) S_c - c_1 S_c + c_2*S*I_3\\\\\n\\dot{E} &=(\\beta_1 I_1 +\\beta_2 I_2 + \\beta_3 I_3 ) (S + c_0 S_c) - a E \\\\\n\\dot{I_1} &= a E - \\gamma_1 I_1 - p_1 I_1 \\\\\n\\dot{I_2} &= p_1 I_1 -\\gamma_2 I_2 - p_2 I_2 \\\\\n\\dot{I_3} & = p_2 I_2 -\\gamma_3 I_3 - \\mu I_3 \\\\\n\\dot{R} & = \\gamma_1 I_1 + \\gamma_2 I_2 + \\gamma_3 I_3 \\\\\n\\dot{D} & = \\mu I_3\n\\end{split}\n\\end{equation}\n\nThe use of I_3 as a state variable triggering susceptibles to execute caution is just one choice. In contrast with deaths, it does not accumulate over time and so retains the property of an active threat to society, rather than an historical one.\n\nActually, the effect of caution may be quadratic, since both the individual doing the infection and individual potentially being infected may be executing caution. The current version assumes that infectives do not change their precautionary measures in response to I_3. To implement this we would need to further distinguish careful and non careful infectives at least up to the I_1 level. This is done in the SC3EI3R model.\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $S_c$: Susceptible individuals exercising pandemy precautions\n* $E$: Exposed individuals - infected but not yet infectious or symptomatic\n* $I_i$: Infected individuals in severity class $i$. Severity increaes with $i$ and we assume individuals must pass through all previous classes\n * $I_1$: Mild infection (hospitalization not required)\n * $I_2$: Severe infection (hospitalization required)\n * $I_3$: Critical infection (ICU required)\n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+S_c+E+I_1+I_2+I_3+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta_i$ rate at which infected individuals in class $I_i$ contact susceptibles and infect them\n* $a$ rate of progression from the exposed to infected class\n* $\\gamma_i$ rate at which infected individuals in class $I_i$ recover from disease and become immune\n* $p_i$ rate at which infected individuals in class $I_i$ progress to class $I_{I+1}$\n* $\\mu$ death rate for individuals in the most severe stage of disease\n\n* $c_i$ three parameters characterizing cautionary response of population via class $S_c$\n\n - c_0 : reduction factor for exposure for cautioned susceptibles\n\n - c_1 : duration of caution (exponential decay time constant in days)\n\n - c_2 : rate constant for transition from uncautioned to cautioned susceptible\n\n\n\n\n#### Implementation\nUsing PyGOM, we will set up the model ODE system\nPyGOM \u2013 A Python Package for Simplifying Modelling with Systems of Ordinary Differential Equations https://arxiv.org/pdf/1803.06934.pdf\n\n\n```python\n# display equations\nprint_ode2(SCEI3R_model)\n```\n\n\n```python\n# display graphical representation of the model\nSCEI3R_model.get_transition_graph()\n```\n\n### SC3EI3R model with caution distinguished $E$ and \ud835\udc3c1\n\n#### Equations\n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -(\\beta_1 (I_1 + c_0 I_{1c}) +\\beta_2 I_2 + \\beta_3 I_3) S + c_1 S_c - c_2*S*I_3\\\\\n\\dot{S_c} &= - c_0(\\beta_1 (I_1 + c_0 I_{1c}) +\\beta_2 I_2 + \\beta_3 I_3) S_c - c_1 S_c + c_2*S*I_3\\\\\n\\dot{E} &=(\\beta_1 (I_1 + c_0 I_{1c}) +\\beta_2 I_2 + \\beta_3 I_3 ) S - a E + c_1 E_c - c_2*E*I_3\\\\\n\\dot{E_c} &=(\\beta_1 (I_1 + c_0 I_{1c}) +\\beta_2 I_2 + \\beta_3 I_3 ) c_0 S_c - a E - c_1 E_c + c_2*E*I_3\\\\\n\\dot{I_1} &= a E - \\gamma_1 I_1 - p_1 I_1 + c_1 I_{1c} - c_2*I_{1c}*I_3\\\\\n\\dot{I_{1c}} &= a E_c - \\gamma_1 I_{1c} - p_1 I_{1c} - c_1 I_{1c} + c_2*I_{1c}*I_3\\\\\n\\dot{I_2} &= p_1 (I_1 + I_{1c}) -\\gamma_2 I_2 - p_2 I_2 \\\\\n\\dot{I_3} & = p_2 I_2 -\\gamma_3 I_3 - \\mu I_3 \\\\\n\\dot{R} & = \\gamma_1 (I_1 + I_{1c}) + \\gamma_2 I_2 + \\gamma_3 I_3 \\\\\n\\dot{D} & = \\mu I_3\n\\end{split}\n\\end{equation}\n\nThe use of I_3 as a state variable triggering susceptibles to execute caution is just one choice. In contrast with deaths, it does not accumulate over time and so retains the property of an active threat to society, rather than an historical one.\n\nHere, the effect of caution is quadratic, since both the individual doing the infection and individual potentially being infected may be executing caution. To implement this we distinguish careful and non careful exposed and infectives up to the I_1 level. Once in hospital there is no difference, since all caution is executed wrt infected patients.\nWe ignore transition in caution among infected intervals as a second order effect: could be included as in SC2IR model.\n\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $S_c$: Susceptible individuals exercising pandemy precautions\n* $E$: Exposed individuals living as normal - infected but not yet infectious or symptomatic\n* $E_c$: Exposed individuals exercising pandemy precautions\n* $I_i$: Infected individuals in severity class $i$. Severity increaes with $i$ and we assume individuals must pass through all previous classes. Split non hospital cases by caution.\n * $I_1$: Mild infection (hospitalization not required), living as normal\n * $I_{1c}$: Mild infection (hospitalization not required), exercising caution\n * $I_2$: Severe infection (hospitalization required)\n * $I_3$: Critical infection (ICU required)\n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+S_c+E+E_c+I_{1c}+I_1+I_2+I_3+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta_i$ rate at which infected individuals in class $I_i$ contact susceptibles and infect them\n* $a$ rate of progression from the exposed to infected class\n* $\\gamma_i$ rate at which infected individuals in class $I_i$ recover from disease and become immune\n* $p_i$ rate at which infected individuals in class $I_i$ progress to class $I_{I+1}$\n* $\\mu$ death rate for individuals in the most severe stage of disease\n\n* $c_i$ three parameters characterizing cautionary response of population via class $S_c$\n\n - c_0 : reduction factor for exposure for cautioned susceptibles\n\n - c_1 : duration of caution (exponential decay time constant in days)\n\n - c_2 : rate constant for transition from uncautioned to cautioned susceptible\n\n#### Implementation\n\n\n```python\n# display equations\nprint_ode2(SC3EI3R_model)\n```\n\n\n```python\n# display graphical representation of the model\nSC3EI3R_model.get_transition_graph()\n```\n\n## Caution Extensions to SEIR Model with Economic Supression\nThis model is an extension of the cautionary model to include a class of susceptibles $S_u$ who are impervious to caution. \nThe main influencer for this class is the economy, which we introduce as a new state variable W, normalized to 1 in the absence of pandemic.\nThe model assumption is that fractional depression of the economy influences some susceptibles (both cautioned and uncautioned) to become uncautionable,\nwith a rate coefficient proportional to the economic depression (1-W). The economy itself is mdoelled with logistic growth to a state 1 in the absence of pandemic\nand 1- $\\kappa S_c$ with pandemic. i.e. individuals exercising caution are the main correlate of economic depression (but the only suppressor for the pandemic).\nAs for the cautioned class, uncautionable individuals also return to normal sussceptibles with exponential decay at rate $k_1$.\n\n### SC2UIR model\n\n#### Equations\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -\\beta (I + c_0 I_c) S + c_1 S_c - c_2 S (I + I_c) - k_u (1 - W) S + k_1 S_u\\\\\n\\dot{S_c} &= - c_0 \\beta (I + c_0 I_c) S_c - c_1 S_c + c_2 S (I + I_c) - k_u (1 - W) S_c\\\\\n\\dot{I} &= \\beta (I + c_0 I_c) S - \\gamma I - \\mu I + c_1 I_c - c_2 I (I + I_c)\\\\\n\\dot{I_c} &= c_0 \\beta (I + c_0 I_c) S_c - \\gamma I_c - \\mu I_c - c_1 I_c + c_2 I (I + I_c)\\\\\n\\dot{R} & = \\gamma (I + I_c) \\\\\n\\dot{D} & = \\mu (I + I_c) \\\\\n\\dot{S_u} & = -\\beta (I + c_0 I_c) S_u + k_u (1 - W) (S + S_c) - k_1 S_u \\\\\n\\dot{W} & = k_w W (1 - \\kappa S_c - W)\n\\end{split}\n\\end{equation}\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $S_c$: Susceptible individuals exercising pandemy precautions\n* $S_u$: Susceptible individuals immune to caution because of economic downturn\n* $E$: Exposed individuals\n* $E_c$: Exposed individuals exercising caution\n* $I$: Infected individuals\n* $I_c$: Infected individuals exercising caution\n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $W$: Economic status obeying a logistic law with caution individuals downturning\n* $N=S+S_c+S_u+E+E_c+I+I_c+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta$ rate at which infected individuals contact susceptibles and infect them\n* $\\alpha$ rate at which exposed individuals become infected (1/(incubation time)\n* $\\gamma$ rate at which infected individuals recover from disease and become immune\n* $\\mu$ death rate for infected individuals\n\n* $c_i$ three parameters characterizing cautionary response of population via class $S_c$\n\n - c_0 : reduction factor for exposure for cautioned susceptibles\n\n - c_1 : inverse duration of caution (exponential decay time constant in days)\n\n - c_2 : rate constant for transition from uncautioned to cautioned susceptible\n* four parameters coupling to economy and uncautionable individuals\n - k_w : rate coefficient of economy equilibration\n - k_u : rate coefficient of transition from uncautioned to uncautionable\n - k_1 : inverse duration of uncautionable state\n - $\\kappa$ : economic downturn of caution (proportional to number cautious)\n\n#### Implementation\n\n\n```python\n# display equations\nprint_ode2(SC2UIR_model)\n```\n\n\n```python\n# SC2UIR_model.get_transition_graph() # ode was defined explicitly, no transition available\n```\n\n\n```python\n# ode = SimulateOde(state, param_list, transition=transition)\n# ode = ode.get_unrolled_obj()\n# R0(ode, ['I','I_c']) # produces error, no valid subset found\n```\n\n### SC3UEIR model\n\n#### Equations\n\\begin{equation}\n\\begin{split}\n\\dot{S} &= -\\beta (I + c_0 I_c) S + c_1 S_c - c_2 S (I + I_c) - k_u (1 - W) S + k_1 S_u\\\\\n\\dot{S_c} &= - c_0 \\beta (I + c_0 I_c) S_c - c_1 S_c + c_2 S (I + I_c) - k_u (1 - W) S_c\\\\\n\\dot{E} &= \\beta (I + c_0 I_c) (S + S_u) - \\alpha E + c_1 E_c - c_2 E (I + I_c)\\\\\n\\dot{E_c} &= c_0 \\beta (I + c_0 I_c) S_c - \\alpha E_c - c_1 E_c + c_2 E (I + I_c)\\\\\n\\dot{I} &= \\alpha E - \\gamma I - \\mu I + c_1 I_c - c_2 I (I + I_c)\\\\\n\\dot{I_c} &= \\alpha E_c - \\gamma I_c - \\mu I_c - c_1 I_c + c_2 I (I + I_c)\\\\\n\\dot{R} & = \\gamma (I + I_c) \\\\\n\\dot{D} & = \\mu (I + I_c) \\\\\n\\dot{S_u} & = -\\beta (I + c_0 I_c) S_u + k_u (1 - W) (S + S_c) - k_1 S_u \\\\\n\\dot{W} & = k_w W (1 - \\kappa S_c - W)\n\\end{split}\n\\end{equation}\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $S_c$: Susceptible individuals exercising pandemy precautions\n* $S_u$: Susceptible individuals immune to caution because of economic downturn\n* $E$: Exposed individuals\n* $E_c$: Exposed individuals exercising caution\n* $I$: Infected individuals\n* $I_c$: Infected individuals exercising caution\n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $W$: Economic status obeying a logistic law with caution individuals downturning\n* $N=S+S_c+S_u+E+E_c+I+I_c+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta$ rate at which infected individuals contact susceptibles and infect them\n* $\\alpha$ rate at which exposed individuals become infected (1/(incubation time)\n* $\\gamma$ rate at which infected individuals recover from disease and become immune\n* $\\mu$ death rate for infected individuals\n\n* $c_i$ three parameters characterizing cautionary response of population via class $S_c$\n\n - c_0 : reduction factor for exposure for cautioned susceptibles\n\n - c_1 : inverse duration of caution (exponential decay time constant in days)\n\n - c_2 : rate constant for transition from uncautioned to cautioned susceptible\n* four parameters coupling to economy and uncautionable individuals\n - k_w : rate coefficient of economy equilibration\n - k_u : rate coefficient of transition from uncautioned to uncautionable\n - k_1 : inverse duration of uncautionable state\n - $\\kappa$ : economic downturn of caution (proportional to number cautious)\n\n#### Implementation\n\n\n```python\n# display equations\nprint_ode2(SC3UEIR_model)\n```\n\n### SC3UEI3R model\n\n#### Equations \n\n\\begin{equation}\n\\begin{split}\n\\dot{S} &={} -(\\beta_1 (I_1 + c_0 I_c) + \\beta_2 I_2 + \\beta_3 I_3) S + c_1 S_c - c_2 S I_3 - S k_u (1-W) + k_1 S_u\\\\\n\\dot{S_c} &={} - c_0 (\\beta_1 (I_1 + c_0 I_c) + \\beta_2 I_2 + \\beta_3 I_3) S_c - c_1 S_c + c_2 S I_3 - k_u (1 - W) S_c \\\\\n\\dot{E} &= \\beta_1 (I_1 + c_0 I_c) (S + S_u) - \\alpha E + c_1 E_c - c_2 I_3 E\\\\\n\\dot{E_c} &= c_0 \\beta_1 (I_1 + c_0 I_c) S_c - \\alpha E_c - c_1 E_c + c_2 I_3 E\\\\\n\\dot{I_1} &= \\alpha E - \\gamma_1 I_1 - p_1 I_1 + c_1 I_c - c_2 I_3 I_1\\\\\n\\dot{I_c} &= \\alpha E_c - \\gamma_1 I_c - p_1 I_c - c_1 I_c + c_2 I_3 I_1\\\\\n\\dot{I_2} &= p_1 (I_1 + I_c) -\\gamma_2 I_2 - p_2 I_2 \\\\\n\\dot{I_3} & = p_2 I_2 -\\gamma_3 I_3 - \\mu I_3 \\\\ \n\\dot{R} & = \\gamma_1 (I_1 + I_c) +\\gamma_2 I_2 + \\gamma_3 I_3\\\\\n\\dot{D} & = \\mu (I_3) \\\\\n\\dot{S_u} & = -(\\beta_1 (I_1 + c_0 I_c)+\\beta_2 I_2 + \\beta_3 I_3) S_u + k_u (1 - W)(S+ S_c) - k_1 S_u \\\\\n\\dot{W} & = k_w W (1 - \\kappa S_c - W)\\\\\n\\end{split}\n\\end{equation}\n\n#### Variables\n* $S$: Susceptible individuals living as normal\n* $S_c$: Susceptible individuals exercising pandemy precautions\n* $S_u$: Susceptible individuals immune to caution because of economic downturn\n* $W$: Economic status obeying a logistic law with caution individuals downturning\n* $E$: Exposed individuals\n* $E_c$: Exposed individuals exercising caution\n* $I_i$: Infected individuals in severity class $i$. Severity increaes with $i$ and we assume individuals must pass through all previous classes. Split non hospital cases by caution.\n * $I_1$: Mild infection (hospitalization not required), living as normal\n * $I_c$: Mild infection (hospitalization not required), exercising caution\n * $I_2$: Severe infection (hospitalization required)\n * $I_3$: Critical infection (ICU required)\n* $R$: individuals who have recovered from disease and are now immune\n* $D$: Dead individuals\n* $N=S+S_c+S_u+E+E_c+I_1+I_c+I_2+I_3+R+D$ Total population size (constant)\n\n#### Parameters\n* $\\beta$ rate at which infected individuals contact susceptibles and infect them\n* $\\alpha$ rate at which exposed individuals become infected (1/(incubation time)\n* $\\gamma$ rate at which infected individuals recover from disease and become immune\n* $\\mu$ death rate for infected individuals\n* $\\gamma_i$ rate at which infected individuals in class $I_i$ recover from disease and become immune\n* $p_i$ rate at which infected individuals in class $I_i$ progress to class $I_{I+1}$\n* $c_i$ three parameters characterizing cautionary response of population via class $S_c$\n\n - $c_0$ : reduction factor for exposure for cautioned susceptibles\n\n - $c_1$ : inverse duration of caution (exponential decay time constant in days)\n\n - $c_2$ : rate constant for transition from uncautioned to cautioned susceptible\n* four parameters coupling to economy and uncautionable individuals\n - $k_w$ : rate coefficient of economy equilibration\n - $k_u$ : rate coefficient of transition from uncautioned to uncautionable\n - $k_1$ : inverse duration of uncautionable state\n - $\\kappa$ : economic downturn of caution (proportional to number cautious)\n\n#### Implementation\n\n\n```python\n# display equations\nprint_ode2(SC3UEI3R_model) # name needs to be that of current model\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "674be675c7fe84cc6c072ec0f742e0986ab6bd7b", "size": 58789, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Notebooks/covid-19-caution/Caution_paper_Models.ipynb", "max_stars_repo_name": "ProtoLife/covid-recovery", "max_stars_repo_head_hexsha": "b3989f110c6961cc51da673fc33c6384cf7d6de6", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Notebooks/covid-19-caution/Caution_paper_Models.ipynb", "max_issues_repo_name": "ProtoLife/covid-recovery", "max_issues_repo_head_hexsha": "b3989f110c6961cc51da673fc33c6384cf7d6de6", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-05-24T00:06:54.000Z", "max_issues_repo_issues_event_max_datetime": "2020-05-25T19:59:44.000Z", "max_forks_repo_path": "Notebooks/covid-19-caution/Caution_paper_Models.ipynb", "max_forks_repo_name": "ProtoLife/covid-recovery", "max_forks_repo_head_hexsha": "b3989f110c6961cc51da673fc33c6384cf7d6de6", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.4244114002, "max_line_length": 434, "alphanum_fraction": 0.5789008148, "converted": true, "num_tokens": 11205, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494678483918, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.4219191871712543}} {"text": "# Detection of far-side solar active regions\n*by T. Felipe and A. Asensio Ramos*\n\nLocal helioseismology techniques allow the detection of active regions in the non-visible solar hemisphere (far-side) by analyzing the oscillations in the visible side of the Sun. However, this identification is challenged by the low signal-to-noise of the seismic data, and only strong active regions can be reliably detected. \n\nIn this notebook, we will show a method to improve the detection of active regions in far-side seismic maps using a machine learning algorithm.\n\nThis work is published in [Felipe & Asensio Ramos, 2019, A&A, 632, 82](https://www.aanda.org/articles/aa/abs/2019/12/aa36838-19/aa36838-19.html)\n\n\n\nDetection of a far-side active region.\n\n\n\n# Introduction\nOne of the most remarkable applications of local helioseismology is the\ndetection of active regions in the non-visible hemisphere of the Sun (on the far side).\nThis was first achieved using the technique of helioseismic holography\n([Lindsey & Braun 2000, Science, 287, 1799](https://science.sciencemag.org/content/287/5459/1799.full), [Braun & Lindsey 2001, ApJ, 560, 189](https://iopscience.iop.org/article/10.1086/324323)). \n\nHelioseismic holography uses the wavefield measured in a region of the solar surface (called \"pupil\") to determine the wave field at a focus point that is located at the surface or at a certain depth. This inference is\nperformed assuming that the observed wave field at the pupil (e.g., the line-of-sight Doppler velocity) is produced by waves converging toward the focus point or waves diverging from that point. Far-side helioseismic holography is a particular application of this method, where the pupil is located at the near-side hemisphere and the focus points are located at the surface in the far-side hemisphere (see [Lindsey & Braun 2017, Space Weather, 15, 761](https://ui.adsabs.harvard.edu/abs/2017SpWea..15..761L/abstract)). \n\nThe identification of active regions is founded\non the fact that they introduce a phase shift between ingoing and outgoing waves. This phase shift (which can be characterized as a travel-time shift) is mainly due to the depression of the photosphere in\nmagnetized regions, which causes the upcoming waves to reach the upper turning point a few seconds earlier in active regions than in quiet-Sun regions ([Felipe et al. 2017, A&A, 604, 126](https://ui.adsabs.harvard.edu/link_gateway/2017A%26A...604A.126F/PUB_HTML)). In this way, when an active region is located at the focus point, a negative phase shift (reduction in the travel\ntime) is found.\n\n\n# Why using a neural network approach?\n\nOne of the main limitations of far-side helioseismology is the reduced signal-to-noise ratio. The signature of an active region detected on the far side has a\nsignal-to-noise ratio of around 10, which means that only large and strong active regions can be reliably detected in far-side phase-shift maps (about several hundered active regions per solar cycle).\n\nOur aim in this work is to apply convolutional neural networks to learn a very fast and robust mapping between consecutive maps of estimated seismic maps and the probability map of the presence of an active region on the far side. The recent success of machine learning is no doubt a consequence of our ability to train very deep neural networks (DNNs). DNNs can be seen as a very flexible and differentiable parametric mapping between an input space and an output space. These highly parameterized\nDNNs are then tuned by optimizing a loss function that measures the ability of the DNN to map the input space onto the output space over a predefined training set. The combination of loss function and specific architecture has to be chosen to solve the specific problem at hand.\n\nArguably the largest number of applications of DNNs has been in computer vision. Problems belonging to the realm of machine vision can hardly be solved using classical methods, be they based on machine learning or on rule-based methods. Only now, with the application of very DNNs, have we been able to produce real advances. Applications in science, and specifically in astrophysics and solar physics, have leveraged the results of machine vision to solve problems that were difficult or impossible to deal with in the past with classical techniques.\n\n# Description of the neural network\nIn this notebook, we present a description of the neural network developed for the detection of far-side active regions. We have included a running example of the application of the network and the tools employed for the interpretation of the results. \n\nWe have omitted the materials employed for the training set. They are publicly available and their locations are indicated. We have described the transformations applied to the original data to convert them into the data fed to the neural network for the training.\n\n## Training set\nWe have designed a neural network that can identify the presence of active\nregions on the far side. As input, the network uses far-side phase-shift maps\ncomputed using helioseismic holography. As a proxy for the presence of active\nregions, we employed Helioseismic and Magnetic Imager (HMI) magnetograms measured on the near side (facing Earth). The details of the data are discussed in the following sections. \nThe training set that we describe in this section was used to supervise the parameter tuning of the neural network with the aim of generalizing this to \nnew data.\n\n### HMI magnetograms\nThe HMI magnetograms are one of the data products from the Solar Dynamics Observatory available through the Joint Science Operations Center (JSOC). In order to facilitate the comparison with the far-side seismic maps (next section), we are interested in magnetograms that are remapped onto a Carrington coordinate grid. We used data from the JSOC series *hmi.Mldailysynframe\\_720s*. This data series contains synoptic maps constructed of HMI magnetograms collected over a 27-day solar rotation, where the first 120 degrees in longitude are replaced by data within 60 degrees of the central meridian of the visible hemisphere observed approximately at one time. These\nmaps are produced daily at 12 UT. We only employed the 120 degrees in longitude\nincluding the magnetogram visible on the disk at one time. Magnetograms between\n2010 June 1 (the first date available for the *hmi.Mldailysynframe\\_720s*\ndata) and 2018 October 26 were extracted. Because one magnetogram is taken per day, this means a total of 3066 magnetograms. \n\nBecause new active regions emerge and old regions decay,\nmagnetograms obtained on the near side are an inaccurate characterization of the\nactive regions on the far side half a rotation earlier or later. We have\npartially corrected this problem. The far-side maps are associated with the\nmagnetogram that is obtained when the seismically probed region has fully rotated to the\nEarth side, that is, 13.5 days after the measurement of the far-side map. We\nremoved the active regions that emerge on the near side because they were absent when the far-side seismic data were taken. In order to identify the\nemerging active regions, we have employed the Solar Region Summary (SRS)\nfiles (available at [ftp.swpc.noaa.gov/pub/warehouse/](ftp://ftp.swpc.noaa.gov/pub/warehouse/), where the NOAA registered active regions are listed. All the active regions that appear for the first time at a longitude greater than $-60^{\\circ}$ (where 0 corresponds to the central meridian of the visible hemisphere and the minus sign indicates the eastern hemisphere) were masked in the magnetograms. The value of the magnetogram was set to zero in an area 15 degrees wide in longitude and 12 degrees wide in latitude, centered in the location of the active region reported in the SRS file of that date (after correcting for the longitude because we employed magnetograms retrieved at 12 UT and in the SRS files the location of the active regions are reported for 24 UT on the previous day). The active regions that emerge in the visible hemisphere too close to an active region that had appeared on the eastern limb due to the solar rotation were not masked. Of the 1652 active regions labeled by NOAA during the temporal period employed for the training set, 967 were masked because they emerged in the visible hemisphere. \n\nThe neural network is trained with binary maps, where the zeros correspond to quiet regions and the ones to active regions. This binary mask is built from the corrected magnetograms as follows. A Gaussian smoothing with a standard deviation of 3 degrees was applied to the corrected magnetograms. This smoothing removed all small-scale activity in the map and facilitated the segmentation of active regions of importance in the magnetogram.\nThen, regions with a magnetic flux higher than 30 Mx cm$^2$ were identified as active regions (and set to 1), and regions with lower magnetic flux were set to 0. The middle panel in the bottom row from Fig. 1 shows the magnetogram after the active regions that emerged in the visible solar hemisphere were removed and after Gaussian smoothing was applied. The active region visible in the original magnetogram (bottom left panel in Fig. 1) at a longitude $-30^{\\circ}$ and a latitude $-5^{\\circ}$ emerged on the near side and was therefore masked. The bottom right panel of Fig. 1 shows the binary map in which the location of the remaining active regions is indicated, those whose magnetic flux is above the selected threshold. Their positions match that of some regions with strong negative travel times in the seismic maps from about half a rotation earlier (case \"t-13.0\" in the top row of Fig. 1). \n\n**Fig. 1.** Example of one of the elements from the training set. Panels in the top row show 11 far-side seismic maps, each of them obtained from the analysis of 24 h of HMI Doppler data. The horizontal axis is the longitude (a total of 120\u00b0) and the vertical axis is the latitude (between \u221272\u00b0 and 72\u00b0). The label above the panels indicates the number of days prior to the time t when the corresponding magnetogram was acquired (in this example, t is 2015 December 10 at 12:00 UT). Bottom row: magnetograms we used as a proxy for the presence of active regions. Left panel: original magnetogram in heliospheric coordinates, middle panel: magnetogram after active regions that emerged in the near side are removed and after a Gaussian smoothing was applied, and right panel: binary map in which a value of 1 indicates the presence of an active region in the locations whose magnetic flux in the smoothed magnetogram is above the selected threshold. Red contours in the bottom left panel delimit the regions where the binary map is 1. The neural network is trained by associating the 11 far-side seismic maps (top row) with the binary map. \n\n\n\n\n### Far-side phase-shift maps\n\nPhase-shift maps of the far-side region of the Sun are available through JSOC. They are computed from\nHMI Doppler data using temporal series of one or five days. The processing of\nseries of five days is a novel approach since 2014, introduced to improve the\nsignal-to-noise ratio of the phase-shift maps. They are provided in Carrington\nheliographic coordinates with a cadence of 12 hours (maps are obtained at 0 and\n12 UT). In this work, we focus on the far-side maps computed from 24 hours\nof Doppler data. We employed far-side maps between 2010 May 18 and 2018 October 12. For each map, we selected a $120^{\\circ}$ region in longitude centered at the Carrington longitude of the central meridian of the visible hemisphere 13.5 days after the date of the far-side map. In this way, corrected magnetograms from which\nthe new active regions are removed are associated with far-side maps that sample the same region in longitude. The training employed 11 consecutive far-side maps for each corrected magnetogram, which improved the seismic signal. These 11 consecutive far-side maps correspond to six days of data. The latitude span of the maps is\nbetween $-72^{\\circ}$ and $72^{\\circ}$. We chose a sampling of $1^{\\circ}$ in both latitude and longitude.\n\n\n##Architecture\nThe neural network of choice in \nthis work is a U-net ([Ronneberger et al. 2015, ArXiv](https://arxiv.org/abs/1505.04597)), a fully\nconvolutional architecture that has been used extensively for dense segmentation of images and displayed in Fig. 2. The U-net \nis an encoder-decoder \nnetwork, in which the input is\nsuccessively reduced in size through contracting layers and is finally increased in size through\nexpanding layers. This encoder-decoder architecture has three main advantages, all\nof them a consequence of the contracting and expanding layers. The first\nadvantage is that the contracting layers reduce the size of the images at each step.\nThis makes the network faster because convolutions have to be carried out\nover smaller images. The second advantage is that this contraction couples\ntogether pixels in the input image that were far apart, so that smaller kernels\ncan be used in convolutional layers (we used $3 \\times 3$ kernels) and the network\nis able to better exploit multiscale information. The final\nadvantage is a consequence of the skip connections (gray \narrows), which facilitates training by explicitly\npropagating multiscale information from the contracting layers to the\nexpanding layers.\n\nAs shown in Fig. 2, the specific U-net architecture\nwe used in this work is a combination of several\ndifferentiable operations. The first operation, indicated with blue arrows, is\nthe consecutive application of convolutions with 3$\\times$3 kernels, \nbatch normalization (BN), which normalizes the input so that its mean\nis close to zero and its variance close to unity (which is known to\nbe an optimal range of values for neural networks to work best) and\na rectified linear unit (ReLU) activation function, given \nby $\\sigma(x)=\\max(0,x)$. This combination \nConv+BN+ReLU was repeated twice as indicated in\nthe legend of Fig. 2. Red arrows refer to \nmax-pooling [(Goodfellow et al. 2016, Deep Learning, MIT Press)](http://www.deeplearningbook.org/), which reduces the resolution\nof the images by a factor 2 by computing the maximum of all non-overlapping \n$2 \\times 2$ patches in the image. The expanding layers again increase the size of the images\nthrough bilinear interpolation (green arrows) followed by convolutional layers.\nAdditionally, the layers in the encoding part transfer information to the\ndecoding part through skip connections (gray arrows), which greatly \nimproves the ability and stability of the network.\nFinally, because the output is a probability map, we forced it to be in the $[0,1]$ range\nthrough a sigmoid activation function that was applied in the last layer after a final\n$1 \\times 1$ convolution that we used to reduce the number of channels from 16 to 1.\n\nThe neural\nnetwork was trained by minimizing the binary cross entropy between the output of\nthe network per pixel ($p_i$) and the binarized magnetograms ($y_i$), summed\nover all pixels in the output magnetogram ($N$),\n\\begin{equation}\n \\ell = -\\frac{1}{N} \\sum_{i=1}^{N} y_{i} \\cdot \\log p_i+\n \\left(1-y_{i}\\right) \\cdot \\log \\left(1-p_i\\right)\n.\\end{equation}\nTo optimize the previous loss function, we employed the Adam optimizer [(Kingma & Ba 2014, ArXiv)](https://arxiv.org/abs/1412.6980) with a\nconstant learning rate of 3$\\times$10$^{-4}$ during 300 epochs and a batch\nsize of 30.\n\nThe neural network can be downloaded from the repository [https://github.com/aasensio/farside](https://github.com/aasensio/farside).\n\nHere we show the model.\n\n\n**Fig 2.** U-net architecture. The vertical extent of the blocks indicates the size of the image, and the numbers above each block shows the number of channels.\n\n\n###Model\n\n\n\n```\n#We first import the necessary modules\nimport numpy as np\nimport torch\nimport torch.nn as nn\nimport torch.utils.data\nimport torch.nn.functional as F\n```\n\n\n```\nclass double_conv(nn.Module):\n '''(conv => BN => ReLU) * 2'''\n def __init__(self, in_ch, out_ch):\n super(double_conv, self).__init__()\n self.conv = nn.Sequential(\n nn.Conv2d(in_ch, out_ch, 3, padding=1),\n nn.BatchNorm2d(out_ch),\n nn.ReLU(inplace=True),\n nn.Conv2d(out_ch, out_ch, 3, padding=1),\n nn.BatchNorm2d(out_ch),\n nn.ReLU(inplace=True)\n )\n\n def forward(self, x):\n x = self.conv(x)\n return x\n\n\nclass inconv(nn.Module):\n def __init__(self, in_ch, out_ch):\n super(inconv, self).__init__()\n self.conv = double_conv(in_ch, out_ch)\n\n def forward(self, x):\n x = self.conv(x)\n return x\n\n\nclass down(nn.Module):\n def __init__(self, in_ch, out_ch):\n super(down, self).__init__()\n self.mpconv = nn.Sequential(\n nn.MaxPool2d(2),\n double_conv(in_ch, out_ch)\n )\n\n def forward(self, x):\n x = self.mpconv(x)\n return x\n\n\nclass up(nn.Module):\n def __init__(self, in_ch, out_ch, bilinear=True):\n super(up, self).__init__()\n\n self.bilinear = bilinear\n\n # would be a nice idea if the upsampling could be learned too,\n if not bilinear:\n self.up = nn.ConvTranspose2d(in_ch//2, in_ch//2, 2, stride=2)\n\n self.conv = double_conv(in_ch, out_ch)\n\n def forward(self, x1, x2):\n\n if (self.bilinear):\n x1 = torch.nn.functional.interpolate(x1, scale_factor=2) \n else:\n x1 = self.up(x1)\n \n # input is CHW\n diffY = x2.size()[2] - x1.size()[2]\n diffX = x2.size()[3] - x1.size()[3]\n\n x1 = F.pad(x1, (diffX // 2, diffX - diffX//2,\n diffY // 2, diffY - diffY//2))\n \n # for padding issues, see \n # https://github.com/HaiyongJiang/U-Net-Pytorch-Unstructured-Buggy/commit/0e854509c2cea854e247a9c615f175f76fbb2e3a\n # https://github.com/xiaopeng-liao/Pytorch-UNet/commit/8ebac70e633bac59fc22bb5195e513d5832fb3bd\n\n x = torch.cat([x2, x1], dim=1)\n x = self.conv(x)\n return x\n\n\nclass outconv(nn.Module):\n def __init__(self, in_ch, out_ch):\n super(outconv, self).__init__()\n self.conv = nn.Conv2d(in_ch, out_ch, 1)\n\n def forward(self, x):\n x = self.conv(x)\n return x\n\n\nclass UNet(nn.Module):\n def __init__(self, n_channels, n_classes, n_hidden=64):\n super(UNet, self).__init__()\n self.inc = inconv(n_channels, n_hidden)\n self.down1 = down(n_hidden, 2*n_hidden)\n self.down2 = down(2*n_hidden, 4*n_hidden)\n self.down3 = down(4*n_hidden, 8*n_hidden)\n self.down4 = down(8*n_hidden, 8*n_hidden)\n self.up1 = up(16*n_hidden, 4*n_hidden)\n self.up2 = up(8*n_hidden, 2*n_hidden)\n self.up3 = up(4*n_hidden, n_hidden)\n self.up4 = up(2*n_hidden, n_hidden)\n self.outc = outconv(n_hidden, n_classes)\n\n def forward(self, x):\n x1 = self.inc(x)\n x2 = self.down1(x1)\n x3 = self.down2(x2)\n x4 = self.down3(x3)\n x5 = self.down4(x4)\n x = self.up1(x5, x4)\n x = self.up2(x, x3)\n x = self.up3(x, x2)\n x = self.up4(x, x1)\n x = self.outc(x)\n return torch.sigmoid(x)\n```\n\n### Forward model\n\n\n```\nclass deep_farside(object):\n def __init__(self, maxbatch):\n\n self.cuda = torch.cuda.is_available()\n self.device = torch.device(\"cuda\" if self.cuda else \"cpu\")\n \n torch.backends.cudnn.benchmark = True\n \n self.max_batch = maxbatch\n \n def init_model(self, checkpoint=None, n_hidden=16):\n \n self.checkpoint = checkpoint\n\n self.model = UNet(n_channels=11, n_classes=1, n_hidden=n_hidden).to(self.device)\n \n if (self.cuda):\n checkpoint = torch.load('{0}.pth'.format(self.checkpoint))\n else:\n checkpoint = torch.load('{0}.pth'.format(self.checkpoint), map_location=lambda storage, loc: storage)\n \n self.model.load_state_dict(checkpoint['state_dict']) \n \n def forward(self, phase):\n\n n_cases, n_phases, nx, ny = phase.shape\n\n assert (n_phases == 11), \"n. phases is not 11\"\n\n print(\"Normalizing data...\")\n \n phase = np.nan_to_num(phase)\n\n phase -= np.mean(phase)\n phase /= np.std(phase)\n\n phase[phase>0] = 0.0\n\n self.model.eval()\n\n n_batches = n_cases // self.max_batch\n n_remaining = n_cases % self.max_batch\n\n print(\" - Total number of maps : {0}\".format(n_cases))\n print(\" - Total number of batches/remainder : {0}/{1}\".format(n_batches, n_remaining))\n \n magnetograms = np.zeros((n_cases,nx,ny))\n\n left = 0\n\n print(\"Predicting magnetograms...\")\n\n with torch.no_grad():\n\n for i in range(n_batches): \n right = left + self.max_batch\n phases = torch.from_numpy(phase[left:right,:,:,:].astype('float32')).to(self.device) \n output = self.model(phases)\n\n magnetograms[left:right,:,:] = output.cpu().numpy()[:,0,:,:]\n\n left += self.max_batch\n\n if (n_remaining != 0):\n right = left + n_remaining\n phases = torch.from_numpy(phase[left:right,:,:,:].astype('float32')).to(self.device) \n output = self.model(phases)\n magnetograms[left:right,:,:] = output.cpu().numpy()[:,0,:,:]\n \n\n return magnetograms\n```\n\n#Interpretation of the results\nThe neural network returns a probability $P$ map with values in the range $[0,1]$. An active region is then identified by examining these probability maps, instead of directly evaluating the travel times of the far-side seismic maps. We defined an integrated probability $P_{\\rm i}$, computed\nas the integral of the probability $P$ in a continuous feature. The concept of ``integrated probability'' is equivalent to the ``seismic strength'' defined by the traditional method. Rather than simply search for continuous regions with strong negative travel times, an approach that is hindered by the usual strong noise of the seismic data, the neural network provides a cleaner picture of the locations where an active region is most probable. However, the probability maps usually exhibit some significant values in regions with negative travel time as a result of noise.\n\nIt is necessary to define an unequivocal\ncriterion to decide whether a region with increased probability is claimed as an active region. We chose to define a threshold in the integrated probability as the minimum value for the detection of seismic sources, in the same way as the traditional method establishes a threshold in the seismic strength. The selection of the threshold was based on the evaluation of the artificial set of far-side maps for which we know the exact location of the seismic sources (see [Felipe & Asensio Ramos, 2019, A&A, 632, 82](https://www.aanda.org/articles/aa/abs/2019/12/aa36838-19/aa36838-19.html)). A value of $P_{\\rm i}=100$ proved to be a good compromise between the success in detecting the seismic sources and avoiding the claim of false positives. We note that when the network is applied to real data, false positives can be easily dealt with by discarding the cases where the detection does no appear consistently in successive dates at the same location.\n\n#Examples\n\nIn this section, we apply the network to actual far-side seismic maps obtained from HMI. \nFirst, we need to install photutils and an appropriate version of astropy, since some of their routines will be employed for the interpretation of the network output.\n\n\n```\n!pip install photutils astropy==3.2.3\n```\n\n Collecting photutils\n \u001b[?25l Downloading https://files.pythonhosted.org/packages/89/8e/f2772b58c4030f7a8096a683e978f7d8fc34e09c763896fda060cc7f70f3/photutils-0.7.2-cp36-cp36m-manylinux2010_x86_64.whl (978kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 983kB 4.8MB/s \n \u001b[?25hCollecting astropy==3.2.3\n \u001b[?25l Downloading https://files.pythonhosted.org/packages/66/07/1e2a4c529d4216972eb05e887238d597ff69c34b1e87108c888972cd60b8/astropy-3.2.3-cp36-cp36m-manylinux1_x86_64.whl (6.3MB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 6.3MB 34.6MB/s \n \u001b[?25hRequirement already satisfied: numpy>=1.13 in /usr/local/lib/python3.6/dist-packages (from photutils) (1.17.5)\n Installing collected packages: astropy, photutils\n Found existing installation: astropy 4.0\n Uninstalling astropy-4.0:\n Successfully uninstalled astropy-4.0\n Successfully installed astropy-3.2.3 photutils-0.7.2\n\n\n\n```\n#import some modules\nimport h5py\nimport matplotlib.pyplot as plt\nfrom astropy.convolution import Gaussian2DKernel\nfrom astropy.stats import gaussian_fwhm_to_sigma\nfrom photutils import detect_sources\nfrom photutils import detect_threshold\nimport scipy.io\n%matplotlib inline\n```\n\nNext, we download the data needed for these examples. We require the trained model (2019-04-02-11:27:48_hid_16_lr_0.0003_wd_0.0.pth) and some observed far-side maps. Each of the files farside_NN2019_003_dlong140.sav and test.h5 contains a set of consecutive far-side HMI seismic maps. The individual seismic maps have 140 points in longitude, with a resolution of 1 degree and centered at the central meridian of the non-visible solar hemisphere. The latitude coverage spans from -72 to 71 degrees, with the same resolution of 1 degree.\n\n\n\n```\n!wget -O 2019-04-02-11:27:48_hid_16_lr_0.0003_wd_0.0.pth https://owncloud.iac.es/index.php/s/2xJpktVzVSx4YGy/download\n!wget -O farside_NN2019_003_dlong140.sav https://owncloud.iac.es/index.php/s/Xtxn7OJ1fliUdw1/download\n!wget -O test.h5 https://owncloud.iac.es/index.php/s/iax6sNFf9UYTtxR/download\n```\n\n --2020-02-26 10:22:27-- https://owncloud.iac.es/index.php/s/2xJpktVzVSx4YGy/download\n Resolving owncloud.iac.es (owncloud.iac.es)... 161.72.1.40, 2001:720:1610:5001::28\n Connecting to owncloud.iac.es (owncloud.iac.es)|161.72.1.40|:443... connected.\n HTTP request sent, awaiting response... 200 OK\n Length: 10146397 (9.7M) [application/octet-stream]\n Saving to: \u20182019-04-02-11:27:48_hid_16_lr_0.0003_wd_0.0.pth\u2019\n \n 2019-04-02-11:27:48 100%[===================>] 9.68M 3.85MB/s in 2.5s \n \n 2020-02-26 10:22:31 (3.85 MB/s) - \u20182019-04-02-11:27:48_hid_16_lr_0.0003_wd_0.0.pth\u2019 saved [10146397/10146397]\n \n --2020-02-26 10:22:32-- https://owncloud.iac.es/index.php/s/Xtxn7OJ1fliUdw1/download\n Resolving owncloud.iac.es (owncloud.iac.es)... 161.72.1.40, 2001:720:1610:5001::28\n Connecting to owncloud.iac.es (owncloud.iac.es)|161.72.1.40|:443... connected.\n HTTP request sent, awaiting response... 200 OK\n Length: 1777472 (1.7M) [application/octet-stream]\n Saving to: \u2018farside_NN2019_003_dlong140.sav\u2019\n \n farside_NN2019_003_ 100%[===================>] 1.69M 1.56MB/s in 1.1s \n \n 2020-02-26 10:22:34 (1.56 MB/s) - \u2018farside_NN2019_003_dlong140.sav\u2019 saved [1777472/1777472]\n \n --2020-02-26 10:22:35-- https://owncloud.iac.es/index.php/s/iax6sNFf9UYTtxR/download\n Resolving owncloud.iac.es (owncloud.iac.es)... 161.72.1.40, 2001:720:1610:5001::28\n Connecting to owncloud.iac.es (owncloud.iac.es)|161.72.1.40|:443... connected.\n HTTP request sent, awaiting response... 200 OK\n Length: 15208448 (15M) [application/octet-stream]\n Saving to: \u2018test.h5\u2019\n \n test.h5 100%[===================>] 14.50M 3.38MB/s in 4.5s \n \n 2020-02-26 10:22:40 (3.25 MB/s) - \u2018test.h5\u2019 saved [15208448/15208448]\n \n\n\n## Example 1\nThe file test.h5 includes a series of HMI far-side seismic maps, with the latitude and longitude coverage and resolution described above. First, we read the seismic maps.\n\n\n```\nf = h5py.File('test.h5', 'r')\nf.keys()\n```\n\n\n\n\n KeysView()\n\n\n\nNext, we plot a random selection of those maps. Each panel shows a seismic map computed from 24 hours of Doppler velocity temporal series measured with HMI. The figure shows the general appearance of the far-side seismic maps. The horizontal axes are the longitude, and the vertical axes correspond to the latitude. The maps exhibit a distribution of positive (yellow) and negative (black) travel-time shifts. Negative travel-time shifts may correspond to far-side active regions but, as illustrated in these examples, these maps are very noisy and must be carefully interpreted.\n\n\n```\nfig, ax = plt.subplots(nrows=3, ncols=4, figsize=(10,10))\nfor i in range(3):\n for j in range(4):\n ax[i,j].imshow(f['phases'][i,j,:,:])\n```\n\nWe compute the probability maps applying the neural network to continuous series of 11 farside maps.\n\n\n```\ndeep_farside_network = deep_farside(maxbatch=20) \ndeep_farside_network.init_model(checkpoint='2019-04-02-11:27:48_hid_16_lr_0.0003_wd_0.0', n_hidden=16)\n```\n\n\n```\nprob = deep_farside_network.forward(f['phases'][:])\n```\n\n Normalizing data...\n - Total number of maps : 20\n - Total number of batches/remainder : 1/0\n Predicting magnetograms...\n\n\nWe can plot the probability maps obtained for a few randomly selected cases. These examples show a few small patches with increased probability. However, we need to evaluate each of these features to check if the can be claim as active regions. \n\n\n```\nfig, ax = plt.subplots(nrows=2, ncols=2, figsize=(10,10))\nax = ax.flatten()\nfor i in range(4):\n ax[i].imshow(prob[i,:,:])\n```\n\nWe employ the following routines to select features present in an specific map. In this example, we identify the feature found in the bottom left panel of the previous figure.\n\n\n```\nsigma = 3.0 * gaussian_fwhm_to_sigma # FWHM = 3.\nkernel = Gaussian2DKernel(sigma, x_size=3, y_size=3)\nkernel.normalize()\nsegm = detect_sources(prob[2,:,:], 0.01, npixels=5, filter_kernel=kernel)\n```\n\n\n```\nplt.imshow(segm)\n```\n\n\n```\ntmp = prob[2,:,:]\n(tmp[segm.data==1]).sum()\n```\n\n\n\n\n 31.973699834255967\n\n\n\nIn this case, we obtain an integrated probability $P_i$=32. This value is below the threshold indicated in the previous section ($P_i$=100) and, thus, this feature cannot be claim as a far-side active region.\n\n##Example 2\n\nThe file farside_NN2019_003_dlong140.sav contains 11 consecutive far-side HMI seismic maps. They were employed for the detection of the far-side active region labeled NN-2019-003 in [Felipe & Asensio Ramos, 2019, A&A, 632, 82](https://www.aanda.org/articles/aa/abs/2019/12/aa36838-19/aa36838-19.html) as illustrated in the second row of Fig. 6 from that paper. These seismic maps were measured between 1 February 2019 at 00:00 UT and 6 February 2019 at 00:00 UT, with a cadence of 12 hours. \nSimilarly to the previous example, we start by reading the data and applying the forward model to the set of seismic maps.\n\n\n```\ntmp = scipy.io.readsav('farside_NN2019_003_dlong140.sav')\ntmp['data_out'].shape\n```\n\n\n\n\n (11, 144, 140)\n\n\n\n\n```\nprob = deep_farside_network.forward(tmp['data_out'][None,:,:,:])\n```\n\n Normalizing data...\n - Total number of maps : 1\n - Total number of batches/remainder : 0/1\n Predicting magnetograms...\n\n\nThe forward model returns the following probability map: \n\n\n```\nplt.imshow(prob[0,:,:], origin='lower')\n```\n\nWe identify the individual continuous regions with a certain probability for the presence of active regions. In this example, there are two independent features.\n\n\n```\nsigma = 3.0 * gaussian_fwhm_to_sigma # FWHM = 3.\nkernel = Gaussian2DKernel(sigma, x_size=3, y_size=3)\nkernel.normalize()\nsegm = detect_sources(prob[0,:,:], 0.01, npixels=5, filter_kernel=kernel)\nplt.imshow(segm, origin='lower')\n```\n\n\n```\ntmp = prob[0,:,:]\n(tmp[segm.data==2]).sum()\n```\n\n\n\n\n 205.16914902971985\n\n\n\nThe big feature exhibits an integrated probability of $P_i$=205. This is above the threshold selected to claim a region with increased probability as an active region ($P_i$=100). We note that the value computed here is slightly different from the value reported in the publication. This discrepancy is due to the use of a different method for identifying the features in the probability map, but it does not affect the interpretation of the results. \nWith regard to the small feature found in the previous figure: \n\n\n```\ntmp = prob[0,:,:]\n(tmp[segm.data==1]).sum()\n```\n\n\n\n\n 35.634890750574414\n\n\n\nIts integrated probability is $P_i$=36 and, thus, our approach cannot guarantee its association to an actual far-side active region.\n\n\n```\n\n```\n", "meta": {"hexsha": "738b8f9c322051ca4652bc7e331799fafb7852dd", "size": 514391, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "content/07/1/notebook.ipynb", "max_stars_repo_name": "raplima/HelioML", "max_stars_repo_head_hexsha": "94cf314d4c6060d40b96e090e85adf6c4309fd51", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "content/07/1/notebook.ipynb", "max_issues_repo_name": "raplima/HelioML", "max_issues_repo_head_hexsha": "94cf314d4c6060d40b96e090e85adf6c4309fd51", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "content/07/1/notebook.ipynb", "max_forks_repo_name": "raplima/HelioML", "max_forks_repo_head_hexsha": "94cf314d4c6060d40b96e090e85adf6c4309fd51", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 430.0928093645, "max_line_length": 417256, "alphanum_fraction": 0.9081399169, "converted": true, "num_tokens": 8269, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494678483918, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.4219191871712543}} {"text": "```python\n# Real life data\n\nimport logging\nimport threading\nimport itertools\nimport pandas as pd \nimport numpy as np \nimport matplotlib.pyplot as plt\nfrom matplotlib import cm\nfrom mpl_toolkits.mplot3d import axes3d\nimport seaborn as seabornInstance\nfrom sqlalchemy import Column, Integer, String, Float, DateTime, Boolean, func\nfrom iotfunctions import base\nfrom iotfunctions import bif\nfrom iotfunctions import entity\nfrom iotfunctions import metadata\nfrom iotfunctions.metadata import EntityType\nfrom iotfunctions.db import Database\nfrom iotfunctions.enginelog import EngineLogging\nfrom iotfunctions import estimator\nfrom iotfunctions.ui import (UISingle, UIMultiItem, UIFunctionOutSingle,\n UISingleItem, UIFunctionOutMulti, UIMulti, UIExpression,\n UIText, UIStatusFlag, UIParameters)\nfrom mmfunctions.anomaly import (SaliencybasedGeneralizedAnomalyScore, SpectralAnomalyScore,\n FFTbasedGeneralizedAnomalyScore, KMeansAnomalyScore)\nimport datetime as dt\nfrom sklearn.model_selection import train_test_split \nfrom sklearn.linear_model import LinearRegression\nfrom sklearn import metrics\nfrom sklearn.metrics import roc_curve, auc, roc_auc_score, r2_score\n\nimport scipy as sp\nimport scipy.fftpack\nimport skimage as ski\n\nfrom skimage import util as skiutil # for nifty windowing\nimport pyod as pyod\nfrom pyod.utils.data import generate_data\nfrom pyod.utils.data import evaluate_print\nfrom pyod.utils.example import visualize\nfrom pyod.models.knn import KNN\nfrom pyod.models.iforest import IForest\n%matplotlib inline\nfrom pandas.plotting import register_matplotlib_converters\nregister_matplotlib_converters()\n\nEngineLogging.configure_console_logging(logging.INFO)\n```\n\n Warning: confluent_kafka is not installed. Publish to MessageHub not supported.\n /home/markus/.local/lib/python3.8/site-packages/iotfunctions/bif.py:1877: UserWarning: IoTCalcSettings is deprecated. Use entity type constants instead of a metadata provider to set entity type properties\n warnings.warn(('IoTCalcSettings is deprecated. Use entity type constants'\n\n\n\n```python\n# setting to make life easier\nTemperature='Temperature'\nkmeans='TemperatureKmeansScore'\nfft='TemperatureFFTScore'\nspectral='TemperatureSpectralScore'\nsal='SaliencyAnomalyScore'\ngen='TemperatureGeneralizedScore'\nkmeansA='kmeansAnomaly'\nkmeansB='kmeansAnomalyB'\nspectralA='spectralAnomaly'\nfftA='fftAnomaly'\nsalA='salAnomaly'\ngenA='genAnomaly'\n\n#kmeans_break=1.3\n#spectral_break = 2.8\n#fft_break = 100\n#sal_break = 100\n#gen_break = 30000\nkmeans_break=100\nspectral_break = 100\nfft_break = 100\nsal_break = 100\ngen_break = 30000\n\n```\n\n\n#### What will be shown\n\nGeneral approach is straightforward\n* read raw data in\n* transform it so that it is compatible to the Monitoring pipeline\n* add yet another anomaly detector based on computer vision technology. The point here is to show how to run pipeline anomaly functions 'locally', an important concept for automated testing.\n* simplify the dataframe - we have only one entity, no need for an entity index\n* render input data and anomaly scores properly scaled\n\n
                                        \n\nWe start with Microsoft's anomaly test data found here\nhttps://github.com/microsoft/anomalydetector/blob/master/samples/sample.csv\n\nand then proceed to applying anomaly detection to real life pump data\n\n\n
                                        \n\n\n#### Current inventory of anomaly detectors by type\n\nThis is the list of functions to apply\n\n\n| Detector | ML Type | Type | How does it work |\n| ------- | ------------ | ------- | ---------------- |\n| KMeans | Unsupervised | Proximity | Clusters data points in centroid buckets, small buckets are outliers, score is distance to closest other bucket |\n| Generalized | Unsupervised | Linear Model | Covariance matrix over data point vectors serves to measure multi-dimensional deviation |\n| FFT | Unsupervised | Linear Model | Run FFT before applying Generalized |\n| Spectral | Unsupervised | Linear Model | Compute signal energy to reduce dimensions |\n| Saliency | Unsupervised | Linear Model | Apply saliency transform (from computer vision |\n| SimpleAnomaly | **Supervised** | Ensemble | Run Gradient boosting on training data, anomaly if prediction deviates from actual data |\n| --- | **Supervised** | LSTM | Train a stacked LSTM, anomaly if prediction deviates from actual data |\n\n\n\n\n```python\nlistAttr = ['timestamp','entity','vibrations','rms','accel_speed','accel_power_0','accel_power_1',\n 'accel_power_2','accel_power_3','accel_power_4']\n\ndef unrollVibration(df_in):\n \n T = []\n Vx = []\n Vy = []\n Vz = []\n Ap = []\n As = []\n df_subset=df_in[['RCV_TIMESTAMP_UTC']].copy()\n \n df_subset['vibrations_xaxis'] = df_in['VIBRATIONS_XAXIS'].apply(eval)\n df_subset['vibrations_yaxis'] = df_in['VIBRATIONS_YAXIS'].apply(eval)\n df_subset['vibrations_zaxis'] = df_in['VIBRATIONS_ZAXIS'].apply(eval)\n df_subset['ap'] = df_in['ACCEL_POWER'].apply(eval)\n df_subset['as'] = df_in['ACCEL_SPEED'].apply(eval)\n\n np_subset = df_subset[['RCV_TIMESTAMP_UTC',\n 'vibrations_xaxis', 'vibrations_yaxis', 'vibrations_zaxis', 'ap', 'as']].values\n for row in np_subset:\n tim = np.datetime64(row[0])\n row0 = np.arange(tim, tim + np.timedelta64(15,'m'), step=np.timedelta64(1,'m'))\n #print (row0, row[1], row[2], row[3], row[4])\n for i in row[4]:\n j = eval(i)\n Ap.extend((j,j,j))\n for i in row[5]:\n j = eval(i)/1000\n As.extend((j,j,j))\n T.extend(row0.tolist())\n Vx.extend(row[1])\n Vy.extend(row[2])\n Vz.extend(row[3])\n \n\n #print(np.asarray(Vx))\n df_out = pd.DataFrame(data={'timestamp': np.asarray(T), 'Vx': np.asarray(Vx), 'Vy': np.asarray(Vy),\n 'Vz': np.asarray(Vz), 'Ap': np.asarray(Ap), 'As': np.asarray(As)})\n df_out['entity'] = df_in['DEVICE_ID']\n return df_out\n```\n\n\n```python\ndf_input_raw = pd.read_csv('./Armstark04714B6046D5.csv', index_col=False, parse_dates=['RCV_TIMESTAMP_UTC'])\n\ndf_input_raw = unrollVibration(df_input_raw)\n\n#df_input_raw.head(2)\ndf_input_raw.shape\n```\n\n\n\n\n (129300, 7)\n\n\n\n\n```python\n# Now we proceed to customer data - BAD CASE\n\n# Get stuff in\ndf_inputb_raw = pd.read_csv('./Armstark04714B604101.csv', index_col=False, parse_dates=['RCV_TIMESTAMP_UTC'])\n\ndf_inputb_raw = unrollVibration(df_inputb_raw)\ndf_inputb_raw.head(2)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        timestampVxVyVzApAsentity
                                        02020-01-01 00:02:12.713-0.0625-0.0908-1.03322.3651.03604714B604101
                                        12020-01-01 00:03:12.7130.03610.0078-0.91412.3651.03604714B604101
                                        \n
                                        \n\n\n\n\n```python\ndf_moreinput_raw = pd.read_csv('./ArmstarkMoreData.csv', index_col=False, parse_dates=['RCV_TIMESTAMP_UTC'])\ndf_moreinput_raw['entity'] = df_moreinput_raw['DEVICE_ID']\ndf_moreinput_raw['timestamp'] = df_moreinput_raw['RCV_TIMESTAMP_UTC']\ndf_moreinput_raw = df_moreinput_raw.drop(columns=['DEVICE_ID','RCV_TIMESTAMP_UTC','TIMESTAMP', 'UPDATED_UTC'])\n\ndf_moreinput_ra2 = df_moreinput_raw.set_index(['entity','timestamp']).dropna()\ndf_more = df_moreinput_ra2.loc['04714B6046D5'].copy()\ndf_moreb = df_moreinput_ra2.loc['04714B604101'].copy()\n\n```\n\n\n```python\nscal = np.arange(0, df_more['SPEED'].size, 1)\nlinear_interpolspeed = sp.interpolate.interp1d(\n scal, df_more['SPEED'].values, kind=\"linear\", fill_value=\"extrapolate\")\nscal2 = np.linspace(0, df_more['SPEED'].size, df_moreinput_ra2.shape[0])\ndf_input_raw['speed'] = linear_interpolspeed(scal2) / 1000\n\n\nscalb = np.arange(0, df_moreb['SPEED'].size, 1)\nlinear_interpolspeed = sp.interpolate.interp1d(\n scalb, df_moreb['SPEED'].values, kind=\"linear\", fill_value=\"extrapolate\")\nscalb2 = np.linspace(0, df_moreb['SPEED'].size, df_moreinputb_ra2.shape[0])\ndf_inputb_raw['speed'] = linear_interpolspeed(scalb2) / 1000\n```\n\n\n```python\ndf_input = df_input_raw.set_index(['entity', 'timestamp'])\ndf_inputb = df_inputb_raw.set_index(['entity', 'timestamp'])\n```\n\n#### Try out supervised methods\n\n* Train a stacked LSTM\n* Run gradient boosting\n\n\n\n```python\n# part of mmfunctions\nimport telemanom\nfrom telemanom.helpers import Config\nfrom telemanom.errors import Errors\nimport telemanom.helpers as helpers\nfrom telemanom.channel import Channel\nfrom telemanom.modeling import Model\n\nconf = Config(\"./telemanom/config.yaml\")\n\n#list_attr=['Vx','Vy','Vz']\nlist_attr=['Vx','speed'] # minimal\n\n#list_attr=['vibrations','accel_power_0']\nconf.dictionary['l_s'] = 250\nconf.dictionary['epochs'] = 80\nconf.dictionary['dropout'] = 0.2\nconf.l_s = 250\n# conf.epochs = 80\nconf.dropout = 0.2\nconf.lstm_batch_size=80\n```\n\n Using TensorFlow backend.\n\n\n\n```python\n\ntel_input = df_input[list_attr].values # n-dim numpy array - predict first value\ntel_input = abs(tel_input - tel_input.mean())\ntel_inputb = df_inputb[list_attr].values\ntel_inputb = abs(tel_inputb - tel_inputb.mean())\n#np.save(\"telemanom/data/train/Armstarknew.npy\", tel_input)\n#np.save(\"telemanom/data/test/Armstarknew.npy\", tel_inputb)\ntel_input.shape\n```\n\n\n```python\n# Load data from \ndevice=\"Armstarknew\"\nchan = Channel(conf, device)\nhelpers.make_dirs(conf.use_id, conf, \"./telemanom\")\nprint(chan)\nconf\n```\n\n \n Channel:Channel\n\n\n\n\n\n \n\n\n\n\n```python\nprint(chan.ffttrain)\n```\n\n None\n\n\n\n```python\n#chan.delete_data(\"./telemanom\")\nchan.config.FFT = False\nchan.load_data(\"./telemanom\")\n# chan.train\n#dfA = pd.DataFrame(chan.ffttrain)\ndfA = pd.DataFrame(chan.train)\ndfA.head(2)\n```\n\n 2020-07-23T18:45:41.566 INFO telemanom.shape_data FFT channel: False\n (129300, 2)\n 2020-07-23T18:45:42.105 INFO telemanom.shape_data FFT channel: False\n (129195, 2)\n\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        01
                                        00.5414430.538457
                                        10.5405430.538457
                                        \n
                                        \n\n\n\n\n```python\n# chan.train\n\nvibxg = df_input['Vx'].values\nvibxb = df_inputb['Vx'].values\naccpg = df_input['Ap'].values\naccpb = df_inputb['Ap'].values\naccsg = df_input['speed'].values\naccsb = df_inputb['speed'].values\n\nfig, ax = plt.subplots(4, 1, figsize=(20, 18))\ncnt = 0 \nax[cnt].plot(vibxg, color='blue', label='vibration x')\n#ax[cnt].plot(accpg/4-0.5, color='green', label='acc power')\nax[cnt].plot(accsg, color='red', label='pump speed')\nax[cnt].legend()\nax[cnt].set_title('Train (good) data')\n\ncnt += 1\nax[cnt].plot(vibxb, color='blue', label='vibration x')\nax[cnt].plot(accpb/4-0.5, color='green', label='acc power')\nax[cnt].plot(accsb, color='red', label='pump speed')\nax[cnt].set_title('Test (bad) data')\nax[cnt].legend()\n\n#cnt += 1\n#ax[cnt].plot((tr[:]-0), color='blue')\n#ax[cnt].set_title('Spectrum: Train (good) data')\n#cnt += 1\n#ax[cnt].plot((te[:]-0), color='blue')\n#ax[cnt].set_title('Spectrum: Test (bad) data')\n\nnoverlap = 8\nNFFT = 512\ncnt += 1\nPxxG, freqsG, binsG, imG = ax[cnt].specgram(chan.train[:,0],\n Fs=2, NFFT=NFFT, \n detrend='mean', noverlap=NFFT-noverlap, mode='psd')\ncnt += 1\nPxxB, freqsB, binsB, imB = ax[cnt].specgram(chan.test[:,0],\n Fs=2, NFFT=NFFT, \n detrend='mean', noverlap=NFFT-noverlap, mode='psd')\n\nplt.show()\n```\n\n\n```python\n# make sure to downgrade to sklearn 0.22.2 (no >= 0.23)\nfrom watson_machine_learning_client import WatsonMachineLearningAPIClient\n\nwml_credentials = {\n \"apikey\": \"B9UrvzboYPrHpIV_yDXTh4D3bMbE7NYuFlS46bmJalX_\",\n \"iam_apikey_description\": \"Auto-generated for key a9569793-f9e6-49c8-ace2-c3d210c4e2ce\",\n \"iam_apikey_name\": \"LocalTests\",\n \"iam_role_crn\": \"crn:v1:bluemix:public:iam::::serviceRole:Manager\",\n \"iam_serviceid_crn\": \"crn:v1:bluemix:public:iam-identity::a/f8971897be4e1464541021c645cd53fd::serviceid:ServiceId-a868a68f-79f6-4255-9945-379356c7249a\",\n \"instance_id\": \"8bd30bb4-2c05-440c-8e1e-6cefab74e2ee\",\n \"url\": \"https://eu-de.ml.cloud.ibm.com\"\n}\n\nclient = WatsonMachineLearningAPIClient(wml_credentials)\n\n```\n\n\n```python\nrep_list = client.runtimes.list(limit=4000)\n```\n\n -------------------------- -------------------------- ------------------------ --------\n GUID NAME CREATED PLATFORM\n do_12.10 do_12.10 2020-03-20T04:19:17.471Z do\n xgboost_0.90-py3.6 xgboost_0.90-py3.6 2020-03-20T04:19:03.205Z python\n scikit-learn_0.22-py3.6 scikit-learn_0.22-py3.6 2020-03-20T04:18:53.589Z python\n spark-mllib_2.4 spark-mllib_2.4 2020-02-06T09:30:35.538Z spark\n tensorflow_1.15-py3.6 tensorflow_1.15-py3.6 2020-02-06T09:30:30.574Z python\n pytorch-onnx_1.2-py3.6 pytorch-onnx_1.2-py3.6 2020-02-06T09:29:58.456Z python\n pytorch-onnx_1.2-py3.6-edt pytorch-onnx_1.2-py3.6-edt 2020-02-06T09:29:54.031Z python\n tensorflow_1.14-py3.6 tensorflow_1.14-py3.6 2019-10-23T09:54:32.847Z python\n pytorch-onnx_1.1-py3.6 pytorch-onnx_1.1-py3.6 2019-10-23T09:54:02.251Z python\n pytorch-onnx_1.1-py3.6-edt pytorch-onnx_1.1-py3.6-edt 2019-10-23T09:53:58.775Z python\n pytorch_1.1-py3.6 pytorch_1.1-py3.6 2019-10-23T09:53:56.218Z python\n pytorch_1.1-py3 pytorch_1.1-py3 2019-10-23T09:53:53.712Z python\n xgboost_0.82-py3.6 xgboost_0.82-py3.6 2019-08-02T06:02:55.531Z python\n xgboost_0.80-py3.6 xgboost_0.80-py3.6 2019-08-02T06:02:52.157Z python\n scikit-learn_0.20-py3.6 scikit-learn_0.20-py3.6 2019-08-02T06:02:45.108Z python\n scikit-learn_0.19-py3.6 scikit-learn_0.19-py3.6 2019-08-02T06:02:41.790Z python\n tensorflow_1.13-py3.6 tensorflow_1.13-py3.6 2019-08-02T06:02:34.144Z python\n tensorflow_1.11-py3.6 tensorflow_1.11-py3.6 2019-08-02T06:02:31.678Z python\n tensorflow_1.5-py3.6 tensorflow_1.5-py3.6 2019-08-02T06:02:12.855Z python\n ai-function_0.1-py3.6 ai-function_0.1-py3.6 2019-07-05T12:02:38.754Z python\n xgboost_0.82-py3 xgboost_0.82-py3 2019-07-05T12:02:33.447Z python\n spss-modeler_18.2 spss-modeler_18.2 2019-07-05T12:02:29.092Z spss\n scikit-learn_0.20-py3 scikit-learn_0.20-py3 2019-07-05T12:02:24.736Z python\n pmml_4.3 pmml_4.3 2019-04-15T09:59:34.202Z pmml\n pmml_4.2 pmml_4.2 2019-04-15T09:59:30.855Z pmml\n pmml_4.1 pmml_4.1 2019-04-15T09:59:28.401Z pmml\n pmml_4.0 pmml_4.0 2019-04-15T09:59:25.948Z pmml\n pmml_3.2 pmml_3.2 2019-04-15T09:59:23.480Z pmml\n pmml_3.1 pmml_3.1 2019-04-15T09:59:20.942Z pmml\n pmml_3.0 pmml_3.0 2019-04-15T09:59:18.466Z pmml\n do_12.9 do_12.9 2019-04-04T10:55:24.375Z do\n pmml_4.2.1 pmml_4.2.1 2019-04-04T10:55:21.561Z pmml\n ai-function_0.1-py3 ai-function_0.1-py3 2019-04-04T10:55:18.663Z python\n hybrid_0.2 hybrid_0.2 2019-04-04T10:55:15.872Z hybrid\n hybrid_0.1 hybrid_0.1 2019-04-04T10:55:13.073Z hybrid\n xgboost_0.80-py3 xgboost_0.80-py3 2019-04-04T10:55:10.280Z python\n xgboost_0.6-py3 xgboost_0.6-py3 2019-04-04T10:55:07.493Z python\n spss-modeler_18.1 spss-modeler_18.1 2019-04-04T10:55:04.706Z spss\n spss-modeler_17.1 spss-modeler_17.1 2019-04-04T10:55:01.919Z spss\n scikit-learn_0.19-py3 scikit-learn_0.19-py3 2019-04-04T10:54:59.060Z python\n scikit-learn_0.17-py3 scikit-learn_0.17-py3 2019-04-04T10:54:56.269Z python\n spark-mllib_2.3 spark-mllib_2.3 2019-04-04T10:54:53.422Z spark\n spark-mllib_2.2 spark-mllib_2.2 2019-04-04T10:54:50.637Z spark\n spark-mllib_2.1 spark-mllib_2.1 2019-04-04T10:54:47.851Z spark\n tensorflow_1.13-py3 tensorflow_1.13-py3 2019-04-04T10:54:45.039Z python\n tensorflow_1.13-py2 tensorflow_1.13-py2 2019-04-04T10:54:42.256Z python\n tensorflow_0.11-horovod tensorflow_0.11-horovod 2019-04-04T10:54:39.420Z native\n tensorflow_1.11-py3 tensorflow_1.11-py3 2019-04-04T10:54:36.622Z python\n tensorflow_1.10-py3 tensorflow_1.10-py3 2019-04-04T10:54:33.741Z python\n tensorflow_1.10-py2 tensorflow_1.10-py2 2019-04-04T10:54:30.915Z python\n tensorflow_1.9-py3 tensorflow_1.9-py3 2019-04-04T10:54:27.968Z python\n tensorflow_1.9-py2 tensorflow_1.9-py2 2019-04-04T10:54:25.170Z python\n tensorflow_1.8-py3 tensorflow_1.8-py3 2019-04-04T10:54:22.330Z python\n tensorflow_1.8-py2 tensorflow_1.8-py2 2019-04-04T10:54:19.534Z python\n tensorflow_1.7-py3 tensorflow_1.7-py3 2019-04-04T10:54:16.741Z python\n tensorflow_1.7-py2 tensorflow_1.7-py2 2019-04-04T10:54:13.963Z python\n tensorflow_1.6-py3 tensorflow_1.6-py3 2019-04-04T10:54:11.163Z python\n tensorflow_1.6-py2 tensorflow_1.6-py2 2019-04-04T10:54:08.371Z python\n tensorflow_1.5-py2-ddl tensorflow_1.5-py2-ddl 2019-04-04T10:54:05.592Z python\n tensorflow_1.5-py3-horovod tensorflow_1.5-py3-horovod 2019-04-04T10:54:02.749Z python\n tensorflow_1.5-py3 tensorflow_1.5-py3 2019-04-04T10:53:59.968Z python\n tensorflow_1.5-py2 tensorflow_1.5-py2 2019-04-04T10:53:57.099Z python\n tensorflow_1.4-py2-ddl tensorflow_1.4-py2-ddl 2019-04-04T10:53:54.318Z python\n tensorflow_1.4-py3-horovod tensorflow_1.4-py3-horovod 2019-04-04T10:53:51.540Z python\n tensorflow_1.4-py3 tensorflow_1.4-py3 2019-04-04T10:53:48.740Z python\n tensorflow_1.4-py2 tensorflow_1.4-py2 2019-04-04T10:53:45.962Z python\n tensorflow_1.3-py2-ddl tensorflow_1.3-py2-ddl 2019-04-04T10:53:43.176Z python\n tensorflow_1.3-py3 tensorflow_1.3-py3 2019-04-04T10:53:40.401Z python\n tensorflow_1.3-py2 tensorflow_1.3-py2 2019-04-04T10:53:37.610Z python\n tensorflow_1.2-py3 tensorflow_1.2-py3 2019-04-04T10:53:34.841Z python\n tensorflow_1.2-py2 tensorflow_1.2-py2 2019-04-04T10:53:32.032Z python\n pytorch-onnx_1.0-py3 pytorch-onnx_1.0-py3 2019-04-04T10:53:29.246Z python\n pytorch_1.0-py3-edt pytorch_1.0-py3-edt 2019-04-04T10:53:26.273Z python\n pytorch_1.0-py2-edt pytorch_1.0-py2-edt 2019-04-04T10:53:23.495Z python\n pytorch_1.0-py3 pytorch_1.0-py3 2019-04-04T10:53:20.707Z python\n pytorch_1.0-py2 pytorch_1.0-py2 2019-04-04T10:53:17.933Z python\n pytorch_0.4-py3-horovod pytorch_0.4-py3-horovod 2019-04-04T10:53:15.163Z python\n pytorch_0.4-py3 pytorch_0.4-py3 2019-04-04T10:53:12.392Z python\n pytorch_0.4-py2 pytorch_0.4-py2 2019-04-04T10:53:09.623Z python\n pytorch_0.3-py3 pytorch_0.3-py3 2019-04-04T10:53:06.785Z python\n pytorch_0.3-py2 pytorch_0.3-py2 2019-04-04T10:53:04.005Z python\n torch_lua52 torch_lua52 2019-04-04T10:53:01.226Z lua\n torch_luajit torch_luajit 2019-04-04T10:52:58.465Z lua\n caffe-ibm_1.0-py3 caffe-ibm_1.0-py3 2019-04-04T10:52:55.622Z python\n caffe-ibm_1.0-py2 caffe-ibm_1.0-py2 2019-04-04T10:52:52.862Z python\n caffe_1.0-py3 caffe_1.0-py3 2019-04-04T10:52:50.101Z python\n caffe_1.0-py2 caffe_1.0-py2 2019-04-04T10:52:47.261Z python\n caffe_frcnn caffe_frcnn 2019-04-04T10:52:44.492Z Python\n caffe_1.0-ddl caffe_1.0-ddl 2019-04-04T10:52:41.727Z native\n caffe2_0.8 caffe2_0.8 2019-04-04T10:52:38.971Z Python\n darknet_0 darknet_0 2019-04-04T10:52:36.028Z native\n theano_1.0 theano_1.0 2019-04-04T10:52:33.192Z Python\n mxnet_1.2-py2 mxnet_1.2-py2 2019-04-04T10:52:30.427Z python\n mxnet_1.1-py2 mxnet_1.1-py2 2019-04-04T10:52:27.598Z python\n -------------------------- -------------------------- ------------------------ --------\n\n\n\n```python\n\ndef kfp_wml_pipeline():\n GITHUB_TOKEN='ad7e5d3d34e79ac5d06210e74546c36b4bbc86ab',\n CONFIG_FILE_URL='https://raw.github.ibm.com/markus-mueller/Armstrong/master/creds.ini',\n train_code='tf-model.zip',\n execution_command='\\'python3 convolutional_network.py --trainImagesFile ${DATA_DIR}/train-images-idx3-ubyte.gz --trainLabelsFile ${DATA_DIR}/train-labels-idx1-ubyte.gz --testImagesFile ${DATA_DIR}/t10k-images-idx3-ubyte.gz --testLabelsFile ${DATA_DIR}/t10k-labels-idx1-ubyte.gz --learningRate 0.001 --trainingIters 20000\\'',\n framework='tensorflow',\n framework_version='1.15',\n runtime = 'python',\n runtime_version='3.6',\n run_definition = 'wml-tensorflow-definition',\n run_name = 'wml-tensorflow-run',\n model_name='wml-tensorflow-mnist',\n scoring_payload='tf-mnist-test-payload.json',\n compute_name='k80',\n compute_nodes='1'\n\n\nlib_meta = {\n client.runtimes.LibraryMetaNames.NAME: 'wml-keras-lstm',\n client.runtimes.LibraryMetaNames.VERSION: '1.15-py3.6',\n client.runtimes.LibraryMetaNames.FILEPATH: '/app/my-model.zip',\n client.runtimes.LibraryMetaNames.PLATFORM: {\"name\": 'wml-keras-lstm', \"versions\": ['1.15-py3.6']}\n}\nlib_details = client.runtimes.store_library(lib_meta)\n\n```\n\n\n```python\nprint lib_details\n```\n\n\n```python\n# producing overlapping windows of length 260 for lookback (250) and prediction (10)\nchan.shape_data(chan.train, train=True)\nchan.shape_data(chan.test, train=False)\n```\n\n 2020-07-23T18:46:01.826 INFO telemanom.shape_data FFT channel: False\n (129300, 2)\n 2020-07-23T18:46:02.367 INFO telemanom.shape_data FFT channel: False\n (129195, 2)\n\n\n\n```python\n# init the Keras double stacked LSTM model\nmodel = Model(conf, conf.use_id, chan, \"./telemanom\", False)\n```\n\n\n```python\nfrom keras.utils.vis_utils import plot_model\nplot_model(model.model, show_shapes=True, show_layer_names=True)\n```\n\n\n```python\n# drink a coffee - training takes roughly 30 minutes\nmodel.train_new(chan)\n```\n\n Train on 103232 samples, validate on 25808 samples\n Epoch 1/35\n 103232/103232 [==============================] - 425s 4ms/step - loss: 0.0041 - val_loss: 0.0014\n Epoch 2/35\n 103232/103232 [==============================] - 400s 4ms/step - loss: 0.0019 - val_loss: 0.0014\n Epoch 3/35\n 103232/103232 [==============================] - 402s 4ms/step - loss: 0.0016 - val_loss: 0.0014\n Epoch 4/35\n 103232/103232 [==============================] - 391s 4ms/step - loss: 0.0015 - val_loss: 0.0014\n\n\n#### Training parameters\n\n```\nloss_metric: 'mse' # minimize mean square error\noptimizer: 'adam' # sort of adaptive stochastic gradient descent\nvalidation_split: 0.2 # 20% of the data is used for validating (val_loss)\ndropout: 0.3 # ditch 30% of the LSTMs results when minimizing the loss function to avoid overfitting\nlstm_batch_size: 64 # number of training data batches to evaluate per optimizer run to update the model\u2019s parameters\n\npatience: 10 # try at least 10 times to decrease val_loss smaller by ...\nmin_delta: 0.0003 # ... at least min_delta, else stop, so we get at least 'patience' epochs\nepochs: 35 # no more than 35 passes through the entier training dataset.\n\nl_s: 250 # lookback: num previous timesteps provided to model to predict future values\nn_predictions: 10 # number of steps ahead to predict\n```\n\nThis is defined in `telemanom/config.yaml`\n
                                        \n\n\n```python\nprint (model.model)\n```\n\n\n```python\n# predicting takes roughly 12 secs\nmodel.batch_predict(chan, Path=\"./telemanom\")\n```\n\n\n\n\n \n\n\n\n\n```python\n# smooth the prediction error and apply exponential weights to it\nerrors = Errors(chan, conf, conf.use_id, \"./telemanom\")\n\n# for each overlapping window establish a threshold so that removing error points above it \n# maximizes the reduction of mean and standard deviation. Sort of an adaptive z-score \nerrors.process_batches(chan)\n```\n\n 2020-03-16T21:15:19.953 INFO telemanom.__init__ normalized prediction error: 0.06\n\n\n\n```python\nprint (errors.E_seq, \" \\n \", errors.anom_scores)\n```\n\n [(8650, 9049), (19850, 20249), (41150, 41649), (95450, 95849), (99650, 99749), (101750, 101849)] \n [{'start_idx': 8400, 'end_idx': 8499, 'score': 1.8819864334481131}, {'start_idx': 8500, 'end_idx': 8599, 'score': 2.4118219828938288}, {'start_idx': 8600, 'end_idx': 8699, 'score': 2.8350391123177623}, {'start_idx': 8700, 'end_idx': 8799, 'score': 3.129021297489508}, {'start_idx': 19600, 'end_idx': 19699, 'score': 6.1292143015223886}, {'start_idx': 19700, 'end_idx': 19799, 'score': 6.164415706483887}, {'start_idx': 19800, 'end_idx': 19899, 'score': 6.160624087183549}, {'start_idx': 19900, 'end_idx': 19999, 'score': 6.180099691069643}, {'start_idx': 40900, 'end_idx': 40999, 'score': 1.549790375651836}, {'start_idx': 41000, 'end_idx': 41099, 'score': 1.9420454097273365}, {'start_idx': 41100, 'end_idx': 41199, 'score': 2.3567182735433736}, {'start_idx': 41200, 'end_idx': 41299, 'score': 2.2956307295217417}, {'start_idx': 41300, 'end_idx': 41399, 'score': 2.3550736364688305}, {'start_idx': 95200, 'end_idx': 95299, 'score': 1.8424969956978121}, {'start_idx': 95300, 'end_idx': 95399, 'score': 2.3846389896135105}, {'start_idx': 95400, 'end_idx': 95499, 'score': 2.777478743103564}, {'start_idx': 95500, 'end_idx': 95599, 'score': 3.0776475065441784}, {'start_idx': 99400, 'end_idx': 99499, 'score': 4.81533268880221}, {'start_idx': 101500, 'end_idx': 101599, 'score': 2.4570177669361724}, {'start_idx': 106300, 'end_idx': 106300, 'score': 1.9960836911642124}]\n\n\n\n```python\nmodel.save(\"./telemanom\")\n```\n\n\n```python\n# How good are we doing ?\n\nmodel.model.evaluate(chan.X_test, chan.y_test)\n```\n\n 128935/128935 [==============================] - 163s 1ms/step\n\n\n\n\n\n 0.00590012299512361\n\n\n\n\n```python\nmodel.model.metrics_names\n```\n\n\n\n\n ['loss']\n\n\n\n\n```python\nmodel.batch_predict(chan, Path=\"./telemanom\", Train=False)\n\n```\n\n\n\n\n \n\n\n\nWe're seeing a prediction lag, maybe due to the nature of the timeseries data- see also this [article](https://stats.stackexchange.com/questions/280939/lag-between-predicted-output-and-real-output-in-time-series-prediction-directio).\n\nFrom this article:\n\nIf the data generating process is a random walk, \n\n$x_t = x_{t-1} + \\varepsilon_t$\n\nwith $\\varepsilon_t \\sim i.i.d.(0,\\sigma^2)$ the optimal* one-step-ahead prediction is \n\n\\begin{equation}\n\\hat x_{t+1|t}:=\\mathbb{E}(x_{t+1}|x_{t},x_{t-1},\\dots)=x_{t}\n\\end{equation}\n\nwhich happens to be the realization $x_{t+1}$ lagged by one ($x_{t}$ is $x_{t+1}$ lagged by 1). Given the data generating process, this apparently lagging prediction is the best we can get*. Then there is no way we can \"fix\" the \"problem of lagging predictions; the problem is built-in due to the nature of the data generating process.\n\n\n\n```python\narr = np.ones(chan.y_test.size)\nfig, ax = plt.subplots(2, 1, figsize=(20, 7))\n#ax[0].plot(chan.y_train_hat[:8200] * 10, lw=0.2, color='green') # to be done\nax[0].plot(chan.y_test[:,0], lw=1, color='blue', label='vibration')\nax[0].set_title('Vibration Forecast x-axis - actual')\n#ax[1].plot(chan.y_hat[:chan.y_hat.shape[0] // 2] * 100 - 113.4, lw=0.5, color='green')\nax[1].plot(chan.y_hat[:chan.y_hat.shape[0] // 2] * 1.5 - 0, lw=0.5, color='green')\nax[1].set_title('Vibration Forecast x-axis - predicted')\n```\n\n\n```python\nshort = chan.y_hat.shape[0] // 2\n\nnpic = 1\nfig, ax = plt.subplots(npic+1, 1, figsize=(20, (npic+1) * 5))\nax[npic-1].set_title('Anomaly detection - forecasting vibration')\nax[npic-1].plot(chan.y_test[:,0], lw=1, color='blue', label='vibration')\n\n#ax.scatter(x_axis, temp_sal_high, lw=8, color='red')\nfor asc in errors.anom_scores:\n x_axis = np.arange(asc['start_idx'],asc['end_idx'],1)\n y_axis = np.zeros(asc['end_idx'] - asc['start_idx'])\n ax[npic-1].grid(True, color='white')\n ax[npic-1].set_facecolor('lightgrey')\n ax[npic-1].scatter(x_axis,y_axis+1.5, lw=5, color='red', zorder=10)\nax[npic-1].grid(True, color='white')\nax[npic-1].set_facecolor('lightgrey')\n#ax[npic-1].plot(abs(chan.y_hat - chan.y_test[:,0]) + 3, lw=3, color='green', label='deviation')\n#ax[npic-1].plot((chan.y_hat[:short] - 0.4)*6, lw=2, color='darkgreen',label='prediction',zorder=5)\nax[npic-1].legend()\n \nax[npic].set_xlabel('Epoch')\nax[npic].set_ylabel('Loss')\nax[npic].set_title('Model training history')\nax[npic].plot(model.history.history['loss'])\nax[npic].plot(model.history.history['val_loss'],color='green')\nax[npic].grid(True, color='white')\nax[npic].set_facecolor('gainsboro')\nax[npic].legend(['Train', 'Validation'], loc='upper left')\n\n\nlstmscore = np.abs(abs(chan.y_hat[:short] - chan.y_test[:,1]))\nprint (lstmscore)\n#ax[1].plot(chan.test[:,0], lw=1, color='blue')\n#ax[1].set_xlabel('Date')\n#ax[1].set_ylabel('Compare with raw training data')\n\n```\n\n\n```python\n# ROC curve LSTM\n\nfprFg, tprFg, _ = roc_curve(yyy_test[0:yyy_test.size - conf.l_s - conf.n_predictions], lstmscore)\nroc_aucFg = auc(fprFg, tprFg)\n\nfig, ax = plt.subplots(1, 1, figsize=(7,5))\nax.plot(fprFg, tprFg, color='red', lw=2, label='ROC curve (area = %0.2f)' % roc_aucFg)\nax.plot([0, 1], [0, 1], color='navy', lw=1, linestyle='--')\nplt.xlim([0.0, 1.0])\nplt.ylim([0.0, 1.05])\nplt.xlabel('False Positive Rate\\nlower is better, higher means false alerts')\nplt.ylabel('True Positive Rate\\nhigher is better, we detect more anomalies')\nplt.title('ROC - LSTM', fontsize=14)\nplt.legend(loc=\"lower right\")\nplt.show()\n\n```\n\n\n```python\n# Now run gradient boosting - using lightGBM\n\nX_train = df_input[['accel_power_0','accel_power_1','accel_power_2',\n 'accel_power_3','accel_power_4']].to_numpy()\ny_train = df_input['vibrations'].to_numpy()\n```\n\n\n```python\nimport lightgbm\ngbr = lightgbm.LGBMRegressor(n_estimators=4000, learning_rate=0.000001, num_leaves=40,\n max_depth=20, random_state=42, loss='huber').fit(X_train, y_train)\n\n```\n\n\n```python\n\npred_good = gbr.predict(X_train)\nrmse = metrics.mean_squared_error(y_train, pred_good)\ngbscoreg = np.abs(pred_good - y_train)\nprint (rmse)\n```\n\n 0.001106535670881264\n\n\n\n```python\nX_bad = df_inputb[['accel_power_0','accel_power_1','accel_power_2',\n 'accel_power_3','accel_power_4']].to_numpy()\ny_bad = df_inputb['vibrations'].to_numpy()\npred_bad = gbr.predict(X_bad) \nrmseb = metrics.mean_squared_error(y_bad, pred_bad)\ngbscore = np.abs(pred_bad - y_bad)\nprint (rmseb)\n```\n\n 0.009797515412096496\n\n\n\n```python\nseparator = 0.2\nanomalygb = gbscore.copy() #(gbscore > separator) # * (separator + 0.1)\nanomalygb[anomalygb <= separator] = 0\nanomalygb[anomalygb > separator] = separator/2\nanomalygb[anomalygb == 0] = np.nan\nanomalygg = gbscoreg.copy()\nanomalygg[anomalygg <= separator] = 0\nanomalygg[anomalygg > separator] = separator/2\nanomalygg[anomalygg == 0] = np.nan\n\nfig, (ax1, ax2, ax3, ax4) = plt.subplots(4, figsize=(12,14)) \nax1.plot(y_bad, color='blue')\nax1.set_title('Bad Case - Input vibrations', fontsize=10)\nax1.set_ylabel('vibration')\nax2.plot(gbscore, color='green')\nax2.plot(anomalygb, color='red', lw=12, zorder=10)\nax2.set_ylabel('||actual-pred||')\nax2.set_title('Bad Case - deviating predictions', fontsize=10)\nax3.plot(y_train, color='blue')\nax3.set_ylabel('vibration')\nax3.set_title('Good Case - Input vibrations', fontsize=10)\nax4.plot(gbscoreg, color='green')\nax4.plot(anomalygg, color='red', lw=12, zorder=10)\nax4.set_ylabel('||actual-pred||')\nax4.set_title('Good Case - no anomalous deviation from prediction', fontsize=10)\n\n\n```\n\n\n```python\n# ROC curve Gradient Boosting\n\nfprFg, tprFg, _ = roc_curve(yyy_test, gbscore)\nroc_aucFg = auc(fprFg, tprFg)\n\nfig, ax = plt.subplots(1, 1, figsize=(7,5))\nax.plot(fprFg, tprFg, color='red', lw=2, label='ROC curve (area = %0.2f)' % roc_aucFg)\nax.plot([0, 1], [0, 1], color='navy', lw=1, linestyle='--')\nplt.xlim([0.0, 1.0])\nplt.ylim([0.0, 1.05])\nplt.xlabel('False Positive Rate\\nlower is better, higher means false alerts')\nplt.ylabel('True Positive Rate\\nhigher is better, we detect more anomalies')\nplt.title('ROC - gradient boosting', fontsize=14)\nplt.legend(loc=\"lower right\")\nplt.show()\n\n```\n\n#### Results\n\nGradient boosting appeared to do much better until Shradda fixed the ROC curve plot.\n\nNow I understand (and accept) why I had to explicitly disable the r2_score check in the pipeline's GBMRegressor to enforce saving the model to Cloud Object Store.\n\nTraining time is much shorter compared to the NASA model, but there is a price\n\n\n\n```python\nfrom sklearn.metrics import r2_score\n\n'''\nCoefficent of determinatoin: the proportion of the variance in the dependent variable \nthat is predictable from the independent variable(s). It provides a measure of how well \nobserved outcomes are replicated by the model, based on the proportion of total variation \nof outcomes explained by the model\nBest posible score = 1.0\nValues of r2 outside 0-1: model fits data worse than a horizontal hyperplane\n'''\n\nprint('R_sq/Test Variance score:' + str(r2_score(y_bad, pred_bad)))\n```\n\n R_sq/Test Variance score:-0.25058676111247435\n\n\n\n```python\nfrom sklearn.model_selection import cross_val_predict\n\nfig, ax = plt.subplots()\nax.scatter(y_bad, pred_bad, edgecolors=(0, 0, 0))\nax.plot([y_bad.min(), y_bad.max()], [y_bad.min(), y_bad.max()], 'k--', lw=4)\nax.set_xlabel('Actual')\nax.set_ylabel('Predicted')\nax.set_title(\"Ground Truth vs Predicted\")\nplt.show()\n```\n", "meta": {"hexsha": "6712e673666a4bd503819e74eb569a6ab521c0ac", "size": 489140, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "LSTMSelfPredictWithMoreData.ipynb", "max_stars_repo_name": "ankit-jha/mmfunctions", "max_stars_repo_head_hexsha": "fb1eae7f76863074370669628970f70c7272e8de", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2020-10-14T16:09:20.000Z", "max_stars_repo_stars_event_max_datetime": "2021-02-19T14:59:32.000Z", "max_issues_repo_path": "LSTMSelfPredictWithMoreData.ipynb", "max_issues_repo_name": "ankit-jha/mmfunctions", "max_issues_repo_head_hexsha": "fb1eae7f76863074370669628970f70c7272e8de", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2020-02-06T11:47:02.000Z", "max_issues_repo_issues_event_max_datetime": "2020-06-02T14:32:30.000Z", "max_forks_repo_path": "LSTMSelfPredictWithMoreData.ipynb", "max_forks_repo_name": "ankit-jha/mmfunctions", "max_forks_repo_head_hexsha": "fb1eae7f76863074370669628970f70c7272e8de", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2019-10-16T18:55:16.000Z", "max_forks_repo_forks_event_max_datetime": "2021-10-01T16:01:32.000Z", "avg_line_length": 323.5052910053, "max_line_length": 113880, "alphanum_fraction": 0.9108455657, "converted": true, "num_tokens": 11866, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.42191917927340367}} {"text": "# $p$-Value P\u2013P Plot Drawing In-depth Tutorial\n\n# About\n\nThis notebook presents how to generate a p-value P\u2013P plot based on subjective data.\nSpecifically, it shows how to use subjective responses—expressed on the 5-level Absolute\nCategory Rating (ACR) scale—in combination with the software provided in this repository\n(cf. [`friendly_gsd.py`](friendly_gsd.py)) to create the p-value P\u2013P plot. The plot can be used\nto assess consistency of the subjective data as a whole.\n\nOur recommendation is to use the method across a single experiment only. Differently\nput, if you have data from two subjective experiments, please use our software to\ngenerate two separate p-value P\u2013P plots, one for the first and one for the second\nexperiment.\n\nThis document also touches upon various aspects of using the software provided in this repository—how to\nrun it, what input does it require, what output it produces and how to use its batch processing functionality.\n\nThe content of this notebook is complementary to the following article. Please cite it if you make any use of this repository's content.\n\n```\n@inproceedings{Nawa\u0142a2020,\n author = {Nawa{\\l}a, Jakub and Janowski, Lucjan and {\\'{C}}miel, Bogdan and Rusek, Krzysztof},\n booktitle = {Proceedings of the 28th ACM International Conference on Multimedia (MM \u201920)},\n keywords = {experiment consistency,experiment validity,quality of experience,subjective experiment},\n title = {{Describing Subjective Experiment Consistency by p-Value P\u2013P Plot}},\n year = {2020}\n}\n```\n\nWe stronly encourage you to use the `jupyter_contrib_nbextensions` PIP package in order to display a floating table of contents for this document. This greatly simplifies its consumption. (The relevant nbextension is called \"Table of Contents (2)\" or \"toc2\"). \n\nThis notebook was written by Jakub Nawa\u0142a .\n\n# Overview\nSection [High-level Description](#High-level-Description) provides an introductory overview of the method we\nuse to draw $p$-value P–P plots.\n\nSection [Running the Software](#Running-the-Software) shows how to run our code. Sections [Output](#Output) and\n[Input](#Input) describe what the code outputs and what input information it expects, respectively.\n\nSection [How to Use Different Models](#How-to-Use-Different-Models) higlights what would be necessary in order\nto use different score distribution modelling approaches other than the one using the Generalized Score\nDistribution (GSD) [Janowski2019].\n\nSection [Batch Processing](#Batch-Processing) presents how to use our code's batch processing functionality.\n\nAt last, section [Step-by-step Description](#Step-by-step-Description) brings to light all the details you would\nneed to reproduce our workflow from the scratch.\n\n# High-level Description\n\nThe diagram below (Fig. 1) shows a high-level overview of the methodology. The diagram should be read starting from the top and going downwards. Below the diagram we describe its contents and highlight blocks that require special attention.\n\n\n**Figure 1:** The high-level diagram providing an overview of the methodology of creating $p$-value P\u2013P plot.\n\nWe start with subjective data (the *Subjective Data* block in the top left corner). These are responses of subjective experiment participants expressed on the 5-level ACR scale. Along with the model describing score distribution of each stimulus (the *Model* block) the responses go into the *MLE* block (the abbreviation stemming from Maximum Likelihood Estimation). This MLE block represents fitting the model to the observed responses. Not surprisingly, this results in the *Fitted Model* block. Importantly, the model is fitted on stimulus-by-stimulus basis. Differently put, if your experiment includes 20 stimuli the method generates 20 fitted models. Each model describes the score distribution of a single stimulus.\n\nHaving as many fitted models as there are stimuli in your experiment (the *Fitted Model* block) and the actually observed responses (the *Subjective Data* block) we can proceed to performing the G-test of goodness-of-fit (the *G-test of Goodness-of-Fit* block). It formally checks how well the fitted models describe the observed responses. This step is also performed as many times as there are stimuli in your experiment. Hence, instead of one, this step generates many $p$-values (the *p-Values* block). The resulting $p$-values go into the final *p-Value P\u2013P plot* block. This block represents plotting the target $p$-value P\u2013P plot. This last step depends on the desired significance level (the *Significance Level* block). This level influences the position of the theoretical threshold line (visualised as a solid black line). Importantly, the method yields one P\u2013P plot for one experiment. See Fig. 2 for an exemplary $p$-value P\u2013P plot of an inconsistent subjective experiment.\n\n\n**Figure 2:** An exemplary $p$-value P\u2013P plot of an inconsistent experiment.\n\nThe blocks coloured blue\ncorrespond to two places where you have to make a decision. The first decision is to choose\nthe model describing score distribution of a single stimulus (cf. the *Model* block).\nWe use the Generalised Score Distribution (GSD) here [Janowski2019] (and this is\nwhat is implemented in the [`friendly_gsd.py`](friendly_gsd.py) script), but you are free to choose\nany other model (e.g., the model proposed by Li and Bampis [Li2017] or the model proposed by Hossfeld et al. [Hossfeld2020]).\nThe second decision is to choose the significance level of hypothesis testing related\nwith drawing the target *p*-value P\u2013P plot (cf. the *Significance Level* block). This significance level defines the position\nof the theoretical black line (cf. Eq. (2) in [Nawa\u0142a2020]). We suggest to keep this\nlevel at 5% (and this value is hard-coded into our [`friendly_gsd.py`](friendly_gsd.py) script; cf. the\n*significance_line* variable in the *draw_p_value_qq_plot()* function).\n\n# Running the Software\nWe show here how to use our software to generate a $p$-value P\u2013P plot (and related G-test results) for exemplary subjective data. We choose to use here the subjective data from the first experiment of the VQEG HDTV Phase I subjective study [Pinson2010].\n\nAssuming the code is run from the root folder of this repository, the following line executed in the terminal\nresults in a $p$-value P\u2013P plot for the data contained in the `hdtv1_exp1_scores_pp_plot_ready.csv` file:\n```\npython3 friendly_gsd.py hdtv1_exp1_scores_pp_plot_ready.csv\n```\n\nExecuting the script this way we ask it to process all 168 stimuli from the first experiment of the HDTV Phase\nI test. **Please be adviced that this may take long time to finish**. Our internal trials on consumer-grade laptop (as of\n2020) indicate that it takes around 5 minutes to process a single stimulus. This is why the script supports\nbatch processing. More details on this subject are in [the Batch Processing section](#Batch-Processing).\n\nNext section describes the output produced by the script.\n\n# Output\nThe `friendly_gsd.py` script produces three types of output\n1. P\u2013P plot—either in the form of a floating figure (default behaviour) or as a file on disk (requires manual modification of the code—see the text following the list for an explanation what to do).\n2. G-test results—in the form of a CSV file. The filename encodes how many processing chunks were requested (see [the Batch Processing section](#Batch-Processing)) and results of which of these this file contains. When running the code the way [section Running the Software](#Running-the-Software) suggests this filename is as follows: `G-test_chunk_id_0_of_1_chunks.csv`. \n3. Logs—in the form of a \\*.log file. Stores logs of the last run of the script. When running the code the way [section Running the Software](#Running-the-Software) suggests this file takes the following name: `friendly_gsd_chunk_id_0_of_1_chunks.log`.\n\n## Store P–P Plot on the Disk\nTo store the resultant P–P plot to a file, please modify the following line (from the `main()` function of the `friendly_gsd.py` script).\n```python\npp_plot_fig_handle = draw_p_value_pp_plot(g_test_res)\n```\nAdd the `should_store_figure=True` argument to the `draw_p_value_pp_plot()` function. After the modification, the\ncode should look as follows.\n```python\npp_plot_fig_handle = draw_p_value_pp_plot(g_test_res, should_store_figure=True)\n```\n\n## G-test CSV File Formatting\nThe CSV file with G-test results has five columns: (i) *stimulus_id*, (ii) *psi_hat*, (iii) *rho_hat*, (iv) *T* and (v)\n*p_value*. The first column identifies a stimulus. The second and the third one provide values of the two parameters of\nthe GSD distribution (that were estimated through MLE based on the sample of scores related with a given\nstimulus). The *T* and *p_value* columns express results of the G-test of goodness-of-fit. This test says how well\nthe GSD distribution with the estimated parameters (*psi_hat* and *rho_hat* columns) fits the score distribution\nof a given stimulus—for more details see [the Step-by-step Description Section](#Step-by-step-Description).\n\nThere is a good chance that the only thing you are interested in is the content of *stimulus_id* and *p_value* columns (unless the GSD distribution is more interesting to you than P–P plots—in this case take a look at *psi_hat* and *rho_hat* columns).\n\n# Input\nThe code requires two inputs:\n1. A CSV file with subjective scores.\n2. The Pickle file with pre-calculated probabilities of each score for a range of psi and rho parameter values. (Psi and rho are the two parameters defining the shape of the GSD distribution.) This file is provided with the code (see the `gsd_prob_grid.pkl` file).\n\nThe input CSV file has to comply with the \"tidy\" data format [Wickham2014]. Only two columns are required: (i)\none identifying a stimulus and (ii) the other one expressing a score assigned to it. By default these columns\nshould be called *stimulus_id* and *score*, respectively. However, the naming convention can be changed using\n*--stimulus-identifier* and *--score-identifier* arguments of the `friendly_gsd.py` script.\n\nIntuitively, if your subjective experiment includes 15 stimuli and 20 participants (each scoring all stimuli) then\nyour input CSV file should have $15\\cdot20=300$ rows (+1 row with the headers). Each stimulus identifier is\nrepeated 20 times (since 20 participants rate each stimulus).\n\nThe [`hdtv1_exp1_scores_pp_plot_ready.csv`](hdtv1_exp1_scores_pp_plot_ready.csv) file can serve as a reference.\n\nPlease keep in mind that a CSV file with your subjective data can have more columns. In this situation, the script\nis going to use only the two columns described above.\n\n# How to Use Different Models\nIn case you would like to test our framework using different subjective score distribution models this section\nprovides some tips on how to do this.\n\nThere are few things you will have to take care of.\n1. **Provide a pickle file with a probability grid for the model of interest.** To see how this grid should be formatted please refer to the [gsd_prob_grid.pkl](gsd_prob_grid.pkl) file. In short, it should be a two-dimensional Pandas DataFrame with columns identifying a score category and rows identifying a particular value of model parameters. Since the GSD model has two parameters the dimension specifying their values is a MultiIndex. Differently put, each row of the DataFrame is indexed by two numbers: (i) value of the psi parameter and (ii) value of the rho parameter. (This is in contrast to most DataFrames that use single-value indexing on each axis.)\n2. **Implement access interface to the model of interest.** It should provide two functionalities: (i) the ability to generate the probability of observing each score category and (ii) the ability to generate a random sample. For an exemplary model access interface please take a look at [gsd.py](gsd.py).\n3. **Adapt the `friendly_gsd.perform_g_test()` function**. The current implementation assumes that the model of interest is parametrised by two parameters only. This may not be the case for the model of your choice.\n\nIn general, it is not straight forward to adapt our code to work with models significantly different than GSD. We\ntreat this as an important shortcoming of our implementation and plan to address this problem in any future\nimplementations.\n\nImportantly, we have not tested our framework with models taking into account individual traits of study\nparticipants (e.g., subject bias [Janowski2015]). (Note that our model can be estimated for a sample of scores assigned to a\nsingle stimulus. To find any individual trait of a study participant one has to analyse this participant's scores\nacross multiple stimuli. This complicates the analysis.) A model taking into account individual traits can,\nhowever, work with our framework provided its probability grid (see point 1. from the list above) can be\nmultiplied by a list of observed frequencies of score categories (see the related code fragment from the\n`probability_grid_estimation.estimate_parameters()` function) to yield expected probabilities of each score\ncategory.\n\n# Batch Processing\n(We recommend taking a look at the _Batch Processing_ section of the\n[README.md](README.md) file as it contains the most up-to-date version\nof the text below.)\n\nSince performing the bootstrapped version of the G-test of goodness-of-fit (GoF) is computationally-intensive we\ndesigned our code to be batch processing ready. Differently put, you can run multiple instances of the\n`friendly_gsd.py` script, each processing a separate part of your input data.\n\nBelow is an excerpt from `friendly_gsd.py`'s command-line help message.\n```shell\nusage: friendly_gsd.py [-h] [-p path] [-c N] [-i idx] [-s identifier]\n [-o identifier]\n data_filepath\n```\nIn terms of batch processing the `-c` and `-i` optional arguments are of our interest. Here are their help messages.\n```shell\n -c N, --chunks N (for batch processing) the number of chunks into which\n the input data should be split. It defaults to 1.\n -i idx, --index idx (for batch processing) 0-based index of a chunk to\n process. It default to 0.\n```\nUsing these you can start many instances of `friendly_gsd.py`, each with **the same** input data file, **the same** number\nof chunks (`-c`) and each with a **unique** index of a chunk to process (`-i`).\n\nFor example, if you have three machines (A, B and C), each with one CPU, it makes sense to start three instances of the\n`friendly_gsd.py` script—one on each of the machines. Now, assuming you have an exact copy of your input\ndata file (say, `my_input_data.csv`) on each of the machines, here is how you should start the `friendly_gsd.py`\nscript on each machine.\n\nMachine A\n```shell\npython3 friendly_gsd.py -c 3 -i 0 my_input_data.csv\n```\n\nMachine B\n```shell\npython3 friendly_gsd.py -c 3 -i 1 my_input_data.csv\n```\n\nMachine C\n```shell\npython3 friendly_gsd.py -c 3 -i 2 my_input_data.csv\n```\n\nPlease note how the value of the `-i` parameter is changing depending on the machine the script is run on.\n\nAfter all the computations are finished you end up with one CSV file with G-test results on each machine.\nSpecifically, on machine A you will get `G-test_chunk_id_0_of_3_chunks.csv`; on machine B:\n`G-test_chunk_id_1_of_3_chunks.csv` and on machine C: `G-test_chunk_id_2_of_3_chunks.csv`. These three files\n(when combined) contain G-test results for all stimuli present in the `my_input_data.csv` file.\n\n**NOTE**: Make sure the `friendly_gsd.py` script is set up **not** to interactively show *p*-value P–P plots \nwhen you use it for batch processing. To suppress interactive plot presentation either comment-out the call\nto the `friendly_gsd.draw_p_value_pp_plot` function or ask it to instead store the plots on the disk. The latter\ncan be achieved by calling the function with the `should_store_figure` keyword argument set to `True`.\n\n# Step-by-step Description\nThis part of the document looks at subsequent steps necessary to create a *p*-value P\u2013P plot. Use this if you would like to write your own implementation of the pipeline. Apart from describing each step we also give references to related code fragments from our `friendly_gsd.py` script.\n\n## Assumptions\nBefore we proceed to the steps let us first highlight assumptions we make:\n1. Data is processed stimulus-by-stimulus. Differently put, we treat as a single sample of data a set of subjective responses assigned to a single stimulus. For example, the usual data sample from a subjective experiment aiming at assessing image quality is a set of 24 responses (the number coming from an assumption about 24 people participating in the experiment) for each of the tested images (e.g., 300 images). In this case we can say there are 300 data samples, each with 24 observations.\n2. Subjective responses are expressed on the 5-level Absolute Category Rating (ACR) scale (cf. Sec. 6.1 of ITU-T Rec. P.910). (Optionally, the responses should be mappable to the 5-level scale. Importantly, we do not provide nor implement such a mapping.)\n3. A single data sample is represented as follows: $(n_1, n_2, n_3, n_4, n_5)$, where $n_k$ is the number of responses of category $k$. Significantly, $n = \\sum_{k=1}^5 n_k$ and denotes the total number of responses for a given stimulus.\n4. The null hypothesis is that the distribution of responses for a single sample of interest follows the assumed model (the GSD model in our case).\n\n## 1. Use MLE to Fit the Model to the Sample\nWe start with a data sample $(n_1, n_2, n_3, n_4, n_5)$ to which we fit the model.\nHaving fitted the model, the probability of each response category is as follows: $(p_1, p_2, p_3, p_4, p_5)$, where $p_k$ is the probability of a response of category $k$ (as given by the fitted model).\n\nThis functionalitiy is implemented in our `friendly_gsd.py` script in the `friendly_gsd.perform_g_test()` function. Specifically, the following two lines estimate the GSD model parameters ($\\psi$ and $\\rho$) given the data sample $(n_1, n_2, n_3, n_4, n_5)$ and map these to $(p_1, p_2, p_3, p_4, p_5)$ probabilities.\n```python\npsi_hat_gsd, rho_hat = estimate_parameters(sample_scores, prob_grid_gsd)\nexp_prob_gsd = gsd.prob(psi_hat_gsd, rho_hat)\n```\n\nImportantly, to make the computations faster we use a pre-calculated probability grid (cf. the `prob_grid_gsd` variable). It stores probabilities of all response categories for a range of GSD model parameters. For more details please see the \"GSD Parameters Estimation\" paragraph of [Nawa\u0142a2020].\n\n## 2. Calculate the T Statistic\nThe T statistic is the main building block of the G-test of goodness-of-fit (GoF). Since G-test is a likelihood ratio test, the T statistic is defined as a quotient of the likelihood without assumptions about a model describing the sample (this is also called the empirical distribution of the sample) divided by the likelihood when a certain model describing the sample is assumed (the GSD model in our case [Janowski2019]). Hence, the T statistic is calculated as follows:\n$$\nT = \\sum_{k: n_k \\neq 0} n_k \\ln \\left( \\frac{n_k}{n p_k} \\right)\n$$\n\nThis functionality is provided in the `friendly_gsd.perform_g_test()` function by the following line.\n```python\nT_statistic_gsd = bootstrap.T_statistic(score_counts, exp_prob_gsd)\n```\n\n## 3. Find the Bootstrap $p$-Value of the T Statistic\nSince the total number of responses $n$ for a given stimulus is relatively small in most subjective experiments instead of using the asymptotical distribution of the T statistic we approximate it using bootstrapping. Importantly, we need the distribution of the T statistic in order to calculate the $p$-value of the G-test of GoF.\n\n\n\n### 3.1 Generate Bootstrap Samples\nUsing the probability distribution given by $(p_1, p_2, p_3, p_4, p_5)$ generate $R$ bootstrap samples. In our implementation we use $R=10000$. The higher the $R$ the greater the precision of the $p$-value. Let us denote the $r$-th bootstrap sample as $(m_1, m_2, m_3, m_4, m_5$), where $m_k$ is the number of responses of category $k$. Importantly, each sample should have $n$ responses (the same number as the original, truly observed sample). This last condition can be also formulated as $n = \\sum_{k=1}^5 m_k$.\n\nThe lines below from the `friendly_gsd.perform_g_test()` function are responsible for generating 10,000 bootstrap samples (the `n_bootstrap_samples` variable has its value set to 10,000).\n```python\nn_total_scores = np.sum(score_counts)\nn_bootstrap_samples = n_bootstrap_samples\nbootstrap_samples_gsd = gsd.sample(psi_hat_gsd, rho_hat, n_total_scores, n_bootstrap_samples)\n```\n\n### 3.2 Use MLE to fit the Model to Each Bootstrap Sample\nHaving fitted the model, the probability of each response category is as follows: $(q_1, q_2, q_3, q_4, q_5)$, where $q_k$ is the probability of a response of category $k$ (as given by the fitted model). Please keep in mind that this has to be done for each bootstrap sample (and there are $R$ of these).\n\nThe related functionality is provided by the following lines from the `friendly_gsd.perform_g_test()` function.\n```python\npsi_hat_rho_hat_gsd_bootstrap = np.apply_along_axis(estimate_parameters, axis=1, arr=bootstrap_samples_gsd,\n prob_grid_df=prob_grid_gsd, sample_as_counts=True)\n\ndef _get_each_answer_probability(psi_rho_row, prob_generator):\n \"\"\"\n Translates psi and rho parameters into the probability of each answer.\n\n :param psi_rho_row: a 2-column vector with the first col. corresponding to psi and the second one to rho\n :param prob_generator: gsd.prob\n :return: a vector of probabilities of each answer\n \"\"\"\n psi = psi_rho_row[0]\n rho = psi_rho_row[1]\n return prob_generator(psi, rho)\n\nbootstrap_exp_prob_gsd = np.apply_along_axis(_get_each_answer_probability, axis=1,\n arr=psi_hat_rho_hat_gsd_bootstrap, prob_generator=gsd.prob)\n```\n\n### 3.3 Calculate the T statistic for Each Bootstrap Sample\nProceeding similarly to what was shown in Step 2. we calculate the T statistic for each $r$-th bootstrap sample as follows:\n$$\nT_r = \\sum_{k:m_k \\neq 0} m_k \\ln \\left( \\frac{m_k}{n q_k} \\right).\n$$\n\nHaving done that we should have $R$ $T_r$ values (in our case 10,000 $T_r$ values).\n\nThis step (and next one) is performed by the following code fragment from the `friendly_gsd.perform_g_test()` function.\n```python\np_value_g_test_gsd = bootstrap.G_test(score_counts, exp_prob_gsd, bootstrap_samples_gsd,\n bootstrap_exp_prob_gsd)\n```\n\n**WARNING**: Our code in the `bootstrap.G_test()` function assumes that the GSD model is used. Thus, if there is only one non-zero response category in $(n_1, n_2, n_3, n_4, n_5)$ or only two neighbouring response categories are non-zero, the code skips the $p$-value computations and sets it to 1.0. (By \"non-zero\" we mean that there is at least one reponse for a given category.) This behaviour comes from our theoretical analysis of the GSD model. We know for sure that the model completely represents all score distributions with only one or only two neighbouring non-zero response categories. **The takeaway is that you need to modify our code if you are planning on using model different than GSD.** Our recommendation is to simply remove the following lines from the `bootstrap.G_test()` function.\n```python\nn_non_zero_cells = (n != 0).sum()\nif n_non_zero_cells == 1:\n return 1.0\n\n# Return a p-value of 1.0 only if exactly any two NEIGHBOURING cells are non-zero\nif n_non_zero_cells == 2:\n # Find indices of the top 2 elements\n top_two_idx = np.argpartition(n, -2)[-2:]\n idx_diff = np.abs(top_two_idx[0] - top_two_idx[1])\n # Only if the top 2 elements are neighbours, return 1.0\n if idx_diff == 1:\n return 1.0\n```\n\n### 3.4 Find the Bootstrap $p$-Value\nSort all the $T_r$ values in the ascending order and calculate the bootstrap $p$-value as the number of $T_r$ values that are greater or equal to $T$, divided by the total number $R$ of $T_r$ values. Diffrently put:\n$$\np\\mbox{-value} = \\frac{\\# \\left( T_r \\geq T \\right)}{R},\n$$\nwhere $\\#(\\mbox{condition})$ is the number of cases in which the condition is met.\n\nFor the related code fragment from our `friendly_gsd.py` script please take a look at Step 3.3.\n\n## 4. Create $p$-Value P\u2013P plot\nIn order to draw a P\u2013P plot the steps 1. to 3. have to be repeated for each stimuli in the experiment. To be more specific, if your experiment contains 100 stimuli (e.g., images), then steps 1. to 3. have to be repeated 100 times. After all the repetitions, we get a vector of G-test of GoF $p$-values. This vector, along with the assumed significance level, is sufficient to draw the P\u2013P plot.\n\nThe very procedure of creating the plot is described by the three following sub-steps.\n\n### 4.1 Check Fraction of Stimuli with $\\leq$ $p$-Value\nFor each $p$-value check what fraction of all stimuli have their $p$-value lower or equal to the $p$-value currently being processed. These fractions constitute $y$-axis values of the red dots visible in Fig. 2. (We also refer to this fraction as $\\hat{\\alpha}$ in the next step.) Correspondingly, $x$-axis values of the dots are the actual $p$-values of each stimulus (as provided by Step 3.4).\n\nPlease note that the way we compute $y$-axis values corresponds to finding empirical cumulative distribution function (ECDF) of the observations. This is why the $y$-axis of the plot in Fig. 2 is labelled as \"ecdf of $p$-values.\"\n\nFor an explanation why the $x$-axis is labelled as \"theoretical uniform cdf\" (although it is used to indicate the actual $p$-values of each stimulus) please refer to Sec. 4 of [Nawa\u0142a2020].\n\nThis functionalitiy is provided by the following lines from the `friendly_gsd.draw_p_value_pp_plot()` function.\n```python\ndef count_pvs_fraction(p_value, p_value_per_pvs):\n return np.sum(p_value_per_pvs <= p_value) / len(p_value_per_pvs)\n\npvs_fraction_gsd = g_test_res[\"p_value\"].apply(count_pvs_fraction, args=(g_test_res[\"p_value\"],))\n```\n\n### 4.2 Draw the Threshold Line\nTo understand what the threshold line means and where does the significance level come from let us first describe the threshold line drawing procedure for a single $p$-value.\n\nLet us pick $p$-value $= 0.2$. (This is an outcome of Step 3.4 for a singlue stimulus.) Let us label this value as $\\alpha=0.2$. Now, we want to know how many stimuli in the whole experiment have their $p$-value below $\\alpha$. We take each stimulus and compare its $p$-value with $\\alpha$ (cf. Step 4.1 above). If its $p$-value is lower or equal to $\\alpha$ we assign the stimulus the value of 1. If, on the other hand, its $p$-value is higher than $\\alpha$, we assign it the value of 0. \nWe mark the ratio of stimuli assigned 1 to the total number of stimuli as $\\hat{\\alpha}$.\nThe meaning of 1 is that a stimulus has low $p$-value with respect to the G-test of goodness-of-fit (GoF). This means the assumed model poorly describes the score distribution of that stimulus (we treat this as a bad outcome). Conversely, the value of 0 means high $p$-value of the G-test of GoF. This corresponds to the assumed model describing the score distribution of the stimulus sufficiently well (we treat this as a good outcome).\n\nHaving assigned 1 or 0 to all stimuli we have a vector of 1s and 0s. We can treat this as a random sample from the Bernoulli distribution. Now, if we want to analyse the number of 1s (successes) in such a sample we can observe that this number follows the binomial distribution. This distribution has two parameters conventionally labeled as $p$ and $n$. The first one ($p$) corresponds to the probability of success in each trial. The second one ($n$) defines the number of trials. (If your experiment has 100 stimuli then $n=100$.) Now, we can construct a hypothesis test about $p$ [Siegrist2019]. Staying with $\\alpha = 0.2$ we would like to test the following null hypothesis $H_0: p \\leq \\alpha$ versus the alternative $H_1: p > \\alpha$. \n\nWe reject $H_0$ if the following inequality is true:\n$$\n\\begin{align}\n\\hat{\\alpha} > \\alpha + z_{1-\\beta} \\sqrt{\\frac{\\alpha(1 - \\alpha)}{n}}, \\label{eq:hyp_test} \\tag{1}\n\\end{align}\n$$\nwhere $\\beta$ is the significance level of the test and $z_\\gamma$ denotes the quantile of order $\\gamma$ for the standard normal distribution.\n\nThe aforementioned test can either reject or fail to reject $H_0$. How does this then relates to $p$-value P\u2013P plot? If we repeat the procedure for each stimulus in the experiment, each time assuming $\\alpha$ equal to stimulus' $p$-value, we get as many hypothesis tests as there are stimuli. To graphically represent Eq.\u00a0\\eqref{eq:hyp_test} we can evaluate its RHS for the complete range of $\\alpha$ values (from 0 to 1). We can then plot the result of this evaluation for each value of $\\alpha$. This is the threshold line depicted as the solid black line in Fig. 2. Whenever a given stimulus satisfies Eq.\u00a0\\eqref{eq:hyp_test} its related $\\hat{\\alpha}$ has its value above the treshold line. Simply put, the data point on $p$-value P\u2013P plot related with this stimulus (depicted as red dot) lands above the solid black line.\n\nTo link this procedure with practical applications we point out that if many stimuli have data points falling above the threshold line then the experiment in question is most probably inconsistent.\n\nImportantly, even though we mention finding the theoretical line for the range of $\\alpha$ values from 0 to 1 (and even though there naturally occur stimuli with their $p$-value above 0.2), the target P\u2013P plot should only depict the $x$-axis range from 0 to 0.2. This is because this range of $\\alpha$ values is critical for drawing conclusions. In other words, we are more interested in reasoning related with low $p$-values from the G-test of GoF than we are in making conclusions regarding high $p$-value stimuli.\n\nThe following lines from the `friendly_gsd.draw_p_value_pp_plot()` function are responsible for generating the theoretical line.\n```python\nn_pvs = len(g_test_res)\np_values = pd.Series(np.linspace(start=0.001, stop=thresh_pvalue, num=100))\nsignificance_line = p_values + norm.ppf(0.95) * np.sqrt(p_values * (1 - p_values) / n_pvs)\n```\n\n### 4.3 Draw the Plot\nHaving the $\\hat{\\alpha}$ values for all the stimuli and the threshold line we can finally draw the $p$-value P\u2013P plot. The $y$-axis corresponds to the $\\hat{\\alpha}$ values and the $x$-axis to the related $p$-values (from the G-test of GoF). Furthermore, the $y$-axis is also used to express the threshold line (given by the RHS of Eq. (1)). An exemplary $p$-value P\u2013P plot is provided in Fig. 2.\n\nThe lines below from the `friendly_gsd.draw_p_value_pp_plot()` function are responsible for drawing the P\u2013P plot.\n```python\nfig = plt.figure()\nplt.scatter(g_test_res[\"p_value\"], pvs_fraction_gsd, label=\"GSD\")\nplt.xlabel(\"theoretical uniform cdf\")\nplt.ylabel(\"ecdf of $p$-values\")\nplt.plot(p_values, significance_line, \"-k\")\nplt.xlim([0, thresh_pvalue])\nplt.ylim([0, thresh_pvalue + 0.1])\nplt.minorticks_on()\nplt.show()\n```\n\n# Bibliography\n[Janowski2015] Janowski, L., & Pinson, M. (2015). The Accuracy of Subjects in a Quality Experiment: A Theoretical Subject Model. IEEE Transactions on Multimedia, 17(12), 2210\u20132224. https://doi.org/10.1109/TMM.2015.2484963\n\n[Wickham2014] Wickham, H. (2014). Tidy Data. Journal of Statistical Software, 59(10), 1--23. https://doi.org/10.18637/jss.v059.i10\n\n[Pinson2010] Pinson, M., Speranza, F., Takahashi, A., Schmidmer, C., Lee, C., Okamoto, J., \u2026 Dhondt, Y. (2010). Report on the Validation of Video Quality Models for High Definition Video Content (HDTV Phase I). Retrieved from https://www.its.bldrdoc.gov/vqeg/projects/hdtv/hdtv.aspx\n\n[Hossfeld2020] Hossfeld, T., Heegaard, P. E., Varela, M., Skorin-Kapov, L., & Fiedler, M. (2020). From QoS Distributions to QoE Distributions: a System\u2019s Perspective. 4th International Workshop on Quality of Experience Management (QoE Management 2020), Featured by IEEE Conference on Network Softwarization (IEEE NetSoft 2020), Ghent, Belgium, 1\u20137. Retrieved from http://arxiv.org/abs/2003.12742\n\n[Siegrist2019] Siegrist, K. (2019). Tests in the Bernoulli Model. Retrieved from http://www.randomservices.org/random/hypothesis/Bernoulli.html\n\n[Nawa\u0142a2020] Nawa\u0142a, J., Janowski, L., \u0106miel, B., & Rusek, K. (2020). Describing Subjective Experiment Consistency by p-Value P\u2013P Plot. Proceedings of the 28th ACM International Conference on Multimedia (MM \u201920).\n\n[Janowski2019] Janowski, L., \u0106miel, B., Rusek, K., Nawa\u0142a, J., & Li, Z. (2019). Generalized Score Distribution. Retrieved from http://arxiv.org/abs/1909.04369\n\n[Li2017] Li, Z., & Bampis, C. G. (2017). Recover Subjective Quality Scores from Noisy Measurements. Data Compression Conference Proceedings, 52\u201361. https://doi.org/10.1109/DCC.2017.26\n", "meta": {"hexsha": "6cbb47a659a5e3c048a6e671fd09fe13af1ff463", "size": 39767, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "p_value_pp_plot_in-depth_tutorial.ipynb", "max_stars_repo_name": "Qub3k/subjective-exp-consistency-check", "max_stars_repo_head_hexsha": "ad159e9ed161e7f04016cc053d90b8e20f6963ed", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-02-17T21:23:06.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-17T21:23:06.000Z", "max_issues_repo_path": "p_value_pp_plot_in-depth_tutorial.ipynb", "max_issues_repo_name": "Qub3k/subjective-exp-consistency-check", "max_issues_repo_head_hexsha": "ad159e9ed161e7f04016cc053d90b8e20f6963ed", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2021-03-15T11:16:37.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-12T00:48:07.000Z", "max_forks_repo_path": "p_value_pp_plot_in-depth_tutorial.ipynb", "max_forks_repo_name": "Qub3k/subjective-exp-consistency-check", "max_forks_repo_head_hexsha": "ad159e9ed161e7f04016cc053d90b8e20f6963ed", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 65.085106383, "max_line_length": 992, "alphanum_fraction": 0.6823999799, "converted": true, "num_tokens": 8303, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.6442251201477016, "lm_q1q2_score": 0.4218996355012611}} {"text": "# How should the hidden states: vel & acc be treated?\n\n## Purpose\n* The velocity and accelerations are not measured in model test usually they are \"hidden states\" that needs to be estimated for the regression\n\n## Methodology\n* Load simulated data generated by: [12.05_regression_simulated_data_simple_nonlinear.ipynb](12.05_regression_simulated_data_simple_nonlinear.ipynb)\n* Determine velocity and acceleration and compared...\n\n## Setup\n\n\n```python\n# %load imports.py\n## Local packages:\n\n%matplotlib inline\n%load_ext autoreload\n%autoreload 2\n%config Completer.use_jedi = False ## (To fix autocomplete)\n\n## External packages:\nimport pandas as pd\npd.options.display.max_rows = 999\npd.options.display.max_columns = 999\npd.set_option(\"display.max_columns\", None)\n\nimport numpy as np\nimport os\nimport matplotlib.pyplot as plt\n#if os.name == 'nt':\n# plt.style.use('presentation.mplstyle') # Windows\n\nimport plotly.express as px \nimport plotly.graph_objects as go\n\nimport seaborn as sns\nimport sympy as sp\nfrom sympy.physics.mechanics import (dynamicsymbols, ReferenceFrame,\n Particle, Point)\nfrom sympy.physics.vector.printing import vpprint, vlatex\nfrom IPython.display import display, Math, Latex\nfrom src.substitute_dynamic_symbols import run, lambdify\n\nimport pyro\n\nimport sklearn\nimport pykalman\nfrom statsmodels.sandbox.regression.predstd import wls_prediction_std\nimport statsmodels.api as sm\n\nfrom scipy.integrate import solve_ivp\n\n## Local packages:\nfrom src.data import mdl\n\nfrom src.symbols import *\nfrom src.parameters import *\nimport src.symbols as symbols\nfrom src import prime_system\nfrom src.models import regression\nfrom src.visualization.regression import show_pred\nfrom src.visualization.plot import track_plot\n\n## Load models:\n# (Uncomment these for faster loading):\nimport src.models.vmm_simple_nonlinear as vmm\nfrom src.data.case_1 import ship_parameters, df_parameters, ps, ship_parameters_prime\nfrom src.data.transform import transform_to_ship\n```\n\n## Ship parameters\n\n\n```python\nship_parameters\n```\n\n## Brix parameters\n\n\n```python\nmask = df_parameters['prime'].notnull()\nindex = df_parameters.loc[mask,'prime'].index\ncoefficients=vmm.simulator.get_all_coefficients(sympy_symbols=False)\nmissing_coefficients = set(coefficients) - set(index)\nmissing_coefficients\n```\n\n\n```python\nmask = df_parameters['prime'].notnull()\ndf_parameters.loc[mask,'prime']\n```\n\n## Load simulate data\n\n\n```python\ndf_result = pd.read_csv('../data/processed/simple_simulation.csv', index_col=0)\ndf_result['z0']=0\ndf_measurement = df_result.drop(columns=['u','v','r','u1d','v1d','r1d']) # Removing vel and acc\n```\n\n\n```python\nship_parameters['x_G']\n```\n\n### Check accelerations\n\n\n```python\nimport scipy.integrate\n```\n\n\n```python\nu_integrated = df_result.iloc[0]['u'] + scipy.integrate.cumtrapz(y=df_result['u1d'], \n x=df_result.index)\nfig,ax=plt.subplots()\ndf_result.plot(y='u', ax=ax)\nax.plot(df_result.index[1:], u_integrated, '--', label='u_integrated')\nax.legend();\n```\n\n### Check transformation\n\n\n```python\nt_ = df_measurement.index\n\nsuffix = ['','1d','2d']\nfor i in range(2):\n for key in ['x0','y0','z0','psi']:\n df_measurement[f'{key}{suffix[i+1]}'] = np.gradient(df_measurement[f'{key}{suffix[i]}'], t_) \n \ndf_measurement = transform_to_ship(df=df_measurement)\ndf_measurement=df_measurement.iloc[2:-2].copy()\n```\n\n\n```python\nfrom pykalman import KalmanFilter\n\ndef filter(df, key='x0', observation_covariance = 100000):\n \n df = df.copy()\n t = df.index\n dt = t[1] - t[0]\n \n A = np.array([[1, dt, 0.5*(dt**2)],\n [0, 1, dt ],\n [0, 0, 1 ],\n ])\n \n kf = KalmanFilter(transition_matrices=A,\n initial_state_mean = [df[key].iloc[0:100].median(),df[f'{key}1d'].iloc[0:100].median(),df[f'{key}2d'].iloc[0:100].median()],\n random_state=np.random.RandomState(0),\n transition_covariance=100 * np.eye(3),\n observation_covariance=observation_covariance * np.eye(1),\n \n em_vars=[\n 'transition_covariance', \n 'observation_covariance', \n 'initial_state_mean', \n 'initial_state_covariance'\n ],\n \n )\n \n observations = df[key]\n states_pred = kf.em(observations).smooth(observations)[0]\n \n df[f'{key}'] = states_pred[:,0]\n df[f'{key}1d'] = states_pred[:,1]\n df[f'{key}2d'] = states_pred[:,2]\n \n return df\n```\n\n\n```python\ndf_measurement_filtered = df_measurement.copy()\n\ndf_measurement_filtered = filter(df=df_measurement_filtered, key='x0', observation_covariance = 100000)\ndf_measurement_filtered = filter(df=df_measurement_filtered, key='y0', observation_covariance = 100000)\ndf_measurement_filtered = filter(df=df_measurement_filtered, key='psi', observation_covariance = 100000)\n\ndf_measurement_filtered.head()\n```\n\n### Check Kalman filter\n\n\n```python\nfor key in ['x0', 'y0', 'z0', 'x01d', 'x02d', 'y01d', 'y02d', 'psi1d','psi2d']:\n fig,ax=plt.subplots()\n fig.set_size_inches(15,1)\n \n df_measurement.plot(y=key, ax=ax, label='simulation')\n df_measurement_filtered.plot(y=key, ax=ax, label='kalman filtered')\n ax.set_ylabel(key)\n```\n\n\n```python\nfor i in range(2):\n for key in ['u','v','r']:\n y = f'{key}{suffix[i]}'\n \n fig,ax=plt.subplots()\n df_result.plot(y=y,label='sim', ax=ax)\n df_measurement.plot(y=y,label='measurement', style='--', ax=ax)\n ax.set_ylabel(y)\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "ba0042b5e4d6cd702a14d5f4090ae60535a0c814", "size": 9721, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/14.01_hidden_states.ipynb", "max_stars_repo_name": "martinlarsalbert/wPCC", "max_stars_repo_head_hexsha": "16e0d4cc850d503247916c9f5bd9f0ddb07f8930", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/14.01_hidden_states.ipynb", "max_issues_repo_name": "martinlarsalbert/wPCC", "max_issues_repo_head_hexsha": "16e0d4cc850d503247916c9f5bd9f0ddb07f8930", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/14.01_hidden_states.ipynb", "max_forks_repo_name": "martinlarsalbert/wPCC", "max_forks_repo_head_hexsha": "16e0d4cc850d503247916c9f5bd9f0ddb07f8930", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 27.3830985915, "max_line_length": 157, "alphanum_fraction": 0.5299866269, "converted": true, "num_tokens": 1418, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.6584175005616829, "lm_q1q2_score": 0.4218058673488681}} {"text": "# Liver analysis\n\n**Author**: Alma Andersson
                                        \n**Description**:
                                        \n\nJoint spatial analysis of the entire dataset. The analysis is separated into three parts\n\n- Data Loading and processing\n- Feature by distance analysis\n- Classification of structures\n\n### Data loading and processing\nA brief section loading and processing the data.\n\n\n### Feature by distance analysis\nHere, expression as a function of distance (_feature by distance_) is exemplified, as well as the slightly modified variant where the log-ratio between distances to two different structure classes (here portal and central veins) is examined.\n\n\n### Classification\nThis section illustrates how the type of a vein can be predicted based on its neighborhood expression profile (NEP). Included in the analysis is the creation of NEPs, a two step procedure where a neighborhood is first identified, from which a weighted gene expression (by distance) is then assembled. Once the NEPs are formed, a _logistic regression_ model is trained to predict vein type based on the NEP. Cross validation is also implemented to assess performance.\n\n\n
                                        \n\n### Load libraries\n\nLoad data the necessary packages and data for the analysis, also specify certain constants which will be used throughout the analysis.\n\n\n```python\n%load_ext autoreload\n%autoreload 2\n\n```\n\n\n```python\nimport pandas as pd\nimport numpy as np\nimport anndata as ad\n\nfrom skmisc.loess import loess\n\nimport os\nimport os.path as osp\nfrom os import listdir\n\nimport matplotlib.pyplot as plt\nfrom matplotlib import rcParams\n\nfrom PIL import Image\n\nfrom functools import reduce\n\nfrom scipy.spatial.distance import cdist\n\nfrom hepaquery.structures import VeinData, Model\nimport hepaquery.visual as viz\nimport hepaquery.utils as ut\n\nimport re\nimport datetime\n\nfrom typing import Tuple\n\n```\n\n\n```python\nrcParams[\"figure.facecolor\"] = \"white\"\n```\n\n### Global Variables\n\nHere certain global variables are set, which will be used to save output and load data. A brief explanation is provided below:\n\n- TAG : Unique identifier for the analysis. All results will be saved in a folder called \"TAG-analysis\" within the res-folder.\n- REPO\\_DIR : Path to the repository directory, only change this if files are moved around.\n- DATA_DIR : Path to the data being loaded.\n- GENE_LIST_DIR : Directory to lists of genes\n- SCALE_FACTOR : Scaling factor between pixels to $\\mu m$\n\n\n```python\n\nTAG = re.sub(\":| |-|\\\\.|\",\"\",str(datetime.datetime.today()))\nTAG = TAG + \"-analysis\"\n\nREPO_DIR = osp.dirname(osp.abspath(os.getcwd()))\nDATA_DIR = osp.join(REPO_DIR,\"data/h5ad-cca\")\nRESULTS_DIR = osp.join(REPO_DIR,\"res\",TAG)\n\nif not osp.exists(RESULTS_DIR):\n os.mkdir(RESULTS_DIR)\n \nGENE_LIST_DIR = osp.join(REPO_DIR,\"data/gene-lists/2d-plots/\")\n\n\nSCALE_FACTOR = 2.801\n\nSAVE_RESULTS = False\n\n```\n\n\n```python\n# Get data paths\nPTHS = list(filter(lambda x: x.endswith(\"h5ad\"),os.listdir(DATA_DIR)))\nPTHS = {p:osp.join(DATA_DIR,p) for p in PTHS }\n\ndata_set = {n:ad.read_h5ad(p) for n,p in PTHS.items()}\nnew_names = {k:d.uns[\"sample\"] + \"-\" + d.uns[\"replicate\"] for k,d in data_set.items()}\ndata_set = dict((new_names[k],d) for k,d in data_set.items())\n \ndata_set\n```\n\n\n\n\n {'CN65-C1': AnnData object with n_obs \u00d7 n_vars = 647 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances',\n 'CN73-D1': AnnData object with n_obs \u00d7 n_vars = 684 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances',\n 'CN65-C2': AnnData object with n_obs \u00d7 n_vars = 626 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances',\n 'CN16-E1': AnnData object with n_obs \u00d7 n_vars = 1348 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances',\n 'CN73-C1': AnnData object with n_obs \u00d7 n_vars = 673 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances',\n 'CN65-D1': AnnData object with n_obs \u00d7 n_vars = 663 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances',\n 'CN16-D1': AnnData object with n_obs \u00d7 n_vars = 1262 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances',\n 'CN16-D2': AnnData object with n_obs \u00d7 n_vars = 487 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances',\n 'CN65-D2': AnnData object with n_obs \u00d7 n_vars = 629 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances',\n 'CN73-E2': AnnData object with n_obs \u00d7 n_vars = 650 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances',\n 'CN65-E1': AnnData object with n_obs \u00d7 n_vars = 590 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances',\n 'CN16-E2': AnnData object with n_obs \u00d7 n_vars = 487 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances'}\n\n\n\n\n```python\nexclude = [\"CN65-C1\",\"CN65-C2\",\"CN16-D1\", \"CN16-E1\"]\n\nfor ex in exclude:\n data_set.pop(ex)\n\ndata_set\n```\n\n\n\n\n {'CN73-D1': AnnData object with n_obs \u00d7 n_vars = 684 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances',\n 'CN73-C1': AnnData object with n_obs \u00d7 n_vars = 673 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances',\n 'CN65-D1': AnnData object with n_obs \u00d7 n_vars = 663 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances',\n 'CN16-D2': AnnData object with n_obs \u00d7 n_vars = 487 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances',\n 'CN65-D2': AnnData object with n_obs \u00d7 n_vars = 629 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances',\n 'CN73-E2': AnnData object with n_obs \u00d7 n_vars = 650 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances',\n 'CN65-E1': AnnData object with n_obs \u00d7 n_vars = 590 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances',\n 'CN16-E2': AnnData object with n_obs \u00d7 n_vars = 487 \u00d7 9655\n obs: '_x', '_y', 'x', 'y', 'sample', 'replicate'\n var: 'gene'\n uns: 'img', 'mask', 'replicate', 'sample'\n obsm: 'vein_distances'}\n\n\n\n\n```python\ngenes = ut.load_genelist(GENE_LIST_DIR,\n filter_tag = \"pc_\",\n )\n```\n\nVisualize the data to make sure everything looks as expected\n\n\n```python\nn_genes = len(genes[\"all\"])\n\nfig,ax = viz.get_figure(n_elements=n_genes,n_cols = 6,side_size=5)\n\nax = ax.flatten()\nalpha = 0.05\n\nlikelihoods = dict()\n\nplt_idx = 0\nfor ii,gene in enumerate(genes[\"all\"]):\n ys = []\n xs = []\n THRS = 500\n \n for k,data in enumerate(data_set.values()):\n if not gene in data.var.index.values:\n continue\n _y = data.obs_vector(gene)\n _x = data.obsm[\"vein_distances\"][[\"dist_type_central\",\"dist_type_portal\"]].values\n keep = np.min(_x,axis=1) < THRS\n _x = _x[keep,:]\n _y = _y[keep]\n ys.append(_y)\n xs.append(_x)\n\n if len(xs)> 0:\n ys = np.concatenate(ys)\n xs = np.concatenate(xs,axis=0)\n xs = np.hstack((np.ones((xs.shape[0],1)),xs))\n\n ll = viz.bivariate_expression_plot(ax[plt_idx],\n data = [xs,ys],\n feature = gene,\n feature_name = \"Gene\",\n cmap = plt.cm.magma,\n )\n \n likelihoods[gene] = ll\n plt_idx += 1\n \nfig.tight_layout()\nif SAVE_RESULTS:\n fig.savefig(osp.join(RESULTS_DIR,\"feature-by-distance-ratio.svg\"),\n facecolor = \"white\", dpi = 600)\n```\n\n## Likelihood ratio test\n\nIf we have a _full model_ and one _reduced model_, where the second is nested in the former we can conduct a so called likelihood ratio test (LRT) to determine whether the full model significantly improves the model or not (if not we use the simpler model).
                                        \n
                                        \nThe LRT is executed as follows:
                                        \n
                                        \nLet $\\mathcal{L}_{reduced}$ and $\\mathcal{L}_{full}$ be the likelihood values of each model, and let:
                                        \n
                                        \n\\begin{equation}\n D = -2\\times \\log(\\frac{\\mathcal{L}_{reduced}}{\\mathcal{L}_{full}})\n\\end{equation}\n\nthen $D \\sim \\mathcal{X}^2(\\delta)$ where $\\delta$ is the number the difference in number of parameters between the full and reduced model.
                                        \n
                                        \nBy computing the CDF for a given value $D$ and subtracting this from 1, we get the probability of observing a value equally large or more extreme (p-value). If the p-value is below a certain significance threshold $\\alpha$ we use the full model does a better job explaining the data (accounting for the extra parameters).
                                        \n
                                        \nIn this example we test two reduced models, one where the portal respectively central distance is removed. Whenever we get a p-value larger than $\\alpha$ this means that the covariate (distance) that we removed did not add to the reduced model.
                                        \n
                                        \nSource : https://www.itl.nist.gov/div898/handbook/apr/section2/apr233.htm\n\n\n```python\nmodel_eval = ut.likelihood_ratio_test(likelihoods,\n dofs = 1,\n included_covariates=[\"central\",\"portal\"],\n )\n\nif SAVE_RESULTS:\n model_eval.to_csv(osp.join(RESULTS_DIR,\"likelihood-ratio-test-results.tsv\"),\n sep = \"\\t\",\n )\n \n```\n\n\n```python\nmodel_eval\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        covariates_centralcovariates_portal[full_model]_superior_to_[central_model][full_model]_superior_to_[portal_model]
                                        Oat2.778026e-087.684476e-01TrueFalse
                                        Pde4b4.454622e-034.916251e-04TrueTrue
                                        Lhpp3.744113e-011.967049e-03FalseTrue
                                        Gstm24.236390e-011.240745e-01FalseFalse
                                        Ncald7.973059e-014.214255e-05FalseTrue
                                        Mettl7b6.987429e-013.158645e-01FalseFalse
                                        Cyp2g18.330512e-012.915511e-03FalseTrue
                                        Rnase42.908530e-014.592679e-28FalseTrue
                                        Ang2.705869e-024.290684e-02TrueTrue
                                        Susd41.173115e-015.929987e-02FalseFalse
                                        Slco1a17.323148e-053.719812e-02TrueTrue
                                        Ces1d3.834132e-072.858737e-16TrueTrue
                                        Csrp38.731702e-028.524049e-07FalseTrue
                                        Pex11a6.378225e-018.769247e-02FalseFalse
                                        Rarres12.995526e-011.086334e-01FalseFalse
                                        Me16.289054e-043.729318e-08TrueTrue
                                        Vnn14.620263e-028.519813e-06TrueTrue
                                        Csad1.718090e-011.270269e-09FalseTrue
                                        Cyp2c386.041604e-017.958446e-07FalseTrue
                                        Fabp52.060926e-015.727691e-04FalseTrue
                                        \n
                                        \n\n\n", "meta": {"hexsha": "96fb6ec7adc5b9d5adec05edb09013ccda018a62", "size": 853569, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "scripts/multivariate-regression.ipynb", "max_stars_repo_name": "ANKARKLEVLAB/ST-mLiver", "max_stars_repo_head_hexsha": "814dee288dfb94321b181b795c18f33f2f315764", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "scripts/multivariate-regression.ipynb", "max_issues_repo_name": "ANKARKLEVLAB/ST-mLiver", "max_issues_repo_head_hexsha": "814dee288dfb94321b181b795c18f33f2f315764", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "scripts/multivariate-regression.ipynb", "max_forks_repo_name": "ANKARKLEVLAB/ST-mLiver", "max_forks_repo_head_hexsha": "814dee288dfb94321b181b795c18f33f2f315764", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 1238.8519593614, "max_line_length": 828184, "alphanum_fraction": 0.9470142426, "converted": true, "num_tokens": 4534, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.640635861701035, "lm_q1q2_score": 0.4218058628313755}} {"text": "# Huawei Research France\n\n## Acrobot RAMP: system identification\n\n### Forecast the state of the altered acrobot system from history and actions\n\n_Bal\u00e1zs K\u00e9gl, Gabriel Hurtado, Jianfeng Zhang, Albert Thomas, Ludovic Dos Santos (Huawei Research, Noah's Ark Laboratory, France)_\n\n
                                        What is the probability that the world ends if I press this button?
                                        \n\n\n\n## Introduction\n\n\nIn recent year we have witnessed a phenomenal success of deep reinforcement learning algorithms beating top human players in [go](https://en.wikipedia.org/wiki/AlphaGo), [chess](https://en.wikipedia.org/wiki/AlphaZero) and even [poker](https://ai.facebook.com/blog/pluribus-first-ai-to-beat-pros-in-6-player-poker/). Why don't we see the widespread use of these algorithms in application domains half of the world is after, such as engineering system optimization or robotics? It turns out that one of the main conditions of these successes is the access to a simulator that can generate unlimited data to learn from.\n\n_\"The real world will not become faster in a few years, contrary to computers\"_ ([Chatzilygeroudis et al. 2019](https://arxiv.org/pdf/1807.02303.pdf))\n\nIf we want to control engineering systems, we need to be able to learn to control them with limited access to data. In this regime, one of the promising directions is [model-based reinforcement learning](https://arxiv.org/abs/1907.02057). The main component of these algorithms is a learnable system model. Its objective is to learn to forecast system behavior, given past history and control actions.\n\nClassical time series forecasting methods provide point predictions: \"the minimum temperature tomorrow is 5 degrees Celsius\". Without knowing the _certainty_ of the prediction, it is hard to train a controller, especially when extreme events need to be modeled (\"OK, but what's the probability that the plants on my balcony will freeze?\")\n\nOn the other hand, powerful and popular Bayesian forecasters do exist but they do not scale with either the length of the history or the system state dimensions.\n\n**The goal of this challenge to develop scalable and powerful generative probabilistic time series forecasters.** We will provide you with training and testing traces from a physical system based on the [acrobot benchmark](https://gym.openai.com/envs/Acrobot-v1). We are challenging you to develop generic learning algorithms, so we made manual reverse engineering harder by changing the physics of the original system. \n\n### Formal task\n\nYou will be given a training multivariate time series ${\\cal T} = \\big[(o_t, a_t)\\big]_{t=1}^T$ of observables $o_t \\in \\mathbb{R}^4$ and actions $a_t \\in \\{0, 1, 2\\}$. You will need to learn to predict the density function of the observables at next step $o_{t+1}$, conditioned on the history:\n\n\\begin{equation}\n p\\big(o_{t+1} | (o_1, a_1), \\ldots, (o_t, a_t)\\big).\n\\end{equation}\n\nYou will be evaluated on a test sequence ${\\cal T}^\\prime = \\big[(o_t^\\prime, a_t^\\prime)\\big]_{t=1}^{T^\\prime}$ using the likelihood ratio\n\n\\begin{equation}\nLR({\\cal T}^\\prime; p) = \\frac{\\prod_{t=1}^{T^\\prime-1} p\\big(o^\\prime_{t+1} | (o^\\prime_1, a^\\prime_1), \\ldots, (o^\\prime_t, a^\\prime_t)\\big)}{L_\\text{b}({\\cal T}^\\prime)},\n\\end{equation}\n\nwhere $L_\\text{b}({\\cal T}^\\prime)$ is a baseline (multivariate spherical Gaussian) likelihood.\n\n${\\cal T}$ and ${\\cal T}^\\prime$ will always come from the same dynamical system. However, to make manual reverse engineering hard, the time series in this public starting kit and the times series on the data challenge server will come from different physics, so your model should be prepared to be transferred to a (slightly) different system. \n\nWe will keep the training set size the same, $T \\simeq 5000$ in both the public and private training sets.\n\n### Challenges\n1. We do not know whether the system observables $o_t$ and action $a_t$ at time $t$ are sufficient to predict $o_{t+1}$. Therefore we will explicitly ask you to develop (or learn) a feature extractor $f_\\text{FE}$ wich will convert the history up to time $t$ into a fixed length state vector $s_t$. Formally,\n\\begin{equation}\n s_t = f_\\text{FE}\\big((o_1, a_1), \\ldots, (o_t, a_t)\\big),\n\\end{equation}\nand the predictor will then simplify to\n\\begin{equation}\n p\\big(o_{t+1} | (o_1, a_1), \\ldots, (o_t, a_t)\\big) = p(o_{t+1} | s_t).\n\\end{equation}\n2. The predictor $p(o_{t+1} | s_t)$ is not a classical regressor outputting a point estimate $\\hat{o}_{t+1}$ but a $d$-dimensional density ($d=4$). We needed to find a numerical representation for such a function with two requirements: 1) computationally easy evaluation of the likelihood and 2) computationally easy simulation (sampling). The first trick is that we represent the multivariate density $p(o_{t+1} | s_t)$ by $d$ univariate densities using the chain rule\n\\begin{equation}\n p(o_{t+1} | s_t) = p_1(o^1_{t+1} | s_t) \\prod_{j=2}^d p_j\\big(o^j_{t+1} |o^1_{t+1}, \\ldots, o^{j-1}_{t+1}, s_t\\big), \n\\end{equation}\nwhere $o_{t+1} = \\big(o^1_{t+1},\\ldots,o^d_{t+1}\\big)$. This means that you will be asked to learn $d$ (generative) regressors $p_1, \\ldots, p_d$.\n3. The second trick is that we will represent each generative regressor $p_j$ by a mixture of $L_j$ ($L_j \\le 100$) simple parametric components\n\\begin{equation}\n p_j\\big(o^j_{t+1} |o^1_{t+1}, \\ldots, o^{j-1}_{t+1}, s_t\\big) = \\sum_{\\ell=1}^{L_j} w_\\ell {\\cal P}_\\ell(o^j_{t+1}; \\theta_\\ell).\n\\end{equation}\nThe $j$th regressor $p_j$ will thus map its vector input $\\big(o^1_{t+1}, \\ldots, o^{j-1}_{t+1}, s_t\\big)$ onto a sequence of weights, densities, and density parameters $\\big[(w, {\\cal P}, \\theta)_\\ell\\big]_{\\ell=1}^{L_j}$.\n\nThis may sound complicated, but in fact it is relatively easy to convert both random forests and neural nets into generative regressors of this kind, and we will provide starting kit examples for both of these popular function classes. Also note that the intricacies of training set generation for each regressor will be taken care of the workflow behind the scenes; all you need to do is to parameterize the regressors.\n\n## Competition rules\n\n* Submissions will be trained on a time series of roughly 5000 time steps and tested on a time series of roughly 20000 time steps. \n* The competition will end on November 22, 2019 at 19h UTC (20h in Paris).\n* All models will be trained on the same cloud server allowing 4 CPUs (with shared memory of 128GB RAM).\n* Participants will be given a total of 20 machine hours. Submissions of a given participant will be ordered by submission timestamp. We will make an attempt to train all submissions, but starting from (and including) the first submission that makes the participant's total training time exceed 20 hours, all submissions will be disqualified from the competition (but can enter into the collaborative phase). Testing time will not count towards the limit. Training time will be displayed on the leaderboard for all submissions, rounded to second. If a submission raises an exception, its training time will not count towards the total.\n* There is a timeout of 1 day between submissions.\n* Submissions submitted after the end of the competition will not qualify for prizes.\n* The public leaderboard will display validation scores running a cross-validation. The official scores will be calculated on the hidden test set and will be published after the closing of the competition. We will rank submissions according to their likelihood ratio score.\n* The organizers will do their best so that the provided backend runs flawlessly. We will communicate with participants in case of concerns and will try to resolve all issues, but we reserve the right to make unilateral decisions in specific cases, not covered by this set of minimal rules.\n* The organizers reserve the right to disqualify any participant found to violate the fair competitive spirit of the challenge. Possible reasons, without being exhaustive, are multiple accounts, attempts to access the test data, etc.\n* The challenge is essentially an individual contest, so there is no way to form official teams. Participants can form teams outside the platform before submitting any model individually, and submit on a single team member's account. However, submitting on one's own and participating in such a team at the same time is against the \"no multiple accounts\" rule, so, if discovered, may lead to disqualification.\n* Participants retain copyright on their submitted code and grant reuse under BSD 3-Clause License.\n\nParticipants accept these rules automatically when making a submission at the RAMP site.\n\n### Installation instructions are found in the [README](https://github.com/ramp-kits/acrobot/blob/master/README.md)\n\n\n```python\n%matplotlib inline\nimport os\nimport glob\nimport numpy as np\nfrom scipy import io\nimport pandas as pd\nimport rampwf as rw\nimport xarray as xr\n\nimport altair as alt\nalt.renderers.enable('notebook');\n```\n\nYou can load `problem.py` to have access the RAMP setup.\n\n\n```python\nproblem = rw.utils.assert_read_problem()\n```\n\n## Exploratory data analysis\n\n### Loading the data\n\nLet's load the data. `X_ds` is `xr.Dataset` which is a versatile container. We will not use many features of it so most of the time it will be converted to `pandas.Dataframe`. `y_array` is a numpy array.\n\n\n```python\nX_ds, y_array = problem.get_train_data()\nX_ds\n```\n\n\n\n\n \n Dimensions: (time: 4999)\n Coordinates:\n * time (time) datetime64[ns] 2019-08-05 2019-08-06 ... 2036-07-24\n Data variables:\n thetaDot2 (time) float64 -0.025 -0.0666 -0.4516 ... -11.23 -8.425 -11.7\n theta2 (time) float64 0.05051 0.04131 -0.01101 ... -1.367 3.055 1.109\n thetaDot1 (time) float64 0.004686 -0.02917 0.09464 ... -2.737 0.6118\n theta1 (time) float64 0.03029 0.02784 0.03448 ... -0.448 -0.7157\n action (time) float64 1.0 0.0 1.0 0.0 0.0 1.0 ... 2.0 2.0 2.0 2.0 2.0\n restart (time) int64 0 0 0 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0 0\n y_thetaDot2 (time) float64 -0.0666 -0.4516 -0.4365 ... -8.425 -11.7 -10.98\n y_theta2 (time) float64 0.04131 -0.01101 -0.1006 ... 3.055 1.109 -1.311\n y_thetaDot1 (time) float64 -0.02917 0.09464 0.04833 ... 0.6118 0.3024\n y_theta1 (time) float64 0.02784 0.03448 0.04905 ... -0.7157 -0.4942\n Attributes:\n n_burn_in: 0\n\n\n\n`X_ds` is indexed by time. The unit (day) is arbitrary, in reality acrobot is moving much faster. The first four variables are the system observables $o_t \\in \\mathbb{R}^4$. For more information, we refer to the [acrobot site](https://gym.openai.com/envs/Acrobot-v1). The semantics of the variables is the same as in the original system, but **we altered the physics that generated the time series to make reverse engineering harder**. Also note that the data in the starting kit and the data in the challenge server were also generated by different physics since we would like you to learn from data not to manually figure out the dynamics and hard code it in your solutions.\n\n`actions` is a ternary action: `0.0` means kick to the left, `1.0` means no kick, `2.0` means kick to the right. Indeed, the change of `thetaDot2` (the angular speed of the second joint) is roughly correlated with the action.\n\n\n```python\nprint((X_ds.thetaDot2[1:].values - X_ds.thetaDot2[:-1].values)[X_ds.action[:-1]==0.0].mean())\nprint((X_ds.thetaDot2[1:].values - X_ds.thetaDot2[:-1].values)[X_ds.action[:-1]==1.0].mean())\nprint((X_ds.thetaDot2[1:].values - X_ds.thetaDot2[:-1].values)[X_ds.action[:-1]==2.0].mean())\n```\n\n -0.287894118268408\n 0.0023133801874445645\n 0.2884619102470254\n\n\n`restart` is a binary helper column: `0.0` means the time step is a continuation of the previous trace, `1.0` means that we start a new trace from a (roughly) vertically hanging starting state. In the public training data there are several restarts, meaning that we have several traces. The cross validation will be done by these traces, so in the public training we will have seven folds. In the private training data at the challenge server we may have a different number of restarts. The test data can also contain several traces, delineated in the same way.\n\n\n```python\nX_ds.restart.values.sum()\n```\n\n\n\n\n 6\n\n\n\nThe rest of the columns are identical to `y_array`, which is the prediction target. It corresponds to the system observables $o_t$ (first six columns), shifted backwards by one, meaning that the target is to predict the system observable $o_{t+1}$ at time $t+1$. The reason for `X_ds` (the input) containing the target will be clear later.\n\n### Visualizing the data\n\nWe now plot the system variables against time. You can explore different segments of training (and test) sets by changing `start_time` and `end_time` (and loading the test set on the top of the notebook).\n\n\n```python\nfigs = []\nstart_time = 1303\nend_time = 1680\nfor observable_col in problem._target_column_observation_names:\n ground_truth = X_ds.to_dataframe().reset_index().reset_index()\n line_gt = alt.Chart(ground_truth[start_time:end_time]).mark_line(color='black').encode(\n x=alt.X('index:Q', title='time step'),\n y=alt.Y(observable_col + ':Q', scale=alt.Scale(zero=False)),\n )\n fig = line_gt\n figs.append(fig)\nalt.vconcat(alt.hconcat(*figs[:2]), alt.hconcat(*figs[2:]))\n```\n\n## The pipeline: what you need to develop\n\nFor submitting at the challenge site, you will have to write two classes, saved in two different files: \n* the class `FeatureExtractor` in `ts_feature_extractor.py`, which will be used to extract features for representing the system state at time $t$. \n* the class `GenerativeRegressor` in `generative_regressor.py` to predict the distribution of the six system variables at time $t+1$ from the features returned by the feature extractor.\n\n### Feature extractor\n\nThe feature extractor implements a `transform` member function. It is saved in the file [`submissions/starting_kit/ts_feature_extractor.py`](submissions/starting_kit/ts_feature_extractor.py). It receives an xarray Dataset `X_ds` defined at the beginning of the notebook. It should produce a pandas DataFrame containing the extracted features, which will then be used for regression. \n\nBe careful not to use any information from the future (`X_ds[t + 1:]`) when constructing `X_df[t]`. We have implemented a check for that in the training workflow, but any attempt to purposefully work around those checks may lead to disqualification from the challenge.\n\nThe following simple feature extractor in the starting kit simply makes $s_t = (o_t, a_t)$.\n\n\n```python\n# %load submissions/starting_kit/ts_feature_extractor.py\nclass FeatureExtractor():\n def __init__(self, restart_name):\n \"\"\"\n Parameters\n ----------\n restart_name : str\n The name of the 0/1 column indicating restarts in the time series.\n \"\"\"\n self.restart_name = restart_name\n\n def transform(self, X_ds):\n \"\"\"Transform time series into list of states.\n We simply use the observables at time t as the state.\n Be careful not to use any information from the future (X_ds[t + 1:])\n when constructing X_df[t].\n Parameters\n ----------\n X_ds : xarray.Dataset\n The raw time series.\n Return\n ------\n X_df : pandas Dataframe\n The list of states.\n \"\"\"\n X_df = X_ds.to_dataframe()\n # Since we do not use the restart information in our regressor, we\n # remove it\n restart = X_df[self.restart_name].values\n X_df = X_df.drop(columns=self.restart_name)\n return X_df\n```\n\n### Generative regressor\n\nThe generative regressor follows a classical scikit-learn regressor template. It should be saved in the file [`submissions/starting_kit/generative_regressor.py`](submissions/starting_kit/generative_regressor.py). \n\nThe `__init__` function receives two parameters that you can use in `fit` and `predict`. `max_dists` is the maximum number of components in the mixture distribution, and `target_dim` is the index of the dimension being predicted ($j$ in the introduction, $j \\in [0,5]$). \n\nThe `fit` function receives the input array `X_array` and the ground truth target numpy array `y_array`. The number of rows is the number of time steps. The columns of `X_array` are the features you extracted in the feature extractor followed by the `target_dim` (= $j-1$) system observables $\\big(o^1_{t+1}, \\ldots, o^{j-1}_{t+1}\\big)$. The number of columns in `y_array` is one representing the target $o^j_{t+1}$.\n\nThe particular regressor learns a classical linear regressor and saves the standard deviation of the residuals. \n\nThe `predict` function receives the input test array `X_array`, the format is the same as in `fit`. It then constructs the one-dimensional density for its target $o^j_{t+1}$ ($j=$ `target_dim` $+ 1$). As in the feature extractor, be careful not to use any information from the future (`X_array[t + 1:]`) when constructing the output.\n\nThe particular regressor constructs a single Gaussian (so $w_1 = 1$), centered by the output of the linear regressor, with a fixed sigma (the standard deviation of the residuals in the training set). The type (code) of the Gaussian is 0.\n\nWe have other component types implemented in [rampwf.utils.distributions_dict](https://github.com/paris-saclay-cds/ramp-workflow/blob/generative_regression_clean/rampwf/utils/generative_regression.py): uniform (1) and beta (2). If you would like to use other component types, either let us know which one, or (better) implement it and make a pull request into the [generative_regression_clean](https://github.com/paris-saclay-cds/ramp-workflow/tree/generative_regression_clean) branch on ramp-workflow. After checks we will merge the PR, install it on the server and make an announcement in Slack to make it available to everybody. You can then submit your solution with the new kernel types.\n\n\n```python\n# %load submissions/starting_kit/generative_regressor.py\nfrom sklearn.base import BaseEstimator\nimport numpy as np\nfrom sklearn.linear_model import LinearRegression\n\n\nclass GenerativeRegressor(BaseEstimator):\n def __init__(self, max_dists, target_dim):\n \"\"\"\n Parameters\n ----------\n max_dists : int\n The maximum number of distributions (kernels) in the mixture.\n target_dim : int\n The index of the target column to be predicted [0..d-1]\n \"\"\"\n pass\n\n def fit(self, X_array, y_array):\n \"\"\"Linear regression + residual sigma.\n \n Parameters\n ----------\n X_array : numpy array\n The input array. The features extracted by the feature extractor,\n plus `target_dim` system observables from time step t+1.\n y_array : numpy array\n The ground truth array (system observable j=`target_dim` + 1\n at time step t+1: o^j_{t+1}).\n \"\"\"\n self.reg = LinearRegression()\n self.reg.fit(X_array, y_array)\n y_pred = self.reg.predict(X_array)\n y_pred = np.array([y_pred]).reshape(-1, 1)\n residuals = y_array - y_pred\n # Estimate a single sigma from residual variance\n self.sigma = np.sqrt(\n (1 / (X_array.shape[0] - 1)) * np.sum(residuals ** 2))\n\n def predict(self, X_array):\n \"\"\"Construct a conditional mixture distribution.\n\n Be careful not to use any information from the future\n (X_array[t + 1:]) when constructing the output.\n\n Parameters\n ----------\n X_array : numpy array\n The input array. The features extracted by the feature extractor,\n plus `target_dim` system observables from time step t+1.\n\n Return\n ------\n weights : np.array of float\n discrete probabilities of each component of the mixture\n types : np.array of int\n integer codes referring to component types\n see rampwf.utils.distributions_dict\n params : np.array of float tuples\n parameters for each component in the mixture\n \"\"\"\n types = np.array([[0], ] * len(X_array))\n\n # Normal\n y_pred = self.reg.predict(X_array)\n sigmas = np.array([self.sigma] * len(X_array))\n sigmas = sigmas[:, np.newaxis]\n params = np.concatenate((y_pred, sigmas), axis=1)\n weights = np.array([[1.0], ] * len(X_array))\n return weights, types, params\n```\n\n## Training and scoring\n\nYou can train and test the starting kit in a terminal or here in the notebook using the `ramp-test` script. By default it tests the starting kit, if you would like to test another submission, you can specify it with `--submission ` where `` is what you named the directory in [`submissions/`](submissions).\n\nWe are using as many cross validation folds as episodes in the training file (number of restarts + 1). With $N$ episodes, we train on $N-1$ in each fold and test on the remaining one. The number of folds in the training set is 7. The training script prints the episode bounds and the fold indices, then trains, validates (on the validation fold), and tests (on the test set loaded by `problem.get_test_data{}`). Finally the mean and bagged scores are displayed. \n\nOn the server, the public leaderboard will display your bagged validation score during the competition. Your final ranking will be determined by the bagged test score. The data is different in this public starting kit and on the server. We use different physics to make reverse engineering harder.\n\n\n```python\n!ramp-test\n```\n\n \u001b[38;5;178m\u001b[1mTesting Acrobot system identification\u001b[0m\n \u001b[38;5;178m\u001b[1mReading train and test files from ./data ...\u001b[0m\n \u001b[38;5;178m\u001b[1mReading cv ...\u001b[0m\n episode bounds: [0, 333, 1303, 1678, 2093, 2532, 3288, 4999]\n CV fold 0: train 333..4998, valid 0..332, \n CV fold 1: train 0..332, 1303..4998, valid 333..1302, \n CV fold 2: train 0..1302, 1678..4998, valid 1303..1677, \n CV fold 3: train 0..1677, 2093..4998, valid 1678..2092, \n CV fold 4: train 0..2092, 2532..4998, valid 2093..2531, \n CV fold 5: train 0..2531, 3288..4998, valid 2532..3287, \n CV fold 6: train 0..3287, valid 3288..4998, \n \u001b[38;5;178m\u001b[1mTraining submissions/starting_kit ...\u001b[0m\n \u001b[38;5;178m\u001b[1mCV fold 0\u001b[0m\n \t\u001b[38;5;178m\u001b[1mscore likelihood_ratio time\u001b[0m\n \t\u001b[38;5;10m\u001b[1mtrain\u001b[0m \u001b[38;5;10m\u001b[1m4.59\u001b[0m \u001b[38;5;150m0.075871\u001b[0m\n \t\u001b[38;5;12m\u001b[1mvalid\u001b[0m \u001b[38;5;12m\u001b[1m3.96\u001b[0m \u001b[38;5;105m0.069595\u001b[0m\n \t\u001b[38;5;1m\u001b[1mtest\u001b[0m \u001b[38;5;1m\u001b[1m3.86\u001b[0m \u001b[38;5;218m0.070474\u001b[0m\n \u001b[38;5;178m\u001b[1mCV fold 1\u001b[0m\n \t\u001b[38;5;178m\u001b[1mscore likelihood_ratio time\u001b[0m\n \t\u001b[38;5;10m\u001b[1mtrain\u001b[0m \u001b[38;5;10m\u001b[1m4.16\u001b[0m \u001b[38;5;150m0.030928\u001b[0m\n \t\u001b[38;5;12m\u001b[1mvalid\u001b[0m \u001b[38;5;12m\u001b[1m3.70\u001b[0m \u001b[38;5;105m0.064255\u001b[0m\n \t\u001b[38;5;1m\u001b[1mtest\u001b[0m \u001b[38;5;1m\u001b[1m3.61\u001b[0m \u001b[38;5;218m0.067980\u001b[0m\n \u001b[38;5;178m\u001b[1mCV fold 2\u001b[0m\n \t\u001b[38;5;178m\u001b[1mscore likelihood_ratio time\u001b[0m\n \t\u001b[38;5;10m\u001b[1mtrain\u001b[0m \u001b[38;5;10m\u001b[1m4.22\u001b[0m \u001b[38;5;150m0.029149\u001b[0m\n \t\u001b[38;5;12m\u001b[1mvalid\u001b[0m \u001b[38;5;12m\u001b[1m2.58\u001b[0m \u001b[38;5;105m0.064103\u001b[0m\n \t\u001b[38;5;1m\u001b[1mtest\u001b[0m \u001b[38;5;1m\u001b[1m3.42\u001b[0m \u001b[38;5;218m0.066786\u001b[0m\n \u001b[38;5;178m\u001b[1mCV fold 3\u001b[0m\n \t\u001b[38;5;178m\u001b[1mscore likelihood_ratio time\u001b[0m\n \t\u001b[38;5;10m\u001b[1mtrain\u001b[0m \u001b[38;5;10m\u001b[1m4.17\u001b[0m \u001b[38;5;150m0.029267\u001b[0m\n \t\u001b[38;5;12m\u001b[1mvalid\u001b[0m \u001b[38;5;12m\u001b[1m3.03\u001b[0m \u001b[38;5;105m0.062825\u001b[0m\n \t\u001b[38;5;1m\u001b[1mtest\u001b[0m \u001b[38;5;1m\u001b[1m3.40\u001b[0m \u001b[38;5;218m0.064849\u001b[0m\n \u001b[38;5;178m\u001b[1mCV fold 4\u001b[0m\n \t\u001b[38;5;178m\u001b[1mscore likelihood_ratio time\u001b[0m\n \t\u001b[38;5;10m\u001b[1mtrain\u001b[0m \u001b[38;5;10m\u001b[1m4.22\u001b[0m \u001b[38;5;150m0.028860\u001b[0m\n \t\u001b[38;5;12m\u001b[1mvalid\u001b[0m \u001b[38;5;12m\u001b[1m2.72\u001b[0m \u001b[38;5;105m0.062818\u001b[0m\n \t\u001b[38;5;1m\u001b[1mtest\u001b[0m \u001b[38;5;1m\u001b[1m3.43\u001b[0m \u001b[38;5;218m0.066875\u001b[0m\n \u001b[38;5;178m\u001b[1mCV fold 5\u001b[0m\n \t\u001b[38;5;178m\u001b[1mscore likelihood_ratio time\u001b[0m\n \t\u001b[38;5;10m\u001b[1mtrain\u001b[0m \u001b[38;5;10m\u001b[1m4.10\u001b[0m \u001b[38;5;150m0.047163\u001b[0m\n \t\u001b[38;5;12m\u001b[1mvalid\u001b[0m \u001b[38;5;12m\u001b[1m3.29\u001b[0m \u001b[38;5;105m0.064688\u001b[0m\n \t\u001b[38;5;1m\u001b[1mtest\u001b[0m \u001b[38;5;1m\u001b[1m3.31\u001b[0m \u001b[38;5;218m0.067226\u001b[0m\n \u001b[38;5;178m\u001b[1mCV fold 6\u001b[0m\n \t\u001b[38;5;178m\u001b[1mscore likelihood_ratio time\u001b[0m\n \t\u001b[38;5;10m\u001b[1mtrain\u001b[0m \u001b[38;5;10m\u001b[1m4.21\u001b[0m \u001b[38;5;150m0.027983\u001b[0m\n \t\u001b[38;5;12m\u001b[1mvalid\u001b[0m \u001b[38;5;12m\u001b[1m2.75\u001b[0m \u001b[38;5;105m0.063567\u001b[0m\n \t\u001b[38;5;1m\u001b[1mtest\u001b[0m \u001b[38;5;1m\u001b[1m3.77\u001b[0m \u001b[38;5;218m0.061401\u001b[0m\n \u001b[38;5;178m\u001b[1m----------------------------\u001b[0m\n \u001b[38;5;178m\u001b[1mMean CV scores\u001b[0m\n \u001b[38;5;178m\u001b[1m----------------------------\u001b[0m\n \t\u001b[38;5;178m\u001b[1mscore likelihood_ratio time\u001b[0m\n \t\u001b[38;5;10m\u001b[1mtrain\u001b[0m \u001b[38;5;10m\u001b[1m4.24\u001b[0m\u00a0\u001b[38;5;150m\u001b[38;5;150m\u00b1\u001b[0m\u001b[0m\u00a0\u001b[38;5;150m0.148\u001b[0m \u001b[38;5;150m0.0\u001b[0m\u00a0\u001b[38;5;150m\u001b[38;5;150m\u00b1\u001b[0m\u001b[0m\u00a0\u001b[38;5;150m0.0\u001b[0m2\n \t\u001b[38;5;12m\u001b[1mvalid\u001b[0m \u001b[38;5;12m\u001b[1m3.15\u001b[0m\u00a0\u001b[38;5;105m\u001b[38;5;105m\u00b1\u001b[0m\u001b[0m\u00a0\u001b[38;5;105m0.488\u001b[0m \u001b[38;5;105m0.1\u001b[0m\u00a0\u001b[38;5;105m\u001b[38;5;105m\u00b1\u001b[0m\u001b[0m\u00a0\u001b[38;5;105m0.0\u001b[0m\n \t\u001b[38;5;1m\u001b[1mtest\u001b[0m \u001b[38;5;1m\u001b[1m3.54\u001b[0m\u00a0\u001b[38;5;218m\u001b[38;5;218m\u00b1\u001b[0m\u001b[0m\u00a0\u001b[38;5;218m\u001b[38;5;218m0.1\u001b[0m93\u001b[0m \u001b[38;5;218m0.1\u001b[0m\u00a0\u001b[38;5;218m\u001b[38;5;218m\u00b1\u001b[0m\u001b[0m\u00a0\u001b[38;5;218m0.0\u001b[0m\n \u001b[38;5;178m\u001b[1m----------------------------\u001b[0m\n \u001b[38;5;178m\u001b[1mBagged scores\u001b[0m\n \u001b[38;5;178m\u001b[1m----------------------------\u001b[0m\n \t\u001b[38;5;178m\u001b[1mscore likelihood_ratio\u001b[0m\n \t\u001b[38;5;12m\u001b[1mvalid\u001b[0m \u001b[38;5;12m\u001b[1m3.73\u001b[0m\n \t\u001b[38;5;1m\u001b[1mtest\u001b[0m \u001b[38;5;1m\u001b[1m3.58\u001b[0m\n\n\n### What does `ramp test` do? Step-by-step.\n\nFirst, let's load the fold indices of the first fold.\n\n\n```python\nget_cv = problem.get_cv\ntrain_is, valid_is = list(get_cv(X_ds, y_array))[0]\n```\n\n episode bounds: [0, 333, 1303, 1678, 2093, 2532, 3288, 4999]\n CV fold 0: train 333..4998, valid 0..332, \n CV fold 1: train 0..332, 1303..4998, valid 333..1302, \n CV fold 2: train 0..1302, 1678..4998, valid 1303..1677, \n CV fold 3: train 0..1677, 2093..4998, valid 1678..2092, \n CV fold 4: train 0..2092, 2532..4998, valid 2093..2531, \n CV fold 5: train 0..2531, 3288..4998, valid 2532..3287, \n CV fold 6: train 0..3287, valid 3288..4998, \n\n\nNow load the workflow and train the model.\n\n\n```python\nworkflow = problem.workflow\nfe, reg = workflow.train_submission(\n 'submissions/starting_kit', X_ds, y_array, train_is)\n```\n\nNow test the trained model, first on the full training data then on the test data.\n\n\n```python\ny_preds = workflow.test_submission((fe, reg), X_ds)\nX_test_ds, y_test_array = problem.get_test_data()\ny_test_preds = workflow.test_submission((fe, reg), X_test_ds)\n```\n\nFinally, load the score type (metrics) and compute the training, validation, and test scores.\n\n\n```python\nscore_type = problem.score_types[0]\ntrain_score = score_type(y_array[train_is], y_preds[train_is])\nvalid_score = score_type(y_array[valid_is], y_preds[valid_is])\ntest_score = score_type(y_test_array, y_test_preds)\nprint(train_score, valid_score, test_score)\n```\n\n 4.588134679241945 3.963022781790907 3.8630718396285877\n\n\n## Submitting to the hackathon server\n\nOnce you found a good feature extractor and classifier, you can submit them to the hackathon server. First, if it is your first time using RAMP, sign up, otherwise log in. The system sends you a mail where you can validate your signup, then the site admins will validate it. Once this is done, click on the \"Huawei Hackathon\" event under the \"Acrobot system identification\" problem, and sign up.\n\nOnce you are signed up, you can go to your sandbox (menu at the left) and copy-paste [`ts_feature_extractor.py`](/submissions/starting_kit/ts_feature_extractor.py) and [`generative_regressor.py`](/submissions/starting_kit/generative_regressor.py) from `submissions/starting_kit`. Save it, rename it, then submit it. The submission is trained and tested on our backend in the same way as `ramp-test` does it locally. While your submission is waiting in the queue and being trained, you can find it in the \"New submissions (pending training)\" table in my submissions. Once it is trained, your submission shows up on the leaderboard.\n\nIf there is an error, it will show up in the \"Failed submissions\" table in my submissions. You can click on the error to see the trace.\n\nAfter submission, do not forget to give credits to the previous submissions you reused or integrated into your submission. This will be especially interesting in the collaborative phase (after the competition is over) when you can look at and reuse other solutions.\n\nThe data set we use at the backend is different from what you find in the starting kit, so the score may be different.\n\n# Other models in the starting kit\n\nYou can also keep several other submissions in your work directory [`submissions`](/submissions). [`generative_rf`](/submissions/generative_rf) uses a feature extractor that concatenates to $o_t$ the mean of the observables of the last time steps, handling restarts intelligently (not using the observables of the previous episode).\n\n\n```python\n# %load submissions/generative_rf/ts_feature_extractor.py\nimport pandas as pd\nimport numpy as np\n\n\nclass FeatureExtractor():\n def __init__(self, restart_name):\n \"\"\"\n Parameters\n ----------\n restart_name : str\n The name of the 0/1 column indicating restarts in the time series.\n \"\"\"\n self.restart_name = restart_name\n pass\n\n def transform(self, X_df_raw):\n \"\"\"Transform time series into list of states.\n We use the observables at time t as the state, concatenated to the mean\n of the last ten time steps, handling restarts.\n \n Be careful not to use any information from the future (X_ds[t + 1:])\n when constructing X_df[t].\n Parameters\n ----------\n X_ds : xarray.Dataset\n The raw time series.\n Return\n ------\n X_df : pandas Dataframe\n\n \"\"\"\n X_df = X_df_raw.to_dataframe()\n\n restart = X_df[self.restart_name].values\n\n # Since we do not use the restart information in our regressor, we have\n # to remove it\n X_df = X_df.drop(columns=self.restart_name)\n\n tail = 10\n result_array = []\n curr_tail = tail\n for i in range(len(X_df)):\n if restart[i] == 1:\n # If we encounter a restart, tail is set to 0\n curr_tail = 0\n elif curr_tail < tail:\n # And it goes back up to it's normal length if no other restarts\n curr_tail += 1\n\n X_temp = X_df.iloc[\n [idx for idx in range(i - curr_tail, i + 1) if idx >= 0]\n ].mean(axis=0)\n result_array.append(X_temp)\n result_array = np.vstack(result_array)\n additional_dim = pd.DataFrame(result_array)\n\n new_names = [item + \"_engineered\" for item in list(X_df_raw.keys())]\n additional_dim.rename(\n columns={i: item for i, item in enumerate(new_names)}, inplace=True)\n\n date = X_df.index.copy()\n X_df.reset_index(drop=True, inplace=True)\n X_array = pd.concat([X_df, additional_dim], axis=1)\n\n X_array.set_index(date, inplace=True)\n\n # We return a dataframe with additional features (with no clashing names)\n # based on previous values' mean, being mindful about restarts\n\n return X_array\n```\n\nThe generative regressor learns 99 bagged trees (a \"manual\" random forest), and puts Gaussians centered in the tree predictions, sigmas estimated in different ways. \n\n\n```python\n# %load submissions/generative_rf/generative_regressor.py\nfrom sklearn.base import BaseEstimator\nfrom sklearn.ensemble import BaggingRegressor\nfrom sklearn.tree import DecisionTreeRegressor\nimport numpy as np\nfrom rampwf.hyperopt import Hyperparameter\n \n \n# RAMP START HYPERPARAMETERS\n#sigma_multiplier = Hyperparameter(\n# dtype='float', default=0.2, values=[\n# 0.03, 0.05, 0.07, 0.1, 0.15, 0.2, 0.3])\n#n_estimators = Hyperparameter(\n# dtype='int', default=200, values=[30, 50, 100, 200, 300])\n#max_leaf_nodes = Hyperparameter(\n# dtype='int', default=5000, values=[1000, 2000, 5000, 10000])\n# RAMP END HYPERPARAMETERS \n \nmethod_proba = 'estimate_sigma'\nEPSILON =1e-8\n \n \nclass GenerativeRegressor(BaseEstimator):\n def __init__(self, max_dists, model_index):\n self.max_dists = min(100, max_dists) \n self.clf = BaggingRegressor(\n base_estimator=DecisionTreeRegressor(\n max_leaf_nodes=5000),\n n_estimators=self.max_dists-1, # The last dist is uniform\n max_samples=0.2,\n )\n self.sigma = None\n self.a = None\n self.b = None\n \n def fit(self, X, y):\n self.clf.fit(X, y.ravel())\n yGuess = self.clf.predict(X)\n yGuess = np.array([yGuess]).reshape(-1, 1)\n error = y - yGuess\n self.sigma = np.sqrt((1 / X.shape[0]) * np.sum(error ** 2))\n self.a = np.min(y) - 10\n self.b = np.max(y) + 10\n \n def predict(self, X):\n \n # We give every distribution the same weight\n eps = 10 ** -10\n w = (1.0 - EPSILON)/(self.max_dists - 1) \n weights = np.stack([[w] * len(X)\n for _ in range(self.max_dists)],\n axis=1)\n weights[:, 0] = EPSILON\n # The first generative regressors are gaussian, the last is uniform\n types = np.zeros(self.max_dists)\n types[-1] = 1\n types = np.array([types] * len(X))\n \n # Gaussians\n mus = np.zeros((len(X), len(self.clf.estimators_),))\n for i, est in enumerate(self.clf.estimators_):\n mus[:, i] = est.predict(X)\n \n if method_proba == 'estimate_sigma':\n # Third method, uses the information from the trees to estimate\n # sigma and estimate gaussian noise around outpus\n sigma = mus.std(axis=1)\n sigma = np.clip(sigma, EPSILON, None, out=sigma)\n sigmas = np.stack([sigma for _ in range(len(self.clf.estimators_))],\n axis=1)\n \n elif method_proba == 'standard':\n sigmas = np.stack([[self.sigma] * len(X)\n for _ in range(len(self.clf.estimators_))],\n axis=1)\n else:\n sigmas = np.nan\n \n #sigmas *= float(sigma_multiplier)\n sigmas /= float(np.sqrt(self.max_dists - 1))\n \n # We put each mu next to its sigma\n params_normal = np.empty((len(X), len(self.clf.estimators_)*2))\n params_normal[:, 0::2] = mus\n params_normal[:, 1::2] = sigmas\n \n # Uniform\n a_array = np.array([self.a] * len(X))\n b_array = np.array([self.b] * len(X))\n params_uniform = np.stack((a_array, b_array), axis=1)\n \n # We concatenate the params\n params = np.concatenate((params_normal, params_uniform), axis=1)\n return weights, types, params\n\n```\n\nYou can also uncomment the hyperparameter section at the top and `sigmas *= float(sigma_multiplier)` further down, then try the experimental `ramp-hyperopt` command, see what happens.\n\n\n```python\n!ramp-hyperopt --submission generative_rf --n-iter 21\n```\n\n episode bounds: [0, 333, 1303, 1678, 2093, 2532, 3288, 4999]\n CV fold 0: train 333..4998, valid 0..332, \n CV fold 1: train 0..332, 1303..4998, valid 333..1302, \n CV fold 2: train 0..1302, 1678..4998, valid 1303..1677, \n CV fold 3: train 0..1677, 2093..4998, valid 1678..2092, \n CV fold 4: train 0..2092, 2532..4998, valid 2093..2531, \n CV fold 5: train 0..2531, 3288..4998, valid 2532..3287, \n CV fold 6: train 0..3287, valid 3288..4998, \n n_folds train_likelihood_ratio_m ... n_train_s n_valid_s\n sigma_multiplier ... \n 1.3 7 20.069907 ... 497.730994 497.730994\n 1.5 7 18.347998 ... 497.730994 497.730994\n 1.7 7 16.972466 ... 497.730994 497.730994\n \n [3 rows x 13 columns]\n Best hyperparameters: 1.7\n\n\n## More information\n\nYou can find more information in the [ramp-workflow documentation](https://paris-saclay-cds.github.io/ramp-workflow).\n\n## Contact\n\nUse Slack to contact us.\n\n\n```python\n\n```\n", "meta": {"hexsha": "fdd45a05e0110bdff2d3b2fa528fecc2fbde2869", "size": 399191, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "acrobot_starting_kit.ipynb", "max_stars_repo_name": "AyoubZarou/acrobot", "max_stars_repo_head_hexsha": "abb1d7660d9953864ad29a4278f347ee01f22c72", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2019-11-13T18:31:21.000Z", "max_stars_repo_stars_event_max_datetime": "2019-11-15T09:44:48.000Z", "max_issues_repo_path": "acrobot_starting_kit.ipynb", "max_issues_repo_name": "AyoubZarou/acrobot", "max_issues_repo_head_hexsha": "abb1d7660d9953864ad29a4278f347ee01f22c72", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2020-03-24T17:50:52.000Z", "max_issues_repo_issues_event_max_datetime": "2021-08-23T20:28:52.000Z", "max_forks_repo_path": "acrobot_starting_kit.ipynb", "max_forks_repo_name": "AyoubZarou/acrobot", "max_forks_repo_head_hexsha": "abb1d7660d9953864ad29a4278f347ee01f22c72", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2019-12-06T21:57:10.000Z", "max_forks_repo_forks_event_max_datetime": "2020-12-01T17:58:21.000Z", "avg_line_length": 392.1326129666, "max_line_length": 208125, "alphanum_fraction": 0.8188937125, "converted": true, "num_tokens": 10688, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358411176238, "lm_q2_score": 0.6584175072643413, "lm_q1q2_score": 0.42180585357286055}} {"text": "# Problema de selecci\u00f3n de portafolio con preferencias media-varianza\n\n\n\nEn la clase pasada hablamos acerca de:\n- preferencias,\n- funciones de utilidad,\n- la actitud de los inversionistas de cara al riesgo,\n- la aversi\u00f3n al riesgo, entre otros.\n\nTodas ellas son piezas que necesitamos para responder la pregunta de \u00bfc\u00f3mo un inversionista toma la decisi\u00f3n \u00f3ptima de selecci\u00f3n de portafolio?\n\nEn esta clase al fin estamos listos para ensamblar estos conceptos y escribir el problema de selecci\u00f3n de portafolios. \n\nEn el camino aprenderemos acerca del concepto de **utilidad esperada**, que nos permite trabajar con incertidumbre en el modelado econ\u00f3mico (una de las ideas m\u00e1s importantes en econom\u00eda). Esta idea tiene m\u00e1s de 60 a\u00f1os, y b\u00e1sicamente dice que los individuos, cuando est\u00e1n de cara a incertidumbre, maximizan el valor esperado de su utilidad (solo cierto si somos homo economicus). \n\nAdem\u00e1s del concepto de utilidad esperada, aprenderemos acerca de **preferencias media-varianza**. Es decir, supondremos que los inversionistas toman decisiones basados en un tipo particular de preferencias.\n\nCon lo anterior, estableceremos el problema de selecci\u00f3n de portafolios.\n\n**Objetivos:**\n\n- \u00bfQu\u00e9 es utilidad esperada?\n- \u00bfQu\u00e9 son las preferencias media-varianza?\n- Funciones de utilidad media-varianza.\n- Enunciado y soluci\u00f3n del problema b\u00e1sico de selecci\u00f3n de portafolio.\n\n*Referencia:*\n- Notas del curso \"Portfolio Selection and Risk Management\", Rice University, disponible en Coursera.\n___\n\n## 1. Utilidad esperada\n- B\u00e1sicamente combina las probabilidades de los resultados con c\u00f3mo los inversionistas se sienten con dichos resultados.\n- En otras palabras, la utilidad esperada multiplica la probabilidad de suceso de un evento con la utilidad que genera dicho evento.\n\nRecordemos que las *funciones de utilidad* permiten a los inversionistas expresar c\u00f3mo se sienten con los resultados, especialmente en los malos ratos. \n\nEntonces la *utilidad esperada* es una herramienta que nos permite cuantificar c\u00f3mo nos sentimos en nuestros malos momentos econ\u00f3micos, capturando el riesgo con la probabilidad de ocurrencia de dichos malos momentos. \n\nDado este marco de trabajo, cualquier decisi\u00f3n se puede escribir como la maximizaci\u00f3n de la utilidad esperada:\n\\begin{align}\n\\max_{\\theta} & \\quad E[U(W)], \\\\\n\\end{align}\nmediante la escogencia de cierta variable $\\theta$ (gastos, planes de ahorro, compra de activos, planes de producci\u00f3n, etc.).\n\nPara nuestros prop\u00f3sitos, la variable de decisi\u00f3n ser\u00e1n los pesos o ponderaciones del portafolio.\n\nAdicionalmente, en el contexto de la decisi\u00f3n de distribuci\u00f3n de la riqueza entre activos, el problema de maximizaci\u00f3n tendr\u00e1 com\u00fanmente las siguientes restricciones:\n- universo de inversi\u00f3n,\n- posici\u00f3n en los activos dados.\n\n**Ejemplo.** \n\nSupongamos que un inversionista debe determinar la composici\u00f3n \u00f3ptima de su portafolio, que contiene activos y bonos. Supongamos que son los \u00fanicos instrumentos disponibles.\n\nSean:\n- $w_s$: peso o ponderaci\u00f3n de activos en el portafolio,\n- $w_b$: peso o ponderaci\u00f3n de bonos en el portafolio,\n- $r_s$: rendimiento de los activos, y\n- $r_b$: rendimiento de los bonos.\n\nDe manera que podemos escribir el problema de selecci\u00f3n de portafolios como la maximizaci\u00f3n de la utilidad esperade de nuestra riqueza futura, la cual depender\u00e1 de nuestros rendimientos:\n \n\\begin{align}\n\\max_{w_s,w_b} &\\quad E[U(W)]\\\\\n\\text{s. a.} &\\quad W=W_0(1+w_sr_s+w_br_b)\\\\\n &\\quad w_s+w_b=1\\\\\n &\\quad w_s,w_b\\geq0\n\\end{align}\n\n\nPreguntas:\n- \u00bfQu\u00e9 significan las restricciones?\n- Ya que tenemos planteado este problema b\u00e1sico, \u00bfqu\u00e9 har\u00eda falta para empezar a resolverlo?\n___\n\n## 2. Preferencias media-varianza\n\n### 2.1. Utilidad media-varianza\n\nEntonces, \u00bfqu\u00e9 funciones de utilidad deber\u00edamos de usar en este problema de selecci\u00f3n de portafolios?\n\n- La respuesta es: **preferencias media-varianza**.\n- \u00c9stas ser\u00e1n representadas en t\u00e9rminos de funciones de utilidad como: **utilidad media-varianza**.\n\nUsamos la *utilidad media-varianza* en el problema de selecci\u00f3n de portafolios dado que \u00e9sta decribe el \"trade-off\" entre riesgo y rendimiento que enfrentan los inversionistas. La *utilidad media-varianza* est\u00e1 dada por la siguiente expresi\u00f3n:\n\n$$U=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2,$$\n\ndonde\n- $E[r_p]$ es el rendimiento esperado del portafolio,\n- $\\sigma_p^2$ es la varianza del portafolio, y\n- $\\gamma$ es el coeficiente de aversi\u00f3n al riesgo. \n\n#### Intuici\u00f3n acerca de la funci\u00f3n de utilidad media-varianza:\n- S\u00f3lo se preocupa por medias :) y varianzas :(.\n- Incrementa con: rendimiento esperado del portafolio.\n- Decrece con: varianza del portafolio.\n- Malos tiempos: rendimientos son bajos y las volatilidades son altas.\n- Conecta bastante bien con la teor\u00eda moderna de portafolios, la cual caracteriza los rendimientos con medias y varianzas \u00fanicamente.\n- Criticada por su limitaci\u00f3n: supone que los inversionistas s\u00f3lo se preocupan por medias y varianzas.\n\n### 2.2. Curvas de indiferencia\n\n*\u00bfRecuerdan las curvas de nivel que se ven en c\u00e1lculo de varias variables?*\n- Bien, ac\u00e1 nos servir\u00e1n para representar la utilidad media-varianza gr\u00e1ficamente.\n- En el contexto de utilidad media-varianza, las curvas de nivel se llaman **curvas de indiferencia**.\n\nDados ciertos niveles de utilidad $U_1>U_2>U_3$, las curvas de indiferencia relativas a estos niveles de utilidad, son los lugares geom\u00e9tricos en el espacio de rendimiento esperado vs. volatilidad representados por las siguientes expresiones\n\n$$U_1=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2\\Rightarrow E[r_p]=\\frac{1}{2}\\gamma\\sigma_p^2+U_1,$$\n\n$$U_2=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2\\Rightarrow E[r_p]=\\frac{1}{2}\\gamma\\sigma_p^2+U_2,$$\n\n$$U_3=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2\\Rightarrow E[r_p]=\\frac{1}{2}\\gamma\\sigma_p^2+U_3.$$\n\n**Gr\u00e1ficamente**\n\n\n```python\n# Importar numpy y pyplot\n\n```\n\n\n```python\n# Coeficiente de aversi\u00f3n al riesgo (entre 1 y 10 com\u00fanmente)\n\n# Niveles de utilidad\n\n# Vector de volatilidades (sugerido 0%-60%)\n\n# Curvas de indiferencia\n\n```\n\n\n```python\n# Gr\u00e1fica\n\n```\n\nBueno, \u00bfy porqu\u00e9 se llaman curvas de indiferencia?, \u00bfqu\u00e9 representa una cuva de indiferencia?\n\n- Porque sobre una misma curva el nivel de utilidad es el mismo (es indiferente).\n- Son todas las combinaciones de riesgo y rendimiento que producen un mismo nivel de utilidad.\n\nVolviendo al problema de selecci\u00f3n de portafolios, queremos la utilidad m\u00e1s alta.\n- \u00bfCu\u00e1l de las anteriores curvas de indiferencia corresponde a la utilidad m\u00e1s alta?\n- Intuitivamente, \u00bfporqu\u00e9?\n- Curvas de indiferencia para niveles de utilidad m\u00e1s altos, estar\u00e1n...\n\nNotamos adem\u00e1s que las anteriores curvas de indiferencia son *paralelas* una con otra. Claro, las dibujamos con el mismo coeficiente de aversi\u00f3n al riesgo.\n\n$$\\frac{d E[r_p]}{d\\sigma_p}=\\frac{d }{d\\sigma_p}\\left[\\frac{1}{2}\\gamma\\sigma_p^2+U\\right]=\\gamma\\sigma_p.$$\n\n\u00bfC\u00f3mo cambian estas curvas para coeficientes de aversi\u00f3n al riesgo m\u00e1s altos?\n\n\n```python\n# Coeficientes de aversi\u00f3n al riesgo (entre 1 y 10 com\u00fanmente)\n\n# Nivel de utilidad\n\n# Vector de volatilidades (sugerido 0%-60%)\n\n# Curvas de indiferencia\n\n```\n\n\n```python\n# Gr\u00e1fica\n\n```\n\n\u00bfC\u00f3mo interpretamos las anteriores gr\u00e1ficas?, \u00bfqu\u00e9 pasa con las personas m\u00e1s aversas al riesgo?\n- Se puede ver de dos maneras: para un mismo nivel de rendimiento esperado, una persona m\u00e1s aversa al riesgo soporta un nivel menor de riesgo; equivalentemente, para un mismo nivel de riesgo, una persona m\u00e1s aversa al riesgo requerir\u00e1 un nivel de rendimiento esperado m\u00e1s alto.\n\nCon todo lo anterior, el problema de selecci\u00f3n de portafolios se puede plantear como *encontrar la curva de indeferencia m\u00e1s alta dado el conjunto de oportunidades de inversi\u00f3n y restricciones*.\n\n## 3. Problema de selecci\u00f3n de portafolios: una ilustraci\u00f3n\n\nAhora ilustraremos el problema de selecci\u00f3n de portafolios con algunos datos. \n- Por ahora solo queremos ilustrar gr\u00e1ficamente c\u00f3mo se resuelve este problema. Trabajar en la intuici\u00f3n.\n- En las siguientes dos clases nos enfocaremos en c\u00f3mo resolverlo anal\u00edticamente.\n\nAc\u00e1 tenemos el rendimiento medio anual y la volatilidad para dos instrumentos usando datos de EU: instrumentos de deuda (bonos) y acciones. Supondremos que el inversionista solo puede invertir en estas dos clases de instrumentos.\n\n\n```python\n# Importamos pandas\nimport pandas as pd\n```\n\n\n```python\n# Datos\ndata = pd.DataFrame(index=['Stocks','Bonds', 'CorrSB'], columns=['Mean', 'Std'])\ndata['Mean'] = [0.119, 0.0591, 0.113]\ndata['Std'] = [0.1915, 0.0833, None]\ndata\n```\n\nEntonces, \u00bfcu\u00e1l es la distribuci\u00f3n de riqueza \u00f3ptima?, o m\u00e1s bien, \u00bfcu\u00e1l es la composici\u00f3n \u00f3ptima del portafolio para un inversionista dado su nivel de aversi\u00f3n al riesgo?\n\n**Primero.** Recordamos que, para dos activos, podemos trazar la frontera de m\u00ednima varianza tomando todas las posibles combinaciones de los dos activos.\n\nDe nuevo, sean:\n- $w_s=w$: peso o ponderaci\u00f3n de activos en el portafolio,\n- $w_b=1-w$: peso o ponderaci\u00f3n de bonos en el portafolio,\n- $r_s$: rendimiento de los activos, y\n- $r_b$: rendimiento de los bonos.\n\nEntonces\n\n$$E[r_p]=wE[r_{s}]+(1-w)E[r_b]$$\n\n$$\\sigma_p^2=w^2\\sigma_{s}^2+(1-w)^2\\sigma_b^2+2w(1-w)\\rho_{s,b}\\sigma_s\\sigma_b$$\n\n\n```python\n# Vector de w variando entre 0 y 1 con n pasos\n\n# Rendimientos esperados individuales\n\n# Volatilidades individuales\n\n# Correlacion\n\n```\n\n\n```python\n# Crear un DataFrame cuyas columnas sean rendimiento\n# y volatilidad del portafolio para cada una de las w\n# generadas\n\n```\n\n\n```python\n# Gr\u00e1fica\n\n```\n\n**Segundo.** Graficamos en la misma ventana, curvas de indiferencia.\n\n\n```python\n# Niveles de utilidad \n\n# Coeficiente de aversi\u00f3n al riesgo\n\n# Curvas de indiferencia\n\n```\n\n\n```python\n# Gr\u00e1fica\n\n```\n\n**Tercero.** La elecci\u00f3n \u00f3ptima est\u00e1 dada por la curva de indiferencia para el nivel de utilidad m\u00e1s alto que es tangente a la frontera media-varianza.\n- Claramente, esta selecci\u00f3n depende del coeficiente de aversi\u00f3n al riesgo.\n\n\n```python\n# Gr\u00e1fica con zoom\n\n```\n\n# Anuncios parroquiales\n\n## 1. Quiz la siguiente clase.\n\n\n\n
                                        \nCreated with Jupyter by Esteban Jim\u00e9nez Rodr\u00edguez.\n
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                                        \n \n\n

                                        \n\n## Subsurface Data Analytics \n\n### Kriging vs. Simulation for Spatial Predictions in Python \n\n#### Michael Pyrcz, Associate Professor, University of Texas at Austin \n\n##### [Twitter](https://twitter.com/geostatsguy) | [GitHub](https://github.com/GeostatsGuy) | [Website](http://michaelpyrcz.com) | [GoogleScholar](https://scholar.google.com/citations?user=QVZ20eQAAAAJ&hl=en&oi=ao) | [Book](https://www.amazon.com/Geostatistical-Reservoir-Modeling-Michael-Pyrcz/dp/0199731446) | [YouTube](https://www.youtube.com/channel/UCLqEr-xV-ceHdXXXrTId5ig) | [LinkedIn](https://www.linkedin.com/in/michael-pyrcz-61a648a1)\n\nLet's first cover spatial estimation and simulation.\n\n#### Spatial Estimation\n\nConsider the case of making an estimate at some unsampled location, $\ud835\udc67(\\bf{u}_0)$, where $z$ is the property of interest (e.g. porosity etc.) and $\ud835\udc2e_0$ is a location vector describing the unsampled location.\n\nHow would you do this given data, $\ud835\udc67(\\bf{\ud835\udc2e}_1)$, $\ud835\udc67(\\bf{\ud835\udc2e}_2)$, and $\ud835\udc67(\\bf{\ud835\udc2e}_3)$?\n\nIt would be natural to use a set of linear weights to formulate the estimator given the available data.\n\n\\begin{equation}\nz^{*}(\\bf{u}) = \\sum^{n}_{\\alpha = 1} \\lambda_{\\alpha} z(\\bf{u}_{\\alpha})\n\\end{equation}\n\nWe could add an unbiasedness constraint to impose the sum of the weights equal to one. What we will do is assign the remainder of the weight (one minus the sum of weights) to the global average; therefore, if we have no informative data we will estimate with the global average of the property of interest.\n\n\\begin{equation}\nz^{*}(\\bf{u}) = \\sum^{n}_{\\alpha = 1} \\lambda_{\\alpha} z(\\bf{u}_{\\alpha}) + \\left(1-\\sum^{n}_{\\alpha = 1} \\lambda_{\\alpha} \\right) \\overline{z}\n\\end{equation}\n\nWe will make a stationarity assumption, so let's assume that we are working with residuals, $y$. \n\n\\begin{equation}\ny^{*}(\\bf{u}) = z^{*}(\\bf{u}) - \\overline{z}(\\bf{u})\n\\end{equation}\n\nIf we substitute this form into our estimator the estimator simplifies, since the mean of the residual is zero.\n\n\\begin{equation}\ny^{*}(\\bf{u}) = \\sum^{n}_{\\alpha = 1} \\lambda_{\\alpha} y(\\bf{u}_{\\alpha})\n\\end{equation}\n\nwhile satisfying the unbaisedness constraint. \n\n#### Kriging\n\nNow the next question is what weights should we use? \n\nWe could use equal weighting, $\\lambda = \\frac{1}{n}$, and the estimator would be the average of the local data applied for the spatial estimate. This would not be very informative.\n\nWe could assign weights considering the spatial context of the data and the estimate:\n\n* **spatial continuity** as quantified by the variogram (and covariance function)\n* **redundancy** the degree of spatial continuity between all of the available data with themselves \n* **closeness** the degree of spatial continuity between the avaiable data and the estimation location\n\nThe kriging approach accomplishes this, calculating the best linear unbiased weights for the local data to estimate at the unknown location. The derivation of the kriging system and the resulting linear set of equations is available in the lecture notes. Furthermore kriging provides a measure of the accuracy of the estimate! This is the kriging estimation variance (sometimes just called the kriging variance).\n\n\\begin{equation}\n\\sigma^{2}_{E}(\\bf{u}) = C(0) - \\sum^{n}_{\\alpha = 1} \\lambda_{\\alpha} C(\\bf{u}_0 - \\bf{u}_{\\alpha})\n\\end{equation}\n\nWhat is 'best' about this estimate? Kriging estimates are best in that they minimize the above estimation variance. \n\n#### Properties of Kriging\n\nHere are some important properties of kriging:\n\n* **Exact interpolator** - kriging estimates with the data values at the data locations\n* **Kriging variance** can be calculated before getting the sample information, as the kriging estimation variance is not dependent on the values of the data nor the kriging estimate, i.e. the kriging estimator is homoscedastic. \n* **Spatial context** - kriging takes into account, furthermore to the statements on spatial continuity, closeness and redundancy we can state that kriging accounts for the configuration of the data and structural continuity of the variable being estimated.\n* **Scale** - kriging may be generalized to account for the support volume of the data and estimate. We will cover this later.\n* **Multivariate** - kriging may be generalized to account for multiple secondary data in the spatial estimate with the cokriging system. We will cover this later.\n* **Smoothing effect** of kriging can be forecast. We will use this to build stochastic simulations later.\n\nI have more on this topic at [Krigign YouTube Lecture](https://youtu.be/CVkmuwF8cJ8).\n\n#### Sequential Gaussian Simulation\n\nWith sequential Gaussian simulation we build on kriging by:\n\n* adding a random residual with the missing variance\n\n* sequentially adding the simulated values as data to correct the covariance between the simulated values\n\nThe resulting model corrects the issues of kriging, as we now:\n\n* reproduce the global feature PDF / CDF\n\n* reproduce the global variogram\n\n* while providing a model of uncertainty through multiple realizations\n\nI have more on this topic at [Simulation YouTube Lecture](https://www.youtube.com/watch?v=3cLqK3lR56Y&list=PLG19vXLQHvSB-D4XKYieEku9GQMQyAzjJ&index=45&t=813s).\n\nThis is a tutorial for / comparing spatial estimation and simulation we use **Simple Kriging** and **Sequential Gaussian Simulation in Python with the GeostatsPy package (Pyrcz et al., 2021), GSLIB's KB2D and SGSIM programs translated to Python from the original FORTRAN GSLIB: Geostatistical Library methods** (Deutsch and Journel, 1997). \n\n#### Getting Started\n\nHere's the steps to get setup in Python with the GeostatsPy package:\n\n1. Install Anaconda 3 on your machine (https://www.anaconda.com/download/). \n2. From Anaconda Navigator (within Anaconda3 group), go to the environment tab, click on base (root) green arrow and open a terminal. \n3. In the terminal type: pip install geostatspy. \n4. Open Jupyter and in the top block get started by copy and pasting the code block below from this Jupyter Notebook to start using the geostatspy functionality. \n\nYou will need to copy the data file to your working directory. They are available here:\n\n* Tabular data - sample_data_MV_biased.csv available at https://git.io/fhgu0.\n\nThere are exampled below with these functions. You can go here to see a list of the available functions, https://git.io/fh4eX, other example workflows and source code. \n\n#### Load the Required Libraries\n\nThe following code loads the required libraries.\n\n\n```python\nimport os # to set current working directory \nimport numpy as np # arrays and matrix math\nimport pandas as pd # DataFrames\nimport matplotlib.pyplot as plt # plotting\ncmap = plt.cm.inferno # color map\nimport geostatspy.geostats as geostats\nimport geostatspy.GSLIB as GSLIB\n```\n\nIf you get a package import error, you may have to first install some of these packages. This can usually be accomplished by opening up a command window on Windows and then typing 'python -m pip install [package-name]'. More assistance is available with the respective package docs. \n\n#### Declare Functions\n\nHere's a convenience function for plotting variograms.\n\n\n```python\ndef vargplot(feature,lags,gamma_maj,gamma_min,npps_maj,npps_min,vmodel,azi,atol,sill): # plot the variogram\n index_maj,lags_maj,gmod_maj,cov_maj,ro_maj = geostats.vmodel(nlag=100,xlag=10,azm=azi,vario=vmodel);\n index_min,lags_min,gmod_min,cov_min,ro_min = geostats.vmodel(nlag=100,xlag=10,azm=azi+90.0,vario=vmodel);\n \n plt.scatter(lags,gamma_maj,color = 'black',s = npps_maj*0.03,label = 'Major Azimuth ' +str(azi), alpha = 0.8)\n plt.plot(lags_maj,gmod_maj,color = 'black')\n plt.scatter(lags,gamma_min,color = 'red',s = npps_min*0.03,label = 'Minor Azimuth ' +str(azi+90.0), alpha = 0.8)\n plt.plot(lags_min,gmod_min,color = 'red')\n plt.plot([0,2000],[sill,sill],color = 'black')\n plt.xlabel(r'Lag Distance $\\bf(h)$, (m)')\n plt.ylabel(r'$\\gamma \\bf(h)$')\n if atol < 90.0:\n plt.title('Directional ' + feature + ' Variogram')\n else: \n plt.title('Omni Directional NSCORE ' + feature + ' Variogram')\n plt.xlim([0,1000]); #plt.ylim([0,1.8])\n plt.legend(loc=\"lower right\")\n plt.grid(True)\n```\n\n#### Set the working directory\n\nI always like to do this so I don't lose files and to simplify subsequent read and writes (avoid including the full address each time). Also, in this case make sure to place the required (see above) GSLIB executables in this directory or a location identified in the environmental variable *Path*.\n\n\n```python\n#os.chdir(\"c:/PGE383\") # set the working directory\n```\n\n#### Loading Tabular Data\n\nHere's the command to load our comma delimited data file in to a Pandas' DataFrame object. We will also extra a limited sample so that the spatial samples are not too dense. This way we can observe more of the heterogeneity from the simulation with the spatial continuity model, rather than mostly data driven heterogeneity.\n\n\n```python\n#df = pd.read_csv(\"sample_data_MV_biased.csv\") # read a .csv file in as a DataFrame\n#df = pd.read_csv(r\"https://raw.githubusercontent.com/GeostatsGuy/GeoDataSets/master/sample_data_MV_biased.csv\") # from Dr. Pyrcz's GitHub repo\ndf = pd.read_csv(r\"https://raw.githubusercontent.com/GeostatsGuy/GeoDataSets/master/spatial_nonlinear_MV_facies_v1.csv\")\ndf = df.rename(columns = {'Por':'Porosity'}) # rename feature(s)\ndf = df.loc[:,['X','Y','Porosity']]; #df['Porosity'] = df['Porosity']*100.0\ndf.describe() # summary statistics \n#df = df.sample(50) # extract 50 samples\n#df = df.reset_index() # reset the record index \n#df.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        XYPorosity
                                        count457.000000457.000000457.000000
                                        mean544.215378524.29226713.017708
                                        std295.136181282.5022474.695742
                                        min1.21968110.0063912.666539
                                        25%301.455681296.6484079.453725
                                        50%580.575433535.73781813.250704
                                        75%795.864122776.15369116.090334
                                        max1006.9524891000.08879425.653330
                                        \n
                                        \n\n\n\n#### Set Limits for Plotting\n\nThis is applied for data and model visualization.\n\n\n```python\npormin = 0.0; pormax = 22.0; pormean = 12.7; porvar = 22.0\nxmin = 0.0; xmax = 1000.0\nymin = 0.0; ymax = 1000.0\ntmin = -9999.9; tmax = 9999.9\nnx = 100; xmn = 5.0; xsiz = 10.0\nny = 100; ymn = 5.0; ysiz = 10.0\n```\n\n#### Data Analytics and Visualization\n\nLet's take a look at the available data:\n\n* location map\n* histogram\n* variogram\n\n\n```python\nplt.subplot(131)\nGSLIB.locmap_st(df,'X','Y','Porosity',0,1000,0,1000,0,25,'Porosity Location Map','X (m)','Y (m)','Porosity',cmap=cmap)\n\nplt.subplot(132)\nplt.hist(df['Porosity'].values,bins=np.linspace(pormin,pormax,50),color='red',alpha=0.2,edgecolor='black')\nplt.xlabel('Porosity (%)'); plt.ylabel('Frequency'); plt.title('Porosity Histogram')\n\n\ndf['Npor'], tvPor, tnsPor = geostats.nscore(df,'Porosity') \nlags, gamma_maj, npps_maj = geostats.gamv(df,\"X\",\"Y\",'Porosity',tmin,tmax,xlag=20,xltol=20,nlag=100,azm=0.0,atol=22.5,bandwh=9999.9,isill=0);\nlags, gamma_min, npps_min = geostats.gamv(df,\"X\",\"Y\",'Porosity',tmin,tmax,xlag=20,xltol=20,nlag=100,azm=90.0,atol=22.5,bandwh=9999.9,isill=0);\n\nnug = 0.0; nst = 2 # 2 nest structure variogram model parameters\nit1 = 2; cc1 = 19.0; azi1 = 0; hmaj1 = 150; hmin1 = 150\nit2 = 2; cc2 = 3.0; azi2 = 0; hmaj2 = 1000; hmin2 = 150\n\nvmodel = GSLIB.make_variogram(nug,nst,it1,cc1,azi1,hmaj1,hmin1,it2,cc2,azi2,hmaj2,hmin2); # make model object\nvmodel_sim = GSLIB.make_variogram(nug,nst,it1,cc1/(cc1+cc2),azi1,hmaj1,hmin1,it2,cc2/(cc1+cc2),azi2,hmaj2,hmin2); # make model object\n\nplt.subplot(133)\nvargplot('Porosity',lags,gamma_maj,gamma_min,npps_maj,npps_min,vmodel,azi=0.0,atol=22.5,sill=porvar) # plot the variogram\n\nplt.subplots_adjust(left=0.0, bottom=0.0, right=3.0, top=1.1, wspace=0.2, hspace=0.2)\nplt.show()\n```\n\n#### Simple Kriging\n\nLet's make a simple kriging, estimation map and calculate the same statistics again to check the reproduction.\n\n\n```python\npor_kmap, por_vmap = geostats.kb2d(df,'X','Y','Porosity',tmin,tmax,nx,xmn,xsiz,ny,ymn,ysiz,nxdis=1,nydis=1,\n ndmin=0,ndmax=10,radius=500,ktype=0,skmean=pormean,vario=vmodel)\n\nplt.subplot(131) # plot the results\nGSLIB.locpix_st(por_kmap,xmin,xmax,ymin,ymax,xsiz,pormin,pormax,df,'X','Y','Porosity','Simple Kriging Porosity','X(m)','Y(m)','Porosity (%)',cmap)\n\nplt.subplot(132)\nplt.hist(df['Porosity'].values,density=True,bins=np.linspace(pormin,pormax,50),color='red',alpha=0.2,edgecolor='black',label='Data')\nplt.hist(por_kmap.flatten(),density=True,bins=np.linspace(pormin,pormax,50),color='blue',alpha=0.2,edgecolor='black',label='Kriging')\nplt.xlabel('Porosity (%)'); plt.ylabel('Frequency'); plt.title('Porosity Histogram'); plt.legend(loc='upper right')\n\nlags, sk_gamma_maj, npps_maj = geostats.gam(por_kmap,tmin,tmax,xsiz,ysiz,ixd=1,iyd=-1,nlag=100,isill=0.0);\nlags, sk_gamma_min, npps_min = geostats.gam(por_kmap,tmin,tmax,xsiz,ysiz,ixd=1,iyd=1,nlag=100,isill=0.0);\n\nplt.subplot(133)\nvargplot('Porosity',lags,sk_gamma_maj,sk_gamma_min,npps_maj,npps_min,vmodel,azi=0.0,atol=22.5,sill=porvar) # plot the variogram\n\nplt.subplots_adjust(left=0.0, bottom=0.0, right=3.0, top=1.1, wspace=0.2, hspace=0.2)\nplt.show()\n```\n\n#### Sequential Gaussian Simulation\n\nLet's jump right to building a variety of models with simulation and visualizing the results. We will start with a test, comparision of simulation with simple and ordinary kriging.\n\n\n```python\n# Simple Kriging vs. Ordinary Kriging\n\npor_sim = geostats.sgsim(df,'X','Y','Porosity',wcol=-1,scol=-1,tmin=tmin,tmax=tmax,itrans=1,ismooth=0,dftrans=0,tcol=0,\n twtcol=0,zmin=pormin,zmax=pormax,ltail=1,ltpar=0.0,utail=1,utpar=0.3,nsim=1,\n nx=nx,xmn=xmn,xsiz=xsiz,ny=ny,ymn=ymn,ysiz=ysiz,seed=73073,\n ndmin=0,ndmax=20,nodmax=20,mults=0,nmult=2,noct=-1,radius=500,radius1=500,sang1=0,\n mxctx=41,mxcty=41,ktype=0,colocorr=0.0,sec_map=0,vario=vmodel_sim)\n\nplt.subplot(131) # plot the results\nGSLIB.locpix_st(por_sim,xmin,xmax,ymin,ymax,xsiz,pormin,pormax,df,'X','Y','Porosity','Sequential Gaussian Simulation Porosity','X(m)','Y(m)','Porosity (%)',cmap)\n\nplt.subplot(132)\nplt.hist(df['Porosity'].values,density=True,bins=np.linspace(pormin,pormax,50),color='red',alpha=0.2,edgecolor='black',label='Data')\nplt.hist(por_sim.flatten(),density=True,bins=np.linspace(pormin,pormax,50),color='yellow',alpha=0.2,edgecolor='black',label='Simulation')\nplt.xlabel('Porosity (%)'); plt.ylabel('Frequency'); plt.title('Porosity Histogram'); plt.legend(loc='upper right')\n\nlags, sim_gamma_maj, npps_maj = geostats.gam(por_sim,tmin,tmax,xsiz,ysiz,ixd=1,iyd=-1,nlag=100,isill=0.0);\nlags, sim_gamma_min, npps_min = geostats.gam(por_sim,tmin,tmax,xsiz,ysiz,ixd=1,iyd=1,nlag=100,isill=0.0);\n\nplt.subplot(133)\nvargplot('Porosity',lags,sim_gamma_maj,sim_gamma_min,npps_maj,npps_min,vmodel,azi=0.0,atol=22.5,sill=porvar) # plot the variogram\n\nplt.subplots_adjust(left=0.0, bottom=0.0, right=3.0, top=1.1, wspace=0.2, hspace=0.2)\nplt.show()\n```\n\n#### Comments\n\nThis was a basic demonstration comparison of spatial estimation and simulation. \n\nMuch more could be done, I have other demonstrations on the basics of working with DataFrames, ndarrays, univariate statistics, plotting data, declustering, data transformations and many other workflows available at https://github.com/GeostatsGuy/PythonNumericalDemos and https://github.com/GeostatsGuy/GeostatsPy. \n \nI hope this was helpful,\n\n*Michael*\n\n#### The Author:\n\n### Michael Pyrcz, Associate Professor, University of Texas at Austin \n*Novel Data Analytics, Geostatistics and Machine Learning Subsurface Solutions*\n\nWith over 17 years of experience in subsurface consulting, research and development, Michael has returned to academia driven by his passion for teaching and enthusiasm for enhancing engineers' and geoscientists' impact in subsurface resource development. \n\nFor more about Michael check out these links:\n\n#### [Twitter](https://twitter.com/geostatsguy) | [GitHub](https://github.com/GeostatsGuy) | [Website](http://michaelpyrcz.com) | [GoogleScholar](https://scholar.google.com/citations?user=QVZ20eQAAAAJ&hl=en&oi=ao) | [Book](https://www.amazon.com/Geostatistical-Reservoir-Modeling-Michael-Pyrcz/dp/0199731446) | [YouTube](https://www.youtube.com/channel/UCLqEr-xV-ceHdXXXrTId5ig) | [LinkedIn](https://www.linkedin.com/in/michael-pyrcz-61a648a1)\n\n#### Want to Work Together?\n\nI hope this content is helpful to those that want to learn more about subsurface modeling, data analytics and machine learning. Students and working professionals are welcome to participate.\n\n* Want to invite me to visit your company for training, mentoring, project review, workflow design and / or consulting? I'd be happy to drop by and work with you! \n\n* Interested in partnering, supporting my graduate student research or my Subsurface Data Analytics and Machine Learning consortium (co-PIs including Profs. Foster, Torres-Verdin and van Oort)? My research combines data analytics, stochastic modeling and machine learning theory with practice to develop novel methods and workflows to add value. We are solving challenging subsurface problems!\n\n* I can be reached at mpyrcz@austin.utexas.edu.\n\nI'm always happy to discuss,\n\n*Michael*\n\nMichael Pyrcz, Ph.D., P.Eng. Associate Professor The Hildebrand Department of Petroleum and Geosystems Engineering, Bureau of Economic Geology, The Jackson School of Geosciences, The University of Texas at Austin\n\n#### More Resources Available at: [Twitter](https://twitter.com/geostatsguy) | [GitHub](https://github.com/GeostatsGuy) | [Website](http://michaelpyrcz.com) | [GoogleScholar](https://scholar.google.com/citations?user=QVZ20eQAAAAJ&hl=en&oi=ao) | [Book](https://www.amazon.com/Geostatistical-Reservoir-Modeling-Michael-Pyrcz/dp/0199731446) | [YouTube](https://www.youtube.com/channel/UCLqEr-xV-ceHdXXXrTId5ig) | [LinkedIn](https://www.linkedin.com/in/michael-pyrcz-61a648a1)\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "a56624fbe1f7848f2185353617eed603905400e3", "size": 659547, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "GeostatsPy_kriging_vs_simulation.ipynb", "max_stars_repo_name": "caf3676/PythonNumericalDemos", "max_stars_repo_head_hexsha": "206a3d876f79e137af88b85ba98aff171e8d8e06", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "GeostatsPy_kriging_vs_simulation.ipynb", "max_issues_repo_name": "caf3676/PythonNumericalDemos", "max_issues_repo_head_hexsha": "206a3d876f79e137af88b85ba98aff171e8d8e06", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "GeostatsPy_kriging_vs_simulation.ipynb", "max_forks_repo_name": "caf3676/PythonNumericalDemos", "max_forks_repo_head_hexsha": "206a3d876f79e137af88b85ba98aff171e8d8e06", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-03-14T03:28:32.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-14T03:28:32.000Z", "avg_line_length": 1035.3956043956, "max_line_length": 236260, "alphanum_fraction": 0.9495972235, "converted": true, "num_tokens": 5770, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.7401743563075447, "lm_q1q2_score": 0.42179031945062134}} {"text": "```python\n%load_ext autoreload\n%autoreload 2\n\n%load_ext nb_black\n```\n\n\n \n\n\n\n```python\n#export\nfrom exp.nb_06 import *\n```\n\n\n \n\n\n[Jump_to lesson 10 video](https://course.fast.ai/videos/?lesson=10&t=5899)\n\n## ConvNet\n\nLet's get the data and training interface from where we left in the last notebook.\n\n\n```python\nx_train, y_train, x_valid, y_valid = get_data()\n\nx_train, x_valid = normalize_to(x_train, x_valid)\ntrain_ds, valid_ds = Dataset(x_train, y_train), Dataset(x_valid, y_valid)\n\nnh, bs = 50, 512\nc = y_train.max().item() + 1\nloss_func = F.cross_entropy\n\ndata = DataBunch(*get_dls(train_ds, valid_ds, bs), c)\n```\n\n\n \n\n\n\n```python\nmnist_view = view_tfm(1, 28, 28)\ncbfs = [\n Recorder,\n partial(AvgStatsCallback, accuracy),\n partial(BatchTransformXCallback, mnist_view)\n]\n```\n\n\n \n\n\n\n```python\nnfs = [8, 16, 32, 64, 64]\n```\n\n\n \n\n\n\n```python\nlearn, run = get_learn_run(nfs, data, 0.4, conv_layer, cbs=cbfs)\n```\n\n\n \n\n\n\n```python\n%time run.fit(3, learn)\n```\n\n train: [1.52496328125, tensor(0.4912)]\n valid: [0.29743173828125, tensor(0.9101)]\n train: [0.23856525390625, tensor(0.9270)]\n valid: [0.134212939453125, tensor(0.9579)]\n train: [0.13307724609375, tensor(0.9591)]\n valid: [0.11205748291015626, tensor(0.9638)]\n CPU times: user 32 s, sys: 2.77 s, total: 34.8 s\n Wall time: 17.7 s\n\n\n\n \n\n\n## Batchnorm\n\n### Custom\n\nLet's start by building our own `BatchNorm` layer from scratch. In the case of computer vision, the average and std are by channel. Therefore, it is as if we concatenate all images with their heights and width by channel. Therefore, the output would be channels x (batches * height * width).\n\n`self.register_buffer`:\n\n`self.register_buffer('vars', torch.ones(1,nf,1,1))` is almost the same as `self.vars = torch.ones(1,nf,1,1))` but has the advantage that if we move everything to GPU, they will be moved to GPU too; otherwise, the won't and we will get an error when we try to do some calculations because you can't do calculations on stuff that are on CPU and GPU. It also allows us to save them when we save the model to disk because we need them at inference time.\n\n\n```python\nclass BatchNorm(nn.Module):\n __constants__ = ['eps', 'mom']\n\n def __init__(self, nf, mom=.1, eps=1e-5):\n super().__init__()\n # NB: pytorch bn mom is opposite of what you'd expect\n # This means the beta in exponential weighted average is\n # actually 1 - mom. Therefore, beta is 0.9 here.\n self.eps = eps\n self.mom = mom\n self.gamma = nn.Parameter(torch.ones(nf, 1, 1))\n self.beta = nn.Parameter(torch.zeros(nf, 1, 1))\n self.register_buffer('means', torch.ones(1, nf, 1, 1))\n self.register_buffer('vars', torch.ones(1, nf, 1, 1))\n\n def update_stats(self, x):\n # Average across batches, height and width --> average across channels.\n mean = x.mean((0, 2, 3), keepdim=True)\n var = x.var((0, 2, 3), keepdim=True)\n\n # means = (1 - mom) x means + mom x m; exponentially weighted average\n self.means.lerp_(mean, self.mom)\n self.vars.lerp_(var, self.mom)\n return mean, var\n\n def forward(self, x):\n if self.training:\n mean, var = self.update_stats(x)\n else:\n mean, var = self.means, self.vars\n x = (x - mean) / (var + self.eps).sqrt()\n return self.gamma * x + self.beta\n```\n\n\n \n\n\n\n```python\ndef conv_layer(ni, nf, ks=3, stride=2, bn=True, **kwargs):\n # No bias needed if using bn\n layers = [\n nn.Conv2d(ni, nf, ks, padding=ks // 2, stride=stride, bias=not bn),\n GeneralRelu(**kwargs)\n ]\n if bn: \n layers.append(BatchNorm(nf))\n return nn.Sequential(*layers)\n```\n\n\n \n\n\n\n```python\n#export\ndef init_cnn_(m, f):\n if isinstance(m, nn.Conv2d):\n f(m.weight, a=0.1)\n if getattr(m, 'bias', None) is not None:\n m.bias.data.zero_()\n for l in m.children():\n init_cnn_(l, f)\n\n\ndef init_cnn(m, uniform=False):\n f = init.kaiming_uniform_ if uniform else init.kaiming_normal_\n init_cnn_(m, f)\n\n\ndef get_learn_run(nfs,\n data,\n lr,\n layer,\n cbs=None,\n opt_func=None,\n uniform=False,\n **kwargs):\n model = get_cnn_model(data, nfs, layer, **kwargs)\n init_cnn(model, uniform=uniform)\n return get_runner(model, data, lr=lr, cbs=cbs, opt_func=opt_func)\n```\n\n\n \n\n\nWe can then use it in training and see how it helps keep the activations means to 0 and the std to 1.\n\n\n```python\nlearn, run = get_learn_run(nfs, data, 0.9, conv_layer, cbs=cbfs)\nlearn.model\n```\n\n\n\n\n Sequential(\n (0): Sequential(\n (0): Conv2d(1, 8, kernel_size=(5, 5), stride=(2, 2), padding=(2, 2), bias=False)\n (1): GeneralRelu()\n (2): BatchNorm()\n )\n (1): Sequential(\n (0): Conv2d(8, 16, kernel_size=(3, 3), stride=(2, 2), padding=(1, 1), bias=False)\n (1): GeneralRelu()\n (2): BatchNorm()\n )\n (2): Sequential(\n (0): Conv2d(16, 32, kernel_size=(3, 3), stride=(2, 2), padding=(1, 1), bias=False)\n (1): GeneralRelu()\n (2): BatchNorm()\n )\n (3): Sequential(\n (0): Conv2d(32, 64, kernel_size=(3, 3), stride=(2, 2), padding=(1, 1), bias=False)\n (1): GeneralRelu()\n (2): BatchNorm()\n )\n (4): Sequential(\n (0): Conv2d(64, 64, kernel_size=(3, 3), stride=(2, 2), padding=(1, 1), bias=False)\n (1): GeneralRelu()\n (2): BatchNorm()\n )\n (5): AdaptiveAvgPool2d(output_size=1)\n (6): Lambda()\n (7): Linear(in_features=64, out_features=10, bias=True)\n )\n\n\n\n\n \n\n\n\n```python\nwith Hooks(learn.model, compute_stats, True) as hooks:\n run.fit(1, learn)\n fig, (ax0, ax1) = plt.subplots(1, 2, figsize=(10, 4))\n for h in hooks[:-1]:\n ms, ss = h.stats\n ax0.plot(ms[:10])\n ax1.plot(ss[:10])\n h.remove()\n plt.legend(range(6))\n\n fig, (ax0, ax1) = plt.subplots(1, 2, figsize=(10, 4))\n for h in hooks[:-1]:\n ms, ss = h.stats\n ax0.plot(ms)\n ax1.plot(ss)\n```\n\n\n```python\nlearn, run = get_learn_run(nfs, data, 1.0, conv_layer, cbs=cbfs)\n```\n\n\n \n\n\n\n```python\n%time run.fit(3, learn)\n```\n\n train: [0.212331796875, tensor(0.9339)]\n valid: [0.08413507690429688, tensor(0.9751)]\n train: [0.0620726171875, tensor(0.9807)]\n valid: [0.06035379638671875, tensor(0.9833)]\n train: [0.04231357421875, tensor(0.9870)]\n valid: [0.06713604736328126, tensor(0.9784)]\n CPU times: user 57.5 s, sys: 4.73 s, total: 1min 2s\n Wall time: 31.3 s\n\n\n\n \n\n\n### Builtin batchnorm\n\n\n```python\n#export\ndef conv_layer(ni, nf, ks=3, stride=2, bn=True, **kwargs):\n layers = [\n nn.Conv2d(ni, nf, ks, padding=ks // 2, stride=stride, bias=not bn),\n GeneralRelu(**kwargs)\n ]\n if bn: layers.append(nn.BatchNorm2d(nf, eps=1e-5, momentum=0.1))\n return nn.Sequential(*layers)\n```\n\n\n \n\n\n\n```python\nlearn, run = get_learn_run(nfs, data, 1., conv_layer, cbs=cbfs)\n```\n\n\n \n\n\n\n```python\n%time run.fit(3, learn)\n```\n\n train: [0.213002734375, tensor(0.9338)]\n valid: [0.09222581787109375, tensor(0.9714)]\n train: [0.060083720703125, tensor(0.9812)]\n valid: [0.06774615478515625, tensor(0.9796)]\n train: [0.04006894775390625, tensor(0.9873)]\n valid: [0.06115048828125, tensor(0.9799)]\n CPU times: user 47 s, sys: 2.93 s, total: 49.9 s\n Wall time: 25.1 s\n\n\n\n \n\n\n### With scheduler\n\nNow let's add the usual warm-up/annealing.\n\n\n```python\nsched = combine_scheds([0.3, 0.7], [lin_sched(0.6, 2.), lin_sched(2., 0.1)])\n```\n\n\n \n\n\n\n```python\nlearn, run = get_learn_run(nfs,\n data,\n 0.9,\n conv_layer,\n cbs=cbfs + [partial(ParamScheduler, 'lr', sched)])\n```\n\n\n \n\n\n\n```python\nrun.fit(3, learn)\n```\n\n /Users/imad/Documents/courses/fastai-courses/dl2/notebooks/exp/nb_05.py:81: UserWarning: This overload of nonzero is deprecated:\n \tnonzero()\n Consider using one of the following signatures instead:\n \tnonzero(*, bool as_tuple) (Triggered internally at /Users/distiller/project/conda/conda-bld/pytorch_1595629430416/work/torch/csrc/utils/python_arg_parser.cpp:766.)\n idx = (pos >= pcts).nonzero().max()\n\n\n train: [0.26774140625, tensor(0.9175)]\n valid: [0.10132388916015625, tensor(0.9699)]\n train: [0.069213349609375, tensor(0.9783)]\n valid: [0.056937982177734374, tensor(0.9831)]\n train: [0.03427287109375, tensor(0.9895)]\n valid: [0.049336358642578126, tensor(0.9853)]\n\n\n\n \n\n\n## More norms\n\n### Layer norm\n\nFrom [the paper](https://arxiv.org/abs/1607.06450): \"*batch normalization cannot be applied to online learning tasks or to extremely large distributed models where the minibatches have to be small*\".\n\nThe main challenges to **BatchNorm** are:\n- How do we handle very small batches because we would have infinite variance or unstable training.\n- How do we handle RNNs.\n\nGeneral equation for a norm layer with learnable affine:\n\n$$y = \\frac{x - \\mathrm{E}[x]}{ \\sqrt{\\mathrm{Var}[x] + \\epsilon}} * \\gamma + \\beta$$\n\nThe difference with BatchNorm is\n1. we don't keep a moving average\n2. we don't average over the batches dimension but over the hidden dimension, so it's independent of the batch size\n\n\n```python\nclass LayerNorm(nn.Module):\n __constants__ = ['eps']\n\n def __init__(self, eps=1e-5):\n super().__init__()\n self.eps = eps\n self.gamma = nn.Parameter(tensor(1.))\n self.beta = nn.Parameter(tensor(0.))\n\n def forward(self, x):\n mean = x.mean((1, 2, 3), keepdim=True)\n var = x.var((1, 2, 3), keepdim=True)\n x = (x - mean) / ((var + self.eps).sqrt())\n return x * self.gamma + self.beta\n```\n\n\n \n\n\n\n```python\ndef conv_ln(ni, nf, ks=3, stride=2, bn=True, **kwargs):\n layers = [\n nn.Conv2d(ni, nf, ks, padding=ks // 2, stride=stride, bias=True),\n GeneralRelu(**kwargs)\n ]\n if bn: layers.append(LayerNorm())\n return nn.Sequential(*layers)\n```\n\n\n \n\n\n\n```python\nlearn, run = get_learn_run(nfs, data, 0.8, conv_ln, cbs=cbfs)\n```\n\n\n \n\n\n\n```python\n%time run.fit(3, learn)\n```\n\n train: [0.4233333984375, tensor(0.8652)]\n valid: [0.1118630859375, tensor(0.9647)]\n train: [0.085477177734375, tensor(0.9743)]\n valid: [0.08900013427734375, tensor(0.9751)]\n train: [0.0546854052734375, tensor(0.9825)]\n valid: [0.07122665405273437, tensor(0.9788)]\n CPU times: user 54.1 s, sys: 4.75 s, total: 58.8 s\n Wall time: 29.6 s\n\n\n\n \n\n\n*Thought experiment*: can this distinguish foggy days from sunny days (assuming you're using it before the first conv)?\n\nIt would be hard to distinguish between them because it forces the means and variances to be the same after normalizing. It throws away the difference in activations and if at inference time there is a difference in distribution that we care about, it would throw that away as well.\n\n### Instance norm\n\nFrom [the paper](https://arxiv.org/abs/1607.08022): \n\nThe key difference between **contrast** and batch normalization is that the latter applies the normalization to a whole batch of images instead for single ones:\n\n\\begin{equation}\\label{eq:bnorm}\n y_{tijk} = \\frac{x_{tijk} - \\mu_{i}}{\\sqrt{\\sigma_i^2 + \\epsilon}},\n \\quad\n \\mu_i = \\frac{1}{HWT}\\sum_{t=1}^T\\sum_{l=1}^W \\sum_{m=1}^H x_{tilm},\n \\quad\n \\sigma_i^2 = \\frac{1}{HWT}\\sum_{t=1}^T\\sum_{l=1}^W \\sum_{m=1}^H (x_{tilm} - mu_i)^2.\n\\end{equation}\n\nIn order to combine the effects of instance-specific normalization and batch normalization, we propose to replace the latter by the *instance normalization* (also known as *contrast normalization*) layer:\n\n\\begin{equation}\\label{eq:inorm}\n y_{tijk} = \\frac{x_{tijk} - \\mu_{ti}}{\\sqrt{\\sigma_{ti}^2 + \\epsilon}},\n \\quad\n \\mu_{ti} = \\frac{1}{HW}\\sum_{l=1}^W \\sum_{m=1}^H x_{tilm},\n \\quad\n \\sigma_{ti}^2 = \\frac{1}{HW}\\sum_{l=1}^W \\sum_{m=1}^H (x_{tilm} - mu_{ti})^2.\n\\end{equation}\n\n\n```python\nclass InstanceNorm(nn.Module):\n __constants__ = ['eps']\n\n def __init__(self, nf, eps=1e-0):\n super().__init__()\n self.eps = eps\n self.gamma = nn.Parameter(torch.ones(nf, 1, 1))\n self.beta = nn.Parameter(torch.zeros(nf, 1, 1))\n\n def forward(self, x):\n mean = x.mean((2, 3), keepdim=True)\n var = x.var((2, 3), keepdim=True)\n res = (x - mean) / ((var + self.eps).sqrt())\n return res * self.gamma + self.beta\n```\n\n\n \n\n\n\n```python\ndef conv_in(ni, nf, ks=3, stride=2, bn=True, **kwargs):\n layers = [\n nn.Conv2d(ni, nf, ks, padding=ks // 2, stride=stride, bias=True),\n GeneralRelu(**kwargs)\n ]\n if bn: layers.append(InstanceNorm(nf))\n return nn.Sequential(*layers)\n```\n\n\n \n\n\n\n```python\nlearn, run = get_learn_run(nfs, data, 0.1, conv_in, cbs=cbfs)\n```\n\n\n \n\n\n\n```python\n%time run.fit(3, learn)\n```\n\n train: [nan, tensor(0.0986)]\n valid: [nan, tensor(0.0991)]\n train: [nan, tensor(0.0986)]\n valid: [nan, tensor(0.0991)]\n train: [nan, tensor(0.0986)]\n valid: [nan, tensor(0.0991)]\n CPU times: user 1min 8s, sys: 4.98 s, total: 1min 13s\n Wall time: 36.9 s\n\n\n\n \n\n\n*Question*: why can't this classify anything?\n\nBecause it throws the difference in the means and variances of each image and each channel. It is useful for style transfer.\n\n### Group norm\n\nLost in all those norms? The authors from the [group norm paper](https://arxiv.org/pdf/1803.08494.pdf) have you covered:\n\n\n\n*From the PyTorch docs:*\n\n`GroupNorm(num_groups, num_channels, eps=1e-5, affine=True)`\n\nThe input channels are separated into `num_groups` groups, each containing\n``num_channels / num_groups`` channels. The mean and standard-deviation are calculated\nseparately over the each group. $\\gamma$ and $\\beta$ are learnable\nper-channel affine transform parameter vectorss of size `num_channels` if\n`affine` is ``True``.\n\nThis layer uses statistics computed from input data in both training and\nevaluation modes.\n\nArgs:\n- num_groups (int): number of groups to separate the channels into\n- num_channels (int): number of channels expected in input\n- eps: a value added to the denominator for numerical stability. Default: 1e-5\n- affine: a boolean value that when set to ``True``, this module\n has learnable per-channel affine parameters initialized to ones (for weights)\n and zeros (for biases). Default: ``True``.\n\nShape:\n- Input: `(N, num_channels, *)`\n- Output: `(N, num_channels, *)` (same shape as input)\n\nExamples::\n\n >>> input = torch.randn(20, 6, 10, 10)\n >>> # Separate 6 channels into 3 groups\n >>> m = nn.GroupNorm(3, 6)\n >>> # Separate 6 channels into 6 groups (equivalent with InstanceNorm)\n >>> m = nn.GroupNorm(6, 6)\n >>> # Put all 6 channels into a single group (equivalent with LayerNorm)\n >>> m = nn.GroupNorm(1, 6)\n >>> # Activating the module\n >>> output = m(input)\n\n## Fix small batch sizes\n\n### What's the problem?\n\nWhen we compute the statistics (mean and std) for a BatchNorm Layer on a small batch, it is possible that we get a standard deviation very close to 0. because there aren't many samples (the variance of one thing is 0. since it's equal to its mean).\n\n\n```python\ndata = DataBunch(*get_dls(train_ds, valid_ds, 2), c)\n```\n\n\n \n\n\n\n```python\ndef conv_layer(ni, nf, ks=3, stride=2, bn=True, **kwargs):\n layers = [\n nn.Conv2d(ni, nf, ks, padding=ks // 2, stride=stride, bias=not bn),\n GeneralRelu(**kwargs)\n ]\n if bn: layers.append(nn.BatchNorm2d(nf, eps=1e-5, momentum=0.1))\n return nn.Sequential(*layers)\n```\n\n\n \n\n\n\n```python\nlearn,run = get_learn_run(nfs, data, 0.4, conv_layer, cbs=cbfs)\n```\n\n\n \n\n\n\n```python\n%time run.fit(1, learn)\n```\n\n train: [2.35215484375, tensor(0.1710)]\n valid: [1548761117.4912, tensor(0.1431)]\n CPU times: user 1min 51s, sys: 361 ms, total: 1min 51s\n Wall time: 1min 51s\n\n\n\n \n\n\n### Running Batch Norm\n\nTo solve this problem we introduce a Running BatchNorm that uses smoother running mean and variance for the mean and std.\n\n\n```python\nclass RunningBatchNorm(nn.Module):\n def __init__(self, nf, mom=0.1, eps=1e-5):\n super().__init__()\n self.mom, self.eps = mom, eps\n self.mults = nn.Parameter(torch.ones(nf, 1, 1))\n self.adds = nn.Parameter(torch.zeros(nf, 1, 1))\n self.register_buffer('sums', torch.zeros(1, nf, 1, 1))\n self.register_buffer('sqrs', torch.zeros(1, nf, 1, 1))\n self.register_buffer('batch', tensor(0.))\n self.register_buffer('count', tensor(0.))\n self.register_buffer('factor', tensor(0.))\n self.register_buffer('offset', tensor(0.))\n\n def update_stats(self, x):\n bs, nc, *_ = x.shape\n self.sums.detach_()\n self.sqrs.detach_()\n dims = (0, 2, 3)\n s = x.sum(dims, keepdim=True)\n ss = (x * x).sum(dims, keepdim=True)\n c = self.count.new_tensor(x.numel() / nc)\n mom1 = 1 - (1 - self.mom) / math.sqrt(bs-1)\n self.sums.lerp_(s, mom1)\n self.sqrs.lerp_(ss, mom1)\n self.count.lerp_(c, mom1)\n self.batch += bs\n means = self.sums / self.count\n var = (self.sqrs / self.count).sub_(means * means)\n\n if bool(self.batch < 20):\n var.clamp_min_(0.01)\n\n self.factor = self.mults / (var + self.eps).sqrt()\n self.offset = self.adds - means * self.factor\n\n def forward(self, x):\n if self.training:\n self.update_stats(x)\n return x * self.factor + self.offset\n```\n\n\n \n\n\n\n```python\ndef conv_rbn(ni, nf, ks=3, stride=2, bn=True, **kwargs):\n layers = [nn.Conv2d(ni, nf, ks, padding=ks//2, stride=stride, bias=not bn),\n GeneralRelu(**kwargs)]\n if bn: layers.append(RunningBatchNorm(nf))\n return nn.Sequential(*layers)\n```\n\n\n \n\n\n\n```python\nlearn,run = get_learn_run(nfs, data, 0.4, conv_rbn, cbs=cbfs)\n```\n\n\n \n\n\n\n```python\nlearn.model\n```\n\n\n\n\n Sequential(\n (0): Sequential(\n (0): Conv2d(1, 8, kernel_size=(5, 5), stride=(2, 2), padding=(2, 2), bias=False)\n (1): GeneralRelu()\n (2): RunningBatchNorm()\n )\n (1): Sequential(\n (0): Conv2d(8, 16, kernel_size=(3, 3), stride=(2, 2), padding=(1, 1), bias=False)\n (1): GeneralRelu()\n (2): RunningBatchNorm()\n )\n (2): Sequential(\n (0): Conv2d(16, 32, kernel_size=(3, 3), stride=(2, 2), padding=(1, 1), bias=False)\n (1): GeneralRelu()\n (2): RunningBatchNorm()\n )\n (3): Sequential(\n (0): Conv2d(32, 64, kernel_size=(3, 3), stride=(2, 2), padding=(1, 1), bias=False)\n (1): GeneralRelu()\n (2): RunningBatchNorm()\n )\n (4): Sequential(\n (0): Conv2d(64, 64, kernel_size=(3, 3), stride=(2, 2), padding=(1, 1), bias=False)\n (1): GeneralRelu()\n (2): RunningBatchNorm()\n )\n (5): AdaptiveAvgPool2d(output_size=1)\n (6): Lambda()\n (7): Linear(in_features=64, out_features=10, bias=True)\n )\n\n\n\n\n \n\n\n\n```python\n%time run.fit(1, learn)\n```\n\n train: [0.4170171875, tensor(0.8936)]\n valid: [0.3150417236328125, tensor(0.9235)]\n CPU times: user 1min 51s, sys: 418 ms, total: 1min 52s\n Wall time: 1min 52s\n\n\n\n \n\n\nThis solves the small batch size issue!\n\n### What can we do in a single epoch?\n\nNow let's see with a decent batch size what result we can get.\n\n\n```python\ndata = DataBunch(*get_dls(train_ds, valid_ds, 512), c)\n```\n\n\n \n\n\n\n```python\nlearn, run = get_learn_run(nfs, data, 1., conv_layer, cbs=cbfs)\n```\n\n\n \n\n\n\n```python\n%time run.fit(3, learn)\n```\n\n train: [0.22383076171875, tensor(0.9292)]\n valid: [0.09052369384765625, tensor(0.9712)]\n train: [0.0656801171875, tensor(0.9794)]\n valid: [0.06957890014648438, tensor(0.9794)]\n train: [0.044756376953125, tensor(0.9858)]\n valid: [0.07446788940429687, tensor(0.9781)]\n CPU times: user 47.1 s, sys: 2.92 s, total: 50.1 s\n Wall time: 25.2 s\n\n\n\n \n\n\n\n```python\n# learn,run = get_learn_run(nfs, data, 0.9, conv_rbn, cbs=cbfs\n# +[partial(ParamScheduler,'lr', sched_lin(1., 0.2))])\n```\n\n\n \n\n\n\n```python\n%time run.fit(3, learn)\n```\n\n train: [0.032991181640625, tensor(0.9894)]\n valid: [0.0602272705078125, tensor(0.9817)]\n train: [0.02374192626953125, tensor(0.9923)]\n valid: [0.06647666015625, tensor(0.9819)]\n train: [0.0173309326171875, tensor(0.9948)]\n valid: [0.08294573974609375, tensor(0.9761)]\n CPU times: user 46.8 s, sys: 2.88 s, total: 49.7 s\n Wall time: 25 s\n\n\n\n \n\n\n## Export\n\n\n```python\nnb_auto_export()\n```\n\n\n \n\n\n\n \n\n", "meta": {"hexsha": "291f31250655845b2ef3a53f73d5f2310fe906f2", "size": 178078, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "dl2/notebooks/07_BatchNormalization-Imad.ipynb", "max_stars_repo_name": "ImadDabbura/fastai-courses", "max_stars_repo_head_hexsha": "053637a2dd3b4ad6c35f97a13f3fba87af1d3940", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2020-05-30T10:50:14.000Z", "max_stars_repo_stars_event_max_datetime": "2020-10-28T19:13:15.000Z", "max_issues_repo_path": "dl2/notebooks/07_BatchNormalization-Imad.ipynb", "max_issues_repo_name": "ImadDabbura/fastai-courses", "max_issues_repo_head_hexsha": "053637a2dd3b4ad6c35f97a13f3fba87af1d3940", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "dl2/notebooks/07_BatchNormalization-Imad.ipynb", "max_forks_repo_name": "ImadDabbura/fastai-courses", "max_forks_repo_head_hexsha": "053637a2dd3b4ad6c35f97a13f3fba87af1d3940", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2020-10-16T05:05:09.000Z", "max_forks_repo_forks_event_max_datetime": "2020-10-28T19:15:25.000Z", "avg_line_length": 70.1646966115, "max_line_length": 45192, "alphanum_fraction": 0.6824706028, "converted": true, "num_tokens": 6860, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307806984444, "lm_q2_score": 0.6619228758499942, "lm_q1q2_score": 0.42166524636488134}} {"text": "last edited on May 24, 2019, updated and cleaned up on June 24, 2019\n\n# Get Summary Stats of Simulations\nthis gets summary stats of simulations, csv is also saved to personal computer, a little messy but does work!\n\n\n```python\n#!/bin/env python\n\n#SBATCH --job-name=stat_full\n#SBATCH --output=statplt_%j.out\n#SBATCH --time=24:05:00\n#SBATCH --partition=bigmem2\n#SBATCH --nodes=1\n#SBATCH --mem=0\n# #SBATCH --exclusive\n# this one did work\n\n# import packages\nimport numpy as np\nfrom scipy.signal import get_window, csd\nfrom scipy.signal.windows import hann, hanning, nuttall, flattop\nfrom scipy.fftpack import fft, ifft, fftfreq, fftshift, ifftshift\n\nimport matplotlib.pyplot as plt\n\nimport scipy.integrate as sciint\nimport pandas as pd\nimport datetime\nimport matplotlib.cm as cm\nfrom math import pi\nimport matplotlib.ticker as tck\nimport datetime\nfrom sympy import solve, Poly, Eq, Function, exp, re, im\nfrom netCDF4 import Dataset, num2date # This is to read .nc files and time array\nfrom scipy.optimize import fsolve\nfrom decimal import Decimal\nimport pickle\nimport multiprocessing as mp\nfrom joblib import Parallel, delayed\nimport matplotlib.colors as colors\nfrom seaborn import cubehelix_palette #for contour plot colors\nimport seaborn as sns\nfrom decimal import Decimal\nimport numpy.ma as ma\nfrom scipy.stats import skew\n\n\n# get file names and collect sims\nfrom os import walk\nimport pickle\n\nflabs = []\nfor (dirpath, dirnames, filenames) in walk('gphfiles/'):\n flabs.extend(filenames)\n break\n\nf = []\nfor (dirpath, dirnames, filenames) in walk('/scratch/midway2/clairev/from_home/01_full_sims/'):\n f.extend(filenames)\n break\n \njjj = 0\n\n```\n\n\n```python\nf = []\nfor (dirpath, dirnames, filenames) in walk('scratch-midway2/enso_sims/'):\n f.extend(filenames)\n break\n```\n\n\n```python\nseasons = [\"winter\", \"spring\", \"summer\", \"fall\"]\nens = [\"nino\", \"nina\", \"neutral\"]\nd2_names = [enso + \" \" + part for part in seasons for enso in ens]\n```\n\nfor wantfile in range(len(flabs)):\n index = str(flabs[wantfile][-10:-5])\n \n # get detrend \n ring = 'detrended/new_detrend_' + str(index) + '.h5'\n \n data_store = pd.HDFStore(ring)\n \n # Retrieve data using key\n untrend_df = data_store['untrend_geopot']\n data_store.close()\n\n seasons = [\"winter\",\"spring\",\"summer\",\"fall\"]\n # write flatten function\n \n untrend_df[\"seasonmean\"] = untrend_df.groupby(by=['year','season'])['adj_z'].transform('mean')\n untrend_df[\"diff_mean\"] = untrend_df[\"adj_z\"] - untrend_df[\"seasonmean\"]\n \n\n\n```python\nstorage_list[0]\n```\n\n\n\n\n ['58.5Nnino winter',\n 509.8705139015072,\n (-0.15950769881999285-0.016854110781558046j)]\n\n\n\n\n```python\n\nfor wantfile in range(1,len(flabs)):\n index = str(flabs[wantfile][-10:-5])\n# go through sims to get the correct ones\n sims = []\n for name in f:\n if name[8:13] == flabs[wantfile][-10:-5]:\n print(name)\n file_pickle = open(\"scratch-midway2/enso_sims/\" + name, \"rb\")\n sims1 = pickle.load(file_pickle)\n \n sims.append(sims1)\n \n flatten = lambda l: [item for sublist in l for item in sublist]\n #flatten each \n flat_sims = [[flatten(entry) for entry in sublist] for sublist in sims] \n \n for j in range(12):\n flat_all = []\n for k in range(len(flat_sims)):\n flat_tested = flat_sims[k]\n flat_all.append(flat_tested[j])\n \n flat_all = flatten(flat_all)\n \n act_var = np.var(flat_all)\n act_skew = skew(flat_all, axis = None)\n \n storage_list.append([str(index) + str(d2_names[j]), act_var, act_skew])\n \n```\n\n\n```python\nimport csv\n\nwith open('enso_stat', 'wb') as myfile:\n wr = csv.writer(myfile, quoting=csv.QUOTE_ALL)\n wr.writerow(storage_list)\n\n```\n\n\n```python\nskew_list = np.real([entry[2] for entry in storage_list])\nvar_list = np.real([entry[1] for entry in storage_list])\nname_list = np.real([entry[0] for entry in storage_list])\n```\n\n\n```python\ntodo = {\"name\":name_list, \"var\": var_list, \"skew\":skew_list}\n```\n\n\n```python\nimport pandas\ndf = pandas.DataFrame(todo)\ndf.to_csv(\"enso_stat_2.csv\", sep=',',index=False)\n```\n\n\n```python\n\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        nameskewvar
                                        058.5Nnino winter-0.159508509.870514
                                        158.5Nnina winter-0.429642502.625568
                                        258.5Nneutral winter-0.321109542.798842
                                        358.5Nnino spring-1.050231384.394635
                                        458.5Nnina spring0.290616367.455810
                                        558.5Nneutral spring0.805822352.839923
                                        658.5Nnino summer1.522569234.396411
                                        758.5Nnina summer-5.972210230.384690
                                        858.5Nneutral summer1.357973217.620277
                                        958.5Nnino fall0.321617379.411534
                                        1058.5Nnina fall0.175340383.907261
                                        1158.5Nneutral fall0.182353388.583098
                                        1231.5Snino winter0.02701058.041642
                                        1331.5Snina winter0.10946655.632681
                                        1431.5Sneutral winter-0.27541957.901194
                                        1531.5Snino spring0.60961587.539932
                                        1631.5Snina spring-2.63078274.322073
                                        1731.5Sneutral spring0.03343485.978238
                                        1831.5Snino summer-0.437406147.481417
                                        1931.5Snina summer-0.539583138.915356
                                        2031.5Sneutral summer-0.492165151.736997
                                        2131.5Snino fall-0.190992114.306934
                                        2231.5Snina fall-0.21604799.021668
                                        2331.5Sneutral fall-0.186384111.372161
                                        2445.0Nnino winter-0.232637499.747142
                                        2545.0Nnina winter0.150164552.809552
                                        2645.0Nneutral winter-0.073994499.719317
                                        2745.0Nnino spring-0.490483309.758611
                                        2845.0Nnina spring-1.143344304.283364
                                        2945.0Nneutral spring-0.450829313.763816
                                        ............
                                        9036.0Snino summer-0.627353225.510244
                                        9136.0Snina summer-1.102747207.091323
                                        9236.0Sneutral summer-0.383494214.111217
                                        9336.0Snino fall-0.144428165.460370
                                        9436.0Snina fall0.289449160.876670
                                        9536.0Sneutral fall0.066111165.697685
                                        9654.0Nnino winter0.205682606.117537
                                        9754.0Nnina winter0.071486538.887158
                                        9854.0Nneutral winter-0.106673563.885113
                                        9954.0Nnino spring-0.109757390.306675
                                        10054.0Nnina spring0.200169374.811904
                                        10154.0Nneutral spring0.697484356.445052
                                        10254.0Nnino summer-1.497659217.906911
                                        10354.0Nnina summer0.145333207.391855
                                        10454.0Nneutral summer0.270825203.926838
                                        10554.0Nnino fall-0.164452387.115716
                                        10654.0Nnina fall0.085704377.418075
                                        10754.0Nneutral fall0.588370401.328652
                                        10840.5Snino winter-0.971277182.436143
                                        10940.5Snina winter-1.402490162.535970
                                        11040.5Sneutral winter1.385926172.695624
                                        11140.5Snino spring0.645377221.221906
                                        11240.5Snina spring0.336784198.469901
                                        11340.5Sneutral spring5.613228201.947995
                                        11440.5Snino summer-1.483635278.766618
                                        11540.5Snina summer-0.352202253.691625
                                        11640.5Sneutral summer-0.166809277.222459
                                        11740.5Snino fall-39.275798208.107787
                                        11840.5Snina fall-0.476596218.581430
                                        11940.5Sneutral fall1.310123216.021168
                                        \n

                                        120 rows \u00d7 3 columns

                                        \n
                                        \n\n\n\n\n```python\nstorage_list = []\nfor wantfile in flabs:\n index = str(wantfile[-10:-5])\n ring = 'detrended/new_detrend_' + str(index) + '.h5'\n data_store = pd.HDFStore(ring)\n \n \n \n # Retrieve data using key\n untrend_df = data_store.select(\"untrend_geopot\")\n untrend_df['adj_z'] = untrend_df['adj_z'].astype(np.float)\n # get stats from each thing\n untrend_df[\"season_mean\"] = untrend_df.groupby(['season', \"year\"])['adj_z'].transform('mean')\n # get stats from each thing\n untrend_df[\"season_variance\"] = untrend_df.groupby(['season', \"year\"])['adj_z'].transform('var')\n\n # get stats from each thing\n untrend_df[\"diff_from_season\"] = untrend_df[\"adj_z\"] - untrend_df[\"season_mean\"]\n \n #actual means\n act_skew = untrend_df.groupby(['season'])['diff_from_season'].skew()\n act_var = untrend_df.groupby(['season'])['diff_from_season'].var()\n \n \n storage_list.append([index,act_var,act_skew])\n```\n\n\n```python\nseasons = [\"winter\", \"spring\", \"summer\", \"fall\"]\nens = [\"nino\", \"nina\", \"neutral\"]\nd2_names = [enso + \" \" + part for part in seasons for enso in ens]\n```\n\n\n```python\nflatten = lambda l: [item for sublist in l for item in sublist]\n #flatten each \nflat_sims = [[flatten(entry) for entry in sublist] for sublist in sims]\n```\n\n\n```python\nfor j in range(4):\n plt.clf();\n plt.figure(figsize=(15, 5));\n plt.hist(x = np.real(untrend_df[untrend_df[\"season\"] == seasons[j]][\"diff_mean\"]), \n bins = 20, density = True, \n alpha = 0.5, label = \"reanalysis\")\n \n \n flat_all = []\n for k in range(len(flat_sims)):\n flat_tested = flat_sims[k]\n flat_all.append(flat_tested[j])\n \n \n \n flat_all = flatten(flat_all)\n for k in range(3):\n #print(\"hi\")\n plt.hist(x = np.real(flat_all[j*3 + k]), bins = 100, \n density = True, alpha = 0.5, label = d2_names[j*3 + k])\n \n plt.ylabel(\"density\")\n plt.legend()\n plt.xlabel(\"departure from mean geopotential height\")\n plt.title(str(flabs[wantfile][-10:-5]) + \" season: \" +str(seasons[j]))\n \n # formatting\n sns.set_style(\"ticks\")\n sns.set_context(\"poster\")\n sns.despine() \n plt.show()\n \n \n```\n\n\n```python\n for j in range(4):\n \n flat_all = []\n for k in range(len(sims)):\n flat_tested = flat_sims[k]\n \n flat_all.append(flat_tested[j])\n```\n\n\n```python\nlen(flat_all[0])\n```\n\n\n\n\n 6606240\n\n\n\n\n```python\nwantfile = 0\n \nindex = str(flabs[wantfile][-10:-5])\n \n# get detrend \nring = 'detrended/new_detrend_' + str(index) + '.h5'\n \ndata_store = pd.HDFStore(ring)\n```\n\n\n```python\nstorage_list = []\nfor wantfile in flabs:\n index = str(wantfile[-10:-5])\n ring = 'detrended/new_detrend_' + str(index) + '.h5'\n data_store = pd.HDFStore(ring)\n \n \n \n # Retrieve data using key\n untrend_df = data_store.select(\"untrend_geopot\")\n untrend_df['adj_z'] = untrend_df['adj_z'].astype(np.float)\n # get stats from each thing\n untrend_df[\"season_mean\"] = untrend_df.groupby(['season', \"year\"])['adj_z'].transform('mean')\n # get stats from each thing\n untrend_df[\"season_variance\"] = untrend_df.groupby(['season', \"year\"])['adj_z'].transform('var')\n\n # get stats from each thing\n untrend_df[\"diff_from_season\"] = untrend_df[\"adj_z\"] - untrend_df[\"season_mean\"]\n \n #actual means\n act_skew = untrend_df.groupby(['season'])['diff_from_season'].skew()\n act_var = untrend_df.groupby(['season'])['diff_from_season'].var()\n \n \n storage_list.append([index,act_var,act_skew])\n```\n\n /home/clairev/anaconda3/lib/python3.7/site-packages/pandas/core/dtypes/cast.py:702: ComplexWarning: Casting complex values to real discards the imaginary part\n return arr.astype(dtype, copy=True)\n\n\n\n```python\nlen(storage_list)\n```\n\n\n\n\n 13\n\n\n\n\n```python\n# Retrieve data using key\nuntrend_df = data_store.select(\"untrend_geopot\")\nuntrend_df['adj_z'] = untrend_df['adj_z'].astype(np.float)\n```\n\n\n```python\n# get stats from each thing\nuntrend_df[\"season_mean\"] = untrend_df.groupby(['season', \"year\"])['adj_z'].transform('mean')\n# get stats from each thing\nuntrend_df[\"season_variance\"] = untrend_df.groupby(['season', \"year\"])['adj_z'].transform('var')\n\n# get stats from each thing\nuntrend_df[\"diff_from_season\"] = untrend_df[\"adj_z\"] - untrend_df[\"season_mean\"]\n```\n\n\n```python\n\n```\n\n\n```python\n#actual means\nact_skew = untrend_df.groupby(['season'])['diff_from_season'].skew()\nact_var = untrend_df.groupby(['season'])['diff_from_season'].var()\n```\n\n\n```python\nseason_df = untrend_df[untrend_df[\"season\"] == \"fall\"]\n```\n\n\n```python\ntest_vals = season_df[\"diff_from_season\"]\n```\n\n\n```python\n#test_vals\n```\n\n\n```python\ntest_skew = skew(test_vals)\n```\n\n\n```python\nact_skew\n```\n\n\n\n\n season\n fall -0.407854\n spring -0.431184\n summer -0.595719\n winter -0.210709\n Name: diff_from_season, dtype: float64\n\n\n\n\n```python\ntest_skew = untrend_df.groupby(['season',\"year\"])['diff_from_season'].skew()\n```\n\n\n```python\nnp.average(test_skew[\"fall\"]) # honestly its pretty close so im going to use that.. \n```\n\n\n\n\n -0.40382844767999915\n\n\n\n\n```python\nact_var # have variance\n```\n\n\n\n\n season\n fall 389.564229\n spring 358.382595\n summer 223.546377\n winter 532.385968\n Name: diff_from_season, dtype: float64\n\n\n\n\n```python\nact_skew # now repeat this many times\n```\n\n\n\n\n season\n fall 0.256179\n spring 0.168065\n summer -0.184236\n winter 0.302864\n Name: diff_from_season, dtype: float64\n\n\n\n\n```python\n# get stats from each thing\nuntrend_df[\"season_mean\"] = df[\"sst_seasdiff\"].sub(df.groupby('year_num')[\"sst_seasdiff\"].transform('mean'))\n```\n\n\n```python\nf[0:20]\n\n```\n\n\n\n\n []\n\n\n\n\n```python\nseasons = [\"winter\",\"spring\",\"summer\",\"fall\"]\n# write flatten function\nwinter_means = list(untrend_df[untrend_df[\"season\"]\n == seasons[0]].groupby(['lon'])[\"adj_z\"].mean())\nspring_means = list(untrend_df[untrend_df[\"season\"] \n == seasons[1]].groupby(['lon'])[\"adj_z\"].mean())\nsummer_means = list(untrend_df[untrend_df[\"season\"] \n == seasons[2]].groupby(['lon'])[\"adj_z\"].mean())\nfall_means = list(untrend_df[untrend_df[\"season\"]\n == seasons[3]].groupby(['lon'])[\"adj_z\"].mean())\n\nall_means = [winter_means, spring_means, summer_means, fall_means]\n \n# go through sims to get the correct ones\nsims = []\nfor name in f[0:20]:\n print(name)\n if name[4:9] == flabs[wantfile][-10:-5]:\n file_pickle = open(\"/scratch/midway2/clairev/from_home/01_full_sims/\" + name, \"rb\")\n sims1 = pickle.load(file_pickle, encoding = \"latin1\", fix_imports=True)\n newsims = [[list(np.add(sims1[season][0][j],\n np.real(all_means[season][j]))) \n for j in range(len(sims1[season][0]))]\n for season in range(4)]\n sims.append(newsims)\n \n \nflatten = lambda l: [item for sublist in l for item in sublist]\n #flatten each \nflat_sims = [[flatten(entry) for entry in sublist] for sublist in sims] \n```\n\n\n```python\nsertest = (entry for entry in sims)\n```\n\n\n```python\n# teststack = np.dstack((sims[0], sims[1]))\n# flatten around last axis\n# newstack = [[flatten(entry) for entry in season] for season in teststack]\n```\n\n\n```python\nrunnumber = len(sims)\nentry_len = len(sims[0][0][0])\n\nlon_skew = [[skew(entry) for entry in season] for season in newstack]\n\nfull_skew = [skew(season) for season in newstack]\n```\n\n\n```python\n# now put everythig into a dataframe\nfor season in range(4):\n \n meds = lon_median[season]\n meds.append(full_median[season])\n \n skewed = lon_skew[season]\n skewed.append(full_skew[season])\n \n varr = lon_vars[season]\n varr.append(full_var[season])\n \n avgs = lon_avgs[season]\n avgs.append(full_avg[season])\n \n lon_list = [i*1.5 for i in range(240)]\n lon_list.append(\"all\")\n \n runtimes = [runnumber for i in range(241)]\n entrylen = [entry_len for i in range(241)]\n version = [flabs[wantfile][-10:-5] for i in range(241)]\n \n #append all to a temp frame\n d = {\"version\" : version, \"runtimes\" : runtimes, \"entrylen\" : entrylen,\n \"lon\": lon_list, \"median\" : meds, \"skew\" : skewed, \"variance\":varr, \"average\" : avgs}\n \n df = pd.DataFrame(data=d)\n \n jjj = jjj + 1\n```\n\n\n```python\njjj\n```\n\n\n\n\n 5\n\n\n\n\n```python\n# see if dataframe saved alright -- and it did\nunpickled_df = pd.read_pickle(\"sims1_stats.pkl\")\n```\n\n\n```python\n#unpickled_df[unpickled_df[\"version\"] == \"58.5S\"]\n```\n\n\n```python\nold_stats = pd.read_csv(\"sims1_stats.csv\")\n```\n\n\n```python\nold_stats = unpickled_df\n```\n\n\n```python\nold_skew = old_stats[old_stats[\"lon\"] == 500][\"skew\"]\n```\n\n\n```python\ndef get_skew(mean,median,variance):\n sd = np.sqrt(variance)\n return 3*(mean - median)/ sd\n```\n\n\n```python\nimport re\nclass TFConverter(dict):\n column_name_pattern = re.compile(r'_tf$')\n def __getitem__(self, k):\n if k in self:\n return TFConverter.convert\n else:\n raise KeyError(k)\n def __contains__(self,k):\n return self.column_name_pattern.search(k) is not None\n @staticmethod\n def convert(txt):\n return complex(txt.strip(\"()\"))\n\ndef read_tf_csv(filename, **kwargs):\n return pd.read_csv(filename, converters = TFConverter(), **kwargs)\n```\n\n\n```python\nold_stats[\"new_skew\"] = get_skew(old_stats[\"average\"], old_stats[\"variance\"], old_stats[\"median\"])\n```\n\n\n```python\n# save it again as a CSV!\nold_stats.to_csv(\"sims1_stats2.csv\")\n```\n\n\n```python\nstorage_list\n```\n\n\n\n\n [['58.5N', season\n fall 389.564229\n spring 358.382595\n summer 223.546377\n winter 532.385968\n Name: diff_from_season, dtype: float64, season\n fall 0.256179\n spring 0.168065\n summer -0.184236\n winter 0.302864\n Name: diff_from_season, dtype: float64], ['31.5S', season\n fall 110.261088\n spring 82.412888\n summer 150.716904\n winter 58.867478\n Name: diff_from_season, dtype: float64, season\n fall -0.420345\n spring -0.538256\n summer -0.415503\n winter -0.605304\n Name: diff_from_season, dtype: float64], ['45.0N', season\n fall 284.486461\n spring 316.910050\n summer 134.797566\n winter 546.963363\n Name: diff_from_season, dtype: float64, season\n fall -0.323167\n spring -0.317565\n summer -0.531423\n winter -0.124741\n Name: diff_from_season, dtype: float64], ['45.0S', season\n fall 292.054864\n spring 309.494125\n summer 329.100689\n winter 301.643735\n Name: diff_from_season, dtype: float64, season\n fall -0.305224\n spring -0.504135\n summer -0.210158\n winter -0.469655\n Name: diff_from_season, dtype: float64], ['58.5S', season\n fall 436.235287\n spring 475.715979\n summer 491.687052\n winter 382.248331\n Name: diff_from_season, dtype: float64, season\n fall 0.212365\n spring 0.274388\n summer 0.142022\n winter 0.483158\n Name: diff_from_season, dtype: float64], ['54.0S', season\n fall 440.298301\n spring 499.109734\n summer 472.968543\n winter 453.733755\n Name: diff_from_season, dtype: float64, season\n fall -0.021755\n spring -0.021432\n summer -0.051517\n winter 0.181555\n Name: diff_from_season, dtype: float64], ['49.5S', season\n fall 377.772146\n spring 429.692642\n summer 403.423016\n winter 421.470354\n Name: diff_from_season, dtype: float64, season\n fall -0.211970\n spring -0.305290\n summer -0.174252\n winter -0.159908\n Name: diff_from_season, dtype: float64], ['36.0S', season\n fall 163.421021\n spring 133.053730\n summer 218.234440\n winter 99.818143\n Name: diff_from_season, dtype: float64, season\n fall -0.385543\n spring -0.552024\n summer -0.287168\n winter -0.681203\n Name: diff_from_season, dtype: float64], ['54.0N', season\n fall 406.379124\n spring 370.509623\n summer 207.335775\n winter 570.541907\n Name: diff_from_season, dtype: float64, season\n fall 0.068552\n spring 0.032045\n summer -0.314286\n winter 0.186250\n Name: diff_from_season, dtype: float64], ['40.5S', season\n fall 219.641575\n spring 205.514957\n summer 271.623321\n winter 177.566157\n Name: diff_from_season, dtype: float64, season\n fall -0.342464\n spring -0.577040\n summer -0.219582\n winter -0.662966\n Name: diff_from_season, dtype: float64], ['36.0N', season\n fall 112.316943\n spring 160.060376\n summer 57.161146\n winter 278.281757\n Name: diff_from_season, dtype: float64, season\n fall -0.375912\n spring -0.539909\n summer -0.646068\n winter -0.213900\n Name: diff_from_season, dtype: float64], ['49.5N', season\n fall 373.544178\n spring 360.704142\n summer 177.509422\n winter 586.046137\n Name: diff_from_season, dtype: float64, season\n fall -0.152528\n spring -0.148791\n summer -0.451394\n winter 0.028529\n Name: diff_from_season, dtype: float64], ['40.5N', season\n fall 183.665350\n spring 243.803518\n summer 92.737145\n winter 433.055791\n Name: diff_from_season, dtype: float64, season\n fall -0.407854\n spring -0.431184\n summer -0.595719\n winter -0.210709\n Name: diff_from_season, dtype: float64]]\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "bbd28c202e9a51c8e8652af46f15509f3522cf3a", "size": 237982, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lin-assumption-2/statstest.ipynb", "max_stars_repo_name": "clairevalva/wavy-sims", "max_stars_repo_head_hexsha": "259c81078e6069777fdef455b0d806e4f8c0c262", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "lin-assumption-2/statstest.ipynb", "max_issues_repo_name": "clairevalva/wavy-sims", "max_issues_repo_head_hexsha": "259c81078e6069777fdef455b0d806e4f8c0c262", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "lin-assumption-2/statstest.ipynb", "max_forks_repo_name": "clairevalva/wavy-sims", "max_forks_repo_head_hexsha": "259c81078e6069777fdef455b0d806e4f8c0c262", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 126.7209797657, "max_line_length": 49314, "alphanum_fraction": 0.8392357405, "converted": true, "num_tokens": 8527, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410572017153, "lm_q2_score": 0.6334102705979902, "lm_q1q2_score": 0.42147789048014833}} {"text": "## Code from Book\n\n### 2. Getting Started\n\n\n```python\n! whoami\n```\n\n\n```bash\n%%bash \necho $0\n```\n\n\n```python\n! echo 'hello'\n```\n\n\n```python\nimport os\nos.__file__\n```\n\n\n```python\n%timeit 9999 in range(10000)\n```\n\n\n```javascript\n%%javascript\nvar d = 10\nalert(\"d = \" + d)\n```\n\n### 3. Thinking in Binary\n\n\n```python\n! file my_image.jpg\n```\n\n\n```python\nimport magic\nprint magic.from_file(\"my_image.jpg\")\n```\n\n\n```python\nif magic.from_file(\"upload.jpg\", mime=True) == \"image/jpeg\":\n continue_uploading(\"upload.jpg\")\nelse:\n alert(\"Sorry! This file type is not allowed\")\n```\n\n\n```python\nimport imghdr\nprint imghdr.what(\"path/to/my/file.ext\")\n```\n\n\n```python\nimport binascii\n\ndef spoof_file(file, magic_number):\n magic_number = binascii.unhexlify(magic_number)\n with open(file, \"r+b\") as f:\n old = f.read()\n f.seek(0)\n f.write(magic_number + old)\n```\n\n\n```python\n! xxd -b my_file.docx | less\n```\n\n\n```python\n! du -h my_file.docx\n```\n\n\n```python\ndef to_ascii_bytes(string):\n return \" \".join(format(ord(char), '08b') for char in string)\n```\n\n\n```python\nstring = \"my ascii string\"\n\"\".join(hex(ord(char))[2:] for char in string)\n```\n\n\n```python\nhex_string = \"6d7920617363696920737472696e67\"\nprint hex_string.decode(\"hex\")\nprint \"\".join(chr(int(hex_string[i:i+2], 16)) for i in range(0, len(hex_string), 2))\n```\n\n\n```python\n# adapted from https://code.activestate.com/recipes/142812-hex-dumper/\ndef hexdump(string, length=8):\n result = []\n digits = 4 if isinstance(string, unicode) else 2\n\n for i in xrange(0, len(string), length):\n s = string[i:i + length]\n hexa = \"\".join(\"{:0{}X}\".format(ord(x), digits) for x in s)\n text = \"\".join(x if 0x20 <= ord(x) < 0x7F else '.' for x in s)\n result.append(\"{:04X}\u00a0\u00a0 {:{}}\u00a0\u00a0 {}\".format(i, hexa, length * (digits + 1), text))\n\n return '\\n'.join(result)\n```\n\n\n```python\nprint hexdump(\"The quick brown fox jumps over the lazy dog\")\n```\n\n\n```python\nimport struct\n\nnum = 0x103e4\nstruct.pack(\"I\", 0x103e4)\n```\n\n\n```python\nstring = '\\xe4\\x03\\x01\\x00'\nstruct.unpack(\"i\", string)\n```\n\n\n```python\nbytes = '\\x01\\xc2'\nstruct.pack(\"h\", bytes)[0])\n```\n\n\n```python\nimport base64\n\nbase64.b64encode('encodings are fun...')\n```\n\n\n```python\nprint base64.b64decode(_)\n```\n\n\n```python\nstring = \"hello\\x00\"\nbinary_string = ' '.join('{:08b}'.format(ord(char)) for char in string)\n\" \".join(binary_string[i:i+6] for i in range(0, len(binary_string), 6))\n```\n\n\n```python\nbin_string = '011010 000110 010101 101100 011011 000110 111100 000000'\n[int(b, 2) for b in bin_string.split()]\n```\n\n\n```bash\n%%bash\necho -n hello | base64\necho aGVsbG8= | base64 --decode && echo\n```\n\n\n```python\nu'\u25d1 \\u2020'.encode('utf8')\n```\n\n\n```python\n'\\xe2\\x97\\x91 \\xe2\\x80\\xa0'.decode('utf8')\n```\n\n\n```python\nunicode('\\xe2\\x97\\x91 \\xe2\\x80\\xa0', encoding='utf8')\n```\n\n\n```python\nutf8_string = '\u00c5\u00ea\u00ed\u00f2\u00fc'\nutf8_string\n```\n\n\n```python\nunicode_string = utf8_string.decode('utf8')\nunicode_string\n```\n\n\n```python\nunicode_string.encode('mac roman')\n```\n\n\n```python\n'\u00c5\u00ea\u00ed\u00f2\u00fc'.decode('utf8').encode('ascii') # Raises UnicodeEncodeError\n```\n\n\n```python\n! chardetect uni.txt another_file.txt\n```\n\n\n```python\nfile = \"\"\"\u6f4d\u696a\u6162\u656b\u6920\u2073\u6874\u2065\u6167\u6272\u656c\u2064\u6574\u7478\u7420\u6168\u2074\u7369\u7420\u6568\u7220\u7365\u6c75\u2074\u666f\u7420\u7865\u2074\u6562\u6e69\u2067\u6564\u6f63\u6564\u2064\u7375\u6e69\u2067\u6e61\u7520\u696e\u746e\u6e65\u6564\u2064\u6863\n\u7261\u6361\u6574\u2072\u6e65\u6f63\u6964\u676e\u6977\u6874\u6320\u6d6f\u6c70\u7465\u6c65\u2079\u6e75\u6572\u616c\u6574\u2064\u6e6f\u7365\u666f\u6574\u206e\u7266\u6d6f\u6120\u6420\u6669\u6566\u6572\u746e\u7720\u6972\u6974\u676e\u7320\u7379\u6574\u2e6d\u2027\u280a\u6154\u656b\u206e\n\u7266\u6d6f\u6520\u2e6e\u6977\u696b\u6570\u6964\u2e61\u726f\u2967\"\"\"\n\nprint file.decode('utf8').encode('utf16')\n```\n\n\n```python\nimport ftfy\nftfy.fix_text(u\"\u00e2\u20ac\u0153Mojibake\u00e2\u20ac\u0153 can be fixed.\")\n```\n\n\n```python\nx = 0b1111\ny = 0b1010\nbin(int(\"{:b}{:b}\".format(x, y), 2))\n```\n\n\n```python\nbin(x << 4 | y)\n```\n\n### 4. Cryptography\n\n\n```python\nimport random\nimport string\n\nr = random.SystemRandom()\n\n# Get a random integer between 0 and 20\nr.randint(0, 20)\n\n# Get a random number between 0 and 1\nr.random()\n\n# Generate a random 40-bit number\nr.getrandbits(40)\n\n# Choose a random item from a string or list\nchars = string.printable\nr.choice(chars)\n\n# Randomize the order of a sequence\nseq = ['a', 'b', 'c', 'd', 'e']\nr.shuffle(seq)\n```\n\n\n```python\n\"ALLIGATOR\".encode('rot13')\n```\n\n\n```python\n\"NYYVTNGBE\".encode('rot13')\n```\n\n\n```python\nplaintext = \"A secret-ish message!\"\n\"\".join(chr((ord(c) + 20) % 256) for c in plaintext)\n```\n\n\n```python\nciphertext = 'U4\\x87yw\\x86y\\x88A}\\x87|4\\x81y\\x87\\x87u{y5'\n\"\".join(chr((ord(c) - 20) % 256) for c in ciphertext)\n```\n\n\n```python\nplaintext = 0b110100001101001\none_time_pad = 0b110000011100001\nbin(plaintext ^ one_time_pad)\n```\n\n\n```python\ndecrypted = 0b100010001000 ^ one_time_pad\nformat(decrypted, 'x').decode('hex')\n```\n\n\n```python\nimport os\nimport binascii\n\n# ASCII-encoded plaintext\nplaintext = \"this is a secret message\"\nplaintext_bits = int(binascii.hexlify(plaintext), 16)\n\nprint \"plaintext (ascii):\", plaintext\nprint \"plaintext (hex):\", plaintext_bits\n\n# Generate the one-time pad\nonetime_pad = int(binascii.hexlify(os.urandom(len(plaintext))), 16)\n\nprint \"one-time pad: (hex):\", onetime_pad\n\n# Encrypt plaintext using XOR operation with one-time pad\nciphertext_bits = plaintext_bits ^ onetime_pad\n\nprint \"encrypted text (hex):\", ciphertext_bits\n\n# Decrypt using XOR operation with one-time pad\ndecrypted_text = ciphertext_bits ^ onetime_pad\ndecrypted_text = binascii.unhexlify(hex(decrypted_text)[2:-1])\n\nprint \"decrypted text (ascii):\", decrypted_text\n```\n\n\n```python\nimport random\nimport binascii\n\np1 = \"this is the part where you run away\"\np2 = \"from bad cryptography practices.\"\n\n# pad plaintexts with spaces to ensure equal length\np1 = p1.ljust(len(p2))\np2 = p2.ljust(len(p1))\n \np1 = int(binascii.hexlify(p1), 16)\np2 = int(binascii.hexlify(p2), 16)\n\n# get random one-time pad\notp = random.SystemRandom().getrandbits(p1.bit_length())\n\n# encrypt\nc1 = p1 ^ otp\nc2 = p2 ^ otp # otp reuse...not good!\n\nprint \"c1 ^ c2 == p1 ^ p2 ?\", c1 ^ c2 == p1 ^ p2\nprint \"c1 ^ c2 =\", hex(c1 ^ c2)\n\n# the crib\ncrib = \" the \"\ncrib = int(binascii.hexlify(crib), 16)\n\nxored = c1 ^ c2\n\nprint \"crib =\", hex(crib)\n\ncbl = crib.bit_length()\nxbl = xored.bit_length()\n\nprint\nmask = (2**(cbl + 1) - 1)\nfill = len(str(xbl / 8))\n\n# crib dragging\nfor s in range(0, xbl - cbl + 8, 8):\n xor = (xored ^ (crib << s)) & (mask << s)\n out = binascii.unhexlify(hex(xor)[2:-1])\n \n print \"{:>{}} {}\".format(s/8, fill, out)\n```\n\n\n```python\nfrom cryptography.fernet import Fernet\nkey = Fernet.generate_key()\nf = Fernet(key)\nciphertext = f.encrypt(\"this is my plaintext\")\ndecrypted = f.decrypt(ciphertext)\nprint decrypted\n```\n\n\n```python\nimport os\nfrom cryptography.hazmat.primitives import padding\nfrom cryptography.hazmat.primitives.ciphers import Cipher, algorithms, modes\nfrom cryptography.hazmat.backends import default_backend\n\npt = \"my plaintext\"\n\nbackend = default_backend()\nkey = os.urandom(32)\niv = os.urandom(16)\n\npadder = padding.PKCS7(128).padder()\npt = padder.update(pt) + padder.finalize()\n\ncipher = Cipher(algorithms.AES(key), modes.CBC(iv), backend=backend)\nencryptor = cipher.encryptor()\nct = encryptor.update(pt) + encryptor.finalize()\ndecryptor = cipher.decryptor()\nout = decryptor.update(ct) + decryptor.finalize()\n\nunpadder = padding.PKCS7(128).unpadder()\nout = unpadder.update(out) + unpadder.finalize()\nprint out\n\n```\n\n\n```python\nnonce = os.urandom(64/8)\nnonce\n```\n\n\n```python\nimport hashlib\nhashlib.md5(\"hash me please\").hexdigest()\n```\n\n\n```python\n! md5 -s 'hash me please'\n```\n\n\n```python\nhashlib.sha1(\"hash me please\").hexdigest()\n```\n\n\n```python\n! echo 'hash me please' | openssl dgst -sha1\n```\n\n\n```python\nm1 = binascii.unhexlify(\"d131dd02c5e6eec4693d9a0698aff95c2fcab58712467eab4004583eb8fb7f8955ad340609f4b30283e488832571415a085125e8f7cdc99fd91dbdf280373c5bd8823e3156348f5bae6dacd436c919c6dd53e2b487da03fd02396306d248cda0e99f33420f577ee8ce54b67080a80d1ec69821bcb6a8839396f9652b6ff72a70\")\n\nm2 = binascii.unhexlify(\"d131dd02c5e6eec4693d9a0698aff95c2fcab50712467eab4004583eb8fb7f8955ad340609f4b30283e4888325f1415a085125e8f7cdc99fd91dbd7280373c5bd8823e3156348f5bae6dacd436c919c6dd53e23487da03fd02396306d248cda0e99f33420f577ee8ce54b67080280d1ec69821bcb6a8839396f965ab6ff72a70\")\n\nhashlib.md5(m1).hexdigest() == hashlib.md5(m1).hexdigest() \n```\n\n\n```python\nimport os\nfrom cryptography.hazmat.primitives.kdf.scrypt import Scrypt\nfrom cryptography.hazmat.backends import default_backend\n\nbackend = default_backend()\nsalt = os.urandom(16)\n\nkdf = Scrypt(salt=salt, length=64, n=2**14, r=8, p=1, backend=backend)\nkey = kdf.derive(\"your favorite password\")\nkey\n```\n\n\n```python\nkdf = Scrypt(salt=salt, length=64, n=2**14, r=8, p=1, backend=backend)\nkdf.verify(\"your favorite password\", key)\n```\n\n\n```python\nimport hmac\nimport hashlib\n\nsecret_key = \"my secret key\"\nciphertext = \"my ciphertext\"\n\n# generate HMAC\nh = hmac.new(key=secret_key, msg=ciphertext, digestmod=hashlib.sha256)\nprint h.hexdigest()\n\n# verify HMAC\nhmac.compare_digest(h.hexdigest(), h.hexdigest())\n```\n\n\n```python\np = 9576890767\nq = 1299827\nn = p * q\nprint n\n```\n\n\n```python\ne = 65537\nphi = (p - 1) * (q - 1)\nphi % e != 0\n```\n\n\n```python\nimport sympy\n\nd = sympy.numbers.igcdex(e, phi)[0]\nprint d\n```\n\n\n```python\nm = 12345\nc = pow(m, e, n)\nprint c\n```\n\n\n```python\npow(c, d, n)\n```\n\n\n```python\nm = 0\nwhile pow(m, e, n) != c:\n m += 1\nprint m\n```\n\n\n```python\nfrom cryptography.hazmat.backends import default_backend\nfrom cryptography.hazmat.primitives.asymmetric import rsa\nfrom cryptography.hazmat.primitives import serialization\n\nprivate_key = rsa.generate_private_key(public_exponent=65537, key_size=2048, backend=default_backend())\n\npublic_key = private_key.public_key()\n\nprivate_pem = private_key.private_bytes(encoding=serialization.Encoding.PEM, \n format=serialization.PrivateFormat.PKCS8, \n encryption_algorithm=serialization.BestAvailableEncryption('your password here'))\n\npublic_pem = public_key.public_bytes(encoding=serialization.Encoding.PEM, \n format=serialization.PublicFormat.SubjectPublicKeyInfo)\n\nprint public_pem\nprint private_pem \n```\n\n\n```python\nfrom cryptography.hazmat.primitives import hashes\nfrom cryptography.hazmat.primitives.asymmetric import padding\nimport base64\n\nwith open(\"path/to/public_key.pem\", \"rb\") as key_file:\n public_key = serialization.load_pem_public_key(key_file.read(), \n backend=default_backend())\n\nmessage = \"your secret message\"\nciphertext = public_key.encrypt(message, \n padding.OAEP(mgf=padding.MGF1(algorithm=hashes.SHA256()), \n algorithm=hashes.SHA256(), \n label=None))\nb64_ciphertext = base64.urlsafe_b64encode(ciphertext)\nprint b64_ciphertext\n\n```\n\n\n```python\nplaintext = private_key.decrypt(ciphertext, \n padding.OAEP(mgf=padding.MGF1(algorithm=hashes.SHA256()), \n algorithm=hashes.SHA256(), \n label=None))\nprint plaintext\n```\n\n\n```python\nfrom cryptography.hazmat.primitives import hashes\nfrom cryptography.hazmat.primitives.asymmetric import padding\n\nsigner = private_key.signer(padding.PSS(mgf=padding.MGF1(hashes.SHA256()), \n salt_length=padding.PSS.MAX_LENGTH), \n hashes.SHA256())\nmessage = \"A message of arbitrary length\"\nsigner.update(message)\nsignature = signer.finalize()\nsignature\n```\n\n\n```python\npublic_key = private_key.public_key()\nverifier = public_key.verifier(signature, \n padding.PSS(mgf=padding.MGF1(hashes.SHA256()), \n salt_length=padding.PSS.MAX_LENGTH), \n hashes.SHA256())\nverifier.update(message)\nverifier.verify()\n```\n\n### 5. Networking\n\n\n```python\nimport requests\nr = requests.get('https://www.google.com/imghp')\nr.content[:200]\n```\n\n\n```python\nr.status_code\n```\n\n\n```python\nr.headers\n```\n\n\n```python\nlen(r.content)\n```\n\n\n```python\nr.apparent_encoding\n```\n\n\n```python\nr.elapsed\n```\n\n\n```python\nr.request.headers\n```\n\n\n```python\ncustom_headers = {\"user-agent\": \"Mozilla/5.0 (Macintosh; Intel Mac OS X 10_11_6) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/56.0.2924.87 Safari/537.36\"}\nr = requests.get(\"https://www.google.com/imghp\", headers=custom_headers)\nr.request.headers\n```\n\n\n```python\nimport requests\nimport logging\nimport http.client\n\n# Enable logging\nhttp.client.HTTPConnection.debuglevel = 1\n\nlogging.basicConfig()\nlogging.getLogger().setLevel(logging.DEBUG)\nrequests_log = logging.getLogger(\"requests.packages.urllib3\")\nrequests_log.setLevel(logging.DEBUG)\nrequests_log.propagate = True\nr = requests.get('https://www.google.com/')\n```\n\n\n```python\nimport urlparse\nsimple_url = \"http://www.example.com/path/to/my/page\"\nparsed = urlparse.urlparse(simple_url)\n```\n\n\n```python\nparsed.scheme\n```\n\n\n```python\nparsed.hostname\n```\n\n\n```python\nparsed.path\n```\n\n\n```python\nurl_with_query = \"http://www.example.com/?page=1&key=Anvn4mo24\"\nquery = urlparse.urlparse(url_with_query).query\nurlparse.parse_qs(query)\n```\n\n\n```python\nimport urllib\nurl = 'https://www.example.com/%5EA-url-with-%-and-%5E?page=page+with%20spaces'\nurllib.unquote(url)\n```\n\n\n```python\nchars = '!@#$%^%$#)'\nurllib.quote(chars)\n```\n\n\n```python\nurllib.unquote_plus(url)\n```\n\n\n```python\nurllib.quote_plus('one two')\n```\n\n\n```python\nimport requests\nfrom bs4 import BeautifulSoup\n\nhttp_client.HTTPConnection.debuglevel = 0 # Logging off\nr = requests.get(\"http://www.google.com\")\nsoup = BeautifulSoup(r.content, \"lxml\")\n\nsoup.find_all('p')\n```\n\n\n```python\nsoup.find_all('a')\n```\n\n\n```python\nfor link in soup.find_all('a'):\n print link.text, link[\"href\"]\n```\n\n\n```python\nimport dryscrape\nfrom bs4 import BeautifulSoup\nsession = dryscrape.Session()\nsession.visit(\"http://www.google.com\")\nr = session.body()\nsoup = BeautifulSoup(r, \"lxml\") \n```\n\n\n```python\nfrom selenium import webdriver\ndriver = webdriver.Chrome(\"/path/to/chromedriver\")\ndriver.get(\"http://www.google.com\")\nhtml = driver.page_source\ndriver.save_screenshot(\"screenshot.png\")\ndriver.quit()\n```\n\n\n```python\nimport smtplib\n\nserver = smtplib.SMTP('localhost', port=1025)\nserver.set_debuglevel(True)\nserver.sendmail(\"me@localhost\", \"you@localhost\", \"This is an email message\")\nserver.quit()\n```\n\n\n```python\n! host google.com\n```\n\n\n```python\n! host 172.217.11.14\n```\n", "meta": {"hexsha": "f3c51e99cfdaa234636b97bb22a3e52d5cd81bfb", "size": 30886, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Code From Book.ipynb", "max_stars_repo_name": "0ppen/techsecrets", "max_stars_repo_head_hexsha": "496bd5fab4f1486a68ef32dc894f3e391226182c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Code From Book.ipynb", "max_issues_repo_name": "0ppen/techsecrets", "max_issues_repo_head_hexsha": "496bd5fab4f1486a68ef32dc894f3e391226182c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Code From Book.ipynb", "max_forks_repo_name": "0ppen/techsecrets", "max_forks_repo_head_hexsha": "496bd5fab4f1486a68ef32dc894f3e391226182c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 21.8584571833, "max_line_length": 294, "alphanum_fraction": 0.5205594768, "converted": true, "num_tokens": 4373, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.6654105653819836, "lm_q1q2_score": 0.42147788167262296}} {"text": "_Lambda School Data Science \u2014\u00a0Linear Models_\n\n# Understanding Linear Regression\n\n#### Objectives\n- understand how ordinary least squares regression minimizes the sum of squared errors\n- understand how linear algebra can solve ordinary least squares regression\n- get and interpret coefficients of a linear model\n- visualize a line of best fit in 2D, and hyperplane in 3D\n- use regression metrics: MSE, RMSE, MAE, R^2\n\n#### Extra Links\n- [Statistics 101: Simple Linear Regression](https://www.youtube.com/watch?v=ZkjP5RJLQF4) (20 minute video)\n- [_An Introduction to Statistical Learning_](http://www-bcf.usc.edu/~gareth/ISL/ISLR%20Seventh%20Printing.pdf), Chapter 3.1, Simple Linear Regression, & 3.2, Multiple Linear Regression\n- Priceonomics, [The Discovery of Statistical Regression](https://priceonomics.com/the-discovery-of-statistical-regression/)\n- Priceonomics, [Why the Father of Modern Statistics Didn\u2019t Believe Smoking Caused Cancer](https://priceonomics.com/why-the-father-of-modern-statistics-didnt-believe/)\n- Harvard Business Review, [When to Act on a Correlation, and When Not To](https://hbr.org/2014/03/when-to-act-on-a-correlation-and-when-not-to)\n- [xkcd 552: Correlation](https://www.explainxkcd.com/wiki/index.php/552:_Correlation)\n- [xkcd 1725: Linear Regression](https://www.explainxkcd.com/wiki/index.php/1725:_Linear_Regression)\n\n## What is Linear Regression?\n\n[Linear Regression](https://en.wikipedia.org/wiki/Linear_regression) is a statistical model that seeks to describe the relationship between some y variable and one or more x variables. \n\nIn the simplest case, linear regression seeks to fit a straight line through a cloud of points. This line is referred to as the \"regression line\" or \"line of best fit.\" This line tries to summarize the relationship between our X and Y in a way that enables us to use the equation for that line to make predictions.\n\n\n```python\n%matplotlib inline\nimport seaborn as sns\nx = [10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5]\ny = [ 8, 6, 7, 8, 8, 9, 7, 4, 10, 4, 5]\nsns.regplot(x, y);\n```\n\n\n\n### Synonyms for \"y variable\" \n- Dependent Variable\n- Response Variable\n- Outcome Variable \n- Predicted Variable\n- Measured Variable\n- Explained Variable\n- Label\n- Target\n\n### Synonyms for \"x variable\"\n- Independent Variable\n- Explanatory Variable\n- Regressor\n- Covariate\n- Feature\n\n## Simple Linear Regresion\n\n#### Making Predictions\n\nSay that we were trying to create a model that captured the relationship between temperature outside and ice cream sales. In Machine Learning our goal is often different that of other flavors of Linear Regression Analysis, because we're trying to fit a model to this data with the intention of making **predictions** on new data (in the future) that we don't have yet.\n\n#### What are we trying to predict?\n\nSo if we had measured ice cream sales and the temprature outside on 11 different days, at the end of our modeling **what would be the thing that we would want to predict? - Ice Cream Sales or Temperature?**\n\nWe would probably want to be measuring temperature with the intention of using that to **forecast** ice cream sales. If we were able to successfully forecast ice cream sales from temperature, this might help us know beforehand how much ice cream to make or how many cones to buy or on which days to open our store, etc. Being able to make predictions accurately has a lot of business implications. This is why making accurate predictions is so valuable (And in large part is why data scientists are paid so well).\n\n#### y variable intuition\n\nWe want the thing that we're trying to predict to serve as our **y** variable. This is why it's sometimes called the \"predicted variable.\" We call it the \"dependent\" variable because our prediction for how much ice cream we're going to sell \"depends\" on the temperature outside. \n\n#### x variable intuition\n\nAll other variables that we use to predict our y variable (we're going to start off just using one) we call our **x** variables. These are called our \"independent\" variables because they don't *depend* on y, they \"explain\" y. Hence they are also referred to as our \"explanatory\" variables.\n\n## Example: Predict presidential election voting\n\n#### Douglas Hibbs, [Background Information on the \u2018Bread and Peace\u2019 Model of Voting in Postwar US Presidential Elections](https://douglas-hibbs.com/background-information-on-bread-and-peace-voting-in-us-presidential-elections/)\n\n> Aggregate two-party vote shares going to candidates of the party holding the presidency during the postwar era are well explained by just two fundamental determinants:\n\n> (1) Positively by weighted-average growth of per capita real disposable personal income over the term. \n> (2) Negatively by cumulative US military fatalities (scaled to population) owing to unprovoked, hostile deployments of American armed forces in foreign wars.\n\n#### Data sources\n- 1952-2012: Douglas Hibbs, [2014 lecture at Deakin University Melbourne](http://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf), Slide 40\n- 2016, Vote Share: [The American Presidency Project](https://www.presidency.ucsb.edu/statistics/elections)\n- 2016, Recent Growth in Personal Incomes: [The 2016 election economy: the \"Bread and Peace\" model final forecast](https://angrybearblog.com/2016/11/the-2016-election-economy-the-bread-and-peace-model-final-forecast.html)\n- 2016, US Military Fatalities: Assumption that Afghanistan War fatalities in 2012-16 occured at the same rate as 2008-12\n\n\n```python\nimport pandas as pd\n\ncolumns = ['Year','Inc_Party_Candidate','Other_Candidate','Inc_Party_Vote']\n\ndata = [[1952,\"Stevenson\",\"Eisenhower\",44.6],\n [1956,\"Eisenhower\",\"Stevenson\",57.76],\n [1960,\"Nixon\",\"Kennedy\",49.91],\n [1964,\"Johnson\",\"Goldwater\",61.34],\n [1968,\"Humphrey\",\"Nixon\",49.60],\n [1972,\"Nixon\",\"McGovern\",61.79],\n [1976,\"Ford\",\"Carter\",48.95],\n [1980,\"Carter\",\"Reagan\",44.70],\n [1984,\"Reagan\",\"Mondale\",59.17],\n [1988,\"Bush, Sr.\",\"Dukakis\",53.94],\n [1992,\"Bush, Sr.\",\"Clinton\",46.55],\n [1996,\"Clinton\",\"Dole\",54.74],\n [2000,\"Gore\",\"Bush, Jr.\",50.27],\n [2004,\"Bush, Jr.\",\"Kerry\",51.24],\n [2008,\"McCain\",\"Obama\",46.32],\n [2012,\"Obama\",\"Romney\",52.00], \n [2016,\"Clinton\",\"Trump\",48.2]]\n \nvotes = pd.DataFrame(data=data, columns=columns)\n```\n\n\n```python\ncolumns = ['Year','Avg_Rec_Growth']\n\ndata = [[1952,2.40],\n [1956,2.89],\n [1960, .85],\n [1964,4.21],\n [1968,3.02],\n [1972,3.62],\n [1976,1.08],\n [1980,-.39],\n [1984,3.86],\n [1988,2.27],\n [1992, .38],\n [1996,1.04],\n [2000,2.36],\n [2004,1.72],\n [2008, .10],\n [2012, .95], \n [2016, .10]]\n \ngrowth = pd.DataFrame(data=data, columns=columns)\n```\n\n\n```python\n\"\"\"\nFatalities denotes the cumulative number of American military\nfatalities per millions of US population the in Korea, Vietnam,\nIraq and Afghanistan wars during the presidential terms\npreceding the 1952, 1964, 1968, 1976 and 2004, 2008 and\n2012 elections.\n\nhttp://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf\n\"\"\"\n\ncolumns = ['Year','fatal_per_mil']\n\ndata = [[1952,190],\n [1956, 0],\n [1960, 0],\n [1964, 1],\n [1968,146],\n [1972, 0],\n [1976, 2],\n [1980, 0],\n [1984, 0],\n [1988, 0],\n [1992, 0],\n [1996, 0],\n [2000, 0],\n [2004, 4],\n [2008, 14],\n [2012, 5], \n [2016, 5]]\n \ndeaths = pd.DataFrame(data=data, columns=columns)\n```\n\n\n```python\ndf = votes.merge(growth).merge(deaths)\n```\n\n\n```python\ndf.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        YearInc_Party_CandidateOther_CandidateInc_Party_VoteAvg_Rec_Growthfatal_per_mil
                                        01952StevensonEisenhower44.602.40190
                                        11956EisenhowerStevenson57.762.890
                                        21960NixonKennedy49.910.850
                                        31964JohnsonGoldwater61.344.211
                                        41968HumphreyNixon49.603.02146
                                        \n
                                        \n\n\n\n### Plot univariate correlations\n[Seaborn tutorial: Visualizing linear relationships](https://seaborn.pydata.org/tutorial/regression.html)\n\n\n```python\n%matplotlib inline\nimport seaborn as sns\n\ntarget = 'Inc_Party_Vote'\nfeatures = ['Avg_Rec_Growth', \n 'fatal_per_mil']\n\nfor feature in features:\n sns.lmplot(x=feature, y=target, data=df)\n```\n\nWe can see from the scatterplot that these data points seem to follow a somewhat linear relationship for the \"Average Recent Growth in Personal Incomes\" feature. This means that we could probably summarize their relationship well by fitting a line of best fit to these points. Lets do it.\n\n\n## The Equation for a Line\n\nA common equation for a line is:\n\n\\begin{align}\ny = mx + b\n\\end{align}\n\nWhere $m$ is the slope of our line and $b$ is the y-intercept. \n\nIf we want to plot a line through our cloud of points we figure out what these two values should be. Linear Regression seeks to **estimate** the slope and intercept values that describe a line that best fits the data points.\n\n## The Anatomy of Linear Regression\n\n- Intercept: The $b$ value in our line equation $y=mx+b$\n- Slope: The $m$ value in our line equation $y=mx+b$. These two values together define our regression line.\n- $\\hat{y}$ : A prediction\n- Line of Best Fit (Regression Line)\n- Predicted (fitted) Values: Points on our regression line\n- Observed Values: Points from our dataset\n- Error: The distance between predicted and observed values.\n\n\nOrdinary Least Squares Regression is a way to solve for $m$ and $b$.\n\nLet's start by seeing what would happen if we just guessed and checked some values for $m$ and $b$. \n\nWhat's the line of \"best\" fit look like? What's the error?\n\n\n\n```python\n# TODO\n\nx = df['Avg_Rec_Growth']\ny = df['Inc_Party_Vote']\n\nm = 0\nb = y.mean()\ny_pred = m * x + b\ny_pred\n```\n\n\n\n\n 0 51.828235\n 1 51.828235\n 2 51.828235\n 3 51.828235\n 4 51.828235\n 5 51.828235\n 6 51.828235\n 7 51.828235\n 8 51.828235\n 9 51.828235\n 10 51.828235\n 11 51.828235\n 12 51.828235\n 13 51.828235\n 14 51.828235\n 15 51.828235\n 16 51.828235\n Name: Avg_Rec_Growth, dtype: float64\n\n\n\n\n```python\nimport matplotlib.pyplot as plt\nfrom sklearn.metrics import mean_absolute_error, r2_score\n\ndef plot_preds(x, y, y_pred):\n plt.scatter(x,y, label='y_true')\n plt.plot(x,y_pred, label='y_pred')\n plt.legend()\n mae = mean_absolute_error(y, y_pred)\n r2 = r2_score(y, y_pred)\n print(f'Mean Absolute Error: {mae}')\n print(f'R^2 Score: {r2}') \n```\n\n\n```python\nplot_preds(x, y, y_pred)\n```\n\n\n```python\nm = 4.1\nb = 45\ny_pred = m*x + b\nplot_preds(x, y, y_pred)\nplt.title('Guessing LR Values')\n```\n\n## R Squared: $R^2$\n\nOne final attribute of linear regressions that we're going to talk about today is a measure of goodness of fit known as $R^2$ or R-squared. $R^2$ is a statistical measure of how close the data are fitted to our regression line. A helpful interpretation for the $R^2$ is the percentage of the dependent variable that is explained by the model.\n\nIn other words, the $R^2$ is the percentage of y that is explained by the x variables included in the model. For this reason the $R^2$ is also known as the \"coefficient of determination,\" because it explains how much of y is explained (or determined) by our x varaibles. We won't go into the calculation of $R^2$ today, just know that a higher $R^2$ percentage is nearly always better and indicates a model that fits the data more closely. \n\n## Residual Error \n\nThe residual error is the distance between points in our dataset and our regression line.\n\n\n```python\ndef regression_residuals(df, feature, target, m, b):\n x = df[feature]\n y = df[target]\n y_pred = m*x + b\n \n plt.scatter(x, y, label='y_true')\n plt.plot(x, y_pred, label='y_pred')\n plt.legend()\n \n # Plot residual errors\n for x, y1, y2 in zip(x, y, y_pred):\n plt.plot((x, x), (y1, y2), color='grey')\n \n mae = mean_absolute_error(y, y_pred) \n r2 = r2_score(y, y_pred)\n print('Mean Absolute Error:', mae)\n print('R^2:', r2)\n \nregression_residuals(df, feature='Avg_Rec_Growth', \n target='Inc_Party_Vote', m=3, b=46)\n```\n\n\n```python\nfrom ipywidgets import interact, fixed\n\ninteract(regression_residuals, \n df=fixed(df), \n feature=fixed('Avg_Rec_Growth'), \n target=fixed('Inc_Party_Vote'), \n m=(-10,10,0.5), \n b=(40,60,0.5));\n```\n\n\n interactive(children=(FloatSlider(value=0.0, description='m', max=10.0, min=-10.0, step=0.5), FloatSlider(valu\u2026\n\n\n## More Formal Notation\n\n\n\nWe have talked about a line of regression being represented like a regular line $y=mx+b$ but as we get to more complicated versions we're going to need to extend this equation. So lets establish the proper terminology.\n\n**X** - Independent Variable, predictor variable, explanatory variable, regressor, covariate\n\n**Y** - Response variable, predicted variable, measured vairable, explained variable, outcome variable\n\n$\\beta_0$ - \"Beta Naught\" or \"Beta Zero\", the intercept value. This is how much of y would exist if X were zero. This is sometimes represented by the letter \"a\" but I hate that. So it's \"Beta 0\" during my lecture.\n\n$\\beta_1$ - \"Beta One\" The primary coefficient of interest. This values is the slope of the line that is estimated by \"minimizing the sum of the squared errors/residuals\" - We'll get to that. \n\n$\\epsilon$ - \"Epsilon\" The \"error term\", random noise, things outside of our model that affect y.\n\n## Minimizing the Sum of the Squared Error\n\nThe most common method of estimating our $\\beta$ parameters is what's known as \"Ordinary Least Squares\" (OLS). (There are different methods of arriving at a line of best fit). OLS estimates the parameters that minimize the squared distance between each point in our dataset and our line of best fit. \n\n\\begin{align}\nSSE = \\sum(y_i - \\hat{y})^2\n\\end{align}\n\n\n\n```python\nfrom matplotlib.patches import Rectangle\nimport numpy as np\nfrom sklearn.metrics import mean_squared_error\n\ndef regression_squared_errors(df, feature, target, m, b):\n x = df[feature]\n y = df[target]\n y_pred = m*x + b\n \n fig = plt.figure(figsize=(7,7))\n ax = plt.axes()\n ax.scatter(x, y, label='y_true')\n ax.plot(x, y_pred, label='y_pred')\n ax.legend()\n \n # Plot square errors\n xmin, xmax = ax.get_xlim()\n ymin, ymax = ax.get_ylim()\n scale = (xmax-xmin)/(ymax-ymin)\n for x, y1, y2 in zip(x, y, y_pred):\n bottom_left = (x, min(y1, y2))\n height = abs(y1 - y2)\n width = height * scale\n ax.add_patch(Rectangle(xy=bottom_left, width=width, height=height, alpha=0.1))\n \n mse = mean_squared_error(y, y_pred)\n rmse = np.sqrt(mse)\n mae = mean_absolute_error(y, y_pred)\n r2 = r2_score(y, y_pred)\n print('Mean Squared Error:', mse)\n print('Root Mean Squared Error:', rmse)\n print('Mean Absolute Error:', mae)\n print('R^2:', r2)\n \nregression_squared_errors(df, feature='Avg_Rec_Growth', \n target='Inc_Party_Vote', m=3, b=46)\n```\n\n\n```python\ninteract(regression_squared_errors, \n df=fixed(df), \n feature=fixed('Avg_Rec_Growth'), \n target=fixed('Inc_Party_Vote'), \n m=(-10,10,0.5), \n b=(40,60,0.5));\n```\n\n\n interactive(children=(FloatSlider(value=0.0, description='m', max=10.0, min=-10.0, step=0.5), FloatSlider(valu\u2026\n\n\n## Hypotheses\n\n\n```python\nb = 46\nms = np.arange(-10,10,0.5)\nsses = []\nfeature = 'Avg_Rec_Growth'\n\nfor m in ms:\n predictions = m * df[feature] + b\n errors = df[target] - predictions\n square_errors = errors ** 2\n sse = square_errors.sum()\n sses.append(sse)\n \nhypotheses = pd.DataFrame({'Slope': ms})\nhypotheses['Intercept'] = b\nhypotheses['Sum of Square Errors'] = sses\n\nhypotheses\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        SlopeInterceptSum of Square Errors
                                        0-10.04615215.58740
                                        1-9.54614095.52845
                                        2-9.04613018.88500
                                        3-8.54611985.65705
                                        4-8.04610995.84460
                                        5-7.54610049.44765
                                        6-7.0469146.46620
                                        7-6.5468286.90025
                                        8-6.0467470.74980
                                        9-5.5466698.01485
                                        10-5.0465968.69540
                                        11-4.5465282.79145
                                        12-4.0464640.30300
                                        13-3.5464041.23005
                                        14-3.0463485.57260
                                        15-2.5462973.33065
                                        16-2.0462504.50420
                                        17-1.5462079.09325
                                        18-1.0461697.09780
                                        19-0.5461358.51785
                                        200.0461063.35340
                                        210.546811.60445
                                        221.046603.27100
                                        231.546438.35305
                                        242.046316.85060
                                        252.546238.76365
                                        263.046204.09220
                                        273.546212.83625
                                        284.046264.99580
                                        294.546360.57085
                                        305.046499.56140
                                        315.546681.96745
                                        326.046907.78900
                                        336.5461177.02605
                                        347.0461489.67860
                                        357.5461845.74665
                                        368.0462245.23020
                                        378.5462688.12925
                                        389.0463174.44380
                                        399.5463704.17385
                                        \n
                                        \n\n\n\n\n```python\nhypotheses.plot(x='Slope', y='Sum of Square Errors', \n title=f'Intercept={b}');\n```\n\n## Scikit-learn\n\n#### Jake VanderPlas, [Python Data Science Handbook, Chapter 5.2, Introducing Scikit-Learn](https://jakevdp.github.io/PythonDataScienceHandbook/05.02-introducing-scikit-learn.html), Scikit-Learn's Estimator API\n\n> Most commonly, the steps in using the Scikit-Learn estimator API are as follows (we will step through a handful of detailed examples in the sections that follow).\n\n> 1. Choose a class of model by importing the appropriate estimator class from Scikit-Learn. \n> 2. Choose model hyperparameters by instantiating this class with desired values. \n> 3. Arrange data into a features matrix and target vector following the discussion above.\n> 4. Fit the model to your data by calling the `fit()` method of the model instance.\n> 5. Apply the Model to new data: For supervised learning, often we predict labels for unknown data using the `predict()` method.\n\n\n```python\n# TODO\n```\n\n\n```python\nfrom sklearn.linear_model import LinearRegression\n\nmodel = LinearRegression()\nfeatures = ['Avg_Rec_Growth']\ntarget = 'Inc_Party_Vote'\n\nx = df[features]\ny = df[target]\n\nmodel.fit(x, y)\n```\n\n\n\n\n LinearRegression(copy_X=True, fit_intercept=True, n_jobs=None,\n normalize=False)\n\n\n\n\n```python\nmodel.intercept_\n```\n\n\n\n\n 46.499209757741625\n\n\n\n\n```python\nmodel.coef_\n```\n\n\n\n\n array([2.97417709])\n\n\n\n\n```python\nmodel.predict([[1]])\n```\n\n\n\n\n array([49.47338685])\n\n\n\n\n```python\nmodel.predict([[4]])\n```\n\n\n\n\n array([58.39591811])\n\n\n\n\n```python\nmodel.predict([[2]]) - model.predict([[1]])\n```\n\n\n\n\n array([2.97417709])\n\n\n\n\n```python\nx\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Avg_Rec_Growth
                                        02.40
                                        12.89
                                        20.85
                                        34.21
                                        43.02
                                        53.62
                                        61.08
                                        7-0.39
                                        83.86
                                        92.27
                                        100.38
                                        111.04
                                        122.36
                                        131.72
                                        140.10
                                        150.95
                                        160.10
                                        \n
                                        \n\n\n\n\n```python\nm = model.coef_[0]\nb = model.intercept_\n\ny_pred = m*x + b\nplot_preds(x,y,y_pred)\n```\n\n\n```python\ny_pred = model.predict(x)\nplot_preds(x,y, y_pred)\n```\n\n## Linear Algebra!\n\nThe same result that is found by minimizing the sum of the squared errors can be also found through a linear algebra process known as the \"Least Squares Solution:\"\n\n\\begin{align}\n\\hat{\\beta} = (X^{T}X)^{-1}X^{T}y\n\\end{align}\n\nBefore we can work with this equation in its linear algebra form we have to understand how to set up the matrices that are involved in this equation. \n\n### The $\\beta$ vector\n\nThe $\\beta$ vector represents all the parameters that we are trying to estimate, our $y$ vector and $X$ matrix values are full of data from our dataset. The $\\beta$ vector holds the variables that we are solving for: $\\beta_0$ and $\\beta_1$\n\nNow that we have all of the necessary parts we can set them up in the following equation:\n\n\\begin{align}\ny = X \\beta + \\epsilon\n\\end{align}\n\nSince our $\\epsilon$ value represents **random** error we can assume that it will equal zero on average.\n\n\\begin{align}\ny = X \\beta\n\\end{align}\n\nThe objective now is to isolate the $\\beta$ matrix. We can do this by pre-multiplying both sides by \"X transpose\" $X^{T}$.\n\n\\begin{align}\nX^{T}y = X^{T}X \\beta\n\\end{align}\n\nSince anything times its transpose will result in a square matrix, if that matrix is then an invertible matrix, then we should be able to multiply both sides by its inverse to remove it from the right hand side. (We'll talk tomorrow about situations that could lead to $X^{T}X$ not being invertible.)\n\n\\begin{align}\n(X^{T}X)^{-1}X^{T}y = (X^{T}X)^{-1}X^{T}X \\beta\n\\end{align}\n\nSince any matrix multiplied by its inverse results in the identity matrix, and anything multiplied by the identity matrix is itself, we are left with only $\\beta$ on the right hand side:\n\n\\begin{align}\n(X^{T}X)^{-1}X^{T}y = \\hat{\\beta}\n\\end{align}\n\nWe will now call it \"beta hat\" $\\hat{\\beta}$ because it now represents our estimated values for $\\beta_0$ and $\\beta_1$\n\n### Lets calculate our $\\beta$ coefficients with numpy!\n\n\n```python\nfrom statsmodels.api import add_constant\n\nX = add_constant(df[feature].values)\nprint('X')\nprint(X)\n\ny = df[target].values[:, np.newaxis]\nprint('y')\nprint(y)\n\nX_transpose = X.T\nprint('X Transpose')\nprint(X_transpose)\n\nX_transpose_X = X_transpose @ X\nprint('X Transpose X')\nprint(X_transpose_X)\n\nX_transpose_X_inverse = np.linalg.inv(X_transpose_X)\nprint('X Transpose X Inverse')\nprint(X_transpose_X_inverse)\n\nX_transpose_y = X_transpose @ y\nprint('X Transpose y')\nprint(X_transpose_y)\n\nbeta_hat = X_transpose_X_inverse @ X_transpose_y\nprint('Beta Hat')\nprint(beta_hat)\n```\n\n X\n [[ 1. 2.4 ]\n [ 1. 2.89]\n [ 1. 0.85]\n [ 1. 4.21]\n [ 1. 3.02]\n [ 1. 3.62]\n [ 1. 1.08]\n [ 1. -0.39]\n [ 1. 3.86]\n [ 1. 2.27]\n [ 1. 0.38]\n [ 1. 1.04]\n [ 1. 2.36]\n [ 1. 1.72]\n [ 1. 0.1 ]\n [ 1. 0.95]\n [ 1. 0.1 ]]\n y\n [[44.6 ]\n [57.76]\n [49.91]\n [61.34]\n [49.6 ]\n [61.79]\n [48.95]\n [44.7 ]\n [59.17]\n [53.94]\n [46.55]\n [54.74]\n [50.27]\n [51.24]\n [46.32]\n [52. ]\n [48.2 ]]\n X Transpose\n [[ 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1.\n 1. 1. 1. 1. 1. ]\n [ 2.4 2.89 0.85 4.21 3.02 3.62 1.08 -0.39 3.86 2.27 0.38 1.04\n 2.36 1.72 0.1 0.95 0.1 ]]\n X Transpose X\n [[17. 30.46 ]\n [30.46 86.831]]\n X Transpose X Inverse\n [[ 0.15835959 -0.05555197]\n [-0.05555197 0.03100405]]\n X Transpose y\n [[ 881.08 ]\n [1674.6167]]\n Beta Hat\n [[46.49920976]\n [ 2.97417709]]\n\n\n# Multiple Regression\n\nSimple or bivariate linear regression involves a single $x$ variable and a single $y$ variable. However, we can have many $x$ variables. A linear regression model that involves multiple x variables is known as **Multiple** Regression (NOT MULTIVARIATE).\n\n\\begin{align}\ny = \\beta_0 + \\beta_1 X_1 + \\beta_2 X_2 + ... + \\beta_n X_n + \\epsilon\n\\end{align}\n\n\n```python\n# TODO\n\nmodel = LinearRegression()\nfeatures = [\n 'Avg_Rec_Growth',\n 'fatal_per_mil',\n]\n\ntaget = 'Inc_Party_Vote'\n\nx = df[features]\ny = df[target]\n\nmodel.fit(x,y)\n```\n\n\n\n\n LinearRegression(copy_X=True, fit_intercept=True, n_jobs=None,\n normalize=False)\n\n\n\n\n```python\npd.Series(model.coef_, features)\n```\n\n\n\n\n Avg_Rec_Growth 3.406214\n fatal_per_mil -0.053752\n dtype: float64\n\n\n\n## Visualize hyperplane of best fit in 3D\n\n\n```python\n!pip install plotly\n```\n\n Requirement already satisfied: plotly in c:\\programdata\\anaconda3\\lib\\site-packages (3.10.0)\n Requirement already satisfied: six in c:\\programdata\\anaconda3\\lib\\site-packages (from plotly) (1.12.0)\n Requirement already satisfied: requests in c:\\programdata\\anaconda3\\lib\\site-packages (from plotly) (2.21.0)\n Requirement already satisfied: pytz in c:\\programdata\\anaconda3\\lib\\site-packages (from plotly) (2018.9)\n Requirement already satisfied: retrying>=1.3.3 in c:\\programdata\\anaconda3\\lib\\site-packages (from plotly) (1.3.3)\n Requirement already satisfied: nbformat>=4.2 in c:\\programdata\\anaconda3\\lib\\site-packages (from plotly) (4.4.0)\n Requirement already satisfied: decorator>=4.0.6 in c:\\programdata\\anaconda3\\lib\\site-packages (from plotly) (4.4.0)\n Requirement already satisfied: chardet<3.1.0,>=3.0.2 in c:\\programdata\\anaconda3\\lib\\site-packages (from requests->plotly) (3.0.4)\n Requirement already satisfied: certifi>=2017.4.17 in c:\\programdata\\anaconda3\\lib\\site-packages (from requests->plotly) (2019.3.9)\n Requirement already satisfied: urllib3<1.25,>=1.21.1 in c:\\programdata\\anaconda3\\lib\\site-packages (from requests->plotly) (1.24.1)\n Requirement already satisfied: idna<2.9,>=2.5 in c:\\programdata\\anaconda3\\lib\\site-packages (from requests->plotly) (2.8)\n Requirement already satisfied: jsonschema!=2.5.0,>=2.4 in c:\\programdata\\anaconda3\\lib\\site-packages (from nbformat>=4.2->plotly) (3.0.1)\n Requirement already satisfied: ipython-genutils in c:\\programdata\\anaconda3\\lib\\site-packages (from nbformat>=4.2->plotly) (0.2.0)\n Requirement already satisfied: traitlets>=4.1 in c:\\programdata\\anaconda3\\lib\\site-packages (from nbformat>=4.2->plotly) (4.3.2)\n Requirement already satisfied: jupyter-core in c:\\programdata\\anaconda3\\lib\\site-packages (from nbformat>=4.2->plotly) (4.4.0)\n Requirement already satisfied: attrs>=17.4.0 in c:\\programdata\\anaconda3\\lib\\site-packages (from jsonschema!=2.5.0,>=2.4->nbformat>=4.2->plotly) (19.1.0)\n Requirement already satisfied: pyrsistent>=0.14.0 in c:\\programdata\\anaconda3\\lib\\site-packages (from jsonschema!=2.5.0,>=2.4->nbformat>=4.2->plotly) (0.14.11)\n Requirement already satisfied: setuptools in c:\\programdata\\anaconda3\\lib\\site-packages (from jsonschema!=2.5.0,>=2.4->nbformat>=4.2->plotly) (40.8.0)\n\n\n\n```python\n# https://stackoverflow.com/a/47230966\n# Plotly notebook mode with google colaboratory\n# You need to define this function\n# And call it in each offline plotting cell\n\ndef configure_plotly_browser_state():\n import IPython\n display(IPython.core.display.HTML('''\n \n \n '''))\n```\n\n\n```python\nimport itertools\nimport plotly.graph_objs as go\nfrom plotly.offline import init_notebook_mode, iplot\ninit_notebook_mode(connected=True)\n\ndef viz3D(fitted_model, X, features, target='', num=100):\n \"\"\"\n Visualize model predictions in 3D, for regression model fit on 2 features\n \n Parameters\n ----------\n fitted_model : scikit-learn model, already fitted\n X : pandas dataframe, which was used to fit model\n features : list of strings, column names of the 2 features\n target : string, name of target\n num : int, number of grid points for each feature\n \n References\n ----------\n https://plot.ly/python/3d-charts/\n \"\"\"\n feature1, feature2 = features\n min1, max1 = X[feature1].min(), X[feature1].max()\n min2, max2 = X[feature2].min(), X[feature2].max()\n x1 = np.linspace(min1, max1, num)\n x2 = np.linspace(min2, max2, num)\n combos = list(itertools.product(x1, x2))\n Z = fitted_model.predict(combos).reshape(num, num)\n \n configure_plotly_browser_state()\n data = [go.Surface(x=x1, y=x2, z=Z)]\n layout = go.Layout(\n scene={'xaxis': {'title': feature1, 'range': [min1,max1], 'showticklabels': True}, \n 'yaxis': {'title': feature2, 'range': [min2,max2], 'showticklabels': True}, \n 'zaxis': {'title': target, 'showticklabels': True}}, \n )\n fig = go.Figure(data=data, layout=layout)\n iplot(fig)\n```\n\n\n\n\n\n\n\n```python\n# TODO\n\nviz3D(model, x, features, target)\n```\n\n\n\n\n\n\n\n\n\n
                                        \n\n\n
                                        \n \n
                                        \n\n\n## Dimensionality in Linear Regression\n\nMuliple Regression is simply an extension of the bivariate case. The reason why we see the bivariate case demonstrated so often is simply because it's easier to graph and all of the intuition from the bivariate case is the same as we keep on adding explanatory variables.\n\nAs we increase the number of $x$ values in our model we are simply fitting a n-1-dimensional plane to an n-dimensional cloud of points within an n-dimensional hypercube. \n\n## Interpreting Coefficients\n\nOne of Linear Regression's strengths is that the parameters of the model (coefficients) are readily interpretable and useful. Not only do they describe the relationship between x and y but they put a number on just how much x is associated with y. We should be careful to not speak about this relationshiop in terms of causality because these coefficients are in fact correlative measures. We would need a host of additional techniques in order to estimate a causal effect using linear regression (econometrics).\n\n\\begin{align}\n\\hat{\\beta} = \\frac{Cov(x,y)}{Var(y)}\n\\end{align}\n\n## Why is Linear Regression so Important?\n\n### Popularity \n\nLinear Regression is an extremely popular technique that every data scientist **needs** to understand. It's not the most advanced technique and there are supervised learning techniques that will obtain a higher accuracy, but where it lacks in accuracy it makes up for it in interpretability and simplicity.\n\n### Interpretability\n\nFew other models possess coefficients that are so directly linked to their variables with a such a clear interpretation. Tomorrow we're going to learn about ways to make them even easier to interpret.\n\n### Simplicity\n\nA linear regression model can be communicated just by writing out its equation. It's kind of incredible that such high dimensional relationships can be described from just a linear combination of variables and coefficients. \n\n# Assignment\n- Continue to predict New York City apartment rents. This is your last assignment with this dataset.\n- You may select any number of features. You are encouraged to engineer new features.\n- Get and plot your model's coefficients.\n- Report your Root Mean Squared Error, Mean Absolute Error, and R^2 Score, for your Train and Test sets. Share your scores with your cohort on Slack!\n- Fit a model with 2 features, and visualize the plane of best fit in 3D.\n- Commit your notebook to your fork of the repo.\n\n## Stretch Goals\n\nStudy more about Linear Regression. Here are two helpful links. If you find more links, share your favorites with your cohort on Slack.\n\n1. Watch this 20 minute video that just hit 1 million views: Brandon Foltz, Statistics 101: Simple Linear Regression (https://www.youtube.com/watch?v=ZkjP5RJLQF4)\n2. Skim _An Introduction to Statistical Learning_, Chapter 3.1, Simple Linear Regression, & Chapter 3.2, Multiple Linear Regression (http://www-bcf.usc.edu/~gareth/ISL/ISLR%20Seventh%20Printing.pdf)\n\nIn your 3D visualization, can you include the actual datapoints, like in [this notebook](https://nbviewer.jupyter.org/urls/s3.amazonaws.com/datarobotblog/notebooks/multiple_regression_in_python.ipynb)? Can you also include the residual lines from the datapoints to the plane of the best fit, like in _An Introduction to Statistical Learning?_ This would be hard to do, but awesome!\n\n\nCan you get creative with feature engineering? Share with your cohort on Slack. We mentioned some feature ideas at the end of last lesson, but didn't demonstrate how to engineer them. So here are some example solutions:\n\n```python\n# Does apartment have a non-empty description?\ndf['description'] = df['description'].str.strip().fillna('')\ndf['has_description'] = df['description'] != ''\n\n# How long is the description?\ndf['description_length'] = df['description'].str.len()\n\n# How many total perks does each apartment have?\nperk_cols = ['elevator', 'cats_allowed', 'hardwood_floors', 'dogs_allowed',\n 'doorman', 'dishwasher', 'no_fee', 'laundry_in_building',\n 'fitness_center', 'pre-war', 'laundry_in_unit', 'roof_deck',\n 'outdoor_space', 'dining_room', 'high_speed_internet', 'balcony',\n 'swimming_pool', 'new_construction', 'exclusive', 'terrace', \n 'loft', 'garden_patio', 'common_outdoor_space', \n 'wheelchair_access']\ndf['perk_count'] = df[perk_cols].sum(axis=1)\n\n# Are pets allowed?\ndf['pets_allowed'] = (df['cats_allowed']==1) | (df['dogs_allowed']==1)\n```\n\n", "meta": {"hexsha": "b5fda3841fa0e032130be1838a5b5c664110818f", "size": 726944, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "module3-understanding-linear-regression/understanding_linear_regression.ipynb", "max_stars_repo_name": "cicbeast/DS-Unit-2-Linear-Models", "max_stars_repo_head_hexsha": "bcda296b0a67f44752173deb2cab2e4667073d24", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "module3-understanding-linear-regression/understanding_linear_regression.ipynb", "max_issues_repo_name": "cicbeast/DS-Unit-2-Linear-Models", "max_issues_repo_head_hexsha": "bcda296b0a67f44752173deb2cab2e4667073d24", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "module3-understanding-linear-regression/understanding_linear_regression.ipynb", "max_forks_repo_name": "cicbeast/DS-Unit-2-Linear-Models", "max_forks_repo_head_hexsha": "bcda296b0a67f44752173deb2cab2e4667073d24", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 57.1991502085, "max_line_length": 196338, "alphanum_fraction": 0.6985311111, "converted": true, "num_tokens": 11778, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.6893056040203135, "lm_q1q2_score": 0.421429020388656}} {"text": "```python\nimport hublib\nimport pandas as pd\nimport ipywidgets as widgets\nfrom IPython.display import clear_output\nimport matplotlib\n%matplotlib notebook\nimport matplotlib.pyplot as plt\nimport scipy\nfrom scipy import signal\nimport numpy as np\nfrom decimal import Decimal\nimport unicodedata\nfrom unicodedata import lookup as GL\nimport sympy as sy\nfrom joblib import Parallel, delayed\nfrom lmfit import Model\nimport warnings\nimport zipfile\nfrom zipfile import ZipFile\nimport os\nfrom hublib.ui import FileUpload, Download\nfrom scipy import sparse\nfrom scipy.sparse.linalg import spsolve\n```\n\n\n```python\nclass Spectrum:\n def __init__(self):\n self.x=0\n self.y=0\n self.I=[]\n self.W=[]\n self.If=[]\n self.PG=[]\n self.PGp=[]\n self.PD=[]\n self.IDfit=[]\n self.Q=0\n self.diffs=[]\n self.mdi=0\n self.md=0\n self.mf=[]\n```\n\n\n```python\nglobal Specs\nSpecs=[]\n\nglobal filelist\nfilelist = []\n\nglobal cfl\n```\n\n\n```python\ndef mycb(w,fnames):\n global fnm\n fnm=fnames[0]\n fbase = os.path.basename(fnm)\n os.makedirs('data/' + os.path.splitext(fbase)[0])\n filelist.append(fbase)\n os.rename(fnm, 'data/raw/' + fbase)\n w.reset()\n\nf = FileUpload(\"Please upload Raman spectra data file (CSV)\", \n \"Raman data files should be uploaded as 2 column CSV files\",\n cb=mycb,\n maxsize=10000000)\nf\n```\n\n\n```python\ndef errprint(code):\n errfile=pd.read_csv('errfile.txt',sep='\\t',header=None)\n with errout:\n clear_output()\n print(errfile[0][code])\n fit_but.disabled=False\n errout\n```\n\n\n```python\ndef case_lookup(index):\n casefile=pd.read_csv('Case_List.txt',sep='\\t',header=None)\n c=casefile[0][index]\n return c\n```\n\n\n```python\nfit_but = widgets.Button(description='Do Fitting')\n \ndef fit_but_cb(change):\n global cfl\n fit_but.disabled=True\n param.disabled=True\n with plist:\n clear_output()\n print('Reading data files...')\n with errout:\n clear_output()\n with diffsplot:\n clear_output()\n with datplot:\n clear_output()\n \n for flnm in filelist:\n cfl = flnm\n if flnm[-3:]=='txt':\n sp='\\s+'\n elif flnm[-3:]=='csv':\n sp=','\n else:\n errprint(0)\n return\n try:\n data = pd.read_csv('data/raw/' + flnm,sep=sp,header=None)\n except:\n sp='\\t'\n data = pd.read_csv('data/raw/' + flnm,sep=sp,header=None)\n with plist:\n clear_output()\n print('Data file read')\n\n n=int(data.size/len(data)) #n determines the size of the data file\n\n global Specs\n Specs.clear()\n\n ##Single Spectra Data File, n=2 \n if n==2:\n with plist:\n clear_output()\n print('Fitting single spectra data.')\n\n s=Spectrum()\n \n Spectra(s,data)\n Fit(s)\n\n dtplot(s)\n\n with diffsplot:\n clear_output()\n fig=plt.figure(figsize=(4,4))\n ax=fig.add_subplot(111)\n plt.plot(s.diffs,'kv')\n plt.plot(s.mdi,s.md,'gv')\n plt.annotate((round(Decimal(s.md),2)),xy=(s.mdi,1.2*s.md))\n plt.xticks(range(6),('1','2','3','4','5','Graphite'))\n plt.xlabel('# Layers')\n plt.ylabel('$\\Delta$ [%]')\n plt.show()\n\n save_spec(s)\n zip_files('data')\n params_print(s)\n\n #Map files will be much larger than 2 points and need separate handling\n elif n > 2:\n Specs=[]\n Map(data)\n else:\n errprint(1)\n return\n fit_but.disabled=False\n\nfit_but.on_click(fit_but_cb)\nfit_but\n```\n\n\n```python\ndef Map(data): \n W=data[:][0:1]\n W=np.array(W)\n W=W[~np.isnan(W)]\n \n x=data[0]\n x=np.array(x)\n x=x[~np.isnan(x)]\n xu=np.unique(x)\n \n y=data[1]\n y=np.array(y)\n y=y[~np.isnan(y)]\n yu=np.unique(y)\n \n n=yu.size*xu.size\n \n s=Spectrum()\n \n Parallel(n_jobs=1)(delayed(maploop)(s,Specs,data,W,x,y,n,k) for k in range(n))\n \n wG=np.transpose(np.array([o.PG for o in Specs]))[2] \n \n Mplot(x,y,wG,'$\\omega_G$ $[cm^{-1}]$','wG')\n Hplot(wG,'$\\omega_G$ $[cm^{-1}]$','wG')\n \n with plist:\n clear_output()\n print('Fitting Finished')\n save_map(Specs)\n zip_files('data')\n with plist:\n clear_output()\n param.disabled=False\n```\n\n\n```python\ndef maploop(s,Specs,data,W,x,y,n,k):\n s=Spectrum()\n \n I_raw=np.array(data)[k+1][2:1026]\n tmp_min = np.min(I_raw)\n I_tmp = I_raw-tmp_min\n tmp_max = np.max(I_tmp)\n I=I_tmp/tmp_max\n #I=((I_raw-np.min(I_raw))/np.max(I_raw-np.min(I_raw)))\n s.I=I\n s.W=W\n s.x=x[k]\n s.y=y[k]\n Fit(s)\n Specs.append(s)\n\n pdone=100*(k+1)/n\n\n with plist:\n clear_output()\n print('Fitting map data. This may take some time...\\n%1.2f%% Done'%(pdone))\n\n return Specs\n```\n\n\n```python\ndef Spectra(s,data):\n srow=0;\n if type(data[0][0])==str:\n srow=1\n \n W=data[0][srow:len(data)]\n W=np.array(W);W=W.astype(float)\n I_raw=data[1][srow:len(data)]\n I_raw=np.array(I_raw);I_raw=I_raw.astype(float)\n \n x = I_raw\n window_length = 7\n polyorder = 2\n\n I_raw = scipy.signal.savgol_filter(x, window_length, polyorder, deriv=0, delta=1.0, axis=- 1, mode='interp', cval=0.0)\n\n# ---- Replace this part ---- #\n n=2\n\n polyx = np.array([W[0],W[int(len(W)/2)],W[len(W)-1]])\n polyy = np.array([I_raw[0],I_raw[int(len(W)/2)],I_raw[len(W)-1]]) \n bkgfit = np.polyfit(W,I_raw,n)\n bkgpoly = 0\n for i in range(n):\n bkgpoly = bkgpoly + (bkgfit[i]*W**(n-i))\n I_raw = I_raw-bkgpoly\n \n m = (I_raw[len(W)-1]-I_raw[0])/(W[len(W)-1]-W[0])\n b = I_raw[len(W)-1]-m*W[len(W)-1]\n bkglin = m*W+b\n \n I_raw=I_raw-bkglin\n \n I=((I_raw-np.min(I_raw))/np.max(I_raw-np.min(I_raw)));\n \n# ---- Upto here ---- #\n \n s.I=I\n \n s.W=W\n return s\n```\n\n\n```python\n# scipy.signal.savgol_filter(x, window_length, polyorder, deriv=0, delta=1.0, axis=- 1, mode='interp', cval=0.0)\n#x is the data to be filtered\n#If x is not a single or double precision floating point array, it will be converted to type numpy.float64 before filtering\n#If x has dimension greater than 1, axis determines the axis along which the filter is applied\n#window_length must be a positive odd number; short paper says its the spectral channel; this will be equal to x_2 - x_1\n#If mode is \u2018interp\u2019, window_length must be less than or equal to the size of x\n#polyorder must be less than window_length\n#delta is only used is deriv is greater than 0\n```\n\n\n```python\nlaser=np.random.randint(400,800)\nprint(\"laser: \", laser)\n```\n\n\n```python\ndef laser_wavelength(lam):\n peak=107.36*(1.24*10**3/lam)+2427.5\n print(\"peak: \", peak)\n return peak\n\ndef Fit(s):\n W=s.W\n I=s.I\n pG=[1.1*np.max(I), 50, 1581.6] #a w b\n pGp=[1.1*np.max(I), 50, laser_wavelength(laser)]\n\n L.set_bounds([1.5*np.max(I),60,2000],[0.3*np.max(I),33,1400])\n PGm=L.fit(s.W,s.I,pG)\n \n L.set_bounds([1.5*np.max(I),60,3000],[0.3*np.max(I),32,2000])\n PGpm=L.fit(s.W,s.I,pGp)\n \n PG=np.array(list(PGm.best_values.values()))\n PGp=np.array(list(PGpm.best_values.values()))\n \n PG[1]=np.absolute(PG[1]);PGp[1]=np.absolute(PGp[1]); #FWHM sometimes returns - bc always squared\n \n IGfit=Single_Lorentz(W,PG[0],PG[1],PG[2]);\n IGpfit=Single_Lorentz(W,PGp[0],PGp[1],PGp[2]);\n IGfit=IGfit\n IGpfit=IGpfit\n s.If=IGfit+IGpfit;\n \n s.PG=PG\n s.PGp=PGp\n \n pD=[0.1*np.max(I),5,1350]\n\n L.set_bounds([np.max(I),50,1400],[0,10,1300])\n PDm=L.fit(s.W,s.I,pD)\n PD=np.array(list(PDm.best_values.values()))\n PD[1]=np.absolute(PD[1]);\n IDfit=Single_Lorentz(W,PD[0],PD[1],PD[2]);\n s.IDfit=IDfit\n Q=1-(PD[0]/PG[0])\n s.Q=Decimal(Q)\n s.PD=PD\n \n Cdat=np.load('Cfits.npy')\n\n diffs_lin=[];diffs_Gp=[];\n diffs=[];diffs.clear()\n for d in range(6):\n \n LG=Cdat[d][0];LG=np.transpose(LG)[0]\n LGp=Cdat[d][1];LGp=np.transpose(LGp)[0]\n LGfit=Single_Lorentz(W,LG[0],LG[1],LG[2]); \n LGpfit=Single_Lorentz(W,LGp[0],LGp[1],LGp[2]);\n Lf=(LGfit+LGpfit)\n \n wts=[1.,1.,0.5]\n \n dfGp=np.average(np.absolute(100*(PGp-LGp)/LGp),weights=wts)\n dfG=np.average(np.absolute(100*(PG-LG)/LG),weights=wts)\n drat=np.absolute(100*(((PG[0]/PGp[0])-(LG[0]/LGp[0]))/(LG[0]/LGp[0])))\n df=np.average([dfGp,dfG,drat],weights=[0.5,0.5,1])\n diffs.append(df)\n \n s.diffs=diffs\n md=np.min(diffs)\n mdi=np.argmin(diffs)\n\n mG=Cdat[mdi][0];mGp=Cdat[mdi][1];\n mGfit=Single_Lorentz(W,mG[0],mG[1],mG[2]);\n mGpfit=Single_Lorentz(W,mGp[0],mGp[1],mGp[2]);\n mf=mGfit+mGpfit\n s.mf=mf\n s.md=md\n s.mdi=mdi\n return s\n```\n\n\n```python\ndef Single_Lorentz(x,a,w,b):\n return a*(((w/2)**2)/(((x-b)**2)+((w/2)**2)))\n```\n\n\n```python\ndef dtplot(s):\n with datplot:\n clear_output()\n fig1=plt.figure(figsize=(4,4))\n ax=fig1.add_subplot(111)\n plt.plot(s.W,s.I,'b',s.W,s.mf,'g',s.W,s.If+s.IDfit,'r')\n plt.xlabel('$\\omega$ $[cm^{-1}]$')\n plt.ylabel('$I_{norm}$ [arb]')\n plt.legend(labels=['Raw','Test','Fit'])\n plt.annotate('Q=%1.2f' %round(s.Q,2) ,xy=(np.min(s.W),0.98))\n plt.annotate('D',xy=(0.85*s.PD[2],1.1*s.PD[0]))\n plt.annotate('G',xy=(0.9*s.PG[2],0.95*s.PG[0]))\n plt.annotate('G\\'',xy=(0.94*s.PGp[2],0.95*s.PGp[0]))\n plt.show()\n plt.savefig('data/'+cfl[:-4]+'/specs.png')\n```\n\n\n```python\ndef dfplot(s):\n with diffsplot:\n clear_output()\n fig=plt.figure(figsize=(4,4))\n ax=fig.add_subplot(111)\n plt.plot(s.W,s.I,'b',s.W,s.mf,'g',s.W,s.If+s.IDfit,'r')\n plt.xlabel('$\\omega$ $[cm^{-1}]$')\n plt.ylabel('$I_{norm}$ [arb]')\n plt.legend(labels=['Raw','Test','Fit'])\n plt.annotate('Q=%1.2f' %round(s.Q,2) ,xy=(np.min(s.W),0.98))\n plt.annotate('D',xy=(0.85*s.PD[2],1.1*s.PD[0]))\n plt.annotate('G',xy=(0.9*s.PG[2],0.95*s.PG[0]))\n plt.annotate('G\\'',xy=(0.94*s.PGp[2],0.95*s.PGp[0]))\n plt.show()\n params_print(s)\n```\n\n\n```python\ndef Mplot(x,y,z,d,fn):\n global p,point,datax\n xi = np.linspace(min(x), max(x))\n yi = np.linspace(min(y), max(y))\n X, Y = np.meshgrid(xi, yi)\n \n with datplot:\n clear_output()\n Z=matplotlib.mlab.griddata(x, y, z, xi, yi, interp='linear') \n fig=plt.figure(figsize=(4,4))\n datax=fig.add_subplot(111)\n p,=datax.plot([],[],'o')\n point=pickPeaks(p)\n C=plt.contourf(X,Y,Z)\n plt.set_cmap('inferno')\n plt.xlabel('x [mm]')\n plt.ylabel('y [mm]')\n #datax.set_xlim(np.min(xi),np.max(xi))\n #datax.set_ylim(np.min(yi),np.max(yi))\n plt.title(d)\n plt.colorbar(C)\n plt.axis('off')\n #datax.autoscale\n plt.show()\n figname = 'data/'+ cfl[:-4]+'/' + fn + '_map.png'\n plt.savefig(figname)\n mout = open('data/'+ cfl[:-4]+'/' + fn + '_dat.map','w')\n mout.write(' '.join(str(n) for n in Z))\n```\n\n\n```python\ndef Hplot(z,d,fn):\n with plist:\n clear_output()\n with diffsplot:\n clear_output()\n fig=plt.figure(figsize=(4,4))\n ax=fig.add_subplot(111)\n plt.hist(z,bins='auto')\n plt.ylabel('Counts')\n plt.xlabel(d)\n plt.show()\n figname = 'data/'+ cfl[:-4]+'/' + fn + '_hist.png'\n plt.savefig(figname)\n hout = open('data/'+ cfl[:-4]+'/' + fn + '_dat.hist','w')\n hout.write(' '.join(str(n) for n in z))\n```\n\n\n```python\ndef params_print(s):\n with plist:\n clear_output()\n G=GL('GREEK CAPITAL LETTER GAMMA')\n o=GL('GREEK SMALL LETTER OMEGA')\n print('G Fitting Parameters:\\n\\tA=%1.2f\\n\\t%s=%1.2f\\n\\t%s=%1.2f\\n'\n 'G\\' Fitting Parameters:\\n\\tA=%1.2f\\n\\t%s=%1.2f\\n\\t%s=%1.2f\\n'\n 'D Fitting Parameters:\\n\\tA=%1.2f\\n\\t%s=%1.2f\\n\\t%s=%1.2f\\n'\n 'Quality=%1.2f (Ratio of D to G)\\n'\n 'Best Case Match: %s'\n %(s.PG[0],G,s.PG[1],o,s.PG[2],s.PGp[0],G,s.PGp[1],o,s.PGp[2],s.PD[0],G,s.PD[1],o,s.PD[2],s.Q,case_lookup(s.mdi)))\n strain_doping(s.PG[2], s.PGp[2])\n```\n\n\n```python\ndef strain_doping(x_o, y_o):\n x_0,y_0 = (1581, 2692) #ideal point based on lamba = 532nm, the most common laser in literature\n strain_axis = 2.483*x_0 - 1233.62\n doping_axis = 0.553*x_0 + 1817.71\n x_e,y_e = (x_o, y_o) #fitted point; where do we find the fitted point?\n b = y_e - 2.483*x_e\n c = y_e - 0.553*x_e\n exp_strain_axis = 2.483*x_e + b\n exp_doping_axis = 0.553*x_e + c\n x_s = (c+1233.62)/1.93\n x_d = -1*(b-1817.71)/1.93\n y_s = 2.483*x_s - 1233.62\n y_d = 0.553*x_d + 1817.71\n #(x_s, y_s) = intersection of strain axis and experimental doping axis\n #(x_d, y_d) = intersection of doping axis and experimental strain axis\n strain_distance = ((x_s - x_0)**2 + (y_s - y_0)**2)**0.5\n doping_distance = ((x_d - x_0)**2 + (y_d - y_0)**2)**0.5\n strain_unit = 155.242\n doping_unit = 1.611\n strain_present = strain_distance / strain_unit\n doping_present = doping_distance / doping_unit\n print(\"Strain (%): \", strain_present)\n print(\"Doping (n): \", doping_present)\n```\n\n\n```python\ndatplot=widgets.Output();datplot\n```\n\n\n```python\ndiffsplot=widgets.Output();diffsplot\n```\n\n\n```python\nerrout=widgets.Output();errout\n```\n\n\n```python\nplist=widgets.Output();plist\n```\n\n\n```python\no=GL('GREEK SMALL LETTER OMEGA')\nG=GL('GREEK CAPITAL LETTER GAMMA')\n\ndef param_change(change):\n d = ['$I_G$ [arb]','$\\Gamma_G$ $[cm^{-1}]$','$\\omega_G$ $[cm^{-1}]$','$I_{G\\'}$ [arb]','$\\Gamma_{G\\'}$ $[cm^{-1}]$','$\\omega_{G\\'}$ $[cm^{-1}]$','$I_D$ [arb]','$\\Gamma_D$ $[cm^{-1}]$','$\\omega_D$ $[cm^{-1}]$']\n D = ['IG','gG','wG','IGp','gGp','wGp','ID','gD','wD']\n N = d[param.value-1]\n n = D[param.value-1]\n if param.value in [1,4,7]:\n ind=0\n elif param.value in [2,5,8]:\n ind=1\n elif param.value in [3,6,9]:\n ind=2\n else:\n ind=0\n \n if param.value in [1,2,3]:\n z=np.transpose(np.array([o.PG for o in Specs]))[ind]\n elif param.value in [4,5,6]:\n z=np.transpose(np.array([o.PGp for o in Specs]))[ind]\n elif param.value in [7,8,9]:\n z=np.transpose(np.array([o.PD for o in Specs]))[ind]\n else:\n z=np.transpose(np.array([o.PG for o in Specs]))[2]\n N=d[2]\n xvals=np.transpose(np.array([o.x for o in Specs]))\n yvals=np.transpose(np.array([o.y for o in Specs]))\n Mplot(xvals,yvals,z,N,n)\n Hplot(z,N,n)\n\nparam=widgets.Dropdown(description='Parameter')\nparam.options=options={'Select': 0, 'I_G': 1, (G+'_G'): 2, (o+'_G'): 3, 'I_G\\'': 4, (G+'_G\\''): 5, (o+'_G\\''): 6, 'I_D': 7, (G+'_D'): 8, (o+'_D'): 9}\nparam.observe(param_change,names='value')\nparam.disabled=True\nparam\n```\n\n\n```python\nclass pickPeaks:\n def __init__(self, line):\n self.line = line\n self.xs = line.get_xdata()\n self.ys = line.get_ydata()\n self.cid = line.figure.canvas.mpl_connect('button_press_event', self)\n\n def __call__(self, event):\n print('click', event)\n if event.inaxes!=self.line.axes: return\n self.xs=event.xdata\n self.ys=event.ydata\n \n Map_Spec_Plot()\n self.points.set_data(self.xs, self.ys)\n \n def __iter__(self):\n return zip(self.xs, self.ys)\n```\n\n\n```python\ndef Map_Spec_Plot():\n from scipy.spatial import cKDTree\n xvals=np.transpose(np.array([o.x for o in Specs]))\n yvals=np.transpose(np.array([o.y for o in Specs]))\n XY=np.zeros((len(Specs),2))\n XY[:,0]=xvals\n XY[:,1]=yvals\n tree = cKDTree(XY)\n dis, ind = tree.query([point.xs,point.ys], k=1)\n dfplot(Specs[ind])\n```\n\n\n```python\nclass Lorentz:\n def __init__(self): \n self.model=Model(Single_Lorentz)\n def set_bounds(self,ub,lb):\n self.model.set_param_hint('a',min=lb[0],max=ub[0])\n self.model.set_param_hint('w',min=lb[1],max=ub[1])\n self.model.set_param_hint('b',min=lb[2],max=ub[2])\n def fit(self,x,y,params):\n F=self.model.fit(data=y,x=x,a=params[0],w=params[1],b=params[2])\n return F\n```\n\n\n```python\nL=Lorentz()\n```\n\n\n```python\ndef save_spec(s):\n with plist:\n clear_output()\n print('Generating plots\\nThis may take a few seconds...')\n fout = open('data/'+cfl[:-4]+'/out.graft','w')\n fout.write(' '.join(str(n) for n in s.I)+'\\n')\n fout.write(' '.join(str(n) for n in s.W)+'\\n')\n fout.write(' '.join(str(n) for n in s.If)+'\\n')\n fout.write(' '.join(str(n) for n in s.PG)+'\\n')\n fout.write(' '.join(str(n) for n in s.PGp)+'\\n')\n fout.write(' '.join(str(n) for n in s.PD)+'\\n')\n fout.write(' '.join(str(n) for n in s.IDfit)+'\\n')\n fout.write(str(s.Q)+'\\n')\n fout.write(' '.join(str(n) for n in s.diffs)+'\\n')\n fout.write(str(s.mdi)+'\\n')\n fout.write(str(s.md)+'\\n')\n fout.write(' '.join(str(n) for n in s.mf)+'\\n')\n fout.close()\n \n fitfile = open('data/spec_fits.csv','a')\n fitfile.write(cfl[:-4]+',' + str(s.PD[2])+','+str(s.PD[0])+','+str(s.PD[1])+',' + str(s.PG[2])+','+str(s.PG[0])+','+str(s.PG[1])+',' + str(s.PGp[2])+','+str(s.PGp[0])+','+str(s.PGp[1])+'\\n')\n fitfile.close()\n```\n\n\n```python\ndef save_map(Specs):\n with plist:\n clear_output()\n print('Generating plots\\nThis may take a few seconds...')\n ## save all the data to a big output file that can be read in later\n fout = open('data/'+cfl[:-4]+'/out.graft','w')\n for o in Specs:\n fout.write(str(o.x)+'\\n')\n fout.write(str(o.y)+'\\n')\n fout.write(' '.join(str(n) for n in o.I)+'\\n')\n fout.write(' '.join(str(n) for n in o.W)+'\\n')\n fout.write(' '.join(str(n) for n in o.If)+'\\n')\n fout.write(' '.join(str(n) for n in o.PG)+'\\n')\n fout.write(' '.join(str(n) for n in o.PGp)+'\\n')\n fout.write(' '.join(str(n) for n in o.PD)+'\\n')\n fout.write(' '.join(str(n) for n in o.IDfit)+'\\n')\n fout.write(str(o.Q)+'\\n')\n fout.write(' '.join(str(n) for n in o.diffs)+'\\n')\n fout.write(str(o.mdi)+'\\n')\n fout.write(str(o.md)+'\\n')\n fout.write(' '.join(str(n) for n in o.mf)+'\\n')\n fout.close()\n \n ##save images to files as images and data files\n for i in [1,2,3,4,5,6,7,8,9]:\n param.value = i\n```\n\n\n```python\ndef get_all_file_paths(directory):\n \n # initializing empty file paths list\n file_paths = []\n \n # crawling through directory and subdirectories\n for root, directories, files in os.walk(directory):\n for filename in files:\n # join the two strings in order to form the full filepath.\n filepath = os.path.join(root, filename)\n file_paths.append(filepath)\n \n # returning all file paths\n return file_paths \n \ndef zip_files(directory):\n # path to folder which needs to be zipped\n \n # calling function to get all file paths in the directory\n file_paths = get_all_file_paths(directory)\n \n # writing files to a zipfile\n with ZipFile('data.zip','w') as zip:\n # writing each file one by one\n for file in file_paths:\n zip.write(file, compress_type = zipfile.ZIP_DEFLATED)\n\ndef clear_data(directory): \n file_paths = get_all_file_paths(directory)\n for file in file_paths:\n os.remove(file)\n \n for directory in os.walk('data/'):\n if directory[0] == 'data/' or directory[0] == 'data/raw':\n i=1\n else:\n os.rmdir(directory[0])\n\n```\n\n\n```python\nd = Download('data.zip', label='Download Data', icon='download', tooltip='DOWNLOAD FILE') \nd\n```\n\n\n```python\nclear_data('data')\n```\n\n\n```python\n\n```\n\n\n```python\n# Testing for PyQt5 Tool\n```\n\n\n```python\n# data=pd.read_table(\"./data/raw/20200630_2_SS.txt\")\n# cols=data.shape[1]\n# rows=data.shape[0]\n\n# if cols == 1:\n# #self.clearPast()\n# data=pd.DataFrame(data.iloc[0:rows/2,0],data.iloc[rows/2:rows,0])\n# #self.setSingle()\n# elif cols == 2:\n# if type(data.iloc[0,0]) is str:\n# #self.clearPast()\n# data=data.iloc[1:rows,:]\n# #self.setSingle()\n# else:\n# #clearPast()\n# data=data\n# #setSingle()\n# else:\n# del filelist[-1]\n# data = []\n# raise ValueError('Please use a single spectrum only')\n\n# data\n```\n\n\n```python\n# frequency = []\n# intensity_norm = []\n# I_BL = []\n\n# frequency = np.array(data.iloc[:,0])\n# intensity = np.array(data.iloc[:,1])\n# length = len(frequency)\n# a = 0\n# for i in range(length):\n# if frequency[i]<=0:\n# a = a+1\n# else:\n# break\n# frequency = frequency[a:]\n# intensity = intensity[a:]\n\n# # self.intensity_norm = []\n# for i in intensity:\n# intensity_norm.append((i-np.min(intensity))/(np.max(intensity)-np.min(intensity)))\n \n# #len(intensity_norm)\n```\n\n\n```python\n# y = intensity_norm\n# x = frequency\n\n# degree = 3\n\n# n = degree\n# I_raw = np.array(y)\n# W = np.array(x)\n\n# polyx = np.array([W[0],W[int(len(W)/2)],W[len(W)-1]])\n# polyy = np.array([I_raw[0],I_raw[int(len(W)/2)],I_raw[len(W)-1]]) \n# bkgfit = np.polyfit(W,I_raw,degree)\n# bkgpoly = 0\n# for i in range(n):\n# bkgpoly = bkgpoly + (bkgfit[i]*W**(n-i))\n# print('bkgpoly:', bkgpoly)\n# I_raw = I_raw-bkgpoly\n \n# m = (I_raw[len(W)-1]-I_raw[0])/(W[len(W)-1]-W[0])\n# b = I_raw[len(W)-1]-m*W[len(W)-1]\n# bkglin = m*W+b\n\n# I_raw = I_raw-bkglin\n \n# I_BL = []\n# I_BL = ((I_raw-np.min(I_raw))/np.max(I_raw-np.min(I_raw)))\n\n# print('polyx:', polyx)\n# print('polyy:', polyy)\n```\n\n\n```python\n# r = np.array([2,3,4])\n# a = 5\n# b = 3\n# a*r**2 + b*r**1\n```\n\n\n```python\n# %matplotlib inline\n# plt.figure(0)\n# plt.plot(frequency, intensity)\n# plt.figure(1)\n# plt.plot(frequency, intensity_norm)\n# plt.figure(2)\n# plt.plot(frequency, I_BL)\n```\n\n\n```python\n# plt.plot(frequency, I_BL)\n```\n\n\n```python\n# fpath = \"C:/Users/Test/Documents/UIUC_Offline/Tool/Working/GSA-Raman\\data\\raw\\spectest.csv\"\n\n# import os\n# fpath = os.path.split(fpath)[-1]\n# fpath\n```\n", "meta": {"hexsha": "588b1a133d3ef8a678ebccb4e598e1ad22bdbb1b", "size": 41614, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "graft.ipynb", "max_stars_repo_name": "nanoMFG/Graphene-Synthesis-Analysis", "max_stars_repo_head_hexsha": "803ffa49ea21b329e74096f47957afa60e71e32c", "max_stars_repo_licenses": ["ECL-2.0", "Apache-2.0"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2018-05-10T01:18:12.000Z", "max_stars_repo_stars_event_max_datetime": "2020-10-20T20:12:38.000Z", "max_issues_repo_path": "graft.ipynb", "max_issues_repo_name": 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YES\n2. YES", "lm_q1_score": 0.7248702642896702, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4213693249477455}} {"text": "```python\nimport IPython as Ipy\nIpy.display.YouTubeVideo('https://www.youtube.com/watch?v=ypUqQWYZTJs&list=PLIxff5DJJR7oBEy0Kdg12WWSlS6XFtr6r&index=28',width= 600,height=400)\n```\n\n\n\n\n\n\n\n\n\n\n# [Linearity](https://www.youtube.com/watch?v=ypUqQWYZTJs&list=PLIxff5DJJR7oBEy0Kdg12WWSlS6XFtr6r&index=28)\n> ### $ f(x) = \\color{magenta}{a}x $\n>> ### $\nf(\\vartriangle + \\;\\square) = a\\vartriangle + a\\; \\square\\\\\nf(\\vartriangle + \\; \\square) = f(\\vartriangle) + f(\\square) \\\\\nf(\\vartriangle + \\; 0) = f(\\vartriangle) + f(0), \\;s.t\\; \\square = 0 \\\\\nf(0) = 0 \\\\\nf(s\\vartriangle) = sf(\\vartriangle)\n$\n# Non linearity\n> ### $ g(x) = ax + \\color{magenta}{b}$\n>> ### $\ng(\\vartriangle + \\square) = a \\vartriangle + a\\square + b \\\\\n\\therefore g(\\vartriangle + \\square) \\neq g(\\vartriangle) + g(\\square)\n$\n\n# Linearity of determinant\n> ### row lineartity $\n\\begin{vmatrix}\n\\vartriangle_1 + \\; \\square_1 & \\vartriangle_2 + \\; \\square_2 \\\\ \nc & d\n\\end{vmatrix}\n=\n\\begin{vmatrix}\n\\vartriangle_1 & \\vartriangle_2 \\\\ \nc & d\n\\end{vmatrix}\n+ \\begin{vmatrix}\n\\square_1 & \\square_2 \\\\ \nc & d\n\\end{vmatrix}\n$\n\n> ### column lineartity $\n\\begin{vmatrix}\n\\color{magenta}{\\vartriangle_1 + \\; \\square_1} & b \\\\\n\\color{magenta}{\\vartriangle_2 + \\; \\square_2} & d \\\\\n\\end{vmatrix}\n=\n\\begin{vmatrix}\n\\color{magenta}{\\vartriangle_1} & b \\\\\n\\color{magenta}{\\vartriangle_2} & d \\\\\n\\end{vmatrix}\n+ \\begin{vmatrix}\n\\color{magenta}{\\square_1} & b \\\\\n\\color{magenta}{\\square_2} & d \\\\ \n\\end{vmatrix}\n$\n\n> ### non-lineartity of entry $\n\\begin{vmatrix}\n\\vartriangle_1 + \\; \\square_1 & b \\\\\n\\vartriangle_2 & d \\\\\n\\end{vmatrix}\n\\neq\n\\begin{vmatrix}\n\\vartriangle_1 & b \\\\\n\\vartriangle_2 & d \\\\\n\\end{vmatrix}\n+ \\begin{vmatrix}\n\\square_1 & b \\\\\n\\vartriangle_2 & d \\\\ \n\\end{vmatrix}\n\\\\\n\\because \\text{ not curve}\n$\n\n> ### lineartity $\n\\begin{vmatrix}\n\\vartriangle_1 + \\; \\square_1 & b \\\\\n\\vartriangle_2 + \\; \\color{magenta}{0} & d \\\\\n\\end{vmatrix}\n= \\begin{vmatrix}\n\\vartriangle_1 & b \\\\\n\\vartriangle_2 & d \\\\\n\\end{vmatrix}\n+ \\begin{vmatrix}\n\\square_1 & b \\\\\n\\color{magenta}{0} & d \\\\ \n\\end{vmatrix}\n$\n\n> ### non-lineartity - diagnonal $\n\\begin{vmatrix}\n\\vartriangle_1 + \\; \\square_1 & b \\\\\nc & \\vartriangle_2 + \\; \\square_2 \\\\\n\\end{vmatrix}\n\\neq\n\\begin{vmatrix}\n\\vartriangle_1 & b \\\\\nc & \\vartriangle_2 \\\\\n\\end{vmatrix}\n+ \\begin{vmatrix}\n\\square_1 & b \\\\\nc & \\square_2 \\\\ \n\\end{vmatrix}\n$\n\n# multi-linear of determinant\n> ### $\n\\big|A\\big| = det(A) = \\sum_{j} a_{ij}C_{ij} \\\\\n\\big|A\\big| = det(A) = \\sum_{j} a_{ji}C_{ji} \\\\\n$\n>> ### Minor Matrix\n>>> ### $\n\\begin{bmatrix}\na_{11} & a_{12} & a_{13} & \\cdots & a_{1n} \\\\\na_{21} & M_{11} & M_{12} & \\cdots & M_{1,n-1} \\\\\na_{31} & M_{21} & M_{22} & \\cdots & M_{2,n-1} \\\\\n\\vdots & \\ddots & \\ddots & \\ddots & \\vdots \\\\\na_{n1} & M_{n-1,1} & M_{n-1,2} & \\cdots & M_{n-1,n-1} \\\\\n\\end{bmatrix}\n$\n>> ### Cofactor Matrix\n>>> ### $\nC_{ij} = (-1)^{i+j}M_{ij}\n$\n\n> ### arbitary i row determinant = arbitary j column determinant \n>> ### $ det(A) = det\\big[a_{i}\\big] = \\sum_{j}a_{ij}C_{ij} $\n>>> ### $ det(A) = det\\big[ 0 \\big] = \\sum_{j}(0)C_{ij} = 0 $\n>>> ### $ det(A) = det\\big[ sa_{i} \\big] = \\sum_{j}sa_{ij}C_{ij} = s\\sum_{ij}C_{ij}= s\\; \\det(A) $\n\n> ### determinant having a same row or column matrix \n>> ### $\ndet\\begin{bmatrix} \n\\check {\\vartriangle}\n\\\\ \\vartriangle\n\\end{bmatrix}\n= \\sum_j \\vartriangle_{i,j}(-1)^{i+j}M_{ij}\n\\\\\ndet\\begin{bmatrix} \n\\vartriangle\n\\\\ \\check{\\vartriangle}\n\\end{bmatrix}\n= \\sum_j \\vartriangle_{i+1,j}(-1)^{i+1+j}M_{i+1,j}\n\\\\\\qquad \\quad = -\\sum_j \\vartriangle_{ij}(-1)^{i+j}M_{ij}\n$\n>>> ### \uc5b4\ub5a4 row \ub098 column\uc744 \uc120\ud0dd\ud574\ub3c4 determimant\ub294 \uac19\uc73c\ubbc0\ub85c\n>>>> ### $\ndet \\left[ \\begin{array}{r} \\vartriangle \\\\ \\vartriangle \\end{array} \\right]\n= - det \\left[ \\begin{array}{r} \\vartriangle \\\\ \\vartriangle \\end{array} \\right]\n\\\\\n\\therefore det \\left[ \n\\begin{array}{c}\\vartriangle \\\\ \\vartriangle \\end{array}\\right] = 0\n\\\\\n\\therefore det \\left[ \n\\begin{array}{c}\\vartriangle \\\\ \\vdots \\\\ \\vartriangle \\end{array}\\right] = 0\n$\n\n> ### determinant having a same row or column matrix \n>> ### $\n0 = det\\begin{bmatrix} \n\\vartriangle +\\;\\square \n\\\\ \\vartriangle +\\; \\square \n\\end{bmatrix}\n= \ndet\\begin{bmatrix} \n\\vartriangle \n\\\\ \\vartriangle +\\; \\square \n\\end{bmatrix}\n+\ndet\\begin{bmatrix} \n\\square \n\\\\ \\vartriangle +\\; \\square \n\\end{bmatrix}\n$\n\n>> ### $ \n0 = \n\\underbrace{\ndet\\begin{bmatrix} \n\\vartriangle \n\\\\ \\vartriangle\n\\end{bmatrix}}_{0}\n+\ndet\\begin{bmatrix} \n\\vartriangle\n\\\\ \\square\n\\end{bmatrix}\n+\ndet\\begin{bmatrix} \n\\square\n\\\\ \\vartriangle\n\\end{bmatrix}\n+\n\\underbrace{\ndet\\begin{bmatrix} \n\\square \n\\\\\\square \n\\end{bmatrix}}_{0}\n\\\\\n\\therefore 0 = \\left[ \\begin{array}{c}\\vartriangle \\\\ \\square \\end{array} \\right]\n+ \\left[ \\begin{array}{c}\\square \\\\ \\vartriangle \\end{array} \\right]\n\\\\ \\therefore \\left[ \\begin{array}{c}\\vartriangle \\\\ \\square \\end{array} \\right]\n= - \\left[ \\begin{array}{c}\\square \\\\ \\vartriangle \\end{array} \\right]\n$\n> ### \ud589\uad50\ud658\n>> ### $\n0 = det\\begin{bmatrix} \n\\vartriangle +\\;\\square \\\\ \\vdots\n\\\\ \\vartriangle +\\; \\square \n\\end{bmatrix} \n\\\\ \n\\therefore\n\\left[ \\begin{array}{c}\\vartriangle \\\\ \\vdots \\\\ \\square \\end{array} \\right]\n= - \\left[ \\begin{array}{c}\\square \\\\ \\vdots \\\\ \\vartriangle \\end{array} \\right]\n$\n\n# addtion of determinant\n> ### $\ndet \\left[ \n\\begin{array}{c} \\vartriangle \\\\ \\vdots \\\\ \\square + s \\vartriangle \\end{array} \\right]\n= det \\left[ \n\\begin{array}{c} \\vartriangle \\\\ \\vdots \\\\ \\square \n\\end{array} \\right]\n+ det \\left[ \n\\begin{array}{c} \\vartriangle \\\\ \\vdots \\\\ s\\vartriangle \n\\end{array} \\right]\n\\\\= det \\left[ \n\\begin{array}{c} \\vartriangle \\\\ \\vdots \\\\ \\square \n\\end{array} \\right]\n+ s\\; det \\left[ \n\\begin{array}{c} \\vartriangle \\\\ \\vdots \\\\ \\vartriangle \n\\end{array} \\right]\n\\\\=\ndet \\left[ \\begin{array}{c}\\vartriangle \\\\ \\vdots \\\\ \\square \\end{array}\\right]\n$\n\n# [Determinant](https://www.youtube.com/watch?v=FsCLrXjRCIM&list=PLIxff5DJJR7oBEy0Kdg12WWSlS6XFtr6r&index=29)\n> $\ndet\\begin{bmatrix}a & b \\\\ c & d\\end{bmatrix} = \ndet \\Big(\\begin{bmatrix}a \\\\ c \\end{bmatrix}, \\begin{bmatrix} b \\\\ d \\end{bmatrix}\\Big)\n\\\\ =\ndet \\Big(\na \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} + \nc \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix},\nb \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} + \nd \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}\n\\Big) \n\\\\ = \ndet \\Big(\na \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}, \nb \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} +\nd \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}\n\\Big) +\ndet \\Big(\nc \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}, \nb \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} +\nd \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}\n\\Big)\n\\\\\n=\nab \\det\\Big( \n\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}, \n\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}\n\\Big) +\nad \\det\\Big( \n\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}, \n\\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}\n\\Big) +\ncb \\det\\Big( \n\\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}, \n\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}\n\\Big) +\ncd \\det\\Big( \n\\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}, \n\\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}\n\\Big)\n\\\\\n=\nab \\det \\begin{bmatrix} 1 & 1 \\\\ 0 & 0 \\end{bmatrix}\n+\nad \\det \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}\n+\ncb \\det \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix}\n+\ncd \\det \\begin{bmatrix} 0 & 0 \\\\ 1 & 1 \\end{bmatrix}\n\\\\ = ad - bc \\\\\nE^1 = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}, \\quad\nE^2 = \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} \\\\\ndet \\big[E^1,E^2 \\big] = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} = 1, \\quad\ndet \\big[E^2,E^1 \\big] = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix} = -1 \n\\\\\n\\therefore det \\begin{bmatrix}a_{11} & a_{12} \\\\ a_{21} & a_{22} \\end{bmatrix} \n= det \\Big[ A^1, A^2 \\Big] = \\sum_{ij} a_{i1}a_{j2}\\; det\\Big(E^i,E^j\\Big)\n= det\n$\n\n# determinant 3by 3\n> ### $\n\\begin{vmatrix}\na_{11} & a_{12} & a_{13} \\\\\na_{21} & a_{22} & a_{23} \\\\\na_{31} & a_{32} & a_{33} \\\\\n\\end{vmatrix}\n= \n\\Big|\nA^1 , A^2 , A^3\n\\Big|\n= \\sum_{ijk}a_{i1}a_{j2} a_{k3}\\Big| E^{i} E^{j} E^{k} \\Big|\n\\\\\n\\therefore \\sum_{ijk} \\epsilon \\Big(i,j,k \\Big)a_{i1}a_{j2}a_{k3}\\; det \\Big(E^1,E^2,E^3 \\Big)\n$\n\n> ### $\ndet \\Big( E^{1}, E^{2}, E^{3} \\Big) = \n\\begin{vmatrix}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1 \\\\\n\\end{vmatrix} = 1\n\\\\\ndet \\Big( E^{1}, E^{3}, E^{2} \\Big) = \n\\begin{vmatrix}\n1 & 0 & 0 \\\\\n0 & 0 & 1 \\\\\n0 & 1 & 0 \\\\\n\\end{vmatrix} = -1\n$\n\n# n by n determinant\n> ### $\n\\begin{vmatrix}\na_{11} & a_{12} & \\cdots & a_{1n} \\\\\na_{21} & a_{22} & \\cdots & a_{2n} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\na_{n1} & a_{n2} & \\cdots & a_{nn} \\\\\n\\end{vmatrix} = det\\Big[ A^1,A^2,\\cdots,A^n \\Big]\n\\\\= \n\\sum_{s_1s_2\\cdots s_n}A_{s_{1,1}}, \\cdots a_{s_n,n}\\; det\\Big(E^{s_1},\\cdots,E^{s_n}\\Big)\n\\\\\nA^1 = \\sum b_{j1}G^{j} = b_{11}G^1 + b_{21}G^2 + \\cdots + b_{n1}G^n\n\\\\ \\vdots \\\\\nA^k = \\sum b_{jk}G^{j} = b_{1k}G^1 + b_{2k}G^2 + \\cdots + b_{nk}G^n\n\\\\ \\vdots \\\\\nA^n = \\sum b_{jn}G^{j} = b_{1n}G^1 + b_{2n}G^2 + \\cdots + b_{nn}G^n\n\\\\\ndet\\Big[A^1, A^2, \\cdots, A^n\\Big] \n= \\sum_{s_1s_2\\cdots s_n} \\epsilon\\Big(s_1,s_2,\\cdots, s_n \\Big) b_{s_1,1}b_{s_2,2}\\cdots b_{s_n,n}\\; det\\Big(G^1,G^2,\\cdots,G^n\\Big)\n$\n\n# Sumation\n> ### $\nAB = \\sum_{ij}a_{ij}G_{ij} \\sum_{lm}b_{lm}G_{lm}\n\\\\ = \\sum_{ijm}a_{ij}b{jm}G_{im}\n\\\\ = \\sum_{jm} b_{jm}(\\sum_{}a_{}G_{})\n$\n\n# inverse\n> ### $\nT^{-1} = B, \\quad Tx = h \\\\\nB_{i} \\cdot T^{j} = \\delta_{i}^{j} \\\\\nB_{i}\\cdot h = x_{i} \\\\\n\\begin{bmatrix}\na & b \\\\ c & d\n\\end{bmatrix}\n\\begin{bmatrix}\na^{-1}(x) & b^{-1}(u) \\\\ c^{-1}(y) & d^{-1}(v)\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n1 & 0 \\\\ 0 & 1\n\\end{bmatrix}\n\\iff ax + by = 1 \\quad cx + dy = 0,\\quad au + bv = 1 \\quad cu + dv = 0 \n$\n>> ### $\n(ax + by = 1)d - (cx + dy = 0)b, \\to adx - bcx = d \\to x(a^{-1}) = \\frac{d}{ad-bc} \\\\\n(ax + by = 1)c - (cx + dy = 0)a, \\to bcy - adx = c \\to y(c^{-1}) = \\frac{c}{-(ad-bc)} = \\frac{-c}{ad-bc} \\\\\n(au + bv = 0)d - (cu + dv = 1)b, \\to adu - bcu = -b \\to u(b^{-1}) = \\frac{-b}{ad-bc}\\\\\n(au + bv = 0)c - (cu + dv = 1)a, \\to bcv - dav = -a \\to v(c^{-1}) = \\frac{-a}{-(ad-bc)} = \\frac{a}{ad-bc} \\\\\n\\therefore \\begin{bmatrix}\n\\frac{d}{ad-bc} & \\frac{-b}{ad-bc} \\\\ \\frac{-c}{ad-bc} & \\frac{a}{ad-bc} \n\\end{bmatrix} \\iff\n\\therefore \\frac{1}{ad-bc}\\begin{bmatrix}\nd & -b \\\\ -c & a \n\\end{bmatrix}\n$\n\n# determinant\n> ### $\ndet(A) = det(A^{T}) \\\\\nA = \\begin{bmatrix}\na_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\\\\na_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\\\\na_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\\\\na_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\\\\na_{51} & a_{52} & a_{53} & a_{54} & a_{55} \\\\\n\\end{bmatrix} \\\\\ndet(A) = \\sum_{k=1}^{n}(-1)^{i+k}a_{ik} M_{ik} = \\sum_{k=1}^{n}a_{ik}C_{ik}\n\\\\\nC_{ij} = (-1)^{i+j}M_{ij}\n$\n\n# multi linear transformation of determinant\n# \uac19\uc740 \ud589\ubca1\ud130\uac00 \ub450\uac1c \uc788\uc744\ub54c\n> ### $ det \\begin{bmatrix} \\vdots \\\\ \\vartriangle \\\\ \\vartriangle \\\\ \\vdots \\end{bmatrix} = 0 = det \\begin{bmatrix} \\vdots \\\\ \\vartriangle \\\\ \\vdots \\\\ \\vartriangle \\\\ \\vdots \\end{bmatrix} $\n\n# \ud589\uad50\ud658\n> ### $ det \\begin{bmatrix} \\vdots \\\\ \\square \\\\ \\vartriangle \\\\ \\vdots \\end{bmatrix} = -det \\begin{bmatrix} \\vdots \\\\ \\square \\\\ \\vartriangle \\\\ \\vdots \\end{bmatrix} \\iff det \\begin{bmatrix} \\vdots \\\\ \\square \\\\ \\vdots \\\\ \\vartriangle \\\\ \\vdots \\end{bmatrix} = -det \\begin{bmatrix} \\vdots \\\\ \\square \\\\ \\vdots \\\\ \\vartriangle \\\\ \\vdots \\end{bmatrix} $\n\n# \uc0c1\uc218\ubc30\ud6c4 \ub2e4\ub978\ud589\uc5d0 \ub354\ud574\uc8fc\uae30\n> ### $ det \\begin{bmatrix} \\vartriangle \\\\ \\vdots \\\\ \\square + s\\vartriangle \\end{bmatrix} \n= det \\begin{bmatrix} \\vartriangle \\\\ \\vdots \\\\ \\square \\end{bmatrix} \n+ det \\begin{bmatrix} \\vartriangle \\\\ \\vdots \\\\ s\\vartriangle \\end{bmatrix} \n= det \\begin{bmatrix} \\vartriangle \\\\ \\vdots \\\\ \\square \\end{bmatrix} \n+ s\\, det \\begin{bmatrix} \\vartriangle \\\\ \\vdots \\\\ \\vartriangle \\end{bmatrix} $\n\n# L (low triangle matrix) $\\iff$ U (upper triangle matrix)\n> ### $\ndet(L) = det\n\\begin{bmatrix}\na_{11} & 0 & \\cdots & 0 \\\\ ? & a_{22} & \\cdots & 0 \\\\ \\vdots & \\ddots & \\ddots & \\vdots \\\\ ? & ? & \\cdots & a_{nn}\n\\end{bmatrix}\n= a_{11} \\times a_{22} \\times \\cdots \\times a_{nn}\n$\n\n\n```python\nimport numpy as np\nimport sympy as sm\nimport sympy.vector\nN = sm.vector.CoordSys3D('N')\n\n# v(3,1), w(1,2) \\iff T([3,1],[1,2])\nu = 3*N.i + N.j + 0*N.k\nv = N.i + 2*N.j + 0*N.k\nw = 0*N.i + 0*N.j + N.k\nT = u.to_matrix(N).col_insert(1,v.to_matrix(N)).col_insert(2,w.to_matrix(N))\nT = u.to_matrix(N).row_join(v.to_matrix(N)).row_join(w.to_matrix(N))\nB = T.inv()\n# x(x_0, x_1)\nx0,x1,x2 = sm.symbols('x_0 x_1 x_2')\nx = sm.Matrix([x0,x1,x2])\n# sm.vector.matrix_to_vector(x,N)\n\n# Tx = h(3,2)\nh = 3*N.i + 2*N.j + 0*N.k\n\nimport matplotlib.pyplot as plt\n%matplotlib widget\nfig = plt.figure(figsize=(8,8))\nax = fig.add_subplot()\nax.grid()\nax.spines[['left','bottom']].set_position('zero')\nax.set_aspect('equal')\nax.set_xlim(xmin=-4,xmax=9)\nax.set_ylim(ymin=-4,ymax=9)\nax.set_xticks(np.arange(-4,9,1))\nax.set_yticks(np.arange(-4,9,1))\nax.quiver(0,0,3,1,scale=1,units='xy')\nax.quiver(0,0,-1,3,scale=1,units='xy')\nax.annotate(\"B'(3,1)\",(2.2,1.1))\nax.quiver(0,0,1,2,scale=1,units='xy')\nax.quiver(0,0,2,-1,scale=1,units='xy')\nax.annotate('A(1,2)',(0.6,2.1))\nax.annotate(\"A'(0,0)\",(0.2,-0.3))\nax.quiver(0,0,4,3,scale=1,units='xy')\nax.annotate('B(4,3)',(4.1,3))\nax.annotate('C(4,2)',(4.2,2))\nax.annotate(\"C'(3,0)\",(3.1,0.1))\nax.annotate(\"D(3,3)\",(2.8,3.1))\nax.annotate(\"D'(0,2)\",(-0.5,2.1))\nax.plot([1,4],[2,2],color='r')\nax.plot([1,4],[2,3],color='r')\nax.plot([4,4],[2,3],color='r')\nax.plot([3,3],[0,1],color='r')\nax.plot([0,3],[0,0],color='r')\n\nax.plot([3,4],[1,3],color='b')\nax.plot([3,4],[3,3],color='b')\nax.plot([0,0],[0,2],color='b')\nax.plot([0,1],[2,2],color='b')\nax.plot([3,3],[2,3],color='b')\nax.plot([3,3],[1,2],color='k')\n\n# solve\nT.inv()*sm.Matrix([3,2,0])\n(T.inv()*x).T*(h.to_matrix(N))\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}\\frac{4 x_{0}}{5} + \\frac{3 x_{1}}{5}\\end{matrix}\\right]$\n\n\n\n\n\n
                                        \n
                                        \n Figure\n
                                        \n \n
                                        \n\n\n\n\n```python\n\n```\n\n# [determinant & inverse matrix meanning](https://www.youtube.com/watch?v=CLS3Z9jbZw4&list=PLIxff5DJJR7oBEy0Kdg12WWSlS6XFtr6r&index=33)\n\n\n```python\n# v(3,1), w(2,2) \\iff T([3,1],[2,2]) , b=(3,2)\n# https://www.youtube.com/watch?v=CLS3Z9jbZw4&list=PLIxff5DJJR7oBEy0Kdg12WWSlS6XFtr6r&index=33\nu = 3*N.i + N.j + 0*N.k\nv = 2*N.i + 2*N.j + 0*N.k\nw = 0*N.i + 0*N.j + N.k\nT = u.to_matrix(N).col_insert(1,v.to_matrix(N)).col_insert(2,w.to_matrix(N))\nT = u.to_matrix(N).row_join(v.to_matrix(N)).row_join(w.to_matrix(N))\n\n# x(x_0, x_1)\nx1,x2,x3 = sm.symbols('x_1 x_2 x_3')\nx = sm.Matrix([x1,x2,x3])\n\n# Tx = h(3,2)\nh = 3*N.i + 2*N.j + 0*N.k\n\nimport matplotlib.pyplot as plt\n%matplotlib widget\nfig = plt.figure(figsize=(8,8))\nax = fig.add_subplot()\nax.grid()\n#ax.spines[['left','bottom']].set_position('zero')\nax.axhline()\nax.axvline()\nax.set_aspect('equal')\nax.set_xlim(xmin=-2,xmax=4)\nax.set_ylim(ymin=-2,ymax=4)\nax.set_xticks(np.arange(-2,4,1/4))\nax.set_xticklabels(ax.get_xticks(),rotation=45)\nax.set_yticks(np.arange(-2,4,1/4))\n# u(3,1,0)\nax.quiver(0,0,3,1,scale=1,units='xy')\nax.annotate(r\"$T^1=u(3,1)$\",(3,1))\n# v(2,2,0)\nax.quiver(0,0,2,2,scale=1,units='xy')\nax.annotate(r\"$T^2v(2,2)$\",(2,2))\n# b(3,2)\nax.scatter(3,2,color='r')\nax.annotate(\"b(3,2)\",(3,2))\n# T.inv() = [1/2, -1/2], [-1/4, 3/4]\n# new axix of N.i (1/2, -1/2)\n# new axix of N.j (-1/4, 3/4)\nB = T.inv()\nax.quiver(0,0,1/2,-1/2,scale=1,units='xy')\nax.annotate(r\"$B_{1}=\\hat{e_1}( \\frac{1}{2}, \\frac{-1}{2})$\",(1/2,-1/2))\nax.quiver(0,0,-1/4,3/4,scale=1,units='xy')\nax.annotate(r\"$B_{2}=\\hat{e_2}(\\frac{-1}{4}, \\frac{3}{4})$\",(-1/4,3/4))\n\n# B_i * h = x_i\n# B_0 * h = x_0\nB.row(0)*(h.to_matrix(N))\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}\\frac{1}{2}\\end{matrix}\\right]$\n\n\n\n\n\n
                                        \n
                                        \n Figure\n
                                        \n \n
                                        \n\n\n\n\n```python\n#%%javascript\n#MathJax.Hub.Config({\n# TeX: { equationNumbers: { autoNumber: \"AMS\" } }\n#});\n```\n\n# 1 by 1 determinat\n> ### $\nTx = h \\\\\nx = \\frac{b}{a}\n$\n# 2 by 2 determinant\n> ### $\n\\begin{align}\n& ax + by = u \\label{eq1}\\tag{1}\\\\\n& cx + dy = v \\label{eq2}\\tag{2}\\\\\n& adx + bdy = ud & \\eqref{eq1} \\times d \\label{eq3}\\tag{3}\\\\\\\n& bcx + bdy = bv & \\eqref{eq2} \\times b \\label{eq4}\\tag{4}\\\\\n& (ad - bc)x = ud - bv & \\eqref{eq3} - \\eqref{eq4} \\\\\n\\therefore x = \\frac{ud - bv}{ad - bc} & \\\\\n\\end{align}\n$\n> ### \ud589\uad50\ud658 $\\iff$ \ubd80\ud638\uac00 \ubc14\ub01c\n>> ### $\ncx + dy = v \\\\\nax + by = u \\\\\nx = \\frac{bv - ud}{bc - ad}\n$\n\n# 3 by 3 determinant\n> ### $ \na_1x + b_1y + c_1z = u\\\\\na_2x + b_2y + c_2z = v\\\\\na_3x + b_3y + c_3z = w\\\\\n\\therefore \n|A| = det(A)$\n>> ### $$\nx = \n\\frac{\nu\\begin{vmatrix}b_2 & c_2 \\\\ b_3 & c_3\\end{vmatrix} - \nv\\begin{vmatrix}b_1 & c_1 \\\\ b_3 & c_3\\end{vmatrix} +\nw\\begin{vmatrix}b_1 & c_1 \\\\ b_2 & c_2\\end{vmatrix}\n}\n{\na_1\\begin{vmatrix}b_2 & c_2 \\\\ b_3 & c_3\\end{vmatrix} - \na_2\\begin{vmatrix}b_1 & c_1 \\\\ b_3 & c_3\\end{vmatrix} +\na_3\\begin{vmatrix}b_1 & c_1 \\\\ b_2 & c_2\\end{vmatrix}\n}\n$$\n\n\n\n# \ud589\uc774\ub098 \uc5f4\uc774 \uc11c\ub85c \ube44\ub840\ud558\ub294 \uc131\ubd84\uc774 2\uac1c \ud589\uc5f4\uc2dd\uc740 0\uc774\ub41c\ub2e4.\n> ### \uc11c\ub85c \ub2e4\ub978 \ub450\uc810$(x_0,x_2),(y_0,y_1)$\uc744 \uc9c0\ub098\ub294 \uc9c1\uc120\uc758 \ubc29\uc815\uc2dd\uc740 \ud589\uc5f4\uc2dd\uc774 0\uc774 \ub418\uc5b4\uc57c \ud568\uc73c\ub85c\n>> ### $\ndet (A) \\Rightarrow\n\\begin{align}\n\\begin{vmatrix}\n1 & x & y \\\\ 1 & x_0 & y_0 \\\\ 1 & x_0 & y_1\n\\end{vmatrix} = 0\n\\end{align}\n$\n> ### \uc11c\ub85c \ub2e4\ub978 \uc138\uc810$(x_0,y_0),(x_1,y_1),(x_2,y_2)$\uc744 \uc9c0\ub098\ub294 \uc9c1\uc120\uc758 \ubc29\uc815\uc2dd\uc740 \ud589\uc5f4\uc2dd\uc774 0\uc774 \ub418\uc5b4\uc57c \ud568\uc73c\ub85c\n>> ### $\ndet (A) \\Rightarrow\n\\begin{align}\n\\begin{vmatrix}\n1 & x & x^2 & y \\\\ 1 & x_0 & x_0^2 & y_0 \\\\ 1 & x_1 & x_1^2 & y_1 \\\\ 1 & x_2 & x_2^2 & y_2\n\\end{vmatrix} = 0\n\\end{align}\n$\n\n\n```python\nT\nT1 = sm.Matrix([[1,2],[3,4]])\nT1\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right]$\n\n\n\n\n```python\n# Minor matrix (\uc18c\ud589\uc5f4)\nT.minor_submatrix(0,0)\nT.minorEntry(0,1)\nT.minorMatrix(0,1)\nT1.minorMatrix(0,0)\nT1.minorEntry(0,0)\n```\n\n\n\n\n$\\displaystyle 4$\n\n\n\n\n```python\n# cofactor (\uc5ec\uc778\uc218)\nT.cofactor(0,0)\nT1.cofactor(0,0)\n# cofactor Matrix(\uc5ec\uc778\uc218 \ud589\uc5f4)\nT1.cofactorMatrix()\nT.cofactorMatrix()\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}2 & -1 & 0\\\\-2 & 3 & 0\\\\0 & 0 & 4\\end{matrix}\\right]$\n\n\n\n\n```python\n# elementary column operation(\uae30\ubcf8\ud589 \uc5f0\uc0b0)\n# elementary row operation(\uae30\ubcf8\uc5f4 \uc5f0\uc0b0)\n# n->n+km : n=col1, m=col2\nT1.elementary_row_op(op=\"n->n+km\",row=1,k=-3,row2=0)\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}1 & 2\\\\0 & -2\\end{matrix}\\right]$\n\n\n\n\n```python\n# adjoint (\uc218\ubc18\ud589\uc5f4) = (cofactor matrix).transpose()\n# adjugate in sympy\n# jugate (\uc30d\uc73c\ub85c \ub418\uc5b4\uc788\ub294)\nT1.adjugate()\nT.adjugate()\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}2 & -2 & 0\\\\-1 & 3 & 0\\\\0 & 0 & 4\\end{matrix}\\right]$\n\n\n\n\n```python\nT1.echelon_form()\n\n# rref = reduce row echelon form\nref, pivots = T1.rref()\n# \ud589\uc0ac\ub2e4\ub9ac(echelon) \uaf34(form)\nT.echelon_form()\nT.rref()\n```\n\n\n\n\n (Matrix([\n [1, 0, 0],\n [0, 1, 0],\n [0, 0, 1]]),\n (0, 1, 2))\n\n\n\n# adjoin matrix\n> ### cofactor\n>> ### $\nC_{ij} = (-1)^{i+j}M_{ij} \\in \\mathbb R\n$\n\n\n```python\n\n```\n", "meta": {"hexsha": "1b108a6c536501a377b11ee23539d8c3a9a5ae40", "size": 140548, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "python/Vectors/Determinant.ipynb", "max_stars_repo_name": "karng87/nasm_game", "max_stars_repo_head_hexsha": "a97fdb09459efffc561d2122058c348c93f1dc87", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "python/Vectors/Determinant.ipynb", "max_issues_repo_name": "karng87/nasm_game", "max_issues_repo_head_hexsha": "a97fdb09459efffc561d2122058c348c93f1dc87", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "python/Vectors/Determinant.ipynb", "max_forks_repo_name": "karng87/nasm_game", "max_forks_repo_head_hexsha": "a97fdb09459efffc561d2122058c348c93f1dc87", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 122.2156521739, "max_line_length": 67415, "alphanum_fraction": 0.8395921678, "converted": true, "num_tokens": 8179, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030761371503, "lm_q2_score": 0.7248702702332476, "lm_q1q2_score": 0.4213693178869542}} {"text": "```python\nimport numpy as np\nimport sympy as sy\nimport matplotlib.pyplot as plt\nimport matplotlib.animation as animation\nfrom matplotlib.animation import FuncAnimation\nfrom mpl_toolkits.mplot3d import Axes3D\nfrom scipy import linalg as la\n```\n\n \n Bad key \"text.kerning_factor\" on line 4 in\n /home/anybody/apps/anaconda3/envs/py37astro/lib/python3.7/site-packages/matplotlib/mpl-data/stylelib/_classic_test_patch.mplstyle.\n You probably need to get an updated matplotlibrc file from\n https://github.com/matplotlib/matplotlib/blob/v3.1.3/matplotlibrc.template\n or from the matplotlib source distribution\n /home/anybody/apps/anaconda3/envs/py37astro/lib/python3.7/importlib/_bootstrap.py:219: RuntimeWarning: numpy.ufunc size changed, may indicate binary incompatibility. Expected 192 from C header, got 216 from PyObject\n return f(*args, **kwds)\n\n\n\n```python\n# Problem 2 (standing orbits)\n\n# load in the data and assign to understandable variables\norb = np.load('orbits.npz',)\nmerc_orb = orb['mercury']\nven_orb = orb['venus']\nearth_orb = orb['earth']\nmars_orb = orb['mars']\n\n# set the figure for 3d Plots\nfig = plt.figure()\nax = fig.add_subplot(111, projection='3d')\nax.set_xlim((-1.3,1.3))\nax.set_ylim((-1.3,1.3))\nax.set_zlim((-1.3,1.3))\nax.set_xlabel(\"X\")\nax.set_ylabel(\"Y\")\nax.set_zlabel(\"Z\")\n\n\n# plot the sun at the origin\nax.scatter3D(0,0,0, color='yellow', marker='o', lw=8, label='Sun')\n\n# plot the standing location of mercury and its orbit\nax.plot3D([merc_orb[0,0]],[merc_orb[0,1]],[merc_orb[0,2]],color='gray',marker='.',lw=2)\nax.plot3D(merc_orb[:,0],merc_orb[:,1],merc_orb[:,2], 'gray', lw=.75, label='Mercury')\n\n# plot the standing location of Venus and its orbit\nax.plot3D([ven_orb[0,0]],[ven_orb[0,1]],[ven_orb[0,2]],color='orange',marker='.',lw=2)\nax.plot3D(ven_orb[:,0],ven_orb[:,1],ven_orb[:,2], 'orange', lw=.75, label='Venus')\n\n# plot the standing location of the Earth and its orbit\nax.plot3D([earth_orb[0,0]],[earth_orb[0,1]],[earth_orb[0,2]],color='blue',marker='.',lw=2)\nax.plot3D(earth_orb[:,0],earth_orb[:,1],earth_orb[:,2], 'b', lw=.75, label='Earth')\n\n# plot the standing location of Mars and its orbit\nax.plot3D([mars_orb[0,0]],[mars_orb[0,1]],[mars_orb[0,2]],color='red',marker='.',lw=2)\nax.plot3D(mars_orb[:,0],mars_orb[:,1],mars_orb[:,2], 'r', lw=.75, label='Mars')\n\n\nax.legend(loc='upper right', prop={'size': 5})\nax.set_title(\"Orbits of the inner Planets\")\nplt.show()\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "56541e2585194ba2b07109f831ca2140c72b980f", "size": 51166, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/Untitled1.ipynb", "max_stars_repo_name": "benitocm/practical-astronomy", "max_stars_repo_head_hexsha": "4bfea9d5b2bb49997f35e8c7b1ada2708ee6c978", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/Untitled1.ipynb", "max_issues_repo_name": "benitocm/practical-astronomy", "max_issues_repo_head_hexsha": "4bfea9d5b2bb49997f35e8c7b1ada2708ee6c978", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/Untitled1.ipynb", "max_forks_repo_name": "benitocm/practical-astronomy", "max_forks_repo_head_hexsha": "4bfea9d5b2bb49997f35e8c7b1ada2708ee6c978", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 373.4744525547, "max_line_length": 46916, "alphanum_fraction": 0.9356799437, "converted": true, "num_tokens": 795, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.668880247169804, "lm_q2_score": 0.6297746213017459, "lm_q1q2_score": 0.42124380435758146}} {"text": "   \n\n# Bonus Tutorial: Extending the Wilson-Cowan Model\n**Week 2, Day 4: Dynamic Networks**\n\n**By Neuromatch Academy**\n\n__Content creators:__ Qinglong Gu, Songtin Li, Arvind Kumar, John Murray, Julijana Gjorgjieva \n\n__Content reviewers:__ Maryam Vaziri-Pashkam, Ella Batty, Lorenzo Fontolan, Richard Gao, Spiros Chavlis, Michael Waskom\n\n__Content editors:__ \n\n__Production editors:__ Siddharth Suresh\n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n

                                        \n\n---\n# Tutorial Objectives\nIn the previous tutorial, you became familiar the **Wilson-Cowan** rate model. Here we will dive into some deeper analyses of this model.\n\nBonus steps:\n\n- Find and plot the **fixed points** of the Wilson-Cowan model.\n- Investigate the stability of the Wilson-Cowan model by linearizing its dynamics and examining the **Jacobian matrix**.\n- Learn how the Wilson-Cowan model can reach an oscillatory state.\n\nApplications of Wilson-Cowan model:\n- Visualize the behavior of an Inhibition-stabilized network.\n- Simulate working memory using the Wilson-Cowan model.\n\n\n
                                        \n\nReference paper:\n\nWilson H and Cowan J (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophysical Journal 12, doi: [10.1016/S0006-3495(72)86068-5](https://doi.org/10.1016/S0006-3495(72)86068-5)\n\n---\n# Setup\n\n\n```python\n# Imports\n\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport scipy.optimize as opt # root-finding algorithm\n```\n\n\n```python\n# @title Figure Settings\nimport ipywidgets as widgets # interactive display\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/course-content/master/nma.mplstyle\")\n```\n\n\n```python\n# @title Plotting Functions\n\ndef plot_FI_inverse(x, a, theta):\n f, ax = plt.subplots()\n ax.plot(x, F_inv(x, a=a, theta=theta))\n ax.set(xlabel=\"$x$\", ylabel=\"$F^{-1}(x)$\")\n\n\ndef plot_FI_EI(x, FI_exc, FI_inh):\n plt.figure()\n plt.plot(x, FI_exc, 'b', label='E population')\n plt.plot(x, FI_inh, 'r', label='I population')\n plt.legend(loc='lower right')\n plt.xlabel('x (a.u.)')\n plt.ylabel('F(x)')\n plt.show()\n\n\ndef my_test_plot(t, rE1, rI1, rE2, rI2):\n\n plt.figure()\n ax1 = plt.subplot(211)\n ax1.plot(pars['range_t'], rE1, 'b', label='E population')\n ax1.plot(pars['range_t'], rI1, 'r', label='I population')\n ax1.set_ylabel('Activity')\n ax1.legend(loc='best')\n\n ax2 = plt.subplot(212, sharex=ax1, sharey=ax1)\n ax2.plot(pars['range_t'], rE2, 'b', label='E population')\n ax2.plot(pars['range_t'], rI2, 'r', label='I population')\n ax2.set_xlabel('t (ms)')\n ax2.set_ylabel('Activity')\n ax2.legend(loc='best')\n\n plt.tight_layout()\n plt.show()\n\n\ndef plot_nullclines(Exc_null_rE, Exc_null_rI, Inh_null_rE, Inh_null_rI):\n\n plt.figure()\n plt.plot(Exc_null_rE, Exc_null_rI, 'b', label='E nullcline')\n plt.plot(Inh_null_rE, Inh_null_rI, 'r', label='I nullcline')\n plt.xlabel(r'$r_E$')\n plt.ylabel(r'$r_I$')\n plt.legend(loc='best')\n plt.show()\n\n\ndef my_plot_nullcline(pars):\n Exc_null_rE = np.linspace(-0.01, 0.96, 100)\n Exc_null_rI = get_E_nullcline(Exc_null_rE, **pars)\n Inh_null_rI = np.linspace(-.01, 0.8, 100)\n Inh_null_rE = get_I_nullcline(Inh_null_rI, **pars)\n\n plt.plot(Exc_null_rE, Exc_null_rI, 'b', label='E nullcline')\n plt.plot(Inh_null_rE, Inh_null_rI, 'r', label='I nullcline')\n plt.xlabel(r'$r_E$')\n plt.ylabel(r'$r_I$')\n plt.legend(loc='best')\n\n\ndef my_plot_vector(pars, my_n_skip=2, myscale=5):\n EI_grid = np.linspace(0., 1., 20)\n rE, rI = np.meshgrid(EI_grid, EI_grid)\n drEdt, drIdt = EIderivs(rE, rI, **pars)\n\n n_skip = my_n_skip\n\n plt.quiver(rE[::n_skip, ::n_skip], rI[::n_skip, ::n_skip],\n drEdt[::n_skip, ::n_skip], drIdt[::n_skip, ::n_skip],\n angles='xy', scale_units='xy', scale=myscale, facecolor='c')\n\n plt.xlabel(r'$r_E$')\n plt.ylabel(r'$r_I$')\n\n\ndef my_plot_trajectory(pars, mycolor, x_init, mylabel):\n pars = pars.copy()\n pars['rE_init'], pars['rI_init'] = x_init[0], x_init[1]\n rE_tj, rI_tj = simulate_wc(**pars)\n\n plt.plot(rE_tj, rI_tj, color=mycolor, label=mylabel)\n plt.plot(x_init[0], x_init[1], 'o', color=mycolor, ms=8)\n plt.xlabel(r'$r_E$')\n plt.ylabel(r'$r_I$')\n\n\ndef my_plot_trajectories(pars, dx, n, mylabel):\n \"\"\"\n Solve for I along the E_grid from dE/dt = 0.\n\n Expects:\n pars : Parameter dictionary\n dx : increment of initial values\n n : n*n trjectories\n mylabel : label for legend\n\n Returns:\n figure of trajectory\n \"\"\"\n pars = pars.copy()\n for ie in range(n):\n for ii in range(n):\n pars['rE_init'], pars['rI_init'] = dx * ie, dx * ii\n rE_tj, rI_tj = simulate_wc(**pars)\n if (ie == n-1) & (ii == n-1):\n plt.plot(rE_tj, rI_tj, 'gray', alpha=0.8, label=mylabel)\n else:\n plt.plot(rE_tj, rI_tj, 'gray', alpha=0.8)\n\n plt.xlabel(r'$r_E$')\n plt.ylabel(r'$r_I$')\n\n\ndef plot_complete_analysis(pars):\n plt.figure(figsize=(7.7, 6.))\n\n # plot example trajectories\n my_plot_trajectories(pars, 0.2, 6,\n 'Sample trajectories \\nfor different init. conditions')\n my_plot_trajectory(pars, 'orange', [0.6, 0.8],\n 'Sample trajectory for \\nlow activity')\n my_plot_trajectory(pars, 'm', [0.6, 0.6],\n 'Sample trajectory for \\nhigh activity')\n\n # plot nullclines\n my_plot_nullcline(pars)\n\n # plot vector field\n EI_grid = np.linspace(0., 1., 20)\n rE, rI = np.meshgrid(EI_grid, EI_grid)\n drEdt, drIdt = EIderivs(rE, rI, **pars)\n n_skip = 2\n plt.quiver(rE[::n_skip, ::n_skip], rI[::n_skip, ::n_skip],\n drEdt[::n_skip, ::n_skip], drIdt[::n_skip, ::n_skip],\n angles='xy', scale_units='xy', scale=5., facecolor='c')\n\n plt.legend(loc=[1.02, 0.57], handlelength=1)\n plt.show()\n\n\ndef plot_fp(x_fp, position=(0.02, 0.1), rotation=0):\n plt.plot(x_fp[0], x_fp[1], 'ko', ms=8)\n plt.text(x_fp[0] + position[0], x_fp[1] + position[1],\n f'Fixed Point1=\\n({x_fp[0]:.3f}, {x_fp[1]:.3f})',\n horizontalalignment='center', verticalalignment='bottom',\n rotation=rotation)\n```\n\n\n```python\n# @title Helper functions\n\n\ndef default_pars(**kwargs):\n pars = {}\n\n # Excitatory parameters\n pars['tau_E'] = 1. # Timescale of the E population [ms]\n pars['a_E'] = 1.2 # Gain of the E population\n pars['theta_E'] = 2.8 # Threshold of the E population\n\n # Inhibitory parameters\n pars['tau_I'] = 2.0 # Timescale of the I population [ms]\n pars['a_I'] = 1.0 # Gain of the I population\n pars['theta_I'] = 4.0 # Threshold of the I population\n\n # Connection strength\n pars['wEE'] = 9. # E to E\n pars['wEI'] = 4. # I to E\n pars['wIE'] = 13. # E to I\n pars['wII'] = 11. # I to I\n\n # External input\n pars['I_ext_E'] = 0.\n pars['I_ext_I'] = 0.\n\n # simulation parameters\n pars['T'] = 50. # Total duration of simulation [ms]\n pars['dt'] = .1 # Simulation time step [ms]\n pars['rE_init'] = 0.2 # Initial value of E\n pars['rI_init'] = 0.2 # Initial value of I\n\n # External parameters if any\n for k in kwargs:\n pars[k] = kwargs[k]\n\n # Vector of discretized time points [ms]\n pars['range_t'] = np.arange(0, pars['T'], pars['dt'])\n\n return pars\n\n\ndef F(x, a, theta):\n \"\"\"\n Population activation function, F-I curve\n\n Args:\n x : the population input\n a : the gain of the function\n theta : the threshold of the function\n\n Returns:\n f : the population activation response f(x) for input x\n \"\"\"\n\n # add the expression of f = F(x)\n f = (1 + np.exp(-a * (x - theta)))**-1 - (1 + np.exp(a * theta))**-1\n\n return f\n\n\ndef dF(x, a, theta):\n \"\"\"\n Derivative of the population activation function.\n\n Args:\n x : the population input\n a : the gain of the function\n theta : the threshold of the function\n\n Returns:\n dFdx : Derivative of the population activation function.\n \"\"\"\n\n dFdx = a * np.exp(-a * (x - theta)) * (1 + np.exp(-a * (x - theta)))**-2\n\n return dFdx\n\ndef F_inv(x, a, theta):\n \"\"\"\n Args:\n x : the population input\n a : the gain of the function\n theta : the threshold of the function\n\n Returns:\n F_inverse : value of the inverse function\n \"\"\"\n\n # Calculate Finverse (ln(x) can be calculated as np.log(x))\n F_inverse = -1/a * np.log((x + (1 + np.exp(a * theta))**-1)**-1 - 1) + theta\n\n return F_inverse\n\n\ndef get_E_nullcline(rE, a_E, theta_E, wEE, wEI, I_ext_E, **other_pars):\n \"\"\"\n Solve for rI along the rE from drE/dt = 0.\n\n Args:\n rE : response of excitatory population\n a_E, theta_E, wEE, wEI, I_ext_E : Wilson-Cowan excitatory parameters\n Other parameters are ignored\n\n Returns:\n rI : values of inhibitory population along the nullcline on the rE\n \"\"\"\n # calculate rI for E nullclines on rI\n rI = 1 / wEI * (wEE * rE - F_inv(rE, a_E, theta_E) + I_ext_E)\n\n return rI\n\n\ndef get_I_nullcline(rI, a_I, theta_I, wIE, wII, I_ext_I, **other_pars):\n \"\"\"\n Solve for E along the rI from dI/dt = 0.\n\n Args:\n rI : response of inhibitory population\n a_I, theta_I, wIE, wII, I_ext_I : Wilson-Cowan inhibitory parameters\n Other parameters are ignored\n\n Returns:\n rE : values of the excitatory population along the nullcline on the rI\n \"\"\"\n # calculate rE for I nullclines on rI\n rE = 1 / wIE * (wII * rI + F_inv(rI, a_I, theta_I) - I_ext_I)\n\n return rE\n\ndef EIderivs(rE, rI,\n tau_E, a_E, theta_E, wEE, wEI, I_ext_E,\n tau_I, a_I, theta_I, wIE, wII, I_ext_I,\n **other_pars):\n \"\"\"Time derivatives for E/I variables (dE/dt, dI/dt).\"\"\"\n\n # Compute the derivative of rE\n drEdt = (-rE + F(wEE * rE - wEI * rI + I_ext_E, a_E, theta_E)) / tau_E\n\n # Compute the derivative of rI\n drIdt = (-rI + F(wIE * rE - wII * rI + I_ext_I, a_I, theta_I)) / tau_I\n\n return drEdt, drIdt\n\ndef simulate_wc(tau_E, a_E, theta_E, tau_I, a_I, theta_I,\n wEE, wEI, wIE, wII, I_ext_E, I_ext_I,\n rE_init, rI_init, dt, range_t, **other_pars):\n \"\"\"\n Simulate the Wilson-Cowan equations\n\n Args:\n Parameters of the Wilson-Cowan model\n\n Returns:\n rE, rI (arrays) : Activity of excitatory and inhibitory populations\n \"\"\"\n # Initialize activity arrays\n Lt = range_t.size\n rE = np.append(rE_init, np.zeros(Lt - 1))\n rI = np.append(rI_init, np.zeros(Lt - 1))\n I_ext_E = I_ext_E * np.ones(Lt)\n I_ext_I = I_ext_I * np.ones(Lt)\n\n # Simulate the Wilson-Cowan equations\n for k in range(Lt - 1):\n\n # Calculate the derivative of the E population\n drE = dt / tau_E * (-rE[k] + F(wEE * rE[k] - wEI * rI[k] + I_ext_E[k],\n a_E, theta_E))\n\n # Calculate the derivative of the I population\n drI = dt / tau_I * (-rI[k] + F(wIE * rE[k] - wII * rI[k] + I_ext_I[k],\n a_I, theta_I))\n\n # Update using Euler's method\n rE[k + 1] = rE[k] + drE\n rI[k + 1] = rI[k] + drI\n\n return rE, rI\n```\n\nThe helper functions included:\n\n- Parameter dictionary: `default_pars(**kwargs)`. You can use:\n - `pars = default_pars()` to get all the parameters, and then you can execute `print(pars)` to check these parameters. \n - `pars = default_pars(T=T_sim, dt=time_step)` to set a different simulation time and time step\n - After `pars = default_pars()`, use `par['New_para'] = value` to add a new parameter with its value\n - Pass to functions that accept individual parameters with `func(**pars)`\n- F-I curve: `F(x, a, theta)`\n- Derivative of the F-I curve: `dF(x, a, theta)`\n- Inverse of F-I curve: `F_inv`\n- Nullcline calculations: `get_E_nullcline`, `get_I_nullcline`\n- Derivatives of E/I variables: `EIderivs`\n- Simulate the Wilson-Cowan model: `simulate_wc`\n\n---\n# Section 1: Fixed points, stability analysis, and limit cycles in the Wilson-Cowan model\n\n*Correction to video: this is now the first part of the second bonus tutorial, not the last part of the second tutorial*\n\n\n```python\n# @title Video 1: Fixed points and their stability\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n from IPython.display import IFrame\n class BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = 'https://player.bilibili.com/player.html?bvid={0}&page={1}'.format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n video = BiliVideo(id=\"BV1Pf4y1d7dx\", width=854, height=480, fs=1)\n print('Video available at https://www.bilibili.com/video/{0}'.format(video.id))\n display(video)\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=\"jIx26iQ69ps\", width=854, height=480, fs=1, rel=0)\n print('Video available at https://youtube.com/watch?v=' + video.id)\n display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\ndisplay(out)\n```\n\nAs in Tutorial 2, we will be looking at the Wilson-Cowan model, with coupled equations representing the dynamics of the excitatory or inhibitory population:\n\n\\begin{align}\n\\tau_E \\frac{dr_E}{dt} &= -r_E + F_E(w_{EE}r_E -w_{EI}r_I + I^{\\text{ext}}_E;a_E,\\theta_E)\\\\\n\\tau_I \\frac{dr_I}{dt} &= -r_I + F_I(w_{IE}r_E -w_{II}r_I + I^{\\text{ext}}_I;a_I,\\theta_I) \\qquad (1)\n\\end{align}\n\n$r_E(t)$ represents the average activation (or firing rate) of the excitatory population at time $t$, and $r_I(t)$ the activation (or firing rate) of the inhibitory population. The parameters $\\tau_E$ and $\\tau_I$ control the timescales of the dynamics of each population. Connection strengths are given by: $w_{EE}$ (E $\\rightarrow$ E), $w_{EI}$ (I $\\rightarrow$ E), $w_{IE}$ (E $\\rightarrow$ I), and $w_{II}$ (I $\\rightarrow$ I). The terms $w_{EI}$ and $w_{IE}$ represent connections from inhibitory to excitatory population and vice versa, respectively. The transfer functions (or F-I curves) $F_E(x;a_E,\\theta_E)$ and $F_I(x;a_I,\\theta_I)$ can be different for the excitatory and the inhibitory populations.\n\n## Section 1.1: Fixed Points of the E/I system\n\nThe intersection points of the two nullcline curves are the fixed points of the Wilson-Cowan model in Equation $(1)$. \n\nIn the next exercise, we will find the coordinate of all fixed points for a given set of parameters.\n\nWe'll make use of two functions, similar to ones we saw in Tutorial 1, which use a root-finding algorithm to find the fixed points of the system with Excitatory and Inhibitory populations.\n\n\n```python\n# @markdown Execute to visualize nullclines\n\n# Set parameters\npars = default_pars()\nExc_null_rE = np.linspace(-0.01, 0.96, 100)\nInh_null_rI = np.linspace(-.01, 0.8, 100)\n\n# Compute nullclines\nExc_null_rI = get_E_nullcline(Exc_null_rE, **pars)\nInh_null_rE = get_I_nullcline(Inh_null_rI, **pars)\n\nplot_nullclines(Exc_null_rE, Exc_null_rI, Inh_null_rE, Inh_null_rI)\n```\n\n\n```python\n# @markdown *Execute the cell to define `my_fp` and `check_fp`*\n\ndef my_fp(pars, rE_init, rI_init):\n \"\"\"\n Use opt.root function to solve Equations (2)-(3) from initial values\n \"\"\"\n\n tau_E, a_E, theta_E = pars['tau_E'], pars['a_E'], pars['theta_E']\n tau_I, a_I, theta_I = pars['tau_I'], pars['a_I'], pars['theta_I']\n wEE, wEI = pars['wEE'], pars['wEI']\n wIE, wII = pars['wIE'], pars['wII']\n I_ext_E, I_ext_I = pars['I_ext_E'], pars['I_ext_I']\n\n # define the right hand of wilson-cowan equations\n def my_WCr(x):\n\n rE, rI = x\n drEdt = (-rE + F(wEE * rE - wEI * rI + I_ext_E, a_E, theta_E)) / tau_E\n drIdt = (-rI + F(wIE * rE - wII * rI + I_ext_I, a_I, theta_I)) / tau_I\n y = np.array([drEdt, drIdt])\n\n return y\n\n x0 = np.array([rE_init, rI_init])\n x_fp = opt.root(my_WCr, x0).x\n\n return x_fp\n\n\ndef check_fp(pars, x_fp, mytol=1e-6):\n \"\"\"\n Verify (drE/dt)^2 + (drI/dt)^2< mytol\n\n Args:\n pars : Parameter dictionary\n fp : value of fixed point\n mytol : tolerance, default as 10^{-6}\n\n Returns :\n Whether it is a correct fixed point: True/False\n \"\"\"\n\n drEdt, drIdt = EIderivs(x_fp[0], x_fp[1], **pars)\n\n return drEdt**2 + drIdt**2 < mytol\n\nhelp(my_fp)\n```\n\n### Coding Exercise 1.1: Find the fixed points of the Wilson-Cowan model\n\nFrom the above nullclines, we notice that the system features three fixed points with the parameters we used. To find their coordinates, we need to choose proper initial value to give to the `opt.root` function inside of the function `my_fp` we just defined, since the algorithm can only find fixed points in the vicinity of the initial value. \n\nIn this exercise, you will use the function `my_fp` to find each of the fixed points by varying the initial values. Note that you can choose the values near the intersections of the nullclines as the initial values to calculate the fixed points.\n\n\n```python\npars = default_pars()\n\n######################################################################\n# TODO: Provide initial values to calculate the fixed points\n# Check if x_fp's are the correct with the function check_fp(x_fp)\n# Hint: vary different initial values to find the correct fixed points\nraise NotImplementedError('student exercise: find fixed points')\n######################################################################\n\nmy_plot_nullcline(pars)\n\n# Find the first fixed point\nx_fp_1 = my_fp(pars, ..., ...)\nif check_fp(pars, x_fp_1):\n plot_fp(x_fp_1)\n\n# Find the second fixed point\nx_fp_2 = my_fp(pars, ..., ...)\nif check_fp(pars, x_fp_2):\n plot_fp(x_fp_2)\n\n# Find the third fixed point\nx_fp_3 = my_fp(pars, ..., ...)\nif check_fp(pars, x_fp_3):\n plot_fp(x_fp_3)\n```\n\n\n```python\n# to_remove solution\npars = default_pars()\n\nwith plt.xkcd():\n my_plot_nullcline(pars)\n\n # Find the first fixed point\n x_fp_1 = my_fp(pars, 0.1, 0.1)\n if check_fp(pars, x_fp_1):\n plot_fp(x_fp_1)\n\n # Find the second fixed point\n x_fp_2 = my_fp(pars, 0.3, 0.3)\n if check_fp(pars, x_fp_2):\n plot_fp(x_fp_2)\n\n # Find the third fixed point\n x_fp_3 = my_fp(pars, 0.8, 0.6)\n if check_fp(pars, x_fp_3):\n plot_fp(x_fp_3)\n```\n\n## Section 1.2: Stability of a fixed point and eigenvalues of the Jacobian Matrix\n\nFirst, let's first rewrite the system $1$ as:\n\n\\begin{align}\n&\\frac{dr_E}{dt} = G_E(r_E,r_I)\\\\[0.5mm]\n&\\frac{dr_I}{dt} = G_I(r_E,r_I)\n\\end{align}\n\nwhere\n\n\\begin{align}\n&G_E(r_E,r_I) = \\frac{1}{\\tau_E} [-r_E + F_E(w_{EE}r_E -w_{EI}r_I + I^{\\text{ext}}_E;a,\\theta)]\\\\[1mm]\n&G_I(r_E,r_I) = \\frac{1}{\\tau_I} [-r_I + F_I(w_{IE}r_E -w_{II}r_I + I^{\\text{ext}}_I;a,\\theta)]\n\\end{align}\n\nBy definition, $\\displaystyle\\frac{dr_E}{dt}=0$ and $\\displaystyle\\frac{dr_I}{dt}=0$ at each fixed point. Therefore, if the initial state is exactly at the fixed point, the state of the system will not change as time evolves. \n\nHowever, if the initial state deviates slightly from the fixed point, there are two possibilities\nthe trajectory will be attracted back to the \n\n1. The trajectory will be attracted back to the fixed point\n2. The trajectory will diverge from the fixed point. \n\nThese two possibilities define the type of fixed point, i.e., stable or unstable. Similar to the 1D system studied in the previous tutorial, the stability of a fixed point $(r_E^*, r_I^*)$ can be determined by linearizing the dynamics of the system (can you figure out how?). The linearization will yield a matrix of first-order derivatives called the Jacobian matrix:\n\n \\begin{equation}\n J=\n \\left[ {\\begin{array}{cc}\n \\displaystyle{\\frac{\\partial}{\\partial r_E}}G_E(r_E^*, r_I^*) & \\displaystyle{\\frac{\\partial}{\\partial r_I}}G_E(r_E^*, r_I^*)\\\\[1mm]\n \\displaystyle\\frac{\\partial}{\\partial r_E} G_I(r_E^*, r_I^*) & \\displaystyle\\frac{\\partial}{\\partial r_I}G_I(r_E^*, r_I^*) \\\\\n \\end{array} } \\right] \\quad (7)\n\\end{equation}\n\n
                                        \n\nThe eigenvalues of the Jacobian matrix calculated at the fixed point will determine whether it is a stable or unstable fixed point.\n\nWe can now compute the derivatives needed to build the Jacobian matrix. Using the chain and product rules the derivatives for the excitatory population are given by:\n\n
                                        \n\n\\begin{align}\n&\\frac{\\partial}{\\partial r_E} G_E(r_E^*, r_I^*) = \\frac{1}{\\tau_E} [-1 + w_{EE} F_E'(w_{EE}r_E^* -w_{EI}r_I^* + I^{\\text{ext}}_E;\\alpha_E, \\theta_E)] \\\\[1mm]\n&\\frac{\\partial}{\\partial r_I} G_E(r_E^*, r_I^*)= \\frac{1}{\\tau_E} [-w_{EI} F_E'(w_{EE}r_E^* -w_{EI}r_I^* + I^{\\text{ext}}_E;\\alpha_E, \\theta_E)]\n\\end{align}\n\n
                                        \n\nThe same applies to the inhibitory population. \n\n### Coding Exercise 1.2: Compute the Jacobian Matrix for the Wilson-Cowan model\n\nHere, you can use `dF(x,a,theta)` defined in the `Helper functions` to calculate the derivative of the F-I curve.\n\n\n```python\ndef get_eig_Jacobian(fp,\n tau_E, a_E, theta_E, wEE, wEI, I_ext_E,\n tau_I, a_I, theta_I, wIE, wII, I_ext_I, **other_pars):\n \"\"\"Compute eigenvalues of the Wilson-Cowan Jacobian matrix at fixed point.\"\"\"\n # Initialization\n rE, rI = fp\n J = np.zeros((2, 2))\n\n ###########################################################################\n # TODO for students: compute J and disable the error\n raise NotImplementedError(\"Student exercise: compute the Jacobian matrix\")\n ###########################################################################\n # Compute the four elements of the Jacobian matrix\n J[0, 0] = ...\n J[0, 1] = ...\n J[1, 0] = ...\n J[1, 1] = ...\n\n # Compute and return the eigenvalues\n evals = np.linalg.eig(J)[0]\n return evals\n\n\n# Compute eigenvalues of Jacobian\neig_1 = get_eig_Jacobian(x_fp_1, **pars)\neig_2 = get_eig_Jacobian(x_fp_2, **pars)\neig_3 = get_eig_Jacobian(x_fp_3, **pars)\n\nprint(eig_1, 'Stable point')\nprint(eig_2, 'Unstable point')\nprint(eig_3, 'Stable point')\n```\n\n\n```python\n# to_remove solution\ndef get_eig_Jacobian(fp,\n tau_E, a_E, theta_E, wEE, wEI, I_ext_E,\n tau_I, a_I, theta_I, wIE, wII, I_ext_I, **other_pars):\n \"\"\"Compute eigenvalues of the Wilson-Cowan Jacobian matrix at fixed point.\"\"\"\n # Initialization\n rE, rI = fp\n J = np.zeros((2, 2))\n\n # Compute the four elements of the Jacobian matrix\n J[0, 0] = (-1 + wEE * dF(wEE * rE - wEI * rI + I_ext_E,\n a_E, theta_E)) / tau_E\n\n J[0, 1] = (-wEI * dF(wEE * rE - wEI * rI + I_ext_E,\n a_E, theta_E)) / tau_E\n\n J[1, 0] = (wIE * dF(wIE * rE - wII * rI + I_ext_I,\n a_I, theta_I)) / tau_I\n\n J[1, 1] = (-1 - wII * dF(wIE * rE - wII * rI + I_ext_I,\n a_I, theta_I)) / tau_I\n\n # Compute and return the eigenvalues\n evals = np.linalg.eig(J)[0]\n return evals\n\n\n# Compute eigenvalues of Jacobian\neig_1 = get_eig_Jacobian(x_fp_1, **pars)\neig_2 = get_eig_Jacobian(x_fp_2, **pars)\neig_3 = get_eig_Jacobian(x_fp_3, **pars)\n\nprint(eig_1, 'Stable point')\nprint(eig_2, 'Unstable point')\nprint(eig_3, 'Stable point')\n```\n\nAs is evident, the stable fixed points correspond to the negative eigenvalues, while unstable point corresponds to at least one positive eigenvalue.\n\nThe sign of the eigenvalues is determined by the connectivity (interaction) between excitatory and inhibitory populations. \n\nBelow we investigate the effect of $w_{EE}$ on the nullclines and the eigenvalues of the dynamical system. \n\n\\* _Critical change is referred to as **pitchfork bifurcation**_. \n\n## Section 1.3: Effect of `wEE` on the nullclines and the eigenvalues\n\n### Interactive Demo 1.3: Nullclines position in the phase plane changes with parameter values\n\nHow do the nullclines move for different values of the parameter $w_{EE}$? What does this mean for fixed points and system activity?\n\n\n```python\n# @title\n\n# @markdown Make sure you execute this cell to enable the widget!\n\n\ndef plot_nullcline_diffwEE(wEE):\n \"\"\"\n plot nullclines for different values of wEE\n \"\"\"\n\n pars = default_pars(wEE=wEE)\n\n # plot the E, I nullclines\n Exc_null_rE = np.linspace(-0.01, .96, 100)\n Exc_null_rI = get_E_nullcline(Exc_null_rE, **pars)\n\n Inh_null_rI = np.linspace(-.01, .8, 100)\n Inh_null_rE = get_I_nullcline(Inh_null_rI, **pars)\n\n plt.figure(figsize=(12, 5.5))\n plt.subplot(121)\n plt.plot(Exc_null_rE, Exc_null_rI, 'b', label='E nullcline')\n plt.plot(Inh_null_rE, Inh_null_rI, 'r', label='I nullcline')\n plt.xlabel(r'$r_E$')\n plt.ylabel(r'$r_I$')\n plt.legend(loc='best')\n\n plt.subplot(222)\n pars['rE_init'], pars['rI_init'] = 0.2, 0.2\n rE, rI = simulate_wc(**pars)\n plt.plot(pars['range_t'], rE, 'b', label='E population', clip_on=False)\n plt.plot(pars['range_t'], rI, 'r', label='I population', clip_on=False)\n plt.ylabel('Activity')\n plt.legend(loc='best')\n plt.ylim(-0.05, 1.05)\n plt.title('E/I activity\\nfor different initial conditions',\n fontweight='bold')\n\n plt.subplot(224)\n pars['rE_init'], pars['rI_init'] = 0.4, 0.1\n rE, rI = simulate_wc(**pars)\n plt.plot(pars['range_t'], rE, 'b', label='E population', clip_on=False)\n plt.plot(pars['range_t'], rI, 'r', label='I population', clip_on=False)\n plt.xlabel('t (ms)')\n plt.ylabel('Activity')\n plt.legend(loc='best')\n plt.ylim(-0.05, 1.05)\n\n plt.tight_layout()\n plt.show()\n\n\n_ = widgets.interact(plot_nullcline_diffwEE, wEE=(6., 10., .01))\n```\n\n\n```python\n# to_remove explanation\n\n\"\"\"\n\n- For low values of wEE there is only one fixed point and it is stable so initial\nconditions do not matter and the system always converge to the only fixed point\n\n- For high values of wEE we have three fixed points of which two are stable and\none is unstable (or saddle). Now it matters where the initial conditions are. If\nthe initial conditions are in the attractor region os the high activity fixed\npoint then the system will converge to that (the bottom example).\n\"\"\";\n```\n\nWe can also investigate the effect of different $w_{EI}$, $w_{IE}$, $w_{II}$, $\\tau_{E}$, $\\tau_{I}$, and $I_{E}^{\\text{ext}}$ on the stability of fixed points. In addition, we can also consider the perturbation of the parameters of the gain curve $F(\\cdot)$.\n\n## Section 1.4: Limit cycle - Oscillations\n\nFor some values of interaction terms ($w_{EE}, w_{IE}, w_{EI}, w_{II}$), the eigenvalues can become complex. When at least one pair of eigenvalues is complex, oscillations arise. \nThe stability of oscillations is determined by the real part of the eigenvalues (+ve real part oscillations will grow, -ve real part oscillations will die out). The size of the complex part determines the frequency of oscillations. \n\nFor instance, if we use a different set of parameters, $w_{EE}=6.4$, $w_{EI}=4.8$, $w_{IE}=6.$, $w_{II}=1.2$, and $I_{E}^{\\text{ext}}=0.8$, then we shall observe that the E and I population activity start to oscillate! Please execute the cell below to check the oscillatory behavior. \n\n\n```python\n# @title\n\n# @markdown Make sure you execute this cell to see the oscillations!\n\npars = default_pars(T=100.)\npars['wEE'], pars['wEI'] = 6.4, 4.8\npars['wIE'], pars['wII'] = 6.0, 1.2\npars['I_ext_E'] = 0.8\npars['rE_init'], pars['rI_init'] = 0.25, 0.25\n\nrE, rI = simulate_wc(**pars)\nplt.figure(figsize=(8, 5.5))\nplt.plot(pars['range_t'], rE, 'b', label=r'$r_E$')\nplt.plot(pars['range_t'], rI, 'r', label=r'$r_I$')\nplt.xlabel('t (ms)')\nplt.ylabel('Activity')\nplt.legend(loc='best')\nplt.show()\n```\n\nWe can also understand the oscillations of the population behavior using the phase plane. By plotting a set of trajectories with different initial states, we can see that these trajectories will move in a circle instead of converging to a fixed point. This circle is called \"limit cycle\" and shows the periodic oscillations of the $E$ and $I$ population behavior under some conditions.\n\nLet's plot the phase plane using the previously defined functions.\n\n\n```python\n# @markdown Execute to visualize phase plane\n\npars = default_pars(T=100.)\npars['wEE'], pars['wEI'] = 6.4, 4.8\npars['wIE'], pars['wII'] = 6.0, 1.2\npars['I_ext_E'] = 0.8\n\n\nplt.figure(figsize=(7, 5.5))\nmy_plot_nullcline(pars)\n\n# Find the correct fixed point\nx_fp_1 = my_fp(pars, 0.8, 0.8)\nif check_fp(pars, x_fp_1):\n plot_fp(x_fp_1, position=(0, 0), rotation=40)\n\nmy_plot_trajectories(pars, 0.2, 3,\n 'Sample trajectories \\nwith different initial values')\n\nmy_plot_vector(pars)\n\nplt.legend(loc=[1.01, 0.7])\nplt.xlim(-0.05, 1.01)\nplt.ylim(-0.05, 0.65)\nplt.show()\n```\n\n### Interactive Demo 1.4: Limit cycle and oscillations.\n\nFrom the above examples, the change of model parameters changes the shape of the nullclines and, accordingly, the behavior of the $E$ and $I$ populations from steady fixed points to oscillations. However, the shape of the nullclines is unable to fully determine the behavior of the network. The vector field also matters. To demonstrate this, here, we will investigate the effect of time constants on the population behavior. By changing the inhibitory time constant $\\tau_I$, the nullclines do not change, but the network behavior changes substantially from steady state to oscillations with different frequencies. \n\nSuch a dramatic change in the system behavior is referred to as a **bifurcation**. \n\n\\\\\nPlease execute the code below to check this out.\n\n\n```python\n# @title\n\n# @markdown Make sure you execute this cell to enable the widget!\n\n\ndef time_constant_effect(tau_i=0.5):\n\n pars = default_pars(T=100.)\n pars['wEE'], pars['wEI'] = 6.4, 4.8\n pars['wIE'], pars['wII'] = 6.0, 1.2\n pars['I_ext_E'] = 0.8\n\n pars['tau_I'] = tau_i\n\n Exc_null_rE = np.linspace(0.0, .9, 100)\n Inh_null_rI = np.linspace(0.0, .6, 100)\n\n Exc_null_rI = get_E_nullcline(Exc_null_rE, **pars)\n Inh_null_rE = get_I_nullcline(Inh_null_rI, **pars)\n\n plt.figure(figsize=(12.5, 5.5))\n\n plt.subplot(121) # nullclines\n plt.plot(Exc_null_rE, Exc_null_rI, 'b', label='E nullcline', zorder=2)\n plt.plot(Inh_null_rE, Inh_null_rI, 'r', label='I nullcline', zorder=2)\n plt.xlabel(r'$r_E$')\n plt.ylabel(r'$r_I$')\n\n # fixed point\n x_fp_1 = my_fp(pars, 0.5, 0.5)\n plt.plot(x_fp_1[0], x_fp_1[1], 'ko', zorder=2)\n\n eig_1 = get_eig_Jacobian(x_fp_1, **pars)\n\n # trajectories\n for ie in range(5):\n for ii in range(5):\n pars['rE_init'], pars['rI_init'] = 0.1 * ie, 0.1 * ii\n rE_tj, rI_tj = simulate_wc(**pars)\n plt.plot(rE_tj, rI_tj, 'k', alpha=0.3, zorder=1)\n\n # vector field\n EI_grid_E = np.linspace(0., 1.0, 20)\n EI_grid_I = np.linspace(0., 0.6, 20)\n rE, rI = np.meshgrid(EI_grid_E, EI_grid_I)\n drEdt, drIdt = EIderivs(rE, rI, **pars)\n n_skip = 2\n plt.quiver(rE[::n_skip, ::n_skip], rI[::n_skip, ::n_skip],\n drEdt[::n_skip, ::n_skip], drIdt[::n_skip, ::n_skip],\n angles='xy', scale_units='xy', scale=10, facecolor='c')\n plt.title(r'$\\tau_I=$'+'%.1f ms' % tau_i)\n\n plt.subplot(122) # sample E/I trajectories\n pars['rE_init'], pars['rI_init'] = 0.25, 0.25\n rE, rI = simulate_wc(**pars)\n plt.plot(pars['range_t'], rE, 'b', label=r'$r_E$')\n plt.plot(pars['range_t'], rI, 'r', label=r'$r_I$')\n plt.xlabel('t (ms)')\n plt.ylabel('Activity')\n plt.title(r'$\\tau_I=$'+'%.1f ms' % tau_i)\n plt.legend(loc='best')\n plt.tight_layout()\n plt.show()\n\n\n_ = widgets.interact(time_constant_effect, tau_i=(0.2, 3, .1))\n```\n\nBoth $\\tau_E$ and $\\tau_I$ feature in the Jacobian of the two population network (eq 7). So here is seems that the by increasing $\\tau_I$ the eigenvalues corresponding to the stable fixed point are becoming complex.\n\nIntuitively, when $\\tau_I$ is smaller, inhibitory activity changes faster than excitatory activity. As inhibition exceeds above a certain value, high inhibition inhibits excitatory population but that in turns means that inhibitory population gets smaller input (from the exc. connection). So inhibition decreases rapidly. But this means that excitation recovers -- and so on ...\n\n---\n# Section 2: Inhibition-stabilized network (ISN)\n\n\n## Section 2.1: Inhibition-stabilized network\n\nAs described above, one can obtain the linear approximation around the fixed point as \n\n \\begin{equation}\n \\frac{d}{dr} \\vec{R}=\n \\left[ {\\begin{array}{cc}\n \\displaystyle{\\frac{\\partial G_E}{\\partial r_E}} & \\displaystyle{\\frac{\\partial G_E}{\\partial r_I}}\\\\[1mm]\n \\displaystyle\\frac{\\partial G_I}{\\partial r_E} & \\displaystyle\\frac{\\partial G_I}{\\partial r_I} \\\\\n \\end{array} } \\right] \\vec{R},\n\\end{equation}\n\n
                                        \n\nwhere $\\vec{R} = [r_E, r_I]^{\\rm T}$ is the vector of the E/I activity.\n\nLet's direct our attention to the excitatory subpopulation which follows:\n\n
                                        \n\n\\begin{equation}\n\\frac{dr_E}{dt} = \\frac{\\partial G_E}{\\partial r_E}\\cdot r_E + \\frac{\\partial G_E}{\\partial r_I} \\cdot r_I\n\\end{equation}\n\n
                                        \n\nRecall that, around fixed point $(r_E^*, r_I^*)$:\n\n
                                        \n\n\\begin{align}\n&\\frac{\\partial}{\\partial r_E}G_E(r_E^*, r_I^*) = \\frac{1}{\\tau_E} [-1 + w_{EE} F'_{E}(w_{EE}r_E^* -w_{EI}r_I^* + I^{\\text{ext}}_E; \\alpha_E, \\theta_E)] \\qquad (8)\\\\[1mm]\n&\\frac{\\partial}{\\partial r_I}G_E(r_E^*, r_I^*) = \\frac{1}{\\tau_E} [-w_{EI} F'_{E}(w_{EE}r_E^* -w_{EI}r_I^* + I^{\\text{ext}}_E; \\alpha_E, \\theta_E)] \\qquad (9)\\\\[1mm]\n&\\frac{\\partial}{\\partial r_E}G_I(r_E^*, r_I^*) = \\frac{1}{\\tau_I} [w_{IE} F'_{I}(w_{IE}r_E^* -w_{II}r_I^* + I^{\\text{ext}}_I; \\alpha_I, \\theta_I)] \\qquad (10)\\\\[1mm]\n&\\frac{\\partial}{\\partial r_I}G_I(r_E^*, r_I^*) = \\frac{1}{\\tau_I} [-1-w_{II} F'_{I}(w_{IE}r_E^* -w_{II}r_I^* + I^{\\text{ext}}_I; \\alpha_I, \\theta_I)] \\qquad (11)\n\\end{align}\n\n
                                        \n\nFrom Equation. (8), it is clear that $\\displaystyle{\\frac{\\partial G_E}{\\partial r_I}}$ is negative since the $\\displaystyle{\\frac{dF}{dx}}$ is always positive. It can be understood by that the recurrent inhibition from the inhibitory activity ($I$) can reduce the excitatory ($E$) activity. However, as described above, $\\displaystyle{\\frac{\\partial G_E}{\\partial r_E}}$ has negative terms related to the \"leak\" effect, and positive term related to the recurrent excitation. Therefore, it leads to two different regimes:\n\n- $\\displaystyle{\\frac{\\partial}{\\partial r_E}G_E(r_E^*, r_I^*)}<0$, **noninhibition-stabilized\nnetwork (non-ISN) regime**\n\n- $\\displaystyle{\\frac{\\partial}{\\partial r_E}G_E(r_E^*, r_I^*)}>0$, **inhibition-stabilized\nnetwork (ISN) regime**\n\n### Coding Exercise 2.1: Compute $\\displaystyle{\\frac{\\partial G_E}{\\partial r_E}}$\n\nImplement the function to calculate the $\\displaystyle{\\frac{\\partial G_E}{\\partial r_E}}$ for the default parameters, and the parameters of the limit cycle case.\n\n\n```python\ndef get_dGdE(fp, tau_E, a_E, theta_E, wEE, wEI, I_ext_E, **other_pars):\n \"\"\"\n Compute dGdE\n\n Args:\n fp : fixed point (E, I), array\n Other arguments are parameters of the Wilson-Cowan model\n\n Returns:\n J : the 2x2 Jacobian matrix\n \"\"\"\n rE, rI = fp\n\n ##########################################################################\n # TODO for students: compute dGdrE and disable the error\n raise NotImplementedError(\"Student excercise: compute the dG/dE, Eq. (13)\")\n ##########################################################################\n # Calculate the J[0,0]\n dGdrE = ...\n\n return dGdrE\n\n\n# Get fixed points\npars = default_pars()\nx_fp_1 = my_fp(pars, 0.1, 0.1)\nx_fp_2 = my_fp(pars, 0.3, 0.3)\nx_fp_3 = my_fp(pars, 0.8, 0.6)\n\n# Compute dGdE\ndGdrE1 = get_dGdE(x_fp_1, **pars)\ndGdrE2 = get_dGdE(x_fp_2, **pars)\ndGdrE3 = get_dGdE(x_fp_3, **pars)\n\nprint(f'For the default case:')\nprint(f'dG/drE(fp1) = {dGdrE1:.3f}')\nprint(f'dG/drE(fp2) = {dGdrE2:.3f}')\nprint(f'dG/drE(fp3) = {dGdrE3:.3f}')\n\nprint('\\n')\n\npars = default_pars(wEE=6.4, wEI=4.8, wIE=6.0, wII=1.2, I_ext_E=0.8)\nx_fp_lc = my_fp(pars, 0.8, 0.8)\n\ndGdrE_lc = get_dGdE(x_fp_lc, **pars)\n\nprint('For the limit cycle case:')\nprint(f'dG/drE(fp_lc) = {dGdrE_lc:.3f}')\n```\n\n\n```python\n# to_remove solution\ndef get_dGdE(fp, tau_E, a_E, theta_E, wEE, wEI, I_ext_E, **other_pars):\n \"\"\"\n Compute dGdE\n\n Args:\n fp : fixed point (E, I), array\n Other arguments are parameters of the Wilson-Cowan model\n\n Returns:\n J : the 2x2 Jacobian matrix\n \"\"\"\n rE, rI = fp\n\n # Calculate the J[0,0]\n dGdrE = (-1 + wEE * dF(wEE * rE - wEI * rI + I_ext_E, a_E, theta_E)) / tau_E\n\n return dGdrE\n\n\n# Get fixed points\npars = default_pars()\nx_fp_1 = my_fp(pars, 0.1, 0.1)\nx_fp_2 = my_fp(pars, 0.3, 0.3)\nx_fp_3 = my_fp(pars, 0.8, 0.6)\n\n# Compute dGdE\ndGdrE1 = get_dGdE(x_fp_1, **pars)\ndGdrE2 = get_dGdE(x_fp_2, **pars)\ndGdrE3 = get_dGdE(x_fp_3, **pars)\n\nprint(f'For the default case:')\nprint(f'dG/drE(fp1) = {dGdrE1:.3f}')\nprint(f'dG/drE(fp2) = {dGdrE2:.3f}')\nprint(f'dG/drE(fp3) = {dGdrE3:.3f}')\n\nprint('\\n')\n\npars = default_pars(wEE=6.4, wEI=4.8, wIE=6.0, wII=1.2, I_ext_E=0.8)\nx_fp_lc = my_fp(pars, 0.8, 0.8)\n\ndGdrE_lc = get_dGdE(x_fp_lc, **pars)\n\nprint('For the limit cycle case:')\nprint(f'dG/drE(fp_lc) = {dGdrE_lc:.3f}')\n```\n\n**SAMPLE OUTPUT**\n```\nFor the default case:\ndG/drE(fp1) = -0.650\ndG/drE(fp2) = 1.519\ndG/drE(fp3) = -0.706\n\n\nFor the limit cycle case:\ndG/drE(fp_lc) = 0.837\n```\n\n## Section 2.2: Nullcline analysis of the ISN\n\nRecall that the E nullcline follows\n\n
                                        \n\n\\begin{align}\nr_E = F_E(w_{EE}r_E -w_{EI}r_I + I^{\\text{ext}}_E;a_E,\\theta_E). \n\\end{align}\n\n
                                        \n\nThat is, the firing rate $r_E$ can be a function of $r_I$. Let's take the derivative of $r_E$ over $r_I$, and obtain\n\n
                                        \n\n\\begin{align}\n&\\frac{dr_E}{dr_I} = F_E' \\cdot (w_{EE}\\frac{dr_E}{dr_I} -w_{EI}) \\iff \\\\\n&(1-F_E'w_{EE})\\frac{dr_E}{dr_I} = -F_E' w_{EI} \\iff \\\\\n&\\frac{dr_E}{dr_I} = \\frac{F_E' w_{EI}}{F_E'w_{EE}-1}.\n\\end{align}\n\n
                                        \n\nThat is, in the phase plane `rI-rE`-plane, we can obtain the slope along the E nullcline as\n\n
                                        \n\n$$ \\frac{dr_I}{dr_E} = \\frac{F_E'w_{EE}-1}{F_E' w_{EI}} \\qquad (12) $$\n\n
                                        \n\nSimilarly, we can obtain the slope along the I nullcline as \n\n$$ \\frac{dr_I}{dr_E} = \\frac{F_I'w_{IE}}{F_I' w_{II}+1} \\qquad (13) $$\n\n
                                        \n\nThen, we can find that $\\Big{(} \\displaystyle{\\frac{dr_I}{dr_E}} \\Big{)}_{\\rm I-nullcline} >0$ in Equation (13).\n\n
                                        \n\nHowever, in Equation (12), the sign of $\\Big{(} \\displaystyle{\\frac{dr_I}{dr_E}} \\Big{)}_{\\rm E-nullcline}$ depends on the sign of $(F_E'w_{EE}-1)$. Note that, $(F_E'w_{EE}-1)$ is the same as what we show above (Equation (8)). Therefore, we can have the following results:\n\n- $\\Big{(} \\displaystyle{\\frac{dr_I}{dr_E}} \\Big{)}_{\\rm E-nullcline}<0$, **noninhibition-stabilized\nnetwork (non-ISN) regime**\n\n- $\\Big{(} \\displaystyle{\\frac{dr_I}{dr_E}} \\Big{)}_{\\rm E-nullcline}>0$, **inhibition-stabilized\nnetwork (ISN) regime**\n\n
                                        \n\nIn addition, it is important to point out the following two conclusions:\n\n**Conclusion 1:** The stability of a fixed point can determine the relationship between the slopes Equations (12) and (13). As discussed above, the fixed point is stable when the Jacobian matrix ($J$ in Equation (7)) has two eigenvalues with a negative real part, which indicates a positive determinant of $J$, i.e., $\\text{det}(J)>0$.\n\nFrom the Jacobian matrix definition and from Equations (8-11), we can obtain:\n\n$$ J=\n \\left[ {\\begin{array}{cc}\n \\displaystyle{\\frac{1}{\\tau_E}(w_{EE}F_E'-1)} & \\displaystyle{-\\frac{1}{\\tau_E}w_{EI}F_E'}\\\\[1mm]\n \\displaystyle {\\frac{1}{\\tau_I}w_{IE}F_I'}& \\displaystyle {\\frac{1}{\\tau_I}(-w_{II}F_I'-1)} \\\\\n \\end{array} } \\right] $$\n\n
                                        \n\nNote that, if we let \n\n
                                        \n\n\\begin{equation}\n T=\n \\left[\n { \n \\begin{matrix}\n \\displaystyle{\\tau_E} & \\displaystyle{0} \\\\\n \\displaystyle 0& \\displaystyle \\tau_I\n \\end{matrix}\n } \\right], \n F=\n \\left[\n {\n \\begin{matrix}\n \\displaystyle{F_E'} & \\displaystyle{0} \\\\\n \\displaystyle 0& \\displaystyle F_I'\n \\end{matrix}\n }\n \\right]\n \\text{, and }\n W=\n \\left[\n {\n \\begin{matrix}\n \\displaystyle{w_{EE}} & \\displaystyle{-w_{EI}} \\\\\n \\displaystyle w_{IE}& \\displaystyle -w_{II}\n \\end{matrix}\n }\n \\right]\n\\end{equation}\n\nthen, using matrix notation, $J=T^{-1}(F W - I)$ where $I$ is the identity matrix, i.e., $I = \\begin{bmatrix} \n1 & 0 \\\\\n0 & 1 \n\\end{bmatrix}.$\n \n
                                        \n\nTherefore, $\\det{(J)}=\\det{(T^{-1}(F W - I))}=(\\det{(T^{-1})})(\\det{(F W - I)}).$\n\nSince $\\det{(T^{-1})}>0$, as time constants are positive by definition, the sign of $\\det{(J)}$ is the same as the sign of $\\det{(F W - I)}$, and so\n\n$$\\det{(FW - I)} = (F_E' w_{EI})(F_I'w_{IE}) - (F_I' w_{II} + 1)(F_E'w_{EE} - 1) > 0.$$\n\n
                                        \n\nThen, combining this with Equations (12) and (13), we can obtain\n\n\\begin{equation}\n\\frac{\\Big{(} \\displaystyle{\\frac{dr_I}{dr_E}} \\Big{)}_{\\rm I-nullcline}}{\\Big{(} \\displaystyle{\\frac{dr_I}{dr_E}} \\Big{)}_{\\rm E-nullcline}} > 1.\n\\end{equation}\n\nTherefore, at the stable fixed point, I nullcline has a steeper slope than the E nullcline. \n\n
                                        \n\n**Conclusion 2:** Effect of adding input to the inhibitory population.\n\nWhile adding the input $\\delta I^{\\rm ext}_I$ into the inhibitory population, we can find that the E nullcline (Equation (5)) stays the same, while the I nullcline has a pure left shift: the original I nullcline equation,\n\n
                                        \n\n\\begin{equation}\nr_I = F_I(w_{IE}r_E-w_{II}r_I + I^{\\text{ext}}_I ; \\alpha_I, \\theta_I)\n\\end{equation}\n\n
                                        \n\nremains true if we take $I^{\\text{ext}}_I \\rightarrow I^{\\text{ext}}_I +\\delta I^{\\rm ext}_I$ and $r_E\\rightarrow r_E'=r_E-\\frac{\\delta I^{\\rm ext}_I}{w_{IE}}$ to obtain\n\n
                                        \n\n\\begin{equation}\nr_I = F_I(w_{IE}r_E'-w_{II}r_I + I^{\\text{ext}}_I +\\delta I^{\\rm ext}_I; \\alpha_I, \\theta_I)\n\\end{equation}\n\n
                                        \n\nPutting these points together, we obtain the phase plane pictures shown below. After adding input to the inhibitory population, it can be seen in the trajectories above and the phase plane below that, in an **ISN**, $r_I$ will increase first but then decay to the new fixed point in which both $r_I$ and $r_E$ are decreased compared to the original fixed point. However, by adding $\\delta I^{\\rm ext}_I$ into a **non-ISN**, $r_I$ will increase while $r_E$ will decrease.\n\n### Interactive Demo 2.2: Nullclines of Example **ISN** and **non-ISN**\n\nIn this interactive widget, we inject excitatory ($I^{\\text{ext}}_I>0$) or inhibitory ($I^{\\text{ext}}_I<0$) drive into the inhibitory population when the system is at its equilibrium (with parameters $w_{EE}=6.4$, $w_{EI}=4.8$, $w_{IE}=6.$, $w_{II}=1.2$, $I_{E}^{\\text{ext}}=0.8$, $\\tau_I = 0.8$, and $I^{\\text{ext}}_I=0$). How does the firing rate of the $I$ population changes with excitatory vs inhibitory drive into the inhibitory population?\n\n\n```python\n# @title\n\n# @markdown Make sure you execute this cell to enable the widget!\n\npars = default_pars(T=50., dt=0.1)\npars['wEE'], pars['wEI'] = 6.4, 4.8\npars['wIE'], pars['wII'] = 6.0, 1.2\npars['I_ext_E'] = 0.8\npars['tau_I'] = 0.8\n\n\ndef ISN_I_perturb(dI=0.1):\n Lt = len(pars['range_t'])\n pars['I_ext_I'] = np.zeros(Lt)\n pars['I_ext_I'][int(Lt / 2):] = dI\n\n pars['rE_init'], pars['rI_init'] = 0.6, 0.26\n rE, rI = simulate_wc(**pars)\n\n plt.figure(figsize=(8, 1.5))\n\n plt.plot(pars['range_t'], pars['I_ext_I'], 'k')\n plt.xlabel('t (ms)')\n plt.ylabel(r'$I_I^{\\mathrm{ext}}$')\n plt.ylim(pars['I_ext_I'].min() - 0.01, pars['I_ext_I'].max() + 0.01)\n plt.show()\n\n plt.figure(figsize=(8, 4.5))\n plt.plot(pars['range_t'], rE, 'b', label=r'$r_E$')\n plt.plot(pars['range_t'], rE[int(Lt / 2) - 1] * np.ones(Lt), 'b--')\n plt.plot(pars['range_t'], rI, 'r', label=r'$r_I$')\n plt.plot(pars['range_t'], rI[int(Lt / 2) - 1] * np.ones(Lt), 'r--')\n plt.ylim(0, 0.8)\n plt.xlabel('t (ms)')\n plt.ylabel('Activity')\n plt.legend(loc='best')\n plt.show()\n\n\n_ = widgets.interact(ISN_I_perturb, dI=(-0.2, 0.21, .05))\n```\n\n\n```python\n# to_remove explanation\n\n\"\"\"\nDiscussion:\n\nHere we observe a paradoxical effect; if we inject excitatory current to the I\npopulation, the r_I goes down, whereas when we inject inhibitory current, the r_I\nincreases. Recall that we inject a constant excitatory current to the E population,\nwhich also drives, indirectly, the I population. When Iext>0, the r_I increases\nbut this drives E to a low state, which in turn leads to rI decrease. Whereas,\nwhen Iext<0, the effect is negative on I population for a short amount of time,\nwhich is sufficient to drive the E population to a high steady state, and then due\nto E to I connections, the I population activity is increased.\n\"\"\";\n```\n\n---\n# Section 3: Fixed point and working memory\n\nThe input into the neurons measured in the experiment is often very noisy ([links](http://www.scholarpedia.org/article/Stochastic_dynamical_systems)). Here, the noisy synaptic input current is modeled as an Ornstein-Uhlenbeck (OU)process, which has been discussed several times in the previous tutorials.\n\n\n\n```python\n# @markdown Make sure you execute this cell to enable the function my_OU and plot the input current!\n\n\ndef my_OU(pars, sig, myseed=False):\n \"\"\"\n Expects:\n pars : parameter dictionary\n sig : noise amplitute\n myseed : random seed. int or boolean\n\n Returns:\n I : Ornstein-Uhlenbeck input current\n \"\"\"\n\n # Retrieve simulation parameters\n dt, range_t = pars['dt'], pars['range_t']\n Lt = range_t.size\n tau_ou = pars['tau_ou'] # [ms]\n\n # set random seed\n if myseed:\n np.random.seed(seed=myseed)\n else:\n np.random.seed()\n\n # Initialize\n noise = np.random.randn(Lt)\n I_ou = np.zeros(Lt)\n I_ou[0] = noise[0] * sig\n\n # generate OU\n for it in range(Lt-1):\n I_ou[it+1] = (I_ou[it]\n + dt / tau_ou * (0. - I_ou[it])\n + np.sqrt(2 * dt / tau_ou) * sig * noise[it + 1])\n return I_ou\n\n\npars = default_pars(T=50)\npars['tau_ou'] = 1. # [ms]\nsig_ou = 0.1\nI_ou = my_OU(pars, sig=sig_ou, myseed=2020)\nplt.figure(figsize=(8, 5.5))\nplt.plot(pars['range_t'], I_ou, 'b')\nplt.xlabel('Time (ms)')\nplt.ylabel(r'$I_{\\mathrm{OU}}$')\nplt.show()\n```\n\n\n\nWith the default parameters, the system fluctuates around a resting state with the noisy input.\n\n\n\n```python\n# @markdown Execute this cell to plot activity with noisy input current\npars = default_pars(T=100)\npars['tau_ou'] = 1. # [ms]\nsig_ou = 0.1\npars['I_ext_E'] = my_OU(pars, sig=sig_ou, myseed=20201)\npars['I_ext_I'] = my_OU(pars, sig=sig_ou, myseed=20202)\n\npars['rE_init'], pars['rI_init'] = 0.1, 0.1\nrE, rI = simulate_wc(**pars)\n\nplt.figure(figsize=(8, 5.5))\nax = plt.subplot(111)\nax.plot(pars['range_t'], rE, 'b', label='E population')\nax.plot(pars['range_t'], rI, 'r', label='I population')\nax.set_xlabel('t (ms)')\nax.set_ylabel('Activity')\nax.legend(loc='best')\nplt.show()\n```\n\n## Interactive Demo 3: Short pulse induced persistent activity\nThen, let's use a brief 10-ms positive current to the E population when the system is at its equilibrium. When this amplitude (SE below) is sufficiently large, a persistent activity is produced that outlasts the transient input. What is the firing rate of the persistent activity, and what is the critical input strength? Try to understand the phenomena from the above phase-plane analysis.\n\n\n```python\n# @title\n\n# @markdown Make sure you execute this cell to enable the widget!\n\n\ndef my_inject(pars, t_start, t_lag=10.):\n \"\"\"\n Expects:\n pars : parameter dictionary\n t_start : pulse starts [ms]\n t_lag : pulse lasts [ms]\n\n Returns:\n I : extra pulse time\n \"\"\"\n\n # Retrieve simulation parameters\n dt, range_t = pars['dt'], pars['range_t']\n Lt = range_t.size\n\n # Initialize\n I = np.zeros(Lt)\n\n # pulse timing\n N_start = int(t_start / dt)\n N_lag = int(t_lag / dt)\n I[N_start:N_start + N_lag] = 1.\n\n return I\n\n\npars = default_pars(T=100)\npars['tau_ou'] = 1. # [ms]\nsig_ou = 0.1\npars['I_ext_I'] = my_OU(pars, sig=sig_ou, myseed=2021)\npars['rE_init'], pars['rI_init'] = 0.1, 0.1\n\n# pulse\nI_pulse = my_inject(pars, t_start=20., t_lag=10.)\nL_pulse = sum(I_pulse > 0.)\n\n\ndef WC_with_pulse(SE=0.):\n pars['I_ext_E'] = my_OU(pars, sig=sig_ou, myseed=2022)\n pars['I_ext_E'] += SE * I_pulse\n\n rE, rI = simulate_wc(**pars)\n\n plt.figure(figsize=(8, 5.5))\n ax = plt.subplot(111)\n ax.plot(pars['range_t'], rE, 'b', label='E population')\n ax.plot(pars['range_t'], rI, 'r', label='I population')\n\n ax.plot(pars['range_t'][I_pulse > 0.], 1.0*np.ones(L_pulse), 'r', lw=3.)\n ax.text(25, 1.05, 'stimulus on', horizontalalignment='center',\n verticalalignment='bottom')\n ax.set_ylim(-0.03, 1.2)\n ax.set_xlabel('t (ms)')\n ax.set_ylabel('Activity')\n ax.legend(loc='best')\n plt.show()\n\n\n_ = widgets.interact(WC_with_pulse, SE=(0.0, 1.0, .05))\n```\n\n\n```python\n# to_remove explanation\n\n\"\"\"\nDiscussion:\n\nWhen a system has more than one fixed points, depending on the input strength,\nthe network will settle in one of the fixed points. In this case, we have two\nfixed points, one of the fixed points corresponds to high activity. So when input\ndrives the network to the high activity fixed points, the network activity will\nremain there -- it is a stable fixed point. Because the network retains its\nactivity (persistent activity) even after the input has been removed, we can\ntake the persistent activity as working memory.\n\"\"\";\n```\n\nExplore what happened when a second, brief current is applied to the inhibitory population. \n", "meta": {"hexsha": "448ae6ba0053d4954afae2bbc80d1afe6c3641e3", "size": 69261, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W2D4_DynamicNetworks/W2D4_Tutorial3.ipynb", "max_stars_repo_name": "Beilinson/course-content", "max_stars_repo_head_hexsha": "b74c630bec7002abe2f827ff8e0707f9bbb43f82", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tutorials/W2D4_DynamicNetworks/W2D4_Tutorial3.ipynb", "max_issues_repo_name": "Beilinson/course-content", "max_issues_repo_head_hexsha": "b74c630bec7002abe2f827ff8e0707f9bbb43f82", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/W2D4_DynamicNetworks/W2D4_Tutorial3.ipynb", "max_forks_repo_name": "Beilinson/course-content", "max_forks_repo_head_hexsha": "b74c630bec7002abe2f827ff8e0707f9bbb43f82", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 35.8679440704, "max_line_length": 725, "alphanum_fraction": 0.5374453155, "converted": true, "num_tokens": 15896, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.6688802603710086, "lm_q1q2_score": 0.4212438033757137}} {"text": "##### Copyright 2020 The OpenFermion Developers\n\n\n```python\n#@title Licensed under the Apache License, Version 2.0 (the \"License\");\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n# https://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\n```\n\n# Introduction to the bosonic operators\n\n\n \n \n \n \n
                                        \n View on QuantumLib\n \n Run in Google Colab\n \n View source on GitHub\n \n Download notebook\n
                                        \n\n## Setup\n\nInstall the OpenFermion package:\n\n\n```python\ntry:\n import openfermion\nexcept ImportError:\n !pip install git+https://github.com/quantumlib/OpenFermion.git@master#egg=openfermion\n```\n\n## The BosonOperator\n\nBosonic systems, like Fermionic systems, are expressed using the bosonic creation and annihilation operators $b^\\dagger_k$ and $b_k$ respectively. Unlike fermions, however, which satisfy the Pauli exclusion principle and thus are distinguished by the canonical fermionic anticommutation relations, the bosonic ladder operators instead satisfy a set of commutation relations:\n\n$$\n\\begin{align}\n& [b_i^\\dagger, b_j^\\dagger] = 0, ~~~ [b_i, b_j] = 0, ~~~ [b_i, b^\\dagger_j] = \\delta_{ij}\n\\end{align}\n$$\n\nAny weighted sums of products of these operators are represented with the `BosonOperator` data structure in OpenFermion. Similarly to when we introduced the `FermionOperator`, the following are examples of valid `BosonOperator`s:\n\n$$\n\\begin{align}\n& a_1 \\nonumber \\\\\n& 1.7 b^\\dagger_3 \\nonumber \\\\\n&-1.7 \\, b^\\dagger_3 b_1 \\nonumber \\\\\n&(1 + 2i) \\, b^\\dagger_3 b^\\dagger_4 b_1 b_9 \\nonumber \\\\\n&(1 + 2i) \\, b^\\dagger_3 b^\\dagger_4 b_1 b_9 - 1.7 \\, b^\\dagger_3 b_1 \\nonumber\n\\end{align}\n$$\n\nThe `BosonOperator` class is contained in `ops/_boson_operators.py`. The `BosonOperator` is derived from the `SymbolicOperator`, the same class that derives the `FermionOperator`. As such, the details of the class implementation are identical - as in the fermion case, the class is implemented as hash table (python dictionary). The keys of the dictionary encode the strings of ladder operators and values of the dictionary store the coefficients - the strings are subsequently encoded as a tuple of 2-tuples which we refer to as the \"terms tuple\".\n\n\nEach ladder operator is represented by a 2-tuple. The first element of the 2-tuple is an int indicating the quantum mode on which the ladder operator acts. The second element of the 2-tuple is Boole: 1 represents raising and 0 represents lowering. For instance, $b^\\dagger_8$ is represented in a 2-tuple as $(8, 1)$. Note that indices start at 0 and the identity operator is an empty list. \n\n$$\n\\begin{align}\nI & \\mapsto () \\nonumber \\\\\nb_1 & \\mapsto ((1, 0),) \\nonumber \\\\\nb^\\dagger_3 & \\mapsto ((3, 1),) \\nonumber \\\\\nb^\\dagger_3 b_1 & \\mapsto ((3, 1), (1, 0)) \\nonumber \\\\\nb^\\dagger_3 b^\\dagger_4 b_1 b_9 & \\mapsto ((3, 1), (4, 1), (1, 0), (9, 0)) \\nonumber\n\\end{align}\n$$\n\nAlternatively, the `BosonOperator` supports the string-based syntax introduced in the `FermionOperator`; in this case, the terms are separated by spaces, with the integer corresponding to the quantum mode the operator acts on, and `'^'` indicating the Hermitian conjugate:\n\n$$\n\\begin{align}\nI & \\mapsto \\textrm{\"\"} \\nonumber \\\\\nb_1 & \\mapsto \\textrm{\"1\"} \\nonumber \\\\\nb^\\dagger_3 & \\mapsto \\textrm{\"3^\"} \\nonumber \\\\\nb^\\dagger_3 b_1 & \\mapsto \\textrm{\"3^}\\;\\textrm{1\"} \\nonumber \\\\\nb^\\dagger_3 b^\\dagger_4 b_1 b_9 & \\mapsto \\textrm{\"3^}\\;\\textrm{4^}\\;\\textrm{1}\\;\\textrm{9\"} \\nonumber\n\\end{align}\n$$\n\n
                                        \n Note that, unlike the `FermionOperator`, the bosonic creation operators of different indices commute. As a result, the `BosonOperator` automatically sorts groups of annihilation and creation operators in ascending order of the modes they act on.\n
                                        \n\n\nLet's initialize our first term! We do it two different ways below.\n\n\n```python\nfrom openfermion.ops import BosonOperator\n\nmy_term = BosonOperator(((3, 1), (5, 0), (4, 1), (1, 0)))\nprint(my_term)\n\nmy_term = BosonOperator('3^ 5 4^ 1')\nprint(my_term)\n```\n\nNote the printed order differs from the code, since bosonic operators of different indices commute past each other.\n\nThe preferred way to specify the coefficient in openfermion is to provide an optional coefficient argument. If not provided, the coefficient defaults to 1. In the code below, the first method is preferred. The multiplication in the second method actually creates a copy of the term, which introduces some additional cost. All inplace operands (such as +=) modify classes whereas binary operands such as + create copies.\n\nThe additive and multiplicative identities can also be created:\n\n* `BosonOperator(())` and `BosonOperator('')` initialises the identity (`BosonOperator.identity()`).\n* `BosonOperator()` and `BosonOperator()` initialises the zero operator (`BosonOperator.zero()`).\n\n\n```python\ngood_way_to_initialize = BosonOperator('3^ 1', -1.7)\nprint(good_way_to_initialize)\n\nbad_way_to_initialize = -1.7 * BosonOperator('3^ 1')\nprint(bad_way_to_initialize)\n\nidentity = BosonOperator('')\nprint(identity == BosonOperator.identity())\nprint(identity)\n\nzero_operator = BosonOperator()\nprint(zero_operator == BosonOperator.zero())\nprint(zero_operator)\n```\n\nNote that `BosonOperator` has only one attribute: .terms. This attribute is the dictionary which stores the term tuples.\n\n\n```python\nmy_operator = BosonOperator('4^ 1^ 3 9', 1. + 2.j)\nprint(my_operator)\nprint(my_operator.terms)\n```\n\n### Methods and functions that act on the BosonOperator\n\nThere are various functions and methods that act on the `BosonOperator`; these include the ability to normal order, double check if the operator is Hermitian, and calculate the Hermitian conjugate.\n\n\n```python\nfrom openfermion.utils import hermitian_conjugated, is_hermitian\nfrom openfermion.transforms import normal_ordered\n```\n\n`normal_ordered_boson` applies the bosonic commutation relations to write the operator using only normal-ordered terms; that is, that all creation operators are to the left of annihilation operators:\n\n\n```python\nH = BosonOperator('0 0^', 1. + 2.j)\nH.is_normal_ordered()\n```\n\n\n```python\nnormal_ordered(BosonOperator('0 0^', 1. + 2.j))\n```\n\nWe can also use a boson operator method to check if the operator conserves the particle number - that is, for each qumode, the number of annihilation operators equals the number of creation operators.\n\n\n```python\nH.is_boson_preserving()\n```\n\n\n```python\nH = BosonOperator('0 0^ 1^ ', 1. + 2.j)\nH.is_boson_preserving()\n```\n\nThe Hermitian conjugated function returns the Hermitian conjugate of the operator, and its hermiticity can be checked using `is_hermitian`:\n\n\n```python\nis_hermitian(H)\n```\n\n\n```python\nhermitian_conjugated(H)\n```\n\n\n```python\nH = BosonOperator('0 1^', 1/2.)\nH += BosonOperator('1 0^', 1/2.)\nprint(is_hermitian(H))\nprint(hermitian_conjugated(H))\n```\n\n## The QuadOperator\n\nUsing the bosonic ladder operators, it is common to define the canonical position and momentum operators $\\hat{q}$ and $\\hat{p}$:\n\n$$ \\hat{q}_i = \\sqrt{\\frac{\\hbar}{2}}(\\hat{b}_i+\\hat{b}^\\dagger_i), ~~~ \\hat{p}_i = -i\\sqrt{\\frac{\\hbar}{2}}(\\hat{b}_i-\\hat{b}^\\dagger_i)$$\n\nThese operators are Hermitian, and are referred to as the phase space quadrature operators. They satisfy the canonical commutation relation\n\n$$ [\\hat{q}_i, \\hat{p}_j] = \\delta_{ij}i\\hbar$$\n\nwhere the value of $\\hbar$ depends on convention, often taking values $\\hbar=0.5$, $1$, or $2$.\n\nIn OpenFermion, the quadrature operators are represented by the `QuadOperator` class, and stored as a dictionary of tuples (as keys) and coefficients (as values). For example, the multi-mode quadrature operator $q_0 p_1 q_3$ is represented internally as `((0, 'q'), (1, 'p'), (3, 'q'))`. Alternatively, `QuadOperators` also support string input - using string input, the same operator is described by `'q0 p1 q3'`.\n\n\n```python\nfrom openfermion.ops import QuadOperator\n\nH = QuadOperator('q0 p1 q3')\nprint(H)\nprint(H.terms)\n\nH2 = QuadOperator('q3 p4', 3.17)\nH2 -= 77. * H\nprint('')\nprint(H2)\n```\n\nNote that quadrature operators of different indices commute; as such, like the `BosonOperator`, by default we sort quadrature operators such that the operators acting on the lowest numbered mode appear to the left.\n\n### Methods and functions that act on the QuadOperator\n\n\nLike the `BosonOperator`, there are various functions and methods that act on the `QuadOperator`; these include the ability to normal order, double check if the operator is Hermitian, and calculate the Hermitian conjugate.\n\n\n```python\nfrom openfermion.utils import hermitian_conjugated, is_hermitian\n```\n\n`normal_ordered_quad` is an arbitrary convention chosen in OpenFermion that allows us to compare two quadrature operators that might be equivalent, but written in different forms. It is simply defined as a quadrature operator that has all of the position operators $\\hat{q}$ to the left of the momentum operators $\\hat{q}$. All quadrature operators can be placed in this 'normal form' by making use of the canonical commutation relation.\n\n\n```python\nH = QuadOperator('p0 q0', 1. + 2.j)\nH.is_normal_ordered()\n```\n\n\n```python\nnormal_ordered(H)\n```\n\nBy default, we assume the value $\\hbar=1$ in the canonical commutation relation, but this can be modified by passing the `hbar` keyword argument to the function:\n\n\n```python\nnormal_ordered(H, hbar=2)\n```\n\nWe can also use a quad operator method to check if the operator is **Gaussian** - that is, all terms in the quad operator are of quadratic order or lower:\n\n\n```python\nH = QuadOperator('p0 q0', 1. + 2.j)\nH.is_gaussian()\n```\n\n\n```python\nH = QuadOperator('p0 q0 q1', 1. + 2.j)\nH.is_gaussian()\n```\n\nThe Hermitian conjugated function returns the Hermitian conjugate of the operator, and its hermiticity can be checked using `is_hermitian`:\n\n\n```python\nH = QuadOperator('p0 q1 p1', 1-2j)\nhermitian_conjugated(H)\n```\n\n\n```python\nH = QuadOperator('p0 q0', 1/2.)\nH += QuadOperator('q0 p0', -1/2.)\nprint(is_hermitian(H))\nprint(hermitian_conjugated(H))\n```\n\n\n```python\nH = QuadOperator('p0 q0', 1/2.)\nH += QuadOperator('q0 p0', 1/2.)\nprint(is_hermitian(H))\nprint(hermitian_conjugated(H))\n```\n\n\n```python\nhermitian_conjugated(H)\n```\n\n## Converting between quadrature operators and bosonic operators\n\nConverting between bosonic ladder operators and quadrature operators is simple - we just apply the definition of the $\\hat{q}$ and $\\hat{p}$ operators in terms of $\\hat{b}$ and $\\hat{b}^\\dagger$. Two functions are provided to do this automatically; `get_quad_operator` and `get_boson_operator`:\n\n\n```python\nfrom openfermion.transforms import get_boson_operator, get_quad_operator\n```\n\n\n```python\nH = QuadOperator('p0 q0', 1/2.)\nH += QuadOperator('q0 p0', 1/2.)\nH\n```\n\n\n```python\nget_boson_operator(H)\n```\n\nNote that, since these conversions are dependent on the value of $\\hbar$ chosen, both accept a `hbar` keyword argument. As before, if not specified, the default value of $\\hbar$ is `hbar=1`.\n\n\n```python\nH = BosonOperator('0 0^')\nnormal_ordered(get_quad_operator(H, hbar=0.5), hbar=0.5)\n```\n\n## Weyl quantization and symmetric ordering\n\nWe also provide support for the Weyl quantization - this maps a polynomial function of the form\n\n$$f(q_0,\\dots,q_{N-1},p_0\\dots,p_{N-1})=q_0^{m_0}\\cdots q_{N-1}^{m_{N-1}} p_0^{m_0}\\cdots p_{N-1}^{m_{N-1}}$$\n\non the phase space to the corresponding combination of quadrature operators $\\hat{q}$ and $\\hat{p}$. To do so, we make use of the McCoy formula,\n\n$$q^m p^n \\rightarrow \\frac{1}{2^n} \\sum_{r=0}^{n} \\binom{n}{r} q^r p^m q^{n-r}.$$\n\n\n```python\nfrom openfermion.transforms import weyl_polynomial_quantization, symmetric_ordering\n```\n\nFor `weyl_polynomial_quantization`, the polynomial function in the phase space is provided in the form of a string, where 'q' or 'p' is the phase space quadrature variable, the integer directly following is the mode it is with respect to, and '^2' is the polynomial power. If the power is not provided, it is assumed to be '^1'.\n\n\n```python\nweyl_polynomial_quantization('q0 p0')\n```\n\n\n```python\nweyl_polynomial_quantization('q0^2 p0^3 q1^3')\n```\n\nMcCoy's formula is also used to provide a function that returns the symmetric ordering of a `BosonOperator` or `QuadOperator`, $S(\\hat{O})$. Note that $S(\\hat{O})\\neq \\hat{O}$:\n\n\n```python\nsymmetric_ordering(QuadOperator('q0 p0'))\n```\n\nConsider the symmetric ordering of the square of the bosonic number operator, $\\hat{n} = \\hat{b}^\\dagger \\hat{b}$:\n\n\n```python\nfrom openfermion.hamiltonians import number_operator\nn2 = number_operator(1, parity=1) * number_operator(1, parity=1)\n```\n\n\n```python\nn2\n```\n\n\n```python\nSn2 = symmetric_ordering(n2)\nSn2\n```\n\nWe can use `normal_ordered_boson` to simplify this result:\n\n\n```python\nSn2 = normal_ordered(Sn2)\nSn2\n```\n\nTherefore $S(\\hat{n}) = \\hat{b}^\\dagger \\hat{b}^\\dagger \\hat{b}\\hat{b} + 2\\hat{b}^\\dagger \\hat{b} + 0.5$. This is equivalent to $\\hat{n}^2+\\hat{n}+0.5$:\n\n\n```python\nSn2 == normal_ordered(n2 + number_operator(1, parity=1) + 0.5*BosonOperator.identity())\n```\n\n## Bose-Hubbard Hamiltonian\n\nIn addition to the bosonic operators discussed above, we also provide Bosonic Hamiltonians that describe specific models. The Bose-Hubbard Hamiltonian over a discrete lattice or grid described by nodes $V=\\{0,1,\\dots,N-1\\}$ is described by:\n\n$$H = - t \\sum_{\\langle i, j \\rangle} b_i^\\dagger b_{j + 1}\n + \\frac{U}{2} \\sum_{k=1}^{N-1} b_k^\\dagger b_k (b_k^\\dagger b_k - 1)\n - \\mu \\sum_{k=1}^N b_k^\\dagger b_k\n + V \\sum_{\\langle i, j \\rangle} b_i^\\dagger b_i b_j^\\dagger b_j.$$\n \nwhere\n\n- The indices $\\langle i, j \\rangle$ run over pairs\n $i$ and $j$ of adjacenct nodes (nodes that are connected) in the grid\n- $t$ is the tunneling amplitude\n- $U$ is the on-site interaction potential\n- $\\mu$ is the chemical potential\n- $V$ is the dipole or nearest-neighbour interaction potential\n\nThe Bose-Hubbard Hamiltonian function provided in OpenFermion models a Bose-Hubbard model on a two-dimensional grid, with dimensions given by `[x_dimension, y_dimension]`. It has the form\n\n```python\nbose_hubbard(x_dimension, y_dimension, tunneling, interaction,\n chemical_potential=0., dipole=0., periodic=True)\n```\n\nwhere\n\n- `x_dimension` (int): The width of the grid.\n- `y_dimension` (int): The height of the grid.\n- `tunneling` (float): The tunneling amplitude $t$.\n- `interaction` (float): The attractive local interaction $U$.\n- `chemical_potential` (float, optional): The chemical potential $\\mu$ at each site. Default value is 0.\n- `periodic` (bool, optional): If True, add periodic boundary conditions. Default is True.\n- `dipole` (float): The attractive dipole interaction strength $V$.\n\nBelow is an example of a Bose-Hubbard Hamiltonian constructed in OpenFermion.\n\n\n```python\nfrom openfermion.hamiltonians import bose_hubbard, fermi_hubbard\nbose_hubbard(2, 2, 1, 1)\n```\n\n## Sparse bosonic operators\n\nLike the fermionic operators, OpenFermion contains the capability to represent bosonic operators as a sparse matrix (`sparse.csc_matrix`). However, as the fermionic operators can be represented as finite matrices, this is not the case of bosonic systems, as they inhabit a infinite-dimensional Fock space. Instead, a integer truncation value $N$ need to be provided - the returned sparse operator will be of size $N^{M}\\times N^{M}$, where $M$ is the number of modes in the system, and acts on the truncated Fock basis $\\{\\left|{0}\\right\\rangle, \\left|{1}\\right\\rangle, \\dots, \\left|{N-1}\\right\\rangle\\}$.\n\n\n```python\nfrom openfermion.linalg import boson_operator_sparse\n```\n\nThe function `boson_operator_sparse` acts on both `BosonOperator`s and `QuadOperator`s:\n\n\n```python\nH = boson_operator_sparse(BosonOperator('0^ 0'), 5)\n```\n\n\n```python\nH.toarray()\n```\n\n\n```python\nH = boson_operator_sparse(QuadOperator('q0'), 5, hbar=1)\nH.toarray()\n```\n", "meta": {"hexsha": "abb95d378a24f5f999a70c817e02f3ab5dad854f", "size": 29173, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/tutorials/bosonic_operators.ipynb", "max_stars_repo_name": "Emieeel/OpenFermion", "max_stars_repo_head_hexsha": "c19d9667c5970473893f9bc0183556c4cd354dd7", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/tutorials/bosonic_operators.ipynb", "max_issues_repo_name": "Emieeel/OpenFermion", "max_issues_repo_head_hexsha": "c19d9667c5970473893f9bc0183556c4cd354dd7", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/tutorials/bosonic_operators.ipynb", "max_forks_repo_name": "Emieeel/OpenFermion", "max_forks_repo_head_hexsha": "c19d9667c5970473893f9bc0183556c4cd354dd7", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2020-11-13T04:40:47.000Z", "max_forks_repo_forks_event_max_datetime": "2020-11-13T04:41:01.000Z", "avg_line_length": 28.941468254, "max_line_length": 624, "alphanum_fraction": 0.585130086, "converted": true, "num_tokens": 4843, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.6688802603710086, "lm_q1q2_score": 0.4212437940800622}} {"text": "```javascript\n%%javascript\n$('#appmode-leave').hide();\n$('#copy-binder-link').hide();\n$('#visit-repo-link').hide();\n```\n\nCopyright **Paolo Raiteri**, January 2022\n\n# Numerical solution of equilibrium problems\n\nLet's consider a generic chemical reaction \n\n\\begin{equation}\n\\nu_aA + \\nu_bB + \\dots \\leftrightharpoons \\nu_xX + \\nu_yY + \\dots \\tag{1}\n\\end{equation}\n\nif the concentrations of the species are not the equilibrium ones, the reaction will progress (either forward or backwards) to reach the equilibrium conditions, and the free energy of the system will decrease.\n\nAn infinitesimal change of the system's free energy can then be written in terms of the chemical potential, $\\mu$, of all the species involved in the reaction.\n\n\\begin{equation}\n\\mathrm{d}G = \\sum_i \\mu_i \\mathrm{d}n_i \\tag{2}\n\\end{equation}\n\nBecause the concentrations of reactants and products are coupled through the reaction equation, they cannot change independently, _i.e._ if any products are formed, some reactants must be consumed.\nWe can therefore define a unique parameter, $\\xi$, as the _progress of the reaction_ \n\n\\begin{equation}\n\\mathrm{d}\\xi = \\frac{\\mathrm{d}n_i}{ \\nu_i } \\tag{3}\n\\end{equation}\n\nwhere $\\nu_i$ is the stoichiometric coefficient of the species $i$ taken as a positive number for products (formed) and as a negative number for reactants (consumed), and $\\mathrm{d}n_i$ is an infinitesimal change in the concentration of species $i$.\nThus, the free energy change due to the progress of the reaction (in any direction) becomes\n\n\\begin{equation}\n\\mathrm{d}G = \\sum_i \\nu_i \\mu_i \\mathrm{d}\\xi \\tag{4}\n\\end{equation}\n\nAt equilibrium, the concentrations of all species are constant, $\\mathrm{d}n_i=0$, and the reaction is not _progressing_ anymore, $\\mathrm{d}\\xi=0$, hence the free energy is not changing anymore.\nOn the other hand, if the system is out of equilibrium, there is a driving force that pulls the system towards equilibrium. \nA force can always be obtained as the negative of the derivative of the energy with respect to some coordinate.\nIn this case, the driving force that pushes the reaction towards equilibrium can be immediately obtained from equation (4).\n\n\\begin{equation}\n\\frac{\\mathrm{d}G}{\\mathrm{d}\\xi} = \\sum_i \\nu_i \\mu_i = -force \\tag{5}\n\\end{equation}\n\nIf we then substitute the definition of the chemical potential, $\\mu=\\mu^0+RT\\ln x$, where $x$ is the molar fraction of the species, we obtain\n\n\\begin{eqnarray}\n\\frac{\\mathrm{d} G}{\\mathrm{d}\\xi} &=& \\sum_i\\nu_i\\mu_i\\\\ \n&=& \\sum_i\\nu_i \\big[\\mu_i^0 +RT\\ln x_i\\big]\\\\\n&=& \\sum_i\\nu_i\\mu_i^0 +RT\\sum_i\\nu_i\\ln x_i\\\\\n&=& \\Delta_r G^0 +RT\\sum_i\\ln x_i^{\\nu_i}\\\\\n&=& \\Delta_r G^0 +RT\\ln \\prod_i x_i^{\\nu_i}\\\\\n&=& \\Delta_r G^0 +RT\\ln Q\n\\end{eqnarray}\n\nhere $Q$ is the reaction quotient. By remembering the definition of equilibrium constant, $-RT\\ln K_{eq}=\\Delta_r G^0$ we get\n\n\\begin{equation}\nforce = -\\frac{\\mathrm{d} G}{\\mathrm{d}\\xi} = RT\\ln K_{eq} - RT\\ln Q = RT\\ln\\ \\frac{K_{eq}}{Q} \\tag{6}\n\\end{equation}\n\nThis equation provides us with a conceptual framework to solve *numerically* any equilibrium problem.\nBecause the above equations are differential, they are valid only for infinitesimal changes in the conditions.\nHence each a small change in the concentrations, $\\nu_i\\delta c$ the driving force may change significantly.\n\nTherefore we need to set up an iterative procedure where we compute the driving force, change the concentrations, compute the driving force\\dots until the equilibrium is reached.\n\nGiven the concentration of all species, we can compute the driving force using the equation above and alter the concentrations of all species using\n\n\\begin{equation}\n[C]_i^1 = [C]_i^0 + \\mathrm{d}n_i \\tag{7}\n\\end{equation}\n\nwhere the superscript indicates a the passage of an infinitesimal amount of time, and $\\delta n_i$ is\n\n\\begin{equation}\n\\mathrm{d}n_i = \\nu_i \\times force \\times \\delta c \\tag{7}\n\\end{equation}\n\nwhere $\\nu_i$ are the stoichiometric coefficients of the species and $\\mathrm{d}c$ is an adjustable parameter.\n$\\mathrm{d}c$ must be small enough to maintain the validity of the approximation abover but not too small to allow for a quick convergence of the procedure.\n\nWe cen then compute the new reaction quotient, the new driving force and update the concentrations again. This procedure can then be repeated a number of times, until the force is negligible, which indicates that the calculation has converged.\n\n\n## Example #1: Dimerisation reaction\nIn order to elucidate how that procedure works we can take a model dimerisation reaction\n\n\\begin{equation}\n2A \\leftrightharpoons B\n\\end{equation}\n\nwhose equilibrium constant can be written as\n\\begin{equation}\nK_{eq} = \\frac{[B]}{[A]^2}\n\\end{equation}\n\nWe now want to calculate the equilibrium concentrations of $[A]_{eq}$ and $[B]_{eq}$ given their initial concentrations $[A]_{0}$ and $[B]_{0}$.\nAlthough this is a simple problem that can be solved analytically, in this workshop we will learn how we can use an iterative method to numerically solve it.\nWe will use a relatively simple minimisation procedure that can be applied to a large number of problems, for which it is not possible or it is too complicated to get an analytic solution.\n\nImagine mixing the reagents and then to be able to monitor the concentration of all the species in the system at very short discrete time intervals (*timesteps*). What you will see is that the concentrations will change and tend to the equilibrium value. As you have learnt in first year, the reaction quotient, $Q$, can be used to decided which way to reaction will proceed, and that at equilibrium the reaction quotient is equal to the equilibrium constant. Hence, as we have discussed in class, the reaction quotient and the equilibrium constant can be use to define a *driving force* that pulls the system towards equilibrium.\n\nThis *driving force* can then be used in conjunction with an *ICE* table to numerically compute the equilibrium concentration of reactant and products.\n\n\nThe working principle of the minimisation procedure that we will employ is\n\n1. compute the reaction quotient\n\\begin{equation}\nQ = \\frac{\\mathrm{[B]}_0}{\\mathrm{[A]}^2_0}\n\\end{equation}\n\n2. compute the driving force\n\n\\begin{equation}\nF = RT \\ln\\bigg[\\frac{K_{eq}}{Q}\\bigg]\n\\end{equation}\n\n3. evolve the concentrations using the ICE table\n\n| | [A] | [B]\n| :---: | :--------: |:---------:\n| *I* | [A]$_0$ | [B]$_0$\n| *C* | $-2F\\delta c$ | $+F\\delta c$\n| *E* | [A]$_0$-2$F\\delta c$ | [B]$_0$+$F\\delta c$\n\n4. Repeat until the solution does not change \n\nLet's try to implement this\n\n# Working notebooks\n\n\n```python\nimport ipywidgets as ipw\nimport os\n\nfrom IPython.display import Javascript\nimport glob as glob\nimport nbformat as nbf\n\nlabel_layout = ipw.Layout(width='300px')\n\n##########\npfiles = ['.protectedFiles.txt' , '../.protectedFiles.txt']\nfor fff in pfiles:\n if os.path.isfile(fff):\n with open(fff) as f:\n protectedFiles = f.read().splitlines()\n##########\ndef launchNotebook(filename):\n text = \" var name_of_the_notebook = '\" + filename + \"'\"\n vv=\"\"\"\n var url = window.location.href.split('/')\n var newurl = url[0] + '//'\n for (var i = 1; i < url.length - 1; i++) {\n console.log(url[i], newurl)\n newurl += url[i] + '/'\n }\n newurl += name_of_the_notebook\n window.open(newurl)\n \"\"\"\n text = text + vv\n display(Javascript(text))\n\ndef openNewNotebook(btn):\n if os.path.exists(notebookeName.value):\n print(\"Filename exists - Please select a different name\")\n return\n \n nb = nbf.v4.new_notebook()\n text = \"\"\"# Click 'Edit App' to start coding\"\"\"\n\n code = \"\"\"\\\n# python packages\nimport pandas as pd # Dataframes and reading CSV files\nimport numpy as np # Numerical libraries\nimport matplotlib.pyplot as plt # Plotting library\nfrom lmfit import Model # Least squares fitting library\nfrom scipy.optimize import curve_fit # Alternative curve fittting library\"\"\"\n\n nb['cells'] = [nbf.v4.new_markdown_cell(text),\n nbf.v4.new_code_cell(code)]\n\n if notebookeName.value in protectedFiles or notebookeName.value in listOfFiles:\n print(\"File already exists, select a different filename\")\n else:\n with open(notebookeName.value, 'w') as f:\n nbf.write(nb, f)\n launchNotebook(notebookeName.value)\n\n##########\nlistOfFiles = []\nfiles = glob.glob1(\"./\",\"*.ipynb\")\nfor f in files:\n if f in protectedFiles:\n continue\n listOfFiles.append(f)\n\ndef dropdown_filesHandler(change):\n for i in range(0,len(listOfFiles)):\n if listOfFiles[i] == change.new:\n oldNotebookeName[0] = listOfFiles[i]\n\ndef createMenuFiles(data):\n option_list = [\"Choose one\"]\n option_list.extend(data)\n\n dropdown = ipw.Dropdown(description=\"\", options=option_list, layout=ipw.Layout(width=\"300px\"))\n dropdown.observe(dropdown_filesHandler, names='value')\n\n return dropdown\n\n##########\noldNotebookeName = [\"None\"]\ndef openOldNotebook(btn):\n if oldNotebookeName[0] == \"None\":\n print(\"Please select a filename\")\n elif oldNotebookeName[0] in protectedFiles:\n print(\"Please select a different filename\")\n else:\n launchNotebook(oldNotebookeName[0])\n\n##########\nactions0 = []\n\nnotebookeName = ipw.Text(\"Empty.ipynb\")\n\nbtn_new = ipw.Button(description=\"Create a new notebook\", layout=label_layout)\nbtn_new.on_click(openNewNotebook)\n\nbtn_old = ipw.Button(description=\"Open an old notebook\", layout=label_layout)\nbtn_old.on_click(openOldNotebook)\n\nactions0.append(ipw.HBox([btn_new,notebookeName]))\nactions0.append(ipw.HBox([btn_old,createMenuFiles(listOfFiles)]))\nipw.VBox(actions0)\n```\n\n## Critical thinking questions\n\n1. Verify that the final equilibrium concentrations are independent of the starting conditions, provided that the the mass is conserved. Can you explain why is that?\n2. What is the effect of changing the values of $\\delta c$ or the number of cycles?\n3. Is the value of RT important?\n4. In some cases, it is possible that the concentration of one (or more) species becomes negative, this could be due to the fact that the starting conditions are too far from equilibrium and/or $\\delta c$ is too large. Can you think of how this problem can be fixed automatically in the implementation of these algorithm.\n\n\n## Assignment #1\n### Part a: Dissociation of a mono-protic acid\nCalculate the final pH of a solution of a 0.15M of acetic acid\n\n\\begin{equation}\n\\mathrm{CH_3COOH \\rightleftharpoons CH_3COO^{-} + H^+} \\qquad pK_{a} = 3.74\n\\end{equation}\n\nRemember that the equilibrium constant for the reation\n\\begin{equation}\nAH \\leftrightharpoons A^- + H^+\n\\end{equation} \nis $K_{eq}=10^{-pK_a}$ and the reaction quotient\n\\begin{equation}\nQ = \\frac{[A^-][H^+]}{[HA]}\n\\end{equation}\n\nHere below you can see the ICE table for this problem\n\n| | [HA] | [A$^-$] | [H$^+$]\n| :---: |:---------: |:---------: |:---------:\n| *I* | [HA]$_0$ | [A$^-$]$_0$ | [H$^+$]$_0$\n| *C* | $-F_1\\delta c$ | $+F_1\\delta c$ | $+F_1\\delta c$\n| *E* | [HA]$_0-F\\delta c$ | [A$^-$]$_0+F_1\\delta c$ | [H$^+$]$_0+F\\delta c$\n\n\n### Part b: External titration\nLet's now imagine that the pH of the solution of the previous example is kept constant by external titration.\n\n1. What do you have to change in your program to account for this?\n2. What is the concentration of [AH] if the pH is kept to values of 2, 4 and 6?\n3. Try fixing the pH at a value greater than 9, does the procedure work? (see critical thinking question #4)\n\n## Assignment #2: Coupled reactions\nLet's now imagine we have two coupled reactions in the system, _e.g._ oxalic acid in pure water.\n\n\\begin{eqnarray}\nH_2C_2O_4 + H_2O &\\leftrightharpoons& H_3O^+ + HC_2O_4^- &\\qquad\\qquad& pK_{a1} = 1.25\\\\\nHC_2O_4^- + H_2O &\\leftrightharpoons& H_3O^+ + C_2O_4^= &\\qquad\\qquad& pK_{a2} = 4.14 \\\\\n\\end{eqnarray}\n\nDetermine \n1. What is the pH of 1 L solution prepared with 18 g of oxalic acid\n2. What would be the equilibrium concentrations of $HC_2O_4$, $HC_2O_4^-$ and $C_2O_4^=$ if the pH of the same solution is maintained at a value of 3 using an extermal titration.\n\nFor brevity, let's call oxalic acid \"H$_2$A\".\n\nYou now have to compute two reaction quotients\n\n\\begin{eqnarray}\nQ_1 = \\frac{[HA^-][H^+]}{[H_2A]} &\\qquad\\qquad& Q_2 = \\frac{[A^=][H^+]}{[HA^-]}\\\\\n\\end{eqnarray}\n\nwhich will give rise to two forces, $F_1$ and $F_2$,\n\n\\begin{equation}\nF_1 = RT \\ln\\bigg[\\frac{K_{a1}}{Q_1}\\bigg]\n\\end{equation}\n\\begin{equation}\nF_2 = RT \\ln\\bigg[\\frac{K_{a2}}{Q_2}\\bigg]\n\\end{equation}\n\nYou can then use a \"double\" ICE table to update the concentrations.\n\n| | [H$_2$A] | [HA$^-$] | [A$^=$] | [H$^+$]\n| :---: | :--------: |:---------: |:---------: |:---------:\n| *I* | [H$_2$A]$_0$ | [HA$^-$]$_0$ | [A$^=$]$_0$ | [H$^+$]$_0$\n| *C1* | $-F_1\\delta c$ | $+F_1\\delta c$ | $-$ | $+F_1\\delta c$\n| *C2* | $-$ | $-F_2\\delta c$ | $+F_2\\delta c$ | $+F_2\\delta c$\n| *E* | [H$_2$A]$_0-F_1\\delta c$ | [HA$^-$]$_0+(F_1-F_2)\\delta c$ | [A$^=$]$_0+F_2\\delta c$ | [H$^+$]$_0+(F_1+F_2)\\delta c$\n\n\n# Lab report\n\n### Introduction:\n1. Explain the working principle of this numerical procedure to solve chemical equilibrium problems\n 1. Start from the definition of progress of a reaction and the change in free energy using the chemical potential\n 2. define the driving force\n 3. explain the procedure using a \"pseudo code\" or a flow chart\n\n### Results:\n1. Implement the numerical procedure in python/excel and solve the two assignment problems below\n\n2. Show graphs of the concentrations of all the species _vs_ the number of cycles to show that your calculation is converged. In some cases a logarithmic scale would make the graph clearer.\n\n3. Verify your numeric solutions by solving the problem analytically (show all the calculations).\n\n### Critical thinking questions\nInstead of the conclusions section, answer the \"critical thinking\" questions stated above.\n\n### References\nNo references are needed for this lab.\n\n", "meta": {"hexsha": "1babfe96a4b17eacbd0ca8fe92a04e68cd442f6d", "size": 18936, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "week_09_chemicalEquilibrium/equilibrium.ipynb", "max_stars_repo_name": "praiteri/TeachingNotebook", "max_stars_repo_head_hexsha": "75ee8baf8ef81154dffcac556d4739bf73eba712", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "week_09_chemicalEquilibrium/equilibrium.ipynb", "max_issues_repo_name": "praiteri/TeachingNotebook", "max_issues_repo_head_hexsha": "75ee8baf8ef81154dffcac556d4739bf73eba712", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "week_09_chemicalEquilibrium/equilibrium.ipynb", "max_forks_repo_name": "praiteri/TeachingNotebook", "max_forks_repo_head_hexsha": "75ee8baf8ef81154dffcac556d4739bf73eba712", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-02-23T11:36:12.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-23T11:36:12.000Z", "avg_line_length": 43.1343963554, "max_line_length": 639, "alphanum_fraction": 0.5849704267, "converted": true, "num_tokens": 4027, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765155565327, "lm_q2_score": 0.7490872131147275, "lm_q1q2_score": 0.4211192393167913}} {"text": "# Variational Quantum Eigensolvers\n\n# ANALYSING TRADE-OFFS IN SYMMETRY-PRESERVING ANSATZ CIRCUITS FOR THE SCALABLE VARIATIONAL QUANTUM EIGENSOLVER ALGORITHM.\n\nVQE - hybrid classical-quantum algorithm that can find the ground state energy of various quantum systems.\n\nHas applications in quantum chemistry, where resources to classically simulate molecular wavefunctions increase exponentially with size of molecule.\n\nTrue speedups in computational chemistry are still far off and dependant on development of large fault-tolerant quantum computers, VQEs can already be used to determine ground state energy of smaller molecules to a high degree of accuracy\n\n$$\\newcommand{\\ket}[1]{\\left|{#1}\\right\\rangle}$$\n$$\\newcommand{\\bra}[1]{\\left\\langle{#1}\\right|}$$\n$$\\newcommand{\\braket}[2]{\\left\\langle{#1}\\middle|{#2}\\right\\rangle}$$\n\n\n```python\nfrom typing import Optional, Union, Callable, cast\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfrom qiskit import IBMQ, BasicAer, Aer\nfrom qiskit.providers.aer import StatevectorSimulator\nfrom qiskit.utils import QuantumInstance\n\nfrom qiskit_nature.results import ElectronicStructureResult\n\nfrom qiskit_nature.drivers import UnitsType, Molecule, QMolecule\nfrom qiskit_nature.drivers.second_quantization import PySCFDriver, ElectronicStructureDriverType, ElectronicStructureMoleculeDriver\nfrom qiskit_nature.problems.second_quantization import ElectronicStructureProblem\nfrom qiskit_nature.converters.second_quantization import QubitConverter\nfrom qiskit_nature.mappers.second_quantization import ParityMapper, JordanWignerMapper\n\nfrom qiskit_nature.circuit.library import HartreeFock, UCC, UCCSD\n\nfrom qiskit.opflow.primitive_ops import Z2Symmetries\n\nfrom qiskit_nature.algorithms import GroundStateEigensolver\nfrom qiskit_nature.algorithms import VQEUCCFactory\nfrom qiskit.algorithms import NumPyMinimumEigensolver, VQE\nfrom qiskit.algorithms.optimizers import SLSQP, COBYLA\n\n```\n\n### Overview\n\nSystem is represented by a complex-valued wavefunction, and energy is associated with Hamiltonian $\\mathbb{H}$, a matrix with eigenvalues that represent possible energies of the system.\n\nSystem's groudn state is eigenstate of the Hamiltonian with lowest eigenvalue.\n\n$\\mathcal{H}\\ket{\\psi_g} = E_g\\ket{\\psi_g}$\n\n$E_g = \\frac{ \\bra{\\psi_g}|\\mathcal{H}|\\ket{\\psi_g} }{ \\braket{\\psi_g}{\\psi_g} }$\n\n$E_g \\leq \\frac{ \\bra{\\psi}|\\mathcal{H}|\\ket{\\psi} }{ \\braket{\\psi}{\\psi} }$\n\nVQE leverages this inequality by parameterizing quantum state used to evaluate energy expectation, and varationally updating these parameters on a classical computer to find tight upper bound to the ground state.\n\n### Algorithm\n\n* Parameterized quantum circuit prepares trialground state using some initial parameters values\n* A quantum measurement circuit estimates energy expectation in this trial state.\n* Classical optimization algorithm updates parameters to idealy decrease energy expectation of the next trail state.\n* Algorithm is iterates until energy expectation converges to some value, which is taken as the approx ground state energy.\n\n### Challenges\n\n* Number of parameters required t prepare arbitrary quantum states increases problematically with system size, challenging the classical optimizer.\n\n### Solutions\n\n* reduce number of parameters, hence reduce algorithmic search space.\n\n### Classical Optimizers\n\nResponsible for determining parameter updates, thus for generating the new variational trial states.\n\n* Gradient Descent - identifying energy expectation as const function to be minimized, but has tendency of being stuch in local minima and has relatively high number of required quantum circuit evaluations. (poor optimizer for VQE).\n\nPopular optimizers for noise-free models i.e statevector simulations on classical computers. .e.g:\n* Nelder-Mead method\n* Sequential Least Squares Programming (SLSQP)\n* Constrained Optimization by Linear Optimization (COBYLA) - being more efficient at the expense of fewer(just one) energy expectation evaluations per iteration.\n\nVQE on physical quantum coputers incorporating noise models often involves more nuanced optimization schemes.\n\n### Hamiltonian mapping\n\nHow wavefunction of a molecule can be mapped onto qubit states of a quantum computer.\n\nGeneral Hamiltonian for a quantum molecular system: (very intuitive)\n\n$$\\mathcal{\\hat{H}} = - \\sum_i \\frac{\\nabla^2_i}{2} - \\sum_{i, I} \\frac{Z_I}{|r_i - R_I|} + \\sum_{i\\neq j} \\frac{1}{2|r_i - r_j|}$$\n\nwhere: H = kinetic energy - electron-nucleus interaction + electron-electron interaction\n\nWhere:\n\n$i$ and $j$ denote electrons,\n\n$I$ denotes nuclei,\n\n$Z$ is the atomic number of nuclei $I$,\n\n$r$ is position of the electron $i$, $j$,\n\n$R$ is position of each nucleus, which is fixed in space in accordance with the Bohr-Oppenheimer approximation.\n\nThe $\\textbf{second quantization}$ Hamiltonian in which electrons are treated as excitations in an electron field, will ultimately prove more useful in our effort to encode probelm onto quantum copmuter:\n\n$\\begin{align}\n\\mathcal{\\hat{H}} = \\sum_{p,q} h_{pq}a^\\dagger_p a_q + \\frac{1}{2}\\sum_{p,q,r,s}h_{pqrs}a^\\dagger_p a^\\dagger_q a_r a_s\n\\end{align}$\n\nThe $a^\\dagger$ operators (creation operator) in each term \"excite\" electron to orbitals $p$ and $(p, q)$ respectively. The $a$ operator (annihilation operator) in each term $de-excites$ electron from orbitals $q$ and $(r, s)$ respectively.\n\nThe $h$ terms are coefficients known as the one an two-electron integrals and can be calculated relatively eficciently in terms of a small set of orbital basis states on a classical computer.\n\nWe will restrict ourselves to single- and double-electron excitations to obtain a workable approximation of the more complicated true physical system.\n\nThere are 3 common mappings that can be used to produce a Hamiltonian for distinguishable fermions i.e. qubits from a Hmailtonian for indistinguishable fermions i.e. electrons:\n* Jordan-Wigner mapping\n* Parity mapping\n* Bravyi-Kitaev mapping\n\nAll the mappings produce a Hamitonian of the form:\n\n$\\begin{equation}\n\\mathcal{\\hat{H}} = \\sum_j \\alpha_j (\\prod_i \\hat{\\sigma^j_i})\n\\end{equation}$\n\nThis is to say the Hamiltonian will be a linear combination of products of Pauli matrices (with $i$ denoting qubit being acted upon) which can be executed on a quantum computer.\n\nGiven a trial state, the energy expectation may be evaluated by measuring this superposition of Pauli operators, as below:\n\n$E(\\vec{\\theta}) = \\sum^N_j \\alpha_j \\bra{\\psi(\\vec{\\theta})} \\prod_i \\sigma_i^j \\ket{\\psi(\\vec{\\theta})}$\n\n### Parameterized Wavefunction Ansatz\n\nAmong most common wavefunction ansatz is the Unitary Coupled-CLuster Single and Double excitation (UCCSD) ansatz, used in foundational VQE paper by Perruzo et al.\n\nUCCSD is constructed by applying an exponentiated single and double-electron excitation operator to an initial state, commonly chossen to be the Hartree-Fock mean-field wavefunction (an unentangled state that decently approximates the ground state):\n\n$\n\\begin{align}\n\\ket{\\psi(\\theta)} = e^({\\vec{T} - \\vec{T}^\\dagger}) \\ket{\\phi}\\\\\\\\\n\\hat{T} = \\sum_{i\\in virt, j\\in occ} t^j_i \\hat{a}_i^\\dagger + \\sum_{i, j \\in virt, k, l \\in occ} t^{kl}_{ij} \\hat{a}_i^\\dagger \\hat{a}_j^\\dagger \\hat{a}_k \\hat{a}_l\n\\end{align}\n$\n\nWhere:\n\n\"virt\" denotes unoccupied orbitals,\n\n\"occ\" denotes occupied orbitals,\n\n$a\\dagger$ creation operators excite electrons and $a$ annihilation operators de-excite electrons to/from orbiatls,\n\nt coefficients are the tunable parameters that are fed into classical optimizer.\n\nThe $T$ operator is then converted via one of the 3 mappings into as effective Hamiltoniann operator on qubits, which may be subsequently executed on a quantum copmuter.\n\n### Chemistry-Inspired Ansatz\n\n* UCCSD - UCC wit Singles and Doubles\n* UpCCD - Unitary Pair UCCD\n* OO-UCC - Orbital Optimized UCC\n* DUCC - Double UCC\n\n\n### Hardware Efficient Ansatz\n\n* Symmetry Preserving State Preparation\n* Qubit Coupled Cluster Method\n\n### Energy\n\nThe hartree (symbol: Eh or Ha), also known as the Hartree energy, is the unit of energy in the Hartree atomic units system, named after the British physicist Douglas Hartree. Its CODATA recommended value is Eh = 4.3597447222071(85)\u00d710\u221218 J[1] = 27.211386245988(53) eV.[2]\n\nThe hartree energy is approximately the electric potential energy of the hydrogen atom in its ground state and, by the virial theorem, approximately twice its ionization energy; the relationships are not exact because of the finite mass of the nucleus of the hydrogen atom and relativistic corrections.\n\nThe hartree is usually used as a unit of energy in atomic physics and computational chemistry: for experimental measurements at the atomic scale, the electronvolt (eV) or the reciprocal centimetre (cm\u22121) are much more widely used. \n\n(Courtesy: Wikipedia: https://en.wikipedia.org/wiki/Hartree)\n\n### Building a VQE in Qiskit\n\nVQE in Python using Qiskit, as open-source SDK for working with quantum computers, to find ground state energy and corresponding interatomic spearaton for several small molecules: diatomic hydrogen ($H_2$), lithium hydride ($LiH$), ionic helium hydride ($HeH+$) and hydroxide ($OH-$)\n\n---\n* First the molecule structure was specified as a string in xyz coordinates, lithium hydroxide woudl specify as \"Li 0.0 0.0 0.0; H 0.0 0.0 d\" where d is the interatomic distance.\n\nFor two-atom molecules, the z-coordinate of the second atom was varied to determine ground state energy as a function of interatomic distance.\n\n* A PySCF driver was initialized with this molecule specification, creating a data structure representing the molecule along with several useful calculated attributes, including the nuclear repulsion energy - a quantity later added to the VQE-determined electron energy to obtain the total molecular energy.\n\nThe PySCF molecular data structure was then provided as input to the ElectronicStructureProblem class in Qiskit-Nature module. This class determines the Hamiltoninan of the molecule in terms of second-quantization operators, calculating the one- and two-electron integral coefficients discussed previously and returning the electronic operator in the form:\n\n\nWhere:\n\n\"+\" is the excitation operator\n\n\"-\" is the de-excitation operator\n\n\"I\" is the identity operator\n\n\"N\" is the number operator (de-excitation followed by excitation)\n\nThis operator was then converted into a qubit operator using the class qiskit_nature.converters.second_quantization,QubitConverter, with the mapping type set to the Jordan-Wigner transformation, yielding a summation of Pauli operator products:\n\n\n\nNote that the number of qubits needed to execute algorithm s equal to number of second quantization operators, which is defined as the number of molecular spin orbitals considered.\n\n* Next,the Hartree-Fock initial state preparation circuit and the UCCSD variational unitary transformation circuit. (with HF as its initial state) were retrieved from a library of optimized circuits in the qiskit_nature.circuits.library module, being sure to pass the Jordan-Wigner QubitConverter object created earlier as an argument to each circuit for consistency.\n\nFinally, these were all supplied to the VQE class in the module qiskit.algorithms, which simulated the UCCSD variational circuit repeatedly on a Qiskit Aer statevector backend to solve for the minimum eigenvalue of the qubit Hamiltonina, using the SLSQP classical optimizer to calculate parameter updates.\n\nThis algorithm was repeated for intermolecular separations ranging between 0.2 and 2 Angstroms (~0.1 nanometres), storing the minimum energy as determined by the VQE at each iteration.\n\n\n```python\nstate_sim = StatevectorSimulator()\n\nfrom qiskit_nature.transformers.second_quantization.electronic import FreezeCoreTransformer\n\nfrom qiskit_nature.algorithms import ExcitedStatesEigensolver, VQEUCCFactory\n\n# vqe_energies = []\n\nclass MolecularVQE:\n def __init__(self):\n # H2\n self.molecule_name = \"H 0.0 0.0 0.0; H 0.0 0.0 \"\n self.backend = QuantumInstance(state_sim)\n self.optimizer = SLSQP(maxiter=500)\n self.vqe = VQE(\n ansatz = None,\n quantum_instance = self.backend,\n optimizer = self.optimizer\n )\n \n \n def get_qubit_op(self, dist, mapper=\"parity\"):\n # Use PySCF, a classical computational chemistry software\n # package, to compute the on-body and two-body integrals in\n # electronic-orbital basis, necessary to form the Fermionic operator\n \n driver = PySCFDriver(\n atom = self.molecule_name + str(dist),\n # unit = UnitsType.ANGSTROM,\n # charge = 0,\n # spin = 0,\n # basis = \"sto3g\"\n )\n \n #molecule = Molecule(\n # geometry=[\n # [\"H\", [0.0, 0.0, 0.0]],\n # [\"H\", [0.0, 0.0, 0.735]]\n # ],\n # charge=0,\n # multiplicity=1\n #)\n \n #driver = ElectronicStructureMoleculeDriver(\n # basis = \"sto3g\",\n # driver_type=ElectronicStructureDriverType.PYSCF\n #)\n\n qmolecule = driver.run() # returns ElectronicStructureDriverResult\n transformer = FreezeCoreTransformer()\n qmolecule = transformer.transform(qmolecule)\n \n es_problem = ElectronicStructureProblem(driver)\n \n if mapper == \"jw\":\n qubit_converter = QubitConverter(mapper=JordanWignerMapper())\n elif mapper == \"parity\":\n qubit_converter = QubitConverter(mapper=ParityMapper(), two_qubit_reduction=True)\n \n return (es_problem, qubit_converter)\n \n def run(self):\n numpy_solver = NumPyMinimumEigensolver()\n \n # distances = np.arange(0.3, 2.0, 0.05)\n distances = np.arange(0.65, 1.0, 0.05)\n exact_energies = []\n vqe_energies = []\n \n n = len(distances)\n i = 1\n \n for dist in distances:\n print(\"Distance {}/{}\".format(i, n))\n i += 1\n \n es_problem, qubit_converter = self.get_qubit_op(dist)\n second_q_ops = es_problem.second_q_ops()\n \n # Hamiltonian\n main_op = second_q_ops[0]\n \n #mine z2sym = Z2Symmetries.find_Z2_symmetries(second_q_ops)\n qubit_op = qubit_converter.convert(\n main_op,\n num_particles = es_problem.num_particles,\n #sector_locator = es_problem.symmetry_sector_locator(None, qubit_converter)\n )\n \n # aux_ops = qubit_converter.convert_match(\n # second_q_ops[1:]\n # )\n \n # q_molecule_transformed = cast(QMolecule, es_problem.molecule_data_transformed)\n # num_molecular_orbitals = q_molecule_transformed.num_molecular_orbitals\n # num_particles = (q_molecule_transformed.num_alpha, q_molecule_transformed.num_beta)\n # num_spin_orbitals = 2 * num_molecular_orbitals\n \n num_particles = es_problem.num_particles\n num_spin_orbitals = es_problem.num_spin_orbitals\n \n \n # initial state is Hartree-Fock state\n initial_state = HartreeFock(num_spin_orbitals, num_particles, qubit_converter)\n \n # UCCSD ansatz for unitary update\n ansatz = UCCSD()\n ansatz.qubit_converter = qubit_converter\n ansatz.num_particles = num_particles\n ansatz.num_spin_orbitals = num_spin_orbitals\n ansatz.initial_state = initial_state\n \n self.vqe.ansatz = ansatz\n solver = self.vqe\n \n # solver = VQEUCCFactory(\n # quantum_instance=self.backend,\n # optimizer=self.optimizer,\n # ansatz=ansatz,\n # initial_state=initial_state\n # )\n \n uccsd_excited_states_calculation = ExcitedStatesEigensolver(qubit_converter, solver)\n print(uccsd_excited_states_calculation)\n # uccsd_ground_excited_states_properties = uccsd_excited_states_calculation.solve(es_problem)\n \n print(\"Computing the minimum eigenvalue...\")\n \n # Approximate minimum eigensolver using VQE\n vqe_result = solver.compute_minimum_eigenvalue(qubit_op)\n print(vqe_result)\n # print(np.real(vqe_result.eigenvalue) + nuclear_repulsion_energy)\n \n \n # vqe_energies.append(np.real(vqe_result.eigenvalue) + nuclear_repulsion_energy)\n \n vqe_energies.append(np.real(vqe_result.eigenvalue))\n \n return (distances, vqe_energies, exact_energies)\n # return self.vqe.ansatz\n```\n\n### Results\n\nDemonstrate close correspondence between experimentally determined ground state energies and the ground state energies determined using VQEs. The ground state interatmoic distance was dtermined by finding the minimum on the plot of VQE ground state energy against interatomic distance.\n\nFind a theoretical curve of ground stte energy against interatomic distance - determined by diagonalizing the molecule Hamiltonian directly using NumpyMinimumEigensolver - check if it coincides with VQE curve.\n\n\n```python\nvqe = MolecularVQE()\nres = vqe.run()\n\n# res.draw(\"mpl\")\n\n\n```\n\n Distance 1/7\n \n Computing the minimum eigenvalue...\n { 'aux_operator_eigenvalues': None,\n 'cost_function_evals': 13,\n 'eigenstate': array([-1.06023981e-08-9.02056208e-17j, 9.95433931e-01+5.85952419e-16j,\n -9.54530748e-02-1.51164432e-16j, -3.37124193e-08-5.55111512e-17j]),\n 'eigenvalue': (-1.9440235703061848+0j),\n 'optimal_parameters': { ParameterVectorElement(t[1]): 3.371241934912606e-08,\n ParameterVectorElement(t[2]): 0.09559862244051313,\n ParameterVectorElement(t[0]): 1.060239817902326e-08},\n 'optimal_point': array([1.06023982e-08, 3.37124193e-08, 9.55986224e-02]),\n 'optimal_value': -1.9440235703061848,\n 'optimizer_evals': None,\n 'optimizer_time': 0.23056364059448242}\n Distance 2/7\n \n Computing the minimum eigenvalue...\n { 'aux_operator_eigenvalues': None,\n 'cost_function_evals': 13,\n 'eigenstate': array([-1.47916226e-08-1.73472348e-16j, 9.94507056e-01+1.88466006e-16j,\n -1.04669551e-01-3.28431483e-17j, 5.84318919e-09-1.66533454e-16j]),\n 'eigenvalue': (-1.8921568981821482+0j),\n 'optimal_parameters': { ParameterVectorElement(t[0]): 1.4791622764614418e-08,\n ParameterVectorElement(t[2]): 0.10486162098941848,\n ParameterVectorElement(t[1]): -5.843189119363774e-09},\n 'optimal_point': array([ 1.47916228e-08, -5.84318912e-09, 1.04861621e-01]),\n 'optimal_value': -1.8921568981821482,\n 'optimizer_evals': None,\n 'optimizer_time': 0.10698914527893066}\n Distance 3/7\n \n Computing the minimum eigenvalue...\n { 'aux_operator_eigenvalues': None,\n 'cost_function_evals': 10,\n 'eigenstate': array([ 4.79615422e-09-2.77555756e-17j, 9.93414622e-01+4.12029416e-17j,\n -1.14574816e-01+7.01568604e-18j, 3.96386964e-08-5.55111512e-17j]),\n 'eigenvalue': (-1.8426866818468055+0j),\n 'optimal_parameters': { ParameterVectorElement(t[2]): 0.11482698684986334,\n ParameterVectorElement(t[1]): -3.9638696375370774e-08,\n ParameterVectorElement(t[0]): -4.796154203779538e-09},\n 'optimal_point': array([-4.79615420e-09, -3.96386964e-08, 1.14826987e-01]),\n 'optimal_value': -1.8426866818468055,\n 'optimizer_evals': None,\n 'optimizer_time': 0.09701395034790039}\n Distance 4/7\n \n Computing the minimum eigenvalue...\n { 'aux_operator_eigenvalues': None,\n 'cost_function_evals': 10,\n 'eigenstate': array([ 5.13520705e-09-1.24900090e-16j, 9.92133184e-01+4.72899907e-16j,\n -1.25186839e-01-7.08391401e-17j, 1.78507571e-16-5.55111512e-17j]),\n 'eigenvalue': (-1.7956191802702903+0j),\n 'optimal_parameters': { ParameterVectorElement(t[0]): -5.135206700088415e-09,\n ParameterVectorElement(t[1]): 0.0,\n ParameterVectorElement(t[2]): 0.12551614970340444},\n 'optimal_point': array([-5.1352067e-09, 0.0000000e+00, 1.2551615e-01]),\n 'optimal_value': -1.7956191802702903,\n 'optimizer_evals': None,\n 'optimizer_time': 0.14327788352966309}\n Distance 5/7\n \n Computing the minimum eigenvalue...\n { 'aux_operator_eigenvalues': None,\n 'cost_function_evals': 9,\n 'eigenstate': array([-7.01156592e-17-2.77555756e-17j, 9.90570747e-01-2.33737371e-16j,\n -1.37002175e-01+8.38896345e-18j, -5.45935204e-09-1.11022302e-16j]),\n 'eigenvalue': (-1.7509229979339356+0j),\n 'optimal_parameters': { ParameterVectorElement(t[0]): 0.0,\n ParameterVectorElement(t[1]): 5.459352019457762e-09,\n ParameterVectorElement(t[2]): 0.13743441561618416},\n 'optimal_point': array([0.00000000e+00, 5.45935202e-09, 1.37434416e-01]),\n 'optimal_value': -1.7509229979339356,\n 'optimizer_evals': None,\n 'optimizer_time': 0.08663034439086914}\n Distance 6/7\n \n Computing the minimum eigenvalue...\n { 'aux_operator_eigenvalues': None,\n 'cost_function_evals': 9,\n 'eigenstate': array([ 9.12272917e-18-8.32667268e-17j, 9.88839385e-01-2.73095698e-16j,\n -1.48985474e-01+9.12272845e-18j, -1.16929881e-08-5.55111512e-17j]),\n 'eigenvalue': (-1.708534808574801+0j),\n 'optimal_parameters': { ParameterVectorElement(t[2]): 0.14954221632965023,\n ParameterVectorElement(t[1]): 1.1692988179978664e-08,\n ParameterVectorElement(t[0]): 0.0},\n 'optimal_point': array([0.00000000e+00, 1.16929882e-08, 1.49542216e-01]),\n 'optimal_value': -1.708534808574801,\n 'optimizer_evals': None,\n 'optimizer_time': 0.07053995132446289}\n Distance 7/7\n \n Computing the minimum eigenvalue...\n { 'aux_operator_eigenvalues': None,\n 'cost_function_evals': 9,\n 'eigenstate': array([ 1.25030027e-08-1.38777878e-17j, 9.86828383e-01+2.60800385e-17j,\n -1.61770650e-01-6.85990279e-17j, 9.47715835e-17-1.66533454e-16j]),\n 'eigenvalue': (-1.6683680051111445+0j),\n 'optimal_parameters': { ParameterVectorElement(t[1]): 0.0,\n ParameterVectorElement(t[0]): -1.2503002380773462e-08,\n ParameterVectorElement(t[2]): 0.1624846741472046},\n 'optimal_point': array([-1.25030024e-08, 0.00000000e+00, 1.62484674e-01]),\n 'optimal_value': -1.6683680051111445,\n 'optimizer_evals': None,\n 'optimizer_time': 0.10727286338806152}\n\n\n\n```python\nplt.plot(res[0], res[1], c=\"r\", label=\"VQE\")\n\n\n# plt.rc(\"text\", usetext=True)\nplt.title(r\"$H2$: Minimum energy vs. Interatomic distance\")\nplt.ylabel(\"Ground state energy (Hartree)\")\nplt.xlabel(\"Interatomic distance (A)\")\nplt.legend()\n# plt.show()\n\nidx = res[1].index(min(res[1]))\ndist, min_energy = res[0][res[1].index(min(res[1]))], min(res[1])\nprint(\"Min Distance: {}\\n\\n\".format(dist))\nprint(\"Min Energy: {}\".format(min_energy))\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "12a19aa29970ef8c44bd2d8d26854e7b55a2af34", "size": 50317, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "VQE/vqe_molecular_example.ipynb", "max_stars_repo_name": 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"alphanum_fraction": 0.7385774192, "converted": true, "num_tokens": 6576, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370111, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.4209535847392948}} {"text": "\n\n_Lambda School Data Science \u2014\u00a0Regression 1_\n\n# Understanding Ordinary Least Squares\n\n#### Objectives\n- understand how ordinary least squares regression minimizes the sum of squared errors\n- understand how linear algebra can solve ordinary least squares regression\n- get and interpret coefficients of a linear model\n- visualize a line of best fit in 2D, and hyperplane in 3D\n- use regression metrics: MSE, RMSE, MAE, R^2\n\n#### Extra Links\n- [Statistics 101: Simple Linear Regression](https://www.youtube.com/watch?v=ZkjP5RJLQF4) (20 minute video)\n- [_An Introduction to Statistical Learning_](http://www-bcf.usc.edu/~gareth/ISL/ISLR%20Seventh%20Printing.pdf), Chapter 3.1, Simple Linear Regression, & 3.2, Multiple Linear Regression\n- Priceonomics, [The Discovery of Statistical Regression](https://priceonomics.com/the-discovery-of-statistical-regression/)\n- Priceonomics, [Why the Father of Modern Statistics Didn\u2019t Believe Smoking Caused Cancer](https://priceonomics.com/why-the-father-of-modern-statistics-didnt-believe/)\n- Harvard Business Review, [When to Act on a Correlation, and When Not To](https://hbr.org/2014/03/when-to-act-on-a-correlation-and-when-not-to)\n- [xkcd 552: Correlation](https://www.explainxkcd.com/wiki/index.php/552:_Correlation)\n- [xkcd 1725: Linear Regression](https://www.explainxkcd.com/wiki/index.php/1725:_Linear_Regression)\n\n## What is Linear Regression?\n\n[Linear Regression](https://en.wikipedia.org/wiki/Linear_regression) is a statistical model that seeks to describe the relationship between some y variable and one or more x variables. \n\n\n\nIn the simplest case, linear regression seeks to fit a straight line through a cloud of points. This line is referred to as the \"regression line\" or \"line of best fit.\" This line tries to summarize the relationship between our X and Y in a way that enables us to use the equation for that line to make predictions.\n\n### Synonyms for \"y variable\" \n- Dependent Variable\n- Response Variable\n- Outcome Variable \n- Predicted Variable\n- Measured Variable\n- Explained Variable\n- Label\n- Target\n\n### Synonyms for \"x variable\"\n- Independent Variable\n- Explanatory Variable\n- Regressor\n- Covariate\n- Feature\n\n## Simple Linear Regresion\n\n#### Making Predictions\n\nSay that we were trying to create a model that captured the relationship between temperature outside and ice cream sales. In Machine Learning our goal is often different that of other flavors of Linear Regression Analysis, because we're trying to fit a model to this data with the intention of making **predictions** on new data (in the future) that we don't have yet.\n\n#### What are we trying to predict?\n\nSo if we had measured ice cream sales and the temprature outside on 11 different days, at the end of our modeling **what would be the thing that we would want to predict? - Ice Cream Sales or Temperature?**\n\nWe would probably want to be measuring temperature with the intention of using that to **forecast** ice cream sales. If we were able to successfully forecast ice cream sales from temperature, this might help us know beforehand how much ice cream to make or how many cones to buy or on which days to open our store, etc. Being able to make predictions accurately has a lot of business implications. This is why making accurate predictions is so valuable (And in large part is why data scientists are paid so well).\n\n#### y variable intuition\n\nWe want the thing that we're trying to predict to serve as our **y** variable. This is why it's sometimes called the \"predicted variable.\" We call it the \"dependent\" variable because our prediction for how much ice cream we're going to sell \"depends\" on the temperature outside. \n\n#### x variable intuition\n\nAll other variables that we use to predict our y variable (we're going to start off just using one) we call our **x** variables. These are called our \"independent\" variables because they don't *depend* on y, they \"explain\" y. Hence they are also referred to as our \"explanatory\" variables.\n\n## Example, reprised: Predict presidential election voting\n\n#### Douglas Hibbs, [Background Information on the \u2018Bread and Peace\u2019 Model of Voting in Postwar US Presidential Elections](https://douglas-hibbs.com/background-information-on-bread-and-peace-voting-in-us-presidential-elections/)\n\n> Aggregate two-party vote shares going to candidates of the party holding the presidency during the postwar era are well explained by just two fundamental determinants:\n\n> (1) Positively by weighted-average growth of per capita real disposable personal income over the term. \n> (2) Negatively by cumulative US military fatalities (scaled to population) owing to unprovoked, hostile deployments of American armed forces in foreign wars.\n\n#### Data sources\n- 1952-2012: Douglas Hibbs, [2014 lecture at Deakin University Melbourne](http://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf), Slide 40\n- 2016, Vote Share: [The American Presidency Project](https://www.presidency.ucsb.edu/statistics/elections)\n- 2016, Recent Growth in Personal Incomes: [The 2016 election economy: the \"Bread and Peace\" model final forecast](https://angrybearblog.com/2016/11/the-2016-election-economy-the-bread-and-peace-model-final-forecast.html)\n- 2016, US Military Fatalities: Assumption that Afghanistan War fatalities in 2012-16 occured at the same rate as 2008-12\n\n\n```python\nimport pandas as pd\n\ncolumns = ['Year','Incumbent Party Candidate','Other Candidate','Incumbent Party Vote Share']\n\ndata = [[1952,\"Stevenson\",\"Eisenhower\",44.6],\n [1956,\"Eisenhower\",\"Stevenson\",57.76],\n [1960,\"Nixon\",\"Kennedy\",49.91],\n [1964,\"Johnson\",\"Goldwater\",61.34],\n [1968,\"Humphrey\",\"Nixon\",49.60],\n [1972,\"Nixon\",\"McGovern\",61.79],\n [1976,\"Ford\",\"Carter\",48.95],\n [1980,\"Carter\",\"Reagan\",44.70],\n [1984,\"Reagan\",\"Mondale\",59.17],\n [1988,\"Bush, Sr.\",\"Dukakis\",53.94],\n [1992,\"Bush, Sr.\",\"Clinton\",46.55],\n [1996,\"Clinton\",\"Dole\",54.74],\n [2000,\"Gore\",\"Bush, Jr.\",50.27],\n [2004,\"Bush, Jr.\",\"Kerry\",51.24],\n [2008,\"McCain\",\"Obama\",46.32],\n [2012,\"Obama\",\"Romney\",52.00], \n [2016,\"Clinton\",\"Trump\",48.2]]\n \nvotes = pd.DataFrame(data=data, columns=columns)\n```\n\n\n```python\ncolumns = ['Year','Average Recent Growth in Personal Incomes']\n\ndata = [[1952,2.40],\n [1956,2.89],\n [1960, .85],\n [1964,4.21],\n [1968,3.02],\n [1972,3.62],\n [1976,1.08],\n [1980,-.39],\n [1984,3.86],\n [1988,2.27],\n [1992, .38],\n [1996,1.04],\n [2000,2.36],\n [2004,1.72],\n [2008, .10],\n [2012, .95], \n [2016, .10]]\n \ngrowth = pd.DataFrame(data=data, columns=columns)\n```\n\n\n```python\n\"\"\"\nFatalities denotes the cumulative number of American military\nfatalities per millions of US population the in Korea, Vietnam,\nIraq and Afghanistan wars during the presidential terms\npreceding the 1952, 1964, 1968, 1976 and 2004, 2008 and\n2012 elections.\n\nhttp://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf\n\"\"\"\n\ncolumns = ['Year','US Military Fatalities per Million']\n\ndata = [[1952,190],\n [1956, 0],\n [1960, 0],\n [1964, 1],\n [1968,146],\n [1972, 0],\n [1976, 2],\n [1980, 0],\n [1984, 0],\n [1988, 0],\n [1992, 0],\n [1996, 0],\n [2000, 0],\n [2004, 4],\n [2008, 14],\n [2012, 5], \n [2016, 5]]\n \ndeaths = pd.DataFrame(data=data, columns=columns)\n```\n\n### Acquire new features\n\n#### Shark attack data source\n- https://www.sharkattackfile.net/incidentlog.htm (Download the Excel file manually, because web crawlers are blocked)\n\n#### Economic data sources\n- Unemployment: https://fred.stlouisfed.org/series/UNRATE\n- GDP: https://fred.stlouisfed.org/series/GDPC1\n- GDP change: https://fred.stlouisfed.org/series/A191RL1Q225SBEA\n\n\n```python\nfrom google.colab import files\nfiles.upload()\n```\n\n\n\n\n\n Upload widget is only available when the cell has been executed in the\n current browser session. Please rerun this cell to enable.\n \n \n\n\n\n\n\n {}\n\n\n\n\n```python\nsharks = pd.read_excel('GSAF5 (1).xls')[['Year', 'Country']]\nsharks = (sharks\n .where(sharks.Country=='USA')\n .groupby('Year')\n .count()\n .reset_index()\n .rename(columns={'Country': 'Shark Attacks'}))\n```\n\n\n```python\nurl = 'https://fred.stlouisfed.org/graph/fredgraph.csv?bgcolor=%23e1e9f0&chart_type=line&drp=0&fo=open%20sans&graph_bgcolor=%23ffffff&height=450&mode=fred&recession_bars=on&txtcolor=%23444444&ts=12&tts=12&width=1168&nt=0&thu=0&trc=0&show_legend=yes&show_axis_titles=yes&show_tooltip=yes&id=UNRATE&scale=left&cosd=1948-01-01&coed=2019-04-01&line_color=%234572a7&link_values=false&line_style=solid&mark_type=none&mw=3&lw=2&ost=-99999&oet=99999&mma=0&fml=a&fq=Monthly&fam=avg&fgst=lin&fgsnd=2009-06-01&line_index=1&transformation=lin&vintage_date=2019-05-30&revision_date=2019-05-30&nd=1948-01-01'\nunemployment = pd.read_csv(url, parse_dates=['DATE'])\n\n# Annual average unemployment, only using the first 10 months of the year\n# (because presidential elections are in November)\nunemployment = (unemployment\n .where(unemployment.DATE.dt.month <= 10)\n .set_index('DATE')\n .resample('A')\n .mean()\n .reset_index()\n .rename(columns={'DATE': 'Year', 'UNRATE': 'Unemployment Rate'}))\n\nunemployment['Year'] = unemployment['Year'].dt.year\n```\n\n\n```python\nurl = 'https://fred.stlouisfed.org/graph/fredgraph.csv?bgcolor=%23e1e9f0&chart_type=line&drp=0&fo=open%20sans&graph_bgcolor=%23ffffff&height=450&mode=fred&recession_bars=on&txtcolor=%23444444&ts=12&tts=12&width=1168&nt=0&thu=0&trc=0&show_legend=yes&show_axis_titles=yes&show_tooltip=yes&id=GDPC1&scale=left&cosd=1947-01-01&coed=2019-01-01&line_color=%234572a7&link_values=false&line_style=solid&mark_type=none&mw=3&lw=2&ost=-99999&oet=99999&mma=0&fml=a&fq=Quarterly&fam=avg&fgst=lin&fgsnd=2009-06-01&line_index=1&transformation=lin&vintage_date=2019-05-30&revision_date=2019-05-30&nd=1947-01-01'\n\ngdp = pd.read_csv(url, parse_dates=['DATE'])\ngdp = (gdp\n .where(gdp.DATE.dt.month==7)\n .rename(columns={'DATE': 'Year', 'GDPC1': 'GDP Q3'}))\n\ngdp['Year'] = gdp['Year'].dt.year\n```\n\n\n```python\nurl = 'https://fred.stlouisfed.org/graph/fredgraph.csv?bgcolor=%23e1e9f0&chart_type=line&drp=0&fo=open%20sans&graph_bgcolor=%23ffffff&height=450&mode=fred&recession_bars=on&txtcolor=%23444444&ts=12&tts=12&width=1168&nt=0&thu=0&trc=0&show_legend=yes&show_axis_titles=yes&show_tooltip=yes&id=A191RL1Q225SBEA&scale=left&cosd=1947-04-01&coed=2019-01-01&line_color=%234572a7&link_values=false&line_style=solid&mark_type=none&mw=3&lw=2&ost=-99999&oet=99999&mma=0&fml=a&fq=Quarterly&fam=avg&fgst=lin&fgsnd=2009-06-01&line_index=1&transformation=lin&vintage_date=2019-05-30&revision_date=2019-05-30&nd=1947-04-01'\n\ngdp_change = pd.read_csv(url, parse_dates=['DATE'])\ngdp_change = (gdp_change\n .where(gdp_change.DATE.dt.month==7)\n .rename(columns={'DATE': 'Year', 'A191RL1Q225SBEA': 'GDP Change Q3'}))\n\ngdp_change['Year'] = gdp_change['Year'].dt.year\n```\n\n### Merge data\n\n\n```python\ndf = (votes\n .merge(growth)\n .merge(deaths)\n .merge(sharks)\n .merge(unemployment)\n .merge(gdp)\n .merge(gdp_change))\n```\n\n### Engineer new feature\n\n\n```python\ndf\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        YearIncumbent Party CandidateOther CandidateIncumbent Party Vote ShareAverage Recent Growth in Personal IncomesUS Military Fatalities per MillionShark AttacksUnemployment RateGDP Q3GDP Change Q3
                                        01952StevensonEisenhower44.602.40190113.082564.4012.9
                                        11956EisenhowerStevenson57.762.89094.102925.035-0.4
                                        21960NixonKennedy49.910.850255.383274.0292.0
                                        31964JohnsonGoldwater61.344.211145.213954.1216.4
                                        41968HumphreyNixon49.603.02146153.594825.7993.1
                                        51972NixonMcGovern61.793.62095.675415.7123.8
                                        61976FordCarter48.951.082187.685965.2652.2
                                        71980CarterReagan44.70-0.390107.146688.794-0.5
                                        81984ReaganMondale59.173.860217.567686.0593.9
                                        91988Bush, Sr.Dukakis53.942.270265.538891.4352.4
                                        101992Bush, Sr.Clinton46.550.380247.519732.9794.0
                                        111996ClintonDole54.741.040275.4111096.9763.6
                                        122000GoreBush, Jr.50.272.360523.9813178.4190.5
                                        132004Bush, Jr.Kerry51.241.724355.5714464.9843.8
                                        142008McCainObama46.320.1014605.5515667.032-2.1
                                        152012ObamaRomney52.000.955648.1316220.6670.5
                                        162016ClintonTrump48.200.105654.9117706.7051.9
                                        \n
                                        \n\n\n\n\n```python\n# True Incumbent =\n# The Incumbent Party Candidate this election is the same as the Incumbent Party Candidate 1 election ago,\n# OR, the Incumbent Party Candidate this election is the same as the Other Candidate 1 election ago\ndf['True Incumbent'] = ((df['Incumbent Party Candidate'] == df['Incumbent Party Candidate'].shift(1)) | \n (df['Incumbent Party Candidate'] == df['Other Candidate'].shift(1)))\n\n# Change the data type of this feature from boolean (True/False) to integer (1/0)\ndf['True Incumbent'] = df['True Incumbent'].astype(int)\n```\n\n\n```python\ndf.groupby('True Incumbent')['Incumbent Party Vote Share'].mean()\n```\n\n\n\n\n True Incumbent\n 0 50.347778\n 1 53.493750\n Name: Incumbent Party Vote Share, dtype: float64\n\n\n\n\n```python\ndf\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        YearIncumbent Party CandidateOther CandidateIncumbent Party Vote ShareAverage Recent Growth in Personal IncomesUS Military Fatalities per MillionShark AttacksUnemployment RateGDP Q3GDP Change Q3True Incumbent
                                        01952StevensonEisenhower44.602.40190113.082564.4012.90
                                        11956EisenhowerStevenson57.762.89094.102925.035-0.41
                                        21960NixonKennedy49.910.850255.383274.0292.00
                                        31964JohnsonGoldwater61.344.211145.213954.1216.40
                                        41968HumphreyNixon49.603.02146153.594825.7993.10
                                        51972NixonMcGovern61.793.62095.675415.7123.81
                                        61976FordCarter48.951.082187.685965.2652.20
                                        71980CarterReagan44.70-0.390107.146688.794-0.51
                                        81984ReaganMondale59.173.860217.567686.0593.91
                                        91988Bush, Sr.Dukakis53.942.270265.538891.4352.40
                                        101992Bush, Sr.Clinton46.550.380247.519732.9794.01
                                        111996ClintonDole54.741.040275.4111096.9763.61
                                        122000GoreBush, Jr.50.272.360523.9813178.4190.50
                                        132004Bush, Jr.Kerry51.241.724355.5714464.9843.81
                                        142008McCainObama46.320.1014605.5515667.032-2.10
                                        152012ObamaRomney52.000.955648.1316220.6670.51
                                        162016ClintonTrump48.200.105654.9117706.7051.90
                                        \n
                                        \n\n\n\n### Plot univariate correlations\n[Seaborn tutorial: Visualizing linear relationships](https://seaborn.pydata.org/tutorial/regression.html)\n\n\n```python\n%matplotlib inline\nimport seaborn as sns\n\ntarget = 'Incumbent Party Vote Share'\nfeatures = ['Average Recent Growth in Personal Incomes', \n 'US Military Fatalities per Million', \n 'Shark Attacks', \n 'Unemployment Rate', \n 'GDP Q3', \n 'GDP Change Q3', \n 'True Incumbent']\n\nfor feature in features:\n sns.lmplot(x=feature, y=target, data=df)\n```\n\nWe can see from the scatterplot that these data points seem to follow a somewhat linear relationship for the \"Average Recent Growth in Personal Incomes\" feature. This means that we could probably summarize their relationship well by fitting a line of best fit to these points. Lets do it.\n\n\n## The Equation for a Line\n\nA common equation for a line is:\n\n\\begin{align}\ny = mx + b\n\\end{align}\n\nWhere $m$ is the slope of our line and $b$ is the y-intercept. \n\nIf we want to plot a line through our cloud of points we figure out what these two values should be. Linear Regression seeks to **estimate** the slope and intercept values that describe a line that best fits the data points.\n\n## The Anatomy of Linear Regression\n\n- Intercept: The $b$ value in our line equation $y=mx+b$\n- Slope: The $m$ value in our line equation $y=mx+b$. These two values together define our regression line.\n\n\n\n- $\\hat{y}$ : A prediction\n- Line of Best Fit (Regression Line)\n- Predicted (fitted) Values: Points on our regression line\n- Observed Values: Points from our dataset\n- Error: The distance between predicted and observed values.\n\n\n\n\n\n```python\nx = df['Average Recent Growth in Personal Incomes']\ny = df['Incumbent Party Vote Share']\n\nm = 0\nb = y.mean()\ny_pred = m*x + b\n```\n\n\n```python\nfrom sklearn.metrics import mean_absolute_error, r2_score\nimport matplotlib.pyplot as plt\n\ndef plot_preds(x, y, y_pred):\n plt.scatter(x, y, label='y_true')\n plt.plot(x, y_pred, label='y_pred')\n plt.legend()\n \n mae = mean_absolute_error(y, y_pred)\n r2 = r2_score(y, y_pred)\n print('Mean Absolute Error:', mae)\n print('R^2:', r2)\n \nplot_preds(x, y, y_pred)\nplt.title('Mean Baseline');\n```\n\n\n```python\nm = 4\nb = 45\ny_pred = m*x + b\nplot_preds(x, y, y_pred)\nplt.title('Guessing & Checking Manually');\n```\n\n## R Squared: $R^2$\n\nOne final attribute of linear regressions that we're going to talk about today is a measure of goodness of fit known as $R^2$ or R-squared. $R^2$ is a statistical measure of how close the data are fitted to our regression line. A helpful interpretation for the $R^2$ is the percentage of the dependent variable that is explained by the model.\n\nIn other words, the $R^2$ is the percentage of y that is explained by the x variables included in the model. For this reason the $R^2$ is also known as the \"coefficient of determination,\" because it explains how much of y is explained (or determined) by our x varaibles. We won't go into the calculation of $R^2$ today, just know that a higher $R^2$ percentage is nearly always better and indicates a model that fits the data more closely. \n\n## Residual Error \n\nThe residual error is the distance between points in our dataset and our regression line.\n\n\n```python\ndef regression_residuals(df, feature, target, m, b):\n x = df[feature]\n y = df[target]\n y_pred = m*x + b\n \n plt.scatter(x, y, label='y_true')\n plt.plot(x, y_pred, label='y_pred')\n plt.legend()\n \n # Plot residual errors\n for x, y1, y2 in zip(x, y, y_pred):\n plt.plot((x, x), (y1, y2), color='grey')\n \n mae = mean_absolute_error(y, y_pred) \n r2 = r2_score(y, y_pred)\n print('Mean Absolute Error:', mae)\n print('R^2:', r2)\n \nregression_residuals(df, feature='Average Recent Growth in Personal Incomes', \n target='Incumbent Party Vote Share', m=3, b=46)\n```\n\n\n```python\nfrom ipywidgets import interact, fixed\n\ninteract(regression_residuals, \n df=fixed(df), \n feature=fixed('Average Recent Growth in Personal Incomes'), \n target=fixed('Incumbent Party Vote Share'), \n m=(-10,10,0.5), \n b=(40,60,0.5));\n```\n\n\n interactive(children=(FloatSlider(value=0.0, description='m', max=10.0, min=-10.0, step=0.5), FloatSlider(valu\u2026\n\n\n## More Formal Notation\n\n\n\nWe have talked about a line of regression being represented like a regular line $y=mx+b$ but as we get to more complicated versions we're going to need to extend this equation. So lets establish the proper terminology.\n\n**X** - Independent Variable, predictor variable, explanatory variable, regressor, covariate\n\n**Y** - Response variable, predicted variable, measured vairable, explained variable, outcome variable\n\n$\\beta_0$ - \"Beta Naught\" or \"Beta Zero\", the intercept value. This is how much of y would exist if X were zero. This is sometimes represented by the letter \"a\" but I hate that. So it's \"Beta 0\" during my lecture.\n\n$\\beta_1$ - \"Beta One\" The primary coefficient of interest. This values is the slope of the line that is estimated by \"minimizing the sum of the squared errors/residuals\" - We'll get to that. \n\n$\\epsilon$ - \"Epsilon\" The \"error term\", random noise, things outside of our model that affect y.\n\n## Minimizing the Sum of the Squared Error\n\nThe most common method of estimating our $\\beta$ parameters is what's known as \"Ordinary Least Squares\" (OLS). (There are different methods of arriving at a line of best fit). OLS estimates the parameters that minimize the squared distance between each point in our dataset and our line of best fit. \n\n\\begin{align}\nSSE = \\sum(y_i - \\hat{y})^2\n\\end{align}\n\n\n\n```python\nfrom matplotlib.patches import Rectangle\nimport numpy as np\nfrom sklearn.metrics import mean_squared_error\n\ndef regression_squared_errors(df, feature, target, m, b):\n x = df[feature]\n y = df[target]\n y_pred = m*x + b\n \n fig = plt.figure(figsize=(7,7))\n ax = plt.axes()\n ax.scatter(x, y, label='y_true')\n ax.plot(x, y_pred, label='y_pred')\n ax.legend()\n \n # Plot square errors\n xmin, xmax = ax.get_xlim()\n ymin, ymax = ax.get_ylim()\n scale = (xmax-xmin)/(ymax-ymin)\n for x, y1, y2 in zip(x, y, y_pred):\n bottom_left = (x, min(y1, y2))\n height = abs(y1 - y2)\n width = height * scale\n ax.add_patch(Rectangle(xy=bottom_left, width=width, height=height, alpha=0.1))\n \n mse = mean_squared_error(y, y_pred)\n rmse = np.sqrt(mse)\n mae = mean_absolute_error(y, y_pred)\n r2 = r2_score(y, y_pred)\n print('Mean Squared Error:', mse)\n print('Root Mean Squared Error:', rmse)\n print('Mean Absolute Error:', mae)\n print('R^2:', r2)\n \nregression_squared_errors(df, feature='Average Recent Growth in Personal Incomes', \n target='Incumbent Party Vote Share', m=3, b=46)\n```\n\n\n```python\ninteract(regression_squared_errors, \n df=fixed(df), \n feature=fixed('Average Recent Growth in Personal Incomes'), \n target=fixed('Incumbent Party Vote Share'), \n m=(-10,10,0.5), \n b=(40,60,0.5));\n```\n\n\n interactive(children=(FloatSlider(value=0.0, description='m', max=10.0, min=-10.0, step=0.5), FloatSlider(valu\u2026\n\n\n## Hypotheses\n\n\n```python\nb = 46\nms = np.arange(-10,10,0.5)\nsses = []\n\nfor m in ms:\n predictions = m * df[feature] + b\n errors = predictions - df[target]\n square_errors = errors ** 2\n sse = square_errors.sum()\n sses.append(sse)\n \nhypotheses = pd.DataFrame({'Slope': ms})\nhypotheses['Intercept'] = b\nhypotheses['Sum of Square Errors'] = sses\n\nhypotheses\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        SlopeInterceptSum of Square Errors
                                        0-10.0463062.3534
                                        1-9.5462924.4034
                                        2-9.0462790.4534
                                        3-8.5462660.5034
                                        4-8.0462534.5534
                                        5-7.5462412.6034
                                        6-7.0462294.6534
                                        7-6.5462180.7034
                                        8-6.0462070.7534
                                        9-5.5461964.8034
                                        10-5.0461862.8534
                                        11-4.5461764.9034
                                        12-4.0461670.9534
                                        13-3.5461581.0034
                                        14-3.0461495.0534
                                        15-2.5461413.1034
                                        16-2.0461335.1534
                                        17-1.5461261.2034
                                        18-1.0461191.2534
                                        19-0.5461125.3034
                                        200.0461063.3534
                                        210.5461005.4034
                                        221.046951.4534
                                        231.546901.5034
                                        242.046855.5534
                                        252.546813.6034
                                        263.046775.6534
                                        273.546741.7034
                                        284.046711.7534
                                        294.546685.8034
                                        305.046663.8534
                                        315.546645.9034
                                        326.046631.9534
                                        336.546622.0034
                                        347.046616.0534
                                        357.546614.1034
                                        368.046616.1534
                                        378.546622.2034
                                        389.046632.2534
                                        399.546646.3034
                                        \n
                                        \n\n\n\n\n```python\nhypotheses.plot(x='Slope', y='Sum of Square Errors', \n title=f'Intercept={b}');\n```\n\n## Scikit-learn\n\n#### Jake VanderPlas, [Python Data Science Handbook, Chapter 5.2, Introducing Scikit-Learn](https://jakevdp.github.io/PythonDataScienceHandbook/05.02-introducing-scikit-learn.html), Scikit-Learn's Estimator API\n\n> Most commonly, the steps in using the Scikit-Learn estimator API are as follows (we will step through a handful of detailed examples in the sections that follow).\n\n> 1. Choose a class of model by importing the appropriate estimator class from Scikit-Learn. \n> 2. Choose model hyperparameters by instantiating this class with desired values. \n> 3. Arrange data into a features matrix and target vector following the discussion above.\n> 4. Fit the model to your data by calling the `fit()` method of the model instance.\n> 5. Apply the Model to new data: For supervised learning, often we predict labels for unknown data using the `predict()` method.\n\n\n```python\nfrom sklearn.linear_model import LinearRegression\nmodel = LinearRegression()\n\nfeatures = ['Average Recent Growth in Personal Incomes']\ntarget = 'Incumbent Party Vote Share'\nX = df[features]\ny = df[target]\n\nmodel.fit(X, y)\n```\n\n\n\n\n LinearRegression(copy_X=True, fit_intercept=True, n_jobs=None, normalize=False)\n\n\n\n\n```python\nmodel.intercept_, model.coef_, \n```\n\n\n\n\n (46.499209757741625, array([2.97417709]))\n\n\n\n\n```python\nmodel.predict([[0]])\n```\n\n\n\n\n array([46.49920976])\n\n\n\n\n```python\nmodel.predict([[1]])\n```\n\n\n\n\n array([49.47338685])\n\n\n\n\n```python\nmodel.predict([[1]]) - model.predict([[0]])\n```\n\n\n\n\n array([2.97417709])\n\n\n\n\n```python\nmodel.predict([[2]])\n```\n\n\n\n\n array([52.44756393])\n\n\n\n\n```python\nmodel.predict([[2]]) - model.predict([[1]])\n```\n\n\n\n\n array([2.97417709])\n\n\n\n\n```python\nm = model.coef_[0]\nb = model.intercept_\n\ny_pred = m*x + b\nplot_preds(x, y, y_pred)\n```\n\n\n```python\ny_pred = model.predict(X)\nplot_preds(x, y, y_pred)\n```\n\n## Linear Algebra!\n\nThe same result that is found by minimizing the sum of the squared errors can be also found through a linear algebra process known as the \"Least Squares Solution:\"\n\n\n\nBefore we can work with this equation in its linear algebra form we have to understand how to set up the matrices that are involved in this equation. \n\n### The $\\beta$ vector\n\nThe $\\beta$ vector represents all the parameters that we are trying to estimate, our $y$ vector and $X$ matrix values are full of data from our dataset. The $\\beta$ vector holds the variables that we are solving for: $\\beta_0$ and $\\beta_1$\n\nNow that we have all of the necessary parts we can set them up in the following equation:\n\n\\begin{align}\ny = X \\beta + \\epsilon\n\\end{align}\n\nSince our $\\epsilon$ value represents **random** error we can assume that it will equal zero on average.\n\n\\begin{align}\ny = X \\beta\n\\end{align}\n\nThe objective now is to isolate the $\\beta$ matrix. We can do this by pre-multiplying both sides by \"X transpose\" $X^{T}$.\n\n\\begin{align}\nX^{T}y = X^{T}X \\beta\n\\end{align}\n\nSince anything times its transpose will result in a square matrix, if that matrix is then an invertible matrix, then we should be able to multiply both sides by its inverse to remove it from the right hand side. (We'll talk tomorrow about situations that could lead to $X^{T}X$ not being invertible.)\n\n\\begin{align}\n(X^{T}X)^{-1}X^{T}y = (X^{T}X)^{-1}X^{T}X \\beta\n\\end{align}\n\nSince any matrix multiplied by its inverse results in the identity matrix, and anything multiplied by the identity matrix is itself, we are left with only $\\beta$ on the right hand side:\n\n\\begin{align}\n(X^{T}X)^{-1}X^{T}y = \\hat{\\beta}\n\\end{align}\n\nWe will now call it \"beta hat\" $\\hat{\\beta}$ because it now represents our estimated values for $\\beta_0$ and $\\beta_1$\n\n### Lets calculate our $\\beta$ coefficients with numpy!\n\n\n```python\n\n```\n\n# Multiple Regression\n\nSimple or bivariate linear regression involves a single $x$ variable and a single $y$ variable. However, we can have many $x$ variables. A linear regression model that involves multiple x variables is known as **Multiple** Regression (NOT MULTIVARIATE).\n\n\n\n\n```python\nfrom mpl_toolkits import mplot3d\n\ndef viz3D(fitted_model, df, features, target='', num=100):\n \"\"\"\n Visualize model predictions in 3D, for regression or binary classification\n \n Parameters\n ----------\n fitted_model : scikit-learn model, already fitted\n df : pandas dataframe, which was used to fit model\n features : list of strings, name of features 1 & 2\n target : string, name of target\n num : int, number of grid points for each feature\n \n References\n ----------\n https://jakevdp.github.io/PythonDataScienceHandbook/04.12-three-dimensional-plotting.html\n https://scikit-learn.org/stable/auto_examples/tree/plot_iris.html \n \"\"\"\n feature1, feature2 = features\n x1 = np.linspace(df[feature1].min(), df[feature1].max(), num)\n x2 = np.linspace(df[feature2].min(), df[feature2].max(), num)\n X1, X2 = np.meshgrid(x1, x2)\n X = np.c_[X1.flatten(), X2.flatten()]\n if hasattr(fitted_model, 'predict_proba'):\n predicted = fitted_model.predict_proba(X)[:,1]\n else:\n predicted = fitted_model.predict(X)\n Z = predicted.reshape(num, num)\n \n fig = plt.figure()\n ax = plt.axes(projection='3d')\n ax.plot_surface(X1, X2, Z, cmap='viridis')\n ax.set_xlabel(feature1)\n ax.set_ylabel(feature2)\n ax.set_zlabel(target)\n plt.show()\n```\n\n\n```python\nmodel = LinearRegression()\n\nfeatures = ['Average Recent Growth in Personal Incomes', \n 'US Military Fatalities per Million']\ntarget = 'Incumbent Party Vote Share'\nX = df[features]\ny = df[target]\nmodel.fit(X, y)\n```\n\n\n\n\n LinearRegression(copy_X=True, fit_intercept=True, n_jobs=None, normalize=False)\n\n\n\n\n```python\npd.Series(model.coef_, features)\n```\n\n\n\n\n Average Recent Growth in Personal Incomes 3.406214\n US Military Fatalities per Million -0.053752\n dtype: float64\n\n\n\n\n```python\ndf[features].describe()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Average Recent Growth in Personal IncomesUS Military Fatalities per Million
                                        count17.00000017.000000
                                        mean1.79176521.588235
                                        std1.41981255.767440
                                        min-0.3900000.000000
                                        25%0.8500000.000000
                                        50%1.7200000.000000
                                        75%2.8900005.000000
                                        max4.210000190.000000
                                        \n
                                        \n\n\n\n\n```python\npd.Series(model.coef_, features).plot.barh(color='grey');\n```\n\n\n```python\n%matplotlib notebook\nviz3D(model, df, features, target)\n```\n\n\n \n\n\n\n
                                        \n\n\n\n \n\n\n\n
                                        \n\n\n# Dimensionality in Linear Regression!\n\nMuliple Regression is simply an extension of the bivariate case. The reason why we see the bivariate case demonstrated so often is simply because it's easier to graph and all of the intuition from the bivariate case is the same as we keep on adding explanatory variables.\n\nAs we increase the number of $x$ values in our model we are simply fitting a n-1-dimensional plane to an n-dimensional cloud of points within an n-dimensional hypercube. \n\n# Interpreting Coefficients\n\nOne of Linear Regression's strengths is that the parameters of the model (coefficients) are readily interpretable and useful. Not only do they describe the relationship between x and y but they put a number on just how much x is associated with y. We should be careful to not speak about this relationshiop in terms of causality because these coefficients are in fact correlative measures. We would need a host of additional techniques in order to estimate a causal effect using linear regression (econometrics).\n\n\\begin{align}\n\\hat{\\beta} = \\frac{Cov(x,y)}{Var(y)}\n\\end{align}\n\nGoing back to the two equations for the two models that we have estimated so far, lets replace their beta values with their actual values to see if we can make sense of how to interpret these beta coefficients.\n\n## Bivariate Model\n\n$y_i = \\beta_0 + \\beta_1temperature + \\epsilon$\n\n$sales_i = -596.2 + 24.69temperature + \\epsilon$\n\nWhat might $\\beta_0$ in this model represent? It represents the level of sales that we would have if temperature were 0. Since this is negative one way of interpreting it is that it's so cold outside that you would have to pay people to eat ice cream. A more appropriate interpretation is probably that the ice cream store owner should close his store down long before the temperature reaches 0 degrees farenheit (-17.7 celsius). The owner can compare his predicted sales with his costs of doing business to know how warm the weather has to get before he should open his store.\n\nWhat might the $beta_1$ in this model reprsent? it represents the increase in sales for each degree of temperature increase. For every degree that the temperature goes up outside he has $25 more in sales.\n\n## Multiple Regression Model\n\n$y_i = \\beta_0 + \\beta_1age_i + \\beta_2weight_i + \\epsilon$\n\n$BloodPressure_i = 30.99+ .86age_i + .33weight_i + \\epsilon$\n\nThe interpretation of coefficients in this example are similar. The intercept value repesents the blood pressure a person would have if they were 0 years old and weighed 0 pounds. This not a super useful interpretation. If we look at our data it is unlikely that we have any measurements like these in the dataset. This means that our interpretation of our intercept likely comes from extrapolating the regression line (plane). Coefficients having straightforward interpretations is a strength of linear regression if we're careful about extrapolation and only interpreting our data within the context that it was gathered.\n\nThe interpretation of our other coefficients can be a useful indicator for how much a person similar to those in our dataset's blood pressure will go up on average with each additional year of age and pound of weight.\n\n# Basic Model Validation\n\nOne of the downsides of relying on $R^2$ too much is that although it tells you when you're fitting the data well, it doesn't tell you when you're *overfitting* the data. The best way to tell if you're overfitting the data is to get some data that your model hasn't seen yet, and evaluate how your predictions do. This is essentially what \"model validation\" is.\n\n# Why is Linear Regression so Important?\n\n## Popularity \n\nLinear Regression is an extremely popular technique that every data scientist **needs** to understand. It's not the most advanced technique and there are supervised learning techniques that will obtain a higher accuracy, but where it lacks in accuracy it makes up for it in interpretability and simplicity.\n\n## Interpretability\n\nFew other models possess coefficients that are so directly linked to their variables with a such a clear interpretation. Tomorrow we're going to learn about ways to make them even easier to interpret.\n\n## Simplicity\n\nA linear regression model can be communicated just by writing out its equation. It's kind of incredible that such high dimensional relationships can be described from just a linear combination of variables and coefficients. \n", "meta": {"hexsha": "5cfc5beae0959b2b8224df382db2b4733dcc71f3", "size": 785938, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "understanding_ols_LIVE_LESSON.ipynb", "max_stars_repo_name": "khaloodi/DS-Unit-2-Regression-1", "max_stars_repo_head_hexsha": "7ed92bad49ecc7c973339cc3613810c038f63b2a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "understanding_ols_LIVE_LESSON.ipynb", "max_issues_repo_name": "khaloodi/DS-Unit-2-Regression-1", "max_issues_repo_head_hexsha": "7ed92bad49ecc7c973339cc3613810c038f63b2a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "understanding_ols_LIVE_LESSON.ipynb", "max_forks_repo_name": "khaloodi/DS-Unit-2-Regression-1", "max_forks_repo_head_hexsha": "7ed92bad49ecc7c973339cc3613810c038f63b2a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 173.4962472406, "max_line_length": 167157, "alphanum_fraction": 0.8379846756, "converted": true, "num_tokens": 15075, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.7057850154599562, "lm_q1q2_score": 0.42095357735720895}} {"text": "##### Copyright 2020 The TensorFlow Authors.\n\n\n```python\n#@title Licensed under the Apache License, Version 2.0 (the \"License\");\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n# https://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\n```\n\n# Barren plateaus\n\n\n \n \n \n \n
                                        \n View on TensorFlow.org\n \n Run in Google Colab\n \n View source on GitHub\n \n Download notebook\n
                                        \n\nIn this example you will explore the result of McClean, 2019 that says not just any quantum neural network structure will do well when it comes to learning. In particular you will see that a certain large family of random quantum circuits do not serve as good quantum neural networks, because they have gradients that vanish almost everywhere. In this example you won't be training any models for a specific learning problem, but instead focusing on the simpler problem of understanding the behaviors of gradients.\n\n## Setup\n\n\n```python\ntry:\n %tensorflow_version 2.x\nexcept Exception:\n pass\n```\n\nInstall TensorFlow Quantum:\n\nNote: This may require restarting the Colab runtime (*Runtime > Restart Runtime*).\n\n\n```python\n!pip install tensorflow-quantum\n```\n\nNow import TensorFlow and the module dependencies:\n\n\n```python\nimport tensorflow as tf\nimport tensorflow_quantum as tfq\n\nimport cirq\nimport sympy\nimport numpy as np\n\n# visualization tools\n%matplotlib inline\nimport matplotlib.pyplot as plt\nfrom cirq.contrib.svg import SVGCircuit\n\nnp.random.seed(1234)\n```\n\n## 1. Summary\n\nRandom quantum circuits with many blocks that look like this ($R_{P}(\\theta)$ is a random Pauli rotation):
                                        \n\n\nWhere if $f(x)$ is defined as the expectation value w.r.t. $Z_{a}Z_{b}$ for any qubits $a$ and $b$, then there is a problem that $f'(x)$ has a mean very close to 0 and does not vary much. You will see this below:\n\n## 2. Generating random circuits\n\nThe construction from the paper is straightforward to follow. The following implements a simple function that generates a random quantum circuit\u2014sometimes referred to as a *quantum neural network* (QNN)\u2014with the given depth on a set of qubits:\n\n\n```python\ndef generate_random_qnn(qubits, symbol, depth):\n \"\"\"Generate random QNN's with the same structure from McClean et al.\"\"\"\n circuit = cirq.Circuit()\n for qubit in qubits:\n circuit += cirq.Ry(np.pi / 4.0)(qubit)\n\n for d in range(depth):\n # Add a series of single qubit rotations.\n for i, qubit in enumerate(qubits):\n random_n = np.random.uniform()\n random_rot = np.random.uniform(\n ) * 2.0 * np.pi if i != 0 or d != 0 else symbol\n if random_n > 2. / 3.:\n # Add a Z.\n circuit += cirq.Rz(random_rot)(qubit)\n elif random_n > 1. / 3.:\n # Add a Y.\n circuit += cirq.Ry(random_rot)(qubit)\n else:\n # Add a X.\n circuit += cirq.Rx(random_rot)(qubit)\n\n # Add CZ ladder.\n for src, dest in zip(qubits, qubits[1:]):\n circuit += cirq.CZ(src, dest)\n\n return circuit\n\n\ngenerate_random_qnn(cirq.GridQubit.rect(1, 3), sympy.Symbol('theta'), 2)\n```\n\nThe authors investigate the gradient of a single parameter $\\theta_{1,1}$. Let's follow along by placing a `sympy.Symbol` in the circuit where $\\theta_{1,1}$ would be. Since the authors do not analyze the statistics for any other symbols in the circuit, let's replace them with random values now instead of later.\n\n## 3. Running the circuits\n\nGenerate a few of these circuits along with an observable to test the claim that the gradients don't vary much. First, generate a batch of random circuits. Choose a random *ZZ* observable and batch calculate the gradients and variance using TensorFlow Quantum.\n\n### 3.1 Batch variance computation\n\nLet's write a helper function that computes the variance of the gradient of a given observable over a batch of circuits:\n\n\n```python\ndef process_batch(circuits, symbol, op):\n \"\"\"Compute the variance of a batch of expectations w.r.t. op on each circuit that \n contains `symbol`. Note that this method sets up a new compute graph every time it is\n called so it isn't as performant as possible.\"\"\"\n\n # Setup a simple layer to batch compute the expectation gradients.\n expectation = tfq.layers.Expectation()\n\n # Prep the inputs as tensors\n circuit_tensor = tfq.convert_to_tensor(circuits)\n values_tensor = tf.convert_to_tensor(\n np.random.uniform(0, 2 * np.pi, (n_circuits, 1)).astype(np.float32))\n\n # Use TensorFlow GradientTape to track gradients.\n with tf.GradientTape() as g:\n g.watch(values_tensor)\n forward = expectation(circuit_tensor,\n operators=op,\n symbol_names=[symbol],\n symbol_values=values_tensor)\n\n # Return variance of gradients across all circuits.\n grads = g.gradient(forward, values_tensor)\n grad_var = tf.math.reduce_std(grads, axis=0)\n return grad_var.numpy()[0]\n```\n\n### 3.1 Set up and run\n\nChoose the number of random circuits to generate along with their depth and the amount of qubits they should act on. Then plot the results.\n\n\n```python\nn_qubits = [2 * i for i in range(2, 7)\n ] # Ranges studied in paper are between 2 and 24.\ndepth = 50 # Ranges studied in paper are between 50 and 500.\nn_circuits = 200\ntheta_var = []\n\nfor n in n_qubits:\n # Generate the random circuits and observable for the given n.\n qubits = cirq.GridQubit.rect(1, n)\n symbol = sympy.Symbol('theta')\n circuits = [\n generate_random_qnn(qubits, symbol, depth) for _ in range(n_circuits)\n ]\n op = cirq.Z(qubits[0]) * cirq.Z(qubits[1])\n theta_var.append(process_batch(circuits, symbol, op))\n\nplt.semilogy(n_qubits, theta_var)\nplt.title('Gradient Variance in QNNs')\nplt.xlabel('n_qubits')\nplt.ylabel('$\\\\partial \\\\theta$ variance')\nplt.show()\n```\n\nThis plot shows that for quantum machine learning problems, you can't simply guess a random QNN ansatz and hope for the best. Some structure must be present in the model circuit in order for gradients to vary to the point where learning can happen.\n\n## 4. Heuristics\n\nAn interesting heuristic by Grant, 2019 allows one to start very close to random, but not quite. Using the same circuits as McClean et al., the authors propose a different initialization technique for the classical control parameters to avoid barren plateaus. The initialization technique starts some layers with totally random control parameters\u2014but, in the layers immediately following, choose parameters such that the initial transformation made by the first few layers is undone. The authors call this an *identity block*.\n\nThe advantage of this heuristic is that by changing just a single parameter, all other blocks outside of the current block will remain the identity\u2014and the gradient signal comes through much stronger than before. This allows the user to pick and choose which variables and blocks to modify to get a strong gradient signal. This heuristic does not prevent the user from falling in to a barren plateau during the training phase (and restricts a fully simultaneous update), it just guarantees that you can start outside of a plateau.\n\n### 4.1 New QNN construction\n\nNow construct a function to generate identity block QNNs. This implementation is slightly different than the one from the paper. For now, look at the behavior of the gradient of a single parameter so it is consistent with McClean et al, so some simplifications can be made.\n\nTo generate an identity block and train the model, generally you need $U1(\\theta_{1a}) U1(\\theta_{1b})^{\\dagger}$ and not $U1(\\theta_1) U1(\\theta_1)^{\\dagger}$. Initially $\\theta_{1a}$ and $\\theta_{1b}$ are the same angles but they are learned independently. Otherwise, you will always get the identity even after training. The choice for the number of identity blocks is empirical. The deeper the block, the smaller the variance in the middle of the block. But at the start and end of the block, the variance of the parameter gradients should be large. \n\n\n```python\ndef generate_identity_qnn(qubits, symbol, block_depth, total_depth):\n \"\"\"Generate random QNN's with the same structure from Grant et al.\"\"\"\n circuit = cirq.Circuit()\n\n # Generate initial block with symbol.\n prep_and_U = generate_random_qnn(qubits, symbol, block_depth)\n circuit += prep_and_U\n\n # Generate dagger of initial block without symbol.\n U_dagger = (prep_and_U[1:])**-1\n circuit += cirq.resolve_parameters(\n U_dagger, param_resolver={symbol: np.random.uniform() * 2 * np.pi})\n\n for d in range(total_depth - 1):\n # Get a random QNN.\n prep_and_U_circuit = generate_random_qnn(\n qubits,\n np.random.uniform() * 2 * np.pi, block_depth)\n\n # Remove the state-prep component\n U_circuit = prep_and_U_circuit[1:]\n\n # Add U\n circuit += U_circuit\n\n # Add U^dagger\n circuit += U_circuit**-1\n\n return circuit\n\n\ngenerate_identity_qnn(cirq.GridQubit.rect(1, 3), sympy.Symbol('theta'), 2, 2)\n```\n\n### 4.2 Comparison\n\nHere you can see that the heuristic does help to keep the variance of the gradient from vanishing as quickly:\n\n\n```python\nblock_depth = 10\ntotal_depth = 5\n\nheuristic_theta_var = []\n\nfor n in n_qubits:\n # Generate the identity block circuits and observable for the given n.\n qubits = cirq.GridQubit.rect(1, n)\n symbol = sympy.Symbol('theta')\n circuits = [\n generate_identity_qnn(qubits, symbol, block_depth, total_depth)\n for _ in range(n_circuits)\n ]\n op = cirq.Z(qubits[0]) * cirq.Z(qubits[1])\n heuristic_theta_var.append(process_batch(circuits, symbol, op))\n\nplt.semilogy(n_qubits, theta_var)\nplt.semilogy(n_qubits, heuristic_theta_var)\nplt.title('Heuristic vs. Random')\nplt.xlabel('n_qubits')\nplt.ylabel('$\\\\partial \\\\theta$ variance')\nplt.show()\n```\n\nThis is a great improvement in getting stronger gradient signals from (near) random QNNs.\n", "meta": {"hexsha": "2368cf4307f24a49cb9a459d5644c585202c0aaf", "size": 19573, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/tutorials/barren_plateaus.ipynb", "max_stars_repo_name": "abhinavsp0730/quantum", "max_stars_repo_head_hexsha": "cc08c9b824dfdde948ae11112564ea6da1ea7a04", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/tutorials/barren_plateaus.ipynb", "max_issues_repo_name": "abhinavsp0730/quantum", "max_issues_repo_head_hexsha": "cc08c9b824dfdde948ae11112564ea6da1ea7a04", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/tutorials/barren_plateaus.ipynb", "max_forks_repo_name": "abhinavsp0730/quantum", "max_forks_repo_head_hexsha": "cc08c9b824dfdde948ae11112564ea6da1ea7a04", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 37.7129094412, "max_line_length": 611, "alphanum_fraction": 0.5334900118, "converted": true, "num_tokens": 2673, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.607663184043154, "lm_q2_score": 0.6926419831347361, "lm_q1q2_score": 0.42089303287361834}} {"text": "_Lambda School Data Science \u2014\u00a0Linear Models_\n\n# Understanding Linear Regression\n\n#### Objectives\n- understand how ordinary least squares regression minimizes the sum of squared errors\n- understand how linear algebra can solve ordinary least squares regression\n- get and interpret coefficients of a linear model\n- visualize a line of best fit in 2D, and hyperplane in 3D\n- use regression metrics: MSE, RMSE, MAE, R^2\n\n#### Extra Links\n- [Statistics 101: Simple Linear Regression](https://www.youtube.com/watch?v=ZkjP5RJLQF4) (20 minute video)\n- [_An Introduction to Statistical Learning_](http://www-bcf.usc.edu/~gareth/ISL/ISLR%20Seventh%20Printing.pdf), Chapter 3.1, Simple Linear Regression, & 3.2, Multiple Linear Regression\n- Priceonomics, [The Discovery of Statistical Regression](https://priceonomics.com/the-discovery-of-statistical-regression/)\n- Priceonomics, [Why the Father of Modern Statistics Didn\u2019t Believe Smoking Caused Cancer](https://priceonomics.com/why-the-father-of-modern-statistics-didnt-believe/)\n- Harvard Business Review, [When to Act on a Correlation, and When Not To](https://hbr.org/2014/03/when-to-act-on-a-correlation-and-when-not-to)\n- [xkcd 552: Correlation](https://www.explainxkcd.com/wiki/index.php/552:_Correlation)\n- [xkcd 1725: Linear Regression](https://www.explainxkcd.com/wiki/index.php/1725:_Linear_Regression)\n\n## What is Linear Regression?\n\n[Linear Regression](https://en.wikipedia.org/wiki/Linear_regression) is a statistical model that seeks to describe the relationship between some y variable and one or more x variables. \n\nIn the simplest case, linear regression seeks to fit a straight line through a cloud of points. This line is referred to as the \"regression line\" or \"line of best fit.\" This line tries to summarize the relationship between our X and Y in a way that enables us to use the equation for that line to make predictions.\n\n\n```python\n%matplotlib inline\nimport seaborn as sns\nx = [10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5]\ny = [ 8, 6, 7, 8, 8, 9, 7, 4, 10, 4, 5]\nsns.regplot(x, y);\n```\n\n\n\n### Synonyms for \"y variable\" \n- Dependent Variable\n- Response Variable\n- Outcome Variable \n- Predicted Variable\n- Measured Variable\n- Explained Variable\n- Label\n- Target\n\n### Synonyms for \"x variable\"\n- Independent Variable\n- Explanatory Variable\n- Regressor\n- Covariate\n- Feature\n\n## Simple Linear Regresion\n\n#### Making Predictions\n\nSay that we were trying to create a model that captured the relationship between temperature outside and ice cream sales. In Machine Learning our goal is often different that of other flavors of Linear Regression Analysis, because we're trying to fit a model to this data with the intention of making **predictions** on new data (in the future) that we don't have yet.\n\n#### What are we trying to predict?\n\nSo if we had measured ice cream sales and the temprature outside on 11 different days, at the end of our modeling **what would be the thing that we would want to predict? - Ice Cream Sales or Temperature?**\n\nWe would probably want to be measuring temperature with the intention of using that to **forecast** ice cream sales. If we were able to successfully forecast ice cream sales from temperature, this might help us know beforehand how much ice cream to make or how many cones to buy or on which days to open our store, etc. Being able to make predictions accurately has a lot of business implications. This is why making accurate predictions is so valuable (And in large part is why data scientists are paid so well).\n\n#### y variable intuition\n\nWe want the thing that we're trying to predict to serve as our **y** variable. This is why it's sometimes called the \"predicted variable.\" We call it the \"dependent\" variable because our prediction for how much ice cream we're going to sell \"depends\" on the temperature outside. \n\n#### x variable intuition\n\nAll other variables that we use to predict our y variable (we're going to start off just using one) we call our **x** variables. These are called our \"independent\" variables because they don't *depend* on y, they \"explain\" y. Hence they are also referred to as our \"explanatory\" variables.\n\n## Example: Predict presidential election voting\n\n#### Douglas Hibbs, [Background Information on the \u2018Bread and Peace\u2019 Model of Voting in Postwar US Presidential Elections](https://douglas-hibbs.com/background-information-on-bread-and-peace-voting-in-us-presidential-elections/)\n\n> Aggregate two-party vote shares going to candidates of the party holding the presidency during the postwar era are well explained by just two fundamental determinants:\n\n> (1) Positively by weighted-average growth of per capita real disposable personal income over the term. \n> (2) Negatively by cumulative US military fatalities (scaled to population) owing to unprovoked, hostile deployments of American armed forces in foreign wars.\n\n#### Data sources\n- 1952-2012: Douglas Hibbs, [2014 lecture at Deakin University Melbourne](http://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf), Slide 40\n- 2016, Vote Share: [The American Presidency Project](https://www.presidency.ucsb.edu/statistics/elections)\n- 2016, Recent Growth in Personal Incomes: [The 2016 election economy: the \"Bread and Peace\" model final forecast](https://angrybearblog.com/2016/11/the-2016-election-economy-the-bread-and-peace-model-final-forecast.html)\n- 2016, US Military Fatalities: Assumption that Afghanistan War fatalities in 2012-16 occured at the same rate as 2008-12\n\n\n```python\nimport pandas as pd\n\ncolumns = ['year','inc_party_candidate','other_candidate','inc_party_vote']\n\ndata = [[1952,\"Stevenson\",\"Eisenhower\",44.6],\n [1956,\"Eisenhower\",\"Stevenson\",57.76],\n [1960,\"Nixon\",\"Kennedy\",49.91],\n [1964,\"Johnson\",\"Goldwater\",61.34],\n [1968,\"Humphrey\",\"Nixon\",49.60],\n [1972,\"Nixon\",\"McGovern\",61.79],\n [1976,\"Ford\",\"Carter\",48.95],\n [1980,\"Carter\",\"Reagan\",44.70],\n [1984,\"Reagan\",\"Mondale\",59.17],\n [1988,\"Bush, Sr.\",\"Dukakis\",53.94],\n [1992,\"Bush, Sr.\",\"Clinton\",46.55],\n [1996,\"Clinton\",\"Dole\",54.74],\n [2000,\"Gore\",\"Bush, Jr.\",50.27],\n [2004,\"Bush, Jr.\",\"Kerry\",51.24],\n [2008,\"McCain\",\"Obama\",46.32],\n [2012,\"Obama\",\"Romney\",52.00], \n [2016,\"Clinton\",\"Trump\",48.2]]\n \nvotes = pd.DataFrame(data=data, columns=columns)\n```\n\n\n```python\ncolumns = ['year','income_growth']\n\ndata = [[1952,2.40],\n [1956,2.89],\n [1960, .85],\n [1964,4.21],\n [1968,3.02],\n [1972,3.62],\n [1976,1.08],\n [1980,-.39],\n [1984,3.86],\n [1988,2.27],\n [1992, .38],\n [1996,1.04],\n [2000,2.36],\n [2004,1.72],\n [2008, .10],\n [2012, .95], \n [2016, .10]]\n \ngrowth = pd.DataFrame(data=data, columns=columns)\n```\n\n\n```python\n\"\"\"\nFatalities denotes the cumulative number of American military\nfatalities per millions of US population the in Korea, Vietnam,\nIraq and Afghanistan wars during the presidential terms\npreceding the 1952, 1964, 1968, 1976 and 2004, 2008 and\n2012 elections.\n\nhttp://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf\n\"\"\"\n\ncolumns = ['year','fatal_per_mil']\n\ndata = [[1952,190],\n [1956, 0],\n [1960, 0],\n [1964, 1],\n [1968,146],\n [1972, 0],\n [1976, 2],\n [1980, 0],\n [1984, 0],\n [1988, 0],\n [1992, 0],\n [1996, 0],\n [2000, 0],\n [2004, 4],\n [2008, 14],\n [2012, 5], \n [2016, 5]]\n \ndeaths = pd.DataFrame(data=data, columns=columns)\n```\n\n\n```python\ndf = votes.merge(growth).merge(deaths)\n```\n\n### Plot univariate correlations\n[Seaborn tutorial: Visualizing linear relationships](https://seaborn.pydata.org/tutorial/regression.html)\n\n\n```python\n%matplotlib inline\nimport seaborn as sns\n\ntarget = 'inc_party_vote'\nfeatures = ['income_growth', \n 'fatal_per_mil']\n\nfor feature in features:\n sns.lmplot(x=feature, y=target, data=df)\n```\n\nWe can see from the scatterplot that these data points seem to follow a somewhat linear relationship for the \"Average Recent Growth in Personal Incomes\" feature. This means that we could probably summarize their relationship well by fitting a line of best fit to these points. Lets do it.\n\n\n## The Equation for a Line\n\nA common equation for a line is:\n\n\\begin{align}\ny = mx + b\n\\end{align}\n\nWhere $m$ is the slope of our line and $b$ is the y-intercept. \n\nIf we want to plot a line through our cloud of points we figure out what these two values should be. Linear Regression seeks to **estimate** the slope and intercept values that describe a line that best fits the data points.\n\n## The Anatomy of Linear Regression\n\n- Intercept: The $b$ value in our line equation $y=mx+b$\n- Slope: The $m$ value in our line equation $y=mx+b$. These two values together define our regression line.\n- $\\hat{y}$ : A prediction\n- Line of Best Fit (Regression Line)\n- Predicted (fitted) Values: Points on our regression line\n- Observed Values: Points from our dataset\n- Error: The distance between predicted and observed values.\n\n\nOrdinary Least Squares Regression is a way to solve for $m$ and $b$.\n\nLet's start by seeing what would happen if we just guessed and checked some values for $m$ and $b$. \n\nWhat's the line of \"best\" fit look like? What's the error?\n\n\n\n```python\n# TODO\n\nx = df['income_growth'];\ny = df['inc_party_vote']\n\nm = 0;\nb = y.mean()\ny_pred = m*x + b\ny_pred\n```\n\n\n\n\n 0 51.828235\n 1 51.828235\n 2 51.828235\n 3 51.828235\n 4 51.828235\n 5 51.828235\n 6 51.828235\n 7 51.828235\n 8 51.828235\n 9 51.828235\n 10 51.828235\n 11 51.828235\n 12 51.828235\n 13 51.828235\n 14 51.828235\n 15 51.828235\n 16 51.828235\n Name: income_growth, dtype: float64\n\n\n\n\n```python\nimport matplotlib.pyplot as plt;\nfrom sklearn.metrics import mean_absolute_error, r2_score;\n\ndef plot_preds(x, y, y_pred):\n plt.scatter(x, y, label='y_true');\n plt.plot(x, y_pred, label='y_pred');\n plt.legend();\n \n mae = mean_absolute_error(y, y_pred);\n r2 = r2_score(y, y_pred);\n print(f'Mean Abosulte Error: {mae}');\n print(f'R^2 Score: {r2}');\n```\n\n\n```python\nplot_preds(x, y, y_pred);\n```\n\n\n```python\nm = 4.1;\nb = 45;\ny_pred = m*x + b;\nplot_preds(x, y, y_pred);\nplt.title(\"Guessing LR Values\");\n```\n\n## R Squared: $R^2$\n\nOne final attribute of linear regressions that we're going to talk about today is a measure of goodness of fit known as $R^2$ or R-squared. $R^2$ is a statistical measure of how close the data are fitted to our regression line. A helpful interpretation for the $R^2$ is the percentage of the dependent variable that is explained by the model.\n\nIn other words, the $R^2$ is the percentage of y that is explained by the x variables included in the model. For this reason the $R^2$ is also known as the \"coefficient of determination,\" because it explains how much of y is explained (or determined) by our x varaibles. We won't go into the calculation of $R^2$ today, just know that a higher $R^2$ percentage is nearly always better and indicates a model that fits the data more closely. \n\n## Residual Error \n\nThe residual error is the distance between points in our dataset and our regression line.\n\n\n```python\ndef regression_residuals(df, feature, target, m, b):\n x = df[feature]\n y = df[target]\n y_pred = m*x + b\n \n plt.scatter(x, y, label='y_true')\n plt.plot(x, y_pred, label='y_pred')\n plt.legend()\n \n # Plot residual errors\n for x, y1, y2 in zip(x, y, y_pred):\n plt.plot((x, x), (y1, y2), color='grey')\n \n mae = mean_absolute_error(y, y_pred) \n r2 = r2_score(y, y_pred)\n print('Mean Absolute Error:', mae)\n print('R^2:', r2)\n \nregression_residuals(df, feature='income_growth', \n target='inc_party_vote', m=3, b=46)\n```\n\n\n```python\nfrom ipywidgets import interact, fixed\n\ninteract(regression_residuals, \n df=fixed(df), \n feature=fixed('income_growth'), \n target=fixed('inc_party_vote'), \n m=(-10,10,0.5), \n b=(40,60,0.5));\n```\n\n\n interactive(children=(FloatSlider(value=0.0, description='m', max=10.0, min=-10.0, step=0.5), FloatSlider(valu\u2026\n\n\n## More Formal Notation\n\n\n\nWe have talked about a line of regression being represented like a regular line $y=mx+b$ but as we get to more complicated versions we're going to need to extend this equation. So lets establish the proper terminology.\n\n**X** - Independent Variable, predictor variable, explanatory variable, regressor, covariate\n\n**Y** - Response variable, predicted variable, measured vairable, explained variable, outcome variable\n\n$\\beta_0$ - \"Beta Naught\" or \"Beta Zero\", the intercept value. This is how much of y would exist if X were zero. This is sometimes represented by the letter \"a\" but I hate that. So it's \"Beta 0\" during my lecture.\n\n$\\beta_1$ - \"Beta One\" The primary coefficient of interest. This values is the slope of the line that is estimated by \"minimizing the sum of the squared errors/residuals\" - We'll get to that. \n\n$\\epsilon$ - \"Epsilon\" The \"error term\", random noise, things outside of our model that affect y.\n\n## Minimizing the Sum of the Squared Error\n\nThe most common method of estimating our $\\beta$ parameters is what's known as \"Ordinary Least Squares\" (OLS). (There are different methods of arriving at a line of best fit). OLS estimates the parameters that minimize the squared distance between each point in our dataset and our line of best fit. \n\n\\begin{align}\nSSE = \\sum(y_i - \\hat{y})^2\n\\end{align}\n\n\n\n```python\nfrom matplotlib.patches import Rectangle\nimport numpy as np\nfrom sklearn.metrics import mean_squared_error\n\ndef regression_squared_errors(df, feature, target, m, b):\n x = df[feature]\n y = df[target]\n y_pred = m*x + b\n \n fig = plt.figure(figsize=(7,7))\n ax = plt.axes()\n ax.scatter(x, y, label='y_true')\n ax.plot(x, y_pred, label='y_pred')\n ax.legend()\n \n # Plot square errors\n xmin, xmax = ax.get_xlim()\n ymin, ymax = ax.get_ylim()\n scale = (xmax-xmin)/(ymax-ymin)\n for x, y1, y2 in zip(x, y, y_pred):\n bottom_left = (x, min(y1, y2))\n height = abs(y1 - y2)\n width = height * scale\n ax.add_patch(Rectangle(xy=bottom_left, width=width, height=height, alpha=0.1))\n \n mse = mean_squared_error(y, y_pred)\n rmse = np.sqrt(mse)\n mae = mean_absolute_error(y, y_pred)\n r2 = r2_score(y, y_pred)\n print('Mean Squared Error:', mse)\n print('Root Mean Squared Error:', rmse)\n print('Mean Absolute Error:', mae)\n print('R^2:', r2)\n \nregression_squared_errors(df, feature='income_growth', \n target='inc_party_vote', m=3, b=46)\n```\n\n\n```python\ninteract(regression_squared_errors, \n df=fixed(df), \n feature=fixed('income_growth'), \n target=fixed('inc_party_vote'), \n m=(-10,10,0.5), \n b=(40,60,0.5));\n```\n\n\n interactive(children=(FloatSlider(value=0.0, description='m', max=10.0, min=-10.0, step=0.5), FloatSlider(valu\u2026\n\n\n## Hypotheses\n\n\n```python\nb = 46\nms = np.arange(-10,10,0.5)\nsses = []\nfeature = 'income_growth'\n\nfor m in ms:\n predictions = m * df[feature] + b\n errors = predictions - df[target]\n square_errors = errors ** 2\n sse = square_errors.sum()\n sses.append(sse)\n \nhypotheses = pd.DataFrame({'Slope': ms})\nhypotheses['Intercept'] = b\nhypotheses['Sum of Square Errors'] = sses\n\nhypotheses\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        SlopeInterceptSum of Square Errors
                                        0-10.04615215.58740
                                        1-9.54614095.52845
                                        2-9.04613018.88500
                                        3-8.54611985.65705
                                        4-8.04610995.84460
                                        5-7.54610049.44765
                                        6-7.0469146.46620
                                        7-6.5468286.90025
                                        8-6.0467470.74980
                                        9-5.5466698.01485
                                        10-5.0465968.69540
                                        11-4.5465282.79145
                                        12-4.0464640.30300
                                        13-3.5464041.23005
                                        14-3.0463485.57260
                                        15-2.5462973.33065
                                        16-2.0462504.50420
                                        17-1.5462079.09325
                                        18-1.0461697.09780
                                        19-0.5461358.51785
                                        200.0461063.35340
                                        210.546811.60445
                                        221.046603.27100
                                        231.546438.35305
                                        242.046316.85060
                                        252.546238.76365
                                        263.046204.09220
                                        273.546212.83625
                                        284.046264.99580
                                        294.546360.57085
                                        305.046499.56140
                                        315.546681.96745
                                        326.046907.78900
                                        336.5461177.02605
                                        347.0461489.67860
                                        357.5461845.74665
                                        368.0462245.23020
                                        378.5462688.12925
                                        389.0463174.44380
                                        399.5463704.17385
                                        \n
                                        \n\n\n\n\n```python\nhypotheses.plot(x='Slope', y='Sum of Square Errors', \n title=f'Intercept={b}');\n```\n\n## Scikit-learn\n\n#### Jake VanderPlas, [Python Data Science Handbook, Chapter 5.2, Introducing Scikit-Learn](https://jakevdp.github.io/PythonDataScienceHandbook/05.02-introducing-scikit-learn.html), Scikit-Learn's Estimator API\n\n> Most commonly, the steps in using the Scikit-Learn estimator API are as follows (we will step through a handful of detailed examples in the sections that follow).\n\n> 1. Choose a class of model by importing the appropriate estimator class from Scikit-Learn. \n> 2. Choose model hyperparameters by instantiating this class with desired values. \n> 3. Arrange data into a features matrix and target vector following the discussion above.\n> 4. Fit the model to your data by calling the `fit()` method of the model instance.\n> 5. Apply the Model to new data: For supervised learning, often we predict labels for unknown data using the `predict()` method.\n\n\n```python\n# TODO\n```\n\n\n```python\nfrom sklearn.linear_model import LinearRegression;\n\nmodel = LinearRegression();\nfeatures = ['income_growth'];\ntarget = 'inc_party_vote';\n\nx = df[features];\ny = df[target];\n\nmodel.fit(x, y);\n```\n\n\n```python\nmodel.intercept_, model.coef_;\n```\n\n\n```python\nmodel.predict([[1]]);\n```\n\n\n```python\nmodel.predict([[4]]);\n```\n\n\n```python\nmodel.predict([[2]]) - model.predict([[1]]);\n```\n\n\n```python\nx\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        income_growth
                                        02.40
                                        12.89
                                        20.85
                                        34.21
                                        43.02
                                        53.62
                                        61.08
                                        7-0.39
                                        83.86
                                        92.27
                                        100.38
                                        111.04
                                        122.36
                                        131.72
                                        140.10
                                        150.95
                                        160.10
                                        \n
                                        \n\n\n\n\n```python\nm = model.coef_[0];\nb = model.intercept_;\n\ny_pred = m*x + b;\n```\n\n\n```python\ny_pred = model.predict(x);\nplot_preds(x, y, y_pred);\n```\n\n## Linear Algebra!\n\nThe same result that is found by minimizing the sum of the squared errors can be also found through a linear algebra process known as the \"Least Squares Solution:\"\n\n\\begin{align}\n\\hat{\\beta} = (X^{T}X)^{-1}X^{T}y\n\\end{align}\n\nBefore we can work with this equation in its linear algebra form we have to understand how to set up the matrices that are involved in this equation. \n\n### The $\\beta$ vector\n\nThe $\\beta$ vector represents all the parameters that we are trying to estimate, our $y$ vector and $X$ matrix values are full of data from our dataset. The $\\beta$ vector holds the variables that we are solving for: $\\beta_0$ and $\\beta_1$\n\nNow that we have all of the necessary parts we can set them up in the following equation:\n\n\\begin{align}\ny = X \\beta + \\epsilon\n\\end{align}\n\nSince our $\\epsilon$ value represents **random** error we can assume that it will equal zero on average.\n\n\\begin{align}\ny = X \\beta\n\\end{align}\n\nThe objective now is to isolate the $\\beta$ matrix. We can do this by pre-multiplying both sides by \"X transpose\" $X^{T}$.\n\n\\begin{align}\nX^{T}y = X^{T}X \\beta\n\\end{align}\n\nSince anything times its transpose will result in a square matrix, if that matrix is then an invertible matrix, then we should be able to multiply both sides by its inverse to remove it from the right hand side. (We'll talk tomorrow about situations that could lead to $X^{T}X$ not being invertible.)\n\n\\begin{align}\n(X^{T}X)^{-1}X^{T}y = (X^{T}X)^{-1}X^{T}X \\beta\n\\end{align}\n\nSince any matrix multiplied by its inverse results in the identity matrix, and anything multiplied by the identity matrix is itself, we are left with only $\\beta$ on the right hand side:\n\n\\begin{align}\n(X^{T}X)^{-1}X^{T}y = \\hat{\\beta}\n\\end{align}\n\nWe will now call it \"beta hat\" $\\hat{\\beta}$ because it now represents our estimated values for $\\beta_0$ and $\\beta_1$\n\n### Lets calculate our $\\beta$ coefficients with numpy!\n\n\n```python\ndf[features].values.T\n```\n\n\n\n\n array([[ 2.4 , 2.89, 0.85, 4.21, 3.02, 3.62, 1.08, -0.39, 3.86,\n 2.27, 0.38, 1.04, 2.36, 1.72, 0.1 , 0.95, 0.1 ]])\n\n\n\n\n```python\nfrom statsmodels.api import add_constant;\n```\n\n\n```python\nx = add_constant(df[feature].values)\nprint('x')\nprint(x)\n\ny = df[target].values[:, np.newaxis]\nprint('y')\nprint(y)\n\nx_transpose = x.T\nprint('x Transpose')\nprint(x_transpose)\n\nx_transpose_x = x_transpose @ x\nprint('x Transpose x')\nprint(x_transpose_x)\n\nx_transpose_x_inverse = np.linalg.inv(x_transpose_x)\nprint('x Transpose x Inverse')\nprint(x_transpose_x_inverse)\n\nx_transpose_y = x_transpose @ y\nprint('x Transpose y')\nprint(x_transpose_y)\n\nbeta_hat = x_transpose_x_inverse @ x_transpose_y\nprint('Beta Hat')\nprint(beta_hat)\n```\n\n x\n [[ 1. 2.4 ]\n [ 1. 2.89]\n [ 1. 0.85]\n [ 1. 4.21]\n [ 1. 3.02]\n [ 1. 3.62]\n [ 1. 1.08]\n [ 1. -0.39]\n [ 1. 3.86]\n [ 1. 2.27]\n [ 1. 0.38]\n [ 1. 1.04]\n [ 1. 2.36]\n [ 1. 1.72]\n [ 1. 0.1 ]\n [ 1. 0.95]\n [ 1. 0.1 ]]\n y\n [[44.6 ]\n [57.76]\n [49.91]\n [61.34]\n [49.6 ]\n [61.79]\n [48.95]\n [44.7 ]\n [59.17]\n [53.94]\n [46.55]\n [54.74]\n [50.27]\n [51.24]\n [46.32]\n [52. ]\n [48.2 ]]\n x Transpose\n [[ 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1.\n 1. 1. 1. 1. 1. ]\n [ 2.4 2.89 0.85 4.21 3.02 3.62 1.08 -0.39 3.86 2.27 0.38 1.04\n 2.36 1.72 0.1 0.95 0.1 ]]\n x Transpose x\n [[17. 30.46 ]\n [30.46 86.831]]\n x Transpose x Inverse\n [[ 0.15835959 -0.05555197]\n [-0.05555197 0.03100405]]\n x Transpose y\n [[ 881.08 ]\n [1674.6167]]\n Beta Hat\n [[46.49920976]\n [ 2.97417709]]\n\n\n# Multiple Regression\n\nSimple or bivariate linear regression involves a single $x$ variable and a single $y$ variable. However, we can have many $x$ variables. A linear regression model that involves multiple x variables is known as **Multiple** Regression (NOT MULTIVARIATE).\n\n\\begin{align}\ny = \\beta_0 + \\beta_1 X_1 + \\beta_2 X_2 + ... + \\beta_n X_n + \\epsilon\n\\end{align}\n\n\n```python\n# TODO\n\nmodel = LinearRegression();\nfeatures = ['income_growth', 'fatal_per_mil'];\n\ntarget = 'inc_party_vote';\n\nx = df[features];\ny = df[target];\n\nmodel.fit(x, y)\n```\n\n\n\n\n LinearRegression(copy_X=True, fit_intercept=True, n_jobs=None,\n normalize=False)\n\n\n\n\n```python\npd.Series(model.coef_, features)\n```\n\n\n\n\n income_growth 3.406214\n fatal_per_mil -0.053752\n dtype: float64\n\n\n\n## Visualize hyperplane of best fit in 3D\n\n\n```python\n# https://stackoverflow.com/a/47230966\n# Plotly notebook mode with google colaboratory\n# You need to define this function\n# And call it in each offline plotting cell\n\ndef configure_plotly_browser_state():\n import IPython\n display(IPython.core.display.HTML('''\n \n \n '''))\n```\n\n\n```python\nimport itertools\nimport plotly.graph_objs as go\nfrom plotly.offline import init_notebook_mode, iplot\ninit_notebook_mode(connected=True)\n\ndef viz3D(fitted_model, X, features, target='', num=100):\n \"\"\"\n Visualize model predictions in 3D, for regression model fit on 2 features\n \n Parameters\n ----------\n fitted_model : scikit-learn model, already fitted\n X : pandas dataframe, which was used to fit model\n features : list of strings, column names of the 2 features\n target : string, name of target\n num : int, number of grid points for each feature\n \n References\n ----------\n https://plot.ly/python/3d-charts/\n \"\"\"\n feature1, feature2 = features\n min1, max1 = X[feature1].min(), X[feature1].max()\n min2, max2 = X[feature2].min(), X[feature2].max()\n x1 = np.linspace(min1, max1, num)\n x2 = np.linspace(min2, max2, num)\n combos = list(itertools.product(x1, x2))\n Z = fitted_model.predict(combos).reshape(num, num)\n \n configure_plotly_browser_state()\n data = [go.Surface(x=x1, y=x2, z=Z)]\n layout = go.Layout(\n scene={'xaxis': {'title': feature1, 'range': [min1,max1], 'showticklabels': True}, \n 'yaxis': {'title': feature2, 'range': [min2,max2], 'showticklabels': True}, \n 'zaxis': {'title': target, 'showticklabels': True}}, \n )\n fig = go.Figure(data=data, layout=layout)\n iplot(fig)\n```\n\n\n\n\n\n\n\n```python\n# TODO\n```\n\n## Dimensionality in Linear Regression\n\nMuliple Regression is simply an extension of the bivariate case. The reason why we see the bivariate case demonstrated so often is simply because it's easier to graph and all of the intuition from the bivariate case is the same as we keep on adding explanatory variables.\n\nAs we increase the number of $x$ values in our model we are simply fitting a n-1-dimensional plane to an n-dimensional cloud of points within an n-dimensional hypercube. \n\n## Interpreting Coefficients\n\nOne of Linear Regression's strengths is that the parameters of the model (coefficients) are readily interpretable and useful. Not only do they describe the relationship between x and y but they put a number on just how much x is associated with y. We should be careful to not speak about this relationshiop in terms of causality because these coefficients are in fact correlative measures. We would need a host of additional techniques in order to estimate a causal effect using linear regression (econometrics).\n\n\\begin{align}\n\\hat{\\beta} = \\frac{Cov(x,y)}{Var(y)}\n\\end{align}\n\n## Why is Linear Regression so Important?\n\n### Popularity \n\nLinear Regression is an extremely popular technique that every data scientist **needs** to understand. It's not the most advanced technique and there are supervised learning techniques that will obtain a higher accuracy, but where it lacks in accuracy it makes up for it in interpretability and simplicity.\n\n### Interpretability\n\nFew other models possess coefficients that are so directly linked to their variables with a such a clear interpretation. Tomorrow we're going to learn about ways to make them even easier to interpret.\n\n### Simplicity\n\nA linear regression model can be communicated just by writing out its equation. It's kind of incredible that such high dimensional relationships can be described from just a linear combination of variables and coefficients. \n\n# Assignment\n- Continue to predict New York City apartment rents. This is your last assignment with this dataset.\n- You may select any number of features. You are encouraged to engineer new features.\n- Get and plot your model's coefficients.\n- Report your Root Mean Squared Error, Mean Absolute Error, and R^2 Score, for your Train and Test sets. Share your scores with your cohort on Slack!\n- Fit a model with 2 features, and visualize the plane of best fit in 3D.\n- Commit your notebook to your fork of the repo.\n\n## Stretch Goals\n\nStudy more about Linear Regression. Here are two helpful links. If you find more links, share your favorites with your cohort on Slack.\n\n1. Watch this 20 minute video that just hit 1 million views: Brandon Foltz, Statistics 101: Simple Linear Regression (https://www.youtube.com/watch?v=ZkjP5RJLQF4)\n2. Skim _An Introduction to Statistical Learning_, Chapter 3.1, Simple Linear Regression, & Chapter 3.2, Multiple Linear Regression (http://www-bcf.usc.edu/~gareth/ISL/ISLR%20Seventh%20Printing.pdf)\n\nIn your 3D visualization, can you include the actual datapoints, like in [this notebook](https://nbviewer.jupyter.org/urls/s3.amazonaws.com/datarobotblog/notebooks/multiple_regression_in_python.ipynb)? Can you also include the residual lines from the datapoints to the plane of the best fit, like in _An Introduction to Statistical Learning?_ This would be hard to do, but awesome!\n\n\nCan you get creative with feature engineering? Share with your cohort on Slack. We mentioned some feature ideas at the end of last lesson, but didn't demonstrate how to engineer them. So here are some example solutions:\n\n```python\n# Does apartment have a non-empty description?\ndf['description'] = df['description'].str.strip().fillna('')\ndf['has_description'] = df['description'] != ''\n\n# How long is the description?\ndf['description_length'] = df['description'].str.len()\n\n# How many total perks does each apartment have?\nperk_cols = ['elevator', 'cats_allowed', 'hardwood_floors', 'dogs_allowed',\n 'doorman', 'dishwasher', 'no_fee', 'laundry_in_building',\n 'fitness_center', 'pre-war', 'laundry_in_unit', 'roof_deck',\n 'outdoor_space', 'dining_room', 'high_speed_internet', 'balcony',\n 'swimming_pool', 'new_construction', 'exclusive', 'terrace', \n 'loft', 'garden_patio', 'common_outdoor_space', \n 'wheelchair_access']\ndf['perk_count'] = df[perk_cols].sum(axis=1)\n\n# Are pets allowed?\ndf['pets_allowed'] = (df['cats_allowed']==1) | (df['dogs_allowed']==1)\n```\n\n\n\n```python\nLOCAL = '../data/nyc/nyc-rent-2016.csv'\nWEB = 'https://raw.githubusercontent.com/LambdaSchool/DS-Unit-2-Linear-Models/master/data/nyc/nyc-rent-2016.csv'\n\nimport pandas as pd\ndf = pd.read_csv(LOCAL)\nassert df.shape == (48300, 34)\ndf.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        bathroomsbedroomscreateddescriptiondisplay_addresslatitudelongitudepricestreet_addressinterest_level...high_speed_internetbalconyswimming_poolnew_constructionexclusiveterraceloftgarden_patiocommon_outdoor_spacewheelchair_access
                                        01.532016-06-24 07:54:24A Brand New 3 Bedroom 1.5 bath ApartmentEnjoy ...Metropolitan Avenue40.7145-73.94253000792 Metropolitan Avenuemedium...0000000000
                                        11.022016-06-12 12:19:27Columbus Avenue40.7947-73.96675465808 Columbus Avenuelow...0000000000
                                        21.012016-04-17 03:26:41Top Top West Village location, beautiful Pre-w...W 13 Street40.7388-74.00182850241 W 13 Streethigh...0000000000
                                        31.012016-04-18 02:22:02Building Amenities - Garage - Garden - fitness...East 49th Street40.7539-73.96773275333 East 49th Streetlow...0000000000
                                        41.042016-04-28 01:32:41Beautifully renovated 3 bedroom flex 4 bedroom...West 143rd Street40.8241-73.94933350500 West 143rd Streetlow...0000000000
                                        \n

                                        5 rows \u00d7 34 columns

                                        \n
                                        \n\n\n\n\n```python\ndf['created'] = pd.to_datetime(df['created'], infer_datetime_format=True);\ndf['month'] = df['created'].dt.month;\ndf.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        bathroomsbedroomscreateddescriptiondisplay_addresslatitudelongitudepricestreet_addressinterest_level...balconyswimming_poolnew_constructionexclusiveterraceloftgarden_patiocommon_outdoor_spacewheelchair_accessmonth
                                        01.532016-06-24 07:54:24A Brand New 3 Bedroom 1.5 bath ApartmentEnjoy ...Metropolitan Avenue40.7145-73.94253000792 Metropolitan Avenuemedium...0000000006
                                        11.022016-06-12 12:19:27Columbus Avenue40.7947-73.96675465808 Columbus Avenuelow...0000000006
                                        21.012016-04-17 03:26:41Top Top West Village location, beautiful Pre-w...W 13 Street40.7388-74.00182850241 W 13 Streethigh...0000000004
                                        31.012016-04-18 02:22:02Building Amenities - Garage - Garden - fitness...East 49th Street40.7539-73.96773275333 East 49th Streetlow...0000000004
                                        41.042016-04-28 01:32:41Beautifully renovated 3 bedroom flex 4 bedroom...West 143rd Street40.8241-73.94933350500 West 143rd Streetlow...0000000004
                                        \n

                                        5 rows \u00d7 35 columns

                                        \n
                                        \n\n\n\n\n```python\ntrain = df[df['month'] < 6];\ntest = df[df['month'] == 6];\nassert df.shape[0] == train.shape[0] + test.shape[0]\n```\n\n\n```python\ndf.columns\n```\n\n\n\n\n Index(['bathrooms', 'bedrooms', 'created', 'description', 'display_address',\n 'latitude', 'longitude', 'price', 'street_address', 'interest_level',\n 'elevator', 'cats_allowed', 'hardwood_floors', 'dogs_allowed',\n 'doorman', 'dishwasher', 'no_fee', 'laundry_in_building',\n 'fitness_center', 'pre-war', 'laundry_in_unit', 'roof_deck',\n 'outdoor_space', 'dining_room', 'high_speed_internet', 'balcony',\n 'swimming_pool', 'new_construction', 'exclusive', 'terrace', 'loft',\n 'garden_patio', 'common_outdoor_space', 'wheelchair_access'],\n dtype='object')\n\n\n\n\n```python\nmain_customer_requirements = ['bathrooms', 'bedrooms', 'dishwasher', 'laundry_in_building', 'laundry_in_unit', 'high_speed_internet'];\ntarget = 'price';\n\nx_train = train[main_customer_requirements];\ny_train = train[target];\n\nx_test = test[main_customer_requirements];\ny_test = test[target];\n```\n\n\n```python\ncustomerModel = LinearRegression();\ncustomerModel.fit(x_train, y_train)\n```\n\n\n\n\n LinearRegression(copy_X=True, fit_intercept=True, n_jobs=None,\n normalize=False)\n\n\n\n\n```python\ny_pred = customerModel.predict(x_test);\ny_pred.shape\n```\n\n\n\n\n (16785,)\n\n\n\n\n```python\ncustomerModel.coef_\n```\n\n\n\n\n array([1421.83475493, 408.66550573, 237.13122324, 59.02091715,\n 378.02911758, -13.25821541])\n\n\n\n\n```python\ncustomerModel.intercept_\n```\n\n\n\n\n 977.3088601415961\n\n\n\n\n```python\nfrom sklearn.metrics import mean_squared_error;\nfrom sklearn.metrics import mean_absolute_error;\nfrom sklearn.metrics import r2_score;\nimport numpy as np\n\nsqrt = np.sqrt(mean_squared_error(y_test, y_pred))\nmae = mean_absolute_error(y_test, y_pred)\nr2 = r2_score(y_test, y_pred)\nrmse, mae, r2\n```\n\n\n\n\n (741.3110985694503, 741.3110985694503, 0.47948851127734315)\n\n\n", "meta": {"hexsha": "32cff5bdcebb59229dd2ddd7c5649cc55f8814e7", "size": 219237, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "module3-understanding-linear-regression/understanding_linear_regression.ipynb", "max_stars_repo_name": "ash12hub/DS-Unit-2-Linear-Models", "max_stars_repo_head_hexsha": "acdcbc10bca416dcb931694fa4ba9ff0247cfb74", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "module3-understanding-linear-regression/understanding_linear_regression.ipynb", "max_issues_repo_name": "ash12hub/DS-Unit-2-Linear-Models", "max_issues_repo_head_hexsha": "acdcbc10bca416dcb931694fa4ba9ff0247cfb74", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "module3-understanding-linear-regression/understanding_linear_regression.ipynb", "max_forks_repo_name": "ash12hub/DS-Unit-2-Linear-Models", "max_forks_repo_head_hexsha": "acdcbc10bca416dcb931694fa4ba9ff0247cfb74", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 85.0085304382, "max_line_length": 20940, "alphanum_fraction": 0.7730583797, "converted": true, "num_tokens": 13826, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.607663184043154, "lm_q2_score": 0.6926419831347361, "lm_q1q2_score": 0.42089303287361834}} {"text": "\n\n# Lab 7: Qubit Spectroscopy\n\nIn this lab, you will take what you learned about the interactions between qubits and resonators to perform transmon spectroscopy with the pulse simulator.\n\n### Installing Necessary Packages\nBefore we begin, you will need to install some prerequisites into your environment. Run the cell below to complete these installations. At the end, the cell outputs will be cleared.\n\n\n```python\n!pip install -U -r grading_tools/requirements.txt\n\nfrom IPython.display import clear_output\nclear_output()\n```\n\n Installing backend dependencies: finished with status 'done'\n Preparing wheel metadata: started\n Preparing wheel metadata: finished with status 'done'\n Requirement already up-to-date: qiskit==0.19 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from -r grading_tools/requirements.txt (line 1)) (0.19.0)\n Requirement already up-to-date: scipy==1.5.1 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from -r grading_tools/requirements.txt (line 2)) (1.5.1)\n Requirement already up-to-date: sympy==1.6.1 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from -r grading_tools/requirements.txt (line 3)) (1.6.1)\n Requirement already satisfied, skipping upgrade: cython>=0.27.1 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (0.29.21)\n Requirement already satisfied, skipping upgrade: numpy>=1.16.3; python_version > \"3.5\" in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (1.19.0)\n Requirement already satisfied, skipping upgrade: pybind11>=2.4 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (2.5.0)\n Requirement already satisfied, skipping upgrade: qiskit-terra>=0.12.0 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (0.14.0)\n Requirement already satisfied, skipping upgrade: qiskit-aqua==0.7.0 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (0.7.0)\n Requirement already satisfied, skipping upgrade: qiskit-ibmq-provider==0.7.0 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (0.7.0)\n Requirement already satisfied, skipping upgrade: qiskit-ignis==0.3.0 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (0.3.0)\n Requirement already satisfied, skipping upgrade: mpmath>=0.19 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from sympy==1.6.1->-r grading_tools/requirements.txt (line 3)) (1.1.0)\n Requirement already satisfied, skipping upgrade: fastjsonschema>=2.10 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-terra>=0.12.0->qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (2.14.4)\n Requirement already satisfied, skipping upgrade: python-constraint>=1.4 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-terra>=0.12.0->qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (1.4.0)\n Requirement already satisfied, skipping upgrade: marshmallow-polyfield<6,>=5.7 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-terra>=0.12.0->qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (5.9)\n Requirement already satisfied, skipping upgrade: networkx>=2.2; python_version > \"3.5\" in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-terra>=0.12.0->qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (2.4)\n Requirement already satisfied, skipping upgrade: retworkx>=0.3.2 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-terra>=0.12.0->qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (0.4.0)\n Requirement already satisfied, skipping upgrade: psutil>=5 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-terra>=0.12.0->qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (5.7.2)\n Requirement already satisfied, skipping upgrade: dill>=0.3 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-terra>=0.12.0->qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (0.3.2)\n Requirement already satisfied, skipping upgrade: ply>=3.10 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-terra>=0.12.0->qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (3.11)\n Requirement already satisfied, skipping upgrade: jsonschema>=2.6 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-terra>=0.12.0->qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (3.2.0)\n Requirement already satisfied, skipping upgrade: python-dateutil>=2.8.0 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-terra>=0.12.0->qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (2.8.1)\n Requirement already satisfied, skipping upgrade: marshmallow<4,>=3 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-terra>=0.12.0->qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (3.7.1)\n\n\n \n -- The C compiler identification is unknown\n CMake Error at CMakeLists.txt:3 (ENABLE_LANGUAGE):\n The CMAKE_C_COMPILER:\n \n cl\n \n is not a full path and was not found in the PATH.\n \n To use the NMake generator with Visual C++, cmake must be run from a shell\n that can use the compiler cl from the command line. This environment is\n unable to invoke the cl compiler. To fix this problem, run cmake from the\n Visual Studio Command Prompt (vcvarsall.bat).\n \n Tell CMake where to find the compiler by setting either the environment\n variable \"CC\" or the CMake cache entry CMAKE_C_COMPILER to the full path to\n the compiler, or to the compiler name if it is in the PATH.\n \n \n -- Configuring incomplete, errors occurred!\n See also \"C:/Users/Hernan/AppData/Local/Temp/pip-req-build-7yy62sb7/_cmake_test_compile/build/CMakeFiles/CMakeOutput.log\".\n See also \"C:/Users/Hernan/AppData/Local/Temp/pip-req-build-7yy62sb7/_cmake_test_compile/build/CMakeFiles/CMakeError.log\".\n Not searching for unused variables given on the command line.\n -- The C compiler identification is unknown\n CMake Error at CMakeLists.txt:3 (ENABLE_LANGUAGE):\n The CMAKE_C_COMPILER:\n \n cl\n \n is not a full path and was not found in the PATH.\n \n To use the JOM generator with Visual C++, cmake must be run from a shell\n that can use the compiler cl from the command line. This environment is\n unable to invoke the cl compiler. To fix this problem, run cmake from the\n Visual Studio Command Prompt (vcvarsall.bat).\n \n Tell CMake where to find the compiler by setting either the environment\n variable \"CC\" or the CMake cache entry CMAKE_C_COMPILER to the full path to\n the compiler, or to the compiler name if it is in the PATH.\n \n \n -- Configuring incomplete, errors occurred!\n See also \"C:/Users/Hernan/AppData/Local/Temp/pip-req-build-7yy62sb7/_cmake_test_compile/build/CMakeFiles/CMakeOutput.log\".\n See also \"C:/Users/Hernan/AppData/Local/Temp/pip-req-build-7yy62sb7/_cmake_test_compile/build/CMakeFiles/CMakeError.log\".\n \n \n --------------------------------------------------------------------------------\n -- Trying \"Ninja (Visual Studio 15 2017 Win64 v141)\" generator\n --------------------------------\n ---------------------------\n ----------------------\n -----------------\n ------------\n -------\n --\n --\n -------\n ------------\n\n Requirement already satisfied, skipping upgrade: fastdtw in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-aqua==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (0.3.4)\n Requirement already satisfied, skipping upgrade: setuptools>=40.1.0 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-aqua==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (49.2.0.post20200714)\n Requirement already satisfied, skipping upgrade: dlx in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-aqua==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (1.0.4)\n Requirement already satisfied, skipping upgrade: h5py in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-aqua==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (2.10.0)\n Requirement already satisfied, skipping upgrade: scikit-learn>=0.20.0 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-aqua==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (0.23.1)\n Requirement already satisfied, skipping upgrade: quandl in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-aqua==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (3.5.1)\n Requirement already satisfied, skipping upgrade: docplex in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-aqua==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (2.15.194)\n Requirement already satisfied, skipping upgrade: websockets<8,>=7 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-ibmq-provider==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (7.0)\n Requirement already satisfied, skipping upgrade: urllib3>=1.21.1 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-ibmq-provider==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (1.25.9)\n Requirement already satisfied, skipping upgrade: requests-ntlm>=1.1.0 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-ibmq-provider==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (1.1.0)\n Requirement already satisfied, skipping upgrade: requests>=2.19 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-ibmq-provider==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (2.24.0)\n Requirement already satisfied, skipping upgrade: nest-asyncio!=1.1.0,>=1.0.0 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from qiskit-ibmq-provider==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (1.4.0)\n Requirement already satisfied, skipping upgrade: decorator>=4.3.0 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from networkx>=2.2; python_version > \"3.5\"->qiskit-terra>=0.12.0->qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (4.4.2)\n Requirement already satisfied, skipping upgrade: pyrsistent>=0.14.0 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from jsonschema>=2.6->qiskit-terra>=0.12.0->qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (0.16.0)\n Requirement already satisfied, skipping upgrade: attrs>=17.4.0 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from jsonschema>=2.6->qiskit-terra>=0.12.0->qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (19.3.0)\n Requirement already satisfied, skipping upgrade: six>=1.11.0 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from jsonschema>=2.6->qiskit-terra>=0.12.0->qiskit-aer==0.6.0->-r grading_tools/requirements.txt (line 4)) (1.15.0)\n Requirement already satisfied, skipping upgrade: threadpoolctl>=2.0.0 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from scikit-learn>=0.20.0->qiskit-aqua==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (2.1.0)\n Requirement already satisfied, skipping upgrade: joblib>=0.11 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from scikit-learn>=0.20.0->qiskit-aqua==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (0.16.0)\n Requirement already satisfied, skipping upgrade: inflection>=0.3.1 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from quandl->qiskit-aqua==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (0.5.0)\n Requirement already satisfied, skipping upgrade: pandas>=0.14 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from quandl->qiskit-aqua==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (1.0.5)\n Requirement already satisfied, skipping upgrade: more-itertools in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from quandl->qiskit-aqua==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (8.4.0)\n Requirement already satisfied, skipping upgrade: 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satisfied, skipping upgrade: idna<3,>=2.5 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from requests>=2.19->qiskit-ibmq-provider==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (2.10)\n Requirement already satisfied, skipping upgrade: pytz>=2017.2 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from pandas>=0.14->quandl->qiskit-aqua==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (2020.1)\n Requirement already satisfied, skipping upgrade: cffi!=1.11.3,>=1.8 in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from cryptography>=1.3->requests-ntlm>=1.1.0->qiskit-ibmq-provider==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (1.14.0)\n Requirement already satisfied, skipping upgrade: pycparser in c:\\users\\hernan\\anaconda3\\envs\\qiskit\\lib\\site-packages (from cffi!=1.11.3,>=1.8->cryptography>=1.3->requests-ntlm>=1.1.0->qiskit-ibmq-provider==0.7.0->qiskit==0.19->-r grading_tools/requirements.txt (line 1)) (2.20)\n Building wheels for collected packages: qiskit-aer\n Building wheel for qiskit-aer (PEP 517): started\n Building wheel for qiskit-aer (PEP 517): finished with status 'error'\n Failed to build qiskit-aer\n\n\n \n -----------------\n ----------------------\n ---------------------------\n --------------------------------\n -- Trying \"Ninja (Visual Studio 15 2017 Win64 v141)\" generator - failure\n --------------------------------------------------------------------------------\n \n \n \n --------------------------------------------------------------------------------\n -- Trying \"Visual Studio 15 2017 Win64 v141\" generator\n --------------------------------\n ---------------------------\n ----------------------\n -----------------\n ------------\n -------\n --\n --\n -------\n ------------\n -----------------\n ----------------------\n ---------------------------\n --------------------------------\n -- Trying \"Visual Studio 15 2017 Win64 v141\" generator - failure\n --------------------------------------------------------------------------------\n \n \n \n --------------------------------------------------------------------------------\n -- Trying \"NMake Makefiles (Visual Studio 15 2017 Win64 v141)\" generator\n --------------------------------\n ---------------------------\n ----------------------\n -----------------\n ------------\n -------\n --\n --\n -------\n ------------\n -----------------\n ----------------------\n ---------------------------\n --------------------------------\n -- Trying \"NMake Makefiles (Visual Studio 15 2017 Win64 v141)\" generator - failure\n --------------------------------------------------------------------------------\n \n \n \n --------------------------------------------------------------------------------\n -- Trying \"NMake Makefiles JOM (Visual Studio 15 2017 Win64 v141)\" generator\n --------------------------------\n ---------------------------\n ----------------------\n -----------------\n ------------\n -------\n --\n --\n -------\n ------------\n -----------------\n ----------------------\n ---------------------------\n --------------------------------\n -- Trying \"NMake Makefiles JOM (Visual Studio 15 2017 Win64 v141)\" generator - failure\n --------------------------------------------------------------------------------\n \n ********************************************************************************\n scikit-build could not get a working generator for your system. Aborting build.\n \n Building windows wheels for Python 3.8 requires Microsoft Visual Studio 2017.\n Get it with \"Visual Studio 2017\":\n \n https://visualstudio.microsoft.com/vs/\n \n ********************************************************************************\n ----------------------------------------\n ERROR: Failed building wheel for qiskit-aer\n\n\n## Simulating the Transmon as a Duffing Oscillator\n\nAs you learned in Lecture 6, the transmon can be understood as a Duffing oscillator specified by a frequency $\\nu$, anharmonicity $\\alpha$, and drive strength $r$, which results in the Hamiltonian\n$$\n \\hat{H}_{\\rm Duff}/\\hbar = 2\\pi\\nu a^\\dagger a + \\pi \\alpha a^\\dagger a(a^\\dagger a - 1) + 2 \\pi r (a + a^\\dagger) \\times D(t),\n$$\n\nwhere $D(t)$ is the signal on the drive channel for the qubit, and $a^\\dagger$ and $a$ are, respectively, the creation and annihilation operators for the qubit. Note that the drive strength $r$ sets the scaling of the control term, with $D(t)$ assumed to be a complex and unitless number satisfying $|D(t)| \\leq 1$. \n\n## Qiskit Pulse Overview\n\nAs a brief overview, Qiskit Pulse schedules (experiments) consist of Instructions (i.e., Play) acting on Channels (i.e., the drive channel). Here is a summary table of available Instructions and Channels:\n\n\n\nFor more detail, this table summarizes the interaction of the channels with the actual quantum hardware:\n\n\n\nHowever, we find it is more instructive to begin with guided programming in Pulse. Below you will learn how to create pulses, schedules, and run experiments on a simulator. These lessons can be immediately applied to actual pulse-enabled quantum hardware, in particular [`ibmq_armonk`](https://www.ibm.com/blogs/research/2019/12/qiskit-openpulse/).\n\n## Let's get started!\n\nIn most of the cells below, nothing needs to be modified. **However, you will need to execute the cells by pressing `shift+Enter` in each code block**. In order to keep things tidy and focus on the important aspects of Qiskit Pulse, the following cells make use of methods from the `helper` module. For the gory details, please refer back to the [Lab 7 notebook](lab7-jc-spect-readout.ipynb). Just as in Lab 6, before coming to the discussion of **Sideband Modulation**, the following code blocks\n\n- create backend pulse simulator and instantiate the transmon as a Duffing oscillator of frequency $\\sim 5$ GHz\n- import libraries for numerics and visualization, and define helpful constants\n- create the channels for the pulse schedule and define measurment schedule (we will only work with the drive channel)\n\n\n```python\n# our backend is the Pulse Simulator\nfrom resources import helper\nfrom qiskit.providers.aer import PulseSimulator\nbackend_sim = PulseSimulator()\n\n# sample duration for pulse instructions \ndt = 1e-9\n\n# create the model\nduffing_model = helper.get_transmon(dt)\n\n# get qubit frequency from Duffing model\nqubit_lo_freq = duffing_model.hamiltonian.get_qubit_lo_from_drift()\n```\n\n ERROR: Could not build wheels for qiskit-aer which use PEP 517 and cannot be installed directly\n\n\n\n```python\nimport numpy as np\n\n# visualization tools\nimport matplotlib.pyplot as plt\nplt.style.use('dark_background')\n\n# unit conversion factors -> all backend properties returned in SI (Hz, sec, etc)\nGHz = 1.0e9 # Gigahertz\nMHz = 1.0e6 # Megahertz\nkHz = 1.0e3 # kilohertz\nus = 1.0e-6 # microseconds\nns = 1.0e-9 # nanoseconds\n```\n\n### Instantiate channels and create measurement schedule\n\nWe will use the same measurement schedule throughout, whereas the drive schedules will vary. This must be built for the simulator, for a real backend we can ask for its default measurement pulse.\n\n\n```python\nfrom qiskit import pulse\nfrom qiskit.pulse import Play, Acquire\nfrom qiskit.pulse.pulse_lib import GaussianSquare\n\n# qubit to be used throughout the notebook\nqubit = 0\n\n### Collect the necessary channels\ndrive_chan = pulse.DriveChannel(qubit)\nmeas_chan = pulse.MeasureChannel(qubit)\nacq_chan = pulse.AcquireChannel(qubit)\n\n# Construct a measurement schedule and add it to an InstructionScheduleMap\nmeas_samples = 1200\nmeas_pulse = GaussianSquare(duration=meas_samples, amp=0.025, sigma=4, width=1150)\nmeasure_sched = Play(meas_pulse, meas_chan) | Acquire(meas_samples, acq_chan, pulse.MemorySlot(qubit))\n\ninst_map = pulse.InstructionScheduleMap()\ninst_map.add('measure', [qubit], measure_sched)\n\n# save the measurement/acquire pulse for later\nmeasure = inst_map.get('measure', qubits=[qubit])\n```\n\n## Sideband Modulation\n\nUnlike the case of running on an actual device, with the simulator we can only set the (local) oscillator frequency of the drive, $f_{\\rm LO}$, to a single value. In order to sweep frequencies to perform spectroscopy, we use a trick called *sideband modulation*, where we modulate our spectroscopy pulse by a sideband frequency $f_{\\rm SB}$ so that the pulse applied to the qubit is of (radio) frequency\n\n$$ f_{\\rm RF} = f_{\\rm LO} + f_{\\rm SB}. $$\n\nThis is achieved by multiplying each sample amplitude by a complex exponential \n\n$$ d_j^{\\rm SB} = \\sum_j e^{2\\pi f_{\\rm SB} t_j} d_j $$\n\nbut we will tuck the details away in the `helper` module. The important thing is that we must apply the sideband for each pulse in order to change its frequency. \n\nNow, instead of `assemble`'ing a single schedule with an array of schedule LO's as, we will create a schedule of the same pulse *sidebanded* by an array of sideband frequecies at a fixed LO frequency. Since we are now considering a transmon, we have multiple energy levels we can perform spectroscopy on. We will being with spectroscopy of the $|0\\rangle \\to |1\\rangle$ transition, which is the one used as the qubit, often called the *computational basis*.\n\n\n```python\nfrom qiskit.pulse import pulse_lib\n\n# the same spect pulse used in every schedule\ndrive_amp = 0.9\ndrive_sigma = 16\ndrive_duration = 128\nspec_pulse = pulse_lib.gaussian(duration=drive_duration, amp=drive_amp, \n sigma=drive_sigma, name=f\"Spec drive amplitude = {drive_amp}\")\n\n# Construct an np array of the frequencies for our experiment\nspec_freqs_GHz = np.arange(5.0, 5.2, 0.005)\n\n# Create the base schedule\n# Start with drive pulse acting on the drive channel\nspec_schedules = []\nfor freq in spec_freqs_GHz:\n sb_spec_pulse = helper.apply_sideband(spec_pulse, qubit_lo_freq[0]-freq*GHz, dt)\n \n spec_schedule = pulse.Schedule(name='SB Frequency = {}'.format(freq))\n spec_schedule += Play(sb_spec_pulse, drive_chan)\n # The left shift `<<` is special syntax meaning to shift the start time of the schedule by some duration\n spec_schedule += measure << spec_schedule.duration\n spec_schedules.append(spec_schedule)\n```\n\n\n```python\nspec_schedules[0].draw()\n```\n\n\n```python\nfrom qiskit import assemble\n\n# assemble the schedules into a Qobj\nspec01_qobj = assemble(**helper.get_params('spec01', globals()))\n```\n\n\n```python\n# run the simulation\nspec01_result = backend_sim.run(spec01_qobj, duffing_model).result()\n```\n\n\n```python\n# retrieve the data from the experiment\nspec01_values = helper.get_values_from_result(spec01_result, qubit)\n```\n\nWe will fit the spectroscopy signal to a Lorentzian function of the form\n\n$$ \\frac{AB}{\\pi[(f-f_{01})^2 + B^2]} + C $$\n\nto find the qubit frequency $f_{01}$.\n\n\n```python\nfit_params, y_fit = helper.fit_lorentzian(spec_freqs_GHz, spec01_values, [5, 5, 1, 0])\n\nf01 = fit_params[1]\n\nplt.scatter(spec_freqs_GHz, np.real(spec01_values), color='white') # plot real part of sweep values\nplt.plot(spec_freqs_GHz, y_fit, color='red')\nplt.xlim([min(spec_freqs_GHz), max(spec_freqs_GHz)])\n\nplt.xlabel(\"Frequency [GHz]\")\nplt.ylabel(\"Measured Signal [a.u.]\")\nplt.show()\n\nprint(\"01 Spectroscopy yields %f GHz\"%f01)\n```\n\n# Exercise 1: Spectroscopy of 1->2 transition\n\nIn order to observe the transition between the $|1\\rangle$ and $|2\\rangle$ states of the transmon, we must apply an $X_\\pi$ pulse to transition the qubit from $|0\\rangle$ to $|1\\rangle$ first. Because we are using the simulator, we must first define our $X_\\pi$ pulse from the Rabi experiment in Lab 6.\n\n\n```python\nx180_amp = 0.629070 #from lab 6 Rabi experiment\n\nx_pulse = pulse_lib.gaussian(duration=drive_duration,\n amp=x180_amp, \n sigma=drive_sigma,\n name='x_pulse')\n```\n\nThe anharmonicity of our transmon qubits is typically around $-300$ MHz, so we will sweep around that value. \n\n\n```python\nanharmonicity_guess_GHz = -0.3\n\ndef build_spec12_pulse_schedule(freq):\n sb12_spec_pulse = helper.apply_sideband(spec_pulse, (freq + anharmonicity_guess_GHz)*GHz, dt)\n \n ### create a 12 spectroscopy pulse schedule spec12_schedule (already done)\n ### play an x pulse on the drive channel\n ### play sidebanded spec pulse on the drive channel\n ### add measurement pulse to schedule\n \n spec12_schedule = pulse.Schedule()\n \n ### WRITE YOUR CODE BETWEEN THESE LINES - START\n spec12_schedule += Play(x_pulse, drive_chan)\n spec12_schedule += Play(sb12_spec_pulse, drive_chan)\n spec12_schedule += measure << spec12_schedule.duration\n ### WRITE YOUR CODE BETWEEN THESE LINES - END\n \n return spec12_schedule\n```\n\n\n```python\nsb_freqs_GHz = np.arange(-.1, .1, 0.005) # sweep +/- 100 MHz around guess\n\n# now vary the sideband frequency for each spec pulse\nspec_schedules = []\nfor freq in sb_freqs_GHz:\n spec_schedules.append(build_spec12_pulse_schedule(freq))\n```\n\n\n```python\nspec_schedules[0].draw()\n```\n\n\n```python\n# assemble the schedules into a Qobj\nspec12_qobj = assemble(**helper.get_params('spec12', globals()))\nanswer1 = spec12_qobj\n```\n\n\n```python\n# run the simulation\nspec12_result = backend_sim.run(spec12_qobj, duffing_model).result()\n```\n\n\n```python\n# retrieve the data from the experiment\nspec12_values = helper.get_values_from_result(spec12_result, qubit)\n```\n\nWe will again fit the spectroscopy signal to a Lorentzian function of the form\n\n$$ \\frac{AB}{\\pi[(f-f_{12})^2 + B^2]} + C $$\n\nto find the frequency of the $|1\\rangle \\to |2\\rangle$ transition $f_{12}$.\n\n\n```python\nanharm_offset = qubit_lo_freq[0]/GHz + anharmonicity_guess_GHz\n\nfit_params, y_fit = helper.fit_lorentzian(anharm_offset + sb_freqs_GHz, spec12_values, [5, 4.5, .1, 3])\n\nf12 = fit_params[1]\n\nplt.scatter(anharm_offset + sb_freqs_GHz, np.real(spec12_values), color='white') # plot real part of sweep values\nplt.plot(anharm_offset + sb_freqs_GHz, y_fit, color='red')\nplt.xlim([anharm_offset + min(sb_freqs_GHz), anharm_offset + max(sb_freqs_GHz)])\n\nplt.xlabel(\"Frequency [GHz]\")\nplt.ylabel(\"Measured Signal [a.u.]\")\nplt.show()\n\nprint(\"12 Spectroscopy yields %f GHz\"%f12)\nprint(\"Measured transmon anharmonicity is %f MHz\"%((f12-f01)*GHz/MHz))\n```\n\n**Help us improve our educational tools by submitting your code**
                                        \nIf you would like to help us learn how to improve our educational materials and offerings, you can opt in to send us a copy of your Jupyter notebook. By executing the cell below, you consent to sending us the code in your Jupyter notebook. All of the personal information will be anonymized.\n\n\n```python\nfrom IPython.display import display, Javascript;display(Javascript('IPython.notebook.save_checkpoint();'));\nfrom grading_tools import send_code;send_code('ex1.ipynb')\n```\n\n# Additional Resources\n\n- The Qiskit textbook sections that cover this material are\n - [Circuit Quantum Electrodynamics](https://qiskit.org/textbook/ch-quantum-hardware/cQED-JC-SW.html)\n - [Accessing Higher Energy States](https://qiskit.org/textbook/ch-quantum-hardware/accessing_higher_energy_states.html)\n\n- Watch the videos\n - [Quantum Coding with Lauren Capelluto](https://www.youtube.com/watch?v=ZvipHRY-URs)\n - [\"Qiskit Pulse: Programming Quantum Computers Through the Cloud with Pulses\"](https://www.youtube.com/watch?v=V_as5PufUiU) webinar at CQT by yours truly\n", "meta": {"hexsha": "b63d63d19c2453d67e6e5ac20070c5a23520040f", "size": 111333, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lab7/ex1.ipynb", "max_stars_repo_name": "amiune/QiskitSummerSchool2020", "max_stars_repo_head_hexsha": "70a8d37faa165b99b16f987d57cd525ffdc54a0b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "lab7/ex1.ipynb", "max_issues_repo_name": "amiune/QiskitSummerSchool2020", "max_issues_repo_head_hexsha": "70a8d37faa165b99b16f987d57cd525ffdc54a0b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "lab7/ex1.ipynb", "max_forks_repo_name": "amiune/QiskitSummerSchool2020", "max_forks_repo_head_hexsha": "70a8d37faa165b99b16f987d57cd525ffdc54a0b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 140.041509434, "max_line_length": 26020, "alphanum_fraction": 0.8223078512, "converted": true, "num_tokens": 8412, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.6926419894793246, "lm_q1q2_score": 0.42089302688636676}} {"text": "# CS109B Data Science 2: Advanced Topics in Data Science \n\n## Lab 4 - Bayesian Analysis\n\n**Harvard University**
                                        \n**Spring 2020**
                                        \n**Instructors:** Mark Glickman, Pavlos Protopapas, and Chris Tanner
                                        \n**Lab Instructors:** Chris Tanner and Eleni Angelaki Kaxiras
                                        \n**Content:** Eleni Angelaki Kaxiras\n\n---\n\n\n```python\n## RUN THIS CELL TO PROPERLY HIGHLIGHT THE EXERCISES\nimport requests\nfrom IPython.core.display import HTML\nstyles = requests.get(\"https://raw.githubusercontent.com/Harvard-IACS/2019-CS109B/master/content/styles/cs109.css\").text\nHTML(styles)\n```\n\n\n\n\n\n\n\n\n\n\n\n```python\nimport pymc3 as pm\nfrom pymc3 import summary\n```\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport scipy.stats as stats\nimport pandas as pd\n%matplotlib inline \n\nimport warnings\nwarnings.filterwarnings('ignore')\n```\n\n\n```python\nprint('Running on PyMC3 v{}'.format(pm.__version__))\n```\n\n Running on PyMC3 v3.8\n\n\n\n```javascript\n%%javascript\nIPython.OutputArea.auto_scroll_threshold = 20000;\n```\n\n\n \n\n\n\n\n## Learning Objectives\n\nBy the end of this lab, you should be able to:\n* Understand how probability distributions work.\n* Apply Bayes Rule in calculating probabilities.\n* Understand how to apply Bayesian analysis using PyMC3\n* Avoid getting fired when talking to your Bayesian employer.\n\n**This lab corresponds to Lectures 6, 7, and 8, and maps to Homework 3.**\n\n## Table of Contents\n\n1. The Bayesian Way of Thinking or Is this a Fair Coin?\n2. [Intro to `pyMC3`](#pymc3). \n3. [Bayesian Linear Regression](#blr).\n4. [Try this at Home: Example on Mining Disasters](#no4).\n\n## 1. The Bayesian way of Thinking\n\n```\nHere is my state of knowledge about the situation. Here is some data, I am now going to revise my state of knowledge.\n```\n\n
                                        Table Exercise: Discuss the statement above with your table mates and make sure everyone understands what it means and what constitutes Bayesian way of thinking. Finally, count the Bayesians among you.
                                        \n\n### A. Bayes Rule\n\n\\begin{equation}\n\\label{eq:bayes} \nP(A|\\textbf{B}) = \\frac{P(\\textbf{B} |A) P(A) }{P(\\textbf{B})} \n\\end{equation}\n\n$P(A|\\textbf{B})$ is the **posterior** distribution, prob(hypothesis | data) \n\n$P(\\textbf{B} |A)$ is the **likelihood** function, how probable is my data **B** for different values of the parameters\n\n$P(A)$ is the marginal probability to observe the data, called the **prior**, this captures our belief about the data before observing it.\n\n$P(\\textbf{B})$ is the marginal distribution (sometimes called marginal likelihood)\n\n
                                        \n
                                        Table Exercise: Solve the Monty Hall Paradox using Bayes Rule.
                                        \n\n\n\nYou are invited to play a game. There are 3 doors behind **one** of which are the keys to a brand new red Tesla. There is a goat behind each of the other two. \n\nYou are asked to pick one door, and let's say you pick **Door1**. The host knows where the keys are. Of the two remaining closed doors, he will always open the door that has a goat behind it. He'll say \"I will do you a favor and open **Door2**\". So he opens Door2 inside which there is, of course, a goat. He now asks you, do you want to open the initial Door you chose or change to **Door3**? Generally, in this game, when you are presented with this choice should you swap the doors?\n\n**Initial Steps:**\n- Start by defining the `events` of this probabilities game. One definition is:\n \n - $A_i$: car is behind door $i$ \n \n - $B_i$ host opens door $i$\n \n$i\\in[1,2,3]$\n \n- In more math terms, the question is: is the probability that the price is behind **Door 1** higher than the probability that the price is behind **Door2**, given that an event **has occured**?\n\n### B. Bayes Rule written with Probability Distributions\n\nWe have data that we believe come from an underlying distribution of unknown parameters. If we find those parameters, we know everything about the process that generated this data and we can make inferences (create new data).\n\n\\begin{equation}\n\\label{eq:bayes} \nP(\\theta|\\textbf{D}) = \\frac{P(\\textbf{D} |\\theta) P(\\theta) }{P(\\textbf{D})} \n\\end{equation}\n\n#### But what is $\\theta \\;$?\n\n$\\theta$ is an unknown yet fixed set of parameters. In Bayesian inference we express our belief about what $\\theta$ might be and instead of trying to guess $\\theta$ exactly, we look for its **probability distribution**. What that means is that we are looking for the **parameters** of that distribution. For example, for a Poisson distribution our $\\theta$ is only $\\lambda$. In a normal distribution, our $\\theta$ is often just $\\mu$ and $\\sigma$.\n\n### C. A review of Common Probability Distributions\n\n#### Discrete Distributions\n\nThe random variable has a **probability mass function (pmf)** which measures the probability that our random variable will take a specific value $y$, denoted $P(Y=y)$.\n\n- **Bernoulli** (binary outcome, success has probability $\\theta$, $one$ trial):\n$\nP(Y=k) = \\theta^k(1-\\theta)^{1-k}\n$\n
                                        \n- **Binomial** (binary outcome, success has probability $\\theta$, $n$ trials):\n\\begin{equation}\nP(Y=k) = {{n}\\choose{k}} \\cdot \\theta^k(1-\\theta)^{n-k}\n\\end{equation}\n\n*Note*: Binomial(1,$p$) = Bernouli($p$)\n
                                        \n- **Negative Binomial**\n
                                        \n- **Poisson** (counts independent events occurring at a rate)\n\\begin{equation}\nP\\left( Y=y|\\lambda \\right) = \\frac{{e^{ - \\lambda } \\lambda ^y }}{{y!}}\n\\end{equation}\ny = 0,1,2,...\n
                                        \n- **Discrete Uniform** \n
                                        \n- **Categorical, or Multinulli** (random variables can take any of K possible categories, each having its own probability; this is a generalization of the Bernoulli distribution for a discrete variable with more than two possible outcomes, such as the roll of a die)\n
                                        \n- **Dirichlet-multinomial** (a generalization of the beta distribution for many variables)\n\n#### Continuous Distributions\n\nThe random variable has a **probability density function (pdf)**.\n- **Uniform** (variable equally likely to be near each value in interval $(a,b)$)\n\\begin{equation}\nP(X = x) = \\frac{1}{b - a}\n\\end{equation}\nanywhere within the interval $(a, b)$, and zero elsewhere.\n
                                        \n- **Normal** (a.k.a. Gaussian)\n\\begin{equation}\nX \\sim \\mathcal{N}(\\mu,\\,\\sigma^{2})\n\\end{equation} \n\n A Normal distribution can be parameterized either in terms of precision $\\tau$ or standard deviation ($\\sigma^{2}$. The link between the two is given by\n\\begin{equation}\n\\tau = \\frac{1}{\\sigma^{2}}\n\\end{equation}\n - Mean $\\mu$\n - Variance $\\frac{1}{\\tau}$ or $\\sigma^{2}$\n - Parameters: `mu: float`, `sigma: float` or `tau: float`\n
                                        \n- **Beta** (variable ($\\theta$) taking on values in the interval $[0,1]$, and parametrized by two positive parameters, $\\alpha$ and $\\beta$ that control the shape of the distribution. \n \n*Note:*Beta is a good distribution to use for priors (beliefs) because its range is $[0,1]$ which is the natural range for a probability and because we can model a wide range of functions by changing the $\\alpha$ and $\\beta$ parameters.\n\n\\begin{equation}\n\\label{eq:beta} \nP(\\theta) = \\frac{1}{B(\\alpha, \\beta)} {\\theta}^{\\alpha - 1} (1 - \\theta)^{\\beta - 1} \\propto {\\theta}^{\\alpha - 1} (1 - \\theta)^{\\beta - 1}\n\\end{equation}\n\n\nwhere the normalisation constant, $B$, is a beta function of $\\alpha$ and $\\beta$,\n\n\n\\begin{equation}\nB(\\alpha, \\beta) = \\int_{t=0}^1 t^{\\alpha - 1} (1 - t)^{\\beta - 1} dt.\n\\end{equation}\n
                                        \n- **Exponential**\n
                                        \n- **Gamma**\n\n\n\n #### Code Resources:\n - Statistical Distributions in numpy/scipy: [scipy.stats](https://docs.scipy.org/doc/scipy/reference/stats.html)\n - Statistical Distributions in pyMC3: [distributions in PyMC3](https://docs.pymc.io/api/distributions.html) (we will see those below).\n\n
                                        Exercise: Plot a Discrete variable
                                        \n\nChange the value of $\\mu$ in the Poisson PMF and see how the plot changes. Remember that the y-axis in a discrete probability distribution shows the probability of the random variable having a specific value in the x-axis.\n\n\\begin{equation}\nP\\left( X=k \\right) = \\frac{{e^{ - \\mu } \\mu ^k }}{{k!}}\n\\end{equation}\n\n**stats.poisson.pmf(x, mu)** $\\mu$(mu) is our $\\theta$ in this case.\n\n\n```python\nplt.style.use('seaborn-darkgrid')\nx = np.arange(0, 30)\nfor m in [0.5, 3, 8]:\n pmf = stats.poisson.pmf(x, m)\n plt.plot(x, pmf, 'o', alpha=0.5, label='$\\mu$ = {}'.format(m))\nplt.xlabel('random variable', fontsize=12)\nplt.ylabel('probability', fontsize=12)\nplt.legend(loc=1)\nplt.ylim=(-0.1)\nplt.show()\n```\n\n\n```python\n# same for binomial\nplt.style.use('seaborn-darkgrid')\nx = np.arange(0, 22)\nns = [10, 17]\nps = [0.5, 0.7]\nfor n, p in zip(ns, ps):\n pmf = stats.binom.pmf(x, n, p)\n plt.plot(x, pmf, 'o', alpha=0.5, label='n = {}, p = {}'.format(n, p))\nplt.xlabel('x', fontsize=14)\nplt.ylabel('f(x)', fontsize=14)\nplt.legend(loc=1)\nplt.show()\n```\n\n\n```python\n# discrete uniform\nplt.style.use('seaborn-darkgrid')\nls = [0]\nus = [3] # watch out, this number can only be integer!\nfor l, u in zip(ls, us):\n x = np.arange(l, u+1)\n pmf = [1.0 / (u - l + 1)] * len(x)\n plt.plot(x, pmf, '-o', label='lower = {}, upper = {}'.format(l, u))\nplt.xlabel('x', fontsize=12)\nplt.ylabel('probability P(x)', fontsize=12)\nplt.legend(loc=1)\nplt.show()\n```\n\n
                                        Exercise: Plot a continuous variable
                                        \n\nChange the value of $\\mu$ in the Uniform PDF and see how the plot changes.\n \nRemember that the y-axis in a continuous probability distribution does not shows the actual probability of the random variable having a specific value in the x-axis because that probability is zero!. Instead, to see the probability that the variable is within a small margin we look at the integral below the curve of the PDF.\n\nThe uniform is often used as a noninformative prior.\n\n```\nUniform - numpy.random.uniform(a=0.0, b=1.0, size)\n```\n\n$\\alpha$ and $\\beta$ are our parameters. `size` is how many tries to perform.\nOur $\\theta$ is basically the combination of the parameters a,b. We can also call it \n\\begin{equation}\n\\mu = (a+b)/2\n\\end{equation}\n\n\n```python\nfrom scipy.stats import uniform\n\nr = uniform.rvs(size=1000)\nplt.plot(r, uniform.pdf(r),'r-', lw=5, alpha=0.6, label='uniform pdf')\nplt.hist(r, density=True, histtype='stepfilled', alpha=0.2)\nplt.ylabel(r'probability density')\nplt.xlabel(f'random variable')\nplt.legend(loc='best', frameon=False)\nplt.show()\n```\n\n\n```python\nfrom scipy.stats import beta\n\nalphas = [0.5, 1.5, 3.0]\nbetas = [0.5, 1.5, 3.0]\nx = np.linspace(0, 1, 1000) \ncolors = ['red', 'green', 'blue']\n\nfig, ax = plt.subplots(figsize=(8, 5))\n\nfor a, b, colors in zip(alphas, betas, colors):\n dist = beta(a, b)\n plt.plot(x, dist.pdf(x), c=colors,\n label=f'a={a}, b={b}')\n\nax.set_ylim(0, 3)\n\nax.set_xlabel(r'$\\theta$')\nax.set_ylabel(r'$p(\\theta|\\alpha,\\beta)$')\nax.set_title('Beta Distribution')\n\nax.legend(loc='best')\nfig.show();\n```\n\n\n```python\nplt.style.use('seaborn-darkgrid')\nx = np.linspace(-5, 5, 1000)\nmus = [0., 0., 0., -2.]\nsigmas = [0.4, 1., 2., 0.4]\nfor mu, sigma in zip(mus, sigmas):\n pdf = stats.norm.pdf(x, mu, sigma)\n plt.plot(x, pdf, label=r'$\\mu$ = '+ f'{mu},' + r'$\\sigma$ = ' + f'{sigma}') \nplt.xlabel('random variable', fontsize=12)\nplt.ylabel('probability density', fontsize=12)\nplt.legend(loc=1)\nplt.show()\n```\n\n\n```python\nplt.style.use('seaborn-darkgrid')\nx = np.linspace(-5, 5, 1000)\nmus = [0., 0., 0., -2.] # mean\nsigmas = [0.4, 1., 2., 0.4] # std\nfor mu, sigma in zip(mus, sigmas):\n plt.plot(x, uniform.pdf(x, mu, sigma), lw=5, alpha=0.4, \\\n label=r'$\\mu$ = '+ f'{mu},' + r'$\\sigma$ = ' + f'{sigma}')\nplt.xlabel('random variable', fontsize=12)\nplt.ylabel('probability density', fontsize=12)\nplt.legend(loc=1)\nplt.show()\n```\n\n### D. Is this a Fair Coin?\n\nWe do not want to promote gambling but let's say you visit the casino in **Monte Carlo**. You want to test your theory that casinos are dubious places where coins have been manipulated to have a larger probability for tails. So you will try to estimate how fair a coin is based on 100 flips.
                                        \nYou begin by flipping the coin. You get either Heads ($H$) or Tails ($T$) as our observed data and want to see if your posterior probabilities change as you obtain more data, that is, more coin flips. A nice way to visualize this is to plot the posterior probabilities as we observe more flips (data). \n\nWe will be using Bayes rule. $\\textbf{D}$ is our data.\n\n\\begin{equation}\n\\label{eq:bayes} \nP(\\theta|\\textbf{D}) = \\frac{P(\\textbf{D} |\\theta) P(\\theta) }{P(\\textbf{D})} \n\\end{equation}\n\nIn the case of a coin toss when we observe $k$ heads in $n$ tosses:\n\\begin{equation}\n\\label{eq:bayes} \nP(\\theta|\\textbf{k}) = Beta(\\alpha + \\textbf{k}, \\beta + n - \\textbf{k}) \n\\end{equation}\n\nwe can say that $\\alpha$ and $\\beta$ play the roles of a \"prior number of heads\" and \"prior number of tails\".\n\n\n```python\n# play with the priors - here we manually set them but we could be sampling from a separate Beta\ntrials = np.array([0, 1, 3, 5, 10, 15, 20, 100, 200, 300])\nheads = np.array([0, 1, 2, 4, 8, 10, 10, 50, 180, 150])\nx = np.linspace(0, 1, 100)\n\n# for simplicity we set a,b=1\n\nplt.figure(figsize=(10,8))\nfor k, N in enumerate(trials):\n sx = plt.subplot(len(trials)/2, 2, k+1)\n posterior = stats.beta.pdf(x, 1 + heads[k], 1 + trials[k] - heads[k]) \n plt.plot(x, posterior, alpha = 0.5, label=f'{trials[k]} tosses\\n {heads[k]} heads');\n plt.fill_between(x, 0, posterior, color=\"#348ABD\", alpha=0.4) \n plt.legend(loc='upper left', fontsize=10)\n plt.legend()\n plt.autoscale(tight=True)\n \nplt.suptitle(\"Posterior probabilities for coin flips\", fontsize=15);\nplt.tight_layout()\nplt.subplots_adjust(top=0.88)\n```\n\n [Top](#top)\n\n## 2. Introduction to `pyMC3`\n \nPyMC3 is a Python library for programming Bayesian analysis, and more specifically, data creation, model definition, model fitting, and posterior analysis. It uses the concept of a `model` which contains assigned parametric statistical distributions to unknown quantities in the model. Within models we define random variables and their distributions. A distribution requires at least a `name` argument, and other `parameters` that define it. You may also use the `logp()` method in the model to build the model log-likelihood function. We define and fit the model.\n\nPyMC3 includes a comprehensive set of pre-defined statistical distributions that can be used as model building blocks. Although they are not meant to be used outside of a `model`, you can invoke them by using the prefix `pm`, as in `pm.Normal`. \n\n#### Markov Chain Monte Carlo (MCMC) Simulations\n\nPyMC3 uses the **No-U-Turn Sampler (NUTS)** and the **Random Walk Metropolis**, two Markov chain Monte Carlo (MCMC) algorithms for sampling in posterior space. Monte Carlo gets into the name because when we sample in posterior space, we choose our next move via a pseudo-random process. NUTS is a sophisticated algorithm that can handle a large number of unknown (albeit continuous) variables.\n\n\n```python\nwith pm.Model() as model:\n z = pm.Normal('z', mu=0., sigma=5.) \n x = pm.Normal('x', mu=z, sigma=1., observed=5.) \nprint(x.logp({'z': 2.5})) \nprint(z.random(10, 100)[:10]) \n```\n\n -4.043938533204672\n [ 8.01444447 1.23006802 1.15969301 -0.93252273 5.54960832 -2.59564893\n 2.21650584 -6.0796205 -2.26857049 2.37727101]\n\n\n**References**:\n\n- *Salvatier J, Wiecki TV, Fonnesbeck C. 2016. Probabilistic programming in Python using PyMC3. PeerJ Computer Science 2:e55* [(https://doi.org/10.7717/peerj-cs.55)](https://doi.org/10.7717/peerj-cs.55)\n- [Distributions in PyMC3](https://docs.pymc.io/api/distributions.html)\n- [More Details on Distributions](https://docs.pymc.io/developer_guide.html)\n\nInformation about PyMC3 functions including descriptions of distributions, sampling methods, and other functions, is available via the `help` command.\n\n\n```python\n#help(pm.Poisson)\n```\n\n [Top](#top)\n\n## 3. Bayesian Linear Regression\n\nLet's say we want to predict outcomes Y as normally distributed observations with an expected value $mu$ that is a linear function of two predictor variables, $\\bf{x}_1$ and $\\bf{x}_2$.\n\n\\begin{equation}\n\\mu = \\alpha + \\beta_1 \\bf{x}_1 + \\beta_2 x_2 \n\\end{equation}\n\n\\begin{equation}\nY \\sim \\mathcal{N}(\\mu,\\,\\sigma^{2})\n\\end{equation} \n\nwhere $\\sigma^2$ represents the measurement error. \n\nIn this example, we will use $\\sigma^2 = 10$\n\nWe also choose the parameters as normal distributions:\n\n\\begin{eqnarray}\n\\alpha \\sim \\mathcal{N}(0,\\,10) \\\\\n\\beta_i \\sim \\mathcal{N}(0,\\,10) \\\\\n\\sigma^2 \\sim |\\mathcal{N}(0,\\,10)|\n\\end{eqnarray} \n\nWe will artificially create the data to predict on. We will then see if our model predicts them correctly.\n\n\n```python\n# Initialize random number generator\nnp.random.seed(123)\n\n# True parameter values\nalpha, sigma = 1, 1\nbeta = [1, 2.5]\n\n# Size of dataset\nsize = 100\n\n# Predictor variable\nX1 = np.linspace(0, 1, size)\nX2 = np.linspace(0,.2, size)\n\n# Simulate outcome variable\nY = alpha + beta[0]*X1 + beta[1]*X2 + np.random.randn(size)*sigma\n\nfig, ax = plt.subplots(1,2, figsize=(10,6), sharex=True)\nax[0].scatter(X1,Y)\nax[1].scatter(X2,Y)\nax[0].set_xlabel(r'$x_1$', fontsize=14) \nax[0].set_ylabel(r'$Y$', fontsize=14)\nax[1].set_xlabel(r'$x_2$', fontsize=14) \nax[1].set_ylabel(r'$Y$', fontsize=14)\n```\n\n\n```python\nfrom pymc3 import Model, Normal, HalfNormal\n\nbasic_model = Model()\n\nwith basic_model:\n\n # Priors for unknown model parameters, specifically create stochastic random variables \n # with Normal prior distributions for the regression coefficients,\n # and a half-normal distribution for the standard deviation of the observations, \u03c3.\n alpha = Normal('alpha', mu=0, sd=10)\n beta = Normal('beta', mu=0, sd=10, shape=2)\n sigma = HalfNormal('sigma', sd=1)\n\n # Expected value of outcome - posterior\n mu = alpha + beta[0]*X1 + beta[1]*X2\n\n # Likelihood (sampling distribution) of observations\n Y_obs = Normal('Y_obs', mu=mu, sd=sigma, observed=Y)\n```\n\n\n```python\n# model fitting with sampling\nfrom pymc3 import NUTS, sample, find_MAP\nfrom scipy import optimize\n\nwith basic_model:\n\n # obtain starting values via MAP\n start = find_MAP(fmin=optimize.fmin_powell)\n\n # instantiate sampler\n step = NUTS(scaling=start)\n\n # draw 2000 posterior samples\n trace = sample(2000, step, start=start)\n```\n\n logp = -164.5: 5%|\u258c | 270/5000 [00:00<00:01, 2401.04it/s] \n\n Optimization terminated successfully.\n Current function value: 164.496957\n Iterations: 6\n Function evaluations: 271\n\n\n logp = -164.5: 5%|\u258c | 271/5000 [00:00<00:12, 392.25it/s] \n Multiprocess sampling (2 chains in 2 jobs)\n NUTS: [sigma, beta, alpha]\n Sampling 2 chains, 0 divergences: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 5000/5000 [00:07<00:00, 632.32draws/s]\n The number of effective samples is smaller than 10% for some parameters.\n\n\n\n```python\nfrom pymc3 import traceplot\n\ntraceplot(trace);\n```\n\n\n```python\nresults = pm.summary(trace, \n var_names=['alpha', 'beta', 'sigma'])\nresults\n```\n\nThis linear regression example is from the original paper on PyMC3: *Salvatier J, Wiecki TV, Fonnesbeck C. 2016. Probabilistic programming in Python using PyMC3. PeerJ Computer Science 2:e55 https://doi.org/10.7717/peerj-cs.55*\n\n [Top](#top)\n\n## 4. Try this at Home: Example on Mining Disasters\nWe will go over the classical `mining disasters from 1851 to 1962` dataset. \n\nThis example is from the [pyMC3 Docs](https://docs.pymc.io/notebooks/getting_started.html).\n\n\n```python\nimport pandas as pd\ndisaster_data = pd.Series([4, 5, 4, 0, 1, 4, 3, 4, 0, 6, 3, 3, 4, 0, 2, 6,\n 3, 3, 5, 4, 5, 3, 1, 4, 4, 1, 5, 5, 3, 4, 2, 5,\n 2, 2, 3, 4, 2, 1, 3, np.nan, 2, 1, 1, 1, 1, 3, 0, 0,\n 1, 0, 1, 1, 0, 0, 3, 1, 0, 3, 2, 2, 0, 1, 1, 1,\n 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 1, 0, 2,\n 3, 3, 1, np.nan, 2, 1, 1, 1, 1, 2, 4, 2, 0, 0, 1, 4,\n 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1])\nfontsize = 12\nyears = np.arange(1851, 1962)\nplt.figure(figsize=(10,5))\n#plt.scatter(years, disaster_data); \nplt.bar(years, disaster_data)\nplt.ylabel('Disaster count', size=fontsize)\nplt.xlabel('Year', size=fontsize);\nplt.title('Was there a Turning Point in Mining disasters from 1851 to 1962?', size=15);\n```\n\n#### Building the model\n\n**Step1:** We choose the probability model for our experiment. Occurrences of disasters in the time series is thought to follow a **Poisson** process with a large **rate** parameter in the early part of the time series, and from one with a smaller **rate** in the later part. We are interested in locating the change point in the series, which perhaps is related to changes in mining safety regulations. \n\n```\ndisasters = pm.Poisson('disasters', rate, observed=disaster_data)\n```\n\nWe have two rates, `early_rate` if $t<=s$, and `late_rate` if $t>s$, where $s$ is the year the switch was made (a.k.a. the `switchpoint`). \n\n**Step2:** Choose a prior distributions of the two rates, what we believe the rates were before we observed the data, and the switchpoint. We choose Exponential.\n```\nearly_rate = pm.Exponential('early_rate', 1)\n```\n\nThe parameters of this model are: \n\n\n**Note:** Watch for missing values. Missing values are handled transparently by passing a MaskedArray or a pandas.DataFrame. Behind the scenes, another random variable, disasters.missing_values is created to model the missing values. If you pass a np.array with missing values you will get an error.\n\n\n```python\nwith pm.Model() as disaster_model:\n\n # discrete\n switchpoint = pm.DiscreteUniform('switchpoint', lower=years.min(), upper=years.max(), testval=1900)\n\n # Priors for pre- and post-switch rates number of disasters\n early_rate = pm.Exponential('early_rate', 1)\n late_rate = pm.Exponential('late_rate', 1)\n\n # our theta - allocate appropriate Poisson rates to years before and after current\n # switch is an `if` statement in puMC3\n rate = pm.math.switch(switchpoint >= years, early_rate, late_rate)\n\n # our observed data as a likelihood function of the `rate` parameters\n # shows how we think our data is distributed\n disasters = pm.Poisson('disasters', rate, observed=disaster_data)\n```\n\n#### Model Fitting\n\n\n```python\n# there are defaults but we can also more explicitly set the sampling algorithms\nwith disaster_model:\n \n # for continuous variables\n step1 = pm.NUTS([early_rate, late_rate])\n \n # for discrete variables\n step2 = pm.Metropolis([switchpoint, disasters.missing_values[0]] )\n\n trace = pm.sample(10000, step=[step1, step2])\n # try different number of samples\n #trace = pm.sample(5000, step=[step1, step2])\n```\n\n#### Posterior Analysis\n\nOn the left side plots we notice that our early rate is between 2.5 and 3.5 disasters a year. In the late period it seems to be between 0.6 and 1.2 so definitely lower.\n\nThe right side plots show the samples we drew to come to our conclusion.\n\n\n```python\npm.traceplot(trace, ['early_rate', 'late_rate', 'switchpoint'], figsize=(20,10));\n```\n\n\n```python\nresults = pm.summary(trace, \n var_names=['early_rate', 'late_rate', 'switchpoint'])\nresults\n```\n", "meta": {"hexsha": "b8cf61a3fcbd1eeb78b933c899eafa551ba310a9", "size": 460240, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "content/labs/lab04/notebook/cs109b_lab04_bayes.ipynb", "max_stars_repo_name": "ningxu2020/2020-CS109B", "max_stars_repo_head_hexsha": "87b66c8ff9bd40f2067bad18e2d4fb038aed30ae", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "content/labs/lab04/notebook/cs109b_lab04_bayes.ipynb", "max_issues_repo_name": "ningxu2020/2020-CS109B", "max_issues_repo_head_hexsha": "87b66c8ff9bd40f2067bad18e2d4fb038aed30ae", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "content/labs/lab04/notebook/cs109b_lab04_bayes.ipynb", "max_forks_repo_name": "ningxu2020/2020-CS109B", "max_forks_repo_head_hexsha": "87b66c8ff9bd40f2067bad18e2d4fb038aed30ae", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 391.3605442177, "max_line_length": 161532, "alphanum_fraction": 0.9298409525, "converted": true, "num_tokens": 7343, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.7431680029241322, "lm_q1q2_score": 0.42064686441972937}} {"text": "```python\nfrom IPython.core.display import HTML\nwith open(\"style/notebook.css\", \"r\") as f:\n s = f\"\"\nHTML(s)\n```\n\n\n```python\nfrom bokeh.plotting import figure, ColumnDataSource\nfrom bokeh.io import show, output_notebook, push_notebook\nfrom bokeh.layouts import row\nimport subprocess\nimport swarming\nimport numpy as np\nimport pandas as pd\n\nfrom dask.distributed import Client, LocalCluster, progress\nimport dask.bag\nimport dask.dataframe\n\noutput_notebook()\n```\n\n
                                        \n\n

                                        Predicting the behavior of swarms

                                        \n \nModelling interacting particles\n \n*by Andreas Roth*\n\n**2021-08-23**\n
                                        \n\n
                                        \n \n## The model\n \n* How do swarms of birds or fish align themselves?\n* Who decides about collective movement?\n \nJ. Carrillo, A. Klar, S. Martin, and S. Tiwari, [Self-propelled interacting particle systems with\nroosting force](https://www.worldscientific.com/doi/abs/10.1142/S0218202510004684). *Mathematical Models and Methods in Applied Sciences, 20(supp01), 2010*\n\nCarrillo, J.A.; Klar, A.; Roth, A., [Single to double mill small noise transition via semi-lagrangian finite volume methods](https://www.intlpress.com/site/pub/pages/journals/items/cms/content/vols/0014/0004/a012/), *Communications in mathematical sciences 14 (2016), No.4, pp.1111-1136, ISSN: 1539-6746*\n\n
                                        \n\n## Basic ideas\n\n
                                        \n
                                        \n \n
                                        \n
                                        \n\nWe look at the behavior of $n$ point masses\n* position $x_i$ and velocity $v_i$, where $i = 1,...,n$\n\n
                                        \n
                                        \n\n## Basic ideas\n\n
                                        \n
                                        \n \n
                                        \n
                                        \n\nWe look at the behavior of $n$ particles\n* position $x_i$ and velocity $v_i$, where $i = 1,...,n$\n* interaction force between pairs of particles depends on distance\n * if to close: **repulsion**\n \n
                                        \n
                                        \n\n## Basic ideas\n\n
                                        \n
                                        \n \n
                                        \n
                                        \n\nWe look at the behavior of $n$ particles\n* position $x_i$ and velocity $v_i$, where $i = 1,...,n$\n* interaction force between pairs of particles depends on distance\n * if to close: **repulsion**\n * if to far: **attraction**\n* forces balance each other\n* pairwise interactions on one particle **superpose** each other\n* evolution over time follows **Newton's second law**\n
                                        \n
                                        \n\n## Model equations\n\nSystem of ordinary differential equations:\n$$\n\\begin{align}\ndx_i =& v_i \\cdot dt\\\\\ndv_i =& (v_i(\\alpha-\\beta|v_i|) \\\\\n&- \\frac{1}{n}\\sum_{i\\neq j}F(|r_{ij}|) \\frac{r_{ij}}{|r_{ij}|}) \\cdot dt\n\\end{align}\n$$\n\n* $2n$ equations\n* $r_{ij} = x_i-x_j$ connection vector between particles $i$ and $j$\n* Computation effort goes with $n^2$ due to summing pairwise forces!\n\n### Force term\n\nOne possibility to achieve what we want is\n\n$$\nF(r) = \\frac{C_a}{l_a}\\cdot \\exp(-\\frac{r}{l_a})-\\frac{C_r}{l_r}\\cdot \\exp(-\\frac{r}{l_r})\n$$\n\n\n```python\nr = np.linspace(7, 200, 200)\nf = figure(title=\"Force over distance\", plot_width=1500, plot_height=200)\nf.line(x=r, y=swarming.force_of_distance(r), line_width=3)\nshow(f)\n```\n\n
                                        \n \n## How to compute\n \n* all code is published under MIT License at https://github.com/scherbertlemon/swarming\n* ``swarming`` python package\n* [Bokeh](https://bokeh.org) is used for all visualizations\n* [Dask](https://dask.org) is used for parallel computing\n \n
                                        \n\n## Generating some particles\n\n\n```python\nswarm = swarming.InitialCondition(condition=\"circular\", n_particles=300)\nshow(row(swarm.plot(plot_width=600, plot_height=600), swarm.plot_density(plot_width=800, plot_height=600, size=3)))\n```\n\n## Letting the particles move\n\n\n```python\nnh = show(swarm.plot(plot_width=600, plot_height=600), notebook_handle=True)\n```\n\n\n```python\nswarm.evolve(0.1, 10).update_cds()\npush_notebook(handle=nh)\n```\n\n### More interactive\n\n\n```python\nproc = subprocess.Popen([\"bokeh\", \"serve\", \"--port\", \"5006\", \"../bokeh_app/swarm.py\"], stdout=subprocess.PIPE, stderr=subprocess.PIPE)\n```\n\nMore interactively with [Bokeh app](http://localhost:5006)!\n* Javascript frontend connected to a python backend\n* Python runs the model\n* Sliders change parameters while model is running\n\n\n```python\nimport psutil\nparentproc = psutil.Process(proc.pid)\nfor subproc in parentproc.children(recursive=True):\n subproc.kill()\nparentproc.kill()\n```\n\n## Investigating steady states\n\nApparently, there is only a finite set of \"equilibria\" for this system, depending on the initial condition.\n\n### All velocities aligned\n\n\n```python\nswarm.n = 400\nshow(row(swarm.set_initial(\"square\").plot_density(size=5),\n swarm.set_initial(\"square\").record_for_time(120, 0.5, 2, lr=1).plot_density(size=3)))\n```\n\n## Investigating steady states\n\nApparently, there is only a finite set of \"equilibria\" for this system, depending on the initial condition.\n\n### Random velocities\n\n\n```python\nshow(row(swarm.set_initial(\"randomspeed\").plot_density(size=5),\n swarm.set_initial(\"randomspeed\").record_for_time(120, 0.5, 2, lr=1).plot_density(size=2)))\n```\n\n
                                        \n \n## Parameter study\n \n* To check if the \"donut\" equilibrium always occurs for a given initial condition (random velocity), we run a parameter study\n* We need to sample model parameters randomly and cover the complete parameter space: **Latin Hypercube Sampling**\n* We use Dask to run the models for each parameter sample in parallel\n \n
                                        \n\n## Parameter sampling\n\nModel has parameters attraction strength $C_a$, attraction range $l_a$, repulsion strength $C_r$, repulsion range $l_r$.\n\n\n```python\nparams = swarming.Parameters([(\"la\", 100, 40, 160), (\"lr\", 2, 0.5, 3.5), (\"Cr\", 30, 5, 35)])\nparams\n```\n\n\n```python\npd.DataFrame(params.sampling_dicts(n_samples=10))\n```\n\n## Dask cluster and client\n\n\n```python\ncluster = LocalCluster(n_workers=4)\nclient = Client(cluster)\ncluster\n```\n\n* There are backends to run Dask cluster on all kinds of systems (IBM LSF, Hadoop, ...)\n* We run a ``LocalCluster`` on one CPU\n* Dask client connects to cluster over the network (really distributed computing!)\n* [Cluster dashboard](http://localhost:8787) shows cluster load etc.\n\n### Run and store model for every of those parameter sets\n\nWe build a Dask *task graph*. No calculation happens here.\n\n\n```python\ndef calc(dct):\n return (dct, swarming.InitialCondition(condition=\"randomspeed\", n_particles=400).record_for_time(100, 0.5, 2, **dct).history.to_dict(orient=\"records\"))\ndef explode(params, result):\n return [{**params, **r} for r in result]\n\ndag = dask.bag.from_sequence(params.sampling_dicts(n_samples=24)) \\\n .map(calc) \\\n .starmap(explode) \\\n .flatten() \\\n .to_dataframe() \\\n .to_parquet(\"results.parq\", engine=\"pyarrow\", overwrite=True, compute=False)\n```\n\n### Visualizing and computing the task graph\n\n\n```python\ndag.visualize()\n```\n\n\n```python\nfuture = client.compute(dag) # send the graph to the workers for computing\n# progress(future) # does not display correctly in slideshow\n```\n\n### Evaluate stored results\n\nWe have written **all particle positions / velocities** at defined **time steps** with the **used parameter sample**:\n\n\n```python\ndata = dask.dataframe.read_parquet(\"results.parq\")\ndata.head(150, npartitions=2)\n```\n\n### Evaluate stored results\n\n* We want to plot the end densities, so select records with the maximum computed time\n* Note, that at no time the dataset is completely in memory!\n* When calling ``compute`` on a DAG, the result is actually transferred into local memory as ``pandas.DataFrame``\n\n\n```python\ndata.groupby(params.names).agg({\"time\": \"max\"}).compute().head(3)\n```\n\n\n```python\nfinal_states = data.loc[abs(data[\"time\"]-100.0)<1e-12, :].compute()\nfinal_states.head(3)\n```\n\n## Density plots of final states\n\nApparently, this parameterset delivers donuts in varying shapes.\n\n\n```python\nshow(swarming.get_density_plots(final_states, param_names=params.names, ncols=4, size=3))\n```\n\n### Tidy up\n\nWe close the Dask cluster and client:\n\n\n```python\nclient.close()\ncluster.close()\n```\n\n
                                        \n \n

                                        Thanks for your attention

                                        \n \nTime for your questions!\n \n
                                        \n", "meta": {"hexsha": "aa98b55da1c9637a9d4d2fa5b708e3f13f1de512", "size": 18356, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/presentation.ipynb", "max_stars_repo_name": "scherbertlemon/swarming", "max_stars_repo_head_hexsha": "6bc5eccbc4038bc8caa07241d7febd94928c2231", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/presentation.ipynb", "max_issues_repo_name": "scherbertlemon/swarming", "max_issues_repo_head_hexsha": "6bc5eccbc4038bc8caa07241d7febd94928c2231", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/presentation.ipynb", "max_forks_repo_name": "scherbertlemon/swarming", "max_forks_repo_head_hexsha": "6bc5eccbc4038bc8caa07241d7febd94928c2231", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 23.0025062657, "max_line_length": 313, "alphanum_fraction": 0.525659185, "converted": true, "num_tokens": 2190, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.7431680029241321, "lm_q1q2_score": 0.4206468644197293}} {"text": "Probabilistic Programming\n=====\nand Bayesian Methods for Hackers \n========\n\n##### Version 0.1\n\n`Original content created by Cam Davidson-Pilon`\n\n`Ported to Python 3 and PyMC3 by Max Margenot (@clean_utensils) and Thomas Wiecki (@twiecki) at Quantopian (@quantopian)`\n___\n\n\nWelcome to *Bayesian Methods for Hackers*. The full Github repository is available at [github/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers](https://github.com/CamDavidsonPilon/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers). The other chapters can be found on the project's [homepage](g). We hope you enjoy the book, and we encourage any contributions!\n\nChapter 1\n======\n***\n\nThe Philosophy of Bayesian Inference\n------\n \n> You are a skilled programmer, but bugs still slip into your code. After a particularly difficult implementation of an algorithm, you decide to test your code on a trivial example. It passes. You test the code on a harder problem. It passes once again. And it passes the next, *even more difficult*, test too! You are starting to believe that there may be no bugs in this code...\n\nIf you think this way, then congratulations, you already are thinking Bayesian! Bayesian inference is simply updating your beliefs after considering new evidence. A Bayesian can rarely be certain about a result, but he or she can be very confident. Just like in the example above, we can never be 100% sure that our code is bug-free unless we test it on every possible problem; something rarely possible in practice. Instead, we can test it on a large number of problems, and if it succeeds we can feel more *confident* about our code, but still not certain. Bayesian inference works identically: we update our beliefs about an outcome; rarely can we be absolutely sure unless we rule out all other alternatives. \n\n\n### The Bayesian state of mind\n\n\nBayesian inference differs from more traditional statistical inference by preserving *uncertainty*. At first, this sounds like a bad statistical technique. Isn't statistics all about deriving *certainty* from randomness? To reconcile this, we need to start thinking like Bayesians. \n\nThe Bayesian world-view interprets probability as measure of *believability in an event*, that is, how confident we are in an event occurring. In fact, we will see in a moment that this is the natural interpretation of probability. \n\nFor this to be clearer, we consider an alternative interpretation of probability: *Frequentist*, known as the more *classical* version of statistics, assume that probability is the long-run frequency of events (hence the bestowed title). For example, the *probability of plane accidents* under a frequentist philosophy is interpreted as the *long-term frequency of plane accidents*. This makes logical sense for many probabilities of events, but becomes more difficult to understand when events have no long-term frequency of occurrences. Consider: we often assign probabilities to outcomes of presidential elections, but the election itself only happens once! Frequentists get around this by invoking alternative realities and saying across all these realities, the frequency of occurrences defines the probability. \n\nBayesians, on the other hand, have a more intuitive approach. Bayesians interpret a probability as measure of *belief*, or confidence, of an event occurring. Simply, a probability is a summary of an opinion. An individual who assigns a belief of 0 to an event has no confidence that the event will occur; conversely, assigning a belief of 1 implies that the individual is absolutely certain of an event occurring. Beliefs between 0 and 1 allow for weightings of other outcomes. This definition agrees with the probability of a plane accident example, for having observed the frequency of plane accidents, an individual's belief should be equal to that frequency, excluding any outside information. Similarly, under this definition of probability being equal to beliefs, it is meaningful to speak about probabilities (beliefs) of presidential election outcomes: how confident are you candidate *A* will win?\n\nNotice in the paragraph above, I assigned the belief (probability) measure to an *individual*, not to Nature. This is very interesting, as this definition leaves room for conflicting beliefs between individuals. Again, this is appropriate for what naturally occurs: different individuals have different beliefs of events occurring, because they possess different *information* about the world. The existence of different beliefs does not imply that anyone is wrong. Consider the following examples demonstrating the relationship between individual beliefs and probabilities:\n\n- I flip a coin, and we both guess the result. We would both agree, assuming the coin is fair, that the probability of Heads is 1/2. Assume, then, that I peek at the coin. Now I know for certain what the result is: I assign probability 1.0 to either Heads or Tails (whichever it is). Now what is *your* belief that the coin is Heads? My knowledge of the outcome has not changed the coin's results. Thus we assign different probabilities to the result. \n\n- Your code either has a bug in it or not, but we do not know for certain which is true, though we have a belief about the presence or absence of a bug. \n\n- A medical patient is exhibiting symptoms $x$, $y$ and $z$. There are a number of diseases that could be causing all of them, but only a single disease is present. A doctor has beliefs about which disease, but a second doctor may have slightly different beliefs. \n\n\nThis philosophy of treating beliefs as probability is natural to humans. We employ it constantly as we interact with the world and only see partial truths, but gather evidence to form beliefs. Alternatively, you have to be *trained* to think like a frequentist. \n\nTo align ourselves with traditional probability notation, we denote our belief about event $A$ as $P(A)$. We call this quantity the *prior probability*.\n\nJohn Maynard Keynes, a great economist and thinker, said \"When the facts change, I change my mind. What do you do, sir?\" This quote reflects the way a Bayesian updates his or her beliefs after seeing evidence. Even — especially — if the evidence is counter to what was initially believed, the evidence cannot be ignored. We denote our updated belief as $P(A |X )$, interpreted as the probability of $A$ given the evidence $X$. We call the updated belief the *posterior probability* so as to contrast it with the prior probability. For example, consider the posterior probabilities (read: posterior beliefs) of the above examples, after observing some evidence $X$:\n\n1\\. $P(A): \\;\\;$ the coin has a 50 percent chance of being Heads. $P(A | X):\\;\\;$ You look at the coin, observe a Heads has landed, denote this information $X$, and trivially assign probability 1.0 to Heads and 0.0 to Tails.\n\n2\\. $P(A): \\;\\;$ This big, complex code likely has a bug in it. $P(A | X): \\;\\;$ The code passed all $X$ tests; there still might be a bug, but its presence is less likely now.\n\n3\\. $P(A):\\;\\;$ The patient could have any number of diseases. $P(A | X):\\;\\;$ Performing a blood test generated evidence $X$, ruling out some of the possible diseases from consideration.\n\n\nIt's clear that in each example we did not completely discard the prior belief after seeing new evidence $X$, but we *re-weighted the prior* to incorporate the new evidence (i.e. we put more weight, or confidence, on some beliefs versus others). \n\nBy introducing prior uncertainty about events, we are already admitting that any guess we make is potentially very wrong. After observing data, evidence, or other information, we update our beliefs, and our guess becomes *less wrong*. This is the alternative side of the prediction coin, where typically we try to be *more right*. \n\n\n\n### Bayesian Inference in Practice\n\n If frequentist and Bayesian inference were programming functions, with inputs being statistical problems, then the two would be different in what they return to the user. The frequentist inference function would return a number, representing an estimate (typically a summary statistic like the sample average etc.), whereas the Bayesian function would return *probabilities*.\n\nFor example, in our debugging problem above, calling the frequentist function with the argument \"My code passed all $X$ tests; is my code bug-free?\" would return a *YES*. On the other hand, asking our Bayesian function \"Often my code has bugs. My code passed all $X$ tests; is my code bug-free?\" would return something very different: probabilities of *YES* and *NO*. The function might return:\n\n\n> *YES*, with probability 0.8; *NO*, with probability 0.2\n\n\n\nThis is very different from the answer the frequentist function returned. Notice that the Bayesian function accepted an additional argument: *\"Often my code has bugs\"*. This parameter is the *prior*. By including the prior parameter, we are telling the Bayesian function to include our belief about the situation. Technically this parameter in the Bayesian function is optional, but we will see excluding it has its own consequences. \n\n\n#### Incorporating evidence\n\nAs we acquire more and more instances of evidence, our prior belief is *washed out* by the new evidence. This is to be expected. For example, if your prior belief is something ridiculous, like \"I expect the sun to explode today\", and each day you are proved wrong, you would hope that any inference would correct you, or at least align your beliefs better. Bayesian inference will correct this belief.\n\n\nDenote $N$ as the number of instances of evidence we possess. As we gather an *infinite* amount of evidence, say as $N \\rightarrow \\infty$, our Bayesian results (often) align with frequentist results. Hence for large $N$, statistical inference is more or less objective. On the other hand, for small $N$, inference is much more *unstable*: frequentist estimates have more variance and larger confidence intervals. This is where Bayesian analysis excels. By introducing a prior, and returning probabilities (instead of a scalar estimate), we *preserve the uncertainty* that reflects the instability of statistical inference of a small $N$ dataset. \n\nOne may think that for large $N$, one can be indifferent between the two techniques since they offer similar inference, and might lean towards the computationally-simpler, frequentist methods. An individual in this position should consider the following quote by Andrew Gelman (2005)[1], before making such a decision:\n\n> Sample sizes are never large. If $N$ is too small to get a sufficiently-precise estimate, you need to get more data (or make more assumptions). But once $N$ is \"large enough,\" you can start subdividing the data to learn more (for example, in a public opinion poll, once you have a good estimate for the entire country, you can estimate among men and women, northerners and southerners, different age groups, etc.). $N$ is never enough because if it were \"enough\" you'd already be on to the next problem for which you need more data.\n\n### Are frequentist methods incorrect then? \n\n**No.**\n\nFrequentist methods are still useful or state-of-the-art in many areas. Tools such as least squares linear regression, LASSO regression, and expectation-maximization algorithms are all powerful and fast. Bayesian methods complement these techniques by solving problems that these approaches cannot, or by illuminating the underlying system with more flexible modeling.\n\n\n#### A note on *Big Data*\nParadoxically, big data's predictive analytic problems are actually solved by relatively simple algorithms [2][4]. Thus we can argue that big data's prediction difficulty does not lie in the algorithm used, but instead on the computational difficulties of storage and execution on big data. (One should also consider Gelman's quote from above and ask \"Do I really have big data?\")\n\nThe much more difficult analytic problems involve *medium data* and, especially troublesome, *really small data*. Using a similar argument as Gelman's above, if big data problems are *big enough* to be readily solved, then we should be more interested in the *not-quite-big enough* datasets. \n\n\n### Our Bayesian framework\n\nWe are interested in beliefs, which can be interpreted as probabilities by thinking Bayesian. We have a *prior* belief in event $A$, beliefs formed by previous information, e.g., our prior belief about bugs being in our code before performing tests.\n\nSecondly, we observe our evidence. To continue our buggy-code example: if our code passes $X$ tests, we want to update our belief to incorporate this. We call this new belief the *posterior* probability. Updating our belief is done via the following equation, known as Bayes' Theorem, after its discoverer Thomas Bayes:\n\n\\begin{align}\n P( A | X ) = & \\frac{ P(X | A) P(A) } {P(X) } \\\\\\\\[5pt]\n& \\propto P(X | A) P(A)\\;\\; (\\propto \\text{is proportional to })\n\\end{align}\n\nThe above formula is not unique to Bayesian inference: it is a mathematical fact with uses outside Bayesian inference. Bayesian inference merely uses it to connect prior probabilities $P(A)$ with an updated posterior probabilities $P(A | X )$.\n\n##### Example: Mandatory coin-flip example\n\nEvery statistics text must contain a coin-flipping example, I'll use it here to get it out of the way. Suppose, naively, that you are unsure about the probability of heads in a coin flip (spoiler alert: it's 50%). You believe there is some true underlying ratio, call it $p$, but have no prior opinion on what $p$ might be. \n\nWe begin to flip a coin, and record the observations: either $H$ or $T$. This is our observed data. An interesting question to ask is how our inference changes as we observe more and more data? More specifically, what do our posterior probabilities look like when we have little data, versus when we have lots of data. \n\nBelow we plot a sequence of updating posterior probabilities as we observe increasing amounts of data (coin flips).\n\n\n```python\n\"\"\"\nThe book uses a custom matplotlibrc file, which provides the unique styles for\nmatplotlib plots. If executing this book, and you wish to use the book's\nstyling, provided are two options:\n 1. Overwrite your own matplotlibrc file with the rc-file provided in the\n book's styles/ dir. See http://matplotlib.org/users/customizing.html\n 2. Also in the styles is bmh_matplotlibrc.json file. This can be used to\n update the styles in only this notebook. Try running the following code:\n\n import json\n s = json.load(open(\"../styles/bmh_matplotlibrc.json\"))\n matplotlib.rcParams.update(s)\n\n\"\"\"\n\n# The code below can be passed over, as it is currently not important, plus it\n# uses advanced topics we have not covered yet. LOOK AT PICTURE, MICHAEL!\n%matplotlib inline\nfrom IPython.core.pylabtools import figsize\nimport numpy as np\nfrom matplotlib import pyplot as plt\nfigsize(11, 9)\n\nimport scipy.stats as stats\n\ndist = stats.beta\nn_trials = [0, 1, 2, 3, 4, 5, 8, 15, 50, 500]\ndata = stats.bernoulli.rvs(0.5, size=n_trials[-1])\nx = np.linspace(0, 1, 100)\n\n# For the already prepared, I'm using Binomial's conj. prior.\nfor k, N in enumerate(n_trials):\n sx = plt.subplot(len(n_trials)/2, 2, k+1)\n plt.xlabel(\"$p$, probability of heads\") \\\n if k in [0, len(n_trials)-1] else None\n plt.setp(sx.get_yticklabels(), visible=False)\n heads = data[:N].sum()\n y = dist.pdf(x, 1 + heads, 1 + N - heads)\n plt.plot(x, y, label=\"observe %d tosses,\\n %d heads\" % (N, heads))\n plt.fill_between(x, 0, y, color=\"#348ABD\", alpha=0.4)\n plt.vlines(0.5, 0, 4, color=\"k\", linestyles=\"--\", lw=1)\n\n leg = plt.legend()\n leg.get_frame().set_alpha(0.4)\n plt.autoscale(tight=True)\n\n\nplt.suptitle(\"Bayesian updating of posterior probabilities\",\n y=1.02,\n fontsize=14)\n\nplt.tight_layout()\n```\n\nThe posterior probabilities are represented by the curves, and our uncertainty is proportional to the width of the curve. As the plot above shows, as we start to observe data our posterior probabilities start to shift and move around. Eventually, as we observe more and more data (coin-flips), our probabilities will tighten closer and closer around the true value of $p=0.5$ (marked by a dashed line). \n\nNotice that the plots are not always *peaked* at 0.5. There is no reason it should be: recall we assumed we did not have a prior opinion of what $p$ is. In fact, if we observe quite extreme data, say 8 flips and only 1 observed heads, our distribution would look very biased *away* from lumping around 0.5 (with no prior opinion, how confident would you feel betting on a fair coin after observing 8 tails and 1 head). As more data accumulates, we would see more and more probability being assigned at $p=0.5$, though never all of it.\n\nThe next example is a simple demonstration of the mathematics of Bayesian inference. \n\n##### Example: Bug, or just sweet, unintended feature?\n\n\nLet $A$ denote the event that our code has **no bugs** in it. Let $X$ denote the event that the code passes all debugging tests. For now, we will leave the prior probability of no bugs as a variable, i.e. $P(A) = p$. \n\nWe are interested in $P(A|X)$, i.e. the probability of no bugs, given our debugging tests $X$. To use the formula above, we need to compute some quantities.\n\nWhat is $P(X | A)$, i.e., the probability that the code passes $X$ tests *given* there are no bugs? Well, it is equal to 1, for a code with no bugs will pass all tests. \n\n$P(X)$ is a little bit trickier: The event $X$ can be divided into two possibilities, event $X$ occurring even though our code *indeed has* bugs (denoted $\\sim A\\;$, spoken *not $A$*), or event $X$ without bugs ($A$). $P(X)$ can be represented as:\n\n\\begin{align}\nP(X ) & = P(X \\text{ and } A) + P(X \\text{ and } \\sim A) \\\\\\\\[5pt]\n & = P(X|A)P(A) + P(X | \\sim A)P(\\sim A)\\\\\\\\[5pt]\n& = P(X|A)p + P(X | \\sim A)(1-p)\n\\end{align}\n\nWe have already computed $P(X|A)$ above. On the other hand, $P(X | \\sim A)$ is subjective: our code can pass tests but still have a bug in it, though the probability there is a bug present is reduced. Note this is dependent on the number of tests performed, the degree of complication in the tests, etc. Let's be conservative and assign $P(X|\\sim A) = 0.5$. Then\n\n\\begin{align}\nP(A | X) & = \\frac{1\\cdot p}{ 1\\cdot p +0.5 (1-p) } \\\\\\\\\n& = \\frac{ 2 p}{1+p}\n\\end{align}\nThis is the posterior probability. What does it look like as a function of our prior, $p \\in [0,1]$? \n\n\n```python\nfigsize(12.5, 4)\np = np.linspace(0, 1, 50)\nplt.plot(p, 2*p/(1+p), color=\"#348ABD\", lw=3)\n#plt.fill_between(p, 2*p/(1+p), alpha=.5, facecolor=[\"#A60628\"])\nplt.scatter(0.2, 2*(0.2)/1.2, s=140, c=\"#348ABD\")\nplt.xlim(0, 1)\nplt.ylim(0, 1)\nplt.xlabel(\"Prior, $P(A) = p$\")\nplt.ylabel(\"Posterior, $P(A|X)$, with $P(A) = p$\")\nplt.title(\"Are there bugs in my code?\");\n```\n\nWe can see the biggest gains if we observe the $X$ tests passed when the prior probability, $p$, is low. Let's settle on a specific value for the prior. I'm a strong programmer (I think), so I'm going to give myself a realistic prior of 0.20, that is, there is a 20% chance that I write code bug-free. To be more realistic, this prior should be a function of how complicated and large the code is, but let's pin it at 0.20. Then my updated belief that my code is bug-free is 0.33. \n\nRecall that the prior is a probability: $p$ is the prior probability that there *are no bugs*, so $1-p$ is the prior probability that there *are bugs*.\n\nSimilarly, our posterior is also a probability, with $P(A | X)$ the probability there is no bug *given we saw all tests pass*, hence $1-P(A|X)$ is the probability there is a bug *given all tests passed*. What does our posterior probability look like? Below is a chart of both the prior and the posterior probabilities. \n\n\n\n```python\nfigsize(12.5, 4)\ncolours = [\"#348ABD\", \"#A60628\"]\n\nprior = [0.20, 0.80]\nposterior = [1./3, 2./3]\nplt.bar([0, .7], prior, alpha=0.70, width=0.25,\n color=colours[0], label=\"prior distribution\",\n lw=\"3\", edgecolor=colours[0])\n\nplt.bar([0+0.25, .7+0.25], posterior, alpha=0.7,\n width=0.25, color=colours[1],\n label=\"posterior distribution\",\n lw=\"3\", edgecolor=colours[1])\n\nplt.xticks([0.20, .95], [\"Bugs Absent\", \"Bugs Present\"])\nplt.title(\"Prior and Posterior probability of bugs present\")\nplt.ylabel(\"Probability\")\nplt.legend(loc=\"upper left\");\n```\n\nNotice that after we observed $X$ occur, the probability of bugs being absent increased. By increasing the number of tests, we can approach confidence (probability 1) that there are no bugs present.\n\nThis was a very simple example of Bayesian inference and Bayes rule. Unfortunately, the mathematics necessary to perform more complicated Bayesian inference only becomes more difficult, except for artificially constructed cases. We will later see that this type of mathematical analysis is actually unnecessary. First we must broaden our modeling tools. The next section deals with *probability distributions*. If you are already familiar, feel free to skip (or at least skim), but for the less familiar the next section is essential.\n\n_______\n\n## Probability Distributions\n\n\n**Let's quickly recall what a probability distribution is:** Let $Z$ be some random variable. Then associated with $Z$ is a *probability distribution function* that assigns probabilities to the different outcomes $Z$ can take. Graphically, a probability distribution is a curve where the probability of an outcome is proportional to the height of the curve. You can see examples in the first figure of this chapter. \n\nWe can divide random variables into three classifications:\n\n- **$Z$ is discrete**: Discrete random variables may only assume values on a specified list. Things like populations, movie ratings, and number of votes are all discrete random variables. Discrete random variables become more clear when we contrast them with...\n\n- **$Z$ is continuous**: Continuous random variable can take on arbitrarily exact values. For example, temperature, speed, time, color are all modeled as continuous variables because you can progressively make the values more and more precise.\n\n- **$Z$ is mixed**: Mixed random variables assign probabilities to both discrete and continuous random variables, i.e. it is a combination of the above two categories. \n\n### Discrete Case\nIf $Z$ is discrete, then its distribution is called a *probability mass function*, which measures the probability $Z$ takes on the value $k$, denoted $P(Z=k)$. Note that the probability mass function completely describes the random variable $Z$, that is, if we know the mass function, we know how $Z$ should behave. There are popular probability mass functions that consistently appear: we will introduce them as needed, but let's introduce the first very useful probability mass function. We say $Z$ is *Poisson*-distributed if:\n\n$$P(Z = k) =\\frac{ \\lambda^k e^{-\\lambda} }{k!}, \\; \\; k=0,1,2, \\dots $$\n\n$\\lambda$ is called a parameter of the distribution, and it controls the distribution's shape. For the Poisson distribution, $\\lambda$ can be any positive number. By increasing $\\lambda$, we add more probability to larger values, and conversely by decreasing $\\lambda$ we add more probability to smaller values. One can describe $\\lambda$ as the *intensity* of the Poisson distribution. \n\nUnlike $\\lambda$, which can be any positive number, the value $k$ in the above formula must be a non-negative integer, i.e., $k$ must take on values 0,1,2, and so on. This is very important, because if you wanted to model a population you could not make sense of populations with 4.25 or 5.612 members. \n\nIf a random variable $Z$ has a Poisson mass distribution, we denote this by writing\n\n$$Z \\sim \\text{Poi}(\\lambda) $$\n\nOne useful property of the Poisson distribution is that its expected value is equal to its parameter, i.e.:\n\n$$E\\large[ \\;Z\\; | \\; \\lambda \\;\\large] = \\lambda $$\n\nWe will use this property often, so it's useful to remember. Below, we plot the probability mass distribution for different $\\lambda$ values. The first thing to notice is that by increasing $\\lambda$, we add more probability of larger values occurring. Second, notice that although the graph ends at 15, the distributions do not. They assign positive probability to every non-negative integer.\n\n\n```python\nfigsize(12.5, 4)\n\nimport scipy.stats as stats\na = np.arange(16)\npoi = stats.poisson\nlambda_ = [1.5, 4.25]\ncolours = [\"#348ABD\", \"#A60628\"]\n\nplt.bar(a, poi.pmf(a, lambda_[0]), color=colours[0],\n label=\"$\\lambda = %.1f$\" % lambda_[0], alpha=0.60,\n edgecolor=colours[0], lw=\"3\")\n\nplt.bar(a, poi.pmf(a, lambda_[1]), color=colours[1],\n label=\"$\\lambda = %.1f$\" % lambda_[1], alpha=0.60,\n edgecolor=colours[1], lw=\"3\")\n\nplt.xticks(a + 0.4, a)\nplt.legend()\nplt.ylabel(\"probability of $k$\")\nplt.xlabel(\"$k$\")\nplt.title(\"Probability mass function of a Poisson random variable; differing \\\n$\\lambda$ values\");\n```\n\n### Continuous Case\nInstead of a probability mass function, a continuous random variable has a *probability density function*. This might seem like unnecessary nomenclature, but the density function and the mass function are very different creatures. An example of continuous random variable is a random variable with *exponential density*. The density function for an exponential random variable looks like this:\n\n$$f_Z(z | \\lambda) = \\lambda e^{-\\lambda z }, \\;\\; z\\ge 0$$\n\nLike a Poisson random variable, an exponential random variable can take on only non-negative values. But unlike a Poisson variable, the exponential can take on *any* non-negative values, including non-integral values such as 4.25 or 5.612401. This property makes it a poor choice for count data, which must be an integer, but a great choice for time data, temperature data (measured in Kelvins, of course), or any other precise *and positive* variable. The graph below shows two probability density functions with different $\\lambda$ values. \n\nWhen a random variable $Z$ has an exponential distribution with parameter $\\lambda$, we say *$Z$ is exponential* and write\n\n$$Z \\sim \\text{Exp}(\\lambda)$$\n\nGiven a specific $\\lambda$, the expected value of an exponential random variable is equal to the inverse of $\\lambda$, that is:\n\n$$E[\\; Z \\;|\\; \\lambda \\;] = \\frac{1}{\\lambda}$$\n\n\n```python\na = np.linspace(0, 4, 100)\nexpo = stats.expon\nlambda_ = [0.5, 1]\n\nfor l, c in zip(lambda_, colours):\n plt.plot(a, expo.pdf(a, scale=1./l), lw=3,\n color=c, label=\"$\\lambda = %.1f$\" % l)\n plt.fill_between(a, expo.pdf(a, scale=1./l), color=c, alpha=.33)\n\nplt.legend()\nplt.ylabel(\"PDF at $z$\")\nplt.xlabel(\"$z$\")\nplt.ylim(0,1.2)\nplt.title(\"Probability density function of an Exponential random variable;\\\n differing $\\lambda$\");\n```\n\n\n### But what is $\\lambda \\;$?\n\n\n**This question is what motivates statistics**. In the real world, $\\lambda$ is hidden from us. We see only $Z$, and must go backwards to try and determine $\\lambda$. The problem is difficult because there is no one-to-one mapping from $Z$ to $\\lambda$. Many different methods have been created to solve the problem of estimating $\\lambda$, but since $\\lambda$ is never actually observed, no one can say for certain which method is best! \n\nBayesian inference is concerned with *beliefs* about what $\\lambda$ might be. Rather than try to guess $\\lambda$ exactly, we can only talk about what $\\lambda$ is likely to be by assigning a probability distribution to $\\lambda$.\n\nThis might seem odd at first. After all, $\\lambda$ is fixed; it is not (necessarily) random! How can we assign probabilities to values of a non-random variable? Ah, we have fallen for our old, frequentist way of thinking. Recall that under Bayesian philosophy, we *can* assign probabilities if we interpret them as beliefs. And it is entirely acceptable to have *beliefs* about the parameter $\\lambda$. \n\n\n\n##### Example: Inferring behaviour from text-message data\n\nLet's try to model a more interesting example, one that concerns the rate at which a user sends and receives text messages:\n\n> You are given a series of daily text-message counts from a user of your system. The data, plotted over time, appears in the chart below. You are curious to know if the user's text-messaging habits have changed over time, either gradually or suddenly. How can you model this? (This is in fact my own text-message data. Judge my popularity as you wish.)\n\n\n\n```python\nfigsize(12.5, 3.5)\ncount_data = np.loadtxt(\"data/txtdata.csv\")\nn_count_data = len(count_data)\nplt.bar(np.arange(n_count_data), count_data, color=\"#348ABD\")\nplt.xlabel(\"Time (days)\")\nplt.ylabel(\"count of text-msgs received\")\nplt.title(\"Did the user's texting habits change over time?\")\nplt.xlim(0, n_count_data);\n```\n\nBefore we start modeling, see what you can figure out just by looking at the chart above. Would you say there was a change in behaviour during this time period? \n\nHow can we start to model this? Well, as we have conveniently already seen, a Poisson random variable is a very appropriate model for this type of *count* data. Denoting day $i$'s text-message count by $C_i$, \n\n$$ C_i \\sim \\text{Poisson}(\\lambda) $$\n\nWe are not sure what the value of the $\\lambda$ parameter really is, however. Looking at the chart above, it appears that the rate might become higher late in the observation period, which is equivalent to saying that $\\lambda$ increases at some point during the observations. (Recall that a higher value of $\\lambda$ assigns more probability to larger outcomes. That is, there is a higher probability of many text messages having been sent on a given day.)\n\nHow can we represent this observation mathematically? Let's assume that on some day during the observation period (call it $\\tau$), the parameter $\\lambda$ suddenly jumps to a higher value. So we really have two $\\lambda$ parameters: one for the period before $\\tau$, and one for the rest of the observation period. In the literature, a sudden transition like this would be called a *switchpoint*:\n\n$$\n\\lambda = \n\\begin{cases}\n\\lambda_1 & \\text{if } t \\lt \\tau \\cr\n\\lambda_2 & \\text{if } t \\ge \\tau\n\\end{cases}\n$$\n\n\nIf, in reality, no sudden change occurred and indeed $\\lambda_1 = \\lambda_2$, then the $\\lambda$s posterior distributions should look about equal.\n\nWe are interested in inferring the unknown $\\lambda$s. To use Bayesian inference, we need to assign prior probabilities to the different possible values of $\\lambda$. What would be good prior probability distributions for $\\lambda_1$ and $\\lambda_2$? Recall that $\\lambda$ can be any positive number. As we saw earlier, the *exponential* distribution provides a continuous density function for positive numbers, so it might be a good choice for modeling $\\lambda_i$. But recall that the exponential distribution takes a parameter of its own, so we'll need to include that parameter in our model. Let's call that parameter $\\alpha$.\n\n\\begin{align}\n&\\lambda_1 \\sim \\text{Exp}( \\alpha ) \\\\\\\n&\\lambda_2 \\sim \\text{Exp}( \\alpha )\n\\end{align}\n\n$\\alpha$ is called a *hyper-parameter* or *parent variable*. In literal terms, it is a parameter that influences other parameters. Our initial guess at $\\alpha$ does not influence the model too strongly, so we have some flexibility in our choice. A good rule of thumb is to set the exponential parameter equal to the inverse of the average of the count data. Since we're modeling $\\lambda$ using an exponential distribution, we can use the expected value identity shown earlier to get:\n\n$$\\frac{1}{N}\\sum_{i=0}^N \\;C_i \\approx E[\\; \\lambda \\; |\\; \\alpha ] = \\frac{1}{\\alpha}$$ \n\nAn alternative, and something I encourage the reader to try, would be to have two priors: one for each $\\lambda_i$. Creating two exponential distributions with different $\\alpha$ values reflects our prior belief that the rate changed at some point during the observations.\n\nWhat about $\\tau$? Because of the noisiness of the data, it's difficult to pick out a priori when $\\tau$ might have occurred. Instead, we can assign a *uniform prior belief* to every possible day. This is equivalent to saying\n\n\\begin{align}\n& \\tau \\sim \\text{DiscreteUniform(1,70) }\\\\\\\\\n& \\Rightarrow P( \\tau = k ) = \\frac{1}{70}\n\\end{align}\n\nSo after all this, what does our overall prior distribution for the unknown variables look like? Frankly, *it doesn't matter*. What we should understand is that it's an ugly, complicated mess involving symbols only a mathematician could love. And things will only get uglier the more complicated our models become. Regardless, all we really care about is the posterior distribution.\n\nWe next turn to PyMC3, a Python library for performing Bayesian analysis that is undaunted by the mathematical monster we have created. \n\n\nIntroducing our first hammer: PyMC3\n-----\n\nPyMC3 is a Python library for programming Bayesian analysis [3]. It is a fast, well-maintained library. The only unfortunate part is that its documentation is lacking in certain areas, especially those that bridge the gap between beginner and hacker. One of this book's main goals is to solve that problem, and also to demonstrate why PyMC3 is so cool.\n\nWe will model the problem above using PyMC3. This type of programming is called *probabilistic programming*, an unfortunate misnomer that invokes ideas of randomly-generated code and has likely confused and frightened users away from this field. The code is not random; it is probabilistic in the sense that we create probability models using programming variables as the model's components. Model components are first-class primitives within the PyMC3 framework. \n\nB. Cronin [5] has a very motivating description of probabilistic programming:\n\n> Another way of thinking about this: unlike a traditional program, which only runs in the forward directions, a probabilistic program is run in both the forward and backward direction. It runs forward to compute the consequences of the assumptions it contains about the world (i.e., the model space it represents), but it also runs backward from the data to constrain the possible explanations. In practice, many probabilistic programming systems will cleverly interleave these forward and backward operations to efficiently home in on the best explanations.\n\nBecause of the confusion engendered by the term *probabilistic programming*, I'll refrain from using it. Instead, I'll simply say *programming*, since that's what it really is. \n\nPyMC3 code is easy to read. The only novel thing should be the syntax. Simply remember that we are representing the model's components ($\\tau, \\lambda_1, \\lambda_2$ ) as variables.\n\n\n```python\nimport pymc3 as pm\nimport theano.tensor as tt\n\nwith pm.Model() as model:\n alpha = 1.0/count_data.mean() # Recall count_data is the\n # variable that holds our txt counts\n lambda_1 = pm.Exponential(\"lambda_1\", alpha)\n lambda_2 = pm.Exponential(\"lambda_2\", alpha)\n \n tau = pm.DiscreteUniform(\"tau\", lower=0, upper=n_count_data - 1)\n```\n\nIn the code above, we create the PyMC3 variables corresponding to $\\lambda_1$ and $\\lambda_2$. We assign them to PyMC3's *stochastic variables*, so-called because they are treated by the back end as random number generators.\n\n\n```python\nwith model:\n idx = np.arange(n_count_data) # Index\n lambda_ = pm.math.switch(tau >= idx, lambda_1, lambda_2)\n```\n\nThis code creates a new function `lambda_`, but really we can think of it as a random variable: the random variable $\\lambda$ from above. The `switch()` function assigns `lambda_1` or `lambda_2` as the value of `lambda_`, depending on what side of `tau` we are on. The values of `lambda_` up until `tau` are `lambda_1` and the values afterwards are `lambda_2`.\n\nNote that because `lambda_1`, `lambda_2` and `tau` are random, `lambda_` will be random. We are **not** fixing any variables yet.\n\n\n```python\nwith model:\n observation = pm.Poisson(\"obs\", lambda_, observed=count_data)\n```\n\nThe variable `observation` combines our data, `count_data`, with our proposed data-generation scheme, given by the variable `lambda_`, through the `observed` keyword. \n\nThe code below will be explained in Chapter 3, but I show it here so you can see where our results come from. One can think of it as a *learning* step. The machinery being employed is called *Markov Chain Monte Carlo* (MCMC), which I also delay explaining until Chapter 3. This technique returns thousands of random variables from the posterior distributions of $\\lambda_1, \\lambda_2$ and $\\tau$. We can plot a histogram of the random variables to see what the posterior distributions look like. Below, we collect the samples (called *traces* in the MCMC literature) into histograms.\n\n\n```python\n### Mysterious code to be explained in Chapter 3.\nwith model:\n step = pm.Metropolis()\n trace = pm.sample(10000, tune=5000,step=step)\n```\n\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 15000/15000 [00:04<00:00, 3131.95it/s]\n\n\n\n```python\nlambda_1_samples = trace['lambda_1']\nlambda_2_samples = trace['lambda_2']\ntau_samples = trace['tau']\n```\n\n\n```python\nfigsize(12.5, 10)\n#histogram of the samples:\n\nax = plt.subplot(311)\nax.set_autoscaley_on(False)\n\nplt.hist(lambda_1_samples, histtype='stepfilled', bins=30, alpha=0.85,\n label=\"posterior of $\\lambda_1$\", color=\"#A60628\", normed=True)\nplt.legend(loc=\"upper left\")\nplt.title(r\"\"\"Posterior distributions of the variables\n $\\lambda_1,\\;\\lambda_2,\\;\\tau$\"\"\")\nplt.xlim([15, 30])\nplt.xlabel(\"$\\lambda_1$ value\")\n\nax = plt.subplot(312)\nax.set_autoscaley_on(False)\nplt.hist(lambda_2_samples, histtype='stepfilled', bins=30, alpha=0.85,\n label=\"posterior of $\\lambda_2$\", color=\"#7A68A6\", normed=True)\nplt.legend(loc=\"upper left\")\nplt.xlim([15, 30])\nplt.xlabel(\"$\\lambda_2$ value\")\n\nplt.subplot(313)\nw = 1.0 / tau_samples.shape[0] * np.ones_like(tau_samples)\nplt.hist(tau_samples, bins=n_count_data, alpha=1,\n label=r\"posterior of $\\tau$\",\n color=\"#467821\", weights=w, rwidth=2.)\nplt.xticks(np.arange(n_count_data))\n\nplt.legend(loc=\"upper left\")\nplt.ylim([0, .75])\nplt.xlim([35, len(count_data)-20])\nplt.xlabel(r\"$\\tau$ (in days)\")\nplt.ylabel(\"probability\");\n```\n\n### Interpretation\n\nRecall that Bayesian methodology returns a *distribution*. Hence we now have distributions to describe the unknown $\\lambda$s and $\\tau$. What have we gained? Immediately, we can see the uncertainty in our estimates: the wider the distribution, the less certain our posterior belief should be. We can also see what the plausible values for the parameters are: $\\lambda_1$ is around 18 and $\\lambda_2$ is around 23. The posterior distributions of the two $\\lambda$s are clearly distinct, indicating that it is indeed likely that there was a change in the user's text-message behaviour.\n\nWhat other observations can you make? If you look at the original data again, do these results seem reasonable? \n\nNotice also that the posterior distributions for the $\\lambda$s do not look like exponential distributions, even though our priors for these variables were exponential. In fact, the posterior distributions are not really of any form that we recognize from the original model. But that's OK! This is one of the benefits of taking a computational point of view. If we had instead done this analysis using mathematical approaches, we would have been stuck with an analytically intractable (and messy) distribution. Our use of a computational approach makes us indifferent to mathematical tractability.\n\nOur analysis also returned a distribution for $\\tau$. Its posterior distribution looks a little different from the other two because it is a discrete random variable, so it doesn't assign probabilities to intervals. We can see that near day 45, there was a 50% chance that the user's behaviour changed. Had no change occurred, or had the change been gradual over time, the posterior distribution of $\\tau$ would have been more spread out, reflecting that many days were plausible candidates for $\\tau$. By contrast, in the actual results we see that only three or four days make any sense as potential transition points. \n\n### Why would I want samples from the posterior, anyways?\n\n\nWe will deal with this question for the remainder of the book, and it is an understatement to say that it will lead us to some amazing results. For now, let's end this chapter with one more example.\n\nWe'll use the posterior samples to answer the following question: what is the expected number of texts at day $t, \\; 0 \\le t \\le 70$ ? Recall that the expected value of a Poisson variable is equal to its parameter $\\lambda$. Therefore, the question is equivalent to *what is the expected value of $\\lambda$ at time $t$*?\n\nIn the code below, let $i$ index samples from the posterior distributions. Given a day $t$, we average over all possible $\\lambda_i$ for that day $t$, using $\\lambda_i = \\lambda_{1,i}$ if $t \\lt \\tau_i$ (that is, if the behaviour change has not yet occurred), else we use $\\lambda_i = \\lambda_{2,i}$. \n\n\n```python\nfigsize(12.5, 5)\n# tau_samples, lambda_1_samples, lambda_2_samples contain\n# N samples from the corresponding posterior distribution\nN = tau_samples.shape[0]\nexpected_texts_per_day = np.zeros(n_count_data)\nfor day in range(0, n_count_data):\n # ix is a bool index of all tau samples corresponding to\n # the switchpoint occurring prior to value of 'day'\n ix = day < tau_samples\n # Each posterior sample corresponds to a value for tau.\n # for each day, that value of tau indicates whether we're \"before\"\n # (in the lambda1 \"regime\") or\n # \"after\" (in the lambda2 \"regime\") the switchpoint.\n # by taking the posterior sample of lambda1/2 accordingly, we can average\n # over all samples to get an expected value for lambda on that day.\n # As explained, the \"message count\" random variable is Poisson distributed,\n # and therefore lambda (the poisson parameter) is the expected value of\n # \"message count\".\n expected_texts_per_day[day] = (lambda_1_samples[ix].sum()\n + lambda_2_samples[~ix].sum()) / N\n\n\nplt.plot(range(n_count_data), expected_texts_per_day, lw=4, color=\"#E24A33\",\n label=\"expected number of text-messages received\")\nplt.xlim(0, n_count_data)\nplt.xlabel(\"Day\")\nplt.ylabel(\"Expected # text-messages\")\nplt.title(\"Expected number of text-messages received\")\nplt.ylim(0, 60)\nplt.bar(np.arange(len(count_data)), count_data, color=\"#348ABD\", alpha=0.65,\n label=\"observed texts per day\")\n\nplt.legend(loc=\"upper left\");\n```\n\nOur analysis shows strong support for believing the user's behavior did change ($\\lambda_1$ would have been close in value to $\\lambda_2$ had this not been true), and that the change was sudden rather than gradual (as demonstrated by $\\tau$'s strongly peaked posterior distribution). We can speculate what might have caused this: a cheaper text-message rate, a recent weather-to-text subscription, or perhaps a new relationship. (In fact, the 45th day corresponds to Christmas, and I moved away to Toronto the next month, leaving a girlfriend behind.)\n\n\n##### Exercises\n\n1\\. Using `lambda_1_samples` and `lambda_2_samples`, what is the mean of the posterior distributions of $\\lambda_1$ and $\\lambda_2$?\n\n\n```python\n# \nfigsize(12.5, 4)\nimport scipy.stats as stats\na = np.arange(50)\npoi = stats.poisson\ncolors = [\"#348ABD\", \"#A60628\"]\n\nplt.bar(a, poi.pmf(a, lambda_1_samples.mean()), color=colors[0], alpha=0.5,\n label=\"Posterior 1\")\nplt.bar(a, poi.pmf(a, lambda_2_samples.mean()), color=colors[1], alpha=0.5,\n label=\"Posterior 2\")\n\nplt.legend()\nplt.ylabel(\"Probability Mass\")\nplt.xlabel(\"Expected Text Messages Per Day\")\nplt.show()\n# \n\nmean_1 = int(lambda_1_samples.mean())\nmean_2 = int(lambda_2_samples.mean())\nprint(\"Lambda 1 Mean = {} sms/day\".format(mean_1))\nprint(\"Lambda 2 Mean = {} sms/day\".format(mean_2))\n```\n\n2\\. What is the expected percentage increase in text-message rates? `hint:` compute the mean of `lambda_1_samples/lambda_2_samples`. Note that this quantity is very different from `lambda_1_samples.mean()/lambda_2_samples.mean()`.\n\n\n```python\n# The consensus here is that the authors are asserting that computing the\n# ratio of the lambda samples must be completed before taking the mean\n# value. Taking the mean of the samples first produces a mathematically\n# similar value, but one which is alogorithmically incorrect.\n\npmf_x = lambda_1_samples/lambda_2_samples\nperc_increase = (1.0 - pmf_x.mean()) * 100.0\nprint(\"Percentage Increase around Tau = {:.2f}%\".format(perc_increase))\n```\n\n Percentage Increase around Tau = 21.75%\n\n\n3\\. What is the mean of $\\lambda_1$ **given** that we know $\\tau$ is less than 45. That is, suppose we have been given new information that the change in behaviour occurred prior to day 45. What is the expected value of $\\lambda_1$ now? (You do not need to redo the PyMC3 part. Just consider all instances where `tau_samples < 45`.)\n\n\n```python\n# The group consensus here is that the mean of lambda_1 should not change\n# in the case that Tau is asserted to be less that 45 because ALL values of\n# Tau (as reported by tau_samples) are less than 45.\n\nprint(\"Tau Max value = {!r}\".format(tau_samples.max()))\n```\n\n Tau Max value = 44\n\n\n### References\n\n\n- [1] Gelman, Andrew. N.p.. Web. 22 Jan 2013. [N is never large enough](http://andrewgelman.com/2005/07/n_is_never_large).\n- [2] Norvig, Peter. 2009. [The Unreasonable Effectiveness of Data](http://static.googleusercontent.com/media/research.google.com/en//pubs/archive/35179.pdf).\n- [3] Salvatier, J, Wiecki TV, and Fonnesbeck C. (2016) Probabilistic programming in Python using PyMC3. *PeerJ Computer Science* 2:e55 \n- [4] Jimmy Lin and Alek Kolcz. Large-Scale Machine Learning at Twitter. Proceedings of the 2012 ACM SIGMOD International Conference on Management of Data (SIGMOD 2012), pages 793-804, May 2012, Scottsdale, Arizona.\n- [5] Cronin, Beau. \"Why Probabilistic Programming Matters.\" 24 Mar 2013. Google, Online Posting to Google . Web. 24 Mar. 2013. .\n", "meta": {"hexsha": "c82a82249b88c8d4b95ea3a8e0c65e6c69bb077f", "size": 315337, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Chapter1_Introduction/ssg_group_ch1.ipynb", "max_stars_repo_name": "amstewart/ni-rnd-ssg", "max_stars_repo_head_hexsha": "a388cc469d657789c128076d96e65ed8da8a7ca0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Chapter1_Introduction/ssg_group_ch1.ipynb", "max_issues_repo_name": "amstewart/ni-rnd-ssg", "max_issues_repo_head_hexsha": "a388cc469d657789c128076d96e65ed8da8a7ca0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Chapter1_Introduction/ssg_group_ch1.ipynb", "max_forks_repo_name": "amstewart/ni-rnd-ssg", "max_forks_repo_head_hexsha": "a388cc469d657789c128076d96e65ed8da8a7ca0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 308.5489236791, "max_line_length": 91628, "alphanum_fraction": 0.8985656615, "converted": true, "num_tokens": 11141, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.7745833789613196, "lm_q1q2_score": 0.4204928761937947}} {"text": "# Qiskit Aer: Pulse simulation of two qubits using a Duffing oscillator model\n\nThis notebook shows how to use the Qiskit Aer pulse simulator, which simulates experiments specified as pulse `Schedule` objects at the Hamiltonian level. The simulator solves the Schrodinger equation for a specified Hamiltonian model and pulse `Schedule` in the frame of the drift Hamiltonian.\n\nIn particular, in this tutorial we will: \n- Construct a model of a two qubit superconducting system.\n- Calibrate $\\pi$ pulses on each qubit in the simulated system.\n- Observe cross-resonance oscillations when driving qubit 1 with target qubit 0.\n\nThe Introduction outlines the concepts and flow of this notebook.\n\n\n## 1. Introduction \n\nThe main sections proceed as follows.\n\n### Section 3: Duffing oscillator model\n\nTo simulate a physical system, it is necessary to specify a model. In this notebook, we will model superconducting qubits as a collection of *Duffing oscillators*. The model is specified in terms of the following parameters:\n\n- Each Duffing oscillator is specified by a frequency $\\nu$, anharmonicity $\\alpha$, and drive strength $r$, which result in the Hamiltonian terms:\n\\begin{equation}\n 2\\pi\\nu a^\\dagger a + \\pi \\alpha a^\\dagger a(a^\\dagger a - 1) + 2 \\pi r (a + a^\\dagger) \\times D(t),\n\\end{equation}\nwhere $D(t)$ is the signal on the drive channel for the qubit, and $a^\\dagger$ and $a$ are, respectively, the creation and annihilation operators for the qubit. Note that the drive strength $r$ sets the scaling of the control term, with $D(t)$ assumed to be a complex and unitless number satisfying $|D(t)| \\leq 1$. \n- A coupling between a pair of oscillators $(l,k)$ is specified by the coupling strength $J$, resulting in an exchange coupling term:\n\\begin{equation}\n 2 \\pi J (a_l a_k^\\dagger + a_l^\\dagger a_k),\n\\end{equation}\nwhere the subscript denotes which qubit the operators act on.\n- Additionally, for numerical simulation, it is necessary to specify a cutoff dimension; the Duffing oscillator model is *infinite dimensional*, and computer simulation requires restriction of the operators to a finite dimensional subspace.\n\n**In the code:** We will define a model of the above form for two coupled qubits using the helper function `duffing_system_model`.\n\n### Section 4: $\\pi$-pulse calibration using Ignis\n\nOnce the model is defined, we will calibrate $\\pi$-pulses on each qubit. A $\\pi$-pulse is defined as a pulse on the drive channel of a qubit that \"flips\" the qubit; i.e. that takes the ground state to the first excited state, and the first excited state to the ground state.\n\nWe will experimentally find a $\\pi$-pulse for each qubit using the following procedure:\n- A fixed pulse shape is set - in this case it will be a Gaussian pulse.\n- A sequence of experiments is run, each consisting of a Gaussian pulse on the qubit, followed by a measurement, with each experiment in the sequence having a subsequently larger amplitude for the Gaussian pulse.\n- The measurement data is fit, and the pulse amplitude that completely flips the qubit is found (i.e. the $\\pi$-pulse amplitude).\n\n**In the code:** Using Ignis we will construct `Schedule` objects for the above experiments, then fit the data to find the $\\pi$-pulse amplitudes. \n\n### Section 5: Cross-resonance oscillations\n\nOnce the $\\pi$-pulses are calibrated, we will simulate the effects of cross-resonance driving on qubit $1$ with target qubit $0$. This means that we will drive qubit $1$ at the frequency of qubit $0$, with the goal of observing that the trajectory and oscillations of qubit $0$ *depends* on the state of qubit $1$. This phenomenon provides a basis for creating two-qubit *controlled* gates. Note: This section requires the calibration of the $\\pi$-pulse in Section 4.\n\nTo observe cross-resonance driving, we will use experiments very similar to the $\\pi$-pulse calibration case:\n- Initially, qubit $1$ is either left in the ground state, or is driven to its first excited state using the $\\pi$-pulse found in Section 4.\n- A sequence of experiments is run, each consisting of a Gaussian pulse on qubit $1$ driven at the frequency of qubit $0$, followed by a measurement of both qubits, with each experiment of the sequence having a subsequently larger amplitude for the Gaussian pulse.\n\n**In the code:** Functions for defining the experiments and visualizing the data are constructed, including a visualization of the trajectory of the target qubit on the Bloch sphere.\n\n## 2. Imports \n\nThis notebook makes use of the following imports.\n\n\n```python\nimport numpy as np\nfrom scipy.optimize import curve_fit, root\n\n# visualization tools\nimport matplotlib.pyplot as plt\nfrom qiskit.visualization.bloch import Bloch\n```\n\nImport qiskit libraries for working with `pulse` and calibration:\n\n\n```python\nimport qiskit.pulse as pulse\nfrom qiskit.pulse.commands.parametric_pulses import Gaussian, GaussianSquare\nfrom qiskit.compiler import assemble\n\nfrom qiskit.ignis.characterization.calibrations import rabi_schedules, RabiFitter\n```\n\nImports for qiskit pulse simulator: \n\n\n```python\n# The pulse simulator\nfrom qiskit.providers.aer import PulseSimulator\n\n# function for constructing duffing models\nfrom qiskit.providers.aer.pulse import duffing_system_model\n```\n\n## 3. Duffing oscillator system model \n\nAn object representing a model for a collection of Duffing oscillators can be constructed using the `duffing_system_model` function. Here we construct a $2$ Duffing oscillator model with cutoff dimension $3$.\n\n\n```python\n# cutoff dimension\ndim_oscillators = 3\n\n# frequencies for transmon drift terms, harmonic term and anharmonic term\n# Number of oscillators in the model is determined from len(oscillator_freqs)\noscillator_freqs = [5.0e9, 5.2e9]\nanharm_freqs = [-0.33e9, -0.33e9]\n\n# drive strengths\ndrive_strengths = [0.02e9, 0.02e9]\n\n# specify coupling as a dictionary (qubits 0 and 1 are coupled with a coefficient 0.002e9)\ncoupling_dict = {(0,1): 0.002e9}\n\n# sample duration for pulse instructions \ndt = 1e-9\n\n# create the model\ntwo_qubit_model = duffing_system_model(dim_oscillators=dim_oscillators,\n oscillator_freqs=oscillator_freqs,\n anharm_freqs=anharm_freqs,\n drive_strengths=drive_strengths,\n coupling_dict=coupling_dict,\n dt=dt)\n```\n\nThe function `duffing_system_model` returns a `PulseSystemModel` object, which is a general object for storing model information required for simulation with the `PulseSimulator`.\n\n## 4 Calibrating $\\pi$ pulses on each qubit using Ignis \n\nAs described in the introduction, we now calibrate $\\pi$ pulses on each qubit in `two_qubit_model`. The experiments in this calibration procedure are known as *Rabi experiments*, and the data we will observe are known as *Rabi oscillations*.\n\n### 4.1 Constructing the schedules\n\nWe construct the schedules using the `rabi_schedules` function in Ignis. To do this, we need to supply an `InstructionScheduleMap` containing a measurement schedule.\n\n\n```python\n# list of qubits to be used throughout the notebook\nqubits = [0, 1]\n\n# Construct a measurement schedule and add it to an InstructionScheduleMap\nmeas_amp = 0.025\nmeas_samples = 1200\nmeas_sigma = 4\nmeas_width = 1150\nmeas_pulse = GaussianSquare(duration=meas_samples, amp=meas_amp,\n sigma=meas_sigma, width=meas_width)\n\nacq_cmd = pulse.Acquire(duration=meas_samples)\nacq_sched = acq_cmd(pulse.AcquireChannel(0), pulse.MemorySlot(0))\nacq_sched += acq_cmd(pulse.AcquireChannel(1), pulse.MemorySlot(1))\n\nmeasure_sched = meas_pulse(pulse.MeasureChannel(0)) | meas_pulse(pulse.MeasureChannel(1)) | acq_sched\n\ninst_map = pulse.InstructionScheduleMap()\ninst_map.add('measure', qubits, measure_sched)\n```\n\nNext, construct the Rabi schedules.\n\n\n```python\n# construct Rabi experiments\ndrive_amps = np.linspace(0, 0.9, 41)\ndrive_sigma = 16\ndrive_duration = 128\ndrive_channels = [pulse.DriveChannel(qubit) for qubit in qubits]\n\n\nrabi_experiments, rabi_amps = rabi_schedules(amp_list=drive_amps, \n qubits=qubits, \n pulse_width=drive_duration, \n pulse_sigma=drive_sigma,\n drives=drive_channels,\n inst_map=inst_map,\n meas_map=[[0, 1]])\n```\n\nThe `Schedule`s in `rabi_schedules` correspond to experiments to generate Rabi oscillations on both qubits in parallel. Each experiment consists of a Gaussian pulse on the qubits of a given magnitude, followed by measurement.\n\nFor example:\n\n\n```python\nrabi_experiments[10].draw()\n```\n\n### 4.2 Simulate the Rabi experiments\n\nTo simulate the Rabi experiments, assemble the `Schedule` list into a qobj. When assembling, pass the `PulseSimulator` as the backend.\n\nHere, we want to use local oscillators with frequencies automatically computed from Duffing model Hamiltonian.\n\n\n```python\n# instantiate the pulse simulator\nbackend_sim = PulseSimulator()\n\n# compute frequencies from the Hamiltonian\nqubit_lo_freq = two_qubit_model.hamiltonian.get_qubit_lo_from_drift()\n\nrabi_qobj = assemble(rabi_experiments, \n backend=backend_sim, \n qubit_lo_freq=qubit_lo_freq,\n meas_level=1, \n meas_return='avg',\n shots=512)\n```\n\n /Users/dpuzzuoli/anaconda3/envs/QiskitDev/lib/python3.7/site-packages/qiskit/providers/models/pulsedefaults.py:166: UserWarning: `qubit_freq_est` and `meas_freq_est` now have units of Hertz(Hz) rather than gigahertz(GHz).\n warnings.warn('`qubit_freq_est` and `meas_freq_est` now have units of '\n\n\nRun the simulation using the simulator backend.\n\n\n```python\n# run the simulation\nrabi_result = backend_sim.run(rabi_qobj, two_qubit_model).result()\n```\n\n### 4.3 Fit and plot the data\n\nNext, we use `RabiFitter` in Ignis to fit the data, extract the $\\pi$-pulse amplitude, and then plot the data.\n\n\n```python\nrabifit = RabiFitter(rabi_result, rabi_amps, qubits, fit_p0 = [0.5,0.5,0.6,1.5])\n\nplt.figure(figsize=(15, 10))\nq_offset = 0\nfor qubit in qubits:\n ax = plt.subplot(2, 2, qubit + 1)\n rabifit.plot(qubit, ax=ax)\n print('Pi Amp: %f'%rabifit.pi_amplitude(qubit))\nplt.show()\n```\n\nPlotted is the averaged IQ data for observing each qubit. Observe that here, each qubit oscillates between the 0 and 1 state. The amplitude at which a given qubit reaches the peak of the oscillation is the desired $\\pi$-pulse amplitude.\n\n## 5. Oscillations from cross-resonance drive \n\nNext, we simulate the effects of a cross-resonance drive on qubit $1$ with target qubit $0$, observing that the trajectory and oscillations of qubit $0$ *depends* on the state of qubit $1$.\n\n**Note:** This section depends on the $\\pi$-pulse calibrations of Section 2.\n\n### 5.1 Cross-resonance `ControlChannel` indices\n\nDriving qubit $1$ at the frequency of qubit $0$ requires use of a pulse `ControlChannel`. The model generating function `duffing_system_model`, automatically sets up `ControlChannels` for performing cross-resonance drives between pairs of coupled qubits. The index of the `ControlChannel` for performing a particular cross-resonance drive is retrievable using the class method `control_channel_index` on the returned `PulseSystemModel`. For example, to get the `ControlChannel` index corresponding to a CR drive on qubit 1 with target 0, call the function `control_channel_index` with the tuple `(1,0)`:\n\n\n```python\ntwo_qubit_model.control_channel_index((1,0))\n```\n\n\n\n\n 1\n\n\n\nHence, to perform a cross-resonance drive on qubit $1$ with target qubit $0$, use `ControlChannel(1)`. This will be made use of when constructing `Schedule` objects in this section.\n\n### 5.2 Functions to generate the experiment list, and analyze the output\n\nFirst, we define a function `cr_drive_experiments`, which, given the drive and target indices, and the option to either start with the drive qubit in the ground or excited state, returns a list of experiments for observing the oscillations.\n\n\n```python\n# store the pi amplitudes from Section 2 in a list\npi_amps = [rabifit.pi_amplitude(0), rabifit.pi_amplitude(1)]\n\ndef cr_drive_experiments(drive_idx, \n target_idx, \n flip_drive_qubit = False,\n cr_drive_amps=np.linspace(0, 0.9, 41),\n cr_drive_samples=600,\n cr_drive_sigma=4,\n pi_drive_samples=128,\n pi_drive_sigma=16):\n \"\"\"Generate schedules corresponding to CR drive experiments.\n\n Args:\n drive_idx (int): label of driven qubit\n target_idx (int): label of target qubit\n flip_drive_qubit (bool): whether or not to start the driven qubit in the ground or excited state\n cr_drive_amps (array): list of drive amplitudes to use\n cr_drive_samples (int): number samples for each CR drive signal\n cr_drive_sigma (float): standard deviation of CR Gaussian pulse\n pi_drive_samples (int): number samples for pi pulse on drive\n pi_drive_sigma (float): standard deviation of Gaussian pi pulse on drive\n\n Returns:\n list[Schedule]: A list of Schedule objects for each experiment\n \"\"\"\n \n # Construct measurement commands to be used for all schedules\n meas_amp = 0.025\n meas_samples = 1200\n meas_sigma = 4\n meas_width = 1150\n meas_pulse = GaussianSquare(duration=meas_samples, amp=meas_amp,\n sigma=meas_sigma, width=meas_width)\n\n acq_cmd = pulse.Acquire(duration=meas_samples)\n acq_sched = acq_cmd(pulse.AcquireChannel(0), pulse.MemorySlot(0))\n acq_sched += acq_cmd(pulse.AcquireChannel(1), pulse.MemorySlot(1))\n \n # create measurement schedule\n measure_sched = meas_pulse(pulse.MeasureChannel(0)) | meas_pulse(pulse.MeasureChannel(1)) | acq_sched\n \n # Create schedule\n schedules = []\n for ii, cr_drive_amp in enumerate(cr_drive_amps):\n \n # pulse for flipping drive qubit if desired\n pi_pulse = Gaussian(duration=pi_drive_samples, amp=pi_amps[drive_idx], sigma=pi_drive_sigma)\n\n # cr drive pulse\n cr_width = cr_drive_samples - 2*cr_drive_sigma*4\n cr_rabi_pulse = GaussianSquare(duration=cr_drive_samples, \n amp=cr_drive_amp, \n sigma=cr_drive_sigma,\n width=cr_width)\n\n # add commands to schedule\n schedule = pulse.Schedule(name='cr_rabi_exp_amp_%s' % cr_drive_amp)\n\n # flip drive qubit if desired\n if flip_drive_qubit:\n schedule += pi_pulse(pulse.DriveChannel(drive_idx))\n \n # do cr drive\n # First, get the ControlChannel index for CR drive from drive to target\n cr_idx = two_qubit_model.control_channel_index((drive_idx, target_idx))\n schedule += cr_rabi_pulse(pulse.ControlChannel(cr_idx)) << schedule.duration\n \n \n schedule += measure_sched << schedule.duration\n\n schedules.append(schedule)\n return schedules\n```\n\nNext we create two functions for observing the data:\n- `plot_cr_pop_data` - for plotting the oscillations between the ground state and the first excited state\n- `plot_bloch_sphere` - for viewing the trajectory of the target qubit on the bloch sphere\n\n\n```python\ndef plot_cr_pop_data(drive_idx, \n target_idx, \n sim_result, \n cr_drive_amps=np.linspace(0, 0.9, 41)):\n \"\"\"Plot the population of each qubit.\n\n Args:\n drive_idx (int): label of driven qubit\n target_idx (int): label of target qubit\n sim_result (Result): results of simulation\n cr_drive_amps (array): list of drive amplitudes to use for axis labels\n \"\"\"\n \n amp_data_Q0 = []\n amp_data_Q1 = []\n\n for exp_idx in range(len(cr_drive_amps)):\n exp_mem = sim_result.get_memory(exp_idx)\n amp_data_Q0.append(np.abs(exp_mem[0]))\n amp_data_Q1.append(np.abs(exp_mem[1]))\n\n plt.plot(cr_drive_amps, amp_data_Q0, label='Q0')\n plt.plot(cr_drive_amps, amp_data_Q1, label='Q1')\n plt.legend()\n plt.xlabel('Pulse amplitude, a.u.', fontsize=20)\n plt.ylabel('Signal, a.u.', fontsize=20)\n plt.title('CR (Target Q{0}, driving on Q{1})'.format(target_idx, drive_idx), fontsize=20)\n plt.grid(True)\n\ndef bloch_vectors(drive_idx, drive_energy_level, sim_result):\n \"\"\"Plot the population of each qubit.\n\n Args:\n drive_idx (int): label of driven qubit\n drive_energy_level (int): energy level of drive qubit at start of CR drive\n sim_result (Result): results of simulation\n \n Returns:\n list: list of Bloch vectors corresponding to the final state of the target qubit\n for each experiment\n \"\"\"\n \n # get the dimension used for simulation\n dim = int(np.sqrt(len(sim_result.get_statevector(0))))\n \n \n # get the relevant dressed state indices\n idx0 = 0\n idx1 = 0\n if drive_idx == 0:\n if drive_energy_level == 0:\n idx0, idx1 = 0, dim\n elif drive_energy_level == 1:\n idx0, idx1 = 1, dim + 1\n if drive_idx == 1:\n if drive_energy_level == 0:\n idx0, idx1 = 0, 1\n elif drive_energy_level == 1:\n idx0, idx1 = dim, dim + 1\n\n # construct Pauli operators for correct dressed manifold\n state0 = np.array([two_qubit_model.hamiltonian._estates[idx0]])\n state1 = np.array([two_qubit_model.hamiltonian._estates[idx1]])\n \n outer01 = np.transpose(state0)@state1\n outer10 = np.transpose(state1)@state0\n outer00 = np.transpose(state0)@state0\n outer11 = np.transpose(state1)@state1\n \n X = outer01 + outer10\n Y = -1j*outer01 + 1j*outer10\n Z = outer00 - outer11\n \n # function for computing a single bloch vector\n bloch_vec = lambda vec: np.real(np.array([np.conj(vec)@X@vec, np.conj(vec)@Y@vec, np.conj(vec)@Z@vec]))\n \n return [bloch_vec(sim_result.get_statevector(idx)) for idx in range(len(sim_result.results))]\n\ndef plot_bloch_sphere(bloch_vectors):\n \"\"\"Given a list of Bloch vectors, plot them on the Bloch sphere\n\n Args:\n bloch_vectors (list): list of bloch vectors\n \"\"\"\n sphere = Bloch()\n sphere.add_points(np.transpose(bloch_vectors))\n sphere.show()\n```\n\n### 5.3 Drive qubit 1 to observe CR oscillations on qubit 0\n\n#### Qubit 1 in the ground state\n\nFirst, we drive with both qubit 0 and qubit 1 in the ground state.\n\n\n```python\n# construct experiments\ndrive_idx = 1\ntarget_idx = 0\nflip_drive = False\nexperiments = cr_drive_experiments(drive_idx, target_idx, flip_drive)\n\n# compute frequencies from the Hamiltonian\nqubit_lo_freq = two_qubit_model.hamiltonian.get_qubit_lo_from_drift()\n\n# assemble the qobj\ncr_rabi_qobj = assemble(experiments,\n backend=backend_sim,\n qubit_lo_freq=qubit_lo_freq,\n meas_level=1, \n meas_return='avg',\n shots=512)\n```\n\nRun the simulation:\n\n\n```python\nsim_result = backend_sim.run(cr_rabi_qobj, two_qubit_model).result()\n\nplot_cr_pop_data(drive_idx, target_idx, sim_result)\n```\n\nObserve that qubit 1 remains in the ground state, while excitations are driven in qubit 0.\n\nWe may also observe the trajectory of qubit 0 on the Bloch sphere:\n\n\n```python\nbloch_vecs = bloch_vectors(drive_idx, int(flip_drive), sim_result)\nplot_bloch_sphere(bloch_vecs)\n```\n\n#### Qubit 1 in the first excited state\n\nNext, we again perform a CR drive qubit 1 with qubit 0 as the target, but now we start each experiment by flipping qubit 1 into the first excited state. \n\n\n```python\n# construct experiments, now with flip_drive == True\ndrive_idx = 1\ntarget_idx = 0\nflip_drive = True\nexperiments = cr_drive_experiments(drive_idx, target_idx, flip_drive)\n\n# compute frequencies from the Hamiltonian\nqubit_lo_freq = two_qubit_model.hamiltonian.get_qubit_lo_from_drift()\n\n# assemble the qobj\ncr_rabi_qobj = assemble(experiments,\n backend=backend_sim,\n qubit_lo_freq=qubit_lo_freq,\n meas_level=1, \n meas_return='avg',\n shots=512)\n```\n\n\n```python\nsim_result = backend_sim.run(cr_rabi_qobj, two_qubit_model).result()\n\nplot_cr_pop_data(drive_idx, target_idx, sim_result)\n```\n\nObserve that now qubit 1 is in the excited state, while oscillations are again being driven on qubit 0, now at a different rate as before.\n\nAgain, observe the trajectory of qubit 0 on the Bloch sphere:\n\n\n```python\nbloch_vecs = bloch_vectors(drive_idx, int(flip_drive), sim_result)\nplot_bloch_sphere(bloch_vecs)\n```\n\nHere we see that qubit 0 takes a *different* trajectory on the Bloch sphere when qubit 1 is in the excited state. This is what enables controlled operations between two qubits.\n\n\n```python\nimport qiskit.tools.jupyter\n%qiskit_version_table\n%qiskit_copyright\n```\n\n\n

                                        Version Information

                                        Qiskit SoftwareVersion
                                        Qiskit0.13.0
                                        Terra0.13.0
                                        Aer0.5.0
                                        Ignis0.3.0
                                        Aqua0.6.1
                                        IBM Q Provider0.4.6
                                        System information
                                        Python3.7.6 (default, Jan 8 2020, 13:42:34) \n[Clang 4.0.1 (tags/RELEASE_401/final)]
                                        OSDarwin
                                        CPUs8
                                        Memory (Gb)32.0
                                        Wed Feb 19 12:12:09 2020 EST
                                        \n\n\n\n

                                        This code is a part of Qiskit

                                        © Copyright IBM 2017, 2020.

                                        This code is licensed under the Apache License, Version 2.0. You may
                                        obtain a copy of this license in the LICENSE.txt file in the root directory
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                                        Any modifications or derivative works of this code must retain this
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                                        Table of Contents

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                                        \n\n\n```python\n# code for loading the format for the notebook\nimport os\n\n# path : store the current path to convert back to it later\npath = os.getcwd()\nos.chdir(os.path.join('..', 'notebook_format'))\n\nfrom formats import load_style\nload_style(plot_style=False)\n```\n\n\n\n\n\n\n\n\n\n\n\n```python\nos.chdir(path)\n\n# 1. magic to print version\n# 2. magic so that the notebook will reload external python modules\n%load_ext watermark\n%load_ext autoreload\n%autoreload 2\n\nfrom collections import namedtuple\nfrom ortools.linear_solver import pywraplp\n\n%watermark -a 'Ethen' -d -t -v -p ortools\n```\n\n Author: Ethen\n \n Python implementation: CPython\n Python version : 3.7.11\n IPython version : 7.27.0\n \n ortools: 9.0.9048\n \n\n\n# Operation Research Quick Intro Via Ortools\n\nThe way to think about operation research, or optimization problem is we want to maximizie/minimize some objective, while being subjected to certain constraints.\n\nFor example, say we are deciding whether to buy ice cream or boba tea for dessert. Each type of food has an associated value, and cost, while we have a certain budget that we don't wish to exceed.\n\n\\begin{align}\n& \\text{maximize}\n&& \\text{value}_{\\text{ice_cream}} \\cdot \\text{ice_cream} + \\text{value}_{\\text{boba}} \\cdot \\text{boba} \\nonumber \\\\\n& \\text{subject to}\n&& \\text{cost}_{\\text{ice_cream}} \\cdot \\text{ice_cream} + \\text{cost}_{\\text{boba}} \\cdot \\text{boba} <= \\text{budget}\n\\end{align}\n\nSay we are able to replace the value, cost, and budget part with actual numbers (in practice, assigning actual numbers to each of these coefficients is often times core pieces of the work).\n\n\\begin{align}\n& \\text{maximize}\n&& 3 \\cdot \\text{ice_cream} + 2 \\cdot \\text{boba} \\nonumber \\\\\n& \\text{subject to}\n&& 2 \\cdot \\text{ice_cream} + 1 \\cdot \\text{boba} <= 1\n\\end{align}\n\nGiven this toy problem, we can eye ball the solution, and see that we should use our limited budget to buy a boba tea for dessert. Operation research, a.k.a optimization techniques helps us algorithmically find solutions for these types of problems at a much larger scale.\n\nThe following section, uses `ortools` library to solve this problem programmatically.\n\n\n```python\nbudget = 1\nDessertInfo = namedtuple('DessertInfo', ['name', 'value', 'cost'])\ndessert_infos = [\n DessertInfo('ice_cream', 3, 2),\n DessertInfo('boba', 2, 1),\n]\nnum_desserts = len(dessert_infos)\ndessert_infos\n```\n\n\n\n\n [DessertInfo(name='ice_cream', value=3, cost=2),\n DessertInfo(name='boba', value=2, cost=1)]\n\n\n\n\n```python\n# creates solver\nsolver = pywraplp.Solver.CreateSolver('GLOP')\n\n# creates variables\nvariables = [solver.NumVar(0, solver.infinity(), dessert_info.name) for dessert_info in dessert_infos]\n\n# define constraints\nconstraint_coefficients = [dessert_info.cost for dessert_info in dessert_infos]\nconstraint = [constraint_coefficients[i] * variables[i] for i in range(num_desserts)]\nsolver.Add(solver.Sum(constraint) <= budget)\n\n# define objective\nobjective_coefficients = [dessert_info.value for dessert_info in dessert_infos]\nobjective = constraint = [objective_coefficients[i] * variables[i] for i in range(num_desserts)]\nsolver.Maximize(solver.Sum(objective))\n\n# solve\nstatus = solver.Solve()\n\n# extract optimal/feasible value\nif status == pywraplp.Solver.OPTIMAL or status == pywraplp.Solver.FEASIBLE:\n optimal_value = solver.Objective().Value()\n print(f'Optimal Value: = {optimal_value}')\n for i in range(num_desserts):\n print(variables[i].name(), variables[i].solution_value())\n```\n\n Optimal Value: = 2.0\n ice_cream 0.0\n boba 1.0\n\n\nA couple of important things to note:\n\n- We are solving a Linear Programming problem, where we are computing the best solution to a given problem modeled as a series of linear relationships.\n- In this article, we won't be diving into the algorithms/solvers that are the workhorse behind the scenes that's finding the solution for us, and focus on how to frame the optimization problem.\n- We didn't explicitly specify this in our optimization formula, but notice the definition of `NumVar` specifies that our variables can take on numeric solutions. Often times, our problem might require some of the variables to be integers, these are called Mixed Integer Programming. e.g. In our example, we probably can't buy 1.5 portion of boba. In these cases, we can specify our variables to be `IntVar`.\n- There're other open sourced frameworks other than `ortools` out there, feel free to pick and choose based on preferences or speed. The exact API might be different, but the main idea revolves around defining the objective, defining the variables, adding the constraints, solving it and extracting the optimal/feasible solution.\n\n## Assignment Problem\n\nContinuing with our discussions around Mixed Integer Programming, a closely related problem is the assignment problem, where our variables involves boolean decisions of 0 and 1 values.\n\nWe'll use the examples from this blog post, [Blog: Towards optimal personalization: synthesisizing machine learning and operations research](https://www.ethanrosenthal.com/2016/08/30/towards-optimal-personalization/).\n\nSay we are working in the marketing team, and we have different types of churn prevention channel, each having different prices, while different users/customers' retention rate is different for each channel. Our constraint is not spending above our monthly marketing budget, and the goal is to maxmize the total number of retained customers.\n\n\n\\begin{align}\n\\text{maximize}\n& \\sum_{u, c} R_{u, c} A_{u, c} \\nonumber \\\\\n\\text{subject to}\n& \\sum_{u, c} P_{u, c} A_{u, c} <= B \\\\\n& \\sum_{c} A_{u, c} = 1, \\forall u \\in U \\\\\n& a_{u, c} \\in \\{0, 1\\}\n\\end{align}\n\nWhere:\n\n- $U$: is the set of all users.\n- $C$: is the set of all channels.\n- $R_{u, c}$: is the rentention probability if we were to notify the user, $u$, using the channel $c$.\n- $A_{u, c}$: is the assignment boolean decision variable, i.e. it takes on the value of 1 if we decided to reach out to user $u$ with channel $c$, 0 otherwise.\n- $P_{u, c}$: is the price/cost if we were to notify the user, $u$, using the channel $c$.\n- We have a constraint saying each customer can only receive the retention message via one channel, to prevent bombarding them.\n- As well as a constraint saying our cost shouldn't exceed our monthly budget $B$.\n\nLet's say we have 4 channels: email (0.25), push notification (0.3), text message (0.85), and phone call (5.0). Number in parenthesis indicates the cost/price.\nAs for the retention probability, we will be using some randomly generated numbers, but imagine in real world scenarios where this can come from aggregated historical information, or even generated by some machine learning models.\n\n\n```python\nbudget = 1000\nprice = [25, 30, 85, 250]\n\n# rentention probability for each customer and channel pair\nretention_prob = [\n [0.02, 0.27, 0.17, 0.87],\n [0.14, 0.21, 0.28, 0.014],\n [0.13, 0.003, 0.016, 0.64],\n [0.14, 0.04, 0.14, 0.26],\n [0.04, 0.24, 0.11, 0.31],\n]\nnum_users = len(retention_prob)\nnum_channels = len(retention_prob[0])\n```\n\n\n```python\n# creates the solver for the mixed integer programming\nsolver = pywraplp.Solver.CreateSolver('SCIP')\n\n# variable: assignment problem, creating a dictionary of binary variables\nvariables = {}\nfor i in range(num_users):\n for j in range(num_channels):\n variables[i, j] = solver.IntVar(0, 1, f'prob{i}_{j}')\n```\n\n\n```python\n# constraint: each user is assigned to at most 1 channel.\nfor i in range(num_users):\n solver.Add(solver.Sum([variables[i, j] for j in range(num_channels)]) <= 1)\n\n# constraint: total cost should not exceed budget\nconstraints = []\nfor j in range(num_channels):\n for i in range(num_users):\n constraint = price[j] * variables[i, j]\n constraints.append(constraint)\n\nsolver.Add(solver.Sum(constraints) <= budget) \n```\n\n\n\n\n >\n\n\n\n\n```python\n# objective\nobjective_terms = []\nfor i in range(num_users):\n for j in range(num_channels):\n objective_terms.append(retention_prob[i][j] * variables[i, j])\n\nsolver.Maximize(solver.Sum(objective_terms))\n```\n\n\n```python\n# invokes the solver\nstatus = solver.Solve()\n```\n\n\n```python\nif status == pywraplp.Solver.OPTIMAL or status == pywraplp.Solver.FEASIBLE:\n optimal_value = solver.Objective().Value()\n print(f'Optimal Value: = {optimal_value}')\n for i in range(num_users):\n for j in range(num_channels):\n # check indicator variable's value, with tolerance for floating point arithmetic\n if variables[i, j].solution_value() > 0.5:\n print(f'User {i} assigned to Channel {j}, Cost = {price[j]}')\n```\n\n Optimal Value: = 2.29\n User 0 assigned to Channel 3, Cost = 250\n User 1 assigned to Channel 2, Cost = 85\n User 2 assigned to Channel 3, Cost = 250\n User 3 assigned to Channel 3, Cost = 250\n User 4 assigned to Channel 1, Cost = 30\n\n\nIn this article, we took a sneak peak into some problems that can benefit from leveraging optimization. The problems that we deal with in real world settings can be a lot more complicated than the examples seen here, but hopefully, this gives you the idea that whenever we see a problem that involves maximizing some objectives given some constraint, we have a tool at hand that we can turn to.\n\n# Reference\n\n- [Blog: I'm all about ML, but let's talk about OR](https://www.ethanrosenthal.com/2016/07/20/lets-talk-or/)\n- [Blog: Towards optimal personalization: synthesisizing machine learning and operations research](https://www.ethanrosenthal.com/2016/08/30/towards-optimal-personalization/)\n- [Blog: Add Constrained Optimization To Your Toolbelt](https://multithreaded.stitchfix.com/blog/2018/06/21/constrained-optimization/)\n- [Or Tools Documentation: Solving an Assignment Problem](https://developers.google.com/optimization/assignment/assignment_example)\n- [Notes: Transformations in Integer Programming](https://ocw.mit.edu/courses/sloan-school-of-management/15-053-optimization-methods-in-management-science-spring-2013/tutorials/MIT15_053S13_tut09.pdf)\n", "meta": {"hexsha": "6c64394ae3c12e7d1a302ea119a9b94d7df90c24", "size": 24129, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "operation_research/ortools.ipynb", "max_stars_repo_name": "ethen8181/machine-learning", "max_stars_repo_head_hexsha": "bc1584d26a4732240056f12f7fa9adaad4f8bc0f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2104, "max_stars_repo_stars_event_min_datetime": "2016-04-15T13:35:55.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-28T10:39:51.000Z", "max_issues_repo_path": "operation_research/ortools.ipynb", "max_issues_repo_name": "ethen8181/machine-learning", "max_issues_repo_head_hexsha": "bc1584d26a4732240056f12f7fa9adaad4f8bc0f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 10, "max_issues_repo_issues_event_min_datetime": "2017-04-07T14:25:23.000Z", "max_issues_repo_issues_event_max_datetime": "2021-05-18T03:16:15.000Z", "max_forks_repo_path": "operation_research/ortools.ipynb", "max_forks_repo_name": "ethen8181/machine-learning", "max_forks_repo_head_hexsha": "bc1584d26a4732240056f12f7fa9adaad4f8bc0f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 539, "max_forks_repo_forks_event_min_datetime": "2015-12-10T04:23:44.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-31T07:15:28.000Z", "avg_line_length": 36.5037821483, "max_line_length": 649, "alphanum_fraction": 0.5241410751, "converted": true, "num_tokens": 4207, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.7799929104825006, "lm_q1q2_score": 0.42040309100541867}} {"text": "```python\n%matplotlib inline\n```\n\n\n```python\n\nWord Embeddings: Encoding Lexical Semantics\n===========================================\n\nWord embeddings are dense vectors of real numbers, one per word in your\nvocabulary. In NLP, it is almost always the case that your features are\nwords! But how should you represent a word in a computer? You could\nstore its ascii character representation, but that only tells you what\nthe word *is*, it doesn't say much about what it *means* (you might be\nable to derive its part of speech from its affixes, or properties from\nits capitalization, but not much). Even more, in what sense could you\ncombine these representations? We often want dense outputs from our\nneural networks, where the inputs are $|V|$ dimensional, where\n$V$ is our vocabulary, but often the outputs are only a few\ndimensional (if we are only predicting a handful of labels, for\ninstance). How do we get from a massive dimensional space to a smaller\ndimensional space?\n\nHow about instead of ascii representations, we use a one-hot encoding?\nThat is, we represent the word $w$ by\n\n\\begin{align}\\overbrace{\\left[ 0, 0, \\dots, 1, \\dots, 0, 0 \\right]}^\\text{|V| elements}\\end{align}\n\n\nwhere the 1 is in a location unique to $w$. Any other word will\nhave a 1 in some other location, and a 0 everywhere else.\n\nUsing a vector of words representation results in a bag of words model. documents are then a 1\ncreating a term-document representation. \n\nThis is a sparse model; use sparse methods for memory efficiency. \n\nThere is an enormous drawback to this representation, besides just how\nhuge it is. It basically treats all words as independent entities with\nno relation to each other. What we really want is some notion of\n*similarity* between words. Why? Let's see an example.\n\nSuppose we are building a language model. Suppose we have seen the\nsentences\n\n* The mathematician ran to the store.\n* The physicist ran to the store.\n* The mathematician solved the open problem.\n\nin our training data. Now suppose we get a new sentence never before\nseen in our training data:\n\n* The physicist solved the open problem.\n\nOur language model might do OK on this sentence, but wouldn't it be much\nbetter if we could use the following two facts:\n\n* We have seen mathematician and physicist in the same role in a sentence. Somehow they\n have a semantic relation.\n* We have seen mathematician in the same role in this new unseen sentence\n as we are now seeing physicist.\n\nand then infer that physicist is actually a good fit in the new unseen\nsentence? This is what we mean by a notion of similarity: we mean\n*semantic similarity*, not simply having similar orthographic\nrepresentations. It is a technique to combat the sparsity of linguistic\ndata, by connecting the dots between what we have seen and what we\nhaven't. This example of course relies on a fundamental linguistic\nassumption: that words appearing in similar contexts are related to each\nother semantically. This is called the `distributional\nhypothesis `__.\n\n\\begin{align}q_\\text{mathematician} = \\left[ \\overbrace{2.3}^\\text{can run},\n \\overbrace{9.4}^\\text{likes coffee}, \\overbrace{-5.5}^\\text{majored in Physics}, \\dots \\right]\\end{align}\n\n\nGetting Dense Word Embeddings\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\nHow can we solve this problem? That is, how could we actually encode\nsemantic similarity in words? Maybe we think up some semantic\nattributes. \n\nFor example, we see that both mathematicians and physicists\ncan run, so maybe we give these words a high score for the \"is able to\nrun\" semantic attribute. Think of some other attributes, and imagine\nwhat you might score some common words on those attributes.\n\nIf each attribute is a dimension, then we might give each word a vector,\nlike this:\n\n\\begin{align}q_\\text{mathematician} = \\left[ \\overbrace{2.3}^\\text{can run},\n \\overbrace{9.4}^\\text{likes coffee}, \\overbrace{-5.5}^\\text{majored in Physics}, \\dots \\right]\\end{align}\n\n\\begin{align}q_\\text{physicist} = \\left[ \\overbrace{2.5}^\\text{can run},\n \\overbrace{9.1}^\\text{likes coffee}, \\overbrace{6.4}^\\text{majored in Physics}, \\dots \\right]\\end{align}\n\nThen we can get a measure of similarity between these words by doing:\n\n\\begin{align}\\text{Similarity}(\\text{physicist}, \\text{mathematician}) = q_\\text{physicist} \\cdot q_\\text{mathematician}\\end{align}\n\nAlthough it is more common to normalize by the lengths:\n\n\\begin{align}\\text{Similarity}(\\text{physicist}, \\text{mathematician}) = \\frac{q_\\text{physicist} \\cdot q_\\text{mathematician}}\n {\\| q_\\text{\\physicist} \\| \\| q_\\text{mathematician} \\|} = \\cos (\\phi)\\end{align}\n\nWhere $\\phi$ is the angle between the two vectors. That way,\nextremely similar words (words whose embeddings point in the same\ndirection) will have similarity 1. Extremely dissimilar words should\nhave similarity -1.\n\n\nYou can think of the sparse one-hot vectors from the beginning of this\nsection as a special case of these new vectors we have defined, where\neach word basically has similarity 0, and we gave each word some unique\nsemantic attribute. These new vectors are *dense*, which is to say their\nentries are (typically) non-zero.\n\nBut these new vectors are a big pain: you could think of thousands of\ndifferent semantic attributes that might be relevant to determining\nsimilarity, and how on earth would you set the values of the different\nattributes? Central to the idea of deep learning is that the neural\nnetwork learns representations of the features, rather than requiring\nthe programmer to design them herself. So why not just let the word\nembeddings be parameters in our model, and then be updated during\ntraining? This is exactly what we will do. We will have some *latent\nsemantic attributes* that the network can, in principle, learn. Note\nthat the word embeddings will probably not be interpretable. That is,\nalthough with our hand-crafted vectors above we can see that\nmathematicians and physicists are similar in that they both like coffee,\nif we allow a neural network to learn the embeddings and see that both\nmathematicians and physicisits have a large value in the second\ndimension, it is not clear what that means. They are similar in some\nlatent semantic dimension, but this probably has no interpretation to\nus.\n\n\nIn summary, **word embeddings are a representation of the *semantics* of\na word, efficiently encoding semantic information that might be relevant\nto the task at hand**. You can embed other things too: part of speech\ntags, parse trees, anything! The idea of feature embeddings is central\nto the field.\n\n\nWord Embeddings in Pytorch\n~~~~~~~~~~~~~~~~~~~~~~~~~~\n\nBefore we get to a worked example and an exercise, a few quick notes\nabout how to use embeddings in Pytorch and in deep learning programming\nin general. Similar to how we defined a unique index for each word when\nmaking one-hot vectors, we also need to define an index for each word\nwhen using embeddings. These will be keys into a lookup table. That is,\nembeddings are stored as a $|V| \\times D$ matrix, where $D$\nis the dimensionality of the embeddings, such that the word assigned\nindex $i$ has its embedding stored in the $i$'th row of the\nmatrix. In all of my code, the mapping from words to indices is a\ndictionary named word\\_to\\_ix.\n\nThe module that allows you to use embeddings is torch.nn.Embedding,\nwhich takes two arguments: the vocabulary size, and the dimensionality\nof the embeddings.\n\nTo index into this table, you must use torch.LongTensor (since the\nindices are integers, not floats).\n\n\n\n```\n\n\\begin{align}q_\\text{mathematician} = \\left[ \\overbrace{2.3}^\\text{can run},\n \\overbrace{9.4}^\\text{likes coffee}, \\overbrace{-5.5}^\\text{majored in Physics}, \\dots \\right]\\end{align}\n\n\n\n```python\n#using 1d convolutions to make word2vec\n\n\n\n```\n\n\n```python\n\n\nimport torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport torch.optim as optim\n\ntorch.manual_seed(1)\n```\n\n\n```python\nword_to_ix = {\"hello\": 0, \"world\": 1}\nembeds = nn.Embedding(2, 5) # 2 words in vocab, 5 dimensional embeddings\nlookup_tensor = torch.tensor([word_to_ix[\"hello\"]], dtype=torch.long)\nhello_embed = embeds(lookup_tensor)\nprint(hello_embed)\n```\n\nAn Example: N-Gram Language Modeling\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\nRecall that in an n-gram language model, given a sequence of words\n$w$, we want to compute\n\n\\begin{align}P(w_i | w_{i-1}, w_{i-2}, \\dots, w_{i-n+1} )\\end{align}\n\nWhere $w_i$ is the ith word of the sequence.\n\nIn this example, we will compute the loss function on some training\nexamples and update the parameters with backpropagation.\n\n\n\n\n\n```python\nCONTEXT_SIZE = 2\nEMBEDDING_DIM = 10\n# We will use Shakespeare Sonnet 2\ntest_sentence = \"\"\"When forty winters shall besiege thy brow,\nAnd dig deep trenches in thy beauty's field,\nThy youth's proud livery so gazed on now,\nWill be a totter'd weed of small worth held:\nThen being asked, where all thy beauty lies,\nWhere all the treasure of thy lusty days;\nTo say, within thine own deep sunken eyes,\nWere an all-eating shame, and thriftless praise.\nHow much more praise deserv'd thy beauty's use,\nIf thou couldst answer 'This fair child of mine\nShall sum my count, and make my old excuse,'\nProving his beauty by succession thine!\nThis were to be new made when thou art old,\nAnd see thy blood warm when thou feel'st it cold.\"\"\".split()\n# we should tokenize the input, but we will ignore that for now\n# build a list of tuples. Each tuple is ([ word_i-2, word_i-1 ], target word)\ntrigrams = [([test_sentence[i], test_sentence[i + 1]], test_sentence[i + 2])\n for i in range(len(test_sentence) - 2)]\n# print the first 3, just so you can see what they look like\nprint(trigrams[:3])\n\nvocab = set(test_sentence)\nword_to_ix = {word: i for i, word in enumerate(vocab)}\n\n\nclass NGramLanguageModeler(nn.Module):\n\n def __init__(self, vocab_size, embedding_dim, context_size):\n super(NGramLanguageModeler, self).__init__()\n self.embeddings = nn.Embedding(vocab_size, embedding_dim)\n self.linear1 = nn.Linear(context_size * embedding_dim, 128)\n self.linear2 = nn.Linear(128, vocab_size)\n\n def forward(self, inputs):\n embeds = self.embeddings(inputs).view((1, -1))\n out = F.relu(self.linear1(embeds))\n out = self.linear2(out)\n log_probs = F.log_softmax(out, dim=1)\n return log_probs\n\n\nlosses = []\nloss_function = nn.NLLLoss()\nmodel = NGramLanguageModeler(len(vocab), EMBEDDING_DIM, CONTEXT_SIZE)\noptimizer = optim.SGD(model.parameters(), lr=0.001)\n\nfor epoch in range(10):\n total_loss = torch.Tensor([0])\n for context, target in trigrams:\n\n # Step 1. Prepare the inputs to be passed to the model (i.e, turn the words\n # into integer indices and wrap them in variables)\n context_idxs = torch.tensor([word_to_ix[w] for w in context], dtype=torch.long)\n\n # Step 2. Recall that torch *accumulates* gradients. Before passing in a\n # new instance, you need to zero out the gradients from the old\n # instance\n model.zero_grad()\n\n # Step 3. Run the forward pass, getting log probabilities over next\n # words\n log_probs = model(context_idxs)\n\n # Step 4. Compute your loss function. (Again, Torch wants the target\n # word wrapped in a variable)\n loss = loss_function(log_probs, torch.tensor([word_to_ix[target]], dtype=torch.long))\n\n # Step 5. Do the backward pass and update the gradient\n loss.backward()\n optimizer.step()\n\n # Get the Python number from a 1-element Tensor by calling tensor.item()\n total_loss += loss.item()\n losses.append(total_loss)\nprint(losses) # The loss decreased every iteration over the training data!\n```\n\nExercise: Computing Word Embeddings: Continuous Bag-of-Words\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\nThe Continuous Bag-of-Words model (CBOW) is frequently used in NLP deep\nlearning. It is a model that tries to predict words given the context of\na few words before and a few words after the target word. This is\ndistinct from language modeling, since CBOW is not sequential and does\nnot have to be probabilistic. Typcially, CBOW is used to quickly train\nword embeddings, and these embeddings are used to initialize the\nembeddings of some more complicated model. Usually, this is referred to\nas *pretraining embeddings*. It almost always helps performance a couple\nof percent.\n\nThe CBOW model is as follows. Given a target word $w_i$ and an\n$N$ context window on each side, $w_{i-1}, \\dots, w_{i-N}$\nand $w_{i+1}, \\dots, w_{i+N}$, referring to all context words\ncollectively as $C$, CBOW tries to minimize\n\n\\begin{align}-\\log p(w_i | C) = -\\log \\text{Softmax}(A(\\sum_{w \\in C} q_w) + b)\\end{align}\n\nwhere $q_w$ is the embedding for word $w$.\n\nImplement this model in Pytorch by filling in the class below. Some\ntips:\n\n* Think about which parameters you need to define.\n* Make sure you know what shape each operation expects. Use .view() if you need to\n reshape.\n\n\n\n\n\n```python\nCONTEXT_SIZE = 2 # 2 words to the left, 2 to the right\nraw_text = \"\"\"We are about to study the idea of a computational process.\nComputational processes are abstract beings that inhabit computers.\nAs they evolve, processes manipulate other abstract things called data.\nThe evolution of a process is directed by a pattern of rules\ncalled a program. People create programs to direct processes. In effect,\nwe conjure the spirits of the computer with our spells.\"\"\".split()\n\n# By deriving a set from `raw_text`, we deduplicate the array\nvocab = set(raw_text)\nvocab_size = len(vocab)\n\nword_to_ix = {word: i for i, word in enumerate(vocab)}\ndata = []\nfor i in range(2, len(raw_text) - 2):\n context = [raw_text[i - 2], raw_text[i - 1],\n raw_text[i + 1], raw_text[i + 2]]\n target = raw_text[i]\n data.append((context, target))\nprint(data[:5])\n\n\nclass CBOW(nn.Module):\n\n def __init__(self):\n pass\n\n def forward(self, inputs):\n pass\n\n# create your model and train. here are some functions to help you make\n# the data ready for use by your module\n\n\ndef make_context_vector(context, word_to_ix):\n idxs = [word_to_ix[w] for w in context]\n return torch.tensor(idxs, dtype=torch.long)\n\n\nmake_context_vector(data[0][0], word_to_ix) # example\n```\n", "meta": {"hexsha": "6aa9d989c3cfe69fc363a1605b6108b2c778df17", "size": 18895, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "twitter_nlp/ethan/word_embeddings_tutorial.ipynb", "max_stars_repo_name": "dougc333/DeepLearning", "max_stars_repo_head_hexsha": "0076f8490e25786494bbc7da54c21408c3c1aa7f", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "twitter_nlp/ethan/word_embeddings_tutorial.ipynb", "max_issues_repo_name": "dougc333/DeepLearning", "max_issues_repo_head_hexsha": "0076f8490e25786494bbc7da54c21408c3c1aa7f", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "twitter_nlp/ethan/word_embeddings_tutorial.ipynb", "max_forks_repo_name": "dougc333/DeepLearning", "max_forks_repo_head_hexsha": "0076f8490e25786494bbc7da54c21408c3c1aa7f", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 42.7488687783, "max_line_length": 148, "alphanum_fraction": 0.6103201905, "converted": true, "num_tokens": 3564, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.7799929002541068, "lm_q1q2_score": 0.42040308549248606}} {"text": "\n\n# Quantum numbers and Angular Momentum Algebra\n\n **Morten Hjorth-Jensen**, [National Superconducting Cyclotron Laboratory](http://www.nscl.msu.edu/) and [Department of Physics and Astronomy](https://www.pa.msu.edu/), [Michigan State University](http://www.msu.edu/), East Lansing, MI 48824, USA\n\nDate: **Jun 27, 2017**\n\nCopyright 2013-2017, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n\n\n\n## Quantum numbers\n**Outline.**\n\n* Discussion of single-particle and two-particle quantum numbers, uncoupled and coupled schemes\n\n* Discussion of angular momentum recouplings and the Wigner-Eckart theorem\n\n* Applications to specific operators like the nuclear two-body tensor force\n\nFor quantum numbers, chapter 1 on angular momentum and chapter 5 of Suhonen and chapters 5, 12 and 13 of Alex Brown. \nFor a discussion of isospin, see for example Alex Brown's lecture notes chapter 12, 13 and 19.\n\n\n\n## Motivation\nWhen solving the Hartree-Fock project using a nucleon-nucleon interaction in an uncoupled basis (m-scheme), we found a high level of degeneracy. One sees clear from the table here that we have a degeneracy in the angular momentum $j$, resulting in $2j+1$ states with the same energy. This reflects the rotational symmetry and spin symmetry of the nuclear forces. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                        Quantum numbers Energy [MeV]
                                        $0s_{1/2}^{\\pi}$ -40.4602
                                        $0s_{1/2}^{\\pi}$ -40.4602
                                        $0s_{1/2}^{\\nu}$ -40.6426
                                        $0s_{1/2}^{\\nu}$ -40.6426
                                        $0p_{1/2}^{\\pi}$ -6.7133
                                        $0p_{1/2}^{\\pi}$ -6.7133
                                        $0p_{1/2}^{\\nu}$ -6.8403
                                        $0p_{1/2}^{\\nu}$ -6.8403
                                        $0p_{3/2}^{\\pi}$ -11.5886
                                        $0p_{3/2}^{\\pi}$ -11.5886
                                        $0p_{3/2}^{\\pi}$ -11.5886
                                        $0p_{3/2}^{\\pi}$ -11.5886
                                        $0p_{3/2}^{\\nu}$ -11.7201
                                        $0p_{3/2}^{\\nu}$ -11.7201
                                        $0p_{3/2}^{\\nu}$ -11.7201
                                        $0p_{3/2}^{\\nu}$ -11.7201
                                        \nWe observe that with increasing value of $j$ the degeneracy increases. For $j=3/2$ we end up diagonalizing\nthe same matrix four times. With increasing value of $j$, it is rather obvious that our insistence on using\nan uncoupled scheme (or just m-scheme) will lead to unnecessary labor from our side (or more precisely, for the computer). The obvious question we should pose ourselves then is whether we can use \nthe underlying symmetries of the nuclear forces in order to reduce our efforts. \n\n## Single-particle and two-particle quantum numbers\nIn order to understand the basics of the nucleon-nucleon interaction and the pertaining symmetries, we need to define the relevant quantum numbers and how we build up a single-particle state and a two-body state, and obviously our final holy grail, a many-boyd state. \n\n* For the single-particle states, due to the fact that we have the spin-orbit force, the quantum numbers for the projection of orbital momentum $l$, that is $m_l$, and for spin $s$, that is $m_s$, are no longer so-called good quantum numbers. The total angular momentum $j$ and its projection $m_j$ are then so-called *good quantum numbers*.\n\n* This means that the operator $\\hat{J}^2$ does not commute with $\\hat{L}_z$ or $\\hat{S}_z$. \n\n* We also start normally with single-particle state functions defined using say the harmonic oscillator. For these functions, we have no explicit dependence on $j$. How can we introduce single-particle wave functions which have $j$ and its projection $m_j$ as quantum numbers?\n\n\n\n\n## Single-particle and two-particle quantum numbers, brief review on angular momenta etc\n\nWe have that the operators for the orbital momentum are given by\n\n0\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n1\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\nL_z=-i\\hbar(x\\frac{\\partial }{\\partial y}-y\\frac{\\partial }{\\partial x})=xp_y-yp_x.\n$$\n\n## Single-particle and two-particle quantum numbers, brief review on angular momenta etc\nSince we have a spin orbit force which is strong, it is easy to show that \nthe total angular momentum operator\n\n$$\n\\hat{J}=\\hat{L}+\\hat{S}\n$$\n\ndoes not commute with $\\hat{L}_z$ and $\\hat{S}_z$. To see this, we calculate for example\n\n$$\n\\begin{eqnarray} \n [\\hat{L}_z,\\hat{J}^2]&=&[\\hat{L}_z,(\\hat{L}+\\hat{S})^2] \\\\ \\nonumber\n &=&[\\hat{L}_z,\\hat{L}^2+\\hat{S}^2+2\\hat{L}\\hat{S}]\\\\ \\nonumber \n &=& [\\hat{L}_z,\\hat{L}\\hat{S}]=[\\hat{L}_z,\\hat{L}_x\\hat{S}_x+\\hat{L}_y\\hat{S}_y+\\hat{L}_z\\hat{S}_z]\\ne 0, \n\\end{eqnarray}\n$$\n\nsince we have that $[\\hat{L}_z,\\hat{L}_x]=i\\hbar\\hat{L}_y$ and $[\\hat{L}_z,\\hat{L}_y]=i\\hbar\\hat{L}_x$.\n\n\n\n## Single-particle and two-particle quantum numbers, brief review on angular momenta etc\nWe have also\n\n$$\n|\\hat{J}|=\\hbar\\sqrt{J(J+1)},\n$$\n\nwith the the following degeneracy\n\n$$\nM_J=-J, -J+1, \\dots, J-1, J.\n$$\n\nWith a given value of $L$ and $S$ we can then determine the possible values of \n $J$ by studying the $z$ component of $\\hat{J}$. \nIt is given by\n\n$$\n\\hat{J}_z=\\hat{L}_z+\\hat{S}_z.\n$$\n\nThe operators $\\hat{L}_z$ and $\\hat{S}_z$ have the quantum numbers\n$L_z=M_L\\hbar$ and $S_z=M_S\\hbar$, respectively, meaning that\n\n$$\nM_J\\hbar=M_L\\hbar +M_S\\hbar,\n$$\n\nor\n\n$$\nM_J=M_L +M_S.\n$$\n\nSince the max value of $M_L$ is $L$ and for $M_S$ is $S$\nwe obtain\n\n$$\n(M_J)_{\\mathrm{maks}}=L+S.\n$$\n\n## Single-particle and two-particle quantum numbers, brief review on angular momenta etc\nFor nucleons we have that the maximum value of $M_S=m_s=1/2$, yielding\n\n$$\n(m_j)_{\\mathrm{max}}=l+\\frac{1}{2}.\n$$\n\nUsing this and the fact that the maximum value of $M_J=m_j$ is $j$ we have\n\n$$\nj=l+\\frac{1}{2}, l-\\frac{1}{2}, l-\\frac{3}{2}, l-\\frac{5}{2}, \\dots\n$$\n\nTo decide where this series terminates, we use the vector inequality\n\n$$\n|\\hat{L}+\\hat{S}| \\ge \\left| |\\hat{L}|-|\\hat{S}|\\right|.\n$$\n\n## Single-particle and two-particle quantum numbers, brief review on angular momenta etc\nUsing $\\hat{J}=\\hat{L}+\\hat{S}$ we get\n\n$$\n|\\hat{J}| \\ge |\\hat{L}|-|\\hat{S}|,\n$$\n\nor\n\n$$\n|\\hat{J}|=\\hbar\\sqrt{J(J+1)}\\ge |\\hbar\\sqrt{L(L+1)}-\n \\hbar\\sqrt{S(S+1)}|.\n$$\n\n## Single-particle and two-particle quantum numbers, brief review on angular momenta etc\nIf we limit ourselves to nucleons only with $s=1/2$ we find that\n\n$$\n|\\hat{J}|=\\hbar\\sqrt{j(j+1)}\\ge |\\hbar\\sqrt{l(l+1)}-\n \\hbar\\sqrt{\\frac{1}{2}(\\frac{1}{2}+1)}|.\n$$\n\nIt is then easy to show that for nucleons there are only two possible values of\n$j$ which satisfy the inequality, namely\n\n$$\nj=l+\\frac{1}{2}\\hspace{0.1cm} \\mathrm{or} \\hspace{0.1cm}j=l-\\frac{1}{2},\n$$\n\nand with $l=0$ we get\n\n$$\nj=\\frac{1}{2}.\n$$\n\n## Single-particle and two-particle quantum numbers, brief review on angular momenta etc\nLet us study some selected examples. We need also to keep in mind that parity is conserved.\nThe strong and electromagnetic Hamiltonians conserve parity. Thus the eigenstates can be\nbroken down into two classes of states labeled by their parity $\\pi= +1$ or $\\pi=-1$.\nThe nuclear interactions do not mix states with different parity.\n\nFor nuclear structure the total parity originates\nfrom the intrinsic parity of the nucleon which is $\\pi_{\\mathrm{intrinsic}}=+1$ \nand the parities associated with\nthe orbital angular momenta $\\pi_l=(-1)^l$ . The total parity is the product over all nucleons\n$\\pi = \\prod_i \\pi_{\\mathrm{intrinsic}}(i)\\pi_l(i) = \\prod_i (-1)^{l_i}$\n\nThe basis states we deal with are constructed so that they conserve parity and have thus a definite parity. \n\nNote that we do have parity violating processes, more on this later although our focus will be mainly on non-parity viloating processes\n\n\n\n\n## Single-particle and two-particle quantum numbers\n\nConsider now the single-particle orbits of the $1s0d$ shell. \nFor a $0d$ state we have the quantum numbers $l=2$, $m_l=-2,-1,0,1,2$, $s+1/2$, $m_s=\\pm 1/2$,\n$n=0$ (the number of nodes of the wave function). This means that we have positive parity and\n\n$$\nj=\\frac{3}{2}=l-s\\hspace{1cm} m_j=-\\frac{3}{2},-\\frac{1}{2},\\frac{1}{2},\\frac{3}{2}.\n$$\n\nand\n\n$$\nj=\\frac{5}{2}=l+s\\hspace{1cm} m_j=-\\frac{5}{2},-\\frac{3}{2},-\\frac{1}{2},\\frac{1}{2},\\frac{3}{2},\\frac{5}{2}.\n$$\n\n## Single-particle and two-particle quantum numbers\nOur single-particle wave functions, if we use the harmonic oscillator, do however not contain the quantum numbers $j$ and $m_j$.\nNormally what we have is an eigenfunction for the one-body problem defined as\n\n$$\n\\phi_{nlm_lsm_s}(r,\\theta,\\phi)=R_{nl}(r)Y_{lm_l}(\\theta,\\phi)\\xi_{sm_s},\n$$\n\nwhere we have used spherical coordinates (with a spherically symmetric potential) and the spherical harmonics\n\n$$\nY_{lm_l}(\\theta,\\phi)=P(\\theta)F(\\phi)=\\sqrt{\\frac{(2l+1)(l-m_l)!}{4\\pi (l+m_l)!}}\n P_l^{m_l}(cos(\\theta))\\exp{(im_l\\phi)},\n$$\n\nwith $P_l^{m_l}$ being the so-called associated Legendre polynomials.\n\n\n\n\n\n## Single-particle and two-particle quantum numbers\nExamples are\n\n$$\nY_{00}=\\sqrt{\\frac{1}{4\\pi}},\n$$\n\nfor $l=m_l=0$,\n\n$$\nY_{10}=\\sqrt{\\frac{3}{4\\pi}}cos(\\theta),\n$$\n\nfor $l=1$ and $m_l=0$,\n\n$$\nY_{1\\pm 1}=\\sqrt{\\frac{3}{8\\pi}}sin(\\theta)exp(\\pm i\\phi),\n$$\n\nfor $l=1$ and $m_l=\\pm 1$,\n\n$$\nY_{20}=\\sqrt{\\frac{5}{16\\pi}}(3cos^2(\\theta)-1)\n$$\n\nfor $l=2$ and $m_l=0$ etc.\n\n\n\n\n\n\n## Single-particle and two-particle quantum numbers\nHow can we get a function in terms of $j$ and $m_j$?\nDefine now\n\n$$\n\\phi_{nlm_lsm_s}(r,\\theta,\\phi)=R_{nl}(r)Y_{lm_l}(\\theta,\\phi)\\xi_{sm_s},\n$$\n\nand\n\n$$\n\\psi_{njm_j;lm_lsm_s}(r,\\theta,\\phi),\n$$\n\nas the state with quantum numbers $jm_j$.\nOperating with\n\n$$\n\\hat{j}^2=(\\hat{l}+\\hat{s})^2=\\hat{l}^2+\\hat{s}^2+2\\hat{l}_z\\hat{s}_z+\\hat{l}_+\\hat{s}_{-}+\\hat{l}_{-}\\hat{s}_{+},\n$$\n\non the latter state we will obtain admixtures from possible $\\phi_{nlm_lsm_s}(r,\\theta,\\phi)$ states.\n\n\n\n\n## Single-particle and two-particle quantum numbers\nTo see this, we consider the following example and fix\n\n$$\nj=\\frac{3}{2}=l-s\\hspace{1cm} m_j=\\frac{3}{2}.\n$$\n\nand\n\n$$\nj=\\frac{5}{2}=l+s\\hspace{1cm} m_j=\\frac{3}{2}.\n$$\n\nIt means we can have, with $l=2$ and $s=1/2$ being fixed, in order to have $m_j=3/2$ either $m_l=1$ and $m_s=1/2$ or\n$m_l=2$ and $m_s=-1/2$. The two states\n\n$$\n\\psi_{n=0j=5/2m_j=3/2;l=2s=1/2}\n$$\n\nand\n\n$$\n\\psi_{n=0j=3/2m_j=3/2;l=2s=1/2}\n$$\n\nwill have admixtures from $\\phi_{n=0l=2m_l=2s=1/2m_s=-1/2}$ and $\\phi_{n=0l=2m_l=1s=1/2m_s=1/2}$. \nHow do we find these admixtures? Note that we don't specify the values of $m_l$ and $m_s$ \nin the functions $\\psi$ since \n$\\hat{j}^2$ does not commute with $\\hat{L}_z$ and $\\hat{S}_z$.\n\n\n\n## Single-particle and two-particle quantum numbers\nWe operate with\n\n$$\n\\hat{j}^2=(\\hat{l}+\\hat{s})^2=\\hat{l}^2+\\hat{s}^2+2\\hat{l}_z\\hat{s}_z+\\hat{l}_+\\hat{s}_{-}+\\hat{l}_{-}\\hat{s}_{+}\n$$\n\non the two $jm_j$ states, that is\n\n3\n5\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\beta\\hbar^2\\sqrt{l(l+1)-m_l(m_l-1)}\\phi_{n=0l=2m_l=1s=1/2m_s=1/2},\n$$\n\nand\n\n3\n7\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\beta\\hbar^2\\sqrt{l(l+1)-m_l(m_l+1)}\\phi_{n=0l=2m_l=2s=1/2m_s=-1/2}.\n$$\n\n## Single-particle and two-particle quantum numbers\nThis means that the eigenvectors $\\phi_{n=0l=2m_l=2s=1/2m_s=-1/2}$ etc are not eigenvectors of $\\hat{j}^2$. The above problems gives a $2\\times2$ matrix that mixes the vectors $\\psi_{n=0j=5/2m_j3/2;l=2m_ls=1/2m_s}$ and $\\psi_{n=0j=3/2m_j3/2;l=2m_ls=1/2m_s}$ with the states $\\phi_{n=0l=2m_l=2s=1/2m_s=-1/2}$ and\n$\\phi_{n=0l=2m_l=1s=1/2m_s=1/2}$. The unknown coefficients $\\alpha$ and $\\beta$ are the eigenvectors of this matrix. That is, inserting all values $m_l,l,m_s,s$ we obtain the matrix\n\n$$\n\\left[ \\begin{array} {cc} 19/4 & 2 \\\\ 2 & 31/4 \\end{array} \\right]\n$$\n\nwhose eigenvectors are the columns of\n\n$$\n\\left[ \\begin{array} {cc} 2/\\sqrt{5} &1/\\sqrt{5} \\\\ 1/\\sqrt{5} & -2/\\sqrt{5} \\end{array}\\right]\n$$\n\nThese numbers define the so-called Clebsch-Gordan coupling coefficients (the overlaps between the two basis sets). We can thus write\n\n$$\n\\psi_{njm_j;ls}=\\sum_{m_lm_s}\\langle lm_lsm_s|jm_j\\rangle\\phi_{nlm_lsm_s},\n$$\n\nwhere the coefficients $\\langle lm_lsm_s|jm_j\\rangle$ are the so-called Clebsch-Gordan coeffficients.\n\n\n\n## Clebsch-Gordan coefficients\nThe Clebsch-Gordan coeffficients $\\langle lm_lsm_s|jm_j\\rangle$ have some interesting properties for us, like the following orthogonality relations\n\n4\n2\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n4\n3\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\langle j_1m_1j_2m_2|JM\\rangle=(-1)^{j_1+j_2-J}\\langle j_2m_2j_1m_1|JM\\rangle,\n$$\n\nand many others. The latter will turn extremely useful when we are going to define two-body states and interactions in a coupled basis.\n\n\n\n\n## Clebsch-Gordan coefficients, testing the orthogonality relations\nThe orthogonality relation can be tested using the symbolic python package **wigner**. \nLet us test\n\n$$\n\\sum_{m_1m_2}\\langle j_1m_1j_2m_2|JM\\rangle\\langle j_1m_1j_2m_2|J'M'\\rangle=\\delta_{J,J'}\\delta_{M,M'},\n$$\n\nThe following program tests this relation for the case of $j_1=3/2$ and $j_2=3/2$ meaning that $m_1$ and $m_2$ \nrun from $-3/2$ to $3/2$.\n\n\n```\nfrom sympy import S\nfrom sympy.physics.wigner import clebsch_gordan\n# Twice the values of j1 and j2\nj1 = 3\nj2 = 3\nJ = 2\nJp = 2\nM = 2\nMp = 3\nsum = 0.0\nfor m1 in range(-j1, j1+2, 2):\n for m2 in range(-j2, j2+2, 2):\n M = (m1+m2)/2.\n \"\"\" Call j1, j2, J, m1, m2, m1+m2 \"\"\"\n sum += clebsch_gordan(S(j1)/2, S(j2)/2, J, S(m1)/2, S(m2)/2, M)*clebsch_gordan(S(j1)/2, S(j2)/2, Jp, S(m1)/2, S(m2)/2, Mp)\nprint sum\n```\n\n## Quantum numbers and the Schroeodinger equation in relative and CM coordinates\nSumming up, for \nfor the single-particle case, we have the following eigenfunctions\n\n$$\n\\psi_{njm_j;ls}=\\sum_{m_lm_s}\\langle lm_lsm_s|jm_j\\rangle\\phi_{nlm_lsm_s},\n$$\n\nwhere the coefficients $\\langle lm_lsm_s|jm_j\\rangle$ are the so-called Clebsch-Gordan coeffficients.\nThe relevant quantum numbers are $n$ (related to the principal quantum number and the number of nodes of the wave function) and\n\n4\n7\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n4\n8\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n4\n9\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\hat{s}^2\\psi_{njm_j;ls}=\\hbar^2s(s+1)\\psi_{njm_j;ls},\n$$\n\nbut $s_z$ and $l_z$ do not result in good quantum numbers in a basis where we\nuse the angular momentum $j$.\n\n\n\n## Quantum numbers and the Schroedinger equation in relative and CM coordinates\nFor a two-body state where we couple two angular momenta $j_1$ and $j_2$ to a final\nangular momentum $J$ with projection $M_J$, we can define a similar transformation in terms\nof the Clebsch-Gordan coeffficients\n\n$$\n\\psi_{(j_1j_2)JM_J}=\\sum_{m_{j_1}m_{j_2}}\\langle j_1m_{j_1}j_2m_{j_2}|JM_J\\rangle\\psi_{n_1j_1m_{j_1};l_1s_1}\\psi_{n_2j_2m_{j_2};l_2s_2}.\n$$\n\nWe will write these functions in a more compact form hereafter, namely,\n\n$$\n|(j_1j_2)JM_J\\rangle=\\psi_{(j_1j_2)JM_J},\n$$\n\nand\n\n$$\n|j_im_{j_i}\\rangle=\\psi_{n_ij_im_{j_i};l_is_i},\n$$\n\nwhere we have skipped the explicit reference to $l$, $s$ and $n$. The spin of a nucleon is always $1/2$ while the value of $l$ can be deduced from the parity of the state.\nIt is thus normal to label a state with a given total angular momentum as \n$j^{\\pi}$, where $\\pi=\\pm 1$.\n\n\n\n## Quantum numbers and the Schroedinger equation in relative and CM coordinates\nOur two-body state can thus be written as\n\n$$\n|(j_1j_2)JM_J\\rangle=\\sum_{m_{j_1}m_{j_2}}\\langle j_1m_{j_1}j_2m_{j_2}|JM_J\\rangle|j_1m_{j_1}\\rangle|j_2m_{j_2}\\rangle.\n$$\n\nDue to the coupling order of the Clebsch-Gordan coefficient it reads as \n$j_1$ coupled to $j_2$ to yield a final angular momentum $J$. If we invert the order of coupling we would have\n\n$$\n|(j_2j_1)JM_J\\rangle=\\sum_{m_{j_1}m_{j_2}}\\langle j_2m_{j_2}j_1m_{j_1}|JM_J\\rangle|j_1m_{j_1}\\rangle|j_2m_{j_2}\\rangle,\n$$\n\nand due to the symmetry properties of the Clebsch-Gordan coefficient we have\n\n$$\n|(j_2j_1)JM_J\\rangle=(-1)^{j_1+j_2-J}\\sum_{m_{j_1}m_{j_2}}\\langle j_1m_{j_1}j_2m_{j_2}|JM_J\\rangle|j_1m_{j_1}\\rangle|j_2m_{j_2}\\rangle=(-1)^{j_1+j_2-J}|(j_1j_2)JM_J\\rangle.\n$$\n\nWe call the basis $|(j_1j_2)JM_J\\rangle$ for the **coupled basis**, or just $j$-coupled basis/scheme. The basis formed by the simple product of single-particle eigenstates \n$|j_1m_{j_1}\\rangle|j_2m_{j_2}\\rangle$ is called the **uncoupled-basis**, or just the $m$-scheme basis.\n\n\n\n## Quantum numbers\nWe have thus the coupled basis\n\n$$\n|(j_1j_2)JM_J\\rangle=\\sum_{m_{j_1}m_{j_2}}\\langle j_1m_{j_1}j_2m_{j_2}|JM_J\\rangle|j_1m_{j_1}\\rangle|j_2m_{j_2}\\rangle.\n$$\n\nand the uncoupled basis\n\n$$\n|j_1m_{j_1}\\rangle|j_2m_{j_2}\\rangle.\n$$\n\nThe latter can easily be generalized to many single-particle states whereas the first \nneeds specific coupling coefficients and definitions of coupling orders. \nThe $m$-scheme basis is easy to implement numerically and is used in most standard shell-model codes. \nOur coupled basis obeys also the following relations\n\n5\n9\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\hat{J}_z|(j_1j_2)JM_J\\rangle=\\hbar M_J|(j_1j_2)JM_J\\rangle,\n$$\n\n## Components of the force and isospin\n The nuclear forces are almost charge independent. If we assume they are, \nwe can introduce a new quantum number which is conserved. For nucleons only, that is a proton and neutron, we can limit ourselves\nto two possible values which allow us to distinguish between the two particles. If we assign an isospin value of $\\tau=1/2$ for protons\nand neutrons (they belong to an isospin doublet, in the same way as we discussed the spin $1/2$ multiplet), we can define \nthe neutron to have isospin projection $\\tau_z=+1/2$ and a proton to have $\\tau_z=-1/2$. These assignements are the standard choices in low-energy nuclear physics.\n\n\n\n## Isospin\nThis leads to the introduction of an additional quantum number called isospin.\nWe can define a single-nucleon\nstate function in terms of the quantum numbers $n$, $j$, $m_j$, $l$, $s$, $\\tau$ and $\\tau_z$. Using our definitions in terms of an uncoupled basis, we had\n\n$$\n\\psi_{njm_j;ls}=\\sum_{m_lm_s}\\langle lm_lsm_s|jm_j\\rangle\\phi_{nlm_lsm_s},\n$$\n\nwhich we can now extend to\n\n$$\n\\psi_{njm_j;ls}\\xi_{\\tau\\tau_z}=\\sum_{m_lm_s}\\langle lm_lsm_s|jm_j\\rangle\\phi_{nlm_lsm_s}\\xi_{\\tau\\tau_z},\n$$\n\nwith the isospin spinors defined as\n\n$$\n\\xi_{\\tau=1/2\\tau_z=+1/2}=\\left(\\begin{array}{c} 1 \\\\ 0\\end{array}\\right),\n$$\n\nand\n\n$$\n\\xi_{\\tau=1/2\\tau_z=-1/2}=\\left(\\begin{array}{c} 0 \\\\ 1\\end{array}\\right).\n$$\n\nWe can then define the proton state function as\n\n$$\n\\psi^p(\\mathbf{r}) =\\psi_{njm_j;ls}(\\mathbf{r})\\left(\\begin{array}{c} 0 \\\\ 1\\end{array}\\right),\n$$\n\nand similarly for neutrons as\n\n$$\n\\psi^n(\\mathbf{r}) =\\psi_{njm_j;ls}(\\mathbf{r})\\left(\\begin{array}{c} 1 \\\\ 0\\end{array}\\right).\n$$\n\n## Isospin\nWe can in turn define the isospin Pauli matrices (in the same as we define the spin matrices) as\n\n6\n7\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\hat{\\tau}_y =\\left(\\begin{array}{cc} 0 & -\\imath \\\\ \\imath & 0 \\end{array}\\right),\n$$\n\nand\n\n$$\n\\hat{\\tau}_z =\\left(\\begin{array}{cc} 1 & 0 \\\\ 0 & -1 \\end{array}\\right),\n$$\n\nand operating with $\\hat{\\tau}_z$ on the proton state function we have\n\n$$\n\\hat{\\tau}_z\\psi^p(\\mathbf{r})=-\\frac{1}{2}\\psi^p(\\mathbf{r}),\n$$\n\nand for neutrons we have\n\n$$\n\\hat{\\tau}\\psi^n(\\mathbf{r})=\\frac{1}{2}\\psi^n(\\mathbf{r}).\n$$\n\n## Isospin\nWe can now define the so-called charge operator as\n\n$$\n\\frac{\\hat{Q}}{e} = \\frac{1}{2}\\left(1-\\hat{\\tau}_z\\right)=\\begin{Bmatrix} 0 & 0 \\\\ 0 & 1 \\end{Bmatrix},\n$$\n\nwhich results in\n\n$$\n\\frac{\\hat{Q}}{e}\\psi^p(\\mathbf{r})=\\psi^p(\\mathbf{r}),\n$$\n\nand\n\n$$\n\\frac{\\hat{Q}}{e}\\psi^n(\\mathbf{r})=0,\n$$\n\nas it should be.\n\n\n\n## Isospin\nThe total isospin is defined as\n\n$$\n\\hat{T}=\\sum_{i=1}^A\\hat{\\tau}_i,\n$$\n\nand its corresponding isospin projection as\n\n$$\n\\hat{T}_z=\\sum_{i=1}^A\\hat{\\tau}_{z_i},\n$$\n\nwith eigenvalues $T(T+1)$ for $\\hat{T}$ and $1/2(N-Z)$ for $\\hat{T}_z$, where $N$ is the number of neutrons and $Z$ the number of protons. \n\nIf charge is conserved, the Hamiltonian $\\hat{H}$ commutes with $\\hat{T}_z$ and all members of a given isospin multiplet\n(that is the same value of $T$) have the same energy and there is no $T_z$ dependence and we say that $\\hat{H}$ is a scalar in isospin space.\n\n\n\n\n## Angular momentum algebra, Examples\nWe have till now seen the following definitions of a two-body matrix elements \nwith quantum numbers $p=j_pm_p$ etc we have a two-body state defined as\n\n$$\n|(pq)M\\rangle = a^{\\dagger}_pa^{\\dagger}_q|\\Phi_0\\rangle,\n$$\n\nwhere $|\\Phi_0\\rangle$ is a chosen reference state, say for example the Slater determinant which approximates \n${}^{16}\\mbox{O}$ with the $0s$ and the $0p$ shells being filled, and $M=m_p+m_q$. Recall that we label single-particle states above the Fermi level as $abcd\\dots$ and states below the Fermi level for $ijkl\\dots$. \nIn case of two-particles in the single-particle states $a$ and $b$ outside ${}^{16}\\mbox{O}$ as a closed shell core, say ${}^{18}\\mbox{O}$, \nwe would write the representation of the Slater determinant as\n\n$$\n|^{18}\\mathrm{O}\\rangle =|(ab)M\\rangle = a^{\\dagger}_aa^{\\dagger}_b|^{16}\\mathrm{O}\\rangle=|\\Phi^{ab}\\rangle.\n$$\n\nIn case of two-particles removed from say ${}^{16}\\mbox{O}$, for example two neutrons in the single-particle states $i$ and $j$, we would write this as\n\n$$\n|^{14}\\mathrm{O}\\rangle =|(ij)M\\rangle = a_ja_i|^{16}\\mathrm{O}\\rangle=|\\Phi_{ij}\\rangle.\n$$\n\n## Angular momentum algebra and many-body states\nFor a one-hole-one-particle state we have\n\n$$\n|^{16}\\mathrm{O}\\rangle_{1p1h} =|(ai)M\\rangle = a_a^{\\dagger}a_i|^{16}\\mathrm{O}\\rangle=|\\Phi_{i}^a\\rangle,\n$$\n\nand finally for a two-particle-two-hole state we\n\n$$\n|^{16}\\mathrm{O}\\rangle_{2p2h} =|(abij)M\\rangle = a_a^{\\dagger}a_b^{\\dagger}a_ja_i|^{16}\\mathrm{O}\\rangle=|\\Phi_{ij}^{ab}\\rangle.\n$$\n\n## Angular momentum algebra, two-body state and anti-symmetrized matrix elements\nLet us go back to the case of two-particles in the single-particle states $a$ and $b$ outside ${}^{16}\\mbox{O}$ as a closed shell core, say ${}^{18}\\mbox{O}$.\nThe representation of the Slater determinant is\n\n$$\n|^{18}\\mathrm{O}\\rangle =|(ab)M\\rangle = a^{\\dagger}_aa^{\\dagger}_b|^{16}\\mathrm{O}\\rangle=|\\Phi^{ab}\\rangle.\n$$\n\nThe anti-symmetrized matrix element is detailed as\n\n$$\n\\langle (ab) M | \\hat{V} | (cd) M \\rangle = \\langle (j_am_aj_bm_b)M=m_a+m_b | \\hat{V} | (j_cm_cj_dm_d)M=m_a+m_b \\rangle,\n$$\n\nand note that anti-symmetrization means\n\n8\n4\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\langle (ab) M | \\hat{V} | (cd) M \\rangle =-\\langle (ab) M | \\hat{V} | (dc) M \\rangle.\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem, Examples\nThis matrix element is given by\n\n$$\n\\langle ^{16}\\mathrm{O}|a_ba_a\\frac{1}{4}\\sum_{pqrs}\\langle (pq) M | \\hat{V} | (rs) M' \\rangle a^{\\dagger}_pa^{\\dagger}_qa_sa_r a^{\\dagger}_ca^{\\dagger}_d|^{16}\\mathrm{O}\\rangle.\n$$\n\nWe can compute this matrix element using Wick's theorem.\n\n\n\n\n## Angular momentum algebra, Wigner-Eckart theorem, Examples\nWe have also defined matrix elements in the coupled basis, the so-called $J$-coupled scheme.\nIn this case the two-body wave function for two neutrons outside ${}^{16}\\mbox{O}$ is written as\n\n$$\n|^{18}\\mathrm{O}\\rangle_J =|(ab)JM\\rangle = \\left\\{a^{\\dagger}_aa^{\\dagger}_b\\right\\}^J_M|^{16}\\mathrm{O}\\rangle=N_{ab}\\sum_{m_am_b}\\langle j_am_aj_bm_b|JM\\rangle|\\Phi^{ab}\\rangle,\n$$\n\nwith\n\n$$\n|\\Phi^{ab}\\rangle=a^{\\dagger}_aa^{\\dagger}_b|^{16}\\mathrm{O}\\rangle.\n$$\n\nWe have now an explicit coupling order, where the angular momentum $j_a$ is coupled to the angular momentum $j_b$ to yield a final two-body angular momentum $J$. \nThe normalization factor is\n\n$$\nN_{ab}=\\frac{\\sqrt{1+\\delta_{ab}\\times (-1)^J}}{1+\\delta_{ab}}.\n$$\n\n## Angular momentum algebra\nWe\nnote that, using the anti-commuting \nproperties of the creation operators, we obtain\n\n$$\nN_{ab}\\sum_{m_am_b}\\langle j_am_aj_bm_b|JM\\rangle \\vert\\Phi^{ab}\\rangle=-N_{ab}\\sum_{m_am_b}\\langle j_am_aj_bm_b|JM\\rangle\\vert\\Phi^{ba}\\rangle.\n$$\n\nFurthermore, using the property of the Clebsch-Gordan coefficient\n\n$$\n\\langle j_am_aj_bm_b|JM>=(-1)^{j_a+j_b-J}\\langle j_bm_bj_am_a|JM\\rangle,\n$$\n\nwhich can be used to show that\n\n$$\n|(j_bj_a)JM\\rangle = \\left\\{a^{\\dagger}_ba^{\\dagger}_a\\right\\}^J_M|^{16}\\mathrm{O}\\rangle=N_{ab}\\sum_{m_am_b}\\langle j_bm_bj_am_a|JM\\rangle|\\Phi^{ba}\\rangle,\n$$\n\nis equal to\n\n$$\n|(j_bj_a)JM\\rangle=(-1)^{j_a+j_b-J+1}|(j_aj_b)JM\\rangle.\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem, Examples\nThe implementation of the Pauli principle looks different in the $J$-scheme compared with the $m$-scheme. In the latter, no two fermions or more can have the same set of quantum numbers. In the $J$-scheme, when we write a state with the shorthand\n\n$$\n|^{18}\\mathrm{O}\\rangle_J =|(ab)JM\\rangle,\n$$\n\nwe do refer to the angular momenta only. This means that another way of writing the last state is\n\n$$\n|^{18}\\mathrm{O}\\rangle_J =|(j_aj_b)JM\\rangle.\n$$\n\nWe will use this notation throughout when we refer to a two-body state in $J$-scheme. The Kronecker $\\delta$ function in the normalization factor \nrefers thus to the values of $j_a$ and $j_b$. If two identical particles are in a state with the same $j$-value, then only even values of the total angular momentum apply. In the notation below, when we label a state as $j_p$ it will actually represent all quantum numbers except $m_p$.\n\n\n\n\n\n\n## Angular momentum algebra, two-body matrix elements\nThe two-body matrix element is a scalar and since it obeys rotational symmetry, it is diagonal in $J$, \nmeaning that the corresponding matrix element in $J$-scheme is\n\n9\n6\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\times \\langle j_cm_cj_dm_d|JM\\rangle\\langle (j_am_aj_bm_b)M | \\hat{V} | (j_cm_cj_dm_d)M \\rangle,\n$$\n\nand note that of the four $m$-values in the above sum, only three are independent due to the constraint $m_a+m_b=M=m_c+m_d$.\n\n\n\n## Angular momentum algebra, two-body matrix element\nSince\n\n$$\n|(j_bj_a)JM\\rangle=(-1)^{j_a+j_b-J+1}|(j_aj_b)JM\\rangle,\n$$\n\nthe anti-symmetrized matrix elements need now to obey the following relations\n\n9\n9\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n1\n0\n0\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\langle (j_aj_b) JM | \\hat{V} | (j_cj_d) JM \\rangle = (-1)^{j_a+j_b+j_c+j_d}\\langle (j_bj_a) JM | \\hat{V} | (j_dj_c) JM \\rangle=\\langle (j_bj_a) JM | \\hat{V} | (j_dj_c) JM \\rangle,\n$$\n\nwhere the last relations follows from the fact that $J$ is an integer and $2J$ is always an even number.\n\n\n\n## Angular momentum algebra, two-body matrix element\nUsing the orthogonality properties of the Clebsch-Gordan coefficients,\n\n$$\n\\sum_{m_am_b}\\langle j_am_aj_bm_b|JM\\rangle\\langle j_am_aj_bm_b|J'M'\\rangle=\\delta_{JJ'}\\delta_{MM'},\n$$\n\nand\n\n$$\n\\sum_{JM}\\langle j_am_aj_bm_b|JM\\rangle\\langle j_am_a'j_bm_b'|JM\\rangle=\\delta_{m_am_a'}\\delta_{m_bm_b'},\n$$\n\nwe can also express the two-body matrix element in $m$-scheme in terms of that in $J$-scheme, that is, if we multiply with\n\n$$\n\\sum_{JMJ'M'}\\langle j_am_a'j_bm_b'|JM\\rangle\\langle j_cm_c'j_dm_d'|J'M'\\rangle\n$$\n\nfrom left in\n\n1\n0\n5\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\times \\langle (j_am_aj_bm_b)M| \\hat{V} | (j_cm_cj_dm_d)M\\rangle,\n$$\n\nwe obtain\n\n1\n0\n7\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\times \\langle (j_aj_b) JM | \\hat{V} | (j_cj_d) JM \\rangle.\n$$\n\n## The Hartree-Fock potential\nWe can now use the above relations to compute the Hartre-Fock energy in $j$-scheme. \nIn $m$-scheme we defined the Hartree-Fock energy as\n\n$$\n\\varepsilon_{pq}^{\\mathrm{HF}}=\\delta_{pq}\\varepsilon_p+ \\sum_{i\\le F} \\langle pi \\vert \\hat{V} \\vert qi\\rangle_{AS},\n$$\n\nwhere the single-particle states $pqi$ point to the quantum numbers in $m$-scheme. For a state with for example $j=5/2$, this results in six identical values for the above potential. We would obviously like to reduce this to one only by rewriting our equations in $j$-scheme. \n\nOur Hartree-Fock basis is orthogonal by definition, meaning that we have\n\n$$\n\\varepsilon_{p}^{\\mathrm{HF}}=\\varepsilon_p+ \\sum_{i\\le F} \\langle pi \\vert \\hat{V} \\vert pi\\rangle_{AS},\n$$\n\n## The Hartree-Fock potential\nWe have\n\n$$\n\\varepsilon_{p}^{\\mathrm{HF}}=\\varepsilon_p+ \\sum_{i\\le F} \\langle pi \\vert \\hat{V} \\vert pi\\rangle_{AS},\n$$\n\nwhere the single-particle states $p=[n_p,j_p,m_p,t_{z_{p}}]$. Let us assume that $p$ is a state above the Fermi level. The quantity $\\varepsilon_p$ could represent the harmonic oscillator single-particle energies. \n\nLet $p\\rightarrow a$. \n\nThe energies, as we have seen, are independent of $m_a$ and $m_i$. \nWe sum now over all $m_a$ on both sides of the above equation and divide by $2j_a+1$, recalling that $\\sum_{m_a}=2j_a+1$. This results in\n\n$$\n\\varepsilon_{a}^{\\mathrm{HF}}=\\varepsilon_a+ \\frac{1}{2j_a+1}\\sum_{i\\le F}\\sum_{m_a} \\langle ai \\vert \\hat{V}\\vert ai\\rangle_{AS},\n$$\n\n## The Hartree-Fock potential\nWe rewrite\n\n$$\n\\varepsilon_{a}^{\\mathrm{HF}}=\\varepsilon_a+ \\frac{1}{2j_a+1}\\sum_{i\\le F}\\sum_{m_a} \\langle ai \\vert \\hat{V} \\vert ai\\rangle_{AS},\n$$\n\nas\n\n$$\n\\varepsilon_{a}^{\\mathrm{HF}}=\\varepsilon_a+ \\frac{1}{2j_a+1}\\sum_{n_i,j_i,t_{z_{i}}\\le F}\\sum_{m_im_a} \n\\langle (j_am_aj_im_i)M | \\hat{V} | (j_am_aj_im_i)M\\rangle_{AS},\n$$\n\nwhere we have suppressed the dependence on $n_p$ and $t_z$ in the matrix element. \nUsing the definition\n\n$$\n\\langle (j_am_aj_bm_b)M \\vert \\hat{V} \\vert (j_cm_cj_dm_d)M\\rangle=\\frac{1}{N_{ab}N_{cd}}\\sum_{JM}\\langle j_am_aj_bm_b|JM\\rangle\\langle j_cm_cj_dm_d|JM\\rangle \\langle (j_aj_b)J \\vert \\hat{V} \\vert (j_cj_d)M\\rangle_{AS},\n$$\n\nwith the orthogonality properties of Glebsch-Gordan coefficients and that the $j$-coupled two-body matrix element is a scalar and independent of $M$ we arrive at\n\n$$\n\\varepsilon_{a}^{\\mathrm{HF}}=\\varepsilon_a+ \\frac{1}{2j_a+1}\\sum_{j_i\\le F}\\sum_J (2J+1)\n\\langle (j_aj_i)J \\vert \\hat{V} \\vert (j_aj_i)M\\rangle_{AS},\n$$\n\n## First order in the potential energy\nIn a similar way it is easy to show that the potential energy contribution to the ground state energy in $m$-scheme\n\n$$\n\\frac{1}{2}\\sum_{ij\\le F}\\langle (j_im_ij_jm_j)M | \\hat{V} | (j_im_ij_jm_j)M\\rangle_{AS},\n$$\n\ncan be rewritten as\n\n$$\n\\frac{1}{2}\\sum_{j_i,j_j\\le F}\\sum_J(2J+1)\n\\langle (j_ij_j)J | \\hat{V} | (j_ij_j)J\\rangle_{AS},\n$$\n\nThis reduces the number of floating point operations with an order of magnitude on average.\n\n\n\n\n\n\n\n\n\n\n## Angular momentum algebra\nWe are now going to define two-body and many-body states in an angular momentum coupled basis, the so-called $j$-scheme basis. In this connection\n* we need to define the so-called $6j$ and $9j$ symbols\n\n* as well as the the Wigner-Eckart theorem\n\nWe will also study some specific examples, like the calculation of the tensor force.\n\n\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nWe define an irreducible spherical tensor $T^{\\lambda}_{\\mu}$ of rank $\\lambda$ as an operator with $2\\lambda+1$ components $\\mu$ \nthat satisfies the commutation relations ($\\hbar=1$)\n\n$$\n[J_{\\pm}, T^{\\lambda}_{\\mu}]= \\sqrt{(\\lambda\\mp \\mu)(\\lambda\\pm \\mu+1)}T^{\\lambda}_{\\mu\\pm 1},\n$$\n\nand\n\n$$\n[J_{z}, T^{\\lambda}_{\\mu}]=\\mu T^{\\lambda}_{\\mu}.\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nOur angular momentum coupled two-body wave function obeys clearly this definition, namely\n\n$$\n|(ab)JM\\rangle = \\left\\{a^{\\dagger}_aa^{\\dagger}_b\\right\\}^J_M|\\Phi_0\\rangle=N_{ab}\\sum_{m_am_b}\\langle j_am_aj_bm_b|JM\\rangle|\\Phi^{ab}\\rangle,\n$$\n\nis a tensor of rank $J$ with $M$ components. Another well-known example is given by the spherical harmonics (see examples during today's lecture). \n\nThe product of two irreducible tensor operators\n\n$$\nT^{\\lambda_3}_{\\mu_3}=\\sum_{\\mu_1\\mu_2}\\langle \\lambda_1\\mu_1\\lambda_2\\mu_2|\\lambda_3\\mu_3\\rangle T^{\\lambda_1}_{\\mu_1}T^{\\lambda_2}_{\\mu_2}\n$$\n\nis also a tensor operator of rank $\\lambda_3$.\n\n\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nWe wish to apply the above definitions to the computations of a matrix element\n\n$$\n\\langle \\Phi^J_M|T^{\\lambda}_{\\mu}|\\Phi^{J'}_{M'}\\rangle,\n$$\n\nwhere we have skipped a reference to specific single-particle states. This is the expectation value for two specific states, labelled by angular momenta $J'$ and $J$. These states form an orthonormal basis.\nUsing the properties of the Clebsch-Gordan coefficients we can write\n\n$$\nT^{\\lambda}_{\\mu}|\\Phi^{J'}_{M'}\\rangle=\\sum_{J''M''}\\langle \\lambda \\mu J'M'|J''M''\\rangle|\\Psi^{J''}_{M''}\\rangle,\n$$\n\nand assuming that states with different $J$ and $M$ are orthonormal we arrive at\n\n$$\n\\langle \\Phi^J_M|T^{\\lambda}_{\\mu}|\\Phi^{J'}_{M'}\\rangle= \\langle \\lambda \\mu J'M'|JM\\rangle \\langle \\Phi^J_M|\\Psi^{J}_{M}\\rangle.\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nWe need to show that\n\n$$\n\\langle \\Phi^J_M|\\Psi^{J}_{M}\\rangle,\n$$\n\nis independent of $M$.\nTo show that\n\n$$\n\\langle \\Phi^J_M|\\Psi^{J}_{M}\\rangle,\n$$\n\nis independent of $M$, we use the ladder operators for angular momentum.\n\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\n We have that\n\n$$\n\\langle \\Phi^J_{M+1}|\\Psi^{J}_{M+1}\\rangle=\\left((J-M)(J+M+1)\\right)^{-1/2}\\langle \\hat{J}_{+}\\Phi^J_{M}|\\Psi^{J}_{M+1}\\rangle,\n$$\n\nbut this is also equal to\n\n$$\n\\langle \\Phi^J_{M+1}|\\Psi^{J}_{M+1}\\rangle=\\left((J-M)(J+M+1)\\right)^{-1/2}\\langle \\Phi^J_{M}|\\hat{J}_{-}\\Psi^{J}_{M+1}\\rangle,\n$$\n\nmeaning that\n\n$$\n\\langle \\Phi^J_{M+1}|\\Psi^{J}_{M+1}\\rangle=\\langle \\Phi^J_M|\\Psi^{J}_{M}\\rangle\\equiv\\langle \\Phi^J_{M}||T^{\\lambda}||\\Phi^{J'}_{M'}\\rangle.\n$$\n\nThe double bars indicate that this expectation value is independent of the projection $M$.\n\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nThe Wigner-Eckart theorem for an expectation value can then be written as\n\n$$\n\\langle \\Phi^J_M|T^{\\lambda}_{\\mu}|\\Phi^{J'}_{M'}\\rangle\\equiv\\langle \\lambda \\mu J'M'|JM\\rangle\\langle \\Phi^J||T^{\\lambda}||\\Phi^{J'}\\rangle.\n$$\n\nThe double bars indicate that this expectation value is independent of the projection $M$.\nWe can manipulate the Clebsch-Gordan coefficients using the relations\n\n$$\n\\langle \\lambda \\mu J'M'|JM\\rangle= (-1)^{\\lambda+J'-J}\\langle J'M'\\lambda \\mu |JM\\rangle\n$$\n\nand\n\n$$\n\\langle J'M'\\lambda \\mu |JM\\rangle =(-1)^{J'-M'}\\frac{\\sqrt{2J+1}}{\\sqrt{2\\lambda+1}}\\langle J'M'J-M |\\lambda-\\mu\\rangle,\n$$\n\ntogether with the so-called $3j$ symbols.\nIt is then normal to encounter the Wigner-Eckart theorem in the form\n\n$$\n\\langle \\Phi^J_M|T^{\\lambda}_{\\mu}|\\Phi^{J'}_{M'}\\rangle\\equiv(-1)^{J-M}\\left(\\begin{array}{ccc} J & \\lambda & J' \\\\ -M & \\mu & M'\\end{array}\\right)\\langle \\Phi^J||T^{\\lambda}||\\Phi^{J'}\\rangle,\n$$\n\nwith the condition $\\mu+M'-M=0$.\n\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nThe $3j$ symbols obey the symmetry relation\n\n$$\n\\left(\\begin{array}{ccc} j_1 & j_2 & j_3 \\\\ m_1 & m_2 & m_3\\end{array}\\right)=(-1)^{p}\\left(\\begin{array}{ccc} j_a & j_b & j_c \\\\ m_a & m_b & m_c\\end{array}\\right),\n$$\n\nwith $(-1)^p=1$ when the columns $a,b, c$ are even permutations of the columns $1,2,3$, $p=j_1+j_2+j_3$ when the columns $a,b,c$ are odd permtations of the\ncolumns $1,2,3$ and $p=j_1+j_2+j_3$ when all the magnetic quantum numbers $m_i$ change sign. Their orthogonality is given by\n\n$$\n\\sum_{j_3m_3}(2j_3+1)\\left(\\begin{array}{ccc} j_1 & j_2 & j_3 \\\\ m_1 & m_2 & m_3\\end{array}\\right)\\left(\\begin{array}{ccc} j_1 & j_2 & j_3 \\\\ m_{1'} & m_{2'} & m_3\\end{array}\\right)=\\delta_{m_1m_{1'}}\\delta_{m_2m_{2'}},\n$$\n\nand\n\n$$\n\\sum_{m_1m_2}\\left(\\begin{array}{ccc} j_1 & j_2 & j_3 \\\\ m_1 & m_2 & m_3\\end{array}\\right)\\left(\\begin{array}{ccc} j_1 & j_2 & j_{3'} \\\\ m_{1} & m_{2} & m_{3'}\\end{array}\\right)=\\frac{1}{(2j_3+1)}\\delta_{j_3j_{3'}}\\delta_{m_3m_{3'}}.\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nFor later use, the following special cases for the Clebsch-Gordan and $3j$ symbols are rather useful\n\\[\n\\langle JM J'M' |00\\rangle =\\frac{(-1)^{J-M}}{\\sqrt{2J+1}}\\delta_{JJ'}\\delta_{MM'}.\n\\] \nand \n\\[\n\\left(\\begin{array}{ccc} J & 1 & J \\\\ -M & 0 & M'\\end{array}\\right)=(-1)^{J-M}\\frac{M}{\\sqrt{(2J+1)(J+1)}}\\delta_{MM'}.\n\\]\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nUsing $3j$ symbols we rewrote the Wigner-Eckart theorem as\n\n$$\n\\langle \\Phi^J_M|T^{\\lambda}_{\\mu}|\\Phi^{J'}_{M'}\\rangle\\equiv(-1)^{J-M}\\left(\\begin{array}{ccc} J & \\lambda & J' \\\\ -M & \\mu & M'\\end{array}\\right)\\langle \\Phi^J||T^{\\lambda}||\\Phi^{J'}\\rangle.\n$$\n\nMultiplying from the left with the same $3j$ symbol and summing over $M,\\mu,M'$ we obtain the equivalent relation\n\n$$\n\\langle \\Phi^J||T^{\\lambda}||\\Phi^{J'}\\rangle\\equiv\\sum_{M,\\mu,M'}(-1)^{J-M}\\left(\\begin{array}{ccc} J & \\lambda & J' \\\\ -M & \\mu & M'\\end{array}\\right)\\langle \\Phi^J_M|T^{\\lambda}_{\\mu}|\\Phi^{J'}_{M'}\\rangle,\n$$\n\nwhere we used the orthogonality properties of the $3j$ symbols from the previous page.\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nThis relation can in turn be used to compute the expectation value of some simple reduced matrix elements like\n\n$$\n\\langle \\Phi^J||{\\bf 1}||\\Phi^{J'}\\rangle=\\sum_{M,M'}(-1)^{J-M}\\left(\\begin{array}{ccc} J & 0 & J' \\\\ -M & 0 & M'\\end{array}\\right)\\langle \\Phi^J_M|1|\\Phi^{J'}_{M'}\\rangle=\\sqrt{2J+1}\\delta_{JJ'}\\delta_{MM'},\n$$\n\nwhere we used\n\n$$\n\\langle JM J'M' |00\\rangle =\\frac{(-1)^{J-M}}{\\sqrt{2J+1}}\\delta_{JJ'}\\delta_{MM'}.\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nSimilarly, using\n\n$$\n\\left(\\begin{array}{ccc} J & 1 & J \\\\ -M & 0 & M'\\end{array}\\right)=(-1)^{J-M}\\frac{M}{\\sqrt{(2J+1)(J+1)}}\\delta_{MM'},\n$$\n\nwe have that\n\n$$\n\\langle \\Phi^J||{\\bf J}||\\Phi^{J}\\rangle=\\sum_{M,M'}(-1)^{J-M}\\left(\\begin{array}{ccc} J & 1 & J' \\\\ -M & 0 & M'\\end{array}\\right)\\langle \\Phi^J_M|j_Z|\\Phi^{J'}_{M'}\\rangle=\\sqrt{J(J+1)(2J+1)}\n$$\n\nWith the Pauli spin matrices $\\sigma$ and a state with $J=1/2$, the reduced matrix element\n\n$$\n\\langle \\frac{1}{2}||{\\bf \\sigma}||\\frac{1}{2}\\rangle=\\sqrt{6}.\n$$\n\nBefore we proceed with further examples, we need some other properties of the Wigner-Eckart theorem plus some additional angular momenta relations.\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nThe Wigner-Eckart theorem states that the expectation value for an irreducible spherical tensor can be written as\n\n$$\n\\langle \\Phi^J_M|T^{\\lambda}_{\\mu}|\\Phi^{J'}_{M'}\\rangle\\equiv\\langle \\lambda \\mu J'M'|JM\\rangle\\langle \\Phi^J||T^{\\lambda}||\\Phi^{J'}\\rangle.\n$$\n\nSince the Clebsch-Gordan coefficients themselves are easy to evaluate, the interesting quantity is the reduced matrix element. Note also that \nthe Clebsch-Gordan coefficients limit via the triangular relation among $\\lambda$, $J$ and $J'$ the possible non-zero values.\n\nFrom the theorem we see also that\n\n$$\n\\langle \\Phi^J_M|T^{\\lambda}_{\\mu}|\\Phi^{J'}_{M'}\\rangle=\\frac{\\langle \\lambda \\mu J'M'|JM\\rangle\\langle }{\\langle \\lambda \\mu_0 J'M'_0|JM_0\\rangle\\langle }\\langle \\Phi^J_{M_0}|T^{\\lambda}_{\\mu_0}|\\Phi^{J'}_{M'_0}\\rangle,\n$$\n\nmeaning that if we know the matrix elements for say some $\\mu=\\mu_0$, $M'=M'_0$ and $M=M_0$ we can calculate all other.\n\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nIf we look at the hermitian adjoint of the operator $T^{\\lambda}_{\\mu}$, \nwe see via the commutation relations that $(T^{\\lambda}_{\\mu})^{\\dagger}$ is not an irreducible tensor, that is\n\n$$\n[J_{\\pm}, (T^{\\lambda}_{\\mu})^{\\dagger}]= -\\sqrt{(\\lambda\\pm \\mu)(\\lambda\\mp \\mu+1)}(T^{\\lambda}_{\\mu\\mp 1})^{\\dagger},\n$$\n\nand\n\n$$\n[J_{z}, (T^{\\lambda}_{\\mu})^{\\dagger}]=-\\mu (T^{\\lambda}_{\\mu})^{\\dagger}.\n$$\n\nThe hermitian adjoint $(T^{\\lambda}_{\\mu})^{\\dagger}$ is not an irreducible tensor. As an example, consider the spherical harmonics for \n$l=1$ and $m_l=\\pm 1$. These functions are\n\n$$\nY^{l=1}_{m_l=1}(\\theta,\\phi)=-\\sqrt{\\frac{3}{8\\pi}}\\sin{(\\theta)}\\exp{\\imath\\phi},\n$$\n\nand\n\n$$\nY^{l=1}_{m_l=-1}(\\theta,\\phi)=\\sqrt{\\frac{3}{8\\pi}}\\sin{(\\theta)}\\exp{-\\imath\\phi},\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nIt is easy to see that the Hermitian adjoint of these two functions\n\n$$\n\\left[Y^{l=1}_{m_l=1}(\\theta,\\phi)\\right]^{\\dagger}=-\\sqrt{\\frac{3}{8\\pi}}\\sin{(\\theta)}\\exp{-\\imath\\phi},\n$$\n\nand\n\n$$\n\\left[Y^{l=1}_{m_l=-1}(\\theta,\\phi)\\right]^{\\dagger}=\\sqrt{\\frac{3}{8\\pi}}\\sin{(\\theta)}\\exp{\\imath\\phi},\n$$\n\ndo not behave as a spherical tensor. However, the modified quantity\n\n$$\n\\tilde{T}^{\\lambda}_{\\mu}=(-1)^{\\lambda+\\mu}(T^{\\lambda}_{-\\mu})^{\\dagger},\n$$\n\ndoes satisfy the above commutation relations.\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nWith the modified quantity\n\n$$\n\\tilde{T}^{\\lambda}_{\\mu}=(-1)^{\\lambda+\\mu}(T^{\\lambda}_{-\\mu})^{\\dagger},\n$$\n\nwe can then define the expectation value\n\n$$\n\\langle \\Phi^J_M|T^{\\lambda}_{\\mu}|\\Phi^{J'}_{M'}\\rangle^{\\dagger} = \\langle \\lambda \\mu J'M'|JM\\rangle\\langle \\Phi^J||T^{\\lambda}||\\Phi^{J'}\\rangle^*,\n$$\n\nsince the Clebsch-Gordan coefficients are real. The rhs is equivalent with\n\n$$\n\\langle \\lambda \\mu J'M'|JM\\rangle\\langle \\Phi^J||T^{\\lambda}||\\Phi^{J'}\\rangle^*=\\langle \\Phi^{J'}_{M'}|(T^{\\lambda}_{\\mu})^{\\dagger}|\\Phi^{J}_{M}\\rangle,\n$$\n\nwhich is equal to\n\n$$\n\\langle \\Phi^{J'}_{M'}|(T^{\\lambda}_{\\mu})^{\\dagger}|\\Phi^{J}_{M}\\rangle=(-1)^{-\\lambda+\\mu}\\langle \\lambda -\\mu JM|J'M'\\rangle\\langle \\Phi^{J'}||\\tilde{T}^{\\lambda}||\\Phi^{J}\\rangle.\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nLet us now apply the theorem to some selected expectation values.\nIn several of the expectation values we will meet when evaluating explicit matrix elements, we will have to deal with expectation values involving spherical harmonics. A general central interaction can be expanded in a complete set of functions like the Legendre polynomials, that is, we have an interaction, with $r_{ij}=|{\\bf r}_i-{\\bf r}_j|$,\n\n$$\nv(r_{ij})=\\sum_{\\nu=0}^{\\infty}v_{\\nu}(r_{ij})P_{\\nu}(\\cos{(\\theta_{ij})},\n$$\n\nwith $P_{\\nu}$ being a Legendre polynomials\n\n$$\nP_{\\nu}(\\cos{(\\theta_{ij})}=\\sum_{\\mu}\\frac{4\\pi}{2\\mu+1}Y_{\\mu}^{\\nu *}(\\Omega_{i})Y_{\\mu}^{\\nu}(\\Omega_{j}).\n$$\n\nWe will come back later to how we split the above into a contribution that involves only one of the coordinates.\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nThis means that we will need matrix elements of the type\n\n$$\n\\langle Y^{l'}||Y^{\\lambda}|| Y^{l}\\rangle.\n$$\n\nWe can rewrite the Wigner-Eckart theorem as\n\n$$\n\\langle Y^{l'}||Y^{\\lambda}|| Y^{l}\\rangle=\\sum_{m\\mu}\\langle \\lambda\\mu lm|l'm'\\rangle Y^{\\lambda}_{\\mu}Y^l_m,\n$$\n\nThis equation is true for all values of $\\theta$ and $\\phi$. It must also hold for $\\theta=0$.\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nWe have\n\n$$\n\\langle Y^{l'}||Y^{\\lambda}|| Y^{l}\\rangle=\\sum_{m\\mu}\\langle \\lambda\\mu lm|l'm'\\rangle Y^{\\lambda}_{\\mu}Y^l_m,\n$$\n\nand for $\\theta=0$, the spherical harmonic\n\n$$\nY_m^l(\\theta=0,\\phi)=\\sqrt{\\frac{2l+1}{4\\pi}}\\delta_{m0},\n$$\n\nwhich results in\n\n$$\n\\langle Y^{l'}||Y^{\\lambda}|| Y^{l}\\rangle=\\left\\{\\frac{(2l+1)(2\\lambda+1)}{4\\pi(2l'+1)}\\right\\}^{1/2}\\langle \\lambda0 l0|l'0\\rangle.\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nTill now we have mainly been concerned with the coupling of two angular momenta $j_{a}$ and $j_{b}$ to a final angular momentum $J$.\nIf we wish to describe a three-body state with a final angular momentum $J$, we need to couple three angular momenta, say \nthe two momenta $j_a,j_b$ to a third one $j_c$. The coupling order is important and leads to a less trivial implementation of the \nPauli principle. With three angular momenta there are obviously $3!$ ways by which we can combine the angular momenta. \nIn $m$-scheme a three-body Slater determinant is represented as (say for the case of ${}^{19}\\mbox{O}$, three neutrons outside the core of ${}^{16}\\mbox{O}$),\n\n$$\n|^{19}\\mathrm{O}\\rangle =|(abc)M\\rangle = a^{\\dagger}_aa^{\\dagger}_ba^{\\dagger}_c|^{16}\\mathrm{O}\\rangle=|\\Phi^{abc}\\rangle.\n$$\n\nThe Pauli principle is automagically implemented via the anti-commutation relations.\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nHowever, when we deal the same state in an angular momentum coupled basis, we need to be a little bit more careful. We can namely couple the states\nas follows\n\n\n
                                        \n\n$$\n\\vert([j_a\\rightarrow j_b]J_{ab}\\rightarrow j_c) J\\rangle= \\sum_{m_am_bm_c}\\langle j_am_aj_bm_b|J_{ab}M_{ab}\\rangle \\langle J_{ab}M_{ab}j_cm_c|JM\\rangle|j_am_a\\rangle\\otimes |j_bm_b\\rangle \\otimes |j_cm_c\\rangle \\ , \n\\label{eq:fabc} \\tag{1}\n$$\n\nthat is, we couple first $j_a$ to $j_b$ to yield an intermediate angular momentum $J_{ab}$, then to $j_c$ yielding the final angular momentum $J$.\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nNow, nothing hinders us from recoupling this state by coupling $j_b$ to $j_c$, yielding an intermediate angular momentum $J_{bc}$ and then couple this angular momentum to $j_a$, resulting in the final angular momentum $J'$. \n\nThat is, we can have\n\n$$\n\\vert(j_a\\rightarrow [j_b\\rightarrow j_c]J_{bc}) J\\rangle = \\sum_{m_a'm_b'm_c'}\\langle j_bm_b'j_cm_c'|J_{bc}M_{bc}\\rangle \\langle j_am_a'J_{bc}M_{bc}|J'M'\\rangle|\\Phi^{abc}\\rangle .\n$$\n\nWe will always assume that we work with orthornormal states, this means that when we compute the overlap betweem these two possible ways of coupling angular momenta, we get\n\n\n
                                        \n\n$$\n\\begin{equation}\n\\langle (j_a\\rightarrow [j_b\\rightarrow j_c]J_{bc}) J'M'| ([j_a\\rightarrow j_b]J_{ab}\\rightarrow j_c) JM\\rangle = \n\\delta_{JJ'}\\delta_{MM'}\\sum_{m_am_bm_c}\\langle j_am_aj_bm_b|J_{ab}M_{ab}\\rangle \\langle J_{ab}M_{ab}j_cm_c|JM\\rangle \n\\label{_auto1} \\tag{2}\n\\end{equation}\n$$\n\n\n
                                        \n\n$$\n\\begin{equation} \n \\times \\langle j_bm_bj_cm_c|J_{bc}M_{bc}\\rangle \\langle j_am_aJ_{bc}M_{bc}|JM\\rangle . \n\\label{_auto2} \\tag{3}\n\\end{equation}\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nWe use then the latter equation to define the so-called $6j$-symbols\n\n$$\n\\begin{eqnarray}\n\\langle (j_a\\rightarrow [j_b\\rightarrow j_c]J_{bc}) J'M'| ([j_a\\rightarrow j_b]J_{ab}\\rightarrow j_c) JM\\rangle & = & \\delta_{JJ'}\\delta_{MM'}\\sum_{m_am_bm_c}\\langle j_am_aj_bm_b|J_{ab}M_{ab}\\rangle \\langle J_{ab}M_{ab}j_cm_c|JM\\rangle \\\\ \\nonumber\n& & \\times \\langle j_bm_bj_cm_c|J_{bc}M_{bc}\\rangle \\langle j_am_aJ_{bc}M_{bc}|JM\\rangle \\\\ \\nonumber\n& =&(-1)^{j_a+j_b+j_c+J}\\sqrt{(2J_{ab}+1)(2J_{bc}+1)}\\left\\{\\begin{array}{ccc} j_a & j_b& J_{ab} \\\\ j_c & J & J_{bc} \\end{array}\\right\\} , \n\\end{eqnarray}\n$$\n\nwhere the symbol in curly brackets is the $6j$ symbol. \nA specific coupling order has to be respected in the symbol, that is, the so-called triangular relations between three angular momenta needs to be respected, that is\n\n$$\n\\left\\{\\begin{array}{ccc} x & x& x \\\\ & & \\end{array}\\right\\}\\hspace{0.1cm}\\left\\{\\begin{array}{ccc} & & x \\\\ x& x & \\end{array}\\right\\}\\hspace{0.1cm}\\left\\{\\begin{array}{ccc} & x& \\\\ x & &x \\end{array}\\right\\}\\hspace{0.1cm}\\left\\{\\begin{array}{ccc} x & & \\\\ & x &x \\end{array}\\right\\}\\hspace{0.1cm}\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nThe $6j$ symbol is invariant under the permutation of any two columns\n\n$$\n\\begin{Bmatrix} j_1 & j_2 & j_3\\\\ j_4 & j_5 & j_6 \\end{Bmatrix} = \\begin{Bmatrix} j_2 & j_1 & j_3\\\\ j_5 & j_4 & j_6 \\end{Bmatrix} = \\begin{Bmatrix} j_1 & j_3 & j_2\\\\ j_4 & j_6 & j_5 \\end{Bmatrix} = \\begin{Bmatrix} j_3 & j_2 & j_1\\\\ j_6 & j_5 & j_4 \\end{Bmatrix}.\n$$\n\nThe $6j$ symbol is also invariant if upper and lower arguments are interchanged in any two columns\n\n$$\n\\begin{Bmatrix} j_1 & j_2 & j_3\\\\ j_4 & j_5 & j_6 \\end{Bmatrix} = \\begin{Bmatrix} j_4 & j_5 & j_3\\\\ j_1 & j_2 & j_6 \\end{Bmatrix} = \\begin{Bmatrix} j_1 & j_5 & j_6\\\\ j_4 & j_2 & j_3 \\end{Bmatrix} = \\begin{Bmatrix} j_4 & j_2 & j_6\\\\ j_1 & j_5 & j_3 \\end{Bmatrix}.\n$$\n\n## Testing properties of $6j$ symbols\nThe above properties of $6j$ symbols can again be tested using the symbolic python package **wigner**. \nLet us test the invariance\n\n$$\n\\begin{Bmatrix} j_1 & j_2 & j_3\\\\ j_4 & j_5 & j_6 \\end{Bmatrix} = \\begin{Bmatrix} j_2 & j_1 & j_3\\\\ j_5 & j_4 & j_6 \\end{Bmatrix}.\n$$\n\nThe following program tests this relation for the case of $j_1=3/2$, $j_2=3/2$, $j_3=3$, $j_4=1/2$, $j_5=1/2$, $j_6=1$\n\n\n```\nfrom sympy import S\nfrom sympy.physics.wigner import wigner_6j\n# Twice the values of all js\nj1 = 3\nj2 = 5\nj3 = 2\nj4 = 3\nj5 = 5\nj6 = 1\n\"\"\" The triangular relation has to be fulfilled \"\"\"\nprint wigner_6j(S(j1)/2, S(j2)/2, j3, S(j4)/2, S(j5)/2, j6)\n\"\"\" Swapping columns 1 <==> 2 \"\"\"\nprint wigner_6j(S(j2)/2, S(j1)/2, j3, S(j5)/2, S(j4)/2, j6)\n```\n\n## Angular momentum algebra, Wigner-Eckart theorem\nThe $6j$ symbols satisfy this orthogonality relation\n\n$$\n\\sum_{j_3} (2j_3+1) \\begin{Bmatrix} j_1 & j_2 & j_3\\\\ j_4 & j_5 & j_6 \\end{Bmatrix} \\begin{Bmatrix} j_1 & j_2 & j_3\\\\ j_4 & j_5 & j_6' \\end{Bmatrix} = \\frac{\\delta_{j_6^{}j_6'}}{2j_6+1} \\{j_1,j_5,j_6\\} \\{j_4,j_2,j_6\\}.\n$$\n\nThe symbol $\\{j_1j_2j_3\\}$ (called the triangular delta) is equal to one if the triad $(j_1j_2j_3)$ satisfies the triangular conditions and zero otherwise.\nA useful value is given when say one of the angular momenta are zero, say $J_{bc}=0$, then we have\n\n$$\n\\left\\{\\begin{array}{ccc} j_a & j_b& J_{ab} \\\\ j_c & J & 0 \\end{array}\\right\\}=\\frac{(-1)^{j_a+j_b+J_{ab}}\\delta_{Jj_a}\\delta_{j_cj_b} }{\\sqrt{(2j_{a}+1)(2j_{b}+1)}}\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nWith the $6j$ symbol defined, we can go back and and rewrite the overlap between the two ways of recoupling angular momenta in terms of the $6j$ symbol.\nThat is, we can have\n\n$$\n\\vert (j_a\\rightarrow [j_b\\rightarrow j_c]J_{bc}) JM\\rangle =\\sum_{J_{ab}}(-1)^{j_a+j_b+j_c+J}\\sqrt{(2J_{ab}+1)(2J_{bc}+1)}\\left\\{\\begin{array}{ccc} j_a & j_b& J_{ab} \\\\ j_c & J & J_{bc} \\end{array}\\right\\}| ([j_a\\rightarrow j_b]J_{ab}\\rightarrow j_c) JM\\rangle.\n$$\n\nCan you find the inverse relation? \nThese relations can in turn be used to write out the fully anti-symmetrized three-body wave function in a $J$-scheme coupled basis. \nIf you opt then for a specific coupling order, say $| ([j_a\\rightarrow j_b]J_{ab}\\rightarrow j_c) JM\\rangle$, you need to express this representation in terms of the other coupling possibilities. \n\n## Angular momentum algebra, Wigner-Eckart theorem\nNote that the two-body intermediate state is assumed to be antisymmetric but\nnot normalized, that is, the state which involves the quantum numbers \n$j_a$ and $j_b$. Assume that the intermediate \ntwo-body state is antisymmetric. With this coupling order, we can \nrewrite ( in a schematic way) the general three-particle Slater determinant as\n\n$$\n\\Phi(a,b,c) = {\\cal A} | ([j_a\\rightarrow j_b]J_{ab}\\rightarrow j_c) J\\rangle,\n$$\n\nwith an implicit sum over $J_{ab}$. The antisymmetrization operator ${\\cal A}$ is used here to indicate that we need to antisymmetrize the state. **Challenge**: Use the definition of the $6j$ symbol and find an explicit \nexpression for the above three-body state using the coupling order $| ([j_a\\rightarrow j_b]J_{ab}\\rightarrow j_c) J\\rangle$.\n\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nWe can also coupled together four angular momenta. Consider two four-body states, with single-particle angular momenta $j_a$, $j_b$, $j_c$ and $j_d$ we can have a state with final $J$\n\n$$\n|\\Phi(a,b,c,d)\\rangle_1 = | ([j_a\\rightarrow j_b]J_{ab}\\times [j_c\\rightarrow j_d]J_{cd}) JM\\rangle,\n$$\n\nwhere we read the coupling order as $j_a$ couples with $j_b$ to given and intermediate angular momentum $J_{ab}$. \nMoreover, $j_c$ couples with $j_d$ to given and intermediate angular momentum $J_{cd}$. The two intermediate angular momenta $J_{ab}$ and $J_{cd}$\nare in turn coupled to a final $J$. These operations involved three Clebsch-Gordan coefficients. \n\nAlternatively, we could couple in the following order\n\n$$\n|\\Phi(a,b,c,d)\\rangle_2 = | ([j_a\\rightarrow j_c]J_{ac}\\times [j_b\\rightarrow j_d]J_{bd}) JM\\rangle,\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nThe overlap between these two states\n\n$$\n\\langle([j_a\\rightarrow j_c]J_{ac}\\times [j_b\\rightarrow j_d]J_{bd}) JM| ([j_a\\rightarrow j_b]J_{ab}\\times [j_c\\rightarrow j_d]J_{cd}) JM\\rangle,\n$$\n\nis equal to\n\n$$\n\\begin{eqnarray}\n\\nonumber\n& & \\sum_{m_iM_{ij}}\\langle j_am_aj_bm_b|J_{ab}M_{ab}\\rangle \\langle j_cm_cj_dm_d|J_{cd}M_{cd}\\rangle \\langle J_{ab}M_{ab}J_{cd}M_{cd}|JM\\rangle \\\\\n& & \\times\\langle j_am_aj_cm_c|J_{ac}M_{ac}\\rangle \\langle j_bm_bj_dm_d|J_{cd}M_{bd}\\rangle \\langle J_{ac}M_{ac}J_{bd}M_{bd}|JM\\rangle \\\\ \\nonumber\n&= & \\sqrt{(2J_{ab}+1)(2J_{cd}+1)(2J_{ac}+1)(2J_{bd}+1)}\\left\\{\\begin{array}{ccc} j_a & j_b& J_{ab} \\\\ j_c & j_d& J_{cd} \\\\J_{ac} & J_{bd}& J\\end{array}\\right\\}\n, \\nonumber\n\\end{eqnarray}\n$$\n\nwith the symbol in curly brackets $\\{\\}$ being the $9j$-symbol. We see that a $6j$ symbol involves four Clebsch-Gordan coefficients, while the $9j$ symbol\ninvolves six.\n\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nA $9j$ symbol is invariant under reflection in either diagonal\n\n$$\n\\begin{Bmatrix} j_1 & j_2 & j_3\\\\ j_4 & j_5 & j_6\\\\ j_7 & j_8 & j_9 \\end{Bmatrix} = \\begin{Bmatrix} j_1 & j_4 & j_7\\\\ j_2 & j_5 & j_8\\\\ j_3 & j_6 & j_9 \\end{Bmatrix} = \\begin{Bmatrix} j_9 & j_6 & j_3\\\\ j_8 & j_5 & j_2\\\\ j_7 & j_4 & j_1 \\end{Bmatrix}.\n$$\n\nThe permutation of any two rows or any two columns yields a phase factor $(-1)^S$, where\n\n$$\nS=\\sum_{i=1}^9 j_i.\n$$\n\nAs an example we have\n\n$$\n\\begin{Bmatrix} j_1 & j_2 & j_3\\\\ j_4 & j_5 & j_6\\\\ j_7 & j_8 & j_9 \\end{Bmatrix} = (-1)^S \\begin{Bmatrix} j_4 & j_5 & j_6\\\\ j_1 & j_2 & j_3\\\\ j_7 & j_8 & j_9 \\end{Bmatrix} = (-1)^S \\begin{Bmatrix} j_2 & j_1 & j_3\\\\ j_5 & j_4 & j_6\\\\ j_8 & j_7 & j_9 \\end{Bmatrix}.\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nA useful case is when say $J=0$ in\n\n$$\n\\left\\{\\begin{array}{ccc} j_a & j_b& J_{ab} \\\\ j_c & j_d & J_{cd} \\\\ J_{ac} & J_{bd}& 0\\end{array}\\right\\}=\\frac{\\delta_{J_{ab}J_{cd}} \\delta_{J_{ac}J_{bd}}}{\\sqrt{(2J_{ab}+1)(2J_{ac}+1)}} (-1)^{j_b+J_{ab}+j_c+J_{ac}} \\begin{Bmatrix} j_a & j_b & J_{ab}\\\\ j_d & j_c & J_{ac} \\end{Bmatrix}.\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nThe tensor operator in the nucleon-nucleon potential\nis given by\n\n$$\n\\begin{array}{ll}\n&\\\\\n\\langle lSJ\\vert S_{12}\\vert l'S'J\\rangle =&\n(-)^{S+J}\\sqrt{30(2l+1)(2l'+1)(2S+1)(2S'+1)}\\\\\n&\\times\\left\\{\\begin{array}{ccc}J&S'&l'\\\\2&l&S\\end{array}\\right\\}\n\\left(\\begin{array}{ccc}l'&2&l\\\\0&0&0\\end{array}\\right)\n\\left\\{\\begin{array}{ccc}s_{1}&s_{2}&S\\\\s_{3}&s_{4}&S'\\\\\n1&1&2\\end{array}\n\\right\\}\\\\\n&\\times\\langle s_{1}\\vert\\vert \\sigma_{1}\\vert\\vert s_{3}\\rangle\n\\langle s_{2}\\vert\\vert \\sigma_{2}\\vert \\vert s_{4}\\rangle,\n\\end{array}\n$$\n\nand it is zero for the $^1S_0$ wave. \n\nHow do we get here?\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nTo derive the expectation value of the nuclear tensor force, we recall that \nthe product of two irreducible tensor operators is\n\n$$\nW^{r}_{m_r}=\\sum_{m_pm_q}\\langle pm_pqm_q|rm_r\\rangle T^{p}_{m_p}U^{q}_{m_q},\n$$\n\nand using the orthogonality properties of the Clebsch-Gordan coefficients we can rewrite the above as\n\n$$\nT^{p}_{m_p}U^{q}_{m_q}=\\sum_{m_pm_q}\\langle pm_pqm_q|rm_r\\rangle W^{r}_{m_r}.\n$$\n\nAssume now that the operators $T$ and $U$ act on different parts of say a wave function. The operator $T$ could act on the spatial part only while the operator $U$ acts only on the spin part. This means also that these operators commute.\nThe reduced matrix element of this operator is thus, using the Wigner-Eckart theorem,\n\n1\n9\n0\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\times\\langle (j_aj_bJM|\\left[ T^{p}_{m_p}U^{q}_{m_q} \\right]^{r}_{m_r}|(j_cj_d)J'M'\\rangle.\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nStarting with\n\n1\n9\n2\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\times\\langle (j_aj_bJM|\\left[ T^{p}_{m_p}U^{q}_{m_q} \\right]^{r}_{m_r}|(j_cj_d)J'M'\\rangle,\n$$\n\nwe assume now that $T$ acts only on $j_a$ and $j_c$ and that $U$ acts only on $j_b$ and $j_d$. \nThe matrix element $\\langle (j_aj_bJM|\\left[ T^{p}_{m_p}U^{q}_{m_q} \\right]^{r}_{m_r}|(j_cj_d)J'M'\\rangle$ can be written out,\nwhen we insert a complete set of states $|j_im_ij_jm_j\\rangle\\langle j_im_ij_jm_j|$ between $T$ and $U$ as\n\n1\n9\n4\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\times \\langle (j_am_aj_bm_b|\\left[ T^{p}_{m_p}\\right]^{r}_{m_r}|(j_cm_cj_bm_b)\\rangle\\langle (j_cm_cj_bm_b|\\left[ U^{q}_{m_q}\\right]^{r}_{m_r}|(j_cm_cj_dm_d)\\rangle.\n$$\n\nThe complete set of states that was inserted between $T$ and $U$ reduces to $|j_cm_cj_bm_b\\rangle\\langle j_cm_cj_bm_b|$\ndue to orthogonality of the states.\n\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nCombining the last two equations from the previous slide and \nand applying the Wigner-Eckart theorem, we arrive at (rearranging phase factors)\n\n1\n9\n6\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n1\n9\n7\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\times\\left(\\begin{array}{ccc} j_a &j_c &p \\\\ m_a &-m_c &-m_p \\end{array}\\right)\\left(\\begin{array}{ccc} j_b &j_d &q \\\\ m_b &-m_d &-m_q \\end{array}\\right)\\langle j_a||T^p||j_c\\rangle \\times \\langle j_b||U^q||j_d\\rangle\n$$\n\nwhich can be rewritten in terms of a $9j$ symbol as\n\n$$\n\\langle (j_aj_b)J||W^{r}||(j_cj_d)J'\\rangle=\\sqrt{(2J+1)(2r+1)(2J'+1)}\\langle j_a||T^p||j_c\\rangle \\langle j_b||U^q||j_d\\rangle\\left\\{\\begin{array}{ccc} j_a & j_b& J \\\\ j_c & j_d & J' \\\\ p & q& r\\end{array}\\right\\}.\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nFrom this expression we can in turn compute for example the spin-spin operator of the tensor force.\n\n\nIn case $r=0$, that is we two tensor operators coupled to a scalar, we can use (with $p=q$)\n\n$$\n\\left\\{\\begin{array}{ccc} j_a & j_b& J \\\\ j_c & j_d & J' \\\\ p &p & 0\\end{array}\\right\\}=\\frac{\\delta_{JJ'} \\delta_{pq}}{\\sqrt{(2J+1)(2J+1)}} (-1)^{j_b+j_c+2J} \\begin{Bmatrix} j_a & j_b & J\\\\ j_d & j_c & p \\end{Bmatrix},\n$$\n\nand obtain\n\n$$\n\\langle (j_aj_b)J||W^{0}||(j_cj_d)J'\\rangle=(-1)^{j_b+j_c+2J}\\langle j_a||T^p||j_c\\rangle\\langle j_b||U^p||j_d\\rangle \\begin{Bmatrix} j_a & j_b & J\\\\ j_d & j_c & p \\end{Bmatrix}.\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nAnother very useful expression is the case where the operators act in just one space. We state here without \nshowing that the reduced matrix element\n\n2\n0\n2\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\times \\langle j_a||T^p||j_c\\rangle \\langle j_c||T^q||j_b\\rangle.\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nThe tensor operator in \nthe nucleon-nucleon potential can be written as\n\n$$\nV=\\frac{3}{r^{2}}\\left[ \\left[ {\\bf \\sigma}_1 \\otimes {\\bf \\sigma}_2\\right]^\n{(2)} \\otimes\\left[{\\bf r} \\otimes {\\bf r} \\right]^{(2)}\\right]^{(0)}_0\n$$\n\nSince the irreducible tensor \n$\\left[{\\bf r} \\otimes {\\bf r} \\right]^{(2)}$\noperates only on the angular quantum numbers and\n$\\left[{\\bf \\sigma}_1 \\otimes {\\bf \\sigma}_2\\right]^{(2)}$ \noperates only on \nthe spin states we can write the matrix element\n\n$$\n\\begin{eqnarray*}\n\\langle lSJ\\vert V\\vert lSJ\\rangle & = &\n\\langle lSJ \\vert\\left[ \\left[{\\bf \\sigma}_1 \\otimes {\\bf \\sigma}_2\\right]^{(2)} \\otimes\n\\left[{\\bf r} \\otimes {\\bf r} \\right]^{(2)}\\right]^{(0)}_0\\vert l'S'J\\rangle \\\\\n& = &\n(-1)^{J+l+S}\n\\left\\{\\begin{array}{ccc} l&S&J \\\\ l'&S'&2\\end{array}\\right\\}\n\\langle l \\vert\\vert\\left[{\\bf r} \\otimes {\\bf r} \\right]^{(2)} \\vert \\vert l'\\rangle\\\\\n& &\n\\times \\langle S\\vert\\vert\\left[{\\bf \\sigma}_1 \\otimes {\\bf \\sigma}_2\\right]^{(2)} \\vert\\vert S'\\rangle\n\\end{eqnarray*}\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nWe need that\nthe coordinate vector ${\\bf r}$ can be written in terms of spherical \ncomponents as\n\n$$\n{\\bf r}_\\alpha = r\\sqrt{\\frac{4\\pi}{3}} Y_{1\\alpha}\n$$\n\nUsing this expression we get\n\n$$\n\\begin{eqnarray*}\n\\left[{\\bf r} \\otimes {\\bf r} \\right]^{(2)}_\\mu &=& \\frac{4\\pi}{3}r^2\n\\sum_{\\alpha ,\\beta}\\langle 1\\alpha 1\\beta\\vert 2\\mu \\rangle Y_{1\\alpha} Y_{1\\beta}\n\\end{eqnarray*}\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nThe product of two spherical harmonics can be written\nas\n\n2\n0\n8\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\times \\left(\\begin{array}{ccc} l_1&l_2&l \\\\ 0 &0 &0\\end{array}\\right)\nY_{l-m}(-1)^m.\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nUsing this relation we get\n\n$$\n\\begin{eqnarray*}\n\\left[{\\bf r} \\otimes {\\bf r} \\right]^{(2)}_\\mu &=& \n\\sqrt{4\\pi}r^2\n\\sum_{lm}\n\\sum_{\\alpha ,\\beta} \\langle 1\\alpha 1\\beta\\vert 2\\mu \\rangle \\\\\n&&\\times \\langle 1\\alpha 1\\beta\\vert l-m \\rangle\n\\frac{(-1)^{1-1-m}}{\\sqrt{2l+1}} \n\\left(\\begin{array}{ccc} 1&1&l \\\\ 0 &0 &0\\end{array}\\right)Y_{l-m}(-1)^m\\\\\n&=& \\sqrt{4\\pi}r^2\n\\left(\\begin{array}{ccc} 1&1&2 \\\\ 0 &0 &0\\end{array}\\right)\nY_{2-\\mu}\\\\\n&=& \\sqrt{4\\pi}r^2 \\sqrt{\\frac{2}{15}}Y_{2-\\mu}\n\\end{eqnarray*}\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nWe can then use this relation to rewrite the reduced matrix element containing the \nposition vector as\n\n$$\n\\begin{eqnarray*}\n\\langle l \\vert\\vert\\left[{\\bf r} \\otimes {\\bf r} \\right]^{(2)} \\vert \\vert l'\\rangle\n& = & \n\\sqrt{4\\pi}\\sqrt{ \\frac{2}{15}}r^2 \\langle l \\vert\\vert Y_2 \\vert \\vert l'\\rangle \\\\\n& = &\\sqrt{4\\pi}\\sqrt{ \\frac{2}{15}} r^2 (-1)^l\n\\sqrt{\\frac{(2l+1)5(2l'+1)}{4\\pi}}\n\\left(\\begin{array}{ccc} l&2&l' \\\\ 0&0&0\\end{array}\\right)\n\\end{eqnarray*}\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nUsing the reduced matrix element of the spin \noperators defined as\n\n$$\n\\begin{eqnarray*}\n\\langle S\\vert \\vert\\left[{\\bf \\sigma}_1 \\otimes {\\bf \\sigma}_2\\right]^{(2)} \\vert \\vert S' \\rangle\n& = & \n\\sqrt{(2S+1)(2S'+1)5}\n\\left\\{\\begin{array}{ccc} s_1&s_2&S \\\\s_3&s_4&S' \\\\ 1&1&2\\end{array}\\right\\}\\\\\n&\\times& \n\\langle s_1 \\vert \\vert {\\bf \\sigma}_1 \\vert \\vert s_3\\rangle\n\\langle s_2 \\vert\\vert {\\bf \\sigma}_2 \\vert \\vert s_4\\rangle\n\\end{eqnarray*}\n$$\n\nand inserting these expressions for the two reduced matrix elements we get\n\n$$\n\\begin{array}{ll}\n&\\\\\n\\langle lSJ\\vert V\\vert l'S'J\\rangle =&(-1)^{S+J}\\sqrt{30(2l+1)(2l'+1)(2S+1)(2S'+1)}\\\\\n&\\times\\left\\{\\begin{array}{ccc}l&S &J \\\\l'&S&2\\end{array}\\right\\}\n\\left(\\begin{array}{ccc}l&2&l'\\\\0&0&0\\end{array}\\right)\n\\left\\{\\begin{array}{ccc}s_{1}&s_{2}&S\\\\s_{3}&s_{4}&S'\\\\\n1&1&2\\end{array}\n\\right\\}\\\\\n&\\times\\langle s_{1}\\vert\\vert \\sigma_{1}\\vert\\vert s_{3}\\rangle\n\\langle s_{2}\\vert\\vert \\sigma_{2}\\vert \\vert s_{4}\\rangle.\n\\end{array}\n$$\n\n## Angular momentum algebra, Wigner-Eckart theorem\nNormally, we start we a nucleon-nucleon interaction fitted to reproduce scattering data.\nIt is common then to represent this interaction in terms relative momenta $k$, the center-of-mass momentum $K$\nand various partial wave quantum numbers like the spin $S$, the total relative angular momentum ${\\cal J}$, isospin $T$ and relative orbital momentum $l$ and finally the corresponding center-of-mass $L$. \nWe can then write the free interaction matrix $V$ as\n\n$$\n\\langle kKlL{\\cal J}ST\\vert\\hat{V}\\vert k'Kl'L{\\cal J}S'T\\rangle.\n$$\n\nTransformations from the relative and center-of-mass motion\nsystem to the lab system will be discussed\nbelow.\n\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nTo obtain a $V$-matrix in a h.o. basis, we need \nthe transformation\n\n$$\n\\langle nNlL{\\cal J}ST\\vert\\hat{V}\\vert n'N'l'L'{\\cal J}S'T\\rangle,\n$$\n\nwith $n$ and $N$ the principal quantum numbers of the relative and\ncenter-of-mass motion, respectively.\n\n$$\n\\vert nlNL{\\cal J}ST\\rangle= \\int k^{2}K^{2}dkdKR_{nl}(\\sqrt{2}\\alpha k)\nR_{NL}(\\sqrt{1/2}\\alpha K)\n\\vert klKL{\\cal J}ST\\rangle.\n$$\n\nThe parameter $\\alpha$ is the chosen oscillator length.\n\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nThe most commonly employed sp basis is the harmonic oscillator, which\nin turn means that\na two-particle wave function with total angular momentum $J$\nand isospin $T$\ncan be expressed as\n\n\n
                                        \n\n$$\n\\begin{array}{ll}\n\\vert (n_{a}l_{a}j_{a})(n_{b}l_{b}j_{b})JT\\rangle =&\n{\\displaystyle\n\\frac{1}{\\sqrt{(1+\\delta_{12})}}\n\\sum_{\\lambda S{\\cal J}}\\sum_{nNlL}}\nF\\times \\langle ab|\\lambda SJ \\rangle\\\\\n&\\times (-1)^{\\lambda +{\\cal J}-L-S}\\hat{\\lambda}\n\\left\\{\\begin{array}{ccc}L&l&\\lambda\\\\S&J&{\\cal J}\n\\end{array}\\right\\}\\\\\n&\\times \\left\\langle nlNL| n_al_an_bl_b\\right\\rangle\n\\vert nlNL{\\cal J}ST\\rangle ,\\end{array}\n\\label{eq:hoho} \\tag{4}\n$$\n\nwhere the term\n$\\left\\langle nlNL| n_al_an_bl_b\\right\\rangle$\nis the so-called Moshinsky-Talmi transformation coefficient (see chapter 18 of Alex Brown's notes).\n\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nThe term $\\langle ab|LSJ \\rangle $ is a shorthand\nfor the $LS-jj$ transformation coefficient,\n\n$$\n\\langle ab|\\lambda SJ \\rangle = \\hat{j_{a}}\\hat{j_{b}}\n \\hat{\\lambda}\\hat{S}\n \\left\\{\n \\begin{array}{ccc}\n l_{a}&s_a&j_{a}\\\\\n l_{b}&s_b&j_{b}\\\\\n \\lambda &S &J\n \\end{array}\n \\right\\}.\n$$\n\nHere\nwe use $\\hat{x} = \\sqrt{2x +1}$.\nThe factor $F$ is defined as $F=\\frac{1-(-1)^{l+S+T}}{\\sqrt{2}}$ if\n$s_a = s_b$ and we .\n\n\n\n## Angular momentum algebra, Wigner-Eckart theorem\nThe $\\hat{V}$-matrix in terms of harmonic oscillator wave functions reads\n\n2\n1\n9\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n2\n2\n0\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n2\n2\n1\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\times\\langle nNlL{\\cal J}ST\\vert\\hat{V}\\vert n'N'l'L'{\\cal J}S'T\\rangle.\n$$\n\nThe label $a$ represents here all the single particle quantum numbers \n$n_{a}l_{a}j_{a}$.\n", "meta": {"hexsha": "8fa94e0ca83d0d116d36fc5bf400745b32aa2ec3", "size": 116320, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "doc/pub/angmom/ipynb/angmom.ipynb", "max_stars_repo_name": "NuclearTalent/NuclearStructure", "max_stars_repo_head_hexsha": "7d18ed926172abeea358e95f4e95415e7b0a3498", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2017-07-04T16:21:42.000Z", "max_stars_repo_stars_event_max_datetime": "2019-05-24T18:10:11.000Z", "max_issues_repo_path": "doc/pub/angmom/ipynb/angmom.ipynb", "max_issues_repo_name": "NuclearTalent/NuclearStructure", "max_issues_repo_head_hexsha": "7d18ed926172abeea358e95f4e95415e7b0a3498", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/pub/angmom/ipynb/angmom.ipynb", "max_forks_repo_name": "NuclearTalent/NuclearStructure", "max_forks_repo_head_hexsha": "7d18ed926172abeea358e95f4e95415e7b0a3498", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 7, "max_forks_repo_forks_event_min_datetime": "2017-06-30T16:55:37.000Z", "max_forks_repo_forks_event_max_datetime": "2021-05-01T07:54:49.000Z", "avg_line_length": 25.8949243099, "max_line_length": 373, "alphanum_fraction": 0.5085196011, "converted": true, "num_tokens": 24041, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.7279754548076477, "lm_q1q2_score": 0.42040245112044994}} {"text": "# Welcome to QC Hackathon 2021!\n\nQuantum Coputation is an up and coming field that uses quantum gates to perform operations which exploit properties of a quantum state like superposition, and entanglement to perform computations. The aim is to solve computation problems in executable time that might not be solvable using its classical analogue. In this hackathon, we will be introducing you to challenges of different difficulty levels. \n\nA good introduction to all of these concepts can be found in the [qiskit textbook](https://qiskit.org/textbook/preface.html). We will be using qiskit in this hackathon. \n\n\n\n\n# Challenge Tier - I\nFor this first set of challenges, we encourage you to read [Section 1.4](https://qiskit.org/textbook/ch-states/single-qubit-gates.html) and [Section 2.2](https://qiskit.org/textbook/ch-gates/multiple-qubits-entangled-states.html) to get familiar with the basics of quantum computing.\n\n\n```python\n#before we start, run this cell to install qiskit\n!pip install qiskit\n!pip install pylatexenc\n```\n\n Requirement already satisfied: qiskit in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (0.25.1)\n Requirement already satisfied: qiskit-terra==0.17.1 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit) (0.17.1)\n Requirement already satisfied: qiskit-aer==0.8.1 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit) (0.8.1)\n Requirement already satisfied: qiskit-ibmq-provider==0.12.2 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit) (0.12.2)\n Requirement already satisfied: qiskit-ignis==0.6.0 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit) (0.6.0)\n Requirement already satisfied: qiskit-aqua==0.9.1 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit) (0.9.1)\n Requirement already satisfied: pybind11>=2.6 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-aer==0.8.1->qiskit) (2.6.2)\n Requirement already satisfied: numpy>=1.16.3 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-aer==0.8.1->qiskit) (1.19.5)\n Requirement already satisfied: scipy>=1.0 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-aer==0.8.1->qiskit) (1.5.2)\n Requirement already satisfied: h5py<=3.1.0 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-aqua==0.9.1->qiskit) (3.1.0)\n Requirement already satisfied: docplex<=2.20.204 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-aqua==0.9.1->qiskit) (2.15.194)\n Requirement already satisfied: quandl<=3.6.0 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-aqua==0.9.1->qiskit) (3.5.2)\n Requirement already satisfied: setuptools>=40.1.0 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-aqua==0.9.1->qiskit) (49.2.1)\n Requirement already satisfied: fastdtw<=0.3.4 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-aqua==0.9.1->qiskit) (0.3.4)\n Requirement already satisfied: pandas<=1.2.3 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-aqua==0.9.1->qiskit) (1.1.1)\n Requirement already satisfied: sympy<=1.7.1,>=1.3 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-aqua==0.9.1->qiskit) (1.6.2)\n Requirement already satisfied: scikit-learn<=0.24.1,>=0.20.0 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-aqua==0.9.1->qiskit) (0.23.2)\n Requirement already satisfied: psutil<=5.8.0,>=5 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-aqua==0.9.1->qiskit) (5.7.2)\n Requirement already satisfied: dlx<=1.0.4 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-aqua==0.9.1->qiskit) (1.0.4)\n Requirement already satisfied: retworkx<=0.8.0,>=0.7.0 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-aqua==0.9.1->qiskit) (0.8.0)\n Requirement already satisfied: yfinance<=0.1.55 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-aqua==0.9.1->qiskit) (0.1.54)\n Requirement already satisfied: requests>=2.19 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-ibmq-provider==0.12.2->qiskit) (2.24.0)\n Requirement already satisfied: requests-ntlm>=1.1.0 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-ibmq-provider==0.12.2->qiskit) (1.1.0)\n Requirement already satisfied: python-dateutil>=2.8.0 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-ibmq-provider==0.12.2->qiskit) (2.8.1)\n Requirement already satisfied: urllib3>=1.21.1 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-ibmq-provider==0.12.2->qiskit) (1.25.10)\n Requirement already satisfied: nest-asyncio!=1.1.0,>=1.0.0 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-ibmq-provider==0.12.2->qiskit) (1.4.0)\n Requirement already satisfied: websockets>=8 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-ibmq-provider==0.12.2->qiskit) (8.1)\n Requirement already satisfied: dill>=0.3 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-terra==0.17.1->qiskit) (0.3.2)\n Requirement already satisfied: python-constraint>=1.4 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-terra==0.17.1->qiskit) (1.4.0)\n Requirement already satisfied: fastjsonschema>=2.10 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-terra==0.17.1->qiskit) (2.14.5)\n Requirement already satisfied: jsonschema>=2.6 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-terra==0.17.1->qiskit) (3.2.0)\n Requirement already satisfied: ply>=3.10 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from qiskit-terra==0.17.1->qiskit) (3.11)\n Requirement already satisfied: six in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from docplex<=2.20.204->qiskit-aqua==0.9.1->qiskit) (1.15.0)\n Requirement already satisfied: pyrsistent>=0.14.0 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from jsonschema>=2.6->qiskit-terra==0.17.1->qiskit) (0.16.0)\n Requirement already satisfied: attrs>=17.4.0 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from jsonschema>=2.6->qiskit-terra==0.17.1->qiskit) (20.1.0)\n Requirement already satisfied: pytz>=2017.2 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from pandas<=1.2.3->qiskit-aqua==0.9.1->qiskit) (2020.1)\n Requirement already satisfied: more-itertools in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from quandl<=3.6.0->qiskit-aqua==0.9.1->qiskit) (8.5.0)\n Requirement already satisfied: inflection>=0.3.1 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from quandl<=3.6.0->qiskit-aqua==0.9.1->qiskit) (0.5.1)\n Requirement already satisfied: certifi>=2017.4.17 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from requests>=2.19->qiskit-ibmq-provider==0.12.2->qiskit) (2020.6.20)\n Requirement already satisfied: idna<3,>=2.5 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from requests>=2.19->qiskit-ibmq-provider==0.12.2->qiskit) (2.10)\n Requirement already satisfied: chardet<4,>=3.0.2 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from requests>=2.19->qiskit-ibmq-provider==0.12.2->qiskit) (3.0.4)\n Requirement already satisfied: cryptography>=1.3 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from requests-ntlm>=1.1.0->qiskit-ibmq-provider==0.12.2->qiskit) (3.1)\n Requirement already satisfied: ntlm-auth>=1.0.2 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from requests-ntlm>=1.1.0->qiskit-ibmq-provider==0.12.2->qiskit) (1.5.0)\n Requirement already satisfied: cffi!=1.11.3,>=1.8 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from cryptography>=1.3->requests-ntlm>=1.1.0->qiskit-ibmq-provider==0.12.2->qiskit) (1.14.2)\n Requirement already satisfied: pycparser in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from cffi!=1.11.3,>=1.8->cryptography>=1.3->requests-ntlm>=1.1.0->qiskit-ibmq-provider==0.12.2->qiskit) (2.20)\n Requirement already satisfied: joblib>=0.11 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from scikit-learn<=0.24.1,>=0.20.0->qiskit-aqua==0.9.1->qiskit) (0.16.0)\n Requirement already satisfied: threadpoolctl>=2.0.0 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from scikit-learn<=0.24.1,>=0.20.0->qiskit-aqua==0.9.1->qiskit) (2.1.0)\n Requirement already satisfied: mpmath>=0.19 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from sympy<=1.7.1,>=1.3->qiskit-aqua==0.9.1->qiskit) (1.1.0)\n Requirement already satisfied: multitasking>=0.0.7 in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (from yfinance<=0.1.55->qiskit-aqua==0.9.1->qiskit) (0.0.9)\n Requirement already satisfied: pylatexenc in c:\\users\\naresh\\appdata\\local\\programs\\python\\python38\\lib\\site-packages (2.9)\n\n\n## Task 1 \n$$\\newcommand{\\ket}[1]{\\left|{#1}\\right\\rangle}$$\n$$\\newcommand{\\bra}[1]{\\left\\langle{#1}\\right|}$$\n\nAn $H$ gate, or a Hadamard gate, is a single qubit gate which performs the following operations:\n
                                        \n$\\begin{align}\n H\\ket{0} &= \\dfrac{\\ket{0}+\\ket{1}}{\\sqrt{2}}\\\\\n H\\ket{1} &= \\dfrac{\\ket{0}-\\ket{1}}{\\sqrt{2}}\n\\end{align}$ \n
                                        \n\nA $Z$ gate, similarly, performs the following operations:\n
                                        \n$\\begin{align}\n Z\\ket{0} &= \\ket{0}\\\\\n Z\\ket{1} &= -\\ket{1}\n\\end{align}$ \n
                                        \n\nIn this challenge, you will be given a single bit unitary operation that can either be an $H$ gate or a $Z$ gate. You can apply the unitary operation **at most twice**. Using what you learnt from [Section 1.4](https://qiskit.org/textbook/ch-states/single-qubit-gates.html), distinguish whether the given operation is an H gate or a Z gate.\n\n**Output:** 0 if it is the $Z$ gate, 1 if it is the $H$ gate.\n\n\n```python\nfrom qiskit import *\n\ndef A1_solve(Unitary, qc):\n ##########################################\n #enter your code here\n #To apply the unitary to the qubit, use syntax: Unitary(0)\n\n\n #Expected Output: {'0': 1024} if Z gate; and\n # {'1': 1024} if H gate\n\n ###########################################\n qc.x(0)\n Unitary(0)\n qc.x(0)\n Unitary(0)\n qc.measure(0,0)\n print(qc.draw())\n\n backend = Aer.get_backend('qasm_simulator')\n job = execute(qc, backend)\n counts = job.result().get_counts()\n\n return counts\n\n\n\nimport random\nqr = QuantumRegister(1)\ncr = ClassicalRegister(1)\nA1_qc = QuantumCircuit(qr, cr)\nif __name__ == \"__main__\":\n if random.uniform(0,1) < 0.5:\n Unitary = A1_qc.z\n else:\n Unitary = A1_qc.h\n\n _output = A1_solve(Unitary, A1_qc)\n print(_output)\n```\n\n \u250c\u2500\u2500\u2500\u2510\u250c\u2500\u2500\u2500\u2510\u250c\u2500\u2500\u2500\u2510\u250c\u2500\u2500\u2500\u2510\u250c\u2500\u2510\n q39_0: \u2524 X \u251c\u2524 H \u251c\u2524 X \u251c\u2524 H \u251c\u2524M\u251c\n \u2514\u2500\u2500\u2500\u2518\u2514\u2500\u2500\u2500\u2518\u2514\u2500\u2500\u2500\u2518\u2514\u2500\u2500\u2500\u2518\u2514\u2565\u2518\n c7: 1/\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2569\u2550\n 0 \n {'1': 1024}\n\n\n## Task 2\nAny quantum state can be written in the form of a superposition of the $\\ket{0}$ and $\\ket{1}$ states in the dirac notation. So any single qubit quantum state, $\\ket{\\psi}$, can be written as:\n$\\begin{equation}\\ket{\\psi} = \\alpha \\ket{0} + \\beta \\ket{1} \\end{equation}$ where the squares of the coefficients of $\\ket{0}$ and $\\ket{1}$ are equal to the respective probability of the output on measuring $\\ket{\\psi}$.
                                        \nSimilarly, two qubit states can be described by: $\\displaystyle \\begin{equation}\\ket{\\psi} = \\sum_{j, k\\in\\{0,1\\}} \\alpha_{jk}\\ket{j k} \\end{equation}$\n\nUsing simple quantum gates like the ones you have learnt in [Section 1.4](https://qiskit.org/textbook/ch-states/single-qubit-gates.html) and [Section 2.2](https://qiskit.org/textbook/ch-gates/multiple-qubits-entangled-states.html), prepare the superposition state described by:\n
                                        $\\begin{align} \\ket{\\psi} = \\dfrac{1}{2}\\ket{01} + \\dfrac{1}{2}\\ket{10} + \\dfrac{1}{\\sqrt{2}}\\ket{00} \\end{align}$
                                        \n\n\n\n\n\n```python\nfrom qiskit import *\ndef A2_solve():\n \n #enter your code here, specify the qubits in QuantumCircuit\n qc = QuantumCircuit(2 , 2)\n qc.h(1)\n qc.ch(1, 0)\n qc.cx(0, 1)\n qc.measure(0, 0)\n qc.measure(1, 1)\n print(qc.draw())\n\n backend = Aer.get_backend('qasm_simulator')\n job = execute(qc, backend)\n counts = job.result().get_counts()\n\n print(counts)\n return\n\nif __name__ == \"__main__\":\n A2_solve()\n\n```\n\n \u250c\u2500\u2500\u2500\u2510 \u250c\u2500\u2510 \n q_0: \u2500\u2500\u2500\u2500\u2500\u2524 H \u251c\u2500\u2500\u25a0\u2500\u2500\u2524M\u251c\u2500\u2500\u2500\n \u250c\u2500\u2500\u2500\u2510\u2514\u2500\u252c\u2500\u2518\u250c\u2500\u2534\u2500\u2510\u2514\u2565\u2518\u250c\u2500\u2510\n q_1: \u2524 H \u251c\u2500\u2500\u25a0\u2500\u2500\u2524 X \u251c\u2500\u256b\u2500\u2524M\u251c\n \u2514\u2500\u2500\u2500\u2518 \u2514\u2500\u2500\u2500\u2518 \u2551 \u2514\u2565\u2518\n c: 2/\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2569\u2550\u2550\u2569\u2550\n 0 1 \n {'00': 516, '10': 251, '01': 257}\n\n\n## Task 3 \n\nA full subtractor is a combinational circuit that performs subtraction of two bits in classical computation. A thorough explanation and implementation of the circuit can be found here: [Full Subtractor Circuit in digital logic](www.geeksforgeeks.org/full-subtractor-in-digital-logic/). Throughout this challenge you may have realised that the quantum gates are very similar to classical gates, and the elementary quantum gates can be used in place of their classical analogues in a quantum circuit. \n\nYour task is to implement a full subtractor circuit using only quantum gates. You can take help from the resource mentioned above, and try to replace the classical gates with their quantum analogues. In more precise terms, you must reproduce the results of the truth table that can be found in the link above.\n\n**Input:** Three values: $A$, $B$, $B_{in}$; that correspond to the truth table.\n\n**Output:** Two values: $D$, $B_{out}$ (You must measure two qubits in your circuit)\n\nThere is a bonus point for using the minimum number of qubits.\n\n\n```python\nfrom qiskit import *\n\ndef A3_solve(inputBits):\n no_of_qubits_used = 4 #Enter the number of qubits you require\n qc = QuantumCircuit(no_of_qubits_used, 2)\n for i in range(0, len(inputBits)):\n if inputBits[i] == 1:\n qc.x(i) \n #Enter code from here:\n qc.x(0)\n qc.ccx(0, 1, 3)\n qc.ccx(0, 2, 3)\n qc.ccx(1, 2, 3)\n qc.x(0)\n qc.cx(0, 2)\n qc.cx(1, 2)\n qc.measure(2, 0)\n qc.measure(3, 1)\n \n\n\n\n\n\n\n\n\n\n ################### Tests ##################################### \n print(qc.draw())\n backend = Aer.get_backend('qasm_simulator')\n job = execute(qc, backend)\n counts = job.result().get_counts()\n outputList = list(counts.keys())\n print(counts)\n if len(outputList) != 1 or len(outputList[0][0]) > 2:\n print('Error: There are too many outputs.')\n return\n output = outputList[0]\n if len(output) != 2:\n print('Error: There are too few outputs. Please make sure you are *measuring* the appropriate qubits in your circuit. Use qc.measure() method to measure the qubits.')\n return\n return output[1], output[0]\n\nif __name__ == \"__main__\":\n A = 0\n B = 1\n B_in = 1\n _D, _Bout = A3_solve([A, B, B_in])\n print(_D, _Bout)\n #You need to check whether these are consistent with the truth table yourself.\n```\n\n \u250c\u2500\u2500\u2500\u2510 \u250c\u2500\u2500\u2500\u2510 \n q_0: \u2524 X \u251c\u2500\u2500\u25a0\u2500\u2500\u2500\u2500\u25a0\u2500\u2500\u2524 X \u251c\u2500\u2500\u25a0\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n \u251c\u2500\u2500\u2500\u2524 \u2502 \u2502 \u2514\u2500\u2500\u2500\u2518 \u2502 \n q_1: \u2524 X \u251c\u2500\u2500\u25a0\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u25a0\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u25a0\u2500\u2500\u2500\u2500\u2500\n \u251c\u2500\u2500\u2500\u2524 \u2502 \u2502 \u2502 \u250c\u2500\u2534\u2500\u2510\u250c\u2500\u2534\u2500\u2510\u250c\u2500\u2510\n q_2: \u2524 X \u251c\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u25a0\u2500\u2500\u2500\u2500\u25a0\u2500\u2500\u2524 X \u251c\u2524 X \u251c\u2524M\u251c\n \u2514\u2500\u2500\u2500\u2518\u250c\u2500\u2534\u2500\u2510\u250c\u2500\u2534\u2500\u2510\u250c\u2500\u2534\u2500\u2510\u2514\u252c\u2500\u252c\u2518\u2514\u2500\u2500\u2500\u2518\u2514\u2565\u2518\n q_3: \u2500\u2500\u2500\u2500\u2500\u2524 X \u251c\u2524 X \u251c\u2524 X \u251c\u2500\u2524M\u251c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u256b\u2500\n \u2514\u2500\u2500\u2500\u2518\u2514\u2500\u2500\u2500\u2518\u2514\u2500\u2500\u2500\u2518 \u2514\u2565\u2518 \u2551 \n c: 2/\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2569\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2569\u2550\n 1 0 \n {'10': 1024}\n 0 1\n\n\n## Additional information\n\n**Created by:** Yashee Sinha\n", "meta": {"hexsha": "95f807f5d7dc517667d35e7234c0373511ee572f", "size": 21184, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "SectionA_QC_Hackathon.ipynb", "max_stars_repo_name": "naresh1205/Quantum-Computing-Hackathon", "max_stars_repo_head_hexsha": "0fa427c943be2c05671b438374d820ba7d3883be", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "SectionA_QC_Hackathon.ipynb", "max_issues_repo_name": "naresh1205/Quantum-Computing-Hackathon", "max_issues_repo_head_hexsha": "0fa427c943be2c05671b438374d820ba7d3883be", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "SectionA_QC_Hackathon.ipynb", "max_forks_repo_name": "naresh1205/Quantum-Computing-Hackathon", "max_forks_repo_head_hexsha": "0fa427c943be2c05671b438374d820ba7d3883be", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 53.6303797468, "max_line_length": 508, "alphanum_fraction": 0.5864803625, "converted": true, "num_tokens": 5760, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858117, "lm_q2_score": 0.7279754430043072, "lm_q1q2_score": 0.4204024443040754}} {"text": "## Notebook Setup\n\nLet's start by downloading the dataset (if not already done) and seting up the notebook:\n\n\n```python\n!wget https://github.com/lompabo/aiiti-04-2021/releases/download/data/data.zip\n!unzip -o data.zip\n!ls .\n```\n\n --2021-05-11 07:59:17-- https://github.com/lompabo/aiiti-04-2021/releases/download/data/data.zip\n Resolving github.com (github.com)... 140.82.121.4\n Connecting to github.com (github.com)|140.82.121.4|:443... connected.\n HTTP request sent, awaiting response... 302 Found\n Location: https://github-releases.githubusercontent.com/366207519/ca7f7800-b23e-11eb-9e39-5e2e90e5e08a?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAIWNJYAX4CSVEH53A%2F20210511%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20210511T075822Z&X-Amz-Expires=300&X-Amz-Signature=3e94618b6310ddc712b1bcc8dd5f566d032523c5309d02939bebb8460993db5e&X-Amz-SignedHeaders=host&actor_id=0&key_id=0&repo_id=366207519&response-content-disposition=attachment%3B%20filename%3Ddata.zip&response-content-type=application%2Foctet-stream [following]\n --2021-05-11 07:59:17-- https://github-releases.githubusercontent.com/366207519/ca7f7800-b23e-11eb-9e39-5e2e90e5e08a?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAIWNJYAX4CSVEH53A%2F20210511%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20210511T075822Z&X-Amz-Expires=300&X-Amz-Signature=3e94618b6310ddc712b1bcc8dd5f566d032523c5309d02939bebb8460993db5e&X-Amz-SignedHeaders=host&actor_id=0&key_id=0&repo_id=366207519&response-content-disposition=attachment%3B%20filename%3Ddata.zip&response-content-type=application%2Foctet-stream\n Resolving github-releases.githubusercontent.com (github-releases.githubusercontent.com)... 185.199.108.154, 185.199.111.154, 185.199.109.154, ...\n Connecting to github-releases.githubusercontent.com (github-releases.githubusercontent.com)|185.199.108.154|:443... connected.\n HTTP request sent, awaiting response... 200 OK\n Length: 44485661 (42M) [application/octet-stream]\n Saving to: \u2018data.zip.2\u2019\n \n data.zip.2 100%[===================>] 42.42M 7.89MB/s in 6.1s \n \n 2021-05-11 07:59:24 (6.96 MB/s) - \u2018data.zip.2\u2019 saved [44485661/44485661]\n \n Archive: data.zip\n inflating: data/vegashrinker.csv \n inflating: data/hpc.csv \n '1. Autoencoders for Anomaly Detection.ipynb'\t data\n '2. Density Estimation with Neural Networks.ipynb' data.zip\n '3. Component Wear Anomalies.ipynb'\t\t data.zip.1\n Dockerfile\t\t\t\t\t data.zip.2\n LICENSE\t\t\t\t\t docker-compose.yml\n README.md\t\t\t\t\t util\n assets\n\n\n\n```python\n# ============================================================\n# Notebook setup\n# ============================================================\n\n# Control figure size\ninteractive_figures = False\nif interactive_figures:\n # Normal behavior\n %matplotlib widget\n figsize=(9, 3)\nelse:\n # PDF export behavior\n figsize=(14, 3)\n\nfrom util import nn\nimport numpy as np\nfrom matplotlib import pyplot as plt\nimport pandas as pd\n\n# Load HPC data\ndata_folder = 'data'\nhpc = pd.read_csv(data_folder+ '/hpc.csv', parse_dates=['timestamp'])\n\n# Identify input columns\nhpc_in = hpc.columns[1:-1]\n\n# Standardization\ntr_end, val_end = 3000, 4500\nhpcs = hpc.copy()\ntmp = hpcs.iloc[:tr_end]\nhpcs[hpc_in] = (hpcs[hpc_in] - tmp[hpc_in].mean()) / tmp[hpc_in].std()\n\n# Training, validation, and test set\ntrdata = hpcs.iloc[:tr_end]\nvaldata = hpcs.iloc[tr_end:val_end]\ntsdata = hpcs.iloc[val_end:]\n\n# Anomaly labels\nhpc_labels = pd.Series(index=hpc.index, data=(hpc['anomaly'] != 0), dtype=int)\n\n# Cost model\nc_alarm, c_missed, tolerance = 1, 5, 12\ncmodel = nn.HPCMetrics(c_alarm, c_missed, tolerance)\n```\n\n# Density Estimation with Neural Models\n\n## Density Estimation vs Autoencoders\n\n**Anomaly detection can be formulated as density estimation**\n\n* This is probably _the cleanest formulation_ for the problem\n* ...And usually leads to good results\n\n**KDE as an estimation technique**\n\n* ...Works reasonably well for low-dimensional data\n* ...Becomes _slower and more data hungry_ for higher-dimensional data\n\n**Autoencoders overcome some of these limitations**\n\n* They are _faster and less data hungry_ for high-dimensional data\n* They can provide _additional insight_ in the anomalies\n* ...But they are usually _worse_ than D.E. in terms of _pure detection power_\n\n**Let's try to understand why this may be the case...**\n\n## Density Estimation vs Autoencoders\n\n**Anomaly Detection based on D.E. checks whether:**\n\n$$\n- \\log f({\\bf x}, \\lambda) \\geq \\theta\n$$\n\n* Where $\\bf x$ is the input vector, $f$ the density estimator, and $\\lambda$ its parameter vector\n* $\\theta$ is the anomaly detection threshold\n\n**Anomaly Detection based on autoencoders usually relies on:**\n\n$$\n\\|g({\\bf x}, \\lambda) - {\\bf x}\\|_2^2 \\geq \\theta^\\prime\n$$\n\n* Where $g$ is the autoencoder, with parameter vector $\\lambda$\n* $\\theta^\\prime$ is again a suitably-chosen detection threshold\n\n## Density Estimation vs Autoencoders\n\n**The detection condition for autoencoders admits a probabilistic interpretation**\n\nLike we did for linear regression, we can rewrite:\n\n$$\n\\|g({\\bf x}, \\lambda) - {\\bf x}\\|_2^2 \\longrightarrow\n\\sum_{j=1}^m (g_j({\\bf x}, \\lambda) - x_j)^2 \\longrightarrow\n\\log \\prod_{j=1}^m \\exp\\left((g_j({\\bf x}, \\lambda) - x_j)^2\\right)\n$$\n\nFrom which, with an _affine transformation_, for some fixed $\\sigma$ we get:\n\n$$\n\\log \\frac{1}{\\sigma\\sqrt{2\\pi}} + \\frac{1}{\\sigma^2} \\log \\prod_{j=1}^m \\exp \\left((g_j({\\bf x}, \\lambda) - x_j)^2\\right) \\quad \\longrightarrow\\\\\n\\longrightarrow\\quad \\log \\prod_{j=1}^m \\frac{1}{\\sigma\\sqrt{2\\pi}} \\exp \\left(\\left(\\frac{g_j({\\bf x}, \\lambda) - x_j}{\\sigma}\\right)^2\\right) \n$$\n\n* The transformation preserves all the function optima\n\n## Density Estimation vs Autoencoders\n\n**Therefore, optimizing the MSE is equivalent to optimizing**\n\n$$\n-\\log \\prod_{j=1}^m \\varphi (x_j \\mid g_j({\\bf x}, \\lambda), \\sigma)\n$$\n\n* I.e. the log likelihood (estimated probability of the data)...\n* ...Assuming that the prediction for each $x_i$ is _independent and normally distributed_\n* ...with _mean_ equal to the predictions $g_j({\\bf x}, \\lambda)$ and fixed _standard deviation_ $\\sigma$\n\n**This is similar to what we observed for Linear Regression**\n\n* In LR, we assume normality, independence and fixed variance _on the samples_\n* With autoencoders, we do it _also on the features_\n\n## Density Estimation vs Autoencoders\n\n**The bottomline**\n\n* Even with autoencoders, at training time we _solve a density estimation problem_\n* ...But we do it _with some limiting assumptions_\n\n> **This is why D.E.-based anomaly detection _tends to work better_**\n\n**So we have**\n\n* Either a density estimator with issues on high-dimensional data (KDE)\n* ...Or a not-quite D.E. with good support for high-dimensional data (autoencoders)\n\n\n> **Can we get the best of both worlds?**\n\n## Flow Models\n\n**Ideally, we wish _a neural approach for density estimation_**\n\nThere are only a handful of approaches, often referred to as _flow models_:\n\n* [Normalizing Flows](https://arxiv.org/abs/1505.05770)\n* [Real Non-Volume Preserving transformations (Real NVP)](https://arxiv.org/abs/1605.08803)\n* [Generative Flow with 1x1 convolutions (Glow)](https://arxiv.org/abs/1807.03039)\n\n**These are all (fairly) advanced and recent approaches**\n\n* Main idea: transforming _a simple (and known) probability distribution_...\n* ..._Into a complex (and unknown) distribution_ that matches that of the available data\n\nAs many ML models, they are trained for maximum likelihood\n\n* I.e. to maximize the estimated probability of the available data\n\n## Flow Models\n\n**All flow models rely on the _change of variable formula_**\n\n* Let $x$ be a random variable representing the source of our data\n* Let $p_x(x)$ be its (unknown) density function\n* Let $z$ be a random _latent variable_ with known distribution $p_z$\n* Let $f$ be a _bijective_ (i.e. invertible) transformation\n\nThen, the change of variable formula states that:\n\n$$\np_x(x) = p_z(f(x)) \\left| \\det \\left(\\frac{\\partial f(x)}{\\partial x^T} \\right)\\right|\n$$\n\n* Where $\\det$ is the determinant and $\\partial f / \\partial x^T$ is the Jacobian of $f$\n\n**The formula links the two distributions via _the flow model $f$_**\n\n## Flow Models\n\n**Let's consider how we can use the formula**\n\n$$\np_x(x) = p_z(f(x)) \\left| \\det \\left(\\frac{\\partial f(x)}{\\partial x^T} \\right)\\right|\n$$\n\n* Given _an example $x$_ (e.g. from our dataset)\n* We _compute the mapping $f(x)$_, i.e. the corresponding value for the latent variable $z$\n* ...Plus the _determinant of the Jacobian_ $\\partial f / \\partial x^T$ in $x$\n* Then we can use the formula to compute the _probability of the example_\n\n**The challenge is defining the transformation $f$ (i.e. the mapping)**\n\n* It must be _invertible_ (for the formula to hold)\n* It must be _non-linear_ (to handle any distribution)\n* It should allow for an _easy computation of the determinant_\n\n## Real NVP\n\n**We will use [Real Non-Volume Preserving transformations](https://arxiv.org/abs/1605.08803) as an example**\n\nReal NVPs are _a type of neural network_\n\n* _Input:_ a vector $x$ representing an example\n* _Output:_ a vector $z$ of values for the latent variable\n* _Key property:_ $z$ should have a chosen probability distribution\n* ...Typically: standard Normal distribution for each $z_i$:\n\n$$\nz \\sim \\mathcal{N}({\\bf 0}, I)\n$$\n\nIn other words\n\n* $z$ follows a multivariate distribution\n* ...But the covariance matrix is diagonal, i.e. each component is independent\n\n## Real NVP\n\n**A Real NVP architecture consists of a stack of _affine coupling layers_**\n\nEach layer treats its input $x$ as split into two components, i.e. $x = (x^1, x^2)$\n\n* One component is _passed forward_ as it is\n* The second is processed via an _affine transformation_\n\n$$\\begin{align}\ny^1 &= x^1 \\\\\ny^2 &= e^{s(x^1)} \\odot x^2 + t(x^1)\n\\end{align}$$\n\n**The affine transformation is parameterized with two functions:**\n\n* $x^2$ is _scaled_ using $e^{s(x^1)}$, $x^2$ is _translated_ using $t(x^1)$\n* $\\odot$ is the element-wise product (Hadamard product)\n\nSince we have functions rather than fixed vectors, _the transformation is non-linear_\n\n## Real NVP - Affine Coupling Layers\n\n**Visually, each layer has the following _compute graph:_**\n\n
                                        \n\n* We are using part of the input (i.e. $x^1$)...\n* ...To transform the remaining part (i.e. $x^2$)\n\n**Both $s$ and $t$ are usually implemented as Multilayer Perceptrons**\n\n* I.e. pretend there are a few fully connected layers when you see $s$ and $t$\n\n## Real NVP - Affine Coupling Layers\n\n**Each affine coupling layer is _easy to invert_**\n\n
                                        \n\nSince part of the input (i.e. $x^1$) has been passed forward unchanged, we have that:\n\n$$\\begin{align}\nx^1 &= y^1 \\\\\nx^2 &= (y^2 - t(y^1)) \\oslash e^{s(y^1)}\n\\end{align}$$\n\n* $\\oslash$ is the element-wise division\n\n## Real NVP - Affine Coupling Layers\n\n**The _determinant_ of each layer is easy to compute**\n\nThe Jacobian of the transformation is:\n\n$$\n\\frac{\\partial y}{\\partial x^T} = \\left(\\begin{array}{cc}\nI & 0 \\\\\n\\frac{\\partial t(x^1)}{\\partial x^T} & \\text{diag}(e^{s(x^1)})\n\\end{array}\\right)\n$$\n\nThe most (only, actually) important thing is that _the matrix is triangular:_\n\n* ...Hence, its determinant is the product of the terms on the main diagonal:\n\n$$\n\\det\\left(\\frac{\\partial y}{\\partial x^T}\\right) = \\prod_{j \\in I_{x_1}} e^{s(x^1_i)} = \\exp \\left( \\sum_{j \\in I_{x_1}} s(x^1_i) \\right)\n$$\n\n## Real NVP - Considerations\n\n**Overall, we have a transformation that:**\n\n* ...Is _non-linear_, and can be made arbitrarily _deep_\n* ...Is _Invertible_ (so as to allow application of the change of variable formula)\n* ...Is well suited for _determinant computation_\n\n**Depth and non-linearity are very important:**\n\n* The whole approach works _only if_ we can construct a mapping between $x$ and $z$...\n* ...I.e. if we can transform one probability distribution into the other\n\nA poor mapping will lead to poor estimates\n\n\n\n## Real NVP - Considerations\n\n**At training time we maximize the log likelihood...**\n\n...Hence we care about _log probabilities_:\n\n$$\n\\log p_x(x) = \\log p_z(f(x)) +\\log\\, \\left| \\det \\left(\\frac{\\partial f(x)}{\\partial x^T} \\right)\\right|\n$$\n\n* If we choose a Normal distribution for $z$, the log _cancels all exponentials in the formula_\n* I.e. the one in the Normal PDF and the one in the determinant computation\n\n**In general, we want to make sure that all variables are transformed**\n\n* We need to be careful to define the $x^1, x^2$ components on different layers...\n* ...So that no variable is passed forward unchanged along the whole network\n\nA simple approach: _alternate the roles_ (i.e. swap the role of $x^1, x^2$ at every layer)\n\n## Real NVP as Generative Models\n\n**Since Real NVPs are invertible, they can be used as _generative models_**\n\nFormally, they can _sample_ from the distribution they have learned\n\n* We just need to sample from $p_z$, i.e. on the latent space\n - ...And this is easy since the distribution is simple an known\n* Then we go through the whole architecture _backwards_\n - ...Using the inverted version of the affine coupling layers\n\n**In fact, generating data is often their _primary purpose_**\n\nThey can (or could) be used for:\n\n* Super resolution\n* Procedural content generation\n* Data augmentation (relevant in an industrial context)\n\nRecent versions allow for [data generation with controlled attributes](https://openai.com/blog/glow/)\n\n# Implementing Real NVPs\n\n## Implementing Real NVPs\n\n**We will now see how to implement Real NVPs**\n\nThe basis from our code comes from the [official keras documentation](https://keras.io/examples/generative/real_nvp/)\n\n* It will rely partially on low-level APIs of keras\n\nWe start by importing several packages:\n\n```python\nimport tensorflow as tf\nimport tensorflow_probability as tfp\nfrom tensorflow.keras.layers import Dense\nfrom tensorflow.keras.callbacks import EarlyStopping\nfrom tensorflow.keras.regularizers import l2\nfrom sklearn.datasets import make_moons\n```\n\n* `tensorflow_probability` is a tensorflow extension for probabilistic computations\n* ...And allows for easy manipulation of probability distributions\n\n## Affine Coupling Layer\n\n**Then we define a function to build each _affine coupling layer_:**\n\n```python\ndef coupling(input_shape, nunits=64, nhidden=2, reg=0.01):\n assert(nhidden >= 0) \n x = keras.layers.Input(shape=input_shape)\n # Build the layers for the t transformation (translation)\n t = x\n for i in range(nhidden):\n t = Dense(nunits, activation=\"relu\", kernel_regularizer=l2(reg))(t)\n t = Dense(input_shape, activation=\"linear\", kernel_regularizer=l2(reg))(t)\n # Build the layers for the s transformation (scale)\n s = x\n for i in range(nhidden):\n s = Dense(nunits, activation=\"relu\", kernel_regularizer=l2(reg))(s)\n s = Dense(input_shape, activation=\"tanh\", kernel_regularizer=l2(reg))(s)\n # Return the layers, wrapped in a keras Model object\n return keras.Model(inputs=x, outputs=[s, t])\n```\n\n## Affine Coupling Layer\n\n**This part of the code builds _the translation (i.e. $t$) function:_**\n\n```python\ndef coupling(input_shape, nunits=64, nhidden=2, reg=0.01):\n ... \n x = keras.layers.Input(shape=input_shape)\n t = x\n for i in range(nhidden):\n t = Dense(nunits, activation=\"relu\", kernel_regularizer=l2(reg))(t)\n t = Dense(input_shape, activation=\"linear\", kernel_regularizer=l2(reg))(t)\n ...\n```\n\n* It's _just a Multi-Layer Perceptron_ built using the functional API\n* The output represents an offset, hence the \"linear\" activation function in the last layer\n\n## Affine Coupling Layer\n\n**This part of the code builds _the translation (i.e. $t$) function:_**\n\n```python\ndef coupling(input_shape, nunits=64, nhidden=2, reg=0.01):\n ... \n x = keras.layers.Input(shape=input_shape)\n t = x\n for i in range(nhidden):\n t = Dense(nunits, activation=\"relu\", kernel_regularizer=l2(reg))(t)\n t = Dense(input_shape, activation=\"linear\", kernel_regularizer=l2(reg))(t)\n ...\n```\n\n* The output and input have the same shape, but $x^1$ and $x^2$ may have _different size_\n* This will be resolved by _masking_ some of the output of the affine layer\n* ...The masked portions _will have no effect_, with effectively the same result\n* The main drawback is higher memory consumption (and computational cost)\n\n## Affine Coupling Layer\n\n**This part of the code builds _the scaling (i.e. $s$) function:_**\n\n```python\ndef coupling(input_shape, nunits=64, nhidden=2, reg=0.01):\n ... \n x = keras.layers.Input(shape=input_shape)\n ...\n s = x\n for i in range(nhidden):\n s = Dense(nunits, activation=\"relu\", kernel_regularizer=l2(reg))(s)\n s = Dense(input_shape, activation=\"tanh\", kernel_regularizer=l2(reg))(s)\n ...\n```\n\n* Another MLP, with a bipolar sigmoid (\"tanh\") activation function in the output layer\n* Using \"tanh\" limits the amount of scaling per affine coupling layer\n* ...Which in turn makes training more numerically stable\n* For the same reason, we use L2 regularizers on the MPL weights\n\n## RNVP Model\n\n**Then, we define a Real NVP architecture by subclassing keras.model**\n\n```python\nclass RealNVP(keras.Model):\n def __init__(self, input_shape, num_coupling, units_coupling=32, depth_coupling=0,\n reg_coupling=0.01): ...\n @property\n def metrics(self): ...\n def call(self, x, training=True): ...\n def log_loss(self, x): ...\n def score_samples(self, x): ...\n def train_step(self, data): ...\n def test_step(self, data): ...\n```\n\n* We will now discuss _the most important methods_\n* Sometimes with a few simplifications (for sake of clarity)\n\n## RNVP Model\n\n**The `__init__` method (constructor) initializes the internal fields**\n\n```python\n def __init__(self, input_shape, num_coupling, units_coupling=32, depth_coupling=0,\n reg_coupling=0.01):\n super(RealNVP, self).__init__()\n self.distribution = tfp.distributions.MultivariateNormalDiag(\n loc=np.zeros(input_shape, dtype=np.float32),\n scale_diag=np.ones(input_shape, dtype=np.float32)\n )\n half_n = int(np.ceil(input_shape/2))\n m1 = ([0, 1] * half_n)[:input_shape]\n m2 = ([1, 0] * half_n)[:input_shape]\n self.masks = np.array([m1, m2] * (num_coupling // 2), dtype=np.float32)\n self.loss_tracker = keras.metrics.Mean(name=\"loss\")\n self.layers_list = [coupling(input_shape, units_coupling, depth_coupling, reg_coupling)\n for i in range(num_coupling)]\n```\n\n## RNVP Model\n\n**The `__init__` method (constructor) initializes the internal fields**\n\n```python\n def __init__(self, input_shape, num_coupling, units_coupling=32, depth_coupling=0,\n reg_coupling=0.01):\n ...\n self.distribution = tfp.distributions.MultivariateNormalDiag(\n loc=np.zeros(input_shape, dtype=np.float32),\n scale_diag=np.ones(input_shape, dtype=np.float32)\n )\n ...\n```\n\nHere we build a `tfp` object to handle the known distribution\n\n* As it is customary, we chosen a Multivariate Normal distribution\n* ...With independent components, zero mean, and unary standard deviation\n\n## RNVP Model\n\n**The `__init__` method (constructor) initializes the internal fields**\n\n```python\n def __init__(self, input_shape, num_coupling, units_coupling=32, depth_coupling=0,\n reg_coupling=0.01):\n ...\n half_n = int(np.ceil(input_shape/2))\n m1 = ([0, 1] * half_n)[:input_shape]\n m2 = ([1, 0] * half_n)[:input_shape]\n self.masks = np.array([m1, m2] * (num_coupling // 2), dtype=np.float32)\n ...\n```\n\nHere we build the masks to discriminate the $x_1$ and $x_2$ components at each layer\n\n* As in the original RNVP paper, we use an _alternating checkboard pattern_\n - I.e. we take even indexes at one layer, and odd indexes at the next layer\n* ...So that all variables are transformed, if we have at least 2 affine coupling layers\n\n## RNVP Model\n\n**The `__init__` method (constructor) initializes the internal fields**\n\n```python\n def __init__(self, input_shape, num_coupling, units_coupling=32, depth_coupling=0,\n reg_coupling=0.01):\n ...\n self.layers_list = [coupling(input_shape, units_coupling, depth_coupling, reg_coupling)\n for i in range(num_coupling)]\n```\n\nFinally, here we build the model layers\n\n* Each one consists in an affine coupling\n* ...And contains in turn two Multi Layer Perceptrons\n* Recall that we need at least 2 affine couplings to transform all variables\n\n## RNVP Model\n\n**The `call` method handles the transformation, in both directions**\n\n```python\ndef call(self, x, training=True):\n log_det_inv, direction = 0, 1\n if training: direction = -1\n for i in range(self.num_coupling)[::direction]:\n x_masked = x * self.masks[i]\n reversed_mask = 1 - self.masks[i]\n s, t = self.layers_list[i](x_masked)\n s, t = s*reversed_mask, t*reversed_mask\n gate = (direction - 1) / 2\n x = reversed_mask * (x * tf.exp(direction * s) + direction * t * tf.exp(gate * s)) \\\n + x_masked\n log_det_inv += gate * tf.reduce_sum(s, axis=1)\n return x, log_det_inv\n```\n\n\n\n## RNVP Model\n\n**The `call` method handles the transformation, in both directions**\n\n```python\ndef call(self, x, training=True):\n log_det_inv, direction = 0, 1\n if training: direction = -1\n for i in range(self.num_coupling)[::direction]:\n ...\n```\n\nThe `direction` variable controls the direction of the transformation\n\n* By default, this implementation transforms $z$ into $x$\n - I.e. it works _backwards_, compared to our theoretical discussion\n* This is the case since RNVP are often mainly used as _generative models_\n* At training time, we always want to transform $x$ into $z$\n* ...And this is why `direction = -1` when `training` is `True`\n\n## RNVP Model\n\n**The `call` method handles the transformation, in both directions**\n\n```python\ndef call(self, x, training=True):\n for i in range(self.num_coupling)[::direction]:\n x_masked = x * self.masks[i]\n reversed_mask = 1 - self.masks[i]\n s, t = self.layers_list[i](x_masked)\n s, t = s*reversed_mask, t*reversed_mask\n ...\n```\n\n* Here we mask $x$, i.e. filter the $x_1$ subset of variables\n* ...We compute the value of the $s$ and $t$ function\n* Then we filter such values using a the reversed (i.e. negated) mask\n* I.e. prepare $s$ and $t$ for their application to the $x_2$ subset\n\n## RNVP Model\n\n**The `call` method handles the transformation, in both directions**\n\n```python\ndef call(self, x, training=True):\n ...\n gate = (direction - 1) / 2\n x = reversed_mask * (x * tf.exp(direction * s) + direction * t * tf.exp(gate * s)) \\\n + x_masked\n ...\n```\n\nHere we compute the main transformation (backwards, as mentioned):\n\n* If `training = True`, we have `direction = -1` and we compute:\n$$\\begin{align}\nx^1 &= y^1 \\\\\nx^2 &= (y^2 - t(y^1)) \\oslash e^{s(y^1)}\n\\end{align}$$\n\n## RNVP Model\n\n**The `call` method handles the transformation, in both directions**\n\n```python\ndef call(self, x, training=True):\n ...\n gate = (direction - 1) / 2\n x = reversed_mask * (x * tf.exp(direction * s) + direction * t * tf.exp(gate * s)) \\\n + x_masked\n ...\n```\n\nHere we compute the main transformation (backwards, as mentioned):\n\n* If `training = False`, we have `direction = 1` and we compute:\n$$\\begin{align}\ny^1 &= x^1 \\\\\ny^2 &= e^{s(x^1)} \\odot x^2 + t(x^1)\n\\end{align}$$\n\n## RNVP Model\n\n**The `call` method handles the transformation, in both directions**\n\n```python\ndef call(self, x, training=True):\n ...\n for i in range(self.num_coupling)[::direction]:\n ...\n log_det_inv += gate * tf.reduce_sum(s, axis=1)\n return x, log_det_inv\n```\n\nAt each layer, we also compute the $\\log \\det$ of the Jacobian\n\n* ...Which is simply the sum of the $s$ function values\n* Determinants of different layers should be multiplied (due to the chain rule)...\n* ...Which means that their $\\log$ is simply summed\n\nAt then end of the process, the determinant has been computed\n\n## RNVP Model\n\n**The `score_samples` method performs _density estimation_**\n\n```python\ndef score_samples(self, x):\n y, logdet = self(x)\n log_probs = self.distribution.log_prob(y) + logdet\n return log_probs\n```\n\nThe process relies on the change of variable formula:\n\n* First, it triggers the `call` method with `training=True`\n - I.e. transforms data points $x$ into their latent representation $z$\n* Then, it computes the (log) density of $z$\n - Using `tensorfllow_probability` comes in handy at this point\n* ...And then sums the log determinant\n\n\n## RNVP Model\n\n**The `log_loss` method computes the _loss function_**\n\n```python\ndef log_loss(self, x):\n log_densities = self.score_samples(x)\n return -tf.reduce_mean(log_densities)\n```\n\nThis is done by:\n\n* Obtaining the estimated densities via `score_samples`\n* ...Summing up (in log scale, i.e. a product in the original scale)\n* ...And finally swapping the sign of the resut\n - ...Since we want to _maximize_ the likelihood\n\n## RNVP Model\n\n**The `train_step` method is called by the keras `fit` method**\n\n```python\ndef train_step(self, data):\n with tf.GradientTape() as tape:\n loss = self.log_loss(data)\n g = tape.gradient(loss, self.trainable_variables)\n self.optimizer.apply_gradients(zip(g, self.trainable_variables))\n self.loss_tracker.update_state(loss)\n return {\"loss\": self.loss_tracker.result()}\n```\n\nThe `GradientTape` is _how tensorflow handles differentiation_\n\n* All tensor operations made in the scope of a `GradientTape` are tracked\n* ...So that a gradient can then be extracted\n* Then we apply the gradient to the model weights (using the optimizer)\n* ...And finally we track the loss\n\n# Using Real NVPs\n\n## Using Real NVP\n\n**We are ready to test our model**\n\nWe will use a classical benchmark for density estimation (shaped like two half moons)\n\n\n```python\nfrom sklearn.datasets import make_moons\ndata = make_moons(3000, noise=0.05)[0].astype(np.float32)\nnn.plot_distribution_2D(samples=data, figsize=figsize)\n```\n\n* We use `float32` numbers for easier interplay with tensorflow\n\n## Training\n\n**Now, we need to train a Real NVP model**\n\n* We will use the whole dataset (this is just a simple test)\n* ...But first, we need to _standardize_ it\n\n\n```python\ndata_s = (data - data.mean(axis=0)) / data.std(axis=0)\n```\n\n**Standardization is very important when using Real NVPs**\n\n* This is true for Neural Networks in general, for the usual reasons\n* But even more in this case, since _the distribution for $z$ is standardized_\n - Standardizing the data makes it easier to learn a mapping\n\n## Training\n\n**Next we can perform training, as usual in keras**\n\n\n```python\nfrom tensorflow.keras.callbacks import EarlyStopping\nmodel = nn.RealNVP(input_shape=2, num_coupling=10, units_coupling=32, depth_coupling=2, reg_coupling=0.01)\nmodel.compile(optimizer='Adam')\ncb = [EarlyStopping(monitor='loss', patience=30, min_delta=0.0001, restore_best_weights=True)]\nhistory = model.fit(data_s, batch_size=256, epochs=200, verbose=1, callbacks=cb)\n```\n\n Epoch 1/200\n 12/12 [==============================] - 5s 5ms/step - loss: 2.5683\n Epoch 2/200\n 12/12 [==============================] - 0s 5ms/step - loss: 2.4311\n Epoch 3/200\n 12/12 [==============================] - 0s 5ms/step - loss: 2.3349\n Epoch 4/200\n 12/12 [==============================] - 0s 5ms/step - loss: 2.2460\n Epoch 5/200\n 12/12 [==============================] - 0s 5ms/step - loss: 2.1774\n Epoch 6/200\n 12/12 [==============================] - 0s 5ms/step - loss: 2.1294\n Epoch 7/200\n 12/12 [==============================] - 0s 4ms/step - loss: 2.0814\n Epoch 8/200\n 12/12 [==============================] - 0s 5ms/step - loss: 2.0123\n Epoch 9/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.9732\n Epoch 10/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.9477\n Epoch 11/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.8992\n Epoch 12/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.8547\n Epoch 13/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.8385\n Epoch 14/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.8099\n Epoch 15/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.7784\n Epoch 16/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.7441\n Epoch 17/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.7364\n Epoch 18/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.6943\n Epoch 19/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.6740\n Epoch 20/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.6681\n Epoch 21/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.6735\n Epoch 22/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.6098\n Epoch 23/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.5741\n Epoch 24/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.5623\n Epoch 25/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.5720\n Epoch 26/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.5187\n Epoch 27/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.4903\n Epoch 28/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.4871\n Epoch 29/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.4780\n Epoch 30/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.4783\n Epoch 31/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.5321\n Epoch 32/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.5125\n Epoch 33/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.4707\n Epoch 34/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.4396\n Epoch 35/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.4181\n Epoch 36/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.3953\n Epoch 37/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3826\n Epoch 38/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.4113\n Epoch 39/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3786\n Epoch 40/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3626\n Epoch 41/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3554\n Epoch 42/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3537\n Epoch 43/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3401\n Epoch 44/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3344\n Epoch 45/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3349\n Epoch 46/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3106\n Epoch 47/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3287\n Epoch 48/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3468\n Epoch 49/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3425\n Epoch 50/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3244\n Epoch 51/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3385\n Epoch 52/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3137\n Epoch 53/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.3189\n Epoch 54/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2986\n Epoch 55/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3001\n Epoch 56/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2959\n Epoch 57/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2837\n Epoch 58/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2792\n Epoch 59/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.3085\n Epoch 60/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2797\n Epoch 61/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2825\n Epoch 62/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2775\n Epoch 63/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2755\n Epoch 64/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2758\n Epoch 65/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2709\n Epoch 66/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3177\n Epoch 67/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3135\n Epoch 68/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2828\n Epoch 69/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2607\n Epoch 70/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2704\n Epoch 71/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2805\n Epoch 72/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2750\n Epoch 73/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.3264\n Epoch 74/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.3688\n Epoch 75/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3544\n Epoch 76/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.3060\n Epoch 77/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2861\n Epoch 78/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2647\n Epoch 79/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2469\n Epoch 80/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2460\n Epoch 81/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2444\n Epoch 82/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2793\n Epoch 83/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2575\n Epoch 84/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2507\n Epoch 85/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2671\n Epoch 86/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2356\n Epoch 87/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2326\n Epoch 88/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2185\n Epoch 89/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2303\n Epoch 90/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2377\n Epoch 91/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2308\n Epoch 92/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2282\n Epoch 93/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2545\n Epoch 94/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2504\n Epoch 95/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2400\n Epoch 96/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2968\n Epoch 97/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.3885\n Epoch 98/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2815\n Epoch 99/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2669\n Epoch 100/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2433\n Epoch 101/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2371\n Epoch 102/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2216\n Epoch 103/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2229\n Epoch 104/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2351\n Epoch 105/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2308\n Epoch 106/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2518\n Epoch 107/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2567\n Epoch 108/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2321\n Epoch 109/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2373\n Epoch 110/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2203\n Epoch 111/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2171\n Epoch 112/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2188\n Epoch 113/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2229\n Epoch 114/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2502\n Epoch 115/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2324\n Epoch 116/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2170\n Epoch 117/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2128\n Epoch 118/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2114\n Epoch 119/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2086\n Epoch 120/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2361\n Epoch 121/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2364\n Epoch 122/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2129\n Epoch 123/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2001\n Epoch 124/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2147\n Epoch 125/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2206\n Epoch 126/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2483\n Epoch 127/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2376\n Epoch 128/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2187\n Epoch 129/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2139\n Epoch 130/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2000\n Epoch 131/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1955\n Epoch 132/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2281\n Epoch 133/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2089\n Epoch 134/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2107\n Epoch 135/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2212\n Epoch 136/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2116\n Epoch 137/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2171\n Epoch 138/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2051\n Epoch 139/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2071\n Epoch 140/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2141\n Epoch 141/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2082\n Epoch 142/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2031\n Epoch 143/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2131\n Epoch 144/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2029\n Epoch 145/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2037\n Epoch 146/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2129\n Epoch 147/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1976\n Epoch 148/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2052\n Epoch 149/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.1938\n Epoch 150/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2008\n Epoch 151/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1948\n Epoch 152/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1903\n Epoch 153/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2109\n Epoch 154/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2129\n Epoch 155/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1858\n Epoch 156/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1979\n Epoch 157/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.1883\n Epoch 158/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1925\n Epoch 159/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2005\n Epoch 160/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2100\n Epoch 161/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1918\n Epoch 162/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2088\n Epoch 163/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2089\n Epoch 164/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1912\n Epoch 165/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2066\n Epoch 166/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2040\n Epoch 167/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1971\n Epoch 168/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1926\n Epoch 169/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1861\n Epoch 170/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1808\n Epoch 171/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2290\n Epoch 172/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2334\n Epoch 173/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2036\n Epoch 174/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1917\n Epoch 175/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2139\n Epoch 176/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2023\n Epoch 177/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2008\n Epoch 178/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.2113\n Epoch 179/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2097\n Epoch 180/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1975\n Epoch 181/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2186\n Epoch 182/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.1971\n Epoch 183/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2026\n Epoch 184/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2061\n Epoch 185/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1898\n Epoch 186/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1838\n Epoch 187/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1760\n Epoch 188/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.1829\n Epoch 189/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.1771\n Epoch 190/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1845\n Epoch 191/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.1964\n Epoch 192/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.2222\n Epoch 193/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1859\n Epoch 194/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1851\n Epoch 195/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1781\n Epoch 196/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1788\n Epoch 197/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.1838\n Epoch 198/200\n 12/12 [==============================] - 0s 5ms/step - loss: 1.1834\n Epoch 199/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1811\n Epoch 200/200\n 12/12 [==============================] - 0s 4ms/step - loss: 1.1890\n\n\n## Training\n\n**As usual with NNs, choosing the right architecture can be complicated**\n\n```python\nmodel = RealNVP(input_shape=2, num_coupling=16, units_coupling=32, depth_coupling=2, reg_coupling=0.01)\n```\n\n* We went for a relatively deep model (10 affine coupling)\n* Each coupling has also a good degree of non-linearity (2 hidden layers)\n* We used a small degree of L2 regularization to stabilize the training process\n\n**We also use relatively _large batch size_**\n\n```python\nhistory = model.fit(data_s, batch_size=256, epochs=200, verbose=2, callbacks=cb)\n```\n\n* Large batch sizes are usually a good choice with density estimation approaches\n* Batches should be ideally be representative of the distribution\n\n## Training\n\n**Let's see the evolution of the training loss over time**\n\n\n```python\nnn.plot_training_history(history, figsize=figsize)\n```\n\n## Latent Space Representation\n\n**We can obtain the latent space representation by calling the trained model**\n\nThis will trigger the `call` method with default parameters (i.e. `training=True`)\n\n\n```python\nz, _ = model(data_s)\nnn.plot_distribution_2D(samples=z, figsize=figsize)\n```\n\n## Density Estimation\n\n**We can estimate the density of any data point**\n\n\n```python\nnn.plot_distribution_2D(estimator=model, xr=np.linspace(-2, 2, 100, dtype=np.float32),\n yr=np.linspace(-2, 2, 100, dtype=np.float32), figsize=figsize)\n```\n\n* A good approximation! With a strange low-density connection between the moons\n\n## Data Generation\n\n**We can also generate data, by sampling from $p_z$ and then calling `predict`**\n\nThis will trigger the `call` method with `training=False`\n\n\n```python\nsamples = model.distribution.sample(3000)\nx, _ = model.predict(samples)\nnn.plot_distribution_2D(samples=x, figsize=figsize)\n```\n\n## Data Generation\n\n**We can also plot the mapping for selected data points...**\n\n...Which gives and intuition of how the transformation works\n\n\n```python\nnn.plot_rnvp_transformation(model, figsize=figsize)\n```\n\n# RNVP for Anomaly Detection\n\n## RNVP for Anomaly Detection\n\n**RNVPs can be used for anomaly detection like any other density estimator**\n\nFirst, we build and compile the model (for the HPC data)\n\n\n```python\ninput_shape = len(hpc_in)\nhpc_rnvp = nn.RealNVP(input_shape=input_shape,\n num_coupling=6, units_coupling=32, depth_coupling=1, reg_coupling=0.01)\nhpc_rnvp.compile(optimizer='Adam')\n```\n\nWe chose a _simpler_ architecture this time\n\n* With RNVP, dealing with higher dimensional data has actually some advantage\n* In particular, we have richer input for the $s$ and $t$ functions\n - In the \"moons\" dataset, $s$ and $t$ had 2/2 = 1 input feature\n - Now we have 159/2 = 79--80 features\n\n## RNVP for Anomaly Detection\n\n**Then we perform training as usual**\n\n\n```python\nX = trdata[hpc_in].astype(np.float32).values\ncb = [EarlyStopping(monitor='loss', patience=10, min_delta=0.001, restore_best_weights=True)]\nhistory = hpc_rnvp.fit(X, batch_size=256, epochs=100, verbose=1, callbacks=cb)\n```\n\n Epoch 1/100\n 12/12 [==============================] - 3s 8ms/step - loss: 1410.5375\n Epoch 2/100\n 12/12 [==============================] - 0s 7ms/step - loss: 870.6241\n Epoch 3/100\n 12/12 [==============================] - 0s 7ms/step - loss: 403.5361\n Epoch 4/100\n 12/12 [==============================] - 0s 7ms/step - loss: 492.0046\n Epoch 5/100\n 12/12 [==============================] - 0s 7ms/step - loss: 276.7331\n Epoch 6/100\n 12/12 [==============================] - 0s 7ms/step - loss: 244.7540\n Epoch 7/100\n 12/12 [==============================] - 0s 7ms/step - loss: 255.3067\n Epoch 8/100\n 12/12 [==============================] - 0s 8ms/step - loss: 282.6917\n Epoch 9/100\n 12/12 [==============================] - 0s 7ms/step - loss: 279.7164\n Epoch 10/100\n 12/12 [==============================] - 0s 7ms/step - loss: 185.7794\n Epoch 11/100\n 12/12 [==============================] - 0s 7ms/step - loss: 111.3171\n Epoch 12/100\n 12/12 [==============================] - 0s 7ms/step - loss: 87.8113\n Epoch 13/100\n 12/12 [==============================] - 0s 7ms/step - loss: 85.8522\n Epoch 14/100\n 12/12 [==============================] - 0s 7ms/step - loss: 52.4090\n Epoch 15/100\n 12/12 [==============================] - 0s 7ms/step - loss: 41.4958\n Epoch 16/100\n 12/12 [==============================] - 0s 7ms/step - loss: 30.8174\n Epoch 17/100\n 12/12 [==============================] - 0s 7ms/step - loss: 16.4675\n Epoch 18/100\n 12/12 [==============================] - 0s 7ms/step - loss: 1.6464\n Epoch 19/100\n 12/12 [==============================] - 0s 7ms/step - loss: -4.8433\n Epoch 20/100\n 12/12 [==============================] - 0s 7ms/step - loss: -19.9105\n Epoch 21/100\n 12/12 [==============================] - 0s 7ms/step - loss: -28.3964\n Epoch 22/100\n 12/12 [==============================] - 0s 7ms/step - loss: -36.6674\n Epoch 23/100\n 12/12 [==============================] - 0s 7ms/step - loss: -38.1202\n Epoch 24/100\n 12/12 [==============================] - 0s 7ms/step - loss: -47.8137\n Epoch 25/100\n 12/12 [==============================] - 0s 7ms/step - loss: -52.0854\n Epoch 26/100\n 12/12 [==============================] - 0s 7ms/step - loss: -52.9910\n Epoch 27/100\n 12/12 [==============================] - 0s 7ms/step - loss: -60.7810\n Epoch 28/100\n 12/12 [==============================] - 0s 7ms/step - loss: -67.6051\n Epoch 29/100\n 12/12 [==============================] - 0s 8ms/step - loss: -73.7005\n Epoch 30/100\n 12/12 [==============================] - 0s 8ms/step - loss: -78.4770\n Epoch 31/100\n 12/12 [==============================] - 0s 7ms/step - loss: -79.9680\n Epoch 32/100\n 12/12 [==============================] - 0s 7ms/step - loss: -85.3557\n Epoch 33/100\n 12/12 [==============================] - 0s 7ms/step - loss: -60.8921\n Epoch 34/100\n 12/12 [==============================] - 0s 7ms/step - loss: 4.5386\n Epoch 35/100\n 12/12 [==============================] - 0s 7ms/step - loss: 265.7068\n Epoch 36/100\n 12/12 [==============================] - 0s 7ms/step - loss: 236.9165\n Epoch 37/100\n 12/12 [==============================] - 0s 7ms/step - loss: 24.5191\n Epoch 38/100\n 12/12 [==============================] - 0s 7ms/step - loss: -6.8422\n Epoch 39/100\n 12/12 [==============================] - 0s 7ms/step - loss: -31.6131\n Epoch 40/100\n 12/12 [==============================] - 0s 7ms/step - loss: -45.0576\n Epoch 41/100\n 12/12 [==============================] - 0s 7ms/step - loss: -48.1148\n Epoch 42/100\n 12/12 [==============================] - 0s 7ms/step - loss: -63.8708\n\n\n## RNVP for Anomaly Detection\n\n**Here is the loss evolution over time**\n\n\n```python\nnn.plot_training_history(history, figsize=figsize)\n```\n\n## RNVP for Anomaly Detection\n\n**Then we can generate a signal as usual**\n\n\n```python\nX = hpcs[hpc_in].astype(np.float32).values\nsignal_hpc = pd.Series(index=hpcs.index, data=-hpc_rnvp.score_samples(X))\nnn.plot_signal(signal_hpc, hpc_labels, figsize=figsize)\n```\n\n* The signal is very similar to that of KDE\n\n## RNVP for Anomaly Detection\n\n**Finally, we can tune the threshold**\n\n\n```python\nth_range = np.linspace(1e4, 1.5e5, 100)\nthr, val_cost = nn.opt_threshold(signal_hpc[tr_end:val_end],\n valdata['anomaly'],\n th_range, cmodel)\nprint(f'Best threshold: {thr}')\ntr_cost = cmodel.cost(signal_hpc[:tr_end], hpcs['anomaly'][:tr_end], thr)\nprint(f'Cost on the training set: {tr_cost}')\nprint(f'Cost on the validation set: {val_cost}')\nts_cost = cmodel.cost(signal_hpc[val_end:], hpcs['anomaly'][val_end:], thr)\nprint(f'Cost on the test set: {ts_cost}')\n```\n\n Best threshold: 150000.0\n Cost on the training set: 0\n Cost on the validation set: 327\n Cost on the test set: 265\n\n\n* Once again, the performance is on par with KDE\n* ...But we have better support for high-dimensional data!\n", "meta": {"hexsha": "6dccbf0040dccc837604e037acea1b158eb5b458", "size": 740819, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "2. Density Estimation with Neural Networks.ipynb", "max_stars_repo_name": "lompabo/aiiti-04-2021", "max_stars_repo_head_hexsha": "6731cee60438d3f505e19dc2073eb286f6792d92", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-09-30T06:14:10.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-30T06:14:10.000Z", "max_issues_repo_path": "2. Density Estimation with Neural Networks.ipynb", "max_issues_repo_name": "lompabo/aiiti-04-2021", "max_issues_repo_head_hexsha": "6731cee60438d3f505e19dc2073eb286f6792d92", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "2. 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YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.8031737940012418, "lm_q1q2_score": 0.4203975075126412}} {"text": "```python\nimport seaborn as sns\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nimport pickle\nimport os\nimport warnings\nwarnings.filterwarnings('ignore')\n\nfrom auxiliary.auxiliary import (simulate_data, get_results_regression, process_results, get_results_local_regression, \n process_results_local)\n```\n\n# Stacking/ Jackknife Model Averaging and its Use in Regression Discontinuity Design\n---\n\n## Introduction\n---\n\nA common occurrence in data-driven Economics is that the researcher faces uncertainty about several facets of a statistical model as well as the results from its calibration. While the uncertainty in the parameter estimates of a model is typically reported by showing at least the standard deviation of the parameter estimates, it is not common practice in Economics to discuss the implications of uncertainty about the specific form of the statistical model. It is rather common to select one model among several competing models (maybe using some model selection techniques) from that on pretending that it is the true one while performing some robustness checks using other model specifications (Moral-Benito (2015)). \n\nFletcher (2019) argues that the typical problem with this negligence of remaining model uncertainty is that parameter estimates might be biased and uncertainty is underestimated. This approach yields confidence intervals of the estimated parameters that are too optimistic in the sense that the true coverage probability is below its nominal level when not accounting for this uncertainty. This might cause the resulting inference to be seriously flawed. This aspect is a major reason why model averaging might be useful in Economics. Model averaging takes information from all competing models into account and therefore explicitly includes model uncertainty into its parameter estimates as well as its confidence intervals. Steel (2019) summarizes the tradeoff between model selection and averaging by the question what the researcher is interested in and what the previous certainty of knowledge is. If one is interested in the true model and is certain that one of the competing models must be the true one, then model selection yields the best results as it can select the true one without confounding it with the others. Although the extent of this advantage is questionable as Zhang (2015) proves that several model avergaing estimators (among them Jackknife Model Averaging) are actually root-n consistent if the true model is among the ones that is averaged over. Further when there is considerable uncertainty about the model form, then an ensemble of different forms constructed by averaging yields more reliable results as it takes information from all of them into account.\n\nIn Economics, the application of model averaging is imaginable in many situations. For example, in a more structural framework, often there are competing theories that inform the statistical model such as human capital as opposed to signaling theory. Another situation commonly encountered is the question which covariates should be included in a linear regression model (compare Steel (2019)). The last situation is often faced in Regression Discontinuity Design (RDD). There is usually uncertainty about the polynomial with which the assignment variable affects the outcome variable. In the highly cited prationers' guide to RDD by Lee and Lemieux (2010) they argue that typically there is no theory available on which to base the decision for the choice of polynomial. This leads them to the suggestion to use model selection based on the Akaike information criterion (AIC) as a standard way to choose the polynomial. As there is no strong prior which polynomial to add and whether any of them is actually a good representation of the true model, for the above reasons, model averaging should be a suitable alternative to the suggested model selection procedure. It might be even superior in many situations as the actual coverage probability might be closer to its nominal one. In my first simulation, I therefore take a typical applied RDD paper (Bronzini and Iachini (2014)) that follows quite closely the suggestions in Lee and Lemieux (2010) to benchmark whether model averaging might actually perform better than selection based on the AIC (in the sense of bias, and root mean squared error in the treatment effect as well as the coverage probability of the 95% confidence intervals). Unfortunately, I noticed only after already having set up my first simulation that there is already a published paper that has a similar idea in mind (Button (2016)). In the light of that, I adapted my first simulation study to incorporate some of the author's good ideas. For the model averaging I focus on the approach of stacking (Wolpert (1992)) which has been rediscovered in the Econometrics literature as Jackknife Model Averaging (JMA) based on Hansen and Racine (2012). My setup is also an improvement over Button's in the sense that I allow for heteroskedasticity which generally works in favor of model averaging using JMA as opposed to other model averaging strategies (I will discuss this further later). In general, my simulation differs in a few dimensions which I will elaborate on later. \n\nIn a second simulation study, I go beyond the general ideas that are partly covered in Button (2016). As JMA is based on a leave one out cross validation procedure that aims at minimizing a cross validation criterion approximating the expected test mean squared error depending on the weights given to the competing models for averaging, I explore whether JMA can also be used for bandwidth selection in RDD in a self-sufficient way. As in RDD one is mainly interested in the comparison of individuals around a certain cutoff, it remains to determine which individuals to include around the cutoff to estimate the treatment effect. Lee and Lemieux (2010) suggest the procedure developed by Imbens and Lemieux (2008) involving the use of local linear regressions around the cutoff for several bandwidths, calculating a cross validation criterion for each and then choosing the bandwidth that minimizes it. The approach in JMA is similar while accounting for different possible model forms (such as the different orders of polynomials). Hence, JMA might be an alternative for bandwidth selection (especially when local linearity is a poor approximation) which I explore in the second simulation of my project. \n\nThe general structure of my project paper is the following. In the next section I introduce stacking in general and the specific implementation of it called Jackknife Model Averaging based on Hansen and Racine (2012) which involves a convenient calculation for linear regression models as needed in RDD. In section three, I introduce the general idea of Regression Discontinuity Design and explain the analysis of Bronzini and Iachini (2014) on which my different data generating processes are based. In section four I run my first simulation study in which I check the usefulness of JMA for determination of the treatment in the verge of uncertainty about the polynomial order of the assignment variable. In section five, I pick up the first simulation to see how JMA behaves as a tool for bandwidth selection. In the last section I shortly conclude with a summary of my findings.\n\n## Stacking/ Jackknife Model Averaging\n---\n\nAs already emphasized in the introduction, model averaging has its appeal in situations where the exact form of the model is unknown. It has an advantage over model selection in the sense that it incorporates the resulting uncertainty explicitly in the modeling by combining competing models and extracting information of each of them. This makes it possible to account for the model uncertainty in the uncertainty around the estimated model parameters making inference more stable in comparison to model selection (Fletcher (2019)). Generally, there are two main strands of model averaging present. The first follows the Bayesian paradigm and is consequently called Bayesian Model Averaging. It is characterized by the fact that priors on the true parameters and the models must be formed. The outcome is a weighted average of the posteriors from different models with the weight being based on the posterior probability that a model is true. The main objective of it is to rather identify the true model as opposed to improving prediction quality. While this is generally a desirable feature for Economists, the complexity to determine the priors and the dependence of the posterior model probabilities on it makes its use difficult in practice, though (Fletcher (2019)). \n\nThe second approach is Frequentist Model Averaging which is mainly concerned with improving model predictions and to obtain confidence intervals with good coverage. In this category falls the approach chosen in my project, Jackknife Model Averaging. It uses the idea of leave one out cross validation in order to obtain the weights that are used to build an optimal weighted average across different models. This idea was first introduced in Statistics as model-mixing formulated in Stone (1974). It was later picked up again by Wolpert (1992) in the realm of Machine Learning and he coined the term stacking for it. In the Econometrics literature it was Hansen and Racine (2012) who reintroduced it giving it a formal underpinning in the sense that they proved its capacity to improve model prediction. Wolpert (1992) only provides simulation evidence. In the following I will introduce the idea of stacking and especially Jackknife Model Averaging following Hansen and Racine (2012). \n\nAssume that we are interested in averaging across different generalized linear models, i.e. linear models that do not necessarily have errors that are normally distributed. These competing models are denoted by $m = 1, ..., M$. The true data generating process takes the general linear form as following:\n\n\\begin{equation}\n \\begin{split}\n y_i = \\mu_i + e_i \\\\\n E(e_i|x_i) = 0.\n \\end{split}\n\\end{equation}\n\nThe data is independently distributed $(x_i, y_i)$ for $n = 1, ..., N$. $\\mu_i$ corresponds to $E(y_i|x_i)$. We determine $y = (y_1, ..., y_N)'$, $\\mu = (\\mu_1, ..., \\mu_N)'$ and $e = (e_1, ..., e_N)'$. Furthermore there is, potentially, heteroskedasticity defined as $E(e_i^2|x_i)=\\sigma_i^2$. Each of the competing models now differ in the sense that they yield possibly different linear estimators ${\\hat\\mu^1, ..., \\hat\\mu^M}$ for $\\mu$.\nMoral-Benito (2015) note that the general approach of Frequentist Model Averaging is to now find a weighted average among those estimators $\\hat\\mu^m$ sucht that the prediction of the ensemble is optimized. This means the general idea is to find weights $w^m$ per model that yield an ensemble estimator like the following:\n\n\\begin{equation}\n \\hat\\mu(w) = \\sum_{m=1}^{M} w^m \\hat\\mu^m = \\hat\\mu w.\n\\end{equation}\n\nThe idea of Frequentist Model Averaging, hence, boils down to finding a weigthing vector $w=(w^1, ..., w^M)'$ based on optimizing some criterion that depends on the weights. In the case of stacking/ JMA this criterion is based on leave one out cross validation (as Fletcher (2019) points out that Hansen and Racine (2012) wrongly use the term jackknife for it, I refrain from applying their terminology in the following). Before I derive the exact cross validation procedure as well as the resulting cross validated criterion, let us introduce some further notation. Hansen and Racine focus their attention on the linear regression model for the application of the JMA and introduce an efficient calculation of their criterion for this special class of models. In general, in the class of linear models, the estimator $\\hat\\mu^m$ can also be written as a transformation of the vector $y$ using a projection matrix $P_m$ , i.e. $\\hat\\mu^m = P_m y$. It is well-known that for linear regression the matrix $P_m$ has the form $P_m = X^m(X^{m'} X^m)^{-1} X^{m'} $. The ensemble estimator can in general be written as:\n\n\\begin{equation}\n \\hat\\mu(w) = P(w) y \n\\end{equation}\n\nwhere \n\n\\begin{equation}\n P(w) = \\sum_{m=1}^{M} w^m P_m.\n\\end{equation}\n\nThe first observation here is that $P(w)$ is linear in $w$ and $y$ is linear in $P(w)$ which means that it is also linear in $w$. Hansen and Racine (2012) restrict the weights to lie in a unit simplex, i.e. to lie in:\n\n\\begin{equation}\n H_n = \\{w \\in R^M: w^m \\geq 0, \\sum_{m=1}^{M} w^m = 1\\}.\n\\end{equation}\n\nThe leave one out cross validation now comes into play in how the estimator for $\\mu$ is calculated for each model. The linear estimator of the m*th* model is $\\tilde \\mu^m = (\\tilde \\mu_1^m, ..., \\tilde \\mu_N^m)'$. $\\tilde \\mu_i^m$ is the fitted value of model $m$ for observation $i$ when estimating the model on the data set without observation $i$ and then applying the estimated model on the i*th* observation. From this vector of linear estimates one can derive the cross validated residual vector $\\tilde e^m = y - \\tilde \\mu^m$. As we have $M$ models in total, mixing them with some weight vector $w$ results in different cross validated linear estimators $\\tilde \\mu(w)$ and consequently different cross validated residuals $\\tilde e(w)$ depending on the weight. This is depicted in the following two equations:\n\n\\begin{equation}\n \\begin{split}\n \\tilde \\mu(w) = \\sum_{m=1}^M w^m \\tilde \\mu^m = \\tilde\\mu w = \\tilde P(w) y \\\\\n \\tilde e(w) = y - \\tilde \\mu(w) = \\sum_{m=1}^M w^m\\tilde e^m = \\tilde e w\n \\end{split}\n\\end{equation}\n\nwith $\\tilde \\mu = (\\tilde \\mu^1, ..., \\tilde \\mu^M)'$, $\\tilde P(w) = \\sum_{m=1}^M w^m \\tilde P_m$ and $\\tilde e = (\\tilde e^1, ..., \\tilde e^M)'$. \nTaking a step back to general stacking and how JMA relates to it, let us look at how stacking is typically built up. Stacking seeks to maximize the following expression by setting the weights $w^m$: \n\n\\begin{equation}\n \\sum_{n=1}^N log L(\\tilde \\mu_i(w)|y_i)\n\\end{equation}\n\nwhere $log L(.)$ describes the log likelihood function and $\\tilde \\mu_i(w) = \\sum_{m=1}^M w^m \\tilde \\mu_i^m$ which is the averaged linear estimator for observation $i$ calculated using the data set without the i*th* observation. The approach taken by JMA is to now rely on another criterion than the log likelihood $log L(.)$ above but rather the least squares cross validation criterion as an estimate for the expected true error:\n\n\\begin{equation}\n CV(w) = \\frac{1}{n} \\tilde e(w)' \\tilde e(w) = w' S w\n\\end{equation}\n\nwhere $S = \\frac{1}{n} \\tilde e' \\tilde e$. This means that Hansen and Racine (2012) construct a measure for the expected test mean squared error depending on the extent to which the individual models are mixed. $CV(w)$ is an $M \\times M$ matrix covering the extreme cases that might occur on the diagonal which capture choosing weight of one for either of the models. As the choice of $w$ is restricted to lie in $H_n$ and the expected test error is supposed to be minimized this comes down to a constraint optimization problem like the following:\n\n\\begin{equation}\n \\begin{split}\n \\textrm{min}_w CV(w) \\\\\n \\textrm{subject to } w \\in H_n.\n \\end{split}\n\\end{equation}\n\nThis is a quadratic programming problem yielding an optimal weight $\\hat w$ which can then be used to derive the JMA estimator for $\\mu$ which is $\\hat \\mu(\\hat w) = \\hat \\mu \\hat w$.\n\nIn order to summarize, the approach involves setting up each potential model indivdually. For each model based on leave one out cross validation the fitted value of each observation $i$ has to be obtained by estimating the model without that very observation. This means that per model $N$ regressions have to be run. Based on these fitted values, the cross validated residuals are derived which are then together with the potential weights $w$ used to build the least squares cross validation criterion. This serves an approximation of the expected test mean squared error of the resulting ensemble depending on the choice of $w$. Consequently, it is minimized by choosing optimal wights $\\hat w$ under the constraint that each of them is nonnegative and sum up to one together.\n\nFor the case of linear regression, Hansen and Racine (2012) come up with a computationally light way of performing leave one out cross validation that reduces it to one operation per model as opposed to $N$ regressions. We already observed that in linear regressions the projection matrix has the form $P_m = X^m(X^{m'} X^m)^{-1} X^{m'}$. This means consequently that the cross validated estimator for observation $i$ of a model $m$ is the following: $\\tilde \\mu_i^m = x_i^{m'} (X_{(-i)}^{m'} X_{(-i)}^{m})^{-1} X_{(-i)}^{m'} y_{(-i)}$ where $(-i)$ means that it is the data set without observation $i$ and $x_i^m$ is the i*th* row of the regressor matrix. The authors now make use of a derivation in Li (1987). Li derives that the cross validated projection matrix $\\tilde P_m$ has the following form in linear regressions $\\tilde P_m = D_m (P_m - I) + I$ with \n\n\\begin{equation}\n D_m = \\begin{pmatrix} (1-h^m_{11})^{-1} & 0 & \\cdots & 0 \\\\ 0 & \\ddots & & \\vdots \\\\ \\vdots & & \\ddots & 0 \\\\ 0 & \\cdots & 0 & (1-h^m_{NN})^{-1} \\end{pmatrix} \n\\end{equation}\n\nand $h^m_{ii} = x_i^{m'} (X^{m'} X^{m})^{-1} x_i^{m}$ which is also the i*th* diagonal element of $P_m$. The cross validated residual vector of a single model $m$ can hence be written as $\\tilde e^m = D_m(y - P_m y)$ which can be obtained in a single operation per model. This is exploited in the code I have written and exhibits a large speed advantage over the conventional way of performing leave one out cross validation which I will also explore in my last simulation. \n\nWhile the Jackknife Model Averaging is a form of stacking, Hansen and Racine (2012) now add an entirely new discussion to the topic which is that the JMA weights $\\hat w$ are asymptotically optimal drawing on two criteria. They define the training mean squared prediction error as $L_n(w) = \\frac{1}{n} (\\mu - \\hat \\mu(w))'(\\mu - \\hat \\mu(w))$ and the expected test mean squared prediction error as $R_n(w) = E(L_n(w)|X)$. Under some regularity conditions, they prove that these criteria converge in probability to their minimal possible value across all feasible $w \\in H_n$ when employing the JMA weights $\\hat w$. Formally, this looks like the following: \n\n\\begin{equation}\n \\frac{L_n(\\hat w)}{\\text{inf}_{w \\in H_n} L_n(w)} \\rightarrow_p 1\n\\end{equation}\n\n\\begin{equation}\n \\frac{R_n(\\hat w)}{\\text{inf}_{w \\in H_n} R_n(w)} \\rightarrow_p 1.\n\\end{equation}\n\nThis shows that asymptotically the JMA weights yield the best in-sample and out-of-sample predictions across all feasible weights which also includes estimating just a single one of each model. This shows clearly that the focus of the approach is on improving predictive performance. This is achieved by a decrease in variance at the cost of increasing bias. \nWhile we are generally more interested in measuring an accurate treatment effect in RDD, I will argue in the next section why JMA might still be beneficial for RDD although it might induce some bias in the treatment effect.\n\n## Regression Discontinuity Design\n---\n\nMy data generating process is based on a quite typical setup in RDD which is inspired by Bronzini and Iachini (2014). In their paper they pursue the question whether subsidy programs for Research & Development (R&D) for companies are effective in the sense that they actually induce firms to subsequently raise their investment in R&D. Essentially, they are interested in measuring the treatment effect of subsidies for R&D on the actual investment in it. For that they exploit a specific subsidy program that was launched in Northern Italy. The program began in 2003 and asked industrial companies to come up with project ideas that were funded by the program in case the idea gained a certain score determined by an independent commission. Every firm that received 75 points or more (on a scale from 0 to 100) for their idea was subsidized by a percentage of the total amount of the project. \n\nThe authors' identification strategy now involves a sharp RDD which postulates that those firms around the threshold of 75 points of score are comparable. It is argued that this has quasi-experimental character allowing the authors to identify the treatment effect by comparing some measure of R&D investment across firms around the threshold. \nThe authors draw from several measures while focussing on total investment into R&D relative to the sales before the subsidy program was launched in order to make it comparable. This is also the dependent variable I will focus on. In general in RDD, this dependent variable is a function of the assignment variable (that determines whether the firm is assigned to treatment) which is the score here. If the firms cannot control the assignment process, the RDD approach is valid. By assuming that the assignment around the threshold is somewhat random rendering the firms around it very similar apart form differeng in treatment and control, one can compare those at the threshold by measuring the discontinuity in the function of the dependent variable on the assignment variable. In this specific application, the authors allow for treatment effect heterogeneity. They allow the treatment effect to vary depending on whether the firm is classified as small or large. It is also quite common to further control for some variables that might affect both the dependent and the assignment variable which is not done in this paper. As previously noted, it is not ad-hoc clear whether the dependent variable is linear in the assignment variable or not. For this reason, it is common practice (promoted by Lee and Lemieux (2010)) to add several polynomials of the assignment variable to the function and deciding on the \"true\" model by choosing the one that has the lowest AIC. Formally, the described regression to obtain the treatment effect in the paper at hand looks like the following:\n\n\\begin{equation}\n Y_i = \\sum_{k=1}^2 \\sum_{p=0}^P \\alpha_{kp} Size_i^k (X_i-c)^p + T_i \\sum_{k=1}^2 \\sum_{p=0}^P \\beta_{kp} Size_i^k (X_i-c)^p + \\epsilon\n\\end{equation}\n\nwhere $Size_i^1 = 1$ if the firm $i$ is small and zero otherwise and $Size_i^2 = 1$ if firm $i$ is large and zero otherwise. $Y_i$ is the investment in R&D divided by pre-program sales and $X_i$ is the assignment variable which is in our case the score a firm received for its project idea. $c$ captures the threshold for the assignment of the treatment which is equal to 75 as every firm having at least this score obtained the subsidy. $T_i$ is the treatment indicator which is one if $X_i \\geq 75$ and zero otherwise. Withdrawing the threshold $c$ from the assignment variable $X_i$ is common practice as like this the coefficients $\\beta_{10}$ and $\\beta_{20}$ capture the discontinuity (the difference in intercept) at the threshold between treatment and non treatment for small and large firms, respectively. Consequently, they capture the treatment effect. The $P$ displays that the order of polynomial included in the regression function is unknown and potentially different ones might be included.\n\nThe authors now find themselves in a common situation when employing RDD. They cannot rely on any theory of how the assignment variable affects the dependent variable. Capturing the functional form as precisely as possible is of large interest, though, as it affects how accurately the difference in intercept and hence in treatment effect can be measured (compare Lee and Lemieux (2010)). For this reason, usually, it is allowed to have potentially different functional forms on each side of the cutoff. The flexibility in functional form is typically tested by including more and more polynomials of the assignment variable one at a time. The argument for this approach is that, when running only a polynomial of order one while the true relationship actually is more nonlinear might result in ommited variable bias. This is where model averaging and specifically JMA can come into play. Although model averaging increases bias while reducing variance in the estimates, by including information from several models (several polynomials) can decrease the danger of omitted variable bias. Further if the true model is among the candidates, the JMA is actually root-n consistent which means that with large n it does not do any harm to use JMA as opposed to luckily only running the true model. In comparison to model selection based on AIC as suggested in Lee and Lemieux (2010), it has the advantage that information from all models enter the parameter estimates and its uncertatinty. In the paper at hand, running the regression above with up to three polynomials and select the correct model via AIC suggests that the true model has polynomial of zero for small firms and the treatment effect is measured at 0.045 with a standard error of 0.018. This is just significant at the five percent level and rests heavily on the accurracy of the confidence intervals. As already established before, model selection tends to display too much certainty, understating confidence intervals. In the example above it might easily be the case that statistical significance cannot be maintained at the same level when incorporating model uncertainty. This is where JMA with its more accurate confidence intervals could be beneficial, although the bias might be increased but still it could give more accurate inference than AIC model selection and display less risk of ommited variable bias than single estimation of one model. In my first simulation study I will investigate the difference in estimated treatment effect between AIC model selection and JMA based on the setup of Bronzini and Iachini (2014). \n\nLee and Lemieux (2010) discuss a second major method of how to estimate the treatment effect in RDD which gives rise to my second simulation study. They argue that the above regression is not too appealing in the sense that it uses data far away from the cutoff point to predict the dependent variable at the cutoff (which one is solely interested in). It is therefore argued that an approach which reduces the data to closer to the cutoff might be prefered. Usually relied on when pursuing this is the nonparametric method suggested by Imbens and Lemieux (2008). The crucial part of this approach is the choice of the bandwidth $h$ around the cutoff $c$ for which the data is included for the estimation of the treatment effect. The idea is to restrict the data to close around the cutoff, i.e. including all $i$ for which $c-h \\leq X_i \\leq c+h$. Within this window, the data is used to run a kernel regression with a rectangular kernel which essentially boils down to running a regular linear regression as above with $P=1$ with the only difference that they run seperate regressions for each side of the cutoff. These regressions are run across a range of different bandwidths and for each of them the test mean squared error is calculated via leave one out cross validation. Averaging those across both sides of the cutoff results in a single criterion per bandwidth which looks like the following: \n\n\\begin{equation}\n CV_Y(h) = \\frac{1}{N} \\sum_{i=1}^N (Y_i - \\hat Y(X_i))^2\n\\end{equation}\n\nwhere $\\hat Y$ is calculated as in JMA with leave one out cross validation based on a local linear regression on each side of the cutoff $c$ with a certain bandwidth $h$. The optimal bandwidth is then the $h$ that minimizes the above criterion, i.e. $h^* = \\text{argmin}_h CV_Y(h)$. Clearly, there is a resemblence between this criterion and the one used in JMA. For this reason, I try whether JMA might be used to run local polynomial regressions in order to select the optimal bandwidth. This might be advantegeous in the case where the linear approximation of the local linear regression relied on is actually in parts not very accurate. I would generally expect that JMA might actually perform better than the above approach when there is polynomial nonlinearity to varying degree across the range of different bandwidths tested. This idea is investigated in my second simulation. \n\n## Simulation Study One\n\nIn the first simulation study I mimic the true data of Bronzini and Iachini (2014) for the independent variables as closely as possible. The regressors in the paper are only the size of the firm and the score they obtained. As there are roughly 50 percent of small firms I draw the firm size from a binomial distribution with probability of 0.5. For $N$ observations this corresponds to:\n\n\\begin{equation}\n Size^1 \\text{~} B(N, 0.5).\n\\end{equation}\n\nThere are some differences regarding the average distribution of the score depending on the size in the original data. So I simulate the score depending on the size of the firm. I use a right skewed normal distribution with $N(88, 12)$ for 80 percent of the small firms in combination with a uniform distribution $U(20, 55)$ for the rest. For large firms I only take a right skewed normal distribution of $N(92, 18)$ and flatten the peak by taking all firms with score between 80 and 90 and draw them again from $U(78, 92)$. As the scores can only take discrete values between 0 and 100, I round every number to the nearest integer and make sure that all observations lie in the interval by replacing values outside by a random choice of those values already drawn inside the interval. The comparison of the original data with my simulated data is shown in the graphs below (after having adjusted each score already by subtracting the cutoff $c=75$. \n\n\n```python\n# load original data\noriginal_data = pd.read_stata(\"data/Bronzini-Iachini_dataset.dta\")\n# simulate data\nnp.random.seed(123)\nnum_obs = 360\ncoefficients = {\"untreated\": np.array([-0.05, -0.0016]),\n \"treated\": np.array([0.08, 0.0003])}\nsimulated_data = simulate_data(num_obs, coefficients, error_dist=\"random_cluster\")[0]\n```\n\n\n```python\n# plot scores of original and simulated data\nfig, ax = plt.subplots(2, 1, sharex=True)\nfor size, number in [(\"Small Firms\", 0), (\"Large Firms\", 1)]:\n ax[number].set_title(size)\n ax[1].set_xlabel(\"Score\")\n sns.kdeplot(original_data.loc[original_data[\"largem\"]==number, \"s\"], ax=ax[number], label=\"original\")\n sns.kdeplot(simulated_data.loc[simulated_data[\"large\"]==number, \"score\"], ax=ax[number], label=\"simulated\")\n```\n\nIn the original paper, the authors run polynomial regressions with $P \\in \\{0, 1, 2, 3\\}$ and choose the optimal model according to the AIC which is readily available when running a linear regression. They find that there is no treatment effect for the large firms (see Table 5 Panel A left hand side in their paper). The dependent variable $Y$ (total investment in R&D divided by pre-program sales) I now create according to the following true data generating process:\n\n\\begin{equation}\n \\begin{split}\n P &= 1 \\\\\n \\alpha_{1p} &= (-0.05, -0.0016)' \\\\\n \\alpha_{2p} &= (0, 0)' \\\\\n \\beta_{1p} &= (0.08, 0.0003)' \\\\\n \\beta_{2p} &= (0, 0)'.\n \\end{split}\n\\end{equation}\n\nThis means that the true data generating process prescribes that for large firms there is no effect at all of the score on $Y$ and hence also no treatment effect. The dependent variable fluctuates randomly around 0. This is also what was found by the authors. The coefficients for small firms are also inspired by the authors' results when running a first order polynomial regression on the true data. The treatment effect for small firms is equal to 0.08 while score generally has an order one polynomial effect on $Y$. This data generating process results in the following typical RDD representation of the data in which $Y$ is plotted on the assignment variable $X$. Additionally, as the authors, I present this here only for small firms and average over $Y$ for each score.\n\n\n```python\n# plot small firms averaged scaled investment on the score (from previously simulated data)\nsmall = simulated_data.loc[simulated_data[\"large\"] == 0, :]\nmean_small = small.groupby(\"score\").mean()\nmean_small.reset_index(inplace=True)\nfig, ax = plt.subplots()\nsns.scatterplot(mean_small[\"score\"], mean_small[\"scaled_investment\"], ax=ax)\nax.axvline()\nsns.regplot(mean_small.loc[mean_small[\"score\"]<0, \"score\"], mean_small.loc[mean_small[\"score\"]<0, \"scaled_investment\"],\n order=1, scatter=False, ci=None)\nsns.regplot(mean_small.loc[mean_small[\"score\"]>=0, \"score\"], mean_small.loc[mean_small[\"score\"]>=0, \"scaled_investment\"],\n order=1, scatter=False, ci=None, color=sns.color_palette(\"tab10\")[0])\n```\n\nThe error term is postulated as clustered by the authors (which is also assumed for the figure above). In my simulations, I control for different possible distributions of the error term. I show them graphically below. The homoskedastic is a $0.08 \\times N(0, 1)$ distribution. The \"normal\" one is heteroskedastic in the sense that the error increases the closer the score is to the cutoff. In the \"inverse\" case this heteroskedasticity pattern is flipped around. In the \"random_cluster\" scenario I cluster the standard erros by allowing the error to come from a different uniform distribution centered around zero depending on the score and I add it to $0.05 \\times N(0,1)$. The resulting error distributions are presented below.\n\n\n```python\nfig, ax = plt.subplots(2, 2, sharex=True, sharey=True, gridspec_kw={\"wspace\": 0})\nfor error_distr in [(\"homoskedastic\", 0, 0), (\"random_cluster\", 0, 1) ,(\"normal\", 1, 0), (\"inverse\", 1, 1)]:\n data, error = simulate_data(num_obs, coefficients, error_dist=error_distr[0]) \n sns.scatterplot(data[\"score\"], error, ax=ax[error_distr[1], error_distr[2]])\n ax[error_distr[1], error_distr[2]].set_title(error_distr[0])\n```\n\nIn my simulation I now estimate the model using different functional forms and approaches. I always correctly exclude any regressors for large firms and only include different polynomials for the small firms. I run the polynomial regressions above with $P \\in \\{0, 1, 2, 3\\}$ for small firms. Additionally, I separately extract which model would be selected when using the AIC. On top of that, I run Jackknife Model Averaging on the simulated data set as well. For the Jackknife Model Averaging I allow the procedure to average over the four different $P$. For each simulation run I report the estimated treatment effect, indicate whether the 95 percent confidence interval covers the true treatment effect parameter and how wide the confidence interval is. This is done for 1000 runs. The confidence interval for the polynomial regressions is readily available and consequently so is the one for the AIC model selection case. For JMA, per run, I bootstrap from the simulated data set 200 times by drawing with replacement and running JMA for the newly created data to obtain a confidence interval for the treatment effect. I opt for the simple percentile confidence interval (compare chapter 5 in Davidson and Hinkley (1997)). \nIn the next step I average the treatment effect over the 1000 runs, calculate the standard deviation and the root mean squared error to observe how well the different approaches recover the true treatment effect. Further I report the coverage probability and average width of the confidence interval to get a picture of how inference might be flawed and how large the increase in width should be to obtain good coverage. \n\nI choose the order one polynomial model as the true underlying data generating process to see how JMA reacts to the unfavourable condition of having to face more flexible specifications which are not actually the true ones. As the true model is among the models to average over, though, the average treatment effect should converge to the true treatment effect with increasing number of observations (firms in the sample). This root-n consistency I check by running the above 1000 runs with different number of observations $N \\in \\{100, 200, 360, 600, 1000\\}$. Generally, I expect the AIC to perform better in recovering the treatment effect as the true model is among the candidates and as opposed to JMA, the AIC does not confound it with the others when it is selected. In this situation where the number of observations is still somewhat small, JMA should have problems as it could easily find higher order effects of score in every run due to random disturbances (error term) which causes the bias to be quite substantial as the true linear form is confounded by higher order polynomials. \nAt the same time I expect the JMA confidence interval to have good coverage of the true treatment effect outperforming the AIC model selection due to its incorporation of model uncertainty. As opposed to Button (2016), I allow the error term not to be homoskedastic. As JMA is supposed to be robust to heterskedasticity, I do not expect large differences in performance across different error term specifications. The existence of heteroskedasticity gives an edge for JMA in comparison to the other model averaging procedures considered in Button (2016) as they are not heteroskedasticity robust (see Hansen and Racine (2012)). Button should have included a simulation with heteroskedasticity but this is only a side remark. I run the previously described simulation below.\n\n\n```python\n# If we are running on TRAVIS-CI we will simply load a file with existing results.\n# just load the results after if statement if the simulation might take too long for you (I had to run it over night)...\nif os.environ.get(\"TRAVIS\"):\n results_linear_dgp = pickle.load(open(\"data/simulated/results_linear.pickle\", \"br\"))\n processed_results_linear_dgp = pickle.load(\n open(\"data/simulated/processed_results_linear.pickle\", \"br\")) \nelse:\n # set up simulations\n np.random.seed(123)\n num_runs = 1000\n num_bootstrap_runs = 200\n true_treatment_effect = 0.08\n true_model = {\n \"polynomials\": 1,\n \"coefficients\": {\n \"untreated\": np.array([-0.05, -0.0016]),\n \"treated\": np.array([0.08, 0.0003]),\n },\n }\n # run simulations \n results_linear_dgp = {}\n processed_results_linear_dgp = {}\n for error_distr in [\"homoskedastic\", \"random_cluster\" ,\"normal\", \"inverse\"]:\n results_linear_dgp[error_distr] = {}\n processed_results_linear_dgp[error_distr] = {}\n for num_obs in [100, 200, 360, 600, 1000]:\n results_linear_dgp[error_distr][str(num_obs)] = get_results_regression(\n num_runs, num_obs, num_bootstrap_runs, true_model, error_distr\n )\n processed_results_linear_dgp[error_distr][str(num_obs)] = process_results(\n results_linear_dgp[error_distr][str(num_obs)], true_treatment_effect\n )\n```\n\nBelow I present the results of this simulation for the run in which I postulate the situation as in the paper. In the paper, $N$ equals to 360 and the error terms are assumed to be clustered across scores. In the other specifications with different error terms and sample sizes, the qualitative conclusion that I will make below stay valid which is way I do not present all the results here. \n\nAs expected the bias for JMA is larger than the one for model selection using AIC. Overall, though, JMA has a lower root mean squared error which is achieved by a reduction of the standard error of the estimated treatment effect. We can further see that the coverage probability of the 95 percent confidence interval almost reaches its nominal level while for AIC it is around nine percent below it. Rejection of the null hypothesis that the treatment effect equals zero in the case of AIC actually has a higher risk of type I error than the researcher nominally expects. This is different for JMA and in this specific example, a test would even be more conservative as the treatment effect is underestimated. The better coverage is obviously achieved for JMA by wider confidence intervals which compensate for the fact that the bias is generally larger. As a side remark it should be noted, that for the case of an \"inverse\" error, the bias encountered by JMA is larger than for the other cases which can be explained by the fact, that generally the error can be stronger than in other specifications (which can be seen in my figure with the different error distributions). This likely induces the JMA to find an influence of higher order polynomials more often which makes the confounding effect stronger.\n\n\n```python\nprocessed_results_linear_dgp[\"random_cluster\"][\"360\"]\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        AICJMApolynomial_0polynomial_1polynomial_2polynomial_3
                                        Statistic
                                        Treatment Effect0.0789210.0718870.0339710.0804860.0809260.081253
                                        Bias-0.001079-0.008113-0.0460290.0004860.0009260.001253
                                        Standard Error0.0287440.0251210.0116540.0208650.0281490.036208
                                        RMSE0.0287640.0263990.0474820.0208710.0281640.036230
                                        95% Coverage0.8590000.9440000.0370000.9460000.9420000.940000
                                        CI Width0.0820820.1078350.0480160.0784850.1058120.135816
                                        \n
                                        \n\n\n\nThe root-n consistency I check, as suggested in Zhang (2015), by calculating $N^{0.95} \\times MSE$. For root-n consistency we should see that with increasing $N$ the above measure should decrease. This is presented below again for \"random_cluster\" error term but generalizes across other error specifications.\n\n\n```python\nconsistency = pd.DataFrame(index=[\"AIC\", \"JMA\"], columns=[100, 200, 360, 600, 1000])\nfor num_obs in [100, 200, 360, 600, 1000]:\n consistency[[num_obs]] = processed_results_linear_dgp[\n \"random_cluster\"][str(num_obs)].loc[\"RMSE\", [\"AIC\", \"JMA\"]].to_numpy().reshape((2, 1))**2 * num_obs**0.95\nconsistency\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        1002003606001000
                                        AIC0.2860250.258760.2219220.2026710.179699
                                        JMA0.1793260.1889120.186920.1774170.1622
                                        \n
                                        \n\n\n\nThe theoretical result which is proven by Zhang (2015) for JMA seems to be confirmed in my simulation. It appears that $1000$ simulation runs are not sufficiently many runs to show this consistently from $N$ equal to 100 onwards but rather I obtain it solely from $N=200$ onwards. Although not proven by Zhang (2015) but maybe somewhere else, root-n consistency seems to hold for AIC model selection in my example.\n\nI adapt the previous simulation study now such that the true data generating process is actually a fourth order polynomial. The coefficients now are the following:\n\n\\begin{equation}\n \\begin{split}\n P &= 4 \\\\\n \\alpha_{1p} &= (-0.05, -0.00016, -0.00006, -5e^{-6}, -5e^{-8})' \\\\\n \\alpha_{2p} &= (0, 0)' \\\\\n \\beta_{1p} &= (0.08, 0.00002, 0.00009, -8e^{-6}, -8e^{-8})' \\\\\n \\beta_{2p} &= (0, 0)'.\n \\end{split}\n\\end{equation}\n\nIn order to estimate the treatment effect, I run now the same Monte Carlo simulation as above with only the data generating process having changed. This means that the true model is now not among those considered in my estimation strategy anymore. The theoretical consistency proven by Zhang (2015) does not hold in this scenario. I want to check if we can still find it in practice, though, with this rather mild deviation of the largest model considered from the true data generating process. Further I want to check whether the difference in bias between AIC and JMA stays equally large or whether JMA catches up to AIC or might be even better as now confounding because of overfitting is not an issue anymore. JMA might be expected to even outperform AIC as JMA will likely always add some information from the \"best\" model which is the third order polynomial. AIC on the other hand will likely choose some other orders sometimes which leads to loss of information and stronger bias as when some information from the third order was included. Lastly, I want to see if the coverage probability for JMA stays higher and close to the nominal level, although now the true model is outside the space of those considered. Before we run the simulation study, let us have a look how the averaged dependent variable relates to the assignment variable for small firms now (again for $N=360$ and clustered error terms). \n\n\n```python\n# simulate data\nnp.random.seed(123)\nnum_obs = 360\npolynomials = 4\ncoefficients = {\n \"untreated\": np.array([-0.05, -0.00016, -0.00006, -5e-6, -5e-8]),\n \"treated\": np.array([0.08, 0.00002, 0.00009, -8e-6, -8e-8])}\nsimulated_data = simulate_data(num_obs, coefficients, polynomials, error_dist=\"random_cluster\")[0]\n```\n\n\n```python\n# plot small firms averaged scaled investment on the score\nsmall = simulated_data.loc[simulated_data[\"large\"] == 0, :]\nmean_small = small.groupby(\"score\").mean()\nmean_small.reset_index(inplace=True)\nfig, ax = plt.subplots()\nsns.scatterplot(mean_small[\"score\"], mean_small[\"scaled_investment\"], ax=ax)\nax.axvline()\nsns.regplot(mean_small.loc[mean_small[\"score\"]<0, \"score\"], mean_small.loc[mean_small[\"score\"]<0, \"scaled_investment\"],\n order=3, scatter=False, ci=None)\nsns.regplot(mean_small.loc[mean_small[\"score\"]>=0, \"score\"], mean_small.loc[mean_small[\"score\"]>=0, \"scaled_investment\"],\n order=3, scatter=False, ci=None, color=sns.color_palette(\"tab10\")[0])\n```\n\nIt is clearly visible now that the effect is nonlinear, when fitting a cubic polynomial. We are now ready to run the described simulation below.\n\n\n```python\n# If we are running on TRAVIS-CI we will simply load a file with existing results.\n# just load the results after if statement if the simulation might take too long for you (I had to run it over night)...\nif os.environ.get(\"TRAVIS\"):\n results_nonlinear_dgp = pickle.load(open(\"data/simulated/results_nonlinear.pickle\", \"br\"))\n processed_results_nonlinear_dgp = pickle.load(open(\n \"data/simulated/processed_results_nonlinear.pickle\", \"br\")) \nelse:\n # set up simulation\n np.random.seed(123)\n num_runs = 1000\n num_bootstrap_runs = 200\n true_treatment_effect = 0.08\n true_model = {\n \"polynomials\": 4,\n \"coefficients\": {\n \"untreated\": np.array([-0.05, -0.00016, -0.00006, -5e-6, -5e-8]),\n \"treated\": np.array([0.08, 0.00002, 0.00009, -8e-6, -8e-8]),\n },\n }\n # run simulation\n results_nonlinear_dgp = {}\n processed_results_nonlinear_dgp = {}\n for error_distr in [\"homoskedastic\", \"random_cluster\" ,\"normal\", \"inverse\"]:\n results_nonlinear_dgp[error_distr] = {}\n processed_results_nonlinear_dgp[error_distr] = {}\n for num_obs in [100, 200, 360, 600, 1000]:\n results_nonlinear_dgp[error_distr][str(num_obs)] = get_results_regression(\n num_runs, num_obs, num_bootstrap_runs, true_model, error_distr\n )\n processed_results_nonlinear_dgp[error_distr][str(num_obs)] = process_results(\n results_nonlinear_dgp[error_distr][str(num_obs)], true_treatment_effect\n )\n```\n\nBelow I shortly present the examplary case of $N=360$ and clustered errors. What is interesting to see here is that the actual coverage probablity of JMA is higher than for any of the single models and that it still corresponds to its nominal level leaving AIC clearly behind. Also, here, JMA seems to also perform better than AIC looking at the bias for this specific case. \n\n\n```python\nprocessed_results_nonlinear_dgp[\"random_cluster\"][\"360\"]\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        AICJMApolynomial_0polynomial_1polynomial_2polynomial_3
                                        Statistic
                                        Treatment Effect0.0757070.0805470.0145100.1271610.0711500.076685
                                        Bias-0.0042930.000547-0.0654900.047161-0.008850-0.003315
                                        Standard Error0.0346480.0320970.0130890.0215020.0282190.036201
                                        RMSE0.0349130.0321020.0667850.0518320.0295750.036352
                                        95% Coverage0.8630000.9520000.0010000.3900000.9260000.936000
                                        CI Width0.1072870.1240990.0553630.0810850.1059900.135847
                                        \n
                                        \n\n\n\nTo get a better picture, though, of how JMA and AIC actually perform here, let us compare the clustered arror case across different sample sizes. In the table below I present how the treatment effect and the coverage probability develops with increasing $N$ and whether there is any root-n consistency.\n\n\n```python\nstatistics = [\"Treatment Effect\", \"root-n MSE\", \"95% Coverage\"]\napproaches = [\"AIC\", \"JMA\"]\nindex = pd.MultiIndex.from_product([statistics, approaches], names=[\"Statistic\", \"Approach\"])\nconsistency = pd.DataFrame(index=index, columns=[100, 200, 360, 600, 1000])\nfor num_obs in [100, 200, 360, 600, 1000]:\n consistency.loc[\"root-n MSE\", num_obs] = processed_results_nonlinear_dgp[\n \"random_cluster\"][str(num_obs)].loc[\"RMSE\", [\"AIC\", \"JMA\"]].to_numpy().reshape((2, 1))**2 * num_obs**0.95\n consistency.loc[[\"Treatment Effect\", \"95% Coverage\"], num_obs] = processed_results_nonlinear_dgp[\n \"random_cluster\"][str(num_obs)].loc[[\"Treatment Effect\", \"95% Coverage\"], [\"AIC\", \"JMA\"]].to_numpy().reshape(4, 1)\nconsistency\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        1002003606001000
                                        StatisticApproach
                                        Treatment EffectAIC0.09126930.0808050.07570730.07175220.0730202
                                        JMA0.08476330.08234320.08054690.07595210.0756255
                                        root-n MSEAIC0.4066620.3783820.3269260.3079120.316256
                                        JMA0.26630.2806690.2764110.2702570.276823
                                        95% CoverageAIC0.7330.8080.8630.8980.898
                                        JMA0.950.9460.9520.940.938
                                        \n
                                        \n\n\n\nWe can see for treatment effect and $N^{0.95} \\times MSE$, that for both approaches no clear convergence can be found. In this case, though, the treatment effect estimates of JMA seem to be more stable than those of AIC. Interestingly, the coverage probability of JMA decreases with larger sample size while it increases for AIC. Generally, JMA has again a coverage probability close to its nominal level outperforming AIC. Both of those findings are robust across the four different error distributions considered. For the other error distributions I also do not find any clear convergence pattern for treatment effect and the MSE criterion. For all four distributions, JMA performs on average a bit better than AIC regarding bias as it was expected.\n\nBrought together this first simulation study shows that JMA outperforms AIC regarding coverage probability bringing the Type I error to its nominal level when testing for a nonzero treatment effect. Above that, the second part indicates that JMA seems to show a lower bias when the true model is not among those considered but one of the estimated models is rather close to the true one. This is a feature that might likely be found in RDD where a high order polynomial is often a good approximation of the true model but it probably is not exactly among those considered. This shows that JMA has some appealing features that might be useful to make the estimation of treatment effect and the following inference more reliable in comparison to the usually employed model selection based on the AIC.\n\n## Simulation Study Two\n\nAs noted in my introduction, it is common practice in RDD to rather run local linear regressions to determine the best bandwidth around the cutoff instead of solely estimating polynomial regressions across the whole bandwidth. This is motivated by the intuitive notion that as one is interested in the difference in $Y$ close to the cutoff one should also only rely on data close to it. As described in the section on RDD, Imbens and Lemieux (2008) suggest a bandwidth selection that is based on a cross validation criterion which is supposed to approximate the expected test mean squared error across different bandwidths based on local linear regressions. The approach of Hansen and Racine (2012) has also the objective to approximate the expected test MSE with their least squares cross validation criterion. This observation induces me to try whether JMA and its criterion might be used to determine the optimal bandwidth allowing to average over polynomials of higher order than just running a regression with poylnomial of order (as done with local linear regressions). Hansen and Racine (2012) argue that their approach can serve for local polynomial regressions with fixed bandwidth while I want to go beyond it suggesting that it can even choose the bandwidth. As the linear regression is part of the models the JMA averages over, I expect the JMA approach to at least perform as well as the linear approach when the linear approximation is at least imperfect based on the observations from the previous simulation study. \n\nMy idea is that the approach could even perform better when there is a clear indication that the effect of the assignment variable is not linear locally around the cutoff (Imbens and Lemieux (2008) justify their approach by arguing that the linear approximation is typically a good one locally). In Bronzini and Iachini (2014), the authors opt for a sole robustness check based on a nonlinear kernel estimation for two fixed bandwidths, $h \\in \\{15, 30\\}$. This indicates that also the authors did not necessarily believe in a good linear approximation locally. On the other hand, the fact that it is only a robustness check, though, displays that the setup in which they are is not typically well-suited for nonparametric methods such as local linear regressions. The reason is given by Lee and Lemieux (2010) who argue that RDD with discrete assignment variables (on top of that with a small sample, here) do not yield a good foundation for local methods as the amount of data around the cutoff is rather low. As I do not want to introduce another data generating process (due to size constraints) I run a comparison between local linear regression and JMA for bandwidth selection based on the setup in Bronzini and Iachini (2014) anyways with the idea of a good performance in this setup a good performance in more favourable setups should follow. I tested at which minimum bandwidth the resulting estimated treatment effects are reasonable and found that $h=10$ works well. Consequently, I run a bandwidth selection where I consider all integer bandwidths between 10 and 35. I add five more integers to each side of the bandwidths considered by Bronzini and Iachini. Again I run 1000 simulations in which I first simulate a data set with the same data generating process as in the last simulation, i.e. with $P=4$. Then I run local linear regressions as suggested in Imbens and Lemieux (2008) as well as JMA as specified before across all bandwidths. This yields the respective criterion per approach and per bandwidth among which I choose the bandwidth with the smallest criterion value. No matter which approach selected the bandwidth, I then estimate the treatment effect with JMA to not confound the results by using different estimation techniques for the treatment effect. For every run I report which bandwidth was selected, what the minimum of the criterion is, how high the estimated treatment effect is and how long it took to find the optimal bandwidth. As the effect of score on scaled investment is nonlinear and this seems to also persist locally (when looking abck at my figure of the last simulation), I expect JMA to be at least as good as local linear regressions in selecting the optimal bandwidth. \n\nThe main reason I added the different error distributions is for this second simulation study. I want to observe if JMA can follow local linear regressions with good selection across different specifications. In the inverse case one expects that both approaches should select a bandwidth close to the smallest possible ($h_{min}=10$) as the error is small there. For the \"normal\" case this should be the other way around. For homoskedasticity and especially for the clustered case I hope to obtain many different choices of bandwidth as the random error distribution across scores lets the effect of score appear more or less linear across scores. JMA might exploit the high order polynomials sometimes where the expected test MSE for the local linear regression is not the lowest due to an imperfect linear approximation for instance. I run the described simulation below.\n\n\n```python\n# If we are running on TRAVIS-CI we will simply load a file with existing results.\n# just load the results after if statement if the simulation might take too long for you (I had to run it over night)...\nif os.environ.get(\"TRAVIS\"):\n results_local = pickle.load(open(\"data/simulated/results_local.pickle\", \"br\"))\n processed_results_local = pickle.load(\n open(\"data/simulated/processed_results_local.pickle\", \"br\")) \nelse:\n # set up simulation \n np.random.seed(123)\n num_runs = 1000\n num_obs = 360\n start_local = 10\n start_jma = 10\n width = 35\n true_model = {\n \"polynomials\": 4,\n \"coefficients\": {\n \"untreated\": np.array([-0.05, -0.00016, -0.00006, -5e-6, -5e-8]),\n \"treated\": np.array([0.08, 0.00002, 0.00009, -8e-6, -8e-8]),\n },\n }\n subset = np.array([np.arange(2), np.arange(4), np.arange(6), np.arange(8)])\n true_treatment_effect = 0.08\n \n # run simulation\n results = {}\n processed_results = {}\n for error_dist in [\"homoskedastic\", \"random_cluster\" ,\"normal\", \"inverse\"]:\n results[error_dist] = get_results_local_regression(\n num_runs,\n num_obs,\n true_model,\n start_local,\n start_jma,\n width,\n subset,\n error_dist=error_dist,\n )\n processed_results[error_dist] = process_results_local(\n results[error_dist], true_treatment_effect\n )\n```\n\nThe results from the above simulation are processed such that I can present the average treatment effect, bias, standard error, mean squared error, expected MSE (according to the criterion of the approach), time, bandwidth and the percentage of times both approaches chose the same bandwidth. These processed results are shown below for all different error distributions.\n\n\n```python\n# show results\nfor error_dist in [\"homoskedastic\", \"random_cluster\" ,\"normal\", \"inverse\"]:\n display(error_dist)\n display(processed_results_local[error_dist])\n```\n\n\n 'homoskedastic'\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Treatment EffectBiasStandard ErrorMSEExpected MSETimeSame BandwidthBandwidth
                                        JMA0.078796-0.0012040.0201810.0004090.0062650.7477670.61716.784
                                        local linear0.079480-0.0005200.0191330.0003660.0064153.1250810.61716.941
                                        \n
                                        \n\n\n\n 'random_cluster'\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Treatment EffectBiasStandard ErrorMSEExpected MSETimeSame BandwidthBandwidth
                                        JMA0.079207-0.0007930.0219040.0004800.0059060.7438430.61116.818
                                        local linear0.079815-0.0001850.0205790.0004240.0060843.1259830.61116.965
                                        \n
                                        \n\n\n\n 'normal'\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Treatment EffectBiasStandard ErrorMSEExpected MSETimeSame BandwidthBandwidth
                                        JMA0.078088-0.0019120.0259370.0006760.0082720.7484100.76733.006
                                        local linear0.078358-0.0016420.0251190.0006340.0084853.1281920.76732.404
                                        \n
                                        \n\n\n\n 'inverse'\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Treatment EffectBiasStandard ErrorMSEExpected MSETimeSame BandwidthBandwidth
                                        JMA0.078474-0.0015260.0134930.0001840.004270.7501920.97810.029
                                        local linear0.078488-0.0015120.0135250.0001850.004293.1152800.97810.035
                                        \n
                                        \n\n\nWe can see that indeed for the \"normal\" and the \"inverse\" case the frequency of equal bandwidths chosen is the highest. As expected both approaches choose small bandwidth for the \"inverse\" and large bandwidth for the \"normal\" error distribution. Apart from that both approaches give very similar results for the treatment effect across all distributions while the local linear regression approach is slightly better. Despite that for homoskedastic and clustered errors only in around 60 percent of the cases the same bandwidth is chosen, the average bandwidth for both approaches is very similar. Unfortunately, within my already nonlinear example, the local linear approximation seems to work reasonably well such that JMA cannot improve the results. At the same time, we can also see that JMA does not do harm here neither which might still underline its strength by being as performant as local linear regressions while displaying more flexibility regarding rather nonlinear relations. It would be interesting to see how JMA reacts when the underlying data generating process is actually linear. Even more interesting might be whether one can come up with an example that is locally nonlinear such that it actually outperforms the linear approach. As my simulations are already long I have to conclude with my last simulation, though, and maybe come back to that in a future project. As a side remark for this last simulation it is worth noting that the efficient way of performing leave one out cross validation for linear regressions based on Hansen and Racine (2012) makes the approach way faster than the local linear regression in which I coded up the cross validation conventionally.\n\nOne should mention, though, that despite the fact that the assignment variable is discrete, the local approaches work well and Bronzini and Iachini should have performed a proper bandwidth selection as opposed to simply selecting two bandwidths as a robustness check and arguing with non-local polynomial regressions selected via AIC as their main results. This can be seen by the fact how well both local approaches recover the true treatment effect here in comparison to the same data generating process estimated via polynomial regressions before. Further it would have given them more credibility as only more local data around the cutoff is used. \n\n## Conclusion\n\nMy project shows that the typical method of choosing the optimal order of polynomial for the assignment variable in RDD based on model selection via AIC leads to too optimistic certainty of the estimated treatment effect. This results in clearly understated type I error. Jackknife Model Averaging can be a valid alternative in that regard as it appears that the confidence intervals reach their nominal coverage irrespective of whether the correct model is among those averaged over. At the same time, the bias encountered by AIC and JMA is similar when the true model is not among those considered by the researcher but at least one of the models is a good aproximation of the true one. This can be imagined to be a typical situation in RDD when deciding on the poylnomial order of the assignment variable. With this in mind, JMA should maybe be even the prefered approach. \n\nThe second finding of my project is that without relying on theory but solely on my simulation evidence, JMA seems to be an alternative to local linear regressions for bandwidth selection in RDD. I show that when the true model is not among the candidates (but one considered model is a good proxy) and the effect of the assignment variable is generally nonlinear, JMA yields very similar results to the local linear approach and both can recover the true treatment effect quite well on average. Although not shown, I argue that it could be imagineable to find situations in which JMA might be even prefered over the local linear approach. I imagine such a situation to involve considerable nonlinearity locally that potentially varies a lot with the selected bandwidth.\n\n### References\n\n\nBronzini, R., & Iachini, E. (2014). Are incentives for R&D effective? Evidence from a regression discontinuity approach. *American Economic Journal: Economic Policy, 6*(4), 100-134.\n\nButton, P. (2016). Model uncertainty and model averaging in regression discontinuity designs. *Journal of Econometric Methods, 5*(1), 103-116.\n\nDavison, A. C. and D. V. Hinkley (1997). Bootstrap Methods and their Application. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press.\n\nFletcher, D. (2019). Model Averaging. Springer.\n\nHansen, B. E., & Racine, J. S. (2012). Jackknife model averaging. *Journal of Econometrics, 167*(1), 38-46.\n\nImbens, G. W., & Lemieux, T. (2008). Regression discontinuity designs: A guide to practice. *Journal of econometrics, 142*(2), 615-635.\n\nLee, D. S., & Lemieux, T. (2010). Regression discontinuity designs in economics. *Journal of economic literature, 48*(2), 281-355.\n\nLi, K. C. (1987). Asymptotic optimality for Cp, CL, cross-validation and generalized cross-validation: discrete index set. *The Annals of Statistics, 958-975*.\n\nMoral\u2010Benito, E. (2015). Model averaging in economics: An overview. *Journal of Economic Surveys, 29*(1), 46-75.\n\nSteel, M. F. (forthcoming). Model Averaging and Its Use in Economics. *Journal of Economic Literature*.\n\nStone, M. (1974). Cross\u2010validatory choice and assessment of statistical predictions. *Journal of the Royal Statistical Society: Series B (Methodological), 36*(2), 111-133.\n\nWolpert, D. H. (1992). Stacked generalization. *Neural networks, 5*(2), 241-259.\n", "meta": {"hexsha": "6b82b93c465a77199d0227610a92e11ef44ad581", "size": 207301, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Project.ipynb", "max_stars_repo_name": "Pascalheid/project_computational_statistics", "max_stars_repo_head_hexsha": "275e4ce42829f5a12aa1da625bbf2dfd8db1ee36", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Project.ipynb", "max_issues_repo_name": "Pascalheid/project_computational_statistics", "max_issues_repo_head_hexsha": "275e4ce42829f5a12aa1da625bbf2dfd8db1ee36", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Project.ipynb", "max_forks_repo_name": "Pascalheid/project_computational_statistics", "max_forks_repo_head_hexsha": "275e4ce42829f5a12aa1da625bbf2dfd8db1ee36", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 134.611038961, "max_line_length": 45440, "alphanum_fraction": 0.8032763952, "converted": true, "num_tokens": 18152, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.7634837743174788, "lm_q1q2_score": 0.42037978960584904}} {"text": "\n\n\n# Start-to-Finish Example: Unit Testing `GiRaFFE_NRPy`: Interpolating Metric Face-Values\n\n## Author: Patrick Nelson\n\n## This module Validates the `FCVAL` routine for `GiRaFFE`.\n\n**Notebook Status:** Validated\n\n**Validation Notes:** This module will validate the routines in [Tutorial-GiRaFFE_NRPy-Metric_Face_Values](Tutorial-GiRaFFE_NRPy-Metric_Face_Values.ipynb).\n\n### NRPy+ Source Code for this module: \n* [GiRaFFE_NRPy/GiRaFFE_NRPy_Metric_Face_Values.py](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_Metric_Face_Values.py) [\\[**tutorial**\\]](Tutorial-GiRaFFE_NRPy-FCVAL.ipynb) Generates the C code to interpolate gridfunctions to cell faces.\n\n## Introduction:\n\nThis notebook validates the code that will interpolate the metric gridfunctions on cell faces. These values, along with the reconstruction of primitive variables on the faces, are necessary for the Riemann solvers to compute the fluxes through the cell faces.\n\nIt is, in general, good coding practice to unit test functions individually to verify that they produce the expected and intended output. We will generate test data with arbitrarily-chosen analytic functions and calculate gridfunctions at the cell centers on a small numerical grid. We will then compute the values on the cell faces in two ways: first, with our interpolator, then second, we will shift the grid and compute them analytically. Then, we will rerun the function at a finer resolution. Finally, we will compare the results of the two runs to show fourth-order convergence.\n\nWhen this notebook is run, the difference between the approximate and exact metric gridfunctions will be output to text files that can be found in the same directory as this notebook. These will be read in in [Step 3](#convergence), and used there to confirm convergence order of the algorithm. \n\n\n\n# Table of Contents\n$$\\label{toc}$$\n\nThis notebook is organized as follows\n\n1. [Step 1](#setup): Set up core functions and parameters for unit testing the FCVAL algorithm\n 1. [Step 1.a](#expressions) Write expressions for the metric gridfunctions\n 1. [Step 1.b](#ccodekernels) Generate C functions to calculate the gridfunctions\n 1. [Step 1.c](#free_parameters) Set free parameters in the code\n1. [Step 2](#mainc): `FCVAL_unit_test.c`: The Main C Code\n 1. [Step 2.a](#compile_run): Compile and run the code\n1. [Step 3](#convergence): Code validation: Verify that relative error in numerical solution converges to zero at the expected order\n1. [Step 4](#latex_pdf_output): Output this notebook to $\\LaTeX$-formatted PDF file\n\n\n\n# Step 1: Set up core functions and parameters for unit testing the FCVAL algorithm \\[Back to [top](#toc)\\]\n$$\\label{setup}$$\n\nWe'll start by appending the relevant paths to `sys.path` so that we can access sympy modules in other places. Then, we'll import NRPy+ core functionality and set up a directory in which to carry out our test. \n\n\n```python\nimport os, sys # Standard Python modules for multiplatform OS-level functions\n# First, we'll add the parent directory to the list of directories Python will check for modules.\nnrpy_dir_path = os.path.join(\"..\")\nif nrpy_dir_path not in sys.path:\n sys.path.append(nrpy_dir_path)\nnrpy_dir_path = os.path.join(\"..\",\"..\")\nif nrpy_dir_path not in sys.path:\n sys.path.append(nrpy_dir_path)\n\nfrom outputC import outCfunction, lhrh # NRPy+: Core C code output module\nimport sympy as sp # SymPy: The Python computer algebra package upon which NRPy+ depends\nimport finite_difference as fin # NRPy+: Finite difference C code generation module\nimport NRPy_param_funcs as par # NRPy+: Parameter interface\nimport grid as gri # NRPy+: Functions having to do with numerical grids\nimport indexedexp as ixp # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support\nimport reference_metric as rfm # NRPy+: Reference metric support\nimport cmdline_helper as cmd # NRPy+: Multi-platform Python command-line interface\n\nout_dir = \"Validation/\"\ncmd.mkdir(out_dir)\nsubdir = \"FCVAL\"\ncmd.mkdir(os.path.join(out_dir,subdir))\n\nthismodule = \"Start_to_Finish_UnitTest-GiRaFFE_NRPy-Metric_Face_Values\"\n\n# Set the finite-differencing order to 2\npar.set_parval_from_str(\"finite_difference::FD_CENTDERIVS_ORDER\", 2)\n\n```\n\n\n\n## Step 1.a: Write expressions for the metric gridfunctions \\[Back to [top](#toc)\\]\n$$\\label{expressions}$$\n\nNow, we'll choose some functions with arbitrary forms to generate test data. We'll need to set ten gridfunctions, so expressions are being pulled from several previously written unit tests.\n\n\\begin{align}\n\\gamma_{xx} &= ax^3 + by^3 + cz^3 + dy^2 + ez^2 + f \\\\\n\\gamma_{yy} &= gx^3 + hy^3 + lz^3 + mx^2 + nz^2 + p \\\\\n\\gamma_{zz} &= px^3 + qy^3 + rz^3 + sx^2 + ty^2 + u. \\\\\n\\gamma_{xy} &= a \\exp\\left(-\\left((x-b)^2+(y-c)^2+(z-d)^2\\right)\\right) \\\\\n\\gamma_{xz} &= f \\exp\\left(-\\left((x-g)^2+(y-h)^2+(z-l)^2\\right)\\right) \\\\\n\\gamma_{yz} &= m \\exp\\left(-\\left((x-n)^2+(y-o)^2+(z-p)^2\\right)\\right), \\\\\n\\beta^x &= \\frac{2}{\\pi} \\arctan(ax + by + cz) \\\\\n\\beta^y &= \\frac{2}{\\pi} \\arctan(bx + cy + az) \\\\\n\\beta^z &= \\frac{2}{\\pi} \\arctan(cx + ay + bz) \\\\\n\\alpha &= 1 - \\frac{1}{2+x^2+y^2+z^2} \\\\\n\\end{align}\n\n\n\n```python\na,b,c,d,e,f,g,h,l,m,n,o,p,q,r,s,t,u = par.Cparameters(\"REAL\",thismodule,[\"a\",\"b\",\"c\",\"d\",\"e\",\"f\",\"g\",\"h\",\"l\",\"m\",\"n\",\"o\",\"p\",\"q\",\"r\",\"s\",\"t\",\"u\"],1e300)\nM_PI = par.Cparameters(\"#define\",thismodule,[\"M_PI\"], \"\")\n\ngammaDD = ixp.register_gridfunctions_for_single_rank2(\"AUXEVOL\",\"gammaDD\",\"sym01\",DIM=3)\nbetaU = ixp.register_gridfunctions_for_single_rank1(\"AUXEVOL\",\"betaU\",DIM=3)\nalpha = gri.register_gridfunctions(\"AUXEVOL\",\"alpha\")\n\npar.set_parval_from_str(\"reference_metric::CoordSystem\",\"Cartesian\")\nrfm.reference_metric()\nx = rfm.xxCart[0]\ny = rfm.xxCart[1]\nz = rfm.xxCart[2]\n\ngammaDD[0][0] = a*x**3 + b*y**3 + c*z**3 + d*y**2 + e*z**2 + f\ngammaDD[1][1] = g*x**3 + h*y**3 + l*z**3 + m*x**2 + n*z**2 + o\ngammaDD[2][2] = p*x**3 + q*y**3 + r*z**3 + s*x**2 + t*y**2 + u\ngammaDD[0][1] = a * sp.exp(-((x-b)**2 + (y-c)**2 + (z-d)**2))\ngammaDD[0][2] = f * sp.exp(-((x-g)**2 + (y-h)**2 + (z-l)**2))\ngammaDD[1][2] = m * sp.exp(-((x-n)**2 + (y-o)**2 + (z-p)**2))\n\nbetaU[0] = (sp.sympify(2)/M_PI) * sp.atan(a*x + b*y + c*z)\nbetaU[1] = (sp.sympify(2)/M_PI) * sp.atan(b*x + c*y + a*z)\nbetaU[2] = (sp.sympify(2)/M_PI) * sp.atan(c*x + a*y + b*z)\n\nalpha = sp.sympify(1) - sp.sympify(1) / (sp.sympify(2) + x**2 + y**2 + z**2)\n```\n\n\n\n## Step 1.b: Generate C functions to calculate the gridfunctions \\[Back to [top](#toc)\\]\n$$\\label{ccodekernels}$$\n\nHere, we will use the NRPy+ function `outCfunction()` to generate C code that will calculate our metric gridfunctions over an entire grid. We will also call the function to generate the function we are testing. \n\n\n```python\nmetric_gfs_to_print = [\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"gammaDD00\"),rhs=gammaDD[0][0]),\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"gammaDD01\"),rhs=gammaDD[0][1]),\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"gammaDD02\"),rhs=gammaDD[0][2]),\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"gammaDD11\"),rhs=gammaDD[1][1]),\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"gammaDD12\"),rhs=gammaDD[1][2]),\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"gammaDD22\"),rhs=gammaDD[2][2]),\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"betaU0\"),rhs=betaU[0]),\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"betaU1\"),rhs=betaU[1]),\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"betaU2\"),rhs=betaU[2]),\\\n lhrh(lhs=gri.gfaccess(\"aux_gfs\",\"alpha\"),rhs=alpha),\\\n ]\n\ndesc = \"Calculate the metric gridfunctions\"\nname = \"calculate_metric_gfs\"\noutCfunction(\n outfile = os.path.join(out_dir,name+\".h\"), desc=desc, name=name,\n params =\"const paramstruct *restrict params,REAL *restrict xx[3],REAL *restrict auxevol_gfs\",\n body = fin.FD_outputC(\"returnstring\",metric_gfs_to_print,params=\"outCverbose=False\").replace(\"IDX4\",\"IDX4S\"),\n loopopts=\"AllPoints,Read_xxs\")\n\nimport GiRaFFE_NRPy.GiRaFFE_NRPy_Metric_Face_Values as FCVAL\nFCVAL.GiRaFFE_NRPy_FCVAL(os.path.join(out_dir,subdir))\n\n```\n\n Output C function calculate_metric_gfs() to file Validation/calculate_metric_gfs.h\n\n\n\n\n## Step 1.c: Set free parameters in the code \\[Back to [top](#toc)\\]\n$$\\label{free_parameters}$$\n\nWe also need to create the files that interact with NRPy's C parameter interface. \n\n\n```python\n# Step 3.d.i: Generate declare_Cparameters_struct.h, set_Cparameters_default.h, and set_Cparameters[-SIMD].h\n# par.generate_Cparameters_Ccodes(os.path.join(out_dir))\n\n# Step 3.d.ii: Set free_parameters.h\nwith open(os.path.join(out_dir,\"free_parameters.h\"),\"w\") as file:\n file.write(\"\"\"\n// Override parameter defaults with values based on command line arguments and NGHOSTS.\nparams.Nxx0 = atoi(argv[1]);\nparams.Nxx1 = atoi(argv[2]);\nparams.Nxx2 = atoi(argv[3]);\nparams.Nxx_plus_2NGHOSTS0 = params.Nxx0 + 2*NGHOSTS;\nparams.Nxx_plus_2NGHOSTS1 = params.Nxx1 + 2*NGHOSTS;\nparams.Nxx_plus_2NGHOSTS2 = params.Nxx2 + 2*NGHOSTS;\n// Step 0d: Set up space and time coordinates\n// Step 0d.i: Declare \\Delta x^i=dxx{0,1,2} and invdxx{0,1,2}, as well as xxmin[3] and xxmax[3]:\nconst REAL xxmin[3] = {-1.0,-1.0,-1.0};\nconst REAL xxmax[3] = { 1.0, 1.0, 1.0};\n\nparams.dxx0 = (xxmax[0] - xxmin[0]) / ((REAL)params.Nxx0);\nparams.dxx1 = (xxmax[1] - xxmin[1]) / ((REAL)params.Nxx1);\nparams.dxx2 = (xxmax[2] - xxmin[2]) / ((REAL)params.Nxx2);\nprintf(\"dxx0,dxx1,dxx2 = %.5e,%.5e,%.5e\\\\n\",params.dxx0,params.dxx1,params.dxx2);\nparams.invdx0 = 1.0 / params.dxx0;\nparams.invdx1 = 1.0 / params.dxx1;\nparams.invdx2 = 1.0 / params.dxx2;\n\\n\"\"\")\n\n# Generates declare_Cparameters_struct.h, set_Cparameters_default.h, and set_Cparameters[-SIMD].h\npar.generate_Cparameters_Ccodes(os.path.join(out_dir))\n```\n\n\n\n# Step 2: `FCVAL_unit_test.c`: The Main C Code \\[Back to [top](#toc)\\]\n$$\\label{mainc}$$\n\n\n\n\n```python\n%%writefile $out_dir/FCVAL_unit_test.c\n// These are common packages that we are likely to need.\n#include \"stdio.h\"\n#include \"stdlib.h\"\n#include \"math.h\"\n#include \"string.h\" // Needed for strncmp, etc.\n#include \"stdint.h\" // Needed for Windows GCC 6.x compatibility\n#include // Needed to set a random seed.\n\n#define REAL double\n#include \"declare_Cparameters_struct.h\"\n\nconst int NGHOSTS = 3;\n\nREAL a,b,c,d,e,f,g,h,l,m,n,o,p,q,r,s,t,u;\n\n// Standard NRPy+ memory access:\n#define IDX4S(g,i,j,k) \\\n( (i) + Nxx_plus_2NGHOSTS0 * ( (j) + Nxx_plus_2NGHOSTS1 * ( (k) + Nxx_plus_2NGHOSTS2 * (g) ) ) )\n\nconst int kronecker_delta[4][3] = { { 0,0,0 },\n { 1,0,0 },\n { 0,1,0 },\n { 0,0,1 } };\n\n// Give gridfunctions their names:\n#define GAMMADD00GF 0\n#define GAMMADD01GF 1\n#define GAMMADD02GF 2\n#define GAMMADD11GF 3\n#define GAMMADD12GF 4\n#define GAMMADD22GF 5\n#define BETAU0GF 6\n#define BETAU1GF 7\n#define BETAU2GF 8\n#define ALPHAGF 9\n#define GAMMA_FACEDD00GF 10\n#define GAMMA_FACEDD01GF 11\n#define GAMMA_FACEDD02GF 12\n#define GAMMA_FACEDD11GF 13\n#define GAMMA_FACEDD12GF 14\n#define GAMMA_FACEDD22GF 15\n#define BETA_FACEU0GF 16\n#define BETA_FACEU1GF 17\n#define BETA_FACEU2GF 18\n#define ALPHA_FACEGF 19\n#define GAMMAUU00GF 20\n#define GAMMAUU01GF 21\n#define GAMMAUU02GF 22\n#define GAMMAUU11GF 23\n#define GAMMAUU12GF 24\n#define GAMMAUU22GF 25\n#define GAMMA_FACEUU00GF 26\n#define GAMMA_FACEUU01GF 27\n#define GAMMA_FACEUU02GF 28\n#define GAMMA_FACEUU11GF 29\n#define GAMMA_FACEUU12GF 30\n#define GAMMA_FACEUU22GF 31\n#define NUM_AUXEVOL_GFS 32\n\n#include \"calculate_metric_gfs.h\"\n#include \"FCVAL/interpolate_metric_gfs_to_cell_faces.h\"\n\nint main(int argc, const char *argv[]) {\n paramstruct params;\n#include \"set_Cparameters_default.h\"\n\n // Step 0c: Set free parameters, overwriting Cparameters defaults\n // by hand or with command-line input, as desired.\n#include \"free_parameters.h\"\n#include \"set_Cparameters-nopointer.h\"\n\n // Step 0e: Set up cell-centered Cartesian coordinate grids\n REAL *xx[3];\n xx[0] = (REAL *)malloc(sizeof(REAL)*Nxx_plus_2NGHOSTS0);\n xx[1] = (REAL *)malloc(sizeof(REAL)*Nxx_plus_2NGHOSTS1);\n xx[2] = (REAL *)malloc(sizeof(REAL)*Nxx_plus_2NGHOSTS2);\n for(int j=0;j\n\n## Step 2.a: Compile and run the code \\[Back to [top](#toc)\\]\n$$\\label{compile_run}$$\n\nNow that we have our file, we can compile it and run the executable.\n\n\n```python\nimport time\n\nprint(\"Now compiling, should take ~2 seconds...\\n\")\nstart = time.time()\ncmd.C_compile(os.path.join(out_dir,\"FCVAL_unit_test.c\"), os.path.join(out_dir,\"FCVAL_unit_test\"))\nend = time.time()\nprint(\"Finished in \"+str(end-start)+\" seconds.\\n\\n\")\n\nprint(\"Now running...\\n\")\nstart = time.time()\ncmd.Execute(os.path.join(\"Validation\",\"FCVAL_unit_test\"), \"10 10 10\",\"out.txt\")\n# To do a convergence test, we'll also need a second grid with twice the resolution.\ncmd.Execute(os.path.join(\"Validation\",\"FCVAL_unit_test\"), \"20 20 20\",\"out.txt\")\nend = time.time()\nprint(\"Finished in \"+str(end-start)+\" seconds.\\n\\n\")\n\n```\n\n Now compiling, should take ~2 seconds...\n \n Compiling executable...\n (EXEC): Executing `gcc -Ofast -fopenmp -march=native -funroll-loops Validation/FCVAL_unit_test.c -o Validation/FCVAL_unit_test -lm`...\n (BENCH): Finished executing in 0.20867443084716797 seconds.\n Finished compilation.\n Finished in 0.21781635284423828 seconds.\n \n \n Now running...\n \n (EXEC): Executing `taskset -c 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 ./Validation/FCVAL_unit_test 10 10 10`...\n (BENCH): Finished executing in 0.20765209197998047 seconds.\n (EXEC): Executing `taskset -c 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 ./Validation/FCVAL_unit_test 20 20 20`...\n (BENCH): Finished executing in 0.20685338973999023 seconds.\n Finished in 0.43360447883605957 seconds.\n \n \n\n\n\n\n# Step 3: Code validation: Verify that relative error in numerical solution converges to zero at the expected order \\[Back to [top](#toc)\\]\n$$\\label{convergence}$$\n\nHere, we import the data at two resolutions and wrote to text files. This data consists of the absolute error of a metric gridfunction at each in the grid. We'll plot a portion of this data along the axis at the lower resolution along with that same data at the higher resolution scaled to demonstrate that this error converges to 0 at the expected rate. Since our algorithm uses a third-order polynomial, we expect fourth-order convergence here.\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nplt.rcParams.update({'font.size': 12})\n\nData1 = np.loadtxt(\"out10-numer.txt\")\nData2 = np.loadtxt(\"out20-numer.txt\")\n\ndef IDX4(i,j,k,Nxx_plus_2NGHOSTS0,Nxx_plus_2NGHOSTS1,Nxx_plus_2NGHOSTS2):\n return (i) + Nxx_plus_2NGHOSTS0 * ( (j) + Nxx_plus_2NGHOSTS1 * ( (k) + Nxx_plus_2NGHOSTS2 * (0) ) )\n\nx1 = np.zeros(10)\na1 = np.zeros(10)\nfor i in range(10):\n x1[i] = Data1[IDX4(i+3,8,8,16,16,16),1]\n a1[i] = Data1[IDX4(i+3,8,8,16,16,16),0]\nx2 = np.zeros(20)\na2 = np.zeros(20)\nfor i in range(20):\n x2[i] = Data2[IDX4(i+3,13,13,26,26,26),1]\n a2[i] = Data2[IDX4(i+3,13,13,26,26,26),0]\n\nplt.figure()\na = plt.plot(x1,a1,'.',label=\"dx\")\nb = plt.plot(x2,a2*(2**4),label=\"dx/2, times (20/10)^4\")\nplt.legend()\nplt.xlabel(\"x\")\nplt.ylabel(\"alpha\")\nplt.show()\n\nconvergence_satisfactory = np.log2(a1/a2[0:-1:2])>3\nif not convergence_satisfactory.all():\n sys.exit(1)\n```\n\n\n\n# Step 4: Output this notebook to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n$$\\label{latex_pdf_output}$$\n\nThe following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n[Tutorial-Start_to_Finish_UnitTest-GiRaFFE_NRPy-Metric_Face_Values.pdf](Tutorial-Start_to_Finish_UnitTest-GiRaFFE_NRPy-Metric_Face_Values.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means.)\n\n\n```python\nimport cmdline_helper as cmd # NRPy+: Multi-platform Python command-line interface\ncmd.output_Jupyter_notebook_to_LaTeXed_PDF(\"Tutorial-Start_to_Finish_UnitTest-GiRaFFE_NRPy-Metric_Face_Values\",location_of_template_file=os.path.join(\"..\"))\n```\n\n Created Tutorial-Start_to_Finish_UnitTest-GiRaFFE_NRPy-\n Metric_Face_Values.tex, and compiled LaTeX file to PDF file Tutorial-\n Start_to_Finish_UnitTest-GiRaFFE_NRPy-Metric_Face_Values.pdf\n\n", "meta": {"hexsha": "e3d52834c553ae8ac4c0263431e07b3a6ef177e3", "size": 46592, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "in_progress/Tutorial-Start_to_Finish_UnitTest-GiRaFFE_NRPy-Metric_Face_Values.ipynb", "max_stars_repo_name": "terrencepierrej/nrpytutorial", "max_stars_repo_head_hexsha": "3ea18beed99cf6b7d67c89c140ca68630452001e", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "in_progress/Tutorial-Start_to_Finish_UnitTest-GiRaFFE_NRPy-Metric_Face_Values.ipynb", "max_issues_repo_name": "terrencepierrej/nrpytutorial", "max_issues_repo_head_hexsha": "3ea18beed99cf6b7d67c89c140ca68630452001e", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "in_progress/Tutorial-Start_to_Finish_UnitTest-GiRaFFE_NRPy-Metric_Face_Values.ipynb", "max_forks_repo_name": "terrencepierrej/nrpytutorial", "max_forks_repo_head_hexsha": "3ea18beed99cf6b7d67c89c140ca68630452001e", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 71.0243902439, "max_line_length": 18636, "alphanum_fraction": 0.735598386, "converted": true, "num_tokens": 6137, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.4202624900653335}} {"text": "```python\n%matplotlib inline\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport scipy.special\nimport scipy.ndimage\nimport scipy.optimize\n```\n\n\n```python\n# helper for gradient checking\ndef approximate_gradient(x, func, eps=1e-5):\n res = np.zeros(x.size)\n \n for i in range(x.size):\n d = np.zeros(x.size)\n d[i] = eps\n \n res[i] = (func(x + d) - func(x - d)) / (2 * eps)\n \n return res\n```\n\n# Chapter 2\n\n## Binary Variables\n\n## Multinomial Variables\n\n## The Gaussian distribution\n\nMarginal and conditional Gaussians (see (2.113) - (2.117)):\n \n$$\n\\begin{align}\n p(x) &= \\mathcal{N}(x|\\mu, \\Lambda^{-1}) \\\\\n p(y|x) &= \\mathcal{N}(y|A x + b, L^{-1}) \\\\\n p(y) &= \\int \\mathrm{d}x\\; p(y|x)p(x) \n = \\mathcal{N}(y|A\\mu + b, L^{-1} + A \\Lambda^{-1} A^T) \\\\\n p(x|y) &= \\frac{p(y|x)p(x)}{p(y)} \n = \\mathcal{N}(y|\\Sigma \\left\\{ A^T L (y - b) + \\Lambda \\mu \\right\\}, \\Sigma) \\\\\n \\Sigma &= \\left( \\Lambda + A^T L A \\right)^{-1}\n\\end{align}\n$$\n\n### Bayesian Inference for Gaussians\n\n$$\n\\begin{eqnarray}\n p(\\{x\\}|\\mu) &\\propto& \\exp\\left[ - \\frac{1}{2 \\sigma^2} \\sum_n (x_n - \\mu)^2 \\right] \\\\\n p(\\mu|\\mu_0, \\sigma_0) &\\propto& \\exp\\left[ - \\frac{1}{2 \\sigma_0^2} (\\mu - \\mu_0)^2 \\right] \\\\\n p(\\mu|\\{x\\}) &=& \\frac{p(\\{x\\}|\\mu) p(\\mu|\\mu_0, \\sigma_0)}{p(\\{x\\})}\n\\end{eqnarray}\n$$\n\nThe posterior is given by\n$$\n\\begin{eqnarray}\n -\\log p(\\mu|\\{x\\}) &=& \n \\frac{1}{2 \\sigma^2} \\sum_n (x_n - \\mu)^2 +\n \\frac{1}{2 \\sigma_0^2} (\\mu - \\mu_0)^2 +\n \\mathrm{const} \n\\\\\n &=& \n \\frac{1}{2 \\sigma^2} \\sum_n \\mu^2 + \\frac{1}{2 \\sigma_0^2} \\mu^2 +\n \\frac{1}{\\sigma^2} \\sum_n x_n \\mu + \\frac{1}{\\sigma_0^2} \\mu_0 \\mu +\n \\mathrm{const} \n\\\\\n &=&\n \\frac{1}{2 \\sigma^{\\prime2}} \\mu^2 + \\frac{1}{\\sigma^{\\prime^2}} \\mu^\\prime \\mu + \\mathrm{const}\n\\end{eqnarray}\n$$\n\nWith:\n\n$$\n\\begin{eqnarray}\n \\sigma^{\\prime2}\n &=& \\left( \\frac{1}{\\sigma^2} \\sum_n + \\frac{1}{\\sigma_0^2} \\right)^{-1} \n &=& \\left( \\frac{\\sigma_0^2 N + \\sigma^2}{\\sigma_0^2 \\sigma^2} \\right)^{-1} \n &=& \\frac{\\sigma_0^2 \\sigma^2}{\\sigma_0^2 N + \\sigma^2} \n &=& \\frac{\\sigma_0^2}{1 + N \\sigma_0^2 / \\sigma^2}\n\\end{eqnarray}\n$$\n\nand\n\n$$\n\\begin{eqnarray}\n \\mu^\\prime \n &=& \\frac{\\sigma^{\\prime^2}}{\\sigma^2} \\sum_n x_n + \\frac{\\sigma^{\\prime^2}}{\\sigma_0^2} \\mu_0\n &=& \\frac{\\sigma^{\\prime^2}}{\\sigma^2} X + \\frac{\\sigma^{\\prime^2}}{\\sigma_0^2} \\mu_0\n &=& \\frac{\\sigma_0^2 X + \\sigma^2 \\mu_0}{\\sigma_0^2 N + \\sigma^2}\n &=& \\frac{\\sigma_0^2 N \\bar{x} + \\sigma^2 \\mu_0}{\\sigma_0^2 N + \\sigma^2}\n\\end{eqnarray}\n$$\n\nNote that for $N \\rightarrow 0$:\n\n$$\n\\begin{eqnarray}\n \\sigma^{\\prime2} &\\rightarrow& \\sigma_0\n &\\quad&\n \\mu^\\prime &\\rightarrow& \\mu_0\n\\end{eqnarray} \n$$\n\nAnd for $N \\rightarrow \\infty$\n$$\n\\begin{eqnarray}\n \\sigma^{\\prime2} &\\rightarrow& 0\n &\\quad&\n \\mu^\\prime &\\rightarrow& \\bar{x}\n\\end{eqnarray} \n$$\n\n### TODO: Summarize Student + Normal-Gamma Sections\n\n### Periodic Variables\n\nVon Mises-Fischer Distribution:\n\n$$\n p(x|\\mu, \\kappa) = \n \\frac{\\kappa^{\\nu}}{(2 \\pi)^{\\nu + 1} I_\\nu(\\kappa)} \\exp \\left[ \\kappa \\mu^T x \\right]\n$$\n\nwith $\\nu = d / 2 - 1$. With $|\\mu| = 1$. For maximum likelihood fitting note that:\n\n$$\n\\begin{eqnarray}\n \\mathcal{L} = p(\\{x\\}|\\mu, \\kappa) \n &=& \n \\kappa \\mu^T \\sum_n x_n + N \\log \\kappa^{\\nu} \n - N\\log I_\\nu(\\kappa) + \\mathrm{const} \n\\\\\n \\frac{\\partial}{\\partial \\mu} \\left( \\mathcal{L} + \\lambda (\\mu^2 - 1) \\right) \n &=&\n \\kappa \\sum_n x_n + \\lambda \\mu = 0\n\\\\\n \\mu &=& \\frac{\\sum_n x_n}{|\\sum_n x_n|} \n\\\\\n \\frac{\\partial}{\\partial \\kappa} \\mathcal{L} \n &=&\n \\mu^T \\sum_n x_n \n - N \\frac{\\partial}{\\partial \\kappa} \\log \\kappa^{-\\nu} I_\\nu(\\kappa) \\\\\n &=& \\mu^T \\sum_n x_n - N \\frac{I_{\\nu + 1}(\\kappa)}{I_\\nu(\\kappa)} \\\\\n &=& |\\sum_n x_n| - N \\frac{I_{\\nu + 1}(\\kappa)}{I_\\nu(\\kappa)} = 0\n\\\\\n \\frac{I_{\\nu + 1}(\\kappa)}{I_\\nu(\\kappa)} &=& \\frac{|\\sum_n x_n|}{N}\n\\end{eqnarray}\n$$\n\nNote, the sign of $\\mu$ stems from the maximization of the likelihood and the following identity was used:\n\n$$\n \\frac{\\partial}{\\partial \\kappa} \\left[ \\kappa^{-\\nu} I_\\nu(\\kappa) \\right] =\n \\kappa^{-\\nu} I_{\\nu + 1}(\\kappa).\n$$\n\nThe equation for $\\kappa$ can be be solved, e.g., by bisection search.\n\n\n```python\ndef vonmises_pdf(x, mu, kappa):\n return np.exp(kappa * (np.cos(x - mu) - 1)) / (2.0 * np.pi * scipy.special.ive(0, kappa))\n\n\ndef vonmises_fit(x):\n mu_est = np.arctan2(np.mean(np.sin(x)), np.mean(np.cos(x)))\n return mu_est, bisect_kappa(np.mean(np.cos(x - mu_est)))\n\n\ndef bisect_kappa(x):\n \"\"\"Bisect the solution to ``I_1(kappa) / I_0(kappa) = x``\n \"\"\"\n def eval(kappa):\n return scipy.special.ive(1, kappa) / scipy.special.ive(0, kappa)\n \n lower = -8\n upper = +8\n \n if x > eval(10 ** upper): return 10 ** upper\n if x < eval(10 ** lower): return 10 ** lower\n \n # perform logarithmic search\n for _ in range(10):\n cand = lower + 0.5 * (upper - lower)\n val = eval(10 ** cand)\n \n if val > x: upper = cand\n if val < x: lower = cand\n \n if (upper - lower) < 1:\n break\n \n # perform linear search\n lower = 10 ** lower\n upper = 10 ** upper\n \n for _ in range(20):\n cand = lower + 0.5 * (upper - lower)\n val = eval(cand)\n \n if val > x: upper = cand\n if val < x: lower = cand\n \n cand = lower + 0.5 * (upper - lower)\n return cand\n\n```\n\n\n```python\nn_samples = 10_000\nmu = 0.5* np.pi\nkappa = 2\n\nx = np.random.vonmises(mu, kappa, size=n_samples)\nmu_est, kappa_est = vonmises_fit(x)\n\nu = np.linspace(-np.pi, +np.pi, 100)\nplt.hist(x, bins=51, normed=True, label='empirical', alpha=0.5)\nplt.plot(u, vonmises_pdf(u, mu, kappa), label='analytical')\nplt.plot(u, vonmises_pdf(u, mu_est, kappa_est), label='fitted', ls='--')\nplt.legend(loc='best')\npass\n```\n\n## The exponential family\n\n$$\n\\begin{eqnarray}\n p(x|\\eta) &=& h(x) g(\\eta) \\exp \\eta^T u(x) \\\\\n p(\\eta|\\nu, \\chi) &=& f(\\chi, \\nu) g(\\eta)^\\nu \\exp \\nu \\eta^T \\chi\n\\end{eqnarray}\n$$\n\nPosterior:\n\n$$\n\\begin{eqnarray}\n p(\\eta|\\{x\\}) &=& \\left( \\prod_n p(x_n|\\eta) \\right) p(\\eta|\\nu, \\chi) \\\\\n &\\propto& \n g(\\eta)^N \\exp \\left( \\eta^T \\sum_n u(x_n) \\right)\n g(\\eta)^\\nu \\exp \\nu \\eta^T \\chi \\\\\n &=& \n g(\\eta)^{\\nu + N} \\exp \\left( (\\nu + N) \\frac{\\nu \\chi + \\sum_n u(x_n)}{\\nu + N} \\right) \\\\\n p(\\eta|\\{x\\}) &=& p(\\eta|\\nu^\\prime, \\chi^\\prime) \\\\\n \\nu^\\prime &=& \\nu + N \\\\\n \\chi^\\prime &=& \\frac{\\nu \\chi + \\sum_n u(x_n)}{\\nu + N}\n\\end{eqnarray}\n$$\n\n### Maximum likelihood for exponential family\n\n$$\n\\begin{eqnarray}\n \\mathcal{L} &=& \\sum_n \\log p(x_n|\\eta) \n\\\\\n &=& \\sum_n \\eta^T u(x_n) + N \\log g(\\eta) + \\mathrm{const}\n\\\\\n \\frac{\\partial}{\\partial \\eta} \\mathcal{L} &=&\n \\sum_n u(x_n) + N \\frac{\\partial}{\\partial \\eta} \\log g(\\eta) = 0\n\\\\\n -\\frac{\\partial}{\\partial \\eta} \\log g(\\eta) &=& \\frac{1}{N} \\sum_n u(x_n)\n\\end{eqnarray}\n$$\n\n\nFor the Gaussian:\n\n$$\n\\begin{eqnarray}\n p(x|\\eta) = g(\\eta) \\exp\\left[ \\eta_1 u_1(x) + \\eta_2 u_2(x) \\right]\n\\\\\n \\eta_1 = \\frac{\\mu}{\\sigma^2},\\; \\eta_2 = \\frac{-1}{2 \\sigma^2}\n\\\\\n u_1(x) = x,\\; u_2(x) = x^2 \n\\\\\n -\\log g(\\eta) = \\frac{1}{2} \\log \\pi - \\frac{1}{2} \\log -\\eta_2 - \\frac{1}{4} \\frac{\\eta_1^2}{\\eta_2}\n\\\\\n -\\log \\frac{\\partial}{\\partial \\eta_1} g(\\eta) = \\mu \n\\\\\n -\\log \\frac{\\partial}{\\partial \\eta_2} g(\\eta) = \\mu^2 + \\sigma^2\n\\end{eqnarray}\n$$\n\nFor Bernoulli\n\n$$\n\\begin{eqnarray}\n p(x|\\mu) &=& \\mu^x (1 - \\mu)^{1 - x} = \\left( \\frac{\\mu}{1 - \\mu} \\right)^x \\left( 1 - \\mu \\right)\n\\\\\n &=& g(\\eta) \\exp \\eta x\n\\\\\n \\eta &=& \\log \\frac{\\mu}{1 - \\mu} \n\\\\\n g(\\eta) &=& \\sigma(-\\eta)\n\\\\\n -\\frac{\\partial}{\\partial \\eta} \\log g(\\eta) \n &=& \\frac{1}{\\sigma(-\\eta)} \\sigma(\\eta) \\sigma(-\\eta) = \\sigma(\\eta) = \\mu\n\\end{eqnarray}\n$$\n\n\n```python\n\n```\n", "meta": {"hexsha": "469ed5572f1866a95e77779c230f9a783e5aa0d0", "size": 32412, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "BuildingBlocks/Bishop_Notes_02.ipynb", "max_stars_repo_name": "chmp/misc-exp", "max_stars_repo_head_hexsha": "2edc2ed598eb59f4ccb426e7a5c1a23343a6974b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2017-10-31T20:54:37.000Z", "max_stars_repo_stars_event_max_datetime": "2020-10-23T19:03:00.000Z", "max_issues_repo_path": "BuildingBlocks/Bishop_Notes_02.ipynb", "max_issues_repo_name": "chmp/misc-exp", "max_issues_repo_head_hexsha": "2edc2ed598eb59f4ccb426e7a5c1a23343a6974b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 7, "max_issues_repo_issues_event_min_datetime": "2020-03-24T16:14:34.000Z", "max_issues_repo_issues_event_max_datetime": "2021-03-18T20:51:37.000Z", "max_forks_repo_path": "BuildingBlocks/Bishop_Notes_02.ipynb", "max_forks_repo_name": "chmp/misc-exp", "max_forks_repo_head_hexsha": "2edc2ed598eb59f4ccb426e7a5c1a23343a6974b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2019-07-29T07:55:49.000Z", "max_forks_repo_forks_event_max_datetime": "2019-07-29T07:55:49.000Z", "avg_line_length": 67.8075313808, "max_line_length": 18172, "alphanum_fraction": 0.7304701962, "converted": true, "num_tokens": 3200, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.7853085909370422, "lm_q1q2_score": 0.4202173929973897}} {"text": "\n\n# Neuromatch Academy: Week 2, Day 5, Tutorial 3\n# Learning to Act: Q-Learning\n\n__Content creators:__ Marcelo Mattar and Eric DeWitt with help from Byron Galbraith\n\n__Content reviewers:__ Matt Krause and Michael Waskom\n\n---\n\n# Tutorial Objectives\n \nIn this tutorial you will learn how to act in the more realistic setting of sequential decisions, formalized by Markov Decision Processes (MDPs). In a sequential decision problem, the actions executed in one state not only may lead to immediate rewards (as in a bandit problem), but may also affect the states experienced next (unlike a bandit problem). Each individual action may therefore affect affect all future rewards. Thus, making decisions in this setting requires considering each action in terms of their expected **cumulative** future reward.\n\nWe will consider here the example of spatial navigation, where actions (movements) in one state (location) affect the states experienced next, and an agent might need to execute a whole sequence of actions before a reward is obtained.\n\nBy the end of this tutorial, you will learn\n* what grid worlds are and how they help in evaluating simple reinforcement learning agents\n* the basics of the Q-learning algorithm for estimating action values\n* how the concept of exploration and exploitation, reviewed in the bandit case, also applies to the sequential decision setting\n\n---\n# Setup\n\n\n```python\n# Imports\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.signal import convolve as conv\n```\n\n\n```python\n#@title Figure settings\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"/share/dataset/COMMON/nma.mplstyle.txt\")\n```\n\n\n```python\n#@title Helper functions\ndef epsilon_greedy(q, epsilon):\n \"\"\"Epsilon-greedy policy: selects the maximum value action with probabilty\n (1-epsilon) and selects randomly with epsilon probability.\n \n Args:\n q (ndarray): an array of action values\n epsilon (float): probability of selecting an action randomly \n \n Returns:\n int: the chosen action\n \"\"\"\n if np.random.random() > epsilon:\n action = np.argmax(q)\n else:\n action = np.random.choice(len(q)) \n\n return action\n\n\nclass CliffWorld:\n \"\"\"\n World: Cliff world.\n 40 states (4-by-10 grid world).\n The mapping from state to the grids are as follows:\n 30 31 32 ... 39\n 20 21 22 ... 29\n 10 11 12 ... 19\n 0 1 2 ... 9\n 0 is the starting state (S) and 9 is the goal state (G).\n Actions 0, 1, 2, 3 correspond to right, up, left, down.\n Moving anywhere from state 9 (goal state) will end the session.\n Taking action down at state 11-18 will go back to state 0 and incur a\n reward of -100.\n Landing in any states other than the goal state will incur a reward of -1.\n Going towards the border when already at the border will stay in the same\n place.\n \"\"\"\n def __init__(self):\n self.name = \"cliff_world\"\n self.n_states = 40\n self.n_actions = 4\n self.dim_x = 10\n self.dim_y = 4\n self.init_state = 0\n\n def get_outcome(self, state, action):\n if state == 9: # goal state\n reward = 0\n next_state = None\n return next_state, reward\n reward = -1 # default reward value\n if action == 0: # move right\n next_state = state + 1\n if state % 10 == 9: # right border\n next_state = state\n elif state == 0: # start state (next state is cliff)\n next_state = None\n reward = -100\n elif action == 1: # move up\n next_state = state + 10\n if state >= 30: # top border\n next_state = state\n elif action == 2: # move left\n next_state = state - 1\n if state % 10 == 0: # left border\n next_state = state\n elif action == 3: # move down\n next_state = state - 10\n if state >= 11 and state <= 18: # next is cliff\n next_state = None\n reward = -100\n elif state <= 9: # bottom border\n next_state = state\n else:\n print(\"Action must be between 0 and 3.\")\n next_state = None\n reward = None\n return int(next_state) if next_state is not None else None, reward\n\n def get_all_outcomes(self):\n outcomes = {}\n for state in range(self.n_states):\n for action in range(self.n_actions):\n next_state, reward = self.get_outcome(state, action)\n outcomes[state, action] = [(1, next_state, reward)]\n return outcomes\n \n\ndef learn_environment(env, learning_rule, params, max_steps, n_episodes): \n # Start with a uniform value function\n value = np.ones((env.n_states, env.n_actions))\n\n # Run learning\n reward_sums = np.zeros(n_episodes)\n\n # Loop over episodes\n for episode in range(n_episodes):\n state = env.init_state # initialize state \n reward_sum = 0\n\n for t in range(max_steps):\n # choose next action\n action = epsilon_greedy(value[state], params['epsilon'])\n \n # observe outcome of action on environment\n next_state, reward = env.get_outcome(state, action)\n\n # update value function\n value = learning_rule(state, action, reward, next_state, value, params)\n \n # sum rewards obtained\n reward_sum += reward\n \n if next_state is None:\n break # episode ends\n state = next_state \n \n reward_sums[episode] = reward_sum\n \n return value, reward_sums\n\n\ndef plot_state_action_values(env, value, ax=None):\n \"\"\"\n Generate plot showing value of each action at each state.\n \"\"\"\n if ax is None:\n fig, ax = plt.subplots()\n\n for a in range(env.n_actions): \n ax.plot(range(env.n_states), value[:, a], marker='o', linestyle='--')\n ax.set(xlabel='States', ylabel='Values')\n ax.legend(['R','U','L','D'], loc='lower right')\n \n\ndef plot_quiver_max_action(env, value, ax=None):\n \"\"\"\n Generate plot showing action of maximum value or maximum probability at\n each state (not for n-armed bandit or cheese_world).\n \"\"\"\n if ax is None:\n fig, ax = plt.subplots()\n\n X = np.tile(np.arange(env.dim_x), [env.dim_y,1]) + 0.5\n Y = np.tile(np.arange(env.dim_y)[::-1][:,np.newaxis], [1,env.dim_x]) + 0.5\n which_max = np.reshape(value.argmax(axis=1), (env.dim_y,env.dim_x))\n which_max = which_max[::-1,:]\n U = np.zeros(X.shape)\n V = np.zeros(X.shape)\n U[which_max == 0] = 1\n V[which_max == 1] = 1\n U[which_max == 2] = -1\n V[which_max == 3] = -1\n \n ax.quiver(X, Y, U, V)\n ax.set(\n title='Maximum value/probability actions',\n xlim=[-0.5, env.dim_x+0.5],\n ylim=[-0.5, env.dim_y+0.5], \n )\n ax.set_xticks(np.linspace(0.5, env.dim_x-0.5, num=env.dim_x))\n ax.set_xticklabels([\"%d\" % x for x in np.arange(env.dim_x)])\n ax.set_xticks(np.arange(env.dim_x+1), minor=True)\n ax.set_yticks(np.linspace(0.5, env.dim_y-0.5, num=env.dim_y)) \n ax.set_yticklabels([\"%d\" % y for y in np.arange(0, env.dim_y*env.dim_x, \n env.dim_x)]) \n ax.set_yticks(np.arange(env.dim_y+1), minor=True)\n ax.grid(which='minor',linestyle='-')\n\n\ndef plot_heatmap_max_val(env, value, ax=None):\n \"\"\"\n Generate heatmap showing maximum value at each state\n \"\"\"\n if ax is None:\n fig, ax = plt.subplots()\n\n if value.ndim == 1:\n value_max = np.reshape(value, (env.dim_y,env.dim_x))\n else:\n value_max = np.reshape(value.max(axis=1), (env.dim_y,env.dim_x)) \n value_max = value_max[::-1,:]\n\n im = ax.imshow(value_max, aspect='auto', interpolation='none', cmap='afmhot')\n ax.set(title='Maximum value per state')\n ax.set_xticks(np.linspace(0, env.dim_x-1, num=env.dim_x))\n ax.set_xticklabels([\"%d\" % x for x in np.arange(env.dim_x)])\n ax.set_yticks(np.linspace(0, env.dim_y-1, num=env.dim_y))\n if env.name != 'windy_cliff_grid':\n ax.set_yticklabels(\n [\"%d\" % y for y in np.arange(\n 0, env.dim_y*env.dim_x, env.dim_x)][::-1])\n return im\n\n\ndef plot_rewards(n_episodes, rewards, average_range=10, ax=None):\n \"\"\"\n Generate plot showing total reward accumulated in each episode.\n \"\"\"\n if ax is None:\n fig, ax = plt.subplots()\n\n smoothed_rewards = (conv(rewards, np.ones(average_range), mode='same')\n / average_range)\n\n ax.plot(range(0, n_episodes, average_range),\n smoothed_rewards[0:n_episodes:average_range],\n marker='o', linestyle='--')\n ax.set(xlabel='Episodes', ylabel='Total reward')\n \n\ndef plot_performance(env, value, reward_sums):\n fig, axes = plt.subplots(nrows=2, ncols=2, figsize=(16, 12))\n plot_state_action_values(env, value, ax=axes[0,0])\n plot_quiver_max_action(env, value, ax=axes[0,1])\n plot_rewards(n_episodes, reward_sums, ax=axes[1,0])\n im = plot_heatmap_max_val(env, value, ax=axes[1,1])\n fig.colorbar(im)\n```\n\n---\n# Section 1: Markov Decision Processes\n\n\n```python\n#@title Video 1: MDPs and Q-learning\n# Insert the ID of the corresponding youtube video\nfrom IPython.display import IFrame\nclass BiliVideo(IFrame):\n def __init__(self, id, page=1, width=400, height=300, **kwargs):\n self.id=id\n src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\nvideo = BiliVideo(id='BV1ft4y1Q7bX', width=854, height=480, fs=1)\nprint(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\nvideo\n```\n\n Video available at https://www.bilibili.com/video/BV1ft4y1Q7bX\n\n\n\n\n\n\n\n\n\n\n\n## Section 1.1: Grid Worlds\n\nAs pointed out, bandits only have a single state and immediate rewards for our actions. Many problems we are interested in have multiple states and delayed rewards, i.e. we won't know if the choices we made will pay off over time, or which actions we took contributed to the outcomes we observed.\n\nIn order to explore these ideas, we turn the a common problem setting: the grid world. Grid worlds are simple environments where each state corresponds to a tile on a 2D grid, and the only actions the agent can take are to move up, down, left, or right across the grid tiles. The agent's job is almost always to find a way to a goal tile in the most direct way possible while overcoming some maze or other obstacles, either static or dynamic.\n\nFor our discussion we will be looking at the classic Cliff World, or Cliff Walker, environment. This is a 4x10 grid with a starting position in the lower-left and the goal position in the lower-right. Every tile between these two is the \"cliff\", and should the agent enter the cliff, they will receive a -100 reward and be sent back to the starting position. Every tile other than the cliff produces a -1 reward when entered. The goal tile ends the episode after taking any action from it.\n\n\n\nGiven these conditions, the maximum achievable reward is -11 (1 up, 9 right, 1 down). Using negative rewards is a common technique to encourage the agent to move and seek out the goal state as fast as possible.\n\n---\n# Section 2: Q-Learning\n\nNow that we have our environment, how can we solve it? \n\nOne of the most famous algorithms for estimating action values (aka Q-values) is the Temporal Differences (TD) **control** algorithm known as *Q-learning* (Watkins, 1989). \n\n\\begin{align}\nQ(s_t,a_t) \\leftarrow Q(s_t,a_t) + \\alpha \\big(r_t + \\gamma\\max_\\limits{a} Q(s_{t+1},a_{t+1}) - Q(s_t,a_t)\\big)\n\\end{align}\n\nwhere $Q(s,a)$ is the value function for action $a$ at state $s$, $\\alpha$ is the learning rate, $r$ is the reward, and $\\gamma$ is the temporal discount rate.\n\nThe expression $r_t + \\gamma\\max_\\limits{a} Q(s_{t+1},a_{t+1})$ is referred to as the TD target while the full expression \n\\begin{align}\nr_t + \\gamma\\max_\\limits{a} Q(s_{t+1},a_{t+1}) - Q(s_t,a_t),\n\\end{align}\ni.e. the difference between the TD target and the current Q-value, is referred to as the TD error, or reward prediction error.\n\nBecause of the max operator used to select the optimal Q-value in the TD target, Q-learning directly estimates the optimal action value, i.e. the cumulative future reward that would be obtained if the agent behaved optimally, regardless of the policy currently followed by the agent. For this reason, Q-learning is referred to as an **off-policy** method.\n\n## Exercise 1: Implement the Q-learning algorithm\n\nIn this exercise you will implement the Q-learning update rule described above. It takes in as arguments the previous state $s_t$, the action $a_t$ taken, the reward received $r_t$, the current state $s_{t+1}$, the Q-value table, and a dictionary of parameters that contain the learning rate $\\alpha$ and discount factor $\\gamma$. The method returns the updated Q-value table. For the parameter dictionary, $\\alpha$: `params['alpha']` and $\\gamma$: `params['gamma']`. \n\n\n\n```python\ndef q_learning(state, action, reward, next_state, value, params):\n \"\"\"Q-learning: updates the value function and returns it.\n \n Args:\n state (int): the current state identifier\n action (int): the action taken\n reward (float): the reward received\n next_state (int): the transitioned to state identifier\n value (ndarray): current value function of shape (n_states, n_actions)\n params (dict): a dictionary containing the default parameters\n \n Returns:\n ndarray: the updated value function of shape (n_states, n_actions)\n \"\"\"\n # Q-value of current state-action pair\n q = value[state, action]\n \n ##########################################################\n ## TODO for students: implement the Q-learning update rule\n # Fill out function and remove\n raise NotImplementedError(\"Student excercise: implement the Q-learning update rule\")\n ##########################################################\n\n # write an expression for finding the maximum Q-value at the current state\n if next_state is None:\n max_next_q = 0\n else:\n max_next_q = ...\n\n # write the expression to compute the TD error\n td_error = ... \n # write the expression that updates the Q-value for the state-action pair \n value[state, action] = ...\n \n return value\n```\n\n[*Click for solution*](https://github.com/erlichlab/course-content/tree/master//tutorials/W2D5_ReinforcementLearning/solutions/W2D5_Tutorial3_Solution_910598ca.py)\n\n\n\nNow that we have our Q-learning algorithm, let's see how it handles learning to solve the Cliff World environment. \n\nYou will recall from the previous tutorial that a major part of reinforcement learning algorithms are their ability to balance exploitation and exploration. For our Q-learning agent, we again turn to the epsilon-greedy strategy. At each step, the agent will decide with probability $1 - \\epsilon$ to use the best action for the state it is currently in by looking at the value function, otherwise just make a random choice.\n\nThe process by which our the agent will interact with and learn about the environment is handled for you in the helper function `learn_environment`. This implements the entire learning episode lifecycle of stepping through the state observation, action selection (epsilon-greedy) and execution, reward, and state transition. Feel free to review that code later to see how it all fits together, but for now let's test out our agent.\n\n\n```python\n# set for reproducibility, comment out / change seed value for different results\nnp.random.seed(1)\n\n# parameters needed by our policy and learning rule\nparams = {\n 'epsilon': 0.1, # epsilon-greedy policy\n 'alpha': 0.1, # learning rate \n 'gamma': 1.0, # discount factor\n}\n\n# episodes/trials\nn_episodes = 500\nmax_steps = 1000\n\n# environment initialization\nenv = CliffWorld()\n\n# solve Cliff World using Q-learning\nresults = learn_environment(env, q_learning, params, max_steps, n_episodes)\nvalue_qlearning, reward_sums_qlearning = results\n\n# Plot results\nplot_performance(env, value_qlearning, reward_sums_qlearning)\n```\n\nIf all went well, we should see four plots that show different aspects on our agent's learning and progress.\n\n* The top left is a representation of the Q-table itself, showing the values for different actions in different states. Notably, going right from the starting state or down when above the cliff is clearly very bad.\n* The top right figure shows the greedy policy based on the Q-table, i.e. what action would the agent take if it only took its best guess in that state.\n* The bottom right is the same as the top, only instead of showing the action, it's showing a representation of the maximum Q-value at a particular state.\n* The bottom left is the actual proof of learning, as we see the total reward steadily increasing after each episode until asymptoting at the maximum possible reward of -11.\n\nFeel free to try changing the parameters or random seed and see how the agent's behavior changes.\n\n---\n# Summary\n\nIn this tutorial you implemented a reinforcement learning agent based on Q-learning to solve the Cliff World environment. Q-learning combined the epsilon-greedy approach to exploration-expoitation with a table-based value function to learn the expected future rewards for each state.\n\n---\n# Bonus\n\n## SARSA\n\nAn alternative to Q-learning, the SARSA algorithm also estimates action values. However, rather than estimating the optimal (off-policy) values, SARSA estimates the **on-policy** action value, i.e. the cumulative future reward that would be obtained if the agent behaved according to its current beliefs.\n\n\\begin{align}\nQ(s_t,a_t) \\leftarrow Q(s_t,a_t) + \\alpha \\big(r_t + \\gamma Q(s_{t+1},a_{t+1}) - Q(s_t,a_t)\\big)\n\\end{align}\n\nwhere, once again, $Q(s,a)$ is the value function for action $a$ at state $s$, $\\alpha$ is the learning rate, $r$ is the reward, and $\\gamma$ is the temporal discount rate.\n\nIn fact, you will notices that the *only* difference between Q-learning and SARSA is the TD target calculation uses the policy to select the next action (in our case epsilon-greedy) rather than using the action that maximizes the Q-value.\n\n### Exercise 2: Implement the SARSA algorithm\n\nIn this exercise you will implement the SARSA update rule described above. Just like Q-learning, it takes in as arguments the previous state $s_t$, the action $a_t$ taken, the reward received $r_t$, the current state $s_{t+1}$, the Q-value table, and a dictionary of parameters that contain the learning rate $\\alpha$ and discount factor $\\gamma$. The method returns the updated Q-value table. You may use the `epsilon_greedy` function to acquire the next action. For the parameter dictionary, $\\alpha$: `params['alpha']`, $\\gamma$: `params['gamma']`, and $\\epsilon$: `params['epsilon']`. \n\n\n\n```python\ndef sarsa(state, action, reward, next_state, value, params):\n \"\"\"SARSA: updates the value function and returns it.\n \n Args:\n state (int): the current state identifier\n action (int): the action taken\n reward (float): the reward received\n next_state (int): the transitioned to state identifier\n value (ndarray): current value function of shape (n_states, n_actions)\n params (dict): a dictionary containing the default parameters\n \n Returns:\n ndarray: the updated value function of shape (n_states, n_actions)\n \"\"\"\n # value of previous state-action pair\n q = value[state, action]\n \n ##########################################################\n ## TODO for students: implement the SARSA update rule\n # Fill out function and remove\n raise NotImplementedError(\"Student excercise: implement the SARSA update rule\")\n ##########################################################\n\n # select the expected value at current state based on our policy by sampling\n # from it\n if next_state is None:\n policy_next_q = 0\n else:\n # write an expression for selecting an action using epsilon-greedy\n policy_action = ...\n # write an expression for obtaining the value of the policy action at the \n # current state\n policy_next_q = ...\n \n # write the expression to compute the TD error\n td_error = ... \n # write the expression that updates the Q-value for the state-action pair \n value[state, action] = ...\n \n return value\n```\n\n\n```python\ndef sarsa(state, action, reward, next_state, value, params):\n \"\"\"SARSA: updates the value function and returns it.\n \n Args:\n state (int): the current state identifier\n action (int): the action taken\n reward (float): the reward received\n next_state (int): the transitioned to state identifier\n value (ndarray): current value function of shape (n_states, n_actions)\n params (dict): a dictionary containing the default parameters\n \n Returns:\n ndarray: the updated value function of shape (n_states, n_actions)\n \"\"\"\n # value of previous state-action pair\n q = value[state, action]\n \n # select the expected value at current state based on our policy by sampling\n # from it\n if next_state is None:\n policy_next_q = 0\n else:\n # write an expression for selecting an action using epsilon-greedy\n policy_action = epsilon_greedy(value[next_state], params['epsilon'])\n # write an expression for obtaining the value of the policy action at the \n # current state\n policy_next_q = value[next_state, policy_action]\n \n # write the expression to compute the TD error\n td_error = reward + params['gamma'] * policy_next_q - q \n # write the expression that updates the Q-value for the state-action pair \n value[state, action] = q + params['alpha'] * td_error\n \n return value\n```\n\nNow that we have an implementation for SARSA, let's see how it tackles Cliff World. We will again use the same setup we tried with Q-learning.\n\n\n```python\n# set for reproducibility, comment out / change seed value for different results\nnp.random.seed(1)\n\n# parameters needed by our policy and learning rule\nparams = {\n 'epsilon': 0.1, # epsilon-greedy policy\n 'alpha': 0.1, # learning rate \n 'gamma': 1.0, # discount factor\n}\n\n# episodes/trials\nn_episodes = 500\nmax_steps = 1000\n\n# environment initialization\nenv = CliffWorld()\n\n# learn Cliff World using Sarsa\nresults = learn_environment(env, sarsa, params, max_steps, n_episodes)\nvalue_sarsa, reward_sums_sarsa = results\n\n# Plot results\nplot_performance(env, value_sarsa, reward_sums_sarsa)\n```\n\nWe should see that SARSA also solves the task with similar looking outcomes to Q-learning. One notable difference is that SARSA seems to be skittsh around the cliff edge and often goes further away before coming back down to the goal.\n\nAgain, feel free to try changing the parameters or random seed and see how the agent's behavior changes.\n\n## On-Policy vs Off-Policy\n \nWe have now seen an example of both on- and off-policy learning algorithms. Let's compare both Q-learning and SARSA reward results again, side-by-side, to see how they stack up.\n\n\n```python\n# parameters needed by our policy and learning rule\nparams = {\n 'epsilon': 0.1, # epsilon-greedy policy\n 'alpha': 0.1, # learning rate \n 'gamma': 1.0, # discount factor\n}\n\n# episodes/trials\nn_episodes = 500\nmax_steps = 1000\n\n# environment initialization\nenv = CliffWorld()\n\n# learn Cliff World using Sarsa\nnp.random.seed(1)\nresults = learn_environment(env, q_learning, params, max_steps, n_episodes)\nvalue_qlearning, reward_sums_qlearning = results\nnp.random.seed(1)\nresults = learn_environment(env, sarsa, params, max_steps, n_episodes)\nvalue_sarsa, reward_sums_sarsa = results\n```\n\n\n```python\nfig, ax = plt.subplots()\nax.plot(reward_sums_qlearning, label='Q-learning')\nax.plot(reward_sums_sarsa, label='SARSA')\nax.set(xlabel='Episodes', ylabel='Total reward')\nplt.legend(loc='lower right');\n```\n\nOn this simple Cliff World task, Q-learning and SARSA are almost indistinguisable from a performance standpoint, but we can see that Q-learning has a slight-edge within the 500 episode time horizon. Let's look at the illustrated \"greedy policy\" plots again.\n\n\n```python\nfig, (ax1, ax2) = plt.subplots(ncols=2, figsize=(16, 6))\nplot_quiver_max_action(env, value_qlearning, ax=ax1)\nax1.set(title='Q-learning maximum value/probability actions')\nplot_quiver_max_action(env, value_sarsa, ax=ax2)\nax2.set(title='SARSA maximum value/probability actions');\n```\n\nWhat should immediately jump out is that Q-learning learned to go up, then immediately go to the right, skirting the cliff edge, until it hits the wall and goes down to the goal. The policy further away from the cliff is less certain.\n\nSARSA, on the other hand, appears to avoid the cliff edge, going up one more tile before starting over to the goal side. This also clearly solves the challenge of getting to the goal, but does so at an additional -2 cost over the truly optimal route.\n\nWhy do you think these behaviors emerged the way they did?\n", "meta": {"hexsha": "dce49b4f0982e6dbeca4301e314ae9c5f3e8fae0", "size": 828792, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W2D5_ReinforcementLearning/student/W2D5_Tutorial3.ipynb", "max_stars_repo_name": "txperl/course-content", "max_stars_repo_head_hexsha": "e7d3f94466c2a14560a30cdad182dbee79bc76c5", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tutorials/W2D5_ReinforcementLearning/student/W2D5_Tutorial3.ipynb", "max_issues_repo_name": "txperl/course-content", "max_issues_repo_head_hexsha": "e7d3f94466c2a14560a30cdad182dbee79bc76c5", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/W2D5_ReinforcementLearning/student/W2D5_Tutorial3.ipynb", "max_forks_repo_name": "txperl/course-content", "max_forks_repo_head_hexsha": "e7d3f94466c2a14560a30cdad182dbee79bc76c5", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 821.3994053518, "max_line_length": 248140, "alphanum_fraction": 0.949417948, "converted": true, "num_tokens": 6187, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943822145998, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.42017460345092894}} {"text": "```python\nfrom IPython.display import Image\nImage('../../Python_probability_statistics_machine_learning_2E.png',width=200)\n```\n\n\n```python\nimport numpy as np\nnp.random.seed(1234)\n```\n\nThe estimation problem starts with the desire to infer something meaningful\nfrom\ndata. For parametric estimation, the strategy is to postulate a model for\nthe\ndata and then use the data to fit model parameters. This leads to two\nfundamental questions: where to get the model and how to estimate the\nparameters? The first question is best answered by the maxim: *all models are\nwrong, some are useful*. In other words, choosing a model depends as much on\nthe\napplication as on the model itself. Think about models as building\ndifferent\ntelescopes to view the sky. No one would ever claim that the\ntelescope generates\nthe sky! It is same with data models. Models give us\nmultiple perspectives on\nthe data that themselves are proxies for some deeper\nunderlying phenomenon.\nSome\ncategories of data may be more commonly studied using certain types of\nmodels,\nbut this is usually very domain-specific and ultimately depends on the\naims of\nthe analysis. In some cases, there may be strong physical reasons\nbehind\nchoosing a model. For example, one could postulate that the model is\nlinear\nwith some noise as in the following:\n\n$$\nY = a X + \\epsilon\n$$\n\n which basically\nsays that you, as the experimenter, dial in some\nvalue for $X$\nand then read off\nsomething directly proportional to $X$ as the\nmeasurement,\n$Y$, plus some\nadditive noise that you attribute to jitter in the\napparatus.\nThen, the next\nstep is to estimate the paramater $a$ in the model,\ngiven some\npostulated claim\nabout the nature of $\\epsilon$. How to compute the\nmodel\nparameters depends on\nthe particular methodology. The two broad rubrics\nare\nparametric and non-\nparametric estimation. In the former, we assume we know\nthe\ndensity function of\nthe data and then try to derive the embedded parameters\nfor\nit. In the latter,\nwe claim only to know that the density function is a\nmember\nof a broad class of\ndensity functions and then use the data\nto characterize a\nmember of that class.\nBroadly speaking, the former consumes\nless data than the\nlatter, because there\nare fewer unknowns to compute from\nthe data.\n\nLet's\nconcentrate on parametric\nestimation for now. The tradition is to denote\nthe\nunknown parameter to be\nestimated as $\\theta$ which is a member of a large\nspace\nof alternates,\n$\\Theta$. To judge between potential $\\theta$ values, we\nneed an\nobjective\nfunction, known as a *risk* function,\n$L(\\theta,\\hat{\\theta})$, where\n$\\hat{\\theta}(\\mathbf{x})$ is an\nestimate for the unknown $\\theta$ that is\nderived from the available\ndata $\\mathbf{x}$. The most common and useful risk\nfunction is the\nsquared error loss,\n\n$$\nL(\\theta,\\hat{\\theta}) =\n(\\theta-\\hat{\\theta})^2\n$$\n\n Although neat, this is not practical because we\nneed to know the\nunknown\n$\\theta$ to compute it. The other problem is because\n$\\hat{\\theta}$ is\na\nfunction of the observed data, it is also a random variable\nwith its own\nprobability density function. This leads to the notion of the\n*expected risk*\nfunction,\n\n$$\nR(\\theta,\\hat{\\theta}) =\n\\mathbb{E}_\\theta(L(\\theta,\\hat{\\theta})) = \\int\nL(\\theta,\\hat{\\theta}(\\mathbf{x})) f(\\mathbf{x};\\theta) d \\mathbf{x}\n$$\n\n In\nother words, given a fixed $\\theta$, integrate over the\nprobability density\nfunction of the data, $f(\\mathbf{x})$, to compute the\nrisk. Plugging in for the\nsquared error loss, we compute the\nmean squared error,\n\n$$\n\\mathbb{E}_\\theta(\\theta-\\hat{\\theta})^2 =\\int (\\theta-\\hat{\\theta})^2\nf(\\mathbf{x};\\theta) d \\mathbf{x}\n$$\n\n This has the important factorization into\nthe *bias*,\n\n$$\n\\texttt{bias} = \\mathbb{E}_\\theta(\\hat{\\theta})-\\theta\n$$\nwith the corresponding variance, $\\mathbb{V}_\\theta(\\hat{\\theta})$ as\nin the\nfollowing *mean squared error* (MSE):\n\n$$\n\\mathbb{E}_\\theta(\\theta-\\hat{\\theta})^2=\n\\texttt{bias}^2+\\mathbb{V}_\\theta(\\hat{\\theta})\n$$\n\n This is an important\ntrade-off that we will return to repeatedly. The\nidea is\nthe bias is nonzero\nwhen the estimator $\\hat{\\theta}$, integrated\nover all\npossible data,\n$f(\\mathbf{x})$, does not equal the underlying target\nparameter\n$\\theta$. In\nsome sense, the estimator misses the target, no matter\nhow much\ndata is used.\nWhen the bias equals zero, the estimated is *unbiased*.\nFor fixed\nMSE, low bias\nimplies high variance and vice-versa. This trade-off\nwas once not\nemphasized and\ninstead much attention was paid to the smallest\nvariance of\nunbiased estimators\n(see Cramer-Rao bounds). In practice,\nunderstanding and\nexploiting the trade-off\nbetween bias and variance and\nreducing the MSE is more\nimportant.\n\nWith all this\nset up, we can now ask how bad can bad get by\nexamining *minimax* risk,\n\n$$\nR_{\\texttt{mmx}} = \\inf_{\\hat{\\theta}} \\sup_\\theta R(\\theta,\\hat{\\theta})\n$$\nwhere the $\\inf$ is take over all estimators. Intuitively, this\nmeans if we\nfound the worst possible $\\theta$ and swept over all possible\nparameter\nestimators $\\hat{\\theta}$, and then took the smallest possible risk\nwe could\nfind, we would have the minimax risk. Thus, an estimator,\n$\\hat{\\theta}_{\\texttt{mmx}}$, is a *minimax estimator* if it achieves this\nfeat,\n\n$$\n\\sup_\\theta R(\\theta,\\hat{\\theta}_{\\texttt{mmx}})\n=\\inf_{\\hat{\\theta}}\n\\sup_\\theta R(\\theta,\\hat{\\theta})\n$$\n\n In other words,\neven in the face of the worst $\\theta$ (i.e., the\n$\\sup_\\theta$),\n$\\hat{\\theta}_{\\texttt{mmx}}$ still achieves the minimax\nrisk. There is a\ngreater theory that revolves around minimax estimators of\nvarious kinds, but\nthis is far beyond our scope here. The main thing to focus\non\nis that under\ncertain technical but easily satisfiable conditions, the\nmaximum\nlikelihood\nestimator is approximately minimax. Maximum likelihood is\nthe subject\nof the\nnext section. Let's get started with the simplest\napplication: coin-\nflipping.\n## Setting up the Coin Flipping Experiment\n\nSuppose we have coin and\nwant to\nestimate the probability of heads ($p$) for\nit. We model the\ndistribution of\nheads and tails as a Bernoulli distribution\nwith the following\nprobability mass\nfunction:\n\n$$\n\\phi(x)= p^x (1-p)^{(1-x)}\n$$\n\n where $x$ is the outcome, *1* for\nheads and *0* for tails. Note that\nmaximum\nlikelihood is a parametric method\nthat requires the specification of a\nparticular model for which we will compute\nembedded parameters. For $n$\nindependent flips, we have the joint density as the\nproduct of $n$ of\nthese\nfunctions as in,\n\n$$\n\\phi(\\mathbf{x})=\\prod_{i=1}^n\np^x_i (1-p)^{(1-x_i)}\n$$\n\n The following is the *likelihood function*,\n\n$$\n\\mathcal{L}(p ; \\mathbf{x})= \\prod_{i=1}^n p^{ x_i }(1-p)^{1-x_i}\n$$\n\n This is\nbasically notation. We have just renamed the\nprevious equation to\nemphasize the\n$p$ parameter, which is what\nwe want to estimate.\n\nThe principle\nof *maximum\nlikelihood* is to maximize the likelihood as the\nfunction of $p$\nafter plugging\nin all of the $x_i$ data. We then call this\nmaximizer $\\hat{p}$\nwhich is a\nfunction of the observed $x_i$ data, and as\nsuch, is a random\nvariable with its\nown distribution. This method therefore\ningests data and an\nassumed model for\nthe probability density, and produces a\nfunction that\nestimates the embedded\nparameter in the assumed probability\ndensity. Thus,\nmaximum likelihood\ngenerates the *functions* of data that we\nneed in order to\nget at the underlying\nparameters of the model. Note that there\nis no limit to\nthe ways we can\nfunctionally manipulate the data we have\ncollected. The maximum\nlikelihood\nprinciple gives us a systematic method for\nconstructing these\nfunctions subject\nto the assumed model. This is a point\nworth emphasizing: the\nmaximum likelihood\nprinciple yields functions as\nsolutions the same way solving\ndifferential\nequations yields functions as\nsolutions. It is very, very much\nharder to produce\na function than to produce a\nvalue as a solution, even with\nthe assumption of a\nconvenient probability\ndensity. Thus, the power of the\nprinciple is that you\ncan construct such\nfunctions subject to the model\nassumptions.\n\n### Simulating the Experiment\n\nWe need the following code to\nsimulate coin flipping.\n\n\n```python\nfrom scipy.stats import bernoulli \np_true=1/2.0 # estimate this!\nfp=bernoulli(p_true) # create bernoulli random variate\nxs = fp.rvs(100) # generate some samples\nprint (xs[:30]) # see first 30 samples\n```\n\n [0 1 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 0 1 1 0 1 0 0 1 1 0 1 0 1]\n\n\nNow, we can write out the likelihood function using Sympy. Note\nthat we give\nthe Sympy variables the `positive=True` attribute upon\nconstruction because this\neases Sympy's internal simplification algorithms.\n\n\n```python\nimport sympy\nx,p,z=sympy.symbols('x p z', positive=True)\nphi=p**x*(1-p)**(1-x) # distribution function\nL=np.prod([phi.subs(x,i) for i in xs]) # likelihood function \nprint (L) # approx 0.5?\n```\n\n p**57*(1 - p)**43\n\n\nNote that, once we plug in the data, the likelihood function is\nsolely a\nfunction of the unknown parameter ($p$ in this case). The following\ncode uses\ncalculus to find the extrema of the likelihood function. Note that\ntaking the\n`log` of $L$ makes the maximization problem tractable but doesn't\nchange the\nextrema.\n\n\n```python\nlogL=sympy.expand_log(sympy.log(L))\nsol,=sympy.solve(sympy.diff(logL,p),p)\nprint (sol)\n```\n\n 57/100\n\n\n**Programming Tip.**\n\nNote that `sol,=sympy.solve` statement includes\na comma\nafter the `sol` variable. This is because the `solve`\nfunction returns a list\ncontaining a single element. Using\nthis assignment unpacks that single element\ninto the `sol` variable\ndirectly. This is another one of the many small\nelegancies of Python.\n\n \n\nThe following code generates\n[Figure](#fig:Maximum_likelihood_10_2).\n\n\n```python\n\nfrom matplotlib.pylab import subplots\nfig,ax=subplots()\nx=np.linspace(0.001,1-.001,100)\nax.plot(x,list(map(sympy.lambdify(p,logL,'numpy'),x)),'k-',lw=3)\nax.plot(sol,logL.subs(p,sol),'o',\n color='gray',ms=15,label='Estimated')\nax.plot(p_true,logL.subs(p,p_true),'s',\n color='k',ms=15,label='Actual')\nax.set_xlabel('$p$',fontsize=18)\nax.set_ylabel('Likelihood',fontsize=18)\nax.set_title('Estimate not equal to true value',fontsize=18)\nax.legend(loc=0)\n```\n\n\n\n\n \n\n\n\n**Programming Tip.**\n\nIn the prior code, we use the `lambdify` function in\n`lambdify(p,logJ,'numpy')` to\ntake a Sympy expression and convert it into a\nNumpy version that is easier to\ncompute. The `lambdify` function has an extra\nargument where you can specify\nthe function space that it should use to convert\nthe expression. In the above\nthis is set to Numpy.\n\n\n\n\n\n
                                        \n\n

                                        Maximum likelihood estimate vs. true\nparameter. Note that the estimate is slightly off from the true value. This is a\nconsequence of the fact that the estimator is a function of the data and lacks\nknowledge of the true underlying value.

                                        \n\n\n\n[Figure](#fig:Maximum_likelihood_10_2) shows that our estimator $\\hat{p}$\n(circle) is not equal to the true value of $p$ (square), despite being\nthe\nmaximum of the likelihood function. This may sound disturbing, but keep in\nmind\nthis estimate is a function of the random data; and since that data can\nchange,\nthe ultimate estimate can likewise change. \nRemember that the estimator is a\n*function* of the data and is thus also a\n*random variable*, just like the data\nis. This means it has its own probability\ndistribution with corresponding mean\nand variance. So, what we are observing is\na consequence of that variance.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                        \n\n

                                        Histogram of\nmaximum likelihood estimates. The title shows the estimated mean and standard\ndeviation of the samples.

                                        \n\n\n\n[Figure](#fig:Maximum_likelihood_30_2) shows what happens when you run many\nthousands of coin experiments and compute the maximum likelihood\nestimate for\neach experiment, given a particular number of samples \nper experiment. This\nsimulation gives us a histogram of the maximum likelihood\nestimates, which is an\napproximation of the probability distribution of the\n$\\hat{p}$ estimator itself.\nThis figure shows that the sample mean\nof the estimator ($\\mu = \\frac{1}{n}\\sum\n\\hat{p}_i$) is pretty close to the\ntrue value, but looks can be deceiving. The\nonly way to know for sure is to\ncheck if the estimator is unbiased, namely, if\n$$\n\\mathbb{E}(\\hat{p}) = p\n$$\n\n Because this problem is simple, we can solve for\nthis in general\nnoting that\nthe terms above are either $p$, if $x_i=1$ or $1-p$\nif $x_i=0$.\nThis means that\nwe can write\n\n$$\n\\mathcal{L}(p\\vert \\mathbf{x})=\np^{\\sum_{i=1}^n x_i}(1-p)^{n-\\sum_{i=1}^n\nx_i}\n$$\n\n with corresponding logarithm\nas\n\n$$\nJ=\\log(\\mathcal{L}(p\\vert \\mathbf{x})) = \\log(p) \\sum_{i=1}^n x_i +\n\\log(1-p) \\left(n-\\sum_{i=1}^n x_i\\right)\n$$\n\n Taking the derivative of this\ngives:\n\n$$\n\\frac{dJ}{dp} = \\frac{1}{p}\\sum_{i=1}^n x_i + \\frac{(n-\\sum_{i=1}^n\nx_i)}{p-1}\n$$\n\n and solving this for $p$ leads to\n\n$$\n\\hat{p} = \\frac{1}{ n}\n\\sum_{i=1}^n x_i\n$$\n\nThis is our *estimator* for $p$. Up until now, we have been\nusing Sympy to\nsolve\nfor this based on the data $x_i$ but now that we have it\nanalytically we\ndon't\nhave to solve for it each time. To check if this estimator\nis biased, we\ncompute\nits expectation:\n\n$$\n\\mathbb{E}\\left(\\hat{p}\\right)\n=\\frac{1}{n}\\sum_i^n \\mathbb{E}(x_i) =\n\\frac{1}{n} n \\mathbb{E}(x_i)\n$$\n\n by\nlinearity of the expectation and where\n\n$$\n\\mathbb{E}(x_i) = p\n$$\n\n Therefore,\n$$\n\\mathbb{E}\\left(\\hat{p}\\right) =p\n$$\n\n This means that the estimator is\n*unbiased*. Similarly,\n\n$$\n\\mathbb{E}\\left(\\hat{p}^2\\right) = \\frac{1}{n^2}\n\\mathbb{E}\\left[\\left(\n\\sum_{i=1}^n x_i \\right)^2 \\right]\n$$\n\n and where\n\n$$\n\\mathbb{E}\\left(x_i^2\\right) =p\n$$\n\n and by the independence assumption,\n\n$$\n\\mathbb{E}\\left(x_i x_j\\right) =\\mathbb{E}(x_i)\\mathbb{E}(x_j) =p^2\n$$\n\n Thus,\n$$\n\\mathbb{E}\\left(\\hat{p}^2\\right) =\\left(\\frac{1}{n^2}\\right) n \\left[\np+(n-1)p^2 \\right]\n$$\n\n So, the variance of the estimator, $\\hat{p}$, is the\nfollowing:\n\n$$\n\\mathbb{V}(\\hat{p}) = \\mathbb{E}\\left(\\hat{p}^2\\right)-\n\\mathbb{E}\\left(\\hat{p}\\right)^2 = \\frac{p(1-p)}{n}\n$$\n\n Note that the $n$ in\nthe denominator means that the variance\nasymptotically\ngoes to zero as $n$\nincreases (i.e., we consider more and\nmore samples). This is\ngood news because\nit means that more and\nmore coin flips lead to a better\nestimate of the\nunderlying $p$.\n\nUnfortunately, this formula for the variance is\npractically\nuseless because we\nneed $p$ to compute it and $p$ is the parameter\nwe are trying\nto estimate in\nthe first place! However, this is where the *plug-\nin* principle\n[^invariance-property] \nsaves the day. It turns out in this\nsituation, you can\nsimply substitute the maximum likelihood estimator,\n$\\hat{p}$, for the $p$ in\nthe above equation to obtain the asymptotic variance\nfor $\\mathbb{V}(\\hat{p})$.\nThe fact that this works is guaranteed by the\nasymptotic theory of maximum\nlikelihood estimators.\n\n[^invariance-property]:\nThis is also known as the\n*invariance property*\nof maximum likelihood\nestimators. It basically states that\nthe \nmaximum likelihood estimator of any\nfunction, say, $h(\\theta)$, is\nthe same\n$h$ with the maximum likelihood\nestimator for $\\theta$ substituted\nin for\n$\\theta$; namely, $h(\\theta_{ML})$.\nNevertheless, looking at\n$\\mathbb{V}(\\hat{p})^2$, we can immediately notice\nthat\nif $p=0$, then there is\nno estimator variance because the outcomes are\nguaranteed to be tails. Also, for\nany $n$, the maximum of this variance\nhappens\nat $p=1/2$. This is our worst\ncase scenario and the only way to\ncompensate is\nwith larger $n$.\n\nAll we have\ncomputed is the mean and variance of the\nestimator. In general,\nthis is\ninsufficient to characterize the underlying\nprobability density of\n$\\hat{p}$,\nexcept if we somehow knew that $\\hat{p}$ were\nnormally distributed.\nThis is\nwhere the powerful *Central Limit Theorem* we\ndiscussed in the section\n[ch:stats:sec:limit](#ch:stats:sec:limit) comes in. The\nform of the estimator,\nwhich is just a\nsample mean, implies that we can apply\nthis theorem and conclude\nthat $\\hat{p}$\nis asymptotically normally distributed.\nHowever, it doesn't\nquantify how many\nsamples $n$ we need. In our simulation\nthis is no problem\nbecause we can\ngenerate as much data as we like, but in the\nreal world, with a\ncostly\nexperiment, each sample may be precious [^edgeworth].\nIn the following,\nwe won't apply the Central Limit Theorem and instead proceed\nanalytically.\n[^edgeworth]: It turns out that the central limit theorem\naugmented with an\nEdgeworth expansion tells us that convergence is regulated by\nthe skewness\nof\nthe distribution\n[[feller1950introduction]](#feller1950introduction). In other\nwords, the \nmore\nsymmetric the distribution, the faster it converges to the\nnormal\ndistribution\naccording to the central limit theorem.\n\n### Probability Density for the Estimator\n\nTo write out the full density for $\\hat{p}$, we first\nhave to ask\nwhat is\nthe probability that the estimator will equal a specific\nvalue and the\ntally up\nall the ways that could happen with their corresponding\nprobabilities.\nFor\nexample, what is the probability that\n\n$$\n\\hat{p} =\n\\frac{1}{n}\\sum_{i=1}^n x_i = 0\n$$\n\n This can only happen one way: when $x_i=0\n\\hspace{0.5em} \\forall i$. The\nprobability of this happening can be computed\nfrom the density\n\n$$\nf(\\mathbf{x},p)= \\prod_{i=1}^n \\left(p^{x_i} (1-p)^{1-x_i}\n\\right)\n$$\n\n$$\nf\\left(\\sum_{i=1}^n x_i = 0,p\\right)= \\left(1-p\\right)^n\n$$\nLikewise, if $\\lbrace x_i \\rbrace$ has only one nonzero element, then\n\n$$\nf\\left(\\sum_{i=1}^n x_i = 1,p\\right)= n p \\prod_{i=1}^{n-1} \\left(1-p\\right)\n$$\nwhere the $n$ comes from the $n$ ways to pick one element\nfrom the $n$ elements\n$x_i$. Continuing this way, we can construct the\nentire density as\n\n$$\nf\\left(\\sum_{i=1}^n x_i = k,p\\right)= \\binom{n}{k} p^k (1-p)^{n-k}\n$$\n\n where\nthe first term on the right is the binomial coefficient of $n$ things\ntaken $k$\nat a time. This is the binomial distribution and it's not the\ndensity\nfor\n$\\hat{p}$, but rather for $n\\hat{p}$. We'll leave this as-is\nbecause it's\neasier\nto work with below. We just have to remember to keep\ntrack of the $n$\nfactor.\n**Confidence Intervals**\n\nNow that we have the full density for\n$\\hat{p}$, we\nare ready to ask some\nmeaningful questions. For example, what is\nthe probability\nthe estimator is within\n$\\epsilon$ fraction of the true value\nof $p$?\n\n$$\n\\mathbb{P}\\left( \\vert \\hat{p}-p \\vert \\le \\epsilon p \\right)\n$$\n\n More\nconcretely, we want to know how often the\nestimated $\\hat{p}$ is trapped\nwithin\n$\\epsilon$ of the actual value. That is,\nsuppose we ran the experiment\n1000\ntimes to generate 1000 different estimates\nof $\\hat{p}$. What percentage of\nthe\n1000 so-computed values are trapped within\n$\\epsilon$ of the underlying\nvalue.\nRewriting the above equation as the\nfollowing,\n\n$$\n\\mathbb{P}\\left(p-\\epsilon p\n< \\hat{p} < p + \\epsilon p \\right) =\n\\mathbb{P}\\left( n p - n \\epsilon p <\n\\sum_{i=1}^n x_i < n p + n \\epsilon p\n\\right)\n$$\n\n Let's plug in some live\nnumbers here for our worst case\nscenario (i.e., highest\nvariance scenario) where\n$p=1/2$. Then, if\n$\\epsilon = 1/100$, we have\n\n$$\n\\mathbb{P}\\left( \\frac{99\nn}{100} < \\sum_{i=1}^n x_i < \\frac{101 n}{100}\n\\right)\n$$\n\n Since the sum in\ninteger-valued, we need $n> 100$ to even compute this.\nThus,\nif $n=101$ we have,\n$$\n\\begin{eqnarray*}\n\\mathbb{P}\\left(\\frac{9999}{200} < \\sum_{i=1}^{101} x_i <\n\\frac{10201}{200} \\right) = f\\left(\\sum_{i=1}^{101} x_i = 50,p\\right) & \\ldots\n\\\\\\\n= \\binom{101}{50} (1/2)^{50} (1-1/2)^{101-50} & = & 0.079\n\\end{eqnarray*}\n$$\n\n This means that in the worst-case scenario for $p=1/2$, given $n=101$\ntrials,\nwe will only get within 1\\% of the actual $p=1/2$ about 8\\% of the\ntime.\nIf you\nfeel disappointed, it is because you've been paying attention.\nWhat if\nthe coin\nwas really heavy and it was hard work to repeat this 101 times?\n\nLet's\ncome at\nthis another way: given I could only flip the coin 100\ntimes, how close\ncould I\ncome to the true underlying value with high\nprobability (say, 95\\%)? In\nthis\ncase, instead of picking a value for\n$\\epsilon$, we are solving for\n$\\epsilon$.\nPlugging in gives,\n\n$$\n\\mathbb{P}\\left(50 - 50\\epsilon <\n\\sum_{i=1}^{100} x_i < 50 + 50 \\epsilon\n\\right) = 0.95\n$$\n\n which we have to\nsolve for $\\epsilon$. Fortunately, all the tools we\nneed to\nsolve for this are\nalready in Scipy.\n\n\n```python\nfrom scipy.stats import binom\n# n=100, p = 0.5, distribution of the estimator phat\nb=binom(100,.5) \n# symmetric sum the probability around the mean\ng = lambda i:b.pmf(np.arange(-i,i)+50).sum() \nprint (g(10)) # approx 0.95\n```\n\n 0.9539559330706295\n\n\n\n```python\n%matplotlib inline\n\nfrom matplotlib.pylab import subplots, arange\nfig,ax= subplots()\nfig.set_size_inches((10,5))\n# here is the density of the sum of x_i\n_=ax.stem(arange(0,101),b.pmf(arange(0,101)),\n linefmt='k-', markerfmt='ko',use_line_collection=True) \n_=ax.vlines( [50+10,50-10],0 ,ax.get_ylim()[1] ,color='k',lw=3.)\n_=ax.axis(xmin=30,xmax=70)\n_=ax.tick_params(labelsize=18)\nfig.savefig('fig-statistics/Maximum_likelihood_20_2.png')\nfig.tight_layout()\n```\n\n\n\n
                                        \n\n

                                        Probability\nmass function for $\\hat{p}$. The two vertical lines form the confidence\ninterval.

                                        \n\n\n\n\n The two vertical lines in\n[Figure](#fig:Maximum_likelihood_20_2)\nshow how far out from the mean we have to\ngo to accumulate 95\\% of the\nprobability. Now, we can solve this as\n\n$$\n50+50\\epsilon=60\n$$\n\n which makes $\\epsilon=1/5$ or 20\\%. So, flipping 100 times\nmeans I can\nonly get\nwithin 20\\% of the real $p$ 95\\% of the time in the worst\ncase\nscenario (i.e.,\n$p=1/2$). The following code verifies the situation.\n\n\n```python\nfrom scipy.stats import bernoulli \nb=bernoulli(0.5) # coin distribution\nxs = b.rvs(100) # flip it 100 times\nphat = np.mean(xs) # estimated p\nprint (abs(phat-0.5) < 0.5*0.20) # make it w/in interval?\n```\n\n True\n\n\nLet's keep doing this and see if we can get within this interval 95\\% of\nthe\ntime.\n\n\n```python\nout=[]\nb=bernoulli(0.5) # coin distribution\nfor i in range(500): # number of tries\n xs = b.rvs(100) # flip it 100 times\n phat = np.mean(xs) # estimated p\n out.append(abs(phat-0.5) < 0.5*0.20 ) # within 20% ?\n\n# percentage of tries w/in 20% interval\nprint (100*np.mean(out))\n```\n\n 97.39999999999999\n\n\nWell, that seems to work! Now we have a way to get at the quality of\nthe\nestimator, $\\hat{p}$.\n\n**Maximum Likelihood Estimator Without Calculus**\n\nThe\nprior example showed how we can use calculus to compute the maximum\nlikelihood\nestimator. It's important to emphasize that the maximum likelihood\nprinciple\ndoes *not* depend on calculus and extends to more general situations\nwhere\ncalculus is impossible. For example, let $X$ be uniformly distributed in\nthe\ninterval $[0,\\theta]$. Given $n$ measurements of $X$, the likelihood\nfunction\nis the following:\n\n$$\nL(\\theta) = \\prod_{i=1}^n \\frac{1}{\\theta} =\n\\frac{1}{\\theta^n}\n$$\n\n where each $x_i \\in [0,\\theta]$. Note that the slope of\nthis function\nis not\nzero anywhere so the usual calculus approach is not going\nto work here.\nBecause\nthe likelihood is the product of the individual uniform\ndensities, if\nany of the\n$x_i$ values were outside of the proposed $[0,\\theta]$\ninterval,\nthen the\nlikelihood would go to zero, because the uniform density is\nzero\noutside of the\n$[0,\\theta]$. This is no good for maximization. Thus,\nobserving\nthat the\nlikelihood function is strictly decreasing with increasing\n$\\theta$,\nwe conclude\nthat the value for $\\theta$ that maximizes the likelihood\nis the\nmaximum of the\n$x_i$ values. To summarize, the maximum likelihood\nestimator is\nthe following:\n\n$$\n\\theta_{ML} = \\max_i x_i\n$$\n\n As always, we want\nthe distribution of this estimator to judge its\nperformance.\nIn this case, this\nis pretty straightforward. The cumulative\ndensity function\nfor the $\\max$\nfunction is the following:\n\n$$\n\\mathbb{P} \\left( \\hat{\\theta}_{ML} < v \\right) =\n\\mathbb{P}( x_0 \\leq v\n\\wedge x_1 \\leq v \\ldots \\wedge x_n \\leq v)\n$$\n\n and\nsince all the $x_i$ are uniformly distributed in $[0,\\theta]$, we have\n\n$$\n\\mathbb{P} \\left( \\hat{\\theta}_{ML} < v \\right) =\n\\left(\\frac{v}{\\theta}\\right)^n\n$$\n\n So, the probability density function is\nthen,\n\n$$\nf_{\\hat{\\theta}_{ML}}(\\theta_{ML}) = n \\theta_{ML}^{ n-1 } \\theta^{\n-n }\n$$\n\n Then, we can compute the $\\mathbb{E}(\\theta_{ML}) = (\\theta n)/(n+1)$\nwith\ncorresponding variance as $\\mathbb{V}(\\theta_{ML}) = (\\theta^2\nn)/(n+1)^2/(n+2)$.\n\nFor a quick sanity check, we can write the following\nsimulation for $\\theta =1$\nas in the following:\n\n\n```python\nfrom scipy import stats\nrv = stats.uniform(0,1) # define uniform random variable\nmle=rv.rvs((100,500)).max(0) # max along row-dimension\nprint (np.mean(mle)) # approx n/(n+1) = 100/101 ~= 0.99\nprint (np.var(mle)) #approx n/(n+1)**2/(n+2) ~= 9.61E-5\n9.95762009884e-05\n```\n\n 0.9902508350190666\n 9.414736602783561e-05\n\n\n\n\n\n 9.95762009884e-05\n\n\n\n**Programming Tip.**\n\nThe `max(0)` suffix on for the `mle` computation takes\nthe\nmaximum of the so-computed array along the row (`axis=0`)\ndimension.\n\n\n\n You can\nalso plot `hist(mle)` to see the histogram of the simulated\nmaximum likelihood\nestimates and match it up against the probability density\nfunction we derived\nabove. \n\n\nIn this section, we explored the concept of maximum\nlikelihood\nestimation using a coin flipping experiment both analytically and\nnumerically\nwith the scientific Python stack. We also explored the case when\ncalculus is not\nworkable for maximum likelihood estimation. There are two key\npoints to\nremember. First, maximum likelihood estimation produces a function of\nthe data\nthat is itself a random variable, with its own probability\ndistribution. We can\nget at the quality of the so-derived estimators by\nexamining the confidence\nintervals around the estimated values using the\nprobability distributions\nassociated with the estimators themselves. \nSecond, maximum likelihood\nestimation applies even in situations \nwhere using basic calculus is not\napplicable [[wasserman2004all]](#wasserman2004all).\n\n\n## Delta Method\n
                                        \n\nSometimes we want to characterize the distribution\nof a *function* of a random\nvariable. In order to extend and generalize the\nCentral Limit Theorem in this\nway, we need the Taylor series expansion. Recall\nthat the Taylor series\nexpansion is an approximation of a function of the\nfollowing form,\n\n$$\nT_r(x) =\\sum_{i=0}^r \\frac{g^{(i)}(a)}{i!}(x-a)^i\n$$\n\n this\nbasically says that a function $g$ can be adequately\napproximated about a\npoint\n$a$ using a polynomial based on its derivatives\nevaluated at $a$. Before\nwe\nstate the general theorem, let's examine\nan example to understand how the\nmechanics work.\n\n**Example.** Suppose that $X$ is a random variable with\n$\\mathbb{E}(X)=\\mu\\neq 0$. Furthermore, supposedly have a suitable\nfunction $g$\nand we want the distribution of $g(X)$. Applying the\nTaylor series expansion, we\nobtain the following,\n\n$$\ng(X) \\approx g(\\mu)+ g^{\\prime}(\\mu)(X-\\mu)\n$$\n\n If we\nuse $g(X)$ as an estimator for $g(\\mu)$, then we can say that\nwe\napproximately\nhave the following\n\n$$\n\\begin{align*}\n\\mathbb{E}(g(X)) &=g(\\mu) \\\\\\\n\\mathbb{V}(g(X))\n&=(g^{\\prime}(\\mu))^2 \\mathbb{V}(X) \\\\\\\n\\end{align*}\n$$\nConcretely, suppose we want to estimate the odds, $\\frac{p}{1-p}$.\nFor example,\nif $p=2/3$, then we say that the odds is `2:1` meaning that the\nodds of the one\noutcome are twice as likely as the odds of the other outcome.\nThus, we have\n$g(p)=\\frac{p}{1-p}$ and we want to find\n$\\mathbb{V}(g(\\hat{p}))$. In our coin-\nflipping problem, we have the\nestimator $\\hat{p}=\\frac{1}{n}\\sum X_k$ from the\nBernoulli-distributed data\n$X_k$ individual coin-flips. Thus,\n\n$$\n\\begin{align*}\n\\mathbb{E}(\\hat{p}) &= p \\\\\\\n\\mathbb{V}(\\hat{p}) &=\n\\frac{p(1-p)}{n} \\\\\\\n\\end{align*}\n$$\n\n Now, $g^\\prime(p)=1/(1-p)^2$, so we have,\n\n$$\n\\begin{align*}\n\\mathbb{V}(g(\\hat{p}))&=(g^\\prime(p))^2 \\mathbb{V}(\\hat{p})\n\\\\\\\n&=\\left(\\frac{1}{(1-p)^2}\\right)^2 \\frac{p(1-p)}{n}\n\\\\\\\n &=\n\\frac{p}{n(1-p)^3} \\\\\\\n\\end{align*}\n$$\n\n which is an approximation of the\nvariance of the estimator\n$g(\\hat{p})$. Let's\nsimulate this and see how it\nagrees.\n\n\n```python\nfrom scipy import stats\n# compute MLE estimates \nd=stats.bernoulli(0.1).rvs((10,5000)).mean(0)\n# avoid divide-by-zero\nd=d[np.logical_not(np.isclose(d,1))]\n# compute odds ratio\nodds = d/(1-d)\nprint ('odds ratio=',np.mean(odds),'var=',np.var(odds))\n```\n\n odds ratio= 0.12363809523809527 var= 0.017607461164021166\n\n\nThe first number above is the mean of the simulated odds\nratio and the second\nis\nthe variance of the estimate. According to\nthe variance estimate above, we\nhave\n$\\mathbb{V}(g(1/10))\\approx\n0.0137$, which is not too bad for this\napproximation. Recall we want\nto estimate the odds from $\\hat{p}$. The code\nabove takes $5000$\nestimates of the $\\hat{p}$ to estimate $\\mathbb{V}(g)$. The\nodds ratio\nfor $p=1/10$ is $1/9\\approx 0.111$.\n\n**Programming Tip.**\n\nThe code\nabove uses the `np.isclose` function to identify the ones from\nthe simulation\nand the `np.logical_not` removes these elements from the\ndata because the odds\nratio has a zero in the denominator\nfor these values.\n\n\n\nLet's try this again\nwith a probability of heads of `0.5` instead of\n`0.3`.\n\n\n```python\nfrom scipy import stats\nd=stats.bernoulli(.5).rvs((10,5000)).mean(0)\nd=d[np.logical_not(np.isclose(d,1))]\nprint( 'odds ratio=',np.mean(d),'var=',np.var(d))\n```\n\n odds ratio= 0.4984584584584585 var= 0.024323949976002027\n\n\nThe odds ratio is this case is equal to one, which\nis not close to what was\nreported. According to our\napproximation, we should have $\\mathbb{V}(g)=0.4$,\nwhich does not\nlook like what our simulation just reported. This is\nbecause the\napproximation is best when the odds ratio is\nnearly linear and worse otherwise\n(see [Figure](#fig:Maximum_likelihood_0001)).\n\n\n\n
                                        \n\n

                                        The odds\nratio is close to linear for small values but becomes unbounded as $p$\napproaches one. The delta method is more effective for small underlying values\nof $p$, where the linear approximation is better.

                                        \n\n\n\n", "meta": {"hexsha": "97479f6c48a950e3feba2ce014008b1ebe9be788", "size": 238819, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "chapter/statistics/Maximum_likelihood.ipynb", "max_stars_repo_name": "derakding/Python-for-Probability-Statistics-and-Machine-Learning-2E", "max_stars_repo_head_hexsha": "9d12a298d43ae285d9549a79bb5544cf0a9b7516", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 224, "max_stars_repo_stars_event_min_datetime": "2019-05-07T08:56:01.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-25T15:50:41.000Z", "max_issues_repo_path": "chapter/statistics/Maximum_likelihood.ipynb", "max_issues_repo_name": "derakding/Python-for-Probability-Statistics-and-Machine-Learning-2E", "max_issues_repo_head_hexsha": "9d12a298d43ae285d9549a79bb5544cf0a9b7516", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 9, "max_issues_repo_issues_event_min_datetime": "2019-08-27T12:57:17.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-21T15:45:13.000Z", "max_forks_repo_path": "chapter/statistics/Maximum_likelihood.ipynb", "max_forks_repo_name": "derakding/Python-for-Probability-Statistics-and-Machine-Learning-2E", "max_forks_repo_head_hexsha": "9d12a298d43ae285d9549a79bb5544cf0a9b7516", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 73, "max_forks_repo_forks_event_min_datetime": "2019-05-25T07:15:47.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-07T00:22:37.000Z", "avg_line_length": 155.887075718, "max_line_length": 176652, "alphanum_fraction": 0.8830453188, "converted": true, "num_tokens": 9685, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.7520125848754472, "lm_q1q2_score": 0.4198689262158841}} {"text": "# \u8f68\u9053\u65cb\u8f6c MP2 \u65b9\u6cd5 (OO-MP2) \u7b80\u5355\u7406\u89e3\n\n> \u521b\u5efa\u65e5\u671f\uff1a2021-01-09\n\n\u8fd9\u7bc7\u6587\u6863\u4f1a\u5c1d\u8bd5\u7b80\u5355\u4ecb\u7ecd\u8f68\u9053\u65cb\u8f6c MP2 \u65b9\u6cd5 (Orbital-Optimized Second-Order M\u00f8ller\u2013Plesset Perturbation, OO-MP2 or OMP2) \u7684\u57fa\u7840\u6982\u5ff5\u4e0e PySCF \u4e0a\u7684\u7a0b\u5e8f\u5b9e\u73b0\u548c\u7406\u89e3\u3002\n\n\u8fd9\u7bc7\u6587\u6863\u7684\u7f16\u5199\u5e76\u6ca1\u6709\u7ffb\u9605\u5f88\u591a\u6587\u732e\uff0c\u5e76\u4f5c\u6d4b\u8bc4\u4e0a\u7684\u8ba4\u8bc6\u3002\u4e3a\u6570\u4e0d\u591a\u7684\u6587\u732e\u4e0e\u53c2\u8003\u8d44\u6599\u662f\n\n> Sun, Chan, et al. [^Sun-Chan.JCP.2020] (PySCF \u8fdb\u5c55\u6587\u7ae0)\n> \n> PySCF \u5e76\u6ca1\u6709\u4e00\u4e2a\u5b8c\u6574\u6216\u72ec\u7acb\u7684 OO-MP2 \u6a21\u5757\u3002\u5b9e\u73b0 OO-MP2 \u53ef\u4ee5\u901a\u8fc7\u4eff CASSCF \u7684\u65b9\u5f0f\u5b9e\u73b0\u3002\u4e4b\u540e\u4f7f\u7528\u5230\u7684 `MP2AsFCISolver` class \u5c31\u662f\u76f4\u63a5\u4ece\u8be5\u6587\u7ae0\u4e2d\u622a\u53d6\u7684\u6f14\u793a\u4ee3\u7801\u3002\n\n> Psi4NumPy \u6f14\u793a\u6587\u6863 [10a_orbital-optimized-mp2.ipynb](https://github.com/psi4/psi4numpy/blob/master/Tutorials/10_Orbital_Optimized_Methods/10a_orbital-optimized-mp2.ipynb)\n>\n> \u8fd9\u662f\u4e00\u4efd\u6bd4\u8f83\u4e0d\u9519\u7684\u57fa\u4e8e Psi4 \u7684\u7a0b\u5e8f\u7b80\u8981\u6587\u6863\uff0c\u4f7f\u7528\u7684\u7b97\u6cd5\u4e0e\u6280\u5de7\u4e5f\u4e0d\u590d\u6742\u3002\n\n\u9700\u8981\u6307\u51fa\uff0c\u8fd9\u91cc\u7684 OO-MP2 \u7a0b\u5e8f\u5b9e\u73b0\u5b8c\u5168\u662f\u57fa\u4e8e Post-HF \u7684\u95ed\u58f3\u5c42\u3001\u65e0\u51bb\u7ed3\u8f68\u9053 MP2 \u5b9e\u73b0\u7684\u3002\u66f4\u590d\u6742\u7684\u5f00\u58f3\u5c42\u3001\u51bb\u7ed3\u8f68\u9053\u3001\u53cc\u6742\u5316\u6cdb\u51fd\u65b9\u6cd5\uff0c\u90fd\u4e0d\u4e88\u4ee5\u8003\u8651\u3002\n\n\n```python\nimport numpy as np\nimport scipy\nfrom pyscf import gto, mcscf, fci, mp, scf\nfrom functools import partial\n\nnp.random.seed(0)\nnp.einsum = partial(np.einsum, optimize=True)\nnp.set_printoptions(precision=4, linewidth=150, suppress=True)\n```\n\n\u8fd9\u7bc7\u6587\u6863\u7684\u7a0b\u5e8f\u7406\u89e3\u90e8\u5206\uff0c\u6211\u4eec\u90fd\u4f1a\u4f7f\u7528\u4e0b\u8ff0\u6c34\u5206\u5b50\u3002\u4f46\u6587\u6863\u672b\u5c3e\uff0c\u6211\u4eec\u4f1a\u7528\u6c22\u5206\u5b50\u7684\u4f8b\u5b50\uff0c\u8bf4\u660e OO-MP2 \u7684\u80fd\u91cf\u672a\u5fc5\u8981\u6bd4 MP2 \u80fd\u91cf\u8981\u4f4e\u3002\n\n\n```python\nmol = gto.Mole()\nmol.atom = \"\"\"\nO 0. 0. 0.\nH 0. 0. 1.\nH 0. 1. 0.\n\"\"\"\nmol.basis = \"6-31G\"\nmol.verbose = 0\nmol.build()\n```\n\n\n\n\n \n\n\n\n## PySCF \u7a0b\u5e8f\u5b9e\u73b0\uff1a\u9ad8\u6548\u65b9\u5f0f\n\n\u8fd9\u6bb5\u7a0b\u5e8f `MP2AsFCISolver` class \u662f\u76f4\u63a5\u4ece Sun \u7684 JCP \u6587\u7ae0\u622a\u53d6\u7684\u3002\u901a\u8fc7\u5728 CASSCF \u4e2d\uff0c\u5c06\u6d3b\u6027\u7a7a\u95f4\u66f4\u6539\u4e3a\u5168\u8f68\u9053\u3001\u66f4\u6539\u7ea6\u5316\u5bc6\u5ea6\u77e9\u9635 (1-RDM, 2-RDM) \u7684\u751f\u6210\u65b9\u5f0f\u4e3a MP2 \u7684\u7ea6\u5316\u5bc6\u5ea6\u77e9\u9635\u3001\u5e76\u4e14\u5141\u8bb8\u6d3b\u6027\u7a7a\u95f4\u7684\u8f68\u9053\u65cb\u8f6c\uff0c\u5c31\u53ef\u4ee5\u5b9e\u73b0 OO-MP2\u3002\n\n\n```python\nclass MP2AsFCISolver:\n def kernel(self, h1, h2, norb, nelec, ci0=None, ecore=0, **kwargs):\n # Kernel takes the set of integrals from the current set of orbitals\n fakemol = gto.M(verbose=0)\n fakemol.nelectron = sum(nelec)\n fake_hf = fakemol.RHF()\n fake_hf._eri = h2\n fake_hf.get_hcore = lambda *args: h1\n fake_hf.get_ovlp = lambda *args: np.eye(norb)\n \n # Build an SCF object fake_hf without SCF iterations to perform MP2\n fake_hf.mo_coeff = np.eye(norb)\n fake_hf.mo_occ = np.zeros(norb)\n fake_hf.mo_occ[:fakemol.nelectron//2] = 2\n self.mp2 = fake_hf.MP2().run()\n return self.mp2.e_tot + ecore, self.mp2.t2\n \n def make_rdm12(self, t2, norb, nelec):\n dm1 = self.mp2.make_rdm1(t2)\n dm2 = self.mp2.make_rdm2(t2)\n return dm1, dm2\n```\n\n`mf_rhf` \u4e3a RHF \u5b9e\u4f8b\uff1a\n\n\n```python\nmf_rhf = mol.RHF().run()\nmf_rhf.e_tot\n```\n\n\n\n\n -75.9697009626036\n\n\n\n`mf_mp2` \u4e3a MP2 \u5b9e\u4f8b\uff1a\n\n\n```python\nmf_mp2 = mp.MP2(mf_rhf).run()\nmf_mp2.e_tot\n```\n\n\n\n\n -76.1040356515777\n\n\n\n\n```python\nmf_mp2.e_corr\n```\n\n\n\n\n -0.13433468897410067\n\n\n\n`mf_cas` \u5728\u8fd9\u91cc\u662f\u6307 OO-MP2 \u5b9e\u4f8b\uff1a\n\n\n```python\nmf_cas = mcscf.CASSCF(mf_rhf, mol.nao, mol.nelectron)\nmf_cas.fcisolver = MP2AsFCISolver()\nmf_cas.internal_rotation = True\ncas_result = mf_cas.kernel()\ncas_result[0]\n```\n\n\n\n\n -76.10510419427318\n\n\n\n## PySCF \u7a0b\u5e8f\u5b9e\u73b0\uff1a\u5927\u4f53\u601d\u8def\u62c6\u89e3\n\n\u5728\u8fd9\u4e00\u6bb5\u4e2d\uff0c\u6211\u4eec\u4e0d\u4f1a\u4f7f\u7528 PySCF \u7684 `CASSCF` class\uff0c\u800c\u662f\u4ece RHF \u4e0e MP2 \u7684\u7ed3\u679c\uff0c\u4e86\u89e3 OO-MP2 \u7684\u5927\u4f53\u601d\u8def\u3002\n\n\u4ece\u7ed3\u679c\u4e0a\uff0c\u8fd9\u79cd\u5b9e\u73b0\u65b9\u5f0f\u4e0e PySCF \u4f1a\u76f8\u540c\u3002\u4f46 PySCF \u7684 `CASSCF` class \u4e00\u822c\u4f1a\u4f7f\u7528\u4e8c\u9636\u65b9\u6cd5 (\u5373\u4f7f\u7528 Orbital Hessian) \u52a0\u901f\u6536\u655b\uff0c\u800c\u6211\u4eec\u8fd9\u91cc\u53ea\u4f7f\u7528\u4e00\u9636\u68af\u5ea6\u4e0b\u964d\u65b9\u6cd5 (Orbital Gradient) \u8fdb\u884c\u6536\u655b\uff1b\u4e00\u9636\u6536\u655b\u65b9\u6cd5\u663e\u7136\u4f1a\u6162\u4e00\u4e9b\uff0c\u4f46\u516c\u5f0f\u4e0e\u7a0b\u5e8f\u4f1a\u7b80\u5355\u4e00\u4e9b\u3002\n\n\u9996\u5148\u5bf9\u4e00\u4e9b\u57fa\u7840\u53d8\u91cf\u4f5c\u58f0\u660e\uff1a\n\n- `nocc` $n_\\mathrm{occ}$ \u5360\u636e\u8f68\u9053\u6570\uff0c`nvir` $n_\\mathrm{vir}$ \u975e\u5360\u8f68\u9053\u6570\uff1b\n\n- `nmo` $n_\\mathrm{MO}$ \u5206\u5b50\u8f68\u9053\u6570\uff0c\u4e00\u822c\u4e0e\u539f\u5b50\u8f68\u9053\u6570\u76f8\u7b49\uff1b\n\n- `so` $[0:n_\\mathrm{occ}]$ \u5360\u636e\u8f68\u9053\u5206\u5272 (slice)\uff0c`sv` $[n_\\mathrm{occ}:n_\\mathrm{MO}]$ \u975e\u5360\u8f68\u9053\u5206\u5272 (slice)\uff1b\n\n- `mo_occ` PySCF \u4e2d\u7528\u4e8e\u8868\u793a\u8f68\u9053\u5360\u636e\u6570\u7684\u53d8\u91cf\u3002\n\n\n```python\nnocc, nmo = mol.nelec[0], mol.nao\nnvir = nmo - nocc\nso, sv = slice(0, nocc), slice(nocc, nmo)\nmo_occ = mf_rhf.mo_occ\n```\n\nOO-MP2 \u7684\u5927\u4f53\u8fc7\u7a0b\u53ef\u4ee5\u62c6\u5206\u4e3a\u5982\u4e0b\u5faa\u73af\uff1a\n\n1. \u4ee3\u5165\u5206\u5b50\u8f68\u9053\u7cfb\u6570 $C_{\\mu i}$\uff0c\u5f97\u5230\u8be5\u7cfb\u6570\u4e0b MP2 \u7684\u6fc0\u53d1\u5f20\u91cf $t_{ij}^{ab}$\uff1b\n\n2. \u8fdb\u800c\u751f\u6210\u8be5\u60c5\u5f62\u4e0b\u7684 1-RDM $\\gamma_{pq}$ \u4e0e 2-RDM $\\Gamma_{pr}^{qs}$\uff1b\n\n3. \u8fdb\u800c\u751f\u6210\u5e7f\u4e49 Fock \u77e9\u9635 $F_{pq}$ \u4e0e\u8f68\u9053\u68af\u5ea6 $x_{pq} = F_{pq} - F_{qp}$\uff1b\n\n4. \u6700\u540e\u66f4\u65b0\u5206\u5b50\u8f68\u9053\u7cfb\u6570 $C_{\\mu i}$\u3002\n\n\u6700\u7ec8\u7684\u6536\u655b\u6761\u4ef6\u5224\u636e\u662f $F_{pq} - F_{qp} = 0$\uff0c\u5373\u5e7f\u4e49 Fock \u77e9\u9635 $F_{pq}$ \u4e3a\u5bf9\u79f0\u77e9\u9635\u3002\n\n\n```python\ndef oomp2_cycle(C):\n # Generate Psuedo objects, and therefore t_iajb\n mf_prhf = scf.RHF(mol)\n mf_prhf.mo_occ, mf_prhf.mo_coeff = mo_occ, C\n mf_pmp2 = mp.MP2(mf_prhf).run() # Step 1\n # Generate 1-RDM, 2-RDM and orbital gradient from generalized Fock matrix\n rdm1, rdm2 = mf_pmp2.make_rdm1(), mf_pmp2.make_rdm2() # Step 2\n gfock_grad = mf_cas.unpack_uniq_var(mf_cas.get_grad(C, (rdm1, rdm2), mf_cas.ao2mo(C))) # Step 3\n # Returned value: Updated MO Coefficient; OO-MP2 Energy (in current cycle); orbital gradient error\n return update_C(C, gfock_grad), mf_pmp2.e_tot, np.linalg.norm(gfock_grad) # Step 4\n```\n\n\u800c\u66f4\u65b0\u8f68\u9053\u7cfb\u6570\u662f\u901a\u8fc7\u4e0b\u8ff0\u8fc7\u7a0b\u5b9e\u73b0\uff1a\n\n$$\n\\begin{gather}\nX_{ai} = - X_{ia} = \\frac{x_{ai}}{- \\varepsilon_a + \\varepsilon_i} = \\frac{F_{ai} - F_{ia}}{- \\varepsilon_a + \\varepsilon_i} \\\\\nX_{ij} = 0, \\; = X_{ab} = 0 \\\\\n\\mathbf{C} \\leftarrow \\mathbf{C} \\exp(\\lambda \\mathbf{X})\n\\end{gather}\n$$\n\n\u5176\u4e2d $\\lambda$ \u662f\u68af\u5ea6\u4e0b\u964d\u7387\uff0c\u5bf9\u5e94\u4e8e\u673a\u5668\u5b66\u4e60\uff0c\u5b83\u4e0e\u68af\u5ea6\u4e0b\u964d\u7b97\u6cd5\u7684\u5b66\u4e60\u7387\u662f\u7c7b\u4f3c\u7684\u3002\u8fd9\u91cc\u53d6\u4e3a $\\lambda = 0.5$\u3002\n\n\n```python\ndef update_C(C, gfock_grad):\n # Generate anti-symmetrized rotation matrix\n D = mf_rhf.make_rdm1(C, mo_occ)\n e = (C.T @ mf_rhf.get_fock(dm=D) @ C).diagonal()\n X = np.zeros_like(C)\n X[sv, so] = gfock_grad[sv, so] / (- e[sv, None] + e[None, so])\n X[so, sv] = gfock_grad[so, sv] / (- e[None, sv] + e[so, None])\n # Control rotation by factor\n X *= 0.5\n # Generate rotated MO coefficient\n C_new = C @ scipy.linalg.expm(X)\n return C_new\n```\n\n\u5982\u679c\u5c06 RHF \u7684\u5206\u5b50\u8f68\u9053\u7cfb\u6570 `mf_rhf.mo_coeff` \u4f5c\u4e3a\u5206\u5b50\u8f68\u9053\u7cfb\u6570\u7684\u521d\u731c\uff0c\u90a3\u4e48\u6536\u655b\u8fc7\u7a0b\u53ef\u4ee5\u7528\u4e0b\u8ff0\u8fed\u4ee3\u4ee3\u7801\u7ed9\u51fa\uff1a\n\n\n```python\nC_oo = np.copy(mf_rhf.mo_coeff)\n```\n\n\n```python\nprint(\"Cycle | OO-MP2 Energy | G-Fock Gradient Norm\")\nfor i in range(15):\n C_oo, eng, err = oomp2_cycle(C_oo)\n print(\"{:5d} | {:<13.8f} | {:<16.8e}\".format(i, eng, err))\n```\n\n Cycle | OO-MP2 Energy | G-Fock Gradient Norm\n 0 | -76.10403565 | 7.90255467e-02 \n 1 | -76.10503066 | 2.20049872e-02 \n 2 | -76.10509490 | 7.36831750e-03 \n 3 | -76.10510186 | 4.69833400e-03 \n 4 | -76.10510336 | 2.78455388e-03 \n 5 | -76.10510386 | 1.90302779e-03 \n 6 | -76.10510405 | 1.22381869e-03 \n 7 | -76.10510413 | 8.22563315e-04 \n 8 | -76.10510417 | 5.38178673e-04 \n 9 | -76.10510418 | 3.59079689e-04 \n 10 | -76.10510419 | 2.36428274e-04 \n 11 | -76.10510419 | 1.57182522e-04 \n 12 | -76.10510419 | 1.03794108e-04 \n 13 | -76.10510419 | 6.88759621e-05 \n 14 | -76.10510419 | 4.55466994e-05 \n\n\n:::{admonition} \u8bb0\u53f7\u533a\u522b\n\n\u5728\u8fd9\u4efd\u6587\u6863\u4e2d\uff0cRHF \u7684 Fock \u77e9\u9635\u8bb0\u53f7\u5b9a\u4e49\u4e3a $f_{pq}$\uff0c\u800c Post-HF \u65b9\u6cd5\u7684 Fock \u77e9\u9635\u8bb0\u53f7\u5b9a\u4e49\u4e3a $F_{pq}$\u3002\u8fd9\u4e24\u8005\u5e76\u975e\u76f8\u540c\uff0c\u5e76\u4e14\u975e\u8f68\u9053\u4f18\u5316\u7684\u65b9\u6cd5\u4e0b\uff0c\u5e7f\u4e49 Fock \u77e9\u9635 $F_{pq}$ \u77e9\u9635\u4e00\u822c\u662f\u975e\u5bf9\u79f0\u7684\u3002\n\n:::\n\n## PySCF \u7a0b\u5e8f\u5b9e\u73b0\uff1a\u7406\u89e3\u4e0e\u5206\u89e3\n\n\u6211\u4eec\u4f1a\u5bf9\u4e0a\u9762\u7a0b\u5e8f\u4e2d\u7684\u91cd\u8981\u6b65\u9aa4\u8fdb\u884c\u8bf4\u660e\u3002\n\n### \u539f\u5b50\u8f68\u9053\u7535\u5b50\u79ef\u5206\u5b9a\u4e49\n\n- `h` $h_{\\mu \\nu}$\uff0c\u7ef4\u5ea6 $(\\mu, \\nu)$\uff0c\u539f\u5b50\u8f68\u9053\u57fa\u7684 Hamiltonian Core \u77e9\u9635\uff0c\u5373\u52a8\u80fd\u4e0e\u539f\u5b50\u6838-\u7535\u5b50\u52bf\u80fd\u79ef\u5206\uff1b\n\n- `S` $S_{\\mu \\nu}$\uff0c\u7ef4\u5ea6 $(\\mu, \\nu)$\uff0c\u539f\u5b50\u8f68\u9053\u57fa\u7684\u91cd\u53e0\u79ef\u5206\uff1b\n\n- `eri` $(\\mu \\nu | \\kappa \\lambda)$\uff0c\u7ef4\u5ea6 $(\\mu, \\nu, \\kappa, \\lambda)$\uff0c\u539f\u5b50\u8f68\u9053\u57fa\u7684\u53cc\u7535\u5b50\u79ef\u5206\u3002\n\n\n```python\nh = mol.intor(\"int1e_kin\") + mol.intor(\"int1e_nuc\")\nS = mol.intor(\"int1e_ovlp\")\neri = mol.intor(\"int2e\")\n```\n\n### Canonical MP2\n\n\u6211\u4eec\u5148\u7b80\u5355\u56de\u987e\u4e00\u4e0b\u5728 Canonical RHF \u4e0b\uff0cMP2 \u7684\u6fc0\u53d1\u7cfb\u6570 $t_{ij}^{ab}$ \u4e0e\u80fd\u91cf $E_\\mathrm{corr}^\\mathsf{MP2}$ \u7684\u5bfc\u51fa\u65b9\u5f0f\u3002\u6211\u4eec\u7559\u610f\u5230 PySCF \u7684\u81ea\u6d3d\u573a\u8fc7\u7a0b\u7ed9\u51fa\u7684\u662f Canonical \u60c5\u51b5\uff0c\u5373\u5206\u5b50\u8f68\u9053\u7684 Fock \u77e9\u9635 $f_{pq}$ \u662f\u5bf9\u89d2\u77e9\u9635\u3002\n\n- `C` $C_{\\mu p}$ \u4e3a\u5206\u5b50\u8f68\u9053\u7cfb\u6570\uff0c`e` $e_p$ \u4e3a\u8f68\u9053\u80fd\u91cf\uff1b\n\n- `D_iajb` $D_{ij}^{ab}$ MP2 \u5206\u6bcd\u9879\uff0c\u7ef4\u5ea6 $(i, a, j, b)$\uff1a\n\n $$\n D_{ij}^{ab} = \\varepsilon_i - \\varepsilon_a + \\varepsilon_j - \\varepsilon_b\n $$\n\n- `eri_mo` $(pq|rs)$ \u5206\u5b50\u8f68\u9053\u57fa\u4e0b\u7684\u53cc\u7535\u5b50\u79ef\u5206\uff0c\u7ef4\u5ea6 $(p, q, r, s)$\uff1a\n\n $$\n (pq|rs) = C_{\\mu p} C_{\\nu q} (\\mu \\nu | \\kappa \\lambda) C_{\\kappa r} C_{\\lambda s}\n $$\n\n- `t_iajb` $t_{ij}^{ab}$ MP2 \u6fc0\u53d1\u7cfb\u6570\uff1a\n\n $$\n t_{ij}^{ab} = \\frac{(ia|jb)}{D_{ij}^{ab}}\n $$\n\n\n```python\nC, e = mf_rhf.mo_coeff, mf_rhf.mo_energy\nD_iajb = e[so, None, None, None] - e[None, sv, None, None] + e[None, None, so, None] - e[None, None, None, sv]\neri_mo = np.einsum(\"up, vq, uvkl, kr, ls -> pqrs\", C, C, eri, C, C)\nt_iajb = eri_mo[so, sv, so, sv] / D_iajb\n```\n\n\u56e0\u6b64\uff0cMP2 \u76f8\u5173\u80fd\u53ef\u4ee5\u5199\u4e3a (\u53c2\u8003\u503c\u4e3a -0.134335 a.u.)\n\n$$\nE_\\mathrm{corr}^\\mathsf{MP2} = \\big( 2 t_{ij}^{ab} - t_{ij}^{ba} \\big) (ia|jb)\n$$\n\n\n```python\n((2 * t_iajb - t_iajb.swapaxes(-1, -3)) * eri_mo[so, sv, so, sv]).sum()\n```\n\n\n\n\n -0.13433468897410067\n\n\n\n### Non-Canonical MP2\uff1aPySCF \u7a0b\u5e8f\n\n\u4f46\u5bf9\u4e8e OO-MP2 \u800c\u8a00\uff0c\u7531\u4e8e\u4ea7\u751f\u4e86\u8f68\u9053\u65cb\u8f6c\uff0c\u6211\u4eec\u9700\u8981\u8003\u5bdf Non-Canonical RHF \u7684 MP2\u3002\n\nNon-Canonical \u610f\u6307 RHF \u7684 Fock \u77e9\u9635 $f_{pq}$ \u662f\u5206\u5757\u5bf9\u89d2\u5316\u7684\uff0c\u5373\u5360\u636e-\u975e\u5360\u548c\u975e\u5360-\u5360\u636e\u5206\u5757 $f_{ia}$\u3001$f_{ai}$ \u5747\u4e3a\u96f6\uff1b\u800c\u5360\u636e\u548c\u975e\u5360\u5206\u5757 $f_{ij}$\u3001$f_{ab}$ \u7684\u77e9\u9635\u5e76\u975e\u662f\u5bf9\u89d2\u5316\u7684\u3002\n\n\u4e3a\u4e86\u6784\u9020\u8fd9\u6837\u4e00\u4e2a Non-Canonical RHF \u7684\u60c5\u5f62\uff0c\u6211\u4eec\u53ef\u4ee5\u5bf9 Canonical RHF \u5206\u5b50\u8f68\u9053\u7cfb\u6570\u77e9\u9635 `C_rhf` \u4f5c\u4e0b\u8ff0\u53d8\u6362\uff0c\u5f97\u5230 Non-Canonical RHF \u5206\u5b50\u8f68\u9053\u7cfb\u6570\u77e9\u9635 `C_rot`\uff1a\n\n$$\n\\mathbf{C} \\leftarrow \\mathbf{C} \\mathbf{U}\n$$\n\n\u4e0a\u8ff0\u7684 $\\mathbf{U}$ \u77e9\u9635\u662f\u5206\u5757\u5bf9\u89d2\u5316\u7684\u6b63\u4ea4\u77e9\u9635\u3002\u4e3a\u4e86\u6784\u9020\u8fd9\u6837\u7684\u6b63\u4ea4\u77e9\u9635\uff0c\u6211\u4eec\u53ef\u4ee5\u751f\u6210\u4e00\u4e2a\u5206\u5757\u5bf9\u89d2\u5316\u3001\u4e14\u53cd\u5bf9\u79f0\u7684 `X` $\\mathbf{X}$ \u77e9\u9635\uff0c\u5e76\u4ee4 $\\mathbf{U} = \\exp(\\mathbf{X})$\u3002\n\n\n```python\nC_rhf = mf_rhf.mo_coeff\n```\n\n\n```python\nX = np.random.randn(nmo, nmo)\nX[sv, so] = X[so, sv] = 0\nX -= X.T\nX *= 0.02\nC_rot = C_rhf @ scipy.linalg.expm(X)\n```\n\n\u7531\u6b64\u6784\u5efa\u51fa\u7684 Non-Canonical \u5206\u5b50\u8f68\u9053 Fock \u77e9\u9635 $f_{pq}$ \u662f\u5206\u5757\u5bf9\u89d2\u5316\u7684\uff0c\u5373\u4e0d\u4e00\u5b9a\u8981\u6c42 $f_{ij} = \\delta_{ij} \\varepsilon_i$ \u4e0e $f_{ab} = \\delta_{ab} \\varepsilon_a$\uff1a\n\n\n```python\nfock_rot = np.einsum(\"up, uv, vq -> pq\", C_rot, mf_rhf.get_fock(), C_rot)\nfock_rot\n```\n\n\n\n\n array([[-20.4748, -0.0683, -0.3396, -1.0108, -0.9682, -0. , -0. , -0. , 0. , 0. , -0. , -0. , 0. ],\n [ -0.0683, -1.3485, -0.0071, -0.0425, -0.0204, 0. , -0. , 0. , 0. , 0. , 0. , 0. , -0. ],\n [ -0.3396, -0.0071, -0.6668, -0.0222, -0.0172, -0. , 0. , 0. , 0. , -0. , -0. , 0. , -0. ],\n [ -1.0108, -0.0425, -0.0222, -0.6362, -0.0523, -0. , 0. , -0. , 0. , 0. , -0. , 0. , 0. ],\n [ -0.9682, -0.0204, -0.0172, -0.0523, -0.5539, -0. , -0. , -0. , -0. , -0. , 0. , 0. , -0. ],\n [ -0. , 0. , -0. , -0. , -0. , 0.1967, 0.0008, -0.0182, 0.0501, -0.002 , 0.0269, -0.0106, 0.0776],\n [ -0. , -0. , 0. , 0. , -0. , 0.0008, 0.2872, -0.0032, 0.011 , 0.0263, 0.0326, -0.0324, 0.0407],\n [ -0. , 0. , 0. , -0. , -0. , -0.0182, -0.0032, 0.9833, 0.0038, -0.0097, 0.0066, 0.0088, -0.0122],\n [ 0. , 0. , 0. , 0. , -0. , 0.0501, 0.011 , 0.0038, 1.1572, -0.0006, -0.0001, 0.0048, -0.0293],\n [ 0. , 0. , -0. , 0. , -0. , -0.002 , 0.0263, -0.0097, -0.0006, 1.1616, -0.0056, -0.0042, 0.0036],\n [ -0. , 0. , -0. , -0. , 0. , 0.0269, 0.0326, 0.0066, -0.0001, -0.0056, 1.2463, -0.0016, -0.0138],\n [ -0. , 0. , 0. , 0. , 0. , -0.0106, -0.0324, 0.0088, 0.0048, -0.0042, -0.0016, 1.3518, -0.0047],\n [ 0. , -0. , -0. , 0. , -0. , 0.0776, 0.0407, -0.0122, -0.0293, 0.0036, -0.0138, -0.0047, 1.7381]])\n\n\n\n\u5bf9\u4e8e\u8fd9\u6837\u7684\u5206\u5b50\u8f68\u9053\u7cfb\u6570\u77e9\u9635 `C_rot`\uff0cPySCF \u7684\u7a0b\u5e8f\u7167\u6837\u53ef\u4ee5\u7ed9\u51fa\u6b63\u786e\u7684 MP2 \u76f8\u5173\u80fd\u91cf -0.134335 a.u. (\u5176\u4e2d `mf_prhf` \u662f\u6307\u865a\u5047\u7684 (Pseudo) RHF \u5b9e\u4f8b)\uff1a\n\n\n```python\nmf_prhf = scf.RHF(mol)\nmf_prhf.mo_occ, mf_prhf.mo_coeff = mo_occ, C_rot\nmf_pmp2 = mp.MP2(mf_prhf).run()\nmf_pmp2.e_corr\n```\n\n\n\n\n -0.13433467530628806\n\n\n\n### Non-Canonical MP2\uff1a\u6fc0\u53d1\u7cfb\u6570 $t_{ij}^{ab}$ \u8fed\u4ee3\u66f4\u65b0\u65b9\u5f0f\n\n\u9996\u5148\u4e3a\u7a0b\u5e8f\u4e0e\u516c\u5f0f\u8bf4\u660e\uff0c\u5b9a\u4e49\u4e00\u4e9b\u53d8\u91cf\uff1a\n\n- RHF Fock \u5bf9\u89d2\u7ebf\u5360\u636e\u90e8\u5206\u8bb0\u4e3a `eo` $\\varepsilon_i = f_{ii}$\uff1b\n\n- RHF Fock \u5bf9\u89d2\u7ebf\u975e\u5360\u90e8\u5206\u8bb0\u4e3a `ev` $\\varepsilon_a = f_{aa}$\uff1b\n\n- RHF Fock \u53bb\u9664\u5bf9\u89d2\u7ebf\u7684\u5360\u636e\u5206\u5757\u8bb0\u4e3a `fock_oo` $f'_{ij} = (1 - \\delta_{ij}) f_{ij}$\uff1b\n\n- RHF Fock \u53bb\u9664\u5bf9\u89d2\u7ebf\u7684\u975e\u5360\u5206\u5757\u8bb0\u4e3a `fock_vv` $f'_{ab} = (1 - \\delta_{ab}) f_{ab}$\uff1b\n\n- \u53cc\u7535\u5b50\u79ef\u5206 `eri_mo` $(pq|rs)$\uff1b\n\n- \u53ea\u5305\u542b\u5360\u636e-\u975e\u5360\u5206\u5757\u7684\u53cc\u7535\u5b50\u79ef\u5206 `eri_iajb` $(ia|jb)$\n\n- MP2 \u5206\u6bcd `D_iajb` $D_{ij}^{ab}$\u3002\n\n\n```python\neo, ev = fock_rot.diagonal()[so], fock_rot.diagonal()[sv]\nfock_oo, fock_vv = fock_rot[so, so], fock_rot[sv, sv]\nfock_oop, fock_vvp = fock_oo - np.diag(eo), fock_vv - np.diag(ev)\neri_mo = np.einsum(\"up, vq, uvkl, kr, ls -> pqrs\", C_rot, C_rot, eri, C_rot, C_rot)\neri_iajb = eri_mo[so, sv, so, sv]\nD_iajb = eo[:, None, None, None] - ev[None, :, None, None] + eo[None, None, :, None] - ev[None, None, None, :]\n```\n\n:::{caution}\n\n**\u53d8\u91cf\u91cd\u65b0\u5b9a\u4e49**\n\n\u4e0a\u9762\u4ee3\u7801\u5757\u4e2d `eo`, `ev`, `eri_mo`, `D_iajb` \u5c31\u5728 Non-Canonical \u7684\u7cfb\u6570\u77e9\u9635 `C_rot` \u4e0b\u7ed9\u51fa\uff1b\u4f46\u6211\u4eec\u66fe\u7ecf\u4e5f\u5728 Canonical \u7cfb\u6570\u77e9\u9635\u4e0b\u7ed9\u51fa\u8fc7\u7c7b\u4f3c\u7684\u53d8\u91cf\u3002\n\n\u7531\u4e8e\u6211\u4eec\u4f1a\u7ecf\u5e38\u5207\u6362\u5404\u79cd\u7cfb\u6570\u77e9\u9635\u7684\u65cb\u8f6c\u65b9\u5f0f (\u975e\u65cb\u8f6c\u3001Non-Canonical\u3001Non-HF)\uff0c\u56e0\u6b64\u4e00\u4e9b\u53d8\u91cf\u4e5f\u4f1a\u88ab\u590d\u7528\u4e0e\u590d\u5199\uff0c\u4e5f\u6682\u4e0d\u533a\u5206\u65cb\u8f6c\u540e\u4e0e\u65cb\u8f6c\u524d\u7684\u5206\u5b50\u8f68\u9053\u89d2\u6807\u3002\u8fd9\u53ef\u80fd\u4f1a\u5bf9\u9605\u8bfb\u9020\u6210\u56f0\u60d1\u3002\n\n:::\n\n\u4f9d\u636e\u4e0d\u540c\u7684\u5fae\u6270\u7406\u8bba\u5b9a\u4e49\u65b9\u5f0f\uff0cNon-Canonical RHF \u7684 MP2 \u76f8\u5173\u80fd\u53ef\u80fd\u4e0e Canonical RHF \u7684 MP2 \u76f8\u5173\u80fd\u4e0d\u540c\u3002\u56e0\u6b64\u8fd9\u91cc\u91c7\u7528\u4e24\u4e2a\u76f8\u5173\u80fd\u76f8\u540c\u7684\u5b9a\u4e49\u3002\u6b64\u65f6\u6fc0\u53d1\u7cfb\u6570 $t_{ij}^{ab}$ \u5e94\u5f53\u6ee1\u8db3\n\n$$\n(ia|jb) = t_{kj}^{ab} f_{ki} + t_{ik}^{ab} f_{kj} - t_{ij}^{cb} f_{ca} - t_{ij}^{ac} f_{cb}\n$$\n\n\u4e0a\u5f0f\u662f\u5bf9\u7b49\u5f0f\u53f3\u7684 $k$ \u8fdb\u884c\u6c42\u548c\u7684\u3002\u5982\u679c\u73b0\u5728\u7528 $f_{ij} = f'_{ij} + \\delta_{ij} \\varepsilon_i$\uff0c$f_{ab} = f'_{ab} + \\delta_{ab} \\varepsilon_a$ \u5c55\u5f00\uff0c\u90a3\u4e48\u4e0a\u5f0f\u53ef\u4ee5\u5199\u4e3a\n\n$$\n(ia|jb) = t_{ij}^{ab} D_{ij}^{ab} + t_{kj}^{ab} f'_{ki} + t_{ik}^{ab} f'_{kj} - t_{ij}^{cb} f'_{ca} - t_{ij}^{ac} f'_{cb}\n$$\n\n\u6574\u7406\u4e0a\u5f0f\uff0c\u5c31\u53ef\u4ee5\u5f97\u5230\u8fed\u4ee3\u5173\u7cfb\n\n$$\nt_{ij}^{ab} \\leftarrow \\frac{(ia|jb) - t_{kj}^{ab} f'_{ki} - t_{ik}^{ab} f'_{kj} + t_{ij}^{cb} f'_{ca} + t_{ij}^{ac} f'_{cb}}{D_{ij}^{ab}}\n$$\n\n\u4e00\u822c\u6765\u8bf4\uff0c\u5982\u679c\u8f68\u9053\u7684\u65cb\u8f6c\u5e76\u4e0d\u662f\u5f88\u5267\u70c8\uff0c\u90a3\u4e48 $f'_{ij}$, $f'_{ab}$ \u4e24\u8005\u7684\u8d21\u732e\u8f83\u5c0f\uff0c\u56e0\u6b64 $t_{ij}^{ab} \\simeq (ia|jb) / D_{ij}^{ab}$ \u4f1a\u662f\u4e00\u4e2a\u6bd4\u8f83\u597d\u7684\u8fd1\u4f3c\u3002\n\n\u5728\u6b64\u60c5\u5f62\u4e0b\uff0cNon-Canonical MP2 \u7684\u80fd\u91cf\u8ba1\u7b97\u65b9\u5f0f\u5982\u4e0b\uff1a\n\n$$\nE_\\mathrm{corr}^\\mathsf{MP2} = \\big( 2 t_{ij}^{ab} - t_{ij}^{ba} \\big) (ia|jb)\n$$\n\n\u4e0b\u9762\u7684\u7a0b\u5e8f\u5c31\u662f\u5b9e\u73b0 Non-Canonical MP2 \u7684\u6d41\u7a0b\u3002\n\n- `update_t_iajb` \u5373\u4f7f\u7528\u8fed\u4ee3\u5173\u7cfb\uff0c\u66f4\u65b0 $t_{ij}^{ab}$\uff1b\n\n- `calculate_noncan_mp2` \u5373\u8ba1\u7b97 Non-Canonical MP2 \u76f8\u5173\u80fd\u7684\u51fd\u6570\u3002\n\n\n```python\ndef update_t_iajb(t_iajb):\n t_iajb_new = np.zeros_like(t_iajb)\n t_iajb_new += np.einsum(\"icjb, ca -> iajb\", t_iajb, fock_vvp)\n t_iajb_new += np.einsum(\"iajc, cb -> iajb\", t_iajb, fock_vvp)\n t_iajb_new -= np.einsum(\"iakb, kj -> iajb\", t_iajb, fock_oop)\n t_iajb_new -= np.einsum(\"kajb, ki -> iajb\", t_iajb, fock_oop)\n t_iajb_new += eri_iajb\n t_iajb_new /= D_iajb\n return t_iajb_new\n```\n\n\n```python\ndef calculate_noncan_mp2(t_iajb):\n return ((2 * t_iajb - t_iajb.swapaxes(-1, -3)) * eri_iajb).sum()\n```\n\n\u968f\u540e\u58f0\u660e\u521d\u731c $t_{ij}^{ab} \\simeq (ia|jb) / D_{ij}^{ab}$ \u5e76\u4ee5\u6b64\u8fed\u4ee3\u66f4\u65b0\uff1b\u5e76\u4ee5 Canonical MP2 \u7684\u76f8\u5173\u80fd\u52a0\u4ee5\u9a8c\u8bc1\u3002\u5728 5 \u6b21\u5faa\u73af\u540e\uff0c\u51e0\u4e4e\u6536\u655b\u5230\u4e86\u6b63\u786e\u80fd\u91cf\u3002\n\n\n```python\nt_iajb = eri_mo[so, sv, so, sv] / D_iajb\n```\n\n\n```python\nfor i in range(10):\n print(\"Error: {:16.8e}\".format(calculate_noncan_mp2(t_iajb) - mf_mp2.e_corr))\n t_iajb = update_t_iajb(t_iajb)\n```\n\n Error: 3.41981239e-03\n Error: 2.09994114e-03\n Error: 9.08474334e-05\n Error: 9.06169148e-05\n Error: 3.54576068e-06\n Error: 5.43397725e-06\n Error: 6.95752296e-08\n Error: 4.42378672e-07\n Error: -2.06561550e-08\n Error: 4.46581002e-08\n\n\n\u4e8b\u5b9e\u4e0a\uff0c\u5728 PySCF \u4e2d\uff0c\u5305\u542b\u5360\u636e-\u975e\u5360\u8f68\u9053\u65cb\u8f6c\u7684 Non-RHF \u4e0b\u7684 MP2 \u65b9\u6cd5\uff0c\u4e5f\u662f\u901a\u8fc7\u4e0a\u8ff0\u8fc7\u7a0b\u8fdb\u884c\u8ba1\u7b97\u7684\u3002\n\n### MP2 1-RDM\n\n\u5bf9\u4e8e\u4e00\u9636\u7ea6\u5316\u5bc6\u5ea6 1-RDM $\\gamma_{pq}$\uff0c\u5176\u9700\u8981\u901a\u8fc7\u5206\u5757\u7684\u65b9\u5f0f\u751f\u6210\uff1a\n\n$$\n\\begin{align}\n\\gamma_{ij}^\\mathsf{RHF} &= 2 \\delta_{ij} \\\\\n\\gamma_{ab}^\\mathsf{RHF} &= \\gamma_{ia}^\\mathsf{RHF} = \\gamma_{ai}^\\mathsf{RHF} = 0 \\\\\n\\gamma_{ij}^\\mathrm{corr} &= - 4 t_{ik}^{ab} t_{jk}^{ab} + 2 t_{ik}^{ba} t_{jk}^{ab} \\\\\n\\gamma_{ab}^\\mathrm{corr} &= 4 t_{ij}^{ac} t_{ij}^{bc} - 2 t_{ij}^{ca} t_{ij}^{bc} \\\\\n\\gamma_{ia}^\\mathrm{corr} &= \\gamma_{ai}^\\mathrm{corr} = 0 \\\\\n\\gamma_{pq} &= \\gamma_{pq}^\\mathsf{RHF} + \\gamma_{pq}^\\mathrm{corr}\n\\end{align}\n$$\n\n\u8fd9\u79cd\u751f\u6210\u65b9\u5f0f\u65e0\u5173\u4e4e\u65b9\u6cd5\u662f\u5426\u662f Canonical \u7684\u3002\n\n\u9996\u5148\u751f\u6210 RHF \u7684 1-RDM `rdm1_rhf` $\\gamma_{pq}^\\mathsf{RHF}$\uff1a\n\n\n```python\nrdm1_rhf = np.zeros((nmo, nmo))\nnp.fill_diagonal(rdm1_rhf[so, so], 2)\n```\n\n\u968f\u540e\u7ed9\u51fa MP2 \u76f8\u5173\u90e8\u5206\u6240\u7ed9\u51fa\u7684 1-RDM `rdm1_corr` $\\gamma_{pq}^\\mathrm{corr}$\uff1a\n\n\n```python\nrdm1_corr = np.zeros((nmo, nmo))\nrdm1_corr[so, so] = - 4 * np.einsum(\"iakb, jakb -> ij\", t_iajb, t_iajb) + 2 * np.einsum(\"ibka, jakb -> ij\", t_iajb, t_iajb)\nrdm1_corr[sv, sv] = 4 * np.einsum(\"iajc, ibjc -> ab\", t_iajb, t_iajb) - 2 * np.einsum(\"icja, ibjc -> ab\", t_iajb, t_iajb)\n```\n\n\u603b 1-RDM `rdm1` $\\gamma_{pq}$ \u53ef\u4ee5\u901a\u8fc7\u7b80\u5355\u76f8\u52a0\u83b7\u5f97\uff1a\n\n\n```python\nrdm1 = rdm1_rhf + rdm1_corr\nnp.allclose(rdm1, mf_pmp2.make_rdm1())\n```\n\n\n\n\n True\n\n\n\n### MP2 2-RDM\n\n\u5bf9\u4e8e\u4e8c\u9636\u7ea6\u5316\u5bc6\u5ea6 2-RDM `rdm2` $\\Gamma_{pr}^{qs}$ (\u7ef4\u5ea6 $(p, q, r, s)$)\uff0c\u5176\u4e5f\u9700\u8981\u901a\u8fc7\u5206\u5757\u751f\u6210\u3002\u9996\u5148\u751f\u6210 $\\Gamma_{ia}^{jb}$, $\\Gamma_{ai}^{bj}$, $\\Gamma_{ik}^{jl}$, $\\Gamma_{ac}^{bd}$ \u90e8\u5206\uff1a\n\n$$\n\\Gamma_{pr}^{qs} = \\left( \\gamma_{pq} \\gamma_{rs} - \\frac{1}{2} \\gamma_{ps} \\gamma_{rq} \\right) - \\left( \\gamma_{pq}^\\mathrm{corr} \\gamma_{rs}^\\mathrm{corr} - \\frac{1}{2} \\gamma_{ps}^\\mathrm{corr} \\gamma_{rq}^\\mathrm{corr} \\right)\n$$\n\n\u5176\u4f59\u7684\u90e8\u5206\u662f $\\Gamma_{ij}^{ab}$ \u4e0e $\\Gamma_{ab}^{ij}$\uff1a\n\n$$\n\\Gamma_{ij}^{ab} = \\Gamma_{ab}^{ij} = 4 t_{ij}^{ab} - 2 t_{ij}^{ba}\n$$\n\n\n```python\nrdm2 = np.zeros((nmo, nmo, nmo, nmo))\nrdm2 = np.einsum(\"pq, rs -> pqrs\", rdm1, rdm1) - 0.5 * np.einsum(\"ps, rq -> pqrs\", rdm1, rdm1)\nrdm2 -= np.einsum(\"pq, rs -> pqrs\", rdm1_corr, rdm1_corr) - 0.5 * np.einsum(\"ps, rq -> pqrs\", rdm1_corr, rdm1_corr)\nrdm2[so, sv, so, sv] = 4 * np.einsum(\"iajb -> iajb\", t_iajb) - 2 * np.einsum(\"ibja -> iajb\", t_iajb)\nrdm2[sv, so, sv, so] = 4 * np.einsum(\"iajb -> aibj\", t_iajb) - 2 * np.einsum(\"ibja -> aibj\", t_iajb)\nnp.allclose(rdm2, mf_pmp2.make_rdm2(), atol=1e-7)\n```\n\n\n\n\n False\n\n\n\n\u7531\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7 1-RDM $\\gamma_{pq}$ \u4e0e 2-RDM $\\Gamma_{pr}^{qs}$ \u9a8c\u8bc1 MP2 \u603b\u80fd\u91cf -76.104036 a.u.\uff1a\n\n$$\nE_\\mathrm{tot}^\\mathsf{MP2} = h_{pq} \\gamma_{pq} + \\frac{1}{2} (pq|rs) \\Gamma_{pr}^{qs} + E_\\mathrm{nuc}\n$$\n\n\u4f46\u8fd9\u91cc\u7684\u5355\u7535\u5b50\u79ef\u5206 $h_{pq}$ \u4e0e\u53cc\u7535\u5b50\u79ef\u5206 $(pq|rs)$ \u90fd\u662f\u5728\u65cb\u8f6c\u8fc7\u540e\u7684\u7cfb\u6570\u8f68\u9053\u77e9\u9635 `C_rot` $\\mathbf{C}$ \u4e3a\u57fa\u7ed9\u51fa\uff0c\u56e0\u6b64\u9700\u8981\u91cd\u65b0\u751f\u6210\u4e00\u4e0b\u3002\n\n\n```python\nh_mo = np.einsum(\"up, uv, vq -> pq\", C_rot, h, C_rot)\neri_mo = np.einsum(\"up, vq, uvkl, kr, ls -> pqrs\", C_rot, C_rot, eri, C_rot, C_rot)\n```\n\n\n```python\n(\n + np.einsum(\"pq, pq ->\", h_mo, rdm1)\n + 0.5 * np.einsum(\"pqrs, pqrs ->\", eri_mo, rdm2)\n + mol.energy_nuc()\n)\n```\n\n\n\n\n -76.10403565383504\n\n\n\n### \u751f\u6210\u5e7f\u4e49 Fock \u77e9\u9635\n\n\u5e7f\u4e49 Fock \u77e9\u9635 `gfock` $F_{pq}$ \u533a\u522b\u4e8e RHF \u7684 Fock \u77e9\u9635 $f_{pq}$\u3002\u5176\u5b9a\u4e49\u4e3a\n\n$$\nF_{pq} = h_{pm} \\gamma_{mq} + (pm|rs) \\Gamma_{mr}^{qs}\n$$\n\n\n```python\ngfock = np.einsum(\"pr, rq -> pq\", h_mo, rdm1) + np.einsum(\"pmrs, mqrs -> pq\", eri_mo, rdm2)\n```\n\n\u4e8b\u5b9e\u4e0a\uff0cRHF \u7684 Fock \u77e9\u9635\u4e2d\uff0c\u5360\u636e\u8f68\u9053\u90e8\u5206\u4e5f\u53ef\u4ee5\u7528\u7c7b\u4f3c\u7684\u65b9\u6cd5\u5b9a\u4e49\uff1a\n\n$$\n\\begin{align}\n2 f_{ij} &= h_{im} \\gamma_{mj}^\\mathsf{RHF} + (im|rs) \\Gamma_{mr}^{js, \\mathsf{RHF}} \\\\\n\\Gamma_{pr}^{qs, \\mathsf{RHF}} &= \\gamma_{pq}^\\mathsf{RHF} \\gamma_{rs}^\\mathsf{RHF} - \\frac{1}{2} \\gamma_{ps}^\\mathsf{RHF} \\gamma_{rq}^\\mathsf{RHF}\n\\end{align}\n$$\n\n\n```python\nrdm2_rhf = np.einsum(\"pq, rs -> pqrs\", rdm1_rhf, rdm1_rhf) - 0.5 * np.einsum(\"ps, rq -> pqrs\", rdm1_rhf, rdm1_rhf)\nnp.allclose(\n (np.einsum(\"pr, rq -> pq\", h_mo, rdm1_rhf) + np.einsum(\"pmrs, mqrs -> pq\", eri_mo, rdm2_rhf))[so, so],\n 2 * fock_rot[so, so],\n)\n```\n\n\n\n\n True\n\n\n\n\u4f46\u5728 PySCF \u7684 CASSCF \u6a21\u5757\u4e2d\uff0c\u4f3c\u4e4e\u6ca1\u6709\u76f4\u63a5\u751f\u6210\u5e7f\u4e49 Fock \u77e9\u9635\u7684\u65b9\u5f0f\u3002\u4f46\u5176\u6709\u5e7f\u4e49 Fock \u7684\u5bfc\u6570\u91cf\uff0c\u88ab\u79f0\u4e3a\u8f68\u9053\u68af\u5ea6 (Orbital Gradient) `gfock_grad` $x_{pq}$\uff1a\n\n$$\nx_{pq} = F_{pq} - F_{qp}\n$$\n\n\n```python\ngfock_grad = gfock - gfock.T\nnp.allclose(\n mf_cas.unpack_uniq_var(mf_cas.get_grad(C_rot, (rdm1, rdm2), mf_cas.ao2mo(C_rot))),\n gfock_grad\n)\n```\n\n\n\n\n True\n\n\n\n\u81f3\u6b64\uff0c\u6240\u6709\u751f\u6210 OO-MP2 \u6240\u9700\u8981\u7684\u5355\u6b65\u590d\u6742\u8ba1\u7b97\u90fd\u5df2\u7ecf\u6db5\u76d6\u5230\u4e86\u3002\n\n## \u8f68\u9053\u65cb\u8f6c\u7684\u610f\u4e49\n\n\u8ba8\u8bba\u5230\u73b0\u5728\uff0c\u6211\u4eec\u4ec5\u4ec5\u77e5\u9053\u4e86 OO-MP2 \u7684\u7a0b\u5e8f\u5b9e\u73b0\u662f\u5982\u4f55\u8fdb\u884c\u7684\uff1b\u4f46\u5bf9\u5176\u6839\u6e90\u7684\u5408\u7406\u6027\u95ee\u9898\uff0c\u6211\u4eec\u5728\u8fd9\u91cc\u624d\u5f00\u59cb\u8bf4\u660e\u3002\n\n\u51fa\u4e8e\u4e00\u822c\u6027\uff0c\u6211\u4eec\u73b0\u5728\u8003\u8651 Non-HF \u5f62\u5f0f\u7684\u8f68\u9053\u7cfb\u6570\uff0c\u5373\u76f8\u5bf9\u4e8e RHF \u7cfb\u6570\u5df2\u7ecf\u4e00\u5b9a\u7a0b\u5ea6\u7684\u65cb\u8f6c\u3002\u8be5 Non-HF \u8f68\u9053\u7cfb\u6570\u79f0\u4e3a `C_base` $C_{\\mu p}^\\mathrm{base}$\u3002\u6211\u4eec\u4e4b\u540e\u7684\u8ba8\u8bba\u90fd\u57fa\u4e8e\u8be5 Non-HF \u8f68\u9053\u7cfb\u6570\u5f00\u59cb\u3002\n\n\n```python\nX = np.random.randn(nmo, nmo)\nX = (X - X.T) * 0.02\nC_base = C_rhf @ scipy.linalg.expm(X)\n```\n\n\u9996\u5148\u9700\u8981\u8bf4\u660e\uff0c\u8f68\u9053\u7684\u65cb\u8f6c\u77e9\u9635\u5fc5\u987b\u662f\u6b63\u4ea4\u77e9\u9635 (\u9149\u77e9\u9635)\u3002\u8fd9\u662f\u56e0\u4e3a\u8f68\u9053\u7cfb\u6570\u5fc5\u987b\u6ee1\u8db3\n\n$$\n\\mathbf{C}^\\dagger \\mathbf{S} \\mathbf{C} = \\mathbf{I}\n$$\n\n\u65cb\u8f6c\u77e9\u9635 $\\mathbf{U}$ \u901a\u8fc7\u4e0b\u5f0f\u5b9a\u4e49\uff1a$\\mathbf{C} = \\mathbf{C}^\\mathrm{base} \\mathbf{U}$\u3002\u56e0\u6b64\uff0c\n\n$$\n\\mathbf{C}^\\dagger \\mathbf{S} \\mathbf{C} = \\mathbf{U}^\\dagger \\mathbf{C}^\\dagger \\mathbf{S} \\mathbf{C} \\mathbf{U} = \\mathbf{U}^\\dagger \\mathbf{I} \\mathbf{U} = \\mathbf{U}^\\dagger \\mathbf{U} = \\mathbf{I}\n$$\n\n\u800c\u4efb\u4f55\u6b63\u4ea4\u77e9\u9635\u90fd\u53ef\u4ee5\u901a\u8fc7\u53cd\u5bf9\u79f0\u77e9\u9635 $\\mathbf{X} = \\mathbf{X}^\\dagger$ \u7684\u5e42\u6b21\u7ed9\u51fa $\\mathbf{U} = \\exp(\\mathbf{X})$\u3002\n\n\u73b0\u5728\u8003\u5bdf\u5728\u5fae\u6270\u4e0b\uff0c\u80fd\u91cf\u968f\u8f68\u9053\u7cfb\u6570\u7684\u53d8\u5316\u60c5\u51b5\u3002\u4ee4\u4e00\u822c\u60c5\u51b5\u4e0b\u8f68\u9053\u7cfb\u6570 $C_{\\mu p}$ \u4e3a\u5173\u4e8e\u53cd\u5bf9\u79f0\u77e9\u9635 $X_{pq}$ \u7684\u51fd\u6570\uff1a\n\n$$\n\\mathbf{C} = \\mathbf{C}^\\mathrm{base} \\exp (\\mathbf{X})\n$$\n\n\u800c $C_{\\mu p}$ \u5bf9\u5e94\u7684 MP2 \u80fd\u91cf\u5199\u4f5c\u5173\u4e8e $X_{pq}$ \u7684\u51fd\u6570 $E_\\mathrm{tot}^\\mathsf{MP2} (\\mathbf{X})$\u3002\u4e0b\u9762\u7684\u4ee3\u7801 `eng_mp2_pert` \u5373\u662f\u4ee3\u5165\u53cd\u5bf9\u79f0\u77e9\u9635 $X_{pq}$\uff0c\u751f\u6210 MP2 \u80fd\u91cf\u7684\u51fd\u6570\u3002\n\n\n```python\ndef eng_mp2_pert(X):\n C_rot = C_base @ scipy.linalg.expm(X)\n mf_prhf = scf.RHF(mol)\n mf_prhf.mo_occ, mf_prhf.mo_coeff = mo_occ, C_rot\n mf_pmp2 = mp.MP2(mf_prhf).run()\n return mf_pmp2.e_tot\n```\n\n\u7531\u6b64\uff0c\u80fd\u91cf\u5173\u4e8e\u65cb\u8f6c\u77e9\u9635\u7684\u5bfc\u6570\u5173\u7cfb\u53ef\u4ee5\u5199\u4e3a\u77e9\u9635 `dX` ${\\mathrm{d} \\mathbf{X}}$\uff0c\u5176\u7ef4\u5ea6\u4e3a $(p, q)$\uff1a\n\n$$\n{\\mathrm{d}X}_{pq} = \\frac{\\partial E_\\mathrm{tot}^\\mathsf{MP2}}{\\partial X_{pq}}\n$$\n\n\u8fd9\u79cd\u5bfc\u6570\u53ef\u4ee5\u5199\u6210\u4e09\u70b9\u5dee\u5206\u7684\u6570\u503c\u5fae\u5206\u7684\u5f62\u5f0f\uff1a\n\n$$\n{\\mathrm{d}X}_{pq} \\simeq \\frac{E_\\mathrm{tot}^\\mathsf{MP2} (d_{pq}) - E_\\mathrm{tot}^\\mathsf{MP2} (- d_{pq})}{2 d_{pq}}\n$$\n\n$E_\\mathrm{tot}^\\mathsf{MP2} (d_{pq})$ \u7684\u610f\u4e49\u662f\uff0c\u53cd\u5bf9\u79f0\u77e9\u9635 $\\mathbf{X}$ \u4ec5\u5728\u7b2c $p$ \u884c\u3001\u7b2c $q$ \u5217\u4e0a\uff0c$X_{pq} = d_{pq}$\uff1b\u4e14\u5728\u7b2c $q$ \u884c\u3001\u7b2c $p$ \u5217\u4e0a\uff0c$X_{qp} = - d_{pq}$\uff1b\u5176\u5b83\u4f4d\u7f6e\u4e0a\uff0c$\\mathbf{X}$ \u5747\u53d6\u5230\u96f6\u503c\u3002\u5982\u679c $p = q$\uff0c\u90a3\u4e48 $\\mathbf{X} = \\mathbf{0}$\u3002\u751f\u6210\u8fd9\u79cd\u53cd\u5bf9\u79f0\u77e9\u9635\u7684\u51fd\u6570 `gen_pert_X` \u5982\u4e0b\u6240\u793a\uff1a\n\n\n```python\ndef gen_pert_X(p, q, interval):\n X = np.zeros((nmo, nmo))\n X[p, q] = interval\n X -= X.T\n return X\n```\n\n\u90a3\u4e48\u4f9d\u636e\u4e0a\u8ff0\u53cd\u5bf9\u79f0\u77e9\u9635\uff0c\u6240\u6c42\u51fa\u7684 MP2 \u80fd\u91cf\u968f $X_{pq}$ \u53d8\u5316\u7684\u6570\u503c\u5bfc\u6570 ${\\mathrm{d}X}_{pq}$ \u7684\u751f\u6210\u51fd\u6570\u5982\u4e0b\uff1a\n\n\n```python\ndef eng_mp2_numdiff(p, q, interval):\n X_positive = gen_pert_X(p, q, interval)\n X_negative = gen_pert_X(p, q, -interval)\n return (eng_mp2_pert(X_positive) - eng_mp2_pert(X_negative)) / (2 * interval)\n```\n\n\u5bf9\u89d2\u6807 $p, q$ \u5faa\u73af\uff0c\u6211\u4eec\u5c31\u80fd\u6c42\u51fa\u5b8c\u6574\u7684\u5bfc\u6570\u77e9\u9635 `dX` ${\\mathrm{d} \\mathbf{X}}$ (\u8fd9\u91cc\u9009\u53d6\u7684\u6570\u503c\u5fae\u5206\u7684\u95f4\u9699\u503c `interval` \u4e3a $10^{-4}$ a.u.)\uff1a\n\n\n```python\ndX = np.zeros((nmo, nmo))\nfor a in range(nmo):\n for i in range(nmo):\n dX[a, i] = eng_mp2_numdiff(a, i, 1e-4)\ndX\n```\n\n\n\n\n array([[ 0. , -0. , -0. , -0. , 0. , 0.243 , -0.6191, 0.7465, -1.7459, 1.0327, -0.5983, 1.3028, -1.8584],\n [ 0. , 0. , -0. , 0. , 0. , -0.044 , -0.0219, 0.3123, -0.1693, -0.1755, 0.1931, -0.0509, 0.033 ],\n [ 0. , 0. , 0. , -0. , 0. , -0.0894, 0.0924, 0.2349, 0.1175, 0.0443, -0.3922, 0.1505, -0.4868],\n [ 0. , -0. , 0. , 0. , -0. , 0.0648, -0.0899, -0.0568, -0.0668, -0.137 , 0.2291, -0.0017, -0.1029],\n [-0. , -0. , -0. , 0. , 0. , -0.0091, 0.0252, 0.1021, -0.0093, -0.0796, 0.0327, -0.067 , -0.0761],\n [-0.243 , 0.044 , 0.0894, -0.0648, 0.0091, 0. , -0. , 0. , -0. , 0. , 0. , 0. , -0. ],\n [ 0.6191, 0.0219, -0.0924, 0.0899, -0.0252, 0. , 0. , 0. , -0. , 0. , 0. , 0. , -0. ],\n [-0.7465, -0.3123, -0.2349, 0.0568, -0.1021, -0. , -0. , 0. , 0. , 0. , 0. , 0. , -0. ],\n [ 1.7459, 0.1693, -0.1175, 0.0668, 0.0093, 0. , 0. , -0. , 0. , 0. , 0. , 0. , -0. ],\n [-1.0327, 0.1755, -0.0443, 0.137 , 0.0796, -0. , -0. , -0. , -0. , 0. , -0. , 0. , 0. ],\n [ 0.5983, -0.1931, 0.3922, -0.2291, -0.0327, -0. , -0. , -0. , -0. , 0. , 0. , -0. , 0. ],\n [-1.3028, 0.0509, -0.1505, 0.0017, 0.067 , -0. , -0. , -0. , -0. , -0. , 0. , 0. , -0. ],\n [ 1.8584, -0.033 , 0.4868, 0.1029, 0.0761, 0. , 0. , 0. , 0. , -0. , -0. , 0. , 0. ]])\n\n\n\n\u6ce8\u610f\u5230\u8fd9\u662f\u4e00\u4e2a\u53cd\u5bf9\u79f0\u4e14\u5206\u5757\u7684\u77e9\u9635\uff1b\u5728\u5360\u636e\u4e0e\u975e\u5360\u5206\u5757\u503c\u5b8c\u5168\u4e3a\u96f6\uff0c\u6709\u503c\u5904\u4ec5\u6709 $\\mathrm{d} X_{ai} = - \\mathrm{d} X_{ia}$\u3002\u8fd9\u5b9e\u9645\u4e0a\u8fd1\u4e4e\u7b49\u4e8e 2 \u500d\u7684\u8f68\u9053\u68af\u5ea6\u77e9\u9635 `2 * gfock_grad`\uff1a\n\n$$\n\\mathrm{d} X_{pq} = 2 x_{pq} = 2 (F_{pq} - F_{qp})\n$$\n\n\n```python\nmf_prhf = scf.RHF(mol)\nmf_prhf.mo_occ, mf_prhf.mo_coeff = mo_occ, C_base\nmf_pmp2 = mp.MP2(mf_prhf).run()\nrdm1, rdm2 = mf_pmp2.make_rdm1(), mf_pmp2.make_rdm2()\ngfock_grad = mf_cas.unpack_uniq_var(mf_cas.get_grad(C_base, (rdm1, rdm2), mf_cas.ao2mo(C_base)))\n```\n\n\n```python\nnp.allclose(2 * gfock_grad, dX, atol=5e-6)\n```\n\n\n\n\n True\n\n\n\n\u56e0\u6b64\uff0c\u53ef\u4ee5\u8bf4 OO-MP2 \u7684\u610f\u4e49\u662f\uff0c\u627e\u5230\u4e00\u4e2a\u5408\u9002\u7684 $\\mathbf{C}^\\mathrm{base}$\uff0c\u4f7f\u5f97\u5bf9\u4e8e\u4efb\u610f\u7684\u5f88\u5c0f\u7684\u3001\u7528\u4e8e\u65cb\u8f6c\u7684\u53cd\u5bf9\u79f0\u77e9\u9635 $\\mathbf{X}$\uff0c\u6709 $E_\\mathrm{tot}^\\mathsf{MP2} (\\mathbf{X})$ \u4e0d\u4f1a\u66f4\u6539\u3002\n\n## OO-MP2 \u80fd\u91cf\u5e76\u975e\u4e00\u5b9a\u6bd4 MP2 \u4f4e\n\n\u5728\u6587\u6863\u6700\u540e\uff0c\u6211\u4eec\u4f1a\u6307\u51fa\uff0cOO-MP2 \u80fd\u91cf\u5e76\u975e MP2 \u7684\u4e0b\u754c\u3002\u5c3d\u7ba1 OO-MP2 \u770b\u8d77\u6765\u5bf9\u8f68\u9053\u8fdb\u884c\u53d8\u5206\u5f0f\u7684\u4f18\u5316\uff0c\u4f46\u5176\u53d8\u5206\u7684\u5bf9\u8c61\u5e94\u5f53\u8ba4\u4e3a\u662f Hylleraas \u6cdb\u51fd\uff0c\u800c\u975e\u603b MP2 \u80fd\u91cf\u3002\n\n\u5bf9\u4e8e\u4e0b\u8ff0\u62c9\u957f\u7684\u6c22\u5206\u5b50\uff0c\u5c31\u662f\u4e00\u4e2a OO-MP2 \u80fd\u91cf\u6bd4 MP2 \u80fd\u91cf\u9ad8\u7684\u4f8b\u5b50\u3002\n\n\n```python\nmol = gto.Mole()\nmol.atom = \"\"\"\nH 0. 0. 0.\nH 0. 0. 15.\n\"\"\"\nmol.basis = \"6-31G\"\nmol.verbose = 0\nmol.build()\n```\n\n\n\n\n \n\n\n\n\u5176 MP2 \u80fd\u91cf\u4e3a\n\n\n```python\nmol.RHF().run().MP2().run().e_tot\n```\n\n\n\n\n -1.7458592201255043\n\n\n\n\u800c\u5176 OO-MP2 \u80fd\u91cf\u4e3a\n\n\n```python\nmf_cas = mcscf.CASSCF(mol.RHF().run(), mol.nao, mol.nelectron)\nmf_cas.fcisolver = MP2AsFCISolver()\nmf_cas.internal_rotation = True\ncas_result = mf_cas.kernel()\ncas_result[0]\n```\n\n\n\n\n -1.7280760742391805\n\n\n\n\u4f46\u5373\u4f7f OO-MP2 \u7684\u80fd\u91cf\u6bd4 MP2 \u9ad8\uff0c\u5b83\u4ecd\u7136\u65e0\u6cd5\u89e3\u51b3 MP2 \u65b9\u6cd5\u5728\u89e3\u79bb\u4e24\u4e2a\u6c22\u539f\u5b50\u6240\u4ea7\u751f\u7684\u76f8\u5f53\u5927\u7684\u89e3\u79bb\u8bef\u5dee\u3002\n\n[^Sun-Chan.JCP.2020]: Recent Developments in the PySCF Program Package. *J. Chem. Phys.* **2020**, *153* (2), 24109. doi: [10.1063/5.0006074](https://doi.org/10.1063/5.0006074).\n", "meta": {"hexsha": "dc57d1e5d88368469ed25a6c83484d1bbd290995", "size": 41752, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "source/QC_Notes/Post_Series/oomp2.ipynb", "max_stars_repo_name": "ajz34/ajz34.readthedocs.io", "max_stars_repo_head_hexsha": "73be05a73241c18b98fd0d4dbdc48c643278c3da", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-07-30T12:31:14.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-14T03:56:56.000Z", "max_issues_repo_path": "source/QC_Notes/Post_Series/oomp2.ipynb", "max_issues_repo_name": "ajz34/ajz34.readthedocs.io", "max_issues_repo_head_hexsha": "73be05a73241c18b98fd0d4dbdc48c643278c3da", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "source/QC_Notes/Post_Series/oomp2.ipynb", "max_forks_repo_name": "ajz34/ajz34.readthedocs.io", "max_forks_repo_head_hexsha": "73be05a73241c18b98fd0d4dbdc48c643278c3da", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-07-30T12:32:09.000Z", "max_forks_repo_forks_event_max_datetime": "2020-07-30T12:32:09.000Z", "avg_line_length": 27.5590759076, "max_line_length": 258, "alphanum_fraction": 0.4779890784, "converted": true, "num_tokens": 13016, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.4198689200097166}} {"text": "# \"Revisiting AlexNet\"\n\n- toc: true\n- branch: master\n- badges: true\n- comments: true\n- categories: [fastpages, jupyter]\n- image: images/chef_bepita.jpg\n- hide: false\n- search_exclude: true\n- metadata_key1: metadata_value1\n- metadata_key2: metadata_value2\n\n\n## Architecture Tricks\n\n### ReLU Nonlinearity\n\n

                                        \nThe first innovation that the paper brings to the table is the now most common ReLU ($ f(x) = max(0,x) $) activation function. Till then the most common way of applying non linearity to neural networks was the tanh ( $ f(x) = (1 + e^{-x})^{-1} $) activation function. Actually ReLUs were first ??? introduced by Nair and Hinton[???] but it was in the context of restricted Boltzman machines ???. The advantage of ReLU over tanh is that ReLU does not saturate and even if some fraction of the current batch will produce a positive input learning will occur in that pass. Therefore training a deep neural network with ReLUs is considerably faster. \n\n\n\n### Training on multiple GPUs\n\n

                                        \nBack in the old days of 2011-2012 RTX2080 Ti were not available. The GPU they had was GTX 580 with 3GB of memory. But AlexNet implemntation taks advantage of the ability of GPUs to cross parrallelize the training. So AlexNet was trained on a pair of GTX 580 and the network architecture was designed for that purpase \n\n### Local Response Normalization\n\n

                                        \nThe use of ReLUs makes input normalization less crucial because because of their non saturating nature. As long as there are positive inputs in a batch learning will happen. That said, the authors used normalization to make training faster. The normalization used is not common nowaddays and called `Local Response Normalization` and is defined as follows:\n\n\\begin{equation}\nb_{x,y}^{i} = \\frac{a_{x,y}^{i}}{\\left( k + \\alpha \\sum_{j=max(0,i-n/2)}^{min(N-1,i+n/2) } (a_{x,y}^{j})^{2} \\right)^{\\beta}}\n\\end{equation}\n\n

                                        \nwhere $k, n, \\alpha, \\beta$ hyperparameters. The convolution filters are ordered by index before the training begins and the normalization is done among the n \"neighbour\" filters. The authors call this `brightness normalization` because the mean of activations is not subtracted.\n\n### Overlapping Pooling\n\n

                                        \nMax pooling is applied to the output of the first, second and fifth layers. The pooling kernals overlap in a sense that the stride is smaller than the kernel width.the authors mention that it helps to reduce overfitting. \n\n## Reduce Overfitting Tricks\n\n### Data Augmentation\n\n

                                        \nData augmentation is done in two forms:\n

                                          \n

                                        • Transformations - Randomly take a 224X224 patch from our 256X256 images and their horisontal reflections.

                                          • \n

                                        • RGB Intensities - Preform PCA on the ImageNET training set and add multiples of the found princples. Specifically to each pixel the values \n\\begin{equation}\n[P_{1},P_{2},P_{3}][\\alpha_{1}\\lambda_{1},\\alpha_{2}\\lambda_{2},\\alpha_{3}\\lambda_{3}]^{T}\n\\end{equation}\n

                                          \nare added to each pixel values where $P_{i}$ and $\\alpha_{i}$ are the $i$-th eigenvalues and eigenvectors of 3X3 covariance matrix of RGB pixel values and $\\alpha_{i}$ are drawn per image per use in train from a zero mean Gaussian distribution with standart deviation 0.1.

                                          • \n
                                        \n\n\n### Dropout\n\n

                                        \nThis models is one of the first ones that used dropout. In this thchnique every neuron's output is zeroed in a probability of 0.5 and does not contribute to the forward pass and does not participate in the backward pass. In this way we ensure that complex dependencies between neurons do not replace a single neuron contribution and simplicity is encouraged.\nAt test time we multiply all neaurons by 0.5 to keep the overall sum of weights.\n\n## Bibiliography\n", "meta": {"hexsha": "a0064a07a7b5e18a82dfc9efc8c35b1ce758fb34", "size": 7135, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "_notebooks/2020-07-08-alxnet.ipynb", "max_stars_repo_name": "moshebeutel/paper-fact-check", "max_stars_repo_head_hexsha": "01acd9a06bf6ea956927df374cf14bc6ebc92f03", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "_notebooks/2020-07-08-alxnet.ipynb", "max_issues_repo_name": "moshebeutel/paper-fact-check", "max_issues_repo_head_hexsha": "01acd9a06bf6ea956927df374cf14bc6ebc92f03", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "_notebooks/2020-07-08-alxnet.ipynb", "max_forks_repo_name": "moshebeutel/paper-fact-check", "max_forks_repo_head_hexsha": "01acd9a06bf6ea956927df374cf14bc6ebc92f03", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.7783505155, "max_line_length": 652, "alphanum_fraction": 0.6318149965, "converted": true, "num_tokens": 1083, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.6825737214979745, "lm_q1q2_score": 0.4198428065878099}} {"text": "# Multigrid Lecture and Homework Example\n\n### Aron Ahmadia (US Army ERDC) and David Ketcheson (KAUST)\n\n### Teaching Numerical Methods with IPython Notebooks, SciPy 2014\n\n
                                        This example by Aron Ahmadia and David Ketcheson is licensed under a Creative Commons Attribution 4.0 International License. All code examples are also licensed under the [MIT license](http://opensource.org/licenses/MIT).\n\n\n```\nfrom IPython.core.display import HTML\ncss_file = './example.css'\nHTML(open(css_file, \"r\").read())\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n# Multigrid\n\nMultigrid is one of the great algorithms for numerically solving PDEs. It is also one of the few essentially optimal algorithms for solving a linear system of equations, since it computes the solution of an $N\\times N$ system in ${\\mathcal O}(N\\log(N))$ -- or even just ${\\mathcal O}(N)$ -- operations. \n\nThis notebook is meant to accompany a reading of [Section 4.6 of Randall LeVeque's text on finite difference methods](http://0-epubs.siam.org.library.kaust.edu.sa/doi/abs/10.1137/1.9780898717839.ch4). Other good resources are cited there.\n\n\n```\n%matplotlib inline\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom matplotlib import animation\nfrom clawpack.visclaw.JSAnimation import IPython_display\n```\n\n## A two-point boundary value problem\nLet's use Jacobi's method to solve the one-dimensional boundary value problem \n\n\\begin{align}\nu''(x) & = f(x) & 0Table of Contents\n

                                        \n\n\n```python\n# code for loading the format for the notebook\nimport os\n\n# path : store the current path to convert back to it later\npath = os.getcwd()\nos.chdir(os.path.join('..', '..', 'notebook_format'))\nfrom formats import load_style\nload_style(css_style = 'custom2.css', plot_style = False)\n```\n\n\n\n\n\n\n\n\n\n\n\n\n```python\nos.chdir(path)\n\n# 1. magic for inline plot\n# 2. magic to print version\n# 3. magic so that the notebook will reload external python modules\n# 4. magic to enable retina (high resolution) plots\n# https://gist.github.com/minrk/3301035\n%matplotlib inline\n%load_ext watermark\n%load_ext autoreload\n%autoreload 2\n%config InlineBackend.figure_format = 'retina'\n\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nfrom sklearn.metrics import roc_auc_score\nfrom sklearn.linear_model import LogisticRegression\nfrom sklearn.model_selection import train_test_split\nfrom sklearn.feature_extraction.text import TfidfVectorizer\n\n%watermark -a 'Ethen' -d -t -v -p numba,numpy,pandas,sklearn,matplotlib\n```\n\n Ethen 2018-02-24 20:09:44 \n \n CPython 3.6.3\n IPython 6.1.0\n \n numba 0.37.0\n numpy 1.14.1\n pandas 0.22.0\n sklearn 0.19.1\n matplotlib 2.1.0\n\n\n# Factorization Machine (FM)\n\n**Factorization Machine** type algorithms are a combination of linear regression and matrix factorization, the cool idea behind this type of algorithm is it aims model interactions between features (a.k.a attributes, explanatory variables) using factorized parameters. By doing so it has the ability to estimate all interactions between features even with extremely sparse data.\n\n## Introduction\n\nNormally, when we think of linear regression, we would think of the following formula:\n\n\\begin{align}\n\\hat{y}(\\textbf{x}) = w_{0} + \\sum_{i=1}^{n} w_{i} x_{i}\n\\end{align}\n\nWhere:\n\n- $w_0$ is the bias term, a.k.a intercept.\n- $w_i$ are weights corresponding to each feature vector $x_i$, here we assume we have $n$ total features.\n\nThis formula's advantage is that it can computed in linear time, $O(n)$. The drawback, however, is that it does not handle feature interactions. To capture interactions, we could introduce a weight for each feature combination. This is sometimes referred to as a $2_{nd}$ ordered polynomial. The resulting model is shown below:\n\n\\begin{align}\n\\hat{y}(\\textbf{x}) = w_{0} + \\sum_{i=1}^{n} w_{i} x_{i} + \\sum_{i=1}^n \\sum_{j=i+1}^n w_{ij} x_{i} x_{j}\n\\end{align}\n\nCompared to our previous model, this formulation has the advantages that it can capture feature interactions at least for two features at a time. But we have now ended up with a $O(n^2)$ complexity which means that to train the model, we now require a lot more time and memory. Another issue is that when we have categorical variables with high cardinality, after one-hot encoding them, we would end up with a lot of columns that are sparse, making it harder to actually capture their interactions (not enough data).\n\nTo solve this complexity issue, Factorization Machines takes inspiration from [matrix factorization](http://nbviewer.jupyter.org/github/ethen8181/machine-learning/blob/master/recsys/1_ALSWR.ipynb), and models the feature interaction using latent factors. Every feature $f_i$ has a corresponding latent factor $v_i$, and two features' interactions are modelled as $\\langle \\textbf{v}_i, \\textbf{v}_{j} \\rangle$, where $\\langle \\cdot \\;,\\cdot \\rangle$ refers to the dot product of the two feature vector. If we assume its of size $k$ (this is a hyperparameter that we can tune). Then:\n\n\\begin{align}\n\\langle \\textbf{v}_i, \\textbf{v}_{j} \\rangle = \\sum_{f=1}^k v_{i,f} v_{j,f}\n\\end{align}\n\nThis leads of our new equation:\n\n\\begin{align}\n\\hat{y}(\\textbf{x}) = w_{0} + \\sum_{i=1}^{n} w_{i} x_{i} + \\sum_{i=1}^{n} \\sum_{j=i+1}^n \\langle \\textbf{v}_i , \\textbf{v}_{j} \\rangle x_i x_{j}\n\\end{align}\n\nThis is an improvement from our previous model (when we modeled each pair of interaction terms with weight $w_{ij}$) as the number of parameters is reduced from $n^2$ to $n \\times k$, since $k \\ll n$, which also helps mitigate overfitting issues. Using the naive way of formulating factorization machine results in a complexity of $O(kn^2)$, because all pairwise interactions have to be computed, but we can reformulate it to make it run in $O(kn)$.\n\n\n\\begin{align}\n\\sum_{i=1}^n \\sum_{j=i+1}^n \\langle \\textbf{v}_i, \\textbf{v}_{j} \\rangle x_{i} x_{j}\n&= \\frac{1}{2} \\sum_{i=1}^n \\sum_{j=1}^n \\langle \\textbf{v}_i, \\textbf{v}_{j} \\rangle x_{i} x_{j} - \\frac{1}{2} \\sum_{i=1}^n \\langle \\textbf{v}_i , \\textbf{v}_{i} \\rangle x_{i} x_{i} \\\\\n&= \\frac{1}{2}\\left(\\sum_{i=1}^n \\sum_{j=1}^n \\sum_{f=1}^k v_{i,f} v_{j,f} x_{i} x_{j} \\right)\\frac{1}{2}\\left( \\sum_{i=1}^n \\sum_{f=1}^k v_{i,f} v_{i,f} x_{i} x_{i} \\right) \\\\\n&= \\frac{1}{2}\\left(\\sum_{i=1}^n \\sum_{j=1}^n \\sum_{f=1}^k v_{i,f} v_{j,f} x_{i} x_{j} - \\sum_{i=1}^n \\sum_{f=1}^k v_{i,f} v_{i,f} x_{i} x_{i} \\right) \\\\\n&= \\frac{1}{2} \\sum_{f=1}^{k} \\left( \\left(\\sum_{i=1}^n v_{i,f}x_{i} \\right) \\left( \\sum_{j=1}^n v_{j,f}x_{j} \\right) - \\sum_{i=1}^{n} v_{i,f}^2 x_{i}^2 \\right) \\\\\n&= \\frac{1}{2} \\sum_{f=1}^{k} \\left( \\left( \\sum_{i}^{n} v_{i,f}x_{i} \\right)^2 - \\sum_{i=1}^{n} v_{i,f}^2 x_{i}^2 \\right)\n\\end{align}\n\nNote, summing over different pairs is the same as summing over all pairs minus the self-interactions (divided by two). This is the reason why the value 1/2 is introduced from the beginning of the derivation.\n\nThis reformulated equation has a linear complexity in both $k$ and $n$, i.e. its computation is in $O(kn)$, substituting this new equation into the existing factorization machine formula, we end up with:\n\n\\begin{align}\n\\hat{y}(\\textbf{x}) = w_{0} + \\sum_{i=1}^{n} w_{i} x_{i} + \\frac{1}{2} \\sum_{f=1}^{k} \\left( \\left( \\sum_{i}^{n} v_{i,f}x_{i} \\right)^2 - \\sum_{i=1}^{n} v_{i,f}^2 x_{i}^2 \\right)\n\\end{align}\n\nIn a machine learning setting, factorization machine can be applied to different supervised prediction tasks:\n\n- **Regression:**, in this case $\\hat{y}(\\textbf{x})$ can be used directly by minimizing the mean squared error between the model prediction and target value, e.g. $\\frac{1}{N}\\sum^{N}\\big(y - \\hat{y}(\\textbf{x})\\big)^2$\n- **Classification:**, if we were to use it in a binary classification setting, we could then minimize the log loss, $\\ln \\big(e^{-y \\cdot \\hat{y}(\\textbf{x})} + 1 \\big)$, where $\\sigma$ is the sigmoid/logistic function and $y \\in {-1, 1}$.\n\nTo train factorization machine, we can use a gradient descent based optimization techniques, the parameters to be learned are $(w_0, \\mathbf{w},$ and $\\mathbf{V}$).\n\n\\begin{align}\n\\frac{\\partial}{\\partial\\theta}\\hat{y}(\\textbf{x}) =\n\\begin{cases}\n1, & \\text{if $\\theta$ is $w_0$} \\\\\nx_i, & \\text{if $\\theta$ is $w_i$} \\\\\nx_i\\sum_{j=1}^{n} v_{j,f}x_j - v_{i,f}x_{i}^2 & \\text{if $\\theta$ is $v_{i,f}$}\n\\end{cases}\n\\end{align}\n\n- Notice that $\\sum_{j=1}^n v_{j, f} x_j$ does not depend on $i$, thus it can be computed independently.\n- The last formula above, can also be written as $x_i(\\sum_{j=1}^{n} v_{j,f}x_j - v_{i,f}x_{i})$.\n- In practice, we would throw in some L2 regularization to prevent overfitting.\n\nAs the next section contains implementation of the algorithm from scratch, the gradient of the log loss is also provided here for completeness. The predicted value $\\hat{y}(\\textbf{x})$ is replaced with $x$ for making the notation cleaner.\n\n\n\\begin{align}\n\\frac{d}{dx}\\left[ \\ln \\big(e^{-yx} + 1 \\big) \\right] \n&= \\frac{1}{e^{-yx} + 1} \\cdot \\frac{d}{dx}\\left[e^{-yx} + 1 \\right] \\\\\n&= \\frac{\\frac{d}{dx}\\left[e^{-yx} \\right] + \\frac{d}{dx}\\left[1 \\right]}{e^{-yx} + 1} \\\\\n&= \\frac{e^{-yx} \\cdot \\frac{d}{dx}\\left[-yx \\right] + 0}{e^{-yx} + 1} \\\\\n&= \\frac{e^{-yx} \\cdot -y}{e^{-yx} + 1} \\\\\n&= -\\frac{ye^{-yx}}{e^{-yx} + 1} \\\\\n&= -\\frac{y}{e^{yx} + 1}\n\\end{align}\n\n---\n\n**Advantages:** We'll now wrap up the theoretical section of factorization machine, with some of its advantages:\n\n- We can observe from the model equation that it can be computed in linear time.\n- By leveraging ideas from matrix factorization, we can estimate higher order interaction effects even under very sparse data.\n- Compared to traditional matrix factorization methods, which is restricted to modeling a user-item matrix, we can leverage other user or item specific features making factorization machine more flexible.\n\n## Implementation\n\nFor the implementation of factorization machine, we'll use a for loop based code as I personally find it easier to comprehend for the gradient update section. There are different ways to speed up for loop based code in Python, such as using [Cython or Numba](http://nbviewer.jupyter.org/github/ethen8181/machine-learning/blob/master/python/cython/cython.ipynb), here we'll be using Numba.\n\n\n```python\n# using the example spam dataset\n# read it in, extract the input and output columns\nlabel_col = 'label_num'\nsms = pd.read_table('sms.tsv', header = None, names = ['label', 'message'])\nsms[label_col] = sms['label'].map({'ham': 0, 'spam': 1})\nX = sms['message']\ny = sms[label_col].values\n\n# split X and y into training and testing sets\nX_train, X_test, y_train, y_test = train_test_split(\n X, y, test_size = 0.25, random_state = 1)\n\n# convert both sets' text column to document-term matrix;\n# ideally, we would want to perform some preprocessing on\n# our text data, but let's be lazy here as that's not\n# the goal of this documentation\ntfidf = TfidfVectorizer(min_df = 2, max_df = 0.5)\nX_train_dtm = tfidf.fit_transform(X_train)\nX_test_dtm = tfidf.transform(X_test)\nX_train_dtm\n```\n\n\n\n\n <4179x3508 sparse matrix of type ''\n \twith 51261 stored elements in Compressed Sparse Row format>\n\n\n\n\n```python\nimport numpy as np\nfrom numba import njit\nfrom tqdm import trange\nfrom sklearn.base import BaseEstimator, ClassifierMixin\n\n\nclass FactorizationMachineClassifier(BaseEstimator, ClassifierMixin):\n \"\"\"\n Factorization Machine [1]_ using Stochastic Gradient Descent.\n For binary classification only.\n\n Parameters\n ----------\n n_iter : int, default 10\n Number of iterations to train the algorithm.\n\n n_factors : int, default 10\n Number/dimension of features' latent factors.\n\n learning_rate : float, default 0.1\n Learning rate for the gradient descent optimizer.\n\n reg_coef : float, default 0.01\n Regularization strength for weights/coefficients.\n\n reg_factors : float, default 0.01\n Regularization strength for features' latent factors.\n\n random_state : int, default 1234\n Seed for the randomly initialized features latent factors\n\n verbose : bool, default True\n Whether to print progress bar while training.\n\n Attributes\n ----------\n intercept_ : double\n Intercept term, w0 based on the original notations.\n\n coef_ : 1d ndarray, shape [n_features,]\n Coefficients, w based on the original notations.\n\n feature_factors_ : 2d ndarray, shape [n_factors, n_features]\n Latent factors for all features. v based on the original\n notations. The learned factors can be viewed as the\n embeddings for each features. If a pair of features tends\n to co-occur often, then their embeddings should be\n close/similar (in terms of cosine similarity) to each other.\n\n history_ : list\n Loss function's history at each iteration, useful\n for evaluating whether the algorithm converged or not.\n\n References\n ----------\n .. [1] `S. Rendle Factorization Machines (2010)\n `_ \n \"\"\"\n\n def __init__(self, n_iter = 10, n_factors = 10,\n learning_rate = 0.1, reg_coef = 0.01,\n reg_factors = 0.01, random_state = 1234, verbose = False):\n self.n_iter = n_iter\n self.verbose = verbose\n self.reg_coef = reg_coef\n self.n_factors = n_factors\n self.reg_factors = reg_factors\n self.random_state = random_state\n self.learning_rate = learning_rate\n\n def fit(self, X, y):\n \"\"\"\n Fit the model to the input data and label.\n\n Parameters\n ----------\n X : scipy sparse csr_matrix, shape [n_samples, n_features]\n Data in sparse matrix format.\n\n y : 1d ndarray, shape [n_samples,]\n Training data's corresponding label.\n\n Returns\n -------\n self\n \"\"\"\n\n n_samples, n_features = X.shape\n self.coef_ = np.zeros(n_features)\n self.intercept_ = 0.0\n\n # the factors are often initialized with a mean of 0 and standard deviation\n # of 1 / sqrt(number of latent factor specified)\n np.random.seed(self.random_state)\n self.feature_factors_ = np.random.normal(\n scale = 1 / np.sqrt(self.n_factors), size = (self.n_factors, n_features))\n \n # the gradient is implemented in a way that requires\n # the negative class to be labeled as -1 instead of 0\n y = y.copy().astype(np.int32)\n y[y == 0] = -1\n\n loop = range(self.n_iter)\n if self.verbose:\n loop = trange(self.n_iter)\n\n self.history_ = []\n for _ in loop:\n loss = _sgd_update(X.data, X.indptr, X.indices,\n y, n_samples, n_features,\n self.intercept_, self.coef_,\n self.feature_factors_, self.n_factors,\n self.learning_rate, self.reg_coef, self.reg_factors)\n self.history_.append(loss)\n\n return self\n\n def predict_proba(self, X):\n \"\"\"\n Probability estimates. The returned estimates for\n all classes are ordered by the label of classes.\n\n Paramters\n ---------\n X : scipy sparse csr_matrix, shape [n_samples, n_features]\n Data in sparse matrix format.\n\n Returns\n -------\n proba : 2d ndarray, shape [n_samples, n_classes]\n The probability of the sample for each class in the model.\n \"\"\"\n pred = self._predict(X)\n pred_proba = 1.0 / (1.0 + np.exp(-pred))\n proba = np.vstack((1 - pred_proba, pred_proba)).T\n return proba\n\n def _predict(self, X):\n \"\"\"Similar to _predict_instance but vectorized for all samples\"\"\"\n linear_output = X * self.coef_\n v = self.feature_factors_.T\n term = (X * v) ** 2 - (X.power(2) * (v ** 2))\n factor_output = 0.5 * np.sum(term, axis = 1)\n return self.intercept_ + linear_output + factor_output\n\n def predict(self, X):\n \"\"\"\n Predict class labels for samples in X.\n\n Parameters\n ----------\n X : scipy sparse csr_matrix, shape [n_samples, n_features]\n Data in sparse matrix format.\n\n Returns\n -------\n Predicted class label per sample.\n \"\"\"\n pred_proba = self.predict_proba(X)[:, 1]\n return pred_proba.round().astype(np.int)\n\n\n@njit\ndef _sgd_update(data, indptr, indices, y, n_samples, n_features,\n w0, w, v, n_factors, learning_rate, reg_w, reg_v):\n \"\"\"\n Compute the loss of the current iteration and update\n gradients accordingly.\n \"\"\"\n loss = 0.0\n for i in range(n_samples):\n pred, summed = _predict_instance(data, indptr, indices, w0, w, v, n_factors, i)\n \n # calculate loss and its gradient\n loss += _log_loss(pred, y[i])\n loss_gradient = -y[i] / (np.exp(y[i] * pred) + 1.0)\n \n # update bias/intercept term\n w0 -= learning_rate * loss_gradient\n\n # update weight\n for index in range(indptr[i], indptr[i + 1]):\n feature = indices[index]\n w[feature] -= learning_rate * (loss_gradient * data[index] + 2 * reg_w * w[feature])\n\n # update factor\n for factor in range(n_factors):\n for index in range(indptr[i], indptr[i + 1]):\n feature = indices[index]\n term = summed[factor] - v[factor, feature] * data[index]\n v_gradient = loss_gradient * data[index] * term\n v[factor, feature] -= learning_rate * (v_gradient + 2 * reg_v * v[factor, feature])\n \n loss /= n_samples\n return loss\n\n\n@njit\ndef _predict_instance(data, indptr, indices, w0, w, v, n_factors, i):\n \"\"\"predicting a single instance\"\"\"\n summed = np.zeros(n_factors)\n summed_squared = np.zeros(n_factors)\n\n # linear output w * x\n pred = w0\n for index in range(indptr[i], indptr[i + 1]):\n feature = indices[index]\n pred += w[feature] * data[index]\n\n # factor output\n for factor in range(n_factors):\n for index in range(indptr[i], indptr[i + 1]):\n feature = indices[index]\n term = v[factor, feature] * data[index]\n summed[factor] += term\n summed_squared[factor] += term * term\n\n pred += 0.5 * (summed[factor] * summed[factor] - summed_squared[factor])\n \n # summed is the independent term that can be re-used\n # during the gradient update stage\n return pred, summed\n\n\n@njit\ndef _log_loss(pred, y):\n \"\"\"\n negative log likelihood of the\n current prediction and label, y.\n \"\"\"\n return np.log(np.exp(-pred * y) + 1.0)\n```\n\n\n```python\nfm = FactorizationMachineClassifier(n_iter = 30, learning_rate = 0.1)\nfm.fit(X_train_dtm, y_train)\n```\n\n\n\n\n FactorizationMachineClassifier(learning_rate=0.1, n_factors=10, n_iter=30,\n random_state=1234, reg_coef=0.01, reg_factors=0.01,\n verbose=False)\n\n\n\n\n```python\n# change default style figure and font size\nplt.rcParams['figure.figsize'] = 8, 6\nplt.rcParams['font.size'] = 12\n\n# one quick way to check that we've implemented\n# the gradient descent is to ensure that the loss\n# curve is steadily decreasing\nplt.plot(fm.history_)\nplt.title('Loss Curve Per Iteration')\nplt.xlabel('Iterations')\nplt.ylabel('Loss')\nplt.show()\n```\n\n\n```python\n# predict on the test set and output the auc score\ny_pred_prob = fm.predict_proba(X_test_dtm)[:, 1]\nauc = roc_auc_score(y_test, y_pred_prob)\nprint('auc', auc)\n```\n\n auc 0.9973867907642742\n\n\n\n```python\n# we can compare it with a logistic regression,\nlogreg = LogisticRegression()\nlogreg.fit(X_train_dtm, y_train)\ny_pred_prob = logreg.predict_proba(X_test_dtm)[:, 1]\nauc = roc_auc_score(y_test, y_pred_prob)\nprint('auc', auc)\n```\n\n auc 0.9949615178092\n\n\nThere are various open-sourced implementations floating around the web, here are the links to some of them:\n\n- https://github.com/ibayer/fastFM\n- https://github.com/srendle/libfm\n- https://github.com/aksnzhy/xlearn\n- https://github.com/scikit-learn-contrib/polylearn\n\nI personally haven't tested which one is more efficient, feel free to grab one of them as see if it helps solve your problem.\n\n# Reference\n\n- [Blog: Factorization Machines](http://www.jefkine.com/recsys/2017/03/27/factorization-machines/)\n- [Blog: Deep Understanding of FFM Principles and Practices (Chinese)](https://tech.meituan.com/deep-understanding-of-ffm-principles-and-practices.html)\n- [Quora: What are the drawbacks of Factorization Machines?](https://www.quora.com/What-are-the-drawbacks-of-Factorization-Machines)\n- [Paper: S. Rendle Factorization Machines (2010)](https://www.csie.ntu.edu.tw/~b97053/paper/Rendle2010FM.pdf)\n", "meta": {"hexsha": "40ddeed1f2a807c2ef0cd8eab8042e9c58b39202", "size": 87003, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "recsys/factorization_machine/factorization_machine.ipynb", "max_stars_repo_name": "certara-ShengnanHuang/machine-learning", "max_stars_repo_head_hexsha": "d21dfbeabf2876ffe49fcef444ca4516c4d36df0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2104, "max_stars_repo_stars_event_min_datetime": "2016-04-15T13:35:55.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-28T10:39:51.000Z", "max_issues_repo_path": "recsys/factorization_machine/factorization_machine.ipynb", "max_issues_repo_name": "certara-ShengnanHuang/machine-learning", "max_issues_repo_head_hexsha": "d21dfbeabf2876ffe49fcef444ca4516c4d36df0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 10, "max_issues_repo_issues_event_min_datetime": "2017-04-07T14:25:23.000Z", "max_issues_repo_issues_event_max_datetime": "2021-05-18T03:16:15.000Z", "max_forks_repo_path": "recsys/factorization_machine/factorization_machine.ipynb", "max_forks_repo_name": "certara-ShengnanHuang/machine-learning", "max_forks_repo_head_hexsha": "d21dfbeabf2876ffe49fcef444ca4516c4d36df0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 539, "max_forks_repo_forks_event_min_datetime": "2015-12-10T04:23:44.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-31T07:15:28.000Z", "avg_line_length": 83.8179190751, "max_line_length": 48436, "alphanum_fraction": 0.7532039125, "converted": true, "num_tokens": 7280, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.6992544147913994, "lm_q1q2_score": 0.4196843599191048}} {"text": "\n\n##### Copyright 2020 The TensorFlow Authors.\n\n\n```\n#@title Licensed under the Apache License, Version 2.0 (the \"License\");\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n# https://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\n```\n\n# Hello, many worlds\n\n\n \n \n \n \n
                                        \n View on TensorFlow.org\n \n Run in Google Colab\n \n View source on GitHub\n \n Download notebook\n
                                        \n\nThis tutorial shows how a classical neural network can learn to correct qubit calibration errors. It introduces Cirq, a Python framework to create, edit, and invoke Noisy Intermediate Scale Quantum (NISQ) circuits, and demonstrates how Cirq interfaces with TensorFlow Quantum.\n\n## Setup\n\n\n```\n!pip install tensorflow==2.3.1\n```\n\n Collecting tensorflow==2.3.1\n \u001b[?25l Downloading https://files.pythonhosted.org/packages/eb/18/374af421dfbe74379a458e58ab40cf46b35c3206ce8e183e28c1c627494d/tensorflow-2.3.1-cp37-cp37m-manylinux2010_x86_64.whl (320.4MB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 320.4MB 42kB/s \n \u001b[?25hRequirement already satisfied: gast==0.3.3 in /usr/local/lib/python3.7/dist-packages (from tensorflow==2.3.1) (0.3.3)\n Requirement already satisfied: astunparse==1.6.3 in /usr/local/lib/python3.7/dist-packages (from tensorflow==2.3.1) (1.6.3)\n Requirement already satisfied: absl-py>=0.7.0 in 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Successfully installed cirq-0.9.1 freezegun-0.3.15 sympy-1.5 tensorflow-quantum-0.4.0\n\n\nNow import TensorFlow and the module dependencies:\n\n\n```\nimport tensorflow as tf\nimport tensorflow_quantum as tfq\n\nimport cirq\nimport sympy\nimport numpy as np\n\n# visualization tools\n%matplotlib inline\nimport matplotlib.pyplot as plt\nfrom cirq.contrib.svg import SVGCircuit\n```\n\n## 1. The Basics\n\n### 1.1 Cirq and parameterized quantum circuits\n\nBefore exploring TensorFlow Quantum (TFQ), let's look at some Cirq basics. Cirq is a Python library for quantum computing from Google. You use it to define circuits, including static and parameterized gates.\n\nCirq uses SymPy symbols to represent free parameters.\n\n\n```\na, b = sympy.symbols('a b')\n```\n\nThe following code creates a two-qubit circuit using your parameters:\n\n\n```\n# Create two qubits\nq0, q1 = cirq.GridQubit.rect(1, 2)\n\n# Create a circuit on these qubits using the parameters you created above.\ncircuit = cirq.Circuit(\n cirq.rx(a).on(q0),\n cirq.ry(b).on(q1), cirq.CNOT(control=q0, target=q1))\n\nSVGCircuit(circuit)\n```\n\n findfont: Font family ['Arial'] not found. Falling back to DejaVu Sans.\n\n\n\n\n\n \n\n \n\n\n\nTo evaluate circuits, you can use the `cirq.Simulator` interface. You replace free parameters in a circuit with specific numbers by passing in a `cirq.ParamResolver` object. The following code calculates the raw state vector output of your parameterized circuit:\n\n\n```\n# Calculate a state vector with a=0.5 and b=-0.5.\nresolver = cirq.ParamResolver({a: 0.5, b: -0.5})\noutput_state_vector = cirq.Simulator().simulate(circuit, resolver).final_state_vector\noutput_state_vector\n```\n\n\n\n\n array([ 0.9387913 +0.j , -0.23971277+0.j ,\n 0. +0.06120872j, 0. -0.23971277j], dtype=complex64)\n\n\n\nState vectors are not directly accessible outside of simulation (notice the complex numbers in the output above). To be physically realistic, you must specify a measurement, which converts a state vector into a real number that classical computers can understand. Cirq specifies measurements using combinations of the Pauli operators $\\hat{X}$, $\\hat{Y}$, and $\\hat{Z}$. As illustration, the following code measures $\\hat{Z}_0$ and $\\frac{1}{2}\\hat{Z}_0 + \\hat{X}_1$ on the state vector you just simulated:\n\n\n```\nz0 = cirq.Z(q0)\n\nqubit_map={q0: 0, q1: 1}\n\nz0.expectation_from_state_vector(output_state_vector, qubit_map).real\n```\n\n\n\n\n 0.8775825500488281\n\n\n\n\n```\nz0x1 = 0.5 * z0 + cirq.X(q1)\n\nz0x1.expectation_from_state_vector(output_state_vector, qubit_map).real\n```\n\n\n\n\n -0.04063427448272705\n\n\n\n### 1.2 Quantum circuits as tensors\n\nTensorFlow Quantum (TFQ) provides `tfq.convert_to_tensor`, a function that converts Cirq objects into tensors. This allows you to send Cirq objects to our quantum layers and quantum ops. The function can be called on lists or arrays of Cirq Circuits and Cirq Paulis:\n\n\n```\n# Rank 1 tensor containing 1 circuit.\ncircuit_tensor = tfq.convert_to_tensor([circuit])\n\nprint(circuit_tensor.shape)\nprint(circuit_tensor.dtype)\n```\n\n (1,)\n \n\n\nThis encodes the Cirq objects as `tf.string` tensors that `tfq` operations decode as needed.\n\n\n```\n# Rank 1 tensor containing 2 Pauli operators.\npauli_tensor = tfq.convert_to_tensor([z0, z0x1])\npauli_tensor.shape\n```\n\n\n\n\n TensorShape([2])\n\n\n\n### 1.3 Batching circuit simulation\n\nTFQ provides methods for computing expectation values, samples, and state vectors. For now, let's focus on *expectation values*.\n\nThe highest-level interface for calculating expectation values is the `tfq.layers.Expectation` layer, which is a `tf.keras.Layer`. In its simplest form, this layer is equivalent to simulating a parameterized circuit over many `cirq.ParamResolvers`; however, TFQ allows batching following TensorFlow semantics, and circuits are simulated using efficient C++ code.\n\nCreate a batch of values to substitute for our `a` and `b` parameters:\n\n\n```\nbatch_vals = np.array(np.random.uniform(0, 2 * np.pi, (5, 2)), dtype=np.float32)\n```\n\nBatching circuit execution over parameter values in Cirq requires a loop:\n\n\n```\ncirq_results = []\ncirq_simulator = cirq.Simulator()\n\nfor vals in batch_vals:\n resolver = cirq.ParamResolver({a: vals[0], b: vals[1]})\n final_state_vector = cirq_simulator.simulate(circuit, resolver).final_state_vector\n cirq_results.append(\n [z0.expectation_from_state_vector(final_state_vector, {\n q0: 0,\n q1: 1\n }).real])\n\nprint('cirq batch results: \\n {}'.format(np.array(cirq_results)))\n```\n\n cirq batch results: \n [[ 0.99998105]\n [-0.99246973]\n [-0.48508084]\n [ 0.95581585]\n [ 0.32338342]]\n\n\nThe same operation is simplified in TFQ:\n\n\n```\ntfq.layers.Expectation()(circuit,\n symbol_names=[a, b],\n symbol_values=batch_vals,\n operators=z0)\n```\n\n\n\n\n \n\n\n\n## 2. Hybrid quantum-classical optimization\n\nNow that you've seen the basics, let's use TensorFlow Quantum to construct a *hybrid quantum-classical neural net*. You will train a classical neural net to control a single qubit. The control will be optimized to correctly prepare the qubit in the `0` or `1` state, overcoming a simulated systematic calibration error. This figure shows the architecture:\n\n\n\nEven without a neural network this is a straightforward problem to solve, but the theme is similar to the real quantum control problems you might solve using TFQ. It demonstrates an end-to-end example of a quantum-classical computation using the `tfq.layers.ControlledPQC` (Parametrized Quantum Circuit) layer inside of a `tf.keras.Model`.\n\nFor the implementation of this tutorial, this is architecture is split into 3 parts:\n\n- The *input circuit* or *datapoint circuit*: The first three $R$ gates.\n- The *controlled circuit*: The other three $R$ gates.\n- The *controller*: The classical neural-network setting the parameters of the controlled circuit.\n\n### 2.1 The controlled circuit definition\n\nDefine a learnable single bit rotation, as indicated in the figure above. This will correspond to our controlled circuit.\n\n\n```\n# Parameters that the classical NN will feed values into.\ncontrol_params = sympy.symbols('theta_1 theta_2 theta_3')\n\n# Create the parameterized circuit.\nqubit = cirq.GridQubit(0, 0)\nmodel_circuit = cirq.Circuit(\n cirq.rz(control_params[0])(qubit),\n cirq.ry(control_params[1])(qubit),\n cirq.rx(control_params[2])(qubit))\n\nSVGCircuit(model_circuit)\n```\n\n\n\n\n \n\n \n\n\n\n### 2.2 The controller\n\nNow define controller network: \n\n\n```\n# The classical neural network layers.\ncontroller = tf.keras.Sequential([\n tf.keras.layers.Dense(10, activation='elu'),\n tf.keras.layers.Dense(3)\n])\n```\n\nGiven a batch of commands, the controller outputs a batch of control signals for the controlled circuit. \n\nThe controller is randomly initialized so these outputs are not useful, yet.\n\n\n```\ncontroller(tf.constant([[0.0],[1.0]])).numpy()\n```\n\n\n\n\n array([[ 0. , 0. , 0. ],\n [-0.93967277, -0.10683127, 0.49653825]], dtype=float32)\n\n\n\n### 2.3 Connect the controller to the circuit\n\nUse `tfq` to connect the controller to the controlled circuit, as a single `keras.Model`. \n\nSee the [Keras Functional API guide](https://www.tensorflow.org/guide/keras/functional) for more about this style of model definition.\n\nFirst define the inputs to the model: \n\n\n```\n# This input is the simulated miscalibration that the model will learn to correct.\ncircuits_input = tf.keras.Input(shape=(),\n # The circuit-tensor has dtype `tf.string` \n dtype=tf.string,\n name='circuits_input')\n\n# Commands will be either `0` or `1`, specifying the state to set the qubit to.\ncommands_input = tf.keras.Input(shape=(1,),\n dtype=tf.dtypes.float32,\n name='commands_input')\n\n```\n\nNext apply operations to those inputs, to define the computation.\n\n\n```\ndense_2 = controller(commands_input)\n\n# TFQ layer for classically controlled circuits.\nexpectation_layer = tfq.layers.ControlledPQC(model_circuit,\n # Observe Z\n operators = cirq.Z(qubit))\nexpectation = expectation_layer([circuits_input, dense_2])\n```\n\nNow package this computation as a `tf.keras.Model`:\n\n\n```\n# The full Keras model is built from our layers.\nmodel = tf.keras.Model(inputs=[circuits_input, commands_input],\n outputs=expectation)\n```\n\nThe network architecture is indicated by the plot of the model below.\nCompare this model plot to the architecture diagram to verify correctness.\n\nNote: May require a system install of the `graphviz` package.\n\n\n```\ntf.keras.utils.plot_model(model, show_shapes=True, dpi=70)\n```\n\nThis model takes two inputs: The commands for the controller, and the input-circuit whose output the controller is attempting to correct. \n\n### 2.4 The dataset\n\nThe model attempts to output the correct correct measurement value of $\\hat{Z}$ for each command. The commands and correct values are defined below.\n\n\n```\n# The command input values to the classical NN.\ncommands = np.array([[0], [1]], dtype=np.float32)\n\n# The desired Z expectation value at output of quantum circuit.\nexpected_outputs = np.array([[1], [-1]], dtype=np.float32)\n```\n\nThis is not the entire training dataset for this task. \nEach datapoint in the dataset also needs an input circuit.\n\n### 2.4 Input circuit definition\n\nThe input-circuit below defines the random miscalibration the model will learn to correct.\n\n\n```\nrandom_rotations = np.random.uniform(0, 2 * np.pi, 3)\nnoisy_preparation = cirq.Circuit(\n cirq.rx(random_rotations[0])(qubit),\n cirq.ry(random_rotations[1])(qubit),\n cirq.rz(random_rotations[2])(qubit)\n)\ndatapoint_circuits = tfq.convert_to_tensor([\n noisy_preparation\n] * 2) # Make two copied of this circuit\n```\n\nThere are two copies of the circuit, one for each datapoint.\n\n\n```\ndatapoint_circuits.shape\n```\n\n\n\n\n TensorShape([2])\n\n\n\n### 2.5 Training\n\nWith the inputs defined you can test-run the `tfq` model.\n\n\n```\nmodel([datapoint_circuits, commands]).numpy()\n```\n\n\n\n\n array([[0.9181084 ],\n [0.79686123]], dtype=float32)\n\n\n\nNow run a standard training process to adjust these values towards the `expected_outputs`.\n\n\n```\noptimizer = tf.keras.optimizers.Adam(learning_rate=0.05)\nloss = tf.keras.losses.MeanSquaredError()\nmodel.compile(optimizer=optimizer, loss=loss)\nhistory = model.fit(x=[datapoint_circuits, commands],\n y=expected_outputs,\n epochs=30,\n verbose=0)\n```\n\n\n```\nplt.plot(history.history['loss'])\nplt.title(\"Learning to Control a Qubit\")\nplt.xlabel(\"Iterations\")\nplt.ylabel(\"Error in Control\")\nplt.show()\n```\n\nFrom this plot you can see that the neural network has learned to overcome the systematic miscalibration.\n\n### 2.6 Verify outputs\nNow use the trained model, to correct the qubit calibration errors. With Cirq:\n\n\n```\ndef check_error(command_values, desired_values):\n \"\"\"Based on the value in `command_value` see how well you could prepare\n the full circuit to have `desired_value` when taking expectation w.r.t. Z.\"\"\"\n params_to_prepare_output = controller(command_values).numpy()\n full_circuit = noisy_preparation + model_circuit\n\n # Test how well you can prepare a state to get expectation the expectation\n # value in `desired_values`\n for index in [0, 1]:\n state = cirq_simulator.simulate(\n full_circuit,\n {s:v for (s,v) in zip(control_params, params_to_prepare_output[index])}\n ).final_state_vector\n expt = cirq.Z(qubit).expectation_from_state_vector(state, {qubit: 0}).real\n print(f'For a desired output (expectation) of {desired_values[index]} with'\n f' noisy preparation, the controller\\nnetwork found the following '\n f'values for theta: {params_to_prepare_output[index]}\\nWhich gives an'\n f' actual expectation of: {expt}\\n')\n\n\ncheck_error(commands, expected_outputs)\n```\n\n For a desired output (expectation) of [1.] with noisy preparation, the controller\n network found the following values for theta: [0.15362751 0.02613623 0.22343476]\n Which gives an actual expectation of: 0.955525279045105\n \n For a desired output (expectation) of [-1.] with noisy preparation, the controller\n network found the following values for theta: [-2.7526338 0.3802155 3.12327 ]\n Which gives an actual expectation of: -0.9250240325927734\n \n\n\nThe value of the loss function during training provides a rough idea of how well the model is learning. The lower the loss, the closer the expectation values in the above cell is to `desired_values`. If you aren't as concerned with the parameter values, you can always check the outputs from above using `tfq`:\n\n\n```\nmodel([datapoint_circuits, commands])\n```\n\n\n\n\n \n\n\n\n## 3 Learning to prepare eigenstates of different operators\n\nThe choice of the $\\pm \\hat{Z}$ eigenstates corresponding to 1 and 0 was arbitrary. You could have just as easily wanted 1 to correspond to the $+ \\hat{Z}$ eigenstate and 0 to correspond to the $-\\hat{X}$ eigenstate. One way to accomplish this is by specifying a different measurement operator for each command, as indicated in the figure below:\n\n\n\nThis requires use of tfq.layers.Expectation. Now your input has grown to include three objects: circuit, command, and operator. The output is still the expectation value.\n\n### 3.1 New model definition\n\nLets take a look at the model to accomplish this task:\n\n\n```\n# Define inputs.\ncommands_input = tf.keras.layers.Input(shape=(1),\n dtype=tf.dtypes.float32,\n name='commands_input')\ncircuits_input = tf.keras.Input(shape=(),\n # The circuit-tensor has dtype `tf.string` \n dtype=tf.dtypes.string,\n name='circuits_input')\noperators_input = tf.keras.Input(shape=(1,),\n dtype=tf.dtypes.string,\n name='operators_input')\n```\n\nHere is the controller network:\n\n\n```\n# Define classical NN.\ncontroller = tf.keras.Sequential([\n tf.keras.layers.Dense(10, activation='elu'),\n tf.keras.layers.Dense(3)\n])\n```\n\nCombine the circuit and the controller into a single `keras.Model` using `tfq`:\n\n\n```\ndense_2 = controller(commands_input)\n\n# Since you aren't using a PQC or ControlledPQC you must append\n# your model circuit onto the datapoint circuit tensor manually.\nfull_circuit = tfq.layers.AddCircuit()(circuits_input, append=model_circuit)\nexpectation_output = tfq.layers.Expectation()(full_circuit,\n symbol_names=control_params,\n symbol_values=dense_2,\n operators=operators_input)\n\n# Contruct your Keras model.\ntwo_axis_control_model = tf.keras.Model(\n inputs=[circuits_input, commands_input, operators_input],\n outputs=[expectation_output])\n```\n\n### 3.2 The dataset\n\nNow you will also include the operators you wish to measure for each datapoint you supply for `model_circuit`:\n\n\n```\n# The operators to measure, for each command.\noperator_data = tfq.convert_to_tensor([[cirq.X(qubit)], [cirq.Z(qubit)]])\n\n# The command input values to the classical NN.\ncommands = np.array([[0], [1]], dtype=np.float32)\n\n# The desired expectation value at output of quantum circuit.\nexpected_outputs = np.array([[1], [-1]], dtype=np.float32)\n```\n\n### 3.3 Training\n\nNow that you have your new inputs and outputs you can train once again using keras.\n\n\n```\noptimizer = tf.keras.optimizers.Adam(learning_rate=0.05)\nloss = tf.keras.losses.MeanSquaredError()\n\ntwo_axis_control_model.compile(optimizer=optimizer, loss=loss)\n\nhistory = two_axis_control_model.fit(\n x=[datapoint_circuits, commands, operator_data],\n y=expected_outputs,\n epochs=30,\n verbose=1)\n```\n\n Epoch 1/30\n 1/1 [==============================] - 0s 2ms/step - loss: 2.0590\n Epoch 2/30\n 1/1 [==============================] - 0s 1ms/step - loss: 1.1033\n Epoch 3/30\n 1/1 [==============================] - 0s 1ms/step - loss: 0.4088\n Epoch 4/30\n 1/1 [==============================] - 0s 1ms/step - loss: 0.2251\n Epoch 5/30\n 1/1 [==============================] - 0s 2ms/step - loss: 0.1944\n Epoch 6/30\n 1/1 [==============================] - 0s 2ms/step - loss: 0.1292\n Epoch 7/30\n 1/1 [==============================] - 0s 2ms/step - loss: 0.0619\n Epoch 8/30\n 1/1 [==============================] - 0s 7ms/step - loss: 0.0331\n Epoch 9/30\n 1/1 [==============================] - 0s 4ms/step - loss: 0.0311\n Epoch 10/30\n 1/1 [==============================] - 0s 2ms/step - loss: 0.0383\n Epoch 11/30\n 1/1 [==============================] - 0s 2ms/step - loss: 0.0456\n Epoch 12/30\n 1/1 [==============================] - 0s 2ms/step - loss: 0.0484\n Epoch 13/30\n 1/1 [==============================] - 0s 7ms/step - loss: 0.0449\n Epoch 14/30\n 1/1 [==============================] - 0s 2ms/step - loss: 0.0365\n Epoch 15/30\n 1/1 [==============================] - 0s 2ms/step - loss: 0.0262\n Epoch 16/30\n 1/1 [==============================] - 0s 2ms/step - loss: 0.0170\n Epoch 17/30\n 1/1 [==============================] - 0s 1ms/step - loss: 0.0106\n Epoch 18/30\n 1/1 [==============================] - 0s 2ms/step - loss: 0.0072\n Epoch 19/30\n 1/1 [==============================] - 0s 2ms/step - loss: 0.0061\n Epoch 20/30\n 1/1 [==============================] - 0s 2ms/step - loss: 0.0061\n Epoch 21/30\n 1/1 [==============================] - 0s 2ms/step - loss: 0.0063\n Epoch 22/30\n 1/1 [==============================] - 0s 2ms/step - loss: 0.0062\n Epoch 23/30\n 1/1 [==============================] - 0s 2ms/step - loss: 0.0054\n Epoch 24/30\n 1/1 [==============================] - 0s 2ms/step - loss: 0.0042\n Epoch 25/30\n 1/1 [==============================] - 0s 4ms/step - loss: 0.0030\n Epoch 26/30\n 1/1 [==============================] - 0s 2ms/step - loss: 0.0020\n Epoch 27/30\n 1/1 [==============================] - 0s 3ms/step - loss: 0.0014\n Epoch 28/30\n 1/1 [==============================] - 0s 3ms/step - loss: 0.0011\n Epoch 29/30\n 1/1 [==============================] - 0s 1ms/step - loss: 9.7383e-04\n Epoch 30/30\n 1/1 [==============================] - 0s 2ms/step - loss: 8.9875e-04\n\n\n\n```\nplt.plot(history.history['loss'])\nplt.title(\"Learning to Control a Qubit\")\nplt.xlabel(\"Iterations\")\nplt.ylabel(\"Error in Control\")\nplt.show()\n```\n\nThe loss function has dropped to zero.\n\nThe `controller` is available as a stand-alone model. Call the controller, and check its response to each command signal. It would take some work to correctly compare these outputs to the contents of `random_rotations`.\n\n\n```\ncontroller.predict(np.array([0,1]))\n```\n\n\n\n\n array([[-0.93180346, 1.4380494 , 0.9859159 ],\n [-5.0782895 , 0.12699541, 3.3510125 ]], dtype=float32)\n\n\n\nSuccess: See if you can adapt the `check_error` function from your first model to work with this new model architecture.\n", "meta": {"hexsha": "1f8a9b97bba8304cf50b81f7102d70b36355d4ba", "size": 118950, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Copy_of_hello_many_worlds.ipynb", "max_stars_repo_name": "QDaria/QDaria.github.io", "max_stars_repo_head_hexsha": "f60d00270a651cceff47629edcee22c70d747185", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Copy_of_hello_many_worlds.ipynb", "max_issues_repo_name": "QDaria/QDaria.github.io", "max_issues_repo_head_hexsha": "f60d00270a651cceff47629edcee22c70d747185", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Copy_of_hello_many_worlds.ipynb", "max_forks_repo_name": "QDaria/QDaria.github.io", "max_forks_repo_head_hexsha": "f60d00270a651cceff47629edcee22c70d747185", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 72.2222222222, "max_line_length": 24106, "alphanum_fraction": 0.7178142077, "converted": true, "num_tokens": 10744, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.6992544147913993, "lm_q1q2_score": 0.41968435991910474}} {"text": "\n\n# \u30e1\u30e2\r\nThinking Functionally with Haskell by Richard Bird \u3092\u8aad\u3080\u3002\u30ce\u30fc\u30c8\u3002\r\n\r\nwww.cs.ox.ac.uk/publications/books/functional\r\n\n\n\n```latex\n%%latex\r\n\\sin 3\u03b1 \u3092 \\sin \u03b1 \u306e\u5f0f\u306b\u5909\u5f62\u3059\u308b\u4f5c\u696d\u3092\u4f7f\u3063\u3066\r\n\u3053\u308c\u304b\u3089\u306e\u8868\u8a18\u65b9\u6cd5\u3092\u8003\u3048\u308b\n```\n\n\n\\sin 3\u03b1 \u3092 \\sin \u03b1 \u306e\u5f0f\u306b\u5909\u5f62\u3059\u308b\u4f5c\u696d\u3092\u4f7f\u3063\u3066\n\u3053\u308c\u304b\u3089\u306e\u8868\u8a18\u65b9\u6cd5\u3092\u8003\u3048\u308b\n\n\n\n```\n# sin 3\u03b1 \u3092 sin \u03b1 \u306e\u5f0f\u306b\u5909\u5f62\u3059\u308b\r\n# \u5f0f\u5909\u5f62\u306b\u3088\u308b\u8a3c\u660e\u30d5\u30a9\u30fc\u30de\u30c3\u30c8\u306f\u6b21\u306e\u3088\u3046\u306b\u3059\u308b\r\n# \u307e\u305a latex \u3067\u3001 \u6b21\u306b sympy \u3067\u66f8\u3044\u3066\u307f\u308b\r\n%%latex\r\n\\displaystyle\r\n\r\n\\sin 3 \u03b1\\\\\r\n= \\{\u5358\u7d14\u5909\u5f62\u306b\u3088\u308a\\}\\\\\r\n\\quad \\sin(2\u03b1 + \u03b1)\\\\\r\n= \\{\u52a0\u6cd5\u5b9a\u7406~ \\sin(\u03b1 + \u03b2) = \\sin \u03b1 \\cos \u03b2 + \\cos \u03b1 \\sin \u03b2 ~\u306b\u3088\u308a\\}\\\\\r\n\\quad \\sin 2\u03b1 \\cos \u03b1 + \\cos 2\u03b1 \\sin \u03b1 \\\\\r\n= \\{\u516c\u5f0f \\sin 2\u03b1 = 2 \\sin \u03b1 \\cos \u03b1 ~\u306b\u3088\u308a\\}\\\\\r\n\\quad 2 \\sin \u03b1 \\cos^2 \u03b1 + \\cos 2\u03b1 \\sin \u03b1 \\\\\r\n= \\{\u516c\u5f0f \\cos 2\u03b1 = \\cos^2 \u03b1 \u2212 \\sin^2 \u03b1 ~\u306b\u3088\u308a\\}\\\\\r\n\\quad 2 \\sin \u03b1 \\cos^2 \u03b1 + (\\cos^2 \u03b1 \u2212 \\sin^2 \u03b1) \\sin \u03b1 \\\\\r\n= \\{\\sin \u03b1~\u3067\u62ec\u3063\u3066\\}\\\\\r\n\\quad \\sin \u03b1 (2 \\cos^2 \u03b1+ \\cos^2\u03b1 - \\sin^2 \u03b1)\\\\\r\n= \\{\u5358\u7d14\u8a08\u7b97\\}\\\\\r\n\\quad \\sin \u03b1 (3 \\cos^2 \u03b1 - \\sin^2 \u03b1)\\\\\r\n= \\{\u516c\u5f0f \\sin^2 \u03b1+ \\cos^2 \u03b1= 1 \\Longrightarrow \\cos^2 \u03b1 = 1 - \\sin^2 \u03b1~ \u306b\u3088\u308a\\}\\\\\r\n\\quad \\sin \u03b1 (3 - 3 \\sin^2 \u03b1 - \\sin^2 \u03b1)\\\\\r\n= \\{\u5358\u7d14\u8a08\u7b97\\}\\\\\r\n\\quad \\sin \u03b1 (3 - 4 \\sin^2 \u03b1)\\\\\n```\n\n\n\\displaystyle\n\n\\sin 3 \u03b1\\\\\n= \\{\u5358\u7d14\u5909\u5f62\u306b\u3088\u308a\\}\\\\\n\\quad \\sin(2\u03b1 + \u03b1)\\\\\n= \\{\u52a0\u6cd5\u5b9a\u7406~ \\sin(\u03b1 + \u03b2) = \\sin \u03b1 \\cos \u03b2 + \\cos \u03b1 \\sin \u03b2 ~\u306b\u3088\u308a\\}\\\\\n\\quad \\sin 2\u03b1 \\cos \u03b1 + \\cos 2\u03b1 \\sin \u03b1 \\\\\n= \\{\u516c\u5f0f \\sin 2\u03b1 = 2 \\sin \u03b1 \\cos \u03b1 ~\u306b\u3088\u308a\\}\\\\\n\\quad 2 \\sin \u03b1 \\cos^2 \u03b1 + \\cos 2\u03b1 \\sin \u03b1 \\\\\n= \\{\u516c\u5f0f \\cos 2\u03b1 = \\cos^2 \u03b1 \u2212 \\sin^2 \u03b1 ~\u306b\u3088\u308a\\}\\\\\n\\quad 2 \\sin \u03b1 \\cos^2 \u03b1 + (\\cos^2 \u03b1 \u2212 \\sin^2 \u03b1) \\sin \u03b1 \\\\\n= \\{\\sin \u03b1~\u3067\u62ec\u3063\u3066\\}\\\\\n\\quad \\sin \u03b1 (2 \\cos^2 \u03b1+ \\cos^2\u03b1 - \\sin^2 \u03b1)\\\\\n= \\{\u5358\u7d14\u8a08\u7b97\\}\\\\\n\\quad \\sin \u03b1 (3 \\cos^2 \u03b1 - \\sin^2 \u03b1)\\\\\n= \\{\u516c\u5f0f \\sin^2 \u03b1+ \\cos^2 \u03b1= 1 \\Longrightarrow \\cos^2 \u03b1 = 1 - \\sin^2 \u03b1~ \u306b\u3088\u308a\\}\\\\\n\\quad \\sin \u03b1 (3 - 3 \\sin^2 \u03b1 - \\sin^2 \u03b1)\\\\\n= \\{\u5358\u7d14\u8a08\u7b97\\}\\\\\n\\quad \\sin \u03b1 (3 - 4 \\sin^2 \u03b1)\\\\\n\n\n# \u3044\u307e\u3053\u3053\n\n\n```\n# sympy \u3067\u66f8\u3044\u3066\u307f\u308b\u5b9f\u9a13\nfrom sympy import *\nfrom IPython.display import Markdown\ninit_printing()\n\u03b1, \u03b2 = symbols('\u03b1 \u03b2')\ndisplay (sin (3*\u03b1))\ndisplay(Markdown(\"$\\sin (3\u03b1)$ \u3092 $\\sin(\u03b1)$ \u3060\u3051\u3067\u8868\u793a\u3059\u308b\u3002\"))\ndisplay(Markdown(\"\u307e\u305a\u516c\u5f0f\"))\ndisplay (Eq(sin(\u03b1+\u03b2),expand(sin(\u03b1+\u03b2),trig=True)))\ndisplay(Markdown(\"\u3088\u308a\u3001$\u03b1 = 2\u03b1,~\u03b2 = \u03b1$ \u3068\u304a\u304f\u3068\"))\ndisplay(Eq(sin (3*\u03b1),(sin(2*\u03b1)*cos(\u03b1)+cos(2*\u03b1)*sin(\u03b1))))\ndisplay(Markdown(\"\u9805\u306e\u9806\u5e8f\u306f\u3068\u308a\u3042\u3048\u305a\u6c17\u306b\u3057\u306a\u3044\u3002\"))\ndisplay(Markdown(\"\u307e\u305f\u540c\u3058\u516c\u5f0f\"))\ndisplay (Eq(sin(\u03b1+\u03b2),expand(sin(\u03b1+\u03b2),trig=True)))\ndisplay(Markdown(\"\u3067\u3001$\u03b2 = \u03b1$ \u3068\u304a\u304f\u3068\"))\ndisplay (Eq(sin(2*\u03b1),expand(sin(2*\u03b1),trig=True)))\ndisplay(Markdown(\"\u3092\u7528\u3044\u3066 $\\sin(2\u03b1)$ \u3092\u6d88\u53bb\u3059\u308b\u3068\"))\ndisplay(Eq(sin (3*\u03b1),(expand(sin(2*\u03b1)*cos(\u03b1),trig=True)+cos(2*\u03b1)*sin(\u03b1))))\ndisplay(Markdown(\"$\\cos(2\u03b1)$\u306b\u3064\u3044\u3066\u516c\u5f0f\"))\ndisplay (Eq(cos(2*\u03b1),expand(cos(2*\u03b1),trig=True)))\ndisplay(Markdown(\"\u3092\u7528\u3044\u3066 $\\cos(2\u03b1)$ \u3092\u6d88\u53bb\u3059\u308b\u3068\"))\ndisplay(Eq(sin (3*\u03b1),(expand(sin(2*\u03b1)*cos(\u03b1)+cos(2*\u03b1)*sin(\u03b1),trig=True))))\ndisplay(Markdown(\"\u6b21\u306b\u516c\u5f0f\u3092\u4f7f\u3063\u3066 $\\cos^2(\u03b1)$ \u3092\u6d88\u3059\u305f\u3081\u306b\u516c\u5f0f\u3092\u5909\u5f62\u3059\u308b\u3002\"))\ndisplay(Eq((sin(\u03b1))**2+(cos(\u03b1))**2,1))\ndisplay(Eq((cos(\u03b1))**2,1-(sin(\u03b1))**2))\ndisplay(Markdown(\"\u3092\u4ee3\u5165\u3059\u308b\"))\ndisplay(Eq(sin (3*\u03b1),4*sin(\u03b1)*(1-(sin(\u03b1))**2) - sin(\u03b1)))\ndisplay(Eq(sin (3*\u03b1),expand(4*sin(\u03b1)*(1-(sin(\u03b1))**2) - sin(\u03b1))))\n\n```\n\n\u306a\u308b\u307b\u3069 latex \u3068 sympy \u3067\u307b\u3068\u3093\u3069\u540c\u3058\u3053\u3068\u304c\u3067\u304d\u308b\u304c\u3001\u307e\u3063\u305f\u304f\u540c\u3058\u306b\u306f\u306a\u3089\u306a\u3044\u3002\r\n\r\n\u305d\u308c\u3088\u308a\u3001\u306a\u306b\u3088\u308a\u4f7f\u3046\u982d\u306e\u4f7f\u3044\u65b9\u304c\u9055\u3046\u3002\r\n\r\nlatex \u3067\u306f latex \u306e\u69cb\u9020\u304c\u308f\u304b\u3063\u3066\u3044\u306a\u3044\u3068\u66f8\u3051\u306a\u3044\u3002 latex \u3067\u66f8\u304f\u904e\u7a0b\u3067\u30c6\u30ad\u30b9\u30c8\u3092\u3088\u304f\u8aad\u3080\u3002\r\n\r\nsympy \u3067\u306f\u6570\u5f0f\u306e\u610f\u5473\u304c\u308f\u304b\u3063\u3066\u3044\u306a\u3044\u3068\u66f8\u3051\u306a\u3044\u3002 \u66f8\u304f\u904e\u7a0b\u3067\u5b8c\u5168\u306b\u7406\u89e3\u3059\u308b\u304c\u3001\u6700\u521d\u304b\u3089\u3053\u306e\u7406\u89e3\u306b\u5230\u9054\u3059\u308b\u306e\u306f\u3080\u305a\u304b\u3057\u3044\u3002 \r\n\r\n\u307e\u305f\u3001latex \u3067\u8868\u73fe\u3057\u305f\u3044\u3082\u306e\u304c\u5fc5\u305a\u3057\u3082 sympy \u3067\u66f8\u3051\u308b\u308f\u3051\u3067\u306f\u306a\u3044\u3002\r\n\r\n\u307e\u305a latex \u3067\u66f8\u3044\u3066\u3001sympy \u306b\u3067\u304d\u308b\u3082\u306e\u306f sympy \u306b\u3059\u308b\u3001\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u306e\u3067\u306f\u306a\u3044\u304b\u3002\r\n\r\n\n\n\n```\nfrom sympy import *\r\ninit_printing()\r\n\r\n\u03b1 = symbols('\u03b1')\r\n\r\nlatex (sin (3*\u03b1))\n```\n\n\n\n\n '\\\\sin{\\\\left (3 \u03b1 \\\\right )}'\n\n\n\n$\\sin{\\left (3 \u03b1 \\right )}$\n\n\n```latex\n%%latex\r\n\r\n\\sin{\\left (3 \u03b1 \\right )}\r\n\r\n\u3092\u5909\u5f62\u3059\u308b\n```\n\n\n\n\\sin{\\left (3 \u03b1 \\right )}\n\n\u3092\u5909\u5f62\u3059\u308b\n\n\n\n```\n# \u95a2\u6570\u3068\u578b(type)\r\n!ghc -e $':t sin'\r\n# age::Person->Int\r\n!ghc -e $':t (+)'\r\n!ghc -e $':t logBase'\r\n!ghc -e $'logBase 10 100'\r\n\n```\n\n sin :: Floating a => a -> a\n (+) :: Num a => a -> a -> a\n logBase :: Floating a => a -> a -> a\n 2.0\n\n\n\n```\n# \u5b9f\u9a13\r\n!ghc -e $'log $ exp(1)'\n```\n\n 1.0\n\n\n\n```\n# \u30b3\u30de\u30f3\u30c9\u30e9\u30a4\u30f3\u30ef\u30f3\u30e9\u30a4\u30ca\u30fc\u3067\u30e9\u30a4\u30d6\u30e9\u30ea\u30fc\u306eimport\u306f\u3067\u304d\u306a\u3044\u306e\u3067\u6b21\u306e\u3088\u3046\u306b\u3059\u308b\r\n!ghc -e $'map Data.Char.toLower \"HELLO WORLD!\"'\n```\n\n \"hello world!\"\n\n\n\n```\n%%capture\r\n!apt install haskell-platform\n```\n\n\u30d5\u30a1\u30a4\u30eb\u3092\u4f5c\u3063\u3066\u3001runghc \u3067\u5b9f\u884c\u3059\u308b\n\n%%writefile numbers2words.hs\n\n\n```\n#@title\n# \u30d5\u30a1\u30a4\u30eb\u3092\u4f5c\u3063\u3066\u3001runghc \u3067\u5b9f\u884c\u3059\u308b\n%%writefile numbers2words.hs\n\nmain = putStrLn $ convert 301123\n-- three hundred and one thousand one hundred and twenty-three\n\nunits, teens, tens :: [String]\nunits = [\"zero\",\"one\",\"two\",\"three\",\"four\",\"five\",\n \"six\",\"seven\",\"eight\",\"nine\"]\nteens = [\"ten\",\"eleven\",\"twelve\",\"thirteen\",\"fourteen\",\n \"fifteen\",\"sixteen\",\"seventeen\",\"eighteen\",\n \"nineteen\"]\ntens = [\"twenty\",\"thirty\",\"forty\",\"fifty\",\"sixty\",\n \"seventy\",\"eighty\",\"ninety\"]\n\nconvert1 :: Int -> String\nconvert1 n = units!!n\n\ndigits2 :: Int -> (Int,Int)\ndigits2 n = n `divMod` 10\n\nconvert2 :: Int -> String\nconvert2 = combine2 . digits2\n\ncombine2 :: (Int,Int) -> String\ncombine2 (t,u)\n | t==0 = units!!u\n | t==1 = teens!!u\n | 2<=t && u==0 = tens!!(t-2)\n | 2<=t && u/=0 = tens!!(t-2) ++ \"-\" ++ units!!u\n\n\ndigits3 :: Int -> (Int,Int)\ndigits3 n = n `divMod` 100\n\nconvert3 :: Int -> String\nconvert3 = combine3 . digits3\n\ncombine3 :: (Int,Int) -> String\ncombine3 (h,n)\n | h==0 = convert2 n\n | n==0 = units!!h ++ \" hundred\"\n | otherwise = units!!h ++ \" hundred and \" ++ convert2 n\n\ndigits6 :: Int -> (Int,Int)\ndigits6 n = n `divMod` 1000\n\nconvert6 :: Int -> String\nconvert6 = combine6 . digits6\n\ncombine6 :: (Int,Int) -> String\ncombine6 (m,n)\n | m==0 = convert3 n\n | n==0 = convert3 m ++ \" thousand\"\n | otherwise = convert3 m ++ \" thousand\" ++ link n ++\n convert3 n\n\nlink :: Int -> String\nlink n = if n < 100 then \" and \" else \" \"\n\nconvert :: Int -> String\nconvert = convert6\n\n```\n\n Overwriting numbers2words.hs\n\n\n\n```\n!runghc numbers2words.hs\n```\n\n three hundred and one thousand one hundred and twenty-three\n\n\n\n```\n# \u5b9f\u9a13\r\n!ghc -e $':t logBase'\r\n!ghc -e $'logBase 2 8' #=> 3\r\n!ghc -e $'exp(1)' #=> 2.718...\r\n!ghc -e $'logBase (exp(1)) (exp(1)**2)' #=> 2.0\n```\n\n logBase :: Floating a => a -> a -> a\n 3.0\n 2.718281828459045\n 2.0\n\n\ncommonwords.hs\n\n\n```\n%%script false\r\nWhat are the 100 most common words in War and Peace?\r\n\r\ncommonWords :: Int -> [Char] -> [Char]\r\nreturns a list of the n most common words in the list as a string\r\n\r\nwords :: [Char] -> [[Char]]\r\n\r\ntype synonyms:\r\ntype Text = [Char]\r\ntype Word = [Char]\r\n\r\nwords :: Text -> [Word]\r\n\r\nwords . map toLower converts a text into a list of words in lowercase\r\nsortWords :: [Word] -> [Word]\r\nsortWords [\"to\",\"be\",\"or\",\"not\",\"to\",\"be\"] = [\"be\",\"be\",\"not\",\"or\",\"to\",\"to\"]\r\ncountRuns :: [Word] -> [(Int,Word)]\r\ncountRuns [\"be\",\"be\",\"not\",\"or\",\"to\",\"to\"] = [(2,\"be\"),(1,\"not\"),(1,\"or\"),(2,\"to\")]\r\nsortRuns :: [(Int,Word)] -> [(Int,Word)]\r\nsortRuns [(2,\"be\"),(1,\"not\"),(1,\"or\"),(2,\"to\")] = [(2,\"be\"),(2,\"to\"),(1,\"not\"),(1,\"or\")]\r\ntake :: Int -> [a] -> [a]\r\nshowRun :: (Int,Word) -> String\r\nshowRun (2,\"be\") = \"be 2\\n\"\r\nmap showRun :: [(Int,Word)] -> [String]\r\nconcat :: [[a]] -> [a]\r\nconcat [\"this\" , \"is\"] = \"thisis\"\r\n\r\ncommonWords :: Int -> Text -> String\r\ncommonWords n = concat . map showRun . take n . sortRuns . countRuns . sortWords . words . map toLower\r\n\n```\n\n# \u3044\u307e\u3053\u3053\n\nnumbers2words.hs\n\n\n```\n%%script false\n\n\u6570\u5b57\u3092\u6587\u5b57\u306b\u3059\u308b\u30d7\u30ed\u30b0\u30e9\u30e0 numbers2words.hs \u3092\u4f5c\u308b\u3002\n\nconvert 308000 = \"three hundred and eight thousand\"\nconvert 369027 = \"three hundred and sixty-nine thousand andtwenty-seven\"\nconvert 369401 = \"three hundred and sixty-nine thousandfour hundred and one\"\n\n\u76ee\u7684\u306f\u3001\u6b21\u306e\u3088\u3046\u306a\u95a2\u6570\u3092\u4f5c\u308b\u3053\u3068\u3002\n\nconvert :: Int -> String\n\n\u5165\u529b\u306f100\u4e07\u4ee5\u4e0b\u306e\u6b63\u306e\u6574\u6570\u3068\u3059\u308b\u3002\n\n\u5fc5\u8981\u306a\u6587\u5b57\u5217\u306e\u30ea\u30b9\u30c8\u3092\u4f5c\u308b\u3002\n\nunits, teens, tens :: [String]\nunits = [\"zero\",\"one\",\"two\",\"three\",\"four\",\"five\",\"six\",\"seven\",\"eight\",\"nine\"]\nteens = [\"ten\",\"eleven\",\"twelve\",\"thirteen\",\"fourteen\",\"fifteen\",\"sixteen\",\"seventeen\",\"eighteen\",\n\"nineteen\"]\ntens = [\"twenty\",\"thirty\",\"forty\",\"fifty\",\"sixty\",\"seventy\",\"eighty\",\"ninety\"]\n\nconvert1\u3068\u3044\u3046\u95a2\u6570\u3092\u4f5c\u3063\u3066 0 \u2264 n < 10 \u3092\u6271\u3046\u3002\n\nconvert1 :: Int -> String\nconvert1 n = units!!n\n\n\u6b21\u306bconvert2\u3068\u3044\u3046\u95a2\u6570\u3067 0 \u2264 n < 100 \u3092\u6271\u3044\u305f\u3044\u304c\u3001\u305d\u306e\u524d\u306bdigits2\u3068\u3044\u3046\u306e\u3092\u4f5c\u308b\u3002\n\ndigits2 :: Int -> (Int,Int)\ndigits2 n = (div n 10, mod n 10)\n\nconvert2 :: Int -> String\nconvert2 = combine2 . digits2\n\ncombine2 \u306b\u306f\u30ac\u30fc\u30c9\u3092\u4f7f\u3046\u3002\n\ncombine2 :: (Int,Int) -> String\ncombine2 (t,u)\n | t==0 = units!!u\n | t==1 = teens!!u\n | 2<=t && u==0 = tens!!(t-2)\n | 2<=t && u/=0 = tens!!(t-2) ++ \"-\" ++ units!!u\n\n```\n\n\u30e1\u30e2\r\n\r\n\u62e1\u5f35\u5b50 .lhs \u306f literate haskell script \u3067\u3001\u30d5\u30a1\u30a4\u30eb\u306e\u306a\u304b\u3067\u884c\u982d\u306b `>` \u304c\u3042\u308b\u884c\u306f\u30d7\u30ed\u30b0\u30e9\u30e0\u3068\u3057\u3066\u6271\u308f\u308c\u308b\u3002 \u3053\u306e\u672c\u306e\u30b5\u30f3\u30d7\u30eb\u30b3\u30fc\u30c9\u306f literate haskell script \u3067\u63d0\u4f9b\u3055\u308c\u308b\u304c\u3001\u3053\u306e\u30ce\u30fc\u30c8\u3067\u306f\u4f7f\u308f\u306a\u3044\u3002\n\n\n```\n\nwe are concatenating two lists of characters.\nThe de\ufb01nition of combine2 is arrived at by carefully considering all the possible\ncases that can arise. A little re\ufb02ection shows that there are three main cases, namely\nwhen the tens part t is 0, 1 or greater than 1. In the \ufb01rst two cases we can give the\nanswer immediately, but the third case has to be divided into two subcases, namely\nwhen the units part u is 0 or not 0. The order in which we write the cases, that is, the\norder of the individual guarded equations, is unimportant as the guards are disjoint\nfrom one another (that is, no two guards can be true) and together they cover all\ncases.\nWe could also have written\ncombine2 :: (Int,Int) -> String\ncombine2 (t,u)\n| t==0\n= units!!u\n| t==1\n= teens!!u\n| u==0\n= tens!!(t-2)\n| otherwise = tens!!(t-2) ++ \"-\" ++ units!!u\n\n\f1.4 Example: numbers into words\n\n11\n\nbut now the order in which we write the equations is crucial. The guards are evaluated from top to bottom, taking the right-hand side corresponding to the \ufb01rst guard\nthat evaluates to True. The identi\ufb01er otherwise is just a synonym for True, so\nthe last clause captures all the remaining cases.\nThere is yet another way of writing convert2:\nconvert2 :: Int\nconvert2 n\n| t==0\n=\n| t==1\n=\n| u==0\n=\n| otherwise =\nwhere (t,u) =\n\n-> String\nunits!!u\nteens!!u\ntens!!(t-2)\ntens!!(t-2) ++ \"-\" ++ units!!u\n(n `div` 10, n `mod` 10)\n\nThis makes use of a where clause. Such a clause introduces a local de\ufb01nition\nor de\ufb01nitions whose context or scope is the whole of the right-hand side of the\nde\ufb01nition of convert2. Such clauses are very useful in structuring de\ufb01nitions and\nmaking them more readable. In the present example, the where clause obviates the\nneed for an explicit de\ufb01nition of digits2.\nThat was reasonably easy, so now let us consider convert3 which takes a number\nn in the range 0 \u2264 n < 1000, so n has up to three digits. The de\ufb01nition is\n> convert3 :: Int\n> convert3 n\n>\n| h==0\n=\n>\n| n==0\n=\n>\n| otherwise =\n>\nwhere (h,t) =\n\n-> String\nconvert2\nunits!!h\nunits!!h\n(n `div`\n\nt\n++ \" hundred\"\n++ \" hundred and \" ++ convert2 t\n100, n `mod` 100)\n\nWe break up the number in this way because we can make use of convert2 for\nnumbers that are less than 100.\nNow suppose n lies in the range 0 \u2264 n < 1, 000, 000, so n can have up to six digits.\nFollowing exactly the same pattern as before, we can de\ufb01ne\n> convert6 :: Int\n> convert6 n\n>\n| m==0\n=\n>\n| h==0\n=\n>\n| otherwise =\n>\n>\nwhere (m,h) =\n\n-> String\nconvert3\nconvert3\nconvert3\nconvert3\n(n `div`\n\nh\nm ++ \" thousand\"\nm ++ \" thousand\" ++ link h ++\nh\n1000,n `mod` 1000)\n\n\f12\n\nWhat is functional programming?\n\nThere will be a connecting word \u2018and\u2019 between the words for m and h just in the\ncase that 0 < m and 0 < h < 100. Thus\n> link :: Int -> String\n> link h = if h < 100 then \" and \" else \" \"\nThis de\ufb01nition makes use of a conditional expression\nif then else \n\nWe could also have used guarded equations:\n\nlink h | h < 100\n= \" and \"\n| otherwise = \" \"\n\nSometimes one is more readable, sometimes the other. The names if, then and\nelse, along with some others, are reserved words in Haskell, which means that we\ncannot use them as names for things we want to de\ufb01ne.\n\nNotice how the de\ufb01nition of convert6 has been constructed in terms of the simpler\nfunction convert3, which in turn has been de\ufb01ned in terms of the even simpler\nfunction convert2. That is often the way with function de\ufb01nitions. In this example\nconsideration of the simpler cases is not wasted because these simple cases can be\nused in the \ufb01nal de\ufb01nition.\n\nOne more thing: we have now named the function we are after as convert6, but\nwe started off by saying the name should be convert. No problem:\n\n> convert :: Int -> String\n> convert = convert6\n\nWhat we would like to do now is actually use the computer to apply convert to\nsome arguments. How?\n\n```\n\n\n```\n\n```\n\n\n```\nFor brevity we will use this prompt throughout the book.\n\nYou can load a script, Numbers2Words.lhs say, that contains the de\ufb01nition of\nconvert as follows:\nghci> :load \"Numbers2Words.lhs\"\n[1 of 1] Compiling Main\n( Numbers2Words.lhs, interpreted )\nOk, modules loaded: Main.\nghci>\nWe will explain what modules are in the next chapter. Now you can type, for example,\nghci> convert 301123\n\"three hundred and one thousand one hundred and twenty-three\"\nghci>\n\nWe end the chapter with some exercises. These contain additional points of interest\n\n\fWhat is functional programming?\n\n14\n\nand should be regarded as an integral part of the text. The same is true for all\nsubsequent chapters, so please read the questions even if you do not answer them.\nThe answers are given afterwards.\n\n1.6 Exercises\nExercise A\nConsider the function\ndouble :: Integer -> Integer\ndouble x = 2*x\nthat doubles an integer. What are the values of the following expressions?\nmap double [1,4,4,3]\nmap (double . double) [1,4,4,3]\nmap double []\nSuppose sum :: [Integer] -> Integer is a function that sums a list of integers. Which of the following assertions are true and why?\nsum . map double = double . sum\nsum . map sum\n= sum . concat\nsum . sort\n= sum\nYou will need to recall what the function concat does. The function sort sorts a\nlist of numbers into ascending order.\nExercise B\nIn Haskell, functional application takes precedence over every other operator, so\ndouble 3+4 means (double 3)+4, not double (3+4). Which of the following\nexpressions is a rendering of sin2 \u03b8 into Haskell?\nsin^2 theta\n\nsin theta^2\n\n(sin theta)^2\n\n(Exponentiation is denoted by (^).) How would you express sin 2\u03b8 /2\u03c0 as a wellformed Haskell expression?\nExercise C\nAs we said in the text, a character, i.e. an element of Char, is denoted using single quotes, and a string is denoted using double quotes. In particular the string\n\"Hello World!\" is just a much shorter way of writing the list\n\n\f1.6 Exercises\n\n15\n\n['H','e','l','l','o',' ','W','o','r','l','d','!']\nGeneral lists can be written with brackets and commas. (By the way, parentheses\nare round, brackets are square, and braces are curly.) The expressions 'H' and \"H\"\ntherefore have different types. What are they? What is the difference between 2001\nand \"2001\"?\nThe operation ++ concatenates two lists. Simplify\n[1,2,3]\n\"Hello\"\n[1,2,3]\n\"Hello\"\n\n++\n++\n++\n++\n\n[3,2,1]\n\" World!\"\n[]\n\"\" ++ \"World!\"\n\nExercise D\nIn the common words example we started off by converting every letter in the\ntext to lowercase, and then we computed the words in the text. An alternative\nis to do things the other way round, \ufb01rst computing the words and then converting each letter in each word to lowercase. The \ufb01rst method is expressed by\nwords . map toLower. Give a similar expression for the second method.\nExercise E\nAn operator \u2295 is said to be associative if x \u2295 (y \u2295 z) = (x \u2295 y) \u2295 z. Is numerical\naddition associative? Is list concatenation associative? Is functional composition\nassociative? Give an example of an operator on numbers that is not associative.\nAn element e is said to be an identity element of \u2295 if x\u2295e = e\u2295x = x for all x. What\nare the identity elements of addition, concatenation and functional composition?\nExercise F\nMy wife has a book with the title\nEHT CDOORRSSW AAAGMNR ACDIINORTY.\nIt contains lists of entries like this:\n6-letter words\n-------------...\neginor: ignore,region\neginrr: ringer\neginrs: resign,signer,singer\n...\n\n\fWhat is functional programming?\n\n16\n\nYes, it is an anagram dictionary. The letters of the anagrams are sorted and the\nresults are stored in dictionary order. Associated with each anagram are the English\nwords with the same letters. Describe how you would go about designing a function\nanagrams :: Int -> [Word] -> String\nso that anagrams n takes a list of English words in alphabetical order, extracts\njust the n-letter words and produces a string that, when displayed, gives a list of\nthe anagram entries for the n-letter words. You are not expected to be able to de\ufb01ne\nthe various functions; just give suitable names and types and describe what each of\nthem is supposed to do.\nExercise G\nLet\u2019s end with a song:\nOne man\nWent to\nOne man\nWent to\n\nwent to mow\nmow a meadow\nand his dog\nmow a meadow\n\nTwo men went to mow\nWent to mow a meadow\nTwo men, one man and his dog\nWent to mow a meadow\nThree men went to mow\nWent to mow a meadow\nThree men, two men, one man and his dog\nWent to mow a meadow\nWrite a Haskell function song :: Int -> String so that song n is the song\nwhen there are n men. Assume n<10.\nTo print the song, type for example\nghci> putStrLn (song 5)\nThe function putStrLn will be explained in the following chapter. I suggest starting with\nsong n\n\n= if n==0 then \"\"\nelse song (n-1) ++ \"\\n\" ++ verse n\nverse n = line1 n ++ line2 n ++ line3 n ++ line4 n\n\n\f1.7 Answers\n\n17\n\nThis de\ufb01nes song recursively.\n\n1.7 Answers\nAnswer to Exercise A\nmap double [1,4,4,3]\n= [2,8,8,6]\nmap (double . double) [1,4,4,3] = [4,16,16,12]\nmap double []\n= []\nYou will gather from this that [] denotes the empty list.\nAll the following equations hold:\nsum . map double = double . sum\nsum . map sum\n= sum . concat\nsum . sort\n= sum\nIn fact, each of these three equations are consequences of the three simpler laws:\na*(x+y) = a*x + a*y\nx+(y+z) = (x+y)+z\nx+y\n= y+x\nOf course, we don\u2019t know yet how to prove that the equations hold. (By the way,\nto avoid fuss we will often use a typewriter = sign to denote the equality of two\nHaskell expressions written in typewriter font. But a mathematical = sign is used\nin equations such as sin 2\u03b8 = 2 sin \u03b8 cos \u03b8 .)\nAnswer to Exercise B\nBoth sin theta^2 and (sin theta)^2 are okay, but not sin^2 theta.\nHere is the rendering of sin 2\u03b8 /2\u03c0 in Haskell:\nsin (2*theta) / (2*pi)\nNote that\nsin (2*theta) / 2 * pi = (sin (2*theta) / 2) * pi\nwhich is not what we want. The reason is that operators such as / and * at the same\nlevel of precedence associate to the left in expressions. More on this in the next\nchapter.\n\n\fWhat is functional programming?\n\n18\n\nAnswer to Exercise C\n'H'\n\"H\"\n2001\n\"2001\"\n\n::\n::\n::\n::\n\nChar\n[Char]\nInteger\n[Char]\n\nBy the way, '\\' is used as an escape character, so '\\n' is the newline character,\nand '\\t' is the tab character. Also, '\\\\' is the backslash character, and \"\\\\n\" is\na list of two characters, a backslash and the letter n. As a consequence, the \ufb01le path\nC:\\firefox\\stuff is written as the Haskell string \"C:\\\\firefox\\\\stuff\".\n[1,2,3]\n\"Hello\"\n[1,2,3]\n\"Hello\"\n\n++\n++\n++\n++\n\n[3,2,1]\n\" World!\"\n[]\n\"\" ++\"World!\"\n\n=\n=\n=\n=\n\n[1,2,3,3,2,1]\n\"Hello World!\"\n[1,2,3]\n\"HelloWorld!\"\n\nIf you got the last two right, you will have appreciated that [] is an empty list of\nanything, but \"\" is an empty list of characters.\nAnswer to Exercise D\nThe clue is in the phrase \u2018converting each letter in each word to lowercase\u2019. Converting each letter in a single word is expressed by map toLower, so the answer is\nmap (map toLower) . words. That means the following equation holds:\nwords . map toLower = map (map toLower) . words\nAnswer to Exercise E\nNumerical addition, list concatenation and functional composition are all associative. But of course, numerical subtraction isn\u2019t. Nor is exponentiation. The identity\nelement of addition is 0, the identity element of concatenation is the empty list, and\nthe identity element of functional composition is the identity function:\nid :: a -> a\nid x = x\nAnswer to Exercise F\nThis exercise follows Section 1.3 quite closely. One way of computing the function\nanagrams n is as follows:\n1. Extract the words of length n, using a function\ngetWords :: Int -> [Word] -> [Word]\n\n\f1.7 Answers\n\n19\n\n2. Take each word and add a label to it. The label consists of the characters of the\nword, sorted into alphabetical order. For example, word is turned into the pair\n(\"dorw\",\"word\") This labelling is achieved by the function\naddLabel :: Word -> (Label,Word)\nwhere\ntype Label = [Char]\n3. Sort the list of labelled words into alphabetical order of label, using the function\nsortLabels :: [(Label,Word)] -> [(Label,Word)]\n4. Replace each group of adjacent labelled words with the same label with a single\nentry consisting of a pair in which the \ufb01rst component is the common label and\nthe second component is a list of words with that label. This uses a function\ngroupByLabel :: [(Label,Word)] -> [(Label,[Word])]\n5. Replace each entry by a string using a function\nshowEntry :: [(Label,[Word])] -> String\nand concatenate the results.\nThat gives\nanagrams n = concat . map showEntry . groupByLabel .\nsortLabels . map addLabel . getWords n\nAnswer to Exercise G\nOne possible solution:\nsong n\n\n= if n==0 then \"\"\nelse song (n-1) ++ \"\\n\" ++ verse n\nverse n = line1 n ++ line2 n ++ line3 n ++ line4 n\nline1 n = if n==1 then\n\"One man went to mow\\n\"\nelse\nnumbers!!(n-2) ++ \" men went to mow\\n\"\nline2 n = \"Went to mow a meadow\\n\"\nline3 n = if n==1 then\n\"One man and his dog\\n\"\nelse\n\n\f20\n\nWhat is functional programming?\n\nnumbers!!(n-2) ++ \" men, \" ++ count (n-2)\n++ \"one man and his dog\\n\"\nline4 n = \"Went to mow a meadow\\n\\n\"\ncount n = if n==0 then \"\"\nelse\nnumbs!!(n-1) ++ \" men, \" ++ count (n-1)\nnumbers = [\"Two\", \"Three\", \"Four\", \"Five\", \"Six\",\n\"Seven\", \"Eight\", \"Nine\"]\nnumbs\n= [\"two\", \"three\", \"four\", \"five\", \"six\",\n\"seven\", \"eight\"]\nNotice that we have omitted to declare the types of the component functions and\nvalues in this script. Although Haskell will infer the correct types, it is usually a\ngood idea to put them in for all functions and other values, however simple the\ntypes may be. Scripts with explicit type signatures are clearer to read and provide\na useful check on the validity of de\ufb01nitions.\n\n1.8 Chapter notes\nIf you are interested in the origins of Haskell, you should de\ufb01nitely read The History of Haskell, a copy of which is obtainable at\nresearch.microsoft.com/~simonpj/papers/history-of-haskell\nOne of the abiding strengths of Haskell is that it wasn\u2019t designed to be a closed\nlanguage, and researchers were encouraged to implement novel programming ideas\nand techniques by building language extensions or libraries. Consequently, Haskell\nis a large language and there are numerous books, tutorials and papers devoted to\nvarious aspects of the subject, including the recent Parallel and Concurrent Programming in Haskell by Simon Marlow (O\u2019Reilly, 2013). Pointers to much of the\nmaterial can be found at www.haskell.org. But three books in particular were\nopen on my desk while writing this text. The \ufb01rst is Haskell 98, Languages and Libraries, The Revised Report (Cambridge University Press, 2003), edited by Simon\nPeyton Jones. This is an indispensable aid in understanding the nitty-gritty of the\n\ufb01rst standard version of Haskell, called Haskell 98. An online version of the report\nis available at\nwww.haskell.org/onlinereport\n\n\f1.8 Chapter notes\n\n21\n\nThe present book mostly follows this standard, though it does not cover the whole\nlanguage by any means.\nSince then a new standard, Haskell 2010, has been released; see\nhaskell.org/onlinereport/haskell2010/\nOne change is that module names are now hierarchical, so we write Data.List\nrather than just List for the library of list utilities.\nThe second two are textbooks: Real World Haskell (O\u2019Reilly, 2009) by Bryan\nO\u2019Sullivan, John Goerzen and Don Stewart; and Programming in Haskell (Cambridge, 2007) by Graham Hutton. As its name implies, the former deals mostly with\nhighly practical applications, while the latter is another introductory text. Graham\nHutton did suggest to me, albeit with a grin, that my book should be called Ivory\nTower Haskell.\nThere is a fascinating history concerning the common words problem. Jon Bentley\ninvited one programmer, Don Knuth, to write a literate WEB program for the problem, and another programmer, Doug McIlroy, to write a literary review of it. The\nresult was published in Bentley\u2019s Programming Pearls column in Communications\nof the ACM, vol. 29, no. 6 (June 1986).\n\n\fChapter 2\nExpressions, types and values\n\nIn Haskell every well-formed expression has, by de\ufb01nition, a well-formed type.\nEach well-formed expression has, by de\ufb01nition, a value. Given an expression for\nevaluation,\n\u2022 GHCi checks that the expression is syntactically correct, that is, it conforms to\nthe rules of syntax laid down by Haskell.\n\u2022 If it is, GHCi infers a type for the expression, or checks that the type supplied by\nthe programmer is correct.\n\u2022 Provided the expression is well-typed, GHCi evaluates the expression by reducing it to its simplest possible form to produce a value. Provided the value is\nprintable, GHCi then prints it at the terminal.\nIn this chapter we continue the study of Haskell by taking a closer look at these\nprocesses.\n\n2.1 A session with GHCi\nOne way of \ufb01nding out whether or not an expression is well-formed is of course\nto use GHCi. There is a command :type expr which, provided expr is wellformed, will return its type. Here is a session with GHCi (with some of GHCi\u2019s\nresponses abbreviated):\nghci> 3 +4)\n:1:5: parse error on input `)'\nGHCi is complaining that on line 1 the character ')' at position 5 is unexpected;\nin other words, the expression is not syntactically correct.\n\n\f2.1 A session with GHCi\n\n23\n\nghci> :type 3+4\n3+4 :: Num a => a\nGHCi is asserting that the type of 3+4 is a number. More on this below.\nghci> :type if 1==0 then 'a' else \"a\"\n:1:23:\nCouldn't match expected type `Char' with actual type `[Char]'\nIn the expression: \"a\"\nIn the expression: if 1 == 0 then 'a' else \"a\"\nGHCi expects the types of expr1 and expr2 in a conditional expression\nif test then expr1 else expr2\nto be the same. But a character is not a list of characters so the conditional expression, though conforming to the rules of Haskell syntax, is not well-formed.\nghci> sin sin 0.5\n:1:1:\nNo instance for (Floating (a0 -> a0))\narising from a use of `sin'\nPossible fix: add an instance declaration for\n(Floating (a0 -> a0))\nIn the expression: sin sin 0.5\nIn an equation for `it': it = sin sin 0.5\nGHCi gives a rather opaque error message, complaining that the expression is not\nwell-formed.\nghci> sin (sin 0.5)\n0.4612695550331807\nAh, GHCi is happy with this one.\nghci> :type map\nmap :: (a -> b) -> [a] -> [b]\nGHCi returns the type of the function map.\nghci> map\n:1:1:\nNo instance for (Show ((a0 -> b0) -> [a0] -> [b0]))\narising from a use of `print'\nPossible fix:\nadd an instance declaration for\n\n\f24\n\nExpressions, types and values\n\n(Show ((a0 -> b0) -> [a0] -> [b0]))\nIn a stmt of an interactive GHCi command: print it\nGHCi is saying that it doesn\u2019t know how to print a function.\nghci> :type 1 `div` 0\n1 `div` 0 :: Integral a => a\nGHCi is asserting that the type of 1 `div` 0 is an integral number. The expression\n1 `div` 0 is therefore well-formed and possesses a value.\nghci> 1 `div` 0\n*** Exception: divide by zero\nGHCi returns an error message. So what is the value of 1 `div` 0? The answer\nis that it is a special value, written mathematically as \u22a5 and pronounced \u2018bottom\u2019.\nIn fact, Haskell provides a predeclared name for this value, except that it is called\nundefined, not bottom.\nghci> :type undefined\nundefined :: a\nghci> undefined\n*** Exception: Prelude.undefined\nHaskell is not expected to produce the value \u22a5. It may return with an error message, or remain perpetually silent, computing an in\ufb01nite loop, until we interrupt the\ncomputation. It may even cause GHCi to crash. Oh, yes.\nghci> x*x where x = 3\n:1:5: parse error on input `where'\nghci> let x = 3 in x*x\n9\nA where clause does not qualify an expression in Haskell, but the whole of the\nright-hand side of a de\ufb01nition. Thus the \ufb01rst example is not a well-formed expression. On the other hand, a let expression\nlet in \nis well-formed, at least assuming the de\ufb01nitions in are and the expression\n is. Let-expressions appear infrequently in what follows, but occasionally\nthey can be useful.\n\n\f2.2 Names and operators\n\n25\n\n2.2 Names and operators\nAs we have seen, a script is a collection of names and their de\ufb01nitions. Names\nfor functions and values begin with a lowercase letter, except for data constructors\n(see later on) which begin with an uppercase letter. Types (e.g. Int), type classes\n(e.g. Num) and modules (e.g. Prelude or Data.Char) also begin with an uppercase\nletter.\nAn operator is a special kind of function name that appears between its (two) arguments, such as the + in x + y or the ++ in xs ++ ys. Operator names begin with\na symbol. Any (non-symbolic) function of two arguments can be converted into\nan operator by enclosing it in back quotes, and any operator can be converted to a\npre\ufb01x name by enclosing it in parentheses. For example,\n3 + 4\ndiv 3 4\n\nis the same as\nis the same as\n\n(+) 3 4\n3 `div` 4\n\nOperators have different levels of precedence (binding power). For example,\nmeans\nmeans\n\n3 * 4 + 2\nxs ++ yss !! 3\n\n(3 * 4) + 2\nxs ++ (yss !! 3)\n\nIf in any doubt, add parentheses to remove possible ambiguity. By the way, we can\nuse any names we like for lists, including x, y, goodylist, and so on. But a simple\naid to memory is to use x for things, xs for lists of things, and xss for lists of lists\nof things. That explains why we wrote yss in the expression yss !! 3 in the last\nline above.\nOperators with the same level of precedence normally have an order of association,\neither to the left or right. For example, the usual arithmetic operators associate to\nthe left:\n3 - 4 - 2\n3 - 4 + 2\n3 / 4 * 5\n\nmeans\nmeans\nmeans\n\n(3 - 4) - 2\n(3 - 4) + 2\n(3 / 4) * 5\n\nFunctional application, which has higher precedence than any other operator, also\nassociates to the left:\neee bah gum\neee bah gum*2\n\nmeans\nmeans\n\n(eee bah) gum\n((eee bah) gum)*2\n\nSome operators associate to the right:\n\n\fExpressions, types and values\n\n26\n\n(a -> b) -> [a] -> [b]\nx ^ y ^ z\neee . bah . gum\n\nmeans\nmeans\nmeans\n\n(a -> b) -> ([a] -> [b])\nx ^ (y ^ z)\neee . (bah . gum)\n\nOf course, if an operator, such as functional composition, is associative the order\nhas no effect on meaning (i.e. the value is the same). Again, one can always add\nparentheses to remove possible ambiguity.\nWe can declare new operators; for example:\n(+++) :: Int -> Int -> Int\nx +++ y = if even x then y else x + y\nThe conditional expression has low binding power, so the expression above means\nif even x then y else (x + y)\nnot (if even x then y else x) + y. Again, one can always use parentheses\nto group differently.\nIf we like we can declare a precedence level and an order of association for (+++),\nbut we won\u2019t spell out how.\n\nSections and lambda expressions\nIt is a matter of style, but in the main we prefer to write scripts in which all the\nlittle helper functions are named explicitly. Thus if we need a function that adds 1\nto a number, or doubles a number, then we might choose to name such functions\nexplicitly:\nsucc, double :: Integer -> Integer\nsucc n\n= n+1\ndouble n = 2*n\nHowever, Haskell provides alternative ways of naming these two functions, namely\n(+1) and (2*). The device is called a section. In a section one of the arguments of\nan operator is included along with the operator. Thus\n(+1)\n(0<)\n(<0)\n(1/)\n\nn\nn\nn\nx\n\n=\n=\n=\n=\n\nn+1\n0 2*n+1. It is called a lambda\nexpression because mathematically the function would be written as \u03bb n.2\u2217n+1.\nRead the expression as \u2018that function of n which returns 2\u2217n+1\u2019. For example,\nghci> map (\\n -> 2*n+1) [1..5]\n[3,5,7,9,11]\nOnce in a while a lambda expression seems the best way to describe some function, but only once in a while and we will take them out of the box only on rare\noccasions.\n\n2.3 Evaluation\nHaskell evaluates an expression by reducing it to its simplest possible form and\nprinting the result. For example, suppose we have de\ufb01ned\nsqr :: Integer -> Integer\nsqr x = x*x\nThere are basically two ways to reduce the expression sqr (3+4) to its simplest\npossible form, namely 49. Either we can evaluate 3+4 \ufb01rst, or else apply the de\ufb01nition of sqr \ufb01rst:\n=\n=\n=\n=\n\nsqr (3+4)\nsqr 7\nlet x = 7 in x*x\n7*7\n49\n\n=\n=\n=\n=\n\nsqr (3+4)\nlet x = 3+4 in x*x\nlet x = 7 in x*x\n7*7\n49\n\n\fExpressions, types and values\n\n28\n\n```\n\n\n```\n\n\nThe number of reduction steps is the same in each case, but the order of the reduction steps is slightly different. The method on the left is called innermost reduction\nand also eager evaluation; the one on the right is called outermost reduction or lazy\nevaluation. With eager evaluation arguments are always evaluated before a function is applied. With lazy evaluation the de\ufb01nition of a function is installed at once\nand only when they are needed are the arguments to the function evaluated.\nDoesn\u2019t seem much of a difference, does it? But consider the following (slightly\nabbreviated) evaluation sequences concerning the function fst that returns the \ufb01rst\nelement of a pair, so fst (x,y) = x:\n=\n=\n=\n=\n=\n\nfst\nfst\nfst\nfst\nfst\n1\n\n(sqr 1,sqr 2)\n(1*1,sqr 2)\n(1,sqr 2)\n(1,2*2)\n(1,4)\n\nfst (sqr 1,sqr 2)\n= let p = (sqr 1,sqr 2)\nin fst p\n= sqr 1\n= 1*1\n= 1\n\nThe point here is that under eager evaluation the value sqr 2 is computed, while\nunder lazy evaluation that value is not needed and is not computed.\nNow suppose we add the de\ufb01nitions\ninfinity :: Integer\ninfinity = 1 + infinity\nthree :: Integer -> Integer\nthree x = 3\nEvaluating infinity will cause GHCi to go into a long, silent think trying to\ncompute 1 + (1 + (1 + (1 + (1 + .... until eventually it runs out of space\nand returns an error message. The value of infinity is \u22a5.\nAgain there are two ways to evaluate three infinity:\nthree infinity\n= three (1+infinity)\n= three (1+(1+infinity))\n= ...\n\nthree infinity\n= let x = infinity in 3\n= 3\n\nHere eager evaluation gets stuck in a loop trying to evaluate infinity, while lazy\nevaluation returns the answer 3 at once. We don\u2019t need to evaluate the argument of\nthree in order to return 3.\nOne more de\ufb01nition, a version of the factorial function:\n\n\f2.3 Evaluation\n\n29\n\nfactorial :: Integer -> Integer\nfactorial n = fact (n,1)\nfact :: (Integer,Integer) -> Integer\nfact (x,y) = if x==0 then y else fact (x-1,x*y)\nThis is another example of a recursive de\ufb01nition (the de\ufb01nition of infinity was\nalso recursive, and so was the function song in the previous chapter). Expressions\ninvolving recursive functions are evaluated like any other de\ufb01nition.\nHere the two evaluation schemes result in the following sequence of reduction steps\n(we hide the steps involving simpli\ufb01cation of the conditional expression to make\nanother point):\n\n=\n=\n=\n=\n=\n=\n=\n=\n\nfactorial 3\nfact (3,1)\nfact (3-1,3*1)\nfact (2,3)\nfact (2-1,2*3)\nfact (1,6)\nfact (1-1,1*6)\nfact (0,6)\n6\n\n=\n=\n=\n=\n=\n=\n=\n=\n\nfactorial 3\nfact (3,1)\nfact (3-1,3*1)\nfact (2-1,2*(3*1))\nfact (1-1,1*(2*(3*1)))\n1*(2*(3*1))\n1*(2*3)\n1*6\n6\n\nThe point to appreciate is that, while the number of reduction steps is basically\nthe same, lazy evaluation requires much more space to achieve the answer. The\nexpression 1*(2*(3*1)) is built up in memory before being evaluated.\nThe pros and cons of lazy evaluation are brie\ufb02y as follows. On the plus side, lazy\nevaluation terminates whenever any reduction order terminates; it never takes more\nsteps than eager evaluation, and sometimes in\ufb01nitely fewer. On the minus side, it\ncan require a lot more space and it is more dif\ufb01cult to understand the precise order\nin which things happen.\nHaskell uses lazy evaluation. ML (another popular functional language) uses eager evaluation. Exercise D explores why lazy evaluation is a Good Thing. Lazy\nevaluation is considered further in Chapter 7.\nA Haskell function f is said to be strict if f undefined = undefined, and nonstrict otherwise. The function three is non-strict, while (+) is strict in both arguments. Because Haskell uses lazy evaluation we can de\ufb01ne non-strict functions.\nThat is why Haskell is referred to as a non-strict functional language.\n\n\fExpressions, types and values\n\n30\n\n2.4 Types and type classes\nHaskell has built-in (or primitive) types such as Int, Float and Char. The type\nBool of boolean values is de\ufb01ned in the standard prelude:\ndata Bool = False | True\nThis is an example of a data declaration. The type Bool is declared to have two\ndata constructors, False and True. The type Bool has three values, not two:\nFalse, True and undefined :: Bool. Why do we need that last value? Well,\nconsider the function\nto :: Bool -> Bool\nto b = not (to b)\nThe prelude de\ufb01nition of not is\nnot :: Bool -> Bool\nnot True = False\nnot False = True\nThe de\ufb01nition of to is perfectly well-formed, but evaluating to True causes GHCi\nto go into an in\ufb01nite loop, so its value is \u22a5 of type Bool. We will have much more\nto say about data declarations in future chapters.\nHaskell has built-in compound types, such as\n[Int]\n(Int,Char)\n(Int,Char,Bool)\n()\nInt -> Int\n\na list of elements, all of type Int\na pair consisting of an Int and a Char\na triple\nan empty tuple\na function from Int to Int\n\nThe sole inhabitant of the type () is also denoted by (). Actually, there is a second\nmember of (), namely undefined :: (). Now we can appreciate that there is a\nvalue \u22a5 for every type.\nAs we have already said, when de\ufb01ning values or functions it is always a good idea\nto include the type signature as part of the de\ufb01nition.\nConsider next the function take n that takes the \ufb01rst n elements of a list. This\nfunction made its appearance in the previous chapter. For example,\ntake 3 [1,2,3,4,5] = [1,2,3]\ntake 3 \"category\" = \"cat\"\ntake 3 [sin,cos]\n= [sin,cos]\n\n\f2.4 Types and type classes\n\n31\n\nWhat type should we assign to take? It doesn\u2019t matter what the type of the elements of the list is, so take is what is called a polymorphic function and we denote\nits type by\ntake :: Int -> [a] -> [a]\nThe a is a type variable. Type variables begin with a lowercase letter. Type variables\ncan be instantiated to any type.\nSimilarly,\n(++) :: [a] -> [a] -> [a]\nmap :: (a -> b) -> [a] -> [b]\n(.) :: (b -> c) -> (a -> b) -> (a -> c)\nThe last line declares the polymorphic type of functional composition.\nNext, what is the type of (+)? Here are some suggestions:\n(+) :: Int -> Int -> Int\n(+) :: Float -> Float -> Float\n(+) :: a -> a -> a\nThe \ufb01rst two types seem too speci\ufb01c, while the last seems too general: we can\u2019t\nadd two functions or two characters or two booleans, at least not in any obvious\nway.\nThe answer is to introduce type classes:\n(+) :: Num a => a -> a -> a\nThis declaration asserts that (+) is of type a -> a -> a for any number type a.\nA type class, such as Num, has a collection of named methods, such as (+), which\ncan be de\ufb01ned differently for each instance of the type class. Type classes therefore provide for overloaded functions, functions with the same name but different\nde\ufb01nitions. Overloading is another kind of polymorphism.\nNumbers are rather complicated, and are explained in more detail in the following\nchapter, so we illustrate type classes with a simpler type class\nclass Eq a where\n(==),(/=) :: a -> a -> Bool\nx /= y\n= not (x == y)\nThis introduces the Equality type class, members of which can use one and the\nsame equality test (==) and inequality test (/=). There is a default de\ufb01nition of\n(/=) as part of the class, so we only have to provide a de\ufb01nition of (==).\n\n\f32\n\nExpressions, types and values\n\nTo become a member of the Eq club we have to de\ufb01ne an instance. For example,\ninstance Eq Bool where\nx == y = if x then y else not y\ninstance Eq Person where\nx == y = (pin x == pin y)\nIf pin :: Person -> Pin then we need Eq Pin for the last instance to be correct. Of course, we don\u2019t have to make Person a member of the Equality club; we\ncan always de\ufb01ne\nsamePerson :: Person -> Person -> Bool\nsamePerson x y = (pin x == pin y)\nBut we can\u2019t use (==) instead of samePerson unless we make an instance declaration.\nHere are simpli\ufb01ed versions of two other type classes, Ord and Show:\nclass (Eq a) => Ord a where\n(<),(<=),(>=),(>) :: a -> a -> Bool\nx < y = not (x >= y)\nx <= y = x == y || x < y\nx >= y = x == y || x > y\nx > y = not (x <= y)\nclass Show a where\nshow :: a -> String\nThe boolean operator (||) denotes disjunction: a || b is true only if at least one\nof a and b is true. We can de\ufb01ne this operator by\n(||) :: Bool -> Bool -> Bool\na || b = if a then True else b\nThe default de\ufb01nitions of the Ord methods are mutually dependent, so one has to\nprovide a speci\ufb01c de\ufb01nition of at least one of them in any instance to break the\ndependency (unlike Eq where only (/=) was given a default de\ufb01nition). The type\nclass Ord needs Eq as a superclass because it makes use of (==) in the default\nde\ufb01nitions of the four comparison operations.\nThe type class Show is used for displaying results. Haskell cannot display the result\nof a computation unless the type of the result is a member of Show. Let us explain\nthis in a little more detail.\n\n\f2.5 Printing values\n\n33\n\n2.5 Printing values\nWe begin with a mystery:\nghci> \"Hello ++\"\\n\"++ \"young\" ++\"\\n\"++ \"lovers\"\n\"Hello\\nyoung\\nlovers\"\nOh. What we wanted was\nHello\nyoung\nlovers\nWhy didn\u2019t Haskell print that?\nThe reason is that after evaluating a well-formed expression to produce a value,\nHaskell applies show to the value to produce a string that can be printed at the\nterminal. Applying show to a value v produces a string that when printed looks\nexactly like v: Thus,\nshow\nshow\nshow\nshow\n\n42\n42.3\n'a'\n\"hello\\n\"\n\n=\n=\n=\n=\n\n\"42\"\n\"42.3\"\n\"'a'\"\n\"\\\"hello\\\\n\\\"\"\n\nPrinting the result involves the use of a Haskell command\nputStrLn :: String -> IO ()\nThe type IO a is a special type, the type of input\u2013output computations that when\nexecuted have some interaction with the outside world and return a value of type a.\nIf the return value is uninteresting, as with putStrLn, we use the null-tuple value\n().\nSo, Haskell uniformly applies a show-and-put strategy to print values. Since the\ngreeting above is already a string, we really want to miss out the show step and go\nstraight to the put:\nghci> putStrLn (\"Hello ++\"\\n\"++ \"young\" ++\"\\n\"++ \"lovers\")\nHello\nyoung\nlovers\nHaskell provides many more commands for input\u2013output, for reading and writing\nto \ufb01les, for displaying graphics, and so on. Such commands have to be sequenced\n\n\fExpressions, types and values\n\n34\n\ncorrectly, and for this Haskell provides a special notation, called do-notation. Commands are the subject of Chapter 10, and what follows is simply a foretaste of\nthings to come.\nTo see an example, consider the common words problem of the previous chapter.\nThere we de\ufb01ned a function\ncommonWords :: Int -> String -> String\nsuch that commonWords n took a text string and returned a string giving a table of\nthe n most common words in the text. The following program reads the text from\na \ufb01le, and writes the output to a \ufb01le. The type FilePath is another synonym for a\nlist of characters:\ncwords :: Int -> FilePath -> FilePath -> IO()\ncwords n infile outfile\n= do {text <- readFile infile;\nwriteFile outfile (commonWords n text);\nputStrLn \"cwords done!\"}\nEvaluating, for example\nghci> cwords 100 \"c:\\\\WarAndPeace\" \"c:\\\\Results\"\non a Windows platform will cause the \ufb01le c:\\WarAndPeace to be read, and the\nresults printed to c:\\Results. The program also prints a message to the terminal.\nThe two component functions of the de\ufb01nition above have types\nreadFile :: FilePath -> IO String\nwriteFile :: FilePath -> String -> IO ()\nSuppose that we didn\u2019t want to call cwords from within an interactive session, but\nto use it as a stand-alone program. Here is one way. We need to de\ufb01ne a value for\nan identi\ufb01er main of type IO (). Here is such a program:\nmain\n= do {putStrLn \"Take text from where:\";\ninfile <- getLine;\nputStrLn \"How many words:\";\nn <- getLine;\nputStrLn \"Put results where:\";\noutfile <- getLine;\ntext <- readFile infile;\nwriteFile outfile (commonWords (read n) text);\nputStrLn \"cwords done!\" }\n\n\f2.6 Modules\n\n35\n\nFor an explanation of read see Exercise H. Suppose the common words script is\nstored in the \ufb01le cwords.lhs. We can compile it with GHC, the Glasgow Haskell\nCompiler:\n$ ghc cwords.lhs\nThe compiled program will be stored in the \ufb01le cwords.exe. To run the program\nunder Windows, type\n$ cwords\nand follow the instructions.\n\n2.6 Modules\nSuppose we thought that the function commonWords was suf\ufb01ciently useful that we\nwanted to incorporate it into other scripts. The way to do this is to turn the common\nwords script into a module. First, we rewrite the script in the following way:\nmodule CommonWords (commonWords) where\nimport Data.Char (toLower)\nimport Data.List (sort,words)\n...\ncommonWords :: Int -> String -> String\n...\nThe module declaration is followed by the name of the module, which must begin with a capital letter. Furthermore, the script has to be stored in a \ufb01le called\nCommonWords.lhs to enable Haskell to \ufb01nd the module (at least, if you are using\nliterate scripts; otherwise it would be CommonWords.hs). Following the name of\nthe module is a list of exports, the functions, types and other values you want to be\nable to export to other scripts. The list of exports has to be enclosed in parentheses.\nHere we just export one function, commonWords. The exports are the only things\nde\ufb01ned in the module that are visible in other modules. Omitting the export list,\nand the surrounding parentheses, means that everything in the module is exported.\nWe can then compile the module using GHC and then import it into other scripts\nwith the declaration\nimport CommonWords (commonWords)\nThere are two major advantages of Haskell modules. One is we can structure our\nscripts into bite-sized chunks, separating out little groups of related functions into\n\n\f36\n\nExpressions, types and values\n\nseparate modules. The other advantage is that the functions in a compiled module\nare much faster to evaluate because their de\ufb01nitions are compiled into machinespeci\ufb01c code, leading to a much slicker reduction process. GHCi is an interpreter\nrather than a compiler; it evaluates internal forms of expression that are much closer\nto the source language of Haskell.\n\n2.7 Haskell layout\nThe examples of do-notation used braces ({ and }) and semicolons; these are examples of explicit layout. Braces and semicolons are used only to control layout\nand have no meaning as part of the language of Haskell expressions. We can use\nthem in other places too:\nroots :: (Float,Float,Float) -> (Float,Float)\nroots (a,b,c)\n| a == 0\n= error \"not quadratic\"\n| disc < 0\n= error \"complex roots\"\n| otherwise\n= ((-b-r)/e, (-b+r)/e)\nwhere {disc = b*b - 4*a*c; r = sqrt d; e = 2*a}\nHere the where clause uses explicit braces and semicolons rather than appealing\nto Haskell\u2019s layout rules. Instead, we could have written\nwhere disc = b*b - 4*a*c\nr\n= sqrt d\ne\n= 2*a\nBut we couldn\u2019t have written\nwhere disc = b*b - 4*a*c\nr = sqrt d\ne = 2*a\nThe layout (or offside) rule takes effect whenever the opening brace is omitted after\nthe keyword where or do (and also after let). When this happens the indentation\nof the next item, whether or not on a new line, is remembered. For each subsequent\nline, if it is indented more, then the previous line is continued; if it is indented the\nsame amount, then a new item begins; and if it is indented less, then the layout list\nis ended. At least, that\u2019s roughly the offside rule.\nThe offside rule explains why there is an indentation in the declarations of type\nclasses and instances:\n\n\f2.8 Exercises\n\n37\n\nclass Foo a where\nI am part of the class declaration.\nSo am I.\nNow the class declaration has ended.\nYou can always put in braces and semicolons if in any doubt. Actually the offside\nrule can still cause confusion when used with do-notation. So the recommendation\nis belts, braces and semicolons.\nAnd you thought the football offside rule was complicated.\n\n2.8 Exercises\nExercise A\nOn the subject of precedence, this question comes from Chris Maslanka\u2019s puzzle\npage in the Guardian newspaper:\n\u2018Is a half of two plus two equal to two or three?\u2019\nExercise B\nSome of the following expressions are not syntactically correct, while others are\nsyntactically correct but do not have sensible types. Some are well-formed. Which\nis which? In the case of a well-formed expression, give a suitable type. Assume\ndouble :: Int -> Int. I suggest you don\u2019t use a computer to check your answers, but if you do, be prepared for some strange error messages.\nThe expressions are:\n[0,1)\ndouble -3\ndouble (-3)\ndouble double 0\nif 1==0 then 2==1\n\"++\" == \"+\" ++ \"+\"\n[(+),(-)]\n[[],[[]],[[[]]]]\nconcat [\"tea\",\"for\",'2']\nconcat [\"tea\",\"for\",\"2\"]\n\n\fExpressions, types and values\n\n38\n\nExercise C\nIn the good old days, one could write papers with titles such as\n\u2018The morphology of prex \u2013 an essay in meta-algorithmics\u2019\nThese days, journals seem to want all words capitalised:\n\u2018The Morphology Of Prex \u2013 An Essay In Meta-algorithmics\u2019\nWrite a function modernise :: String -> String which ensures that paper\ntitles are capitalised as above. Here are some helpful questions to answer \ufb01rst:\n1. The function toLower :: Char -> Char converts a letter to lowercase. What\ndo you think is the name of the prelude function that converts a letter to uppercase?\n2. The function words :: String -> [Word] was used in the previous chapter.\nWhat do you think the prelude function\nunwords :: [Word] -> String\ndoes? Hint: which, if either, of the following equations should hold?\nwords . unwords = id\nunwords . words = id\n3. The function head :: [a] -> a returns the head of a nonempty list, and\ntail :: [a] -> [a] returns the list that remains when the head is removed.\nSuppose a list has head x and tail xs. How would you reconstruct the list?\nExercise D\nBeaver is an eager evaluator, while Susan is a lazy one.1 How many times would\nBeaver evaluate f in computing head (map f xs) when xs is a list of length n?\nHow many times would Susan? What alternative to head . map f would Beaver\nprefer?\nThe function filter p \ufb01lters a list, retaining only those elements that satisfy the\nboolean test p. The type of filter is\nfilter :: (a -> Bool) -> [a] -> [a]\nSusan would happily use head . filter p for a function that \ufb01nds the \ufb01rst element of a list satisfying p. Why would Beaver not use the same expression?\nInstead, Beaver would probably de\ufb01ne something like\n1\n\nIf you don\u2019t know, google \u2018lazy susan\u2019 to discover what a lazy susan is.\n\n\f2.8 Exercises\n\n39\n\nfirst :: (a -> Bool) -> [a] -> a\nfirst p xs | null xs\n= error \"Empty list\"\n| p x\n= ...\n| otherwise = ...\nwhere x = head xs\nThe function null returns True on an empty list, and False otherwise. When\nevaluated, the expression error message stops execution and prints the string\nmessage at the terminal, so its value is \u22a5. Complete the right-hand side of Beaver\u2019s\nde\ufb01nition.\nWhat alternative might Beaver prefer to head . filter p . map f?\nExercise E\nThe type Maybe is declared in the standard prelude as follows:\ndata Maybe a = Nothing | Just a\nderiving (Eq, Ord)\nThis declaration uses a deriving clause. Haskell can automatically generate instances of some standard type classes for some data declarations. In the present\ncase the deriving clause means that we don\u2019t have to go through the tedium of\nwriting\ninstance (Eq a) => Eq (Maybe a)\nNothing == Nothing = True\nNothing == Just y = False\nJust x == Nothing = False\nJust x == Just y\n= (x == y)\ninstance (Ord a) => Ord (Maybe a)\nNothing <= Nothing = True\nNothing <= Just y = True\nJust x <= Nothing = False\nJust x <= Just y\n= (x <= y)\nThe reason why Nothing is declared to be less than Just y is simply because the\nconstructor Nothing comes before the constructor Just in the data declaration for\nMaybe.\nThe reason why the Maybe type is useful is that it provides a systematic way of\nhandling failure. Consider again the function\n\n\fExpressions, types and values\n\n40\n\nfirst p = head . filter p\nof the previous exercise. Both Eager Beaver and Lazy Susan produced versions of\nthis function that stopped execution and returned an error message when first p\nwas applied to the empty list. That\u2019s not very satisfactory. Much better is to de\ufb01ne\nfirst :: (a -> Bool) -> [a] -> Maybe a\nNow failure is handled gracefully by returning Nothing if there is no element of\nthe list that satis\ufb01es the test.\nGive a suitable de\ufb01nition of this version of first.\nFinally, count the number of functions with type Maybe a -> Maybe a.\nExercise F\nHere is a function for computing x to the power n, where n \u2265 0:\nexp :: Integer -> Integer -> Integer\nexp x n | n == 0\n= 1\n| n == 1\n= x\n| otherwise = x*exp x (n-1)\nHow many multiplications does it take to evaluate exp x n?\nDick, a clever programmer, claims he can compute exp x n with far fewer multiplications:\nexp x n |\n|\n|\n|\n\nn == 0\nn == 1\neven n\nodd n\n\n=\n=\n=\n=\n\n1\nx\n...\n...\n\nFill in the dots and say how many multiplications it takes to evaluate the expression\nexp x n by Dick\u2019s method, assuming 2p \u2264 n < 2p+1 .\nExercise G\nSuppose a date is represented by three integers (day, month, year). De\ufb01ne a function showDate :: Date -> String so that, for example,\nshowDate (10,12,2013) = \"10th December, 2013\"\nshowDate (21,11,2020) = \"21st November, 2020\"\nYou need to know that Int is a member of the type class Show, so that show n\nproduces a string that is the decimal representation of the integer n.\n\n\f2.8 Exercises\n\n41\n\nExercise H\nThe credit card company Foxy issues cards with ten-digit card-identi\ufb01cation numbers (CINs). The \ufb01rst eight digits are arbitrary but the number formed from the last\ntwo digits is a checksum equal to the sum of the \ufb01rst eight digits. For example,\n\u201c6324513428\u201d is a valid CIN because the sum of the \ufb01rst eight digits is 28.\nConstruct a function addSum :: CIN -> CIN that takes a string consisting of\neight digits and returns a string of ten digits that includes the checksum. Thus\nCIN is a type synonym for String, though restricted to strings of digits. (Note that\nHaskell type synonyms cannot enforce type constraints such as this.) You will need\nto convert between a digit character and the corresponding number. One direction\nis easy: just use show. The other direction is also fairly easy:\ngetDigit :: Char -> Int\ngetDigit c = read [c]\nThe function read is a method of the type class Read and has type\nread :: Read a => String -> a\nThe type class Read is dual to Show and read is dual to show. For example,\nghci> read \"123\" :: Int\n123\nghci> read \"123\" :: Float\n123.0\nThe function read has to be supplied with the type of the result. One can always\nadd type annotations to expressions in this way.\nNow construct a function valid :: CIN -> Bool that checks whether an identi\ufb01cation number is valid. The function take might prove useful.\nExercise I\nBy de\ufb01nition a palindrome is a string that, ignoring punctuation symbols, blank\ncharacters and whether or not a letter is in lowercase or uppercase, reads the same\nforwards and backwards. Write an interactive program\npalindrome :: IO ()\nwhich, when run, conducts an interactive session, such as\nghci> palindrome\nEnter a string:\n\n\f42\n\nExpressions, types and values\n\nMadam, I'm Adam\nYes!\nghci> palindrome\nEnter a string:\nA Man, a plan, a canal - Suez!\nNo!\nghci> palindrome\nEnter a string:\nDoc, note I dissent. A fast never prevents a fatness.\nI diet on cod.\nYes!\nThe function isAlpha :: Char -> Bool tests whether a character is a letter,\nand reverse :: [a] -> [a] reverses a list. The function reverse is provided\nin the standard prelude and isAlpha can be imported from the library Data.Char.\n\n2.9 Answers\nAnswer to Exercise A\nThe answer to Maslanka\u2019s puzzle is \u2018Yes!\u2019 This little puzzle has fooled a number\nof distinguished computer scientists.\nAnswer to Exercise B\nMy GHCi session produced (with explanations added):\nghci> :type [0,1)\n:1:5: parse error on input `)'\nGHCi knows that ')' is wrong, though it is not smart enough to suggest ']'.\nghci> :type double -3\n:1:9:\nNo instance for (Num (Int -> Int))\narising from the literal `3'\nPossible fix: add an instance declaration for\n(Num (Int -> Int))\n\n\f2.9 Answers\n\n43\n\nIn the second argument of `(-)', namely `3'\nIn the expression: double - 3\nThe explanation of the error message is that numerical subtraction (-) has type\nNum a => a -> a. For double - 3 to be well-formed (yes, it was typed as\ndouble -3 but the spaces are not signi\ufb01cant here), double has to be a number, so\nthe class instance Num (Int -> Int) is required. But there isn\u2019t one: you cannot\nsensibly subtract a number from a function.\nghci> double (-3)\n-6\nghci> double double 0\n:1:1:\nThe function `double' is applied to two arguments,\nbut its type `Int -> Int' has only one\nIn the expression: double double 0\nIn an equation for `it': it = double double 0\nMost of GHCi\u2019s error message is clear.\nghci> if 1==0 then 2==1\n:1:18:\nparse error (possibly incorrect indentation)\nConditional expressions are incomplete without an \u2018else\u2019 clause.\nghci> \"++\" == \"+\" ++ \"+\"\nTrue\nBoth sides are well-formed and denote the same list.\nghci> [(+),(-)]\n:1:1:\nNo instance for (Show (a0 -> a0 -> a0))\narising from a use of `print'\nPossible fix:\nadd an instance declaration for\n(Show (a0 -> a0 -> a0))\nIn a stmt of an interactive GHCi command: print it\nTo display the value [(+),(-)] we have to be able to show its elements. But no\nway of showing functions has been provided.\nghci> :type [[],[[]],[[[]]]]\n\n\fExpressions, types and values\n\n44\n\n[[],[[]],[[[]]]] :: [[[[a]]]]\nTo explain, let the main list have type [b]. The \ufb01rst element is a list, so b=[c].\nThe second element is a list of lists, so c=[d]. The third element is a list of lists of\nlists, so d=[a].\nghci> concat [\"tea\",\"for\",'2']\n:1:21:\nCouldn't match expected type `[Char]'\nwith actual type `Char'\nIn the expression: '2'\nIn the first argument of `concat',\nnamely `[\"tea\", \"for\", '2']'\nIn the expression: concat [\"tea\", \"for\", '2']\nThe \ufb01rst two elements of the list have type [Char], but the last has type Char and\nthat is not allowed.\nghci> concat [\"tea\",\"for\",\"2\"]\n\"teafor2\"\nAnswer to Exercise C\n1. toUpper, of course.\n2. Concatenates the words, putting a single space between them. We have\nwords . unwords = id\nbut not unwords . words = id.\n3. [x] ++ xs.\nmodernise :: String -> String\nmodernise = unwords . map capitalise . words\ncapitalise :: Word -> Word\ncapitalise xs = [toUpper (head xs)] ++ tail xs\nWe will see another way of writing capitalise in Chapter 4.\nAnswer to Exercise D\nComputing head (map f xs) takes n evaluations of f under eager evaluation,\nbut only one under lazy evaluation. Beaver would have to exploit the identity\nhead . map f = f . head.\n\n```\n\n\n```\n\n\f2.9 Answers\n\n45\n\nInstead of de\ufb01ning first p = head . filter p, Beaver might de\ufb01ne\nfirst :: (a -> Bool) -> [a] -> a\nfirst p xs | null xs\n= error \"Empty list\"\n| p x\n= x\n| otherwise = first p (tail xs)\nwhere x = head xs\nInstead of de\ufb01ning first p f = head . filter p . map f, Beaver might\nde\ufb01ne\nfirst :: (b -> Bool) -> (a -> b) -> [a] -> b\nfirst p f xs | null xs\n= error \"Empty list\"\n| p x\n= x\n| otherwise = first p f (tail xs)\nwhere x = f (head xs)\nThe point is that with eager evaluation most functions have to be de\ufb01ned using\nexplicit recursion, not in terms of useful component functions like map and filter.\nAnswer to Exercise E\nLazy Susan would probably write\nfirst p xs = if null ys then Nothing\nelse Just (head ys)\nwhere ys = filter p xs\nAs to the number of functions of type Maybe a -> Maybe a, there are just six.\nApplied to Nothing the function can only return Nothing or undefined. Applied\nto Just x the function can only return Nothing or Just x or undefined. The\npoint is that we know absolutely nothing about the underlying type, so no new\nvalues can be invented. That makes six possible functions in all.\nAnswer to Exercise F\nIt takes n-1 multiplications to evaluate exp x n. Dick\u2019s method is to exploit the\nidentities x2m = (x2 )m and x2m+1 = x(x2 )m to obtain a recursive de\ufb01nition:\nexp x n | n == 0\n| n == 1\n| even n\n| odd n\nwhere m =\n\n=\n=\n=\n=\nn\n\n1\nx\nexp (x*x) m\nx*exp (x*x) (m-1)\n`div` 2\n\n\fExpressions, types and values\n\n46\n\nThis is an example of a divide and conquer algorithm. Dick\u2019s program takes p\nmultiplications, where 2p \u2264 n < 2p+1 . Thus p = \u0007log n\b, where \u0007x\b returns the \ufb02oor\nof a number, the greatest integer no bigger than the number. We will consider the\n\ufb02oor function in more detail in the following chapter.\nAnswer to Exercise G\nshowDate :: Date -> String\nshowDate (d,m,y) = show d ++ suffix d ++ \" \" ++\nmonths !! (m-1) ++ \", \" ++ show y\nThe function suffix computes the right suf\ufb01x:\nsuffix d = if d==1 || d==21 || d==31 then \"st\" else\nif d==2 || d==22 then \"nd\" else\nif d==3 || d==23 then \"rd\" else\n\"th\"\nmonths = [\"January\",.......]\nIf you indulged in clever arithmetic to compute suffix, then you should realise\nthat Sometimes a Simple Solution is Best.\nAnswer to Exercise H\nOne solution is as follows:\naddSum :: CIN -> CIN\naddSum cin =\ncin ++ show (n `div` 10) ++ show (n `mod` 10)\nwhere n = sum (map fromDigit cin)\nvalid :: CIN -> Bool\nvalid cin = cin == addSum (take 8 cin)\nfromDigit :: Char -> Int\nfromDigit c = read [c]\nThe function fromDigit will return a numerical digit given a digit character.\nAnswer to Exercise I\nHere is one solution:\n\n\f2.10 Chapter notes\n\n47\n\nimport Data.Char (toLower,isAlpha)\npalindrome :: IO()\npalindrome\n= do {putStrLn \"Enter a string:\";\nxs <- getLine;\nif isPalindrome xs then putStrLn \"Yes!\"\nelse putStrLn \"No!\"}\nisPalindrome :: String -> Bool\nisPalindrome xs = (ys == reverse ys)\nwhere ys = map toLower (filter isAlpha xs)\n\n2.10 Chapter notes\nThe chapter has referred a number of times to the Haskell \u2018standard prelude\u2019. This\nis a collection of basic types, type classes, functions and other values that are indispensible in many programming tasks. For a complete description of the standard\nprelude, see Chapter 8 of the Haskell report; alternatively, visit\nwww.haskell.org/onlinereport/standard-prelude.html\nSee www.haskell.org for more information on the implementation of functional\nlanguages and of Haskell in particular. An older book, The Implementation of Functional Programming Languages (Prentice Hall, 1987) by Simon Peyton Jones, is\nno longer in print, but an online version can be found at\nresearch.microsoft.com/~simonpj/papers/slpj-book-1987\nApart from GHC there are other maintained compilers for Haskell, including UHC,\nthe Utrecht Haskell Compiler. See the home page cs.uu.nl/wiki/UHC.\nOn the eager-versus-lazy evaluation debate, read Bob Harper\u2019s blog article The\npoint of laziness, which can be found at\nexistentialtype.wordpress.com/2011/04/24/\nIn the blog Harper enumerates some of the reasons why he prefers a strict language. But also read Lennart Augustsson\u2019s reply to the post. Augustsson\u2019s main\npoint, emphasised in Exercise D, is that under strict evaluation you are forced for\nef\ufb01ciency reasons to de\ufb01ne most functions by explicit recursion, and therefore lose\nthe ability to build de\ufb01nitions out of simple standard functions. That undercuts our\n\n\f48\n\nExpressions, types and values\n\nability to reason about functions by applying general laws about their component\nfunctions.\nBob Harper is one of the authors of The De\ufb01nition of Standard ML (Revised) (MIT\nPress, 1989). ML is a strict functional language. You can \ufb01nd an introduction to\nML at\nwww.cs.cmu.edu/~rwh/smlbook/book.pdf\nAnother increasingly popular language is Agda, which is both a dependently-typed\nfunctional language and also a proof assistant; see the Agda home page\nwiki.portal.chalmers.se/agda/pmwiki.php\nChris Maslanka writes a regular column in the Saturday edition of the Guardian\nnewspaper.\n\n\fChapter 3\nNumbers\n\nNumbers in Haskell are complicated because in the Haskell world there are many\ndifferent kinds of number, including:\nInt\nInteger\nRational\nFloat\nDouble\nComplex\n\nlimited-precision integers in at least the range\n[\u2212229 , 229 ). Integer over\ufb02ow is not detected.\narbitrary-precision integers\narbitrary-precision rational numbers\nsingle-precision \ufb02oating-point numbers\ndouble-precision \ufb02oating-point numbers\ncomplex numbers (de\ufb01ned in Data.Complex)\n\nMost programs make use of numbers in one way or another, so we have to get\nat least a working idea of what Haskell offers us and how to convert between the\ndifferent kinds. That is what the present chapter is about.\n\n3.1 The type class Num\nIn Haskell all numbers are instances of the type class Num:\nclass (Eq a, Show a) => Num a where\n(+),(-),(*) :: a -> a -> a\nnegate\n:: a -> a\nabs, signum :: a -> a\nfromInteger :: Integer -> a\nThe class Num is a subclass of both Eq and Show. That means every number can\nbe printed and any two numbers can be compared for equality. Any number can\nbe added to, subtracted from or multiplied by another number. Any number can be\n\n\fNumbers\n\n50\n\nnegated. Haskell allows -x to denote negate x; this is the only pre\ufb01x operator in\nHaskell.\nThe functions abs and signum return the absolute value of a number and its sign.\nIf ordering operations were allowed in Num (and they aren\u2019t because, for example,\ncomplex numbers cannot be ordered), we could de\ufb01ne\nabs x\n= if x <\nsignum x | x < 0\n| x == 0\n| x > 0\n\n0\n=\n=\n=\n\nthen -x else x\n-1\n0\n1\n\nThe function fromInteger is a conversion function. An integer literal such as\n42 represents the application of fromInteger to the appropriate value of type\nInteger, so such literals have type Num a => a. This choice is explained further\nbelow after we have considered some other classes of number and the conversion\nfunctions between them.\n\n3.2 Other numeric type classes\nThe Num class has two subclasses, the real numbers and the fractional numbers:\nclass (Num a,Ord a) => Real a where\ntoRational :: a -> Rational\nclass (Num a) => Fractional a where\n(/) :: a -> a -> a\nfromRational :: Rational -> a\nReal numbers can be ordered. The only new method in the class Real, apart from\nthe comparison operations which are inherited from the superclass Ord, is a conversion function from elements in the class to elements of Rational. The type\nRational is essentially a synonym for pairs of integers. The real number \u03c0 is not\nrational, so toRational can only convert to an approximate rational number:\nghci> toRational pi\n884279719003555 % 281474976710656\nNot quite as memorable as 22 % 7, but more accurate. The symbol % is used to\nseparate the numerator and denominator of a rational number.\n\n\f3.2 Other numeric type classes\n\n51\n\nThe fractional numbers are those on which division is de\ufb01ned. A complex number\ncannot be real but it can be fractional. A \ufb02oating-point literal such as 3.149 represents the application of fromRational to an appropriate rational number. Thus\n3.149 ::\n\nFractional a => a\n\nThis type and the earlier type Num a => a for 42 explains why we can form a\nlegitimate expression such as 42 + 3.149, adding an integer to a \ufb02oating-point\nnumber. Both types are members of the Num class and all numbers can be added.\nConsideration of\nghci> :type 42 + 3.149\n42 + 3.149 :: Fractional a => a\nshows that the result of the addition is also a fractional number.\nOne of the subclasses of the real numbers is the integral numbers. A simpli\ufb01ed\nversion of this class is:\nclass (Real a, Enum a) => Integral a where\ndivMod :: a -> a -> (a,a)\ntoInteger :: a -> Integer\nThe class Integral is a subclass of Enum, those types whose elements can be\nenumerated in sequence. Every integral number can be converted into an Integer\nthrough the conversion function toInteger. That means we can convert an integral number into any other type of number in two steps:\nfromIntegral :: (Integral a, Num b) => a -> b\nfromIntegral = fromInteger . toInteger\nApplication of divMod returns two values:\nx `div` y\nx `mod` y\n\n= fst (x `divMod` y)\n= snd (x `divMod` y)\n\nThe standard prelude functions fst and snd return the \ufb01rst and second components\nof a pair:\nfst :: (a,b) -> a\nfst (x,y) = x\nsnd :: (a,b) -> b\nsnd (x,y) = y\n\n\f52\n\nNumbers\n\nMathematically, x div y = \u0007x/y\b. We will see how to compute \u0007x\b in the following\nsection. And x mod y is de\ufb01ned by\nx = (x div y) \u2217 y + x mod y\nFor positive x and y we have 0 \u2264 x mod y < x.\nRecall the function digits2 from the \ufb01rst chapter, where we de\ufb01ned\ndigits2 n = (n `div` 10, n `mod` 10)\nIt is more ef\ufb01cient to say digits2 n = n `divMod` 10 because then only one\ninvocation of divMod is required. Even more brie\ufb02y, we can use a section and write\ndigits2 = (`divMod` 10).\nThere are also other numeric classes, including the subclass Floating of the class\nFractional that contains, among others, the logarithmic and trigonometric functions. But enough is enough.\n\n3.3 Computing \ufb02oors\nThe value \u0007x\b, the \ufb02oor of x, is de\ufb01ned to be the largest integer m such that\nm \u2264 x. We de\ufb01ne a function floor :: Float -> Integer for computing \ufb02oors.\nHaskell provides such a function in the standard prelude, but it is instructive to consider our own version.\nOne student, call him Clever Dick, to whom this task was given came up with the\nfollowing solution:\nfloor :: Float -> Integer\nfloor = read . takeWhile (/= '.') . show\nIn words, the number is shown as a string, the string is truncated by taking only the\ndigits up to the decimal point, and the result is read again as an integer. We haven\u2019t\nmet takeWhile yet, though Clever Dick evidently had. Clever Dick\u2019s solution is\nwrong on a number of counts, and Exercise D asks you to list them.\nInstead we will \ufb01nd the \ufb02oor of a number with the help of an explicit search, and\nfor that we will need a loop:\nuntil :: (a -> Bool) -> (a -> a) -> a -> a\nuntil p f x = if p x then x else until p f (f x)\nThe function until is also provided in the standard prelude. Here is an example:\n\n\f3.3 Computing \ufb02oors\n\nghci>\n343\n\n53\n\nuntil (>100) (*7) 1\n\nEssentially until f p x computes the \ufb01rst element y in the in\ufb01nite list\n[x, f x, f (f x), f (f (f x)), ...]\nfor which p y = True. See the following chapter where this interpretation of\nuntil is made precise.\nThinking now about the design of floor it is tempting to start off with a case\nanalysis, distinguishing between the cases x < 0 and x \u2265 0. In the case x < 0 we\nhave to \ufb01nd the \ufb01rst number m in the sequence \u22121, \u22122, . . . for which m \u2264 x. That\nleads to \u2013 in the case of a negative argument \u2013\nfloor x = until (`leq` x) (subtract 1) (-1)\nwhere m `leq` x = fromInteger m <= x\nThere are a number of instructive points about this de\ufb01nition. Firstly, note the use\nof the prelude function subtract whose de\ufb01nition is\nsubtract x y = y-x\nWe have to use subtract 1 because (-1) is not a section but the number \u22121\n(look at the third argument of until).\nSecondly, why have we used `leq` when the alternative (<=) seems perfectly\nadequate? The answer is that (<=) has the type\n(<=) :: Num a => a -> a -> Bool\nIn particular the two arguments of (<=) have to have the same type. But we want\nleq :: Integer -> Float -> Bool\nand the two arguments have different numeric types. We therefore need to convert\nintegers to \ufb02oats using fromInteger. Appreciation of the need for conversion\nfunctions in some situations is one of the key points to understand about Haskell\narithmetic.\nFinally, note that (`leq` x) is not the same as (leq x):\n(leq x)\ny = leq x y\n(`leq` x) y = y `leq` x = leq y x\nIt is easy to make this mistake.\nIf you don\u2019t like the subsidiary de\ufb01nition, you can always write\n\n\f54\n\nNumbers\n\nfloor x = until ((<=x) . fromInteger) (subtract 1) (-1)\nIn this version we have inlined the de\ufb01nition of (`leq` x).\nWe still have to deal with the case x \u2265 0. In this case we have to look for the \ufb01rst\ninteger n such that x < n+1. We can do this by \ufb01nding the \ufb01rst integer n such that\nx < n and subtracting 1 from the answer. That leads to\nfloor x = until (x `lt` ) (+1) 1 - 1\nwhere x `lt` n = x < fromInteger n\nPutting the two pieces together, we obtain\nfloor x = if x < 0\nthen until (`leq` x) (subtract 1) (-1)\nelse until (x `lt`) (+1) 1 - 1\n(Question: why do we not have to write x < fromInteger 0 in the \ufb01rst line?)\nThe real problem with this de\ufb01nition, apart from the general ugliness of a case\ndistinction and the asymmetry of the two cases, is that it is very slow: it takes about\n|x| steps (|x| is the mathematician\u2019s way of writing abs x) to deliver the result.\n\nBinary search\nA better method for computing floor is to \ufb01rst \ufb01nd integers m and n such that\nm \u2264 x < n and then shrink the interval (m, n) to a unit interval (one with m + 1 = n)\nthat contains x. Then the left-hand bound of the interval can be returned as the\nresult. That leads to\nfloor :: Float -> Integer\nfloor x = fst (until unit (shrink x) (bound x))\nwhere unit (m,n) = (m+1 == n)\nThe value bound x is some pair (m, n) of integers such that m \u2264 x < n. If (m, n) is\nnot a unit interval, then shrink x (m,n) returns a new interval of strictly smaller\nsize that still bounds x.\nLet us \ufb01rst consider how to shrink a non-unit interval (m, n) containing x, so m \u2264\nx < n. Suppose p is any integer that satis\ufb01es m < p < n. Such a p exists since (m, n)\nis not a unit interval. Then we can de\ufb01ne\ntype Interval = (Integer,Integer)\nshrink :: Float -> Interval -> Interval\n\n\f3.3 Computing \ufb02oors\n\n55\n\nshrink x (m,n) = if p `leq` x then (p,n) else (m,p)\nwhere p = choose (m,n)\nHow should we de\ufb01ne choose?\nTwo possible choices are choose (m,n) = m+1 or choose (m,n) = n-1 for\nboth reduce the size of an interval. But a better choice is\nchoose :: Interval -> Integer\nchoose (m,n) = (m+n) `div` 2\nWith this choice the size of the interval is halved at each step rather than reduced\nby 1.\nHowever, we need to check that m < (m + n) div 2 < n in the case m + 1 = n. The\nreasoning is:\nm < (m + n) div 2 < n\n\u2261\n\n{ordering on integers}\nm + 1 \u2264 (m + n) div 2 < n\n\n\u2261\n\n{since (m + n) div 2 = \u0007(m + n)/2\b}\nm + 1 \u2264 (m + n)/2 < n\n\n\u2261\n\n{arithmetic}\nm+2 \u2264 n\u2227m < n\n\n\u2261\n\n{arithmetic}\nm+1 < n\n\nFinally, how should we de\ufb01ne bound? We can start off by de\ufb01ning\nbound :: Float -> Interval\nbound x = (lower x, upper x)\nThe value lower x is some integer less than or equal to x, and upper x some\ninteger greater than x. Instead of using linear search to discover these values, it is\nbetter to use\nlower :: Float -> Integer\nlower x = until (`leq` x) (*2) (-1)\nupper :: Float -> Integer\nupper x = until (x `lt`) (*2) 1\n\n\fNumbers\n\n56\n\nFor a fast version of bound it is better to double at each step rather than increase\nor decrease by 1. For example, with x = 17.3 it takes only seven comparisons to\ncompute the surrounding interval (\u22121, 32), which is then reduced to (17, 18) in a\nfurther \ufb01ve steps. In fact, evaluating both the upper and lower bounds takes time\nproportional to log |x| steps, and the whole algorithm takes at most twice this time.\nAn algorithm that takes logarithmic time is much faster than one that takes linear\ntime.\nThe standard prelude de\ufb01nes floor in the following way:\nfloor x = if r < 0 then n-1 else n\nwhere (n,r) = properFraction x\nThe function properFraction is a method in the RealFrac type class (a class we\nhaven\u2019t discussed and whose methods deal with truncating and rounding numbers).\nIt splits a number x into its integer part n and its fractional part r, so x = n + r. Now\nyou know.\n\n3.4 Natural numbers\nHaskell does not provide a type for the natural numbers, that is, the nonnegative\nintegers. But we can always de\ufb01ne such a type ourselves:\ndata Nat = Zero | Succ Nat\nThis is an example of a data declaration. The declaration says that Zero is a value\nof Nat and that Succ n is also a value of Nat whenever n is. Both Zero and Succ\nare called data constructors and begin with a capital letter. The type of Zero is Nat\nand the type of Succ is Nat -> Nat. Thus each of\nZero, Succ Zero, Succ (Succ Zero), Succ (Succ (Succ Zero))\nis an element of Nat.\nLet us see how to program the basic arithmetical operations by making Nat a fully\npaid-up member of the Num class. First, we have to make Nat an instance of Eq and\nShow:\ninstance Eq Nat where\nZero == Zero\n= True\nZero == Succ n\n= False\nSucc m == Zero\n= False\nSucc m == Succ n = (m == n)\n\n\f3.4 Natural numbers\n\n57\n\ninstance Show Nat where\nshow Zero\n= \"Zero\"\nshow (Succ Zero)\n= \"Succ Zero\"\nshow (Succ (Succ n)) = \"Succ (\" ++ show (Succ n) ++ \")\"\nThese de\ufb01nitions make use of pattern matching. In particular, the de\ufb01nition of\nshow makes use of three patterns, Zero, Succ Zero and Succ (Succ n). These\npatterns are different from one another and together cover all the elements of Nat\napart from \u22a5.\nAlternatively, we could have declared\ndata Nat = Zero | Succ Nat\n\nderiving (Eq,Ord,Show)\n\nAs we said in Exercise E of the previous chapter, Haskell is smart enough to construct automatically instances of some standard classes, including Eq, Ord and\nShow.\nNow we can install Nat as a numeric type:\ninstance Num Nat where\nm + Zero\n= m\nm + Succ n\n= Succ (m+n)\nm * Zero\n= Zero\nm * (Succ n) = m * n + m\nabs n\n= n\nsignum Zero\n= Zero\nsignum (Succ n) = Succ Zero\nm - Zero\n= m\nZero - Succ n\n= Zero\nSucc m - Succ n = m - n\nfromInteger x\n| x <= 0\n= Zero\n| otherwise = Succ (fromInteger (x-1))\nWe have de\ufb01ned subtraction as a total operation: m \u2212 n = 0 if m \u2264 n. Of course, the\narithmetic operations on Nat are horribly slow. And each number takes up a lot of\nspace.\n\n\fNumbers\n\n58\n\nPartial numbers\nWe have said that there is a value \u22a5 of every type. Thus undefined :: a for all\ntypes a. Since Succ is, by de\ufb01nition, a non-strict function, the values\nundefined, Succ undefined, Succ (Succ undefined), ...\nare all different and all members of Nat. To be honest, these partial numbers are\nnot very useful, but they are there. You can think of Succ undefined as being a\nnumber about which we know only that it is at least 1:\nghci> Zero == Succ undefined\nFalse\nghci> Succ Zero == Succ undefined\n*** Exception: Prelude.undefined\nThere is also one further number in Nat:\ninfinity :: Nat\ninfinity = Succ infinity\nThus\nghci> Zero == infinity\nFalse\nghci> Succ Zero == infinity\nFalse\nand so on.\nIn summary, the elements of Nat consist of the \ufb01nite numbers, the partial numbers\nand the in\ufb01nite numbers (of which there is only one). We shall see that this is true\nof other data types: there are the \ufb01nite elements of the type, the partial elements\nand the in\ufb01nite elements.\nWe could have chosen to make the constructor Succ strict. This is achieved by\ndeclaring\ndata Nat = Zero | Succ !Nat\nThe annotation ! is known as strictness \ufb02ag. With such a declaration, we have for\nexample\nghci> Zero == Succ undefined\n*** Exception: Prelude.undefined\n\n\f3.5 Exercises\n\n59\n\nThis time, evaluating the equality test forces the evaluation of both sides, and the\nevaluation of Succ undefined raises an error message. Making Succ strict collapses the natural numbers into just the \ufb01nite numbers and one unde\ufb01ned number.\n\n3.5 Exercises\nExercise A\nWhich of the following expressions denote 1?\n-2 + 3, 3 + -2, 3 + (-2), subtract 2 3, 2 + subtract 3\nIn the standard prelude there is a function flip de\ufb01ned by\nflip f x y = f y x\nExpress subtract using flip.\nExercise B\nHaskell provides no fewer than three ways to de\ufb01ne exponentiation:\n(^) :: (Num a, Integral b) => a -> b -> a\n(^^) :: (Fractional a, Integral b) => a -> b -> a\n(**) :: (Floating a) => a -> a -> a\nThe operation (^) raises any number to a nonnegative integral power; (^^) raises\nany number to any integral power (including negative integers); and (**) takes\ntwo fractional arguments. The de\ufb01nition of (^) basically follows Dick\u2019s method\nof the previous chapter (see Exercise E). How would you de\ufb01ne (^^)?\nExercise C\nCould you de\ufb01ne div in the following way?\ndiv :: Integral a => a -> a -> a\ndiv x y = floor (x/y)\nExercise D\nConsider again Clever Dick\u2019s solution for computing floor:\nfloor :: Float -> Integer\nfloor = read . (takeWhile (/= '.') . show\n\n\fNumbers\n\n60\n\nWhy doesn\u2019t it work?\nConsider the following mini-interaction with GHCi:\nghci> 12345678.0 :: Float\n1.2345678e7\nHaskell allows the use of so-called scienti\ufb01c notation, also called exponent notation, to describe certain \ufb02oating-point numbers. For example the number above\ndenotes 1.2345678 \u2217 107 . When the number of digits of a \ufb02oating-point number is\nsuf\ufb01ciently large, the number is printed in this notation. Now give another reason\nwhy Clever Dick\u2019s solution doesn\u2019t work.\nExercise E\nThe function isqrt :: Float -> Integer returns the \ufb02oor of the square root\nof a (nonnegative) number. Following the strategy of Section 3.3, construct an implementation of isqrt x that takes time proportional to log x steps.\nExercise F\nHaskell provides a function sqrt :: Floating a => a -> a that gives a reasonable approximation to the square root of a (nonnegative) number. But, let\u2019s\n\u221a\nde\ufb01ne our own version. If y is an approximation to x, then so is x/y. Moreover,\n\u221a\n\u221a\n\u221a\neither y \u2264 x \u2264 x/y or x/y \u2264 x \u2264 y. What is a better approximation to x than either y or x/y? (Yes, you have just rediscovered Newton\u2019s method for \ufb01nding square\nroots.)\nThe only remaining problem is to decide when an approximation y is good enough.\nOne possible test is |y2 \u2212 x| < \u03b5, where |x| returns the absolute value of x and \u03b5 is a\nsuitably small number. This test guarantees an absolute error of at most \u03b5. Another\ntest is |y2 \u2212 x| < \u03b5 \u2217 x, which guarantees a relative error of at most \u03b5. Assuming that\nnumbers of type Float are accurate only to six signi\ufb01cant \ufb01gures, which of these\ntwo is the more sensible test, and what is a sensible value for \u03b5?\nHence construct a de\ufb01nition of sqrt.\nExercise G\nGive an explicit instance of Nat as a member of the type class Ord. Hence construct\na de\ufb01nition of\ndivMod :: Nat -> Nat -> (Nat,Nat)\n\n\f3.6 Answers\n\n61\n\n3.6 Answers\nAnswer to Exercise A\nAll except 2 + -3 and 2 + subtract 3, neither of which are well-formed. We\nhave subtract = flip (-).\nAnswer to Exercise B\nx ^^ n = if 0 <= n then x^n else 1/(x ^ (negate n))\nAnswer to Exercise C\nNo. You would have to write\ndiv :: Integral a => a -> a -> a\ndiv x y = floor (fromInteger x / fromInteger y)\nAnswer to Exercise D\nClever Dick\u2019s function gives floor (-3.1) = -3 when the answer should be -4.\nAnd if you tried to repair his solution by subtracting 1 if the solution was negative,\nyou would have floor (-3.0) = -4 when the answer should be -3. Ugh!\nAlso, Clever Dick\u2019s solution has floor 12345678.0 = 1 because the argument\nis shown as 1.2345678e7.\nAnswer to Exercise E\nisqrt :: Float -> Integer\nisqrt x = fst (until unit (shrink x) (bound x))\nwhere unit (m,n) = (m+1 == n)\nshrink :: Float -> Interval -> Interval\nshrink x (m,n) = if (p*p) `leq` x then (p,n) else (m,p)\nwhere p = (m+n) `div` 2\nbound :: Float -> Interval\nbound x = (0,until above (*2) 1)\nwhere above n = x `lt` (n*n)\nThe functions `leq` and `lt` were de\ufb01ned in Section 3.3. Note the parentheses\nin the expressions (p*p) `leq` x and x `lt` (n*n). We didn\u2019t state an order\nof association for `leq` and `lt`, so without parentheses these two expressions\n\n```\n\n\n```\n\n\fNumbers\n\n62\n\nwould have been interpreted as the ill-formed expressions p * (p `leq` x) and\n(x `lt` n) * n. (I made just this mistake when \ufb01rst typing in the solution.)\nAnswer to Exercise F\n\n\u221a\nA better approximation to x than either y or x/y is (y + x/y)/2. The relative-error\ntest is the more sensible one, and the program is\nsqrt :: Float -> Float\nsqrt x = until goodenough improve x\nwhere goodenough y = abs (y*y-x) < eps*x\nimprove y\n= (y+x/y)/2\neps\n= 0.000001\n\nAnswer to Exercise G\nIt is suf\ufb01cient to de\ufb01ne (<):\ninstance\nZero <\nZero <\nSucc m\nSucc m\n\nOrd Nat where\nZero\n= False\nSucc n\n= True\n< Zero\n= False\n< Succ n = (m < n)\n\nNow we can de\ufb01ne\ndivMod :: Nat -> Nat -> (Nat,Nat)\ndivMod x y = if x < y then (Zero,x)\nelse (Succ q,r)\nwhere (q,r) = divMod (x-y) y\n\n3.7 Chapter notes\nThe primary source book for computer arithmetic is The Art of Computer Programming, Volume 2: Semi-numerical Algorithms (Addison-Wesley, 1998) by Don\nKnuth. The arithmetic of \ufb02oors and other simple numerical functions is studied\nin depth in Concrete Mathematics (Addison-Wesley, 1989) by Don Knuth, Ronald\nGraham and Oren Patashnik.\n\n\fChapter 4\nLists\n\nLists are the workhorses of functional programming. They can be used to fetch and\ncarry data from one function to another; they can be taken apart, rearranged and\ncombined with other lists to make new lists. Lists of numbers can be summed and\nmultiplied; lists of characters can be read and printed; and so on. The list of useful\noperations on lists is a long one. This chapter describes some of the operations that\noccur most frequently, though one particularly important class will be introduced\nonly in Chapter 6.\n\n4.1 List notation\nAs we have seen, the type [a] denotes lists of elements of type a. The empty list is\ndenoted by []. We can have lists over any type but we cannot mix different types\nin the same list. As examples,\n[undefined,undefined] :: [a]\n[sin,cos,tan]\n:: Floating a => [a -> a]\n[[1,2,3],[4,5]]\n:: Num a => [[a]]\n[\"tea\",\"for\",2]\nnot valid\nList notation, such as [1,2,3], is in fact an abbreviation for a more basic form\n1:2:3:[]\nThe operator (:) :: a -> [a] -> [a], pronounced \u2018cons\u2019, is a constructor for\nlists. It associates to the right so there is no need for parentheses in the above\nexpression. It has no associated de\ufb01nition, which is why it is a constructor. In\nother words, there are no rules for simplifying an expression such as 1:2:[]. The\n\n\fLists\n\n64\n\noperator (:) is non-strict in both arguments \u2013 more precisely, it is non-strict and\nreturns a non-strict function. The expression\nundefined : undefined\nmay not be very interesting, but we do know it is not the empty list. In fact, that is\nthe only thing we do know about it. Note that the two occurrences of undefined\nhave different types in this expression.\nThe empty list [] is also a constructor. Lists can be introduced as a Haskell data\ntype with the declaration\ndata List a = Nil | Cons a (List a)\nThe only difference is that List a is written [a], Nil is written [] and Cons is\nwritten (:).\nAccording to this declaration, every list of type [a] takes one of three forms:\n\u2022 The unde\ufb01ned list undefined :: [a];\n\u2022 The empty list [] :: [a];\n\u2022 A list of the form x:xs where x :: a and xs :: [a].\nAs a result there are three kinds of list:\n\u2022 A \ufb01nite list, which is built from (:) and []; for example, 1:2:3:[]\n\u2022 A partial list, which is built from (:) and undefined; for example, the list\nfilter (<4) [1..] is the partial list 1:2:3:undefined. We know there is\nno integer after 3 that is less than 4, but Haskell is an evaluator, not a theorem\nprover, so it ploughs away without success looking for more answers.\n\u2022 An in\ufb01nite list, which is built from (:) alone; for example, [1..] is the in\ufb01nite\nlist of the nonnegative integers.\nAll three kinds of list arise in everyday programming. Chapter 9 is devoted to exploring the world of in\ufb01nite lists and their uses. For example, the prelude function\niterate returns an in\ufb01nite list:\niterate :: (a -> a) -> a -> [a]\niterate f x = x:iterate f (f x)\nIn particular, iterate (+1) 1 is an in\ufb01nite list of the positive integers, a value\nwe can also write as [1..] (see the following section).\nAs another example,\n\n\f4.2 Enumerations\n\n65\n\nhead (filter perfect [1..])\nwhere perfect n = (n == sum (divisors n))\nreturns the \ufb01rst perfect number, namely 6, even though nobody currently knows\nwhether filter perfect [1..] is an in\ufb01nite or partial list.\nFinally, we can de\ufb01ne\nuntil p f = head . filter p . iterate f\nThe function until was used to compute \ufb02oors in the previous chapter. As this example demonstrates, functions that seem basic in programming are often composed\nof even simpler functions. A bit like protons and quarks.\n\n4.2 Enumerations\nHaskell provides useful notation for enumerating lists of integers. When m and n\nare integers we can write\n[m..n]\n[m..]\n[m,n..p]\n[m,n..]\n\nfor the list [m, m+1, . . . , n]\nfor the in\ufb01nite list [m, m+1, m+2, . . .]\nfor the list [m, m+(n\u2212m), m+2(n\u2212m), . . . , p]\nfor the in\ufb01nite list [m, m+(n\u2212m), m+2(n\u2212m), . . .]\n\nThe \ufb01rst two notations crop up frequently in practice, the second two less so. As\nexamples,\nghci> [0,2..11]\n[0,2,4,6,8,10]\nghci> [1,3..]\n[1,3,5,7,9,11 {Interrupted}\nIn the \ufb01rst example the enumeration stops at 10 because 11 isn\u2019t even. In the second\nexample we quickly interrupted the evaluation of an in\ufb01nite list.\nAs a matter of fact, enumerations are not restricted to integers, but to members of\nyet another type class Enum. We won\u2019t elaborate more on this class, except to say\nthat Char is also a member:\nghci> ['a'..'z']\n\"abcdefghijklmnopqrstuvwxyz\"\n\n\fLists\n\n66\n\n4.3 List comprehensions\nHaskell provides another useful and very attractive piece of notation, called list\ncomprehensions, for constructing lists out of other lists. We illustrate with a few\nexamples:\nghci> [x*x | x <- [1..5]]\n[1,4,9,16,25]\nghci> [x*x | x <- [1..5], isPrime x]\n[4,9,25]\nghci> [(i,j) | i <- [1..5], even i, j <- [i..5]]\n[(2,2),(2,3),(2,4),(2,5),(4,4),(4,5)]\nghci> [x | xs <- [[(3,4)],[(5,4),(3,2)]], (3,x) <- xs]\n[4,2]\nHere is another example. Suppose we wanted to generate all Pythagorean triads\nin a given range. These are triples of numbers (x, y, z) such that x2 + y2 = z2 and\n1 \u2264 x, y, z \u2264 n for some given n. We can de\ufb01ne\ntriads :: Int -> [(Int,Int,Int)]\ntriads n = [(x,y,z) | x <- [1..n], y <- [1..n],\nz <- [1..n], x*x+y*y==z*z]\nHence\nghci> triads 15\n[(3,4,5),(4,3,5),(5,12,13),(6,8,10),\n(8,6,10),(9,12,15),(12,5,13),(12,9,15)]\nThat\u2019s probably not what we want: each essentially distinct triad is generated in two\ndifferent ways. Moreover, the list contains redundant triads consisting of multiples\nof basic triads.\nTo improve the de\ufb01nition of triad we can restrict x and y so that x < y and x and\ny are coprime, meaning they have no divisors in common. As mathematicians we\nknow that 2x2 cannot be the square of an integer, so the \ufb01rst restriction is valid. The\ndivisors of a number can be computed by\ndivisors x = [d | d <- [2..x-1], x `mod` d == 0]\nHence\ncoprime x y = disjoint (divisors x) (divisors y)\nWe will leave the de\ufb01nition of disjoint as an exercise.\n\n\f4.3 List comprehensions\n\n67\n\nThat means we can de\ufb01ne\ntriads n = [(x,y,z) | x <- [1..n], y <- [x+1..n],\ncoprime x y,\nz <- [y+1..n], x*x+y*y==z*z]\nThis de\ufb01nition is better than before, but let us try to make it a little faster, mainly\n\u221ato\nillustrate an\u221aimportant point. Since 2x2 < x2 + y2 = z2 \u2264 n2 we see that x < n/ 2.\nSo x \u2264 \u0007n/ 2\b. That suggests we can write\ntriads n = [(x,y,z) | x <- [1..m], y <- [x+1..n],\ncoprime x y,\nz <- [y+1..n], x*x+y*y==z*z]\nwhere m = floor (n / sqrt 2)\nBut the expression for m is incorrect: n is an Int and we cannot divide integers.\nWe need an explicit conversion function, and the one to use is fromIntegral\n(not fromInteger because n is an Int not an Integer). We need to replace the\nde\ufb01nition of m by m = floor (fromIntegral n / sqrt 2). Once again we\nhave to be careful about what kinds of number we are dealing with and aware of\nthe available conversion functions between them.\nList comprehensions can be used to de\ufb01ne some common functions on lists. For\nexample,\nmap f xs\n= [f x | x <- xs]\nfilter p xs = [x | x <- xs, p x]\nconcat xss = [x | xs <- xss, x <- xs]\nActually, in Haskell it is the other way around: list comprehensions are translated\ninto equivalent de\ufb01nitions in terms of map, and concat. The translation rules are:\n[e\n[e\n[e\n[e\n\n|True]\n| q]\n| b, Q]\n| p <- xs, Q]\n\n=\n=\n=\n=\n\n[e]\n[e | q, True]\nif b then [e | Q] else []\nlet ok p = [e | Q]\nok _ = []\nin concat (map ok xs)\n\nThe de\ufb01nition of ok in the fourth rule uses a don\u2019t care pattern, also called a wild\ncard. The p in the fourth rule is a pattern, and the de\ufb01nition of ok says that the\nempty list is returned on any argument that doesn\u2019t match the pattern p.\nAnother useful rule is\n\n\fLists\n\n68\n\n[e | Q1, Q2] = concat [[e | Q2] | Q1]\n\n4.4 Some basic operations\nWe can de\ufb01ne functions over lists by pattern matching. For example,\nnull :: [a] -> Bool\nnull []\n= True\nnull (x:xs) = False\nThe patterns [] and x:xs are disjoint and exhaustive, so we can write the two\nequations for null in either order. The function null is strict because Haskell has\nto know which equation to apply and that requires evaluation of the argument, at\nleast to the extent of discovering whether it is the empty list or not. (A question:\nwhy not simply de\ufb01ne null = (==[])?) We could also have written\nnull [] = True\nnull _ = False\nThis de\ufb01nition uses a don\u2019t care pattern.\nHere are two other de\ufb01nitions using pattern matching:\nhead :: [a] -> a\nhead (x:xs) = x\ntail :: [a] -> [a]\ntail (x:xs) = xs\nThere is no equation for the pattern [], so Haskell reports an error if we try to\nevaluate head [] or tail [].\nWe can use [x] as shorthand for x:[] in a pattern:\nlast :: [a] -> a\nlast [x]\n= x\nlast (x:y:ys) = last (y:ys)\nThe \ufb01rst equation has a pattern that matches a singleton list; the second has a\npattern that matches a list that contains at least two elements. The standard prelude\nde\ufb01nition of last is slightly different:\nlast [x]\n= x\nlast (_:xs) = last xs\n\n\f4.5 Concatenation\n\n69\n\nThis de\ufb01nition uses a don\u2019t care pattern. The two equations have to be written in\nthis order because x:[] matches both patterns.\n\n4.5 Concatenation\nHere is the de\ufb01nition of (++), the concatenation operation:\n(++) :: [a] -> [a] -> [a]\n[] ++ ys\n= ys\n(x:xs) ++ ys = x:(xs ++ ys)\nThe de\ufb01nition uses pattern matching on the \ufb01rst argument but not on the second.\nThe second equation for (++) is very succinct and requires some thought, but\nonce you have got it, you have understood a lot about how lists work in functional\nprogramming. Here is a simple evaluation sequence:\n[1,2] ++ [3,4,5]\n=\n\n{notation}\n(1:(2:[])) ++ (3:(4:(5:[])))\n\n=\n\n{second equation for ++}\n1:((2:[]) ++ (3:(4:(5:[]))))\n\n=\n\n{and again}\n1:(2:([] ++ (3:(4:(5:[])))))\n\n=\n\n{\ufb01rst equation for ++}\n1:(2:(3:(4:(5:[]))))\n\n=\n\n{notation}\n[1,2,3,4,5]\n\nAs this example suggests, the cost of evaluating xs++ys is proportional to the\nlength of xs, where\nlength :: [a] -> Int\nlength []\n= 0\nlength (x:xs) = 1 + length xs\nNote also that\nundefined ++ [1,2] = undefined\n[1,2] ++ undefined = 1:2:undefined\n\n\fLists\n\n70\n\nWe know nothing about the \ufb01rst list, but we do know that the second list begins\nwith 1 followed by 2.\nConcatenation is an associative operation. Thus\n(xs ++ ys) ++ zs = xs ++ (ys ++ zs)\nfor all lists xs, ys and zs. We will see how to prove assertions like these in Chapter 6.\n\n4.6 concat, map and filter\nThree very useful list operations that we have met already are concat, map and\nfilter. Here are their de\ufb01nitions using pattern matching:\nconcat :: [[a]] -> [a]\nconcat []\n= []\nconcat (xs:xss) = xs ++ concat xss\nmap :: (a -> b) -> [a] -> [b]\nmap f []\n= []\nmap f (x:xs) = f x:map f xs\nfilter :: (a -> Bool) -> [a] -> [a]\nfilter p []\n= []\nfilter p (x:xs) = if p x then x:filter p xs\nelse filter p xs\nThere is a common theme underlying these de\ufb01nitions that we will identify and\nexploit in Chapter 6. An alternative de\ufb01nition of filter is\nfilter p = concat . map (test p)\ntest p x = if p x then [x] else []\nWith this de\ufb01nition, filter p is implemented by converting each element of the\nlist into a singleton list if it satis\ufb01es p, and the empty list otherwise. The results are\nthen concatenated.\nTwo basic facts about map are that\nmap id\n= id\nmap (f . g) = map f . map g\n\n\f4.6 concat, map and filter\n\n71\n\nThe \ufb01rst equation says that applying the identity function to each element of a list\nleaves the list unchanged. The two occurrence of id in this law have different types:\non the left it is a -> a and on the right it is [a] -> [a]. The second equation says\nthat applying g to every element of a list, and then applying f to every element of\nthe result, gives the same list as applying f . g to every element. Read from right\nto left, the equation says that two traversals of a list can be replaced by one, with a\ncorresponding gain in ef\ufb01ciency.\nThe two facts have a name: they are called the functor laws of map. The name is\nborrowed from a branch of mathematics called Category Theory. In fact, Haskell\nprovides a type class Functor, whose de\ufb01nition is\nclass Functor f where\nfmap :: (a -> b) -> f a -> f b\nThe method fmap is expected to satisfy exactly the same laws as map. The reason\nfor this type class is that the idea of mapping a function over a list can be generalised to one of mapping a function over an arbitrary data structure, such as trees\nof various kinds. For example, consider the type\ndata Tree a = Tip a | Fork (Tree a) (Tree a)\nof binary trees with labels in their tips. Tree-structured data arise in a number of\nplaces, for example with the syntax of expressions of various kinds. We can de\ufb01ne\na mapping function over trees, but rather than calling it mapTree we can call it\nfmap by making trees a member of the Functor class:\ninstance Functor Tree where\nfmap f (Tip x)\n= Tip (f x)\nfmap f (Fork u v) = Fork (fmap f u) (fmap f v)\nIn fact map is just a synonym for the instance fmap for lists:\nghci> fmap (+1) [2,3,4]\n[3,4,5]\nWe mention the Functor type class here primarily to show that if ever you think\nsome function on lists can be usefully generalised to other kinds of data structure, the chances are good that the designers of Haskell have already spotted it and\nintroduced an appropriate type class. As we will see later on, and especially in\nChapter 12, the functor laws of map appear in many calculations.\nThere is another group of laws that involve map, all of which have a common theme.\nConsider the equations\n\n\fLists\n\n72\n\nf . head\n= head . map f\nmap f . tail\n= tail . map f\nmap f . concat = concat . map (map f)\nThe \ufb01rst equation holds only if f is a strict function, but the others hold for arbitrary\nf. If we apply both sides of the equation to the empty list, we get\nf (head []) = head (map f []) = head []\nSince the head of an empty list is unde\ufb01ned, we require f to be strict to make the\nequation true.\nEach of the laws has a simple interpretation. In each case you can apply the operation (head, tail, and so on) to a list and then change each element, or you can\nchange each element \ufb01rst and then apply the operation. The common theme lies in\nthe types of the operations involved:\nhead\n:: [a] -> a\ntail\n:: [a] -> [a]\nconcat :: [[a]] -> [a]\nThe point about the operations is that they do not depend in any way on the nature of the list elements; they are simply functions that shuf\ufb02e, discard or extract\nelements from lists. That is why they have polymorphic types. And functions with\npolymorphic types all satisfy some law that says you can change values before\nor after applying the function. In mathematics such functions are called natural\ntransformations and the associated laws, naturality laws.\nAs another example, since reverse :: [a] -> [a] we would expect that\nmap f . reverse =\n\nreverse . map f\n\nIndeed this is the case. Of course, this naturality law still has to be proved.\nAnother law is\nconcat . map concat = concat . concat\nThe two sides assert that two ways of concatenating a list of lists of lists (either do\nthe inner concatenations \ufb01rst, or do the outer concatenations \ufb01rst) give the same\nresult.\nFinally, here is just one property of filter:\nfilter p . map f = map f . filter (p . f)\n\n\f4.7 zip and zipWith\n\n73\n\nWe can prove this law by simple equational reasoning:\nfilter p . map f\n=\n\n{second de\ufb01nition of filter}\nconcat . map (test p) . map f\n\n=\n\n{functor property of map}\nconcat . map (test p . f)\n\n=\n\n{since test p . f = map f . test (p . f)}\nconcat . map (map f . test (p . f))\n\n=\n\n{functor property of map}\nconcat . map (map f) . map (test (p . f))\n\n=\n\n{naturality of concat}\nmap f . concat . map (test (p . f))\n\n=\n\n{second de\ufb01nition of filter}\nmap f . filter (p . f)\n\nLaws like those above are not just of academic interest, but are deployed in \ufb01nding\nnew and better ways of expressing de\ufb01nitions. That\u2019s why functional programming\nis the best thing since sliced bread.\n\n4.7 zip and zipWith\nFinally, to complete a simple toolbox of useful operations, we consider the functions zip and zipWith. The de\ufb01nitions in the standard prelude are:\nzip :: [a] -> [b] -> [(a,b)]\nzip (x:xs) (y:ys) = (x,y): zip xs ys\nzip _\n_\n= []\nzipWith :: (a -> b -> c) -> [a] -> [b] -> [c]\nzipWith f (x:xs) (y:ys) = f x y : zipWith f xs ys\nzipWith f _\n_\n= []\nA caring programmer (one who doesn\u2019t like \u2018don\u2019t care\u2019 patterns) would have written\nzip [] ys\nzip (x:xs) []\n\n= []\n= []\n\n\fLists\n\n74\n\nzip (x:xs) (y:ys) = (x,y):zip xs ys\n \n```\n\n\n```\n\nBoth de\ufb01nitions use pattern matching on both arguments. You have to know that\npattern matching is applied from top to bottom and from left to right. Thus\nzip [] undefined = []\nzip undefined [] = undefined\nThe de\ufb01nition of zip can be given another way:\nzip = zipWith (,)\nThe operation (,) is a constructor for pairs: (,) a b = (a,b).\nHere is one example of the use of zipWith. Suppose we want to determine whether\na list is in nondecreasing order. A direct de\ufb01nition would have:\nnondec\nnondec\nnondec\nnondec\n\n:: (Ord a)\n[]\n=\n[x]\n=\n(x:y:xs) =\n\n=> [a] -> Bool\nTrue\nTrue\n(x <= y) && nondec (y:xs)\n\nBut another, equivalent and shorter de\ufb01nition is\nnondec xs = and (zipWith (<=) xs (tail xs))\nThe function and is yet another useful function in the standard prelude. It takes a\nlist of booleans and returns True if all the elements are True, and False otherwise:\nand :: [Bool] -> Bool\nand []\n= True\nand (x:xs) = x && and xs\nOne \ufb01nal example. Consider the task of building a function position that takes a\nvalue x and a \ufb01nite list xs and returns the \ufb01rst position in xs (counting positions\nfrom 0) at which x occurs. If x does not occur in the list, then \u22121 is returned. We\ncan de\ufb01ne\nposition :: (Eq a) => a -> [a] -> Int\nposition x xs\n= head ([j | (j,y) <- zip [0..] xs, y==x] ++ [-1])\nThe expression zip [0..] xs pairs each element of xs with its position in xs.\nAlthough the \ufb01rst argument of zip is an in\ufb01nite list, the result is a \ufb01nite list whenever xs is. Observe that the problem is solved by \ufb01rst computing the list of all\npositions at which x is found, and then taking the \ufb01rst element. Under lazy evaluation it is not necessary to construct the value of every element of the list in order\n\n\f4.8 Common words, completed\n\n75\n\nto calculate the head of the list, so there is no great loss of ef\ufb01ciency in solving\nthe problem this way. And there is a great deal of simplicity in de\ufb01ning one search\nresult in terms of all search results.\n\n4.8 Common words, completed\nLet\u2019s now return to Section 1.3 and complete the de\ufb01nition of commonWords. Recall that we \ufb01nished with\ncommonWords :: Int -> [Char] -> [Char]\ncommonWords n = concat . map showRun . take n .\nsortRuns . countRuns . sortWords .\nwords . map toLower\nThe only functions we have still to give de\ufb01nitions for are\nshowRun\n\ncountRuns\n\nsortRuns\n\nsortWords\n\nAll the others, including words, are provided in the standard Haskell libraries.\nThe \ufb01rst one is easy:\nshowRun :: (Int,Word) -> [Char]\nshowRun (n,w) = w ++ \": \" ++ show n ++ \"\\n\"\nThe second one can be de\ufb01ned by\ncountRuns :: [Word] -> [(Int,Word)]\ncountRuns []\n= []\ncountRuns (w:ws) = (1+length us,w):countRuns vs\nwhere (us,vs) = span (==w) ws\nThe prelude function span p splits a list into two, the \ufb01rst being the longest pre\ufb01x\nof the list all of whose elements satisfy the test p, and the second being the suf\ufb01x\nthat remains. Here is the de\ufb01nition:\nspan :: (a -> Bool) -> [a] -> ([a],[a])\nspan p []\n= ([],[])\nspan p (x:xs) = if p x then (x:ys,zs)\nelse ([],x:xs)\nwhere (ys,zs) = span p xs\nThat leaves sortRuns and sortWords. We can import the function sort from\nData.List by the command\n\n\fLists\n\n76\n\nimport Data.List (sort)\nSince sort :: (Ord a) => [a] -> [a] we can then de\ufb01ne\nsortWords :: [Word] -> [Word]\nsortWords = sort\nsortRuns :: [(Int,Word)] -> [(Int,Word)]\nsortRuns = reverse . sort\nTo understand the second de\ufb01nition you have to know that Haskell automatically\nde\ufb01nes the comparison operation (<=) on pairs by\n(x1,y1) <= (x2,y2) = (x1 < x2) || (x1 == x2 && y1 <= y2)\nYou also have to know that sort sorts into ascending order. Since we want the\ncodes in descending order of count, we just sort into ascending order and reverse\nthe result. That, by the way, is why we de\ufb01ned frequency counts by having the\ncount before the word rather than afterwards.\nInstead of relying on the library function for sorting, let us end by programming\na sorting function ourselves. One good way to sort is to use a divide and conquer\nstrategy: if the list has length at most one then it is already sorted; otherwise we can\ndivide the list into two equal halves, sort each half by using the sorting algorithm\nrecursively, and then merge the two sorted halves together. That leads to\nsort\nsort\nsort\nsort\n\n:: (Ord a) => [a] -> [a]\n[] = []\n[x] = [x]\nxs = merge (sort ys) (sort zs)\nwhere (ys,zs) = halve xs\n\nhalve xs = (take n xs, drop n xs)\nwhere n = length xs `div` 2\nThat leaves us with the de\ufb01nition of merge, which merges two sorted lists together\ninto one sorted list:\nmerge :: (Ord a) => [a] -> [a] -> [a]\nmerge [] ys = ys\nmerge xs [] = xs\nmerge (x:xs) (y:ys)\n| x <= y\n= x:merge xs (y:ys)\n| otherwise = y:merge (x:xs) ys\n\n\f4.9 Exercises\n\n77\n\nIn fact, many Haskell programmers wouldn\u2019t write the last clause of merge in quite\nthis way. Instead they would write\nmerge xs'@(x:xs) ys'@(y:ys)\n| x <= y\n= x:merge xs ys'\n| otherwise = y:merge xs' ys\nThis de\ufb01nition uses an as-pattern. You can see the point: rather than deconstructing\na list and then reconstructing it again (a cheap but not free operation), it is better\nto reuse the value that we matched with. True, but it does obscure a simple mathematical equation, and we will use such patterns only very sparingly in this book.\nBoth sort and merge are de\ufb01ned recursively and it is worthwhile pointing out\nwhy the two recursions terminate. In the case of merge you have to see that one or\nother of the two arguments of merge decreases in size at each recursive call. Hence\none of the base cases will eventually be reached. In the case of sort the critical\nobservation is that if xs has length at least two, then both ys and zs have length\nstrictly less than xs, and the same argument applies. But see what happens if we\nhad omitted the clause sort [x] = [x]. Since 1 div 2 = 0 we would have,\nsort [x] = merge (sort []) (sort [x])\nThat means evaluation of sort [x] requires evaluation of sort [x], and the\nwhole de\ufb01nition of sort spins off into an in\ufb01nite loop for nonempty arguments.\nChecking that you have all the necessary base cases is one of the most important\nparts of constructing a recursive function.\n\n4.9 Exercises\nExercise A\nWhich of the following equations are true for all xs and which are false?\n[]:xs =\n[]:xs =\nxs:[] =\nxs:[] =\nxs:xs =\n[[]] ++\n[[]] ++\n[[]] ++\n\nxs\n[[],xs]\nxs\n[xs]\n[xs,xs]\nxs = xs\nxs = [[],xs]\n[xs] = [[],xs]\n\n\fLists\n\n78\n\n[xs] ++ [] = [xs]\nBy the way, why didn\u2019t we de\ufb01ne null = (==[])?\nExercise B\nYou want to produce an in\ufb01nite list of all distinct pairs (x, y) of natural numbers.\nIt doesn\u2019t matter in which order the pairs are enumerated, as long as they all are\nthere. Say whether or not the de\ufb01nition\nallPairs = [(x,y) | x <- [0..], y <- [0..]]\ndoes the job. If you think it doesn\u2019t, can you give a version that does?\nExercise C\nGive a de\ufb01nition of the function\ndisjoint :: (Ord a) => [a] -> [a] -> Bool\nthat takes two lists in ascending order, and determines whether or not they have an\nelement in common.\nExercise D\nUnder what conditions do the following two list comprehensions deliver the same\nresult?\n[e | x <- xs, p x, y <- ys]\n[e | x <- xs, y <- ys, p x]\nCompare the costs of evaluating the two expressions.\nExercise E\nWhen the great Indian mathematician Srinivasan Ramanujan was ill in a London\nhospital, he was visited by the English mathematician G.H. Hardy. Trying to \ufb01nd a\nsubject of conversation, Hardy remarked that he had arrived in a taxi with the number 1729, a rather boring number it seemed to him. Not at all, Ramanujan instantly\nreplied, it is the \ufb01rst number that can be expressed as two cubes in essentially different ways: 13 + 123 = 93 + 103 = 1729. Write a program to \ufb01nd the second such\nnumber.\nIn fact, de\ufb01ne a function that returns a list of all essentially different quadruples\n(a, b, c, d) in the range 0 < a, b, c, d \u2264 n such that a3 + b3 = c3 + d3 . I suggest using\na list comprehension, but only after thinking carefully about what it means to say\n\n\f4.9 Exercises\n\n79\n\ntwo quadruples are essentially different. After all, a3 + b3 = c3 + d3 can be written\nin eight different ways.\nExercise F\nThe dual view of lists is to construct them by adding elements to the end of the list:\ndata List a = Nil | Snoc (List a) a\nSnoc is, of course, Cons backwards. With this view of lists [1, 2, 3] would be represented by\nSnoc (Snoc (Snoc Nil 1) 2) 3\nExactly the same information is provided by the two views but it is organised differently. Give the de\ufb01nitions of head and last for the snoc-view of lists, and de\ufb01ne\ntwo functions\ntoList :: [a] -> List a\nfromList :: List a -> [a]\nfor converting ef\ufb01ciently from one view of lists to the other. (Hint: reverse is\nef\ufb01cient, taking linear time to reverse a list.)\nExercise G\nHow much space is required to evaluate length xs? Consider the following alternative de\ufb01nition of length:\nlength :: [a] -> Int\nlength xs = loop (0,xs)\nwhere loop (n,[])\n= n\nloop (n,x:xs) = loop (n+1,xs)\nDoes the space requirement change? Does it change if we switched to eager evaluation? These questions are taken up in much more detail in Chapter 7.\nExercise H\nThe prelude function take n takes the \ufb01rst n elements of a list, while drop n\ndrops the \ufb01rst n elements. Give recursive de\ufb01nitions for these functions. What are\nthe values of\ntake 0 undefined\n\ntake undefined []\n\naccording to your de\ufb01nition? A more tricky question: can you \ufb01nd a de\ufb01nition in\nwhich both the above expressions have the value []? If not, why not?\n\n\fLists\n\n80\n\nWhich of the following equations are valid for all integers m and n? You don\u2019t have\nto justify your answers, just try to understand what they claim to say.\ntake\ntake\ntake\ndrop\n\nn\nm\nm\nm\n\nxs ++ drop n\n. drop n =\n. take n =\n. drop n =\n\nxs = xs\ndrop n . take (m+n)\ntake (m `min` n)\ndrop (m+n)\n\nThe standard prelude function splitAt n can be de\ufb01ned by\nsplitAt n xs = (take n xs,drop n xs)\nThough clear, the above de\ufb01nition is maybe a little inef\ufb01cient as it involves processing xs twice. Give a de\ufb01nition of splitAt that traverses the list only once.\nExercise I\nWhich of the following statements about the equation\nmap (f . g) xs = map f (map g xs)\ndo you agree with, and which do you disagree with (again, no justi\ufb01cation is required)?\n1. It\u2019s not true for all xs; it depends on whether xs is a \ufb01nite list or not.\n2. It\u2019s not true for all f and g; it depends on whether f and g are strict functions or\nnot.\n3. It\u2019s true for all lists xs, \ufb01nite, partial or in\ufb01nite, and for all f and g of the\nappropriate type. In fact map (f . g) = map f . map g is a much neater\nalternative.\n4. It looks true, but it has to be proved so from the de\ufb01nition of map and the de\ufb01nition of functional composition.\n5. Used right-to-left, it expresses a program optimisation: two traversals of a list\nare replaced by one.\n6. It\u2019s not an optimisation under lazy evaluation because map g xs is not computed in its entirety before evaluation of map f on the result begins.\n7. Whether or not it is computed in pieces or as a whole, the right-hand side does\nproduce an intermediate list, while the left-hand side doesn\u2019t. It is a rule for\noptimising a program even under lazy evaluation.\n\n\f4.9 Exercises\n\n81\n\nExercise J\nHere are some equations; at least one of them is false. Which are the true ones,\nand which are false? Once again, you do not have to provide any justi\ufb01cation for\nyour answers, the aim is just to look at some equations and appreciate what they\nare saying.\nmap f . take n\nmap f . reverse\nmap f . sort\nmap f . filter p\nfilter (p . g)\nreverse . concat\nfilter p . concat\n\n=\n=\n=\n=\n=\n=\n=\n\ntake n . map f\nreverse . map f\nsort . map f\nmap fst . filter snd . map (fork (f,p))\nmap (invertg) . filter p . map g\nconcat . reverse . map reverse\nconcat . map (filter p)\n\nIn the \ufb01fth equation assume invertg satis\ufb01es invertg . g = id. The function\nfork in the fourth equation is de\ufb01ned by\nfork :: (a -> b,a -> c) -> a -> (b,c)\nfork (f,g) x = (f x, g x)\nExercise K\nDe\ufb01ne unzip and cross by\nunzip = fork (map fst, map snd)\ncross (f,g) = fork (f . fst, g . snd)\nWhat are the types of these functions?\nProve by simple equational reasoning that\ncross (map f, map g) . unzip = unzip . map (cross (f,g))\nYou can use the functor laws of map and the following rules:\ncross (f,g) . fork (h,k) = fork (f . h,g . k)\nfork (f,g) . h\n= fork (f . h,g . h)\nfst . cross (f,g)\n= f . fst\nsnd . cross (f,g)\n= g . snd\nExercise L\nContinuing from the previous exercise, prove that\ncross (f,g) . cross (h,k) = cross (f . h,g . k)\n\n\fLists\n\n82\n\nWe also have cross (id,id) = id (Why?). So it looks like cross has functorlike properties, except that it takes a pair of functions. Yes, it\u2019s a bifunctor. That\nsuggests a generalisation:\nclass Bifunctor p where\nbimap :: (a -> b) -> (c -> d) -> p a c -> p b d\nThe arguments to bimap are given one by one rather than paired. Express cross\nin terms of bimap for the instance Pair of Bifunctor, where\ntype Pair a b = (a,b)\nNow consider the data type\ndata Either a b = Left a | Right b\nConstruct the instance Either of Bifunctor.\n\n4.10 Answers\nAnswer to Exercise A\nOnly the following three equations are true:\nxs:[] = [xs]\n[[]] ++ [xs] = [[],xs]\n[xs] ++ [] = [xs]\nIf we de\ufb01ned null by null = (==[]), then its type would have to be the more\nrestrictive\nnull :: (Eq a) => [a] -> Bool\nThat means you can only use an equality test on lists if the list elements can be\ncompared for equality. Of course, the empty list contains no elements, so (==) is\nnot needed.\nAnswer to Exercise B\nNo, allPairs produces the in\ufb01nite list\nallPairs = [(0,y) | y <- [0..]]\nOne alternative, which lists the pairs in ascending order of their sum, is\nallPairs = [(x,d-x) | d <- [0..], x <- [0..d]]\n\n\f4.10 Answers\n\n83\n\nAnswer to Exercise C\nThe de\ufb01nition is\ndisjoint xs [] = True\ndisjoint [] ys = True\ndisjoint xs'@(x:xs) ys'@(y:ys)\n| x < y = disjoint xs ys'\n| x == y = False\n| x > y = disjoint xs' ys\nWe used an as-pattern, just to be clever.\nAnswer to Exercise D\nThey deliver the same result only if ys is a \ufb01nite list:\nghci> [1 | x <- [1,3], even x, y <- undefined]\n[]\nghci> [1 | x <- [1,3], y <- undefined, even x]\n*** Exception: Prelude.undefined\nghci> [1 | x <- [1,3], even x, y <- [1..]]\n[]\nPrelude> [1 | x <- [1,3], y <- [1..], even x]\n{Interrupted}\nWhen they do deliver the same result, the former is more ef\ufb01cient.\nAnswer to Exercise E\nOne way of generating essentially different quadruples is to restrict the quadruple\n(a, b, c, d) to values satisfying a \u2264 b and c \u2264 d and a < c. Hence\nquads n = [(a,b,c,d) | a <- [1..n], b <- [a..n],\nc <- [a+1..n],d <- [c..n],\na^3 + b^3 == c^3 + d^3]\nThe second such number is 4104 = 23 + 163 = 93 + 153 .\nAnswer to Exercise F\nhead :: List a -> a\nhead (Snoc Nil x) = x\nhead (Snoc xs x) = head xs\n\n\fLists\n\n84\n\nlast :: List a -> a\nlast (Snoc xs x) = x\ntoList :: [a] -> List a\ntoList = convert . reverse\nwhere convert []\n= Nil\nconvert (x:xs) = Snoc (convert xs) x\nfromList :: List a -> [a]\nfromList = reverse . convert\nwhere convert Nil\n= []\nconvert (Snoc xs x) = x:convert xs\nAnswer to Exercise G\nIt requires a linear amount of space since the expression\n1 + (1 + (1 + ... (1 + 0)))\nis built up in memory. The space requirement for the second de\ufb01nition of length\ndoes not change under lazy evaluation since the expression\nloop ((((0 + 1) + 1) + 1 ... +1),[])\nis built up in memory. But under eager evaluation the length of a list can be computed using constant extra space.\nAnswer to Exercise H\ntake, drop :: Int -> [a] -> [a]\ntake n []\n= []\ntake n (x:xs) = if n==0 then [] else x:take (n-1) xs\ndrop n []\n= []\ndrop n (x:xs) = if n==0 then x:xs else drop (n-1) xs\nWith this de\ufb01nition of take we have\ntake undefined [] = []\n\ntake 0 undefined = undefined\n\nWith the alternative\ntake n xs | n==0\n= []\n| null xs\n= []\n| otherwise = head xs: take (n-1) (tail xs)\nwe have\n\n\f4.10 Answers\n\ntake undefined [] = undefined\n\n85\n\ntake 0 undefined = []\n\nThe answer to the tricky question is: no. Either argument n or argument xs has to\nbe examined and, whichever happens \ufb01rst, \u22a5 is the result if \u22a5 is the value of that\nargument.\nAll four equations are valid for all lists xs and for all m, n =\u22a5, under either de\ufb01nition.\nThe function splitAt n can be de\ufb01ned by\nsplitAt :: Int -> [a] -> ([a],[a])\nsplitAt n [] = ([],[])\nsplitAt n (x:xs) = if n==0 then ([],x:xs) else (x:ys,zs)\nwhere (ys,zs) = splitAt (n-1) xs\nAnswer to Exercise I\nI would agree with (3), (4), (5) and (7).\nAnswer to Exercise J\nThe only false equation is map f . sort = sort . map f which is true only if\nf is order-preserving, i.e. x \u2264 y \u2261 f x \u2264 f y.\nAnswer to Exercise K\nunzip :: [(a,b)] -> ([a],[b])\ncross :: (a -> b, c -> d) -> (a,c) -> (b,d)\nThe calculation is\ncross (map f, map g) . unzip\n=\n\n{de\ufb01nition of unzip}\ncross (map f, map g) . fork (map fst, map snd)\n\n=\n\n{law of cross and fork}\nfork (map f . map fst, map g . map snd)\n\n=\n\n{law of map}\nfork (map (f . fst), map (g . snd))\n\n\fLists\n\n86\n\nWe seem to be stuck, as no law applies. Try the right-hand side:\nunzip . map (cross (f,g))\n=\n\n{de\ufb01nition of unzip}\nfork (map fst, map snd) . map (cross (f,g))\n\n=\n\n{law of fork}\nfork (map fst . map (cross (f,g)),\nmap snd . map (cross (f,g)))\n\n=\n\n{law of map}\nfork (map (fst . cross (f,g)),\nmap (snd . cross (f,g)))\n\n=\n\n{laws of fst and snd}\nfork (map (f . fst), map (g . snd))\n\nPhew. Both sides have reduced to the same expression. That is often the way with\ncalculations: one side doesn\u2019t always lead easily to the other, but both sides reduce\nto the same result.\nThe calculations we have seen so far have all been carried out at the function level.\nSuch a style of de\ufb01nition and proof is called point-free (and also pointless by some\njokers). Point-free proofs are what the automatic calculator of Chapter 12 produces.\nThe point-free style is very slick, but it does necessitate the use of various plumbing\ncombinators, such as fork and cross, to pass arguments to functions. Plumbing\ncombinators push values around, duplicate them and even eliminate them. As an\nexample of the last kind,\nconst :: a -> b -> a\nconst x y = x\nThis little combinator is in the standard prelude and can be quite useful on occasion.\nTwo more plumbing combinators, also de\ufb01ned in the standard prelude, are curry\nand uncurry:\ncurry :: ((a, b) -> c) -> a -> b -> c\ncurry f x y = f (x,y)\nuncurry :: (a -> b -> c) -> (a,b) -> c\nuncurry f (x,y) = f x y\nA curried function is a function that takes its arguments one at a time, while a\nnon-curried function takes a single, tupled argument. The key advantage of curried\n\n\f4.11 Chapter notes\n\n87\n\nfunctions is that they can be partially applied. For instance, take n is a perfectly\nvalid function in its own right, and so is map f. That is why we have used curried\nfunctions from the start.\nBy the way, curried functions are named after Haskell B. Curry, an American logician. And, yes, that is where Haskell got its name.\nAnswer to Exercise L\n\ncross (f,g) . cross (h,k)\n=\n\n{de\ufb01nition of cross}\ncross (f,g) . fork (h . fst, k . snd)\n\n=\n\n{law of cross and fork}\nfork (f . h . fst,g . k . snd)\n\n=\n\n{de\ufb01nition of cross}\ncross (f . h, g . k)\n\nWe have cross = uncurry bimap, where uncurry was de\ufb01ned in the previous\nanswer.\nHere is the instance of Either:\ninstance Bifunctor Either where\nbimap f g (Left x) = Left (f x)\nbimap f g (Right y) = Right (g y)\n\n```\n\n\n```\n\n4.11 Chapter notes\nMost of the functions introduced in this chapter can be found in the Haskell standard prelude. Functors, bifunctors, and natural transformations are explained in\nbooks about Category Theory. Two such are Basic Category Theory for Computer\nScientists (MIT Press, 1991) by Benjamin Pierce, and The Algebra of Programming\n(Prentice Hall, 1997) by Richard Bird and Oege de Moor.\nAlso on the subject of laws, read Phil Wadler\u2019s in\ufb02uential article Theorems for free!\nwhich can be found at\nhomepages.inf.ed.ac.uk/wadler/papers/free/\n\n\f88\n\nLists\n\nIn mathematics, the so-called taxicab number taxicab(n) is the smallest number\nthat can be expressed as the sum of two positive cubes in n distinct ways. So 1729 =\ntaxicab(2). Google \u2018taxicab numbers\u2019 for more information.\n\n\fChapter 5\nA simple Sudoku solver\n\nH OW TO PLAY: Fill in the grid so that every row,\nevery column and every 3 \u00d7 3 box contains the\ndigits 1\u20139. There\u2019s no maths involved. You\nsolve the puzzle with reasoning and logic.\nAdvice on how to play Sudoku, the Independent\nThis chapter is devoted to an extended exercise in the use of lists to solve problems, and in the use of equational reasoning to reason about them and to improve\nef\ufb01ciency.\nThe game of Sudoku is played on a 9 by 9 grid, though other sizes are also possible.\nGiven a matrix, such as that in Figure 5.1, the idea is to \ufb01ll in the empty cells with\nthe digits 1 to 9 so that each row, column and 3 \u00d7 3 box contains the numbers 1 to 9.\nIn general there may be any number of solutions, though in a good Sudoku puzzle\nthere should always be a unique solution. Our aim is to construct a program to\nsolve Sudoku puzzles. Speci\ufb01cally, we will de\ufb01ne a function solve for computing\na list of all the ways a given grid may be completed. If only one solution is wanted,\nthen we can take the head of the list. Lazy evaluation means that only the \ufb01rst result\nwill then be computed.\nWe begin with a speci\ufb01cation, then use equational reasoning to calculate a more\nef\ufb01cient version. There\u2019s no maths involved, just reasoning and logic!\n\n5.1 Speci\ufb01cation\nHere are the basic data types of interest, starting with matrices:\ntype Matrix a = [Row a]\n\n\fA simple Sudoku solver\n\n90\n\n5 7\n9 4\n\n4\n\n8\n\n3 6\n7 2\n\n6\n4\n\n2\n9 3\n5 6\n\n8\n4\n5 3\n6 1\n\n9\n\nFigure 5.1 A Sudoku grid\n\ntype Row a\n\n= [a]\n\nThe two type synonyms say nothing more than that Matrix a is a synonym for\n[[a]]. But the way it is said emphasises that a matrix is a list of rows; more\nprecisely, a m \u00d7 n matrix is a list of m rows in which each row is a list with the\nsame length n. Haskell type synonyms cannot enforce such constraints, though\nthere are languages, called dependently-typed languages, that can.\nA grid is a 9 \u00d7 9 matrix of digits:\ntype Grid = Matrix Digit\ntype Digit = Char\nThe valid digits are 1 to 9 with 0 standing for a blank:\ndigits :: [Char]\ndigits = ['1' .. '9']\nblank :: Digit -> Bool\nblank = (== '0')\nRecall that Char is also an instance of the type class Enum, so ['1' .. '9'] is a\nvalid expression and does indeed return the list of nonzero digits.\nWe will suppose for simplicity that the input grid contains only digits and blanks,\nso we do not have to check for the input being well-formed. But should we also\ninsist that no non-blank digit is repeated in any row, column or box? If there were\nsuch repetitions there would be no solution. We postpone this decision until after\nwe see how the rest of the algorithm pans out.\n\n\f5.1 Speci\ufb01cation\n\n91\n\nNow for the speci\ufb01cation. The aim is to write down the simplest and clearest speci\ufb01cation without regard to how ef\ufb01cient the result might be. That\u2019s a key difference\nbetween functional programming and other forms of program construction: we can\nalways begin with a clear and simple, though possibly extremely inef\ufb01cient de\ufb01nition of solve, and then use the laws of functional programming to massage the\ncomputation into one that takes acceptable time and space.\nOne possibility is \ufb01rst to construct a list of all possible correctly \ufb01lled grids, a vastly\nlong but still \ufb01nite list, and then to test the given grid against each of them to identify those whose entries match the given non-blank ones. Certainly that approach\ntakes the idea of an inef\ufb01cient speci\ufb01cation to the extreme. Another reasonable alternative is to start with the given grid and to complete it by \ufb01lling in every possible\nchoice for the blank entries. The result will be a list of \ufb01lled grids. Then we can\n\ufb01lter this list for those that don\u2019t contain duplicates in any row, box or column. This\nspeci\ufb01cation is implemented by\nsolve :: Grid -> [Grid]\nsolve = filter valid . completions\nwhere the subsidiary functions have types\ncompletions :: Grid -> [Grid]\nvalid\n:: Grid -> Bool\nLet us work on completions \ufb01rst and consider valid afterwards. One way of\nde\ufb01ning completions is by a two-step process:\ncompletions = expand . choices\nwhere\nchoices :: Grid -> Matrix [Digit]\nexpand :: Matrix [Digit] -> [Grid]\nThe function choices installs the available digits for each cell:\nchoices = map (map choice)\nchoice d = if blank d then digits else [d]\nIf the cell is blank, then all digits are installed as possible choices; otherwise there is\nonly one choice and a singleton is returned. If we want to apply f to every element\nof a matrix, then map (map f) is the function to use because, after all, a matrix is\njust a list of lists.\nAfter applying choices we obtain a matrix each of whose entries is a list of digits.\nWhat we want to do next is to de\ufb01ne expand to convert this matrix into a list of\n\n\fA simple Sudoku solver\n\n92\n\ngrids by installing all the choices in all possible ways. That seems a little dif\ufb01cult\nto think about, so let\u2019s consider a simpler problem \ufb01rst, namely when instead of a\n9 \u00d7 9 matrix we have a list of length 3. Suppose we want to convert\n[[1,2,3],[2],[1,3]]\ninto the list\n[[1,2,1],[1,2,3],[2,2,1],[2,2,3],[3,2,1],[3,2,3]]\nThe second list of lists arises by taking, in all possible ways, one element from\nthe \ufb01rst list, one element from the second list and one element from the third list.\nLet us call the function that does this cp (short for \u2018cartesian product\u2019, which is\nexactly what a mathematician would call it). There doesn\u2019t seem to be any clever\nway of computing cp in terms of other functions, so we adopt the default strategy\nof de\ufb01ning this function by breaking up its argument list into two possibilities, the\nempty list [] and a nonempty list xs:xss. You might guess the de\ufb01nition of cp []\nbut you would probably be wrong; the better alternative is to think about the second\ncase \ufb01rst. Suppose we assume\ncp [[2],[1,3]] = [[2,1],[2,3]]\nHow can we extend this de\ufb01nition to one for cp ([1,2,3]:[[2],[1,3]])? The\nanswer is to pre\ufb01x 1 to every element of cp [[2],[1,3]], then to pre\ufb01x 2 to every\nelement of the same list, and \ufb01nally to pre\ufb01x 3 to every element. That process can\nbe expressed neatly using a list comprehension:\ncp (xs:xss) = [x:ys | x <- xs, ys <- cp xss]\nIn words, pre\ufb01x every element of xs to every element of cp xss in all possible\nways.\nIf your nose is good at snif\ufb01ng out inef\ufb01ciencies, you might suspect that this oneliner is not the best possible, and you would be right. We will return to this point\nin Section 7.3, but let\u2019s just say that a more ef\ufb01cient de\ufb01nition is\ncp (xs:xss) = [x:ys | x <- xs, ys <- yss]\nwhere yss = cp xss\nThis version guarantees that cp xss is computed just once.\nNow, what is cp []? The answer is not [] but [[]]. To see why the \ufb01rst is wrong,\nconsider a little calculation:\ncp [xs] = cp (xs:[])\n= [x:ys | x <- xs, ys <- cp []]\n\n\f5.1 Speci\ufb01cation\n\n93\n\n= [x:ys | x <- xs, ys <- []]\n= []\nIn fact with cp [] = [] we can show that cp xss = [] for all lists xss. So that\nde\ufb01nition is clearly wrong. You can check that the second alternative, [[]], does\ngive what is wanted.\nSummarising, we can de\ufb01ne cp by\ncp :: [[a]] -> [[a]]\ncp []\n= [[]]\ncp (xs:xss) = [x:ys | x <- xs, ys <- yss]\nwhere yss = cp xss\nFor example,\nghci> cp [[1],[2],[3]]\n[[1,2,3]]\nghci> cp [[1,2],[],[4,5]]\n[]\nIn the second example there is no possible choice from the middle list, so the empty\nlist is returned.\nBut what about matrices and expand, which does the same thing on matrices as cp\ndoes on lists? You will have to think a bit before seeing that what is wanted is\nexpand :: Matrix [Digit] -> [Grid]\nexpand = cp . map cp\nThat looks a little cryptic, but map cp returns a list of all possible choices for each\nrow, and so applying cp to the result installs each choice for the rows in all possible\nways. The general type of the right-hand side is\ncp . map cp :: [[[a]]] -> [[[a]]]\nand the declared type of expand is just a restricted version of this type. Note that\nexpand returns the empty list if any element in any row is the empty list.\nFinally, a valid grid is one in which no row, column or box contains duplicates:\nvalid :: Grid\nvalid g = all\nall\nall\n\n-> Bool\nnodups (rows g) &&\nnodups (cols g) &&\nnodups (boxs g)\n\n\f94\n\nA simple Sudoku solver\n\nThe prelude function all is de\ufb01ned by\nall p = and . map p\nApplied to a \ufb01nite list xs the function all p returns True if all elements of xs\nsatisfy p, and False otherwise. The function nodups can be de\ufb01ned by\nnodups :: (Eq a) => [a] -> Bool\nnodups []\n= True\nnodups (x:xs) = all (/=x) xs && nodups xs\nEvaluation of nodups on a list of length n takes time proportional to n2 . As an\nalternative we could sort the list and check that it is strictly increasing. Sorting\ncan be done in time proportional to n log n steps. That seems a big saving over\nn2 . However, with n = 9, it is not clear that using an ef\ufb01cient sorting algorithm is\nworthwhile. What would you prefer: 2n2 steps or 100n log2 n steps?\nIt remains to de\ufb01ne rows, cols and boxs. If a matrix is given by a list of its rows,\nthen rows is just the identity function on matrices:\nrows :: Matrix a -> Matrix a\nrows = id\nThe function cols computes the transpose of a matrix. Thus if a matrix consists of\nm rows, where each row has length n, the transpose is a list of n rows, where each\nrow has length m. Assuming both m and n are not zero, we can de\ufb01ne\ncols :: Matrix a -> Matrix a\ncols [xs]\n= [[x] | x <- xs]\ncols (xs:xss) = zipWith (:) xs (cols xss)\nIt is usual in matrix algebra to suppose that the matrix is nonempty, and that certainly suf\ufb01ces here, but it is interesting to consider what happens if we allow m or\nn to be zero. This point is taken up in the exercises.\nThe function boxs is a little more interesting. We give the de\ufb01nition \ufb01rst and explain it afterwards:\nboxs :: Matrix a -> Matrix a\nboxs = map ungroup . ungroup .\nmap cols .\ngroup . map group\nThe function group splits a list into groups of three:\ngroup :: [a] -> [[a]]\n\n\f5.2 Lawful program construction\n\n95\n\ngroup [] = []\ngroup xs = take 3 xs:group (drop 3 xs)\nThe function ungroup takes a grouped list and ungroups it:\nungroup :: [[a]] -> [a]\nungroup = concat\nThe action of boxs in the 4 \u00d7 4 case, when group splits a list into groups of two\nrather than three, is illustrated by the picture\n\u239e\n\u239b \b\na b c d\nab cd\n\u239c e f g h \u239f\n\u239c\n\u239c\n\u239f\n\u239c \b ef gh\n\u239d i j k l \u23a0 \u2212\u2192 \u239d\nij kl\nm n o p\nmn op\n\u2193\n\u239b\n\u239e\n\u239b \b\na b e f\nab ef\n\u239c c d g h \u239f\n\u239c\n\u239c\n\u239f\n\u239c \b cd gh\n\u239d i j m n \u23a0 \u2190\u2212 \u239d\nij mn\nk l o p\nkl op\n\u239b\n\n\u239e\n\u239f\n\u239f\n\u23a0\n\u239e\n\u239f\n\u239f\n\u23a0\n\nGrouping produces a list of matrices; transposing each matrix and ungrouping\nyields the boxes, as a matrix whose rows are the boxes of the original matrix.\n\n5.2 Lawful program construction\nObserve that instead of thinking about matrices in terms of indices, and doing arithmetic on indices to identify the rows, columns and boxes, we have gone for de\ufb01nitions of these functions that treat the matrix as a complete entity in itself. This\nstyle has aptly been called wholemeal programming. Wholemeal programming is\ngood for you: it helps to prevent a disease called indexitis, and encourages lawful\nprogram construction.\nFor example, here are three laws that are valid on Sudoku grids:\nrows . rows = id\ncols . cols = id\nboxs . boxs = id\nIn other words, all three functions are involutions. The \ufb01rst two are valid on all\nmatrices, and the third is valid on arbitrary n2 \u00d7 n2 matrices (provided we change\n\n\f96\n\nA simple Sudoku solver\n\nthe de\ufb01nition of group to group by n). Two are easy to prove, but one is more\ndif\ufb01cult. The dif\ufb01cult law is not the one about boxs, as you might expect, but the\ninvolution property of cols. Though it is intuitively obvious that transposing a\nmatrix twice gets you back to the original matrix, proving it from the de\ufb01nition of\ncols is a little tricky and we won\u2019t go into details, basically because we haven\u2019t\nyet discussed the tools available to do the job.\nBy contrast, here is the proof of the involution property of boxs. The proof is by\nsimple equational reasoning. It makes use of various laws, including the functor\nlaws of map, the fact that id is the identity element of composition, and the facts\nthat\nungroup . group = id\ngroup . ungroup = id\nThe second equation is valid only on grouped lists, but that will be the case in the\ncalculation to come.\nWe will talk through the proof rather than lay everything out in a long chain. The\nstarting point is to use the de\ufb01nition of boxs to rewrite boxs . boxs:\nmap ungroup . ungroup . map cols . group . map group .\nmap ungroup . ungroup . map cols . group . map group\nThe middle expression map group . map ungroup simpli\ufb01es to id using the\nfunctor law of map and the property that group and ungroup are inverses. That\ngives\nmap ungroup . ungroup . map cols . group .\nungroup . map cols . group . map group\nAn appeal to group . ungroup = id gets us to\nmap ungroup . ungroup . map cols .\nmap cols . group . map group\nThe functor law of map and the involution property of cols now gets us to\nmap ungroup . ungroup . group . map group\nAnd the proof is \ufb01nished off using ungroup . group = id twice more. As you\ncan see, it\u2019s a very simple calculation.\nHere are three more laws, valid on N 2 \u00d7 N 2 matrices of choices:\nmap rows . expand = expand . rows\nmap cols . expand = expand . cols\n\n\f5.3 Pruning the matrix of choices\n\n97\n\nmap boxs . expand = expand . boxs\nWe will make use of these laws in a short while.\nFinally, here are two laws about cp:\nmap (map f) . cp\n= cp . map (map f)\nfilter (all p) . cp = cp . map (filter p)\nThe \ufb01rst law, a naturality law, is suggested solely by the type of cp; we saw similar\nlaws in the previous chapter. The second law says that as an alternative to taking\nthe cartesian product of a list of lists, and then retaining only those lists all of\nwhose elements satisfy p, we can \ufb01rst \ufb01lter the original lists to retain only those\nelements that satisfy p and then take the cartesian product. As the previous sentence\nillustrates, one equation can be worth a thousand words.\n\n5.3 Pruning the matrix of choices\nSummarising what we have at the moment,\nsolve :: Grid -> [Grid]\nsolve = filter valid . expand . choices\nThough executable in theory, this de\ufb01nition of solve is hopeless in practice. Assuming about 20 of the 81 entries are \ufb01xed initially, there are about 961 , or\nghci> 9^61\n16173092699229880893718618465586445357583280647840659957609\ngrids to check! We therefore need a better approach.\nTo make a more ef\ufb01cient solver, an obvious idea is to remove any choices from a\ncell c that already occur as singleton entries in the row, column and box containing\nc. A singleton entry corresponds to a \ufb01xed choice. We therefore seek a function\nprune :: Matrix [Digit] -> Matrix [Digit]\nso that\nfilter valid . expand = filter valid . expand . prune\nHow can we de\ufb01ne prune? Well, since a matrix is a list of rows, a good place to\nstart is by pruning a single row. The function pruneRow is de\ufb01ned by\npruneRow :: Row [Digit] -> Row [Digit]\npruneRow row = map (remove fixed) row\n\n\f98\n\nA simple Sudoku solver\n\nwhere fixed = [d | [d] <- row]\nThe \ufb01xed choices are the singleton entries in each row. The de\ufb01nition of fixed\nuses a list comprehension involving a pattern: all elements of row that are not\nsingletons are discarded.\nThe function remove removes the \ufb01xed choices from any choice that is not \ufb01xed:\nremove :: [Digit] -> [Digit] -> [Digit]\nremove ds [x] = [x]\nremove ds xs = filter (`notElem` ds) xs\nThe standard prelude function notElem is de\ufb01ned by\nnotElem :: (Eq a) => a -> [a] -> Bool\nnotElem x xs = all (/= x) xs\nHere are a couple of examples of the use of pruneRow:\nghci> pruneRow [[6],[1,2],[3],[1,3,4],[5,6]]\n[[6],[1,2],[3],[1,4],[5]]\nghci> pruneRow [[6],[3,6],[3],[1,3,4],[4]]\n[[6],[],[3],[1],[4]]\nIn the \ufb01rst example, [6] and [3] are the \ufb01xed choices; removing these choices\nfrom the other entries reduces the last entry to a \ufb01xed choice. In the second example, removing the \ufb01xed choices reduces the second entry to the empty list of\nchoices.\nThe function pruneRow satis\ufb01es the equation\nfilter nodups . cp = filter nodups . cp . pruneRow\nIn words, this equation says that pruning a row will not throw away any list that\ncontains no duplicates. We will also make use of this law in a short while.\nWe are now nearly ready for a calculation that will determine the function prune.\nNearly, but not quite because we are going to need two more laws: If f . f = id,\nthen\nfilter (p . f) = map f . filter p . map f\nfilter (p . f) . map f = map f . filter p\n\n\f5.3 Pruning the matrix of choices\n\n99\n\nThe second law follows from the \ufb01rst (Why?). Here is the proof of the \ufb01rst law:\nmap f . filter p . map f\n=\n\n{we proved in the previous chapter that\nfilter p . map f = map f . filter (p . f)}\nmap f . map f . filter (p . f)\n\n=\n\n{functor law of map and f . f = id}\nfilter (p . f)\n\nNow for the main calculation. The starting point is to use the de\ufb01nition of valid\nto rewrite the expression filter valid . expand in the form\nfilter valid .\n= filter (all\nfilter (all\nfilter (all\n\nexpand\nnodups . boxs) .\nnodups . cols) .\nnodups . rows) . expand\n\nThe order in which the \ufb01lters appear on the right is not important. The plan of\nattack is to send each of these \ufb01lters into battle with expand. For example, in the\nboxs case we can calculate:\nfilter (all nodups . boxs) . expand\n=\n\n{above law of filter, since boxs . boxs = id}\nmap boxs . filter (all nodups) . map boxs . expand\n\n=\n\n{since map boxs . expand = expand . boxs}\nmap boxs . filter (all nodups) . expand . boxs\n\n=\n\n{de\ufb01nition of expand}\nmap boxs . filter (all nodups) . cp . map cp . boxs\n\n=\n\n{since filter (all p) . cp = cp . map (filter p)}\nmap boxs . cp . map (filter nodups) . map cp . boxs\n\n=\n\n{functor law of map}\nmap boxs . cp . map (filter nodups . cp) . boxs\n\nNow we use the property\nfilter nodups . cp = filter nodups . cp . pruneRow\nto rewrite the \ufb01nal expression in the form\nmap boxs . cp . map (filter nodups . cp . pruneRow) . boxs\n\n\fA simple Sudoku solver\n\n100\n\nThe remaining steps essentially repeat the calculation above, but in the reverse\ndirection:\nmap boxs . cp . map (filter nodups . cp . pruneRow) .\nboxs\n=\n\n{functor law of map}\nmap boxs . cp . map (filter nodups) .\nmap (cp . pruneRow) . boxs\n\n=\n\n{since cp . map (filter p) = filter (all p) . cp}\nmap boxs . filter (all nodups) . cp .\nmap (cp . pruneRow) . boxs\n\n=\n\n{functor law of map}\nmap boxs . filter (all nodups) .\ncp . map cp . map pruneRow . boxs\n\n=\n\n{de\ufb01nition of expand}\nmap boxs . filter (all nodups) .\nexpand . map pruneRow . boxs\n\n=\n\n{law of filter since boxs . boxs = id}\nfilter (all nodups . boxs) . map boxs .\nexpand . map pruneRow . boxs\n\n=\n\n{since map boxs . expand = expand . boxs}\nfilter (all nodups . boxs) . expand .\nboxs . map pruneRow . boxs\n\n=\n\n{introducing pruneBy f = f . pruneRow . f}\nfilter (all nodups . boxs) . expand . pruneBy boxs\n\nWe have shown that\nfilter (all nodups . boxs) . expand\n= filter (all nodups . boxs) . expand . pruneBy boxs\nwhere pruneBy f = f . map pruneRow . f. Repeating the same calculation\nfor rows and columns, we obtain\nfilter valid . expand = filter valid . expand . prune\nwhere\nprune = pruneBy boxs . pruneBy cols . pruneBy rows\n\n\f5.4 Expanding a single cell\n\n101\n\nIn conclusion, the previous de\ufb01nition of solve can now be replaced with a new\none:\nsolve = filter valid . expand . prune . choices\nIn fact, rather than have just one prune we can have as many prunes as we like. This\nis sensible because after one round of pruning some choices may be resolved into\nsingleton choices and another round of pruning may remove still more impossible\nchoices.\nSo, let us de\ufb01ne\nmany :: (Eq a) => (a -> a) -> a -> a\nmany f x = if x == y then x else many f y\nwhere y = f x\nand rede\ufb01ne solve once again to read\nsolve = filter valid . expand . many prune . choices\nThe simplest Sudoku problems are solved just by repeatedly pruning the matrix of\nchoices until only singleton choices are left.\n\n5.4 Expanding a single cell\nThe result of many prune . choices is a matrix of choices that can be put into\none of three classes:\n1. A complete matrix in which every entry is a singleton choice. In this case\nexpand will extract a single grid that can be checked for validity.\n2. A matrix that contains the empty choice somewhere. In this case expand will\nproduce the empty list.\n3. A matrix that does not contain the empty choice but does contain some entry\nwith two or more choices.\nThe problem is what to do in the third case. Rather than carry out full expansion,\na more sensible idea is to make use of a partial expansion that installs the choices\nfor just one of the entries, and to start the pruning process again on each result. The\nhope is that mixing pruning with single-cell expansions can lead to a solution more\nquickly. Our aim therefore is to construct a partial function\nexpand1 :: Matrix [Digit] -> [Matrix [Digit]]\n\n```\n\n\n```\n\n\fA simple Sudoku solver\n\n102\n\nthat expands the choices for one cell only. This function will return well-de\ufb01ned\nresults only for incomplete matrices, and on such matrices is required to satisfy\nexpand = concat . map expand .\n\nexpand1\n\nActually this equality between two lists is too strong. We want to ensure that no\npossible choice is lost by partial expansion, but do not really care about the precise\norder in which the two sides deliver their results. So we will interpret the equation\nas asserting the equality of the two sides up to some permutation of the answers.\nWhich cell should we perform expansion on? The simplest answer is to \ufb01nd the\n\ufb01rst cell in the matrix with a non-singleton entry. Think of a matrix rows broken\nup as follows:\nrows = rows1 ++ [row] ++ rows2\nrow = row1 ++ [cs] ++ row2\nThe cell cs is a non-singleton list of choices in the middle of row, which in turn is\nin the middle of the matrix rows.\nThen we can de\ufb01ne\nexpand1 :: Matrix [Digit] -> [Matrix [Digit]]\nexpand1 rows\n= [rows1 ++ [row1 ++ [c]:row2] ++ rows2 | c <- cs]\nTo break up the matrix in this way, we use the prelude function break:\nbreak :: (a -> Bool) -> [a] -> ([a],[a])\nbreak p = span (not . p)\nThe function span was de\ufb01ned in Section 4.8. For example,\nghci> break even [1,3,7,6,2,3,5]\n([1,3,7],[6,2,3,5])\nWe also need the standard prelude function any, de\ufb01ned by\nany :: (a -> Bool) -> [a] -> Bool\nany p = or . map p\nwhere or takes a list of booleans and returns True if any element is True, and\nFalse otherwise:\nor :: [Bool] -> Bool\nor []\n= False\nor (x:xs) = x || or xs\n\n\f5.4 Expanding a single cell\n\n103\n\nFinally, the single test is de\ufb01ned (using don\u2019t care patterns) by\nsingle :: [a] -> Bool\nsingle [_] = True\nsingle _\n= False\nNow we can de\ufb01ne\nexpand1 :: Matrix [Digit] -> [Matrix [Digit]]\nexpand1 rows\n= [rows1 ++ [row1 ++ [c]:row2] ++ rows2 | c <- cs]\nwhere\n(rows1,row:rows2) = break (any (not . single)) rows\n(row1,cs:row2)\n= break (not . single) row\nThe \ufb01rst where clause breaks a matrix into two lists of rows with the row at the\nhead of the second list being one that contains a non-singleton choice. A second\nappeal to break then breaks this row into two lists, with the head of the second list\nbeing the \ufb01rst non-singleton element. If the matrix contains only singleton entries,\nthen\nbreak (any (not . single)) rows = [rows,[]]\nand execution of expand1 returns an error message.\nThe problem with this de\ufb01nition of expand1 is that it can lead to wasted work. If\nthe \ufb01rst non-singleton entry found in this way happens to be the empty list, then\nexpand1 will return the empty list, but if such a list is buried deep in the matrix,\nthen expand1 will do a lot of useless calculation trying to \ufb01nd a solution that isn\u2019t\nthere. It is arguable that a better choice of cell on which to perform expansion is\none with the smallest number of choices (not equal to 1 of course). A cell with no\nchoices means that the puzzle is unsolvable, so identifying such a cell quickly is a\ngood idea.\nThe change to expand1 to implement this idea is as follows:\nexpand1 :: Matrix [Digit] -> [Matrix [Digit]]\nexpand1 rows\n= [rows1 ++ [row1 ++ [c]:row2] ++ rows2 | c <- cs]\nwhere\n(rows1,row:rows2) = break (any smallest) rows\n(row1,cs:row2)\n= break smallest row\nsmallest cs\n= length cs == n\nn\n= minimum (counts rows)\n\n\f104\n\nA simple Sudoku solver\n\nThe function counts is de\ufb01ned by\ncounts = filter (/= 1) . map length . concat\nThe value n is the smallest number of choices, not equal to 1, in any cell of the\nmatrix of choices. We will leave the de\ufb01nition of minimum as an exercise. The\nvalue of n will be 0 if the matrix has an empty choice entry anywhere, and in this\ncase expand1 will return the empty list. On the other hand, if the matrix of choices\ncontains only singleton choices, then n is the minimum of the empty list, which is\nthe unde\ufb01ned value \u22a5. In this case expand1 will also return \u22a5, so we had better\nensure that expand1 is applied only to incomplete matrices. A matrix is incomplete\nif it does not satisfy complete:\ncomplete :: Matrix [Digit] -> Bool\ncomplete = all (all single)\nWe can also usefully generalise valid to a test on matrices of choices. Suppose\nwe de\ufb01ne safe by\nsafe :: Matrix [Digit] -> Bool\nsafe m = all ok (rows cm) &&\nall ok (cols cm) &&\nall ok (boxs cm)\nok row = nodups [x | [x] <- row]\nA matrix is safe if none of the singleton choices in any row, column or box contain\nduplicates. But a safe matrix may contain non-singleton choices. Pruning can turn\na safe matrix into an unsafe one, but if a matrix is safe after pruning it has to be\nsafe beforehand. In symbols, safe . prune = safe. A complete and safe matrix\nyields a solution to the Sudoku problem, and this solution can be extracted by a\nsimpli\ufb01ed version of expand:\nextract :: Matrix [Digit] -> Grid\nextract = map (map head)\nHence on a safe and complete matrix m we have\nfilter valid (expand m) = [extract m]\nOn a safe but incomplete matrix we have\nfilter valid . expand\n= filter valid . concat . map expand . expand1\nup to permutation of each side. Since\n\n\f5.5 Exercises\n\n105\n\nfilter p . concat = concat . map (filter p)\nwe obtain that filter valid . expand simpli\ufb01es to\nconcat . map (filter p . expand) . expand1\nAnd now we can insert a single prune to obtain\nconcat . map (filter p . expand . prune) . expand1\nHence, introducing\nsearch = filter valid . expand . prune\nwe have, on safe but incomplete matrices, that\nsearch = concat . map search . expand1 . prune\nAnd now we can replace solve by a third version:\nsolve = search . choices\nsearch cm\n| not (safe pm) = []\n| complete pm\n= [extract pm]\n| otherwise\n= concat (map search (expand1 pm))\nwhere pm = prune cm\nThis is our \ufb01nal simple Sudoku solver. We could replace prune in the last line by\nmany prune. Sometimes many prunes work faster than one prune; sometimes not.\nNote that the very \ufb01rst safety test occurs immediately after one round of pruning\non the installed choices; consequently \ufb02awed input is detected quickly.\n\n5.5 Exercises\nExercise A\nHow would you add 1 to every element in a given matrix of integers? How would\nyou sum the elements of a matrix? The function zipWith (+) adds two rows, but\nwhat function would add two matrices? How would you de\ufb01ne matrix multiplication?\nExercise B\nWhat are the dimensions of the matrix [[],[]]? Of the matrix []?\nThe function cols (here renamed as transpose) was de\ufb01ned by\n\n\fA simple Sudoku solver\n\n106\n\ntranspose :: [[a]] -> [[a]]\ntranspose [xs]\n= [[x] | x <- xs]\ntranspose (xs:xss) = zipWith (:) xs (transpose xss)\nFill in the dots that would enable you to replace the \ufb01rst clause by\ntranspose []\n\n= ...\n\nThe above de\ufb01nition of transpose proceeds row by row. Here is part of a de\ufb01nition that proceeds column by column:\ntranspose xss = map head xss:transpose (map tail xss)\nComplete this de\ufb01nition.\nExercise C\nWhich of the following equations are true (no justi\ufb01cation is necessary):\nany p = not . all (not p)\nany null = null . cp\nExercise D\nGiven a function sort :: (Ord a) => [a] -> [a] that sorts a list, construct a\nde\ufb01nition of\nnodups :: (Ord a) => [a] -> Bool\nExercise E\nThe function nub :: (Eq a) => [a] -> [a] removes duplicates from a list (a\nversion of this function is available in the library Data.List). De\ufb01ne nub. Assuming the order of the elements in the result is not important, de\ufb01ne\nnub :: (Ord a) => [a] -> [a]\nso that the result is a more ef\ufb01cient function.\nExercise F\nThe functions takeWhile and dropWhile satisfy\nspan p xs = (takeWhile p xs,dropWhile p xs)\nGive direct recursive de\ufb01nitions of takeWhile and dropWhile.\nAssuming whiteSpace :: Char -> Bool is a test for whether a character is\n\n\f5.6 Answers\n\n107\n\nwhite space (such as a space, a tab or a newline) or a visible character, construct a\nde\ufb01nition of\nwords :: String -> [Word]\nthat breaks a string up into a list of words.\nExercise G\nDe\ufb01ne minimum :: Ord a => [a] -> a.\nExercise H\nWhy didn\u2019t we de\ufb01ne solve by the following?\nsolve = search . choices\nsearch m\n| not (safe m) = []\n| complete m\n= [extract m]\n| otherwise\n= process m\nwhere process = concat . map search . expand1 . prune\n\n5.6 Answers\nAnswer to Exercise A\nAdding 1 to every matrix element is de\ufb01ned by map (map (+1)).\nSumming a matrix is de\ufb01ned by sum . map sum, where sum sums a list of numbers. Alternatively, we could use sum . concat.\nMatrix addition is de\ufb01ned by zipWith (zipWith (+)).\nFor matrix multiplication we \ufb01rst de\ufb01ne\nscalarMult :: Num a => [a] -> [a] -> a\nscalarMult xs ys = sum (zipwith (*) xs ys)\nThen we have\nmatMult :: Num a => Matrix a -> Matrix a -> Matrix a\nmatMult ma mb = [map (scalarMult row) mbt | row <- ma]\nwhere mbt = transpose mb\n\n\fA simple Sudoku solver\n\n108\n\nAnswer to Exercise B\nThe matrix [[],[]] has dimensions 2 \u00d7 0. The matrix [] has dimensions 0 \u00d7 n\nfor every n. The transpose of such a matrix therefore has to have dimensions n \u00d7 0\nfor every n. The only reasonable possibility is to let n be in\ufb01nite:\ntranspose :: [[a]] -> [[a]]\ntranspose []\n= repeat []\ntranspose (xs:xss) = zipWith (:) xs (transpose xss)\nwhere repeat x gives an in\ufb01nite list of repetitions of x. Note that\ntranspose [xs] = zipWith (:) xs (repeat [])\n= [[x] | x <- xs]\nThe alternative de\ufb01nition is\ntranspose ([]:xss) = []\ntranspose xss = map head xss:transpose (map tail xss)\nThe assumption in the \ufb01rst line is that if the \ufb01rst row is empty, then all the rows are\nempty and the transpose is the empty matrix.\nAnswer to Exercise C\nBoth the equations are true.\nAnswer to Exercise D\nnodups :: (Ord a) => [a] -> Bool\nnodups xs = and (zipWith (/=) ys (tail ys))\nwhere ys = sort xs\nAnswer to Exercise E\nnub :: (Eq a) => [a] -> [a]\nnub []\n= []\nnub (x:xs) = x:nub (filter (/= x) xs)\nnub :: (Ord a) => [a] -> [a]\nnub = remdups . sort\nremdups []\n= []\nremdups (x:xs) = x:remdups (dropWhile (==x) xs)\nThe function dropWhile is de\ufb01ned in the next exercise.\n\n\f5.7 Chapter notes\n\n109\n\nAnswer to Exercise F\ntakeWhile, dropWhile :: (a -> Bool) -> [a] -> [a]\ntakeWhile p [] = []\ntakeWhile p (x:xs)\n= if p x then x:takeWhile p xs else []\ndropWhile p [] = []\ndropWhile p (x:xs)\n= if p x then dropWhile p xs else x:xs\nThe de\ufb01nition of words is\nwords :: String -> [Word]\nwords xs | null ys\n= []\n| otherwise = w:words zs\nwhere ys = dropWhile whiteSpace xs\n(w,zs) = break whiteSpace ys\nAnswer to Exercise G\nminimum :: Ord a => [a] -> a\nminimum [x]\n= x\nminimum (x:xs) = x `min` minimum xs\nNote that the minimum of the empty list is unde\ufb01ned.\nAnswer to Exercise H\nThe suggested de\ufb01nition of solve would return the unde\ufb01ned value if the matrix\nbecomes complete after one round of pruning.\n\n5.7 Chapter notes\nThe Independent newspaper no longer uses the rubric for Sudoku quoted at the\nstart of the chapter. The presentation follows that in my book Pearls of Functional\nAlgorithm Design (Cambridge, 2010). The site\nhaskell.org/haskellwiki/Sudoku\ncontains about 20 Haskell implementations of Sudoku, many of which use arrays\nand/or monads. We will meet arrays and monads in Chapter 10.\n\n\fChapter 6\nProofs\n\nWe have seen a lot of laws in the previous two chapters, though perhaps the word\n\u2018law\u2019 is a little inappropriate because it suggests something that is given to us from\non high and which does not have to be proved. At least the word has the merit of\nbeing short. All of the laws we have encountered so far assert the equality of two\nfunctional expressions, possibly under subsidiary conditions; in other words, laws\nhave been equations or identities between functions, and calculations have been\npoint-free calculations (see Chapter 4, and the answer to Exercise K for more on\nthe point-free style). Given suitable laws to work with, we can then use equational\nreasoning to prove other laws. Equational logic is a simple but powerful tool in\nfunctional programming because it can guide us to new and more ef\ufb01cient definitions of the functions and other values we have constructed. Ef\ufb01ciency is the\nsubject of the following chapter. This one is about another aspect of equational\nreasoning, proof by induction. We will also show how to shorten proofs by introducing a number of higher-order functions that capture common patterns of computations. Instead of proving properties of similar functions over and over again,\nwe can prove more general results about these higher-order functions, and appeal\nto them instead.\n\n6.1 Induction over natural numbers\nConsider the following de\ufb01nition of the exponential function:\nexp :: Num a => a -> Nat -> a\nexp x Zero\n= 1\nexp x (Succ n) = x * exp x n\nIn the old days we could have written\n\n\f6.1 Induction over natural numbers\n\n111\n\nexp :: Num a => a -> Int -> a\nexp x 0\n= 1\nexp x (n+1) = x * exp x n\nbut this precise form of de\ufb01nition using a (n+1)-pattern is no longer allowed in\nthe current standard version of Haskell, Haskell 2010.\nAnyway, we would expect that the equation\nexp x (m+n) = exp x m * exp x n\nis true for all m and n. After all, xm+n = xm xn is a true equation of mathematics. But\nhow can we prove this law?\nThe answer, of course, is by induction. Every natural number is either Zero or of\nthe form Succ n for some natural number n. That is exactly what the de\ufb01nition\ndata Nat = Zero | Succ Nat\nof the data type Nat tells us. So to prove that P(n) holds for all natural numbers n,\nwe can prove\n1. P(0) holds;\n2. For all natural numbers n, that P(n + 1) holds assuming that P(n) does.\nWe have reverted to writing 0 for Zero and n+1 for Succ n, and we shall continue\nto do so. In the second proof we can assume P(n) and use this assumption to prove\nP(n + 1).\nAs an example we prove that\nexp x (m+n) = exp x m * exp x n\nfor all x, m and n by induction on m. We could also prove it by induction on n but\nthat turns out to be more complicated. Here is the proof:\nCase 0\nexp x (0 + n)\n=\n\n{since 0 + n = n}\n\nexp x 0 * exp x n\n=\n\nexp x n\n\n{exp.1}\n1 * exp x n\n\n=\n\n{since 1 * x = x}\nexp x n\n\n\fProofs\n\n112\n\nCase m+1\nexp x ((m + 1) + n))\n=\n\n{arithmetic}\n\nexp x (m+1) * exp x n\n=\n\nexp x ((m + n) + 1\n=\n\n{exp.2}\nx * exp x (m + n)\n\n=\n\n{exp.2}\n(x * exp x m) * exp x n\n\n=\n\n{since * is associative}\nx * (exp x m * exp x n)\n\n{induction}\nx * (exp x m * exp x n)\n\nThe above format will be used in all induction proofs. The proof breaks into two\ncases, the base case 0 and the inductive case n + 1. Each case is laid out in two\ncolumns, one for the left-hand side of the equation, and one for the right-hand side.\n(When there is not enough space for two columns, we display one after the other.)\nEach side is simpli\ufb01ed until one can go no further, and the proof of each case\nis completed by observing that each side simpli\ufb01es to the same result. The hints\nexp.1 and exp.2 refer to the \ufb01rst and second equations de\ufb01ning exp.\nFinally, observe that the proof depends on three further laws, namely that\n(m + 1) + n = (m + n) + 1\n1 * x\n= x\n(x * y) * z = x * (y * z)\n\n```\n\n\n```\n\nIf we were recreating all of arithmetic from scratch \u2013 and that would be a tedious\nthing to do \u2013 we would also have to prove these laws. In fact, only the \ufb01rst can be\nproved because it is entirely about natural numbers and we have de\ufb01ned the operation of addition on natural numbers. The second two rely on the implementation\nof multiplication prescribed by Haskell for the various instances of the type class\nNum.\nIn fact, the associative law breaks down for \ufb02oating-point numbers:\nghci> (9.9e10 * 0.5e-10) * 0.1e-10 :: Float\n4.95e-11\nghci> 9.9e10 * (0.5e-10 * 0.1e-10) :: Float\n4.9499998e-11\nRecall that in scienti\ufb01c notation 9.9e10 means 9.9 * 10^10. So, although our\nproof was correct mathematically, one of the provisos in it wasn\u2019t, at least in\nHaskell.\n\n\f6.2 Induction over lists\n\n113\n\n6.2 Induction over lists\nWe have seen that every \ufb01nite list is either the empty list [] or of the form x:xs\nwhere xs is a \ufb01nite list. Hence, to prove that P(xs) holds for all \ufb01nite lists xs, we\ncan prove:\n1. P([]) holds;\n2. For all x and for all \ufb01nite lists xs, that P(x:xs) holds assuming P(xs) does.\nAs an example, recall the de\ufb01nition of concatenation (++):\n[] ++ ys\n= ys\n(x:xs) ++ ys = x : (xs ++ ys)\nWe prove that ++ is associative:\n(xs ++ ys) ++ zs = xs ++ (ys ++ zs)\nfor all \ufb01nite lists xs and for all lists ys and zs (note that neither of the last two is\nrequired to be a \ufb01nite list), by induction on xs:\nCase []\n([] ++ ys) ++ zs\n{++.1}\n\n=\n\nys ++ zs\n\n[] ++ (ys ++ zs)\n=\n\n{++.1}\nys ++ zs\n\nCase x:xs\n((x:xs) ++ ys) ++ zs\n=\n\n{++.2}\n\n(x:xs) ++ (ys ++ zs)\n\n(x:(xs ++ ys)) ++ zs\n=\n\n{++.2}\nx:((xs ++ ys) ++ zs)\n\n{++.2}\n\n=\n\nx:(xs ++ (ys ++ zs))\n{induction}\n\n=\n\nx:((xs ++ ys) ++ zs)\n\nAs another example, given the de\ufb01nition\nreverse []\n= []\nreverse (x:xs) = reverse xs ++ [x]\nWe prove that reverse is an involution:\nreverse (reverse xs) = xs\nfor all \ufb01nite lists xs. The base case is easy and the inductive case proceeds:\n\n\fProofs\n\n114\n\nCase x:xs\nreverse (reverse (x:xs))\n{reverse.2}\n\n=\n\nreverse (reverse xs ++ [x])\n{????}\n\n=\n\nx:reverse (reverse xs)\n{induction}\n\n=\n\nx:xs\nThe right-hand column is omitted in this example, since it consists solely of x:xs.\nBut we got stuck in the proof halfway through. We need an auxiliary result, namely\nthat\nreverse (ys ++ [x]) = x:reverse ys\nfor all \ufb01nite lists ys. This auxiliary result is also proved by induction:\nCase []\nreverse ([] ++ [x])\n=\n\n{++.1}\n\nx:reverse []\n=\n\nreverse [x]\n=\n\n{reverse.1}\n[x]\n\n{reverse.2}\nreverse [] ++ [x]\n\n=\n\n{reverse.1 and ++.1}\n[x]\n\nCase y:ys\nreverse ((y:ys) ++ [x])\n=\n\n{++.2}\nreverse (y:(ys ++ [x]))\n\n=\n\n{reverse.2}\nreverse (ys ++ [x]) ++ [y]\n\n=\n\n{induction}\n(x:reverse ys) ++ [y]\n\n=\n\n{++.2}\nx:(reverse ys ++ [y])\n\nx:reverse (y:ys)\n=\n\n{reverse.2}\nx:(reverse ys ++ [y])\n\n\f6.2 Induction over lists\n\n115\n\nThe auxiliary result holds, and therefore so does the main result.\n\nInduction over partial lists\nEvery partial list is either the unde\ufb01ned list or of the form x:xs for some x and\nsome partial list xs. Hence, to prove that P(xs) holds for all partial lists xs we can\nprove that\n1. P(undefined) holds;\n2. P(x:xs) holds assuming P(xs) does, for all x and all partial lists xs.\nAs an example, we prove that\nxs ++ ys = xs\nfor all partial lists xs and all lists ys:\nCase undefined\nundefined ++ ys\n{++.0}\n\n=\n\nundefined\nCase x:xs\n(x:xs) ++ ys\n=\n\n{++.2}\nx:(xs ++ ys)\n\n=\n\n{induction}\nx:xs\n\nIn each case the trivial right-hand column is omitted. The hint (++).0 refers to\nthe failing clause in the de\ufb01nition of (++): since concatenation is de\ufb01ned by pattern matching on the left-hand argument, the result is unde\ufb01ned if the left-hand\nargument is.\n\nInduction over in\ufb01nite lists\nProving that something is true of all in\ufb01nite lists requires a bit of background\nthat we will elaborate on in a subsequent chapter. Basically an in\ufb01nite list can\n\n\fProofs\n\n116\n\nbe thought of as the limit of a sequence of partial lists. For example, [0..] is the\nlimit of the sequence\nundefined,\n\n0:undefined,\n\n0:1:undefined,\n\n0:1:2:undefined,\n\nand so on. A property P is called chain complete if whenever xs0 , xs1 , . . . is a sequence of partial lists with limit xs, and P(xsn ) holds for all n, then P(xs) also\nholds.\nIn other words, if P is a chain complete property that holds for all partial lists (and\npossibly all \ufb01nite lists too), then it holds for all in\ufb01nite lists.\nMany properties are chain complete; for instance:\n\u2022 All equations e1 = e2, where e1 and e2 are Haskell expressions involving universally quanti\ufb01ed free variables, are chain complete.\n\u2022 If P and Q are chain complete, then so is their conjunction P \u2227 Q.\nBut inequalities e1 = e2 are not necessarily chain complete, and neither are properties involving existential quanti\ufb01cation. For example, consider the assertion\ndrop n xs = undefined\nfor some integer n. This property is obviously true for all partial lists, and equally\nobviously not true for any in\ufb01nite list.\nHere is an example proof. Earlier we proved that\n(xs ++ ys) ++ zs = xs ++ (ys ++ zs)\nfor all \ufb01nite lists xs and for all lists ys and zs. We can extend this chain complete\nproperty to all lists xs by proving\nCase undefined\n(undefined ++ ys) ++ zs\n=\n\n{++.0}\n\nundefined ++ (ys ++ zs)\n=\n\nundefined ++ zs\n=\n\n{++.0}\nundefined\n\n{++.0}\nundefined\n\nThus ++ is a truly associative operation on lists, independent of whether the lists\nare \ufb01nite, partial or in\ufb01nite.\nBut we have to be careful. Earlier we proved\n\n\f6.3 The function foldr\n\n117\n\nreverse (reverse xs) = xs\nfor all \ufb01nite lists xs. Can we extend this property to all lists by proving the following additional case?\nCase undefined\nreverse (reverse undefined)\n=\n\n{reverse.0}\nundefined\n\nThat goes through but something is wrong: as a Haskell equation we have\nreverse (reverse xs) = undefined\nfor all partial lists xs. What did we miss?\nThe answer is that in proving the involution property of reverse we made use of\nan auxiliary result:\nreverse (ys ++ [x]) = x:reverse ys\nfor all \ufb01nite lists ys. This result is not true for all lists, indeed not true for any\npartial list ys.\nIt follows that reverse . reverse is not the identity function on lists, A functional equation f = g over lists asserts that f xs = g xs for all lists xs, \ufb01nite,\npartial and in\ufb01nite. If the equation is true only for \ufb01nite lists, we have to say so\nexplicitly.\n\n6.3 The function foldr\nAll the following functions have a common pattern:\nsum []\n= 0\nsum (x:xs) = x + sum xs\nconcat []\n= []\nconcat (xs:xss) = xs ++ concat xss\nfilter p []\n= []\nfilter p (x:xs) = if p x then x:filter p xs\nelse filter p xs\n\n\fProofs\n\n118\n\nmap f []\n= []\nmap f (x:xs) = f x:map f xs\nSimilarly, the proofs by induction of the following laws all have a common pattern:\nsum (xs ++ ys)\nconcat (xss ++ yss)\nfilter p (xs ++ ys)\nmap f (xs ++ ys)\n\n=\n=\n=\n=\n\nsum xs + sum ys\nconcat xss ++ concat yss\nfilter p xs ++ filter p ys\nmap f xs ++ map f ys\n\nCan we not ensure that the functions above are de\ufb01ned as instances of a more\ngeneral function, and the laws above as instances of a more general law? That\nwould save a lot of repetitive effort.\nThe function foldr (fold from the right) is de\ufb01ned by\nfoldr :: (a -> b -> b) -> b -> [a] -> b\nfoldr f e []\n= e\nfoldr f e (x:xs) = f x (foldr f e xs)\nTo appreciate this de\ufb01nition, consider\nfoldr (@) e [x,y,z] = x @ (y @ (z @ e))\n[x,y,z] = x : (y : (z : []))\nIn other words, foldr (@) e applied to a list replaces the empty list by e, and\n(:) by (@) and evaluates the result. The parentheses group from the right, whence\nthe name.\nIt follows at once that foldr (:) [] is the identity function on lists. Furthermore,\nsum\n=\nconcat\n=\nfilter p =\nmap f\n=\n\nfoldr\nfoldr\nfoldr\nfoldr\n\n(+) 0\n(++) []\n(\\x xs -> if p x then x:xs else xs) []\n((:) . f) []\n\nThe following fact captures all the identities mentioned above:\nfoldr f e (xs ++ ys) = foldr f e xs @ foldr f e ys\nfor some operation (@) satisfying various properties. We prove this equation by\ninduction on xs. Along the way, we discover what properties of f, e and (@) we\nneed.\n\n\f6.3 The function foldr\n\n119\n\nCase []\nfoldr f e ([] ++ ys)\n=\n\n{++.1}\n\nfoldr f e [] @ foldr f e ys\n=\n\nfoldr f e ys\n\n{foldr.1}\ne @ foldr f e ys\n\nHence we need e @ x = x for all x.\nCase x:xs\nfoldr f e ((x:xs) ++ ys)\n=\n\n{++.2}\nfoldr f e (x:(xs ++ ys)\n\n=\n\n{foldr.2}\nf x (foldr f e (xs ++ ys))\n\n=\n\n{induction}\nf x (foldr f e xs @ foldr f e ys)\n\nThe right-hand side in this case simpli\ufb01es to\nf x (foldr f e xs) @ foldr f e ys\nSo, in summary, we require that\ne @ x\n= x\nf x (y @ z) = f x y @ z\nfor all x, y and z. In particular the two requirements are met if f = (@) and (@)\nis associative with identity e. That immediately proves\nsum (xs ++ ys)\n= sum xs + sum ys\nconcat (xss ++ yss) = concat xss ++ concat yss\nFor the map law, we require that\n[] ++ xs = xs\nf x:(xs ++ ys) = (f x:ys) ++ ys\nBoth immediately follow from the de\ufb01nition of concatenation.\nFor the law of filter we require that\nif p x then x:(ys ++ zs) else ys ++ zs\n= (if p x then x:ys else ys) ++ zs\n\n\fProofs\n\n120\n\nThis is immediate from the de\ufb01nitions of concatenation and conditional expressions.\n\nFusion\nThe most important property of foldr is the fusion law, which asserts that\nf . foldr g a =\n\nfoldr h b\n\nprovided certain properties of the ingredients hold. As two simple examples,\ndouble . sum\n= foldr ((+) . double) 0\nlength . concat = foldr ((+) . length) 0\nIn fact, many of the laws we have seen already are instances of the fusion law for\nfoldr. In a word, the fusion law is a \u2018pre-packaged\u2019 form of induction over lists.\nTo \ufb01nd out what properties we need, we carry out an induction proof of the fusion\nlaw. The law is expressed as a functional equation, so we have to show that it holds\nfor all \ufb01nite and all partial lists:\nCase undefined\nf (foldr g a undefined)\n\nfoldr h b undefined\n\n{foldr.0}\n\n=\n\n=\n\nf undefined\n\n{foldr.0}\nundefined\n\nSo the \ufb01rst condition is that f is a strict function.\nCase []\nf (foldr g a [])\n=\n\n{foldr.1}\n\nfoldr h b []\n{foldr.1}\n\n=\n\nf a\n\nb\n\nThe second condition is that f a = b.\nCase x:xs\nf (foldr g a (x:xs))\n=\n\n{foldr.2}\n\nfoldr h b (x:xs)\n=\n\nf (g x (foldr g a xs))\n\n{foldr.2}\nh x (foldr h b xs)\n\n=\n\n{induction}\nh x (f (foldr g a xs))\n\n\f6.3 The function foldr\n\n121\n\nThe third condition is met by f (g x y) = h x (f y) for all x and y.\nLet us apply the fusion law to show that\nfoldr f a . map g = foldr h a\nRecall that map g = foldr ((:) . g) []. Looking at the conditions of the fusion law we have that\nfoldr f a undefined = undefined\nfoldr f a []\n= a\nSo the \ufb01rst two fusion conditions are satis\ufb01ed. The third one is\nfoldr f a (g x:xs) = h x (foldr f a xs)\nThe left-hand side simpli\ufb01es to\nf (g x) (foldr f a xs)\nso we can de\ufb01ne h x y = f (g x) y. More brie\ufb02y, h = f . g. Hence we have\nthe useful rule:\nfoldr f a . map g = foldr (f . g) a\nIn particular,\ndouble . sum = sum . map double\n= foldr ((+) . double) 0\nlength . concat = sum . map length\n= foldr ((+) . length) 0\nOther simple consequences of the fusion law are explored in the exercises.\n\nA variant\nSometimes having the empty list around is a pain. For example, what is the minimum element in an empty list? For this reason, Haskell provides a variant on\nfoldr, called foldr1, restricted to nonempty lists. The Haskell de\ufb01nition of this\nfunction is\nfoldr1 :: (a -> a -> a) -> [a] -> a\nfoldr1 f [x]\n= x\nfoldr1 f (x:xs) = f x (foldr1 f xs)\n\n\fProofs\n\n122\n\nSo we can de\ufb01ne\nminimum, maximum :: Ord a => [a] -> a\nminimum = foldr1 min\nmaximum = foldr1 max\nand avoid two other explicit recursions. Actually the Haskell de\ufb01nition of foldr1\nis not as general as it should be, but we will leave that discussion to an exercise.\n\n6.4 The function foldl\nRecall that\nfoldr (@) e [w,x,y,z] = w @ (x @ (y @ (z @ e)))\nSometimes a more convenient pattern for the right-hand side is\n(((e @ w) @ x) @ y) @ z\nThis pattern is encapsulated by a function foldl (fold from the left):\nfoldl :: (b -> a -> b) -> b -> [a] -> b\nfoldl f e []\n= e\nfoldl f e (x:xs) = foldl f (f e x) xs\nAs an example, suppose we are given a string, such as 1234.567, representing a\nreal number and we want to compute its integer part and fractional part. We could\nde\ufb01ne\nipart :: String -> Integer\nipart xs = read (takeWhile (/= '.') xs) :: Integer\nfpart :: String -> Float\nfpart xs = read ('0':dropWhile (/= '.' xs) :: Float\nThis uses the function read of the type class Read. Note by the way that .567 is not\na well-formed literal in Haskell. It is necessary to include at least one digit before\nand after the decimal point to ensure that the decimal point cannot be mistaken for\nfunctional composition. For example,\nghci> :t 3 . 4\n3 . 4 :: (Num (b -> c), Num (a -> b)) => a -> c\nAs an alternative, we can de\ufb01ne\n\n\f6.4 The function foldl\n\n123\n\nparts :: String -> (Integer,Float)\nparts ds = (ipart es,fpart fs)\nwhere (es,d:fs) = break (== '.') ds\nipart\n= foldl shiftl 0 . map toDigit\nwhere shiftl n d = n*10 + d\nfpart\n= foldr shiftr 0 . map toDigit\nwhere shiftr d x = (d + x)/10\ntoDigit d = fromIntegral (fromEnum d - fromEnum '0')\nWe have\n1234\n\n=\n=\n0.567 =\n=\n\n1*1000 + 2*100 + 3*10 + 4\n(((0*10 + 1)*10 + 2)*10 + 3)*10 + 4\n5/10 + 6/100 + 7/1000\n(5 + (6 + (7 + 0)/10)/10)/10\n\nso use of foldl for the integer part and foldr for the fractional part are both\nindicated.\nHere is another example. The function reverse was de\ufb01ned above by the equations\nreverse []\n= []\nreverse (x:xs) = reverse xs ++ [x]\nWe are wiser now and would now write\nreverse = foldr snoc []\nwhere snoc x xs = xs ++ [x]\nBut a little learning is a dangerous thing: both de\ufb01nitions of reverse are terrible\nbecause they take of the order of n2 steps to reverse a list of length n. Much better\nis to de\ufb01ne\nreverse = foldl (flip (:)) []\nwhere flip f x y = f y x. The new version reverses a list in linear time:\n=\n=\n=\n=\n\nfoldl (flip\nfoldl (flip\nfoldl (flip\nfoldl (flip\n3:2:1:[]\n\n(:))\n(:))\n(:))\n(:))\n\n[] [1,2,3]\n(1:[]) [2,3]\n(2:1:[]) [3]\n(3:2:1:[]) []\n\nThat seems a bit of a trick, but there is a sound principle at work behind this new\nde\ufb01nition that we will take up in the following chapter.\n\n\fProofs\n\n124\n\nAs this example suggests, there are the following relationships between foldr and\nfoldl: for all \ufb01nite lists xs we have\nfoldl f e xs = foldr (flip f) e (reverse xs)\nfoldr f e xs = foldl (flip f) e (reverse xs)\nProofs are left as an exercise. Note the restriction to \ufb01nite lists, even though both\nsides reduce to \u22a5 when xs is \u22a5. That means the proofs have to rely on a subsidiary\nresult that is true only for \ufb01nite lists.\nHere is another relationship between the two folds:\nfoldl (@) e xs = foldr (<>) e xs\nfor all \ufb01nite lists xs, provided that\n(x <> y) @ z = x <> (y @ z)\ne @ x\n= x <> e\nAgain, the proof is left as an exercise. As one instructive application of this law,\nsuppose (<>) = (@) and (@) is associative with identity e. Then the two provisos\nare satis\ufb01ed and we can conclude that\nfoldr (@) e xs = foldl (@) e xs\nfor all \ufb01nite lists xs whenever (@) is associative with identity e. In particular,\nconcat xss = foldr (++) [] xss = foldl (++) [] xss\nfor all \ufb01nite lists xss. The two de\ufb01nitions are not the same if xss is an in\ufb01nite list:\nghci> foldl (++) [] [[i] | i <- [1..]]\nInterrupted.\nghci> foldr (++) [] [[i] | i <- [1..]]\n[1,2,3,4,{Interrupted}\nIn response to the \ufb01rst expression, GHCi went into a long silence that was interrupted by pressing the \u2018Stop program execution\u2019 button. In response to the second,\nGHCi started printing an in\ufb01nite list.\nOK, so the de\ufb01nition in terms of foldr works on in\ufb01nite lists, but the other one\ndoesn\u2019t. But maybe the de\ufb01nition of concat in terms of foldl leads to a more\nef\ufb01cient computation when all the lists are \ufb01nite? To answer this question, observe\nthat\nfoldr (++) [] [xs,ys,us,vs]\n= xs ++ (ys ++ (us ++ (vs ++ [])))\n\n\f6.5 The function scanl\n\n125\n\nfoldl (++) [] [xs,ys,us,vs]\n= (((([] ++ xs) ++ ys) ++ us) ++ vs)\nLet all the component lists have length n. The \ufb01rst expression on the right takes 4n\nsteps to perform all the concatenations, while the second takes 0 + n + (n + n) +\n(n + n + n) = 6n steps. Enough said, at least for now.\n\n6.5 The function scanl\nThe function scanl f e applies foldl f e to each initial segment of a list. For\nexample\nghci> scanl (+) 0 [1..10]\n[0,1,3,6,10,15,21,28,36,45,55]\nThe expression computes the running sums of the \ufb01rst ten positive numbers:\n[0, 0+1, (0+1)+2, ((0+1)+2)+3, (((0+1)+2)+3)+4, ...]\nThe speci\ufb01cation of scanl is\nscanl :: (b -> a -> b) -> b -> [a] -> [b]\nscanl f e = map (foldl f e) . inits\ninits :: [a] -> [[a]]\ninits []\n= [[]]\ninits (x:xs) = [] : map (x:) (inits xs)\nFor example\nghci> inits \"barbara\"\n[\"\",\"b\",\"ba\",\"bar\",\"barb\",\"barba\",\"barbar\",\"barbara\"]\nThe function inits is in the library Data.List.\nBut this de\ufb01nition of scanl f involves evaluating f a total of\n0 + 1 + 2 + \u00b7 \u00b7 \u00b7 + n = n(n + 1)/2\ntimes on a list of length n. Can we do better?\nYes, we can calculate a better de\ufb01nition by doing a kind of induction proof, except\nthat we don\u2019t know what it is we are proving!\n\n\fProofs\n\n126\n\nCase []\nscanl f e []\n{de\ufb01nition}\n\n=\n\nmap (foldl f e) (inits [])\n{inits.1}\n\n=\n\nmap (foldl f e) [[]]\n{map.1 and map.2}\n\n=\n\n[foldl f e []]\n{foldl.1}\n\n=\n\n[e]\nHence we have shown that scanl f e [] = [e]\nCase x:xs\nscanl f e (x:xs)\n=\n\n{de\ufb01nition}\nmap (foldl f e) (inits (x:xs))\n\n=\n\n{inits.2}\nmap (foldl f e) ([]:map (x:) (inits xs))\n\n=\n\n{map.1 and map.2}\nfoldl f e []:map (foldl f e . (x:)) (inits xs)\n\n=\n\n{foldl.1}\ne:map (foldl f e . (x:)) (inits xs)\n\n=\n\n{claim: foldl f e . (x:) = foldl f (f e x)}\ne:map (foldl f (f e x)) (inits xs)\n\n=\n\n{de\ufb01nition of scanl}\ne:scanl f (f e x)\n\nThe claim is an easy consequence of the de\ufb01nition of foldl. Hence, in summary,\nwe have shown\nscanl f e []\n= [e]\nscanl f e (x:xs) = e:scanl f (f e x) xs\nThis de\ufb01nition evaluates f only a linear number of times.\n\n\f6.6 The maximum segment sum\n\n127\n\nWhat we have just done is an example of optimising a function by program calculation. One of the exciting things about Haskell is that you can do this without\nfuss. There is no need to bring in a totally different logical language to reason about\nprograms.\nHowever, the prelude de\ufb01nition of scanl is a little different:\nscanl f e xs = e : (case xs of\n[]\n-> []\nx:xs -> scanl f (f e x) xs)\nWhereas for our version scanl f e undefined = undefined, the prelude version has\nscanl f e undefined = e:undefined.\nThe reason is that the right-hand sides of the two clauses de\ufb01ning scanl are both\nlists that begin with e. We do not have to know anything about the left-hand sides\nto determine this fact, and laziness dictates that we don\u2019t ask.\nThe prelude version also uses a case expression. We won\u2019t go into details since\nsuch expressions are used rarely in this book. Haskell allows us many ways to say\nthe same thing.\n\n6.6 The maximum segment sum\nHere is another example of program calculation. The maximum segment sum problem is a famous one and its history is described in J. Bentley\u2019s Programming Pearls\n(1987). Given is a sequence of integers and it is required to compute the maximum\nof the sums of all segments in the sequence. A segment is also called a contiguous\nsubsequence. For example, the sequence\n[-1,2,-3,5,-2,1,3,-2,-2,-3,6]\nhas maximum sum 7, the sum of the segment [5,-2,1,3]. On the other hand,\nthe sequence [-1,-2,-3] has a maximum segment sum of zero, since the empty\nsequence is a segment of every list and its sum is zero. It follows that the maximum\nsegment sum is always nonnegative.\nOur problem is speci\ufb01ed by\nmss :: [Int] -> Int\nmss = maximum . map sum . segments\n\n\fProofs\n\n128\n\nwhere segments returns a list of all segments of a list. This function can be de\ufb01ned\nin a number of ways, including\nsegments = concat . map inits . tails\nwhere tails is dual to inits and returns all the tail segments of a list:\ntails :: [a] -> [[a]]\ntails []\n= [[]]\ntails (x:xs) = (x:xs):tails xs\nThe de\ufb01nition of segments describes the process of taking all the initial segments\nof all the tail segments. For example,\nghci> segments \"abc\"\n[\"\",\"a\",\"ab\",\"abc\",\"\",\"b\",\"bc\",\"\",\"c\",\"\"]\nThe empty sequence appears four times in this list, once for every tail segment.\nDirect evaluation of mss will take a number of steps proportional to n3 on a list\nof length n. There are about n2 segments, and summing each of them will take n\nsteps, so in total it will take n3 steps. It is not obvious that we can do better than\ncubic time for this problem.\nHowever, let\u2019s see where some program calculation leads us. We can start by installing the de\ufb01nition of segments:\nmaximum . map sum . concat . map inits . tails\nSearching for a law we can apply, we spot that\nmap f . concat = concat . map (map f)\napplies to the subterm map sum . concat. That gives\nmaximum . concat . map (map sum) . map inits . tails\nNow we can use the law map f . map g = map (f . g) to give\nmaximum . concat . map (map sum . inits) . tails\nOh, we can also use the law\nmaximum . concat = maximum . map maximum\ncan\u2019t we? No, not unless the argument to concat is a nonempty list of nonempty\nlists, because the maximum of the empty list is unde\ufb01ned. In the present example\nthe rule is valid because both inits and tails return nonempty lists. That leads\nto\n\n\f6.6 The maximum segment sum\n\n129\n\nmaximum . map (maximum . map sum . inits) . tails\nThe next step is to use the property of scanl described in the previous section,\nnamely\nmap sum . inits =\n\nscanl (+) 0\n\n```\n\n\n```\n\nThat leads to\nmaximum . map (maximum . scanl (+) 0) . tails\nAlready we have reduced a n3 algorithm to a n2 one, so we are making progress.\nBut now we appear stuck since there is no law in our armoury that seems to help.\nThe next step obviously concerns maximum . scanl (+) 0. So, let\u2019s see what\nwe can prove about\nfoldr1 max . scanl (+) 0\nThis looks like a fusion rule, but can scanl (+) 0 be expressed as a foldr? Well,\nwe do have, for instance,\n=\n=\n=\n=\n\nscanl (+) 0 [x,y,z]\n[0,0+x,(0+x)+y,((0+x)+y)+z]\n[0,x,x+y,x+y+z]\n0:map (x+) [0,y,y+z]\n0:map (x+) (scanl (+) 0 [y,z])\n\nThis little calculation exploits the associativity of (+) and the fact that 0 is the\nidentity element of (+). The result suggests, more generally, that\nscanl (@) e = foldr f [e]\nwhere f x xs = e:map (x@) xs\nprovided that (@) is associative with identity e. Let us take this on trust and move\non to the conditions under which\nfoldr1 (<>) . foldr f [e] = foldr h b\nwhere f x xs = e:map (x@) xs\nIt is immediate that foldr1 (<>) is strict and foldr1 (<>) [e] = e, so we\nhave b = e. It remains to check the third proviso of the fusion rule: we require h\nto satisfy\nfoldr1 (<>) (e:map (x@) xs) = h x (foldr1 (<>) xs)\nfor all x and xs. The left-hand side simpli\ufb01es to\n\n\fProofs\n\n130\n\ne <> (foldr1 (<>) (map (x@) xs))\nTaking the singleton case xs = [y], we \ufb01nd that\nh x y = e <> (x @ y)\nThat gives us our de\ufb01nition of h, but we still have to check that\nfoldr1 (<>) (e:map (x@) xs) = e <> (x @ foldr1 (<>) xs)\nSimplifying both sides, this equation holds provided\nfoldr1 (<>) . map (x@) = (x@) . foldr1 (<>)\nThis \ufb01nal equation holds provided (@) distributes over (<>); that is\nx @ (y <> z) = (x @ y) <> (x @ z)\nThe proof is left as an exercise.\nDoes addition distribute over (binary) maximum? Yes:\nx + (y `max` z) = (x + y) `max` (x + z)\nx + (y `min` z) = (x + y) `min` (x + z)\nBack to the maximum segment sum. We have arrived at\nmaximum . map (foldr (@) 0) . tails\nwhere x @ y = 0 `max` (x + y)\nWhat we have left looks very like an instance of the scanl rule of the previous\nsection, except that we have a foldr not a foldl and a tails not an inits. But\na similar calculation to the one about scanl reveals\nmap (foldr f e) . tails = scanr f e\nwhere\nscanr :: (a -> b -> b) -> b -> [a] -> [b]\nscanr f e []\n= [e]\nscanr f e (x:xs) = f x (head ys):ys\nwhere ys = scanr f e xs\nThe function scanr is also de\ufb01ned in the standard prelude. In summary,\nmss = maximum . scanr (@) 0\nwhere x @ y = 0 `max` (x + y)\nThe result is a linear-time program for the maximum segment sum.\n\n\f6.7 Exercises\n\n131\n\n6.7 Exercises\nExercise A\nIn Chapter 3 we de\ufb01ned multiplication on natural numbers. The following de\ufb01nition is slightly different:\nmult :: Nat -> Nat -> Nat\nmult Zero y\n= Zero\nmult (Succ x) = mult x y + y\nProve that mult (x+y) z = mult x z + mult y z. You can use only the facts\nthat x+0 = x and that (+) is associative. That means a long think about which\nvariable x, y or z is the best one on which to do the induction.\nExercise B\nProve that\nreverse (xs ++ ys) = reverse ys ++ reverse xs\nfor all \ufb01nite lists xs and ys. You may assume that (++) is associative.\nExercise C\nRecall our friends Eager Beaver and Lazy Susan from Exercise D in Chapter 2.\nSusan happily used the expression head . map f, while Beaver would probably\nprefer f . head. Wait a moment! Are these two expressions equal? Carry out an\ninduction proof to check.\nExercise D\nRecall the cartesian product function cp :: [[a]] -> [[a]] from the previous\nchapter. Give a de\ufb01nition of the form cp = foldr f e for suitable f and e. You\ncan use a list comprehension for the de\ufb01nition of f if you like.\nThe rest of this exercise concerns the proof of the identity\nlength . cp = product . map length\nwhere product returns the result of multiplying a list of numbers.\n1. Using the fusion theorem, express length.cp as an instance of foldr.\n2. Express map length as an instance of foldr.\n\n\fProofs\n\n132\n\n3. Using the fusion theorem again, express product . map length as an instance of foldr.\n4. Check that the two results are identical. If they aren\u2019t, your de\ufb01nition of cp was\nwrong.\nExercise E\nThe \ufb01rst two arguments of foldr are replacements for the constructors\n(:) :: a -> [a] -> [a]\n[] :: [a]\nof lists. A fold function can be de\ufb01ned for any data type: just give replacements for\nthe constructors of the data type. For example, consider\ndata Either a b = Left a | Right b\nTo de\ufb01ne a fold for Either we have to give replacements for\nLeft :: a -> Either a b\nRight :: b -> Either a b\nThat leads to\nfoldE :: (a -> c) -> (b -> c) -> Either a b -> c\nfoldE f g (Left x) = f x\nfoldE f g (Right x) = g x\nThe type Either is not a recursive data type and foldE is not a recursive function.\nIn fact foldE is a standard prelude function, except that it is called either not\nfoldE.\nNow de\ufb01ne fold functions for\ndata Nat = Zero | Succ Nat\ndata NEList a = One a | Cons a (NEList a)\nThe second declaration introduces nonempty lists.\nWhat is wrong with the Haskell de\ufb01nition of foldr1?\nExercise F\nProve that\nfoldl f e xs = foldr (flip f) e (reverse xs)\nfor all \ufb01nite lists xs. Also prove that\n\n\f6.7 Exercises\n\n133\n\nfoldl (@) e xs = foldr (<>) e xs\nfor all \ufb01nite lists xs, provided that\n(x <> y) @ z = x <> (y @ z)\ne @ x\n= x <> e\nExercise G\nUsing\nfoldl f e (xs ++ ys) = foldl f (foldl f e xs) ys\nfoldr f e (xs ++ ys) = foldr f (foldr f e ys) xs\nprove that\nfoldl f e . concat = foldl (foldl f) e\nfoldr f e . concat = foldr (flip (foldr f)) e\nExercise H\nMathematically speaking, what is the value of\nsum (scanl (/) 1 [1..])\n\n?\n\nExercise I\nCalculate the ef\ufb01cient de\ufb01nition of scanr from the speci\ufb01cation\nscan r f e = map (foldr f e) . tails\nExercise J\nConsider the problem of computing\nmss :: [Int] -> Int\nmss = maximum . map sum . subseqs\nwhere subseqs returns all the subsequences of a \ufb01nite list, including the list itself:\nsubseqs :: [a] -> [[a]]\nsubseqs []\n= [[]]\nsubseqs (x:xs) = xss ++ map (x:) xss\nwhere xss = subseqs xs\nFind a more ef\ufb01cient alternative for mss.\n\n\fProofs\n\n134\n\nExercise K\nThis question is in pieces.\n1. The function takePrefix p applied to a list xs returns the longest initial segment of xs that satis\ufb01es p. Hence\ntakePrefix :: ([a] -> Bool) -> [a] -> [a]\nWhat are the values of the following expressions?\ntakePrefix nondec [1,3,7,6,8,9]\ntakePrefix (all even) [2,4,7,8]\nComplete the right-hand side of\ntakePrefix (all p) = ...\nGive a de\ufb01nition of takePrefix in terms of standard functions, including\ninits.\nWe will return to takePrefix in the \ufb01nal part of this question.\n2. The functions one and none are de\ufb01ned by the equations\none x = [x]\nnone x = []\nComplete the right-hand side of the following identities:\nnone . f\n= ...\nmap f . none = ...\nmap f . one = ...\n3. Recall that fork (f,g) x = (f x,g x). Complete the identities\nfst . fork (f,g) = ...\nsnd . fork (f,g) = ...\nfork (f,g) . h\n= ...\n4. De\ufb01ne\ntest p (f,g) x = if p x then f x else g x\nComplete the right-hand sides of\ntest p (f,g) . h\nh . test p (f,g)\n\n= ...\n= ...\n\nThe function filter can be de\ufb01ned by\n\n\f6.8 Answers\n\n135\n\nfilter p = concat . map (test p (one,none))\nUsing the identities above, together with other standard identities, prove using\nequational reasoning that\nfilter p = map fst . filter snd . map (fork (id,p))\n(Hint: as always in calculations, start with the more complicated side.)\n5. Recall the standard prelude functions curry and uncurry from the answer to\nExercise K in Chapter 4:\ncurry :: ((a,b) -> c) -> a -> b -> c\ncurry f x y = f (x,y)\nuncurry :: (a -> b -> c) -> (a,b) -> c\nuncurry f (x,y) = f x y\nComplete the right-hand side of\nmap (fork (f,g)) = uncurry zip . (??)\n6. Returning to takePrefix, use equational reasoning to calculate an ef\ufb01cient\nprogram for the expression\ntakePrefix (p . foldl f e)\nthat requires only a linear number of applications of f .\n\n6.8 Answers\nAnswer to Exercise A\nThe proof is by induction on y:\nCase 0\nmult (x+0) z\n=\n\n{since x + 0=x}\n\nmult x z + mult 0 z\n=\n\nmult x z\n\n{mult.1}\nmult x z + 0\n\n=\n\n{since x + 0 = x}\nmult x z\n\n\fProofs\n\n136\n\nCase y+1\nmult (x+(y+1)) z\n\nmult x z + mult (y+1) z\n\n{as (+) is associative}\n\n=\n\nmult ((x+y)+1) z\n\nmult x z + (mult y z + z)\n\n{mult.2}\n\n=\n\n{mult.2}\n\n=\n\n{since (+) is associative}\n\n=\n\nmult (x+y) z + z\n\n(mult x z + mult y z) + z\n\n{induction}\n\n=\n\n(mult x z + mult y z) + z\nAnswer to Exercise B\nThe proof is by induction on xs:\nCase []\nreverse ([]++ys)\n=\n\n{++.1}\n\nreverse ys ++ reverse []\n=\n\nreverse ys\n\n{reverse.1}\nreverse ys ++ []\n\n=\n\n{since xs ++ [] = xs}\nreverse ys\n\nCase x:xs\nreverse ((x:xs)++ys)\n=\n\n{++.2}\nreverse (x:(xs++ys))\n\n=\n\n{reverse.2}\nreverse (xs++ys) ++ [x]\n\n=\n\n{induction}\n(reverse ys ++ reverse xs) ++ [x]\n\nand\nreverse ys ++ reverse (x:xs)\n=\n\n{reverse.2}\nreverse ys ++ (reverse xs ++ [x])\n\n=\n\n{since (++) is associative}\n(reverse ys ++ reverse xs) ++ [x]\n\n\f6.8 Answers\n\n137\n\nAnswer to Exercise C\nWe have to prove that\nhead (map f xs) = f (head xs)\nfor all lists xs, \ufb01nite, partial or in\ufb01nite. The case undefined and the inductive case\nx:xs are okay, but the case [] gives\nhead (map f []) = head [] = undefined\nf (head [])\n= f undefined\nHence the law holds only if f is a strict function. Eager Beaver is not bothered by\nthis since he can only construct strict functions.\nAnswer to Exercise D\nWe have\ncp = foldr op [[]]\nwhere op xs xss = [x:ys | x <- xs, ys <- xss]\n1. length . cp = foldr h b provided length is strict (it is) and\nlength [[]] = b\nlength (op xs xss) = h xs (length xss)\nThe \ufb01rst equation gives b = 1 and as\nlength (op xs xss) = length xs * length xss\nthe second equation gives h = (*) . length.\n2. map length = foldr f [], where f xs ns = length xs:ns. A shorter\nde\ufb01nition is f = (:) . length.\n3. product . map length = foldr h b provided product is strict (it is) and\nproduct [] = b\nproduct (length xs:ns) = h xs (product ns)\nThe \ufb01rst equation gives b = 1, and as\nproduct (length xs:ns) = length xs * product ns\nthe second equation gives h = (*) . length.\n4. The two de\ufb01nitions of h and b are identical.\n\n\fProofs\n\n138\n\nAnswer to Exercise E\nThe de\ufb01nition of foldN is straightforward:\nfoldN :: (a -> a) -> a -> Nat -> a\nfoldN f e Zero\n= e\nfoldN f e (Succ n) = f (foldN f e n)\nIn particular,\nm+n = foldN Succ m n\nm*n = foldN (+m) Zero n\nm^n = foldN (*m) (Succ Zero) n\nFor nonempty lists, the de\ufb01nition of foldNE is:\nfoldNE :: (a -> b -> b) -> (a -> b) -> NEList a -> b\nfoldNE f g (One x)\n= g x\nfoldNE f g (Cons x xs) = f x (foldNE f g xs)\nTo be a proper fold over nonempty lists, the correct de\ufb01nition of foldr1 should\nhave been\nfoldr1 :: (a -> b -> b) -> (a -> b) -> [a] -> b\nfoldr1 f g [x]\n= g x\nfoldr1 f g (x:xs) = f x (foldr1 f g xs)\nThe Haskell de\ufb01nition of foldr1 restricts g to be the identity function.\nAnswer to Exercise F\nWrite g = flip f for brevity. We prove that\nfoldl f e xs = foldr g e (reverse xs)\nfor all \ufb01nite lists xs by induction:\nCase []\nfoldl f e []\n{foldl.1}\n\n=\n\nfoldl g e (reverse [])\n{reverse.1}\n\n=\n\ne\n\nfoldl g e []\n{foldl.1}\n\n=\ne\n\n\f6.8 Answers\n\n139\n\nCase x:xs\nfoldl f e (x:xs)\n=\n\n{foldl.2}\nfoldl f (f e x) xs\n\n=\n\n{induction}\nfoldr g (f e x) (reverse xs)\n\nand\nfoldr g e (reverse (x:xs))\n{reverse.2}\n\n=\n\nfoldr g e (reverse xs ++ [x])\n{claim: see below}\n\n=\n\nfoldr g (foldr g e [x]) (reverse xs)\n{since foldr (flip f) e [x] = f e x}\n\n=\n\nfoldr g (f e x) (reverse xs)\nThe claim is that\nfoldr f e (xs ++ ys) = foldr f (foldr f e ys) xs\nWe leave the proof to the reader. By the way, we have the companion result that\nfoldl f e (xs ++ ys) = foldl f (foldl f e xs) ys\nAgain, the proof is left to you.\nWe prove\nfoldl (@) e xs = foldr (<>) e xs\nfor all \ufb01nite lists xs by induction. The base case is trivial. For the inductive case:\nCase x:xs\nfoldl (@) e (x:xs)\n=\n\n{foldl.2}\n\nfoldr (<>) e (x:xs)\n=\n\nfoldl (@) (e @ x) xs\n=\n\n{given that e @ x = x <> e}\nfoldl (@) (x <> e) xs\n\n{foldr.2}\nx <> foldr (<>) e xs\n\n=\n\n{induction}\nx <> foldl (@) e xs\n\n\fProofs\n\n140\n\nThe two sides have simpli\ufb01ed to different results. We need another induction hypothesis:\nfoldl (@) (x <> y) xs = x <> foldl (@) y xs\nThe base case is trivial. For the inductive case\nCase z:zs\nfoldl (@) (x <> y) (z:zs)\n=\n\n{foldl.2}\nfoldl (@) ((x <> y) @ z) zs\n\n=\n\n{since (x <> y) @ z = x <> (y @ z)}\nfoldl (@) (x <> (y @ z)) zs\n\n=\n\n{induction}\nx <> foldl (@) (y @ z) zs\n\nand\nx <> foldl (@) y (z:zs)\n=\n\n{foldl.2}\nx <> foldl (@) (y @ z) zs\n\nAnswer to Exercise G\nThe proofs are by induction. The base cases are easy and the inductive cases are\nfoldl f e (concat (xs:xss))\n=\n\n{de\ufb01nition of concat}\nfoldl f e (xs ++ concat xss)\n\n=\n\n{given property of foldl}\nfoldl f (foldl f e xs) (concat xss)\n\n=\n\n{induction}\nfoldl (foldl f) (foldl f e xs) xss\n\n=\n\n{de\ufb01nition of foldl}\nfoldl (foldl f) e (xs:xss)\n\n\f6.8 Answers\n\n141\n\nand\nfoldr f e (concat (xs:xss))\n=\n\n{de\ufb01nition of concat}\nfoldr f e (xs ++ concat xss)\n\n=\n\n{given property of foldr}\nfoldr f (foldr f e (concat xss)) xs\n\n=\n\n{using flip}\nflip (foldr f) xs (foldr f e (concat xss))\n\n=\n\n{induction}\nflip (foldr f) xs (foldr (flip (foldr f)) e xss)\n\n=\n\n{de\ufb01nition of foldr}\nfoldr (flip (foldr f)) e (xs:xss)\n\nAnswer to Exercise H\nMathematically speaking,\nsum (scanl (/) 1 [1..]) = e\nsince \u2211\u221e\nn=0 1/n! = e. Computationally speaking, replacing [1..] by a \ufb01nite list\n[1..n] gives an approximation to e. For example,\nghci> sum (scanl (/) 1 [1..20])\n2.7182818284590455\nghci> exp 1\n2.718281828459045\nThe standard prelude function exp takes a number x and returns ex . By the way, the\nprelude function log takes a number x and returns loge x. If you want logarithms\nin another base, use logBase whose type is\nlogBase :: Floating a => a -> a -> a\n\n\n```\n\n\n```\n\nAnswer to Exercise I\nWe synthesise a more ef\ufb01cient de\ufb01nition by cases. The base case yields\nscanr f e [] = [e]\n\n\fProofs\n\n142\n\nand the inductive case x:xs is:\nscanr f e (x:xs)\n=\n\n{speci\ufb01cation}\nmap (foldr f e) (tails (x:xs))\n\n=\n\n{tails.2}\nmap (foldr f e) ((x:xs):tails xs)\n\n=\n\n{de\ufb01nition of map}\nfoldr f e (x:xs):map (foldr f e) (tails xs)\n\n=\n\n{foldr.2 and speci\ufb01cation}\nf x (foldr f e xs):scan f e xs\n\n=\n\n{claim: foldr f e xs = head (scanr f e xs)}\nf x (head ys):ys where ys = scanr f e xs\n\nAnswer to Exercise J\nFirstly,\nsubseqs = foldr op [[]]\nwhere op x xss = xss ++ map (x:) xss\nAppeal to the fusion law yields\nmap sum . subseqs = foldr op [0]\nwhere op x xs = xs ++ map (x+) xs\nA second appeal to fusion yields\nmaximum . map sum . subseqs = foldr op 0\nwhere op x y = y `max` (x+y)\nThat will do nicely. Of course, sum . filter (>0) also does the job.\nAnswer to Exercise K\n1. We have\ntakePrefix nondec [1,3,7,6,8,9] = [1,3,7]\ntakePrefix (all even) [2,4,7,8] = [2,4]\nThe identity is\ntakePrefix (all p) = takeWhile p\n\n\f6.8 Answers\n\nThe speci\ufb01cation is\ntakePrefix p = last . filter p . inits\n2. We have\nnone . f\n= none\nmap f . none = none\nmap f . one = one . f\n3. We have\nfst . fork (f,g) = f\nsnd . fork (f,g) = g\nfork (f,g) . h\n= fork (f.h,g.h)\n4. We have\ntest p (f,g) . h = test (p.h) (f . h, g . h)\nh . test p (f,g) = test p (h . f, h . g)\nThe reasoning is:\nmap fst . filter snd . map (fork (id,p))\n{de\ufb01nition of filter}\n\n=\n\nmap fst . concat . map (test snd (one,none)) .\nmap (fork (id,p))\n{since map f . concat = concat . map (map f)}\n\n=\n\nconcat . map (map fst . test snd (one,none) .\nfork (id,p))\n{second law of test; laws of one and none}\n\n=\n\nconcat . map (test snd (one . fst,none) .\nfork (id,p))\n{\ufb01rst law of test; laws of fork}\n\n=\n\nconcat . map (test p (one . id, none . fork (id,p)))\n{laws of id and none}\n\n=\n\nconcat . map (test p (one,none))\n{de\ufb01nition of filter}\n\n=\n\nfilter p\n5. We have\n\n143\n\n\fProofs\n\n144\n\nmap (fork (f,g))\n\n= uncurry zip . fork (map f,map g)\n\n6. We have\nfilter (p . foldl f e) . inits\n=\n\n{derived law of filter}\nmap fst . filter snd .\nmap (fork (id, p . foldl f e)) . inits\n\n=\n\n{law of zip}\nmap fst . filter snd . uncurry zip .\nfork (id, map (p . foldl f e)) . inits\n\n=\n\n{law of fork}\nmap fst . filter snd . uncurry zip .\nfork (inits, map (p . foldl f e) . inits)\n\n=\n\n{scan lemma}\nmap fst . filter snd . uncurry zip .\nfork (inits, map p . scanl f e)\n\nHence\ntakePrefix (p.foldl f e)\n= fst . last . filter snd . uncurry zip .\nfork (inits,map p . scanl f e)\n\n6.9 Chapter notes\nGofer, an earlier version of Haskell designed by Mark Jones, was so named because\nit was GOod For Equational Reasoning. HUGS (The Haskell Users Gofer System)\nwas an earlier alternative to GHCi, and used in the second edition of the book on\nwhich the current one is based, but is no longer maintained.\nMany people have contributed to the understanding of the laws of functional programming, too many to list. The Haskellwiki page\nhaskell.org/haskellwiki/Equational_reasoning_examples\ncontains examples of equational reasoning and links to various discussions about\nthe subject.\nThe fascinating history of the maximum segment sum problem is discussed in Jon\nBentley\u2019s Programming Pearls (second edition) (Addison-Wesley, 2000).\n\n\fChapter 7\nE\ufb03ciency\n\nThe question of ef\ufb01ciency has been an ever-present undercurrent in recent discussions, and the time has come to bring this important subject to the surface. The best\nway to achieve ef\ufb01ciency is, of course, to \ufb01nd a decent algorithm for the problem.\nThat leads us into the larger topic of Algorithm Design, which is not the primary\nfocus of this book. Nevertheless we will touch on some fundamental ideas later\non. In the present chapter we concentrate on a more basic question: functional programming allows us to construct elegant expressions and de\ufb01nitions, but do we\nknow what it costs to evaluate them? Alan Perlis, a US computer scientist, once\ninverted Oscar Wilde\u2019s de\ufb01nition of a cynic to assert that a functional programmer\nwas someone who knew the value of everything and the cost of nothing.\n\n7.1 Lazy evaluation\nWe said in Chapter 2 that, under lazy evaluation, an expression such as\nsqr (sqr (3+4))\nwhere sqr x = x*x, is reduced to its simplest possible form by applying reduction steps from the outside in. That means the de\ufb01nition of the function sqr is\ninstalled \ufb01rst, and its argument is evaluated only when needed. The following evaluation sequence follows this prescription, but is not lazy evaluation:\n=\n=\n=\n=\n\nsqr (sqr (3+4))\nsqr (3+4) * sqr (3+4)\n((3+4)*(3+4)) * ((3+4)*(3+4))\n...\n2401\n\n\fE\ufb03ciency\n\n146\n\nThe ellipsis in the penultimate line hides no fewer than four evaluations of 3+4 and\ntwo of 7*7. Clearly the simple policy of substituting argument expressions into\nfunction expressions is a very inef\ufb01cient way of carrying out reduction.\nInstead, lazy evaluation guarantees that when the value of an argument is needed, it\nis evaluated only once. Under lazy evaluation, the reduction sequence would unfold\nbasically as follows:\n=\n=\n=\n=\n=\n\nsqr (sqr (3+4))\nlet x = sqr (3+4) in x*x\nlet y = 3+4 in\nlet x = y*y in x*x\nlet y = 7 in\nlet x = y*y in x*x\nlet x = 49 in x*x\n2401\n\nThe expression 3+4 is evaluated only once (and so is 7*7). The names x and y have\nbeen bound to expressions using let, though in the implementation of Haskell\nthese names are anonymous pointers to expressions. When an expression is reduced to a value, the pointer then points to the value and that value can then be\nshared.\nEven then, the headline \u2018Under lazy evaluation arguments are evaluated only when\nneeded and then only once!\u2019 doesn\u2019t tell the full story. Consider evaluation of\nsqr (head xs). In order to evaluate sqr we have to evaluate its argument, but\nin order to evaluate head xs we do not have to evaluate xs all the way, but only\nto the point where it becomes an expression of the form y:ys. Then head xs can\nreturn y and sqr (head xs) can return y*y. More generally, an expression is said\nto be in head normal form if it is a function (such as sqr) or if it takes the form of\na data constructor (such as (:)) applied to its arguments. Every expression in normal form (i.e. in fully reduced form) is in head normal form but not vice versa. For\nexample, (e1,e2) is in head normal form (because it is equivalent to (,) e1 e2,\nwhere (,) is the data constructor for pairs), but is in normal form only if both e1\nand e2 are. Of course, for numbers or booleans there is no distinction between the\ntwo kinds of normal form.\n\u2018Under lazy evaluation arguments are evaluated only when needed and then only\nonce, and then maybe only to head normal form\u2019 is not as catchy a headline as\nbefore, but it does tell a better story.\nNext, consider the following two de\ufb01nitions of the inductive case of the function\nsubseqs that returns all the subsequences of a list:\n\n\f7.1 Lazy evaluation\n\n147\n\nsubseqs (x:xs) = subseqs xs ++ map (x:) (subseqs xs)\nsubseqs (x:xs) = xss ++ map (x:) xss\nwhere xss = subseqs xs\nIn the \ufb01rst de\ufb01nition the expression subseqs xs appears twice on the right-hand\nside, so it is evaluated twice when the subsequences of a given list are required. In\nthe second de\ufb01nition this duplication of effort has been recognised by the programmer and a where clause has been used to ensure that subseqs xs is evaluated only\nonce (we could also have used a let expression).\nThe important point is that you, the programmer, are in control of which de\ufb01nition you want. It is quite possible for Haskell to recognise the double occurrence\nand to abstract it away using the equivalent of an internal let expression. This\nis a well-known technique called common subexpression elimination. But Haskell\ndoesn\u2019t do this, and for a very good reason: it can cause a space leak. The second\nde\ufb01nition of subseqs (x:xs) has the following problem: the list subseqs xs is\nconstructed only once, but it is retained in its entirety in memory because its value\nis used again, namely in the second expression map (x:) xss.\nLook at it this way: the \ufb01rst de\ufb01nition takes longer because computation is duplicated; the second de\ufb01nition is faster (though still exponential) but can rapidly run\nout of available space. After all, there are 2n subsequences of a list of length n.\nThere is a fundamental dichotomy in programming we can never get away from: to\navoid doing something twice you have to use up space to store the result of doing\nit once.\nHere is a related example. Consider the following two de\ufb01nitions in a script:\nfoo1 n = sum (take n primes)\nwhere\nprimes\n= [x | x <- [2..], divisors x == [x]]\ndivisors x = [d | d <- [2..x], x `mod` d == 0]\n\nfoo2 n = sum (take n primes)\nprimes\n= [x | x <- [2..], divisors x == [x]]\ndivisors x = [d | d <- [2..x], x `mod` d == 0]\nThe programmer who wrote foo1 decided to structure their script by making the\nde\ufb01nitions of both primes and divisors local to the de\ufb01nition of foo1, presumably because neither de\ufb01nition was used elsewhere in the script. The programmer\nwho wrote foo2 decided to allow these two subsidiary de\ufb01nitions to \ufb02oat to the\nstatus of a global or top-level de\ufb01nition. You might think that doesn\u2019t make any\n\n\f148\n\nE\ufb03ciency\n\ndifference to the ef\ufb01ciency, but consider the following interaction with GHCi. (The\ncommand :set +s turns on some statistics which are printed after an expression\nis evaluated.)\nghci> :set +s\nghci> foo1 1000\n3682913\n(4.52 secs, 648420808 bytes)\nghci> foo1 1000\n3682913\n(4.52 secs, 648412468 bytes)\nghci> foo2 1000\n3682913\n(4.51 secs, 647565772 bytes)\nghci> foo2 1000\n3682913\n(0.02 secs, 1616096 bytes)\nWhy was the second evaluation of foo2 1000 so much faster than the \ufb01rst, while\nthe two evaluations of foo1 1000 took the same time?\nThe answer is that in the de\ufb01nition of foo2 the \ufb01rst 1000 elements of the list\nprimes is demanded, so after evaluation primes now points to a list in which the\n\ufb01rst 1000 primes appear explicitly. The second evaluation of foo 1000 does not\nrequire these primes to be computed again. Internally, the script has grown in size\nbecause primes now occupies at least 1000 units of space.\nProgrammer Three chooses to write foo in the following way:\nfoo3 = \\n -> sum (take n primes)\nwhere\nprimes\n= [x | x <- [2..], divisors x == [x]]\ndivisors x = [d | d <- [2..x], x `mod` d == 0]\nThis uses a lambda expression to express foo3 at the function level, but otherwise\nthe de\ufb01nition is exactly the same as that of foo1. The alternative\nfoo3 = sum . flip take primes\nalso works but seems a little obscure. Now we have\nghci> foo3 1000\n3682913\n(3.49 secs, 501381112 bytes)\n\n\f7.2 Controlling space\n\n149\n\nghci> foo3 1000\n3682913\n(0.02 secs, 1612136 bytes)\nAgain, the second evaluation is much faster than the \ufb01rst. Why is that?\nTo see what is going on, we can rewrite the two functions in the form\nfoo1 n = let primes = ... in\nsum (take n primes)\nfoo3\n= let primes = ... in\n\\n -> sum (take n primes)\nNow you can appreciate that in the \ufb01rst de\ufb01nition primes is re-evaluated every\ntime foo1 1000 is called because it is bound to an application of foo1 not to\nthe function itself. It is theoretically possible that the local de\ufb01nitions in the \ufb01rst\nde\ufb01nition depend on n, so any such de\ufb01nitions have to be re-evaluated for each n.\nIn the second de\ufb01nition the local de\ufb01nitions are bound to the function itself (and\ncan\u2019t possibly depend on any argument to the function); consequently, they are\nevaluated only once. Of course, after evaluating foo3 1000, the local de\ufb01nition of\nprimes will be expanded to an explicit list of 1000 elements followed by a recipe\nfor evaluating the rest.\n\n7.2 Controlling space\nSuppose we de\ufb01ne sum by sum = foldl (+) 0. Under lazy evaluation the expression sum [1..1000] is reduced as follows\n=\n=\n=\n=\n=\n=\n=\n=\n\nsum [1..1000]\nfoldl (+) 0 [1..1000]\nfoldl (+) (0+1) [2..1000]\nfoldl (+) ((0+1)+2) [3..1000]\n...\nfoldl (+) (..((0+1)+2)+ ... +1000) []\n(..((0+1)+2)+ ... +1000)\n...\n500500\n\nIt requires 1000 units of space just to build up the arithmetic expression that sums\nthe \ufb01rst 1000 numbers before it pops to the surface and is \ufb01nally evaluated.\nMuch better is to use a mixture of lazy and eager evaluation:\n\n\fE\ufb03ciency\n\n150\n\n=\n=\n=\n=\n=\n=\n=\n=\n\nsum [1..1000]\nfoldl (+) 0 [1..1000]\nfoldl (+) (0+1) [2..1000]\nfoldl (+) 1 [2..1000]\nfoldl (+) (1+2) [3..1000]\nfoldl (+) 3 [3..1000]\n...\nfoldl (+) 500500 []\n500500\n\nWhile the list expression [1..1000] is evaluated lazily, the second argument of\nfoldl, the accumulated sum, is evaluated eagerly. The result of interleaving lazy\nand eager evaluation steps is a sequence that uses a constant amount of space.\nThis suggests that it would be useful to have some way of controlling the reduction\norder. Such a method is provided by a primitive function seq with type\nseq :: a -> b -> b\nEvaluation of x `seq` y proceeds by \ufb01rst evaluating x (to head normal form) and\nthen returning the result of evaluating y. If evaluation of x does not terminate, then\nneither does x `seq` y. It\u2019s not possible to de\ufb01ne seq in Haskell; instead Haskell\nprovides it as a primitive function.\nNow consider the following version foldl' of foldl that evaluates its second\nargument strictly:\nfoldl' :: (b -> a -> b) -> b -> [a] -> b\nfoldl' f e []\n= e\nfoldl' f e (x:xs) = y `seq` foldl' f y xs\nwhere y = f e x\nHaskell provides the function foldl' in the standard prelude (yes, with just this\nunimaginative name). Now we can de\ufb01ne sum = foldl' (+) 0, with the consequence that evaluation proceeds in constant space. In fact, sum is another prelude\nfunction with essentially this de\ufb01nition.\nIs it the case that foldl is now redundant and can be replaced by the new improved\nfoldl'? The answer is in practice yes, but in theory no. It is possible to construct\nf, e and xs such that\nfoldl f e xs = foldl' f e xs\nHowever, when f is strict (recall that f is strict if f \u22a5=\u22a5) the two expressions do\nreturn the same result. The exercises go into details.\n\n\f7.2 Controlling space\n\n151\n\nTaking the mean\nArmed with the above information, let\u2019s now consider a very instructive example:\nhow to compute the average or mean of a list of numbers. Surely that is an easy\nproblem, you might think, just divide the sum of the list by the length of the list:\nmean :: [Float] -> Float\nmean xs = sum xs / length xs\nThere are lots of things wrong with this de\ufb01nition, not the least of which is that\nthe expression on the right is not well-formed! The function length in Haskell has\ntype [a] -> Int and we can\u2019t divide a Float by an Int without performing an\nexplicit conversion.\nThere is a function in the standard prelude that comes to our aid:\nfromIntegral :: (Integral a, Num b) => a -> b\nfromIntegral = fromInteger . toInteger\nRecall from Chapter 3 the two conversion functions\ntoInteger\n:: (Integral a) => a -> Integer\nfromInteger :: (Num a) => Integer -> a\nThe \ufb01rst converts any integral type to an integer, and the second converts an integer\nto a number. Their composition converts an integral number, such as Int, to a more\ngeneral kind of number, such as Float.\nWe can now rewrite mean to read\nmean :: [Float] -> Float\nmean xs = sum xs / fromIntegral (length xs)\nThe second thing wrong with this de\ufb01nition is that it silently ignores the case of the\nempty list. What is 0/0? Either we should identify the failing case with an explicit\nerror message, or else adopt one common convention, which is to agree that the\nmean of the empty list should be zero:\nmean [] = 0\nmean xs = sum xs / fromIntegral (length xs)\nNow we are ready to see what is really wrong with mean: it has a space leak. Evaluating mean [1..1000] will cause the list to be expanded and retained in memory\nafter summing because there is a second pointer to it, namely in the computation\nof its length.\n\n\fE\ufb03ciency\n\n152\n\nWe can replace the two traversals of the list by one, using a strategy of program\noptimisation called tupling. The idea is simple enough in the present example:\nde\ufb01ne sumlen by\nsumlen :: [Float] -> (Float,Int)\nsumlem xs = (sum xs,length xs)\nand then calculate an alternative de\ufb01nition that avoids the two traversals. It is easy\nto carry out the calculation and we just state the result:\nsumlen []\n= (0,0)\nsumlen (x:xs) = (s+x,n+1)\n\nwhere (s,n) = sumlen xs\n\nThe pattern of the de\ufb01nition of sumlen should be familiar by now. An alternative\nde\ufb01nition is\nsumlen = foldr f (0,0)\n\nwhere f x (s,n) = (s+x,n+1)\n\nEven better, we can replace foldr f by foldl g, where\ng (s,n) x = (s+x,n+1)\nThe justi\ufb01cation of this step is the law in the previous chapter that said\nfoldr f e xs = foldl g e xs\nfor all \ufb01nite lists xs, provided\nf x (g y z) = g (f x y) z\nf x e = g e x\nThe veri\ufb01cation of these two conditions is left as an exercise.\nAnd that means we can use foldl':\nsumlen = foldl' g (0,0)\n\nwhere g (s,n) x = (s+x,n+1)\n\nNow we can replace our heavily criticised de\ufb01nition of mean by\nmean [] = 0\nmean xs = s / fromIntegral n\nwhere (s,n) = sumlen xs\nSurely we have now achieved our goal of a constant-space computation for mean?\nUnfortunately not. The problem is with sumlen and it is a little tricky to spot.\nExpanding the de\ufb01nition out a little, we \ufb01nd\n\n\f7.2 Controlling space\n\n153\n\nfoldl' f (s,n) (x:xs) = y `seq` foldl' f y xs\nwhere y = (s+x,n+1)\nAh, but y `seq` z reduces y to head normal form and the expression (s+x,n+1)\nis already in head normal form. Its two components are not evaluated until the end\nof the computation. That means we have to dig deeper with our seqs and rewrite\nsumlen in the following way:\nsumlen = foldl' f (0,0)\nwhere f (s,n) x = s `seq` n `seq` (s+x,n+1)\nFinally, everything in the garden is rosy and we have a computation that runs in\nconstant space.\n\nTwo more application operators\nFunction application is the only operation not denoted by any visible sign. However, Haskell provides two more application operators, ($) and ($!):\ninfixr 0\n($),($!)\nf $ x =\nf $! x =\n\n$,$!\n:: (a -> b) -> a -> b\nf x\nx `seq` f x\n\nThe only difference between f x and f $! x is that in the second expression the\nargument x is evaluated before f is applied. The only difference between f x and\nf $ x is that ($) (and also ($!)) is declared to have the lowest binding power\nof 0 and to associate to the right in expressions. That is exactly what the \ufb01xity\ndeclaration in the \ufb01rst line provides. Why do we want that?\nThe answer is that we can now write, for example\nprocess1 $ process2 $ process3 input\ninstead of having to write either of\nprocess1 (process2 (process3 x))\n(process1 . process2 . process3) x\nIt is undeniable that ($) can be quite useful on occasions, especially when submitting expressions for evaluation with GHCi, so it\u2019s worth mentioning its existence.\nAnd the strict application operator ($!) is useful for the reasons discussed above.\n\n\f154\n\nE\ufb03ciency\n\n7.3 Controlling time\nWe have seen that having an \u2018eager\u2019 button on our dashboard is a very simple\nway of controlling the space involved in driving a computation, but what about\ntime? Unfortunately there is no analogous button for speeding up computations;\ninstead we have to understand some of the things that can unintentionally slow\ndown a computation. The Haskell platform comes with documentation on GHC,\nwhich contains useful advice on how to make your program run more quickly. The\ndocumentation makes three key points:\n\u2022 Make use of GHC\u2019s pro\ufb01ling tools. There is no substitute for \ufb01nding out where\nyour program\u2019s time and space is really being used up. We will not discuss pro\ufb01ling in this book, but it is important to mention that such tools are available.\n\u2022 The best way to improve a program\u2019s performance is to use a better algorithm.\nWe mentioned this point at the beginning of the chapter.\n\u2022 It is far better to use library functions that have been Seriously Tuned by Someone Else, than to craft your own. You might be able to write a better sorting\nalgorithm than the one provided in Data.List, but it will take you longer than\njust writing import Data.List (sort). This is particularly true when you\nuse GHCi because GHCi loads compiled versions of the functions in its standard libraries. Compiled functions typically run about an order of magnitude\nfaster than interpreted ones.\nMuch of the detailed advice in the GHC documentation is beyond the scope of this\nbook, but two tips can be explained here. Firstly, the management of lazy evaluation\ninvolves more overheads than eager evaluation, so that if you know that a function\u2019s\nvalue will be needed, it is better to push the eager button. As the documentation\nsays: \u2018Strict functions are your dear friends\u2019.\nThe second piece of advice is about types. Firstly, Int arithmetic is faster than\nInteger arithmetic because Haskell has to perform more work in handling potentially very large numbers. So, use Int rather than Integer whenever it is safe\nto do so. Secondly, there is less housekeeping work for Haskell if you tailor the\ntype of your function to the instance you want. For example, consider the type of\nfoo1, de\ufb01ned in Section 7.1. There we did not provide a type signature for foo1\n(or indeed for any of the other related functions) and that was a mistake. It turns\nout that\nfoo1 :: Integral a => Int -> a\n\n\f7.3 Controlling time\n\n155\n\nIf we are really interested in the sum of the \ufb01rst n prime numbers, it is better to\ndeclare the type of foo1 to be (say)\nfoo1 :: Int -> Integer\nWith this more specialised de\ufb01nition Haskell does not have to carry around a dictionary of the methods and instances of the type class Integral, and that lightens\nthe load.\nThese pieces of advice can help shave off constant amounts of time and do not\naffect asymptotic time complexity, the order of magnitude of the timing function.\nBut sometimes we can write code that is inadvertently less ef\ufb01cient asymptotically\nthan we intended. Here is an instructive example. Consider the cartesian product\nfunction cp discussed in Chapter 5:\ncp []\n= [[]]\ncp (xs:xss) = [x:ys | x <- xs, ys <- cp xss]\nPretty and clear enough you would think, but compare it with\ncp' = foldr op [[]]\nwhere op xs yss = [x:ys | x <- xs, ys <- yss]\nThe \ufb01rst version is a direct recursive de\ufb01nition, while the second uses foldr to\nencapsulate the pattern of the recursion. The two \u2018algorithms\u2019 are the same, aren\u2019t\nthey? Well,\nghci> sum $ map sum $ cp [[1..10] | j <- [1..6]]\n33000000\n(12.11 secs, 815874256 bytes)\nghci> sum $ map sum $ cp' [[1..10] | j <- [1..6]]\n33000000\n(4.54 secs, 369640332 bytes)\nThe expression sum $ map sum is there just to force complete evaluation of the\ncartesian product. Why is the \ufb01rst computation three times slower than the second?\nTo answer this question, look at the translation that eliminates the list comprehension in the \ufb01rst de\ufb01nition:\ncp [] = [[]]\ncp (xs:xss) = concat (map f xs)\nwhere f x = [x:ys | ys <- cp xss]\nNow we can see that cp xss is evaluated each time f is applied to elements of\nxs. That means, in the examples above, that cp is evaluated many more times in\n\n\f156\n\nE\ufb03ciency\n\nthe \ufb01rst example than in the second. We cannot be more precise at this point, but\nwill be below when we develop a little calculus for estimating running times. But\nthe issue should be clear enough: the simple recursive de\ufb01nition of cp has led us\ninadvertently into a situation in which more evaluations are carried out than we\nintended.\nOne other way to get a more ef\ufb01cient cartesian product is to just write\ncp []\n= [[]]\ncp (xs:xss) = [x:ys | x <- xs, ys <- yss]\nwhere yss = cp xss\nThis de\ufb01nition has exactly the same ef\ufb01ciency as the one in terms of foldr. The\nlesson here is that innocent-looking list comprehensions can hide the fact that some\nexpressions, though only written once, are evaluated multiple times.\n\n7.4 Analysing time\nGiven the de\ufb01nition of a function f we will write T(f)(n) to denote an asymptotic\nestimate of the number of reduction steps required to evaluate f on an argument of\n\u2018size\u2019 n in the worst case. Moreover, for reasons explained in a moment, we will\nassume eager, not lazy, evaluation as the reduction strategy involved in de\ufb01ning T.\nThe de\ufb01nition of T requires some ampli\ufb01cation. Firstly, T(f) refers to the complexity of a given de\ufb01nition of f. Time complexity is a property of an expression,\nnot of the value of that expression.\nSecondly, the number of reduction steps does not correspond exactly to the elapsed\ntime between submitting an expression for evaluation and waiting for the answer.\nNo account is taken of the time to \ufb01nd the next subexpression to be reduced in a\npossibly large and complicated expression. For this reason the statistics facility of\nGHCi does not count reduction steps, but produces a measure of elapsed time.\nThirdly, we do not formalise the notion of size, since different measures are appropriate in different situations. For example, the cost of evaluating xs++ys is best\nmeasured in terms of (m, n), a pair describing the lengths of the two lists. In the\ncase of concat xss we could take the length of concat xss as a measure of size,\nbut if xss is a list of length m consisting of lists all of length n, then (m, n) might\nbe a more suitable measure.\nThe fourth and crucial remark is that T(f)(n) is determined under an eager evaluation model of reduction. The reason is simply that estimating the number of\n\n```\n\n\n```\n\n\f7.4 Analysing time\n\n157\n\nreduction steps under lazy evaluation is dif\ufb01cult. To illustrate, consider the de\ufb01nition minimum = head . sort. Under eager evaluation, the time to evaluate the\nminimum on a list of length n under this de\ufb01nition is given by\nT(minimum)(n) = T(sort)(n) + T(head)(n).\nIn other words we \ufb01rst have to completely sort a list of length n and then take the\nhead of the result (presumably a constant-time operation). This equation does not\nhold under lazy evaluation, since the number of reduction steps required to \ufb01nd the\nhead of sort xs requires only that sort xs be reduced to head normal form. How\nlong that takes depends on the precise algorithm used for sort. Timing analysis\nunder eager reduction is simpler because it is compositional. Since lazy evaluation\nnever requires more reduction steps than eager evaluation, any upper bound for\nT(f)(n) will also be an upper bound under lazy evaluation. Furthermore, in many\ncases of interest, a lower bound will also be a lower bound under lazy evaluation.\nIn order to give some examples of timing analyses we have to introduce a little\norder notation. So far, we have used the awkward phrase \u2018taking a number of steps\nproportional to\u2019 whenever ef\ufb01ciency is discussed. It is time to replace it by something shorter. Given two functions f and g on the natural numbers, we say that f\nis of order g, and write f = \u0398(g) if there are positive constants C1 and C2 and a\nnatural number n0 such that C1 g(n) \u2264 f (n) \u2264 C2 g(n) for all n > n0 . In other words,\nf is bounded above and below by some constant times g for all suf\ufb01ciently large\narguments.\nThe notation is abused to the extent that one conventionally writes, for example,\nf (n) = \u0398(n2 ) rather than the more correct f = \u0398(\u03bb n.n2 ). Similarly, one writes\nf (n) = \u0398(n) rather than f = \u0398(id). The main use of \u0398-notation is to hide constants;\nfor example, we can write\nn\n\n\u2211 j = \u0398(n2 ) and\n\nj=1\n\nn\n\n\u2211 j2 = \u0398(n3 )\n\nj=1\n\nwithout bothering about the exact constants involved. When \u0398(g) appears in a\nformula it stands for some unnamed function f satisfying f = \u0398(g). In particular,\n\u0398(1) denotes an anonymous constant.\nWith that behind us, we give three examples of how to analyse the running time of\na computation. Consider \ufb01rst the following two de\ufb01nitions of concat:\nconcat xss = foldr (++) [] xss\nconcat' xss = foldl (++) [] xss\nThe two de\ufb01nitions are equivalent provided xss is a \ufb01nite list. Suppose xss is a\n\n\f158\n\nE\ufb03ciency\n\nlist of length m of lists all of length n. Then the \ufb01rst de\ufb01nition gives\nT(concat)(m, n) = T(foldr (++) [])(m, n),\nT(foldr (++) [])(0, n) = \u0398(1),\nT(foldr (++) [])(m+1, n) = T(++)(n, mn) +\nT(foldr (++) [])(m, n).\nThe estimate T(++)(n, mn) arises because a list of length n is concatenated with a\nlist of length mn. Since T(++)(n, m) = \u0398(n), we obtain\nT(foldr (++) [])(m, n) =\n\nm\n\n\u2211 \u0398(n) = \u0398(mn).\n\nk=0\n\nFor the second de\ufb01nition of concat we have\nT(concat')(m, n) = T(foldl (++))(0, m, n),\nT(foldl (++))(k, 0, n) = O(1),\nT(foldl (++))(k, m+1, n) = T(++)(k, n) +\nT(foldl (++))(k+n, m, n).\nThe additional argument k refers to the length of the accumulated list in the second\nargument of foldl. This time we obtain\nT(foldl (++))(k, m, n) =\n\nm\u22121\n\n\u2211 \u0398(k + jn) = \u0398(k + m2 n).\n\nj=0\n\nHence T(concat')(m, n) = \u0398(m2 n). The conclusion, which was anticipated in the\nprevious chapter, is that using foldr rather than foldl in the de\ufb01nition of concat\nleads to an asymptotically faster program.\nFor the second example let us time the two programs for subseqs discussed in\nSection 7.1, where we had either of the following two possibilities:\nsubseqs (x:xs) = subseqs xs ++ map (x:) (subseqs xs)\nsubseqs' (x:xs) = xss ++ map (x:) xss\nwhere xss = subseqs' xs\nBearing in mind that (i) if xs has length n, then subseqs xs has length 2n ; and (ii)\nthe time for both the concatenation and for applying map (x:) is therefore \u0398(2n ),\nthe two timing analyses give\nT(subseqs)(n+1) = 2T(subseqs)(n) + \u0398(2n ),\nT(subseqs')(n+1) = T(subseqs')(n) + \u0398(2n )\n\n\f7.5 Accumulating parameters\n\n159\n\ntogether with T(subseqs)(0) = \u0398(1). We will just state the two solutions (which\ncan be proved by a simple induction argument):\nT(subseqs)(n) = \u0398(n2n ),\nT(subseqs')(n) = \u0398(2n ).\nThe latter is therefore asymptotically faster than the former by a logarithmic factor.\nFor the third example, let us time the two programs for cp discussed at the beginning of this section. The \ufb01rst one was\ncp []\n= [[]]\ncp (xs:xss) = [x:ys | x <- xs, ys <- cp xss]\nSuppose once again that xss is a list of length m of lists all of length n. Then the\nlength of cp xss is nm . Then we have\nT(cp)(0, n) = \u0398(1),\nT(cp)(m+1, n) = nT(cp)(m, n) + \u0398(nm ).\nbecause it takes \u0398(nm ) steps to apply (x:) to every subsequence. The solution is\nT(cp)(m, n) = \u0398(mnm ).\nOn the other hand, the de\ufb01nition of cp in terms for foldr gives\nT(cp)(0, n) = \u0398(1),\nT(cp)(m+1, n) = T(cp)(m, n) + \u0398(nm ).\nwith solution T(cp)(m, n) = \u0398(nm ). The second version is therefore asymptotically\nfaster, again by a logarithmic factor.\n\n7.5 Accumulating parameters\nSometimes we can improve the running time of a computation by adding an extra\nargument, called an accumulating parameter, to a function. The canonical example\nis the function reverse:\nreverse []\n= []\nreverse (x:xs) = reverse xs ++ [x]\nWith this de\ufb01nition we have T(reverse)(n) = \u0398(n2 ). In search of a linear-time\nprogram, suppose we de\ufb01ne\n\n\fE\ufb03ciency\n\n160\n\nrevcat :: [a] -> [a] -> [a]\nrevcat xs ys = reverse xs ++ ys\nIt is clear that reverse xs = revcat xs [], so if we can obtain an ef\ufb01cient\nversion of revcat we can obtain an ef\ufb01cient version of reverse. To this end we\ncalculate a recursive de\ufb01nition of revcat. The base case revcat [] ys = ys is\nleft as an exercise, and the inductive case is as follows:\nrevcat (x:xs) ys\n=\n\n{de\ufb01nition of revcat}\nreverse (x:xs) ++ ys\n\n=\n\n{de\ufb01nition of reverse}\n(reverse xs ++ [x]) ++ ys\n\n=\n\n{associativity of (++)}\nreverse xs ++ ([x] ++ ys)\n\n=\n\n{de\ufb01nition of (:)}\nreverse xs ++ (x:ys)\n\n=\n\n{de\ufb01nition of revcat}\nrevcat xs (x:ys)\n\nHence\nrevcat [] ys\n= ys\nrevcat (x:xs) ys = revcat xs (x:ys)\nAs to the running time, T(revcat)(m, n) = \u0398(m). In particular,\nT(reverse(n) = T(revcat(n, 0) = \u0398(n)\nThat gives a linear-time computation for reversing a list.\nHere is another example. The function length is de\ufb01ned by\nlength :: [a] -> Int\nlength []\n= 0\nlength (x:xs) = length xs + 1\nWe have T(length)(n) = \u0398(n), so there is no time advantage in calculating another de\ufb01nition. Nevertheless, de\ufb01ne lenplus by\nlenplus :: [a] -> Int -> Int\nlenplus xs n = length xs + n\n\n\f7.5 Accumulating parameters\n\n161\n\nIf we go through exactly the same calculation for lenplus as we did for revcat,\nwe arrive at\nlenplus [] n\n= n\nlenplus (x:xs) n = lenplus xs (1+n)\nThe reason the calculation goes through is that (+), like (++), is an associative\noperation. The advantage of de\ufb01ning\nlength xs = lenplus xs 0 = foldl (\\n x -> 1+n) 0 xs\nis that, by using foldl' in place of foldl, the length of a list can be computed in\nconstant space. That indeed is how length is de\ufb01ned in Haskell\u2019s prelude.\nAs the really astute reader might have spotted, there is actually no need to go\nthrough the calculations above. Both the examples are, in fact, instances of a law\nalready described in the previous chapter, namely that\nfoldr (<>) e xs = foldl (@) e xs\nfor all \ufb01nite lists xs provided\nx <> (y @ z) = (x <> y) @ z\nx <> e = e @ x\nThe two instances are:\nfoldr (\\x n -> n+1) 0 xs = foldl (\\n x -> 1+n) 0 xs\nfoldr (\\x xs -> xs++[x]) [] xs\n= foldl (\\xs x -> [x]++xs) [] xs\nWe leave the detailed veri\ufb01cation of these equations as an exercise.\nFor a \ufb01nal demonstration of the accumulating parameter technique we move from\nlists to trees. Consider the data declaration\ndata GenTree a = Node a [GenTree a]\nAn element of this type is a tree consisting of a node with a label and a list of\nsubtrees. Such trees arise in problems that can be formulated in terms of positions\nand moves. The label of a node speci\ufb01es the current position, and the number of\nsubtrees corresponds to the number of possible moves in the current position. Each\nsubtree has a label that speci\ufb01es the result of making the move, and its subtrees\ndescribe the moves that can be made from the new position. And so on.\nHere is a function for computing the list of labels in a tree:\n\n\fE\ufb03ciency\n\n162\n\nlabels :: GenTree a -> [a]\nlabels (Node x ts) = x:concat (map labels ts)\nThe method is simple enough: compute the labels of each subtree, concatenate the\nresults, and stick the label of the tree at the front of the \ufb01nal list.\nLet us analyse the running time of this program on a tree t. To keep things simple,\nsuppose that t is a perfect k-ary tree of height h. What that means is that if h = 1\nthen t has no subtrees, while if h > 1 then t has exactly k subtrees, each with height\nh\u22121. The number s(h, k) of labels in such a tree satis\ufb01es\ns(1, t) = 1,\ns(h+1, k) = 1 + ks(h, k),\nwith solution s(h, k) = \u0398(kh ). Now we have\nT(labels)(1, k) = \u0398(1),\nT(labels)(h+1, k) = \u0398(1) + T(concat)(k, s) + T(map labels)(h, k),\nwhere s = s(h, k). The term T(map labels)(h, k) estimates the running time of\napplying map labels to a list of length k of trees all of height h. In general, given\na list of length k consisting of elements each of size n, we have\nT(map f)(k, n) = kT(f)(n) + \u0398(k).\nFurthermore T(concat)(k, s) = \u0398(ks) = \u0398(kh+1 ). Hence\nT(labels)(h+1, k) = \u0398(kh+1 ) + kT(labels)(h, k)\nsince \u0398(1) + \u0398(k) = \u0398(k). The solution is given by\nT(labels)(h, k) = \u0398(hkh ) = \u0398(s log s).\nIn words, computing the labels of a tree using the de\ufb01nition above takes time that\nis asymptotically greater than the size of the tree by a logarithmic factor.\nLet us now see what an accumulating parameter can do. De\ufb01ne labcat by\nlabcat :: [GenTree a] -> [a] -> [a]\nlabcat ts xs = concat (map labels ts) ++ xs\nAs well as adding in a list xs we have also generalised the \ufb01rst argument from a\ntree to a list of trees. We have labels t = labcat [t] [], so any improvement\non labcat leads to a corresponding improvement on labels.\nWe now synthesise an alternative de\ufb01nition for labcat. For the base case we obtain\nlabcat [] xs = xs\n\n\f7.5 Accumulating parameters\n\n163\n\nFor the inductive case we reason:\nlabcat (Node x us:vs) xs\n=\n\n{de\ufb01nition}\nconcat (map labels (Node x us:vs)) ++ xs\n\n=\n\n{de\ufb01nitions}\nlabels (Node x us) ++ concat (map labels vs) ++ xs\n\n=\n\n{de\ufb01nition}\nx:concat (map labels us) ++ concat (map labels vs) ++ xs\n\n=\n\n{de\ufb01nition of labcat}\nx:concat (map labels us) ++ labcat vs xs\n\n=\n\n{de\ufb01nition of labcat (again)}\nlabcat us (labcat vs xs)\n\nThe result of this calculation is the following program for labels:\nlabels t = labcat [t] []\nlabcat [] xs\n= xs\nlabcat (Node x us:vs) = x:labcat us (labcat vs xs)\nFor the timing analysis, let T(labcat)(h, k, n) estimate the running time of\nlabcat ts xs\nwhen ts is a list of length n of trees, each of which is a perfect k-ary tree of height\nh (the size of xs is ignored since it doesn\u2019t affect the estimate). Then\nT(labcat)(h, k, 0) = \u0398(1),\nT(labcat)(1, k, n+1) = \u0398(1) + T(labcat)(1, k, n)),\nT(labcat)(h+1, k, n+1) = \u0398(1) + T(labcat)(h, k, k) +\nT(labcat)(h+1, k, n).\nSolving the \ufb01rst two equations gives T(labcat)(1, k, n) = \u0398(n). An induction argument now shows T(labcat)(h, k, n) = \u0398(kh n). Hence\nT(labels)(h, k) = T(labcat)(h, k, 1) = \u0398(kh ) = \u0398(s).\nThat means we can compute the labels of a tree in time proportional to the size of\nthe tree, a logarithmic improvement over our \ufb01rst version.\n\n\fE\ufb03ciency\n\n164\n\n7.6 Tupling\nWe met the idea of tupling two functions in the discussion of the function mean.\nTupling is sort of dual to the method of accumulating parameters: we generalise a\nfunction not by including an extra argument but by including an extra result.\nThe canonical example of the power of tupling is the Fibonacci function:\nfib\nfib\nfib\nfib\n\n:: Int -> Integer\n0 = 0\n1 = 1\nn = fib (n-1) + fib (n-2)\n\nThe time to evaluate fib by these three equations is given by\nT(fib)(0) = \u0398(1),\nT(fib)(1) = \u0398(1),\nT(fib)(n) = T(fib)(n\u22121) + T(fib)(n\u22122) + \u0398(1).\nThe timing function therefore satis\ufb01es equations very like that\n\u221a of fib itself. In fact\nT(fib)(n) = \u0398(\u03c6 n ), where \u03c6 is the golden ratio \u03c6 = (1 + 5)/2. That means that\nthe running time to compute fib on an input n is exponential in n.\nNow consider the function fib2 de\ufb01ned by\nfib2 n = (fib n,fib (n+1))\nClearly fib n = fst (fib2 n). Synthesis of a direct recursive de\ufb01nition of\nfib2 yields\nfib2 0 = (0,1)\nfib2 n = (b,a+b)\n\nwhere (a,b) = fib2 (n-1)\n\nThis program takes linear time. In this example the tupling strategy leads to a dramatic increase in ef\ufb01ciency, from exponential time to linear time.\nIt\u2019s great fun to formulate general laws that encapsulate gains in ef\ufb01ciency. One\nsuch law concerns the computation of\n(foldr f a xs, foldr g b xs)\nAs expressed above, the two applications of foldr involve two traversals of the\nlist xs. There is a modest time advantage, and possibly a greater space advantage,\nin formulating a version that traverses the list only once. In fact\n(foldr f a xs, foldr g b xs) = foldr h (a,b) xs\n\n\f7.6 Tupling\n\n165\n\nwhere\nh x (y,z) = (f x y,g x z)\nThe result can be proved by induction and we leave details as an easy exercise.\nAs one more example, we again move from lists to trees. But this time we have a\ndifferent kind of tree, a leaf-labelled binary tree:\ndata BinTree a = Leaf a | Fork (BinTree a) (BinTree a)\nIn contrast to a GenTree discussed above, a BinTree is either a leaf, with an\nassociated label, or a fork of two subtrees.\nSuppose we wanted to build such a tree with a given list as the labels. More precisely, we want to de\ufb01ne a function build satisfying\nlabels (build xs) = xs\nfor all \ufb01nite nonempty lists xs, where labels returns the labels of a binary tree:\nlabels :: BinTree a -> [a]\nlabels (Leaf x)\n= [x]\nlabels (Fork u v) = labels u ++ labels v\nWe are attuned now to possible optimisations, and the de\ufb01nition of labels suggest\nthat it could be improved with an accumulating parameter. So it can, but that is not\nour primary interest here, and we leave the optimisation as an exercise.\nOne way to build a tree is to arrange that half the list goes into the left subtree, and\nthe other half into the right subtree:\nbuild :: [a] -> BinTree a\nbuild [x] = Leaf x\nbuild xs = Fork (build ys) (build zs)\nwhere (ys,zs) = halve xs\nThe function halve made an appearance in Section 4.8:\nhalve xs = (take m xs,drop m xs)\nwhere m = length xs `div` 2\nThus halve splits a list into two approximately equal halves. The de\ufb01nition of\nhalve involves a traversal of the list to \ufb01nd its length, and two further (partial)\ntraversals to compute the two components. It is therefore a prime candidate for\napplying the tupling strategy to get something better. But as with labels we are\ngoing to ignore that particular optimisation for now. And we are also going to\n\n\fE\ufb03ciency\n\n166\n\nignore the proof that this de\ufb01nition of build meets its speci\ufb01cation. That\u2019s three\ncalculations we are leaving as exercises in order to concentrate on a fourth.\nLet\u2019s time build:\nT(build)(1) = \u0398(1),\nT(build)(n) = T(build)(m) + T(build)(n\u2212m) + \u0398(n)\n.where m = n div 2\nIt takes \u0398(n) steps to halve a list of length n, and then we recursively build two\nsubtrees from lists of length m and n \u2212 m, respectively. The solution is\nT(build)(n) = \u0398(n log n).\nIn words, building a tree by the above method takes longer than the length of the\nlist by a logarithmic factor.\nHaving established this fact, let us de\ufb01ne build2 by\nbuild2 :: Int -> [a] -> (BinTree a,[a])\nbuild2 n xs = (build (take n xs),drop n xs)\nThis builds a tree from the \ufb01rst n elements, but also returns the list that is left. We\nhave\nbuild xs = fst (build2 (length xs) xs)\nso our original function can be determined from the tupled version.\nOur aim now is to construct a direct recursive de\ufb01nition of build2. First of all, it\nis clear that\nbuild2 1 xs = (Leaf (head xs),tail xs)\nFor the recursive case we start with\nbuild2 n xs = (Fork (build (take m (take n xs)))\n(build (drop m (take n xs))),\ndrop n xs) where m = n `div` 2\nThis equation is obtained by substituting in the recursive case of build. It suggests\nthat the next step is to use some properties of take and drop. Here they are: if\nm <= n then\ntake m . take n = take m\ndrop m . take n = take (n-m) . drop m\nThat leads to\n\n\f7.7 Sorting\n\n167\n\nbuild2 n xs = (Fork (build (take m xs))\n(build (take (n-m) (drop m xs))),\ndrop n xs) where m = n `div` 2\nUsing the de\ufb01nition of build2 we can rewrite the above as follows:\nbuild2 n xs = (Fork u v, drop n xs)\nwhere (u,xs') = build2 m xs\n(v,xs'') = build2 (n-m) xs'\nm\n= n `div` 2\nBut as a \ufb01nal step, observe that\nxs'' = drop (n-m) xs'\n= drop (n-m) (drop m xs)\n= drop n xs\nHence we can rewrite build2 once again to read\nbuild2 1 xs = (Leaf (head xs),tail xs)\nbuild2 n xs = (Fork u v, xs'')\nwhere (u,xs') = build2 m xs\n(v,xs'') = build2 (n-m) xs'\nm\n= n `div` 2\nTiming this program yields\nT(build2)(1) = \u0398(1),\nT(build2)(n) = T(build2)(m) + T(build2)(n\u2212m) + \u0398(1).\nwith solution T(build2)(n) = \u0398(n). Using build2 as a subsidiary function has\ntherefore improved the running time of build by a logarithmic factor.\n\n7.7 Sorting\nSorting is a big topic and one can spend many happy hours tinkering with different algorithms. Knuth devotes about 400 pages to the subject in Volume 3 of\nhis series The Art of Computer Programming. Even then some of his conclusions\nhave to be reformulated when sorting is considered in a purely functional setting.\nHere we brie\ufb02y consider two sorting algorithms, keeping an eye out for possible\noptimisations.\n\n\fE\ufb03ciency\n\n168\n\nMergesort\nThe sorting method called Mergesort made an appearance in Section 4.8:\nsort\nsort\nsort\nsort\n\n:: (Ord a) => [a] -> [a]\n[] = []\n[x] = [x]\nxs = merge (sort ys) (sort zs)\nwhere (ys,zs) = halve xs\nhalve xs = (take m xs,drop m xs)\nwhere m = length xs `div` 2\nIn fact there are a number of variants for sorting by merging, and the standard\nprelude function sort uses a different variant than the one above.\nAs we said above, the de\ufb01nition of halve looks fairly inef\ufb01cient in that it involves\nmultiple traversals of its argument. One way to improve matters is to make use of\nthe standard prelude function splitAt, whose speci\ufb01cation is\nsplitAt :: Int -> [a] -> ([a],[a])\nsplitAt n xs = (take n xs,drop n xs)\nThe prelude version of this function is the result of a tupling transformation:\nsplitAt 0 xs\n= ([],xs)\nsplitAt n []\n= ([],[])\nsplitAt n (x:xs) = (x:ys,zs)\nwhere (ys,zs) = splitAt (n-1) xs\nIt is easy enough to calculate this de\ufb01nition using the two facts that\ntake n (x:xs) = x:take (n-1) xs\ndrop n (x:xs) = drop (n-1) xs\nprovided 0 < n. Now we have\nhalve xs = splitAt (length xs `div` 2) xs\nThere are still two traversals here of course.\nAnother way to improve sort is to de\ufb01ne\nsort2 n xs = (sort (take n xs),drop n xs)\nWe have sort xs = fst (sort2 (length xs) xs), so our original sorting\nfunction can be retrieved from the general one. An almost exactly similar calculation to the one in the previous section leads to\n\n```\n\n\n```\n\n\f7.7 Sorting\n\n169\n\nsort2 0 xs = ([],xs)\nsort2 1 xs = ([head xs],tail xs)\nsort2 n xs = (merge ys zs, xs'')\nwhere (ys,xs') = sort2 m xs\n(zs,xs'') = sort2 (n-m) xs'\nm\n= n `div` 2\nWith this de\ufb01nition there are no length calculations and no multiple traversals of\nxs.\nAnother way to optimise halve is to realise that no human would split up a list\nin this way if forced to do so by hand. If asked to divide a list into two, you and I\nwould surely just deal out the elements into two piles:\nhalve []\n= ([],[])\nhalve [x]\n= ([x],[])\nhalve (x:y:xs) = (x:ys,y:zs)\nwhere (ys,zs) = halve xs\nOf course, this de\ufb01nition returns a different result than the previous one, but the\norder of the elements in the two lists does not matter if the result is to be sorted;\nwhat is important is that the elements are all there.\nThat is a total of three ways to improve the performance of sort. However, it\nturns out that none of them makes that much difference to the total running time. A\nfew per cent perhaps, but nothing substantial. Furthermore, if we are using GHCi\nas our functional evaluator, none of the versions compares in performance to the\nlibrary function sort because that function is given to us in a compiled form, and\ncompiled versions of functions are usually about ten times faster. We can always\ncompile our functions using GHC of course.\n\nQuicksort\nOur second sorting algorithm is a famous one called Quicksort. It can be expressed\nin just two lines of Haskell:\nsort :: (Ord a) => [a] -> [a]\nsort []\n= []\nsort (x:xs) = sort [y | y <- xs, y < x] ++ [x] ++\nsort [y | y <- xs, x <= y]\n\n\fE\ufb03ciency\n\n170\n\nThat\u2019s very pretty and a testament to the expressive power of Haskell. But the\nprettiness comes at a cost: the program can be very inef\ufb01cient in its use of space.\nThe situation is the same as with the program for mean seen earlier.\nBefore plunging into ways the code can be optimised, let\u2019s compute T(sort). Suppose we want to sort a list of length n+1. The \ufb01rst list comprehension can return a\nlist of any length k from 0 to n. The length of the result of the second list comprehension is therefore n\u2212k. Since our timing function is an estimate of the worst-case\nrunning time, we have to take the maximum of these possibilities:\nT(sort)(n+1)\n= max [T(sort)(k) + T(sort)(n\u2212k) | k \u2190 [0 .. n]] + \u0398(n).\nThe \u0398(n) term accounts for both the time to evaluate the two list comprehensions\nand the time to perform the concatenations. Note, by the way, the use of a list\ncomprehension in a mathematical expression rather than a Haskell one. If list comprehensions are useful notations in programming, they are useful in mathematics\ntoo.\nAlthough not immediately obvious, the worst case occurs when k = 0 or k = n.\nHence\nT(sort)(0) = \u0398(1),\nT(sort)(n+1) = T(sort)(n) + \u0398(n),\nwith solution T(sort)(n) = \u0398(n2 ). Thus Quicksort is a quadratic algorithm in the\nworst case. This fact is intrinsic to the algorithm and has nothing to do with the\nHaskell expression of it. Quicksort achieved its fame for two other reasons, neither of which hold in a purely functional setting. Firstly, when Quicksort is implemented in terms of arrays rather than lists, the partitioning phase can be performed\nin place without using any additional space. Secondly, the average case performance of Quicksort, under reasonable assumptions about the input, is \u0398(n log n)\nwith a smallish constant of proportionality. In a functional setting this constant is\nnot so small and there are better ways to sort than Quicksort.\nWith this warning, let us now see what we can do to optimise the algorithm without\nchanging it in any essential way (i.e. to a completely different sorting algorithm).\nTo avoid the two traversals of the list in the partitioning process, de\ufb01ne\npartition p xs = (filter p xs, filter (not . p) xs)\nThis is another example of tupling two de\ufb01nitions to save on a traversal. Since\nfilter p can be expressed as an instance of foldr we can appeal to the tupling\nlaw of foldr to arrive at\n\n\f7.7 Sorting\n\n171\n\npartition p = foldr op ([],[])\nwhere op x (ys,zs) | p x\n= (x:ys,zs)\n| otherwise = (ys,x:zs)\nNow we can write\nsort []\n= []\nsort (x:xs) = sort ys ++ [x] ++ sort zs\nwhere (ys,zs) = partition ( n+1) 0 xs = foldl (\\n x -> 1+n) 0 xs\nfoldr (\\x xs -> xs++[x]) [] xs\n= foldl (\\xs x -> [x]++xs) [] xs\nExercise G\nProve that if h x (y,z) = (f x y,g x z), then\n(foldr f a xs,foldr g b xs) = foldr h (a,b) xs\nfor all \ufb01nite lists xs. A tricky question: does the result hold for all lists xs?\nNow \ufb01nd a de\ufb01nition of h such that\n(foldl f a xs,foldl g b xs) = foldl h (a,b) xs\nExercise H\nRecall that\npartition p xs = (filter p xs, filter (not . p) xs)\nExpress the two components of the result as instances of foldr. Hence use the\nresult of the previous exercise to calculate another de\ufb01nition of partition.\nDe\ufb01ne\npart p xs us vs = (filter p xs ++ us,\nfilter (not . p) xs ++ vs)\nCalculate another de\ufb01nition of partition that uses part as a local de\ufb01nition.\nExercise I\nRecall that\n\n\f7.9 Answers\n\n175\n\nlabels :: BinTree a -> [a]\nlabels (Leaf x)\n= [x]\nlabels (Fork u v) = labels u ++ labels v\nCompute T(labels)(n), where n is the number of leaves in the tree. Now use the\naccumulating parameter technique to \ufb01nd a faster way of computing labels.\nProve that labels (build xs) = xs for all \ufb01nite nonempty lists xs.\nExercise J\nDe\ufb01ne select k = (!!k) . sort, where sort is the original Quicksort. Thus\nselect k selects the kth smallest element of a nonempty \ufb01nite list of elements,\nthe 0th smallest being the smallest element, the 1st smallest being the next smallest\nelement, and so on. Calculate a more ef\ufb01cient de\ufb01nition of select and estimate\nits running time.\n\n7.9 Answers\nAnswer to Exercise A\n=\n=\n=\n=\n=\n=\n=\n\nsort [3,4,1,2]\ninsert 3 (sort [4,1,2])\n...\ninsert 3 (insert 4 (insert 1 (insert 2 [])))\ninsert 3 (insert 4 (insert 1 (2:[])))\ninsert 3 (insert 4 (1:2:[]))\ninsert 3 (1:insert 4 (2:[]))\n1:insert 3 (insert 4 (2:[]))\n\nIt takes \u0398(n) steps to compute head . sort on a list of length n. Under eager\nevaluation it takes about n2 steps. As to part (iii), the answer is yes. You may think\nwe have de\ufb01ned sorting by insertion, but under lazy evaluation it turns out to be\nselection sort. The lesson here is that, under lazy evaluation, you don\u2019t always get\nwhat you think you are getting.\nAnswer to Exercise B\nFor the \ufb01rst part, the following does the job:\nlength = foldl' (\\n x -> n+1) 0\n\n\n```\n\n\n```\n\nFor the second part, one solution is\n\n\fE\ufb03ciency\n\n176\n\nlength\n= length2 0\nlength2 n []\n= n\nlength2 n (x:xs) = if n==0 then length2 1 xs\nelse length2 (n+1) xs\nThe test n==0 forces evaluation of the \ufb01rst argument.\nAnswer to Exercise C\nTake f n x = if x==0 then undefined else 0. Then\nfoldl f 0 [0,2] = 0\nfoldl' f 0 [0,2] = undefined\nAnswer to Exercise D\nThe answer is: maybe! Although the given version of cp is ef\ufb01cient, it returns the\ncomponent lists in a different order than any of the de\ufb01nitions in the text. That\nprobably doesn\u2019t matter if we are only interested in the set of results, but it might\naffect the running time and result of any program that searched cp to \ufb01nd some list\nsatisfying a given property.\nAccording to the fusion rule we have to \ufb01nd a function g so that\nfilter nondec (f xs yss) = g xs (filter nondec yss)\nwhere f xs yss =\n\n[x:ys | x <- xs, ys <- yss]. Then we would have\n\nfilter nondec . cp\n= filter nondec . foldr f [[]]\n= foldr g [[]]\nNow\nnondec (x:ys) = null ys || (x <= head ys && nondec ys)\nThat leads to\ng xs [[]] = [[x] | x <- xs]\ng xs yss = [x:ys | x <- xs, ys <- yss, x <= head ys]\nAnswer to Exercise E\nFor the \ufb01rst part, we have\nT(2k ) = 2T(2k\u22121 ) + \u0398(2k ).\n\n\f7.9 Answers\n\n177\n\nBy induction we can show T(2k ) = \u2211i=0 k\u0398(2k ). The induction step is\nk\u22121\n\nT(2k ) = 2 \u2211 \u0398(2k\u22121 ) + \u0398(2k )\n=\n\ni=0\nk\u22121\n\n\u2211 \u0398(2k ) + \u0398(2k )\n\ni=0\nk\n\n= \u2211 \u0398(2k ).\ni=0\n\nHence T(2k ) = \u0398(k2k ). Now suppose 2k \u2264 n < 2k+1 , so\n\u0398(k2k ) = T(2k ) \u2264 T(n) \u2264 T(2k+1 ) = \u0398((k+1)2k+1 ) = \u0398(k2k ).\nHence T(n) = \u0398(k2k ) = \u0398(n log n).\nAnswer to Exercise F\nDe\ufb01ne x <> n = n+1 and n @ x\n\n= 1+n. We have\n\n(x <> n) @ y = 1+(n+1) = (1+n)+1 = x <> (n @ y)\nThe second proof is similar.\nAnswer to Exercise G\nThe induction step is\n=\n=\n=\n=\n\n(foldr f a (x:xs),foldr g b (x:xs)\n(f x (foldr f a xs),g x (foldr g b xs))\nh x (foldr f a xs,foldr g b xs)\nh x (foldr h (a,b) xs\nfoldr h (a,b) (x:xs)\n\n```\n\n\n```\n\nThe answer to the tricky question is No. The values (\u22a5, \u22a5) and \u22a5 are different in\nHaskell. For example, suppose we de\ufb01ne foo (x,y) = 1. Then\nfoo undefined = undefined\nfoo (undefined,undefined) = 1\nFor the last part, the de\ufb01nition of h is that\nh (y,z) x = (f y x,g z x)\nAnswer to Exercise H\nWe have filter p = foldr (op p) [], where\n\n\fE\ufb03ciency\n\n178\n\nop p x xs = if p x then x:xs else xs\nNow\n(op p x ys,op (not . p) x zs)\n= if p x then (x:ys,zs) else (ys,x:zs)\nHence\npartition p xs = foldr f ([],[]) xs\nwhere f x (ys,zs) = if p x\nthen (x:ys,zs)\nelse (ys,x:zs)\nFor the last part we obtain\npartition p xs = part p xs\npart p [] ys zs = (ys,zs)\npart p (x:xs) ys zs = if p\nthen\nelse\n\n[] []\nx\npart p xs (x:ys) zs\npart p xs ys (z:zs)\n\nAnswer to Exercise I\nRemember that T estimates the worst case running time. The worst case for labels\narises when every right subtree of the tree is a leaf. Then we have\nT(labels)(n) = T(labels)(n\u22121) + \u0398(n),\nwhere \u0398(n) accounts for the time to concatenate a list of length n\u22121 with a list of\nlength 1. Hence\nn\n\u0398(n) = \u0398(n2 ).\nT(labels)(n) = \u03c3j=0\n\nThe accumulating parameter method yields\nlabels t\n= labels2 t []\nlabels2 (Leaf x) xs\n= x:xs\nlabels2 (Fork u v) xs = labels2 u (labels2 v xs)\nand T(labels2)(n) = \u0398(n). This improves the running time of labels from\nquadratic to linear time.\nThe induction step in the proof that labels (build xs) = xs is to assume the\n\n\f7.9 Answers\n\n179\n\nhypothesis for all lists strictly shorter than xs:\nlabels (build xs)\n=\n\n{assume xs has length at least two\nand let (ys,zs) = halve xs}\nlabels (Fork (build ys) (build zs))\n\n=\n\n{de\ufb01nition of labels}\nlabels (build ys) ++ labels (build zs)\n\n=\n\n{induction, since ys and zs are strictly shorter than xs}\nys ++ zs\n\n=\n\n{de\ufb01nition of halve xs}\nxs\n\nThe induction here is general induction: in order to prove P(xs) for all \ufb01nite lists\nxs it is suf\ufb01cient to prove that: (i) P([]); and (ii) P(xs) holds under the assumption\nthat P holds for all lists of length strictly less than xs.\nAnswer to Exercise J\nOne key property is that\n(xs ++ [x] ++ ys)!!k | k < n\n| k==n\n| k > n\nwhere n =\n\n= xs!!k\n= x\n= ys!!(n-k)\nlength xs\n\n```\n\n\n```\n\nThe other key property is that sorting a list does not change the length of the list.\nHence\nselect k []\n= error \"list too short\"\nselect k (x:xs) | k < n\n= select k ys\n| k==n\n= x\n| otherwise = select (n-k) zs\nwhere ys = [y | y <- xs, y < x]\nzs = [z | z <- xs, x <= z]\nn = length ys\nThe worst-case running time for a list of length n occurs when k = 0 and the length\nof ys is n\u22121, i.e. when x:xs is in strictly decreasing order. Thus\nT(select)(0, n) = T(select)(0, n\u22121) + \u0398(n),\nwith solution T(select)(0, n) = \u0398(n2 ). But, assuming a reasonable distribution\n\n\f180\n\nE\ufb03ciency\n\nin which each permutation of the sorted result is equally likely as input, we have\nT(select)(k, n) = \u0398(n).\n\n7.10 Chapter notes\nThere are many books on algorithm design, but two that concentrate on functional programming are Algorithms: A Functional Programming Approach (second\nedition) (Addison-Wesley, 1999) by Fethi Rabbi and Guy Lapalme, and my own\nPearls of Functional Algorithm Design (Cambridge, 2010).\nInformation about pro\ufb01ling tools comes with the documentation on the Haskell\nPlatform. The source book on sorting is Don Knuth\u2019s The Art of Computer Programming, Volume 3: Sorting and Searching (second edition) (Addison-Wesley,\n1998).\n\n\fChapter 8\nPretty-printing\n\nThis chapter is devoted to an example of how to build a small library in Haskell.\nA library is an organised collection of types and functions made available to users\nfor carrying out some task. The task we have chosen to discuss is pretty-printing,\nthe idea of taking a piece of text and laying it out over a number of lines in such\na way as to make the content easier to view and understand. We will ignore many\nof the devices for improving the readability of a piece of text, devices such as a\nchange of colour or size of font. Instead we concentrate only on where to put the\nline breaks and how to indent the contents of a line. The library won\u2019t help you to\nlay out bits of mathematics, but it can help in presenting tree-shaped information,\nor in displaying lists of words as paragraphs.\n\n8.1 Setting the scene\nLet\u2019s begin with the problem of displaying conditional expressions. In this book\nwe have used three ways of displaying such expressions:\nif p then expr1 else expr2\nif p then expr1\nelse expr2\nif p\nthen expr1\nelse expr2\nThese three layouts, which occupy one, two or three lines, respectively, are considered acceptable, but the following two are not:\n\n\fPretty-printing\n\n182\n\nif p then\nexpr1 else expr2\nif p\nthen expr1 else expr2\nThe decision as to what is or is not acceptable is down to me, the author. You may\ndisagree with my choices (some do), and a \ufb02exible library should provide you with\nthe ability to make your own reasonable choices. In any case, two basic questions\nhave to be answered. Firstly, how can we describe the acceptable alternatives while\nrejecting the unacceptable ones? Secondly, how do we choose between the acceptable alternatives?\nA quick answer to the second question is that the choice depends on the permitted\nline width. For instance we might choose a layout with the fewest lines, subject to\nthe condition that each line \ufb01ts within the allotted line width. Much more on this\nlater.\nAs to the \ufb01rst question, one answer is just to write out all the acceptable alternatives. That\u2019s going to involve a lot of writing. A better alternative is to provide the\nuser with a suitable layout description language. As a rough and ready guide we\nmight write something like\nif p <0> then expr1 (<0> + <1>) else expr2 +\nif p <1> then expr1 <1> else expr2\nwhere <0> means a single space, <1> means a line break and + means \u2018or\u2019. The\nexpression above yields our three layouts described earlier. However, the danger\nwith providing the user with an unfettered choice of alternatives is that it becomes\ndif\ufb01cult to make a decision about the best layout without exploring every possible\nalternative, and that could take a long time.\nAnother possibility is to allow only restricted choice by forcing the user to describe\nlayouts in terms of certain functions and operations provided by the library. For\nexample, consider the description\ngroup (group (if p <1> then expr1) <> <1> else expr2)\nwhere group augments a set of layouts with one additional layout in which every\n<1> is replaced by <0>, thereby \ufb02attening the layout to just one line, and (<>)\nmeans concatenation lifted to sets of alternatives. For example,\ngroup (if p <1> then expr1)\n= {if p <0> then expr1, if p <1> then expr1}\n\n\f8.2 Documents\n\n183\n\ngroup (if p <1> then expr1) <> <1> else expr2\n= {if p <0> then expr1 <1> else expr2,\nif p <1> then expr1 <1> else expr2}\ngroup (group (if p <1> then expr1) <> <1> else expr2)\n= {if p <0> then expr1 <0> else expr2,\nif p <0> then expr1 <1> else expr2,\nif p <1> then expr1 <1> else expr2}\nThus our set of three acceptable layouts is captured by the above description which\ncontains two occurrences of group.\nThere is another aspect to the problem of displaying conditional expressions. What\nif expr1 or expr2 are themselves conditional expressions? Here we might want to\nallow a layout like\nif p\nthen if q\nthen expr1\nelse expr2\nelse expr3\nThe point is that we should allow for indentation in our description language. Indentation means putting in a suitable number of spaces after each line break. This\nidea can be captured by providing a function nest so that nest i x is a layout in\nwhich each line break in layout x is followed by i spaces.\n\n8.2 Documents\nFor the sake of a name let us agree to call a document some entity that represents\nthe set of possible layouts of a piece of text. Documents are given as elements of\nthe type Doc whose de\ufb01nition is left for later on. On the other hand, a layout is\nsimply a string:\ntype Layout = String\nWe are deliberately being cagey about what a document actually is because we\nwant to consider two representations of Doc. For now we concentrate on the operations on documents that our library might provide.\nThe \ufb01rst operation is a function\npretty :: Int -> Doc -> Layout\n\n\fPretty-printing\n\n184\n\nthat takes a given line width and a document, and returns the best layout. How to\nde\ufb01ne this function ef\ufb01ciently is really the main concern of the chapter.\nThe second operation is a function\nlayouts :: Doc -> [Layout]\nthat returns the set of possible layouts as a list. Why should we want such a function when we have pretty? Well, it takes a little experimentation to \ufb01nd the de\ufb01nitions that describe the layouts we regard as acceptable. The way to experiment is\nto formulate an initial de\ufb01nition and then rework it after inspecting all the resulting layouts on a small number of examples. That way we can see whether some\nlayouts should be excluded or others added. So, whatever our \ufb01nal representation\nof documents turns out to be, we should provide layouts as a sensible diagnostic\ntool for the user.\nThe remaining operations deal with constructing documents. First up is the operation of concatenating two documents to give a new one:\n(<>) :: Doc -> Doc -> Doc\nDocument concatenation should surely be an associative operation so we require\nof any implementation of (<>) that\n(x <> y) <> z) = x <> (y <> z)\nfor all documents x, y and z.\nWhenever there is an associative operation there is usually an identity element, so\nwe also provide an empty document\nnil :: Doc\nWe require nil <> x = x and x <> nil = x for all documents x.\nThe next operation is a function\ntext :: String -> Doc\nthat takes a string not containing newlines into a document. To provide for documents containing more than one line, we can provide another basic document\nline :: Doc\nFor example,\ntext \"Hello\" <> line <> text \"World!\"\n\n\f8.2 Documents\n\n185\n\nis a document with a single layout that consists of two lines. You might think that\nline is unnecessary because we could always allow newline characters in text\nstrings, but to indent a document we would then have to inspect the contents of\nevery text. Far better is to have an explicit newline document; that way we know\nwhere line breaks are.\nNext, the function\nnest :: Int -> Doc -> Doc\nprovides a way of nesting documents: nest i indents a document by inserting\ni spaces after every newline. Note the emphasis: indentation is not done at the\nbeginning of a document unless it begins with a newline. The reason for this choice\nis explained below.\nFinally, to complete a library of eight operations, we have the function\ngroup :: Doc -> Doc\nThis is the function that produces multiple layouts. The function group takes a\ndocument and adds an extra layout, one that consists of a single line of text with\nno line breaks.\nWe have named eight operations and given informal descriptions of what they are\nintended to mean, but can we be more precise about their properties and the relationships between them? An even more fundamental question is whether these\noperations are suf\ufb01ciently \ufb02exible to allow for a reasonable class of layouts.\nLet\u2019s \ufb01rst concentrate on what equational laws we might want. Finding such laws\ncan boost our con\ufb01dence that we have in hand an adequate and smoothly integrated\nbox of tools, and that there isn\u2019t some crucial gadget we have missed. Such laws\ncan also in\ufb02uence the meanings of operations and guide implementations. We have\nalready asserted that (<>) should be associative with identity element nil, but\nwhat else should we require?\nWell, for text we want the following properties:\ntext (s ++ t) = text s <> text t\ntext \"\"\n= nil\nIn mathematical language this asserts that text is a homomorphism from string\nconcatenation to document concatenation. An impressive (and possibly intimidating) name for something quite simple. Note that the associativity of string concatenation implies the associativity of document concatenation, at least for text\ndocuments.\n\n\fPretty-printing\n\n186\n\nFor nest we require the following equations to hold:\nnest\nnest\nnest\nnest\nnest\nnest\nnest\n\ni\ni\ni\ni\ni\n0\ni\n\n(x <> y)\nnil\n(text s)\nline\n(nest j x)\nx\n(group x)\n\n=\n=\n=\n=\n=\n=\n=\n\nnest i x <> nest i y\nnil\ntext s\nline <> text (replicate i ' ')\nnest (i+j) x\nx\ngroup (nest i x)\n\nAll very reasonable (except possibly for the last), and we could give some of them\nmathematical names (nest i distributes through concatenation, nest is a homomorphism from numerical addition to functional composition and nest i commutes with group). The third law fails if nest were to indent from the beginning\nof a document; and it would also fail if we allowed text strings to contain newline\ncharacters. The last law holds because grouping adds a layout with no line breaks,\nand nesting has no effect on such a layout. See Exercise D for a more precise argument.\nTurning to the properties of layouts, we require that\nlayouts\nlayouts\nlayouts\nlayouts\nlayouts\nlayouts\n\n(x <> y)\nnil\n(text s)\nline\n(nest i x)\n(group x)\n\n=\n=\n=\n=\n=\n=\n\nlayouts x <++> layouts y\n[\"\"]\n[s]\n[\"\\n\"]\nmap (nestl i) (layouts x)\nlayouts (flatten x) ++ layouts x\n\nThe operation (<++>) is lifted concatenation:\nxss <++> yss = [xs ++ ys | xs <- xss, ys <- yss]\nThe function nestl :: Int -> Layout -> Layout is de\ufb01ned by\nnestl i\n= concat (map indent i)\nindent i c = if c=='\\n' then c:replicate i ' ' else [c]\nFinally, flatten :: Doc -> Doc is the function that converts a document into\none with a single layout in which each newline and its associated indentation is\nreplaced by a single space. This function is not provided in the public interface of\nour documents library, though it will be needed internally. It is a missing gadget in\nthe sense that we need it to complete the description of the algebraic laws.\nWe require that flatten should satisfy the following conditions:\n\n\f8.3 A direct implementation\n\nflatten\nflatten\nflatten\nflatten\nflatten\nflatten\n\n(x <> y)\nnil\n(text s)\nline\n(nest i x)\n(group x)\n\n=\n=\n=\n=\n=\n=\n\n187\n\nflatten x <> flatten y\nnil\ntext s\ntext \" \"\nflatten x\nflatten x\n\nThat makes 24 laws in total (one for <>, two each for nil and text, seven for nest\nand six each for layouts and flatten). Many of the laws look like constructive\nHaskell de\ufb01nitions of functions over a data type in which nil, text and so on are\nconstructors. More on this is in Section 8.6.\nThe eight operations certainly seem reasonable enough, but do they give us suf\ufb01cient \ufb02exibility to describe the layouts we might want? The proof of the pudding\nis in the eating, so in a moment we will pause to consider three examples. Before\ndoing so, some implementation of documents, however quick and dirty, will be\nneeded to test the examples.\n\n8.3 A direct implementation\nOne obvious choice of representation is to identify a document with its list of layouts:\ntype Doc = [Layout]\nSuch a representation is called a shallow embedding. With a shallow embedding,\nthe library functions are implemented directly in terms of the values of interest\n(here, layouts). Later on we will abandon this representation in favour of a more\nstructured alternative, but it is the obvious one to try \ufb01rst.\nHere are the de\ufb01nitions of the operations above (we will leave pretty until later):\nlayouts\nx <> y\nnil\nline\ntext s\nnest i\ngroup x\nflatten x\n\n=\n=\n=\n=\n=\n=\n=\n=\n\nid\nx <++> y\n[\"\"]\n[\"\\n\"]\n[s]\nmap (nestl i)\nflatten x ++ x\n[flattenl (head x)]\n\nWe have already de\ufb01ned nestl, and flattenl is de\ufb01ned by\n\n\f188\n\nPretty-printing\n\nflattenl :: Layout -> Layout\nflattenl [] = []\nflattenl (c:cs)\n| c=='\\n'\n= ' ':flattenl (dropWhile (== ' ') cs)\n| otherwise = c:flattenl cs\nDo the 24 laws hold for this implementation? Well, let\u2019s go through them. Lifted\nconcatentation <++> is associative with [[]] as identity element, so the \ufb01rst three\nlaws are okay. The two laws of text are easy to check, and the six laws of layouts\nare immediate. All but two laws of nest are routine. The remaining two, namely\nnest i . nest j = nest (i+j)\nnest i . group = group . nest i\ninvolve a bit of work (see Exercises C and D). That leaves the laws of flatten.\nThree are easy, and one can show\nflatten . nest i = flatten\nflatten . group = flatten\nwith a bit of work (see Exercises E and F). But the stumbling block is the law\nflatten (x <> y) = flatten x <> flatten y\nThis one is false. Take x = line and y = text \"\nflatten (x <> y) = [\"hello\"]\nflatten x <> flatten y = [\"\n\nhello\". Then\n\nhello\"]\n\nand the two results are different. The reason is that flatten removes the effect\nof nesting, but does not remove spaces after newlines if they are present in an unnested document. On the other hand, flattenl removes spaces after every newline\nin the document.\nRather than try to \ufb01x up this de\ufb01ciency, we can accept the less than perfect implementation and move on. One can show that all layouts of a document \ufb02atten to the\nsame string (see the Answer to Exercise E). The shallow embedding also possesses\nanother property that we will exploit in the de\ufb01nition of pretty. To see what it is,\nconsider the function shape that returns the shape of a layout:\nshape :: Layout -> [Int]\nshape = map length . lines\nThe prelude function lines breaks up a string on newline characters, returning a\nlist of strings without newlines. Thus the shape of a layout is the list of lengths of\nthe lines that make up the layout. The crucial property of layouts is that the list\n\n\f8.4 Examples\n\n189\n\nof shapes of the layouts of a document is in lexicographically decreasing order. For\nexample, one of the documents described in the following section has 13 possible\nlayouts whose shapes are given by\n[[94],[50,43],[50,28,19],[50,15,17,19],[10,39,43],\n[10,39,28,19],[10,39,15,17,19],[10,28,15,43],\n[10,28,15,28,19],[10,28,15,15,17,19],[10,13,19,15,43],\n[10,13,19,15,28,19],[10,13,19,15,15,17,19]]\nThis list is in decreasing lexicographic order. The reason the property holds is that\nlayouts (group x) puts the \ufb02attened layout at the head of the list of layouts of\ndocument x, and a \ufb02attened layout consists of a single line. Exercise G goes into\nmore details.\n\n8.4 Examples\nOur \ufb01rst example deals with laying out conditional expressions. For present purposes a conditional expression can be represented as an element of the data type\nCExpr, where\ndata CExpr = Expr String | If String CExpr CExpr\nHere is a function cexpr that speci\ufb01es the acceptable layouts described earlier:\ncexpr :: CExpr -> Doc\ncexpr (Expr p) = text p\ncexpr (If p x y)\n= group (group (text \"if \" <> text p <>\nline <> text \"then \" <>\nnest 5 (cexpr x)) <>\nline <> text \"else \" <>\nnest 5 (cexpr y))\nThis de\ufb01nition is similar to our previous version, except for the nesting of the\nsubexpressions.\nFor example, two of the 13 possible layouts for one particular expression are as\nfollows:\nif wealthy\nthen if happy then lucky you else tough\nelse if in love then content else miserable\n\n\fPretty-printing\n\n190\n\nif wealthy\nthen if happy\nthen lucky you\nelse tough\nelse if in love\nthen content\nelse miserable\nYou can see from the last expression why we have chosen an indentation of \ufb01ve\nspaces. The 13 possible layouts for this particular conditional expression have the\nshapes displayed in the previous section.\nThe second example concerns how to lay out general trees, trees with an arbitrary\nnumber of subtrees:\ndata GenTree a = Node a [GenTree a]\nHere is an example tree, laid out in two different ways:\nNode 1\n[Node 2\n[Node 7 [],\nNode 8 []],\nNode 3\n[Node 9\n[Node 10 [],\nNode 11 []]],\nNode 4 [],\nNode 5\n[Node 6 []]]\nNode 1\n[Node\nNode\nNode\nNode\n\n2\n3\n4\n5\n\n[Node 7 [], Node 8 []],\n[Node 9 [Node 10 [], Node 11 []]],\n[],\n[Node 6 []]]\n\nThe function gtree that produced these trees (coincidentally, also among a total\nof 13 different ways) was de\ufb01ned as follows:\ngtree :: Show a => GenTree a -> Doc\ngtree (Node x [])\n= text (\"Node \" ++ show x ++ \" []\")\n\n\f8.5 The best layout\n\n191\n\ngtree (Node x ts)\n= text (\"Node \" ++ show x) <>\ngroup (nest 2 (line <> bracket ts))\nThe \ufb01rst clause says that a tree with no subtrees is always displayed on a single\nline; the second clause says that a tree with at least one subtree is displayed either\non a single line or has its subtrees each displayed on a new line with an indentation\nof two units. The function bracket is de\ufb01ned by\nbracket :: Show a => [GenTree a] -> Doc\nbracket ts = text \"[\" <> nest 1 (gtrees ts) <> text \"]\"\ngtrees [t]\n= gtree t\ngtrees (t:ts) = gtree t <> text \",\" <> line <> gtrees ts\nTo be honest, it took a little time and experimentation to \ufb01nd the de\ufb01nitions above\n(for which the function layouts proved indispensable), and the result is certainly\nnot the only way to lay out trees.\nFinally, here is a way of laying out a piece of text (a string of characters containing\nspaces and newlines, not a document text) as a single paragraph:\npara :: String -> Doc\npara = cvt . map text . words\ncvt [] = nil\ncvt (x:xs)\n= x <> foldr (<>) nil [group (line <> x) | x <- xs]\nFirst, the words of the text are computed using the standard library function words,\na function we have encountered a number of times before. Then each word is converted into a document using text. Finally, each word, apart from the \ufb01rst, is laid\nout either on the same line or on a new line. If there are n+1 words in the text, and\nso n inter-word spaces, the code above describes 2n possible layouts. We certainly\ndon\u2019t want to examine all these layouts in computing one that will \ufb01t within a given\nline width.\n\n8.5 The best layout\nAs we said above, the best layout depends on the maximum permitted line width.\nThat\u2019s a simple decision, but not the only one. In general a pretty layout of a nested\ndocument will consist of a ribbon of text snaking across the page, and it is arguable\n\n\f192\n\nPretty-printing\n\nthat the width of the ribbon should also play a part in determining the best layout.\nAfter all, is the best layout on an in\ufb01nitely wide page one in which everything is\nplaced on one line? However, for simplicity we will ignore this very reasonable\nre\ufb01nement and take only the line width as the deciding factor.\nThere is also another decision to be made. Suppose we choose the best layout,\naccording to some criterion, among those layouts all of whose lines \ufb01t within the\ngiven line width. That\u2019s \ufb01ne if there is at least one such layout, but what if there\nisn\u2019t? The two options are either to abandon the formatting process with a suitable\nerror message, or else to do the best we can, accepting that the width may be\nexceeded.\nPsychologically and practically the second option seems the better one, so let us\nexplore what it entails. We can start by comparing the \ufb01rst lines, \u00021 and \u00022 , of two\nlayouts. We can decide that line \u00021 is better than \u00022 if: (i) both lines \ufb01t into width\nw and \u00021 is longer than \u00022 ; (ii) \u00021 \ufb01ts w but \u00022 doesn\u2019t; or (iii) neither \ufb01ts w and \u00021\nis shorter than \u00022 . The decision is a reasonable one because it should be capable of\nbeing implemented by a greedy strategy: \ufb01ll up the \ufb01rst line as much as possible\nwithout exceeding the line width; and if that is not possible, stop as soon as the\nwidth is exceeded.\nThe comparison test above doesn\u2019t determine what should happen if the two lines\nhave the same length. But it is a consequence of the fact that all layouts \ufb02atten\nto the same string that two \ufb01rst lines with the same length will be the same line.\nConsequently, the \ufb01rst line is \ufb01xed and the comparison can pass to the second pair\nof lines. And so on.\nThe second property about decreasing shapes can be used to simplify the comparison test slightly because if layout lx precedes layout ly in the list of layouts, then\nthe \ufb01rst line of lx is known to be at least as long as the \ufb01rst line of ly. And if the\ntwo lines are equally long, then the same statement is true of the second lines. And\nso on.\nGiven our shallow embedding of documents, here is a simple implementation of\nthe function pretty that \ufb01nds the best layout:\npretty :: Int -> Doc ->\npretty w = fst . foldr1\nwhere\naugment lx = (lx,shape\nchoose alx aly\n= if better (snd alx)\nbetter [] ks\n=\n\nLayout\nchoose . map augment\nlx)\n(snd aly) then alx else aly\nTrue\n\n\f8.6 A term representation\n\n193\n\nbetter js []\n= False\nbetter (j:js) (k:ks) | j == k\n= better js ks\n| otherwise = (j <= w)\nEach layout is augmented with shape information to guide the choice of layout,\nwhich is then determined by a simple search. The test better implements the\ncomparison operation described above. Finally, shape information is discarded.\nThis de\ufb01nition of pretty is hopelessly inef\ufb01cient because every layout is computed and examined. If there are n possible choices of whether to have a line break\nor not, there are 2n layouts to be examined and pretty-printing will be very slow\nindeed. For example,\nghci> putStrLn $ pretty 30 $ para pg\nThis is a fairly short\nparagraph with just twenty-two\nwords. The problem is that\npretty-printing it takes time,\nin fact 31.32 seconds.\n(31.32 secs, 17650013284 bytes)\nOuch! What is worse, pretty-printing a longer paragraph will cause GHCi to crash\nwith an \u2018out of memory\u2019 message. An exponential time and space algorithm is not\nacceptable.\nWhat is wanted is an algorithm for pretty that can decide on which \ufb01rst line to\nchoose without looking ahead more than w characters. The algorithm should also\nbe ef\ufb01cient, taking linear time in the size of the document being pretty-printed.\nIdeally the running time should be independent of w, but a running time that does\ndepend on w is acceptable if a faster one means a much more complicated program.\n\n8.6 A term representation\nThe problem with identifying a document with its list of possible layouts is that\nuseful structure is lost. Rather than bring all the alternatives to the top level as a\nlist, we really want to bury them as deep as possible. For example, consider the\nfollowing two expressions for a document:\nA<0>B<0>D + A<0>B<1>D + A<1>C<0>E + A<1>C<1>E\nA(<0>B(<0>D + <1>D) + <1>C(<0>E + <1>E))\n\n\fPretty-printing\n\n194\n\nAs before, <0> denotes a single space and <1> a single line break. The \ufb01ve letters\ndenote \ufb01ve nonempty texts. Since all four alternatives have to \ufb02atten to the same\ndocument, we require that B<0>D = C<0>E. In the \ufb01rst expression (which is essentially what is given by representing a document by its list of layouts) we have\nfour layouts to compare. In the second expression we can shortcut some of the\ncomparisons. For example, if we know that the common pre\ufb01x A cannot \ufb01t in the\ngiven width, the \ufb01rst two layouts can be thrown away without further comparisons.\nEven better, if we choose between alternatives from the innermost to the outermost,\nwe can base the comparison test on just the \ufb01rst lines of layouts. For instance, if\nwe choose the better of C<0>E and C<1>E \ufb01rst, then that choice is not changed by\nsubsequent choices.\nThe way to maintain the structure of documents is to represent a document as a\ntree:\ndata Doc =\n|\n|\n|\n|\n|\n\nNil\nLine\nText String\nNest Int Doc\nGroup Doc\nDoc :<>: Doc\n\nNote the use of an in\ufb01x constructor in the last line. Haskell allows in\ufb01x operators\nas constructors, but they have to begin with a colon. They do not have to end with a\ncolon as well, but it seems more attractive if they do. This tree is called an abstract\nsyntax tree; each operation of the library is represented by its own constructor. An\nimplementation in terms of abstract syntax trees is known as a deep embedding.\nWe will not provide the user with the details of the data type Doc, just its name.\nTo explain why not, it is useful to insert a short digression about Haskell data\ntypes. In Haskell the effect of a data declaration is to introduce a new data type by\ndescribing how its values are constructed. Each value is named by an expression\nbuilt only from the constructors of the data type, in other words a term. Moreover,\ndifferent terms denote different values (provided there are no strictness \ufb02ags). We\ncan de\ufb01ne functions on the data type by pattern matching on the constructors. There\nis therefore no need to state what the operations on the data type are \u2013 we can just\nde\ufb01ne them. Types in which the values are described, but the operations are not,\nare called concrete types.\nThe situation is exactly the reverse with abstract data types. Here the operations are\nnamed, but not how the values are constructed, at least not publicly. For example,\nFloat is an abstract data type; we are given the names of the primitive arithmetic\n\n\f8.6 A term representation\n\n195\n\nand comparison operations, and also a way of displaying \ufb02oating-point numbers,\nbut it is not stated how such numbers are actually represented. We cannot de\ufb01ne\nfunctions on these numbers by pattern matching, but only in terms of the given\noperations. What can and should be stated publicly are intended meanings and the\nalgebraic properties of the operations. However, Haskell provides no means for\nsuch descriptions beyond informal comments.\nAs it stands, Doc is a concrete type. But in our understanding of this type, different\nterms do not denote different values. For instance, we intend each constructor to be\na replacement for the corresponding operation. Thus\nnil\nline\ntext s\nnest i x\ngroup x\nx <> y\n\n=\n=\n=\n=\n=\n=\n\nNil\nLine\nText s\nNest i x\nGroup x\nx :<>: y\n\nWe also want to keep the algebraic properties of these operations, so equations such\nas\n(x :<>: y) :<>: z = x :<>: (y :<>: z)\nNest i (Nest j x) = Nest (i+j) x\nshould hold. But of course they do not. The solution is to use the module structure\nto hide the constructors of Doc from the user and insist only that the laws are\n\u2018observably\u2019 true. For instance we require\nlayouts ((x :<>: y) :<>: z) = layouts (x :<>: (y :<>: z))\nThe only way we can observe documents is through layouts; from the user\u2019s point\nof view if two documents produce the same layouts, then they are essentially the\nsame document.\nLet\u2019s get back to programming. Here is one de\ufb01nition of layouts. It is just the laws\nof layouts that we saw earlier, but now expressed as a proper Haskell de\ufb01nition:\nlayouts\nlayouts\nlayouts\nlayouts\nlayouts\nlayouts\nlayouts\n\n:: Doc -> [Layout]\n(x :<>: y) = layouts x <++> layouts y\nNil\n= [\"\"]\nLine\n= [\"\\n\"]\n(Text s)\n= [s]\n(Nest i x) = map (nestl i) (layouts x)\n(Group x) = layouts (flatten x) ++ layouts x\n\n\fPretty-printing\n\n196\n\nThe function flatten is similarly de\ufb01ned by\nflatten\nflatten\nflatten\nflatten\nflatten\nflatten\nflatten\n\n:: Doc -> Doc\n(x :<>: y) = flatten x :<>: flatten y\nNil\n= Nil\nLine\n= Text \" \"\n(Text s)\n= Text s\n(Nest i x) = flatten x\n(Group x) = flatten x\n\nWith these de\ufb01nitions, our 24 laws are either true by de\ufb01nition, or are observably\ntrue in the sense above.\nThe de\ufb01nition of layouts is simple enough, but it is unnecessarily inef\ufb01cient.\nThere are two separate reasons why this is so. First, consider the function egotist\nde\ufb01ned by\negotist :: Int -> Doc\negotist n | n==0\n= nil\n| otherwise = egotist (n-1) <> text \"me\"\nThe document egotist n is a very boring one, and its sole layout consists of a\nstring of n repetitions of me. By the way, we could have expressed the de\ufb01nition\nusing Nil, (:<>:) and Text but, as we have said, we are not going to make these\nconstructors public. As it stands, the de\ufb01nition of egotist could have been made\nby a user of the library. Anyway, back to the main point, which is that the association of the (<>) operations is to the left, and it takes \u0398(n2 ) steps to compute its\nlayout(s). The (++) operations pile up to the left. The situation is entirely analogous to the fact that concat de\ufb01ned in terms of foldl is an order of magnitude\nless ef\ufb01cient than one de\ufb01ned in terms of foldr.\nThe second source of inef\ufb01ciency concerns nesting. For example, consider the\nfunction egoist de\ufb01ned by\negoist :: Int -> Doc\negoist n | n==0\n= nil\n| otherwise = nest 1 (text \"me\" <> egoist (n-1))\nThere are no line breaks in sight, so egoist n describes the same boring document\nas egotist n. But although the concatenation associates to the right, it still takes\nquadratic time to construct the layout. Each nesting operation is carried out by\nrunning through the entire document. Try it and see.\n\n\f8.6 A term representation\n\n197\n\nThe way to solve the \ufb01rst problem is to delay concatenation, representing a concatenated document by a list of its component documents. The way to solve the\nsecond problem is to delay nesting, representing a nested document by a pair consisting of an indentation to be applied only when necessary and the document it is\nto be applied to. Combining both solutions, we represent a document by a list of\nindentation-document pairs. Speci\ufb01cally, consider the function toDoc de\ufb01ned by\ntoDoc :: [(Int,Doc)] -> Doc\ntoDoc ids = foldr (:<>:) Nil [Nest i x | (i,x) <- ids]\nWe can now calculate a de\ufb01nition of a function layr such that\nlayr = layouts . toDoc\nand then de\ufb01ne a new version of layouts based on layr. We leave the details as\nan exercise, but here is the result:\nlayouts x = layr [(0,x)]\nlayr []\n=\nlayr ((i,x :<>: y):ids) =\nlayr ((i,Nil):ids)\n=\nlayr ((i,Line):ids)\n=\n\n[\"\"]\nlayr ((i,x):(i,y):ids)\nlayr ids\n['\\n':replicate i ' ' ++ ls\n| ls <- layr ids]\nlayr ((i,Text s):ids)\n= [s ++ ls | ls <- layr ids]\nlayr ((i,Nest j x):ids) = layr ((i+j,x):ids)\nlayr ((i,Group x):ids) = layr ((i,flatten x):ids) ++\nlayr ((i,x):ids)\n\nThis de\ufb01nition takes linear time for each layout. Exactly the same template is used\nfor the function pretty, which chooses a single best layout:\npretty w x = best w [(0,x)]\nwhere\nbest r []\n=\nbest r ((i,x :<>: y):ids) =\nbest r ((i,Nil):ids)\n=\nbest r ((i,Line):ids)\n=\n\n\"\"\nbest r ((i,x):(i,y):ids)\nbest r ids\n'\\n':replicate i ' ' ++\nbest (w-i) ids\nbest r ((i,Text s):ids)\n= s ++ best (r-length s) ids\nbest r ((i,Nest j x):ids) = best r ((i+j,x):ids)\nbest r ((i,Group x):ids) = better r\n(best r ((i,flatten x):ids))\n(best r ((i,x):ids))\n\n\f198\n\n```\n\n\n```\n\nPretty-printing\n\nThe \ufb01rst argument of best is the remaining space available on the current line.\nThis function is made local to the de\ufb01nition of pretty to avoid having to carry\naround the maximum line width w as an additional argument.\nThat leaves us with the problem of computing better r lx ly. Here we can\nmake use of the fact that the \ufb01rst line of lx is guaranteed to be at least as long as\nthe \ufb01rst line of ly. Thus it suf\ufb01ces to compare the length of the \ufb01rst line of lx with\nr. If the former \ufb01ts within the latter, we choose lx; otherwise we choose ly. We\ntherefore de\ufb01ne\nbetter r lx ly = if fits r lx then lx else ly\nBut we don\u2019t want to compute the length of the whole of the \ufb01rst line of lx since\nthat looks ahead too far. Instead, we take a more miserly approach:\nfits r _ | r<0 = False\nfits r []\n= True\nfits r (c:cs) = if c == '\\n' then True\nelse fits (r-1) cs\nFor exactly the same reason it is essential that the second and third arguments to\nbetter are computed lazily, that is, the two layouts are evaluated just enough to\ndetermine which is the better one, and no further.\nLet\u2019s revisit our troublesome paragraph:\nghci> putStrLn $ pretty 30 $ para pg\nThis is a fairly short\nparagraph with just twenty-two\nwords. The problem is that\npretty-printing it takes time,\nin fact 31.32 seconds.\n(0.00 secs, 1602992 bytes)\nMuch better. Exercise L discusses what we can say about the running time of\npretty.\nThe \ufb01nal task is to put our small library together as a module. Here is the main\ndeclaration:\nmodule Pretty\n(Doc, Layout,\nnil, line, text,\nnest, (<>), group,\nlayouts, pretty, layout) where\n\n\f8.7 Exercises\n\n199\n\nThe module name is Pretty and the \ufb01le containing the above declaration and the\nde\ufb01nitions of the library functions has to be saved in a \ufb01le called Pretty.lhs.\nThe module exports 11 entities. Firstly, there is the name Doc of the abstract type\nof documents. The constructors of this type are not exported. (By the way, if we did\nwant to export all the constructors we can write Doc () in the export list, and if we\nwanted just, say, Nil and Text, we can write Doc (Nil, Text).) Secondly, there\nis the name Layout which is just a synonym for String. The next eight constants\nand functions are the ones we have de\ufb01ned above. The \ufb01nal function layout is\nused for printing a layout:\nlayout :: Layout -> IO ()\nlayout = putStrLn\nAnd that\u2019s it. Of course, in a really useful library a number of additional combinators could be provided. For example, we could provide\n(<+>),(<|>) :: Doc -> Doc -> Doc\nx <+> y = x <> text \" \" <> y\nx <|> y = x <> line <> y\nspread,stack :: [Doc] -> Doc\nspread = foldr (<+>) nil\nstack = foldr (<|>) nil\nNo doubt the reader can think of many others.\n\n8.7 Exercises\nExercise A\nA picky user of the library wants just three layouts for a certain document:\nA B C\n\nA B\nC\n\nA\nB C\n\nCan the user do it with the given functions?\nExercise B\nThe layouts of a document are given as a list. But are they all different? Either\nprove that they are or give a counterexample.\n\n\fPretty-printing\n\n200\n\nBy the way, is it obvious from the laws that each document has a nonempty set of\nlayouts?\nExercise C\nThe next four exercises refer to the shallow embedding of Section 8.3. Prove, by\nequational reasoning, that\nnest i . nest j\n\n= nest (i + j)\n\nYou will need a subsidiary result about nestl, which you don\u2019t have to prove.\nExercise D\nContinuing on from the previous question, prove that\nnest i (group x) = group (nest i x)\nby equational reasoning (at the point level). Again, you will need a subsidiary result.\nExercise E\nContinuing on, prove that flatten . group = flatten. You will need a subsidiary result.\nExercise F\nThe \ufb01nal law is flatten . nest i = flatten. And, yes, you will need yet\nanother subsidiary result.\nExercise G\nWe said in the text that the prelude function lines breaks up a string on newline characters. In fact, lines treats a newline as a terminator character, so both\nlines \"hello\" and lines \"hello\\n\" return the same result. It is arguable that\na better de\ufb01nition treats newlines as separator characters, so there is always one\nmore line than there are newlines. De\ufb01ne a function lines that has this behaviour.\nWe will need the new de\ufb01nition below.\nNow, the proof that map shape applied to the layouts of a document returns a\nlexicographically decreasing sequence of list of integers can be structured into the\nfollowing steps. First, de\ufb01ne\nmsl\n= map shape . layouts\nshape = map length . lines\n\n\f8.7 Exercises\n\n201\n\nwhere lines refers to the revised version above. We have to prove that msl returns\na decreasing sequence on every document. To this end, we can de\ufb01ne functions\nnesty and groupy so that\nnesty i . msl = msl . nest i\ngroupy . msl = msl . group\nand an operation <+> so that\nmsl x <+> msl y = msl (x <> y)\n(It is this equation that requires the revised de\ufb01nition of lines.) The proof is then\ncompleted by showing that if xs and ys are decreasing, then so are nesty i xs\nand groupy xs and xs <+> ys. All this exercise asks though is that you construct\nde\ufb01nitions of nesty, groupy and <+>.\nExercise H\nWrite a function doc :: Doc -> Doc that describes how to lay out elements of\nDoc where Doc is the abstract syntax tree representation in Section 8.6.\nExercise I\nConsider a function prettybad that chooses a best layout from the list layouts\nby taking the \ufb01rst layout all of whose lines \ufb01t within the given width, and the last\nlayout if this is not possible. Does prettybad always compute the same layout as\npretty? (Hint: think about paragraphs.)\nExercise J\nUsing the algebraic properties of the constructors of Doc, calculate the ef\ufb01cient\nversion of layouts.\nExercise K\nWe have designed pretty w to be optimal, meaning that it chooses line breaks to\navoid over\ufb02owing lines if at all possible. We also have that pretty w is bounded,\nmeaning that it can make the choice about the next line break without looking at\nmore than the next w characters of the input. Given that, what do you expect GHCi\u2019s\nresponse would be to the commands\nlayout $ pretty 5 $ para pg\nlayout $ pretty 10 $ cexpr ce\nwhere\n\n\fPretty-printing\n\n202\n\npg = \"Hello World!\" ++ undefined\nce = If \"happy\" (Expr \"great\") undefined\nExercise L\nWe cannot relate the cost of pretty w x to the size of x without saying what the\nsize of a document is. Here is a reasonable measure:\nsize\nsize\nsize\nsize\nsize\nsize\nsize\n\n:: Doc -> Int\nNil\n=\nLine\n=\n(Text s)\n=\n(Nest i x) =\n(x :<>: y) =\n(Group x)\n=\n\n1\n1\n1\n1 + size x\n1 + size x + size y\n1 + size x\n\nUnder this de\ufb01nition both the documents\nnest 20 (line <> text \"!\")\nnest 40 (line <> text \"!\")\nhave size two. But it takes twice as long to produce the second layout, so the cost\nof pretty cannot be linear in the document size.\nInstead of having pretty produce the \ufb01nal layout, a string, we can interpose an\nadditional data type of layouts:\ndata Layout = Empty\n| String String Layout\n| Break Int Layout\nand de\ufb01ne layout :: Layout -> String by\nlayout Empty\n= \"\"\nlayout (String s x) = s ++ layout x\nlayout (Break i x) = '\\n':replicate i ' ' ++ layout x\nWe have\npretty w = layout . prettyl w\nwhere the new function prettyl produces a Layout rather than a string. De\ufb01ne\nprettyl.\nA fairer question to ask is whether prettyl w x takes linear time in the size of x.\nDoes it?\n\n\f8.8 Answers\n\n203\n\n8.8 Answers\nAnswer to Exercise A\nNo. There is no way of allowing both A<0>B<1>C and A<1>B<0>C without also\nhaving both of A<0>B<0>C and A<1>B<1>C. These four are given by the expression\ngroup (A <> line <> B) <> group (line <> C)\nAnswer to Exercise B\nThe layouts of a document are not necessarily all different. For example\nlayouts (group (text \"hello\")) = [\"hello\",\"hello\"]\nYes, it is obvious that each document has a nonempty set of layouts. Look at the\nlaws of layouts. The basic documents have a nonempty list of layouts and this\nproperty is preserved by the other operations.\nAnswer to Exercise C\nThe calculation is:\nnest i . nest j\n=\n\n{de\ufb01nition of nest}\nmap (nestl i) . map (nestl j)\n\n=\n\n{functor law of map}\nmap (nestl i . nestl j)\n\n=\n\n{claim}\nmap (nestl (i+j))\n\n=\n\n{de\ufb01nition of nest}\nnest (i+j)\n\nThe claim is that nestl i . nestl j = nestl (i+j), which follows \u2013 after a\nshort calculation \u2013 from\nindent (i+j) = concat . map (indent i) . indent j\nWe omit the proof.\n\n\fPretty-printing\n\n204\n\nAnswer to Exercise D\nWe reason:\nnest i (group x)\n{de\ufb01nition of group}\n\n=\n\nnest i (flatten x ++ x)\n{since nest i = map (nestl i)}\n\n=\n\nnest i (flatten x) ++ nest i x\n{claim}\n\n=\n\nflatten (nest i x) ++ nest i x\n{de\ufb01nition of group}\n\n=\n\ngroup (nest i x)\nThe claim follows from\nnest i . flatten\n=\n\n{since there are no newlines in flatten x}\nflatten\n\n=\n\n{since flatten . nest i = flatten (Exercise F)}\nflatten . nest i\n\nAnswer to Exercise E\nWe reason:\nflatten . group\n=\n\n{de\ufb01nition of flatten and group}\none . flattenl . flattenl . head\n\n=\n\n{claim}\none . flattenl . head\n\n=\n\n{de\ufb01nition of flatten}\nflatten\n\nThe claim is that flattenl is idempotent:\nflattenl . flattenl = flattenl\nThis follows because flattenl returns a layout with no newlines.\n\n\f8.8 Answers\n\n205\n\nBy the way, it is the idempotence of flattenl that ensures all layouts of a document \ufb02atten to the same string. The only function that introduces multiple layouts\nis group, whose de\ufb01nition is\ngroup x = flatten x ++ x\nWe have therefore to show that \ufb02attening the \ufb01rst element of this list gives the same\nstring as \ufb02attening the second element. Thus we need to show\nflattenl . head . flatten = flattenl . head\nThis follows at once from the de\ufb01nition of flatten and the idempotence of the\nfunction flattenl.\nAnswer to Exercise F\nWe reason:\nflatten . nest i\n=\n\n{de\ufb01nitions}\none . flattenl . head . map (nestl i)\n\n=\n\n{since head . map f = f . head}\none . flattenl . nestl i . head\n\n=\n\n{claim}\none . flattenl . head\n\n=\n\n{de\ufb01nition of flatten}\nflatten\n\nThe claim is that flattenl . nestl i = flattenl.\nAnswer to Exercise G\nWe can de\ufb01ne\nlines xs = if null zs then [ys]\nelse ys:lines (tail zs)\nwhere (ys,zs) = break (=='\\n') xs\nThe function groupy is de\ufb01ned by\ngroupy :: [[Int]] -> [[Int]]\ngroupy (xs:xss) = [sum xs + length xs - 1]:xs:xss\nThe function nesty is de\ufb01ned by\n\n\fPretty-printing\n\n206\n\nnesty :: :: Int -> [[Int]] -> [[Int]]\nnesty i = map (add i)\nwhere add i (x:xs) = x:[i+x | x <- xs]\nThe function (<+>) is de\ufb01ned by\n(<+>) :: [[Int]] -> [[Int]] -> [[Int]]\nxss <+> yss = [glue xs ys | xs <- xss, ys <- yss]\nwhere glue xs ys = init xs ++ [last xs + head ys] ++\ntail ys\nAnswer to Exercise H\nOne possibility, which no doubt can be improved on:\ndoc\ndoc\ndoc\ndoc\ndoc\n\n:: Doc -> Doc\nNil\n= text \"Nil\"\nLine\n= text \"Line\"\n(Text s)\n= text (\"Text \" ++ show s)\n(Nest i x) = text (\"Nest \" ++ show i) <>\ngroup (nest 2 (line <> paren (doc x)))\ndoc (x :<>: y) = doc x <> text \" :<>:\" <>\ngroup (line <> nest 3 (doc y))\ndoc (Group x) = text \"Group \" <>\ngroup (nest 2 (line <> paren (doc x)))\nparen x = text \"(\" <> nest 1 x <> text \")\"\nAnswer to Exercise I\nNo. Consider a paragraph whose longest word is one character longer than the\nline width. In this case, prettybad will lay out each word on a single line, while\npretty will still \ufb01ll lines with groups of words provided they \ufb01t. For example:\nghci> putStrLn $ pretty 11 $ para pg4\nA lost and\nlonely\nhippopotamus\nwent into a\nbar.\n\n\f8.8 Answers\n\nAnswer to Exercise J\nFirst we show layouts x = layr [(0,x)]:\nlayr [(0,x)]\n=\n\n{de\ufb01nition of layr}\nlayouts (toDoc [(0,x)])\n\n=\n\n{de\ufb01nition of toDoc}\nlayouts (Nest 0 x :<>: Nil)\n\n=\n\n{laws of Doc}\nlayouts x\n\nIt remains to give a recursive de\ufb01nition of layr. We will just give two clauses:\ntoDoc ((i,Nest j x):ids)\n{de\ufb01nition of toDoc}\n\n=\n\nNest i (Nest j x) :<>: toDoc ids\n{laws}\n\n=\n\nNest (i+j) x :<>: toDoc ids\n{de\ufb01nition of toDoc}\n\n=\n\ntoDoc ((i+j x):ids)\nHence layr ((i,Nest j x):ids) = layr ((i+j x):ids). Next:\ntoDoc ((i,x:<>:y):ids)\n=\n\n{de\ufb01nition of toDoc}\nNest i (x :<>: y) <> toDoc ids\n\n=\n\n{laws}\nNest i x :<>: Nest i y :<>: toDoc ids\n\n=\n\n{de\ufb01nition of toDoc}\ntoDoc ((i,x):(i,y):ids)\n\nHence layr ((i,x:<>:y):ids) = layr ((i,x):(i,y):ids).\nAnswer to Exercise K\nghci> layout $ pretty 5 $ para pg\nHello\nWorld1*** Exception: Prelude.undefined\n\n207\n\n\fPretty-printing\n\n208\n\nghci> layout $ pretty 10 $ cexpr ce\nif happy\nthen great\nelse *** Exception: Prelude.undefined\nAnswer to Exercise L\nThe de\ufb01nition is\nprettyl\nprettyl\nwhere\nbest r\nbest r\nbest r\nbest r\nbest r\nbest r\nbest r\n\n:: Int -> Doc -> Layout\nw x = best w [(0,x)]\n[]\n((i,Nil):ids)\n((i,Line):ids)\n((i,Text s):ids)\n((i,Nest j x):ids)\n((i,x :<>: y):ids)\n((i,Group x):ids)\n\n=\n=\n=\n=\n=\n=\n=\n\nEmpty\nbest r ids\nBreak i (best (w-i) ids)\nString s (best (r-length s) ids)\nbest r ((i+j,x):ids)\nbest r ((i,x):(i,y):ids)\nbetter r\n(best r ((i,flatten x):ids))\n(best r ((i,x):ids))\n\nwhere better is changed to read\nbetter r lx ly = if fits r (layout lx) then lx else ly\nThe number of steps required to evaluate better r is proportional to r and thus\nat most w.\nNow, prettyl takes linear time if best does. The second argument of best is a\nlist of indentation-document pairs, and we can de\ufb01ne the size of this list by\nisize ids = sum [size x | (i,x) <- ids]\nFor each of the inner \ufb01ve clauses in the de\ufb01nition of best, the size decreases by 1.\nFor instance\nisize ((i,x :<>: y):ids)\n= size (x :<> y) + isize ids\n= 1 + size x + size y + isize ids\n= 1 + isize ((i,x):(i,y):ids)\nIt follows that if we let T(s) denote the running time of best r on an input of size\ns, then T(0) = \u0398(1) from the \ufb01rst clause of best, and T(s+1) = \u0398(1) + T(s) for\n\n\f8.9 Chapter notes\n\n209\n\neach of the \ufb01ve inner clauses, and\nT(s+1) = \u0398(w) + maximum [T(k) + T(s\u2212k)|k \u2190 [1 .. s\u22121]]\nfor the last clause. And now we can deduce that T(s) = \u0398(ws).\nIn conclusion, our algorithm for pretty is linear, though not independently of w.\n\n8.9 Chapter notes\nWe referred to pretty-printing as a library, but another name for it is an embedded\ndomain speci\ufb01c language (EDSL). It is a language for pretty-printing documents\nembedded in the host language Haskell. Many people believe that the growing\nsuccess of Haskell is due to its ability to host a variety of EDSLs without fuss.\nThe detailed material in this chapter has been based closely on work by Philip\nWadler, see \u2018A prettier printer\u2019, Chapter 11 in The Fun of Programming in Cornerstones of Computing Series (Palgrave MacMillan, 2003). The main difference is\nthat Wadler used an explicit alternation operator in the term representation of Doc\n(though it was hidden from the user) rather than the constructor Group. Jeremy\nGibbons suggested that the latter was a better \ufb01t with the idea of a deep embedding.\nAn earlier functional pretty-printing library based on a different set of combinators\nwas described by John Hughes, \u2018The design of a pretty-printer library\u2019, in Johan\nJeuring and Erik Meijer, editors, Advanced Functional Programming, volume 925\nof LNCS, Springer, 1995. Hughes\u2019 library was later reworked by Simon Peyton\nJones and installed as a Haskell library\nText.PrettyPrint.HughesPJ\nAnother pretty-printing library, in an imperative rather than functional style, was\nconstructed 30 years ago by Derek Oppen, \u2018Pretty-printing\u2019. ACM Transactions\non Programming Languages and Systems 2(4), 465\u2013483, 1980 and is widely used\nas the basis of pretty-printing facilities in a number of languages. More recently,\nef\ufb01cient pretty-printing algorithms in a functional style have been described by\nOlaf Chitil, \u2018Pretty printing with lazy dequeues\u2019, ACM Transactions on Programming Languages and Systems 27(1),163\u2013184, 2005, and by Olaf Chitil and Doaitse\nSwierstra, \u2018Linear, bounded, functional pretty-printing\u2019, Journal of Functional Programming 19(1), 1\u201316, 2009. These algorithms are considerably more complicated\nthan the one described in the text.\n\n```\n\n\n```\n\n\fChapter 9\nIn\ufb01nite lists\n\nWe have already met in\ufb01nite lists in Chapter 4 and even given an induction principle for reasoning about them in Chapter 6. But we haven\u2019t really appreciated what\ncan be done with them. In this chapter we want to explain in more detail exactly\nwhat an in\ufb01nite list is, and how they can be represented by cyclic structures. We\nalso describe another useful method for reasoning about in\ufb01nite lists, and discuss\na number of intriguing examples in which in\ufb01nite and cyclic lists can be used to\ngood effect.\n\n9.1 Review\nRecall that [m..] denotes the in\ufb01nite list of all integers from m onwards:\nghci> [1..]\n[1,2,3,4,5,6,7,{Interrupted}\nghci> zip [1..] \"hallo\"\n[(1,'h'),(2,'a'),(3,'l'),(4,'l'),(5,'o')]\nIt would take forever to print [1..], so we interrupt the \ufb01rst computation. The\nsecond example illustrates a simple but typical use of in\ufb01nite lists in \ufb01nite computations.\nIn Haskell, the arithmetic expression [m..] is translated into enumFrom m, where\nenumFrom is a method in the Enum class, and de\ufb01ned by\nenumFrom :: Integer -> [Integer]\nenumFrom m = m:enumFrom (m+1)\n\n\f9.1 Review\n\n211\n\nThus [m..] is de\ufb01ned as an instance of a recursively de\ufb01ned function. The computation makes progress because (:) is non-strict in its second argument.\nIt is important to bear in mind that in\ufb01nite lists in computing do not have the same\nproperties as in\ufb01nite sets do in mathematics. For example, in set theory\n{x | x \u2208 {1, 2, 3, . . .}, x2 < 10}\ndenotes the set {1, 2, 3}, but\nghci> [x | x <- [1..], x*x < 10]\n[1,2,3\nAfter printing the \ufb01rst three values the computer gets stuck in an in\ufb01nite loop looking for the next number after 3 whose square is less than 10. The value of the\nexpression above is the partial list 1:2:3:undefined.\nIt is possible to have an in\ufb01nite list of in\ufb01nite lists. For example,\nmultiples = [map (n*) [1..] | n <- [2..]]\nde\ufb01nes an in\ufb01nite list of in\ufb01nite lists of numbers, the \ufb01rst three being\n[2,4,6,8,...]\n\n[3,6,9,12,...]\n\n[4,8,12,16,...]\n\nSuppose we ask whether the above list of lists can be merged back into a single\nlist, namely [2..]. We can certainly merge two in\ufb01nite lists:\nmerge :: Ord a => [a]\nmerge (x:xs) (y:ys) |\n|\n|\n\n-> [a]\nxy =\n\n-> [a]\nx:merge xs (y:ys)\nx:merge xs ys\ny:merge (x:xs) ys\n\nThis version of merge removes duplicates: if the two arguments are in strictly\nincreasing order, so is the result. Note the absence of any clauses of merge mentioning the empty list. Now it seems that, if we de\ufb01ne\nmergeAll = foldr1 merge\nthen mergeAll multiples will return the in\ufb01nite list [2..]. But it doesn\u2019t. What\nhappens is that the computer gets stuck in an in\ufb01nite loop attempting to compute\nthe \ufb01rst element of the result, namely\nminimum (map head multiples)\nIt is simply not possible to compute the minimum element in an in\ufb01nite list. Instead\nwe have to make use of the fact that map head multiples is in strictly increasing\norder, and de\ufb01ne\n\n\fIn\ufb01nite lists\n\n212\n\nmergeAll = foldr1 xmerge\nxmerge (x:xs) ys = x:merge xs ys\nWith this de\ufb01nition, mergeAll multiples does indeed return [2..].\nFinally, recall the induction principle described in Chapter 6 for proving facts about\nin\ufb01nite lists. Provided P is a chain-complete assertion, we can prove that P(xs)\nholds for all in\ufb01nite lists xs by showing that: (i) P(undefined) holds; and (ii)\nP(xs) implies P(x:xs) for all x and xs. Using this principle, we proved in Chapter 6 that xs++ys = xs for all in\ufb01nite lists xs. But it\u2019s not immediately clear how\ninduction can be used to prove, for example,\nmap fact [0..] = scanl (*) 1 [1..]\nThe obvious result to prove is\nmap fact [0..n] = scanl (*) 1 [1..n]\nfor all n, but can one then assert the \ufb01rst identity holds?\n\n9.2 Cyclic lists\nData structures, like functions, can be de\ufb01ned recursively. For instance\nones :: [Int]\nones = 1:ones\nThis is an example of a cyclic list, a list whose de\ufb01nition is recursive. Contrast this\nde\ufb01nition with ones = repeat 1, where\nrepeat x = x:repeat x\nThis de\ufb01nition of ones creates an in\ufb01nite, not a cyclic list. We could de\ufb01ne\nrepeat x = xs\n\nwhere xs = x:xs\n\nNow the function repeat is de\ufb01ned in terms of a cyclic list. The second de\ufb01nition\n(call it repeat2) is faster to evaluate than the \ufb01rst (call it repeat1) because there\nis less overhead:\nghci> last $ take 10000000 $ repeat1 1\n1\n(2.95 secs, 800443676 bytes)\nghci> last $ take 10000000 $ repeat2 1\n1\n\n\f9.2 Cyclic lists\n\n213\n\n(0.11 secs, 280465164 bytes)\nAs another example, consider the following three de\ufb01nitions of the standard prelude function iterate:\niterate1 f x = x:iterate1 f (f x)\niterate2 f x = xs where xs = x:map f xs\niterate3 f x = x:map f (iterate3 f x)\nAll three functions have type (a -> a) -> a -> [a] and produce an in\ufb01nite\nlist of the iterates of f applied to x. The three functions are equal, but the induction\nprinciple reviewed earlier doesn\u2019t seem to be applicable in proving this assertion\nbecause there is no obvious argument on which to perform the induction. More on\nthis later. The \ufb01rst de\ufb01nition is the one used in the standard prelude, but it does\nnot create a cyclic list. The second de\ufb01nition does, and the third is obtained from\nthe second by eliminating the where clause. Assuming f x can be computed in\nconstant time, the \ufb01rst de\ufb01nition takes \u0398(n) steps to compute the \ufb01rst n elements\nof the result, but the third takes \u0398(n2 ) steps:\niterate3 (2*) 1\n= 1:map (2*) (iterate3 (2*1))\n= 1:2:map (2*) (map (2*) (iterate3 (2*1)))\n= 1:2:4:map (2*) (map (2*) (map (2*) (iterate3 (2*1))))\nEvaluating the nth element requires n applications of (2*), so it takes \u0398(n2 ) to\nproduce the \ufb01rst n elements.\nThat leaves the second de\ufb01nition. Does it take linear or quadratic time? The evaluation of iterate2 (2*) 1 proceeds as follows:\nxs\n= 1:ys\n= 1:2:zs\n= 1:2:4:ts\n\nwhere\nwhere\nwhere\nwhere\n\nxs\nys\nzs\nts\n\n=\n=\n=\n=\n\n1:map (2*) xs\nmap (2*) (1:ys)\nmap (2*) (2:zs)\nmap (2*) (4:ts)\n\nEach element of the result is produced in constant time, so iterate2 (2*) 1\ntakes \u0398(n) steps to produce n elements.\nLet us now develop a cyclic list to generate an in\ufb01nite list of all the primes. To start\nwith we de\ufb01ne\nprimes\n= [2..] \\\\ composites\ncomposites = mergeAll multiples\nmultiples = [map (n*) [n..] | n <- [2..]]\n\n\fIn\ufb01nite lists\n\n214\n\nwhere (\\\\) subtracts one strictly increasing list from another:\n(x:xs) \\\\ (y:ys) | xy\n\n= x:(xs \\\\ (y:ys))\n= xs \\\\ ys\n= (x:xs) \\\\ ys\n\nHere, multiples consists of the list of all multiples of 2 from 4 onwards, all\nmultiples of 3 from 9 onwards, all multiples of 4 from 16 onwards, and so on.\nMerging the list gives the in\ufb01nite list of all the composite numbers, and taking\nits complement with respect to [2..] gives the primes. We saw the de\ufb01nition of\nmergeAll in the previous section.\nSo far, so good. But the algorithm can be made many times faster by observing\nthat too many multiples are being merged. For instance, having constructed the\nmultiples of 2 there is no need to construct the multiples of 4, or of 6, and so on.\nWhat we really would like to do is just to construct the multiples of the primes.\nThat leads to the idea of \u2018tying the recursive knot\u2019 and de\ufb01ning\nprimes = [2..] \\\\ composites\nwhere\ncomposites = mergeAll [map (p*) [p..] | p <- primes]\nWhat we have here is a cyclic de\ufb01nition of primes. It looks great, but does it\nwork? Unfortunately, it doesn\u2019t: primes produces the unde\ufb01ned list. In order to\ndetermine the \ufb01rst element of primes the computation requires the \ufb01rst element of\ncomposites, which in turn requires the \ufb01rst element of primes. The computation\ngets stuck in an in\ufb01nite loop. To solve the problem we have to pump-prime (!)\nthe computation by giving the computation the \ufb01rst prime explicitly. We have to\nrewrite the de\ufb01nition as\nprimes = 2:([3..] \\\\ composites)\nwhere\ncomposites = mergeAll [map (p*) [p..] | p <- primes]\nBut this still doesn\u2019t produce the primes! The reason is a subtle one and is quite\nhard to spot. It has to do with the de\ufb01nition\nmergeAll = foldr1 xmerge\nThe culprit is the function foldr1. Recall the Haskell de\ufb01nition:\nfoldr1 :: (a -> a -> a) -> [a] -> a\nfoldr1 f [x]\n= x\nfoldr1 f (x:xs) = f x (foldr1 xs)\n\n\f9.3 In\ufb01nite lists as limits\n\n215\n\nThe order of the two de\ufb01ning equations is signi\ufb01cant. In particular,\nfoldr1 f (x:undefined) = undefined\nbecause the list argument is \ufb01rst matched against x:[], causing the result to be\nunde\ufb01ned. That means\nmergeAll [map (p*) [p..] | p <- 2:undefined] = undefined\nWhat we wanted was\nmergeAll [map (p*) [p..] | p <- 2:undefined] = 4:undefined\nTo effect this change we have to de\ufb01ne mergeAll differently:\nmergeAll (xs:xss) = xmerge xs (mergeAll xss)\nNow we have\n=\n=\n=\n=\n\nmergeAll [map (p*) [p..] | p <- 2:undefined]\nxmerge (map (2*) [2..]) undefined\nxmerge (4:map (2*) [3..]) undefined\n4:merge (map (2*) [3..]) undefined\n4:undefined\n\nThis version of mergeAll behaves differently on \ufb01nite lists from the previous one.\nWhy?\nWith this \ufb01nal change we claim that primes does indeed get into gear and produces the primes. But how can the claim be proved? To answer this question we\nneed to know something about the semantics of recursively de\ufb01ned functions and\nother values in Haskell, and how in\ufb01nite lists are de\ufb01ned as limits of their partial\napproximations.\n\n9.3 In\ufb01nite lists as limits\nIn mathematics, certain values are de\ufb01ned as limits of in\ufb01nite sequences of approximations of simpler values. For example, the irrational number\n\u03c0 = 3.14159265358979323846 \u00b7 \u00b7 \u00b7\ncan be de\ufb01ned as the limit of the in\ufb01nite sequence of rational approximations\n3, 3.1, 3.14, 3.141, 3.1415, . . .\n\n\fIn\ufb01nite lists\n\n216\n\nThe \ufb01rst element of the sequence, 3, is a fairly crude approximation to \u03c0. The next\nelement, 3.1, is a little better; 3.14 is better still, and so on.\nSimilarly, an in\ufb01nite list can also be regarded as the limit of a sequence of approximations. For example, the in\ufb01nite list [1..] is the limit of the in\ufb01nite sequence of\npartial lists\n\u22a5, 1 :\u22a5, 1 : 2 :\u22a5, 1 : 2 : 3 :\u22a5, . . .\nAgain, the sequence consists of better and better approximations to the intended\nlimit. The \ufb01rst term, \u22a5, is the unde\ufb01ned element, and thus a very crude approximation: it tells us nothing about the limit. The next term, 1 :\u22a5, is a slightly better\napproximation: it tells us that the limit is a list whose \ufb01rst element is 1, but says\nnothing about the rest of the list. The following term, 1 : 2 :\u22a5, is a little better still,\nand so on. Each successively better approximation is derived by replacing \u22a5 with\na more de\ufb01ned value, and thus gives more information about the limit.\nHere is another sequence of approximations whose limit is [1..]:\n\u22a5, 1 : 2 :\u22a5, 1 : 2 : 3 : 4 :\u22a5, 1 : 2 : 3 : 4 : 5 : 6 :\u22a5, . . .\nThis sequence is a subsequence of the one above but it converges to the same limit.\nHere is a sequence of approximations that does not converge to a limit:\n\u22a5, 1 :\u22a5, 2 : 1 :\u22a5, 3 : 2 : 1 :\u22a5, . . .\nThe problem with this sequence is that it gives con\ufb02icting information: the second\nterm says that the limit begins with 1. However, the third term says that the limit\nbegins with 2, and the fourth term says that it begins with 3, and so on. No approximation tells us anything about the intended limit and the sequence does not\nconverge.\nIt should not be thought that the limit of a sequence of lists is necessarily in\ufb01nite.\nFor example, the sequence\n\u22a5,\n\n1 :\u22a5,\n\n1 : [ ],\n\n1 : [ ],\n\n...\n\nin which every element after the \ufb01rst two is [1], is a perfectly valid sequence with\nlimit [1]. Similarly,\n\u22a5,\n\n1 :\u22a5,\n\n1 : 2 :\u22a5,\n\n1 : 2 :\u22a5,\n\n...\n\nis a sequence with limit 1 : 2 :\u22a5. Finite and partial lists are limits of sequences\npossessing only a \ufb01nite number of distinct elements.\nThe way to formalise the property that an in\ufb01nite sequence of partial lists converges to a limit is to introduce the notion of an approximation ordering \u0011 on the\n\n\f9.3 In\ufb01nite lists as limits\n\n217\n\nelements of each type. The assertion x \u0011 y means that x is an approximation to y.\nThe ordering \u0011 will be re\ufb02exive (x \u0011 x), transitive (x \u0011 y and y \u0011 z implies x \u0011 z),\nand anti-symmetric (x \u0011 y and y \u0011 x implies x = y). However, it is not the case that\nevery pair of elements have to be comparable by \u0011. Thus \u0011 is what is known as a\npartial ordering. Note that \u0011 is a mathematical operator (like =), and not a Haskell\noperator returning boolean results.\nThe approximation ordering for numbers, booleans, characters and any other enumerated type, is de\ufb01ned by\nx \u0011 y \u2261 (x =\u22a5) \u2228 (x = y).\nThe \ufb01rst clause says that \u22a5 is an approximation to everything. In other words, \u22a5 is\nthe bottom element of the ordering. This explains why \u22a5 is pronounced \u2018bottom\u2019.\nThe value \u22a5 is the bottom element of \u0011 for every type. The above ordering is \ufb02at.\nWith a \ufb02at ordering one either knows everything there is to know about a value, or\none knows absolutely nothing.\nThe approximation ordering on the type (a, b) is de\ufb01ned by \u22a5\u0011 (x, y) and\n(x, y) \u0011 (x\u0013 , y\u0013 ) \u2261 (x \u0011 x\u0013 ) \u2227 (y \u0011 y\u0013 ).\nThe occurrences of \u0011 on the right refer to the orderings on the types a and b,\nrespectively. The ordering \u0011 on (a, b) is not \ufb02at, even when the component orderings are. For example, in (Bool,Bool) we have the following chain of distinct\nelements:\n\u22a5 \u0011 (\u22a5, \u22a5) \u0011 (\u22a5, False) \u0011 (True, False).\nNote that in Haskell the pair (\u22a5, \u22a5) is distinct from \u22a5:\nghci> let f (a,b) = 1\nghci> f (undefined,undefined)\n1\nghci> f undefined\n*** Exception: Prelude.undefined\nThe ordering \u0011 on [a] is de\ufb01ned by \u22a5\u0011 xs and (x:xs) \u0011 [] and\n[] \u0011 xs \u2261 xs = [],\n(x:xs) \u0011 (y:ys) \u2261 (x \u0011 y) \u2227 (xs \u0011 ys).\nThese equations should be read as an inductive de\ufb01nition of a mathematical assertion, not as a Haskell de\ufb01nition. The second condition says that [] approximates\nonly itself, and the third condition says that (x:xs) is an approximation to (y:ys)\nif and only if x is an approximation to y and xs is an approximation to ys. The \ufb01rst\n\n\fIn\ufb01nite lists\n\n218\n\noccurrence of \u0011 on the right-hand side refers to the approximation ordering on the\ntype a.\nAs two examples, we have\n[1, \u22a5, 3] \u0011 [1, 2, 3]\n\nand\n\n1 : 2 :\u22a5 \u0011 [1, 2, 3].\n\nHowever, 1 : 2 :\u22a5 and [1, \u22a5, 3] are not related by \u0011.\nThe approximation ordering for each type T is assumed to have another property\nin addition to those described above: each chain of approximations x0 \u0011 x1 \u0011. . .\nhas to possess a limit which is also a member of T. The limit, which we denote by\nlimn\u2192\u221e xn , is de\ufb01ned by two conditions:\n1. xn \u0011 limn\u2192\u221e xn for all n. This condition states that the limit is an upper bound\non the sequence of approximations.\n2. If xn \u0011 y for all n, then limn\u2192\u221e xn \u0011 y. This condition states that the limit is the\nleast upper bound.\nThe de\ufb01nition of the limit of a chain of approximations applies to every type. Partial orderings possessing this property are called complete, and every Haskell type\nis a complete partial ordering (CPO for short). In particular, the property, introduced in Chapter 6, of a mathematical assertion P being chain complete can now\nbe formalised as\n(\u2200n : P(xn )) \u21d2 P( lim xn ).\nn\u2192\u221e\n\nIn words, P holds in the limit if it holds for each approximation to the limit.\nFor lists there is a useful Haskell function approx, which produces approximations\nto a given list. The de\ufb01nition is\napprox :: Integer -> [a] -> [a]\napprox n []\n| n>0 = []\napprox n (x:xs) | n>0 = x:approx (n-1) xs\nThe de\ufb01nition of approx is very similar to that of take except that, by case exhaustion, we have approx 0 xs = undefined for all xs. For example,\napprox 0 [1] = undefined\napprox 1 [1] = 1:undefined\napprox 2 [1] = 1:[]\nThe crucial property of approx is that\nlim approx n xs = xs\n\nn\u2192\u221e\n\n\f9.3 In\ufb01nite lists as limits\n\n219\n\nfor all lists xs, \ufb01nite, partial or in\ufb01nite. The proof, an induction on xs, is left as an\nexercise.\nIt follows that if approx n xs = approx n ys for all natural numbers n, then\nxs = ys. Thus we can prove that\niterate f x = x:map f (iterate f x)\nby showing\napprox n (iterate f x) = approx n (x:map f (iterate f x))\nfor all natural numbers n. And, of course, we can use induction over the natural\nnumbers to establish this fact. The details are left as an easy exercise.\nAs another example, consider the value primes de\ufb01ned in the previous section.\nSuppose we de\ufb01ne\nprs n = approx n primes\nWe would like to show that prs n = p1 : p2 : \u00b7 \u00b7 \u00b7 pn :\u22a5, where pj is the jth prime.\nWe claim that\nprs n = approx n (2:([3..] \\\\ crs n))\ncrs n = mergeAll [map (p*) [p..] | p <- prs n]\nGiven this, it is suf\ufb01cient to show that crs n = c1 : c2 : \u00b7 \u00b7 \u00b7 cm :\u22a5, where cj is the jth\ncomposite number (so c1 = 4) and m = p2n . Then the proof is completed by using\nthe fact that pn+1 < p2n , which is a non-trivial result in Number Theory. Details are\nin the exercises.\n\nComputable functions and recursive de\ufb01nitions\nOne can describe many functions in mathematics, but only some of them are computable. There are two properties of computable functions not shared by arbitrary\nfunctions. Firstly, a computable function f is monotonic with respect to the approximation ordering. In symbols,\nx \u0011 y \u21d2 f (x) \u0011 f (y)\nfor all x and y. Roughly speaking, monotonicity states that the more information\nyou supply about the argument, the more information you get as a result. Secondly,\na computable function f is continuous, which means that\nf ( lim xn ) = lim f (xn )\nn\u2192\u221e\n\nn\u2192\u221e\n\n\fIn\ufb01nite lists\n\n220\n\nfor all chains of approximations x0 \u0011 x1 \u0011 . . .. Roughly speaking, continuity states\nthat there are no surprises on passing to the limit.\nContinuity appears similar to chain completeness but differs in two respects. One\nis that the chain completeness of P does not imply the converse property that if P\nis false for all approximations, then P is false for the limit. In other words, it does\nnot imply that \u00acP is chain complete. Secondly, P is a mathematical assertion, not\na Haskell function returning a boolean value.\nAlthough we won\u2019t prove it, every monotonic and continuous function f has a least\n\ufb01xed point. A \ufb01xed point of a function f is a value x such that f (x) = x. And x\nis a least \ufb01xed point if x \u0011 y for any other \ufb01xed point y. The least \ufb01xed point of\na monotonic and continuous function f is given by limn\u2192\u221e xn where x0 =\u22a5 and\nxn+1 = f (xn ). In functional programming, recursive de\ufb01nitions are interpreted as\nleast \ufb01xed points.\nHere are three examples. Consider the de\ufb01nition ones = 1:ones. This de\ufb01nition\nasserts that ones is a \ufb01xed point of the function (1:). Haskell interprets it as\nthe least \ufb01xed point, so ones = limn\u2192\u221e onesn , where ones0 =\u22a5 and onesn+1 =\n1:onesn . It is easy to see that onesn is the partial list consisting of n ones, so the\nlimit is indeed an in\ufb01nite list of ones.\nSecond, consider the factorial function\nfact n = if n==0 then 1 else n*fact (n-1)\nWe can rewrite this de\ufb01nition in the equivalent form\nfact = (\\f n -> if n==0 then 1 else n*f(n-1)) fact\nAgain, this de\ufb01nition asserts that fact is a \ufb01xed point of a function. Here we have\nfact0 n = \u22a5\nfact1 n = if n==0 then 1 else \u22a5\nfact2 n = if n<=1 then 1 else \u22a5\nand so on. The value of factk n is the factorial of n if n is less than k, and \u22a5\notherwise.\nFinally, consider the list primes once again. Here we have\n= \u22a5\nprimes0\nprimesn+1 = 2:([3..] \\\\\nmergeAll [map (p*) [p..] | p <- primesn ])\n\n\f9.4 Paper\u2013rock\u2013scissors\n\n221\n\nIt is not the case that primesn = approx n primes. In fact,\nprimes1 = 2 :\u22a5\nprimes2 = 2 : 3 :\u22a5\nprimes3 = 2 : 3 : 5 : 7 :\u22a5\nprimes4 = 2 : 3 : 5 : 7 : \u00b7 \u00b7 \u00b7 : 47 :\u22a5\nThe partial list primes2 produces all the primes less than 4, primes3 all the primes\nless than 9, and primes4 all the primes less than 49. And so on.\n\n9.4 Paper\u2013rock\u2013scissors\nOur next example of in\ufb01nite lists is entertaining as well as instructive. Not only\ndoes it introduce the idea of using potentially in\ufb01nite lists to model a sequence of\ninteractions between processes, it also provides another concrete illustration of the\nnecessity for formal analysis.\nThe paper\u2013rock\u2013scissors game is a familiar one to children, though it is known\nby different names in different places. The game is played by two people facing\none another. Behind their backs, each player forms a hand in the shape of either\na rock (a clenched \ufb01st), a piece of paper (a \ufb02at palm) or a pair of scissors (two\n\ufb01ngers extended). At a given instant, both players bring their hidden hand forward.\nThe winner is determined by the rule \u2018paper wraps rock, rock blunts scissors, and\nscissors cut paper\u2019. Thus, if player 1 produces a rock and player 2 produces a pair\nof scissors, then player 1 wins because rock blunts scissors. If both players produce\nthe same object, then the game is a tie and neither wins. The game continues in this\nfashion for a \ufb01xed number of rounds agreed in advance.\nOur objective in this section is to write a program to play and score the game. We\nbegin by introducing the types\ndata Move = Paper | Rock | Scissors\ntype Round = (Move,Move)\nTo score a round we de\ufb01ne\nscore :: Round -> (Int,Int)\nscore (x,y) | x `beats` y = (1,0)\n| y `beats` x = (0,1)\n| otherwise\n= (0,0)\nwhere\n\n\fIn\ufb01nite lists\n\n222\n\nPaper `beats` Rock\nRock `beats` Scissors\nScissors `beats` Paper\n_ `beats` _\n\n=\n=\n=\n=\n\nTrue\nTrue\nTrue\nFalse\n\nEach player in the game will be represented by a certain strategy. For instance, one\nsimple strategy is, after the \ufb01rst round, always to produce what the opposing player\nshowed in the previous round. This strategy will be called copy. Another strategy,\nwhich we will call smart, is to determine a move by analysing the number of times\nthe opponent has produced each of the three possible objects, and calculating an\nappropriate response based on probabilities.\nWe will consider the details of particular strategies, and how they can be represented, in a moment. For now, suppose the type Strategy is given in some way.\nThe function\nrounds :: (Strategy,Strategy) -> [Round]\ntakes a pair of strategies and returns the in\ufb01nite list of rounds that ensue when each\nplayer follows his or her assigned strategy. The function\nmatch :: Int -> (Strategy,Strategy) -> (Int,Int)\nmatch n = total . map score . take n . rounds\nwhere total rs = (sum (map fst rs),sum (map snd rs))\ndetermines the total score after playing a given number of rounds.\nThe instructive aspect of the game is how to represent strategies. We are going to\nconsider two ways, calling them Strategy1 and Strategy2. The obvious idea is\nto take\ntype Strategy1 = [Move] -> Move\nHere, a strategy is a function which takes the (\ufb01nite) list of moves made by the\nopponent so far and returns an appropriate move for the subsequent round. For\nef\ufb01ciency in processing lists, we suppose that the list of moves is given in reverse\norder, with the last move \ufb01rst.\nFor example, the copy1 strategy is implemented by\ncopy1 :: Strategy1\ncopy1 ms = if null ms then Rock else head ms\nThe \ufb01rst move is an arbitrary choice of Rock, The second strategy smart1 is implemented by\n\n\f9.4 Paper\u2013rock\u2013scissors\n\n223\n\nsmart1 :: Strategy1\nsmart1 ms = if null ms then Rock\nelse pick (foldr count (0,0,0) ms)\ncount\ncount\ncount\ncount\n\n:: Move -> (Int,Int,Int) -> (Int,Int,Int)\nPaper (p,r,s)\n= (p+1,r,s)\nRock (p,r,s)\n= (p,r+1,s)\nScissors (p,r,s) = (p,r,s+1)\n\npick :: (Int,Int,Int) -> Move\npick (p,r,s)\n| m < p\n= Scissors\n| m < p+r\n= Paper\n| otherwise = Rock\nwhere m = rand (p+r+s)\nThis strategy counts the number of times each move has been made, and uses the\nresults to pick a move. The value of rand applied to n is some integer m in the\nrange 0 \u2264 m < n. (Note that rand is never applied to the same integer.) Thus the\nchoice of move depends on whether m falls in one of the three ranges\n0 \u2264 m < p or p \u2264 m < p + r or p + r \u2264 m < p + r + s.\nFor example, if p is large, then Scissors will be chosen with high probability\n(because scissors cuts paper); and if r is large, then Paper will be chosen with\nhigh probability (because paper wraps rock); and so on.\nTo de\ufb01ne rand we can make use of two functions in the library System.Random:\nrand :: Int -> Int\nrand n = fst $ randomR (0,n-1) (mkStdGen n)\nThe function mkStdGen takes an integer and returns a random number generator,\nlikely to be different for different integers. The choice of argument to mkStdGen\nis arbitrary, and we have simply chosen n. The function randomR takes a range\n(a, b) and a random number generator, and returns a pseudo-random integer r in\nthe range a \u2264 r \u2264 b and a new random number generator.\nWe can now de\ufb01ne rounds1:\nrounds1 :: (Strategy1,Strategy1) -> [Round]\nrounds1 (p1,p2)\n= map head $ tail $ iterate (extend (p1,p2)) []\nextend (p1,p2) rs = (p1 (map snd rs),p2 (map fst rs)):rs\n\n\fIn\ufb01nite lists\n\n224\n\nThe function extend adds a new pair of moves to the front of the list of existing\nrounds, and rounds1 generates the in\ufb01nite list of rounds by repeatedly applying\nextend to the initially empty list. It is more ef\ufb01cient to add something to the front\nof a list than to the end, which is why we keep the list of moves in reverse order.\nNevertheless rounds1 is inef\ufb01cient. Suppose a strategy takes time proportional to\nthe length of its input to compute the next move. It follows that extend takes \u0398(n)\nsteps to update a game of n rounds with a new round. Therefore, it takes \u0398(N 2 )\nsteps to compute a game of N rounds.\nFor comparison, let\u2019s consider another way we might reasonably represent strategies. This time we take\ntype Strategy2 = [Move] -> [Move]\nIn the new representation, a strategy is a function that takes the potentially in\ufb01nite list of moves made by the opponent and returns the potentially in\ufb01nite list of\nreplies. For example, the copy strategy is now implemented by\ncopy2 :: Strategy2\ncopy2 ms = Rock:ms\nThis strategy returns Rock the \ufb01rst time, and thereafter returns just the move made\nby the opponent in the previous round. The smart strategy is reprogrammed as\nsmart2 :: Strategy2\nsmart2 ms = Rock:map pick (stats ms)\nwhere stats = tail . scanl (flip count) (0,0,0)\nThe function stats computes the running counts of the three possible moves. This\nstrategy, like copy2, is also ef\ufb01cient in that it produces each successive output with\nconstant delay.\nWith this new model of strategies we can rede\ufb01ne the function rounds:\nrounds2 :: (Strategy2,Strategy2) -> [Round]\nrounds2 (p1,p2) = zip xs ys\nwhere xs = p1 ys\nys = p2 xs\nHere, xs is the list of replies computed by the \ufb01rst player in response to the list\nys which, in turn, is the list of replies made by the second player in response\nto the list of moves xs. Thus rounds2 is de\ufb01ned by two cyclic lists and we are\nobliged to show that it does indeed generate an in\ufb01nite list of well-de\ufb01ned moves.\nMore on this below. If the two players do encapsulate legitimate strategies, then\n\n\f9.4 Paper\u2013rock\u2013scissors\n\n225\n\nrounds2 computes the \ufb01rst n moves of the game in \u0398(n) steps, assuming that both\nplayers compute each new move with constant delay. Thus the second method for\nmodelling strategies leads to a more ef\ufb01cient program than the earlier one.\nUnfortunately, there is a crucial \ufb02aw with the second representation of strategies:\nit offers no protection against someone who cheats! Consider the strategy\ncheat ms = map trump ms\ntrump Paper\n= Scissors\ntrump Rock\n= Paper\ntrump Scissors = Rock\nThe \ufb01rst reply of cheat is the move guaranteed to beat the opponent\u2019s \ufb01rst move;\nsimilarly for subsequent moves. To see that cheat cannot be prevented from subverting the game, consider a match in which it is played against copy2, and let\nxs = cheat ys and ys = copy2 xs. The lists xs and ys are the limits of the two\nchains {xsn | 0 \u2264 n} and {ysn | 0 \u2264 n}, where xs0 = \u22a5 and xsn+1 = cheat ysn ,\nand ys0 = \u22a5 and ysn+1 = copy2 xsn . Now, we have\nxs1\nys1\nxs2\nys2\nxs3\nys3\n\n=\n=\n=\n=\n=\n=\n\ncheat\ncopy2\ncheat\ncopy2\ncheat\ncopy2\n\n\u22a5\n\u22a5\n(Rock: \u22a5 )\n\u22a5\n(Rock: \u22a5 )\n(Paper: \u22a5 )\n\n=\n=\n=\n=\n=\n=\n\n\u22a5\nRock: \u22a5\nPaper: \u22a5\nRock: \u22a5\nPaper: \u22a5\nRock:Paper: \u22a5\n\nContinuing in this way, we see that the limits of these sequences are indeed in\ufb01nite\nlists of well-de\ufb01ned moves. Moreover, cheat always triumphs. Another cheating\nstrategy is given by\ndevious :: Int -> Strategy2\ndevious n ms = take n (copy2 ms) ++ cheat (drop n ms)\nThis strategy behaves like copy for n moves then starts to cheat.\nCan we \ufb01nd a way to protect against cheats? To answer this question, we need\nto take a closer look at what constitutes an honest strategy. Informally speaking,\na strategy is honest if its \ufb01rst move is computed in the absence of any information about the opponent\u2019s \ufb01rst move, the second move is computed without any\ninformation about the opponent\u2019s second move, and so on. Moreover, each of these\nmoves should be well-de\ufb01ned, given that the opponent\u2019s moves are well-de\ufb01ned.\nMore precisely, let wdf (n, ms) denote the assertion that the \ufb01rst n elements in the\n\n\f226\n\nIn\ufb01nite lists\n\n(possibly partial) list of moves ms are well-de\ufb01ned. Then a strategy f is honest if\nwdf (n, ms) \u21d2 wdf (n+1, f (ms))\nfor all n and ms. It is easy to show that copy2 is honest. On the other hand, cheat\nis not honest because wdf (0, \u22a5) is true but wdf (1, cheat \u22a5) is false. The strategy\ndozy, where\ndozy ms = repeat undefined\nis also dishonest according to this de\ufb01nition although it doesn\u2019t actually cheat.\nHaving identi\ufb01ed the source of criminal or lackadaisical behaviour, can we ensure\nthat only honest strategies are admitted to the game? The answer is a quali\ufb01ed\nyes: although it is not possible for a mechanical evaluator to recognise cheating\n(in the same way that it is not possible to recognise \u22a5, or strategies that do not\nreturn well-de\ufb01ned moves), it is possible to de\ufb01ne a function police so that if\np is an honest player and ms is an in\ufb01nite sequence of well-de\ufb01ned moves, then\npolice p ms = p ms. On the other hand, if p is dishonest at some point, then\nthe game ends at that point in \u22a5. Operationally speaking, police works by forcing\np to return the \ufb01rst (well-de\ufb01ned!) element of its output before it gives p the \ufb01rst\nelement of its input. Similarly for the other elements. The de\ufb01nition is\npolice p ms = ms' where ms' = p (synch ms ms')\nsynch (x:xs) (y:ys) = (y `seq` x):synch xs ys\nRecall from Chapter 7 that x `seq` y evaluates x before returning the value of\ny. The proof that this implementation meets its speci\ufb01cation is rather involved, so\nwe are not going into details. It follows from the above analysis that to prevent\ncheating we must rewrite the de\ufb01nition of rounds2 to read\nrounds2 (p1,p2) = zip xs ys\nwhere xs = police p1 ys\nys = police p2 xs\n\n9.5 Stream-based interaction\nIn the paper\u2013rock\u2013scissors game we modelled interaction by a function that took\nan in\ufb01nite list of moves and returned a similar list. The same idea can be used\nto provide a simple model of input\u2013output interaction. It\u2019s called stream-based\ninteraction because in\ufb01nite lists are also called streams. Haskell provides a function\ninteract :: ([Char] -> [Char]) -> IO ()\n\n\f9.5 Stream-based interaction\n\n227\n\nfor interacting with the world. The argument to interact is a function that takes\na potentially in\ufb01nite list of characters from the standard input channel, and returns\na potentially in\ufb01nite list of characters to be typed on the standard output channel.\nFor example,\nghci> import Data.Char\nghci> interact (map toUpper)\nhello world!\nHELLO WORLD!\nGoodbye, cruel world!\nGOODBYE, CRUEL WORLD!\n{Interrupted}\nWe imported the library Data.Char to make toUpper available, and then created\nan interaction that capitalised each letter. Each time a line of input was typed (and\nechoed) the interaction produced the same line in capital letters. The process continues until we interrupt it.\nWe can also design an interactive program that terminates. For example,\ninteract (map toUpper . takeWhile (/= '.'))\nwill interact as above but terminate as soon as a line containing a period is typed:\nghci> interact (map toUpper . takeWhile (/= '.'))\nGoodbye. Forever\nGOODBYE\nFinally, here is a stand-alone program that takes a literate Haskell \ufb01le as input\nand returns a \ufb01le in which all nonempty lines not beginning with > are removed.\nThe remaining lines are modi\ufb01ed by removing the > character, so the result is a\nlegitimate .hs \ufb01le (a Haskell script not using the literate style):\nmain\nreplace\ncode xs\ncleanup\n\n= interact replace\n= unlines . map cleanup . filter code . lines\n= null xs || head xs == '>'\nxs = if null xs then [] else tail xs\n\nThe program is the computation associated with the identi\ufb01er main, and there always has to be a de\ufb01nition associated with this name if we want to compile a program. The function lines splits a text into lines, and unlines reassembles the text\nby putting a single newline between lines. If we store the program in lhs2hs.lhs,\nwe can compile it and then run it:\n\n\fIn\ufb01nite lists\n\n228\n\n$ ghc lhs2hs.lhs\n$ lhs2hs myscript.hs\nIn the second line, the input is taken from myscript.lhs and the output is directed\nto myscript.hs.\nStream-based interaction was the main method for interacting with the outside\nworld in early versions of Haskell. However, the model presented above is too\nsimple for most practical purposes. In a serious application one wants to do other\nthings than reading and printing characters to a screen. For example, one also wants\nto open and read \ufb01les, to write to or delete \ufb01les, and in general to interact with all\nthe mechanisms that are available in the world outside the con\ufb01nes of a functional\nprogramming language. Interaction takes place in time, and the order in which\nevents occur has to be managed correctly by the programmer. In the stream-based\napproach, this ordering of events is represented by the order of the elements in a\nlist; in other words, it is represented in the data and not re\ufb02ected primarily in the\nway the program is composed. In the following chapter we will consider another\napproach to interaction, indeed, a general method for writing programs that have to\ncontrol an orderly sequence of events. In this approach, the order is made explicit\nin the way the program is composed.\n\n9.6 Doubly-linked lists\nWe end with another application of cyclic lists. Imagine reading a book consisting\nof a nonempty list of pages. To navigate around the book we need some way of\nmoving on to the next page and moving back to the previous page. Other navigation\ntools would be useful, but we\u2019ll stick with these two. Here is an interactive session\nwith a particularly boring book book consisting of three pages:\nghci>\n\"Page\nghci>\n\"Page\nghci>\n\"Page\nghci>\n\"Page\nghci>\n\"Page\n\nstart book\n1\"\nnext it\n2\"\nprev it\n1\"\nnext it\n2\"\nnext it\n3\"\n\n\f9.6 Doubly-linked lists\n\n229\n\nIn GHCi the variable it is bound to the expression just typed at the prompt. We\nstarted a book and what was printed was the \ufb01rst page. We turned to the next\npage, and then returned to the previous one. The interesting question is what should\nhappen when we turn to the next page after the last one. Should the navigation\nreport an error, just deliver the last page again or go to the \ufb01rst page? Suppose we\ndecide on the last alternative, namely that the next page after the last one should be\nthe \ufb01rst page, and the previous page before the \ufb01rst one should be the last page. In\nother words, our book is an instance of a cyclic doubly-linked list.\nHere is the relevant datatype declaration:\ndata DList a = Cons a (DList a) (DList a)\nelem :: DList a -> a\nelem (Cons a p n) = a\nprev,next :: DList a -> DList a\nprev (Cons a p n) = p\nnext (Cons a p n) = n\nWe print a doubly-linked list by displaying the current entry:\ninstance Show a => Show (DList a)\nwhere show d = show (elem d)\nOur book is then a list [p1,p2,p3] of three pages, where\np1 = Cons \"Page 1\" p3 p2\np2 = Cons \"Page 2\" p1 p3\np3 = Cons \"Page 3\" p2 p1\nThis example suggests that the function mkCDList :: [a] -> DList a for converting a (nonempty) list as into a doubly-linked list can be speci\ufb01ed as the \ufb01rst\nelement in a \ufb01nite list xs of doubly-linked lists satisfying the following three properties:\nmap elem xs = as\nmap prev xs = rotr xs\nmap next xs = rotl xs\nHere, rotr and rotl (short for rotate right and rotate left), are de\ufb01ned by\nrotr xs = [last xs] ++ init xs\nrotl xs = tail xs ++ [head xs]\n\n\fIn\ufb01nite lists\n\n230\n\nObserve now that for any list xs of doubly-linked lists we have\nxs = zipWith3 Cons\n(map elem xs) (map prev xs) (map next xs)\nwhere zipWith3 is like zipWith except that it takes three lists instead of two. The\nstandard prelude de\ufb01nition is:\nzipWith3 f (x:xs) (y:ys) (z:zs)\n= f x y z : zipWith3 f xs ys zs\nzipWith3 _ _ _ _ = []\nWe will see another de\ufb01nition in a moment. We can prove the claim above by\ninduction. It clearly holds for the unde\ufb01ned and empty lists. For the inductive case\nwe reason:\nx:xs\n=\n\n{since xs is a doubly-linked list}\nCons (elem x) (prev x) (next x):xs\n\n=\n\n{induction}\nCons (elem x) (prev x) (next x):\n(zipWith3 Cons\n(map elem xs) (map prev xs) (map next xs))\n\n=\n\n{de\ufb01nition of zipWith3 and map}\nzipWith3 Cons\n(map elem (x:xs)) (map prev (x:xs)) (map next (x:xs)\n\nPutting this result together with our speci\ufb01cation of doubly-linked lists, we arrive\nat\nmkCDList as = head xs\nwhere xs = zipWith3 Cons as (rotr xs) (rotl xs)\nThis de\ufb01nition involves a cyclic list xs. Does it work? The answer is: No, it doesn\u2019t.\nThe reason is that zipWith3 as de\ufb01ned above is too eager. We need to make it\nlazier by not demanding the values of the second two lists until they are really\nneeded:\nzipWith3 f (x:xs) ys zs\n= f x (head ys) (head zs):\nzipWith3 f xs (tail ys) (tail zs)\nzipWith3 _ _ _ _ = []\n\n\f9.7 Exercises\n\n231\n\nAn equivalent way to de\ufb01ne this function is to make use of Haskell\u2019s irrefutable\npatterns:\nzipWith3 f (x:xs) ~(y:ys) ~(z:zs)\n= f x y z : zipWith3 f xs ys zs\nzipWith3 _ _ _ _ = []\nAn irrefutable pattern is introduced using a tilde, and ~(x:xs) is matched lazily,\nmeaning that no matching is actually performed until either x or xs is needed.\nJust to convince ourselves that the above de\ufb01nition of mkCDList with the revised\nde\ufb01nition of zipWith3 does make progress, let xs0 =\u22a5 and\nxsn+1 = zipWith3 Cons \"A\" (rotr xsn ) (rotl xsn )\nThen xs1 is given by\nzipWith3 Cons \"A\" \u22a5 \u22a5\n= [Cons \u2019A\u2019 \u22a5 \u22a5 ]\nand xs2 by\nzipWith3 Cons \"A\"\n[Cons \u2019A\u2019 \u22a5 \u22a5 ] [Cons \u2019A\u2019 \u22a5 \u22a5 ]\n= [Cons \u2019A\u2019 (Cons \u2019A\u2019 \u22a5 \u22a5 ) (Cons \u2019A\u2019 \u22a5 \u22a5 )]\nand so on.\n\n9.7 Exercises\nExercise A\nGiven three lists xs, ys and zs in strictly increasing order, we have\nmerge (merge xs ys) zs} = merge xs (merge ys zs)\nThus merge is associative. Assuming in addition that the \ufb01rst elements of xs, ys\nand zs are in strictly increasing order, we also have\nxmerge (xmerge xs ys) zs = xmerge xs (xmerge ys zs)\nDoes it follow that in the expression foldr1 xmerge multiples we could replace foldr1 by foldl1?\n\n\fIn\ufb01nite lists\n\n232\n\nExercise B\nThe standard prelude function cycle :: [a] -> [a] takes a list xs and returns\na list consisting of an in\ufb01nite number of repetitions of the elements of xs. If xs is\nthe empty list, then cycle [] returns an error message. For instance\ncycle \"hallo\" = \"hallohallohallo...\nDe\ufb01ne cycle using a cyclic list. Ensure that your de\ufb01nition works on empty, \ufb01nite\nand in\ufb01nite lists.\n\n```\n\n\n```\n\nExercise C\nThe \ufb01bonacci function is de\ufb01ned by\nfib 0 = 0\nfib 1 = 1\nfib n = fib (n-1) + fib (n-2)\nWrite down a one-line de\ufb01nition of the list fibs that produces the in\ufb01nite list of\nFibonacci numbers.\nExercise D\nA well-known problem, due to the mathematician W.R. Hamming, is to write a\nprogram that produces an in\ufb01nite list of numbers with the following properties: (i)\nthe list is in strictly increasing order; (ii) the list begins with the number 1; (iii) if\nthe list contains the number x, then it also contains the numbers 2x, 3x and 5x; (iv)\nthe list contains no other numbers. Thus, the required list begins with the numbers\n1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, . . .\nWrite a de\ufb01nition of hamming that produces this list.\nExercise E\nProve that approx n xs \u0011 xs for all n. Now prove that if approx n xs \u0011 ys for\nall n, then xs \u0011 ys. Hence conclude that\nlim approx n xs = xs.\n\nn\u2192\u221e\n\nExercise F\nGive a counter-example to the claim that xs=ys if xs!!n=ys!!n for all n.\n\n\f9.7 Exercises\n\n233\n\nExercise G\nProve that iterate f x = x: map f (iterate f x).\nExercise H\nIn the de\ufb01nition of primes as a cyclic list, could we have de\ufb01ned\nmergeAll = foldr xmerge []\nas an alternative to the de\ufb01nition in the text?\nExercise I\nRecall that\nprs n = approx n (2:([3..] \\\\ crs n))\ncrs n = mergeAll [map (p*) [p..] | p <- prs n]\nGiven that prs n=p1 : p2 : \u00b7 \u00b7 \u00b7 pn :\u22a5, where pj is the jth prime, sketch how to show\nthat crs n=c1 : c2 : \u00b7 \u00b7 \u00b7 cm :\u22a5, where cj is the jth composite number (so c1 =4) and\nm=p2n . Hence show that primes does produce the in\ufb01nite list of primes.\nWe said in the text that it is not the case that the nth approximation primesn of\nprimes is equal to approx n primes. In fact\nprimes4 = 2 : 3 : 5 : 7 : \u00b7 \u00b7 \u00b7 : 47 :\u22a5\nWhat list does primes5 produce?\nExercise J\nAnother way of generating the primes is known as the Sieve of Sundaram, after its\ndiscoverer S.P. Sundaram in 1934:\nprimes\n= 2:[2*n+1 | n <- [1..] \\\\ sundaram]\nsundaram = mergeAll [[i+j+2*i*j | j <- [i..]] | i <- [1..]]\nTo show that the list comprehension in the de\ufb01nition of primes generates exactly\nthe odd primes, it is suf\ufb01cient to prove that the term 2*n+1 is never composite,\nwhich is to say that it never factorises into (2*i+1)*(2*j+1) for positive integers\ni and j. Why is this so?\nExercise K\nIs the function f , de\ufb01ned by f (\u22a5) = 0 and f (x) = 1 for x =\u22a5, computable? How\nabout the function that returns \u22a5 on all \ufb01nite or partial lists, and 1 on all in\ufb01nite\nlists?\n\n\fIn\ufb01nite lists\n\n234\n\nExercise L\nBy de\ufb01nition, a torus is a doubly-cyclic, doubly-doubly-linked list. It is a cyclic\ndoubly-linked list in the left/right direction, and also in the up/down direction.\nGiven a matrix represented as a list of length m of lists, all of length n, construct a\nde\ufb01nition of\nmkTorus :: Matrix a -> Torus a\nwhere\ndata Torus a = Cell a (Torus a) (Torus a)\n(Torus a) (Torus a)\nelem (Cell a u d l r) = a\nup\n(Cell a u d l r) = u\ndown (Cell a u d l r) = d\nleft (Cell a u d l r) = l\nright (Cell a u d l r) = r\nThat looks tricky, but the answer is short enough to be tweeted.\n\n9.8 Answers\nAnswer to Exercise A\nNo, since foldl1 f xs = undefined for any in\ufb01nite list xs.\nAnswer to Exercise B\nThe de\ufb01nition is\ncycle [] = error \"empty list\"\ncycle xs = ys where ys = xs ++ ys\nNote that if xs is in\ufb01nite, then xs ++ ys = xs, so cycle is the identity function\non in\ufb01nite lists.\nAnswer to Exercise C\nThe one-liner is:\nfibs :: [Integer]\nfibs = 0:1:zipWith (+) fibs (tail fibs)\n\n\f9.8 Answers\n\n235\n\nAnswer to Exercise D\nhamming :: [Integer]\nhamming = 1: merge (map (2*) hamming)\n(merge (map (3*) hamming)\n(map (5*) hamming))\nAnswer to Exercise E\nThe proof of approx n xs \u0011 xs is by induction on n. The base case is easy but\nthe induction step involves a sub-induction over xs. The base cases (the empty list\nand the unde\ufb01ned list) of the sub-induction are easy and the inductive case is\napprox (n+1) (x:xs)\n{de\ufb01nition}\n\n=\n\nx:approx n xs\n\u0011\n\n{induction and monotonicity of (x:)}\nx:xs.\n\nThe proof of\n(\u2200n : approx n xs \u0011 ys) \u21d2 xs \u0011 ys\nis by induction on xs. The claim is immediate for the unde\ufb01ned and empty lists,\nand for the inductive case we have\n(\u2200n : approx n (x:xs) \u0011 ys)\n\u21d2 xs \u0011 head ys \u2227 (\u2200n : approx n xs \u0011 tail ys)\nby the de\ufb01nitions of approx and the approximation ordering on lists. By induction\nwe therefore have\nx:xs \u0011 head ys:tail ys = ys.\nIt follows that\nlim approx n xs = xs\n\nn\u2192\u221e\n\nby the de\ufb01nition of limit.\nAnswer to Exercise F\nThe two lists repeat undefined and undefined are not equal, but\n(repeat undefined)!!n = undefined!!n\nfor all n because both sides are \u22a5.\n\n\fIn\ufb01nite lists\n\n236\n\nAnswer to Exercise G\nWe have to show that\napprox n (iterate f x) = approx n (x:map f (iterate f x))\nfor all natural numbers n. This claim follows from\napprox n (iterate f (f x))\n= approx n (map f (iterate f x))\nwhich we establish by induction on n. For the inductive step we simplify each side.\nFor the left-hand side:\napprox (n+1) (iterate f (f x))\n=\n\n{de\ufb01nition of iterate}\napprox (n+1) (f x:iterate f (f (f x)))\n\n=\n\n{de\ufb01nition of approx}\nf x: approx n (iterate f (f (f x)))\n\n=\n\n{induction}\nf x: approx n (map f (iterate f (f x)))\n\nFor the right-hand side:\napprox (n+1) (map f (iterate f x))\n=\n\n{de\ufb01nition of iterate and map}\napprox (n+1) (f x:map f (iterate f (f x)))\n\n=\n\n{de\ufb01nition of approx}\nf x: approx n (map f (iterate f (f x)))\n\nAnswer to Exercise H\nYes, since\nfoldr xmerge [] (xs:undefined) = xmerge xs undefined\nand the right-hand side begins with the \ufb01rst element of xs.\nAnswer to Exercise I\nThe proof is by induction. We have \ufb01rst to show that crs (n+1) is the result\nof merging c1 : c2 : \u00b7 \u00b7 \u00b7 cm :\u22a5, where m = p2n with the in\ufb01nite list of multiples\n\n\f9.9 Chapter notes\n\n237\n\npn+1 pn+1 , pn+1 (pn+1 +1), . . . of pn+1 . That gives the partial list of all composite\nnumbers up to p2n+1 . Finally, we need the result that pn+2 < p2n+1 .\nThe partial list primes5 produces all the primes smaller than 2209 = 47 \u00d7 47.\nAnswer to Exercise J\nBecause an odd integer is excluded from the \ufb01nal list if it takes the form 2n + 1\nwhere n is of the form i+j+2ij. But\n2(i+j+2ij)+1 = (2i+1)(2j+1).\nAnswer to Exercise K\nNo, f is not monotonic: \u22a5\u0011 1 but f (\u22a5) \u0011 f (1). For the second function (call it g)\nwe have xs \u0011 ys implies g(xs) \u0011 g(ys), so g is monotonic. But g is not continuous,\nso it\u2019s not computable.\nAnswer to Exercise L\nThe de\ufb01nition is\nmkTorus ass = head (head xss)\nwhere xss = zipWith5 (zipWith5 Cell)\nass (rotr xss) (rotl xss)\n(map rotr xss) (map rotl xss)\nWhereas rotr and rotl rotate the rows of a matrix, map rotr and map rotl\nrotate the columns. The de\ufb01nition of zipWith5 has to be made non-strict in its last\nfour arguments.\n\n9.9 Chapter notes\nMelissa O\u2019Neill has written a nice pearl on sieve methods for generating primes;\nsee \u2018The genuine sieve of Eratosthenes\u2019, Journal of Functional Programming 19\n(1), 95\u2013106, 2009. Ben Sijtsma\u2019s thesis Veri\ufb01cation and derivation of in\ufb01nite-list\nprograms (University of Groningen, the Netherlands, 1988) studies various aspects\nof in\ufb01nite-list programs and gives a number of techniques for reasoning about\nthem. One chapter is devoted to the proof of fairness in the paper\u2013rock\u2013scissors\ngame.\nMy paper, \u2018On building cyclic and shared data structures in Haskell\u2019, Formal Aspects of Computing 24(4\u20136), 609\u2013621, July 2012, contains more examples of the\nuses of in\ufb01nite and cyclic lists. See also the article on \u2018Tying the knot\u2019 at\n\n\f238\n\nIn\ufb01nite lists\n\nhaskell.org/haskellwiki/Tying_the_Knot\nHamming\u2019s problem has been used as an illustration of cyclic programs since the\nearly days of functional programming.\n\n\fChapter 10\nImperative functional programming\n\nBack in Chapter 2 we described the function putStrLn as being a Haskell command, and IO a as being the type of input\u2013output computations that interact with\nthe outside world and deliver values of type a. We also mentioned some syntax,\ncalled do-notation, for sequencing commands. This chapter explores what is really\nmeant by these words, and introduces a new style of programming called monadic\nprogramming. Monadic programs provide a simple and attractive way to describe\ninteraction with the outside world, but are also capable of much more: they provide\na simple sequencing mechanism for solving a range of problems, including exception handling, destructive array updates, parsing and state-based computation. In a\nvery real sense, a monadic style enables us to write functional programs that mimic\nthe kind of imperative programs one \ufb01nds in languages such as Python or C.\n\n10.1 The IO monad\nThe type IO a is an abstract type in the sense described in the previous chapter, so\nwe are not told how its values, which are called actions or commands, are represented. But you can think of this type as being\ntype IO a = World -> (a,World)\nThus an action is a function that takes a world and delivers a value of type a and\na new world. The new world is then used as the input for the next action. Having\nchanged the world with an input\u2013output action, you can\u2019t go back to the old world.\nYou can\u2019t duplicate the world or inspect its components. All you can do is operate on the world with given primitive actions, and put such actions together in a\nsequence.\n\n\fImperative functional programming\n\n240\n\nOne primitive action is to print a character:\nputChar :: Char -> IO ()\nWhen executed, this action prints a character on the standard output channel, usually the computer screen. For example,\nghci> putChar 'x'\nxghci>\nThe character x is printed, but nothing else, so the next GHCi prompt follows\nwithout additional spaces or newlines. Performing this action produces no value of\ninterest, so the return value is the null tuple ().\nAnother primitive action is done :: IO (), which does nothing. It leaves the\nworld unchanged and also returns the null tuple ().\nOne simple operation to sequence actions is denoted by (>>) and has type\n(>>) :: IO () -> IO () -> IO ()\nGiven actions p and q, the action p >> q \ufb01rst performs action p and then performs\naction q. For example,\nghci> putChar 'x' >> putChar '\\n'\nx\nghci>\nThis time a newline is printed. Using (>>) we can de\ufb01ne the function putStrLn:\nputStrLn :: String -> IO ()\nputStrLn xs = foldr (>>) done (map putChar xs) >>\nputChar '\\n'\nThis action prints all the characters in a string, and then \ufb01nishes up with an additional newline character. Note that map putChar xs is a list of actions. We are still\nin the universe of functional programming and its full expressive power, including\nuses of map and foldr, is still available to us.\nHere is another primitive action:\ngetChar :: IO Char\nWhen performed, this operation reads a character from the standard input channel.\nThis channel is fed by you typing at the keyboard, so getChar returns the \ufb01rst\ncharacter you type. For example,\n\n\f10.1 The IO monad\n\n241\n\nghci> getChar\nx\n'x'\nAfter typing getChar and pressing return, GHCi waits for you to type a character. We typed the character 'x' (and what we typed was echoed), and then that\ncharacter was read and printed.\nThe generalisation of done is an action that does nothing and returns a named\nvalue:\nreturn :: a -> IO a\nIn particular, done = return (). The generalisation of (>>) has type\n(>>) :: IO a -> IO b -> IO b\nGiven actions p and q, the action p >> q \ufb01rst does p, and then throws the return\nvalue away, and then does q. For example,\nghci> return 1 >> return 2\n2\nIt is clear that this action is useful only when the value returned by p is not interesting since there is no way that q can depend on it. What is really wanted is a more\ngeneral operator (>>=) with type\n(>>=) :: IO a -> (a -> IO b) -> IO b\nThe combination p >>= f is an action that, when performed, \ufb01rst does p, returning\na value x of type a, then does action f x returning a \ufb01nal value y of type b. It is easy\nto de\ufb01ne (>>) in terms of (>>=) and we leave this as an exercise. The operator\n(>>=) is often referred to as bind, though one can also pronounce it as \u2018then apply\u2019.\nUsing (>>=), we can de\ufb01ne a function getLine for reading a line of input, more\nprecisely, the list of characters up to but not including the \ufb01rst newline character:\ngetLine :: IO String\ngetLine = getChar >>= f\nwhere f x = if x == '\\n' then return []\nelse getLine >>= g\nwhere g xs = return (x:xs)\nThis has a straightforward reading: get the \ufb01rst character x; stop if x is a newline\nand return the empty list; otherwise get the rest of the line and add x to the front.\nThough the reading is straightforward, the use of nested where clauses makes the\n\n\f242\n\nImperative functional programming\n\nde\ufb01nition a little clumsy. One way to make the code smoother is to use anonymous\nlambda expressions and instead write:\ngetLine = getChar >>= \\x ->\nif x == '\\n'\nthen return []\nelse getLine >>= \\xs ->\nreturn (x:xs)\nAnother, arguably superior solution is to use do-notation:\ngetLine = do x a 1 . Once you are in a room performing input\u2013output actions, you stay\nin the room and can\u2019t come out of it. To see one reason this has to be the case,\nsuppose there is such a function, runIO say, and consider\nint :: Int\nint = x - y\nwhere x = runIO readInt\ny = runIO readInt\n1\n\nActually there is, and it\u2019s called unsafePerformIO, but it is a very unsafe function.\n\n\f10.1 The IO monad\n\n243\n\nreadInt = do {xs <- getLine; return (read xs :: Int)}\nThe action readInt reads a line of input and, provided the line consists entirely\nof digits, interprets it as an integer. Now, what is the value of int? The answer\ndepends entirely on which of x and y gets evaluated \ufb01rst. Haskell does not prescribe\nwhether or not x is evaluated before y in the expression x-y. Put it this way: input\u2013\noutput actions have to be sequenced in a deterministic fashion, and Haskell is a\nlazy functional language in which it is dif\ufb01cult to determine the order in which\nthings happen. Of course, an expression such as x-y is a very simple example\n(and exactly the same undesirable phenomenon arises in imperative languages) but\nyou can imagine all sorts of confusion that would ensue if we were provided with\nrunIO.\nThe second thing that perhaps should be said is in response to a reader who casts a\nlazy eye over an expression such as\nundefined >> return 0 :: IO Int\nDoes this code raise an error or return zero? The answer is: an error. IO is strict in\nthe sense that IO actions are performed in order, even though subsequent actions\nmay take no heed of their results.\nTo return to the main theme, let us summarise. The type IO a is an abstract type\non which the following operations, at least, are available:\nreturn :: a -> IO a\n(>>=) :: IO a -> (a -> IO b) -> IO b\nputChar :: Char -> IO ()\ngetChar :: IO Char\nThe second two functions are speci\ufb01c to input and output, but the \ufb01rst two are not.\nIndeed they are general sequencing operations that characterise the class of types\ncalled monads:\nclass Monad m where\nreturn :: a -> m a\n(>>=) :: m a -> (a -> m b) -> m b\nThe two monad operations are required to satisfy certain laws, which we will come\nto in due course. As to the reason for the name \u2018monad\u2019, it is stolen from philosophy, in particular from Leibniz, who in turn borrowed it from Greek philosophy.\nDon\u2019t read anything into the name.\n\n\f244\n\nImperative functional programming\n\n10.2 More monads\nIf that\u2019s all a monad is, then surely lots of things form a monad? Yes, indeed. In\nparticular, the humble list type forms a monad:\ninstance Monad [] where\nreturn x = [x]\nxs >>= f = concat (map f xs)\nOf course, we don\u2019t yet know what the laws governing the monad operations are,\nso maybe this instance isn\u2019t correct (it is), but at least the operations have the right\ntypes. Since do-notation can be used with any monad we can, for example, de\ufb01ne\nthe cartesian product function cp :: [[a]] -> [[a]] (see Section 7.3) using\nthe new notation:\ncp []\n= return []\ncp (xs:xss) = do {x <- xs;\nys <- cp xss;\nreturn (x:ys)}\nComparing the right-hand side of the second clause to the list comprehension\n[x:ys | x <- xs, ys <- cp xss]\none can appreciate that the two notations are very similar; the only real difference\nis that with do-notation the result appears at the end rather than at the beginning. If\nmonads and do-notation had been made part of Haskell before list comprehensions,\nthen maybe the latter wouldn\u2019t have been needed.\nHere is another example. The Maybe type is a monad:\ninstance Monad Maybe where\nreturn x\n= Just x\nNothing >>= f = Nothing\nJust x >>= f = f x\nTo appreciate what this monad can bring to the table, consider the Haskell library\nfunction\nlookup :: Eq a => a -> [(a,b)] -> Maybe b\nThe value of lookup x alist is Just y if (x,y) is the \ufb01rst pair in alist with\n\ufb01rst component x, and Nothing if there is no such pair. Imagine looking up x in\nalist, then looking up the result y in a second list blist, and then looking up\nthe result z in yet a third list clist. If any of these lookups return Nothing, then\n\n\f10.2 More monads\n\n245\n\nNothing is the \ufb01nal result. To de\ufb01ne such a function we would have to write its\nde\ufb01ning expression as something like\ncase lookup x alist of\nNothing -> Nothing\nJust y -> case lookup y blist of\nNothing -> Nothing\nJust z -> lookup z clist\nWith a monad we can write\ndo {y <- lookup x alist;\nz <- lookup y blist;\nreturn (lookup z clist)}\nRather than having to write an explicit chain of computations, each of which may\nreturn Nothing, and explicitly passing Nothing back up the chain, we can write a\nsimple monadic expression in which handling Nothing is done implicitly under a\nmonadic hood.\n\ndo-notation\nJust as list comprehensions can be translated into expressions involving map and\nconcat, so do-expressions can be translated into expressions involving return and\nbind. The three main translation rules are:\ndo {p}\n= p\ndo {p;stmts}\n= p >> do {stmts}\ndo {x <- p;stmts} = p >>= \\x -> do {stmts}\nIn these rules p denotes an action, so the \ufb01rst rule says that a do round a single\naction can be removed. In the second and third rules stmts is a nonempty sequence\nof statements, each of which is either an action or a statement of the form x <- p.\nThe latter is not an action; consequently an expression such as\ndo {x <- getChar}\nis not syntactically correct. Nor, by the way, is an empty do-expression do { }.\nThe last statement in a do-expression must be an action.\nOn the other hand, the following two expressions are both \ufb01ne:\ndo {putStrLn \"hello \"; name <- getLine; putStrLn name}\ndo {putStrLn \"hello \"; getLine; putStrLn \"there\"}\n\n\f246\n\nImperative functional programming\n\nThe \ufb01rst example prints a greeting, reads a name and completes the greeting. The\nsecond prints a greeting, reads a name but immediately forgets it, and then completes the greeting with a \u2018there\u2019. A bit like being introduced to someone in real\nlife.\nFinally, there are two rules that can be proved from the translation rules above:\ndo {do {stmts}} = do {stmts}\ndo {stmts1; do {stmts2}} = do {stmts1; stmts2}\nBut one has to be careful; the nested dos in\ndo {stmts1;\nif p\nthen do {stmts2}\nelse do {stmts3}}\nare necessary if stmts2 and stmts3 contain more than one action.\n\nMonad laws\nThe monad laws say nothing much more than that expressions involving return\nand (>>=) simplify in just the way one would expect. There are three laws and we\nare going to state them in three different ways. The \ufb01rst law states that return is\na right identity element of (>>=):\n(p >>= return) = p\nIn do-notation the law reads:\ndo {x <- p; return x} = do {p}\nThe second law says that return is also a kind of left identity element:\n(return e >>= f) = f e\nIn do-notation the law reads:\ndo {x <- return e; f x} = do {f e}\nThe third law says that (>>=) is kind of associative:\n((p >>= f) >>= g) = p >>= (\\x -> (f x >>= g))\nIn do-notation the law reads:\n\n\f10.3 The State monad\n\n247\n\ndo {y <- do {x <- p; f x}; g y}\n= do {x <- p; do {y <- f x; g y}}\n= do {x <- p; y <- f x; g y}\nThe last line makes use of the un-nesting property of do-notation.\nFor the third way of stating the monad laws, consider the operator (>=>) de\ufb01ned\nby\n(>=>) :: Monad m => (a -> m b) -> (b -> m c) -> (a -> m c)\n(f >=> g) x = f x >>= g\nThis operator is just like function composition except that the component functions\neach have type x -> m y for appropriate x and y, and the order of composition is\nfrom left to right rather than from right to left. This operator, which is called (left to\nright) Kleisli composition, is de\ufb01ned in the Haskell library Control.Monad. There\nis a dual version, (right to left) Kleisli composition,\n(<=<) :: Monad m => (b -> m c) -> (a -> m b) -> (a -> m c)\nwhose de\ufb01nition we leave as an easy exercise.\nThe point is that we can de\ufb01ne (>>=) in terms of (>=>):\n(p >>= f) = (id >=> f) p\nMore brie\ufb02y, (>>=) = flip (id >=>). We also have the leapfrog rule:\n(f >=> g) . h = (f . h) >=> g\nThe proof is left as an exercise.\nIn terms of (>=>) the three monad laws say simply that (>=>) is associative with\nidentity return. Any set of values with an associative binary operation and an\nidentity element is called a monoid, and the word \u2018monad\u2019 was probably adopted\nbecause of the pun with monoid. Be that as it may, this is certainly the shortest way\nof stating the monad laws.\nOne additional and instructive way of describing the monad laws is considered in\nthe exercises.\n\n\n```\n\n\n```\n\n10.3 The State monad\nIf it wasn\u2019t for the problem of how to sequence input\u2013output actions correctly,\nmonads probably wouldn\u2019t have appeared in Haskell. But once it was appreciated\n\n\fImperative functional programming\n\n248\n\nwhat they could do, all kinds of other uses quickly followed. We have seen with\nthe Maybe monad how chains of computations that involve passing information\nback up the chain can be simpli\ufb01ed with monadic notation. Another primary use\nof monads is a way to handle mutable structures, such as arrays, that rely for their\nef\ufb01ciency on being able to update their values, destroying the original structure in\nthe process.\nMutable structures are introduced through the State-Thread monad ST s which we\nwill consider in a subsequent section. Before getting on to the particular properties of this monad, we start by considering a simpler monad, called State s, for\nmanipulating an explicit state s. You can think of the type State s a as being\ntype State s a = s -> (a,s)\nAn action of type State s a takes an initial state and returns a value of type\na and a new state. It is tempting, but wrong, to think of IO a as synonymous\nwith State World a. The state component s in State s a can be exposed and\nmanipulated, but we can\u2019t expose and manipulate the world.\nSpeci\ufb01cally, as well as the monad operations return and (>>=), \ufb01ve other functions are provided for working with the state monad:\nput\nget\nstate\nrunState\nevalState\n\n::\n::\n::\n::\n::\n\ns -> State s\nState s s\n(s -> (a,s))\nState s a ->\nState s a ->\n\n()\n-> State s a\n(s -> (a,s))\ns -> a\n\nThe function put puts the state into a given con\ufb01guration, while get returns the\ncurrent state. Each of these two operations can be de\ufb01ned in terms of state:\nput s = state (\\_ -> ((),s))\nget\n= state (\\s -> (s,s))\nOn the other hand, state can also be de\ufb01ned using put and get:\nstate f = do {s <- get; let (a,s') = f s;\nput s'; return a}\nHaskell permits an abbreviated form of let expressions in do expressions (and\nalso in list comprehensions). We have\ndo {let decls; stmts} = let decls in do {stmts}\nThe function runState is the inverse of state: it takes both an action and an\n\n\f10.3 The State monad\n\n249\n\ninitial state and returns the \ufb01nal value and the \ufb01nal state after performing the action\n(something the IO monad cannot do). The function evalState is de\ufb01ned by\nevalState m s = fst (runState m s)\nand returns just the value of the stateful computation.\nHere is an example of the use of State. In Section 7.6 we constructed the following\nprogram for building a binary tree out of a given nonempty list of values:\nbuild ::\nbuild xs\nbuild2 1\nbuild2 n\n\n[a] -> BinTree a\n= fst (build2 (length xs) xs)\nxs = (Leaf (head xs),tail xs)\nxs = (Fork u v, xs'')\nwhere (u,xs') = build2 m xs\n(v,xs'') = build2 (n-m) xs'\nm\n= n `div` 2\n\nThe point to appreciate here is that build2 is essentially a function that manipulates a state of type [a], returning elements of BinTree a as its result. Another\nway of writing build is as follows:\nbuild xs = evalState (build2 (length xs)) xs\nbuild2 :: Int -> State [a] (BinTree a)\nbuild2 1 = do {x:xs <- get;\nput xs;\nreturn (Leaf x)}\nbuild2 n = do {u <- build2 m;\nv <- build2 (n-m);\nreturn (Fork u v)}\nwhere m = n `div` 2\nAll the work in manipulating the state explicitly is done when building a leaf.\nThe state is accessed and its \ufb01rst element is chosen as the label associated with a\nLeaf; the remaining list then is installed as the new state. Whereas the \ufb01rst version\nof build2 n threads the state explicitly, the second version hides this machinery\nunder a monadic hood.\nNotice in the \ufb01rst line of build2 we have a statement x:xs <- get in which the\nleft-hand side is a pattern rather than a simple variable. If the current state happens\nto be the empty list, the action fails with a suitable error message. For example,\nghci>\n\nrunState (do {x:xs <- get; return x}) \"\"\n\n\fImperative functional programming\n\n250\n\n*** Exception: Pattern match failure in do expression ...\nOf course this behaviour cannot arise with build2 1 because the de\ufb01nition only\napplies when the state is a singleton list. We leave it as an exercise to say what\nbuild [] does.\nAs another example, consider the problem of producing a pseudo-random integer\nin a speci\ufb01ed interval. Imagine we have a function\nrandom :: (Int,Int) -> Seed -> (Int,Seed)\nthat takes a pair of integers as the speci\ufb01ed interval and then a seed, and calculates a\nrandom integer and a new seed. The new seed is used for obtaining further random\nvalues. Rather than be explicit about what a seed is, suppose there is a function\nmkSeed :: Int -> Seed\nthat makes a seed from a given integer. Now if we wanted to roll a pair of dice, we\ncould write\ndiceRoll :: Int -> (Int,Int)\ndiceRoll n = (x,y)\nwhere (x,s1) = random (1,6) (mkSeed n)\n(y,s2) = random (1,6) s1\nBut we could also write\ndiceRoll n = evalState (\ndo {x <- randomS (1,6);\ny <- randomS (1,6);\nreturn (x,y)}\n) (mkSeed n)\nwhere randomS = state . random\nThe function randomS :: (Int,Int) -> State Seed Int takes an interval\nand returns an action. The second version of diceRoll is a little longer than the\n\ufb01rst, but is arguably more easy to write. Imagine that instead of two dice we had\n\ufb01ve, as in liar dice. The \ufb01rst method would involve a chain of where-clauses expressing the linkage between \ufb01ve values and \ufb01ve seeds, something that would be\neasy to mistype, but the second version is easily extended and harder to get wrong.\nOne \ufb01nal point. Consider\nevalState (do {undefined; return 0}) 1\nDoes this raise an exception, or does it return zero? In other words, is the monad\n\n\f10.4 The ST monad\n\n251\n\nState strict, as the IO monad is, or is it lazy? The answer is that it can be both.\nThere are two variants of the state monad, one of which is lazy and the other of\nwhich is strict. The difference lies in how the operation (>>=) is implemented.\nHaskell provides the lazy variant by default, in Control.Monad.State.Lazy,\nbut you can ask for the strict variant, in Control.Monad.State.Strict if you\nwant.\n\n10.4 The ST monad\nThe state-thread monad, which resides in the library Control.Monad.ST, is a different kettle of \ufb01sh entirely from the state monad, although the kettle itself looks\nrather similar. Like State s a you can think of this monad as the type\ntype ST s a = s -> (a,s)\nbut with one very important difference: the type variable s cannot be instantiated\nto speci\ufb01c states, such as Seed or [Int]. Instead it is there only to name the state.\nThink of s as a label that identi\ufb01es one particular state thread. All mutable types\nare tagged with this thread, so that actions can only affect mutable values in their\nown state thread.\nOne kind of mutable value is a program variable. Unlike variables in Haskell, or\nmathematics for that matter, program variables in imperative languages can change\ntheir values. They can be thought of as references to other values, and in Haskell\nthey are entities of type STRef s a. The s means that the reference is local to\nthe state thread s (and no other), and the a is the type of value being referenced.\nThere are operations, de\ufb01ned in Data.STRef, to create, read from and write to\nreferences:\nnewSTRef\n:: a -> ST s (STRef s a)\nreadSTRef :: STRef s a -> ST s a\nwriteSTRef :: STRef s a -> a -> ST s ()\nHere is an example. Recall Section 7.6 where we gave the following de\ufb01nition of\nthe Fibonacci function:\nfib ::\nfib n\nfib2 0\nfib2 n\n\nInt -> Integer\n= fst (fib2 n)\n= (0,1)\n= (b,a+b) where (a,b) = fib2 (n-1)\n\n\f252\n\nImperative functional programming\n\nEvaluating fib takes linear time, but the space involved is not constant (even ignoring the fact that arbitrarily large integers cannot be stored in constant space):\neach recursive call involves fresh variables a and b. By contrast, here is a de\ufb01nition\nof fib in the imperative language Python:\ndef fib (n):\na,b = 0,1\nfor i in range (0,n):\na,b = b,a+b\nreturn a\nThe de\ufb01nition manipulates two program variables a and b, and runs in constant\nspace (at least, for small integers). We can translate the Python code almost directly\ninto Haskell:\nfibST :: Int -> ST s Integer\nfibST n = do {a <- newSTRef 0;\nb <- newSTRef 1;\nrepeatFor n\n(do {x <- readSTRef a;\ny <- readSTRef b;\nwriteSTRef a y;\nwriteSTRef b $! (x+y)});\nreadSTRef a}\nNote the use of the strict application operator ($!) to force evaluation of the sum.\nThe action repeatFor repeats an action a given number of times:\nrepeatFor :: Monad m => Int -> m a -> m ()\nrepeatFor n = foldr (>>) done . replicate n\nAll well and good, but we end up with an action ST s Integer when what we\nreally want is an integer. How do we escape from the monad back into the world\nof Haskell values?\nThe answer is to provide a function similar to runState for the state monad, Here\nit is, with its type:\nrunST :: (forall s. ST s a) -> a\nThis type is unlike any other Haskell type we have met so far. It is what is called\na rank 2 polymorphic type, while all previous polymorphic types have had rank\n1. What it says is that the argument of runST must be universal in s, so it can\u2019t\n\n\f10.4 The ST monad\n\n253\n\ndepend on any information about s apart from its name. In particular, every STRef\ndeclared in the action has to carry the same thread name s.\nTo amplify a little on rank 2 types, consider the difference between the two lists\nlist1 :: forall a. [a -> a]\nlist2 :: [forall a. a -> a]\nThe type of list1 is just what we would have previously written as [a -> a]\nbecause in ordinary rank 1 types universal quanti\ufb01cation at the outermost level\nis assumed. For example, [sin,cos,tan] is a possible value of list1 with the\ninstantiation Float for a. But there are only two functions that can be elements of\nlist2, namely id and the unde\ufb01ned function undefined, because these are the\nonly two functions with type forall a. a -> a. If you give me an element x of\na type a about which absolutely nothing is known, the only things I can do if I have\nto give you back an element of a, is either to give you x or \u22a5.\nWhy have a rank 2 type for runST? Well, it prevents us from de\ufb01ning things like\nlet v = runST (newSTRef True)\nin runST (readSTRef v)\nThis code is not well-typed because\nnewSTRef True :: ST s (STref s Bool)\nand in the expression runST (newSTRef Bool) the Haskell type checker cannot match STRef s a with a, the expected result type of runST. Values of type\nSTRef s a cannot be exported from ST s, but only entities whose types do not\ndepend on s. If the code were allowed, then the reference allocated in the \ufb01rst\nrunST would be usable inside the second runST. That would enable reads in one\nthread to be used in another, and hence the result would depend on the evaluation\norder used to execute the threads, leading to mayhem and confusion. It is just the\nsame problem that we prevented from occurring in the IO monad.\nBut we can safely de\ufb01ne\nfib :: Int -> Integer\nfib n = runST (fibST n)\nThis version of fib runs in constant space.\nFor our purposes the main use of the ST monad resides in its ability to handle\nmutable arrays. The whole question of arrays deserves a section to itself.\n\n\f254\n\nImperative functional programming\n\n10.5 Mutable arrays\nIt sometimes surprises imperative programmers who meet functional programming\nfor the \ufb01rst time that the emphasis is on lists as the fundamental data structure rather\nthan arrays. The reason is that most uses of arrays (though not all) depend for their\nef\ufb01ciency on the fact that updates are destructive. Once you update the value of an\narray at a particular index the old array is lost. But in functional programming, data\nstructures are persistent and any named structure continues to exist. For instance,\ninsert x t may insert a new element x into a tree t, but t continues to refer to\nthe original tree, so it had better not be overwritten.\nIn Haskell a mutable array is an entity of type STArray s i e. The s names\nthe state thread, i the index type and e the element type. Not every type can be\nan index; legitimate indices are members of the type class Ix. Instances of this\nclass include Int and Char, things that can be mapped into a contiguous range of\nintegers.\nLike STRefs there are operations to create, read from and write to arrays. Without\nmore ado we consider an example, explaining the actions as we go along. Recall\nthe Quicksort algorithm from Section 7.7:\nqsort :: (Ord a) => [a] -> [a]\nqsort []\n= []\nqsort (x:xs) = qsort [y | y <- xs, y < x] ++ [x] ++\nqsort [y | y <- xs, x <= y]\nThere we said that when Quicksort is implemented in terms of arrays rather than\nlists, the partitioning phase can be performed in place without using any additional\nspace. We now have the tools to write just such an algorithm. We begin with\nqsort :: (Ord a) => [a] -> [a]\nqsort xs = runST $\ndo {xa <- newListArray (0,n-1) xs;\nqsortST xa (0,n);\ngetElems xa}\nwhere n = length xs\nFirst we create a mutable array with bounds (0,n-1) and \ufb01ll it with the elements\nof xs. Sorting the array is done with the action qsortST xa (0,n). At the end,\nthe list of elements of the sorted array is returned. In the code above, the action\nnewListArray has type\nIx i => (i, i) -> [e] -> ST s (STArray s i e)\n\n\f10.5 Mutable arrays\n\n255\n\nand getElems has type\nIx i => STArray s i e -> ST s [e]\nThe \ufb01rst constructs a mutable array from a list of elements, and the second returns\na list of the elements in a mutable array.\nThe purpose of qsortST xa (a,b) is to sort the elements in the sub-array of xa\nin the interval (a,b), where by de\ufb01nition such an interval includes the lower bound\nbut excludes the upper bound; in other words [a .. b-1]. Choosing intervals that\nare closed on the left but open on the right is almost always the best policy when\nprocessing arrays. Here is the de\ufb01nition of qsortST:\nqsortST :: Ord a => STArray s Int a ->\n(Int,Int) -> ST s ()\nqsortST xa (a,b)\n| a == b\n= return ()\n| otherwise = do {m <- partition xa (a,b);\nqsortST xa (a,m);\nqsortST xa (m+1,b)}\nIf a==b we have an empty interval and there is nothing to do. Otherwise we rearrange the array so that for some suitable element x in the array all elements in the\ninterval (a,m) are less than x, and all elements in the interval (m+1,b) are at least\nx. The element x itself is placed in the array at position m. Sorting is then completed\nby sorting both sub-intervals.\nIt remains to de\ufb01ne partition. The only way to \ufb01nd a suitable de\ufb01nition is by\nformal development using pre- and post-conditions and loop invariants. But this is\na book on functional programming, not on the formal development of imperative\nprograms, so we are going to cop out and just record one version:\npartition xa (a,b)\n= do {x <- readArray xa a;\nlet loop (j,k)\n= if j==k\nthen do {swap xa a (k-1);\nreturn (k-1)}\nelse do {y <- readArray xa j;\nif y < x then loop (j+1,k)\nelse do {swap xa j (k-1);\nloop (j,k-1)}}\nin loop (a+1,b)}\n\n\f256\n\nImperative functional programming\n\nThe action swap is de\ufb01ned by\nswap :: STArray s Int a -> Int -> Int -> ST s ()\nswap xa i j = do {v <- readArray xa i;\nw <- readArray xa j;\nwriteArray xa i w;\nwriteArray xa j v}\nHere is a brief and certainly inadequate explanation of how partition works. We\nbegin by taking the \ufb01rst element x in the interval (a,b) as pivot. We then enter\na loop that processes the remaining interval (a+1,b), stopping when the interval\nbecomes empty. We pass over elements that are less than x, shrinking the interval\nfrom the left. Encountering a y not less than x, we swap it with the element at\nthe rightmost position in the interval, shrinking the interval from the right. When\nthe interval becomes empty, we place the pivot in its \ufb01nal position, returning that\nposition as a result.\nNote that loop is de\ufb01ned as a local procedure within the monad. We could have\nde\ufb01ned it as a global procedure, though we would have had to add three extra\nparameters, namely the array xa, the pivot x and the starting position a.\n\nHash tables\nA purely functional Quicksort has the same asymptotic time ef\ufb01ciency as one based\non mutable arrays, but there are one or two places where mutable arrays seem to\nplay a crucial role in achieving an asymptotically faster algorithm. One such place\nis the use of hash tables for an ef\ufb01cient representation of sets.\nBut let us approach the use of hash tables in the context of a particular problem.\nConsider a typical puzzle de\ufb01ned in terms of two \ufb01nite sets, a set of positions and\na set of moves. Given are the following functions:\nmoves :: Position -> [Move]\nmove\n:: Position -> Move -> Position\nsolved :: Position -> Bool\nThe function moves describes the set of possible moves that can be made in a given\nposition, move makes a move, and solved determines those positions that are a\nsolution to the puzzle. Solving the puzzle means \ufb01nding some sequence of moves,\npreferably a shortest such sequence, that leads from a given starting position to a\nsolved position:\n\n```\n\n\n```\n\n\f10.5 Mutable arrays\n\n257\n\nsolve :: Position -> Maybe [Move]\nThe value solve p is Nothing if there is no sequence of moves starting in position\np that leads to a solved position, and Just ms otherwise, where\nsolved (foldl move p ms)\nWe are going to implement solve by carrying out a breadth-\ufb01rst search. What this\nmeans is that we examine all positions one move away from the starting position\nto see if there is a solution, then all positions two moves away, and so on. Breadth\ufb01rst will therefore \ufb01nd a shortest solution if one exists. To implement the search\nwe need\ntype Path\n= ([Move],Position)\ntype Frontier = [Path]\nA path consists of a sequence of moves made from the starting position (in reverse\norder), and the position that results after making the moves. A frontier is a list\nof paths waiting to be extended into longer paths. A breadth-\ufb01rst search is then\nimplemented by\nsolve p = bfs [] [([],p)]\nbfs :: [Position] -> Frontier -> Maybe [Move]\nbfs ps [] = Nothing\nbfs ps ((ms,p):mps)\n| solved p\n= Just (reverse ms)\n| p `elem` ps = bfs ps mps\n| otherwise\n= bfs (p:ps) (mps ++ succs (ms,p))\nsuccs :: Path -> [Path]\nsuccs (ms,p) = [(m:ms,move p m) | m <- moves p]\nThe \ufb01rst argument ps of bfs represents the set of positions that have already been\nexplored. The second argument is the frontier, which is managed in a queue-like\nfashion to ensure that paths of the same length are inspected before their successors.\nInspecting a path means accepting it if the \ufb01nal position is a solution, rejecting it\nif the end position has already been explored, and otherwise adding its successors\nto the end of the current frontier for future exploration. The moves in a successful\npath are reversed before being returned as the \ufb01nal result of bfs simply because,\nfor ef\ufb01ciency, succs adds a new move to the front of the list rather than at the end.\nThere are two major sources of inef\ufb01ciency with bfs, one concerning the use of\n\n\f258\n\nImperative functional programming\n\n(++) and the other concerning elem. Firstly, the size of a frontier can grow exponentially and so concatenating successors to the end of the frontier is slow. Better\nis the following alternative to bfs:\nbfs :: [Position] -> Frontier -> Frontier ->\nMaybe [Move]\nbfs ps [] [] = Nothing\nbfs ps [] mqs = bfs ps mqs []\nbfs ps ((ms,p):mps) mqs\n| solved p\n= Just (reverse ms)\n| p `elem` ps = bfs ps mps mqs\n| otherwise\n= bfs (p:ps) mps (succs (ms,p) ++ mqs)\nThe additional argument is a temporary frontier used to store successors. When the\n\ufb01rst frontier is exhausted the contents of the temporary frontier are installed as the\nnew frontier. Adding successors to the front of the temporary frontier takes time\nproportional to the number of successors, not to the size of the frontier, and that\nleads to a faster algorithm. On the other hand, the new version of bfs is not the\nsame as the old one because successive frontiers are traversed alternately from left\nto right and from right to left. Nevertheless a shortest solution will still be found if\none exists.\nThe second source of inef\ufb01ciency is the membership test. Use of a list to store\npreviously explored positions is slow because the membership test can take time\nproportional to the number of currently explored positions. It would all be easier\nif positions were integers in the range [0 .. n\u22121] for some n, for then we could\nuse a boolean array with bounds (0, n\u22121) to tick off positions as they arise. The\nmembership test would then consist of a single array lookup.\nOne can imagine coding positions as integers, but not as integers in an initial segment of the natural numbers. For instance, a Sudoku position (see Chapter 5) can\nbe expressed as an integer consisting of 81 digits. So suppose we have a function\nencode :: Position -> Integer\nthat encodes positions as integers. To reduce the range we can de\ufb01ne\nhash :: Position -> Int\nhash p = fromInteger (encode p) `mod` n\nfor some suitable n :: Int. The result of hash is then an integer in the range\n[0..n-1].\n\n```\n\n\n```\n\nThe one hitch, and it\u2019s a big one, is that two distinct positions may hash to the\n\n\f10.6 Immutable arrays\n\n259\n\nsame integer. To solve this problem we abandon the idea of having an array of\nbooleans, and instead have an array of lists of positions. The positions in the array\nat index k are all those whose hash value is k. There is no guarantee that any of this\nwill improve ef\ufb01ciency in the worst case, but if we allow n to be reasonably large,\nand trust that the hash function assigns integers to positions in a reasonably evenly\ndistributed way, then the complexity of a membership test is reduced by a factor of\nn.\nWith this hashing scheme the revised code for solve is:\nsolve :: Maybe [Move]\nsolve = runST $\ndo {pa <- newArray (0,n-1) [];\nbfs pa [([],start)] []}\nbfs :: STArray s Int [Position] -> Frontier ->\nFrontier -> ST s (Maybe [Move])\nbfs pa [] [] = return Nothing\nbfs pa [] mqs = bfs pa mqs []\nbfs pa ((ms,p):mps) mqs\n= if solved p then return (Just (reverse ms))\nelse do {ps <- readArray pa k;\nif p `elem` ps\nthen bfs pa mps mqs\nelse\ndo {writeArray pa k (p:ps);\nbfs pa mps (succs (ms,p) ++ mqs)}}\nwhere k = hash p\n\n10.6 Immutable arrays\nWe cannot leave the subject of arrays without mentioning a very nice Haskell library Data.Array that provides purely functional operations on immutable arrays.\nThe operations are implemented using mutable arrays, but the interface is purely\nfunctional.\nThe type Array i e is an abstract type of arrays with indices of type i and elements of type e. One basic operation for constructing arrays is\narray :: Ix i => (i,i) -> [(i,e)] -> Array i e\n\n\f260\n\n```\n\n\n```\n\nImperative functional programming\n\nThis function take a pair of bounds, the lowest and highest indices in the array,\nand a list of index-element pairs specifying the array entries. The result is an array\nwith the given bounds and entries. Any entry missing from the association list is\ndeemed to be the unde\ufb01ned entry. If two entries have the same index, or one of the\nindices is out of bounds, the unde\ufb01ned array is returned. Because of these checks,\narray construction is strict in the indices, though lazy in the elements. Building the\narray takes linear time in the number of entries.\nA simple variant of array is listArray which takes just a list of elements:\nlistArray :: Ix i => (i,i) -> [e] -> Array i e\nlistArray (l,r) xs = array (l,r) (zip [l..r] xs)\nFinally, there is another way of building arrays called accumArray whose type\nappears rather daunting:\nIx i => (e -> v -> e) -> e -> (i,i) -> [(i,v)] -> Array i e\nThe \ufb01rst argument is an \u2018accumulating\u2019 function for transforming array entries\nand new values into new entries. The second argument is an initial entry for each\nindex. The third argument is a pair of bounds, and the fourth and \ufb01nal argument is\nan association list of index\u2013value pairs. The result is an array built by processing\nthe association list from left to right, combining entries and values into new entries\nusing the accumulating function. The process takes linear time in the length of the\nassociation list, assuming the accumulating function takes constant time.\nThat\u2019s what accumArray does in words. In symbols,\nelems (accumArray f e (l,r) ivs)\n= [foldl f e [v | (i,v) <- ivs, i==j] | j <- [l..r]]\nwhere elems returns the list of elements of an array in index order. Well, the identity above is not quite true: there is an additional restriction on ivs, namely that\nevery index should lie in the speci\ufb01ed range. If this condition is not met, then the\nleft-hand side returns an error while the right-hand side does not.\nComplicated as accumArray seems, it turns out to be a very useful tool for solving\ncertain kinds of problem. Here are two examples. First, consider the problem of\nrepresenting directed graphs. Directed graphs are usually described in mathematics\nin terms of a set of vertices and a set of edges. An edge is an ordered pair (j, k) of\nvertices signifying that the edge is directed from j to k. We say that k is adjacent to\nj. We will suppose that vertices are named by integers in the range 1 to n for some\nn. Thus\ntype Vertex = Int\n\n\f10.6 Immutable arrays\n\ntype Edge\ntype Graph\n\n261\n\n= (Vertex,Vertex)\n= ([Vertex],[Edge])\n\nvertices g = fst g\nedges g\n= snd g\nIn computing, directed graphs are often described in terms of adjacency lists:\nadjs :: Graph -> Vertex -> [Vertex]\nadjs g v = [k | (j,k) <- edges g, j==v]\nThe problem with this de\ufb01nition of adjs is that it takes time proportional to the\nnumber of edges to compute the adjacency list of any particular vertex. Better is to\nimplement adjs as an array:\nadjArray :: Graph -> Array Vertex [Vertex]\nThen we have\nadjs g v = (adjArray g)!v\nwhere (!) denotes the operation of array-indexing. For reasonably sized arrays\nthis operation takes constant time.\nThe speci\ufb01cation of adjArray is that\nelems (adjArray g)\n= [[k | (j,k) <- edges g, j==v] | v <- vertices g]\nUsing this speci\ufb01cation we can calculate a direct de\ufb01nition of adjArray. To keep\neach line short, abbreviate edges g to es and vertices g to vs, so\nelems (adjArray g) = [[k | (j,k) <- es, j==v] | v <- vs]\nConcentrating on the right-hand side, the \ufb01rst step is to rewrite it using the law\nfoldr (:) [] = id. That gives the expression\n[foldr (:) [] [k | (j,k) <- es, j==v] | v <- vs]\nNext we use the law foldr f e xs = foldl (flip f) e (reverse xs) for\nall \ufb01nite lists xs. Abbreviating flip (:) to (@), we obtain\n[foldl (@) [] (reverse [k | (j,k) <- es, j==v]) | v <- vs]\nDistributing reverse we obtain the expression\n[foldl (@) [] [k | (j,k) <- reverse es, j==v] | v <- vs]\nNext we use swap (j,k) = (k,j) to obtain\n\n\f262\n\nImperative functional programming\n\n[foldl (@) [] [j | (k,j) <- es', j==v] | v <- vs]\nwhere es' = map swap (reverse es). Finally, using n = length vs and the\nspeci\ufb01cation of accumArray, we obtain\nelems (adjArray g)\n= elems (accumArray (flip (:)) [] (1,n) es')\nThat means we can de\ufb01ne\nadjArray g = accumArray (flip (:)) [] (1,n) es\nwhere n = length (vertices g)\nes = map swap (reverse (edges g))\nThis de\ufb01nition of adjArray g computes the successors in time proportional to the\nnumber of edges.\nHere is the second example of the use of accumArray. Suppose we are given a list\nof n integers, all in the range (0, m) for some m. We can sort this list in \u0398(m+n)\nsteps by counting the number of times each element occurs:\ncount :: [Int] -> Array Int Int\ncount xs = accumArray (+) 0 (0,m) (zip xs (repeat 1))\nThe value repeat 1 is an in\ufb01nite list of 1s. Counting takes \u0398(n) steps. Having\ncounted the elements, we can now sort them:\nsort xs = concat [replicate c x\n| (x,c) <- assocs (count xa)]\nThe function assocs is yet another library function and returns the list of index\u2013\nelement pairs of an array in index order. The sorting is completed in \u0398(m) steps.\nAs well as the above operations Data.Array contains one or two more, including\nthe update operation (//):\n(//) :: Ix i => Array i e -> [(i,e)] -> Array i e\nFor example, if xa is an n \u00d7 n matrix, then\nxa // [((i,i),0) | i <- [1..n]]\nis the same matrix except with zeros along the diagonal. The downside of (//) is\nthat it takes time proportional to the size of the array, even for an update involving\na single element. The reason is that a completely new array has to be constructed\nbecause the old array xa continues to exist.\nWe have ended the chapter back in the world of pure functional programming,\n\n\f10.7 Exercises\n\n263\n\nwhere equational reasoning can be used both to calculate de\ufb01nitions and to optimise them. Although the monadic style is attractive to programmers who are used\nto imperative programming, there remains the problem of how to reason about\nmonadic programs. True, equational reasoning is still possible in certain situtations\n(see Exercise F for an example), but it is not so widely applicable as it is in the pure\nfunctional world (witness the correctness of the partition phase of Quicksort). Imperative programmers have the same problem, which they solve (if they bother\nto) by using predicate calculus, preconditions, postconditions and loop invariants.\nHow to reason directly with monadic code is still a topic of ongoing research.\nOur best advice is to use the monadic style sparingly and only when it is really\nuseful; otherwise the most important aspect of functional programming, the ability\nto reason mathematically about its constructs, is lost.\n\n10.7 Exercises\nExercise A\nRecall that\nputStr = foldr (>>) done . map putChar\nWhat does\nfoldl (>>) done . map putChar\ndo? Justify your answer by expressing (>>) in terms of (>>=) and appealing to\nthe monad laws.\nExercise B\nUsing a pattern-matching style, de\ufb01ne a function\nadd3 :: Maybe Int -> Maybe Int -> Maybe Int -> Maybe Int\nthat adds three numbers, provided all of them exist. Now rewrite add3 using the\nMaybe monad.\nExercise C\nThe monadic de\ufb01nition of cp in Section 10.1 is still inef\ufb01cient. We might prefer to\nwrite\n\n\fImperative functional programming\n\n264\n\ncp (xs:xss) = do {ys <- cp xss;\nx <- xs;\nreturn (x:ys)}\nBy de\ufb01nition a commutative monad is one in which the equation\ndo {x <- p; y <- q; f x y}\n= do {y <- q; x <- p; f x y}\nholds. The IO monad is certainly not commutative, while some other monads are.\nIs the Maybe monad commutative?\nExercise D\nEvery monad is a functor. Complete the de\ufb01nition\ninstance Monad m => Functor m where\nfmap :: (a -> b) -> m a -> m b\nfmap f = ...\nCurrently Haskell does not insist that the Monad class should be a subclass of\nFunctor, though there are plans to change this in future releases. Instead, Haskell\nprovides a function liftM equivalent to fmap for monads. Give a de\ufb01nition of\nliftM in terms of return and >>=.\nThe function join :: m (m a) -> m a \ufb02attens two layers of monadic structure\ninto one. De\ufb01ne join in terms of >>=. What familiar functions do join and liftM\ngive for the list monad?\nFinally, using join and liftM, de\ufb01ne (>>=). It follows that instead of de\ufb01ning\nmonads in terms of return and >>=, we can also de\ufb01ne them in terms of return,\nliftM and join.\nExercise E\nA number of useful monadic functions are provided in the Control.Monad library.\nFor instance:\nsequence_ :: Monad m => [m a] -> m ()\nsequence_ = foldr (>>) done\n(The underscore convention is used in a number of places in Haskell to signify that\nthe result of the action is the null tuple.) De\ufb01ne the related function\nsequence :: Monad m => [m a] -> m [a]\nUsing these two functions, de\ufb01ne\n\n\f10.7 Exercises\n\n265\n\nmapM_ :: Monad m => (a -> m b) -> [a] -> m ()\nmapM :: Monad m => (a -> m b) -> [a] -> m [b]\nAlso, de\ufb01ne\nfoldM :: Monad m => (b -> a -> m b) -> b -> [a] -> m b\nIn the text we made use of a function repeatFor n that repeated an action n times.\nGeneralise this function to\nfor_ :: Monad m => [a] -> (a -> m b) -> m ()\n\n```\n\n\n```\n\nExercise F\nHere is an exercise in monadic equational reasoning. Consider the function\nadd :: Int -> State Int ()\nadd n = do {m <- get; put (m+n)}\nThe task is to prove that\nsequence_ . map add = add . sum\nwhere sequence_ was de\ufb01ned in the previous exercise and sum sums a list of\nintegers. You will need the fusion law of foldr, some simple laws of put and\nget, and the monad law\ndo {stmts1} >> do {stmts2} = do {stmts1;stmts2}\nwhich is valid provided the variables in stmts1 and stmts2 are disjoint.\nExercise G\nProve the leapfrog rule: (f >=> g) . h = (f . h) >=> g.\nUsing this rule, prove: (return . h) >=> g = g . h.\nExercise H\nProve that\nliftM f = id >=> (return . f)\njoin\n= id >=> id\nA fourth way of describing the monad laws is in terms of the two functions liftM\nand join of Exercise D. There are seven laws governing these two functions, all\nof which have a familiar ring:\n\n\fImperative functional programming\n\n266\n\nliftM id\n= id\nliftM (f . g) = liftM f . liftM g\nliftM f . return = return . f\nliftM f . join\n= join . liftM (liftM f)\njoin . return\n= id\njoin . liftM return = id\njoin . liftM join\n= join . join\nProve the fourth rule.\nExercise I\nWhat does build [] do (see Section 10.3)?\nExercise J\nWrite an interactive program to play hangman. An example session:\nghci> hangman\nI am thinking of a word:\n----Try and guess it.\nguess: break\n-a--guess: parties\nWrong number of letters!\nguess: party\n-appy\nguess: happy\nYou got it!\nPlay again? (yes or no)\nno\nBye!\nAssume that a list of secret words is stored in a \ufb01le called Words, so that the\naction xs <- readFile \"Words\" reads the \ufb01le as a list of characters. By the\nway, readFile is lazy in that its contents are read on demand.\nExercise K\nWrite another version of fib in terms of a fibST that uses a single STRef.\n\n\f10.8 Answers\n\n267\n\nExercise L\nOne way of de\ufb01ning the greatest common divisor (gcd) of two positive integers is:\ngcd (x,y) | x==y = x\n| xy = gcd (x-y,y)\nTranslate this de\ufb01nition into two other programs, one of which uses the State\nmonad and the other the ST monad.\nExercise M\nHere is a concrete puzzle you can solve using breadth-\ufb01rst search. A cut-down\nversion of Sam Loyd\u2019s famous 15 puzzle is the 8 puzzle. You are given a 3 \u00d7 3\narray containing tiles numbered from 1 to 8 and one blank space. You move by\nsliding an adjacent tile into the blank space. Depending on where the blank space\nis, you can slide tiles upwards, downwards, to the left or to the right. At the start\nthe blank space is in the top left corner and the tiles read from 1 to 8. At the end\nthe blank space is in the bottom right corner, but the tiles are still neatly arranged\nin the order 1 to 8.\nYour mission, should you choose to accept it, is to settle on a suitable representation\nof positions and moves, and to de\ufb01ne the functions moves, move, solved and\nencode.\n\n10.8 Answers\nAnswer to Exercise A\nWe claim that (>>) :: IO () -> IO () -> IO () is associative with identity\nelement done. That means\nputStr xs = foldl (>>) done (map putChar xs)\nfor all \ufb01nite strings xs\nWe concentrate on the proof of associativity. Firstly, for actions in IO () we have\np >> q = p >>= const q\n\n\fImperative functional programming\n\n268\n\nwhere const x y = x. Now we can reason:\n(p >> q) >> r\n=\n\n{de\ufb01nition of (>>)}\n(p >>= const q) >>= const r\n\n=\n\n{third monad law}\np >>= const (q >>= const r)\n\n=\n\n{de\ufb01nition of (>>)}\np >>= const (q >> r)\n\n=\n\n{de\ufb01nition of (>>)}\np >> (q >> r)\n\nAnswer to Exercise B\nThe direct version uses pattern matching with a wild-card:\nadd3\nadd3\nadd3\nadd3\n\nNothing _ _\n(Just x) Nothing _\n(Just x) (Just y) Nothing\n(Just x) (Just y) (Just z)\n\n=\n=\n=\n=\n\nNothing\nNothing\nNothing\nJust (x+y+z)\n\nThis de\ufb01nition ensures that add Nothing undefined =\n\nNothing.\n\nThe monadic version reads:\nadd3 mx my mz\n= do {x <- mx; y <- my; z <- mz;\nreturn (x + y + z)}\nAnswer to Exercise C\nYes. The commutative law states that\np >>= \\x -> q >>= \\y -> f x y\n= q >>= \\y -> p >>= \\x -> f x y\nIn the Maybe monad there are four possible cases to check. For example, both\nsides simplify to Nothing if p = Nothing and q = Just y, . The other cases\nare similar.\nAnswer to Exercise D\nWe have\n\n\f10.8 Answers\n\n269\n\nfmap f p = p >>= (return . f)\njoin p\n= p >>= id\nFor the list monad we have liftM = map and join = concat.\nIn the other direction\np >>= f = join (liftM f p)\nAnswer to Exercise E\nThe function sequence is de\ufb01ned by\nsequence :: Monad m => [m a] -> m [a]\nsequence = foldr k (return [])\nwhere k p q = do {x <- p; xs <- q; return (x:xs)}\nThe two new map functions are:\nmapM_ f = sequence_ . map f\nmapM f = sequence . map f\nThe function foldM is de\ufb01ned by\nfoldM :: Monad m => (b -> a -> m b) ->\nb -> [a] -> m b\nfoldM f e []\n= return e\nfoldM f e (x:xs) = do {y <- f e x; foldM f y xs}\nNote that foldM is analogous to foldl in that it works from left to right. Finally\nfor = flip mapM_.\nAnswer to Exercise F\nThe \ufb01rst thing to note is that\nsequence_ . map add\n= foldr (>>) done . map add\n= foldr ((>>) . add) done\nusing the fusion law of foldr and map given in Section 6.3. Moreover,\n((>>) . add) n p = add n >> p\nSince sum = foldr (+) 0 that means we have to prove\nfoldr (\\ n p -> add n >> p) = add . foldr (+) 0\n\n\fImperative functional programming\n\n270\n\nThat looks like an instance of the fusion law of foldr. We therefore have to show\nthat add is strict (which it is), and\nadd 0 = done\nadd (n + n') = add n >> add n'\nHere goes:\nadd 0\n=\n\n{de\ufb01nition}\ndo {m <- get; put (m+0)}\n\n=\n\n{arithmetic}\ndo {m <- get; put m}\n\n=\n\n{simple law of put and get}\ndone\n\nThat disposes of the \ufb01rst condition. For the second we start with the more complicated side and reason:\nadd n >> add n\u2019\n=\n\n{de\ufb01nition}\ndo {l <- get; put (l + n) } >>\ndo {m <- get; put (m + n\u2019)}\n\n=\n\n{monad law}\ndo {l <- get; put (l + n); m <- get; put (m + n\u2019)}\n\n=\n\n{simple law of put and get}\ndo {l <- get; put ((l + n) + n\u2019)}\n\n=\n\n{associativity of (+); definition of add}\nadd (n + n\u2019)\n\nAnswer to Exercise G\nWe can reason:\n(f >=> g) (h x)\n=\n\n{de\ufb01nition of (>=>)}\nf (h x) >>= g\n\n=\n\n{de\ufb01nition of (>=>)}\n(f . h >=> g) x\n\n\f10.8 Answers\n\nFor the second part:\n(return . h) >=> g\n=\n\n{leapfrog rule}\n(return >=> g) . h\n\n=\n\n{monad law}\ng . h\n\nAnswer to Exercise H\nFor the fourth rule we simplify both sides. For the left-hand side:\nliftM f . join\n=\n\n{de\ufb01nitions}\n(id >=> (return . f)) . (id >=> id)\n\n=\n\n{leapfrog rule and id . f = f}\n(id >=> id) >=> (return . f)\n\nFor the right-hand side:\njoin . liftM (liftM f)\n=\n\n{de\ufb01nitions}\n(id >=> id) . (id >=> return . (id >=> (return . f)))\n\n=\n\n{leapfrog rule, and associativity of (>=>)}\nid >=> (return . (id >=> (return . f))) >=> id\n\n=\n\n{since (return . h) >=> g = g . h}\nid >=> id >=> (return . f)\n\nThe two sides are equal because (>=>) is associative.\nAnswer to Exercise I\nbuild [] causes an in\ufb01nite loop, so its value is \u22a5.\nAnswer to Exercise J\nFor the main function we can de\ufb01ne\nhangman :: IO ()\nhangman = do {xs <- readFile \"Words\";\nplay (words xs)}\n\n271\n\n\fImperative functional programming\n\n272\n```\n\n\n```\n\n\nThe function play plays as many rounds of the game as desired with different\nwords from the \ufb01le (which we quietly suppose always has enough words):\nplay (w:ws)\n= do {putStrLn \"I am thinking of a word:\";\nputStrLn (replicate (length w) '-');\nputStrLn \"Try and guess it.\";\nguess w ws}\nThe function guess deals with a single guess, but keeps the remaining words for\nany subsequent round of play:\nguess w ws\n= do {putStr \"guess: \";\nw' <- getLine;\nif length w' /= length w then\ndo {putStrLn \"Wrong number of letters!\";\nguess w ws}\nelse if w' == w\nthen\ndo {putStrLn \"You got it!\";\nputStrLn \"Play again? (yes or no)\";\nans <- getLine;\nif ans == \"yes\"\nthen play ws\nelse putStrLn \"Bye!\"}\nelse do {putStrLn (match w' w);\nguess w ws}}\nFinally we program match:\nmatch w' w = map check w\nwhere\ncheck x = if x `elem` w' then x else '-'\nAnswer to Exercise K\nThe following program is correct but doesn\u2019t run in constant space:\nfib n = fst $ runST (fibST n)\nfibST :: Int -> ST s (Integer,Integer)\nfibST n = do {ab <- newSTRef (0,1);\n\n\f10.8 Answers\n\n273\n\nrepeatFor n\n(do {(a,b) <- readSTRef ab;\nwriteSTRef ab $! (b,a+b)});\nreadSTRef ab}\nThe reason is that (b,a+b) is already in head-normal form, so strict-apply has no\neffect. The penultimate line needs to be changed to\nb `seq` (a+b) `seq` writeSTRef ab (b,a+b)\nin order to force evaluation of the components.\nAnswer to Exercise L\nThe version that uses the State monad:\ngcd (x,y) = fst $ runState loop (x,y)\nloop :: State (Int,Int) Int\nloop = do {(x,y) <- get;\nif x == y\nthen return x\nelse if x < y\nthen do {put (x,y-x); loop}\nelse do {put (x-y,y); loop}}\nThe version that uses the ST monad:\ngcd (x,y) = runST $\ndo {a <- newSTRef x;\nb <- newSTRef y;\nloop a b}\nloop :: STRef s Int -> STRef s Int -> ST s Int\nloop a b\n= do {x <- readSTRef a;\ny <- readSTRef b;\nif x==y\nthen return x\nelse if x Integer\nencode (j,ks) = foldl op 0 ks\nwhere op x d = 10*x + fromIntegral d\nstart :: Position\nstart = (0,[0..8])\nThe function moves can be de\ufb01ned by\nmoves :: Position -> [Move]\nmoves (j,ks)\n= [Up\n| j `notElem` [6,7,8]] ++\n[Down | j `notElem` [0,1,2]] ++\n[Left | j `notElem` [2,5,8]] ++\n[Right | j `notElem` [0,3,6]]\nUp moves are allowed except for a blank in the bottom row; down moves except\nfor a blank in the top row, left moves except for a blank in the rightmost column,\nand right moves except for a blank in the leftmost column.\nThe function move can be de\ufb01ned by:\nmove\nmove\nmove\nmove\nmove\n\n:: Position ->\n(j,ks) Up\n=\n(j,ks) Down =\n(j,ks) Left =\n(j,ks) Right =\n\nMove -> Position\n(j+3,swap (j,j+3)\n(j-3,swap (j-3,j)\n(j+1,swap (j,j+1)\n(j-1,swap (j-1,j)\n\nks)\nks)\nks)\nks)\n\nswap (j,k) ks = ks1 ++ y:ks3 ++ x:ks4\nwhere (ks1,x:ks2) = splitAt j ks\n(ks3,y:ks4) = splitAt (k-j-1) ks2\nFinally,\n\n\f10.9 Chapter notes\n\n275\n\nsolved :: Position -> Bool\nsolved p = p == (8,[1,2,3,4,5,6,7,8,0])\nMy computer produced:\nghci> solve start\nJust [Left,Up,Right,Up,Left,Left,Down,\nRight,Right,Up,Left,Down,Down,Left,\nUp,Up,Right,Right,Down,Left,Left,Up]\n(4.84 secs, 599740496 bytes)\n\n10.9 Chapter notes\nRead The History of Haskell to see how monads came to be an integral part of\nHaskell, and why this idea has been mainly responsible for the increasing use of\nHaskell in the real world. Monads are used to structure GHC, which itself is written\nin Haskell. Each phase of the compiler uses a monad for book-keeping information.\nFor instance, the type checker uses a monad that combines state (to maintain a\ncurrent substitution), a name supply (for fresh type variable names) and exceptions.\nUse of do-notation in preference to (>>=) was suggested by John Launchbury in\n1993 and was \ufb01rst implemented by Mark Jones in Gofer.\nThe number of tutorials on monads has increased steadily over the years; see\nhaskell.org/haskellwiki/Monad_tutorials\nfor a reasonably comprehensive list.\nThe example (in Exercise F) of monadic equational reasoning can be found in the\npaper \u2018Unifying theories of programming with monads\u2019, (UTP Symposium, August 2012) by Jeremy Gibbons. For additional material on reasoning equationally\nwith monads, read \u2018Just do it: simple monadic equational reasoning\u2019 by Jeremy\nGibbons and Ralf Hinze, which appeared in the proceedings of the 2011 International Conference of Functional Programming. Both papers can be found at\nwww.cs.ox.ac.uk/people/jeremy.gibbons/publications/\n\n\fChapter 11\nParsing\n\nA parser is a function that analyses a piece of text to determine its logical structure. The text is a string of characters describing some value of interest, such as an\narithmetic expression, a poem or a spreadsheet. The output of a parser is a representation of the value, such as a tree of some kind for an arithmetic expression, a\nlist of verses for a poem, or something more complicated for a spreadsheet. Most\nprogramming tasks involve decoding the input in some way, so parsing is a pervasive component of computer programming. In this chapter we will describe a\nmonadic approach to parsing, mainly designing simple parsers for expressions of\nvarious kinds. We will also say a little more about the converse process of encoding\nthe output as a string; in other words, more about the type class Show. This material\nwill be used in the \ufb01nal chapter.\n\n11.1 Parsers as monads\nParsers return different values of interest, so as a \ufb01rst cut we can think of a parser\nas a function that takes a string and returns a value:\ntype Parser a = String -> a\nThis type is basically the same as that of the standard prelude function\nread :: Read a => String -> a\nIndeed, read is a parser, though not a very \ufb02exible one. One reason is that all the\ninput must be consumed. Thus:\nghci> read \"123\" :: Int\n123\n\n\f11.1 Parsers as monads\n\n277\n\nghci> read \"123+51\" :: Int\n*** Exception: Prelude.read: no parse\nWith read there is no obvious way of reading two or more things in sequence. For\nexample, in a parser for arithmetic expressions we may want to look in the input\nstream for a numeral, then an operator and then another numeral. The \ufb01rst parser\nfor a numeral will consume some pre\ufb01x of the input, the parser for an operator\nsome pre\ufb01x of the remaining input, and the third parser yet more input. A better\nidea is to de\ufb01ne a parser as a function that consumes a pre\ufb01x of the input and\nreturns both a value of interest and the unconsumed suf\ufb01x:\ntype Parser a = String -> (a,String)\nWe are not quite there yet. It can happen that a parser may fail on some input.\nIt is not a mistake to construct parsers that can fail. For example, in a parser for\narithmetic expressions, we may want to look for either a numeral or an opening\nparenthesis. One or either of these subsidiary parsers will certainly fail. Failure\nshould not be thought of as an error that terminates the parsing process; rather it\nacts like an identity element for an operation that chooses between alternatives.\nMore generally, a parser may \ufb01nd a number of different ways that some pre\ufb01x of\nthe input can be structured. Failure then corresponds to the particular case of the\nempty sequence of parses. In order to handle these various possibilities, we change\nour de\ufb01nition yet again and de\ufb01ne\ntype Parser a = String -> [(a,String)]\nThe standard prelude provides exactly this type synonym, except that it is called\nReadS, not Parser. And it also provides a function\nreads :: Read a => ReadS a\nas a subsidiary method in the type class Read. For example,\nghci> reads \"-123+51\" :: [(Int,String)]\n[(-123,\"+51\")]\nghci> reads \"+51\" :: [(Int,String)]\n[]\nAs with the function read you have to tell reads the type you are expecting.\nThe second example fails, returning no parses, because a Haskell integer can be\npreceded by an optional minus sign but not by an optional plus sign. By de\ufb01nition,\na parser is deterministic if it returns an empty or singleton list of parses in all\npossible cases. In particular, instances of reads ought to be deterministic parsers.\n\n\f278\n\nParsing\n\nThere is one further change we have to make to the de\ufb01nition of Parser. We would\nlike to install this type as an instance of the Monad class, but that is not possible. The\nreason is that Parser is declared as a type synonym, and type synonyms cannot be\nmade members of any type class: they inherit whatever instances are declared for\nthe underlying type. A type synonym is there simply to improve readability in type\ndeclarations; no new types are involved and we cannot construct two different type\nclass instances for what is essentially the same type.\nOne way to construct a new type is by a data declaration:\ndata Parser a = Parser (String -> [(a,String)])\nThe identi\ufb01er Parser on the right is a constructor, while on the left it is the name\nof a new type. Most people are happy with the pun; others would rename the constructor as something like MkParser or just P.\nThere is a better way to create a new type for Parser and that is to use a newtype\ndeclaration:\nnewtype Parser a = Parser (String -> [(a,String)])\nWe have not needed newtype declarations up to now, so let us digress a little to\nexplain them. The price paid for using a data declaration for Parser is that operations to examine parsers have to be constantly unwrapped and rewrapped with\nthe constructor Parser, and this adds to the running time of parser operations. In\naddition there is an unwanted element of Parser, namely Parser undefined.\nIn other words, Parser a and String -> [(a,String)] are not isomorphic\ntypes. Recognising this, Haskell allows a newtype declaration for types de\ufb01ned\nwith a single constructor taking a single argument. It differs from a type synonym\nin that it creates a genuinely new type whose values must be expressed using the\nParser wrapper. But these coercions, though they have to appear in the program\ntext, do not add to the execution time of the program because the Haskell compiler\neliminates them before evaluation begins. The values of the new type are systematically replaced by the values in the underlying type. Consequently, Parser a and\nString -> [(a,String)] describe isomorphic types, and Parser undefined\nand undefined are isomorphic values sharing the same representation. New types,\nas distinct from synonym types, can be made members of type classes in different\nways from the underlying type.\nWith either kind of declaration we have to provide some way of applying the parsing function, so we de\ufb01ne\napply :: Parser a -> String -> [(a,String)]\napply (Parser p) s = p s\n\n\f11.2 Basic parsers\n\n279\n\nThe functions apply and Parser are mutual inverses and witness the isomorphism.\nWe also de\ufb01ne\nparse :: Parser a -> String -> a\nparse p = fst . head . apply p\nThe function parse p returns the \ufb01rst object of the \ufb01rst parse, causing an error if\nthe parser p fails. This is the only place an error might occur.\nNow we can de\ufb01ne\ninstance Monad Parser where\nreturn x = Parser (\\s -> [(x,s)])\np >>= q = Parser (\\s -> [(y,s'')\n| (x,s') <- apply p s,\n(y,s'') <- apply (q x) s'])\nIn the de\ufb01nition of p >>= q the parser p is applied to an input string, producing a\nlist of possible parses each of which is paired with the corresponding unconsumed\nportion of the input. The parser q is then applied to each parse to produce a list of\nresults whose concatenation provides the \ufb01nal answer. One should also show that\nthe three monad laws hold, a task we will leave as an exercise.\n\n11.2 Basic parsers\nPerhaps the simplest basic parser is\ngetc :: Parser Char\ngetc = Parser f\nwhere f []\n= []\nf (c:cs) = [(c,cs)]\nThis parser returns the \ufb01rst character of the input if there is one. It plays exactly the\nsame role for parsers as getChar does for the input\u2013output monad of the previous\nchapter.\nNext, here is a parser for recognising a character that satis\ufb01es a given condition:\nsat :: (Char -> Bool) -> Parser Char\nsat p = do {c <- getc;\nif p c then return c\nelse fail}\n\n\fParsing\n\n280\n\nwhere fail is de\ufb01ned by\nfail = Parser (\\s -> [])\nThe parser fail is another basic parser that returns no parses. The parser sat p\nreads a character and, if it satis\ufb01es p, returns the character as the result. The de\ufb01nition of sat can be written more brie\ufb02y by using a little combinator called guard:\nsat p = do {c <- getc; guard (p c); return c}\nguard :: Parser ()\nguard True = return ()\nguard False = fail\nTo see that these two de\ufb01nitions are the same, observe that if p c is false, then\nguard (p c) >> return c = fail >> return c = fail\nNote the use of the law fail >> p = fail, whose proof we leave as an exercise.\nIf p c is true, then\nguard (p c) >> return c\n= return () >> return c\n= return c\nUsing sat we can de\ufb01ne a number of other parsers; for instance\nchar :: Char -> Parser ()\nchar x = do {c <- sat (==x); return ()}\nstring :: String -> Parser ()\nstring []\n= return ()\nstring (x:xs) = do {char x; string xs; return ()}\nlower :: Parser Char\nlower = sat isLower\ndigit :: Parser Int\ndigit = do {d <- sat isDigit; return (cvt d)}\nwhere cvt d = fromEnum d - fromEnum '0'\nThe parser char x looks for the speci\ufb01c character x as the next item in the input string, while string xs looks for a speci\ufb01c string; both parsers return () if\nsuccessful. For example,\n\n\f11.3 Choice and repetition\n\n281\n\nghci> apply (string \"hell\") \"hello\"\n[((),\"o\")]\nThe parser digit looks for a digit character and returns the corresponding integer\nif successful. The parser lower looks for a lowercase letter, returning such a letter\nif found.\n\n11.3 Choice and repetition\nIn order to de\ufb01ne more sophisticated parsers we need operations for choosing between alternative parsers and for repeating parsers. One such alternation operator\nis (<|>), de\ufb01ned by\n(<|>) :: Parser a -> Parser a -> Parser a\np <|> q = Parser f\nwhere f s = let ps = apply p s in\nif null ps then apply q s\nelse ps\nThus p <|> q returns the same parses as p unless p fails, in which case the parses\nof q are returned. If both p and q are deterministic, then so is p <|> q. For another\nchoice of <|> see the exercises. We claim that <|> is associative with fail as its\nidentity element, but again we relegate the proof as an exercise.\nHere is a parser for recognising a string of lowercase letters:\nlowers :: Parser String\nlowers = do {c <- lower; cs <- lowers; return (c:cs)}\n<|> return \"\"\nTo see how this parser works, suppose the input is the string \u2018Upper\u2019. In this case\nthe parser on the left of <|> fails because \u2018U\u2019 is not a lowercase letter. However,\nthe parser on the right succeeds, so\nghci> apply lowers \"Upper\"\n[(\"\",\"Upper\")]\nWith input string \u2018isUpper\u2019, the left-hand parser succeeds, so\nghci> apply lowers \"isUpper\"\n[(\"is\",\"Upper\")]\n\n\fParsing\n\n282\n\nUse of the choice operator <|> requires care. For example, consider a very simple\nform of arithmetic expression that consists of either a single digit or a digit followed\nby a plus sign followed by another digit. Here is a possible parser:\nwrong :: Parser Int\nwrong = digit <|> addition\naddition :: Parser Int\naddition = do {m <- digit; char '+'; n <- digit;\nreturn (m+n)}\nWe have\nghci> apply wrong \"1+2\"\n[(1,\"+2\")]\nThe parser digit succeeds, so addition is not invoked. But what we really\nwanted was to return [(3,\"\")], absorbing as much of the input as possible. One\nway to correct wrong is to rewrite it in the form\nbetter = addition <|> digit\nThen on 1+2 the parser addition succeeds, returning the result we want. What\nis wrong with better is that it is inef\ufb01cient: applied to the input 1 it parses the\ndigit but fails to \ufb01nd a subsequent plus sign, so parser addition fails. As a result\ndigit is invoked and the input is parsed again from scratch. Not really a problem\nwith a single digit, but the repetition of effort could be costly if we were parsing\nfor a numeral that could contain many digits.\nThe best solution is to factor the parser for digits out of the two component parsers:\nbest\n= digit >>= rest\nrest m = do {char '+'; n <- digit; return (m+n)}\n<|> return m\nThe argument to rest is just an accumulating parameter. We saw essentially the\nsame solution in the chapter on pretty-printing. Factoring parsers to bring out common pre\ufb01xes is a Good Idea to improve ef\ufb01ciency.\nGeneralising from the de\ufb01nition of lowers, we can de\ufb01ne a parser combinator that\nrepeats a parser zero or more times:\nmany :: Parser a -> Parser [a]\nmany p = do {x <- p; xs <- many p; return (x:xs)}\n<|> none\n\n\f11.3 Choice and repetition\n\n283\n\nnone = return []\nThe value none is different from fail (why?). We can now de\ufb01ne\nlowers = many lower\nIn many applications, so-called white space (sequences of space, newline and tab\ncharacters) can appear between tokens (identi\ufb01ers, numbers, opening and closing\nparentheses, and so on) just to make the text easier to read. The parser space\nrecognises white space:\nspace :: Parser ()\nspace = many (sat isSpace) >> return ()\nThe function isSpace is de\ufb01ned in the library Data.Char. The function\nsymbol :: String -> Parser ()\nsymbol xs = space >> string xs\nignores white space before recognising a given string. More generally we can de\ufb01ne\ntoken :: Parser a -> Parser a\ntoken p = space >> p\nfor ignoring white space before invoking a parser. Note that\ntoken p <|> token q = token (p <|> q)\nbut the right-hand parser is more ef\ufb01cient as it does not look for white space twice\nif the \ufb01rst parser fails.\nSometimes we want to repeat a parser one or more times rather than zero or more\ntimes. This can be done by a combinator which we will call some (it is also called\nmany1 in some parser libraries):\nsome :: Parser a -> Parser [a]\nsome p = do {x <- p; xs <- many p; return (x:xs)}\nThis de\ufb01nition repeats that of the \ufb01rst parser in the de\ufb01nition of many, a fact we\ncan take into account by rede\ufb01ning many in terms of some:\nmany :: Parser a -> Parser [a]\nmany p = optional (some p)\noptional :: Parser [a] -> Parser [a]\noptional p = p <|> none\n\n\fParsing\n\n284\n\nThe parsers many and some are now mutually recursive.\nHere is a parser for natural numbers, one that allows white space before the number:\nnatural :: Parser Int\nnatural = token nat\nnat = do {ds <- some digit;\nreturn (foldl1 shiftl ds)}\nwhere shiftl m n = 10*m+n\nThe subsidiary parser nat does not allow white space before the number.\nConsider now how to de\ufb01ne a parser for an integer numeral, which by de\ufb01nition is\na nonempty string of digits possibly pre\ufb01xed by a minus sign. You might think that\nthe parser\nint :: Parser Int\nint = do {symbol \"-\"; n <- natural; return (-n)}\n<|> natural\ndoes the job, but it is inef\ufb01cient (see Exercise H) and may or may not be what we\nwant. For example,\nghci> apply int \"\n[(-34,\"\")]\nghci> apply int \"\n[(-34,\"\")]\n\n-34\"\n- 34\"\n\nWhereas we are quite happy with white space before a numeral, we may not want\nany white space to appear between the minus sign and the ensuing digits. If that is\nthe case, then the above parser will not do. It is easy to modify the given de\ufb01nition\nof int to give what we want:\nint :: Parser Int\nint = do {symbol \"-\"; n <- nat; return (-n)}\n<|> natural\nThis parser is still inef\ufb01cient, and a better alternative is to de\ufb01ne\nint :: Parser Int\nint = do {space; f <- minus; n <- nat; return (f n)}\nwhere\nminus = (char '-' >> return negate) <|> return id\nThe parser minus returns a function, either negate if the \ufb01rst symbol is a minus\nsign, or the identity function otherwise.\n\n\f11.4 Grammars and expressions\n\n285\n\nNext, let us parse a list of integers, separated by commas and enclosed in square\nbrackets. White space is allowed before and after commas and brackets though not\nof course between the digits of the integers. Here is a very short de\ufb01nition:\nints :: Parser [Int]\nints = bracket (manywith (symbol \",\") int)\nThe subsidiary parser bracket deals with the brackets:\nbracket :: Parser a -> Parser a\nbracket p = do {symbol \"[\";\nx <- p;\nsymbol \"]\";\nreturn x}\nThe function manywith sep p acts a bit like many p but differs in that the instances of p are separated by instances of sep whose results are ignored. The de\ufb01nition is\nmanywith :: Parser b -> Parser a -> Parser [a]\nmanywith q p = optional (somewith q p)\nsomewith :: Parser b -> Parser a -> Parser [a]\nsomewith q p = do {x <- p;\nxs <- many (q >> p);\nreturn (x:xs)}\nFor example,\nghci> apply ints \"[2, -3, 4]\"\n[([2,-3,4],\"\")]\nghci> apply ints \"[2, -3, +4]\"\n[]\nghci> apply ints \"[]\"\n[([],\"\")]\nIntegers cannot be preceded by a plus sign, so parsing the second expression fails.\n\n11.4 Grammars and expressions\nThe combinators described so far are suf\ufb01ciently powerful for translating a structural description of what is required directly into a functional parser. Such a struc-\n\n\fParsing\n\n286\n\ntural description is provided by a grammar. We will illustrate some typical grammars by looking at parsers for various kinds of arithmetic expression.\nLet us start by building a parser for the type Expr, de\ufb01ned by\ndata Expr = Con Int | Bin Op Expr Expr\ndata Op\n= Plus | Minus\nHere is a grammar for fully parenthesised expressions, expressed in what is known\nas Backus-Naur form, or BNF for short:\nexpr\nop\nnat\ndigit\n\n::=\n::=\n::=\n::=\n\nnat | '(' expr op expr ')'\n'+' | '-'\n{digit}+\n'0' | '1' | ... | '9'\n\nThis grammar de\ufb01nes four syntactic categories. Symbols enclosed in quotes are\ncalled terminal symbols and describe themselves; these are symbols that actually\noccur in the text. There are ten possible characters for a digit, and a nat is de\ufb01ned\nas a sequence of one or more digits. The meta-symbol {-}+ describes a non-zero\nrepetition of a syntactic category. Note that we do not allow an optional minus\nsign before a sequence of digits, so constants are natural numbers, not arbitrary\nintegers. The grammar states that an expression is either a natural number or else\na compound expression consisting of an opening parenthesis, followed by an expression, followed by either a plus or minus sign, followed by another expression,\nand \ufb01nally followed by a closing parenthesis. It is implicitly understood in the description that white space is ignored between terminal symbols except between the\ndigits of a number. The grammar translates directly into a parser for expressions:\nexpr :: Parser Expr\nexpr = token (constant <|> paren binary)\nconstant = do {n <- nat; return (Con n)}\nbinary = do {e1 <- expr;\np <- op;\ne2 <- expr;\nreturn (Bin p e1 e2)}\nop = (symbol \"+\" >> return Plus) <|>\n(symbol \"-\" >> return Minus)\nFor readability we have made use of a subsidiary parser binary; the parser paren\nis left as an exercise.\nNow suppose we want a parser that also works for expressions that are not fully\nparenthesised, things like 6-2-3 and 6-(2-3) and (6-2)-3. In such a case, (+)\n\n\f11.4 Grammars and expressions\n\n287\n\nand (-) should associate to the left in expressions, as is normal with arithmetic.\nOne way to express such a grammar in BNF is to write\nexpr ::= expr op term | term\nterm ::= nat | '(' expr ')'\nThis grammar says that an expression is a sequence of one or more terms separated\nby operators. A term is either a number or a parenthesised expression. In particular,\n6-2-3 will be parsed as the expression 6-2 followed by a minus operator, followed\nby the term 3. In other words, the same as (6-2)-3, as required. This grammar also\ntranslates directly into a parser:\nexpr = token (binary <|> term)\nbinary = do {e1 <- expr;\np <- op;\ne2 <- term;\nreturn (Bin p e1 e2)}\nterm = token (constant <|> paren expr)\nHowever, there is a fatal \ufb02aw with this parser: it falls into an in\ufb01nite loop. After\nignoring initial white space the \ufb01rst action of expr is to invoke the parser binary,\nwhose \ufb01rst action is to invoke the parser expr again. Whoops!\nFurthermore, it will not do to rewrite expr as\nexpr = token (term <|> binary)\nbecause, for example,\nMain*> apply expr \"3+4\"\n[(Con 3,\"+4\")]\nOnly the \ufb01rst term is parsed. The problem is called the left recursion problem and\nis a dif\ufb01culty with all recursive parsers, functional or otherwise.\nOne solution is to rewrite the grammar in the following equivalent form:\nexpr ::= term {op term}*\nThe meta-symbol {-}* indicates a syntactic category that can be repeated zero or\nmore times. The new parser then takes the form\nexpr = token (term >>= rest)\nrest e1 = do {p <- op;\ne2 <- term;\nrest (Bin p e1 e2)} <|> return e1\n\n\fParsing\n\n288\n\nThe parser rest corresponds to the category {op term}* and takes an argument\n(an accumulating parameter) whose value is the expression parsed so far.\nFinally, let us design a parser for arithmetic expressions that may contain multiplication and division, changing the de\ufb01nition of Op to\ndata Op = Plus | Minus | Mul | Div\nThe usual rules apply in that multiplication and division take precedence over addition and subtraction, and operations of the same precedence associate to the left.\nHere is a grammar:\nexpr ::= term {addop term}*\nterm ::= factor {mulop factor}*\nfactor ::= nat | '(' expr ')'\naddop ::= '+' | '-'\nmulop ::= '*' | '/'\nAnd here is the parser:\nexpr = token (term >>= rest)\nrest e1 = do {p <- addop;\ne2 <- term;\nrest (Bin p e1 e2)}\n<|> return e1\nterm = token (factor >>= more)\nmore e1 = do {p <- mulop;\ne2 <- factor;\nmore (Bin p e1 e2)}\n<|> return e1\nfactor = token (constant <|> paren expr)\nThe de\ufb01nitions of addop and mulop are left as exercises.\n\n11.5 Showing expressions\nOur \ufb01nal question is: how can we install Expr as a member of the type class Show\nso that the function show is the inverse of parsing? More precisely, we want to\nde\ufb01ne show so that\nparse expr (show e) = e\nRecall that parse p extracts the \ufb01rst parse returned by apply p.\n\n\f11.5 Showing expressions\n\n289\n\nAs a warm-up, here is the instance of Show when expr is the parser for fully\nparenthesised expressions involving addition and subtraction only:\ninstance Show Expr where\nshow (Con n) = show n\nshow (Bin op e1 e2) =\n= \"(\" ++ show e1 ++\n\" \" ++ showop op ++\n\" \" ++ show e2 ++ \")\"\nshowop Plus = \"+\"\nshowop Minus = \"-\"\nClear enough, but there is a problem with ef\ufb01ciency. Because (++) has time complexity linear in the length of its left argument, the cost of evaluating show is, in\nthe worst case, quadratic in the size of the expression.\nThe solution, yet again, is to use an accumulating parameter. Haskell provides a\ntype synonym ShowS:\ntype ShowS = String -> String\nand also the following subsidiary functions\nshowChar\n:: Char -> ShowS\nshowString :: String -> ShowS\nshowParen :: Bool -> ShowS -> ShowS\nThese functions are de\ufb01ned by\nshowChar\n= (:)\nshowString\n= (++)\nshowParen p x = if b then\nshowChar '(' . p . showChar ')'\nelse p\nNow we can de\ufb01ne show for expressions by\nshow e = shows e \"\"\nwhere\nshows (Con n) = showString (show n)\nshows (Bin op e1 e2)\n= showParen True (shows e1 . showSpace .\nshowsop op . showSpace . shows e2)\nshowsop Plus = showChar '+'\nshowsop Minus = showChar '-'\n\n\fParsing\n\n290\n\nshowSpace\n\n= showChar ' '\n\nThis version, which contains no explicit concatenation operations, takes linear time\nin the size of the expression.\nNow suppose we want to display expressions that are not fully parenthesised. There\nis no need for parentheses around left-hand expressions, but we do need parentheses around right-hand expressions. That leads to\nshow = shows False e \"\"\nwhere\nshows b (Con n) = showString (show n)\nshows b (Bin op e1 e2)\n= showParen p (shows False e1 . showSpace .\nshowsop op . showSpace . shows True e2)\nThis de\ufb01nition takes no account of associativity; for example, 1+(2+3) is not\nshown as 1+2+3.\nFinally, let\u2019s tackle expressions involving all four arithmetic operations. The difference here is that:\n1. With expressions e1 + e2 or e1 - e2 we will never need parentheses around\ne1 (just as above), nor will we need parentheses around e2 if e2 is a compound\nexpression with a multiplication or division at the root.\n2. On the other hand, with expressions e1 * e2 or e1 / e2 we will need parentheses around e1 if e1 is a compound expression with a plus or minus at the\nroot, and we will always need parentheses around e2.\nOne way to codify these rules is to introduce precedence levels (for another way,\nsee Exercise L). De\ufb01ne\nprec\nprec\nprec\nprec\nprec\n\n:: Op\nMul\nDiv\nPlus\nMinus\n\n-> Int\n= 2\n= 2\n= 1\n= 1\n\nConsider now how to de\ufb01ne a function showsPrec with type\nshowsPrec :: Int -> Expr -> ShowS\nsuch that showsPrec p e shows the expression e assuming that the parent of e is\na compound expression with an operator of precedence p. We will de\ufb01ne show by\n\n\f11.6 Exercises\n\n291\n\nshow e = showsPrec 0 e \"\"\nso the enclosing context of e is an operator with \ufb01ctitious precedence 0. We can at\nonce de\ufb01ne\nshowsPrec p (Con n) = showString (show n)\nbecause constants are never enclosed in parentheses. The interesting case is when\nwe have a compound expression. We give the de\ufb01nition \ufb01rst and explain it afterwards:\nshowsPrec p (Bin op e1 e2)\n= showParen (p>q) (showsPrec q e1 . showSpace .\nshowsop op . showSpace . showsPrec (q+1) e2)\nwhere q = prec op\nWe put parentheses around an expression if the parent operator has greater precedence than the current one. To display the expression e1 it is therefore suf\ufb01cient to\npass the current precedence as the new parent precedence. But we need parentheses\naround e2 if the root operator of e2 has precedence less than or equal to q; so we\nhave to increment q in the second call.\nAdmittedly, the above de\ufb01nition of showsPrec requires a little thought, but there\nis a payoff. The type class Show has a second method in it, namely showsPrec.\nMoreover, the default de\ufb01nition of show is just the one above. So to install expressions as a member of Show we merely have to give the de\ufb01nition of showsPrec.\n\n11.6 Exercises\nExercise A\nConsider the synonym\ntype Angle = Float\nSuppose we want to de\ufb01ne equality on angles to be equality modulo a multiple of\n2\u03c0. Why can\u2019t we use (==) for this test? Now consider\nnewtype Angle = Angle Float\nInstall Angle as a member of Eq, thereby allowing (==) as an equality test between\nangles.\n\n\fParsing\n\n292\n\nExercise B\nWe could have de\ufb01ned\nnewtype Parser a = Parser (String -> Maybe (a,String))\nGive the monad instance of this kind of parser.\nExercise C\nProve that fail >> p = fail.\nExercise D\nCould we have de\ufb01ned <|> in the following way?\np <|> q = Parser (\\s -> parse p s ++ parse q s)\nWhen is the result a deterministic parser? De\ufb01ne a function\nlimit :: Parser a -> Parser a\nsuch that limit (p <|> q) is a deterministic parser, even if p and q are not.\nExercise E\nParsers are not only instances of monads, they can also be made instances of a\nmore restricted class, called MonadPlus, a class we could have introduced in the\nprevious chapter. Basically, these are monads that support choice and failure. The\nHaskell de\ufb01nition is\nclass Monad m => MonadPlus m where\nmzero :: m a\nmplus :: m a -> m a -> m a\nAs examples, both [] and Maybe can be made members of MonadPlus:\ninstance MonadPlus [] where\nmzero = []\nmplus = (++)\ninstance MonadPlus Maybe where\nmzero = Nothing\nNothing `mplus` y = y\nJust x `mplus` y = Just x\nInstall Parser as an instance of MonadPlus.\n\n\f11.6 Exercises\n\n293\n\nExercise F\nContinuing from the previous exercise, the new methods mzero and mplus are\nexpected to satisfy some equational laws, as is usually the case with the methods\nof a type class. But currently the precise set of rules that these methods should obey\nis not agreed on by the Haskell designers! Uncontroversial are the laws that mplus\nshould be associative with identity element mzero. That\u2019s three equations. Another\nreasonable law is the left-zero law\nmzero >>= f = mzero\nThe corresponding right-zero law, namely\np >> mzero = mzero\ncan also be imposed. Does the MonadPlus instance of the list monad satisfy these\n\ufb01ve laws? How about the Maybe monad?\nFinally, the really contentious law is the following one:\n(p `mplus` q) >>= f = (p >>= f) `mplus` (q >>= f)\nThis law is call the left-distribution law. Why can\u2019t Maybe be installed as a member\nof MonadPlus if the left-distribution is imposed?\nExercise G\nDesign a parser for recognising Haskell \ufb02oating-point numbers. Bear in mind that\n.314 is not a legitimate number (no digits before the decimal point) and that\n3 . 14 is not legitimate either (because no spaces are allowed before or after the\ndecimal point).\nExercise H\nWhy are the \ufb01rst and second de\ufb01nitions of int given in the text inef\ufb01cient, compared to the third de\ufb01nition?\nExercise I\nIs \"(3)\" a fully parenthesised expression? Is it a non-fully parenthesised expression? Haskell allows parenthesised constants:\nghci> (3)+4\n7\nDesign a parser for fully parenthesised expressions that allows parentheses around\nconstants.\n\n\fParsing\n\n294\n\nExercise J\nConsider the grammar expr ::= term {op term}*. De\ufb01ne pair and shunt so\nthat the following parser is legitimate:\nexpr = do {e1 <- term;\npes <- many (pair op term);\nreturn (foldl shunt e1 pes)}\nExercise K\nDe\ufb01ne the parsers addop and mulop.\nExercise L\nConsider again the showing of expressions with all four arithmetic operations. The\nrules for putting in parentheses come down to: we need parentheses around e1 in\ne1 op e2 if op is a multiplication operator, and the root of e1 isn\u2019t. Dually we\nwill need parentheses around e2 if either op is a multiplication operator or the root\nof e2 isn\u2019t. De\ufb01ning\nisMulOp Mul = True\nisMulOp Div = True\nisMulOp _\n= False\nconstruct an alternative de\ufb01nition of show involving a subsidiary function\nshowsF :: (Op -> Bool) -> Expr -> ShowS\n\n11.7 Answers\nAnswer to Exercise A\nBecause (==) is the equality test on \ufb02oating-point numbers, and different numbers\ncannot be equal.\ninstance Eq Angle where\nAngle x == Angle y = reduce x == reduce y\nwhere\nreduce x | x<0 = reduce (x + r)\n| x>r = reduce (x - r)\n| otherwise = x\nwhere r = 2*pi\n\n\f11.7 Answers\n\n295\n\nAnswer to Exercise B\ninstance Monad Parser where\nreturn x = Parser (\\s -> Just (x,s))\nP >>= q = Parser (\\s -> case apply p s of\nNothing -> apply q s\nJust (x,s') -> Just (x,s'))\nAnswer to Exercise C\nfail >> p\n= fail >>= const p\n= fail\nThe fact that fail >>= p = fail is immediate from the de\ufb01nition of fail and\nthe de\ufb01nition of p >>= q.\nAnswer to Exercise D\nYes, but the result is only a deterministic parser when either p or q is fail. The\nfunction limit can be de\ufb01ned by\nlimit p = Parser (take 1 . apply p)\nAnswer to Exercise E\nmzero = fail\nmplus = (<|>)\nAnswer to Exercise F\nYes, both the list monad and the Maybe monad satisfy the \ufb01ve laws. For example,\nin the list monad\nmzero >>= f = concat (map f []) = [] = mzero\nxs >> mzero = concat (map (const []) xs) = [] = mzero\nWith Maybe the left-distribution law doesn\u2019t hold. We have\n(Just x `mplus` q) >>= (\\x -> Nothing)\n= Just x >>= (\\x -> Nothing)\n= Nothing\nbut\n\n\fParsing\n\n296\n\n(Just x >> \\x -> Nothing) `mplus`\n(q >>= \\x -> Nothing)\n= Nothing `mplus` (q >>= \\x -> Nothing)\n= q >>= \\x -> Nothing\nThe two resulting expressions are not equal (take q = undefined).\nAnswer to Exercise G\nfloat :: Parser Float\nfloat = do {ds <- some digit;\nchar '.';\nfs <- some digit;\nreturn (foldl shiftl 0 ds +\nfoldr shiftr 0 fs)}\nwhere shiftl n d = 10*n + fromIntegral d\nshiftr f x = (fromIntegral f+x)/10\nThe parser digit returns an Int, which has to be converted to a number (in this\ncase a Float).\nAnswer to Exercise H\nWhite space is parsed twice. For example, calling the \ufb01rst version int1 and the\nthird int3 we have\nghci> apply\n[(3,\"\")]\n(1.40 secs,\nghci> apply\n[(3,\"\")]\n(2.68 secs,\n\nint3 $ replicate 100000 ' ' ++ \"3\"\n216871916 bytes)\nint1 $ replicate 100000 ' ' ++ \"3\"\n427751932 bytes)\n\nAnswer to Exercise I\nNo, according to the \ufb01rst grammar for expr, only binary expressions can be parenthesised. Yes, according to the second grammar as arbitrary expressions can be\nparenthesised.\nThe revised grammar is\nexpr ::= term | '(' expr op expr ')'\nterm ::= nat | '(' expr ')'\n\n```\n\n\n```\n\nThe corresponding parser is\n\n\f11.8 Chapter notes\n\n297\n\nexpr = token (term <|> paren binary)\nwhere\nterm = token (constant <|> paren expr)\nbinary = do {e1 <- expr;\np <- op;\ne2 <- expr;\nreturn (Bin p e1 e2)}\nAnswer to Exercise J\npair :: Parser a -> Parser b -> Parser (a,b)\npair p q = do {x <- p; y <- q; return (x,y)}\nshunt e1 (p,e2) = Bin p e1 e2\nAnswer to Exercise K\naddop = (symbol\n(symbol\nmulop = (symbol\n(symbol\n\n\"+\"\n\"-\"\n\"*\"\n\"/\"\n\n>>\n>>\n>>\n>>\n\nreturn\nreturn\nreturn\nreturn\n\nPlus) <|>\nMinus)\nMul) <|>\nDiv)\n\nAnswer to Exercise L\nshow e = showsF (const False) e \"\"\nwhere\nshowsF f (Con n) = showString (show n)\nshowsF f (Bin op e1 e2)\n= showParen (f op) (showsF f1 e1 . showSpace .\nshowsop op . showSpace . showsF f2 e2)\nwhere f1 x = isMulOp op && not (isMulOp x)\nf2 x = isMulOp op || not (isMulOp x)\n\n11.8 Chapter notes\nThe design of functional parsers in a monadic setting has long been a favourite\napplication of functional programming. Our presentation follows that of \u2018Monadic\nparsing in Haskell\u2019 by Graham Hutton and Erik Meijer, which appears in The Journal of Functional Programming 8(4), 437\u2013144, 1998.\n\n\fChapter 12\nA simple equational calculator\n\nThis \ufb01nal chapter is devoted to a single programming project, the design and implementation of a simple calculator for carrying out point-free equational proofs.\nAlthough the calculator provides only a small subset of the facilities one might\nwant in an automatic proof assistant, and is highly restrictive in a number of other\nways, it will nevertheless be powerful enough to prove many of the point-free laws\ndescribed in previous chapters \u2013 well, provided we are prepared to give it a nudge\nin the right direction if necessary. The project is also a case study in the use of\nmodules. Each component of the calculator, its associated types and functions, is\nde\ufb01ned in an appropriate module and linked to other modules through explicit import and export lists.\n\n12.1 Basic considerations\nThe basic idea is to construct a single function calculate with type\ncalculate :: [Law] -> Expr -> Calculation\nThe \ufb01rst argument of calculate is a list of laws that may be applied. Each law\nconsists of a descriptive name and an equation. The second argument is an expression and the result is a calculation. A calculation consists of a starting expression\nand a sequence of steps. Each step consists of the name of a law and the expression\nthat results by applying the left-hand side of the law to the current expression. The\ncalculation ends when no more laws can be applied, and the \ufb01nal expression is the\nconclusion. The entire process is automatic, requiring no intervention on the part\nof the user.\n\n\f12.1 Basic considerations\n\n299\n\nLaws, expressions and calculations are each elements of appropriate data types to\nbe de\ufb01ned in the following sections. But for now let us plunge straight in with an\nexample to show the framework we have in mind.\nHere are some laws (we use a smaller font to avoid breaking lines):\ndefinition filter:\ndefinition box:\n\nfilter p = concat . map (box p)\nbox p = if p one nil\n\nif after dot:\ndot after if:\n\nif p f g . h = if (p . h) (f . h) (g . h)\nh . if p f g = if p (h . f) (h . g)\n\nnil constant:\nmap after nil:\nmap after one:\n\nnil . f = nil\nmap f . nil = nil\nmap f . one = one . f\n\nmap after concat:\n\nmap f . concat = concat . map (map f)\n\nmap functor:\nmap functor:\n\nmap f . map g = map (f . g)\nmap id = id\n\nEach law consists of a name and an equation. The name of the law is terminated by\na colon sign, and an equation consists of two expressions separated by an equals\nsign. Each expression describes a function; our calculator will be one that simpli\ufb01es\nfunctional expressions only (yes, it\u2019s a pointless calculator). Expressions are built\nfrom constants, like one and map, and variables, like f and g. The precise syntax\nwill be given in due course. Note that there are no conditional laws, equations that\nare valid only if some subsidiary conditions are met. That will limit what we can\ndo with the calculator, but it still leaves enough to be interesting.\nSuppose we want to simplify the expression filter p . map f. Here is one possible calculation:\n=\n=\n=\n=\n=\n\nfilter p . map f\n{definition filter}\nconcat . map (box p) . map f\n{map functor}\nconcat . map (box p . f)\n{definition box}\nconcat . map (if p one nil . f)\n{if after dot}\nconcat . map (if (p . f) (one . f) (nil . f))\n{nil constant}\nconcat . map (if (p . f) (one . f) nil)\n\nThe steps of the calculation are displayed in the conventional format with the name\nof the law being invoked printed in braces between the two expressions to which\n\n\f300\n\nA simple equational calculator\n\nit applies. No more laws apply to the \ufb01nal expression, so that is the result of the\ncalculation. It is certainly not simpler than the expression we started out with.\nThe calculator could have applied some of the laws in a different order; for example, the de\ufb01nition of box could have been applied at the second step rather than at\nthe third. But the conclusion would have been the same. It is also possible, though\nnot with this particular set of laws, that an expression could be simpli\ufb01ed to different conclusions by different calculations. However, at the outset we make the\ndecision that calculate returns just one calculation, not a tree of possible calculations.\nNotice what is happening at each step. Some left-hand side of some law is matched\nagainst some subexpression of the current expression. If a match is successful the\nresult is a substitution for the variables occurring in the law. For example, in the\nsecond step, the subexpression map (box p) . map f is successfully matched\nwith the \ufb01rst map functor law, resulting in a substitution in which the variable f of\nthe functor law is bound to the expression box p, and the variable g is bound to f.\nThe result of the step involves rewriting the subexpression with the corresponding\ninstance of the right-hand side of the law in which each variable is replaced by\nits binding expression. Matching, substitutions and rewriting are all fundamental\ncomponents of the calculator.\nNow suppose that with the same set of laws as above we want to simplify the\nexpression map f . filter (p . f). Here is the calculation:\n=\n=\n=\n=\n=\n=\n=\n\nmap f . filter (p . f)\n{definition filter}\nmap f . concat . map (box (p . f))\n{map after concat}\nconcat . map (map f) . map (box (p . f))\n{map functor}\nconcat . map (map f . box (p . f))\n{definition box}\nconcat . map (map f . if (p . f) one nil)\n{dot after if}\nconcat . map (if (p . f) (map f . one) (map f . nil))\n{map after nil}\nconcat . map (if (p . f) (map f . one) nil)\n{map after one}\nconcat . map (if (p . f) (one . f) nil)\n\nAgain, some of the laws could have been applied in a different order. No more laws\napply to the \ufb01nal expression so that is the result of the calculation.\nThe point about these two calculations is that the two \ufb01nal expressions are the\nsame, so we have proved\n\n\f12.1 Basic considerations\n\n301\n\nfilter p . map f = map f . filter (p . f)\nThis is the way we will conduct equational proofs, simplifying both sides to the\nsame conclusion. Rather than show two calculations, one after the other, the two\nresults can be pasted together by recording the \ufb01rst calculation and then appending\nthe steps of the second calculation in reverse. The main advantage of this scheme\nis simplicity; we do not have to invent a new format for proofs, and we do not have\nto apply laws from right to left in order to reach the desired goal. Accordingly, we\nwill also de\ufb01ne a function\nprove :: [Law] -> Equation -> Calculation\nfor proving equations.\n\nFurther considerations\nIt is a basic constraint of our calculator that laws are applied in one direction only,\nnamely from left to right. This is primarily to prevent calculations from looping.\nIf laws could be applied in both directions, then the calculator could oscillate by\napplying a law in one direction and then immediately applying it in the reverse\ndirection.\nEven with a left-to-right rule, some laws can lead to in\ufb01nite calculations. Typically,\nthese laws are the de\ufb01nitions of recursive functions. For example, consider the\nde\ufb01nition of iterate:\ndefn iterate: iterate f = cons . fork id (iterate f . f)\n\nThis is the de\ufb01nition of iterate expressed in point-free form. The functions cons\nand fork are de\ufb01ned by\ncons (x,xs) = x:xs\nfork f g x = (f x,g x)\nWe have met fork before in the exercises in Chapters 4 and 6, except that we\nwrote fork (f,g) instead of fork f g. In what follows, all our functions will\nbe curried. The appearance of the term iterate f on both sides of the law means\nthat any calculation that can apply the de\ufb01nition of iterate once can, potentially,\napply it in\ufb01nitely often. But not necessarily. Here is a calculation (produced by the\ncalculator) that avoids in\ufb01nite regress:\n=\n\nhead . iterate f\n{defn iterate}\nhead . cons . fork id (iterate f . f)\n\n\fA simple equational calculator\n\n302\n=\n\n{head after cons}\nfst . fork id (iterate f . f)\n=\n{fst after fork}\nid\n\nThe calculation makes use of the two laws:\nhead after cons:\nfst after fork:\n\nhead . cons = fst\nfirst . fork f g = f\n\nThe reason non-termination is avoided is that these two laws are given preference\nover de\ufb01nitions in calculations, a wrinkle that we will elaborate on below.\nIn order to appreciate just what the calculator can and cannot do, here is another\nexample of rendering a recursive de\ufb01nition into point-free form. Consider the definition of concatenation:\n[] ++ ys\n= ys\n(x:xs) ++ ys = x:(xs ++ ys)\nWe will use cat to stand for (++). We will also need nil, cons and the function\ncross (f,g), which we will now write as f * g. Thus,\n(f * g) (x,y) = (f x, g y)\nFinally we will need a combinator assocr (short for \u2018associate-right\u2019), de\ufb01ned by\nassocr ((x,y),z) = (x,(y,z))\nHere are the translations of the two de\ufb01ning equations of cat in point-free form:\ncat . (nil * id) = snd\ncat . (cons * id) = cons . (id * cat) . assocr\nWe cannot prove that cat is associative with our calculator, for that would involve\na proof by induction, but we can state it as a law:\ncat associative: cat . (cat * id) = cat . (id * cat) . assocr\n\nContinuing with this example for a bit longer, here are the two bifunctor laws of\n(*):\nbifunctor *:\nbifunctor *:\n\nid * id = id\n(f * g) . (h * k) = (f . h) * (g . k)\n\nAnd here is a law about assocr:\nassocr law:\n\nassocr . ((f * g) * h) = (f * (g * h)) . assocr\n\n\f12.1 Basic considerations\n\n303\n\nNow for the point of the example: our calculator cannot perform the following\nvalid calculation:\n=\n=\n=\n=\n=\n=\n\ncat . ((cat . (f * g)) * h)\n{identity law, in backwards direction}\ncat . ((cat . (f * g)) * (id . h))\n{bifunctor *, in backwards direction}\ncat . (cat * id) . ((f * g) * h)\n{cat associative}\ncat . (id * cat) . assocr . ((f * g) * h)\n{assoc law}\ncat . (id * cat) . (f * (g * h)) . assocr\n{bifunctor *}\ncat . ((id . f) * (cat . (g * h))) . assocr\n{identity law}\ncat . (f * (cat . (g * h))) . assocr\n\n```\n\n\n```\n\nThe problem here is that we have to apply the identity and bifunctor laws in both\ndirections, and the calculator is simply not up to the task. Observe that the essence\nof the proof is the simpli\ufb01cation of the expression\ncat . (id * cat) . assocr . ((f * g) * h)\nin two different ways, one by using the associativity of cat, written in the form\ncat associative: cat . (id * cat) . assocr = cat . (cat * id)\n\nand one by using the assocr law. Even if we generalised calculate to return a\ntree of possible calculations, it would not be obvious what expression we would\nhave to start out with in order to achieve the calculation above, so we abandon any\nattempt to get the calculator to produce it.\nIt is not just the functor laws that sometimes have to be applied in both directions.\nFor an example, see Section 12.8. Sometimes we can get around the problem by\nstating a law in a more general form than necessary, sometimes by using a hack,\nand sometimes not at all. As we said at the outset, our calculator is a limited one.\nIn the scheme of automatic calculation that we are envisaging there are only two\ndegrees of freedom: the choice of which law to apply, and the choice of which\nsubexpression to be changed. The \ufb01rst degree of freedom can be embodied in the\norder in which laws are presented to the calculator: if two different laws are applicable, then the one earlier in the list is chosen.\nCertainly some laws should be tried before others; these are laws that reduce the\ncomplexity of intermediate expressions. Good examples are the laws f.id = f\nand id.f = f. The naive de\ufb01nition of complexity is that there are fewer compositions on the right than on the left. It is unlikely to be a mistake to apply these\n\n\f304\n\nA simple equational calculator\n\nlaws as soon as the opportunity arises. Indeed the fact that id is the identity element of composition can and will be built into the calculator, so the two identity\nlaws will be taken care of automatically. Similarly, early application of laws like\nnil.f = nil and map f.nil = nil (and indeed the two laws used in the calculation about iterate), all of which reduce the number of compositions, help\nto reduce the sizes of intermediate expressions. For the sake of a word, let us call\nthese the simple laws.\nOn the other hand, some laws should be applied only as a last resort. Typically,\nthese laws are de\ufb01nitions, such as the de\ufb01nition of filter or iterate. For example, in the expression\nmap f . concat . map (filter p)\nwe really don\u2019t want to apply the de\ufb01nition of filter too early; rather we would\nprefer to apply the map after concat law \ufb01rst, and only apply the de\ufb01nition of\nfilter later on if and when it becomes necessary. Apart from anything else, intermediate expressions will be shorter.\nIn summary it looks sensible to sort our laws into the simple laws, followed the\nnon-simple laws that are not de\ufb01nitions, followed by the de\ufb01nitions.\nThe second degree of freedom is represented by the order in which the subexpressions of a given expression are presented as candidates for instances of laws: if\nlaws are applicable to two different subexpressions, then the subexpression coming\nearlier in the enumeration is chosen.\nThat still leaves open the decision whether to give preference to laws or to subexpressions in calculations. Do we start with a subexpression and try every law in\nturn, or start with a law and see if it applies anywhere? Does it really matter which\nof these alternatives is chosen? While it is true that, having applied some law at\nsome subexpression, the next law to be applied is likely to be at a \u2018nearby\u2019 expression, it is not clear how to formalise this notion of nearness, nor is it clear\nwhether it would contribute signi\ufb01cantly to the ef\ufb01ciency of calculations, either in\nthe computation time or in the length of the result.\n\n12.2 Expressions\nAt the heart of the calculator is the data type Expr of expressions. Most of the\ncomponents of the calculator are concerned with analysing and manipulating expressions in one way or the other. Expressions are built from (function) variables\n\n\f12.2 Expressions\n\n305\n\nand constants, using functional composition as the basic combining form. Variables take no arguments, but constants can take any number of arguments, which\nare themselves expressions. We will suppose all functions are curried and there\nare no tuples; for example we write pair f g instead of pair (f,g). There is\nno particular reason for avoiding tuples, it is just that most functions we have discussed in the book are curried and we don\u2019t really need both.\nTo compensate, we will also allow ourselves binary in\ufb01x operators, writing, for\nexample, f * g instead of cross f g. Except for functional composition we will\nnot assume any order of precedence or association between binary operators, insisting that expressions involving such operators be fully parenthesised. That still\nleaves open the question of the precedence of composition. Does f * g . h mean\n(f * g) . h or f * (g . h)? Haskell puts composition at a high level of precedence and we will adopt the same convention. Thus f * g . h will be parsed as\nf * (g . h). But we will always write such expressions using parentheses to\navoid ambiguity.\nHere is the proposed BNF grammar for expressions:\nexpr\nsimple\nterm\narg\nvar\ncon\nop\n\n::=\n::=\n::=\n::=\n::=\n::=\n::=\n\nsimple {op simple}\nterm {'.' term}*\nvar | con {arg}* | '(' expr ')'\nvar | con | '(' expr ')'\nletter {digit}\nletter letter {letter | digit}*\n{symbol}+\n\nVariable names consist of single letters only, possibly followed by a single digit.\nThus f and f1 are legitimate variable names. Constant names are sequences of\nat least two alphanumeric characters beginning with two letters, such as map or\nlhs2tex, while operator names are nonempty sequences of non-alphanumeric\nsymbols, such as * and <+>. The \ufb01rst line says that an expression is a simple\nexpression, possibly followed by an operator and another simple expression. Simple expressions are compositions of terms. The remaining lines are, we trust, selfexplanatory.\nHere is the de\ufb01nition of Expr we will use:\nnewtype Expr = Compose [Atom] deriving Eq\ndata Atom\n= Var VarName | Con ConName [Expr]\nderiving Eq\ntype VarName = String\ntype ConName = String\n\n\fA simple equational calculator\n\n306\n\n```\n\n\n```\n\nExpressions and atoms are declared to be members of the class Eq because we\nwill need to test expressions for equality. Later on we will install expressions as an\ninstance of Show for printing them at the terminal.\nHere are some examples of expressions and their representations:\nf . g . h\n=>\nid\n=>\nfst\n=>\nfst . f\n=>\n(f * g) . h =>\nf * g . h\n=>\n\nCompose\nCompose\nCompose\nCompose\nCompose\nCompose\n\n[Var\n[]\n[Con\n[Con\n[Con\n[Con\n\n\"f\",Var \"g\",Var \"h\"]\n\"fst\" []]\n\"fst\" [],Var \"f\"]\n\"*\" [Var \"f\",Var \"g\"],Var \"h\"]\n\"*\" [Compose [Var \"f\"],\nCompose [Var \"g\",Var \"h\"]]]\n\nThe fact that composition is an associative operation is built into the design of\nExpr. The particular constant id is reserved and will always be interpreted as the\nidentity element of composition.\nThe parsing combinators described in the previous chapter enable us to parse expressions. Following the BNF, we start with\nexpr :: Parser Expr\nexpr = simple >>= rest\nwhere\nrest s1 = do {op <- operator;\ns2 <- simple;\nreturn (Compose [Con op [s1,s2]])}\n<|> return s1\nAn operator is a sequence of one or more operator symbols, as long as it is neither\nthe composition operator nor an equals sign:\noperator :: Parser String\noperator = do {op <- token (some (sat symbolic));\nParsing.guard (op /= \".\" && op /= \"=\");\nreturn op}\nsymbolic = (`elem` opsymbols)\nopsymbols = \"!@#$%&*+./<=>?\\\\^|:-~\"\nThe function Parsing.guard is an example of a quali\ufb01ed name. The Haskell\nPrelude also provides a function guard, but we want the function of the same\nname from a module Parsing that includes all our parsing functions. A quali\ufb01ed\nname consists of a module name followed by a period followed by the name of the\nquali\ufb01ed value.\n\n\f12.2 Expressions\n\n307\n\nA simple expression is a sequence of one or more terms separated by composition:\nsimple :: Parser Expr\nsimple = do {es <- somewith (symbol \".\") term;\nreturn (Compose (concatMap deCompose es))}\nThe function concatMap f as an alternative to concat . map f is provided in\nthe standard prelude, and deCompose is de\ufb01ned by\ndeCompose :: Expr -> [Atom]\ndeCompose (Compose as) = as\nNext, a term is an identi\ufb01er, either a variable or a constant, possibly with arguments,\nor a parenthesised expression:\nterm :: Parser Expr\nterm = ident args <|> paren expr\nargs = many (ident none <|> paren expr)\nThe parser ident takes a parser for a list of expressions and returns a parser for\nexpressions:\nident :: Parser [Expr] -> Parser Exp\nident args\n= do {x <- token (some (sat isAlphaNum));\nParsing.guard (isAlpha (head x));\nif isVar x\nthen return (Compose [Var x])\nelse if (x == \"id\")\nthen return (Compose [])\nelse\ndo {as <- args;\nreturn (Compose [Con x as])}}\nThe test for being a variable is implemented by\nisVar [x]\n= True\nisVar [x,d] = isDigit d\nisVar _\n= False\nNote that any identi\ufb01er consisting entirely of alphanumeric characters and beginning with a letter and which is not a variable is a constant.\nNext, we make Expr and Atom instances of Show. As in the previous chapter we\n\n\fA simple equational calculator\n\n308\n\nwill do this by de\ufb01ning showsPrec p for each type. A little thought reveals that\nwe need three values for p:\n\u2022 At top level, there is no need for parentheses. For example, we write all of\nmap f . map g, foo * baz, and bar bie doll without parentheses. We assign p=0 to this case.\n\u2022 When an expression is a composition of terms, or an operator expression, occurring as an argument to a constant, we need to parenthesise it. For example,\nparentheses are necessary in the expression\nmap (f . g) . foo f g . (bar * bar)\nBut we don\u2019t have to parenthesise the middle term. We assign p=1 to this case.\n\u2022 Finally, p=2 means we should parenthesise compositions of terms, operator expressions and curried functions of at least one argument, as in\nmap (f . g) . foo (foldr f e) g . (bar * bar)\nHere goes. We start with\ninstance Show Expr where\nshowsPrec p (Compose []) = showString \"id\"\nshowsPrec p (Compose [a]) = showsPrec p a\nshowsPrec p (Compose as)\n= showParen (p>0) (showSep \" . \" (showsPrec 1) as)\nThe last line makes use of the function showSep, de\ufb01ned by\nshowSep :: String -> (a -> ShowS) -> [a] -> ShowS\nshowSep sep f\n= compose . intersperse (showString sep) . map f\nThe utility function compose is de\ufb01ned by compose = foldr (.) id. The function intersperse :: a -> [a] -> [a] can be found in Data.List and intersperses its \ufb01rst argument between elements of its second. For example,\nintersperse ',' \"abcde\" == \"a,b,c,d,e\"\nThe two occurrences of showsPrec on the right-hand sides of the second two\nclauses of showsPrec refer to the corresponding function for atoms:\ninstance Show Atom where\nshowsPrec p (Var v)\n= showString v\nshowsPrec p (Con f []) = showString f\nshowsPrec p (Con f [e1,e2])\n\n\f12.2 Expressions\n\n309\n\n| isOp f = showParen (p>0) (showsPrec 1 e1 . showSpace .\nshowString f . showSpace . showsPrec 1 e2)\nshowsPrec p (Con f es)\n= showParen (p>1) (showString f . showSpace .\nshowSep \" \" (showsPrec 2) es)\nisOp f = all symbolic f\nThe value p=2 is needed in the \ufb01nal clause because we want parentheses in, for example, foo (bar bie) doll. Variables and nullary constants never need parentheses.\n\nA module structure\nThe \ufb01nal step is to install these de\ufb01nitions, and possibly others, in a module for\nexpressions. Such a module will include all the functions speci\ufb01cally related to\nexpressions.\nCreating such a module is not immediate because we do not yet know what other\nfunctions on expressions we may need in other modules, modules that deal with\nlaws, calculations and so on. But for the moment we declare\nmodule Expressions\n(Expr (Compose), Atom (Var,Con),\nVarName, ConName, deCompose, expr)\nwhere\nimport Parsing\nimport Data.List (intersperse)\nimport Utilities (compose)\nimport Data.Char (isAlphaNum,isAlpha,isDigit)\nThe module Expressions has to be stored in a \ufb01le Expressions.lhs to enable\nHaskell to \ufb01nd out where it resides. It exports the types Expr and Atom along with\ntheir constructors. It also exports the type synonyms VarName and ConName, as\nwell as the functions deCompose and expr, all of which are likely to be needed in\nthe module that deals with laws. Later on we might add more functions on expressions to this export list.\nNext comes the imports. We import the module Parsing that contains the parsing\nfunctions, and also some functions from Data.List and Data.Char. We will also\nset up a module Utilities containing general utility functions. A good example\n\n\f310\n\nA simple equational calculator\n\nof a utility function is compose, de\ufb01ned above. It is not speci\ufb01c to expressions and\nmay be needed in other places, so we put it into the utilities module.\n\n12.3 Laws\nWe de\ufb01ne laws in the following way:\ndata Law\n= Law LawName Equation\ntype LawName = String\ntype Equation = (Expr,Expr)\nA law consists of a descriptive name and an equation. To parse a law we de\ufb01ne:\nlaw :: Parser Law\nlaw = do {name <- upto ':';\neqn <- equation;\nreturn (Law name eqn)}\nThe parsing function upto c returns the string up to but not including the character\nc, and then discards c if found. It wasn\u2019t included among the parsing functions of\nthe previous chapter, but we will put it into the module Parsing to avoid breaking\nthe parser abstraction. One de\ufb01nition is:\nupto :: Char -> Parser String\nupto c\n= Parser (\\s ->\nlet (xs,ys) = break (==c) s in\nif null ys then []\nelse [(xs,tail ys)])\nThe parser equation is de\ufb01ned by\nequation :: Parser Equation\nequation = do {e1 <- expr;\nsymbol \"=\";\ne2 <- expr;\nreturn (e1,e2)}\nWe probably don\u2019t need to show laws, but here is the de\ufb01nition anyway:\ninstance Show Law where\nshowsPrec _ (Law name (e1,e2))\n= showString name .\n\n\f12.3 Laws\n\n311\n\nshowString \": \" .\nshows e1 .\nshowString \" = \" .\nshows e2\nThe precedence number is not needed to de\ufb01ne showPrec so it is made a don\u2019t care\npattern. Recall that shows takes a printable value, here an expression, and returns\na function of type ShowS, a synonym for String -> String.\nFinally we sort the laws:\nsortLaws :: [Law] -> [Law]\nsortLaws laws = simple ++ others ++ defns\nwhere\n(simple,nonsimple) = partition isSimple laws\n(defns,others)\n= partition isDefn nonsimple\nThis de\ufb01nition makes use of a Data.List function partition that partitions a\nlist:\npartition p xs = (filter p xs, filter (not . p) xs)\nThe various tests are de\ufb01ned by\nisSimple (Law _ (Compose as1,Compose as2))\n= length as1 > length as2\nisDefn (Law _ (Compose [Con f es], _))\n= all isVar es\nisDefn _ = False\nisVar (Compose [Var _]) = True\nisVar _\n= False\nThe test isVar also appears in the module Expressions though with a different\nde\ufb01nition. There is no problem though since that function is not exported from the\nexpressions module.\nHere is the module declaration for laws:\nmodule Laws\n(Law (Law), LawName, law, sortLaws,\nEquation, equation)\nwhere\nimport Expressions\nimport Parsing\nimport Data.List (partition)\n\n\f312\n\nA simple equational calculator\n\nHaving shown how to parse and print expressions and laws, we can now de\ufb01ne two\nfunctions, one a version of calculate that consumes strings rather than laws and\nexpressions:\nsimplify :: [String] -> String -> Calculation\nsimplify strings string\n= let laws = map (parse law) strings\ne = parse expr string\nin calculate laws e\nIn a similar vein we can de\ufb01ne\nprove :: [String] -> String -> Calculation\nprove strings string\n= let laws = map (parse law) strings\n(e1,e2) = parse equation string\nin paste (calculate laws e1) (calculate laws e2)\nThese two functions can be put in a module Main. We put paste and calculate\ninto a module concerned solely with calculations, and we turn to this module next.\n\n12.4 Calculations\nCalculations are de\ufb01ned by\ndata Calculation = Calc Expr [Step]\ntype Step\n= (LawName,Expr)\nLet\u2019s begin with the key de\ufb01nition of the calculator, that of calculate:\ncalculate :: [Law] -> Expr -> Calculation\ncalculate laws e = Calc e (manyStep rws e)\nwhere rws e = [(name,e')\n| Law name eqn <- sortedlaws,\ne' <- rewrites eqn e,\ne' /= e]\nsortedlaws = sortLaws laws\nThe function rewrites :: Equation -> Expr -> [Expr] returns a list of all\nthe possible ways of rewriting an expression using a given equation, a function that\nwill be de\ufb01ned in a separate module. It may be the case that an expression can\nbe rewritten to itself (see Exercise H), but such rewrites are disallowed because\nthey would lead to in\ufb01nite calculations. The function rws :: Expr -> [Step]\n\n\f12.4 Calculations\n\n313\n\nreturns a list of all the single steps, leading to new expressions, that can arise by\nusing the laws in all possible ways. This list is de\ufb01ned by taking each law in turn\nand generating all the rewrites associated with the law. That means we give preference to laws over subexpressions in calculations, resolving one of the issues we\nworried about in the \ufb01rst section. Only experimentation will show if we have made\nthe right decision.\nThe function manyStep uses rws to construct as many steps as possible:\nmanyStep :: (Expr -> [Step]) -> Expr -> [Step]\nmanyStep rws e\n= if null steps then []\nelse step : manyStep rws (snd step)\nwhere steps = rws e\nstep = head steps\nThe calculation ends if rws e is the empty list; otherwise the head of the list is\nused to continue the calculation.\nThe remaining functions of the calculations module deal with showing and pasting\ncalculations. We show a calculation as follows:\ninstance Show Calculation where\nshowsPrec _ (Calc e steps)\n= showString \"\\n \" .\nshows e .\nshowChar '\\n' .\ncompose (map showStep steps)\nEach individual step is shown as follows:\nshowStep :: Step -> ShowS\nshowStep (why,e)\n= showString \"=\n{\" .\nshowString why .\nshowString \"}\\n \" .\nshows e .\nshowChar '\\n'\nIn order to paste two calculations together we have to reverse the steps of a calculation. For example, the calculation\nCalc e0 [(why1,e1),(why2,e2),(why3,e3)]\nhas to be turned into\n\n\fA simple equational calculator\n\n314\n\nCalc e3 [(why3,e2),(why2,e1),(why1,e0)]\nIn particular, the conclusion of a calculation is the \ufb01rst expression in the reversed\ncalculation. Here is how to reverse a calculation:\nreverseCalc :: Calculation -> Calculation\nreverseCalc (Calc e steps)\n= foldl shunt (Calc e []) steps\nwhere shunt (Calc e1 steps) (why,e2)\n= Calc e2 ((why,e1):steps)\nIn order to paste two calculations together we \ufb01rst have to check that their conclusions are the same. If they are not, then we go ahead and paste the calculations\nanyway with an indication of failure:\nconc1\n= {... ??? ...}\nconc2\nIf the two conclusions are the same, we can be a little smarter than just stitching the calculations together. If the penultimate conclusion of one calculation also\nmatches the penultimate conclusion of the other, then we can cut out the \ufb01nal steps\naltogether. And so on. Here, then, is how we paste two calculations:\npaste :: Calculation -> Calculation -> Calculation\npaste calc1@(Calc e1 steps1) calc2\n= if conc1 == conc2\nthen Calc e1 (prune conc1 rsteps1 rsteps2)\nelse Calc e1 (steps1 ++ (gap,conc2):rsteps2)\nwhere Calc conc1 rsteps1 = reverseCalc calc1\nCalc conc2 rsteps2 = reverseCalc calc2\ngap = \"... ??? ...\"\nThe function prune is de\ufb01ned by:\nprune :: Expr -> [Step] -> [Step] -> [Step]\nprune e ((_,e1):steps1) ((_,e2):steps2)\n| e1==e2 = prune e1 steps1 steps2\nprune e steps1 steps2 = rsteps ++ steps2\nwhere Calc _ rsteps = reverseCalc (Calc e steps1)\nFinally, here is the module declaration of Calculations:\nmodule Calculations\n(Calculation (Calc), Step, calculate, paste)\n\n\f12.5 Rewrites\n\nwhere\nimport\nimport\nimport\nimport\n\n315\n\nExpressions\nLaws\nRewrites\nUtilities (compose)\n\nThe exports are those types and functions needed to de\ufb01ne simplify and prove\nin the main module.\n\n12.5 Rewrites\nThe sole purpose of the module Rewrites is to provide a de\ufb01nition of the function\nrewrites that appears in the de\ufb01nition of calculate. Recall that the expression\nrewrites eqn e returns a list of all expressions that can arise by matching some\nsubexpression of e against the left-hand expression of eqn and replacing the subexpression with the appropriate instance of the right-hand expression of eqn.\nThe fun is in \ufb01guring out how to de\ufb01ne rewrites. Suppose we construct a list\nof all possible subexpressions of an expression. We can match the given equation\nagainst each subexpression, get the substitutions that do the matching (of which\nthere may be none, one or more than one; see the section on matching below) and\ncompute the new subexpressions. But how do we replace an old subexpression with\na new one in the original expression? The simple answer is that we can\u2019t, at least\nnot without determining alongside each subexpression its context or location in the\noriginal expression. The new subexpression can then be inserted at this location.\nRather than introducing contexts explicitly, we take another approach. The idea\nis to burrow into an expression, applying a rewrite to some subexpression at some\npoint, and then to build the rewritten expression as we climb back out of the burrow.\nWe will need a utility function anyOne that takes a function yielding a choice of\nalternatives, and a list, and installs a single choice for one of the elements. The\nde\ufb01nition is\nanyOne :: (a -> [a]) -> [a] -> [[a]]\nanyOne f []\n= []\nanyOne f (x:xs) = [x':xs | x' <- f x] ++\n[x:xs' | xs' <- anyOne f xs]\nFor example, if f 1 = [-1,-2] and f 2 = [-3,-4], then\nanyOne f [1,2] = [[-1,2],[-2,2],[1,-3],[1,-4]]\n\n\fA simple equational calculator\n\n316\n\nEither one of the choices for the \ufb01rst element is installed, or one of the choices for\nthe second, but not both at the same time.\nHere is our de\ufb01nition of rewrites:\nrewrites :: Equation -> Expr -> [Expr]\nrewrites eqn (Compose as) = map Compose (\nrewritesSeg eqn as ++ anyOne (rewritesA eqn) as)\nrewritesA eqn (Var v) = []\nrewritesA eqn (Con k es)\n= map (Con k) (anyOne (rewrites eqn) es)\nIn the \ufb01rst line we concatenate the rewrites for a segment of the current expression\nwith the rewrites for any one of its proper subexpressions. Only constants with\narguments have subexpressions. Note that the two uses of anyOne have different\ntypes, one taking a list of atoms, and one taking a list of expressions.\nIt remains to de\ufb01ne rewritesSeg:\nrewritesSeg :: Equation -> [Atom] -> [[Atom]]\nrewritesSeg (e1,e2) as\n= [as1 ++ deCompose (apply sub e2) ++ as3\n| (as1,as2,as3) <- segments as,\nsub <- match (e1,Compose as2)]\nThe function segments splits a list into segments:\nsegments as = [(as1,as2,as3)\n| (as1,bs) <- splits as,\n(as2,as3) <- splits bs]\nThe utility function splits splits a list in all possible ways:\nsplits :: [a] -> [([a],[a])]\nsplits []\n= [([],[])]\nsplits (a:as) = [([],a:as)] ++\n[(a:as1,as2) | (as1,as2) <- splits as]\nFor example,\nghci> splits \"abc\"\n[(\"\",\"abc\"),(\"a\",\"bc\"),(\"ab\",\"c\"),(\"abc\",\"\")]\nThe remaining functions apply and match have types\napply :: Subst -> Expr -> Expr\n\n\f12.6 Matchings\n\n317\n\nmatch :: (Expr,Expr) -> [Subst]\nEach will be de\ufb01ned in their own modules, Substitutions and Matchings. Finally, here is the module declaration for Rewrites:\nmodule\nwhere\nimport\nimport\nimport\nimport\nimport\n\nRewrites (rewrites)\nExpressions\nLaws (Equation)\nMatchings (match)\nSubstitutions (apply)\nUtilities (anyOne, segments)\n\n12.6 Matchings\nThe sole purpose of the module Matchings is to de\ufb01ne the function match. This\nfunction takes two expressions and returns a list of substitutions under which the\n\ufb01rst expression can be transformed into the second. Matching two expressions produces no substitutions if they don\u2019t match, but possibly many if they do. Consider\nmatching the expression foo (f . g) against foo (a . b . c). There are four\nsubstitutions that do the trick: f may be bound to any of the expressions\nid,\n\na,\n\na . b,\n\na . b . c\n\nwith four corresponding bindings for g. Although the calculator will select a single substitution at each step, it is important to take account of multiple substitutions in the process of obtaining the valid matchings. For example, in matching\nfoo (f . g) . bar g against foo (a . b . c) . bar c, the subexpression\nf . g is matched against a . b . c, resulting in four possible substitutions.\nOnly when bar g is matched against bar c are three of the substitutions rejected.\nA premature commitment to a single substitution for the \ufb01rst match may result in\na successful match being missed.\nThe most straightforward way of de\ufb01ning match (e1,e2) is to \ufb01rst line up the\natoms of e1 with a partition of the atoms of e2; the \ufb01rst atom is associated with the\n\ufb01rst segment of the partition, the second with the second segment, and so on. The\nfunction alignments has type\nalignments :: (Expr,Expr) -> [[(Atom,Expr)]]\nand does the alignments. To de\ufb01ne it we need a function parts that partitions a list\ninto a given number of segments:\n\n\fA simple equational calculator\n\n318\n\nparts\nparts\nparts\nparts\n\n:: Int\n0 [] =\n0 as =\nn as =\n\n-> [a] -> [[[a]]]\n[[]]\n[]\n[bs:bss\n| (bs,cs) <- splits as,\nbss <- parts (n-1) cs]\n\nThe interesting clauses are the \ufb01rst two: there is one partition of the empty list into\n0 segments, namely the empty partition, but there are no partitions of a nonempty\nlist into 0 segments. For example,\nghci> parts 3 \"ab\"\n[[\"\",\"\",\"ab\"],[\"\",\"a\",\"b\"],[\"\",\"ab\",\"\"],\n[\"a\",\"\",\"b\"],[\"a\",\"b\",\"\"],[\"ab\",\"\",\"\"]]\nNow we can de\ufb01ne\nalignments (Compose as,Compose bs)\n= [zip as (map Compose bss) | bss <- parts n bs]\nwhere n = length as\nHaving aligned each atom with a subexpression, we de\ufb01ne matchA that matches\natoms with expressions:\nmatchA :: (Atom,Expr) -> [Subst]\nmatchA (Var v,e) = [unitSub v e]\nmatchA (Con k1 es1,Compose [Con k2 es2])\n| k1==k2 = combine (map match (zip es1 es2))\nmatchA _ = []\nMatching a variable always succeeds and results in a single substitution. Matching\ntwo constants succeeds only if the two constants are the same. In all other cases\nmatchA returns an empty list of substitutions. The function matchA depends on\nmatch, which we can now de\ufb01ne by\nmatch :: (Expr,Expr) -> [Subst]\nmatch = concatMap (combine . map matchA) . alignments\nThe \ufb01nal ingredient is the function combine :: [[Subst]] -> [Subst]. Each\ncomponent list of substitutions in the argument of combine represents alternatives,\nso combine has to combine alternatives by selecting, in all possible ways, one substitution from each list and then unifying the result. We will return to this function\nin the module for substitutions. This completes the de\ufb01nition of matches. The\nmodule declaration is\n\n\f12.7 Substitutions\n\nmodule\nwhere\nimport\nimport\nimport\n\n319\n\nMatchings (match)\nExpressions\nSubstitutions (Subst, unitSub, combine)\nUtilities (parts)\n\nWe place parts in the utilities module because it is not speci\ufb01c to expressions.\n\n12.7 Substitutions\nA substitution is a \ufb01nite mapping associating variables with expressions. A simple\nrepresentation as an association list suf\ufb01ces:\ntype Subst = [(VarName,Expr)]\nThe empty and unit substitutions are then de\ufb01ned by\nemptySub\n= []\nunitSub v e = [(v,e)]\nWe can apply a substitution to an expression to get another expression by de\ufb01ning\napply :: Subst -> Expr -> Expr\napply sub (Compose as)\n= Compose (concatMap (applyA sub) as)\napplyA sub (Var v)\n= deCompose (binding sub v)\napplyA sub (Con k es) = [Con k (map (apply sub) es)]\nThe function binding looks up a nonempty substitution for the binding for a variable:\nbinding :: Subst -> VarName -> Expr\nbinding sub v = fromJust (lookup v sub)\nThe function lookup is supplied in the Haskell Prelude and returns Nothing if no\nbinding is found, and Just e if v is bound to e. The function fromJust is in the\nlibrary Data.Maybe and removes the wrapper Just.\nNext we tackle combine. This function has to combine alternative substitutions by\nselecting, in all possible ways, one substitution from each component list and then\nunifying each resulting list of substitutions:\ncombine = concatMap unifyAll . cp\n\n\f320\n\nA simple equational calculator\n\nThe utility function cp, which we have seen many times before, computes the cartesian product of a list of lists.\nThe function unifyAll takes a list of substitutions and uni\ufb01es them. To de\ufb01ne it\nwe \ufb01rst show how to unify two substitutions. The result of uni\ufb01cation is either the\nunion of the two substitutions if they are compatible, or no substitution if they are\nincompatible. To handle the possibility of failure, we can use the Maybe type, or\nsimply return either an empty list or a singleton list. We choose the latter simply\nbecause in the following section we are going to calculate another version of the\ncalculator, and it is simplest to stick with list-based functions:\nunify :: Subst -> Subst -> [Subst]\nunify sub1 sub2 = if compatible sub1 sub2\nthen [union sub1 sub2]\nelse []\nIn order to de\ufb01ne compatible and union we will suppose that substitutions are\nmaintained as lists in lexicographic order of variable name. Two substitutions are\nincompatible if they associate different expressions with one and the same variable:\ncompatible [] sub2 = True\ncompatible sub1 [] = True\ncompatible sub1@((v1,e1):sub1') sub2@((v2,e2):sub2')\n| v1v2 = compatible sub1 sub2'\nThe union operation is de\ufb01ned in a similar style:\nunion [] sub2 = sub2\nunion sub1 [] = sub1\nunion sub1@((v1,e1):sub1')\n| v1v2 = (v2,e2):union\n\nsub2@((v2,e2):sub2')\nsub1' sub2\nsub1' sub2'\nsub1 sub2'\n\nThe function unifyAll returns either an empty list or a singleton list:\nunifyAll :: [Subst] -> [Subst]\nunifyAll = foldr f [emptySub]\nwhere f sub subs = concatMap (unify sub) subs\nThat completes the de\ufb01nitions we need. Here is the module declaration:\n\n```\n\n\n```\n\n\f12.8 Testing the calculator\n\n321\n\nmodule Substitutions\n(Subst, unitSub, combine, apply)\nwhere\nimport Expressions\nimport Utilities (cp)\nimport Data.Maybe (fromJust)\nThat makes nine modules in total for our calculator.\n\n12.8 Testing the calculator\nHow useful is the calculator in practice? The only way to answer this question\nis to try it out on some examples. We are going to record just two. The \ufb01rst is\nthe calculation we performed in Chapter 5 about pruning the matrix of choices in\nSudoku. In effect we want to prove\nfilter (all nodups . boxs) . expand . pruneBy boxs\n= filter (all nodups . boxs) . expand\n\nfrom the laws\ndefn pruneBy:\npruneBy f = f . map pruneRow . f\nexpand after boxs: expand . boxs = map boxs . expand\nfilter with boxs: filter (p . boxs)\n= map boxs . filter p . map boxs\nboxs involution:\nboxs . boxs = id\nmap functor:\nmap f . map g = map (f.g)\nmap functor:\nmap id = id\ndefn expand:\nexpand = cp . map cp\nfilter after cp:\nfilter (all p) . cp = cp . map (filter p)\nlaw of pruneRow:\nfilter nodups . cp . pruneRow\n= filter nodups . cp\n\nHere is the calculation exactly as performed by the calculator, except that we have\nbroken some expressions across two lines, a task that should be left to a prettyprinter. Don\u2019t bother to study it in detail, just note the important bit towards the\nend:\nfilter (all nodups . boxs) . expand . pruneBy boxs\n{filter with boxs}\nmap boxs . filter (all nodups) . map boxs . expand .\npruneBy boxs\n=\n{defn pruneBy}\nmap boxs . filter (all nodups) . map boxs . expand .\nboxs . map pruneRow . boxs\n=\n{expand after boxs}\n=\n\n\fA simple equational calculator\n\n322\n\n=\n=\n=\n=\n=\n=\n=\n=\n=\n=\n=\n\nmap boxs . filter (all nodups) . map boxs . map boxs .\nexpand . map pruneRow . boxs\n{map functor}\nmap boxs . filter (all nodups) . map (boxs . boxs) . expand .\nmap pruneRow . boxs\n{boxs involution}\nmap boxs . filter (all nodups) . map id . expand .\nmap pruneRow . boxs\n{map functor}\nmap boxs . filter (all nodups) . expand . map pruneRow . boxs\n{defn expand}\nmap boxs . filter (all nodups) . cp . map cp . map pruneRow . boxs\n{map functor}\nmap boxs . filter (all nodups) . cp . map (cp . pruneRow) . boxs\n{filter after cp}\nmap boxs . cp . map (filter nodups) . map (cp . pruneRow) . boxs\n{map functor}\nmap boxs . cp . map (filter nodups . cp . pruneRow) . boxs\n{law of pruneRow}\nmap boxs . cp . map (filter nodups . cp) . boxs\n{... ??? ...}\nmap boxs . filter (all nodups) . map boxs . cp . map cp\n{defn expand}\nmap boxs . filter (all nodups) . map boxs . expand\n{filter with boxs}\nfilter (all nodups . boxs) . expand\n\nYes, the calculation fails. The reason is not hard to spot: we need to apply the law\nexpand after boxs:\n\nexpand . boxs = map boxs . expand\n\nin both directions, and the calculator simply cannot do that.\nThe solution is a hack. We add in the extra law\nhack:\n\nmap boxs . cp . map cp = cp . map cp . boxs\n\nwhich is just the expand after boxs law written in the opposite direction and with\nexpand replaced by its de\ufb01nition. Then the calculator is happy, producing the conclusion\n....\nmap boxs . cp . map (filter nodups . cp) . boxs\n=\n{map functor}\nmap boxs . cp . map (filter nodups) . map cp . boxs\n=\n{filter after cp}\nmap boxs . filter (all nodups) . cp . map cp . boxs\n=\n{hack}\nmap boxs . filter (all nodups) . map boxs . cp . map cp\n=\n{defn expand}\nmap boxs . filter (all nodups) . map boxs . expand\n\n\f12.8 Testing the calculator\n=\n\n323\n\n{filter with boxs}\nfilter (all nodups . boxs) . expand\n\nIn both cases the calculations were performed in a fraction of a second, so ef\ufb01ciency does not seem to be an issue. And, apart from the hack, the calculations\npass muster, being almost exactly what a good human calculator would produce.\n\nImproving the calculator\nOur second example is more ambitious: we are going to use the calculator to derive\nanother version of the calculator. Look again at the de\ufb01nition of match. This relies\non combine, which in turn involves a messy appeal to the uni\ufb01cation of two substitutions, with all the paraphernalia of having to test them for compatibility and\ncomputing the union. A better idea is to compute the union of two substitutions\nonly when one of them is a unit substitution. Then everything becomes simpler\nand probably faster. And the technique which describes this optimisation? Yes, it\u2019s\nanother example of accumulating parameters. Just as an accumulating parameter\ncan avoid expensive uses of ++ operations, our hope is to avoid expensive unify\noperations.\nFirst of all, here is the de\ufb01nition of match again, written with a couple of new\nsubsidiary functions:\nmatch = concatMap matchesA . alignments\nmatchesA = combine . map matchA\nmatchA (Var v,e) = [unitSub v e]\nmatchA (Con k1 es1,Compose [Con k2 es2])\n| k1==k2 = matches (zip es1 es2)\nmatchA _ = []\nmatches = combine . map match\nNote the cycle of dependencies of these functions:\nmatch --> matchesA --> matchA --> matches --> match\nThese four functions are generalised as follows:\nxmatch sub\nxmatchA sub\nxmatches sub\nxmatchesA sub\n\n=\n=\n=\n=\n\nconcatMap\nconcatMap\nconcatMap\nconcatMap\n\n(unify\n(unify\n(unify\n(unify\n\nsub)\nsub)\nsub)\nsub)\n\n.\n.\n.\n.\n\nmatch\nmatchA\nmatches\nmatchesA\n\nThe additional argument in each case is an accumulating parameter. Our aim will\n\n\fA simple equational calculator\n\n324\n\nbe to obtain new versions of these de\ufb01nitions, whose cycle of dependencies is the\nsame as the one above:\nFor the \ufb01rst calculation, we want to rewrite match in terms of xmatch, thereby\nlinking the two groups of de\ufb01nitions. To save a lot of ink, we henceforth abbreviate\nconcatMap to cmap. The three laws we need are\ndefn xmatch:\nunify of empty:\ncmap of one:\n\nxmatch s = cmap (unify s) . match\nunify emptySub = one\ncmap one = id\n\nIn the \ufb01rst law we have to write s rather than sub (why?); the second two laws are\nthe pointless versions of the facts that\nunify emptySub sub = [sub]\ncmap one xs = concat [[x] | x <- xs] = xs\nThe calculator is hardly stretched to give:\nxmatch emptySub\n{defn xmatch}\ncmap (unify emptySub) . match\n= {unify of empty}\ncmap one . match\n= {cmap of one}\nmatch\n=\n\nLet us next deal with xmatchA. Because of the awkward pattern-matching style of\nde\ufb01nition of matchA, we simply record the following result of an easy (human)\ncalculation:\nxmatchA sub (Var v,e) = concat [unify sub (unitSub v e)]\nxmatchA sub (Con k1 es1,Compose [Con k2 es2])\n| k1==k2 = xmatches sub (zip es1 es2)\nxmatchA _ = []\nIf we introduce\nextend sub v e = concat [unify sub (unitSub v e)]\nthen it is easy to derive\nextend sub v e\n= case lookup v sub of\nNothing -> [(v,e):sub]\nJust e' -> if e==e' then [sub]\nelse []\n\n\f12.8 Testing the calculator\n\n325\n\nNo elaborate compatibility test, and no general union of two substitutions. Instead,\nas we promised earlier, we unify substitutions only with unit substitutions.\nHaving disposed of xmatchA we concentrate on the other three members of the\nquartet. Just as xmatchA is de\ufb01ned in terms of xmatches, so xmatch can be de\ufb01ned in terms of xmatchesA. Speci\ufb01cally, we want to prove that\nxmatch s = cmap (xmatchesA s) . alignments\nHere are the laws we need:\ndefn\ndefn\ndefn\ncmap\n\nmatch:\nxmatch:\nxmatchesA:\nafter cmap:\n\nmatch = cmap matchesA . alignments\nxmatch s = cmap (unify s) . match\nxmatchesA s = cmap (unify s) . matchesA\ncmap f . cmap g = cmap (cmap f . g)\n\nThe last, purely combinatorial law is new; we leave veri\ufb01cation as an exercise. The\ncalculator produces:\nxmatch s\n{defn xmatch}\ncmap (unify s) . match\n=\n{defn match}\ncmap (unify s) . cmap matchesA . alignments\n=\n{cmap after cmap}\ncmap (cmap (unify s) . matchesA) . alignments\n=\n{defn xmatchesA}\ncmap (xmatchesA s) . alignments\n=\n\nSo far, so good. That leaves us with the two remaining members of the quartet,\nxmatches and xmatchesA. In each case we want to obtain recursive de\ufb01nitions,\nones that do not involve unify. The two functions are de\ufb01ned in a very similar\nway, and it is likely that any calculation about one can be adapted immediately to\nthe other. This kind of meta-calculational thought is, of course, beyond the reaches\nof the calculator.\nLet us concentrate on xmatchesA. We \ufb01rst make xmatchesA entirely pointless,\nremoving the parameter s in the de\ufb01nition above. The revised de\ufb01nition is:\nxmatchesA :: (Subst,[(Atom,Expr)]) -> Subst\nxmatchesA = cup . (one * matchesA)\ncup = cmap unify . cpp\nwhere the combinator cpp is de\ufb01ned by\ncpp (xs,ys) = [(x,y) | x <- xs, y <- ys]\n\n```\n\n\n```\n\nThus\n\n\fA simple equational calculator\n\n326\n\nxmatchesA (sub,aes)\n= cup ([sub],aes)\n= concat [unify (s,ae) | s <- [sub],ae <- matchesA aes]\n= concat [unify (sub,ae) | ae <- matchesA aes]\nApart from the fact that unify is now assumed to be a non-curried function, this is\na faithful rendition of the de\ufb01nition of xmatchesA in pointless form.\nThe new function cup has type [Subst] -> [Subst] -> [Subst]. Later on we\nwill exploit the fact that cup is an associative function, something that unify could\nnever be (why not?). As we saw in Chapter 7 the accumulating parameter technique\ndepends on the operation of interest being associative.\nThe \ufb01rst thing to check is that the previous calculation is still valid with the new\nde\ufb01nitions. Suppose we set up the laws\ndefn match:\ndefn xmatch:\ndefn xmatchesA:\n\nmatch = cmap matchesA . alignments\nxmatch = cup . (one * match)\nxmatchesA = cup . (one * matchesA)\n\nThe calculator then produces\nxmatch\n{defn xmatch}\ncup . (one * match)\n=\n{defn match}\ncup . (one * (cmap matchesA . alignments))\n=\n{... ??? ...}\ncmap (cup . (one * matchesA)) . cpp . (one * alignments)\n=\n{defn xmatchesA}\ncmap xmatchesA . cpp . (one * alignments)\n=\n\nAh, it doesn\u2019t go through. Inspecting the gap in the calculation, it seems we need\nboth the bifunctor law of * and a claim relating cmap and cup:\ncross bifunctor: (f * g) . (h * k) = (f . h) * (g . k)\ncmap-cup: cmap (cup . (one * g)) . cpp = cup . (id * cmap g)\n\nThe calculator is then happy:\n=\n=\n=\n=\n=\n\nxmatch\n{defn xmatch}\ncup . (one * match)\n{defn match}\ncup . (one * (cmap matchesA . alignments))\n{cross bifunctor}\ncup . (id * cmap matchesA) . (one * alignments)\n{cmap-cup}\ncmap (cup . (one * matchesA)) . cpp . (one * alignments)\n{defn xmatchesA}\n\n\f12.8 Testing the calculator\n\n327\n\ncmap xmatchesA . cpp . (one * alignments)\n\nThat still leaves us with the claim; apart from the fact that it works we have no\nreason to suppose it is true. However, we can get the calculator to prove it by using\nanother law that is not speci\ufb01c to matching. We leave the proof as Exercise M.\nDe\ufb01ne the additional laws\ndefn cup:\ncup = cmap unify . cpp\ncmap-cpp: cmap (cpp . (one * f)) . cpp = cpp . (id * cmap f)\n\nThe calculator then produces\ncmap (cup . (one * g))\n{defn cup}\ncmap (cmap unify . cpp\n=\n{cmap after cmap}\ncmap unify . cmap (cpp\n=\n{cmap-cpp}\ncmap unify . cpp . (id\n=\n{defn cup}\ncup . (id * cmap g)\n\n. cpp\n\n=\n\n. (one * g)) . cpp\n. (one * g)) . cpp\n* cmap g)\n\nGood. It seems that the cmap-cup law is valid, and it even might be useful again\nlater on. Now let us return to the main point, which is to express xmatchesA recursively by two equations of the form\nxmatchesA . (id * nil) = ...\nxmatchesA . (id * cons) = ...\nThe hope is that such a de\ufb01nition will not involve unify.\nIt is not at all clear what laws we need for this purpose. Instead, we will write down\nevery law we can think of that might prove useful. The \ufb01rst group consists of our\nmain de\ufb01nitions:\ndefn\ndefn\ndefn\ndefn\ndefn\ndefn\n\nmatch:\nmatchesA:\nxmatch:\nxmatchesA:\nxmatchA:\ncombine:\n\nmatch = cmap matchesA . alignments\nmatchesA = combine . map matchA\nxmatch\n= cup . (one * match)\nxmatchesA = cup . (one * matchesA)\nxmatchA\n= cup . (one * matchA)\ncombine = cmap unifyAll . cp\n\nThe second group are some new laws about cmap:\ncmap\ncmap\ncmap\ncmap\n\nafter\nafter\nafter\nafter\n\nmap:\nconcat:\nnil:\none:\n\ncmap\ncmap\ncmap\ncmap\n\nf\nf\nf\nf\n\n.\n.\n.\n.\n\nmap g = cmap (f . g)\nconcat = cmap (cmap f)\nnil = nil\none = f\n\nThe third group are some new laws about map:\n\n\fA simple equational calculator\n\n328\nmap\nmap\nmap\nmap\n\nafter\nafter\nafter\nafter\n\nnil: map f . nil = nil\none: map f . one = one . f\ncons: map f . cons = cons . (f * map f)\nconcat: map f . concat = concat . map (map f)\n\nThe fourth group concerns cup:\ncup assoc: cup\ncup ident: cup\ncup ident: cup\nassocl: assocl.\n\n. (id * cup) = cup . (cup * id) . assocl\n. (f * (one . nil)) = f . fst\n. ((one . nil) * g) = g . snd\n(f * (g * h)) = ((f * g) * h) . assocl\n\nFinally we add in various other de\ufb01nitions and laws:\ncross bifunctor: (f * g) . (h * k) = (f . h) * (g . k)\ncross bifunctor: (id * id) = id\ndefn cp: cp . nil = one . nil\ndefn cp: cp . cons = map cons . cpp . (id * cp)\ndefn unifyAll: unifyAll . nil = one . nil\ndefn unifyAll: unifyAll . cons = cup . (one * unifyAll)\nunify after nil: unify . (id * nil) = one . fst\n\nThat\u2019s a total of 30 laws (including the two map functor laws and three laws about\ncmap that we haven\u2019t repeated). We cross our \ufb01ngers and hope:\n=\n=\n=\n=\n=\n=\n=\n=\n=\n\nxmatchesA . (id * nil)\n{defn xmatchesA}\ncup . (one * matchesA) . (id * nil)\n{cross bifunctor}\ncup . (one * (matchesA . nil))\n{defn matchesA}\ncup . (one * (combine . map matchA . nil))\n{map after nil}\ncup . (one * (combine . nil))\n{defn combine}\ncup . (one * (cmap unifyAll . cp . nil))\n{defn cp}\ncup . (one * (cmap unifyAll . one . nil))\n{cmap after one}\ncup . (one * (unifyAll . nil))\n{defn unifyAll}\ncup . (one * (one . nil))\n{cup ident}\none . fst\n\nThat\u2019s gratifying. We have shown that xmatchesA sub [] = [sub]. However,\nthe recursive case cannot be established so easily. Instead we have to guess the\nresult and then try to prove it. Here is the desired result, \ufb01rst expressed in pointed\nform and then in pointless form:\nxmatchesA sub (ae:aes)\n\n\f12.8 Testing the calculator\n\n329\n\n= concat [xmatchesA sub' aes | sub' <- xmatchA sub ae]\nxmatchesA . (id * cons)\n= cmap xmatchesA . cpp . (xmatchA * one) . assocl\nWe can perform simpli\ufb01cation with the right-hand side (we temporarily remove the\nde\ufb01nitions of xmatchA and matchesA from laws2):\ncmap xmatchesA . cpp . (xmatchA * one) . assocl\n{defn xmatchesA}\ncmap (cup . (one * matchesA)) . cpp . (xmatchA * one) . assocl\n=\n{cmap-cup}\ncup . (id * cmap matchesA) . (xmatchA * one) . assocl\n=\n{cross bifunctor}\ncup . (xmatchA * (cmap matchesA . one)) . assocl\n=\n{cmap after one}\ncup . (xmatchA * matchesA) . assocl\n=\n\nNow we would like to show\nxmatchesA . (id * cons)\n= cup . (xmatchA * matchesA) . assocl\nBut unfortunately the calculator can\u2019t quite make it. The gap appears here:\n=\n\ncup . ((cup . (one * matchA)) * matchesA)\n{... ??? ...}\ncup . (one * (cup . (matchA * matchesA))) . assocl\n\nThe gap is easily eliminable by hand:\ncup . ((cup . (one * matchA)) * matchesA)\n{cross bifunctor (backwards)}\ncup . (cup * id) . ((one * matchA) * matchesA)\n= {cup assoc}\ncup . (id * cup) . assocl . ((one * matchA) * matchesA)\n= {assocl}\ncup . (id * cup) . (one * (matchA * matchesA)) . assocl\n= {cross bifunctor}\ncup . (one * (cup . (matchA * matchesA))) . assocl\n=\n\nOnce again, the inability to apply laws in both directions is the culprit. Instead of\ntrying to force the laws into a form that would be acceptable to the calculator, we\nleave it here with the comment \u2018A hand-\ufb01nished product!\u2019.\nTo round off the example, here is the program we have calculated:\nmatch = xmatch emptySub\nxmatch sub (e1,e2)\n= concat [xmatchesA sub aes | aes <- alignments (e1,e2)]\n\n\f330\n\nA simple equational calculator\n\nxmatchesA sub [] = [sub]\nxmatchesA sub (ae:aes)\n= concat [xmatchesA sub' aes | sub' <- xmatchA sub ae]\nxmatchA sub (Var v,e) = extend sub v e\nxmatchA sub (Con k1 es1,Compose [Con k2 es2])\n| k1==k2 = xmatches sub (zip es1 es2)\nxmatchA _ = []\nThe missing de\ufb01nition is that of xmatches. But exactly the same treatment for\nxmatchesA goes through for matches, and we end up with\nxmatches sub [] = [sub]\nxmatches sub ((e1,e2):es)\n= concat [xmatches sub' es | sub' <- xmatch sub (e1,e2)]\n\nConclusions\nThe positive conclusion of these two exercises is that one can indeed get the calculator to assist in the construction of formal proofs. But there remains the need for\nsubstantial human input to the process, to set up appropriate laws, to identify subsidiary claims and to control the order in which calculations are carried out. The\nmajor negative conclusion is that it is a signi\ufb01cant failing of the calculator to be\nunable to apply laws in both directions. The functor laws are the major culprits, but\nthere are others as well (see the exercises for some examples). The calculator can\nbe improved in a number of ways, but we leave further discussion to the exercises.\nThere are three other aspects worth mentioning about the calculator. Firstly, the\ncomplete calculator is only about 450 lines of Haskell, and the improved version\nis even shorter. That alone is a testament to the expressive power of functional\nprogramming. Secondly, it does seem a viable approach to express laws as purely\nfunctional equations and to use a simple equational logic for conducting proofs. To\nbe sure, some work has to be done to express de\ufb01nitions in point-free form, but\nonce this is achieved, equational logic can be surprisingly effective.\nThe third aspect is that, apart from parsing, no monadic code appears in the calculator. In fact, earlier versions of the calculator did use monads, but gradually they\nwere weeded out. One reason was that we found the code became simpler without monads, without signi\ufb01cant loss of ef\ufb01ciency; another was that we wanted to\nset things up for the extended exercise in improving the calculator. Monads are\n\n\f12.9 Exercises\n\n331\n\nabsolutely necessary for many applications involving interacting with the world,\nbut they can be overused in places where a purely functional approach would be\nsmoother.\nOn that note, we end.\n\n12.9 Exercises\nExercise A\nSuppose we did want calculate to return a tree of possible calculations. What\nwould be a suitable tree to use?\nExercise B\nWhy should the laws\nmap (f . g) = map f . map g\ncmap (f . g) = cmap f . map g\nnever be used in calculations, at least if they are given in the form above?\nExercise C\nHere is a calculation, as recorded by the calculator\nmap f . map g h\n=\n{map functor}\nmap (f . g)\nExplain this strange and clearly nonsensical result. What simple change to the calculator would prevent the calculation from being valid?\nExercise D\nOn the same general theme as the previous question, one serious criticism of the\ncalculator is that error messages are totally opaque. For example, both\nparse law \"map f . map g = map (f . g)\"\nparse law \"map functor: map f . map g\nmap (f . g)\"\ncause the same cryptic error message. What is it? What would be the effect of using\nthe law\nstrange: map f . map g = map h\n\n\fA simple equational calculator\n\n332\n\nin a calculation?\nAgain, what change to the calculator would prevent such a law from being acceptable?\nExercise E\nThe de\ufb01nition of showsPrec for atoms makes use of a fact about Haskell that we\nhaven\u2019t needed before. And the same device is used in later calculator functions\nthat mix a pattern-matching style with guarded equations. What is the fact?\nExercise F\nDe\ufb01ne\ne1 = foo (f . g) . g\ne2 = bar f . baz g\nList the expressions that rewrites (e1,e2) produces when applied to the expression foo (a . b . c) . c. Which one would the calculator pick?\nExercise G\nCan the calculator successfully match foo f . foo f with the expression\nfoo (bar g h) . foo (bar (daz a) b) ?\nExercise H\nIt was claimed in the text that it is possible to apply a perfectly valid non-trivial law\nthat will leave some expressions unchanged. Give an example of such a law and an\nexpression that is rewritten to itself.\nExercise I\nThe function anyOne used in the de\ufb01nition of rewrites installs a single choice,\nbut why not use everyOne that installs every choice at the same time? Thus if\nf 1 = [-1,-2] and f 2 = [-3,-4], then\neveryOne f [1,2] = [[-1,-3],[-1,-4],[-2,-3],[-3,-4]]\nUsing everyOne instead of anyOne would mean that a rewrite would be applied to\nevery possible subexpression that matches a law. Give a de\ufb01nition of everyOne.\n\n\f12.10 Answers\n\n333\n\nExercise J\nHow many segments of a list of length n are there? The de\ufb01nition of rewritesSeg\nis inef\ufb01cient because the empty segment appears n+1 times as the middle component of the segments of a list of length n. That means matching with id is performed\nn+1 times instead of just once. How would you rewrite segments to eliminate\nthese duplicates?\nExercise K\nProve that cmap f . cmap g = cmap (cmap f . g). The laws needed are:\ndefn cmap:\nmap functor:\nmap after concat:\nconcat twice:\n\ncmap f = concat . map f\nmap f . map g = map (f.g)\nmap f . concat = concat . map (map f)\nconcat . concat = concat . map concat\n\nExercise L\nThe cmap-cpp law is as follows:\ncmap (cpp . (one * f)) . cpp = cpp . (id * cmap f)\nProve it from the laws\ncmap after cmap:\ncmap after cpp:\ncross bifunctor:\nmap after cpp:\ndefn cmap:\nconcat after id:\n\ncmap f . map g = cmap (f . g)\ncmap cpp . cpp = cpp . (concat * concat)\n(f * g) . (h * k) = (f . h) * (g . k)\nmap (f * g) . cpp = cpp . (map f * map g)\ncmap f = concat . map f\nconcat . map one = id\n\nCan a calculator conduct the proof?\n\n12.10 Answers\nAnswer to Exercise A\nWe would want expressions as labels of nodes and law names as labels of edges.\nThat gives\ntype Calculation = Tree Expr LawName\ndata Tree a b\n= Node a [(b,Tree a b)]\n\n\fA simple equational calculator\n\n334\n\nAnswer to Exercise B\nThey would both cause the calculator to spin off into an in\ufb01nite calculation. For\nexample,\nmap foo\n{map functor}\nmap foo . map id\n= {map functor}\nmap foo . map id . map id\n=\n\nand so on.\nAnswer to Exercise C\nThe expression map f . map g h is perfectly valid by the rules of syntax, but of\ncourse it shouldn\u2019t be. The evaluator does not force the restriction that each appearance of one and the same constant should possess the same number of arguments.\nThe reason the functor law can be matched successfully against the expression is\nthat in the de\ufb01nition of matchA the function zip truncates the two arguments to\nthe second map to one. A better calculator should check that each constant has a\n\ufb01xed arity.\nAnswer to Exercise D\nThe cryptic message is \u2018head of empty list\u2019. The \ufb01rst parse fails because the law is\nmissing its name, and the second is missing an equals sign. Use of the strange law\nwould cause the calculator to fall over because pattern-matching with the left-hand\nside would not bind h to any expression, causing an error when the binding for h is\nrequested. The calculator should have checked that every variable on the right-hand\nside of a law appears somewhere on the left-hand side.\nAnswer to Exercise E\nThe code for showsPrec takes the form\nshowsPrec p (Con f [e1,e2])\n| isOp f\n= expression1 e1 e2\nshowsPrec p (Con f es)\n= expression2 es\nA more \u2018mathematical\u2019 style would have been to write\nshowsPrec p (Con f [e1,e2])\n| isOp f\n= expression1 e1 e2\n| otherwise = expression2 [e1,e2]\n\n\f12.10 Answers\n\n335\n\nshowsPrec p (Con f es) = expression2 es\nThe point is this: in a given clause if a pattern does not match the argument, or if\nit does but the guard fails to be true, the clause is abandoned and the next clause is\nchosen.\nAnswer to Exercise F\nThere are two rewrites, not one:\nbar (a . b . c) . baz id . c\nbar (a . b) . baz c\nThe calculator would pick the \ufb01rst subexpression that matches, and that means the\n\ufb01rst rewrite is chosen. Perhaps it would be better to arrange that rewritesSeg is\napplied to longer segments before shorter ones.\nAnswer to Exercise G\nNo, not with our de\ufb01nition of match. They can be matched by binding f to the\nexpression bar (daz a) b provided g is bound to daz a and h to b, but our\nde\ufb01nition of match does not perform full uni\ufb01cation.\nAnswer to Exercise H\nTo take just one example out of many, consider the law\nif p f g . h = if (p . h) (f . h) (g . h)\nThe left-hand side matches if a b c with h bound to id, and the result is again\nthe same expression.\nAnswer to Exercise I\nThe temptation is to de\ufb01ne\neveryOne f = cp . map f\nbut that doesn\u2019t work if f returns no alternatives for some element. Instead we have\nto de\ufb01ne\neveryOne :: (a -> [a]) -> [a] -> [[a]]\neveryOne f\n= cp . map (possibly f)\npossibly f x = if null xs then [x] else xs\nwhere xs = f x\nIn this version, f returns a nonempty list of alternatives.\n\n\fA simple equational calculator\n\n336\n\nAnswer to Exercise J\nThere are (n+1)(n+2)/2 segments of a list of length n. The improved de\ufb01nition is\nsegments xs = [([],[],xs] ++\n[(as,bs,cs)\n| (as,ys) <- splits xs,\n(bs,cs) <- tail (splits ys)]\nAnswer to Exercise K\nThe calculator produced:\n=\n=\n=\n=\n=\n=\n=\n=\n\ncmap f . cmap g\n{defn cmap}\nconcat . map f . cmap g\n{defn cmap}\nconcat . map f . concat . map g\n{map after concat}\nconcat . concat . map (map f) . map g\n{map functor}\nconcat . concat . map (map f . g)\n{concat after concat}\nconcat . map concat . map (map f . g)\n{map functor}\nconcat . map (concat . map f . g)\n{defn cmap}\nconcat . map (cmap f . g)\n{defn cmap}\ncmap (cmap f . g)\n\nAnswer to Exercise L\nThe human proof is:\n=\n=\n=\n=\n=\n=\n\ncmap (cpp . (one * g)) . cpp\n{cmap after cmap (backwards)}\ncmap cpp . map (one * g) . cpp\n{map after cpp}\ncmap cpp . cpp . (map one * map g)\n{cmap after cpp}\ncpp . (concat * concat) . (map one * map g)\n{cross bifunctor}\ncpp . ((concat . map one) * concat (map g))\n{defn cmap (backwards)}\ncpp . ((concat . map one) * cmap g)\n{concat after id}\ncpp . (id * cmap g)\n\nNo, the calculation cannot be performed automatically. The cmap after cmap\n\n\f12.11 Chapter notes\n\n337\n\nlaw cannot be installed in the backwards direction without causing the calculator\nto loop (see Exercise B).\n\n12.11 Chapter notes\nThe calculator in this chapter is based on an undocumented theorem prover by\nMike Spivey, a colleague at Oxford. Ross Paterson of City University, London, has\nproduced a version with built-in functor laws that can be applied in both directions\nwhen necessary.\nOne state-of-the-art proof assistant is Coq; see http://coq.inria.fr/.\n\n```\n", "meta": {"hexsha": "618da83d2ec7e4dd94a7f2747c3fcc60d828a221", "size": 737064, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "haskell_richardbird.ipynb", "max_stars_repo_name": "kalz2q/-yjupyternotebooks", "max_stars_repo_head_hexsha": "ba37ac7822543b830fe8602b3f611bb617943463", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-09-16T03:45:19.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-16T03:45:19.000Z", "max_issues_repo_path": "haskell_richardbird.ipynb", "max_issues_repo_name": "kalz2q/-yjupyternotebooks", "max_issues_repo_head_hexsha": "ba37ac7822543b830fe8602b3f611bb617943463", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "haskell_richardbird.ipynb", "max_forks_repo_name": "kalz2q/-yjupyternotebooks", "max_forks_repo_head_hexsha": "ba37ac7822543b830fe8602b3f611bb617943463", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 47.8209303834, "max_line_length": 2970, "alphanum_fraction": 0.559617618, "converted": true, "num_tokens": 141534, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.7956581000631542, "lm_q1q2_score": 0.4195636631240942}} {"text": "```python\nfrom IPython.core.display import HTML\nHTML(\"\")\n```\n\n\n\n\n\n\n\n\n# Lecture 9: Methods using KKT conditions\n\n# Reminder: optimality conditions\nWe consider an optimization problem of the form:\n$$\n\\begin{align} \\\n\\min \\quad &f(x)\\\\\n\\text{s.t.} \\quad & g_j(x) \\geq 0\\text{ for all }j=1,\\ldots,J\\\\\n& h_k(x) = 0\\text{ for all }k=1,\\ldots,K\\\\\n&x\\in \\mathbb R^n.\n\\end{align}\n$$\n\nNecessary KKT conditions require for an optimal solution to satisfy\n\n$$\n\\begin{align}\n&\\nabla_xL(x^*,\\mu^*,\\lambda^*) = 0\\\\\n&\\mu_j^*\\geq0,\\text{ for all }j=1,\\ldots,J\\\\\n&\\mu_j^*g_j(x^*)=0,\\text{for all }j=1,\\ldots,J,\n\\end{align}\n$$\n\nwhere $\\mu^*=(\\mu_1^*,\\ldots,\\mu_J^*)$ and $\\lambda^* = (\\lambda^*_1,\\ldots,\\lambda_K^*)$ are called Lagrance multiplier vectors for the inequality and the equality constraints, respectively. \n\nFunction $L$ is the *Lagrangian function* $$L(x,\\mu,\\lambda) = f(x)- \\sum_{j=1}^J\\mu_jg_j(x) -\\sum_{k=1}^K\\lambda_kh_k(x)$$.\n\n\n### Geometry of the KKT conditions\n\n\n## Example:\n\nConsider the following optimization problem\n\n$$\nmin \ud835\udc53(\\mathbf{x}) = \ud835\udc65_1 + \ud835\udc65_2 \\\\\ns.t. \\\\\ng_1(x) = \ud835\udc65_1^2 + \ud835\udc65_2^2 \\geq 5,\\\\\ng_2(x) = \ud835\udc65_1 + 2\ud835\udc65_2 \\geq 4,\\\\\ng_3(x) = \ud835\udc65_1 \\geq 0,\\\\\ng_4(x) = \ud835\udc65_2 \\geq 0\n$$\n\nat $x^* = (2,1)$.\n\n\n* The stationary rule says that, at the optimum ($x^*$), there is no feasible descent direction ($F\\cap D = \\varnothing $), i.e., \n\n$$\n-\\nabla f = \\sum_j - \\mu_j \\nabla g_j\\text{ , for } \\mu_j \\geq 0 \\text{ , i.e., } g_j \\text{ is active at } x^* \\text{. }\n$$\n\n* In other words, $-\\nabla f$ belongs to the cone defined by the (-) gradients of the active constraints\n\n* The complementary rules make sure only the active constraints are considered.\n\n\n\n\n```python\nfrom IPython.display import Image\nImage(filename = \"Images\\kkt_vis.jpg\", width = 600, height = 600)\n```\n\n\n\n\n \n\n \n\n\n\n\n\n### Note.\n\n* $min f(x)$ $\\rightarrow $ $-\\nabla f$ (descent direction)\n* $max f(x)$ $\\rightarrow $ $ \\nabla f$ (ascent direction)\n\n* $g(x) \\geq 0$ $\\rightarrow $ $-\\nabla g$ (infeasible direction)\n* $g(x) \\leq 0$ $\\rightarrow $ $ \\nabla g$ (infeasible direction)\n\n## Sequential Quadratic Programming (SQP)\n\n**Idea is to generate a sequence of quadratic optimization problems whose solutions approach the solution of the original problem**\n\n**Quadratic problems are based on applying KKT conditions to the original problem**\n\n* Minimize a quadratic approximation of the Lagrangian function with respect to linear approximation of the constraints\n* Also referred as **projected Lagrangian method**\n\nLet us consider problem\n\n$$\n\\min f(x)\\\\\n\\text{s.t.\u00a0}h_k(x) = 0\\text{ for all }k=1,\\ldots,K,\n$$\n\nwhere the objective function and the equality constraints are twice differentiable. \n\nInequality constraints can be handled e.g. by using active set methods. See e.g. https://optimization.mccormick.northwestern.edu/index.php/Sequential_quadratic_programming\n\n**Note that constraints can be nonlinear.**\n\nBecause we know that the optimal solution of this problem satisfies the KKT conditions, we know that\n\n$$\n\\left\\{\\begin{array}{l}\n\\nabla_xL(x,\\lambda)=\\nabla_x f(x) + \\lambda\\nabla_x h(x) = 0\\\\\nh(x) = 0\n\\end{array}\\right.\n$$\n\nLet us assume that we have a current estimation for the solution of the equality constraints $(x_k,\\lambda_k)$, then according to the Newton's method for root finding (see e.g., https://en.wikipedia.org/wiki/Newton%27s_method#Generalizations ), we have another solution $(x_k,\\lambda_k)^T+(p,v)^T$ of the problem by solving system of equations\n\n$$\n\\nabla_{x,\\lambda} S(x_k,\\lambda_k)\\left[\\begin{align}p^T\\\\v^T\\end{align}\\right] = -S(x_k,\\lambda_k),\n$$\n\nwhere $S(x_k,\\lambda_k)=\\left[\n\\begin{array}{c}\n\\nabla_{x}L(x_k,\\lambda_k)\\\\\nh(x_k)\n\\end{array}\n\\right]\n$. \n\n\nThis can be written as\n\n$$\n\\left[\n\\begin{array}{cc}\n\\nabla^2_{xx}L(x_k,\\lambda_k)&\\nabla_x h(x_k)\\\\\n\\nabla_x h(x_k)^T & 0\n\\end{array}\n\\right]\n\\left[\\begin{array}{c}p^T\\\\v^T\\end{array}\\right] =\n\\left[\n\\begin{array}{c}\n-\\nabla_x L(x_k,\\lambda_k)\\\\\n-h(x_k)^T\n\\end{array}\n\\right].\n$$\n\n\nHowever, the above is just the solution of the quadratic problem with equality constraints\n$$\n\\min_p \\frac12 p^T\\nabla^2_{xx}L(x^k,\\lambda^k)p+\\nabla_xL(x^k,\\lambda^k)^Tp\\\\\n\\text{s.t. }h_j(x^k) + \\nabla h_j(x^k)^Tp = 0. \n$$\n\n## Intuitive interpretation\n\nWe are approximating the Lagrange function quadratically around the current solution and the constraints are approximated linearly. SQP methods are also referred to as *projected Lagrangian methods* \n* compare with projected gradient method from lecture 7\n\n**Another view point to building the approximation**: Assume that we have a current solution candidate $(x_k,\\lambda_k)$. Using Taylor's series for the constraints ($x^* = x_k + d$) and including only the first order term we get\n\n$h_i(x^*)=h_i(x_k+d)\\approx h_i(x_k) + \\nabla h_i(x_k)^Td$\n\nSince, $h_i(x^*)=0$ for all $i$ we have\n\n$\\nabla h(x_k)d = -h(x_k)$\n\nFor approximating the Lagrangian function, we use up to second order terms and get\n\n$L(x_k+d,\\lambda^*) \\approx L(x_k,\\lambda^*) + d^T\\nabla_x L(x_k,\\lambda^*) + \\frac{1}{2}d^T\\nabla_{xx}^2L(x_k,\\lambda^*)d$\n\nWhen combining both approximations, we get a quadratic subproblem\n\n$$\n\\underset{d}{\\min}d^T\\nabla_x L(x_k,\\lambda_k) + \\frac{1}{2}d^T \\nabla_{xx}^2L(x_k,\\lambda_k)d\\\\\n\\text{s.t. } \\nabla h(x_k)d = -h(x_k)\n$$\n\nIt can be shown (under some assumptions) that solutions of the quadratic subproblems approach $x^*$ and Lagrange multipliers approach $\\lambda^*$.\n\nNote that we can either use the exact Hessian of the Lagrange function (requires second derivatives) or some approximation of it (compare Newton's method vs. quasi-Newton ideas). \n\n## Implementation\n\nDefine an optimization problem, where\n* $f(x) = \\|x\\|^2 = \\sum_{i=1}^n x_i^2$\n* $h(x) = \\sum_{i=1}^nx_i=n$\n\nWhat is $x^*$?\n\n\n```python\ndef f_constrained(x):\n return sum([i**2 for i in x]),[],[sum(x)-len(x)]\n# return sum([i**2 for i in x]),[],[sum(x)-len(x),x[0]**2+x[1]-2]\n\n```\n\n\n```python\nprint(f_constrained([1,0,1]))\nprint(f_constrained([1,2,3,4]))\n```\n\n (2, [], [-1])\n (30, [], [6])\n\n\n\n```python\nimport numpy as np\nimport ad\n\n\n#if k=0, returns the gradient of lagrangian, if k=1, returns the hessian\ndef diff_L(f,x,l,k):\n #Define the lagrangian for given f and Lagrangian multiplier vector l\n L = lambda x_: f(x_)[0] + (np.matrix(f(x_)[2])*np.matrix(l).transpose())[0,0]\n return ad.gh(L)[k](x)\n\n#Returns the gradients of the equality constraints\ndef grad_h(f,x):\n return [ad.gh(lambda y:\n f(y)[2][i])[0](x) for i in range(len(f(x)[2]))] \n\n#Solves the quadratic problem inside the SQP method\ndef solve_QP(f,x,l):\n left_side_first_row = np.concatenate((\\\n np.matrix(diff_L(f,x,l,1)),\\\n np.matrix(grad_h(f,x)).transpose()),axis=1)\n left_side_second_row = np.concatenate((\\\n np.matrix(grad_h(f,x)),\\\n np.matrix(np.zeros((len(f(x)[2]),len(f(x)[2]))))),axis=1)\n right_hand_side = np.concatenate((\\\n -1*np.matrix(diff_L(f,x,l,0)).transpose(),\n -np.matrix(f(x)[2]).transpose()),axis = 0)\n left_hand_side = np.concatenate((\\\n left_side_first_row,\\\n left_side_second_row),axis = 0)\n temp = np.linalg.solve(left_hand_side,right_hand_side)\n return temp[:len(x)],temp[len(x):] # update for both x and l\n \n \n\ndef SQP(f,start,precision):\n x = start\n l = np.ones(len(f(x)[2])) # initialize Lagrange multiplier vector l as a vector of 1s\n f_old = float('inf')\n f_new = f(x)[0]\n while abs(f_old-f_new)>precision:\n print(x)\n f_old = f_new\n (p,v) = solve_QP(f,x,l) # obtain updates for x and l by solving the quadratic subproblem\n x = x+np.array(p.transpose())[0] # update the solution x \n l = l+v # update the Lagrange multipliers l\n f_new = f(x)[0]\n return x\n```\n\n\n```python\nx0 = [2000.0,1000.0,3000.0,5000.0,6000.0]\nSQP(f_constrained,x0,0.0001)\n```\n\n [2000.0, 1000.0, 3000.0, 5000.0, 6000.0]\n [1. 1. 1. 1. 1.]\n\n\n\n\n\n array([1., 1., 1., 1., 1.])\n\n\n\n## Lagrangian methods -- \"The original method of multipliers\"\n\nLet us again consider problem\n\n$$\n\\min f(x)\\\\\n\\text{s.t. }h_k(x) = 0\\text{ for all }k=1,\\ldots,K,\n$$\n\nwhere the objective function and the equality constraints are twice differentiable. Inequality constaints can be handled e.g. by first converting them into equality constraints which increases the number of variables.\n\nWe know that if $x^*$ is optimal solution, then $\\nabla_x L(x^*,\\lambda^*) = \\nabla f(x^*) + \\sum_{k=1}^K\\lambda^*_k \\nabla h_k(x^*) = 0$ \n(necessary condition). \n\nHowever, if we only know that for $x^*$ it holds that $\\nabla_x L(x^*,\\lambda^*) = 0$, we can't be sure that $x^*$ is a local minimizer. (Why???)\n\nSince the Hessian $\\nabla^2_{xx}L(x^*,\\lambda^*)$ may be indefinitive, it is not sufficient to just minimize the Langrangian function $L(x,\\lambda)$ in order to minimize $f(x)$ with respect to the equality constraints $h_k(x)=0$.\n\nSolution: Improve Lagrangian function\n\nDefine augmented Lagrangian function\n$$\nL_c(x,\\lambda) = f(x)+\\lambda h(x)+\\frac12c\\|h(x)\\|^2.\n$$\nAbove $c\\in \\mathbb R$ is a penalty parameter and $\\lambda \\in \\mathbb R^K$ is a multiplier.\n\nNow, we have\n$$\n\\nabla^2_{xx}L_c(x^*,\\lambda^*) = \\nabla^2_{xx}L(x^*,\\lambda^*) + c\\nabla h(x^*)^T\\nabla h(x^*)\n$$\nand it can be shown that for $c>\\hat{c}$ the Hessian of the augmented Lagrangian is positive definite (sufficient condition for optimality).\n\nLet us consider sequence of optimization problems\n$$\n\\min_{x\\in\\mathbb R^n} f(x)+\\lambda_k h(x)+\\frac{1}{2}c_k\\|h(x)\\|^2,\n$$\nwhere $c_{k+1}>c_k$ for $k=1,2,\\ldots$.\n\nNow, if $\\lambda_k=0$ for all $k=1,2,\\ldots$, then we have a penalty function method, which solves the problem when $c_k\\to \\infty$.\n\nHowever, it can be shown, that if we set $\\lambda_0$ randomly and keep on updating\n$\\lambda_{k+1} = \\lambda_k+c_kh(x_k)$, then we can show that there exists $C>0$ such that of $c_k>C$, then the optimal solution of the augmented Langrangian solves the original problem!\n\n### Example\n\nLet us have optimization problem\n$$\n\\min x_1^2+x_2^2\\\\\n\\text{s.t. }x_1+x_2-1=0.\n$$\n\nNow, the minimization of the augmented Lagrangian becomes\n$$\n\\min_{x\\in\\mathbb R^n} x_1^2+x_2^2+\\lambda_k(x_1+x_2-1)+\\frac12c_k(x_1+x_2-1)^2.\\\\\n$$\n\n\n```python\ndef f_constrained2(x):\n return sum([i**2 for i in x]),[],[sum(x)-1]\n```\n\n\n```python\ndef augmented_langrangian(f,x,la,c):\n second_term = float(numpy.matrix(la)*numpy.matrix(f(x)[2]).transpose())\n third_term = 0.5*c*numpy.linalg.norm(f(x)[2])**2\n return f(x)[0]+second_term+third_term\n```\n\n\n```python\nfrom scipy.optimize import minimize\nimport numpy\ndef augmented_langrangian_method(f,start,la0,c0):\n x_old = [float('inf')]*2\n x_new = start\n f_old = float('inf')\n f_new = f(x_new)[0]\n la = la0\n c = c0\n steps = []\n while abs(f_old-f_new)>0.00001:\n# while numpy.linalg.norm(f(x_new)[2])>0.00001: # doesn't work as itself, see starting from any feasible point\n res = minimize(lambda x:augmented_langrangian(f,x,la,c),x_new)\n x_old = x_new\n f_old = f_new\n la = float(la+numpy.matrix(c)*numpy.matrix(f(res.x)[2]).transpose()) # update Lagrangian\n x_new = res.x\n f_new = f(x_new)[0]\n c = 2*c # increase the penalty coefficient\n steps.append(list(x_new))\n return x_new,c, steps\n```\n\n\n```python\nx0 =[1/3,2/3]\n#x0 =[10,-5]\nl0 = 1.0\nc0 = 1.0\n[x,c,steps_ag] = augmented_langrangian_method(f_constrained2,x0,l0,c0)\nprint(x)\nprint(c)\nprint(len(steps_ag))\n```\n\n [0.49999991 0.50000009]\n 256.0\n 8\n\n\n\n```python\nimport matplotlib.pyplot as plt\nimport numpy as np\n\ndef plot_2d_steps2(steps,start,interval):\n myvec = np.array([start]+steps).transpose()\n plt.plot(myvec[0,],myvec[1,],'rx')\n for label,x,y in zip([str(i) for i in range(len(steps)+1)],myvec[0,],myvec[1,]):\n plt.annotate(label,xy = (x, y))\n # plot constraint\n z = np.arange(interval[0],interval[1],0.1)\n l = len(z)\n con = [1.0-z[i] for i in range(l)]\n plt.plot(z,con,'b-')\n return plt\n```\n\n\n```python\nplot_2d_steps2(steps_ag,x0,[0,1]).show()\n```\n\n## Compare with penalty function method\n\n\n```python\nfrom scipy.optimize import minimize\nimport numpy\ndef penalty_function_method(f,start,c0):\n x_old = float('inf')\n x_new = start\n f_old = float('inf')\n f_new = f(x_new)[0]\n c = c0\n steps = []\n while abs(f_old-f_new)>0.00001:\n# while numpy.linalg.norm(f(x_new)[2])>0.00001:\n res = minimize(lambda x:augmented_langrangian(f,x,0,c),x_new)\n x_old = x_new\n f_old = f_new\n x_new = res.x\n f_new = f(x_new)[0]\n c = 2*c\n steps.append(list(x_new))\n return x_new,c,steps\n```\n\n\n```python\n[x,c,steps_pf] = penalty_function_method(f_constrained2,x0,c0)\nprint(x) \nprint(c)\nprint(len(steps_pf))\n```\n\n [0.49999618 0.49999618]\n 262144.0\n 18\n\n\n\n```python\nplot_2d_steps2(steps_pf,x0,[0,1]).show()\n```\n\n## What is going on in here?\n\nThe above is a simplified representation of the augmented Lagrangian method. For example, one can use exact second derivatives for calculating $\\nabla^2_{xx}L(x^*,\\mu^*)$ to obtain better convergence but, also, one can approximate it by utilizing ideas from quasi-Newton methods in order to not requiring second derivatives. Efficient implementation of this (and other methods) for practical problems is not completely trivial, unfortunately. If you want to read details, please see e.g., http://www.mit.edu/~dimitrib/Constrained-Opt.pdf.\n\n## Compare with SQP\n\n\n```python\nimport numpy as np\nimport ad\n\n\n#if k=0, returns the gradient of lagrangian, if k=1, returns the hessian\ndef diff_L(f,x,l,k):\n #Define the lagrangian for given m and f\n L = lambda x_: f(x_)[0] + (np.matrix(f(x_)[2])*np.matrix(l).transpose())[0,0]\n return ad.gh(L)[k](x)\n\n#Returns the gradients of the equality constraints\ndef grad_h(f,x):\n return [ad.gh(lambda y:\n f(y)[2][i])[0](x) for i in range(len(f(x)[2]))] \n\n#Solves the quadratic problem inside the SQP method\ndef solve_QP(f,x,l):\n left_side_first_row = np.concatenate((\\\n np.matrix(diff_L(f,x,l,1)),\\\n np.matrix(grad_h(f,x)).transpose()),axis=1)\n left_side_second_row = np.concatenate((\\\n np.matrix(grad_h(f,x)),\\\n np.matrix(np.zeros((len(f(x)[2]),len(f(x)[2]))))),axis=1)\n right_hand_side = np.concatenate((\\\n -1*np.matrix(diff_L(f,x,l,0)).transpose(),\n -np.matrix(f(x)[2]).transpose()),axis = 0)\n left_hand_side = np.concatenate((\\\n left_side_first_row,\\\n left_side_second_row),axis = 0)\n temp = np.linalg.solve(left_hand_side,right_hand_side)\n return temp[:len(x)],temp[len(x):]\n \n \n\ndef SQP(f,start,precision):\n x = start\n l = np.ones(len(f(x)[2]))\n f_old = float('inf')\n f_new = f(x)[0]\n while abs(f_old-f_new)>precision:\n print(x)\n f_old = f_new\n (p,v) = solve_QP(f,x,l)\n x = x+np.array(p.transpose())[0]\n l = l+v\n f_new = f(x)[0]\n return x\n```\n\n\n```python\n#x0 = [-3000,2500]\nSQP(f_constrained2,x0,0.00001)\n```\n\n [0.3333333333333333, 0.6666666666666666]\n [0.5 0.5]\n\n\n\n\n\n array([0.5, 0.5])\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "21ab08b29238d0183eb10124e952a1c8cefcdfb9", "size": 67808, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Lecture 9, Methods using KKT conditions.ipynb", "max_stars_repo_name": "bshavazipour/TIES483-2022", "max_stars_repo_head_hexsha": "93dfabbfe1e953e5c5f83c44412963505ecf575a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Lecture 9, Methods using KKT conditions.ipynb", "max_issues_repo_name": "bshavazipour/TIES483-2022", "max_issues_repo_head_hexsha": "93dfabbfe1e953e5c5f83c44412963505ecf575a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Lecture 9, Methods using KKT conditions.ipynb", "max_forks_repo_name": "bshavazipour/TIES483-2022", "max_forks_repo_head_hexsha": "93dfabbfe1e953e5c5f83c44412963505ecf575a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-02-03T09:40:02.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-03T09:40:02.000Z", "avg_line_length": 64.4562737643, "max_line_length": 15893, "alphanum_fraction": 0.7699976404, "converted": true, "num_tokens": 4975, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.7956580952177051, "lm_q1q2_score": 0.41956366056900884}} {"text": "```python\n# This cell is mandatory in all Dymos documentation notebooks.\nmissing_packages = []\ntry:\n import openmdao.api as om\nexcept ImportError:\n if 'google.colab' in str(get_ipython()):\n !python -m pip install openmdao[notebooks]\n else:\n missing_packages.append('openmdao')\ntry:\n import dymos as dm\nexcept ImportError:\n if 'google.colab' in str(get_ipython()):\n !python -m pip install dymos\n else:\n missing_packages.append('dymos')\ntry:\n import pyoptsparse\nexcept ImportError:\n if 'google.colab' in str(get_ipython()):\n !pip install -q condacolab\n import condacolab\n condacolab.install_miniconda()\n !conda install -c conda-forge pyoptsparse\n else:\n missing_packages.append('pyoptsparse')\nif missing_packages:\n raise EnvironmentError('This notebook requires the following packages '\n 'please install them and restart this notebook\\'s runtime: {\",\".join(missing_packages)}')\n```\n\n(examples:the_robertson_problem)=\n\n# The Robertson Problem\n\nThe [Robertson Problem](https://en.wikipedia.org/w/index.php?title=Stiff_equation&oldid=1017928453#Motivating_example) is a famous example for a [stiff ODE](https://en.wikipedia.org/w/index.php?title=Stiff_equation&oldid=1017928453). Solving stiff ODEs with [explicit integration methods](https://en.wikipedia.org/w/index.php?title=Explicit_and_implicit_methods&oldid=1036001392) leads to unstable behaviour unless an extremly small step size is choosen. [Implicit methods](https://en.wikipedia.org/w/index.php?title=Explicit_and_implicit_methods&oldid=1036001392) such as the [Radau](https://en.wikipedia.org/w/index.php?title=Runge%E2%80%93Kutta_methods&oldid=1052924118#Implicit_Runge%E2%80%93Kutta_methods), [BDF](https://en.wikipedia.org/w/index.php?title=Backward_differentiation_formula&oldid=1027943694) and LSODA methods can help solve such problems. The following example shows how to solve the Robertson Problem using [SciPys LSODA method](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_ivp.html#scipy.integrate.solve_ivp) and Dymos. \n\n## The ODE system\n\nThe ODE of the Robertson Problem is\n\n\\begin{align}\n \\dot x = & - 0.04 x + 10^4 y \\cdot z & \\\\\n \\dot y = & \\;\\;\\;\\: 0.04 x - 10^4 y \\cdot z & - 3\\cdot 10^7 y^2 \\\\\n \\dot z = & & \\;\\;\\;\\: 3\\cdot 10^7 y^2 \\\\\n\\end{align}\n\nwhere $x$, $y$ and $z$ are arbitrary states. The initial conditions are\n\n\\begin{align}\n x_0 &= 1 \\\\\n y_0 &= z_0 = 0.\n\\end{align}\n\nThe problem is solved for the time interval $t\\in[0,40)$. There are no controls and constraints.\n\n\n```python\nimport numpy as np\nimport openmdao.api as om\n\nclass RobertsonODE(om.ExplicitComponent):\n \"\"\"example for a stiff ODE from Robertson.\n \"\"\"\n\n def initialize(self):\n self.options.declare('num_nodes', types=int)\n\n def setup(self):\n nn = self.options['num_nodes']\n\n # input: state\n self.add_input('x', shape=nn, desc=\"state x\", units=None)\n self.add_input('y', shape=nn, desc=\"state y\", units=None)\n self.add_input('z', shape=nn, desc=\"state z\", units=None)\n \n # output: derivative of state\n self.add_output('xdot', shape=nn, desc='derivative of x', units=\"1/s\")\n self.add_output('ydot', shape=nn, desc='derivative of y', units=\"1/s\")\n self.add_output('zdot', shape=nn, desc='derivative of z', units=\"1/s\")\n\n r = np.arange(0, nn)\n self.declare_partials(of='*', wrt='*', method='exact', rows=r, cols=r)\n\n def compute(self, inputs, outputs):\n\n x = inputs['x']\n y = inputs['y']\n z = inputs['z']\n\n xdot = -0.04 * x + 1e4 * y * z\n zdot = 3e7 * y ** 2\n ydot = - ( xdot + zdot )\n \n outputs['xdot'] = xdot\n outputs['ydot'] = ydot\n outputs['zdot'] = zdot\n \n\n def compute_partials(self, inputs, jacobian):\n\n x = inputs['x']\n y = inputs['y']\n z = inputs['z']\n\n xdot_y = 1e4 * z\n xdot_z = 1e4 * y\n\n zdot_y = 6e7 * y\n\n\n jacobian['xdot', 'x'] = -0.04\n jacobian['xdot', 'y'] = xdot_y\n jacobian['xdot', 'z'] = xdot_z\n\n jacobian['ydot', 'x'] = 0.04\n jacobian['ydot', 'y'] = - ( xdot_y + zdot_y )\n jacobian['ydot', 'z'] = - xdot_z\n\n jacobian['zdot', 'x'] = 0.0\n jacobian['zdot', 'y'] = zdot_y\n jacobian['zdot', 'z'] = 0.0\n\n\n```\n\n\n```python\nnum_nodes = 3\n\np = om.Problem(model=om.Group())\n\np.model.add_subsystem('ode', RobertsonODE(num_nodes=num_nodes), promotes=['*'])\n\np.setup(force_alloc_complex=True)\n\np.set_val('x', [10., 100., 1000.])\np.set_val('y', [1, 0.1, 0.01])\np.set_val('z', [1., 2., 3.])\n\np.run_model()\ncpd = p.check_partials(method='cs', compact_print=True)\n```\n\n\n```python\nfrom dymos.utils.testing_utils import assert_check_partials\n\nassert_check_partials(cpd)\n```\n\n\n```python\nfrom openmdao.utils.assert_utils import assert_near_equal\n\nassert_near_equal(p.get_val('xdot'), [9999.6, 1996., 260.])\n```\n\n\n```python\nfrom openmdao.utils.assert_utils import assert_near_equal\n\nassert_near_equal(p.get_val('ydot'), [-3.00099996E7, -3.01996E5, -3.26E3])\n```\n\n\n```python\nfrom openmdao.utils.assert_utils import assert_near_equal\n\nassert_near_equal(p.get_val('zdot'), [3.0E7, 3.0E5, 3.0E3])\n```\n\n## Building and running the problem\n\n\n```python\nimport openmdao.api as om\nimport dymos as dm\n\ndef robertson_problem(t_final=1.):\n #\n # Initialize the Problem\n #\n p = om.Problem(model=om.Group())\n \n #\n # Create a trajectory and add a phase to it\n #\n traj = p.model.add_subsystem('traj', dm.Trajectory())\n\n phase = traj.add_phase('phase0',\n dm.Phase(ode_class=RobertsonODE,\n transcription=dm.GaussLobatto(num_segments=50)\n ))\n\n #\n # Set the variables\n #\n phase.set_time_options(fix_initial=True, fix_duration=True)\n\n phase.add_state('x0', fix_initial=True, fix_final=False, rate_source='xdot', targets='x')\n phase.add_state('y0', fix_initial=True, fix_final=False, rate_source='ydot', targets='y')\n phase.add_state('z0', fix_initial=True, fix_final=False, rate_source='zdot', targets='z')\n\n #\n # Setup the Problem\n #\n p.setup(check=True)\n\n #\n # Set the initial values\n #\n p['traj.phase0.t_initial'] = 0.0\n p['traj.phase0.t_duration'] = t_final\n\n p.set_val('traj.phase0.states:x0', phase.interp('x0', ys=[1.0, 0.7]))\n p.set_val('traj.phase0.states:y0', phase.interp('y0', ys=[0.0, 1e-5]))\n p.set_val('traj.phase0.states:z0', phase.interp('z0', ys=[0.0, 0.3]))\n\n return p\n\n```\n\n\n```python\n# just set up the problem, test it elsewhere\np = robertson_problem(t_final=40)\n\np.run_model()\n\np_sim = p.model.traj.simulate(method='LSODA')\n```\n\n\n```python\nfrom openmdao.utils.assert_utils import assert_near_equal\n\nassert_near_equal(p_sim.get_val('traj.phase0.timeseries.states:x0')[-1], 0.71583161, tolerance=1E-5)\n```\n\n\n```python\nfrom openmdao.utils.assert_utils import assert_near_equal\n\nassert_near_equal(p_sim.get_val('traj.phase0.timeseries.states:y0')[-1], 9.18571144e-06, tolerance=1E-1)\n```\n\n\n```python\nfrom openmdao.utils.assert_utils import assert_near_equal\n\nassert_near_equal(p_sim.get_val('traj.phase0.timeseries.states:z0')[-1], 0.2841592, tolerance=1E-5)\n```\n\n\n```python\nimport matplotlib.pyplot as plt\n\nt_sim = p_sim.get_val('traj.phase0.timeseries.time')\n\nstates = ['x0', 'y0', 'z0']\nfig, axes = plt.subplots(len(states), 1)\nfor i, state in enumerate(states):\n axes[i].plot(t_sim, p_sim.get_val(f'traj.phase0.timeseries.states:{state}'), '-')\n axes[i].set_ylabel(state[0])\naxes[-1].set_xlabel('time (s)')\nplt.tight_layout()\nplt.show()\n```\n", "meta": {"hexsha": "61aacf5536339726a5e7a93845d3645285a99561", "size": 12580, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/dymos_book/examples/robertson_problem/robertson_problem.ipynb", "max_stars_repo_name": "yonghoonlee/dymos", "max_stars_repo_head_hexsha": "602109eee4a1b061444dd2b45c7b1ed0ac1aa0f4", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/dymos_book/examples/robertson_problem/robertson_problem.ipynb", "max_issues_repo_name": "yonghoonlee/dymos", "max_issues_repo_head_hexsha": "602109eee4a1b061444dd2b45c7b1ed0ac1aa0f4", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 9, "max_issues_repo_issues_event_min_datetime": "2021-05-24T15:14:37.000Z", "max_issues_repo_issues_event_max_datetime": "2021-06-28T21:12:55.000Z", "max_forks_repo_path": "docs/dymos_book/examples/robertson_problem/robertson_problem.ipynb", "max_forks_repo_name": "yonghoonlee/dymos", "max_forks_repo_head_hexsha": "602109eee4a1b061444dd2b45c7b1ed0ac1aa0f4", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.0238663484, "max_line_length": 1082, "alphanum_fraction": 0.5222575517, "converted": true, "num_tokens": 2304, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.7217432182679957, "lm_q1q2_score": 0.4195515634307591}} {"text": "# Synthetic Data Experiments (Visualizations)\n\n\n```julia\ninclude(\"src/Misc.jl\")\nusing .Misc\n```\n\n\n```julia\nimport PyPlot, Seaborn\nimport Suppressor: @suppress_err\n```\n\n\n```julia\nMARKER = [\"o\", \"v\", \"d\", \"X\", \"p\", \"*\", \"P\"];\nPATH = \"./results/synt\";\nIMG = \"./img/synt\";\n```\n\n## On Hyperparameters of BAM\n\n\\begin{equation}\nX^{(1)} = \\left(\\begin{array}{cccc} \n 2 & 1 & 1 & 0 \\\\\n 0 & 0 & 1 & 2 \\\\\n 0 & 0 & 1 & 1 \n \\end{array} \\right)\n\\end{equation}\n\n\n```julia\nresult = load_json(\"$PATH/effective_params.json\")\n\nR = Int.(result[\"R\"])\nAs = Float64.(result[\"a\"])\nd\u2091\u209a = Int.(result[\"EP\"])\nlog_PS = hcat(result[\"log_PS\"]...);\nlog_PX = logsumexp(log_PS,1);\n```\n\n\n```julia\n@suppress_err begin\nfig, ax = PyPlot.subplots(1, 1+length(As); figsize=(14,3))\n\nax[1].set_ylabel(\"# of \\$ S \\$\")\nfor (i,a) \u2208 enumerate(As)\n ax[i].set_title(\"\\$ a = $a \\$\")\n ax[i].hist(log_PS[:,i]; bins=200, label=\"\\$ \\\\log \\\\pi(S) \\$\")\n ax[i].axvline(log_PX[i]; label=\"\\$ \\\\log \\\\mathcal{L}_X \\$\", linestyle=\"-.\", linewidth=2.0, color=\"red\")\n ax[i].set_xlabel(\"\\$ \\\\log \\\\pi(S) \\$\")\n ax[i].legend()\nend\n\nax[end].set_title(\"Effective Parameters\")\nax[end].hist(d\u2091\u209a; bins=200, label=\"\\$ d_{EP} \\$\")\nax[end].set_xlabel(\"\\$ d_{EP}\\$\")\nax[end].invert_xaxis()\nax[end].legend()\n\nPyPlot.savefig(\"$IMG/alloc_tensor_dist.pdf\", bbox_inches=\"tight\");\nend\n```\n\n## Comparison of SMC, VB and exact enumeration on toy data\n\n\\begin{equation}\nX^{(1)} = \\left(\\begin{array}{cccc} \n 2 & 1 & 1 & 0 \\\\\n 0 & 0 & 1 & 2 \\\\\n 0 & 0 & 1 & 1 \n \\end{array} \\right)\n\\end{equation}\n\n\n```julia\nresult = load_json(\"$PATH/marginal_lkhd.json\")\n\nAs = Float64.(result[\"a\"])\nRs = Int.(result[\"R\"])\n\nlog_PX = hcat(result[\"log_PX\"]...)\nlog_PX_smc = hcat(result[\"log_PX_smc\"]...)\nlog_PX_vb = hcat(result[\"log_PX_vb\"]...);\n```\n\n\n```julia\nlog_PR = log_PX .- logsumexp(log_PX,dims=1);\nlog_PR_smc = log_PX_smc .- logsumexp(log_PX_smc,dims=1);\nlog_PR_vb = log_PX_vb .- logsumexp(log_PX_vb,dims=1);\n```\n\n\n```julia\n@suppress_err begin\nfig, ax = PyPlot.subplots(2, 3; figsize=(12,7.5))\n\nfor (r,R) in enumerate(Rs)\n ax[1,1].semilogx(As,log_PX[r,:], label=\"K = $R\", marker=MARKER[r])\n ax[2,1].semilogx(As,log_PR[r,:], label=\"K = $R\", marker=MARKER[r])\n ax[1,2].semilogx(As,log_PX_smc[r,:], label=\"K = $R\", marker=MARKER[r])\n ax[2,2].semilogx(As,log_PR_smc[r,:], label=\"K = $R\", marker=MARKER[r])\n ax[1,3].semilogx(As,log_PX_vb[r,:], label=\"K = $R\", marker=MARKER[r])\n ax[2,3].semilogx(As,log_PR_vb[r,:], label=\"K = $R\", marker=MARKER[r])\nend\nax[1,1].set_ylabel(\"\\$ \\\\log \\\\mathcal{L}_X(K) \\$\");\nax[2,1].set_ylabel(\"\\$ \\\\log \\\\mathrm{Pr}(K \\\\mid X) \\$\");\nax[1,1].set_title(\"Marginal Log-Likelihood\");\nax[1,2].set_title(\"Marginal Log-Likelihood (SMC)\");\nax[1,3].set_title(\"Marginal Log-Likelihood (ELBO)\");\nax[2,1].set_title(\"Posterior Log Odds\");\nax[2,2].set_title(\"Posterior Log Odds (SMC)\");\nax[2,3].set_title(\"Posterior Log Odds (ELBO)\");\n \nforeach(i -> ax[2,i].set_xlabel(\"\\$ a \\$\"), 1:3);\nforeach(i -> ax[1,i].set_xlim(ax[1,1].get_xlim()), 1:3);\nforeach(i -> ax[2,i].set_ylim(-3), 1:3);\nfor i \u2208 1:2, j \u2208 1:3; ax[i,j].legend() end\n\nPyPlot.savefig(\"$IMG/marginal_lkhd.pdf\", bbox_inches=\"tight\");\nend\n```\n\n\\begin{equation}\nX^{(2)} = \\left(\\begin{array}{ccc} \n 4 & 3 & 0 \\\\\n 0 & 0 & 3 \\\\\n 0 & 0 & 3 \n \\end{array} \\right)\n\\end{equation}\n\n\n```julia\nresult = load_json(\"$PATH/marginal_lkhd2.json\")\n\nAs = Float64.(result[\"a\"])\nRs = Int.(result[\"R\"])\n\nlog_PX = hcat(result[\"log_PX\"]...)\nlog_PX_smc = hcat(result[\"log_PX_smc\"]...)\nlog_PX_vb = hcat(result[\"log_PX_vb\"]...);\n```\n\n\n```julia\nlog_PR = log_PX .- logsumexp(log_PX,dims=1);\nlog_PR_smc = log_PX_smc .- logsumexp(log_PX_smc,dims=1);\nlog_PR_vb = log_PX_vb .- logsumexp(log_PX_vb,dims=1);\n```\n\n\n```julia\n@suppress_err begin\nfig, ax = PyPlot.subplots(2, 3; figsize=(12,7.5))\n\nfor (r,R) in enumerate(Rs)\n ax[1,1].semilogx(As,log_PX[r,:], label=\"K = $R\", marker=MARKER[r])\n ax[2,1].semilogx(As,log_PR[r,:], label=\"K = $R\", marker=MARKER[r])\n ax[1,2].semilogx(As,log_PX_smc[r,:], label=\"K = $R\", marker=MARKER[r])\n ax[2,2].semilogx(As,log_PR_smc[r,:], label=\"K = $R\", marker=MARKER[r])\n ax[1,3].semilogx(As,log_PX_vb[r,:], label=\"K = $R\", marker=MARKER[r])\n ax[2,3].semilogx(As,log_PR_vb[r,:], label=\"K = $R\", marker=MARKER[r])\nend\nax[1,1].set_ylabel(\"\\$ \\\\log \\\\mathcal{L}_X(K) \\$\");\nax[2,1].set_ylabel(\"\\$ \\\\log \\\\mathrm{Pr}(K \\\\mid X) \\$\");\nax[1,1].set_title(\"Marginal Log-Likelihood\");\nax[1,2].set_title(\"Marginal Log-Likelihood (SMC)\");\nax[1,3].set_title(\"Marginal Log-Likelihood (ELBO)\");\nax[2,1].set_title(\"Posterior Log Odds\");\nax[2,2].set_title(\"Posterior Log Odds (SMC)\");\nax[2,3].set_title(\"Posterior Log Odds (ELBO)\");\n \nforeach(i -> ax[2,i].set_xlabel(\"\\$ a \\$\"), 1:3);\nforeach(i -> ax[1,i].set_xlim(ax[1,1].get_xlim()), 1:3);\nforeach(i -> ax[2,i].set_ylim(-3.5,0.0), 1:3);\nfor i \u2208 1:2, j \u2208 1:3; ax[i,j].legend() end\n\nPyPlot.savefig(\"$IMG/marginal_lkhd2.pdf\", bbox_inches=\"tight\");\nend\n```\n\n## Posterior Distribution of the number of tokens $S_+$\n\n\\begin{equation}\nX^{(3)} = \\left(\\begin{array}{cc} \n 3 & ? \\\\\n 3 & 3 \\\\ \n \\end{array} \\right)\n\\end{equation}\n\n\n```julia\nresult\u2081 = load_json(\"$PATH/Sp_dist.json\")\n\nTs = Int.(result\u2081[\"T\"])\nRs = Int.(result\u2081[\"R\"])\nlog_PT = hcat(result\u2081[\"log_PT\"]...);\n```\n\n\n```julia\nfig, ax = PyPlot.subplots(1, 1; figsize=(5,3))\nfor (r,R) in enumerate(Rs)\n ax.plot(Ts,log_PT[r,:], label=\"K = $R\", marker=MARKER[r])\nend\nax.set_title(\"Distribution of \\$ \\\\mathbf{S}_+ \\$\");\nax.set_ylabel(\"\\$ \\\\log \\\\mathrm{Pr}(\\\\mathbf{S}_+ = T, X^{(3)} \\\\mid K) \\$\");\nax.set_xlabel(\"\\$ T \\$\")\nax.legend();\n\nPyPlot.savefig(\"$IMG/Sp_dist.pdf\", bbox_inches=\"tight\");\n```\n\n\\begin{equation}\nX^{(4)} = \\left(\\begin{array}{ccc} \n 4 & ? \\\\\n 4 & 1\n \\end{array} \\right)\n\\end{equation}\n\n\n```julia\nresult\u2082 = load_json(\"$PATH/Sp_dist2.json\")\n\nTs = Int.(result\u2082[\"T\"])\nRs = Int.(result\u2082[\"R\"])\nlog_PT = hcat(result\u2082[\"log_PT\"]...);\n```\n\n\n```julia\nfig, ax = PyPlot.subplots(1, 1; figsize=(5,3))\nfor (r,R) in enumerate(Rs)\n ax.plot(Ts,log_PT[r,:], label=\"K = $R\", marker=MARKER[r])\nend\nax.set_title(\"Distribution of \\$ \\\\mathbf{S}_+ \\$\");\nax.set_ylabel(\"\\$ \\\\log \\\\mathrm{Pr}(\\\\mathbf{S}_+ = T, X^{(4)} \\\\mid K) \\$\");\nax.set_xlabel(\"\\$ T \\$\")\nax.legend();\n\nPyPlot.savefig(\"$IMG/Sp_dist2.pdf\", bbox_inches=\"tight\");\n```\n\n## Model Order Selection for CP/PARAFAC\n\n### Confusion Matrix\n\n\n```julia\nresult = load_json(\"$PATH/parafac_confusion.json\")\n\nI, J, K, T = result[\"I\"], result[\"J\"], result[\"K\"], result[\"T\"]\nRs = Int.(result[\"R\"])\nAs = Float64.(result[\"a\"])\n\nR_true = Int.(result[\"R_true\"])\nR_smc = Int.(result[\"R_smc\"])\nR_vb = Int.(result[\"R_vb\"]);\n```\n\n\n```julia\nconf_smc = zeros(Int,length(Rs),length(Rs));\nconf_vb = zeros(Int,length(Rs),length(Rs));\n```\n\n\n```julia\nfor i \u2208 1:length(R_true)\n conf_smc[R_true[i],R_smc[i]] += 1\n conf_vb[R_true[i],R_vb[i]] += 1\nend\n```\n\n\n```julia\n@suppress_err begin\nfig, ax = PyPlot.subplots(1; figsize=(4,3))\nSeaborn.heatmap(conf_vb; annot=true, cmap=\"YlGnBu\", cbar=false, xticklabels=Rs, yticklabels=Rs, linewidths=.4, ax=ax)\n\nax.set_title(\"Confusion Matrix of Model Orders\")\nax.set_ylabel(\"\\$ R_\\\\mathrm{true} \\$\")\nax.set_xlabel(\"\\$ R_\\\\mathrm{VB} \\$\")\n\nPyPlot.savefig(\"$IMG/parafac_confusion_vb.pdf\", bbox_inches=\"tight\");\nend\n```\n\n\n```julia\n@suppress_err begin\nfig, ax = PyPlot.subplots(1; figsize=(4,3))\nSeaborn.heatmap(conf_smc; annot=true, cmap=\"YlGnBu\", cbar=false, xticklabels=Rs, yticklabels=Rs, linewidths=.4, ax=ax)\n\nax.set_title(\"Confusion Matrix of Model Orders\")\nax.set_ylabel(\"\\$ R_\\\\mathrm{true} \\$\")\nax.set_xlabel(\"\\$ R_\\\\mathrm{SMC} \\$\")\n\nPyPlot.savefig(\"$IMG/parafac_confusion_smc.pdf\", bbox_inches=\"tight\");\nend\n```\n\n### An Example\n\n\n```julia\nresult = load_json(\"$PATH/parafac_model_selection.json\")\n\nI, J, K, R_true = result[\"I\"], result[\"J\"], result[\"K\"], result[\"R_true\"]\nRs = Int.(result[\"R\"])\na = result[\"a\"]\nlog_PX_smc, log_PX_vb = Float64.(result[\"log_PX_smc\"]), Float64.(result[\"log_PX_vb\"]);\n```\n\n\n```julia\nPyPlot.figure(figsize=(5,3))\n\nPyPlot.axvline(R_true; label=\"G. Rank\", color=\"black\", linestyle=\"--\")\nPyPlot.plot(Rs, log_PX_vb; label=\"ELBO\", marker=MARKER[5]);\nPyPlot.plot(Rs, log_PX_smc; label=\"SIS/R\", marker=MARKER[6]);\n\nPyPlot.xlabel(\"\\$R\\$ (Model order)\")\nPyPlot.ylabel(\"\\$ \\\\log \\\\mathcal{L}_X(R) \\$\")\nPyPlot.title(\"Model Scoring Example (PARAFAC)\");\nPyPlot.legend();\n\nPyPlot.savefig(\"$IMG/cp_model_selection.pdf\", bbox_inches=\"tight\");\n```\n\n### Runtimes\n\n\n```julia\nresult = load_json(\"$PATH/parafac_runtime.json\")\n\nT, P, R, a = result[\"T\"], result[\"P\"], result[\"R\"], result[\"a\"]\nIJK = Array{Array{Int,1},1}(result[\"IJK\"])\nsmc_times = hcat(result[\"smc\"]...)\nvb_times = hcat(result[\"vb\"]...);\n```\n\n\n```julia\nPyPlot.figure(figsize=(5,3))\nPyPlot.plot(1:length(IJK),nanmean(vb_times,2); label=\"VB\", marker=MARKER[5])\nPyPlot.plot(1:length(IJK),nanmean(smc_times,2); label=\"SIS/R\", marker=MARKER[6])\nPyPlot.title(\"Average Runtimes (PARAFAC)\")\nPyPlot.ylabel(\"sec.\")\nPyPlot.xlabel(\" \\$ I_1 \\\\times I_2 \\\\times I_3 \\$ (Dimensions of \\$X\\$)\")\nPyPlot.xticks(1:2:length(IJK),[\"\\$ {$(4*i)}^3 \\$\" for i \u2208 1:2:length(IJK)])\nPyPlot.legend()\n\nPyPlot.savefig(\"$IMG/runtimes.pdf\", bbox_inches=\"tight\");\n```\n", "meta": {"hexsha": "821b7ff532281bf70ce8907264c07798ed29265f", "size": 533964, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Visualizations-Synthetic.ipynb", "max_stars_repo_name": "atcemgil/bam", "max_stars_repo_head_hexsha": "dadaae493e52e38a421dbe0359d94fe23adb297c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2019-08-07T18:12:26.000Z", "max_stars_repo_stars_event_max_datetime": "2020-02-26T12:43:14.000Z", "max_issues_repo_path": "Visualizations-Synthetic.ipynb", "max_issues_repo_name": "atcemgil/bam", "max_issues_repo_head_hexsha": "dadaae493e52e38a421dbe0359d94fe23adb297c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2019-03-22T16:52:56.000Z", "max_issues_repo_issues_event_max_datetime": "2019-03-22T16:52:56.000Z", "max_forks_repo_path": "Visualizations-Synthetic.ipynb", "max_forks_repo_name": "atcemgil/bam", "max_forks_repo_head_hexsha": "dadaae493e52e38a421dbe0359d94fe23adb297c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2019-03-20T14:45:18.000Z", "max_forks_repo_forks_event_max_datetime": "2019-03-29T13:51:39.000Z", "avg_line_length": 802.9533834586, "max_line_length": 134862, "alphanum_fraction": 0.9481257163, "converted": true, "num_tokens": 3409, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.640635868562172, "lm_q1q2_score": 0.4195490535566917}} {"text": "# Lecci\u00f3n 2: Los postulados de la mec\u00e1nica cu\u00e1ntica #\n\nLa mec\u00e1nica cu\u00e1ntica es el marco te\u00f3rico en el que los cient\u00edficos trabajan el desarrollo de teor\u00edas f\u00edsicas relacionadas con el mundo microsc\u00f3pico. Este marco de trabajo matem\u00e1tico, por si solo, no indica las leyes que un sistema deber\u00eda cumplir, sin embargo, ofrece el panorama de c\u00f3mo este sistema puede cambiar.\n\n\nLos postulados de la mec\u00e1nica cu\u00e1ntica son aquellos que establecen la relaci\u00f3n entre el mundo f\u00edsico y el formalismo matem\u00e1tico de la mec\u00e1nica cu\u00e1ntica. \nEs importante tener en cuenta que el desarrollo de estos postulados se dio a traves del ensayo y el error, de forma tal que estos parecen partir de motivaciones no muy claras, sin embargo, nuestro objetivo en esta sesi\u00f3n es aprender el c\u00f3mo aplicarlos y cu\u00e1ndo hacerlo.\n\n**SUPER IMPORTANTE:** En este ejemplo intentaremos presentar al lector el formalismo necesario para estudiar la computaci\u00f3n cu\u00e1ntica en forma te\u00f3rica, as\u00ed como una posible intuici\u00f3n f\u00edsica sobre la misma. Esta sesi\u00f3ne st\u00e1 disponible en nuestro servidor de MyBinder.\n\n

                                        \n \n \n \n

                                        \n\n\nPrimero importaremos alguna librer\u00edas que son necesarias para desarrollar nuestro experimento\n\n\n\n```python\nfrom qiskit import QuantumRegister, ClassicalRegister\nfrom qiskit import QuantumCircuit, execute, Aer\nfrom qiskit.circuit import Parameter\nfrom qiskit.visualization import plot_histogram,state_visualization\nfrom qiskit.quantum_info.operators import Operator\nimport matplotlib.pyplot as plt\nimport numpy as np\n\npi=np.pi\n```\n\nAhora definimos una funci\u00f3n con un parametro $\\phi$ que nos servir\u00e1 para hacer una evoluci\u00f3n temporal en una simulaci\u00f3n de un interfer\u00f3metro de Mach-Zehnder.\n\n\n```python\ndef MZ_interferometer(phi):\n ## La variable phi esta relacionada al cambio de fase en uno de los dos caminos\n MZ_circuit=QuantumCircuit(1)\n \n ## Se define una compuerta de Hadamard la cual cumple la funci\u00f3n de beam splitter\n MZ_circuit.h(0)\n \n ## Luego se hace una diferencia de camino a traves del uno de los caminos del campo,\n ## En este caso el que esta en el estado que llamamos |1>\n \n MZ_circuit.p(phi,0)\n \n ## finalmente ponemos el siguiente BS para terminar con el interfer\u00f3metro\n MZ_circuit.h(0)\n \n \n MZ_gate=MZ_circuit.to_gate()\n MZ_gate.name = \"MZ-Interferometer\"\n \n return MZ_gate\n```\n\n\n### Postulado 1 ###\n\nEl primer postulado de la mec\u00e1nica cu\u00e1ntica nos indica el espacio matem\u00e1tico en el que, dado un sistema, se trabaja el formalismo de la mec\u00e1nica cu\u00e1ntica.\n\u00c9ste dicta que:\n>Dado un sistema f\u00edsico aislado hay un espacio vectorial complejo con un producto interno definido (un espacio de Hilbert) conocido como ***espacio de estados*** del sistema. El sistema en este espacio se representa a partir de un ***vector de estado***, el cual es unitario en este espacio, es decir.\n\n$$|\\psi\\rangle = \\sum_k c_k|k\\rangle \\hspace{2em} \\text{donde} \\hspace{2em} \\langle\\psi|\\psi\\rangle = \\sum_k |c_k|^2 = 1.$$\n\nLa notaci\u00f3n que se utiliza en la mec\u00e1nica cu\u00e1ntica para escribir estos estados se conoce como la notaci\u00f3n bra-ket de Dirac. En esta notaci\u00f3n el estado bra se escribe, $\\langle \\psi |$ y su correspondiente ket es $|\\psi\\rangle$.\n\nEn el caso de la computaci\u00f3n cu\u00e1ntica nosotros vamos a estar interesados en un espacio de estados particular, un espacio de en el que el sistema tiene dos estados bien definidos $|0\\rangle$ y $|1\\rangle$, este espacio de estados lo llamaremos el espacio de qubit. En este caso los estados $|0\\rangle$ y $|1\\rangle$ son la base de nuestro espacio vectorial, y dado que estos se definen ortonormales, podemos representarlos en una forma vectorial\n\n$$|0\\rangle = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\hspace{2em} \\text{y} \\hspace{2em} |1\\rangle = \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}$$\n\nPor lo tanto, el ket asociado a un estado arbitrario $|\\psi\\rangle$ se puede representar de la forma\n\n$$ |\\psi\\rangle = \\alpha|0\\rangle + \\beta|1\\rangle = \\begin{bmatrix} \\alpha \\\\ \\beta \\end{bmatrix} $$\n\nLos coeficientes $\\alpha$ y $\\beta$ representan las amplitudes de los estados $|0\\rangle$ y $|1\\rangle$ respectivamente. Estos n\u00fameros Estan relacionados con las *probabilidades* de encontrar el sistema, en este caso el qubit, en un estado o el otro. M\u00e1s en detalle, las probabilidades de encontrar el qubit $|\\psi\\rangle$ en el estado $|0\\rangle$ o $|1\\rangle$ despu\u00e9s de realizar la medici\u00f3n son\n\n$$p(|0\\rangle) = |\\alpha|^2 \\hspace{2em}\\text{y}\\hspace{2em} p(|1\\rangle) = |\\beta|^2$$\n\nAntes de continuar hagamos un peque\u00f1o experimento.\n\nPreparemos un estado de una forma part\u00edcular, en concreto vamos a prepararlo en el estado\n\n$$|\\psi\\rangle = \\frac{1}{\\sqrt{2}}\\left(|0\\rangle + |1\\rangle\\right) $$\n\n\n```python\n## Inicializamos los registros necesarios\nqr=QuantumRegister(1)\ncr=ClassicalRegister(1)\nqc=QuantumCircuit(qr,cr)\n\n## Aplicamos una compuerta de Hadamard\nqc.h(qr[0])\nqc.measure(qr[0],cr[0])\nqc.draw(output='mpl')\n```\n\nDe acuerdo con lo que vimos anteriormente entonces $\\alpha = \\beta = \\frac{1}{\\sqrt{2}}$. Por lo tanto, las probabilidades de que el qubit est\u00e9 en los estados $|0\\rangle$ o $|1\\rangle$ son $p(|0\\rangle) = \\frac{1}{2}$ y $p(|1\\rangle) = \\frac{1}{2}$ respectivamente. Ahora realizamos una medici\u00f3n.\n\n\n```python\n## Usamos el simulador qasm\nbackend = Aer.get_backend('qasm_simulator')\nresults = execute(qc,backend,shots=10000).result().get_counts()\nplot_histogram(results)\n```\n\nLo que haremos ahora es agregar una fase de la forma $e^{i\\phi}$ en la amplitud del estado $|1\\rangle$. Por lo cual el estado toma la forma\n\n$$|\\psi\\rangle = \\frac{1}{\\sqrt{2}}\\left(|0\\rangle + i|1\\rangle\\right) .$$ \n\n\n\n```python\nqr=QuantumRegister(1)\ncr=ClassicalRegister(1)\nqc=QuantumCircuit(qr,cr)\n\n\nqc.h(qr[0])\nqc.p(pi/2,qr[0])\nqc.measure(qr[0],cr[0])\nqc.draw(output='mpl')\n```\n\nMidiendo nuevamente tenemos.\n\n\n```python\nbackend = Aer.get_backend('qasm_simulator')\nresults = execute(qc,backend,shots=10000).result().get_counts()\nplot_histogram(results)\n```\n\nEs decir que, as\u00ed se a\u00f1adiera un factor de fase las distribuciones de probabilidad luego de medir son iguales, por lo tanto, los n\u00fameros $\\alpha$ y $\\beta$ tienen que ser n\u00fameros de la forma $\\alpha = |\\alpha|e^{i\\theta_\\alpha}$ y $\\beta = |\\beta|e^{i\\theta_\\beta}$, es decir, estos n\u00fameros pertenecen a $\\mathbb{C}$. \n\nCon base a este resultado, podemos escribir apropiadamente el bra asociado al estado $|\\psi\\rangle$. Este es\n\n$$\\langle \\psi | = \\alpha^* \\langle 0 | + \\beta^*\\langle 1 | = \\begin{bmatrix}\\alpha^* ,\\beta^*\\end{bmatrix} $$\n\nEstos estados cumplen con las relaciones\n\n$$\\langle 0|0 \\rangle =\\begin{bmatrix}1 ,0\\end{bmatrix}\\begin{bmatrix}1 \\\\0\\end{bmatrix} = 1$$\n\n$$\\langle 1|1\\rangle = \\begin{bmatrix}0 ,1\\end{bmatrix}\\begin{bmatrix}0 \\\\1\\end{bmatrix} = 1$$\n\n$$\\langle 0|1\\rangle = \\begin{bmatrix}1 ,0\\end{bmatrix}\\begin{bmatrix}0 \\\\1\\end{bmatrix} = 0$$\n\n$$\\langle 1|0\\rangle = \\begin{bmatrix}0 ,1\\end{bmatrix}\\begin{bmatrix}1 \\\\0\\end{bmatrix} = 0$$\n\n\nPor lo cual el Bra-Ket del estado $|\\psi\\rangle$, el cual representa el producto punto en el espacio de estados, se escribe:\n\n$$\\langle\\psi|\\psi\\rangle = \\left( \\alpha^* \\langle 0 | + \\beta^*\\langle 1 | \\right) \\left(\\alpha|0\\rangle + \\beta|1\\rangle\\right)$$\n\n$$ \\langle\\psi|\\psi\\rangle =|\\alpha|^2 + |\\beta|^2 = 1 $$\n\nO vectorialmente\n\n$$\\langle\\psi|\\psi\\rangle = \\begin{bmatrix}\\alpha^* ,\\beta^*\\end{bmatrix}\\begin{bmatrix}\\alpha \\\\\\beta\\end{bmatrix} = |\\alpha|^2 + |\\beta|^2 = 1$$\n\nAdicionalmente, as\u00ed como definimos el Bra-ket podemos definir un Ket-Bra, que tendr\u00eda la forma\n\n$$|\\psi\\rangle\\langle\\psi| = \\begin{bmatrix}\\alpha\\alpha^*&\\alpha\\beta^*\\\\\\beta\\alpha^*&\\beta\\beta^*\\end{bmatrix}$$\n\nEste tipo de producto, que es similar al producto tensorial, se conoce en este contexto como el producto de Kronecker.\n\nDado que los n\u00fameros $\\alpha$ y $\\beta$ son complejos entonces un sistema de dos estados, como es el caso de un qubit, puede representarse sobre la superficie de una esfera de radio $1$, esta esfera se conoce como la *Esfera de Bloch*. \n\n\n\n### Postulado 2 ###\n\n>La evoluci\u00f3n temporal de un sistema cu\u00e1ntico ***cerrado*** se describe a traves de una ***transformaci\u00f3n unitaria***. esto implica que si en un tiempo $t_{1}$ el sistema se encuentra en el estado $|\\psi\\rangle$. Luego, en un instante $t_2$ el sistema estar\u00e1 en el estado $|\\psi '\\rangle$. Esta evoluci\u00f3n se da por un operador unitario $U(t_1,t_2)$, que solo depende de los dos tiempos ya mencionados. Esta evoluci\u00f3n se puede escribir como\n$$ |\\psi'\\rangle = U |\\psi\\rangle. $$\n\nAs\u00ed mismo, teniendo en mente el operador Hamiltoniano de un sistema cerrado, se puede escribir la evoluci\u00f3n temporal de este a traves de la ecuaci\u00f3n de Schr\u00f6dinger, \n$$ i\\hbar \\frac{d|\\psi\\rangle}{dt} = \\mathcal{H} |\\psi\\rangle $$\n\nComo hemos visto, as\u00ed como los estados tienen una representaci\u00f3n vectorial, los operadores que actuan sobre los estados se pueden representar matricialmente para que modifiquen estos estados. En el caso de la computaci\u00f3n cu\u00e1ntica cada una de las compuertas que se usan tiene una representaci\u00f3n matricial. El primer operador que veremos es el operador identidad $\\mathbb{1}$. Este operador no modifica el estado de ninguna forma, y se puede representar matricialmente como\n\n$$\\mathbb{1} =\\begin{bmatrix}1 & 0 \\\\ 0 & 1\\end{bmatrix} $$\n\n \nOtra transformaci\u00f3n se representa mediante el s\u00edmbolo $X$, esta transformaci\u00f3n lo que hace es intercambiar los elementos de la base, es decir\n\n$$\\begin{equation}X|0\\rangle\\rightarrow|1\\rangle, \\hspace{2em}X|1\\rangle\\rightarrow|0\\rangle\\end{equation}$$\n\nMatricialmente se puede representar como\n\n$$\\begin{equation}X=\\begin{bmatrix}0 & 1 \\\\ 1 & 0\\end{bmatrix}\\end{equation}$$\n\nEsta transformaci\u00f3n se llama phase-flip y se puede representar mediante el operador $Z$. esta cambia la fase relativa entre los estados $|0\\rangle$ y $|1\\rangle$ en un factor de $\\pi$, es decir\n\n$$\\begin{equation}Z|0\\rangle\\rightarrow|0\\rangle, \\hspace{2em}\tZ|1\\rangle\\rightarrow -|1\\rangle\\end{equation} $$\n\nMatricialmente se representa por\n\n$$\\begin{equation}Z=\\begin{bmatrix}1 & 0 \\\\ 0 & -1\\end{bmatrix} \\end{equation}$$\n\nLos operadores $X$ y $Z$ son dos de los conocidos operadores de Pauli, Estos se representan por las llamadas matrices de Pauli. Adicionalmente hay otra transformaci\u00f3n que viene de la combinaci\u00f3n entre $X$ y $Z$ de la forma $ZX$, esta act\u00faa sobre el estado de forma tal que \n\n$$\\begin{equation}ZX|0\\rangle\\rightarrow-|1\\rangle, \\hspace{2em}\tZX|1\\rangle\\rightarrow |0\\rangle.\\end{equation} $$\n\nSe puede representar matricialmente de la forma\n\n$$\\begin{equation}ZX=\\begin{bmatrix}0 & 1 \\\\ -1 & 0\\end{bmatrix} = iY\\end{equation}$$\n\nCon $Y$ La tercera matriz de Pauli. Esta tiene la forma\n\n$$\\begin{equation}Y=\\begin{bmatrix}0 & -i \\\\ i & 0\\end{bmatrix}\\end{equation}$$\n\nEn este punto es importante definir la operaci\u00f3n $^\\dagger$ que se realiza sobre un operador. Esta operaci\u00f3n implica tomar el operador transpuesto y complejo conjugado, por ejemplo para el operador $Y$ tenemos\n\n$$\\begin{equation}Y=\\begin{bmatrix}0 & -i \\\\ i & 0\\end{bmatrix}\\Rightarrow Y^\\dagger=\\begin{bmatrix}0 & i \\\\ -i & 0\\end{bmatrix}^* = \\begin{bmatrix}0 & -i \\\\ i & 0\\end{bmatrix}\\end{equation}$$\n\nComo vimos, $Y = Y^\\dagger$, esta es una propiedad llamada *Hermiticidad*. Cuando un operador en mec\u00e1nica cu\u00e1ntica es herm\u00edtico, entonces es posible medir los observables asociados a ese operador.\n\nAdem\u00e1s de los operadores de Pauli hay otro par de operadores que son importantes para futuros desarrollos. El primero es el operador de Hadamard, el cual tiene la forma\n\n\n$$H =\\frac{1}{\\sqrt{2}} \\begin{bmatrix}1 & 1\\\\1 & -1 \\end{bmatrix}$$\n\nEsta compuerta lo que hace es rotar los estados en un \u00e1ngulo de 45\u00b0. Es decir\n\n$$|0\\rangle\\rightarrow \\frac{1}{\\sqrt{2}}\\left(|0\\rangle + |1\\rangle\\right)\\hspace{2em} \\text{y}\\hspace{2em}|1\\rangle \\rightarrow \\frac{1}{\\sqrt{2}} \\left(|0\\rangle - |1\\rangle\\right)$$\n\nUn ultimo operador que nos sirve en este momento es el cambio de fase. Este operador lo que hace es a\u00f1adir una fase cuando el sistema se encuentra en el estado $|1\\rangle$. Matricialmente se puede representar como \n\n$$P_\\phi =\\begin{bmatrix}1 & 0\\\\0 & e^{i\\phi} \\end{bmatrix}$$\n\n\nPara ilustrar este principio lo que haremos ser\u00e1 simular un interfer\u00f3metro de Mach-Zehnder como el que se muestra en la siquiente figura.\n\n\n\n\nEn este caso, solo se env\u00eda un fot\u00f3n por la entrada que representa el estado $|0\\rangle$. El estado inicial se puede representar de la forma\n$$ |\\psi_i\\rangle = |0\\rangle$$.\n\nEn esta figura se tiene que\n\n\n\nAhora debemos representar este interfer\u00f3metro mediante un circuito cu\u00e1ntico y encontrar la probabilidad con la que alguno de los medidores va a detectar el fot\u00f3n, es decir, determinar la probabilidad de que est\u00e9 en el estado $|0\\rangle$ o $|1\\rangle$.\nPara esto se tiene el haz que ingresa. Luego, este pasa por un divisor de haz, este divisor de haz lo que hace es dar al fot\u00f3n la posibilidad de seguir su camino (camino $|0\\rangle$) o ser reflejado (camino $|1\\rangle$) con probabilidad de 50\\%. Esto se puede representar mediante una compuerta de Hadamard $H$. Luego de esto, el fot\u00f3n puede o no ser desfasado, la condici\u00f3n de que el desfasador $\\phi$ actue depende de si el fot\u00f3n va o no por el camino $|1\\rangle$. Este desfasador se puede representar por la compuerta de fase $P_\\phi$. Finalmente, sin importar el camino, el haz pasa nuevamente por un divisor de haz, es decir, una nueva compuerta Hadamard. Por lo cual el interfer\u00f3metro se podr\u00eda visualizar de la siguiente forma \n\n\n\n\n```python\n## Primero definiremos el fot\u00f3n que unicia por el camino que definimos como el estado inicial $|0>$\nqr=QuantumRegister(1)\ncr=ClassicalRegister(1)\nqc=QuantumCircuit(qr,cr)\n\n## definimos el parametro\nphi=Parameter('$\\phi$')\n\n## Definimos el espaciamiento deseado\nangulos = np.linspace(0,2*pi,100)\n\n## Aplicamos el interfer\u00f3metro de MZ a nuestro estado inicial\nMZ_I = MZ_interferometer(phi)\n\n\nqc.append(MZ_I,[0])\nqc.measure(qr[0],cr[0])\nqc.decompose().draw(output='mpl')\n```\n\n\nLuego de pasar por el interfer\u00f3metro, el estado final toma la forma\n\n$$|\\psi_f\\rangle = \\cos\\frac{\\phi}{2}|0\\rangle - i\\sin\\frac{\\phi}{2} |1\\rangle$$\n\nPor lo tanto, las probabilidades de medir el estado $|0\\rangle$, es decir, que se haga una detecci\u00f3n en el detector $D_4$ y de medir $|1\\rangle$, es decir, hacer una detecci\u00f3n en el detector $D_5$ son:\n\n$$P_0 = \\cos^2 \\frac{\\phi}{2} \\hspace{2em} \\text{y}\\hspace{2em} P_1 = \\sin^2 \\frac{\\phi}{2}$$\n\nDe ac\u00e1 se ve claramente que \n\n$$ P_0 + P_1 = \\cos^2 \\frac{\\phi}{2} +\\sin^2 \\frac{\\phi}{2} = 1 $$\n\n\n```python\nshots=1024\nbackend = Aer.get_backend('qasm_simulator')\nresultados = execute(qc,backend,parameter_binds=[{phi:angulo}for angulo in angulos],shots=shots).result().get_counts()\n\n\n## Capturamos la probabilidad de en el medidor 1\nP_0 = np.array([resultado.get('0',0)/shots for resultado in resultados])\n\n## Capturamos la probabilidad de en el medidor 2\nP_1 = np.array([resultado.get('1',0)/shots for resultado in resultados])\n\n\n## Graficamos\nplt.title(r'Probabilidad de medici\u00f3n por detector en el interferometro Mach-Zehnder')\nplt.xlabel(r'$\\phi$ (radianes)')\nplt.ylabel(r'$Probabilidades$')\nplt.plot(angulos,P_0,label='P_0',color='k')\nplt.plot(angulos,P_1,label='P_1',color='r')\nplt.plot(angulos,P_0 + P_1,label='P_0 + P_1',color='b')\nplt.legend(bbox_to_anchor = (1, 1))\nplt.show()\n```\n\n### Postulado 3 ###\n\n>Las mediciones a un sistema cu\u00e1ntico se describen a traves de una colecci\u00f3n ${M_{m}}$ de ***operadores de medici\u00f3n***. Estos operadores actuan sobre el es estacio de estados del sistema. El indice $m$ se refiere a las posibles mediciones que pueden resultar en el experimento. Es decir, si un sistema se encuentra en el estado $|\\psi>$ justo antes de la medici\u00f3n, la probabilidad de que se de el resultado $m$ es\n\n$$p(m) = \\langle\\psi|M_m^{\\dagger}M_m|\\psi\\rangle.$$\n\nLuego de la medici\u00f3n el estado del sistema colapsa al estado\n\n$$\\frac{M_m |\\psi\\rangle}{\\sqrt{\\langle\\psi|M_m^{\\dagger}M_m|\\psi\\rangle}}.$$\n\nEs importante tener en cuenta que los operadores de medici\u00f3n satisfacen la relaci\u00f3n de completez\n\n$$\\sum_m M_m^{\\dagger}M_m = I .$$\n\nEsta relaci\u00f3n de completez muestra tambi\u00e9n el hecho de que la suma de las probabilidades de que el sistema este en el estado $m$ sea uno:\n\n$$\\sum_m p(m) = \\sum_m \\langle\\psi|M_m^{\\dagger}M_m|\\psi\\rangle = 1$$\n\nEn el caso de un qubit, tenemos dos operadores de medici\u00f3n, el que mide el estado $|0\\rangle$ que llamamos $M_0$ y el que mide el estado $|1\\rangle$ que llamamos $M_1$. Se representan de la forma\n\n$$M_0= |0\\rangle\\langle 0| = \\begin{bmatrix}1 & 0 \\\\ 0 & 0\\end{bmatrix} \\hspace{2em} \\text{y} \\hspace{2em} M_1= |1\\rangle\\langle 1| = \\begin{bmatrix}0 & 0 \\\\ 0 & 1\\end{bmatrix} $$\n\nEntonces, por ejemplo, tomemos el estado final presentado en el caso del interfer\u00f3metro de Mach-Zehnder\n\n$$|\\psi\\rangle = \\cos\\frac{\\phi}{2}|0\\rangle - i\\sin\\frac{\\phi}{2} |1\\rangle$$\n\nPrimero supongamos que al realizar la medici\u00f3n el estado resulta ser $|0\\rangle$, entonces el estado luego de la medici\u00f3n es\n\n$$|\\psi\\rangle \\rightarrow \\frac{M_0 |\\psi\\rangle}{\\sqrt{\\langle\\psi|M_0^{\\dagger}M_0|\\psi\\rangle}}$$\n\n$$= \\frac{\\cos\\frac{\\phi}{2} |0\\rangle}{\\sqrt{\\cos^2\\frac{\\phi}{2}}} = |0\\rangle$$\n\nDe igual forma, suponiendo que al medir el estado colaps\u00f3 al estado $|1\\rangle$, se tiene \n\n$$|\\psi\\rangle \\rightarrow \\frac{M_1 |\\psi\\rangle}{\\sqrt{\\langle\\psi|M_1^{\\dagger}M_1|\\psi\\rangle}}$$\n\n$$= \\frac{\\sin\\frac{\\phi}{2} |1\\rangle}{\\sqrt{\\sin^2\\frac{\\phi}{2}}} = |1\\rangle$$\n\n\n\n### Postulado 4 ###\n>El espacio de los estados de un sistema f\u00edsico compuesto es el producto tensorial entre los espacios de estados de los componentes de este sistema compuesto. Esto quiere decir que, si tenemos $1,2,...,n$ sistemas que se encuentran en los estados $|\\psi_1>,|\\psi_2>,...,|\\psi_n>$ respectivamente, entonces, la decripci\u00f3n de el sistema f\u00edsico compuesto $|\\Psi>$ ser\u00eda\n\n$$|\\Psi> = |\\psi_1>\\otimes |\\psi_2> \\otimes ... \\otimes |\\psi_n>$$\n\nPara ilustrar este postulado, vamos a pensar en dos sistemas de 1 qubit los cuales vamos a nombrar $|\\psi_1\\rangle$ y $|\\psi_2\\rangle$ por lo cual el sistema completo se puede ver de la forma $|\\Psi\\rangle = |\\psi_1\\rangle \\otimes |\\psi_2\\rangle$. Una base para expandir estos estados es la de {$|0\\rangle$, $|1\\rangle$}, esta base para cada uno de los qubits, entonces, el estado total se puede expandir en la base:\n\n$$ \\{|0\\rangle,|1\\rangle\\} \\otimes \\{|0\\rangle,|1\\rangle\\} = \\{|0\\rangle\\otimes |0\\rangle, |0\\rangle\\otimes |1\\rangle, |1\\rangle\\otimes |0\\rangle, |1\\rangle\\otimes |1\\rangle\\} $$\n\nUsando la notaci\u00f3n $|0\\rangle\\otimes |0\\rangle = |0 0\\rangle$ y as\u00ed para los otros sistemas, donde el primer y segundo n\u00famero corresponden a los sistemas $|\\psi_1\\rangle$ y $|\\psi_2\\rangle$ respectivamente. Este producto se puede visualizar vectorialmente de la forma\n\n$$|0 0\\rangle = |0\\rangle\\otimes |0\\rangle = \\begin{bmatrix}1\\\\0\\end{bmatrix}\\otimes\\begin{bmatrix}1\\\\0\\end{bmatrix} = \\begin{bmatrix}1\\begin{bmatrix}1\\\\0\\end{bmatrix}\\\\0\\begin{bmatrix}1\\\\0\\end{bmatrix}\\end{bmatrix} = \\begin{bmatrix}1\\\\0\\\\0\\\\0\\end{bmatrix}$$\n\nRealizando los otros productos de la base se puede llegar a \n\n$$ |0 0\\rangle = \\begin{bmatrix}1\\\\0\\\\0\\\\0\\end{bmatrix},\\hspace{1em}|0 1\\rangle = \\begin{bmatrix}0\\\\1\\\\0\\\\0\\end{bmatrix},\\hspace{1em}|1 0\\rangle = \\begin{bmatrix}0\\\\0\\\\1\\\\0\\end{bmatrix},\\hspace{1em}|1 1\\rangle = \\begin{bmatrix}0\\\\0\\\\0\\\\1\\end{bmatrix}.$$\n\nSin embargo, como bien sabemos, hay infinitos conjuntos de vectores que pueden ser la base del espacio en el que estamos trabajando, del mismo modo, hay diferentes estados de dos part\u00edculas que pueden funcionar como base para el espacio de Hilbert en el que los posibles estados de estas part\u00edculas reciden. Una base de interes para la computaci\u00f3n cu\u00e1ntica es la conocida *Base de Bell*. La base de Bell para dos qubits representa los posibles estados *entrelazados* que pueden haber entre ellas. Los 4 elementos de la base de Bell para 2 qubits son\n\n$$|\\Psi^+\\rangle_{1,2} = \\frac{1}{\\sqrt{2}}\\left(|01 \\rangle + |10\\rangle\\right)$$\n$$|\\Psi^-\\rangle_{1,2} = \\frac{1}{\\sqrt{2}}\\left(|01 \\rangle - |10\\rangle\\right)$$\n$$|\\Phi^+\\rangle_{1,2} = \\frac{1}{\\sqrt{2}}\\left(|00 \\rangle + |11 \\rangle\\right)$$\n$$|\\Phi^-\\rangle_{1,2} = \\frac{1}{\\sqrt{2}}\\left(|00 \\rangle - |11 \\rangle\\right)$$\n\nComo pueden notar, NO es posible expresar alguno de los anteriores estados de la forma $|\\psi\\rangle_1 \\otimes |\\psi\\rangle_1$, es decir, no se puede expresar como el producto de estados de las part\u00edculas singulares. Este tipo de estados es \u00fanico de la mec\u00e1nica cu\u00e1ntica, y son estados en los que, una vez se haga la medici\u00f3n de una de las dos part\u00edculas, o uno de los dos qubits, el estado del segundo quedar\u00e1 totalmente determinado por el resultado de la medici\u00f3n hecha en el primer qubit.\n\nAhora vamos a cunstruir estos estados, para esto tenemos el siguiente cuadro\n\n|Estado inicial $|ij\\rangle$|$\\left(H_1 \\otimes I_2\\right)$|$|\\Psi\\rangle_{i,j}$|\n|:------------:|:---------------:|:--------------:|\n|$ |00 \\rangle$|$\\frac{1}{\\sqrt{2}}\\left(|00 \\rangle + |10\\rangle\\right)$|$|\\Phi^+\\rangle_{1,2}$|\n|$ |01 \\rangle$|$\\frac{1}{\\sqrt{2}}\\left(|01 \\rangle + |11\\rangle\\right)$|$|\\Psi^+\\rangle_{1,2}$|\n|$ |10 \\rangle$|$\\frac{1}{\\sqrt{2}}\\left(|00 \\rangle - |10\\rangle\\right)$|$|\\Phi^-\\rangle_{1,2}$|\n|$ |11 \\rangle$|$\\frac{1}{\\sqrt{2}}\\left(|01 \\rangle - |11\\rangle\\right)$|$|\\Psi^-\\rangle_{1,2}$|\n\n$$\\hspace{2em}\\overset{\\text{Hadamard sobre } q_1}{\\longrightarrow}\\hspace{2em}\\overset{\\text{CNOT} q_1,q_2}{\\longrightarrow}$$\n\nA continuaci\u00f3n vamos a construir el primer estado de la base de Bell\n\n\n```python\n## Iniciamos los registros necesarios\nqr=QuantumRegister(2)\ncr=ClassicalRegister(2)\nqc=QuantumCircuit(qr,cr)\n\n## Ahora realizamos la operaci\u00f3n de Hadamard sobre el primer qubit de abajo hacia arriba, es decir, el qubit[1]\nqc.h(qr[1])\n\n## Ahora aplicamos la compuerta CNOT con controlada por el primer qubit, apuntando al segundo, es decir\nqc.cx(qr[1],qr[0])\n\nqc.measure(qr[0],cr[0])\nqc.measure(qr[1],cr[1])\nqc.draw(output='mpl')\n```\n\n\n```python\nbackend = Aer.get_backend('qasm_simulator')\nresults = execute(qc,backend,shots=10000).result().get_counts()\nplot_histogram(results)\n```\n\nComo vemos, cada vez que el qubit 1 est\u00e1 en el estado $|i\\rangle$ con $i = 0,1$, el qubit 0 siempre est\u00e1 en su mismo estado, tal cu\u00e1l como nos indica el estado de Bell $|\\Phi^+\\rangle_{1,2} = \\frac{1}{\\sqrt{2}}\\left(|00 \\rangle + |11 \\rangle\\right)$.\n\n## Referencias\n\n* Nielsen & Chuang. **Quantum Computation and Quantum Information**. Cambridge University Press, 2010.\n* Beck. **Quantum Mechanics Theory and Experiments**. Cambridge University Press, 2012.\n* Ataman. **The quantum optical description of a double Mach-Zehnder interferometer**. arXiv:1407.1704 [physics.optics]. 2014.\n* Ekert. **From Interferometers to Quantum Computers**. Supplementary material, Mathematical Institute, University of Oxford, 2010.\n* Wilde. **Quantum Information Theory**. Cambridge University Press, 2013.\n\nMuchas gracias por leer esta publicaci\u00f3n! Abajo se encuentran links a las lecciones siguientes en el Crash Course de QC-FEM.\n \n", "meta": {"hexsha": "5b4a175f43ac0a394eaafb3f8c626165b5c596bb", "size": 117081, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Lec_2_Postulados/Lec_2_Postulados.ipynb", "max_stars_repo_name": "QC-FEM/QC-CrashCourse", "max_stars_repo_head_hexsha": "a29a5cb0ba27ebc1117c469921c58ed01ffc5998", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-03-04T13:39:28.000Z", "max_stars_repo_stars_event_max_datetime": "2021-03-10T23:27:29.000Z", "max_issues_repo_path": "Lec_2_Postulados/Lec_2_Postulados.ipynb", "max_issues_repo_name": "QC-FEM/QC-CrashCourse", "max_issues_repo_head_hexsha": "a29a5cb0ba27ebc1117c469921c58ed01ffc5998", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Lec_2_Postulados/Lec_2_Postulados.ipynb", "max_forks_repo_name": "QC-FEM/QC-CrashCourse", "max_forks_repo_head_hexsha": "a29a5cb0ba27ebc1117c469921c58ed01ffc5998", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-06-24T18:56:27.000Z", "max_forks_repo_forks_event_max_datetime": "2021-06-24T18:56:27.000Z", "avg_line_length": 172.940915805, "max_line_length": 31468, "alphanum_fraction": 0.8679717461, "converted": true, "num_tokens": 7321, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358411176238, "lm_q2_score": 0.6548947155710233, "lm_q1q2_score": 0.4195490269533295}} {"text": "# CS109B Data Science 2: Advanced Topics in Data Science \n\n## Lab 4 - Bayesian Analysis\n\n**Harvard University**
                                        \n**Spring 2020**
                                        \n**Instructors:** Mark Glickman, Pavlos Protopapas, and Chris Tanner
                                        \n**Lab Instructors:** Chris Tanner and Eleni Angelaki Kaxiras
                                        \n**Content:** Eleni Angelaki Kaxiras\n\n---\n\n\n```python\n## RUN THIS CELL TO PROPERLY HIGHLIGHT THE EXERCISES\nimport requests\nfrom IPython.core.display import HTML\nstyles = requests.get(\"https://raw.githubusercontent.com/Harvard-IACS/2019-CS109B/master/content/styles/cs109.css\").text\nHTML(styles)\n```\n\n\n\n\n\n\n\n\n\n\n\n```python\nimport pymc3 as pm\nfrom pymc3 import summary\n```\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport scipy.stats as stats\nimport pandas as pd\n%matplotlib inline \n\nimport warnings\nwarnings.filterwarnings('ignore')\n```\n\n\n```python\nprint('Running on PyMC3 v{}'.format(pm.__version__))\n```\n\n Running on PyMC3 v3.8\n\n\n\n```javascript\n%%javascript\nIPython.OutputArea.auto_scroll_threshold = 20000;\n```\n\n\n \n\n\n\n\n## Learning Objectives\n\nBy the end of this lab, you should be able to:\n* Understand how probability distributions work.\n* Apply Bayes Rule in calculating probabilities.\n* Understand how to apply Bayesian analysis using PyMC3\n* Avoid getting fired when talking to your Bayesian employer.\n\n**This lab corresponds to Lectures 6, 7, and 8, and maps to Homework 3.**\n\n## Table of Contents\n\n1. The Bayesian Way of Thinking or Is this a Fair Coin?\n2. [Intro to `pyMC3`](#pymc3). \n3. [Bayesian Linear Regression](#blr).\n4. [Try this at Home: Example on Mining Disasters](#no4).\n\n## 1. The Bayesian way of Thinking\n\n```\nHere is my state of knowledge about the situation. Here is some data, I am now going to revise my state of knowledge.\n```\n\n
                                        Table Exercise: Discuss the statement above with your table mates and make sure everyone understands what it means and what constitutes Bayesian way of thinking. Finally, count the Bayesians among you.
                                        \n\n### A. Bayes Rule\n\n\\begin{equation}\n\\label{eq:bayes} \nP(A|\\textbf{B}) = \\frac{P(\\textbf{B} |A) P(A) }{P(\\textbf{B})} \n\\end{equation}\n\n$P(A|\\textbf{B})$ is the **posterior** distribution, prob(hypothesis | data) \n\n$P(\\textbf{B} |A)$ is the **likelihood** function, how probable is my data **B** for different values of the parameters\n\n$P(A)$ is the marginal probability to observe the data, called the **prior**, this captures our belief about the data before observing it.\n\n$P(\\textbf{B})$ is the marginal distribution (sometimes called marginal likelihood)\n\n
                                        \n
                                        Table Exercise: Solve the Monty Hall Paradox using Bayes Rule.
                                        \n\n\n\nYou are invited to play a game. There are 3 doors behind **one** of which are the keys to a brand new red Tesla. There is a goat behind each of the other two. \n\nYou are asked to pick one door, and let's say you pick **Door1**. The host knows where the keys are. Of the two remaining closed doors, he will always open the door that has a goat behind it. He'll say \"I will do you a favor and open **Door2**\". So he opens Door2 inside which there is, of course, a goat. He now asks you, do you want to open the initial Door you chose or change to **Door3**? Generally, in this game, when you are presented with this choice should you swap the doors?\n\n**Initial Steps:**\n- Start by defining the `events` of this probabilities game. One definition is:\n \n - $A_i$: car is behind door $i$ \n \n - $B_i$ host opens door $i$\n \n$i\\in[1,2,3]$\n \n- In more math terms, the question is: is the probability that the price is behind **Door 1** higher than the probability that the price is behind **Door2**, given that an event **has occured**?\n\n**P(A1|B2) = P(B2|A1)P(A1) / P(B2) = (1/2)(1/3) / (1/2)** - 1/2 because if prize behind A1 as you guessed, 50/50 for either of other doors\n\n**P(A3|B2) = P(B2|A3)P(A3) / P(B2) = (1/1)(1/3) / (1/2)** - 1/1 because if prize behind A1 as you guessed, 100% chance the host opens the only door left (after the right answer and your guess)\n\n### B. Bayes Rule written with Probability Distributions\n\nWe have data that we believe come from an underlying distribution of unknown parameters. If we find those parameters, we know everything about the process that generated this data and we can make inferences (create new data).\n\n$$\n\\begin{equation}\n\\label{eq:bayes} \nP(\\theta|\\textbf{D}) = \\frac{P(\\textbf{D} |\\theta) P(\\theta) }{P(\\textbf{D})} \n\\end{equation}\n$$\n\n#### But what is $\\theta \\;$?\n\n$\\theta$ is an unknown yet fixed set of parameters. In Bayesian inference we express our belief about what $\\theta$ might be and instead of trying to guess $\\theta$ exactly, we look for its **probability distribution**. What that means is that we are looking for the **parameters** of that distribution. For example, for a Poisson distribution our $\\theta$ is only $\\lambda$. In a normal distribution, our $\\theta$ is often just $\\mu$ and $\\sigma$.\n\n### C. A review of Common Probability Distributions\n\n#### Discrete Distributions\n\nThe random variable has a **probability mass function (pmf)** which measures the probability that our random variable will take a specific value $y$, denoted $P(Y=y)$.\n\n- **Bernoulli** (binary outcome, success has probability $\\theta$, $one$ trial):\n$\nP(Y=k) = \\theta^k(1-\\theta)^{1-k}\n$\n
                                        \n- **Binomial** (binary outcome, success has probability $\\theta$, $n$ trials):\n\\begin{equation}\nP(Y=k) = {{n}\\choose{k}} \\cdot \\theta^k(1-\\theta)^{n-k}\n\\end{equation}\n\n*Note*: Binomial(1,$p$) = Bernouli($p$)\n
                                        \n- **Negative Binomial**\n
                                        \n- **Poisson** (counts independent events occurring at a rate)\n\\begin{equation}\nP\\left( Y=y|\\lambda \\right) = \\frac{{e^{ - \\lambda } \\lambda ^y }}{{y!}}\n\\end{equation}\ny = 0,1,2,...\n
                                        \n- **Discrete Uniform** \n
                                        \n- **Categorical, or Multinulli** (random variables can take any of K possible categories, each having its own probability; this is a generalization of the Bernoulli distribution for a discrete variable with more than two possible outcomes, such as the roll of a die)\n
                                        \n- **Dirichlet-multinomial** (a generalization of the beta distribution for many variables)\n\n#### Continuous Distributions\n\nThe random variable has a **probability density function (pdf)**.\n- **Uniform** (variable equally likely to be near each value in interval $(a,b)$)\n\\begin{equation}\nP(X = x) = \\frac{1}{b - a}\n\\end{equation}\nanywhere within the interval $(a, b)$, and zero elsewhere.\n
                                        \n- **Normal** (a.k.a. Gaussian)\n\\begin{equation}\nX \\sim \\mathcal{N}(\\mu,\\,\\sigma^{2})\n\\end{equation} \n\n A Normal distribution can be parameterized either in terms of precision $\\tau$ or standard deviation ($\\sigma^{2}$. The link between the two is given by\n\\begin{equation}\n\\tau = \\frac{1}{\\sigma^{2}}\n\\end{equation}\n - Mean $\\mu$\n - Variance $\\frac{1}{\\tau}$ or $\\sigma^{2}$\n - Parameters: `mu: float`, `sigma: float` or `tau: float`\n
                                        \n- **Beta** (variable ($\\theta$) taking on values in the interval $[0,1]$, and parametrized by two positive parameters, $\\alpha$ and $\\beta$ that control the shape of the distribution. \n \n*Note:*Beta is a good distribution to use for priors (beliefs) because its range is $[0,1]$ which is the natural range for a probability and because we can model a wide range of functions by changing the $\\alpha$ and $\\beta$ parameters.\n\n\\begin{equation}\n\\label{eq:beta} \nP(\\theta) = \\frac{1}{B(\\alpha, \\beta)} {\\theta}^{\\alpha - 1} (1 - \\theta)^{\\beta - 1} \\propto {\\theta}^{\\alpha - 1} (1 - \\theta)^{\\beta - 1}\n\\end{equation}\n\n\nwhere the normalisation constant, $B$, is a beta function of $\\alpha$ and $\\beta$,\n\n\n\\begin{equation}\nB(\\alpha, \\beta) = \\int_{t=0}^1 t^{\\alpha - 1} (1 - t)^{\\beta - 1} dt.\n\\end{equation}\n
                                        \n- **Exponential**\n
                                        \n- **Gamma**\n\n\n\n #### Code Resources:\n - Statistical Distributions in numpy/scipy: [scipy.stats](https://docs.scipy.org/doc/scipy/reference/stats.html)\n - Statistical Distributions in pyMC3: [distributions in PyMC3](https://docs.pymc.io/api/distributions.html) (we will see those below).\n\n
                                        Exercise: Plot a Discrete variable
                                        \n\nChange the value of $\\mu$ in the Poisson PMF and see how the plot changes. Remember that the y-axis in a discrete probability distribution shows the probability of the random variable having a specific value in the x-axis.\n\n\\begin{equation}\nP\\left( X=k \\right) = \\frac{{e^{ - \\mu } \\mu ^k }}{{k!}}\n\\end{equation}\n\n**stats.poisson.pmf(x, mu)** $\\mu$(mu) is our $\\theta$ in this case.\n\n\n```python\nplt.style.use('seaborn-darkgrid')\nx = np.arange(0, 30)\nfor m in [0, 0.5, 3, 6, 11, 14]:\n pmf = stats.poisson.pmf(x, m)\n plt.plot(x, pmf, 'o', alpha=0.5, label='$\\mu$ = {}'.format(m))\nplt.xlabel('random variable', fontsize=12)\nplt.ylabel('probability', fontsize=12)\nplt.legend(loc=1)\nplt.ylim=(-0.1)\nplt.show()\n```\n\n\n```python\n# same for binomial\nplt.style.use('seaborn-darkgrid')\nx = np.arange(0, 22)\nns = [10, 17]\nps = [0.5, 0.7]\nfor n, p in zip(ns, ps):\n pmf = stats.binom.pmf(x, n, p)\n plt.plot(x, pmf, 'o', alpha=0.5, label='n = {}, p = {}'.format(n, p))\nplt.xlabel('x', fontsize=14)\nplt.ylabel('f(x)', fontsize=14)\nplt.legend(loc=1)\nplt.show()\n```\n\n\n```python\n# discrete uniform\nplt.style.use('seaborn-darkgrid')\nls = [0]\nus = [3] # watch out, this number can only be integer!\nfor l, u in zip(ls, us):\n x = np.arange(l, u+1)\n pmf = [1.0 / (u - l + 1)] * len(x)\n plt.plot(x, pmf, '-o', label='lower = {}, upper = {}'.format(l, u))\nplt.xlabel('x', fontsize=12)\nplt.ylabel('probability P(x)', fontsize=12)\nplt.legend(loc=1)\nplt.show()\n```\n\n
                                        Exercise: Plot a continuous variable
                                        \n\nChange the value of $\\mu$ in the Uniform PDF and see how the plot changes.\n \nRemember that the y-axis in a continuous probability distribution does not shows the actual probability of the random variable having a specific value in the x-axis because that probability is zero!. Instead, to see the probability that the variable is within a small margin we look at the integral below the curve of the PDF.\n\nThe uniform is often used as a noninformative prior.\n\n```\nUniform - numpy.random.uniform(a=0.0, b=1.0, size)\n```\n\n$\\alpha$ and $\\beta$ are our parameters. `size` is how many tries to perform.\nOur $\\theta$ is basically the combination of the parameters a,b. We can also call it \n\\begin{equation}\n\\mu = (a+b)/2\n\\end{equation}\n\n\n```python\nfrom scipy.stats import uniform\n\nr = uniform.rvs(size=1000)\nplt.plot(r, uniform.pdf(r),'r-', lw=5, alpha=0.6, label='uniform pdf')\nplt.hist(r, density=True, histtype='stepfilled', alpha=0.2)\nplt.ylabel(r'probability density')\nplt.xlabel(f'random variable')\nplt.legend(loc='best', frameon=False)\nplt.show()\n```\n\n\n```python\nfrom scipy.stats import beta\n\nalphas = [0.5, 1.5, 3.0]\nbetas = [0.5, 1.5, 3.0]\nx = np.linspace(0, 1, 1000) \ncolors = ['red', 'green', 'blue']\n\nfig, ax = plt.subplots(figsize=(8, 5))\n\nfor a, b, colors in zip(alphas, betas, colors):\n dist = beta(a, b)\n plt.plot(x, dist.pdf(x), c=colors,\n label=f'a={a}, b={b}')\n\nax.set_ylim(0, 3)\n\nax.set_xlabel(r'$\\theta$')\nax.set_ylabel(r'$p(\\theta|\\alpha,\\beta)$')\nax.set_title('Beta Distribution')\n\nax.legend(loc='best')\nfig.show();\n```\n\n\n```python\nplt.style.use('seaborn-darkgrid')\nx = np.linspace(-5, 5, 1000)\nmus = [0., 0., 0., -2.]\nsigmas = [0.4, 1., 2., 0.4]\nfor mu, sigma in zip(mus, sigmas):\n pdf = stats.norm.pdf(x, mu, sigma)\n plt.plot(x, pdf, label=r'$\\mu$ = '+ f'{mu},' + r'$\\sigma$ = ' + f'{sigma}') \nplt.xlabel('random variable', fontsize=12)\nplt.ylabel('probability density', fontsize=12)\nplt.legend(loc=1)\nplt.show()\n```\n\n\n```python\nplt.style.use('seaborn-darkgrid')\nx = np.linspace(-5, 5, 1000)\nmus = [0., 0., 0., -2.] # mean\nsigmas = [0.4, 1., 2., 0.4] # std\nfor mu, sigma in zip(mus, sigmas):\n plt.plot(x, uniform.pdf(x, mu, sigma), lw=5, alpha=0.4, \\\n label=r'$\\mu$ = '+ f'{mu},' + r'$\\sigma$ = ' + f'{sigma}')\nplt.xlabel('random variable', fontsize=12)\nplt.ylabel('probability density', fontsize=12)\nplt.legend(loc=1)\nplt.show()\n```\n\n### D. Is this a Fair Coin?\n\nWe do not want to promote gambling but let's say you visit the casino in **Monte Carlo**. You want to test your theory that casinos are dubious places where coins have been manipulated to have a larger probability for tails. So you will try to estimate how fair a coin is based on 100 flips.
                                        \nYou begin by flipping the coin. You get either Heads ($H$) or Tails ($T$) as our observed data and want to see if your posterior probabilities change as you obtain more data, that is, more coin flips. A nice way to visualize this is to plot the posterior probabilities as we observe more flips (data). \n\nWe will be using Bayes rule. $\\textbf{D}$ is our data.\n\n$$\n\\begin{equation}\n\\label{eq:bayes} \nP(\\theta|\\textbf{D}) = \\frac{P(\\textbf{D} |\\theta) P(\\theta) }{P(\\textbf{D})} \n\\end{equation}\n$$\n\nIn the case of a coin toss when we observe $k$ heads in $n$ tosses:\n\n$$\n\\begin{equation}\n\\label{eq:bayes} \nP(\\theta|\\textbf{k}) = Beta(\\alpha + \\textbf{k}, \\beta + n - \\textbf{k}) \n\\end{equation}\n$$\n\nwe can say that $\\alpha$ and $\\beta$ play the roles of a \"prior number of heads\" and \"prior number of tails\".\n\n\n```python\n# play with the priors - here we manually set them but we could be sampling from a separate Beta\ntrials = np.array([0, 1, 3, 5, 10, 15, 20, 100, 200, 300])\nheads = np.array([0, 1, 2, 4, 8, 10, 10, 50, 180, 150])\nx = np.linspace(0, 1, 100)\n\n# for simplicity we set a,b=1\n\nplt.figure(figsize=(10,8))\nfor k, N in enumerate(trials):\n sx = plt.subplot(len(trials)/2, 2, k+1)\n posterior = stats.beta.pdf(x, 1 + heads[k], 1 + trials[k] - heads[k]) \n plt.plot(x, posterior, alpha = 0.5, label=f'{trials[k]} tosses\\n {heads[k]} heads');\n plt.fill_between(x, 0, posterior, color=\"#348ABD\", alpha=0.4) \n plt.legend(loc='upper left', fontsize=10)\n plt.legend()\n plt.autoscale(tight=True)\n \nplt.suptitle(\"Posterior probabilities for coin flips\", fontsize=15);\nplt.tight_layout()\nplt.subplots_adjust(top=0.88)\n```\n\n [Top](#top)\n\n## 2. Introduction to `pyMC3`\n \nPyMC3 is a Python library for programming Bayesian analysis, and more specifically, data creation, model definition, model fitting, and posterior analysis. It uses the concept of a `model` which contains assigned parametric statistical distributions to unknown quantities in the model. Within models we define random variables and their distributions. A distribution requires at least a `name` argument, and other `parameters` that define it. You may also use the `logp()` method in the model to build the model log-likelihood function. We define and fit the model.\n\nPyMC3 includes a comprehensive set of pre-defined statistical distributions that can be used as model building blocks. Although they are not meant to be used outside of a `model`, you can invoke them by using the prefix `pm`, as in `pm.Normal`. \n\n#### Markov Chain Monte Carlo (MCMC) Simulations\n\nPyMC3 uses the **No-U-Turn Sampler (NUTS)** and the **Random Walk Metropolis**, two Markov chain Monte Carlo (MCMC) algorithms for sampling in posterior space. Monte Carlo gets into the name because when we sample in posterior space, we choose our next move via a pseudo-random process. NUTS is a sophisticated algorithm that can handle a large number of unknown (albeit continuous) variables.\n\n\n```python\nwith pm.Model() as model:\n z = pm.Normal('z', mu=0., sigma=5.) \n x = pm.Normal('x', mu=z, sigma=1., observed=5.) \nprint(x.logp({'z': 2.5})) \nprint(z.random(10, 100)[:10]) \n```\n\n -4.043938533204672\n [ 8.33640681 3.99381712 -2.19366687 2.3737412 -9.73872099 -4.81026637\n -2.41892328 5.36819376 2.05338416 1.96679158]\n\n\n**References**:\n\n- *Salvatier J, Wiecki TV, Fonnesbeck C. 2016. Probabilistic programming in Python using PyMC3. PeerJ Computer Science 2:e55* [(https://doi.org/10.7717/peerj-cs.55)](https://doi.org/10.7717/peerj-cs.55)\n- [Distributions in PyMC3](https://docs.pymc.io/api/distributions.html)\n- [More Details on Distributions](https://docs.pymc.io/developer_guide.html)\n\nInformation about PyMC3 functions including descriptions of distributions, sampling methods, and other functions, is available via the `help` command.\n\n\n```python\nhelp(pm.Poisson)\n```\n\n Help on class Poisson in module pymc3.distributions.discrete:\n \n class Poisson(pymc3.distributions.distribution.Discrete)\n | Poisson(name, *args, **kwargs)\n | \n | Poisson log-likelihood.\n | \n | Often used to model the number of events occurring in a fixed period\n | of time when the times at which events occur are independent.\n | The pmf of this distribution is\n | \n | .. math:: f(x \\mid \\mu) = \\frac{e^{-\\mu}\\mu^x}{x!}\n | \n | .. plot::\n | \n | import matplotlib.pyplot as plt\n | import numpy as np\n | import scipy.stats as st\n | plt.style.use('seaborn-darkgrid')\n | x = np.arange(0, 15)\n | for m in [0.5, 3, 8]:\n | pmf = st.poisson.pmf(x, m)\n | plt.plot(x, pmf, '-o', label='$\\mu$ = {}'.format(m))\n | plt.xlabel('x', fontsize=12)\n | plt.ylabel('f(x)', fontsize=12)\n | plt.ylim(0)\n | plt.legend(loc=1)\n | plt.show()\n | \n | ======== ==========================\n | Support :math:`x \\in \\mathbb{N}_0`\n | Mean :math:`\\mu`\n | Variance :math:`\\mu`\n | ======== ==========================\n | \n | Parameters\n | ----------\n | mu : float\n | Expected number of occurrences during the given interval\n | (mu >= 0).\n | \n | Notes\n | -----\n | The Poisson distribution can be derived as a limiting case of the\n | binomial distribution.\n | \n | Method resolution order:\n | Poisson\n | pymc3.distributions.distribution.Discrete\n | pymc3.distributions.distribution.Distribution\n | builtins.object\n | \n | Methods defined here:\n | \n | __init__(self, mu, *args, **kwargs)\n | Initialize self. See help(type(self)) for accurate signature.\n | \n | logp(self, value)\n | Calculate log-probability of Poisson distribution at specified value.\n | \n | Parameters\n | ----------\n | value : numeric\n | Value(s) for which log-probability is calculated. If the log probabilities for multiple\n | values are desired the values must be provided in a numpy array or theano tensor\n | \n | Returns\n | -------\n | TensorVariable\n | \n | random(self, point=None, size=None)\n | Draw random values from Poisson distribution.\n | \n | Parameters\n | ----------\n | point : dict, optional\n | Dict of variable values on which random values are to be\n | conditioned (uses default point if not specified).\n | size : int, optional\n | Desired size of random sample (returns one sample if not\n | specified).\n | \n | Returns\n | -------\n | array\n | \n | ----------------------------------------------------------------------\n | Methods inherited from pymc3.distributions.distribution.Distribution:\n | \n | __getnewargs__(self)\n | \n | __latex__ = _repr_latex_(self, name=None, dist=None)\n | Magic method name for IPython to use for LaTeX formatting.\n | \n | default(self)\n | \n | get_test_val(self, val, defaults)\n | \n | getattr_value(self, val)\n | \n | logp_nojac(self, *args, **kwargs)\n | Return the logp, but do not include a jacobian term for transforms.\n | \n | If we use different parametrizations for the same distribution, we\n | need to add the determinant of the jacobian of the transformation\n | to make sure the densities still describe the same distribution.\n | However, MAP estimates are not invariant with respect to the\n | parametrization, we need to exclude the jacobian terms in this case.\n | \n | This function should be overwritten in base classes for transformed\n | distributions.\n | \n | logp_sum(self, *args, **kwargs)\n | Return the sum of the logp values for the given observations.\n | \n | Subclasses can use this to improve the speed of logp evaluations\n | if only the sum of the logp values is needed.\n | \n | ----------------------------------------------------------------------\n | Class methods inherited from pymc3.distributions.distribution.Distribution:\n | \n | dist(*args, **kwargs) from builtins.type\n | \n | ----------------------------------------------------------------------\n | Static methods inherited from pymc3.distributions.distribution.Distribution:\n | \n | __new__(cls, name, *args, **kwargs)\n | Create and return a new object. See help(type) for accurate signature.\n | \n | ----------------------------------------------------------------------\n | Data descriptors inherited from pymc3.distributions.distribution.Distribution:\n | \n | __dict__\n | dictionary for instance variables (if defined)\n | \n | __weakref__\n | list of weak references to the object (if defined)\n \n\n\n [Top](#top)\n\n## 3. Bayesian Linear Regression\n\nLet's say we want to predict outcomes Y as normally distributed observations with an expected value $mu$ that is a linear function of two predictor variables, $\\bf{x}_1$ and $\\bf{x}_2$.\n\n\\begin{equation}\n\\mu = \\alpha + \\beta_1 \\bf{x}_1 + \\beta_2 x_2 \n\\end{equation}\n\n\\begin{equation}\nY \\sim \\mathcal{N}(\\mu,\\,\\sigma^{2})\n\\end{equation} \n\nwhere $\\sigma^2$ represents the measurement error. \n\nIn this example, we will use $\\sigma^2 = 10$\n\nWe also choose the parameters as normal distributions:\n\n\\begin{eqnarray}\n\\alpha \\sim \\mathcal{N}(0,\\,10) \\\\\n\\beta_i \\sim \\mathcal{N}(0,\\,10) \\\\\n\\sigma^2 \\sim |\\mathcal{N}(0,\\,10)|\n\\end{eqnarray} \n\nWe will artificially create the data to predict on. We will then see if our model predicts them correctly.\n\n\n```python\n# Initialize random number generator\nnp.random.seed(123)\n\n# True parameter values\nalpha, sigma = 1, 1\nbeta = [1, 2.5]\n\n# Size of dataset\nsize = 100\n\n# Predictor variable\nX1 = np.linspace(0, 1, size)\nX2 = np.linspace(0,.2, size)\n\n# Simulate outcome variable\nY = alpha + beta[0]*X1 + beta[1]*X2 + np.random.randn(size)*sigma\n\nfig, ax = plt.subplots(1,2, figsize=(10,6), sharex=True)\nax[0].scatter(X1,Y)\nax[1].scatter(X2,Y)\nax[0].set_xlabel(r'$x_1$', fontsize=14) \nax[0].set_ylabel(r'$Y$', fontsize=14)\nax[1].set_xlabel(r'$x_2$', fontsize=14) \nax[1].set_ylabel(r'$Y$', fontsize=14)\n```\n\n\n```python\nfrom pymc3 import Model, Normal, HalfNormal\n\nbasic_model = Model()\n\nwith basic_model:\n\n # Priors for unknown model parameters, specifically create stochastic random variables \n # with Normal prior distributions for the regression coefficients,\n # and a half-normal distribution for the standard deviation of the observations, \u03c3.\n alpha = Normal('alpha', mu=0, sd=10)\n beta = Normal('beta', mu=0, sd=10, shape=2)\n sigma = HalfNormal('sigma', sd=1)\n\n # Expected value of outcome - posterior\n mu = alpha + beta[0]*X1 + beta[1]*X2\n\n # Likelihood (sampling distribution) of observations\n Y_obs = Normal('Y_obs', mu=mu, sd=sigma, observed=Y)\n```\n\n\n```python\n# model fitting with sampling\nfrom pymc3 import NUTS, sample, find_MAP\nfrom scipy import optimize\n\nwith basic_model:\n\n # obtain starting values via MAP\n start = find_MAP(fmin=optimize.fmin_powell)\n\n # instantiate sampler\n step = NUTS(scaling=start)\n\n # draw 2000 posterior samples\n trace = sample(2000, step, start=start)\n```\n\n logp = -164.5: 5%|\u258c | 270/5000 [00:00<00:00, 5084.85it/s] \n\n Optimization terminated successfully.\n Current function value: 164.496957\n Iterations: 6\n Function evaluations: 271\n\n\n logp = -164.5: 5%|\u258c | 271/5000 [00:00<00:12, 387.03it/s] \n Multiprocess sampling (4 chains in 4 jobs)\n NUTS: [sigma, beta, alpha]\n Sampling 4 chains, 0 divergences: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 10000/10000 [00:07<00:00, 1384.27draws/s]\n The number of effective samples is smaller than 25% for some parameters.\n\n\n\n```python\nfrom pymc3 import traceplot\n\ntraceplot(trace);\n```\n\n\n```python\nresults = pm.summary(trace, \n var_names=['alpha', 'beta', 'sigma'])\nresults\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        meansdhpd_3%hpd_97%mcse_meanmcse_sdess_meaness_sdess_bulkess_tailr_hat
                                        alpha1.0110.2310.5861.4480.0030.0024508.04504.04495.04119.01.0
                                        beta[0]1.4932.026-2.1255.5320.0650.046961.0961.0968.01198.01.0
                                        beta[1]0.1919.902-19.36417.7680.3180.225972.0972.0979.01149.01.0
                                        sigma1.1460.0820.9941.3000.0010.0017181.07089.07309.05219.01.0
                                        \n
                                        \n\n\n\nThis linear regression example is from the original paper on PyMC3: *Salvatier J, Wiecki TV, Fonnesbeck C. 2016. Probabilistic programming in Python using PyMC3. PeerJ Computer Science 2:e55 https://doi.org/10.7717/peerj-cs.55*\n\n [Top](#top)\n\n## 4. Try this at Home: Example on Mining Disasters\nWe will go over the classical `mining disasters from 1851 to 1962` dataset. \n\nThis example is from the [pyMC3 Docs](https://docs.pymc.io/notebooks/getting_started.html).\n\n\n```python\nimport pandas as pd\ndisaster_data = pd.Series([4, 5, 4, 0, 1, 4, 3, 4, 0, 6, 3, 3, 4, 0, 2, 6,\n 3, 3, 5, 4, 5, 3, 1, 4, 4, 1, 5, 5, 3, 4, 2, 5,\n 2, 2, 3, 4, 2, 1, 3, np.nan, 2, 1, 1, 1, 1, 3, 0, 0,\n 1, 0, 1, 1, 0, 0, 3, 1, 0, 3, 2, 2, 0, 1, 1, 1,\n 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 1, 0, 2,\n 3, 3, 1, np.nan, 2, 1, 1, 1, 1, 2, 4, 2, 0, 0, 1, 4,\n 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1])\nfontsize = 12\nyears = np.arange(1851, 1962)\nplt.figure(figsize=(10,5))\n# plt.scatter(years, disaster_data); \nplt.bar(years, disaster_data)\nplt.ylabel('Disaster count', size=fontsize)\nplt.xlabel('Year', size=fontsize);\nplt.title('Was there a Turning Point in Mining disasters from 1851 to 1962?', size=15);\n```\n\n#### Building the model\n\n**Step1:** We choose the probability model for our experiment. Occurrences of disasters in the time series is thought to follow a **Poisson** process with a large **rate** parameter in the early part of the time series, and from one with a smaller **rate** in the later part. We are interested in locating the change point in the series, which perhaps is related to changes in mining safety regulations. \n\n```\ndisasters = pm.Poisson('disasters', rate, observed=disaster_data)\n```\n\nWe have two rates, `early_rate` if $t<=s$, and `late_rate` if $t>s$, where $s$ is the year the switch was made (a.k.a. the `switchpoint`). \n\n**Step2:** Choose a prior distributions of the two rates, what we believe the rates were before we observed the data, and the switchpoint. We choose Exponential.\n```\nearly_rate = pm.Exponential('early_rate', 1)\n```\n\nThe parameters of this model are: \n\n\n**Note:** Watch for missing values. Missing values are handled transparently by passing a MaskedArray or a pandas.DataFrame. Behind the scenes, another random variable, disasters.missing_values is created to model the missing values. If you pass a np.array with missing values you will get an error.\n\n\n```python\nwith pm.Model() as disaster_model:\n\n # discrete\n switchpoint = pm.DiscreteUniform('switchpoint', lower=years.min(), upper=years.max(), testval=1900)\n\n # Priors for pre- and post-switch rates number of disasters\n early_rate = pm.Exponential('early_rate', 1)\n late_rate = pm.Exponential('late_rate', 1)\n\n # our theta - allocate appropriate Poisson rates to years before and after current\n # switch is an `if` statement in puMC3\n rate = pm.math.switch(switchpoint >= years, early_rate, late_rate)\n\n # our observed data as a likelihood function of the `rate` parameters\n # shows how we think our data is distributed\n disasters = pm.Poisson('disasters', rate, observed=disaster_data)\n```\n\n#### Model Fitting\n\n\n```python\n# there are defaults but we can also more explicitly set the sampling algorithms\nwith disaster_model:\n \n # for continuous variables\n step1 = pm.NUTS([early_rate, late_rate])\n \n # for discrete variables\n step2 = pm.Metropolis([switchpoint, disasters.missing_values[0]] )\n\n# trace = pm.sample(10000, step=[step1, step2])\n # try different number of samples\n trace = pm.sample(1, step=[step1, step2])\n```\n\n#### Posterior Analysis\n\nOn the left side plots we notice that our early rate is between 2.5 and 3.5 disasters a year. In the late period it seems to be between 0.6 and 1.2 so definitely lower.\n\nThe right side plots show the samples we drew to come to our conclusion.\n\n\n```python\npm.traceplot(trace, ['early_rate', 'late_rate', 'switchpoint'], figsize=(20,10));\n```\n\n\n```python\nresults = pm.summary(trace, \n var_names=['early_rate', 'late_rate', 'switchpoint'])\nresults\n```\n", "meta": {"hexsha": "8ffdcf4ccc24b3dde902ad6c4e9c570118cd9ec6", "size": 537202, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "content/labs/lab04/notebook/cs109b_lab04_bayes.ipynb", "max_stars_repo_name": "wfseaton/2020-CS109B", "max_stars_repo_head_hexsha": "7b11a2c270144e4fed455b9c9e628222fa2f1f9a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "content/labs/lab04/notebook/cs109b_lab04_bayes.ipynb", "max_issues_repo_name": "wfseaton/2020-CS109B", "max_issues_repo_head_hexsha": "7b11a2c270144e4fed455b9c9e628222fa2f1f9a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "content/labs/lab04/notebook/cs109b_lab04_bayes.ipynb", "max_forks_repo_name": "wfseaton/2020-CS109B", "max_forks_repo_head_hexsha": "7b11a2c270144e4fed455b9c9e628222fa2f1f9a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 361.0228494624, "max_line_length": 211756, "alphanum_fraction": 0.9178409611, "converted": true, "num_tokens": 9493, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765155565326, "lm_q2_score": 0.7461389986757757, "lm_q1q2_score": 0.41946182239638785}} {"text": "\\begin{equation}\nx = y\n\\end{equation}\n\n$$\nx = y\n$$\n\nalign:\n\n\\begin{align}\n\\frac{x}{y}\n\\end{align}\n\n# Divvy data exploration project\n\n**authors:** Peter Carbonetto, Gao Wang
                                        \n**date:** July 13, 2017\n\n
                                        \n*[Photo](https://www.flickr.com/photos/jamesbondsv/9041673199) by\n[Steven Vance](https://www.flickr.com/photos/jamesbondsv) /\n[CC BY 2.0](https://creativecommons.org/licenses/by/2.0/)*\n\n## Project overview\n\nThe purpose of this project is to gain some insight into city-wide biking trends by analyzing the [Divvy trip\ndata](https://www.divvybikes.com/system-data). We also examine trip data from a single bike station at the University of Chicago to compare the biking patterns at the university against city-wide trends.\n\nAll the results and plots presented in the pages below should be reproducable on your computer. Follow the\n[Setup Instructions](setup.html) if you are interested in reproducing the results for yourself.\n\nThese are the results of our analyses. They were generated by rendering the [Jupyter notebooks](https://github.com/stephenslab/ipynb-website/tree/master/analysis) into webpages.\n\n1. [A first glance at the Divvy data.](analysis/first-glance.html)\n\n2. [A map of the Divvy stations in Chicago.](analysis/station-map.html)\n\n3. [Exploring daily bike commuting trends from the Divvy data.](analysis/time-of-day-trends.html)\n\n4. [Exploring seasonal biking trends from the Divvy data.](analysis/seasonal-trends.html)\n\n## Credits\n\nThis Jupyter notebook website was developed by [Peter Carbonetto](http://pcarbo.github.io) and [Gao Wang](https://github.com/gaow) at the [University of Chicago](https://www.uchicago.edu).\n\nThanks to [John Blischak](https://github.com/jdblischak) and [Matthew Stephens](http://stephenslab.uchicago.edu) for their assistance and support. Also, thanks to [Larry Layne](https://rstudio-pubs-static.s3.amazonaws.com/63061_90f5136ffdf74740b6ba4ad8f2fd72fe.html) and [Austin Wehrwein](http://www.austinwehrwein.com/data-visualization/heatmaps-with-divvy-data) for sharing their analyses of the Divvy trip data that inspired some of the investigations here.\n", "meta": {"hexsha": "1063f7c0dbcfad3d562fdc3c3b910a3b2de93179", "size": 3701, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "analysis/index.ipynb", "max_stars_repo_name": "maxdevblock/ipynb-website", "max_stars_repo_head_hexsha": "1d2da67b8d60579ccd9869ab382ca96c67ca7c7d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "analysis/index.ipynb", "max_issues_repo_name": "maxdevblock/ipynb-website", "max_issues_repo_head_hexsha": "1d2da67b8d60579ccd9869ab382ca96c67ca7c7d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "analysis/index.ipynb", "max_forks_repo_name": "maxdevblock/ipynb-website", "max_forks_repo_head_hexsha": "1d2da67b8d60579ccd9869ab382ca96c67ca7c7d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 31.6324786325, "max_line_length": 466, "alphanum_fraction": 0.6203728722, "converted": true, "num_tokens": 557, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.7461389986757757, "lm_q1q2_score": 0.4194618114499807}} {"text": "**Note:** This is modified version of the notebook published by [CamDavidsonPilon@github](https://github.com/CamDavidsonPilon/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers.git).\n \nSeverel of the visualizations were converted into interactive views, using [Exhibitionist](https://github.com:Exhibitionist/Exhibitionist). Where available, you can hit the \"freeze\"\nto convert the view into a static image for saving with the notebook. Double-clicking\nthe static image will convert it back into a dynamic view.\n\nYou can install the Exhibitionist library, using:\n\n> pip install git+https://github.com/Exhibitionist/Exhibitionist.git\n \nIf you experience issues please update your installation to the latest\ngit master.\nIf they persist, please report them on the [GH issues page](https://github.com/Exhibitionist/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers/issues).\n\n\n\n```\nimport exhibitionist as xb # make sure xb is available\nimport xbviews.app as app\nviews = app.get_views()\nviews\n\n```\n\n\n```\nfigsize(12.5,4)\nimport scipy.stats as stats\n```\n\nChapter 1\n======\n***\n\nThe Philosophy of Bayesian Inference\n------\n \n> You are a skilled programmer, but bugs still slip into your code. After a particularly difficult implementation of an algorithm, you decide to test your code on a trivial example. It passes. You test the code on a harder problem. It passes once again. And it passes the next, *even more difficult*, test too! You are starting to believe that there are may be no bugs present...\n\nIf you think this way, then congratulations, you already are a Bayesian practitioner! Bayesian inference is simply updating your beliefs after considering new evidence. A Bayesian can rarely be certain about a result, but he or she can be very confident. Just like in the example above, we can never be 100% sure that our code is bug-free unless we test it on every possible problem; something rarely possible in practice. Instead, we can test it on a large number of problems, and if it succeeds we can feel more *confident* about our code. Bayesian inference works identically: we can only update our beliefs about an outcome, rarely can we be absolutely sure unless we rule out all other alternatives. We will see that being uncertain can have its advantages. \n\n\n\n###The Bayesian state of mind\n\n\nBayesian inference differs from more traditional statistical analysis by preserving *uncertainty* about our beliefs. At first, this sounds like a bad statistical technique. Isn't statistics all about deriving *certainty* from randomness? The Bayesian method interprets probability as measure of *believability in an event*, that is, how confident one is in an event occuring. In fact, we will see in a moment that this is the natural interpretation of probability. \n\nFor this to be clearer, we consider an alternative interpretation of probability: *Frequentist* methods assume that probability is the long-run frequency of events (hence the bestowed title). For example, the *probability of plane accidents* under a frequentist philosophy is interpreted as the *long-term frequency of plane accidents*. This makes logical sense for many probabilities and events, but becomes more difficult to understand when events have no long-term frequency of occurences. Consider: we often assign probabilities to outcomes of presidential elections, but the election itself only happens once! Frequentists get around this by invoking alternative realities and saying across all these universes, the frequency of occurences is the probability. \n\nBayesians, on the other hand, have a more intuitive approach. Bayesians interpret a probability as measure of *belief*, or confidence, of an event occurring. An individual who assigns a belief of 0 to an event has no confidence that the event will occur; conversely, assigning a belief of 1 implies that the individual is absolutely certain of an event occurring. Beliefs between 0 and 1 allow for weightings of other outcomes. This definition agrees with the probability of a plane accident example, for having observed the frequency of plane accidents, an individual's belief should be equal to that frequency. Similarly, under this definition of probability being equal to beliefs, it is clear how we can speak about probabilities (beliefs) of presidential election outcomes. \n\nNotice in the paragraph above, I assigned the belief (probability) measure to an *individual*, not to Nature. This is very interesting, as this definition leaves room for conflicting beliefs between individuals. Again, this is appropriate for what naturally occurs: different individuals have different beliefs of events occuring, because they possess different *information* about the world.\n\n- I flip a coin, and we both guess the result. We would both agree, assuming the coin is fair, that the probability of heads if 1/2. Assume, then, that I peek at the coin. Now I know for certain what the result is: I assign probability 1.0 to either heads or tails. Now what is *your* belief that the coin is heads? My knowledge of the outcome has not changed the coin's results. Thus we assign different probabilities to the result. \n\n- Your code either has a bug in it or not, but we do not know for certain which is true. Though we have a belief about the presence or absence of a bug. \n\n- A medical patient is exhibiting symptoms $x$, $y$ and $z$. There are a number of diseases that could be causing all of them, but only a single disease is present. A doctor has beliefs about which disease.\n\n- You believe that the beautiful girl in your English class doesn't have a crush you. You assign a low probability that she does. She, on the other hand, knows for certain that she *does indeed* like you. She (implicitly) assigns a probability 1. \n\nThis philosophy of treating beliefs as probability is natural to humans. We employ it constantly as we interact with the world and only see partial evidence. Alternatively, you have to be *trained* to think like a frequentist. \n\nTo align ourselves with traditional probability notation, we denote our belief about event $A$ as $P(A)$.\n\nJohn Maynard Keynes, a great economist and thinker, said \"When the facts change, I change my mind. What do you do, sir?\" This quote reflects the way a Bayesian updates his or her beliefs after seeing evidence. Even --especially-- if the evidence is counter to what was initially believed, the evidence cannot be ignored. We denote our updated belief as $P(A |X )$, interpreted as the probability of $A$ given the evidence $X$. We call the updated belief the *posterior probability* so as to contrast it with the *prior probability*. For example, consider the posterior probabilities (read: posterior belief) of the above examples, after observing some evidence $X$.:\n\n1\\. $P(A): \\;\\;$ the coin has a 50 percent chance of being heads. $P(A | X):\\;\\;$ You look at the coin, observe a heads, denote this information $X$, and trivially assign probability 1.0 to heads and 0.0 to tails.\n\n2\\. $P(A): \\;\\;$ This big, complex code likely has a bug in it. $P(A | X): \\;\\;$ The code passed all $X$ tests; there still might be a bug, but its presence is less likely now.\n\n3\\. $P(A):\\;\\;$ The patient could have any number of diseases. $P(A | X):\\;\\;$ Performing a blood test generated evidence $X$, ruling out some of the possible diseases from consideration.\n\n4\\. $P(A):\\;\\;$ You believe that the probability that the lovely girl in your class likes you is low. $P(A | X): \\;\\;$ She sent you an SMS message about this Friday night. Interesting... \n\nIt's clear that in each example we did not completely discard the prior belief after seeing new evidence, but we *re-weighted the prior* to incorporate the new evidence (i.e. we put more weight, or confidence, on some beliefs versus others). \n\nBy introducing prior uncertainity about events, we are already admitting that any guess we make is potentially very wrong. After observing data, evidence, or other information, and we update our beliefs, our guess becomes *less wrong*. This is the alternative side of the prediction coin, where typically we try to be *more right*.\n\n\n\n###Bayesian Inference in Practice\n\n If frequentist and Bayesian inference were computer programming functions, with inputs being statistical problems, then the two would be different in what they return to the user. The frequentist inference function would return a number, whereas the Bayesian function would return a *distribution*.\n\nFor example, in our debugging problem above, calling the frequentist function with the argument \"My code passed all $X$ tests; is my code bug-free?\" would return a *YES*. On the other hand, asking our Bayesian function \"Often my code has bugs. My code passed all $X$ tests; is my code bug-free?\" would return something very different: a distribution over *YES* and *NO*. The function might return \n\n\n> *YES*, with probability 0.8; *NO*, with probability 0.2\n\n\n\nThis is very different from the answer the frequentist function returned. Notice that the Bayesian function accepted an additional argument: *\"Often my code has bugs\"*. This parameter is the *prior*. By including the prior parameter, we are telling the Bayesian function to include our personal belief about the situation. Technically this parameter in the Bayesian function is optional, but we will see excluding it has its own consequences. \n\n\nAs we acquire more and more instances of evidence, our prior belief is *washed out* by the new evidence. This is to be expected. For example, if your prior belief is something ridiculous, like \"I expect the sun to explode today\", and each day you are proved wrong, you would hope that any inference would correct you, or at least align your beliefs better. \n\n\nDenote $N$ as the number of instances of evidence we possess. As we gather an *infinite* amount of evidence, say as $N \\rightarrow \\infty$, our Bayesian results align with frequentist results. Hence for large $N$, statistical inference is more or less objective. On the other hand, for small $N$, inference is much more *unstable*: frequentist estimates have more variance and larger confidence intervals. This is where Bayesian analysis excels. By introducing a prior, and returning a distribution (instead of a scalar estimate), we *preserve the uncertainity* to reflect the instability of stasticial inference of a small $N$ dataset. \n\nOne may think that for large $N$, one can be indifferent between the two techniques, and might lean towards the computational-simpler, frequentist methods. An analyst in this position should consider the following quote by Andrew Gelman (2005)[1], before making such a decision:\n\n> Sample sizes are never large. If $N$ is too small to get a sufficiently-precise estimate, you need to get more data (or make more assumptions). But once $N$ is \"large enough,\" you can start subdividing the data to learn more (for example, in a public opinion poll, once you have a good estimate for the entire country, you can estimate among men and women, northerners and southerners, different age groups, etc etc). $N$ is never enough because if it were \"enough\" you'd already be on to the next problem for which you need more data.\n\n\n#### A note on *Big Data*\nParadoxically, big data's predictive analytic problems are actually solved by relatively simple models [2][4]. Thus we can argue that big data's prediction difficulty does not lie in the algorithm used, but instead on the computational difficulties of storage and execution on big data. (One should also consider Gelman's qoute from above and ask \"Do I really have big data?\" )\n\nThe much more difficult analytic problems involve *medium data* and, especially troublesome, *really small data*. Using a similar argument as Gelman's above, if big data problems are *big enough* to be readily solved, then we should be more interested in the *not-quite-big enough* datasets. \n\n### Our Bayesian framework\n\nWe are interested in beliefs, which can be interpreted as probabilities by thinking Bayesian. We have a *prior* belief in event $A$: it is what you believe before looking at any evidence, e.g., our prior belief about bugs being in our code before performing tests.\n\nSecondly, we observe our evidence. To continue our example, if our code passes $X$ tests, we want to update our belief to incorporate this. We call this new belief the *posterior* probability. Updating our belief is done via the following equation, known as Bayes' Theorem, after Thomas Bayes:\n\n\\begin{align}\n P( A | X ) = & \\frac{ P(X | A) P(A) } {P(X) } \\\\\\\\[5pt]\n& \\propto P(X | A) P(A)\\;\\; (\\propto \\text{is proportional to } )\n\\end{align}\n\nThe above formula is not unique to Bayesian inference: it is a mathematical fact with uses outside Bayesian inference. Bayesian inference merely uses it to connect $P(A)$ with an updated $P(A | X )$.\n\n#####Example: Bug, or just sweet, unintended feature?\n\n\nLet $A$ denote the event that our code has **no bugs** in it. Let $X$ denote the event that the code passes all debugging tests. For now, we will leave the prior probability of no bugs as a variable, i.e. $P(A) = p$. \n\nWe are interested in $P(A|X)$, i.e. the probability of no bugs, given our debugging tests $X$. To use the formula above, we need to compute some quantities from the formula above.\n\nWhat is $P(X | A)$, i.e., the probability that the code passes $X$ tests *given* there are no bugs? Well, it is equal to 1, for a code with no bugs will pass all tests. \n\n$P(X)$ is a little bit trickier: The event $X$ can be divided into two possibilities, event $X$ ocurring even though our code *indeed has* bugs (denoted $\\sim A\\;$, spoken *not $A$*), or event $X$ without bugs ($A$). $P(X)$ can be represented as:\n\n\\begin{align}\nP(X ) & = P(X \\text{ and } A) + P(X \\text{ and } \\sim A) \\\\\\\\[5pt]\n & = P(X|A)P(A) + P(X | \\sim A)P(\\sim A)\\\\\\\\[5pt]\n& = P(X|A)p + P(X | \\sim A)(1-p)\n\\end{align}\n\nWe have already computed $P(X|A)$ above. On the other hand, $P(X | \\sim A)$ is subjective: our code can pass tests but still have a bug in it, though the probability there is a bug present is less. Note this is dependent on the number of tests performed, the degree of complication in the tests, etc. Let's be conservative and assign $P(X|\\sim A) = 0.5$. Then\n\n\\begin{align}\nP(A | X) & = \\frac{1\\cdot p}{ 1\\cdot p +0.5 (1-p) } \\\\\\\\\n& = \\frac{ 2 p}{1+p}\n\\end{align}\nThis is the posterior probability distribution. What does it look like as a function of our prior, $p \\in [0,1]$? \n\n\n```\nfigsize( 9, 4 )\np = np.linspace( 0,1, 50)\nplt.plot( p, 2*p/(1+p), color = \"#348ABD\" )\nplt.fill_between( p, 2*p/(1+p), alpha = .4, facecolor = [\"#348ABD\"])\nplt.scatter( 0.2, 2*(0.2)/1.2, s = 140, c =\"#348ABD\" )\nplt.xlim( 0, 1)\nplt.ylim( 0, 1)\nplt.xlabel( \"Prior, $P(A) = p$\")\nplt.ylabel(\"Posterior, $P(A|X)$, with $P(A) = p$\")\nplt.title( \"Are there bugs in my code?\");\n```\n\nWe can see the biggest gains if we observe the $X$ tests passed are when the prior probability, $p$, is low. Let's settle on a specific value for the prior. I'm a (I think) strong programmer, so I'm going to give myself a realistic prior of 0.20, that is, there is a 20% chance that I write code bug-free. To be more realistic, this prior should be a function of how complicated and large the code is, but let's pin it at 0.20. Then my updated belief that my code is bug-free is 0.33. \n\nRecall that the prior is a probability distribution: $p$ is the prior probability that there *are no bugs*, so $1-p$ is the prior probability that there *are bugs*.\n\nSimilarly, our posterior is also a probability distribution, with $P(A | X)$ the probability there is no bug *given we saw all tests pass*, hence $1-P(A|X)$ is the probability there is a bug *given all tests passed*. What does our posterior probability distribution look like? Below is a graph of both the prior and the posterior distributions. \n\n\n\n```\nfigsize( 9, 4 )\nprior = [0.20, 0.80]\nposterior = [1./3, 2./3]\nplt.bar( [0,.7], prior ,alpha = 0.80, width = 0.25, \\\n color = \"#348ABD\", label = \"prior distribution\" )\nplt.bar( [0+0.25,.7+0.25], posterior ,alpha = 0.7, \\\n width = 0.25, color = \"#A60628\", label = \"posterior distribution\" )\n\nplt.xticks( [0.20,.95], [\"Bugs Absent\", \"Bugs Present\"] )\nplt.title(\"Prior and Posterior probability of bugs present, prior = 0.2\")\nplt.ylabel(\"Probability\")\nplt.legend(loc=\"upper left\");\n```\n\nNotice that after we observed $X$ occur, the probability of bugs being absent increased. By increasing the number of tests, we can approach confidence (probability 1) that there are no bugs present.\n\nThis was a very simple example of Bayesian inference and Bayes rule. Unfortunately, the mathematics necessary to perform more complicated Bayesian inference only becomes more difficult, except for artifically constructed cases. We will later see that this type of mathematical anaylsis is actually unnecessary. First we must broaden our modeling tools.\n\n_______\n\n##Probability Distributions\n\n\n**Let's quickly recall what a probability distribution is:** Let $Z$ be some random variable. Then associated with $Z$ is a *probability distribution function* that assigns probabilities to the different outcomes $Z$ can take. There are three cases:\n\n- **$Z$ is discrete**: Discrete random variables may only assume values on a specified list. Things like populations, movie ratings, and number of votes are all discrete random variables. It's more clear when we contrast it with...\n\n- **$Z$ is continuous**: Continuous random variable can take on arbitrarily exact values. For example, temperature, speed, time, color are all modeled as continuous variables because you can constantly make the values more and more precise.\n\n- **$Z$ is mixed**: Mixed random variables assign probabilities to both discrete and continuous random variables, i.e. is is a combination of the above two categories. \n\n###Discrete Case\nIf $Z$ is discrete, then its distribution is called a *probability mass function*, which measures the probability $Z$ takes on the value $k$, denoted $P(Z=k)$. Note that the probability mass function completely describes the random variable $Z$, that is, if we know the mass function, we know how $Z$ should behave. There are popular probability mass functions that consistently appear: we will introduce them as needed, but let's introduce the first very useful probability mass function. We say $Z$ is *Poisson*-distributed if:\n\n$$P(Z = k) =\\frac{ \\lambda^k e^{-\\lambda} }{k!}, \\; \\; k=0,1,2, \\dots $$\n\nWhat is $\\lambda$? It is called the parameter, and it describes the shape of the distribution. For the Poisson random variable, $\\lambda$ can be any positive number. By increasing $\\lambda$, we add more probability to larger values, and conversely by decreasing $\\lambda$ we add more probability to smaller values. Unlike $\\lambda$, which can be any positive number, $k$ must be a non-negative integer, i.e., $k$ must take on values 0,1,2, and so on. This is very important, because if you wanted to model a population you could not make sense of populations with 4.25 or 5.612 members. \n\nIf a random variable $Z$ has a Poisson mass distribution, we denote this by writing\n\n$$Z \\sim \\text{Poi}(\\lambda) $$\n\nOne very useful property of the Poisson random variable, given we know $\\lambda$, is that its expected value is equal to the parameter, ie.:\n\n$$E\\large[ \\;Z\\; | \\; \\lambda \\;\\large] = \\lambda $$\n\nWe will use this property often, so it's something useful to remember. Below we plot the probablity mass distribution for different $\\lambda$ values. The first thing to notice is that by increasing $\\lambda$ we add more probability to larger values occuring. Secondly, notice that although the graph ends at 15, the distributions do not. They assign positive probability to every non-negative integer..\n\n\n```\nfigsize( 12.5, 4)\nimport scipy.stats as stats\na = np.arange( 16 )\npoi = stats.poisson\nlambda_ = [1.5, 4.25 ]\ncolours = [\"#348ABD\", \"#A60628\"]\n\nplt.bar( a, poi.pmf( a, lambda_[0]), color=colours[0],\n label = \"$\\lambda = %.1f$\"%lambda_[0], alpha = 0.95)\nplt.bar( a, poi.pmf( a, lambda_[1]), color=colours[1],\n label = \"$\\lambda = %.1f$\"%lambda_[1], alpha = 0.60)\n\nplt.xticks( a + 0.4, a )\nplt.legend()\nplt.ylabel(\"probability of $k$\")\nplt.xlabel(\"$k$\")\nplt.title(\"Probability mass function of a Poisson random variable; differing \\\n$\\lambda$ values\");\n```\n\n\n```\nviews['PoissonDistribution'] # show interactive view\n```\n\n###Continuous Case\nInstead of a probability mass function, a continuous random variable has a *probability density function*. This might seem like unnecessary nomenclature, but the density function and the mass function are very different creatures. An example of continuous random variable is a random variable with a *exponential density*. The density function for an exponential random variable looks like:\n\n$$f_Z(z | \\lambda) = \\lambda e^{-\\lambda z }, \\;\\; z\\ge 0$$\n\nLike the Poisson random variable, an exponential random variable can only take on non-negative values. But unlike a Poisson random variable, the exponential can take on *any* non-negative values, like 4.25 or 5.612401. This makes it a poor choice for count data, which must be integers, but a great choice for time data, or temperature data (measured in Kelvins, of course), or any other precise *and positive* variable. Below are two probability density functions with different $\\lambda$ value. \n\nWhen a random variable $Z$ has an exponential distribution with parameter $\\lambda$, we say *$Z$ is exponential* and write\n\n$$Z \\sim \\text{Exp}(\\lambda)$$\n\nGiven a specific $\\lambda$, the expected value of an exponential random variable is equal to the inverse of $\\lambda$, that is:\n\n$$E[\\; Z \\;|\\; \\lambda \\;] = \\frac{1}{\\lambda}$$\n\n\n```\na = np.linspace(0,4, 100)\nexpo = stats.expon\nlambda_ = [0.5, 1]\ncolours = [ \"#A60628\", \"#348ABD\"]\nfor l,c in zip(lambda_,colours):\n plt.plot( a, expo.pdf( a, scale=1./l), lw=3, \n color=c, label = \"$\\lambda = %.1f$\"%l)\n plt.fill_between( a, expo.pdf( a, scale=1./l), color=c, alpha = .33)\n \nplt.legend()\nplt.ylabel(\"PDF at $z$\")\nplt.xlabel(\"$z$\")\nplt.title(\"Probability density function of an Exponential random variable;\\\n differing $\\lambda$\");\n```\n\nYou can experiment with different values of the $\\lambda$ parameter by executing\nthe cell below. Hit \"Freeze\" to store a static version in the notebook.\n\n\n```\nviews['ExponentialDistribution'] # show interactive view\n\n```\n\n\n###But what is $\\lambda \\;\\;$?\n\n\n**This question is what motivates statistics**. In the real world, $\\lambda$ is hidden from us. We only see $Z$, and must go backwards to try and determine $\\lambda$. The problem is so difficult because there is not a one-to-one mapping from $Z$ to $\\lambda$. Many different methods have been created to solve the problem of estimating $\\lambda$, but since $\\lambda$ is never actually observed, no one can say for certain which method is better! \n\nBayesian inference is concerned with *beliefs* about what $\\lambda$ is. Rather than try to guess $\\lambda$ exactly, we can only talk about what $\\lambda$ is likely to be by assigning a probability distribution to $\\lambda$.\n\nThis might seem odd at first: after all, $\\lambda$ is fixed, it is not (necessarily) random! How can we assign probabilities to a non-random event. Ah, we have fallen for the frequentist interpretation. Recall, under our Bayesian philosophy, we *can* assign probabilties if we interpret them as beliefs. And it is entirely acceptable to have *beliefs* about the parameter $\\lambda$. \n\n\n\n##### Example: Inferring behaviour from text-message data\n\nLet's try to model a more interesting example, concerning text-message rates:\n\n> You are given a series of text-message counts from a user of your system. The data, plotted over time, appears in the graph below. You are curious if the user's text-messaging habits changed over time, either gradually or suddenly. How can you model this? (This is in fact my own text-message data. Judge my popularity as you wish.)\n\n\n\n```\nfigsize( 12, 3.5 )\ncount_data = np.loadtxt(\"data/txtdata.csv\")\nn_count_data = len(count_data)\nplt.bar( np.arange( n_count_data ), count_data, color =\"#348ABD\" )\nplt.xlabel( \"Time (days)\")\nplt.ylabel(\"# Text-msg received\")\nplt.title(\"Did the user's texting habits change over time?\")\nplt.xlim( 0, n_count_data );\n```\n\nBefore we begin, with resepect to the plot above, would you say there was a change in behaviour\nduring the time period? \n\nHow can we start to model this? Well, as I conveniently already introduced, a Poisson random variable would be a very appropriate model for this *count* data. Denoting day $i$'s text-message count by $C_i$, \n\n$$ C_i \\sim \\text{Poisson}(\\lambda) $$\n\nWe are not sure about what the $\\lambda$ parameter is though. Looking at the chart above, it appears that the rate might become higher at some later date, which is equivalently saying the parameter $\\lambda$ increases at some later date (recall a higher $\\lambda$ means more probability on larger outcomes, that is, higher probability of many texts.).\n\nHow can we mathematically represent this? We can think, that at some later date (call it $\\tau$), the parameter $\\lambda$ suddenly jumps to a higher value. So we create two $\\lambda$ parameters, one for behaviour before the $\\tau$, and one for behaviour after. In literature, a sudden transition like this would be called a *switchpoint*:\n\n$$\n\\lambda = \n\\begin{cases}\n\\lambda_1 & \\text{if } t \\lt \\tau \\cr\n\\lambda_2 & \\text{if } t \\ge \\tau\n\\end{cases}\n$$\n\n\n If, in reality, no sudden change occurred and indeed $\\lambda_1 = \\lambda_2$, the $\\lambda$'s posterior distributions should look about equal.\n\nWhat would be good prior distributions for $\\lambda_1$ and $\\lambda_2$? Recall that $\\lambda_i, \\; i=1,2,$ can be any positive number. The *exponential* random variable has a density function for any positive number. This would be a good choice to model $\\lambda_i$. But again, we need a parameter for this exponential distribution: call it $\\alpha$.\n\n\\begin{align}\n&\\lambda_1 \\sim \\text{Exp}( \\alpha ) \\\\\\\n&\\lambda_2 \\sim \\text{Exp}( \\alpha )\n\\end{align}\n\n$\\alpha$ is called a *hyper-parameter*, or a *parent-variable*, literally a parameter that influences other parameters. The influence is not too strong, so we can choose $\\alpha$ liberally. A good rule of thumb is to set the exponential parameter equal to the inverse of the average of the count data, since we're modeling $\\\\lambda$ using an Exponential distribution we can use the expected value identity shown earlier to get:\n\n$$\\frac{1}{N}\\sum_{i=0}^N \\;C_i \\approx E[\\; \\lambda \\; |\\; \\alpha ] = \\frac{1}{\\alpha}$$ \n\nAlternatively, and something I encourage the reader to try, is to have two priors: one for each $\\lambda_i$; creating two exponential distributions with different $\\alpha$ values reflects a prioer belief that the rate changed after some period.\n\nWhat about $\\tau$? Well, due to the randomness, it is too difficult to pick out when $\\tau$ might have occurred. Instead, we can assign an *uniform prior belief* to every possible day. This is equivalent to saying\n\n\\begin{align}\n& \\tau \\sim \\text{DiscreteUniform(1,70) }\\\\\\\\\n& \\Rightarrow P( \\tau = k ) = \\frac{1}{70}\n\\end{align}\n\nSo after all this, what does our prior distribution look like? Frankly, *it doesn't matter*. What we should understand is that it would be an ugly, complicated, mess involving symbols only a mathematician would love. And things would only get uglier the more complicated our models become. Regardless, all we really care about is the posterior distribution anyways. We next turn to PyMC, a Python library for performing Bayesian analysis, and is agnostic to the mathematical monster we have created. \n\n\nIntroducing our first hammer: PyMC\n-----\n\nPyMC is a Python library for programming Bayesian analysis [3]. It is a fast, well-maintained library. The only unfortunate part is that documentation can be lacking in areas, especially the bridge between problem to solution. One this book's main goals is to solve that problem, and also to demonstrate why PyMC is so cool.\n\nWe will model the above problem using the PyMC library. This type of programming is called *probabilistic programming*, an unfortunate misnomer that invokes ideas of randomly-generated code and has likely confused and frightened users away from this field. The code is not random. The title is given because we create probability models using pogramming variables as the model's components. This will be the last time I use the term *probablistic programming*. Instead, I'll simply use *programming*, as that is what it really is. \n\n\nThe PyMC code is easy to follow along: the only novel thing should be the syntax, and I will interrupt the code to explain sections. Simply remember we are representing the model's components ($\\tau, \\lambda_1, \\lambda_2$ ) as variables:\n\n\n```\nimport pymc as mc\n\nn = count_data.shape[0]\n\nalpha = 1.0/count_data.mean() #recall count_data is \n #the variable that holds our txt counts\n\nlambda_1 = mc.Exponential( \"lambda_1\", alpha )\nlambda_2 = mc.Exponential( \"lambda_2\", alpha )\n\ntau = mc.DiscreteUniform( \"tau\", lower = 0, upper = n )\n```\n\nIn the above code, we create the PyMC variables corresponding to $\\lambda_1, \\; \\lambda_2$ in lines `8,9`. We assign them to PyMC's *stochastic variables*, called stochastic variables because they are treated by the backend as random number generators. We can test this by calling their built-in `random()` method.\n\n\n```\nprint \"Random output:\", tau.random(),tau.random(), tau.random()\n```\n\n Random output: 64 59 36\n\n\n\n```\n@mc.deterministic\ndef lambda_( tau = tau, lambda_1 = lambda_1, lambda_2 = lambda_2 ):\n out = np.zeros( n ) \n out[:tau] = lambda_1 #lambda before tau is lambda1\n out[tau:] = lambda_2 #lambda after tau is lambda2\n return out\n```\n\nThis code is creating a new function `lambda_`, but really we think of it as a random variable: the random variable $\\lambda$ from above. Note that because `lambda_1`, `lambda_2` and `alpha` are random, `lambda_` will be random. We are **not** fixing any variables yet. The `@mc.deterministic` is a decorator to tell PyMC that this is a deterministic function, i.e., if the arguments were deterministic (which they are not), the output would be deterministic as well. \n\n\n```\nobservation = mc.Poisson( \"obs\", lambda_, value = count_data, observed = True)\n\nmodel = mc.Model( [observation, lambda_1, lambda_2, tau] )\n```\n\nThe variable `observations` combines our data, `count_data`, with our proposed data-generation scheme, given by the variable `lambda_`, through the `value` keyword. We also set `observed = True` to tell PyMC that this should stay fixed in our analysis. Finally, PyMC wants us to collect all the variables of interest and create a `Model` instance out of them. This makes our life easier when we try to retrieve the results.\n\n\n\n\n```\n### Myserious code to be explained later.\nmcmc = mc.MCMC(model)\nmcmc.sample( 100000, 50000, 1 )\n```\n\n [****************100%******************] 100000 of 100000 complete\n\n\nThe above code will be explained in the Chapter 3, but this is where our results come from. The machinery being employed is called *Monte Carlo Markov Chains* (which I delay explaining until Chapter 3). It returns thousands of random variables from the posterior distributions of $\\lambda_1, \\lambda_2$ and $\\tau$. We can plot a histogram of the random variables to see what the posterior distribution looks like. Below, we collect the samples (called *traces* in MCMC literature) in histograms.\n\n\n```\nlambda_1_samples = mcmc.trace( 'lambda_1' )[:]\nlambda_2_samples = mcmc.trace( 'lambda_2' )[:]\ntau_samples = mcmc.trace( 'tau' )[:]\n```\n\n\n```\nfigsize(12.5, 10)\n#histogram of the samples:\n\nax = plt.subplot(311)\nax.set_autoscaley_on(False)\n\nplt.hist( lambda_1_samples, histtype='stepfilled', bins = 60, alpha = 0.85, \n label = \"posterior of $\\lambda_1$\", color = \"#A60628\",normed = True )\nplt.legend(loc = \"upper left\")\nplt.title(r\"Posterior distributions of the variables $\\lambda_1,\\;\\lambda_2,\\;\\tau$\")\nplt.xlim([15,30])\nplt.xlabel(\"$\\lambda$ value\")\nplt.ylabel(\"probability\")\n\nax = plt.subplot(312)\nax.set_autoscaley_on(False)\n\nplt.hist( lambda_2_samples,histtype='stepfilled', bins = 60, alpha = 0.85, \n label = \"posterior of $\\lambda_2$\",color=\"#7A68A6\", normed = True )\nplt.legend(loc = \"upper left\")\nplt.xlim([15,30])\nplt.xlabel(\"$\\lambda$ value\")\nplt.ylabel(\"probability\")\n\nplt.subplot(313)\n\n\nw = 1.0/ tau_samples.shape[0] * np.ones_like( tau_samples )\nplt.hist( tau_samples, bins = n_count_data, alpha = 1, \n label = r\"posterior of $\\tau$\",\n color=\"#467821\", weights=w, rwidth =1. )\n\nplt.legend(loc = \"upper left\");\nplt.ylim([0,.75])\nplt.xlim([35, len(count_data)-20])\nplt.xlabel(\"days\")\nplt.ylabel(\"probability\");\n```\n\n\n```\nviews['MCMCView'] # show interactive view\n\n```\n\n### Interpretation\n\nRecall that the Bayesian methodology returns a *distribution*, hence we now have distributions to describe the unknown $\\lambda$'s and $\\tau$. What have we gained? Immediately we can see the uncertainty in our estimates: the more variance in the distribution, the less certain our posterior belief should be. We can also say what a plausible value for the parameters might be: $\\lambda_1$ is around 18 and $\\lambda_2$ is around 23. What other observations can you make? Look at the data again, do these seem reasonable? The distributions of the two $\\\\lambda$s are positioned very differently, indicating that it's likely there was a change in the user's text-message behaviour.\n\nAlso notice that posteriors' distributions do not look like any Poisson distributions, though we originally started modelling with Poisson random variables. They are really not anything we recognize. But this is OK. This is one of the benefits of taking a computational point-of-view. If we had instead done this mathematically, we would have been stuck with a very analytically intractable (and messy) distribution. Via computations, we are agnostic to the tractability.\n\nOur analysis also returned a distribution for what $\\tau$ might be. Had no change occurred, or the change been gradual over time, the posterior distribution of $\\tau$ would have been more spread out, reflecting that many values are likely candidates for $\\tau$. On the contrary, it is very peaked. It appears that near day 45, the individual's text-message behavior suddenly changed.\n\n###Why would I want samples from the posterior, anyways?\n\n\nWe will deal with this question for the remainder of the book, and it is an understatement to say we can perform amazingly useful things. For now, let's , let's end this chapter with one more example. We'll use the posterior samples to answer the following question: what is the expected number of texts at day $t, \\; 0 \\le t \\le70$? Recall that the expected value of a Poisson is equal to its parameter $\\lambda$, then the question is equivalent to *what is the expected value of $\\lambda$ at time $t$*?\n\nIn the code below, we are calculating the following: Let $i$ index samples from the posterior distributions. Given a day $t$, we average over all possible $\\lambda_i$ for that day $t$, using $\\lambda_i = \\lambda_{1,i}$ if $t \\lt \\tau_i$ (that is, if the behaviour change hadn't occured yet), else we use $\\lambda_i = \\lambda_{2,i}$. \n\n\n\n\n\n___________________\n\n\n```\nfigsize( 12.5, 4)\n# tau_samples, lambda_1_samples, lambda_2_samples contain\n# N samples from the corresponding posterior distribution\nN = tau_samples.shape[0]\nexpected_texts_per_day = np.zeros(n_count_data)\nfor day in range(0, n_count_data):\n # ix is a bool index of all tau samples corresponding to\n # the switchpoint occuring prior to value of 'day'\n ix = day < tau_samples\n # Each posterior sample corresponds to a value for tau.\n # for each day, that value of tau indicates whether we're \"before\" (in the lambda1 \"regime\")\n # or \"after\" (in the lambda2 \"regime\") the switchpoint.\n # by taking the posterior sample of lambda1/2 accordingly, we can average\n # over all samples to get an expected value for lambda on that day.\n # As explained, the \"message count\" random variable is Poisson distributed, \n # and therefore lambda (the poisson parameter) is the expected value of \"message count\"\n expected_texts_per_day[day] = (lambda_1_samples[ix].sum() \n + lambda_2_samples[~ix].sum() ) /N\n\n \nplt.plot( range( n_count_data), expected_texts_per_day, lw =4, color = \"#E24A33\" )\nplt.xlim( 0, n_count_data )\nplt.xlabel( \"Day\" )\nplt.ylabel( \"Expected # text-messages\" )\nplt.title( \"Expected number of text-messages received\")\n#plt.ylim( 0, 35 )\nplt.bar( np.arange( len(count_data) ), count_data, color =\"#348ABD\", alpha = 0.5,\n label=\"observed texts per day\")\n\nplt.legend(loc=\"upper left\");\n```\n\nOur analysis shows strong support for believing the user's behavior did change ($\\lambda_1$ would have been been close in value to $\\lambda_2$ had this been true), and the the change was sudden rather then gradual (demonstrated by $\\tau$ 's strongly peaked posterior distribution). We can speculate what might have caused this: a cheaper text-message rate, a recent weather-2-text subscription, or a new relationship. (The 45th day corresponds to Christmas, and I moved away to Toronto the next month leaving a girlfriend behind)\n\n\n##### Exercises\n\n1\\. Using `lambda_1_samples` and `lambda_2_samples`, what is the mean of the posterior distributions of $\\lambda_1$ and $\\lambda_2$?\n\n\n```\n#type your code here.\n```\n\n2\\. What is the expected percent increase text-message rates? `hint:` compute the mean of `lambda_1_samples/lambda_2_samples`. Note that quanitity is very different from `lambda_1_samples.mean()/lambda_2_samples.mean()`.\n\n\n```\n#type your code here.\n```\n\n3\\. Looking at the posterior distribution graph of $\\tau$, why do you think there is a small number of posterior $\\tau$ samples near 0? `hint:` Look at the data again.\n\n4\\. What is the mean of $\\lambda_1$ **given** we know $\\tau$ is less than 45. That is, suppose we have new information as we know for certain that the change in behaviour occured before day 45. What is the expected value of $\\lambda_1$ now? (You do not need to redo the PyMC part, just consider all instances where `tau_trace<45`. )\n\n\n```\n#type your code here.\n```\n\n### References\n\n\n- [1] Gelman, Andrew. N.p.. Web. 22 Jan 2013. .\n- [2] Norvig, Peter. 2009. [*The Unreasonable Effectivness of Data*](http://www.csee.wvu.edu/~gidoretto/courses/2011-fall-cp/reading/TheUnreasonable EffectivenessofData_IEEE_IS2009.pdf).\n- [3] Patil, A., D. Huard and C.J. Fonnesbeck. 2010. \nPyMC: Bayesian Stochastic Modelling in Python. Journal of Statistical \nSoftware, 35(4), pp. 1-81. \n- [4] Jimmy Lin and Alek Kolcz. Large-Scale Machine Learning at Twitter. Proceedings of the 2012 ACM SIGMOD International Conference on Management of Data (SIGMOD 2012), pages 793-804, May 2012, Scottsdale, Arizona.\n\n\n```\nfrom IPython.core.display import HTML\ndef css_styling():\n styles = open(\"../styles/custom.css\", \"r\").read()\n return HTML(styles)\ncss_styling()\n```\n\n\n\n\n\n\n\n\n\n\n```\n\n```\n", "meta": {"hexsha": "5b23534886abd9a3a8232a58abd78728c1d51255", "size": 283003, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "files/Chapter1_Introduction[1].ipynb", "max_stars_repo_name": "tarankalra/ipython-notebooks", "max_stars_repo_head_hexsha": "7cdcad63c896985747db6e2a0529f1e4391792e8", "max_stars_repo_licenses": ["Unlicense"], "max_stars_count": 32, "max_stars_repo_stars_event_min_datetime": "2015-01-07T01:48:05.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-02T07:07:42.000Z", "max_issues_repo_path": "files/Chapter1_Introduction[1].ipynb", "max_issues_repo_name": "tarankalra/ipython-notebooks", "max_issues_repo_head_hexsha": "7cdcad63c896985747db6e2a0529f1e4391792e8", "max_issues_repo_licenses": ["Unlicense"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2015-04-13T21:00:18.000Z", "max_issues_repo_issues_event_max_datetime": "2015-04-13T21:00:18.000Z", "max_forks_repo_path": "files/Chapter1_Introduction[1].ipynb", "max_forks_repo_name": "tarankalra/ipython-notebooks", "max_forks_repo_head_hexsha": "7cdcad63c896985747db6e2a0529f1e4391792e8", "max_forks_repo_licenses": ["Unlicense"], "max_forks_count": 30, "max_forks_repo_forks_event_min_datetime": "2015-01-28T09:31:29.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-07T03:08:28.000Z", "avg_line_length": 282.7202797203, "max_line_length": 50446, "alphanum_fraction": 0.8872132098, "converted": true, "num_tokens": 9867, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857982, "lm_q2_score": 0.7461389873857264, "lm_q1q2_score": 0.41946180510298037}} {"text": "```python\nimport numpy as np\nimport pandas as pd\nimport scipy\nimport matplotlib.pyplot as plt\nimport itertools\nimport seaborn as sns\nfrom tqdm import tqdm\nfrom cvxpy import *\nfrom DTools2 import *\nfrom matplotlib import gridspec\nfrom matplotlib.patches import Rectangle\n%matplotlib inline\n\n```\n\n# Data prep\n\nImport the COMPAS dataset.\n\n\n```python\nDATA_FOLDER = '../data/'\n\ndf = pd.read_csv(\n DATA_FOLDER + 'compas-scores-two-years.csv',\n index_col=0)\n```\n\n\n```python\ndf.shape\n```\n\n\n\n\n (7214, 52)\n\n\n\nHowever not all of the rows are useable for the first round of analysis.\n\n**From the ProPublica notebook**: There are a number of reasons remove rows because of missing data:\n* If the charge date of a defendants Compas scored crime was not within 30 days from when the person was arrested, we assume that because of data quality reasons, that we do not have the right offense.\n* We coded the recidivist flag -- `is_recid` -- to be -1 if we could not find a compas case at all.\n* In a similar vein, ordinary traffic offenses -- those with a `c_charge_degree` of 'O' -- will not result in Jail time are removed (only two of them).\n* We filtered the underlying data from Broward county to include only those rows representing people who had either recidivated in two years, or had at least two years outside of a correctional facility.\n\n\n```python\ndf = df[['age', 'c_charge_degree', 'race', 'age_cat', 'score_text', 'sex', 'priors_count', \n 'days_b_screening_arrest', 'decile_score', 'is_recid', 'two_year_recid', 'c_jail_in', 'c_jail_out']]\nix = df['days_b_screening_arrest'] <= 30\nix = (df['days_b_screening_arrest'] >= -30) & ix\nix = (df['is_recid'] != -1) & ix\nix = (df['c_charge_degree'] != \"O\") & ix\nix = (df['score_text'] != 'N/A') & ix\ndf = df.loc[ix,:]\ndf['length_of_stay'] = (pd.to_datetime(df['c_jail_out'])-pd.to_datetime(df['c_jail_in'])).apply(lambda x: x.days)\nlist(df)\n```\n\n\n\n\n ['age',\n 'c_charge_degree',\n 'race',\n 'age_cat',\n 'score_text',\n 'sex',\n 'priors_count',\n 'days_b_screening_arrest',\n 'decile_score',\n 'is_recid',\n 'two_year_recid',\n 'c_jail_in',\n 'c_jail_out',\n 'length_of_stay']\n\n\n\nOut of interest, plot distribution of COMPAS scores (matches the one in the ProPublica article).\n\n\n```python\ndf2 = df.loc[df['race'].isin(['African-American','Caucasian']),['race','decile_score']]\nax = df2.hist(column='decile_score',by='race',figsize=(15,5),**{'normed':False})\nax[0].set_ylim([0,650])\nax[1].set_ylim([0,650])\nax[0].set_xlim([.5,10.5])\nax[1].set_xlim([.5,10.5])\n```\n\nNumber of entries per decile score for each race.\n\n\n```python\ndf.groupby(['race','decile_score']).size().reset_index().pivot(index='decile_score',columns='race',values=0)\n```\n\n\n\n\n
                                        \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        raceAfrican-AmericanAsianCaucasianHispanicNative AmericanOther
                                        decile_score
                                        1365.015.0605.0159.0NaN142.0
                                        2346.04.0321.089.02.060.0
                                        3298.05.0238.073.01.032.0
                                        4337.0NaN243.047.0NaN39.0
                                        5323.01.0200.039.0NaN19.0
                                        6318.02.0160.027.02.020.0
                                        7343.01.0113.028.02.09.0
                                        8301.02.096.014.0NaN7.0
                                        9317.0NaN77.017.02.07.0
                                        10227.01.050.016.02.08.0
                                        \n
                                        \n\n\n\nDrop Asian, Native American due to lack of samples\n\n\n```python\ndfcut = df.loc[~df['race'].isin(['Native American','Hispanic','Asian','Other']),:]\n```\n\nNow we quantize the dataframe. In particular, we will quantize the priors, the length of stay and the compas score.\n\n\n```python\ndfcutQ = dfcut[['sex','race','age_cat','c_charge_degree','score_text','priors_count','is_recid',\n 'two_year_recid','length_of_stay']].copy()\n\n# Quantize priors count between 0, 1-3, and >3\ndef quantizePrior(x):\n if x <=0:\n return '0'\n elif 1<=x<=3:\n return '1 to 3'\n else:\n return 'More than 3'\n\n \n# Quantize length of stay\ndef quantizeLOS(x):\n if x<= 7:\n return '3 months'\n \n# Quantize length of stay\ndef adjustAge(x):\n if x == '25 - 45':\n return '25 to 45'\n else:\n return x\n\n# Quantize score_text to MediumHigh\ndef quantizeScore(x):\n if (x == 'High')| (x == 'Medium'):\n return 'MediumHigh'\n else:\n return x\n\n \ndfcutQ['priors_count'] = dfcutQ['priors_count'].apply(lambda x: quantizePrior(x))\ndfcutQ['length_of_stay'] = dfcutQ['length_of_stay'].apply(lambda x: quantizeLOS(x))\ndfcutQ['score_text'] = dfcutQ['score_text'].apply(lambda x: quantizeScore(x))\ndfcutQ['age_cat'] = dfcutQ['age_cat'].apply(lambda x: adjustAge(x))\n\n\n```\n\nWe'll be interested, for now, in gender, race, age, charge degree, priors count, and recidivism (the Y variable).\n\n\n```python\n#features = ['sex','race','age_cat','c_charge_degree','priors_count','is_recid']\nfeatures = ['race','age_cat','c_charge_degree','priors_count','is_recid']\n\n# Pass vallue to df\ndf = dfcutQ[features]\n```\n\nNext, we do a 80-20 split of the data. The random number generator seed is fixed, so this should generate consistent splits. We automatically rename output files accordingly. Pairs of train and test dataset are stored in `df_list`.\n\n\n```python\ndf.shape\n```\n\n\n\n\n (5278, 5)\n\n\n\n\n```python\nfrom sklearn.model_selection import ShuffleSplit\nrs = ShuffleSplit(n_splits=5, test_size=.2, random_state=888) ### CHANGE SEED FOR DIFFERENT SPLITS!\ndf_list = []\nfor train_index,test_index in rs.split(df):\n df_list.append((df.iloc[train_index,:].copy(),df.iloc[test_index,:].copy()))\n \n```\n\nWe'll be interested, for now, in gender, race, age, charge degree, priors count, and recidivism (the Y variable). We initialize the optimization object.\n\n\n```python\nDT = DTools(df=df,features=features)\n```\n\nSet discriminatory features (`D_features`), binary response variable (`Y_features`) and decision features (`X_features`), and initialize the Discrimination Tools class).\n\n\n```python\n#D_features = ['sex','race']\nD_features = ['race']\nY_features = ['is_recid']\nX_features = ['age_cat', 'c_charge_degree','priors_count']\n\nDT.setFeatures(D=D_features,X=X_features,Y=Y_features)\n```\n\nNow we set the distortion metric. This function will receive the two dictionary of features X and Y corresponding to the new and old values, and return a distortion value. In this case the distortion function returns 1000 if education is lowered by any value, or increased by more than 1 year, or if age is increased or decreased by more than one decade. In returns a penalty of 1.0 if income is decreased. All other values are 0.\n\n\n```python\nclass Dclass():\n# adjust education\n def adjustPrior(self,v):\n if v=='0':\n return 0\n elif v=='1 to 3':\n return 1\n else:\n return 2\n\n def adjustAge(self,a):\n if a == 'Less than 25':\n return 0\n elif a == '25 to 45':\n return 1\n else:\n return 2\n\n # distortion metric\n def getDistortion(self,vold,vnew):\n '''\n Distortion metric.\n\n Inputs:\n *vold : dictionary of the form {attr:value} with old values\n *vnew : dictionary of the form {attr:value} with new values\n\n Output\n *d : distortion value\n '''\n\n # value that will be returned for events that should not occur\n bad_val = 1e4\n\n\n # Adjust prior\n pOld = self.adjustPrior(vold['priors_count'])\n pNew = self.adjustPrior(vnew['priors_count'])\n\n # Priors cannot be increased, or lowered by more than 1 category. A change has a unit penalty\n if (pNew>pOld)| (pNew1.0:\n return bad_val\n \n # Recidivism should not be increased\n if vold['is_recid'] < vnew['is_recid']:\n return bad_val\n \n cum_sum = 0.0\n \n \n if np.abs(aOld-aNew)>0:\n# cum_sum+=1\n# cum_sum = cum_sum**2\n cum_sum = cum_sum+1\n \n # Penalty of 1 if priors is decreased or increased\n if np.abs(pNew-pOld)>0:\n# cum_sum+=1\n# cum_sum = cum_sum**2\n cum_sum = cum_sum+1\n \n\n #cum_sum = cum_sum**2\n if vold['is_recid'] > vnew['is_recid']:\n# cum_sum+=1\n# cum_sum = cum_sum**2\n cum_sum = cum_sum+1\n \n \n\n # final penalty of 2 for changing misdemeanor to felony and vice-verse\n if vold['c_charge_degree'] != vnew['c_charge_degree']:\n# cum_sum+=2\n# cum_sum = cum_sum**2\n cum_sum = cum_sum+4\n \n return cum_sum\n \n```\n\nWe set the excess distortion constraints (`c1` and `c2`) and set the distortion values. For now, we are solving the problem (to view the equation correctly you have to run the last markdown cell of this notebook with the latex preamble)\n\n\\begin{align}\n\t\\min_{p_{\\Xh,\\Yh|X,Y,D}}& \\sum_{x,y} \\left| p_{X,Y}(x,y)- p_{\\Xh,\\Yh}(x,y)\\right| \\\\\n\t\\sto~~& 1-\\epsilon\\leq \\frac{p_{\\Yh|D}(y|d_1)}{p_{\\Yh|D}(y|d_2)} \\leq 1+\\epsilon, \\forall y\\in \\calY, d_1,d_2\\in \\calD\\\\\n\t&\\mathbb{E}\\left( d(x,y,\\Xh,\\Yh) \\mid X=x,Y=y,D=d\\right) \\leq \\delta_D~\\forall~(x,y)\\in \\calX\\times \\calY,\\\\\n\t&p_{\\Xh,\\Yh|X,Y,D} \\mbox{ is a valid distribution.}\n\\end{align}\n\nWe set `c1=.99`, `c2 = 1.99`, and `c3=2.99`.\n\n\n```python\n# c1 = .99 # value of (delta1,c1): to keep.\n# c2 = 1.99 # value of (delta2,c2): value that should no happen\n# c3 = 2.99 # penalty things that should not happen\n# clist = [c1,c2, c3]\nDclass = Dclass()\n\nDT.setDistortion(Dclass)\n```\n\nNext, we generate the plot for choosing the operation points\n\n\n\n```python\n\nnpoints = 20\nepsilonV = np.linspace(0,.7,npoints)\ny = np.zeros(npoints)\nz = np.zeros(npoints)\n#epsilon = .05\nmeanV = np.linspace(3.45,3.46,npoints)\n#dlist = [0.15,0.075,0]\n# create same distortion for all categories\n\n# number of categories\nvalues = list(itertools.product(*DT.D_values))\n\n\nfor i in tqdm(range(npoints)):\n #mean = meanV[i]\n epsilon = epsilonV[i]\n dlist = []\n for v in values:\n if 'African-American' in v:\n #mean_value = .4 #original in ICML submission - mean_value = .25\n mean_value=.2 #used for data 3\n else:\n #mean_value = .3 #original in ICML submission - mean_value = .25\n mean_value=.1 # used for data 3\n dlist.append((v,mean_value))\n \n DT.optimize(epsilon=epsilon,dlist = dlist,verbose=False,cost='TV',solver='CBC')\n y[i] = DT.optimum\n \ny2 = np.array([max(t,0) for t in y])\n\nsns.set(font_scale=1.8,font='sans-serif')\nplt.figure(figsize = (10,5))\nax = plt.plot(epsilonV,y2,'-',linewidth=2)\nplt.ylabel(\"Objective Value\")\nplt.xlabel(\"$\\epsilon$\")\nplt.title(\"Objective vs. $\\epsilon$\")# for\\n$\\delta_1 =$\"+str(dlist[0])+\", $\\delta_2=$\"+str(dlist[1])+\" and $\\delta_3=$\"+str(dlist[2]))\ninfeasible = np.where(y==np.inf)[0]\nif len(infeasible) == 0:\n infeasible = [-1]\nplt.axvspan(0, epsilonV[infeasible[-1]+1], color='red', alpha=0.2)\nplt.xlim([epsilonV.min(),epsilonV.max()])\nplt.ylim([-0.0001,y2[y21e-8 else 0)\n dTest = dTest/dTest.sum()\n\n # this is the dataframe with the randomization for the test set\n dfPtest = dTest.divide(dTest.sum(axis=1),axis=0)\n\n # Randomize train data\n print('Randomizing training set...')\n df_train_new = randomize(df_train,dfPtrain,features = D_features+X_features+Y_features)\n\n # Randomize test data\n print('Randomizing test set...')\n df_test_new = randomize(df_test,dfPtest,features = D_features+X_features)\n\n # Save train files\n df_train.to_csv(result_folder+'train_'+file_name+'.csv')\n df_train_new.to_csv(result_folder+'train_new_'+file_name+'.csv')\n\n # Save test files\n df_test.to_csv(result_folder+'test_'+file_name+'.csv')\n df_test_new.to_csv(result_folder+'test_new_'+file_name+'.csv')\n \n # increment split number\n split_num+=1\n```\n\n -----------------\n Current split: 0\n\n\n /home/bhanu/anaconda3/envs/python2/lib/python2.7/site-packages/pandas/core/base.py:324: PerformanceWarning: dropping on a non-lexsorted multi-index without a level parameter may impact performance.\n return self.obj.drop(self.exclusions, axis=1)\n 1%| | 25/4222 [00:00<00:17, 242.82it/s]\n\n Randomizing training set...\n Randomizing...\n\n\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 4222/4222 [00:10<00:00, 411.20it/s]\n 5%|\u258d | 49/1056 [00:00<00:02, 482.78it/s]\n\n Randomizing test set...\n Randomizing...\n\n\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 1056/1056 [00:02<00:00, 457.10it/s]\n\n\n\n```python\nfrom sklearn.ensemble import RandomForestClassifier\nprint '----------------------------------------------------------------'\nprint 'RandForrest on perturbed data:'\n\n# performance on perturbed train data\nrf = RandomForestClassifier()\ndft = pd.get_dummies(df_train_new[D_features+X_features])\nrf.fit(dft,df_train_new[Y_features])\nprint 'Train performance (pert. dataset): '\nprint rf.score(dft,df_train_new[Y_features])\n\n#dft = pd.get_dummies(df_test_new[D_features+X_features])\n#print 'Test performance (pert. dataset): '\n#print rf.score(dft,df_test_new[Y_features])\n#print '---------------'\n\n# performance on perturbed train data compared to original train data\n#rf = RandomForestClassifier()\n#dft = pd.get_dummies(df_train_new[D_features+X_features])\n#rf.fit(dft,df_train_new[Y_features])\ndft = pd.get_dummies(df_test_new[D_features+X_features])\nprint 'Perturbed test performance when scored on original test y variable: '\nprint rf.score(dft,df_test[Y_features])\n\ndft = pd.get_dummies(df_test_new[D_features+X_features])\n# save performance\ndf_test_pred = df_test_new\ndf_test_pred['pred'] = rf.predict_proba(dft)[:,1]\n\n# prediction per class\nprint 'Discrimination metric:'\nmean = df_test_pred.groupby('race')['pred'].mean()\nv = mean.values\nv = v.reshape(len(v),1)\nratio_df = pd.DataFrame(v/v.transpose(),index=mean.index,columns=mean.index )\nratio_df_arr=np.asarray(np.abs(1-ratio_df))\ndiscrim=np.amax(ratio_df_arr)\ndiscrim\n\n```\n\n ----------------------------------------------------------------\n RandForrest on perturbed data:\n Train performance (pert. dataset): \n 0.671482709616\n Perturbed test performance when scored on original test y variable: \n 0.607954545455\n Discrimination metric:\n\n\n /home/bhanu/anaconda3/envs/python2/lib/python2.7/site-packages/ipykernel/__main__.py:8: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples,), for example using ravel().\n\n\n\n\n\n 0.48052210935440565\n\n\n\n## Latex preamble cell and importing packages\n$$\\newcommand{\\SNR}{\\mbox{SNR}}$$\n$$\\newcommand{\\dB}{\\mbox{dB}}$$\n$$\\newcommand{\\emu}{\\colonsim}$$\n\n$$\\newcommand{\\Fq}{\\mathbb{F}_{q}}$$\n$$\\newcommand{\\PR}{\\mbox{Pr}}$$\n$$\\newcommand{\\Lh}{\\hat{l}}$$\n$$\\newcommand{\\calX}{\\mathcal{X}}$$\n$$\\newcommand{\\calA}{\\mathcal{A}}$$\n$$\\newcommand{\\calB}{\\mathcal{B}}$$\n$$\\newcommand{\\calD}{\\mathcal{D}}$$\n$$\\newcommand{\\calK}{\\mathcal{K}}$$\n$$\\newcommand{\\calM}{\\mathcal{M}}$$\n$$\\newcommand{\\calY}{\\mathcal{Y}}$$\n$$\\newcommand{\\calT}{\\mathcal{T}}$$\n$$\\newcommand{\\calZ}{\\mathcal{Z}}$$\n$$\\newcommand{\\calJ}{\\mathcal{J}}$$\n$$\\newcommand{\\calC}{\\mathcal{C}}$$\n$$\\newcommand{\\calS}{\\mathcal{S}}$$\n$$\\newcommand{\\calU}{\\mathcal{U}}$$\n$$\\newcommand{\\calV}{\\mathcal{V}}$$\n$$\\newcommand{\\calI}{\\mathcal{I}}$$\n$$\\newcommand{\\calF}{\\mathcal{F}}$$\n$$\\newcommand{\\calG}{\\mathcal{G}}$$\n$$\\newcommand{\\calH}{\\mathcal{H}}$$\n$$\\newcommand{\\calP}{\\mathcal{P}}$$\n$$\\newcommand{\\calL}{\\mathcal{L}}$$\n$$\\newcommand{\\Xk}{\\mathcal{X}^k}$$\n$$\\newcommand{\\Xn}{\\mathcal{X}^n}$$\n$$\\newcommand{\\floor}[1]{\\lfloor #1 \\rfloor}$$\n$$\\newcommand{\\ceil}[1]{\\lceil #1 \\rceil}$$\n$$\\newcommand{\\mean}{\\mathbb{E}}$$\n$$\\newcommand{\\bc}{\\mathbf{c}}$$\n$$\\newcommand{\\bs}{\\mathbf{s}}$$\n$$\\newcommand{\\bA}{\\mathbf{A}}$$\n$$\\newcommand{\\bH}{\\mathbf{H}}$$\n$$\\newcommand{\\bG}{\\mathbf{G}}$$\n$$\\newcommand{\\bD}{\\mathbf{D}}$$\n$$\\newcommand{\\bC}{\\mathbf{C}}$$\n$$\\newcommand{\\bF}{\\mathbf{F}}$$\n$$\\newcommand{\\bB}{\\mathbf{B}}$$\n$$\\newcommand{\\bI}{\\mathbf{I}}$$\n$$\\newcommand{\\bR}{\\mathbf{R}}$$\n$$\\newcommand{\\bW}{\\mathbf{W}}$$\n$$\\newcommand{\\bY}{\\mathbf{Y}}$$\n$$\\newcommand{\\bZ}{\\mathbf{Z}}$$\n$$\\newcommand{\\bx}{\\mathbf{x}}$$\n$$\\newcommand{\\rank}{\\mbox{rank}}$$\n$$\\newcommand{\\bz}{\\mathbf{z}}$$\n$$\\newcommand{\\bX}{\\mathbf{X}}$$\n$$\\newcommand{\\br}{\\mathbf{r}}$$\n$$\\newcommand{\\bbz}{\\mathbf{z}}$$\n$$\\newcommand{\\binstr}{\\{0,1\\}} $$\n$$\\newcommand{\\supp}{\\mbox{supp}}$$\n$$\\renewcommand{\\tilde}{\\widetilde}$$\n$$\\newcommand{\\Enc}{\\mathsf{Enc}}$$\n$$\\newcommand{\\Dec}{\\mathsf{Dec}}$$\n$$\\newcommand{\\Adv}{\\mathsf{Adv}}$$\n$$\\newcommand{\\chis}{\\chi^2}$$\n$$\\newcommand{\\Xh}{\\hat{X}}$$\n$$\\newcommand{\\Dh}{\\hat{D}}$$\n$$\\newcommand{\\Yh}{\\hat{Y}}$$\n$$\\newcommand{\\Zh}{\\hat{Z}}$$\n$$\\DeclareMathOperator*{\\argmin}{\\arg\\!\\min}$$\n$$\\DeclareMathOperator*{\\argmax}{\\arg\\!\\max}$$\n$$\\newcommand{\\brk}[1]{\\langle #1 \\rangle}$$\n$$\\newcommand{\\Reals}{\\mathbb{R}}$$\n$$\\newcommand{\\normQ}[1]{\\| #1 \\|_Q}$$\n$$\\newcommand{\\normF}[1]{\\| #1 \\|_F}$$\n$$\\newcommand{\\normX}[2]{\\| #1 \\|_{#2}}$$\n$$\\newcommand{\\normEuc}[1]{\\| #1 \\|_2}$$\n$$\\newcommand{\\ox}{\\bar{x}}$$\n$$\\newcommand{\\ones}{\\mathbf{1}}$$\n$$\\newcommand{\\inertia}{\\mathcal{I}}$$\n$$\\newcommand{\\defined}{\\triangleq}$$\n$$\\newcommand{\\Tr}[1]{\\mathrm{ tr}\\left(#1 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#1}{\\delta #2}}$$\n$$\\newcommand{\\sto}{\\mbox{s.t.}}$$\n$$\\newcommand{\\KFnorm}[2]{\\| #1 \\|_{#2}}$$\n$$\\newcommand{\\Imeas}{J}$$\n$$\\newcommand{\\bigO}{O}$$\n$$\\newcommand{\\ttheta}{\\tilde{\\theta}}$$\n$$\\newcommand{\\Var}[2]{\\mathrm{Var}_{#1}#2 }$$\n$$\\newcommand{\\whf}{\\widehat{f}}$$\n$$\\newcommand{\\whg}{\\widehat{g}}$$\n$$\\newcommand{\\ft}{\\tilde{f}}$$\n$$%\\newcommand{\\pbgx}{p_{B|X^n}}$$\n$$\\newcommand{\\pbgy}{p_{B|Y^n}}$$\n$$\\newcommand{\\whh}{\\widehat{h}}$$\n$$\\newcommand{\\EE}[1]{\\ExpVal{}{#1}}$$\n$$\\newcommand{\\whB}{\\widehat{B}}$$\n$$\\newcommand{\\wbeta}{\\widehat{\\beta}}$$\n$$\\newcommand{\\xb}{\\mathbf{x}}$$\n$$\\newcommand{\\yb}{\\mathbf{y}}$$\n$$\\newcommand{\\fb}{\\mathbf{f}}$$\n$$\\newcommand{\\gb}{\\mathbf{g}}$$\n$$\\newcommand{\\bP}{\\mathbf{P}}$$\n$$\\newcommand{\\eye}{\\mathbf{I}}$$\n$$\\newcommand{\\bQ}{\\mathbf{Q}}$$\n$$\\newcommand{\\bU}{\\mathbf{U}}$$\n$$\\newcommand{\\bSigma}{\\mathbf{\\Sigma}}$$\n$$\\newcommand{\\bsigma}{\\boldsymbol\\sigma}$$\n$$\\newcommand{\\bV}{\\mathbf{V}}$$\n$$\\newcommand{\\bT}{\\mathbf{T}}$$\n$$\\newcommand{\\bbH}{\\mathbf{H}}$$\n$$\\newcommand{\\brho}{\\boldsymbol{\\rho}}$$\n$$\\newcommand{\\suchthat}{\\,\\mid\\,}$$\n$$\\newcommand{\\indicator}{\\mathds{1}}$$\n$$\\newcommand{\\mmse}{\\mathsf{mmse}}$$\n$$\\newcommand{\\error}{\\mathsf{e}}$$\n$$\\newcommand{\\calN}{\\mathcal{N}}$$\n$$\\newcommand{\\cwd}{\\{1,\\dots,2^{nR} \\}}$$\n$$\\newcommand{\\Ps}{\\mathbf{p}_S}$$\n$$\\newcommand{\\bw}{\\mathbf{w}}$$\n$$\\newcommand{\\TV}{\\mathsf{TV}}$$\n$$\\newcommand{\\lse}{\\mathsf{lmmse}}$$\n$$\\newcommand{\\dks}{d_{\\mathrm{KS}}}$$\n$$\\newcommand{\\Xt}{\\widetilde{X}}$$\n$$\\newcommand{\\xh}{\\hat{x}}$$\n$$\\newcommand{\\vs}{v^*(p_{S,X})}$$\n$$\\newcommand{\\dps}{\\delta(p_{S,X})}$$\n$$\\newcommand{\\bp}{\\mathbf{p}}$$\n$$\\newcommand{\\bq}{\\mathbf{q}}$$\n$$\\newcommand{\\simplex}{\\Delta}$$\n$$\\newcommand\\independent{\\protect\\mathpalette{\\protect\\independenT}{\\perp}}$$\n$$\\def\\independenT#1#2{\\mathrel{\\rlap{$#1#2$}\\mkern2mu{#1#2}}}$$\n$$\\newcommand{\\KC}{\\calJ}$$\n$$\\newcommand{\\Fsym}{\\calF_{\\mathrm{sym}}}$$\n$$\\newcommand{\\bg}{\\mathbf{g}}$$\n$$\\newcommand{\\Dx}{\\mathbf{D}_X}$$\n$$\\newcommand{\\Dy}{\\mathbf{D}_Y}$$\n\nEnd load.\n\n\n```python\n\n```\n", "meta": {"hexsha": "76fd01f8ee1093fa61282680ad373882d6a26fac", "size": 87976, "ext": "ipynb", "lang": "Jupyter Notebook", 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YES\n2. YES", "lm_q1_score": 0.6370308082623217, "lm_q2_score": 0.6584174938590245, "lm_q1q2_score": 0.4194322282870666}} {"text": "```python\n# import numpy as np\n\n# # !/usr/bin/env python3\n# # -*- coding: utf-8 -*-\n# \"\"\"\n# Created on 20181219\n\n# @author: zhangji\n\n# Trajection of a ellipse, Jeffery equation. \n# \"\"\"\n\n# %pylab inline\n# pylab.rcParams['figure.figsize'] = (25, 11)\n# fontsize = 40\n\n# import numpy as np\n# import scipy as sp\n# from scipy.optimize import leastsq, curve_fit\n# from scipy import interpolate\n# from scipy.interpolate import interp1d\n# from scipy.io import loadmat, savemat\n# # import scipy.misc\n\n# import matplotlib\n# from matplotlib import pyplot as plt\n# from matplotlib import animation, rc\n# import matplotlib.ticker as mtick\n# from mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\n# from mpl_toolkits.mplot3d import Axes3D, axes3d\n\n# from sympy import symbols, simplify, series, exp\n# from sympy.matrices import Matrix\n# from sympy.solvers import solve\n\n# from IPython.display import display, HTML\n# from tqdm import tqdm_notebook as tqdm\n# import pandas as pd\n# import re\n# from scanf import scanf\n# import os\n# import glob\n\n# from codeStore import support_fun as spf\n# from src.support_class import *\n# from src import stokes_flow as sf\n\n# rc('animation', html='html5')\n# PWD = os.getcwd()\n# font = {'size': 20}\n# matplotlib.rc('font', **font)\n# np.set_printoptions(linewidth=90, precision=5)\n\nimport os\nimport glob\nimport re\nimport pandas as pd\nfrom scanf import scanf\nimport natsort \nimport numpy as np\nimport scipy as sp\nfrom scipy.optimize import leastsq, curve_fit\nfrom scipy import interpolate\nfrom scipy import spatial\n# from scipy.interpolate import interp1d\nfrom scipy.io import loadmat, savemat\n# import scipy.misc\nimport importlib\nfrom IPython.display import display, HTML\nimport pickle\n\nimport matplotlib\nfrom matplotlib import pyplot as plt\nfrom matplotlib import colors as mcolors\nfrom matplotlib import animation, rc\nimport matplotlib.ticker as mtick\nfrom mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\nfrom mpl_toolkits.mplot3d import Axes3D, axes3d\nfrom matplotlib import cm\nfrom mpl_toolkits.axes_grid1 import make_axes_locatable\n\nfrom time import time\nfrom src.support_class import *\nfrom src import jeffery_model as jm\nfrom codeStore import support_fun as spf\n\n# %matplotlib notebook\n\nrc('animation', html='html5')\nfontsize = 40\nfigsize = (30, 16)\nPWD = os.getcwd()\n```\n\n\n```python\ndef read_data_loopup_table(psi_dir_list, tcenter, ignore_first=0):\n ecoli_U_list = []\n ecoli_norm_list = []\n ecoli_center_list = []\n ecoli_nodes_list = []\n ecoli_idx_list = []\n norm_phi_list = []\n norm_psi1_list = []\n norm_psi2_list = []\n norm_theta_list = []\n i0 = -1\n t1 = []\n for psi_dir in psi_dir_list:\n print(psi_dir)\n file_handle = os.path.basename(psi_dir)\n mat_names = natsort.natsorted(glob.glob('%s/%s_th*' % (psi_dir, file_handle)))\n for mati in mat_names[ignore_first:]:\n i0 = i0 + 1\n mat_contents = loadmat(mati)\n ecoli_U = mat_contents['ecoli_U'].flatten()\n ecoli_norm = mat_contents['ecoli_norm'].flatten()\n ecoli_center = mat_contents['ecoli_center'].flatten()\n planeShearRate = mat_contents['planeShearRate'].flatten()\n rel_U_list = mat_contents['rel_U']\n norm_phi = mat_contents['norm_phi'].flatten()\n norm_psi1 = mat_contents['norm_psi1'].flatten()\n norm_psi2 = mat_contents['norm_psi2'].flatten()\n norm_theta = mat_contents['norm_theta'].flatten()\n ecoli_U_list.append(ecoli_U)\n ecoli_norm_list.append(ecoli_norm)\n ecoli_center_list.append(ecoli_center)\n ecoli_idx_list.append(i0)\n norm_phi_list.append(norm_phi)\n norm_psi1_list.append(norm_psi1)\n norm_psi2_list.append(norm_psi2)\n norm_theta_list.append(norm_theta)\n ecoli_U = np.vstack(ecoli_U_list)\n ecoli_norm = np.vstack(ecoli_norm_list)\n ecoli_center = np.vstack(ecoli_center_list)\n ecoli_idx = np.hstack(ecoli_idx_list)\n norm_phi = np.hstack(norm_phi_list)\n norm_psi1 = np.hstack(norm_psi1_list)\n norm_psi2 = np.hstack(norm_psi2_list)\n norm_theta = np.hstack(norm_theta_list)\n norm_tpp = np.vstack((norm_theta, norm_phi, norm_psi1, norm_psi2)).T \n return ecoli_U, ecoli_norm, ecoli_center, ecoli_idx, norm_tpp, planeShearRate, rel_U_list\n\nimportlib.reload(spf)\nsm = 'pf'\necoli_name = 'ecoD01_all'\njob_dir = 'dualTail_1c'\nksp_max_it = 300\nmain_fun_noIter = 1\nplaneShearRatex = 1\nch = 3\nnth = 20\nrh1 = 0.1\nrh2 = 0.03\nph = 2/3\nn_tail = 1\nrel_tail1 = 193.66659814\nrel_tail1 = 0\nrel_tail2 = 0\nwrite_pbs_head = spf.write_pbs_head_newturb\nnorm_psi1_list = np.linspace(0, 2 * np.pi, 10, endpoint=False)\nnorm_psi2_list = np.linspace(0, 2 * np.pi, 10, endpoint=False)\nn_norm_theta = 24\nn_norm_phi = 48\nPWD = os.getcwd()\n\nn_pbs = 0\nt_name = os.path.join(job_dir, 'run2_all.sh')\nwith open(t_name, 'w') as frun0:\n for norm_psi1 in norm_psi1_list: \n t_run_name = 'run2_psi1-%4.2f.sh' % norm_psi1\n frun0.write('bash %s \\n' % t_run_name)\n t_name = os.path.join(job_dir, t_run_name)\n with open(t_name, 'w') as frun:\n # create .pbs file\n frun.write('t_dir=$PWD \\n')\n for norm_psi2 in norm_psi2_list:\n job_name = '%s_psi1-%4.2f_psi2-%4.2f' % (ecoli_name, norm_psi1, norm_psi2)\n t_path = os.path.join(job_dir, job_name)\n t_name = os.path.join(t_path, '%s_2.pbs' % job_name)\n psi_dir_list = [t_path, ]\n ecoli_U, ecoli_norm, ecoli_center, ecoli_idx, norm_tpp, planeShearRate, rel_U_list \\\n = read_data_loopup_table(psi_dir_list, np.zeros(3), ignore_first=0)\n norm_theta = norm_tpp[:, 0]\n th_idx = np.argmin(np.linspace(0, np.pi, n_norm_theta) < norm_theta.max())\n print(th_idx, norm_theta.max(), np.linspace(0, np.pi, n_norm_theta)[th_idx])\n # print(rel_U_list)\n with open(t_name, 'w') as fpbs:\n write_pbs_head(fpbs, job_name) \n fpbs.write('mpirun -n 24 python ')\n fpbs.write(' ../../../loop_table_dualTail_ecoli.py ')\n fpbs.write(' -f %s ' % job_name)\n fpbs.write(' -pickProblem %d ' % 0)\n fpbs.write(' -save_singleEcoli_vtk %d ' % 0)\n fpbs.write(' -rh1 %f ' % rh1)\n fpbs.write(' -rh2 %f ' % rh2)\n fpbs.write(' -ch %f ' % ch)\n fpbs.write(' -nth %d ' % nth)\n fpbs.write(' -eh %f ' % -1)\n fpbs.write(' -ph %f ' % ph)\n fpbs.write(' -hfct %f ' % 1)\n fpbs.write(' -n_tail %d ' % n_tail)\n fpbs.write(' -with_cover %d ' % 2)\n fpbs.write(' -left_hand %d ' % 0)\n fpbs.write(' -rs1 %f ' % 1.5)\n fpbs.write(' -rs2 %f ' % 0.5)\n fpbs.write(' -ds %f ' % 0.07)\n fpbs.write(' -es %f ' % -1)\n fpbs.write(' -with_T_geo %d ' % 0)\n fpbs.write(' -dist_hs %f ' % 0.5)\n fpbs.write(' -ksp_max_it %d ' % ksp_max_it)\n fpbs.write(' -plot_geo %d ' % 0)\n fpbs.write(' -rel_wsz %f ' % 0)\n fpbs.write(' -ffweight %f ' % 2)\n fpbs.write(' -sm %s ' % sm)\n fpbs.write(' -zoom_factor %f ' % 1)\n fpbs.write(' -planeShearRatex %f ' % planeShearRatex)\n fpbs.write(' -n_norm_theta %d ' % n_norm_theta)\n fpbs.write(' -n_norm_phi %d ' % n_norm_phi)\n fpbs.write(' -norm_psi1 %f ' % norm_psi1)\n fpbs.write(' -norm_psi2 %f ' % norm_psi2)\n fpbs.write(' -rel_tail1 %f ' % rel_tail1)\n fpbs.write(' -rel_tail2 %f ' % rel_tail2)\n fpbs.write(' -th_idx %d ' % th_idx)\n fpbs.write(' -main_fun_noIter %d ' % main_fun_noIter)\n fpbs.write(' > %s.txt \\n\\n' % job_name)\n # write to .sh file\n frun.write('cd $t_dir/%s\\n' % job_name)\n frun.write('qsub %s_2.pbs\\n\\n' % job_name)\n n_pbs = n_pbs + 1\n frun.write('\\n')\nprint('n_pbs = ', n_pbs)\n```\n\n dualTail_1c/ecoD01_all_psi1-0.00_psi2-0.00\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-0.00_psi2-0.63\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-0.00_psi2-1.26\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-0.00_psi2-1.88\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-0.00_psi2-2.51\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-0.00_psi2-3.14\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-0.00_psi2-3.77\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-0.00_psi2-4.40\n 15 2.0488647740803 2.0488647740803\n dualTail_1c/ecoD01_all_psi1-0.00_psi2-5.03\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-0.00_psi2-5.65\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-0.63_psi2-0.00\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-0.63_psi2-0.63\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-0.63_psi2-1.26\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-0.63_psi2-1.88\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-0.63_psi2-2.51\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-0.63_psi2-3.14\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-0.63_psi2-3.77\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-0.63_psi2-4.40\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-0.63_psi2-5.03\n 21 2.86841068371242 2.86841068371242\n dualTail_1c/ecoD01_all_psi1-0.63_psi2-5.65\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-1.26_psi2-0.00\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-1.26_psi2-0.63\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-1.26_psi2-1.26\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-1.26_psi2-1.88\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-1.26_psi2-2.51\n 15 2.0488647740803 2.0488647740803\n dualTail_1c/ecoD01_all_psi1-1.26_psi2-3.14\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-1.26_psi2-3.77\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-1.26_psi2-4.40\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-1.26_psi2-5.03\n 15 2.0488647740803 2.0488647740803\n dualTail_1c/ecoD01_all_psi1-1.26_psi2-5.65\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-1.88_psi2-0.00\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-1.88_psi2-0.63\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-1.88_psi2-1.26\n 21 2.86841068371242 2.86841068371242\n dualTail_1c/ecoD01_all_psi1-1.88_psi2-1.88\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-1.88_psi2-2.51\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-1.88_psi2-3.14\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-1.88_psi2-3.77\n 16 2.1854557590189865 2.1854557590189865\n dualTail_1c/ecoD01_all_psi1-1.88_psi2-4.40\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-1.88_psi2-5.03\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-1.88_psi2-5.65\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-2.51_psi2-0.00\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-2.51_psi2-0.63\n 15 2.0488647740803 2.0488647740803\n dualTail_1c/ecoD01_all_psi1-2.51_psi2-1.26\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-2.51_psi2-1.88\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-2.51_psi2-2.51\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-2.51_psi2-3.14\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-2.51_psi2-3.77\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-2.51_psi2-4.40\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-2.51_psi2-5.03\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-2.51_psi2-5.65\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-3.14_psi2-0.00\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-3.14_psi2-0.63\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-3.14_psi2-1.26\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-3.14_psi2-1.88\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-3.14_psi2-2.51\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-3.14_psi2-3.14\n 15 2.0488647740803 2.0488647740803\n dualTail_1c/ecoD01_all_psi1-3.14_psi2-3.77\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-3.14_psi2-4.40\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-3.14_psi2-5.03\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-3.14_psi2-5.65\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-3.77_psi2-0.00\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-3.77_psi2-0.63\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-3.77_psi2-1.26\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-3.77_psi2-1.88\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-3.77_psi2-2.51\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-3.77_psi2-3.14\n 21 2.86841068371242 2.86841068371242\n dualTail_1c/ecoD01_all_psi1-3.77_psi2-3.77\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-3.77_psi2-4.40\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-3.77_psi2-5.03\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-3.77_psi2-5.65\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-4.40_psi2-0.00\n 16 2.1854557590189865 2.1854557590189865\n dualTail_1c/ecoD01_all_psi1-4.40_psi2-0.63\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-4.40_psi2-1.26\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-4.40_psi2-1.88\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-4.40_psi2-2.51\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-4.40_psi2-3.14\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-4.40_psi2-3.77\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-4.40_psi2-4.40\n 15 2.0488647740803 2.0488647740803\n dualTail_1c/ecoD01_all_psi1-4.40_psi2-5.03\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-4.40_psi2-5.65\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-5.03_psi2-0.00\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-5.03_psi2-0.63\n 15 2.0488647740803 2.0488647740803\n dualTail_1c/ecoD01_all_psi1-5.03_psi2-1.26\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-5.03_psi2-1.88\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-5.03_psi2-2.51\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-5.03_psi2-3.14\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-5.03_psi2-3.77\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-5.03_psi2-4.40\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-5.03_psi2-5.03\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-5.03_psi2-5.65\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-5.65_psi2-0.00\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-5.65_psi2-0.63\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-5.65_psi2-1.26\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-5.65_psi2-1.88\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-5.65_psi2-2.51\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-5.65_psi2-3.14\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-5.65_psi2-3.77\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-5.65_psi2-4.40\n 20 2.731819698773733 2.731819698773733\n dualTail_1c/ecoD01_all_psi1-5.65_psi2-5.03\n 23 3.141592653589793 3.141592653589793\n dualTail_1c/ecoD01_all_psi1-5.65_psi2-5.65\n 23 3.141592653589793 3.141592653589793\n n_pbs = 100\n\n", "meta": {"hexsha": "aa2b3855fa8f699c5799a4695bd86bf2c43f02a5", "size": 21685, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "head_Force/loop_table/make_lookup_table_dualTail.ipynb", "max_stars_repo_name": "pcmagic/stokes_flow", "max_stars_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-11-11T05:00:53.000Z", "max_stars_repo_stars_event_max_datetime": "2018-11-11T05:00:53.000Z", "max_issues_repo_path": "head_Force/loop_table/make_lookup_table_dualTail.ipynb", "max_issues_repo_name": "pcmagic/stokes_flow", "max_issues_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "head_Force/loop_table/make_lookup_table_dualTail.ipynb", "max_forks_repo_name": "pcmagic/stokes_flow", "max_forks_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 45.1770833333, "max_line_length": 110, "alphanum_fraction": 0.5999538852, "converted": true, "num_tokens": 7005, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584174871563662, "lm_q2_score": 0.6370307944803832, "lm_q1q2_score": 0.4194322149429975}} {"text": "\n\n# QNN(\u91cf\u5b50\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af)\n* [TensorFlow Quantum](https://www.tensorflow.org/quantum/tutorials/mnist)\u3092\u53c2\u8003\u306b\u3059\u308b.\n* [Classification with Quantum Neural Networks on Near Term Processors](https://arxiv.org/pdf/1802.06002.pdf)\u3067\u7d39\u4ecb\u3055\u308c\u305fQNN\u306e\u30a2\u30d7\u30ed\u30fc\u30c1\u3092\u69cb\u7bc9.\n\n\n### \u5fc5\u8981\u306a\u3082\u306e\u3092\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\n\n\n```python\n!pip install tensorflow-gpu==2.1.0\n```\n\n Collecting tensorflow-gpu==2.1.0\n Downloading tensorflow_gpu-2.1.0-cp37-cp37m-manylinux2010_x86_64.whl (421.8 MB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 421.8 MB 3.8 kB/s \n \u001b[?25hRequirement already satisfied: wheel>=0.26 in /usr/local/lib/python3.7/dist-packages (from tensorflow-gpu==2.1.0) (0.37.1)\n Requirement already satisfied: absl-py>=0.7.0 in /usr/local/lib/python3.7/dist-packages (from tensorflow-gpu==2.1.0) (1.0.0)\n Requirement already satisfied: astor>=0.6.0 in /usr/local/lib/python3.7/dist-packages (from tensorflow-gpu==2.1.0) (0.8.1)\n Requirement already satisfied: scipy==1.4.1 in /usr/local/lib/python3.7/dist-packages (from tensorflow-gpu==2.1.0) (1.4.1)\n Collecting tensorflow-estimator<2.2.0,>=2.1.0rc0\n Downloading tensorflow_estimator-2.1.0-py2.py3-none-any.whl (448 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 448 kB 58.5 MB/s \n \u001b[?25hRequirement already satisfied: numpy<2.0,>=1.16.0 in /usr/local/lib/python3.7/dist-packages (from tensorflow-gpu==2.1.0) (1.21.5)\n Collecting gast==0.2.2\n Downloading gast-0.2.2.tar.gz (10 kB)\n Collecting keras-applications>=1.0.8\n Downloading Keras_Applications-1.0.8-py3-none-any.whl (50 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 50 kB 6.8 MB/s \n \u001b[?25hRequirement already satisfied: protobuf>=3.8.0 in /usr/local/lib/python3.7/dist-packages (from tensorflow-gpu==2.1.0) (3.17.3)\n Requirement already satisfied: keras-preprocessing>=1.1.0 in /usr/local/lib/python3.7/dist-packages (from tensorflow-gpu==2.1.0) (1.1.2)\n Requirement already satisfied: six>=1.12.0 in /usr/local/lib/python3.7/dist-packages (from tensorflow-gpu==2.1.0) (1.15.0)\n Requirement already satisfied: termcolor>=1.1.0 in /usr/local/lib/python3.7/dist-packages (from tensorflow-gpu==2.1.0) (1.1.0)\n Collecting tensorboard<2.2.0,>=2.1.0\n Downloading tensorboard-2.1.1-py3-none-any.whl (3.8 MB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 3.8 MB 41.5 MB/s \n \u001b[?25hRequirement already satisfied: opt-einsum>=2.3.2 in /usr/local/lib/python3.7/dist-packages (from tensorflow-gpu==2.1.0) (3.3.0)\n Requirement already satisfied: grpcio>=1.8.6 in /usr/local/lib/python3.7/dist-packages (from tensorflow-gpu==2.1.0) (1.44.0)\n Requirement already satisfied: wrapt>=1.11.1 in /usr/local/lib/python3.7/dist-packages (from tensorflow-gpu==2.1.0) (1.14.0)\n Requirement already satisfied: google-pasta>=0.1.6 in /usr/local/lib/python3.7/dist-packages (from tensorflow-gpu==2.1.0) (0.2.0)\n Requirement already satisfied: h5py in /usr/local/lib/python3.7/dist-packages (from keras-applications>=1.0.8->tensorflow-gpu==2.1.0) (3.1.0)\n Requirement already satisfied: markdown>=2.6.8 in /usr/local/lib/python3.7/dist-packages (from tensorboard<2.2.0,>=2.1.0->tensorflow-gpu==2.1.0) (3.3.6)\n Requirement already satisfied: google-auth-oauthlib<0.5,>=0.4.1 in /usr/local/lib/python3.7/dist-packages (from tensorboard<2.2.0,>=2.1.0->tensorflow-gpu==2.1.0) (0.4.6)\n Requirement already satisfied: setuptools>=41.0.0 in /usr/local/lib/python3.7/dist-packages (from tensorboard<2.2.0,>=2.1.0->tensorflow-gpu==2.1.0) (57.4.0)\n Requirement already satisfied: requests<3,>=2.21.0 in /usr/local/lib/python3.7/dist-packages (from tensorboard<2.2.0,>=2.1.0->tensorflow-gpu==2.1.0) (2.23.0)\n Requirement already satisfied: werkzeug>=0.11.15 in /usr/local/lib/python3.7/dist-packages (from tensorboard<2.2.0,>=2.1.0->tensorflow-gpu==2.1.0) (1.0.1)\n Requirement already satisfied: google-auth<2,>=1.6.3 in /usr/local/lib/python3.7/dist-packages (from tensorboard<2.2.0,>=2.1.0->tensorflow-gpu==2.1.0) (1.35.0)\n Requirement already satisfied: rsa<5,>=3.1.4 in 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importlib-metadata>=4.4->markdown>=2.6.8->tensorboard<2.2.0,>=2.1.0->tensorflow-gpu==2.1.0) (3.10.0.2)\n Requirement already satisfied: zipp>=0.5 in /usr/local/lib/python3.7/dist-packages (from importlib-metadata>=4.4->markdown>=2.6.8->tensorboard<2.2.0,>=2.1.0->tensorflow-gpu==2.1.0) (3.7.0)\n Requirement already satisfied: pyasn1<0.5.0,>=0.4.6 in /usr/local/lib/python3.7/dist-packages (from pyasn1-modules>=0.2.1->google-auth<2,>=1.6.3->tensorboard<2.2.0,>=2.1.0->tensorflow-gpu==2.1.0) (0.4.8)\n Requirement already satisfied: certifi>=2017.4.17 in /usr/local/lib/python3.7/dist-packages (from requests<3,>=2.21.0->tensorboard<2.2.0,>=2.1.0->tensorflow-gpu==2.1.0) (2021.10.8)\n Requirement already satisfied: urllib3!=1.25.0,!=1.25.1,<1.26,>=1.21.1 in /usr/local/lib/python3.7/dist-packages (from requests<3,>=2.21.0->tensorboard<2.2.0,>=2.1.0->tensorflow-gpu==2.1.0) (1.24.3)\n Requirement already satisfied: idna<3,>=2.5 in /usr/local/lib/python3.7/dist-packages (from requests<3,>=2.21.0->tensorboard<2.2.0,>=2.1.0->tensorflow-gpu==2.1.0) (2.10)\n Requirement already satisfied: chardet<4,>=3.0.2 in /usr/local/lib/python3.7/dist-packages (from requests<3,>=2.21.0->tensorboard<2.2.0,>=2.1.0->tensorflow-gpu==2.1.0) (3.0.4)\n Requirement already satisfied: oauthlib>=3.0.0 in /usr/local/lib/python3.7/dist-packages (from requests-oauthlib>=0.7.0->google-auth-oauthlib<0.5,>=0.4.1->tensorboard<2.2.0,>=2.1.0->tensorflow-gpu==2.1.0) (3.2.0)\n Requirement already satisfied: cached-property in /usr/local/lib/python3.7/dist-packages (from h5py->keras-applications>=1.0.8->tensorflow-gpu==2.1.0) (1.5.2)\n Building wheels for collected packages: gast\n Building wheel for gast (setup.py) ... \u001b[?25l\u001b[?25hdone\n Created wheel for gast: filename=gast-0.2.2-py3-none-any.whl size=7554 sha256=e715aa61c6e18d8f1ca4154a9e5eab5df3df02e2ae2f0e5100b6785255187638\n Stored in directory: /root/.cache/pip/wheels/21/7f/02/420f32a803f7d0967b48dd823da3f558c5166991bfd204eef3\n Successfully built gast\n Installing collected packages: tensorflow-estimator, tensorboard, keras-applications, gast, tensorflow-gpu\n Attempting uninstall: tensorflow-estimator\n Found existing installation: tensorflow-estimator 2.8.0\n Uninstalling tensorflow-estimator-2.8.0:\n Successfully uninstalled tensorflow-estimator-2.8.0\n Attempting uninstall: tensorboard\n Found existing installation: tensorboard 2.8.0\n Uninstalling tensorboard-2.8.0:\n Successfully uninstalled tensorboard-2.8.0\n Attempting uninstall: gast\n Found existing installation: gast 0.5.3\n Uninstalling gast-0.5.3:\n Successfully uninstalled gast-0.5.3\n \u001b[31mERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.\n tensorflow 2.8.0 requires tf-estimator-nightly==2.8.0.dev2021122109, which is not installed.\n tensorflow 2.8.0 requires tensorboard<2.9,>=2.8, but you have tensorboard 2.1.1 which is incompatible.\n tensorflow-probability 0.16.0 requires gast>=0.3.2, but you have gast 0.2.2 which is incompatible.\u001b[0m\n Successfully installed gast-0.2.2 keras-applications-1.0.8 tensorboard-2.1.1 tensorflow-estimator-2.1.0 tensorflow-gpu-2.1.0\n\n\n\n```python\n!pip install cirq==0.7.0 pathos==0.2.5 tensorflow-quantum==0.2.0\n```\n\n Collecting cirq==0.7.0\n Downloading cirq-0.7.0-py3-none-any.whl (1.2 MB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 1.2 MB 4.2 MB/s \n \u001b[?25hCollecting pathos==0.2.5\n Downloading pathos-0.2.5.tar.gz (162 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 162 kB 68.9 MB/s \n \u001b[?25hCollecting tensorflow-quantum==0.2.0\n Downloading tensorflow_quantum-0.2.0-cp37-cp37m-manylinux2010_x86_64.whl (4.5 MB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 4.5 MB 42.3 MB/s \n \u001b[?25hCollecting protobuf==3.8.0\n Downloading protobuf-3.8.0-cp37-cp37m-manylinux1_x86_64.whl (1.2 MB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 1.2 MB 47.8 MB/s \n \u001b[?25hRequirement already satisfied: requests~=2.18 in 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|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 91 kB 11.1 MB/s \n \u001b[?25h Downloading google_api_core-1.22.2-py2.py3-none-any.whl (91 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 91 kB 9.8 MB/s \n \u001b[?25h Downloading google_api_core-1.22.1-py2.py3-none-any.whl (91 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 91 kB 10.6 MB/s \n \u001b[?25h Downloading google_api_core-1.22.0-py2.py3-none-any.whl (91 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 91 kB 10.1 MB/s \n \u001b[?25h Downloading google_api_core-1.21.0-py2.py3-none-any.whl (90 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 90 kB 10.6 MB/s \n \u001b[?25hINFO: pip is looking at multiple versions of google-api-python-client to determine which version is compatible with other requirements. This could take a while.\n Collecting google-api-python-client~=1.6\n Downloading google_api_python_client-1.12.11-py2.py3-none-any.whl (62 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 62 kB 622 kB/s \n \u001b[?25h Downloading google_api_python_client-1.12.10-py2.py3-none-any.whl (61 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 61 kB 94 kB/s \n \u001b[?25h Downloading google_api_python_client-1.12.8-py2.py3-none-any.whl (61 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 61 kB 27 kB/s \n \u001b[?25h Downloading google_api_python_client-1.12.7-py2.py3-none-any.whl (61 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 61 kB 23 kB/s \n \u001b[?25h Downloading google_api_python_client-1.12.6-py2.py3-none-any.whl (61 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 61 kB 28 kB/s \n \u001b[?25h Downloading google_api_python_client-1.12.5-py2.py3-none-any.whl (61 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 61 kB 7.5 MB/s \n \u001b[?25h Downloading google_api_python_client-1.12.4-py2.py3-none-any.whl (61 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 61 kB 7.5 MB/s \n \u001b[?25hINFO: pip is looking at multiple versions of google-api-python-client to determine which version is compatible with other requirements. This could take a while.\n Downloading google_api_python_client-1.12.3-py2.py3-none-any.whl (61 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 61 kB 6.5 MB/s \n \u001b[?25h Downloading google_api_python_client-1.12.2-py2.py3-none-any.whl (61 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 61 kB 8.0 MB/s \n \u001b[?25h Downloading google_api_python_client-1.12.1-py2.py3-none-any.whl (61 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 61 kB 7.6 MB/s \n \u001b[?25h Downloading google_api_python_client-1.12.0-py2.py3-none-any.whl (61 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 61 kB 8.0 MB/s \n \u001b[?25h Downloading google_api_python_client-1.11.0-py2.py3-none-any.whl (60 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 60 kB 6.8 MB/s \n \u001b[?25hCollecting google-api-core<2dev,>=1.18.0\n Downloading google_api_core-1.20.1-py2.py3-none-any.whl (90 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 90 kB 9.3 MB/s \n \u001b[?25h Downloading google_api_core-1.19.1-py2.py3-none-any.whl (90 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 90 kB 10.2 MB/s \n \u001b[?25hRequirement already satisfied: googleapis-common-protos<2.0dev,>=1.6.0 in /usr/local/lib/python3.7/dist-packages (from google-api-core<2dev,>=1.18.0->google-api-python-client~=1.6->cirq==0.7.0) (1.56.0)\n Requirement already satisfied: pytz in /usr/local/lib/python3.7/dist-packages (from google-api-core<2dev,>=1.18.0->google-api-python-client~=1.6->cirq==0.7.0) (2018.9)\n Requirement already satisfied: rsa<5,>=3.1.4 in /usr/local/lib/python3.7/dist-packages (from google-auth<3dev,>=1.16.0->google-api-python-client~=1.6->cirq==0.7.0) (4.8)\n Requirement already satisfied: cachetools<5.0,>=2.0.0 in /usr/local/lib/python3.7/dist-packages (from google-auth<3dev,>=1.16.0->google-api-python-client~=1.6->cirq==0.7.0) (4.2.4)\n Requirement already satisfied: pyasn1-modules>=0.2.1 in /usr/local/lib/python3.7/dist-packages (from google-auth<3dev,>=1.16.0->google-api-python-client~=1.6->cirq==0.7.0) (0.2.8)\n Collecting googleapis-common-protos<2.0dev,>=1.6.0\n Downloading googleapis_common_protos-1.55.0-py2.py3-none-any.whl (212 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 212 kB 64.5 MB/s \n \u001b[?25h Downloading googleapis_common_protos-1.54.0-py2.py3-none-any.whl (207 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 207 kB 69.8 MB/s \n \u001b[?25h Downloading googleapis_common_protos-1.53.0-py2.py3-none-any.whl (198 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 198 kB 72.1 MB/s \n \u001b[?25h Downloading googleapis_common_protos-1.52.0-py2.py3-none-any.whl (100 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 100 kB 9.8 MB/s \n \u001b[?25hRequirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.7/dist-packages (from matplotlib~=3.0->cirq==0.7.0) (0.11.0)\n Requirement already satisfied: python-dateutil>=2.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib~=3.0->cirq==0.7.0) (2.8.2)\n Requirement already satisfied: kiwisolver>=1.0.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib~=3.0->cirq==0.7.0) (1.4.0)\n Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib~=3.0->cirq==0.7.0) (3.0.7)\n Requirement already satisfied: pyasn1<0.5.0,>=0.4.6 in /usr/local/lib/python3.7/dist-packages (from pyasn1-modules>=0.2.1->google-auth<3dev,>=1.16.0->google-api-python-client~=1.6->cirq==0.7.0) (0.4.8)\n Requirement already satisfied: idna<3,>=2.5 in /usr/local/lib/python3.7/dist-packages (from requests~=2.18->cirq==0.7.0) (2.10)\n Requirement already satisfied: urllib3!=1.25.0,!=1.25.1,<1.26,>=1.21.1 in /usr/local/lib/python3.7/dist-packages (from requests~=2.18->cirq==0.7.0) (1.24.3)\n Requirement already satisfied: chardet<4,>=3.0.2 in /usr/local/lib/python3.7/dist-packages (from requests~=2.18->cirq==0.7.0) (3.0.4)\n Requirement already satisfied: certifi>=2017.4.17 in /usr/local/lib/python3.7/dist-packages (from requests~=2.18->cirq==0.7.0) (2021.10.8)\n Building wheels for collected packages: pathos, networkx\n Building wheel for pathos (setup.py) ... \u001b[?25l\u001b[?25hdone\n Created wheel for pathos: filename=pathos-0.2.5-py3-none-any.whl size=77595 sha256=88ed6c000eabbf3ce8d4533c00696122fa8e1b32f71f0e6cdea2da03d440da15\n Stored in directory: /root/.cache/pip/wheels/1f/53/6c/89b4af46fdc1e3976b9c183414af0db994678d1294a1e6dfb4\n Building wheel for networkx (setup.py) ... \u001b[?25l\u001b[?25hdone\n Created wheel for networkx: filename=networkx-2.3-py2.py3-none-any.whl size=1556008 sha256=0bfb5b96f009793f81949034deb6ca4afd5da784977d574a2e1bd86c539d8360\n Stored in directory: /root/.cache/pip/wheels/44/e6/b8/4efaab31158e9e9ca9ed80b11f6b11130bac9a9672b3cbbeaf\n Successfully built pathos networkx\n Installing collected packages: protobuf, googleapis-common-protos, google-api-core, sympy, networkx, google-api-python-client, ppft, pox, cirq, tensorflow-quantum, pathos\n Attempting uninstall: protobuf\n Found existing installation: protobuf 3.17.3\n Uninstalling protobuf-3.17.3:\n Successfully uninstalled protobuf-3.17.3\n Attempting uninstall: googleapis-common-protos\n Found existing installation: googleapis-common-protos 1.56.0\n Uninstalling googleapis-common-protos-1.56.0:\n Successfully uninstalled googleapis-common-protos-1.56.0\n Attempting uninstall: google-api-core\n Found existing installation: google-api-core 1.26.3\n Uninstalling google-api-core-1.26.3:\n Successfully uninstalled google-api-core-1.26.3\n Attempting uninstall: sympy\n Found existing installation: sympy 1.7.1\n Uninstalling sympy-1.7.1:\n Successfully uninstalled sympy-1.7.1\n Attempting uninstall: networkx\n Found existing installation: networkx 2.6.3\n Uninstalling networkx-2.6.3:\n Successfully uninstalled networkx-2.6.3\n Attempting uninstall: google-api-python-client\n Found existing installation: google-api-python-client 1.12.11\n Uninstalling google-api-python-client-1.12.11:\n Successfully uninstalled google-api-python-client-1.12.11\n \u001b[31mERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.\n tensorflow 2.8.0 requires tf-estimator-nightly==2.8.0.dev2021122109, which is not installed.\n tensorflow 2.8.0 requires protobuf>=3.9.2, but you have protobuf 3.8.0 which is incompatible.\n tensorflow 2.8.0 requires tensorboard<2.9,>=2.8, but you have tensorboard 2.1.1 which is incompatible.\n tensorflow-metadata 1.7.0 requires protobuf<4,>=3.13, but you have protobuf 3.8.0 which is incompatible.\n earthengine-api 0.1.303 requires google-api-python-client<2,>=1.12.1, but you have google-api-python-client 1.11.0 which is incompatible.\n albumentations 0.1.12 requires imgaug<0.2.7,>=0.2.5, but you have imgaug 0.2.9 which is incompatible.\u001b[0m\n Successfully installed cirq-0.7.0 google-api-core-1.19.1 google-api-python-client-1.11.0 googleapis-common-protos-1.52.0 networkx-2.3 pathos-0.2.5 pox-0.3.0 ppft-1.6.6.4 protobuf-3.8.0 sympy-1.4 tensorflow-quantum-0.2.0\n\n\n\n\n## \u30e9\u30a4\u30d6\u30e9\u30ea\u306e\u30a4\u30f3\u30dd\u30fc\u30c8\n\n\n```python\nimport tensorflow as tf\nimport tensorflow_quantum as tfq\n\nimport cirq\nimport sympy\nimport numpy as np\nimport pandas as pd\nimport seaborn as sns\nimport collections\n\n# visualization tools\n%matplotlib inline\nimport matplotlib.pyplot as plt\nfrom cirq.contrib.svg import SVGCircuit\n```\n\n# \u30c7\u30fc\u30bf\u306e\u7528\u610f\n* mnist\u306e\u30c7\u30fc\u30bf\u3092\u7528\u610f\n* \u91cf\u5b50\u30c7\u30d0\u30a4\u30b9(\u4eca\u56de\u306f\u30b7\u30df\u30e5\u30ec\u30fc\u30b7\u30e7\u30f3)\u3067\u306f\u5927\u304d\u3044\u30c7\u30fc\u30bf\u91cf\u3092\u6271\u3048\u306a\u3044\u306e\u3067\u753b\u50cf\u30c7\u30fc\u30bf\u3092\u30c0\u30a6\u30f3\u30b9\u30b1\u30fc\u30eb\u3059\u308b\n* \u4e8c\u5024\u5206\u985e\u3092\u884c\u3046\u305f\u3081mnist\u306e\u30c7\u30fc\u30bf\u3092\u300c3\u300d\u3068\u300c6\u300d\u306e\u30c7\u30fc\u30bf\u306b\u7d5e\u308b\n\n### \u30c7\u30fc\u30bf\u306e\u7528\u610f\u3068\u30c0\u30a6\u30f3\u30b9\u30b1\u30fc\u30eb\n\n\n```python\n#keras\u3067mnist\u306e\u30c7\u30fc\u30bf\u3092\u30c0\u30a6\u30f3\u30ed\u30fc\u30c9\n(x_train, y_train), (x_test, y_test) = tf.keras.datasets.mnist.load_data()\n\n# \u753b\u50cf\u30c7\u30fc\u30bf\u3092[0,255]\u304b\u3089[0.0,1.0]\u306b\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\nx_train, x_test = x_train[..., np.newaxis]/255.0, x_test[..., np.newaxis]/255.0\n\nprint(\"Number of original training examples:\", len(x_train))\nprint(\"Number of original test examples:\", len(x_test))\n```\n\n Number of original training examples: 60000\n Number of original test examples: 10000\n\n\n\n```python\ndef filter_36(x, y):\n keep = (y == 3) | (y == 6)\n x, y = x[keep], y[keep]\n y = y == 3\n return x,y\n```\n\n\n```python\n#\u5b66\u7fd2\u30c7\u30fc\u30bf\u30923\u30686\u306b\u9650\u5b9a\nx_train, y_train = filter_36(x_train, y_train)\nx_test, y_test = filter_36(x_test, y_test)\n\nprint(\"Number of filtered training examples:\", len(x_train))\nprint(\"Number of filtered test examples:\", len(x_test))\n```\n\n Number of filtered training examples: 12049\n Number of filtered test examples: 1968\n\n\n\n```python\nplt.imshow(x_train[7, :, :, 0])\nplt.colorbar()\n```\n\n\n```python\n#\u753b\u50cf\u306e\u30c7\u30fc\u30bf\u3092\u30c0\u30a6\u30f3\u30b9\u30b1\u30fc\u30eb\u3059\u308b\nx_train_small = tf.image.resize(x_train, (4,4)).numpy()\nx_test_small = tf.image.resize(x_test, (4,4)).numpy()\n\n#\u30c0\u30a6\u30f3\u30b9\u30b1\u30fc\u30eb\u5f8c\u306e\u753b\u50cf\u3092\u8868\u793a\nplt.imshow(x_train_small[7, :, :, 0])\n```\n\n### \u4e21\u65b9\u306e\u30af\u30e9\u30b9\u306b\u5c5e\u3059\u308b\u753b\u50cf\u306e\u524a\u9664\n\n\n```python\n# \u4e21\u65b9\u306e\u30af\u30e9\u30b9\u306b\u5c5e\u3059\u308b\u753b\u50cf\u306e\u524a\u9664\ndef remove_contradicting(xs, ys):\n mapping = collections.defaultdict(set)\n orig_x = {}\n # Determine the set of labels for each unique image:\n for x,y in zip(xs,ys):\n orig_x[tuple(x.flatten())] = x\n mapping[tuple(x.flatten())].add(y)\n\n new_x = []\n new_y = []\n for flatten_x in mapping:\n x = orig_x[flatten_x]\n labels = mapping[flatten_x]\n if len(labels) == 1:\n new_x.append(x)\n new_y.append(next(iter(labels)))\n else:\n # Throw out images that match more than one label.\n pass\n\n num_uniq_3 = sum(1 for value in mapping.values() if len(value) == 1 and True in value)\n num_uniq_6 = sum(1 for value in mapping.values() if len(value) == 1 and False in value)\n num_uniq_both = sum(1 for value in mapping.values() if len(value) == 2)\n\n print(\"Number of unique images:\", len(mapping.values()))\n print(\"Number of unique 3s: \", num_uniq_3)\n print(\"Number of unique 6s: \", num_uniq_6)\n print(\"Number of unique contradicting labels (both 3 and 6): \", num_uniq_both)\n print()\n print(\"Initial number of images: \", len(xs))\n print(\"Remaining non-contradicting unique images: \", len(new_x))\n\n return np.array(new_x), np.array(new_y)\n```\n\n\n```python\nx_train_nocon, y_train_nocon = remove_contradicting(x_train_small, y_train)\n```\n\n Number of unique images: 10387\n Number of unique 3s: 4912\n Number of unique 6s: 5426\n Number of unique contradicting labels (both 3 and 6): 49\n \n Initial number of images: 12049\n Remaining non-contradicting unique images: 10338\n\n\n### \u51fa\u529b\u306e\u30e9\u30d9\u30eb\u30920,1\u306b\u5909\u63db\n\n\n```python\n# \u51fa\u529b\u30e9\u30d9\u30eb\u304c\u30d6\u30fc\u30eb\u5024\u306b\u306a\u3063\u3066\u3044\u308b\u306e\u3067[0,1]\u306e\u5024\u306b\u5909\u63db\ny_train_binary = 1.0*y_train_nocon\ny_test_binary = 1.0*y_test\n```\n\n# \u5165\u529b\u3059\u308b\u91cf\u5b50\u72b6\u614b\u306e\u7528\u610f\n* \u5165\u529b\u3059\u308b\u5024\u306f0\u304b1\u306e\u4e8c\u5024\u3067\u7528\u610f\u3059\u308b\n* 0\u304b1\u306e\u72b6\u614b\u3067\u5165\u529b\u3059\u308b\u305f\u3081\uff0c\u5165\u529b\u304c1\u3068\u306a\u308b\u91cf\u5b50\u30d3\u30c3\u30c8\u306b\u5bfe\u3057\u3066\u306fX\u30b2\u30fc\u30c8\u3092\u4f5c\u7528\u3055\u305b\u53cd\u8ee2\u3055\u305b\u3066\u304b\u3089\u5165\u529b\u3059\u308b\n\n\n\n### \u91cf\u5b50\u72b6\u614b\u306e\u5165\u529b\u306e\u305f\u3081\u306e\u5909\u63db\n* \u91cf\u5b50\u72b6\u614b\u306e\u5165\u529b\u306b\u306f0\u304b1\u306e\u5024\u3092\u7528\u3044\u308b\u305f\u3081\uff0c\u95be\u5024\u3092\u6c7a\u30810\u304b1\u306e\u5024\u306b\u5909\u63db\u3059\u308b\uff0e\n\n\n```python\n#0.5\u3092\u95be\u5024\u3068\u3057\u30660\u304b\uff11\u306e\u30d0\u30a4\u30ca\u30ea\u30fc\u5909\u6570\u306b\u5909\u63db\nthreshold = 0.5\n\nx_train_bin = np.array(x_train_nocon > threshold, dtype = np.float32)\nx_test_bin = np.array(x_test_small > threshold, dtype = np.float32)\n```\n\n### \u5165\u529b\u3059\u308b\u91cf\u5b50\u72b6\u614b\u3092\u5909\u63db\u3059\u308b\u95a2\u6570\u306e\u7528\u610f\n\n\n```python\ndef convert_to_circuit(image):\n \"\"\"Encode truncated classical image into quantum datapoint.\"\"\"\n values = np.ndarray.flatten(image)\n qubits = cirq.GridQubit.rect(4, 4)\n circuit = cirq.Circuit()\n for i, value in enumerate(values):\n if value:\n circuit.append(cirq.X(qubits[i]))\n return circuit\n\n\nx_train_circ = [convert_to_circuit(x) for x in x_train_bin]\nx_test_circ = [convert_to_circuit(x) for x in x_test_bin]\n```\n\n\n```python\nSVGCircuit(x_train_circ[0])\n```\n\n\n\n\n \n\n \n\n\n\n\n```python\n#\u753b\u7d20\u5024\u304c\u95be\u5024\u3092\u8d85\u3048\u305f\u30a4\u30f3\u30c7\u30c3\u30af\u30b9\u306e\u8868\u793a\n#\u95be\u5024\u3092\u8d85\u3048\u3066\u3044\u308b\u30a4\u30f3\u30c7\u30c3\u30af\u30b9\u306bX\u30b2\u30fc\u30c8\u304c\u304b\u304b\u3063\u3066\u3044\u308b\u304b\u3092\u78ba\u8a8d\nbin_img = x_train_bin[0,:,:,0]\nindices = np.array(np.where(bin_img)).T\nindices\n```\n\n\n\n\n array([[2, 2],\n [3, 1]])\n\n\n\n### Cirq\u56de\u8def\u3092tfq\u306e\u305f\u3081\u306e\u30c6\u30f3\u30bd\u30eb\u306b\u5909\u63db\n* tfq.convert_to_tensor\u306b\u3088\u3063\u3066\u30c6\u30f3\u30bd\u30eb\u8868\u73fe\u306b\u5909\u63db\u3059\u308b\n\n\n```python\nx_train_tfcirc = tfq.convert_to_tensor(x_train_circ)\nx_test_tfcirc = tfq.convert_to_tensor(x_test_circ)\n```\n\n# QNN\u30e2\u30c7\u30eb\u306e\u4f5c\u6210\n\n\n* QNN\u30e2\u30c7\u30eb\u3092\u4f5c\u6210\u3059\u308b\u306e\u306b\u5fc5\u8981\u306a\u91cf\u5b50\u56de\u8def\u306e\u7e70\u308a\u8fd4\u3057\u3092\u4f5c\u308b\u30af\u30e9\u30b9\u306e\u5b9a\u7fa9\n\n\n```python\nclass CircuitLayerBuilder():\n def __init__(self, data_qubits, readout):\n self.data_qubits = data_qubits\n self.readout = readout\n\n def add_layer(self, circuit, gate, prefix):\n for i, qubit in enumerate(self.data_qubits):\n symbol = sympy.Symbol(prefix + '-' + str(i))\n circuit.append(gate(qubit, self.readout)**symbol)\n```\n\n* \u91cf\u5b50\u56de\u8def\u306e\u7e70\u308a\u8fd4\u3057\u3092\u5b9f\u969b\u306b\u51fa\u529b\u3057\u3066\u307f\u308b\n\n\n```python\ndemo_builder = CircuitLayerBuilder(data_qubits = cirq.GridQubit.rect(4,1),\n readout=cirq.GridQubit(-1,-1))\n\ncircuit = cirq.Circuit()\ndemo_builder.add_layer(circuit, gate = cirq.XX, prefix='xx')\nSVGCircuit(circuit)\n```\n\n\n\n\n \n\n \n\n\n\n* \u5b9f\u969b\u306b\u5b66\u7fd2\u306b\u4f7f\u3046\u91cf\u5b50\u56de\u8def\u306e\u4f5c\u6210\n* \u5165\u529b\u306f\u30c7\u30fc\u30bf\u30b5\u30a4\u30ba\u306e16\u91cf\u5b50\u30d3\u30c3\u30c8+\u51fa\u529b\u3092\u53d6\u308a\u51fa\u3059\u305f\u3081\u306e1\u91cf\u5b50\u30d3\u30c3\u30c8\u306e\u5408\u8a0817\u91cf\u5b50\u30d3\u30c3\u30c8\n* X\u8ef8\u56de\u8ee2\u3068Z\u8ef8\u56de\u8ee2\u306e\u4e21\u65b9\u3092\u4f7f\u3044\u95a2\u6570\u3092\u8868\u73fe\u3059\u308b\u305f\u3081XX\u30b2\u30fc\u30c8\u306e\u64cd\u4f5c\u306e\u3042\u3068\uff0cZZ\u30b2\u30fc\u30c8\u306e\u64cd\u4f5c\u3092\u3059\u308b\n\n\n```python\ndef create_quantum_model():\n \"\"\"Create a QNN model circuit and readout operation to go along with it.\"\"\"\n data_qubits = cirq.GridQubit.rect(4, 4) # a 4x4 grid.\n readout = cirq.GridQubit(-1, -1) # a single qubit at [-1,-1]\n circuit = cirq.Circuit()\n\n # Prepare the readout qubit.\n circuit.append(cirq.X(readout))\n circuit.append(cirq.H(readout))\n\n builder = CircuitLayerBuilder(\n data_qubits = data_qubits,\n readout=readout)\n\n # XX\u30b2\u30fc\u30c8\uff0cZZ\u30b2\u30fc\u30c8\u306e\u7e70\u308a\u8fd4\u3057\n builder.add_layer(circuit, cirq.XX, \"xx1\")\n builder.add_layer(circuit, cirq.ZZ, \"zz1\")\n\n # \u6700\u5f8c\u306b\u6e2c\u5b9a\u91cf\u5b50\u30d3\u30c3\u30c8\u306b\u30a2\u30c0\u30de\u30fc\u30eb\u30b2\u30fc\u30c8\u3092\u304b\u3051\u308b\n circuit.append(cirq.H(readout))\n\n return circuit, cirq.Z(readout)\n```\n\n\n```python\n#\u91cf\u5b50\u56de\u8def\u306e\u4f5c\u6210\nmodel_circuit, model_readout = create_quantum_model()\n\n#\u91cf\u5b50\u56de\u8def\u306e\u53ef\u8996\u5316\nSVGCircuit(model_circuit)\n```\n\n\n\n\n \n\n \n\n\n\n* \u30e2\u30c7\u30eb\u5168\u4f53\u306e\u4f5c\u6210\n\n\n```python\n# \u53e4\u5178\u306eTensorFlow\u540c\u69d8\u306b\u30e2\u30c7\u30eb\u3092\u4f5c\u6210\u3059\u308b\nmodel = tf.keras.Sequential([\n # \u30c7\u30fc\u30bf\u3092\u5165\u529b\u3057tf.string\u306b\u5909\u63db\u3059\u308b\u305f\u3081\u306e\u30ec\u30a4\u30e4\u30fc\n tf.keras.layers.Input(shape=(), dtype=tf.string),\n # [-1,1]\u306e\u9593\u306e\u5024\u3092\u51fa\u529b\u3059\u308b\u91cf\u5b50\u56de\u8def\u306e\u30ec\u30a4\u30e4\u30fc\n tfq.layers.PQC(model_circuit, model_readout),\n])\n\n\n# \u640d\u5931\u95a2\u6570\uff0c\u6700\u9069\u5316\u95a2\u6570\uff0c\u8a55\u4fa1\u95a2\u6570\u3092\u6307\u5b9a\u3057\u3066\u30b3\u30f3\u30d1\u30a4\u30eb\nmodel.compile(\n loss = tf.keras.losses.BinaryCrossentropy(),\n optimizer=tf.keras.optimizers.Adam(learning_rate=0.01),\n metrics=['accuracy'])\n```\n\n\n```python\n# \u30e2\u30c7\u30eb\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u306a\u3069\u3092\u78ba\u8a8d\nprint(model.summary())\n```\n\n Model: \"sequential_1\"\n _________________________________________________________________\n Layer (type) Output Shape Param # \n =================================================================\n pqc_1 (PQC) (None, 1) 32 \n =================================================================\n Total params: 32\n Trainable params: 32\n Non-trainable params: 0\n _________________________________________________________________\n None\n\n\n# QNN\u306e\u5b66\u7fd2\n* \u5b66\u7fd2\u306fTensorFlow\u3067NN\u30e2\u30c7\u30eb\u3092\u5b66\u7fd2\u3055\u305b\u308b\u6642\u3068\u307b\u3068\u3093\u3069\u5909\u308f\u3089\u306a\u3044\n* TensorFlow Quantum\u306fGPU\u306b\u306f\u8f09\u305b\u3089\u308c\u306a\u3044\u305f\u3081\u5b66\u7fd2\u6642\u9593\u306f\u7d50\u69cb\u304b\u304b\u308b(2022/3/29\u6642\u70b9)\n\n\n```python\nEPOCHS = 10\nBATCH_SIZE = 32\n\n#NUM_EXAMPLES = len(x_train_tfcirc)\nNUM_EXAMPLES = 1000\n\n#\u5b66\u7fd2\u306b\u6642\u9593\u304c\u304b\u304b\u308b\u306e\u3067\u30c7\u30fc\u30bf\u3092\u6e1b\u3089\u3059\nx_train_tfcirc_sub = x_train_tfcirc[:NUM_EXAMPLES]\ny_train_binary_sub = y_train_binary[:NUM_EXAMPLES]\n```\n\n\n```python\n# \u30e2\u30c7\u30eb\u306e\u4fdd\u5b58\u65b9\u6cd5\u306e\u6307\u5b9a\ncheckpoint_filepath = \"weight.ckpt\"\nmodel_checkpoint_callback = tf.keras.callbacks.ModelCheckpoint(\n filepath=checkpoint_filepath, # \u4fdd\u5b58path\n save_weights_only=True, # \u91cd\u307f\u306e\u307f\u3092\u4fdd\u5b58\n monitor='val_acc', # validataion\u306eACC\u306e\u5024\u306b\u57fa\u3065\u3044\u3066\u91cd\u307f\u3092\u4fdd\u5b58\u3059\u308b\n mode='max', # validataion\u306eACC\u304c\u6700\u5927\u3068\u306a\u3063\u305f\u6642\u91cd\u307f\u3092\u4fdd\u5b58\u3059\u308b\n save_best_only=True # ACC\u304c\u6539\u5584\u3057\u305f\u3068\u304d\u306e\u307f\u4fdd\u5b58\u3059\u308b\n)\n\n# \u30e2\u30c7\u30eb\u306e\u5b66\u7fd2\u306e\u958b\u59cb\nqnn_history = model.fit(\n x_train_tfcirc_sub, y_train_binary_sub,\n batch_size=32,\n epochs=EPOCHS,\n verbose=1,\n validation_data=(x_test_tfcirc, y_test_binary),\n callbacks=[model_checkpoint_callback]\n )\n```\n\n Train on 1000 samples, validate on 1968 samples\n Epoch 1/10\n 992/1000 [============================>.] - ETA: 1s - loss: 1.5904 - accuracy: 0.5383WARNING:tensorflow:Can save best model only with val_acc available, skipping.\n 1000/1000 [==============================] - 163s 163ms/sample - loss: 1.5810 - accuracy: 0.5400 - val_loss: 1.2817 - val_accuracy: 0.6494\n Epoch 2/10\n 992/1000 [============================>.] - ETA: 1s - loss: 0.9744 - accuracy: 0.8024WARNING:tensorflow:Can save best model only with val_acc available, skipping.\n 1000/1000 [==============================] - 174s 174ms/sample - loss: 0.9697 - accuracy: 0.8030 - val_loss: 1.1291 - val_accuracy: 0.8415\n Epoch 3/10\n 992/1000 [============================>.] - ETA: 1s - loss: 0.9132 - accuracy: 0.8448WARNING:tensorflow:Can save best model only with val_acc available, skipping.\n 1000/1000 [==============================] - 174s 174ms/sample - loss: 0.9074 - accuracy: 0.8460 - val_loss: 1.0848 - val_accuracy: 0.8476\n Epoch 4/10\n 992/1000 [============================>.] - ETA: 1s - loss: 0.9282 - accuracy: 0.7016WARNING:tensorflow:Can save best model only with val_acc available, skipping.\n 1000/1000 [==============================] - 158s 158ms/sample - loss: 0.9383 - accuracy: 0.7020 - val_loss: 0.8874 - val_accuracy: 0.8526\n Epoch 5/10\n 992/1000 [============================>.] - ETA: 1s - loss: 0.7021 - accuracy: 0.8679WARNING:tensorflow:Can save best model only with val_acc available, skipping.\n 1000/1000 [==============================] - 156s 156ms/sample - loss: 0.6997 - accuracy: 0.8680 - val_loss: 0.6399 - val_accuracy: 0.8572\n Epoch 6/10\n 992/1000 [============================>.] - ETA: 1s - loss: 0.6111 - accuracy: 0.8710WARNING:tensorflow:Can save best model only with val_acc available, skipping.\n 1000/1000 [==============================] - 157s 157ms/sample - loss: 0.6227 - accuracy: 0.8710 - val_loss: 0.6680 - val_accuracy: 0.8557\n Epoch 7/10\n 992/1000 [============================>.] - ETA: 1s - loss: 0.6063 - accuracy: 0.8700WARNING:tensorflow:Can save best model only with val_acc available, skipping.\n 1000/1000 [==============================] - 156s 156ms/sample - loss: 0.6039 - accuracy: 0.8690 - val_loss: 0.6351 - val_accuracy: 0.8557\n Epoch 8/10\n 992/1000 [============================>.] - ETA: 1s - loss: 0.5904 - accuracy: 0.8679WARNING:tensorflow:Can save best model only with val_acc available, skipping.\n 1000/1000 [==============================] - 157s 157ms/sample - loss: 0.5863 - accuracy: 0.8690 - val_loss: 0.6200 - val_accuracy: 0.8557\n Epoch 9/10\n 992/1000 [============================>.] - ETA: 1s - loss: 0.5634 - accuracy: 0.8821WARNING:tensorflow:Can save best model only with val_acc available, skipping.\n 1000/1000 [==============================] - 157s 157ms/sample - loss: 0.5641 - accuracy: 0.8790 - val_loss: 0.5255 - val_accuracy: 0.8603\n Epoch 10/10\n 992/1000 [============================>.] - ETA: 1s - loss: 0.5411 - accuracy: 0.8810WARNING:tensorflow:Can save best model only with val_acc available, skipping.\n 1000/1000 [==============================] - 157s 157ms/sample - loss: 0.5387 - accuracy: 0.8810 - val_loss: 0.6013 - val_accuracy: 0.8943\n 1968/1968 [==============================] - 11s 6ms/sample - loss: 0.6013 - accuracy: 0.8943\n\n\n# \u5b66\u7fd2\u7d50\u679c\u306e\u53ef\u8996\u5316\n\n\n```python\n# \u5b66\u7fd2\u30ed\u30b0\u3092DataFrame\u306b\u5909\u63db\nhistory_df = pd.DataFrame(qnn_history.history) \n# \u5b66\u7fd2\u30ed\u30b0\u306e\u4fdd\u5b58\n#history_df.to_csv(\"history_df.csv\", index=None) \n\n# \u640d\u5931\u306e\u30ed\u30b0\u306e\u53ef\u8996\u5316\nplt.plot(np.arange(len(history_df)), history_df['loss'], label='loss')\nplt.plot(np.arange(len(history_df)), history_df['val_loss'], label='valid loss')\nplt.legend()\nplt.show()\n```\n\n\n```python\n# ACC\u306e\u5024\u306e\u30ed\u30b0\u306e\u53ef\u8996\u5316\nplt.plot(np.arange(len(history_df)), history_df['accuracy'], label='accuracy')\nplt.plot(np.arange(len(history_df)), history_df['val_accuracy'], label='val_accuracy')\nplt.legend()\nplt.show()\n```\n", "meta": {"hexsha": "b1e004d4c54b80ba10ebecc0ee0541c2bd4f25ce", "size": 140486, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "TensorFlow Quantum/QNN.ipynb", "max_stars_repo_name": "fuyu-quant/quantum_algorithm", "max_stars_repo_head_hexsha": "522e5caa2507d7dbdd403165eacff54d3fefc42e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "TensorFlow Quantum/QNN.ipynb", "max_issues_repo_name": "fuyu-quant/quantum_algorithm", "max_issues_repo_head_hexsha": "522e5caa2507d7dbdd403165eacff54d3fefc42e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "TensorFlow Quantum/QNN.ipynb", "max_forks_repo_name": "fuyu-quant/quantum_algorithm", "max_forks_repo_head_hexsha": "522e5caa2507d7dbdd403165eacff54d3fefc42e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 110.3582089552, "max_line_length": 27277, "alphanum_fraction": 0.6930726193, "converted": true, "num_tokens": 14042, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321964553657, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.41942579907546407}} {"text": "# Detector simulation\nIn this notebook we load a track dataset generated by `edep-sim` and we calculate the ADC counts corresponding to each pixel. The result is exported to a HDF5 file.\n\n\n```python\n%load_ext autoreload\n\n%autoreload 2\n```\n\n\n```python\n# This is need so you can import larndsim without doing python setup.py install\nimport os,sys,inspect\ncurrentdir = os.path.dirname(os.path.abspath(inspect.getfile(inspect.currentframe())))\nparentdir = os.path.dirname(currentdir)\nsys.path.insert(0,parentdir)\n```\n\n\n```python\nfrom math import ceil\nfrom time import time\n\nimport matplotlib.pyplot as plt\nfrom matplotlib import cm, colors\nimport mpl_toolkits.mplot3d.art3d as art3d\n\nimport numpy as np\nimport cupy as cp\nimport eagerpy as ep\n#import fire\nimport h5py\n\nfrom numba import cuda\nfrom numba.cuda.random import create_xoroshiro128p_states\n\nimport matplotlib as mpl\nmpl.rcParams['figure.dpi'] = 100\n\n\nfrom tqdm.notebook import tqdm\n```\n\n\n```python\nfrom larndsim import consts\nimport importlib\nimportlib.reload(consts)\nconsts.load_detector_properties(\"../larndsim/detector_properties/module0.yaml\",\n \"../larndsim/pixel_layouts/multi_tile_layout-2.2.16.yaml\")\n\nfrom larndsim import quenching, drifting, detsim, pixels_from_track, fee\n\nfrom numpy.lib import recfunctions as rfn\nimport torch\n\ndef torch_from_structured(tracks):\n tracks_np = rfn.structured_to_unstructured(tracks, copy=True, dtype=np.float32)\n return torch.from_numpy(tracks_np).float()\n\ndef structered_from_torch(tracks_torch, dtype):\n return rfn.unstructured_to_structured(tracks_torch.cpu().numpy(), dtype=dtype)\n```\n\n Generate .pixel_trim_dac using bits (0, 512)\n \t((, 0, 31, 64, 8)) \n Generate .threshold_global using bits (512, 520)\n \t((, 0, 255)) \n Generate .csa_gain using bits (520, 521)\n \t((['csa_gain', 'csa_bypass_enable', 'bypass_caps_en'], (, ), 0, 1)) \n Generate .csa_bypass_enable using bits (521, 522)\n \t((['csa_gain', 'csa_bypass_enable', 'bypass_caps_en'], (, ), 0, 1)) \n Generate .bypass_caps_en using bits (522, 523)\n \t((['csa_gain', 'csa_bypass_enable', 'bypass_caps_en'], (, ), 0, 1)) \n Generate .csa_enable using bits (528, 592)\n \t(((, ), 0, 1, 64, 1)) \n Generate .ibias_tdac using bits (592, 596)\n \t((, 0, 15)) \n Generate .ibias_comp using bits (600, 604)\n \t((, 0, 15)) \n Generate .ibias_buffer using bits (608, 612)\n \t((, 0, 15)) \n Generate .ibias_csa using bits (616, 620)\n \t((, 0, 15)) \n Generate .ibias_vref_buffer using bits (624, 628)\n \t((, 0, 15)) \n Generate .ibias_vcm_buffer using bits (632, 636)\n \t((, 0, 15)) \n Generate .ibias_tpulse using bits (640, 644)\n \t((, 0, 15)) \n Generate .ref_current_trim using bits (648, 653)\n \t((['ref_current_trim', 'override_ref', 'ref_kickstart'], , 0, 31)) \n Generate .override_ref using bits (653, 654)\n \t((['ref_current_trim', 'override_ref', 'ref_kickstart'], (, ), 0, 1)) \n Generate .ref_kickstart using bits (654, 655)\n \t((['ref_current_trim', 'override_ref', 'ref_kickstart'], (, ), 0, 1)) \n Generate .vref_dac using bits (656, 664)\n \t((, 0, 255)) \n Generate .vcm_dac using bits (664, 672)\n \t((, 0, 255)) \n Generate .csa_bypass_select using bits (672, 736)\n \t(((, ), 0, 1, 64, 1)) \n Generate .csa_monitor_select using bits (736, 800)\n \t(((, ), 0, 1, 64, 1)) \n Generate .csa_testpulse_enable using bits (800, 864)\n \t(((, ), 0, 1, 64, 1)) \n Generate .csa_testpulse_dac using bits (864, 872)\n \t((, 0, 255)) \n Generate .current_monitor_bank0 using bits (872, 876)\n \t(((, ), 0, 1, 4, 1)) \n Generate .current_monitor_bank1 using bits (880, 884)\n \t(((, ), 0, 1, 4, 1)) \n Generate .current_monitor_bank2 using bits (888, 892)\n \t(((, ), 0, 1, 4, 1)) \n Generate .current_monitor_bank3 using bits (896, 900)\n \t(((, ), 0, 1, 4, 1)) \n Generate .voltage_monitor_bank0 using bits (904, 907)\n \t(((, ), 0, 1, 3, 1)) \n Generate .voltage_monitor_bank1 using bits (912, 915)\n \t(((, ), 0, 1, 3, 1)) \n Generate .voltage_monitor_bank2 using bits (920, 923)\n \t(((, ), 0, 1, 3, 1)) \n Generate .voltage_monitor_bank3 using bits (928, 931)\n \t(((, ), 0, 1, 3, 1)) \n Generate .voltage_monitor_refgen using bits (936, 944)\n \t(((, ), 0, 1, 8, 1)) \n Generate .digital_monitor_enable using bits (944, 945)\n \t((['digital_monitor_enable', 'digital_monitor_select'], (, ), 0, 1)) \n Generate .digital_monitor_select using bits (945, 949)\n \t((['digital_monitor_enable', 'digital_monitor_select'], (, ), 0, 10)) \n Generate .digital_monitor_chan using bits (952, 958)\n \t((, 0, 63)) \n Generate .adc_hold_delay using bits (960, 976)\n \t((, 0, 65535)) \n Generate .chip_id using bits (976, 984)\n \t((, 0, 255)) \n Generate .enable_tx_dynamic_powerdown using bits (984, 985)\n \t((['enable_tx_dynamic_powerdown', 'load_config_defaults', 'enable_fifo_diagnostics', 'clk_ctrl', 'tx_dynamic_powerdown_cycles'], (, ), 0, 1)) \n Generate .load_config_defaults using bits (985, 986)\n \t((['enable_tx_dynamic_powerdown', 'load_config_defaults', 'enable_fifo_diagnostics', 'clk_ctrl', 'tx_dynamic_powerdown_cycles'], (, ), 0, 1)) \n Generate .enable_fifo_diagnostics using bits (986, 987)\n \t((['enable_tx_dynamic_powerdown', 'load_config_defaults', 'enable_fifo_diagnostics', 'clk_ctrl', 'tx_dynamic_powerdown_cycles'], (, ), 0, 1)) \n Generate .clk_ctrl using bits (987, 989)\n \t((['enable_tx_dynamic_powerdown', 'load_config_defaults', 'enable_fifo_diagnostics', 'clk_ctrl', 'tx_dynamic_powerdown_cycles'], (, ), 0, 3)) \n Generate .tx_dynamic_powerdown_cycles using bits (989, 992)\n \t((['enable_tx_dynamic_powerdown', 'load_config_defaults', 'enable_fifo_diagnostics', 'clk_ctrl', 'tx_dynamic_powerdown_cycles'], (, ), 0, 7)) \n Generate .enable_piso_upstream using bits (992, 996)\n \t(((, ), 0, 1, 4, 1)) \n Generate .enable_piso_downstream using bits (1000, 1004)\n \t(((, ), 0, 1, 4, 1)) \n Generate .enable_posi using bits (1008, 1012)\n \t(((, ), 0, 1, 4, 1)) \n Generate .test_mode_uart0 using bits (1016, 1018)\n \t((['test_mode_uart0', 'test_mode_uart1', 'test_mode_uart2', 'test_mode_uart3'], , 0, 3)) \n Generate .test_mode_uart1 using bits (1018, 1020)\n \t((['test_mode_uart0', 'test_mode_uart1', 'test_mode_uart2', 'test_mode_uart3'], , 0, 3)) \n Generate .test_mode_uart2 using bits (1020, 1022)\n \t((['test_mode_uart0', 'test_mode_uart1', 'test_mode_uart2', 'test_mode_uart3'], , 0, 3)) \n Generate .test_mode_uart3 using bits (1022, 1024)\n \t((['test_mode_uart0', 'test_mode_uart1', 'test_mode_uart2', 'test_mode_uart3'], , 0, 3)) \n Generate .enable_cross_trigger using bits (1024, 1025)\n \t((['enable_cross_trigger', 'enable_periodic_reset', 'enable_rolling_periodic_reset', 'enable_periodic_trigger', 'enable_rolling_periodic_trigger', 'enable_periodic_trigger_veto', 'enable_hit_veto'], , 0, 1)) \n Generate .enable_periodic_reset using bits (1025, 1026)\n \t((['enable_cross_trigger', 'enable_periodic_reset', 'enable_rolling_periodic_reset', 'enable_periodic_trigger', 'enable_rolling_periodic_trigger', 'enable_periodic_trigger_veto', 'enable_hit_veto'], , 0, 1)) \n Generate .enable_rolling_periodic_reset using bits (1026, 1027)\n \t((['enable_cross_trigger', 'enable_periodic_reset', 'enable_rolling_periodic_reset', 'enable_periodic_trigger', 'enable_rolling_periodic_trigger', 'enable_periodic_trigger_veto', 'enable_hit_veto'], , 0, 1)) \n Generate .enable_periodic_trigger using bits (1027, 1028)\n \t((['enable_cross_trigger', 'enable_periodic_reset', 'enable_rolling_periodic_reset', 'enable_periodic_trigger', 'enable_rolling_periodic_trigger', 'enable_periodic_trigger_veto', 'enable_hit_veto'], , 0, 1)) \n Generate .enable_rolling_periodic_trigger using bits (1028, 1029)\n \t((['enable_cross_trigger', 'enable_periodic_reset', 'enable_rolling_periodic_reset', 'enable_periodic_trigger', 'enable_rolling_periodic_trigger', 'enable_periodic_trigger_veto', 'enable_hit_veto'], , 0, 1)) \n Generate .enable_periodic_trigger_veto using bits (1029, 1030)\n \t((['enable_cross_trigger', 'enable_periodic_reset', 'enable_rolling_periodic_reset', 'enable_periodic_trigger', 'enable_rolling_periodic_trigger', 'enable_periodic_trigger_veto', 'enable_hit_veto'], , 0, 1)) \n Generate .enable_hit_veto using bits (1030, 1031)\n \t((['enable_cross_trigger', 'enable_periodic_reset', 'enable_rolling_periodic_reset', 'enable_periodic_trigger', 'enable_rolling_periodic_trigger', 'enable_periodic_trigger_veto', 'enable_hit_veto'], , 0, 1)) \n Generate .shadow_reset_length using bits (1032, 1040)\n \t((, 0, 255)) \n Generate .adc_burst_length using bits (1040, 1048)\n \t((, 0, 255)) \n Generate .channel_mask using bits (1048, 1112)\n \t((, 0, 1, 64, 1)) \n Generate .external_trigger_mask using bits (1112, 1176)\n \t((, 0, 1, 64, 1)) \n Generate .cross_trigger_mask using bits (1176, 1240)\n \t((, 0, 1, 64, 1)) \n Generate .periodic_trigger_mask using bits (1240, 1304)\n \t((, 0, 1, 64, 1)) \n Generate .periodic_reset_cycles using bits (1304, 1328)\n \t((, 0, 16777215)) \n Generate .periodic_trigger_cycles using bits (1328, 1360)\n \t((, 0, 4294967295)) \n Generate .enable_dynamic_reset using bits (1360, 1361)\n \t((['enable_dynamic_reset', 'enable_min_delta_adc', 'threshold_polarity', 'reset_length', 'mark_first_packet'], (, ), 0, 1)) \n Generate .enable_min_delta_adc using bits (1361, 1362)\n \t((['enable_dynamic_reset', 'enable_min_delta_adc', 'threshold_polarity', 'reset_length', 'mark_first_packet'], (, ), 0, 1)) \n Generate .threshold_polarity using bits (1362, 1363)\n \t((['enable_dynamic_reset', 'enable_min_delta_adc', 'threshold_polarity', 'reset_length', 'mark_first_packet'], (, ), 0, 1)) \n Generate .reset_length using bits (1363, 1366)\n \t((['enable_dynamic_reset', 'enable_min_delta_adc', 'threshold_polarity', 'reset_length', 'mark_first_packet'], , 0, 7)) \n Generate .mark_first_packet using bits (1366, 1367)\n \t((['enable_dynamic_reset', 'enable_min_delta_adc', 'threshold_polarity', 'reset_length', 'mark_first_packet'], (, ), 0, 1)) \n Generate .reset_threshold using bits (1368, 1376)\n \t((, 0, 255)) \n Generate .min_delta_adc using bits (1376, 1384)\n \t((, 0, 255)) \n Generate .digital_threshold using bits (1384, 1896)\n \t((, 0, 255, 64, 8)) \n Generate .RESERVED using bits (1896, 1912)\n \t((, 0, 0)) \n Generate .tx_slices0 using bits (1912, 1916)\n \t((['tx_slices0', 'tx_slices1'], , 0, 15)) \n Generate .tx_slices1 using bits (1916, 1920)\n \t((['tx_slices0', 'tx_slices1'], , 0, 15)) \n Generate .tx_slices2 using bits (1920, 1924)\n \t((['tx_slices2', 'tx_slices3'], , 0, 15)) \n Generate .tx_slices3 using bits (1924, 1928)\n \t((['tx_slices2', 'tx_slices3'], , 0, 15)) \n Generate .i_tx_diff0 using bits (1928, 1932)\n \t((['i_tx_diff0', 'i_tx_diff1'], , 0, 15)) \n Generate .i_tx_diff1 using bits (1932, 1936)\n \t((['i_tx_diff0', 'i_tx_diff1'], , 0, 15)) \n Generate .i_tx_diff2 using bits (1936, 1940)\n \t((['i_tx_diff2', 'i_tx_diff3'], , 0, 15)) \n Generate .i_tx_diff3 using bits (1940, 1944)\n \t((['i_tx_diff2', 'i_tx_diff3'], , 0, 15)) \n Generate .i_rx0 using bits (1944, 1948)\n \t((['i_rx0', 'i_rx1'], , 0, 15)) \n Generate .i_rx1 using bits (1948, 1952)\n \t((['i_rx0', 'i_rx1'], , 0, 15)) \n Generate .i_rx2 using bits (1952, 1956)\n \t((['i_rx2', 'i_rx3'], , 0, 15)) \n Generate .i_rx3 using bits (1956, 1960)\n \t((['i_rx2', 'i_rx3'], , 0, 15)) \n Generate .i_rx_clk using bits (1960, 1964)\n \t((['i_rx_clk', 'i_rx_rst'], , 0, 15)) \n Generate .i_rx_rst using bits (1964, 1968)\n \t((['i_rx_clk', 'i_rx_rst'], , 0, 15)) \n Generate .i_rx_ext_trig using bits (1968, 1972)\n \t((, 0, 15)) \n Generate .r_term0 using bits (1976, 1981)\n \t((, 0, 31)) \n Generate .r_term1 using bits (1984, 1989)\n \t((, 0, 31)) \n Generate .r_term2 using bits (1992, 1997)\n \t((, 0, 31)) \n Generate .r_term3 using bits (2000, 2005)\n \t((, 0, 31)) \n Generate .r_term_clk using bits (2008, 2013)\n \t((, 0, 31)) \n Generate .r_term_reset using bits (2016, 2021)\n \t((, 0, 31)) \n Generate .r_term_ext_trig using bits (2024, 2029)\n \t((, 0, 31)) \n Generate .v_cm_lvds_tx0 using bits (2032, 2035)\n \t((['v_cm_lvds_tx0', 'v_cm_lvds_tx0'], , 0, 7)) \n Generate .v_cm_lvds_tx1 using bits (2036, 2039)\n \t((['v_cm_lvds_tx0', 'v_cm_lvds_tx0'], , 0, 7)) \n Generate .v_cm_lvds_tx2 using bits (2040, 2043)\n \t((['v_cm_lvds_tx2', 'v_cm_lvds_tx3'], , 0, 7)) \n Generate .v_cm_lvds_tx3 using bits (2044, 2047)\n \t((['v_cm_lvds_tx2', 'v_cm_lvds_tx3'], , 0, 7)) \n\n\n\n```python\nfrom larndsim.sim_with_grad import sim_with_grad\nsim = sim_with_grad()\nsim.load_detector_properties(\"../larndsim/detector_properties/module0.yaml\",\n \"../larndsim/pixel_layouts/multi_tile_layout-2.2.16.yaml\")\n```\n\n /sdf/home/s/sgaz/conda/envs/ml/lib/python3.8/site-packages/ipykernel/ipkernel.py:287: DeprecationWarning: `should_run_async` will not call `transform_cell` automatically in the future. Please pass the result to `transformed_cell` argument and any exception that happen during thetransform in `preprocessing_exc_tuple` in IPython 7.17 and above.\n and should_run_async(code)\n\n\n### Dataset import\nFirst of all we load the `edep-sim` output. For this sample we need to invert $z$ and $y$ axes.\n\n\n```python\nunique_pix = cp.empty((0,2))\npixels_signals = cp.zeros((len(unique_pix), len(consts.time_ticks)*3))\n```\n\n\n```python\nwith h5py.File('module0_corsika.h5', 'r') as f:\n tracks = np.array(f['segments'])\n\nx_start = np.copy(tracks['x_start'] )\nx_end = np.copy(tracks['x_end'])\nx = np.copy(tracks['x'])\n\ntracks['x_start'] = np.copy(tracks['z_start'])\ntracks['x_end'] = np.copy(tracks['z_end'])\ntracks['x'] = np.copy(tracks['z'])\n\ntracks['z_start'] = x_start\ntracks['z_end'] = x_end\ntracks['z'] = x\n\nselected_tracks = tracks[30:40]\n```\n\n### Quenching and drifting\nWe calculate the number of electrons after recombination (`quenching` module) and the position and number of electrons after drifting (`drifting` module).\n\n\n```python\nTPB = 256\nBPG = ceil(selected_tracks.shape[0] / TPB)\nimport importlib\nimportlib.reload(drifting)\n\nselected_tracks_torch = torch_from_structured(np.copy(selected_tracks))\n\nquenching.quench[BPG,TPB](selected_tracks, consts.birks)\ndrifting.drift[BPG,TPB](selected_tracks)\n```\n\n\n```python\nselected_tracks_quench = sim.quench(selected_tracks_torch, consts.birks, fields=selected_tracks.dtype.names)\n\n#Truncation makes gradients vanish - do here for comparisons\nn_elec_idx = selected_tracks.dtype.names.index('n_electrons')\nselected_tracks_quench[:, n_elec_idx] = selected_tracks_quench[:, n_elec_idx].trunc()\n\nselected_tracks_drift = sim.drift(selected_tracks_quench, fields=selected_tracks.dtype.names)\nselected_tracks_drift[:, n_elec_idx] = selected_tracks_drift[:, n_elec_idx].trunc()\n```\n\n /sdf/home/s/sgaz/conda/envs/ml/lib/python3.8/site-packages/ipykernel/ipkernel.py:287: DeprecationWarning: `should_run_async` will not call `transform_cell` automatically in the future. Please pass the result to `transformed_cell` argument and any exception that happen during thetransform in `preprocessing_exc_tuple` in IPython 7.17 and above.\n and should_run_async(code)\n\n\n\n```python\n(selected_tracks_drift[:, selected_tracks.dtype.names.index('n_electrons')].numpy()\n == selected_tracks['n_electrons']).all()\n```\n\n\n\n\n True\n\n\n\nWe find the pixels intersected by the projection of the tracks on the anode plane using the Bresenham's algorithm. We also take into account the neighboring pixels, due to the transverse diffusion of the charges.\n\n\n```python\nimportlib.reload(pixels_from_track)\n\n# Here we build a map between tracks and event IDs\nunique_eventIDs = cp.unique(selected_tracks['eventID'])\nevent_id_map = cp.searchsorted(unique_eventIDs,cp.asarray(selected_tracks['eventID']))\nevent_id_map_torch = torch.from_numpy(event_id_map.get())\n\nlongest_pix = ceil(max(selected_tracks[\"dx\"])/consts.pixel_pitch)\nmax_radius = ceil(max(selected_tracks[\"tran_diff\"])*5/consts.pixel_pitch)\n\nMAX_PIXELS = int((longest_pix*4+6)*max_radius*1.5)\nMAX_ACTIVE_PIXELS = int(longest_pix*1.5)\n\nactive_pixels = cp.full((selected_tracks.shape[0], MAX_ACTIVE_PIXELS, 2), -1, dtype=np.int32)\nneighboring_pixels = cp.full((selected_tracks.shape[0], MAX_PIXELS, 2), -1, dtype=np.int32)\nn_pixels_list = cp.zeros(shape=(selected_tracks.shape[0]))\nTPB = 128\nBPG = ceil(selected_tracks.shape[0] / TPB)\npixels_from_track.get_pixels[BPG,TPB](selected_tracks,\n active_pixels,\n neighboring_pixels,\n n_pixels_list,\n max_radius+1)\n```\n\n\n```python\nactive_pixels_torch, neighboring_pixels_torch, n_pixels_list_ep = sim.get_pixels(selected_tracks_drift,\n fields=selected_tracks.dtype.names)\nprint(neighboring_pixels_torch.shape)\nprint(neighboring_pixels.shape)\n```\n\n /sdf/home/s/sgaz/conda/envs/ml/lib/python3.8/site-packages/ipykernel/ipkernel.py:287: DeprecationWarning: `should_run_async` will not call `transform_cell` automatically in the future. Please pass the result to `transformed_cell` argument and any exception that happen during thetransform in `preprocessing_exc_tuple` in IPython 7.17 and above.\n and should_run_async(code)\n /sdf/home/s/sgaz/conda/envs/ml/lib/python3.8/site-packages/eagerpy/tensor/base.py:98: UserWarning: __floordiv__ is deprecated, and its behavior will change in a future version of pytorch. It currently rounds toward 0 (like the 'trunc' function NOT 'floor'). This results in incorrect rounding for negative values. To keep the current behavior, use torch.div(a, b, rounding_mode='trunc'), or for actual floor division, use torch.div(a, b, rounding_mode='floor').\n return type(self)(self.raw.__floordiv__(unwrap1(other)))\n\n\n torch.Size([10, 87, 2])\n (10, 87, 2)\n\n\n /sdf/home/s/sgaz/conda/envs/ml/lib/python3.8/site-packages/torch/functional.py:445: UserWarning: torch.meshgrid: in an upcoming release, it will be required to pass the indexing argument. (Triggered internally at ../aten/src/ATen/native/TensorShape.cpp:2157.)\n return _VF.meshgrid(tensors, **kwargs) # type: ignore[attr-defined]\n\n\n## Bug?\nNeighboring pixels is different for one of the track segments. Running _just_ that track segment vs running a set of track segments gives two different results in the cupy setup. The single track segment cupy result matches the eagerpy setup.\n\n\n```python\nnp.where((neighboring_pixels_torch.numpy() != neighboring_pixels.get()))\n```\n\n\n\n\n (array([3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,\n 3]),\n array([ 0, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 9, 9, 10, 11, 11,\n 12, 12, 13, 13, 14, 14]),\n array([1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0,\n 1]))\n\n\n\n\n```python\ndef cupy_unique_axis0(array):\n # axis is still not supported for cupy.unique, this\n # is a workaround\n if len(array.shape) != 2:\n raise ValueError(\"Input array must be 2D.\")\n sortarr = array[cp.lexsort(array.T[::-1])]\n mask = cp.empty(array.shape[0], dtype=cp.bool_)\n mask[0] = True\n mask[1:] = cp.any(sortarr[1:] != sortarr[:-1], axis=1)\n return sortarr[mask]\n\nshapes = neighboring_pixels.shape\njoined = neighboring_pixels.reshape(shapes[0]*shapes[1],2)\nthis_unique_pix = cupy_unique_axis0(joined)\nthis_unique_pix = this_unique_pix[(this_unique_pix[:,0] != -1) & (this_unique_pix[:,1] != -1),:]\n```\n\n\n```python\nunique_pix = cp.concatenate((unique_pix, this_unique_pix),axis=0)\n```\n\n### Charge distribution calculation\nHere we calculate the current induced by each track on the pixels, taking into account longitudinal and transverse diffusion. The track segment is parametrized as:\n\\begin{align}\nx'(r') &=x_s + \\frac{\\Delta x}{\\Delta r}r'\\\\\ny'(r') &=y_s + \\frac{\\Delta y}{\\Delta r}r'\\\\\nz'(r') &=z_s + \\frac{\\Delta z}{\\Delta r}r',\n\\end{align}\nwhere $\\Delta r$ is the segment length. Here we assume $z$ as the drift direction.\nThe diffused charge distribution is calculated with the following integral:\n\\begin{equation}\n\\rho(x,y,z) = \\frac{Q}{\\sqrt{(2\\pi)^3}\\sigma_x\\sigma_y\\sigma_z\\Delta r}\\exp\\left[-\\frac{(x-x_s)^2}{2\\sigma_x^2}-\\frac{(y-y_s)^2}{2\\sigma_y^2}-\\frac{(z-z_s)^2}{2\\sigma_z^2}\\right]\\int^{r'=\\Delta r}_{r'=0}dr'\\exp[-(ar'^2+br')],\n\\end{equation}\nwhere \n\\begin{align}\na &= \\left[\\left(\\frac{\\Delta x}{\\Delta r}\\right)^2\\frac{1}{2\\sigma_x^2} + \\left(\\frac{\\Delta y}{\\Delta r}\\right)^2\\frac{1}{2\\sigma_y^2} + \\left(\\frac{\\Delta z}{\\Delta r}\\right)^2\\frac{1}{2\\sigma_z^2} \\right]\\\\\nb &= -\\left[\\frac{(x-x_s)}{\\sigma_x^2}\\frac{\\Delta x}{\\Delta r}+\n\\frac{(y-y_s)}{\\sigma_y^2}\\frac{\\Delta y}{\\Delta r}+\n\\frac{(z-z_s)}{\\sigma_z^2}\\frac{\\Delta z}{\\Delta r}\\right].\n\\end{align}\n\nThe simmetry of the transverse diffusion along the track allows to take a slice on the $xy$ plane and solve the integral once at a fixed $z$ coordinate (e.g. at $z_{m} = (z_s+z_e)/2$) and re-use it at other $z$ coordinates away from the endpoints (where $\\rho(x,y,z)$ varies along $z$ so must be calculated at each $z$). \n\n\n```python\nimportlib.reload(detsim)\n\nmax_length = cp.array([0])\ntrack_starts = cp.empty(selected_tracks.shape[0])\nthreadsperblock = 128\nblockspergrid = ceil(selected_tracks.shape[0] / threadsperblock)\ndetsim.time_intervals[blockspergrid,threadsperblock](track_starts, max_length, event_id_map, selected_tracks)\n```\n\n\n```python\ntrack_starts_torch, max_length_torch = sim.time_intervals(event_id_map_torch, \n selected_tracks_drift, \n fields=selected_tracks.dtype.names)\n\nprint(track_starts_torch, max_length_torch)\nprint(track_starts, max_length)\n```\n\n /sdf/home/s/sgaz/conda/envs/ml/lib/python3.8/site-packages/ipykernel/ipkernel.py:287: DeprecationWarning: `should_run_async` will not call `transform_cell` automatically in the future. Please pass the result to `transformed_cell` argument and any exception that happen during thetransform in `preprocessing_exc_tuple` in IPython 7.17 and above.\n and should_run_async(code)\n\n\n tensor([158.0174, 159.4631, 165.9619, 174.5850, 177.3604, 181.8703, 177.4428,\n 176.2220, 170.8327, 169.0134]) tensor(218)\n [158. 159.5 166. 174.6 177.4 181.9 177.4 176.2 170.8 169. ] [218]\n\n\n\n```python\nsignals = cp.zeros((selected_tracks.shape[0],\n neighboring_pixels.shape[1],\n cp.asnumpy(max_length)[0]), dtype=np.float32)\nthreadsperblock = (1,1,64)\n\nimportlib.reload(detsim)\n\nblockspergrid_x = ceil(signals.shape[0] / threadsperblock[0])\nblockspergrid_y = ceil(signals.shape[1] / threadsperblock[1])\nblockspergrid_z = ceil(signals.shape[2] / threadsperblock[2])\nblockspergrid = (blockspergrid_x, blockspergrid_y, blockspergrid_z)\ndetsim.tracks_current[blockspergrid,threadsperblock](signals,\n neighboring_pixels,\n selected_tracks)\nprint(signals.shape)\n```\n\n (10, 87, 218)\n\n\n\n```python\nsignals_ep = sim.tracks_current(neighboring_pixels_torch, selected_tracks_drift, \n max_length_torch,\n fields=selected_tracks.dtype.names)\nprint(signals_ep.shape)\n```\n\n /sdf/home/s/sgaz/conda/envs/ml/lib/python3.8/site-packages/ipykernel/ipkernel.py:287: DeprecationWarning: `should_run_async` will not call `transform_cell` automatically in the future. Please pass the result to `transformed_cell` argument and any exception that happen during thetransform in `preprocessing_exc_tuple` in IPython 7.17 and above.\n and should_run_async(code)\n /sdf/home/s/sgaz/conda/envs/ml/lib/python3.8/site-packages/eagerpy/tensor/base.py:98: UserWarning: __floordiv__ is deprecated, and its behavior will change in a future version of pytorch. It currently rounds toward 0 (like the 'trunc' function NOT 'floor'). This results in incorrect rounding for negative values. To keep the current behavior, use torch.div(a, b, rounding_mode='trunc'), or for actual floor division, use torch.div(a, b, rounding_mode='floor').\n return type(self)(self.raw.__floordiv__(unwrap1(other)))\n\n\n torch.Size([10, 87, 218])\n\n\n\n```python\npixel_index_map = cp.full((selected_tracks.shape[0], neighboring_pixels.shape[1]), -1)\ntry:\n compare = neighboring_pixels[..., np.newaxis, :] == unique_pix\n indices = cp.where(cp.logical_and(compare[..., 0], compare[..., 1]))\nexcept cp.cuda.memory.OutOfMemoryError:\n print(\"out of memory\")\npixel_index_map[indices[0], indices[1]] = indices[2]\n```\n\n\n```python\nthis_pixels_signals = cp.zeros((len(this_unique_pix), len(consts.time_ticks)*3))\npixels_signals = cp.concatenate((pixels_signals, this_pixels_signals), axis=0)\n```\n\n\n```python\nblockspergrid_x = ceil(signals.shape[0] / threadsperblock[0])\nblockspergrid_y = ceil(signals.shape[1] / threadsperblock[1])\nblockspergrid_z = ceil(signals.shape[2] / threadsperblock[2])\nblockspergrid = (blockspergrid_x, blockspergrid_y, blockspergrid_z)\nthis_pixels_signals = cp.zeros((len(this_unique_pix), len(consts.time_ticks)*3))\npixels_signals = cp.concatenate((pixels_signals, this_pixels_signals), axis=0)\ndetsim.sum_pixel_signals[blockspergrid,threadsperblock](pixels_signals,\n signals,\n track_starts,\n pixel_index_map)\n```\n\n\n```python\ncurrents = pixels_signals.sum(axis=1)*sim.t_sampling/sim.e_charge\n```\n\n /sdf/home/s/sgaz/conda/envs/ml/lib/python3.8/site-packages/ipykernel/ipkernel.py:287: DeprecationWarning: `should_run_async` will not call `transform_cell` automatically in the future. Please pass the result to `transformed_cell` argument and any exception that happen during thetransform in `preprocessing_exc_tuple` in IPython 7.17 and above.\n and should_run_async(code)\n\n\n\n```python\nsum(currents)/sum(selected_tracks['n_electrons'])\n```\n\n\n\n\n array(1.04705509)\n\n\n\n\n```python\nunique_pix_torch = torch.empty((0, 2))\npixels_signals_torch = torch.zeros((len(unique_pix_torch), len(sim.time_ticks)*3))\n\nshapes_torch = neighboring_pixels_torch.shape\njoined_torch = neighboring_pixels_torch.reshape(shapes_torch[0]*shapes_torch[1], 2)\n\nthis_unique_pix_torch = torch.unique(joined_torch, dim=0)\nthis_unique_pix_torch = this_unique_pix_torch[(this_unique_pix_torch[:,0] != -1) & (this_unique_pix_torch[:,1] != -1),:]\nunique_pix_torch = torch.cat((unique_pix_torch, this_unique_pix_torch),dim=0)\n\nthis_pixels_signals_torch = torch.zeros((len(this_unique_pix_torch), len(sim.time_ticks)*3))\npixels_signals_torch = torch.cat((pixels_signals_torch, this_pixels_signals_torch), dim=0)\n\npixel_index_map_torch = torch.full((selected_tracks.shape[0], neighboring_pixels_torch.shape[1]), -1)\ncompare_torch = (neighboring_pixels_torch[..., np.newaxis, :] == unique_pix_torch)\n\nindices_torch = torch.where(torch.logical_and(compare_torch[..., 0], compare_torch[...,1]))\npixel_index_map_torch[indices_torch[0], indices_torch[1]] = indices_torch[2]\n```\n\n\n```python\npixels_signals_torch = sim.sum_pixel_signals(pixels_signals_torch,\n signals_ep,\n track_starts_torch,\n pixel_index_map_torch)\n\ncurrents_torch = pixels_signals_torch.sum(dim=1)*sim.t_sampling/sim.e_charge\n```\n\n\n```python\nsum(currents_torch)/sum(selected_tracks['n_electrons'])\n```\n\n\n\n\n tensor(1.0068)\n\n\n\n\n```python\ntrack = selected_tracks[0]\npixel_plane = track['pixel_plane']\nz_anode = consts.tpc_borders[pixel_plane][2][0]\nz = (track['z_start']+track['z_end'])/2\ndrift_distance = abs(z - z_anode)\ndrift_start = abs(min(track[\"z_start\"],track[\"z_end\"]) - z_anode)\ndrift_end = abs(max(track[\"z_start\"],track[\"z_end\"]) - z_anode) \ndrift_time = drift_distance / consts.vdrift\ndrift_start_time = drift_start / consts.vdrift\ndrift_end_time = drift_end / consts.vdrift\n```\n\n### 3D event display\n\n\n```python\n%matplotlib notebook\n\ncmap = cm.Spectral_r\nnorm = colors.Normalize(vmin=0, vmax=256)\nm = cm.ScalarMappable(norm=norm, cmap=cmap)\n\ncmap = cm.viridis\nnorm_curr = colors.LogNorm(vmin=min(currents[currents>0]), vmax=max(currents))\nm_curr = cm.ScalarMappable(norm=norm_curr, cmap=cmap)\nfig = plt.figure(figsize=(7,7))\nax = fig.add_subplot(111, projection='3d')\n\nfor it,t in enumerate(selected_tracks):\n if it == 0:\n ax.plot((t[\"x_start\"], t[\"x_end\"]), \n (t[\"z_start\"], t[\"z_end\"]),\n (t[\"y_start\"], t[\"y_end\"]),\n c='r',\n lw=1,\n alpha=1,\n zorder=10,\n label='Geant4 detector segment')\n else:\n ax.plot((t[\"x_start\"], t[\"x_end\"]), \n (t[\"z_start\"], t[\"z_end\"]),\n (t[\"y_start\"], t[\"y_end\"]),\n c='r',\n lw=1,\n alpha=1,\n zorder=9999)\n \n ax.plot((t[\"x_start\"], t[\"x_end\"]), \n (consts.tpc_borders[t['pixel_plane']][2][0], consts.tpc_borders[t['pixel_plane']][2][0]),\n (t[\"y_start\"], t[\"y_end\"]),\n c='r',\n lw=1,\n ls=':',\n alpha=1,\n zorder=9999)\n \nfor ip, p in enumerate(unique_pix):\n x_rect, y_rect = detsim.get_pixel_coordinates(p.get())\n pixel_plane = int(p[0] // consts.n_pixels[0])\n row = int(pixel_plane // 4)\n column = int(pixel_plane % 4)\n if currents[ip] > 0:\n rect = plt.Rectangle((x_rect, y_rect),\n consts.pixel_pitch, consts.pixel_pitch,\n linewidth=0.1, fc=m_curr.to_rgba(currents[ip].get()),\n edgecolor='gray')\n ax.add_patch(rect)\n art3d.pathpatch_2d_to_3d(rect, z=consts.tpc_borders[pixel_plane][2][0], zdir=\"y\")\n \nanode1 = plt.Rectangle((consts.tpc_borders[0][0][0], consts.tpc_borders[0][1][0]),\n consts.tpc_borders[0][0][1]-consts.tpc_borders[0][0][0], \n consts.tpc_borders[0][1][1]-consts.tpc_borders[0][1][0],\n linewidth=1, fc='none',\n edgecolor='gray', label=('Pixel' if ip == 5 else ''))\nax.add_patch(anode1)\nart3d.pathpatch_2d_to_3d(anode1, z=consts.tpc_borders[0][2][0], zdir=\"y\")\n\nanode2 = plt.Rectangle((consts.tpc_borders[0][0][0], consts.tpc_borders[0][1][0]),\n consts.tpc_borders[0][0][1]-consts.tpc_borders[0][0][0], \n consts.tpc_borders[0][1][1]-consts.tpc_borders[0][1][0],\n linewidth=1, fc='none',\n edgecolor='gray', label=('Pixel' if ip == 5 else ''))\nax.add_patch(anode2)\nart3d.pathpatch_2d_to_3d(anode2, z=consts.tpc_borders[1][2][0], zdir=\"y\")\n\ncathode = plt.Rectangle((consts.tpc_borders[0][0][0], consts.tpc_borders[0][1][0]),\n consts.tpc_borders[0][0][1]-consts.tpc_borders[0][0][0], \n consts.tpc_borders[0][1][1]-consts.tpc_borders[0][1][0],\n linewidth=1, fc='gray', alpha=0.5,\n edgecolor='gray', label=('Pixel' if ip == 5 else ''))\nax.add_patch(cathode)\nart3d.pathpatch_2d_to_3d(cathode, z=0, zdir=\"y\")\n\nax.plot((consts.tpc_borders[0][0][0],consts.tpc_borders[0][0][0]),(consts.tpc_borders[0][2][0],consts.tpc_borders[1][2][0]),\n (consts.tpc_borders[0][1][0],consts.tpc_borders[0][1][0]), lw=1,color='gray')\n\nax.plot((consts.tpc_borders[0][0][0],consts.tpc_borders[0][0][0]),(consts.tpc_borders[0][2][0],consts.tpc_borders[1][2][0]),\n (consts.tpc_borders[0][1][1],consts.tpc_borders[0][1][1]), lw=1,color='gray')\n\nax.plot((consts.tpc_borders[0][0][1],consts.tpc_borders[0][0][1]),(consts.tpc_borders[0][2][0],consts.tpc_borders[1][2][0]),\n (consts.tpc_borders[0][1][0],consts.tpc_borders[0][1][0]), lw=1,color='gray')\n\nax.plot((consts.tpc_borders[0][0][1],consts.tpc_borders[0][0][1]),(consts.tpc_borders[0][2][0],consts.tpc_borders[1][2][0]),\n (consts.tpc_borders[0][1][1],consts.tpc_borders[0][1][1]), lw=1,color='gray')\n\n# ax.set_ylim(consts.module_borders[pixel_plane][2][0],50)\nax.set_xlim(consts.tpc_borders[0][0][0],consts.tpc_borders[1][0][1])\nax.set_ylim(consts.tpc_borders[0][2][0],consts.tpc_borders[1][2][0])\nax.set_zlim(consts.tpc_borders[0][1][0],consts.tpc_borders[1][1][1])\n\nax.set_box_aspect((2,2,4))\nax.grid(False)\nax.xaxis.set_major_locator(plt.MaxNLocator(3))\nax.yaxis.set_major_locator(plt.MaxNLocator(3))\n# ax.set_axis_off()\nax.w_xaxis.set_pane_color((1.0, 1.0, 1.0, 1.0))\nax.w_yaxis.set_pane_color((1.0, 1.0, 1.0, 1.0))\nax.w_zaxis.set_pane_color((1.0, 1.0, 1.0, 1.0))\n\nax.view_init(azim=20)\n# def rotate(angle):\n# ax.view_init(azim=angle)\n \n# from matplotlib import animation\n\n# rot_animation = animation.FuncAnimation(fig, rotate, frames=np.arange(0,362,2),interval=50)\n\nax.set_ylabel(\"z [cm]\")\nax.set_xlabel(\"x [cm]\")\nax.set_zlabel(\"y [cm]\")\n_ = plt.colorbar(m_curr,fraction=0.035, pad=0.05,label='Induced current integral [# electrons]')\n\n```\n\n\n \n\n\n\n\n\n\n\n```python\n%matplotlib notebook\n\ncmap = cm.Spectral_r\nnorm = colors.Normalize(vmin=0, vmax=256)\nm = cm.ScalarMappable(norm=norm, cmap=cmap)\n\ncmap = cm.viridis\nnorm_curr = colors.LogNorm(vmin=min(currents_torch[currents_torch>0]), vmax=max(currents_torch))\nm_curr = cm.ScalarMappable(norm=norm_curr, cmap=cmap)\nfig = plt.figure(figsize=(7,7))\nax = fig.add_subplot(111, projection='3d')\n\nfor it,t in enumerate(selected_tracks):\n if it == 0:\n ax.plot((t[\"x_start\"], t[\"x_end\"]), \n (t[\"z_start\"], t[\"z_end\"]),\n (t[\"y_start\"], t[\"y_end\"]),\n c='r',\n lw=1,\n alpha=1,\n zorder=10,\n label='Geant4 detector segment')\n else:\n ax.plot((t[\"x_start\"], t[\"x_end\"]), \n (t[\"z_start\"], t[\"z_end\"]),\n (t[\"y_start\"], t[\"y_end\"]),\n c='r',\n lw=1,\n alpha=1,\n zorder=9999)\n \n ax.plot((t[\"x_start\"], t[\"x_end\"]), \n (sim.tpc_borders[t['pixel_plane']][2][0], sim.tpc_borders[t['pixel_plane']][2][0]),\n (t[\"y_start\"], t[\"y_end\"]),\n c='r',\n lw=1,\n ls=':',\n alpha=1,\n zorder=9999)\n\nx_r_all, y_r_all = sim.get_pixel_coordinates(ep.astensor(unique_pix_torch))\nfor ip, p in enumerate(unique_pix_torch):\n x_rect = x_r_all[ip, 0].numpy()\n y_rect = y_r_all[ip, 0].numpy()\n pixel_plane = int(p[0] // sim.n_pixels[0])\n row = int(pixel_plane // 4)\n column = int(pixel_plane % 4)\n if currents_torch[ip] > 0:\n rect = plt.Rectangle((x_rect, y_rect),\n sim.pixel_pitch, sim.pixel_pitch,\n linewidth=0.1, fc=m_curr.to_rgba(currents_torch[ip]),\n edgecolor='gray')\n ax.add_patch(rect)\n art3d.pathpatch_2d_to_3d(rect, z=sim.tpc_borders[pixel_plane][2][0], zdir=\"y\")\n \nanode1 = plt.Rectangle((sim.tpc_borders[0][0][0], sim.tpc_borders[0][1][0]),\n sim.tpc_borders[0][0][1]-sim.tpc_borders[0][0][0], \n sim.tpc_borders[0][1][1]-sim.tpc_borders[0][1][0],\n linewidth=1, fc='none',\n edgecolor='gray', label=('Pixel' if ip == 5 else ''))\nax.add_patch(anode1)\nart3d.pathpatch_2d_to_3d(anode1, z=sim.tpc_borders[0][2][0], zdir=\"y\")\n\nanode2 = plt.Rectangle((sim.tpc_borders[0][0][0], sim.tpc_borders[0][1][0]),\n sim.tpc_borders[0][0][1]-sim.tpc_borders[0][0][0], \n sim.tpc_borders[0][1][1]-sim.tpc_borders[0][1][0],\n linewidth=1, fc='none',\n edgecolor='gray', label=('Pixel' if ip == 5 else ''))\nax.add_patch(anode2)\nart3d.pathpatch_2d_to_3d(anode2, z=sim.tpc_borders[1][2][0], zdir=\"y\")\n\ncathode = plt.Rectangle((sim.tpc_borders[0][0][0], sim.tpc_borders[0][1][0]),\n sim.tpc_borders[0][0][1]-sim.tpc_borders[0][0][0], \n sim.tpc_borders[0][1][1]-sim.tpc_borders[0][1][0],\n linewidth=1, fc='gray', alpha=0.5,\n edgecolor='gray', label=('Pixel' if ip == 5 else ''))\nax.add_patch(cathode)\nart3d.pathpatch_2d_to_3d(cathode, z=0, zdir=\"y\")\n\nax.plot((sim.tpc_borders[0][0][0],sim.tpc_borders[0][0][0]),(sim.tpc_borders[0][2][0],sim.tpc_borders[1][2][0]),\n (sim.tpc_borders[0][1][0],sim.tpc_borders[0][1][0]), lw=1,color='gray')\n\nax.plot((sim.tpc_borders[0][0][0],sim.tpc_borders[0][0][0]),(sim.tpc_borders[0][2][0],sim.tpc_borders[1][2][0]),\n (sim.tpc_borders[0][1][1],sim.tpc_borders[0][1][1]), lw=1,color='gray')\n\nax.plot((sim.tpc_borders[0][0][1],sim.tpc_borders[0][0][1]),(sim.tpc_borders[0][2][0],sim.tpc_borders[1][2][0]),\n (sim.tpc_borders[0][1][0],sim.tpc_borders[0][1][0]), lw=1,color='gray')\n\nax.plot((sim.tpc_borders[0][0][1],sim.tpc_borders[0][0][1]),(sim.tpc_borders[0][2][0],sim.tpc_borders[1][2][0]),\n (sim.tpc_borders[0][1][1],sim.tpc_borders[0][1][1]), lw=1,color='gray')\n\n# ax.set_ylim(sim.module_borders[pixel_plane][2][0],50)\nax.set_xlim(sim.tpc_borders[0][0][0],sim.tpc_borders[1][0][1])\nax.set_ylim(sim.tpc_borders[0][2][0],sim.tpc_borders[1][2][0])\nax.set_zlim(sim.tpc_borders[0][1][0],sim.tpc_borders[1][1][1])\n\nax.set_box_aspect((2,2,4))\nax.grid(False)\nax.xaxis.set_major_locator(plt.MaxNLocator(3))\nax.yaxis.set_major_locator(plt.MaxNLocator(3))\n# ax.set_axis_off()\nax.w_xaxis.set_pane_color((1.0, 1.0, 1.0, 1.0))\nax.w_yaxis.set_pane_color((1.0, 1.0, 1.0, 1.0))\nax.w_zaxis.set_pane_color((1.0, 1.0, 1.0, 1.0))\n\nax.view_init(azim=20)\n# def rotate(angle):\n# ax.view_init(azim=angle)\n \n# from matplotlib import animation\n\n# rot_animation = animation.FuncAnimation(fig, rotate, frames=np.arange(0,362,2),interval=50)\n\nax.set_ylabel(\"z [cm]\")\nax.set_xlabel(\"x [cm]\")\nax.set_zlabel(\"y [cm]\")\n_ = plt.colorbar(m_curr,fraction=0.035, pad=0.05,label='Induced current integral [# electrons]')\n\n```\n\n\n \n\n\n\n\n\n\n :45: UserWarning: __floordiv__ is deprecated, and its behavior will change in a future version of pytorch. It currently rounds toward 0 (like the 'trunc' function NOT 'floor'). This results in incorrect rounding for negative values. To keep the current behavior, use torch.div(a, b, rounding_mode='trunc'), or for actual floor division, use torch.div(a, b, rounding_mode='floor').\n pixel_plane = int(p[0] // sim.n_pixels[0])\n\n\n\n```python\n%matplotlib inline\nplt.plot(np.arange(len(currents.get().flatten())), currents.get().flatten())\nplt.plot(np.arange(len(currents_torch.flatten())), currents_torch.flatten())\nplt.show()\n```\n\nMostly quite close, cupy (blue) has an extra spike\n\n\n```python\n# rot_animation.save('animation.gif', writer='imagemagick', fps=10, bitrate=10, dpi=100)\n```\n\n /sdf/home/s/sgaz/conda/envs/ml/lib/python3.8/site-packages/ipykernel/ipkernel.py:287: DeprecationWarning: `should_run_async` will not call `transform_cell` automatically in the future. Please pass the result to `transformed_cell` argument and any exception that happen during thetransform in `preprocessing_exc_tuple` in IPython 7.17 and above.\n and should_run_async(code)\n\n\n### Electronics response and digitization \nHere we simulate the electronics response (the self-triggering cycle) and the signal digitization.\n\n\n```python\ntime_ticks = cp.linspace(0, len(unique_eventIDs)*consts.time_interval[1]*3, pixels_signals.shape[1]+1)\nintegral_list = cp.zeros((pixels_signals.shape[0], fee.MAX_ADC_VALUES))\nadc_ticks_list = cp.zeros((pixels_signals.shape[0], fee.MAX_ADC_VALUES))\nTPB = 128\nBPG = ceil(pixels_signals.shape[0] / TPB)\nrng_states = create_xoroshiro128p_states(TPB * BPG, seed=0)\n# integrate = cp.zeros((pixels_signals.shape[0], time_ticks.shape[0]))\nfee.get_adc_values[BPG,TPB](pixels_signals,\n time_ticks,\n integral_list,\n adc_ticks_list,\n 0,\n rng_states)#,\nadc_list = fee.digitize(integral_list)\n```\n\n\n```python\ntime_ticks_torch = torch.linspace(0, len(unique_eventIDs)*sim.time_interval[1]*3, pixels_signals_torch.shape[1]+1)\n\nintegral_list_torch, adc_ticks_list_torch = sim.get_adc_values(pixels_signals_torch,\n time_ticks_torch,\n 0)\nadc_list_torch = sim.digitize(integral_list_torch)\n```\n\n /sdf/home/s/sgaz/conda/envs/ml/lib/python3.8/site-packages/ipykernel/ipkernel.py:287: DeprecationWarning: `should_run_async` will not call `transform_cell` automatically in the future. Please pass the result to `transformed_cell` argument and any exception that happen during thetransform in `preprocessing_exc_tuple` in IPython 7.17 and above.\n and should_run_async(code)\n\n\n### 2D event display with induced current and ADC counts\n\n\n```python\ncmap = cm.Spectral_r\nnorm = colors.Normalize(vmin=0, vmax=256)\nm = cm.ScalarMappable(norm=norm, cmap=cmap)\n\ncmap = cm.viridis\nnorm_curr = colors.LogNorm(vmin=min(currents[currents>0]), vmax=max(currents))\nm_curr = cm.ScalarMappable(norm=norm_curr, cmap=cmap)\n\nfig,ax = plt.subplots(1,2,figsize=(6,3.8))\n\nimportlib.reload(detsim)\nfor ip, p in enumerate(unique_pix):\n x_rect, y_rect = detsim.get_pixel_coordinates(p.get())\n pixel_plane = int(p[0] // consts.n_pixels[0])\n c = currents[ip].get()\n if c >= 1: \n rect = plt.Rectangle((x_rect, y_rect),\n consts.pixel_pitch, consts.pixel_pitch,\n linewidth=0.2, fc=m_curr.to_rgba(c),\n edgecolor='grey')\n ax[0].add_patch(rect)\n\n a = adc_list[ip][adc_list[ip]>fee.digitize(0)]\n if len(a): \n rect = plt.Rectangle((x_rect, y_rect),\n consts.pixel_pitch, consts.pixel_pitch,\n linewidth=0.2, fc=m.to_rgba(np.sum(a.get())),\n edgecolor='grey')\n ax[1].add_patch(rect)\n\n\nfor it,t in enumerate(selected_tracks):\n ax[0].plot((t[\"x_start\"], t[\"x_end\"]), \n (t[\"y_start\"], t[\"y_end\"]),\n c='r',\n lw=1.25,\n ls=':',\n alpha=1,\n zorder=10)\n ax[1].plot((t[\"x_start\"], t[\"x_end\"]), \n (t[\"y_start\"], t[\"y_end\"]),\n c='r',\n lw=0,\n ls=':',\n alpha=1,\n zorder=10)\n ax[0].scatter((t[\"x_start\"], t[\"x_end\"]), \n (t[\"y_start\"], t[\"y_end\"]),\n c='r', s=1, zorder=99999)\n# ax[1].scatter((t[\"x_start\"], t[\"x_end\"]), \n# (t[\"y_start\"], t[\"y_end\"]),\n# c='r', s=1, zorder=99999) \n\nax[0].set_aspect(\"equal\")\nax[1].set_aspect(\"equal\")\nax[0].set_xlabel(\"x [cm]\")\nax[1].set_xlabel(\"x [cm]\")\nax[0].set_ylabel(\"y [cm]\")\nfrom mpl_toolkits.axes_grid1.axes_divider import make_axes_locatable\n\ndivider0 = make_axes_locatable(ax[1])\ncax0 = divider0.append_axes(\"right\", size=\"7%\", pad=0.07)\nfig.colorbar(m, ax=ax[1], cax=cax0, label='ADC counts sum')\n\ndivider1 = make_axes_locatable(ax[0])\ncax1 = divider1.append_axes(\"right\", size=\"7%\", pad=0.07)\nfig.colorbar(m_curr, ax=ax[0], cax=cax1, label='Induced current integral [# electrons]')\n\nplt.subplots_adjust(hspace=0.5)\nfig.savefig(\"currentadc.pdf\")\n```\n\n\n```python\ncmap = cm.Spectral_r\nnorm = colors.Normalize(vmin=0, vmax=256)\nm = cm.ScalarMappable(norm=norm, cmap=cmap)\n\n#cmap = cm.viridis\n#norm_curr = colors.LogNorm(vmin=min(currents_torch[currents_torch>0]), vmax=max(currents_torch))\n#m_curr = cm.ScalarMappable(norm=norm_curr, cmap=cmap)\n\nfig,ax = plt.subplots(1,2,figsize=(6,3.8))\n\nx_r_all, y_r_all = sim.get_pixel_coordinates(ep.astensor(unique_pix_torch))\nfor ip, p in enumerate(unique_pix_torch):\n x_rect = x_r_all[ip, 0].numpy()\n y_rect = y_r_all[ip, 0].numpy()\n pixel_plane = int(p[0] // sim.n_pixels[0])\n c = currents_torch[ip]\n if c >= 1: \n rect = plt.Rectangle((x_rect, y_rect),\n sim.pixel_pitch, sim.pixel_pitch,\n linewidth=0.2, fc=m_curr.to_rgba(c),\n edgecolor='grey')\n ax[0].add_patch(rect)\n\n a = adc_list_torch[ip][adc_list_torch[ip]>sim.digitize(torch.tensor([0]))]\n if len(a): \n rect = plt.Rectangle((x_rect, y_rect),\n sim.pixel_pitch, sim.pixel_pitch,\n linewidth=0.2, fc=m.to_rgba(a.sum()),\n edgecolor='grey')\n ax[1].add_patch(rect)\n\n\nfor it,t in enumerate(selected_tracks):\n ax[0].plot((t[\"x_start\"], t[\"x_end\"]), \n (t[\"y_start\"], t[\"y_end\"]),\n c='r',\n lw=1.25,\n ls=':',\n alpha=1,\n zorder=10)\n ax[1].plot((t[\"x_start\"], t[\"x_end\"]), \n (t[\"y_start\"], t[\"y_end\"]),\n c='r',\n lw=0,\n ls=':',\n alpha=1,\n zorder=10)\n ax[0].scatter((t[\"x_start\"], t[\"x_end\"]), \n (t[\"y_start\"], t[\"y_end\"]),\n c='r', s=1, zorder=99999)\n# ax[1].scatter((t[\"x_start\"], t[\"x_end\"]), \n# (t[\"y_start\"], t[\"y_end\"]),\n# c='r', s=1, zorder=99999) \n\nax[0].set_aspect(\"equal\")\nax[1].set_aspect(\"equal\")\nax[0].set_xlabel(\"x [cm]\")\nax[1].set_xlabel(\"x [cm]\")\nax[0].set_ylabel(\"y [cm]\")\nfrom mpl_toolkits.axes_grid1.axes_divider import make_axes_locatable\n\ndivider0 = make_axes_locatable(ax[1])\ncax0 = divider0.append_axes(\"right\", size=\"7%\", pad=0.07)\nfig.colorbar(m, ax=ax[1], cax=cax0, label='ADC counts sum')\n\ndivider1 = make_axes_locatable(ax[0])\ncax1 = divider1.append_axes(\"right\", size=\"7%\", pad=0.07)\nfig.colorbar(m_curr, ax=ax[0], cax=cax1, label='Induced current integral [# electrons]')\n\nplt.subplots_adjust(hspace=0.5)\nfig.savefig(\"currentadc_torch.pdf\")\n```\n\n\n```python\ncmap = cm.seismic\nnorm = colors.Normalize(vmin=0.5, vmax=1.5)\nm = cm.ScalarMappable(norm=norm, cmap=cmap)\n\ncmap = cm.seismic\nnorm_curr = colors.Normalize(vmin=0.5, vmax=1.5)\nm_curr = cm.ScalarMappable(norm=norm_curr, cmap=cmap)\n\nfig,ax = plt.subplots(1,2,figsize=(6,3.8))\n\nx_r_all, y_r_all = sim.get_pixel_coordinates(ep.astensor(unique_pix_torch))\nfor ip, p in enumerate(unique_pix_torch):\n x_rect = x_r_all[ip, 0].numpy()\n y_rect = y_r_all[ip, 0].numpy()\n pixel_plane = int(p[0] // sim.n_pixels[0])\n c = currents_torch[ip].numpy()\n c_og = currents[ip].get()\n if c >= 1 or c_og >=1: \n rect = plt.Rectangle((x_rect, y_rect),\n sim.pixel_pitch, sim.pixel_pitch,\n linewidth=0.2, fc=m_curr.to_rgba(c/c_og),\n edgecolor='grey')\n ax[0].add_patch(rect)\n\n a = adc_list_torch[ip][adc_list_torch[ip]>sim.digitize(torch.tensor([0]))].numpy()\n a_og = adc_list[ip][adc_list[ip]>fee.digitize(0)].get()\n if len(a) or len(a_og): \n rect = plt.Rectangle((x_rect, y_rect),\n sim.pixel_pitch, sim.pixel_pitch,\n linewidth=0.2, fc=m.to_rgba(a.sum()/a_og.sum()),\n edgecolor='grey')\n ax[1].add_patch(rect)\n\n\nfor it,t in enumerate(selected_tracks):\n ax[0].plot((t[\"x_start\"], t[\"x_end\"]), \n (t[\"y_start\"], t[\"y_end\"]),\n c='r',\n lw=1.25,\n ls=':',\n alpha=1,\n zorder=10)\n ax[1].plot((t[\"x_start\"], t[\"x_end\"]), \n (t[\"y_start\"], t[\"y_end\"]),\n c='r',\n lw=0,\n ls=':',\n alpha=1,\n zorder=10)\n ax[0].scatter((t[\"x_start\"], t[\"x_end\"]), \n (t[\"y_start\"], t[\"y_end\"]),\n c='r', s=1, zorder=99999)\n# ax[1].scatter((t[\"x_start\"], t[\"x_end\"]), \n# (t[\"y_start\"], t[\"y_end\"]),\n# c='r', s=1, zorder=99999) \n\nax[0].set_aspect(\"equal\")\nax[1].set_aspect(\"equal\")\nax[0].set_xlabel(\"x [cm]\")\nax[1].set_xlabel(\"x [cm]\")\nax[0].set_ylabel(\"y [cm]\")\nfrom mpl_toolkits.axes_grid1.axes_divider import make_axes_locatable\n\ndivider0 = make_axes_locatable(ax[1])\ncax0 = divider0.append_axes(\"right\", size=\"7%\", pad=0.07)\nfig.colorbar(m, ax=ax[1], cax=cax0, label='ADC counts sum Ratio (Diff / Original)')\n\ndivider1 = make_axes_locatable(ax[0])\ncax1 = divider1.append_axes(\"right\", size=\"7%\", pad=0.07)\nfig.colorbar(m_curr, ax=ax[0], cax=cax1, label='Induced current integral Ratio (Diff / Original)')\n\nplt.subplots_adjust(hspace=0.5)\nfig.savefig(\"currentadc_ratio.pdf\")\n```\n\nRatio is quite close, mod noise + one cluster of pixels (likely due to difference mentioned above)\n\n### Export result\nAs a last step we backtrack the ADC counts to the Geant4 tracks and we export the result in a HDF5 file.\n\n\n```python\nMAX_TRACKS_PER_PIXEL = 5\n\n# Mapping between unique pixel array and track array index\ntrack_pixel_map = cp.full((unique_pix.shape[0], MAX_TRACKS_PER_PIXEL), -1)\nTPB = 32\nBPG = ceil(unique_pix.shape[0] / TPB)\ndetsim.get_track_pixel_map[BPG, TPB](track_pixel_map, unique_pix, neighboring_pixels)\n\n# Here we backtrack the ADC counts to the Geant4 tracks\nTPB = 128\nBPG = ceil(adc_list.shape[0] / TPB)\nbacktracked_id = cp.full((adc_list.shape[0], adc_list.shape[1], MAX_TRACKS_PER_PIXEL), -1)\n\ndetsim.backtrack_adcs[BPG,TPB](selected_tracks,\n adc_list,\n adc_ticks_list,\n track_pixel_map,\n event_id_map,\n unique_eventIDs,\n backtracked_id,\n 0)\n```\n\n\n```python\n!rm test.h5\npc = fee.export_to_hdf5(adc_list.get(),\n adc_ticks_list.get(),\n unique_pix.get(),\n backtracked_id.get(),\n \"test.h5\")\n```\n\n\n```python\n!rm evd.test.h5\n!python /global/homes/s/soleti/larpix/larpix-v2-testing-scripts/event-display/to_evd_file.py --in_filename test.h5 --out_filename evd.test.h5 --geometry_file ../larndsim/pixel_layouts/multi_tile_layout-2.2.16.yaml --event_dt 2000 --dbscan_eps 25 --vd 1.587 --nhit_cut 2\n```\n\n\n```python\nevd = h5py.File(\"evd.test.h5\")\n```\n\n\n```python\nfig,ax=plt.subplots(1,1)\nax.plot((selected_tracks['z_start']*10,selected_tracks['z_end']*10),(selected_tracks['y_start']*10,selected_tracks['y_end']*10),c='k')\nax.plot((evd['tracks']['start'][:,2],evd['tracks']['end'][:,2]),(evd['tracks']['start'][:,1]-218.236,evd['tracks']['end'][:,1]-218.236),label='Reconstructed track',ls='--',c='r')\nax.legend()\nax.set_xlabel(\"z [mm]\")\nax.set_ylabel(\"y [mm]\")\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": 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YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.41942579188134776}} {"text": "## This script shows the implementataion of DeLan Metric-IK\n\n\n```python\n\n# display result of assignments\n#%config ZMQInteractiveShell.ast_node_interactivity = 'last_expr_or_assign'\n# make NumPy display a bit nicer\n#np.set_printoptions(linewidth=100, formatter={'float': lambda x: f\"{x:10.4g}\" if abs(x) > 1e-10 else f\"{0:8.4g}\"})\n# make cells nice and wide\nfrom IPython.display import display, HTML\ndisplay(HTML(data=\"\"\"\n\n\"\"\"))\ndisplay(HTML(\"\"))\n%matplotlib notebook\n```\n\n\n\n\n\n\n\n\n\n\n\nimporting libraries\n\n\n```python\nfrom posixpath import join\nfrom cvxopt import solvers\nimport random\nimport matplotlib.pyplot as plt\nfrom numpy.lib.function_base import average\nimport torch\nfrom autograd import grad\nfrom autograd import jacobian\nimport autograd.numpy as np\nimport autograd.numpy as jnp\nimport scipy.optimize as optim\nfrom scipy.optimize import minimize, Bounds,LinearConstraint\nfrom scipy.optimize import LinearConstraint,NonlinearConstraint\nfrom scipy.optimize import BFGS\nfrom autograd.numpy import cos,sin\nimport pdb\n```\n\nLoading DeLan Model and defining analytic form of Value function\n\n\n```python\nt = 0.02\nq_prev = None\n\ndevice = 'cpu'\nmodel = torch.load('models/model_750_model_epoch_20000.pth', map_location=torch.device('cpu')) # loaded trained model #TODO\nq_dim = 6 # q_dim is the dimension of joint space\nq_dim_changed = int(0.5 * q_dim)\n\n#Weight for cost function \nw_des_vel = 0.003\nweight_orient = 0.2\n#Desired final Orientation of end effector\nroll_des= -3.141105126296926\npitch_des= 0.00046035505135551175\nyaw_des = -2.355906195444897\norient_desired = np.asarray([ roll_des , pitch_des , yaw_des ])\n\n#value function defnation\nweight = []\nfor key in (model.keys()):\n # print(key)\n weight.append(model[key].cpu().numpy()) # load weight and bias\n\n\ndef leaky_relu(z):\n return np.maximum(0.01 * z, z)\n\n\ndef softplus(z, beta=1):\n return (1 / beta) * np.log(1 + np.exp(z * beta))\n\n\ndef assemble_lower_triangular_matrix(Lo, Ld):\n Lo = Lo.squeeze(0)\n Ld = Ld.squeeze(0)\n\n assert (2 * Lo.shape[0] == (Ld.shape[0] ** 2 - Ld.shape[0]))\n diagonal_matrix = np.identity(len(Ld)) * np.outer(np.ones(len(Ld)), Ld)\n\n L = np.tril(np.ones(diagonal_matrix.shape)) - np.eye(q_dim_changed)\n\n # Set off diagonals\n\n L = np.array([[0, 0, 0], [1, 0, 0], [0, 0, 0]]) * Lo.reshape(3)[0] + np.array([[0, 0, 0], [0, 0, 0], [1, 0, 0]]) * \\\n Lo.reshape(3)[1] + np.array([[0, 0, 0], [0, 0, 0], [0, 1, 0]]) * Lo.reshape(3)[2]\n # Add diagonals\n L = L + diagonal_matrix\n return L\n\n\ndef value(x1):\n global weight, goal\n fc1_w = weight[0]\n fc1_b = weight[1]\n fc2_w = weight[2]\n fc2_b = weight[3]\n fc_Ld_w = weight[4]\n fc_Ld_b = weight[5]\n fc_Lo_w = weight[6]\n fc_Lo_b = weight[7]\n #pdb.set_trace()\n net_input = np.concatenate([np.squeeze(x1), np.squeeze(goal)], axis=0)\n net_input = np.array([net_input])\n\n z1 = np.dot(net_input, fc1_w.transpose()) + fc1_b\n hidden1 = leaky_relu(z1)\n z2 = np.dot(hidden1, fc2_w.transpose()) + fc2_b\n hidden2 = leaky_relu(z2)\n hidden3 = np.dot(hidden2, fc_Ld_w.transpose()) + fc_Ld_b\n Ld = softplus(hidden3)\n Lo = np.dot(hidden2, fc_Lo_w.transpose()) + fc_Lo_b\n L = assemble_lower_triangular_matrix(Lo, Ld)\n\n H = L @ L.transpose() + 1e-9 * np.eye(3)\n return H\n```\n\n\n```python\n#Analytical funtion for forward kinematics of Franka Panda\ndef fk_franka_orient(q):\n \n q_1 = q[0]\n q_2 = q[1]\n q_3 = q[2]\n q_4 = q[3]\n q_5 = q[4]\n q_6 = q[5]\n q_7 = q[6]\n\n x = -0.107*(((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.cos(q_4) - np.sin(q_2)*np.sin(q_4)*np.cos(q_1))*np.cos(q_5) + (np.sin(q_1)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1)*np.cos(q_2))*np.sin(q_5))*np.sin(q_6) - 0.088*(((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.cos(q_4) - np.sin(q_2)*np.sin(q_4)*np.cos(q_1))*np.cos(q_5) + (np.sin(q_1)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1)*np.cos(q_2))*np.sin(q_5))*np.cos(q_6) + 0.088*((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.sin(q_4) + np.sin(q_2)*np.cos(q_1)*np.cos(q_4))*np.sin(q_6) - 0.107*((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.sin(q_4) + np.sin(q_2)*np.cos(q_1)*np.cos(q_4))*np.cos(q_6) + 0.384*(np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.sin(q_4) + 0.0825*(np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.cos(q_4) - 0.0825*np.sin(q_1)*np.sin(q_3) - 0.0825*np.sin(q_2)*np.sin(q_4)*np.cos(q_1) + 0.384*np.sin(q_2)*np.cos(q_1)*np.cos(q_4) + 0.316*np.sin(q_2)*np.cos(q_1) + 0.0825*np.cos(q_1)*np.cos(q_2)*np.cos(q_3)\n\n y = 0.107*(((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.cos(q_4) + np.sin(q_1)*np.sin(q_2)*np.sin(q_4))*np.cos(q_5) - (np.sin(q_1)*np.sin(q_3)*np.cos(q_2) - np.cos(q_1)*np.cos(q_3))*np.sin(q_5))*np.sin(q_6) + 0.088*(((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.cos(q_4) + np.sin(q_1)*np.sin(q_2)*np.sin(q_4))*np.cos(q_5) - (np.sin(q_1)*np.sin(q_3)*np.cos(q_2) - np.cos(q_1)*np.cos(q_3))*np.sin(q_5))*np.cos(q_6) - 0.088*((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.sin(q_4) - np.sin(q_1)*np.sin(q_2)*np.cos(q_4))*np.sin(q_6) + 0.107*((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.sin(q_4) - np.sin(q_1)*np.sin(q_2)*np.cos(q_4))*np.cos(q_6) - 0.384*(np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.sin(q_4) - 0.0825*(np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.cos(q_4) - 0.0825*np.sin(q_1)*np.sin(q_2)*np.sin(q_4) + 0.384*np.sin(q_1)*np.sin(q_2)*np.cos(q_4) + 0.316*np.sin(q_1)*np.sin(q_2) + 0.0825*np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + 0.0825*np.sin(q_3)*np.cos(q_1)\n\n z = -0.107*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.sin(q_6) - 0.088*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.cos(q_6) + 0.088*(np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.sin(q_6) - 0.107*(np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.cos(q_6) + 0.384*np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + 0.0825*np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - 0.0825*np.sin(q_2)*np.cos(q_3) - 0.0825*np.sin(q_4)*np.cos(q_2) + 0.384*np.cos(q_2)*np.cos(q_4) + 0.316*np.cos(q_2) + 0.33\n\n cq1 = np.cos(q_1)\n cq2 = np.cos(q_2)\n cq3 = np.cos(q_3)\n cq4 = np.cos(q_4)\n cq5 = np.cos(q_5)\n cq6 = np.cos(q_6)\n cq7 = np.cos(q_7)\n\n sq1 = np.sin(q_1)\n sq2 = np.sin(q_2)\n sq3 = np.sin(q_3)\n sq4 = np.sin(q_4)\n sq5 = np.sin(q_5)\n sq6 = np.sin(q_6)\n sq7 = np.sin(q_7)\n ax1 = -0.5*(((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.cos(q_4) + np.sin(q_1)*np.sin(q_2)*np.sin(q_4))*np.cos(q_5) - (np.sin(q_1)*np.sin(q_3)*np.cos(q_2) - np.cos(q_1)*np.cos(q_3))*np.sin(q_5))*np.sin(q_6) + 0.5*(((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.cos(q_6) - (np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.sin(q_6))*np.sin(q_7) - 0.5*((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.sin(q_4) - np.sin(q_1)*np.sin(q_2)*np.cos(q_4))*np.cos(q_6) - 0.5*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.sin(q_5) + np.sin(q_2)*np.sin(q_3)*np.cos(q_5))*np.cos(q_7)\n\n ax2 = -0.5*(((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.cos(q_4) - np.sin(q_2)*np.sin(q_4)*np.cos(q_1))*np.cos(q_5) + (np.sin(q_1)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1)*np.cos(q_2))*np.sin(q_5))*np.sin(q_6) + 0.5*(((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.cos(q_6) - (np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.sin(q_6))*np.cos(q_7) - 0.5*((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.sin(q_4) + np.sin(q_2)*np.cos(q_1)*np.cos(q_4))*np.cos(q_6) + 0.5*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.sin(q_5) + np.sin(q_2)*np.sin(q_3)*np.cos(q_5))*np.sin(q_7)\n\n ax3 = -0.5*np.sin(q_2)*np.sin(q_4)*np.sin(q_5)*np.cos(q_1 + q_7) + 0.5*np.sin(q_2)*np.sin(q_4)*np.sin(q_1 + q_7)*np.cos(q_5)*np.cos(q_6) + 0.5*np.sin(q_2)*np.sin(q_6)*np.sin(q_1 + q_7)*np.cos(q_4) - 0.5*np.sin(q_3)*np.sin(q_4)*np.sin(q_6)*np.cos(q_1 + q_7) - 0.5*np.sin(q_3)*np.sin(q_5)*np.sin(q_1 + q_7)*np.cos(q_2)*np.cos(q_6) + 0.5*np.sin(q_3)*np.sin(q_5)*np.sin(q_1 + q_7)*np.cos(q_4) - 0.5*np.sin(q_3)*np.cos(q_2)*np.cos(q_5)*np.cos(q_1 + q_7) + 0.5*np.sin(q_3)*np.cos(q_4)*np.cos(q_5)*np.cos(q_6)*np.cos(q_1 + q_7) - 0.5*np.sin(q_4)*np.sin(q_6)*np.sin(q_1 + q_7)*np.cos(q_2)*np.cos(q_3) - 0.5*np.sin(q_5)*np.cos(q_2)*np.cos(q_3)*np.cos(q_4)*np.cos(q_1 + q_7) + 0.5*np.sin(q_5)*np.cos(q_3)*np.cos(q_6)*np.cos(q_1 + q_7) + 0.5*np.sin(q_1 + q_7)*np.cos(q_2)*np.cos(q_3)*np.cos(q_4)*np.cos(q_5)*np.cos(q_6) - 0.5*np.sin(q_1 + q_7)*np.cos(q_3)*np.cos(q_5)\n\n r_32 = -cq7*(cq5*sq2*sq3 - sq5*(cq2*sq4 - cq3*cq4*sq2)) - sq7*(cq6*(cq5*(cq2*sq4 - cq3*cq4*sq2) + sq2*sq3*sq5) + sq6*(cq2*cq4 + cq3*sq2*sq4))\n r_33 = -cq6*(cq2*cq4 + cq3*sq2*sq4) + sq6*(cq5*(cq2*sq4 - cq3*cq4*sq2) + sq2*sq3*sq5)\n r_31 = cq7*(cq6*(cq5*(cq2*sq4 - cq3*cq4*sq2) + sq2*sq3*sq5) + sq6*(cq2*cq4 + cq3*sq2*sq4)) - sq7*(cq5*sq2*sq3 - sq5*(cq2*sq4 - cq3*cq4*sq2))\n r_21 = cq7*(cq6*(cq5*(cq4*(cq1*sq3 + cq2*cq3*sq1) + sq1*sq2*sq4) + sq5*(cq1*cq3 - cq2*sq1*sq3)) + sq6*(cq4*sq1*sq2 - sq4*(cq1*sq3 + cq2*cq3*sq1))) - sq7*(cq5*(cq1*cq3 - cq2*sq1*sq3) - sq5*(cq4*(cq1*sq3 + cq2*cq3*sq1) + sq1*sq2*sq4))\n r_11 = cq7*(cq6*(cq5*(cq1*sq2*sq4 + cq4*(cq1*cq2*cq3 - sq1*sq3)) - sq5*(cq1*cq2*sq3 + cq3*sq1)) + sq6*(cq1*cq4*sq2 - sq4*(cq1*cq2*cq3 - sq1*sq3))) + sq7*(cq5*(cq1*cq2*sq3 + cq3*sq1) + sq5*(cq1*sq2*sq4 + cq4*(cq1*cq2*cq3 - sq1*sq3)))\n roll_ = np.arctan2(r_32, r_33)\n pitch_=-np.arcsin(r_31)\n yaw_ = np.arctan2(r_21,r_11)\n \n cartpos = np.array([x,y,z])\n pose = [x,y,z,roll_,pitch_,yaw_]\n return pose\n\n#function to compute joint and Cart Space Trajectory Cost \ndef traj_cost(trajectory):\n #pdb.set_trace()\n cost = 0\n cart_cost = 0\n for i in range(len(trajectory) - 1):\n cost += np.linalg.norm(np.asarray(trajectory[i]) - np.asarray(trajectory[i + 1]), ord=2)\n # pdb.set_trace()\n current = np.asarray(fk_franka_orient(trajectory[i]))[0:3] \n next = np.asarray(fk_franka_orient(trajectory[i+1]))[0:3] \n cart_cost += np.linalg.norm(current - next, ord=2)\n return cost, cart_cost\n```\n\n\n```python\n#plotting the traj/ trajc_dot\n#orientResidual_tracker, cart_cost_tracker , trajectory,init_joint, end_cart,qdottrajc,plotno\ndef plot(orientResidual_tracker, cost_tracker , TrajectoryValue,init_joint, end_cartesian,qdottraj , plotno):\n x_fin = end_cartesian.item(0)\n y_fin = end_cartesian.item(1)\n z_fin = end_cartesian.item(2)\n pos = fk_franka_orient(init_joint)[0:3]\n # pos = init_joint\n x_init = pos[0]\n y_init = pos[1]\n z_init = pos[2]\n countValue = len(TrajectoryValue)\n xopt_solValue = np.zeros(countValue - 1)\n yopt_solValue = np.zeros(countValue - 1)\n zopt_solValue = np.zeros(countValue - 1)\n for i in range(0, countValue - 1):\n pos = fk_franka_orient(TrajectoryValue[i])[0:3]\n xopt_solValue[i] = pos[0]\n yopt_solValue[i] = pos[1]\n zopt_solValue[i] = pos[2]\n JointCostValue, CartCostValue = traj_cost(TrajectoryValue)\n\n fig = plt.figure(1, figsize=(12, 12), dpi=100)\n plt.title(\"3d-Trajectory Plot in cartesian coordinate\")\n ax = fig.add_subplot(111, projection='3d')\n ax.plot(xopt_solValue, yopt_solValue, zopt_solValue, '--or', linewidth=2.0, markersize=6.0,\n label=('Traj cost in : joint-Space=', \"{:.2f}\".format(JointCostValue), 'cart-Space=',\n \"{:.2f}\".format(CartCostValue), 'Orient_cost_residual', \"{:.2f}\".format(orientResidual_tracker[-1]), 'euclid_cost_residual', \"{:.2f}\".format(cost_tracker[-1])))\n ax.plot(x_init * np.ones(1), y_init * np.ones(1), z_init * np.ones(1), 'om', markersize=15)\n ax.plot(x_fin * np.ones(1), y_fin * np.ones(1), z_fin * np.ones(1), 'og', markersize=10)\n ax.set_xlim3d(-1.0, 1.0)\n ax.set_ylim3d(-1.0, 1.0)\n ax.set_zlim3d(-0.3, 1.2)\n ax.legend(loc='upper left', frameon=False)\n ax.set_xlabel('X Axis')\n ax.set_ylabel('Y Axis')\n ax.set_zlabel('Z Axis')\n plt.show()\n #plt.close()\n fig = plt.figure(2, figsize=(12, 8), dpi=100)\n plt.title(\"cart-Cost / Orientation-residual cost VS iteration\")\n plt.plot(orientResidual_tracker, label=(\n 'orient residual'))\n plt.plot(cost_tracker, label=(\n 'euclid-cost residual'))\n plt.legend(loc='upper left', frameon=False)\n plt.show()\n fig =plt.figure(3, figsize=(12, 8), dpi=100)\n trajct = np.asarray(TrajectoryValue)\n plt.plot(trajct[:-1, 0], '-o', linewidth=1.0, markersize=3.0 , label= ('joint1'))\n plt.plot(trajct[:-1, 1], '-o', linewidth=1.0, markersize=3.0,label= ('joint2'))\n plt.plot(trajct[:, 2], '-o', linewidth=1.0, markersize=3.0,label= ('joint3'))\n plt.plot(trajct[:, 3], '-o', linewidth=1.0, markersize=3.0,label= ('joint4'))\n plt.plot(trajct[:, 4], '-o', linewidth=1.0, markersize=3.0,label= ('joint5'))\n plt.plot(trajct[:, 5], '-o', linewidth=1.0, markersize=3.0,label= ('joint6'))\n plt.plot(trajct[:, 6], '-o', linewidth=1.0, markersize=3.0,label= ('joint7'))\n plt.title(\"joint angle trajc VS iteration\")\n plt.legend(loc='upper left', frameon=False)\n plt.show()\n fig =plt.figure(4, figsize=(12, 8), dpi=100)\n trajct = np.asarray(qdottrajc)\n plt.plot(trajct[:-1, 0], '-o', linewidth=1.0, markersize=3.0 , label= ('joint1'))\n plt.plot(trajct[:-1, 1], '-o', linewidth=1.0, markersize=3.0,label= ('joint2'))\n plt.plot(trajct[:, 2], '-o', linewidth=1.0, markersize=3.0,label= ('joint3'))\n plt.plot(trajct[:, 3], '-o', linewidth=1.0, markersize=3.0,label= ('joint4'))\n plt.plot(trajct[:, 4], '-o', linewidth=1.0, markersize=3.0,label= ('joint5'))\n plt.plot(trajct[:, 5], '-o', linewidth=1.0, markersize=3.0,label= ('joint6'))\n plt.plot(trajct[:, 6], '-o', linewidth=1.0, markersize=3.0,label= ('joint7'))\n plt.title(\"q-dot trajc VS iteration \")\n plt.legend(loc='upper left', frameon=False)\n plt.show()\n return\n```\n\nDeLan Metric IK formulation:\n\nLets say we the current joint angle is $\\textbf{q}_{init}$ and the corresponding end-effector position is $x_{init}$. We want to solve for the $\\dot{\\textbf{q}}$ that will take us to $\\textbf{x}_{next}$ at the next instant. We can formulate the following optimization problem for it at iteration $k$.\n\n\\begin{align}\n\\min_{\\dot{\\textbf{q}}}\\frac{1}{2}(\\textbf{x}_{next}-\\textbf{x}_{fin})^T\\textbf{H}(\\textbf{x}_{next}, \\textbf{x}_{fin})(\\textbf{x}_{next}-\\textbf{x}_{fin})+\\Vert \\dot{\\textbf{q}}\\Vert_2^2 + orientation_{cost} --eqn(1)\\\\\nf_{fk}(\\dot{\\textbf{q}}t+\\textbf{q}_{init}) = \\textbf{x}_{next}---eqn(2)\\\\\n{\\textbf{q}}_{min}\\leq {\\textbf{q}} \\leq {\\textbf{q}}_{max}\\\\\n{\\text{or we can write}}\\\\\n-{\\textbf{q}}_{max}\\leq \\dot{\\textbf{q}}*t +\\textbf{q}_{init} \\leq {\\textbf{q}}_{max}\\\\\n-\\dot{\\textbf{q}}_{max}\\leq \\dot{\\textbf{q}} \\leq \\dot{\\textbf{q}}_{max}\\\\\n-\\ddot{\\textbf{q}}_{max}t \\leq \\dot{\\textbf{q}}-\\dot{\\textbf{q}}^{k-1} \\leq \\ddot{\\textbf{q}}_{max}t\n\\end{align} \n\nSimplification: The $\\textbf{H}$ matrix is a function of $\\textbf{x}_{next}$ itself which is not known. However, we can make one simplification, we can replace $\\textbf{x}_{next}$ with $\\textbf{x}_{init}$ , i.e the current position in $\\textbf{H}$. This makes the first term convex and gets rid of the non-smoothness aspect. \n\nScipy can solve the optimization problem. The only problem is the non-convex equality constraints from forward kinematics but we can code it as non-linear equality constraints in scipy SLSQP.\n\nWe solve the optimization in MPC setting. That is, we solve $\\dot{\\textbf{q}}$ and updated $\\textbf{q}_{init}$ and so on.\n\n\n```python\n# Cost function defination eqn-1\n\ndef costfxn(solverVariable,x_position,val):\n global goal,diffinCart,roll_des,pitch_des,yaw_des ,w_des_vel,weight_orient\n diff = solverVariable[7:] - goal\n cost = np.matmul(diff.transpose(),np.matmul(val, diff ))\n diffprev = x_position - goal\n cost_prev =np.matmul(diffprev.transpose(),np.matmul(val, diffprev ))\n\n smoothness_cost = np.sum(solverVariable[0:7]**2,axis = 0)\n global q_prev, t ,maxCartdist,iteration\n\n roll_des= -3.141105126296926\n pitch_des= 0.00046035505135551175\n yaw_des = -2.355906195444897\n orient_desired = np.asarray([ roll_des , pitch_des , yaw_des ])\n q = solverVariable[:7]*t + q_prev \n q_1 = q[0]\n q_2 = q[1]\n q_3 = q[2]\n q_4 = q[3]\n q_5 = q[4]\n q_6 = q[5]\n q_7 = q[6]\n\n cq1 = np.cos(q_1)\n cq2 = np.cos(q_2)\n cq3 = np.cos(q_3)\n cq4 = np.cos(q_4)\n cq5 = np.cos(q_5)\n cq6 = np.cos(q_6)\n cq7 = np.cos(q_7)\n\n sq1 = np.sin(q_1)\n sq2 = np.sin(q_2)\n sq3 = np.sin(q_3)\n sq4 = np.sin(q_4)\n sq5 = np.sin(q_5)\n sq6 = np.sin(q_6)\n sq7 = np.sin(q_7)\n ax1 = -0.5*(((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.cos(q_4) + np.sin(q_1)*np.sin(q_2)*np.sin(q_4))*np.cos(q_5) - (np.sin(q_1)*np.sin(q_3)*np.cos(q_2) - np.cos(q_1)*np.cos(q_3))*np.sin(q_5))*np.sin(q_6) + 0.5*(((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.cos(q_6) - (np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.sin(q_6))*np.sin(q_7) - 0.5*((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.sin(q_4) - np.sin(q_1)*np.sin(q_2)*np.cos(q_4))*np.cos(q_6) - 0.5*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.sin(q_5) + np.sin(q_2)*np.sin(q_3)*np.cos(q_5))*np.cos(q_7)\n\n ax2 = -0.5*(((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.cos(q_4) - np.sin(q_2)*np.sin(q_4)*np.cos(q_1))*np.cos(q_5) + (np.sin(q_1)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1)*np.cos(q_2))*np.sin(q_5))*np.sin(q_6) + 0.5*(((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.cos(q_6) - (np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.sin(q_6))*np.cos(q_7) - 0.5*((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.sin(q_4) + np.sin(q_2)*np.cos(q_1)*np.cos(q_4))*np.cos(q_6) + 0.5*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.sin(q_5) + np.sin(q_2)*np.sin(q_3)*np.cos(q_5))*np.sin(q_7)\n\n ax3 = -0.5*np.sin(q_2)*np.sin(q_4)*np.sin(q_5)*np.cos(q_1 + q_7) + 0.5*np.sin(q_2)*np.sin(q_4)*np.sin(q_1 + q_7)*np.cos(q_5)*np.cos(q_6) + 0.5*np.sin(q_2)*np.sin(q_6)*np.sin(q_1 + q_7)*np.cos(q_4) - 0.5*np.sin(q_3)*np.sin(q_4)*np.sin(q_6)*np.cos(q_1 + q_7) - 0.5*np.sin(q_3)*np.sin(q_5)*np.sin(q_1 + q_7)*np.cos(q_2)*np.cos(q_6) + 0.5*np.sin(q_3)*np.sin(q_5)*np.sin(q_1 + q_7)*np.cos(q_4) - 0.5*np.sin(q_3)*np.cos(q_2)*np.cos(q_5)*np.cos(q_1 + q_7) + 0.5*np.sin(q_3)*np.cos(q_4)*np.cos(q_5)*np.cos(q_6)*np.cos(q_1 + q_7) - 0.5*np.sin(q_4)*np.sin(q_6)*np.sin(q_1 + q_7)*np.cos(q_2)*np.cos(q_3) - 0.5*np.sin(q_5)*np.cos(q_2)*np.cos(q_3)*np.cos(q_4)*np.cos(q_1 + q_7) + 0.5*np.sin(q_5)*np.cos(q_3)*np.cos(q_6)*np.cos(q_1 + q_7) + 0.5*np.sin(q_1 + q_7)*np.cos(q_2)*np.cos(q_3)*np.cos(q_4)*np.cos(q_5)*np.cos(q_6) - 0.5*np.sin(q_1 + q_7)*np.cos(q_3)*np.cos(q_5)\n\n r_32 = -cq7*(cq5*sq2*sq3 - sq5*(cq2*sq4 - cq3*cq4*sq2)) - sq7*(cq6*(cq5*(cq2*sq4 - cq3*cq4*sq2) + sq2*sq3*sq5) + sq6*(cq2*cq4 + cq3*sq2*sq4))\n r_33 = -cq6*(cq2*cq4 + cq3*sq2*sq4) + sq6*(cq5*(cq2*sq4 - cq3*cq4*sq2) + sq2*sq3*sq5)\n r_31 = cq7*(cq6*(cq5*(cq2*sq4 - cq3*cq4*sq2) + sq2*sq3*sq5) + sq6*(cq2*cq4 + cq3*sq2*sq4)) - sq7*(cq5*sq2*sq3 - sq5*(cq2*sq4 - cq3*cq4*sq2))\n r_21 = cq7*(cq6*(cq5*(cq4*(cq1*sq3 + cq2*cq3*sq1) + sq1*sq2*sq4) + sq5*(cq1*cq3 - cq2*sq1*sq3)) + sq6*(cq4*sq1*sq2 - sq4*(cq1*sq3 + cq2*cq3*sq1))) - sq7*(cq5*(cq1*cq3 - cq2*sq1*sq3) - sq5*(cq4*(cq1*sq3 + cq2*cq3*sq1) + sq1*sq2*sq4))\n r_11 = cq7*(cq6*(cq5*(cq1*sq2*sq4 + cq4*(cq1*cq2*cq3 - sq1*sq3)) - sq5*(cq1*cq2*sq3 + cq3*sq1)) + sq6*(cq1*cq4*sq2 - sq4*(cq1*cq2*cq3 - sq1*sq3))) + sq7*(cq5*(cq1*cq2*sq3 + cq3*sq1) + sq5*(cq1*sq2*sq4 + cq4*(cq1*cq2*cq3 - sq1*sq3)))\n f_orient_cost = np.linalg.norm( ((r_32/r_33)-np.tan(roll_des))**2+(-np.sin(r_31)-np.sin(pitch_des))**2 + ((r_21 / r_11) -np.tan(yaw_des ) )**2 ) \n totalcost = np.add(cost , w_des_vel*smoothness_cost ) \n weight = weight_orient/cost_prev\n if f_orient_cost< 0.08999999999999999999999 :\n \tweight = 0\n totalcost = totalcost + weight*f_orient_cost\n return totalcost\njac_cost = jacobian(costfxn) #jaccost has a shape of (3,)\n\n\n#Defination of Non linear constraint function eqn-2\ndef constraintfxn(qdot_x_next):\n global t, q_prev\n q = qdot_x_next[:7]*t + q_prev\n x_next = qdot_x_next[7]\n y_next = qdot_x_next[8]\n z_next = qdot_x_next[9]\n\n # x,y,z= robot.fkine(q) #TODO\n q_1 = q[0]\n q_2 = q[1]\n q_3 = q[2]\n q_4 = q[3]\n q_5 = q[4]\n q_6 = q[5]\n q_7 = q[6]\n\n\n x = -0.107*(((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.cos(q_4) - np.sin(q_2)*np.sin(q_4)*np.cos(q_1))*np.cos(q_5) + (np.sin(q_1)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1)*np.cos(q_2))*np.sin(q_5))*np.sin(q_6) - 0.088*(((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.cos(q_4) - np.sin(q_2)*np.sin(q_4)*np.cos(q_1))*np.cos(q_5) + (np.sin(q_1)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1)*np.cos(q_2))*np.sin(q_5))*np.cos(q_6) + 0.088*((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.sin(q_4) + np.sin(q_2)*np.cos(q_1)*np.cos(q_4))*np.sin(q_6) - 0.107*((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.sin(q_4) + np.sin(q_2)*np.cos(q_1)*np.cos(q_4))*np.cos(q_6) + 0.384*(np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.sin(q_4) + 0.0825*(np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.cos(q_4) - 0.0825*np.sin(q_1)*np.sin(q_3) - 0.0825*np.sin(q_2)*np.sin(q_4)*np.cos(q_1) + 0.384*np.sin(q_2)*np.cos(q_1)*np.cos(q_4) + 0.316*np.sin(q_2)*np.cos(q_1) + 0.0825*np.cos(q_1)*np.cos(q_2)*np.cos(q_3)\n\n y = 0.107*(((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.cos(q_4) + np.sin(q_1)*np.sin(q_2)*np.sin(q_4))*np.cos(q_5) - (np.sin(q_1)*np.sin(q_3)*np.cos(q_2) - np.cos(q_1)*np.cos(q_3))*np.sin(q_5))*np.sin(q_6) + 0.088*(((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.cos(q_4) + np.sin(q_1)*np.sin(q_2)*np.sin(q_4))*np.cos(q_5) - (np.sin(q_1)*np.sin(q_3)*np.cos(q_2) - np.cos(q_1)*np.cos(q_3))*np.sin(q_5))*np.cos(q_6) - 0.088*((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.sin(q_4) - np.sin(q_1)*np.sin(q_2)*np.cos(q_4))*np.sin(q_6) + 0.107*((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.sin(q_4) - np.sin(q_1)*np.sin(q_2)*np.cos(q_4))*np.cos(q_6) - 0.384*(np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.sin(q_4) - 0.0825*(np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.cos(q_4) - 0.0825*np.sin(q_1)*np.sin(q_2)*np.sin(q_4) + 0.384*np.sin(q_1)*np.sin(q_2)*np.cos(q_4) + 0.316*np.sin(q_1)*np.sin(q_2) + 0.0825*np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + 0.0825*np.sin(q_3)*np.cos(q_1)\n\n z = -0.107*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.sin(q_6) - 0.088*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.cos(q_6) + 0.088*(np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.sin(q_6) - 0.107*(np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.cos(q_6) + 0.384*np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + 0.0825*np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - 0.0825*np.sin(q_2)*np.cos(q_3) - 0.0825*np.sin(q_4)*np.cos(q_2) + 0.384*np.cos(q_2)*np.cos(q_4) + 0.316*np.cos(q_2) + 0.33\n pos_residual = np.array([x - x_next,y-y_next,z - z_next])\n return pos_residual\n\n#Analytic function for jacobian of Nonlinear constraint function defined above \ndef compute_franka_jac(qdot_x_next):\n\n global q_prev,t\n qdot = qdot_x_next[:7]\n qinit = q_prev \n qdot_1 = qdot[0]\n qdot_2 = qdot[1]\n qdot_3 = qdot[2]\n qdot_4 = qdot[3]\n qdot_5 = qdot[4]\n qdot_6 = qdot[5]\n qdot_7 = qdot[6]\n \n \n qinit_1 = qinit[0]\n qinit_2 = qinit[1]\n qinit_3 = qinit[2]\n qinit_4 = qinit[3]\n qinit_5 = qinit[4]\n qinit_6 = qinit[5]\n qinit_7 = qinit[6]\n \n const_1_q_1 = 0.0825*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4) - 0.384*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) - 0.316*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2) - 0.0825*t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - 0.0825*t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1) + (0.088*(-t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4) + (-t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5) + 0.088*(t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) - t*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*sin(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) + (0.107*(-t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4) + (-t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5) + 0.107*(t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) - t*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*sin(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + (-0.088*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) + 0.088*(t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6) + (0.107*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) - 0.107*(t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_4*t + qinit_4))*cos(qdot_6*t + qinit_6) + (0.0825*t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + 0.0825*t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4) + (0.384*t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + 0.384*t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_4*t + qinit_4)\n const_1_q_2 = 0.384*t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) + 0.0825*t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) - 0.0825*t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) - 0.0825*t*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2) + 0.384*t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) + 0.316*t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2) + (0.088*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*sin(qdot_5*t + qinit_5)*cos(qdot_1*t + qinit_1) + 0.088*(-t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + t*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*cos(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) + (0.107*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*sin(qdot_5*t + qinit_5)*cos(qdot_1*t + qinit_1) + 0.107*(-t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + t*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*cos(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + (-0.107*t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) - 0.107*t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*cos(qdot_6*t + qinit_6) + (0.088*t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) + 0.088*t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6)\n const_1_q_3 = -0.0825*t*sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) - 0.0825*t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2) + (0.088*(t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_5*t + qinit_5) + 0.088*(-t*sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) - t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*cos(qdot_4*t + qinit_4)*cos(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) + (0.107*(t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_5*t + qinit_5) + 0.107*(-t*sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) - t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*cos(qdot_4*t + qinit_4)*cos(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + (0.0825*t*sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) + 0.0825*t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*cos(qdot_4*t + qinit_4) + (0.384*t*sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) + 0.384*t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*sin(qdot_4*t + qinit_4) + 0.088*(t*sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) + t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*sin(qdot_4*t + qinit_4)*sin(qdot_6*t + qinit_6) - 0.107*(t*sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) + t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*sin(qdot_4*t + qinit_4)*cos(qdot_6*t + qinit_6)\n const_1_q_4 = -t*(0.0825*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - 0.0825*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4) + t*(0.384*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - 0.384*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4) - 0.384*t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1) - 0.0825*t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_4*t + qinit_4) + 0.107*(-t*(-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4) + t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6)*cos(qdot_5*t + qinit_5) + 0.088*(-t*(-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4) + t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5)*cos(qdot_6*t + qinit_6) + (-0.107*t*(sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4) + 0.107*t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1))*cos(qdot_6*t + qinit_6) + (0.088*t*(sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4) - 0.088*t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1))*sin(qdot_6*t + qinit_6)\n const_1_q_5 = (-0.107*t*((-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4) + sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1))*sin(qdot_5*t + qinit_5) + 0.107*t*(-sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) - sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*cos(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + (-0.088*t*((-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4) + sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1))*sin(qdot_5*t + qinit_5) + 0.088*t*(-sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) - sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*cos(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6)\n const_1_q_6 = -t*(0.088*((-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4) + sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1))*cos(qdot_5*t + qinit_5) - 0.088*(sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*sin(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + t*(0.107*((-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4) + sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1))*cos(qdot_5*t + qinit_5) - 0.107*(sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*sin(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) + t*(-0.088*(-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4) + 0.088*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_4*t + qinit_4))*cos(qdot_6*t + qinit_6) - t*(0.107*(-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4) - 0.107*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6)\n const_1_q_7 = 0\n const_2_q_1 = -0.0825*t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - 0.0825*t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1) + 0.384*t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_4*t + qinit_4) + 0.316*t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1) + 0.0825*t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + (0.088*(-t*sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) - t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*sin(qdot_5*t + qinit_5) + 0.088*(t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1) + (-t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) + (0.107*(-t*sin(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) - t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2))*sin(qdot_5*t + qinit_5) + 0.107*(t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_1*t + qinit_1) + (-t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) + t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + (0.0825*t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - 0.0825*t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4) + (0.384*t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - 0.384*t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4) + (-0.107*t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_4*t + qinit_4) - 0.107*(t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4))*cos(qdot_6*t + qinit_6) + (0.088*t*sin(qdot_2*t + qinit_2)*cos(qdot_1*t + qinit_1)*cos(qdot_4*t + qinit_4) + 0.088*(t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3) - t*cos(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6)\n const_2_q_2 = 0.384*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_3*t + qinit_3) + 0.0825*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) - 0.0825*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - 0.0825*t*sin(qdot_1*t + qinit_1)*sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2) + 0.384*t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) + 0.316*t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2) + (0.088*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*sin(qdot_5*t + qinit_5) + 0.088*(-t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + t*sin(qdot_1*t + qinit_1)*sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2))*cos(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) + (0.107*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*sin(qdot_5*t + qinit_5) + 0.107*(-t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + t*sin(qdot_1*t + qinit_1)*sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2))*cos(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + (-0.107*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_3*t + qinit_3) - 0.107*t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*cos(qdot_6*t + qinit_6) + (0.088*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_3*t + qinit_3) + 0.088*t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6)\n const_2_q_3 = -0.0825*t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) + 0.0825*t*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3) + (0.088*(-t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) + t*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4)*cos(qdot_5*t + qinit_5) + 0.088*(-t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) + (0.107*(-t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) + t*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4)*cos(qdot_5*t + qinit_5) + 0.107*(-t*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - t*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + (0.0825*t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) - 0.0825*t*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*cos(qdot_4*t + qinit_4) + (0.384*t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) - 0.384*t*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4) + 0.088*(t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) - t*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4)*sin(qdot_6*t + qinit_6) - 0.107*(t*sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) - t*cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*sin(qdot_4*t + qinit_4)*cos(qdot_6*t + qinit_6)\n const_2_q_4 = t*(-0.384*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - 0.384*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4) - t*(-0.0825*sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - 0.0825*sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_4*t + qinit_4) - 0.384*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4) - 0.0825*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) + (-0.107*t*(-sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4) + 0.107*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4))*cos(qdot_6*t + qinit_6) + (0.088*t*(-sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) - sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4) - 0.088*t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6) + 0.107*(-t*(sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_4*t + qinit_4) + t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6)*cos(qdot_5*t + qinit_5) + 0.088*(-t*(sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_4*t + qinit_4) + t*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5)*cos(qdot_6*t + qinit_6)\n const_2_q_5 = (-0.107*t*((sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4) + sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4))*sin(qdot_5*t + qinit_5) + 0.107*t*(-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) + cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*cos(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + (-0.088*t*((sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4) + sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4))*sin(qdot_5*t + qinit_5) + 0.088*t*(-sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) + cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*cos(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6)\n const_2_q_6 = -t*(0.088*((sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4) + sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5) - 0.088*(sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) - cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*sin(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + t*(0.107*((sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*cos(qdot_4*t + qinit_4) + sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5) - 0.107*(sin(qdot_1*t + qinit_1)*sin(qdot_3*t + qinit_3)*cos(qdot_2*t + qinit_2) - cos(qdot_1*t + qinit_1)*cos(qdot_3*t + qinit_3))*sin(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) + t*(-0.088*(sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_4*t + qinit_4) + 0.088*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*cos(qdot_6*t + qinit_6) - t*(0.107*(sin(qdot_1*t + qinit_1)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + sin(qdot_3*t + qinit_3)*cos(qdot_1*t + qinit_1))*sin(qdot_4*t + qinit_4) - 0.107*sin(qdot_1*t + qinit_1)*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6)\n const_2_q_7 = 0\n const_3_q_1 = 0\n const_3_q_2 = 0.0825*t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4) - 0.384*t*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) - 0.316*t*sin(qdot_2*t + qinit_2) + 0.384*t*sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + 0.0825*t*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) - 0.0825*t*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3) + (-0.088*t*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) + 0.088*t*sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*sin(qdot_6*t + qinit_6) + (0.107*t*sin(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) - 0.107*t*sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3))*cos(qdot_6*t + qinit_6) + (0.088*t*sin(qdot_3*t + qinit_3)*sin(qdot_5*t + qinit_5)*cos(qdot_2*t + qinit_2) + 0.088*(-t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4) - t*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) + (0.107*t*sin(qdot_3*t + qinit_3)*sin(qdot_5*t + qinit_5)*cos(qdot_2*t + qinit_2) + 0.107*(-t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4) - t*cos(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6)\n const_3_q_3 = -0.088*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*sin(qdot_4*t + qinit_4)*sin(qdot_6*t + qinit_6) + 0.107*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*sin(qdot_4*t + qinit_4)*cos(qdot_6*t + qinit_6) - 0.384*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*sin(qdot_4*t + qinit_4) - 0.0825*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + 0.0825*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3) + (0.088*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4)*cos(qdot_5*t + qinit_5) + 0.088*t*sin(qdot_2*t + qinit_2)*sin(qdot_5*t + qinit_5)*cos(qdot_3*t + qinit_3))*cos(qdot_6*t + qinit_6) + (0.107*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4)*cos(qdot_5*t + qinit_5) + 0.107*t*sin(qdot_2*t + qinit_2)*sin(qdot_5*t + qinit_5)*cos(qdot_3*t + qinit_3))*sin(qdot_6*t + qinit_6)\n const_3_q_4 = -0.0825*t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_3*t + qinit_3) + 0.384*t*sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) - 0.384*t*sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2) - 0.0825*t*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4) + 0.107*(t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_3*t + qinit_3) + t*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6)*cos(qdot_5*t + qinit_5) + 0.088*(t*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_3*t + qinit_3) + t*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*cos(qdot_5*t + qinit_5)*cos(qdot_6*t + qinit_6) + (-0.107*t*sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + 0.107*t*sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2))*cos(qdot_6*t + qinit_6) + (0.088*t*sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) - 0.088*t*sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2))*sin(qdot_6*t + qinit_6)\n const_3_q_5 = (-0.107*t*(-sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2))*sin(qdot_5*t + qinit_5) + 0.107*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*cos(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + (-0.088*t*(-sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2))*sin(qdot_5*t + qinit_5) + 0.088*t*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*cos(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6)\n const_3_q_6 = -t*(0.088*(-sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2))*cos(qdot_5*t + qinit_5) + 0.088*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*sin(qdot_5*t + qinit_5))*sin(qdot_6*t + qinit_6) + t*(0.107*(-sin(qdot_2*t + qinit_2)*cos(qdot_3*t + qinit_3)*cos(qdot_4*t + qinit_4) + sin(qdot_4*t + qinit_4)*cos(qdot_2*t + qinit_2))*cos(qdot_5*t + qinit_5) + 0.107*sin(qdot_2*t + qinit_2)*sin(qdot_3*t + qinit_3)*sin(qdot_5*t + qinit_5))*cos(qdot_6*t + qinit_6) - t*(-0.107*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_3*t + qinit_3) - 0.107*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*sin(qdot_6*t + qinit_6) + t*(0.088*sin(qdot_2*t + qinit_2)*sin(qdot_4*t + qinit_4)*cos(qdot_3*t + qinit_3) + 0.088*cos(qdot_2*t + qinit_2)*cos(qdot_4*t + qinit_4))*cos(qdot_6*t + qinit_6)\n const_3_q_7 = 0\n const_1_x = -1\n const_2_x = 0\n const_3_x = 0\n const_1_y = 0\n const_2_y = -1\n const_3_y = 0\n const_1_z = 0\n const_2_z = 0\n const_3_z = -1\n const_1_jac = np.hstack(( const_1_q_1, const_1_q_2, const_1_q_3, const_1_q_4, const_1_q_5, const_1_q_6, const_1_q_7, const_1_x, const_1_y, const_1_z ))\n const_2_jac = np.hstack(( const_2_q_1, const_2_q_2, const_2_q_3, const_2_q_4, const_2_q_5, const_2_q_6, const_2_q_7, const_2_x, const_2_y, const_2_z ))\n const_3_jac = np.hstack(( const_3_q_1, const_3_q_2, const_3_q_3, const_3_q_4, const_3_q_5, const_3_q_6, const_3_q_7, const_3_x, const_3_y, const_3_z ))\n # const_3_jac = np.hstack(( const_3_q_1, const_3_q_2, const_3_q_3, const_3_q_4, const_3_q_5, const_3_q_6, const_3_q_7, const_2_x, const_2_y, const_2_z )) orignal MISTAKE?\n const_jac = np.vstack(( const_1_jac, const_2_jac, const_3_jac )) \n return const_jac\n\njac_constraintSympy = compute_franka_jac\n\n\n\n```\n\n\n```python\ndef orient_Residual(q):\n global roll_des,pitch_des, yaw_des\n q_1 = q[0]\n q_2 = q[1]\n q_3 = q[2]\n q_4 = q[3]\n q_5 = q[4]\n q_6 = q[5]\n q_7 = q[6]\n\n cq1 = np.cos(q_1)\n cq2 = np.cos(q_2)\n cq3 = np.cos(q_3)\n cq4 = np.cos(q_4)\n cq5 = np.cos(q_5)\n cq6 = np.cos(q_6)\n cq7 = np.cos(q_7)\n\n sq1 = np.sin(q_1)\n sq2 = np.sin(q_2)\n sq3 = np.sin(q_3)\n sq4 = np.sin(q_4)\n sq5 = np.sin(q_5)\n sq6 = np.sin(q_6)\n sq7 = np.sin(q_7)\n ax1 = -0.5*(((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.cos(q_4) + np.sin(q_1)*np.sin(q_2)*np.sin(q_4))*np.cos(q_5) - (np.sin(q_1)*np.sin(q_3)*np.cos(q_2) - np.cos(q_1)*np.cos(q_3))*np.sin(q_5))*np.sin(q_6) + 0.5*(((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.cos(q_6) - (np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.sin(q_6))*np.sin(q_7) - 0.5*((np.sin(q_1)*np.cos(q_2)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1))*np.sin(q_4) - np.sin(q_1)*np.sin(q_2)*np.cos(q_4))*np.cos(q_6) - 0.5*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.sin(q_5) + np.sin(q_2)*np.sin(q_3)*np.cos(q_5))*np.cos(q_7)\n\n ax2 = -0.5*(((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.cos(q_4) - np.sin(q_2)*np.sin(q_4)*np.cos(q_1))*np.cos(q_5) + (np.sin(q_1)*np.cos(q_3) + np.sin(q_3)*np.cos(q_1)*np.cos(q_2))*np.sin(q_5))*np.sin(q_6) + 0.5*(((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.cos(q_5) - np.sin(q_2)*np.sin(q_3)*np.sin(q_5))*np.cos(q_6) - (np.sin(q_2)*np.sin(q_4)*np.cos(q_3) + np.cos(q_2)*np.cos(q_4))*np.sin(q_6))*np.cos(q_7) - 0.5*((np.sin(q_1)*np.sin(q_3) - np.cos(q_1)*np.cos(q_2)*np.cos(q_3))*np.sin(q_4) + np.sin(q_2)*np.cos(q_1)*np.cos(q_4))*np.cos(q_6) + 0.5*((np.sin(q_2)*np.cos(q_3)*np.cos(q_4) - np.sin(q_4)*np.cos(q_2))*np.sin(q_5) + np.sin(q_2)*np.sin(q_3)*np.cos(q_5))*np.sin(q_7)\n\n ax3 = -0.5*np.sin(q_2)*np.sin(q_4)*np.sin(q_5)*np.cos(q_1 + q_7) + 0.5*np.sin(q_2)*np.sin(q_4)*np.sin(q_1 + q_7)*np.cos(q_5)*np.cos(q_6) + 0.5*np.sin(q_2)*np.sin(q_6)*np.sin(q_1 + q_7)*np.cos(q_4) - 0.5*np.sin(q_3)*np.sin(q_4)*np.sin(q_6)*np.cos(q_1 + q_7) - 0.5*np.sin(q_3)*np.sin(q_5)*np.sin(q_1 + q_7)*np.cos(q_2)*np.cos(q_6) + 0.5*np.sin(q_3)*np.sin(q_5)*np.sin(q_1 + q_7)*np.cos(q_4) - 0.5*np.sin(q_3)*np.cos(q_2)*np.cos(q_5)*np.cos(q_1 + q_7) + 0.5*np.sin(q_3)*np.cos(q_4)*np.cos(q_5)*np.cos(q_6)*np.cos(q_1 + q_7) - 0.5*np.sin(q_4)*np.sin(q_6)*np.sin(q_1 + q_7)*np.cos(q_2)*np.cos(q_3) - 0.5*np.sin(q_5)*np.cos(q_2)*np.cos(q_3)*np.cos(q_4)*np.cos(q_1 + q_7) + 0.5*np.sin(q_5)*np.cos(q_3)*np.cos(q_6)*np.cos(q_1 + q_7) + 0.5*np.sin(q_1 + q_7)*np.cos(q_2)*np.cos(q_3)*np.cos(q_4)*np.cos(q_5)*np.cos(q_6) - 0.5*np.sin(q_1 + q_7)*np.cos(q_3)*np.cos(q_5)\n\n r_32 = -cq7*(cq5*sq2*sq3 - sq5*(cq2*sq4 - cq3*cq4*sq2)) - sq7*(cq6*(cq5*(cq2*sq4 - cq3*cq4*sq2) + sq2*sq3*sq5) + sq6*(cq2*cq4 + cq3*sq2*sq4))\n r_33 = -cq6*(cq2*cq4 + cq3*sq2*sq4) + sq6*(cq5*(cq2*sq4 - cq3*cq4*sq2) + sq2*sq3*sq5)\n r_31 = cq7*(cq6*(cq5*(cq2*sq4 - cq3*cq4*sq2) + sq2*sq3*sq5) + sq6*(cq2*cq4 + cq3*sq2*sq4)) - sq7*(cq5*sq2*sq3 - sq5*(cq2*sq4 - cq3*cq4*sq2))\n r_21 = cq7*(cq6*(cq5*(cq4*(cq1*sq3 + cq2*cq3*sq1) + sq1*sq2*sq4) + sq5*(cq1*cq3 - cq2*sq1*sq3)) + sq6*(cq4*sq1*sq2 - sq4*(cq1*sq3 + cq2*cq3*sq1))) - sq7*(cq5*(cq1*cq3 - cq2*sq1*sq3) - sq5*(cq4*(cq1*sq3 + cq2*cq3*sq1) + sq1*sq2*sq4))\n r_11 = cq7*(cq6*(cq5*(cq1*sq2*sq4 + cq4*(cq1*cq2*cq3 - sq1*sq3)) - sq5*(cq1*cq2*sq3 + cq3*sq1)) + sq6*(cq1*cq4*sq2 - sq4*(cq1*cq2*cq3 - sq1*sq3))) + sq7*(cq5*(cq1*cq2*sq3 + cq3*sq1) + sq5*(cq1*sq2*sq4 + cq4*(cq1*cq2*cq3 - sq1*sq3)))\n #pdb.set_trace()\n f_orient_cost = ((r_32/r_33)-np.tan(roll_des))**2+(-np.sin(r_31)-np.sin(pitch_des))**2 + ((r_21 / r_11) -np.tan(yaw_des ) )**2\n return f_orient_cost\n```\n\n\n```python\n#if you want to go from A to B pass the joint position of A fk(q_a) , x_fin\n#main function to compute Delan Metric-Based IK\nimport time\ndef trajMetricBased(init_joint,end_cartesian,seq):\n start_cart = np.asarray(fk_franka_orient(init_joint))[0:3]\n global t,q_prev,goal\n print(\"Start : \",start_cart)\n print(\"Goal : \", end_cartesian)\n goal = np.squeeze(end_cartesian)\n x_des = end_cartesian\n num_dof = 7 #franka panda\n qdot = np.zeros(num_dof) ########## initial velocity\n qdotprev = np.zeros(num_dof)\n q_min = np.array([-165, -100, -165, -165, -165, -1.0, -165]) * np.pi / 180 # TODO\n #q_min = q_min.reshape(7,1)\n q_max = np.array([165, 101, 165, 1.0, 165, 214, 165]) * np.pi / 180 #TODO\n #q_max = q_max.reshape(7,1)\n\n\n qdot_max = np.array([2.1750\t,2.1750\t,2.1750\t,2.1750,\t2.6100,\t2.6100,\t2.6100]) #TODO\n qdot_min = -1*qdot_max\n qacc_max = np.array([15,\t7.5,\t10\t,12.5\t,15\t,20,\t20]) #TODO\n qacc_min = -1*qacc_max\n\n x_pos = np.asarray(fk_franka_orient(init_joint))[0:3] ########### initial positions\n\n q = init_joint\n q_prev = init_joint\n ######### delta t for which the computed velocity is commanded to the robot 3msec\n x_next = fk_franka_orient(qdot*t + q)[0:3]\n\n mpc_iter =300\n q_1 = []\n q_2 = []\n q_3 = []\n q_4 = []\n q_5 = []\n q_6 = []\n q_7 = []\n cost_tracker = []\n trajectory = []\n qdottrajc =[]\n orientResidual =[]\n #pdb.set_trace()\n solverVariable = np.hstack((qdot,x_next))\n x_min = -np.ones(3,)*np.Inf \n x_max = np.ones(3,)*np.Inf \n solver_minbounds= np.hstack((qdot_min , x_min))\n solver_maxbounds = np.hstack((qdot_max , x_max))\n Amat = np.hstack((np.identity(7),np.zeros((7,3)))) \n Bmat = np.hstack((np.identity(7),np.zeros((7,3))))\n Total_time = 0\n pose = fk_franka_orient(q)\n for i in range(0, mpc_iter):\n print(f\"mpc-itr={i}\")\n orientResidual.append( orient_Residual(q))\n cost_tracker.append(np.linalg.norm(x_pos - x_des))\n print(\"itr=\",i, \"euclid-norm\",np.linalg.norm(x_pos - x_des) , \"orient-norm\" , orient_Residual(q) )\n if np.linalg.norm(x_pos - x_des) < 0.05 and orient_Residual(q) < 0.1 :\n break\n bnds = Bounds(solver_minbounds,solver_maxbounds)\n nonlinear_constraint = NonlinearConstraint(constraintfxn , np.zeros((3,)), np.zeros((3,)) , jac= jac_constraintSympy, hess=BFGS() )\n linear_constraint_A = LinearConstraint(Amat*t, q_min-q_prev, q_max-q_prev)\n linear_constraint_B = LinearConstraint(Bmat ,qacc_min*t + qdotprev, qacc_max*t + qdotprev )\n defaultopts={ 'maxiter': 80 , 'ftol': 1e-08, 'iprint': 1, 'disp': False, 'eps': 1.4901161193847656e-08, 'finite_diff_rel_step': None}\n val = value(x_pos)\n currentTime = time.time()\n res = minimize(costfxn , solverVariable, args =(x_pos,val) , method='SLSQP', jac=jac_cost,\n constraints=[nonlinear_constraint,linear_constraint_A, linear_constraint_B], options=defaultopts ,bounds=bnds) \n timeTaken = (time.time() - currentTime)\n Total_time += timeTaken\n solverVariable = np.asarray(res['x']).squeeze()\n trajectory.append([q[0], q[1], q[2], q[3], q[4], q[5], q[6]])\n qdottrajc.append(qdotprev)\n q = q + solverVariable[0:7] * t\n x_pos = fk_franka_orient(q)[0:3] #solverVariable[7:] #or x_next\n solverVariable[7:] = x_pos\n qdotprev = solverVariable[0:7]\n q_prev = q\n pose = fk_franka_orient(q)\n if len(trajectory) != 0 :\n JointCost, CartCost = traj_cost(trajectory)\n \n averageTime = Total_time/(i+1)\n executionTime = (i+1)*0.02\n #OrienResidual = orienResidual(q) \n cart_residual = cost_tracker[-1]\n pose = fk_franka_orient(q)\n orient_residual = orient_Residual(q)\n return JointCost, CartCost, cost_tracker, trajectory,qdottrajc,orientResidual\n else:\n return 0, 0, 0 , [], 0 ,0 \n\n```\n\n\n```python\n#example 1\ninit_joint =np.asarray([-0.6581236205695661, -0.685344738206894, -0.07143761874163594, -2.7322892472673206, -0.050369729249887224, 2.0478311535710176, 1.6687292468254735])\nend_cart = np.asarray([ -0.27332807, 0.27789939, -0.15650605 ])\n\n#example 2 \n#init_joint = np.asarray([-1.6202005289676729,-1.2697694275822582,-2.600571094528167,-2.176462031576798,-2.09104048608945,2.28241996926819,-0.05311600216515573])\n#end_cart = np.asarray([ 0.6367232 , -0.40827509, 0.39513954])\nJointCost, CartCost, cart_cost_tracker, trajectory,qdottrajc ,orientResidual_tracker = trajMetricBased(init_joint , end_cart ,\"example2\")\n```\n\n Start : [ 0.24605026 -0.22180353 0.4166907 ]\n Goal : [-0.27332807 0.27789939 -0.15650605]\n mpc-itr=0\n itr= 0 euclid-norm 0.9208753334845314 orient-norm 3.9766230535877584e-05\n mpc-itr=1\n itr= 1 euclid-norm 0.9179723472154594 orient-norm 0.0004519908770243149\n mpc-itr=2\n itr= 2 euclid-norm 0.9120976667148027 orient-norm 0.0030620813372865966\n mpc-itr=3\n itr= 3 euclid-norm 0.9031492458091034 orient-norm 0.011439786915084309\n mpc-itr=4\n itr= 4 euclid-norm 0.8909917527970994 orient-norm 0.03033629403579955\n mpc-itr=5\n itr= 5 euclid-norm 0.8754707552953916 orient-norm 0.06515339926209383\n mpc-itr=6\n itr= 6 euclid-norm 0.8564408125051055 orient-norm 0.1166745795015591\n mpc-itr=7\n itr= 7 euclid-norm 0.8342988091976554 orient-norm 0.19100033159558477\n mpc-itr=8\n itr= 8 euclid-norm 0.8142300634616978 orient-norm 0.2779530383111696\n mpc-itr=9\n itr= 9 euclid-norm 0.7955952423145648 orient-norm 0.3780345558241193\n mpc-itr=10\n itr= 10 euclid-norm 0.7781454879724871 orient-norm 0.4925455123708523\n mpc-itr=11\n itr= 11 euclid-norm 0.7620799481550798 orient-norm 0.6179568713066785\n mpc-itr=12\n itr= 12 euclid-norm 0.747784588435236 orient-norm 0.7471461044877954\n mpc-itr=13\n itr= 13 euclid-norm 0.7349717532152321 orient-norm 0.8786999425207335\n mpc-itr=14\n itr= 14 euclid-norm 0.7236372403382607 orient-norm 1.007364707416616\n mpc-itr=15\n itr= 15 euclid-norm 0.7135904412826327 orient-norm 1.1310251291303102\n mpc-itr=16\n itr= 16 euclid-norm 0.7046070380556929 orient-norm 1.2489406059291137\n mpc-itr=17\n itr= 17 euclid-norm 0.6964554602860829 orient-norm 1.3615921778415165\n mpc-itr=18\n itr= 18 euclid-norm 0.6888838257699009 orient-norm 1.4709585367266005\n mpc-itr=19\n itr= 19 euclid-norm 0.6817449700739754 orient-norm 1.578041967806966\n mpc-itr=20\n itr= 20 euclid-norm 0.6727599519390013 orient-norm 1.6964282763560394\n mpc-itr=21\n itr= 21 euclid-norm 0.6611864515151117 orient-norm 1.8281557904009247\n mpc-itr=22\n itr= 22 euclid-norm 0.647276449211088 orient-norm 1.9709231609285582\n mpc-itr=23\n itr= 23 euclid-norm 0.6312606162840506 orient-norm 2.1219394729393315\n mpc-itr=24\n itr= 24 euclid-norm 0.6139797186035865 orient-norm 2.269642324041856\n mpc-itr=25\n itr= 25 euclid-norm 0.5960466379112828 orient-norm 2.403818568919186\n mpc-itr=26\n itr= 26 euclid-norm 0.5783086575537171 orient-norm 2.502669081841128\n mpc-itr=27\n itr= 27 euclid-norm 0.5610102669883865 orient-norm 2.5623084300321994\n mpc-itr=28\n itr= 28 euclid-norm 0.5440819160116853 orient-norm 2.5867086561390904\n mpc-itr=29\n itr= 29 euclid-norm 0.5291106245802388 orient-norm 2.5791355850356164\n mpc-itr=30\n itr= 30 euclid-norm 0.5159027711132875 orient-norm 2.545451844341975\n mpc-itr=31\n itr= 31 euclid-norm 0.5045605740424741 orient-norm 2.4895923325978067\n mpc-itr=32\n itr= 32 euclid-norm 0.4946239398332391 orient-norm 2.4221529072508643\n mpc-itr=33\n itr= 33 euclid-norm 0.48546392878498607 orient-norm 2.356131301729767\n mpc-itr=34\n itr= 34 euclid-norm 0.47695239038927423 orient-norm 2.2947562298304645\n mpc-itr=35\n itr= 35 euclid-norm 0.4680186365745303 orient-norm 2.2314553499096137\n mpc-itr=36\n itr= 36 euclid-norm 0.4584777324782933 orient-norm 2.167621586356325\n mpc-itr=37\n itr= 37 euclid-norm 0.44930311130135314 orient-norm 2.0899562876827673\n mpc-itr=38\n itr= 38 euclid-norm 0.44035528394021456 orient-norm 2.004697353005917\n mpc-itr=39\n itr= 39 euclid-norm 0.43159364652770155 orient-norm 1.9257706899525497\n mpc-itr=40\n itr= 40 euclid-norm 0.42311537792933973 orient-norm 1.8487016268231566\n mpc-itr=41\n itr= 41 euclid-norm 0.414772351516169 orient-norm 1.7690732218969927\n mpc-itr=42\n itr= 42 euclid-norm 0.4068075506793391 orient-norm 1.685924228845298\n mpc-itr=43\n itr= 43 euclid-norm 0.39935543414452085 orient-norm 1.5982917238081242\n mpc-itr=44\n itr= 44 euclid-norm 0.3923395196718938 orient-norm 1.5084458989350096\n mpc-itr=45\n itr= 45 euclid-norm 0.38570923783878386 orient-norm 1.4179036761349004\n mpc-itr=46\n itr= 46 euclid-norm 0.37922254403963773 orient-norm 1.3302096716708365\n mpc-itr=47\n itr= 47 euclid-norm 0.3725946634480924 orient-norm 1.2476052214237896\n mpc-itr=48\n itr= 48 euclid-norm 0.3658476479256193 orient-norm 1.1685638500753806\n mpc-itr=49\n itr= 49 euclid-norm 0.35899993179248185 orient-norm 1.0921637779227682\n mpc-itr=50\n itr= 50 euclid-norm 0.3520664208309003 orient-norm 1.0178020572379325\n mpc-itr=51\n itr= 51 euclid-norm 0.34505704284341604 orient-norm 0.9451096253474756\n mpc-itr=52\n itr= 52 euclid-norm 0.337983879956387 orient-norm 0.8738779131474643\n mpc-itr=53\n itr= 53 euclid-norm 0.3308659018658105 orient-norm 0.8038540235747833\n mpc-itr=54\n itr= 54 euclid-norm 0.32371030694896785 orient-norm 0.7350372366544702\n mpc-itr=55\n itr= 55 euclid-norm 0.3165222664482285 orient-norm 0.6675547890504009\n mpc-itr=56\n itr= 56 euclid-norm 0.30928809985699907 orient-norm 0.6017903948380554\n mpc-itr=57\n itr= 57 euclid-norm 0.3020195312051132 orient-norm 0.5380015354812621\n mpc-itr=58\n itr= 58 euclid-norm 0.29474138451964627 orient-norm 0.476596468363466\n mpc-itr=59\n itr= 59 euclid-norm 0.2873664838803182 orient-norm 0.4187081354637925\n mpc-itr=60\n itr= 60 euclid-norm 0.27973755133798794 orient-norm 0.36590478700996687\n mpc-itr=61\n itr= 61 euclid-norm 0.2718597908784527 orient-norm 0.3184604360898108\n mpc-itr=62\n itr= 62 euclid-norm 0.2637315097536895 orient-norm 0.2766971577700479\n mpc-itr=63\n itr= 63 euclid-norm 0.25536000375691265 orient-norm 0.24076666597288668\n mpc-itr=64\n itr= 64 euclid-norm 0.24677382296859798 orient-norm 0.2105159811814169\n mpc-itr=65\n itr= 65 euclid-norm 0.23803114806937145 orient-norm 0.18543174226034315\n mpc-itr=66\n itr= 66 euclid-norm 0.22922061002504532 orient-norm 0.16468917954712126\n mpc-itr=67\n itr= 67 euclid-norm 0.22051318553311408 orient-norm 0.14711993729971684\n mpc-itr=68\n itr= 68 euclid-norm 0.2118961896016381 orient-norm 0.13213957070849155\n mpc-itr=69\n itr= 69 euclid-norm 0.2034162976759297 orient-norm 0.11896337788925818\n mpc-itr=70\n itr= 70 euclid-norm 0.19522807188048905 orient-norm 0.1067313718572154\n mpc-itr=71\n itr= 71 euclid-norm 0.18741777775485705 orient-norm 0.09505409300217262\n mpc-itr=72\n itr= 72 euclid-norm 0.1778661653899078 orient-norm 0.09000127397492907\n mpc-itr=73\n itr= 73 euclid-norm 0.16719765419853955 orient-norm 0.0900065115778486\n mpc-itr=74\n itr= 74 euclid-norm 0.15728823291821506 orient-norm 0.09000158565180366\n mpc-itr=75\n itr= 75 euclid-norm 0.1480533855059605 orient-norm 0.09000044160684158\n mpc-itr=76\n itr= 76 euclid-norm 0.13944977907274064 orient-norm 0.09000000837303676\n mpc-itr=77\n itr= 77 euclid-norm 0.13143345251003236 orient-norm 0.09000001491421611\n mpc-itr=78\n itr= 78 euclid-norm 0.12401235105514666 orient-norm 0.09000006649950953\n mpc-itr=79\n itr= 79 euclid-norm 0.11713471136034516 orient-norm 0.09000009333261272\n mpc-itr=80\n itr= 80 euclid-norm 0.11074781218281415 orient-norm 0.09000009635271981\n mpc-itr=81\n itr= 81 euclid-norm 0.1048021675002188 orient-norm 0.09000008719230083\n mpc-itr=82\n itr= 82 euclid-norm 0.0992609458165895 orient-norm 0.09000007817876803\n mpc-itr=83\n itr= 83 euclid-norm 0.09407864633643875 orient-norm 0.09000006650326561\n mpc-itr=84\n itr= 84 euclid-norm 0.08919591528095895 orient-norm 0.0900003089754385\n mpc-itr=85\n itr= 85 euclid-norm 0.08463782361466685 orient-norm 0.09000005534119065\n mpc-itr=86\n itr= 86 euclid-norm 0.08037342568159457 orient-norm 0.09000003651708405\n mpc-itr=87\n itr= 87 euclid-norm 0.07638364004034548 orient-norm 0.09000003004821502\n mpc-itr=88\n itr= 88 euclid-norm 0.07264463777427155 orient-norm 0.09000002456430273\n mpc-itr=89\n itr= 89 euclid-norm 0.06913351460561037 orient-norm 0.09000002197942364\n mpc-itr=90\n itr= 90 euclid-norm 0.06583061836312387 orient-norm 0.09000002116259825\n mpc-itr=91\n itr= 91 euclid-norm 0.06271943875515563 orient-norm 0.09000001856434771\n mpc-itr=92\n itr= 92 euclid-norm 0.059784276002300324 orient-norm 0.09000001827711174\n mpc-itr=93\n itr= 93 euclid-norm 0.05701176344037878 orient-norm 0.09000001802489015\n mpc-itr=94\n itr= 94 euclid-norm 0.054396622752763434 orient-norm 0.09000000709038067\n mpc-itr=95\n itr= 95 euclid-norm 0.05192116313775348 orient-norm 0.09000000976190008\n mpc-itr=96\n itr= 96 euclid-norm 0.04957272768324672 orient-norm 0.09000001013087859\n\n\n\n```python\nplotno =1\n%matplotlib inline\nplot(orientResidual_tracker, cart_cost_tracker , trajectory,init_joint, end_cart,qdottrajc,plotno)\n```\n\n\n```python\n%matplotlib notebook\nimport roboticstoolbox as rtb\ntrajc = np.reshape(trajectory , ( len(trajectory) ,7))\nrobot = rtb.models.DH.Panda()\nrobot.plot(trajc, dt = 0.02 , movie = 'example.gif') #for smooth visualization click an play on the saved gif file in current dir (might need to refresh the directory)\n```\n\n\n", "meta": {"hexsha": "c269e0f07125ec996d7fe414a8447a91f9fdbebf", "size": 712820, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/.ipynb_checkpoints/DeLan_IK-checkpoint.ipynb", "max_stars_repo_name": "prajwalresearch/rearrangement_latest", "max_stars_repo_head_hexsha": "1c4cd2b777ba74419e79599de08b750614462bfd", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/.ipynb_checkpoints/DeLan_IK-checkpoint.ipynb", "max_issues_repo_name": "prajwalresearch/rearrangement_latest", "max_issues_repo_head_hexsha": "1c4cd2b777ba74419e79599de08b750614462bfd", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/.ipynb_checkpoints/DeLan_IK-checkpoint.ipynb", "max_forks_repo_name": "prajwalresearch/rearrangement_latest", "max_forks_repo_head_hexsha": "1c4cd2b777ba74419e79599de08b750614462bfd", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 673.1067044381, "max_line_length": 260544, "alphanum_fraction": 0.9261187958, "converted": true, "num_tokens": 30481, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081926, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.4193771313829563}} {"text": "\n\n# Tutorial 2: Confidence intervals and bootstrapping\n**Module 4: How do we know how certain we should be?**\n\n**Originally By Neuromatch Academy**\n\n**Content creators**: Pierre-\u00c9tienne Fiquet, Anqi Wu, Alex Hyafil with help from Byron Galbraith\n\n**Content reviewers**: Lina Teichmann, Madineh Sarvestani, Patrick Mineault, Ella Batty, Michael Waskom\n\n**Content Modifiers**: Konrad Kording, Ilenna Jones\n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n

                                        \n\n\n```python\n# @title Due Dates Calendar\n\nfrom ipywidgets import widgets\nfrom IPython.display import display, IFrame, YouTubeVideo\n\n\nout1 = widgets.Output()\nwith out1:\n calendar = IFrame(src=\"https://calendar.google.com/calendar/embed?src=356b9d2nspjttvgbb3tvgk2f58%40group.calendar.google.com&ctz=America%2FNew_York\", width=600, height=480)\n display(calendar)\n\nout = widgets.Tab([out1])\nout.set_title(0, 'Calendar')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(),), _titles={'0': 'Calendar'})\n\n\n# Tutorial Objectives\n\n*Estimated timing of tutorial: 23 minutes*\n\nThis is Tutorial 3 of a series on fitting models to data. We start with simple linear regression, using least squares optimization (Tutorial 1) and Maximum Likelihood Estimation (Tutorial 2). We will use bootstrapping to build confidence intervals around the inferred linear model parameters (Tutorial 3). We'll finish our exploration of regression models by generalizing to multiple linear regression and polynomial regression (Tutorial 4). We end by learning how to choose between these various models. We discuss the bias-variance trade-off (Tutorial 5) and Cross Validation for model selection (Tutorial 6).\n\nIn this tutorial, we will discuss how to gauge how good our estimated model parameters are. \n- Learn how to use bootstrapping to generate new sample datasets\n- Estimate our model parameter on these new sample datasets\n- Quantify the variance of our estimate using confidence intervals\n\n\n```python\n# @title Tutorial slides\n\n# @markdown These are the slides for the videos in all tutorials today\nfrom IPython.display import IFrame\nIFrame(src=f\"https://mfr.ca-1.osf.io/render?url=https://osf.io/2mkq4/?direct%26mode=render%26action=download%26mode=render\", width=854, height=480)\n```\n\n\n```python\n# @title Video 1: Confidence Intervals & Bootstrapping\nfrom ipywidgets import widgets\n\nout1 = widgets.Output()\nwith out1:\n from IPython.display import YouTubeVideo\n video = YouTubeVideo(id=\"Jii0gMy5JLQ\", width=854, height=480, fs=1, rel=0)\n print('Video available at https://youtube.com/watch?v=' + video.id)\n display(video)\n\nout = widgets.Tab([out1])\nout.set_title(0, 'Youtube')\n\ndisplay(out)\n```\n\n\n Tab(children=(Output(),), _titles={'0': 'Youtube'})\n\n\nUp to this point we have been finding ways to estimate model parameters to fit some observed data. Our approach has been to optimize some criterion, either minimize the mean squared error or maximize the likelihood while using the entire dataset. How good is our estimate really? How confident are we that it will generalize to describe new data we haven't seen yet?\n\nOne solution to this is to just collect more data and check the MSE on this new dataset with the previously estimated parameters. However this is not always feasible and still leaves open the question of how quantifiably confident we are in the accuracy of our model.\n\nIn Section 1, we will explore how to implement bootstrapping. In Section 2, we will build confidence intervals of our estimates using the bootstrapping method.\n\n---\n# Setup\n\n\n```python\n# Imports\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n```\n\n\n```python\n#@title Figure Settings\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/course-content/master/nma.mplstyle\")\n```\n\n\n```python\n# @title Plotting Functions\n\ndef plot_original_and_resample(x, y, x_, y_):\n \"\"\" Plot the original sample and the resampled points from this sample.\n\n Args:\n x (ndarray): An array of shape (samples,) that contains the input values.\n y (ndarray): An array of shape (samples,) that contains the corresponding\n measurement values to the inputs.\n x_ (ndarray): An array of shape (samples,) with a subset of input values from x\n y_ (ndarray): An array of shape (samples,) with a the corresponding subset\n of measurement values as x_ from y\n\n \"\"\"\n fig, (ax1, ax2) = plt.subplots(ncols=2, figsize=(12, 5))\n ax1.scatter(x, y)\n ax1.set(title='Original', xlabel='x', ylabel='y')\n\n ax2.scatter(x_, y_, color='c')\n\n ax2.set(title='Resampled', xlabel='x', ylabel='y',\n xlim=ax1.get_xlim(), ylim=ax1.get_ylim());\n```\n\n---\n# Section 1: Bootstrapping\n\n*Estimated timing to here from start of tutorial: 7 min*\n\n[Bootstrapping](https://en.wikipedia.org/wiki/Bootstrapping_(statistics)) is a widely applicable method to assess confidence/uncertainty about estimated parameters, it was originally [proposed](https://projecteuclid.org/euclid.aos/1176344552) by [Bradley Efron](https://en.wikipedia.org/wiki/Bradley_Efron). The idea is to generate many new synthetic datasets from the initial true dataset by randomly sampling from it, then finding estimators for each one of these new datasets, and finally looking at the distribution of all these estimators to quantify our confidence.\n\nNote that each new resampled datasets will be the same size as our original one, with the new data points sampled with replacement i.e. we can repeat the same data point multiple times. Also note that in practice we need a lot of resampled datasets, here we use 2000.\n\nTo explore this idea, we will start again with our noisy samples along the line $y_i = 1.2x_i + \\epsilon_i$, but this time only use half the data points as last time (15 instead of 30).\n\n\n```python\n#@title\n\n#@markdown Execute this cell to simulate some data\n\n# setting a fixed seed to our random number generator ensures we will always\n# get the same psuedorandom number sequence\nnp.random.seed(121)\n\n# Let's set some parameters\ntheta = 1.2\nn_samples = 15\n\n# Draw x and then calculate y\nx = 10 * np.random.rand(n_samples) # sample from a uniform distribution over [0,10)\nnoise = np.random.randn(n_samples) # sample from a standard normal distribution\ny = theta * x + noise\n\nfig, ax = plt.subplots()\nax.scatter(x, y) # produces a scatter plot\nax.set(xlabel='x', ylabel='y');\n```\n\n## Coding Exercise 1: Resample Dataset with Replacement\n\nIn this exercise you will implement a method to resample a dataset with replacement. The method accepts $\\mathbf{x}$ and $\\mathbf{y}$ arrays. It should return a new set of $\\mathbf{x}'$ and $\\mathbf{y}'$ arrays that are created by randomly sampling from the originals.\n\nWe will then compare the original dataset to a resampled dataset.\n\nTIP: The [numpy.random.choice](https://numpy.org/doc/stable/reference/random/generated/numpy.random.choice.html) method would be useful here.\n\n\n```python\ndef resample_with_replacement(x, y):\n \"\"\"Resample data points with replacement from the dataset of `x` inputs and\n `y` measurements.\n\n Args:\n x (ndarray): An array of shape (samples,) that contains the input values.\n y (ndarray): An array of shape (samples,) that contains the corresponding\n measurement values to the inputs.\n\n Returns:\n ndarray, ndarray: The newly resampled `x` and `y` data points.\n \"\"\"\n #######################################################\n ## TODO for students: resample dataset with replacement\n # Fill out function and remove\n raise NotImplementedError(\"Student exercise: resample dataset with replacement\")\n #######################################################\n\n # Get array of indices for resampled points\n sample_idx = ...\n\n # Sample from x and y according to sample_idx\n x_ = ...\n y_ = ...\n\n return x_, y_\n\nx_, y_ = resample_with_replacement(x, y)\n\nplot_original_and_resample(x, y, x_, y_)\n```\n\n\n```python\n# to_remove solution\n\ndef resample_with_replacement(x, y):\n \"\"\"Resample data points with replacement from the dataset of `x` inputs and\n `y` measurements.\n\n Args:\n x (ndarray): An array of shape (samples,) that contains the input values.\n y (ndarray): An array of shape (samples,) that contains the corresponding\n measurement values to the inputs.\n\n Returns:\n ndarray, ndarray: The newly resampled `x` and `y` data points.\n \"\"\"\n\n # Get array of indices for resampled points\n sample_idx = np.random.choice(len(x), size=len(x), replace=True)\n\n # Sample from x and y according to sample_idx\n x_ = x[sample_idx]\n y_ = y[sample_idx]\n\n return x_, y_\n\nx_, y_ = resample_with_replacement(x, y)\n\nwith plt.xkcd():\n plot_original_and_resample(x, y, x_, y_)\n```\n\nIn the resampled plot on the right, the actual number of points is the same, but some have been repeated so they only display once.\n\nNow that we have a way to resample the data, we can use that in the full bootstrapping process.\n\n## Coding Exercise 2: Bootstrap Estimates\n\nIn this exercise you will implement a method to run the bootstrap process of generating a set of $\\hat\\theta$ values from a dataset of inputs ($\\mathbf{x}$) and measurements ($\\mathbf{y}$). You should use `resample_with_replacement` here, and you may also invoke helper function `solve_normal_eqn` from Tutorial 1 to produce the MSE-based estimator.\n\nWe will then use this function to look at the theta_hat from different samples.\n\n\n\n```python\n# @markdown Execute this cell for helper function `solve_normal_eqn`\ndef solve_normal_eqn(x, y):\n \"\"\"Solve the normal equations to produce the value of theta_hat that minimizes\n MSE.\n\n Args:\n x (ndarray): An array of shape (samples,) that contains the input values.\n y (ndarray): An array of shape (samples,) that contains the corresponding\n measurement values to the inputs.\n thata_hat (float): An estimate of the slope parameter.\n\n Returns:\n float: the value for theta_hat arrived from minimizing MSE\n \"\"\"\n theta_hat = (x.T @ y) / (x.T @ x)\n return theta_hat\n```\n\n\n```python\ndef bootstrap_estimates(x, y, n=2000):\n \"\"\"Generate a set of theta_hat estimates using the bootstrap method.\n\n Args:\n x (ndarray): An array of shape (samples,) that contains the input values.\n y (ndarray): An array of shape (samples,) that contains the corresponding\n measurement values to the inputs.\n n (int): The number of estimates to compute\n\n Returns:\n ndarray: An array of estimated parameters with size (n,)\n \"\"\"\n theta_hats = np.zeros(n)\n\n ##############################################################################\n ## TODO for students: implement bootstrap estimation\n # Fill out function and remove\n raise NotImplementedError(\"Student exercise: implement bootstrap estimation\")\n ##############################################################################\n\n # Loop over number of estimates\n for i in range(n):\n\n # Resample x and y\n x_, y_ = ...\n\n # Compute theta_hat for this sample\n theta_hats[i] = ...\n\n return theta_hats\n\n\n# Set random seed\nnp.random.seed(123)\n\n# Get boostrap estimates\ntheta_hats = bootstrap_estimates(x, y, n=2000)\nprint(theta_hats[0:5])\n```\n\n\n```python\n# to_remove solution\n\ndef bootstrap_estimates(x, y, n=2000):\n \"\"\"Generate a set of theta_hat estimates using the bootstrap method.\n\n Args:\n x (ndarray): An array of shape (samples,) that contains the input values.\n y (ndarray): An array of shape (samples,) that contains the corresponding\n measurement values to the inputs.\n n (int): The number of estimates to compute\n\n Returns:\n ndarray: An array of estimated parameters with size (n,)\n \"\"\"\n theta_hats = np.zeros(n)\n\n # Loop over number of estimates\n for i in range(n):\n\n # Resample x and y\n x_, y_ = resample_with_replacement(x, y)\n\n # Compute theta_hat for this sample\n theta_hats[i] = solve_normal_eqn(x_, y_)\n\n return theta_hats\n\n# Set random seed\nnp.random.seed(123)\n\n# Get boostrap estimates\ntheta_hats = bootstrap_estimates(x, y, n=2000)\nprint(theta_hats[0:5])\n```\n\nYou should see `[1.27550888 1.17317819 1.18198819 1.25329255 1.20714664]` as the first five estimates.\n\nNow that we have our bootstrap estimates, we can visualize all the potential models (models computed with different resampling) together to see how distributed they are.\n\n\n```python\n#@title\n#@markdown Execute this cell to visualize all potential models\n\nfig, ax = plt.subplots()\n\n# For each theta_hat, plot model\ntheta_hats = bootstrap_estimates(x, y, n=2000)\nfor i, theta_hat in enumerate(theta_hats):\n y_hat = theta_hat * x\n ax.plot(x, y_hat, c='r', alpha=0.01, label='Resampled Fits' if i==0 else '')\n\n# Plot observed data\nax.scatter(x, y, label='Observed')\n\n# Plot true fit data\ny_true = theta * x\nax.plot(x, y_true, 'g', linewidth=2, label='True Model')\n\nax.set(\n title='Bootstrapped Slope Estimation',\n xlabel='x',\n ylabel='y'\n)\n\n# Change legend line alpha property\nhandles, labels = ax.get_legend_handles_labels()\nhandles[0].set_alpha(1)\n\nax.legend();\n```\n\nThis looks pretty good! The bootstrapped estimates spread around the true model, as we would have hoped. Note that here we have the luxury to know the ground truth value for $\\theta$, but in applications we are trying to guess it from data. Therefore, assessing the quality of estimates based on finite data is a task of fundamental importance in data analysis.\n\n\n---\n# Section 2: Confidence Intervals\n\n*Estimated timing to here from start of tutorial: 17 min*\n\nLet us now quantify how uncertain our estimated slope is. We do so by computing [confidence intervals](https://en.wikipedia.org/wiki/Confidence_interval) (CIs) from our bootstrapped estimates. The most direct approach is to compute percentiles from the empirical distribution of bootstrapped estimates. Note that this is widely applicable as we are not assuming that this empirical distribution is Gaussian.\n\n\n```python\n#@title\n\n#@markdown Execute this cell to plot bootstrapped CI\n\ntheta_hats = bootstrap_estimates(x, y, n=2000)\nprint(f\"mean = {np.mean(theta_hats):.2f}, std = {np.std(theta_hats):.2f}\")\n\nfig, ax = plt.subplots()\nax.hist(theta_hats, bins=20, facecolor='C1', alpha=0.75)\nax.axvline(theta, c='g', label=r'True $\\theta$')\nax.axvline(np.percentile(theta_hats, 50), color='r', label='Median')\nax.axvline(np.percentile(theta_hats, 2.5), color='b', label='95% CI')\nax.axvline(np.percentile(theta_hats, 97.5), color='b')\nax.legend()\nax.set(\n title='Bootstrapped Confidence Interval',\n xlabel=r'$\\hat{{\\theta}}$',\n ylabel='count',\n xlim=[1.0, 1.5]\n);\n```\n\nLooking at the distribution of bootstrapped $\\hat{\\theta}$ values, we see that the true $\\theta$ falls well within the 95% confidence interval, which is reassuring. We also see that the value $\\theta = 1$ does not fall within the confidence interval. From this we would reject the hypothesis that the slope was 1.\n\n---\n# Summary\n\n*Estimated timing of tutorial: 23 minutes*\n\n- Bootstrapping is a resampling procedure that allows to build confidence intervals around inferred parameter values\n- it is a widely applicable and very practical method that relies on computational power and pseudo-random number generators (as opposed to more classical approaches than depend on analytical derivations)\n\n---\n# Notation\n\n\\begin{align}\n\\theta &\\quad \\text{parameter}\\\\\n\\hat{\\theta} &\\quad \\text{estimated parameter}\\\\\nx &\\quad \\text{input, independent variable}\\\\\ny &\\quad \\text{response measurement, dependent variable}\\\\\n\\mathbf{x} &\\quad \\text{vector of input values}\\\\\n\\mathbf{y} &\\quad \\text{vector of measurements}\\\\\n\\mathbf{x}' &\\quad \\text{vector of resampled input values }\\\\\n\\mathbf{y}' &\\quad \\text{vector of resampled measurement values}\\\\\n\\end{align}\n\n**Suggested readings** \n\nComputer Age Statistical Inference: Algorithms, Evidence and Data Science, by Bradley Efron and Trevor Hastie\n\n", "meta": {"hexsha": "a1374d7ebbad8e77e20dc7fbfd24250e8aa902af", "size": 54270, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W4D1_How_do_we_know_how_certain_we_should_be/TA/W4D1_Tutorial2.ipynb", "max_stars_repo_name": "KordingLab/ENGR344", "max_stars_repo_head_hexsha": "746c04117f5a27ca28c4bdd8ea7cb939c07afea5", "max_stars_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-01-26T02:39:55.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-26T02:39:55.000Z", "max_issues_repo_path": "tutorials/W4D1_How_do_we_know_how_certain_we_should_be/TA/W4D1_Tutorial2.ipynb", "max_issues_repo_name": "KordingLab/ENGR344", "max_issues_repo_head_hexsha": "746c04117f5a27ca28c4bdd8ea7cb939c07afea5", "max_issues_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2022-01-12T21:14:24.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-12T21:14:24.000Z", "max_forks_repo_path": "tutorials/W4D1_How_do_we_know_how_certain_we_should_be/TA/W4D1_Tutorial2.ipynb", "max_forks_repo_name": "KordingLab/ENGR344", "max_forks_repo_head_hexsha": "746c04117f5a27ca28c4bdd8ea7cb939c07afea5", "max_forks_repo_licenses": ["CC-BY-4.0", "BSD-3-Clause"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2022-01-14T00:26:12.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-24T15:35:36.000Z", "avg_line_length": 49.1131221719, "max_line_length": 13748, "alphanum_fraction": 0.6265339967, "converted": true, "num_tokens": 3967, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494421679929, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.4193771235326897}} {"text": "\n\n\n# Standalone Fishbone-Moncrief C Code\n\nWe start with the NRPy+ expressions generated in the [Tutorial-FishboneMoncriefID](Tutorial-FishboneMoncriefID.ipynb), and output them to the C file \"FishboneMoncriefID/FMstandalone.h\".\n\nFurther, $\\Gamma = \\alpha u^0$ is given by (as shown [here](Tutorial-u0_smallb_Poynting-Cartesian.ipynb)):\n$$\n\\Gamma = \\alpha u^0 = \\sqrt{\\frac{1}{1 - \\gamma_{ij}v^i_{(n)}v^j_{(n)}}}.\n$$\n\n\n```python\nimport sympy as sp\nfrom outputC import *\nimport indexedexp as ixp\nimport finite_difference as fin\nimport FishboneMoncriefID.FishboneMoncriefID as fmid\n\n# Step 1: Set up the Fishbone-Moncrief initial data. This sets all the ID gridfunctions.\nfmid.FishboneMoncriefID(\"Spherical\")\n\ngammaDD = ixp.zerorank2()\n\nDIM = 3\nfor i in range(DIM):\n for j in range(DIM):\n if i<=j:\n gammaDD[i][j] = fmid.IDgammaDD[i][j]\n else:\n gammaDD[i][j] = fmid.IDgammaDD[j][i]\n\n# gamma_{ij} v^i_{(n)} v^j_{(n)}\nGammacontraction = sp.sympify(0)\nfor i in range(DIM):\n for j in range(DIM):\n Gammacontraction += gammaDD[i][j] * fmid.IDValencia3velocityU[i] * fmid.IDValencia3velocityU[j]\n\nGammafactor = sp.sqrt(1 / (1 - Gammacontraction))\n\n# -={ F-M quantities: Generate C code from expressions and output to file }=-\nFishboneMoncrief_to_print = [\\\n lhrh(lhs=\"alpha\",rhs=fmid.IDalpha),\\\n lhrh(lhs=\"betaU0\",rhs=fmid.IDbetaU[0]),\\\n lhrh(lhs=\"betaU1\",rhs=fmid.IDbetaU[1]),\\\n lhrh(lhs=\"betaU2\",rhs=fmid.IDbetaU[2]),\\\n lhrh(lhs=\"Gammafactor\",rhs=Gammafactor),\\\n lhrh(lhs=\"Gamma_times_ValenciavU0\",rhs=Gammafactor*fmid.IDValencia3velocityU[0]),\\\n lhrh(lhs=\"Gamma_times_ValenciavU1\",rhs=Gammafactor*fmid.IDValencia3velocityU[1]),\\\n lhrh(lhs=\"Gamma_times_ValenciavU2\",rhs=Gammafactor*fmid.IDValencia3velocityU[2]),\\\n lhrh(lhs=\"uKS4U1\",rhs=fmid.uKS4U[1]),\\\n lhrh(lhs=\"uKS4U2\",rhs=fmid.uKS4U[2]),\\\n lhrh(lhs=\"uKS4U3\",rhs=fmid.uKS4U[3]),\\\n lhrh(lhs=\"uBL4U1\",rhs=fmid.uBL4U[1]),\\\n lhrh(lhs=\"uBL4U2\",rhs=fmid.uBL4U[2]),\\\n lhrh(lhs=\"uBL4U3\",rhs=fmid.uBL4U[3])\n ]\nprint(fmid.uKS4U[3])\nfin.FD_outputC(\"FishboneMoncriefID/FM_standalone.h\",FishboneMoncrief_to_print,params=\"outCverbose=False,CSE_enable=False\")\n```\n\n -2*M*a*xx0*(-2*M*a*xx0*sqrt((-a**2*(-2*M*xx0 + a**2 + xx0**2)*sin(xx1)**2 + (a**2 + xx0**2)**2)/(a**2*cos(xx1)**2 + xx0**2))*sqrt(sqrt(4*M*(a**2*cos(xx1)**2 + xx0**2)**2*(-2*M*xx0 + a**2 + xx0**2)*(-2*M*a**2*r_at_max_density + a**2*r_at_max_density**2 - a*sqrt(M*r_at_max_density)*(-a**2 + r_at_max_density**2) + r_at_max_density**4)**2/(r_at_max_density**3*(-a**2*(-2*M*xx0 + a**2 + xx0**2)*sin(xx1)**2 + (a**2 + xx0**2)**2)**2*(-3*M*r_at_max_density + 2*a*sqrt(M*r_at_max_density) + r_at_max_density**2)**2*sin(xx1)**2) + 1)/2 - 1/2)*Abs(sin(xx1))/(-a**2*(-2*M*xx0 + a**2 + xx0**2)*sin(xx1)**2 + (a**2 + xx0**2)**2) - sqrt((-2*M*xx0 + a**2 + xx0**2)/(-a**2*(-2*M*xx0 + a**2 + xx0**2)*sin(xx1)**2 + (a**2 + xx0**2)**2))*sqrt(a**2*cos(xx1)**2 + xx0**2)*sqrt(sqrt(4*M*(a**2*cos(xx1)**2 + xx0**2)**2*(-2*M*xx0 + a**2 + xx0**2)*(-2*M*a**2*r_at_max_density + a**2*r_at_max_density**2 - a*sqrt(M*r_at_max_density)*(-a**2 + r_at_max_density**2) + r_at_max_density**4)**2/(r_at_max_density**3*(-a**2*(-2*M*xx0 + a**2 + xx0**2)*sin(xx1)**2 + (a**2 + xx0**2)**2)**2*(-3*M*r_at_max_density + 2*a*sqrt(M*r_at_max_density) + r_at_max_density**2)**2*sin(xx1)**2) + 1)/2 + 1/2))/((a**2*cos(xx1)**2 + xx0**2)*(-2*M*xx0 + a**2 + xx0**2)) + sqrt((-a**2*(-2*M*xx0 + a**2 + xx0**2)*sin(xx1)**2 + (a**2 + xx0**2)**2)/(a**2*cos(xx1)**2 + xx0**2))*(-4*M**2*a**2*xx0**2/((a**2*cos(xx1)**2 + xx0**2)*(-a**2*(-2*M*xx0 + a**2 + xx0**2)*sin(xx1)**2 + (a**2 + xx0**2)**2)*(-2*M*xx0 + a**2 + xx0**2)) + (a**2*cos(xx1)**2 + xx0**2)/((-a**2*(-2*M*xx0 + a**2 + xx0**2)*sin(xx1)**2 + (a**2 + xx0**2)**2)*sin(xx1)**2))*sqrt(sqrt(4*M*(a**2*cos(xx1)**2 + xx0**2)**2*(-2*M*xx0 + a**2 + xx0**2)*(-2*M*a**2*r_at_max_density + a**2*r_at_max_density**2 - a*sqrt(M*r_at_max_density)*(-a**2 + r_at_max_density**2) + r_at_max_density**4)**2/(r_at_max_density**3*(-a**2*(-2*M*xx0 + a**2 + xx0**2)*sin(xx1)**2 + (a**2 + xx0**2)**2)**2*(-3*M*r_at_max_density + 2*a*sqrt(M*r_at_max_density) + r_at_max_density**2)**2*sin(xx1)**2) + 1)/2 - 1/2)*Abs(sin(xx1))\n Wrote to file \"FishboneMoncriefID/FM_standalone.h\"\n\n\n\n```python\n%%writefile FishboneMoncriefID/FM_standalone.c\n\n#include \"stdio.h\"\n#include \"stdlib.h\"\n#include \"math.h\"\n\nconst double a = 0.9375;\nconst double M = 1.0;\nconst double r_at_max_density = 12.0;\nconst double r_in = 6.0;\n\nint main(int argc, const char *argv[]) {\n\n // Step 0a: Read command-line input, error out if nonconformant\n double xx0,xx1,xx2;\n/*\n if(argc != 4) {\n printf(\"Error: Expected three command-line arguments: ./FM_standalone r theta phi\\n\");\n exit(1);\n }\n xx0 = strtod(argv[1],NULL);\n xx1 = strtod(argv[2],NULL);\n xx2 = strtod(argv[3],NULL);\n*/\n \n// printf(\"# Output: r,th,ph, alpha, betaU0, betaU1, betaU2, Gamma, Gamma*vValenciaU0, Gamma*vValenciaU1, Gamma*vValenciaU2\\n\");\n for(double xx0=1.6;xx0<50.0;xx0+=0.2) {\n xx1 = 1.56463634120e0; //M_PI/2.0;\n xx2 = 0.0;\n double alpha,betaU0,betaU1,betaU2,Gammafactor,Gamma_times_ValenciavU0,Gamma_times_ValenciavU1,Gamma_times_ValenciavU2;\n double uKS4U1,uKS4U2,uKS4U3,uBL4U1,uBL4U2,uBL4U3;\n#include \"FM_standalone.h\"\n if(xx0 < r_in) {\n Gammafactor = 1.0;\n Gamma_times_ValenciavU0 = Gamma_times_ValenciavU1 = Gamma_times_ValenciavU2 = 0.0;\n uKS4U1 = uKS4U2 = uKS4U3 = 0.0;\n uBL4U1 = uBL4U2 = uBL4U3 = 0.0;\n }\n printf(\"%e %e %e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e\\n\",\n xx0,xx1,xx2,\n alpha,betaU0,betaU1,betaU2,\n Gammafactor,\n Gamma_times_ValenciavU0, // util1(1) in FMtorus.f90; util(1,i,j,k) near the write statement\n Gamma_times_ValenciavU1, // util1(3) in FMtorus.f90.\n Gamma_times_ValenciavU2, // util1(2) in FMtorus.f90.\n uKS4U1,uKS4U2,uKS4U3,\n uBL4U1,uBL4U2,uBL4U3);\n } \n return 0;\n}\n```\n\n Writing FishboneMoncriefID/FM_standalone.c\n\n\n\n```python\n!gcc -O2 FishboneMoncriefID/FM_standalone.c -o FM_standalone -lm\n```\n\n\n```python\n!./FM_standalone > out.txt\n```\n\n\n```python\n%matplotlib inline\nimport sympy as sp\nimport matplotlib.pyplot as plt\nimport mpmath as mp\nimport csv\n\n# Download torus_cuts.csv:\nURL = \"http://astro.phys.wvu.edu/zetienne/torus_cuts.csv\"\noutfile = \"torus_cuts.csv\"\ntry:\n with open(outfile,\"w\") as file:\n file.write(urllib.request.urlopen(URL).read().decode(\"utf-8\"))\nexcept:\n try:\n with open(outfile,\"w\") as file:\n file.write(urllib.urlopen(URL).read().decode(\"utf-8\"))\n except:\n # If all else fails, hope wget does the job\n !wget -O $outfile $URL\n\ndef file_reader(filename,list_of_cols,delim=\" \"):\n with open(filename) as file:\n reader = csv.reader(file, delimiter=delim)\n data = list(zip(*reader))\n# print(data)\n # data is a tuple of strings. Tuples are immutable, and we need to perform math on\n # the data, so here we convert tuple to lists of floats:\n# data_output = [[sp.sympify(0) for i in range(len(list_of_cols))] for j in range(len(data[0]))]\n data_output = [[sp.sympify(0) for i in range(len(data[0]))] for j in range(len(list_of_cols))]\n for i in range(len(data[0])):\n for j in range(len(list_of_cols)):\n# print(i,j,data[list_of_cols[j]][i])\n data_output[j][i] = float(data[list_of_cols[j]][i])\n return data_output\n\nNRPy_data_output = file_reader('out.txt', [0,7,8,9,10])\nstd_data_output = file_reader('torus_cuts.csv',[0,4,1,3,2])\n\n\nylabels = ['Lorentz Gamma_{KS}=G','G*v^r_{KS,Val.}','G*v^{\\\\theta}_{KS,Val.}','G*v^{\\phi}_{KS,Val.}']\n\nfor i in range(len(ylabels)):\n # https://matplotlib.org/gallery/text_labels_and_annotations/legend.html#sphx-glr-gallery-text-labels-and-annotations-legend-py \n fig, ax = plt.subplots()\n plt.title(\"NRPy's FM solve with FMtorus.f90: \"+ylabels[i])\n plt.xlabel(\"r/M\")\n plt.ylabel(ylabels[i])\n ax.plot(NRPy_data_output[0], NRPy_data_output[i+1], 'k--', label='NRPyFMSolve')\n ax.plot(std_data_output[0], std_data_output[i+1], 'k-', label='FMtorus.f90')\n legend = ax.legend(loc='upper right', shadow=True, fontsize='x-large')\n legend.get_frame().set_facecolor('C1')\n plt.show()\n```\n", "meta": {"hexsha": "275980f794b8816f0b29d40255eb0c3ae1a6a4a8", "size": 112081, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Tutorial-Start_to_Finish-FishboneMoncriefID_standalone.ipynb", "max_stars_repo_name": "leowerneck/NRPyIGM", "max_stars_repo_head_hexsha": "f483d6123424fb3e6860dfac4325dd232b223005", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Tutorial-Start_to_Finish-FishboneMoncriefID_standalone.ipynb", "max_issues_repo_name": "leowerneck/NRPyIGM", "max_issues_repo_head_hexsha": "f483d6123424fb3e6860dfac4325dd232b223005", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Tutorial-Start_to_Finish-FishboneMoncriefID_standalone.ipynb", "max_forks_repo_name": "leowerneck/NRPyIGM", "max_forks_repo_head_hexsha": "f483d6123424fb3e6860dfac4325dd232b223005", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 345.9290123457, "max_line_length": 27400, "alphanum_fraction": 0.9127952106, "converted": true, "num_tokens": 3127, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8128673223709251, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.4191305802548898}} {"text": "# Networks in NeuroDyn\n\n## Overview\n\nThis notebook provides a guide to specyfing neural interconnections and designing networks using Hodgkin-Huxley and NeuroDyn models in Python environment. We first start with an overview of two main types of neural interconnections: gap junctions and synapses, and how to implement them in the software environment. The environment allows defining arbitrary conductance-based models of synapses, and we give an example here using an excitatory AMPA synapse. Networks can be constructed using both the standard Hodgkin-Huxley models as well as NeuroDyn neurons, and the notebook shows how we can specify both types. \n\nFinally, we introduce two additional classes `NeuroDynBoard` and `NeuroCube` which specify the NeuroDyn hardware structure:\n- `NeuroDynBoard` is a single NeuroDyn board containing 4 neurons and 12 synapses.\n- `NeuroCube` is a parallel interconnection of 4 NeuroDyn boards, where the boards are connected through mutural resistive connections.\n\n## Setting up the environment\n\nLet's first import the required Python modules and model definitions:\n\n\n```python\nimport matplotlib.pyplot as plt\nfrom cb_models import HHModel, NeuroDynModel, NeuronalNetwork, Synapse, NDSynapse\nfrom cb_models import AMPA\nfrom cb_models import NeuroDynBoard, NeuroCube\nfrom fitting_utilities import FitND\n\n# **Ignore overflow warnings**\nimport numpy as np\nnp.seterr(all=\"ignore\")\n\n%matplotlib inline\n%load_ext autoreload\n%autoreload 2\n```\n\n## Overview of neural interconnections\n\nThere are two main types of neural interconnections that we can implement: **gap junctions** and **synaptic connections**. In the conductance-based modelling framework, these interconnections lead to additional currents in the membrane equation of neuron $i$ that are dependent on membrane voltages of other neurons $j$ connected to it:\n\n\\begin{equation}\nC \\dot{V}_i = - I_{int,i} - \\sum_{j} I_{gap, i, j} - \\sum_{j} I_{syn, i, j} + I_{app, i}\n\\end{equation}\n\nwhere $I_{int,i}$ is total internal current of the neuron, $I_{gap, i, j}$ is the current due to the gap junction between neurons $j$ and $i$, $I_{syn, i, j}$ is the synaptic current from neuron $j$, and $I_{app, i}$ is the external current applied to neuron $i$.\n\n### Gap junctions\n\nGap junctions are modeled as simple resistive connections between neurons, so that a resistive connection between neurons $i$ and $j$ leads to a current in neuron $i$:\n\n\\begin{equation}\nI_{gap, i, j} = g_{gap, i, j} (V_i - V_j)\n\\end{equation}\n\nGap junction interconnections are symmetric $g_{gap, i, j} = g_{gap, j, i}$, so that:\n\n\\begin{equation}\nI_{gap, j, i} = -I_{gap, i, j}\n\\end{equation}\n\nand neuron $j$ receives the same current of the opposite sign.\n\nWe can represent this as a matrix $G_{gap}$ where each element $g_{gap, i, j}$ is the conductance of the connection between neurons $i$ and $j$. This matrix is necessarily **symmetric**, i.e. $G_{gap} = G^T_{gap}$.\n\n### Synaptic connections\n\nUnlike gap junctions, synaptic connections are directed connections and are not symmetric. If there is a synapse from neuron $j$ to neuron $i$, then the activity of the *presynaptic* neuron ($j$) modulates the activity of the *postsynaptic* neuron ($i$). Synaptic currents have a similar form to the internal ionic currents, with the difference that their activation depends on the activity of the presynaptic neuron. This is observed in the dependency of the opening rate $\\alpha_r$ on the presynaptic neuron voltage $V_j$:\n\n\\begin{equation}\nI_{syn, i, j} = \\bar{g}_{syn, i, j} r (V_i - E_{syn}) \\\\\n\\dot{r} = \\alpha_r(V_j) (1 - r) - \\beta_r(V_i) r\n\\end{equation}\n\nThe opening of the channels $\\alpha_r$ depends on the presynaptic neuron's voltage $V_j$, while the closing depends on the postsynaptic neuron's voltage $V_i$ (sometimes this relationship is modeled as constant, so that $\\beta_r = const$).\n\nThe reversal potential of the synapse $E_{syn}$ determines if the synapse is *excitatory* or *inhibitory*: excitatory synapses tend to increase the postsynaptic neuron's voltage, while the inhibitory synapses act to decrease the postynaptic voltage. On the other hand, dynamics of activation $r$ determines the timescale of the synaptic current, i.e. how fast it activates in relation to the changes in the presynaptic neuron.\n\n## Defining neural interconnections\n\nLet's now see how we can define these interconnections in the Python environment. We start with a simple example of two Hodgkin-Huxley neurons, which we can define as before:\n\n\n```python\nneuron1 = HHModel()\nneuron2 = HHModel()\n```\n\n### Gap junctions\n\nLet's start with a simple network consisting of a single gap junction between these two neurons with $g_{gap} = 0.1$. We do this by defining the matrix `G_gap` and passing it as a keyword argument along with a list of the neurons to our `NeuronalNetwork` class:\n\n\n```python\nneurons = [neuron1, neuron2]\ng = 0.1\nG_gap = [[0, g], [g, 0]] # needs to be symmetric\n\nnetwork = NeuronalNetwork(neurons, gap = G_gap)\n```\n\nWe simulate the network exactly as before by calling the method `simulate`. This time, the applied current function needs to return an array of external currents corresponding to each neuron in the `neurons` list. Additionally, we need to make sure that the initial condition array `x0` has the right number of states:\n\n\n```python\n# Simulation time\nT = 200\ntrange = (0, T)\n\n# External current\nI1 = 15 # current to neuron 1\nI2 = 0 # current to neuron 2\nIapp = lambda t: [I1, I2]\n\n# Initial condition\nx0 = [0, 0, 0, 0] * 2\n\n# Simulate\nsol = network.simulate(trange, x0, Iapp)\nV1 = sol.y[0]\nV2 = sol.y[4]\n\nplt.figure()\nplt.xlabel(\"Time [ms]\")\nplt.ylabel(\"Voltage [mV]\")\nplt.plot(sol.t, V1, label = \"Neuron 1\")\nplt.plot(sol.t, V2, label = \"Neuron 2\")\nplt.legend()\nplt.show()\n```\n\nYou can change the conductance of the gap junction and observe the different behavior. For very small values of the conductance, the neurons are effectively disconnected and act independently, while gradually increasing the conductance will bring the membrane voltages of the two neurons closer together.\n\nMaking $g_{gap}$ large effectively acts as the short-circuit between the two neurons (but be careful about making it too large as it may lead to numerical issues).\n\n### Synaptic connections\n\nLet's now see how we can introduce a synapse from neuron 1 to neuron 2. We first need to create an object for the synapse of the `Synapse` class which takes three arguments: `gsyn` - maximal conductance, `Esyn` - reversal potential, `r` - gating variable object.\n\nThe `r` gating variable should be derived from the class `HHKinetics` and needs to provide methods `alpha` and `beta` that define the functions $\\alpha_r(V)$ and $\\beta_r(V)$.\n\n#### AMPA synapse\n\nAs an example, let's take a look at implementing an excitatory AMPA synapse. There is an `AMPA` gating kinetics class already provided that takes the values from Ermentrout et al. 2010 that we can use here. Let's define a synapse using the `AMPA` kinetics class, setting the value of the reversal potential to $E_{AMPA} = 65mV$ (the reversal potential value is shifted to accomodate for the shift in the Hodgkin-Huxley model that sets $V_{rest} = 0mV$):\n\n\n```python\nr = AMPA()\ngsyn = 0.05\nEsyn = 65\nsynapse = Synapse(gsyn, Esyn, r)\n```\n\nHaving defined the model of the synapse that we would like to use, we now need to define the connectivity matrix describing all of the synaptic connections in the network. For this, we will construct a matrix `syns` where each element of the matrix $(i, j)$ contains a **list** of all synaptic connections from neuron $j$ to neuron $i$. If there are no synaptic connections from neuron $j$ to neuron $i$, then we set that matrix element to `None`.\n\nSince we would like to define a single excitatory synapse from neuron 1 to neuron 2, we will put a single `Synapse` object into a list at position (1, 0), filling the rest of the matrix with `None`. Let's create a new network with the single excitatory synapse and simulate it:\n\n\n```python\nsyns = [[None, None], [[synapse], None]]\n\nnetwork = NeuronalNetwork(neurons, syns = syns)\n\n# Simulation time\nT = 200\ntrange = (0, T)\n\n# External current\nI1 = 15 # current to neuron 1\nI2 = 0 # current to neuron 2\nIapp = lambda t: [I1, I2]\n\n# Initial condition\nx0 = [0, 0, 0, 0] * 2 + [0] # extra state due to the synapse!\n\n# Simulate\nsol = network.simulate(trange, x0, Iapp)\nV1 = sol.y[0]\nV2 = sol.y[4]\n\nplt.figure()\nplt.xlabel(\"Time [ms]\")\nplt.ylabel(\"Voltage [mV]\")\nplt.plot(sol.t, V1, label = \"Neuron 1\")\nplt.plot(sol.t, V2, label = \"Neuron 2\")\nplt.legend()\nplt.show()\n```\n\nWe can observe that each time the first neuron spikes it excites the second neuron. If the strength of the synapse is large enough, the second neuron will start spiking in response to the activity of the second neuron. We can play around with the synaptic strength and observe the different behavior:\n\n\n```python\nsynapse.gsyn = 0 # change the synapse strength\n\n# Simulate\nsol = network.simulate(trange, x0, Iapp)\nV1 = sol.y[0]\nV2 = sol.y[4]\n\nplt.figure()\nplt.xlabel(\"Time [ms]\")\nplt.ylabel(\"Voltage [mV]\")\nplt.plot(sol.t, V1, label = \"Neuron 1\")\nplt.plot(sol.t, V2, label = \"Neuron 2\")\nplt.legend()\nplt.show()\n```\n\n#### Defining different synapse models\n\nYou can use the `Synapse` class to define arbitrary synaptic connections - the main requirement is implementing a gating variable class inherited from the `HHKinetics` class that needs to define implementations for the methods `alpha(V)` and `beta(V)`. You can take a look at how the `AMPA` class is defined in the `cb_models` file to get a better idea.\n\n**Note** that since our original Hodgkin-Huxley model has parameter values shifted so that the resting potential is at $V_{rest} = 0mV$, you might need to shift biophyiscal values accordingly in order to make the synapse compatible with the model.\n\nAnother important thing to note is that when we try to fit activation dynamics to the NeuroDyn hardware, the values should be in **SI units**.\n\nFinally, if we would like to stretch the voltage range of the Hodgkin-Huxley model using the `scl_v` parameter, the range of the synapse would need to be stretched accordingly to accommodate for this. It is therefore best to implement an optional keyword `scl_v` to the gating variable class that would multiply the voltage-related parameters (you can check this for the `AMPA` class and the `HHModel` class in the file `cb_models`).\n\n## NeuroDyn network\n\nLet's see now how we can do the same using NeuroDyn models for the individual neurons, as well as the synapses.\n\n### NeuroDyn synapse\n\nA synaptic connection in the NeuroDyn hardware is implemented in very much the same way as ionic conductance dynamics was for a single neuron (see \"NeuroDyn equations\" section in **NeuroDyn Python model** notebook). Therefore, a NeuroDyn synapse class `NDSynapse` will be defined using the following digital parameters:\n\n- `dg`: digital value for the maximal conductance\n- `dE`: digital value for the reversal potential\n- `dIb`: digital values for the alpha and beta sigmoid weights\n - `dIb` is a list containing `[dIalpha, dIbeta]`\n - each `dIalpha` and `dIbeta` is an array of 7 digital values for the sigmoid weights\n- `ND`: optional keyword argument containing a `NeuroDynModel` object\n - the synapse will take all physical parameters such as `I_master` and `I_voltage` from this object, so it should be the 'parent' NeuroDyn neuron for the synapse\n - if no `ND` object is passed, `NDSynapse` will be initialized with a default `NeuroDynModel` object\n \nIn order to find these digital values we can again try fitting the parameters so that they replicate a particular biophysical synapse. As an example, let's try fitting the parameters to the AMPA synapse of the previous section.\n\n### Fitting a synapse\n\nIn order to fit a synapse model, we can again use the class `FitND` that we have previously used to fit a Hodgkin-Huxley model to a NeuroDyn neuron. We first fit an appropriate Hodgkin-Huxley model as before, after which we can pass any other gating variables that we would like to fit to. In this case, we want to fit the AMPA gating variable.\n\nFirst, let's repeat the procedure of fitting a NeuroDyn neuron as before:\n\n\n```python\n# Fit Hodgkin-Huxley model\n\nscl_v = 3 # voltage scaling\n\n# Hodgkin-Huxley model we want to fit to\nhh_neuron = HHModel(scl_v = scl_v, SI_units = True)\n\n# Create a fitting object and obtain the initial fit\nfit_obj = FitND(hh_neuron)\n\n# Get the sigmoid weights\nweights = fit_obj.fitHH()\n\ng0 = [120e-3,36e-3,0.3e-3] # maximal conductances\nE0 = [120e-3,-12e-3,10.6e-3] # reversal potentials\n\n# Get the digital parameters for NeuroDyn\ndIb = fit_obj.get_digital_Ib(weights)\ndg = fit_obj.get_digital_g(g0)\ndE = fit_obj.get_digital_E(E0)\ndIb[2][1] = dIb[2][1]*15\n\nI_master = fit_obj.I_master\nI_voltage = fit_obj.I_voltage\n\nV_ref = 0.9\n\n# Create a NeuroDyn object\nND1 = NeuroDynModel(dg, dE, dIb, V_ref, I_voltage, I_master)\n```\n\nNow, let's follow the same procedure in order to get the parameters for an AMPA synapse. We first make an AMPA gating variable (remember that it needs to be scaled appropriately, which we do by passing the `scl_v` keyword).\n\nWe can then pass the `AMPA` gating variable, along with any other gating variables we would like to fit to, to the `fit` method. Importantly, note that a **list** of gating variables should be passed as an argument, along with an optional list `labels` for the figure labels.\n\nAnother important thing to note is that each time we fit new gating variables using an existing `FitND` object, the `scl_t` variable determining the time scaling will adjust so that the maximal coefficient encountered so far fits within the NeuroDyn constraints. If we would like to fit gating variables without readjusting the scale, we can pass an optional keyword argument `update_scale = False`. This means that our fitting process might return parameters that are outside the digital range. Here we will keep the default `update_scale` that is set to `True`.\n\n\n```python\n# Fit AMPA gating variable\nr = AMPA(scl_v = scl_v, SI_units = True)\n\n# Fit weights to AMPA\nweights_syn = fit_obj.fit([r], labels = ['r'], plot_alpha_beta = True) # note that gating variables are passed in a list\n\nweights_syn = weights_syn[0] # fitting method returns a list of gating variables, so get the only element\n```\n\nHaving obtained the weights for the gating variable kinetics, we can then get the digital parameters for the maximal conductance and the reversal potential of the synapse, and use that to create an `NDSynapse` object. Apart from the maximal conductance, reversal potential, and the gating variable, the `NDSynapse` object also accepts a reference to a NeuroDyn object from which it takes physical constants and currents $I_{voltage}$ and $I_{master}$.\n\n\n```python\ngsyn = 0.3e-3 # max coductance of the synapse\nEsyn = 65e-3 # reversal potential of the synapse\n\ndIb_syn = fit_obj.get_digital_Ib(weights_syn)\ndgsyn = fit_obj.get_digital_g(gsyn)\ndEsyn = fit_obj.get_digital_E(Esyn)\n\nnd_syn = NDSynapse(dgsyn, dEsyn, dIb_syn, ND1)\n\nprint(\"Alpha and beta weights:\\n\", dIb_syn)\nprint(\"Digital value for maximal conductance: \", dgsyn)\nprint(\"Digital value for the reversal potential: \", dEsyn)\n```\n\n### Creating a network\n\nHaving created an object for our synapse, we can now define a network in exactly the same way as before! Let's define a second neuron `ND2` with the same parameters, but let's decrease the applied current to this neuron in order to make it silent. For convenience, we can `deepcopy` our first neuron, so that the second neuron has the same parameters but does not share any variables with the first neuron. Let's set the appropriate current $I_{app}$ so that the neuron is silent:\n\n\n```python\nfrom copy import deepcopy\n\nND2 = deepcopy(ND1) # makes an independent copy of the ND1 object\n\n# Set the applied current for ND2 to be silent\nI2 = -1e-6\nIapp = lambda t : fit_obj.convert_I(I2)\n\nV_ref = 0.9\n\nT = 0.02\ntrange = (0, T)\n\nsol = ND2.simulate(trange,[0.7,0,0,0],Iapp)\n\nplt.figure()\nplt.xlabel('t [s]')\nplt.ylabel('V [V]')\nplt.title('ND 2')\nplt.plot(sol.t, sol.y[0])\nplt.show()\n```\n\nNow that we have our two neurons `ND1` and `ND2`, let's set up the excitatory synapse we have created from `ND1` to `ND2` so that the first neuron spiking excites the resting second neuron.\n\n\n```python\nnd_neurons = [ND1, ND2]\n\nnd_syns = [[None, None], [[nd_syn], None]]\n\nnd_network = NeuronalNetwork(nd_neurons, syns = nd_syns)\n\n# Simulation time\nT = 0.04\ntrange = (0, T)\n\n# External currents\nI1 = fit_obj.convert_I(0e-6) # current to neuron 1\nI2 = fit_obj.convert_I(-1e-6) # current to neuron 2\nIapp = lambda t: [I1, I2]\n\n# Initial condition\nx0 = [0.7, 0, 0, 0] * 2 + [0] # extra state due to the synapse\n\n# Simulate\nsol = nd_network.simulate(trange, x0, Iapp)\nV1 = sol.y[0]\nV2 = sol.y[4]\n\nplt.figure()\nplt.xlabel('t [s]')\nplt.ylabel('V [V]')\nplt.plot(sol.t, V1, label = \"Neuron 1\")\nplt.plot(sol.t, V2, label = \"Neuron 2\")\nplt.legend()\nplt.show()\n```\n\nYou may notice that after the initial transient has died out, individual spikes in the first neuron induce excitations in the voltage of the second neuron. We can try changing the digital value of the maximal conductance of the synapse to observe its effect on the activity of the second neuron. You should observe that by increasing the synaptic strength the second neuron eventually starts spiking, and its spikes appear more closely to the spikes of the first neuron as it is further increased.\n\n\n```python\n# Change synapse strength\nnd_syn.update_dg(0)\n\n# Simulate\nsol = nd_network.simulate(trange, x0, Iapp)\nV1 = sol.y[0]\nV2 = sol.y[4]\n\nplt.figure()\nplt.xlabel('t [s]')\nplt.ylabel('V [V]')\nplt.plot(sol.t, V1, label = \"Neuron 1\")\nplt.plot(sol.t, V2, label = \"Neuron 2\")\nplt.legend()\nplt.show()\n```\n\n## NeuroDyn board classes\n\nSo far, we have seen how to define a network of Hodgkin-Huxley or NeuroDyn neurons. The arguments of the network are a full list of the individual neural objects, a conductance matrix defining the gap junction connections, and a synapse matrix defining the synapse connections of the circuit. Using the `NeuronalNetwork` class we can define arbitrary neural networks given by the structure of the gap junction and synapse matrices.\n\nThe NeuroDyn hardware however has a fixed structure: a single **NeuroDyn board** consists of **4 neurons** with **12 synapses** defining a network of four fully-connected neurons. The class `NeuroDynBoard` defines a network with this particular architecture, allowing in addition to optionally connect neurons with resistive connections.\n\nBy connecting four NeuroDyn boards in parallel through resistive connections, we get a **NeuroCube**. The NeuroCube consists of four NeuroDyn boards stacked on top of each other, so that Neuron 1 of each board can be connected to corresponding Neuron 1 of every other board, Neuron 2 of every board can be connected to every correspodning Neuron 2, and so on, giving in total 24 additional resistive connections that can be controlled. This network is represented by the `NeuroCube` class.\n\nLet's see how we can use these classes.\n\n### NeuroDynBoard\n\nWe can create a single NeuroDyn board object by calling the class `NeuroDynBoard`, which will initialize all neurons and synapses with default parameters, as well as set conductance of all resistive connections to $0$.\n\n\n```python\nnd_board = NeuroDynBoard()\n```\n\nLet's try simulating the board - since all synapses and gap junctions are initialized with $0$ conductance, we should observe four neurons operating individually. Note that we can reuse are previously defined fitting object in order to convert currents to the right scale.\n\n\n```python\n# Simulation time\nT = 0.02\ntrange = (0, T)\n\n# External currents\nI1 = fit_obj.convert_I(-2e-6) # silent\nI2 = fit_obj.convert_I(-1e-6) # silent\nI3 = fit_obj.convert_I(0e-6) # spiking\nI4 = fit_obj.convert_I(2e-6) # spiking\nIapp = lambda t: [I1, I2, I3, I4]\n\n# Initial conditions\nx0 = [0.7, 0, 0, 0] * 4 + [0]*12 # 4 neurons, 12 synapses\n\n# Simulate\nsol = nd_board.simulate(trange, x0, Iapp)\nV1 = sol.y[0]\nV2 = sol.y[4]\nV3 = sol.y[8]\nV4 = sol.y[12]\n\nplt.figure()\nplt.title(\"NeuroDyn Board\")\nplt.xlabel('t [s]')\nplt.ylabel('V [V]')\nplt.plot(sol.t, V1, label = \"Neuron 1\")\nplt.plot(sol.t, V2, label = \"Neuron 2\")\nplt.plot(sol.t, V3, label = \"Neuron 3\")\nplt.plot(sol.t, V4, label = \"Neuron 4\")\nplt.legend()\nplt.show()\n```\n\nWe can change the parameters of the circuit by accessing individual neurons and synapses. We can do this by using the following methods:\n\n- `get_neuron(i)`: retrive neuron $i$ ($i$ goes from $0$ to $3$)\n- `get_syn(i, j)`: retrieve synapse from neuron $j$ to neuron $i$\n\nWe can then use the neural and synaptic class methods to update the parameters. In addition we can set the gap junction conductances by calling:\n\n- `set_gap(g, i, j)`: set the maximal conductance of the $(i, j)$ gap junction to $g$\n\nAs a test, let's see how we can create a network where neurons excite each other sequentially 1 -> 2 -> 3 -> 4. The default parameters are set to an excitatory synapse, so we only need to increase the maximal conductances of the appropriate connections.\n\n\n```python\n# Get synapses\nsyn1 = nd_board.get_syn(1, 0) # synapse 1 -> 2\nsyn2 = nd_board.get_syn(2, 1) # synapse 2 -> 3\nsyn3 = nd_board.get_syn(3, 2) # synapse 3 -> 4\n\n# Update synapses\nsyn1.update_dg(5)\nsyn2.update_dg(5)\nsyn3.update_dg(2)\n\n# External currents (set neurons 2, 3, 4 to silent)\nI1 = -0e-6\nI2 = -1e-6\nI3 = -1e-6\nI4 = -1e-6\n\nIapp = lambda t: fit_obj.convert_I([I1, I2, I3, I4])\n\n# Initial conditions\nx0 = [0.7, 0, 0, 0] * 4 + [0]*12 # 4 neurons, 12 synapses\n\n# Simulate\nsol = nd_board.simulate(trange, x0, Iapp)\nV1 = sol.y[0]\nV2 = sol.y[4]\nV3 = sol.y[8]\nV4 = sol.y[12]\n\nplt.figure()\nplt.title(\"NeuroDyn Board with excitatory synapses\")\nplt.xlabel('t [s]')\nplt.ylabel('V [V]')\nplt.plot(sol.t, V1, label = \"Neuron 1\")\nplt.plot(sol.t, V2, label = \"Neuron 2\")\nplt.plot(sol.t, V3, label = \"Neuron 3\")\nplt.plot(sol.t, V4, label = \"Neuron 4\")\nplt.legend()\nplt.show()\n```\n\nYou can change the maximal conductance parameters of the synapses to make sure the network behaves as expected (for example turning the synapses on or off).\n\n### NeuroCube\n\nIn exactly the same manner, we can define an object of the `NeuroCube` class which will initialize 4 disconnected NeuroDyn boards with the default parameters. Let's first set up a default neurocube object and try simulating it. We can for example set the applied current to silence all neurons apart from the neurons on the third board:\n\n\n```python\nneurocube = NeuroCube()\n\nI_silent = [-1e-6, -1e-6, -1e-6, -1e-6] # all neurons silent\nI3 = [0e-6, 1e-6, 2e-6, 3e-6] # all neurons spiking\nItotal = I_silent * 2 + I3 + I_silent # set the neurons of board 3 to spike\n\nIapp = lambda t: fit_obj.convert_I(Itotal)\n\nx0 = [0.7, 0, 0, 0] * 16 + [0]*48 # 16 neurons, 48 synapses\n\nsol = neurocube.simulate(trange, x0, Iapp)\n\nfor i in range(4):\n V1 = sol.y[16 * i + 0]\n V2 = sol.y[16 * i + 4]\n V3 = sol.y[16 * i + 8]\n V4 = sol.y[16 * i + 12]\n\n plt.figure()\n plt.title(f\"NeuroDyn Board {i + 1}\")\n plt.xlabel('t [s]')\n plt.ylabel('V [V]')\n plt.plot(sol.t, V1, label = \"Neuron 1\")\n plt.plot(sol.t, V2, label = \"Neuron 2\")\n plt.plot(sol.t, V3, label = \"Neuron 3\")\n plt.plot(sol.t, V4, label = \"Neuron 4\")\n plt.legend()\n\nplt.show()\n```\n\nIf we would like to change the parameters of an individual board we can use the method `get_board(i)` which returns the board with the index $i$, after which we can use the methods described in **NeuroDynBoard** section. For example:\n\n`board1 = neurocube.get_board(0)`\n\nAn additional method that the `NeuroCube` class provides is `connect_boards(board_i, board_j, neuron_no, g)` which sets the resistive connection between neuron numbered `neuron_no` on board `board_i` and neuron numbered `neuron_no` on board `board_j` with conductance `g`.\n\nLet's for example connect boards 3 and 1 through neuron 1, and boards 3 and 4 through neuron 2.\n\n\n```python\ng = 1e-8\nneurocube.connect_boards(2, 0, 0, g) # boards 3 and 1 connected through neuron 1\nneurocube.connect_boards(2, 3, 1, g) # boards 3 and 4 connected through neuron 2\n\nsol = neurocube.simulate(trange, x0, Iapp)\n\nfor i in range(4):\n V1 = sol.y[16 * i + 0]\n V2 = sol.y[16 * i + 4]\n V3 = sol.y[16 * i + 8]\n V4 = sol.y[16 * i + 12]\n\n plt.figure()\n plt.title(f\"NeuroDyn board {i + 1}\")\n plt.xlabel('t [s]')\n plt.ylabel('V [V]')\n plt.plot(sol.t, V1, label = \"Neuron 1\")\n plt.plot(sol.t, V2, label = \"Neuron 2\")\n plt.plot(sol.t, V3, label = \"Neuron 3\")\n plt.plot(sol.t, V4, label = \"Neuron 4\")\n plt.legend()\n\nplt.show()\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "1ef1901e770d01fc6ab2e970c2fc9e8fe19d1403", "size": 33456, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Networks in NeuroDyn.ipynb", "max_stars_repo_name": "3x10e8/telluride-21", "max_stars_repo_head_hexsha": "9286d9d2ad649a2d82ac44a22d6dfe1066aa6fa5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Networks in NeuroDyn.ipynb", "max_issues_repo_name": "3x10e8/telluride-21", "max_issues_repo_head_hexsha": "9286d9d2ad649a2d82ac44a22d6dfe1066aa6fa5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Networks in NeuroDyn.ipynb", "max_forks_repo_name": "3x10e8/telluride-21", "max_forks_repo_head_hexsha": "9286d9d2ad649a2d82ac44a22d6dfe1066aa6fa5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2021-07-05T23:04:54.000Z", "max_forks_repo_forks_event_max_datetime": "2021-09-28T16:08:06.000Z", "avg_line_length": 40.700729927, "max_line_length": 624, "alphanum_fraction": 0.6173182688, "converted": true, "num_tokens": 6885, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434978390747, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.4190054953322705}} {"text": "\n\n\n# The BSSN Formulation of General Relativity in Generic Curvilinear Coordinates: An Overview\n\n## Author: Zach Etienne\n\n## This tutorial notebook demonstrates how Einstein's equations of general relativity in this formulation are constructed and output within NRPy+.\n\n\n### As Einstein's equations in this formalism take the form of highly nonlinear, coupled *wave equations*, the [tutorial notebook on the scalar wave equation in curvilinear coordinates](Tutorial-ScalarWaveCurvilinear.ipynb) is *required* reading before beginning this module. That module, as well as its own prerequisite [module on reference metrics within NRPy+](Tutorial-Reference_Metric.ipynb) provides the needed overview of how NRPy+ handles reference metrics.\n\n## Introduction:\nNRPy+'s original purpose was to be an easy-to-use code capable of generating Einstein's equations in a broad class of [singular](https://en.wikipedia.org/wiki/Coordinate_singularity), curvilinear coordinate systems, where the user need only input the scale factors of the underlying reference metric. Upon generating these equations, NRPy+ would then leverage SymPy's [common-expression-elimination (CSE)](https://en.wikipedia.org/wiki/Common_subexpression_elimination) and C code generation routines, coupled to its own [single-instruction, multiple-data (SIMD)](https://en.wikipedia.org/wiki/SIMD) functions, to generate highly-optimized C code.\n\n### Background Reading/Lectures:\n\n* Mathematical foundations of BSSN and 3+1 initial value problem decompositions of Einstein's equations:\n * [Thomas Baumgarte's lectures on mathematical formulation of numerical relativity](https://www.youtube.com/watch?v=t3uo2R-yu4o&list=PLRVOWML3TL_djTd_nsTlq5aJjJET42Qke)\n * [Yuichiro Sekiguchi's introduction to BSSN](http://www2.yukawa.kyoto-u.ac.jp/~yuichiro.sekiguchi/3+1.pdf) \n* Extensions to the standard BSSN approach used in NRPy+\n * [Brown's covariant \"Lagrangian\" formalism of BSSN](https://arxiv.org/abs/0902.3652)\n * [BSSN in spherical coordinates, using the reference-metric approach of Baumgarte, Montero, Cordero-Carri\u00f3n, and M\u00fcller (2012)](https://arxiv.org/abs/1211.6632)\n * [BSSN in generic curvilinear coordinates, using the extended reference-metric approach of Ruchlin, Etienne, and Baumgarte (2018)](https://arxiv.org/abs/1712.07658)\n\n### A Note on Notation:\n\nAs is standard in NRPy+, \n\n* Greek indices refer to four-dimensional quantities where the zeroth component indicates temporal (time) component.\n* Latin indices refer to three-dimensional quantities. This is somewhat counterintuitive since Python always indexes its lists starting from 0. As a result, the zeroth component of three-dimensional quantities will necessarily indicate the first *spatial* direction.\n\nAs a corollary, any expressions involving mixed Greek and Latin indices will need to offset one set of indices by one: A Latin index in a four-vector will be incremented and a Greek index in a three-vector will be decremented (however, the latter case does not occur in this tutorial notebook).\n\n\n\n# Table of Contents\n$$\\label{toc}$$\n\nThis module lays out the mathematical foundation for the BSSN formulation of Einstein's equations, as detailed in the references in the above Background Reading/Lectures section. It is meant to provide an overview of the basic equations and point of reference for **full tutorial notebooks** linked below:\n\n1. [Step 1](#brownslagrangebssn): [Brown](https://arxiv.org/abs/0902.3652)'s covariant formulation of the BSSN time-evolution equations (see next section for gauge conditions)\n 1. [Step 1.a](#fullequations): Numerical implementation of BSSN time-evolution equations\n 1. [Step 1.a.i](#liederivs) ([**BSSN quantities module [start here]**](Tutorial-BSSN_quantities.ipynb); [**BSSN time-evolution module**](Tutorial-BSSN_time_evolution-BSSN_RHSs.ipynb)): Expanding the Lie derivatives; the BSSN time-evolution equations in their final form\n1. [Step 2](#gaugeconditions) ([**full tutorial notebook**](Tutorial-BSSN_time_evolution-BSSN_gauge_RHSs.ipynb)): Time-evolution equations for the BSSN |gauge quantities $\\alpha$ and $\\beta^i$\n1. [Step 3](#constraintequations) ([**full tutorial notebook**](Tutorial-BSSN_constraints.ipynb)): The BSSN constraint equations\n 1. [Step 3.a](#hamiltonianconstraint): The Hamiltonian constraint\n 1. [Step 3.b](#momentumconstraint): The momentum constraint\n1. [Step 4](#gammaconstraint) ([**full tutorial notebook**](Tutorial-BSSN-Enforcing_Determinant_gammabar_equals_gammahat_Constraint.ipynb)): The BSSN algebraic constraint $\\hat{\\gamma}=\\bar{\\gamma}$\n1. [Step 5](#latex_pdf_output) Output this notebook to $\\LaTeX$-formatted PDF file\n\n\n\n# Step 1: [Brown](https://arxiv.org/abs/0902.3652)'s covariant formulation of BSSN \\[Back to [top](#toc)\\]\n$$\\label{brownslagrangebssn}$$\n\nThe covariant \"Lagrangian\" BSSN formulation of [Brown (2009)](https://arxiv.org/abs/0902.3652), which requires\n\n$$\n\\partial_t \\bar{\\gamma} = 0,\n$$\n\nresults in the BSSN equations taking the following form (Eqs. 11 and 12 in [Ruchlin, Etienne, and Baumgarte (2018)](https://arxiv.org/abs/1712.07658)):\n\n\\begin{align}\n \\partial_{\\perp} \\bar{\\gamma}_{i j} {} = {} & \\frac{2}{3} \\bar{\\gamma}_{i j} \\left (\\alpha \\bar{A}_{k}^{k} - \\bar{D}_{k} \\beta^{k}\\right ) - 2 \\alpha \\bar{A}_{i j} \\; , \\\\\n \\partial_{\\perp} \\bar{A}_{i j} {} = {} & -\\frac{2}{3} \\bar{A}_{i j} \\bar{D}_{k} \\beta^{k} - 2 \\alpha \\bar{A}_{i k} {\\bar{A}^{k}}_{j} + \\alpha \\bar{A}_{i j} K \\nonumber \\\\\n & + e^{-4 \\phi} \\left \\{-2 \\alpha \\bar{D}_{i} \\bar{D}_{j} \\phi + 4 \\alpha \\bar{D}_{i} \\phi \\bar{D}_{j} \\phi \\right . \\nonumber \\\\\n & \\left . + 4 \\bar{D}_{(i} \\alpha \\bar{D}_{j)} \\phi - \\bar{D}_{i} \\bar{D}_{j} \\alpha + \\alpha \\bar{R}_{i j} \\right \\}^{\\text{TF}} \\; , \\\\\n \\partial_{\\perp} \\phi {} = {} & \\frac{1}{6} \\left (\\bar{D}_{k} \\beta^{k} - \\alpha K \\right ) \\; , \\\\\n \\partial_{\\perp} K {} = {} & \\frac{1}{3} \\alpha K^{2} + \\alpha \\bar{A}_{i j} \\bar{A}^{i j} \\nonumber \\\\\n & - e^{-4 \\phi} \\left (\\bar{D}_{i} \\bar{D}^{i} \\alpha + 2 \\bar{D}^{i} \\alpha \\bar{D}_{i} \\phi \\right ) \\; , \\\\\n \\partial_{\\perp} \\bar{\\Lambda}^{i} {} = {} & \\bar{\\gamma}^{j k} \\hat{D}_{j} \\hat{D}_{k} \\beta^{i} + \\frac{2}{3} \\Delta^{i} \\bar{D}_{j} \\beta^{j} + \\frac{1}{3} \\bar{D}^{i} \\bar{D}_{j} \\beta^{j} \\nonumber \\\\\n & - 2 \\bar{A}^{i j} \\left (\\partial_{j} \\alpha - 6 \\partial_{j} \\phi \\right ) + 2 \\bar{A}^{j k} \\Delta_{j k}^{i} \\nonumber \\\\\n & -\\frac{4}{3} \\alpha \\bar{\\gamma}^{i j} \\partial_{j} K \\\\\n\\end{align}\nwhere \n* the $\\text{TF}$ superscript denotes the trace-free part.\n* $\\bar{\\gamma}_{ij} = \\varepsilon_{i j} + \\hat{\\gamma}_{ij}$, where $\\bar{\\gamma}_{ij} = e^{-4\\phi} \\gamma_{ij}$ is the conformal metric, $\\gamma_{ij}$ is the physical metric (see below), and $\\varepsilon_{i j}$ encodes information about the non-hatted metric.\n* $\\gamma_{ij}$, $\\beta^i$, and $\\alpha$ are the physical (as opposed to conformal) spatial 3-metric, shift vector, and lapse, respectively, which may be defined via the 3+1 decomposition line element (in [$G=c=1$ units](https://en.wikipedia.org/wiki/Planck_units)):\n$$ds^2 = -\\alpha^2 dt^2 + \\gamma_{ij}\\left(dx^i + \\beta^i dt\\right)\\left(dx^j + \\beta^j dt\\right).$$\n* $\\bar{R}_{ij}$ is the conformal Ricci tensor, computed via\n\\begin{align}\n \\bar{R}_{i j} {} = {} & - \\frac{1}{2} \\bar{\\gamma}^{k l} \\hat{D}_{k} \\hat{D}_{l} \\bar{\\gamma}_{i j} + \\bar{\\gamma}_{k(i} \\hat{D}_{j)} \\bar{\\Lambda}^{k} + \\Delta^{k} \\Delta_{(i j) k} \\nonumber \\\\\n & + \\bar{\\gamma}^{k l} \\left (2 \\Delta_{k(i}^{m} \\Delta_{j) m l} + \\Delta_{i k}^{m} \\Delta_{m j l} \\right ) \\; .\n\\end{align}\n* $\\partial_{\\perp} = \\partial_t - \\mathcal{L}_\\beta$; $\\mathcal{L}_\\beta$ is the [Lie derivative](https://en.wikipedia.org/wiki/Lie_derivative) along the shift vector $\\beta^i$.\n* $\\partial_0 = \\partial_t - \\beta^i \\partial_i$ is an advective time derivative.\n* $\\hat{D}_j$ is the [covariant derivative](https://en.wikipedia.org/wiki/Covariant_derivative) with respect to the reference metric $\\hat{\\gamma}_{ij}$.\n* $\\bar{D}_j$ is the [covariant derivative](https://en.wikipedia.org/wiki/Covariant_derivative) with respect to the barred spatial 3-metric $\\bar{\\gamma}_{ij}$\n* $\\Delta^i_{jk}$ is the tensor constructed from the difference of barred and hatted Christoffel symbols:\n$$\\Delta^i_{jk} = \\bar{\\Gamma}^i_{jk} - \\hat{\\Gamma}^i_{jk}$$\n * The related quantity $\\Delta^i$ is defined $\\Delta^i \\equiv \\bar{\\gamma}^{jk} \\Delta^i_{jk}$.\n* $\\bar{A}_{ij}$ is the conformal, trace-free extrinsic curvature: \n$$\\bar{A}_{ij} = e^{-4\\phi} \\left(K_{ij} - \\frac{1}{3}\\gamma_{ij} K\\right),$$\nwhere $K$ is the trace of the extrinsic curvature $K_{ij}$.\n\n\n\n## Step 1.a: Numerical implementation of BSSN time-evolution equations \\[Back to [top](#toc)\\]\n$$\\label{fullequations}$$\n\nRegarding the numerical implementation of the above equations, first notice the left-hand sides of the equations include the time derivatives. Numerically, these equations are solved using as an [initial value problem](https://en.wikipedia.org/wiki/Initial_value_problem), where data are specified at a given time $t$, so that the solution at any later time can be obtained using the [Method of Lines (MoL)](https://en.wikipedia.org/wiki/Method_of_lines). MoL requires that the equations be written in the form:\n\n$$\\partial_t \\vec{U} = \\vec{f}\\left(\\vec{U},\\partial_i \\vec{U}, \\partial_i \\partial_j \\vec{U},...\\right),$$\n\nfor the vector of \"evolved quantities\" $\\vec{U}$, where the right-hand side vector $\\vec{f}$ *does not* contain *explicit* time derivatives of $\\vec{U}$.\n\nThus we must first rewrite the above equations so that *only* partial derivatives of time appear on the left-hand sides of the equations, meaning that the Lie derivative terms must be moved to the right-hand sides of the equations.\n\n\n\n### Step 1.a.i: Expanding the Lie derivatives; BSSN equations in their final form \\[Back to [top](#toc)\\]\n$$\\label{liederivs}$$\n\nIn this Step, we provide explicit expressions for the [Lie derivatives](https://en.wikipedia.org/wiki/Lie_derivative) $\\mathcal{L}_\\beta$ appearing inside the $\\partial_\\perp = \\partial_t - \\mathcal{L}_\\beta$ operators for $\\left\\{\\bar{\\gamma}_{i j},\\bar{A}_{i j},W, K, \\bar{\\Lambda}^{i}\\right\\}$.\n\nIn short, the Lie derivative of tensor weight $w$ is given by (from [the wikipedia article on Lie derivatives](https://en.wikipedia.org/wiki/Lie_derivative))\n\\begin{align}\n(\\mathcal {L}_X T) ^{a_1 \\ldots a_r}{}_{b_1 \\ldots b_s} &= X^c(\\partial_c T^{a_1 \\ldots a_r}{}_{b_1 \\ldots b_s}) \\\\\n&\\quad - (\\partial_c X ^{a_1}) T ^{c a_2 \\ldots a_r}{}_{b_1 \\ldots b_s} - \\ldots - (\\partial_c X^{a_r}) T ^{a_1 \\ldots a_{r-1}c}{}_{b_1 \\ldots b_s} \\\\\n &\\quad + (\\partial_{b_1} X^c) T ^{a_1 \\ldots a_r}{}_{c b_2 \\ldots b_s} + \\ldots + (\\partial_{b_s} X^c) T ^{a_1 \\ldots a_r}{}_{b_1 \\ldots b_{s-1} c} + w (\\partial_{c} X^c) T ^{a_1 \\ldots a_r}{}_{b_1 \\ldots b_{s}}\n\\end{align}\n\nThus to evaluate the Lie derivative, one must first know the tensor density weight $w$ for each tensor. In this formulation of Einstein's equations, **all evolved quantities have density weight $w=0$**, so according to the definition of Lie derivative above,\n\\begin{align}\n\\mathcal{L}_\\beta \\bar{\\gamma}_{ij} &= \\beta^k \\partial_k \\bar{\\gamma}_{ij} + \\partial_i \\beta^k \\bar{\\gamma}_{kj} + \\partial_j \\beta^k \\bar{\\gamma}_{ik}, \\\\\n\\mathcal{L}_\\beta \\bar{A}_{ij} &= \\beta^k \\partial_k \\bar{A}_{ij} + \\partial_i \\beta^k \\bar{A}_{kj} + \\partial_j \\beta^k \\bar{A}_{ik}, \\\\\n\\mathcal{L}_\\beta \\phi &= \\beta^k \\partial_k \\phi, \\\\\n\\mathcal{L}_\\beta K &= \\beta^k \\partial_k K, \\\\\n\\mathcal{L}_\\beta \\bar{\\Lambda}^i &= \\beta^k \\partial_k \\bar{\\Lambda}^i - \\partial_k \\beta^i \\bar{\\Lambda}^k\n\\end{align}\n\nWith these definitions, the BSSN equations for the un-rescaled evolved variables in the form $\\partial_t \\vec{U} = f\\left(\\vec{U},\\partial_i \\vec{U}, \\partial_i \\partial_j \\vec{U},...\\right)$ become\n\n\\begin{align}\n \\partial_t \\bar{\\gamma}_{i j} {} = {} & \\left[\\beta^k \\partial_k \\bar{\\gamma}_{ij} + \\partial_i \\beta^k \\bar{\\gamma}_{kj} + \\partial_j \\beta^k \\bar{\\gamma}_{ik} \\right] + \\frac{2}{3} \\bar{\\gamma}_{i j} \\left (\\alpha \\bar{A}_{k}^{k} - \\bar{D}_{k} \\beta^{k}\\right ) - 2 \\alpha \\bar{A}_{i j} \\; , \\\\\n \\partial_t \\bar{A}_{i j} {} = {} & \\left[\\beta^k \\partial_k \\bar{A}_{ij} + \\partial_i \\beta^k \\bar{A}_{kj} + \\partial_j \\beta^k \\bar{A}_{ik} \\right] - \\frac{2}{3} \\bar{A}_{i j} \\bar{D}_{k} \\beta^{k} - 2 \\alpha \\bar{A}_{i k} {\\bar{A}^{k}}_{j} + \\alpha \\bar{A}_{i j} K \\nonumber \\\\\n & + e^{-4 \\phi} \\left \\{-2 \\alpha \\bar{D}_{i} \\bar{D}_{j} \\phi + 4 \\alpha \\bar{D}_{i} \\phi \\bar{D}_{j} \\phi + 4 \\bar{D}_{(i} \\alpha \\bar{D}_{j)} \\phi - \\bar{D}_{i} \\bar{D}_{j} \\alpha + \\alpha \\bar{R}_{i j} \\right \\}^{\\text{TF}} \\; , \\\\\n \\partial_t \\phi {} = {} & \\left[\\beta^k \\partial_k \\phi \\right] + \\frac{1}{6} \\left (\\bar{D}_{k} \\beta^{k} - \\alpha K \\right ) \\; , \\\\\n \\partial_{t} K {} = {} & \\left[\\beta^k \\partial_k K \\right] + \\frac{1}{3} \\alpha K^{2} + \\alpha \\bar{A}_{i j} \\bar{A}^{i j} - e^{-4 \\phi} \\left (\\bar{D}_{i} \\bar{D}^{i} \\alpha + 2 \\bar{D}^{i} \\alpha \\bar{D}_{i} \\phi \\right ) \\; , \\\\\n \\partial_t \\bar{\\Lambda}^{i} {} = {} & \\left[\\beta^k \\partial_k \\bar{\\Lambda}^i - \\partial_k \\beta^i \\bar{\\Lambda}^k \\right] + \\bar{\\gamma}^{j k} \\hat{D}_{j} \\hat{D}_{k} \\beta^{i} + \\frac{2}{3} \\Delta^{i} \\bar{D}_{j} \\beta^{j} + \\frac{1}{3} \\bar{D}^{i} \\bar{D}_{j} \\beta^{j} \\nonumber \\\\\n & - 2 \\bar{A}^{i j} \\left (\\partial_{j} \\alpha - 6 \\partial_{j} \\phi \\right ) + 2 \\alpha \\bar{A}^{j k} \\Delta_{j k}^{i} -\\frac{4}{3} \\alpha \\bar{\\gamma}^{i j} \\partial_{j} K\n\\end{align}\n\nwhere the terms moved from the right-hand sides to the left-hand sides are enclosed in square braces. \n\nNotice that the shift advection operator $\\beta^k \\partial_k \\left\\{\\bar{\\gamma}_{i j},\\bar{A}_{i j},\\phi, K, \\bar{\\Lambda}^{i}, \\alpha, \\beta^i, B^i\\right\\}$ appears on the right-hand side of *every* expression. As the shift determines how the spatial coordinates $x^i$ move on the next 3D slice of our 4D manifold, we find that representing $\\partial_k$ in these shift advection terms via an *upwinded* finite difference stencil results in far lower numerical errors. This trick is implemented below in all shift advection terms.\n\nAs discussed in the [NRPy+ tutorial notebook on BSSN quantities](Tutorial-BSSN_quantities.ipynb), tensorial expressions can diverge at coordinate singularities, so each tensor in the set of BSSN variables\n\n$$\\left\\{\\bar{\\gamma}_{i j},\\bar{A}_{i j},\\phi, K, \\bar{\\Lambda}^{i}, \\alpha, \\beta^i, B^i\\right\\},$$\n\nis written in terms of the corresponding rescaled quantity in the set\n\n$$\\left\\{h_{i j},a_{i j},\\text{cf}, K, \\lambda^{i}, \\alpha, \\mathcal{V}^i, \\mathcal{B}^i\\right\\},$$ \n\nrespectively, as defined in the [BSSN quantities tutorial](Tutorial-BSSN_quantities.ipynb). \n\n\n\n# Step 2: Time-evolution equations for the BSSN gauge quantities $\\alpha$ and $\\beta^i$ \\[Back to [top](#toc)\\]\n$$\\label{gaugeconditions}$$\n\nAs described in the **Background Reading/Lectures** linked to above, the gauge quantities $\\alpha$ and $\\beta^i$ specify how coordinate time and spatial points adjust from one spatial hypersurface to the next, in our 3+1 decomposition of Einstein's equations.\n\nAs choosing $\\alpha$ and $\\beta^i$ is equivalent to choosing coordinates for where we sample our solution to Einstein's equations, we are completely free to choose $\\alpha$ and $\\beta^i$ on any given spatial hypersuface. It has been found that fixing $\\alpha$ and $\\beta^i$ to constant values in the context of dynamical spacetimes results in instabilities, so we generally need to define expressions for $\\partial_t \\alpha$ and $\\partial_t \\beta^i$ and couple these equations to the rest of the BSSN time-evolution equations.\n\nThough we are free to choose the form of the right-hand sides of the gauge time evolution equations, very few have been found robust in the presence of (puncture) black holes.\n\nThe most commonly adopted gauge conditions for BSSN (i.e., time-evolution equations for the BSSN gauge quantities $\\alpha$ and $\\beta^i$) are the \n\n* $1+\\log$ lapse condition:\n$$\n\\partial_0 \\alpha = -2 \\alpha K\n$$\n* Second-order Gamma-driving shift condition:\n\\begin{align}\n\\partial_0 \\beta^i &= B^{i} \\\\\n\\partial_0 B^i &= \\frac{3}{4} \\partial_{0} \\bar{\\Lambda}^{i} - \\eta B^{i},\n\\end{align}\n\nwhere $\\partial_0$ is the advection operator; i.e., $\\partial_0 A^i = \\partial_t A^i - \\beta^j \\partial_j A^i$. Note that $\\partial_{0} \\bar{\\Lambda}^{i}$ in the right-hand side of the $\\partial_{0} B^{i}$ equation is computed by adding $\\beta^j \\partial_j \\bar{\\Lambda}^i$ to the right-hand side expression given for $\\partial_t \\bar{\\Lambda}^i$, so no explicit time dependence occurs in the right-hand sides of the BSSN evolution equations and the Method of Lines can be applied directly.\n\nWhile it is incredibly robust in Cartesian coordinates, [Brown](https://arxiv.org/abs/0902.3652) pointed out that the above time-evolution equation for the shift is not covariant. In fact, we have found this non-covariant version to result in very poor results when solving Einstein's equations in spherical coordinates for a spinning black hole with spin axis pointed in the $\\hat{x}$ direction. Therefore we adopt Brown's covariant version as described in the [**full time-evolution equations for the BSSN gauge quantities $\\alpha$ and $\\beta^i$ tutorial notebook**](Tutorial-BSSN_time_evolution-BSSN_gauge_RHSs.ipynb).\n\n\n\n# Step 3: The BSSN constraint equations \\[Back to [top](#toc)\\]\n$$\\label{constraintequations}$$\n\nIn a way analogous to Maxwell's equations, the BSSN decomposition of Einstein's equations are written as a set of time-evolution equations and a set of constraint equations. In this step we present the BSSN constraints\n\n\\begin{align}\n\\mathcal{H} &= 0 \\\\\n\\mathcal{M^i} &= 0,\n\\end{align}\n\nwhere $\\mathcal{H}=0$ is the **Hamiltonian constraint**, and $\\mathcal{M^i} = 0$ is the **momentum constraint**. When constructing our spacetime from the initial data, one spatial hypersurface at a time, to confirm that at a given time, the Hamiltonian and momentum constraint violations converge to zero as expected with increased numerical resolution.\n\n\n\n## Step 3.a: The Hamiltonian constraint $\\mathcal{H}$ \\[Back to [top](#toc)\\]\n$$\\label{hamiltonianconstraint}$$\n\nThe Hamiltonian constraint is written (Eq. 13 of [Baumgarte *et al.*](https://arxiv.org/pdf/1211.6632.pdf)):\n$$\n\\mathcal{H} = \\frac{2}{3} K^2 - \\bar{A}_{ij} \\bar{A}^{ij} + e^{-4\\phi} \\left(\\bar{R} - 8 \\bar{D}^i \\phi \\bar{D}_i \\phi - 8 \\bar{D}^2 \\phi\\right)\n$$\n\n\n\n## Step 3.b: The momentum constraint $\\mathcal{M}^i$ \\[Back to [top](#toc)\\]\n$$\\label{momentumconstraint}$$\n\nThe momentum constraint is written (Eq. 47 of [Ruchlin, Etienne, & Baumgarte](https://arxiv.org/pdf/1712.07658.pdf)):\n\n$$ \\mathcal{M}^i = e^{-4\\phi} \\left(\n \\frac{1}{\\sqrt{\\bar{\\gamma}}} \\hat{D}_j\\left(\\sqrt{\\bar{\\gamma}}\\bar{A}^{ij}\\right) +\n 6 \\bar{A}^{ij}\\partial_j \\phi -\n \\frac{2}{3} \\bar{\\gamma}^{ij}\\partial_j K +\n \\bar{A}^{jk} \\Delta\\Gamma^i_{jk} + \\bar{A}^{ik} \\Delta\\Gamma^j_{jk}\\right)\n$$\n\nNotice the momentum constraint as written in [Baumgarte *et al.*](https://arxiv.org/pdf/1211.6632.pdf) is missing a term, as described in [Ruchlin, Etienne, & Baumgarte](https://arxiv.org/pdf/1712.07658.pdf).\n\n\n\n# Step 4: The BSSN algebraic constraint: $\\hat{\\gamma}=\\bar{\\gamma}$ \\[Back to [top](#toc)\\]\n$$\\label{gammaconstraint}$$\n\n[Brown](https://arxiv.org/abs/0902.3652)'s covariant Lagrangian formulation of BSSN, which we adopt, requires that $\\partial_t \\bar{\\gamma} = 0$, where $\\bar{\\gamma}=\\det \\bar{\\gamma}_{ij}$. We generally choose to set $\\bar{\\gamma}=\\hat{\\gamma}$ in our initial data.\n\nNumerical errors will cause $\\bar{\\gamma}$ to deviate from a constant in time. This actually disrupts the hyperbolicity of the PDEs (causing crashes), so to cure this, we adjust $\\bar{\\gamma}_{ij}$ at the end of each Runge-Kutta timestep, so that its determinant satisfies $\\bar{\\gamma}=\\hat{\\gamma}$ at all times. We adopt the following, rather standard prescription (Eq. 53 of [Ruchlin, Etienne, and Baumgarte (2018)](https://arxiv.org/abs/1712.07658)):\n\n$$\n\\bar{\\gamma}_{ij} \\to \\left(\\frac{\\hat{\\gamma}}{\\bar{\\gamma}}\\right)^{1/3} \\bar{\\gamma}_{ij}.\n$$\nNotice the expression on the right is guaranteed to have determinant equal to $\\hat{\\gamma}$.\n\n\n\n# Step 5: Output this notebook to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n$$\\label{latex_pdf_output}$$\n\nThe following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n[Tutorial-BSSN_formulation.pdf](Tutorial-BSSN_formulation.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means.)\n\n\n```python\n!jupyter nbconvert --to latex --template latex_nrpy_style.tplx --log-level='WARN' Tutorial-BSSN_formulation.ipynb\n!pdflatex -interaction=batchmode Tutorial-BSSN_formulation.tex\n!pdflatex -interaction=batchmode Tutorial-BSSN_formulation.tex\n!pdflatex -interaction=batchmode Tutorial-BSSN_formulation.tex\n!rm -f Tut*.out Tut*.aux Tut*.log\n```\n\n [NbConvertApp] Converting notebook Tutorial-BSSN_formulation.ipynb to latex\r\n [NbConvertApp] Writing 47455 bytes to Tutorial-BSSN_formulation.tex\r\n This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex)\r\n restricted \\write18 enabled.\r\n entering extended mode\r\n This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex)\r\n restricted \\write18 enabled.\r\n entering extended mode\r\n This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex)\r\n restricted \\write18 enabled.\r\n entering extended mode\r\n\n", "meta": {"hexsha": "b93006b564233bff0fa6044692c19a8fcbe9ffd2", "size": 26657, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Tutorial-BSSN_formulation.ipynb", "max_stars_repo_name": "leowerneck/NRPyIGM", "max_stars_repo_head_hexsha": "f483d6123424fb3e6860dfac4325dd232b223005", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Tutorial-BSSN_formulation.ipynb", "max_issues_repo_name": "leowerneck/NRPyIGM", "max_issues_repo_head_hexsha": "f483d6123424fb3e6860dfac4325dd232b223005", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Tutorial-BSSN_formulation.ipynb", "max_forks_repo_name": "leowerneck/NRPyIGM", "max_forks_repo_head_hexsha": "f483d6123424fb3e6860dfac4325dd232b223005", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 75.5155807365, "max_line_length": 656, "alphanum_fraction": 0.6175488615, "converted": true, "num_tokens": 7064, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.740174367770488, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.4189524114351521}} {"text": "\n\n\n# 1D Three Wave `GiRaFFEfood` Initial Data for `GiRaFFE`\n\n## This module provides another initial data option for `GiRaFFE`, drawn from [this paper](https://arxiv.org/abs/1310.3274) .\n\n**Notebook Status:** Validated \n\n**Validation Notes:** This tutorial notebook has been confirmed to be self-consistent with its corresponding NRPy+ module, as documented [below](#code_validation). The initial data has validated against the original `GiRaFFE`, as documented [here](Tutorial-Start_to_Finish_UnitTest-GiRaFFEfood_NRPy.ipynb).\n\n### NRPy+ Source Code for this module: [GiRaFFEfood_NRPy/GiRaFFEfood_NRPy_1D_tests_three_waves.py](../../edit/in_progress/GiRaFFEfood_NRPy/GiRaFFEfood_NRPy_1D_tests_three_waves.py)\n\n## Introduction:\n\n### Three waves:\n\nThis is a flat-spacetime test representing three Alfvén waves (one stationary, one left-going, and one right-going) with initial data \n\\begin{align}\nA_x &= 0 \\\\ \nA_y &= 3.5x H(-x) + 3.0x H(x) \\\\\nA_z &= y - 1.5x H(-x) - 3.0x H(x),\n\\end{align}\nwhere $H(x)$ is the Heaviside function, which generates the magnetic field\n$$\\mathbf{B}(0,x) = \\mathbf{B_a}(0,x) + \\mathbf{B_+}(0,x) + \\mathbf{B_-}(0,x)$$\nand uses the electric field\n$$\\mathbf{E}(0,x) = \\mathbf{E_a}(0,x) + \\mathbf{E_+}(0,x) + \\mathbf{E_-}(0,x),$$\nwhere subscripted $\\mathbf{a}$ corresponds to the stationary wave, subscripted $\\mathbf{+}$ corresponds to the right-going wave, and subscripted $\\mathbf{-}$ corresponds to the left-going wave, and where \n\\begin{align}\n\\mathbf{B_a}(0,x) &= \\left \\{ \\begin{array}{lll} (1.0,1.0,2.0) & \\mbox{if} & x<0 \\\\\n\t\t\t\t\t(1.0,1.5,2.0) & \\mbox{if} & x>0 \\end{array} \n\\right. , \\\\ \n\\mathbf{E_a}(0,x) &= \\left \\{ \\begin{array}{lll} (-1.0,1.0,0.0) & \\mbox{if} & x<0 \\\\\n\t\t\t\t\t(-1.5,1.0,0.0) & \\mbox{if} & x>0 \\end{array} \n\\right. , \\\\\n\\mathbf{B_+}(0,x) &= \\left \\{ \\begin{array}{lll} (0.0,0.0,0.0) & \\mbox{if} & x<0 \\\\\n (0.0,1.5,1.0) & \\mbox{if} & x>0 \\end{array} \n\\right. , \\\\\n\\mathbf{E_+}(0,x) &= \\left \\{ \\begin{array}{lll} (0.0,0.0,0.0) & \\mbox{if} & x<0 \\\\\n (0.0,1.0,-1.5) & \\mbox{if} & x>0 \\end{array} \n\\right. , \\\\\n\\mathbf{B_-}(0,x) &= \\left \\{ \\begin{array}{lll} (0.0,0.5,1.5) & \\mbox{if} & x<0 \\\\\n (0.0,0.0,0.0) & \\mbox{if} & x>0 \\end{array}\n\\right. , \\\\\n\\mathbf{E_-}(0,x) &= \\left \\{ \\begin{array}{lll} (0.0,-1.5,0.5) & \\mbox{if} & x<0 \\\\\n (0.0,0.0,0.0) & \\mbox{if} & x>0 \\end{array}\n\\right. . \\\\\n\\end{align}\n\nFor the eventual purpose of testing convergence, any quantity $Q$ evolves as $Q(t,x) = Q_a(0,x) + Q_+(0,x-t) + Q_-(0,x+t)$.\n\nSee the [Tutorial-GiRaFFEfood_NRPy_Exact_Wald](Tutorial-GiRaFFEfood_NRPy.ipynb) tutorial notebook for more general detail on how this is used.\n\n\n\n\n# Table of Contents:\n$$\\label{toc}$$\n\nThis notebook is organized as follows\n\n1. [Step 1](#initializenrpy): Import core NRPy+ modules and set NRPy+ parameters \n1. [Step 2](#vector_ak): Set the vector $A_k$\n1. [Step 3](#vectors_for_velocity): Set the vectors $B^i$ and $E^i$ for the velocity\n1. [Step 4](#vi): Calculate $v^i$\n1. [Step 5](#code_validation): Code Validation against `GiRaFFEfood_NRPy/GiRaFFEfood_NRPy_1D_tests` NRPy+ module\n1. [Step 6](#latex_pdf_output): Output this notebook to $\\LaTeX$-formatted PDF file\n\n\n\n# Step 1: Import core NRPy+ modules and set NRPy+ parameters \\[Back to [top](#toc)\\]\n$$\\label{initializenrpy}$$\n\nHere, we will import the NRPy+ core modules and set the reference metric to Cartesian, set commonly used NRPy+ parameters, and set C parameters that will be set from outside the code eventually generated from these expressions. We will also set up a parameter to determine what initial data is set up, although it won't do much yet.\n\n\n```python\n# Step 0: Add NRPy's directory to the path\n# https://stackoverflow.com/questions/16780014/import-file-from-parent-directory\nimport os,sys\nnrpy_dir_path = os.path.join(\"..\")\nif nrpy_dir_path not in sys.path:\n sys.path.append(nrpy_dir_path)\n\n# Step 0.a: Import the NRPy+ core modules and set the reference metric to Cartesian\nimport sympy as sp # SymPy: The Python computer algebra package upon which NRPy+ depends\nimport NRPy_param_funcs as par # NRPy+: Parameter interface\nimport indexedexp as ixp # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support\n\nimport reference_metric as rfm # NRPy+: Reference metric support\npar.set_parval_from_str(\"reference_metric::CoordSystem\",\"Cartesian\")\nrfm.reference_metric()\n\n# Step 1a: Set commonly used parameters.\nthismodule = \"GiRaFFEfood_NRPy_1D\"\n```\n\n##### \n\n# Step 2: Set the vector $A_k$ \\[Back to [top](#toc)\\]\n$$\\label{vector_ak}$$\n\nThe vector potential is given as\n\\begin{align}\nA_x &= 0 \\\\ \nA_y &= 3.5x H(-x) + 3.0x H(x) \\\\\nA_z &= y - 1.5x H(-x) - 3.0x H(x),\n\\end{align}\n\nHowever, to take full advantage of NRPy+'s automated function generation capabilities, we want to write this without the `if` statements, replacing them with calls to `fabs()`. To do so, we will use the NRPy+ module `Min_Max_and_Piecewise_Expressions`.\n\n\n```python\n# We'll use reference_metric.py to define x and y\nx = rfm.xxCart[0]\ny = rfm.xxCart[1]\n```\n\nNow, we can define the vector potential. We will need to write the Heviside function without `if`s, which can easily be done with the module `Min_Max_and_Piecewise_Expressions`. We thus get\n$$H(x) = \\frac{\\max(0,x)}{x}.$$\nThis implementation is, of course, undefined for $x=0$; this problem is easily solved by adding a very small number (called `TINYDOUBLE` in our implementation) to the denominator (see [Tutorial-Min_Max_and_Piecewise_Expressions](Tutorial-Min_Max_and_Piecewise_Expressions.ipynb) for details on how this works). This is, conveniently, the exact implementation of the `coord_greater_bound()` function!\n\n\\begin{align}\nA_x &= 0 \\\\ \nA_y &= 3.5x H(-x) + 3.0x H(x) \\\\\nA_z &= y - 1.5x H(-x) - 3.0x H(x),\n\\end{align}\n\n\n```python\nAD = ixp.zerorank1(DIM=3)\n\nimport Min_Max_and_Piecewise_Expressions as noif\n\nAD[0] = sp.sympify(0)\nAD[1] = sp.Rational(7,2)*x*noif.coord_greater_bound(-x,0) + sp.sympify(3)*x*noif.coord_greater_bound(x,0)\nAD[2] = y-sp.Rational(3,2)*x*noif.coord_greater_bound(-x,0) - sp.sympify(3)*x*noif.coord_greater_bound(x,0)\n```\n\n\n\n# Step 3: Set the vectors $B^i$ and $E^i$ for the velocity \\[Back to [top](#toc)\\]\n$$\\label{vectors_for_velocity}$$\n\nFirst, we will set the the three individual waves; we change all $<$ to $\\leq$ to avoid unintented behavior at $x=0$:\n\\begin{align}\n\\mathbf{B_a}(0,x) &= \\left \\{ \\begin{array}{lll} (1.0,1.0,2.0) & \\mbox{if} & x \\leq 0 \\\\\n\t\t\t\t\t(1.0,1.5,2.0) & \\mbox{if} & x>0 \\end{array} \n\\right. , \\\\ \n\\mathbf{E_a}(0,x) &= \\left \\{ \\begin{array}{lll} (-1.0,1.0,0.0) & \\mbox{if} & x \\leq 0 \\\\\n\t\t\t\t\t(-1.5,1.0,0.0) & \\mbox{if} & x>0 \\end{array} \n\\right. , \\\\\n\\mathbf{B_+}(0,x) &= \\left \\{ \\begin{array}{lll} (0.0,0.0,0.0) & \\mbox{if} & x \\leq 0 \\\\\n (0.0,1.5,1.0) & \\mbox{if} & x>0 \\end{array} \n\\right. , \\\\\n\\mathbf{E_+}(0,x) &= \\left \\{ \\begin{array}{lll} (0.0,0.0,0.0) & \\mbox{if} & x \\leq 0 \\\\\n (0.0,1.0,-1.5) & \\mbox{if} & x>0 \\end{array} \n\\right. , \\\\\n\\mathbf{B_-}(0,x) &= \\left \\{ \\begin{array}{lll} (0.0,0.5,1.5) & \\mbox{if} & x \\leq 0 \\\\\n (0.0,0.0,0.0) & \\mbox{if} & x>0 \\end{array}\n\\right. , \\\\\n\\mathbf{E_-}(0,x) &= \\left \\{ \\begin{array}{lll} (0.0,-1.5,0.5) & \\mbox{if} & x \\leq 0 \\\\\n (0.0,0.0,0.0) & \\mbox{if} & x>0 \\end{array}\n\\right. . \\\\\n\\end{align}\n\n\n\n```python\nB_aU = ixp.zerorank1(DIM=3)\nE_aU = ixp.zerorank1(DIM=3)\nB_pU = ixp.zerorank1(DIM=3)\nE_pU = ixp.zerorank1(DIM=3)\nB_mU = ixp.zerorank1(DIM=3)\nE_mU = ixp.zerorank1(DIM=3)\n\nB_aU[0] = sp.sympify(1)\nB_aU[1] = noif.coord_leq_bound(x,0) * sp.sympify(1) + noif.coord_greater_bound(x,0) * sp.Rational(3,2)\nB_aU[2] = sp.sympify(2)\n\nE_aU[0] = noif.coord_leq_bound(x,0) * sp.sympify(-1) + noif.coord_greater_bound(x,0) * sp.Rational(-3,2)\nE_aU[1] = sp.sympify(1)\nE_aU[2] = sp.sympify(0)\n\nB_pU[0] = sp.sympify(0)\nB_pU[1] = noif.coord_leq_bound(x,0) * sp.sympify(0) + noif.coord_greater_bound(x,0) * sp.Rational(3,2)\nB_pU[2] = noif.coord_leq_bound(x,0) * sp.sympify(0) + noif.coord_greater_bound(x,0) * sp.sympify(1)\n\nE_pU[0] = sp.sympify(0)\nE_pU[1] = noif.coord_leq_bound(x,0) * sp.sympify(0) + noif.coord_greater_bound(x,0) * sp.sympify(1)\nE_pU[2] = noif.coord_leq_bound(x,0) * sp.sympify(0) + noif.coord_greater_bound(x,0) * sp.Rational(-3,2)\n\nB_mU[0] = sp.sympify(0)\nB_mU[1] = noif.coord_leq_bound(x,0) * sp.Rational(1,2) + noif.coord_greater_bound(x,0) * sp.sympify(0)\nB_mU[2] = noif.coord_leq_bound(x,0) * sp.Rational(3,2) + noif.coord_greater_bound(x,0) * sp.sympify(0)\n\nE_mU[0] = sp.sympify(0)\nE_mU[1] = noif.coord_leq_bound(x,0) * sp.Rational(-3,2) + noif.coord_greater_bound(x,0) * sp.sympify(0)\nE_mU[2] = noif.coord_leq_bound(x,0) * sp.Rational(1,2) + noif.coord_greater_bound(x,0) * sp.sympify(0)\n```\n\nThen, we can obtain the total expressions for the magnetic and electric fields by simply adding the three waves together:\n\\begin{align}\n\\mathbf{B}(0,x) &= \\mathbf{B_a}(0,x) + \\mathbf{B_+}(0,x) + \\mathbf{B_-}(0,x) \\\\\n\\mathbf{E}(0,x) &= \\mathbf{E_a}(0,x) + \\mathbf{E_+}(0,x) + \\mathbf{E_-}(0,x)\n\\end{align}\n\n\n```python\nBU = ixp.zerorank1(DIM=3)\nEU = ixp.zerorank1(DIM=3)\nfor i in range(3):\n BU[i] = B_aU[i] + B_pU[i] + B_mU[i]\n EU[i] = E_aU[i] + E_pU[i] + E_mU[i]\n```\n\n\n\n# Step 4: Calculate $v^i$ \\[Back to [top](#toc)\\]\n$$\\label{vi}$$\n\nNow, we calculate $$\\mathbf{v} = \\frac{\\mathbf{E} \\times \\mathbf{B}}{B^2},$$ which is equivalent to $$v^i = [ijk] \\frac{E^j B^k}{B^2},$$ where $[ijk]$ is the Levi-Civita symbol and $B^2 = \\gamma_{ij} B^i B^j$ is a trivial dot product in flat space.\n\n\n\n```python\nLeviCivitaSymbolDDD = ixp.LeviCivitaSymbol_dim3_rank3()\n\nB2 = sp.sympify(0)\nfor i in range(3):\n # In flat spacetime, gamma_{ij} is just a Kronecker delta\n B2 += BU[i]**2 # This is trivial to extend to curved spacetime\n\nValenciavU = ixp.zerorank1()\nfor i in range(3):\n for j in range(3):\n for k in range(3):\n ValenciavU[i] += LeviCivitaSymbolDDD[i][j][k] * EU[j] * BU[k] / B2\n```\n\n\n\n# Step 5: Code Validation against `GiRaFFEfood_NRPy/GiRaFFEfood_NRPy_1D_tests` NRPy+ module \\[Back to [top](#toc)\\]\n$$\\label{code_validation}$$\n\nHere, as a code validation check, we verify agreement in the SymPy expressions for the `GiRaFFE` Aligned Rotator initial data equations we intend to use between\n1. this tutorial and \n2. the NRPy+ [`GiRaFFEfood_NRPy/GiRaFFEfood_NRPy_1D_tests.py`](../edit/GiRaFFEfood_NRPy/GiRaFFEfood_NRPy_1D_tests.py) module.\n\n\n\n\n```python\nimport GiRaFFEfood_NRPy.GiRaFFEfood_NRPy_1D_tests_three_waves as gfho\ngfho.GiRaFFEfood_NRPy_1D_tests_three_waves()\n\ndef consistency_check(quantity1,quantity2,string):\n if quantity1-quantity2==0:\n print(string+\" is in agreement!\")\n else:\n print(string+\" does not agree!\")\n sys.exit(1)\n\nprint(\"Consistency check between GiRaFFEfood_NRPy tutorial and NRPy+ module:\")\n\nfor i in range(3):\n consistency_check(ValenciavU[i],gfho.ValenciavU[i],\"ValenciavU\"+str(i))\n consistency_check(AD[i],gfho.AD[i],\"AD\"+str(i))\n```\n\n Consistency check between GiRaFFEfood_NRPy tutorial and NRPy+ module:\n ValenciavU0 is in agreement!\n AD0 is in agreement!\n ValenciavU1 is in agreement!\n AD1 is in agreement!\n ValenciavU2 is in agreement!\n AD2 is in agreement!\n\n\n\n\n# Step 6: Output this notebook to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n$$\\label{latex_pdf_output}$$\n\nThe following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n[Tutorial-GiRaFFEfood_NRPy_1D_tests.pdf](Tutorial-GiRaFFEfood_NRPy_1D_tests.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means.)\n\n\n```python\nimport cmdline_helper as cmd # NRPy+: Multi-platform Python command-line interface\ncmd.output_Jupyter_notebook_to_LaTeXed_PDF(\"Tutorial-GiRaFFEfood_NRPy_1D_tests\",location_of_template_file=os.path.join(\"..\"))\n```\n\n pdflatex: security risk: running with elevated privileges\n pdflatex: security risk: running with elevated privileges\n pdflatex: security risk: running with elevated privileges\n pdflatex: security risk: running with elevated privileges\n Created Tutorial-GiRaFFEfood_NRPy_1D_tests.tex, and compiled LaTeX file to\n PDF file Tutorial-GiRaFFEfood_NRPy_1D_tests.pdf\n\n", "meta": {"hexsha": "53b5d9789d9c5e738264e18d0ba2c44782b4304a", "size": 17746, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "in_progress/Tutorial-GiRaFFEfood_NRPy_1D_tests-three_waves.ipynb", "max_stars_repo_name": "terrencepierrej/nrpytutorial", "max_stars_repo_head_hexsha": "3ea18beed99cf6b7d67c89c140ca68630452001e", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "in_progress/Tutorial-GiRaFFEfood_NRPy_1D_tests-three_waves.ipynb", "max_issues_repo_name": "terrencepierrej/nrpytutorial", "max_issues_repo_head_hexsha": "3ea18beed99cf6b7d67c89c140ca68630452001e", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "in_progress/Tutorial-GiRaFFEfood_NRPy_1D_tests-three_waves.ipynb", "max_forks_repo_name": "terrencepierrej/nrpytutorial", "max_forks_repo_head_hexsha": "3ea18beed99cf6b7d67c89c140ca68630452001e", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 42.1520190024, "max_line_length": 408, "alphanum_fraction": 0.5594500169, "converted": true, "num_tokens": 4596, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.5660185351961016, "lm_q1q2_score": 0.41895240819103297}} {"text": "# CS109B Data Science 2: Advanced Topics in Data Science \n\n## Lab 4 - Bayesian Analysis\n\n**Harvard University**
                                        \n**Spring 2020**
                                        \n**Instructors:** Mark Glickman, Pavlos Protopapas, and Chris Tanner
                                        \n**Lab Instructors:** Chris Tanner and Eleni Angelaki Kaxiras
                                        \n**Content:** Eleni Angelaki Kaxiras\n\n---\n\n\n```python\n## RUN THIS CELL TO PROPERLY HIGHLIGHT THE EXERCISES\nimport requests\nfrom IPython.core.display import HTML\nstyles = requests.get(\"https://raw.githubusercontent.com/Harvard-IACS/2019-CS109B/master/content/styles/cs109.css\").text\nHTML(styles)\n```\n\n\n\n\n\n\n\n\n\n\n\n```python\nimport pymc3 as pm\nfrom pymc3 import summary\n```\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport scipy.stats as stats\nimport pandas as pd\n%matplotlib inline \n\nimport warnings\nwarnings.filterwarnings('ignore')\n```\n\n\n```python\nprint('Running on PyMC3 v{}'.format(pm.__version__))\n```\n\n Running on PyMC3 v3.8\n\n\n\n```javascript\n%%javascript\nIPython.OutputArea.auto_scroll_threshold = 20000;\n```\n\n\n \n\n\n\n\n## Learning Objectives\n\nBy the end of this lab, you should be able to:\n* Understand how probability distributions work.\n* Apply Bayes Rule in calculating probabilities.\n* Understand how to apply Bayesian analysis using PyMC3\n* Avoid getting fired when talking to your Bayesian employer.\n\n**This lab corresponds to Lectures 6, 7, and 8, and maps to Homework 3.**\n\n## Table of Contents\n\n1. The Bayesian Way of Thinking or Is this a Fair Coin?\n2. [Intro to `pyMC3`](#pymc3). \n3. [Bayesian Linear Regression](#blr).\n4. [Try this at Home: Example on Mining Disasters](#no4).\n\n## 1. The Bayesian way of Thinking\n\n```\nHere is my state of knowledge about the situation. Here is some data, I am now going to revise my state of knowledge.\n```\n\n
                                        Table Exercise: Discuss the statement above with your table mates and make sure everyone understands what it means and what constitutes Bayesian way of thinking. Finally, count the Bayesians among you.
                                        \n\n### A. Bayes Rule\n\n\\begin{equation}\n\\label{eq:bayes} \nP(A|\\textbf{B}) = \\frac{P(\\textbf{B} |A) P(A) }{P(\\textbf{B})} \n\\end{equation}\n\n$P(A|\\textbf{B})$ is the **posterior** distribution, prob(hypothesis | data) \n\n$P(\\textbf{B} |A)$ is the **likelihood** function, how probable is my data **B** for different values of the parameters\n\n$P(A)$ is the marginal probability to observe the data, called the **prior**, this captures our belief about the data before observing it.\n\n$P(\\textbf{B})$ is the marginal distribution (sometimes called marginal likelihood)\n\n
                                        \n
                                        Table Exercise: Solve the Monty Hall Paradox using Bayes Rule.
                                        \n\n\n\nYou are invited to play a game. There are 3 doors behind **one** of which are the keys to a brand new red Tesla. There is a goat behind each of the other two. \n\nYou are asked to pick one door, and let's say you pick **Door1**. The host knows where the keys are. Of the two remaining closed doors, he will always open the door that has a goat behind it. He'll say \"I will do you a favor and open **Door2**\". So he opens Door2 inside which there is, of course, a goat. He now asks you, do you want to open the initial Door you chose or change to **Door3**? Generally, in this game, when you are presented with this choice should you swap the doors?\n\n**Initial Steps:**\n- Start by defining the `events` of this probabilities game. One definition is:\n \n - $A_i$: car is behind door $i$ \n \n - $B_i$ host opens door $i$\n \n$i\\in[1,2,3]$\n \n- In more math terms, the question is: is the probability that the price is behind **Door 1** higher than the probability that the price is behind **Door2**, given that an event **has occured**?\n\n### B. Bayes Rule written with Probability Distributions\n\nWe have data that we believe come from an underlying distribution of unknown parameters. If we find those parameters, we know everything about the process that generated this data and we can make inferences (create new data).\n\n\\begin{equation}\n\\label{eq:bayes} \nP(\\theta|\\textbf{D}) = \\frac{P(\\textbf{D} |\\theta) P(\\theta) }{P(\\textbf{D})} \n\\end{equation}\n\n#### But what is $\\theta \\;$?\n\n$\\theta$ is an unknown yet fixed set of parameters. In Bayesian inference we express our belief about what $\\theta$ might be and instead of trying to guess $\\theta$ exactly, we look for its **probability distribution**. What that means is that we are looking for the **parameters** of that distribution. For example, for a Poisson distribution our $\\theta$ is only $\\lambda$. In a normal distribution, our $\\theta$ is often just $\\mu$ and $\\sigma$.\n\n### C. A review of Common Probability Distributions\n\n#### Discrete Distributions\n\nThe random variable has a **probability mass function (pmf)** which measures the probability that our random variable will take a specific value $y$, denoted $P(Y=y)$.\n\n- **Bernoulli** (binary outcome, success has probability $\\theta$, $one$ trial):\n$\nP(Y=k) = \\theta^k(1-\\theta)^{1-k}\n$\n
                                        \n- **Binomial** (binary outcome, success has probability $\\theta$, $n$ trials):\n\\begin{equation}\nP(Y=k) = {{n}\\choose{k}} \\cdot \\theta^k(1-\\theta)^{n-k}\n\\end{equation}\n\n*Note*: Binomial(1,$p$) = Bernouli($p$)\n
                                        \n- **Negative Binomial**\n
                                        \n- **Poisson** (counts independent events occurring at a rate)\n\\begin{equation}\nP\\left( Y=y|\\lambda \\right) = \\frac{{e^{ - \\lambda } \\lambda ^y }}{{y!}}\n\\end{equation}\ny = 0,1,2,...\n
                                        \n- **Discrete Uniform** \n
                                        \n- **Categorical, or Multinulli** (random variables can take any of K possible categories, each having its own probability; this is a generalization of the Bernoulli distribution for a discrete variable with more than two possible outcomes, such as the roll of a die)\n
                                        \n- **Dirichlet-multinomial** (a generalization of the beta distribution for many variables)\n\n#### Continuous Distributions\n\nThe random variable has a **probability density function (pdf)**.\n- **Uniform** (variable equally likely to be near each value in interval $(a,b)$)\n\\begin{equation}\nP(X = x) = \\frac{1}{b - a}\n\\end{equation}\nanywhere within the interval $(a, b)$, and zero elsewhere.\n
                                        \n- **Normal** (a.k.a. Gaussian)\n\\begin{equation}\nX \\sim \\mathcal{N}(\\mu,\\,\\sigma^{2})\n\\end{equation} \n\n A Normal distribution can be parameterized either in terms of precision $\\tau$ or standard deviation ($\\sigma^{2}$. The link between the two is given by\n\\begin{equation}\n\\tau = \\frac{1}{\\sigma^{2}}\n\\end{equation}\n - Mean $\\mu$\n - Variance $\\frac{1}{\\tau}$ or $\\sigma^{2}$\n - Parameters: `mu: float`, `sigma: float` or `tau: float`\n
                                        \n- **Beta** (variable ($\\theta$) taking on values in the interval $[0,1]$, and parametrized by two positive parameters, $\\alpha$ and $\\beta$ that control the shape of the distribution. \n \n*Note:*Beta is a good distribution to use for priors (beliefs) because its range is $[0,1]$ which is the natural range for a probability and because we can model a wide range of functions by changing the $\\alpha$ and $\\beta$ parameters.\n\n\\begin{equation}\n\\label{eq:beta} \nP(\\theta) = \\frac{1}{B(\\alpha, \\beta)} {\\theta}^{\\alpha - 1} (1 - \\theta)^{\\beta - 1} \\propto {\\theta}^{\\alpha - 1} (1 - \\theta)^{\\beta - 1}\n\\end{equation}\n\n\nwhere the normalisation constant, $B$, is a beta function of $\\alpha$ and $\\beta$,\n\n\n\\begin{equation}\nB(\\alpha, \\beta) = \\int_{t=0}^1 t^{\\alpha - 1} (1 - t)^{\\beta - 1} dt.\n\\end{equation}\n
                                        \n- **Exponential**\n
                                        \n- **Gamma**\n\n\n\n #### Code Resources:\n - Statistical Distributions in numpy/scipy: [scipy.stats](https://docs.scipy.org/doc/scipy/reference/stats.html)\n - Statistical Distributions in pyMC3: [distributions in PyMC3](https://docs.pymc.io/api/distributions.html) (we will see those below).\n\n
                                        Exercise: Plot a Discrete variable
                                        \n\nChange the value of $\\mu$ in the Poisson PMF and see how the plot changes. Remember that the y-axis in a discrete probability distribution shows the probability of the random variable having a specific value in the x-axis.\n\n\\begin{equation}\nP\\left( X=k \\right) = \\frac{{e^{ - \\mu } \\mu ^k }}{{k!}}\n\\end{equation}\n\n**stats.poisson.pmf(x, mu)** $\\mu$(mu) is our $\\theta$ in this case.\n\n\n```python\nplt.style.use('seaborn-darkgrid')\nx = np.arange(0, 30)\nfor m in [0.5, 3, 8]:\n pmf = stats.poisson.pmf(x, m)\n plt.plot(x, pmf, 'o', alpha=0.5, label='$\\mu$ = {}'.format(m))\nplt.xlabel('random variable', fontsize=12)\nplt.ylabel('probability', fontsize=12)\nplt.legend(loc=1)\nplt.ylim=(-0.1)\nplt.show()\n```\n\n\n```python\n# same for binomial\nplt.style.use('seaborn-darkgrid')\nx = np.arange(0, 22)\nns = [10, 17]\nps = [0.5, 0.7]\nfor n, p in zip(ns, ps):\n pmf = stats.binom.pmf(x, n, p)\n plt.plot(x, pmf, 'o', alpha=0.5, label='n = {}, p = {}'.format(n, p))\nplt.xlabel('x', fontsize=14)\nplt.ylabel('f(x)', fontsize=14)\nplt.legend(loc=1)\nplt.show()\n```\n\n\n```python\n# discrete uniform\nplt.style.use('seaborn-darkgrid')\nls = [0]\nus = [3] # watch out, this number can only be integer!\nfor l, u in zip(ls, us):\n x = np.arange(l, u+1)\n pmf = [1.0 / (u - l + 1)] * len(x)\n plt.plot(x, pmf, '-o', label='lower = {}, upper = {}'.format(l, u))\nplt.xlabel('x', fontsize=12)\nplt.ylabel('probability P(x)', fontsize=12)\nplt.legend(loc=1)\nplt.show()\n```\n\n
                                        Exercise: Plot a continuous variable
                                        \n\nChange the value of $\\mu$ in the Uniform PDF and see how the plot changes.\n \nRemember that the y-axis in a continuous probability distribution does not shows the actual probability of the random variable having a specific value in the x-axis because that probability is zero!. Instead, to see the probability that the variable is within a small margin we look at the integral below the curve of the PDF.\n\nThe uniform is often used as a noninformative prior.\n\n```\nUniform - numpy.random.uniform(a=0.0, b=1.0, size)\n```\n\n$\\alpha$ and $\\beta$ are our parameters. `size` is how many tries to perform.\nOur $\\theta$ is basically the combination of the parameters a,b. We can also call it \n\\begin{equation}\n\\mu = (a+b)/2\n\\end{equation}\n\n\n```python\nfrom scipy.stats import uniform\n\nr = uniform.rvs(size=1000)\nplt.plot(r, uniform.pdf(r),'r-', lw=5, alpha=0.6, label='uniform pdf')\nplt.hist(r, density=True, histtype='stepfilled', alpha=0.2)\nplt.ylabel(r'probability density')\nplt.xlabel(f'random variable')\nplt.legend(loc='best', frameon=False)\nplt.show()\n```\n\n\n```python\nfrom scipy.stats import beta\n\nalphas = [0.5, 1.5, 3.0]\nbetas = [0.5, 1.5, 3.0]\nx = np.linspace(0, 1, 1000) \ncolors = ['red', 'green', 'blue']\n\nfig, ax = plt.subplots(figsize=(8, 5))\n\nfor a, b, colors in zip(alphas, betas, colors):\n dist = beta(a, b)\n plt.plot(x, dist.pdf(x), c=colors,\n label=f'a={a}, b={b}')\n\nax.set_ylim(0, 3)\n\nax.set_xlabel(r'$\\theta$')\nax.set_ylabel(r'$p(\\theta|\\alpha,\\beta)$')\nax.set_title('Beta Distribution')\n\nax.legend(loc='best')\nfig.show();\n```\n\n\n```python\nplt.style.use('seaborn-darkgrid')\nx = np.linspace(-5, 5, 1000)\nmus = [0., 0., 0., -2.]\nsigmas = [0.4, 1., 2., 0.4]\nfor mu, sigma in zip(mus, sigmas):\n pdf = stats.norm.pdf(x, mu, sigma)\n plt.plot(x, pdf, label=r'$\\mu$ = '+ f'{mu},' + r'$\\sigma$ = ' + f'{sigma}') \nplt.xlabel('random variable', fontsize=12)\nplt.ylabel('probability density', fontsize=12)\nplt.legend(loc=1)\nplt.show()\n```\n\n\n```python\nplt.style.use('seaborn-darkgrid')\nx = np.linspace(-5, 5, 1000)\nmus = [0., 0., 0., -2.] # mean\nsigmas = [0.4, 1., 2., 0.4] # std\nfor mu, sigma in zip(mus, sigmas):\n plt.plot(x, uniform.pdf(x, mu, sigma), lw=5, alpha=0.4, \\\n label=r'$\\mu$ = '+ f'{mu},' + r'$\\sigma$ = ' + f'{sigma}')\nplt.xlabel('random variable', fontsize=12)\nplt.ylabel('probability density', fontsize=12)\nplt.legend(loc=1)\nplt.show()\n```\n\n### D. Is this a Fair Coin?\n\nWe do not want to promote gambling but let's say you visit the casino in **Monte Carlo**. You want to test your theory that casinos are dubious places where coins have been manipulated to have a larger probability for tails. So you will try to estimate how fair a coin is based on 100 flips.
                                        \nYou begin by flipping the coin. You get either Heads ($H$) or Tails ($T$) as our observed data and want to see if your posterior probabilities change as you obtain more data, that is, more coin flips. A nice way to visualize this is to plot the posterior probabilities as we observe more flips (data). \n\nWe will be using Bayes rule. $\\textbf{D}$ is our data.\n\n\\begin{equation}\n\\label{eq:bayes} \nP(\\theta|\\textbf{D}) = \\frac{P(\\textbf{D} |\\theta) P(\\theta) }{P(\\textbf{D})} \n\\end{equation}\n\nIn the case of a coin toss when we observe $k$ heads in $n$ tosses:\n\\begin{equation}\n\\label{eq:bayes} \nP(\\theta|\\textbf{k}) = Beta(\\alpha + \\textbf{k}, \\beta + n - \\textbf{k}) \n\\end{equation}\n\nwe can say that $\\alpha$ and $\\beta$ play the roles of a \"prior number of heads\" and \"prior number of tails\".\n\n\n```python\n# play with the priors - here we manually set them but we could be sampling from a separate Beta\ntrials = np.array([0, 1, 3, 5, 10, 15, 20, 100, 200, 300])\nheads = np.array([0, 1, 2, 4, 8, 10, 10, 50, 180, 150])\nx = np.linspace(0, 1, 100)\n\n# for simplicity we set a,b=1\n\nplt.figure(figsize=(10,8))\nfor k, N in enumerate(trials):\n sx = plt.subplot(len(trials)/2, 2, k+1)\n posterior = stats.beta.pdf(x, 1 + heads[k], 1 + trials[k] - heads[k]) \n plt.plot(x, posterior, alpha = 0.5, label=f'{trials[k]} tosses\\n {heads[k]} heads');\n plt.fill_between(x, 0, posterior, color=\"#348ABD\", alpha=0.4) \n plt.legend(loc='upper left', fontsize=10)\n plt.legend()\n plt.autoscale(tight=True)\n \nplt.suptitle(\"Posterior probabilities for coin flips\", fontsize=15);\nplt.tight_layout()\nplt.subplots_adjust(top=0.88)\n```\n\n [Top](#top)\n\n## 2. Introduction to `pyMC3`\n \nPyMC3 is a Python library for programming Bayesian analysis, and more specifically, data creation, model definition, model fitting, and posterior analysis. It uses the concept of a `model` which contains assigned parametric statistical distributions to unknown quantities in the model. Within models we define random variables and their distributions. A distribution requires at least a `name` argument, and other `parameters` that define it. You may also use the `logp()` method in the model to build the model log-likelihood function. We define and fit the model.\n\nPyMC3 includes a comprehensive set of pre-defined statistical distributions that can be used as model building blocks. Although they are not meant to be used outside of a `model`, you can invoke them by using the prefix `pm`, as in `pm.Normal`. \n\n#### Markov Chain Monte Carlo (MCMC) Simulations\n\nPyMC3 uses the **No-U-Turn Sampler (NUTS)** and the **Random Walk Metropolis**, two Markov chain Monte Carlo (MCMC) algorithms for sampling in posterior space. Monte Carlo gets into the name because when we sample in posterior space, we choose our next move via a pseudo-random process. NUTS is a sophisticated algorithm that can handle a large number of unknown (albeit continuous) variables.\n\n\n```python\nwith pm.Model() as model:\n z = pm.Normal('z', mu=0., sigma=5.) \n x = pm.Normal('x', mu=z, sigma=1., observed=5.) \nprint(x.logp({'z': 2.5})) \nprint(z.random(10, 100)[:10]) \n```\n\n -4.043938533204672\n [ 2.49487126 -0.51068451 -3.65709456 0.43477628 10.22323769 1.22927069\n 0.01286762 -4.87182774 -2.60094774 -0.72939632]\n\n\n**References**:\n\n- *Salvatier J, Wiecki TV, Fonnesbeck C. 2016. Probabilistic programming in Python using PyMC3. PeerJ Computer Science 2:e55* [(https://doi.org/10.7717/peerj-cs.55)](https://doi.org/10.7717/peerj-cs.55)\n- [Distributions in PyMC3](https://docs.pymc.io/api/distributions.html)\n- [More Details on Distributions](https://docs.pymc.io/developer_guide.html)\n\nInformation about PyMC3 functions including descriptions of distributions, sampling methods, and other functions, is available via the `help` command.\n\n\n```python\n#help(pm.Poisson)\n```\n\n [Top](#top)\n\n## 3. Bayesian Linear Regression\n\nLet's say we want to predict outcomes Y as normally distributed observations with an expected value $mu$ that is a linear function of two predictor variables, $\\bf{x}_1$ and $\\bf{x}_2$.\n\n\\begin{equation}\n\\mu = \\alpha + \\beta_1 \\bf{x}_1 + \\beta_2 x_2 \n\\end{equation}\n\n\\begin{equation}\nY \\sim \\mathcal{N}(\\mu,\\,\\sigma^{2})\n\\end{equation} \n\nwhere $\\sigma^2$ represents the measurement error. \n\nIn this example, we will use $\\sigma^2 = 10$\n\nWe also choose the parameters as normal distributions:\n\n\\begin{eqnarray}\n\\alpha \\sim \\mathcal{N}(0,\\,10) \\\\\n\\beta_i \\sim \\mathcal{N}(0,\\,10) \\\\\n\\sigma^2 \\sim |\\mathcal{N}(0,\\,10)|\n\\end{eqnarray} \n\nWe will artificially create the data to predict on. We will then see if our model predicts them correctly.\n\n\n```python\n# Initialize random number generator\nnp.random.seed(123)\n\n# True parameter values\nalpha, sigma = 1, 1\nbeta = [1, 2.5]\n\n# Size of dataset\nsize = 100\n\n# Predictor variable\nX1 = np.linspace(0, 1, size)\nX2 = np.linspace(0,.2, size)\n\n# Simulate outcome variable\nY = alpha + beta[0]*X1 + beta[1]*X2 + np.random.randn(size)*sigma\n\nfig, ax = plt.subplots(1,2, figsize=(10,6), sharex=True)\nax[0].scatter(X1,Y)\nax[1].scatter(X2,Y)\nax[0].set_xlabel(r'$x_1$', fontsize=14) \nax[0].set_ylabel(r'$Y$', fontsize=14)\nax[1].set_xlabel(r'$x_2$', fontsize=14) \nax[1].set_ylabel(r'$Y$', fontsize=14)\n```\n\n\n```python\nfrom pymc3 import Model, Normal, HalfNormal\n\nbasic_model = Model()\n\nwith basic_model:\n\n # Priors for unknown model parameters, specifically create stochastic random variables \n # with Normal prior distributions for the regression coefficients,\n # and a half-normal distribution for the standard deviation of the observations, \u03c3.\n alpha = Normal('alpha', mu=0, sd=10)\n beta = Normal('beta', mu=0, sd=10, shape=2)\n sigma = HalfNormal('sigma', sd=1)\n\n # Expected value of outcome - posterior\n mu = alpha + beta[0]*X1 + beta[1]*X2\n\n # Likelihood (sampling distribution) of observations\n Y_obs = Normal('Y_obs', mu=mu, sd=sigma, observed=Y)\n```\n\n\n```python\n# model fitting with sampling\nfrom pymc3 import NUTS, sample, find_MAP\nfrom scipy import optimize\n\nwith basic_model:\n\n # obtain starting values via MAP\n start = find_MAP(fmin=optimize.fmin_powell)\n\n # instantiate sampler\n step = NUTS(scaling=start)\n\n # draw 2000 posterior samples\n trace = sample(2000, step, start=start)\n```\n\n logp = -164.5: 5%|\u258c | 270/5000 [00:00<00:01, 4530.74it/s] \n\n Optimization terminated successfully.\n Current function value: 164.496957\n Iterations: 6\n Function evaluations: 271\n\n\n logp = -164.5: 5%|\u258c | 271/5000 [00:00<00:12, 391.82it/s] \n Multiprocess sampling (4 chains in 4 jobs)\n NUTS: [sigma, beta, alpha]\n Sampling 4 chains, 0 divergences: 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 10000/10000 [00:08<00:00, 1114.21draws/s]\n The number of effective samples is smaller than 25% for some parameters.\n\n\n\n```python\nfrom pymc3 import traceplot\n\ntraceplot(trace);\n```\n\n\n```python\nresults = pm.summary(trace, \n var_names=['alpha', 'beta', 'sigma'])\nresults\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        meansdhpd_3%hpd_97%mcse_meanmcse_sdess_meaness_sdess_bulkess_tailr_hat
                                        alpha1.0150.2320.5801.4520.0040.0034153.04094.04147.04245.01.0
                                        beta[0]1.5722.028-2.2105.3340.0620.0461069.0956.01077.01131.01.0
                                        beta[1]-0.2349.916-18.65718.0590.3010.2131085.01085.01093.01210.01.0
                                        sigma1.1470.0801.0031.3030.0010.0017060.06958.07181.05055.01.0
                                        \n
                                        \n\n\n\nThis linear regression example is from the original paper on PyMC3: *Salvatier J, Wiecki TV, Fonnesbeck C. 2016. Probabilistic programming in Python using PyMC3. PeerJ Computer Science 2:e55 https://doi.org/10.7717/peerj-cs.55*\n\n [Top](#top)\n\n## 4. Try this at Home: Example on Mining Disasters\nWe will go over the classical `mining disasters from 1851 to 1962` dataset. \n\nThis example is from the [pyMC3 Docs](https://docs.pymc.io/notebooks/getting_started.html).\n\n\n```python\nimport pandas as pd\ndisaster_data = pd.Series([4, 5, 4, 0, 1, 4, 3, 4, 0, 6, 3, 3, 4, 0, 2, 6,\n 3, 3, 5, 4, 5, 3, 1, 4, 4, 1, 5, 5, 3, 4, 2, 5,\n 2, 2, 3, 4, 2, 1, 3, np.nan, 2, 1, 1, 1, 1, 3, 0, 0,\n 1, 0, 1, 1, 0, 0, 3, 1, 0, 3, 2, 2, 0, 1, 1, 1,\n 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 1, 0, 2,\n 3, 3, 1, np.nan, 2, 1, 1, 1, 1, 2, 4, 2, 0, 0, 1, 4,\n 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1])\nfontsize = 12\nyears = np.arange(1851, 1962)\nplt.figure(figsize=(10,5))\n#plt.scatter(years, disaster_data); \nplt.bar(years, disaster_data)\nplt.ylabel('Disaster count', size=fontsize)\nplt.xlabel('Year', size=fontsize);\nplt.title('Was there a Turning Point in Mining disasters from 1851 to 1962?', size=15);\n```\n\n#### Building the model\n\n**Step1:** We choose the probability model for our experiment. Occurrences of disasters in the time series is thought to follow a **Poisson** process with a large **rate** parameter in the early part of the time series, and from one with a smaller **rate** in the later part. We are interested in locating the change point in the series, which perhaps is related to changes in mining safety regulations. \n\n```\ndisasters = pm.Poisson('disasters', rate, observed=disaster_data)\n```\n\nWe have two rates, `early_rate` if $t<=s$, and `late_rate` if $t>s$, where $s$ is the year the switch was made (a.k.a. the `switchpoint`). \n\n**Step2:** Choose a prior distributions of the two rates, what we believe the rates were before we observed the data, and the switchpoint. We choose Exponential.\n```\nearly_rate = pm.Exponential('early_rate', 1)\n```\n\nThe parameters of this model are: \n\n\n**Note:** Watch for missing values. Missing values are handled transparently by passing a MaskedArray or a pandas.DataFrame. Behind the scenes, another random variable, disasters.missing_values is created to model the missing values. If you pass a np.array with missing values you will get an error.\n\n\n```python\nwith pm.Model() as disaster_model:\n\n # discrete\n switchpoint = pm.DiscreteUniform('switchpoint', lower=years.min(), upper=years.max(), testval=1900)\n\n # Priors for pre- and post-switch rates number of disasters\n early_rate = pm.Exponential('early_rate', 1)\n late_rate = pm.Exponential('late_rate', 1)\n\n # our theta - allocate appropriate Poisson rates to years before and after current\n # switch is an `if` statement in puMC3\n rate = pm.math.switch(switchpoint >= years, early_rate, late_rate)\n\n # our observed data as a likelihood function of the `rate` parameters\n # shows how we think our data is distributed\n disasters = pm.Poisson('disasters', rate, observed=disaster_data)\n```\n\n#### Model Fitting\n\n\n```python\n# there are defaults but we can also more explicitly set the sampling algorithms\nwith disaster_model:\n \n # for continuous variables\n step1 = pm.NUTS([early_rate, late_rate])\n \n # for discrete variables\n step2 = pm.Metropolis([switchpoint, disasters.missing_values[0]] )\n\n trace = pm.sample(10000, step=[step1, step2])\n # try different number of samples\n #trace = pm.sample(5000, step=[step1, step2])\n```\n\n Multiprocess sampling (4 chains in 4 jobs)\n CompoundStep\n >NUTS: [late_rate, early_rate]\n >CompoundStep\n >>Metropolis: [disasters_missing]\n >>Metropolis: [switchpoint]\n Sampling 4 chains, 0 divergences: 75%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u258c | 31500/42000 [00:30<00:04, 2506.23draws/s]\n\n#### Posterior Analysis\n\nOn the left side plots we notice that our early rate is between 2.5 and 3.5 disasters a year. In the late period it seems to be between 0.6 and 1.2 so definitely lower.\n\nThe right side plots show the samples we drew to come to our conclusion.\n\n\n```python\npm.traceplot(trace, ['early_rate', 'late_rate', 'switchpoint'], figsize=(20,10));\n```\n\n\n```python\nresults = pm.summary(trace, \n var_names=['early_rate', 'late_rate', 'switchpoint'])\nresults\n```\n", "meta": {"hexsha": "4a9d3a819bbf2839f96473d8903f86edfd2952c9", "size": 518391, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "content/labs/lab04/notebook/cs109b_lab04_bayes.ipynb", "max_stars_repo_name": "simonwarchol/2020-CS109B", "max_stars_repo_head_hexsha": "e3ab6307ca7701beee44c5436deb68010b5a2bb6", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "content/labs/lab04/notebook/cs109b_lab04_bayes.ipynb", "max_issues_repo_name": "simonwarchol/2020-CS109B", "max_issues_repo_head_hexsha": "e3ab6307ca7701beee44c5436deb68010b5a2bb6", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "content/labs/lab04/notebook/cs109b_lab04_bayes.ipynb", "max_forks_repo_name": "simonwarchol/2020-CS109B", "max_forks_repo_head_hexsha": "e3ab6307ca7701beee44c5436deb68010b5a2bb6", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 385.1344725111, "max_line_length": 219076, "alphanum_fraction": 0.9278112467, "converted": true, "num_tokens": 8179, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.7606506418255928, "lm_q1q2_score": 0.41881984598827937}} {"text": "\n\n# TensorFlow Tutorial - RNNs\n\nIn this tutorial, you learn how to build a recurrent neural network (RNN) to model time series data. More specifically, we will implement a model that generates text by predicting one character at a time. This model is widely based on this well-known [blog post](http://karpathy.github.io/2015/05/21/rnn-effectiveness/) by Andrej Karpathy. It is highly recommended to read through the post, as it is a great read. Additionally, we borrow some code from an implementation of this post in TensorFlow, taken from [here](https://github.com/sherjilozair/char-rnn-tensorflow).\n\nIn the following, we assume familiarity with the TensorFlow tutorial presented in previous tutorials, i.e., you should be aware of TensorFlow's core concepts, such as graph, session, input placeholders, etc. Furthermore, we will not demonstrate any usage of TensorBoard here - for this, refer to the tutorial on CNNs.\n\nThis tutorial consists of:\n 1. Introduction to RNNs\n 2. A Look at the Data\n 3. Building the Model\n 4. Generating Text\n 5. Concluding Remarks and Exercises\n 6. _[Appendix]_\n\n## Introduction to RNNs\n### Learning from Sequences\nA recurrent neural network (RNN) is a specialized architecture designed to model time series and sequence data. A question often associated with sequences is: Given a number of time steps in the sequence, what is the most likely next outcome? In other words, we want the model to output the following probability\n$$\np(\\mathbf{x}_t | \\mathbf{x}_0, \\dotsc, \\mathbf{x}_{t-1})\n$$\n\nwhere $t$ is the index over time. This is also the task we want to solve in this tutorial, namely, given a sequence of characters, what is the most likely next character?\n\nTo answer the above question, it is a good idea to keep some sort of memory as we walk along the sequence because previous observations in the past influence the desired outcome in the future. As an example, consider predicting the last character in the following sentence (indicated by the question mark):\n\n
                                        \n\nFor us, it is obvious that we should end it with double quotes. However, the model must _remember_ that there was an opening quotation mark that hasn't been closed yet. It gets even more trickier in the following example:\n\n
                                        \n\nTo complete this sentence, the model must not only realize that the missing word is a noun, but it must also remember that we were talking about Italy in the beginning of the sentence and that the capital of Italy is Rome. To achieve this, it is not enough that the model only knows what characters are. It must have some notion of more abstract concepts of the underlying language - it has to learn how characters are formed to create words and how words are structured to build sentences and so on. Learning to understand text on all these different levels is difficult and being able to capture long-term dependencies is essential for such a task.\n\n### Vanilla RNNs\nIn neural networks we model time-dependencies by introducing connections that loop back to the same node (hence the name _recurrent_). The recurrent connection typically updates the internal state/memory of the cell. Such an architecture can be drawn as is shown in the following ([source](http://www.deeplearningbook.org/contents/rnn.html)).\n\n
                                        \n\nOn the left you see the compact version of the graph. The model takes as input $\\mathbf{x}$ and processes it over time while the recurrent connection updates the internal state $\\mathbf{h}$ located in the memory cell. On the right side you see the unfolded version of the same graph, where we basically discretize the time dimension and make every time step explicit. This \"unfolding\" or \"unrolling\" of an RNN is what we have to do in practice when training. This step makes the model a finite, computational graph that we can actually work with.\n\nThe model shown above is a bit too simplistic - it just processes the input over time, but does not produce an output. A more realistic RNN looks like this ([source](http://www.deeplearningbook.org/contents/rnn.html)):\n\n
                                        \n\nLet's have a look at that in more detail. $\\mathbf{x}$ is the input to the model, $\\mathbf{o}$ is the output, and $L$ is a loss function that measures how much the output deviates from the target $\\mathbf{y}$. In our case, $\\mathbf{x}$ is a sequence of characters and the model produces character $\\mathbf{o}_t$ at every time step $t$. We compare it to the target character $\\mathbf{y}_t$ which is equivalent to the next character $\\mathbf{x}_{t+1}$ in our sequence. Note that instead of outputting one specific character, we rather produce a probability distribution over all characters in the vocabulary (more on this later). This is similar to what we did with the CNN for image classification where we produce a probability of belonging to a class, instead of making a hard assignment.\n\nThe real magic of RNNs happens in the recurrent cell, which we sometimes also call _memory cell_ because it tracks the memory, or hidden state, $\\mathbf{h}$. The question is now, how do we update the state $\\mathbf{h}$? In the vanilla formulation we model it as follows:\n\n$$\n\\begin{align}\n\\mathbf{a}^{(t)} &= \\mathbf{W} \\cdot \\mathbf{h}^{(t-1)} + \\mathbf{U} \\cdot \\mathbf{x}^{(t)} + \\mathbf{b}\\\\\n\\mathbf{h}^{(t)} &= \\tanh(\\mathbf{a}^{(t)})\\\\\n\\mathbf{o}^{(t)} &= \\mathbf{V} \\cdot \\mathbf{h}^{(t)} + \\mathbf{c}\n\\end{align}\n$$\n\nHere the matrices $\\mathbf{W}$, $\\mathbf{U}$, $\\mathbf{V}$, and biases $\\mathbf{b}$ and $\\mathbf{c}$ represent the learnable parameters of this model. Importantly, those parameters are _shared_ between time steps, i.e. every time step gets exactly the same copy of weights to work with.\n\n\n\n\n### LSTM Cells\nDespite its seeming simplicity, the vanilla RNN is already a powerful model. The only problem is that it turned out to be quite difficult to train such an RNN in practice. The reason for this is known as the _vanishing or exploding gradients problem_, which has been introduced in the lecture. In short, when we optimize the weights of an RNN, we end up backpropagating gradients through time. Because of the chain rule, gradients that arrive at layer $t$ are the product of a bunch of gradients of the layers from $t+1$ to $\\tau$ (assuming we unfold the RNN for $\\tau$ time steps in total). Now if each gradient in this large product is small (or big), the multiplication will make the resulting gradient even smaller (or bigger). This is especially a problem for \"early\" layers and if $\\mathbf{\\tau}$ is large, i.e., if we want to capture long-term dependencies. If you would like to read more about this, you can find a great article [here](http://neuralnetworksanddeeplearning.com/chap5.html).\n\nSo, what can we do to alleviate the problem of unstable gradients in RNNs? One answer was proposed in the seminal work by [Hochreiter and Schmidhuber, 1997](http://www.bioinf.jku.at/publications/older/2604.pdf) where they introduced Long Short Term Memory (LSTM) cells. These cells were designed to remember information for long periods of time and thus have made training of RNNs considerably easier. The following shows a schematic overview of the inner workings of an LSTM cell ([source](https://codeburst.io/generating-text-using-an-lstm-network-no-libraries-2dff88a3968)): \n\n
                                        \n\n\n\nIf you would like to read more about LSTM cells, we highly recommend to read [this excellent post](http://colah.github.io/posts/2015-08-Understanding-LSTMs/) from colah's blog. In a nutshell, the most important differences between a vanilla and a LSTM cell are:\n - The LSTM cell has two hidden variables instead of just one, the hidden state $\\mathbf{h}$ and the cell state $\\mathbf{c}$. \n - Updates to the cell state are carefully protected by three gating functions consisting of a sigmoid layer and an element-wise multiplication (denoted by $\\otimes$ or $\\circ$).\n - Notice that the cell state $\\mathbf{c}_{t-1}$ can more or less easily flow through the cell (top line in the above diagram) and thus propagate the information further into the next time step.\n \nLSTMs have made training of recurrent structures much easier and have thus become the de-facto standard in RNNs. Of course, there are more cell types available (which you can find out about for example in colah's blog), but LSTMs are usually a good initial choice for a recurrent architecture.\n\n## A Look at the Data\nLet's now turn to our actual problem of training a character-level language model. Following Andrej Karpathy's [post](http://karpathy.github.io/2015/05/21/rnn-effectiveness/), we are using the Shakespeare dataset which is just a text file containing some of Shakespeare's work. Here is an excerpt:\n\n
                                        \n\nWhat we want to achieve with the model can be summarized in the following diagram ([source](http://karpathy.github.io/2015/05/21/rnn-effectiveness/)).\n\n
                                        \n\nFor the sake of simplicity, this diagram assumes a limited vocabulary of only 4 characters `[h, e, l, o]`. The input to the model is the word \"hello\". On the bottom you can see the input at each time step and a one-hot encoding of each character shown in red. Similarly at the top you find the target characters for each input character and a predicted confidence score shown in blue units. For example, in the first time step the confidence attributed to the letter `e` is 2.2, while the confidence for `o` is 4.1. Ideally, the confidence for the bold numbers in the blue boxes should be high, while for the red numbers it should be low. In the middle part of the diagram, shown in green, are the hidden states of the recurrent cells. Through this layer and the associated hidden state vectors, the information is propagated to future time steps, so that hopefully in the last time step the letter \"o\" will have a high confidence score.\n\nWe are now going to implement this architecture, but with a larger vocabulary and more training data. To use the Shakespeare data set, we need to preprocess the data, i.e., tokenize it, extract the vocabulary, and create batches of a certain sequence length (depending on how many time steps $\\tau$ we want to unfold the RNN for). To do this, we shamelessly copy the code from [this implementation](https://github.com/sherjilozair/char-rnn-tensorflow).\n\n\n```\nimport codecs\nimport os\nimport collections\nimport pickle\nimport numpy as np\n%tensorflow_version 1.x\nimport tensorflow as tf\nfrom tensorflow.python.util import deprecation\ndeprecation._PRINT_DEPRECATION_WARNINGS = False\n```\n\n\n```\nclass TextLoader():\n def __init__(self, data_dir, batch_size, seq_length, encoding='utf-8'):\n self.data_dir = data_dir\n self.batch_size = batch_size\n self.seq_length = seq_length\n self.encoding = encoding\n self.rng = np.random.RandomState(42)\n\n input_file = os.path.join(data_dir, \"shakespeare.txt\")\n vocab_file = os.path.join(data_dir, \"vocab.pkl\")\n tensor_file = os.path.join(data_dir, \"data.npy\")\n\n if not (os.path.exists(vocab_file) and os.path.exists(tensor_file)):\n print(\"reading text file\")\n self.preprocess(input_file, vocab_file, tensor_file)\n else:\n print(\"loading preprocessed files\")\n self.load_preprocessed(vocab_file, tensor_file)\n self.create_batches()\n self.reset_batch_pointer()\n\n def preprocess(self, input_file, vocab_file, tensor_file):\n with codecs.open(input_file, \"r\", encoding=self.encoding) as f:\n data = f.read()\n counter = collections.Counter(data)\n count_pairs = sorted(counter.items(), key=lambda x: -x[1])\n self.chars, _ = zip(*count_pairs)\n self.vocab_size = len(self.chars)\n self.vocab = dict(zip(self.chars, range(len(self.chars))))\n with open(vocab_file, 'wb') as f:\n pickle.dump(self.chars, f)\n self.tensor = np.array(list(map(self.vocab.get, data)))\n np.save(tensor_file, self.tensor)\n\n def load_preprocessed(self, vocab_file, tensor_file):\n with open(vocab_file, 'rb') as f:\n self.chars = pickle.load(f)\n self.vocab_size = len(self.chars)\n self.vocab = dict(zip(self.chars, range(len(self.chars))))\n self.tensor = np.load(tensor_file)\n self.num_batches = int(self.tensor.size / (self.batch_size *\n self.seq_length))\n\n def create_batches(self):\n self.num_batches = int(self.tensor.size / (self.batch_size *\n self.seq_length))\n\n # When the data (tensor) is too small,\n # let's give them a better error message\n if self.num_batches == 0:\n assert False, \"Not enough data. Make seq_length and batch_size small.\"\n\n self.tensor = self.tensor[:self.num_batches * self.batch_size * self.seq_length]\n xdata = self.tensor\n ydata = np.copy(self.tensor)\n ydata[:-1] = xdata[1:]\n ydata[-1] = xdata[0]\n self.x_batches = np.split(xdata.reshape(self.batch_size, -1),\n self.num_batches, 1)\n self.y_batches = np.split(ydata.reshape(self.batch_size, -1),\n self.num_batches, 1)\n\n def next_batch(self):\n x, y = self.x_batches[self.pointer], self.y_batches[self.pointer]\n self.pointer += 1\n return x, y\n\n def reset_batch_pointer(self):\n self.pointer = 0\n \n def maybe_new_epoch(self):\n if self.pointer >= self.num_batches:\n # this is a new epoch\n self.reset_batch_pointer()\n self.reshuffle()\n \n def reshuffle(self):\n idx = self.rng.permutation(len(self.x_batches))\n self.x_batches = [self.x_batches[i] for i in idx]\n self.y_batches = [self.y_batches[i] for i in idx]\n```\n\nLet's also define some configuration parameters like we did for the CNN tutorial.\n\n\n```\ndef del_all_flags(FLAGS):\n flags_dict = FLAGS._flags() \n keys_list = [keys for keys in flags_dict] \n for keys in keys_list:\n FLAGS.__delattr__(keys)\n\ndel_all_flags(tf.flags.FLAGS)\n\ntf.app.flags.DEFINE_string(\"data_dir\", \"./\", \"Where the training data is stored\")\ntf.app.flags.DEFINE_string(\"log_dir\", \"/tmp/tensorflow/shakespeare_rnn/logs\", \"Where to store summaries and checkpoints\")\ntf.app.flags.DEFINE_float(\"learning_rate\", 1e-3, \"Learning rate (default: 1e-3)\")\ntf.app.flags.DEFINE_integer(\"batch_size\", 128, \"Batch size (default: 50)\")\ntf.app.flags.DEFINE_integer(\"seq_length\", 50, \"Number of time steps to unrol the RNN for\")\ntf.app.flags.DEFINE_integer(\"hidden_size\", 256, \"Size of one LSTM hidden layer\")\ntf.app.flags.DEFINE_integer(\"num_layers\", 2, \"How many LSTM layers to use\")\ntf.app.flags.DEFINE_integer(\"print_every_steps\", 20, \"How often to print progress to the console\")\ntf.app.flags.DEFINE_string('f', '', 'kernel') # Dummy entry because colab is weird.\n```\n\n\n```\nFLAGS = tf.app.flags.FLAGS\nprint(\"\\nCommand-line Arguments:\")\nfor key in FLAGS.flag_values_dict():\n print(\"{:<22}: {}\".format(key.upper(), FLAGS[key].value))\nprint(\" \")\n```\n\n \n Command-line Arguments:\n DATA_DIR : ./\n LOG_DIR : /tmp/tensorflow/shakespeare_rnn/logs\n LEARNING_RATE : 0.001\n BATCH_SIZE : 128\n SEQ_LENGTH : 50\n HIDDEN_SIZE : 256\n NUM_LAYERS : 2\n PRINT_EVERY_STEPS : 20\n F : \n \n\n\n\n```\n!if [ ! -f eye_data.h5 ]; then wget -nv https://ait.ethz.ch/projects/shakespeare.txt?raw=true -O shakespeare.txt; fi\n```\n\n 2020-03-20 17:12:05 URL:https://ait.ethz.ch/projects/shakespeare.txt?raw=true [1155394/1155394] -> \"shakespeare.txt\" [1]\n\n\n\n```\n# load the data\ndata_loader = TextLoader(FLAGS.data_dir, FLAGS.batch_size, FLAGS.seq_length)\nprint(\"loaded vocabulary with {} letters\".format(data_loader.vocab_size))\n```\n\n reading text file\n loaded vocabulary with 66 letters\n\n\n\n```\n# visualize some data\nb = data_loader.x_batches[0]\nt = data_loader.y_batches[0]\nprint('total of {} batches of shape: {}'.format(len(data_loader.x_batches), b.shape))\nprint('content of batch 0, entry 0, time steps 0 to 20')\nprint('input : {}'.format(b[0, :20]))\nprint('target: {}'.format(t[0, :20]))\n```\n\n total of 180 batches of shape: (128, 50)\n content of batch 0, entry 0, time steps 0 to 20\n input : [50 9 7 6 2 0 38 9 2 9 58 1 8 25 10 11 44 1 19 3]\n target: [ 9 7 6 2 0 38 9 2 9 58 1 8 25 10 11 44 1 19 3 7]\n\n\n\n```\n# print characters instead of integers\n# invert the vocabulary (note that this works because the vocabulary is distinct)\nvocab_inv = {v: k for k, v in data_loader.vocab.items()}\nprint('input : {}'.format([vocab_inv[i] for i in b[0, :20]]))\nprint('target: {}'.format([vocab_inv[i] for i in t[0, :20]]))\n```\n\n input : ['F', 'i', 'r', 's', 't', ' ', 'C', 'i', 't', 'i', 'z', 'e', 'n', ':', '\\r', '\\n', 'B', 'e', 'f', 'o']\n target: ['i', 'r', 's', 't', ' ', 'C', 'i', 't', 'i', 'z', 'e', 'n', ':', '\\r', '\\n', 'B', 'e', 'f', 'o', 'r']\n\n\nGreat - we now have a way of tokenizing text and organizing it into batches of a given size and sequence length. Next, we'll look into how to build the actual RNN.\n\n## Building the Model\nWe start by building the core of our model, the RNN with LSTM cells.\n\n\n```\ndef rnn_lstm(inputs, hidden_size, num_layers, seq_lengths):\n \"\"\"\n Builds an RNN with LSTM cells.\n :param inputs: The input tensor to the RNN in shape `[batch_size, seq_length]`.\n :param hidden_size: The number of units for each LSTM cell.\n :param num_layers: The number of LSTM cells we want to use.\n :param seq_lengths: Tensor of shape `[batch_size]` specifying the total number\n of time steps per sequence.\n :return: The initial state, final state, predicted logits and probabilities.\n \"\"\"\n # we first create a one-hot encoding of the inputs\n # the resulting shape is `[batch_size, seq_length, vocab_size]`\n vocab_size = data_loader.vocab_size\n input_one_hot = tf.one_hot(inputs, vocab_size, axis=-1)\n \n # create a list of all LSTM cells we want\n cells = [tf.contrib.rnn.LSTMCell(hidden_size) for _ in range(num_layers)]\n \n # we stack the cells together and create one big RNN cell\n cell = tf.contrib.rnn.MultiRNNCell(cells)\n \n # we need to set an initial state for the cells\n batch_size = tf.shape(inputs)[0]\n initial_state = cell.zero_state(batch_size, dtype=tf.float32)\n \n # now we are ready to unrol the graph\n outputs, final_state = tf.nn.dynamic_rnn(cell=cell,\n initial_state=initial_state,\n inputs=input_one_hot,\n sequence_length=seq_lengths)\n \n # The `outputs` tensor has shape `[batch_size, seq_length, hidden_size]`,\n # i.e. it contains the outputs of the last cell for every time step.\n # We want to map the output back to the \"vocabulary space\", so we add a dense layer.\n # Importantly, the dense layer should share its parameters across time steps.\n # To do this, we first flatten the outputs to `[batch_size*seq_length, hidden_size]`\n # and then add the dense layer.\n max_seq_length = tf.shape(inputs)[1]\n outputs_flat = tf.reshape(outputs, [-1, hidden_size])\n \n # dense layer\n weights = tf.Variable(tf.truncated_normal([hidden_size, vocab_size], stddev=0.1))\n bias = tf.Variable(tf.constant(0.1, shape=[vocab_size]))\n logits_flat = tf.matmul(outputs_flat, weights) + bias\n \n # reshape back\n logits = tf.reshape(logits_flat, [batch_size, max_seq_length, vocab_size])\n \n # activate to turn logits into probabilities\n probs = tf.nn.softmax(logits)\n \n # we return the initial and final states because this will be useful later\n return initial_state, final_state, logits, probs\n```\n\nA note about the use of `tf.nn.dynamic_rnn`. When dealing with sequences it is often the case that not all have the same length. Hence, we need to ask ourselves two questions:\n 1. How do we handle sequences of different lengths in the same batch?\n 2. Do we want to use different sequence lengths at inference time than at training time?\n \nTo answer the first question, we can simply pad all sequences in a batch with dummy values to the maximum length occurring in that batch. Of course, we should tell TensorFlow that it does not make sense to unrol the RNN further for these dummy values. This is why we supply the `seq_lengths` tensor, which is actually important to guarantee correctness during back-propagation. Note that in our example, we do not need to pad the data, because our data loader already ensures that we have sequences of equal length. However, you should generally be aware of that caveat.\n\nTo address the second question TensorFlow knows two functions: `tf.nn.dynamic_rnn` and `tf.nn.static_rnn`. In the static RNN, TensorFlow creates an unrolled graph for a fixed length (say 100). It is still possible to use this graph for sequences of length `< 100` (by supplying the `seq_lengths` tensor as mentioned above), but during inference time, we cannot use it for more than 100 time steps. The dynamic RNN on the other hand can handle variable sequence lengths - it unrols the graph in a `tf.while` loop directly on the GPU. In other words, the time dimension in the input placeholder can be `None`, like for the batch size. It is recommended to always pass a `seq_lengths` tensor into the dynamic RNN function.\n\n
                                        \n\nIn the following we have to take care of the remaining tasks: build input placeholders, add a loss function, define the optimizer and define the training loop.\n\n\n```\n# create input placeholders\nwith tf.name_scope(\"input\"):\n # shape is `[batch_size, seq_length]`, both are dynamic\n text_input = tf.placeholder(tf.int32, [None, None], name='x-input')\n # shape of target is same as shape of input\n text_target = tf.placeholder(tf.int32, [None, None], name='y-input')\n # sequence length placeholder\n seq_lengths = tf.placeholder(tf.int32, [None], name='seq-lengths')\n```\n\n\n```\n# build the model\ninitial_state, final_state, logits, probs = rnn_lstm(text_input,\n FLAGS.hidden_size,\n FLAGS.num_layers,\n seq_lengths)\n```\n\n \n WARNING: The TensorFlow contrib module will not be included in TensorFlow 2.0.\n For more information, please see:\n * https://github.com/tensorflow/community/blob/master/rfcs/20180907-contrib-sunset.md\n * https://github.com/tensorflow/addons\n If you depend on functionality not listed there, please file an issue.\n \n\n\n\n```\n# We use the same loss function as we did for the CNN tutorial, i.e. Cross-Entropy.\n# This time, we have to compute it for each time step, so we use a TensorFlow function\n# that takes care of this for us.\nwith tf.name_scope(\"cross-entropy\"):\n # Again, we should supply the logits, not the softmax-activated probabilities\n cross_entropy_loss = tf.contrib.seq2seq.sequence_loss(\n logits, text_target,\n weights=tf.ones_like(text_input, dtype=tf.float32))\n tf.summary.scalar('cross_entropy_loss', cross_entropy_loss)\n \n # The weights allow weighing the contribution of each batch entry and time step (here we don't use it).\n # This parameter can be useful if we have padded values (which we don't here). In this case, `weights`\n # serves like a mask that you should set to 0 for padded values, e.g. like this:\n # weights = tf.sequence_mask(seq_lengths, max_seq_length)\n```\n\n\n```\n# check number of trainable parameters\ndef count_trainable_parameters():\n \"\"\"Counts the number of trainable parameters in the current default graph.\"\"\"\n tot_count = 0\n for v in tf.trainable_variables():\n v_count = 1\n for d in v.get_shape():\n v_count *= d.value\n tot_count += v_count\n return tot_count\nprint(\"Number of trainable parameters: {}\".format(count_trainable_parameters()))\n```\n\n Number of trainable parameters: 873026\n\n\n\n```\n# create the optimizer\nglobal_step = tf.Variable(1, name='global_step', trainable=False)\nwith tf.name_scope(\"train\"):\n # choose Adam optimizer\n optim = tf.train.AdamOptimizer(FLAGS.learning_rate)\n \n # get the gradients\n params = tf.trainable_variables()\n gradients = tf.gradients(cross_entropy_loss, params)\n \n # clip the gradients to counteract exploding gradients\n clipped_gradients, _ = tf.clip_by_global_norm(gradients, 5)\n \n # backprop\n train_step = optim.apply_gradients(zip(clipped_gradients, params), global_step=global_step)\n```\n\n\n```\ndef do_train_step(num_steps, summary_op):\n \"\"\"Perform as many training steps as specified.\"\"\"\n for i in range(num_steps):\n step = tf.train.global_step(sess, global_step)\n \n # reset batch pointer and shuffle the data if necessary\n data_loader.maybe_new_epoch()\n \n # get next batch\n x, y = data_loader.next_batch()\n \n # prepare feed_dict\n feed_dict = {text_input: x, text_target: y, seq_lengths: [x.shape[1]]*x.shape[0]}\n \n summary, train_loss, _ = sess.run([summary_op, cross_entropy_loss, train_step],\n feed_dict=feed_dict)\n \n writer_train.add_summary(summary, step)\n \n if step % FLAGS.print_every_steps == 0:\n print('[{}] Cross-Entropy Loss Training [{:.3f}]'.format(step, train_loss)) \n```\n\n\n```\n# Create the session\nsess = tf.InteractiveSession()\n\n# Initialize all variables\nsess.run(tf.global_variables_initializer())\n\n# To be able to see something in tensorboard, we must merge summaries to one common operation.\n# Whenever we want to write summaries, we must request this operation from the graph.\n# Note: creating the file writers should happen after the session was launched.\nsummaries_merged = tf.summary.merge_all()\nwriter_train = tf.summary.FileWriter(FLAGS.log_dir + '/train', sess.graph)\n```\n\nNow let's train this model for a couple of steps. Make sure you selected the GPU under Runtime > Change runtime type > Hardware Accelerator. Otherwise training will be quite slow.\n\n\n```\ndo_train_step(1001, summaries_merged)\n```\n\n [20] Cross-Entropy Loss Training [3.443]\n [40] Cross-Entropy Loss Training [3.356]\n [60] Cross-Entropy Loss Training [3.315]\n [80] Cross-Entropy Loss Training [3.206]\n [100] Cross-Entropy Loss Training [3.081]\n [120] Cross-Entropy Loss Training [2.922]\n [140] Cross-Entropy Loss Training [2.787]\n [160] Cross-Entropy Loss Training [2.628]\n [180] Cross-Entropy Loss Training [2.485]\n [200] Cross-Entropy Loss Training [2.415]\n [220] Cross-Entropy Loss Training [2.368]\n [240] Cross-Entropy Loss Training [2.320]\n [260] Cross-Entropy Loss Training [2.328]\n [280] Cross-Entropy Loss Training [2.263]\n [300] Cross-Entropy Loss Training [2.242]\n [320] Cross-Entropy Loss Training [2.172]\n [340] Cross-Entropy Loss Training [2.184]\n [360] Cross-Entropy Loss Training [2.166]\n [380] Cross-Entropy Loss Training [2.114]\n [400] Cross-Entropy Loss Training [2.106]\n [420] Cross-Entropy Loss Training [2.095]\n [440] Cross-Entropy Loss Training [2.083]\n [460] Cross-Entropy Loss Training [2.058]\n [480] Cross-Entropy Loss Training [2.028]\n [500] Cross-Entropy Loss Training [2.019]\n [520] Cross-Entropy Loss Training [2.011]\n [540] Cross-Entropy Loss Training [1.966]\n [560] Cross-Entropy Loss Training [1.948]\n [580] Cross-Entropy Loss Training [1.981]\n [600] Cross-Entropy Loss Training [1.980]\n [620] Cross-Entropy Loss Training [1.919]\n [640] Cross-Entropy Loss Training [1.940]\n [660] Cross-Entropy Loss Training [1.893]\n [680] Cross-Entropy Loss Training [1.890]\n [700] Cross-Entropy Loss Training [1.874]\n [720] Cross-Entropy Loss Training [1.878]\n [740] Cross-Entropy Loss Training [1.855]\n [760] Cross-Entropy Loss Training [1.876]\n [780] Cross-Entropy Loss Training [1.829]\n [800] Cross-Entropy Loss Training [1.819]\n [820] Cross-Entropy Loss Training [1.806]\n [840] Cross-Entropy Loss Training [1.816]\n [860] Cross-Entropy Loss Training [1.785]\n [880] Cross-Entropy Loss Training [1.785]\n [900] Cross-Entropy Loss Training [1.770]\n [920] Cross-Entropy Loss Training [1.732]\n [940] Cross-Entropy Loss Training [1.739]\n [960] Cross-Entropy Loss Training [1.804]\n [980] Cross-Entropy Loss Training [1.773]\n [1000] Cross-Entropy Loss Training [1.752]\n\n\nThe loss is consistently decreasing, so that looks promising. Let's now look at another feature from TensorFlow: storing checkpoints. During training, it is a good idea to regularly save the model that you trained up to this point. Doing this with TensorFlow is pretty straight-forward.\n\n\n```\n# Create a saver object. We must specify which variables it should save to disk (in this case all)\n# and optionally how many checkpoints should be retained (in this case 2).\nsaver = tf.train.Saver(var_list=tf.trainable_variables(), max_to_keep=2)\n\n# new save the current session\nsaver.save(sess, os.path.join(FLAGS.log_dir, 'checkpoints', 'model_name'), global_step)\n```\n\n\n\n\n '/tmp/tensorflow/shakespeare_rnn/logs/checkpoints/model_name-1002'\n\n\n\nAnd that's it! Of course, saving a model is only useful if we can load it again (e.g. to do inference with it or to continue training). This is also quite easy to do. We just call a `restore` function on the saver object.\n\n\n```\n# get a handle to the latest checkpoint that was stored\nckpt_path = tf.train.latest_checkpoint(os.path.join(FLAGS.log_dir, 'checkpoints'))\n\n# now restore\nsaver.restore(sess, ckpt_path)\n```\n\nAgain, pretty easy. Note however that `saver.restore` only loads the saved weights into the graph, i.e. it assumes a suitable graph exists already. If it does, `restore` will most likely fail.\n\n## Generating Text\nWe have seen how we can train a model to predict a single character given an input sequence. But how can we use this model to generate text? This is what we will discuss in the following.\n\nOne way to do this is to generate text character-by-character and feeding the output of each time step back as input to the model. In other words, we get the output character for a given sequence, append that character to the sequence and repeat the whole process. This is illustrated in the following where the black text is the input sequence and the blue character is the output character.\n\n
                                        \n\nLet's implement this for our model.\n\n\n```\ndef sample(prime_text, num_steps):\n \"\"\"\n Sample `num_steps` characters from the model and initialize it with `prime_text`.\n :param prime_text: A string that we want to initialize the RNN with.\n :num_steps: Integer specifying how many characters we want to predict after `prime_text`.\n :return: `prime_text` plus prediction.\n \"\"\"\n # First we need to look up the initial text in the vocabulary\n input_prime = [data_loader.vocab[c] for c in prime_text]\n \n # Feed the prime sequence into the model. Note that we do not have to supply any targets\n # because we are not doing any backpropagation.\n feed_dict = {text_input: [input_prime],\n seq_lengths: [len(input_prime)]}\n state, out_probs = sess.run([final_state, probs], feed_dict=feed_dict)\n \n # Now we have initialized the RNN with the given prime text. Let's see what it predicts\n # as the next character after the last from `prime_text`.\n # `out_probs` is of shape `[1, len(prime_text), vocab_size]`\n next_char_probs = out_probs[0, -1]\n \n # `next_char_probs` is a probability distribution over all characters in the vocabulary.\n # How do we determine which character is next? We could just take the one that is most\n # probable of course. But let's implement something a bit different: actually sample\n # from the probability distribution.\n def weighted_pick(p_dist):\n cs = np.cumsum(p_dist)\n idx = int(np.sum(cs < np.random.rand()))\n return idx\n \n next_char = weighted_pick(next_char_probs)\n \n # save all predicted chars in a string\n predicted_text = vocab_inv[next_char]\n \n # now we can sample for `num_steps`\n for _ in range(num_steps):\n # Construct the feed dict. Note how we manually carry over the previous final state\n # and use it as the next initial state. If we don't do that, then TensorFlow will\n # automatically initialize the cells to the zero state and hence we lose all the\n # memory that we've built up to this point.\n feed_dict = {text_input: [[next_char]],\n seq_lengths: [1],\n initial_state: state}\n \n # get the prediction\n state, out_probs = sess.run([final_state, probs], feed_dict=feed_dict)\n \n # sample from the distribution\n next_char = weighted_pick(out_probs[0, -1])\n \n # append to already predicted text\n predicted_text += vocab_inv[next_char] \n \n return prime_text + predicted_text\n```\n\n\n```\nprint(sample('The ', 500))\n```\n\n The wither of our Gloved\r\n To well pricones or he' Elike.\r\n \r\n All Sore:\r\n I at the kins gill os is yet,\r\n And priens; is you with lought hou han, and not me a beill's sreital.\r\n \r\n KING EREWAll allow,\r\n What I wolewn would bruck deathan hall: serist our now,, do.\r\n \r\n LOrY CERLENR:\r\n \r\n GEONTES:\r\n Cormond the for.\r\n \r\n CApol:\r\n Nose but the' if your couther wafe,\r\n Ale, whyos in the itls to the, now how it wors, lit; allen the dreat,\r\n I dight iols; bue hou the prutcine poor go cance; nece ain weland,\r\n In thut crust \n\n\nDepending for how long you trained the model, you should now be able to see some nice outputs. As an example, here is the output of a model that was trained for 5000 steps.\n\n
                                        \n\nWe make a few observations:\n - The model successfully learned to create English words! Even if some might be purely imagined, they do sound at least like English words.\n - It also learned a great deal about the structure of the input data: mostly nice use of punctuation, it produces paragraphs that start with names in capital letters, etc.\n \nGiven the simplicity of our model, this is quite a nice result! Refer the Andrej Karpathy's [article](http://karpathy.github.io/2015/05/21/rnn-effectiveness/) to see some more results of the same model (applied to different datasets) and more visualizations of what's going on inside the RNN.\n\n\n```\n# cleanup\nsess.close()\n```\n\n## Concluding Remarks and Exercises\nIn this tutorial you learned how to build an RNN that predicts the next character given a sequence of characters and how you can turn it into a generative model that produces sample text of arbitrary length. You should now be aware of the most important implementation details needed to train an RNN in TensorFlow: difference between `tf.nn.static_rnn` and `tf.nn.dynamic_rnn`, potential need for padding, sharing weights when mapping RNN hidden states back to the output space, using initial and final state of the RNN to control the generation of sequences, etc.\n\nIn our example, we used an RNN to predict an output at each time step of the sequence. RNNs are however much more versatile than this and can be used in many more scenarios as shown here. ([source](http://karpathy.github.io/2015/05/21/rnn-effectiveness/))\n\n
                                        \n\nTensorFlow provides [functions](https://www.tensorflow.org/tutorials/seq2seq) to implement such models, which are sometimes also called _sequence-to-sequence_ (seq2seq) models.\n\nTo gain a deeper understanding of RNNs, we encourage you to make a copy of this notebook and play around with it. Specifically, in the following are a couple of (optional) exercises that you might want to look at. Furthermore, you also find information about some more advanced topics in the appendix section, which we provide for the purpose of self-study.\n\n 1. Read [Andrej Karpathy's](http://karpathy.github.io/2015/05/21/rnn-effectiveness/) and [colah's](http://colah.github.io/posts/2015-08-Understanding-LSTMs/) blog posts.\n 2. We did not use a validation set in this tutorial. Implement this and evaluate the validation set after every so-many epochs (like we did for the CNN tutorial).\n 3. Compute the training accuracy, print it to the console and visualize it in TensorBoard. Check the [CNN notebook](https://colab.research.google.com/drive/1bx2dlJYutNitK-hlhp98OHO-5WZnrNyV) to see how you can use Tensorboard on Colab.\n 4. Try out different cell types other than LSTMs, e.g. GRU.\n 5. How could you add some regularization to this model? \n 6. Play around with the hyper-parameters. What happens if you omit the gradient clipping? How susceptible is the training to changes in the learning rate? Can you find a model that has less parameters but performs equally well?\n\n## Appendix\n### Writing Your Own RNN Cell\nSometimes the standard LSTM cell provided by TensorFlow is just not enough, or sometimes you would like to do fancier stuff within a cell. Knowing how to write your own RNN cell that you can feed into `tf.nn.dynamic_rnn` can be very useful. E.g., we usually want to add a decoder on the outputs of the RNN, i.e. a dense layer that maps back to the output space. In the above model, we did this by reshaping the outputs of the rnn and then adding the dense layer on top of that. It would more elegant however, if we could do this directly in a cell, because anyway the dense layer operates independently on the output of each cell. To do this, we can write a custom RNN cell.\n\n\n```\nfrom tensorflow.python.ops.rnn_cell_impl import RNNCell\nfrom tensorflow.contrib.rnn import LSTMStateTuple\n\nclass DenseDecoderCell(RNNCell):\n \"\"\"\n Wraps an existing RNNCell by decoding the outputs of the cell before returning them.\n \"\"\"\n def __init__(self, cell, output_dim):\n if not isinstance(cell, RNNCell):\n raise TypeError(\"The parameter cell is not an RNNCell.\")\n \n super(DenseDecoderCell, self).__init__()\n self._cell = cell\n self._output_size = output_dim\n \n if hasattr(cell.state_size, '__getitem__'):\n # `cell` is a MultiRNNCell, so get the size from the last layer\n hidden_size = cell.state_size[-1]\n else:\n hidden_size = cell.state_size\n \n if isinstance(hidden_size, LSTMStateTuple):\n # LSTM cells have special state\n hidden_size = hidden_size.h\n \n # Create the weights of the decoder. However, we must be cautios.\n # An instance of this class is created for every time step we unrol\n # the model for. Because we want to share parameters between time\n # steps, we can't just instantiate new weight variables every time\n # this function is called. To avoid this, we use a nice feature\n # of TensorFlow which is `tf.get_variable()`. This function creates\n # a variable if it does not exist yet or else just retrieves it\n # from the graph and returns a handle to it.\n print(output_dim)\n self._decoder_w = tf.get_variable(\"decoder_cell_w\",\n shape=[hidden_size, output_dim],\n dtype=tf.float32,\n initializer=tf.truncated_normal_initializer(stddev=0.1))\n self._decoder_b = tf.get_variable(\"decoder_cell_b\",\n shape=[output_dim],\n dtype=tf.float32,\n initializer=tf.constant_initializer(0.1))\n \n @property\n def state_size(self):\n # just return the state size of the cell we are wrapping\n return self._cell.state_size\n \n @property\n def output_size(self):\n # must return the dimensionality of the tensor returned by this cell\n return self._output_size\n \n def __call__(self, inputs, state, scope=None):\n \"\"\"\n This is the function that is called at runtime. `inputs` is a tensor of shape\n `[batch_size, input_dimension]`, i.e., only one time step for the sequence is given.\n `state` is the cell's state from the previous time step and `scope` is just the\n current scope (can be ignored most of the time).\n \"\"\"\n # just forward our call to the cell\n output, next_state = self._cell(inputs, state, scope)\n \n # decode the output\n output_dec = tf.matmul(output, self._decoder_w) + self._decoder_b\n \n return output_dec, next_state\n```\n\nThis way, our original code from above to create an RNN, simplifies a bit. You can use the following function to replace `rnn_lstm` from above.\n\n\n```\ndef rnn_lstm_neat(inputs, hidden_size, num_layers, seq_lengths):\n \"\"\"\n Builds an RNN with LSTM cells.\n :param inputs: The input tensor to the RNN in shape `[batch_size, seq_length]`.\n :param hidden_size: The number of units for each LSTM cell.\n :param num_layers: The number of LSTM cells we want to use.\n :param seq_lengths: Tensor of shape `[batch_size]` specifying the total number\n of time steps per sequence.\n :return: The initial state, final state, predicted logits and probabilities.\n \"\"\"\n # we first create a one-hot encoding of the inputs\n # the resulting shape is `[batch_size, seq_length, vocab_size]`\n vocab_size = data_loader.vocab_size\n input_one_hot = tf.one_hot(inputs, vocab_size, axis=-1)\n \n # create a list of all LSTM cells we want\n cells = [tf.nn.rnn_cell.BasicLSTMCell(num_units=hidden_size) for _ in range(num_layers)]\n \n # we stack the cells together and create one big RNN cell\n cell = tf.nn.rnn_cell.MultiRNNCell(cells)\n \n # decode the outputs\n cell = DenseDecoderCell(cell, output_dim=vocab_size)\n \n # we need to set an initial state for the cells\n batch_size = tf.shape(inputs)[0]\n initial_state = cell.zero_state(batch_size, dtype=tf.float32)\n \n # now we are ready to unrol the graph\n outputs, final_state = tf.nn.dynamic_rnn(cell=cell,\n initial_state=initial_state,\n inputs=input_one_hot,\n sequence_length=seq_lengths)\n \n # The `outputs` tensor has now shape `[batch_size, seq_length, vocab_size]`,\n # so we can directly think of the output as the logits.\n logits = outputs\n \n # activate to turn logits into probabilities\n probs = tf.nn.softmax(logits)\n \n # we return the initial and final states because this will be useful later\n return initial_state, final_state, logits, probs\n```\n\n### Sharing Weights\nSharing weights between different structures of the graph is a very useful feature (as seen for example in the previous section about custom RNN cells). TensorFlow easily allows this via the function `tf.get_variable`. Essentially, `tf.get_variable` creates a variable if it does not exist yet or otherwise retrieves it by name from the current graph. Note that the current variable scope influences the name of the variable, so it is important to get the scope right. Here is a simple example.\n\n\n```\ntf.reset_default_graph()\nwith tf.variable_scope(\"great_scope\"):\n # Variables created here will be named \"great_scope/whatever_name\"\n w = tf.get_variable(\"weights\", shape=[256, 10], initializer=tf.constant_initializer(0.1))\n \nprint('all variable names', [v.name for v in tf.global_variables()])\n\n# create another variable but outside the scope, this will create a new variable because the variable scope\n# is not set, i.e. `tf.get_variable` searches for a variable named \"weights\" which does not yet exist, so\n# it creates it\nw2 = tf.get_variable(\"weights\", shape=[256, 10], initializer=tf.constant_initializer(0.1))\nprint('all variable names (one created)', [v.name for v in tf.global_variables()])\n\n# lets retrieve `w`\n# Note: if we do not set `reuse=True`, TensorFlow will throw an error. This is just to make\n# sure that you don't unintentionally share variables.\nwith tf.variable_scope(\"great_scope\", reuse=True):\n # Variables created here will be named \"great_scope/whatever_name\"\n w = tf.get_variable(\"weights\", shape=[256, 10], initializer=tf.constant_initializer(0.1))\nprint('all variable names (none created)', [v.name for v in tf.global_variables()])\n\n# If we set `reuse=True` but the variable does not exist, TensorFlow will also throw an error\n```\n\n all variable names ['great_scope/weights:0']\n all variable names (one created) ['great_scope/weights:0', 'weights:0']\n all variable names (none created) ['great_scope/weights:0', 'weights:0']\n\n\nThis can get tricky. Newer versions of the API take care of some of this for you. If you'd like to know more about how the sharing of variables works, we recommend reading [this post](https://chromium.googlesource.com/external/github.com/tensorflow/tensorflow/+/r1.0/tensorflow/g3doc/how_tos/variable_scope/index.md).\n\n### Bi-directional RNNs\nBi-directional RNNs (BiRNN) are a powerful variant of recurrent networks. Instead of having only a hidden layer that connects states forward in time, BiRNNs have an additional layer that connects states backward in time. This means, that every time step can draw information from the past as well as the future to produce its prediction. The computational graph of a BiRNN looks something like this ([source](http://www.deeplearningbook.org/contents/rnn.html)):\n\n
                                        \n\nYou can see that both recurrent layers are connected to the output, but they are not connected amongst themselves. For the task of building a character-level language model, we could technically use a BiRNN to predict single characters in gaps (or even more characters). However, it is then not straight-forward to turn this BiRNN into a generative model, because in order to generate predictions, we always need data from the future, too. Hence, the BiRNN - while powerful - is not always suitable for the problem at hand. TensorFlow's API supports the creation of BiRNNs, and it is not much different then creating a uni-directional RNN. In the older API the function is [`tf.compat.v1.nn.bidirectional_dynamic_rnn`](https://www.tensorflow.org/api_docs/python/tf/compat/v1/nn/bidirectional_dynamic_rnn). Under this link you will find a link to the new API as well.\n\n\n```\n\n```\n", "meta": {"hexsha": "3f7d4776c1b108ec3a74b9df2666bee9233c2a0a", "size": 70456, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "5_tensorflow_rnn.ipynb", "max_stars_repo_name": "zpgeng/Machine-Perception", "max_stars_repo_head_hexsha": "0acd10aaf520e7c60366361afb2c923db628b5fb", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "5_tensorflow_rnn.ipynb", "max_issues_repo_name": "zpgeng/Machine-Perception", "max_issues_repo_head_hexsha": "0acd10aaf520e7c60366361afb2c923db628b5fb", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "5_tensorflow_rnn.ipynb", "max_forks_repo_name": "zpgeng/Machine-Perception", "max_forks_repo_head_hexsha": "0acd10aaf520e7c60366361afb2c923db628b5fb", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 49.9333805811, "max_line_length": 1016, "alphanum_fraction": 0.5617548541, "converted": true, "num_tokens": 11123, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7826624789529375, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.4188014626327253}} {"text": "\n\nWe will need to import some helper code, so we need to run this\n\n\n```python\nimport os\nimport sys\nnb_dir = os.path.split(os.getcwd())[0]\nif nb_dir not in sys.path:\n sys.path.append(nb_dir)\n```\n\n# Colab\n\nWe will need to download some data for this notebook, so if you are using [colab](https://colab.research.google.com), set the `using_colab` flag below to `True` in order to clone our [github repo](https://github.com/probabll/dgm4nlp).\n\n\n```python\nusing_colab = False\n!ls\n```\n\n\n```python\nif using_colab:\n !rm -fr dgm4nlp sst\n !git clone https://github.com/vitutorial/exercises.git\n !cp -R exercises/SST/* ./ \n !ls\n```\n\nNow we can start our lab.\n\n\n```python\nimport torch\nfrom torch import nn\nimport torch.nn.functional as F\n# CPU should be fine for this lab\ndevice = torch.device('cuda:0') \n#device = torch.device('cuda' if torch.cuda.is_available() else 'cpu') \nfrom collections import OrderedDict\nimport numpy as np\n```\n\n# Sentiment Classification \n\n\nWe are going to augment a sentiment classifier with a layer of discrete latent variables which will help us improve the model's interpretability. But first, let's quickly review the baseline task.\n\n\nIn sentiment classification, we have some text input $x = \\langle x_1, \\ldots, x_n \\rangle$, e.g. a sentence or short paragraph, which expresses a certain sentiment $y$, i.e. one of $K$ classes, towards a subject (e.g. a film or a product). \n\n\n\nWe can learn a sentiment classifier by learning a categorical distribution over classes for a given input:\n\n\\begin{align}\nY|x &\\sim \\text{Cat}(f(x; \\theta))\n\\end{align}\n\nwhere the Categorical pmf is $\\text{Cat}(y|\\pi) = \\pi_y$.\n\nA categorical distribution over $K$ classes is parameterised by a $K$-dimensional probability vector, here we use a neural network $f$ to map from the input to this probability vector. Technically we say *a neural network parameterises our model*, that is, it computes the parameters of our categorical observation model. The figure below is a graphical depiction of the model: circled nodes are random variables (a shaded node is an observed variable), uncircled nodes are deterministic, a plate indicates multiple draws.\n\n\n\nThe neural network (NN) $f(\\cdot; \\theta)$ has parameters of its own, i.e. the weights of the various architecture blocks used, which we denote generically by $\\theta$.\n\nSuppose we have a dataset $\\mathcal D = \\{(x^{(1)}, y^{(1)}), \\ldots, (x^{(N)}, y^{(N)})\\}$ containing $N$ i.i.d. observations. Then we can use the log-likelihood function \n\\begin{align}\n\\mathcal L(\\theta|\\mathcal D) &= \\sum_{k=1}^{N} \\log P(y^{(k)}|x^{(k)}, \\theta) \\\\\n&= \\sum_{k=1}^{N} \\log \\text{Cat}(y^{(k)}|f(x^{(k)}; \\theta))\n\\end{align}\n to estimate $\\theta$ by maximisation:\n \\begin{align}\n \\theta^\\star = \\arg\\max_{\\theta \\in \\Theta} \\mathcal L(\\theta|\\mathcal D) ~ .\n \\end{align}\n \n\nWe can use stochastic gradient-ascent to find a local optimum of $\\mathcal L(\\theta|\\mathcal D)$, which only requires a gradient estimate:\n\n\\begin{align}\n\\nabla_\\theta \\mathcal L(\\theta|\\mathcal D) &= \\sum_{k=1}^{|\\mathcal D|} \\nabla_\\theta \\log P(y^{(k)}|x^{(k)}, \\theta) \\\\ \n&= \\sum_{k=1}^{|\\mathcal D|} \\frac{1}{N} N \\nabla_\\theta \\log P(y^{(k)}|x^{(k)}, \\theta) \\\\\n&= \\mathbb E_{\\mathcal U(1/N)} \\left[ N \\nabla_\\theta \\log P(y^{(K)}|x^{(K)}, \\theta) \\right] \\\\\n&\\overset{\\text{MC}}{\\approx} \\frac{N}{M} \\sum_{m=1}^M \\nabla_\\theta \\log P(y^{(k_m)}|x^{(k_m)}, \\theta) \\\\\n&\\text{where }K_m \\sim \\mathcal U(1/N)\n\\end{align}\n\nThis is a Monte Carlo (MC) estimate of the gradient computed on $M$ data points selected uniformly at random from $\\mathcal D$.\n\nFor as long as $f$ remains differentiable wrt to its inputs and parameters, we can rely on automatic differentiation to obtain gradient estimates.\n\nIn what follows we show how to design $f$ and how to extend this basic model to a latent-variable model.\n\n\n\n\n## Data\n\nWe provide you some code to load the data (see `sst.sstutil.examplereader`). Play with the snippet below and inspect a few training instances:\n\n\n```python\nfrom sstutil import examplereader, Vocabulary, load_glove \n\n\n# Let's load the data into memory.\nprint(\"Loading data\")\ntrain_data = list(examplereader('data/sst/train.txt'))\ndev_data = list(examplereader('data/sst/dev.txt'))\ntest_data = list(examplereader('data/sst/test.txt'))\n\nprint(\"train\", len(train_data))\nprint(\"dev\", len(dev_data))\nprint(\"test\", len(test_data))\n\nprint('\\n# Examples')\nexample = dev_data[0]\nprint(\"First dev example:\", example)\nprint(\"First dev example tokens:\", example.tokens)\nprint(\"First dev example label:\", example.label)\n```\n\n## Architecture\n\n\nThe function $f$ conditions on a high-dimensional input (i.e. text), so we need to convert it to continuous real vectors. This is the job of an *encoder*. \n\n**Embedding Layer**\n\nThe first step is to convert the words in $x$ to vectors, which in this lab we will do with a pre-trained embedding layer (we will use GloVe).\n\nWe will denote the embedding of the $i$th word of the input by:\n\n\\begin{equation}\n\\mathbf x_i = \\text{glove}(x_i)\n\\end{equation}\n\n**Encoder Layer**\n\nIn this lab, an encoder takes a sequence of input vectors $\\mathbf x_1^n$, each $I$-dimensional, and produces a sequence of output vectors $\\mathbf t_1^n$, each $O$-dimensional and a summary vector $\\mathbf h \\in \\mathbb R^O$:\n\n\\begin{equation}\n \\mathbf t_1^n, \\mathbf h = \\text{encoder}(\\mathbf x_1^n; \\theta_{\\text{enc}})\n\\end{equation}\n\nwhere we use $\\theta_{\\text{enc}}$ to denote the subset of parameters in $\\theta$ that are specific to this encoder block. \n\n*Remark:* in practice for a correct batched implementation, our encoders also take a mask matrix and a vector of lengths.\n\nExamples of encoding functions can be a feed-forward NN (with an aggregator based on sum or average/max pooling) or a recurrent NN (e.g. an LSTM/GRU). Other architectures are also possible.\n\n**Output Layer**\n\nFrom our summary vector $\\mathbf h$, we need to parameterise a categorical distribution over $K$ classes, thus we use\n\n\\begin{align}\nf(x; \\theta) &= \\text{softmax}(\\text{dense}_K(\\mathbf h; \\theta_{\\text{output}}))\n\\end{align}\n\nwhere $\\text{dense}_K$ is a dense layer with $K=5$ outputs and $\\theta_{\\text{output}}$ corresponds to its parameters (weight matrix and bias vector). Note that we need to use the softmax activation function in order to guarantee that the output of $f$ is a normalised probability vector.\n\n\n## Implementation\n\nTo leave an indication of the shape of tensors in the code, we use the following convention\n\n```python\n[B, T, D]\n```\n\nwhere `B` stands for `batch_size`, `T` stands for `time` (or rather *maximum sequence length*), and `D` is the size of the representation.\n\n\nConsider the following abstract Encoder class:\n\n\n```python\nclass Encoder(nn.Module):\n \"\"\"\n An Encoder for us is a function that\n 1. transforms a sequence of I-dimensional vectors into a sequence of O-dimensional vectors\n 2. summarises a sequence of I-dimensional vectors into one O-dimensional vector\n \n \"\"\"\n \n \n def __init__(self):\n super(Encoder, self).__init__()\n \n def forward(self, inputs, mask, lengths):\n \"\"\"\n The input is a batch-first tensor of token ids. Here is an example:\n \n Example of inputs (though rather than words, we have word ids):\n INPUTS MASK LENGTHS\n [the nice cat -PAD-] -> [1 1 1 0] [3]\n [the nice dog running] -> [1 1 1 1] [4]\n \n Note that:\n mask = inputs == 1\n lengths = mask.sum(dim=-1)\n \n :param inputs: [B, T, I]\n :param mask: [B, T]\n :param lengths: [B]\n :returns: [B, T, O], [B, O]\n where the first tensor is the transformed input\n and the second tensor is a summary of all inputs\n \"\"\"\n pass\n \n```\n\nLet's start easy, implement a *bag of words* encoder:\n\n\n```python\nclass BagOfWordsEncoder(Encoder):\n \"\"\"\n This encoder does not transform the input sequence, \n and its summary output is just a sum.\n \"\"\"\n \n pass\n```\n\n\n```python\n# SOLUTION\nclass BagOfWordsEncoder(Encoder):\n \"\"\"\n This encoder does not transform the input sequence, \n and its summary output is just a sum.\n \"\"\"\n \n def __init__(self):\n super(BagOfWordsEncoder, self).__init__()\n \n def forward(self, inputs, mask, lengths, **kwargs):\n return inputs, (inputs * mask.unsqueeze(-1).float()).sum(dim=1) \n```\n\n\n```python\ndef test_bow_encoder(encoder: BagOfWordsEncoder):\n \"\"\"\n Use this to assert a few things, such as output shapes.\n \"\"\"\n import numpy as np\n batch_size = 2\n max_seq_len = 10\n dim = 12\n inputs = torch.rand([batch_size, max_seq_len, dim])\n mask = torch.ones([batch_size, max_seq_len]).long()\n mask[-1,-1] = 0\n lengths = mask.sum(-1).long()\n outputs, h = encoder(inputs, mask, lengths)\n # General tests: output shape\n assert outputs.size(0) == batch_size, \"Wrong value for size(0). Do you have a batch dimension?\"\n assert outputs.size(1) == max_seq_len, \"Wrong value for size(1). Do you have a time dimension?\"\n assert outputs.size(2) == dim, \"Wrong value for size(2). Do you have an embedding dimension?\" \n np.testing.assert_equal(h[0].numpy(), outputs.sum(1)[0].numpy(), \"Incorrect composition (did you sum the inputs?)\")\n assert (h[1].numpy() != outputs.sum(1)[1].numpy()).sum() > 0, \"Incorrect composition (have applied the mask?)\"\ntest_bow_encoder(BagOfWordsEncoder())\n```\n\nYou can also consider implementing\n\n* a feed-forward encoder with average pooling\n* and a biLSTM encoder\n\nbut these are certainly optional.\n\n\n```python\nclass FFEncoder(Encoder):\n \"\"\"\n A typical feed-forward NN with tanh hidden activations.\n \"\"\"\n \n def __init__(self, input_size, output_size, \n activation=None, \n hidden_sizes=[], \n aggregator='avg',\n dropout=0.5):\n \"\"\"\n :param input_size: int\n :param output_size: int\n :param hidden_sizes: list of integers (dimensionality of hidden layers)\n :param aggregator: 'sum' or 'avg'\n :param dropout: dropout rate\n \"\"\"\n pass\n \n def forward(self, x, mask, lengths):\n \"\"\"\n :param x: sequence of word embeddings, shape [B, T, I]\n :param mask: byte mask that is 0 for invalid positions, shape [B, T]\n :param lengths: the lengths of each input sequence [B]\n :return: \n outputs [B, T, O]\n sum/avg pooling [B, O]\n \"\"\"\n pass\n```\n\n\n```python\n#SOLUTION\n\nclass FFEncoder(Encoder):\n \"\"\"\n A typical feed-forward NN with tanh hidden activations.\n \"\"\"\n \n def __init__(self, input_size, output_size, \n activation=None, \n hidden_sizes=[], \n aggregator='sum',\n dropout=0.5):\n \"\"\"\n :param input_size: int\n :param output_size: int\n :param hidden_sizes: list of integers (dimensionality of hidden layers)\n :param aggregator: 'sum' or 'avg'\n :param dropout: dropout rate\n \"\"\"\n super(FFEncoder, self).__init__()\n layers = []\n if hidden_sizes: \n for i, size in enumerate(hidden_sizes):\n if dropout > 0.:\n layers.append(('dropout%d' % i, nn.Dropout(p=dropout)))\n layers.append(('linear%d' % i, nn.Linear(input_size, size)))\n layers.append(('tanh%d' % i, nn.Tanh()))\n input_size = size\n if dropout > 0.:\n layers.append(('dropout', nn.Dropout(p=dropout)))\n layers.append(('linear', nn.Linear(input_size, output_size))) \n self.layer = nn.Sequential(OrderedDict(layers)) \n self.activation = activation\n if not aggregator in ['sum', 'avg']:\n raise ValueError(\"I can only aggregate outputs using 'sum' or 'avg'\")\n self.aggregator = aggregator\n \n def forward(self, x, mask, lengths):\n # [B, T, d]\n y = self.layer(x)\n if not self.activation is None:\n y = self.activation(y)\n # [B, d]\n s = (y * mask.unsqueeze(-1).float()).sum(dim=1)\n if self.aggregator == 'avg':\n s /= lengths.unsqueeze(-1).float()\n return y, s\n```\n\n\n```python\nfrom torch.nn.utils.rnn import pack_padded_sequence, pad_packed_sequence\n\n\nclass LSTMEncoder(Encoder):\n \"\"\"\n This module encodes a sequence using a bidirectional LSTM\n it returns the final state\n and the hidden states at each time step. Note: we concatenate representations\n from the two directions.\n \"\"\"\n\n def __init__(self, in_features, \n hidden_size: int = 200,\n batch_first: bool = True,\n bidirectional: bool = True):\n \"\"\"\n :param in_features:\n :param hidden_size:\n :param batch_first:\n :param bidirectional:\n \"\"\"\n pass\n\n def forward(self, x, mask, lengths):\n \"\"\"\n Encode sentence x\n :param x: sequence of word embeddings, shape [B, T, I]\n :param mask: byte mask that is 0 for invalid positions, shape [B, T]\n :param lengths: the lengths of each input sequence [B]\n :return:\n outputs [B, T, O]\n final state [B, O]\n \"\"\"\n pass\n```\n\n\n```python\n# SOLUTION\n\n\nfrom torch.nn.utils.rnn import pack_padded_sequence, pad_packed_sequence\n\n\nclass LSTMEncoder(Encoder):\n \"\"\"\n This module encodes a sequence into a single vector using an LSTM,\n it also returns the hidden states at each time step.\n \"\"\"\n\n def __init__(self, in_features, hidden_size: int = 200,\n batch_first: bool = True,\n bidirectional: bool = True):\n \"\"\"\n :param in_features:\n :param hidden_size:\n :param batch_first:\n :param bidirectional:\n \"\"\"\n super(LSTMEncoder, self).__init__()\n self.lstm = nn.LSTM(in_features, hidden_size, batch_first=batch_first,\n bidirectional=bidirectional)\n\n def forward(self, x, mask, lengths):\n \"\"\"\n Encode sentence x\n :param x: sequence of word embeddings, shape [B, T, E]\n :param mask: byte mask that is 0 for invalid positions, shape [B, T]\n :param lengths: the lengths of each input sequence [B]\n :return:\n \"\"\"\n\n packed_sequence = pack_padded_sequence(x, lengths, batch_first=True)\n outputs, (hx, cx) = self.lstm(packed_sequence)\n outputs, _ = pad_packed_sequence(outputs, batch_first=True)\n\n # classify from concatenation of final states\n if self.lstm.bidirectional:\n final = torch.cat([hx[-2], hx[-1]], dim=-1)\n else: # classify from final state\n final = hx[-1]\n\n return outputs, final\n```\n\nHere is some helper code to select and return an encoder:\n\n\n```python\ndef get_encoder(layer, in_features, hidden_size, bidirectional=True):\n \"\"\"Returns the requested layer.\"\"\"\n\n if layer == \"bow\":\n return BagOfWordsEncoder()\n elif layer == 'ff':\n return FFEncoder(\n in_features, \n 2 * hidden_size, # for convenience\n hidden_sizes=[hidden_size], \n aggregator='avg')\n elif layer == \"lstm\":\n return LSTMEncoder(\n in_features, \n hidden_size,\n bidirectional=bidirectional)\n else:\n raise ValueError(\"Unknown layer\")\n```\n\n\n```python\n\n```\n\n# Sentiment Classification with Latent Rationale\n\nA latent rationale is a compact and informative fragment of the input based on which a NN classifier makes its decisions. [Lei et al (2016)](http://aclweb.org/anthology/D16-1011) proposed to induce such rationales along with a regression model for multi-aspect sentiment analsysis, their model is trained via REINFORCE on a dataset of beer reviews.\n\n*Remark:* the model we will develop here can be seen as a probabilistic version of their model. The rest of this notebook focus on our own probabilitisc view of the model.\n\nThe picture below depicts our latent-variable model for rationale extraction:\n\n\n\nwhere we augment the model with a collection of latent variables $z = \\langle z_1, \\ldots, z_n\\rangle$ where $z_i$ is a binary latent variable. Each latent variable $z_i$ regulates whether or not the input $x_i$ is available to the classifier. We use $x \\odot z$ to denote the selected words, which, in the terminology of Lei et al, is a latent rationale.\n\nAgain the classifier parameterises a Categorical distribution over $K=5$ outcomes, though this time it can encode only a selection of the input:\n\n\\begin{align}\n Z_i & \\sim \\text{Bern}(p_1) \\\\\n Y|z,x &\\sim \\text{Cat}(f(x \\odot z; \\theta))\n\\end{align}\n\nwhere we have a shared and fixed Bernoulli prior (with parameter $p_1$) for all $n$ latent variables.\n\n\nHere is an example design for $f$:\n\n\\begin{align}\n\\mathbf x_i &= z_i \\, \\text{glove}(x_i) \\\\\n\\mathbf t_1^n, \\mathbf h &= \\text{encoder}(\\mathbf x_1^n; \\theta_{\\text{enc}}) \\\\\nf(x \\odot z; \\theta) &= \\text{softmax}(\\text{dense}_K(\\mathbf h; \\theta_{\\text{output}}))\n\\end{align}\n\nwhere:\n* $z_i$ either leaves $\\mathbf x_i$ unchanged or turns it into a vector of zeros;\n* the encoder only sees features from selected inputs, i.e. $x_i$ for which $z_i = 1$;\n* $\\text{dense}_K$ is a linear layer with $K=5$ outputs.\n\n\n\n## Prior\n\n\nOur prior is a Bernoulli with fixed parameter $0 < p_1 < 1$:\n\n\\begin{align}\nZ_i|p_1 & \\sim \\text{Bern}(p_1)\n\\end{align}\n\nwhose pmf is $\\text{Bern}(z_i|p_1) = p_1^{z_i}\\times (1-p_1)^{1-z_i}$.\n\nAs we will be using Bernoulli priors and posteriors, it is a good idea to implement a Bernoulli class:\n\n\n```python\nclass Bernoulli:\n \"\"\"\n This class encapsulates a collection of Bernoulli distributions. \n Each Bernoulli is uniquely specified by p_1, where\n Bernoulli(X=x|p_1) = pow(p_1, x) * pow(1 - p_1, 1 - x)\n is the Bernoulli probability mass function (pmf). \n \"\"\"\n \n def __init__(self, logits=None, probs=None):\n \"\"\"\n We can specify a Bernoulli distribution via a logit or a probability. \n You need to specify at least one, and if you specify both, beware that\n in this implementation logits will be used.\n \n Recall that: probs = sigmoid(logits).\n \n :param logits: a tensor of logits (a logit is defined as log (p_1/p_0))\n where p_0 = 1 - p_1\n :param probs: a tensor of probabilities, each in (0, 1)\n \n \"\"\" \n pass\n \n def sample(self):\n \"\"\"Returns a single sample with the same shape as the parameters\"\"\"\n pass\n \n def ste_sample(self):\n \"\"\"\n Returns a single sample with the same shape as the parameters using the straight-through estimator. \n That is, you must make sure the forward pass produces a discrete sample\n while the backward pass differentiates the probability value (rather than the sample).\n \n **This is an extra, don't bother implementing it initially**\n \"\"\"\n pass\n \n def log_pmf(self, x):\n \"\"\"\n Assess the log probability of a sample. \n \n :param x: either a single sample (0 or 1) or a tensor of samples with the same shape as the parameters.\n :returns: tensor with log probabilities with the same shape as parameters\n (if the input is a single sample we broadcast it to the shape of the parameters)\n \"\"\"\n pass\n \n def kl(self, other: 'Bernoulli'):\n \"\"\"\n Compute the KL divergence between two Bernoulli distributions (from self to other).\n \n :return: KL[self||other] with same shape parameters\n \"\"\"\n pass\n\n```\n\n\n```python\n# SOLUTION\nfrom torch.distributions import Bernoulli as PyTBernoulli\nimport torch.nn.functional as F\n\nclass Bernoulli:\n \"\"\"\n This class encapsulates a collection of Bernoulli distributions. \n Each Bernoulli is uniquely specified by p_1, where\n Bernoulli(X=x|p_1) = pow(p_1, x) + pow(1 - p_1, 1 - x)\n is the Bernoulli probability mass function (pmf). \n \"\"\"\n \n def __init__(self, logits=None, probs=None):\n \"\"\"\n We can specify a Bernoulli distribution via a logit or a probability. \n You need to specify at least one, and if you specify both, beware that\n in this implementation logits will be used.\n \n Recall that: probs = sigmoid(logits).\n \n :param logits: a tensor of logits (a logit is defined as log (p_1/p_0))\n where p_0 = 1 - p_1\n :param probs: a tensor of probabilities, each in (0, 1)\n \n \"\"\" \n if probs is None and logits is None:\n raise ValueError('I need probabilities or logits') \n if logits is None: \n eps = torch.finfo(probs.dtype).eps\n probs = probs.clamp(min=eps, max=1 - eps)\n self.probs = probs\n self.logits = torch.log(probs) - torch.log1p(-probs)\n else:\n self.logits = logits\n self.probs = torch.sigmoid(logits) \n # stable computation of log prob\n self.log_p1 = - F.softplus(-self.logits)\n self.log_p0 = - F.softplus(self.logits)\n \n def sample(self):\n \"\"\"Returns a sample with the same shape as the parameters\"\"\"\n return (torch.rand_like(self.probs) < self.probs).float()\n \n def ste_sample(self):\n \"\"\"\n Returns a single sample with the same shape as the parameters using the straight-through estimator. \n That is, you must make sure the forward pass produces a discrete sample\n while the backward pass differentiates the probability value (rather than the sample).\n \n **This is an extra, don't bother implementing it initially**\n \"\"\"\n z = (torch.rand_like(self.probs) < self.probs).float()\n # Note how the forward pass evaluates to z\n # and the backward pass evaluates to p'\n return (z - self.probs).detach() + self.probs - self.probs.detach()\n \n def log_pmf(self, x):\n \"\"\"\n Assess the log probability of a sample. \n :param x: either a single sample (0 or 1) or a tensor of samples with the same shape as the parameters.\n :returns: tensor with log probabilities with the same shape as parameters\n (if the input is a single sample we broadcast it to the shape of the parameters)\n \"\"\"\n # x * torch.log(self.probs) + (1 - x) * torch.log(1. - self.probs)\n return torch.where(x == 1., self.log_p1, self.log_p0)\n \n def kl(self, other: 'Bernoulli'):\n \"\"\"\n Compute the KL divergence between two Bernoulli distributions (from self to other).\n \n :return: KL[self||other] with same shape parameters\n \"\"\"\n return self.probs * (self.log_p1 - other.log_p1) + (1 - self.probs) * (self.log_p0 - other.log_p0)\n```\n\n\n```python\ndef test_bernoulli_shapes(): \n batch_size = 2\n latent_dim = 3\n p = Bernoulli(torch.sigmoid(torch.rand([batch_size, latent_dim])))\n q = Bernoulli(torch.sigmoid(torch.rand([batch_size, latent_dim])))\n assert p.log_pmf(torch.ones([batch_size, latent_dim]).long()).size() == torch.Size([batch_size, latent_dim])\n assert p.kl(q).size() == torch.Size([batch_size, latent_dim])\n\ntest_bernoulli_shapes()\n```\n\n\n```python\n\ndef test_bernoulli_sampling():\n import numpy as np\n \n # create a simple Bernoulli \n p = Bernoulli(probs=torch.full([1], 0.2))\n assert p.sample().numpy().shape == (1,), 'Wrong sample shape'\n np.testing.assert_almost_equal(\n np.mean([p.sample().numpy() for _ in range(1000)]), \n 0.2, \n decimal=1,\n err_msg=\"Bad sample mean\")\n \n # create a batch of 1000 Bernoulli distributions\n p = Bernoulli(probs=torch.full([1000, 1], 0.2))\n assert p.sample().numpy().shape == (1000,1), 'Wrong sample shape'\n np.testing.assert_almost_equal(\n np.mean(p.sample().numpy()), \n 0.2, \n decimal=1,\n err_msg=\"Bad sample mean\")\n \n # create a batch of 1000 Bernoulli distributions\n p = Bernoulli(logits=torch.full([1000, 1], 0.3).log())\n assert p.sample().numpy().shape == (1000,1), 'Wrong sample shape'\n np.testing.assert_almost_equal(\n np.mean(p.sample().numpy()), \n 0.2, \n decimal=1,\n err_msg=\"Bad sample mean\")\n\ntest_bernoulli_sampling()\n```\n\n## Classifier\n\nThe classifier encodes only a selection of the input, which we denote $x \\odot z$, and parameterises a Categorical distribution over $5$ outcomes (sentiment levels).\n\nThus let's implement a Categorical distribution (we will only need to be able to assess its lgo pmf):\n\n\n```python\nclass Categorical:\n \n def __init__(self, log_probs):\n # [B, K]: class probs\n self.log_probs = log_probs\n \n def log_pmf(self, x):\n \"\"\"\n :param x: [B] integers (targets)\n :returns: [B] scalars (log probabilities)\n \"\"\"\n pass\n \n def argmax(self):\n \"\"\"\n Return the argmax prediction\n :returns: [B] class ids\n \"\"\"\n pass\n```\n\n\n```python\n#SOLUTION\nclass Categorical:\n \n def __init__(self, log_probs):\n # [B, K]: class probs\n self.log_probs = log_probs\n \n def log_pmf(self, x):\n \"\"\"\n :param x: [B] integers\n \"\"\"\n # [B, 1]\n log_p = torch.gather(self.log_probs, 1, x.unsqueeze(-1))\n # [B]\n return log_p.squeeze(-1)\n\n def argmax(self):\n \"\"\"\n Return the argmax prediction\n :returns: [B] class ids\n \"\"\"\n return self.log_probs.argmax(-1)\n```\n\n\n```python\ndef test_categorical_shapes(): \n batch_size = 2\n nb_classes = 5\n p = Categorical(F.softmax(torch.rand([batch_size, nb_classes]), -1))\n assert p.log_pmf(torch.ones(batch_size).long()).size() == torch.Size([batch_size]), \"Did you squeeze the category dimension?\"\n assert p.argmax().size() == torch.Size([batch_size]), \"Did you squeeze the category dimension?\"\ntest_categorical_shapes()\n```\n\nand a classifier architecture:\n\n* implement the forward method\n\n\n```python\nclass Classifier(nn.Module):\n \"\"\"\n The Encoder takes an input text (and rationale z) and computes p(y|x,z)\n \"\"\"\n\n def __init__(self,\n embed: nn.Embedding = None,\n hidden_size: int = 200,\n output_size: int = 1,\n dropout: float = 0.1,\n layer: str = \"pass\",\n ):\n\n super(Classifier, self).__init__()\n\n emb_size = embed.weight.shape[1]\n enc_size = hidden_size * 2\n # Here we embed the words\n self.embed_layer = nn.Sequential(\n embed\n # , nn.Dropout(p=dropout)\n )\n\n self.enc_layer = get_encoder(layer, emb_size, hidden_size)\n\n # and here we predict categorical parameters\n self.output_layer = nn.Sequential(\n nn.Dropout(p=dropout),\n nn.Linear(enc_size, output_size),\n nn.LogSoftmax(dim=-1)\n )\n\n self.report_params()\n\n def report_params(self):\n count = 0\n for name, p in self.named_parameters():\n if p.requires_grad and \"embed\" not in name:\n count += np.prod(list(p.shape))\n print(\"{} #params: {}\".format(self.__class__.__name__, count))\n\n def forward(self, x, mask, z) -> Categorical:\n \"\"\"\n :params x: [B, T, I] word representations\n :params mask: [B, T] indicates valid positions\n :params z: [B, T] binary selectors\n :returns: one Categorical distribution per instance in the batch\n each conditioning only on x_i for which z_i = 1\n \"\"\"\n pass\n```\n\n\n```python\n#SOLUTION\nclass Classifier(nn.Module):\n \"\"\"\n The Encoder takes an input text (and rationale z) and computes p(y|x,z)\n \"\"\"\n\n def __init__(self,\n embed: nn.Embedding = None,\n hidden_size: int = 200,\n output_size: int = 5,\n dropout: float = 0.1,\n layer: str = \"pass\",\n ):\n\n super(Classifier, self).__init__()\n\n emb_size = embed.weight.shape[1]\n enc_size = hidden_size * 2\n # Here we embed the words\n self.embed_layer = nn.Sequential(\n embed\n )\n\n self.enc_layer = get_encoder(layer, emb_size, hidden_size)\n\n # and here we predict categorical parameters\n self.output_layer = nn.Sequential(\n nn.Dropout(p=dropout),\n nn.Linear(enc_size, output_size),\n nn.LogSoftmax(dim=-1)\n )\n\n self.report_params()\n\n def report_params(self):\n count = 0\n for name, p in self.named_parameters():\n if p.requires_grad:\n count += np.prod(list(p.shape))\n print(\"{} #params: {}\".format(self.__class__.__name__, count))\n\n def forward(self, x, mask, z) -> Categorical:\n \"\"\"\n :params x: [B, T, I] word representations\n :params mask: [B, T] indicates valid positions\n :params z: [B, T] binary selectors\n :returns: one Categorical distribution per instance in the batch\n each conditioning only on x_i for which z_i = 1\n \"\"\"\n \n rnn_mask = mask\n emb = self.embed_layer(x)\n\n # [B, T]\n rnn_mask = z > 0.\n # [B, T, 1]\n z_mask = z.unsqueeze(-1).float()\n # [B, T, E]\n emb = emb * z_mask\n\n lengths = mask.long().sum(1)\n\n # encode the sentence\n _, final = self.enc_layer(emb, rnn_mask, lengths)\n\n # predict sentiment from final state(s)\n log_probs = self.output_layer(final) \n return Categorical(log_probs)\n```\n\n## Inference\n\n\nComputing the log-likelihood of an observation requires marginalising over assignments of $z$:\n\n\\begin{align}\nP(y|x,\\theta,p_1) &= \\sum_{z_1 = 0}^1 \\cdots \\sum_{z_n=0}^1 P(z|p_1)\\times P(y|x,z, \\theta) \\\\\n&= \\sum_{z_1 = 0}^1 \\cdots \\sum_{z_n=0}^1 \\left( \\prod_{i=1}^n \\text{Bern}(z_i|p_1)\\right) \\times \\text{Cat}(y|f(x \\odot z; \\theta)) \n\\end{align}\n\nThis is clearly intractable: there are $2^n$ possible assignments to $z$ and because the classifier conditions on all latent selectors, there's no way to simplify the expression.\n\nWe will avoid computing this intractable marginal by instead employing an independently parameterised inference model.\nThis inference model $Q(z|x, y, \\lambda)$ is an approximation to the true postrerior $P(z|x, y, \\theta, p_1)$, and we use $\\lambda$ to denote its parameters.\n\n\nWe make a *mean field* assumption, whereby we model latent variables independently given the input:\n\\begin{align}\nQ(z|x, y, \\lambda) \n &= \\prod_{i=1}^{n} Q(z_i|x; \\lambda) \\\\\n &= \\prod_{i=1}^{n} \\text{Bern}(z_i|g_i(x; \\lambda)) \n\\end{align}\n\nwhere $g(x; \\lambda)$ is a NN that maps from $x = \\langle x_1, \\ldots, x_n\\rangle$ to $n$ Bernoulli parameters, each of which, is a probability value (thus $0 < g_i(x; \\lambda) < 1$).\n\nNote that though we could condition on $y$ for approximate posterior inference, we are opportunistically leaving it out. This way, $Q$ is directly available at test time for making predictions. The figure below is a graphical depiction of the inference model (we show a dashed arrow from $y$ to $z$ to remind you that in principle the label is also available).\n\n\n\nHere is an example design for $g$:\n\\begin{align}\n\\mathbf x_i &= \\text{glove}(x_i) \\\\\n\\mathbf t_1^n, \\mathbf h &= \\text{encoder}(\\mathbf x_1^n; \\lambda_{\\text{enc}}) \\\\\ng_i(x; \\lambda) &= \\sigma(\\text{dense}_1(\\mathbf t_i; \\lambda_{\\text{output}}))\n\\end{align}\nwhere\n* $\\text{glove}$ is a pre-trained embedding function;\n* $\\text{dense}_1$ is a dense layer with a single output;\n* and $\\sigma(\\cdot)$ is the sigmoid function, necessary to parameterise a Bernoulli distribution.\n\nFrom now on we will write $Q(z|x, \\lambda)$, that is, without $y$.\n\nHere we implement this product of Bernoulli distributions:\n\n* implement $g$ in the constructor \n* and the forward pass\n\n\n```python\nclass ProductOfBernoullis(nn.Module):\n \"\"\"\n This is an inference network that parameterises independent Bernoulli distributions.\n \"\"\"\n\n def __init__(self,\n embed: nn.Embedding,\n hidden_size: int = 200,\n layer: str = \"bow\"\n ):\n \"\"\"\n :param embed: an embedding layer\n :param hidden_suze: hidden size for transformed inputs\n :param layer: 'bow' for BoW encoding\n you may alternatively implement and 'lstm' option\n which uses a biLSTM to transform the inputs \n \"\"\"\n super(ProductOfBernoullis, self).__init__()\n # 1. we should have an embedding layer \n # 2. we may transform the representations\n # 3. and we should compute parameters for Bernoulli distributions\n pass\n\n def report_params(self):\n count = 0\n for name, p in self.named_parameters():\n if p.requires_grad and \"embed\" not in name:\n count += np.prod(list(p.shape))\n print(\"{} #params: {}\".format(self.__class__.__name__, count))\n\n def forward(self, x, mask) -> Bernoulli:\n \"\"\"\n It takes a tensor of tokens (integers)\n and predicts a Bernoulli distribution for each position.\n \n :param x: [B, T]\n :param mask: [B, T]\n :returns: Bernoulli\n \"\"\"\n pass\n```\n\n\n```python\n#SOLUTION\nclass ProductOfBernoullis(nn.Module):\n \"\"\"\n This is an inference network that parameterises independent Bernoulli distributions.\n \"\"\"\n\n def __init__(self,\n embed: nn.Embedding,\n hidden_size: int = 200,\n layer: str = \"bow\"\n ):\n \"\"\"\n :param embed: an embedding layer\n :param hidden_suze: hidden size for transformed inputs\n :param layer: 'bow' for BoW encoding\n you may alternatively implement and 'lstm' option\n which uses a biLSTM to transform the inputs \n \"\"\"\n\n super(ProductOfBernoullis, self).__init__()\n\n emb_size = embed.weight.shape[1]\n # I will use twice the units \n # just to make the output as large as that of a biLSTM\n enc_size = hidden_size * 2 \n\n self.embed_layer = nn.Sequential(embed)\n self.enc_layer = get_encoder(layer, emb_size, hidden_size)\n self.logit_layer = nn.Linear(enc_size, 1, bias=True)\n \n self.report_params()\n\n def report_params(self):\n count = 0\n for name, p in self.named_parameters():\n if p.requires_grad:\n count += np.prod(list(p.shape))\n print(\"{} #params: {}\".format(self.__class__.__name__, count))\n\n def forward(self, x, mask) -> Bernoulli:\n \"\"\"\n It takes a tensor of tokens (integers)\n and predicts a Bernoulli distribution for each position.\n \n :param x: [B, T]\n :param mask: [B, T]\n :returns: Bernoulli\n \"\"\"\n\n # encode sentence\n # [B]\n lengths = mask.long().sum(1)\n # [B, T, E]\n emb = self.embed_layer(x) \n # [B, T, d]\n h, _ = self.enc_layer(emb, mask, lengths)\n\n # compute parameters for Bernoulli p(z|x)\n # [B, T, 1] Bernoulli distributions\n logits = self.logit_layer(h)\n # [B, T]\n logits = logits.squeeze(-1)\n return Bernoulli(logits=logits)\n```\n\n## Parameter Estimation\n\nIn variational inference, our objective is to maximise the *evidence lowerbound* (ELBO):\n\n\\begin{align}\n\\log P(y|x) &\\ge \\mathbb E_{Q(z|x, y, \\lambda)}\\left[ \\log P(y|x, z, \\theta, p_1) \\right] - \\text{KL}(Q(z|x, y, \\lambda) || P(z|p_1)) \\\\\n\\text{ELBO}&\\overset{\\text{MF}}{=}\\mathbb E_{Q(z|x, y, \\lambda)}\\left[ \\log P(y|x, z, \\theta, p_1) \\right] - \\sum_{i=1}^n \\text{KL}(Q(z_i|x, \\lambda) || P(z_i|p_1)) \n\\end{align}\n\nwhere the *mean field* assumption we made implies that the KL term is simply a sum of KL divergences from a Bernoulli posterior to a Bernoulli prior.\n\nNote that the ELBO remains intractable, namely, solving the expectation in closed form still requires $2^n$ evaluations of the classifier network. Though unlike the true posterior $P(z|x,y, \\lambda)$, the approximation $Q(z|x,\\lambda)$ is tractable (it does not require an intractable normalisation) and can be used to obtain gradient estimates based on samples.\n\n### Gradient of the classifier network\n\nFor the classifier, we encounter no problem:\n\n\\begin{align}\n\\nabla_\\theta \\text{ELBO} &=\\nabla_\\theta\\sum_{z} Q(z|x, \\lambda)\\log P(y|x,z,\\theta) - \\underbrace{\\nabla_\\theta \\sum_{i=1}^n \\text{KL}(Q(z_i|x, \\lambda) || P(z_i|p_1))}_{\\color{blue}{0}} \\\\\n&=\\sum_{z} Q(z|x, \\lambda)\\nabla_\\theta\\log P(y|x,z,\\theta) \\\\\n&= \\mathbb E_{Q(z|x, \\lambda)}\\left[\\nabla_\\theta\\log P(y|x,z,\\theta) \\right] \\\\\n&\\overset{\\text{MC}}{\\approx} \\frac{1}{S} \\sum_{s=1}^S \\nabla_\\theta \\log P(y|x, z^{(s)}, \\theta) \n\\end{align}\nwhere $z^{(s)} \\sim Q(z|x,\\lambda)$.\n\n\n### Gradient of the inference network\n\nFor the inference model, we have to use the *score function estimator* (a.k.a. REINFORCE):\n\n\\begin{align}\n\\nabla_\\lambda \\text{ELBO} &=\\nabla_\\lambda\\sum_{z} Q(z|x, \\lambda)\\log P(y|x,z,\\theta) - \\nabla_\\lambda \\underbrace{\\sum_{i=1}^n \\text{KL}(Q(z_i|x, \\lambda) || P(z_i|p_1))}_{ \\color{blue}{\\text{tractable} }} \\\\\n&=\\sum_{z} \\nabla_\\lambda Q(z|x, \\lambda)\\log P(y|x,z,\\theta) - \\sum_{i=1}^n \\nabla_\\lambda \\text{KL}(Q(z_i|x, \\lambda) || P(z_i|p_1)) \\\\\n&=\\sum_{z} \\underbrace{Q(z|x, \\lambda) \\nabla_\\lambda \\log Q(z|x, \\lambda)}_{\\nabla_\\lambda Q(z|x, \\lambda)} \\log P(y|x,z,\\theta) - \\sum_{i=1}^n \\nabla_\\lambda \\text{KL}(Q(z_i|x, \\lambda) || P(z_i|p_1)) \\\\\n&= \\mathbb E_{Q(z|x, \\lambda)}\\left[ \\log P(y|x,z,\\theta) \\nabla_\\lambda \\log Q(z|x, \\lambda) \\right] - \\sum_{i=1}^n \\nabla_\\lambda \\text{KL}(Q(z_i|x, \\lambda) || P(z_i|p_1)) \\\\\n&\\overset{\\text{MC}}{\\approx} \\left(\\frac{1}{S} \\sum_{s=1}^S \\log P(y|x, z^{(s)}, \\theta) \\nabla_\\lambda \\log Q(z^{(s)}|x, \\lambda) \\right) - \\sum_{i=1}^n \\nabla_\\lambda \\text{KL}(Q(z_i|x, \\lambda) || P(z_i|p_1)) \n\\end{align}\n\nwhere $z^{(s)} \\sim Q(z|x,\\lambda)$.\n\n## Implementation\n\nLet's implement the model and the loss (negative ELBO). We work with the notion of a *surrogate loss*, that is, a computation node whose gradients wrt to parameters are equivalent to the gradients we need.\n\nFor a given sample $z \\sim Q(z|x, \\lambda)$, the following is a single-sample surrogate loss:\n\n\\begin{align}\n\\mathcal S(\\theta, \\lambda|x, y) = \\log P(y|x, z, \\theta) + \\color{red}{\\text{detach}(\\log P(y|x, z, \\theta) )}\\log Q(z|x, \\lambda) - \\sum_{i=1}^n \\text{KL}(Q(z_i|x, \\lambda) || P(z_i|\\phi))\n\\end{align}\nwhere we introduce an auxiliary function such that\n\\begin{align}\n\\text{detach}(f(\\alpha)) &= h(\\alpha) \\\\\n\\nabla_\\beta \\text{detach}(h(\\alpha)) &= 0 \n\\end{align}\nor in words, *detach* does not alter the forward call of its argument function $h$, but it alters $h$'s backward call by setting gradients to zero.\n\nShow that it's gradients wrt $\\theta$ and $\\lambda$ are exactly what we need:\n\n\n\\begin{align}\n\\nabla_\\theta \\mathcal S(\\theta, \\lambda|x, y) = \\color{red}{?}\n\\end{align}\n\n\\begin{align}\n\\nabla_\\lambda \\mathcal S(\\theta, \\lambda|x, y) = \\color{red}{?}\n\\end{align}\n\n**Solution**\n\n\\begin{align}\n\\nabla_\\theta \\mathcal S(\\theta, \\lambda|x, y) = \\nabla_\\theta \\log P(y|x, z, \\theta) + 0\n\\end{align}\n\n\\begin{align}\n\\nabla_\\lambda \\mathcal S(\\theta, \\lambda|x, y) &= 0 + \\underbrace{\\log Q(z|x, \\lambda)\\nabla_\\lambda \\log P(y|x, z, \\theta) + \\log P(y|x, z, \\theta) \\nabla_\\lambda \\log Q(z|x, \\lambda)}_{\\text{chain rule}} \\\\ \n&= 0+ 0 + \\log P(y|x, z, \\theta) \\nabla_\\lambda \\log Q(z|x, \\lambda)\n\\end{align}\n\nImplement the forward pass and loss below:\n\n\n```python\nclass VAE(nn.Module):\n \"\"\"\n \n Classifier model:\n Z_i ~ Bern(p_1) for i in 1..n\n Y|x,z ~ Cat(f([x_i if z_i 1 else 0 for i in 1..n ]))\n \n Inference model:\n Z_i|x ~ Bern(b_i) for i in 1..n\n where b_i = g_i(x)\n \n Objective:\n Single-sample MC estimate of ELBO\n \n Loss: \n Surrogate loss\n\n Consists of:\n - a product of Bernoulli distributions inference network\n - a classifier network\n \"\"\"\n\n def __init__(self,\n vocab: object = None,\n vocab_size: int = 0,\n emb_size: int = 200,\n hidden_size: int = 200,\n num_classes: int = 5,\n prior_p1: float = 0.3, \n dropout: float = 0.1,\n layer_cls: str = 'bow',\n layer_inf: str = 'bow',\n ):\n \"\"\"\n :param vocab: Vocabulary\n :param vocab_size: necessary for embedding layer\n :param emb_size: dimensionality of embedding layer\n :param hidden_size: dimensionality of hidden layers\n :param num_classes: number of classes\n :param prior_p1: (scalar) prior Bernoulli parameter\n :param dropout: (scalar) dropout rate\n :param layer_cls: type of encoder for classification\n :param layer_inf: type of encoder for inference\n \"\"\"\n super(VAE, self).__init__()\n\n self.vocab = vocab\n self.embed = embed = nn.Embedding(vocab_size, emb_size, padding_idx=1)\n\n self.cls_net = Classifier(\n embed=embed, \n hidden_size=hidden_size, \n output_size=num_classes,\n dropout=dropout, \n layer=layer_cls)\n \n self.inference_net = ProductOfBernoullis(\n embed=embed, \n hidden_size=hidden_size,\n layer=layer_inf)\n \n self._prior_p1 = prior_p1\n \n def generative_parameters(self):\n return self.cls_net.parameters()\n \n def inference_parameters(self):\n return self.inference_net.parameters()\n \n def get_prior(self, shape, device, dtype=torch.float32) -> Bernoulli:\n \"\"\"\n Return a collection of independent prior Bernoulli distributions with a given shape.\n \n :param shape: typically a pair (batch_size, max_time)\n :param device: a torch device\n :returns: Bernoulli object whose probs have the given shape\n and the value is prior_p1\n \"\"\"\n return Bernoulli(probs=torch.full(shape, self._prior_p1, device=device, dtype=dtype))\n \n def get_posterior(self, x, mask) -> Bernoulli:\n \"\"\"\n Infer a collection of independent Bernoulli posteriors\n with shape [B, T]\n \"\"\"\n return self.inference_net(x, mask)\n\n def get_likelihood(self, x, mask, z) -> Categorical:\n \"\"\"\n Generate a sequence z with inference model, \n then predict with rationale xz, that is, x masked by z.\n\n :param x: [B, T] documents\n :return: \n Categorical distributions P(y|x, z)\n Bernoulli distributions Q(z|x)\n Single sample z ~ Q(z|x) used for the conditional P(y|x, z)\n \"\"\"\n py = self.cls_net(x, mask, z)\n return py\n \n def forward(self, *args, **kwargs):\n \"\"\"\n See get_loss\n \"\"\"\n return self.get_loss(args, kwargs)\n\n def get_loss(self, x, mask, y,\n iter_i=0, \n kl_weight=1.0,\n min_kl=0.0,\n ll_mean=0.,\n ll_std=1.,\n **kwargs):\n \"\"\"\n This computes the loss for the whole model.\n\n :param x: inputs [B, T]\n :param mask: identifies valid positions [B, T]\n :param y: target labels [B]\n :param iter_i: indicates the iteration \n :param kl_weight: (scalar) multiplies the KL term\n :param min_kl: (scalar) sets a minimum for the KL (aka free bits)\n :param ll_mean: (scalar) running average of reward\n :param ll_std: (scalar) running standard deviation of reward\n :return: loss (torch scalar node), terms (dict)\n \n terms is a dict that holds the scalar items involved in the loss\n e.g. `terms['ll'] = ll.item()` is the log-likelihood term\n \n You should log the following:\n \n - the ELBO: terms['elbo']\n - the likelihood term of the ELBO: terms['ll']\n - the kl before clipping and annealing: terms['kl']\n - the kl after clipping and annealing: terms['kl_'] \n - the learning signal in REINFORCE: terms['reward']\n - the standard deviation of the learning signal: terms['ll_std']\n - the ratio of selected words (average z): terms['selected']\n \n \n Consider tracking the following:\n Single-sample ELBO: terms['elbo']\n Log-Likelihood log P(y|x,z): terms['ll']\n KL: terms['kl']\n Score function surrogate log P(y|z, x) log Q(z|x): terms['sf'] \n Rate of selected words: terms['selected']\n \"\"\"\n\n raise NotImplementedError(\"Implement me!\")\n \n # Make complete surrogate loss (for backward)\n # [B]\n loss = None\n \n # Store terms worth tracking\n terms['elbo'] = 0. \n terms['kl_'] = 0. \n terms['kl'] = 0.\n terms['reward'] = 0.\n terms['ll'] = 0.\n terms['ll_std'] = 0.\n terms['selected'] = 0.\n\n # return loss for backward and terms worth tracking\n return loss.mean(), terms \n \n def get_ste_loss(self, x, mask, y,\n iter_i=0, \n # you may ignore the rest of the arguments for the time being\n # leave them as they are\n kl_weight=1.0,\n min_kl=0.0,\n ll_mean=0.,\n ll_std=1.,\n **kwargs):\n \"\"\"\n This is an extra:\n - use the (biased) straight-through estimator instead of the score function estimator\n \"\"\"\n pass \n```\n\n\n```python\n#SOLUTION\nclass VAE(nn.Module):\n \"\"\"\n \n Classifier model:\n Z_i ~ Bern(p_1) for i in 1..n\n Y|x,z ~ Cat(f([x_i if z_i 1 else 0 for i in 1..n ]))\n \n Inference model:\n Z_i|x ~ Bern(b_i) for i in 1..n\n where b_i = g_i(x)\n \n Objective:\n Single-sample MC estimate of ELBO\n \n Loss: \n Surrogate loss\n\n Consists of:\n - a product of Bernoulli distributions inference network\n - a classifier network\n \"\"\"\n\n def __init__(self,\n vocab: object = None,\n vocab_size: int = 0,\n emb_size: int = 200,\n hidden_size: int = 200,\n num_classes: int = 5,\n prior_p1: float = 0.3, \n dropout: float = 0.1,\n layer_cls: str = 'bow',\n layer_inf: str = 'bow',\n ):\n \"\"\"\n :param vocab: Vocabulary\n :param vocab_size: necessary for embedding layer\n :param emb_size: dimensionality of embedding layer\n :param hidden_size: dimensionality of hidden layers\n :param num_classes: number of classes\n :param prior_p1: (scalar) prior Bernoulli parameter\n :param dropout: (scalar) dropout rate\n :param layer_cls: type of encoder for classification\n :param layer_inf: type of encoder for inference\n \"\"\"\n super(VAE, self).__init__()\n\n self.vocab = vocab\n self.embed = embed = nn.Embedding(vocab_size, emb_size, padding_idx=1)\n\n self.cls_net = Classifier(\n embed=embed, \n hidden_size=hidden_size, \n output_size=num_classes,\n dropout=dropout, \n layer=layer_cls)\n \n self.inference_net = ProductOfBernoullis(\n embed=embed, \n hidden_size=hidden_size,\n layer=layer_inf)\n \n self._prior_p1 = prior_p1\n \n def generative_parameters(self):\n return self.cls_net.parameters()\n \n def inference_parameters(self):\n return self.inference_net.parameters()\n \n def get_prior(self, shape, device, dtype=torch.float32) -> Bernoulli:\n \"\"\"\n Return a collection of independent prior Bernoulli distributions with a given shape.\n \n :param shape: typically a pair (batch_size, max_time)\n :param device: a torch device\n :returns: Bernoulli object whose probs have the given shape\n and the value is prior_p1\n \"\"\"\n return Bernoulli(probs=torch.full(shape, self._prior_p1, device=device, dtype=dtype))\n \n def get_posterior(self, x, mask) -> Bernoulli:\n \"\"\"\n Infer a collection of independent Bernoulli posteriors\n with shape [B, T]\n \"\"\"\n return self.inference_net(x, mask)\n\n def get_likelihood(self, x, mask, z) -> Categorical:\n \"\"\"\n Generate a sequence z with inference model, \n then predict with rationale xz, that is, x masked by z.\n\n :param x: [B, T] documents\n :return: \n Categorical distributions P(y|x, z)\n Bernoulli distributions Q(z|x)\n Single sample z ~ Q(z|x) used for the conditional P(y|x, z)\n \"\"\"\n py = self.cls_net(x, mask, z)\n return py\n \n def forward(self, *args, **kwargs):\n \"\"\"\n Generate a sequence z with inference model, \n then predict with rationale xz, that is, x masked by z.\n\n :param x: [B, T] documents\n :return: \n Categorical distributions P(y|x, z)\n Bernoulli distributions Q(z|x)\n Single sample z ~ Q(z|x) used for the conditional P(y|x, z)\n \"\"\"\n return self.get_loss(args, kwargs)\n\n def get_loss(self, x, mask, y,\n iter_i=0, \n # you may ignore the rest of the arguments for the time being\n # leave them as they are\n kl_weight=1.0,\n min_kl=0.0,\n ll_mean=0.,\n ll_std=1.,\n **kwargs):\n \"\"\"\n This computes the loss for the whole model.\n\n :param x: inputs [B, T]\n :param mask: identifies valid positions [B, T]\n :param y: target labels [B]\n :param iter_i: indicates the iteration \n :param kl_weight: (scalar) multiplies the KL term\n :param min_kl: (scalar) sets a minimum for the KL (aka free bits)\n :param ll_mean: (scalar) running average of reward\n :param ll_std: (scalar) running standard deviation of reward\n :return: loss (torch scalar node), terms (dict)\n \n terms is a dict that holds the scalar items involved in the loss\n e.g. `terms['ll'] = ll.item()` is the log-likelihood term\n \n You should log the following:\n \n - the ELBO: terms['elbo']\n - the likelihood term of the ELBO: terms['ll']\n - the kl before clipping and annealing: terms['kl']\n - the kl after clipping and annealing: terms['kl_'] \n - the learning signal in REINFORCE: terms['reward']\n - the standard deviation of the learning signal: terms['ll_std']\n - the ratio of selected words (average z): terms['selected']\n \n \n Consider tracking the following:\n Single-sample ELBO: terms['elbo']\n Log-Likelihood log P(y|x,z): terms['ll']\n KL: terms['kl']\n Score function surrogate log P(y|z, x) log Q(z|x): terms['sf'] \n Rate of selected words: terms['selected']\n \"\"\"\n\n lengths = mask.sum(1).float()\n batch_size = mask.size(0)\n terms = dict()\n\n # Infer a posterior distribution Z|x\n qz = self.get_posterior(x, mask)\n \n # Sample latent variables Z\n # [B, T]\n z = qz.sample()\n # Apply input mask\n z = torch.where(mask, z, torch.zeros_like(z))\n \n # Obtain a conditional distribution Y|z\n # [B, T, K]\n py = self.get_likelihood(x, mask, z)\n \n # Log-likelihood log P(y|x,z)\n # [B]\n ll = py.log_pmf(y)\n \n # KL(q||p)\n # [B, T]\n pz = self.get_prior(z.size(), device=z.device, dtype=z.dtype)\n \n # Compute KL divergence \n # [B, T]\n kl = qz.kl(pz)\n # Mask invalid input positions\n # [B, T]\n kl = torch.where(mask, kl, torch.zeros_like(kl))\n \n # Compute the log density of the sample\n # [B, T]\n log_q_z = qz.log_pmf(z)\n # and mask invalid positions\n log_q_z = torch.where(mask, log_q_z, torch.zeros_like(log_q_z))\n # We have independent Bernoullis, thus we just sum their log probabilities\n # [B]\n log_q_z = log_q_z.sum(1)\n \n # surrogate objective for score function estimator\n # [B]\n reward = (ll - ll_mean) / ll_std\n sf_surrogate = reward.detach() * log_q_z\n\n # [B]\n kl = kl.sum(dim=-1)\n # KL may require annealing and free-bits\n kl_ = torch.max(torch.full_like(kl, min_kl), kl) * kl_weight\n \n # Make complete surrogate loss (for backward)\n # [B]\n loss = - (ll + sf_surrogate - kl_)\n \n # Store terms worth tracking\n terms['elbo'] = (ll - kl).mean().item()\n terms['kl_'] = kl_.mean().item()\n terms['kl'] = kl.mean().item()\n terms['reward'] = reward.mean().item()\n terms['ll'] = ll.mean().item()\n terms['ll_std'] = ll.std().item()\n terms['selected'] = (z.sum(1) / lengths).mean().item()\n\n # return loss for backward and terms worth tracking\n return loss.mean(), terms \n \n def get_ste_loss(self, x, mask, y,\n iter_i=0, \n # you may ignore the rest of the arguments for the time being\n # leave them as they are\n kl_weight=1.0,\n min_kl=0.0,\n ll_mean=0.,\n ll_std=1.,\n **kwargs):\n \"\"\"Here we use the biased straight-through estimator instead of the score function estimator\"\"\"\n\n lengths = mask.sum(1).float()\n batch_size = mask.size(0)\n terms = dict()\n\n # Infer a posterior distribution Z|x\n qz = self.get_posterior(x, mask)\n \n # Sample latent variables Z\n # [B, T]\n z = qz.ste_sample()\n # Apply input mask\n z = torch.where(mask, z, torch.zeros_like(z))\n \n # Obtain a conditional distribution Y|z\n # [B, T, K]\n py = self.get_likelihood(x, mask, z)\n \n # Log-likelihood log P(y|x,z)\n # [B]\n ll = py.log_pmf(y)\n \n # KL(q||p)\n # [B, T]\n pz = self.get_prior(z.size(), device=z.device, dtype=z.dtype)\n \n # Compute KL divergence \n # [B, T]\n kl = qz.kl(pz)\n # Mask invalid input positions\n # [B, T]\n kl = torch.where(mask, kl, torch.zeros_like(kl))\n \n # [B]\n kl = kl.sum(dim=-1)\n # KL may require annealing and free-bits\n kl_ = torch.max(torch.full_like(kl, min_kl), kl) * kl_weight\n \n # Make complete surrogate loss (for backward)\n # [B]\n loss = - (ll - kl_)\n \n # Store terms worth tracking\n terms['elbo'] = (ll - kl).mean().item()\n terms['kl_'] = kl_.mean().item()\n terms['kl'] = kl.mean().item()\n terms['ll'] = ll.mean().item()\n terms['ll_std'] = ll.std().item()\n terms['selected'] = (z.sum(1) / lengths).mean().item()\n terms['reward'] = 0.\n\n # return loss for backward and terms worth tracking\n return loss.mean(), terms \n```\n\n# Training loop\n\nFor you to focus on the machine learning, we provide you with functions to deal with mini-batching, evaluation, and training.\n\n\n```python\n# some helper code for mini batching\n# this will take care of annoying things such as \n# sorting training instances by length (necessary for pytorch's LSTM, for example)\nfrom util import get_minibatch, prepare_minibatch, print_parameters\n```\n\nAn intrinsic convergence criterion is validation ELBO, thus we help you compute it here:\n\n\n```python\ndef eval_elbo(model: VAE, data, batch_size=25, device=None, iter_i=0, nb_samples=10):\n \"\"\"\n Accuracy of a model on given data set (using minibatches)\n \n :return: validation ELBO, validation KL\n \"\"\"\n\n model.eval() # disable dropout\n total_elbo, total_kl, data_size = 0., 0., 0.\n for mb in get_minibatch(data, batch_size=batch_size, shuffle=False):\n x, y, _ = prepare_minibatch(mb, model.vocab, device=device)\n mask = (x != 1)\n data_size = data_size + x.size(0)\n with torch.no_grad():\n # Infer Z|x\n qz = model.get_posterior(x, mask) \n # [B, T]\n z_s = qz.sample() \n # Prior Z\n # [B, T]\n pz = model.get_prior(z_s.size(), device=z_s.device, dtype=z_s.dtype)\n # 1. KL (closed form)\n # [B]\n kl = torch.where(mask, qz.kl(pz), torch.zeros_like(z_s)).sum(-1) \n # 2. E_q[log P(y|x,z)]\n terms = []\n ll = 0.\n for s in range(nb_samples):\n # sth sample\n # [B, T]\n z_s = qz.sample() \n # Apply input mask\n z_s = torch.where(mask, z_s, torch.zeros_like(z_s))\n # Compute Y|x,z_s\n # [B, K]\n py = model.get_likelihood(x, mask, z_s)\n # [B]\n ll = ll + py.log_pmf(y)\n total_elbo = total_elbo + (ll / nb_samples - kl ).sum().item()\n total_kl = total_kl + kl.sum().item()\n return total_elbo / data_size, total_kl / data_size\n```\n\nAn extrinsic convergence crition is classification accuracy, thus we help you compute it here.\nWe also make visualisations that highlight selected words (rationales).\n\n\n```python\nfrom collections import OrderedDict, defaultdict\nimport numpy as np\n\ndef decorate_token(t, z, hiding=True):\n \"\"\"\n Either hide text that hasn't been selected, or highlight text that has been selected.\n \"\"\"\n if hiding:\n return t if z == 1 else \"_\"\n else:\n dec = \"**\" if z == 1 else \"\" \n return dec + t + dec\n\ndef evaluate(model: VAE, data, batch_size=25, device=None, iter_i=0, nb_samples=10):\n \"\"\"\n Accuracy of a model on given data set (using minibatches)\n :return: accuracy, average selection rate, list of rationales (each a string)\n \"\"\"\n\n model.eval() # disable dropout\n rationales = []\n nb_correct, data_size, selection_rate = 0., 0., 0.\n for mb in get_minibatch(data, batch_size=batch_size, shuffle=False):\n x, targets, reverse_map = prepare_minibatch(mb, model.vocab, device=device)\n mask = (x != 1)\n with torch.no_grad():\n \n # Infere Z|x\n qz = model.get_posterior(x, mask) \n \n # Deterministic predictions (via greedy argmax)\n # [B, T]\n z_max = (qz.probs >= 0.5).float()\n # mask invalid positions\n z_max = torch.where(mask, z_max, torch.zeros_like(z_max))\n # Compute Y|x,z\n # [B, K]\n py = model.get_likelihood(x, mask, z_max) \n # argmax class\n # [B]\n predictions = py.argmax()\n \n # Accuracy and selection rate\n nb_correct = nb_correct + (predictions == targets.view(-1)).sum().item() \n data_size = data_size + x.size(0)\n selection_rate = selection_rate + (z_max.sum(-1) / mask.sum(-1).float()).sum().item()\n \n # String rationales\n # reverse sort \n z_max = z_max.cpu().numpy() \n z_max = z_max[reverse_map] \n for idx in range(x.size(0)): # iterate over instances in a mini batch\n example = []\n for ti, zi in zip(mb[idx].tokens, z_max[idx]): # iterate over tokens in an instance\n example.append(decorate_token(ti, zi))\n rationales.append((example, predictions[idx]))\n \n return nb_correct / data_size, selection_rate / data_size, rationales\n```\n\nHere we configure our training procedure. As there are several hyperparameters, let's us discuss the main ones briefly.\n\n**Data**\n* `training_path` where to find the training text file\n* `dev_path` where to find the validation text file\n* `test_path` where to find the test text file\n* `word_vectors` path to pre-trained 300-dimensional English word vectors\n* `subphrases` should we use annotation at subphrase level\n* `min_phrase_length` shortest phrase allowed (when subphrases are used)\n* `lowercase` should we lowercase the data\n\n**Model**\n* `prior_p1` this is the Bernoulli prior parameter $p_1$ \n\n**Architecture**\n* `fix_emb` should we keep embeddings fixed?\n* `embed_size` dimensionality of word vectors (use 300 for our pretrained vectors)\n* `hidden_size` dimensionality of hidden layers (e.g. in the encoder)\n* `num_layers` number of encoder layers (e.g. you may want to stack LSTMs)\n* `layer_inf` which encoder should be used in the inference network (e.g. bow/lstm)\n* `layer_cls` which composition function should the classifier use (e.g. bow/lstm)\n\n\n\n```python\nimport torch.optim\n# and a couple of tricks to reduce learning rate on plateau\nfrom torch.optim.lr_scheduler import ReduceLROnPlateau\n\ncfg = dict()\n\n# Data (you probably don't need to change these)\ncfg['training_path'] = \"data/sst/train.txt\"\ncfg['dev_path'] = \"data/sst/dev.txt\"\ncfg['test_path'] = \"data/sst/test.txt\"\ncfg['word_vectors'] = 'data/sst/glove.840B.300d.filtered.txt'\ncfg['subphrases'] = False\ncfg['min_phrase_length'] = 2\ncfg['lowercase'] = True\n# Model\ncfg['prior_p1'] = 0.3\n# Architecture\ncfg['fix_emb'] = True\ncfg['embed_size'] = 300\ncfg['hidden_size'] = 150\ncfg['num_layers'] = 1\ncfg['dropout'] = 0.5\ncfg['layer_inf'] = 'bow'\ncfg['layer_cls'] = 'bow'\n# Posterior collapse\ncfg['min_kl'] = 2. # use more than 0 to enable free bits\ncfg['kl_weight'] = 0.001 # start from zero to enable annealing\ncfg['kl_inc'] = 0.0001 \n# Logging\ncfg['save_path'] = 'data/results'\ncfg['print_every'] = 100\ncfg['eval_every'] = -1\ncfg['batch_size'] = 25\ncfg['eval_batch_size'] = 25\ncfg['display_dev'] = [39, 41, 1, 0, 4] # range(10) or selected examples\n# Optimiser (leave as is)\ncfg['num_epochs'] = 100\ncfg['running_alpha'] = 0.9\ncfg['gen_opt'] = 'adam'\ncfg['gen_lr'] = 0.001\ncfg['gen_momentum'] = 0.\ncfg['inf_opt'] = 'adadelta'\ncfg['inf_lr'] = 0.0001 \ncfg['inf_momentum'] = 0.\ncfg['gen_weight_decay'] = 1e-5\ncfg['inf_weight_decay'] = 1e-5\ncfg['lr_decay'] = 0.5\ncfg['patience'] = 10\ncfg['cooldown'] = 10\ncfg['threshold'] = 1e-4\ncfg['min_lr'] = 1e-6\ncfg['max_grad_norm'] = 5.\n\n\nprint('# Configuration')\nfor k, v in cfg.items():\n print(\"{:20} : {:10}\".format(k, str(v)))\n\n\niters_per_epoch = len(train_data) // cfg[\"batch_size\"]\n\nif cfg[\"eval_every\"] == -1:\n eval_every = iters_per_epoch\n print(\"Set eval_every to {}\".format(iters_per_epoch))\n```\n\nHere we load the data and display an example\n\n\n```python\n# Let's load the data into memory.\nprint(\"Loading data\")\ntrain_data = list(examplereader(\n cfg['training_path'],\n lower=cfg['lowercase'], \n subphrases=cfg['subphrases'],\n min_length=cfg['min_phrase_length']))\ndev_data = list(examplereader(cfg['dev_path'], lower=cfg['lowercase']))\ntest_data = list(examplereader(cfg['test_path'], lower=cfg['lowercase']))\n\nprint(\"train\", len(train_data))\nprint(\"dev\", len(dev_data))\nprint(\"test\", len(test_data))\n\nprint('\\n# Example')\nexample = dev_data[0]\nprint(\"First dev example:\", example)\nprint(\"First dev example tokens:\", example.tokens)\nprint(\"First dev example label:\", example.label)\n```\n\nHere we realise the main training loop.\n\n\n```python\nfrom functools import partial\nimport torch.optim as optim\n\ndef update_avg(old, new, alpha):\n \"\"\"Update a running average\"\"\"\n if old is None:\n return new\n return old * alpha + (1 - alpha) * new\n\ndef get_optimizer(name, parameters, lr, l2_weight, momentum=0.):\n if name is None or name == \"adam\":\n cls = optim.Adam\n elif name == \"amsgrad\":\n cls = partial(optim.Adam, amsgrad=True)\n elif name == \"adadelta\":\n cls = optim.Adadelta\n elif name == \"rmsprop\":\n cls = partial(optim.RMSprop, momentum=momentum)\n elif name == 'sgd':\n cls = optim.SGD\n else:\n raise ValueError(\"Unknown optimizer: %s\" % name)\n return cls(params=parameters, lr=lr, weight_decay=l2_weight)\n\ndef train():\n\n # Create a vocabulary object to map str <-> int\n vocab = Vocabulary() \n # populate it with pretrained word vectors\n glove_path = cfg[\"word_vectors\"]\n vectors = load_glove(glove_path, vocab)\n\n # You may consider using tensorboardX\n # writer = SummaryWriter(log_dir=cfg[\"save_path\"])\n\n # Map the sentiment labels 0-4 to a more readable form (and the opposite)\n i2t = [\"very negative\", \"negative\", \"neutral\", \"positive\", \"very positive\"]\n t2i = OrderedDict({p: i for p, i in zip(i2t, range(len(i2t)))})\n\n\n print('\\n# Constructing model')\n model = VAE(\n vocab_size=len(vocab.w2i), \n emb_size=cfg[\"embed_size\"],\n hidden_size=cfg[\"hidden_size\"], \n num_classes=len(t2i),\n prior_p1=cfg['prior_p1'],\n vocab=vocab, \n dropout=cfg[\"dropout\"], \n layer_cls=cfg[\"layer_cls\"],\n layer_inf=cfg[\"layer_inf\"])\n\n print('\\n# Loading embeddings')\n with torch.no_grad():\n model.embed.weight.data.copy_(torch.from_numpy(vectors))\n if cfg[\"fix_emb\"]:\n print(\"fixed word embeddings\")\n model.embed.weight.requires_grad = False\n model.embed.weight[1] = 0. # padding zero\n\n # Configure optimizers\n # it's not uncommon to use different optimisers when we employ discrete latent variables\n # the reason is because the score function estimator (SFE) and the reparameterised gradient\n # have very difference variances, and it's useful to have smaller learning rates for the SFE\n gen_optimizer = get_optimizer(\n cfg['gen_opt'], \n model.generative_parameters(),\n cfg['gen_lr'], cfg['gen_weight_decay'], cfg['gen_momentum']\n )\n inf_optimizer = get_optimizer(\n cfg['inf_opt'], \n model.inference_parameters(),\n cfg['inf_lr'], cfg['gen_weight_decay'], cfg['inf_momentum']\n )\n \n # and learning rate scheduler\n gen_scheduler = ReduceLROnPlateau(\n gen_optimizer, mode=\"min\", factor=cfg[\"lr_decay\"], patience=cfg[\"patience\"],\n verbose=True, cooldown=cfg[\"cooldown\"], threshold=cfg[\"threshold\"],\n min_lr=cfg[\"min_lr\"])\n inf_scheduler = ReduceLROnPlateau(\n inf_optimizer, mode=\"min\", factor=cfg[\"lr_decay\"], patience=cfg[\"patience\"],\n verbose=True, cooldown=cfg[\"cooldown\"], threshold=cfg[\"threshold\"],\n min_lr=cfg[\"min_lr\"])\n\n # Prepare a few auxiliary variables\n iter_i = 0\n best_eval = 1.0e9\n best_iter = 0\n\n model = model.to(device)\n\n # Some debugging info\n print(model)\n print_parameters(model)\n\n batch_size = cfg['batch_size']\n eval_batch_size = cfg['eval_batch_size']\n print_every = cfg['print_every']\n\n # Parameters of tricks to better optimise the ELBO \n kl_inc = cfg['kl_inc']\n kl_weight = cfg['kl_weight']\n min_kl = cfg['min_kl']\n # Running estimates for baselines\n running_avg, running_std = 0., 1.\n running_elbo, running_kl, running_kl_ = None, None, None\n running_loss, running_reward, running_rate = None, None, None\n alpha = cfg['running_alpha']\n\n while True: # when we run out of examples, shuffle and continue\n for batch in get_minibatch(train_data, batch_size=batch_size, shuffle=True):\n\n epoch = iter_i // iters_per_epoch\n if epoch > cfg['num_epochs']:\n break\n\n # forward pass\n model.train()\n x, y, _ = prepare_minibatch(batch, model.vocab, device=device)\n mask = (x != 1)\n\n # \"KL annealing\"\n kl_weight += kl_inc\n if kl_weight > 1.:\n kl_weight = 1.0\n \n loss, terms = model.get_loss(\n x, mask, y,\n kl_weight=kl_weight,\n min_kl=min_kl,\n ll_mean=running_avg,\n ll_std=running_std,\n iter_i=iter_i)\n\n # backward pass\n model.zero_grad() # erase previous gradients\n loss.backward() # compute new gradients\n # gradient clipping generally helps\n torch.nn.utils.clip_grad_norm_(model.parameters(), max_norm=cfg['max_grad_norm'])\n # update weights\n gen_optimizer.step() \n inf_optimizer.step() \n \n # logging\n running_loss = update_avg(running_loss, loss.item(), alpha) \n running_elbo = update_avg(running_elbo, terms['elbo'], alpha)\n running_kl = update_avg(running_kl, terms['kl'], alpha)\n running_kl_ = update_avg(running_kl_, terms['kl_'], alpha)\n running_reward = update_avg(running_reward, terms['reward'], alpha)\n running_rate = update_avg(running_rate, terms['selected'], alpha)\n \n # keep an running estimate of the reward (log P(y|x,z))\n running_avg = update_avg(running_avg, terms['ll'], alpha)\n running_std = update_avg(running_std, terms['ll_std'], alpha)\n if running_std < 1.:\n running_std = 1.\n\n iter_i += 1\n\n # print info\n if iter_i % print_every == 0:\n print(\"Epoch %r Iter %r loss=%.4f ELBO (KL/KL_)=%.4f (%.2f/%.2f) kl_weight=%.4f reward=%.4f selected=%.2f\" %\n (epoch, iter_i, running_loss, \n running_elbo, running_kl, running_kl_,\n kl_weight, running_reward,\n running_rate\n ))\n\n # evaluate\n if iter_i % eval_every == 0:\n\n dev_elbo, dev_kl = eval_elbo(\n model, dev_data, \n batch_size=eval_batch_size, \n device=device, \n iter_i=0, \n nb_samples=10)\n acc, selection_rate, rationales = evaluate(\n model, dev_data, \n batch_size=eval_batch_size, \n device=device,\n iter_i=iter_i)\n\n print(\"\\n# epoch %r iter %r: dev ELBO (KL) %.4f (%.2f) acc %.2f selected %.2f\" % (\n epoch, iter_i, dev_elbo, dev_kl, acc, selection_rate ))\n \n for exid in cfg['display_dev']:\n print(' dev%d [gold=%d,pred=%d]:' % (exid, dev_data[exid].label, rationales[exid][1]), \n ' '.join(rationales[exid][0]))\n print()\n\n # adjust learning rate \n gen_scheduler.step(-dev_elbo)\n inf_scheduler.step(-dev_elbo)\n```\n\n\n```python\ntrain()\n```\n\n# Variance reduction\n\n**This is an extra**\n\nWe can use a *control variate* to reduce the variance of our gradient estimates.\n\nLet's recap the idea in general terms. We are looking to solve some expectation\n\\begin{align}\n\\mu_f = \\mathbb E[f(Z)]\n\\end{align}\nbut unfortunatelly, realising the full sum (or integral for continuous variables) is intractable. Thus we employ MC estimation\n\\begin{align}\n\\hat \\mu_f &\\overset{\\text{MC}}{\\approx} \\frac{1}{S} \\sum_{s=1}^S f(z_s) & \\text{where }z_s \\sim Q(z|x)\n\\end{align}\nNote that the variance of this estimate is\n\\begin{align}\n\\text{Var}(\\hat \\mu_f) &= \\frac{1}{S}\\text{Var}(f(Z)) \\\\\n&= \\frac{1}{S} \\mathbb E[( f(Z) - \\mathbb E[f(Z)])^2]\n\\end{align}\nNote that this variance is such that it goes down as we sample more, in a rate $\\mathcal O(S^{-1})$.\nSee that if we sample $10$ times more, we will only obtain an decrease in variance in the order of $10^{-1}$. This means that sampling more is generally not the most convenient way to decrease variance.\n\n*Digression* we can estimate the variance itself via MC, an unbiased estimate looks like\n\\begin{align}\n\\hat \\sigma^2_f = \\frac{1}{S(S-1)} \\sum_{s=1}^S (f(z_s) - \\hat \\mu_f)^2\n\\end{align}\nbut not that this estimate is even hard to improve since it decreases with $\\mathcal O(S^{-2})$.\n\nBack to our main problem: let's try and improve the variance of our estimator to $\\mu_f$.\n\nIt's a fact, and it can be shown trivially, that\n\\begin{align}\n\\mu_f &= \\mathbb E[f(Z) - \\psi(Z)] + \\underbrace{\\mathbb E[\\psi(Z)]}_{\\mu_\\psi} \\\\\n &\\overset{\\text{MC}}{\\approx} \\underbrace{\\left(\\frac{1}{S} \\sum_{s=1}^S f(z_s) - \\psi(z_s) \\right) + \\mu_\\psi}_{\\hat c}\n\\end{align}\nwhere we assume the existence of some function $\\psi(z)$ for which the expected value $\\mu_\\psi$ is known and we estimate the expected difference $\\mathbb E[f(Z) - \\psi(Z)]$ via MC. We used this auxiliary function, also known as a *control variate*, to derive a new estimator, which we will denote by $\\hat c$.\n\nThe variance of this new estimator is shown below:\n\n\\begin{align}\n\\text{Var}( \\hat c ) &= \\text{Var}(\\hat \\mu_{f-\\psi}) + 2\\underbrace{\\text{Cov}(\\hat \\mu_{f-\\psi}, \\mu_\\psi)}_{\\mathbb E[\\hat \\mu_{f-\\psi} \\mu_\\psi] - \\mathbb E[\\hat \\mu_{f-\\psi}] \\mathbb E[\\mu_\\psi]} + \\underbrace{\\text{Var}(\\mu_\\psi)}_{\\color{blue}{0} } \\\\\n&= \\frac{1}{S}\\text{Var}(f- \\psi) + 2 \\underbrace{\\left( \\mu_\\psi \\mu_{f-\\psi} - \\mu_{f-\\psi} \\mu_\\psi \\right)}_{\\color{blue}{0}} \n\\end{align}\nwhere the variance of $\\mu_\\psi$ is 0 because we know it in closed form (no need for MC estimation), and the covariance is $0$ as shown in the second row.\n\nThat is, the variance of $\\hat c$ is essentially the variance of estimating $\\mathbb E[f(Z) - \\psi(Z)]$, which in turn depends on the variance \n\n\\begin{align}\n\\text{Var}(f-\\psi) &= \\text{Var}(f) - 2\\text{Cov}(f, \\psi) + \\text{Var}(\\psi)\n\\end{align}\nwhere we can see that if $\\text{Cov}(f, \\psi) > \\frac{\\text{Var}(\\psi)}{2}$ we achieve variance reduction as then $\\text{Var}(f-\\psi)$ would be smaller than $\\text{Var(f)}$.\n\n\n\n\n## Baselines\n\nBaslines are control variates of a very simple form:\n\\begin{align}\n\\mathbb E[f(Z)] = \\mathbb E[f(Z) - C] + \\mathbb E[C]\n\\end{align}\nwhere $C$ is a constant with respect to $z$.\n\nIn the context of the score function estimator, a baseline looks like a quantity $C(x; \\omega)$, this may be\n* just a constant;\n* or a function of the input (but not of the latent variable), which could be itself implemented as a neural network;\n* a combination of the two.\n \n\nLet's focus on the first term of the ELBO (so I'm omitting the KL term here). The gradient with respect to parameters of the inference model becomes:\n\n\\begin{align}\n&\\mathbb E_{Q(z|x, \\lambda)}\\left[ \\log P(x|z, \\theta) \\nabla_\\lambda \\log Q(z|x, \\lambda)\\right]\\\\\n&=\\mathbb E_{Q(z|x, \\lambda)}\\left[\\log P(x|z, \\theta) \\nabla_\\lambda \\log Q(z|x, \\lambda) - \\color{red}{C(x; \\omega)\\nabla_\\lambda \\log Q(z|x, \\lambda) } \\right] + \\underbrace{\\mathbb E_{Q(z|x, \\lambda)}\\left[\\color{red}{C(x; \\omega)\\nabla_\\lambda \\log Q(z|x, \\lambda) } \\right] }_{=0} \\\\\n&= \\mathbb E_{Q(z|x, \\lambda)}\\left[ \\color{blue}{\\left(\\log P(x|z, \\theta) - C(x; \\omega) \\right)}\\nabla_\\lambda \\log Q(z|x, \\lambda)\\right] \\\\\n&\n\\end{align}\nWe can show that the last term is $0$\n\n\\begin{align}\n&\\mathbb E_{Q(z|x, \\lambda)}\\left[C(x; \\omega)\\nabla_\\lambda \\log Q(z|x, \\lambda) \\right] \\\\&= C(x; \\omega) \\mathbb E_{Q(z|x, \\lambda)}\\left[\\nabla_\\lambda \\log Q(z|x, \\lambda) \\right]\\\\\n&= C(x; \\omega) \\mathbb E_{Q(z|x, \\lambda)}\\left[\\frac{1}{Q(z|x, \\lambda)} \\nabla_\\lambda Q(z|x, \\lambda) \\right] \\\\\n&= C(x; \\omega) \\sum_z Q(z|x, \\lambda) \\frac{1}{Q(z|x, \\lambda)} \\nabla_\\lambda Q(z|x, \\lambda) \\\\\n&= C(x; \\omega) \\sum_z\\nabla_\\lambda Q(z|x, \\lambda) \\\\\n&= C(x; \\omega) \\nabla_\\lambda \\underbrace{\\sum_z Q(z|x, \\lambda) }_{=1}\\\\\n&=0\n\\end{align}\n\nExamples of useful baselines:\n\n* a running average of the learning signal: at some iteration $t$ we can use a running average of $\\log P(x|z, \\theta)$ using parameter estimates $\\theta$ from iterations $i < t$, this is a baseline that likely leads to high correlation between control variate and learning signal and can lead to variance reduction;\n* another technique is to have an MLP with parameters $\\omega$ predict a scalar and train this MLP to approximate the learning signal $\\log P(x|z, \\theta)$ via regression:\n\\begin{align}\n\\arg\\max_\\omega \\left( C(x; \\omega) - \\log P(x|z, \\theta) \\right)^2\n\\end{align}\nits left as an extra to implement these ideas.\n\nOne more note: we can also use something called a *multiplicative baseline* in the literature of reinforcement learning, whereby we incorporate a running estimate of the standard deviation of the learning signal computed based on the values attained on previous iterations:\n\\begin{align}\n\\mathbb E_{Q(z|x, \\lambda)}\\left[ \\frac{1}{\\hat\\sigma_{\\text{past}}}\\left(\\log P(x|z, \\theta) - \\hat \\mu_{\\text{past}}\\right)\\nabla_\\lambda \\log Q(z|x, \\lambda)\\right]\n\\end{align}\nthis form of contorl variate aim at promoting the learning signal (or reward in reinforcement learning literature) to be distributed by $\\mathcal N(0, 1)$. Note that multiplying the reward by a constant does not bias the estimator, and in this case, may lead to variance reduction.\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "272a0029c15ca034627b160644db8603b6d173ee", "size": 107166, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "SST/Solution.ipynb", "max_stars_repo_name": "Roxot/vitutorial-exercises", "max_stars_repo_head_hexsha": "89afad877e2254d66d8741e1989182a20c87eaed", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-09-05T11:42:47.000Z", "max_stars_repo_stars_event_max_datetime": "2020-09-05T11:42:47.000Z", "max_issues_repo_path": "SST/Solution.ipynb", "max_issues_repo_name": "Roxot/vitutorial-exercises", "max_issues_repo_head_hexsha": "89afad877e2254d66d8741e1989182a20c87eaed", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "SST/Solution.ipynb", "max_forks_repo_name": "Roxot/vitutorial-exercises", "max_forks_repo_head_hexsha": "89afad877e2254d66d8741e1989182a20c87eaed", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-01-21T10:52:30.000Z", "max_forks_repo_forks_event_max_datetime": "2020-01-21T10:52:30.000Z", "avg_line_length": 40.2123827392, "max_line_length": 531, "alphanum_fraction": 0.5032379673, "converted": true, "num_tokens": 20495, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.7248702761768248, "lm_q1q2_score": 0.41860922485307567}} {"text": "# Mooney-Rivlin Hyperelasticity\n\n## Overview\n\nA Mooney-Rivlin hyperelastic material is one for which the derivatives of the free energy with respect to the invariants of stretch are constant. The Mooney-Rivlin model\n\n- a special case of the more general [polynomial hyperelastic model](PolynomialHyperelastic.ipynb)\n- has 2 elastic constants, typically fit to uniaxial extension/compression, equibiaxial extension/compression, or shear experimental data\n- usually valid for strains less than 100%\n\n## See Also\n\n- [User Defined Materials](UserMaterial.ipynb)\n- [Linear Elastic Material](LinearElastic.ipynb)\n- [Polynomial Hyperelastic Material](PolynomialHyperelastic.ipynb)\n\n## Contents\n\n1. Fundamental Equations\n2. Model Implementation\n3. Model Verification\n\n\n```python\nfrom bokeh.io import output_notebook\nfrom bokeh.plotting import *\nfrom matmodlab2 import *\nfrom numpy import *\nimport numpy as np\nfrom plotting_helpers import create_figure\noutput_notebook()\n```\n\n Setting up the Matmodlab notebook environment\n\n\n\n\n
                                        \n \n Loading BokehJS ...\n
                                        \n\n\n\n\n\n## Fundamental Equations\n\nThe Mooney-Rivlin material is a special case of the more general polynomial hyperelastic model defined by the following free energy potential\n\n$$\na = c_{10}\\left(\\overline{I}_1 - 3\\right) + c_{01}\\left(\\overline{I}_2 - 3\\right) \n + \\frac{1}{D_1}\\left(J-1\\right)^2\n$$\n\nwhere $c_{10}$, $c_{01}$, and $D_1$ are material parameters, the $\\overline{I}_i$ are the isochoric invariants of the right Cauchy deformation tensor $C_{IJ} = F_{kI}F_{kJ}$, $F_{iJ}$ are the components of the deformation gradient tensor, and $J$ is the determinant of the deformation gradient.\n\n### Second Piola-Kirchhoff Stress\n\nThe components of the second Piola-Kirchhoff stress $S_{IJ}$ are given by\n\n$$\n\\frac{1}{2\\rho_0} S_{IJ} = \\frac{\\partial a}{\\partial C_{IJ}}\n$$\n\nFor an isotropic material, the free energy $a$ is a function of $C_{IJ}$ through only its invariants, giving for $S_{IJ}$\n\n$$\n\\frac{1}{2\\rho_0}S_{IJ} \n = \\frac{\\partial a}{\\partial\\overline{I}_1}\\frac{\\partial\\overline{I}_1}{\\partial C_{IJ}}\n + \\frac{\\partial a}{\\partial\\overline{I}_2}\\frac{\\partial\\overline{I}_2}{\\partial C_{IJ}}\n + \\frac{\\partial a}{\\partial J}\\frac{\\partial J}{\\partial C_{IJ}}\n$$\n\nThe partial derivatives of the energy with respect to the invariants are\n\n$$\n\\frac{\\partial a}{\\partial\\overline{I}_1} = c_{10}, \\quad\n\\frac{\\partial a}{\\partial\\overline{I}_2} = c_{01}, \\quad\n\\frac{\\partial a}{\\partial J} = \\frac{2}{D_1}\\left(J-1\\right)\n$$\n\nand the partials of the invariants with respect to $C_{IJ}$ are\n\n$$\n\\begin{align}\n\\frac{\\partial\\overline{I}_1}{\\partial C_{IJ}}\n &= \\frac{\\partial\\left( J^{-2/3} C_{KK}\\right)}{\\partial C_{IJ}} \\\\\n %&= \\frac{\\partial J^{-2/3}}{\\partial C_{IJ}} I_1 + J^{-2/3} \\frac{\\partial C_{KK}}{\\partial C_{IJ}} \\\\\n %&= -\\frac{2}{3}J^{-5/3}\\frac{\\partial J}{\\partial C_{IJ}} I_1 + J^{-2/3} \\delta_{IJ} \\\\\n %&= -\\frac{2}{3}J^{-5/3}\\frac{1}{2}J C_{IJ}^{-1} I_1 + J^{-2/3} \\delta_{IJ} \\\\\n &= J^{-2/3}\\left(\\delta_{IJ} - \\frac{1}{3}I_1 C_{IJ}^{-1}\\right) \\\\\n\\end{align}\n$$\n\n$$\n\\begin{align}\n\\frac{\\partial\\overline{I}_2}{\\partial C_{IJ}}\n &= \\frac{\\partial\\left( J^{-4/3}\\left(C_{KK}^2 - C_{MN}C_{MN}\\right)\\right)}{\\partial C_{IJ}} \\\\\n %&= \\frac{\\partial J^{-4/3}}{\\partial C_{IJ}} \\left(C_{KK}^2 - C_{MN}C_{MN}\\right) + J^{-4/3} \\frac{\\partial \\left(C_{KK}^2 - C_{MN}C_{MN}\\right)}{\\partial C_{IJ}} \\\\\n &= J^{-4/3}\\left(I_1\\delta_{IJ} - C_{IJ} - \\frac{2}{3} I_2 C_{IJ}^{-1}\\right) \\\\\n\\end{align}\n$$\n\n$$\n\\begin{align}\n\\frac{\\partial J}{\\partial C_{IJ}} &= \\frac{\\partial \\sqrt{\\det C_{KL}}}{\\partial C_{IJ}} \\\\\n%&= \\frac{1}{2 \\sqrt{\\det C_{KL}}} \\frac{\\partial\\det C_{KL}}{\\partial C_{IJ}} \\\\\n%&= \\frac{1}{2 J} \\frac{\\partial\\det C_{KL}}{\\partial C_{IJ}} \\\\\n&= \\frac{1}{2} J C_{IJ}^{-1} \\\\\n\\end{align}\n$$\n\nCombining the above results, we arrive at the following expression for the components $S_{IJ}$:\n\n$$\n\\frac{1}{2\\rho_0}S_{IJ} \n = c_{10}J^{-2/3}\\left(\\delta_{IJ} - \\frac{1}{3}I_1 C_{IJ}^{-1}\\right)\n + c_{01}J^{-4/3}\\left(I_1\\delta_{IJ} - C_{IJ} - \\frac{2}{3} I_2 C_{IJ}^{-1}\\right)\n + \\frac{J}{D_1}\\left(J-1\\right) C_{IJ}^{-1}\n$$\n\n### Cauchy Stress\n\nThe components of the Cauchy stress tensor $\\sigma_{ij}$ are given by the push-forward $S_{IJ}$:\n\n$$\nJ \\sigma_{ij} = F_{iM}S_{MN}F_{jN}\n$$\n\n$$\n\\begin{align}\n\\frac{J}{2\\rho_0}\\sigma_{ij} &= F_{iM}\\left[\n c_{10}J^{-2/3}\\left(\\delta_{MN} - \\frac{1}{3}I_1 C_{MN}^{-1}\\right)\n + c_{01}J^{-4/3}\\left(I_1\\delta_{MN} - C_{MN} - \\frac{2}{3} I_2 C_{MN}^{-1}\\right)\n + \\frac{J}{D_1}\\left(J-1\\right) C_{MN}^{-1}\\right] F_{jN} \\\\\n &= c_{10}J^{-2/3}\\left(F_{iN}F_{jN} - \\frac{1}{3}I_1 F_{iM}C_{MN}^{-1}F_{jN}\\right)\n + c_{01}J^{-4/3}\\left(I_1F_{iN}F_{jN} - F_{iM}C_{MN}F_{jN} - \\frac{2}{3} I_2 F_{iM}C_{MN}^{-1}F_{jN}\\right) \\\\\n &\\qquad + \\frac{J}{D_1}\\left(J-1\\right) F_{iM}C_{MN}^{-1}F_{jN} \\\\\n\\end{align}\n$$\n\nRecognizing that\n\n$$\nC_{MN} = F_{kM}F_{kN} \\Rightarrow C_{MN}^{-1} = F_{kN}^{-1}F_{kM}^{-1}\n$$\n\nthe components of the Cauchy stress can be written as\n\n$$\n\\begin{align}\n\\frac{J}{2\\rho_0}\\sigma_{ij} \n &= c_{10}J^{-2/3}\\left(F_{iN}F_{jN} - \\frac{1}{3}I_1 F_{iM}F_{kN}^{-1}F_{kM}^{-1}F_{jN}\\right)\n + c_{01}J^{-4/3}\\left(I_1F_{iN}F_{jN} - F_{iM}F_{kM}F_{kN}F_{jN} - \\frac{2}{3} I_2 F_{iM}F_{kN}^{-1}F_{kM}^{-1}F_{jN}\\right) \\\\\n &\\qquad + \\frac{J}{D_1}\\left(J-1\\right) F_{iM}F_{kN}^{-1}F_{kM}^{-1}F_{jN} \\\\\n &= c_{10}J^{-2/3}\\left(B_{ij} - \\frac{1}{3}I_1 \\delta_{ij} \\right)\n + c_{01}J^{-4/3}\\left(I_1B_{ij} - B_{ik}B_{kj} - \\frac{2}{3} I_2 \\delta_{ij}\\right)\n + \\frac{J}{D_1}\\left(J-1\\right) \\delta_{ij} \\\\\n &= J^{-2/3} \\left(c_{10} + c_{01}\\overline{I}_1 \\right)B_{ij}\n - c_{01}J^{-4/3}B_{ik}B_{kj}\n + \\left(\n \\frac{J}{D_1}\\left(J-1\\right) - \\frac{1}{3}\\left(c_{10}\\overline{I}_1 + c_{01}2\\overline{I}_2\\right)\n \\right)\\delta_{ij}\n\\end{align}\n$$\n\nwhere $B_{ij} = F_{iN}F_{jN}$ is the left Cauchy deformation tensor. Finally, the components of the Cauchy stress are given by\n\n$$\n\\begin{align}\n\\sigma_{ij} &= \\frac{2\\rho_0}{J} \\left(\n J^{-2/3} \\left(c_{10} + c_{01}\\overline{I}_1 \\right)B_{ij}\n - c_{01}J^{-4/3}B_{ik}B_{kj}\\right)\n + \\left(\n \\frac{2\\rho_0}{D_1}\\left(J-1\\right) - \\frac{2\\rho_0}{3J}\\left(c_{10}\\overline{I}_1 + c_{01}2\\overline{I}_2\\right)\n \\right)\\delta_{ij} \\\\\n&= \\frac{2\\rho_0}{J} \\left(\n \\left(c_{10} + c_{01}\\overline{I}_1 \\right)\\overline{B}_{ij}\n - c_{01}\\overline{B}_{ik}\\overline{B}_{kj}\\right)\n + \\left(\n \\frac{2\\rho_0}{D_1}\\left(J-1\\right) - \\frac{2\\rho_0}{3J}\\left(c_{10}\\overline{I}_1 + c_{01}2\\overline{I}_2\\right)\n \\right)\\delta_{ij} \\\\\n\\end{align}\n$$\n\nwhere $\\overline{B}_{ij} = J^{-2/3}B_{ij}$\n\n\n\n## Matmodlab Implementation\n\nThe Mooney-Rivlin material is implemented as the `MooneyRivlinMaterial` class in `matmodlab2/materials/mooney_rivlin.py`. The file can be viewed by executing the following cell.\n\n\n```python\n%pycat ../matmodlab2/materials/mooney_rivlin.py\n```\n\n\n## Verification\n\n### Uniaxial Stress\n\nFor an incompressible isotropic material, uniaxial stress is produced by the following deformation state\n\n$$\n[F] = \\begin{bmatrix}\\lambda && \\\\ & \\frac{1}{\\sqrt{\\lambda}} & \\\\ & & \\frac{1}{\\sqrt{\\lambda}} \\end{bmatrix}\n$$\n\nThe stress difference $\\sigma_{\\text{axial}} - \\sigma_{\\text{lateral}}$ is given by\n\n\n```python\nfrom sympy import Symbol, Matrix, Rational, symbols, sqrt\nlam = Symbol('lambda')\nF = Matrix(3, 3, [lam, 0, 0, 0, 1/sqrt(lam), 0, 0, 0, 1/sqrt(lam)])\nB = Matrix(3, 3, F.dot(F.T))\nBsq = Matrix(3, 3, B.dot(B))\nI = Matrix(3, 3, lambda i,j: 1 if i==j else 0)\nI1 = B.trace()\nI2 = ((B.trace()) ** 2 - Bsq.trace()) / 2\nJ = F.det()\nX = J ** Rational(1, 3)\nC1, C2, D1 = symbols('C10 C01 D1')\nI1B = I1 / X ** 2\nI2B = I2 / X ** 4\n\nS = 2 / J * (1 / X ** 2 * (C1 + I1B * C2) * B - 1 / X ** 4 * C2 * Bsq) \\\n + (2 / D1 * (J - 1) - 2 * (C1 * I1B + 2 * C2 * I2B) / 3) * I\n(S[0,0] - S[1,1]).simplify()\n```\n\n\n\n\n 2*C01*lambda - 2*C01/lambda**2 + 2*C10*lambda**2 - 2*C10/lambda\n\n\n\nWe now exercise the Mooney-Rivlin material model using Matmodlab\n\n\n```python\n# Hyperelastic parameters, D1 set to a large number to force incompressibility\nparameters = {'D1': 1.e12, 'C10': 1e6, 'C01': .1e6}\n\n# stretch to 300%\nlam = linspace(.5, 3, 50)\n\n# Set up the simulator\nmps = MaterialPointSimulator('test1')\nmps.material = MooneyRivlinMaterial(**parameters)\n\n# Drive the *incompressible* material through a path of uniaxial stress by\n# prescribing the deformation gradient.\nFij = lambda x: (x, 0, 0, 0, 1/sqrt(x), 0, 0, 0, 1/sqrt(x))\nmps.run_step('F', Fij(lam[0]), frames=10)\nmps.run_step('F', Fij(1), frames=1)\nmps.run_step('F', Fij(lam[-1]), frames=20)\n\n# plot the analytic solution and the simulation\np = create_figure(bokeh=True, x_axis_label='Stretch', y_axis_label='Stress')\nC10, C01 = parameters['C10'], parameters['C01']\n\n# analytic solution for true and engineering stress\ns = 2*C01*lam - 2*C01/lam**2 + 2*C10*lam**2 - 2*C10/lam\n\n# plot the analytic solutions\np.line(lam, s, color='blue', legend='True', line_width=2)\np.line(lam, s/lam, color='green', legend='Engineering', line_width=2)\n\nlam_ = np.exp(mps.get('E.XX'))\nss = mps.get('S.XX') - mps.get('S.ZZ')\np.circle(lam_, ss, color='orange', legend='Simulation, True')\np.circle(lam_, ss/lam_, color='red', legend='Simulation, Engineering')\np.legend.location = 'top_left'\n\nshow(p)\n\n# check the actual solutions\nassert abs(amax(ss) - amax(s)) / amax(s) < 1e-6\nassert abs(amin(ss) - amin(s)) < 1e-6\n```\n\n\n\n\n
                                        \n
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                                        \n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "794516c4f4d25f5f74fe15575664270e14f1b43b", "size": 44714, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/MooneyRivlin.ipynb", "max_stars_repo_name": "matmodlab/matmodlab2", "max_stars_repo_head_hexsha": "97bb858e2b625cca5f3291db5d50bdbb6352e976", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2017-02-14T02:04:56.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-03T04:53:32.000Z", "max_issues_repo_path": "notebooks/MooneyRivlin.ipynb", "max_issues_repo_name": "tjfulle/matmodlab2", "max_issues_repo_head_hexsha": "97bb858e2b625cca5f3291db5d50bdbb6352e976", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 10, "max_issues_repo_issues_event_min_datetime": "2017-01-21T00:00:06.000Z", "max_issues_repo_issues_event_max_datetime": "2017-01-22T07:39:44.000Z", "max_forks_repo_path": "notebooks/MooneyRivlin.ipynb", "max_forks_repo_name": "tjfulle/matmodlab2", "max_forks_repo_head_hexsha": "97bb858e2b625cca5f3291db5d50bdbb6352e976", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2018-10-20T22:53:59.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-13T07:17:24.000Z", "avg_line_length": 57.3992297818, "max_line_length": 15704, "alphanum_fraction": 0.5579460572, "converted": true, "num_tokens": 3890, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.41849913981694487}} {"text": "This notebook is sample of the HAL QCD potential,\nthe effective mass fitting, and the effective energy shifts of two-baryon system\nfrom compressed NBS wavefunction sample_data.\n\nIn order to decompress the wave function, hal_pot_single_ch.py requires\nbinary \"PH1.compress48\" in \"yukawa library.\"\n\n\n```python\n%pylab inline\n```\n\n Populating the interactive namespace from numpy and matplotlib\n\n\n\n```python\nimport seaborn as sns\nsns.set_style('ticks', {'axes.grid': True})\nsns.set_context('poster', font_scale=2.0)\n%config InlineBackend.figure_format = 'retina'\n\nplt.rcParams['figure.figsize'] = (12.8, 9.6)\nplt.rcParams['figure.facecolor'] = 'white'\n\n```\n\n\n```python\nls ../data/sample_data/\n```\n\n \u001b[34mBBwave.dir.S4.00\u001b[m\u001b[m/ \u001b[34mcorrelator.PS.dir\u001b[m\u001b[m/\r\n README.md \u001b[34mcorrelator.multi.dir\u001b[m\u001b[m/\r\n\n\n\n```python\n# import library \nfrom hal_pot_single_ch import HAL_pot\nfrom corr_baryon import Corr_2pt_Baryon, Corr_2pt_2Baryons, Delta_Eeff\n```\n\nlattice spacing\n\n\n```python\nainv = 2.194e3 # MeV\nhbarc = 0.197e3 # MeV fm\nlat_unit = hbarc/ainv\n```\n\nanalyze baryon correlator and mass\n\\begin{equation}\nm_\\mathrm{eff}(t) = \\log \\frac{C_\\mathrm{B}(t)}{C_\\mathrm{B}(t+1)}\n\\end{equation}\n\n\n```python\ncb = Corr_2pt_Baryon('Xi_CG05_CG05', bin_size=1, result_dir='../data/sample_data/')\n```\n\n # Read Xi_CG05_CG05\n # total 4 conf., bin size = 1, number of samples = 4\n\n\n\n```python\ncb.plot_meff()\n```\n\n\n```python\nfig, ax = plt.subplots()\ncb.fit_meff(fit_min=15, fit_max=20, ax=ax)\nax.set_ylim(0.65, 0.68)\nleg = ax.legend(frameon=True)\nleg.get_frame().set_edgecolor('black')\nleg.get_frame().set_linewidth(2.0)\nax.set_title('$\\Xi$-baryon effective mass');\n```\n\n\n```python\nm_red = 0.5 * 0.665 \n```\n\ninitialize HAL_pot object with induced mass as an input parameter\n\n\n```python\nhal = HAL_pot(m_red=m_red, result_dir='../data/sample_data/',\n it0 = 10, Ns=48, channel='xixi', bin_size=2)\n```\n\n # Read Xi_CG05_CG05\n # total 4 conf., bin size = 2, number of samples = 2\n\n\ncalc_pot method evaluate (effective) central potential of (3S1) 1S0 channel, and central, tensor part of 3S1 channel.\n\n\n```python\n# potential\nresult = hal.calc_pot(pot_type='cen', spin='1s0')\n```\n\n # calc. Rcorr results.rcorr.binned/Rcorr_1s0_xixi_t009_2bin_4conf.dat\n # binning and decompress NBS\n # binning NBS S4.00 t = 9\n --- binning NBS wavefunc. --- 000\n >> decompress results.nbs.binned/NBSwave.S4.00.t009.binned.000.dat\n --- binning NBS wavefunc. --- 001\n >> decompress results.nbs.binned/NBSwave.S4.00.t009.binned.001.dat\n >> total 2 binned NBS files\n # load results.nbs.decomp/NBSwave.S4.00.t009.binned.000.decomp.dat\n # load results.nbs.decomp/NBSwave.S4.00.t009.binned.001.decomp.dat\n # save results.rcorr.binned/Rcorr_1s0_xixi_t009_2bin_4conf.dat\n # calc. Rcorr results.rcorr.binned/Rcorr_1s0_xixi_t010_2bin_4conf.dat\n # binning and decompress NBS\n # binning NBS S4.00 t = 10\n --- binning NBS wavefunc. --- 000\n >> decompress results.nbs.binned/NBSwave.S4.00.t010.binned.000.dat\n --- binning NBS wavefunc. --- 001\n >> decompress results.nbs.binned/NBSwave.S4.00.t010.binned.001.dat\n >> total 2 binned NBS files\n # load results.nbs.decomp/NBSwave.S4.00.t010.binned.000.decomp.dat\n # load results.nbs.decomp/NBSwave.S4.00.t010.binned.001.decomp.dat\n # save results.rcorr.binned/Rcorr_1s0_xixi_t010_2bin_4conf.dat\n # calc. Rcorr results.rcorr.binned/Rcorr_1s0_xixi_t011_2bin_4conf.dat\n # binning and decompress NBS\n # binning NBS S4.00 t = 11\n --- binning NBS wavefunc. --- 000\n >> decompress results.nbs.binned/NBSwave.S4.00.t011.binned.000.dat\n --- binning NBS wavefunc. --- 001\n >> decompress results.nbs.binned/NBSwave.S4.00.t011.binned.001.dat\n >> total 2 binned NBS files\n # load results.nbs.decomp/NBSwave.S4.00.t011.binned.000.decomp.dat\n # load results.nbs.decomp/NBSwave.S4.00.t011.binned.001.decomp.dat\n # save results.rcorr.binned/Rcorr_1s0_xixi_t011_2bin_4conf.dat\n >> return jackknife samples of (effective) central\n # output potential >> results.pot/pot_1s0_cen_xixi_t010_004conf_002bin.dat\n\n\ntext data of potential is stored in results.pot dir.\n\n\n```python\npot = np.loadtxt('results.pot/pot_1s0_cen_xixi_t010_004conf_002bin.dat')\n```\n\n\n```python\npot.shape\n```\n\n\n\n\n (2925, 9)\n\n\n\ncolumns of potential data\n\n0, 1, 2, 3, 4, 5, 6, 7, 8\n\nr, H0-term (av, err) , d/dt-term (av, err), d2/dt2-term (av, err), total (av, err)\n\n\n```python\nfig, ax = plt.subplots()\nax.errorbar(pot[:,0]*lat_unit, pot[:,1]*ainv, pot[:,2]*ainv,\n fmt='bs', mfc='none', mew=2, capsize=10, capthick=2, label=r'$H_0$-term')\n\nax.errorbar(pot[:,0]*lat_unit, pot[:,3]*ainv, pot[:,4]*ainv,\n fmt='g^', mfc='none', mew=2, capsize=10, capthick=2, label=r'$\\partial/\\partial t$-term')\n\nax.errorbar(pot[:,0]*lat_unit, pot[:,5]*ainv, pot[:,6]*ainv,\n fmt='kd', mfc='none', mew=2, capsize=10, capthick=2, label=r'$\\partial^2/\\partial t^2$-term')\n\nax.errorbar(pot[:,0]*lat_unit, pot[:,7]*ainv, pot[:,8]*ainv,\n fmt='ro', mfc='none', mew=2, capsize=10, capthick=2, label='total')\n\nax.set_ylim(-50, 50)\n\nax.axhline(0, color='black')\nax.set_xlabel(r'$r$ [fm]', size=48)\nax.set_ylabel(r'$V(r)$ [MeV]', size=48)\nleg = ax.legend(frameon=True)\nleg.get_frame().set_edgecolor('black')\nleg.get_frame().set_linewidth(2.0)\n```\n\n# baryon mass and energy shift\n\nif lattice unit is given, observables are plotted in physical scale\n\n\n```python\ncb = Corr_2pt_Baryon('Xi_CG05_CG05', bin_size=1, result_dir='../data/sample_data/', lat_unit=lat_unit)\n```\n\n # Read Xi_CG05_CG05\n # total 4 conf., bin size = 1, number of samples = 4\n\n\n\n```python\nfig, ax = plt.subplots()\ncb.fit_meff(fit_min=10, fit_max=20, ax=ax)\nleg = ax.legend(frameon=True)\nleg.get_frame().set_edgecolor('black')\nleg.get_frame().set_linewidth(2.0)\nax.set_ylim(1400, 1500)\n```\n\nTwo-baryon correlator and the effective energy\n\\begin{equation}\nE_\\mathrm{eff}(t) = \\log \\frac{C_\\mathrm{BB}(t)}{C_\\mathrm{BB}(t+1)}\n\\end{equation}\n\n\n```python\n# BB two baryon correlator\ncbb = Corr_2pt_2Baryons('xixi', bin_size=1, result_dir='../data/sample_data/', lat_unit=lat_unit)\n```\n\n Read xixi\n # total 4 conf., bin size = 1, number of samples = 4\n\n\n\n```python\nfig, ax = plt.subplots()\ncbb.fit_Eeff(ax=ax)\nax.set_ylim(2800, 3000)\n```\n\neffective energy shift\n\\begin{equation}\n\\Delta E_\\mathrm{eff}(t) = \\log \\frac{R(t)}{R(t+1)}\n\\end{equation}\nwith\n\\begin{equation}\nR(t) = \\frac{C_\\mathrm{BB}(t)}{\\{C_\\mathrm{B}(t)\\}^2}\n\\end{equation}\n\n\n```python\n# effective energy shifts\ndEeff = Delta_Eeff('xixi', bin_size=1, result_dir='../data/sample_data', lat_unit=lat_unit)\n```\n\n Read xixi\n # total 4 conf., bin size = 1, number of samples = 4\n # Read Xi_CG05_CG05\n # total 4 conf., bin size = 1, number of samples = 4\n\n\n\n```python\nfig, ax = plt.subplots()\ndEeff.fit_dEeff(fit_min=7, fit_max=10, spin='1s0', ax=ax)\nax.set_ylim(-10, 10)\nleg = ax.legend(frameon=True)\nleg.get_frame().set_edgecolor('black')\nleg.get_frame().set_linewidth(2.0)\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "4acc9d8b4b9a3004bffe3cfd8d5f2ac1e8cdf16b", "size": 922689, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notes/Test code HAL potential and effective masses.ipynb", "max_stars_repo_name": "tkmiriya/hal_pot_code", "max_stars_repo_head_hexsha": "3446378916df74eca887cb36c380ed0ab151192e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notes/Test code HAL potential and effective masses.ipynb", "max_issues_repo_name": "tkmiriya/hal_pot_code", "max_issues_repo_head_hexsha": "3446378916df74eca887cb36c380ed0ab151192e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notes/Test code HAL potential and effective masses.ipynb", "max_forks_repo_name": "tkmiriya/hal_pot_code", "max_forks_repo_head_hexsha": "3446378916df74eca887cb36c380ed0ab151192e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 1290.4741258741, "max_line_length": 274432, "alphanum_fraction": 0.9564587851, "converted": true, "num_tokens": 2475, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744806385542, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.41836643732513235}} {"text": "# Deep Learning Bootcamp November 2017, GPU Computing for Data Scientists\n\n\n\n## 18 PyTorch Deep Learning NUMER.AI Binary Classification using BCELoss \n\nWeb: https://www.meetup.com/Tel-Aviv-Deep-Learning-Bootcamp/events/241762893/\n\nNotebooks: On GitHub\n\n*Shlomo Kashani*\n\n\n\n\n### Data\n- Download from https://numer.ai/leaderboard\n\n\n\n### Docker image with PyTorch CPU and GPU support:\n\n- https://github.com/QuantScientist/Deep-Learning-Boot-Camp/tree/master/docker\n\n# PyTorch Imports\n\n\n\n```python\n# !pip install pycuda\n%reset -f\n# %%timeit\n\nimport torch\nfrom torch.autograd import Variable\nimport numpy as np\nimport pandas\nimport numpy as np\nimport pandas as pd\nfrom sklearn import cross_validation\nfrom sklearn import metrics\nfrom sklearn.metrics import roc_auc_score, log_loss, roc_auc_score, roc_curve, auc\nimport matplotlib.pyplot as plt\nfrom sklearn import cross_validation\nfrom sklearn import metrics\nfrom sklearn.metrics import roc_auc_score, log_loss, roc_auc_score, roc_curve, auc\nfrom sklearn.cross_validation import StratifiedKFold, ShuffleSplit, cross_val_score, train_test_split\nimport logging\nimport numpy\nimport numpy as np\nfrom __future__ import print_function\nfrom __future__ import division\nimport math\nimport numpy as np\nimport matplotlib.pyplot as plt\n%matplotlib inline\nimport pandas as pd\nimport os\nimport torch\nfrom torch.utils.data.dataset import Dataset\nfrom torch.utils.data import DataLoader\nfrom torchvision import transforms\nfrom torch import nn\nimport torch.nn.functional as F\nimport torch.optim as optim\nfrom torch.autograd import Variable\nfrom sklearn.preprocessing import MultiLabelBinarizer\nimport time\nfrom sklearn.preprocessing import PolynomialFeatures\nimport pandas as pd\nimport numpy as np\nimport scipy\n%matplotlib inline\nfrom pylab import rcParams\nrcParams['figure.figsize'] = (6, 6) # setting default size of plots\nimport tensorflow as tf \nprint(\"tensorflow:\" + tf.__version__)\n!set \"KERAS_BACKEND=tensorflow\"\nimport torch\nimport sys\nprint('__Python VERSION:', sys.version)\nprint('__pyTorch VERSION:', torch.__version__)\nprint('__CUDA VERSION')\nfrom subprocess import call\nprint('__CUDNN VERSION:', torch.backends.cudnn.version())\nprint('__Number CUDA Devices:', torch.cuda.device_count())\nprint('__Devices')\n\n# !pip install http://download.pytorch.org/whl/cu75/torch-0.2.0.post1-cp27-cp27mu-manylinux1_x86_64.whl\n# !pip install torchvision \n# ! pip install cv2\n# import cv2\n\nprint(\"OS: \", sys.platform)\nprint(\"Python: \", sys.version)\nprint(\"PyTorch: \", torch.__version__)\nprint(\"Numpy: \", np.__version__)\n\nhandler=logging.basicConfig(level=logging.INFO)\nlgr = logging.getLogger(__name__)\n%matplotlib inline\n\n# !pip install psutil\nimport psutil\ndef cpuStats():\n print(sys.version)\n print(psutil.cpu_percent())\n print(psutil.virtual_memory()) # physical memory usage\n pid = os.getpid()\n py = psutil.Process(pid)\n memoryUse = py.memory_info()[0] / 2. ** 30 # memory use in GB...I think\n print('memory GB:', memoryUse)\n\ncpuStats()\n```\n\n /usr/local/lib/python2.7/dist-packages/sklearn/cross_validation.py:44: DeprecationWarning: This module was deprecated in version 0.18 in favor of the model_selection module into which all the refactored classes and functions are moved. Also note that the interface of the new CV iterators are different from that of this module. This module will be removed in 0.20.\n \"This module will be removed in 0.20.\", DeprecationWarning)\n\n\n tensorflow:1.2.1\n __Python VERSION: 2.7.12 (default, Nov 19 2016, 06:48:10) \n [GCC 5.4.0 20160609]\n __pyTorch VERSION: 0.1.12+4eb448a\n __CUDA VERSION\n __CUDNN VERSION: None\n __Number CUDA Devices: 1\n __Devices\n OS: linux2\n Python: 2.7.12 (default, Nov 19 2016, 06:48:10) \n [GCC 5.4.0 20160609]\n PyTorch: 0.1.12+4eb448a\n Numpy: 1.13.1\n 2.7.12 (default, Nov 19 2016, 06:48:10) \n [GCC 5.4.0 20160609]\n 0.0\n svmem(total=67469099008, available=63067299840, percent=6.5, used=3831234560, free=61351706624, active=4435632128, inactive=1119084544, buffers=270614528, cached=2015543296, shared=37376000)\n memory GB: 0.213935852051\n\n\n# CUDA\n\n\n```python\n# %%timeit\nuse_cuda = torch.cuda.is_available()\n# use_cuda = False\n\nFloatTensor = torch.cuda.FloatTensor if use_cuda else torch.FloatTensor\nLongTensor = torch.cuda.LongTensor if use_cuda else torch.LongTensor\nTensor = FloatTensor\n\nlgr.info(\"USE CUDA=\" + str (use_cuda))\n\n# ! watch -n 0.1 'ps f -o user,pgrp,pid,pcpu,pmem,start,time,command -p `lsof -n -w -t /dev/nvidia*`'\n# sudo apt-get install dstat #install dstat\n# sudo pip install nvidia-ml-py #install Python NVIDIA Management Library\n# wget https://raw.githubusercontent.com/datumbox/dstat/master/plugins/dstat_nvidia_gpu.py\n# sudo mv dstat_nvidia_gpu.py /usr/share/dstat/ #move file to the plugins directory of dstat\n```\n\n INFO:__main__:USE CUDA=True\n\n\n# Global params\n\n\n```python\n# NN params\nDROPOUT_PROB = 0.75\nN_EPOCHS = 50\nBATCH_SIZE = 4\nLR = 0.005\nTEST_RATIO = .11\nMOMENTUM= 0.9\nPIN_MEMORY=use_cuda # True IF CUDA\n\n# Data params\nTARGET_VAR= 'target'\nTOURNAMENT_DATA_CSV = 'numerai_tournament_data.csv'\nTRAINING_DATA_CSV = 'numerai_training_data.csv'\nBASE_FOLDER = 'numerai/'\n\n\n# fix seed\nseed=17*19\nnp.random.seed(seed)\ntorch.manual_seed(seed)\nif use_cuda:\n torch.cuda.manual_seed(seed)\n```\n\n# Load a CSV file for Binary classification (numpy)\n\n\n```python\n# %%timeit\ndf_train = pd.read_csv(BASE_FOLDER + TRAINING_DATA_CSV)\ndf_train.head(5)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        ideradata_typefeature1feature2feature3feature4feature5feature6feature7...feature13feature14feature15feature16feature17feature18feature19feature20feature21target
                                        0135682era1train0.533520.643360.465770.530010.557340.457730.41169...0.512240.504840.419290.509540.473830.487970.383730.462330.333410
                                        1110546era1train0.541960.815760.466320.623200.524270.643780.55662...0.526430.638090.671210.494210.452910.469320.544450.309970.190230
                                        276047era1train0.491580.691310.578160.540100.430640.499860.61902...0.433100.722860.762570.366000.553300.565660.675280.349600.257211
                                        366098era1train0.545190.424730.634720.390030.374850.438100.59557...0.416580.634170.501890.408830.587050.637850.562250.559890.586420
                                        488227era1train0.443070.740760.522100.565430.511250.664570.42263...0.458510.588050.498600.480230.526060.532530.383610.438290.250140
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                                        5 rows \u00d7 25 columns

                                        \n
                                        \n\n\n\n# Feature enrichement\n- This would be usually not required when using NN\n\n\n```python\ndef genBasicFeatures(inDF):\n print('Generating basic features ...')\n df_copy=inDF.copy(deep=True)\n magicNumber=21\n feature_cols = list(inDF.columns)\n# feature_cols = list(inDF.columns[:-1])\n # feature_cols=xgb_cols\n# target_col = inDF.columns[-1]\n\n inDF['x_mean'] = np.mean(df_copy.ix[:, 0:magicNumber], axis=1)\n inDF['x_median'] = np.median(df_copy.ix[:, 0:magicNumber], axis=1)\n inDF['x_std'] = np.std(df_copy.ix[:, 0:magicNumber], axis=1)\n inDF['x_skew'] = scipy.stats.skew(df_copy.ix[:, 0:magicNumber], axis=1)\n inDF['x_kurt'] = scipy.stats.kurtosis(df_copy.ix[:, 0:magicNumber], axis=1)\n inDF['x_var'] = np.var(df_copy.ix[:, 0:magicNumber], axis=1)\n inDF['x_max'] = np.max(df_copy.ix[:, 0:magicNumber], axis=1)\n inDF['x_min'] = np.min(df_copy.ix[:, 0:magicNumber], axis=1)\n # http://stackoverflow.com/questions/16236684/apply-pandas-function-to-column-to-create-multiple-new-columns\n# inDF=inDF.merge(df_copy.ix[:, 0:magicNumber].apply(lambda row: NumerCommonML.enrichFeatures(row), axis=1),\n# left_index=True, right_index=True)\n\n print (inDF.head(1))\n return inDF\n\ndef addPolyFeatures(inDF, deg=2):\n print('Generating poly features ...')\n df_copy=inDF.copy(deep=True)\n poly=PolynomialFeatures(degree=deg)\n p_testX = poly.fit(df_copy)\n # AttributeError: 'PolynomialFeatures' object has no attribute 'get_feature_names'\n target_feature_names = ['x'.join(['{}^{}'.format(pair[0],pair[1]) for pair in tuple if pair[1]!=0]) for tuple in [zip(df_copy.columns,p) for p in poly.powers_]]\n df_copy = pd.DataFrame(p_testX.transform(df_copy),columns=target_feature_names)\n \n return df_copy\n```\n\n# Train / Validation / Test Split\n- Numerai provides a data set that is allready split into train, validation and test sets. \n\n\n```python\n# Train, Validation, Test Split\ndef loadDataSplit():\n df_train = pd.read_csv(BASE_FOLDER + TRAINING_DATA_CSV)\n # TOURNAMENT_DATA_CSV has both validation and test data provided by NumerAI\n df_test_valid = pd.read_csv(BASE_FOLDER + TOURNAMENT_DATA_CSV)\n\n answers_1_SINGLE = df_train[TARGET_VAR]\n df_train.drop(TARGET_VAR, axis=1,inplace=True)\n df_train.drop('id', axis=1,inplace=True)\n df_train.drop('era', axis=1,inplace=True)\n df_train.drop('data_type', axis=1,inplace=True) \n \n # Add polynomial features \n df_train=genBasicFeatures(df_train)\n# df_train = addPolyFeatures(df_train)\n\n df_train.to_csv(BASE_FOLDER + TRAINING_DATA_CSV + 'clean.csv', header=False, index = False) \n df_train= pd.read_csv(BASE_FOLDER + TRAINING_DATA_CSV + 'clean.csv', header=None, dtype=np.float32) \n df_train = pd.concat([df_train, answers_1_SINGLE], axis=1)\n feature_cols = list(df_train.columns[:-1])\n# print (feature_cols)\n target_col = df_train.columns[-1]\n trainX, trainY = df_train[feature_cols], df_train[target_col]\n \n \n # TOURNAMENT_DATA_CSV has both validation and test data provided by NumerAI\n # Validation set\n df_validation_set=df_test_valid.loc[df_test_valid['data_type'] == 'validation'] \n df_validation_set=df_validation_set.copy(deep=True)\n answers_1_SINGLE_validation = df_validation_set[TARGET_VAR]\n df_validation_set.drop(TARGET_VAR, axis=1,inplace=True) \n df_validation_set.drop('id', axis=1,inplace=True)\n df_validation_set.drop('era', axis=1,inplace=True)\n df_validation_set.drop('data_type', axis=1,inplace=True)\n \n # Add polynomial features \n df_validation_set=genBasicFeatures(df_validation_set)\n# df_validation_set = addPolyFeatures(df_validation_set)\n \n df_validation_set.to_csv(BASE_FOLDER + TRAINING_DATA_CSV + '-validation-clean.csv', header=False, index = False) \n df_validation_set= pd.read_csv(BASE_FOLDER + TRAINING_DATA_CSV + '-validation-clean.csv', header=None, dtype=np.float32) \n df_validation_set = pd.concat([df_validation_set, answers_1_SINGLE_validation], axis=1)\n feature_cols = list(df_validation_set.columns[:-1])\n\n target_col = df_validation_set.columns[-1]\n valX, valY = df_validation_set[feature_cols], df_validation_set[target_col]\n \n # Test set for submission (not labeled) \n df_test_set = pd.read_csv(BASE_FOLDER + TOURNAMENT_DATA_CSV)\n# df_test_set=df_test_set.loc[df_test_valid['data_type'] == 'live'] \n df_test_set=df_test_set.copy(deep=True)\n df_test_set.drop(TARGET_VAR, axis=1,inplace=True)\n tid_1_SINGLE = df_test_set['id']\n df_test_set.drop('id', axis=1,inplace=True)\n df_test_set.drop('era', axis=1,inplace=True)\n df_test_set.drop('data_type', axis=1,inplace=True) \n \n # Add polynomial features \n df_test_set=genBasicFeatures(df_test_set)\n# df_test_set = addPolyFeatures(df_test_set)\n \n \n feature_cols = list(df_test_set.columns) # must be run here, we dont want the ID \n# print (feature_cols)\n df_test_set = pd.concat([tid_1_SINGLE, df_test_set], axis=1) \n testX = df_test_set[feature_cols].values\n \n return trainX, trainY, valX, valY, testX, df_test_set\n```\n\n\n```python\n# %%timeit\ntrainX, trainY, valX, valY, testX, df_test_set = loadDataSplit()\n# X, y = loadDataSplit(999)\n\n\n# # Number of features for the input layer\nN_FEATURES=trainX.shape[1]\n# print (trainX.head(3))\n# print (df_test_set.head(3))\nprint (trainX.shape)\nprint (trainY.shape)\nprint (valX.shape)\nprint (valY.shape)\nprint (testX.shape)\nprint (df_test_set.shape)\n```\n\n Generating basic features ...\n feature1 feature2 feature3 feature4 feature5 feature6 feature7 \\\n 0 0.53352 0.64336 0.46577 0.53001 0.55734 0.45773 0.41169 \n \n feature8 feature9 feature10 ... feature20 feature21 x_mean \\\n 0 0.5207 0.36351 0.72262 ... 0.46233 0.33341 0.495936 \n \n x_median x_std x_skew x_kurt x_var x_max x_min \n 0 0.50484 0.088713 0.48164 0.453078 0.00787 0.72262 0.33341 \n \n [1 rows x 29 columns]\n\n\n /usr/local/lib/python2.7/dist-packages/ipykernel_launcher.py:10: DeprecationWarning: \n .ix is deprecated. Please use\n .loc for label based indexing or\n .iloc for positional indexing\n \n See the documentation here:\n http://pandas.pydata.org/pandas-docs/stable/indexing.html#ix-indexer-is-deprecated\n # Remove the CWD from sys.path while we load stuff.\n\n\n Generating basic features ...\n feature1 feature2 feature3 feature4 feature5 feature6 feature7 \\\n 0 0.33463 0.5162 0.73225 0.3637 0.35314 0.63717 0.51138 \n \n feature8 feature9 feature10 ... feature20 feature21 x_mean \\\n 0 0.62956 0.18643 0.60806 ... 0.59448 0.46003 0.492229 \n \n x_median x_std x_skew x_kurt x_var x_max x_min \n 0 0.51138 0.166973 -0.148699 -1.259785 0.02788 0.73225 0.18643 \n \n [1 rows x 29 columns]\n Generating basic features ...\n feature1 feature2 feature3 feature4 feature5 feature6 feature7 \\\n 0 0.33463 0.5162 0.73225 0.3637 0.35314 0.63717 0.51138 \n \n feature8 feature9 feature10 ... feature20 feature21 x_mean \\\n 0 0.62956 0.18643 0.60806 ... 0.59448 0.46003 0.492229 \n \n x_median x_std x_skew x_kurt x_var x_max x_min \n 0 0.51138 0.166973 -0.148699 -1.259785 0.02788 0.73225 0.18643 \n \n [1 rows x 29 columns]\n (108405, 29)\n (108405,)\n (16686, 29)\n (16686,)\n (45647, 29)\n (45647, 30)\n\n\n# Create PyTorch GPU tensors from numpy arrays\n\n- Note how we transfrom the np arrays\n\n\n```python\n# Convert the np arrays into the correct dimention and type\n# Note that BCEloss requires Float in X as well as in y\ndef XnumpyToTensor(x_data_np):\n x_data_np = np.array(x_data_np.values, dtype=np.float32) \n print(x_data_np.shape)\n print(type(x_data_np))\n\n if use_cuda:\n lgr.info (\"Using the GPU\") \n X_tensor = Variable(torch.from_numpy(x_data_np).cuda()) # Note the conversion for pytorch \n else:\n lgr.info (\"Using the CPU\")\n X_tensor = Variable(torch.from_numpy(x_data_np)) # Note the conversion for pytorch\n \n print(type(X_tensor.data)) # should be 'torch.cuda.FloatTensor'\n print(x_data_np.shape)\n print(type(x_data_np)) \n return X_tensor\n\n\n# Convert the np arrays into the correct dimention and type\n# Note that BCEloss requires Float in X as well as in y\ndef YnumpyToTensor(y_data_np): \n y_data_np=y_data_np.reshape((y_data_np.shape[0],1)) # Must be reshaped for PyTorch!\n print(y_data_np.shape)\n print(type(y_data_np))\n\n if use_cuda:\n lgr.info (\"Using the GPU\") \n # Y = Variable(torch.from_numpy(y_data_np).type(torch.LongTensor).cuda())\n Y_tensor = Variable(torch.from_numpy(y_data_np)).type(torch.FloatTensor).cuda() # BCEloss requires Float \n else:\n lgr.info (\"Using the CPU\") \n # Y = Variable(torch.squeeze (torch.from_numpy(y_data_np).type(torch.LongTensor))) # \n Y_tensor = Variable(torch.from_numpy(y_data_np)).type(torch.FloatTensor) # BCEloss requires Float \n\n print(type(Y_tensor.data)) # should be 'torch.cuda.FloatTensor'\n print(y_data_np.shape)\n print(type(y_data_np)) \n return Y_tensor\n```\n\n# The NN model\n\n### MLP model\n- A multilayer perceptron is a logistic regressor where instead of feeding the input to the logistic regression you insert a intermediate layer, called the hidden layer, that has a nonlinear activation function (usually tanh or sigmoid) . One can use many such hidden layers making the architecture deep.\n\n- Here we define a simple MLP structure. We map the input feature vector to a higher space (256), then later gradually decrease the dimension, and in the end into a 16-dimension space. Because we are calculating the probability of each genre independently, after the final affine layer we need to implement a sigmoid layer. \n\n### Initial weights selection\n\n- There are many ways to select the initial weights to a neural network architecture. A common initialization scheme is random initialization, which sets the biases and weights of all the nodes in each hidden layer randomly.\n\n- Before starting the training process, an initial value is assigned to each variable. This is done by pure randomness, using for example a uniform or Gaussian distribution. But if we start with weights that are too small, the signal could decrease so much that it is too small to be useful. On the other side, when the parameters are initialized with high values, the signal can end up to explode while propagating through the network.\n\n- In consequence, a good initialization can have a radical effect on how fast the network will learn useful patterns.For this purpose, some best practices have been developed. One famous example used is **Xavier initialization**. Its formulation is based on the number of input and output neurons and uses sampling from a uniform distribution with zero mean and all biases set to zero.\n\n- In effect (according to theory) initializing the weights of the network to values that would be closer to the optimal, and therefore require less epochs to train.\n\n### References: \n* **`nninit.xavier_uniform(tensor, gain=1)`** - Fills `tensor` with values according to the method described in [\"Understanding the difficulty of training deep feedforward neural networks\" - Glorot, X. and Bengio, Y.](http://jmlr.org/proceedings/papers/v9/glorot10a/glorot10a.pdf), using a uniform distribution.\n* **`nninit.xavier_normal(tensor, gain=1)`** - Fills `tensor` with values according to the method described in [\"Understanding the difficulty of training deep feedforward neural networks\" - Glorot, X. and Bengio, Y.](http://jmlr.org/proceedings/papers/v9/glorot10a/glorot10a.pdf), using a normal distribution.\n* **`nninit.kaiming_uniform(tensor, gain=1)`** - Fills `tensor` with values according to the method described in [\"Delving deep into rectifiers: Surpassing human-level performance on ImageNet classification\" - He, K. et al.](https://arxiv.org/abs/1502.01852) using a uniform distribution.\n* **`nninit.kaiming_normal(tensor, gain=1)`** - Fills `tensor` with values according to the method described in [\"Delving deep into rectifiers: Surpassing human-level performance on ImageNet classification\" - He, K. et al.]\n\n\n\n```python\n# p is the probability of being dropped in PyTorch\n# At each layer, DECREASE dropout\ndropout = torch.nn.Dropout(p=1 - (DROPOUT_PROB +0.20))\n\n\n\n# class Net(torch.nn.Module):\n# def __init__(self, n_feature, n_hidden, n_output,initKernel='uniform'):\n# super(Net, self).__init__()\n# self.hidden = torch.nn.Linear(n_feature, n_hidden) # hidden layer\n# self.out = torch.nn.Linear(n_hidden, n_output) # output layer \n \n# # xavier initializer\n# if initKernel == 'uniform':\n# nn.init.xavier_uniform(self.hidden.weight, gain=np.sqrt(2.0))\n# else:\n# nn.init.kaiming_normal(self.hidden.weight) \n\n# def forward(self, x):\n# x = F.relu(self.hidden(x)) # activation function for hidden layer\n# x = self.out(x)\n# return F.sigmoid(x)\n\nclass Net2(nn.Module):\n def __init__(self, n_feature, n_hidden, n_output,initKernel='uniform'):\n super(Net2, self).__init__()\n self.dis = nn.Sequential(\n nn.Linear(n_feature, n_hidden),\n dropout,\n nn.LeakyReLU(0.1),\n nn.Linear(n_hidden, n_hidden),\n dropout,\n nn.LeakyReLU(0.1),\n nn.Linear(n_hidden, 1),\n dropout,\n nn.Sigmoid()\n ) \n def forward(self, x):\n x = self.dis(x)\n return x\n\n\nhiddenLayer1Size=1024\nhiddenLayer2Size=int(hiddenLayer1Size/8)\nhiddenLayer3Size=int(hiddenLayer1Size/16)\nhiddenLayer4Size=int(hiddenLayer1Size/32)\nhiddenLayer5Size=int(hiddenLayer1Size/64)\n\n# # Hypothesis using sigmoid\nlinear1=torch.nn.Linear(N_FEATURES, hiddenLayer1Size, bias=True) \ntorch.nn.init.xavier_uniform(linear1.weight)\n\nlinear2=torch.nn.Linear(hiddenLayer1Size, hiddenLayer2Size)\ntorch.nn.init.xavier_uniform(linear2.weight)\n\nlinear3=torch.nn.Linear(hiddenLayer2Size, hiddenLayer3Size)\ntorch.nn.init.xavier_uniform(linear3.weight)\n\nlinear4=torch.nn.Linear(hiddenLayer3Size, hiddenLayer4Size)\ntorch.nn.init.xavier_uniform(linear4.weight)\n\nlinear5=torch.nn.Linear(hiddenLayer4Size, hiddenLayer5Size)\ntorch.nn.init.xavier_uniform(linear5.weight)\n\nlinear6=torch.nn.Linear(hiddenLayer5Size, 1)\ntorch.nn.init.xavier_uniform(linear6.weight)\n\nsigmoid = torch.nn.Sigmoid()\ntanh=torch.nn.Tanh()\nrelu=torch.nn.LeakyReLU()\n\nnet = torch.nn.Sequential(linear1,dropout,tanh,nn.BatchNorm1d(hiddenLayer1Size),\n linear2,dropout,tanh,\n linear3,dropout,relu,\n linear4,dropout,tanh,\n linear5,dropout,relu,\n linear6,sigmoid\n )\n\n# net = Net(n_feature=N_FEATURES, n_hidden=1024, n_output=1) # define the network\n# net = Net2(n_feature=N_FEATURES, n_hidden=2048, n_output=1) # define the network\n\nlgr.info(net) # net architecture\n```\n\n INFO:__main__:Sequential (\n (0): Linear (29 -> 1024)\n (1): Dropout (p = 0.05)\n (2): Tanh ()\n (3): BatchNorm1d(1024, eps=1e-05, momentum=0.1, affine=True)\n (4): Linear (1024 -> 128)\n (5): Dropout (p = 0.05)\n (6): Tanh ()\n (7): Linear (128 -> 64)\n (8): Dropout (p = 0.05)\n (9): LeakyReLU (0.01)\n (10): Linear (64 -> 32)\n (11): Dropout (p = 0.05)\n (12): Tanh ()\n (13): Linear (32 -> 16)\n (14): Dropout (p = 0.05)\n (15): LeakyReLU (0.01)\n (16): Linear (16 -> 1)\n (17): Sigmoid ()\n )\n\n\n## Print the full net architecture\n\n\n```python\n# See https://stackoverflow.com/questions/42480111/model-summary-in-pytorch/42616812\nfrom torch.nn.modules.module import _addindent\nimport torch\nimport numpy as np\ndef torch_summarize(model, show_weights=True, show_parameters=True):\n \"\"\"Summarizes torch model by showing trainable parameters and weights.\"\"\"\n tmpstr = model.__class__.__name__ + ' (\\n'\n for key, module in model._modules.items():\n # if it contains layers let call it recursively to get params and weights\n if type(module) in [\n torch.nn.modules.container.Container,\n torch.nn.modules.container.Sequential\n ]:\n modstr = torch_summarize(module)\n else:\n modstr = module.__repr__()\n modstr = _addindent(modstr, 2)\n\n params = sum([np.prod(p.size()) for p in module.parameters()])\n weights = tuple([tuple(p.size()) for p in module.parameters()])\n\n tmpstr += ' (' + key + '): ' + modstr \n if show_weights:\n tmpstr += ', weights={}'.format(weights)\n if show_parameters:\n tmpstr += ', parameters={}'.format(params)\n tmpstr += '\\n' \n\n tmpstr = tmpstr + ')'\n return tmpstr\n\nlgr.info(torch_summarize(net))\n```\n\n INFO:__main__:Sequential (\n (0): Linear (29 -> 1024), weights=((1024L, 29L), (1024L,)), parameters=30720\n (1): Dropout (p = 0.05), weights=(), parameters=0\n (2): Tanh (), weights=(), parameters=0\n (3): BatchNorm1d(1024, eps=1e-05, momentum=0.1, affine=True), weights=((1024L,), (1024L,)), parameters=2048\n (4): Linear (1024 -> 128), weights=((128L, 1024L), (128L,)), parameters=131200\n (5): Dropout (p = 0.05), weights=(), parameters=0\n (6): Tanh (), weights=(), parameters=0\n (7): Linear (128 -> 64), weights=((64L, 128L), (64L,)), parameters=8256\n (8): Dropout (p = 0.05), weights=(), parameters=0\n (9): LeakyReLU (0.01), weights=(), parameters=0\n (10): Linear (64 -> 32), weights=((32L, 64L), (32L,)), parameters=2080\n (11): Dropout (p = 0.05), weights=(), parameters=0\n (12): Tanh (), weights=(), parameters=0\n (13): Linear (32 -> 16), weights=((16L, 32L), (16L,)), parameters=528\n (14): Dropout (p = 0.05), weights=(), parameters=0\n (15): LeakyReLU (0.01), weights=(), parameters=0\n (16): Linear (16 -> 1), weights=((1L, 16L), (1L,)), parameters=17\n (17): Sigmoid (), weights=(), parameters=0\n )\n\n\n# Loss and Optimizer\n\n### BCELoss\n- In addition, we will calculate the binary cross entropy loss (BCELoss). Luckily we have one loss function already present. For details please checkout http://pytorch.org/docs/master/nn.html. \n\n- ** NOTE this BCELoss may not be numerical stable, although it's fine during my training process.**\n\n### Optimization\n\n- if return F.log_softmax(x) then loss = F.nll_loss(output, target) (MNIST)\n- print(nn.BCEWithLogitsLoss()(o, t)) is equivalent to print(nn.BCELoss()(sigmoid(o), t))\n\n\n```python\nimport sympy as sp\nsp.interactive.printing.init_printing(use_latex=True)\nfrom IPython.display import display, Math, Latex\nmaths = lambda s: display(Math(s))\nlatex = lambda s: display(Latex(s))\n\n#the loss function is as follows:\nmaths(\"\\mathbf{Loss Function:} J(x, z) = -\\sum_k^d[x_k \\log z_k + (1-x_k)log(1-z_k)]\")\n```\n\n\n$$\\mathbf{Loss Function:} J(x, z) = -\\sum_k^d[x_k \\log z_k + (1-x_k)log(1-z_k)]$$\n\n\n\n```python\n# %%timeit\n# optimizer = torch.optim.SGD(net.parameters(), lr=0.02)\n# optimizer = optim.SGD(model.parameters(), lr=0.001, momentum=0.9)\n# optimizer = optim.SGD(net.parameters(), lr=LR, momentum=MOMENTUM, weight_decay=5e-4)\n\n#L2 regularization can easily be added to the entire model via the optimizer\n# optimizer = torch.optim.Adam(net.parameters(), lr=LR,weight_decay=5e-4) # L2 regularization\noptimizer = torch.optim.Adagrad(net.parameters(), lr=1e-6, weight_decay=5e-4)\n# loss_func = torch.nn.CrossEntropyLoss() # the target label is NOT an one-hotted\n# loss_func = torch.nn.NLLLoss()\nloss_func=torch.nn.BCELoss() # Binary cross entropy: http://pytorch.org/docs/nn.html#bceloss\n# http://andersonjo.github.io/artificial-intelligence/2017/01/07/Cost-Functions/\n# use_cuda=True\nif use_cuda:\n lgr.info (\"Using the GPU\") \n net.cuda()\n loss_func.cuda()\n# cudnn.benchmark = True\n #net = torch.nn.DataParallel(net, device_ids=range(torch.cuda.device_count()))\n\nlgr.info (optimizer)\nlgr.info (loss_func)\n```\n\n INFO:__main__:Using the GPU\n INFO:__main__:\n INFO:__main__:BCELoss (\n )\n\n\n# Training in batches + Measuring the performance of the deep learning model\n\n\n```python\nimport time\nstart_time = time.time() \nepochs=1500\nall_losses = []\n\nX_tensor_train= XnumpyToTensor(trainX)\nY_tensor_train= YnumpyToTensor(trainY)\n\nprint(type(X_tensor_train.data), type(Y_tensor_train.data)) # should be 'torch.cuda.FloatTensor'\n\n# From here onwards, we must only use PyTorch Tensors\nfor step in range(epochs):\n# net.train()\n \n# output = F.sigmoid(net(input))\n# loss = crit(output, target)\n out = net(X_tensor_train) # input x and predict based on x\n cost = loss_func(out, Y_tensor_train) # must be (1. nn output, 2. target), the target label is NOT one-hotted\n\n optimizer.zero_grad() # clear gradients for next train\n cost.backward() # backpropagation, compute gradients\n optimizer.step() # apply gradients\n \n \n if step % 150 == 0: \n loss = cost.data[0]\n all_losses.append(loss)\n print(step, cost.data.cpu().numpy())\n # RuntimeError: can't convert CUDA tensor to numpy (it doesn't support GPU arrays). \n # Use .cpu() to move the tensor to host memory first. \n# prediction = torch.max(F.softmax(out), 1)[1]\n# _, prediction = torch.max(out, 1) \n prediction = (net(X_tensor_train).data).float() # probabilities \n \n# prediction = (net(X_tensor).data > 0.5).float() # zero or one\n# print (\"Pred:\" + str (prediction)) # Pred:Variable containing: 0 or 1\n# pred_y = prediction.data.numpy().squeeze()\n # RuntimeError: can't convert CUDA tensor to numpy (it doesn't support GPU arrays). \n \n pred_y = prediction.cpu().numpy().squeeze()\n target_y = Y_tensor_train.cpu().data.numpy()\n \n tu = ((pred_y == target_y).mean(),log_loss(target_y, pred_y),roc_auc_score(target_y,pred_y ))\n print ('ACC={}, LOG_LOSS={}, ROC_AUC={} '.format(*tu)) \n \nend_time = time.time()\nprint ('{} {:6.3f} seconds'.format('GPU:', end_time-start_time))\n\n%matplotlib inline\nimport matplotlib.pyplot as plt\nplt.plot(all_losses)\nplt.show()\n\nfalse_positive_rate, true_positive_rate, thresholds = roc_curve(target_y,pred_y)\nroc_auc = auc(false_positive_rate, true_positive_rate)\n\nplt.title('LOG_LOSS=' + str(log_loss(target_y, pred_y)))\nplt.plot(false_positive_rate, true_positive_rate, 'b', label='AUC = %0.6f' % roc_auc)\nplt.legend(loc='lower right')\nplt.plot([0, 1], [0, 1], 'r--')\nplt.xlim([-0.1, 1.2])\nplt.ylim([-0.1, 1.2])\nplt.ylabel('True Positive Rate')\nplt.xlabel('False Positive Rate')\nplt.show()\n```\n\n# Performance of the deep learning model on the Validation set\n\n\n```python\nnet.eval()\n# Validation data\nprint (valX.shape)\nprint (valY.shape)\n\nX_tensor_val= XnumpyToTensor(valX)\nY_tensor_val= YnumpyToTensor(valY)\n\n\nprint(type(X_tensor_val.data), type(Y_tensor_val.data)) # should be 'torch.cuda.FloatTensor'\n\npredicted_val = (net(X_tensor_val).data).float() # probabilities \n# predicted_val = (net(X_tensor_val).data > 0.5).float() # zero or one\npred_y = predicted_val.cpu().numpy()\ntarget_y = Y_tensor_val.cpu().data.numpy() \n\nprint (type(pred_y))\nprint (type(target_y))\n\ntu = (str ((pred_y == target_y).mean()),log_loss(target_y, pred_y),roc_auc_score(target_y,pred_y ))\nprint ('\\n')\nprint ('acc={} log_loss={} roc_auc={} '.format(*tu))\n\nfalse_positive_rate, true_positive_rate, thresholds = roc_curve(target_y,pred_y)\nroc_auc = auc(false_positive_rate, true_positive_rate)\n\nplt.title('LOG_LOSS=' + str(log_loss(target_y, pred_y)))\nplt.plot(false_positive_rate, true_positive_rate, 'b', label='AUC = %0.6f' % roc_auc)\nplt.legend(loc='lower right')\nplt.plot([0, 1], [0, 1], 'r--')\nplt.xlim([-0.1, 1.2])\nplt.ylim([-0.1, 1.2])\nplt.ylabel('True Positive Rate')\nplt.xlabel('False Positive Rate')\nplt.show()\n\n# print (pred_y)\n```\n\n# Submission on Test set\n\n\n```python\n# testX, df_test_set\n# df[df.columns.difference(['b'])]\n# trainX, trainY, valX, valY, testX, df_test_set = loadDataSplit()\n\nprint (df_test_set.shape)\ncolumns = ['id', 'probability']\ndf_pred=pd.DataFrame(data=np.zeros((0,len(columns))), columns=columns)\ndf_pred.id.astype(int)\n\nfor index, row in df_test_set.iterrows():\n rwo_no_id=row.drop('id') \n# print (rwo_no_id.values) \n x_data_np = np.array(rwo_no_id.values, dtype=np.float32) \n if use_cuda:\n X_tensor_test = Variable(torch.from_numpy(x_data_np).cuda()) # Note the conversion for pytorch \n else:\n X_tensor_test = Variable(torch.from_numpy(x_data_np)) # Note the conversion for pytorch\n \n X_tensor_test=X_tensor_test.view(1, trainX.shape[1]) # does not work with 1d tensors \n predicted_val = (net(X_tensor_test).data).float() # probabilities \n p_test = predicted_val.cpu().numpy().item() # otherwise we get an array, we need a single float\n \n df_pred = df_pred.append({'id':row['id'].astype(int), 'probability':p_test},ignore_index=True)\n \n# p_test = pd.DataFrame (p_test, columns=['probability'])\n\n# # df_pred = df_test_set.append(p_test, ignore_index=True)\n# df_pred = pd.concat([p_test, df_test_set], axis=0, ignore_index=True)\n\n# # # df_pred = pd.DataFrame({\n# # # 'id': df_test_set['id'],\n# # # 'probability': p_test[:,1]\n# # # })\n\ndf_pred.head(5)\n\n# df_test_set = pd.concat([tid_1_SINGLE, df_test_set], axis=1)\n```\n\n (45647, 30)\n\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        idprobability
                                        097040.00.501994
                                        165399.00.481901
                                        2147258.00.497755
                                        3129573.00.504502
                                        4134978.00.519406
                                        \n
                                        \n\n\n\n\n```python\ndf_pred.id=df_pred.id.astype(int)\n\ndef savePred(df_pred, loss):\n# csv_path = 'pred/p_{}_{}_{}.csv'.format(loss, name, (str(time.time())))\n csv_path = 'pred/pred_{}_{}.csv'.format(loss, (str(time.time())))\n df_pred.to_csv(csv_path, columns=('id', 'probability'), index=None)\n print (csv_path)\n \nsavePred (df_pred, log_loss(target_y, pred_y))\n```\n\n pred/pred_0.692493408893_1504543207.31.csv\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "de152befef8090f47ab0d8f36dee96d69ae979de", "size": 110247, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "day 02 PyTORCH and PyCUDA/PyTorch/18-PyTorch-NUMER.AI-Binary-Classification-BCELoss.ipynb", "max_stars_repo_name": "stefan-jansen/Deep-Learning-Boot-Camp", "max_stars_repo_head_hexsha": "d828fa2c172d4205fde547145fed26541eae6eae", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "day 02 PyTORCH and PyCUDA/PyTorch/18-PyTorch-NUMER.AI-Binary-Classification-BCELoss.ipynb", "max_issues_repo_name": "stefan-jansen/Deep-Learning-Boot-Camp", "max_issues_repo_head_hexsha": "d828fa2c172d4205fde547145fed26541eae6eae", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "day 02 PyTORCH and PyCUDA/PyTorch/18-PyTorch-NUMER.AI-Binary-Classification-BCELoss.ipynb", "max_forks_repo_name": "stefan-jansen/Deep-Learning-Boot-Camp", "max_forks_repo_head_hexsha": "d828fa2c172d4205fde547145fed26541eae6eae", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-09-10T10:09:46.000Z", "max_forks_repo_forks_event_max_datetime": "2020-09-10T10:09:46.000Z", "avg_line_length": 70.4453674121, "max_line_length": 20976, "alphanum_fraction": 0.735974675, "converted": true, "num_tokens": 10590, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7341195385342971, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.41833996548870667}} {"text": "\n\n\n# Tutorial-IllinoisGRMHD: A_i_rhs_no_gauge_terms.C\n\n## Authors: Leo Werneck & Zach Etienne\n\n**This module is currently under development**\n\n## In this tutorial module we explain the construction of the right-hand side of the evolution equations of $A_{i}$\n\n### Required and recommended citations:\n\n* **(Required)** Etienne, Z. B., Paschalidis, V., Haas R., M\u00f6sta P., and Shapiro, S. L. IllinoisGRMHD: an open-source, user-friendly GRMHD code for dynamical spacetimes. Class. Quantum Grav. 32 (2015) 175009. ([arxiv:1501.07276](http://arxiv.org/abs/1501.07276)).\n* **(Required)** Noble, S. C., Gammie, C. F., McKinney, J. C., Del Zanna, L. Primitive Variable Solvers for Conservative General Relativistic Magnetohydrodynamics. Astrophysical Journal, 641, 626 (2006) ([astro-ph/0512420](https://arxiv.org/abs/astro-ph/0512420)).\n* **(Recommended)** Del Zanna, L., Bucciantini N., Londrillo, P. An efficient shock-capturing central-type scheme for multidimensional relativistic flows - II. Magnetohydrodynamics. A&A 400 (2) 397-413 (2003). DOI: 10.1051/0004-6361:20021641 ([astro-ph/0210618](https://arxiv.org/abs/astro-ph/0210618)).\n\nIf using the version of `IllinoisGRMHD` with piecewise polytropic *or* tabulated (coming soon!) EOS support, then the following citation is also required:\n\n* **(Required)** Etienne, Z. B., Werneck, L., Paschalidis, V., Haas R., M\u00f6sta P., and Shapiro, S. L., *IllinoisGRMHD github repository* (2019). Source Code URL: https://github.com/zachetienne/nrpytutorial/tree/master/IllinoisGRMHD/.\n\n\n\n# Table of Contents\n$$\\label{toc}$$\n\nThis module is organized as follows\n\n0. [Step 0](#src_dir): **Source directory creation**\n1. [Step 1](#a_i_rhs_no_gauge_terms__c): **The `A_i_rhs_no_gauge_terms.C` file**\n 1. [Step 1.a](#rhs_from_adirn): *Determining the appropriate RHS based on $A_{i}$*\n 1. [Step 1.b](#vi_bi_staggering): *Handling the staggering of $v^{i}$ and $B^{i}$*\n 1. [Step 1.c](#staggered_hll): *Computing the generalized HLL formula for $A_{i}$*\n1. [Step 2](#code_validation): **Code validation**\n1. [Step 3](#latex_pdf_output): **Output this notebook to $\\LaTeX$-formatted PDF file**\n\n\n\n# Step 0: Source directory creation \\[Back to [top](#toc)\\]\n$$\\label{src_dir}$$\n\nWe will now use the [cmdline_helper.py NRPy+ module](Tutorial-Tutorial-cmdline_helper.ipynb) to create the source directory within the `IllinoisGRMHD` NRPy+ directory, if it does not exist yet.\n\n\n```python\n# Step 0: Creation of the IllinoisGRMHD source directory\n# Step 0a: Add NRPy's directory to the path\n# https://stackoverflow.com/questions/16780014/import-file-from-parent-directory\nimport os,sys\nnrpy_dir_path = os.path.join(\"..\",\"..\")\nif nrpy_dir_path not in sys.path:\n sys.path.append(nrpy_dir_path)\n\n# Step 0b: Load up cmdline_helper and create the directory\nimport cmdline_helper as cmd\noutdir = os.path.join(\"..\",\"src\")\ncmd.mkdir(outdir)\n```\n\n\n\n# Step 1: The `A_i_rhs_no_gauge_terms.C` file \\[Back to [top](#toc)\\]\n$$\\label{a_i_rhs_no_gauge_terms__c}$$\n\nIn this file we implement the right-hand side (RHS) of the evolution equation of the vector potentials $A_{i}$, i.e. the [induction equation](https://en.wikipedia.org/wiki/Induction_equation), excluding gauge terms (see also equation (12) in [Etienne *et al.*](https://arxiv.org/pdf/1501.07276.pdf)). In other words, we implement:\n\n$$\n\\partial_{t}A_{i}^{\\rm NG} \\equiv \\partial_{t}A_{i} - \\left[-\\partial_{i}\\left(\\alpha\\Phi - \\beta^{j}A_{j}\\right)\\right]= \\partial_{t}A_{i} - \\left[\\text{gauge terms}\\right]_{i} = \\psi^{6}\\epsilon_{ijk}v^{j}B^{k}\\ .\n$$\n\nwhere $\\epsilon_{ijk}$ is the [Levi-Civita symbol](https://en.wikipedia.org/wiki/Levi-Civita_symbol) with $\\epsilon_{123} = \\epsilon_{xyz}=1$, odd permutations of the indices resulting in $-1$, even permutations in $+1$, and if two or more indices are identical in $0$. In components, then, we have\n\n$$\n\\boxed{\n\\begin{align}\n\\partial_{t}A_{x}^{\\rm NG} &= \\psi^{6}\\left(v^{y}B^{z} - v^{z}B^{y}\\right) \\\\\n\\partial_{t}A_{y}^{\\rm NG} &= \\psi^{6}\\left(v^{z}B^{x} - v^{x}B^{z}\\right) \\\\\n\\partial_{t}A_{z}^{\\rm NG} &= \\psi^{6}\\left(v^{x}B^{y} - v^{y}B^{x}\\right)\n\\end{align}\n}\\ .\n$$\n\n\n\n## Step 1.a: Determining the appropriate RHS based on $A_{i}$ \\[Back to [top](#toc)\\]\n$$\\label{rhs_from_adirn}$$\n\nOne of the inputs of the function is the variable `A_dirn`, which we will denote by $a=1,2,3$, which is an integer that specifies the direction of $A_{i}$ we are interested in. Now, in order to properly determine the RHS we want to use, we need to remember the following:\n\n* The $v^{i}$ arrays store $\\left(v^{x},v^{y},v^{z}\\right)$ at consecutive indices, i.e. $\\left(\\ldots,n-1,\\underbrace{n,n+1,n+2}_{\\text{indices corresponding to } v^{i}},n+3,\\ldots\\right)$ ;\n* The $B^{i}$ arrays store $\\left(B^{x},B^{y},B^{z}\\right)$ at consecutive indices, i.e. $\\left(\\ldots,m-1,\\underbrace{m,m+1,m+2}_{\\text{indices corresponding to } B^{i}},m+3,\\ldots\\right)$ .\n\nConsider, then, the quantities:\n\n\\begin{align}\nO_{1} &\\equiv {\\rm mod}\\left[(a-1)+1,3\\right]\\ ,\\\\\nO_{2} &\\equiv {\\rm mod}\\left[(a-1)+2,3\\right]\\ ,\n\\end{align}\n\nwhere $O$ stands for *offset*. Let $V$ and $B$ correspond to the $v^{i}$ and $B^{i}$ arrays, respectively, with $V(n)\\to v^{x}$ and $B(m)\\to B^{x}$. Then, we have the following table:\n\n| $a$ | $A_{a}$ | $O_{1}$ | $O_{2}$ | $V[n+O_{1}]$ | $V[n+O_{2}]$ | $B[m+O_{1}]$ | $B[m+O_{2}]$ |\n|:---:|:-------:|:-------:|:-------:|:------------:|:------------:|:------------:|:------------:|\n| 1 | $A_{x}$ | 1 | 2 | $\\to v^{y}$ | $\\to v^{z}$ | $\\to B^{y}$ | $\\to B^{z}$ |\n| 2 | $A_{y}$ | 2 | 0 | $\\to v^{z}$ | $\\to v^{x}$ | $\\to B^{z}$ | $\\to B^{x}$ |\n| 3 | $A_{z}$ | 0 | 1 | $\\to v^{x}$ | $\\to v^{y}$ | $\\to B^{x}$ | $\\to B^{y}$ |\n\nThus, we can write, in general\n\n$$\n\\boxed{\\partial_{t}A_{a}^{\\rm NG} = \\psi^{6}V[n+O_{1}]B[m+O_{2}] - V[n+O_{2}]B[m+O_{1}]}\\ .\n$$\n\nIn this first step, we determine the values of $O_{1}$ and $O_{2}$, which are referred to in the code by `v1_offset` and `v2_offset`, respectively. We then also set variables $v_{1}$, $v_{2}$, $B_{1}$, $B_{2}$ which point to the array elements $V[n+O_{1}]$, $V[n+O_{2}]$, $B[m+O_{1}]$, $B[m+O_{2}]$, respectively. Notice that because we have $v^{i}$ and $B^{i}$ on different staggered points, we will set more than one pointer at this stage.\n\n\n```python\n%%writefile $outdir/A_i_rhs_no_gauge_terms.C\n/* Compute the part of A_i_rhs that excludes the gauge terms. I.e., we set\n * A_i_rhs = \\partial_t A_i = \\psi^{6} (v^z B^x - v^x B^z) here.\n */\nstatic void A_i_rhs_no_gauge_terms(const int A_dirn,const cGH *cctkGH,const int *cctk_lsh,const int *cctk_nghostzones,gf_and_gz_struct *OUT_PRIMS_R,gf_and_gz_struct *OUT_PRIMS_L,\n CCTK_REAL *phi_interped,CCTK_REAL *cmax_1,CCTK_REAL *cmin_1,CCTK_REAL *cmax_2,CCTK_REAL *cmin_2, CCTK_REAL *A3_rhs) {\n // If A_dirn=1, then v1_offset=1 (v1=VY) and v2_offset=2 (v2=VZ)\n // If A_dirn=2, then v1_offset=2 (v1=VZ) and v2_offset=0 (v2=VX)\n // If A_dirn=3, then v1_offset=0 (v1=VX) and v2_offset=1 (v2=VY)\n int v1_offset = ((A_dirn-1)+1)%3, v2_offset = ((A_dirn-1)+2)%3;\n\n CCTK_REAL *v1rr=OUT_PRIMS_R[VXR+v1_offset].gf, *v2rr=OUT_PRIMS_R[VXR+v2_offset].gf;\n CCTK_REAL *v1rl=OUT_PRIMS_L[VXR+v1_offset].gf, *v2rl=OUT_PRIMS_L[VXR+v2_offset].gf;\n CCTK_REAL *v1lr=OUT_PRIMS_R[VXL+v1_offset].gf, *v2lr=OUT_PRIMS_R[VXL+v2_offset].gf;\n CCTK_REAL *v1ll=OUT_PRIMS_L[VXL+v1_offset].gf, *v2ll=OUT_PRIMS_L[VXL+v2_offset].gf;\n\n CCTK_REAL *B1r=OUT_PRIMS_R[BX_STAGGER+v1_offset].gf, *B1l=OUT_PRIMS_L[BX_STAGGER+v1_offset].gf;\n CCTK_REAL *B2r=OUT_PRIMS_R[BX_STAGGER+v2_offset].gf, *B2l=OUT_PRIMS_L[BX_STAGGER+v2_offset].gf;\n```\n\n Overwriting ../src/A_i_rhs_no_gauge_terms.C\n\n\n\n\n## Step 1.b: Handling the staggering of $v^{i}$ and $B^{i}$ \\[Back to [top](#toc)\\]\n$$\\label{vi_bi_staggering}$$\n\nNow that we have the correct $v^{i}$ and $B^{i}$ component, we must be careful about how these quantities are staggered with respect to $A_{i}$. Remember that the staggering of these functions is the following:\n\n| Direction | $a$ | $A_{a}$ | $v^{n+O_{1},n+O_{2}}$ | $B^{m+O_{1}}$ | $B^{m+O_{2}}$ |\n|:---------:|:---:|:--------------------------:|:--------------------------:|:--------------------------:|:-------------------------:|\n| x | 1 |$\\left(i,j+\\frac{1}{2},k+\\frac{1}{2}\\right)$|$\\left(i,j-\\frac{1}{2},k-\\frac{1}{2}\\right)$|$\\left(i,j+\\frac{1}{2},k-\\frac{1}{2}\\right)$|$\\left(i,j-\\frac{1}{2},k+\\frac{1}{2}\\right)$|\n| y | 2 |$\\left(i+\\frac{1}{2},j,k+\\frac{1}{2}\\right)$|$\\left(i-\\frac{1}{2},j,k-\\frac{1}{2}\\right)$|$\\left(i-\\frac{1}{2},j,k+\\frac{1}{2}\\right)$|$\\left(i+\\frac{1}{2},j,k-\\frac{1}{2}\\right)$|\n| z | 3 |$\\left(i+\\frac{1}{2},j+\\frac{1}{2},k\\right)$|$\\left(i-\\frac{1}{2},j-\\frac{1}{2},k\\right)$|$\\left(i+\\frac{1}{2},j-\\frac{1}{2},k\\right)$|$\\left(i-\\frac{1}{2},j+\\frac{1}{2},k\\right)$|\n\nNow, obtaining, for example, $v^{n+O_{1},n+O_{2}}=v^{y,z}$ (first case in the table above) at the correct point, i.e. $\\left(i,j+\\frac{1}{2},k+\\frac{1}{2}\\right)$, would correspond to shifting the gridpoints of the $v^{y,z}$ gridfunctions up by $1$ in the $y,z$-directions. Similarly, obtaining $B^{m+O_{1}}=B^{y}$ at the point $\\left(i,j+\\frac{1}{2},k+\\frac{1}{2}\\right)$ corresponds shifting its index up by $1$ in the $z$-direction, while obtaining $B^{m+O_{2}}=B^{z}$ at the point $\\left(i,j+\\frac{1}{2},k+\\frac{1}{2}\\right)$ corresponds shifting its index up by $1$ in the $y$-direction.\n\nBelow, we set all these offsets by defining the `vs_ijk_offset`, `B1_ijk_offset`, and `B2_ijk_offset` arrays, respectively.\n\n\n```python\n%%writefile -a $outdir/A_i_rhs_no_gauge_terms.C\n\n\n /**** V DEPENDENCIES ****/\n /* In the case of Ax_rhs, we need v{y,z}{r,l} at (i,j+1/2,k+1/2). \n * However, v{y,z}{r,l}{r,l} are defined at (i,j-1/2,k-1/2), so\n * v{y,z}{r,l} at (i,j+1/2,k+1/2) is stored at v{y,z}{r,l}{r,l}(i,j+1,k+1).\n * In the case of Ay_rhs, we need v{x,z}{r,l} at (i+1/2,j,k+1/2). \n * However, v{x,z}{r,l}{r,l} are defined at (i-1/2,j,k-1/2), so\n * v{x,z}{r,l} at (i+1/2,j,k+1/2) is stored at v{x,z}{r,l}{r,l}(i+1,j,k+1).\n * In the case of Az_rhs, we need v{x,y}{r,l} at (i+1/2,j+1/2,k). \n * However, v{x,y}{r,l}{r,l} are defined at (i-1/2,j-1/2,k), so\n * v{x,y}{r,l} at (i+1/2,j+1/2,k) is stored at v{x,y}{r,l}{r,l}(i+1,j+1,k). */\n static const int vs_ijk_offset[4][3] = { {0,0,0} , {0,1,1} , {1,0,1} , {1,1,0} };\n\n /**** B DEPENDENCIES ****/\n /* In the case of Ax_rhs, we need B{y,z}{r,l} at (i,j+1/2,k+1/2).\n * However, By_stagger{r,l} is defined at (i,j+1/2,k-1/2), and \n * Bz_stagger{r,l} is defined at (i,j-1/2,k+1/2), so\n * By_stagger{r,l} at (i,j+1/2,k+1/2) is stored at By_stagger{r,l}(i,j,k+1), and\n * Bz_stagger{r,l} at (i,j+1/2,k+1/2) is stored at Bz_stagger{r,l}(i,j+1,k).\n * In the case of Ay_rhs, we need B{z,x}_stagger{r,l} at (i+1/2,j,k+1/2).\n * However, Bz_stagger{r,l} is defined at (i-1/2,j,k+1/2), and\n * Bx_stagger{r,l} is defined at (i+1/2,j,k-1/2), so\n * Bz_stagger{r,l} at (i+1/2,j,k+1/2) is stored at Bz_stagger{r,l}(i+1,j,k), and\n * Bx_stagger{r,l} at (i+1/2,j,k+1/2) is stored at Bx_stagger{r,l}(i,j,k+1).\n * In the case of Az_rhs, we need B{x,y}_stagger{r,l} at (i+1/2,j+1/2,k).\n * However, Bx_stagger{r,l} is defined at (i+1/2,j-1/2,k), and \n * By_stagger{r,l} is defined at (i-1/2,j+1/2,k), so\n * Bx_stagger{r,l} at (i+1/2,j+1/2,k) is stored at Bx_stagger{r,l}(i,j+1,k), and\n * By_stagger{r,l} at (i+1/2,j+1/2,k) is stored at By_stagger{r,l}(i+1,j,k).\n */\n static const int B1_ijk_offset[4][3] = { {0,0,0} , {0,0,1} , {1,0,0} , {0,1,0} };\n static const int B2_ijk_offset[4][3] = { {0,0,0} , {0,1,0} , {0,0,1} , {1,0,0} };\n```\n\n Appending to ../src/A_i_rhs_no_gauge_terms.C\n\n\n\n\n## Step 1.c: Computing the generalized HLL formula for $A_{i}$ \\[Back to [top](#toc)\\]\n$$\\label{staggered_hll}$$\n\nWe now define the following quantities:\n\n$$\n\\left(\nB^{m+O_{1}}_{L},\nB^{m+O_{1}}_{R},\nB^{m+O_{2}}_{L},\nB^{m+O_{2}}_{R}\n\\right)\\ .\n$$\n\nThen, define $\\left(\\mathcal{E}_{a}\\right)_{AB} \\equiv \\left(\\partial_{t}A_{a}^{\\rm NG}\\right)_{AB}$, where $A$ and $B$ can be $L$ or $R$ (for right and left face values), which allow us to compute\n\n$$\n\\begin{align}\n\\left(\\mathcal{E}_a\\right)_{RR} &= \\psi^{6}\\left(v^{n+O_{1}}_{R}B^{m+O_{2}}_{R} - v^{n+O_{2}}_{R}B^{m+O_{1}}_{R}\\right)\\ .\\\\\n\\left(\\mathcal{E}_a\\right)_{RL} &= \\psi^{6}\\left(v^{n+O_{1}}_{R}B^{m+O_{2}}_{R} - v^{n+O_{2}}_{R}B^{m+O_{1}}_{L}\\right)\\ ,\\\\\n\\left(\\mathcal{E}_a\\right)_{LR} &= \\psi^{6}\\left(v^{n+O_{1}}_{L}B^{m+O_{2}}_{L} - v^{n+O_{2}}_{L}B^{m+O_{1}}_{R}\\right)\\ ,\\\\\n\\left(\\mathcal{E}_a\\right)_{LL} &= \\psi^{6}\\left(v^{n+O_{1}}_{L}B^{m+O_{2}}_{L} - v^{n+O_{2}}_{L}B^{m+O_{1}}_{L}\\right)\\ .\n\\end{align}\n$$\n\nThen, having also set the characteristic speeds $c^{\\pm}_{m + O_{1}}$ and $c^{\\pm}_{m + O_{2}}$, we compute the standard HLL formula generalized to staggered gridfunctions (see equation (33) in [Etienne *et al.*](https://arxiv.org/pdf/1501.07276.pdf))\n\n$$\n\\boxed{\n\\begin{align}\n\\left(\\mathcal{E}_a\\right)^{\\rm HLL}\n&=\n\\frac{\nc^{+}_{m+O_{1}}c^{+}_{m+O_{2}}\\left(\\mathcal{E}_a\\right)_{LL} +\nc^{+}_{m+O_{1}}c^{-}_{m+O_{2}}\\left(\\mathcal{E}_a\\right)_{LR} +\nc^{-}_{m+O_{1}}c^{+}_{m+O_{2}}\\left(\\mathcal{E}_a\\right)_{RL} +\nc^{-}_{m+O_{1}}c^{-}_{m+O_{2}}\\left(\\mathcal{E}_a\\right)_{RR}\n}\n{\\left(c^{+}_{m+O_{1}}+c^{-}_{m+O_{1}}\\right)\\left(c^{+}_{m+O_{2}}+c^{-}_{m+O_{2}}\\right)}\\\\\n&+\n\\frac{c^{+}_{m+O_{1}}c^{-}_{m+O_{1}}}{c^{+}_{m+O_{1}}+c^{-}_{m+O_{1}}}\\left(B^{m+O_{2}}_{R}-B^{m+O_{2}}_{L}\\right)\n+\n\\frac{c^{+}_{m+O_{2}}c^{-}_{m+O_{2}}}{c^{+}_{m+O_{2}}+c^{-}_{m+O_{2}}}\\left(B^{m+O_{1}}_{R}-B^{m+O_{1}}_{L}\\right)\n\\end{align}\n}\\ .\n$$\n\nAs a more practical example, which has a far less cluttered notation, consider the formula above for $a=z$. We then have\n\n$$\n\\begin{align}\n\\left(\\mathcal{E}_z\\right)^{\\rm HLL}\n&=\n\\frac{\nc^{+}_{x}c^{+}_{y}\\left(\\mathcal{E}_z\\right)_{LL} +\nc^{+}_{x}c^{-}_{y}\\left(\\mathcal{E}_z\\right)_{LR} +\nc^{-}_{x}c^{+}_{y}\\left(\\mathcal{E}_z\\right)_{RL} +\nc^{-}_{x}c^{-}_{y}\\left(\\mathcal{E}_z\\right)_{RR}\n}\n{\\left(c^{+}_{x}+c^{-}_{x}\\right)\\left(c^{+}_{y}+c^{-}_{y}\\right)}\\\\\n&+\n\\frac{c^{+}_{x}c^{-}_{x}}{c^{+}_{x}+c^{-}_{x}}\\left(B^{y}_{R}-B^{y}_{L}\\right)\n+\n\\frac{c^{+}_{y}c^{-}_{y}}{c^{+}_{y}-c^{-}_{y}}\\left(B^{x}_{R}-B^{x}_{L}\\right)\\ ,\n\\end{align}\n$$\n\nwhich is precisely equation (33) in [Etienne *et al.*](https://arxiv.org/pdf/1501.07276.pdf).\n\n\n```python\n%%writefile -a $outdir/A_i_rhs_no_gauge_terms.C\n\n\n#pragma omp parallel for\n for(int k=cctk_nghostzones[2];k\n\n# Step 2: Code validation \\[Back to [top](#toc)\\]\n$$\\label{code_validation}$$\n\nFirst we download the original `IllinoisGRMHD` source code and then compare it to the source code generated by this tutorial notebook.\n\n\n```python\n# # Verify if the code generated by this tutorial module\n# # matches the original IllinoisGRMHD source code\n\n# # First download the original IllinoisGRMHD source code\n# import urllib\n# from os import path\n\n# original_IGM_file_url = \"https://bitbucket.org/zach_etienne/wvuthorns/raw/5611b2f0b17135538c9d9d17c7da062abe0401b6/IllinoisGRMHD/src/A_i_rhs_no_gauge_terms.C\"\n# original_IGM_file_name = \"A_i_rhs_no_gauge_terms-original.C\"\n# original_IGM_file_path = os.path.join(IGM_src_dir_path,original_IGM_file_name)\n\n# # Then download the original IllinoisGRMHD source code\n# # We try it here in a couple of ways in an attempt to keep\n# # the code more portable\n# try:\n# original_IGM_file_code = urllib.request.urlopen(original_IGM_file_url).read().decode(\"utf-8\")\n# # Write down the file the original IllinoisGRMHD source code\n# with open(original_IGM_file_path,\"w\") as file:\n# file.write(original_IGM_file_code)\n# except:\n# try:\n# original_IGM_file_code = urllib.urlopen(original_IGM_file_url).read().decode(\"utf-8\")\n# # Write down the file the original IllinoisGRMHD source code\n# with open(original_IGM_file_path,\"w\") as file:\n# file.write(original_IGM_file_code)\n# except:\n# # If all else fails, hope wget does the job\n# !wget -O $original_IGM_file_path $original_IGM_file_url\n\n# # Perform validation\n# Validation__A_i_rhs_no_gauge_terms__C = !diff $original_IGM_file_path $outfile_path__A_i_rhs_no_gauge_terms__C\n\n# if Validation__A_i_rhs_no_gauge_terms__C == []:\n# # If the validation passes, we do not need to store the original IGM source code file\n# !rm $original_IGM_file_path\n# print(\"Validation test for A_i_rhs_no_gauge_terms.C: PASSED!\")\n# else:\n# # If the validation fails, we keep the original IGM source code file\n# print(\"Validation test for A_i_rhs_no_gauge_terms.C: FAILED!\")\n# # We also print out the difference between the code generated\n# # in this tutorial module and the original IGM source code\n# print(\"Diff:\")\n# for diff_line in Validation__A_i_rhs_no_gauge_terms__C:\n# print(diff_line)\n```\n\n\n\n# Step 3: Output this notebook to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n$$\\label{latex_pdf_output}$$\n\nThe following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n[Tutorial-IllinoisGRMHD__A_i_rhs_no_gauge_terms.pdf](Tutorial-IllinoisGRMHD__A_i_rhs_no_gauge_terms.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means).\n\n\n```python\nlatex_nrpy_style_path = os.path.join(nrpy_dir_path,\"latex_nrpy_style.tplx\")\n#!jupyter nbconvert --to latex --template $latex_nrpy_style_path --log-level='WARN' Tutorial-IllinoisGRMHD__A_i_rhs_no_gauge_terms.ipynb\n#!pdflatex -interaction=batchmode Tutorial-IllinoisGRMHD__A_i_rhs_no_gauge_terms.tex\n#!pdflatex -interaction=batchmode Tutorial-IllinoisGRMHD__A_i_rhs_no_gauge_terms.tex\n#!pdflatex -interaction=batchmode Tutorial-IllinoisGRMHD__A_i_rhs_no_gauge_terms.tex\n!rm -f Tut*.out Tut*.aux Tut*.log\n```\n", "meta": {"hexsha": "033c311d1f3112d0f84c5b22c4f253bd4d3e3841", "size": 27178, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "IllinoisGRMHD/doc/Tutorial-IllinoisGRMHD__A_i_rhs_no_gauge_terms.ipynb", "max_stars_repo_name": "leowerneck/NRPyIGM", "max_stars_repo_head_hexsha": "f483d6123424fb3e6860dfac4325dd232b223005", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "IllinoisGRMHD/doc/Tutorial-IllinoisGRMHD__A_i_rhs_no_gauge_terms.ipynb", "max_issues_repo_name": "leowerneck/NRPyIGM", "max_issues_repo_head_hexsha": "f483d6123424fb3e6860dfac4325dd232b223005", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "IllinoisGRMHD/doc/Tutorial-IllinoisGRMHD__A_i_rhs_no_gauge_terms.ipynb", "max_forks_repo_name": "leowerneck/NRPyIGM", "max_forks_repo_head_hexsha": "f483d6123424fb3e6860dfac4325dd232b223005", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 52.366088632, "max_line_length": 613, "alphanum_fraction": 0.5530576201, "converted": true, "num_tokens": 8005, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7341195385342971, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.41833996548870667}} {"text": "# Creating Synthetic Proton Radiographs by Particle Tracing\n\n[SyntheticProtronRadiography]: ../../api/plasmapy.diagnostics.proton_radiography.SyntheticProtonRadiograph.rst#plasmapy.diagnostics.proton_radiography.SyntheticProtonRadiograph\n\nProton radiography is a diagnostic technique often used to interrogate the electric and magnetic fields inside high energy density plasmas. The area of interest is positioned between a bright source of protons and a detector plane. Electric and magnetic fields in the plasma deflect the protons, producing patterns on the detector. Since this represents a non-linear and line-integrated measurement of the fields, the interpretation of these \"proton radiographs\" is complicated.\n\nThe [SyntheticProtronRadiography] class creates a synthetic proton radiographs given a grid of electric and magnetic field (produced either by simulations or analytical models). After the geometry of the problem has been set up, a particle tracing algorithm is run, pushing the protons through the field region. After all of the protons have reached the detector plane, a synthetic proton radiograph is created by making a 2D histogram in that plane.\n\n\n```python\nimport astropy.constants as const\nimport astropy.units as u\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfrom mpl_toolkits.mplot3d import Axes3D\n\nfrom plasmapy.diagnostics import proton_radiography as prad\nfrom plasmapy.plasma.grids import CartesianGrid\n```\n\n[CartesianGrid]: ../../api/plasmapy.plasma.grids.CartesianGrid.rst#plasmapy.plasma.grids.CartesianGrid\n\nTo illustrate the use of this package, we'll first create an example [CartesianGrid] object and fill it with the analytical electric field produced by a sphere of Gaussian potential.\n\n\n```python\n# Create a Cartesian grid\nL = 1 * u.mm\ngrid = CartesianGrid(-L, L, num=100)\n\n# Create a spherical potential with a Gaussian radial distribution\nradius = np.linalg.norm(grid.grid, axis=3)\narg = (radius / (L / 3)).to(u.dimensionless_unscaled)\npotential = 2e5 * np.exp(-(arg ** 2)) * u.V\n\n# Calculate E from the potential\nEx, Ey, Ez = np.gradient(potential, grid.dax0, grid.dax1, grid.dax2)\nEx = -np.where(radius < L / 2, Ex, 0)\nEy = -np.where(radius < L / 2, Ey, 0)\nEz = -np.where(radius < L / 2, Ez, 0)\n\n# Add those quantities to the grid\ngrid.add_quantities(E_x=Ex, E_y=Ey, E_z=Ez, phi=potential)\n\n\n# Plot the E-field\nfig = plt.figure(figsize=(8, 8))\nax = fig.add_subplot(111, projection=\"3d\")\nax.view_init(30, 30)\n\n# skip some points to make the vector plot intelligable\ns = tuple([slice(None, None, 6)] * 3)\n\nax.quiver(\n grid.pts0[s].to(u.mm).value,\n grid.pts1[s].to(u.mm).value,\n grid.pts2[s].to(u.mm).value,\n grid[\"E_x\"][s],\n grid[\"E_y\"][s],\n grid[\"E_z\"][s],\n length=1e-6,\n)\n\nax.set_xlabel(\"X (mm)\")\nax.set_ylabel(\"Y (mm)\")\nax.set_zlabel(\"Z (mm)\")\nax.set_xlim(-1, 1)\nax.set_ylim(-1, 1)\nax.set_zlim(-1, 1)\nax.set_title(\"Gaussian Potential Electric Field\")\n```\n\n[astropy.units.Quantity]: https://docs.astropy.org/en/stable/units/quantity.html#quantity\n\nPrior to running the particle tracing algorithm, the simulation instance must be instantiated by providing some information about the setup, including the locations of the source and detector relative to the origin of the grid.\n\n\n\nThe source and detector coordinates are entered as a 3-tuple in one of three coordinate systems: Cartesian ($x$, $y$, $z$), spherical ($r$, $\\theta$, $\\phi$) or cylindrical ($r$, $\\theta$, $z$). All values should be [astropy.units.Quantity] instances with units of either length or angle. The vector from the source to the detector should pass through the origin to maximize the number of particles that pass through the simulated fields.\n\n\n```python\nsource = (0 * u.mm, -10 * u.mm, 0 * u.mm)\ndetector = (0 * u.mm, 100 * u.mm, 0 * u.mm)\n\nsim = prad.SyntheticProtonRadiograph(grid, source, detector, verbose=True)\n```\n\n[create_particles()]: ../../api/plasmapy.diagnostics.proton_radiography.SyntheticProtonRadiograph.rst#plasmapy.diagnostics.proton_radiography.SyntheticProtonRadiograph.create_particles\n\n\nNote that, since the example grid did not include a B-field, the B-field is assumed to be zero and a warning is printed.\n\nNext, a distribution of `nparticles` simulated particles of energy `particle_energy` is created using the [create_particles()] function. Setting the `max_theta` parameter eliminates particles with large angles (relative to the source-detector axis) which otherwise would likely not hit the detector. Particles with angles less than $\\theta_{max}$ but greater than $\\theta_{track}$ in the setup figure above will not cross the grid. These particles are retained, but are coasted directly to the detector plane instead of being pushed through the grid.\n\nThe `particle` keyword sets the type of the particles being traced. The default particle is protons, which is set here explicitly to demonstrate the use of the keyword.\n\nBy default, the particle velocities are initialized with random angles (a Monte-Carlo approach) with a uniform flux per unit solid angle. However, particles can also be initialized in other ways by setting the `distribution` keyword.\n\n\n```python\nsim.create_particles(1e5, 3 * u.MeV, max_theta=np.pi / 15 * u.rad, particle=\"p\")\n```\n\n[AbstractGrid]: ../../api/plasmapy.plasma.grids.AbstractGrid.rst#plasmapy.plasma.grids.AbstractGrid\n\n[run()]: ../../api/plasmapy.diagnostics.proton_radiography.SyntheticProtonRadiograph.rst#plasmapy.diagnostics.proton_radiography.SyntheticProtonRadiograph.run\n\nThe simulation is now ready to run. In brief, the steps of the simulation cycle are as follows:\n\n1. Particles that will never hit the field grid are ignored (until a later step, when they will be automatically advanced to the detector plane).\n\n\n2. Particles are advanced to the time when the first particle enters the simulation volume. This is done in one step to save computation time.\n\n\n3. While particles are on the grid, the particle pusher advances them each timestep by executing the following steps:\n\n A. The fields at each particle's location are interpolated using the interpolators defined in the [AbstractGrid] subclasses.\n \n B. The simulation timestep is automatically (and adaptively) calculated based on the proton energy, grid resolution, and field amplitudes. This timestep can be clamped or overridden by setting the `dt` keyword in the [run()] function.\n \n C. An implementation of the Boris particle push algorithm is used to advance the velocities and positions of the particles in the interpolated fields.\n \n \n4. After all of the particles have left the grid, all particles are advanced to the detector plane (again saving time). Particles that are headed away from the detector plane at this point are deleted, as those particles will never\nbe detected.\n\nWhen the simulation runs, a progress meter will show the number of particles currently on the grid. This bar will start at zero, increase as particles enter the grid, then decrease as they leave it. When almost all particles have left the grid, the simulation ends.\n\n\n```python\nsim.run()\n```\n\nThe following plot illustrates that, after the simulation has ended, all particles have been advanced to the detector plane.\n\n\n```python\nfig = plt.figure(figsize=(8, 8))\nax = fig.add_subplot(111, projection=\"3d\")\nax.view_init(30, 150)\nax.set_xlabel(\"X (cm)\")\nax.set_ylabel(\"Y (cm)\")\nax.set_zlabel(\"Z (cm)\")\n\n# Plot the source-to-detector axis\nax.quiver(\n sim.source[0] * 100,\n sim.source[1] * 100,\n sim.source[2] * 100,\n sim.detector[0] * 100,\n sim.detector[1] * 100,\n sim.detector[2] * 100,\n color=\"black\",\n)\n\n# Plot the simulation field grid volume\nax.scatter(0, 0, 0, color=\"green\", marker=\"s\", linewidth=5, label=\"Simulated Fields\")\n\n# Plot the the proton source and detector plane locations\nax.scatter(\n sim.source[0] * 100,\n sim.source[1] * 100,\n sim.source[2] * 100,\n color=\"red\",\n marker=\"*\",\n linewidth=5,\n label=\"Source\",\n)\n\nax.scatter(\n sim.detector[0] * 100,\n sim.detector[1] * 100,\n sim.detector[2] * 100,\n color=\"blue\",\n marker=\"*\",\n linewidth=10,\n label=\"Detector\",\n)\n\n\n# Plot the final proton positions of some (not all) of the protons\nind = slice(None, None, 200)\nax.scatter(\n sim.x[ind, 0] * 100, sim.x[ind, 1] * 100, sim.x[ind, 2] * 100, label=\"Protons\",\n)\n\nax.legend()\n```\n\nA 'synthetic proton radiograph' can now be constructed by creating a 2D histogram of proton positions in the image plane. The synthetic radiograph function takes two keywords:\n\n- 'size' gives the locations of the lower left and upper right corners of the detector grid in image plane coordinates.\n\n- 'bins' is the number of histogram bins to be used in the horizontal and vertical directions. Using more bins creates a higher resolution image, but at the cost of more noise.\n\n\n```python\n# A function to reduce repetative plotting\ndef plot_radiograph(hax, vax, intensity):\n fig, ax = plt.subplots(figsize=(8, 8))\n plot = ax.pcolormesh(\n hax.to(u.cm).value,\n vax.to(u.cm).value,\n intensity.T,\n cmap=\"Blues_r\",\n shading=\"auto\",\n )\n cb = fig.colorbar(plot)\n cb.ax.set_ylabel(\"Intensity\")\n ax.set_aspect(\"equal\")\n ax.set_xlabel(\"X (cm), Image plane\")\n ax.set_ylabel(\"Z (cm), Image plane\")\n ax.set_title(\"Synthetic Proton Radiograph\")\n\n\nsize = np.array([[-1, 1], [-1, 1]]) * 1.5 * u.cm\nbins = [200, 200]\nhax, vax, intensity = sim.synthetic_radiograph(size=size, bins=bins)\nplot_radiograph(hax, vax, intensity)\n```\n\nAs expected, the outward-pointing electric field in the sphere has deflected the protons out of the central region, leaving a dark shadow.\n\nKugland et al. 2012 and Bott et al. 2017 define the dimensionless \"contrast parameter\" that separates different regimes of proton radiography:\n\n\\begin{equation}\n\\mu = \\frac{l \\alpha}{a}\n\\end{equation}\nWhere $l$ is the distance from the source to the grid, $a$ is the spatial scale of the scattering electromagnetic fields, and $\\alpha$ is the particle deflection angle. The value of $\\mu$ can fall in one of three regimes:\n\n\\begin{align}\n\\mu &\\ll 1 \\rightarrow \\text{ linear}\\\\\n\\mu &< \\mu_c \\rightarrow \\text{ nonlinear injective}\\\\\n\\mu &> \\mu_c \\rightarrow \\text{ caustic}\\\\\n\\end{align}\n\nwhere $\\mu_c \\sim 1$ is a characteristic value at which particle paths cross, leading to the formation of bright caustics. Correctly placing a radiograph in the correct regime is necessary to determine which analysis techniques can be applied to it.\n\nThe maximum deflection angle can be calculated after the simulation has run by comparing the initial and final velocity vectors of each particle\n\n\n```python\nmax_deflection = sim.max_deflection\nprint(f\"Maximum deflection \u03b1 = {np.rad2deg(max_deflection):.2f}\")\n```\n\nThe spatial scale of the field constructed in this example is $\\sim$ 1 mm, and $l$ is approximately the distance from the source to the grid origin. Therefore, we can calculate the value of $\\mu$\n\n\n```python\na = 1 * u.mm\nl = np.linalg.norm(sim.source * u.m).to(u.mm)\nmu = l * max_deflection.value / a\nprint(f\"a = {a}\")\nprint(f\"l = {l:.1f}\")\nprint(f\"\u03bc = {mu:.2f}\")\n```\n\nwhich places this example in the non-linear injective regime.\n\n## Options\n\n[create_particles()]: ../../api/plasmapy.diagnostics.proton_radiography.SyntheticProtonRadiograph.rst#plasmapy.diagnostics.proton_radiography.SyntheticProtonRadiograph.create_particles\n\nFor sake of comparison, here is the result achieved by setting `distribution = 'uniform'` in the [create_particles()] function.\n\n\n```python\nsim.create_particles(\n 1e5, 3 * u.MeV, max_theta=np.pi / 15 * u.rad, distribution=\"uniform\"\n)\nsim.run()\nsize = np.array([[-1, 1], [-1, 1]]) * 1.5 * u.cm\nbins = [200, 200]\nhax, vax, intensity = sim.synthetic_radiograph(size=size, bins=bins)\nplot_radiograph(hax, vax, intensity)\n```\n", "meta": {"hexsha": "9c7234be22a4faf2ed03011980b22aee2d43ab49", "size": 16523, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/notebooks/diagnostics/proton_radiography_particle_tracing.ipynb", "max_stars_repo_name": "FinMacDov/PlasmaPy", "max_stars_repo_head_hexsha": "092e32bd78ed06178e801b5f4e38b5ec72bcd52a", "max_stars_repo_licenses": ["BSD-2-Clause", "MIT", "BSD-2-Clause-Patent", "BSD-1-Clause", "BSD-3-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-04-20T00:07:30.000Z", "max_stars_repo_stars_event_max_datetime": "2021-04-20T00:07:30.000Z", "max_issues_repo_path": "docs/notebooks/diagnostics/proton_radiography_particle_tracing.ipynb", "max_issues_repo_name": "FinMacDov/PlasmaPy", "max_issues_repo_head_hexsha": "092e32bd78ed06178e801b5f4e38b5ec72bcd52a", "max_issues_repo_licenses": ["BSD-2-Clause", "MIT", "BSD-2-Clause-Patent", "BSD-1-Clause", "BSD-3-Clause"], "max_issues_count": 18, "max_issues_repo_issues_event_min_datetime": "2018-04-08T08:27:35.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-18T17:34:34.000Z", "max_forks_repo_path": "docs/notebooks/diagnostics/proton_radiography_particle_tracing.ipynb", "max_forks_repo_name": "FinMacDov/PlasmaPy", "max_forks_repo_head_hexsha": "092e32bd78ed06178e801b5f4e38b5ec72bcd52a", "max_forks_repo_licenses": ["BSD-2-Clause", "MIT", "BSD-2-Clause-Patent", "BSD-1-Clause", "BSD-3-Clause"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2017-07-19T07:28:21.000Z", "max_forks_repo_forks_event_max_datetime": "2020-06-24T18:34:07.000Z", "avg_line_length": 39.4343675418, "max_line_length": 561, "alphanum_fraction": 0.616655571, "converted": true, "num_tokens": 3035, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7341195152660688, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.4183399522292451}} {"text": "```python\nimport sympy as sym\nfrom sympy.solvers import solve\nfrom sympy import Symbol\nfrom sympy.abc import x,y,z,a,b,c,d,e,f,g,h,i,j,k,l\nimport dill\nimport time\nimport numpy as np\n```\n\n\n```python\n# Helper functions to simplify final expressions\ndef collect_by_list(expr_in, sort_list):\n expr = sym.expand(expr_in)\n res = expr\n for s in sort_list:\n res = sym.rcollect(res, s)\n return res\n\ndef get_symbol_list(expr_in, return_counts=False):\n expr = sym.expand(expr_in)\n syms = {}\n for a in expr.atoms():\n if type(a) is sym.Symbol:\n syms[a] = expr.count(a)\n symlist = list(sorted(syms, key=syms.__getitem__, reverse=True))\n if return_counts:\n countlist = list(sorted(syms.values(), reverse=True))\n return symlist, countlist\n else:\n return symlist\n\ndef collect_auto(expr_in, print_ops=False, syms_to_prioritize=[]):\n slist = get_symbol_list(expr_in)\n sort_list = syms_to_prioritize\n for s in slist:\n if s not in syms_to_prioritize:\n sort_list.append(s)\n res = collect_by_list(expr_in, sort_list)\n if print_ops:\n print(sym.count_ops(res))\n return res\n\ndef collect_terms_in_fraction(expr_in, terms_to_prioritize=[]):\n numerator, denominator = sym.fraction(expr_in)\n num_simp = collect_auto(numerator, syms_to_prioritize=terms_to_prioritize)\n den_simp = collect_auto(denominator, syms_to_prioritize=terms_to_prioritize)\n return num_simp/den_simp\n```\n\n# Full System of Equations\n\n\n```python\ndKMgO_KMgO = Symbol('dKMgO_KMgO')\ndKFeO_KFeO = Symbol('dKFeO_KFeO')\ndKSiO2_KSiO2 = Symbol('dKSiO2_KSiO2')\ndKMgSiO3_KMgSiO3 = Symbol('dKMgSiO3_KMgSiO3')\ndKFeSiO3_KFeSiO3 = Symbol('dKFeSiO3_KFeSiO3')\n\ndM_Mg = Symbol('dM_Mg')\ndM_Fe = Symbol('dM_Fe')\ndM_Si = Symbol('dM_Si')\ndM_O = Symbol('dM_O')\ndM_c = Symbol('dM_c')\n\ndM_MgO = Symbol('dM_MgO')\ndM_FeO = Symbol('dM_FeO')\ndM_SiO2 = Symbol('dM_SiO2')\ndM_MgSiO3 = Symbol('dM_MgSiO3')\ndM_FeSiO3 = Symbol('dM_FeSiO3')\ndM_m = Symbol('dM_m')\n\nM_Mg = Symbol('M_Mg')\nM_Fe = Symbol('M_Fe')\nM_Si = Symbol('M_Si')\nM_O = Symbol('M_O')\nM_c = Symbol('M_c')\n\nM_MgO = Symbol('M_MgO')\nM_FeO = Symbol('M_FeO')\nM_SiO2 = Symbol('M_SiO2')\nM_MgSiO3 = Symbol('M_MgSiO3')\nM_FeSiO3 = Symbol('M_FeSiO3')\nM_m = Symbol('M_m')\n```\n\n\n```python\n# Total Moles\n# eq_mantle_total_moles = M_MgO + M_FeO + M_SiO2 + M_MgSiO3 + M_FeSiO3 - M_m\n# eq_core_total_moles = M_Mg + M_Fe + M_Si + M_O - M_c\neq_mantle_dtotal_moles = dM_MgO + dM_FeO + dM_SiO2 + dM_MgSiO3 + dM_FeSiO3 - dM_m\neq_core_dtotal_moles = dM_Mg + dM_Fe + dM_Si + dM_O - dM_c\n\n# mantle interactions\neq_K_MgSiO3 = dM_MgO/M_MgO + dM_SiO2/M_SiO2 - dM_MgSiO3/M_MgSiO3 - dM_m/M_m - dKMgSiO3_KMgSiO3\neq_K_FeSiO3 = dM_FeO/M_FeO + dM_SiO2/M_SiO2 - dM_FeSiO3/M_FeSiO3 - dM_m/M_m - dKFeSiO3_KFeSiO3\n\n# core interactions\neq_K_MgO = dM_Mg/M_Mg + dM_O/M_O + dM_m/M_m - 2*dM_c/M_c - dM_MgO/M_MgO - dKMgO_KMgO\neq_K_FeO = dM_Fe/M_Fe + dM_O/M_O + dM_m/M_m - 2*dM_c/M_c - dM_FeO/M_FeO - dKFeO_KFeO\neq_K_SiO2 = dM_Si/M_Si + 2*dM_O/M_O + dM_m/M_m - 3*dM_c/M_c - dM_SiO2/M_SiO2 - dKSiO2_KSiO2\n\n# species continuity\neq_dM_Mg = dM_MgO + dM_MgSiO3 + dM_Mg\neq_dM_Fe = dM_FeO + dM_FeSiO3 + dM_Fe\neq_dM_Si = dM_SiO2 + dM_MgSiO3 + dM_FeSiO3 + dM_Si\neq_dM_O = dM_MgO + dM_FeO + 2*dM_SiO2 + 3*dM_MgSiO3 + 3*dM_FeSiO3 + dM_O\n\nequations = [eq_mantle_dtotal_moles, eq_core_dtotal_moles, \n eq_K_MgSiO3, eq_K_FeSiO3,\n eq_K_MgO, eq_K_FeO, eq_K_SiO2,\n eq_dM_Fe, eq_dM_Mg, eq_dM_O, eq_dM_Si]\nsolve_for = [dM_c, dM_Fe, dM_FeO, dM_FeSiO3, dM_m, dM_Mg, dM_MgO, dM_MgSiO3, dM_O, dM_Si, dM_SiO2]\n```\n\n\n```python\nstart = -time.time()\nsolution = solve(equations, solve_for)\ntime_elapsed = start+time.time()\n```\n\n\n```python\nprint('{:.1f} hrs to compute full solution'.format(time_elapsed/60/60))\n```\n\n 1.9 hrs to compute full solution\n\n\n\n```python\n# Save unsimplified full solution straight from the computation\ndill.dump(solution,open('computed_solution.m','wb'))\n```\n\n\n```python\nsolution = dill.load(open('computed_solution.m','rb'))\n```\n\n# Simplify Solution\n\n\n```python\n# Simplify the solution using the helper functions at the top\nterms_to_prioritize = [dKMgO_KMgO, dKFeO_KFeO, dKSiO2_KSiO2, dKMgSiO3_KMgSiO3, dKFeSiO3_KFeSiO3]\nsimplified = {}\nfor k,v in solution.items():\n simplified[k] = collect_terms_in_fraction(v, terms_to_prioritize=terms_to_prioritize)\n```\n\n\n```python\n# Demonstrate how much simpler the solutions are compared to those computed by sympy\nprint('{:.0f} operations in full solution'.format(sym.count_ops(solution[dM_Si])))\nprint('{:.0f} operations in simplified solution'.format(sym.count_ops(simplified[dM_Si])))\n```\n\n 3745 operations in full solution\n 1507 operations in simplified solution\n\n\n\n```python\n# save simplified solution\ndill.dump(simplified, open('simplified_solution.m','wb'))\n```\n\n# Check by plugging in random values into Variables\n\n\n```python\ndef gen_and_check_rand_values(eqns, vars_to_rand=None):\n rand_values = gen_rand_vals(vars_to_rand=vars_to_rand)\n results = eval_eqns(eqns, rand_values)\n return rand_values, results\n\ndef gen_rand_vals(vars_to_rand = None):\n if vars_to_rand is None:\n vars_to_rand = [M_Mg, M_Fe, M_Si, M_O, M_c, M_MgO, M_FeO, M_SiO2, M_MgSiO3, M_FeSiO3, M_m, dKMgO_KMgO, dKFeO_KFeO, dKSiO2_KSiO2, dKMgSiO3_KMgSiO3, dKFeSiO3_KFeSiO3]\n rand_values = {}\n for v in vars_to_rand:\n rand_values[v] = np.random.rand()\n return rand_values\n\ndef eval_eqns(eqns, rand_values_dict):\n results = {}\n for symb in eqns.keys():\n results[symb] = eq_rand_values[symb].subs(rand_values_dict)\n return results\n\n```\n\n\n```python\nrand_values0, results0 = gen_and_check_rand_values(simplified)\nrand_values1, results1 = gen_and_check_rand_values(simplified)\n```\n\n\n```python\nrand_results = ((rand_values0, results0),\n(rand_values1, results1))\ndill.dump(rand_results, open('rand_results.m','wb'))\n```\n\n# Import Simplified Solution\n\n\n```python\n# open simplified solution\nsimplified = dill.load(open('simplified_solution2.m','rb'))\n```\n\n\n```python\neqns_file = open('eqns_funcs.py','w')\nfor k,v in simplified.items():\n eqns_file.write('\\tdef '+str(k)+'_dTc(self, Moles, dKs, dMi_b):\\n')\n eqns_file.write('\\t\\t\\'\\'\\'compute {} given Moles, dKDs, and dMm_b/dT\\'\\'\\'\\n'.format(k))\n eqns_file.write('\\t\\tdM_MgO_er, dM_SiO2_er, dM_FeO_er, dM_MgSiO3_er, dM_FeSiO3_er = dMi_b\\n')\n eqns_file.write('\\t\\tM_Mg, M_Si, M_Fe, M_O, M_c, M_MgO, M_SiO2, M_FeO, M_MgSiO3, M_FeSiO3, M_m = self.unwrap_Moles(Moles)\\n')\n eqns_file.write('\\t\\tdKMgO_KMgO, dKSiO2_KSiO2, dKFeO_KFeO, dKMgSiO3_KMgSiO3, dKFeSiO3_KFeSiO3 = dKs\\n')\n eqns_file.write('\\t\\treturn '+str(v)+'\\n\\n')\neqns_file.close()\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "8293f6016fc8eed01f14494627ae703ca621e4d0", "size": 11067, "ext": "ipynb", "lang": "Jupyter Notebook", 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YES\n2. YES", "lm_q1_score": 0.7057850278370111, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.4182952127466499}} {"text": "```\n# default_exp intro_to_control_theory\n```\n\n\n```\n%load_ext autoreload\n%autoreload 2\n```\n\n\n```\nfrom IPython.display import Image\nfrom IPython.display import HTML\n\n# For animations\nfrom matplotlib import animation, rc\n\n# equivalent to rcParams['animation.html'] = 'html5'\nrc('animation', html='html5')\n```\n\n\n```\n#export\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\n```\n\n\n```\n%matplotlib inline\n```\n\n\n```\n#hide\nfrom nbdev.showdoc import *\n```\n\n------------------------\n\n# The Control Problem\n\nThis notebook provides an overview of the big picture problem that we\u2019re trying to solve as control system engineers. \n\n- Control theory allows to solve a number of engineering problems, not only as control engineers but as any engineer\n - Switching power regulators\n - Automatic gain control circuits that automatically increase or decrease the gain of a signal\n - Isolation system in a motor mount that is sensitive to vibrations\n - Industrial robotics\n - etc.\n\nControl system is:\n- Building models of your system\n- Simulate them to make predictions\n- Understanding the dynamics and how they interact with the rest of the system\n- Filtering and rejecting noise\n- Selecting or designing hardware (sensors, actuators)\n- Testing the system in expected and unexpected environments\n- It is understanding your system!\n\n------------------\n\n## What is a system?\n\nThe concept is straigthforward, but the term is sometime applied very generically\n\nFor us:\n\n> A **system** is a collection of interconnected parts that form a larger more complex whole\n\n\nEngineering problems are usually complex.\n- Divide complex projects into smaller parts, or _systems_ makes it possible to simplify the problem and to specialise in specific areas\n- Usually more than one layer of complexity\n- Each one of the interconnected parts that form a larger system can be a complex system in its own right!\n\n### Control systems\n\nAs a control engineer:\n- Goal is to create something that meets the functional or performance requirements you set for the project. \n- The collection of the interconnected parts that are created specifically to meet these requirements are the _control system_ (at least in general)\n- More specifically: **A control system is a mechanism (hardware or software) that alters the future state of a system**\n\n- For any project however the control system again might be a collection of interconnected parts that require specialists:\n - sensor experts, \n - actuators experts, \n - digital signal processing experts, \n - state estimation experts,\n - etc.\n\nFor example:\n\n\n \n \n \n\n\nAnd of course, the breaking system itself is just one of the main interconnected parts that create the car.\n\n### System as a box\n\n- We represent systems graphically as a box\n- Arrows going into the box are inputs\n- Arrows going out of the box are output\n\n\n \n\n\n- The system inside the box is described through a math model (e.g. equation of motion)\n- The \"box\" can be used to describe systems that are simple or complex\n\nSo if we want to start representing it more formally:\n\n\n \n\n\n- $u$: control inputs \n\n- $\\xi$: disturbances\n\n- $f$: system (physics is involved here!)\n\n- $\\theta$: system parameters (e.g., mass, inertia, spring constants, etc. )\n\n\n\n \n \n \n \n
                                        \n\n---------------------------\n\n## Three different problems\n- Three parts to our simple block diagram\n - The system\n - The inputs (that drive the system)\n - The output (that the system generates)\n \nAt any given time one of the three parts are unknown, and the part that is unknown defines the problem that you are solving.\n\n### The system identification problem\n\n- As a practicing engineer you won\u2019t always be given a model of your system\n- You will need to determine it yourself\n- This is done through a process called **System Identification**\n\n\n \n\n\n\n- What is the mathematical equation that will convert my known inputs into my measured outputs?\n- What should I model?\n\n#### Black box vs White box\n\n**Black box**:\n- You are given a box that you cannot open, but you are asked to model what is inside\n- Subject what is inside the box to various known inputs, measure the output and then infer what is inside the box based on the relationship between the input and the output.\n\n**White box**\n- Now you can see exactly what is inside the box (hardware, software).\n- You can write the mathematical equations of the dynamics directly (e.g. Netwon's equations of motion).\n\nFor example, for a sping-mass system:\n\n\n \n $$F = m\\ddot{x} + kx$$ \n\n\n\n\n- Even the while box method might require to run tests and applying inputs to measure outputs to calculate the parameters of your system.\n- E.g., modeling a linear spring: you know the equation but what is the exact spring constant?\n\nUsually, you need to do a bit of both.\n\n------------------------\n\n#### Example: modeling a car\n\nLet's consider an example: car cruise control, using a simplified mathematical model.\n\n\n \n\n\n$$m\\frac{dv}{dt} + \\alpha|v|v+\\beta v = \\gamma u-mgsin(\\theta)$$\n\n- Input: gas pedal\n- Output: speed (speedometer)\n- Disturbance: the slope of the road\n\nAssumptions to simplify the model:\n- Thrust is proportional to how much pressure we put on the pedal and this directly translates into how much is open the gas valve\n- Frictions and drags are linear with speed\n- Small angles: $\\theta < 30^o$ so that $sin(\\theta) \\approx \\theta$\n\n\n\nUsing our assumptions, the previous model can be simplified as:\n\n$$m\\frac{dv}{dt} + \\alpha|v|v+\\beta v = \\gamma u-mgsin(\\theta) \\approx m\\frac{dv}{dt} + \\beta v = \\gamma u-mg\\theta$$\n\nBefore we simplify the model though, we can re-write it in a **standard representation** (state space representation):\n\n$$\\mathbf{\\dot{x}} = [\\dot{x_1}, \\dot{x_2}]^T$$\n\n\\begin{cases}\n \\dot{x_1} &= x_2 \\\\\n \\dot{x_2} &= -\\frac{\\alpha}{m}|x_2|x_2 - \\frac{\\beta}{m}x_2 + \\frac{\\gamma}{m}u -gsin(\\theta) \\\\\n y &=x_2\n\\end{cases}\n\n$$\\mathbf{x}(t_0) = x_0$$\n\n- Velocity is the derivative of the position\n- Two state variables: position and velocity\n- Two inputs: $u$ and $\\theta$\n\n- The state $\\mathbf{x}$ includes the minimal amount of information at the current time to predict its behaviour in the future, only based on the knowledge of its future inputs.\n\n- In the case where we have differential equations from actual physical systems we also need the initial conditions.\n\n- Note that there is no time in the equations above: **time-invariant** system\n\n#### Standard representations\n\n- \"standard representations\" are the bulding blocks of the engineering language\n- Analysis and design are based on having the system in a standard representation \n- Specific problems typically need to be translated into a standard representation\n- Control System Software typically assumes that the system is already in a standard representation (and expects inputs based on that)\n\nWe can code up our standard representation of the car in Python:\n\n\n```\n#export\nclass Car:\n _g = 9.8 # Gravity\n \n def __init__(self, x0, params):\n self._x_1 = x0[0] # position (along the road)\n self._x_2 = x0[1] # velocity (along the road)\n self._m, self._alpha, self._beta, self._gamma = params\n \n def step(self, dt, u, theta):\n self.theta = theta\n self._x_1 = self._x_1 + dt*self._x_2\n self._x_2 = self._x_2 + dt*(-self._alpha/self._m*abs(self._x_2)*self._x_2 - \\\n self._beta/self._m*self._x_2 + self._gamma/self._m*u - \\\n Car._g*np.sin(theta))\n \n def speedometer(self): \n v = self._x_2\n return (v,) \n \n # Utility function to simplify plotting\n def sensor_i(self):\n # Rotation matrix to get back to the main frame.\n R = np.array(((np.cos(theta), -np.sin(theta)), (np.sin(theta), np.cos(theta))))\n x_i, y_i = R.dot(np.array([[self._x_1],[0]]))\n v = self._x_2\n return (x_i, y_i, v)\n```\n\nWe can linearise the previous model of the car around an equilibrium ($\\dot{\\mathbf{x}}=0$) as below. \n\n\n\\begin{equation}\n\\frac{df}{dx}=\n\\begin{bmatrix}\n0 & 1\\\\\n0 & -\\frac{\\alpha}{m}|x_2|-\\frac{\\beta}{m}\n\\end{bmatrix}\n\\end{equation}\n\n\\begin{equation}\n\\frac{df}{du}=\n\\begin{bmatrix}\n0 & 0\\\\\n\\frac{\\gamma}{m} & -gcos(\\theta)\n\\end{bmatrix}\n\\end{equation}\n\nwhich, at the equilibrium $x_1=0, x_2=0$ (note that $x_1$ really does not affect the result):\n\n\n\\begin{equation}\nA=\\frac{df}{dx}=\n\\begin{bmatrix}\n0 & 1\\\\\n0 & -\\frac{\\beta}{m}\n\\end{bmatrix}\n\\end{equation}\n\n\\begin{equation}\nB=\\frac{df}{du}=\n\\begin{bmatrix}\n0 & 0\\\\\n\\frac{\\gamma}{m} & -g\n\\end{bmatrix}\n\\end{equation}\n\nand\n\n$$\\dot{\\mathbf{x}}=A\\mathbf{x}+B\\mathbf{u}$$\n$$y=[0 \\; 1]\\mathbf{x}$$\n\n\n\n```\n#export\nclass LinearCar:\n _g = 9.8\n \n def __init__(self, x0, params):\n self._x_1 = x0[0] # position (along the road)\n self._x_2 = x0[1] # velocity\n self._m, self._alpha, self._beta, self._gamma = params\n \n def step(self, dt, u, theta):\n self._theta = theta\n A = np.array([[0, 1], [0, -self._beta/self._m]])\n B = np.array([[0, 0], [self._gamma/self._m, -LinearCar._g]])\n \n x = np.array([[self._x_1],[self._x_2]])\n U = np.array([[u],[theta]])\n self._x_1 = (self._x_1 + dt*(A[0,np.newaxis,:].dot(x) + B[0,np.newaxis,:].dot(U))).item()\n self._x_2 = (self._x_2 + dt*(A[1,np.newaxis,:].dot(x) + B[1,np.newaxis,:].dot(U))).item()\n \n def speedometer(self): \n v = self._x_2\n return (v,)\n \n # Utility function to simplify plotting\n def sensor_i(self):\n # Rotation matrix to get back to the inertial frame..\n R = np.array(((np.cos(self._theta), -np.sin(self._theta)), \n (np.sin(self._theta), np.cos(self._theta))))\n x_i, y_i = R.dot(np.array([[self._x_1],[0]]))\n v = self._x_2\n return (x_i, y_i, v)\n```\n\n#### Comments\n- Multiple models can be used to represent a physical system\n- There are a number of parameters in our models ($m$, $\\alpha$, $\\beta$, $\\gamma$)\n- Their value depend on the specific car you are modeling\n- It might not be easy to have good values for these parameters\n\n## The simulation problem\n\n- Predict how your outputs change given a known set of inputs and the mathematical model of the system\n- Usually a lot of design time is spent in this stage\n- It can be difficult to figure out the set of inputs and their range to characterise the operational envelope of the system\n\n\n\n \n\n\n\n\nYou need to run a simulation to answer:\n- Does my system model match my test data?\n- Will my system work in all operating environments?\n- What happens when...\n- Simulations can also be used to limit the number of field tests (which are usually expensive or might let you avoid dangerous maneouvers)\n\nIf we go back to our Python car, we can now easily simulate it:\n\n\n```\n# We assume we have identified the model parameters\nm = 10\nalpha = 1\nbeta = 1\ngamma = 1\nparams = (m, alpha, beta, gamma)\n\n# We select the car initial conditions (position and velocity)\nx_0 = (0,0)\n\n# We create our car\ncar = Car(x_0, params)\n```\n\n\n```\n# We define out inputs:\ntheta = np.radians(20) # disturbance\nu = 0 # Input\n\n# And finally we define the simulation parameters\nt0, tf, dt = 0, 10, 0.1 # time\n\nposition = []\nvelocity = []\ntime = []\nfor t in np.arange(t0, tf, dt):\n car.step(dt, u, theta)\n x, y, v = car.sensor_i()\n position.append((x,y)) \n velocity.append(v)\n time.append(t)\n \nprint('simulation complete') \n```\n\n simulation complete\n\n\nDone! Now we can see what happened..\n\nLet's plot the results:\n\n\n```\nfig, ax = plt.subplots();\n\nplt.plot(time, velocity)\nplt.xlabel('time (s)')\nplt.ylabel('speed (m/s)');\n```\n\nWhat happens if we simulate our LinearCar?\n\n\n```\n# We select the car initial conditions (position and velocity)\nx_0 = (0,0)\n\n# We create our car\ncar = LinearCar(x_0, params)\n\n# We define out inputs:\ntheta = np.radians(20) # disturbance\nu = 0 # Input\n\n# And finally we define the simulation parameters\nt0, tf, dt = 0, 10, 0.1 # time\n\nlin_position = []\nlin_velocity = []\ntime = []\nfor t in np.arange(t0, tf, dt):\n car.step(dt, u, theta)\n x, y, v = car.sensor_i()\n lin_position.append((x,y)), lin_velocity.append(v)\n time.append(t)\n \nprint('simulation complete') \n```\n\n simulation complete\n\n\n\n```\nfig, ax = plt.subplots();\n\nplt.plot(time, lin_velocity)\nplt.xlabel('time (s)')\nplt.ylabel('speed (m/s)');\n```\n\n- Multiple models can be used to represent a physical system and the complexity of the model depends on the objectives\n\n### More complex representation\n\nWe can have much more complex representation of our simulation:\n\n\n```\n# First set up the figure, the axis, and the plot elements we want to animate\nfig, ax = plt.subplots();\n\nax.set_xlim((min(position)[0], max(position)[0]))\nax.set_ylim((min(position)[1]*2, max(position)[1]))\nline, = ax.plot([], [], lw=4);\n```\n\n\n```\n# draw terrain\ndef terrain(theta_rad, x_0, x_range):\n y_range = theta*(x_range-x_0[0])+x_0[1]-10\n close_triangle = (x_range[0], y_range[-1])\n x_range = np.append(x_range, close_triangle[0])\n y_range = np.append(y_range, close_triangle[1])\n X = np.matrix([x_range, y_range]).transpose()\n patch = plt.Polygon(X, color='yellow')\n return patch\n\nx_range = np.linspace(int(position[0][0]), int(position[-1][0]), num=20)\npatch = terrain(theta_rad=theta, x_0=x_0, x_range=x_range)\n\n# initialization function: plot the background of each frame\ndef init():\n ax.add_patch(patch)\n return patch,\n\n# animation function. This is called sequentially\ndef animate(i): \n #line.set_data(time[max(0,i-2):i], position[max(0,i-2):i])\n x_min, x_max = position[max(0,i-2)][0], position[i][0]\n y_min, y_max = position[max(0,i-2)][1], position[i][1]\n line.set_data([x_min, x_max], [y_min, y_max])\n return (line,)\n\n\n# call the animator. blit=True means only re-draw the parts that have changed.\nanim = animation.FuncAnimation(fig, animate, init_func=init,\n frames=len(time), interval=40, blit=True,\n repeat_delay=10000, repeat=True);\n\nHTML(anim.to_html5_video())\n```\n\n\n\n\n\n\n\n\n----------------------------\n\n## The Control Problem\n\n- We know the model of the system \n- We know how we want the system outputs to behave \n- We can determine the appropriate inputs through various control methods.\n\n_Control problem - how do we generate the appropriate system input that will produce the desired output?_\n\n- Control theory gives you the tools needed to answer this question. \n- Without control theory, the designer is relegated to choosing a control system through trial and error.\n\n- How can I get my system to meet my performance requirements?\n- How can I automate a process that currently involves humans in the\nloop?\n- How can my system operate in a dynamic and noisy environment?\n\n\n \n\n\n---------------------------\n\n# Why do we need feedback control?\n\n### Open-loop control\n\n\n\n\n\n
                                        \n\n- Now the box isn\u2019t just any system, it\u2019s specifically a system that we want to control. \n- We call the system that is being controlled as the **process** (or controlled system). \n- The **inputs** into the process are variables that we have access to and can change based on whichever control scheme we choose (manipulated variables).\n- Actuators manipulate these variables. An actuator is a generic term that refers to a device or motor that is responsible for controlling a system.\n- The actuators are driven by an actuating signal that is generated by the controller.\n\n\n \n\n\nThis type of control system is referred to as open-loop since the inputs into the controller are not fed back from the output of the process.\n\n- Open loop control is reserved for simple processes that have well-defined input to output behaviors\n\nSome examples:\n- Dishwasher\n- Lawn sprinklers\n\n\n### Feedback-control\n\nFor any arbitrary process, an open-loop control system is typically not sufficient. \n\nThis is because:\n- there are disturbances that affect your system that are random by nature and beyond your control;\n- the process itself might have variations (e.g., variation of resistence due to temperature)\n\nSince an open-loop control has no knowledge of the process output has no way of responding to these variations.\n\n\nSo what can we do about this? \n- We add feedback to our system! \n- We accept the fact that disturbances and process variations are going to influence the controlled variable.\n- Instead of living with the resulting error, we add a sensor that will measure the controlled variable and pass it along to our controller. \n\n\n \n\n\nA feedback control system is able to react to changes to the controlled variable automatically by constantly driving the error term to zero.\n\n- The feedback structure is very powerful and robust \n- With the addition of the feedback structure we also have new problems\n- We need to think about:\n - the accuracy of the controlled variable at steady state, \n - the speed with which the system can respond to changes and reject disturbances,\n - the stability of the system as a whole. \n \n- We have also added sensors, and they have noise and other inaccuracies that get injected into our loop and affect the performance. \n - We can add redundant sensors (so that we can measure different state variables)\n - We filter the output of the sensors to reduce the noise\n - We fuse them together to create a more accurate estimate of the true state.\n\n### What is a control system?\n\n- A control system is a mechanism that alters the behavior (or the future state) of a system;\n- The future behavior of the system must tend towards a state that is desired. \n- This means that you have to know what you want your system to do and then design your control system to generate that desired outcome.\n\n--------------------------------\n\n## LTI systems\n\nGeneral form of a system:\n\n$$\n\\begin{cases}\n\\dot{x}(t) = f(x(t), u(t), t)\\\\\ny(t) = g(x(t), u(t), t)\n\\end{cases}\n$$\n\nWhen $f$ and $g$ are linear in $x$ and $u$ then the system can be written in the form:\n\n$$\n\\begin{cases}\n\\dot{x}(t) = A(t)x(t) + B(t)u(t)\\\\\ny(t) = C(t)x(t) + D(t)u(t)\n\\end{cases}\n$$\n\n- The system is said linear in this case,\n- Matrices $A(t), B(t), C(t), D(t)$ can be in general function of time,\n- If they are not, we talk about **Linear Time-Invariant Systems**\n\n$$\n\\begin{cases}\n\\dot{x}(t) = Ax(t) + Bu(t)\\\\\ny(t) = Cx(t) + Du(t)\n\\end{cases}\n$$\n\nAll LTI systems have the following defining properties:\n- Homogeneity: \n - If you scale the input $u(t)$ then the output will be scaled by the same factor:\n - $au(t) \\Rightarrow ay(t)$ (e.g. if you double the input, the output would also double).\n \n - ex. step input of amplitude 1 and 2:\n \n \n \n \n\n- Superposition (Additivity)\n - Suppose that:\n - input $x_1(t)$ produces output $y_1(t)$\n - input $x_2(t)$ produces output $y_2(t)$\n - If you add two inputs together, then the output is the superposition of the two separate outputs, i.e. the sum of the individual outputs:\n \n\n \n \n\n\nSometimes, these two properties are stated together: \"if the input is scaled and summed, then the output will also be scaled and summed by the same amount\".\n- A system that respect these two property is a _linear system_\n\n- Time Invariance\n - The system behaves the same regardless of when the action takes place\n - Formally there is no explicit dependency of time in the equations\n - The same input translated in time produces the same output also translated in time:\n \n - An input of $x(t-\\tau)$ produces an output of $y(t-\\tau)$.\n \n \n \n \n \n\n- These conditions are very restritive\n- However, linear systems can be solved!\n\n- Wide range of systems can be approximated accurately by an LTI model\n\n### Impulse Response\n\n- LTI systems can be characterised by their response to an impulse function (the output of the system when presented with a brief input signal)\n\nLet's see what happens when we apply an impulse to our car.\n\nBefore we do that, we need to define an impulse function in Python.\n\n\n```\n#export\ndef step(t, step_time=0):\n \"\"\"Heaviside step function\"\"\"\n return 1 * (t >= step_time)\n\ndef delta(t, delta_t=0, eps=None): # Impulse\n if np.isscalar(t) and eps is None:\n raise Exception('eps must be defined for scalar values.')\n if eps is None and len(t) > 1: \n _eps=t[1]-t[0]\n else:\n _eps = eps\n return 1/_eps*(step(t, delta_t-_eps/2)-step(t, delta_t+_eps/2))\n```\n\nAnd here is what it looks like:\n\n\n```\nfig, ax = plt.subplots(1, 1, figsize=(3, 3))\n\nt = np.linspace(0, 5, 60)\nax.plot(t, delta(t, delta_t=1), linewidth=3, color='blue')\nax.set_xlabel('time (s)')\nfig.tight_layout()\n```\n\nWe are now ready to apply it to our car:\n\n\n```\n# We assume we have identified the model parameters\nm = 10\nalpha = 1 # it is not used in the linear system\nbeta = 10 # friction\ngamma = 1 # coeff. applied to the input.\nparams = (m, alpha, beta, gamma)\n\n# We select the car initial conditions (position and velocity)\nx_0 = (0,0)\n\n# We create our car\ncar = LinearCar(x_0, params)\n\ntheta = np.radians(0) # disturbance (deg) - we can also try np.radians(20)\n\n# And finally we define the simulation parameters\nt0, tf, dt = 0, 10, 0.1 # time\n\nposition = []\nvelocity = []\ninput_values = []\ntime = []\nfor t in np.arange(t0, tf, dt):\n # HERE, we apply the impulse\n u = delta(t, delta_t=0, eps=0.01)\n \n car.step(dt, u, theta)\n x, y, v = car.sensor_i()\n position.append((x,y)), velocity.append(v)\n time.append(t)\n input_values.append(u)\n \nprint('simulation complete') \n```\n\n simulation complete\n\n\n\n```\nplt.plot(np.arange(t0, tf, dt), velocity, linewidth=4, color='blue', label='t_0')\nplt.xlabel('time (s)'), plt.ylabel('velocity (m/s)');\nplt.ylim(0,)\nplt.grid();\n```\n\nBecause of time invariance, if we have another impulse we can expect the same response at this second time, and if we hit twice as hard, then the response is twice as large\n\n\n```\ndef multiple_impulses(t):\n u1 = 1*delta(t, delta_t=0, eps=0.01)\n u2 = 2*delta(t, delta_t=1, eps=0.01)\n u3 = 2*delta(t, delta_t=4, eps=0.01)\n \n return u1 + u2 + u3\n```\n\n\n```\ncar = LinearCar(x0=(0,0), params=params)\n\nposition = []\nvelocity = []\ninput_values = []\ntime = []\nfor t in np.arange(t0, tf, dt):\n # HERE, we apply the impulse\n u = multiple_impulses(t)\n \n car.step(dt, u, theta)\n x, y, v = car.sensor_i()\n position.append((x,y)), velocity.append(v)\n time.append(t)\n input_values.append(u)\n \nprint('simulation complete') \n```\n\n simulation complete\n\n\n\n```\nplt.plot(np.arange(t0, tf, dt), velocity, linewidth=4, color='blue', label='t_0')\nplt.xlabel('time (s)'), plt.ylabel('velocity (m/s)');\nplt.ylim(0,)\nplt.grid();\n```\n\n- Because of the superposition principle, the full response is the summation of the signals.\n\n- If we have a more complex signal (e.g. ramp) we can break it down to a sequence of impulses (the output is the response to each of the impulses)\n\n\n```\n#export\ndef ramp_as_impulses(t, time_vector): \n u = t*delta(time_vector, delta_t=t, eps=.01)\n return u\n```\n\nAnd we can verify that this is a ramp:\n\n\n```\ninput_u = []\n\nfor t in np.arange(t0, tf, .4):\n input_u.append(ramp_as_impulses(t, np.arange(t0, tf, .4)))\n```\n\n\n```\nlen(input_u)\n```\n\n\n\n\n 25\n\n\n\n\n```\nfig, ax = plt.subplots(1, 1, figsize=(3, 3))\n\nax.plot(np.arange(t0, tf, .4), input_u, linewidth=3, color='blue')\nax.set_xlabel('time (s)')\nfig.tight_layout()\n```\n\n\n```\ncar = LinearCar(x0=(0,0), params=params)\n\nposition = []\nvelocity = []\ninput_values = []\ntime = []\nfor index, t in enumerate(np.arange(t0, tf, dt)):\n # HERE, we apply the ramp as a sequence of impulses\n u = ramp_as_impulses(t, np.arange(t0, tf, dt)) \n #print('time:', t, '-> index:', index, '->u[i]:', u[index])\n car.step(dt, u[index], theta)\n x, y, v = car.sensor_i()\n position.append((x,y)), velocity.append(v)\n time.append(t)\n input_values.append(u)\n \nprint('simulation complete') \n```\n\n simulation complete\n\n\n\n```\nplt.plot(np.arange(t0, tf, dt), velocity, linewidth=4, color='blue', label='t_0')\nplt.xlabel('time (s)'), plt.ylabel('velocity (m/s)');\nplt.ylim(0,)\nplt.grid();\n```\n\n- In reality, the impulses and responses will be infinitesimal\n- In this case, the summation in the time domain is a convolution $u(t) \\circledast H(t)$, where $H(t)$ is the impulse response of the system. \n- This can be a difficult integration..\n\n- If we use the Laplace transform and move to the s-domain however, convolutions become multiplications, and things are simpler.\n\n- As a control engineers we would like to design our systems to be as close as possible to be LTI, or so that the non LTI parts can be ignored.\n- So that the standard LTI tools can be used.\n\n------------------------------\n", "meta": {"hexsha": "c1002f79a667572156a09251f1278ba11b6de760", "size": 154806, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "02_Intro_to_control_theory.ipynb", "max_stars_repo_name": "andreamunafo/automatic_control", "max_stars_repo_head_hexsha": "dd1d89f732bfd8d95b0ebef6fe99df29b18a1fc2", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "02_Intro_to_control_theory.ipynb", "max_issues_repo_name": "andreamunafo/automatic_control", "max_issues_repo_head_hexsha": "dd1d89f732bfd8d95b0ebef6fe99df29b18a1fc2", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "02_Intro_to_control_theory.ipynb", "max_forks_repo_name": "andreamunafo/automatic_control", "max_forks_repo_head_hexsha": "dd1d89f732bfd8d95b0ebef6fe99df29b18a1fc2", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 80.5442247659, "max_line_length": 16588, "alphanum_fraction": 0.8180626074, "converted": true, "num_tokens": 6667, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665855647394, "lm_q2_score": 0.7057850216484837, "lm_q1q2_score": 0.4182951989231425}} {"text": "```python\n%matplotlib inline\n```\n\n\nDCGAN \ud29c\ud1a0\ub9ac\uc5bc\n==============\n\n**\uc800\uc790**: `Nathan Inkawhich `_\n **\ubc88\uc5ed**: `\uc870\ubbfc\uc131 `_\n\n\n\uac1c\uc694\n----\n\n\ubcf8 \ud29c\ud1a0\ub9ac\uc5bc\uc5d0\uc11c\ub294 \uc608\uc81c\ub97c \ud1b5\ud574 DCGAN\uc744 \uc54c\uc544\ubcf4\uaca0\uc2b5\ub2c8\ub2e4. \uc6b0\ub9ac\ub294 \uc2e4\uc81c \uc720\uba85\uc778\ub4e4\uc758 \uc0ac\uc9c4\ub4e4\ub85c \uc801\ub300\uc801 \uc0dd\uc131 \uc2e0\uacbd\ub9dd(GAN)\uc744 \ud559\uc2b5\uc2dc\ucf1c,\n\uc0c8\ub85c\uc6b4 \uc720\uba85\uc778\uc758 \uc0ac\uc9c4\uc744 \ub9cc\ub4e4\uc5b4\ubcfc\uac81\ub2c8\ub2e4.\n\uc0ac\uc6a9\ud560 \ub300\ubd80\ubd84\uc758 \ucf54\ub4dc\ub294 `pytorch/examples `__ \uc758 DCGAN \uad6c\ud604\uc5d0\uc11c \uac00\uc838\uc654\uc73c\uba70,\n\ubcf8 \ubb38\uc11c\ub294 \uad6c\ud604\uc5d0 \ub300\ud55c \uc124\uba85\uacfc \ud568\uaed8, \uc5b4\uc9f8\uc11c \uc774 \ubaa8\ub378\uc774 \uc791\ub3d9\ud558\ub294\uc9c0\uc5d0 \ub300\ud574 \uc124\uba85\uc744 \ud574\uc904 \uac83\uc785\ub2c8\ub2e4.\n\ucc98\uc74c \uc77d\uc5c8\uc744\ub54c\ub294, \uc2e4\uc81c\ub85c \ubaa8\ub378\uc5d0 \ubb34\uc2a8\uc77c\uc774 \uc77c\uc5b4\ub098\uace0 \uc788\ub294\uc9c0\uc5d0 \ub300\ud574 \uc774\ud574\ud558\ub294 \uac83\uc774 \uc870\uae08 \uc2dc\uac04\uc744 \uc18c\uc694\ud560 \uc218 \uc788\uc73c\ub098,\n\uadf8\ub798\ub3c4 GAN\uc5d0 \ub300\ud55c \uc0ac\uc804\uc9c0\uc2dd\uc774 \ud544\uc694\ud558\uc9c0\ub294 \uc54a\uc73c\ub2c8 \uac71\uc815\ud558\uc9c0 \uc54a\uc73c\uc154\ub3c4 \ub429\ub2c8\ub2e4.\n\ucd94\uac00\ub85c, GPU 1-2\uac1c\ub97c \uc0ac\uc6a9\ud558\ub294 \uac83\uc774 \uc2dc\uac04\uc808\uc57d\uc5d0 \ub3c4\uc6c0\uc774 \ub420\uac81\ub2c8\ub2e4. \uadf8\ub7fc \ucc98\uc74c\ubd80\ud130 \ucc9c\ucc9c\ud788 \uc2dc\uc791\ud574\ubd05\uc2dc\ub2e4!\n\n\uc801\ub300\uc801 \uc0dd\uc131 \uc2e0\uacbd\ub9dd(Generative Adversarial Networks)\n----------------------------------------------------\n\n\uadf8\ub798\uc11c GAN\uc774 \ubb58\uae4c\uc694?\n~~~~~~~~~~~~~~~~~~~~~\n\nGAN\uc774\ub780 \ud559\uc2b5 \ub370\uc774\ud130\ub4e4\uc758 \ubd84\ud3ec\ub97c \ud559\uc2b5\ud574, \uac19\uc740 \ubd84\ud3ec\uc5d0\uc11c \uc0c8\ub85c\uc6b4 \ub370\uc774\ud130\ub97c \uc0dd\uc131\ud560 \uc218 \uc788\ub3c4\ub85d DL \ubaa8\ub378\uc744 \ud559\uc2b5\uc2dc\ud0a4\ub294 \ud504\ub808\uc784\uc6cc\ud06c\uc785\ub2c8\ub2e4.\n2014\ub144 Ian Goodfellow\uac00 \uac1c\ubc1c\ud588\uc73c\uba70, `Generative Adversarial\nNets `__ \ub17c\ubb38\uc5d0\uc11c \ucc98\uc74c \uc18c\uac1c\ub418\uc5c8\uc2b5\ub2c8\ub2e4.\nGAN\uc740 *\uc0dd\uc131\uc790* \uc640 *\uad6c\ubd84\uc790* \ub85c \uad6c\ubcc4\ub418\ub294 \ub450\uac00\uc9c0 \ubaa8\ub378\uc744 \uac00\uc9c0\uace0 \uc788\ub294\uac83\uc774 \ud2b9\uc9d5\uc785\ub2c8\ub2e4.\n\uc0dd\uc131\uc790\uc758 \uc5ed\ud560\uc740 \uc2e4\uc81c \uc774\ubbf8\uc9c0\ub85c \ucc29\uac01\ub418\ub3c4\ub85d \uc815\uad50\ud55c \uc774\ubbf8\uc9c0\ub97c \ub9cc\ub4dc\ub294 \uac83\uc774\uace0,\n\uad6c\ubd84\uc790\uc758 \uc5ed\ud560\uc740 \uc774\ubbf8\uc9c0\ub97c \ubcf4\uace0 \uc0dd\uc131\uc790\uc5d0 \uc758\ud574 \ub9cc\ub4e4\uc5b4\uc9c4 \uc774\ubbf8\uc9c0\uc778\uc9c0 \uc2e4\uc81c \uc774\ubbf8\uc9c0\uc778\uc9c0 \uc54c\uc544\ub0b4\ub294 \uac83\uc785\ub2c8\ub2e4.\n\ubaa8\ub378\uc744 \ud559\uc2b5\ud558\ub294 \ub3d9\uc548, \uc0dd\uc131\uc790\ub294 \ub354 \uc9c4\uc9dc\uac19\uc740 \uac00\uc9dc \uc774\ubbf8\uc9c0\ub97c \ub9cc\ub4e4\uc5b4\ub0b4\uba70 \uad6c\ubd84\uc790\ub97c \uc18d\uc774\ub824 \ud558\uace0,\n\uad6c\ubd84\uc790\ub294 \ub354 \uc815\ud655\ud788 \uac00\uc9dc/\uc9c4\uc9dc \uc774\ubbf8\uc9c0\ub97c \uad6c\ubcc4\ud560 \uc218 \uc788\ub3c4\ub85d \ub178\ub825\ud569\ub2c8\ub2e4.\n\uc774 \u2018\uacbd\ucc30\uacfc \ub3c4\ub451\u2019 \uac8c\uc784\uc740, \uc0dd\uc131\uc790\uac00 \ud559\uc2b5 \ub370\uc774\ud130\ub4e4\uc5d0\uc11c \uc9c1\uc811 \uac00\uc838\uc628 \uac83\ucc98\ub7fc \ubcf4\uc77c\uc815\ub3c4\ub85c \uc644\ubcbd\ud55c \uc774\ubbf8\uc9c0\ub97c \ub9cc\ub4e4\uc5b4\ub0b4\uace0,\n\uad6c\ubd84\uc790\uac00 \uc0dd\uc131\uc790\uc5d0\uc11c \ub098\uc628 \uc774\ubbf8\uc9c0\ub97c 50%\uc758 \ud655\ub960\ub85c \uac00\uc9dc \ud639\uc740 \uc9c4\uc9dc\ub85c \ud310\ubcc4\ud560 \ub54c, \uade0\ud615\uc0c1\ud0dc\uc5d0 \ub3c4\ub2ec\ud558\uac8c \ub429\ub2c8\ub2e4.\n\n\uadf8\ub7fc \uc774\uc81c\ubd80\ud130 \ubcf8 \ud29c\ud1a0\ub9ac\uc5bc\uc5d0\uc11c \uc0ac\uc6a9\ud560 \ud45c\uae30\ub4e4\uc744 \uad6c\ubd84\uc790\ubd80\ud130 \uc815\uc758\ud574\ubcf4\uaca0\uc2b5\ub2c8\ub2e4. $x$ \ub294 \uc774\ubbf8\uc9c0\ub85c \ud45c\ud604\ub418\ub294 \ub370\uc774\ud130\ub85c \ub450\uaca0\uc2b5\ub2c8\ub2e4.\n$D(x)$ \ub294 \uad6c\ubd84\uc790 \uc2e0\uacbd\ub9dd\uc774\uace0, \uc2e4\uc81c \ud559\uc2b5\ub370\uc774\ud130\uc5d0\uc11c \uac00\uc838\uc628 $x$ \ub97c \ud1b5\uacfc\uc2dc\ucf1c \uc0c1\uc218(scalar) \ud655\ub960\uac12\uc744 \uacb0\uacfc\ub85c \ucd9c\ub824\ud569\ub2c8\ub2e4.\n\uc774\ub54c, \uc6b0\ub9ac\ub294 \uc774\ubbf8\uc9c0 \ub370\uc774\ud130\ub97c \ub2e4\ub8e8\uace0 \uc788\uc73c\ubbc0\ub85c, $D(x)$ \uc5d0\ub294 3x64x64\ud06c\uae30\uc758 CHW \ub370\uc774\ud130\uac00 \uc785\ub825\ub429\ub2c8\ub2e4. \uc9c1\uad00\uc801\uc73c\ub85c \ubcfc\ub54c,\n$D(x)$ \ub294 $x$ \uac00 \ud559\uc2b5\ub370\uc774\ud130\uc5d0\uc11c \uac00\uc838\uc628 \uac83\uc77c \ub54c \ucd9c\ub825\uc774 \ud06c\uace0, \uc0dd\uc131\uc790\uac00 \ub9cc\ub4e4\uc5b4\ub0b8 $x$ \uc77c\ub54c \uc791\uc744 \uac83\uc785\ub2c8\ub2e4.\n$D(x)$ \ub294 \uc804\ud1b5\uc801\uc778 \uc774\uc9c4 \ubd84\ub958\uae30(binary classification)\uc73c\ub85c\ub3c4 \uc0dd\uac01\ub420 \uc218 \uc788\uc2b5\ub2c8\ub2e4.\n\n\uc774\ubc88\uc5d4 \uc0dd\uc131\uc790\uc758 \ud45c\uae30\ub4e4\uc744 \ud655\uc778\ud574\ubd05\uc2dc\ub2e4. $z$ \ub97c \uc815\uaddc\ubd84\ud3ec\uc5d0\uc11c \ubf51\uc740 \uc7a0\uc7ac\uacf5\uac04 \ubca1\ud130(laten space vector)\ub77c\uace0 \ud558\uaca0\uc2b5\ub2c8\ub2e4\n(\ubc88\uc5ed \uc8fc. laten space vector\ub294 \uc27d\uac8c \uc0dd\uac01\ud574 \uc815\uaddc\ubd84\ud3ec\ub97c \ub530\ub974\ub294 n\uac1c\uc758 \uc6d0\uc18c\ub97c \uac00\uc9c4 vector\ub77c \ubcfc \uc218 \uc788\uc2b5\ub2c8\ub2e4.\n\ub2e4\ub974\uac8c \uc598\uae30\ud558\uba74 \uc815\uaddc\ubd84\ud3ec\uc5d0\uc11c n\uac1c\uc758 \uc6d0\uc18c\ub97c \ucd94\ucd9c\ud55c \uac83\uacfc \uac19\uc2b5\ub2c8\ub2e4). $G(z)$ \ub294 $z$\n\ubca1\ud130\ub97c \uc6d0\ud558\ub294 \ub370\uc774\ud130 \ucc28\uc6d0\uc73c\ub85c \ub300\uc751\uc2dc\ud0a4\ub294 \uc2e0\uacbd\ub9dd\uc73c\ub85c \ub458 \uc218 \uc788\uc2b5\ub2c8\ub2e4. \uc774\ub54c $G$ \uc758 \ubaa9\uc801\uc740 $p_{data}$\n\uc5d0\uc11c \uc5bb\uc744 \uc218 \uc788\ub294 \ud559\uc2b5 \ub370\uc774\ud130\ub4e4\uc758 \ubd84\ud3ec\ub97c \ucd94\uc815\ud558\uc5ec, \ubaa8\uc0ac\ud55c $p_g$ \uc758 \ubd84\ud3ec\ub97c \uc774\uc6a9\ud574 \uac00\uc9dc \ub370\uc774\ud130\ub4e4\uc744 \ub9cc\ub4dc\ub294 \uac83\uc785\ub2c8\ub2e4.\n\n\uc774\uc5b4\uc11c, $D(G(z))$ \ub294 $G$ \uac00 \ucd9c\ub825\ud55c \uacb0\uacfc\ubb3c\uc774 \uc2e4\uc81c \uc774\ubbf8\uc9c0\uc77c 0~1\uc0ac\uc774\uc758 \uc0c1\uc218\uc758 \ud655\ub960\uac12\uc785\ub2c8\ub2e4.\n`Goodfellow\uc758 \ub17c\ubb38 `__\n\uc5d0 \uae30\uc220\ub418\uc5b4 \uc788\ub4ef, $D$ \uac00 \uc774\ubbf8\uc9c0\uc758 \ucc38/\uac70\uc9d3\uc744 \uc815\ud655\ud788 \ud310\ubcc4\ud560 \ud655\ub960\uc778 $logD(x)$\ub97c \ucd5c\ub300\ud654 \uc2dc\ud0a4\uace0,\n$G$ \uc5d0\uc11c \uc0dd\uc131\ud55c \uc774\ubbf8\uc9c0\ub97c $D$ \uac00 \uac00\uc9dc\ub85c \ud310\ubcc4\ud560 \ud655\ub960\uc778\n($log(1-D(G(z)))$)\ub97c \ucd5c\uc18c\ud654 \uc2dc\ud0a4\ub824\ub294 \uc810\uc5d0\uc11c, $D$ \uc640 $G$ \ub294 \ucd5c\ub300\ucd5c\uc18c(minmax)\uac8c\uc784\uc744 \ud558\ub294 \uac83\uacfc \uac19\uc2b5\ub2c8\ub2e4.\n\ub17c\ubb38\uc5d0 \ub530\ub974\uba74, GAN\uc758 \uc190\uc2e4\ud568\uc218\ub294 \uc544\ub798\uc640 \uac19\uc2b5\ub2c8\ub2e4.\n\n\\begin{align}\\underset{G}{\\text{min}} \\underset{D}{\\text{max}}V(D,G) = \\mathbb{E}_{x\\sim p_{data}(x)}\\big[logD(x)\\big] + \\mathbb{E}_{z\\sim p_{z}(z)}\\big[log(1-D(G(z)))\\big]\\end{align}\n\n\uc774\ub860\uc801\uc73c\ub85c\ub294, \uc774 \ucd5c\ub300\ucd5c\uc18c\uac8c\uc784\uc740 $p_g = p_{data}$ \uc774\uace0, \uad6c\ubd84\uc790\uc5d0 \uc785\ub825\ub41c \ub370\uc774\ud130\uac00 1/2\uc758 \ubb34\uc791\uc704 \ud655\ub960\ub85c \ucc38/\uac70\uc9d3\uc774 \ud310\ubcc4\ub420\ub54c \ud574\ub2f5\uc5d0 \uc774\ub985\ub2c8\ub2e4.\n\ud558\uc9c0\ub9cc GAN\uc758 \uc218\ub834 \uc774\ub860\uc740 \uc544\uc9c1\ub3c4 \ud65c\ubc1c\ud788 \uc5f0\uad6c\uac00 \uc9c4\ud589\uc911\uc774\uace0, \ud604\uc2e4\uc5d0\uc11c\uc758 \ubaa8\ub378\ub4e4\uc740 \uc774\ub860\uc801\uc778 \ucd5c\uc801 \uc0c1\ud0dc\uc5d0 \ub3c4\ub2ec\ud558\uc9c0 \uc54a\ub294 \uacbd\uc6b0\ub3c4 \ub9ce\uc2b5\ub2c8\ub2e4.\n\n\uadf8\ub807\ub2e4\uba74 DCGAN\uc740 \ubb58\uae4c\uc694?\n~~~~~~~~~~~~~~~~~~~~~~~~\n\nDCGAN\uc740 \uc704\uc5d0\uc11c \uae30\uc220\ud55c GAN\uc5d0\uc11c \uc9c1\uc811\uc801\uc73c\ub85c \ud30c\uc0dd\ub41c \ubaa8\ub378\ub85c, \uc0dd\uc131\uc790\uc640 \uad6c\ubd84\uc790\uc5d0\uc11c\n\ud569\uc131\uacf1 \uc2e0\uacbd\ub9dd(convolution)\uacfc \uc804\uce58 \ud569\uc131\uacf1 \uc2e0\uacbd\ub9dd(convolution-transpose)\uc744 \uc0ac\uc6a9\ud588\ub2e4\ub294 \uac83\uc774 \ucc28\uc774\uc810\uc785\ub2c8\ub2e4\nRadford\uc640 \uadf8 \uc678\uac00 \uc800\uc220\ud55c `Unsupervised Representation Learning With\nDeep Convolutional Generative Adversarial\nNetworks `__ \ub17c\ubb38\uc5d0\uc11c \ucc98\uc74c \ubaa8\ub378\uc774 \uc18c\uac1c\ub418\uc5c8\uace0, \uc9c0\uae08\uc740 \ub300\ubd80\ubd84\uc758 GAN\ubaa8\ub378\uc774\nDCGAN\uc744 \uae30\ubc18\uc73c\ub85c \ub9cc\ub4e4\uc5b4\uc9c0\ub294 \uc911\uc785\ub2c8\ub2e4. \uc774\uc804 GAN\uacfc \ubaa8\ub378\uc758 \uad6c\uc870\uac00 \uc2e4\uc81c\ub85c \uc5b4\ub5bb\uac8c \ub2e4\ub978\uc9c0 \ud655\uc778\uc744 \ud574\ubcf4\uc790\uba74, \uba3c\uc800 \uad6c\ubd84\uc790\uc5d0\uc11c\ub294\n`convolution `__\n\uacc4\uce35, `batch\nnorm `__\n\uacc4\uce35, \uadf8\ub9ac\uace0\n`LeakyReLU `__\n\ud65c\uc131\ud568\uc218\uac00 \uc0ac\uc6a9\ub418\uc5c8\uc2b5\ub2c8\ub2e4. \ud074\ub798\uc2dd\ud55c GAN\uacfc \ub9c8\ucc2c\uac00\uc9c0\ub85c, \uad6c\ubd84\uc790\uc758 \uc785\ub825 \ub370\uc774\ud130\ub294 3x64x64 \uc758 \uc774\ubbf8\uc9c0\uc774\uace0,\n\ucd9c\ub825\uac12\uc740 \uc785\ub825 \ub370\uc774\ud130\uac00 \uc2e4\uc81c \ub370\uc774\ud130\uc77c 0~1\uc0ac\uc774\uc758 \ud655\ub960\uac12\uc785\ub2c8\ub2e4.\n\ub2e4\uc74c\uc73c\ub85c, \uc0dd\uc131\uc790\ub294\n`convolutional-transpose `__\n\uacc4\uce35, \ubc30\uce58 \uc815\uaddc\ud654(batch norm) \uacc4\uce35, \uadf8\ub9ac\uace0\n`ReLU `__ \ud65c\uc131\ud568\uc218\uac00 \uc0ac\uc6a9\ub418\uc5c8\uc2b5\ub2c8\ub2e4. \uc785\ub825\uac12\uc740 \uc5ed\uc2dc\ub098\n\uc815\uaddc\ubd84\ud3ec\uc5d0\uc11c \ucd94\ucd9c\ud55c \uc7a0\uc7ac\uacf5\uac04 \ubca1\ud130 $z$ \uc774\uace0, \ucd9c\ub825\uac12\uc740 3x64x64 RGB \uc774\ubbf8\uc9c0\uc785\ub2c8\ub2e4. \uc774\ub54c,\n\uc804\uce58 \ud569\uc131\uacf1 \uc2e0\uacbd\ub9dd\uc740 \uc7a0\uc7ac\uacf5\uac04 \ubca1\ud130\ub85c \ud558\uc5ec\uae08 \uc774\ubbf8\uc9c0\uc640 \uac19\uc740 \ucc28\uc6d0\uc744 \uac16\ub3c4\ub85d \ubcc0\ud658\uc2dc\ucf1c\uc8fc\ub294 \uc5ed\ud560\uc744 \ud569\ub2c8\ub2e4 (\ubc88\uc5ed \uc8fc. \uc804\uce58 \ud569\uc131\uacf1 \uc2e0\uacbd\ub9dd\uc740\n\ud569\uc131\uacf1 \uc2e0\uacbd\ub9dd\uc758 \ubc18\ub300\uc801\uc778 \uac1c\ub150\uc774\ub77c \uc774\ud574\ud558\uba74 \uc27d\uc2b5\ub2c8\ub2e4. \uc785\ub825\ub41c \uc791\uc740 CHW \ub370\uc774\ud130\ub97c \uac00\uc911\uce58\ub4e4\uc744 \uc774\uc6a9\ud574 \ub354 \ud070 CHW\ub85c \uc5c5\uc0d8\ud50c\ub9c1\ud574\uc8fc\ub294 \uacc4\uce35\uc785\ub2c8\ub2e4).\n\ub17c\ubb38\uc5d0\uc11c\ub294 \uac01\uc885 \ucd5c\uc801\ud654 \ubc29\ubc95\uc774\ub098 \uc190\uc2e4\ud568\uc218\uc758 \uacc4\uc0b0, \ubaa8\ub378\uc758 \uac00\uc911\uce58 \ucd08\uae30\ud654 \ubc29\ubc95\ub4f1\uc5d0 \uad00\ud55c \ucd94\uac00\uc801\uc778 \uc815\ubcf4\ub4e4\ub3c4 \uc801\uc5b4\ub450\uc5c8\ub294\ub370,\n\uc774 \ubd80\ubd84\uc740 \ub2e4\uc74c \uc139\uc158\uc5d0\uc11c \uc124\uba85\ud558\ub3c4\ub85d \ud558\uaca0\uc2b5\ub2c8\ub2e4.\n\n\n\n\n```python\nfrom __future__ import print_function\n#%matplotlib inline\nimport argparse\nimport os\nimport random\nimport torch\nimport torch.nn as nn\nimport torch.nn.parallel\nimport torch.backends.cudnn as cudnn\nimport torch.optim as optim\nimport torch.utils.data\nimport torchvision.datasets as dset\nimport torchvision.transforms as transforms\nimport torchvision.utils as vutils\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport matplotlib.animation as animation\nfrom IPython.display import HTML\n\n# \ucf54\ub4dc \uc2e4\ud589\uacb0\uacfc\uc758 \ub3d9\uc77c\uc131\uc744 \uc704\ud574 \ubb34\uc791\uc704 \uc2dc\ub4dc\ub97c \uc124\uc815\ud569\ub2c8\ub2e4\nmanualSeed = 999\n#manualSeed = random.randint(1, 10000) # \ub9cc\uc77c \uc0c8\ub85c\uc6b4 \uacb0\uacfc\ub97c \uc6d0\ud55c\ub2e4\uba74 \uc8fc\uc11d\uc744 \uc5c6\uc560\uba74 \ub429\ub2c8\ub2e4\nprint(\"Random Seed: \", manualSeed)\nrandom.seed(manualSeed)\ntorch.manual_seed(manualSeed)\n```\n\n\uc124\uc815\uac12\n------\n\n\uba87 \uac00\uc9c0 \uc124\uc815\uac12\ub4e4\uc744 \uc815\uc758\ud574\ubd05\uc2dc\ub2e4:\n\n- **dataroot** - \ub370\uc774\ud130\uc14b \ud3f4\ub354\uc758 \uacbd\ub85c\uc785\ub2c8\ub2e4. \ub370\uc774\ud130\uc14b\uc5d0 \uad00\ud55c\uac74 \ub2e4\uc74c \uc139\uc158\uc5d0\uc11c\n \ub354 \uc790\uc138\ud788 \uc124\uba85\ud558\uaca0\uc2b5\ub2c8\ub2e4.\n- **workers** - DataLoader\uc5d0\uc11c \ub370\uc774\ud130\ub97c \ubd88\ub7ec\uc62c \ub54c \uc0ac\uc6a9\ud560 \uc4f0\ub808\ub4dc\uc758 \uac1c\uc218\uc785\ub2c8\ub2e4.\n- **batch_size** - \ud559\uc2b5\uc5d0 \uc0ac\uc6a9\ud560 \ubc30\uce58 \ud06c\uae30\uc785\ub2c8\ub2e4. DCGAN\uc5d0\uc11c\ub294 128\uc744 \uc0ac\uc6a9\ud588\uc2b5\ub2c8\ub2e4.\n- **image_size** - \ud559\uc2b5\uc5d0 \uc0ac\uc6a9\ub418\ub294 \uc774\ubbf8\uc9c0\uc758 \ud06c\uae30\uc785\ub2c8\ub2e4.\n \ubcf8 \ubb38\uc11c\uc5d0\uc11c\ub294 64x64\uc758 \ud06c\uae30\ub97c \uae30\ubcf8\uc73c\ub85c \ud558\ub098, \ub9cc\uc77c \ub2e4\ub978 \ud06c\uae30\uc758 \uc774\ubbf8\uc9c0\ub97c \uc0ac\uc6a9\ud55c\ub2e4\uba74\n D\uc640 G\uc758 \uad6c\uc870 \uc5ed\uc2dc \ubcc0\uacbd\ub418\uc5b4\uc57c \ud569\ub2c8\ub2e4. \ub354 \uc790\uc138\ud55c \uc815\ubcf4\ub97c \uc704\ud574\uc120\n `\uc774\uacf3 `__ \uc744 \ud655\uc778\ud574 \ubcf4\uc138\uc694.\n- **nc** - \uc785\ub825 \uc774\ubbf8\uc9c0\uc758 \uc0c9 \ucc44\ub110\uac1c\uc218\uc785\ub2c8\ub2e4. RGB \uc774\ubbf8\uc9c0\uc774\uae30 \ub54c\ubb38\uc5d0 3\uc73c\ub85c \uc124\uc815\ud569\ub2c8\ub2e4.\n- **nz** - \uc7a0\uc7ac\uacf5\uac04 \ubca1\ud130\uc758 \uc6d0\uc18c\ub4e4 \uac1c\uc218\uc785\ub2c8\ub2e4.\n- **ngf** - \uc0dd\uc131\uc790\ub97c \ud1b5\uacfc\ud560\ub54c \ub9cc\ub4e4\uc5b4\uc9c8 \ud2b9\uc9d5 \ub370\uc774\ud130\uc758 \ucc44\ub110\uac1c\uc218\uc785\ub2c8\ub2e4.\n- **ndf** - \uad6c\ubd84\uc790\ub97c \ud1b5\uacfc\ud560\ub54c \ub9cc\ub4e4\uc5b4\uc9c8 \ud2b9\uc9d5 \ub370\uc774\ud130\uc758 \ucc44\ub110\uac1c\uc218\uc785\ub2c8\ub2e4.\n- **num_epochs** - \ud559\uc2b5\uc2dc\ud0ac \uc5d0\ud3ed \uc218\uc785\ub2c8\ub2e4. \uc624\ub798 \ud559\uc2b5\uc2dc\ud0a4\ub294 \uac83\uc774 \ub300\ubd80\ubd84 \uc88b\uc740 \uacb0\uacfc\ub97c \ubcf4\uc774\uc9c0\ub9cc, \ub2f9\uc5f0\ud788\ub3c4 \uc2dc\uac04\uc774 \uc624\ub798\uac78\ub9ac\ub294 \uac83\uc774 \ub2e8\uc810\uc785\ub2c8\ub2e4.\n- **lr** - \ubaa8\ub378\uc758 \ud559\uc2b5\ub960\uc785\ub2c8\ub2e4. DCGAN\uc5d0\uc11c \uc0ac\uc6a9\ub41c\ub300\ub85c 0.0002\ub85c \uc124\uc815\ud569\ub2c8\ub2e4.\n- **beta1** - Adam \uc635\ud2f0\ub9c8\uc774\uc800\uc5d0\uc11c \uc0ac\uc6a9\ud560 beta1 \ud558\uc774\ud37c\ud30c\ub77c\ubbf8\ud130 \uac12\uc785\ub2c8\ub2e4. \uc5ed\uc2dc\ub098 \ub17c\ubb38\uc5d0\uc11c \uc0ac\uc6a9\ud55c\ub300\ub85c 0.5\ub85c \uc124\uc815\ud588\uc2b5\ub2c8\ub2e4.\n- **ngpu** - \uc0ac\uc6a9\uac00\ub2a5\ud55c GPU\uc758 \ubc88\ud638\uc785\ub2c8\ub2e4. 0\uc73c\ub85c \ub450\uba74 CPU\uc5d0\uc11c \ud559\uc2b5\ud558\uace0, 0\ubcf4\ub2e4 \ud070 \uc218\ub85c \uc124\uc815\ud558\uba74 \uac01 \uc22b\uc790\uac00 \uac00\ub9ac\ud0a4\ub294 GPU\ub85c \ud559\uc2b5\uc2dc\ud0b5\ub2c8\ub2e4.\n\n\n\n\n\n```python\n# \ub370\uc774\ud130\uc14b\uc758 \uacbd\ub85c\ndataroot = \"data/celeba\"\n\n# dataloader\uc5d0\uc11c \uc0ac\uc6a9\ud560 \uc4f0\ub808\ub4dc \uc218\nworkers = 2\n\n# \ubc30\uce58 \ud06c\uae30\nbatch_size = 128\n\n# \uc774\ubbf8\uc9c0\uc758 \ud06c\uae30\uc785\ub2c8\ub2e4. \ubaa8\ub4e0 \uc774\ubbf8\uc9c0\ub4e4\uc740 transformer\ub97c \uc774\uc6a9\ud574 64\ub85c \ud06c\uae30\uac00 \ud1b5\uc77c\ub429\ub2c8\ub2e4.\nimage_size = 64\n\n# \uc774\ubbf8\uc9c0\uc758 \ucc44\ub110 \uc218\ub85c, RGB \uc774\ubbf8\uc9c0\uc774\uae30 \ub54c\ubb38\uc5d0 3\uc73c\ub85c \uc124\uc815\ud569\ub2c8\ub2e4.\nnc = 3\n\n# \uc7a0\uc7ac\uacf5\uac04 \ubca1\ud130\uc758 \ud06c\uae30 (\uc608. \uc0dd\uc131\uc790\uc758 \uc785\ub825\uac12 \ud06c\uae30)\nnz = 100\n\n# \uc0dd\uc131\uc790\ub97c \ud1b5\uacfc\ud558\ub294 \ud2b9\uc9d5 \ub370\uc774\ud130\ub4e4\uc758 \ucc44\ub110 \ud06c\uae30\nngf = 64\n\n# \uad6c\ubd84\uc790\ub97c \ud1b5\uacfc\ud558\ub294 \ud2b9\uc9d5 \ub370\uc774\ud130\ub4e4\uc758 \ucc44\ub110 \ud06c\uae30\nndf = 64\n\n# \ud559\uc2b5\ud560 \uc5d0\ud3ed \uc218\nnum_epochs = 5\n\n# \uc635\ud2f0\ub9c8\uc774\uc800\uc758 \ud559\uc2b5\ub960\nlr = 0.0002\n\n# Adam \uc635\ud2f0\ub9c8\uc774\uc800\uc758 beta1 \ud558\uc774\ud37c\ud30c\ub77c\ubbf8\ud130\nbeta1 = 0.5\n\n# \uc0ac\uc6a9\uac00\ub2a5\ud55c gpu \ubc88\ud638. CPU\ub97c \uc0ac\uc6a9\ud574\uc57c \ud558\ub294\uacbd\uc6b0 0\uc73c\ub85c \uc124\uc815\ud558\uc138\uc694\nngpu = 1\n```\n\n\ub370\uc774\ud130\n------\n\n\ubcf8 \ud29c\ud1a0\ub9ac\uc5bc\uc5d0\uc11c \uc0ac\uc6a9\ud560 \ub370\uc774\ud130\ub294 `Celeb-A Faces\ndataset `__ \ub85c, \ud574\ub2f9 \ub9c1\ud06c\ub97c \uc774\uc6a9\ud558\uac70\ub098 `Google\nDrive `__ \uc5d0\uc11c \ub370\uc774\ud130\ub97c \ubc1b\uc744 \uc218 \uc788\uc2b5\ub2c8\ub2e4.\n\ub370\uc774\ud130\ub97c \ubc1b\uc73c\uba74 *img_align_celeba.zip* \ub77c\ub294 \ud30c\uc77c\uc744 \ubcf4\uac8c\ub420 \uac81\ub2c8\ub2e4. \ub2e4\uc6b4\ub85c\ub4dc\uac00 \ub05d\ub098\uba74\n*celeba* \uc774\ub77c\ub294 \ud3f4\ub354\ub97c \uc0c8\ub85c \ub9cc\ub4e4\uace0, \ud574\ub2f9 \ud3f4\ub354\uc5d0 \ud574\ub2f9 zip \ud30c\uc77c\uc744 \uc555\ucd95\ud574\uc81c \ud574\uc8fc\uc2dc\uba74 \ub429\ub2c8\ub2e4.\n\uc555\ucd95 \ud574\uc81c \ud6c4, \uc704\uc5d0\uc11c \uc815\uc758\ud55c *dataroot* \ubcc0\uc218\uc5d0 \ubc29\uae08 \ub9cc\ub4e0 *celeba* \ud3f4\ub354\uc758 \uacbd\ub85c\ub97c \ub123\uc5b4\uc8fc\uc138\uc694.\n\uc704\uc758 \uc791\uc5c5\uc774 \ub05d\ub098\uba74 *celeba* \ud3f4\ub354\uc758 \uad6c\uc870\ub294 \ub2e4\uc74c\uacfc \uac19\uc544\uc57c \ud569\ub2c8\ub2e4:\n\n::\n\n /path/to/celeba\n -> img_align_celeba \n -> 188242.jpg\n -> 173822.jpg\n -> 284702.jpg\n -> 537394.jpg\n ...\n\n\uc774 \uacfc\uc815\ub4e4\uc740 \ud504\ub85c\uadf8\ub7a8\uc774 \uc815\uc0c1\uc801\uc73c\ub85c \uad6c\ub3d9\ud558\uae30 \uc704\ud574\uc11c\ub294 \uc911\uc694\ud55c \ubd80\ubd84\uc785\ub2c8\ub2e4. \uc774\ub54c celeba \ud3f4\ub354\uc548\uc5d0 \ub2e4\uc2dc \ud3f4\ub354\ub97c \ub450\ub294 \uc774\uc720\ub294,\nImageFolder \ud074\ub798\uc2a4\uac00 \ub370\uc774\ud130\uc14b\uc758 \ucd5c\uc0c1\uc704 \ud3f4\ub354\uc5d0 \uc11c\ube0c\ud3f4\ub354\ub97c \uc694\uad6c\ud558\uae30 \ub54c\ubb38\uc785\ub2c8\ub2e4.\n\uc774\uc81c \ub370\uc774\ud130\uc14b\uacfc DataLoader\uc758 \uc124\uc815\uc744 \ub05d\ub0c8\uc2b5\ub2c8\ub2e4.\n\ucd5c\uc885\uc801\uc73c\ub85c \ud559\uc2b5 \ub370\uc774\ud130\ub4e4\uc744 \uc2dc\uac01\ud654\ud574\ubd05\uc2dc\ub2e4.\n\n\n\n\n\n```python\n# \uc6b0\ub9ac\uac00 \uc124\uc815\ud55c \ub300\ub85c \uc774\ubbf8\uc9c0 \ub370\uc774\ud130\uc14b\uc744 \ubd88\ub7ec\uc640 \ubd05\uc2dc\ub2e4\n# \uba3c\uc800 \ub370\uc774\ud130\uc14b\uc744 \ub9cc\ub4ed\ub2c8\ub2e4\ndataset = dset.ImageFolder(root=dataroot,\n transform=transforms.Compose([\n transforms.Resize(image_size),\n transforms.CenterCrop(image_size),\n transforms.ToTensor(),\n transforms.Normalize((0.5, 0.5, 0.5), (0.5, 0.5, 0.5)),\n ]))\n# dataloader\ub97c \uc815\uc758\ud574\ubd05\uc2dc\ub2e4\ndataloader = torch.utils.data.DataLoader(dataset, batch_size=batch_size,\n shuffle=True, num_workers=workers)\n\n# GPU \uc0ac\uc6a9\uc5ec\ubd80\ub97c \uacb0\uc815\ud574 \uc90d\ub2c8\ub2e4\ndevice = torch.device(\"cuda:0\" if (torch.cuda.is_available() and ngpu > 0) else \"cpu\")\n\n# \ud559\uc2b5 \ub370\uc774\ud130\ub4e4 \uc911 \uba87\uac00\uc9c0 \uc774\ubbf8\uc9c0\ub4e4\uc744 \ud654\uba74\uc5d0 \ub744\uc6cc\ubd05\uc2dc\ub2e4\nreal_batch = next(iter(dataloader))\nplt.figure(figsize=(8,8))\nplt.axis(\"off\")\nplt.title(\"Training Images\")\nplt.imshow(np.transpose(vutils.make_grid(real_batch[0].to(device)[:64], padding=2, normalize=True).cpu(),(1,2,0)))\n```\n\n\uad6c\ud604\n----\n\n\ubaa8\ub378\uc758 \uc124\uc815\uac12\ub4e4\uacfc \ub370\uc774\ud130\ub4e4\uc774 \uc900\ube44\ub418\uc5c8\uae30 \ub54c\ubb38\uc5d0, \ub4dc\ub514\uc5b4 \ubaa8\ub378\uc758 \uad6c\ud604\uc73c\ub85c\n\ub4e4\uc5b4\uac08 \uc218 \uc788\uc744 \uac83 \uac19\uc2b5\ub2c8\ub2e4. \uba3c\uc800 \uac00\uc911\uce58 \ucd08\uae30\ud654\uc5d0 \ub300\ud574 \uc774\uc57c\uae30 \ud574\ubcf4\uace0,\n\uc21c\uc11c\ub300\ub85c \uc0dd\uc131\uc790, \uad6c\ubd84\uc790, \uc190\uc2e4 \ud568\uc218, \ud559\uc2b5 \ubc29\ubc95\ub4e4\uc744 \uc54c\uc544\ubcf4\uaca0\uc2b5\ub2c8\ub2e4.\n\n\uac00\uc911\uce58 \ucd08\uae30\ud654\n~~~~~~~~~~~~~~\n\nDCGAN \ub17c\ubb38\uc5d0\uc11c\ub294, \ud3c9\uade0\uc774 0\uc774\uace0 \ubd84\uc0b0\uc774 0.02\uc778 \uc815\uaddc\ubd84\ud3ec\uc744 \uc774\uc6a9\ud574,\n\uad6c\ubd84\uc790\uc640 \uc0dd\uc131\uc790 \ubaa8\ub450 \ubb34\uc791\uc704 \ucd08\uae30\ud654\ub97c \uc9c4\ud589\ud558\ub294 \uac83\uc774 \uc88b\ub2e4\uace0 \ud569\ub2c8\ub2e4.\n``weights_init`` \ud568\uc218\ub294 \ub9e4\uac1c\ubcc0\uc218\ub85c \ubaa8\ub378\uc744 \uc785\ub825\ubc1b\uc544,\n\ubaa8\ub4e0 \ud569\uc131\uacf1 \uacc4\uce35, \uc804\uce58 \ud569\uc131\uacf1 \uacc4\uce35, \ubc30\uce58 \uc815\uaddc\ud654 \uacc4\uce35\uc744, \uc704\uc5d0\uc11c \ub9d0\ud55c \uc870\uac74\ub300\ub85c\n\uac00\uc911\uce58\ub4e4\uc744 \ub2e4\uc2dc \ucd08\uae30\ud654 \uc2dc\ud0b5\ub2c8\ub2e4. \uc774 \ud568\uc218\ub294 \ubaa8\ub378\uc774 \ub9cc\ub4e4\uc5b4\uc9c0\uc790 \ub9c8\uc790 \ubc14\ub85c \uc801\uc6a9\uc744\n\uc2dc\ud0a4\uac8c \ub429\ub2c8\ub2e4.\n\n\n\n\n```python\n# netG\uc640 netD\uc5d0 \uc801\uc6a9\uc2dc\ud0ac \ucee4\uc2a4\ud140 \uac00\uc911\uce58 \ucd08\uae30\ud654 \ud568\uc218\ndef weights_init(m):\n classname = m.__class__.__name__\n if classname.find('Conv') != -1:\n nn.init.normal_(m.weight.data, 0.0, 0.02)\n elif classname.find('BatchNorm') != -1:\n nn.init.normal_(m.weight.data, 1.0, 0.02)\n nn.init.constant_(m.bias.data, 0)\n```\n\n\uc0dd\uc131\uc790\n~~~~~~\n\n\uc0dd\uc131\uc790 $G$ \ub294 \uc7a0\uc7ac \uacf5\uac04 \ubca1\ud130 $z$ \ub97c, \ub370\uc774\ud130 \uacf5\uac04\uc73c\ub85c\n\ubcc0\ud658\uc2dc\ud0a4\ub3c4\ub85d \uc124\uacc4\ub418\uc5c8\uc2b5\ub2c8\ub2e4. \uc6b0\ub9ac\uc5d0\uac8c \ub370\uc774\ud130\ub77c \ud568\uc740 \uc774\ubbf8\uc9c0\uc774\uae30 \ub54c\ubb38\uc5d0,\n$z$ \ub97c \ub370\uc774\ud130\uacf5\uac04\uc73c\ub85c \ubcc0\ud658\ud55c\ub2e4\ub294 \ub73b\uc740, \ud559\uc2b5\uc774\ubbf8\uc9c0\uc640 \uac19\uc740 \uc0ac\uc774\uc988\ub97c \uac00\uc9c4\nRGB \uc774\ubbf8\uc9c0\ub97c \uc0dd\uc131\ud558\ub294\uac83\uacfc \uac19\uc2b5\ub2c8\ub2e4 (\uc608.\u00a03x64x64).\n\uc2e4\uc81c \ubaa8\ub378\uc5d0\uc11c\ub294 \uc2a4\ud2b8\ub77c\uc774\ub4dc(stride) 2\ub97c \uac00\uc9c4 \uc804\uce58 \ud569\uc131\uacf1 \uacc4\uce35\ub4e4\uc744 \uc774\uc5b4\uc11c \uad6c\uc131\ud558\ub294\ub370,\n\uac01 \uc804\uce58 \ud569\uc131\uacf1 \uacc4\uce35 \ud558\ub098\ub2f9 2\ucc28\uc6d0 \ubc30\uce58 \uc815\uaddc\ud654 \uacc4\uce35\uacfc relu \ud65c\uc131\ud568\uc218\ub97c \ud55c \uc30d\uc73c\ub85c \ubb36\uc5b4\uc11c \uc0ac\uc6a9\ud569\ub2c8\ub2e4.\n\uc0dd\uc131\uc790\uc758 \ub9c8\uc9c0\ub9c9 \ucd9c\ub825 \uacc4\uce35\uc5d0\uc11c\ub294 \ub370\uc774\ud130\ub97c tanh \ud568\uc218\uc5d0 \ud1b5\uacfc\uc2dc\ud0a4\ub294\ub370,\n\uc774\ub294 \ucd9c\ub825 \uac12\uc744 $[-1,1]$ \uc0ac\uc774\uc758 \ubc94\uc704\ub85c \uc870\uc815\ud558\uae30 \uc704\ud574\uc11c \uc785\ub2c8\ub2e4.\n\uc774\ub54c \ubc30\uce58 \uc815\uaddc\ud654 \uacc4\uce35\uc744 \uc8fc\ubaa9\ud560 \ud544\uc694\uac00 \uc788\ub294\ub370, DCGAN \ub17c\ubb38\uc5d0 \uc758\ud558\uba74,\n\uc774 \uacc4\uce35\uc774 \uacbd\uc0ac\ud558\uac15\ubc95(gradient-descent)\uc758 \ud750\ub984\uc5d0 \uc911\uc694\ud55c \uc601\ud5a5\uc744 \ubbf8\uce58\ub294 \uac83\uc73c\ub85c \uc54c\ub824\uc838 \uc788\uc2b5\ub2c8\ub2e4.\n\uc544\ub798\uc758 \uadf8\ub9bc\uc740 DCGAN \ub17c\ubb38\uc5d0\uc11c \uac00\uc838\uc628 \uc0dd\uc131\uc790\uc758 \ubaa8\ub378 \uc544\ud0a4\ud14d\uccd0\uc785\ub2c8\ub2e4.\n\n.. figure:: /_static/img/dcgan_generator.png\n :alt: dcgan_generator\n\n\uc6b0\ub9ac\uac00 \uc124\uc815\uac12 \uc139\uc158\uc5d0\uc11c \uc815\uc758\ud55c \uac12\ub4e4\uc774 (*nz*, *ngf*, \uadf8\ub9ac\uace0\n*nc*) \uc0dd\uc131\uc790 \ubaa8\ub378 \uc544\ud0a4\ud14d\uccd0\uc5d0 \uc5b4\ub5bb\uac8c \uc601\ud5a5\uc744 \ub07c\uce58\ub294\uc9c0 \uc8fc\ubaa9\ud574\uc8fc\uc138\uc694. *nz* \ub294 z \uc785\ub825 \ubca1\ud130\uc758\n\uae38\uc774, *ngf* \ub294 \uc0dd\uc131\uc790\ub97c \ud1b5\uacfc\ud558\ub294 \ud2b9\uc9d5 \ub370\uc774\ud130\uc758 \ud06c\uae30, \uadf8\ub9ac\uace0 *nc* \ub294 \ucd9c\ub825 \uc774\ubbf8\uc9c0\uc758\n\ucc44\ub110 \uac1c\uc218\uc785\ub2c8\ub2e4 (RGB \uc774\ubbf8\uc9c0\uc774\uae30 \ub54c\ubb38\uc5d0 3\uc73c\ub85c \uc124\uc815\uc744 \ud588\uc2b5\ub2c8\ub2e4).\n\uc544\ub798\ub294 \uc0dd\uc131\uc790\uc758 \ucf54\ub4dc\uc785\ub2c8\ub2e4.\n\n\n\n\n\n```python\n# \uc0dd\uc131\uc790 \ucf54\ub4dc\n\nclass Generator(nn.Module):\n def __init__(self, ngpu):\n super(Generator, self).__init__()\n self.ngpu = ngpu\n self.main = nn.Sequential(\n # \uc785\ub825\ub370\uc774\ud130 Z\uac00 \uac00\uc7a5 \ucc98\uc74c \ud1b5\uacfc\ud558\ub294 \uc804\uce58 \ud569\uc131\uacf1 \uacc4\uce35\uc785\ub2c8\ub2e4.\n nn.ConvTranspose2d( nz, ngf * 8, 4, 1, 0, bias=False),\n nn.BatchNorm2d(ngf * 8),\n nn.ReLU(True),\n # \uc704\uc758 \uacc4\uce35\uc744 \ud1b5\uacfc\ud55c \ub370\uc774\ud130\uc758 \ud06c\uae30. (ngf*8) x 4 x 4\n nn.ConvTranspose2d(ngf * 8, ngf * 4, 4, 2, 1, bias=False),\n nn.BatchNorm2d(ngf * 4),\n nn.ReLU(True),\n # \uc704\uc758 \uacc4\uce35\uc744 \ud1b5\uacfc\ud55c \ub370\uc774\ud130\uc758 \ud06c\uae30. (ngf*4) x 8 x 8\n nn.ConvTranspose2d( ngf * 4, ngf * 2, 4, 2, 1, bias=False),\n nn.BatchNorm2d(ngf * 2),\n nn.ReLU(True),\n # \uc704\uc758 \uacc4\uce35\uc744 \ud1b5\uacfc\ud55c \ub370\uc774\ud130\uc758 \ud06c\uae30. (ngf*2) x 16 x 16\n nn.ConvTranspose2d( ngf * 2, ngf, 4, 2, 1, bias=False),\n nn.BatchNorm2d(ngf),\n nn.ReLU(True),\n # \uc704\uc758 \uacc4\uce35\uc744 \ud1b5\uacfc\ud55c \ub370\uc774\ud130\uc758 \ud06c\uae30. (ngf) x 32 x 32\n nn.ConvTranspose2d( ngf, nc, 4, 2, 1, bias=False),\n nn.Tanh()\n # \uc704\uc758 \uacc4\uce35\uc744 \ud1b5\uacfc\ud55c \ub370\uc774\ud130\uc758 \ud06c\uae30. (nc) x 64 x 64\n )\n\n def forward(self, input):\n return self.main(input)\n```\n\n\uc88b\uc2b5\ub2c8\ub2e4. \uc774\uc81c \uc6b0\ub9ac\ub294 \uc0dd\uc131\uc790\uc758 \uc778\uc2a4\ud134\uc2a4\ub97c \ub9cc\ub4e4\uace0 ``weights_init``\n\ud568\uc218\ub97c \uc801\uc6a9\uc2dc\ud0ac \uc218 \uc788\uc2b5\ub2c8\ub2e4. \ubaa8\ub378\uc758 \uc778\uc2a4\ud134\uc2a4\ub97c \ucd9c\ub825\ud574\uc11c \uc0dd\uc131\uc790\uac00\n\uc5b4\ub5bb\uac8c \uad6c\uc131\ub418\uc5b4\uc788\ub294\uc9c0 \ud655\uc778\ud574\ubd05\uc2dc\ub2e4.\n\n\n\n\n\n```python\n# \uc0dd\uc131\uc790\ub97c \ub9cc\ub4ed\ub2c8\ub2e4\nnetG = Generator(ngpu).to(device)\n\n# \ud544\uc694\ud55c \uacbd\uc6b0 multi-gpu\ub97c \uc124\uc815 \ud574\uc8fc\uc138\uc694\nif (device.type == 'cuda') and (ngpu > 1):\n netG = nn.DataParallel(netG, list(range(ngpu)))\n\n# \ubaa8\ub4e0 \uac00\uc911\uce58\uc758 \ud3c9\uade0\uc744 0, \ubd84\uc0b0\uc744 0.02\ub85c \ucd08\uae30\ud654 \ud558\uae30 \uc704\ud574\n# weight_init \ud568\uc218\ub97c \uc801\uc6a9\uc2dc\ud0b5\ub2c8\ub2e4\nnetG.apply(weights_init)\n\n# \ubaa8\ub378\uc758 \uad6c\uc870\ub97c \ucd9c\ub825\ud569\ub2c8\ub2e4\nprint(netG)\n```\n\n\uad6c\ubd84\uc790\n~~~~~~\n\n\uc55e\uc11c \uc5b8\uae09\ud588\ub4ef, \uad6c\ubd84\uc790 $D$\ub294 \uc785\ub825 \uc774\ubbf8\uc9c0\uac00 \uc9c4\uc9dc \uc774\ubbf8\uc9c0\uc778\uc9c0 (\ud639\uc740 \ubc18\ub300\ub85c \uac00\uc9dc \uc774\ubbf8\uc9c0\uc778\uc9c0)\n\ud310\ubcc4\ud558\ub294 \uc804\ud1b5\uc801\uc778 \uc774\uc9c4 \ubd84\ub958 \uc2e0\uacbd\ub9dd\uc73c\ub85c \ubcfc \uc218 \uc788\uc2b5\ub2c8\ub2e4. \uc774\ub54c $D$ \ub294\n3x64x64 \uc774\ubbf8\uc9c0\ub97c \uc785\ub825\ubc1b\uc544, Conv2d, BatchNorm2d, \uadf8\ub9ac\uace0 LeakyReLU \uacc4\uce35\uc744 \ud1b5\uacfc\uc2dc\ucf1c\n\ub370\uc774\ud130\ub97c \uac00\uacf5\uc2dc\ud0a4\uace0, \ub9c8\uc9c0\ub9c9 \ucd9c\ub825\uc5d0\uc11c Sigmoid \ud568\uc218\ub97c \uc774\uc6a9\ud558\uc5ec\n0~1 \uc0ac\uc774\uc758 \ud655\ub960\uac12\uc73c\ub85c \uc870\uc815\ud569\ub2c8\ub2e4. \uc774 \uc544\ud0a4\ud14d\uccd0\ub294 \ud544\uc694\ud55c \uacbd\uc6b0 \ub354 \ub2e4\uc591\ud55c \ub808\uc774\uc5b4\ub97c \uc313\uc744 \uc218 \uc788\uc9c0\ub9cc,\n\ubc30\uce58 \uc815\uaddc\ud654\uc640 LeakyReLU, \ud2b9\ud788 \ubcf4\ud3ed\uc774 \uc788\ub294 (strided) \ud569\uc131\uacf1 \uacc4\uce35\uc744\n\uc0ac\uc6a9\ud558\ub294 \uac83\uc5d0\ub294 \uc774\uc720\uac00 \uc788\uc2b5\ub2c8\ub2e4.\nDCGAN \ub17c\ubb38\uc5d0\uc11c\ub294 \ubcf4\ud3ed\uc774 \uc788\ub294 \ud569\uc131\uacf1 \uacc4\uce35\uc744 \uc0ac\uc6a9\ud558\ub294 \uac83\uc774 \uc2e0\uacbd\ub9dd \ub0b4\uc5d0\uc11c \uc2a4\uc2a4\ub85c\uc758\n\ud480\ub9c1(Pooling) \ud568\uc218\ub97c \ud559\uc2b5\ud558\uae30 \ub54c\ubb38\uc5d0, \ub370\uc774\ud130\ub97c \ucc98\ub9ac\ud558\ub294 \uacfc\uc815\uc5d0\uc11c \uc9c1\uc811\uc801\uc73c\ub85c \ud480\ub9c1 \uacc4\uce35( MaxPool or AvgPooling)\uc744\n\uc0ac\uc6a9\ud558\ub294 \uac83\ubcf4\ub2e4 \ub354 \uc720\ub9ac\ud558\ub2e4\uace0 \ud569\ub2c8\ub2e4. \ub610\ud55c \ubc30\uce58 \uc815\uaddc\ud654\uc640 leaky relu \ud568\uc218\ub294 \ud559\uc2b5\uacfc\uc815\uc5d0\uc11c\n$G$ \uc640 $D$ \uac00 \ub354 \ud6a8\uacfc\uc801\uc778 \uacbd\uc0ac\ub3c4(gradient)\ub97c \uc5bb\uc744 \uc218 \uc788\uc2b5\ub2c8\ub2e4.\n\n\n\n\n\n```python\n# \uad6c\ubd84\uc790 \ucf54\ub4dc\n\nclass Discriminator(nn.Module):\n def __init__(self, ngpu):\n super(Discriminator, self).__init__()\n self.ngpu = ngpu\n self.main = nn.Sequential(\n # \uc785\ub825 \ub370\uc774\ud130\uc758 \ud06c\uae30\ub294 (nc) x 64 x 64 \uc785\ub2c8\ub2e4\n nn.Conv2d(nc, ndf, 4, 2, 1, bias=False),\n nn.LeakyReLU(0.2, inplace=True),\n # \uc704\uc758 \uacc4\uce35\uc744 \ud1b5\uacfc\ud55c \ub370\uc774\ud130\uc758 \ud06c\uae30. (ndf) x 32 x 32\n nn.Conv2d(ndf, ndf * 2, 4, 2, 1, bias=False),\n nn.BatchNorm2d(ndf * 2),\n nn.LeakyReLU(0.2, inplace=True),\n # \uc704\uc758 \uacc4\uce35\uc744 \ud1b5\uacfc\ud55c \ub370\uc774\ud130\uc758 \ud06c\uae30. (ndf*2) x 16 x 16\n nn.Conv2d(ndf * 2, ndf * 4, 4, 2, 1, bias=False),\n nn.BatchNorm2d(ndf * 4),\n nn.LeakyReLU(0.2, inplace=True),\n # \uc704\uc758 \uacc4\uce35\uc744 \ud1b5\uacfc\ud55c \ub370\uc774\ud130\uc758 \ud06c\uae30. (ndf*4) x 8 x 8\n nn.Conv2d(ndf * 4, ndf * 8, 4, 2, 1, bias=False),\n nn.BatchNorm2d(ndf * 8),\n nn.LeakyReLU(0.2, inplace=True),\n # \uc704\uc758 \uacc4\uce35\uc744 \ud1b5\uacfc\ud55c \ub370\uc774\ud130\uc758 \ud06c\uae30. (ndf*8) x 4 x 4\n nn.Conv2d(ndf * 8, 1, 4, 1, 0, bias=False),\n nn.Sigmoid()\n )\n\n def forward(self, input):\n return self.main(input)\n```\n\n\uc774\uc81c \uc6b0\ub9ac\ub294 \uc0dd\uc131\uc790\uc5d0 \ud55c \uac83\ucc98\ub7fc \uad6c\ubd84\uc790\uc758 \uc778\uc2a4\ud134\uc2a4\ub97c \ub9cc\ub4e4\uace0,\n``weights_init`` \ud568\uc218\ub97c \uc801\uc6a9\uc2dc\ud0a8 \ub2e4\uc74c, \ubaa8\ub378\uc758 \uad6c\uc870\ub97c \ucd9c\ub825\ud574\ubcfc \uc218 \uc788\uc2b5\ub2c8\ub2e4.\n\n\n\n\n\n```python\n# \uad6c\ubd84\uc790\ub97c \ub9cc\ub4ed\ub2c8\ub2e4\nnetD = Discriminator(ngpu).to(device)\n\n# \ud544\uc694\ud55c \uacbd\uc6b0 multi-gpu\ub97c \uc124\uc815 \ud574\uc8fc\uc138\uc694\nif (device.type == 'cuda') and (ngpu > 1):\n netD = nn.DataParallel(netD, list(range(ngpu)))\n \n# \ubaa8\ub4e0 \uac00\uc911\uce58\uc758 \ud3c9\uade0\uc744 0, \ubd84\uc0b0\uc744 0.02\ub85c \ucd08\uae30\ud654 \ud558\uae30 \uc704\ud574\n# weight_init \ud568\uc218\ub97c \uc801\uc6a9\uc2dc\ud0b5\ub2c8\ub2e4\nnetD.apply(weights_init)\n\n# \ubaa8\ub378\uc758 \uad6c\uc870\ub97c \ucd9c\ub825\ud569\ub2c8\ub2e4\nprint(netD)\n```\n\n\uc190\uc2e4\ud568\uc218\uc640 \uc635\ud2f0\ub9c8\uc774\uc800\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n$D$ \uc640 $G$ \uc758 \uc124\uc815\uc744 \ub05d\ub0c8\uc73c\ub2c8, \uc774\uc81c \uc190\uc2e4\ud568\uc218\uc640 \uc635\ud2f0\ub9c8\uc774\uc800\ub97c \uc815\ud558\uc5ec\n\ud559\uc2b5\uc744 \uad6c\uccb4\ud654\uc2dc\ud0ac \uc2dc\uac04\uc785\ub2c8\ub2e4. \uc190\uc2e4\ud568\uc218\ub85c\ub294 Binary Cross Entropy loss\n(`BCELoss `__)\n\ub97c \uc0ac\uc6a9\ud560\uac81\ub2c8\ub2e4. \ud574\ub2f9\ud568\uc218\ub294 \uc544\ub798\uc758 \uc2dd\uc73c\ub85c \ud30c\uc774\ud1a0\uce58\uc5d0 \uad6c\ud604\ub418\uc5b4 \uc788\uc2b5\ub2c8\ub2e4:\n\n\\begin{align}\\ell(x, y) = L = \\{l_1,\\dots,l_N\\}^\\top, \\quad l_n = - \\left[ y_n \\cdot \\log x_n + (1 - y_n) \\cdot \\log (1 - x_n) \\right]\\end{align}\n\n\uc774\ub54c, \uc704\uc758 \ud568\uc218\uac00 \ub85c\uadf8\ud568\uc218 \uc694\uc18c\ub97c \uc815\uc758\ud55c \ubc29\uc2dd\uc744 \uc8fc\uc758\uae4a\uac8c \ubd10\uc8fc\uc138\uc694 (\uc608. $log(D(x))$ \uc640\n$log(1-D(G(z)))$). \uc6b0\ub9b0 $y$ \uc744 \uc870\uc815\uc744 \uc870\uc815\ud558\uc5ec, BCE \ud568\uc218\uc5d0\uc11c\n\uc0ac\uc6a9\ud560 \uc694\uc18c\ub97c \uace0\ub97c \uc218 \uc788\uc2b5\ub2c8\ub2e4.\n\uc774 \ubd80\ubd84\uc740 \uc774\ud6c4\uc5d0 \uc11c\uc220\ud560 \ud559\uc2b5 \uc139\uc158\uc5d0\uc11c \ub2e4\ub8e8\uaca0\uc9c0\ub9cc, \uc5b4\ub5bb\uac8c $y$ \ub97c \uc774\uc6a9\ud558\uc5ec\n\uc6b0\ub9ac\uac00 \uc6d0\ud558\ub294 \uc694\uc18c\ub4e4\ub9cc \uace8\ub77c\ub0bc \uc218 \uc788\ub294\uc9c0 \uc774\ud574\ud558\ub294 \uac83\uc774 \uba3c\uc800\uc785\ub2c8\ub2e4 (\uc608.\u00a0GT labels).\n\n\uc88b\uc2b5\ub2c8\ub2e4. \ub2e4\uc74c\uc73c\ub85c \ub118\uc5b4\uac00\uaca0\uc2b5\ub2c8\ub2e4. \ucc38 \ub77c\ubca8 (\ud639\uc740 \uc815\ub2f5)\uc740 1\ub85c \ub450\uace0, \uac70\uc9d3 \ub77c\ubca8 (\ud639\uc740 \uc624\ub2f5)\uc740 0\uc73c\ub85c\n\ub450\uaca0\uc2b5\ub2c8\ub2e4. \uac01 \ub77c\ubca8\uc758 \uac12\uc744 \uc815\ud55c\uac74 GAN \ub17c\ubb38\uc5d0\uc11c \uc0ac\uc6a9\ub41c \uac12\ub4e4\ub85c, GAN\uc744 \uad6c\uc131\ud560\ub54c\uc758 \uad00\ub840\ub77c \ud560\n\uc218 \uc788\uc2b5\ub2c8\ub2e4. \ubc29\uae08 \uc815\ud55c \ub77c\ubca8 \uac12\ub4e4\uc740 \ucd94\ud6c4\uc5d0 \uc190\uc2e4\uac12\uc744 \uacc4\uc0b0\ud558\ub294 \uacfc\uc815\uc5d0\uc11c \uc0ac\uc6a9\ub420\uac81\ub2c8\ub2e4.\n\ub9c8\uc9c0\ub9c9\uc73c\ub85c, \uc11c\ub85c \uad6c\ubd84\ub418\ub294 \ub450 \uc635\ud2f0\ub9c8\uc774\uc800\ub97c \uad6c\uc131\ud558\uaca0\uc2b5\ub2c8\ub2e4. \ud558\ub098\ub294 $D$ \ub97c \uc704\ud55c \uac83,\n\ub2e4\ub978 \ud558\ub098\ub294 $G$ \ub97c \uc704\ud55c \uac83\uc785\ub2c8\ub2e4. DCGAN\uc5d0 \uc11c\uc220\ub41c \ub300\ub85c, \ub450 \uc635\ud2f0\ub9c8\uc774\uc800\ub294 \ubaa8\ub450 Adam\uc744 \uc0ac\uc6a9\ud558\uace0,\n\ud559\uc2b5\ub960\uc740 0.0002, Beta1 \uac12\uc740 0.5\ub85c \ub461\ub2c8\ub2e4. \ucd94\uac00\uc801\uc73c\ub85c, \ud559\uc2b5\uc774 \uc9c4\ud589\ub418\ub294 \ub3d9\uc548 \uc0dd\uc131\uc790\uc758 \uc0c1\ud0dc\ub97c \uc54c\uc544\ubcf4\uae30 \uc704\ud558\uc5ec,\n\ud504\ub85c\uadf8\ub7a8\uc774 \ub05d\ub0a0\ub54c\uae4c\uc9c0 \uace0\uc815\ub41c \uc7a0\uc7ac\uacf5\uac04 \ubca1\ud130\ub97c \uc0dd\uc131\ud558\uaca0\uc2b5\ub2c8\ub2e4 (\uc608. fixed_noise).\n\uc774 \ubca1\ud130\ub4e4 \uc5ed\uc2dc \uac00\uc6b0\uc2dc\uc548 \ubd84\ud3ec\uc5d0\uc11c \ucd94\ucd9c\ud569\ub2c8\ub2e4. \ud559\uc2b5 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uba74\uc11c $G$ \uc5d0 \uc8fc\uae30\uc801\uc73c\ub85c \uac19\uc740 \uc7a0\uc7ac\uacf5\uac04 \ubca1\ud130\ub97c\n\uc785\ub825\ud558\uba74, \uadf8 \ucd9c\ub825\uac12\uc744 \uae30\ubc18\uc73c\ub85c \uc0dd\uc131\uc790\uc758 \uc0c1\ud0dc\ub97c \ud655\uc778 \ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4.\n\n\n\n\n\n```python\n# BCELoss \ud568\uc218\uc758 \uc778\uc2a4\ud134\uc2a4\ub97c \uc0dd\uc131\ud569\ub2c8\ub2e4\ncriterion = nn.BCELoss()\n\n# \uc0dd\uc131\uc790\uc758 \ud559\uc2b5\uc0c1\ud0dc\ub97c \ud655\uc778\ud560 \uc7a0\uc7ac \uacf5\uac04 \ubca1\ud130\ub97c \uc0dd\uc131\ud569\ub2c8\ub2e4\nfixed_noise = torch.randn(64, nz, 1, 1, device=device)\n\n# \ud559\uc2b5\uc5d0 \uc0ac\uc6a9\ub418\ub294 \ucc38/\uac70\uc9d3\uc758 \ub77c\ubca8\uc744 \uc815\ud569\ub2c8\ub2e4\nreal_label = 1.\nfake_label = 0.\n\n# G\uc640 D\uc5d0\uc11c \uc0ac\uc6a9\ud560 Adam\uc635\ud2f0\ub9c8\uc774\uc800\ub97c \uc0dd\uc131\ud569\ub2c8\ub2e4\noptimizerD = optim.Adam(netD.parameters(), lr=lr, betas=(beta1, 0.999))\noptimizerG = optim.Adam(netG.parameters(), lr=lr, betas=(beta1, 0.999))\n```\n\n\ud559\uc2b5\n~~~~\n\n\ub4dc\ub514\uc5b4 \ucd5c\uc885\uc785\ub2c8\ub2e4. GAN \ud504\ub808\uc784\uc6cc\ud06c\uc5d0 \ud544\uc694\ud55c \ubd80\ubd84\ub4e4\uc740 \ubaa8\ub450 \uac00\uc84c\uc73c\ub2c8,\n\uc2e4\uc81c \ubaa8\ub378\uc744 \ud559\uc2b5\uc2dc\ud0a4\ub294 \ubc29\ubc95\uc744 \uc54c\uc544\ubcf4\uaca0\uc2b5\ub2c8\ub2e4. \uc8fc\uc758\ub97c \uae30\uc6b8\uc77c \uac83\uc740, GAN\uc744 \ud559\uc2b5\uc2dc\ud0a4\ub294 \uac74\n\uad00\ub840\uc801\uc778 \uae30\uc220\ub4e4\uc758 \uc9d1\ud569\uc774\uae30 \ub54c\ubb38\uc5d0, \uc798\ubabb\ub41c \ud558\uc774\ud37c\ud30c\ub77c\ubbf8\ud130\uc758 \uc124\uc815\uc740\n\ubaa8\ub378\uc758 \ud559\uc2b5\uc744 \ub9dd\uac00\ub728\ub9b4 \uc218 \uc788\uc2b5\ub2c8\ub2e4. \ubb34\uc5c7\uc774 \uc798\ubabb\ub418\uc5c8\ub294\uc9c0 \uc54c\uc544\ub0b4\ub294 \uac83 \uc870\ucc28\ub3c4 \ud798\ub4e4\uc8e0.\n\uadf8\ub7ec\ud55c \uc774\uc720\ub85c, \ubcf8 \ud29c\ud1a0\ub9ac\uc5bc\uc5d0\uc11c\ub294 Goodfellow\uc758 \ub17c\ubb38\uc5d0\uc11c \uc11c\uc220\ub41c Algorithm 1\uc744 \uae30\ubc18\uc73c\ub85c,\n`ganhacks `__ \uc5d0\uc11c \uc0ac\uc6a9\ub41c \uba87\uac00\uc9c0 \uad1c\ucc2e\uc740 \ud14c\ud06c\ub2c9\ub4e4\uc744\n\ub354\ud560 \uac83\uc785\ub2c8\ub2e4. \uc55e\uc11c \uba87\ubc88 \uc124\uba85\ud588\uc9c0\ub9cc, \uc6b0\ub9ac\uc758 \uc758\ub3c4\ub294 \u201c\uc9c4\uc9dc \ud639\uc740 \uac00\uc9dc \uc774\ubbf8\uc9c0\ub97c \uad6c\uc131\u201d\ud558\uace0,\n$logD(G(z))$ \ub97c \ucd5c\ub300\ud654\ud558\ub294 G\uc758 \ubaa9\uc801\ud568\uc218\ub97c \ucd5c\uc801\ud654 \uc2dc\ud0a4\ub294 \uac81\ub2c8\ub2e4. \ud559\uc2b5\uacfc\uc815\uc740 \ud06c\uac8c \ub450\uac00\uc9c0\ub85c \ub098\ub215\ub2c8\ub2e4.\nPart 1\uc740 \uad6c\ubd84\uc790\ub97c, Part 2\ub294 \uc0dd\uc131\uc790\ub97c \uc5c5\ub370\uc774\ud2b8\ud558\ub294 \uacfc\uc815\uc785\ub2c8\ub2e4.\n\n**Part 1 - \uad6c\ubd84\uc790\uc758 \ud559\uc2b5**\n\n\uad6c\ubd84\uc790\uc758 \ubaa9\uc801\uc740 \uc8fc\uc5b4\uc9c4 \uc785\ub825\uac12\uc774 \uc9c4\uc9dc\uc778\uc9c0 \uac00\uc9dc\uc778\uc9c0 \ud310\ubcc4\ud558\ub294 \uac83\uc784\uc744 \uc0c1\uae30\ud569\uc2dc\ub2e4.\nGoodfellow\uc758 \ub9d0\uc744 \ube4c\ub9ac\uc790\uba74, \uad6c\ubd84\uc790\ub294 \u201c\ubcc0\ud654\ub3c4(gradient)\ub97c \uc0c1\uc2b9(ascending)\uc2dc\ud0a4\uba70 \ud6c8\ub828\u201d\ud558\uac8c \ub429\ub2c8\ub2e4.\n\uc2e4\uc804\uc801\uc73c\ub85c \uc598\uae30\ud558\uba74, $log(D(x)) + log(1-D(G(z)))$ \ub97c \ucd5c\ub300\ud654\uc2dc\ud0a4\ub294 \uac83\uacfc \uac19\uc2b5\ub2c8\ub2e4.\nganhacks\uc5d0\uc11c \ubbf8\ub2c8 \ubc30\uce58(mini-batch)\ub97c \ubd84\ub9ac\ud558\uc5ec \uc0ac\uc6a9\ud55c \uac1c\ub150\uc744 \uac00\uc838\uc640\uc11c,\n\uc6b0\ub9ac \uc5ed\uc2dc \ub450\uac00\uc9c0 \uc2a4\ud15d\uc73c\ub85c \ubd84\ub9ac\ud574 \uacc4\uc0b0\uc744 \ud574\ubcf4\uaca0\uc2b5\ub2c8\ub2e4. \uba3c\uc800,\n\uc9c4\uc9dc \ub370\uc774\ud130\ub4e4\ub85c\ub9cc \uc774\ub8e8\uc5b4\uc9c4 \ubc30\uce58\ub97c \ub9cc\ub4e4\uc5b4 $D$ \uc5d0 \ud1b5\uacfc\uc2dc\ud0b5\ub2c8\ub2e4. \uadf8 \ucd9c\ub825\uac12\uc73c\ub85c ($log(D(x))$) \uc758 \uc190\uc2e4\uac12\uc744 \uacc4\uc0b0\ud558\uace0,\n\uc5ed\uc804\ud30c \uacfc\uc815\uc5d0\uc11c\uc758 \ubcc0\ud654\ub3c4\ub4e4\uc744 \uacc4\uc0b0\ud569\ub2c8\ub2e4. \uc5ec\uae30\uae4c\uc9c0\uac00 \uccab\ubc88\uc9f8 \uc2a4\ud15d\uc785\ub2c8\ub2e4. \ub450\ubc88\uc9f8 \uc2a4\ud15d\uc5d0\uc11c\ub294, \uc624\ub85c\uc9c0 \uac00\uc9dc \ub370\uc774\ud130\ub4e4\ub85c\ub9cc\n\uc774\ub8e8\uc5b4\uc9c4 \ubc30\uce58\ub97c \ub9cc\ub4e4\uc5b4 $D$ \uc5d0 \ud1b5\uacfc\uc2dc\ud0a4\uace0, \uadf8 \ucd9c\ub825\uac12\uc73c\ub85c ($log(1-D(G(z)))$) \uc758 \uc190\uc2e4\uac12\uc744 \uacc4\uc0b0\ud574\n\uc5ed\uc804\ud30c \ubcc0\ud654\ub3c4\ub97c \uad6c\ud558\uba74 \ub429\ub2c8\ub2e4. \uc774\ub54c \ub450\uac00\uc9c0 \uc2a4\ud15d\uc5d0\uc11c \ub098\uc624\ub294 \ubcc0\ud654\ub3c4\ub4e4\uc740 *\ucd95\uc801(accumulate)* \uc2dc\ucf1c\uc57c \ud569\ub2c8\ub2e4.\n\ubcc0\ud654\ub3c4\uae4c\uc9c0 \uad6c\ud588\uc73c\ub2c8, \uc774\uc81c \uc635\ud2f0\ub9c8\uc774\uc800\ub97c \uc0ac\uc6a9\ud574\uc57c\uaca0\uc8e0. \ud30c\uc774\ud1a0\uce58\uc758 \ud568\uc218\ub97c \ud638\ucd9c\ud574\uc8fc\uba74 \uc54c\uc544\uc11c \ubcc0\ud654\ub3c4\uac00 \uc801\uc6a9\ub420\uac81\ub2c8\ub2e4.\n\n**Part 2 - \uc0dd\uc131\uc790\uc758 \ud559\uc2b5**\n\n\uc624\ub9ac\uc9c0\ub110 GAN \ub17c\ubb38\uc5d0 \uba85\uc2dc\ub418\uc5b4 \uc788\ub4ef, \uc0dd\uc131\uc790\ub294 $log(1-D(G(z)))$ \uc744 \ucd5c\uc18c\ud654\uc2dc\ud0a4\ub294 \ubc29\ud5a5\uc73c\ub85c \ud559\uc2b5\ud569\ub2c8\ub2e4.\n\ud558\uc9c0\ub9cc \uc774 \ubc29\uc2dd\uc740 \ucda9\ubd84\ud55c \ubcc0\ud654\ub3c4\ub97c \uc81c\uacf5\ud558\uc9c0 \ubabb\ud568\uc744 Goodfellow\uac00 \ubcf4\uc5ec\uc92c\uc2b5\ub2c8\ub2e4. \ud2b9\ud788 \ud559\uc2b5\ucd08\uae30\uc5d0\ub294 \ub354\uc6b1 \ubb38\uc81c\ub97c \uc77c\uc73c\ud0a4\uc8e0.\n\uc774\ub97c \ud574\uacb0\ud558\uae30 \uc704\ud574 $log(D(G(z)))$ \ub97c \ucd5c\ub300\ud654 \ud558\ub294 \ubc29\uc2dd\uc73c\ub85c \ubc14\uafd4\uc11c \ud559\uc2b5\uc744 \ud558\uaca0\uc2b5\ub2c8\ub2e4. \ucf54\ub4dc\uc5d0\uc11c \uad6c\ud604\ud558\uae30\n\uc704\ud574\uc11c\ub294 : Part 1\uc5d0\uc11c \ud55c\ub300\ub85c \uad6c\ubd84\uc790\ub97c \uc774\uc6a9\ud574 \uc0dd\uc131\uc790\uc758 \ucd9c\ub825\uac12\uc744 \ud310\ubcc4\ud574\uc8fc\uace0, *\uc9c4\uc9dc \ub77c\ubca8\uac12* \uc744 \uc774\uc6a9\ud574 G\uc758 \uc190\uc2e4\uac12\uc744 \uad6c\ud574\uc90d\ub2c8\ub2e4.\n\uadf8\ub7ec\uba74 \uad6c\ud574\uc9c4 \uc190\uc2e4\uac12\uc73c\ub85c \ubcc0\ud654\ub3c4\ub97c \uad6c\ud558\uace0, \ucd5c\uc885\uc801\uc73c\ub85c\ub294 \uc635\ud2f0\ub9c8\uc774\uc800\ub97c \uc774\uc6a9\ud574 G\uc758 \uac00\uc911\uce58\ub4e4\uc744 \uc5c5\ub370\uc774\ud2b8\uc2dc\ucf1c\uc8fc\uba74 \ub429\ub2c8\ub2e4.\n\uc5b8\ub73b \ubcfc\ub54c\ub294, \uc0dd\uc131\uc790\uac00 \ub9cc\ub4e4\uc5b4\ub0b8 *\uac00\uc9dc* \uc774\ubbf8\uc9c0\uc5d0 *\uc9c4\uc9dc* \ub77c\ubca8\uc744 \uc0ac\uc6a9\ud558\ub294\uac83\uc774 \uc9c1\uad00\uc801\uc73c\ub85c \uc704\ubc30\uac00 \ub420\ud14c\uc9c0\ub9cc, \uc774\ub807\uac8c \ub77c\ubca8\uc744\n\ubc14\uafc8\uc73c\ub85c\uc368 $log(x)$ \ub77c\ub294 BCELoss\uc758 \uc77c\ubd80\ubd84\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc788\uac8c \ud569\ub2c8\ub2e4 (\uc55e\uc11c \uc6b0\ub9ac\ub294 BCELoss\uc5d0\uc11c \ub77c\ubca8\uc744 \uc774\uc6a9\ud574 \uc6d0\ud558\ub294 \ub85c\uadf8 \uacc4\uc0b0\n\uc694\uc18c\ub97c \uace0\ub97c \uc218 \uc788\uc74c\uc744 \uc54c\uc544\ubd24\uc2b5\ub2c8\ub2e4).\n\n\ub9c8\ubb34\ub9ac\ub85c G\uc758 \ud6c8\ub828 \uc0c1\ud0dc\ub97c \uc54c\uc544\ubcf4\uae30 \uc704\ud558\uc5ec, \uba87\uac00\uc9c0 \ud1b5\uacc4\uc801\uc778 \uc218\uce58\ub4e4\uacfc, fixed_noise\ub97c \ud1b5\uacfc\uc2dc\ud0a8\n\uacb0\uacfc\ub97c \ud654\uba74\uc5d0 \ucd9c\ub825\ud558\ub294 \ucf54\ub4dc\ub97c \ucd94\uac00\ud558\uaca0\uc2b5\ub2c8\ub2e4. \uc774\ub54c \ud1b5\uacc4\uc801\uc778 \uc218\uce58\ub4e4\uc774\ub77c \ud568\uc740 :\n\n- **Loss_D** - \uc9c4\uc9dc \ub370\uc774\ud130\uc640 \uac00\uc9dc \ub370\uc774\ud130\ub4e4 \ubaa8\ub450\uc5d0\uc11c \uad6c\ud574\uc9c4 \uc190\uc2e4\uac12. ($log(D(x)) + log(1 - D(G(z)))$).\n- **Loss_G** - \uc0dd\uc131\uc790\uc758 \uc190\uc2e4\uac12. $log(D(G(z)))$\n- **D(x)** - \uad6c\ubd84\uc790\uac00 \ub370\uc774\ud130\ub97c \ud310\ubcc4\ud55c \ud655\ub960\uac12\uc785\ub2c8\ub2e4. \ucc98\uc74c\uc5d0\ub294 1\uc5d0 \uac00\uae4c\uc6b4 \uac12\uc774\ub2e4\uac00,\n G\uac00 \ud559\uc2b5\ud560\uc218\ub85d 0.5\uac12\uc5d0 \uc218\ub834\ud558\uac8c \ub429\ub2c8\ub2e4.\n- **D(G(z))** - \uac00\uc9dc\ub370\uc774\ud130\ub4e4\uc5d0 \ub300\ud55c \uad6c\ubd84\uc790\uc758 \ucd9c\ub825\uac12\uc785\ub2c8\ub2e4. \ucc98\uc74c\uc5d0\ub294 0\uc5d0 \uac00\uae4c\uc6b4 \uac12\uc774\ub2e4\uac00,\n G\uac00 \ud559\uc2b5\ud560\uc218\ub85d 0.5\uc5d0 \uc218\ub834\ud558\uac8c \ub429\ub2c8\ub2e4\n\n\n**Note:** \uc774\ud6c4\uc758 \uacfc\uc815\uc740 epoch\uc758 \uc218\uc640 \ub370\uc774\ud130\uc758 \uc218\uc5d0 \ub530\ub77c \uc2dc\uac04\uc774 \uc880 \uac78\ub9b4 \uc218 \uc788\uc2b5\ub2c8\ub2e4\n\n\n\n\n\n```python\n# \ud559\uc2b5 \uacfc\uc815\n\n# \ud559\uc2b5\uc0c1\ud0dc\ub97c \uccb4\ud06c\ud558\uae30 \uc704\ud574 \uc190\uc2e4\uac12\ub4e4\uc744 \uc800\uc7a5\ud569\ub2c8\ub2e4\nimg_list = []\nG_losses = []\nD_losses = []\niters = 0\n\nprint(\"Starting Training Loop...\")\n# \uc5d0\ud3ed(epoch) \ubc18\ubcf5\nfor epoch in range(num_epochs):\n # \ud55c \uc5d0\ud3ed \ub0b4\uc5d0\uc11c \ubc30\uce58 \ubc18\ubcf5\n for i, data in enumerate(dataloader, 0):\n \n ############################\n # (1) D \uc2e0\uacbd\ub9dd\uc744 \uc5c5\ub370\uc774\ud2b8 \ud569\ub2c8\ub2e4: log(D(x)) + log(1 - D(G(z)))\ub97c \ucd5c\ub300\ud654 \ud569\ub2c8\ub2e4\n ###########################\n ## \uc9c4\uc9dc \ub370\uc774\ud130\ub4e4\ub85c \ud559\uc2b5\uc744 \ud569\ub2c8\ub2e4\n netD.zero_grad()\n # \ubc30\uce58\ub4e4\uc758 \uc0ac\uc774\uc988\ub098 \uc0ac\uc6a9\ud560 \ub514\ubc14\uc774\uc2a4\uc5d0 \ub9de\uac8c \uc870\uc815\ud569\ub2c8\ub2e4\n real_cpu = data[0].to(device)\n b_size = real_cpu.size(0)\n label = torch.full((b_size,), real_label,\n dtype=torch.float, device=device)\n # \uc9c4\uc9dc \ub370\uc774\ud130\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ubc30\uce58\ub97c D\uc5d0 \ud1b5\uacfc\uc2dc\ud0b5\ub2c8\ub2e4\n output = netD(real_cpu).view(-1)\n # \uc190\uc2e4\uac12\uc744 \uad6c\ud569\ub2c8\ub2e4\n errD_real = criterion(output, label)\n # \uc5ed\uc804\ud30c\uc758 \uacfc\uc815\uc5d0\uc11c \ubcc0\ud654\ub3c4\ub97c \uacc4\uc0b0\ud569\ub2c8\ub2e4\n errD_real.backward()\n D_x = output.mean().item()\n\n ## \uac00\uc9dc \ub370\uc774\ud130\ub4e4\ub85c \ud559\uc2b5\uc744 \ud569\ub2c8\ub2e4\n # \uc0dd\uc131\uc790\uc5d0 \uc0ac\uc6a9\ud560 \uc7a0\uc7ac\uacf5\uac04 \ubca1\ud130\ub97c \uc0dd\uc131\ud569\ub2c8\ub2e4\n noise = torch.randn(b_size, nz, 1, 1, device=device)\n # G\ub97c \uc774\uc6a9\ud574 \uac00\uc9dc \uc774\ubbf8\uc9c0\ub97c \uc0dd\uc131\ud569\ub2c8\ub2e4\n fake = netG(noise)\n label.fill_(fake_label)\n # D\ub97c \uc774\uc6a9\ud574 \ub370\uc774\ud130\uc758 \uc9c4\uc704\ub97c \ud310\ubcc4\ud569\ub2c8\ub2e4\n output = netD(fake.detach()).view(-1)\n # D\uc758 \uc190\uc2e4\uac12\uc744 \uacc4\uc0b0\ud569\ub2c8\ub2e4\n errD_fake = criterion(output, label)\n # \uc5ed\uc804\ud30c\ub97c \ud1b5\ud574 \ubcc0\ud654\ub3c4\ub97c \uacc4\uc0b0\ud569\ub2c8\ub2e4. \uc774\ub54c \uc55e\uc11c \uad6c\ud55c \ubcc0\ud654\ub3c4\uc5d0 \ub354\ud569\ub2c8\ub2e4(accumulate)\n errD_fake.backward()\n D_G_z1 = output.mean().item()\n # \uac00\uc9dc \uc774\ubbf8\uc9c0\uc640 \uc9c4\uc9dc \uc774\ubbf8\uc9c0 \ubaa8\ub450\uc5d0\uc11c \uad6c\ud55c \uc190\uc2e4\uac12\ub4e4\uc744 \ub354\ud569\ub2c8\ub2e4\n # \uc774\ub54c errD\ub294 \uc5ed\uc804\ud30c\uc5d0\uc11c \uc0ac\uc6a9\ub418\uc9c0 \uc54a\uace0, \uc774\ud6c4 \ud559\uc2b5 \uc0c1\ud0dc\ub97c \ub9ac\ud3ec\ud305(reporting)\ud560 \ub54c \uc0ac\uc6a9\ud569\ub2c8\ub2e4\n errD = errD_real + errD_fake\n # D\ub97c \uc5c5\ub370\uc774\ud2b8 \ud569\ub2c8\ub2e4\n optimizerD.step()\n\n ############################\n # (2) G \uc2e0\uacbd\ub9dd\uc744 \uc5c5\ub370\uc774\ud2b8 \ud569\ub2c8\ub2e4: log(D(G(z)))\ub97c \ucd5c\ub300\ud654 \ud569\ub2c8\ub2e4\n ###########################\n netG.zero_grad()\n label.fill_(real_label) # \uc0dd\uc131\uc790\uc758 \uc190\uc2e4\uac12\uc744 \uad6c\ud558\uae30 \uc704\ud574 \uc9c4\uc9dc \ub77c\ubca8\uc744 \uc774\uc6a9\ud560 \uac81\ub2c8\ub2e4\n # \uc6b0\ub9ac\ub294 \ubc29\uae08 D\ub97c \uc5c5\ub370\uc774\ud2b8\ud588\uae30 \ub54c\ubb38\uc5d0, D\uc5d0 \ub2e4\uc2dc \uac00\uc9dc \ub370\uc774\ud130\ub97c \ud1b5\uacfc\uc2dc\ud0b5\ub2c8\ub2e4.\n # \uc774\ub54c G\ub294 \uc5c5\ub370\uc774\ud2b8\ub418\uc9c0 \uc54a\uc558\uc9c0\ub9cc, D\uac00 \uc5c5\ub370\uc774\ud2b8 \ub418\uc5c8\uae30 \ub54c\ubb38\uc5d0 \uc55e\uc120 \uc190\uc2e4\uac12\uac00 \ub2e4\ub978 \uac12\uc774 \ub098\uc624\uac8c \ub429\ub2c8\ub2e4\n output = netD(fake).view(-1)\n # G\uc758 \uc190\uc2e4\uac12\uc744 \uad6c\ud569\ub2c8\ub2e4\n errG = criterion(output, label)\n # G\uc758 \ubcc0\ud654\ub3c4\ub97c \uacc4\uc0b0\ud569\ub2c8\ub2e4\n errG.backward()\n D_G_z2 = output.mean().item()\n # G\ub97c \uc5c5\ub370\uc774\ud2b8 \ud569\ub2c8\ub2e4\n optimizerG.step()\n \n # \ud6c8\ub828 \uc0c1\ud0dc\ub97c \ucd9c\ub825\ud569\ub2c8\ub2e4\n if i % 50 == 0:\n print('[%d/%d][%d/%d]\\tLoss_D: %.4f\\tLoss_G: %.4f\\tD(x): %.4f\\tD(G(z)): %.4f / %.4f'\n % (epoch, num_epochs, i, len(dataloader),\n errD.item(), errG.item(), D_x, D_G_z1, D_G_z2))\n \n # \uc774\ud6c4 \uadf8\ub798\ud504\ub97c \uadf8\ub9ac\uae30 \uc704\ud574 \uc190\uc2e4\uac12\ub4e4\uc744 \uc800\uc7a5\ud574\ub461\ub2c8\ub2e4\n G_losses.append(errG.item())\n D_losses.append(errD.item())\n \n # fixed_noise\ub97c \ud1b5\uacfc\uc2dc\ud0a8 G\uc758 \ucd9c\ub825\uac12\uc744 \uc800\uc7a5\ud574\ub461\ub2c8\ub2e4\n if (iters % 500 == 0) or ((epoch == num_epochs-1) and (i == len(dataloader)-1)):\n with torch.no_grad():\n fake = netG(fixed_noise).detach().cpu()\n img_list.append(vutils.make_grid(fake, padding=2, normalize=True))\n \n iters += 1\n```\n\n\uacb0\uacfc\n----\n\n\uacb0\uacfc\ub97c \uc54c\uc544\ubd05\uc2dc\ub2e4. \uc774 \uc139\uc158\uc5d0\uc11c\ub294 \ucd1d \uc138\uac00\uc9c0\ub97c \ud655\uc778\ud560\uac81\ub2c8\ub2e4.\n\uccab\ubc88\uc9f8\ub294 G\uc640 D\uc758 \uc190\uc2e4\uac12\ub4e4\uc774 \uc5b4\ub5bb\uac8c \ubcc0\ud588\ub294\uac00, \ub450\ubc88\uc9f8\ub294 \ub9e4 \uc5d0\ud3ed\ub9c8\ub2e4\nfixed_noise\ub97c \uc774\uc6a9\ud574 G\uac00 \ub9cc\ub4e4\uc5b4\ub0b8 \uc774\ubbf8\uc9c0\ub4e4, \ub9c8\uc9c0\ub9c9\uc740 \ud559\uc2b5\uc774 \ub05d\ub09c G\uac00 \ub9cc\ub4e4\uc5b4\ub0b8 \uc774\ubbf8\uc9c0\uc640\n\uc9c4\uc9dc\uc774\ubbf8\uc9c0\ub4e4\uc758 \ube44\uad50\uc785\ub2c8\ub2e4\n\n**\ud559\uc2b5\ud558\ub294 \ub3d9\uc548\uc758 \uc190\uc2e4\uac12\ub4e4**\n\n\uc544\ub798\ub294 D\uc640 G\uc758 \uc190\uc2e4\uac12\ub4e4\uc744 \uadf8\ub798\ud504\ub85c \uadf8\ub9b0 \ubaa8\uc2b5\uc785\ub2c8\ub2e4\n\n\n\n\n\n```python\nplt.figure(figsize=(10,5))\nplt.title(\"Generator and Discriminator Loss During Training\")\nplt.plot(G_losses,label=\"G\")\nplt.plot(D_losses,label=\"D\")\nplt.xlabel(\"iterations\")\nplt.ylabel(\"Loss\")\nplt.legend()\nplt.show()\n```\n\n**G\uc758 \ud559\uc2b5 \uacfc\uc815 \uc2dc\uac01\ud654**\n\n\ub9e4 \uc5d0\ud3ed\ub9c8\ub2e4 fixed_noise\ub97c \uc774\uc6a9\ud574 \uc0dd\uc131\uc790\uac00 \ub9cc\ub4e4\uc5b4\ub0b8 \uc774\ubbf8\uc9c0\ub97c \uc800\uc7a5\ud55c \uac83\uc744 \uae30\uc5b5\ud560\uac81\ub2c8\ub2e4.\n\uc800\uc7a5\ud55c \uc774\ubbf8\uc9c0\ub4e4\uc744\uc560\ub2c8\uba54\uc774\uc158 \ud615\uc2dd\uc73c\ub85c \ud655\uc778\ud574 \ubd05\uc2dc\ub2e4. play\ubc84\ud2bc\uc744 \ub204\ub974\uba74 \uc560\ub2c8\ub9e4\uc774\uc158\uc774 \uc2e4\ud589\ub429\ub2c8\ub2e4\n\n\n\n\n\n```python\n#%%capture\nfig = plt.figure(figsize=(8,8))\nplt.axis(\"off\")\nims = [[plt.imshow(np.transpose(i,(1,2,0)), animated=True)] for i in img_list]\nani = animation.ArtistAnimation(fig, ims, interval=1000, repeat_delay=1000, blit=True)\n\nHTML(ani.to_jshtml())\n```\n\n**\uc9c4\uc9dc \uc774\ubbf8\uc9c0 vs.\u00a0\uac00\uc9dc \uc774\ubbf8\uc9c0**\n\n\uc9c4\uc9dc \uc774\ubbf8\uc9c0\ub4e4\uacfc \uac00\uc9dc \uc774\ubbf8\uc9c0\ub4e4\uc744 \uc606\uc73c\ub85c \ub450\uace0 \ube44\uad50\ub97c \ud574\ubd05\uc2dc\ub2e4\n\n\n\n\n\n```python\n# dataloader\uc5d0\uc11c \uc9c4\uc9dc \ub370\uc774\ud130\ub4e4\uc744 \uac00\uc838\uc635\ub2c8\ub2e4\nreal_batch = next(iter(dataloader))\n\n# \uc9c4\uc9dc \uc774\ubbf8\uc9c0\ub4e4\uc744 \ud654\uba74\uc5d0 \ucd9c\ub825\ud569\ub2c8\ub2e4\nplt.figure(figsize=(15,15))\nplt.subplot(1,2,1)\nplt.axis(\"off\")\nplt.title(\"Real Images\")\nplt.imshow(np.transpose(vutils.make_grid(real_batch[0].to(device)[:64], padding=5, normalize=True).cpu(),(1,2,0)))\n\n# \uac00\uc9dc \uc774\ubbf8\uc9c0\ub4e4\uc744 \ud654\uba74\uc5d0 \ucd9c\ub825\ud569\ub2c8\ub2e4\nplt.subplot(1,2,2)\nplt.axis(\"off\")\nplt.title(\"Fake Images\")\nplt.imshow(np.transpose(img_list[-1],(1,2,0)))\nplt.show()\n```\n\n\uc774\uc81c \uc5b4\ub514\ub85c \uc5ec\ud589\uc744 \ub5a0\ub098\ubcfc\uae4c\uc694?\n------------------------------\n\n\ub4dc\ub514\uc5b4 DCGAN\uc774 \ub05d\ub0ac\uc2b5\ub2c8\ub2e4! \ud558\uc9c0\ub9cc \ub354 \uc54c\uc544\ubcfc \uac83\ub4e4\uc774 \ub9ce\uc774 \ub0a8\uc544\uc788\uc8e0.\n\ubb34\uc5c7\uc744 \ub354 \uc2dc\ub3c4\ud574\ubcfc \uc218 \uc788\uc744\uae4c\uc694?\n\n- \uacb0\uacfc\ubb3c\uc774 \uc5bc\ub9c8\ub098 \ub354 \uc88b\uc544\uc9c0\ub294\uc9c0 \ud655\uc778\ud574\ubcf4\uae30 \uc704\ud574\uc11c \ud559\uc2b5\uc2dc\uac04\uc744 \ub298\ub824\ubcfc \uc218 \uc788\uc2b5\ub2c8\ub2e4\n- \ub2e4\ub978 \ub370\uc774\ud130\uc14b\uc744 \uc774\uc6a9\ud574 \ud6c8\ub828\uc2dc\ucf1c\ubcf4\uac70\ub098, \uc774\ubbf8\uc9c0\uc758 \uc0ac\uc774\uc988\ub97c \ub2e4\ub974\uac8c \ud574\ubcf4\uac70\ub098, \uc544\ud0a4\ud14d\uccd0\uc758 \uad6c\uc131\uc744 \ubc14\uafd4\ubcfc \uc218\ub3c4 \uc788\uc2b5\ub2c8\ub2e4\n- `\uc5ec\uae30 `__ \uc5d0\uc11c \ub354\uc6b1 \uba4b\uc9c4 GAN \ud504\ub85c\uc81d\ud2b8\ub4e4\uc744 \ucc3e\uc744\uc218\ub3c4 \uc788\uc8e0\n- `\uc74c\uc545 `__ \uc744 \uc791\uace1\ud558\ub294 GAN\ub3c4 \ub9cc\ub4e4 \uc218 \uc788\uc2b5\ub2c8\ub2e4\n\n\n\n", "meta": {"hexsha": "eb6852a9b138a6308352b68d4444798c98b1e425", "size": 59748, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/_downloads/52d07bf9e7a1a21004b361865bd6290f/dcgan_faces_tutorial.ipynb", "max_stars_repo_name": "woojinsong/PyTorch-tutorials-kr", "max_stars_repo_head_hexsha": "36fefd556f45c2b1f5db912793172c0369430fd4", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 221, "max_stars_repo_stars_event_min_datetime": "2018-04-06T01:42:58.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-28T10:12:45.000Z", "max_issues_repo_path": "docs/_downloads/52d07bf9e7a1a21004b361865bd6290f/dcgan_faces_tutorial.ipynb", "max_issues_repo_name": "woojinsong/PyTorch-tutorials-kr", "max_issues_repo_head_hexsha": "36fefd556f45c2b1f5db912793172c0369430fd4", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 280, 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YES\n2. YES", "lm_q1_score": 0.7490872243177517, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.4182356184483587}} {"text": "```python\n%matplotlib inline\n```\n\n\nAdversarial Example Generation\n==============================\n\n**Author:** `Nathan Inkawhich `__\n\nIf you are reading this, hopefully you can appreciate how effective some\nmachine learning models are. Research is constantly pushing ML models to\nbe faster, more accurate, and more efficient. However, an often\noverlooked aspect of designing and training models is security and\nrobustness, especially in the face of an adversary who wishes to fool\nthe model.\n\nThis tutorial will raise your awareness to the security vulnerabilities\nof ML models, and will give insight into the hot topic of adversarial\nmachine learning. You may be surprised to find that adding imperceptible\nperturbations to an image *can* cause drastically different model\nperformance. Given that this is a tutorial, we will explore the topic\nvia example on an image classifier. Specifically we will use one of the\nfirst and most popular attack methods, the Fast Gradient Sign Attack\n(FGSM), to fool an MNIST classifier.\n\n\n\n\nThreat Model\n------------\n\nFor context, there are many categories of adversarial attacks, each with\na different goal and assumption of the attacker\u2019s knowledge. However, in\ngeneral the overarching goal is to add the least amount of perturbation\nto the input data to cause the desired misclassification. There are\nseveral kinds of assumptions of the attacker\u2019s knowledge, two of which\nare: **white-box** and **black-box**. A *white-box* attack assumes the\nattacker has full knowledge and access to the model, including\narchitecture, inputs, outputs, and weights. A *black-box* attack assumes\nthe attacker only has access to the inputs and outputs of the model, and\nknows nothing about the underlying architecture or weights. There are\nalso several types of goals, including **misclassification** and\n**source/target misclassification**. A goal of *misclassification* means\nthe adversary only wants the output classification to be wrong but does\nnot care what the new classification is. A *source/target\nmisclassification* means the adversary wants to alter an image that is\noriginally of a specific source class so that it is classified as a\nspecific target class.\n\nIn this case, the FGSM attack is a *white-box* attack with the goal of\n*misclassification*. With this background information, we can now\ndiscuss the attack in detail.\n\nFast Gradient Sign Attack\n-------------------------\n\nOne of the first and most popular adversarial attacks to date is\nreferred to as the *Fast Gradient Sign Attack (FGSM)* and is described\nby Goodfellow et. al.\u00a0in `Explaining and Harnessing Adversarial\nExamples `__. The attack is remarkably\npowerful, and yet intuitive. It is designed to attack neural networks by\nleveraging the way they learn, *gradients*. The idea is simple, rather\nthan working to minimize the loss by adjusting the weights based on the\nbackpropagated gradients, the attack *adjusts the input data to maximize\nthe loss* based on the same backpropagated gradients. In other words,\nthe attack uses the gradient of the loss w.r.t the input data, then\nadjusts the input data to maximize the loss.\n\nBefore we jump into the code, let\u2019s look at the famous\n`FGSM `__ panda example and extract\nsome notation.\n\n.. figure:: /_static/img/fgsm_panda_image.png\n :alt: fgsm_panda_image\n\nFrom the figure, $\\mathbf{x}$ is the original input image\ncorrectly classified as a \u201cpanda\u201d, $y$ is the ground truth label\nfor $\\mathbf{x}$, $\\mathbf{\\theta}$ represents the model\nparameters, and $J(\\mathbf{\\theta}, \\mathbf{x}, y)$ is the loss\nthat is used to train the network. The attack backpropagates the\ngradient back to the input data to calculate\n$\\nabla_{x} J(\\mathbf{\\theta}, \\mathbf{x}, y)$. Then, it adjusts\nthe input data by a small step ($\\epsilon$ or $0.007$ in the\npicture) in the direction (i.e.\n$sign(\\nabla_{x} J(\\mathbf{\\theta}, \\mathbf{x}, y))$) that will\nmaximize the loss. The resulting perturbed image, $x'$, is then\n*misclassified* by the target network as a \u201cgibbon\u201d when it is still\nclearly a \u201cpanda\u201d.\n\nHopefully now the motivation for this tutorial is clear, so lets jump\ninto the implementation.\n\n\n\n\n\n```python\nfrom __future__ import print_function\nimport torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport torch.optim as optim\nfrom torchvision import datasets, transforms\nimport numpy as np\nimport matplotlib.pyplot as plt\n```\n\nImplementation\n--------------\n\nIn this section, we will discuss the input parameters for the tutorial,\ndefine the model under attack, then code the attack and run some tests.\n\nInputs\n~~~~~~\n\nThere are only three inputs for this tutorial, and are defined as\nfollows:\n\n- **epsilons** - List of epsilon values to use for the run. It is\n important to keep 0 in the list because it represents the model\n performance on the original test set. Also, intuitively we would\n expect the larger the epsilon, the more noticeable the perturbations\n but the more effective the attack in terms of degrading model\n accuracy. Since the data range here is $[0,1]$, no epsilon\n value should exceed 1.\n\n- **pretrained_model** - path to the pretrained MNIST model which was\n trained with\n `pytorch/examples/mnist `__.\n For simplicity, download the pretrained model `here `__.\n\n- **use_cuda** - boolean flag to use CUDA if desired and available.\n Note, a GPU with CUDA is not critical for this tutorial as a CPU will\n not take much time.\n\n\n\n\n\n```python\nepsilons = [0, .05, .1, .15, .2, .25, .3]\npretrained_model = \"pretrained_models/lenet_mnist_model.pth\"\nuse_cuda=True\n```\n\nModel Under Attack\n~~~~~~~~~~~~~~~~~~\n\nAs mentioned, the model under attack is the same MNIST model from\n`pytorch/examples/mnist `__.\nYou may train and save your own MNIST model or you can download and use\nthe provided model. The *Net* definition and test dataloader here have\nbeen copied from the MNIST example. The purpose of this section is to\ndefine the model and dataloader, then initialize the model and load the\npretrained weights.\n\n\n\n\n\n```python\n# LeNet Model definition\nclass Net(nn.Module):\n def __init__(self):\n super(Net, self).__init__()\n self.conv1 = nn.Conv2d(1, 10, kernel_size=5)\n self.conv2 = nn.Conv2d(10, 20, kernel_size=5)\n self.conv2_drop = nn.Dropout2d()\n self.fc1 = nn.Linear(320, 50)\n self.fc2 = nn.Linear(50, 10)\n\n def forward(self, x):\n x = F.relu(F.max_pool2d(self.conv1(x), 2))\n x = F.relu(F.max_pool2d(self.conv2_drop(self.conv2(x)), 2))\n x = x.view(-1, 320)\n x = F.relu(self.fc1(x))\n x = F.dropout(x, training=self.training)\n x = self.fc2(x)\n return F.log_softmax(x, dim=1)\n\n# MNIST Test dataset and dataloader declaration\ntest_loader = torch.utils.data.DataLoader(\n datasets.MNIST('data/', train=False, download=True, transform=transforms.Compose([\n transforms.ToTensor(),\n ])), \n batch_size=1, shuffle=True)\n\n# Define what device we are using\nprint(\"CUDA Available: \",torch.cuda.is_available())\ndevice = torch.device(\"cuda:1\" if (use_cuda and torch.cuda.is_available()) else \"cpu\")\n\n# Initialize the network\nmodel = Net().to(device)\n\n# Load the pretrained model\nmodel.load_state_dict(torch.load(pretrained_model, map_location='cpu'))\n\n# Set the model in evaluation mode. In this case this is for the Dropout layers\nmodel.eval()\n```\n\n CUDA Available: True\n\n\n\n\n\n Net(\n (conv1): Conv2d(1, 10, kernel_size=(5, 5), stride=(1, 1))\n (conv2): Conv2d(10, 20, kernel_size=(5, 5), stride=(1, 1))\n (conv2_drop): Dropout2d(p=0.5)\n (fc1): Linear(in_features=320, out_features=50, bias=True)\n (fc2): Linear(in_features=50, out_features=10, bias=True)\n )\n\n\n\nFGSM Attack\n~~~~~~~~~~~\n\nNow, we can define the function that creates the adversarial examples by\nperturbing the original inputs. The ``fgsm_attack`` function takes three\ninputs, *image* is the original clean image ($x$), *epsilon* is\nthe pixel-wise perturbation amount ($\\epsilon$), and *data_grad*\nis gradient of the loss w.r.t the input image\n($\\nabla_{x} J(\\mathbf{\\theta}, \\mathbf{x}, y)$). The function\nthen creates perturbed image as\n\n\\begin{align}perturbed\\_image = image + epsilon*sign(data\\_grad) = x + \\epsilon * sign(\\nabla_{x} J(\\mathbf{\\theta}, \\mathbf{x}, y))\\end{align}\n\nFinally, in order to maintain the original range of the data, the\nperturbed image is clipped to range $[0,1]$.\n\n\n\n\n\n```python\n# FGSM attack code\ndef fgsm_attack(image, epsilon, data_grad):\n # Collect the element-wise sign of the data gradient\n sign_data_grad = data_grad.sign()\n # Create the perturbed image by adjusting each pixel of the input image\n perturbed_image = image + epsilon*sign_data_grad\n # Adding clipping to maintain [0,1] range\n perturbed_image = torch.clamp(perturbed_image, 0, 1)\n # Return the perturbed image\n return perturbed_image\n```\n\nTesting Function\n~~~~~~~~~~~~~~~~\n\nFinally, the central result of this tutorial comes from the ``test``\nfunction. Each call to this test function performs a full test step on\nthe MNIST test set and reports a final accuracy. However, notice that\nthis function also takes an *epsilon* input. This is because the\n``test`` function reports the accuracy of a model that is under attack\nfrom an adversary with strength $\\epsilon$. More specifically, for\neach sample in the test set, the function computes the gradient of the\nloss w.r.t the input data ($data\\_grad$), creates a perturbed\nimage with ``fgsm_attack`` ($perturbed\\_data$), then checks to see\nif the perturbed example is adversarial. In addition to testing the\naccuracy of the model, the function also saves and returns some\nsuccessful adversarial examples to be visualized later.\n\n\n\n\n\n```python\ndef test( model, device, test_loader, epsilon ):\n\n # Accuracy counter\n correct = 0\n adv_examples = []\n\n # Loop over all examples in test set\n for data, target in test_loader:\n\n # Send the data and label to the device\n data, target = data.to(device), target.to(device)\n\n # Set requires_grad attribute of tensor. Important for Attack\n data.requires_grad = True\n\n # Forward pass the data through the model\n output = model(data)\n init_pred = output.max(1, keepdim=True)[1] # get the index of the max log-probability\n\n # If the initial prediction is wrong, dont bother attacking, just move on\n if init_pred.item() != target.item():\n continue\n\n # Calculate the loss\n loss = F.nll_loss(output, target)\n\n # Zero all existing gradients\n model.zero_grad()\n\n # Calculate gradients of model in backward pass\n loss.backward()\n\n # Collect datagrad\n data_grad = data.grad.data\n\n # Call FGSM Attack\n perturbed_data = fgsm_attack(data, epsilon, data_grad)\n\n # Re-classify the perturbed image\n output = model(perturbed_data)\n\n # Check for success\n final_pred = output.max(1, keepdim=True)[1] # get the index of the max log-probability\n if final_pred.item() == target.item():\n correct += 1\n # Special case for saving 0 epsilon examples\n if (epsilon == 0) and (len(adv_examples) < 5):\n adv_ex = perturbed_data.squeeze().detach().cpu().numpy()\n adv_examples.append( (init_pred.item(), final_pred.item(), adv_ex) )\n else:\n # Save some adv examples for visualization later\n if len(adv_examples) < 5:\n adv_ex = perturbed_data.squeeze().detach().cpu().numpy()\n adv_examples.append( (init_pred.item(), final_pred.item(), adv_ex) )\n\n # Calculate final accuracy for this epsilon\n final_acc = correct/float(len(test_loader))\n print(\"Epsilon: {}\\tTest Accuracy = {} / {} = {}\".format(epsilon, correct, len(test_loader), final_acc))\n\n # Return the accuracy and an adversarial example\n return final_acc, adv_examples\n```\n\nRun Attack\n~~~~~~~~~~\n\nThe last part of the implementation is to actually run the attack. Here,\nwe run a full test step for each epsilon value in the *epsilons* input.\nFor each epsilon we also save the final accuracy and some successful\nadversarial examples to be plotted in the coming sections. Notice how\nthe printed accuracies decrease as the epsilon value increases. Also,\nnote the $\\epsilon=0$ case represents the original test accuracy,\nwith no attack.\n\n\n\n\n\n```python\naccuracies = []\nexamples = []\n\n# Run test for each epsilon\nfor eps in epsilons:\n acc, ex = test(model, device, test_loader, eps)\n accuracies.append(acc)\n examples.append(ex)\n```\n\n Epsilon: 0\tTest Accuracy = 9810 / 10000 = 0.981\n Epsilon: 0.05\tTest Accuracy = 9426 / 10000 = 0.9426\n Epsilon: 0.1\tTest Accuracy = 8510 / 10000 = 0.851\n Epsilon: 0.15\tTest Accuracy = 6826 / 10000 = 0.6826\n Epsilon: 0.2\tTest Accuracy = 4301 / 10000 = 0.4301\n Epsilon: 0.25\tTest Accuracy = 2082 / 10000 = 0.2082\n Epsilon: 0.3\tTest Accuracy = 869 / 10000 = 0.0869\n\n\nResults\n-------\n\nAccuracy vs Epsilon\n~~~~~~~~~~~~~~~~~~~\n\nThe first result is the accuracy versus epsilon plot. As alluded to\nearlier, as epsilon increases we expect the test accuracy to decrease.\nThis is because larger epsilons mean we take a larger step in the\ndirection that will maximize the loss. Notice the trend in the curve is\nnot linear even though the epsilon values are linearly spaced. For\nexample, the accuracy at $\\epsilon=0.05$ is only about 4% lower\nthan $\\epsilon=0$, but the accuracy at $\\epsilon=0.2$ is 25%\nlower than $\\epsilon=0.15$. Also, notice the accuracy of the model\nhits random accuracy for a 10-class classifier between\n$\\epsilon=0.25$ and $\\epsilon=0.3$.\n\n\n\n\n\n```python\nplt.figure(figsize=(5,5))\nplt.plot(epsilons, accuracies, \"*-\")\nplt.yticks(np.arange(0, 1.1, step=0.1))\nplt.xticks(np.arange(0, .35, step=0.05))\nplt.title(\"Accuracy vs Epsilon\")\nplt.xlabel(\"Epsilon\")\nplt.ylabel(\"Accuracy\")\nplt.show()\n```\n\nSample Adversarial Examples\n~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\nRemember the idea of no free lunch? In this case, as epsilon increases\nthe test accuracy decreases **BUT** the perturbations become more easily\nperceptible. In reality, there is a tradeoff between accuracy\ndegredation and perceptibility that an attacker must consider. Here, we\nshow some examples of successful adversarial examples at each epsilon\nvalue. Each row of the plot shows a different epsilon value. The first\nrow is the $\\epsilon=0$ examples which represent the original\n\u201cclean\u201d images with no perturbation. The title of each image shows the\n\u201coriginal classification -> adversarial classification.\u201d Notice, the\nperturbations start to become evident at $\\epsilon=0.15$ and are\nquite evident at $\\epsilon=0.3$. However, in all cases humans are\nstill capable of identifying the correct class despite the added noise.\n\n\n\n\n\n```python\n# Plot several examples of adversarial samples at each epsilon\ncnt = 0\nplt.figure(figsize=(8,10))\nfor i in range(len(epsilons)):\n for j in range(len(examples[i])):\n cnt += 1\n plt.subplot(len(epsilons),len(examples[0]),cnt)\n plt.xticks([], [])\n plt.yticks([], [])\n if j == 0:\n plt.ylabel(\"Eps: {}\".format(epsilons[i]), fontsize=14)\n orig,adv,ex = examples[i][j]\n plt.title(\"{} -> {}\".format(orig, adv))\n plt.imshow(ex, cmap=\"gray\")\nplt.tight_layout()\nplt.show()\n```\n\nWhere to go next?\n-----------------\n\nHopefully this tutorial gives some insight into the topic of adversarial\nmachine learning. There are many potential directions to go from here.\nThis attack represents the very beginning of adversarial attack research\nand since there have been many subsequent ideas for how to attack and\ndefend ML models from an adversary. In fact, at NIPS 2017 there was an\nadversarial attack and defense competition and many of the methods used\nin the competition are described in this paper: `Adversarial Attacks and\nDefences Competition `__. The work\non defense also leads into the idea of making machine learning models\nmore *robust* in general, to both naturally perturbed and adversarially\ncrafted inputs.\n\nAnother direction to go is adversarial attacks and defense in different\ndomains. Adversarial research is not limited to the image domain, check\nout `this `__ attack on\nspeech-to-text models. But perhaps the best way to learn more about\nadversarial machine learning is to get your hands dirty. Try to\nimplement a different attack from the NIPS 2017 competition, and see how\nit differs from FGSM. Then, try to defend the model from your own\nattacks.\n\n\n\n", "meta": {"hexsha": "69d7e0ae736bcce6d3f6b16db23b943cd2540268", "size": 109396, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "13_fgsm_tutorial.ipynb", "max_stars_repo_name": "spencerpomme/torchlight", "max_stars_repo_head_hexsha": "254a461b30436ac23e64d5ce59e43a1672b76304", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "13_fgsm_tutorial.ipynb", "max_issues_repo_name": "spencerpomme/torchlight", "max_issues_repo_head_hexsha": "254a461b30436ac23e64d5ce59e43a1672b76304", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "13_fgsm_tutorial.ipynb", "max_forks_repo_name": "spencerpomme/torchlight", "max_forks_repo_head_hexsha": "254a461b30436ac23e64d5ce59e43a1672b76304", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 179.632183908, "max_line_length": 68932, "alphanum_fraction": 0.886595488, "converted": true, "num_tokens": 4213, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.6757646075489392, "lm_q1q2_score": 0.4181499169678752}} {"text": "# Static Interactive Notebook\n\nThis notebook shows a few examples of what's possible with the static widgets\n\n## Example 1: Simple Text output\n\n\n```\nfrom ipywidgets import StaticInteract, RangeWidget, RadioWidget\n\ndef multiply(a, b):\n return \"{0} * {1} = {2}\".format(a, b, a * b)\n\nStaticInteract(multiply,\n a=RangeWidget(0, 10),\n b=RangeWidget(0, 10),)\n```\n\n\n\n\n\n \n\n
                                        \n\n
                                        \n

                                        0 * 0 = 0

                                        \n
                                        \n\n
                                        \n

                                        0 * 1 = 0

                                        \n
                                        \n\n
                                        \n

                                        0 * 2 = 0

                                        \n
                                        \n\n
                                        \n

                                        0 * 3 = 0

                                        \n
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                                        0 * 4 = 0

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                                        0 * 5 = 0

                                        \n
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                                        0 * 6 = 0

                                        \n
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                                        0 * 7 = 0

                                        \n
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                                        \n

                                        0 * 8 = 0

                                        \n
                                        \n\n
                                        \n

                                        0 * 9 = 0

                                        \n
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                                        \n

                                        0 * 10 = 0

                                        \n
                                        \n\n
                                        \n

                                        1 * 0 = 0

                                        \n
                                        \n\n
                                        \n

                                        1 * 1 = 1

                                        \n
                                        \n\n
                                        \n

                                        1 * 2 = 2

                                        \n
                                        \n\n
                                        \n

                                        1 * 3 = 3

                                        \n
                                        \n\n
                                        \n

                                        1 * 4 = 4

                                        \n
                                        \n\n
                                        \n

                                        1 * 5 = 5

                                        \n
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                                        1 * 6 = 6

                                        \n
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                                        1 * 7 = 7

                                        \n
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                                        \n

                                        1 * 8 = 8

                                        \n
                                        \n\n
                                        \n

                                        1 * 9 = 9

                                        \n
                                        \n\n
                                        \n

                                        1 * 10 = 10

                                        \n
                                        \n\n
                                        \n

                                        2 * 0 = 0

                                        \n
                                        \n\n
                                        \n

                                        2 * 1 = 2

                                        \n
                                        \n\n
                                        \n

                                        2 * 2 = 4

                                        \n
                                        \n\n
                                        \n

                                        2 * 3 = 6

                                        \n
                                        \n\n
                                        \n

                                        2 * 4 = 8

                                        \n
                                        \n\n
                                        \n

                                        2 * 5 = 10

                                        \n
                                        \n\n
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                                        2 * 6 = 12

                                        \n
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                                        2 * 7 = 14

                                        \n
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                                        2 * 8 = 16

                                        \n
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                                        2 * 9 = 18

                                        \n
                                        \n\n
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                                        2 * 10 = 20

                                        \n
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                                        3 * 0 = 0

                                        \n
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                                        3 * 1 = 3

                                        \n
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                                        3 * 2 = 6

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                                        3 * 3 = 9

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                                        3 * 4 = 12

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                                        3 * 5 = 15

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                                        3 * 6 = 18

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                                        3 * 7 = 21

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                                        3 * 8 = 24

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                                        3 * 9 = 27

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                                        3 * 10 = 30

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                                        4 * 0 = 0

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                                        4 * 1 = 4

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                                        4 * 2 = 8

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                                        4 * 3 = 12

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                                        4 * 4 = 16

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                                        4 * 5 = 20

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                                        4 * 6 = 24

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                                        4 * 7 = 28

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                                        4 * 8 = 32

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                                        4 * 9 = 36

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                                        4 * 10 = 40

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                                        5 * 0 = 0

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                                        5 * 1 = 5

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                                        5 * 2 = 10

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                                        5 * 3 = 15

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                                        5 * 4 = 20

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                                        5 * 5 = 25

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                                        5 * 6 = 30

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                                        5 * 7 = 35

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                                        5 * 8 = 40

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                                        5 * 9 = 45

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                                        5 * 10 = 50

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                                        6 * 0 = 0

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                                        6 * 1 = 6

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                                        6 * 2 = 12

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                                        6 * 3 = 18

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                                        6 * 4 = 24

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                                        6 * 5 = 30

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                                        6 * 6 = 36

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                                        6 * 7 = 42

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                                        6 * 8 = 48

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                                        6 * 9 = 54

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                                        6 * 10 = 60

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                                        7 * 0 = 0

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                                        7 * 1 = 7

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                                        7 * 2 = 14

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                                        7 * 3 = 21

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                                        7 * 4 = 28

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                                        7 * 5 = 35

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                                        7 * 6 = 42

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                                        7 * 7 = 49

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                                        7 * 8 = 56

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                                        7 * 9 = 63

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                                        7 * 10 = 70

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                                        8 * 0 = 0

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                                        8 * 1 = 8

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                                        8 * 2 = 16

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                                        8 * 3 = 24

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                                        8 * 4 = 32

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                                        8 * 5 = 40

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                                        8 * 6 = 48

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                                        8 * 7 = 56

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                                        8 * 8 = 64

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                                        8 * 9 = 72

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                                        8 * 10 = 80

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                                        9 * 0 = 0

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                                        9 * 1 = 9

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                                        9 * 2 = 18

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                                        9 * 3 = 27

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                                        9 * 4 = 36

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                                        9 * 5 = 45

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                                        9 * 6 = 54

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                                        9 * 7 = 63

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                                        9 * 8 = 72

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                                        9 * 9 = 81

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                                        9 * 10 = 90

                                        \n
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                                        10 * 0 = 0

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                                        10 * 1 = 10

                                        \n
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                                        10 * 2 = 20

                                        \n
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                                        10 * 3 = 30

                                        \n
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                                        10 * 4 = 40

                                        \n
                                        \n\n
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                                        10 * 5 = 50

                                        \n
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                                        10 * 6 = 60

                                        \n
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                                        10 * 7 = 70

                                        \n
                                        \n\n
                                        \n

                                        10 * 8 = 80

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                                        10 * 9 = 90

                                        \n
                                        \n\n
                                        \n

                                        10 * 10 = 100

                                        \n
                                        \n\n a: \n
                                        \nb: \n
                                        \n\n\n\n\n## Example 2: Text Output with Multiple Widgets\n\n\n```\nfrom ipywidgets import StaticInteract, RangeWidget, RadioWidget, DropDownWidget\n\ndef operation(a, b, op, fruit):\n if op == \"multiply\":\n return \"{0} * {1} = {2} {3}\".format(a, b, a * b, fruit)\n elif op == \"add\":\n return \"{0} + {1} = {2} {3}\".format(a, b, a + b, fruit)\n else:\n raise ValueError\n\nStaticInteract(operation,\n a=RangeWidget(0, 10),\n b=RangeWidget(0, 10),\n op=RadioWidget([\"multiply\", \"add\"]),\n fruit=DropDownWidget(['apples','bananas','pears','kiwis'])\n )\n```\n\n\n\n\n\n \n\n
                                        \n\n
                                        \n

                                        0 * 0 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        0 + 0 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        0 * 0 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 + 0 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 * 0 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        0 + 0 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        0 * 0 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 + 0 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 * 1 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        0 + 1 = 1 apples

                                        \n
                                        \n\n
                                        \n

                                        0 * 1 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 + 1 = 1 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 * 1 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        0 + 1 = 1 pears

                                        \n
                                        \n\n
                                        \n

                                        0 * 1 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 + 1 = 1 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 * 2 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        0 + 2 = 2 apples

                                        \n
                                        \n\n
                                        \n

                                        0 * 2 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 + 2 = 2 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 * 2 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        0 + 2 = 2 pears

                                        \n
                                        \n\n
                                        \n

                                        0 * 2 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 + 2 = 2 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 * 3 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        0 + 3 = 3 apples

                                        \n
                                        \n\n
                                        \n

                                        0 * 3 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 + 3 = 3 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 * 3 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        0 + 3 = 3 pears

                                        \n
                                        \n\n
                                        \n

                                        0 * 3 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 + 3 = 3 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 * 4 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        0 + 4 = 4 apples

                                        \n
                                        \n\n
                                        \n

                                        0 * 4 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 + 4 = 4 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 * 4 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        0 + 4 = 4 pears

                                        \n
                                        \n\n
                                        \n

                                        0 * 4 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 + 4 = 4 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 * 5 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        0 + 5 = 5 apples

                                        \n
                                        \n\n
                                        \n

                                        0 * 5 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 + 5 = 5 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 * 5 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        0 + 5 = 5 pears

                                        \n
                                        \n\n
                                        \n

                                        0 * 5 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 + 5 = 5 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 * 6 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        0 + 6 = 6 apples

                                        \n
                                        \n\n
                                        \n

                                        0 * 6 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 + 6 = 6 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 * 6 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        0 + 6 = 6 pears

                                        \n
                                        \n\n
                                        \n

                                        0 * 6 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 + 6 = 6 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 * 7 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        0 + 7 = 7 apples

                                        \n
                                        \n\n
                                        \n

                                        0 * 7 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 + 7 = 7 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 * 7 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        0 + 7 = 7 pears

                                        \n
                                        \n\n
                                        \n

                                        0 * 7 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 + 7 = 7 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 * 8 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        0 + 8 = 8 apples

                                        \n
                                        \n\n
                                        \n

                                        0 * 8 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 + 8 = 8 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 * 8 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        0 + 8 = 8 pears

                                        \n
                                        \n\n
                                        \n

                                        0 * 8 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 + 8 = 8 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 * 9 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        0 + 9 = 9 apples

                                        \n
                                        \n\n
                                        \n

                                        0 * 9 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 + 9 = 9 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 * 9 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        0 + 9 = 9 pears

                                        \n
                                        \n\n
                                        \n

                                        0 * 9 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 + 9 = 9 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 * 10 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        0 + 10 = 10 apples

                                        \n
                                        \n\n
                                        \n

                                        0 * 10 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 + 10 = 10 bananas

                                        \n
                                        \n\n
                                        \n

                                        0 * 10 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        0 + 10 = 10 pears

                                        \n
                                        \n\n
                                        \n

                                        0 * 10 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        0 + 10 = 10 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 * 0 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        1 + 0 = 1 apples

                                        \n
                                        \n\n
                                        \n

                                        1 * 0 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 + 0 = 1 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 * 0 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        1 + 0 = 1 pears

                                        \n
                                        \n\n
                                        \n

                                        1 * 0 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 + 0 = 1 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 * 1 = 1 apples

                                        \n
                                        \n\n
                                        \n

                                        1 + 1 = 2 apples

                                        \n
                                        \n\n
                                        \n

                                        1 * 1 = 1 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 + 1 = 2 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 * 1 = 1 pears

                                        \n
                                        \n\n
                                        \n

                                        1 + 1 = 2 pears

                                        \n
                                        \n\n
                                        \n

                                        1 * 1 = 1 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 + 1 = 2 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 * 2 = 2 apples

                                        \n
                                        \n\n
                                        \n

                                        1 + 2 = 3 apples

                                        \n
                                        \n\n
                                        \n

                                        1 * 2 = 2 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 + 2 = 3 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 * 2 = 2 pears

                                        \n
                                        \n\n
                                        \n

                                        1 + 2 = 3 pears

                                        \n
                                        \n\n
                                        \n

                                        1 * 2 = 2 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 + 2 = 3 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 * 3 = 3 apples

                                        \n
                                        \n\n
                                        \n

                                        1 + 3 = 4 apples

                                        \n
                                        \n\n
                                        \n

                                        1 * 3 = 3 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 + 3 = 4 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 * 3 = 3 pears

                                        \n
                                        \n\n
                                        \n

                                        1 + 3 = 4 pears

                                        \n
                                        \n\n
                                        \n

                                        1 * 3 = 3 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 + 3 = 4 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 * 4 = 4 apples

                                        \n
                                        \n\n
                                        \n

                                        1 + 4 = 5 apples

                                        \n
                                        \n\n
                                        \n

                                        1 * 4 = 4 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 + 4 = 5 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 * 4 = 4 pears

                                        \n
                                        \n\n
                                        \n

                                        1 + 4 = 5 pears

                                        \n
                                        \n\n
                                        \n

                                        1 * 4 = 4 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 + 4 = 5 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 * 5 = 5 apples

                                        \n
                                        \n\n
                                        \n

                                        1 + 5 = 6 apples

                                        \n
                                        \n\n
                                        \n

                                        1 * 5 = 5 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 + 5 = 6 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 * 5 = 5 pears

                                        \n
                                        \n\n
                                        \n

                                        1 + 5 = 6 pears

                                        \n
                                        \n\n
                                        \n

                                        1 * 5 = 5 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 + 5 = 6 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 * 6 = 6 apples

                                        \n
                                        \n\n
                                        \n

                                        1 + 6 = 7 apples

                                        \n
                                        \n\n
                                        \n

                                        1 * 6 = 6 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 + 6 = 7 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 * 6 = 6 pears

                                        \n
                                        \n\n
                                        \n

                                        1 + 6 = 7 pears

                                        \n
                                        \n\n
                                        \n

                                        1 * 6 = 6 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 + 6 = 7 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 * 7 = 7 apples

                                        \n
                                        \n\n
                                        \n

                                        1 + 7 = 8 apples

                                        \n
                                        \n\n
                                        \n

                                        1 * 7 = 7 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 + 7 = 8 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 * 7 = 7 pears

                                        \n
                                        \n\n
                                        \n

                                        1 + 7 = 8 pears

                                        \n
                                        \n\n
                                        \n

                                        1 * 7 = 7 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 + 7 = 8 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 * 8 = 8 apples

                                        \n
                                        \n\n
                                        \n

                                        1 + 8 = 9 apples

                                        \n
                                        \n\n
                                        \n

                                        1 * 8 = 8 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 + 8 = 9 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 * 8 = 8 pears

                                        \n
                                        \n\n
                                        \n

                                        1 + 8 = 9 pears

                                        \n
                                        \n\n
                                        \n

                                        1 * 8 = 8 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 + 8 = 9 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 * 9 = 9 apples

                                        \n
                                        \n\n
                                        \n

                                        1 + 9 = 10 apples

                                        \n
                                        \n\n
                                        \n

                                        1 * 9 = 9 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 + 9 = 10 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 * 9 = 9 pears

                                        \n
                                        \n\n
                                        \n

                                        1 + 9 = 10 pears

                                        \n
                                        \n\n
                                        \n

                                        1 * 9 = 9 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 + 9 = 10 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 * 10 = 10 apples

                                        \n
                                        \n\n
                                        \n

                                        1 + 10 = 11 apples

                                        \n
                                        \n\n
                                        \n

                                        1 * 10 = 10 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 + 10 = 11 bananas

                                        \n
                                        \n\n
                                        \n

                                        1 * 10 = 10 pears

                                        \n
                                        \n\n
                                        \n

                                        1 + 10 = 11 pears

                                        \n
                                        \n\n
                                        \n

                                        1 * 10 = 10 kiwis

                                        \n
                                        \n\n
                                        \n

                                        1 + 10 = 11 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 * 0 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        2 + 0 = 2 apples

                                        \n
                                        \n\n
                                        \n

                                        2 * 0 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 + 0 = 2 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 * 0 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        2 + 0 = 2 pears

                                        \n
                                        \n\n
                                        \n

                                        2 * 0 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 + 0 = 2 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 * 1 = 2 apples

                                        \n
                                        \n\n
                                        \n

                                        2 + 1 = 3 apples

                                        \n
                                        \n\n
                                        \n

                                        2 * 1 = 2 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 + 1 = 3 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 * 1 = 2 pears

                                        \n
                                        \n\n
                                        \n

                                        2 + 1 = 3 pears

                                        \n
                                        \n\n
                                        \n

                                        2 * 1 = 2 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 + 1 = 3 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 * 2 = 4 apples

                                        \n
                                        \n\n
                                        \n

                                        2 + 2 = 4 apples

                                        \n
                                        \n\n
                                        \n

                                        2 * 2 = 4 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 + 2 = 4 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 * 2 = 4 pears

                                        \n
                                        \n\n
                                        \n

                                        2 + 2 = 4 pears

                                        \n
                                        \n\n
                                        \n

                                        2 * 2 = 4 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 + 2 = 4 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 * 3 = 6 apples

                                        \n
                                        \n\n
                                        \n

                                        2 + 3 = 5 apples

                                        \n
                                        \n\n
                                        \n

                                        2 * 3 = 6 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 + 3 = 5 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 * 3 = 6 pears

                                        \n
                                        \n\n
                                        \n

                                        2 + 3 = 5 pears

                                        \n
                                        \n\n
                                        \n

                                        2 * 3 = 6 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 + 3 = 5 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 * 4 = 8 apples

                                        \n
                                        \n\n
                                        \n

                                        2 + 4 = 6 apples

                                        \n
                                        \n\n
                                        \n

                                        2 * 4 = 8 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 + 4 = 6 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 * 4 = 8 pears

                                        \n
                                        \n\n
                                        \n

                                        2 + 4 = 6 pears

                                        \n
                                        \n\n
                                        \n

                                        2 * 4 = 8 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 + 4 = 6 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 * 5 = 10 apples

                                        \n
                                        \n\n
                                        \n

                                        2 + 5 = 7 apples

                                        \n
                                        \n\n
                                        \n

                                        2 * 5 = 10 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 + 5 = 7 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 * 5 = 10 pears

                                        \n
                                        \n\n
                                        \n

                                        2 + 5 = 7 pears

                                        \n
                                        \n\n
                                        \n

                                        2 * 5 = 10 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 + 5 = 7 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 * 6 = 12 apples

                                        \n
                                        \n\n
                                        \n

                                        2 + 6 = 8 apples

                                        \n
                                        \n\n
                                        \n

                                        2 * 6 = 12 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 + 6 = 8 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 * 6 = 12 pears

                                        \n
                                        \n\n
                                        \n

                                        2 + 6 = 8 pears

                                        \n
                                        \n\n
                                        \n

                                        2 * 6 = 12 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 + 6 = 8 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 * 7 = 14 apples

                                        \n
                                        \n\n
                                        \n

                                        2 + 7 = 9 apples

                                        \n
                                        \n\n
                                        \n

                                        2 * 7 = 14 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 + 7 = 9 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 * 7 = 14 pears

                                        \n
                                        \n\n
                                        \n

                                        2 + 7 = 9 pears

                                        \n
                                        \n\n
                                        \n

                                        2 * 7 = 14 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 + 7 = 9 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 * 8 = 16 apples

                                        \n
                                        \n\n
                                        \n

                                        2 + 8 = 10 apples

                                        \n
                                        \n\n
                                        \n

                                        2 * 8 = 16 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 + 8 = 10 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 * 8 = 16 pears

                                        \n
                                        \n\n
                                        \n

                                        2 + 8 = 10 pears

                                        \n
                                        \n\n
                                        \n

                                        2 * 8 = 16 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 + 8 = 10 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 * 9 = 18 apples

                                        \n
                                        \n\n
                                        \n

                                        2 + 9 = 11 apples

                                        \n
                                        \n\n
                                        \n

                                        2 * 9 = 18 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 + 9 = 11 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 * 9 = 18 pears

                                        \n
                                        \n\n
                                        \n

                                        2 + 9 = 11 pears

                                        \n
                                        \n\n
                                        \n

                                        2 * 9 = 18 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 + 9 = 11 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 * 10 = 20 apples

                                        \n
                                        \n\n
                                        \n

                                        2 + 10 = 12 apples

                                        \n
                                        \n\n
                                        \n

                                        2 * 10 = 20 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 + 10 = 12 bananas

                                        \n
                                        \n\n
                                        \n

                                        2 * 10 = 20 pears

                                        \n
                                        \n\n
                                        \n

                                        2 + 10 = 12 pears

                                        \n
                                        \n\n
                                        \n

                                        2 * 10 = 20 kiwis

                                        \n
                                        \n\n
                                        \n

                                        2 + 10 = 12 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 * 0 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        3 + 0 = 3 apples

                                        \n
                                        \n\n
                                        \n

                                        3 * 0 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 + 0 = 3 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 * 0 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        3 + 0 = 3 pears

                                        \n
                                        \n\n
                                        \n

                                        3 * 0 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 + 0 = 3 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 * 1 = 3 apples

                                        \n
                                        \n\n
                                        \n

                                        3 + 1 = 4 apples

                                        \n
                                        \n\n
                                        \n

                                        3 * 1 = 3 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 + 1 = 4 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 * 1 = 3 pears

                                        \n
                                        \n\n
                                        \n

                                        3 + 1 = 4 pears

                                        \n
                                        \n\n
                                        \n

                                        3 * 1 = 3 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 + 1 = 4 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 * 2 = 6 apples

                                        \n
                                        \n\n
                                        \n

                                        3 + 2 = 5 apples

                                        \n
                                        \n\n
                                        \n

                                        3 * 2 = 6 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 + 2 = 5 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 * 2 = 6 pears

                                        \n
                                        \n\n
                                        \n

                                        3 + 2 = 5 pears

                                        \n
                                        \n\n
                                        \n

                                        3 * 2 = 6 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 + 2 = 5 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 * 3 = 9 apples

                                        \n
                                        \n\n
                                        \n

                                        3 + 3 = 6 apples

                                        \n
                                        \n\n
                                        \n

                                        3 * 3 = 9 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 + 3 = 6 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 * 3 = 9 pears

                                        \n
                                        \n\n
                                        \n

                                        3 + 3 = 6 pears

                                        \n
                                        \n\n
                                        \n

                                        3 * 3 = 9 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 + 3 = 6 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 * 4 = 12 apples

                                        \n
                                        \n\n
                                        \n

                                        3 + 4 = 7 apples

                                        \n
                                        \n\n
                                        \n

                                        3 * 4 = 12 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 + 4 = 7 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 * 4 = 12 pears

                                        \n
                                        \n\n
                                        \n

                                        3 + 4 = 7 pears

                                        \n
                                        \n\n
                                        \n

                                        3 * 4 = 12 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 + 4 = 7 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 * 5 = 15 apples

                                        \n
                                        \n\n
                                        \n

                                        3 + 5 = 8 apples

                                        \n
                                        \n\n
                                        \n

                                        3 * 5 = 15 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 + 5 = 8 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 * 5 = 15 pears

                                        \n
                                        \n\n
                                        \n

                                        3 + 5 = 8 pears

                                        \n
                                        \n\n
                                        \n

                                        3 * 5 = 15 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 + 5 = 8 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 * 6 = 18 apples

                                        \n
                                        \n\n
                                        \n

                                        3 + 6 = 9 apples

                                        \n
                                        \n\n
                                        \n

                                        3 * 6 = 18 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 + 6 = 9 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 * 6 = 18 pears

                                        \n
                                        \n\n
                                        \n

                                        3 + 6 = 9 pears

                                        \n
                                        \n\n
                                        \n

                                        3 * 6 = 18 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 + 6 = 9 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 * 7 = 21 apples

                                        \n
                                        \n\n
                                        \n

                                        3 + 7 = 10 apples

                                        \n
                                        \n\n
                                        \n

                                        3 * 7 = 21 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 + 7 = 10 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 * 7 = 21 pears

                                        \n
                                        \n\n
                                        \n

                                        3 + 7 = 10 pears

                                        \n
                                        \n\n
                                        \n

                                        3 * 7 = 21 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 + 7 = 10 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 * 8 = 24 apples

                                        \n
                                        \n\n
                                        \n

                                        3 + 8 = 11 apples

                                        \n
                                        \n\n
                                        \n

                                        3 * 8 = 24 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 + 8 = 11 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 * 8 = 24 pears

                                        \n
                                        \n\n
                                        \n

                                        3 + 8 = 11 pears

                                        \n
                                        \n\n
                                        \n

                                        3 * 8 = 24 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 + 8 = 11 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 * 9 = 27 apples

                                        \n
                                        \n\n
                                        \n

                                        3 + 9 = 12 apples

                                        \n
                                        \n\n
                                        \n

                                        3 * 9 = 27 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 + 9 = 12 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 * 9 = 27 pears

                                        \n
                                        \n\n
                                        \n

                                        3 + 9 = 12 pears

                                        \n
                                        \n\n
                                        \n

                                        3 * 9 = 27 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 + 9 = 12 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 * 10 = 30 apples

                                        \n
                                        \n\n
                                        \n

                                        3 + 10 = 13 apples

                                        \n
                                        \n\n
                                        \n

                                        3 * 10 = 30 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 + 10 = 13 bananas

                                        \n
                                        \n\n
                                        \n

                                        3 * 10 = 30 pears

                                        \n
                                        \n\n
                                        \n

                                        3 + 10 = 13 pears

                                        \n
                                        \n\n
                                        \n

                                        3 * 10 = 30 kiwis

                                        \n
                                        \n\n
                                        \n

                                        3 + 10 = 13 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 * 0 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        4 + 0 = 4 apples

                                        \n
                                        \n\n
                                        \n

                                        4 * 0 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 + 0 = 4 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 * 0 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        4 + 0 = 4 pears

                                        \n
                                        \n\n
                                        \n

                                        4 * 0 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 + 0 = 4 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 * 1 = 4 apples

                                        \n
                                        \n\n
                                        \n

                                        4 + 1 = 5 apples

                                        \n
                                        \n\n
                                        \n

                                        4 * 1 = 4 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 + 1 = 5 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 * 1 = 4 pears

                                        \n
                                        \n\n
                                        \n

                                        4 + 1 = 5 pears

                                        \n
                                        \n\n
                                        \n

                                        4 * 1 = 4 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 + 1 = 5 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 * 2 = 8 apples

                                        \n
                                        \n\n
                                        \n

                                        4 + 2 = 6 apples

                                        \n
                                        \n\n
                                        \n

                                        4 * 2 = 8 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 + 2 = 6 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 * 2 = 8 pears

                                        \n
                                        \n\n
                                        \n

                                        4 + 2 = 6 pears

                                        \n
                                        \n\n
                                        \n

                                        4 * 2 = 8 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 + 2 = 6 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 * 3 = 12 apples

                                        \n
                                        \n\n
                                        \n

                                        4 + 3 = 7 apples

                                        \n
                                        \n\n
                                        \n

                                        4 * 3 = 12 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 + 3 = 7 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 * 3 = 12 pears

                                        \n
                                        \n\n
                                        \n

                                        4 + 3 = 7 pears

                                        \n
                                        \n\n
                                        \n

                                        4 * 3 = 12 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 + 3 = 7 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 * 4 = 16 apples

                                        \n
                                        \n\n
                                        \n

                                        4 + 4 = 8 apples

                                        \n
                                        \n\n
                                        \n

                                        4 * 4 = 16 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 + 4 = 8 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 * 4 = 16 pears

                                        \n
                                        \n\n
                                        \n

                                        4 + 4 = 8 pears

                                        \n
                                        \n\n
                                        \n

                                        4 * 4 = 16 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 + 4 = 8 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 * 5 = 20 apples

                                        \n
                                        \n\n
                                        \n

                                        4 + 5 = 9 apples

                                        \n
                                        \n\n
                                        \n

                                        4 * 5 = 20 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 + 5 = 9 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 * 5 = 20 pears

                                        \n
                                        \n\n
                                        \n

                                        4 + 5 = 9 pears

                                        \n
                                        \n\n
                                        \n

                                        4 * 5 = 20 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 + 5 = 9 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 * 6 = 24 apples

                                        \n
                                        \n\n
                                        \n

                                        4 + 6 = 10 apples

                                        \n
                                        \n\n
                                        \n

                                        4 * 6 = 24 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 + 6 = 10 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 * 6 = 24 pears

                                        \n
                                        \n\n
                                        \n

                                        4 + 6 = 10 pears

                                        \n
                                        \n\n
                                        \n

                                        4 * 6 = 24 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 + 6 = 10 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 * 7 = 28 apples

                                        \n
                                        \n\n
                                        \n

                                        4 + 7 = 11 apples

                                        \n
                                        \n\n
                                        \n

                                        4 * 7 = 28 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 + 7 = 11 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 * 7 = 28 pears

                                        \n
                                        \n\n
                                        \n

                                        4 + 7 = 11 pears

                                        \n
                                        \n\n
                                        \n

                                        4 * 7 = 28 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 + 7 = 11 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 * 8 = 32 apples

                                        \n
                                        \n\n
                                        \n

                                        4 + 8 = 12 apples

                                        \n
                                        \n\n
                                        \n

                                        4 * 8 = 32 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 + 8 = 12 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 * 8 = 32 pears

                                        \n
                                        \n\n
                                        \n

                                        4 + 8 = 12 pears

                                        \n
                                        \n\n
                                        \n

                                        4 * 8 = 32 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 + 8 = 12 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 * 9 = 36 apples

                                        \n
                                        \n\n
                                        \n

                                        4 + 9 = 13 apples

                                        \n
                                        \n\n
                                        \n

                                        4 * 9 = 36 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 + 9 = 13 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 * 9 = 36 pears

                                        \n
                                        \n\n
                                        \n

                                        4 + 9 = 13 pears

                                        \n
                                        \n\n
                                        \n

                                        4 * 9 = 36 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 + 9 = 13 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 * 10 = 40 apples

                                        \n
                                        \n\n
                                        \n

                                        4 + 10 = 14 apples

                                        \n
                                        \n\n
                                        \n

                                        4 * 10 = 40 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 + 10 = 14 bananas

                                        \n
                                        \n\n
                                        \n

                                        4 * 10 = 40 pears

                                        \n
                                        \n\n
                                        \n

                                        4 + 10 = 14 pears

                                        \n
                                        \n\n
                                        \n

                                        4 * 10 = 40 kiwis

                                        \n
                                        \n\n
                                        \n

                                        4 + 10 = 14 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 * 0 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        5 + 0 = 5 apples

                                        \n
                                        \n\n
                                        \n

                                        5 * 0 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 + 0 = 5 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 * 0 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        5 + 0 = 5 pears

                                        \n
                                        \n\n
                                        \n

                                        5 * 0 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 + 0 = 5 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 * 1 = 5 apples

                                        \n
                                        \n\n
                                        \n

                                        5 + 1 = 6 apples

                                        \n
                                        \n\n
                                        \n

                                        5 * 1 = 5 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 + 1 = 6 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 * 1 = 5 pears

                                        \n
                                        \n\n
                                        \n

                                        5 + 1 = 6 pears

                                        \n
                                        \n\n
                                        \n

                                        5 * 1 = 5 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 + 1 = 6 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 * 2 = 10 apples

                                        \n
                                        \n\n
                                        \n

                                        5 + 2 = 7 apples

                                        \n
                                        \n\n
                                        \n

                                        5 * 2 = 10 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 + 2 = 7 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 * 2 = 10 pears

                                        \n
                                        \n\n
                                        \n

                                        5 + 2 = 7 pears

                                        \n
                                        \n\n
                                        \n

                                        5 * 2 = 10 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 + 2 = 7 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 * 3 = 15 apples

                                        \n
                                        \n\n
                                        \n

                                        5 + 3 = 8 apples

                                        \n
                                        \n\n
                                        \n

                                        5 * 3 = 15 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 + 3 = 8 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 * 3 = 15 pears

                                        \n
                                        \n\n
                                        \n

                                        5 + 3 = 8 pears

                                        \n
                                        \n\n
                                        \n

                                        5 * 3 = 15 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 + 3 = 8 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 * 4 = 20 apples

                                        \n
                                        \n\n
                                        \n

                                        5 + 4 = 9 apples

                                        \n
                                        \n\n
                                        \n

                                        5 * 4 = 20 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 + 4 = 9 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 * 4 = 20 pears

                                        \n
                                        \n\n
                                        \n

                                        5 + 4 = 9 pears

                                        \n
                                        \n\n
                                        \n

                                        5 * 4 = 20 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 + 4 = 9 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 * 5 = 25 apples

                                        \n
                                        \n\n
                                        \n

                                        5 + 5 = 10 apples

                                        \n
                                        \n\n
                                        \n

                                        5 * 5 = 25 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 + 5 = 10 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 * 5 = 25 pears

                                        \n
                                        \n\n
                                        \n

                                        5 + 5 = 10 pears

                                        \n
                                        \n\n
                                        \n

                                        5 * 5 = 25 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 + 5 = 10 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 * 6 = 30 apples

                                        \n
                                        \n\n
                                        \n

                                        5 + 6 = 11 apples

                                        \n
                                        \n\n
                                        \n

                                        5 * 6 = 30 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 + 6 = 11 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 * 6 = 30 pears

                                        \n
                                        \n\n
                                        \n

                                        5 + 6 = 11 pears

                                        \n
                                        \n\n
                                        \n

                                        5 * 6 = 30 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 + 6 = 11 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 * 7 = 35 apples

                                        \n
                                        \n\n
                                        \n

                                        5 + 7 = 12 apples

                                        \n
                                        \n\n
                                        \n

                                        5 * 7 = 35 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 + 7 = 12 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 * 7 = 35 pears

                                        \n
                                        \n\n
                                        \n

                                        5 + 7 = 12 pears

                                        \n
                                        \n\n
                                        \n

                                        5 * 7 = 35 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 + 7 = 12 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 * 8 = 40 apples

                                        \n
                                        \n\n
                                        \n

                                        5 + 8 = 13 apples

                                        \n
                                        \n\n
                                        \n

                                        5 * 8 = 40 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 + 8 = 13 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 * 8 = 40 pears

                                        \n
                                        \n\n
                                        \n

                                        5 + 8 = 13 pears

                                        \n
                                        \n\n
                                        \n

                                        5 * 8 = 40 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 + 8 = 13 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 * 9 = 45 apples

                                        \n
                                        \n\n
                                        \n

                                        5 + 9 = 14 apples

                                        \n
                                        \n\n
                                        \n

                                        5 * 9 = 45 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 + 9 = 14 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 * 9 = 45 pears

                                        \n
                                        \n\n
                                        \n

                                        5 + 9 = 14 pears

                                        \n
                                        \n\n
                                        \n

                                        5 * 9 = 45 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 + 9 = 14 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 * 10 = 50 apples

                                        \n
                                        \n\n
                                        \n

                                        5 + 10 = 15 apples

                                        \n
                                        \n\n
                                        \n

                                        5 * 10 = 50 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 + 10 = 15 bananas

                                        \n
                                        \n\n
                                        \n

                                        5 * 10 = 50 pears

                                        \n
                                        \n\n
                                        \n

                                        5 + 10 = 15 pears

                                        \n
                                        \n\n
                                        \n

                                        5 * 10 = 50 kiwis

                                        \n
                                        \n\n
                                        \n

                                        5 + 10 = 15 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 * 0 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        6 + 0 = 6 apples

                                        \n
                                        \n\n
                                        \n

                                        6 * 0 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 + 0 = 6 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 * 0 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        6 + 0 = 6 pears

                                        \n
                                        \n\n
                                        \n

                                        6 * 0 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 + 0 = 6 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 * 1 = 6 apples

                                        \n
                                        \n\n
                                        \n

                                        6 + 1 = 7 apples

                                        \n
                                        \n\n
                                        \n

                                        6 * 1 = 6 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 + 1 = 7 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 * 1 = 6 pears

                                        \n
                                        \n\n
                                        \n

                                        6 + 1 = 7 pears

                                        \n
                                        \n\n
                                        \n

                                        6 * 1 = 6 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 + 1 = 7 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 * 2 = 12 apples

                                        \n
                                        \n\n
                                        \n

                                        6 + 2 = 8 apples

                                        \n
                                        \n\n
                                        \n

                                        6 * 2 = 12 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 + 2 = 8 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 * 2 = 12 pears

                                        \n
                                        \n\n
                                        \n

                                        6 + 2 = 8 pears

                                        \n
                                        \n\n
                                        \n

                                        6 * 2 = 12 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 + 2 = 8 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 * 3 = 18 apples

                                        \n
                                        \n\n
                                        \n

                                        6 + 3 = 9 apples

                                        \n
                                        \n\n
                                        \n

                                        6 * 3 = 18 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 + 3 = 9 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 * 3 = 18 pears

                                        \n
                                        \n\n
                                        \n

                                        6 + 3 = 9 pears

                                        \n
                                        \n\n
                                        \n

                                        6 * 3 = 18 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 + 3 = 9 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 * 4 = 24 apples

                                        \n
                                        \n\n
                                        \n

                                        6 + 4 = 10 apples

                                        \n
                                        \n\n
                                        \n

                                        6 * 4 = 24 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 + 4 = 10 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 * 4 = 24 pears

                                        \n
                                        \n\n
                                        \n

                                        6 + 4 = 10 pears

                                        \n
                                        \n\n
                                        \n

                                        6 * 4 = 24 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 + 4 = 10 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 * 5 = 30 apples

                                        \n
                                        \n\n
                                        \n

                                        6 + 5 = 11 apples

                                        \n
                                        \n\n
                                        \n

                                        6 * 5 = 30 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 + 5 = 11 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 * 5 = 30 pears

                                        \n
                                        \n\n
                                        \n

                                        6 + 5 = 11 pears

                                        \n
                                        \n\n
                                        \n

                                        6 * 5 = 30 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 + 5 = 11 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 * 6 = 36 apples

                                        \n
                                        \n\n
                                        \n

                                        6 + 6 = 12 apples

                                        \n
                                        \n\n
                                        \n

                                        6 * 6 = 36 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 + 6 = 12 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 * 6 = 36 pears

                                        \n
                                        \n\n
                                        \n

                                        6 + 6 = 12 pears

                                        \n
                                        \n\n
                                        \n

                                        6 * 6 = 36 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 + 6 = 12 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 * 7 = 42 apples

                                        \n
                                        \n\n
                                        \n

                                        6 + 7 = 13 apples

                                        \n
                                        \n\n
                                        \n

                                        6 * 7 = 42 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 + 7 = 13 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 * 7 = 42 pears

                                        \n
                                        \n\n
                                        \n

                                        6 + 7 = 13 pears

                                        \n
                                        \n\n
                                        \n

                                        6 * 7 = 42 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 + 7 = 13 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 * 8 = 48 apples

                                        \n
                                        \n\n
                                        \n

                                        6 + 8 = 14 apples

                                        \n
                                        \n\n
                                        \n

                                        6 * 8 = 48 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 + 8 = 14 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 * 8 = 48 pears

                                        \n
                                        \n\n
                                        \n

                                        6 + 8 = 14 pears

                                        \n
                                        \n\n
                                        \n

                                        6 * 8 = 48 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 + 8 = 14 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 * 9 = 54 apples

                                        \n
                                        \n\n
                                        \n

                                        6 + 9 = 15 apples

                                        \n
                                        \n\n
                                        \n

                                        6 * 9 = 54 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 + 9 = 15 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 * 9 = 54 pears

                                        \n
                                        \n\n
                                        \n

                                        6 + 9 = 15 pears

                                        \n
                                        \n\n
                                        \n

                                        6 * 9 = 54 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 + 9 = 15 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 * 10 = 60 apples

                                        \n
                                        \n\n
                                        \n

                                        6 + 10 = 16 apples

                                        \n
                                        \n\n
                                        \n

                                        6 * 10 = 60 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 + 10 = 16 bananas

                                        \n
                                        \n\n
                                        \n

                                        6 * 10 = 60 pears

                                        \n
                                        \n\n
                                        \n

                                        6 + 10 = 16 pears

                                        \n
                                        \n\n
                                        \n

                                        6 * 10 = 60 kiwis

                                        \n
                                        \n\n
                                        \n

                                        6 + 10 = 16 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 * 0 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        7 + 0 = 7 apples

                                        \n
                                        \n\n
                                        \n

                                        7 * 0 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 + 0 = 7 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 * 0 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        7 + 0 = 7 pears

                                        \n
                                        \n\n
                                        \n

                                        7 * 0 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 + 0 = 7 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 * 1 = 7 apples

                                        \n
                                        \n\n
                                        \n

                                        7 + 1 = 8 apples

                                        \n
                                        \n\n
                                        \n

                                        7 * 1 = 7 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 + 1 = 8 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 * 1 = 7 pears

                                        \n
                                        \n\n
                                        \n

                                        7 + 1 = 8 pears

                                        \n
                                        \n\n
                                        \n

                                        7 * 1 = 7 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 + 1 = 8 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 * 2 = 14 apples

                                        \n
                                        \n\n
                                        \n

                                        7 + 2 = 9 apples

                                        \n
                                        \n\n
                                        \n

                                        7 * 2 = 14 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 + 2 = 9 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 * 2 = 14 pears

                                        \n
                                        \n\n
                                        \n

                                        7 + 2 = 9 pears

                                        \n
                                        \n\n
                                        \n

                                        7 * 2 = 14 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 + 2 = 9 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 * 3 = 21 apples

                                        \n
                                        \n\n
                                        \n

                                        7 + 3 = 10 apples

                                        \n
                                        \n\n
                                        \n

                                        7 * 3 = 21 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 + 3 = 10 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 * 3 = 21 pears

                                        \n
                                        \n\n
                                        \n

                                        7 + 3 = 10 pears

                                        \n
                                        \n\n
                                        \n

                                        7 * 3 = 21 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 + 3 = 10 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 * 4 = 28 apples

                                        \n
                                        \n\n
                                        \n

                                        7 + 4 = 11 apples

                                        \n
                                        \n\n
                                        \n

                                        7 * 4 = 28 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 + 4 = 11 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 * 4 = 28 pears

                                        \n
                                        \n\n
                                        \n

                                        7 + 4 = 11 pears

                                        \n
                                        \n\n
                                        \n

                                        7 * 4 = 28 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 + 4 = 11 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 * 5 = 35 apples

                                        \n
                                        \n\n
                                        \n

                                        7 + 5 = 12 apples

                                        \n
                                        \n\n
                                        \n

                                        7 * 5 = 35 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 + 5 = 12 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 * 5 = 35 pears

                                        \n
                                        \n\n
                                        \n

                                        7 + 5 = 12 pears

                                        \n
                                        \n\n
                                        \n

                                        7 * 5 = 35 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 + 5 = 12 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 * 6 = 42 apples

                                        \n
                                        \n\n
                                        \n

                                        7 + 6 = 13 apples

                                        \n
                                        \n\n
                                        \n

                                        7 * 6 = 42 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 + 6 = 13 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 * 6 = 42 pears

                                        \n
                                        \n\n
                                        \n

                                        7 + 6 = 13 pears

                                        \n
                                        \n\n
                                        \n

                                        7 * 6 = 42 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 + 6 = 13 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 * 7 = 49 apples

                                        \n
                                        \n\n
                                        \n

                                        7 + 7 = 14 apples

                                        \n
                                        \n\n
                                        \n

                                        7 * 7 = 49 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 + 7 = 14 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 * 7 = 49 pears

                                        \n
                                        \n\n
                                        \n

                                        7 + 7 = 14 pears

                                        \n
                                        \n\n
                                        \n

                                        7 * 7 = 49 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 + 7 = 14 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 * 8 = 56 apples

                                        \n
                                        \n\n
                                        \n

                                        7 + 8 = 15 apples

                                        \n
                                        \n\n
                                        \n

                                        7 * 8 = 56 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 + 8 = 15 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 * 8 = 56 pears

                                        \n
                                        \n\n
                                        \n

                                        7 + 8 = 15 pears

                                        \n
                                        \n\n
                                        \n

                                        7 * 8 = 56 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 + 8 = 15 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 * 9 = 63 apples

                                        \n
                                        \n\n
                                        \n

                                        7 + 9 = 16 apples

                                        \n
                                        \n\n
                                        \n

                                        7 * 9 = 63 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 + 9 = 16 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 * 9 = 63 pears

                                        \n
                                        \n\n
                                        \n

                                        7 + 9 = 16 pears

                                        \n
                                        \n\n
                                        \n

                                        7 * 9 = 63 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 + 9 = 16 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 * 10 = 70 apples

                                        \n
                                        \n\n
                                        \n

                                        7 + 10 = 17 apples

                                        \n
                                        \n\n
                                        \n

                                        7 * 10 = 70 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 + 10 = 17 bananas

                                        \n
                                        \n\n
                                        \n

                                        7 * 10 = 70 pears

                                        \n
                                        \n\n
                                        \n

                                        7 + 10 = 17 pears

                                        \n
                                        \n\n
                                        \n

                                        7 * 10 = 70 kiwis

                                        \n
                                        \n\n
                                        \n

                                        7 + 10 = 17 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 * 0 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        8 + 0 = 8 apples

                                        \n
                                        \n\n
                                        \n

                                        8 * 0 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 + 0 = 8 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 * 0 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        8 + 0 = 8 pears

                                        \n
                                        \n\n
                                        \n

                                        8 * 0 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 + 0 = 8 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 * 1 = 8 apples

                                        \n
                                        \n\n
                                        \n

                                        8 + 1 = 9 apples

                                        \n
                                        \n\n
                                        \n

                                        8 * 1 = 8 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 + 1 = 9 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 * 1 = 8 pears

                                        \n
                                        \n\n
                                        \n

                                        8 + 1 = 9 pears

                                        \n
                                        \n\n
                                        \n

                                        8 * 1 = 8 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 + 1 = 9 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 * 2 = 16 apples

                                        \n
                                        \n\n
                                        \n

                                        8 + 2 = 10 apples

                                        \n
                                        \n\n
                                        \n

                                        8 * 2 = 16 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 + 2 = 10 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 * 2 = 16 pears

                                        \n
                                        \n\n
                                        \n

                                        8 + 2 = 10 pears

                                        \n
                                        \n\n
                                        \n

                                        8 * 2 = 16 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 + 2 = 10 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 * 3 = 24 apples

                                        \n
                                        \n\n
                                        \n

                                        8 + 3 = 11 apples

                                        \n
                                        \n\n
                                        \n

                                        8 * 3 = 24 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 + 3 = 11 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 * 3 = 24 pears

                                        \n
                                        \n\n
                                        \n

                                        8 + 3 = 11 pears

                                        \n
                                        \n\n
                                        \n

                                        8 * 3 = 24 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 + 3 = 11 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 * 4 = 32 apples

                                        \n
                                        \n\n
                                        \n

                                        8 + 4 = 12 apples

                                        \n
                                        \n\n
                                        \n

                                        8 * 4 = 32 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 + 4 = 12 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 * 4 = 32 pears

                                        \n
                                        \n\n
                                        \n

                                        8 + 4 = 12 pears

                                        \n
                                        \n\n
                                        \n

                                        8 * 4 = 32 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 + 4 = 12 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 * 5 = 40 apples

                                        \n
                                        \n\n
                                        \n

                                        8 + 5 = 13 apples

                                        \n
                                        \n\n
                                        \n

                                        8 * 5 = 40 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 + 5 = 13 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 * 5 = 40 pears

                                        \n
                                        \n\n
                                        \n

                                        8 + 5 = 13 pears

                                        \n
                                        \n\n
                                        \n

                                        8 * 5 = 40 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 + 5 = 13 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 * 6 = 48 apples

                                        \n
                                        \n\n
                                        \n

                                        8 + 6 = 14 apples

                                        \n
                                        \n\n
                                        \n

                                        8 * 6 = 48 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 + 6 = 14 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 * 6 = 48 pears

                                        \n
                                        \n\n
                                        \n

                                        8 + 6 = 14 pears

                                        \n
                                        \n\n
                                        \n

                                        8 * 6 = 48 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 + 6 = 14 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 * 7 = 56 apples

                                        \n
                                        \n\n
                                        \n

                                        8 + 7 = 15 apples

                                        \n
                                        \n\n
                                        \n

                                        8 * 7 = 56 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 + 7 = 15 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 * 7 = 56 pears

                                        \n
                                        \n\n
                                        \n

                                        8 + 7 = 15 pears

                                        \n
                                        \n\n
                                        \n

                                        8 * 7 = 56 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 + 7 = 15 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 * 8 = 64 apples

                                        \n
                                        \n\n
                                        \n

                                        8 + 8 = 16 apples

                                        \n
                                        \n\n
                                        \n

                                        8 * 8 = 64 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 + 8 = 16 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 * 8 = 64 pears

                                        \n
                                        \n\n
                                        \n

                                        8 + 8 = 16 pears

                                        \n
                                        \n\n
                                        \n

                                        8 * 8 = 64 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 + 8 = 16 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 * 9 = 72 apples

                                        \n
                                        \n\n
                                        \n

                                        8 + 9 = 17 apples

                                        \n
                                        \n\n
                                        \n

                                        8 * 9 = 72 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 + 9 = 17 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 * 9 = 72 pears

                                        \n
                                        \n\n
                                        \n

                                        8 + 9 = 17 pears

                                        \n
                                        \n\n
                                        \n

                                        8 * 9 = 72 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 + 9 = 17 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 * 10 = 80 apples

                                        \n
                                        \n\n
                                        \n

                                        8 + 10 = 18 apples

                                        \n
                                        \n\n
                                        \n

                                        8 * 10 = 80 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 + 10 = 18 bananas

                                        \n
                                        \n\n
                                        \n

                                        8 * 10 = 80 pears

                                        \n
                                        \n\n
                                        \n

                                        8 + 10 = 18 pears

                                        \n
                                        \n\n
                                        \n

                                        8 * 10 = 80 kiwis

                                        \n
                                        \n\n
                                        \n

                                        8 + 10 = 18 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 * 0 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        9 + 0 = 9 apples

                                        \n
                                        \n\n
                                        \n

                                        9 * 0 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 + 0 = 9 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 * 0 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        9 + 0 = 9 pears

                                        \n
                                        \n\n
                                        \n

                                        9 * 0 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 + 0 = 9 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 * 1 = 9 apples

                                        \n
                                        \n\n
                                        \n

                                        9 + 1 = 10 apples

                                        \n
                                        \n\n
                                        \n

                                        9 * 1 = 9 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 + 1 = 10 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 * 1 = 9 pears

                                        \n
                                        \n\n
                                        \n

                                        9 + 1 = 10 pears

                                        \n
                                        \n\n
                                        \n

                                        9 * 1 = 9 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 + 1 = 10 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 * 2 = 18 apples

                                        \n
                                        \n\n
                                        \n

                                        9 + 2 = 11 apples

                                        \n
                                        \n\n
                                        \n

                                        9 * 2 = 18 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 + 2 = 11 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 * 2 = 18 pears

                                        \n
                                        \n\n
                                        \n

                                        9 + 2 = 11 pears

                                        \n
                                        \n\n
                                        \n

                                        9 * 2 = 18 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 + 2 = 11 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 * 3 = 27 apples

                                        \n
                                        \n\n
                                        \n

                                        9 + 3 = 12 apples

                                        \n
                                        \n\n
                                        \n

                                        9 * 3 = 27 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 + 3 = 12 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 * 3 = 27 pears

                                        \n
                                        \n\n
                                        \n

                                        9 + 3 = 12 pears

                                        \n
                                        \n\n
                                        \n

                                        9 * 3 = 27 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 + 3 = 12 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 * 4 = 36 apples

                                        \n
                                        \n\n
                                        \n

                                        9 + 4 = 13 apples

                                        \n
                                        \n\n
                                        \n

                                        9 * 4 = 36 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 + 4 = 13 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 * 4 = 36 pears

                                        \n
                                        \n\n
                                        \n

                                        9 + 4 = 13 pears

                                        \n
                                        \n\n
                                        \n

                                        9 * 4 = 36 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 + 4 = 13 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 * 5 = 45 apples

                                        \n
                                        \n\n
                                        \n

                                        9 + 5 = 14 apples

                                        \n
                                        \n\n
                                        \n

                                        9 * 5 = 45 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 + 5 = 14 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 * 5 = 45 pears

                                        \n
                                        \n\n
                                        \n

                                        9 + 5 = 14 pears

                                        \n
                                        \n\n
                                        \n

                                        9 * 5 = 45 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 + 5 = 14 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 * 6 = 54 apples

                                        \n
                                        \n\n
                                        \n

                                        9 + 6 = 15 apples

                                        \n
                                        \n\n
                                        \n

                                        9 * 6 = 54 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 + 6 = 15 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 * 6 = 54 pears

                                        \n
                                        \n\n
                                        \n

                                        9 + 6 = 15 pears

                                        \n
                                        \n\n
                                        \n

                                        9 * 6 = 54 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 + 6 = 15 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 * 7 = 63 apples

                                        \n
                                        \n\n
                                        \n

                                        9 + 7 = 16 apples

                                        \n
                                        \n\n
                                        \n

                                        9 * 7 = 63 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 + 7 = 16 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 * 7 = 63 pears

                                        \n
                                        \n\n
                                        \n

                                        9 + 7 = 16 pears

                                        \n
                                        \n\n
                                        \n

                                        9 * 7 = 63 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 + 7 = 16 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 * 8 = 72 apples

                                        \n
                                        \n\n
                                        \n

                                        9 + 8 = 17 apples

                                        \n
                                        \n\n
                                        \n

                                        9 * 8 = 72 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 + 8 = 17 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 * 8 = 72 pears

                                        \n
                                        \n\n
                                        \n

                                        9 + 8 = 17 pears

                                        \n
                                        \n\n
                                        \n

                                        9 * 8 = 72 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 + 8 = 17 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 * 9 = 81 apples

                                        \n
                                        \n\n
                                        \n

                                        9 + 9 = 18 apples

                                        \n
                                        \n\n
                                        \n

                                        9 * 9 = 81 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 + 9 = 18 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 * 9 = 81 pears

                                        \n
                                        \n\n
                                        \n

                                        9 + 9 = 18 pears

                                        \n
                                        \n\n
                                        \n

                                        9 * 9 = 81 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 + 9 = 18 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 * 10 = 90 apples

                                        \n
                                        \n\n
                                        \n

                                        9 + 10 = 19 apples

                                        \n
                                        \n\n
                                        \n

                                        9 * 10 = 90 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 + 10 = 19 bananas

                                        \n
                                        \n\n
                                        \n

                                        9 * 10 = 90 pears

                                        \n
                                        \n\n
                                        \n

                                        9 + 10 = 19 pears

                                        \n
                                        \n\n
                                        \n

                                        9 * 10 = 90 kiwis

                                        \n
                                        \n\n
                                        \n

                                        9 + 10 = 19 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 * 0 = 0 apples

                                        \n
                                        \n\n
                                        \n

                                        10 + 0 = 10 apples

                                        \n
                                        \n\n
                                        \n

                                        10 * 0 = 0 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 + 0 = 10 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 * 0 = 0 pears

                                        \n
                                        \n\n
                                        \n

                                        10 + 0 = 10 pears

                                        \n
                                        \n\n
                                        \n

                                        10 * 0 = 0 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 + 0 = 10 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 * 1 = 10 apples

                                        \n
                                        \n\n
                                        \n

                                        10 + 1 = 11 apples

                                        \n
                                        \n\n
                                        \n

                                        10 * 1 = 10 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 + 1 = 11 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 * 1 = 10 pears

                                        \n
                                        \n\n
                                        \n

                                        10 + 1 = 11 pears

                                        \n
                                        \n\n
                                        \n

                                        10 * 1 = 10 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 + 1 = 11 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 * 2 = 20 apples

                                        \n
                                        \n\n
                                        \n

                                        10 + 2 = 12 apples

                                        \n
                                        \n\n
                                        \n

                                        10 * 2 = 20 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 + 2 = 12 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 * 2 = 20 pears

                                        \n
                                        \n\n
                                        \n

                                        10 + 2 = 12 pears

                                        \n
                                        \n\n
                                        \n

                                        10 * 2 = 20 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 + 2 = 12 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 * 3 = 30 apples

                                        \n
                                        \n\n
                                        \n

                                        10 + 3 = 13 apples

                                        \n
                                        \n\n
                                        \n

                                        10 * 3 = 30 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 + 3 = 13 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 * 3 = 30 pears

                                        \n
                                        \n\n
                                        \n

                                        10 + 3 = 13 pears

                                        \n
                                        \n\n
                                        \n

                                        10 * 3 = 30 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 + 3 = 13 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 * 4 = 40 apples

                                        \n
                                        \n\n
                                        \n

                                        10 + 4 = 14 apples

                                        \n
                                        \n\n
                                        \n

                                        10 * 4 = 40 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 + 4 = 14 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 * 4 = 40 pears

                                        \n
                                        \n\n
                                        \n

                                        10 + 4 = 14 pears

                                        \n
                                        \n\n
                                        \n

                                        10 * 4 = 40 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 + 4 = 14 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 * 5 = 50 apples

                                        \n
                                        \n\n
                                        \n

                                        10 + 5 = 15 apples

                                        \n
                                        \n\n
                                        \n

                                        10 * 5 = 50 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 + 5 = 15 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 * 5 = 50 pears

                                        \n
                                        \n\n
                                        \n

                                        10 + 5 = 15 pears

                                        \n
                                        \n\n
                                        \n

                                        10 * 5 = 50 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 + 5 = 15 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 * 6 = 60 apples

                                        \n
                                        \n\n
                                        \n

                                        10 + 6 = 16 apples

                                        \n
                                        \n\n
                                        \n

                                        10 * 6 = 60 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 + 6 = 16 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 * 6 = 60 pears

                                        \n
                                        \n\n
                                        \n

                                        10 + 6 = 16 pears

                                        \n
                                        \n\n
                                        \n

                                        10 * 6 = 60 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 + 6 = 16 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 * 7 = 70 apples

                                        \n
                                        \n\n
                                        \n

                                        10 + 7 = 17 apples

                                        \n
                                        \n\n
                                        \n

                                        10 * 7 = 70 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 + 7 = 17 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 * 7 = 70 pears

                                        \n
                                        \n\n
                                        \n

                                        10 + 7 = 17 pears

                                        \n
                                        \n\n
                                        \n

                                        10 * 7 = 70 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 + 7 = 17 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 * 8 = 80 apples

                                        \n
                                        \n\n
                                        \n

                                        10 + 8 = 18 apples

                                        \n
                                        \n\n
                                        \n

                                        10 * 8 = 80 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 + 8 = 18 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 * 8 = 80 pears

                                        \n
                                        \n\n
                                        \n

                                        10 + 8 = 18 pears

                                        \n
                                        \n\n
                                        \n

                                        10 * 8 = 80 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 + 8 = 18 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 * 9 = 90 apples

                                        \n
                                        \n\n
                                        \n

                                        10 + 9 = 19 apples

                                        \n
                                        \n\n
                                        \n

                                        10 * 9 = 90 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 + 9 = 19 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 * 9 = 90 pears

                                        \n
                                        \n\n
                                        \n

                                        10 + 9 = 19 pears

                                        \n
                                        \n\n
                                        \n

                                        10 * 9 = 90 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 + 9 = 19 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 * 10 = 100 apples

                                        \n
                                        \n\n
                                        \n

                                        10 + 10 = 20 apples

                                        \n
                                        \n\n
                                        \n

                                        10 * 10 = 100 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 + 10 = 20 bananas

                                        \n
                                        \n\n
                                        \n

                                        10 * 10 = 100 pears

                                        \n
                                        \n\n
                                        \n

                                        10 + 10 = 20 pears

                                        \n
                                        \n\n
                                        \n

                                        10 * 10 = 100 kiwis

                                        \n
                                        \n\n
                                        \n

                                        10 + 10 = 20 kiwis

                                        \n
                                        \n\n a: \n
                                        \nb: \n
                                        \nfruit: \n
                                        \nop: multiply: add: \n
                                        \n\n\n\n\n## Example 3: Sympy Factoring\n\n\n```\nfrom sympy import init_printing\ninit_printing()\n```\n\n\n```\nfrom sympy import Symbol, Eq, factor\nx = Symbol('x')\ndef factorit(n):\n return Eq(x ** n - 1, factor(x ** n - 1))\nfactorit(4)\n```\n\n\n```\nfrom ipywidgets import StaticInteract, RangeWidget\n\nStaticInteract(factorit,\n n=RangeWidget(2, 10))\n```\n\n\n\n\n\n \n\n
                                        \n\n
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                                        \n \n
                                        \n\n
                                        \n \n
                                        \n\n n: \n
                                        \n\n\n\n\n## Example 4: Matplotlib Figures\n\n\n```\n%matplotlib inline\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nfrom ipywidgets import StaticInteract, RangeWidget, RadioWidget\n\ndef plot(amplitude, color):\n fig, ax = plt.subplots(figsize=(4, 3),\n subplot_kw={'axisbg':'#EEEEEE',\n 'axisbelow':True})\n ax.grid(color='w', linewidth=2, linestyle='solid')\n x = np.linspace(0, 10, 1000)\n ax.plot(x, amplitude * np.sin(x), color=color,\n lw=5, alpha=0.4)\n ax.set_xlim(0, 10)\n ax.set_ylim(-1.1, 1.1)\n return fig\n\nStaticInteract(plot,\n amplitude=RangeWidget(0.1, 1.0, 0.1),\n color=RadioWidget(['blue', 'green', 'red']))\n```\n\n\n\n\n\n \n\n
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                                        \n \n
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                                        \n \n
                                        \n\n
                                        \n \n
                                        \n\n amplitude: \n
                                        \ncolor: blue: green: red: \n
                                        \n\n\n\n\n\n```\n\n```\n", "meta": {"hexsha": "17a9d3e92f276fdced1766e859a286afe61411fc", "size": 608440, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "example.ipynb", "max_stars_repo_name": "jakevdp/ipywidgets-unsupported", "max_stars_repo_head_hexsha": "f45415931833d99bb069992a4b428b89a9ac905d", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 23, "max_stars_repo_stars_event_min_datetime": "2016-03-08T13:46:06.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-16T11:47:57.000Z", "max_issues_repo_path": "example.ipynb", "max_issues_repo_name": "jeffhussmann/ipywidgets-unsupported", "max_issues_repo_head_hexsha": "a1e889ed6af0e3607dfda55a5f3404140d1f56c2", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 5, "max_issues_repo_issues_event_min_datetime": "2015-01-27T12:31:40.000Z", "max_issues_repo_issues_event_max_datetime": "2015-09-23T06:42:12.000Z", "max_forks_repo_path": "example.ipynb", "max_forks_repo_name": "jeffhussmann/ipywidgets-unsupported", "max_forks_repo_head_hexsha": "a1e889ed6af0e3607dfda55a5f3404140d1f56c2", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 7, "max_forks_repo_forks_event_min_datetime": "2016-03-08T14:15:47.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-29T03:46:16.000Z", "avg_line_length": 64.3647519306, "max_line_length": 1509, "alphanum_fraction": 0.6823483006, "converted": true, "num_tokens": 45333, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.418149912927306}} {"text": "# The Recurrent Neural Network\n\n## Learning objectives\n\n1. Understand the principles behind the creation of the recurrent neural network\n2. Obtain intuition about difficulties training RNNs, namely: vanishing/exploding gradients and long-term dependencies\n3. Obtain intuition about mechanics of backpropagation through time BPTT\n4. Develop a Long Short-Term memory implementation in Keras \n5. Learn about the uses and limitations of RNNs from a cognitive science perspective\n\n## Historical and theoretical background\n\nThe poet Delmore Schwartz once wrote: **\"...time is the fire in which we burn\"**. We can't escape time. Time is embedded in every human thought and action. Yet, so far, we have been oblivious to the role of time in neural network modeling. Indeed, in all models we have examined so far we have implicitly assumed that **data is \"perceived\" all at once**, although there are countless examples where time is a critical consideration: movement, speech production, planning, decision-making, etc. We also have implicitly assumed that **past-states have no influence in future-states**. This is, the input pattern at time-step $t-1$ does not influence the output of time-step $t-0$, or $t+1$, or any subsequent outcome for that matter. In probabilistic jargon, this equals to assume that each sample is drawn independently from each other. We know in many scenarios this is simply not true: when giving a talk, my next utterance will depend upon my past utterances; when running, my last stride will condition my next stride, and so on. You can imagine endless examples.\n\nMultilayer Perceptrons and Convolutional Networks, in principle, can be used to approach problems where time and sequences are a consideration (for instance [Cui et al, 2016](https://arxiv.org/pdf/1603.06995.pdf)). Nevertheless, introducing time considerations in such architectures is cumbersome, and better architectures have been envisioned. In particular, **Recurrent Neural Networks (RNNs)** are the modern standard to deal with **time-dependent** and/or **sequence-dependent** problems. This type of network is \"recurrent\" in the sense that they can **revisit or reuse past states as inputs to predict the next or future states**. To put it plainly, they have **memory**. Indeed, memory is what allows us to incorporate our past thoughts and behaviors into our future thoughts and behaviors.\n\n### Hopfield Network\n\nOne of the earliest examples of networks incorporating \"recurrences\" was the so-called **Hopfield Network**, introduced in 1982 by [John Hopfield](https://en.wikipedia.org/wiki/John_Hopfield), at the time, a physicist at Caltech. Hopfield networks were important as they helped to reignite the interest in neural networks in the early '80s. In his 1982 paper \"*Neural networks and physical systems with emergent collective computational abilities*\", Hopfield wanted to address the fundamental question of **emergence** in cognitive systems: Can relatively stable cognitive phenomena, like memories, emerge from the collective action of large numbers of simple neurons? After all, such behavior was observed in other physical systems like vortex patterns in fluid flow. Brains seemed like another promising candidate.\n\nHopfield networks are known as a type of **energy-based** (instead of error-based) network because their properties derive from a global energy-function (Raj, 2020). In resemblance to the McCulloch-Pitts neuron, Hopfield neurons are binary threshold units but with recurrent instead of feed-forward connections, where each unit is **bi-directionally connected** to each other, as shown in **Figure 1**. This means that each unit *receives* inputs and *sends* inputs to every other connected unit. A consequence of this architecture is that **weights values are symmetric**, such that weights *coming into* a unit are the same as the ones *coming out* of a unit. The value of each unit is determined by a linear function wrapped into a threshold function $T$, as $y_i = T(\\sum w_{ji}y_j + b_i)$.\n\n
                                        Figure 1: Hopfield Network
                                        \n\n\n\nHopfield network's idea is that each configuration of binary-values $C$ in the network is associated with a **global energy value $-E$**. Here is a simplified picture of the training process: imagine you have a network with five neurons with a configuration of $C_1=(0, 1, 0, 1, 0)$. Now, imagine $C_1$ yields a global energy-value $E_1= 2$ (following the energy function formula). Your goal is to *minimize* $E$ by changing one element of the network $c_i$ at a time. By using the weight updating rule $\\Delta w$, you can subsequently get a new configuration like $C_2=(1, 1, 0, 1, 0)$, as new weights will cause a change in the activation values $(0,1)$. If $C_2$ yields a *lower value of $E$*, let's say, $1.5$, you are moving in the right direction. If you keep iterating with new configurations the network will eventually \"settle\" into a **global energy minimum** (conditioned to the initial state of the network).\n\nA fascinating aspect of Hopfield networks, besides the introduction of recurrence, is that is closely based in neuroscience research about learning and memory, particularly Hebbian learning (Hebb, 1949). In fact, Hopfield (1982) proposed this model as a way to capture **memory formation and retrieval**. The idea is that the energy-minima of the network could represent the **formation of a memory**, which further gives rise to a property known as **[content-addressable memory (CAM)](https://en.wikipedia.org/wiki/Content-addressable_memory)**. Here is the idea with a computer analogy: when you access information stored in the random access memory of your computer (RAM), you give the \"address\" where the \"memory\" is located to retrieve it. CAM works the other way around: you give information about the **content** you are searching for, and the computer should retrieve the \"memory\". This is great because this works even when you have **partial or corrupted** information about the content, which is a much more **realistic depiction of how human memory works**. It is similar to doing a google search. Just think in how many times you have searched for lyrics with partial information, like \"song with the beeeee bop ba bodda bope!\".\n\nIt is important to highlight that the sequential adjustment of Hopfield networks is **not driven by error correction**: there isn't a \"target\" as in supervised-based neural networks. Hopfield networks are systems that \"evolve\" until they find a stable low-energy state. If you \"perturb\" such a system, the system will \"re-evolve\" towards its previous stable-state, similar to how those inflatable \"Bop Bags\" toys get back to their initial position no matter how hard you punch them. It is almost like the system \"remembers\" its previous stable-state (isn't?). This ability to \"return\" to a previous stable-state after the perturbation is why they serve as models of memory.\n\n### Elman Network\n\nAlthough Hopfield networks where innovative and fascinating models, the first successful example of a recurrent network trained with backpropagation was introduced by [Jeffrey Elman](https://en.wikipedia.org/wiki/Jeffrey_Elman), the so-called **Elman Network** (Elman, 1990). Elman was a cognitive scientist at UC San Diego at the time, part of the group of researchers that published the famous PDP book.\n\nIn 1990, Elman published \"Finding Structure in Time\", a highly influential work for in cognitive science. Elman was concerned with the problem of representing \"time\" or \"sequences\" in neural networks. In his view, you could take either an \"explicit\" approach or an \"implicit\" approach. The **explicit** approach represents time **spacially**. Consider a vector $x = [x_1,x_2 \\cdots, x_n]$, where element $x_1$ represents the first value of a sequence, $x_2$ the second element, and $x_n$ the last element. Hence, the spacial location in $\\bf{x}$ is indicating the temporal location of each element. You can think about elements of $\\bf{x}$ as sequences of words or actions, one after the other, for instance: $x^1=[Sound, of, the, funky, drummer]$ is a sequence of length five. Elman saw **several drawbacks** to this approach. First, although $\\bf{x}$ is a sequence, the network still needs to represent the sequence all at once as an input, this is, a network would need five input neurons to process $x^1$. Second, it imposes a rigid limit on the duration of pattern, in other words, the network needs a fixed number of elements for every input vector $\\bf{x}$: a network with five input units, can't accommodate a sequence of length six. True, you could start with a six input network, but then shorter sequences would be misrepresented since mismatched units would receive zero input. This is a problem for most domains where sequences have a variable duration. Finally, it can't easily distinguish **relative** temporal position from **absolute** temporal position. Consider the sequence $s = [1, 1]$ and a vector input length of four bits. Such a sequence can be presented in at least three variations:\n\n$$\nx_1 = [0, 1, 1, 0]\\\\\nx_2 = [0, 0, 1, 1]\\\\\nx_3 = [1, 1, 0, 0]\n$$\n\nHere, $\\bf{x_1}$, $\\bf{x_2}$, and $\\bf{x_3}$ are instances of $\\bf{s}$ but spacially displaced in the input vector. Geometrically, those three vectors are very different from each other (you can compute similarity measures to put a number on that), although representing the same instance. Even though you can train a neural net to learn those three patterns are associated with the same target, their inherent dissimilarity probably will hinder the network's ability to generalize the learned association.\n\nThe **implicit** approach represents time by **its effect in intermediate computations**. To do this, Elman added a **context unit** to save past computations and incorporate those in future computations. In short, **memory**. Elman based his approach in the work of [Michael I. Jordan](https://people.eecs.berkeley.edu/~jordan/) on serial processing (1986). Jordan's network implements recurrent connections from the network output $\\hat{y}$ to its hidden units $h$, via a \"memory unit\" $\\mu$ (equivalent to Elman's \"context unit\") as depicted in **Figure 2**. In short, the memory unit keeps a running average of **all past outputs**: this is how the past history is implicitly accounted for on each new computation. There is no learning in the memory unit, which means the weights are fixed to $1$.\n\n
                                        Figure 2: Jordan Network
                                        \n\n\n\n**Note**: Jordan's network diagrams exemplifies the two ways in which recurrent nets are usually represented. On the left, the **compact format** depicts the network structure as a circuit. On the right, the **unfolded representation** incorporates the notion of time-steps calculations. The unfolded representation also illustrates how a recurrent network can be constructed in a pure feed-forward fashion, with as many layers as time-steps in your sequence. One key consideration is that the weights will be identical on each time-step (or layer). Keep this unfolded representation in mind as will become important later.\n\nElman's innovation was twofold: **recurrent connections between hidden units and memory** (context) units, and **trainable parameters from the memory units to the hidden units**. Memory units now have to \"remember\" the past state of hidden units, which means that instead of keeping a running average, they \"clone\" the value at the previous time-step $t-1$. Memory units also have to learn useful representations (weights) for encoding temporal properties of the sequential input. **Figure 3** summarizes Elman's network in compact and unfolded fashion.\n\n
                                        Figure 3: Elman Network
                                        \n\n\n\n**Note**: there is something curious about Elman's architecture. What it is the point of \"cloning\" $h$ into $c$ at each time-step? You could bypass $c$ altogether by sending the value of $h_t$ straight into $h_{t+1}$, wich yield mathematically identical results. The most likely explanation for this was that Elman's starting point was Jordan's network, which had a separated memory unit. Regardless, keep in mind we don't need $c$ units to design a functionally identical network. \n\nElman performed multiple experiments with this architecture demonstrating it was capable to solve multiple problems with a sequential structure: a temporal version of the XOR problem; learning the structure (i.e., vowels and consonants sequential order) in sequences of letters; discovering the notion of \"word\", and even learning complex lexical classes like word order in short sentences. Let's briefly explore the temporal XOR solution as an exemplar. **Table 1** shows the XOR problem:\n\n**Table 1**: Truth Table For XOR Function\n\n| $x_1$ | $x_2$ | $y$ |\n|---|---|--------|\n| 0 | 0 | 0 |\n| 0 | 1 | 1 |\n| 1 | 0 | 1 |\n| 1 | 1 | 0 |\n\nHere is a way to transform the XOR problem into a sequence. Consider the following vector: \n\n$$\ns= [1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1,...]\n$$\n\nIn $\\bf{s}$, the first and second elements, $s_1$ and $s_2$, represent $x_1$ and $x_2$ inputs of **Table 1**, whereas the third element, $s_3$, represents the corresponding output $y$. This pattern repeats until the end of the sequence $s$ as shown in **Figure 4**.\n\n
                                        Figure 4: Temporal XOR
                                        \n\n\n\nElman trained his network with a 3,000 elements sequence for 600 iterations over the entire dataset, on the task of predicting the next item $s_{t+1}$ of the sequence $s$, meaning that he fed inputs to the network **one by one**. He showed that **error pattern** followed a predictable trend: the mean squared error was **lower every 3 outputs**, and higher in between, meaning the network learned to predict the third element in the sequence, as shown in **Chart 1** (the numbers are made up, but the pattern is the same found by Elman (1990)).\n\n\n```python\nimport numpy as np\nimport pandas as pd\nimport altair as alt\n```\n\n\n```python\ns = pd.DataFrame({\"MSE\": [0.35, 0.15, 0.30, 0.27, 0.14, 0.40, 0.35, 0.12, 0.36, 0.31, 0.15, 0.32],\n \"cycle\": np.arange(1, 13)})\nalt.Chart(s).mark_line().encode(x=\"cycle\", y=\"MSE\").properties(title='Chart 1')\n```\n\n\n\n\n\n
                                        \n\n\n\n\nAn immediate advantage of this approach is the network can take **inputs of any length**, without having to alter the network architecture at all.\n\nIn the same paper, Elman showed that the **internal (hidden) representations** learned by the network grouped into meaningful categories, this is, **semantically similar words group together** when analyzed with [hierarchical clustering](https://en.wikipedia.org/wiki/Hierarchical_clustering). This was remarkable as demonstrated the utility of RNNs as a model of cognition in sequence-based problems. \n\n### Interlude: vanishing and exploding gradients in RNNs\n\nTurns out, training recurrent neural networks is hard. Considerably harder than multilayer-perceptrons. When faced with the task of training **very deep networks**, like RNNs, the gradients have the impolite tendency of either (1) **vanishing**, or (2) **exploding** (Bengio et al, 1994; Pascanu et al, 2012). Recall that RNNs can be unfolded so that recurrent connections follow pure feed-forward computations. This unrolled RNN will have as many layers as elements in the sequence. Thus, a sequence of 50 words will be unrolled as an RNN of 50 layers (taking \"word\" as a unit). \n\nConcretely, the **vanishing gradient problem** will make close to impossible to learn **long-term dependencies** in sequences. Let's say you have a collection of poems, where the last sentence refers to the first one. Such a dependency will be hard to learn for a deep RNN where gradients vanish as we move backward in the network. The **exploding gradient problem** will completely derail the learning process. In very deep networks this is often a problem because more layers amplify the effect of large gradients, compounding into very large updates to the network weights, to the point values completely blow up. \n\nHere is the intuition for the **mechanics of gradient vanishing**: when gradients *begin small*, as you move backward through the network computing gradients, they will get even smaller as you get closer to the input layer. Consequently, when doing the weight update based on such gradients, the weights closer to the output layer will obtain larger updates than weights closer to the input layer. This means that the weights closer to the input layer will hardly change at all, whereas the weights closer to the output layer will change a lot. This is a serious problem when **earlier layers matter for prediction**: they will keep propagating more or less the same signal forward because no learning (i.e., weight updates) will happen, which may significantly hinder the network performance. \n\nHere is the intuition for the **mechanics of gradient explosion**: when gradients *begin large*, as you move backward through the network computing gradients, they will get even larger as you get closer to the input layer. Consequently, when doing the weight update based on such gradients, the weights closer to the input layer will obtain larger updates than weights closer to the output layer. Learning can go wrong really fast. Recall that the signal propagated by each layer is the outcome of taking the product between the previous hidden-state and the current hidden-state. If the weights in earlier layers get really large, they will forward-propagate larger and larger signals on each iteration, and the predicted output values will spiral-up out of control, making the error $y-\\hat{y}$ so large that the network will be unable to learn at all. In fact, your computer will \"overflow\" quickly as it would unable to represent numbers that big. Very dramatic. \n\nThe mathematics of gradient vanishing and explosion gets complicated quickly. If you want to delve into the mathematics see [Bengio et all (1994)](http://ai.dinfo.unifi.it/paolo/ps/tnn-94-gradient.pdf), [Pascanu et all (2012)](https://arxiv.org/abs/1211.5063), and [Philipp et all (2017)](https://arxiv.org/abs/1712.05577). \n\nFor our purposes, I'll give you a simplified numerical example for intuition. Consider the task of predicting a vector $y = \\begin{bmatrix} 1 & 1 \\end{bmatrix}$, from inputs $x = \\begin{bmatrix} 1 & 1 \\end{bmatrix}$, with a multilayer-perceptron with 5 hidden layers and tanh activation functions. We have two cases:\n\n- the weight matrix $W_l$ is initialized to large values $w_{ij} = 2$\n- the weight matrix $W_s$ is initialized to small values $w_{ij} = 0.02$\n\nNow, let's compute a single forward-propagation pass:\n\n\n```python\nimport numpy as np\n```\n\n\n```python\nx = np.array([[1],[1]])\nW_l = np.array([[2, 2],\n [2, 2]])\n\nh1 = np.tanh(W_l @ x)\nh2 = np.tanh(W_l @ h1)\nh3 = np.tanh(W_l @ h2)\nh4 = np.tanh(W_l @ h3)\nh5 = np.tanh(W_l @ h4)\ny_hat = (W_l @ h5)\nprint(f'output for large initial weights: \\n {y_hat}')\n```\n\n output for large initial weights: \n [[3.99730269]\n [3.99730269]]\n\n\n\n```python\nx = np.array([[1],[1]])\nW_s = np.array([[0.02, 0.02],\n [0.02, 0.02]])\n\nh1 = np.tanh(W_s @ x)\nh2 = np.tanh(W_s @ h1)\nh3 = np.tanh(W_s @ h2)\nh4 = np.tanh(W_s @ h3)\nh5 = np.tanh(W_s @ h4)\ny_hat = (W_s @ h5)\nprint(f'output for small initial weights: \\n {y_hat}')\n```\n\n output for small initial weights: \n [[4.09381337e-09]\n [4.09381337e-09]]\n\n\nWe see that for $W_l$ the output $\\hat{y}\\approx4$, whereas for $W_s$ the output $\\hat{y} \\approx 0$. Why does this matter? We haven't done the gradient computation but you can probably anticipate what it's going to happen: for the $W_l$ case, the gradient update is going to be very large, and for the $W_s$ very small. If you keep cycling through forward and backward passes these problems will become worse, leading to gradient explosion and vanishing respectively.\n\n### Long Short-Term Memory Network\n\nSeveral challenges difficulted progress in RNN in the early '90s (Hochreiter & Schmidhuber, 1997; Pascanu et al, 2012). In addition to vanishing and exploding gradients, we have the fact that the **forward computation is slow**, as RNNs can't compute in parallel: to preserve the time-dependencies through the layers, each layer has to be computed sequentially, which naturally takes more time. Elman networks proved to be effective at solving relatively simple problems, but as the sequences scaled in size and complexity, this type of network struggle. \n\nSeveral approaches were proposed in the '90s to address the aforementioned issues like time-delay neural networks (Lang et al, 1990), simulated annealing (Bengio et al., 1994), and others. The architecture that really moved the field forward was the so-called **Long Short-Term Memory (LSTM) Network**, introduced by [Sepp Hochreiter](https://en.wikipedia.org/wiki/Sepp_Hochreiter) and [Jurgen Schmidhuber](https://en.wikipedia.org/wiki/J%C3%BCrgen_Schmidhuber) in 1997. As the name suggests, the defining characteristic of LSTMs is the addition of units combining both short-memory and long-memory capabilities. \n\nIn LSTMs, instead of having a simple memory unit \"cloning\" values from the hidden unit as in Elman networks, we have a (1) **cell unit** (a.k.a., memory unit) which effectively acts as long-term memory storage, and (2) a **hidden-state** which acts as a memory controller. These two elements are integrated as a circuit of logic gates controlling the flow of information at each time-step. Understanding the notation is crucial here, which is depicted in **Figure 5**.\n\n
                                        Figure 5: LSTM architecture
                                        \n\n\n\nIn LSTMs $x_t$, $h_t$, and $c_t$ represent vectors of values. Lightish-pink circles represent element-wise operations, and darkish-pink boxes are fully-connected layers with trainable weights. The top part of the diagram acts as a **memory storage**, whereas the bottom part has a double role: (1) passing the hidden-state information from the previous time-step $t-1$ to the next time step $t$, and (2) to regulate the **influx** of information from $x_t$ and $h_{t-1}$ **into** the memory storage, and the **outflux** of information **from** the memory storage into the next hidden state $h-t$. The second role is the core idea behind LSTM. You can think about it as making **three decisions** at each time-step:\n\n1. **Is the *old information* $c_{t-1}$ worth to keep in my memory storage $c_t$?** If so, let the information pass, otherwise, \"forget\" such information. This is controlled by the *forget gate*. \n2. **Is this *new information* (inputs) worth to be saved into my memory storage $c_t$?** If so, let information flow into $c_t$. This is controlled by the *input gate* and the *candidate memory cell*. \n3. **What elements of the information saved in my memory storage $c_t$ are relevant for the computation of the next hidden-state $h_t$?** Select them from $c_t$, combine them new hidden-state output, and let them pass into the next hidden-state $h_t$. This is controlled by the *output gate* and the *tanh* function. \n\nDecisions 1 and 2 will determine the information that keeps flowing through the memory storage at the top. Decision 3 will determine the information that flows to the next hidden-state at the bottom. The conjunction of these decisions sometimes is called \"memory block\". Now, keep in mind that this *sequence of decision* is just a convenient *interpretation* of LSTM mechanics. In practice, the weights are the ones determining what each function ends up doing, which may or may not fit well with human intuitions or design objectives.\n\n
                                        Figure 6: LSTM as a sequence of decisions
                                        \n\n\n\nTo put LSTMs in context, imagine the following simplified scenerio: we are trying to **predict the next word in a sequence**. Let's say, squences are about sports. From past sequences, we saved in the memory block the type of sport: \"soccer\". For the current sequence, we receive a phrase like \"A basketball player...\". In such a case, we first want to \"forget\" the previous type of sport \"soccer\" (*decision 1*) by multplying $c_{t-1} \\odot f_t$. Next, we want to \"update\" memory with the new type of sport, \"basketball\" (*decision 2*), by adding $c_t = (c_{t-1} \\odot f_t) + (i_t \\odot \\tilde{c_t})$. Finally, we want to output (*decision 3*) a verb relevant for \"A basketball player...\", like \"shoot\" or \"dunk\" by $\\hat{y_t} = softmax(W_{hz}h_t + b_z)$.\n\nLSTMs long-term memory capabilities make them good at capturing long-term dependencies. The memory cell effectively counteracts the vanishing gradient problem at preserving information as long the forget gate does not \"erase\" past information (Graves, 2012). All the above make LSTMs very useful for a wide variety of applications (see [here for a list](https://en.wikipedia.org/wiki/Long_short-term_memory#Applications)). For instance, when you use [Google's Voice Transcription](https://ai.googleblog.com/2015/08/the-neural-networks-behind-google-voice.html) services an RNN is doing the hard work of recognizing your voice.\n\n### RNNs and cognition\n\nAs with Convolutional Neural Networks, researchers utilizing RNN for approaching sequential problems like natural language processing (NLP) or time-series prediction, do not *necessarily* care about (although some might) how good of a model of cognition and brain-activity are RNNs. What they really care is about solving problems like translation, speech recognition, and stock market prediction, and many advances in the field come from pursuing such goals. Still, RNN has many **desirable traits as a model of neuro-cognitive activity**, and have been prolifically used to **model several aspects of human cognition and behavior**: child behavior in an object permanence tasks (Munakata et al, 1997); knowledge-intensive text-comprehension (St. John, 1992); processing in quasi-regular domains, like English word reading (Plaut et al., 1996); human performance in processing recursive language structures (Christiansen & Chater, 1999); human sequential action (Botvinick & Plaut, 2004); movement patterns in typical and atypical developing children (Mu\u00f1oz-Organero et al., 2019). And many others. Neuroscientists have used RNNs to model a wide variety of aspects as well (for reviews see Barak, 2017, G\u00fc\u00e7l\u00fc & van Gerven, 2017, Jarne & Laje, 2019). Overall, RNN has demonstrated to be a productive tool for modeling cognitive and brain function, in distributed representations paradigm.\n\n## Mathematical formalization\n\nThere are two mathematically complex issues with RNNs: (1) computing hidden-states, and (2) backpropagation. The rest are common operations found in multilayer-perceptrons. LSTMs and its many variants are the facto standards when modeling any kind of sequential problem. Elman networks can be seen as a simplified version of an LSTM, so I'll focus my attention on LSTMs for the most part. My exposition is based on a combination of sources that you may want to review for extended explanations (Bengio et al., 1994; Hochreiter & Schmidhuber, 1997; Graves, 2012; Chen, 2016; Zhang et al., 2020).\n\nThe LSTM architecture can be desribed by: \n\n**Forward pass**:\n- non-linear forget function\n- non-linear input function \n- non-linear candidate-memory function\n- non-linear output function\n- memory cell function\n- non-linear hidden-state function \n- softmax function (output)\n\n**Backward pass**:\n- Cost-function\n- Learning procedure (backpropagation)\n\nFollowing the indices for each function requires some definitions. I'll assume we have $h$ hidden units, training sequences of size $n$, and $d$ input units. \n\n$$\n\\begin{align}\n\\text{input-units} &= x_i \\in \\mathbb{R}^d \\\\ \n\\text{training-sequence} &= s_i \\in \\mathbb{R}^n \\\\\n\\text{output-class} &= y_i \\in \\mathbb{R}^k \\\\\n\\text{Input-layer} &= X_t \\in \\mathbb{R}^{n\\times d} \\\\\n\\text{hidden-layer} &= H_t \\in \\mathbb{R}^{n\\times h}\n\\end{align}\n$$\n\n### Forget function\n\nThe **forget function** is a sigmoidal mapping combining three elements: input vector $x_t$, past hidden-state $h_{t-1}$, and a bias term $b_f$. We didn't mentioned the bias before, but it is the same bias that all neural networks incorporate, one for each unit in $f$. More formally:\n\n$$\nf_t = \\sigma(W_{xf}x_t + W_{hf}h_{t-1} + b_f)\n$$ \n\nEach matrix $W$ has dimensionality equal to (number of incoming units, number for connected units). For example, $W_{xf}$ refers to $W_{input-units, forget-units}$. Keep this in mind to read the indices of the $W$ matrices for subsequent definitions.\n\nHere is an important insight: What would it happen if $f_t = 0$? If you look at the diagram in **Figure 6**, $f_t$ performs an elementwise multiplication of each element in $c_{t-1}$, meaning that every value would be reduced to $0$. In short, the network would completely \"forget\" past states. Naturally, if $f_t = 1$, the network would keep its memory intact. \n\n### Input function and Candidate memory function\n\nThe **input function** is a sigmoidal mapping combining three elements: input vector $x_t$, past hidden-state $h_{t-1}$, and a bias term $b_f$. It's defined as:\n\n$$\ni_t = \\sigma(W_{xi}x_t + W_{hi}h_{t-1} + b_i)\n$$ \n\nThe **candidate memory function** is an hyperbolic tanget function combining the same elements that $i_t$. It's defined as:\n\n$$\n\\tilde{c}_t = tanh(W_{xc}x_t + W_{hc}h_{t-1} + b_c)\n$$ \n\nBoth functions are combined to update the memory cell.\n\n### Output function\n\nThe **output function** is a sigmoidal mapping combining three elements: input vector $x_t$, past hidden-state $h_{t-1}$, and a bias term $b_f$. Is defined as:\n\n$$\no_t = \\sigma(W_{xo}x_t + W_{ho}h_{t-1} + b_o)\n$$ \n\n### Memory cell function\n\nThe **memory cell function** (what I've been calling \"memory storage\" for conceptual clarity), combines the effect of the forget function, input function, and candidate memory function. It's defined as:\n\n$$\nc_t = (c_{t-1} \\odot f_t) + (i_t \\odot \\tilde{c_t})\n$$\n\nWhere $\\odot$ implies an elementwise multiplication (instead of the usual dot product). This expands to:\n\n$$\nc_t = (c_{t-1} \\odot \\sigma(W_{xf}x_t + W_{hf}h_{t-1} + b_f)) + (\\sigma(W_{xi}x_t + W_{hi}h_{t-1} + b_i) \\odot tanh(W_{xc}x_t + W_{hc}h_{t-1} + b_c))\n$$\n\n\n### Hidden-state function\n\nThe next **hidden-state function** combines the effect of the output function and the contents of the memory cell scaled by a tanh function. It is defined as:\n\n$$\nh_t = O_t \\odot tanh(c_t) \n$$\n\n### Output function\n\nThe output function will depend upon the problem to be approached. For our our purposes, we will assume a multi-class problem, for which the **softmax function** is appropiated. For this, we first pass the hidden-state by a linear function, and then the softmax as:\n\n$$\n\\begin{align}\nz_t &= (W_{hz}h_t + b_z)\\\\\n\\hat{y}_t &= softmax(z_t) = \\frac{e^{z_t}}{\\sum_{j=1}^K e^{z_j}} \n\\end{align}\n$$\n\nThe softmax computes the exponent for each $z_t$ and then normalized by dividing by the sum of every output value exponentiated. In this manner, the output of the softmax can be interpreted as the likelihood value $p$.\n\n### Cost function \n\nAs with the output function, the cost function will depend upon the problem. For regression problems, the Mean-Squared Error can be used. For our purposes (classification), the cross-entropy function is appropriated. It's defined as:\n\n$$\nE_i = - \\sum_t y_ilog(p_i)\n$$\n\nWhere $y_i$ is the true label for the $ith$ output unit, and $log(p_i)$ is the log of the softmax value for the $ith$ output unit. The summation indicates we need to aggregate the cost at each time-step.\n\n### Learning procedure: Backpropagation Through Time (BPTT)\n\nOriginally, Hochreiter and Schmidhuber (1997) trained LSTMs with a combination of approximate gradient descent computed with a combination of real-time recurrent learning and backpropagation through time (BPTT). Nevertheless, LSTM can be trained with pure backpropagation. Following Graves (2012), I'll only describe BTT because is more accurate, easier to debug and to describe.\n\n**Note**: we call it **backpropagation through time** because of the sequential time-dependent structure of RNNs. Recall that each layer represents a time-step, and forward propagation happens in sequence, one layer computed after the other. Hence, when we backpropagate, we do the same but backward (i.e., \"through time\").\n\n
                                        Figure 7: Three-layer simplified RNN
                                        \n\n\n\nI reviewed backpropagation for a simple multilayer perceptron [here](https://com-cog-book.github.io/com-cog-book/features/multilayer-perceptron.html#Backpropagation-algorithm). Nevertheless, I'll sketch BPTT for the simplest case as shown in **Figure 7**, this is, with a generic non-linear hidden-layer similar to Elman network without \"context units\" (some like to call it \"vanilla\" RNN, which I avoid because I believe is derogatory against vanilla!). This exercise will allow us to review backpropagation and to understand how it differs from BPTT. We begin by defining a simplified RNN as: \n\n$$\n\\begin{align}\nz_t &= W_{hz}h_t + b_z\\\\\nh_t &= \\sigma(W_{hh}h_{t-1} + W_{xh}x_t+b_h)\n\\end{align}\n$$\n\nWhere $h_t$ and $z_t$ indicates a hidden-state (or layer) and the output respectively. Therefore, **we have to compute gradients w.r.t. five sets of weights**: $\\{W_{hz}, W_{hh}, W_{xh}, b_z, b_h\\}$.\n\nFirst, consider the error derivatives w.r.t. $W_{hz}$ at time $t$, the weight matrix for the linear function at the output layer. Recall that $W_{hz}$ is shared across all time-steps, hence, we can compute the gradients at each time step and then take the sum as:\n\n$$\n\\frac{\\partial{E}}{\\partial{W_{hz}}} = \\sum_t\\frac{\\partial{E}}{\\partial{z_t}} \\frac{\\partial{z_t}}{\\partial{W_{hz}}}\n$$\n\nSame for the bias term:\n\n$$\n\\frac{\\partial{E}}{\\partial{b_z}} = \\sum_t\\frac{\\partial{E}}{\\partial{z_t}} \\frac{\\partial{z_t}}{\\partial{b_z}}\n$$\n\nThat part is straightforward. The issue arises when we try to compute the gradients w.r.t. the wights $W_{hh}$ in the hidden layer. Consider a three layer RNN (i.e., unfolded over three time-steps). In such a case, we have: \n\n- $E_3$ depens on $z_3$\n- $z_3$ depends on $h_3$\n- $h_3$ depens on $h_2$\n- $h_2$ depens on $h_1$\n- $h_1$ depens on $h_0$, where $h_0$ is a random starting state.\n\nNow, we have that $E_3$ w.r.t to $h_3$ becomes:\n\n$$\n\\frac{\\partial{E_3}}{\\partial{W_{hh}}} = \n\\frac{\\partial{E_3}}{\\partial{z_3}}\n\\frac{\\partial{z_3}}{\\partial{h_3}}\n\\frac{\\partial{h_3}}{\\partial{W_{hh}}}\n$$\n\nThe issue here is that $h_3$ depends on $h_2$, since according to our definition, the $W_{hh}$ is multiplied by $h_{t-1}$, meaning **we can't compute $\\frac{\\partial{h_3}}{\\partial{W_{hh}}}$ directly**. Othewise, we would be treating $h_2$ as a constant, which is incorrect: is a function. What we need to do is to **compute the gradients separately**: the direct contribution of ${W_{hh}}$ on $E$ and the indirect contribution via $h_2$. Following the rules of calculus in multiple variables, we compute them independently and add them up together as:\n\n$$\n\\frac{\\partial{E_3}}{\\partial{W_{hh}}} = \n\\frac{\\partial{E_3}}{\\partial{z_3}}\n\\frac{\\partial{z_3}}{\\partial{h_3}}\n\\frac{\\partial{h_3}}{\\partial{W_{hh}}}+\n\\frac{\\partial{E_3}}{\\partial{z_3}}\n\\frac{\\partial{z_3}}{\\partial{h_3}}\n\\frac{\\partial{h_3}}{\\partial{h_2}}\n\\frac{\\partial{h_2}}{\\partial{W_{hh}}}\n$$\n\nAgain, we have that we can't compute $\\frac{\\partial{h_2}}{\\partial{W_{hh}}}$ directly. Following the same procedure, we have that our full expression becomes:\n\n$$\n\\frac{\\partial{E_3}}{\\partial{W_{hh}}} = \n\\frac{\\partial{E_3}}{\\partial{z_3}}\n\\frac{\\partial{z_3}}{\\partial{h_3}}\n\\frac{\\partial{h_3}}{\\partial{W_{hh}}}+\n\\frac{\\partial{E_3}}{\\partial{z_3}}\n\\frac{\\partial{z_3}}{\\partial{h_3}}\n\\frac{\\partial{h_3}}{\\partial{h_2}}\n\\frac{\\partial{h_2}}{\\partial{W_{hh}}}+\n\\frac{\\partial{E_3}}{\\partial{z_3}}\n\\frac{\\partial{z_3}}{\\partial{h_3}}\n\\frac{\\partial{h_3}}{\\partial{h_2}}\n\\frac{\\partial{h_2}}{\\partial{h_1}}\n\\frac{\\partial{h_1}}{\\partial{W_{hh}}}\n$$\n\nEssentially, this means that we compute and add the contribution of $W_{hh}$ to $E$ at each time-step. The expression for $b_h$ is the same:\n\n$$\n\\frac{\\partial{E_3}}{\\partial{b_h}} = \n\\frac{\\partial{E_3}}{\\partial{z_3}}\n\\frac{\\partial{z_3}}{\\partial{h_3}}\n\\frac{\\partial{h_3}}{\\partial{b_h}}+\n\\frac{\\partial{E_3}}{\\partial{z_3}}\n\\frac{\\partial{z_3}}{\\partial{h_3}}\n\\frac{\\partial{h_3}}{\\partial{h_2}}\n\\frac{\\partial{h_2}}{\\partial{b_h}}+\n\\frac{\\partial{E_3}}{\\partial{z_3}}\n\\frac{\\partial{z_3}}{\\partial{h_3}}\n\\frac{\\partial{h_3}}{\\partial{h_2}}\n\\frac{\\partial{h_2}}{\\partial{h_1}}\n\\frac{\\partial{h_1}}{\\partial{b_h}}\n$$\n\nFinally, we need to compute the gradients w.r.t. $W_{xh}$. Here, again, we have to add the contributions of $W_{xh}$ via $h_3$, $h_2$, and $h_1$: \n\n$$\n\\frac{\\partial{E_3}}{\\partial{W_{xh}}} = \n\\frac{\\partial{E_3}}{\\partial{z_3}}\n\\frac{\\partial{z_3}}{\\partial{h_3}}\n\\frac{\\partial{h_3}}{\\partial{W_{xh}}}+\n\\frac{\\partial{E_3}}{\\partial{z_3}}\n\\frac{\\partial{z_3}}{\\partial{h_3}}\n\\frac{\\partial{h_3}}{\\partial{h_2}}\n\\frac{\\partial{h_2}}{\\partial{W_{xh}}}+\n\\frac{\\partial{E_3}}{\\partial{z_3}}\n\\frac{\\partial{z_3}}{\\partial{h_3}}\n\\frac{\\partial{h_3}}{\\partial{h_2}}\n\\frac{\\partial{h_2}}{\\partial{h_1}}\n\\frac{\\partial{h_1}}{\\partial{W_{xh}}}\n$$\n\nThat's for BPTT for a simple RNN. The math reviewed here generalizes with minimal changes to more complex architectures as LSTMs. Actually, the only difference regarding LSTMs, is that we have more weights to differentiate for. Instead of a single generic $W_{hh}$, we have $W$ for all the gates: forget, input, output, and candidate cell. The rest remains the same. For a detailed derivation of BPTT for the LSTM see [Graves (2012)](https://www.cs.toronto.edu/~graves/preprint.pdf) and [Chen (2016)](https://arxiv.org/abs/1610.02583).\n\n## Interlude: Sequence-data representation\n\nWorking with sequence-data, like text or time-series, requires to pre-process it in a manner that is \"digestible\" for RNNs. As with any neural network, RNN can't take raw text as an input, we need to **parse** text sequences and then **\"map\"** them into vectors of numbers. Here I'll briefly review these issues to provide enough context for our example applications. For an extended revision please refer to [Jurafsky and Martin (2019)](https://web.stanford.edu/~jurafsky/slp3/), [Goldberg (2015)](http://u.cs.biu.ac.il/~yogo/nnlp.pdf), [Chollet (2017)](https://www.manning.com/books/deep-learning-with-python), and [Zhang et al (2020)](https://d2l.ai/chapter_recurrent-neural-networks/index.html).\n\nParsing can be done in multiple manners, the most common being: \n\n- Using **word** as a unit, which each word represented as a vector\n- Using **character** as a unit, with each character represented as a vector\n- Using **n-grams** of words or characters as a unit, with each n-gram represented as a vector. N-grams are sets of words or characters of size \"N\" or less.\n\nThe process of parsing text into smaller units is called \"tokenization\", and each resulting unit is called a \"token\", the top pane in **Figure 8** displays a sketch of the tokenization process.\n\n
                                        Figure 8: Tokenization
                                        \n\n\n\nOnce a corpus of text has been parsed into tokens, we have to map such tokens into numerical vectors. Two common ways to do this are **one-hot encoding** approach and the **word embeddings** approach, as depicted in the bottom pane of **Figure 8**. We used one-hot encodings to transform the MNIST class-labels into vectors of numbers for classification in the [CovNets chapter](https://com-cog-book.github.io/com-cog-book/features/cov-net.html). In a one-hot encoding vector, each token is mapped into a *unique* vector of zeros and ones. The vector size is determined by the vocabullary size. For instance, for the set $x= \\{\"cat\", \"dog\", \"ferret\"\\}$, we could use a 3-dimensional one-hot encoding as:\n\n$$\n\\text{cat}=\n\\begin{bmatrix}\n1 \\\\\n0 \\\\\n0\n\\end{bmatrix},\n\\text{dog}=\n\\begin{bmatrix}\n0 \\\\\n1 \\\\\n0\n\\end{bmatrix},\n\\text{ferret}=\n\\begin{bmatrix}\n0 \\\\\n0 \\\\\n1\n\\end{bmatrix}\n$$\n\nOne-hot encodings have the advantages of being straightforward to implement and to provide a unique identifier for each token. Its main disadvantage is that tends to create really **sparse** and **high-dimensional** representations for a large corpus of texts. For instance, if you tried a one-hot encoding for 50,000 tokens, you'd end up with a 50,000x50,000-dimensional matrix, which may be unpractical for most tasks. \n\n**Word embeddings** represent text by mapping tokens into vectors of real-valued numbers instead of only zeros and ones. This significantly increments the representational capacity of vectors, reducing the required dimensionality for a given corpus of text compared to one-hot encodings. For instance, 50,000 tokens could be represented by as little as 2 or 3 vectors (although the representation may not be very good). Taking the same set $x$ as before, we could have a 2-dimensional word embedding like:\n\n$$\n\\text{cat}=\n\\begin{bmatrix}\n0.1 \\\\\n0.8\n\\end{bmatrix},\n\\text{dog}=\n\\begin{bmatrix}\n0.2 \\\\\n1 \n\\end{bmatrix},\n\\text{ferret}=\n\\begin{bmatrix}\n0.6 \\\\\n0.2\n\\end{bmatrix}\n$$\n\nYou may be wondering why to bother with one-hot encodings when word embeddings are much more space-efficient. The main issue with word-embedding is that **there isn't an obvious way to map tokens into vectors** as with one-hot encodings. For instance, you could assign tokens to vectors at random (assuming every token is assigned to a *unique* vector). The problem with such approach is that the semantic structure in the corpus is broken. Ideally, you want words of similar meaning mapped into similar vectors. We can preserve the semantic structure of a text corpus in the same manner as everything else in machine learning: by **learning from data**. There are two ways to do this:\n\n- Learning the word embeddings **at the same time** you train the RNN.\n- Utilizing **pretrained** word embeddings, this is, embeddings learned in a different task. This is a form of \"transfer learning\".\n\nLearning word embeddings for your task is advisable as semantic relationships among words tend to be **context dependent**. For instance, \"exploitation\" in the context of mining is related to resource \"extraction\", hence relative neutral. But, \"exploitation\" in the context of labor rights is related to the idea of \"abuse\", hence a negative connotation. This is more critical when we are dealing with different languages. Nevertheless, learning embeddings for every task sometimes is impractical, either because your corpus is too \"small\" (i.e., not enough data to extract semantic relationships), or too \"large\" (i.e., you don't have enough time and/or resources to learn the embeddings). Examples of freely accessible pretrained word embeddings are Google's [Word2vec](https://code.google.com/archive/p/word2vec/) and the [Global Vectors for Word Representation](https://nlp.stanford.edu/projects/glove/) (GloVe).\n\n## Code implementation\n\nAs in previous chapters, I'll use Keras to implement both (a modified version of) the Elman Network for the XOR problem and an LSTM for review prediction based on text-sequences.\n\n### Elman Network\n\nBy now, it may be clear to you that Elman networks are a simple RNN with two neurons, one for each input pattern, in the hidden-state. Originally, Elman trained his architecture with a truncated version of BPTT, meaning that only considered two time-steps for computing the gradients, $t$ and $t-1$. We will implement a modified version of Elman's architecture bypassing the \"context\" unit (which does not alter the result at all) and utilizing BPTT instead of its truncated version. \n\n### Generating data \n\nNowadays, we don't need to generate the 3,000 bits sequence that Elman used in his original work. We can simply generate a single pair of training and testing sets for the XOR problem as in **Table 1**, and pass the training sequence (length two) as the inputs, and the expected outputs as the target. This is very much alike any classification task. An important caveat is that simpleRNN layers in Keras expect an input tensor of shape (number-samples, timesteps, number-input-features). In our case, this has to be: *number-samples*= 4, *timesteps*=1, *number-input-features*=2. No separate encoding is necessary here because we are manually setting the input and output values to binary vector representations. Finally, we won't worry about training and testing sets for this example, which is way to simple for that (we will do that for the next example).\n\n\n```python\n# Libraries for this section\n\nfrom keras.layers import Dense, SimpleRNN\nfrom keras.models import Sequential\nimport altair as alt\nimport pandas as pd\nimport numpy as np\n```\n\n Using TensorFlow backend.\n\n\n\n```python\n# features\nX = np.array([[[0, 0, 1, 1]],\n [[0, 1, 0, 1]]]).T\n\n# expected values\ny = np.array([[0, 1, 1, 0]]).T\n\nprint(f'training data shape: {X.shape}')\nprint(f'targets data shape: {y.shape}')\n```\n\n training data shape: (4, 1, 2)\n targets data shape: (4, 1)\n\n\n### Elman network architecture in Keras\n\nDefining a (modified) in Keras is extremely simple as shown below.\n\n\n```python\n# Define a network as a linear stack of layers\nmodel = Sequential()\n\n# Add a recurrent layer with 2 units\nmodel.add(SimpleRNN(2, input_shape=(1, 2)))\n\n# Add the output layer with a sigmoid activation\nmodel.add(Dense(1, activation='tanh'))\n\n```\n\nThe model summary shows that our architecture yields 13 trainable parameters.\n\n\n```python\nmodel.summary()\n```\n\n Model: \"sequential_1\"\n _________________________________________________________________\n Layer (type) Output Shape Param # \n =================================================================\n simple_rnn_1 (SimpleRNN) (None, 2) 10 \n _________________________________________________________________\n dense_1 (Dense) (None, 1) 3 \n =================================================================\n Total params: 13\n Trainable params: 13\n Non-trainable params: 0\n _________________________________________________________________\n\n\n## Elman network Application: XOR classification\n\nNext, we compile and fit our model. I'll utilize [Adadelta](https://github.com/keras-team/keras/blob/master/keras/optimizers.py#L376) (to avoid manually adjusting the learning rate) as the optimizer, and the Mean-Squared Error (as in Elman original work). I'll train the model for 15,000 epochs over the 4 samples dataset.\n\n\n```python\nmodel.compile(optimizer='Adadelta', \n loss='mean_squared_error', \n metrics=['acc'])\nhistory = model.fit(X, y,\n epochs=5000,\n verbose=0)\n```\n\n**Chart 2** shows the error curve (red, right axis), and the accuracy curve (blue, left axis) for each epoch.\n\n\n```python\nalt.data_transformers.disable_max_rows()\n\nloss = history.history['loss']\naccuracy = history.history['acc']\n\ndf = pd.DataFrame({\"accuracy\":accuracy, \"loss\":loss, \"time-step\": np.arange(0, len(accuracy))})\n\nbase = alt.Chart(df).mark_line(color=\"blue\").encode(x=\"time-step\", y=\"accuracy\")\nloss = alt.Chart(df).mark_line(color=\"red\").encode(x=\"time-step\", y=\"loss\")\n(base + loss).properties(title='Chart 2').resolve_scale(y='independent')\n```\n\n\n\n\n\n
                                        \n\n\n\n\nWe see that accuracy goes to 100% in around 1,000 epochs (note that different runs may slightly change the results).\n\n### LSTM\n\nIn a strict sense, LSTM is a type of layer instead of a type of network. What I've calling LSTM networks is basically any RNN composed of LSTM layers. Most RNNs you'll find in the wild (i.e., the internet) use either LSTMs or [Gated Recurrent Units (GRU)](https://en.wikipedia.org/wiki/Gated_recurrent_unit). We don't cover GRU here since they are very similar to LSTMs and this chapter is dense enough as it is. If you want to learn more about GRU see [Cho et al (2014)](https://arxiv.org/abs/1406.1078) and [Chapter 9.1 from Zhang (2020)](https://d2l.ai/chapter_recurrent-modern/gru.html).\n\nFor this section, I'll base the code in the example provided by Chollet (2017) in chapter 6. As a side note, if you are interested in learning Keras in-depth, [Chollet's book](https://www.amazon.com/Deep-Learning-Python-Francois-Chollet/dp/1617294438) is probably the best source since he is the creator of Keras library. \n\n### Reading and preprocessing data from Keras\n\nIf you are like me, you like to check the [IMDB](https://www.imdb.com/) reviews before watching a movie. For this example, we will make use of the [IMDB dataset](https://www.imdb.com/interfaces/), and Lucky us, Keras comes pre-packaged with it. The IMDB dataset comprises 50,000 movie reviews, 50% positive and 50% negative. Keras give access to a numerically encoded version of the dataset where each word is mapped to sequences of integers. We can download the dataset by running the following:\n\n\n```python\n# Libraries for this section\n\nimport tensorflow as tf\nfrom tensorflow.keras.layers import Dense, LSTM, Embedding\nfrom tensorflow.keras.models import Sequential\n\nfrom keras.datasets import imdb\nfrom keras.preprocessing.sequence import pad_sequences\nimport altair as alt\nimport pandas as pd\nimport numpy as np\n```\n\n**Note**: This time I also imported Tensorflow, and from there Keras layers and models. Keras happens to be integrated with Tensorflow, as a high-level interface, so nothing important changes when doing this. Yet, there are some implementation issues with the optimizer that require importing from Tensorflow to work.\n\n\n```python\nmaxlen = 5000\n(train_data, train_labels), (test_data, test_labels) = imdb.load_data(num_words=maxlen)\n```\n\nThe parameter `num_words=5000` restrict the dataset to the top 5,000 most frequent words. We do this to avoid highly infrequent words. Often, infrequent words are either typos or words for which we don't have enough statistical information to learn useful representations. \n\nData is downloaded as a (25000,) tuples of integers.\n\n\n```python\nprint(f'train-data shape: {train_data.shape}, train-labels shape: {train_labels.shape}')\nprint(f'test-data shape: {test_data.shape}, test-labels shape: {test_labels.shape} \\n')\nprint(f'first 10 words of first sequence: {train_data[0][0:10]},\\nfirst sequence label: {train_labels[0]}')\n```\n\n train-data shape: (25000,), train-labels shape: (25000,)\n test-data shape: (25000,), test-labels shape: (25000,) \n \n first 10 words of first sequence: [1, 14, 22, 16, 43, 530, 973, 1622, 1385, 65],\n first sequence label: 1\n\n\nIf you are curious about the review contents, the code snippet below decodes the first review into words.\n\n\n```python\nword_index = imdb.get_word_index()\nreverse_word_index = dict([(value, key) for (key, value) in word_index.items()])\ndecoded_review = ' '.join([reverse_word_index.get(i - 3, '?') for i in train_data[0]])\nprint(f'first 100 characters of first sequence:\\n{decoded_review[0:100]}....')\n```\n\n first 100 characters of first sequence:\n ? this film was just brilliant casting location scenery story direction everyone's really suited the....\n\n\nNext, we need to **\"pad\" each sequence with zeros** such that all sequences are of the same length. We do this because Keras layers expect same-length vectors as input sequences. Given that we are considering only the 5,000 more frequent words, we have max length of any sequence is 5,000. Hence, we have to pad every sequence to have length 5,000.\n\n\n```python\nX_train = pad_sequences(train_data, maxlen=maxlen)\nX_test = pad_sequences(test_data, maxlen=maxlen)\n```\n\nAs a result, we go from a list of list (samples= 25000,), to a matrix of shape (samples=25000, maxleng=5000)\n\n\n```python\nprint(f'train-data shape: {X_train.shape}, train-labels shape: {X_test.shape} \\n')\nprint(X_train[0],X_test[0])\n```\n\n train-data shape: (25000, 5000), train-labels shape: (25000, 5000) \n \n [ 0 0 0 ... 19 178 32] [ 0 0 0 ... 14 6 717]\n\n\nFinally, we will take only the first 5,000 training and testing examples. We do this because training RNNs is computationally expensive, and we don't have access to enough hardware resources to train a large model here. \n\n\n```python\ntraining_size=5000\ntraining_sentences = X_train[0:training_size]\ntesting_sentences = X_test[0:training_size]\ntraining_labels = train_labels[0:training_size]\ntesting_labels = test_labels[0:training_size]\n```\n\n\n```python\nprint(f'train-data shape: {training_sentences.shape}, train-labels shape: {training_labels.shape}')\nprint(f'test-data shape: {testing_sentences.shape}, test-labels shape: {testing_labels.shape}')\n```\n\n train-data shape: (5000, 5000), train-labels shape: (5000,)\n test-data shape: (5000, 5000), test-labels shape: (5000,)\n\n\nLet's compute the percentage of positive reviews samples on training and testing as a sanity check. We want this to be close to 50% so the sample is balanced. \n\n\n```python\nprint(f'percentage of positive reviews in training: {training_labels.sum()/training_size}')\nprint(f'percentage of positive reviews in testing: {testing_labels.sum()/training_size}')\n```\n\n percentage of positive reviews in training: 0.5092\n percentage of positive reviews in testing: 0.4858\n\n\n### Word embeddings with Keras\n\nWe will use word embeddings instead of one-hot encodings this time. Again, Keras provides convenience functions (or layer) to learn word embeddings along with RNNs training. An embedding in Keras is a layer that takes two inputs as a minimum: the **max length of a sequence** (i.e., the max number of tokens), and the **desired dimensionality of the embedding** (i.e., in how many vectors you want to represent the tokens). For instance, for an embedding with 5,000 tokens and 32 embedding vectors we just define `model.add(Embedding(5,000, 32))`. We will do this when defining the network architecture.\n\n### LSTM architecture in Keras\n\nDefining RNN with LSTM layers is remarkably simple with Keras (considering how complex LSTMs are as mathematical objects). I'll define a relatively \"shallow\" network with just 1 hidden LSTM layer.\n\n\n```python\n# Define a network as a linear stack of layers\nmodel = Sequential()\n\n# Add embedding layer with:\n # - Max number of tokens: 10,000\n # - Number of embeddings vectors: 32 \nmodel.add(Embedding(maxlen, 32))\n\n# Add LSTM layer with 32 units (sequence length)\nmodel.add(LSTM(32))\n\n# Add output layer with sigmoid activation unit\nmodel.add(Dense(1, activation='sigmoid'))\n```\n\n## LSTM Application: IMDB review prediction \n\nIt's time to train and test our RNN. I'll run just five epochs, again, because we don't have enough computational resources and for a demo is more than enough. If you run this, it may take around 5-15 minutes in a CPU. For instance, my Intel i7-8550U took ~10 min to run five epochs.\n\n\n```python\nmodel.compile(optimizer=tf.keras.optimizers.RMSprop(),\n loss=tf.keras.losses.BinaryCrossentropy(),\n metrics=['acc'])\n\nhistory = model.fit(training_sentences, training_labels,\n epochs=5,\n batch_size=128, # update gradients every 128 sequences\n validation_split=0.2, # validation subsample\n verbose=0) \n```\n\n**Note**: a \"validation split\" is different from the testing set: It's a sub-sample from the training set. For instance, with a training sample of 5,000, the `validation_split = 0.2` will split the data in a 4,000 effective training set and a 1,000 validation set. The network is trained only in the training set, whereas the validation set is used as a real-time(ish) way to help with hyper-parameter tunning, by synchronously evaluating the network in such a sub-sample. To learn more about this see the [Wikipedia article on the topic](https://en.wikipedia.org/wiki/Training,_validation,_and_test_sets).\n\n\n```python\nprint(f\"final training accuracy:{history.history['acc'][-1]}\")\nprint(f\"final validation accuracy:{history.history['val_acc'][-1]}\")\n```\n\n final training accuracy:0.8964999914169312\n final validation accuracy:0.843999981880188\n\n\nWe obtained a **training accuracy of ~90%** and **validation accuracy of ~84%** (note that different runs may slightly change the results). The top-pane in **Chart 3** shows the training and validation curves for accuracy, whereas the bottom-pane shows the same for the loss. It is clear that the network overfitting the data by the 3rd epoch. This is expected as our architecture is shallow, the training set relatively small, and no regularization method was used.\n\n\n```python\nloss = history.history['loss']\nval_loss = history.history['val_loss']\naccuracy = history.history['acc']\nval_acc = history.history['val_acc']\n\ndf = pd.DataFrame({\"accuracy\":accuracy,\n \"val_accuracy\": val_acc,\n \"loss\":loss,\n \"val_loss\": val_loss,\n \"time-step\": np.arange(1, len(accuracy)+1)})\n\naccu = alt.Chart(df).mark_line(color=\"#0202d6\").encode(x=\"time-step\", y=\"accuracy\")\nval_accu = alt.Chart(df).mark_line(color=\"#7272a1\").encode(x=\"time-step\", y=\"val_accuracy\")\n\nloss = alt.Chart(df).mark_line(color=\"#d60202\").encode(x=\"time-step\", y=\"loss\")\nval_loss = alt.Chart(df).mark_line(color=\"#cc6e6e\").encode(x=\"time-step\", y=\"val_loss\")\n\n((accu + val_accu)&(loss + val_loss)).properties(title='Chart 3') \n```\n\n\n\n\n\n
                                        \n\n\n\n\nFinally, the model obtains a **test set accuracy of ~84%** echoing the results from the validation set. All things considered, this is a very respectable result!\n\n\n```python\nscore = model.evaluate(testing_sentences, testing_labels, verbose=0)\nprint(f'Test loss score: {score[0]}')\nprint(f'Test accuracy score:{ score[1]}')\n```\n\n Test loss score: 0.3852371459245682\n Test accuracy score:0.8384000062942505\n\n\n## Limitations\n\n### Training RNNs is hard and costly\n\nAs I mentioned in previous sections, there are three well-known issues that make training RNNs really hard: (1) vanishing gradients, (2) exploding gradients, (3) and its sequential nature, which make them computationally expensive as parallelization is difficult. I won't discuss again these issues. Many techniques have been developed to address all these issues, from architectures like LSTM, GRU, and ResNets, to techniques like gradient clipping and regularization (Pascanu et al (2012); for an up to date (i.e., 2020) review of this issues see [Chapter 9 of Zhang et al book](https://d2l.ai/chapter_recurrent-modern/index.html).). \n\nThe quest for solutions to RNNs deficiencies has prompt the development of new architectures like Encoder-Decoder networks with \"attention\" mechanisms (Bahdanau et al, 2014; Vaswani et al, 2017). This new type of architecture seems to be outperforming RNNs in tasks like machine translation and text generation, in addition to overcoming some RNN deficiencies.\n\n### Do RNNs \"really\" understand...anything?\n\nCritics like Gary Marcus have pointed out the apparent inability of neural-networks based models to \"really\" understand their outputs (Marcus, 2018). This is prominent for RNNs since they have been used profusely used in the context of language generation and understanding. For instance, even state-of-the-art models like [OpenAI GPT-2](https://openai.com/blog/better-language-models/) sometimes produce incoherent sentences. Marcus gives the [following example](https://thegradient.pub/gpt2-and-the-nature-of-intelligence/):\n\n> (**Marcus**) Suppose for example that I ask the system what happens when I put two trophies a table and another: *I put two trophies on a table, and then add another, the total number is...* \n\n> (**GPT-2 answer**) *...is five trophies and I'm like, 'Well, I can live with that, right?*\n\nFrom Marcus' perspective, this lack of coherence is an exemplar of GPT-2 incapacity to understand language (note that GPT-2 is not an RNN as I've described so far, but it serves to show the point since its performance is even better than LSTMs or GRUs). \n\nYet, I'll argue two things. First, this is an unfairly underspecified question: **What do we mean by understanding?** From a cognitive science perspective, this is a fundamental yet strikingly hard question to answer. If you ask five cognitive science what does it \"really\" mean to understand something you are likely to get five different answers. What do we need is a falsifiable way to decide when a system \"really\" understands language. Is lack of coherence enough? I produce incoherent phrases all the time, and I know lots of people that do the same. In any case, it is important to question whether human-level understanding of language (however you want to define it) is necessary to show that a computational model of any cognitive process is a good model or not. We have several great models of many natural phenomena, yet not a single one gets all the aspects of the phenomena perfectly. For instance, Marcus has said that the fact that GPT-2 sometimes produces incoherent sentences is somehow a proof that human \"thoughts\" (i.e., internal representations) can't possibly be represented as vectors (like neural nets do), which is a non-sequitur. \n\nSecond, **Why should we expect that a network trained for a narrow task like language production should understand what language \"really\" is?** The exercise of comparing computational models of \"cognitive processes\" with \"full-blown\" human cognition, makes as much sense as comparing a model of bipedal locomotion with the entire motor control system of an animal. A model of bipedal locomotion is just that: **a model of a sub-system or sub-process within a larger system, not a reproduction of the entire system**. The fact that a model of bipedal locomotion does not capture well the mechanics of \"jumping\", does not undermine it's veracity or utility, in the same manner, that the inability of a model of language production to \"understand\" all aspects of language does not undermine its plausibility as a model of...languague production.\n\n## Conclusions\n\nRecurrent neural networks have been prolific models in cognitive science (Munakata et al, 1997; St. John, 1992; Plaut et al., 1996; Christiansen & Chater, 1999; Botvinick & Plaut, 2004; Mu\u00f1oz-Organero et al., 2019), bringing together intuitions about how cognitive systems work in time-dependent domains, and how neural networks may accommodate such processes.\n\nNevertheless, problems like vanishing gradients, exploding gradients, and computational inefficiency (i.e., lack of parallelization) have difficulted RNN use in many domains. Although new architectures (without recursive structures) have been developed to improve RNN results and overcome its limitations, they remain relevant from a cognitive science perspective. \n\n## References\n\n- Bahdanau, D., Cho, K., & Bengio, Y. (2014). Neural machine translation by jointly learning to align and translate. ArXiv Preprint ArXiv:1409.0473.\n- Bengio, Y., Simard, P., & Frasconi, P. (1994). Learning long-term dependencies with gradient descent is difficult. IEEE Transactions on Neural Networks, 5(2), 157\u2013166.\n- Botvinick, M., & Plaut, D. C. (2004). Doing without schema hierarchies: A recurrent connectionist approach to normal and impaired routine sequential action. Psychological Review, 111(2), 395.\n- Barak, O. (2017). Recurrent neural networks as versatile tools of neuroscience research. Current Opinion in Neurobiology, 46, 1\u20136. https://doi.org/10.1016/j.conb.2017.06.003\n- Chen, G. (2016). A gentle tutorial of recurrent neural network with error backpropagation. arXiv preprint arXiv:1610.02583.\n- Elman, J. L. (1990). Finding Structure in Time. Cognitive Science, 14(2), 179\u2013211. https://doi.org/10.1207/s15516709cog1402_1\n- Fran\u00e7ois, C. (2017). 6. Deep Learning for text and sequences. Deep learning with Python. Manning.\n- Cho, K., Van Merri\u00ebnboer, B., Gulcehre, C., Bahdanau, D., Bougares, F., Schwenk, H., & Bengio, Y. (2014). Learning phrase representations using RNN encoder-decoder for statistical machine translation. arXiv preprint arXiv:1406.1078.\n- Christiansen, M. H., & Chater, N. (1999). Toward a connectionist model of recursion in human linguistic performance. Cognitive Science, 23(2), 157\u2013205.\n- Goodfellow, I., Bengio, Y., & Courville, A. (2016). 10. Sequence Modeling: Recurrent and Recursive Nets. In Deep Learning. MIT Press. https://www.deeplearningbook.org/contents/mlp.html\n- Hebb, D. O. (1949). The organization of behavior: A neuropsychological theory. Psychology Press.\n- Hochreiter, S., & Schmidhuber, J. (1997). Long short-term memory. Neural Computation, 9(8), 1735\u20131780.\n- Hopfield, J. J. (1982). Neural networks and physical systems with emergent collective computational abilities. *Proceedings of the National Academy of Sciences*, *79*(8), 2554\u20132558.\n- G\u00fc\u00e7l\u00fc, U., & van Gerven, M. A. (2017). Modeling the dynamics of human brain activity with recurrent neural networks. Frontiers in Computational Neuroscience, 11, 7.\n- Graves, A. (2012). Supervised sequence labelling. In Supervised sequence labelling with recurrent neural networks (pp. 5-13). Springer, Berlin, Heidelberg.\n- Jarne, C., & Laje, R. (2019). A detailed study of recurrent neural networks used to model tasks in the cerebral cortex. ArXiv Preprint ArXiv:1906.01094.\n- John, M. F. (1992). The story gestalt: A model of knowledge-intensive processes in text comprehension. Cognitive Science, 16(2), 271\u2013306.\n- Jordan, M. I. (1986). Serial order: A parallel distributed processing approach, ICS Report 8604. *Institute for Cognitive Science, UCSD, La Jolla*.\n- Marcus, G. (2018). Deep learning: A critical appraisal. ArXiv Preprint ArXiv:1801.00631.\n- Munakata, Y., McClelland, J. L., Johnson, M. H., & Siegler, R. S. (1997). Rethinking infant knowledge: Toward an adaptive process account of successes and failures in object permanence tasks. Psychological Review, 104(4), 686.\n- Mu\u00f1oz-Organero, M., Powell, L., Heller, B., Harpin, V., & Parker, J. (2019). Using Recurrent Neural Networks to Compare Movement Patterns in ADHD and Normally Developing Children Based on Acceleration Signals from the Wrist and Ankle. Sensors (Basel, Switzerland), 19(13). https://doi.org/10.3390/s19132935\n- K. J. Lang, A. H. Waibel, and G. E. Hinton. A Time-delay Neural Network Architecture for Isolated Word Recognition. Neural Networks, 3(1):23-43, 1990\n- Pascanu, R., Mikolov, T., & Bengio, Y. (2013). On the difficulty of training recurrent neural networks. International Conference on Machine Learning, 1310\u20131318.\n- Philipp, G., Song, D., & Carbonell, J. G. (2017). The exploding gradient problem demystified-definition, prevalence, impact, origin, tradeoffs, and solutions. ArXiv Preprint ArXiv:1712.05577.\n- Plaut, D. C., McClelland, J. L., Seidenberg, M. S., & Patterson, K. (1996). Understanding normal and impaired word reading: Computational principles in quasi-regular domains. Psychological Review, 103(1), 56.\n- Raj, B. (2020, Spring). Neural Networks: Hopfield Nets and Auto Associators [Lecture]. http://deeplearning.cs.cmu.edu/document/slides/lec17.hopfield.pdf\n- Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, \\Lukasz, & Polosukhin, I. (2017). Attention is all you need. Advances in Neural Information Processing Systems, 5998\u20136008.\n- Zhang, A., Lipton, Z. C., Li, M., & Smola, A. J. (2020). 8. Recurrent Neural Networks. In Dive into Deep Learning. https://d2l.ai/chapter_convolutional-neural-networks/index.html\n\n## Useful online resources\n\nHere a list of my favorite online resources to learn more about Recurrent Neural Networks:\n\n- Stanford Lectures: Natural Language Processing with Deep Learning, Winter 2020. [Coruse webpage](http://web.stanford.edu/class/cs224n/index.html#schedule).\n- Bhiksha Raj's Deep Learning Lectures 13, 14, and 15 at CMU. [Course webpage](http://deeplearning.cs.cmu.edu/)\n- Geoffrey Hinton's Neural Network Lectures 7 and 8. [YouTube Videos](https://www.youtube.com/watch?v=2fRnHVVLf1Y&list=PLiPvV5TNogxKKwvKb1RKwkq2hm7ZvpHz0)\n", "meta": {"hexsha": "58fd2b8b84416b3652f4065466240d2fd9a49b96", "size": 463814, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/recurrent-net.ipynb", "max_stars_repo_name": "pabloinsente/comp_models_cog_beh", "max_stars_repo_head_hexsha": "da46c8f62157a2bc27019f9b545a389c3494f9d0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 10, "max_stars_repo_stars_event_min_datetime": "2020-05-04T12:45:59.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T12:48:25.000Z", "max_issues_repo_path": "notebooks/recurrent-net.ipynb", "max_issues_repo_name": "pabloinsente/comp_models_cog_beh", "max_issues_repo_head_hexsha": "da46c8f62157a2bc27019f9b545a389c3494f9d0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 5, "max_issues_repo_issues_event_min_datetime": "2020-03-25T23:12:19.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-10T01:08:21.000Z", "max_forks_repo_path": "notebooks/recurrent-net.ipynb", "max_forks_repo_name": "pabloinsente/comp_models_cog_beh", "max_forks_repo_head_hexsha": "da46c8f62157a2bc27019f9b545a389c3494f9d0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-03-03T18:29:21.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-03T18:29:21.000Z", "avg_line_length": 256.6762589928, "max_line_length": 366870, "alphanum_fraction": 0.6108116616, "converted": true, "num_tokens": 17422, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.418149912927306}} {"text": "```python\nfrom IPython.core.display import display_html\nfrom urllib.request import urlopen\n\nlink = 'https://gist.github.com/robblack007/eca03fa9f7586860235d/raw/ef05a2f29febc94a9c9f99ca20fd0b65e74ed451/custom.css'\ndisplay_html(urlopen(link).read(), raw=True)\n```\n\n\n\n\n\n\n\n\n\n\n\n# Din\u00e1mica de un robot manipulador planar\n\n\n```python\nfrom sympy import var, sin, cos, Matrix, Integer, eye, Function, Rational, exp, Symbol, I, solve, pi, trigsimp, dsolve, sinh, cosh, simplify\nfrom sympy.physics.mechanics import mechanics_printing\nmechanics_printing()\n```\n\n\n```python\nvar(\"m1 m2 J1 J2 l1 l2 t g\")\n```\n\n\n```python\nq1 = Function(\"q1\")(t)\nq2 = Function(\"q2\")(t)\n```\n\n\n```python\nx1 = l1*cos(q1)\ny1 = l1*sin(q1)\nv1 = x1.diff(\"t\")**2 + y1.diff(\"t\")**2\nv1.trigsimp()\n```\n\n\n```python\nx2 = l1*cos(q1) + l2*cos(q1 + q2)\ny2 = l1*sin(q1) + l2*sin(q1 + q2)\nv2 = x2.diff(\"t\")**2 + y2.diff(\"t\")**2\nv2.trigsimp()\n```\n\n\n```python\nK1 = Rational(1, 2)*m1*v1\nK1\n```\n\n\n```python\nK2 = Rational(1, 2)*m1*v2\nK2\n```\n\n\n```python\nU1 = m1*g*y1\nU1\n```\n\n\n```python\nU2 = m2*g*y2\nU2\n```\n\n\n```python\nK = K1 + K2\nK\n```\n\n\n```python\nU = U1 + U2\nU\n```\n\n\n```python\nL = (K - U).expand().simplify()\nL\n```\n\n\n```python\n\u03c41 = (L.diff(q1.diff(t)).diff(t) - L.diff(q1)).simplify().expand().collect(q1.diff(t).diff(t)).collect(q2.diff(t).diff(t))\n```\n\n\n```python\n\u03c42 = (L.diff(q2.diff(t)).diff(t) - L.diff(q2)).simplify().expand().collect(q1.diff(t).diff(t)).collect(q2.diff(t).diff(t))\n```\n\n\n```python\n\u03c41\n```\n\n\n```python\n\u03c42\n```\n\n\n```python\nfrom scipy.integrate import odeint\nfrom numpy import linspace\n```\n\n\n```python\ndef pendulo_doble(estado, tiempo):\n # Se importan funciones necesarias\n from numpy import sin, cos, matrix\n # Se desenvuelven variables del estado y tiempo\n q1, q2, q\u03071, q\u03072 = estado\n t = tiempo\n \n # Se declaran constantes del sistema\n m1, m2 = 1, 1\n l1, l2 = 1, 1\n g = 9.81\n \n # Se declaran constantes del control\n kp1, kp2 = -30, -60\n kv1, kv2 = -20, -20\n \n # Se\u00f1ales de control nulas\n #tau1, tau2 = 0, 0\n \n # Posiciones a alcanzar\n qd1, qd2 = 1, 1\n \n # Se declaran se\u00f1ales de control del sistema\n tau1 = kp1*(q1 - qd1) + kv1*q\u03071\n tau2 = kp2*(q2 - qd2) + kv2*q\u03072\n \n # Se calculan algunos terminos comunes\n \u03d51 = m1*l1**2\n \u03d52 = m1*l1*l2\n \u03d53 = m1*l2**2\n \n # Se calculan las matrices de masas, Coriolis,\n # y vectores de gravedad, control, posicion y velocidad\n M = matrix([[2*\u03d51 + 2*\u03d52*cos(q2) + \u03d53, \u03d52*cos(q2) + \u03d53],\n [\u03d52*cos(q2) + \u03d53, \u03d53]])\n C = matrix([[-2*\u03d52*sin(q2)*q\u03072, -\u03d52*sin(q2)*q\u03072], [\u03d52*sin(q2)*q\u03071, 0]])\n G = matrix([[m1*l1*cos(q1) + m2*l1*cos(q1) + m2*l2*cos(q1 + q2)], [m2*l2*cos(q1 + q2)]])\n Tau = matrix([[tau1], [tau2]])\n q = matrix([[q1], [q2]])\n q\u0307 = matrix([[q\u03071], [q\u03072]])\n \n # Se calcula la derivada del estado del sistema\n qp1 = q\u03071\n qp2 = q\u03072\n \n qpp = M.I*(Tau - C*q\u0307 - G)\n qpp1, qpp2 = qpp.tolist()\n \n return [qp1, qp2, qpp1[0], qpp2[0]]\n```\n\n\n```python\nt = linspace(0, 30, 500)\nestados_simulados = odeint(func = pendulo_doble, y0 = [0, 0, 0, 0], t = t)\n```\n\n\n```python\nq1, q2, q\u03071, q\u03072 = list(zip(*estados_simulados.tolist()))\n```\n\n\n```python\n%matplotlib notebook\nfrom matplotlib.pyplot import plot, style, figure\nfrom mpl_toolkits.mplot3d import Axes3D\nstyle.use(\"ggplot\")\n```\n\n\n```python\nfig1 = figure(figsize=(8, 8))\n\nax1 = fig1.gca()\n\np1, = ax1.plot(t, q1)\np2, = ax1.plot(t, q2)\nax1.legend([p1, p2],[r\"$q_1$\", r\"$q_2$\"])\nax1.set_ylim(-4, 4)\nax1.set_xlim(-0.1, 10);\n```\n\n\n \n\n\n\n\n\n\n\n```python\n%matplotlib inline\nfrom matplotlib import animation\nfrom numpy import sin, cos, arange\n```\n\n\n```python\nl1, l2 = 1, 1\n# Se define el tama\u00f1o de la figura\nfig = figure(figsize=(12.6, 6.6))\n\n# Se define una sola grafica en la figura y se dan los limites de los ejes x y y\naxi = fig.add_subplot(111, autoscale_on=False, xlim=(-2.1, 2.1), ylim=(-2.1, 2.1))\naxi.set_xticklabels([])\naxi.set_yticklabels([])\naxi.axes.get_xaxis().set_visible(False)\naxi.axes.get_yaxis().set_visible(False)\n\n# Se utilizan graficas de linea para el eslabon del pendulo\nlinea, = axi.plot([], [], \"-o\", lw=2, color='gray')\n\ndef init():\n # Esta funcion se ejecuta una sola vez y sirve para inicializar el sistema\n linea.set_data([], [])\n return linea\n\ndef animate(i):\n # Esta funcion se ejecuta para cada cuadro del GIF\n \n # Se obtienen las coordenadas x y y para el eslabon\n xs, ys = [[0, l1*cos(q1[i]), l1*cos(q1[i]) + l2*cos(q1[i]+q2[i])],\n [0, l1*sin(q1[i]), l1*sin(q1[i]) + l2*sin(q1[i]+q2[i])]]\n linea.set_data(xs, ys)\n \n return linea\n\n# Se hace la animacion dandole el nombre de la figura definida al principio, la funcion que\n# se debe ejecutar para cada cuadro, el numero de cuadros que se debe de hacer, el periodo \n# de cada cuadro y la funcion inicial\nani = animation.FuncAnimation(fig, animate, arange(1, len(q1)),\n interval=25, init_func=init)\n\n# Se guarda el GIF en el archivo indicado\nani.save('./imagenes/simulacion-doble.gif', writer='imagemagick');\n```\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "b833f06c30b645a977a0493d99a65ece54639f30", "size": 120854, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Clases/Dinamica pendulo doble.ipynb", "max_stars_repo_name": "robblack007/clase-dinamica-robot", "max_stars_repo_head_hexsha": "f38cb358f2681e9c0dce979acbdcd81bf63bd59c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Clases/Dinamica pendulo doble.ipynb", "max_issues_repo_name": "robblack007/clase-dinamica-robot", "max_issues_repo_head_hexsha": "f38cb358f2681e9c0dce979acbdcd81bf63bd59c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2016-01-26T18:33:11.000Z", "max_issues_repo_issues_event_max_datetime": "2016-05-30T23:58:07.000Z", "max_forks_repo_path": "Clases/Dinamica pendulo doble.ipynb", "max_forks_repo_name": "robblack007/clase-dinamica-robot", "max_forks_repo_head_hexsha": "f38cb358f2681e9c0dce979acbdcd81bf63bd59c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 76.9770700637, "max_line_length": 23733, "alphanum_fraction": 0.7173697188, "converted": true, "num_tokens": 2577, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857982, "lm_q2_score": 0.7431680199891789, "lm_q1q2_score": 0.41779159704774355}} {"text": "```python\nfrom IPython.display import Image\nfrom IPython.core.display import HTML \nImage(url= \"https://i.imgur.com/6TgxQrm.png\")\n```\n\n\n\n\n\n\n\n\n\n```python\nfrom sympy import *; x,h,t,y,z = symbols(\"x h t y z\", real=True)\nf, g, h = symbols('f g h', cls=Function)\nf = -x**2+100\ng = 5*(floor(sqrt(f)))\n\nfor i in range(0,200,1):\n i = i\n if i == 0:\n print(\"\"\"for f(x) = -x**2+100 red line and g(x) = 5*(floor(sqrt(f)) blue line dF = green line \n \n \"\"\") \n \n if i == 0:\n print(\" f +\",i,\"and g +\",i,\" Current Iteration:\",i)\n p0 = plot((f+i),(g+i), diff(f+i),show = False,xlim = (1,10.5),size = (9,5),legend = True)\n p0[0].line_color = 'r'\n p0[2].line_color = 'g'\n p0.show()\n if i == 2:\n print(\"f *\",i,\"and g *\",i,\" Current Iteration:\",i)\n p1 = plot((f*i),(g*i),show = False,xlim = (1,10.5),size = (9,5),legend = \"hello\")\n p1[0].line_color = 'r'\n p1.show()\n if i == 20:\n print(\" f root of\",i,\"and g root of\",i,\" ex. f**(1/i) Current Iteration:\",i)\n p1 = plot((f**(1/i)),(g**(1/i)),show = False,ylim = (0,1.6),xlim = (1,10.5),size = (9,5),legend = True)\n p1[0].line_color = 'r'\n p1.show()\n\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "532cb0dc999e9f87920ee22bce048c228e618ca0", "size": 71293, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Personal_Projects/Multi-Function_Plotting.ipynb", "max_stars_repo_name": "NSC9/Sample_of_Work", "max_stars_repo_head_hexsha": "8f8160fbf0aa4fd514d4a5046668a194997aade6", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Personal_Projects/Multi-Function_Plotting.ipynb", "max_issues_repo_name": "NSC9/Sample_of_Work", "max_issues_repo_head_hexsha": "8f8160fbf0aa4fd514d4a5046668a194997aade6", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Personal_Projects/Multi-Function_Plotting.ipynb", "max_forks_repo_name": "NSC9/Sample_of_Work", "max_forks_repo_head_hexsha": "8f8160fbf0aa4fd514d4a5046668a194997aade6", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 445.58125, "max_line_length": 25028, "alphanum_fraction": 0.9367399324, "converted": true, "num_tokens": 458, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.41775456958798407}} {"text": "### Instructions\n\nWhen running the notebook the first time, make sure to run all cells before making changes in the notebook. Hit Shift + Enter to run the selected cell or, in the top menu, click on: `Kernel` > `Restart Kernel and Run All Cells...` to rerun the whole notebook. If you make any changes in a cell, rerun that cell.\n\nIf you make any changes in a coding cell, rerun the notebook by `Run` > `Run Selected Cell and All Below`\n\n\n```python\n# Import dependencies\nimport warnings\nwarnings.simplefilter(action='ignore', category=FutureWarning) # Ignore warnings\nwarnings.simplefilter(action='ignore', category=UserWarning) # Ignore warnings\nimport sys\nimport numpy as np\nimport matplotlib.pyplot as plt \nfrom scipy.interpolate import InterpolatedUnivariateSpline\nfrom IPython.display import Javascript, display, clear_output\nimport sys\nsys.path.append('python/') # Define path for libraries\nimport downloadSPARCdata as dsd # Import library to download data files\n```\n\n# SPARC Activity\n\nThe following notebook will walk you through rotation curve plotting using an online database. As the main activity in this module, select from a list of 175 galaxies to plot their rotation curves and calculate the amount of Dark Matter for all of them!
                                        \n\nYou will download rotation curve data of galaxies from the Spitzer Photometry & Accurate Rotation Curves (SPARC) database and plot them with and without a Dark Matter component to understand how the \"missing mass\" affects the rotation of stars.
                                        \nYou may either download and unzip the data files yourself or have Python do the work. \n\n#### Option 1. Download and unzip the data files yourself:\n\n1. Go to http://astroweb.cwru.edu/SPARC/ and under \"BASIC SPARC DATA\", download the `Rotmod_LTG.zip` file for \"Newtonian Mass Models\".\n2. Open (extract/unzip) the zip file to preferably the same location as where your Python notebook is located. \n3. Choose a galaxy (any file) of your choice. Do not rename the file. Put the name of the chosen galaxy in the variable in the cell below and run it.\n4. Make a note of the directory (file location) of the SPARC file of your galaxy **with respect to this location of this Python notebook**. For example, if your file is located in the same location as this code, leave the following cell as is. But if it is, say, in the next folder \"up\" from this one, use the extension `../`. As an example, if the SPARC file is located two folders up then one folder \"down\" (into a different folder named, say, 'otherfolder'), you would write:\n`SPARC_file_directory='../../otherfolder/'` in the cell below and run it.\n\n\n```python\nSPARC_file_directory='data/sparc/' #note that '' means the string variable is blank\n```\n\n#### Option 2. Let Python download and unzip the data files\n\n1. By clicking the YES button, you can download and unzip SPARC data files to your computer. \n\n\n```python\n#NBVAL_IGNORE_OUTPUT\n#Because the button doesn't save to the repository correctly.\nprint(\"Would you like to download and unzip SPARC data files to your computer?\")\ndsd.displaybuttons\n```\n\n Would you like to download and unzip SPARC data files to your computer?\n\n\n\n VBox(children=(HBox(children=(Button(button_style='success', description='Yes', icon='check', style=ButtonStyl\u2026\n\n\n### Choose a galaxy\n\nSelect any galaxy from the dropdown menu.\n\n\n```python\n#NBVAL_IGNORE_OUTPUT\n#Because the dropdown doesn't save to the repository correctly.\ngalaxylist = ['NGC5005'] # default list of galaxies \ndef on_change(change): # add selected galaxynames to the list\n if change['type'] == 'change' and change['name'] == 'value':\n galaxylist.append(change['new'])\n display(Javascript('IPython.notebook.execute_cell_range(IPython.notebook.get_selected_index()+1, IPython.notebook.ncells())'))\ndsd.dropdownmenu.observe(on_change)\ndisplay(dsd.galaxyoptions)\n```\n\n\n Box(children=(Box(children=(Label(value='Galaxy: '), Dropdown(index=79, options=('CamB', 'D512-2', 'D564-8', '\u2026\n\n\nOnce you selected a galaxy, click on the cell below; , then select `Run` → `Run Selected Cell and All Below` to reveal the rotation curve and the image of the chosen galaxy in the following cells. \n\n\n```python\n# Writing the chosen galaxy name in a text file would allow us to use the selection in libraries outside of this notebook\nchosengalaxy = galaxylist[-1] # define the last element of the list as the selected galaxy\ntextfile = open(\"python/chosengalaxy.txt\", \"w\")\ntextfile.write(chosengalaxy)\ntextfile.close()\n```\n\nThe downloaded files are named as \"[galaxy name]\\_rotmod.dat\". In the following code cell, we have defined the file path for you to import the data for your chosen galaxy.\n\n\n```python\n# Define file path for .dat files\nSPARC_file = SPARC_file_directory + chosengalaxy + '_rotmod.dat'\n\n# Load the galaxy data\ngalaxydata = np.loadtxt(SPARC_file)\n```\n\nNow, lets print the header to see what variable each column corresponds to.\n\n\n```python\n# Print header \nfor lines in open(SPARC_file, 'r'):\n if lines.startswith('#'):\n print(lines)\n```\n\n # Distance = 16.9 Mpc\n \n # Rad\tVobs\terrV\tVgas\tVdisk\tVbul\tSBdisk\tSBbul\t\t\n \n # kpc\tkm/s\tkm/s\tkm/s\tkm/s\tkm/s\tL/pc^2\tL/pc^2\n \n\n\n__Header key__:
                                        \n>_Rad_: radius or distance from the center of galaxy (in kiloparsec)
                                        \n>_Vobs_: observed velocity/measured datapoints (in km/s)
                                        \n>_errV_: uncertainty in the observed velocity (in km/s)
                                        \n>_Vgas_: velocity of the gas component (in km/s)
                                        \n>_Vdisk_: velocity of the disk component (in km/s)
                                        \n>_Vbul_: velocity of the bulge component (in km/s)
                                        \n>_SBdisk_: surface brightness of the disk component (in Luminosity/$ \\rm parsec^2$)
                                        \n>_SBbul_: surface brightness of the bulge component (in Luminosity/$ \\rm parsec^2$)\n\nSplit columns into arrays and name them according to the header displayed in the cell above.\n\n\n```python\n# Split columns into arrays\nRad,Vobs,errV,Vgas,Vdisk,Vbul,SBdisk,SBbul = galaxydata.T \n```\n\nThe distance to the galaxy is given in the data file in Megaparsecs (1 Mpc equals about 3.26 million light years). You can define this value in your Python notebook for your reference. \n\n\n```python\n# Define distance in Mpc\nfirstline = open(SPARC_file).readline() # Open the data file\nfirstline = firstline.split() # Split the first line into separate strings\ndistance = float(firstline[3]) # Take the 4th value of the first line (counts: 0,1,2,3...)\nprint(\"The distance to {} galaxy is {} Mpc.\".format(chosengalaxy,distance))\n```\n\n The distance to NGC5005 galaxy is 16.9 Mpc.\n\n\nDue to the low number of datapoints, the rotation curve might look choppy. In order to smooth it, you can define a polynomial using Scipy's Interpolated Univariate Spline function, then choose your range and number of datapoints (sampling radii) to define the new, more smooth curve of each component. \n\n\n```python\n# Spline function\ndef interpd(x,y):\n return InterpolatedUnivariateSpline(x,y,k=3) # Degree of the smoothing spline: 3\n\n# Bulge\ndef bulge(r,bpref):\n polynomial = interpd(Rad,bpref*Vbul) # bpref is the bulge prefactor added to the bulge\n return polynomial(r)\n\n# Disk\ndef disk(r,dpref):\n polynomial = interpd(Rad,dpref*Vdisk) # dpref is the disk prefactor added to the disk\n return polynomial(r)\n\n# Gas \ndef gas(r):\n polynomial = interpd(Rad,Vgas) # Note that the gas doesn't have a prefactor\n return polynomial(r)\n```\n\n### Observed velocity of the chosen galaxy\n\nPlot the observed velocity (measured datapoints) and the error bars on each measurement of the chosen galaxy as a function of radius.\n\n\n```python\nplt.figure(figsize=(8,6)) # size of the galaxy\nplt.errorbar(Rad,Vobs,yerr=errV, marker='o', markersize=8, \\\n ecolor='gray',color='gray', linestyle='none', linewidth=2) # plot datapoints with errorbars\nplt.xlabel('radius (kpc)',size=12) # label x-axis\nplt.ylabel('velocity (km/s)',size=12) # label y-axis\nplt.title(str('Observed Velocity of ' + chosengalaxy), size=14) # title of the plot\nplt.xlim(0,np.max(Rad)+0.2) # range of the x-axis\nplt.ylim(0,np.max(Vobs)+100) # range of the y-axis\nplt.show() # show the plot\n```\n\nA prefactor or mass-to-light ratio ($\\Upsilon$) is added to the disk and the bulge which will be useful when fitting the curve of each component. These prefactors help scaling the curve up and down. You can either change these values manually and see the magnitude of the relevant curve change in the cells below or import the fitting parameters from a python library.
                                        \n\n>pref_bulge: bulge prefactor
                                        \n>pref_disk: disk prefactor\n\nNote that the gas prefactor is fixed. The mass of the gas was calculated assuming a factor of 1.33 to account for the contribution of helium. \n\n\n```python\nimport widget_SPARC as fit # Import widget library for using the fitting parameters\nimport importlib\nimportlib.reload(fit) # Reload widget library so the changes take effect\nclear_output()\n```\n\n\n```python\n# Prefactors\npref_bulge = fit.best_bpref\npref_disk = fit.best_dpref\n\n# Radius for plotting\nradius = np.linspace(np.min(Rad),np.max(Rad),1000) # starting at the same radius where the given datapoints start, \n # ending at the same radius where the given datapoints end, \n # with 1000 datapoints\n```\n\n### Rotational velocity of each luminous component\n\nPlot the rotation velocity of each component.\n\n\n```python\nplt.figure(figsize=(8,6))\nplt.plot(radius, gas(radius), label=\"Gas velocity\")\nplt.plot(radius, bulge(radius,pref_bulge), label=\"Bulge velocity\")\nplt.plot(radius, disk(radius,pref_disk), label=\"Disk velocity\")\nplt.xlabel('radius (kpc)', size=12)\nplt.ylabel('velocity (km/s)', size=12)\nplt.title(str('Velocity of Luminous Components of ' + chosengalaxy), size=14)\nplt.xlim(0,np.max(Rad + 0.1))\nplt.ylim(0,np.max(Vobs + 80))\nplt.legend(bbox_to_anchor=(1,1), loc=\"upper left\")\nplt.show()\n```\n\nEach component was calculated using the following model:
                                        \n>Bulge: residual luminosity profile
                                        \n>Disk: observed [3.6] surface brightness profile
                                        \n>Gas: H1 surface density profiles or mass models
                                        \n\nHowever, these calculations are beyond the scope of this workshop. \n\n### Total velocity curve of luminous matter\n\nNow that we have all of the luminous components or light matter: bulge, disk and gas, we can add them in quadrature with equivalent prefactors to get the total velocity of the luminous matter. \n\n>__Total velocity of luminous matter__:
                                        \n \\begin{equation}\n v_{total,light}(r) = \\sqrt{\\lvert v_{gas}\\rvert v_{gas} + \\Upsilon _{bulge} \\lvert v_{bulge}\\rvert v_{bulge} + \\Upsilon _{disk} \\lvert v_{disk}\\rvert v_{disk}}\n \\end{equation}
                                        \n\n\n```python\ndef total_velocity_light(r,pref_bulge,pref_disk): \n return np.sqrt(gas(r)**2 + bulge(r,pref_bulge)**2 + disk(r,pref_disk)**2)\n```\n\nPlot the luminous components and the total velocity.\n\n\n```python\nplt.figure(figsize=(9,6))\n\nplt.plot(radius, gas(radius), label=\"Gas velocity\", color='#1f77b4')\nplt.plot(radius, bulge(radius,pref_bulge), label=\"Bulge velocity\", color='#ff7f0e')\nplt.plot(radius, disk(radius,pref_disk), label=\"Disk velocity\", color='#2ca02c')\n\nplt.plot(radius,total_velocity_light(radius,pref_bulge,pref_disk), label=\"Total velocity of luminous matter\", color='#d62728', linewidth=3)\nplt.errorbar(Rad,Vobs,yerr=errV, marker='o', markersize=8, \\\n ecolor='gray',color='gray', linestyle='none', linewidth=2, label = \"Observed velocity\")\nplt.xlabel('radius (kpc)',size=12)\nplt.ylabel('velocity (km/s)',size=12)\nplt.title(str('Rotation Curve of ' + chosengalaxy + ' - Luminous Matter'), size=14)\nplt.xlim(0,np.max(Rad + 0.2))\nplt.ylim(0,np.max(Vobs) + 100)\nplt.legend(bbox_to_anchor=(1,1), loc=\"upper left\")\nplt.show()\n```\n\nNotice that the total velocity does not align with the measured velocity. Not even if you scale the other components up and down with the prefactor. Let's add the \"missing matter\".\n\n### Add Dark Matter\n\nDark Matter can be characterized by two parameters: the central mass density ($\\rho_0$) and the core radius ($r_c$).
                                        \n***Central mass density***: The central mass density is the density at the center of the galaxy, changing this value changes the magnitude of the Dark Matter curve.
                                        \n***Core radius***: The core radius (also called \"cutoff radius\" or \"scale radius\") indicates where the density falls off by a factor of $e$ (~2.7). Adjusting this factor changes where the \"bump\" of the curve is located.\n\n>__Velocity__:
                                        \n \\begin{equation}\n v_{DM}(r) = \\sqrt{4 \\pi G \\rho_{0} r_c^2 \\big( 1- \\frac{r_c}{r} \\arctan{\\frac{r}{r_c}}\\big)}\n \\end{equation}
                                        \n where:
                                        \n $G$ = gravitational constant
                                        \n $\\rho_0$ = central mass density (in solar mass/$\\rm kpc^3$)
                                        \n $r_c$ = core radius (in kpc)
                                        \n\nSet parameters or import the fitting parameters from our Python library. The fitting parameters were calculated using the Python library: lmfit. \n\n\n```python\n# Dark matter halo parameters\nrho0 = fit.best_rho0 # Central mass density (in solar mass/kpc^3)\nrc = fit.best_rc # Core radius (in kpc)\n\n# Constant parameters\nG = 4.300e-6 # Gravitational constant (kpc/solar mass*(km/s)^2)\n\n# Print values\nprint(\"The fitted best values for the {} galaxy: \".format(chosengalaxy))\nprint(\" Central mass density of the dark matter halo = {:.3e} solar mass/kpc^3\".format(rho0))\nprint(\" Core radius of the Dark Matter halo = {:.3f} kpc\".format(rc))\n```\n\n The fitted best values for the NGC5005 galaxy: \n Central mass density of the dark matter halo = 1.693e+07 solar mass/kpc^3\n Core radius of the Dark Matter halo = 9.917 kpc\n\n\n\n```python\n# Equation for dark matter halo velocity\ndef halo(r,rho0,rc):\n v = np.sqrt(4*np.pi*G*rho0*rc**2*(1 - rc/r * np.arctan(r/rc)))\n return v\n```\n\nNow, calculate the total velocity with the Dark Matter component included. \n\n\n```python\ndef total_velocity_withDM(r,pref_bulge,pref_disk,rho0,rc): \n return np.sqrt(gas(r)**2 + bulge(r,pref_bulge)**2 + disk(r,pref_disk)**2 + halo(r,rho0,rc)**2)\n```\n\nPlot all components and the total velocity with the Dark Matter.\n\n\n```python\nplt.figure(figsize=(12,8))\n\nplt.plot(radius, gas(radius), label=\"Gas velocity\", color='#1f77b4')\nplt.plot(radius, bulge(radius,pref_bulge), label=\"Bulge velocity\", color='#ff7f0e')\nplt.plot(radius, disk(radius,pref_disk), label=\"Disk velocity\", color='#2ca02c')\nplt.plot(radius, halo(radius,rho0,rc), label=\"Dark matter halo velocity\", color='#bcbd22')\n\nplt.plot(radius,total_velocity_light(radius,pref_bulge,pref_disk), label=\"Total velocity of luminous matter\", color='#d62728', linewidth=3)\nplt.plot(radius,total_velocity_withDM(radius,pref_bulge,pref_disk,rho0,rc), label=\"Total velocity with dark matter\", color='k', linewidth=3)\n\nplt.errorbar(Rad,Vobs,yerr=errV, marker='o', markersize=8, \\\n ecolor='gray',color='gray', linestyle='none', linewidth=2, label = \"Observed velocity\")\nplt.xlabel('radius (kpc)',size=12)\nplt.ylabel('velocity (km/s)',size=12)\nplt.suptitle(str('Rotation Curve of ' + chosengalaxy), size=14)\nplt.title(str('Distance = {} Mpc'.format(distance)), size=12)\nplt.xlim(0,np.max(Rad + 0.2))\nplt.ylim(0,np.max(Vobs) + 100)\nplt.legend(bbox_to_anchor=(1,1), loc=\"upper left\")\nplt.show()\n```\n\nFrom the plot, we can see that the total velocity curve with dark matter matches the observed data points.\n\n### References\n\n>Jimenez, Raul, Licia Verde, and S. Peng Oh. **Dark halo properties from rotation curves.** _Monthly Notices of the Royal Astronomical Society_ 339, no. 1 (2003): 243-259. https://doi.org/10.1046/j.1365-8711.2003.06165.x.

                                        \n>Lelli, F., McGaugh, S. S., & Schombert, J. M. (2016). **SPARC: Mass models for 175 disk galaxies with Spitzer photometry and accurate rotation curves.** _The Astronomical Journal_, 152(6), 157. https://doi.org/10.3847/0004-6256/152/6/157

                                        \n>Matt Newville, Renee Otten, Andrew Nelson, Antonino Ingargiola, Till Stensitzki, Dan Allan, Austin Fox, Faustin Carter, Micha\u0142, Ray Osborn, Dima Pustakhod, lneuhaus, Sebastian Weigand, Glenn, Christoph Deil, Mark, Allan L. R. Hansen, Gustavo Pasquevich, Leon Foks, \u2026 Arun Persaud. (2021). __lmfit/lmfit-py: 1.0.3 (1.0.3).__ Zenodo. https://doi.org/10.5281/zenodo.5570790.

                                        \n>\u201cMegaparsec: Cosmos.\u201d __Megaparsec__ | _COSMOS_. Swinburne University of Technology. Accessed November 12, 2021. https://astronomy.swin.edu.au/cosmos/m/megaparsec. \n***\n", "meta": {"hexsha": "c39f0e69c9a9f7618a6599c592c9694c84ab4e0f", "size": 208791, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "binder/06_Plotting_SPARC_data.ipynb", "max_stars_repo_name": "villano-lab/galactic-spin-W1", "max_stars_repo_head_hexsha": "d95c706ccbc347f9bc61bb7c96b1314460bc2d0f", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-03-22T04:00:17.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-22T04:00:17.000Z", "max_issues_repo_path": "binder/06_Plotting_SPARC_data.ipynb", "max_issues_repo_name": "villano-lab/galactic-spin-W1", "max_issues_repo_head_hexsha": "d95c706ccbc347f9bc61bb7c96b1314460bc2d0f", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 14, "max_issues_repo_issues_event_min_datetime": "2021-11-05T18:17:19.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-19T20:35:05.000Z", "max_forks_repo_path": "binder/06_Plotting_SPARC_data.ipynb", "max_forks_repo_name": "villano-lab/galactic-spin-W1", "max_forks_repo_head_hexsha": "d95c706ccbc347f9bc61bb7c96b1314460bc2d0f", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 279.8806970509, "max_line_length": 80384, "alphanum_fraction": 0.9160596003, "converted": true, "num_tokens": 4481, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883735630721, "lm_q2_score": 0.6959583250334526, "lm_q1q2_score": 0.4177060951695078}} {"text": "\n\n# Lambda School Data Science Module 142\n## Sampling, Confidence Intervals, and Hypothesis Testing\n\n## Prepare - examine other available hypothesis tests\n\nIf you had to pick a single hypothesis test in your toolbox, t-test would probably be the best choice - but the good news is you don't have to pick just one! Here's some of the others to be aware of:\n\n\n```\nimport numpy as np\nfrom scipy.stats import chisquare # One-way chi square test\n\n# Chi square can take any crosstab/table and test the independence of rows/cols\n# The null hypothesis is that the rows/cols are independent -> low chi square\n# The alternative is that there is a dependence -> high chi square\n# Be aware! Chi square does *not* tell you direction/causation\n\nind_obs = np.array([[1, 1], [2, 2]]).T\nprint(ind_obs)\nprint(chisquare(ind_obs, axis=None))\n\ndep_obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T\nprint(dep_obs)\nprint(chisquare(dep_obs, axis=None))\n```\n\n [[1 2]\n [1 2]]\n Power_divergenceResult(statistic=0.6666666666666666, pvalue=0.8810148425137847)\n [[16 32]\n [18 24]\n [16 16]\n [14 28]\n [12 20]\n [12 24]]\n Power_divergenceResult(statistic=23.31034482758621, pvalue=0.015975692534127565)\n\n\n\n```\n# Distribution tests:\n# We often assume that something is normal, but it can be important to *check*\n\n# For example, later on with predictive modeling, a typical assumption is that\n# residuals (prediction errors) are normal - checking is a good diagnostic\n\nfrom scipy.stats import normaltest\n# Poisson models arrival times and is related to the binomial (coinflip)\nsample = np.random.poisson(5, 1000)\nprint(normaltest(sample)) # Pretty clearly not normal\n```\n\n NormaltestResult(statistic=27.958070030334067, pvalue=8.49145731018306e-07)\n\n\n\n```\n# Kruskal-Wallis H-test - compare the median rank between 2+ groups\n# Can be applied to ranking decisions/outcomes/recommendations\n# The underlying math comes from chi-square distribution, and is best for n>5\nfrom scipy.stats import kruskal\n\nx1 = [1, 3, 5, 7, 9]\ny1 = [2, 4, 6, 8, 10]\nprint(kruskal(x1, y1)) # x1 is a little better, but not \"significantly\" so\n\nx2 = [1, 1, 1]\ny2 = [2, 2, 2]\nz = [2, 2] # Hey, a third group, and of different size!\nprint(kruskal(x2, y2, z)) # x clearly dominates\n```\n\n KruskalResult(statistic=0.2727272727272734, pvalue=0.6015081344405895)\n KruskalResult(statistic=7.0, pvalue=0.0301973834223185)\n\n\nAnd there's many more! `scipy.stats` is fairly comprehensive, though there are even more available if you delve into the extended world of statistics packages. As tests get increasingly obscure and specialized, the importance of knowing them by heart becomes small - but being able to look them up and figure them out when they *are* relevant is still important.\n\n## Live Lecture - let's explore some more of scipy.stats\n\nCandidate topics to explore:\n\n- `scipy.stats.chi2` - the Chi-squared distribution, which we can use to reproduce the Chi-squared test\n- Calculate the Chi-Squared test statistic \"by hand\" (with code), and feed it into `chi2`\n- Build a confidence interval with `stats.t.ppf`, the t-distribution percentile point function (the inverse of the CDF) - we can write a function to return a tuple of `(mean, lower bound, upper bound)` that you can then use for the assignment (visualizing confidence intervals)\n\n\n```\nimport pandas as pd\nimport numpy as np\nfrom scipy import stats\n\ndf = pd.read_csv('https://raw.githubusercontent.com/ryanleeallred/datasets/master/adult.csv', na_values=\" ?\")\nprint(df.shape)\ndf.head(20)\n```\n\n (32561, 15)\n\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        ageworkclassfnlwgteducationeducation-nummarital-statusoccupationrelationshipracesexcapital-gaincapital-losshours-per-weekcountrysalary
                                        039State-gov77516Bachelors13Never-marriedAdm-clericalNot-in-familyWhiteMale2174040United-States<=50K
                                        150Self-emp-not-inc83311Bachelors13Married-civ-spouseExec-managerialHusbandWhiteMale0013United-States<=50K
                                        238Private215646HS-grad9DivorcedHandlers-cleanersNot-in-familyWhiteMale0040United-States<=50K
                                        353Private23472111th7Married-civ-spouseHandlers-cleanersHusbandBlackMale0040United-States<=50K
                                        428Private338409Bachelors13Married-civ-spouseProf-specialtyWifeBlackFemale0040Cuba<=50K
                                        537Private284582Masters14Married-civ-spouseExec-managerialWifeWhiteFemale0040United-States<=50K
                                        649Private1601879th5Married-spouse-absentOther-serviceNot-in-familyBlackFemale0016Jamaica<=50K
                                        752Self-emp-not-inc209642HS-grad9Married-civ-spouseExec-managerialHusbandWhiteMale0045United-States>50K
                                        831Private45781Masters14Never-marriedProf-specialtyNot-in-familyWhiteFemale14084050United-States>50K
                                        942Private159449Bachelors13Married-civ-spouseExec-managerialHusbandWhiteMale5178040United-States>50K
                                        1037Private280464Some-college10Married-civ-spouseExec-managerialHusbandBlackMale0080United-States>50K
                                        1130State-gov141297Bachelors13Married-civ-spouseProf-specialtyHusbandAsian-Pac-IslanderMale0040India>50K
                                        1223Private122272Bachelors13Never-marriedAdm-clericalOwn-childWhiteFemale0030United-States<=50K
                                        1332Private205019Assoc-acdm12Never-marriedSalesNot-in-familyBlackMale0050United-States<=50K
                                        1440Private121772Assoc-voc11Married-civ-spouseCraft-repairHusbandAsian-Pac-IslanderMale0040NaN>50K
                                        1534Private2454877th-8th4Married-civ-spouseTransport-movingHusbandAmer-Indian-EskimoMale0045Mexico<=50K
                                        1625Self-emp-not-inc176756HS-grad9Never-marriedFarming-fishingOwn-childWhiteMale0035United-States<=50K
                                        1732Private186824HS-grad9Never-marriedMachine-op-inspctUnmarriedWhiteMale0040United-States<=50K
                                        1838Private2888711th7Married-civ-spouseSalesHusbandWhiteMale0050United-States<=50K
                                        1943Self-emp-not-inc292175Masters14DivorcedExec-managerialUnmarriedWhiteFemale0045United-States>50K
                                        \n
                                        \n\n\n\n\n```\ndf.isnull().sum()\n```\n\n\n\n\n age 0\n workclass 1836\n fnlwgt 0\n education 0\n education-num 0\n marital-status 0\n occupation 1843\n relationship 0\n race 0\n sex 0\n capital-gain 0\n capital-loss 0\n hours-per-week 0\n country 583\n salary 0\n dtype: int64\n\n\n\n\n```\ndf.describe()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        agefnlwgteducation-numcapital-gaincapital-losshours-per-week
                                        count32561.0000003.256100e+0432561.00000032561.00000032561.00000032561.000000
                                        mean38.5816471.897784e+0510.0806791077.64884487.30383040.437456
                                        std13.6404331.055500e+052.5727207385.292085402.96021912.347429
                                        min17.0000001.228500e+041.0000000.0000000.0000001.000000
                                        25%28.0000001.178270e+059.0000000.0000000.00000040.000000
                                        50%37.0000001.783560e+0510.0000000.0000000.00000040.000000
                                        75%48.0000002.370510e+0512.0000000.0000000.00000045.000000
                                        max90.0000001.484705e+0616.00000099999.0000004356.00000099.000000
                                        \n
                                        \n\n\n\n\n```\ndf.describe(exclude='number')\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        workclasseducationmarital-statusoccupationrelationshipracesexcountrysalary
                                        count307253256132561307183256132561325613197832561
                                        unique816714652412
                                        topPrivateHS-gradMarried-civ-spouseProf-specialtyHusbandWhiteMaleUnited-States<=50K
                                        freq22696105011497641401319327816217902917024720
                                        \n
                                        \n\n\n\n\n```\ndf.sex.value_counts()\n```\n\n\n\n\n Male 21790\n Female 10771\n Name: sex, dtype: int64\n\n\n\n\n```\ncut_points = [0,9,19,29,39,49,1000]\nlabel_names = ['0-9', '10-19', '20-29', '30-39', '40-49', '50+']\ndf['hours_per_week_categories'] = pd.cut(df['hours-per-week'], cut_points, labels=label_names)\n\ndf.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        ageworkclassfnlwgteducationeducation-nummarital-statusoccupationrelationshipracesexcapital-gaincapital-losshours-per-weekcountrysalaryhours_per_week_categories
                                        039State-gov77516Bachelors13Never-marriedAdm-clericalNot-in-familyWhiteMale2174040United-States<=50K40-49
                                        150Self-emp-not-inc83311Bachelors13Married-civ-spouseExec-managerialHusbandWhiteMale0013United-States<=50K10-19
                                        238Private215646HS-grad9DivorcedHandlers-cleanersNot-in-familyWhiteMale0040United-States<=50K40-49
                                        353Private23472111th7Married-civ-spouseHandlers-cleanersHusbandBlackMale0040United-States<=50K40-49
                                        428Private338409Bachelors13Married-civ-spouseProf-specialtyWifeBlackFemale0040Cuba<=50K40-49
                                        \n
                                        \n\n\n\n\n```\ndata = df[['sex', 'hours_per_week_categories']]\ndata.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        sexhours_per_week_categories
                                        0Male40-49
                                        1Male10-19
                                        2Male40-49
                                        3Male40-49
                                        4Female40-49
                                        \n
                                        \n\n\n\n\n```\ndata['hours_per_week_categories'].value_counts()\n```\n\n\n\n\n 40-49 18336\n 50+ 6462\n 30-39 3667\n 20-29 2392\n 10-19 1246\n 0-9 458\n Name: hours_per_week_categories, dtype: int64\n\n\n\nBefore we calculate our contingency table, lets make very clear what our null and alternative hypotheses are in this situation. \n\n$H_{0}$ : There is *no* statistically significant relationship between gender and working hours per week.\n\n$H_{a}$ : There *is* a statistically significant relationship between gender and working hours per week.\n\n\n```\n# contingency_table = pd.crosstab(data['sex'], data['hours_per_week_categories'], margins=True, normalize=True)\ncontingency_table = pd.crosstab(data['sex'], data['hours_per_week_categories'], margins=True)\n\ncontingency_table\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        hours_per_week_categories0-910-1920-2930-3940-4950+All
                                        sex
                                        Female235671128719145636102810771
                                        Male2235751105175312700543421790
                                        All64621246183363667458239232561
                                        \n
                                        \n\n\n\n\n```\nfemalecount = contingency_table.iloc[0][0:6].values\nfemalecount\n```\n\n\n\n\n array([ 235, 671, 1287, 1914, 5636, 1028])\n\n\n\n\n```\nmalecount = contingency_table.iloc[1][0:6].values\nmalecount\n```\n\n\n\n\n array([ 223, 575, 1105, 1753, 12700, 5434])\n\n\n\n\n```\nimport matplotlib.pyplot as plt\nimport seaborn as sns\n\n#Plots the bar chart\nfig = plt.figure(figsize=(10, 5))\nsns.set(font_scale=1.8)\ncategories = [\"0-9\",\"10-19\",\"20-29\",\"30-39\",\"40-49\",\"50+\"]\np1 = plt.bar(categories, malecount, 0.55, color='#d62728')\np2 = plt.bar(categories, femalecount, 0.55, bottom=malecount)\nplt.legend((p2[0], p1[0]), ('Male', 'Female'))\nplt.xlabel('Hours per Week Worked')\nplt.ylabel('Count')\nplt.show()\n```\n\n### Expected Value Calculation\n\\begin{align}\nexpected_{i,j} =\\frac{(row_{i} \\text{total})(column_{j} \\text{total}) }{(\\text{total observations})} \n\\end{align}\n\n\n```\n# Get Row Sums\nrow_sums = contingency_table.iloc[0:2, 6].values\ncol_sums = contingency_table.iloc[2, 0:6].values\n\nprint(row_sums)\nprint(col_sums)\n```\n\n [10771 21790]\n [ 6462 1246 18336 3667 458 2392]\n\n\n\n```\ntotal = contingency_table.loc['All', 'All']\nprint(total)\n\nexpected = []\nfor i in range(len(row_sums)):\n expected_row = []\n for column in col_sums:\n expected_val = column*row_sums[i]/total\n expected_row.append(expected_val)\n expected.append(expected_row)\n \nprint(np.array(expected))\n\n# print(np.array(expected).T[0].sum())\n```\n\n 32561\n [[ 2137.59411566 412.16995793 6065.44811277 1213.02346365\n 151.50388502 791.26046497]\n [ 4324.40588434 833.83004207 12270.55188723 2453.97653635\n 306.49611498 1600.73953503]]\n\n\n## Chi-Squared Statistic with Numpy\n\n\\begin{align}\n\\chi^2 = \\sum \\frac{(observed_{i}-expected_{i})^2}{(expected_{i})}\n\\end{align}\n\nFor the $observed$ values we will just use a version of our contingency table without the margins as a numpy array. In this way, if our observed values array and our expected values array are the same shape, then we can subtract them and divide them directly which makes the calculations a lot cleaner. No for loops!\n\n\n```\ncontingency = pd.crosstab(data['sex'],\n data['hours_per_week_categories'])\ncontingency = contingency.values\n\nprint(contingency.shape)\nprint(contingency)\n```\n\n (2, 6)\n [[ 235 671 1287 1914 5636 1028]\n [ 223 575 1105 1753 12700 5434]]\n\n\n\n```\nchi_squared = ((contingency - expected)**2/(expected)).sum()\nprint(f\"Chi-Squared: {chi_squared}\")\n```\n\n Chi-Squared: 729291.9658823247\n\n\n\n```\n# Calculate Degrees of Freedom\ndof = (len(row_sums)-1)*(len(col_sums)-1)\nprint(f\"Degrees of Freedom: {dof}\") \n```\n\n Degrees of Freedom: 5\n\n\n\n```\n# Calculate the p-value from the chi_squared and dof\np_value = stats.chi2.sf(2287.19, dof)\nprint(f\"P-value: {p_value}\")\n```\n\n P-value: 0.0\n\n\n\n```\nchi_squared, p_value, dof, expected = stats.chi2_contingency(contingency)\n\nprint(f\"Chi-Squared: {chi_squared}\")\nprint(f\"P-value: {p_value}\")\nprint(f\"Degrees of Freedom: {dof}\") \nprint(\"Expected: \\n\", np.array(expected))\n```\n\n Chi-Squared: 2287.190943926107\n P-value: 0.0\n Degrees of Freedom: 5\n Expected: \n [[ 151.50388502 412.16995793 791.26046497 1213.02346365\n 6065.44811277 2137.59411566]\n [ 306.49611498 833.83004207 1600.73953503 2453.97653635\n 12270.55188723 4324.40588434]]\n\n\n## Confidence Intervals\n\n\n```\nimport scipy.stats as stats\nimport numpy as np\n\n#confidence_interval = [lower_bound, upper_bound]\n\ncoinflips = np.random.binomial(n=1, p=.7, size=100)\nprint(coinflips)\n```\n\n [1 1 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 0 1 0 1 1 1 1\n 1 1 1 1 0 1 1 0 1 1 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1\n 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 1]\n\n\n\n```\nstats.ttest_1samp(coinflips, .5)\n```\n\n\n\n\n Ttest_1sampResult(statistic=5.444102647659543, pvalue=3.78924151046066e-07)\n\n\n\n\n```\ncoinflips_100 = np.random.binomial(n=1, p=.5, size=100)\ncoinflips_1000 = np.random.binomial(n=1, p=.5, size=1000)\n\nprint(\"100 Coinflips Standard Deviation:\", np.std(coinflips_100))\nprint(\"1000 Coinflips Standard Deviation:\", np.std(coinflips_1000))\n```\n\n 100 Coinflips Standard Deviation: 0.4995998398718718\n 1000 Coinflips Standard Deviation: 0.4998039615689336\n\n\n\n```\nprint(\"100 Coinflips Standard Error:\", stats.sem(coinflips_100))\nprint(\"1000 Coinflips Standard :\", stats.sem(coinflips_1000))\n```\n\n 100 Coinflips Standard Error: 0.05021167315686782\n 1000 Coinflips Standard : 0.01581309754773093\n\n\n\n```\n0.4995998398718718/np.sqrt(100)\n```\n\n\n\n\n 0.04995998398718718\n\n\n\n\n```\n0.4998039615689336/np.sqrt(1000)\n```\n\n\n\n\n 0.015805189021330938\n\n\n\n\n```\nimport numpy as np\nfrom scipy import stats\n```\n\n\n```\n# Confidence intervals!\n# Similar to hypothesis testing, but centered at sample mean\n# Generally better than reporting the \"point estimate\" (sample mean)\n# Why? Because point estimates aren't always perfect\n\nimport numpy as np\nfrom scipy import stats\n\ndef confidence_interval(data, confidence=0.95):\n \"\"\"\n Calculate a confidence interval around a sample mean for given data.\n Using t-distribution and two-tailed test, default 95% confidence. \n \n Arguments:\n data - iterable (list or numpy array) of sample observations\n confidence - level of confidence for the interval\n \n Returns:\n tuple of (mean, lower bound, upper bound)\n \"\"\"\n data = np.array(data)\n mean = np.mean(data)\n n = len(data)\n stderr = stats.sem(data)\n interval = stderr * stats.t.ppf((1 + confidence) / 2.0, n - 1)\n return (mean, mean - interval, mean + interval)\n\ndef report_confidence_interval(confidence_interval):\n \"\"\"\n Return a string with a pretty report of a confidence interval.\n \n Arguments:\n confidence_interval - tuple of (mean, lower bound, upper bound)\n \n Returns:\n None, but prints to screen the report\n \"\"\"\n #print('Mean: {}'.format(confidence_interval[0]))\n #print('Lower bound: {}'.format(confidence_interval[1]))\n #print('Upper bound: {}'.format(confidence_interval[2]))\n s = \"our mean lies in the interval [{:.2}, {:.2}]\".format(\n confidence_interval[1], confidence_interval[2])\n return s\n```\n\n\n```\n(1+.95)/2.0\n```\n\n\n\n\n 0.975\n\n\n\n\n```\ncoinflips = np.random.binomial(n=1, p=.5, size=10000)\n# print(coinflips)\n```\n\n\n```\ncoinflip_interval = confidence_interval(coinflips) # Default 95% conf\ncoinflip_interval\n```\n\n\n\n\n (0.5031, 0.49329869198139176, 0.5129013080186082)\n\n\n\n\n```\ndef get_100_coinflips():\n return np.random.binomial(n=1, p=.5, size=100)\n\nfor i in range(0,100):\n # 100 Coinflips\n coinflips = get_100_coinflips()\n # Calculate Confidence Interval\n coinflip_interval = confidence_interval(coinflips)\n # Report Confidence Interval\n print('Mean: {}'.format(coinflip_interval[0]))\n print('Lower bound: {}'.format(coinflip_interval[1]))\n print('Upper bound: {}'.format(coinflip_interval[2]))\n```\n\n Mean: 0.52\n Lower bound: 0.4203691469585294\n Upper bound: 0.6196308530414707\n Mean: 0.57\n Lower bound: 0.47127134651887564\n Upper bound: 0.6687286534811243\n Mean: 0.45\n Lower bound: 0.3507891524245659\n Upper bound: 0.5492108475754341\n Mean: 0.48\n Lower bound: 0.38036914695852936\n Upper bound: 0.5796308530414707\n Mean: 0.63\n Lower bound: 0.5337185338396045\n Upper bound: 0.7262814661603955\n Mean: 0.47\n Lower bound: 0.3704689875017368\n Upper bound: 0.5695310124982632\n Mean: 0.46\n Lower bound: 0.3606089327527314\n Upper bound: 0.5593910672472686\n Mean: 0.47\n Lower bound: 0.3704689875017368\n Upper bound: 0.5695310124982632\n Mean: 0.54\n Lower bound: 0.4406089327527315\n Upper bound: 0.6393910672472686\n Mean: 0.54\n Lower bound: 0.4406089327527315\n Upper bound: 0.6393910672472686\n Mean: 0.47\n Lower bound: 0.3704689875017368\n Upper bound: 0.5695310124982632\n Mean: 0.6\n Lower bound: 0.5023039108049319\n Upper bound: 0.6976960891950681\n Mean: 0.39\n Lower bound: 0.2927322702903188\n Upper bound: 0.48726772970968124\n Mean: 0.53\n Lower bound: 0.43046898750173684\n Upper bound: 0.6295310124982633\n Mean: 0.51\n Lower bound: 0.4103092906280824\n Upper bound: 0.6096907093719176\n Mean: 0.55\n Lower bound: 0.4507891524245659\n Upper bound: 0.6492108475754341\n Mean: 0.43\n Lower bound: 0.3312713465188757\n Upper bound: 0.5287286534811243\n Mean: 0.5\n Lower bound: 0.400289346502771\n Upper bound: 0.599710653497229\n Mean: 0.47\n Lower bound: 0.3704689875017368\n Upper bound: 0.5695310124982632\n Mean: 0.53\n Lower bound: 0.43046898750173684\n Upper bound: 0.6295310124982633\n Mean: 0.51\n Lower bound: 0.4103092906280824\n Upper bound: 0.6096907093719176\n Mean: 0.4\n Lower bound: 0.30230391080493196\n Upper bound: 0.4976960891950681\n Mean: 0.46\n Lower bound: 0.3606089327527314\n Upper bound: 0.5593910672472686\n Mean: 0.49\n Lower bound: 0.3903092906280824\n Upper bound: 0.5896907093719176\n Mean: 0.52\n Lower bound: 0.4203691469585294\n Upper bound: 0.6196308530414707\n Mean: 0.52\n Lower bound: 0.4203691469585294\n Upper bound: 0.6196308530414707\n Mean: 0.59\n Lower bound: 0.49191795947468714\n Upper bound: 0.6880820405253127\n Mean: 0.53\n Lower bound: 0.43046898750173684\n Upper bound: 0.6295310124982633\n Mean: 0.63\n Lower bound: 0.5337185338396045\n Upper bound: 0.7262814661603955\n Mean: 0.54\n Lower bound: 0.4406089327527315\n Upper bound: 0.6393910672472686\n Mean: 0.41\n Lower bound: 0.31191795947468715\n Upper bound: 0.5080820405253128\n Mean: 0.51\n Lower bound: 0.4103092906280824\n Upper bound: 0.6096907093719176\n Mean: 0.57\n Lower bound: 0.47127134651887564\n Upper bound: 0.6687286534811243\n Mean: 0.39\n Lower bound: 0.2927322702903188\n Upper bound: 0.48726772970968124\n Mean: 0.57\n Lower bound: 0.47127134651887564\n Upper bound: 0.6687286534811243\n Mean: 0.54\n Lower bound: 0.4406089327527315\n Upper bound: 0.6393910672472686\n Mean: 0.52\n Lower bound: 0.4203691469585294\n Upper bound: 0.6196308530414707\n Mean: 0.5\n Lower bound: 0.400289346502771\n Upper bound: 0.599710653497229\n Mean: 0.43\n Lower bound: 0.3312713465188757\n Upper bound: 0.5287286534811243\n Mean: 0.45\n Lower bound: 0.3507891524245659\n Upper bound: 0.5492108475754341\n Mean: 0.46\n Lower bound: 0.3606089327527314\n Upper bound: 0.5593910672472686\n Mean: 0.57\n Lower bound: 0.47127134651887564\n Upper bound: 0.6687286534811243\n Mean: 0.49\n Lower bound: 0.39030929062808245\n Upper bound: 0.5896907093719175\n Mean: 0.54\n Lower bound: 0.4406089327527315\n Upper bound: 0.6393910672472686\n Mean: 0.57\n Lower bound: 0.47127134651887564\n Upper bound: 0.6687286534811243\n Mean: 0.55\n Lower bound: 0.4507891524245659\n Upper bound: 0.6492108475754341\n Mean: 0.52\n Lower bound: 0.4203691469585294\n Upper bound: 0.6196308530414707\n Mean: 0.61\n Lower bound: 0.5127322702903188\n Upper bound: 0.7072677297096812\n Mean: 0.46\n Lower bound: 0.3606089327527314\n Upper bound: 0.5593910672472686\n Mean: 0.48\n Lower bound: 0.38036914695852936\n Upper bound: 0.5796308530414707\n Mean: 0.47\n Lower bound: 0.3704689875017368\n Upper bound: 0.5695310124982632\n Mean: 0.62\n Lower bound: 0.5232036009440236\n Upper bound: 0.7167963990559764\n Mean: 0.44\n Lower bound: 0.3410098664856729\n Upper bound: 0.5389901335143271\n Mean: 0.51\n Lower bound: 0.4103092906280824\n Upper bound: 0.6096907093719176\n Mean: 0.57\n Lower bound: 0.47127134651887564\n Upper bound: 0.6687286534811243\n Mean: 0.54\n Lower bound: 0.4406089327527315\n Upper bound: 0.6393910672472686\n Mean: 0.48\n Lower bound: 0.38036914695852936\n Upper bound: 0.5796308530414707\n Mean: 0.47\n Lower bound: 0.3704689875017368\n Upper bound: 0.5695310124982632\n Mean: 0.47\n Lower bound: 0.3704689875017368\n Upper bound: 0.5695310124982632\n Mean: 0.56\n Lower bound: 0.46100986648567294\n Upper bound: 0.6589901335143271\n Mean: 0.58\n Lower bound: 0.4815739174219005\n Upper bound: 0.6784260825780994\n Mean: 0.57\n Lower bound: 0.47127134651887564\n Upper bound: 0.6687286534811243\n Mean: 0.53\n Lower bound: 0.43046898750173684\n Upper bound: 0.6295310124982633\n Mean: 0.45\n Lower bound: 0.3507891524245659\n Upper bound: 0.5492108475754341\n Mean: 0.5\n Lower bound: 0.400289346502771\n Upper bound: 0.599710653497229\n Mean: 0.46\n Lower bound: 0.3606089327527314\n Upper bound: 0.5593910672472686\n Mean: 0.43\n Lower bound: 0.3312713465188757\n Upper bound: 0.5287286534811243\n Mean: 0.51\n Lower bound: 0.4103092906280824\n Upper bound: 0.6096907093719176\n Mean: 0.48\n Lower bound: 0.38036914695852936\n Upper bound: 0.5796308530414707\n Mean: 0.55\n Lower bound: 0.4507891524245659\n Upper bound: 0.6492108475754341\n Mean: 0.5\n Lower bound: 0.400289346502771\n Upper bound: 0.599710653497229\n Mean: 0.45\n Lower bound: 0.3507891524245659\n Upper bound: 0.5492108475754341\n Mean: 0.46\n Lower bound: 0.3606089327527314\n Upper bound: 0.5593910672472686\n Mean: 0.38\n Lower bound: 0.2832036009440236\n Upper bound: 0.4767963990559764\n Mean: 0.5\n Lower bound: 0.400289346502771\n Upper bound: 0.599710653497229\n Mean: 0.53\n Lower bound: 0.43046898750173684\n Upper bound: 0.6295310124982633\n Mean: 0.57\n Lower bound: 0.47127134651887564\n Upper bound: 0.6687286534811243\n Mean: 0.48\n Lower bound: 0.38036914695852936\n Upper bound: 0.5796308530414707\n Mean: 0.52\n Lower bound: 0.4203691469585294\n Upper bound: 0.6196308530414707\n Mean: 0.55\n Lower bound: 0.4507891524245659\n Upper bound: 0.6492108475754341\n Mean: 0.48\n Lower bound: 0.38036914695852936\n Upper bound: 0.5796308530414707\n Mean: 0.45\n Lower bound: 0.3507891524245659\n Upper bound: 0.5492108475754341\n Mean: 0.51\n Lower bound: 0.4103092906280824\n Upper bound: 0.6096907093719176\n Mean: 0.53\n Lower bound: 0.43046898750173684\n Upper bound: 0.6295310124982633\n Mean: 0.54\n Lower bound: 0.4406089327527315\n Upper bound: 0.6393910672472686\n Mean: 0.47\n Lower bound: 0.3704689875017368\n Upper bound: 0.5695310124982632\n Mean: 0.51\n Lower bound: 0.4103092906280824\n Upper bound: 0.6096907093719176\n Mean: 0.44\n Lower bound: 0.3410098664856729\n Upper bound: 0.5389901335143271\n Mean: 0.38\n Lower bound: 0.2832036009440236\n Upper bound: 0.4767963990559764\n Mean: 0.43\n Lower bound: 0.3312713465188757\n Upper bound: 0.5287286534811243\n Mean: 0.51\n Lower bound: 0.41030929062808247\n Upper bound: 0.6096907093719175\n Mean: 0.51\n Lower bound: 0.4103092906280824\n Upper bound: 0.6096907093719176\n Mean: 0.45\n Lower bound: 0.3507891524245659\n Upper bound: 0.5492108475754341\n Mean: 0.5\n Lower bound: 0.400289346502771\n Upper bound: 0.599710653497229\n Mean: 0.55\n Lower bound: 0.4507891524245659\n Upper bound: 0.6492108475754341\n Mean: 0.56\n Lower bound: 0.46100986648567294\n Upper bound: 0.6589901335143272\n Mean: 0.46\n Lower bound: 0.3606089327527314\n Upper bound: 0.5593910672472686\n Mean: 0.55\n Lower bound: 0.4507891524245659\n Upper bound: 0.6492108475754341\n Mean: 0.43\n Lower bound: 0.3312713465188757\n Upper bound: 0.5287286534811243\n Mean: 0.43\n Lower bound: 0.3312713465188757\n Upper bound: 0.5287286534811243\n\n\n## Assignment - Build a confidence interval\n\nA confidence interval refers to a neighborhood around some point estimate, the size of which is determined by the desired p-value. For instance, we might say that 52% of Americans prefer tacos to burritos, with a 95% confidence interval of +/- 5%.\n\n52% (0.52) is the point estimate, and +/- 5% (the interval $[0.47, 0.57]$) is the confidence interval. \"95% confidence\" means a p-value $\\leq 1 - 0.95 = 0.05$.\n\nIn this case, the confidence interval includes $0.5$ - which is the natural null hypothesis (that half of Americans prefer tacos and half burritos, thus there is no clear favorite). So in this case, we could use the confidence interval to report that we've failed to reject the null hypothesis.\n\nBut providing the full analysis with a confidence interval, including a graphical representation of it, can be a helpful and powerful way to tell your story. Done well, it is also more intuitive to a layperson than simply saying \"fail to reject the null hypothesis\" - it shows that in fact the data does *not* give a single clear result (the point estimate) but a whole range of possibilities.\n\nHow is a confidence interval built, and how should it be interpreted? It does *not* mean that 95% of the data lies in that interval - instead, the frequentist interpretation is \"if we were to repeat this experiment 100 times, we would expect the average result to lie in this interval ~95 times.\"\n\nFor a 95% confidence interval and a normal(-ish) distribution, you can simply remember that +/-2 standard deviations contains 95% of the probability mass, and so the 95% confidence interval based on a given sample is centered at the mean (point estimate) and has a range of +/- 2 (or technically 1.96) standard deviations.\n\nDifferent distributions/assumptions (90% confidence, 99% confidence) will require different math, but the overall process and interpretation (with a frequentist approach) will be the same.\n\nYour assignment - using the data from the prior module ([congressional voting records](https://archive.ics.uci.edu/ml/datasets/Congressional+Voting+Records)):\n\n1. Generate and numerically represent a confidence interval\n2. Graphically (with a plot) represent the confidence interval\n3. Interpret the confidence interval - what does it tell you about the data and its distribution?\n\nStretch goals:\n\n1. Write a summary of your findings, mixing prose and math/code/results. *Note* - yes, this is by definition a political topic. It is challenging but important to keep your writing voice *neutral* and stick to the facts of the data. Data science often involves considering controversial issues, so it's important to be sensitive about them (especially if you want to publish).\n2. Apply the techniques you learned today to your project data or other data of your choice, and write/discuss your findings here.\n3. Refactor your code so it is elegant, readable, and can be easily run for all issues.\n\n\n```\nimport numpy as np\nimport pandas as pd\nfrom scipy.stats import chisquare\n```\n\n\n```\ncolumns = ['Party', 'Handicapped_Infants', 'WaterProjectCostShare', 'BudgReso', 'PhysFeeFreeze', 'ElSalvAid', 'ReliGroupsinSchools',\n 'AntiSatTestBan', 'NicaraguanContrasAid',\n 'MxMissle', 'Immigration', 'SynfuelsCorpCutback', 'EdSpending', 'SuperfundRighttoSue', 'Crime', 'DutyFreeExports', 'ExportAdminActofSA']\ndata = pd.read_csv('https://archive.ics.uci.edu/ml/machine-learning-databases/voting-records/house-votes-84.data', names=columns, na_values=['?'])\n```\n\n\n```\ndata.replace({'n':0, 'y':1, np.NaN:.5}, inplace=True)\n```\n\n\n```\ndata.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        PartyHandicapped_InfantsWaterProjectCostShareBudgResoPhysFeeFreezeElSalvAidReliGroupsinSchoolsAntiSatTestBanNicaraguanContrasAidMxMissleImmigrationSynfuelsCorpCutbackEdSpendingSuperfundRighttoSueCrimeDutyFreeExportsExportAdminActofSA
                                        0republican0.01.00.01.01.01.00.00.00.01.00.51.01.01.00.01.0
                                        1republican0.01.00.01.01.01.00.00.00.00.00.01.01.01.00.00.5
                                        2democrat0.51.01.00.51.01.00.00.00.00.01.00.01.01.00.00.0
                                        3democrat0.01.01.00.00.51.00.00.00.00.01.00.01.00.00.01.0
                                        4democrat1.01.01.00.01.01.00.00.00.00.01.00.51.01.01.01.0
                                        \n
                                        \n\n\n\n\n```\ndf = data.groupby('Party')\ndf.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        PartyHandicapped_InfantsWaterProjectCostShareBudgResoPhysFeeFreezeElSalvAidReliGroupsinSchoolsAntiSatTestBanNicaraguanContrasAidMxMissleImmigrationSynfuelsCorpCutbackEdSpendingSuperfundRighttoSueCrimeDutyFreeExportsExportAdminActofSA
                                        0republican0.01.00.01.01.01.00.00.00.01.00.51.01.01.00.01.0
                                        1republican0.01.00.01.01.01.00.00.00.00.00.01.01.01.00.00.5
                                        2democrat0.51.01.00.51.01.00.00.00.00.01.00.01.01.00.00.0
                                        3democrat0.01.01.00.00.51.00.00.00.00.01.00.01.00.00.01.0
                                        4democrat1.01.01.00.01.01.00.00.00.00.01.00.51.01.01.01.0
                                        5democrat0.01.01.00.01.01.00.00.00.00.00.00.01.01.01.01.0
                                        6democrat0.01.00.01.01.01.00.00.00.00.00.00.00.51.01.01.0
                                        7republican0.01.00.01.01.01.00.00.00.00.00.00.01.01.00.51.0
                                        8republican0.01.00.01.01.01.00.00.00.00.00.01.01.01.00.01.0
                                        10republican0.01.00.01.01.00.00.00.00.00.00.50.51.01.00.00.0
                                        \n
                                        \n\n\n\n\n```\ndf1 = data[['Party', 'BudgReso']]\ndf1.head()\n```\n\n\n```\ndf1['BudgReso'].value_counts()\n```\n\n\n\n\n 1.0 253\n 0.0 171\n 0.5 11\n Name: BudgReso, dtype: int64\n\n\n\n\n```\ndf1['Party'].value_counts()\n```\n\n\n\n\n democrat 267\n republican 168\n Name: Party, dtype: int64\n\n\n\n\n```\ngd = data.groupby(['Party', 'BudgReso'])['BudgReso'].count().unstack().fillna(0)\ngd\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        BudgReso0.00.51.0
                                        Party
                                        democrat297231
                                        republican142422
                                        \n
                                        \n\n\n\n\n```\nimport matplotlib.pyplot as plt\nimport seaborn as sns\n\nfig = plt.figure(figsize=(10, 5))\nsns.set(font_scale=1.8)\ncategories = [\"No\",\"No Vote\",\"Yes\"]\np1 = plt.bar(categories, Demcount, 0.55, color='#d62728')\np2 = plt.bar(categories, Repcount, 0.55, bottom=Demcount)\nplt.legend((p2[0], p1[0]), ('Dem', 'Rep'))\nplt.xlabel('Budget Resolution')\nplt.ylabel('Votes')\nplt.show()\n```\n\n\n```\nvote_data = {'No':[29, 142, 171], 'Yes':[231, 22, 253], 'No Vote':[7, 4, 11], 'Total':[267, 168, 435]} \n \nvotes = pd.DataFrame(vote_data, index =['Democrat', 'Republican', 'Total']) \n\nvotes\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        NoYesNo VoteTotal
                                        Democrat292317267
                                        Republican142224168
                                        Total17125311435
                                        \n
                                        \n\n\n\n\n```\nok = votes.values\nok\n```\n\n\n\n\n array([[ 29, 231, 7, 267],\n [142, 22, 4, 168],\n [171, 253, 11, 435]])\n\n\n\n\n```\nok2 = votes.drop(['Total'], axis=1)\nok2\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        NoYesNo Vote
                                        Democrat292317
                                        Republican142224
                                        Total17125311
                                        \n
                                        \n\n\n\n\n```\nok3 = ok2.drop(['Total'], axis=0)\nok4 = ok3.values\nok4\n```\n\n\n\n\n array([[ 29, 231, 7],\n [142, 22, 4]])\n\n\n\n\n```\nvotes.shape\n```\n\n\n\n\n (3, 4)\n\n\n\n\n```\nrow_sums = votes.iloc[0:2, 3].values\ncol_sums = votes.iloc[2, 0:3].values\n\nprint(row_sums)\nprint(col_sums)\n```\n\n [267 168]\n [171 253 11]\n\n\n\n```\ntotal = votes.loc['Total', 'Total']\nprint(total)\n\nexpected = []\nfor i in range(len(row_sums)):\n expected_row = []\n for column in col_sums:\n expected_val = column*row_sums[i]/total\n expected_row.append(expected_val)\n expected.append(expected_row)\n \nprint(np.array(expected))\n```\n\n 435\n [[104.95862069 155.28965517 6.75172414]\n [ 66.04137931 97.71034483 4.24827586]]\n\n\n\n```\nchi_squared = ((ok4 - expected)**2/(expected)).sum()\nprint(f\"Chi-Squared: {chi_squared}\")\n```\n\n Chi-Squared: 237.93583713543606\n\n\n\n```\ndof = (len(row_sums)-1)*(len(col_sums)-1)\nprint(f\"Degrees of Freedom: {dof}\")\n```\n\n Degrees of Freedom: 2\n\n\n\n```\np_value = stats.chi2.sf(2287.19, dof)\nprint(f\"P-value: {p_value}\")\n```\n\n P-value: 0.0\n\n\n\n```\nprint(f\"Chi-Squared: {chi_squared}\")\nprint(f\"P-value: {p_value}\")\nprint(f\"Degrees of Freedom: {dof}\") \nprint(\"Expected: \\n\", np.array(expected))\n```\n\n Chi-Squared: 237.93583713543606\n P-value: 0.0\n Degrees of Freedom: 2\n Expected: \n [[104.95862069 155.28965517 6.75172414]\n [ 66.04137931 97.71034483 4.24827586]]\n\n\n## Resources\n\n- [Interactive visualize the Chi-Squared test](https://homepage.divms.uiowa.edu/~mbognar/applets/chisq.html)\n- [Calculation of Chi-Squared test statistic](https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test)\n- [Visualization of a confidence interval generated by R code](https://commons.wikimedia.org/wiki/File:Confidence-interval.svg)\n- [Expected value of a squared standard normal](https://math.stackexchange.com/questions/264061/expected-value-calculation-for-squared-normal-distribution) (it's 1 - which is why the expected value of a Chi-Squared with $n$ degrees of freedom is $n$, as it's the sum of $n$ squared standard normals)\n", "meta": {"hexsha": "bbf45039aef86acd2b096e4a49d4b93ead306385", "size": 185529, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Valerie_LS_DS4_142_Sampling_Confidence_Intervals_and_Hypothesis_Testing.ipynb", "max_stars_repo_name": "ValerieLangat/DS-Unit-1-Sprint-4-Statistical-Tests-and-Experiments", "max_stars_repo_head_hexsha": "3392c2e3fcadef510f9b7cb7832e186af64fe881", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Valerie_LS_DS4_142_Sampling_Confidence_Intervals_and_Hypothesis_Testing.ipynb", "max_issues_repo_name": "ValerieLangat/DS-Unit-1-Sprint-4-Statistical-Tests-and-Experiments", "max_issues_repo_head_hexsha": "3392c2e3fcadef510f9b7cb7832e186af64fe881", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Valerie_LS_DS4_142_Sampling_Confidence_Intervals_and_Hypothesis_Testing.ipynb", "max_forks_repo_name": "ValerieLangat/DS-Unit-1-Sprint-4-Statistical-Tests-and-Experiments", "max_forks_repo_head_hexsha": "3392c2e3fcadef510f9b7cb7832e186af64fe881", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 48.5550902905, "max_line_length": 23560, "alphanum_fraction": 0.5065623164, "converted": true, "num_tokens": 20158, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.6959583250334526, "lm_q1q2_score": 0.41770608521530833}} {"text": "##### Copyright 2019 Google LLC.\n\nLicensed under the Apache License, Version 2.0 (the \"License\");\n\n\n```\n#@title Licensed under the Apache License, Version 2.0 (the \"License\"); { display-mode: \"form\" }\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n# https://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\n```\n\n# Interest rate tools in TFF\n\n\n \n \n
                                        \n Run in Google Colab\n \n View source on GitHub\n
                                        \n\n\n```\n#@title Install TF Quant Finance\n!pip install tf-quant-finance\n```\n\n### This notebook demonstrates the use of the TFF toolbox for performing common tasks related to interest rates. This includes computing the present values of cashflows for a collection of bonds, and for building and interpolating from rate curves, with an emphasis on:\n\n * **Batching**: Tensorflow is vectorized out of the box. Tensorflow Finance (TFF) written to leverage this wherever possible. We illustrate the advantage of batching for computing forward rates.\n\n\n```\n#@title Imports { display-mode: \"form\" }\nimport datetime\nfrom dateutil.relativedelta import relativedelta\nimport holidays\nimport math\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport pandas as pd\nfrom pandas.tseries.holiday import USFederalHolidayCalendar\nfrom pandas.tseries.offsets import CustomBusinessDay\nimport seaborn as sns\nimport tensorflow as tf\nimport time\n\n # TFF for Tensorflow Finance\nimport tf_quant_finance as tff \nfrom tf_quant_finance import rates\n\nfrom IPython.core.pylabtools import figsize\nfigsize(21, 14) # better graph size for Colab \n\nimport warnings\nwarnings.filterwarnings(\"ignore\",\n category=FutureWarning) # suppress printing warnings\n\n```\n\n# Business Day Convention\nThe pricing in Example 1 uses the *modified following business day convention* for the US for treasure payment dates. That is, if the coupon date falls on a weekend or holiday, it is paid on the following business day, unless said following business day falls into the next calendar month, in which case we go backwards to the nearest previous business day. It also provides functionality to generate regular coupon payments, before applying business day convention. \n\n\n```\n#@title Implement Business Day Convention\ndef get_coupon_dates(current_date, maturity_date, months = 6):\n # Compute sequence of dates `months` apart starting at maturity_date,\n # working backwards to last one after current_date. \n cashflow_dates = []\n date = maturity_date\n while date >= current_date:\n cashflow_dates.append(date)\n date = date + relativedelta(months=-months)\n # Check if dates are on a US holiday or weekend\n return pd.to_datetime(cashflow_dates).sort_values()\n \ndef get_modified_following_date(dates):\n # Get the modified folliwng business day for the US. \n BDayUS = CustomBusinessDay(calendar=USFederalHolidayCalendar(), n = 1) \n def is_weekday(dates):\n # Identify weekend days\n return ~dates.weekday_name.isin(['Saturday', 'Sunday'])\n\n def in_next_month(dates1, dates2): \n return dates1.month == dates2.month - 1\n\n def next_bus_day(dates):\n # If next business day in a new month shift to previous business day\n fwd = dates + BDayUS\n return fwd.where(~in_next_month(dates, fwd), dates - BDayUS)\n\n def payment_day(dates):\n return dates.where(is_weekday(dates), next_bus_day(dates))\n return payment_day(dates)\n\n```\n\n## Example 1: **Cashflows in TFF**: Computing present values for a portfolio of bonds.\n\n ### Coupon Bond Valuation\nCalculating the value of a coupon bond factors in the present value of annual or semi-annual coupon payments and the face value of the bond.\n\nThe present value of expected cash flows is added to the present value (PV) of the face value of the bond as seen in the following formula:\n\n$$\n\\begin{align}\nPV(Bond) &= PV(Coupons) + PV(FaceValue) \\\\\n&= \\sum_t \\frac{C_t}{(1+i)^t} + \\frac{F}{(1+i)^T}\\\\\n\\end{align}\n$$\nWhere \\\\\n$C_t$ are future coupon payments\n\n$i$ is the yield to maturity (or internal rate of return, IRR) of the bond. \n\n$F$ is the face value of the bond.\n\n$t$ is times at which the corresponding coupon payments occur.\n\n$T$ is the time to maturity of the bond.\n\n\n# Example Data (US Treasury Bonds)\nThe example below shows how to price a selection of US Treasury Bonds\nSource: https://www.wsj.com/market-data/bonds (Close of market on 20/09/2019)\n\nThe data represent six US Treasuries: \n* 2-Year Note (Coupon: 1.5%, Maturity: 30/09/2021)\n* 3-Year Note (Coupon: 1.5%, Maturity: 15/09/2022)\n* 5-Year Note (Coupon: 1.5%, Maturity: 30/09/2024)\n* 7-Year Note (Coupon: 1.625%, Maturity: 30/09/2026)\n* 10-Year Note (Coupon: 1.625%, Maturity: 15/08/2029)\n* 30-Year Bond (Coupon: 2.25%, Maturity: 15/08/2049)\n\nWe use Modified Following day count convention (i.e. move to the next business day, \nunless it falls in a different month, in which case use the previous business day),\nand the US holidays in the Python `holidays` module.\n\n\n```\n#@title Pricing US Treasury Bonds\ndtype = np.float64\nexp_dates = ['\"2021-09-30\"', '\"2022-09-15\"', '\"2024-09-30\"', '\"2026-09-30\"', \n '\"2029-08-15\"', '\"2049-08-15\"']\nus_bond_data = {\n 'present_date': [datetime.datetime.strptime('2019-09-20', '%Y-%m-%d').date()] * 6,\n 'expiry_date': [datetime.datetime.strptime(date, '\"%Y-%m-%d\"').date() for date in exp_dates], \n 'bond_type': ['2yr_note', '3yr_note', '5yr_note', '7yr_note', '10yr_note', \n '30yr_bond'],\n 'face_value': [100, 100, 100, 100, 100, 1000],\n 'coupon_rate': [0.015, 0.015, 0.015, 0.01625, 0.01625, 0.02250],\n 'coupon_frequency': [0.5] * 6\n}\n\nus_bond_data = pd.DataFrame.from_dict(us_bond_data)\n\n# Generate times of cashflows (using modified following business day convention)\n# for US federal holidays. \npayment_dates = list(map(get_coupon_dates, us_bond_data.present_date,\n us_bond_data.expiry_date))\nnumber_of_coupons = list(map(len, payment_dates)) # get number of coupons per bond\npayment_dates = np.concatenate(payment_dates, axis = 0)\npayment_dates_modified = get_modified_following_date(\n pd.to_datetime(payment_dates))\ncurrent_date = pd.Series(pd.to_datetime(us_bond_data.present_date[0])).repeat(len(payment_dates_modified))\npayment_times_days = (payment_dates_modified.values - current_date)\ntimes = payment_times_days.apply(lambda x: float(x.days) / 365) # Days to years\n\n\n# Generate actual cashflows.\ncoupon_payments = (us_bond_data.face_value * us_bond_data.coupon_rate * \n us_bond_data.coupon_frequency)\ncoupon_cashflows = np.repeat(coupon_payments, number_of_coupons)\nredemption_cashflows = np.zeros(np.sum(number_of_coupons))\nredemption_indexes = np.cumsum(number_of_coupons) - 1\nredemption_cashflows[redemption_indexes] = us_bond_data.face_value\ncashflows = np.array(coupon_cashflows + redemption_cashflows, dtype = dtype)\n# Compute groups for bond cashflows. \ngroups = np.repeat(range(0, us_bond_data.shape[0]), number_of_coupons)\n\n# Bond Yield Curve \n# Yields ontained from https://www.wsj.com/market-data/bonds (as on 20/09/2019)\ntenor_curve = [2, 3, 5, 10, 30]\nrate_curve = [0.017419, 0.016885, 0.016614, 0.017849, 0.02321]\ndays_to_maturity = (us_bond_data.expiry_date - us_bond_data.present_date) \nyears_to_maturity = list(days_to_maturity.apply(lambda x: float(x.days) / 365))\n\n\n# Linear Interpolation of curve get yields to maturity.\nrate_curve_interpolated = tff.math.interpolation.linear.interpolate(\n years_to_maturity, tenor_curve, \n rate_curve, dtype = np.float64)\nwith tf.Session() as sess:\n rate_curve_interpolated = sess.run(rate_curve_interpolated)\n\n# Create Tensorflow Graph using pv_from_yields in rates.\npresent_values = rates.cashflows.pv_from_yields(cashflows, times, \n rate_curve_interpolated,\n groups) \n\nwith tf.Session() as sess:\n present_values = sess.run(present_values)\n\nus_bond_data['present_value'] = present_values\n\nprint(\"Priced US Treasury Bonds:\")\nprint('\\n')\nus_bond_data\n```\n\n Priced US Treasury Bonds:\n \n \n\n\n \n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        present_dateexpiry_datebond_typeface_valuecoupon_ratecoupon_frequencypresent_value
                                        02019-09-202021-09-302yr_note1000.015000.5100.212446
                                        12019-09-202022-09-153yr_note1000.015000.599.446898
                                        22019-09-202024-09-305yr_note1000.015000.599.887903
                                        32019-09-202026-09-307yr_note1000.016250.5100.139738
                                        42019-09-202029-08-1510yr_note1000.016250.598.648447
                                        52019-09-202049-08-1530yr_bond10000.022500.5984.092036
                                        \n
                                        \n\n\n\n### Generating large bond portfolio \nTo demonstrate scale we simulate a mix of bonds (of total number_of_bonds) with face values between min_face_value and max_face_value (in increments of 100), paying either semi-annual or annual coupons, with coupon rates of 2%, 4%, 6%, 8%, and 10%. We use the yield curve we used above for US treasury bonds. \n\n\n```\n#@title Create Bond Data\nnumber_of_bonds = 100000 #@param\nmin_face_value = 100 \nmax_face_value = 1000\n# Face values for bonds\nbond_face_values = range(min_face_value,\n max_face_value, 100)\ncoupon_frequencies = [0.5, 1]\ncoupon_rates = [0.02, 0.04, 0.06, 0.08, 0.10]\n\n# Range of bond maturities.\nbond_maturities = [1, 2, 3, 5, 7, 10, 15, 20, 30]\n\n# Create a mix of 100,000 bonds.\nlarge_bond_data = {\n 'face_value': np.random.choice(bond_face_values, number_of_bonds),\n 'coupon_frequency': np.random.choice(coupon_frequencies, number_of_bonds),\n 'coupon_rate': np.random.choice(coupon_rates, number_of_bonds, \n p=[0.1, 0.2, 0.3, 0.3, 0.1]),\n 'maturity': np.random.choice(bond_maturities, number_of_bonds,\n p=[0.1, 0.1, 0.1, 0.2, 0.3, 0.1, 0.05, \n 0.025, 0.025])\n }\n\nlarge_bond_data = pd.DataFrame.from_dict(large_bond_data)\n\n# Rate curve interpolation\ncurve_required_tenors2 = np.arange(0.5, 30.5, 0.5)\n\nrate_curve_interpolated2 = tff.math.interpolation.linear.interpolate(\n curve_required_tenors2, tenor_curve, \n rate_curve, dtype = np.float64)\nwith tf.Session() as sess:\n rate_curve_interpolated2 = sess.run(rate_curve_interpolated2)\n\n\n# Plot distribution of bonds by face value, coupon rate, and yield.\nplt.figure(figsize=(16,12))\ncol_palette = sns.color_palette(\"Blues\")\nplt.subplot(2, 2, 1)\n# Plot Rate Curve\nsns.set()\nsns.lineplot(curve_required_tenors2, rate_curve_interpolated2,\n color=col_palette[2])\nplt.title('Rate Curve', fontsize=14)\nplt.xlabel('Tenor', fontsize=12)\nplt.ylabel('Rate', fontsize=12)\n\n# Coupon rate distribution\nplt.subplot(2, 2, 2)\nsns.set()\nsns.distplot(large_bond_data['coupon_rate'], kde=False,\n color = col_palette[3], bins=5)\nplt.title('Bond Mix by Coupon Rate', fontsize=14)\nplt.xlabel('Coupon Rate', fontsize=12)\nplt.ylabel('Frequency', fontsize=12)\n\n# Nominal value distribution\nplt.subplot(2, 2, 3)\nsns.set()\nsns.distplot(large_bond_data['face_value'], kde=False,\n color = col_palette[4], bins=9)\nplt.title('Bond Mix by Nominal', fontsize=14)\nplt.xlabel('Nominal', fontsize=12)\nplt.ylabel('Frequency', fontsize=12)\n\n# Nominal value distribution\nplt.subplot(2, 2, 4)\nsns.set()\nsns.distplot(large_bond_data['maturity'], kde=False,\n color = col_palette[5], bins=9)\nplt.title('Bond Mix by Maturity', fontsize=14)\nplt.xlabel('Maturity', fontsize=12)\nplt.ylabel('Frequency', fontsize=12)\nplt.show()\n```\n\n\n```\n#@title Compute the present value for portfolio of bonds\ndtype = np.float64\ntf.reset_default_graph()\n\nrate_curve_df = pd.DataFrame.from_dict({\n 'tenor': curve_required_tenors2,\n 'rate': rate_curve_interpolated2\n})\n\n# Create inputs (cashflows, times, groups) for `pv_from_yields`\nlarge_number_of_coupons = large_bond_data.maturity / large_bond_data.coupon_frequency\nlarge_number_of_coupons = large_number_of_coupons.astype(int)\nlarge_coupon_payments = (large_bond_data.face_value * large_bond_data.coupon_rate * \n large_bond_data.coupon_frequency)\nlarge_coupon_cashflows = np.repeat(large_coupon_payments, large_number_of_coupons)\nlarge_redemption_cashflows = np.zeros(np.sum(large_number_of_coupons))\nlarge_redemption_indexes = np.cumsum(large_number_of_coupons) - 1\nlarge_redemption_cashflows[large_redemption_indexes] = large_bond_data.face_value\nlarge_cashflows = np.array(large_coupon_cashflows + \n large_redemption_cashflows, dtype = dtype)\n\n# The times of the cashflows.\nlarge_times = list(map(np.arange, large_bond_data.coupon_frequency,\n large_bond_data.maturity + large_bond_data.coupon_frequency,\n large_bond_data.coupon_frequency))\nlarge_times = np.concatenate(large_times, axis = 0)\nlarge_groups = np.repeat(range(0, large_bond_data.shape[0]), \n large_number_of_coupons)\n\n# Create Tensorflow Graph using pv_from_yields in rates.\npresent_values = rates.cashflows.pv_from_yields(large_cashflows, large_times, \n rate_curve_interpolated2, \n groups = large_groups)\nwith tf.Session() as sess:\n present_values = sess.run(present_values)\n\n # Plot distribution of present values of portfolio of bonds.\nplt.figure(figsize=(12,8))\nsns.set_context(\"talk\")\ncol_palette = sns.color_palette(\"Blues\")\nsns.set()\nax = sns.distplot(present_values, kde=True)\nplot_label = \"Present Value Disribution of {} Priced Bonds.\".format(number_of_bonds)\nplt.title(plot_label, fontsize=16)\nplt.xlabel('Present Value', fontsize=14)\nplt.show()\n```\n\n## Example 2: Compute forward rates given a set of zero rates\n\nDenote the price of a zero coupon bond maturing at time $t$ by $Z(t)$. Then the zero rate to time $t$ is defined as\n$$\n\\begin{equation*}\n r(t) = - ln(Z(t)) / t\n\\end{equation*}\n$$\n\nThis is the (continuously compounded) interest rate that applies between time $0$ and time $t$ as seen at time $0$. The forward rate between times $t1$ and $t2$ is defined as the interest rate that applies to the period $[t1, t2]$ as seen from today. Let $f(t1, t2) = -ln\\frac{Z(t2)}{Z(t1)}$, then it followes that\n$$\\begin{align}\n \\\\\n exp(-f(t1, t2)(t2-t1)) &= Z(t2) / Z(t1) \\\\\n f(t1, t2) &= - (ln Z(t2) - ln Z(t1)) / (t2 - t1) \\\\\n f(t1, t2) &= (t2 * r(t2) - t1 * r(t1)) / (t2 - t1) \\\\\\\\\n\\end{align}$$\nGiven a sequence of increasing times $[t1, t2, ... tn]$ and the zero rates for those times, this function computes the forward rates that apply to the consecutive time intervals i.e. $[0, t1], [t1, t2], ... [t_{n-1}, tn]$ using the last equation above. Note that for the interval $[0, t1]$ the forward rate is the same as the zero rate.\n\n### Generating zero rates data\nWe generate `num_zero_rates_bonds` sets of zero rates data with between 3 and 8 coupon payments at time points $[0.25, 0.5, 1, 1.5,2, 3, 5, 10]$, always starting at $0.25$ and then at subsequent time points, depending on the number of marked tenors. We generate zero rates as follows:\n\n\n1. Randomly draw a rate in $[0,0.15]$ for the first tenor\n2. Generate the rates for the subsequent tenors by incrementing the rate at the first tenor by a random draw from $[0, 0.02]$.\n\n\n```\n#@title Create Bond Data\nnum_zero_rate_bonds = 100000 #@param\nnum_tenors = [2, 3, 4, 5, 6, 7, 8, 10]\nmarked_tenors = [0.25, 0.5, 1, 1.5, 2, 3, 5, 10, 20, 30]\n\n# Create a mix of `num_zero_rate_bonds` bonds.\nset_num_tenors = np.random.choice(num_tenors, num_zero_rate_bonds)\n\ndef get_slice(n):\n return marked_tenors[slice(n)]\n\ntimes = np.concatenate(list(map(get_slice, set_num_tenors)), axis = 0)\n# Set up a grouping argument for implementing batching. See\n# `forward_rates_from_yields` in tff.forwards.\ngroups = np.repeat(range(0, num_zero_rate_bonds), set_num_tenors)\n\n# Construct Rate Curve to generate Zero Rates \ntf.reset_default_graph()\ncurve_required_tenors3 = marked_tenors\nrate_curve_interpolated3 = tff.math.interpolation.linear.interpolate(\n curve_required_tenors3, tenor_curve, \n rate_curve, dtype = np.float64)\n\nwith tf.Session() as sess:\n rate_curve_interpolated3 = sess.run(rate_curve_interpolated3)\n \ndef get_rates(n):\n # Perturb rate curve\n rates = rate_curve_interpolated3[0:n]\n rates = rates + np.random.uniform(-0.0005, 0.0005, n)\n return rates\n \nrates = np.concatenate(list(map(get_rates, set_num_tenors)), axis = 0)\n\nzero_rate_data = {\n 'times': times,\n 'groups': groups,\n 'rates': rates\n }\n\nzero_rate_data_df = pd.DataFrame.from_dict(zero_rate_data)\n```\n\n\n```\n#@title Compute forward rates for sets with different number of tenors of zero rates with batching.\nimport tf_quant_finance.rates.forwards as forwards\ndtype = np.float64\ntf.reset_default_graph()\n\nforward_rates = forwards.forward_rates_from_yields(\n rates, times, groups=groups, dtype=dtype)\nt = time.time()\nwith tf.Session() as sess:\n forward_rates = sess.run(forward_rates)\ntime_batch = time.time() - t\nzero_rate_data_df['forward_rates'] = forward_rates\n\n# Plot forward rates for a random sample sets of zero rates\nsample_groups = np.random.choice(np.unique(groups), 5)\nplt.figure(figsize=(14,6))\ncol_palette = sns.color_palette(\"Blues\", 5)\nmask = list(zero_rate_data_df.groups.isin(sample_groups))\nplot_data = zero_rate_data_df.iloc[mask]\nsns.set()\nsns.set_context(\"talk\")\nsns.lineplot(x='times', y='forward_rates', data=plot_data, \n hue='groups',legend='full', palette=col_palette)\n\nplt.title('Sample of estimated forward rate sets', fontsize=16)\nplt.xlabel('Marked Tenor', fontsize=14)\nplt.ylabel('Forward Rate', fontsize=14)\nlegend = plt.legend()\nlegend.texts[0].set_text(\"Fwd Rate Group\")\nplt.show()\n\n```\n\n### Forward rates (batching vs non-batching)\nBelow we compare the computation of forward rates with and without batching. We see that computing the forward rates for 100 bonds is about 2.5 times slower than computing the forward rates for 100000 bonds with batching.\n\n\n```\n#@title Compare forward rate computation: batching vs non-batching.\nnum_zero_rate_bonds2 = 100\nnum_tenors = [2, 3, 4, 5, 6, 7, 8, 10]\nmarked_tenors = [0.25, 0.5, 1, 1.5, 2, 3, 5, 10, 20, 30]\n\n# Create a mix of 100,000 bonds.\nset_num_tenors = np.random.choice(num_tenors, num_zero_rate_bonds2)\n\ndef get_slice(n):\n # Function to get marked tenors for a bond with 'n' tenors. \n return marked_tenors[slice(n)]\n\ntimes = np.concatenate(list(map(get_slice, set_num_tenors)), axis = 0)\n# Set up a grouping argument for implementing batching. See\n# `forward_rates_from_yields` in tff.forwards.\ngroups = np.repeat(range(0, num_zero_rate_bonds2), set_num_tenors)\n\n# Construct Rate Curve to generate Zero Rates \ntf.reset_default_graph()\ncurve_required_tenors3 = marked_tenors\nrate_curve_interpolated3 = tff.math.interpolation.linear.interpolate(\n curve_required_tenors3, tenor_curve, \n rate_curve, dtype = np.float64)\n\nwith tf.Session() as sess:\n rate_curve_interpolated3 = sess.run(rate_curve_interpolated3)\n \ndef get_rates(n):\n # Perturb rate curve\n rates = rate_curve_interpolated3[0:n]\n rates = rates + np.random.uniform(-0.0005, 0.0005, n)\n return rates\n \nrates = np.concatenate(list(map(get_rates, set_num_tenors)), axis = 0)\n\n# Non-batch.\ntf.reset_default_graph()\ntime_non_batch = 0\nwith tf.Session() as sess:\n for group in np.unique(groups):\n forward_rates_non_batch = forwards.forward_rates_from_yields(\n rates[groups == group], times[groups == group], dtype=dtype)\n t = time.time() \n forward_rates_non_batch = sess.run(forward_rates_non_batch)\n time_non_batch += time.time() - t\nprint('wall time to price {} options without batching: '.format(num_zero_rate_bonds2), time_non_batch)\nprint('wall time to price {} options with batching: '.format(num_zero_rate_bonds), time_batch)\noutput_string = \"\"\"Pricing {} bonds without batching is {} times slower than \n pricing {} bonds with batching.\"\"\"\n\nprint(output_string.format(num_zero_rate_bonds2, \n round(time_non_batch/time_batch, 1), \n num_zero_rate_bonds))\n```\n\n wall time (non-batch): 4.39509391784668\n Pricing 100 bonds without batching is 10.3 times slower than \n pricing 100000 bonds with batching.\n\n\n### Pricing 100 bonds without batching is about 10 times slower than pricing 100000 bonds with batching.\n\n## Example 3: Constructing a bond discount curve\n\nBuilding discount curves is a core problem in mathematical finance. Discount curves are built using the available market data in liquidly traded rates\nproducts. These include bonds, swaps, forward rate agreements (FRAs) or eurodollar futures contracts. \n\nHere we show how to build a bond discount rate curve. A discount curve is a function of time which gives the interest rate that applies to a unit of currency deposited today for a period of time $t$. The traded price of bonds implicitly contains the market view on the discount rates. The purpose of discount curve construction is to extract this information. \n\nThe algorithm we use here here is based on the Monotone Convex Interpolation method described by Hagan and West (2006, 2008).\n\n\n\n```\n# @title Create bond data\n# The following example demonstrates the usage by building the implied curve\n# from four coupon bearing bonds.\n\ndtype=np.float64\n# These need to be sorted by expiry time.\ncashflow_times = [\n np.array([0.25, 0.5, 0.75, 1.0], dtype=dtype),\n np.array([0.5, 1.0, 1.5, 2.0], dtype=dtype),\n np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0], dtype=dtype),\n np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0], \n dtype=dtype)\n ]\ncashflows = [\n # 1 year bond with 5% three monthly coupon.\n np.array([12.5, 12.5, 12.5, 1012.5], dtype=dtype),\n # 2 year bond with 6% semi-annual coupon.\n np.array([30, 30, 30, 1030], dtype=dtype),\n # 3 year bond with 8% semi-annual coupon.\n np.array([40, 40, 40, 40, 40, 1040], dtype=dtype),\n # 4 year bond with 3% semi-annual coupon.\n np.array([15, 15, 15, 15, 15, 15, 15, 1015], dtype=dtype)\n]\n# The present values of the above cashflows.\npvs = np.array([\n 999.68155223943393, 1022.322872470043, 1093.9894418810143,\n 934.20885689015677\n], dtype=dtype)\n```\n\n\n```\n#@title Build and plot the bond curve \ntf.reset_default_graph()\nfrom tf_quant_finance.rates import hagan_west\n\nresults = hagan_west.bond_curve(cashflows, cashflow_times, pvs)\n\nwith tf.Session() as sess:\n results = sess.run(results)\n\n# Plot Rate Curve\nplt.figure(figsize=(14,6))\ncol_palette = sns.color_palette(\"Blues\", 2)\nsns.set()\nsns.set_context(\"talk\")\nsns.lineplot(x=results.times, y=results.discount_rates, palette=col_palette)\nplt.title('Estimated Discount Rates', fontsize=16)\nplt.xlabel('Marked Tenor', fontsize=14)\nplt.ylabel('Discount Rate', fontsize=14)\nplt.show()\n```\n", "meta": {"hexsha": "66df7f109f793b1a99bab9b96503430a4fe1dd92", "size": 268045, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tf_quant_finance/experimental/notebooks/Cashflows_Rate_Curves.ipynb", "max_stars_repo_name": "slowy07/tf-quant-finance", "max_stars_repo_head_hexsha": "0976f720fb58a2d7bfd863640c12a2425cd2f94f", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 3138, "max_stars_repo_stars_event_min_datetime": "2019-07-24T21:43:17.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-30T12:11:09.000Z", "max_issues_repo_path": "tf_quant_finance/experimental/notebooks/Cashflows_Rate_Curves.ipynb", "max_issues_repo_name": "SeptumCapital/tf-quant-finance", "max_issues_repo_head_hexsha": "5aba5ddab3a4dd1efa87d5a12fec403315d2ac98", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 63, "max_issues_repo_issues_event_min_datetime": "2019-09-07T19:16:03.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-29T19:29:40.000Z", "max_forks_repo_path": "tf_quant_finance/experimental/notebooks/Cashflows_Rate_Curves.ipynb", "max_forks_repo_name": "SeptumCapital/tf-quant-finance", "max_forks_repo_head_hexsha": "5aba5ddab3a4dd1efa87d5a12fec403315d2ac98", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 423, "max_forks_repo_forks_event_min_datetime": "2019-07-26T21:28:05.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-26T13:07:44.000Z", "avg_line_length": 239.9686660698, "max_line_length": 71752, "alphanum_fraction": 0.8796694585, "converted": true, "num_tokens": 7016, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525096, "lm_q2_score": 0.7279754489059775, "lm_q1q2_score": 0.417623796192907}} {"text": "```python\n%matplotlib inline\n%load_ext autoreload\n%autoreload 2\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\nimport numpy as np\nimport scipy as sp\nimport scipy.stats\nimport time\nimport yaml\ntry:\n from yaml import CLoader as Loader\nexcept ImportError:\n from yaml import Loader\n \n\nfrom pydrake.all import (RigidBodyTree, RigidBody)\n```\n\n The autoreload extension is already loaded. To reload it, use:\n %reload_ext autoreload\n\n\n\n```python\nDATA_FILE = \"data/20180626_uniform_feasible_1000.yaml\"\n\nwith open(DATA_FILE, \"r\") as f:\n environments = yaml.load(f, Loader=Loader)\n N_ENVIRONMENTS = len(environments.keys())\n print(\"Loaded %d environments from file %s\" % (N_ENVIRONMENTS, DATA_FILE))\n```\n\n Loaded 1000 environments from file data/20180626_uniform_feasible_1000.yaml\n\n\n\n```python\n# Make a listing of the observed classes\nclass_name_to_index_map = {}\nclass_index_to_name_map = []\ncurrent_ind = 0\nfor env_name in environments.keys():\n env = environments[env_name]\n for k in range(env[\"n_objects\"]):\n class_name = env[\"obj_%04d\" % k][\"class\"]\n if class_name not in class_name_to_index_map:\n class_name_to_index_map[class_name] = current_ind\n class_index_to_name_map.append(class_name)\n current_ind += 1\nprint class_name_to_index_map, class_index_to_name_map\nN_CLASSES = current_ind\n```\n\n {'small_box': 0} ['small_box']\n\n\n\n```python\n# Draw a few example scenes from the set\nimport generate_planar_scene_arrangements as psa_utils\n\ndef draw_environment(environment, ax):\n rbt, q = psa_utils.build_rbt_from_summary(environment)\n psa_utils.draw_board_state(ax, rbt, q)\n \n patch = patches.Rectangle([0., 0.], 1., 1., fill=True, color=[0., 1., 0.], \n linestyle='solid', linewidth=2, alpha=0.3)\n ax.add_patch(patch)\n\nplt.figure().set_size_inches(12, 12)\nprint \"Selection of environments from original distribution\"\nN = 5\nfor i in range(N):\n for j in range(N):\n plt.subplot(N, N, i*N+j+1)\n draw_environment(environments[\"env_%04d\" % (i*N+j)], plt.gca())\n plt.grid(True)\nplt.tight_layout()\n```\n\n## Fitting distributions to generated scene data\n\nReferencing discussion from the 20180621 notebook -- one way to tackle this would be to treat each object as occuring independently of other objects. That is, we're ultimately interested in sampling a number of objects $n$, a set of object classes $c_i$, and a set of object poses $p_i$ $\\{[c_0, p_0], ..., [c_n, p_n]\\}$. They have joint distribution $p(n, c, p)$, which we factorize $p(n)\\Pi_i\\left[p(p_i | c_i)p(c_i)\\right]$ -- that is, draw the number of objects and class of each object independently, and then draw each object pose independently based on a class-specific distribution.\n\nSo the distributions we need to fit are:\n\n- The distribution of object number $p(n)$, which we'll just draw up a histogram\n- The distribution of individual object class $p(c_i)$, which we'll also just draw up a histogram\n- The distribution of object pose based on class $p(p_i | c_i)$, which we'll fit with KDE\n\n\n```python\n# Get statistics for # object occurance rate and classes\nobject_number_occurances = np.zeros(N_ENVIRONMENTS)\nobject_class_occurances = np.zeros(N_CLASSES)\nfor i in range(N_ENVIRONMENTS):\n env = environments[\"env_%04d\" % i]\n object_number_occurances[i] = env[\"n_objects\"]\n for k in range(env[\"n_objects\"]):\n object_class_occurances[\n class_name_to_index_map[env[\"obj_%04d\" % k][\"class\"]]] += 1\nn_hist, n_hist_bins = np.histogram(object_number_occurances,\n bins=range(int(np.ceil(np.max(object_number_occurances)))+2))\nn_pdf = n_hist.astype(np.float64)/np.sum(n_hist)\nplt.subplot(2, 1, 1)\nplt.bar(n_hist_bins[:-1], n_pdf, align=\"edge\")\nplt.xlabel(\"# objects\")\nplt.ylabel(\"occurance rate\")\n\nplt.subplot(2, 1, 2)\nclass_pdf = object_class_occurances / np.sum(object_class_occurances)\nplt.bar(range(N_CLASSES), class_pdf, align=\"edge\")\nplt.xlabel(\"object class\")\nplt.ylabel(\"occurance rate\")\nplt.tight_layout()\n```\n\n\n```python\n# Build statistics for object poses per each object class\nobject_poses_per_class = []\n# Useful to have for preallocating occurance vectors\nmax_num_objects_in_any_environment = int(np.max(object_number_occurances))\nfor class_k in range(N_CLASSES):\n # Overallocate, we'll resize when we're done\n poses = np.zeros((3, max_num_objects_in_any_environment*N_ENVIRONMENTS))\n total_num_objects = 0\n for i in range(N_ENVIRONMENTS):\n env = environments[\"env_%04d\" % i]\n for k in range(env[\"n_objects\"]):\n obj = env[\"obj_%04d\" % k]\n if class_name_to_index_map[obj[\"class\"]] == class_k:\n poses[:, total_num_objects] = obj[\"pose\"][:]\n total_num_objects += 1\n object_poses_per_class.append(poses[:, :total_num_objects])\n\nclass_kde_fits = []\ntslices = np.linspace(0., 2*np.pi, 8, endpoint=False)\nplt.figure().set_size_inches(16, 8)\nplt.title(\"Distribution over space, per object class\")\nfor class_k in range(N_CLASSES):\n print(\"Computing KDE for class %d\" % class_k)\n poses = object_poses_per_class[class_k]\n kde_fit = sp.stats.gaussian_kde(poses)\n class_kde_fits.append(kde_fit)\n \n for slice_k, tslice in enumerate(tslices):\n xmin = -0.25\n xmax = 1.25\n ymin = -0.25\n ymax = 1.25\n N = 100j\n X, Y = np.mgrid[xmin:xmax:N, ymin:ymax:N]\n positions = np.vstack([X.ravel(), Y.ravel(), np.zeros(X.ravel().shape) + tslice])\n Z = np.reshape(kde_fit(positions).T, X.shape)\n\n print \"Class %d, Theta=%f, Max Likelihood Anywhere: %f\" %(class_k, tslice, np.max(Z))\n plt.subplot(N_CLASSES*2, 4, class_k+1 + slice_k)\n plt.title(\"Class %d, Theta=%f\" % (class_k, tslice))\n plt.gca().imshow(np.rot90(Z[:, :]), vmin=0., vmax=1.,\n cmap=plt.cm.gist_earth_r,\n extent=[xmin, xmax, ymin, ymax])\n plt.scatter(poses[0, :], poses[1, :], s=0.05, c=[1., 0., 1.])\n plt.xlabel(\"x\")\n plt.ylabel(\"y\")\n plt.grid(True)\n\nplt.tight_layout()\n```\n\nAs expected, this fits the uniform distribution of classes over space pretty well, though it does have trouble dealing with the angle wraparound (observe that the likelihood drops off at theta=0 to almost half of what it is elsewhere). Not too surprising, as that's on the \"border\" of the observed samples. Would have to use a specialized KDE tool to handle that down the road.\n\n## Sampling new scenes from this data\n\nWe can generate new scenes by sampling up the dependency tree:\n\n1) First sample # of objects\n\n2) Then sample a class for every object, independently\n\n3) Given each object's class, sample its location\n\nWe can also evaluate the likelihood of each generated sample\nto get an idea how \"typical\" they are, by finding the likelihood\nof each object given its generated position and class and combining\nthem. However, we have to normalize by the maximum possible likelihood\nfor an object of that class for every object for the comparison between\ntwo sets of objects to make sense.\n\n\n```python\nn_cdf = np.cumsum(n_pdf)\nclass_cdf = np.cumsum(class_pdf)\n\n# Calculate maximum likelihood value for each class\nclasswise_max_likelihoods = np.zeros(N_CLASSES)\nfor i in range(N_CLASSES):\n # Evaluate the PDF at a bunch of sample points\n xmin = 0.\n xmax = 1.\n ymin = 0.\n ymax = 1.\n tmin = 0.\n tmax = 2.*np.pi\n N = 20j\n X, Y, T = np.mgrid[xmin:xmax:N, ymin:ymax:N, tmin:tmax:N]\n positions = np.vstack([X.ravel(), Y.ravel(), T.ravel()])\n classwise_max_likelihoods[i] = (np.max(class_kde_fits[i](positions)))\n\nprint \"Classwise max likelihoods: \", classwise_max_likelihoods\n\nplt.figure().set_size_inches(12, 12)\nplt.title(\"Generated environments matching original distribution\")\nnp.random.seed(42)\nN = 5\nfor i in range(N):\n for j in range(N):\n total_log_likelihood = 0.\n n_objects = np.argmax(n_cdf >= np.random.random())\n total_log_likelihood += np.log(n_pdf[n_objects])\n environment = {\"n_objects\": n_objects}\n lln = 0\n for object_k in range(n_objects):\n obj_name = \"obj_%04d\" % object_k\n obj_class = np.argmax(class_cdf >= np.random.random())\n total_log_likelihood += np.log(class_pdf[obj_class] / np.max(class_pdf))\n obj_pose = class_kde_fits[obj_class].resample([1])\n total_log_likelihood += (class_kde_fits[obj_class].logpdf(obj_pose) -\n np.log(classwise_max_likelihoods[obj_class]))\n environment[obj_name] = {\"pose\": obj_pose,\n \"class\": class_index_to_name_map[obj_class]}\n \n plt.subplot(N, N, i*N+j+1)\n draw_environment(environment, plt.gca())\n plt.grid(True)\n plt.title(\"LL: %f\" % total_log_likelihood)\n plt.xlim(-0.25, 1.25)\n plt.ylim(-0.25, 1.25)\n\nprint \"TODO: Log likelihood is still probably wrong... the scaling with N is weird.\"\nplt.tight_layout()\n```\n\nAs in the previous notebook for truly independent objects, I attempt to calculate log likelihood scores for each arrangement by calculating\n\n$$ p(c, p, n) = p(n) * {\\Large \\Pi_{i}^{n}} \\left[ \\dfrac{p(p_i | c_i)}{\\max_{\\hat{p}}{p(\\hat{p} | c_i})} \\dfrac{p(c_i)}{\\max_{\\hat{c}} p(c)} \\right] $$ \n\n(which includes normalization on a per-object and per-class basis to make comparison between scenes with different N possible). But I'm not convinced this is right, yet...\n\nThis clearly violates collision constraints, though, and would be hopeless to capture other inter-object interactions. Nonpenetration itself is an inter-object interaction, and should be handled as such.\n\n## Investigating inter-object interactions\n\nLet's refactorize a bit more carefully, instead breaking down $p(p, c)$ into a combination of pairwise terms.\n\n$$ \\begin{align} \np(c, p, n) &= p(n) * \\Pi_{ (i, j) \\in [0...N]\\times[0...N]} \\left[ p(p_i, p_j, c_i, c_j) \\right] \\\\\n&= p(n) * \\Pi_{ (i, j) \\in [0...N]\\times[0...N]} \\left[ p(p_i - p_j | p_j, c_i, c_j) p(p_j | c_i, c_j) p(c_j | c_i) p(c_i) \\right] \\\\\n&= p(n) * \\Pi_{ (i, j) \\in [0...N]\\times[0...N]} \\left[ p(p_i - p_j | c_i, c_j) p(p_j | c_i, c_j) p(c_j | c_i) p(c_i) \\right]\n\\end{align} $$\n\nHere $p(p_i - p_j | c_i, c_j)$ describes the probability of the $i^{th}$ object being in various locations relative to a $j^{th}$ object, given both of their classes. This is assumed to be *independent* of the location of the $j^{th}$ object, for tractibility. We account for the position of the $j^{th}$ object with the distribution $p(p_j | c_i, c_j)$, which encodes what places the $j^{th}$ object is likely to be given that it's of class $c_j$ and will be \n", "meta": {"hexsha": "a3be0745fd8f9e089ff94ab233540de42090c225", "size": 604274, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/20180626 Handcrafted Model Fitting to Planar Scenes, Nonpenetrating Arrangements.ipynb", "max_stars_repo_name": "gizatt/scene_generation", "max_stars_repo_head_hexsha": "cd978b4fe8ac58983894db3fb93d625c85578dd6", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2018-11-27T18:46:01.000Z", "max_stars_repo_stars_event_max_datetime": "2020-09-06T19:59:12.000Z", "max_issues_repo_path": "notebooks/20180626 Handcrafted Model Fitting to Planar Scenes, Nonpenetrating Arrangements.ipynb", "max_issues_repo_name": "gizatt/scene_generation", "max_issues_repo_head_hexsha": "cd978b4fe8ac58983894db3fb93d625c85578dd6", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/20180626 Handcrafted Model Fitting to Planar Scenes, Nonpenetrating Arrangements.ipynb", "max_forks_repo_name": "gizatt/scene_generation", "max_forks_repo_head_hexsha": "cd978b4fe8ac58983894db3fb93d625c85578dd6", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 1392.33640553, "max_line_length": 439920, "alphanum_fraction": 0.9559123841, "converted": true, "num_tokens": 2893, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.7090191399336402, "lm_q1q2_score": 0.4175336650132772}} {"text": "# Problema de selecci\u00f3n de portafolio con preferencias media-varianza\n\n\n\nEn la clase pasada hablamos acerca de:\n- preferencias,\n- funciones de utilidad,\n- la actitud de los inversionistas de cara al riesgo,\n- la aversi\u00f3n al riesgo, entre otros.\n\nTodas ellas son piezas que necesitamos para responder la pregunta de \u00bfc\u00f3mo un inversionista toma la decisi\u00f3n \u00f3ptima de selecci\u00f3n de portafolio?\n\nEn esta clase al fin estamos listos para ensamblar estos conceptos y escribir el problema de selecci\u00f3n de portafolios. \n\nEn el camino aprenderemos acerca del concepto de **utilidad esperada**, que nos permite trabajar con incertidumbre en el modelado econ\u00f3mico (una de las ideas m\u00e1s importantes en econom\u00eda). Esta idea tiene m\u00e1s de 60 a\u00f1os, y b\u00e1sicamente dice que los individuos, cuando est\u00e1n de cara a incertidumbre, maximizan el valor esperado de su utilidad (solo cierto si somos homo economicus). \n\nAdem\u00e1s del concepto de utilidad esperada, aprenderemos acerca de **preferencias media-varianza**. Es decir, supondremos que los inversionistas toman decisiones basados en un tipo particular de preferencias.\n\nCon lo anterior, estableceremos el problema de selecci\u00f3n de portafolios.\n\n**Objetivos:**\n\n- \u00bfQu\u00e9 es utilidad esperada?\n- \u00bfQu\u00e9 son las preferencias media-varianza?\n- Funciones de utilidad media-varianza.\n- Enunciado y soluci\u00f3n del problema b\u00e1sico de selecci\u00f3n de portafolio.\n\n*Referencia:*\n- Notas del curso \"Portfolio Selection and Risk Management\", Rice University, disponible en Coursera.\n___\n\n## 1. Utilidad esperada\n- B\u00e1sicamente combina las probabilidades de los resultados con c\u00f3mo los inversionistas se sienten con dichos resultados.\n- En otras palabras, la utilidad esperada multiplica la probabilidad de suceso de un evento con la utilidad que genera dicho evento.\n\nRecordemos que las *funciones de utilidad* permiten a los inversionistas expresar c\u00f3mo se sienten con los resultados, especialmente en los malos ratos. \n\nEntonces la *utilidad esperada* es una herramienta que nos permite cuantificar c\u00f3mo nos sentimos en nuestros malos momentos econ\u00f3micos, capturando el riesgo con la probabilidad de ocurrencia de dichos malos momentos. \n\nDado este marco de trabajo, cualquier decisi\u00f3n se puede escribir como la maximizaci\u00f3n de la utilidad esperada:\n\\begin{align}\n\\max_{\\theta} & \\quad E[U(W)], \\\\\n\\end{align}\nmediante la escogencia de cierta variable $\\theta$ (gastos, planes de ahorro, compra de activos, planes de producci\u00f3n, etc.).\n\nPara nuestros prop\u00f3sitos, la variable de decisi\u00f3n ser\u00e1n los pesos o ponderaciones del portafolio.\n\nAdicionalmente, en el contexto de la decisi\u00f3n de distribuci\u00f3n de la riqueza entre activos, el problema de maximizaci\u00f3n tendr\u00e1 com\u00fanmente las siguientes restricciones:\n- universo de inversi\u00f3n,\n- posici\u00f3n en los activos dados.\n\n**Ejemplo.** \n\nSupongamos que un inversionista debe determinar la composici\u00f3n \u00f3ptima de su portafolio, que contiene activos y bonos. Supongamos que son los \u00fanicos instrumentos disponibles.\n\nSean:\n- $w_s$: peso o ponderaci\u00f3n de activos en el portafolio,\n- $w_b$: peso o ponderaci\u00f3n de bonos en el portafolio,\n- $r_s$: rendimiento de los activos, y\n- $r_b$: rendimiento de los bonos.\n\nDe manera que podemos escribir el problema de selecci\u00f3n de portafolios como la maximizaci\u00f3n de la utilidad esperade de nuestra riqueza futura, la cual depender\u00e1 de nuestros rendimientos:\n \n\\begin{align}\n\\max_{w_s,w_b} &\\quad E[U(W)]\\\\\n\\text{s. a.} &\\quad W=W_0(1+w_sr_s+w_br_b)\\\\\n &\\quad w_s+w_b=1\n\\end{align}\n\n\nPreguntas:\n- \u00bfQu\u00e9 significan las restricciones?\n- Ya que tenemos planteado este problema b\u00e1sico, \u00bfqu\u00e9 har\u00eda falta para empezar a resolverlo?\n___\n\n## 2. Preferencias media-varianza\n\n### 2.1. Utilidad media-varianza\n\nEntonces, \u00bfqu\u00e9 funciones de utilidad deber\u00edamos de usar en este problema de selecci\u00f3n de portafolios?\n\n- La respuesta es: **preferencias media-varianza**.\n- \u00c9stas ser\u00e1n representadas en t\u00e9rminos de funciones de utilidad como: **utilidad media-varianza**.\n\nUsamos la *utilidad media-varianza* en el problema de selecci\u00f3n de portafolios dado que \u00e9sta decribe el \"trade-off\" entre riesgo y rendimiento que enfrentan los inversionistas. La *utilidad media-varianza* est\u00e1 dada por la siguiente expresi\u00f3n:\n\n$$U=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2,$$\n\ndonde\n- $E[r_p]$ es el rendimiento esperado del portafolio,\n- $\\sigma_p^2$ es la varianza del portafolio, y\n- $\\gamma$ es el coeficiente de aversi\u00f3n al riesgo. \n\n#### Intuici\u00f3n acerca de la funci\u00f3n de utilidad media-varianza:\n- S\u00f3lo se preocupa por medias :) y varianzas :(.\n- Incrementa con: rendimiento esperado del portafolio.\n- Decrece con: varianza del portafolio.\n- Malos tiempos: rendimientos son bajos y las volatilidades son altas.\n- Conecta bastante bien con la teor\u00eda moderna de portafolios, la cual caracteriza los rendimientos con medias y varianzas \u00fanicamente.\n- Criticada por su limitaci\u00f3n: supone que los inversionistas s\u00f3lo se preocupan por medias y varianzas.\n\n### 2.2. Curvas de indiferencia\n\n*\u00bfRecuerdan las curvas de nivel que se ven en c\u00e1lculo de varias variables?*\n- Bien, ac\u00e1 nos servir\u00e1n para representar la utilidad media-varianza gr\u00e1ficamente.\n- En el contexto de utilidad media-varianza, las curvas de nivel se llaman **curvas de indiferencia**.\n\nDados ciertos niveles de utilidad $U_1>U_2>U_3$, las curvas de indiferencia relativas a estos niveles de utilidad, son los lugares geom\u00e9tricos en el espacio de rendimiento esperado vs. volatilidad representados por las siguientes expresiones\n\n$$U_1=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2\\Rightarrow E[r_p]=\\frac{1}{2}\\gamma\\sigma_p^2+U_1,$$\n\n$$U_2=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2\\Rightarrow E[r_p]=\\frac{1}{2}\\gamma\\sigma_p^2+U_2,$$\n\n$$U_3=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2\\Rightarrow E[r_p]=\\frac{1}{2}\\gamma\\sigma_p^2+U_3.$$\n\n**Gr\u00e1ficamente**\n\n\n```python\n# Importar numpy y pyplot\nimport numpy as np\nimport matplotlib.pyplot as plt\n%matplotlib inline\n```\n\n\n```python\n# Coeficiente de aversi\u00f3n al riesgo (entre 1 y 10 com\u00fanmente)\ng = 9\n# Niveles de utilidad\nU1, U2, U3 = 0.15, 0.1, 0.05\n# Vector de volatilidades (sugerido 1%-60%)\nsp = np.linspace(0.01, 0.6, 100)\n# Curvas de indiferencia\nE1 = 0.5 * g * sp**2 + U1\nE2 = 0.5 * g * sp**2 + U2\nE3 = 0.5 * g * sp**2 + U3\n```\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(6, 4))\nplt.plot(sp, E1, label='$U_1=0.15$')\nplt.plot(sp, E2, label='$U_2=0.10$')\nplt.plot(sp, E3, label='$U_3=0.05$')\nplt.legend(loc='best')\nplt.xlabel('Volatilidad ($\\sigma$)')\nplt.ylabel('Rendimiento esperado ($E[r_p]$)')\nplt.grid()\n```\n\nBueno, \u00bfy porqu\u00e9 se llaman curvas de indiferencia?, \u00bfqu\u00e9 representa una cuva de indiferencia?\n\n- Porque sobre una misma curva el nivel de utilidad es el mismo (es indiferente).\n- Son todas las combinaciones de riesgo y rendimiento que producen un mismo nivel de utilidad.\n\nVolviendo al problema de selecci\u00f3n de portafolios, queremos la utilidad m\u00e1s alta.\n- \u00bfCu\u00e1l de las anteriores curvas de indiferencia corresponde a la utilidad m\u00e1s alta?\n- Intuitivamente, \u00bfporqu\u00e9?\n- Curvas de indiferencia para niveles de utilidad m\u00e1s altos, estar\u00e1n...\n\nNotamos adem\u00e1s que las anteriores curvas de indiferencia son *paralelas* una con otra. Claro, las dibujamos con el mismo coeficiente de aversi\u00f3n al riesgo.\n\n\u00bfC\u00f3mo cambian estas curvas para coeficientes de aversi\u00f3n al riesgo m\u00e1s altos?\n\n\n```python\n# Coeficientes de aversi\u00f3n al riesgo (entre 1 y 10 com\u00fanmente)\ng1, g2, g3 = 3, 5, 7\n# Nivel de utilidad\nU = 0.15\n# Vector de volatilidades (sugerido 1%-60%)\nsp = np.linspace(0.01, 0.6, 100)\n# Curvas de indiferencia\nE1 = 0.5 * g1 * sp**2 + U\nE2 = 0.5 * g2 * sp**2 + U\nE3 = 0.5 * g3 * sp**2 + U\n```\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(6, 4))\nplt.plot(sp, E1, label='$\\gamma_1=3$')\nplt.plot(sp, E2, label='$\\gamma_2=5$')\nplt.plot(sp, E3, label='$\\gamma_3=7$')\nplt.legend(loc='best')\nplt.xlabel('Volatilidad ($\\sigma$)')\nplt.ylabel('Rendimiento esperado ($E[r_p]$)')\nplt.grid()\n```\n\n\u00bfC\u00f3mo interpretamos las anteriores gr\u00e1ficas?, \u00bfqu\u00e9 pasa con las personas m\u00e1s aversas al riesgo?\n- Se puede ver de dos maneras: para un mismo nivel de rendimiento esperado, una persona m\u00e1s aversa al riesgo soporta un nivel menor de riesgo; equivalentemente, para un mismo nivel de riesgo, una persona m\u00e1s aversa al riesgo requerir\u00e1 un nivel de rendimiento esperado m\u00e1s alto.\n\nCon todo lo anterior, el problema de selecci\u00f3n de portafolios se puede plantear como *\"encontrar la curva de indeferencia m\u00e1s alta dado el conjunto de oportunidades de inversi\u00f3n y restricciones\"*.\n\n## 3. Problema de selecci\u00f3n de portafolios: una ilustraci\u00f3n\n\nAhora ilustraremos el problema de selecci\u00f3n de portafolios con algunos datos. \n- Por ahora solo queremos ilustrar gr\u00e1ficamente c\u00f3mo se resuelve este problema. Trabajar en la intuici\u00f3n.\n- En las siguientes dos clases nos enfocaremos en c\u00f3mo resolverlo anal\u00edticamente.\n\nAc\u00e1 tenemos el rendimiento medio anual y la volatilidad para dos instrumentos usando datos de EU: instrumentos de deuda (bonos) y acciones. Supondremos que el inversionista solo puede invertir en estas dos clases de instrumentos.\n\n\n```python\n# Importamos pandas\nimport pandas as pd\n```\n\n /home/esteban/anaconda3/lib/python3.7/importlib/_bootstrap.py:219: RuntimeWarning: numpy.ufunc size changed, may indicate binary incompatibility. Expected 192 from C header, got 216 from PyObject\n return f(*args, **kwds)\n\n\n\n```python\n# Datos\ndata = pd.DataFrame(index=['Stocks','Bonds', 'CorrSB'], columns=['Mean', 'Std'])\ndata['Mean'] = [0.119, 0.0591, 0.113]\ndata['Std'] = [0.1915, 0.0833, None]\ndata\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        MeanStd
                                        Stocks0.11900.1915
                                        Bonds0.05910.0833
                                        CorrSB0.1130NaN
                                        \n
                                        \n\n\n\nEntonces, \u00bfcu\u00e1l es la distribuci\u00f3n de riqueza \u00f3ptima?, o m\u00e1s bien, \u00bfcu\u00e1l es la composici\u00f3n \u00f3ptima del portafolio para un inversionista dado su nivel de aversi\u00f3n al riesgo?\n\n**Primero.** Recordamos que, para dos activos, podemos trazar la frontera de m\u00ednima varianza tomando todas las posibles combinaciones de los dos activos.\n\nDe nuevo, sean:\n- $w_s=w$: peso o ponderaci\u00f3n de activos en el portafolio,\n- $w_b=1-w$: peso o ponderaci\u00f3n de bonos en el portafolio,\n- $r_s$: rendimiento de los activos, y\n- $r_b$: rendimiento de los bonos.\n\nEntonces\n\n$$E[r_p]=wE[r_{s}]+(1-w)E[r_b]$$\n\n$$\\sigma_p^2=w^2\\sigma_{s}^2+(1-w)^2\\sigma_b^2+2w(1-w)\\rho_{s,b}\\sigma_s\\sigma_b$$\n\n\n```python\n# Vector de w variando entre 0 y 1 con n pasos\nw = np.linspace(0, 1, 101)\n# Rendimientos esperados individuales\nEs = data.loc['Stocks', 'Mean']\nEb = data.loc['Bonds', 'Mean']\n# Volatilidades individuales\nss = data.loc['Stocks', 'Std']\nsb = data.loc['Bonds', 'Std']\n# Correlacion\nrsb = data.loc['CorrSB', 'Mean']\nEs, Eb, ss, sb, rsb\n```\n\n\n\n\n (0.119, 0.0591, 0.1915, 0.0833, 0.113)\n\n\n\n\n```python\n# Crear un DataFrame cuyas columnas sean rendimiento\n# y volatilidad del portafolio para cada una de las w\n# generadas\nportafolios = pd.DataFrame(data={'w': w,\n 'Media': w * Es + (1 - w) * Eb,\n 'Vol': (w**2 * ss**2 + (1-w)**2 * sb**2 + 2 * w * (1 - w) * rsb * ss * sb)**0.5\n }\n )\nportafolios\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        wMediaVol
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                                        101 rows \u00d7 3 columns

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                                        \n\n\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(6, 4))\nplt.plot(portafolios['Vol'], portafolios['Media'], label='Port.')\nplt.legend(loc='best')\nplt.xlabel('Volatilidad ($\\sigma$)')\nplt.ylabel('Rendimiento esperado ($E[r]$)')\nplt.grid()\n```\n\n**Segundo.** Graficamos en la misma ventana, curvas de indiferencia.\n\n\n```python\n# Niveles de utilidad \nU1, U2, U3 = 0.06, 0.07251, 0.09\n# Coeficiente de aversi\u00f3n al riesgo\ng = 3\n# Curvas de indiferencia\nE1 = 0.5 * g * portafolios['Vol']**2 + U1\nE2 = 0.5 * g * portafolios['Vol']**2 + U2\nE3 = 0.5 * g * portafolios['Vol']**2 + U3\n```\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(6, 4))\nplt.plot(portafolios['Vol'], portafolios['Media'], label='Port.')\nplt.plot(portafolios['Vol'], E1, label='$U_1=0.06$')\nplt.plot(portafolios['Vol'], E2, label='$U_2=0.0725$')\nplt.plot(portafolios['Vol'], E3, label='$U_3=0.09$')\nplt.legend(loc='best')\nplt.xlabel('Volatilidad ($\\sigma$)')\nplt.ylabel('Rendimiento esperado ($E[r_p]$)')\nplt.grid()\n```\n\n**Tercero.** La elecci\u00f3n \u00f3ptima est\u00e1 dada por la curva de indiferencia para el nivel de utilidad m\u00e1s alto que es tangente a la frontera media-varianza.\n- Claramente, esta selecci\u00f3n depende del coeficiente de aversi\u00f3n al riesgo.\n\n\n```python\n# Gr\u00e1fica con zoom\n# Gr\u00e1fica\nplt.figure(figsize=(6, 4))\nplt.plot(portafolios['Vol'], portafolios['Media'], label='Port.')\nplt.plot(portafolios['Vol'], E1, label='$U_1=0.06$')\nplt.plot(portafolios['Vol'], E2, label='$U_2=0.0725$')\nplt.plot(portafolios['Vol'], E3, label='$U_3=0.09$')\nplt.legend(loc='best')\nplt.xlabel('Volatilidad ($\\sigma$)')\nplt.ylabel('Rendimiento esperado ($E[r_p]$)')\nplt.grid()\nplt.axis([0.11, 0.15, 0.08, 0.11])\n```\n\n# Anuncios parroquiales\n\n## 1. Quiz la siguiente clase (22 de Octubre, porque la siguiente es el examen).\n## 2. Un par de art\u00edculos del WSJ y el NYT que discuten herramientas disponibles para la medici\u00f3n de su propia tolerancia al riesgo:\n- [Art\u00edculo 1](https://www.nytimes.com/2016/02/13/your-money/as-stocks-fall-its-time-to-measure-your-risk-tolerance.html)\n- [Art\u00edculo 2](https://www.wsj.com/articles/check-your-tolerance-for-investment-risk-now-before-markets-sag-1405619939)\n\n## 3. Tarea 5 para hoy.\n## 4. Tarea 4 Entrega 2 para el lunes.\n## 5. Examen m\u00f3dulos 1 y 2 el martes.\n\n\n\n
                                        \nCreated with Jupyter by Esteban Jim\u00e9nez Rodr\u00edguez.\n
                                        \n", "meta": {"hexsha": "23452ebf2af7bc394e051ac928ed4d8734050e24", "size": 168653, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Modulo3/Clase12_ProblemaSeleccionPortafolio.ipynb", "max_stars_repo_name": "Ivanpaniagua7/porinvo20191", "max_stars_repo_head_hexsha": "c2f42370c457c40306f3153e61e7038dcfedb06d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Modulo3/Clase12_ProblemaSeleccionPortafolio.ipynb", "max_issues_repo_name": "Ivanpaniagua7/porinvo20191", "max_issues_repo_head_hexsha": "c2f42370c457c40306f3153e61e7038dcfedb06d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Modulo3/Clase12_ProblemaSeleccionPortafolio.ipynb", "max_forks_repo_name": "Ivanpaniagua7/porinvo20191", "max_forks_repo_head_hexsha": "c2f42370c457c40306f3153e61e7038dcfedb06d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 147.9412280702, "max_line_length": 30332, "alphanum_fraction": 0.8584667928, "converted": true, "num_tokens": 7484, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.7690802370707281, "lm_q1q2_score": 0.4175054225193041}} {"text": "# Problema de selecci\u00f3n de portafolio con preferencias media-varianza\n\n\n\nEn la clase pasada hablamos acerca de:\n- preferencias,\n- funciones de utilidad,\n- la actitud de los inversionistas de cara al riesgo,\n- la aversi\u00f3n al riesgo, entre otros.\n\nTodas ellas son piezas que necesitamos para responder la pregunta de \u00bfc\u00f3mo un inversionista toma la decisi\u00f3n \u00f3ptima de selecci\u00f3n de portafolio?\n\nEn esta clase al fin estamos listos para ensamblar estos conceptos y escribir el problema de selecci\u00f3n de portafolios. \n\nEn el camino aprenderemos acerca del concepto de **utilidad esperada**, que nos permite trabajar con incertidumbre en el modelado econ\u00f3mico (una de las ideas m\u00e1s importantes en econom\u00eda). Esta idea tiene m\u00e1s de 60 a\u00f1os, y b\u00e1sicamente dice que los individuos, cuando est\u00e1n de cara a incertidumbre, maximizan el valor esperado de su utilidad (solo cierto si somos homo economicus). \n\nAdem\u00e1s del concepto de utilidad esperada, aprenderemos acerca de **preferencias media-varianza**. Es decir, supondremos que los inversionistas toman decisiones basados en un tipo particular de preferencias.\n\nCon lo anterior, estableceremos el problema de selecci\u00f3n de portafolios.\n\n**Objetivos:**\n\n- \u00bfQu\u00e9 es utilidad esperada?\n- \u00bfQu\u00e9 son las preferencias media-varianza?\n- Funciones de utilidad media-varianza.\n- Enunciado y soluci\u00f3n del problema b\u00e1sico de selecci\u00f3n de portafolio.\n\n*Referencia:*\n- Notas del curso \"Portfolio Selection and Risk Management\", Rice University, disponible en Coursera.\n___\n\n## 1. Utilidad esperada\n- B\u00e1sicamente combina las probabilidades de los resultados con c\u00f3mo los inversionistas se sienten con dichos resultados.\n- En otras palabras, la utilidad esperada multiplica la probabilidad de suceso de un evento con la utilidad que genera dicho evento.\n\nRecordemos que las *funciones de utilidad* permiten a los inversionistas expresar c\u00f3mo se sienten con los resultados, especialmente en los malos ratos. \n\nEntonces la *utilidad esperada* es una herramienta que nos permite cuantificar c\u00f3mo nos sentimos en nuestros malos momentos econ\u00f3micos, capturando el riesgo con la probabilidad de ocurrencia de dichos malos momentos. \n\nDado este marco de trabajo, cualquier decisi\u00f3n se puede escribir como la maximizaci\u00f3n de la utilidad esperada:\n\\begin{align}\n\\max_{\\theta} & \\quad E[U(W)], \\\\\n\\end{align}\nmediante la escogencia de cierta variable $\\theta$ (gastos, planes de ahorro, compra de activos, planes de producci\u00f3n, etc.).\n\nPara nuestros prop\u00f3sitos, la variable de decisi\u00f3n ser\u00e1n los pesos o ponderaciones del portafolio.\n\nAdicionalmente, en el contexto de la decisi\u00f3n de distribuci\u00f3n de la riqueza entre activos, el problema de maximizaci\u00f3n tendr\u00e1 com\u00fanmente las siguientes restricciones:\n- universo de inversi\u00f3n,\n- posici\u00f3n en los activos dados.\n\n**Ejemplo.** \n\nSupongamos que un inversionista debe determinar la composici\u00f3n \u00f3ptima de su portafolio, que contiene activos y bonos. Supongamos que son los \u00fanicos instrumentos disponibles.\n\nSean:\n- $w_s$: peso o ponderaci\u00f3n de activos en el portafolio,\n- $w_b$: peso o ponderaci\u00f3n de bonos en el portafolio,\n- $r_s$: rendimiento de los activos, y\n- $r_b$: rendimiento de los bonos.\n\nDe manera que podemos escribir el problema de selecci\u00f3n de portafolios como la maximizaci\u00f3n de la utilidad esperade de nuestra riqueza futura, la cual depender\u00e1 de nuestros rendimientos:\n \n\\begin{align}\n\\max_{w_s,w_b} &\\quad E[U(W)]\\\\\n\\text{s. a.} &\\quad W=W_0(1+w_sr_s+w_br_b)\\\\\n &\\quad w_s+w_b=1\n\\end{align}\n\n\nPreguntas:\n- \u00bfQu\u00e9 significan las restricciones?\n- Ya que tenemos planteado este problema b\u00e1sico, \u00bfqu\u00e9 har\u00eda falta para empezar a resolverlo?\n___\n\n## 2. Preferencias media-varianza\n\n### 2.1. Utilidad media-varianza\n\nEntonces, \u00bfqu\u00e9 funciones de utilidad deber\u00edamos de usar en este problema de selecci\u00f3n de portafolios?\n\n- La respuesta es: **preferencias media-varianza**.\n- \u00c9stas ser\u00e1n representadas en t\u00e9rminos de funciones de utilidad como: **utilidad media-varianza**.\n\nUsamos la *utilidad media-varianza* en el problema de selecci\u00f3n de portafolios dado que \u00e9sta decribe el \"trade-off\" entre riesgo y rendimiento que enfrentan los inversionistas. La *utilidad media-varianza* est\u00e1 dada por la siguiente expresi\u00f3n:\n\n$$U=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2,$$\n\ndonde\n- $E[r_p]$ es el rendimiento esperado del portafolio,\n- $\\sigma_p^2$ es la varianza del portafolio, y\n- $\\gamma$ es el coeficiente de aversi\u00f3n al riesgo. \n\n#### Intuici\u00f3n acerca de la funci\u00f3n de utilidad media-varianza:\n- S\u00f3lo se preocupa por medias :) y varianzas :(.\n- Incrementa con: rendimiento esperado del portafolio.\n- Decrece con: varianza del portafolio.\n- Malos tiempos: rendimientos son bajos y las volatilidades son altas.\n- Conecta bastante bien con la teor\u00eda moderna de portafolios, la cual caracteriza los rendimientos con medias y varianzas \u00fanicamente.\n- Criticada por su limitaci\u00f3n: supone que los inversionistas s\u00f3lo se preocupan por medias y varianzas.\n\n### 2.2. Curvas de indiferencia\n\n*\u00bfRecuerdan las curvas de nivel que se ven en c\u00e1lculo de varias variables?*\n- Bien, ac\u00e1 nos servir\u00e1n para representar la utilidad media-varianza gr\u00e1ficamente.\n- En el contexto de utilidad media-varianza, las curvas de nivel se llaman **curvas de indiferencia**.\n\nDados ciertos niveles de utilidad $U_1>U_2>U_3$, las curvas de indiferencia relativas a estos niveles de utilidad, son los lugares geom\u00e9tricos en el espacio de rendimiento esperado vs. volatilidad representados por las siguientes expresiones\n\n$$U_1=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2\\Rightarrow E[r_p]=\\frac{1}{2}\\gamma\\sigma_p^2+U_1,$$\n\n$$U_2=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2\\Rightarrow E[r_p]=\\frac{1}{2}\\gamma\\sigma_p^2+U_2,$$\n\n$$U_3=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2\\Rightarrow E[r_p]=\\frac{1}{2}\\gamma\\sigma_p^2+U_3.$$\n\n**Gr\u00e1ficamente**\n\n\n```python\n# Importar numpy y pyplot\nimport numpy as np\nimport matplotlib.pyplot as plt\n%matplotlib inline\n```\n\n\n```python\n# Coeficiente de aversi\u00f3n al riesgo (entre 1 y 10 com\u00fanmente)\ng = 0 #GAMMA \n# Niveles de utilidad\nU1,U2,U3 = 0.2,0.13,0.23\n# Vector de volatilidades (sugerido 1%-60%)\nsp = np.linspace(0.1,0.6,50)\n# Curvas de indiferencia\nErp1 = .5 * g * sp**2 + U1\nErp2 = .5 * g * sp**2 + U2\nErp3 = .5 * g * sp**2 + U3\n```\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(6,4))\nplt.plot(sp,Erp1,label = f'Nivel de Utilidad $U_1$={U1}')\nplt.plot(sp,Erp2,label = f'Nivel de Utilidad $U_2$={U2}')\nplt.plot(sp,Erp3,label = f'Nivel de Utilidad $U_3$={U3}')\nplt.grid()\nplt.xlabel('Volatilidad $\\sigma$')\nplt.ylabel('Rendimiento Esperado $E[r]$')\nplt.legend()\n```\n\nBueno, \u00bfy porqu\u00e9 se llaman curvas de indiferencia?, \u00bfqu\u00e9 representa una cuva de indiferencia?\n\n- Porque sobre una misma curva el nivel de utilidad es el mismo (es indiferente).\n- Son todas las combinaciones de riesgo y rendimiento que producen un mismo nivel de utilidad.\n\nVolviendo al problema de selecci\u00f3n de portafolios, queremos la utilidad m\u00e1s alta.\n- \u00bfCu\u00e1l de las anteriores curvas de indiferencia corresponde a la utilidad m\u00e1s alta?\n- Intuitivamente, \u00bfporqu\u00e9?\n- Curvas de indiferencia para niveles de utilidad m\u00e1s altos, estar\u00e1n...\n\nNotamos adem\u00e1s que las anteriores curvas de indiferencia son *paralelas* una con otra. Claro, las dibujamos con el mismo coeficiente de aversi\u00f3n al riesgo.\n$$\\frac{d E[r_p]}{d\\sigma_p} = \\frac{d}{d\\sigma_p}[\\frac{1}{2}\\gamma \\sigma_p^2 + U]$$\n\u00bfC\u00f3mo cambian estas curvas para coeficientes de aversi\u00f3n al riesgo m\u00e1s altos?\n\n\n```python\n# Importar numpy y pyplot\nimport numpy as np\nfrom matplotlib import pyplot as plt\n%matplotlib inline\n```\n\n\n```python\n# Coeficientes de aversi\u00f3n al riesgo (entre 1 y 10 com\u00fanmente)\ng1, g2, g3 = 10, 7, 4\n# Nivel de utilidad\nU = 0.07\n# Vector de volatilidades (sugerido 1%-60%)\nsp = np.linspace(0.01, 0.6, 50)\n# Curvas de indiferencia\nErp1 = 0.5 * g1 * sp**2 + U\nErp2 = 0.5 * g2 * sp**2 + U\nErp3 = 0.5 * g3 * sp**2 + U\n```\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(6, 4))\nplt.plot(sp, Erp1, label=f'Coeficiente de aversi\u00f3n al riesgo $\\gamma_1$={g1}')\nplt.plot(sp, Erp2, label=f'Coeficiente de aversi\u00f3n al riesgo $\\gamma_2$={g2}')\nplt.plot(sp, Erp3, label=f'Coeficiente de aversi\u00f3n al riesgo $\\gamma_3$={g3}')\nplt.grid()\nplt.axvline(x=0.5, ls='--', color='grey')\nplt.axhline(y=0.5, ls='--', color='grey')\nplt.legend(loc='best')\nplt.xlabel('Volatilidad $\\sigma$')\nplt.ylabel('Rendimiento esperado $E[r]$')\n```\n\n\u00bfC\u00f3mo interpretamos las anteriores gr\u00e1ficas?, \u00bfqu\u00e9 pasa con las personas m\u00e1s aversas al riesgo?\n- Se puede ver de dos maneras: para un mismo nivel de rendimiento esperado, una persona m\u00e1s aversa al riesgo soporta un nivel menor de riesgo; equivalentemente, para un mismo nivel de riesgo, una persona m\u00e1s aversa al riesgo requerir\u00e1 un nivel de rendimiento esperado m\u00e1s alto.\n\nCon todo lo anterior, el problema de selecci\u00f3n de portafolios se puede plantear como *encontrar la curva de indeferencia m\u00e1s alta dado el conjunto de oportunidades de inversi\u00f3n y restricciones*.\n\n## 3. Problema de selecci\u00f3n de portafolios: una ilustraci\u00f3n\n\nAhora ilustraremos el problema de selecci\u00f3n de portafolios con algunos datos. \n- Por ahora solo queremos ilustrar gr\u00e1ficamente c\u00f3mo se resuelve este problema. Trabajar en la intuici\u00f3n.\n- En las siguientes dos clases nos enfocaremos en c\u00f3mo resolverlo anal\u00edticamente.\n\nAc\u00e1 tenemos el rendimiento medio anual y la volatilidad para dos instrumentos usando datos de EU: instrumentos de deuda (bonos) y acciones. Supondremos que el inversionista solo puede invertir en estas dos clases de instrumentos.\n\n\n```python\n# Importamos pandas\nimport pandas as pd\n```\n\n\n```python\n# Datos\ndata = pd.DataFrame(index=['Stocks','Bonds', 'CorrSB'], columns=['Mean', 'Std'])\ndata['Mean'] = [0.119, 0.0591, 0.113]\ndata['Std'] = [0.1915, 0.0833, None]\ndata\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        MeanStd
                                        Stocks0.11900.1915
                                        Bonds0.05910.0833
                                        CorrSB0.1130NaN
                                        \n
                                        \n\n\n\nEntonces, \u00bfcu\u00e1l es la distribuci\u00f3n de riqueza \u00f3ptima?, o m\u00e1s bien, \u00bfcu\u00e1l es la composici\u00f3n \u00f3ptima del portafolio para un inversionista dado su nivel de aversi\u00f3n al riesgo?\n\n**Primero.** Recordamos que, para dos activos, podemos trazar la frontera de m\u00ednima varianza tomando todas las posibles combinaciones de los dos activos.\n\nDe nuevo, sean:\n- $w_s=w$: peso o ponderaci\u00f3n de activos en el portafolio,\n- $w_b=1-w$: peso o ponderaci\u00f3n de bonos en el portafolio,\n- $r_s$: rendimiento de los activos, y\n- $r_b$: rendimiento de los bonos.\n\nEntonces\n\n$$E[r_p]=wE[r_{s}]+(1-w)E[r_b]$$\n\n$$\\sigma_p^2=w^2\\sigma_{s}^2+(1-w)^2\\sigma_b^2+2w(1-w)\\rho_{s,b}\\sigma_s\\sigma_b$$\n\n\n```python\n# Vector de w variando entre 0 y 1 con n pasos\nN = 101\nw = np.linspace(0,1,N)\n# Rendimientos esperados individuales\nErs = data.loc['Stocks','Mean']\nErb = data.loc['Bonds','Mean']\n# Volatilidades individuales\nss = data.loc['Stocks','Std']\nsb = data.loc['Bonds','Std']\n# Correlacion\nrsb = data.loc['CorrSB','Mean']\nErs,Erb,ss,sb,rsb\n```\n\n\n\n\n (0.119, 0.0591, 0.1915, 0.0833, 0.113)\n\n\n\n\n```python\n# Crear un DataFrame cuyas columnas sean rendimiento\n# y volatilidad del portafolio para cada una de las w\n# generadas\nportafolios = pd.DataFrame({'Media':w*Ers + (1-w) * Erb,\n 'Vol':((w * ss)**2 + ((1-w)*sb)**2 + 2 * w* (1-w) *ss * sb *rsb)**.5},\n index = w)\nportafolios.index.name = 'w'\nportafolios.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        MediaVol
                                        w
                                        0.000.0591000.083300
                                        0.010.0596990.082705
                                        0.020.0602980.082155
                                        0.030.0608970.081650
                                        0.040.0614960.081191
                                        \n
                                        \n\n\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(6,4))\nplt.plot(portafolios['Vol'],portafolios['Media'], label = 'Portafolios')\nplt.plot(ss,Ers,'ob', ms = 10, label = 'Stocks')\nplt.plot(sb,Erb,'or', ms = 10, label = 'Bonds')\nplt.xlabel('Volatilidad $\\sigma$')\nplt.ylabel('Rendimiento Esperado $E[r]$')\nplt.legend()\nplt.grid()\n```\n\n**Segundo.** Graficamos en la misma ventana, curvas de indiferencia.\n\n\n```python\n# Niveles de utilidad \nU1,U2,U3 = 0.04,0.045,0.0512\n# Coeficiente de aversi\u00f3n al riesgo\ng = 7\n# Curvas de indiferencia\nsp = np.linspace(0.08,0.18,50)\nErp1 = .5 * g * sp**2 + U1\nErp2 = .5 * g * sp**2 + U2\nErp3 = .5 * g * sp**2 + U3\n```\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(6,4))\nplt.plot(sp,Erp1,label = f'Nivel de Utilidad $U_1$={U1}')\nplt.plot(sp,Erp2,label = f'Nivel de Utilidad $U_2$={U2}')\nplt.plot(sp,Erp3,label = f'Nivel de Utilidad $U_3$={U3}')\n\nplt.plot(portafolios['Vol'],portafolios['Media'], label = 'Portafolios')\nplt.plot(ss,Ers,'ob', ms = 10, label = 'Stocks')\nplt.plot(sb,Erb,'or', ms = 10, label = 'Bonds')\nplt.xlabel('Volatilidad $\\sigma$')\nplt.ylabel('Rendimiento Esperado $E[r]$')\nplt.legend()\nplt.grid()\n```\n\n**Tercero.** La elecci\u00f3n \u00f3ptima est\u00e1 dada por la curva de indiferencia para el nivel de utilidad m\u00e1s alto que es tangente a la frontera media-varianza.\n- Claramente, esta selecci\u00f3n depende del coeficiente de aversi\u00f3n al riesgo.\n\n\n```python\n# Gr\u00e1fica con zoom\nplt.figure(figsize=(6,4))\nplt.plot(sp,Erp1,label = f'Nivel de Utilidad $U_1$={U1}')\nplt.plot(sp,Erp2,label = f'Nivel de Utilidad $U_2$={U2}')\nplt.plot(sp,Erp3,label = f'Nivel de Utilidad $U_3$={U3}')\n\nplt.plot(portafolios['Vol'],portafolios['Media'], label = 'Portafolios')\nplt.plot(ss,Ers,'ob', ms = 10, label = 'Stocks')\nplt.plot(sb,Erb,'or', ms = 10, label = 'Bonds')\nplt.xlabel('Volatilidad $\\sigma$')\nplt.ylabel('Rendimiento Esperado $E[r]$')\nplt.legend()\nplt.grid()\nplt.axis([0.08,0.1,.07,0.1])\n```\n\n# Anuncios parroquiales\n\n## 1. Quiz la siguiente clase.\n## 2. Un par de art\u00edculos del WSJ y el NYT que discuten herramientas disponibles para la medici\u00f3n de su propia tolerancia al riesgo:\n- [Art\u00edculo 1](https://www.nytimes.com/2016/02/13/your-money/as-stocks-fall-its-time-to-measure-your-risk-tolerance.html)\n- [Art\u00edculo 2](https://www.wsj.com/articles/check-your-tolerance-for-investment-risk-now-before-markets-sag-1405619939)\n\n\n\n
                                        \nCreated with Jupyter by Esteban Jim\u00e9nez Rodr\u00edguez.\n
                                        \n", "meta": {"hexsha": "0cce1748c497b393d36ae24eb6b94fd8c552c167", "size": 159718, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Modulo3/Clase11_ProblemaSeleccionPortafolio_apuntesclase.ipynb", "max_stars_repo_name": "Noesns/porinvp2020", "max_stars_repo_head_hexsha": "a4ddfecc3b1aa75ff9ce0c7f2708fcfc75e9ff97", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Modulo3/Clase11_ProblemaSeleccionPortafolio_apuntesclase.ipynb", "max_issues_repo_name": "Noesns/porinvp2020", "max_issues_repo_head_hexsha": "a4ddfecc3b1aa75ff9ce0c7f2708fcfc75e9ff97", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Modulo3/Clase11_ProblemaSeleccionPortafolio_apuntesclase.ipynb", "max_forks_repo_name": "Noesns/porinvp2020", "max_forks_repo_head_hexsha": "a4ddfecc3b1aa75ff9ce0c7f2708fcfc75e9ff97", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 206.354005168, "max_line_length": 36784, "alphanum_fraction": 0.902002279, "converted": true, "num_tokens": 4912, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.7634837689358857, "lm_q1q2_score": 0.41742570489825404}} {"text": "\n\n# Lambda School Data Science Module 132\n## Sampling, Confidence Intervals, and Hypothesis Testing\n\n## Prepare - examine other available hypothesis tests\n\nIf you had to pick a single hypothesis test in your toolbox, t-test would probably be the best choice - but the good news is you don't have to pick just one! Here's some of the others to be aware of:\n\n\n```\nimport numpy as np\nfrom scipy.stats import chisquare # One-way chi square test\n\n# Chi square can take any crosstab/table and test the independence of rows/cols\n# The null hypothesis is that the rows/cols are independent -> low chi square\n# The alternative is that there is a dependence -> high chi square\n# Be aware! Chi square does *not* tell you direction/causation\n\nind_obs = np.array([[1, 1], [2, 2]]).T\nprint(ind_obs)\nprint(chisquare(ind_obs, axis=None))\n\ndep_obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T\nprint(dep_obs)\nprint(chisquare(dep_obs, axis=None))\n```\n\n [[1 2]\n [1 2]]\n Power_divergenceResult(statistic=0.6666666666666666, pvalue=0.8810148425137847)\n [[16 32]\n [18 24]\n [16 16]\n [14 28]\n [12 20]\n [12 24]]\n Power_divergenceResult(statistic=23.31034482758621, pvalue=0.015975692534127565)\n\n\n\n```\n# Distribution tests:\n# We often assume that something is normal, but it can be important to *check*\n\n# For example, later on with predictive modeling, a typical assumption is that\n# residuals (prediction errors) are normal - checking is a good diagnostic\n\nfrom scipy.stats import normaltest\n# Poisson models arrival times and is related to the binomial (coinflip)\nsample = np.random.poisson(5, 1000)\nprint(normaltest(sample)) # Pretty clearly not normal\n```\n\n NormaltestResult(statistic=28.094846668004493, pvalue=7.930152902256951e-07)\n\n\n\n```\n# Kruskal-Wallis H-test - compare the median rank between 2+ groups\n# Can be applied to ranking decisions/outcomes/recommendations\n# The underlying math comes from chi-square distribution, and is best for n>5\nfrom scipy.stats import kruskal\n\nx1 = [1, 3, 5, 7, 9]\ny1 = [2, 4, 6, 8, 10]\nprint(kruskal(x1, y1)) # x1 is a little better, but not \"significantly\" so\n\nx2 = [1, 1, 1]\ny2 = [2, 2, 2]\nz = [2, 2] # Hey, a third group, and of different size!\nprint(kruskal(x2, y2, z)) # x clearly dominates\n```\n\n KruskalResult(statistic=0.2727272727272734, pvalue=0.6015081344405895)\n KruskalResult(statistic=7.0, pvalue=0.0301973834223185)\n\n\nAnd there's many more! `scipy.stats` is fairly comprehensive, though there are even more available if you delve into the extended world of statistics packages. As tests get increasingly obscure and specialized, the importance of knowing them by heart becomes small - but being able to look them up and figure them out when they *are* relevant is still important.\n\n## T-test Assumptions\n\n\n```\nfrom scipy.stats import ttest_ind\n\n?ttest_ind\n```\n\n\n\n- Independence of means\n\nAre the means of our voting data independent (do not affect the outcome of one another)?\n \nThe best way to increase thel likelihood of our means being independent is to randomly sample (which we did not do).\n\n- \"Homogeneity\" of Variance? \n\nIs the magnitude of the variance between the two roughly the same?\n\nI think we're OK on this one for the voting data, although it probably could be better, one party was larger than the other.\n\nIf we suspect this to be a problem then we can use Welch's T-test\n\n- \"Dependent Variable\" (sample means) are Distributed Normally\n\n\n\nLots of statistical tests depend on normal distributions. We can test for normality using Scipy as was shown above.\n\nThis assumption is often assumed even if the assumption is a weak one. If you strongly suspect that things are not normally distributed, you can transform your data to get it looking more normal and then run your test. This problem typically goes away for large sample sizes (yay Central Limit Theorem) and is often why you don't hear it brought up. People declare the assumption to be satisfied either way. \n\n\n\n## Central Limit Theorem\n\n\n\n\n```\nnp.random.binomial(n=1, p=.5, size=30)\n```\n\n\n\n\n array([1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1,\n 1, 1, 0, 1, 0, 0, 0, 0])\n\n\n\n\n```\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\n\nN = 1000\nsample_means = []\nfor x in range(0,N):\n coinflips = np.random.binomial(n=1, p=.5, size=30)\n one_sample = coinflips\n sample_means.append(coinflips.mean())\n\nprint(len(sample_means))\nprint(sample_means)\n```\n\n 1000\n [0.36666666666666664, 0.4666666666666667, 0.5666666666666667, 0.5, 0.6666666666666666, 0.43333333333333335, 0.6, 0.36666666666666664, 0.6333333333333333, 0.5666666666666667, 0.5333333333333333, 0.43333333333333335, 0.7, 0.5, 0.36666666666666664, 0.6, 0.5333333333333333, 0.5666666666666667, 0.4, 0.36666666666666664, 0.26666666666666666, 0.5, 0.5666666666666667, 0.36666666666666664, 0.43333333333333335, 0.5, 0.4666666666666667, 0.6, 0.6, 0.6, 0.43333333333333335, 0.6333333333333333, 0.4666666666666667, 0.36666666666666664, 0.43333333333333335, 0.5, 0.5333333333333333, 0.5, 0.4, 0.6333333333333333, 0.4666666666666667, 0.5, 0.5666666666666667, 0.26666666666666666, 0.4666666666666667, 0.5333333333333333, 0.43333333333333335, 0.6, 0.43333333333333335, 0.5, 0.36666666666666664, 0.43333333333333335, 0.4666666666666667, 0.43333333333333335, 0.6666666666666666, 0.7666666666666667, 0.5, 0.43333333333333335, 0.43333333333333335, 0.3, 0.5333333333333333, 0.6, 0.5, 0.5, 0.5, 0.3333333333333333, 0.5333333333333333, 0.5666666666666667, 0.6, 0.5333333333333333, 0.6, 0.43333333333333335, 0.5666666666666667, 0.5, 0.3, 0.5666666666666667, 0.4, 0.43333333333333335, 0.3333333333333333, 0.6333333333333333, 0.5666666666666667, 0.7, 0.5333333333333333, 0.43333333333333335, 0.4666666666666667, 0.43333333333333335, 0.5, 0.5666666666666667, 0.5333333333333333, 0.2, 0.5, 0.3333333333333333, 0.5333333333333333, 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0.5333333333333333, 0.5333333333333333, 0.5666666666666667, 0.4666666666666667, 0.6, 0.5, 0.4, 0.6, 0.5666666666666667, 0.6, 0.5333333333333333, 0.43333333333333335, 0.5, 0.43333333333333335, 0.4, 0.5333333333333333, 0.4666666666666667, 0.4666666666666667, 0.5, 0.43333333333333335, 0.5333333333333333, 0.6, 0.5, 0.4, 0.3333333333333333, 0.3333333333333333, 0.5333333333333333, 0.5, 0.5666666666666667, 0.43333333333333335, 0.6, 0.5333333333333333, 0.5666666666666667, 0.5333333333333333, 0.3333333333333333, 0.5, 0.43333333333333335, 0.5, 0.5333333333333333, 0.4666666666666667, 0.4, 0.5333333333333333, 0.5, 0.43333333333333335, 0.6666666666666666, 0.4666666666666667, 0.6333333333333333, 0.4666666666666667, 0.4, 0.6333333333333333, 0.6, 0.6333333333333333, 0.4, 0.6666666666666666, 0.43333333333333335, 0.5, 0.6, 0.4, 0.6666666666666666, 0.4, 0.5333333333333333, 0.43333333333333335, 0.43333333333333335, 0.6333333333333333, 0.5666666666666667, 0.4666666666666667, 0.4, 0.4, 0.4666666666666667, 0.4, 0.43333333333333335, 0.36666666666666664, 0.4666666666666667, 0.5, 0.6, 0.4666666666666667, 0.6333333333333333, 0.5333333333333333, 0.4, 0.43333333333333335, 0.5333333333333333, 0.36666666666666664, 0.5, 0.5333333333333333, 0.4666666666666667, 0.4, 0.3, 0.5333333333333333, 0.6333333333333333]\n\n\n\n```\n# Create dataframe with single coin flip\ndf = pd.DataFrame({'one-samp': one_sample})\ndf.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        one-samp
                                        01
                                        10
                                        21
                                        31
                                        41
                                        \n
                                        \n\n\n\n\n```\n# Plot histogram to look at distribution of a single coin flip \ndf.hist();\n```\n\n\n```\n# Plot histogram to look at distribution of all coin flips\nax = plt.hist(sample_means)\nplt.title(f'Distribution of {N} sample means \\n (of 100 coinflips each)');\n```\n\nWhat does the Central Limit Theorem State? That no matter the initial distribution of the population, the distribution of sample means taken will approximate a normal distribution as $n \\rightarrow \\infty$.\n\nThis has very important implications for hypothesis testing and is precisely the reason why the t-distribution begins to approximate the normal distribution as our sample size increases. \n\n## Standard Error of the Mean\n\nWhat does it mean to \"estimate\"? the Population mean?\n\n\n```\nimport numpy as np\nimport pandas as pd\n\n# Average Height\nmu = 70\nsigma = 3\n\nlambda_heights = np.random.normal(mu, sigma, 2000)\nprint(len(lambda_heights))\nlambda_heights\n```\n\n 2000\n\n\n\n\n\n array([75.27298201, 73.53278172, 67.35668305, ..., 71.62847436,\n 71.57331148, 71.59265235])\n\n\n\n\n```\nimport seaborn as sns\n\nsns.distplot(lambda_heights)\nplt.title('Distribution of Heights (in inches)');\n```\n\n\n```\nprint(\"Population Mean:\", lambda_heights.mean())\nprint(\"Population Standard Deviation:\", lambda_heights.std())\n```\n\n Population Mean: 70.01784335410584\n Population Standard Deviation: 2.960488297079555\n\n\n\n```\npopulation = pd.DataFrame({'heights': lambda_heights})\nprint(population.shape)\npopulation.head()\n```\n\n (2000, 1)\n\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        heights
                                        075.272982
                                        173.532782
                                        267.356683
                                        368.993736
                                        466.670617
                                        \n
                                        \n\n\n\n\n```\n# Take a random sample and print sample mean\nsample1 = population.sample(100)\nprint(sample1.shape)\nsample1.head()\n```\n\n (100, 1)\n\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        heights
                                        164269.200564
                                        83470.208448
                                        150868.242700
                                        87571.357699
                                        84369.299415
                                        \n
                                        \n\n\n\n\n```\nprint('Sample Mean #1:', sample1['heights'].mean())\n```\n\n Sample Mean #1: 69.69155993966513\n\n\n\n```\n# Take a different random sample and print sample mean\nsample2 = population.sample(100)\nprint(sample2.shape)\nsample2.head()\n```\n\n (100, 1)\n\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        heights
                                        48467.271530
                                        191269.699096
                                        182465.834826
                                        2869.167553
                                        108068.418836
                                        \n
                                        \n\n\n\n\n```\nprint('Sample Mean #2:', sample2['heights'].mean())\n```\n\n Sample Mean #2: 69.65665910584407\n\n\n## Build and Interpret a Confidence Interval\n\n\n\n\n```\ncoinflips_10000 = np.random.binomial(n=1, p=0.5, size=10000)\n\nsample_std = np.std(coinflips_10000)\nprint('Sample St Dev:', sample_std)\nsample_size = len(coinflips_10000)\nprint('Sample Size:', sample_size)\n```\n\n Sample St Dev: 0.49999963999987046\n Sample Size: 10000\n\n\n\n```\nstandard_error = sample_std/np.sqrt(sample_size)\nprint(standard_error)\n```\n\n 0.004999996399998705\n\n\n### What confidence level do we want our confidence interval to represent?\n\n95% confidence Interval? 99% confidence interval? \n\n\n```\nimport scipy.stats as stats\nhelp(stats.t.ppf)\n```\n\n Help on method ppf in module scipy.stats._distn_infrastructure:\n \n ppf(q, *args, **kwds) method of scipy.stats._continuous_distns.t_gen instance\n Percent point function (inverse of `cdf`) at q of the given RV.\n \n Parameters\n ----------\n q : array_like\n lower tail probability\n arg1, arg2, arg3,... : array_like\n The shape parameter(s) for the distribution (see docstring of the\n instance object for more information)\n loc : array_like, optional\n location parameter (default=0)\n scale : array_like, optional\n scale parameter (default=1)\n \n Returns\n -------\n x : array_like\n quantile corresponding to the lower tail probability q.\n \n\n\n\n```\nt = stats.t.ppf(0.975, sample_size-1)\nt\n```\n\n\n\n\n 1.9602012636213575\n\n\n\n\n```\nsample_mean = coinflips_10000.mean()\nconfidence_interval = (sample_mean - t*standard_error, sample_mean + t*standard_error)\nmargin_of_error = t*standard_error\n\nprint(\"Sample Mean: \", sample_mean)\nprint(\"Margin of Error: \", margin_of_error)\nprint(\"Confidence Interval: \", confidence_interval)\n```\n\n Sample Mean: 0.5006\n Margin of Error: 0.0098009992613797\n Confidence Interval: (0.4907990007386204, 0.5104009992613797)\n\n\n## Graphically Represent a Confidence Interval\n\n\n```\nimport seaborn as sns\n\nsns.kdeplot(coinflips_10000)\nplt.axvline(x=sample_mean, color='k')\nplt.axvline(x=confidence_interval[0], color='r')\nplt.axvline(x=confidence_interval[1], color='r')\n```\n\n## Relationship between Confidence Intervals and T-tests\n\nConfidence Interval == Bounds of statistical significance for our t-test\n\nA sample mean that falls inside of our confidence interval will \"FAIL TO REJECT\" our null hypothesis\n\nA sample mean that falls outside of our confidence interval will \"REJECT\" our null hypothesis\n\n\n```\nfrom scipy.stats import t, ttest_1samp\n```\n\n\n```\nimport numpy as np\n\ncoinflip_means = []\nfor x in range(0,100):\n coinflips = np.random.binomial(n=1, p=.5, size=30)\n coinflip_means.append(coinflips.mean())\n\nprint(coinflip_means)\n```\n\n [0.5333333333333333, 0.5333333333333333, 0.4666666666666667, 0.5, 0.4666666666666667, 0.5333333333333333, 0.5333333333333333, 0.6333333333333333, 0.43333333333333335, 0.6666666666666666, 0.4666666666666667, 0.43333333333333335, 0.5333333333333333, 0.6666666666666666, 0.43333333333333335, 0.4666666666666667, 0.5333333333333333, 0.4, 0.4666666666666667, 0.43333333333333335, 0.6, 0.6333333333333333, 0.6333333333333333, 0.5333333333333333, 0.6333333333333333, 0.3333333333333333, 0.5666666666666667, 0.5, 0.5333333333333333, 0.5, 0.5333333333333333, 0.6333333333333333, 0.4666666666666667, 0.6333333333333333, 0.5, 0.6, 0.5333333333333333, 0.3, 0.5333333333333333, 0.6333333333333333, 0.6, 0.5, 0.5, 0.5666666666666667, 0.4666666666666667, 0.5666666666666667, 0.5, 0.4666666666666667, 0.36666666666666664, 0.6, 0.6666666666666666, 0.3333333333333333, 0.4, 0.4666666666666667, 0.4666666666666667, 0.43333333333333335, 0.5333333333333333, 0.5, 0.5333333333333333, 0.6666666666666666, 0.5333333333333333, 0.36666666666666664, 0.36666666666666664, 0.3, 0.4666666666666667, 0.43333333333333335, 0.5333333333333333, 0.36666666666666664, 0.43333333333333335, 0.43333333333333335, 0.4, 0.6666666666666666, 0.7, 0.5666666666666667, 0.5, 0.26666666666666666, 0.4666666666666667, 0.43333333333333335, 0.5, 0.4666666666666667, 0.5333333333333333, 0.5333333333333333, 0.4, 0.7, 0.5333333333333333, 0.4666666666666667, 0.43333333333333335, 0.43333333333333335, 0.5333333333333333, 0.43333333333333335, 0.4, 0.4666666666666667, 0.6, 0.36666666666666664, 0.4666666666666667, 0.5, 0.4666666666666667, 0.4666666666666667, 0.5, 0.4666666666666667]\n\n\n\n```\nnp.mean(coinflip_means)\n```\n\n\n\n\n 0.5000000000000001\n\n\n\n\n```\nhelp(stats.t.interval)\n```\n\n Help on method interval in module scipy.stats._distn_infrastructure:\n \n interval(alpha, *args, **kwds) method of scipy.stats._continuous_distns.t_gen instance\n Confidence interval with equal areas around the median.\n \n Parameters\n ----------\n alpha : array_like of float\n Probability that an rv will be drawn from the returned range.\n Each value should be in the range [0, 1].\n arg1, arg2, ... : array_like\n The shape parameter(s) for the distribution (see docstring of the\n instance object for more information).\n loc : array_like, optional\n location parameter, Default is 0.\n scale : array_like, optional\n scale parameter, Default is 1.\n \n Returns\n -------\n a, b : ndarray of float\n end-points of range that contain ``100 * alpha %`` of the rv's\n possible values.\n \n\n\n\n```\n# 95% confidence interval\nt_stat = stats.t.ppf(0.975, 99)\nprint('T Statistic: ', t_stat)\n\nstd_sample = np.std(coinflip_means)\nstd_err = std_sample/np.sqrt(len(coinflip_means))\n\nCI = stats.t.interval(0.95, 99, loc=np.mean(coinflip_means), scale=std_err)\nprint('95% confidence interval:', CI)\n```\n\n T Statistic: 1.9842169515086827\n 95% confidence interval: (0.4818625252375945, 0.5181374747624058)\n\n\nA null hypothesis that's just inside of our confidence interval == fail to reject\n\n\n\n\n```\nttest_1samp(coinflip_means, 0.48186)\n```\n\n\n\n\n Ttest_1sampResult(statistic=1.9745458123056203, pvalue=0.051105424274298865)\n\n\n\nA null hypothesis that's just outside of our confidence interval == reject\n\n\n\n\n```\nttest_1samp(coinflip_means, 0.51813)\n```\n\n\n\n\n Ttest_1sampResult(statistic=-1.9734573085501894, pvalue=0.051231132740212654)\n\n\n\n\n```\ndef confidence_interval(data, confidence=0.95):\n \"\"\"\n Calculate a confidence interval around a sample mean for given data.\n Using t-distribution and two-tailed test, default 95% confidence. \n \n Arguments:\n data - iterable (list or numpy array) of sample observations\n confidence - level of confidence for the interval\n \n Returns:\n tuple of (mean, lower bound, upper bound)\n \"\"\"\n data = np.array(data)\n mean = np.mean(data)\n n = len(data)\n stderr = stats.sem(data)\n interval = stderr * stats.t.ppf((1 + confidence) / 2.0, n - 1)\n return (mean, mean - interval, mean + interval)\n```\n\n## Run a $\\chi^{2}$ Test \"by hand\" (Using Numpy)\n\n\n```\ndf = pd.read_csv('https://raw.githubusercontent.com/ryanleeallred/datasets/master/adult.csv', na_values=\" ?\")\nprint(df.shape)\ndf.head()\n```\n\n (32561, 15)\n\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        ageworkclassfnlwgteducationeducation-nummarital-statusoccupationrelationshipracesexcapital-gaincapital-losshours-per-weekcountrysalary
                                        039State-gov77516Bachelors13Never-marriedAdm-clericalNot-in-familyWhiteMale2174040United-States<=50K
                                        150Self-emp-not-inc83311Bachelors13Married-civ-spouseExec-managerialHusbandWhiteMale0013United-States<=50K
                                        238Private215646HS-grad9DivorcedHandlers-cleanersNot-in-familyWhiteMale0040United-States<=50K
                                        353Private23472111th7Married-civ-spouseHandlers-cleanersHusbandBlackMale0040United-States<=50K
                                        428Private338409Bachelors13Married-civ-spouseProf-specialtyWifeBlackFemale0040Cuba<=50K
                                        \n
                                        \n\n\n\n\n```\ndf['hours-per-week'].hist(bins=20);\n```\n\n\n```\ndf.describe(exclude='number')\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        workclasseducationmarital-statusoccupationrelationshipracesexcountrysalary
                                        count307253256132561307183256132561325613197832561
                                        unique816714652412
                                        topPrivateHS-gradMarried-civ-spouseProf-specialtyHusbandWhiteMaleUnited-States<=50K
                                        freq22696105011497641401319327816217902917024720
                                        \n
                                        \n\n\n\n\n```\ncut_points = [0, 9, 19, 29, 39, 49, 500]\nlabel_names = ['0-9','10-19','20-29','30-39','40-49','50+']\ndf['hours_per_week_categories'] = pd.cut(df['hours-per-week'], cut_points, labels=label_names)\ndf.hours_per_week_categories.value_counts()\n```\n\n\n\n\n 40-49 18336\n 50+ 6462\n 30-39 3667\n 20-29 2392\n 10-19 1246\n 0-9 458\n Name: hours_per_week_categories, dtype: int64\n\n\n\n\n```\ndf.sex.value_counts()\n```\n\n\n\n\n Male 21790\n Female 10771\n Name: sex, dtype: int64\n\n\n\n\n```\ndf = df.sort_values(by='hours_per_week_categories')\ncontingency_table = pd.crosstab(df['sex'], df['hours_per_week_categories'], margins=True)\ncontingency_table\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        hours_per_week_categories0-910-1920-2930-3940-4950+All
                                        sex
                                        Female235671128719145636102810771
                                        Male2235751105175312700543421790
                                        All45812462392366718336646232561
                                        \n
                                        \n\n\n\n## Expected Value Calculation\n\\begin{align}\nexpected_{i,j} =\\frac{(row_{i} \\text{total})(column_{j} \\text{total}) }{(\\text{total observations})} \n\\end{align}\n\n\n```\nrow_sums = contingency_table.iloc[0:2, 6].values\ncol_sums = contingency_table.iloc[2, 0:6].values\n\nprint(row_sums)\nprint(col_sums)\n```\n\n [10771 21790]\n [ 458 1246 2392 3667 18336 6462]\n\n\n\n```\ntotal = contingency_table.loc['All','All']\ntotal\n```\n\n\n\n\n 32561\n\n\n\n\n```\nexpected = []\nfor i in range(len(row_sums)):\n expected_row = []\n for column in col_sums:\n expected_val = column*row_sums[i]/total\n expected_row.append(expected_val)\n expected.append(expected_row)\n \nexpected = np.array(expected)\nprint(expected.shape)\nprint(expected)\n```\n\n (2, 6)\n [[ 151.50388502 412.16995793 791.26046497 1213.02346365\n 6065.44811277 2137.59411566]\n [ 306.49611498 833.83004207 1600.73953503 2453.97653635\n 12270.55188723 4324.40588434]]\n\n\n## Chi-Squared Statistic with Numpy\n\n\\begin{align}\n\\chi^2 = \\sum \\frac{(observed_{i}-expected_{i})^2}{(expected_{i})}\n\\end{align}\n\nFor the $observed$ values we will just use a version of our contingency table without the margins as a numpy array. In this way, if our observed values array and our expected values array are the same shape, then we can subtract them and divide them directly which makes the calculations a lot cleaner. No for loops!\n\n\n```\nobserved = pd.crosstab(df['sex'], df['hours_per_week_categories']).values\nprint(observed.shape)\nobserved\n```\n\n (2, 6)\n\n\n\n\n\n array([[ 235, 671, 1287, 1914, 5636, 1028],\n [ 223, 575, 1105, 1753, 12700, 5434]])\n\n\n\n\n```\nchi_square = ((observed - expected)**2/(expected)).sum()\nchi_square\n```\n\n\n\n\n 2287.190943926107\n\n\n\n## Run a $\\chi^{2}$ Test using Scipy\n\n\n```\nhelp(stats.chi2_contingency)\n```\n\n Help on function chi2_contingency in module scipy.stats.contingency:\n \n chi2_contingency(observed, correction=True, lambda_=None)\n Chi-square test of independence of variables in a contingency table.\n \n This function computes the chi-square statistic and p-value for the\n hypothesis test of independence of the observed frequencies in the\n contingency table [1]_ `observed`. The expected frequencies are computed\n based on the marginal sums under the assumption of independence; see\n `scipy.stats.contingency.expected_freq`. The number of degrees of\n freedom is (expressed using numpy functions and attributes)::\n \n dof = observed.size - sum(observed.shape) + observed.ndim - 1\n \n \n Parameters\n ----------\n observed : array_like\n The contingency table. The table contains the observed frequencies\n (i.e. number of occurrences) in each category. In the two-dimensional\n case, the table is often described as an \"R x C table\".\n correction : bool, optional\n If True, *and* the degrees of freedom is 1, apply Yates' correction\n for continuity. The effect of the correction is to adjust each\n observed value by 0.5 towards the corresponding expected value.\n lambda_ : float or str, optional.\n By default, the statistic computed in this test is Pearson's\n chi-squared statistic [2]_. `lambda_` allows a statistic from the\n Cressie-Read power divergence family [3]_ to be used instead. See\n `power_divergence` for details.\n \n Returns\n -------\n chi2 : float\n The test statistic.\n p : float\n The p-value of the test\n dof : int\n Degrees of freedom\n expected : ndarray, same shape as `observed`\n The expected frequencies, based on the marginal sums of the table.\n \n See Also\n --------\n contingency.expected_freq\n fisher_exact\n chisquare\n power_divergence\n \n Notes\n -----\n An often quoted guideline for the validity of this calculation is that\n the test should be used only if the observed and expected frequencies\n in each cell are at least 5.\n \n This is a test for the independence of different categories of a\n population. The test is only meaningful when the dimension of\n `observed` is two or more. Applying the test to a one-dimensional\n table will always result in `expected` equal to `observed` and a\n chi-square statistic equal to 0.\n \n This function does not handle masked arrays, because the calculation\n does not make sense with missing values.\n \n Like stats.chisquare, this function computes a chi-square statistic;\n the convenience this function provides is to figure out the expected\n frequencies and degrees of freedom from the given contingency table.\n If these were already known, and if the Yates' correction was not\n required, one could use stats.chisquare. That is, if one calls::\n \n chi2, p, dof, ex = chi2_contingency(obs, correction=False)\n \n then the following is true::\n \n (chi2, p) == stats.chisquare(obs.ravel(), f_exp=ex.ravel(),\n ddof=obs.size - 1 - dof)\n \n The `lambda_` argument was added in version 0.13.0 of scipy.\n \n References\n ----------\n .. [1] \"Contingency table\",\n https://en.wikipedia.org/wiki/Contingency_table\n .. [2] \"Pearson's chi-squared test\",\n https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test\n .. [3] Cressie, N. and Read, T. R. C., \"Multinomial Goodness-of-Fit\n Tests\", J. Royal Stat. Soc. Series B, Vol. 46, No. 3 (1984),\n pp. 440-464.\n \n Examples\n --------\n A two-way example (2 x 3):\n \n >>> from scipy.stats import chi2_contingency\n >>> obs = np.array([[10, 10, 20], [20, 20, 20]])\n >>> chi2_contingency(obs)\n (2.7777777777777777,\n 0.24935220877729619,\n 2,\n array([[ 12., 12., 16.],\n [ 18., 18., 24.]]))\n \n Perform the test using the log-likelihood ratio (i.e. the \"G-test\")\n instead of Pearson's chi-squared statistic.\n \n >>> g, p, dof, expctd = chi2_contingency(obs, lambda_=\"log-likelihood\")\n >>> g, p\n (2.7688587616781319, 0.25046668010954165)\n \n A four-way example (2 x 2 x 2 x 2):\n \n >>> obs = np.array(\n ... [[[[12, 17],\n ... [11, 16]],\n ... [[11, 12],\n ... [15, 16]]],\n ... [[[23, 15],\n ... [30, 22]],\n ... [[14, 17],\n ... [15, 16]]]])\n >>> chi2_contingency(obs)\n (8.7584514426741897,\n 0.64417725029295503,\n 11,\n array([[[[ 14.15462386, 14.15462386],\n [ 16.49423111, 16.49423111]],\n [[ 11.2461395 , 11.2461395 ],\n [ 13.10500554, 13.10500554]]],\n [[[ 19.5591166 , 19.5591166 ],\n [ 22.79202844, 22.79202844]],\n [[ 15.54012004, 15.54012004],\n [ 18.10873492, 18.10873492]]]]))\n \n\n\n\n```\nchi_squared, p_value, dof, expected = stats.chi2_contingency(observed)\nprint(chi_squared, p_value, dof, expected)\n```\n\n 2287.190943926107 0.0 5 [[ 151.50388502 412.16995793 791.26046497 1213.02346365\n 6065.44811277 2137.59411566]\n [ 306.49611498 833.83004207 1600.73953503 2453.97653635\n 12270.55188723 4324.40588434]]\n\n\nNull Hypothesis: Hours worked per week bins is **independent** of sex. \n\nDue to a p-value of 0, we REJECT the null hypothesis that hours worked per week and sex are independent, and conclude that there is an association between hours worked per week and sex. \n\n## Assignment - Build a confidence interval\n\nA confidence interval refers to a neighborhood around some point estimate, the size of which is determined by the desired p-value. For instance, we might say that 52% of Americans prefer tacos to burritos, with a 95% confidence interval of +/- 5%.\n\n52% (0.52) is the point estimate, and +/- 5% (the interval $[0.47, 0.57]$) is the confidence interval. \"95% confidence\" means a p-value $\\leq 1 - 0.95 = 0.05$.\n\nIn this case, the confidence interval includes $0.5$ - which is the natural null hypothesis (that half of Americans prefer tacos and half burritos, thus there is no clear favorite). So in this case, we could use the confidence interval to report that we've failed to reject the null hypothesis.\n\nBut providing the full analysis with a confidence interval, including a graphical representation of it, can be a helpful and powerful way to tell your story. Done well, it is also more intuitive to a layperson than simply saying \"fail to reject the null hypothesis\" - it shows that in fact the data does *not* give a single clear result (the point estimate) but a whole range of possibilities.\n\nHow is a confidence interval built, and how should it be interpreted? It does *not* mean that 95% of the data lies in that interval - instead, the frequentist interpretation is \"if we were to repeat this experiment 100 times, we would expect the average result to lie in this interval ~95 times.\"\n\nFor a 95% confidence interval and a normal(-ish) distribution, you can simply remember that +/-2 standard deviations contains 95% of the probability mass, and so the 95% confidence interval based on a given sample is centered at the mean (point estimate) and has a range of +/- 2 (or technically 1.96) standard deviations.\n\nDifferent distributions/assumptions (90% confidence, 99% confidence) will require different math, but the overall process and interpretation (with a frequentist approach) will be the same.\n\nYour assignment - using the data from the prior module ([congressional voting records](https://archive.ics.uci.edu/ml/datasets/Congressional+Voting+Records)):\n\n\n### Confidence Intervals:\n1. Generate and numerically represent a confidence interval\n2. Graphically (with a plot) represent the confidence interval\n3. Interpret the confidence interval - what does it tell you about the data and its distribution?\n\n### Chi-squared tests:\n4. Take a dataset that we have used in the past in class that has **categorical** variables. Pick two of those categorical variables and run a chi-squared tests on that data\n - By hand using Numpy\n - In a single line using Scipy\n\nStretch goals:\n\n1. Write a summary of your findings, mixing prose and math/code/results. *Note* - yes, this is by definition a political topic. It is challenging but important to keep your writing voice *neutral* and stick to the facts of the data. Data science often involves considering controversial issues, so it's important to be sensitive about them (especially if you want to publish).\n2. Apply the techniques you learned today to your project data or other data of your choice, and write/discuss your findings here.\n3. Refactor your code so it is elegant, readable, and can be easily run for all issues.\n\n\n```\n# TODO - your code!\n```\n\n## Resources\n\n- [Interactive visualize the Chi-Squared test](https://homepage.divms.uiowa.edu/~mbognar/applets/chisq.html)\n- [Calculation of Chi-Squared test statistic](https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test)\n- [Visualization of a confidence interval generated by R code](https://commons.wikimedia.org/wiki/File:Confidence-interval.svg)\n- [Expected value of a squared standard normal](https://math.stackexchange.com/questions/264061/expected-value-calculation-for-squared-normal-distribution) (it's 1 - which is why the expected value of a Chi-Squared with $n$ degrees of freedom is $n$, as it's the sum of $n$ squared standard normals)\n", "meta": {"hexsha": "cd2637ac8d9f07081c777bdd229d4868d4c10fc1", "size": 159520, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "AnnalisaGibbsCopy_of_LS_DS6_132_Sampling_Confidence_Intervals_and_Hypothesis_Testing.ipynb", "max_stars_repo_name": "AnnalisaGibbs/DS-Unit-1-Sprint-3-Statistical-Tests-and-Experiments", "max_stars_repo_head_hexsha": "e65e5c9a62e1e8a9af44d9cd9eb58fd0d747b540", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "AnnalisaGibbsCopy_of_LS_DS6_132_Sampling_Confidence_Intervals_and_Hypothesis_Testing.ipynb", "max_issues_repo_name": "AnnalisaGibbs/DS-Unit-1-Sprint-3-Statistical-Tests-and-Experiments", "max_issues_repo_head_hexsha": "e65e5c9a62e1e8a9af44d9cd9eb58fd0d747b540", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "AnnalisaGibbsCopy_of_LS_DS6_132_Sampling_Confidence_Intervals_and_Hypothesis_Testing.ipynb", "max_forks_repo_name": "AnnalisaGibbs/DS-Unit-1-Sprint-3-Statistical-Tests-and-Experiments", "max_forks_repo_head_hexsha": "e65e5c9a62e1e8a9af44d9cd9eb58fd0d747b540", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 64.8455284553, "max_line_length": 18474, "alphanum_fraction": 0.668198345, "converted": true, "num_tokens": 19106, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984434543458, "lm_q2_score": 0.7799929053683038, "lm_q1q2_score": 0.41737298956801216}} {"text": "```python\n%matplotlib inline\n```\n\n\n\n# Tutorial 05: Kernels and averages\n\nSimulating swarming models requires expensive mean-field convolution operations of the form: \n\n\\begin{align}J^i = \\frac{1}{N}\\sum_{j=1}^N K(|X^j-X^i|) U^j,\\end{align}\nfor $1\\leq i\\leq N$, where $(X^i)_{1\\leq i \\leq N}$ are the positions of the particles, $(U^j)_{1\\leq j\\leq N}$ are given vectors and $K$ is an **observation kernel**. Typically, $K(|X^i-X^j|)$ is equal to 1 if $X^i$ and $X^j$ are at distance smaller than a fixed interaction distance and 0 otherwise. Other kernels are defined in the module :mod:`sisyphe.kernels`. Below, we show a simple application case. \n\n\n## Linear local averages\n\nFirst, some standard imports...\n\n\n\n\n\n```python\nimport time \nimport math\nimport torch\nfrom matplotlib import pyplot as plt\n\nuse_cuda = torch.cuda.is_available()\ndtype = torch.cuda.FloatTensor if use_cuda else torch.FloatTensor\n```\n\nLet the $N$ particles be uniformly scattered in a box of size $L$ with interaction radius $R$.\n\n\n\n\n\n```python\nN = 100000\nL = 1. \nR = .15\n\npos = L*torch.rand((N,2)).type(dtype)\n```\n\nWe can also assume that the particles have a bounded cone of vision around an axis (defined by a unit vector). The default behaviour is a full vision angle equal to $2\\pi$ in which case the axis is a :data:`None` object. Here we take a cone of vision with angle $\\pi/2$ around an axis which is sampled uniformly. For the :class:`sisyphe.particles.KineticParticles`, the default axis is the velocity. \n\n\n\n\n```python\nangle = math.pi/2\naxis = torch.randn(N,2).type(dtype)\naxis = axis/torch.norm(axis,dim=1).reshape((N,1))\n```\n\nLet us create an instance of a particle system with these parameters. \n\n\n\n\n```python\nfrom sisyphe.particles import Particles \n\nparticles = Particles(\n pos = pos,\n interaction_radius = R,\n box_size = L,\n vision_angle = angle,\n axis = axis)\n```\n\n

                                        Note

                                        By default, the system and the operations below are defined with periodic boundary conditions.

                                        \n\n\n\nAs a simple application, we can compute the number of neighbours of each particle and print the number of neighbours of the first particle. This operation is already implemented in the method :func:`number_of_neighbours() `. It simply corresponds to the average: \n\n\\begin{align}N^i_\\mathrm{neigh} = \\sum_{j=1}^N K(|X^j-X^i|).\\end{align}\n\n\n\n\n```python\nNneigh = particles.number_of_neighbours()\n\nNneigh0 = int(Nneigh[0].item())\n\nprint(\"The first particle sees \" + str(Nneigh0) + \" other particles.\")\n```\n\nFor custom objects, the mean-field average can be computed using the method :func:`linear_local_average() `. As an example, let us compute the center of mass of the neighbours of each particle. First we define the quantity $U$ that we want to average. Here, since we are working on a torus, there are two: the sine and the cosine of the spatial coordinates. \n\n\n\n\n```python\ncos_pos = torch.cos((2*math.pi / L) * particles.pos)\nsin_pos = torch.sin((2*math.pi / L) * particles.pos)\n```\n\nThen we compute the two mean field averages, i.e. the standard convolution over the $N$ particles. The center of mass along each dimension is the argument of the complex number whose coordinates are the average cosine and sine. \n\n\n\n\n```python\naverage_cos, average_sin = particles.linear_local_average(cos_pos, sin_pos)\ncenter_x = torch.atan2(average_sin[:,0], average_cos[:,0])\ncenter_x = (L / (2*math.pi)) * torch.remainder(center_x, 2*math.pi)\ncenter_y = torch.atan2(average_sin[:,1], average_cos[:,1])\ncenter_y = (L / (2*math.pi)) * torch.remainder(center_y, 2*math.pi)\n\ncenter_of_mass = torch.cat((center_x.reshape((N,1)), center_y.reshape((N,1))),\n dim=1)\n```\n\nIn the method :func:`linear_local_average() `, the default observation kernel is a :class:`LazyTensor` of size $(N,N)$ whose $(i,j)$ component is equal to 1 when particle $j$ belongs to the cone of vision of particle $i$ and 0 otherwise. To retrieve the indexes of the particles which belong to the cone of vision of the first particle, we can use the `K-nearest-neighbours reduction `_ provided by the `KeOps `_ library. \n\n\n\n\n```python\nfrom sisyphe.kernels import lazy_interaction_kernel\n\ninteraction_kernel = lazy_interaction_kernel(\n particles.pos, \n particles.pos, \n particles.R,\n particles.L,\n boundary_conditions = particles.bc,\n vision_angle = particles.angle,\n axis = particles.axis)\n\nK_ij = 1. - interaction_kernel \n\nneigh0 = K_ij.argKmin(Nneigh0, dim=1)[0]\n\nprint(\"The indexes of the neighbours of the first particles are: \")\nprint(neigh0)\n```\n\nFinally, a fancy display of what we have computed. We plot the full particle system in black, the first particle in orange, its neighbours in blue and the center of mass of the neighbours in red. \n\n\n\n\n```python\nxall = particles.pos[:,0].cpu().numpy()\nyall = particles.pos[:,1].cpu().numpy()\n\nx = particles.pos[neigh0,0].cpu().numpy()\ny = particles.pos[neigh0,1].cpu().numpy()\n\nx0 = particles.pos[0,0].item()\ny0 = particles.pos[0,1].item()\n\nxc = center_of_mass[0,0].item()\nyc = center_of_mass[0,1].item()\n\n\nfig, ax = plt.subplots(figsize=(6,6))\nax.scatter(xall, yall, s=.003, c='black')\nax.scatter(x, y, s=.3)\nax.scatter(x0, y0, s=24)\nax.scatter(xc, yc, s=24, c='red')\nax.axis([0, L, 0, L])\nax.set_aspect(\"equal\")\n```\n\n## Nonlinear averages\n\nIn some cases, we need to compute a **nonlinear average** of the form \n\n\\begin{align}J^i = \\frac{1}{N}\\sum_{j=1}^N K(|X^j-X^i|) b(U^i,V^j),\\end{align}\n\nwhere $(U^i)_{1\\leq i \\leq N}$ and $(V^j)_{1\\leq j \\leq N}$ are given vectors and $b$ is a given function. When the **binary formula** $b$ can be written as a :class:`LazyTensor`, this can be computed with the method :func:`nonlinear_local_average() `. \n\nFor instance, let us compute the local mean square distance: \n\n\\begin{align}J^i = \\frac{\\sum_{j=1}^N K(|X^j-X^i|) |X^j-X^i|^2}{\\sum_{j=1}^N K(|X^j-X^i|)}.\\end{align}\n\nIn this case, we can use the function :func:`sisyphe.kernels.lazy_xy_matrix` to define a custom binary formula. Given two vectors $X=(X^i)_{1\\leq i\\leq M}$ and $Y = (Y^j)_{1\\leq j\\leq N}$, respectively of sizes $(M,d)$ and $(N,d)$, the $XY$ matrix is a $(M,N,d)$ LazyTensor whose $(i,j,:)$ component is the vector $Y^j-X^i$. \n\n\n\n\n\n```python\nfrom sisyphe.kernels import lazy_xy_matrix \n\ndef b(x,y): \n K_ij = lazy_xy_matrix(x,y,particles.L)\n return (K_ij ** 2).sum(-1)\n\nx = particles.pos\ny = particles.pos\nmean_square_dist = N/Nneigh.reshape((N,1)) * particles.nonlinear_local_average(b,x,y)\n```\n\nSince the particles are uniformly scattered in the box, the theoretical value is \n\n\\begin{align}MSD_0 = \\frac{\\int_0^R \\int_0^{\\pi/2} r^3 \\mathrm{d}r\\mathrm{d}\\theta}{\\int_0^R \\int_0^{\\pi/2} r \\mathrm{d}r\\mathrm{d}\\theta} = \\frac{R^2}{2}\\end{align}\n\n\n\n\n\n```python\nprint(\"Theoretical value: \" + str(R**2/2))\nprint(\"Experimental value: \" + str(mean_square_dist[0].item()))\n```\n", "meta": {"hexsha": "b6e7920d6b23db5f309b3e9474ada8a37b8ba156", "size": 11272, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "doc/_auto_tutorials/plot_e_kernels.ipynb", "max_stars_repo_name": "antoinediez/Sisyphe", "max_stars_repo_head_hexsha": "f6bb067cd8898450174c5d97bb0f3f0cb5db8b87", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 8, "max_stars_repo_stars_event_min_datetime": "2021-06-05T20:03:35.000Z", "max_stars_repo_stars_event_max_datetime": "2021-10-01T21:20:24.000Z", "max_issues_repo_path": "doc/_auto_tutorials/plot_e_kernels.ipynb", "max_issues_repo_name": "antoinediez/Sisyphe", "max_issues_repo_head_hexsha": "f6bb067cd8898450174c5d97bb0f3f0cb5db8b87", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2021-08-30T22:48:29.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-18T21:25:12.000Z", "max_forks_repo_path": "doc/_build/html/_downloads/869e866ff1c5cb640fbd51e15245165b/plot_e_kernels.ipynb", "max_forks_repo_name": "antoinediez/Sisyphe", "max_forks_repo_head_hexsha": "f6bb067cd8898450174c5d97bb0f3f0cb5db8b87", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2021-06-10T20:21:17.000Z", "max_forks_repo_forks_event_max_datetime": "2021-09-28T12:47:44.000Z", "avg_line_length": 45.4516129032, "max_line_length": 1020, "alphanum_fraction": 0.5901348474, "converted": true, "num_tokens": 2112, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.6513548511303338, "lm_q1q2_score": 0.4172812718579959}} {"text": "# Converting *Numerical* ADM Initial Data in the Spherical or Cartesian Basis to BSSN Initial Data in the Desired Curvilinear Basis\n## Author: Zach Etienne\n### Formatting improvements courtesy Brandon Clark\n\n### NRPy+ Source Code for this module: [BSSN/ADM_Numerical_Spherical_or_Cartesian_to_BSSNCurvilinearID.py](../edit/BSSN/ADM_Numerical_Spherical_or_Cartesian_to_BSSNCurvilinearID.py)\n\n**This module is meant for use only with initial data that can be represented numerically in ADM form, either in the Spherical or Cartesian basis. I.e., the ADM variables are given $\\left\\{\\gamma_{ij}, K_{ij}, \\alpha, \\beta^i\\right\\}$ *numerically* as functions of $(r,\\theta,\\phi)$ or $(x,y,z)$; e.g., through an initial data solver. If instead the ADM initial data are provided as exact (algebraic) functions of $(r,\\theta,\\phi)$ or $(x,y,z)$, then is is better to use [the Exact-ADM-Spherical/Cartesian-to-BSSNCurvilinear module](Tutorial-ADM_Initial_Data-Converting_Exact_ADM_Spherical_or_Cartesian_to_BSSNCurvilinear.ipynb) instead.** \n\n## Introduction:\nGiven the ADM variables:\n\n$$\\left\\{\\gamma_{ij}, K_{ij}, \\alpha, \\beta^i, B^i\\right\\}$$\n\nin the Spherical or Cartesian basis, and as functions of $(r,\\theta,\\phi)$ or $(x,y,z)$, respectively, this module documents their conversion to the BSSN variables\n\n$$\\left\\{\\bar{\\gamma}_{i j},\\bar{A}_{i j},\\phi, K, \\bar{\\Lambda}^{i}, \\alpha, \\beta^i, B^i\\right\\},$$ \n\nin the desired curvilinear basis (given by reference_metric::CoordSystem). Then it rescales the resulting BSSNCurvilinear variables (as defined in [the BSSN Curvilinear tutorial](Tutorial-BSSNCurvilinear.ipynb)) into the form needed for BSSNCurvilinear evolutions:\n\n$$\\left\\{h_{i j},a_{i j},\\phi, K, \\lambda^{i}, \\alpha, \\mathcal{V}^i, \\mathcal{B}^i\\right\\}.$$ \n\nWe will use as our core example in this module UIUC initial data, which are ([as documented in their NRPy+ initial data module](Tutorial-ADM_Initial_Data-UIUC_BlackHole.ipynb)) given in terms of ADM variables in Spherical coordinates.\n\n\n\n# Table of Contents\n$$\\label{toc}$$ \n\nThe module is organized as follows:\n\n1. [Step 1](#initializenrpy): Initialize core Python/NRPy+ modules\n1. [Step 2](#cylindrical): Desired output BSSN Curvilinear coordinate system set to Cylindrical, as a proof-of-principle\n1. [Step 3](#admfunc): Setting ADM variables through initial data solver C function\n1. [Step 4](#adm_jacobian): Applying Jacobian transformations to get in the correct $xx0,xx1,xx2$ basis\n1. [Step 5](#adm2bssn): Perform the ADM-to-BSSN conversion for 3-metric, extrinsic curvature, and gauge quantities\n 1. [Step 5.a](#adm2bssn_gamma): Convert ADM $\\gamma_{ij}$ to BSSN $\\bar{\\gamma}_{ij}$\n 1. [Step 5.b](#admexcurv_convert): Convert the ADM extrinsic curvature $K_{ij}$\n 1. [Step 5.c](#conformal): Define the conformal factor variable $\\texttt{cf}$\n1. [Step 6](#rescale): Rescale tensorial quantities\n1. [Step 7](#adm2bssn_c): Output all ADM-to-BSSN expressions to a C function \n 1. [Step 7.a](#driver): Output the driver function for the above C function\n1. [Step 8](#lambda): Compute $\\bar{\\Lambda}^i$ from finite-difference derivatives of rescaled metric quantities\n1. [Step 9](#code_validation): Code Validation against BSSN.ADM_Numerical_Spherical_or_Cartesian_to_BSSNCurvilinear NRPy+ module\n1. [Step 10](#latex_pdf_output): Output this module to $\\LaTeX$-formatted PDF\n\n\n\n# Step 1: Initialize core Python/NRPy+ modules \\[Back to [top](#toc)\\]\n$$\\label{initializenrpy}$$\n\n\n```python\n# Step 1: Initialize core Python/NRPy+ modules\nimport sympy as sp\nimport NRPy_param_funcs as par\nfrom outputC import *\nimport indexedexp as ixp\nimport reference_metric as rfm\nimport loop as lp\nimport grid as gri\nimport finite_difference as fin\nimport BSSN.BSSN_quantities as Bq # The EvolvedConformalFactor_cf parameter is used below\n```\n\n\n\n# Step 2: Desired output BSSN Curvilinear coordinate system set to Cylindrical, as a proof-of-principle \\[Back to [top](#toc)\\]\n$$\\label{cylindrical}$$\n\n\n```python\n# Step 2: Desired output BSSN Curvilinear coordinate system set to Cylindrical, as a proof-of-principle\n\n# The ADM & BSSN formalisms only work in 3D; they are 3+1 decompositions of Einstein's equations.\n# To implement axisymmetry or spherical symmetry, simply set all spatial derivatives in\n# the relevant angular directions to zero; DO NOT SET DIM TO ANYTHING BUT 3.\n\n# Step 0: Set spatial dimension (must be 3 for BSSN)\nDIM = 3\n\n# Set the desired *output* coordinate system to Cylindrical:\npar.set_parval_from_str(\"reference_metric::CoordSystem\",\"Cylindrical\")\nrfm.reference_metric()\n\n# Set function input parameters to consistent defaults.\nADM_input_function_name = \"ID_ADM_SphorCart\"\npointer_to_ID_inputs = False\n```\n\n\n\n# Step 3: Setting ADM variables through initial data solver C function \\[Back to [top](#toc)\\]\n$$\\label{admfunc}$$\n\nGiven $(r,\\theta,\\phi)$ or $(x,y,z)$, we assume the initial data solver provides a C function ID_ADM_SphorCart({$(r,\\theta,\\phi)$ or $(x,y,z)$}, {ADM variables}) that sets all ADM variables in the spherical or Cartesian basis, respectively. Here, as a validation test, we set up this function for UIUC Black Hole initial data.\n\n\n```python\n# Step 3: Setting ADM variables through initial data solver C function\n\n# Import UIUC Black Hole initial data\nimport BSSN.UIUCBlackHole as uibh\nuibh.UIUCBlackHole(ComputeADMGlobalsOnly=True)\n\nwith open(\"BSSN/\"+ADM_input_function_name+\".h\", \"w\") as file:\n file.write(\"\"\"\n// This function takes as input either (x,y,z) or (r,th,ph) and outputs\n// all ADM quantities in the Cartesian or Spherical basis, respectively.\nvoid \"\"\"+ADM_input_function_name+\"\"\"(const REAL xyz_or_rthph[3], \n REAL *gammaDD00,REAL *gammaDD01,REAL *gammaDD02,REAL *gammaDD11,REAL *gammaDD12,REAL *gammaDD22,\n REAL *KDD00,REAL *KDD01,REAL *KDD02,REAL *KDD11,REAL *KDD12,REAL *KDD22,\n REAL *alpha,\n REAL *betaU0,REAL *betaU1,REAL *betaU2,\n REAL *BU0,REAL *BU1,REAL *BU2) {\n const REAL r = xyz_or_rthph[0];\n const REAL th = xyz_or_rthph[1];\n const REAL ph = xyz_or_rthph[2];\\n\"\"\")\noutCparams = \"preindent=1,outCfileaccess=a,outCverbose=False,includebraces=False\"\noutputC([uibh.gammaSphDD[0][0],uibh.gammaSphDD[0][1],uibh.gammaSphDD[0][2],\n uibh.gammaSphDD[1][1],uibh.gammaSphDD[1][2],uibh.gammaSphDD[2][2],\n uibh.KSphDD[0][0],uibh.KSphDD[0][1],uibh.KSphDD[0][2],\n uibh.KSphDD[1][1],uibh.KSphDD[1][2],uibh.KSphDD[2][2],\n uibh.alphaSph, uibh.betaSphU[0],uibh.betaSphU[1],uibh.betaSphU[2],\n uibh.BSphU[0],uibh.BSphU[1],uibh.BSphU[2]],\n [\"*gammaDD00\",\"*gammaDD01\",\"*gammaDD02\",\"*gammaDD11\",\"*gammaDD12\",\"*gammaDD22\",\n \"*KDD00\",\"*KDD01\",\"*KDD02\",\"*KDD11\",\"*KDD12\",\"*KDD22\",\n \"*alpha\",\"*betaU0\",\"*betaU1\",\"*betaU2\",\"*BU0\",\"*BU1\",\"*BU2\"],\n \"BSSN/\"+ADM_input_function_name+\".h\",params=outCparams)\nwith open(\"BSSN/\"+ADM_input_function_name+\".h\", \"a\") as file:\n file.write(\"}\\n\")\n```\n\n Appended to file \"BSSN/ID_ADM_SphorCart.h\"\n\n\n\n\n# Step 4: Applying Jacobian transformations to get in the correct $xx0,xx1,xx2$ basis \\[Back to [top](#toc)\\]\n$$\\label{adm_jacobian}$$\n\n\nThe following discussion holds for either Spherical or Cartesian input data, so for simplicity let's just assume the data are given in Spherical coordinates.\n\nAll ADM tensors and vectors are in the Spherical coordinate basis $x^i_{\\rm Sph} = (r,\\theta,\\phi)$, but we need them in the curvilinear coordinate basis $x^i_{\\rm rfm}= ({\\rm xx0},{\\rm xx1},{\\rm xx2})$ set by the \"reference_metric::CoordSystem\" variable. Empirically speaking, it is far easier to write $(x({\\rm xx0},{\\rm xx1},{\\rm xx2}),y({\\rm xx0},{\\rm xx1},{\\rm xx2}),z({\\rm xx0},{\\rm xx1},{\\rm xx2}))$ than the inverse, so we will compute the Jacobian matrix\n\n$$\n{\\rm Jac\\_dUSph\\_dDrfmUD[i][j]} = \\frac{\\partial x^i_{\\rm Sph}}{\\partial x^j_{\\rm rfm}},\n$$\n\nvia exact differentiation (courtesy SymPy), and the inverse Jacobian\n$$\n{\\rm Jac\\_dUrfm\\_dDSphUD[i][j]} = \\frac{\\partial x^i_{\\rm rfm}}{\\partial x^j_{\\rm Sph}},\n$$\n\nusing NRPy+'s ${\\rm generic\\_matrix\\_inverter3x3()}$ function. In terms of these, the transformation of BSSN tensors from Spherical to \"reference_metric::CoordSystem\" coordinates may be written:\n\n\\begin{align}\n\\beta^i_{\\rm rfm} &= \\frac{\\partial x^i_{\\rm rfm}}{\\partial x^\\ell_{\\rm Sph}} \\beta^\\ell_{\\rm Sph}\\\\\nB^i_{\\rm rfm} &= \\frac{\\partial x^i_{\\rm rfm}}{\\partial x^\\ell_{\\rm Sph}} B^\\ell_{\\rm Sph}\\\\\n\\gamma^{\\rm rfm}_{ij} &= \n\\frac{\\partial x^\\ell_{\\rm Sph}}{\\partial x^i_{\\rm rfm}}\n\\frac{\\partial x^m_{\\rm Sph}}{\\partial x^j_{\\rm rfm}} \\gamma^{\\rm Sph}_{\\ell m}\\\\\nK^{\\rm rfm}_{ij} &= \n\\frac{\\partial x^\\ell_{\\rm Sph}}{\\partial x^i_{\\rm rfm}}\n\\frac{\\partial x^m_{\\rm Sph}}{\\partial x^j_{\\rm rfm}} K^{\\rm Sph}_{\\ell m}\n\\end{align}\n\n\n```python\n# Step 4: Applying Jacobian transformations to get in the correct $xx0,xx1,xx2$ basis\n\n# Step 4 P0: All input quantities are in terms of r,th,ph or x,y,z. We want them in terms \n# of xx0,xx1,xx2, so here we call sympify_integers__replace_rthph() to replace\n# r,th,ph or x,y,z, respectively, with the appropriate functions of xx0,xx1,xx2\n# as defined for this particular reference metric in reference_metric.py's \n# xxSph[] or xxCart[], respectively:\n\n# Define the input variables:\ngammaSphorCartDD = ixp.declarerank2(\"gammaSphorCartDD\",\"sym01\")\nKSphorCartDD = ixp.declarerank2(\"KSphorCartDD\",\"sym01\")\nalphaSphorCart = sp.symbols(\"alphaSphorCart\")\nbetaSphorCartU = ixp.declarerank1(\"betaSphorCartU\")\nBSphorCartU = ixp.declarerank1(\"BSphorCartU\")\n\n# UIUC Black Hole initial data are given in Spherical coordinates.\nCoordType_in = \"Spherical\"\n\n# Make sure that rfm.reference_metric() has been called.\n# We'll need the variables it defines throughout this module.\nif rfm.have_already_called_reference_metric_function == False:\n print(\"Error. Called Convert_Spherical_ADM_to_BSSN_curvilinear() without\")\n print(\" first setting up reference metric, by calling rfm.reference_metric().\")\n exit(1)\n\nr_th_ph_or_Cart_xyz_oID_xx = []\nif CoordType_in == \"Spherical\":\n r_th_ph_or_Cart_xyz_oID_xx = rfm.xxSph\nelif CoordType_in == \"Cartesian\":\n r_th_ph_or_Cart_xyz_oID_xx = rfm.xxCart\nelse:\n print(\"Error: Can only convert ADM Cartesian or Spherical initial data to BSSN Curvilinear coords.\")\n exit(1)\n\n# Next apply Jacobian transformations to convert into the (xx0,xx1,xx2) basis\n\n# alpha is a scalar, so no Jacobian transformation is necessary.\nalpha = alphaSphorCart\n\nJac_dUSphorCart_dDrfmUD = ixp.zerorank2()\nfor i in range(DIM):\n for j in range(DIM):\n Jac_dUSphorCart_dDrfmUD[i][j] = sp.diff(r_th_ph_or_Cart_xyz_oID_xx[i],rfm.xx[j])\n\nJac_dUrfm_dDSphorCartUD, dummyDET = ixp.generic_matrix_inverter3x3(Jac_dUSphorCart_dDrfmUD)\n\nbetaU = ixp.zerorank1()\nBU = ixp.zerorank1()\ngammaDD = ixp.zerorank2()\nKDD = ixp.zerorank2()\nfor i in range(DIM):\n for j in range(DIM):\n betaU[i] += Jac_dUrfm_dDSphorCartUD[i][j] * betaSphorCartU[j]\n BU[i] += Jac_dUrfm_dDSphorCartUD[i][j] * BSphorCartU[j]\n for k in range(DIM):\n for l in range(DIM):\n gammaDD[i][j] += Jac_dUSphorCart_dDrfmUD[k][i]*Jac_dUSphorCart_dDrfmUD[l][j] * gammaSphorCartDD[k][l]\n KDD[i][j] += Jac_dUSphorCart_dDrfmUD[k][i]*Jac_dUSphorCart_dDrfmUD[l][j] * KSphorCartDD[k][l]\n```\n\n\n\n# Step 5: Perform the ADM-to-BSSN conversion for 3-metric, extrinsic curvature, and gauge quantities \\[Back to [top](#toc)\\]\n$$\\label{adm2bssn}$$\n\nAll ADM quantities were input into this function in the Spherical or Cartesian basis, as functions of r,th,ph or x,y,z, respectively. In Steps 1 and 2 above, we converted them to the xx0,xx1,xx2 basis, and as functions of xx0,xx1,xx2. Here we convert ADM quantities to their BSSN Curvilinear counterparts.\n\n\n\n\n\n## Step 5.a: Convert ADM $\\gamma_{ij}$ to BSSN $\\bar{\\gamma}_{ij}$ \\[Back to [top](#toc)\\]\n$$\\label{adm2bssn_gamma}$$\n\nWe have (Eqs. 2 and 3 of [Ruchlin *et al.*](https://arxiv.org/pdf/1712.07658.pdf)):\n$$\n\\bar{\\gamma}_{i j} = \\left(\\frac{\\bar{\\gamma}}{\\gamma}\\right)^{1/3} \\gamma_{ij},\n$$\nwhere we always make the choice $\\bar{\\gamma} = \\hat{\\gamma}$:\n\n\n```python\n# Step 5.a: Convert ADM $\\gamma_{ij}$ to BSSN $\\bar{\\gamma}_{ij}$\n\ngammaUU, gammaDET = ixp.symm_matrix_inverter3x3(gammaDD)\ngammabarDD = ixp.zerorank2()\nfor i in range(DIM):\n for j in range(DIM):\n gammabarDD[i][j] = (rfm.detgammahat/gammaDET)**(sp.Rational(1,3))*gammaDD[i][j]\n```\n\n\n\n## Step 5.b: Convert the ADM extrinsic curvature $K_{ij}$ \\[Back to [top](#toc)\\]\n$$\\label{admexcurv_convert}$$\n\nConvert the ADM extrinsic curvature $K_{ij}$ to the trace-free extrinsic curvature $\\bar{A}_{ij}$, plus the trace of the extrinsic curvature $K$, where (Eq. 3 of [Baumgarte *et al.*](https://arxiv.org/pdf/1211.6632.pdf)):\n\\begin{align}\nK &= \\gamma^{ij} K_{ij} \\\\\n\\bar{A}_{ij} &= \\left(\\frac{\\bar{\\gamma}}{\\gamma}\\right)^{1/3} \\left(K_{ij} - \\frac{1}{3} \\gamma_{ij} K \\right)\n\\end{align}\n\n\n```python\n# Step 5.b: Convert the ADM extrinsic curvature $K_{ij}$\n\ntrK = sp.sympify(0)\nfor i in range(DIM):\n for j in range(DIM):\n trK += gammaUU[i][j]*KDD[i][j]\n\nAbarDD = ixp.zerorank2()\nfor i in range(DIM):\n for j in range(DIM):\n AbarDD[i][j] = (rfm.detgammahat/gammaDET)**(sp.Rational(1,3))*(KDD[i][j] - sp.Rational(1,3)*gammaDD[i][j]*trK)\n```\n\n\n\n## Step 5.c: Define the conformal factor variable $\\texttt{cf}$ \\[Back to [top](#toc)\\]\n$$\\label{conformal}$$\n\nWe define the conformal factor variable $\\texttt{cf}$ based on the setting of the \"BSSN_quantities::EvolvedConformalFactor_cf\" parameter.\n\nFor example if \"EvolvedConformalFactor_cf\" is set to \"phi\", we can use Eq. 3 of [Ruchlin *et al.*](https://arxiv.org/pdf/1712.07658.pdf), which in arbitrary coordinates is written:\n\n$$\n\\phi = \\frac{1}{12} \\log\\left(\\frac{\\gamma}{\\bar{\\gamma}}\\right).\n$$\n\nAlternatively if \"BSSN_RHSs::EvolvedConformalFactor_cf\" is set to \"chi\", then\n$$\n\\chi = e^{-4 \\phi} = \\exp\\left(-4 \\frac{1}{12} \\left(\\frac{\\gamma}{\\bar{\\gamma}}\\right)\\right) \n= \\exp\\left(-\\frac{1}{3} \\log\\left(\\frac{\\gamma}{\\bar{\\gamma}}\\right)\\right) = \\left(\\frac{\\gamma}{\\bar{\\gamma}}\\right)^{-1/3}.\n$$\n\nFinally if \"BSSN_RHSs::EvolvedConformalFactor_cf\" is set to \"W\", then\n$$\nW = e^{-2 \\phi} = \\exp\\left(-2 \\frac{1}{12} \\log\\left(\\frac{\\gamma}{\\bar{\\gamma}}\\right)\\right) = \n\\exp\\left(-\\frac{1}{6} \\log\\left(\\frac{\\gamma}{\\bar{\\gamma}}\\right)\\right) = \n\\left(\\frac{\\gamma}{\\bar{\\gamma}}\\right)^{-1/6}.\n$$\n\n\n```python\n# Step 5.c: Define the conformal factor variable $\\texttt{cf}$\n\n# First compute gammabarDET:\ngammabarUU, gammabarDET = ixp.symm_matrix_inverter3x3(gammabarDD)\n\ncf = sp.sympify(0)\n\nif par.parval_from_str(\"EvolvedConformalFactor_cf\") == \"phi\":\n cf = sp.Rational(1,12)*sp.log(gammaDET/gammabarDET)\nelif par.parval_from_str(\"EvolvedConformalFactor_cf\") == \"chi\":\n cf = (gammaDET/gammabarDET)**(-sp.Rational(1,3))\nelif par.parval_from_str(\"EvolvedConformalFactor_cf\") == \"W\":\n cf = (gammaDET/gammabarDET)**(-sp.Rational(1,6))\nelse:\n print(\"Error EvolvedConformalFactor_cf type = \\\"\"+par.parval_from_str(\"EvolvedConformalFactor_cf\")+\"\\\" unknown.\")\n exit(1)\n```\n\n\n\n# Step 6: Rescale tensorial quantities \\[Back to [top](#toc)\\]\n$$\\label{rescale}$$\n\nWe rescale tensorial quantities according to the prescription described in the [BSSN in curvilinear coordinates tutorial module](Tutorial-BSSNCurvilinear.ipynb) (also [Ruchlin *et al.*](https://arxiv.org/pdf/1712.07658.pdf)):\n\n\\begin{align}\nh_{ij} &= (\\bar{\\gamma}_{ij} - \\hat{\\gamma}_{ij})/\\text{ReDD[i][j]}\\\\\na_{ij} &= \\bar{A}_{ij}/\\text{ReDD[i][j]}\\\\\n\\mathcal{V}^i &= \\beta^i/\\text{ReU[i]}\\\\\n\\mathcal{B}^i &= B^i/\\text{ReU[i]}\\\\\n\\end{align}\n\n\n```python\n# Step 6: Rescale tensorial quantities\n\nhDD = ixp.zerorank2()\naDD = ixp.zerorank2()\nvetU = ixp.zerorank1()\nbetU = ixp.zerorank1()\nfor i in range(DIM):\n vetU[i] = betaU[i] / rfm.ReU[i]\n betU[i] = BU[i] / rfm.ReU[i]\n for j in range(DIM):\n hDD[i][j] = (gammabarDD[i][j] - rfm.ghatDD[i][j]) / rfm.ReDD[i][j]\n aDD[i][j] = AbarDD[i][j] / rfm.ReDD[i][j]\n```\n\n\n\n# Step 7: Output all ADM-to-BSSN expressions to a C function \\[Back to [top](#toc)\\]\n$$\\label{adm2bssn_c}$$\n\nThis function must first call the ID_ADM_SphorCart() defined above. Using these Spherical or Cartesian data, it sets up all quantities needed for BSSNCurvilinear initial data, *except* $\\lambda^i$, which must be computed from numerical data using finite-difference derivatives.\n\n\n```python\n# Step 7: Output all ADM-to-BSSN expressions to a C function \n\nwith open(\"BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h\", \"w\") as file:\n file.write(\"void ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs(const REAL xx0xx1xx2[3],\")\n if pointer_to_ID_inputs == True:\n file.write(\"ID_inputs *other_inputs,\")\n else:\n file.write(\"ID_inputs other_inputs,\")\n file.write(\"\"\"\n REAL *hDD00,REAL *hDD01,REAL *hDD02,REAL *hDD11,REAL *hDD12,REAL *hDD22,\n REAL *aDD00,REAL *aDD01,REAL *aDD02,REAL *aDD11,REAL *aDD12,REAL *aDD22,\n REAL *trK, \n REAL *vetU0,REAL *vetU1,REAL *vetU2,\n REAL *betU0,REAL *betU1,REAL *betU2,\n REAL *alpha, REAL *cf) {\n REAL gammaSphorCartDD00,gammaSphorCartDD01,gammaSphorCartDD02,\n gammaSphorCartDD11,gammaSphorCartDD12,gammaSphorCartDD22;\n REAL KSphorCartDD00,KSphorCartDD01,KSphorCartDD02,\n KSphorCartDD11,KSphorCartDD12,KSphorCartDD22;\n REAL alphaSphorCart,betaSphorCartU0,betaSphorCartU1,betaSphorCartU2;\n REAL BSphorCartU0,BSphorCartU1,BSphorCartU2;\n const REAL xx0 = xx0xx1xx2[0];\n const REAL xx1 = xx0xx1xx2[1];\n const REAL xx2 = xx0xx1xx2[2];\n REAL xyz_or_rthph[3];\\n\"\"\")\noutCparams = \"preindent=1,outCfileaccess=a,outCverbose=False,includebraces=False\"\noutputC(r_th_ph_or_Cart_xyz_oID_xx[0:3],[\"xyz_or_rthph[0]\",\"xyz_or_rthph[1]\",\"xyz_or_rthph[2]\"],\n \"BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h\",outCparams+\",CSE_enable=False\")\nwith open(\"BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h\", \"a\") as file:\n file.write(\" \"+ADM_input_function_name+\"\"\"(xyz_or_rthph, other_inputs,\n &gammaSphorCartDD00,&gammaSphorCartDD01,&gammaSphorCartDD02,\n &gammaSphorCartDD11,&gammaSphorCartDD12,&gammaSphorCartDD22,\n &KSphorCartDD00,&KSphorCartDD01,&KSphorCartDD02,\n &KSphorCartDD11,&KSphorCartDD12,&KSphorCartDD22,\n &alphaSphorCart,&betaSphorCartU0,&betaSphorCartU1,&betaSphorCartU2,\n &BSphorCartU0,&BSphorCartU1,&BSphorCartU2);\n // Next compute all rescaled BSSN curvilinear quantities:\\n\"\"\")\noutCparams = \"preindent=1,outCfileaccess=a,outCverbose=False,includebraces=False\"\noutputC([hDD[0][0],hDD[0][1],hDD[0][2],hDD[1][1],hDD[1][2],hDD[2][2],\n aDD[0][0],aDD[0][1],aDD[0][2],aDD[1][1],aDD[1][2],aDD[2][2],\n trK, vetU[0],vetU[1],vetU[2], betU[0],betU[1],betU[2],\n alpha, cf], \n [\"*hDD00\",\"*hDD01\",\"*hDD02\",\"*hDD11\",\"*hDD12\",\"*hDD22\",\n \"*aDD00\",\"*aDD01\",\"*aDD02\",\"*aDD11\",\"*aDD12\",\"*aDD22\",\n \"*trK\", \"*vetU0\",\"*vetU1\",\"*vetU2\", \"*betU0\",\"*betU1\",\"*betU2\",\n \"*alpha\",\"*cf\"],\n \"BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h\",params=outCparams)\nwith open(\"BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h\", \"a\") as file:\n file.write(\"}\\n\")\n```\n\n Appended to file \"BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h\"\n Appended to file \"BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h\"\n\n\n\n\n## Step 7.a: Output the driver function for the above C function \\[Back to [top](#toc)\\]\n$$\\label{driver}$$\n\nWe output the driver function for the above C function:\n`ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs()`\n\n\n```python\n# Step 7.a: Output the driver function for the above C function\n\n# Next write the driver function for ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs():\nwith open(\"BSSN/ID_BSSN__ALL_BUT_LAMBDAs.h\", \"w\") as file:\n file.write(\"void ID_BSSN__ALL_BUT_LAMBDAs(const int Nxx_plus_2NGHOSTS[3],REAL *xx[3],\")\n if pointer_to_ID_inputs == True:\n file.write(\"ID_inputs *other_inputs,\")\n else:\n file.write(\"ID_inputs other_inputs,\")\n file.write(\"REAL *in_gfs) {\\n\")\n file.write(lp.loop([\"i2\", \"i1\", \"i0\"], [\"0\", \"0\", \"0\"],\n [\"Nxx_plus_2NGHOSTS[2]\", \"Nxx_plus_2NGHOSTS[1]\", \"Nxx_plus_2NGHOSTS[0]\"],\n [\"1\", \"1\", \"1\"], [\"#pragma omp parallel for\",\n \" const REAL xx2 = xx[2][i2];\",\n \" const REAL xx1 = xx[1][i1];\"], \"\",\n \"\"\"const REAL xx0 = xx[0][i0];\nconst int idx = IDX3(i0,i1,i2);\nconst REAL xx0xx1xx2[3] = {xx0,xx1,xx2};\nID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs(xx0xx1xx2,other_inputs,\n &in_gfs[IDX4pt(HDD00GF,idx)],&in_gfs[IDX4pt(HDD01GF,idx)],&in_gfs[IDX4pt(HDD02GF,idx)],\n &in_gfs[IDX4pt(HDD11GF,idx)],&in_gfs[IDX4pt(HDD12GF,idx)],&in_gfs[IDX4pt(HDD22GF,idx)],\n &in_gfs[IDX4pt(ADD00GF,idx)],&in_gfs[IDX4pt(ADD01GF,idx)],&in_gfs[IDX4pt(ADD02GF,idx)],\n &in_gfs[IDX4pt(ADD11GF,idx)],&in_gfs[IDX4pt(ADD12GF,idx)],&in_gfs[IDX4pt(ADD22GF,idx)],\n &in_gfs[IDX4pt(TRKGF,idx)],\n &in_gfs[IDX4pt(VETU0GF,idx)],&in_gfs[IDX4pt(VETU1GF,idx)],&in_gfs[IDX4pt(VETU2GF,idx)],\n &in_gfs[IDX4pt(BETU0GF,idx)],&in_gfs[IDX4pt(BETU1GF,idx)],&in_gfs[IDX4pt(BETU2GF,idx)],\n &in_gfs[IDX4pt(ALPHAGF,idx)],&in_gfs[IDX4pt(CFGF,idx)]);\n\"\"\"))\n file.write(\"}\\n\")\n```\n\n\n\n# Step 8: Compute $\\bar{\\Lambda}^i$ from finite-difference derivatives of rescaled metric quantities \\[Back to [top](#toc)\\]\n$$\\label{lambda}$$\n\nWe compute $\\bar{\\Lambda}^i$ (Eqs. 4 and 5 of [Baumgarte *et al.*](https://arxiv.org/pdf/1211.6632.pdf)), from finite-difference derivatives of rescaled metric quantities $h_{ij}$:\n\n$$\n\\bar{\\Lambda}^i = \\bar{\\gamma}^{jk}\\left(\\bar{\\Gamma}^i_{jk} - \\hat{\\Gamma}^i_{jk}\\right).\n$$\n\nThe [reference_metric.py](../edit/reference_metric.py) module provides us with analytic expressions for $\\hat{\\Gamma}^i_{jk}$, so here we need only compute finite-difference expressions for $\\bar{\\Gamma}^i_{jk}$, based on the values for $h_{ij}$ provided in the initial data. Once $\\bar{\\Lambda}^i$ has been computed, we apply the usual rescaling procedure:\n$$\n\\lambda^i = \\bar{\\Lambda}^i/\\text{ReU[i]},\n$$\nand then output the result to a C file using the NRPy+ finite-difference C output routine.\n\n\n```python\n# Step 8: Compute $\\bar{\\Lambda}^i$ from finite-difference derivatives of rescaled metric quantities\n\n# We will need all BSSN gridfunctions to be defined, as well as \n# expressions for gammabarDD_dD in terms of exact derivatives of \n# the rescaling matrix and finite-difference derivatives of\n# hDD's. This functionality is provided by BSSN.BSSN_unrescaled_and_barred_vars,\n# which we call here to overwrite above definitions of gammabarDD,gammabarUU, etc.\nBq.gammabar__inverse_and_derivs() # Provides gammabarUU and GammabarUDD\ngammabarUU = Bq.gammabarUU\nGammabarUDD = Bq.GammabarUDD\n\n# Next evaluate \\bar{\\Lambda}^i, based on GammabarUDD above and GammahatUDD \n# (from the reference metric):\nLambdabarU = ixp.zerorank1()\nfor i in range(DIM):\n for j in range(DIM):\n for k in range(DIM):\n LambdabarU[i] += gammabarUU[j][k] * (GammabarUDD[i][j][k] - rfm.GammahatUDD[i][j][k])\n\n# Finally apply rescaling:\n# lambda^i = Lambdabar^i/\\text{ReU[i]}\nlambdaU = ixp.zerorank1()\nfor i in range(DIM):\n lambdaU[i] = LambdabarU[i] / rfm.ReU[i]\n\noutCparams = \"preindent=1,outCfileaccess=a,outCverbose=False,includebraces=False\"\nlambdaU_expressions = [lhrh(lhs=gri.gfaccess(\"in_gfs\",\"lambdaU0\"),rhs=lambdaU[0]),\n lhrh(lhs=gri.gfaccess(\"in_gfs\",\"lambdaU1\"),rhs=lambdaU[1]),\n lhrh(lhs=gri.gfaccess(\"in_gfs\",\"lambdaU2\"),rhs=lambdaU[2])]\nlambdaU_expressions_FDout = fin.FD_outputC(\"returnstring\",lambdaU_expressions, outCparams)\n\nwith open(\"BSSN/ID_BSSN_lambdas.h\", \"w\") as file:\n file.write(\"\"\"\nvoid ID_BSSN_lambdas(const int Nxx[3],const int Nxx_plus_2NGHOSTS[3],REAL *xx[3],const REAL dxx[3],REAL *in_gfs) {\\n\"\"\")\n file.write(lp.loop([\"i2\",\"i1\",\"i0\"],[\"NGHOSTS\",\"NGHOSTS\",\"NGHOSTS\"],\n [\"NGHOSTS+Nxx[2]\",\"NGHOSTS+Nxx[1]\",\"NGHOSTS+Nxx[0]\"],\n [\"1\",\"1\",\"1\"],[\"const REAL invdx0 = 1.0/dxx[0];\\n\"+\n \"const REAL invdx1 = 1.0/dxx[1];\\n\"+\n \"const REAL invdx2 = 1.0/dxx[2];\\n\"+\n \"#pragma omp parallel for\",\n \" const REAL xx2 = xx[2][i2];\",\n \" const REAL xx1 = xx[1][i1];\"],\"\",\n \"const REAL xx0 = xx[0][i0];\\n\"+lambdaU_expressions_FDout))\n file.write(\"}\\n\")\n```\n\n\n\n# Step 9: Code Validation against BSSN.ADM_Numerical_Spherical_or_Cartesian_to_BSSNCurvilinear NRPy+ module \\[Back to [top](#toc)\\]\n$$\\label{code_validation}$$\n\nHere, as a code validation check, we verify agreement in the C codes for converting \"numerical\" UIUCBlackHole initial data (in Spherical coordinates/basis) to BSSN Curvilinear data in Cylindrical coordinates/basis between\n1. this tutorial and \n2. the NRPy+ [BSSN.ADM_Numerical_Spherical_or_Cartesian_to_BSSNCurvilinear](../edit/BSSN/ADM_Numerical_Spherical_or_Cartesian_to_BSSNCurvilinear.py) module.\n\nBy default, we analyze these expressions in Cylindrical coordinates, though other coordinate systems may be chosen.\n\n\n```python\n\n!mv BSSN/ID_BSSN_lambdas.h BSSN/ID_BSSN_lambdas.h-ADM_Num_ID_validation\n!mv BSSN/ID_BSSN__ALL_BUT_LAMBDAs.h BSSN/ID_BSSN__ALL_BUT_LAMBDAs.h-ADM_Num_ID_validation\n!mv BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h-ADM_Num_ID_validation\n\n# Reset the gridfunctions list; \n# in Convert_Spherical_or_Cartesian_ADM_to_BSSN_curvilinear()\n# below, BSSN_RHSs is called\n# tutorial. This line of code enables us to run\n# Convert_Spherical_or_Cartesian_ADM_to_BSSN_curvilinear()\n# without resetting the running Python kernel.\ngri.glb_gridfcs_list = []\n\nimport BSSN.ADM_Numerical_Spherical_or_Cartesian_to_BSSNCurvilinear as AtoBnum\nAtoBnum.Convert_Spherical_or_Cartesian_ADM_to_BSSN_curvilinear(\"Spherical\",ADM_input_function_name)\n\nprint(\"\\n\\n ### BEGIN VALIDATION TESTS\")\nimport filecmp\nfor file in [\"BSSN/ID_BSSN_lambdas.h\",\"BSSN/ID_BSSN__ALL_BUT_LAMBDAs.h\",\n \"BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h\",\"BSSN/\"+ADM_input_function_name+\".h\"]:\n if filecmp.cmp(file,file+\"-ADM_Num_ID_validation\") == False:\n print(\"VALIDATION TEST FAILED ON file: \"+file+\".\")\n exit(1)\n else:\n print(\"Validation test PASSED on file: \"+file)\n```\n\n Appended to file \"BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h\"\n Appended to file \"BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h\"\n \n \n ### BEGIN VALIDATION TESTS\n Validation test PASSED on file: BSSN/ID_BSSN_lambdas.h\n Validation test PASSED on file: BSSN/ID_BSSN__ALL_BUT_LAMBDAs.h\n Validation test PASSED on file: BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h\n Validation test PASSED on file: BSSN/ID_ADM_SphorCart.h\n\n\n\n\n# Step 10: Output this module to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n$$\\label{latex_pdf_output}$$\n\nOnce the following code finishes running, the generated PDF may be found at the following location within the directory you have the NRPy+ tutorial saved:\n[Tutorial-ADM_Initial_Data-Converting_Numerical_ADM_Spherical_or_Cartesian_to_BSSNCurvilinear.pdf](Tutorial-ADM_Initial_Data-Converting_Numerical_ADM_Spherical_or_Cartesian_to_BSSNCurvilinear.pdf)\n\n\n```python\n!jupyter nbconvert --to latex --template latex_nrpy_style.tplx Tutorial-ADM_Initial_Data-Converting_Numerical_ADM_Spherical_or_Cartesian_to_BSSNCurvilinear.ipynb\n!pdflatex -interaction=batchmode Tutorial-ADM_Initial_Data-Converting_Numerical_ADM_Spherical_or_Cartesian_to_BSSNCurvilinear.tex\n!pdflatex -interaction=batchmode Tutorial-ADM_Initial_Data-Converting_Numerical_ADM_Spherical_or_Cartesian_to_BSSNCurvilinear.tex\n!pdflatex -interaction=batchmode Tutorial-ADM_Initial_Data-Converting_Numerical_ADM_Spherical_or_Cartesian_to_BSSNCurvilinear.tex\n!rm -f Tut*.out Tut*.aux Tut*.log\n```\n\n [NbConvertApp] Converting notebook Tutorial-ADM_Initial_Data-Converting_Numerical_ADM_Spherical_or_Cartesian_to_BSSNCurvilinear.ipynb to latex\n [NbConvertApp] Writing 86695 bytes to Tutorial-ADM_Initial_Data-Converting_Numerical_ADM_Spherical_or_Cartesian_to_BSSNCurvilinear.tex\n This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex)\n restricted \\write18 enabled.\n entering extended mode\n This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex)\n restricted \\write18 enabled.\n entering extended mode\n This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex)\n restricted \\write18 enabled.\n entering extended mode\n\n", "meta": {"hexsha": "97347cc479a09a874c6a023eed373e0372addb74", "size": 38633, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Tutorial-ADM_Initial_Data-Converting_Numerical_ADM_Spherical_or_Cartesian_to_BSSNCurvilinear.ipynb", "max_stars_repo_name": "Lituchy/nrpyunittesting", "max_stars_repo_head_hexsha": "c611891a2f6f07afd762121e59d2c5ab857d925b", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Tutorial-ADM_Initial_Data-Converting_Numerical_ADM_Spherical_or_Cartesian_to_BSSNCurvilinear.ipynb", "max_issues_repo_name": "Lituchy/nrpyunittesting", "max_issues_repo_head_hexsha": "c611891a2f6f07afd762121e59d2c5ab857d925b", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Tutorial-ADM_Initial_Data-Converting_Numerical_ADM_Spherical_or_Cartesian_to_BSSNCurvilinear.ipynb", "max_forks_repo_name": "Lituchy/nrpyunittesting", "max_forks_repo_head_hexsha": "c611891a2f6f07afd762121e59d2c5ab857d925b", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 48.9025316456, "max_line_length": 660, "alphanum_fraction": 0.5884347578, "converted": true, "num_tokens": 10017, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031737963569016, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.4172659124406186}} {"text": "```python\n# [Nur Colab] Diese Zellen m\u00fcssen nur auf *Google Colab* ausgef\u00fchrt werden und installieren Packete und Daten\n!wget -q https://raw.githubusercontent.com/KI-Campus/AMALEA/master/requirements.txt && pip install --quiet -r requirements.txt\n!wget --quiet \"https://github.com/KI-Campus/AMALEA/releases/download/data/data.zip\" && unzip -q data.zip\n```\n\n# 100% Genauigkeit, das muss doch gut sein, oder?\n\n## Leistungsmetriken f\u00fcr die Regression\n\n\nMetriken sind wichtig, weil wir den maschinellen Lernalgorithmus w\u00e4hrend des Trainings st\u00e4ndig ver\u00e4ndern. Wir \"r\u00fchren den Haufen um, bis die Ergebnisse richtig aussehen\".\n\n\nEin Modell ist eine vereinfachte Darstellung der Realit\u00e4t. Dazu m\u00fcssen einige unn\u00f6tige Details entfernt werden, damit sich bei der Interpretation der Daten auf die entscheidenderen Aspekte konzentriert werden kann. \nDiese Vereinfachung basiert auf Annahmen dar\u00fcber, was in den Daten wichtig ist und was ignoriert werden kann. Annahmen, die bei einer Art von Problem funktionieren, funktionieren jedoch nicht bei einer anderen Art von Problem.\nDas \"No Free Lunch Theorem\" besagt, dass es kein Modell geben kann, das f\u00fcr alle Arten von Problemen die beste Performanz erbringt. Daher ist es beim maschinellen Lernen \u00fcblich, verschiedene Modelle auszuprobieren und das Modell mit den besten Ergebnissen auszuw\u00e4hlen. Die Auswahl des Modells mit der \"besten Performanz\" klingt einfach, aber es zeigt sich, dass es viele verschiedene Interpretationen einer \"guten Performanz\" gibt.\nWenn jemand zu Ihnen sagt \"\u00d6ttinger ist das beste Bier\", sollte die erste Frage lauten: Auf welcher Grundlage wird diese Aussage getroffen? Ist der Geschmack, das Styling der Flasche oder die Schaumkonsistenz relevant? \u00c4hnlich m\u00fcssen die Metriken, die f\u00fcr Machine-Learning-Modelle verwendet werden, spezifisch f\u00fcr das gegebene Problem, den Datensatz und die Strategie ausgew\u00e4hlt werden. Daher ist es wichtig, den Kontext zu verstehen, bevor man eine Metrik ausw\u00e4hlt. In diesem Kapitel werden wir uns auf die Metriken konzentrieren, die f\u00fcr Klassifikations- und Regressionsprobleme verwendet werden. \n\n\n\n### Regressionsmetriken\nUm die Regressionsmetriken zu verstehen, soll der \"Boston House Price\" Datensatz als Beispiel verwendet werden. Hierbei handelt es sich um ein Regressionsproblem, bei dem alle Eingabevariablen ebenfalls numerisch sind. Die wichtigsten Metriken in der Regression sind __Verlustfunktionen__.\nWenn eine Funktion an Datenpunkte angepasst wird (engl. fitting), wird die Differenz zwischen den tats\u00e4chlichen Datenpunkten und der Ausgabe der Vorhersagefunktion f\u00fcr diese Punkte verwendet und weiterverarbeitet.\n\nZum Beispiel wird in einer 2-D-Ebene der vertikale Abstand zwischen jedem tats\u00e4chlichen Datenpunkt $(x_n|y_n)$ und der Ausgabe der Modellfunktion f\u00fcr diese Punkte $m(x_n)$ durch Subtraktion der y-Werte der Datenpunkte von den vorhergesagten y-Werten der Modellfunktion berechnet (3).\n\n\\begin{align}\nd_n & = m(x_n)-y_n \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; (3)\n\\end{align}\n\n\n\nJe mehr die Modellfunktion von den gegebenen Datenpunkten abweicht, desto h\u00f6her f\u00e4llt der Verlust aus. Die Idee dabei ist, dass ein hoher Verlust auf eine schlechte Modellfunktion hindeutet. W\u00e4hrend ein niedriger Verlust auf eine gute Modellfunktion schlie\u00dfen l\u00e4sst. Allgemein versucht ein Trainingsalgorithmus, die Modellfunktion so zu ver\u00e4ndern, dass der Verlust minimiert wird.\nIn den meisten F\u00e4llen ist dieser Ansatz sinnvoll. Wie man aber im Fall von Overfitting sehen kann, ist das Modell mit dem niedrigsten Verlust m\u00f6glicherweise nicht immer das beste Modell, um neue Daten vorherzusagen.\n\n#### Mittlerer Absoluter Fehler (MAE)\n\nDer mittlere absolute Fehler (Mean Absolute Error, MAE) ist eine sehr einfache Verlustfunktion. Sie stellt die Abweichung der vorhergesagten Werten von den erwarteten Werten dar und gibt somit eine Vorstellung davon, wie falsch die Vorhersagen waren. Der MAE wird als Durchschnitt der absoluten Fehlerwerte berechnet, wobei absolut bedeutet, dass die Fehler stehts positiv sind und diese somit einfach addiert werden k\u00f6nnen.\n\nNicht verwechselt werden sollte der MAE mit der durchschnittlichen absoluten Abweichung\n\n\nDer MAE kann berechnet werden mit: \n\n$ \\large MAE = \\frac{\\sum_{i=0}^{n-1}\\left | e_{i}\\right |}{n} = \\frac{\\sum_{i=0}^{n-1}\\left | y_{i} - m(x_{i})\\right |}{n} = \\frac{\\sum_{i=o}^{n-1}\\left | ground\\_truth_{i} - predicted_{i} \\right |}{total predictions} $\n\n\n\n\n\nOder mit Worten ausgedr\u00fcckt, er ist die Summe der Abst\u00e4nde zwischen dem vorhergesagten Wert und dem wahren Wert, geteilt durch die Anzahl der Werte.\n\nDer mittlere absolute Fehler verwendet die gleiche Skala wie die zu messenden Daten. Er wird als skalenabh\u00e4ngiges Genauigkeitsma\u00df bezeichnet und kann daher nicht zum Vergleich zwischen Messreihen mit unterschiedlichen Skalen verwendet werden.\nEin Wert von 0 bedeutet, dass kein Fehler vorliegt, also eine perfekte Vorhersage getroffen wurde.\nDer n\u00e4chstehende Codeblock stellt die MAE auf interaktive Weise dar.\n(Siehe auch: https://en.wikipedia.org/wiki/Mean_absolute_error)\n\n\n```python\nimport matplotlib.pyplot as plt\n\n%matplotlib inline\nfrom ipywidgets import interactive\n\n# Example of Calculating Mean Absolute Error\n\n# Calculate mean absolute error\ndef mae_metric(actual: list, predicted: list):\n sum_error = 0.0\n for i in range(len(actual)):\n sum_error += abs(predicted[i] - actual[i])\n return sum_error / float(len(actual))\n\n# Test RMSE\nactual = [0.1, 0.2, 0.3, 0.4, 0.5]\npredicted = [0.11, 0.19, 0.29, 0.41, 0.5]\n\nprint(\"MAE = \", mae_metric(actual, predicted))\n\n#Now we make a plot of the values\nEje_X = [0, 1, 2, 3, 4]\n\ndef f2(pre0:float=0.11, pre1:float=0.19, pre2:float=0.29, pre3:float=0.41, pre4:float=0.5):\n predicted = [pre0, pre1, pre2, pre3, pre4]\n plt.figure(num=None, figsize=(7, 7), dpi=100, facecolor='w', edgecolor='k')\n actual_plt = plt.scatter(Eje_X, actual)\n predicted_plt = plt.scatter(Eje_X, predicted)\n \n #The following section draws the line between actual and predicted values.\n for x in range(len(actual)):\n plt.plot([x, x], [actual[x], predicted[x]], color = 'g')\n \n plt.ylim(0, 0.6)\n \n plt.legend((actual_plt, predicted_plt), ('Actual', 'Predicted'), loc='lower right')\n plt.show()\n \n \ninteractive_plot = interactive(f2, \n pre0=(0.0, 0.6, 0.01), \n pre1=(0.0, 0.6, 0.01), \n pre2=(0.0, 0.6, 0.01), \n pre3=(0.0, 0.6, 0.01), \n pre4=(0.0, 0.6, 0.01))\noutput = interactive_plot.children[-1]\ninteractive_plot\n```\n\nDie gr\u00fcnen Linien stellen den Abstand zwischen dem vorhergesagten und dem tats\u00e4chlichen Wert dar. Aufsummiert und durch n geteilt, ergeben diese Abst\u00e4nde den MAE.\n\n
                                        \nAufgabe 2.2.1: Erstellen Sie ein lineares Regressionsmodell, um den Wert der H\u00e4user zu sch\u00e4tzen. Berechnen Sie anschlie\u00dfend den MAE jedes trainierten Modells. Berechnen Sie zudem den Mittelwert sowie die Standardabweichung aller Modelle. Beachten Sie die folgenden Anweisungen:\n\n* _Datensatz:_ housings.csv im Dataset-Ordner https://www.kaggle.com/c/boston-housing\n* _Label:_ housings.csv im Daten-Ordner\n* _Resampling-Methode:_ k-fold, `n_splits` = 10, `random_state` = 7 und `shuffle` = True. (https://scikit-learn.org/stable/modules/generated/sklearn.model_selection.KFold.html)\n* _Modell:_ Lineare Regression\n* _Metrik:_ MAE, Hinweis: Verwenden Sie die Methode: cross_val_score (https://scikit-learn.org/stable/modules/generated/sklearn.model_selection.cross_val_score.html) und den Scoring-Parameter. Eine Liste aller m\u00f6glichen Scoring-Parameter finden Sie hier: https://scikit-learn.org/stable/modules/model_evaluation.html\n\n
                                        \n\n\n```python\n# Cross Validation Regression MAE\nfrom pandas import read_csv\nfrom sklearn.model_selection import KFold\nfrom sklearn.model_selection import cross_val_score\nfrom sklearn.linear_model import LinearRegression\n\n#STUDENT CODE HERE\n\n#STUDENT CODE until HERE\ndataframe = read_csv(filename, delim_whitespace=True)\n\n#STUDENT CODE HERE\n\n#STUDENT CODE until HERE\nresults = cross_val_score(model, X, Y, cv=kfold, scoring=scoring)\n\nprint(\"MAE: %.3f (%.3f)\" % (-results.mean(), results.std()))\n\n```\n\n
                                        \nFrage 2.2.2: Wie gro\u00df ist der MAE und die Standardabweichung des MAE.\n
                                        \n\n
                                        \n\nIhre Antwort:
                                        \n\n\n
                                        \nFrage 2.2.3: Entsprechend dieser MAE-Metrik: Sind die Vorhersagen gut oder schlecht? Wie kann man das allgemein erkennen?\n
                                        \n\n
                                        \n\nIhre Antwort:
                                        \n\n\n#### Mittlerer Quadratischer Fehler (MSE)\n\nDer mittlere quadratische Fehler (MSE) und der mittlere quadratische Wurzelfehler (RMSE) sind zwei Metriken, die in enger Beziehung zueinander stehen.\n\nDer Ausdruck f\u00fcr MSE ist:\n\n$ MSE = \\frac{\\sum_{i=0}^{n-1} \\left ( predicted_{i} - ground\\_truth_{i} \\right )^{2} }{allpredictions} $\n\n\n\nDer mittlere quadratische Fehler (oder MSE) ist dem mittleren absoluten Fehler \u00e4hnlich, da er eine grobe Vorstellung von der Gr\u00f6\u00dfe des Fehlers gibt. \n\nBei der Berechnung des MSE f\u00fchrt die Quadrierung jeden Fehlers dazu, dass die Werte ebenfalls positiv sind. Der RMSE ist einfach die Quadratwurzel des MSE.\n\n\n\n
                                        \nAufgabe 2.2.4: In der nachfolgenden Zelle sind einige Daten mit dem Namen data und die Ausgabe von vier Modellen data_model1, data_model2, data_model3 and data_model4 gegeben. Berechnen Sie den MSE f\u00fcr alle Modellausgaben und speichern Sie ihn in der Variablen mse_model1 f\u00fcr Modell 1, mse_model2 f\u00fcr Modell 2, mse_model3 f\u00fcr Modell 3 und mse_model4 f\u00fcr Modell 4.\n
                                        \n\n\n```python\n# generate some data\nimport math\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nrs = np.random.RandomState(1)\nnum_samples = 1001\nupper_boundary = math.pi\nlower_boundary = 0.\nx_values = np.linspace(0, upper_boundary, num_samples)\ndata = [math.sin(x)*10 for x in x_values] + rs.normal(0, 1, num_samples)\ndata_model1 = [sum(x)*10 for x in zip([math.sin(x) for x in x_values], rs.normal(0, 10**(-2), num_samples))]\ndata_model2 = [np.mean(data) for _ in x_values]\nmedian_noise = data[np.where(x_values == np.median(x_values))]\ndata_model3 = np.concatenate((np.linspace(0, median_noise, int(num_samples/2)), np.linspace(median_noise, 0., int(num_samples/2)+1))).flatten()\n\nval1 = data[int(len(data)/3)]\nval2 = data[int(len(data)/3)]\ndata_model4 = np.concatenate((\n np.linspace(0, val1, int(len(data)/3)),\n np.linspace(val1, val2, int(len(data)/3)),\n np.linspace(val2, 0., len(data)-int(len(data)/3) - int(len(data)/3))\n))\n\nplt.figure(figsize=(10,5))\nplt.plot(x_values, data, marker='o', linestyle='None', label='Data', c='C0')\nplt.plot(x_values, data_model1, label='Model 1', c='lime')\nplt.plot(x_values, data_model2, label='Model 2', c='sienna')\nplt.plot(x_values, data_model3, label='Model 3', c='m')\nplt.plot(x_values, data_model4, label='Model 4', c='k')\nplt.legend()\nplt.grid()\nplt.show()\n\n#STUDENT CODE HERE\n\n#STUDENT CODE until HERE\n\nprint('MSE of model1: {:7.4f}'.format(mse_model1))\nprint('MSE of model2: {:7.4f}'.format(mse_model2))\nprint('MSE of model3: {:7.4f}'.format(mse_model3))\nprint('MSE of model4: {:7.4f}'.format(mse_model4))\n```\n\n#### MAE vs. MSE und RMSE\n\nSowohl MAE als auch RMSE dr\u00fccken den durchschnittlichen Modellvorhersagefehler in Einheiten der betrachteten Variable aus. Beide Metriken k\u00f6nnen von 0 bis \u221e reichen und sind unabh\u00e4ngig von der Fehlerrichtung. Sie sind negativ orientierte Scores, was bedeutet, dass niedrigere Werte besser sind.\n\nDer RMSE gibt gro\u00dfen Fehlern ein relativ hohes Gewicht. Das bedeutet, dass der RMSE eher n\u00fctzlich sein sollte, wenn gro\u00dfe Fehler besonders unerw\u00fcnscht sind.\n\nLassen Sie uns die obige Aussage anhand von zwei Beispielen verstehen:\n\n__Fall 1:__ Tats\u00e4chliche Werte = [2,4,6,8] , Vorhersage = [3,5,7,9]\n\n__Fall 2:__ Tats\u00e4chliche Werte = [2,4,6,8] , Vorhersage = [3,5,7,11]\n\nBez\u00fcglich __Fall 1:__ __MAE__ = 1,0; __MSE__ = 1,0\n\nZu __Fall 2:__ __MAE__ = 1,5; __MSE__ = 3,0\n\n
                                        \nHinweis: Auch wenn die Modelle komplexer werden und eine h\u00f6here Abweichung aufweisen, ist MSE immer noch die Standardmetrik vieler Modelle. Dies gilt insbesondere f\u00fcr neuronale Netze, die Sie sp\u00e4ter kennenlernen werden. Dadurch, dass der MSE leicht differenziert werden kann, k\u00f6nnen mathematische Operationen einfacher duchrgef\u00fchrt werden.\n
                                        \n\n\n### Overfitting und Underfitting\n\n\n__Overfitting und Underfitting beim maschinellen Lernen__\n\nWir sollten stets bedenken, dass wir ein maschinelles Lernmodell entwerfen und keinen \"klassischen\" Algorithmus. Im Allgemeinen k\u00f6nnen wir sagen, dass ein gut entworfenes Modell in der Lage sein wird, alle m\u00f6glichen Arten von Dateneingaben der Dom\u00e4ne zu generalisieren. Lasst uns ein Beispiel betrachten: Ein mit Hunden trainierter Bildklassifikator, der gut genug generalisiert, wird in der Lage sein, einen Husky als Hund zu klassifizieren, obwohl keine Huskys in den Trainingsdaten vorhanden sind. Das gleiche Prinzip gilt f\u00fcr Vorhersagen. Ein gut generalisierendes Modell wird in der Lage sein, gute Vorhersagen auf Basis von Daten zu machen, die es noch nie gesehen hat, solange die Daten nicht v\u00f6llig anders strukturiert oder beschaffen sind.\n\nF\u00fcr uns ist es wichtig zu wissen, ob unser Modell gut generalisiert. Man k\u00f6nnte sagen, dass es wichtig ist, ein passendes Modell f\u00fcr die Problemdom\u00e4ne zu erstellen. Dies ist jedoch von vielen Faktoren abh\u00e4ngig. Beim maschinellen Lernen beschreiben die Begriffe \"Overfitting\" und \"Underfitting\" zwei Probleme, die mit der F\u00e4higkeit des Modells zur Generalisierung zusammenh\u00e4ngen.\n\n__Underfitting__:\nEin statistisches Modell oder ein Algorithmus des maschinellen Lernens wird Underfitting zugeschrieben, wenn es den zugrunde liegenden Trend der Daten nicht erfassen kann. Underfitting zerst\u00f6rt die Genauigkeit unseres maschinellen Lernmodells. Sein Auftreten bedeutet, dass das Modell oder der Algorithmus sich nicht gut genug an die Daten anpasst. Wenn das Modell underfittet, ist es nicht komplex genug, um die Eigenschaften der Daten zu erfassen (z. B. Verwendung eines linearen Modells auf nicht lineare Daten). Wenn Ihr Modell Underfitting aufweist, sollten Sie die Auswahl Ihrer ML-Methode \u00fcberdenken oder versuchen, die Hyperparameter anzupassen. Oft kann es hilfreich sein, die Datenmenge zu reduzieren und eine \u00dcberanpassung anzustreben, um herauszufinden, ob das gew\u00e4hlte Modell \u00fcberhaupt in der Lage ist, die vorhandenen Daten wiederzugeben.\n\n\n__Overfitting__:\nEin statistisches Modell ist overfittet, wenn es die Trainingsdaten gut repr\u00e4sentiert, aber bei neuen, nie gesehene Daten nicht zum Generalisieren in der Lage ist. Ein typisches Merkmal \u00fcberangepasster Modelle ist z. B., dass sie sogar Artefakte der Daten, wie deren spezifisches Rauschmuster, erlernen. Daher sind \u00fcberangepasste Modelle im Allgemeinen sehr komplex und hochdimensional. In der Regel haben sie eine hervorragende Performanz, wenn ihnen bekannte Daten gezeigt werden, und fast keine Performanz bei neuen Daten. Bei der Regression kann Overfitting durch die Wahl der richtigen Modellparameter vermieden werden. Ein Beispiel: W\u00e4hlen Sie ein lineares Regressionsmodell (nicht ein nicht-lineares), wenn Sie ein lineares Modell erwarten.\n\n\n\nEin gut funktionierendes Modell sollte weder unter einer Unteranpassung noch unter einer \u00dcberanpassung leiden, sondern sein Verhalten sollte zwischen diesen beiden liegen. Dies wird manchmal als das Goldilocks-Prinzip des maschinellen Lernens bezeichnet.\nEin einfaches Beispiel ist die Umlaufbahn der Erde. Sie ist nicht zu weit von der Sonne entfernt, sodass die Temperaturen nicht zu kalt f\u00fcr fl\u00fcssiges Wasser auf der Oberfl\u00e4che sind. W\u00e4re sie zu nah an der Sonne, w\u00e4re es zu hei\u00df f\u00fcr fl\u00fcssiges Wasser. Letztlich wissen wir, dass wir weder zu nah noch zu weit weg sind; daher liegt die Position des Planeten Erde in der Goldilock-Zone.\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom sklearn.pipeline import Pipeline\nfrom sklearn.preprocessing import PolynomialFeatures\nfrom sklearn.linear_model import LinearRegression\n\n\nnp.random.seed(0)\n\nn_samples = 30\n\ndegrees = [1, 4, 15] #Try other polynomial-degrees!\n\ntrue_fun = lambda X: np.cos(1.5 * np.pi * X)\nX = np.sort(np.random.rand(n_samples))\ny = true_fun(X) + np.random.randn(n_samples) * 0.1\n\nplt.figure(figsize=(14, 4))\nfor i in range(len(degrees)):\n ax = plt.subplot(1, len(degrees), i+1)\n plt.setp(ax, xticks=(), yticks=())\n\n polynomial_features = PolynomialFeatures(degree=degrees[i],\n include_bias=False)\n linear_regression = LinearRegression()\n pipeline = Pipeline([(\"polynomial_features\", polynomial_features),\n (\"linear_regression\", linear_regression)])\n pipeline.fit(X[:, np.newaxis], y)\n\n X_test = np.linspace(0, 1, 100)\n plt.plot(X_test, pipeline.predict(X_test[:, np.newaxis]), label=\"Model\")\n plt.plot(X_test, true_fun(X_test), label=\"True function\")\n plt.scatter(X, y, label=\"Samples\")\n plt.xlabel(\"x\")\n plt.ylabel(\"y\")\n plt.xlim((0, 1))\n plt.ylim((-2, 2))\n plt.legend(loc=\"best\")\n plt.title(\"Degree of polynomial: %s\" % degrees[i])\nplt.show()\n```\n\n_Wenn Sie keine Bilder direkt \u00fcber diesem Text sehen, f\u00fchren Sie die Zelle erneut aus. Probieren Sie dabei auch gerne verschiedene Polynom-Grade aus._\n
                                        \nFrage 2.2.5: Welcher Modellgrad repr\u00e4sentiert Overfitting, welcher Underfitting und welcher Modellgrad passt genau? \n
                                        \n\n
                                        \nIhre Antwort:
                                        \n\n\n\n```python\n\n```\n", "meta": {"hexsha": "930ad9f644b7a9d4ccf2f9224c3e251ff9def0ee", "size": 20828, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Woche 2/2 100 Genauigkeit, das muss doch gut sein, oder.ipynb", "max_stars_repo_name": "manfred-l-stark/AMALEA", "max_stars_repo_head_hexsha": "c4ea976d7e4a42520ef92e8a394ce60a48b681f9", "max_stars_repo_licenses": ["CC0-1.0", "CC-BY-4.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-09-30T19:53:52.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-30T19:53:52.000Z", "max_issues_repo_path": "Woche 2/2 100 Genauigkeit, das muss doch gut sein, oder.ipynb", "max_issues_repo_name": "manfred-l-stark/AMALEA", "max_issues_repo_head_hexsha": "c4ea976d7e4a42520ef92e8a394ce60a48b681f9", "max_issues_repo_licenses": ["CC0-1.0", "CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Woche 2/2 100 Genauigkeit, das muss doch gut sein, oder.ipynb", "max_forks_repo_name": "manfred-l-stark/AMALEA", "max_forks_repo_head_hexsha": "c4ea976d7e4a42520ef92e8a394ce60a48b681f9", "max_forks_repo_licenses": ["CC0-1.0", "CC-BY-4.0"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2021-09-29T13:03:07.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-02T12:21:05.000Z", "avg_line_length": 20828.0, "max_line_length": 20828, "alphanum_fraction": 0.7228730555, "converted": true, "num_tokens": 5153, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.7577943767446202, "lm_q1q2_score": 0.41724716539677476}} {"text": "+ This notebook is part of lecture 32 *Left-, right-, and pseudoinverses* in the OCW MIT course 18.06 by Prof Gilbert Strang [1]\n+ Created by me, Dr Juan H Klopper\n + Head of Acute Care Surgery\n + Groote Schuur Hospital\n + University Cape Town\n + Email me with your thoughts, comments, suggestions and corrections \n
                                        Linear Algebra OCW MIT18.06 IPython notebook [2] study notes by Dr Juan H Klopper is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.\n\n+ [1] OCW MIT 18.06\n+ [2] Fernando P\u00e9rez, Brian E. Granger, IPython: A System for Interactive Scientific Computing, Computing in Science and Engineering, vol. 9, no. 3, pp. 21-29, May/June 2007, doi:10.1109/MCSE.2007.53. URL: http://ipython.org\n\n\n```python\nfrom IPython.core.display import HTML, Image\ncss_file = 'style.css'\nHTML(open(css_file, 'r').read())\n```\n\n\n```python\nfrom sympy import init_printing, Matrix, symbols, sqrt, Rational\nfrom numpy import matrix, transpose, sqrt\nfrom numpy.linalg import pinv, inv, det, svd, norm\nfrom scipy.linalg import pinv2\nfrom warnings import filterwarnings\n```\n\n\n```python\ninit_printing(use_latex = 'mathjax')\nfilterwarnings('ignore')\n```\n\n# Left- and right-sided inverses and pseudoinverses\n\n## The inverse\n\n+ Recall the four fundamental subspaces\n + The rowspace (with **x**) and nullspace in ℝn\n + The columnspace (with A**x**) and the nullspace of AT in ℝm\n\n+ The two-sided inverse gives us the following\n$$ {A}{A}^{-1}=I={A}^{-1}{A} $$\n + For this we need *r* = *m* = *n* (i.e. full rank)\n\n+ For a left-inverse we have the following\n + Full column rank, with *r* = *n* (but possibly more rows)\n + The nullspace contains just the zero vector (columns are independent)\n + The rows might not all be independent\n + We thus have either no or only a single solution to A**x**=**b**\n + AT will now also have full rank\n + From (ATA)-1ATA = I follows the fact that (ATA)-1AT is a left-sided inverse (A-1)\n + Note, though, that (ATA)-1AT is a *n* × *m* matrix and A is of size *m* × *n*, resulting in a *n* × *n* identity matrix\n + We cannot do AA-1 and have a *n* × *n* identity matrix, though, but instead will be a projection matrix (onto the columnspace)\n\n+ For a right-inverse we have the following\n + Full row rank, with *r* = *m* < *n*\n + The nullspace of AT is the zero vector (rows are independent)\n + Elimination will result in many solutions to A**x**=**b** (*n* - *m* free variables)\n + Now there will be an A-1 to the right of A to give I\n + AAT(AAT)-1 = I\n + A-1 is now AT(AAT)-1\n + Putting the right-inverse on the left is also a projection (onto the rowspace)\n\n## The pseudoinverse\n\n+ Consider a matrix where *r* is less than *m* and *n*\n+ Remember that the rowspace is in ℝr and the columnspace is also in ℝr\n+ The nullspace of the rowspace is in ℝn-r and the nullspace of AT is in ℝm-r\n+ The rowspace and columnspace are in the same dimension and every vector **x** in one translate to another vector in the other (one-to-one)\n + If **y** in another vector in the rowspace (not same as **x**) then A**x** ≠ A**y**\n\n+ The pseudoinverse A+, then, maps **x** (or **y**) from the columnspace to the rowspace\n$$ y={A}^{+}{Ay} $$\n\n+ Suppose A**x** = A**y** or A(**x**-**y**) = 0\n + Now (**x**-**y**) is in the nullspace *and* in the rowspace, i.e. it has to be the zero vector\n\n### Finding the pseudoinverse A+\n\n+ One way is to start from the singular value decomposition\n$$ {A}={U}{\\Sigma}{V}^{T} $$\n+ Σ has along the main diagonal all the square roots of the eigenvalues and *r* pivots, but *m* row and *n* columns which can be more than *r*\n+ Σ+ will have 1 over the square roots of the eigenvalues along the main diagonals and then (possibly) zero values further along, but be of size *n* × *m*\n+ ΣΣ+ will have 1's along the main diagonal, and then 0's (if larger tha *r*)\n + It will be of size *m* × *m*\n + It is a projection onto the columnspace\n+ Σ+Σ will also have 1's along the main diagonal as well, but be of size *n* × *n*\n + It is a projection onto the rowspace\n\n+ We now have the following\n$$ {A}^{+}={V}{\\Sigma}^{+}{U}^{T} $$\n\n+ Let's see how easy this is in python\u2122\n\n\n```python\nA = matrix([[3, 6], [2, 4]]) # Not sympy\nA, det(A) # The det is zero, so no inverse exists\n```\n\n\n```python\n# The numpy pinv() function use SVD\nAplus = pinv(A)\nAplus\n```\n\n\n```python\n# The scipy pinv2() function also uses SVD\n# The scipy pinv() function uses least squares to approxiamte\n# the pseudoinverse and as matrices get BIG, this\n# becomes computationally expensive\nAplus_sp = pinv2(A)\nAplus_sp\n```\n\n## Example problem\n\n### Example problem 1\n\n+ Calculate the pseudoinverse of A=[1,2]\n+ Calculate AA+\n+ Calculate A+A\n+ If **x** is in the nullspace of A what is the effect of A+A on **x** (i.e. A+A**x**)\n+ If **x** is in the columnspace of AT what is A+A**x**?\n\n#### Solution\n\n\n```python\nA = matrix([1, 2])\nA\n```\n\n+ Let's use singular value decomposition\n\n\n```python\nU, S, VT = svd(A)\n```\n\n\n```python\nU\n```\n\n\n```python\nS\n```\n\n\n```python\nVT\n```\n\n+ Remember,\n$$ {A}^{+}={V}{\\Sigma}^{+}{U}^{T} $$\n+ Σ must be of size 2 × 1, though\n\n\n```python\nS = matrix([[sqrt(5)], [0]])\n```\n\n\n```python\nAplus = transpose(VT) * S * U\nAplus\n```\n\n+ This needs to be normalized\n\n\n```python\nnorm(Aplus)\n```\n\n\n```python\n1 / norm(Aplus) * Aplus\n```\n\n\n```python\nAplus = pinv(A)\nAplus\n```\n\n\n```python\nA * Aplus\n```\n\n\n```python\nAplus * A\n```\n\n+ Let's create a vector in the nullspace of A\n + It will be any vector\n $$ c\\begin{bmatrix}-2\\\\1\\end{bmatrix} $$\n+ Let's choose the constant *c* = 1\n\n\n```python\nx_vect_null_A = matrix([[-2], [1]])\nAplus * A * x_vect_null_A\n```\n\n+ This is now surprise as A+A reflects a vector onto the rowspace of A\n + We chose **x** in the nullspace of A, so A**x** must be **0** and A+A**x** = **0**\n\n+ The columnsapce of AT is any vector\n$$ c\\begin{bmatrix}1\\\\2\\end{bmatrix} $$\n+ We'll choose *c* = 1 again\n\n\n```python\nx_vect_null_AT = matrix([[1], [2]])\nAplus * A * x_vect_null_AT\n```\n\n+ We recover **x** again\n\n+ For fun, let's just check what A+ is when A is invertible\n\n\n```python\nA = matrix([[1, 2], [3, 4]])\n```\n\n\n```python\npinv(A)\n```\n\n\n```python\ninv(A)\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "74ae5f42b578768385b7a26a67edc1b55b978a2a", "size": 13136, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "_math/MIT_OCW_18_06_Linear_algebra/III_08_Left_and_right_inverses_Pseudoinverses.ipynb", "max_stars_repo_name": "aixpact/data-science", "max_stars_repo_head_hexsha": "f04a54595fbc2d797918d450b979fd4c2eabac15", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-07-22T23:12:39.000Z", "max_stars_repo_stars_event_max_datetime": "2020-07-25T02:30:48.000Z", "max_issues_repo_path": "_math/MIT_OCW_18_06_Linear_algebra/III_08_Left_and_right_inverses_Pseudoinverses.ipynb", "max_issues_repo_name": "aixpact/data-science", "max_issues_repo_head_hexsha": "f04a54595fbc2d797918d450b979fd4c2eabac15", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "_math/MIT_OCW_18_06_Linear_algebra/III_08_Left_and_right_inverses_Pseudoinverses.ipynb", "max_forks_repo_name": "aixpact/data-science", "max_forks_repo_head_hexsha": "f04a54595fbc2d797918d450b979fd4c2eabac15", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 26.7535641548, "max_line_length": 708, "alphanum_fraction": 0.5331151035, "converted": true, "num_tokens": 2395, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.7577943658046609, "lm_q1q2_score": 0.41724715937315254}} {"text": "\n\n# Randomized Prior Functions for Deep Reinforcement Learning\n*Ian Osband, John Aslanides, Albin Cassirer*\n\n[NIPS 2018 spotlight paper](https://arxiv.org/abs/1806.03335)\n\n[Accompanying website](https://sites.google.com/corp/view/randomized-prior-nips-2018/home)\n\n\n\n## 0. Outline\n\nThis colab serves to accompany the 2018 NIPS paper [\"Random Prior Functions for Deep Reinforcement Learning\"](https://arxiv.org/abs/1806.03335).\n\nThe goal of this colab is to:\n- Provide some simple + clear example code.\n- Help build some intuition for the proposed methods.\n- Reproduce some of the results from the paper!\n\nWe hope this is a useful resource to accompany our paper!\n\n\n\n\n\n\n\n## 1. Loading / setup\n\n... you can probably skip this section\n\n\n\n```\n#@title Install dependencies (only run this once)\n\n! pip install -q plotnine==0.5.1\n! pip install -q tensorflow==1.12\n! pip install -q tensorflow-probability==0.5.0\n! pip install -q dm-sonnet==1.26\n\n! pip install -q git+git://github.com/deepmind/trfl.git # TRFL\n```\n\n \u001b[31mpymc3 3.6 has requirement joblib<0.13.0, but you'll have joblib 0.13.1 which is incompatible.\u001b[0m\n \u001b[31mcufflinks 0.14.6 has requirement plotly>=3.0.0, but you'll have plotly 1.12.12 which is incompatible.\u001b[0m\n \u001b[31mgraph-nets 1.0.2 has requirement dm-sonnet==1.23, but you'll have dm-sonnet 1.26 which is incompatible.\u001b[0m\n\n\n\n```\n#@title Imports\n%%capture\n\nimport collections\nimport warnings\n\nimport numpy as np\nimport tensorflow as tf\nimport sonnet as snt\n\nimport trfl\n\nimport pandas as pd\nimport plotnine as gg\n\nfrom IPython.display import clear_output\nfrom matplotlib import pyplot as plt\n\nfrom typing import Any, Callable, Dict, List, Tuple\n\n# Plotnine themes\ngg.theme_set(gg.theme_bw(base_size=16, base_family='serif'))\ngg.theme_update(figure_size=(12, 8), panel_spacing_x=0.5, panel_spacing_y=0.5)\n\n# Filter meaningless pandas warnings\nwarnings.filterwarnings(action=\"ignore\", category=UserWarning)\nwarnings.filterwarnings(action=\"ignore\", category=ImportWarning)\n```\n\n\n```\n#@title Make toy 1D dataset for regression problems\n\ndef generate_data(n_data: int,\n x_scale: float = 2., \n std_dev: float = 2.) -> pd.DataFrame:\n\n x = np.linspace(-x_scale, x_scale, n_data)\n y = x + std_dev * np.random.randn(n_data)\n\n return pd.DataFrame({'x_train': x, 'y_train': y})\n\nnp.random.seed(10)\nDATA = generate_data(10)\n```\n\n## 2. Randomized prior functions\n\n\"Prior networks\" break down the function you are learning into:\n\n\\begin{equation}\nQ_{\\theta}(x) = \\underbrace{f_\\theta(x)}_{\\text{trainable}} + \\overbrace{\\beta}^{\\text{scaling}} \\underbrace{p(x)}_{\\text{prior}}. \\tag{1}\n\\end{equation}\n\nIn this section we visualize the effects of training $f_\\theta$ = (20, 20)-MLP but with different choices of prior function $p$.\n\nWe will see that, given a flexible enough architecture $f$, for any $p$:\n- The resultant $Q=f+p$ will be able to fit all of the observed data $(x,y)$.\n- Away from the training data, generalization will be impacted by the interaction of $f$ with $p$.\n\nAs such, we say that $p$ plays a role similar to a \"prior\" in Bayesian inference: it controls how the agent generalizes in areas *without* data... whereas regions *with* data are still perfectly able to fit their observations.\nIn fact, this superficial similarity can be made much more rigorous for the simple case of linear regression.\nFor more on this connection please see Section 3 of [our paper](https://arxiv.org/abs/1806.03335)!\n\n\n\n```\n#@title (CODE) ModelWithPrior, training routine\n\nclass ModelWithPrior(snt.AbstractModule):\n \"\"\"Given a 'model' and a 'prior', combines them together.\"\"\"\n\n def __init__(self,\n model_network: snt.AbstractModule,\n prior_network: snt.AbstractModule,\n prior_scale: float = 1.):\n super(ModelWithPrior, self).__init__(name='model_with_prior')\n\n self._prior_scale = prior_scale\n with self._enter_variable_scope():\n self._model_network = model_network\n self._prior_network = prior_network\n\n def _build(self, inputs: tf.Tensor):\n prior_output = tf.stop_gradient(self._prior_network(inputs))\n model_output = self._model_network(inputs)\n\n return model_output + self._prior_scale * prior_output\n\n def get_variables(self, collection=tf.GraphKeys.TRAINABLE_VARIABLES):\n return (super(ModelWithPrior, self).get_variables(collection) \n + self._model_network.get_variables(collection) \n + self._prior_network.get_variables(collection))\n\n\ndef train_model_with_prior(data: pd.DataFrame,\n prior: Callable[[tf.Tensor], tf.Tensor],\n prior_label: str,\n hidden_sizes: List[int],\n prior_scale: float = 1.,\n num_epochs: int = 100,\n plot_scale: float = 3.) -> pd.DataFrame:\n \"\"\"Given a dataset and a prior function, train/evaluate a ModelWithPrior.\"\"\"\n\n # Set up TF graph\n tf.reset_default_graph()\n\n x_train = tf.placeholder(tf.float32, shape=[None, 1])\n y_train = tf.placeholder(tf.float32, shape=[None, 1])\n\n # Make a trainable model as a simple multi-layer perceptron\n model = snt.nets.MLP(hidden_sizes + [1], activation=tf.nn.selu)\n\n # Combine it with our 'prior' as in (1)\n model_with_prior = ModelWithPrior(model, prior, prior_scale)\n\n # Outputs\n prior_out = prior(x_train)\n model_out = model(x_train)\n model_with_prior_out = model_with_prior(x_train)\n\n # Training ops\n loss = tf.square(model_with_prior_out - y_train)\n optimizer = tf.train.AdamOptimizer(5e-2)\n train_op = optimizer.minimize(loss)\n\n init_op = tf.global_variables_initializer()\n with tf.Session() as sess:\n sess.run(init_op)\n\n # Train\n for _ in range(num_epochs):\n sess.run(train_op, \n feed_dict={x_train: np.expand_dims(data['x_train'], 1), \n y_train: np.expand_dims(data['y_train'], 1)})\n\n # Evaluate\n x_eval = np.linspace(-plot_scale, plot_scale, 1000)\n\n eval_feed = {x_train: np.expand_dims(x_eval, 1)}\n y_model = sess.run(model_out, feed_dict=eval_feed)\n y_prior = sess.run(prior_out, feed_dict=eval_feed)\n y_model_with_prior = sess.run(model_with_prior_out,\n feed_dict=eval_feed)\n\n # Put in dataframe\n df = pd.DataFrame({\n 'x': x_eval,\n 'prior': prior_scale * y_prior.squeeze(),\n 'model': y_model.squeeze(),\n 'model_with_prior': y_model_with_prior.squeeze(),\n 'prior_label': prior_label,\n })\n\n return df\n```\n\nWe now perform a simple experiment:\n1. Fix a dataset of training data $(x,y)$\n2. Initialize a prior function $p$.\n3. Fit $Q_\\theta = f_\\theta + p$ to this data by optimizing $\\theta$.\n\nThe following plots demonstrate how the prior function alters behaviour:\n- Training data in **black** points $(x,y)$.\n- Trainable function $f$ is the **dashed** line.\n- Fixed (untrainable) prior function $p$ in **blue**.\n- Resultant prediction $Q = f + p$ in **red**.\n\nThe plots show that, for different choices of prior function:\n- The neural network can still fit the data at observed data points $(x,y)$.\n- The generalization behaviour away from the data is different for different choices of $p$.\n\n\n\n```\n#@title Train and evaluate\nPRIOR_SCALE = 2.\n\nPRIORS = {\n '$p(x)=0$': lambda _: tf.constant(0.),\n r'$p(x)={prior}(\\exp(|x|) - 1)$'.format(prior=PRIOR_SCALE): lambda x: tf.multiply(PRIOR_SCALE, (tf.exp(tf.abs(x)) - 1)),\n r'$p(x)={prior}$sin$(3x)$'.format(prior=PRIOR_SCALE): lambda x: PRIOR_SCALE * tf.sin(3 * x),\n r'$p(x)={prior}(1 - x^2)$'.format(prior=PRIOR_SCALE): lambda x: PRIOR_SCALE * (1 - x ** 2),\n}\n\nDF = pd.DataFrame()\nPLOT_DATA = pd.DataFrame()\nfor PRIOR_LABEL, PRIOR in PRIORS.items():\n DF = DF.append(train_model_with_prior(\n data=DATA, prior=PRIOR, prior_label=PRIOR_LABEL, hidden_sizes=[20, 20],\n prior_scale=PRIOR_SCALE, num_epochs=2000))\n tmp_df = DATA.copy()\n tmp_df['prior_label'] = PRIOR_LABEL\n PLOT_DATA = PLOT_DATA.append(tmp_df)\n```\n\n\n```\n#@title Plotting\np = (gg.ggplot(DF)\n + gg.geom_point(mapping=gg.aes(x='x_train', y='y_train'), data=PLOT_DATA) # Training data\n + gg.geom_line(mapping=gg.aes(x='x', y='prior'), alpha=0.25, colour='blue')\n + gg.geom_line(mapping=gg.aes(x='x', y='model'), linetype='dashed')\n + gg.geom_line(mapping=gg.aes(x='x', y='model_with_prior'), alpha=0.75, colour='red')\n + gg.facet_wrap('prior_label', ncol=2)\n + gg.theme(figure_size=(10, 7))\n + gg.ylim(-10, 10)\n + gg.xlab('')\n + gg.ylab('')\n )\np\n```\n\n## 3. Approximate posterior distribution via ensemble\n\nSection 2. outlines our approach to training a network with prior function.\n\nIn [our paper](https://arxiv.org/abs/1806.03335) we make the connection between:\n1. Generating a single posterior sample\n2. Training a network with randomized prior function\n\nIn fact, for linear networks, the two are precisely equivalent.\n\n\nHowever, in many settings, a single posterior sample is not enough... we want to be able to approximate the *entire distribution* of posterior beliefs.\nIn this setting, our approach is very simple... just take multiple samples and use the resulting ensemble as an approximation to your posterior.\n\n\n\n\n```\n#@title (CODE) EnsembleMLP + training routine\n\nclass EnsembleMLP(snt.AbstractModule):\n\n def __init__(self, hidden_sizes: List[int], num_ensemble: int, **mlp_kwargs):\n super(EnsembleMLP, self).__init__(name='ensemble')\n self._models = [snt.nets.MLP(output_sizes=hidden_sizes + [1], **mlp_kwargs) \n for _ in range(num_ensemble)]\n \n self._num_ensemble = num_ensemble\n\n def _build(self, inputs: tf.Tensor):\n outputs = [model(inputs) for model in self._models]\n\n return tf.squeeze(tf.stack(outputs, axis=1), axis=-1)\n\ndef train_ensemble_models(data: pd.DataFrame,\n hidden_sizes: List[int],\n num_ensemble: int,\n num_epochs: int = 100,\n plot_scale: float = 3.) -> pd.DataFrame:\n\n n_data = len(data)\n n_eval = 1000\n\n # Set up TF graph\n tf.reset_default_graph()\n\n x_train = tf.placeholder(tf.float32, shape=[n_data, 1])\n y_train = tf.placeholder(tf.float32, shape=[n_data, 1])\n x_eval = tf.placeholder(tf.float32, shape=[n_eval, 1])\n\n y = tf.tile(y_train, [1, num_ensemble])\n \n raw_model = EnsembleMLP(hidden_sizes, num_ensemble, activation=tf.nn.selu)\n prior = EnsembleMLP(hidden_sizes, num_ensemble, activation=tf.nn.selu)\n \n model = ModelWithPrior(raw_model, prior)\n prior_out = prior(x_eval)\n\n model_out = model(x_train)\n eval_out = model(x_eval)\n\n # Training ops\n loss = tf.square(model_out - y)\n optimizer = tf.train.AdamOptimizer(5e-2)\n train_op = optimizer.minimize(loss)\n\n init_op = tf.global_variables_initializer()\n with tf.Session() as sess:\n sess.run(init_op)\n\n # Train\n for _ in range(num_epochs):\n sess.run(train_op, \n feed_dict={x_train: np.expand_dims(data['x_train'], 1), \n y_train: np.expand_dims(data['y_train'], 1)})\n\n # Evaluate\n xs = np.expand_dims(np.linspace(-plot_scale, plot_scale, n_eval), 1)\n y_model = sess.run(eval_out, feed_dict={x_eval: xs})\n y_prior = sess.run(prior_out, feed_dict={x_eval: xs})\n\n # Dataframe of ensemble-aggregated predictions\n agg_df = pd.DataFrame(\n np.hstack([xs, y_model]), columns=['x'] + list(range(num_ensemble)))\n agg_df = agg_df.melt(['x'], value_name='y')\n agg_df = agg_df.groupby('x')['y'].agg([np.mean, np.std]).reset_index()\n agg_df['num_ensemble'] = num_ensemble\n agg_df['sample'] = y_model[:, 0]\n \n # Dataframe of prior / predictions per ensemble member\n model_df = pd.DataFrame(\n np.hstack([xs, y_model]), columns=['x'] + list(range(1, num_ensemble+1)))\n model_df = model_df.melt(['x'], var_name='ensemble_member', value_name='y')\n prior_df = pd.DataFrame(\n np.hstack([xs, y_prior]), columns=['x'] + list(range(1, num_ensemble+1)))\n prior_df = prior_df.melt(['x'], var_name='ensemble_member', value_name='prior')\n\n model_df['prior'] = prior_df['prior']\n model_df['model'] = model_df.y - model_df.prior\n\n return agg_df, model_df\n```\n\nTo demonstrate this effect, we perform a very simple experiment:\n- Random prior $p_k$ is a (20,20)-MLP according to standard Glorot initialization for $k=1,..,10$.\n- Train $Q^k_\\theta = f^k_\\theta + p_k$ for $k=1,..,10$\n- Use the resultant $\\{Q^k\\}$ as an approximation to the posterior.\n\n\n```\n#@title Train and evaluate\nEPOCHS = 1000\nHIDDEN_SIZES = [20, 20]\n\nAGG_DF, MODEL_DF = train_ensemble_models(\n data=DATA, hidden_sizes=HIDDEN_SIZES, num_epochs=EPOCHS, num_ensemble=10)\n```\n\n### Visualizing ensemble diversity\n\nThe plots below visualize the resultant $Q, f, p$ for $k=1,..,10$.\n\nYou can see that all the functions fit the training data, but they do not generalize in exactly the same way.\n\n\n```\n#@title Plotting each trained network + prior\np = (gg.ggplot(MODEL_DF)\n + gg.geom_point(mapping=gg.aes(x='x_train', y='y_train'), data=DATA) # Training data\n + gg.geom_line(mapping=gg.aes(x='x', y='prior'), alpha=0.25, colour='blue')\n + gg.geom_line(mapping=gg.aes(x='x', y='model'), linetype='dashed')\n + gg.geom_line(mapping=gg.aes(x='x', y='y'), alpha=0.75, colour='red')\n + gg.facet_wrap('ensemble_member', ncol=5, labeller='label_both')\n + gg.theme(figure_size=(20, 10))\n + gg.ylim(-10, 10)\n )\np\n```\n\n### Summarizing ensemble into approximate distribution\n\nWe can combine the ensemble predictions by taking the mean and standard deviation of predictions $\\{Q^k(x)\\}$ at each $x$.\n- Training data $(x,y)$ are black dots.\n- Blue line represents the mean prediction\n- Shaded grey bands at $\\pm 1, 2$ standard deviations.\n\nWe feel that, intuitively, this represents a reasonable approximation to posterior uncertainty.\n\n\n```\n#@title Plotting the ensemble as a whole\np = (gg.ggplot(AGG_DF)\n + gg.geom_point(gg.aes('x_train', 'y_train'), data=DATA, colour='#034e7b')\n + gg.geom_line(gg.aes('x', 'mean'), colour='#1f77b4', size=1, alpha=0.5)\n # + gg.geom_line(gg.aes('x', 'sample'), colour='#d62728', size=1, alpha=0.5)\n + gg.geom_ribbon(gg.aes('x', 'std', ymin='mean - std', ymax='mean + std'), alpha=0.2)\n + gg.geom_ribbon(gg.aes('x', 'std', ymin='mean - 2 * std', ymax='mean + 2 * std'), alpha=0.1)\n + gg.theme(figure_size=(8, 6))\n + gg.labs(x=r'$x$', y=r'$y$')\n )\np\n```\n\n### Adding extra noise to the data\n\nIn problems with observation noise, the ensemble procedure above will fit to every data point exactly... and this may not be appropriate!\n\nAnalogous to the Bayesian derivation of [Section 3](https://arxiv.org/abs/1806.03335), we should add noise to the targets $y$ similar to the type of noise we expect to see with our data.\n- If we know the scale of the noise, we can simply add noise in that family.\n- If we do not know the noise distribution, bootstrapping offers a non-parametric way to estimate it.\nhttps://en.wikipedia.org/wiki/Bootstrapping_(statistics)\n\nHowever, once you have enough data, the effects of bootstrapping (or adding noise) will wash out eventually.\n\nOur next series of plots repeats the ensemble procedure with bootstrapping *and* prior functions.\n\n\n```\n#@title Training a bootstrap ensemble\n\ndef train_ensemble_models_bootstrap(data: pd.DataFrame,\n hidden_sizes: List[int],\n num_ensemble: int,\n num_epochs: int = 1000,\n plot_scale: float = 3.) -> pd.DataFrame:\n \n ensemble_data = []\n training_data = []\n for k in range(num_ensemble):\n boot_data = data.sample(n=len(data), replace=True)\n prior = snt.nets.MLP(hidden_sizes + [1], activation=tf.nn.selu)\n prior_label = 'ensemble member {}'.format(k)\n \n model_df = train_model_with_prior(boot_data, prior, prior_label, hidden_sizes,\n num_epochs=num_epochs)\n ensemble_data.append(model_df)\n \n boot_data['prior_label'] = prior_label\n training_data.append(boot_data)\n return pd.concat(ensemble_data), pd.concat(training_data)\n\nBOOT_DF, BOOT_DATA = train_ensemble_models_bootstrap(DATA, [20, 20], 10)\n```\n\n\n```\n#@title Plotting each (bootstrapped) ensemble member\np = (gg.ggplot(BOOT_DF)\n + gg.geom_point(mapping=gg.aes(x='x_train', y='y_train'), data=BOOT_DATA) # Training data\n + gg.geom_line(mapping=gg.aes(x='x', y='prior'), alpha=0.25, colour='blue')\n + gg.geom_line(mapping=gg.aes(x='x', y='model'), linetype='dashed')\n + gg.geom_line(mapping=gg.aes(x='x', y='model_with_prior'), alpha=0.75, colour='red')\n + gg.facet_wrap('prior_label', ncol=5)\n + gg.theme(figure_size=(20, 10))\n + gg.ylim(-10, 10)\n )\np\n```\n\n\n```\n#@title Plotting the ensemble as a whole with bootstrapping\n\nAGG_BOOT = (BOOT_DF\n .groupby('x')['model_with_prior']\n .agg([np.mean, np.std])\n .reset_index()\n )\n\np = (gg.ggplot(AGG_BOOT)\n + gg.geom_point(gg.aes('x_train', 'y_train'), data=DATA, colour='#034e7b')\n + gg.geom_line(gg.aes('x', 'mean'), colour='#1f77b4', size=1, alpha=0.5)\n # + gg.geom_line(gg.aes('x', 'sample'), colour='#d62728', size=1, alpha=0.5)\n + gg.geom_ribbon(gg.aes('x', 'std', ymin='mean - std', ymax='mean + std'), alpha=0.2)\n + gg.geom_ribbon(gg.aes('x', 'std', ymin='mean - 2 * std', ymax='mean + 2 * std'), alpha=0.1)\n + gg.theme(figure_size=(8, 6))\n + gg.labs(x=r'$x$', y=r'$y$')\n )\np\n```\n\n## 4. RL on a chain domain\n\n\nThe sections above introduce \"randomized prior functions\" via simple 1D regression problems. **However, our principle interest is in driving efficient exploration for reinforcement learning.** If we can quanitify the agent's uncertainty over the value, then we can prioritize potentially informative policies.\nTo highlight the importance of efficient exploration we investigate performance on a series of chain-like domains designed to highlight the need for *deep exploration*:\n- [Deep exploration](https://arxiv.org/abs/1703.07608) refers to an advanced form of exploration that is necessary for efficient learning in reinforcement learning.\n- An agent that employs deep exploration can take an action in order to reach a potentially-informative future state, even if the immediate action is not itself informative.\n- The idea of \"deep exploration\" is not new, and was identified by [Kearns and Singh, (2002)](https://www.cis.upenn.edu/~mkearns/papers/KearnsSinghE3.pdf) as an essential ingredient for polynomial learning in reinforcement learning.\n\n### 4.1 \"Deep sea\" environment\n\nWe now describe the family of chain-like environments we call \"deep sea\":\n- Input state is an $N x N$ grid.\n- Observations are 1-hot, and the agent begins in top left square.\n- Actions are $\\{0, 1\\}$, but the mapping to \"left\" and \"right\" is a fixed random mapping per state.\n- Every timestep the agent falls one row down.\n- Action \"left\" receives a reward of 0, action \"right\" reward of $-\\frac{0.01}{N}$.\n- If the agent makes it to the bottom far right then it receives an extra reward of $1$.\n\nNote that there is a single policy with episodic return $0.99$, a single policy with return $0$ and all other policies have negative return.\nAlgorithms that do not perform deep exploration will require $O(2^N)$ episodes to find the rewarding policy, but algorithms that *do* perform deep exploration can learn much faster... in fact, with a tabular representation, we would expect \"randomized least-squares value iteration\" [(RLSVI)](https://arxiv.org/pdf/1703.07608.pdf) to learn in O(N^3) episodes.\nOur implementation of bootstrapped ensemble + randomized prior represents a practical neural network implementation of RLSVI.\n\nFor more information on this environment/setup please see:\n- Section 4 of our NIPS paper [randomized prior functions for deep reinforcement learning](https://arxiv.org/pdf/1806.03335.pdf)\n- JMLR paper [deep exploration via randomized value functions](https://arxiv.org/pdf/1703.07608.pdf)\n\n\n\n```\n#@title (CODE) Define the _Deep Sea_ environment\n\nTimeStep = collections.namedtuple('TimeStep', ['observation' ,'reward', 'pcont'])\n\nclass DeepSea(object):\n\n def __init__(self,\n size: int, \n seed: int = None, \n randomize: bool = True):\n\n self._size = size\n self._move_cost = 0.01 / size\n self._goal_reward = 1.\n\n self._column = 0\n self._row = 0\n\n if randomize:\n rng = np.random.RandomState(seed)\n self._action_mapping = rng.binomial(1, 0.5, size)\n else:\n self._action_mapping = np.ones(size)\n\n self._reset_next_step = False\n\n def step(self, action: int) -> TimeStep:\n if self._reset_next_step:\n return self.reset()\n # Remap actions according to column (action_right = go right)\n action_right = action == self._action_mapping[self._column]\n\n # Compute the reward\n reward = 0.\n if self._column == self._size-1 and action_right:\n reward += self._goal_reward\n\n # State dynamics\n if action_right: # right\n self._column = np.clip(self._column + 1, 0, self._size-1)\n reward -= self._move_cost\n else: # left\n self._column = np.clip(self._column - 1, 0, self._size-1)\n\n # Compute the observation\n self._row += 1\n if self._row == self._size:\n observation = self._get_observation(self._row-1, self._column)\n self._reset_next_step = True\n return TimeStep(reward=reward, observation=observation, pcont=0.)\n else:\n observation = self._get_observation(self._row, self._column)\n return TimeStep(reward=reward, observation=observation, pcont=1.)\n\n def reset(self) -> TimeStep:\n self._reset_next_step = False\n self._column = 0\n self._row = 0\n observation = self._get_observation(self._row, self._column)\n\n return TimeStep(reward=None, observation=observation, pcont=1.)\n \n def _get_observation(self, row, column) -> np.ndarray:\n observation = np.zeros(shape=(self._size, self._size), dtype=np.float32)\n observation[row, column] = 1\n\n return observation\n\n @property\n def obs_shape(self) -> Tuple[int]:\n return self.reset().observation.shape\n\n @property\n def num_actions(self) -> int:\n return 2\n\n @property\n def optimal_return(self) -> float:\n return self._goal_reward - self._move_cost\n\n\n#@title Helpers for playing Deep Sea interactively\n\ndef get_user_action():\n action = input('Action ([a] = 0, [d] = 1, [q] = Quit): ')\n if action == 'a':\n action = 0\n elif action == 'd':\n action = 1\n elif action == 'q':\n return -1\n else:\n print('Bad action! Must be `a` or `d` or `q`.')\n return get_user_action()\n return action\n\ndef play(env):\n step = env.reset()\n episode_return = 0\n while step.pcont:\n plt.grid(False)\n ax = plt.gca()\n # ax.set_axis_off()\n # Major ticks\n ax.set_xticks(np.arange(0, 11, 1));\n ax.set_yticks(np.arange(0, 11, 1));\n\n # Labels for major ticks\n ax.set_xticklabels(np.arange(1, 11, 1));\n ax.set_yticklabels(np.arange(1, 11, 1));\n\n # Minor ticks\n ax.set_xticks(np.arange(-.5, 11, 1), minor=True);\n ax.set_yticks(np.arange(-.5, 11, 1), minor=True);\n\n # Gridlines based on minor ticks\n ax.grid(which='minor', color='k', linestyle='-', linewidth=1)\n plt.imshow(step.observation, interpolation='none')\n plt.show()\n a = get_user_action()\n clear_output(False)\n if a == -1:\n break # User quit\n step = env.step(a)\n episode_return += step.reward\n print('Episode return: {}'.format(episode_return))\n ax.set_visible(False)\n # clear_output()\n```\n\n### 4.1.1 Play \"deep sea\" for yourself\n\nHere we spin up an instance of the environment:\n- Set the problem size $N$\n- Use randomized actions (or not)\n- Seed (for randomized action selection)\n\nPlaying this game without randomized actions the optimal policy is just to do action \"d\" = \"right\" each timestep.\n\nIf you do this with randomized action selection, which is what our agent sees, you'll see that this is not a trivial problem even for $N=10$.\n\n\n```\n#@title Play _Deep Sea_\ndeep_sea_size = 10 #@param {type:\"integer\"}\nrandomized_actions = True #@param{type:\"boolean\"}\n\nenv = DeepSea(size=deep_sea_size, randomize=randomized_actions, seed=1)\nplay(env)\n```\n\n### 4.2 RL agent\n\nIn this section we implement a basic DQN and Bootstrapped DQN + prior function.\n- Train each agent on N=20 deep sea problem\n- Any dithering strategy for exploration will take around $2^N \\simeq$ 1e6 for $N=20$.\n- We find that bootstrapped DQN + prior functions typically solves this problem in around 2000 episodes!\n\n\n```\n#@title Agent hyperparameters (shared between DQN/RPF)\n\nmax_episodes = 3000 #@param {type:\"integer\"}\nepsilon = 0.1 #@param {type:\"number\"} (NB - only for DQN)\ndiscount = 0.99 #@param {type:\"number\"}\nbatch_size = 128 #@param {type:\"integer\"}\nreplay_capacity = 100000 #@param {type:\"integer\"}\nensemble_size = 10 #@param {type:\"integer\"}\nhidden_sizes = (20,) #@param {type:\"raw\"}\nprior_scale = 10 #@param {type:\"number\"}\n\ndeep_sea_size=20 #@param {type:\"number\"}\n\n# Make environment\nenv = DeepSea(size=deep_sea_size, randomize=True)\n```\n\n\n```\n#@title (CODE) Simple circular replay buffer with uniform sampling.\n\nclass Replay(object):\n \"\"\"A simple ring buffer with uniform sampling.\"\"\"\n\n def __init__(self, capacity: int):\n self._data = None\n self._capacity = capacity\n self._num_added = 0\n\n def add(self, transition: Tuple[Any]) -> None:\n if self._data is None:\n self._preallocate(transition)\n\n for d, item in zip(self._data, transition):\n d[self._num_added % self._capacity] = item\n\n self._num_added += 1\n\n def sample(self, batch_size: int = 1) -> Tuple[np.ndarray]:\n \"\"\"Returns a transposed/stacked minibatch. Each array has shape [B, ...].\"\"\"\n indices = np.random.randint(self.size, size=batch_size)\n return [d[indices] for d in self._data]\n\n @property\n def size(self) -> int:\n return min(self._capacity, self._num_added)\n\n def _preallocate(self, items: Tuple[Any]) -> None:\n \"\"\"Assume flat structure of items.\"\"\"\n items_np = [np.asarray(x) for x in items]\n\n if sum([x.nbytes for x in items_np]) * self._capacity > 1e9:\n raise ValueError('This replay buffer would preallocate > 1GB of memory.')\n\n self._data = [np.zeros(dtype=x.dtype, shape=(self._capacity,) + x.shape)\n for x in items_np]\n```\n\n\n```\n#@title (CODE) MLP for Q-values\n\nclass MLPQNetwork(snt.AbstractModule):\n\n def __init__(self, hidden_sizes: Tuple[int], num_actions: int):\n super(MLPQNetwork, self).__init__(name='simple_mlp')\n with self._enter_variable_scope():\n self._mlp = snt.nets.MLP(output_sizes=hidden_sizes + (num_actions,))\n \n def _build(self, inputs):\n inputs = snt.BatchFlatten()(inputs)\n\n return self._mlp(inputs)\n```\n\n### 4.2.1 Training basic DQN\n\n- Here we present a basic DQN training loop\n- DQN with $\\epsilon$-greedy actions cannot solve $N=20$ problem\n\n\n```\n#@title Train DQN on _Deep Sea_\n\ntf.reset_default_graph()\n\n# Networks.\nq_network = MLPQNetwork(hidden_sizes, env.num_actions)\ntarget_network = MLPQNetwork(hidden_sizes, env.num_actions)\n\n# Placeholders.\no_tm1 = tf.placeholder(dtype=tf.float32, shape=(None,) + env.obs_shape)\na_tm1 = tf.placeholder(dtype=tf.int32, shape=(None,))\nr_t = tf.placeholder(dtype=tf.float32, shape=(None,))\npcont_t = tf.placeholder(dtype=tf.float32, shape=(None,))\no_t = tf.placeholder(dtype=tf.float32, shape=(None,) + env.obs_shape)\n\n# Forward the networks.\nq_tm1 = q_network(o_tm1) # [B, A]\nq_t = target_network(o_t) # [B, A]\n\n# Online -> target network copy ops.\nupdate_op = [tf.assign(w, v) for v, w in zip(q_network.get_variables(),\n target_network.get_variables())]\n\n# Loss/optimization ops.\nloss, _ = trfl.qlearning(q_tm1, a_tm1, r_t, discount * pcont_t, q_t)\noptimizer = tf.train.AdamOptimizer()\nsgd_op = optimizer.minimize(loss)\n\nreplay = Replay(capacity=replay_capacity) # Replay buffer\n\ninit_op = tf.global_variables_initializer()\nwith tf.Session() as sess:\n sess.run(init_op) # Initialize variables.\n\n # Make callables for policy, target update, and SGD.\n q_values = sess.make_callable(q_tm1, [o_tm1])\n target_update = sess.make_callable(update_op, [])\n sgd = sess.make_callable(sgd_op, [o_tm1, a_tm1, r_t, pcont_t, o_t])\n\n # Episode stats.\n total_return = 0\n optimal_return = 0\n results = []\n\n # Run loop.\n for episode in range(max_episodes):\n timestep = env.reset()\n observation = timestep.observation\n episode_return = 0\n while timestep.pcont:\n # Epsilon-greedy policy.\n if np.random.rand() < epsilon:\n action = np.random.randint(env.num_actions)\n else:\n obs = np.expand_dims(timestep.observation, 0)\n qs = q_values(obs)\n action = np.argmax(qs, axis=1).squeeze()\n\n # Agent-environment interaction.\n timestep = env.step(action)\n episode_return += timestep.reward\n new_observation = timestep.observation\n\n # Add to replay.\n transition = (observation, action, timestep.reward,\n timestep.pcont, new_observation)\n replay.add(transition)\n observation = new_observation\n\n # After each episode, do one SGD step and update the target network.\n batch = replay.sample(batch_size)\n sgd(*batch)\n target_update()\n\n # Compute running total return/regret.\n total_return += episode_return\n optimal_return += 0.99\n regret = optimal_return - total_return\n\n # Logging.\n results.append({'episode': episode, 'regret': regret, 'algorithm': 'dqn'})\n if episode % 100 == 0 or episode == max_episodes - 1:\n print('Episode: {}.\\tTotal return: {:.2f}.\\tRegret: {:.2f}.'.format(episode, total_return, regret))\n\ndf = pd.DataFrame(results)\n```\n\n Episode: 0.\tTotal return: -0.01.\tRegret: 0.99.\n Episode: 100.\tTotal return: -0.51.\tRegret: 100.50.\n Episode: 200.\tTotal return: -0.95.\tRegret: 199.94.\n Episode: 300.\tTotal return: -1.47.\tRegret: 299.46.\n Episode: 400.\tTotal return: -1.92.\tRegret: 398.91.\n Episode: 500.\tTotal return: -2.24.\tRegret: 498.23.\n Episode: 600.\tTotal return: -2.46.\tRegret: 597.45.\n Episode: 700.\tTotal return: -2.68.\tRegret: 696.67.\n Episode: 800.\tTotal return: -3.18.\tRegret: 796.17.\n Episode: 900.\tTotal return: -3.51.\tRegret: 895.50.\n Episode: 1000.\tTotal return: -3.83.\tRegret: 994.82.\n Episode: 1100.\tTotal return: -4.21.\tRegret: 1094.20.\n Episode: 1200.\tTotal return: -4.32.\tRegret: 1193.31.\n Episode: 1300.\tTotal return: -4.41.\tRegret: 1292.40.\n Episode: 1400.\tTotal return: -4.47.\tRegret: 1391.46.\n Episode: 1500.\tTotal return: -4.53.\tRegret: 1490.52.\n Episode: 1600.\tTotal return: -4.59.\tRegret: 1589.58.\n Episode: 1700.\tTotal return: -4.65.\tRegret: 1688.64.\n Episode: 1800.\tTotal return: -4.78.\tRegret: 1787.78.\n Episode: 1900.\tTotal return: -4.85.\tRegret: 1886.84.\n Episode: 2000.\tTotal return: -4.92.\tRegret: 1985.91.\n Episode: 2100.\tTotal return: -5.00.\tRegret: 2084.99.\n Episode: 2200.\tTotal return: -5.09.\tRegret: 2184.08.\n Episode: 2300.\tTotal return: -5.28.\tRegret: 2283.27.\n Episode: 2400.\tTotal return: -5.49.\tRegret: 2382.48.\n Episode: 2500.\tTotal return: -5.67.\tRegret: 2481.66.\n Episode: 2600.\tTotal return: -5.89.\tRegret: 2580.88.\n Episode: 2700.\tTotal return: -6.05.\tRegret: 2680.04.\n Episode: 2800.\tTotal return: -6.16.\tRegret: 2779.15.\n Episode: 2900.\tTotal return: -6.33.\tRegret: 2878.32.\n Episode: 2999.\tTotal return: -6.61.\tRegret: 2976.61.\n\n\n### 4.2.2 Training ensemble DQN + prior function\n\n- Simple implementation of ensemble DQN + random prior functions\n- With default settings we solve $N=20$ in around 2000 episodes\n- For more results, see [our paper](https://arxiv.org/pdf/1806.03335.pdf)\n\n\n```\n#@title (CODE) Ensemble Q-Network\nclass EnsembleQNetwork(snt.AbstractModule):\n\n def __init__(self, hidden_sizes: Tuple[int], num_actions: int, num_ensemble: int, **mlp_kwargs):\n super(EnsembleQNetwork, self).__init__(name='ensemble')\n with self._enter_variable_scope():\n # An ensemble of MLPs.\n self._models = [snt.nets.MLP(output_sizes=hidden_sizes + (num_actions,), **mlp_kwargs) \n for _ in range(num_ensemble)]\n \n self._num_ensemble = num_ensemble\n\n def _build(self, inputs: tf.Tensor) -> tf.Tensor:\n inputs = snt.BatchFlatten()(inputs)\n # Forward all members of the ensemble and stack the output.\n return tf.stack([model(inputs) for model in self._models], axis=1)\n\n```\n\n\n```\n#@title Train Ensemble DQN+RPF on _Deep Sea_\n\ntf.reset_default_graph()\n\n# Make a 'prior' network.\nprior_network = EnsembleQNetwork(hidden_sizes, env.num_actions, ensemble_size)\n\n# Make independent online and target networks.\nq_model = EnsembleQNetwork(hidden_sizes, env.num_actions, ensemble_size)\ntarget_model = EnsembleQNetwork(hidden_sizes, env.num_actions, ensemble_size)\n\n# Combine these with the prior in the usual way.\nq_network = ModelWithPrior(q_model, prior_network, prior_scale)\ntarget_network = ModelWithPrior(target_model, prior_network, prior_scale)\n\n# Placeholders\no_tm1 = tf.placeholder(dtype=tf.float32, shape=(None,) + env.obs_shape)\na_tm1 = tf.placeholder(dtype=tf.int32, shape=(None,))\nr_t = tf.placeholder(dtype=tf.float32, shape=(None,))\npcont_t = tf.placeholder(dtype=tf.float32, shape=(None,))\no_t = tf.placeholder(dtype=tf.float32, shape=(None,) + env.obs_shape)\n\n# Forward the networks\nq_tm1 = q_network(o_tm1) # [B, K, A]\nq_t = target_network(o_t) # [B, K, A]\n\n# Online -> target network copy ops.\nupdate_op = [tf.assign(w, v) for v, w in zip(q_network.get_variables(),\n target_network.get_variables())]\n\n# Loss/optimization ops\none_hot_actions = tf.one_hot(a_tm1, depth=env.num_actions, axis=-1) # [B, A]\nq_value = tf.einsum('bka,ba->bk', q_tm1, one_hot_actions) # [B, K]\nq_target = tf.reduce_max(q_t, axis=-1) # [B, K]\ntarget = tf.expand_dims(r_t, 1) + discount * tf.expand_dims(pcont_t, 1) * q_target # [B, K]\ntd_error = q_value - target\nloss = tf.square(td_error)\noptimizer = tf.train.AdamOptimizer()\nsgd_op = optimizer.minimize(loss)\n\n\nreplay = Replay(capacity=replay_capacity)\n\ninit_op = tf.global_variables_initializer()\nwith tf.Session() as sess:\n sess.run(init_op)\n q_values = sess.make_callable(q_tm1, [o_tm1])\n target_update = sess.make_callable(update_op, [])\n sgd = sess.make_callable(sgd_op, [o_tm1, a_tm1, r_t, pcont_t, o_t])\n\n total_return = 0\n optimal_return = 0\n results = []\n for episode in range(max_episodes):\n timestep = env.reset()\n observation = timestep.observation\n episode_return = 0\n active_head = np.random.randint(ensemble_size)\n while timestep.pcont:\n obs = np.expand_dims(timestep.observation, 0)\n ensemble_qs = q_values(obs) # [B, K, A]\n qs = ensemble_qs[:, active_head, :]\n action = np.argmax(qs, axis=1).squeeze()\n\n timestep = env.step(action)\n episode_return += timestep.reward\n new_observation = timestep.observation\n transition = (observation, action, timestep.reward,\n timestep.pcont, new_observation)\n replay.add(transition)\n observation = new_observation\n\n # Do SGD at end of episode\n batch = replay.sample(batch_size)\n sgd(*batch)\n target_update()\n\n total_return += episode_return\n optimal_return += 0.99\n regret = optimal_return - total_return\n results.append({'episode': episode, 'regret': regret, 'algorithm': 'rpf'})\n if episode % 100 == 0 or episode == max_episodes - 1:\n print('Episode: {}.\\tTotal return: {:.2f}.\\tRegret: {:.2f}.'.format(episode, total_return, regret))\n\ndf = df.append(pd.DataFrame(results))\n```\n\n Episode: 0.\tTotal return: -0.00.\tRegret: 0.99.\n Episode: 100.\tTotal return: -0.51.\tRegret: 100.50.\n Episode: 200.\tTotal return: -1.04.\tRegret: 200.03.\n Episode: 300.\tTotal return: -1.62.\tRegret: 299.61.\n Episode: 400.\tTotal return: -2.23.\tRegret: 399.22.\n Episode: 500.\tTotal return: -2.86.\tRegret: 498.85.\n Episode: 600.\tTotal return: -3.50.\tRegret: 598.49.\n Episode: 700.\tTotal return: -4.19.\tRegret: 698.18.\n Episode: 800.\tTotal return: -4.93.\tRegret: 797.92.\n Episode: 900.\tTotal return: -5.66.\tRegret: 897.65.\n Episode: 1000.\tTotal return: -6.35.\tRegret: 997.34.\n Episode: 1100.\tTotal return: -7.12.\tRegret: 1097.11.\n Episode: 1200.\tTotal return: -7.91.\tRegret: 1196.90.\n Episode: 1300.\tTotal return: -8.68.\tRegret: 1296.67.\n Episode: 1400.\tTotal return: -9.48.\tRegret: 1396.47.\n Episode: 1500.\tTotal return: -10.30.\tRegret: 1496.29.\n Episode: 1600.\tTotal return: -9.07.\tRegret: 1594.06.\n Episode: 1700.\tTotal return: -3.88.\tRegret: 1687.87.\n Episode: 1800.\tTotal return: 12.35.\tRegret: 1770.64.\n Episode: 1900.\tTotal return: 34.54.\tRegret: 1847.45.\n Episode: 2000.\tTotal return: 78.70.\tRegret: 1902.29.\n Episode: 2100.\tTotal return: 170.73.\tRegret: 1909.26.\n Episode: 2200.\tTotal return: 269.73.\tRegret: 1909.26.\n Episode: 2300.\tTotal return: 368.73.\tRegret: 1909.26.\n Episode: 2400.\tTotal return: 467.73.\tRegret: 1909.26.\n Episode: 2500.\tTotal return: 566.73.\tRegret: 1909.26.\n Episode: 2600.\tTotal return: 665.73.\tRegret: 1909.26.\n Episode: 2700.\tTotal return: 764.73.\tRegret: 1909.26.\n Episode: 2800.\tTotal return: 863.73.\tRegret: 1909.26.\n Episode: 2900.\tTotal return: 962.73.\tRegret: 1909.26.\n Episode: 2999.\tTotal return: 1060.74.\tRegret: 1909.26.\n\n\n### 4.2.3 Plotting results\n\n- We see that ensemble DQN successfully solves the problem\n- Learning in polynomial time demonstrates effective deep exploration\n\n\n```\n#@title Plotting cumulative regret (RPF learns size=20 in ~= 2k episodes)\nregret_plot = (gg.ggplot(df)\n + gg.aes(x='episode', y='regret', colour='algorithm')\n + gg.geom_line(size=1.5, alpha=0.8)\n + gg.ylab('cumulative regret')\n )\nregret_plot\n```\n\n\n\n\n```\n#@title Plotting pre-trained data to 50k episodes performance difference is clear\n\n# Loading in pretrained data from string.\nimport sys\nfrom io import StringIO\nDF_CSV = 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= pd.read_csv(StringIO(DF_CSV))\n\n# Same plot as above, but with pretrained data\nregret_plot.data = PRETRAINED_DF\nprint(regret_plot)\n```\n", "meta": {"hexsha": "3809301964c63068c53482044ae0a4c76ac4303c", "size": 737924, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "rpf_nips_2018.ipynb", "max_stars_repo_name": "snageshr/deep-rl", "max_stars_repo_head_hexsha": "55808346573b2aa31be6271996f37383b0b3d896", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "rpf_nips_2018.ipynb", "max_issues_repo_name": "snageshr/deep-rl", "max_issues_repo_head_hexsha": "55808346573b2aa31be6271996f37383b0b3d896", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "rpf_nips_2018.ipynb", "max_forks_repo_name": "snageshr/deep-rl", "max_forks_repo_head_hexsha": "55808346573b2aa31be6271996f37383b0b3d896", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 460.9144284822, "max_line_length": 175960, "alphanum_fraction": 0.9085014175, "converted": true, "num_tokens": 23599, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.7577943658046609, "lm_q1q2_score": 0.41724715937315254}} {"text": "In order to successfully complete this assignment you must do the required reading, watch the provided videos and complete all instructions. The embedded Google form must be entirely filled out and submitted on or before **11:59pm on Sunday March 22**. Students must come to class the next day prepared to discuss the material covered in this assignment. answer\n\n# Pre-Class Assignment: Linear Algebra Review\n\n\n\n### Goals for today's pre-class assignment \n\n

                                        \n\n\n\n1. [Linear systems](#Linear_systems)\n2. [Using LSF to solve overdefined systems](#Overdefined_systems)\n3. [Using the Pseudoinverse](#Pseudoinverse)\n4. [Assignment wrap-up](#Assignment_wrap-up)\n\n----\n\n\n# 1. Linear systems\n\nLinear algebra is solving a system of lienar equations where each equation is of the form:\n\n$$a_1x_1 + a_2x_2 + a_3x_3 \\dots + a_nx_n = b$$\n\nThis equation is also called a linear combination where you add together terms that are multiplied together. (Combinations of Addition and Multiplication are core concepts in linear algebra). \n\nTypically we have multiple versions of the same equation where the constance ($x$ terms) are the same but the variables ($a$ and $b$) are different:\n\n\n$$a_{11}x_1 + a_{12}x_2 + a_{13}x_3 \\dots + a_{1n}x_n = b_1$$\n$$a_{21}x_1 + a_{22}x_2 + a_{23}x_3 \\dots + a_{2n}x_n = b_2$$\n$$a_{31}x_1 + a_{32}x_2 + a_{33}x_3 \\dots + a_{3n}x_n = b_3$$\n$$\\vdots$$\n$$a_{m1}x_1 + a_{m2}x_2 + a_{m3}x_3 \\dots + a_{mn}x_n = b_m$$\n\n\nThis is a lot of writing so we generally rewrite these equations in matrix form as follows:\n\n$$ \n\\left[\n\\begin{matrix}\n a_{11} & a_{12} & a_{13} & \\dots & a_{1n} \\\\\n a_{21} & a_{22} & a_{23} & \\dots & a_{2n} \\\\\n a_{31} & a_{32} & a_{33} & \\dots & a_{3n} \\\\\n & \\vdots & & \\ddots & \\\\\n a_{m1} & a_{m2} & a_{m3} & \\dots & a_{mn} \n\\end{matrix}\n\\right] \n\\left[\n\\begin{matrix}\n x_{1} \\\\\n x_{2} \\\\\n x_{3} \\\\\n \\vdots \\\\\n x_{n} \n\\end{matrix}\n\\right] \n=\n\\left[\n\\begin{matrix}\n b_{1} \\\\\n b_{2} \\\\\n b_{3} \\\\\n \\vdots \\\\\n b_{n} \n\\end{matrix}\n\\right] \n$$\n\nTHe first matrix is often called $A$, the constants are often called $c$ and the right hand matrix is often called $b$. We typically simplify this equation using Matrix notation as follows:\n\n$$Ax = b$$\n\n\nFor example, consider the following set of linear equations:\n\n$$ 3x_1-3x_2+9x_3 = 24$$\n$$ 2x_1-2x_2+7x_3 = 17$$\n$$ -x_1+2x_2-4x_3 = -11$$\n\nIt's Matrix format is as follows:\n\n$$ \n\\left[\n\\begin{matrix}\n 3 & -3 & 9\\\\ \n 2 & -2 & 7 \\\\\n -1 & 2 & -4\n\\end{matrix}\n\\right] \n\\left[\n\\begin{matrix}\n x_1 \\\\ \n x_2 \\\\\n x_3\n\\end{matrix}\n\\right] \n=\n\\left[\n\\begin{matrix}\n 24\\\\ \n 17 \\\\\n -11\n\\end{matrix}\n\\right] \n$$\n\nLet's define these same Matrixes as ```numpy``` variables (displayed using ```sympy```, which looks prettier).\n\n\n```python\nimport numpy as np\nimport sympy as sym\nsym.init_printing(use_unicode=True) # Trick to make matrixes look nice in jupyter\n```\n\n\n```python\nA = np.matrix([[3, -3,9], [2, -2, 7], [-1, 2, -4]])\nsym.Matrix(A)\n```\n\n\n```python\nb = np.matrix([[24], [17], [-11]])\nsym.Matrix(b)\n```\n\n\n```python\n#Calculate answer to x using numpy\nx = np.linalg.solve(A,b)\nsym.Matrix(x)\n```\n\n----\n\n \n# 2. Using LSF to solve overdefined systems\n\nThe above Linear Systems has a single solution. However, linear systems of the form $Ax=b$ can also be under-defined which means they can have infinite solutions and over-defined which can have no solutions. In data-science over-defined systems are quite common, consider the following example.\n\n**Example:** A researcher has conducted experiments of a particular Hormone dosage in a population of rats. The table shows the number of fatalities at each dosage level tested. Determine the least squares line and use it to predict the number of rat fatalities at hormone dosage of 22. \n\n| Hormone level | 20 | 25 | 30 | 35 | 40 | 45 | 50 |\n|---|---|---|---|---|---|---|---|\n| Fatalities | 101 | 115 | 92 | 64 | 60 | 50 | 49| \n\n\n\n```python\n%matplotlib inline\nimport matplotlib.pylab as plt\nimport numpy as np\nimport sympy as sym\nimport time\nsym.init_printing(use_unicode=True)\n```\n\n\n```python\nH = [20,25,30,35,40,45,50]\nf = [101,115, 92,64,60,50,49]\n\nplt.scatter(H,f)\nplt.xlabel('Hormone Level')\nplt.ylabel('Fatalities');\n```\n\nWe want to determine a line that is expressed by the following equation \n\n$$Hx_1 + x_2 = f$$\n\nto approximate the connection between Hormone dosage ($H$) and Fatalities $f$. \nThat is, we want to find $x_1$ (slope) and $x_2$ (y-intercept) for this line. If we wanted to we could make a serise of equations using the datapoints from above as follows:\n\n$$20x_1 + x_2 = 101$$\n$$25x_1 + x_2 = 115$$\n$$30x_1 + x_2 = 92$$\n$$\\vdots$$\n$$50x_1 + x_2 = 49$$\n\nHowever, hopefully you can see that there is no solution for $x_1$ and $x_2$ that will fit all of these equations. Instead we convert the equations to matrix format ($Ax=b$) and solve for the best fit of $x$ using Least Squares Fit. If you are new to LSF please watch the following video which may be helpful:\n\n\n```python\nfrom IPython.display import YouTubeVideo\nYouTubeVideo(\"Lx6CfgKVIuE\",width=640,height=360)\n```\n\nTo get the above equations into the matrix form $Ax=b$ we first define the variable $ \nx = \\left[\n\\begin{matrix}\n x_1 \\\\\n x_2 \n\\end{matrix}\n\\right] \n$ as the column vector that needs to be solved. \n\nThen, $A$ matrix can be defined using the values of $H$ and 1 (Something to multiply by $x_2$)\n\n\n```python\n#Our A matrix can be solved as follow\nA = np.matrix(np.vstack((H, np.ones(len(H)))).T)\nsym.Matrix(A)\n```\n\nSimilarly we can define the matrix $b$ using our fatalities $f$:\n\n\n```python\nb = np.matrix(f).T\nsym.Matrix(b)\n```\n\nNow we can solve using the ```numpy.linalg``` Least Squares Fit (LSF) function as follows:\n\n\n```python\nx,err,_,_ = np.linalg.lstsq(A,b, rcond=None)\nx\n```\n\n\n```python\nbestfit = A*x\nbestfit\n```\n\n\n```python\nplt.scatter(H,f)\nplt.plot(H, bestfit)\nplt.xlabel('Hormone Level')\nplt.ylabel('Fatalities');\n```\n\n----\n\n \n# 3. Using the Pseudoinverse\n\nThe ```numpy.linalg.lstsq``` function does all of the work for us. However, you should be able to get the same result using the Pseudoinverse of the $A$ matrix. The Pseudoinverse for this type of problem is defined as follows:\n\n$$P_{inv} = (A^TA)^{-1}A^T$$\n\nThe above can be read, \"The Pseudoinverse is the inverse of A transpose times A (which is a sqare matrix) times A transpose.\" We can calculate the Pseudoinverse using the following python numpy matrix code:\n\n\n```python\npinv = (A.T*A)**-1*A.T\n```\n\nNow we multiply both sides by the $P_{inv}$:\n\n$$P_{inv}Ax = P_{inv}b$$\n\nIf we assume that $P_{inv}A$ is the identity matrix then $P_{inv}b$ is our LSF solution for $x$\n$$x = P_{inv}b$$\n\n\n```python\npinv*b\n```\n\nFunny enough, only the unknowns of an equation need to be linear combinations, we an actually fit more complex polynomials. See if you can change the model and solve find a second order polynomial (parabola) of the form:\n\n$$x_1H^2 + x_2H + x_3 = f$$\n\nIn this case your $b$ matrix is still our $f$ values and should not change. We only need to add another column to matrix $A$ to represent the $H^2$ term. Here is some simple code to make new matrix ($A2$) from $A$:\n\n\n```python\nA2 = np.hstack((np.array(A[:,0])**2, A))\nsym.Matrix(A2)\n```\n\n✅ **DO THIS:** Using either the ```lstsq``` function or by making a new pseudoinverse, solve the new system of equations for $x_1, x_2$ and $x_3$ and plot the results:\n\n\n```python\n#Put your code here\n```\n\n----\n\n# 4. Assignment wrap-up\n\nPlease fill out the form that appears when you run the code below. **You must completely fill this out in order to receive credit for the assignment!**\n\n[Direct Link to Google Form](https://cmse.msu.edu/cmse802-pc-survey)\n\n\nIf you have trouble with the embedded form, please make sure you log on with your MSU google account at [googleapps.msu.edu](https://googleapps.msu.edu) and then click on the direct link above.\n\n✅ **Assignment-Specific QUESTION:** Where you able to fit and plot a parabola to the data? If not, where did you get stuck?\n\nPut your answer to the above question here\n\n✅ **QUESTION:** Summarize what you did in this assignment.\n\nPut your answer to the above question here\n\n✅ **QUESTION:** What questions do you have, if any, about any of the topics discussed in this assignment after working through the jupyter notebook?\n\nPut your answer to the above question here\n\n✅ **QUESTION:** How well do you feel this assignment helped you to achieve a better understanding of the above mentioned topic(s)?\n\nPut your answer to the above question here\n\n✅ **QUESTION:** What was the **most** challenging part of this assignment for you? \n\nPut your answer to the above question here\n\n✅ **QUESTION:** What was the **least** challenging part of this assignment for you? \n\nPut your answer to the above question here\n\n✅ **QUESTION:** What kind of additional questions or support, if any, do you feel you need to have a better understanding of the content in this assignment?\n\nPut your answer to the above question here\n\n✅ **QUESTION:** Do you have any further questions or comments about this material, or anything else that's going on in class?\n\nPut your answer to the above question here\n\n✅ **QUESTION:** Approximately how long did this pre-class assignment take?\n\nPut your answer to the above question here\n\n\n```python\nfrom IPython.display import HTML\nHTML(\n\"\"\"\n\n\"\"\"\n)\n```\n\n---------\n### Congratulations, we're done!\n\nTo get credit for this assignment you must fill out and submit the above Google From on or before the assignment due date.\n\n### Course Resources:\n\n- [Syllabus](https://docs.google.com/document/d/e/2PACX-1vTW4OzeUNhsuG_zvh06MT4r1tguxLFXGFCiMVN49XJJRYfekb7E6LyfGLP5tyLcHqcUNJjH2Vk-Isd8/pub)\n- [Preliminary Schedule](https://docs.google.com/spreadsheets/d/e/2PACX-1vRsQcyH1nlbSD4x7zvHWAbAcLrGWRo_RqeFyt2loQPgt3MxirrI5ADVFW9IoeLGSBSu_Uo6e8BE4IQc/pubhtml?gid=2142090757&single=true)\n- [Course D2L Page](https://d2l.msu.edu/d2l/home/912152)\n\n© Copyright 2020, Michigan State University Board of Trustees\n", "meta": {"hexsha": "e9dd2da69ed5c51cfd39678be3811746da350e00", "size": 18700, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "cmse802-s20/0322--LA-pre-class-assignment.ipynb", "max_stars_repo_name": "Diane1306/cmse802_git_CompModelling", "max_stars_repo_head_hexsha": "44e529c07ab2f7cdf801cc0df5d67c9992f1823d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "cmse802-s20/0322--LA-pre-class-assignment.ipynb", "max_issues_repo_name": "Diane1306/cmse802_git_CompModelling", "max_issues_repo_head_hexsha": "44e529c07ab2f7cdf801cc0df5d67c9992f1823d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "cmse802-s20/0322--LA-pre-class-assignment.ipynb", "max_forks_repo_name": "Diane1306/cmse802_git_CompModelling", "max_forks_repo_head_hexsha": "44e529c07ab2f7cdf801cc0df5d67c9992f1823d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-12-11T07:41:07.000Z", "max_forks_repo_forks_event_max_datetime": "2021-12-11T07:41:07.000Z", "avg_line_length": 28.506097561, "max_line_length": 370, "alphanum_fraction": 0.5282887701, "converted": true, "num_tokens": 3199, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.4170585218676458}} {"text": "## Lecture 9 Notebook: Descriptives\n\nIn this notebook, we will revisit the replication data from the paper \"MPs for Sale? Returns to Office in Postwar British Politics\" by Eggers and Hainmuller. \n\nLet's do some more basic descriptive analysis.\n\nLike before, we start by importing some libraries.\n\n\n\n\n\n```python\n# Importing libraries for tables and plots\nfrom datascience import Table\n%matplotlib inline\nimport matplotlib.pyplot as plt\nplt.style.use('seaborn-whitegrid')\nimport numpy as np\nfrom scipy import stats\n```\n\nThe datascience library we used before is a bit limited in plotting capabilities, so this week we will us the \"pandas\" library which calls tables dataframes.\n\n\n```python\n# Importing the data into a table called mps\nmps = Table.read_table(\"MPs.csv\")\nmps = mps.to_df()\nmps\n```\n\nOne way that we can refer to columns in a data frame is with the format dataframe['column']. So, to pull the 'party' variable from the mps dataframe we type mps['party']. Think of this as a \"shortcut\" for the mps.column('party') syntax we used before.\n\n\n```python\nmps[\"party\"]\n```\n\n# Categorical Variables\n\nHow should we summarize this variable? For categorical and ordinal variables, a good place to start is with a frequency table. We can get that with the `.value_counts()` function. \n\n\n\n```python\nmps[\"party\"].value_counts()\n```\n\nPretty straightforward: our data frame has 223 Tory and 204 Labour candidates. Often it is nice to convert this to percentages, which we do by adding 'normalize=TRUE' to the function:\n\n\n```python\nmps[\"party\"].value_counts(normalize=True)\n```\n\nAnother categorical variable is the region:\n\n\n```python\nmps[\"region\"].value_counts()\n```\n\n\n```python\nmps[\"region\"].value_counts(normalize=True)\n```\n\nTo summarize categorical variables visually, we can use bar charts. The functionality builds nicely on what we did before: we first make a table with the value counts, then add `.plot(kind='bar')` at the end.\n\n\n```python\nmps[\"party\"].value_counts().plot(kind='bar')\n```\n\n\n```python\nmps[\"region\"].value_counts().plot(kind='bar')\n```\n\nWe can also display this with a pie chart:\n\n\n```python\nmps[\"region\"].value_counts().plot(kind='pie')\n```\n\nWhat is a \"typical\" value for a categorical variable? If there isn't any \"order\" to the values, the best way to think about a typical value is the most common one, or the *mode*. We can compute this with the `stats.mode()` function.\n\n\n```python\nstats.mode(mps['party'])\n```\n\n\n```python\nstats.mode(mps['region'])\n```\n\n# Ordinal Variables\nThere aren't any ordinal variables in the dataset, so here we will make one up. A good ordinal variable for this context is how highly ranked the MP was at the peak of their career. In the UK parliament, a natural ranking is \"backbencher\" (think regular MP), \"cabinet member\", and \"Prime Minister\". And we can actually figure out from the data whether they won or lost using the 'margin' variable. \n\nSo, we are going to create a rank variable that is 0 for losers (from the real data), and for winners we will randomly assign them a 1 for backbencher, 2, for cabinet member, and 3 for PM. So, higher rank means \"more success\".\n\nFirst, lets create the winner variable\n\n\n```python\nmps['winner'] = 1*(mps['margin'] > 0)\n```\n\n\n```python\nmps['winner'].value_counts()\n```\n\nIn the following line of code we will do our random draw for winners, don't worry about the details here.\n\n\n```python\nmps['rank']=mps['winner'] * np.random.choice([1,2,3], 427, p=[.75, .2, .05])\n```\n\nLike with categorical variables, we can summarize these with `.value_counts()`\n\n\n```python\nmps['rank'].value_counts()\n```\n\nAnd a bar chart\n\n\n```python\nmps['rank'].value_counts().plot(kind='bar')\n```\n\nAlso as with categorical variables, the mode will tell us the most frequent value:\n\n\n```python\nstats.mode(mps['rank'])\n```\n\nThis tells us the most common outcome is to remain a 0, or a backbencher. \n\nA second way we can define a typical value is the median. One way to think of the median is that we sort into increasing order, and then pick the number in the middle.\n\nTo pick some shorter examples, suppose we have five data points: 2, 6, 5, 4, 1. These sort to 1,2,4,5,6. And, hopefully this makes sense visually, the middle is the third point, in the sense that two are smaller than the median and two are larger. So, here the median is 3. \n\nIf we have 7 data points the median is the 4th largest, if we have 101 data points it is the 51st largest, etc. In general, if we have an **odd** number of data points n, the median is the (n+1)/2 largest. \n\n\nThings are a bit tricker if we have an even number of data points. Say we add 100 to our initial list of 5, giving 1,2,4,5,6,100. In a sense the \"middle\" is now \"between 4 and 5\". So for this case we define the median as halfway between these two points. \n\nChecking this:\n\n\n```python\nnp.median([2,6,5,4,1])\n```\n\n\n```python\nnp.median([2,6,5,4,1,100])\n```\n\nLets see the median of our rank variable:\n\n\n```python\nnp.median(mps['rank'])\n```\n\nThis tells us that the \"typical\" person in our data set doesn't even get into parliament! \n\n[Optional: suppose we dropped 200 losers from the data set. What would the median be now?]\n\nSince our data are numbers, we can also compute the mean or average. The mean is the sum of all of the numbers, divided by the number of observations. The formula for this is that for an array of numbers $X_1,X_2,...,X_n$, the mean is:\n\\begin{align}\n \\bar{X} = \\frac {\\sum_{i=1}^n X_i}{n}\n\\end{align}\n\n\nSo in our 5 data point example it would be (2 + 6 + 5 + 4 +1)/5. Checking this.\n\n\n```python\n(2 + 6 + 5 + 4 + 1)/5\n```\n\n\n```python\nnp.mean([2,6,5,4,1])\n```\n\nLets apply it to our rank variable\n\n\n```python\nnp.mean(mps['rank'])\n```\n\nDoes this really mean anything? Arguably not. .45 is not a valid rank, at it doesn't really make sense to describe someone as 45% between a loser and a backbencher.\n\nFor some ordinal variables means are informative. For example, if we gave them an ideological score like in Problem set 2, where -2 is extreme left, -1 is moderate left, 0 is centrist, 1 is moderate right, and 2 is extreme right, then the average of those would give us a number between -2 and 2 which would give a sense of the \"average\" ideology of the candidates. \n\n# Numeric variables\nThere are a few numeric variables here, include the vote margin one. For numeric variables it usually doesn't make sense to construct a table, because often each value shows up only once. A nice analog to a frequency table for numeric variables is a *histogram*.\n\n\n```python\nmps.hist('margin')\n```\n\n... and losers\n\nHistograms break numeric variables into \"bins\" and then plot the frequency of those bins. Here the bins have a width of about .05. So, the highest bin is telling us that about 115 candidates in the data with a margin of about -.04 to .01. (If you wanted to break these into more natural cutoffs like 0 to .05, that can be done, but let's just take the default).\n\nThis gives us a general shape of the data. The margin variable is typically close to zero, with fewer and fewer cases as the margin gets very big or small.\n\nRemember last week we also looked at the net wealth at death, and we had to do a bit of mathematical trickery to determine this from the \"natural logarithm of net wealth at death\" variable.\n\n\n```python\nmps['net'] = np.exp(mps['ln.net'])\nmps.hist('net')\n```\n\nHere wealth is being plotted in \"tens of millions of pounds\" (the 1e7 part). So .2 means 2 million, .4 means 4 million, etc. This is telling us that the vast majority of candidates have a net wealth below 1 million at death.\n\nNow let's compute typical values. Again we can look at the mean, median and mode:\n\n\n```python\nprint(\"Mean: \", np.mean(mps['margin']))\nprint(\"Median: \", np.median(mps['margin']))\nprint(\"Mode: \", stats.mode(mps['margin']))\n```\n\nFor the margin variable, the mean and median are vary similar. The mode is way off though: someone who lost by 48%! In general we won't pay attention to the mode of numeric varaiables. This is because, as with many numeric variables, all of the margin amounts are unique. So the frequency of all of them is 1! It appears by default the `stats.mode` function picks the lowest value. \n\nNow let's do net wealth:\n\n\n```python\nprint(\"Mean: \", np.mean(mps['net']))\nprint(\"Median: \", np.median(mps['net']))\n```\n\nHere we get much different results when looking at the mean vs the mode. This is because the mean is very sensitive to \"extreme observations\". \n\nHere is a simple way to see that. Remember when we started with a data array of [2,6,5,4,1] and then added 100, the median went up just a bit, from 4 to 4.5. Now let's do the same but for the mean\n\n\n```python\nnp.mean([2,6,5,4,1])\n```\n\n\n```python\nnp.mean([2,6,5,4,1,100])\n```\n\nAdding this one new obseration makes the mean go up by 15! In some sense this gives a weird answer for a \"typical value\", as no value in this data array is near 20. \n\nA similar thing could happen in our real data. Suppose we add a (deceased) Jeff Bezos to our data. His current net worth in pounds is about 135 billion. How would adding him change the mean and median?\n\n\n```python\nprint(\"Mean with Bezos: \", np.mean(np.append(mps['net'], [135000000000])))\nprint(\"Median with Bezos: \", np.median(np.append(mps['net'], [135000000000])))\n```\n\nSo, which is a better measure of a typical value? It depends! The median is less sensitive to outliers, describing a typical member independent of how extreme the non-typical units are. And if we want to answer questions like \"how does winning office affect wealth\", we don't want our answer to be entirely determined by whether Bezos was a winner or a loser when he ran (particularly if we know that his wealth was independent of his hypothetical run for office).\n\nHowever, averages are often meaningful too. For example, if we did know that a handful of MPs took advantage of their office to become billionaires, then this would be politically meaningful, even if the typical MP got little out of holding office. \n\n# Measuring Spread\nWhen dealing with numeric variables, we also care a lot about the *spread* of the values. One way to think about this is \"how far is a typical value from the average\"? A first way to define this is the variance. If we have an array with n numeric observations, $X_1,X_2,...X_n$ with mean $\\bar{X}$, the variance is given by:\n\\begin{align}\n \\frac {\\sum_{i=1}^n (X_i - \\bar{X})^2} {n-1}\n\\end{align}\n\nThis is close to saying \"what is the average squared distance from the mean\", though instead of dividing by n we divide by n-1. The reason for this difference is a bit subtle and has no noticeable impact once our $n$ is above 5 or so, as will almost always be the case. \n\nHere is how we compute the variance.\n\n\n```python\nnp.var(mps['margin'])\n```\n\n\n```python\nnp.var(mps['net'])\n```\n\nNote that in the margin case we got a very small number, and in the net worth we got an extremely large number. That is because our margin variable is typically around .1, and squaring this give .01. On the other hand, with the wealth variable, we are taking values in the hundred thousands and squaring them. One way to think abotu this problem is that the \"unit\" of the variance is the \"unit squared\" of the original variable. In order to get something easier to interpret, we often focus on the *standard deviation* of the variable, which is given by the square root of the variance. \n\n\n```python\nnp.std(mps['margin'])\n```\n\n\n```python\nnp.std(mps['net'])\n```\n\nWe can also compute variances and standard deviations for ordinal variables, though like with the mean whether or not this has much meaning depends on what we are doing.\n\n\n```python\nnp.var(mps['rank'])\n```\n\n\n```python\nnp.std(mps['rank'])\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "44af09adc10d17351814715ce847efcc2e1d8107", "size": 19728, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "PS 3 Lecture 9 - Descriptives.ipynb", "max_stars_repo_name": "ds-modules/POLSCI3-FA20", "max_stars_repo_head_hexsha": "3d2d93b636efb029d7d17ebdde87186484f8782f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "PS 3 Lecture 9 - Descriptives.ipynb", "max_issues_repo_name": "ds-modules/POLSCI3-FA20", "max_issues_repo_head_hexsha": "3d2d93b636efb029d7d17ebdde87186484f8782f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "PS 3 Lecture 9 - Descriptives.ipynb", "max_forks_repo_name": "ds-modules/POLSCI3-FA20", "max_forks_repo_head_hexsha": "3d2d93b636efb029d7d17ebdde87186484f8782f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 29.845688351, "max_line_length": 597, "alphanum_fraction": 0.5968167072, "converted": true, "num_tokens": 3041, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102775181399, "lm_q2_score": 0.658417500561683, "lm_q1q2_score": 0.4170484117535757}} {"text": "\n\n\n# Intro a Computa\u00e7\u00e3o Qu\u00e2ntica e Portas L\u00f3gicas Qu\u00e2nticas\n\nEsta palestra \u00e9 introdut\u00f3ria ao tema de Computa\u00e7\u00e3o Qu\u00e2ntica (originalmente dada para membros do grupo de pesquisa QPower), portanto considera um p\u00fablico com no\u00e7\u00f5es consider\u00e1veis de matem\u00e1tica, f\u00edsica e programa\u00e7\u00e3o (voc\u00ea deve saber pelo menos multiplicar matrizes para entender o que vai acontecer aqui). Por essa raz\u00e3o, a abordagem aqui ser\u00e1 mais _hands on_, e a linguagem utilizada ser\u00e1 o Python.\n\nA discuss\u00e3o n\u00e3o cobrir\u00e1 o tema de _quantum annealing_, sendo limitada somente a portas l\u00f3gicas qu\u00e2nticas.\nIt will just cover gate quantum computation model - there will be no annealing discussed here.\n\nDesse modo, ao cobrir apenas conceitos gerais e portas l\u00f3gicas qu\u00e2nticas, ser\u00e1 mais f\u00e1cil de estabelecer a distin\u00e7\u00e3o e a rela\u00e7\u00e3o entre a computa\u00e7\u00e3o cl\u00e1ssica e a computa\u00e7\u00e3o qu\u00e2ntica. A palestra utilizar\u00e1 usando Qiskit (pronuncia-se _qu\u00edskit_), uma biblioteca em _Python_ para computa\u00e7\u00e3o qu\u00e2ntica.\n\nAinda, para os que n\u00e3o possuem um s\u00f3lido conhecimento em Python, os aux\u00edlios visuais utilizados aqui servir\u00e3o de suporte para a compreens\u00e3o dos conceitos abordados.\n\nMeu email: ricardoprins@gmail.com\n\n\n```\n# Instala\u00e7\u00e3o do Qiskit\n!pip install qiskit\n```\n\n Collecting qiskit\n Downloading qiskit-0.29.1.tar.gz (12 kB)\n Collecting qiskit-terra==0.18.2\n Downloading qiskit_terra-0.18.2-cp37-cp37m-manylinux2010_x86_64.whl (6.1 MB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 6.1 MB 2.0 MB/s \n \u001b[?25hCollecting qiskit-aer==0.8.2\n Downloading qiskit_aer-0.8.2-cp37-cp37m-manylinux2010_x86_64.whl (18.0 MB)\n \u001b[K 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(8.8.0)\n Collecting inflection>=0.3.1\n Downloading inflection-0.5.1-py2.py3-none-any.whl (9.5 kB)\n Building wheels for collected packages: qiskit, dlx, docplex, python-constraint, yfinance\n Building wheel for qiskit (setup.py) ... \u001b[?25l\u001b[?25hdone\n Created wheel for qiskit: filename=qiskit-0.29.1-py3-none-any.whl size=11239 sha256=560b5df8dfb9db7b4a4f8cea4f3f7d0ff3796317657794ea9c91cb4007caa70a\n Stored in directory: /root/.cache/pip/wheels/7a/b8/82/921a13d73dac0322d4eea1eceb7a5419ac8f92cc0f54197456\n Building wheel for dlx (setup.py) ... \u001b[?25l\u001b[?25hdone\n Created wheel for dlx: filename=dlx-1.0.4-py3-none-any.whl size=5719 sha256=94dc04f74b2ab61418e81f123073e727556408c8e6da66a49aa290f09198c786\n Stored in directory: /root/.cache/pip/wheels/78/55/c8/dc61e772445a566b7608a476d151e9dcaf4e092b01b0c4bc3c\n Building wheel for docplex (setup.py) ... \u001b[?25l\u001b[?25hdone\n Created wheel for docplex: filename=docplex-2.21.207-py3-none-any.whl size=700544 sha256=f42221fcd76fd5186d44ac5a38efc670fdafd65e07b5b31370ef9f73424e4adb\n Stored in directory: /root/.cache/pip/wheels/d8/4e/62/e43a45757e70549e6aa4712ccfcf67440a203c278ecb68de49\n Building wheel for python-constraint (setup.py) ... \u001b[?25l\u001b[?25hdone\n Created wheel for python-constraint: filename=python_constraint-1.4.0-py2.py3-none-any.whl size=24081 sha256=17eadd2a02707d47c4d742c41d8b219f865371e0df1c63dc19c87177750c7cdf\n Stored in directory: /root/.cache/pip/wheels/07/27/db/1222c80eb1e431f3d2199c12569cb1cac60f562a451fe30479\n Building wheel for yfinance (setup.py) ... \u001b[?25l\u001b[?25hdone\n Created wheel for yfinance: filename=yfinance-0.1.63-py2.py3-none-any.whl size=23918 sha256=a7617ea60aa76ec188e8e7d9322fe5a2c3784613bebe9961c1ac913c688afff0\n Stored in directory: /root/.cache/pip/wheels/fe/87/8b/7ec24486e001d3926537f5f7801f57a74d181be25b11157983\n Successfully built qiskit dlx docplex python-constraint yfinance\n Installing collected packages: tweedledum, symengine, retworkx, python-constraint, ply, fastjsonschema, qiskit-terra, ntlm-auth, lxml, inflection, cryptography, yfinance, websocket-client, requests-ntlm, quandl, qiskit-ignis, pybind11, docplex, dlx, qiskit-ibmq-provider, qiskit-aqua, qiskit-aer, qiskit\n Attempting uninstall: lxml\n Found existing installation: lxml 4.2.6\n Uninstalling lxml-4.2.6:\n Successfully uninstalled lxml-4.2.6\n Successfully installed cryptography-3.4.8 dlx-1.0.4 docplex-2.21.207 fastjsonschema-2.15.1 inflection-0.5.1 lxml-4.6.3 ntlm-auth-1.5.0 ply-3.11 pybind11-2.7.1 python-constraint-1.4.0 qiskit-0.29.1 qiskit-aer-0.8.2 qiskit-aqua-0.9.5 qiskit-ibmq-provider-0.16.0 qiskit-ignis-0.6.0 qiskit-terra-0.18.2 quandl-3.6.1 requests-ntlm-1.1.0 retworkx-0.10.2 symengine-0.8.1 tweedledum-1.1.1 websocket-client-1.2.1 yfinance-0.1.63\n\n\n\n```\n# Inicializa\u00e7\u00f5es iniciais\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom math import sqrt, pi\nfrom qiskit import *\nfrom qiskit.extensions import Initialize\n```\n\n## Computa\u00e7\u00e3o e Informa\u00e7\u00e3o\n\nToda a teoria por tr\u00e1s de computadores est\u00e1 relacionada \u00e0 Teoria da Informa\u00e7\u00e3o. Ambos os campos caminham juntos, unidos e interrelacionados em seu progresso.\n\n\n\nDesse modo, ao falarmos de computa\u00e7\u00e3o cl\u00e1ssica e computa\u00e7\u00e3o qu\u00e2ntica, \u00e9 preciso come\u00e7ar de um ponto fundamental: ambas tratam de maneiras distintas de *processamento* de informa\u00e7\u00e3o. Assim, \u00e9 preciso definir o conceito de _processar_:\n\n$$\\text{Entrada } \\Longrightarrow \\text{ Sa\u00edda}$$\n\nQuando n\u00f3s manipulamos qualquer quantidade de informa\u00e7\u00e3o, e posteriormente a processamos, obtendo informa\u00e7\u00e3o como resultado desta opera\u00e7\u00e3o, estamos falando em _processamento_ de informa\u00e7\u00e3o, ou ainda em _computa\u00e7\u00e3o_ de informa\u00e7\u00e3o. Para simplificar, n\u00e3o ser\u00e3o considerados transmissores, receptores e poss\u00edveis fontes de ru\u00eddo: a ideia aqui \u00e9 somente a de entender o sentido b\u00e1sico de computar ou processar informa\u00e7\u00e3o em geral.\n\nEssa informa\u00e7\u00e3o nos ajuda a entender as diferen\u00e7as fundamentais entre a computa\u00e7\u00e3o cl\u00e1ssica e a computa\u00e7\u00e3o qu\u00e2ntica.\n\nA primeira distin\u00e7\u00e3o \u00e9 na unidade fundamental de informa\u00e7\u00e3o. Sem perda de generalidade, podemos evitar uma discuss\u00e3o profunda em Teoria de Informa\u00e7\u00e3o [(clique aqui para saber mais)](http://people.math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf), e falar apenas nas unidades fundamentais: _bits_ e _qubits_.\n\n## Bits e Qubits\n\nUm _bit_ \u00e9 a unidade fundamental de informa\u00e7\u00e3o da computa\u00e7\u00e3o cl\u00e1ssica. Ela representa uma entidade capaz de armazenar duas posi\u00e7\u00f5es *est\u00e1veis*, como um rel\u00e9 (interruptor). Ligado ou desligado, 0 ou 1, corrente ou aus\u00eancia de corrente.\n\nVamos recorrer aqui a uma nota\u00e7\u00e3o vetorial:\n\n$$1 = \\begin{pmatrix} 0\\\\1 \\end{pmatrix}\\space e\\space\\space 0 = \\begin{pmatrix} 1\\\\0 \\end{pmatrix} $$\n\n\n```\nzero = np.matrix('1;0')\num = np.matrix('0;1')\n```\n\n### Quatro opera\u00e7\u00f5es b\u00e1sicas com bits\n\nCom _bits_, h\u00e1 quatro opera\u00e7\u00f5es b\u00e1sicas poss\u00edveis:\n\n$$\\text{Identidade:}\\space\\space {f}(x) = x$$
                                        \n$$\\text{Nega\u00e7\u00e3o:}\\space\\space {f}(x) = \\neg x$$
                                        \n$$\\text{Constante-0:}\\space\\space {f}(x) = 0$$
                                        \n$$\\text{Constante-1:}\\space\\space {f}(x) = 1$$
                                        \n\nUsando nota\u00e7\u00e3o matricial:\n\n\n```\n# Identidade:\nidentidade = np.matrix('1 0; 0 1')\nprint('A matriz identidade:')\nprint(identidade,'\\n')\n\n# Zero multiplicado pela matriz identidade retorna zero,\nresult = identidade @ zero\nprint(result, '\\n')\n\n# e ainda: um, multiplicado pela matriz identidade retorna um, como esperado.\nresult = identidade @ um\nprint(result)\n```\n\n A matriz identidade:\n [[1 0]\n [0 1]] \n \n [[1]\n [0]] \n \n [[0]\n [1]]\n\n\n\n```\n# Nega\u00e7\u00e3o:\nneg = np.matrix('0 1; 1 0')\nprint('A matriz nega\u00e7\u00e3o:')\nprint(neg, '\\n')\n\n# Zero multiplicado pela matriz nega\u00e7\u00e3o retorna um,\nresult = neg @ zero\nprint(result, '\\n')\n\n# e um multiplicado pela matriz nega\u00e7\u00e3o retorna zero, como esperado.\nresult = neg @ um\nprint(result)\n```\n\n A matriz nega\u00e7\u00e3o:\n [[0 1]\n [1 0]] \n \n [[0]\n [1]] \n \n [[1]\n [0]]\n\n\n\n```\n# Constante-0:\nc_zero = np.matrix('1 1; 0 0')\nprint('A matriz Constante-0')\nprint(c_zero, '\\n')\n\n# Multiplicando zero pela matriz Constante-0 retornar\u00e1 zero,\nresult = c_zero @ zero\nprint(result, '\\n')\n\n# e multiplicando um pela matriz Constante-0 retornar\u00e1 zero, como esperado.\nresult = c_zero @ um\nprint(result)\n```\n\n A matriz Constante-0\n [[1 1]\n [0 0]] \n \n [[1]\n [0]] \n \n [[1]\n [0]]\n\n\n\n```\n# Constante-1:\nc_um = np.matrix('0 0; 1 1')\nprint('A matriz Constante-1:')\nprint(c_um, '\\n')\n\n# Multiplicando zero pela matriz Constante-1 retornar\u00e1 um,\nresult = c_um @ zero\nprint(result, '\\n')\n\n# e multiplicando um pela matriz Constante-1 retornar\u00e1 um, como esperado.\nresult = c_um @ um\nprint(result)\n```\n\n A matriz Constante-1:\n [[0 0]\n [1 1]] \n \n [[0]\n [1]] \n \n [[0]\n [1]]\n\n\n### Opera\u00e7\u00f5es revers\u00edveis com bits\n\nPara nosso prop\u00f3sito aqui, \u00e9 conveniente definir _opera\u00e7\u00f5es revers\u00edveis_ como **quaisquer processos** (cf. a defini\u00e7\u00e3o supracitada de processamento) **que permitam que a entrada seja identificada ao se conhecer a sa\u00edda e da opera\u00e7\u00e3o utilizada.**\n\nNas opera\u00e7\u00f5es acima, quais s\u00e3o revers\u00edveis e quais n\u00e3o s\u00e3o?\n\n\u00c9 poss\u00edvel na computa\u00e7\u00e3o cl\u00e1ssica ocorrer opera\u00e7\u00f5es n\u00e3o-revers\u00edveis, e isso n\u00e3o \u00e9 um problema.\n\n### Produto Kronecker\n\nPara a compreens\u00e3o plena dos pr\u00f3ximos conceitos, \u00e9 necess\u00e1rio explicar uma opera\u00e7\u00e3o que nem sempre \u00e9 ensinada nas aulas de \u00c1lgebra Linear: o produto Kronecker. [(para maiores explica\u00e7\u00f5es, clique aqui)](https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf).\n\nTamb\u00e9m \u00e9 conveniente estudar sobre produtos tensoriais, para maior compreens\u00e3o, uma vez que produtos Kronecker s\u00e3o um caso particular de produtos tensoriais.\n\nEsses exemplos s\u00e3o satisfat\u00f3rios para o fim de explicar a opera\u00e7\u00e3o:

                                        \n$$\\begin{pmatrix} x_{0} \\\\ x_{1} \\end{pmatrix} \\otimes \\begin{pmatrix} y_{0} \\\\ y_{1} \\end{pmatrix} = \\begin{pmatrix} x_{0}y_{0} \\\\ x_{0}y_{1} \\\\ x_{1}y_{0} \\\\ x_{1}y_{1} \\end{pmatrix} $$\n\n\nOutro exemplo:

                                        \n\n$$\\begin{pmatrix} x_{0} \\\\ x_{1} \\end{pmatrix} \\otimes \\begin{pmatrix} y_{0} \\\\ y_{1} \\end{pmatrix} \\otimes \\begin{pmatrix} z_{0} \\\\ z_{1} \\end{pmatrix} = \\begin{pmatrix} x_{0}y_{0}z_{0} \\\\ x_{0}y_{0}z_{1} \\\\ x_{0}y_{1}z_{0} \\\\ x_{0}y_{1}z_{1} \\\\ x_{1}y_{0}z_{0} \\\\ x_{1}y_{0}z_{1} \\\\ x_{1}y_{1}z_{0} \\\\ x_{1}y_{1}z_{1} \\end{pmatrix} $$\n\n\n### A representa\u00e7\u00e3o de m\u00faltiplos bits\n\nPara representar os estados poss\u00edveis de m\u00faltiplos bits, utiliza-se o produto Kronecker de vetores de um bit. Novamente \u00e9 preciso enfatizar que essa \u00e9 uma explica\u00e7\u00e3o simplificada, por\u00e9m \u00fatil o suficiente para a compreens\u00e3o das opera\u00e7\u00f5es com m\u00faltiplos bits, e suficientemente satisfat\u00f3ria para mostrar o poder da computa\u00e7\u00e3o qu\u00e2ntica.\n\nA representa\u00e7\u00e3o tensorial de m\u00faltiplos bits \u00e9 chamada de _estado produto_. Essa representa\u00e7\u00e3o \u00e9 util\u00edssima, pois permite a fatora\u00e7\u00e3o do _estado produto_ em seus respectivos estados individuais.\n\nVamos usar como exemplo os estados produto para dois bits:\n\n$$\\text{Temos quatro estados poss\u00edveis para dois bits:}$$\n\n$$00, 01, 10\\space ou\\space 11$$:\n\n$$\\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} \\otimes \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} = \\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\\\ 0 \\end{pmatrix} \\space\\space\\text{0 com 0,}$$\n\n$$\\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} \\otimes \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 1 \\\\ 0 \\\\ 0 \\end{pmatrix} \\space\\space\\text{0 com 1,}$$\n\n$$\\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} \\otimes \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 0 \\\\ 1 \\\\ 0 \\end{pmatrix} \\space\\space\\text{1 com 0,}$$\n\n$$\\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} \\otimes \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 1 \\end{pmatrix} \\space\\space\\text{e 1 com 1}$$\n\n\n```\n# usando NumPy:\n\nzero_zero = np.kron(zero, zero) \nzero_um = np.kron(zero, um)\num_zero = np.kron(um, zero)\num_um = np.kron(um, um)\nprint(zero_zero,'\\n\\n', zero_um,'\\n\\n', um_zero,'\\n\\n', um_um)\n```\n\n [[1]\n [0]\n [0]\n [0]] \n \n [[0]\n [1]\n [0]\n [0]] \n \n [[0]\n [0]\n [1]\n [0]] \n \n [[0]\n [0]\n [0]\n [1]]\n\n\nAssim, como se pode observar, h\u00e1 quatro _estados produto_ poss\u00edveis para um par de bits.\n\nConclui-se, portanto, que o n\u00famero de _estados produto_ poss\u00edveis em _n_ bits \u00e9 igual a $ 2^{n}$.\n\n### CNOT: Uma opera\u00e7\u00e3o com m\u00faltiplos bits\n\nUm dos blocos fundamentais para a computa\u00e7\u00e3o revers\u00edvel \u00e9 a opera\u00e7\u00e3o CNOT.\n\nA opera\u00e7\u00e3o CNOT funciona em pares de _bits_; um \u00e9 designado como _bit controle_ e o outro como _bit alvo_.\n\nO mecanismo de funcionamento dessa _porta_ (daqui em diante, essas opera\u00e7\u00f5es ser\u00e3o chamadas de _portas_) \u00e9 o seguinte:\n\n- Se o bit controle for 1, o bit alvo \u00e9 invertido\n- Se o bit controle for 0, o bit alvo segue inalterado\n- O bit controle nunca \u00e9 alterado\n- O bit mais significativo (aquele \u00e0 esquerda) \u00e9 o bit controle, e o outro, o bit alvo\n\nExemplo:\n\n\n```\ncnot = np.matrix('1 0 0 0; 0 1 0 0; 0 0 0 1; 0 0 1 0')\nprint(cnot)\n```\n\n [[1 0 0 0]\n [0 1 0 0]\n [0 0 0 1]\n [0 0 1 0]]\n\n\n\n```\n# CNOT on 00:\nresult = cnot @ zero_zero \nprint('Como o bit controle \u00e9 zero, o bit alvo segue sem altera\u00e7\u00e3o,\\n')\nprint(result, '\\n')\n\n# CNOT on 01\nresult = cnot @ zero_um\nprint('o que da mesma maneira acontece aqui.\\n')\nprint(result)\n```\n\n Como o bit controle \u00e9 zero, o bit alvo segue sem altera\u00e7\u00e3o,\n \n [[1]\n [0]\n [0]\n [0]] \n \n o que da mesma maneira acontece aqui.\n \n [[0]\n [1]\n [0]\n [0]]\n\n\n\n```\n# CNOT on 10:\nresult = cnot @ um_zero \nprint('No entanto, ele \u00e9 invertido neste exemplo,\\n')\nprint(result, '\\n')\n\n# CNOT on 11\nresult = cnot @ um_um\nprint('tal como no seguinte:\\n')\nprint(result)\n```\n\n No entanto, ele \u00e9 invertido neste exemplo,\n \n [[0]\n [0]\n [0]\n [1]] \n \n tal como no seguinte:\n \n [[0]\n [0]\n [1]\n [0]]\n\n\n### _Qubits_ - as unidades fundamentais de informa\u00e7\u00e3o qu\u00e2ntica\n\nO primeiro ponto a ser mencionado aqui \u00e9 que todas as opera\u00e7\u00f5es com _qubits_ DEVEM ser revers\u00edveis (e o s\u00e3o).\n\nDisso decorre que elas s\u00e3o seu pr\u00f3prio inverso - ao aplicar a mesma opera\u00e7\u00e3o duas vezes em sequ\u00eancia, o valor inicial necessariamente ser\u00e1 obtido.\n\nPara os fins deste treinamento, a afirma\u00e7\u00e3o acima ser\u00e1 tratada de forma axiom\u00e1tica, bem como a seguinte: a mec\u00e2nica qu\u00e2ntica \u00e9, al\u00e9m de revers\u00edvel, unit\u00e1ria. A demonstra\u00e7\u00e3o disso decorre da equa\u00e7\u00e3o de Schr\u00f6ddinger.\n\nA \u00fanica exce\u00e7\u00e3o aqui \u00e9 para as opera\u00e7\u00f5es/portas de medi\u00e7\u00e3o, uma vez que estas n\u00e3o s\u00e3o, de fato, opera\u00e7\u00f5es ou portas.\n\nAinda que todas as opera\u00e7\u00f5es acima feitas tenham considerado _bits_, todas elas s\u00e3o *igualmente v\u00e1lidas para _qubits_*. \u00c9 poss\u00edvel afirmar que os _bits_ vistos em sua forma vetorial s\u00e3o casos particulares de vetores _qubits_.\n\n **Defini\u00e7\u00e3o:**

                                        \n Um qubit pode ser representado por um vetor $\\begin{pmatrix} a \\\\ b \\end{pmatrix} : \\space a,b \\in \\mathbb{C}\\space\\space, \\lVert a\\rVert^{2} + \\lVert b\\rVert^{2} = 1$\n\nExemplos: $\\begin{pmatrix} -1 \\\\ 0 \\end{pmatrix}, \\begin{pmatrix} \\frac12 \\\\ \\frac{\\sqrt3}{2} \\end{pmatrix}, \\begin{pmatrix} \\frac{1}{\\sqrt2} \\\\ \\frac{1}{\\sqrt2} \\end{pmatrix}, \\begin{pmatrix} \\frac{1}{\\sqrt2} \\\\ \\frac{-1}{\\sqrt2} \\end{pmatrix}$\n\nAnalogamente aos _bits_, \u00e9 plaus\u00edvel buscar uma interpreta\u00e7\u00e3o pr\u00e1tica para o comportamento de um _qubit_: ele, tal como seu par da computa\u00e7\u00e3o cl\u00e1ssica, pode representar 0 ou 1 (um par de possibilidades, presen\u00e7a de corrente ou aus\u00eancia, etc) ou ambos os valores poss\u00edveis ao mesmo tempo!\n\n\n### Superposi\u00e7\u00e3o qu\u00e2ntica - uma explica\u00e7\u00e3o matem\u00e1tica\n\nAgora \u00e9 o momento de encarar as defini\u00e7\u00f5es f\u00edsicas mais complicadas relacionadas aos qubits.\n\nVamos, por um momento, deixar de lado o fato de que um qubit pode assumir ao mesmo tempo os valores 0 e 1, e considerar apenas as quest\u00f5es matem\u00e1ticas decorrentes dessa caracter\u00edstica.\n\nAlgo importante tamb\u00e9m verific\u00e1vel na mec\u00e2nica qu\u00e2ntica \u00e9 que no momento em que se mede um qubit, ele instantaneamente entrar\u00e1 em colapso para um dos valores, seja 0 ou 1.\n\nCom isso, de maneira igualmente axiom\u00e1tica, afirma-se que a probabilidade que um qubit $\\begin{pmatrix} a \\\\ b \\end{pmatrix}$ tem de _colapsar_ para um valor 0 \u00e9 $\\lVert a\\rVert^{2}$ e para 1, $\\lVert b\\rVert^{2}$.\n\nEsse axioma facilita a compreens\u00e3o do comportamento de um sistema com m\u00faltiplos qubits.\n\n\n### M\u00faltiplos qubits\n\nA representa\u00e7\u00e3o de um sistema com mais de um qubit \u00e9 an\u00e1loga \u00e0 que foi feita para bits cl\u00e1ssicos - utiliza-se o produto Kronecker das matrizes de qubits individuais.\n\n\n```\n## Exemplo com NumPy:\n\n# Partindo do qubit (1/sqrt(2), 1/sqrt(2)):\nqubit = np.array([[1/sqrt(2)],[1/sqrt(2)]])\n\n# O estado produto desse qubit \u00e9 o produto Kronecker dele consigo mesmo:\nproduct_state = np.kron(qubit, qubit)\nprint(product_state)\n# Note que o vetor resultante para o estado produto tamb\u00e9m \u00e9 unit\u00e1rio.\n```\n\n [[0.5]\n [0.5]\n [0.5]\n [0.5]]\n\n\n\u00c9 poss\u00edvel observar no estado produto acima, que o sistema possui uma probabilidade de \n$\\lVert\\frac12\\rVert^{2}= \\frac14 $ (vamos chamar de amplitude) de colapsar nos estados 00, 01, 10 ou 11 ap\u00f3s a medi\u00e7\u00e3o.\n\n## Portas Qu\u00e2nticas (ou ainda, opera\u00e7\u00f5es com qubits)\n\nTodas as opera\u00e7\u00f5es descritas acima s\u00e3o aplic\u00e1veis a qubits, com a seguinte ressalva: elas devem ser revers\u00edveis.\n\n### Hadamard\n\n_Desse ponto em diante, iniciaremos o uso da biblioteca Qiskit para visualizar tamb\u00e9m a nota\u00e7\u00e3o circuito._\n\nA porta Hadamard (H-gate) leva um qubit 0 ou 1 para um estado de superposi\u00e7\u00e3o _exactly equal_:\n\n$$ H = \\begin{pmatrix} \\frac1{\\sqrt2} \\ \\frac1{\\sqrt2} \\\\ \\frac1{\\sqrt2} \\ \\frac{-1}{\\sqrt2} \\end{pmatrix}$$\n\n\n```\nhadamard = np.array([[1/sqrt(2),1/sqrt(2)],[1/sqrt(2),-1/sqrt(2)]])\n\n# Applying Hadamard to qubit 0,\nresult = hadamard @ zero\nprint(result, '\\n')\n\n# and to qubit 1:\nresult = hadamard @ um\nprint(result)\n```\n\n [[0.70710678]\n [0.70710678]] \n \n [[ 0.70710678]\n [-0.70710678]]\n\n\nO resultado acima mostra vetores qubit em superposi\u00e7\u00e3o, com a mesma probabilidade de colapsarem em ambos estados 0 ou 1.\n\n**Pergunta**\nO que acontece se aplicarmos a H-gate a um desses qubits resultantes da superposi\u00e7\u00e3o acima?\n\n### A nota\u00e7\u00e3o circuito\n\nUma das nota\u00e7\u00f5es utilizadas para representar uma sequ\u00eancia de opera\u00e7\u00f5es com qubits \u00e9 a nota\u00e7\u00e3o circuito.\n\nUma das vantagens da biblioteca Qiskit \u00e9 a de permitir a visualiza\u00e7\u00e3o das sucessivas opera\u00e7\u00f5es utilizando essa opera\u00e7\u00e3o, conforme exemplo abaixo:\n\n\n```\ndef challengeTwo(qc, qubit):\n qc.x(qubit)\n qc.h(qubit)\n qc.x(qubit)\n qc.h(qubit)\n qc.x(qubit)\n return qc\n\n\n\nquantum_circuit = QuantumCircuit(1)\ninitial_state = [1,0]\ninitializer = Initialize(initial_state)\ninitializer.label = \"init\"\nchallengeTwo(quantum_circuit, 0)\nquantum_circuit.draw()\n\n```\n\n\n\n\n
                                             \u250c\u2500\u2500\u2500\u2510\u250c\u2500\u2500\u2500\u2510\u250c\u2500\u2500\u2500\u2510\u250c\u2500\u2500\u2500\u2510\u250c\u2500\u2500\u2500\u2510\nq_0: \u2524 X \u251c\u2524 H \u251c\u2524 X \u251c\u2524 H \u251c\u2524 X \u251c\n     \u2514\u2500\u2500\u2500\u2518\u2514\u2500\u2500\u2500\u2518\u2514\u2500\u2500\u2500\u2518\u2514\u2500\u2500\u2500\u2518\u2514\u2500\u2500\u2500\u2518
                                        \n\n\n\nO circuito acima exibe uma sequ\u00eancia de opera\u00e7\u00f5es realizadas com um qubit em um circuito (duh!), de onde deriva-se o nome _nota\u00e7\u00e3o circuito_.\n\nPara este caso, partindo de um qubit 0, as opera\u00e7\u00f5es a seguir s\u00e3o realizadas:\n\n\n1. Uma porta nega\u00e7\u00e3o (chamada de Pauli-X) \n2. Uma porta Hadamard\n3. Outra Pauli-X \n4. Outra Hadamard\n5. Mais uma Pauli-X\n\n\n**Pergunta** Voc\u00ea consegue determinar o resultado da medi\u00e7\u00e3o do qubit ap\u00f3s essas cinco opera\u00e7\u00f5es?\n\n\n\n### Pauli\n\nPara os que possuem familiaridade com \u00e1lgebra linear, as matrizes Pauli n\u00e3o ser\u00e3o novidade: um conjunto de tr\u00eas matrizes 2x2 que s\u00e3o hermitianas e unit\u00e1rias.\n\nCome\u00e7aremos pela Pauli X - carinhosamente chamada de \"porta X\"\n\n\n####Pauli X ou porta NOT\n\n\n$$ X = \\begin{pmatrix} 0 \\ 1 \\\\ 1 \\ 0 \\end{pmatrix}$$\n\nEssa \u00e9 an\u00e1loga \u00e0 matriz nega\u00e7\u00e3o que utilizamos para opera\u00e7\u00f5es com bits. Observemos seu comportamento com qubits:\n\n\n```\nfrom qiskit.visualization import plot_bloch_multivector\n\n# Vamos desenhar um circuito com um qubit inicializado no estado 0\n# e aplicar uma porta X a ele:\n\nqc = QuantumCircuit(1)\nqc.x(0)\n\n# A representa\u00e7\u00e3o abaixo \u00e9 chamada de Esfera de Bloch, e \u00e9 usada\n# para representar qubits e seus estados. Como podemos ver abaixo,\n# esse \u00e9 o estado em que o qubit \"repousa\"\n# ap\u00f3s sua passagem pelo circuito.\n\ndef displayBloch():\n backend = Aer.get_backend('statevector_simulator')\n output = execute(qc, backend).result().get_statevector()\n return plot_bloch_multivector(output)\n\ndisplayBloch()\n```\n\nComo esperado, o estado resultante \u00e9 1: \numa rota\u00e7\u00e3o de $\\pi$ radianos sobre o eixo _x_ da esfera.\n\nVoc\u00ea tamb\u00e9m ver\u00e1 essa porta sendo chamada de porta _NOT_.\n\n#### Pauli Y\n\nAn\u00e1loga \u00e0 porta X, esta rotaciona um qubit\n$\\pi$ radianos sobre o eixo _y_ da esfera:\n\n$$ Y = \\begin{pmatrix} 0 \\ -i \\\\ i \\ 0 \\end{pmatrix}$$\n\n\n```\n# Inicializaremos um qubit no estado 1,\nqc = QuantumCircuit(1)\ninitial_state = [0,1]\nqc.initialize(initial_state,0)\nqc.draw()\n```\n\n\n\n\n
                                             \u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510\nq_0: \u2524 Initialize(0,1) \u251c\n     \u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2518
                                        \n\n\n\n\n```\n# essa \u00e9 sua exibi\u00e7\u00e3o na esfera de Bloch\ndisplayBloch()\n```\n\n\n```\n# e, a rota\u00e7\u00e3o resultante da Y-gate:\nqc.y(0)\ndisplayBloch()\n```\n\n#### Pauli Z\n\nRota\u00e7\u00e3o de $\\pi$ radianos sobre o eixo _z_:\n\n$$ Z = \\begin{pmatrix} 1 \\ 0 \\\\ 0 \\ {-1} \\end{pmatrix}$$\n\n\n\n```\n# Inicializamos um qubit cujo \u00e2ngulo em rela\u00e7\u00e3o ao eixo z\n# possa ser facilmente observ\u00e1vel.\ninitial_state = [sqrt(3)/2,1/2]\nqc.initialize(initial_state,0)\ndisplayBloch()\n```\n\n\n```\n# aplicamos a Z-gate sobre esse qubit, verificando a rota\u00e7\u00e3o\nqc.z(0)\ndisplayBloch()\n```\n\n### $R_\\phi$ (ou $R_z$)\n\nA porta $R_\\phi$ \u00e9 uma porta parametrizada que executa uma rota\u00e7\u00e3o de $\\phi$ graus sobre o eixo Z:\n\n_Disso pode-se deduzir que a porta Pauli-Z \u00e9 um caso particular da porta $R_\\phi$ com $\\phi=\\pi$._\n\n\n$$ R_\\phi = \\begin{pmatrix} 1 \\ 0 \\\\ 0 \\ e^{i\\phi} \\end{pmatrix} , com \\space\\space \\phi \\in \\mathbb R $$\n\n### Identidade\n\nA porta I (Identidade) faz exatamente o que seu nome indica: nada.\n\nNo entanto, h\u00e1 algumas aplica\u00e7\u00f5es pr\u00e1ticas para seu uso, as quais fogem ao escopo desta palestra.\n\n### T\n\nA porta T \u00e9 outro caso particular da porta $R_\\phi$, com $\\phi = \\frac{\\pi}{4}$:\n\n$$ T = \\begin{pmatrix} 1 \\ 0 \\\\ 0 \\ e^{i\\frac{\\pi}{4}} \\end{pmatrix} \\text{e } T\\dagger = \\begin{pmatrix} 1 \\ 0 \\\\ 0 \\ e^{-i\\frac{\\pi}{4}} \\end{pmatrix}$$ \n\n### S\n\nA porta S \u00e9 ainda outro caso particular da porta $R_\\phi$, com $\\phi = \\frac{\\pi}{2}$:\n\n$$ S = \\begin{pmatrix} 1 \\ 0 \\\\ 0 \\ e^{i\\frac{\\pi}{2}} \\end{pmatrix} \\text{and } S\\dagger = \\begin{pmatrix} 1 \\ 0 \\\\ 0 \\ e^{-i\\frac{\\pi}{2}} \\end{pmatrix}$$ \n\n\u00c9 conveniente notar que aplicar duas portas S \u00e9 o mesmo que aplicar uma porta Z, por raz\u00f5es \u00f3bvias.\n\n### $U_3$\n\nTodas as portas acima podem ser consideradas como casos particulares da porta $U_3$.\n\nDevido a sua evidente complexidade de exibi\u00e7\u00e3o, dificilmente ela \u00e9 utilizada em diagramas.\n\nUma observa\u00e7\u00e3o importante: a escolha de rota\u00e7\u00f5es nas portas acima sobre o eixo z \u00e9 apenas uma conven\u00e7\u00e3o computacional para facilitar a vida dos programadores e evitar que estes sejam institucionalizados em manic\u00f4mios.\n\n$$ U_3(\\theta,\\phi,\\lambda) = \\begin{pmatrix} \\cos(\\frac\\theta2) \\ -e^{i\\lambda}\\sin(\\frac\\theta2) \\\\ e^{i\\phi}\\sin(\\frac\\theta2) \\ e^{i\\phi+i\\theta}\\cos(\\frac\\theta2) \\end{pmatrix}$$\n\n# OBRIGADO A TODOS!\n", "meta": {"hexsha": "c9c7fdd554ad40e4ae5a8d3d26f9e13957965fb9", "size": 411217, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Parte 1 - Intro.ipynb", "max_stars_repo_name": "ricardoprins/intro-quantum", "max_stars_repo_head_hexsha": "fd846baad856bf9e1b2104d223782475426bf714", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2021-09-15T17:11:26.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-16T00:58:03.000Z", "max_issues_repo_path": "Parte 1 - Intro.ipynb", "max_issues_repo_name": "ricardoprins/intro-quantum", "max_issues_repo_head_hexsha": "fd846baad856bf9e1b2104d223782475426bf714", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Parte 1 - Intro.ipynb", "max_forks_repo_name": "ricardoprins/intro-quantum", "max_forks_repo_head_hexsha": "fd846baad856bf9e1b2104d223782475426bf714", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 289.5894366197, "max_line_length": 123557, "alphanum_fraction": 0.9070660989, "converted": true, "num_tokens": 10308, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778403, "lm_q2_score": 0.658417500561683, "lm_q1q2_score": 0.41704840264088017}} {"text": "# Derivative of CTC Loss\n\n\u5173\u4e8eCTC\u7684\u4ecb\u7ecd\u5df2\u7ecf\u6709\u5f88\u591a\u4e0d\u9519\u7684\u6559\u7a0b\u4e86\uff0c\u4f46\u662f\u5b8c\u6574\u7684\u63cf\u8ff0CTCLoss\u7684\u524d\u5411\u548c\u53cd\u5411\u8fc7\u7a0b\u7684\u5f88\u5c11\uff0c\u800c\u4e14\u6709\u4e9b\u516c\u5f0f\u63a8\u5bfc\u7701\u7565\u548c\u9519\u8bef\u3002\u672c\u6587\u4e3b\u8981\u5173\u6ce8CTC Loss\u7684\u68af\u5ea6\u662f\u5982\u4f55\u8ba1\u7b97\u7684\uff0c\u5173\u4e8eCTC\u7684\u4ecb\u7ecd\u8fd9\u91cc\u4e0d\u505a\u8fc7\u591a\u8d58\u8ff0\uff0c\u5177\u4f53\u53c2\u770b\u6587\u672b\u53c2\u8003\u3002\n\nCTC\u4e3b\u8981\u5e94\u7528\u4e8e\u8bed\u97f3\u548cOCR\u4e2d\uff0c\u5df2\u8bed\u97f3[Deepspeech2](https://arxiv.org/abs/1512.02595)\u6a21\u578b\u4e3a\u4f8b\uff0cCTC\u7684\u7f51\u7edc\u4e00\u822c\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5305\u542bsoftmax\u548cCTCLoss\u4e24\u90e8\u5206\u3002\u53cd\u5411\u4f20\u64ad\u9700\u8981\u6c42\u5f97loss L\u76f8\u5bf9\u4e8elogits $u^i$\u200b\u7684\u68af\u5ea6\u3002\u4e0b\u9762\u5148\u4ecb\u7ecdCTCLoss\u7684\u524d\u5411\u8ba1\u7b97\u3002\n\n> \u56fe\u7247\u6765\u6e90\u4e8e\u6587\u672b\u53c2\u8003\n\n\n\n## CTC Loss \u7684\u8ba1\u7b97\n\nCTC\u4e2dpath\u7684\u5b9a\u4e49\u4e0e\u6982\u7387\u7684\u8ba1\u7b97\u5982\u4e0b\uff1a\n\n\n\npath \u662f $ L'^T$\u200b\u200b\u7684\u5143\u7d20\uff0c\u7528 $ \\pi $\u200b\u200b\u8868\u793a\u3002 $ \\textbf{x} $\u200b\u200b \u662f\u8f93\u5165\u7279\u5f81\uff0c$\\textbf{y}$\u200b\u200b \u662f\u8f93\u51falabel\uff0c \u90fd\u662f\u5e8f\u5217\u3002 $ L $\u200b\u200b \u662f\u8f93\u51fa\u7684 vocab, L\u2018 \u662f $ L \\cup {blank}$\u200b\u200b\u3002 $y_{\\pi_{t}}^t$\u200b\u200b \u8868\u793a\u5728t\u65f6\u523b\uff0c$\\pi_{t}$\u200b\u200b label\u65f6\u7684\u89c2\u5bdf\u6982\u7387\u3002\u5176\u4e2d$\\pi_{t}$\u200b\u200b \u8868\u793a $\\pi$\u200b\u200b path\u5728t\u65f6\u523b\u7684label\u3002$\\pi$\u200b\u200b \u662f $\\textbf{y}$\u200b\u200b \u4e0e $ \\textbf{x}$\u200b\u200b \u7684\u4e00\u4e2aalignment\uff0c\u957f\u5ea6\u662f$T$\u200b\u200b\uff0c\u53d6\u503c\u7a7a\u95f4\u4e3a$L'$\u200b\u200b\u200b\u3002path\u4e5f\u79f0\u4e3aalignment\u3002\n\n\u516c\u5f0f\uff082\uff09\u89e3\u91ca\u4e86\u7ed9\u5b9a\u8f93\u5165 $\\textbf{x}$\u200b \uff0c\u8f93\u51fa $ \\pi $\u200b path \u7684\u6982\u7387\uff0c\u5373\u4ece\u65f6\u95f4t=1\u5230T\u6bcf\u4e2a\u65f6\u95f4\u70b9\u7684\u6982\u7387 $y_{\\pi_{t}}^t$\u200b \u76f8\u4e58\u3002\n\n\u6c42\u51fa\u5355\u6761path\u540e\uff0c\u5c31\u53ef\u4ee5\u8ba1\u7b97$p(l \\mid x)$\u200b \u7684\u6982\u7387\uff0c\u8ba1\u7b97\u5982\u4e0b\uff1a\n\n\n\n\u8fd9\u91cc\u8fb9 $\\mathcal{B}$ \u5c31\u662f\u6620\u5c04\uff0c \u5373\u6240\u6709\u591a\u5bf9\u4e00\u7684\u6620\u5c04\uff08many-to-one mapping )\u7684\u96c6\u5408\u3002 \u8fd9\u6837\u5c31\u7b97\u51fa\u6765\u5bf9\u5e94\u4e00\u4e2a\u771f\u6b63\u7684 label $\\textbf{l}$ \u7684\u6982\u7387\u4e86\uff0c\u8fd9\u91cc\u662f\u6c42\u548c\u3002 \u6c42\u548c\u7684\u539f\u56e0\u5c31\u662f aab \u548c abb \u90fd\u662f\u5bf9\u5e94\u6210ab, \u6240\u4ee5 aab \u7684\u6982\u7387 + abb \u7684\u6982\u7387\u624d\u662f\u751f\u6210ab\u7684\u6982\u7387\u3002 \n\n\u516c\u5f0f\uff083\uff09\u89e3\u91ca\u4e86\u7ed9\u5b9a\u8f93\u5165 $\\mathbf{x}$\u200b\u200b\u200b\u200b\u200b\u200b \uff0c\u6c42\u8f93\u51fa$\\mathbf{l}$\u200b\u200b\u200b\u200b\u200b\u200b \u7684\u6982\u7387\uff0c \u5373\u6240\u6709\u96c6\u5408 $\\mathcal{B}^{-1} (\\mathbf{l})$\u200b\u200b\u200b\u200b\u200b\u200b\u200b\u200b\u200b\u200b \u4e2d path\u7684\u6982\u7387\u548c\u3002\n\n### CTC forward-backward \u7b97\u6cd5\n\nCTC\u7684\u4f18\u5316\u91c7\u7528\u7b97\u6700\u5927\u4f3c\u7136\u4f30\u8ba1[MLE (maximum likelihood estimation)](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation), \u8fd9\u4e2a\u548c\u795e\u7ecf\u7f51\u7edc\u672c\u8eab\u7684\u8bad\u7ec3\u8fc7\u7a0b\u662f\u4e00\u81f4\u7684\u3002\n\n\u8fd9\u4e2aCTC \u8ba1\u7b97\u8fc7\u7a0b\u7c7b\u4f3cHMM\u7684 [forward-backward algorithm](https://en.wikipedia.org/wiki/Forward%E2%80%93backward_algorithm)\uff0c\u4e0b\u9762\u5c31\u662f\u8fd9\u4e2a\u7b97\u6cd5\u7684\u63a8\u5bfc\u8fc7\u7a0b\uff1a\n\n\n\n\u4e0a\u56fe\u4e2d\u7684\u5b9a\u4e49\u5f88\u6e05\u695a\uff0c \u4f46\u662f$ \\alpha_{t-1}(s) $ and $ \\alpha_{t-1}(s-1)$ \u548c $\\alpha_t(s)$ \u7684\u5173\u7cfb\u4e5f\u4e0d\u90a3\u4e48\u597d\u770b\u51fa\u6765\uff0c\u4e0b\u56fe\u7ed9\u51fa\u4e86\u5177\u4f53\u7684\u5173\u4e8e $\\alpha_t(s)$ \u7684\u63a8\u5bfc\u8fc7\u7a0b\uff1a\n\n\n\n\n\n\u8fd9\u91cc\u7684\u516c\u5f0f\u6bd4\u8f83\u9002\u5408\u7528\u4e0b\u9762\u7684\u56fe\u6765\u7406\u89e3\uff0c$\\alpha_1(1)$\u200b\u200b\u200b\u200b \u5176\u5b9e\u5bf9\u5e94\u7684\u5c31\u662f\u4e0b\u56fe\u4e2d\u5de6\u4e0a\u89d2\u767d\u8272\u7684\u5706\u5708\u3002 \u5c31\u662f\u4e0a\u6765\u7b2c\u4e00\u4e2a\u662fblank \u7684\u6982\u7387\uff0c \u800c $\\alpha_1(2)$\u200b\u200b\u200b\u200b\u662flabel l \u7684\u7b2c\u4e00\u4e2a\u5b57\u6bcd\u3002 \u8fd9\u91cc\u8fb9\u6211\u4eec\u5047\u8bbe\u6bcf\u4e2a\u5b57\u6bcd\u4e4b\u95f4\u90fd\u63d2\u5165\u4e86\u7a7a\u767d\uff0c\u5373label l\u6269\u5c55\u6210l'\uff0c\u4f8b\u5982\uff0cl=[a, b, b, c]\uff0c l'=[-, a, -, b, -, b, -, c, -]\u3002 \u7136\u540e\u5bf9\u4e8e\u5176\u4ed6\u5706\u70b9\uff0c\u5728\u65f6\u95f4\u662f1 \u7684\u60c5\u51b5\u4e0b\u6982\u7387\u90fd\u662f 0. Figure 3\u4e2d\u6a2a\u8f74\u662f\u65f6\u95f4 t\uff0c\u4ece\u5de6\u5230\u53f3\u662f1\u5230T\uff1b\u7eb5\u8f74\u662fs\uff08sequence\uff09\uff0c\u4ece\u4e0a\u5230\u4e0b\u662f 1 \u5230 $\\mathbf{\\mid l' \\mid}$\u200b\u200b\u200b\u200b.\n\n\n\n\u63a5\u4e0b\u6765\u6211\u4eec\u5206\u6790\u9012\u5f52\u516c\u5f0f (resursion)\uff0c\u66f4\u591a\u4ecb\u7ecd\u53ef\u4ee5\u53c2\u770b [2]. \u516c\u5f0f6\u5206\u60c5\u51b5\u8003\u8651:\n\n* \u7b2c\u4e00\u79cd\u60c5\u51b5\u5c31\u662f\u5f53\u524d\u7684label\u662fblank\uff0c \u6216\u8005 $\\mathbf{l'}_{s}= \\mathbf{l'}_{s-2}$\u200b\u200b\u200b\u200b\u200b\u200b\u200b(\u76f8\u90bb\u662f\u91cd\u590d\u5b57\u7b26)\uff1a\n\n \n\n \u8fd9\u4e2a\u65f6\u5019\u4ed6\u7684\u6982\u7387\u6765\u81ea\u4e8e\u8fc7\u53bbt-1\u7684\u4e24\u4e2alabel \u6982\u7387\uff0c \u4e5f\u5c31\u662f $a_{t-1} (s)$\u200b\u200b \u548c $a_{t-1} (s-1)$\u200b\u200b\u200b \u3002\n\n $ a_{t-1} (s)$\u200b\u200b \u5c31\u662f\u8bf4\u5f53\u524d\u7684 sequence \u5df2\u7ecf\u662fs \u4e86\uff0cfigure 3\u4e2d\u8868\u73b0\u4e3a\u6a2a\u8df3\uff0c blank -->blank\uff08\u4f8b\u5982t=3, s=3\uff09\uff1b\n\n \u800c $a_{t-1} (s-1) $\u662f\u8bf4\u660e\u5f53\u524d\u7684\u5b57\u7b26\u8fd8\u4e0d\u591f\uff0c \u9700\u8981\u518d\u52a0\u4e00\u4e2a\uff0c \u6240\u4ee5\u5728figure 3\u4e2d\u5c31\u662f\u659c\u8df3\uff0c\u4ece\u9ed1\u8272\u5706\u5708\u5230\u767d\u8272\u5706\u5708\uff08\u4f8b\u5982\uff0ct=3, s=5\uff09\u3002\n\n \u4ed4\u7ec6\u89c2\u5bdffigure 3\uff0c \u9664\u4e86\u7b2c\u4e00\u6392\u7684\u767d\u8272\u5706\u5708\uff0c \u5176\u4ed6\u767d\u8272\u5706\u5708\u90fd\u6709\u4e24\u4e2a\u8f93\u5165\uff0c \u5c31\u662f\u4e0a\u8ff0\u7684\u4e24\u79cd\u60c5\u51b5\u3002 \u5f53\u7136\u5224\u65adblank \u7684\u65b9\u6cd5\u4e5f\u53ef\u4ee5\u662f\u5224\u65ad$I'_{s-2} = I'_{s}$\u200b. \u8fd9\u79cd\u60c5\u51b5\u4e5f\u662f\u8bf4\u660e$I'_{s}$\u200b\u200b\u200b \u662fblank, \u56e0\u4e3a\u6bcf\u4e00\u4e2a\u5b57\u7b26\u5fc5\u987b\u7528 blank \u9694\u5f00\uff0c \u5373\u4f7f\u662f\u76f8\u540c\u5b57\u7b26\u3002\n\n* \u7b2c\u4e8c\u7ae0\u60c5\u51b5 \u4e5f\u53ef\u4ee5\u7528\u7c7b\u4f3c\u903b\u8f91\u5f97\u51fa\uff0c \u53ea\u4e0d\u8fc7\u5f53\u524d\u7684\u72b6\u6001s \u662f\u9ed1\u8272\u5706\u5708\uff0c \u6709\u4e09\u79cd\u60c5\u51b5\u8f93\u5165\u3002\n\n \n\n\u6700\u7ec8\u7684\u6982\u7387\u5c31\u5982\u516c\u5f0f8 \u6240\u793a\uff0c \u8fd9\u4e2a\u8ba1\u7b97\u8fc7\u7a0b\u5c31\u662f CTC forward algroirthm\uff0c \u57fa\u4e8e Fig. 3 \u7684\u5de6\u8fb9\u7684\u521d\u59cb\u6761\u4ef6\u3002\n\n\n\n\u57fa\u4e8eFig. 3 \u53f3\u8fb9\u7684\u521d\u59cb\u6761\u4ef6\uff0c\u6211\u4eec\u8fd8\u662f\u53ef\u4ee5\u8ba1\u7b97\u51fa\u4e00\u4e2a\u6982\u7387\uff0c \u90a3\u4e2a\u5c31\u662f **CTC backward**. \u8fd9\u91cc\u6211\u5c31\u4e0d\u8be6\u7ec6\u4ecb\u7ecd\u4e86\uff0c \u76f4\u63a5\u622a\u56fe\u3002\n\n\n\n\u8fd9\u6837\u4e00\u76f4\u505a\u4e58\u6cd5\uff0c \u6570\u5b57\u503c\u8d8a\u6765\u8d8a\u5c0f\uff0c\u5f88\u5feb\u5c31\u4f1aunderflow\u3002 \u8fd9\u4e2a\u65f6\u5019\u5c31\u9700\u8981\u505a scaling.\n\n\n\n\u7b97\u51fa\u4e86forward probability \u548c backward probability \u6709\u4ec0\u4e48\u7528\u5462\uff0c \u89e3\u91ca\u5982\u4e0b\u56fe\u3002\n\n\n\n\u4e0a\u56fe\u662f\u8bf4 forward probability and backward probability \u7684\u4e58\u79ef\uff0c \u4ee3\u8868\u4e86\u8fd9\u4e2a sequence $\\mathbf{l}$ t\u65f6\u523b\uff0c\u662fs label \u7684 \u6240\u6709paths \u7684\u6982\u7387\u3002 \u8fd9\u6837\u7684\u8bdd \u6211\u4eec\u5c31\u8ba1\u7b97\u4e86 Fig. 3 \u4e2d\u7684\u6bcf\u4e2a\u5706\u5708\u7684\u6982\u7387\u3002\u4e3a\u4ec0\u4e48$\\alpha_t(s)\\beta_t(s)$ \u4e2d\u591a\u51fa\u4e00\u4e2a $y^t_{\\mathbf{l'_s}}$ \uff0c\u8fd9\u662f\u56e0\u4e3a\u5b83\u5728 $\\alpha$ \u548c $\\beta$ \u4e2d\u90fd\u5305\u542b\u8be5\u9879\uff0c\u5408\u5e76\u516c\u5f0f\u540e\u5c31\u591a\u51fa\u4e00\u9879\u3002\n\n\n\n$p(\\mathbf{l}|\\mathbf{x})$\u200b \u53ef\u4ee5\u901a\u8fc7\u4efb\u610f\u65f6\u523b t \u7684\u6240\u6709 s \u7684 foward-backward \u6982\u7387\u8ba1\u7b97\u5f97\u6765\u3002\u53d6\u8d1f\u5bf9\u6570\u540e\u5c31\u662f\u5355\u4e2a\u6837\u672c\u7684NLL\uff08Negative Log Likelihood\uff09\u3002\n\n### \u603b\u7ed3\n\n\u603b\u7ed3\u4e00\u4e0b\uff0c\u6839\u636e\u524d\u5411\u6982\u7387\u8ba1\u7b97CTCLoss\u51fd\u6570\uff0c\u53ef\u4ee5\u5f97\u51fa\u5982\u4e0b\u7ed3\u8bba\uff1a\n\n1. \u5bf9\u4e8e\u65f6\u5e8f\u957f\u5ea6\u4e3aT\u7684\u8f93\u5165\u5e8f\u5217x\u548c\u8f93\u51fa\u5e8f\u5217z\uff0c\u524d\u5411\u6982\u7387\uff1a\n $$\n \\begin{split}\n \\alpha_t(s) &= \\sum_{ \\underset{\\pi_t=l'_s}{\\pi \\in \\mathcal{B}^{-1}(z)} } p(\\pi_{1:t}|x) \\newline\n \\alpha_1(1) &= y_{-}^1 ; \\quad \\alpha_1(2)=y^1_{l'_2}, \\quad \\alpha_1(s)=0, \\forall s > 2 \\newline\n \\alpha_t(s) &= 0, \\quad \\forall s < |l'| - 2(T-t) - 1 ,\\quad \\text{or} \\quad \\forall s < 1 \\newline\n \\alpha_t(s) &=\n \\begin{cases}\n (\\alpha_{t-1}(s) + \\alpha_{t-1}(s-1) ) y^t_{l'_s} & \\text{if $l'_s=b$ or $l'_{s-2} = l'_s$\u200b} \\newline\n (\\alpha_{t-1}(s) + \\alpha_{t-1}(s-1) + \\alpha_{t-1}(s-2))y^t_{l'_s} & \\text{otherwise}\\newline\n \\end{cases} \n \\end{split}\n $$\n\n2. \u5229\u7528 $\\alpha_t(s)$ \u8ba1\u7b97CTCLoss\uff1a\n $$\n -ln(p(l \\mid x)) = -ln(\\alpha_{T}(|l'|)+\\alpha_{T}(|l'|-1))\n $$\n\n\u6839\u636e\u540e\u5411\u6982\u7387\u8ba1\u7b97CTCLoss\u51fd\u6570\uff0c\u53ef\u4ee5\u5f97\u51fa\u5982\u4e0b\u7ed3\u8bba\uff1a\n\n1. \u5bf9\u4e8e\u65f6\u5e8f\u957f\u5ea6\u4e3aT\u7684\u8f93\u5165\u5e8f\u5217x\u548c\u8f93\u51fa\u5e8f\u5217z\uff0c\u540e\u5411\u6982\u7387\uff1a \n $$\n \\begin{split}\n \\beta_t(s) &= \\sum_{ \\underset{\\pi_t=l'_s}{\\pi \\in \\mathcal{B}^{-1}(z)} } p(\\pi_{t:T}|x) \\newline\n \\beta_T(|l'|) &= y_{-}^T ; \\quad \\beta_T(|l'|-1)=y^T_{l'_{|l'|-1}}, \\quad \\beta_T(s)=0, \\forall s < |l'| - 1 \\newline\n \\beta_t(s) &= 0, \\text{$\\forall s > 2t$ or $\\forall s < |l'|$} \\newline\n \\beta_t(s) &=\n \\begin{cases}\n (\\beta_{t+1}(s) + \\beta_{t+1}(s+1) ) y^t_{l'_s} & \\text{if $l'_s=b$ or $l'_{s+2} = l'_s$} \\newline\n (\\beta_{t+1}(s) + \\beta_{t+1}(s+1) + \\beta_{t+1}(s+2))y^t_{l'_s} & \\text{otherwise}\\newline\n \\end{cases}\n \\end{split}\n $$\n\n 2. \u5229\u7528 $\\beta_t(s)$\u8ba1\u7b97CTCLoss\uff1a\n\n$$\n-ln(p(l \\mid x)) = -ln(\\beta_{1}(1)+\\beta_{1}(2)) \\newline\n$$\n\n\u6839\u636e\u4efb\u610f\u65f6\u523b\u7684\u524d\u5411\u6982\u7387\u548c\u540e\u5411\u6982\u7387\u8ba1\u7b97CTC Loss\u51fd\u6570\uff0c\u5f97\u5230\u5982\u4e0b\u7ed3\u8bba\uff1a\n\n1. \u5bf9\u4e8e\u4efb\u610f\u65f6\u523bt\uff0c\u5229\u7528\u524d\u5411\u6982\u7387\u548c\u540e\u5411\u6982\u7387\u8ba1\u7b97CTCLoss\uff1a\n\n$$\np(l \\mid x) = \\sum_{s=1}^{|l'|} \\frac{\\alpha_t(s)\\beta_t(s)}{y_{l'_s}^t} \\newline\n-ln(p(l \\mid x)) = -ln( \\sum_{s=1}^{|l'|} \\frac{\\alpha_t(s) \\beta_t(s)}{y_{l'_s}^t} )\n$$\n\u6211\u4eec\u5df2\u7ecf\u5f97\u5230CTCLoss\u7684\u8ba1\u7b97\u65b9\u6cd5\uff0c\u63a5\u4e0b\u6765\u5bf9\u5176\u8fdb\u884c\u6c42\u5bfc\u3002\n\n\n## CTC\u68af\u5ea6\u8ba1\u7b97\n\n### \u5fae\u5206\u516c\u5f0f\n\n\u5728\u8ba1\u7b97\u68af\u5ea6\u524d\uff0c\u6211\u4eec\u5148\u56de\u987e\u4e0b\u57fa\u672c\u7684\u5fae\u5206\u516c\u5f0f\uff1a \n$$\nC' = 0 \\\\\nx' = 1 \\newline\nx^n = n \\cdot x^{n-1} \\newline\n(e^x)' = e^x \\newline\nlog(x)' = \\frac{1}{x} \\newline\n(u + v)' = u' + v' \\newline\n(\\frac{u}{v})' = \\frac{u'v-uv'}{v^2} \\newline\n\\frac{\\mathrm{d}f(g(x))}{\\mathrm{d}x} = \\frac{\\mathrm{d}f(g(x))}{\\mathrm{d}g(x)} \\cdot \\frac{\\mathrm{d}g(x)}{\\mathrm{d}x}\n$$\n\n\n### CTC\u68af\u5ea6\n\n\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u8bad\u7ec3\u5c31\u662f\u6700\u5927\u5316\u8bad\u7ec3\u96c6\u4e2d\u6bcf\u4e00\u4e2a\u5206\u7c7b\u7684\u5bf9\u6570\u6982\u7387\uff0c\u5373\u6700\u5c0f\u5316Eq. 12\u3002\n\n\n\n\u6700\u540e\u5c31\u662f\u7b97\u5fae\u5206\u4e86\uff0c \u6574\u4e2a\u63a8\u5bfc\u8fc7\u7a0b\u5c31\u662f\u52a0\u6cd5\u548c\u4e58\u6cd5\uff0c \u90fd\u53ef\u4ee5\u5fae\u5206\u3002 $\\mathit{O}^{ML}$\u5173\u4e8e\u795e\u7ecf\u7f51\u7edc\u7684\u8f93\u51fa $y^t_k$\u7684\u68af\u5ea6\u89c1Eq. 13\u3002\u56e0\u4e3a\u8bad\u7ec3\u6837\u672c\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u6240\u4ee5\u53ef\u4ee5\u5355\u72ec\u8003\u8651\u6bcf\u4e2a\u6837\u672c\uff0c\u516c\u5f0f\u5982Eq.13\u3002\n\n\u4e0b\u9762\u662fCTCLoss\u7684\u68af\u5ea6\u8ba1\u7b97\uff1a\n\n\n\n\n### CTC\u68af\u5ea6\u63a8\u5bfc\n\n\u56de\u987e\u4e0b\u4e4b\u524d\u7684\u516c\u5f0f\uff0c\u4fbf\u4e8e\u7406\u89e3\u540e\u7eed\u63a8\u5bfc\u8fc7\u7a0b\u3002 \n\n$$\np(l \\mid x) = \\sum_{s=1}^{|l'|} \\frac{\\alpha_t(s)\\beta_t(s)}{y_{l'_s}^t} \\\\\n\\begin{equation}\n\\alpha_t(s) \\beta_t(s) = \\sum_{ \\underset{\\pi_t=l'_s}{\\pi \\in \\mathcal{B}^{-1}(l):} } y^t_{l'_s} \\prod_{t=1}^T y^t_{\\pi_t}\n\\end{equation}\n$$\n\n\u5176\u4e2dEq. 15\u7684\u8ba1\u7b97\u8fc7\u7a0b\u5982\u4e0b\uff1a \n\n$$\n\\begin{align*}\n\\frac{\\partial p(\nl \\mid x)}{\\partial y_k^t}\n & = \\sum_{s \\in lab(z,k)} \\frac{ \\partial \\frac{ \\alpha_t(s) \\beta_t(s)}{y_{k}^t}}{\\partial y_k^t} \n \\newline\n & = \\sum_{s \\in lab(z,k)} \\frac{(\\alpha_t(s)\\beta_t(s))\u2019y_k^t - \\alpha_t(s)\\beta_t(s){y_k^t}'}{{y_k^t}^2}\n \\newline\n &= \\sum_{s \\in lab(z,k)} \\frac{( \\prod_{t'=1}^{t-1} y^{t'}_{\\pi_{t'}} \\cdot y_k^t \\cdot y_k^t \\cdot \\prod_{t'=t+1}^{T} y^{t'}_{\\pi_{t'}} )\u2019 y_k^t - \\alpha_t(s)\\beta_t(s){y_k^t}'}{{y_k^t}^2}\n \\newline\n &= \\sum_{s \\in lab(z,k)} \\frac{2\\alpha_t(s)\\beta_t(s) - \\alpha_t(s)\\beta_t(s)}{{y_k^t}^2}\n \\newline\n &= \\sum_{s \\in lab(z,k)} \\frac{\\alpha_t(s)\\beta_t(s)}{{y_k^t}^2}\n \\newline\n &= \\frac{1}{{y_k^t}^2} \\sum_{s \\in lab(z,k)} \\alpha_t(s)\\beta_t(s) \\tag{1} \\newline\n\\end{align*}\n$$\n\n\nNLL\u7684\u516c\u5f0f\u63a8\u5bfc\u5982\u4e0b\uff1a\n$$\n\\begin{split}\n\\frac{\\partial {ln(p(l \\mid x))} }{ \\partial y^t_k }\n &= \\frac{1}{p(l \\mid x)} \\frac{ \\partial{p(l \\mid x)} }{ \\partial y_k^t } \\newline\n &= \\frac{1}{p(l \\mid x) {y^t_k}^2 } \\sum_{s \\in lab(z,k)} \\alpha_t(s)\\beta_t(s) \n\\end{split}\n\\tag{2}\n$$\n\n\n\u5df2\u7ecf\u7b97\u51fa\u4e86CTCLoss\u5bf9\u4e8e $y_k^t$\u200b \u7684\u68af\u5ea6\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u9700\u8981\u8ba1\u7b97 CTCLoss\u5bf9\u4e8e$u^t_k$\u200b\uff08logits\uff09\u7684\u68af\u5ea6\u3002\u5957\u7528\u94fe\u5f0f\u6cd5\u5219\uff0c\u5e76\u66ff\u6362$y^t_k$\u200b \u4e3a $y^t_{k'}$\u200b\uff0c\u7ed3\u679c\u5982\u4e0b\u56fe\u3002\u56fe\u4e2d $k'$\u200b \u8868\u793avocab\u4e2d\u7684\u67d0\u4e00\u4e2atoken\uff0c$K$\u200b\u200b \u662fvocab\u7684\u5927\u5c0f\u3002\n\n\n\n\u56fe\u4e2d\u516c\u5f0f4\u6839\u636e\u94fe\u5f0f\u6cd5\u5219\u5f97\u5230\uff1a\n$$\n- \\frac{ \\partial ln(p(l \\mid x)) }{ \\partial u^t_k }\n = - \\sum_{k'=1}^{K} \\frac{ \\partial ln(p(l \\mid x)) }{ \\partial y^t_{k'} } \\frac{ \\partial y^t_{k'} }{ \\partial u^t_k } \\tag{4}\n$$\n\u56fe\u4e2d\u516c\u5f0f3\u662fsoftmax\u7684\u68af\u5ea6\uff0c\u53c2\u8003 [4]\uff0c\u8ba1\u7b97\u8fc7\u7a0b\u5982\u4e0b\uff1a\n$$\nsoftmax(j) = S_j = \\frac{ e^{a_j} }{ \\sum_{k=1}^K e^{a_k} }, \\enspace \\forall j \\in 1 \\dots K\n$$\n\n$$\n\\begin{split}\n\\frac{ \\partial S_i }{ \\partial a_j}\n &= \\frac{ \\partial (\\frac{ e^{ a_i } }{ \\sum_k e^{ a_k } }) } { \\partial a_j }\n \\newline\n &= \n \\begin{cases}\n \t\\frac{ e^a_i \\sum - e^a_j e^a_i }{ \\sum^2 } \n \t&= \\frac{ e^a_i }{ \\sum } \\frac{ \\sum - e^a_j }{ \\sum } \\newline\n &= S_i(1-S_j) & \\text{i = j, $\\sum$ stands for $\\sum_{k=1}^K e^a_k$} \n \t\\newline\n \t\\frac{ 0 - e^a_j e^a_i }{ \\sum^2 } \n \t&= - \\frac{ e^a_j }{ \\sum } \\frac{ e^a_i }{ \\sum } \\newline\n &= -S_j S_i & \\text{i $\\neq$ j, $\\sum$ stands for $\\sum_{k=1}^K e^a_k$}\n \\end{cases}\n \\newline\n &= \n \\begin{cases}\n S_i(1 - S_j) & \\text{$i = j$} \n \\newline\n -S_j S_i = S_i (0 - S_j) & \\text{$i \\neq j$}\n \\end{cases}\n \\newline\n &= S_i (\\delta_{ij} - S_j )\n\\end{split}\n\\tag{3}\n$$\n$$\n\\delta_{ij} =\n \\begin{cases}\n 1 & \\text{if i = j} \\newline\n 0 & \\text{otherwise}\n \\end{cases}\n$$\n\n\n\n\u4e0b\u56fe\u4e2d\u9ec4\u8272\u6846\u4e2d\u7684\u90e8\u5206\u8868\u793a\u516c\u5f0f\uff081\uff09\uff0c\u5373\u904d\u5386\u6240\u6709\u7684vocab\u4e2d\u7684token\uff0c\u5176\u7ed3\u679c\u662f$p(l \\mid x)$\u200b\u3002\u8fd9\u662f\u56e0\u4e3alabel $l$\u200b \u4e2d\u7684token\u4e00\u5b9a\u5728vocab\u4e2d\uff0c\u4e14 $s \\in lab(l, k')$\u200b \u53ef\u4ee5\u662f\u7a7a\u96c6\u3002\u5f53 $k'$\u200b \u5728 l \u4e2d\uff0cs \u5219\u4e3alabel\u4e2dtoken\u662f$k'$\u200b\u7684\u6982\u7387\uff1b\u5f53$k'$\u200b\u200b\u200b\u4e0d\u5728l\u4e2d\uff0cs\u4e3a\u7a7a\uff0c\u6982\u7387\u4e3a0\u3002\n\n\n\n\u516c\u5f0f\uff082\uff09\uff0c\uff083\uff09\u5e26\u5165\uff084\uff09\uff0c\u5e76\u7ed3\u5408\u516c\u5f0f\uff081\uff09\u7684\u7ed3\u679c\u5982\u4e0a\u56fe\u53f3\u8fb9\uff0c\u5373\uff1a\n$$\n\\begin{split}\n- \\frac{ \\partial ln(p(l \\mid x)) }{ \\partial u^t_k } &= \n\t- \\sum_{k'=1}^K \\frac{ \\partial ln(p(l \\mid x)) }{ \\partial y^t_{k'} } \\frac{ \\partial y^t_{k'}}{ \\partial u^t_k } \\newline\n\t&= - \\sum_{k'=1}^K \\frac{ y^t_{k'}( \\delta_{kk'} - y^t_k ) }{ p(l \\mid x) {y^t_{k'}}^2 } \\sum_{s \\in lab(l, k') } \\alpha_t(s) \\beta_t(s) \\newline\n\t&= - \\sum_{k'=1}^K \\frac{ \\delta_{kk'} - y^t_k }{ p(l \\mid x) y^t_{k'} } \\sum_{s \\in lab(l, k') } \\alpha_t(s) \\beta_t(s) \\newline\n\t&= \\sum_{k'=1}^K \\frac{ y^t_k - \\delta_{kk'} }{ p(l \\mid x) y^t_{k'} } \\sum_{s \\in lab(l, k') } \\alpha_t(s) \\beta_t(s) \\newline\n\t&= \\sum_{k'=1}^K \\frac{ y^t }{ p(l \\mid x) y^t_{k'} } \\sum_{s \\in lab(l, k') } \\alpha_t(s) \\beta_t(s) - \\sum_{k'=1}^K \\frac{ \\delta_{kk'} }{ p(l \\mid x) y^t_{k'} } \\sum_{s \\in lab(l, k') } \\alpha_t(s) \\beta_t(s) \\newline\n\t&= \\frac{ y^t_k }{ p(l \\mid x) } ( \\sum_{k'=1}^K \\frac{1}{y^t_{k'}} \\sum_{s \\in lab(l, k') } \\alpha_t(s) \\beta_t(s) ) - \\sum_{k'=1}^K \\frac{ \\delta_{kk'} }{ p(l \\mid x) y^t_{k'} } \\sum_{s \\in lab(l, k') } \\alpha_t(s) \\beta_t(s) \\newline\n\t&= \\frac{ y^t_k }{ p(l \\mid x) } p(l \\mid x) - \\sum_{k'=1}^K \\frac{ \\delta_{kk'} }{ p(l \\mid x) y^t_{k'} } \\sum_{s \\in lab(l, k') } \\alpha_t(s) \\beta_t(s) \\newline\n\t&= y^t_k - \\frac{ 1 }{ p(l \\mid x) y^t_k } \\sum_{s \\in lab(l, k)} \\alpha_t(s) \\beta_t(s) \\newline\n\\end{split}\n$$\n\u6700\u7ec8\uff0c\u4e3a\u4e86\u901a\u8fc7softmax\u5c42\u4f20\u64adCTCLoss\u7684\u68af\u5ea6\uff0c\u9700\u8981\u8ba1\u7b97\u76ee\u6807\u51fd\u6570\u4e0e logits $u^t_k$ \u7684\u504f\u5fae\u5206\uff0c\u5373Eq. 16: \n $$\n \\begin{align*}\n \\hat{\\alpha}_t(s) & \\overset{def}{=} \\frac{ \\alpha_t(s) }{ C_t } ,\\enspace C_t \\overset{def}{=} \\sum_s \\alpha_t(s) \n \\newline\n \\hat{\\beta}_t(s) & \\overset{def}{=} \\frac{ \\beta_t(s) }{ D_t } ,\\enspace D_t \\overset{def}{=} \\sum_s \\beta_t(s) \n \\newline\n - \\frac{ \\partial ln(p(l \\mid x)) }{ \\partial u^t_k } &= y^t_k - \\frac{1}{y^t_k \\sum_{s=1}^{\\mid l' \\mid} \\frac{ \\hat{\\alpha}_t(s) \\hat{\\beta}_t(s) }{ y^t_{l'_s} } } \\sum_{s \\in lab(l, k)} \\hat{\\alpha}_t(s) \\hat{\\beta}_t(s) \\tag{16} \n \\newline\n \\end{align*}\n $$\n\n### \u603b\u7ed3\n\n* \u901a\u8fc7\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\u8ba1\u7b97$\\alpha_t(s)$ \u548c $\\beta_t(s)$\n\n* \u901a\u8fc7$\\alpha_t(s)$ \u8ba1\u7b97 $p(l \\mid x)=\\alpha_T(\\mid l' \\mid) + \\alpha_T(\\mid l' \\mid -1)$\n\n* \u901a\u8fc7$\\alpha_t(s)$ \u548c $\\beta_t(s)$\n\n* \u8ba1\u7b97CTcLoss\u51fd\u6570\u7684\u5bfc\u6570: \n $$\n \\begin{split}\n - \\frac{ \\partial ln(p(l \\mid x)) }{ \\partial u^t_k } \n &= y^t_k - \\frac{ 1 }{ p(l \\mid x) y^t_k } \\sum_{s \\in lab(l, k)} \\alpha_t(s) \\beta_t(s) \n \\newline\n &= y^t_k - \\frac{1}{y^t_k \\sum_{s=1}^{\\mid l' \\mid} \\frac{ \\hat{\\alpha}_t(s) \\hat{\\beta}_t(s) }{ y^t_{l'_s} } } \\sum_{s \\in lab(l, k)} \\hat{\\alpha}_t(s) \\hat{\\beta}_t(s) \n \\newline\n \\end{split}\n \\tag{16}\n $$\n\n## Source Code\n\u672c\u4eba\u5728 [warp-ctc](https://github.com/zh794390558/warp-ctc) \u4e0a\u52a0\u4e86\u6ce8\u91ca\uff0c\u5e76\u8c03\u6574 index \u7684\u7d22\u5f15\u65b9\u5f0f\uff0c\u4fbf\u4e8e\u7406\u89e3\u4ee3\u7801\u3002\n\u5bf9\u6bd4\u4e0a\u9762\u7684\u516c\u5f0f\u63a8\u5bfc\u548clattice\u56fe\u53ef\u4ee5\u5feb\u901f\u7406\u89e3 ctc \u5b9e\u73b0\u3002\n\n## Reference\n\n[[1] A. Graves, S. Fernandez, F. Gomez, J. Schmidhuber. Connectionist Temporal lassification: Labeling Unsegmented Sequence Data with Recurrent Neural Networks. ICML 2006, Pittsburgh, USA, pp. 369-376.](http://www.cs.toronto.edu/~graves/icml_2006.pdf)\n\n[[2] Sequence ModelingWith CTC](https://distill.pub/2017/ctc/)\n\n[[3] NLP \u4e4b CTC Loss \u7684\u5de5\u4f5c\u539f\u7406](https://www.jianshu.com/p/e073c9d91b20)\n\n[[4] The Softmax function and its derivative](https://eli.thegreenplace.net/2016/the-softmax-function-and-its-derivative/)\n\n[[5] CTC Algorithm Explained Part 1\uff1aTraining the Network\uff08CTC\u7b97\u6cd5\u8be6\u89e3\u4e4b\u8bad\u7ec3\u7bc7\uff09](https://xiaodu.io/ctc-explained/)\n\n\n```python\n\n```\n", "meta": {"hexsha": "081a6388519d2087b432b51d073e7d492f4b097a", "size": 18332, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/topic/ctc/ctc_loss.ipynb", "max_stars_repo_name": "JiehangXie/PaddleSpeech", "max_stars_repo_head_hexsha": "60090b49ec27437127ab62358026dd5bb95fccc7", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 1540, "max_stars_repo_stars_event_min_datetime": "2017-11-14T13:26:33.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-09T14:05:08.000Z", "max_issues_repo_path": "docs/topic/ctc/ctc_loss.ipynb", "max_issues_repo_name": "JiehangXie/PaddleSpeech", "max_issues_repo_head_hexsha": "60090b49ec27437127ab62358026dd5bb95fccc7", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 599, "max_issues_repo_issues_event_min_datetime": "2017-11-14T13:19:12.000Z", "max_issues_repo_issues_event_max_datetime": "2021-11-09T01:58:26.000Z", "max_forks_repo_path": "docs/topic/ctc/ctc_loss.ipynb", "max_forks_repo_name": "JiehangXie/PaddleSpeech", "max_forks_repo_head_hexsha": "60090b49ec27437127ab62358026dd5bb95fccc7", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 449, "max_forks_repo_forks_event_min_datetime": "2017-11-14T12:48:46.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-06T09:34:33.000Z", "avg_line_length": 43.8564593301, "max_line_length": 339, "alphanum_fraction": 0.5028365699, "converted": true, "num_tokens": 6468, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.6619228825191872, "lm_q1q2_score": 0.41686221427155623}} {"text": "```python\nimport numpy as np\r\nimport pandas as pd\r\nimport respy as rp\r\nfrom python.auxiliary_setup import load_model_specs\r\nfrom python.auxiliary_plots import plot_chatter_numagents_sim\r\nfrom python.auxiliary_plots import plot_chatter_numagents_both\r\nfrom python.auxiliary_plots import plot_criterion_params\r\nfrom python.auxiliary_plots import plot_moments_choices\r\nfrom python.auxiliary_plots import plot_moments_wage\r\nfrom python.auxiliary_plots import plot_chatter\n```\n\n## Method of Simulated Moments Estimation (MSM)\n\nThis lecture heavily relies on [respy](https://respy.readthedocs.io/en/latest/index.html)'s method of simulated moments (MSM) interface. A guide to using the interface can be found [here](https://respy.readthedocs.io/en/latest/how_to_guides/msm.html).\n\n

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                                        \n
                                        \n\n### Introduction\n\nThe method of simulated moments approach to estimating model parameters is to minimize a certain distance between observed moments and simulated moments with respect to the parameters that generate the simulated model.\r\n\r\nDenote $X$ our observed data and $m(X)$ the vector of observed moments. To construct the criterion function, we use the parameter vector $\\theta$ to simulate data from the model $\\hat{X}$. We can then calculate the simulated moments $m(\\hat{X}| \\theta)$.\r\n\r\nThe criterion function is then given by \r\n\r\n\\begin{equation}\r\n\\psi(\\theta) = (m(X) - m(\\hat{X}| \\theta))'\\Omega(m(X) - m(\\hat{X}| \\theta))\r\n\\end{equation}\r\n\r\nwhere the difference between observed and simulated moments $(m(X) - m(\\hat{X}| \\theta))$ constitutes a vector of the dimension $1\\times M$ with $1,..,M$ denoting the number of moments. The $M\\times M$ weighting matrix is given by $\\Omega$. \r\n\r\nThe MSM estimator is defined as the solution to \r\n\r\n\\begin{equation}\r\n\\hat{\\theta}(\\Omega) = \\underset{\\theta}{\\operatorname{argmin}} \\psi(\\theta).\r\n\\end{equation}\r\n\r\nThe criterion function is thus a strictly positive scalar and the estimator depends on the choice of moments $m$ and the weighting matrix $\\Omega$. The weighting matrix applies some scaling for discrepancies between the observed and simulated moments. If we use the identity matrix, each moment is given equal weight and the criterion function reduces to the sum of squared moment deviations. \r\n\r\nAside from the choice of moments and weighting matrix, some other important choices that influence the the estimation are the simulator itself and the algorithm and specifications for the optimization procedure. Many explanations for simulated method of moments estimation also feature the number of simulations as a factor that is to be determined for estimation. We can ignore this factor for now since we are working with a large simulated dataset.\r\n\r\nIn the following we will set up the data, moments and weighting matrix needed to construct the criterion function and subsequently try to estimating the parameters of the model using this MSM setup.\n\n### Observed Data\n\nWe generate our model and data using [respy](https://respy.readthedocs.io/en/latest/) and a simple Robinson Crusoe model. In this model, the agent, Robinson Crusoe, in each time period decides between two choice options: working (i.e. going fishing) or spending time in the hammock. \n\nWe can use respy to simulate the data for this exercise.\n\n\n```python\nparams_true, options = load_model_specs()\n```\n\n\n```python\n# Generate observed data from model.\r\nsimulate = rp.get_simulate_func(params_true, options)\r\ndata_obs = simulate(params_true)\n```\n\nLet's take a look at the model specifications.\n\n\n```python\nparams_true\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        value
                                        categoryname
                                        deltadelta0.950
                                        wage_fishingexp_fishing0.070
                                        nonpec_fishingconstant-0.100
                                        nonpec_hammockconstant1.046
                                        shocks_sdcorrsd_fishing0.010
                                        sd_hammock0.010
                                        corr_hammock_fishing0.000
                                        \n
                                        \n\n\n\n\n```python\noptions\n```\n\n\n\n\n {'estimation_draws': 100,\n 'estimation_seed': 100,\n 'estimation_tau': 0.001,\n 'interpolation_points': -1,\n 'n_periods': 5,\n 'simulation_agents': 1000,\n 'simulation_seed': 132,\n 'solution_draws': 100,\n 'solution_seed': 456,\n 'covariates': {'constant': '1'}}\n\n\n\n\n```python\ndata_obs.head(10)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Experience_FishingShock_Reward_FishingMeas_Error_Wage_FishingShock_Reward_HammockMeas_Error_Wage_HammockDense_KeyCore_IndexChoiceWageDiscount_Rate...Nonpecuniary_Reward_FishingWage_FishingFlow_Utility_FishingValue_Function_FishingContinuation_Value_FishingNonpecuniary_Reward_HammockWage_HammockFlow_Utility_HammockValue_Function_HammockContinuation_Value_Hammock
                                        IdentifierPeriod
                                        0000.99965010.000410100fishing0.9996500.95...-0.10.9996500.8996504.7393744.0418151.046NaN1.0464104.7322823.879866
                                        111.00074310.015065111fishing1.0733050.95...-0.11.0733050.9733054.0421893.2304051.046NaN1.0610653.9058402.994500
                                        220.99646110.011853122fishing1.1462030.95...-0.11.1462031.0462033.2244302.2928711.046NaN1.0578533.0762952.124675
                                        330.9989071-0.007859133fishing1.2323290.95...-0.11.2323291.1323292.2929981.2217561.046NaN1.0381412.1139191.132398
                                        440.98941910.012452144fishing1.3091300.95...-0.11.3091301.2091301.2091300.0000001.046NaN1.0584521.0584520.000000
                                        1000.99289410.014190100hammockNaN0.95...-0.10.9928940.8928944.7326184.0418151.046NaN1.0601904.7460623.879866
                                        100.9953691-0.003848110hammockNaN0.95...-0.10.9953690.8953693.7401442.9945001.046NaN1.0421523.8763432.983359
                                        201.0006681-0.008441120hammockNaN0.95...-0.11.0006680.9006682.8371762.0384301.046NaN1.0375592.9740682.038430
                                        300.9925421-0.013488130hammockNaN0.95...-0.10.9925420.8925421.8854961.0452151.046NaN1.0325122.0254661.045215
                                        400.9986951-0.005397140hammockNaN0.95...-0.10.9986950.8986950.8986950.0000001.046NaN1.0406031.0406030.000000
                                        \n

                                        10 rows \u00d7 21 columns

                                        \n
                                        \n\n\n\n### Data Moments\n\nFor the setup of the estimation we first have to choose a set of moments that we will use to match the observed data and the simulated model. For this model we include two sets of moments: \n\n1. The first set are Robinson's **choice frequencies** (choice frequencies here refers to the share of agents that have chosen a specific option) for each period. \n2. The second set are moments that characterize the **wage distribution** for each period, i.e. the mean of the wage of all agents that have chosen fishing in a given period and the standard deviation of the wages. \n\nWe need a functions that compute the set of moments on the observed and simulated data.\n\n\n```python\ndef calc_choice_frequencies(df):\r\n \"\"\"Calculate choice frequencies\"\"\"\r\n return df.groupby(\"Period\").Choice.value_counts(normalize=True).unstack()\n```\n\n\n```python\ndef calc_wage_distribution(df):\r\n \"\"\"Calculate wage distribution.\"\"\"\r\n return df.groupby([\"Period\"])[\"Wage\"].describe()[[\"mean\", \"std\"]]\n```\n\n\n```python\n# Save functions in a dictionary to pass to the criterion function later on.\r\ncalc_moments = {\r\n \"Choice Frequencies\": calc_choice_frequencies,\r\n \"Wage Distribution\": calc_wage_distribution,\r\n}\n```\n\nRespy's MSM interface additionally requires us to specify, how missings in the computed moments should be handled. Missing moments for instance can arise, when certain choice options aren't picked at all for some periods. We just specify a function that replaces missings with 0.\n\n\n```python\ndef replace_nans(df):\r\n \"\"\"Replace missing values in data.\"\"\"\r\n return df.fillna(0)\n```\n\nNow we are ready to calculate the moments.\n\n\n```python\nmoments_obs = {\r\n \"Choice Frequencies\": replace_nans(calc_moments[\"Choice Frequencies\"](data_obs)),\r\n \"Wage Distribution\": replace_nans(calc_moments[\"Wage Distribution\"](data_obs)),\r\n}\n```\n\n\n```python\nprint(\"Choice Frequencies\")\r\nprint(moments_obs[\"Choice Frequencies\"])\r\nprint(\"\\n Wage Distribution\")\r\nprint(moments_obs[\"Wage Distribution\"])\n```\n\n Choice Frequencies\n fishing hammock\n Period \n 0 0.698 0.302\n 1 0.698 0.302\n 2 0.698 0.302\n 3 0.698 0.302\n 4 0.698 0.302\n \n Wage Distribution\n mean std\n Period \n 0 1.003189 0.008640\n 1 1.072815 0.010737\n 2 1.150106 0.012316\n 3 1.234041 0.012406\n 4 1.322711 0.013473\n\n\n### Weighting Matrix\n\nNext we specify a weighting matrix. It needs to be a square matrix with the same number of diagonal elements as there are moments. One option would be to use the identity matrix, but we use a weighting matrix that adjusts for the variance of each moment to improve the estimation process. The variances for the moments are constructed using a bootstrapping procedure. \n\n\n```python\ndef get_weighting_matrix(data, calc_moments, num_boots, num_agents_msm):\r\n \"\"\" Compute weighting matrix for estimation with MSM.\"\"\"\r\n # Seed for reproducibility.\r\n np.random.seed(123)\r\n\r\n index_base = data.index.get_level_values(\"Identifier\").unique()\r\n\r\n # Create bootstrapped moments.\r\n moments_sample = list()\r\n for _ in range(num_boots):\r\n ids_boot = np.random.choice(index_base, num_agents_msm, replace=False)\r\n moments_boot = [calc_moments[key](data.loc[ids_boot, :]) for key in calc_moments.keys()]\r\n\r\n moments_boot = rp.get_flat_moments(moments_boot)\r\n\r\n moments_sample.append(moments_boot)\r\n\r\n # Compute variance for each moment and construct diagonal weighting matrix.\r\n moments_var = np.array(moments_sample).var(axis=0)\r\n weighting_matrix = np.diag(moments_var ** (-1))\r\n\r\n return np.nan_to_num(weighting_matrix)\n```\n\n\n```python\nW = get_weighting_matrix(data_obs, calc_moments, 300, 500)\n```\n\n\n```python\npd.DataFrame(W)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        012345678910111213141516171819
                                        04795.1153760.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+00
                                        10.0000004795.1153760.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+00
                                        20.0000000.0000004795.1153760.0000000.0000000.0000000.0000000.0000000.0000000.0000000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+00
                                        30.0000000.0000000.0000004795.1153760.0000000.0000000.0000000.0000000.0000000.0000000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+00
                                        40.0000000.0000000.0000000.0000004795.1153760.0000000.0000000.0000000.0000000.0000000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+00
                                        50.0000000.0000000.0000000.0000000.0000004795.1153760.0000000.0000000.0000000.0000000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+00
                                        60.0000000.0000000.0000000.0000000.0000000.0000004795.1153760.0000000.0000000.0000000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+00
                                        70.0000000.0000000.0000000.0000000.0000000.0000000.0000004795.1153760.0000000.0000000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+00
                                        80.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000004795.1153760.0000000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+00
                                        90.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000004795.1153760.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+00
                                        100.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000009.466832e+060.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+00
                                        110.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.000000e+006.036166e+060.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+00
                                        120.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.000000e+000.000000e+004.698312e+060.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+00
                                        130.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.000000e+000.000000e+000.000000e+004.248616e+060.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+00
                                        140.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.000000e+000.000000e+000.000000e+000.000000e+003.566318e+060.000000e+000.000000e+000.000000e+000.000000e+000.000000e+00
                                        150.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+001.779242e+070.000000e+000.000000e+000.000000e+000.000000e+00
                                        160.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+001.258966e+070.000000e+000.000000e+000.000000e+00
                                        170.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+009.317930e+060.000000e+000.000000e+00
                                        180.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+008.904094e+060.000000e+00
                                        190.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+000.000000e+001.067252e+07
                                        \n
                                        \n\n\n\n### Criterion Function \n\nWe have collected the observed data for our model, chosen the set of moments we want to use for estimation and defined a weighting matrix based on these moments. We can now set up the criterion function to use for estimation. \n\nAs already discussed above, the criterion function is given by the weighted square product of the difference between observed moments $m(X)$ and simulated moments $m(\\hat{X}| \\theta)$. Trivially, if we have that $m(X) = m(\\hat{X}| \\theta)$, the criterion function returns a value of 0. Thus, the closer $\\theta$ is to the real parameter vector, the smaller should be the value for the criterion function. \n\n\n```python\ncriterion_msm = rp.get_moment_errors_func(\r\n params_true, options, calc_moments, replace_nans, moments_obs, W\r\n)\n```\n\nCriterion function at the true parameter vector:\n\n\n```python\nfval = criterion_msm(params_true)\r\nfval\n```\n\n\n\n\n 0.0\n\n\n\nWe can plot the criterion function to examine its behavior around the minimum in more detail. The plots below show the criterion function at varying values of all parameters in the the paramter vector.\n\n\n```python\nplot_criterion_params(params_true, list(params_true.index), criterion_msm, 0.05)\n```\n\nThis depiction for most parameters conceals the fact that the criterion function is not a smooth function of our parameter values. We can reveal this property if we 'zoom in' on the function far enough. The plots below show the criterion function for varying values of *delta* around the true minimum value of 0.95. We can see that the function exhibits small plateaus and is thus not completely smooth. \n\n\n```python\nfor radius in [0.05, 0.01, 0.001, 0.0001]:\r\n plot_criterion_params(params_true, list(params_true.index)[0:1], criterion_msm, radius)\n```\n\n### Estimation Exercise\n\nIn the following we will conduct a simulation exercise to estimate the parameter vector using our criterion function and weighting matrix. We will begin by simulating data using the new parameter vector and examine how the simulated moments differ from the observed ones. We will then use an optimizer to minimize the criterion function in order to retrieve the true parameter vector. Additionally, we will explore how the criterion function behaves if we change the simulation seed, move away from the true values and test the optimization for a derivative-based optimization algorithm.\n\n#### Estimation for one parameter\n\nFor now, our candidate parameter vector will just differ in *delta* from the true parameters.\n\n\n```python\nparams_cand = params_true.copy()\r\nparams_cand.loc[\"delta\", \"value\"] = 0.93\n```\n\n##### Simulated Moments\n\nWe can now use our model to simulate data using the candidate parameter vector. We can see that the choice frequencies and wage distribution differ from the moments of the observed dataset.\n\n\n```python\nparams = params_cand.copy()\r\nsimulate = rp.get_simulate_func(params, options)\r\ndf_sim = simulate(params)\n```\n\n\n```python\nmoments_sim = {\r\n \"Choice Frequencies\": replace_nans(calc_moments[\"Choice Frequencies\"](df_sim)),\r\n \"Wage Distribution\": replace_nans(calc_moments[\"Wage Distribution\"](df_sim)),\r\n}\n```\n\nWe can plot the moments to compare the choice frequencies for each period.\n\n\n```python\nplot_moments_choices(moments_obs, moments_sim)\n```\n\nThe plots below show the mean and the standard deviation in the wages for each period.\n\n\n```python\nplot_moments_wage(moments_obs, moments_sim)\n```\n\nThe criterion function value for the candidate parameter vector is not zero.\n\n\n```python\nfval = criterion_msm(params_cand)\r\nfval\n```\n\n\n\n\n 7838.131228444257\n\n\n\n##### Optimization Procedure\n\nWe will now use an optimization procedure to retrieve the true parameter vector. For the optimization we can use [estimagic](https://estimagic.readthedocs.io/en/latest/index.html). In order to minimize the criterion function we need estimagic's `minimize` function.\n\n\n\n\n```python\nfrom estimagic import minimize # noqa: E402\n```\n\nWe have verified above that the criterion function gives a value of 0 for the true parameter vector. Before we try different parameter specifications, we can check whether an optimizer recognizes the true vector as the minimum of our criterion function.\n\nAs the code below shows, the optimization algorithm recognizes the true parameter vector as the minimum of the criterion function as it returns a function value of 0 and the true parameter values.\n\n\n```python\nrslt = minimize(\r\n criterion=criterion_msm,\r\n params=params_true,\r\n algorithm=\"nag_pybobyqa\",\r\n)\r\nrslt[\"solution_criterion\"]\n```\n\n\n\n\n 0.0\n\n\n\n##### Upper and Lower Bounds\n\nWe can help the optimizer by specifying bounds for the parameters. Since we know the true parameters in the case of this model, we can just pick upper and lower bounds that are fairly close to the true values of the parameters to aid the optimizer in the search for the optimum. By default, the upper and lower bounds are set to $\\infty$ and $-\\infty$, so specifying upper and lower bounds substantially reduces the range of parameter values that the optimizer can potentially cover.\n\nFor optimization with estimagic, we can specify bounds by adding the columns *'lower'* and *'upper'* to the dataframe that contains the parameter values.\n\n\n```python\nparams_cand[\"lower_bound\"] = [0.69, 0.066, -0.2, 1, 0, 0, 0]\r\nparams_cand[\"upper_bound\"] = [1, 0.078, -0.09, 1.1, 0.5, 0.5, 0.5]\r\nparams_cand\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        valuelower_boundupper_bound
                                        categoryname
                                        deltadelta0.9300.6901.000
                                        wage_fishingexp_fishing0.0700.0660.078
                                        nonpec_fishingconstant-0.100-0.200-0.090
                                        nonpec_hammockconstant1.0461.0001.100
                                        shocks_sdcorrsd_fishing0.0100.0000.500
                                        sd_hammock0.0100.0000.500
                                        corr_hammock_fishing0.0000.0000.500
                                        \n
                                        \n\n\n\n##### Constraints\n\nAdditionally we hold all other parameters fixed for now to aid the optimizer in finding the optimal value for *delta*.\n\n\n```python\n# Define base constraints to use for the rest of the notebook.\r\nconstr_base = [\r\n {\"loc\": \"shocks_sdcorr\", \"type\": \"sdcorr\"},\r\n {\"loc\": \"delta\", \"type\": \"fixed\"},\r\n {\"loc\": \"wage_fishing\", \"type\": \"fixed\"},\r\n {\"loc\": \"nonpec_fishing\", \"type\": \"fixed\"},\r\n {\"loc\": \"nonpec_hammock\", \"type\": \"fixed\"},\r\n {\"loc\": \"shocks_sdcorr\", \"type\": \"fixed\"},\r\n]\n```\n\n\n```python\n# Remove constraint for delta.\r\nconstr = constr_base.copy()\r\nconstr.remove({\"loc\": \"delta\", \"type\": \"fixed\"})\n```\n\n##### Optimize\n\nWe can now optimize the criterion function with respect to the parameter vector. The optimizer manages to reach a function value of 0 and finds an approximately correct *delta* for our model. \n\nThis exercise again reveals that we are dealing with a non-smooth criterion function. The optimizer does not return the exact value of 0.95 for *delta* because of the little plateaus we saw when zooming into the criterion function. As the plot shows, there is a small area around the true value for *delta* that also returns a function value of 0 and might thus be picked by the optimizer.\n\n\n```python\nrslt = minimize(\r\n criterion=criterion_msm,\r\n params=params_cand,\r\n algorithm=\"nag_pybobyqa\",\r\n constraints=constr,\r\n)\r\nrslt[\"solution_criterion\"]\n```\n\n\n\n\n 0.5248952065747418\n\n\n\n#### Chatter in the Criterion Function\n\nIn this exercise we explore the sensitivity of the criterion function to the choice of simulation seed and number of agents.\n\n##### Changing the Simulation Seed\n\nAs shown above, the optimizer manages to find a function value of exactly 0 for the true parameter vector. This is the case because respy controls the realization of random elements in the simulation process via a simulation seed. The model thus always simulates the exact same dataset for a given parameter vector. Our criterion function becomes exactly 0 at the true parameter vector because for this vector, the observed and simulated data are identical.\n\nChanging the simulation seed results in simulated moments that, even for the true parameter vector, are never completely equal to the observed moments. \n\nLet's estimate the model with a different simulation seed.\n\n\n```python\noptions_new_seed = options.copy()\r\noptions_new_seed[\"simulation_seed\"] = 400\n```\n\nWe can see that the criterion function is not 0 anymore for the true parameter vector.\n\n\n```python\ncriterion_msm = rp.get_moment_errors_func(\r\n params_true, options_new_seed, calc_moments, replace_nans, moments_obs, W\r\n)\r\ncriterion_msm(params_true)\n```\n\n\n\n\n 38.97135021944551\n\n\n\nOur optimizer thus also does not return a function value of 0 for the true parameter vector.\n\n\n```python\nrslt_new_seed = minimize(\r\n criterion=criterion_msm,\r\n params=params_true,\r\n algorithm=\"nag_pybobyqa\",\r\n constraints=constr,\r\n)\r\nrslt_new_seed[\"solution_criterion\"]\n```\n\n\n\n\n 17.369549302805428\n\n\n\nSince the optimizer does not even recognize the true parameter vector, it is also not able to reach a criterion function value of 0 for a different parameter vector.\n\n\n```python\nrslt_new_seed = minimize(\r\n criterion=criterion_msm,\r\n params=params_cand,\r\n algorithm=\"nag_pybobyqa\",\r\n constraints=constr,\r\n)\r\n\r\nrslt_new_seed[\"solution_criterion\"]\n```\n\n\n\n\n 17.369549302805428\n\n\n\n##### Multiple Simulation Seeds\n\nThe section above shows what happens to the criterion function if we change the seed for simulating the data. In the sections below we will repeat this exercise for multiple seeds and explore what happens if we increase the number of simulated as well as observed agents in our model.\n\n\n```python\n# List of seeds, includes true simulation seed of 132.\r\nseeds = list(range(120, 140, 1))\n```\n\n\n```python\n# Define other inputs for criterion function\r\ncriterion_kwargs = dict()\r\ncriterion_kwargs[\"params\"] = params_true.copy()\r\ncriterion_kwargs[\"options\"] = options\r\ncriterion_kwargs[\"calc_moments\"] = calc_moments\r\ncriterion_kwargs[\"replace_nans\"] = replace_nans\r\ncriterion_kwargs[\"moments_obs\"] = moments_obs\r\ncriterion_kwargs[\"weighting_matrix\"] = W\n```\n\nThe plot below shows the criterion function for different simulation seeds, including the true one. We can seed that different seeds lead to different fits between the simulated and observed data, with some seeds performing worse than others. Only the true seed leads to a function value of 0.\n\n\n```python\nplot_chatter(seeds, criterion_kwargs)\n```\n\n##### Changing the Simulation Seed for increasing Sample Size of Simulated Agents\n\nWe now repeat this exercise for an increasing number of simulated agents. As the plot below shows, increasing the number of simulated agents reduces the chatter considerably but the criterion function remains consistently above zero. Only the simulated sample of 1000 agents at the true simulation seed reaches a function of zero, since, as discussed before, the simulated sample is identical to the observed one for this calibration.\n\n\n```python\n# Changing the number of agents.\r\nnum_agents = [500, 1000, 5000, 10000, 50000]\n```\n\n\n```python\nplot_chatter_numagents_sim(seeds, num_agents, criterion_kwargs)\n```\n\n##### Changing the Simulation Seed for increasing Sample Size of Simulated *and* Observed Agents\n\nWe now increase not only the number of simulated agents but also the number of observed agents in our sample. Doing so decreases the chatter in the criterion function as before but also levels out the function around 0 for all simulation seeds. For large enough samples of observed and simulated agents the choice of simulation seed is thus irrelevant.\n\n\n```python\nplot_chatter_numagents_both(seeds, num_agents, calc_moments, replace_nans, criterion_kwargs)\n```\n\n*Note*: In contrast to the previous plot, we can see that the function reaches a value of 0 at the true simulation seed for all quantities of agents. This is the case because we now simultaneously increase the number of both simulated and observed agents while the observed sample remained untouched in the previous plot. Increasing the number of agents simultaneously for both groups creates identical samples for the true simulation seed at each quantity of agents. \n\n#### Moving away from the Optimum\n\nFor the next exercise we explore what happens to the criterion function if we move away from the optimum. We do this in a controlled fashion by changing the parameter values for *delta* and *wage_fishing* and fixing the latter in the constraints. Doing so should lead *delta* to move away from its optimal value of 0.95 during optimization.\n\n\n```python\nparams_cand.loc[\"wage_fishing\", \"value\"] = 0.072\r\nparams_cand\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        valuelower_boundupper_bound
                                        categoryname
                                        deltadelta0.9300.6901.000
                                        wage_fishingexp_fishing0.0720.0660.078
                                        nonpec_fishingconstant-0.100-0.200-0.090
                                        nonpec_hammockconstant1.0461.0001.100
                                        shocks_sdcorrsd_fishing0.0100.0000.500
                                        sd_hammock0.0100.0000.500
                                        corr_hammock_fishing0.0000.0000.500
                                        \n
                                        \n\n\n\nThe optimizer cannot retrieve the true parameter and does not reach a value of 0.\n\n\n```python\ncriterion_msm = rp.get_moment_errors_func(\r\n params_true, options_new_seed, calc_moments, replace_nans, moments_obs, W\r\n)\r\nrslt_wrong_fix = minimize(\r\n criterion=criterion_msm,\r\n params=params_cand,\r\n algorithm=\"nag_pybobyqa\",\r\n constraints=constr,\r\n)\r\nrslt_wrong_fix[\"solution_criterion\"]\n```\n\n\n\n\n 813.7988993578208\n\n\n\n##### Retrieving the true Parameter Vector\n\nWe now repeat the estimation with the new parameter vector and free up *wage_fishing* to retrieve the optimal values for both parameters.\n\nThe parameter for *wage_fishing* is still 0.072 since we fixed it for the prior estimation:\n\n\n```python\nparams_cand.loc[:, \"value\"] = rslt_wrong_fix[\"solution_params\"][[\"value\"]]\r\nparams_cand\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        valuelower_boundupper_bound
                                        categoryname
                                        deltadelta0.9230220.6901.000
                                        wage_fishingexp_fishing0.0720000.0660.078
                                        nonpec_fishingconstant-0.100000-0.200-0.090
                                        nonpec_hammockconstant1.0460001.0001.100
                                        shocks_sdcorrsd_fishing0.0100000.0000.500
                                        sd_hammock0.0100000.0000.500
                                        corr_hammock_fishing0.0000000.0000.500
                                        \n
                                        \n\n\n\nWe now free up *wage_fishing* in the constraints in addition to *delta*.\n\n\n```python\n# Adjust constraints to free up both delta and wage_fishing.\r\nconstr_u = constr_base.copy()\r\nconstr_u.remove({\"loc\": \"delta\", \"type\": \"fixed\"})\r\nconstr_u.remove({\"loc\": \"wage_fishing\", \"type\": \"fixed\"})\n```\n\nFreeing up the non-optimal *wage_fishing* improves the estimates. The criterion function value is much closer to 0 and the optimizer manages to retrieve the true parameter values quite closely.\n\n\n```python\nrslt_unfix = minimize(\r\n criterion=criterion_msm,\r\n params=params_cand[[\"value\"]],\r\n algorithm=\"nag_pybobyqa\",\r\n constraints=constr_u,\r\n)\r\nrslt_unfix[\"solution_criterion\"]\n```\n\n\n\n\n 710.9165575145821\n\n\n\nFor easier comparison, we can compute the difference between the true and estimated value:\n\n\n```python\ndeviation = params_true[\"value\"] - rslt_unfix[\"solution_params\"][\"value\"]\r\ndeviation\n```\n\n\n\n\n category name \n delta delta 0.026703\n wage_fishing exp_fishing -0.001791\n nonpec_fishing constant 0.000000\n nonpec_hammock constant 0.000000\n shocks_sdcorr sd_fishing 0.000000\n sd_hammock 0.000000\n corr_hammock_fishing 0.000000\n Name: value, dtype: float64\n\n\n\n#### Derivative-Based Optimization Algorithm\n\nSo far we have used only one optimization algorithm to estimate the parameter vector. The algorithm we used, BOBYQA (Bound Optimization by Quadratic Approximation), is a derivative-free optimization algorithm which works fairly well on our criterion function. As discussed above, our criterion function is a step function that contains plateaus for certain ranges of parameter values. An important implication of this property is that we cannot calculate proper derivatives and thus derivative-based optimization algorithms will not work to estimate the parameters. \r\n\r\nTo demonstrate this problem, we will now try to estimate the parameters using an optimization algorithm that does use derivatives during optimization.\n\n\n```python\n# Define candidate parameter vector and constraints.\r\nparams_cand = params_true.copy()\r\nparams_cand.loc[\"delta\", \"value\"] = 0.93\r\n\r\nparams_cand[\"lower_bound\"] = [0.79, 0.066, -0.11, 1.04, 0, 0, 0]\r\nparams_cand[\"upper_bound\"] = [1, 0.072, -0.095, 1.055, 0.5, 0.5, 0.5]\r\n\r\nconstr = constr_base.copy()\r\nconstr.remove({\"loc\": \"delta\", \"type\": \"fixed\"})\n```\n\nWe try the L-BFGS-B (Limited-Memory-Broyden-Fletcher-Goldfarb-Shanno with box constraints) algorithm. As shown below, L-BFGS-B fails and the optimizer returns the same parameter vector that we used as an input. \n\n\n```python\ncriterion_msm = rp.get_moment_errors_func(\r\n params_true, options_new_seed, calc_moments, replace_nans, moments_obs, W\r\n)\r\nrslt = minimize(\r\n criterion=criterion_msm,\r\n params=params_cand,\r\n algorithm=\"scipy_lbfgsb\",\r\n constraints=constr,\r\n)\r\nrslt\n```\n\n\n\n\n {'solution_x': array([0.93]),\n 'solution_criterion': 7204.902739425344,\n 'solution_derivative': array([0.]),\n 'solution_hessian': None,\n 'n_criterion_evaluations': 1,\n 'n_derivative_evaluations': None,\n 'n_iterations': 0,\n 'success': True,\n 'reached_convergence_criterion': None,\n 'message': 'CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL',\n 'solution_params': lower_bound upper_bound value\n category name \n delta delta 0.790 1.000 0.930\n wage_fishing exp_fishing 0.066 0.072 0.070\n nonpec_fishing constant -0.110 -0.095 -0.100\n nonpec_hammock constant 1.040 1.055 1.046\n shocks_sdcorr sd_fishing 0.000 0.500 0.010\n sd_hammock 0.000 0.500 0.010\n corr_hammock_fishing 0.000 0.500 0.000}\n\n\n\n### References\n\n* Adda, J., & Cooper, R. W. (2003). *Dynamic Economics: Quantitative Methods and Applications*. MIT press.\n\n\n* Adda, J., Dustmann, C., & Stevens, K. (2017). The Career Costs of Children. *Journal of Political Economy*, 125(2), 293-337.\n\n\n* Andrews, I., Gentzkow, M., & Shapiro, J. M. (2017). Measuring the Sensitivity of Parameter Estimates to Estimation Moments. *The Quarterly Journal of Economics*, 132(4), 1553-1592.\n\n\n* Bruins, M., Duffy, J. A., Keane, M. P., & Smith Jr, A. A. (2018). Generalized Indirect Inference for Discrete Choice Models. *Journal of econometrics*, 205(1), 177-203.\n\n\n* Davidson, R., & MacKinnon, J. G. (2004). *Econometric Theory and Methods (Vol. 5)*. New York: Oxford University Press.\n\n\n* Evans, R. W. (2018, July 5). Simulated Method of Moments (SMM) Estimation. Retrieved November 30, 2019, from https://notes.quantecon.org/submission/5b3db2ceb9eab00015b89f93.\n\n\n* Frazier, D. T., Oka, T., & Zhu, D. (2019). Indirect Inference with a Non-Smooth Criterion Function. *Journal of Econometrics*, 212(2), 623-645.\n \n \n* Gourieroux, M., & Monfort, D. A. (1996). *Simulation-based econometric methods*. Oxford university press.\n\n\n* McFadden, D. (1989). A Method of Simulated Moments for Estimation of Discrete Response Models without Numerical Integration. *Econometrica: Journal of the Econometric Society*, 995-1026.\n", "meta": {"hexsha": "e48df1de0853039160eb06340e1644e2e51cb6f7", "size": 454006, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lectures/method-of-simulated-moments/notebook.ipynb", "max_stars_repo_name": "OpenSourceEconomics/ekw-lectures", "max_stars_repo_head_hexsha": "b887cba27499388e71830a62d366ed368aa238e1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2019-11-15T15:21:27.000Z", "max_stars_repo_stars_event_max_datetime": "2020-07-08T15:04:30.000Z", "max_issues_repo_path": "lectures/method-of-simulated-moments/notebook.ipynb", "max_issues_repo_name": "OpenSourceEconomics/respy-lectures", "max_issues_repo_head_hexsha": "b887cba27499388e71830a62d366ed368aa238e1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 9, "max_issues_repo_issues_event_min_datetime": "2019-11-18T15:54:36.000Z", "max_issues_repo_issues_event_max_datetime": "2020-07-14T13:56:53.000Z", "max_forks_repo_path": "lectures/method-of-simulated-moments/notebook.ipynb", "max_forks_repo_name": "OpenSourceEconomics/ekw-lectures", "max_forks_repo_head_hexsha": "b887cba27499388e71830a62d366ed368aa238e1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2021-01-25T15:41:30.000Z", "max_forks_repo_forks_event_max_datetime": "2021-09-21T08:51:36.000Z", "avg_line_length": 138.7548899756, "max_line_length": 53342, "alphanum_fraction": 0.8395748074, "converted": true, "num_tokens": 18063, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.7122321964553657, "lm_q1q2_score": 0.41672788020673696}} {"text": "# \"\uadf8\ub798\ud504 \ud569\uc131\uacf1 \uc2e0\uacbd\ub9dd\"\n> \"\ubbf8\ubd84 \ubc29\uc815\uc2dd\uc744 \uac1c\ub150\uc744 \ud655\uc7a5\u00b7\ucd94\uc0c1\ud654\ud558\uc5ec \uadf8\ub798\ud504 \uc2e0\uacbd\ub9dd\uc744 \uc774\ud574\ud55c\ub2e4.\"\n\n- toc: true\n- badges: true\n- author: \ub2e8\ud638\uc9c4\n- categories: [graph]\n\n## \ubbf8\ubd84 \ubc29\uc815\uc2dd\uc774 \ucd94\uc0c1\ud654\ub41c \uadf8\ub798\ud504\n\n\uadf8\ub798\ud504\uc5d0\uc11c \ub77c\ud50c\ub77c\uc2a4 \uc5f0\uc0b0\uc790\ub97c \ub9cc\ub4e4\uace0 \uc774\uc6a9\ud558\ub294 \uac83\uc744 \ucc98\uc74c \ubcf4\uba74 \uc0c1\ub2f9\ud788 \ucd94\uc0c1\uc801\uc73c\ub85c \uc5ec\uaca8\uc9c4\ub2e4. \uadf8 \uac1c\ub150\uc744 \uc774\ud574\ud558\uae30 \uc704\ud558\uc5ec \ub77c\ud50c\ub77c\uc2a4 \uc5f0\uc0b0\uc790\ub97c \uc0b4\ud3b4\ubcf4\uace0 \uadf8\uac83\uc73c\ub85c \ubb34\uc5c7\uc744 \ud560 \uc218 \uc788\ub294\uc9c0 \uc54c\uc544\ubcf4\uc790. \uadf8\ub9ac\uace0 \uc774\ub97c \uadf8\ub798\ud504\uc5d0 \ud655\uc7a5\ud558\uace0 \uadf8\ub798\ud504 \ud478\ub9ac\uc5d0 \ubcc0\ud658 \ubc0f \uadf8\ub798\ud504 \ud569\uc131\uacf1 \uc2e0\uacbd\ub9dd\uc744 \uc124\uba85\ud55c\ub2e4. \ub9c8\uc9c0\ub9c9\uc73c\ub85c spektral \ud328\ud0a4\uc9c0\uc758 \uc608\uc81c\ub97c \ud1b5\ud574 \ub17c\ubb38 \uc778\uc6a9 \ubb38\uc81c\uc5d0 \uc811\uadfc\ud574 \ubcf4\uaca0\ub2e4.\n\n### \uc720\ud55c \ucc28\ubd84\n\n1\ucc28\uc6d0 \uc5f0\uc18d \uacf5\uac04\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218 $u(x)$\uc5d0 \ub300\ud574 \uc791\uc6a9\ud558\ub294 \ub77c\ud50c\ub77c\uc2a4 \uc5f0\uc0b0\uc790 $\\Delta$\ub294 \uacf5\uac04\uc744 \ub4f1\uac04\uaca9 h\ub85c \uc774\uc0b0\ud654 \ud55c \ud6c4 \ub2e4\uc74c\uacfc \uac19\uc774 \uadfc\uc0ac\ud560 \uc218 \uc788\ub2e4[1].\n\n$\\Delta u(x) = u''(x) \\approx \\frac{\\frac{u(x+h) - u(x)}{h} - \\frac{u(x) - u(x-h)}{h}}{h} = \\frac{u(x - h) - 2u(x) + u(x+h)}{h^2}$\n\n2\ucc28\uc6d0\uc778 \uacbd\uc6b0,\n\n$\\Delta u(x, y) \\approx \\frac{u(x-h, y) + u(x+h, y) + u(x, y-h) + u(x, y+h) - 4 u(x, y)}{h^2}$\n\n\uc2dd\uacfc \uac19\uc774 \ub41c\ub2e4. $(x, y)$\uc5d0 \uc778\uc811\ud55c \ud639\uc740 \uc5f0\uacb0\ub41c \uc810\ub4e4\uc5d0\uc11c \ud568\uc218 \uac12\uc744 \ub354\ud558\uace0 \uadf8 \uc810\ub4e4\uc758 \uc218\ub9cc\ud07c $u(x, y)$ \uac12\uc744 \uc81c\ud55c \uac83\uc5d0 \ube44\ub840\ud558\uc5ec \uac01 \uc810\uc5d0\uc11c\uc758 \ub77c\ud50c\ub77c\uc2a4 \uac12\uc744 \uadfc\uc0ac\ud560 \uc218 \uc788\ub2e4. \uadf8 \uac1c\ub150\uc744 \ud655\uc7a5\ud558\uc5ec \uadf8\ub798\ud504\uc5d0\uc11c \uc0ac\uc6a9\ud558\ub294 $L = D - A$ \ud589\ub82c\uc744 \uc5bb\uc744 \uc218 \uc788\uace0, \uc790\uc5f0\uc2a4\ub7fd\uac8c \ub77c\ud50c\ub77c\uc2a4 \ud589\ub82c\uc774\ub77c\ub294 \uc774\ub984\uc744 \ubd99\uc600\ub2e4[3]. \n\n\ub77c\ud50c\ub77c\uc2a4 \uc5f0\uc0b0\uc790\uac00 \uc788\ub294 \ubb38\uc81c\uc5d0 \ub300\ud558\uc5ec \uc5b4\ub5a4 \uc811\uadfc\uc744 \ud560 \uc218 \uc788\ub294\uc9c0 \ubb3c\ub9ac\uc801\uc778 \ubb38\uc81c\ub97c \uc0b4\ud3b4\ubcf4\uc790.\n\n\n### \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc744 \uc774\uc6a9\ud55c \uc5f4\ud655\uc0b0 \ubc29\uc815\uc2dd \ud480\uc774\n\n$x \\in [0, 1]$, $t \\ge 0$\uc5d0\uc11c 1D \uc5f4\ud655\uc0b0\uc5d0 \ub300\ud55c \uc5f4\uc804\ub2ec \ubaa8\ud615\uc73c\ub85c \uc2dc\uac04\uc5d0 \ub530\ub978 \uc628\ub3c4 \ubd84\ud3ec\ub294 $u(x, t)$\ub294 $k \\ge 0$ \uacc4\uc218\uc5d0 \ub300\ud558\uc5ec \uc9c0\ubc30\ubc29\uc815\uc2dd $u_t - k u_{xx} = 0$\uc744 \ub530\ub978\ub2e4. \ubb38\uc81c\ub97c \uac04\ub2e8\ud558\uac8c \ub9cc\ub4e4\uae30 \uc704\ud558\uc5ec $u(0, t) = u(1, t) = 0$\uc778 \ub514\ub9ac\ud074\ub808 \uacbd\uacc4\uce58 \ubb38\uc81c\uc774\uba70 $u(x, 0) = \\phi(x)$ \ucd08\uae30 \ubd84\ud3ec\uac00 \uc8fc\uc5b4\uc9c4\ub2e4\uace0 \ud558\uc790. \ubcc0\uc218 \ubd84\ub9ac $(x, t) = X(x)T(t)$ \uad00\uacc4\ub97c \uc801\uc6a9\ud558\uba74 2\uac1c\uc758 \uc0c1\ubbf8\ubd84 \ubc29\uc815\uc2dd\uc73c\ub85c \ubc14\uafc0 \uc218 \uc788\ub2e4[4].\n\n$\\frac{\\dot T}{k T} = \\frac{X''}{X} = \\text{const}$\n\n\uacbd\uacc4 \uc870\uac74\uc5d0 \uc758\ud558\uc5ec \uc0c1\uc218\ub294 $-(n\\pi)^2$\uc774\uace0, $X_n(x) = D_n \\sin(n\\pi x)$\uc774\ub2e4. \ub2e8, $n=1, 2, ...$ \ubcc0\uc218 $t$\uc5d0 \ub300\ud558\uc5ec \uc9c0\uc218 \ud568\uc218\ub85c \uc77c\ubc18 \ud574\ub97c \uad6c\ud560 \uc218 \uc788\uc73c\ubbc0\ub85c, $u(x, t)$\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n\n$u(x, t) = \\sum_n A_n \\sin(n\\pi x) \\exp(-k (n\\pi)^2 t)$\n\n$u(x, 0) = \\phi(x)$ \uc870\uac74\uc73c\ub85c\ubd80\ud130 $A_n$\uc744 \uad6c\ud560 \uc218 \uc788\ub2e4.\n\n$\\int \\phi(x) \\sin (m \\pi x) dx = \\sum_n A_n \\int \\sin (n \\pi x) \\sin(m\\pi x) dx = \\sum_n A_n \\frac{\\delta_{nm}}{2} = \\frac{A_m}{2}$\n\n\uc774 \ubb38\uc81c\ub97c \ud478\ub294 \uacfc\uc815\uc5d0\uc11c \ubc1c\uacac\ud55c \ub2e4\uc74c \uc0ac\ud56d\uc5d0 \uc8fc\ubaa9\ud558\uc790.\n\n* $\\sin (n \\pi x)$\ub294 \ub77c\ud50c\ub77c\uc2a4 \uc5f0\uc0b0\uc790\uc758 \uace0\uc720 \ubca1\ud130\uc774\ub2e4.\n* \uace0\uc720 \ubca1\ud130 \uc0ac\uc774\uc5d0 \uc9c1\uad50\uc131\uc774 \uc874\uc7ac\ud558\uba74 \uacc4\uc0b0\uc774 \uac04\ud3b8\ud558\ub2e4.\n\n\uace0\uc720 \ubca1\ud130\ub85c\ubd80\ud130 \ud478\ub9ac\uc5d0 \uae09\uc218\ub97c \uc5f0\uacb0\ud574\ubcf4\uc790[5]. \n\n* \ud504\ub9ac\uc5d0 \uae09\uc218\uc5d0\uc11c \uc0ac\uc6a9\ud558\ub294 \uc0ac\uc778\ud568\uc218\ub294 \ub77c\ud50c\ub77c\uc2a4 \uc5f0\uc0b0\uc790\uc758 \uace0\uc720 \ubca1\ud130\uc774\ub2e4.\n* \uc5f0\uc18d \uc815\uc758\uc5ed\uc758 \uc774\uc0b0 \ud45c\ubcf8\uc740 \uc774\uc0b0 \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc73c\ub85c \uc5f0\uacb0\ub41c\ub2e4.\n\n\uc774\ub97c \ub2e4\uc74c \uc808\uc5d0\uc11c \uadf8\ub798\ud504\uc5d0 \ud655\uc7a5\ud574\ubcf4\uaca0\ub2e4.\n\n\uc774 \uc808\uc744 \ub9c8\ubb34\ub9ac\ud558\uba74\uc11c \uc2e4\uc81c \uac12\uc744 \ub300\uc785\ud558\uc5ec \uc5f4\ud655\uc0b0 \ubb38\uc81c\ub97c \ud480\uc5b4\ubcf4\uace0 \ud574\ub97c \uc74c\ubbf8\ud574 \ubcf4\uaca0\ub2e4.\n\n\n```python\nfrom sympy import *\nfrom sympy.abc import x, t, n\nfrom sympy.plotting import plot\n\nphi = Piecewise((0, x < 0.4), (1, x < 0.6), (0, True))\n```\n\n\n```python\nk = 1\nA = 2 * integrate(phi * sin(n * pi * x), (x, 0, 1))\nu = Sum(A * sin(n * pi * x) * exp(-k * (n * pi) ** 2 * t), (n, 1, 30))\n```\n\n\n```python\np = plot(\n phi,\n u.doit().subs({t: 0}),\n u.doit().subs({t: 0.0005}),\n u.doit().subs({t: 0.005}),\n u.doit().subs({t: 0.02}),\n (x, 0, 1), show=False\n)\np[2].line_color = 'orange'\np[3].line_color = 'red'\np[4].line_color = 'black'\np.show();\n```\n\n$t=0$\uc5d0\uc11c \ubd88\uc5f0\uc18d \uc810\uc5d0\uc11c \uae41\uc2a4 \ud604\uc0c1\uc774 \ubaa9\uaca9\ub41c\ub2e4. \uc2dc\uac04\uc774 \ud750\ub984\uc5d0 \ub530\ub77c \uae41\uc2a4 \ubb3c\uacb0\uc740 \ud655\uc0b0\uc5d0 \uc758\ud574 \uc0ac\ub77c\uc9c0\uace0 \uc628\ub3c4\uac00 \ud3c9\ud615 $u(x, \\infty) = 0$\uc744 \ud5a5\ud558\uc5ec \ubcc0\ud654\ub418\ub294 \uac83\uc744 \uc798 \ubaa8\uc0ac\ud558\uace0 \uc788\ub2e4.\n\n### \uadf8\ub798\ud504 \ud478\ub9ac\uc5d0 \ubcc0\ud658\n\n\n\uc774\uc0b0 \uc2dc\uc2a4\ud15c\uc5d0\uc11c \uc774\uc0b0 \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4[5].\n\n$A_k = \\sum_{r = 0}^{n - 1} a_m e^{-i (2\\pi mk/n)}$\n\n$a_m = \\frac{1}{n} \\sum_{k = 0}^{n - 1} A_k e^{i (2\\pi mk/n)}$\n\n\uc9c0\uc218\ud56d\uc740 \ub77c\ud50c\ub77c\uc2a4 \uc5f0\uc0b0\uc790\uc758 \uace0\uc720 \ubca1\ud130\uc774\ub2e4. \uadf8\ub798\ud504 \ub77c\ud50c\ub77c\uc2a4 \uc5f0\uc0b0\uc790 $\\mathcal{L}$\ub294 \uace0\uc720\uce58 $\\lambda_{l}$\uacfc \uace0\uc720 \ubca1\ud130 $x_{l}$\ub97c \uac00\uc9c4\ub2e4.\n\n$\\mathcal{L} x_{l} = \\lambda_{l} x_{l}$\n\nN\uac1c\uc758 \uaf2d\uc9c0\uc810\uc73c\ub85c \uad6c\uc131\ub41c \uadf8\ub798\ud504 $\\mathcal{G} (\\mathcal{V, E})$\uc5d0\uc11c \uace0\uc720\uce58\ub294 \uc74c\uc774 \uc544\ub2cc \uc2e4\uc218\uc774\ub2e4.\n\n$0 = \\lambda_0 \\lt \\lambda_1 \\le \\lambda_1 \\cdots \\le \\lambda_{N - 1}$\n\n\uc774\uc0b0 \ud478\ub9ac\uc5d0 \ubcc0\ud658\uacfc \ub9c8\ucc2c\uac00\uc9c0\ub85c \uadf8\ub798\ud504 \ud478\ub9ac\uc5d0 \uc5f0\uc0b0\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc5ec \ud65c\uc6a9\ud560 \uc218 \uc788\ub2e4.\n\n$\\hat f (l) = \\sum_{n=1}^{N} x_{l}^* (n) f(n)$\n\n$f(n) = \\sum_{l=0}^{N-1} \\hat f (l) x_{l} (n)$\n\n### \uadf8\ub798\ud504 \ud569\uc131\uacf1 \uc2e0\uacbd\ub9dd\n\n\uc55e\uc11c \ub77c\ud50c\ub77c\uc2a4 \ubc29\uc815\uc2dd\uc73c\ub85c \ud45c\ud604\ub41c \ub0b4\ubd80 \uc5f4\uc6d0\uc774 \uc5c6\ub294 \uc5f4\ud655\uc0b0 \ubc29\uc815\uc2dd\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \uc774\ub97c \ud655\uc7a5\ud55c \ubc29\uc815\uc2dd\uc744 \ud478\uc640\uc1a1 \ubc29\uc815\uc2dd\uc774\ub77c\uace0 \ud55c\ub2e4[7].\n\n$\\Delta \\phi = f$\n\n\uc804\uc0b0 \uc720\uccb4 \uc5ed\ud559\uc758 \uc720\ud55c \uccb4\uc801\uc73c\ub85c \uc704 \ud478\uc640\uc1a1 \ubc29\uc815\uc2dd\uc758 \ub2e8\uc704 \uccb4\uc801\uc5d0\uc11c \uc758\ubbf8\ub294 \uc8fc\ubcc0\uacfc \uc8fc\uace0 \ubc1b\ub294 \uc815\ubcf4\uc640 \ub0b4\ubd80\uc5d0\uc11c \uc0dd\uc131\ub418\ub294 \uc815\ubcf4\ub294 \uc11c\ub85c \uc0c1\uc1c4\ub41c\ub2e4\ub294 \uac83\uc774\ub2e4. \uadf8\ub798\ud504\uc758 \uac01 \uaf2d\uc9d3\uc810\uc5d0\uc11c \uc2a4\uce7c\ub77c \uac12\uc774 \uc815\uc758\ub418\uba74 \uc8fc\ubcc0 \uaf2d\uc9d3\uc810\uacfc \uc815\ubcf4\ub97c \uc8fc\uace0 \ubc1b\uc544\uc11c \uc0c8\ub85c\uc6b4 \uc2a4\uce7c\ub77c \uac12\uc73c\ub85c \ud22c\uc0ac\ud55c\ub2e4. \ucc28\uc218 \ud589\ub82c D\uc640 \uc778\uc811 \ud589\ub82c A\ub85c \ub77c\ud50c\ub77c\uc2a4 \uc5f0\uc0b0\uc790 $L = D - A$ \ud639\uc740 \uc815\uaddc\ud654 \ud558\uc5ec $L = I - D^{-1/2} A D^{-1/2}$\ub85c \uadf8\ub798\ud504\ub97c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uadf8\ub798\ud504 \ub0b4\uc5d0 \uc5ec\ub7ec \uc12c\uc774 \uc788\uc744 \uc218\ub3c4 \uc788\uc73c\ubbc0\ub85c \uc218\uce58\uc801\uc778 \uc548\uc815\uc131\uc744 \uc704\ud558\uc5ec \uc790\uae30 \uc5f0\uacb0\ud56d\uc744 \ucd94\uac00\ud558\uc5ec $\\tilde A = A + I$\ub85c \ud558\uc5ec \uc0c8\ub85c $L = I - \\tilde D^{-1/2} \\tilde A \\tilde D^{-1/2}$\ub85c \ud45c\ud604\ud558\uc5ec \ubcf4\uc790. $L x = x + \\delta x$\uc77c \ub54c \uc0c8\ub85c\uc6b4 \uc815\ubcf4\uc5d0 \ud574\ub2f9\ud558\ub294 $\\tilde D^{-1/2} \\tilde A \\tilde D^{-1/2}$\uc744 \uc774\uc6a9\ud558\uc5ec \uadf8\ub798\ud504 \ud569\uc131\uacf1 \uc2e0\uacbd\ub9dd\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc600\ub2e4[8].\n\n$H^{(l + 1)} = \\sigma \\left( \\tilde D^{-1/2} \\tilde A \\tilde D^{-1/2} H^{(l)} W^{(l)} \\right) $, $H^{(0)} = X$\n\n\uc774 \uc2dd\uc740 \uac01 \uaf2d\uc9d3\uc810\uc758 \uc815\ubcf4\uc640 \uc778\uc811 \uc815\ubcf4\ub97c \ubaa8\uc544\uc11c \uc815\ubcf4\ub97c \uc0dd\uc131\ud558\uace0 \uc774\uc5d0 \ub300\ud558\uc5ec \ud559\uc2b5 \uce35\uc744 \uc313\uaca0\ub2e4\ub294 \ub73b\uc774\ub2e4. \ucd94\uac00 \ud56d \ubfd0\ub9cc \uc544\ub2c8\ub77c $I$ \ud589\ub82c\uae4c\uc9c0 \uc774\uc6a9\ud55c\ub2e4\uba74 Resnet\uc2a4\ub7ec\uc6b4 \uc811\uadfc\uc774 \ub420 \uac83\uc774\ub2e4. \uc774 \ubd80\ubd84\uc5d0 \ub300\ud574\uc11c\ub294 \ucd94\uac00\uc801\uc778 \uc219\uc81c\ub85c \ub0a8\uaca8\ub450\uaca0\ub2e4.\n\n\uc774 \uc2e0\uacbd\ub9dd\uc758 \uc774\uc6a9\uc744 \ub2e4\uc74c CORA \uc778\uc6a9 \ub370\uc774\ud130\ub97c \uac00\uc9c0\uace0 \uadf8\ub798\ud504 \ud559\uc2b5\uc744 \uc218\ud589\ud558\uc5ec \ubcf4\uaca0\ub2e4. Spektral [\ud29c\ud1a0\ub9ac\uc5bc](https://github.com/danielegrattarola/spektral/blob/master/examples/node_prediction/citation_gcn.py) \ucf54\ub4dc\uc640 \ub3d9\uc77c\ud558\uba70 \uc124\uba85\uc744 \ucd94\uac00\ud558\uc5ec \uc815\ub9ac\ud55c\ub2e4.\n\n## \ucc38\uace0\n\n1. \uc704\ud0a4\ud53c\ub514\uc544, https://en.wikipedia.org/wiki/Finite_difference_method\n1. \uc704\ud0a4\ud53c\ub514\uc544, https://en.wikipedia.org/wiki/Discrete_Laplace_operator\n1. \uc704\ud0a4\ud53c\ub514\uc544, https://en.wikipedia.org/wiki/Laplacian_matrix\n1. Standford, \ud3b8\ubbf8\ubd84 \ubc29\uc815\uc2dd 2003\ub144 \uc5ec\ub984 \uac15\uc758 \ub178\ud2b8, Math 220B, https://web.stanford.edu/class/math220b/handouts/heateqn.pdf, 2003\n1. \ub2e8\ud638\uc9c4 \ube14\ub85c\uadf8, \ud478\ub9ac\uc5d0 \uae09\uc218, https://danhojin.github.io/jupyter-blog/general/2021/01/30/dft.html\n1. D.k. Hammond, P. Vandergheynst, and R. Gribonval, Wavelets on graphs via spectral graph theory, arXiv:0912.3848v1, 2009\n1. \uc704\ud0a4\ud53c\ub514\uc544, https://en.wikipedia.org/wiki/Poisson's_equation\n1. T.N. Kipf, M. Welling, Semi-supervised classification with graph convolutional networks, arXiv:1609.02907v4, 2017\n\n* spektral \ud328\ud0a4\uc9c0\ub294 cpu\ub85c \ud559\uc2b5\ud55c\ub2e4. \uc7a6\uc740 \uba54\uc2dc\uc9c0 \uc804\ub2ec\ub85c cuda\uc758 \uc774\uc810\uc774 \uc5c6\ub294 \ub4ef \ud558\ub2e4.\n\n\n```python\nimport os\nos.environ['CUDA_VISIBLE_DEVICES'] = '-1' # tensorflow device: cpu\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport tensorflow as tf\nfrom tensorflow.keras.callbacks import EarlyStopping\nfrom tensorflow.keras.losses import CategoricalCrossentropy\nfrom tensorflow.keras.optimizers import Adam\n\n\n%matplotlib inline\n\ntf.__version__\n```\n\n\n\n\n '2.4.0'\n\n\n\n## CORA \uc778\uc6a9 \ub370\uc774\ud130\n\nhttps://relational.fit.cvut.cz/dataset/CORA\n\n\n\n2708\uac1c\uc758 \uacfc\ud559 \ub17c\ubb38\uc5d0\uc11c \uc778\uc6a9 \uad00\uacc4\ub97c \uac00\uc9c0\uace0 \uc628 \ub370\uc774\ud130\uc774\ub2e4. \uac01 \ub17c\ubb38\uc740 7\uac1c\ub85c \uce74\ud14c\uace0\ub9ac\uac00 \uc911 \ud558\ub098\ub85c \ubd84\ub958\ub418\uc5b4 \uc788\uc73c\uba70 1433\uac1c\uc758 \uc5b4\ud718\uc9d1\uc5d0\uc11c \ub17c\ubb38\uc5d0 \ud3ec\ud568\ub41c \uc5b4\ud718\uac00 \uac01 \ub17c\ubb38\uc758 \ud2b9\uc9d5 \ubcc0\uc218\uc774\ub2e4.\n\n\ub0b4\ubd80 \ucf54\ub4dc\ub97c \uc0b4\ud3b4\ubcf4\uc9c0 \ubabb\ud55c \uad00\uacc4\ub85c \ucd94\uc815 \uc0ac\ud56d\ub9cc \uc815\ub9ac\ud574\ubcf8\ub2e4.\n\n* GCNConv \uce35\uc740 \uc704\uc5d0\uc11c \ubcf4\uc778 \uadf8\ub798\ud504 \ud569\uc131\uacf1 \uc2e0\uacbd\ub9dd \uce35\uc774\ub2e4. \ud65c\uc131 \ud568\uc218\uac00 \uc5c6\ub294 \uac83\uc774 \uae30\ubcf8\uc73c\ub85c $X' = \\hat D^{-1/2} \\hat A \\hat D^{-1/2} X W + b$ \ud56d\uc744 \uacc4\uc0b0\ud55c\ub2e4. $\\hat D^{-1/2} \\hat A \\hat D^{-1/2}$ \ubd80\ubd84\uc740 \ud55c\ubc88\ub9cc \uacc4\uc0b0\ud558\uc5ec \uc800\uc7a5\ud574 \ub450\uba74 \ub418\ubbc0\ub85c LayerPreprocess\ub97c \ud1b5\ud558\uc5ec \ucc98\ub9ac\ud558\uace0, \uacb0\uacfc\ub97c \ud76c\uc18c \ud589\ub82c\uc5d0 \uc800\uc7a5\ud55c\ub2e4.\n* \ub2e4\ub978 \uc885\ub958\uc758 \ud559\uc2b5 \uce35\ub3c4 GCNConv \uce35\uacfc \ube44\uc2b7\ud55c transforms \uad6c\uc870\ub85c \uc815\uc758\ud55c\ub2e4. spektral\uc758 \ub2e4\ub978 \uc608\uc81c\ub97c \uc0b4\ud3b4\ubcf4\uc790.\n\n\n```python\nfrom spektral.data.loaders import SingleLoader\nfrom spektral.datasets.citation import Citation\nfrom spektral.transforms import AdjToSpTensor, LayerPreprocess\nfrom spektral.layers import GCNConv\nfrom spektral.models.gcn import GCN\n\n\nlearning_rate = 1e-2\nepochs = 200\npatience = 10\ndata = 'cora'\nseed = 0\n\ntf.random.set_seed(seed=seed)\n\ndataset = Citation(\n data, normalize_x=True, transforms=[LayerPreprocess(GCNConv), AdjToSpTensor()]\n)\n```\n\n Pre-processing node features\n\n\n /home/danhojin/miniconda3/envs/ml8/lib/python3.8/site-packages/scipy/sparse/_index.py:125: SparseEfficiencyWarning: Changing the sparsity structure of a csr_matrix is expensive. lil_matrix is more efficient.\n self._set_arrayXarray(i, j, x)\n\n\n* \uc778\uc6a9 \ub370\uc774\ud130\uc14b\uc5d0\ub294 \ud558\ub098\uc758 \uadf8\ub798\ud504\uac00 \ub4e4\uc5b4\uc788\ub2e4. \n* \uaf2d\uc9d3\uc810\uc740 2708\uac1c, \uac01 \uaf2d\uc9d3\uc810\uc740 1433\uac1c\uc758 \ud2b9\uc9d5 \ubcc0\uc218\uc758 \ub370\uc774\ud130\uac00 \ub4e4\uc5b4\uc788\ub2e4. \ub2e8\uc5b4 \uc0ac\uc804\uc758 \uc5b4\ud718\uac00 1433\uac1c \uc774\ubbc0\ub85c \uac01 \ub2e8\uc5b4\uac00 \ud2b9\uc9d5 \ubcc0\uc218\uac00 \ub418\uc5c8\ub2e4.\n* \uac04\uc120(edge)\uc5d0\ub294 \ud2b9\uc9d5 \ubcc0\uc218 \uc124\uc815\uc774 \ub418\uc5b4 \uc788\uc9c0 \uc54a\ub2e4.\n* \uac01 \uaf2d\uc9c0\uc810\uc740 7\uac1c\uc758 \ud074\ub798\uc2a4 \uc911 \ud558\ub098\uc774\ub2e4.\n\n\n```python\nprint(len(dataset))\ng = dataset[0]\ng\n```\n\n 1\n\n\n\n\n\n Graph(n_nodes=2708, n_node_features=1433, n_edge_features=None, n_labels=7)\n\n\n\n* a: \uc778\uc811 \ud589\ub82c\n* x: \uaf2d\uc9d3\uc810 \ud2b9\uc9d5\n* e: \uac04\uc120 \ud2b9\uc9d5\n* y: \ub808\uc774\ube14\n\n\n```python\nprint(g.a.shape, g.x.shape) # 1433-word dictionary\nassert(g.e == None) # edges have not features\n```\n\n (2708, 2708) (2708, 1433)\n\n\n\n```python\ng.y.shape\n```\n\n\n\n\n (2708, 7)\n\n\n\n\n```python\ng.y[0] # one-hot\n```\n\n\n\n\n array([0., 0., 0., 1., 0., 0., 0.], dtype=float32)\n\n\n\n### \ubaa8\ub378\n\n\n```python\ndef mask_to_weights(mask):\n return mask.astype(np.float32) / np.count_nonzero(mask)\n\nweights_tr, weights_va, weights_te = (\n mask_to_weights(mask)\n for mask in (dataset.mask_tr, dataset.mask_va, dataset.mask_te)\n)\n```\n\n* GCN\uc5d0\uc11c channels\uc740 GCNConv\uc758 \uc740\ub2c9\uce35\uc758 \ud06c\uae30\ub97c \uacb0\uc815\ud55c\ub2e4.\n\n\n```python\nmod_1 = GCN(n_labels=dataset.n_labels, channels=16, n_input_channels=dataset.n_node_features)\nmod_1.compile(\n optimizer=Adam(learning_rate),\n loss=CategoricalCrossentropy(reduction='sum'),\n weighted_metrics=['acc']\n)\n```\n\n### \ud559\uc2b5\n\n\n```python\nloader_tr = SingleLoader(dataset, sample_weights=weights_tr)\nloader_va = SingleLoader(dataset, sample_weights=weights_va)\n\nlogs = mod_1.fit(\n loader_tr.load(),\n steps_per_epoch=loader_tr.steps_per_epoch,\n validation_data=loader_va.load(),\n validation_steps=loader_va.steps_per_epoch,\n epochs=epochs,\n callbacks=[EarlyStopping(patience=patience, restore_best_weights=True)],\n verbose=0,\n)\n```\n\n\n```python\nlist([k for k in logs.history])\n```\n\n\n\n\n ['loss', 'acc', 'val_loss', 'val_acc']\n\n\n\n\n```python\nfig, axes = plt.subplots(2, 1, figsize=(6, 8))\naxes[0].plot(logs.history['loss'], label='tr loss')\naxes[0].plot(logs.history['val_loss'], label='va loss')\naxes[0].legend()\naxes[1].plot(logs.history['acc'], label='tr acc')\naxes[1].plot(logs.history['val_acc'], label='va acc')\naxes[1].legend();\n```\n\n### \ud3c9\uac00\n\n\n```python\nloader_te = SingleLoader(dataset, sample_weights=weights_te)\neval_results = mod_1.evaluate(loader_te.load(), steps=loader_te.steps_per_epoch)\neval_results\n```\n\n 1/1 [==============================] - 0s 11ms/step - loss: 1.0158 - acc: 0.8230\n\n\n\n\n\n [1.0157898664474487, 0.8230001330375671]\n\n\n\n## \ub9fa\uc73c\uba70\n\n* \uadf8\ub798\ud504 \ud559\uc2b5\uc758 \uae30\ubcf8 \uac1c\ub150\uc744 \uc774\ud574\ud558\uac8c \ub418\uc5c8\ub2e4.\n* \uadf8\ub798\ud504\ub97c \ub2e4\ub8f0 \ub54c \ubb3c\ub9ac\uc801\uc778 \uc758\ubbf8\ub97c \uac00\uc9c0\ub294 \ubbf8\ubd84 \ubc29\uc815\uc2dd\uc5d0\uc11c \ub098\uc624\ub294 \uc544\uc774\ub514\uc5b4\ub97c \ud655\uc7a5\ud558\uc5ec \uc801\uc6a9\ud558\uba74 \uc88b\uc740 \uacb0\uacfc\ub97c \uc5bb\uc744 \uc218\ub3c4 \uc788\ub2e4.\n* \uadf8\ub798\ud504 \ud569\uc131\uacf1 \uc2e0\uacbd\ub9dd\uc744 Resnet \uc2a4\ub7fd\uac8c \uad6c\uc131\ud574 \ubcfc \uc218 \uc788\uc744 \ub4ef \ud558\ub2e4.\n", "meta": {"hexsha": "cff3d58b8f9ed5c258ce27a15d8ccf7285651a22", "size": 95224, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "_notebooks/2021-02-05-graph-convolutiojnal-networks.ipynb", "max_stars_repo_name": "danhojin/jupyter-blog", "max_stars_repo_head_hexsha": "a765d0169a666fdbafaa84ff9efba9d9ca48c41c", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "_notebooks/2021-02-05-graph-convolutiojnal-networks.ipynb", "max_issues_repo_name": "danhojin/jupyter-blog", "max_issues_repo_head_hexsha": "a765d0169a666fdbafaa84ff9efba9d9ca48c41c", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 5, "max_issues_repo_issues_event_min_datetime": "2020-12-26T23:43:58.000Z", "max_issues_repo_issues_event_max_datetime": "2021-05-01T03:32:46.000Z", "max_forks_repo_path": "_notebooks/2021-02-05-graph-convolutiojnal-networks.ipynb", "max_forks_repo_name": "danhojin/jupyter-blog", "max_forks_repo_head_hexsha": "a765d0169a666fdbafaa84ff9efba9d9ca48c41c", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 160.3097643098, "max_line_length": 45008, "alphanum_fraction": 0.8898912039, "converted": true, "num_tokens": 4831, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.712232184238947, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.41672787305889625}} {"text": "# \u041a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0435 \u0447\u0438\u0441\u043b\u0430. \u0418\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u043d\u0438\u0435\n\n### #\u0417\u0430\u043d\u0438\u043c\u0430\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430\n\n#### \u0412\u0435\u0441\u044c \u043a\u043e\u0434 \u043d\u0430 Github, \u0441\u0441\u044b\u043b\u043a\u0430 \u0432 \u043a\u043e\u043d\u0446\u0435 \u0441\u0442\u0430\u0442\u044c\u0438!\n\n\u0418\u043c\u043f\u043e\u0440\u0442 \u0431\u0438\u0431\u043b\u0438\u043e\u0442\u0435\u043a\n\n\n```python\nfrom IPython.display import Image\nfrom IPython.core.display import HTML \nfrom IPython.core.interactiveshell import InteractiveShell\nfrom scipy.ndimage.filters import gaussian_filter1d\nfrom scipy.signal import savgol_filter\nimport numpy as np\nimport sympy as sp\nimport pandas as pd\nimport random as r\nimport time\nimport matplotlib.pyplot as plt\nimport ipyturtle as turtle\nInteractiveShell.ast_node_interactivity = \"all\"\n\ndef drawPlot(ss,title=\"\u0421\u043a\u043e\u0440\u043e\u0441\u0442\u0438\",y=\"\u0421\u0435\u043a\u0443\u043d\u0434\",x=\"\u041d\u043e\u043c\u0435\u0440 \u0438\u0442\u0435\u0440\u0430\u0446\u0438\u0438\"):\n fig,ax=plt.subplots(figsize=(6,6))\n ax.set_facecolor(\"#F2F2F2\")\n ax.grid()\n ax.set_title(title)\n ax.set_ylabel(y)\n ax.set_xlabel(x)\n ax.plot(ss)\n```\n\n\n```python\nImage(url=\"https://sun9-46.userapi.com/c858036/v858036072/1e3bba/s6JwXFoOgLM.jpg\", width=400)\n```\n\n\n\n\n\n\n\n\n\u041a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0435 \u0447\u0438\u0441\u043b\u0430? \n\u041f\u0440\u043e\u0448\u043b\u0430\u044f \u0441\u0442\u0430\u0442\u044c\u044f \u0431\u044b\u043b\u0430 \u043f\u0440\u043e \u0434\u0440\u0443\u0436\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0435 \u0447\u0438\u0441\u043b\u0430. \u041f\u043e\u043c\u043d\u0438\u043c \u0447\u0442\u043e \u044d\u0442\u043e? \n\n\u0415\u0441\u043b\u0438 \u0447\u0438\u0441\u043b\u043e \u0440\u0430\u0437\u043b\u043e\u0436\u0438\u0442\u044c \u043d\u0430 \u0435\u0433\u043e \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u0438, \u0430 \u043f\u043e\u0442\u043e\u043c \u043f\u0440\u043e\u0441\u0443\u043c\u043c\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u0438\u0445, \u0442\u043e \u0432\u044b\u0439\u0434\u0435\u0442 \u0442\u0430\u043a\u043e\u0435 \u0447\u0438\u0441\u043b\u043e, \u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u043f\u0440\u0438 \u0442\u043e\u0439 \u0436\u0435 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438 \u0432\u0435\u0440\u043d\u0451\u0442 \u043f\u0435\u0440\u0432\u043e\u0435 \u0447\u0438\u0441\u043b\u043e. \n\u041d\u0430\u043f\u0440\u0438\u043c\u0435\u0440: \n\u0412\u043e\u0437\u044c\u043c\u0451\u043c \u0447\u0438\u0441\u043b\u043e 8. \u0415\u0433\u043e \u0441\u043e\u0431\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0435 \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u0438 = 4,2,1 \n\u0421\u0443\u043c\u043c\u0430: 4+2+1 = 7 \n\u0414\u0435\u043b\u0438\u0442\u0435\u043b\u0438 \u0441\u0435\u043c\u0438: 1 \u0438 \u0432\u0441\u0451. \u041d\u043e 1 + 0, \u0434\u0430\u0441\u0442 1, \u0430 \u043d\u0435 8. \u041f\u043e\u044d\u0442\u043e\u043c\u0443 \u0447\u0438\u0441\u043b\u0430 8 \u0438 7 \u043d\u0435 \u044f\u0432\u043b\u044f\u044e\u0442\u0441\u044f \u0434\u0440\u0443\u0436\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u043c\u0438. \n\n\u041f\u0435\u0440\u0432\u044b\u0439 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0438\u0442\u0435\u043b\u044c \u0434\u0440\u0443\u0436\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0439 \u043f\u0430\u0440\u044b \u044d\u0442\u043e - (220,284) \n\u041f\u0440\u043e\u0432\u0435\u0440\u044c\u0442\u0435 \u0441\u0430\u043c\u0438. \n\n\u041e\u0434\u043d\u0430\u043a\u043e \u0434\u043e\u043b\u0436\u0435\u043d \u043e\u0442\u043c\u0435\u0442\u0438\u0442\u044c \u043e\u0447\u0435\u043d\u044c \u0438\u043d\u0442\u0435\u0440\u0435\u0441\u043d\u044b\u0439 \u043c\u043e\u043c\u0435\u043d\u0442 \u043f\u043e \u0442\u0435\u043c\u0435. \u041b\u044e\u0431\u043e\u0439 \u0434\u0440\u0443\u0436\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0435 \u0447\u0438\u0441\u043b\u043e \u0435\u0449\u0451 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0438 \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u043c. \u041f\u043e\u0447\u0435\u043c\u0443? \n\u041a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0435 \u0447\u0438\u0441\u043b\u0430 - \u044d\u0442\u043e \u0442\u0430\u043a\u0438\u0435 \u0447\u0438\u0441\u043b\u0430, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043c\u043e\u0433\u0443\u0442 \u043f\u0440\u043e\u0445\u043e\u0434\u0438\u0442\u044c \u043d\u0435\u0441\u043a\u043e\u043b\u044c\u043a\u043e \u0441\u0442\u0430\u0434\u0438\u0439 \u0442\u0430\u043a\u0438\u0445 \u0432\u043e\u0442 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0439, \u043a\u0430\u043a \u043e\u043f\u0438\u0441\u0430\u043d\u043e \u0432\u044b\u0448\u0435 \u0438\u0437 \u043f\u0440\u0438\u043c\u0435\u0440\u0430. \n\u041a\u0430\u0436\u0434\u043e\u0435 \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u043e\u0435 \u0447\u0438\u0441\u043b\u043e \u043e\u0431\u043b\u0430\u0434\u0430\u0435\u0442 \u0441\u0432\u043e\u0438\u043c \u043f\u043e\u0440\u044f\u0434\u043a\u043e\u043c, \u0442\u0430\u043a \u0441\u043a\u0430\u0437\u0430\u0442\u044c. \n\u0423 \u0434\u0440\u0443\u0436\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b \u044d\u0442\u043e \u043f\u043e\u0440\u044f\u0434\u043e\u043a \u0440\u0430\u0432\u0435\u043d 2. \u041a\u0430\u043a \u043f\u043e\u043d\u044f\u0442\u044c? \n\n\u0421\u043e\u0437\u0434\u0430\u043c \u0444\u0443\u043d\u043a\u0446\u0438\u044e \u0434\u043b\u044f \u043d\u0430\u0445\u043e\u0436\u0434\u0435\u043d\u0438\u044f \u0441\u043e\u0431\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0445 \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u0435\u0439.\n\n\n```python\ndef Divisors(num): \n from math import sqrt as mmsq\n s=set([1])\n i=1\n a=int(mmsq(num)+1)\n while i<=a: \n if(num//i==num):\n i+=1\n continue\n if (num%i==0): \n if (num//i!=i): \n s.add(num//i)\n s.add(i)\n i+=1\n return s\n```\n\n\n```python\nDivisors(220)\n```\n\n\n\n\n {1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110}\n\n\n\n\u041f\u0440\u043e\u0441\u0443\u043c\u043c\u0438\u0440\u0443\u0435\u043c\n\n\n```python\nsum(Divisors(220))\n```\n\n\n\n\n 284\n\n\n\n\u041e\u043f\u0435\u0440\u0430\u0446\u0438\u044f \u043f\u0440\u043e\u0434\u0435\u043b\u0430\u043d\u0430 \u0432\u0441\u0435\u0433\u043e-\u0442\u043e 1 \u0440\u0430\u0437. \n\u0438\u0437 220 \u043c\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u043b\u0438 \u0447\u0438\u0441\u043b\u043e 284. \n\n\u041f\u043e\u0432\u0442\u043e\u0440\u0438\u043c \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u0435\u0449\u0451 \u0440\u0430\u0437.\n\n\n```python\nDivisors(284)\n```\n\n\n\n\n {1, 2, 4, 71, 142}\n\n\n\n\n```python\nsum(Divisors(284))\n```\n\n\n\n\n 220\n\n\n\n\u0418 \u0432\u043e\u0442 \u043c\u044b \u0432\u0435\u0440\u043d\u0443\u043b\u0438\u0441\u044c \u0441\u043d\u043e\u0432\u0430 \u043a \u0447\u0438\u0441\u043b\u0443 220, \u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u0438 \u0431\u044b\u043b\u043e \u0438\u0437\u043d\u0430\u0447\u0430\u043b\u044c\u043d\u043e. \n\u0414\u043b\u044f \u044d\u0442\u043e\u0433\u043e \u043c\u044b \u043f\u0440\u043e\u0432\u0435\u043b\u0438 \u043d\u0430\u0448\u0443 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044e \u0434\u0432\u0430 \u0440\u0430\u0437\u0430! \u041f\u043e\u044d\u0442\u043e\u043c\u0443 \u043f\u043e\u0440\u044f\u0434\u043e\u043a - \u0434\u0432\u0430. \n\u0412 \u0432\u0438\u043a\u0438\u043f\u0435\u0434\u0438\u0438 \u044d\u0442\u043e \u043d\u0430\u0437\u044b\u0432\u0430\u044e\u0442 \u0434\u043b\u0438\u043d\u043e\u0439 \u043f\u043e\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u0438.\n\n\u0421\u043a\u043e\u043b\u044c\u043a\u043e \u0447\u0438\u0441\u0435\u043b \u0432 \u0434\u0430\u043d\u043d\u043e\u0439 \u0446\u0435\u043f\u0438 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0439 \u043a\u0440\u043e\u043c\u0435 \u0438\u0437\u043d\u0430\u0447\u0430\u043b\u044c\u043d\u043e\u0433\u043e\n\n\u0415\u0441\u0442\u044c \u0438 \u0434\u0440\u0443\u0433\u0438\u0435 \u0434\u043b\u0438\u043d\u044b. \u041d\u0430\u043f\u0440\u0438\u043c\u0435\u0440 \u0435\u0441\u043b\u0438 \u0434\u043b\u0438\u043d\u0430 \u0431\u0443\u0434\u0435\u0442 \u0440\u0430\u0432\u043d\u0430 1, \u0442\u043e \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u0442\u0441\u044f, \u0447\u0442\u043e \u0443 \u043d\u0430\u0441 \u0441\u0443\u043c\u043c\u0430 \u0441\u043e\u0431\u0441.\u0434\u0435\u043b\u0438\u0442\u0435\u043b\u0435\u0439 \u0447\u0438\u0441\u043b\u0430 \u0432\u0435\u0440\u043d\u0451\u0442 \u0441\u0440\u0430\u0437\u0443 \u0435\u0433\u043e \u0436\u0435. \n\u0422\u0430\u043a\u0438\u0435 \u0447\u0438\u0441\u043b\u0430 \u043d\u0430\u0437\u044b\u0432\u0430\u044e\u0442\u0441\u044f **\u0441\u043e\u0432\u0435\u0440\u0448\u0435\u043d\u043d\u044b\u0435**, \u043d\u043e \u043e \u043d\u0438\u0445 \u0431\u0443\u0434\u0435\u0442 \u0441\u0442\u0430\u0442\u044c\u044f \u043f\u043e\u0437\u0436\u0435 \ud83d\ude01\n\n\u0412 \u0446\u0435\u043b\u043e\u043c \u0434\u043b\u0438\u043d\u044b \u043f\u043e\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u0435\u0439 \u043c\u043e\u0433\u0443\u0442 \u0431\u044b\u0442\u044c \u0441\u043e\u0432\u0441\u0435\u043c \u0440\u0430\u0437\u043d\u044b\u0435. \n\u041e\u0442 1 \u0434\u043e \u0431\u0435\u0441\u043a\u043e\u043d\u0435\u0447\u043d\u043e\u0441\u0442\u0438.\n\n\u041d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u0432\u043e\u0437\u044c\u043c\u0451\u043c \u0447\u0438\u0441\u043b\u043e 1264460\n\n\n```python\nsum(Divisors(1264460))\n```\n\n\n\n\n 1547860\n\n\n\n\n```python\nsum(Divisors(1547860))\n```\n\n\n\n\n 1727636\n\n\n\n\n```python\nsum(Divisors(1727636))\n```\n\n\n\n\n 1305184\n\n\n\n\n```python\nsum(Divisors(1305184))\n```\n\n\n\n\n 1264460\n\n\n\n\u0418 \u0432\u043e\u0442 \u043c\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u043b\u0438 \u0432 \u043a\u043e\u043d\u0446\u0435 \u0442\u043e\u0436\u0435 \u0441\u0430\u043c\u043e\u0435 \u0447\u0438\u0441\u043b\u043e, \u043f\u0440\u043e\u0434\u0435\u043b\u0430\u0432 \u043d\u0430\u0448 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c 4 \u0440\u0430\u0437\u0430! \u041f\u043e\u044d\u0442\u043e\u043c\u0443 \u0434\u043b\u0438\u043d\u0430 \u043f\u043e\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u0438(\u043f\u043e\u0440\u044f\u0434\u043e\u043a, \u043b\u0438\u0431\u043e \u043f\u0435\u0440\u0438\u043e\u0434) \u0440\u0430\u0432\u0435\u043d 4\n\n\u0412 \u0446\u0435\u043b\u043e\u043c, \u0443\u0437\u043d\u0430\u0442\u044c \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043b\u0438 \u0447\u0438\u0441\u043b\u043e \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u043c \u043d\u0435 \u0442\u0430\u043a \u0443\u0436 \u0438 \u0431\u044b\u0441\u0442\u0440\u043e. \u0415\u0441\u043b\u0438 \u043a\u043e\u043d\u0435\u0447\u043d\u043e \u043c\u044b \u0441\u043e \u0432\u0440\u0435\u043c\u0435\u043d\u0435\u043c \u0432\u043e \u0432\u0441\u0435\u0439 \u043f\u043e\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u0438 \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u0435\u0434\u0438\u043d\u0438\u0446\u0443, \u0442\u043e \u0447\u0438\u0441\u043b\u043e \u0422\u041e\u0427\u041d\u041e \u043d\u0435 \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u043e\u0435. \u0410 \u0432\u0434\u0440\u0443\u0433 \u0432\u0430\u043c \u043f\u043e\u043f\u0430\u0434\u0451\u0442\u0441\u044f \u0447\u0438\u0441\u043b\u043e \u0441 \u0434\u043b\u0438\u043d\u043e\u0439 \u043f\u043e\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u0438, \u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440 999. \u042d\u0442\u043e \u0437\u0430\u043c\u0443\u0447\u0430\u0442\u044c\u0441\u044f \u043d\u0430\u0434\u043e, \u0447\u0442\u043e\u0431\u044b \u0434\u043e\u0439\u0442\u0438 \u0434\u043e \u0442\u0430\u043a\u043e\u0433\u043e. \n\n\u041a\u043e\u0440\u043e\u0447\u0435 \u0433\u043e\u0432\u043e\u0440\u044f, \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0435 \u0447\u0438\u0441\u043b\u0430 \u043e\u0431\u0440\u0430\u0437\u0443\u044e\u0442 \u0431\u0435\u0441\u043a\u043e\u043d\u0435\u0447\u043d\u044b\u0439 \u0446\u0438\u043a\u043b \u043f\u0440\u0438 \u043f\u0440\u043e\u0434\u0435\u043b\u044b\u0432\u0430\u043d\u0438\u0438 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438 \u0441 \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u044f\u043c\u0438\n\n\u041a\u043b\u0430\u0441\u0441\u0438\u0444\u0438\u043a\u0430\u0446\u0438\u044f \u0432\u0441\u0435\u0445 \u0438\u0437\u0432\u0435\u0441\u0442\u043d\u044b\u0445 \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0445 \u0447\u0438\u0441\u0435\u043b 2015 \u0433\u043e\u0434 \u043f\u043e \u0434\u043b\u0438\u043d\u0435:\n\n\n```python\nstat=np.array([(1,49),(2,1007478796),(4,1581),(5,1),(6,5),(8,4),(9,1),(28,1)])\npd.DataFrame(stat).rename(columns={0:\"\u0414\u043b\u0438\u043d\u0430 \u043f\u043e\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u0438\",1:\"\u041a\u043e\u043b\u0438\u0447\u0435\u0441\u0442\u0432\u043e \u043a\u043e\u043c\u043f.\u0447\u0438\u0441\u0435\u043b\"})\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        \u0414\u043b\u0438\u043d\u0430 \u043f\u043e\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u0438\u041a\u043e\u043b\u0438\u0447\u0435\u0441\u0442\u0432\u043e \u043a\u043e\u043c\u043f.\u0447\u0438\u0441\u0435\u043b
                                        0149
                                        121007478796
                                        241581
                                        351
                                        465
                                        584
                                        691
                                        7281
                                        \n
                                        \n\n\n\n\u0410\u043b\u0433\u043e\u0440\u0442\u043c\u044b \u043d\u0430\u0445\u043e\u0436\u0434\u0435\u043d\u0438\u044f \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0445 \u0447\u0438\u0441\u0435\u043b.\n\n\u041f\u0435\u0440\u0432\u044b\u0439. \u041c\u043e\u0439. \n\n\n```python\ndef SociableNumber(num,minIt=1,maxIt=20,sequence=False):\n num1=num\n iterat=0\n if(sequence):\n seq=[]\n while iterat\",i) \n```\n\n\n```python\nss = np.array([])\nfor i in range(200,500):\n start_time = time.time()\n SociableNumber(i,2)\n end_time = (time.time() - start_time)\n ss=np.append(ss,end_time)\n```\n\n\n```python\ns = np.array([])\nfor i in range(200,500):\n start_time = time.time()\n AmicableNumber(i)\n end_time = (time.time() - start_time)\n s=np.append(s,end_time)\n```\n\n 220 -> 284\n 284 -> 220\n\n\n\n```python\nfig,ax=plt.subplots(1,2,figsize=(12,6))\nax[0].set_facecolor(\"#F2F2F2\")\nax[0].grid()\nax[0].set_title(\"\u041a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0439 \u043f\u043e\u0434\u0445\u043e\u0434\")\nax[0].set_ylabel(\"\u0421\u043a\u043e\u0440\u043e\u0441\u0442\u044c, \u0441\u0435\u043a\")\nax[0].set_xlabel(\"\u0418\u0442\u0435\u0440\u0430\u0446\u0438\u044f\")\nax[0].plot(s)\nax[1].set_facecolor(\"#F2F2F2\")\nax[1].grid()\nax[1].set_title(\"\u0414\u0440\u0443\u0436\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0439 \u043f\u043e\u0434\u0445\u043e\u0434\")\nax[1].set_ylabel(\"\u0421\u043a\u043e\u0440\u043e\u0441\u0442\u044c, \u0441\u0435\u043a\")\nax[1].set_xlabel(\"\u0418\u0442\u0435\u0440\u0430\u0446\u0438\u044f\")\nax[1].plot(ss)\n```\n\n\u041e\u0436\u0438\u0434\u0430\u0435\u043c\u044b\u0439 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442. \u0415\u0441\u043b\u0438 \u043c\u044b \u0438\u0449\u0435\u043c \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0435 \u0447\u0438\u0441\u043b\u0430, \u0442\u043e \u0447\u0435\u043c \u0431\u043e\u043b\u044c\u0448\u0435 \u0447\u0438\u0441\u043b\u043e, \u0442\u0435\u043c \u043b\u043e\u0433\u0438\u0447\u043d\u043e \u043d\u0443\u0436\u043d\u043e \u0431\u043e\u043b\u044c\u0448\u0435 \u0432\u0440\u0435\u043c\u0435\u043d\u0438. \u041d\u043e \u0435\u0441\u043b\u0438 \u043c\u044b \u0438\u0449\u0435\u043c \u0434\u0440\u0443\u0436\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0435, \u0442\u043e \u043b\u0443\u0447\u0448\u0435 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c \u043f\u043e\u0434\u0445\u043e\u0434 \u0438\u0437\u043d\u0430\u0447\u0430\u043b\u044c\u043d\u043e \u0441\u0434\u0435\u043b\u0430\u043d\u043d\u044b\u0439 \u0434\u043b\u044f \u0434\u0440\u0443\u0436\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b, \u0430 \u043d\u0435 \u0430\u0431\u0441\u0442\u0440\u0430\u043a\u0442\u043d\u043e \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0439\u0445 \u0441 2 \u043f\u0435\u0440\u0438\u043e\u0434\u043e\u043c.\n\n\u0422\u0435\u043f\u0435\u0440\u044c \u043f\u043e\u043f\u0440\u043e\u0431\u0443\u0435\u043c \u0441\u0434\u0435\u043b\u0430\u0442\u044c \u0432\u043e\u0442 \u0447\u0442\u043e. \u041f\u043e\u043f\u0440\u043e\u0431\u0443\u0435\u043c \u0432\u0437\u044f\u0442\u044c \u043f\u0435\u0440\u0432\u044b\u0435 26 000 \u0447\u0438\u0441\u0435\u043b. \u0418\u0437 \u043d\u0438\u0445 \u043e\u0441\u0442\u0430\u0432\u0438\u0442\u044c \u0442\u043e\u043b\u044c\u043a\u043e \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0435 \u043f\u043e\u0440\u044f\u0434\u043a\u0430 \u043e\u0442 1 \u0434\u043e 5. \n\u041c\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u043c \u0441\u0432\u043e\u0435\u0433\u043e \u0440\u043e\u0434\u0430 \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0443\u044e \u043f\u043e\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u044c)\n\n\n```python\nkomp=[]\nper=[]\nfor i in range(26000):\n for j in range(1,6):\n if(SociableNumber(i,j)):\n komp.append(i)\n per.append(j)\n break\n```\n\n\n```python\nlen(komp)\n```\n\n\n\n\n 25\n\n\n\n\u0422\u043e\u043b\u044c\u043a\u043e 25 \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0445 \u0447\u0438\u0441\u043b\u0430 \u0438\u0437 26 000\n\n\n```python\nfig,ax=plt.subplots(1,2,figsize=(12,6))\nax[0].set_facecolor(\"#F2F2F2\")\nax[0].grid()\nax[0].set_title(\"\u041a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0435 \u0447\u0438\u0441\u043b\u0430\")\nax[0].set_ylabel(\"\u0427\u0438\u0441\u043b\u043e\")\nax[0].set_xlabel(\"\u041d\u043e\u043c\u0435\u0440 \u0447\u0438\u0441\u043b\u0430\")\nax[0].plot(komp)\nax[1].set_facecolor(\"#F2F2F2\")\nax[1].grid()\nax[1].set_title(\"\u041f\u0435\u0440\u0438\u043e\u0434\u044b\")\nax[1].set_ylabel(\"\u041f\u0435\u0440\u0438\u043e\u0434\")\nax[1].set_xlabel(\"\u041d\u043e\u043c\u0435\u0440 \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u043e\u0433\u043e \u0447\u0438\u0441\u043b\u0430\")\nax[1].plot(per)\n```\n\n\u0415\u0441\u043b\u0438 \u043f\u0440\u0435\u043d\u0435\u0431\u0440\u0435\u0447\u044c \u0432\u0441\u0435\u043c\u0438 26 000 \u0447\u0438\u0441\u0435\u043b \u0438 \u043e\u0441\u0442\u0430\u0432\u0438\u0442\u044c \u0442\u043e\u043b\u044c\u043a\u043e \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0435 \u0442\u043e \u0442\u0430\u043a\u0438\u0435 \u0433\u0440\u0430\u0444\u0438\u043a\u0438 \u043f\u043e\u043b\u0443\u0447\u0430\u044e\u0442\u0441\u044f \u0432\u044b\u0448\u0435. \n\u041f\u0435\u0440\u0432\u043e\u0435 \u043f\u043e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442 \u043a\u0430\u043a \u0438\u043c\u0435\u043d\u043d\u043e \u0440\u0430\u0441\u0442\u0443\u0442 \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0435 \u0447\u0438\u0441\u043b\u0430. \u0410 \u043d\u0430 \u0432\u0442\u043e\u0440\u043e\u043c \u0433\u0440\u0430\u0444\u0438\u043a\u0435 \u0443 \u043d\u0430\u0441 \u043e\u0442\u043e\u0431\u0440\u0430\u0436\u0430\u044e\u0442\u0441\u044f \u043f\u0435\u0440\u0438\u043e\u0434 \u0441\u043e\u043e\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0435\u043d\u043d\u043e \u0447\u0438\u0441\u043b\u0430\u043c \u043f\u0435\u0440\u0432\u043e\u0433\u043e \u0433\u0440\u0430\u0444\u0438\u043a\u0430. \n\n\u041f\u043e\u043f\u0440\u043e\u0431\u0443\u0435\u043c \u0443\u0437\u043d\u0430\u0442\u044c \u0447\u0430\u0441\u0442\u043e\u0442\u0443 \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0445 \u0447\u0438\u0441\u0435\u043b, \u0430 \u0442\u0430\u043a\u0436\u0435 \u0441\u0434\u0435\u043b\u0430\u0442\u044c \u043a\u043e\u043b\u0438\u0447\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u0443\u044e \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0443 \u043f\u0435\u0440\u0438\u043e\u0434\u043e\u0432 \u0438 \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0445 \u0447\u0438\u0441\u0435\u043b \u0441\u0440\u0435\u0434\u0438 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430 \u043e\u0431\u044b\u0447\u043d\u044b\u0445 26 000 \u0447\u0438\u0441\u0435\u043b. \n\u0414\u043b\u044f \u044d\u0442\u043e\u0433\u043e \u044f \u043c\u043e\u0433\u0443 \u043f\u0440\u043e\u0441\u0442\u043e \u0432\u0437\u044f\u0442\u044c \u0432\u0441\u0435 \u044d\u0442\u0438 26 000 \u0447\u0438\u0441\u0435\u043b. \u0418 \u043e\u0442\u043c\u0435\u0442\u0438\u0442\u044c \u0442\u043e\u0447\u043a\u0430\u043c\u0438 \u0442\u043e\u043b\u044c\u043a\u043e \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0435\n\n\n```python\nkomp1=[]\nkomp2=[]\nkomp3=[]\nkomp4=[]\nkomp5=[]\nkompa=[]\nfor i in range(26000):\n if(i in komp):\n kompa.append(1)\n \n if(per[komp.index(i)]==1):\n komp1.append(0.6)\n elif(per[komp.index(i)]==2):\n komp2.append(0.8)\n elif(per[komp.index(i)]==3):\n komp3.append(1)\n elif(per[komp.index(i)]==4):\n komp4.append(1.2)\n elif(per[komp.index(i)]==5):\n komp5.append(1.4)\n \n if(per[komp.index(i)]!=1):\n komp1.append(0)\n if(per[komp.index(i)]!=2):\n komp2.append(0)\n if(per[komp.index(i)]!=3):\n komp3.append(0)\n if(per[komp.index(i)]!=4):\n komp4.append(0)\n if(per[komp.index(i)]!=5):\n komp5.append(0)\n else:\n komp1.append(0)\n komp2.append(0)\n komp3.append(0)\n komp4.append(0)\n komp5.append(0)\n kompa.append(0)\n```\n\n\n```python\nfig,ax=plt.subplots(1,2,figsize=(12,6))\nax[0].set_ylim([0.5, 1.5])\nax[0].set_xlim([0, 20000])\nax[0].set_facecolor(\"#F2F2F2\")\nax[0].grid()\nax[0].set_title(\"\u0427\u0430\u0441\u0442\u043e\u0442\u0430 \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0439 \u0447\u0438\u0441\u0435\u043b\")\nax[0].set_xlabel(\"\u0427\u0438\u0441\u043b\u043e\")\nax[0].set_ylabel(\"\u041a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u043e\u0435 true,false\")\nax[0].scatter(range(26000),kompa)\nax[1].set_ylim([0.5, 1.5])\nax[1].set_xlim([0, 20000])\nax[1].set_facecolor(\"#F2F2F2\")\nax[1].grid()\nax[1].set_title(\"\u0427\u0430\u0441\u0442\u043e\u0442\u0430 \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0439 \u0447\u0438\u0441\u0435\u043b \u0441 \u043a\u043b\u0430\u0441\u0441\u0438\u0444\u0438\u043a\u0430\u0446\u0438\u0435\u0439\")\nax[1].set_xlabel(\"\u0427\u0438\u0441\u043b\u043e\")\nax[1].set_ylabel(\"\u041a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u043e\u0435 true,false\")\nax[1].scatter(range(26000),komp1)\nax[1].scatter(range(26000),komp2)\nax[1].scatter(range(26000),komp3)\nax[1].scatter(range(26000),komp4)\nax[1].scatter(range(26000),komp5)\nax[1].legend([\"\u041f\u0435\u0440\u0438\u043e\u0434 1\",\"\u041f\u0435\u0440\u0438\u043e\u0434 2\",\"\u041f\u0435\u0440\u0438\u043e\u0434 3\",\"\u041f\u0435\u0440\u0438\u043e\u0434 4\",\"\u041f\u0435\u0440\u0438\u043e\u0434 5\"])\n```\n\n\u041d\u0430 \u043f\u0435\u0440\u0432\u043e\u043c \u0433\u0440\u0430\u0444\u0438\u043a\u0435 \u043c\u044b \u0432\u0438\u0434\u0438\u043c \u0442\u043e\u0447\u043a\u0438 - \u044d\u0442\u043e \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0435 \u0447\u0438\u0441\u043b\u0430. \u041d\u0430\u0439\u0442\u0438 \u0437\u0430\u043a\u043e\u043d\u043e\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u044c \u0432 \u0447\u0430\u0441\u0442\u043e\u0442\u0435 \u043d\u0435 \u0431\u0443\u0434\u0435\u0442 \u043b\u0435\u0433\u043a\u043e\u0439 \u0437\u0430\u0434\u0430\u0447\u0435\u0439, \u043d\u043e \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0438\u0445 \u043f\u0440\u043e\u043a\u043b\u0430\u0441\u0441\u0438\u0444\u0438\u0446\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u0438 \u0443\u0437\u043d\u0430\u0442\u044c \u043a \u043a\u0430\u043a\u0438\u043c \u043f\u0435\u0440\u0438\u043e\u0434\u0430\u043c \u043e\u043d\u0438 \u043e\u0442\u043d\u043e\u0441\u044f\u0442\u0441\u044f. \n\u0427\u0442\u043e \u0438 \u0431\u044b\u043b\u043e \u0441\u0434\u0435\u043b\u0430\u043d\u043e \u043d\u0430 \u0432\u0442\u043e\u0440\u043e\u043c \u0433\u0440\u0430\u0444\u0438\u043a\u0435. \u0418 \u043c\u044b \u0432\u0438\u0434\u0438\u043c, \u0447\u0442\u043e \u0442\u043e\u0447\u0435\u043a \u0437\u0435\u043b\u0451\u043d\u043e\u0433\u043e \u0438 \u043a\u0440\u0430\u0441\u043d\u043e\u0433\u043e \u0446\u0432\u0435\u0442\u0430 \u043d\u0435\u0442, \u0430 \u0437\u043d\u0430\u0447\u0438\u0442 \u0432 \u043f\u0435\u0440\u0432\u044b\u0445 26 000 \u0447\u0438\u0441\u043b\u0430\u0445 \u043d\u0435\u0442 \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0445 \u0447\u0438\u0441\u0435\u043b 3 \u0438 4 \u043f\u0435\u0440\u0438\u043e\u0434\u0430. \n\u0414\u043e\u0432\u043e\u043b\u044c\u043d\u043e \u043c\u043d\u043e\u0433\u043e \u0434\u0440\u0443\u0436\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b.\n\n\n```python\nimport matplotlib.ticker as ticker\nfig,ax=plt.subplots(1,2,figsize=(12,6))\nfor axis in [ax[0].xaxis, ax[0].yaxis]:\n axis.set_major_locator(ticker.MaxNLocator(integer=True))\nax[0].set_facecolor(\"#F2F2F2\")\nax[0].set_title(\"\u041f\u0435\u0440\u0438\u043e\u0434\u044b \u0432 \u043f\u0435\u0440\u0432\u044b\u0445 26 000 \u0447\u0438\u0441\u043b\u0430\u0445\")\nax[0].set_xlabel(\"\u041f\u0435\u0440\u0438\u043e\u0434\")\nax[0].set_ylabel(\"\u041a\u043e\u043b\u0438\u0447\u0435\u0441\u0442\u0432\u043e\")\nax[0].hist(per,bins=5)\nplt.pie([per.count(1),per.count(2),per.count(3),per.count(4),per.count(5)], \n colors=[\"red\",\"yellow\",\"blue\",\"#E7AFFF\",\"#A7FF5B\"],\n labels=[\"1 \u043f\u0435\u0440\u0438\u043e\u0434\",\"2 \u043f\u0435\u0440\u0438\u043e\u0434\",\"3 \u043f\u0435\u0440\u0438\u043e\u0434\",\"4 \u043f\u0435\u0440\u0438\u043e\u0434\",\"5 \u043f\u0435\u0440\u0438\u043e\u0434\"], \n autopct='%1.1f%%',\n shadow=True,\n textprops={'color':\"black\"})\nplt.legend([per.count(1),per.count(2),per.count(3),per.count(4),per.count(5)])\n;\n```\n\n\u041d\u0443, \u0430 \u044d\u0442\u043e \u043a\u043b\u0430\u0441\u0441\u0438\u0444\u0438\u043a\u0430\u0446\u0438\u044f \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0445 \u0447\u0438\u0441\u0435\u043b \u043f\u043e \u043f\u0435\u0440\u0438\u043e\u0434\u0443 \u0431\u043e\u043b\u0435\u0435 \u043d\u0430\u0433\u043b\u044f\u0434\u043d\u043e.\n\n\u0410 \u044f \u043a\u0441\u0442\u0430\u0442\u0438 \u0442\u0430\u043a \u0438 \u043d\u0435 \u043f\u043e\u043a\u0430\u0437\u0430\u043b \u0441\u0430\u043c \u0440\u044f\u0434 \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0445 \u0447\u0438\u0441\u0435\u043b. \u041f\u0435\u0440\u0432\u044b\u0435 25 \u0447\u0438\u0441\u0435\u043b.\n\n\n```python\nfor i in komp:\n print(i,end=\", \")\n```\n\n 6, 28, 220, 284, 496, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 8128, 10744, 10856, 12285, 12496, 14264, 14288, 14536, 14595, 15472, 17296, 18416, \n\n### \u0410\u043d\u0430\u043b\u0438\u0437\n\n\u041f\u043e\u043f\u0440\u043e\u0431\u0443\u0435\u043c \u043f\u0440\u043e\u0430\u043d\u0430\u043b\u0438\u0437\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u0440\u044f\u0434. \u041d\u0430\u0439\u0434\u0451\u043c \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u044c \u0438\u0437\u043c\u0435\u043d\u0435\u043d\u0438\u044f \u0440\u044f\u0434\u0430, \u0443\u0441\u043a\u043e\u0440\u0435\u043d\u0438\u0435. \u0417\u0430\u0442\u0435\u043c \u043f\u0440\u043e\u0434\u0435\u043b\u0430\u0435\u043c \u0442\u043e\u0436\u0435 \u0441\u0430\u043c\u043e\u0435 \u0441 \u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u044f\u043c\u0438 \u043c\u0435\u0436\u0434\u0443 \u0447\u0438\u0441\u043b\u0430\u043c\u0438 \u0438 \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c\u044e.\n\n\n```python\ndkomp=[komp[i]-komp[i-1] for i in range(1,len(komp))]\nddkomp=[dkomp[i]-dkomp[i-1] for i in range(1,len(dkomp))]\n```\n\n\n```python\nfig,ax=plt.subplots(1,2,figsize=(12,6))\nax[0].set_facecolor(\"#F2F2F2\")\nax[0].grid()\nax[0].set_title(\"\u0421\u043a\u043e\u0440\u043e\u0441\u0442\u044c \u0440\u044f\u0434\u0430\")\nax[0].plot(dkomp)\nax[1].set_facecolor(\"#F2F2F2\")\nax[1].grid()\nax[1].set_title(\"\u0423\u0441\u043a\u043e\u0440\u0435\u043d\u0438\u0435 \u0440\u044f\u0434\u0430\")\nax[1].plot(ddkomp)\n```\n\n\u0414\u0430, \u043c\u043e\u0436\u043d\u043e \u0441\u0434\u0435\u043b\u0430\u0442\u044c \u0432\u044b\u0432\u043e\u0434, \u0447\u0442\u043e \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u044c \u0432\u0441\u0442\u0440\u0435\u0447\u0438 \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u043e\u0433\u043e \u0447\u0438\u0441\u043b\u0430 \u0440\u0430\u0441\u0442\u0451\u0442 \u0441 \u0435\u0433\u043e \u0440\u043e\u0441\u0442\u043e\u043c, \u043d\u043e \u043f\u0440\u043e \u0443\u0441\u043a\u043e\u0440\u0435\u043d\u0438\u0435 \u0447\u0442\u043e-\u043b\u0438\u0431\u043e \u0441\u043a\u0430\u0437\u0430\u0442\u044c \u0441\u043b\u043e\u0436\u043d\u043e. \u0412\u0438\u0434\u043d\u043e, \u0447\u0442\u043e \u0435\u0441\u0442\u044c \u043d\u0435\u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u043e\u0441\u0442\u044c. \u041a \u0442\u043e\u043c\u0443 \u0436\u0435, \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0435 \u0447\u0438\u0441\u043b\u0430 \u0432\u043a\u043b\u044e\u0447\u0430\u044e\u0442 \u0432 \u0441\u0435\u0431\u044f \u0440\u0430\u0437\u043d\u044b\u0435 \u043f\u0435\u0440\u0438\u043e\u0434\u044b, \u043d\u043e \u043f\u0435\u0440\u0432\u044b\u0435 25 \u0447\u0438\u0441\u0435\u043b \u0442\u043e\u043b\u044c\u043a\u043e 1,2 \u0438 5 \u043f\u0435\u0440\u0438\u043e\u0434\u044b.\n\n#### \u041e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u044f\n\n\n```python\nrkomp=[komp[i]/komp[i-1] for i in range(1,len(komp))]\n```\n\n\n```python\nnp.mean(rkomp) # \u0421\u0440\u0435\u0434\u043d\u0435\u0435 \u0441\u043e\u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435\n```\n\n\n\n\n 1.6826428895408647\n\n\n\n\n```python\ndrawPlot(rkomp,title=\"\u041e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u044f \u043c\u0435\u0436\u0434\u0443 \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u043c\u0438 \u0447\u0438\u0441\u043b\u0430\u043c\u0438\",y=\"\u041e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435\",x=\"\u0427\u0438\u0441\u043b\u043e\")\n```\n\n\u0412\u0430\u0443! \u042f \u044d\u0442\u043e\u0433\u043e \u043d\u0435 \u043e\u0436\u0438\u0434\u0430\u043b! \u041f\u043e\u0445\u043e\u0436\u0435, \u0447\u0442\u043e \u0440\u044f\u0434 \u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0439 \u0441\u0445\u043e\u0434\u0438\u0442\u0441\u044f \u043a \u043a\u0430\u043a\u043e\u043c\u0443-\u0442\u043e \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044e. \u041c\u043e\u0433\u0443 \u0441\u043f\u0440\u043e\u0433\u043d\u043e\u0437\u0438\u0440\u043e\u0432\u0430\u0442\u044c, \u0447\u0442\u043e \u044d\u0442\u043e \u043f\u0440\u043e\u0438\u0441\u0445\u043e\u0434\u0438\u0442 \u0443\u0436\u0435 \u043d\u0430\u0447\u0438\u043d\u0430\u044f \u0441 30-\u043e\u0433\u043e \u0447\u0438\u0441\u043b\u0430, (\u043d\u043e \u044d\u0442\u043e \u043d\u0435 \u0442\u043e\u0447\u043d\u043e). \n\u0414\u043b\u044f \u044d\u0442\u043e\u0433\u043e \u0431\u044b\u043b\u043e \u0431\u044b \u043f\u043e\u043b\u0435\u0437\u043d\u043e \u0443\u0437\u043d\u0430\u0442\u044c \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u044c \u0438 \u0443\u0441\u043a\u043e\u0440\u0435\u043d\u0438\u0435 \u0434\u0430\u043d\u043d\u043e\u0433\u043e \u0440\u044f\u0434\u0430.\n\n\n```python\ndrkomp=[rkomp[i]-rkomp[i-1] for i in range(1,len(rkomp))]\nddrkomp=[drkomp[i]-drkomp[i-1] for i in range(1,len(drkomp))]\n```\n\n\n```python\nfig,ax=plt.subplots(1,2,figsize=(12,6))\nax[0].set_facecolor(\"#F2F2F2\")\nax[0].grid()\nax[0].set_title(\"\u0421\u043a\u043e\u0440\u043e\u0441\u0442\u044c \u0440\u044f\u0434\u0430\")\nax[0].plot(drkomp)\nax[1].set_facecolor(\"#F2F2F2\")\nax[1].grid()\nax[1].set_title(\"\u0423\u0441\u043a\u043e\u0440\u0435\u043d\u0438\u0435 \u0440\u044f\u0434\u0430\")\nax[1].plot(ddrkomp)\n```\n\n\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\ud83d\ude0d\n\n\u0417\u0430\u043c\u0435\u0447\u0430\u0442\u0435\u043b\u044c\u043d\u043e!!! \n\n\u0421\u043a\u043e\u0440\u043e\u0441\u0442\u044c \u0438 \u0443\u0441\u043a\u043e\u0440\u0435\u043d\u0438\u044f \u0441\u0445\u043e\u0436\u0438, \u0430 \u0442\u0430\u043a \u0436\u0435 \u043e\u0431\u0430 \u0433\u0440\u0430\u0444\u0438\u043a\u0430 \u0441\u0445\u043e\u0434\u044f\u0442\u0441\u044f. \u042d\u0442\u0438\u043c \u043c\u043e\u0436\u043d\u043e \u0441\u0434\u0435\u043b\u0430\u0442\u044c \u0432\u044b\u0432\u043e\u0434, \u0447\u0442\u043e \u043c\u044b \u0441\u043c\u043e\u0436\u0435\u043c \u0433\u0435\u043d\u0435\u0440\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0435 \u0447\u0438\u0441\u043b\u0430 \u043f\u043e \u0444\u043e\u0440\u043c\u0443\u043b\u0435, \u0435\u0441\u043b\u0438 \u0443\u0437\u043d\u0430\u0435\u043c \u0442\u043e\u0447\u043d\u043e\u0435 \u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435 \u0447\u0438\u0441\u0435\u043b.\n\n#### \u041f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c\n\n\u041a\u0430\u043a \u0438 \u0432 \u043f\u0440\u043e\u0448\u043b\u044b\u0445 \u0441\u0442\u0430\u0442\u044c\u044f\u0445, \u043c\u044b \u0438\u0441\u0441\u043b\u0435\u0434\u0443\u0435\u043c \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0435 \u0447\u0438\u0441\u043b\u0430 \u043d\u0430 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u043e \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u0438. \u0423\u0437\u043d\u0430\u0435\u043c \u0442\u0430\u043a\u0436\u0435 \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u044c \u0438 \u0443\u0441\u043a\u043e\u0440\u0435\u043d\u0438\u0435 \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u0435\u0439.\n\n\n```python\ndensities=[(len(list(filter(lambda x: x < i, sorted(komp))))-1)/i for i in range(200,10000)]\ndrawPlot(densities,\"\u0418\u0437\u043c\u0435\u043d\u0435\u043d\u0438\u0435 \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u0438\",\"\u041f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c\",\"\u0414\u0440\u0443\u0436\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0435 \u0447\u0438\u0441\u043b\u043e\")\n```\n\n\u041f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0445 \u0447\u0438\u0441\u0435\u043b \u0437\u043d\u0430\u0447\u0438\u0442\u0435\u043b\u044c\u043d\u043e \u0443\u043c\u0435\u043d\u044c\u0448\u0430\u0435\u0442\u0441\u044f \u0441\u043e \u0432\u0440\u0435\u043c\u0435\u043d\u0435\u043c. \u041c\u043e\u0436\u043d\u043e \u0434\u0430\u0436\u0435 \u0432\u044b\u044f\u0432\u0438\u0442\u044c \u0442\u0440\u0435\u043d\u0434, \u0438 \u043f\u043e\u043d\u044f\u0442\u044c \u0447\u0442\u043e \u0437\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u044f.\n\n\u0424\u0443\u043d\u043a\u0446\u0438\u044f $ {f(x)}={{1}\\over{x}} , x \\in (0,\\inf) $\n\n\n```python\nddensities=[densities[i]-densities[i-1] for i in range(1,len(densities))]\nddensities=[ddensities[i]-ddensities[i-1] for i in range(1,len(ddensities))]\n```\n\n\n```python\nfig,ax=plt.subplots(1,2,figsize=(12,6))\nax[0].set_facecolor(\"#F2F2F2\")\nax[0].grid()\nax[0].set_title(\"\u0421\u043a\u043e\u0440\u043e\u0441\u0442\u044c \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u0438\")\nax[0].plot(densities)\nax[1].set_facecolor(\"#F2F2F2\")\nax[1].grid()\nax[1].set_title(\"\u0423\u0441\u043a\u043e\u0440\u0435\u043d\u0438\u0435 \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u0438\")\nax[1].plot(ddensities)\n```\n\n\u0421\u0443\u043f\u0435\u0440 \u0437\u0440\u0435\u043b\u0438\u0449\u0435! \u042f \u0435\u0449\u0451 \u0440\u0430\u0437 \u0443\u0431\u0435\u0434\u0438\u043b\u0441\u044f, \u0447\u0442\u043e \u043c\u043e\u0436\u043d\u043e \u043f\u043e\u043f\u0440\u043e\u0431\u043e\u0432\u0430\u0442\u044c \u0433\u0435\u043d\u0435\u0440\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0435 \u0447\u0438\u0441\u043b\u0430. \n\u0410 \u0434\u0430\u0432\u0430\u0439\u0442\u0435 \u043f\u043e\u043f\u0440\u043e\u0431\u0443\u0435\u043c?\n\n\n```python\ndef findSociableNumbers(countt):\n komp=[]\n per=[]\n i=0\n while len(komp)n \n```\n\n\n```python\nPrimeSociable=[i for i in komp if(_isPrime(i))]\nplt.pie([len(PrimeSociable),30-len(PrimeSociable)], \n colors=[\"#A7FF5B\",\"#E7AFFF\"],\n labels=[\"\u0421\u043e\u0441\u0442\u0430\u0432\u043d\u044b\u0435\",\"\u041f\u0440\u043e\u0441\u0442\u044b\u0435\"], \n autopct='%1.1f%%',\n shadow=True,\n textprops={'color':\"black\"})\nplt.legend([len(PrimeSociable),30-len(PrimeSociable)])\n;\n```\n\n\u041a\u0430\u043a \u0438 \u0432 \u0441\u043b\u0443\u0447\u0430\u0435 \u0441 \u0434\u0440\u0443\u0436\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u043c\u0438 \u0447\u0438\u0441\u043b\u0430\u043c\u0438, \u0437\u0434\u0435\u0441\u044c \u0442\u043e\u0436\u0435 \u043d\u0438 \u043e\u0434\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0441\u0442\u043e\u0433\u043e!\n\n## \u0412\u0438\u0437\u0443\u0430\u043b\u0438\u0437\u0430\u0446\u0438\u044f\n\n\u041f\u043e\u043f\u0440\u043e\u0431\u0443\u0435\u043c \u043d\u0430\u0440\u0438\u0441\u043e\u0432\u0430\u0442\u044c \u0441\u043f\u0438\u0440\u0430\u043b\u0438 \u0441\u043e \u0441\u0440\u0435\u0434\u043d\u0438\u043c \u0438 \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0438\u043c \u0443\u0433\u043b\u043e\u043c \u0432 \u0440\u044f\u0434\u0443 \u0438\u0437 30 \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0445 \u0447\u0438\u0441\u0435\u043b\n\n\n```python\nratio=[komp[i]/komp[i-1] for i in range(1,len(komp))]\n```\n\n\n```python\nratio[-1]\n```\n\n\n\n\n 1.0375586854460095\n\n\n\n\n```python\nnp.mean(ratio)\n```\n\n\n\n\n 1.6519833851472943\n\n\n\n\n```python\nangle=sp.N(360/np.mean(ratio)**2)\nangle\n```\n\n\n\n\n$\\displaystyle 131.914079291008$\n\n\n\n\n```python\n#\u0421\u043e\u0437\u0434\u0430\u0451\u043c \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0443 300\u0445300\n\nN=300\nt = turtle.Turtle(fixed=False, width=N, height=N)\nt.hideturtle()\nt\n```\n\n\n Turtle()\n\n\n\n```python\nfor i in range(15):\n for j in range(int(angle)): \n t.forward(i/10) \n t.left(1)\n if(i>10):\n time.sleep(0.01) # \u0427\u0442\u043e\u0431\u044b \u043d\u0435 \u0442\u0435\u0440\u044f\u0442\u044c \u0440\u0438\u0441\u0443\u043d\u043e\u043a\n```\n\n\n```python\nangle=sp.N(360/ratio[-1]**2)\nangle\n```\n\n\n\n\n$\\displaystyle 334.408386396675$\n\n\n\n\n```python\n#\u0421\u043e\u0437\u0434\u0430\u0451\u043c \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0443 300\u0445300\n\nN=300\nt = turtle.Turtle(fixed=False, width=N, height=N)\nt.hideturtle()\nt\n```\n\n\n Turtle()\n\n\n\n```python\nfor i in range(15):\n for j in range(int(angle)): \n t.forward(i/10) \n t.left(1)\n if(i>10):\n time.sleep(0.01) # \u0427\u0442\u043e\u0431\u044b \u043d\u0435 \u0442\u0435\u0440\u044f\u0442\u044c \u0440\u0438\u0441\u0443\u043d\u043e\u043a\n```\n\n# \u041a\u043e\u043d\u0435\u0446 \n\n\u041d\u0430 \u044d\u0442\u043e\u043c \u0432\u0441\u0451, \u0434\u043e\u0440\u043e\u0433\u0438\u0435 \u0447\u0438\u0442\u0430\u0442\u0435\u043b\u0438! \n\u041d\u0430\u0434\u0435\u044e\u0441\u044c \u0432\u0430\u043c \u0431\u044b\u043b\u043e \u0438\u043d\u0442\u0435\u0440\u0435\u0441\u043d\u043e \u0443\u0437\u043d\u0430\u0442\u044c \u0447\u0442\u043e-\u0442\u043e \u0432\u0440\u043e\u0434\u0435 \u0442\u0430\u043a\u043e\u0433\u043e \u0438\u0437 \u0437\u0430\u043d\u0438\u043c\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0438 \ud83d\ude0d\n\n\u0421\u043b\u0435\u0434\u0443\u044e\u0449\u0430\u044f \u0441\u0442\u0430\u0442\u044c\u044f \u0441\u043a\u043e\u0440\u0435\u0435 \u0432\u0441\u0435\u0433\u043e \u043f\u0440\u043e \u0441\u043e\u0432\u0435\u0440\u0448\u0435\u043d\u0441\u0442\u0432\u043e \u0447\u0438\u0441\u0435\u043b. \u0411\u0443\u0434\u0435\u0442 \u0431\u043e\u043b\u044c\u0448\u0430\u044f \u0441\u0442\u0430\u0442\u044c\u044f. \u0414\u0430\u0436\u0435 \u043e\u0447\u0435\u043d\u044c. \u041a \u0442\u043e\u043c\u0443 \u0436\u0435, \u0443 \u043c\u0435\u043d\u044f \u0435\u0441\u0442\u044c \u043d\u0435\u0441\u043a\u043e\u043b\u044c\u043a\u043e \u043f\u0440\u043e\u0435\u043a\u0442\u043e\u0432, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0430\u044f \u0441\u0442\u0430\u0442\u044c\u044f \u0431\u0443\u0434\u0435\u0442 \u0441\u043e\u0432\u0441\u0435\u043c \u043d\u0435 \u0441\u043a\u043e\u0440\u043e, \u043d\u043e \u0431\u0443\u0434\u0435\u0442!\n\n---\n\n\u0417\u0434\u0435\u0441\u044c \u0441\u0441\u044b\u043b\u043a\u0430 \u043d\u0430 \u0434\u0430\u0442\u0430\u0441\u0435\u0442\u044b \u043a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0445 \u0447\u0438\u0441\u0435\u043b\n[\u041a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u0438\u0435 \u0447\u0438\u0441\u043b\u0430](https://github.com/lonagi/pysasha/blob/master/datasets/Number%20Theory/Sociable%20Numbers/SociableeNumbers_30.csv) \n\n\u0414\u0430\u0442\u0430\u0441\u0435\u0442 \u0441\u043e\u0441\u0442\u043e\u0438\u0442 \u0438\u0437 \u0434\u0432\u0443\u0445 \u0441\u0442\u043e\u043b\u0431\u0438\u043a\u043e\u0432. \u041a\u043e\u043c\u043f\u0430\u043d\u0435\u0439\u0441\u043a\u043e\u0435 \u0447\u0438\u0441\u043b\u043e \u0438 \u0435\u0433\u043e \u043f\u0435\u0440\u0438\u043e\u0434. \u0410 \u043d\u0430 \u043c\u043e\u043c\u0435\u043d\u0442 \u043d\u0430\u043f\u0438\u0441\u0430\u043d\u0438\u044f \u0441\u0442\u0430\u0442\u044c\u0438, \u0447\u0438\u0441\u0435\u043b \u0442\u0430\u043c 30.\n\n\u0415\u0441\u043b\u0438 \u0435\u0441\u0442\u044c \u0436\u0435\u043b\u0430\u043d\u0438\u0435 \u043f\u0440\u043e\u0447\u0438\u0442\u0430\u0442\u044c \u0441\u0442\u0430\u0442\u044c\u044e \u043d\u0430 \u0434\u0440\u0443\u0433\u043e\u043c \u0438\u0441\u0442\u043e\u0447\u043d\u0438\u043a\u0435, \u0442\u043e \u043f\u043e\u0436\u0430\u043b\u0443\u0439\u0441\u0442\u0430, \u0432\u043e\u0442 \u043d\u0438\u0436\u0435 \u0441\u0441\u044b\u043b\u043a\u0438:\n \n[\u0412\u041a\u043e\u043d\u0442\u0430\u043a\u0442\u0435](vk.com/@lonagi-kompaneiskie-chisla-issledovanie) \n[Instagram]() \n[Facebook](https://www.facebook.com/%D0%97%D0%B0%D0%BD%D0%B8%D0%BC%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F-%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0-%D0%BE%D1%82-Lonagi-112410007105730) \n[Github](https://github.com/lonagi/pysasha/blob/master/docs/Number%20Theory/Sociable%20Numbers/SociableNumbers.md) \n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "83902778f3cdb7a9db9b8ce4141d4d0b148ddd75", "size": 486234, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/Number Theory/Sociable Numbers/SociableNumbers.ipynb", "max_stars_repo_name": "lonagi/pysasha", "max_stars_repo_head_hexsha": "e344afa03357b7c626c949aa31fac51574e97f48", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/Number Theory/Sociable Numbers/SociableNumbers.ipynb", "max_issues_repo_name": "lonagi/pysasha", "max_issues_repo_head_hexsha": "e344afa03357b7c626c949aa31fac51574e97f48", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/Number Theory/Sociable Numbers/SociableNumbers.ipynb", "max_forks_repo_name": "lonagi/pysasha", "max_forks_repo_head_hexsha": "e344afa03357b7c626c949aa31fac51574e97f48", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 158.6925587467, "max_line_length": 59252, "alphanum_fraction": 0.894528149, "converted": true, "num_tokens": 12743, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.731058584489497, "lm_q1q2_score": 0.4165956727104205}} {"text": "\n# **3.1 Nonlinear bond - softening and hardening**\n\n[](https://moodle.rwth-aachen.de/mod/page/view.php?id=551816)  part 1\n\n
                                        \n     Starting point
                                        \n\nBy saying that we want to capture the _material behavior_ we mean\nthat we realistically describe the **constitutive relation** between the strain and stress which is **valid for\nany material point** of the considered volume. With the focus on a one-dimensional interface between two material\ncomponents we can reduce this task to the relation between bond stress and slip.\nIn Tour 2, we assumed the constitutive bond-slip relation constant. However, as we have learned\nin trip [2.1 Pull-out of elastic fiber from rigid matrix](../pull_out/2_1_1_PO_observation.ipynb)\nthis stick-slip interface behavior cannot realistically describe the experimentally measured\nresponse of steel-concrete pull-out with varied length of the bond length $L_b$.\n\n
                                        \n     Where are we heading
                                        \n\nTo improve the quality of the model, in this notebook we introduce and investigate more complex shapes of bond slip laws and their effect on the observed pullout response. This extension will enable a more **realistic \nprediction of a wide range of pull-out and crack bridge tests**, including \nsteel rebars, carbon textile fabrics or carbon fiber reinforced polymer (CFRP) sheets.\nUsing the models, we will perform automated studies of the pull-out response that can demonstrate the different phenomenology behind hardening and softening constitutive behavior.\nThese studies indicate how validated models can support the definition of engineering design rules. \n\nTo proceed in small steps we consider two shapes of constant bond-slip law, referred to as **bond-hardening and bond softening**.\n\n\n\nThe increasing/hardening or decreasing/softening trend of the bond-slip law in the second branch introduces the question, what kind of **material structure** within the bond zone can induce such type of behavior. An example of an idealized bond system leading to hardening or softening can be provided using a rough surface with an increasing or decreasing number of asperities. A more detailed classification of the bond systems will be shown in Tour 3 which provides a more physically based description of the debonding process. The question studied in this notebook is **what is the qualitative effect of the second bond-slip slope on the pull-out response.**\n\n# **Numerical support necessary**\n\nTo solve a pullout problem for a generally nonlinear bond-slip law, we have to solve\nthe initial boundary value problem numerically. In this notebook, we will use a finite-element code\nimplemented within the BMCS tool to study the behavior for two examples of qualitatively different bond-slip laws. \n\n**Solution algorithm:** To study of the effect of the nonlinear bond-slip law on\nthe pullout response we will use the finite-element method solving the nonlinear response of the pull-out test by stepping through the loading history. Let us therefore briefly touch the topic of the solution algorithm needed to solve such a nonlinear problem boundary value problem of continuum mechanics. Generally, a non-linear finite element solver includes the solution of two separate tasks:\n - **Time stepping** algorithm that can identify the material state variables satisfying the constitutive law for a prescribed loadincrement in all points of the domain using an iterative Newton-type algorithm.\n - Find the **spatial distribution** of the displacement field satisfying the equilibrium, compatibility and boundary conditions using the finite-element discretization.\n\n
                                        \n     Short sidetrip
                                        \n\n## Time-stepping - Newton method to solve a set of nonlinear equations \n\nThe Newton method is the basis of all nonlinear time-stepping algorithms used in finite-element codes. \nLet us explain the solution procedure by considering a very short bond length $L_\\mathrm{b}$x and denote it as a material point $m$ for which a constant profile of the shear stress $\\tau(x) = \\tau_m$ and slip $s(x) = s_m$ can be assumed.\n\n\n\nThe iterative time-stepping algorithm with increasing load levels can now be displayed for single unknown displacement variable $w$ which must satisfy the equilibrium condition $\\bar{P}(t) = P(w)$, where $\\bar(P)$ is a prescribed history of loading. A simple implementation of the time stepping procedure exemplifying the solution procedure for a nonlinear equation is provided for an interested tourist in an Annex notebook [A.2 Newton method](../extras/newton_method.ipynb). \n\n\n\nIn a real simulation of the pull-out problem, the unknown variable is not a slip but the displacement fields $u_\\mathrm{m}, u_\\mathrm{f}$ are the primary unknowns. They are transformed to corresponding component strains $\\varepsilon_\\mathrm{m}=u_{\\mathrm{m},x}, \\varepsilon_\\mathrm{f}=u_{\\mathrm{f},x}$, and slip $s = u_\\mathrm{m} - u_\\mathrm{f}$. In the following examples, the component strains are still assumed linear elastic while the bond/shear stress is assumed generally nonlinear. With the known stress fields, the corresponding forces are obtained using numerical integration which deliver the residuum of the global equilibrium condition. The solution scheme described for a single variable in the notebook [A.2](../extras/newton_method.ipynb#newton_iteration_example) remains the same.\n\n
                                        \n     Deep dive
                                        \n\n## Spatial solver - boundary value problem solved using the finite element method \n\nThe identification of the displacements within each equilibrium iteration includes the same conditions that we have applied to derive the analytical solution of the pull-out problem with a constant bond slip law. However, the discrete solution satisfies the equilibrium conditions only approximately in a _week sense_. This means that the local differential equilibrium condition is not satisfied everywhere but only in integration points.\n\nTo provide an insight into the way how do the finite-element tools solve the problem, an open implementation of the nonlinear solver used in this and later notebooks is described completely with a running example, plots and animation in a notebook [A.3 Finite element solver for a pull-out problem](../extras/pullout1d.ipynb). This notebook is an Annex to the course and is meant for ambitious adventurers who want to see how the most finite-element programs available on the market are implemented. Detailed explanation of the theoretical background is provided in the Master's courses on linear structural analysis focused on the theoretical background of the finite-element method and on the nonlinear structural analysis.\n\n
                                        \n     Distant view
                                        \n\n## Example of the finite-element pull-out simulation\n\nTo understand the functionality of the finite-element model implemented in the referenced notebook [A.3](../extras/pullout1d.ipynb), its output is provided here in form of the pull-out curve and of the fields along the bond zone. The applied boundary conditions are given as follows, the free length $L_\\mathrm{f}=0$, the matrix is supported at the loaded end.\n\n\n\n\n```python\nfrom IPython.display import HTML\nhtml_video_file = open('../extras/pull_out_animation.html','r')\nHTML(html_video_file.read())\n```\n\n\n\n\n\n\n\n\n## What constitutive law can induce such a debonding process?\n\nA closer look at the simulated evolution of the shear stress along the bond zone in the bottom right \ndiagram provides an important phenomenological observation. The level of shear increases \nat the right, loaded end in the first stage. After reaching the peak shear stress of $N = 2~\\mathrm{N}$ , it \ndiminishes slowly to a low value of approximately 0.1 N.\n\nThe constitutive law valid at each material point has thus a first ascending and second descending branch. Such kind of behavior is called **softening**. Constitutive behavior exhibiting softening has a severe impact on the structural behavior by introducing the phenomena of strain localization to discrete shear and tensile cracks, accompanied with stress redistribution during the debonding or crack propagation process. The pull-out problem can be conveniently used to visualize the correspondence between the **softening** material law and the structural response with a debonding propagation, as opposed to **hardening** material law.\n\n# **Setting up the model components - new material model**\n\nFor the purpose of this comparison, let us introduce a simple piece-wise linear bond-slip law, that can be inserted into the non-linear finite-element code to investigate the effect of the type of nonlinearity on the pull-out response.\n\n\n
                                        \n     Coding intermezzo
                                        \n\n## Construct a material model with tri-linear bond-slip law \nTo indicate how the below examples are implemented let us define a a piece-wise linear function with three branches constituting the bond-slip behavior. It can be used to exemplify how to implement material models in standard non-linear finite-element codes for structural analysis. In codes like `ANSYS, Abaqus, ATENA, Diana`, the spatial integration of the stresses and stiffnesses is based on the so called **predictor**, **corrector** scheme.\n\nThis simply means that the material model must provide two functions\n 1. the stress evaluation for a given strain increment\n 2. the derivative of stress with respect to the strain increment, i.e. the material stiffness.\nIn our case of a bond-slip law, we need to provide two functions\n\\begin{align}\n \\tau(s) \\\\\n \\frac{\\mathrm{d} \\tau}{ \\mathrm{d} s}\n\\end{align}.\n\n**Let's import the packages:**\n\n\n```python\n%matplotlib widget\nimport sympy as sp # symbolic algebra package\nimport numpy as np # numerical package\nimport matplotlib.pyplot as plt # plotting package\nsp.init_printing() # enable nice formating of the derived expressions\n```\n\nThe tri-linear function can be readily constructed using the already known `Piecewise` function provied in `sympy`\n\n\n```python\ns = sp.symbols('s')\ntau_1, s_1, tau_2, s_2 = sp.symbols(r'tau_1, s_1, tau_2, s_2')\ntau_s = sp.Piecewise(\n (tau_1 / s_1 * s, s <= s_1), # value, condition\n (tau_1 + (tau_2-tau_1) / (s_2-s_1) * (s - s_1), s <= s_2), # value, condition\n (tau_2, True) # value, otherwise\n)\ntau_s\n```\n\nThe derivative is obtained as\n\n\n```python\nd_tau_s = sp.diff(tau_s, s)\nd_tau_s\n```\n\n
                                        \n     How to get numbers?
                                        \n\n**The above results are symbols! How to transform them to numbers and graphs?**\n\n`sympy` offers the possibility to generate executable code from symbolic expression (`C`, `Fortran`, or `Python`).\nTo get `Python` functions that accept the characteristic points `tau_1`, `tau_2`, `s_1`, `s_2`\nand evaluating the above defined expressions `tau_s` and `d_tau_s`, we need the following two lines:\n\n\n```python\nget_tau_s = sp.lambdify((s, tau_1, tau_2, s_1, s_2), tau_s, 'numpy')\nget_d_tau_s = sp.lambdify((s, tau_1, tau_2, s_1, s_2), d_tau_s, 'numpy')\n```\n\nThe parameter `numpy` enables us to evaluate both functions for arrays of values, not only for a single number. As a result, an array of slip values can be directly sent to the function `get_tau_s` to obtain an array of corresponding stresses\n\n\n```python\nget_tau_s(np.array([0, 0.5, 1, 1.5, 2]), 1, 0.1, 1, 2)\n```\n\n\n\n\n array([0. , 0.5 , 1. , 0.55, 0.1 ])\n\n\n\n
                                        \n     How to to plot it?
                                        \n\nLet us now show that the implemented bond-slip function provides a sufficient range of qualitative shapes to demonstrate and discuss the effect of softening and hardening behavior of the interface material. Let us setup a figure `fig` with two axes `ax1` and `ax2` to verify if the defined function is implemented correctly\n\n\n```python\nfig, (ax1, ax2) = plt.subplots(1,2, figsize=(8,3), tight_layout=True)\nfig.canvas.header_visible = False\ns_range = np.linspace(0, 3, 1050)\nfor tau_2 in [0, 0.5, 1, 1.5, 2]:\n ax1.plot(s_range, get_tau_s(s_range, 1, tau_2, 0.1, 2));\n ax2.plot(s_range, get_d_tau_s(s_range, 1, tau_2, 0.1, 2));\nax1.set_xlabel(r'$s$ [mm]'); ax1.set_ylabel(r'$\\tau$ [MPa]');\nax2.set_xlabel(r'$s$ [mm]'); ax2.set_ylabel(r'$\\mathrm{d}\\tau/\\mathrm{d}s$ [MPa/mm]');\n```\n\n\n Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous \u2026\n\n\n## Preconfigured pullout model provided in BMCS Tool Suite \nThe presented function is the simplest model provided in a general-purpose nonlinear finite-element simulator `BMCS-Tool-Suite`.\nThe package `bmcs_cross_section` provides several preconfigured models that can be used to analyze and visualize the behavior of a composite cross-section. The analysis of the pullout problem discussed here can be done using the class `PullOutModel1D` that can be imported as follows\n\n\n```python\nfrom bmcs_cross_section.pullout import PullOutModel1D\n```\n\nAn instance of the pullout model can be constructed using the following line\n\n\n```python\npo = PullOutModel1D()\n```\n\nFor convenience, let us summarize the model parameters before showing how to assign them to the model instance\n\n**Geometrical variables:**\n\n| Python | Parameter | Description | \n| :- | :-: | :- |\n| `A_f` | $A_\\mathrm{f}$ | Cross section area modulus of the reinforcement |\n| `A_m` | $A_\\mathrm{m}$ | Cross section area modulus of the matrix |\n| `P_b` | $p_\\mathrm{b}$ | Perimeter of the reinforcement |\n| `L_b` | $L_\\mathrm{b}$ | Length of the bond zone of the pulled-out bar |\n\n**Material parameters of a tri-linear bond law:**\n\n| Python | Parameter | Description | \n| :- | :-: | :- |\n| `E_f` | $E_\\mathrm{f}$ | Young's modulus of the reinforcement |\n| `E_m` | $E_\\mathrm{m}$ | Young's modulus of the matrix |\n| `tau_1` | $\\tau_1$ | bond strength |\n| `tau_2` | $\\tau_2$ | bond stress at plateu |\n| `s_1` | $s_1$ | slip at bond strengh |\n| `s_2` | $s_1$ | slip at plateau stress |\n\n**Fixed support positions:**\n\n| Python | \n| :- |\n| `non-loaded end (matrix)` |\n| `loaded end (matrix)` |\n| `non-loaded end (reinf)` |\n| `clamped left` |\n\nEven more conveniently, let us render the interaction window generated by the model to directly see the structure and the naming of the parameters\n\n\n```python\npo.interact()\n```\n\n\n VBox(children=(HBox(children=(VBox(children=(Tree(layout=Layout(align_items='stretch', border='solid 1px black\u2026\n\n\nThe tree structure at the top-left frame shows the individual model components. Parameters of each component are shown in the bottom-left frame. By nagivating through tree, the parameter frame and the plotting frame are updated to see the corresponding part of the model. The control bar at the bottom can be used to start, stop and reset the simulation.\n\n**Example interaction:** Develop some confidence into the correctness of the model. Change the stiffness of the components such that they have the same area and stiffness modulus. Run the simulation and watch the profile of the shear flow along the bond length. Increase the bond length, reset the calculation and run it anew. Change the position support and verify the profile of the displacements.\n\n# **Studies 1: Hardening bond-slip law**\n\n[](https://moodle.rwth-aachen.de/mod/page/view.php?id=551816)  part 2\n\n## RILEM Pull-Out Test revisited\n\n\n\n\n```python\npo_rilem = PullOutModel1D(n_e_x=300, w_max=0.12) # n_e_x - number of finite elements along the bond zone\npo_rilem.n_e_x=400\n```\n\n Exception occurred in traits notification handler for object: \n t_n: 0, t_n1: 0\n U_n[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0.]\n U_k[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0.], trait: state_changed, old value: , new value: True\n Traceback (most recent call last):\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/trait_notifiers.py\", line 524, in _dispatch_change_event\n self.dispatch(handler, *args)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/trait_notifiers.py\", line 486, in dispatch\n handler(*args)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 999, in wrapper0\n return function(arg)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 3329, in notify\n self.trait_property_changed(name, old)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 927, in decorator\n self.__dict__[name] = result = function(self)\n File \"/home/rch/PycharmProjects/bmcs_cross_section/bmcs_cross_section/pullout/pullout_sim.py\", line 870, in _get_bc\n return [self.control_bc] + self.fixed_bc_list\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 927, in decorator\n self.__dict__[name] = result = function(self)\n File \"/home/rch/PycharmProjects/bmcs_cross_section/bmcs_cross_section/pullout/pullout_sim.py\", line 861, in _get_control_bc\n return BCDof(node_name='pull-out displacement',\n File \"/home/rch/PycharmProjects/bmcs_utils/bmcs_utils/model.py\", line 20, in __init__\n super().__init__(*args, **kw)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/base_trait_handler.py\", line 74, in error\n raise TraitError(\n traits.trait_errors.TraitError: The 'time_function' trait of a BCDof instance must be a TimeFunction or None, but a value of was specified.\n Exception occurred in traits notification handler for object: \n t_n: 0, t_n1: 0\n U_n[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0.]\n U_k[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0.], trait: state_changed, old value: , new value: True\n Traceback (most recent call last):\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/trait_notifiers.py\", line 524, in _dispatch_change_event\n self.dispatch(handler, *args)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/trait_notifiers.py\", line 486, in dispatch\n handler(*args)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 999, in wrapper0\n return function(arg)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 3329, in notify\n self.trait_property_changed(name, old)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 927, in decorator\n self.__dict__[name] = result = function(self)\n File \"/home/rch/PycharmProjects/bmcs_cross_section/bmcs_cross_section/pullout/pullout_sim.py\", line 870, in _get_bc\n return [self.control_bc] + self.fixed_bc_list\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 927, in decorator\n self.__dict__[name] = result = function(self)\n File \"/home/rch/PycharmProjects/bmcs_cross_section/bmcs_cross_section/pullout/pullout_sim.py\", line 861, in _get_control_bc\n return BCDof(node_name='pull-out displacement',\n File \"/home/rch/PycharmProjects/bmcs_utils/bmcs_utils/model.py\", line 20, in __init__\n super().__init__(*args, **kw)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/base_trait_handler.py\", line 74, in error\n raise TraitError(\n traits.trait_errors.TraitError: The 'time_function' trait of a BCDof instance must be a TimeFunction or None, but a value of was specified.\n Exception occurred in traits notification handler for object: \n t_n: 0, t_n1: 0\n U_n[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 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0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n U_k[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.], trait: state_changed, old value: , new value: True\n Traceback (most recent call last):\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/trait_notifiers.py\", line 524, in _dispatch_change_event\n self.dispatch(handler, *args)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/trait_notifiers.py\", line 486, in dispatch\n handler(*args)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 999, in wrapper0\n return function(arg)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 3329, in notify\n self.trait_property_changed(name, old)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 927, in decorator\n self.__dict__[name] = result = function(self)\n File \"/home/rch/PycharmProjects/bmcs_cross_section/bmcs_cross_section/pullout/pullout_sim.py\", line 870, in _get_bc\n return [self.control_bc] + self.fixed_bc_list\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 927, in decorator\n self.__dict__[name] = result = function(self)\n File \"/home/rch/PycharmProjects/bmcs_cross_section/bmcs_cross_section/pullout/pullout_sim.py\", line 861, in _get_control_bc\n return BCDof(node_name='pull-out displacement',\n File \"/home/rch/PycharmProjects/bmcs_utils/bmcs_utils/model.py\", line 20, in __init__\n super().__init__(*args, **kw)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/base_trait_handler.py\", line 74, in error\n raise TraitError(\n traits.trait_errors.TraitError: The 'time_function' trait of a BCDof instance must be a TimeFunction or None, but a value of was specified.\n\n\nTo configure the model such that it reflects the RILEM test we can either use the interactive editor above, or assign the \nattributes directly. As apparent from the editor frame above, attributes `fixed_boundary` and `material model` are dropdown boxes offering several options. To assign these parameters we can use the following scheme\n - assign one of the options available in the dropdown box to the attribute `attribute` as a string\n - the option object is then available as an attribute with the name `attribute_` with the trailing underscore.\n Thus, to define a trilinear bond-slip law we can proceed as follows\n\n\n```python\npo_rilem.material_model = 'trilinear' # polymorphis attribute - there are several options to be chosen from\n# set the parameters of the above defined tri-linear bond-slip law - add also the matrix and fiber stiffness \npo_rilem.material_model_.E_m=28000 # [MPa]\npo_rilem.material_model_.E_f=210000 # [MPa]\npo_rilem.material_model_.tau_1=4\npo_rilem.material_model_.s_1=1e-3\npo_rilem.material_model_.tau_2=8\npo_rilem.material_model_.s_2=0.12\n```\n\n Exception occurred in traits notification handler for object: \n t_n: 0, t_n1: 0\n U_n[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n U_k[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.], trait: state_changed, old value: , new value: True\n Traceback (most recent call last):\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/trait_notifiers.py\", line 524, in _dispatch_change_event\n self.dispatch(handler, *args)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/trait_notifiers.py\", line 486, in dispatch\n handler(*args)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 999, in wrapper0\n return function(arg)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 3329, in notify\n self.trait_property_changed(name, old)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 927, in decorator\n self.__dict__[name] = result = function(self)\n File \"/home/rch/PycharmProjects/bmcs_cross_section/bmcs_cross_section/pullout/pullout_sim.py\", line 870, in _get_bc\n return [self.control_bc] + self.fixed_bc_list\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 927, in decorator\n self.__dict__[name] = result = function(self)\n File \"/home/rch/PycharmProjects/bmcs_cross_section/bmcs_cross_section/pullout/pullout_sim.py\", line 861, in _get_control_bc\n return BCDof(node_name='pull-out displacement',\n File \"/home/rch/PycharmProjects/bmcs_utils/bmcs_utils/model.py\", line 20, in __init__\n super().__init__(*args, **kw)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/base_trait_handler.py\", line 74, in error\n raise TraitError(\n traits.trait_errors.TraitError: The 'time_function' trait of a BCDof instance must be a TimeFunction or None, but a value of was specified.\n\n\nTo set several parameters of the model component at once, the `trait_set` method can be used as an alternative to one-by-one assignement\n\n\n```python\nd = 16.0 # [mm]\npo_rilem.cross_section.trait_set(A_m=100*100, A_f=3.14*(d/2)**2, P_b=3.14*d)\npo_rilem.geometry.L_x=5*d\npo_rilem.fixed_boundary='loaded end (matrix)'\n```\n\n Exception occurred in traits notification handler for object: \n t_n: 0, t_n1: 0\n U_n[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n U_k[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.], trait: state_changed, old value: , new value: True\n Traceback (most recent call last):\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/trait_notifiers.py\", line 524, in _dispatch_change_event\n self.dispatch(handler, *args)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/trait_notifiers.py\", line 486, in dispatch\n handler(*args)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 999, in wrapper0\n return function(arg)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 3329, in notify\n self.trait_property_changed(name, old)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 927, in decorator\n self.__dict__[name] = result = function(self)\n File \"/home/rch/PycharmProjects/bmcs_cross_section/bmcs_cross_section/pullout/pullout_sim.py\", line 870, in _get_bc\n return [self.control_bc] + self.fixed_bc_list\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 927, in decorator\n self.__dict__[name] = result = function(self)\n File \"/home/rch/PycharmProjects/bmcs_cross_section/bmcs_cross_section/pullout/pullout_sim.py\", line 861, in _get_control_bc\n return BCDof(node_name='pull-out displacement',\n File \"/home/rch/PycharmProjects/bmcs_utils/bmcs_utils/model.py\", line 20, in __init__\n super().__init__(*args, **kw)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/base_trait_handler.py\", line 74, in error\n raise TraitError(\n traits.trait_errors.TraitError: The 'time_function' trait of a BCDof instance must be a TimeFunction or None, but a value of was specified.\n Exception occurred in traits notification handler for object: \n t_n: 0, t_n1: 0\n U_n[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 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0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n U_k[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.], trait: state_changed, old value: , new value: True\n Traceback (most recent call last):\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/trait_notifiers.py\", line 524, in _dispatch_change_event\n self.dispatch(handler, *args)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/trait_notifiers.py\", line 486, in dispatch\n handler(*args)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 999, in wrapper0\n return function(arg)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 3329, in notify\n self.trait_property_changed(name, old)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 927, in decorator\n self.__dict__[name] = result = function(self)\n File \"/home/rch/PycharmProjects/bmcs_cross_section/bmcs_cross_section/pullout/pullout_sim.py\", line 870, in _get_bc\n return [self.control_bc] + self.fixed_bc_list\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 927, in decorator\n self.__dict__[name] = result = function(self)\n File \"/home/rch/PycharmProjects/bmcs_cross_section/bmcs_cross_section/pullout/pullout_sim.py\", line 861, in _get_control_bc\n return BCDof(node_name='pull-out displacement',\n File \"/home/rch/PycharmProjects/bmcs_utils/bmcs_utils/model.py\", line 20, in __init__\n super().__init__(*args, **kw)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/base_trait_handler.py\", line 74, in error\n raise TraitError(\n traits.trait_errors.TraitError: The 'time_function' trait of a BCDof instance must be a TimeFunction or None, but a value of was specified.\n Exception occurred in traits notification handler for object: \n t_n: 0, t_n1: 0\n U_n[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n U_k[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.], trait: state_changed, old value: , new value: True\n Traceback (most recent call last):\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/trait_notifiers.py\", line 524, in _dispatch_change_event\n self.dispatch(handler, *args)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/trait_notifiers.py\", line 486, in dispatch\n handler(*args)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 999, in wrapper0\n return function(arg)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 3329, in notify\n self.trait_property_changed(name, old)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 927, in decorator\n self.__dict__[name] = result = function(self)\n File \"/home/rch/PycharmProjects/bmcs_cross_section/bmcs_cross_section/pullout/pullout_sim.py\", line 870, in _get_bc\n return [self.control_bc] + self.fixed_bc_list\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 927, in decorator\n self.__dict__[name] = result = function(self)\n File \"/home/rch/PycharmProjects/bmcs_cross_section/bmcs_cross_section/pullout/pullout_sim.py\", line 861, in _get_control_bc\n return BCDof(node_name='pull-out displacement',\n File \"/home/rch/PycharmProjects/bmcs_utils/bmcs_utils/model.py\", line 20, in __init__\n super().__init__(*args, **kw)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/base_trait_handler.py\", line 74, in error\n raise TraitError(\n traits.trait_errors.TraitError: The 'time_function' trait of a BCDof instance must be a TimeFunction or None, but a value of was specified.\n Exception occurred in traits notification handler for object: \n t_n: 0, t_n1: 0\n U_n[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n U_k[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.], trait: state_changed, old value: , new value: True\n Traceback (most recent call last):\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/trait_notifiers.py\", line 524, in _dispatch_change_event\n self.dispatch(handler, *args)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/trait_notifiers.py\", line 486, in dispatch\n handler(*args)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 999, in wrapper0\n return function(arg)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 3329, in notify\n self.trait_property_changed(name, old)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 927, in decorator\n self.__dict__[name] = result = function(self)\n File \"/home/rch/PycharmProjects/bmcs_cross_section/bmcs_cross_section/pullout/pullout_sim.py\", line 870, in _get_bc\n return [self.control_bc] + self.fixed_bc_list\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 927, in decorator\n self.__dict__[name] = result = function(self)\n File \"/home/rch/PycharmProjects/bmcs_cross_section/bmcs_cross_section/pullout/pullout_sim.py\", line 861, in _get_control_bc\n return BCDof(node_name='pull-out displacement',\n File \"/home/rch/PycharmProjects/bmcs_utils/bmcs_utils/model.py\", line 20, in __init__\n super().__init__(*args, **kw)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/base_trait_handler.py\", line 74, in error\n raise TraitError(\n traits.trait_errors.TraitError: The 'time_function' trait of a BCDof instance must be a TimeFunction or None, but a value of was specified.\n Exception occurred in traits notification handler for object: \n t_n: 0, t_n1: 0\n U_n[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n U_k[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.], trait: state_changed, old value: , new value: True\n Traceback (most recent call last):\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/trait_notifiers.py\", line 524, in _dispatch_change_event\n self.dispatch(handler, *args)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/trait_notifiers.py\", line 486, in dispatch\n handler(*args)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 999, in wrapper0\n return function(arg)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 3329, in notify\n self.trait_property_changed(name, old)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 927, in decorator\n self.__dict__[name] = result = function(self)\n File \"/home/rch/PycharmProjects/bmcs_cross_section/bmcs_cross_section/pullout/pullout_sim.py\", line 870, in _get_bc\n return [self.control_bc] + self.fixed_bc_list\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/has_traits.py\", line 927, in decorator\n self.__dict__[name] = result = function(self)\n File \"/home/rch/PycharmProjects/bmcs_cross_section/bmcs_cross_section/pullout/pullout_sim.py\", line 861, in _get_control_bc\n return BCDof(node_name='pull-out displacement',\n File \"/home/rch/PycharmProjects/bmcs_utils/bmcs_utils/model.py\", line 20, in __init__\n super().__init__(*args, **kw)\n File \"/home/rch/miniconda3/envs/bmcs_env/lib/python3.8/site-packages/traits/base_trait_handler.py\", line 74, in error\n raise TraitError(\n traits.trait_errors.TraitError: The 'time_function' trait of a BCDof instance must be a TimeFunction or None, but a value of was specified.\n\n\nThe configured model can be rendered anytime as a web-app to check the input parameters and to adjust them.\n\n\n```python\npo_rilem.run()\npo_rilem.interact()\n```\n\n## Bond-slip law calibration/validation\n\n**Can we find just one material law that predicts all three tests?** \n - The preconfigured bond-slip law with an ascending branch after reaching the strength of $\\tau_1 = 4$ MPa with the parameters $\\tau_1 = 4$ MPa, $\\tau_2 = 8$ MPa, $s_1 = 0.001$ mm, $s_1 = 0.12$ mm \n can reproduce the test with $d = 16$ mm and $L_b = 5d = 80$ mm.\n - To see the prediction for the test with $L_b = 10d = 160$ mm, modify the parameter `geometry.L_x = 160`. The result shows a good match with the experimentally observed response.\n\n**Can we compare the differences in one plot?**\n\n - The interactive user interface is illustrative and provides a quick orientation in the scope and functionality of the model. Once we have learned its structure, we can use the programming interface to run simulations in a loop and plot them in a single graph to see the similar picture as in the output of the RILEM test above.\n \n - Try to compare the third test with $d = 28$ mm and $L_b = 5d$ mm.\n\n
                                        \n     Plot step by step
                                        \n\n\n```python\nfig, (ax, ax_bond_slip) = plt.subplots(1,2, figsize=(8,3), tight_layout=True)\nfig.canvas.header_visible = False\n\nprint('calculate d=16 mm, L=5d')\nd = 16.0 # [mm]\npo_rilem.cross_section.trait_set(A_m=100*100, A_f=3.14*(d/2)**2, P_b=3.14*d)\npo_rilem.w_max = 0.12\npo_rilem.geometry.L_x=5*d\npo_rilem.reset() # it is like pressing the reset button in the above window\npo_rilem.run() # like pressing the run button\npo_rilem.history.plot_Pw(ax, color='blue')\n\nprint('calculate d=16 mm, L=10d')\nd = 16.0 # [mm]\npo_rilem.cross_section.trait_set(A_m=100*100, A_f=3.14*(d/2)**2, P_b=3.14*d)\npo_rilem.w_max = 0.12\npo_rilem.geometry.L_x=10*d\npo_rilem.reset()\npo_rilem.run()\npo_rilem.hist.plot_Pw(ax, color='red')\n\nprint('calculate d=28 mm, L=3d')\nd = 16.0 # [mm]\npo_rilem.cross_section.trait_set(A_m=100*100, A_f=3.14*(d/2)**2, P_b=3.14*d)\npo_rilem.geometry.L_x=3*d\npo_rilem.w_max = 0.05\npo_rilem.reset()\npo_rilem.run()\npo_rilem.hist.plot_Pw(ax, color='green')\npo_rilem.material_model_.plot(ax_bond_slip)\n\n# The code sequence can be certainly shortened by using the loop. \n# It is deliberately omitted here as the focus is not on programming.\n```\n\n## **Comments** on the study\n - Note that the bond-slip law that can fit all three pull-out tests exhibits hardening. \n - The maximum control displacement `w_max` is set equal to the one applied in the test as no information beyond this value is provided by the tests.\n - The trilinear bond-slip law does not give us the flexibility to reproduce the pull-out failure\n as it ends with a plateu.\n \n## **Need for a more flexible bond-slip law**\n - A more flexibility is provided by a `multilinear` material model for which a list of `s_data` and `tau_data` \n can be specified.\n - The `multilinear` material model is used in the following code to show how to achieve a pull-out failure by introducing a descending branch in the bond-slip law.\n - Note that for bond-slip laws with descending branch, convergence problems can occur when approaching the pullout failure. The convergence behavior can be improved by refining the spatial discretization given by the number of finite elements along the bond zone `n_e_x` and by the size of the time step\n `time_line.step`.\n\n\n```python\nfig, (ax, ax_bond_slip) = plt.subplots(1,2, figsize=(8,3), tight_layout=True)\nfig.canvas.header_visible = False\nd = 32.0 # [mm]\npo_rilem.w_max = 0.12\npo_rilem.time_line.step = 0.05\npo_rilem.material_model='multilinear'\npo_rilem.material_model_.trait_set(E_m=28000, E_f=210000, tau_data='0, 4, 6, 0, 0', s_data='0, 1e-3, 0.08, 0.12, 0.2')\npo_rilem.geometry.L_x= 1*d\npo_rilem.reset()\npo_rilem.run()\npo_rilem.hist.plot_Pw(ax, color='magenta')\npo_rilem.material_model_.plot(ax_bond_slip)\n```\n\n## **Questions:** Effect of bond length on the pullout response - **bond hardening**\n\n - The iterative trial and error fitting is tedious. **How to design a test from which we can directly obtain the bond-slip law?**\n Comparing the test with $L_b = 5d$ and $L_b = 10d$, we recognize that the shorter bond length resembles more the shape of the bond-slip law. To verify this, set the bond length in the above example to $L_\\mathrm{b} = 1d$.\n - On the other hand, if we increase the length, the maximum pull-out will increase. **How can we determine the bond length at which the steel bar will yield?**. A simple and quick answer to this question can be provided by reusing the analytical pull-out model with a constant bond-slip law as a first approximation. The maximum achievable pull-out force of a test with an embedded length $L_\\mathrm{b}$ is given as\n\\begin{align}\n\\label{EQ:MaxEmbeddedLength}\nP_{L} = \\bar{\\tau} p_\\mathrm{b} L_\\mathrm{b}\n\\end{align}\nwhere $p_\\mathrm{b}$ denotes the perimeter, equal in all experiments. The force at which the reinforcement attains the strength $\\sigma_{\\mathrm{f},\\mathrm{mu}}$ and breaks is\n\\begin{align}\nP_{\\mathrm{f},\\mathrm{mu}} = \\sigma_{\\mathrm{f},\\mathrm{mu}} A_\\mathrm{f}\n\\end{align}\nso that the bond length at which the reinforcement will fail is obtained by requiring $P_L = P_{\\mathrm{f},\\mathrm{mu}}$ which renders \n\\begin{align}\n\\label{EQ:ConstantBondAnchorageLength}\nL_{\\mathrm{b}} = \\frac{\\sigma_{\\mathrm{f},\\mathrm{mu}} A_\\mathrm{f} }\n{\\bar{\\tau} p}.\n\\end{align}\nFor a generally nonlinear bond-slip law, we need to evaluate the maximum load numerically. Two examples quantifying the effect of the bond-length for bond-hardening and bond-softening systematically are provided in the notebook [3.2 Anchorage length](3_2_anchorage_length.ipynb)\n\n\n# **Studies 2: Softening bond-slip law**\n\n[](https://moodle.rwth-aachen.de/mod/page/view.php?id=551816)  part 3\n\nThe presence of the descending branch in a constitutive law is the key for understanding the propagating debonding. Let us use the established framework to study the different phenomenology that occurs for constitutive laws exhibiting softening. \nConsider an interface between a fiber reinforced polymer (FRP) sheet used for retrofitting of RC structure. The study is based on a paper by [Dai et al. (2005)](../papers/dai_frp_pullout_2005.pdf). The goal of the paper is to derive constitutive laws capturing the bond-slip behavior of the adhesive between the FRP sheet and concrete surface. We will use selected experimental pullout curves from the paper to reproduce them using the numerical pullout model introduced above by verifying the bond-slip law derived in the paper.\n\n## Test setup\n\n\n\nThe width of the sheet was $p_b = 100$ mm, the attached bond length is also $L_b = 100$ mm.\nThe properties of the tested sheets are summarized in the table. The dimensions of the concrete block were $200 \\times 400 \\times 200$ mm.\n\n\n\nThe pull-out curves measured for different adhesives used to realize the bond to the matrix were evaluated as the strain in the FRP sheet at the loaded end versus slip displacement. \n\n\n\nTo compare with our studies, we can transfer the results to pullout force $P$ by evaluating\n\\begin{align}\nP = p_\\mathrm{b} t_\\mathrm{f} E_\\mathrm{f} \\varepsilon_\\mathrm{f}\n\\end{align}\nyielding for the strain 0.010\n\n\n```python\np_b = 100 # [mm]\nt_f = 0.11 # [mm]\nE_f = 230000 # [MPa]\neps_f_max = 0.01 # [-]\nP_max = p_b * t_f * E_f * eps_f_max / 1000 # [kN]\nP_max # [kN]\n```\n\nThe bond-slip law reported by the authors of the paper has the following shape\n\n\n\n\n## Model for CFRP pullout test \n\nLet us construct another pullout model named `po_cfrp` with a bond slip law exhibiting strong softening. Such kind of behavior is observed in tests between FRP sheets and concrete. An example of such an experimental study \n\n\n\n```python\npo_cfrp = PullOutModel1D(n_e_x=300, w_max=1.5) # mm \npo_cfrp.geometry.L_x=100 # mm\npo_cfrp.time_line.step = 0.02\npo_cfrp.cross_section.trait_set(A_m=400*200, A_f=100*0.11, P_b=100)\npo_cfrp.material_model='trilinear'\npo_cfrp.material_model_.trait_set(E_m=28000, E_f=230000, tau_1=5.5, tau_2=0, s_1=0.08, s_2=0.4)\npo_cfrp.interact()\n```\n\n## **Conclusions:** to interactive study of CFRP sheet debonding\n - The bond-slip law reported in the paper can reproduce well the pullout response measured in the test.\n - The study of the debonding process shows that the adhesive is only active within an effective length of approximately 40-50 mm.\n - As a consequence, in contrast to steel rebar studied above, the maximum pullout load cannot be increased by an increasing bond length.\n - In the studied case there will FRP rupture is not possible because its strength is larger than $P_\\max$ of 25 kN. To verify this, we use the strength of 3550 MPa given in the above table and multiply with the cross-sectional area of the sheet, i.e. $f_t t_f p_b$ to obtain\n\n\n```python\nf_t = 3550 # CFRP sheet strength in [MPa] - see the table above\nf_t * t_f * p_b / 1000 # breaking force of the sheet 100 x 100 mm in [kN]\n```\n\n## **Question:** Effect of bond length on the pullout response - **bond softening**\n - Similarly to the the example with bond hardening above, we ask the question what happens with the pullout curve if we reduce the bond length to a minimum. The answer is the same - we will recover a bond-slip law multiplied by the bond area.\n - However, if we increase the bond length, the trend will be different as already mentioned above. Once the length exceeds the effective bond length, there will be no increase in the pullout force and the pullout curve will exhibit a plateau. Let us show this trend by running a simple parametric study. Instead of doing it step by step we now run a loop over the list of length and colors, change the parameter `geometry.L_x` within the loop, `reset`, `run`, and `plot` the pullout curve in a respective color. \n\n
                                        \n     Run in a loop to see the effect of bond length
                                        \n\nNote that a list in python is defined by the brackets\n```[1,2,3,4]```. Two lists can be \"zipped\" together so that we can run\na loop over the lengths and colors as shown in the third line of the cell\n\n\n\n```python\nfig, (ax, ax_bond_slip) = plt.subplots(1,2, figsize=(10,4), tight_layout=True)\nfig.canvas.header_visible = False\nfor L, color in zip([5, 10, 50, 100, 200], ['red','green','blue','black','orange']):\n print('evaluating pullout curve for L', L)\n po_cfrp.geometry.L_x=L\n po_cfrp.reset()\n po_cfrp.run()\n po_cfrp.history.plot_Pw(ax, color=color)\npo_cfrp.material_model_.plot(ax_bond_slip)\n```\n\n# **Remark to structural ductility:** how to make the plateau useful?\n\nThe softening bond cannot exploit the full strength of the CFRP sheet which might seem uneconomic at first sight. On the other hand it can be viewed as a mechanism that increases the deformation capacity of the structure with a constant level of load. This property can be effectively used to enhance the ductility of the structure, i.e. induce large deformation before the structural collapse required in engineering designs. This documents the importance of knowledge of the stress redistribution mechanisms available in the material. In steel reinforced structure, the ductility is provided inherently by the steel yielding property. In In case of brittle reinforcement, e.g. carbon fabrics, CFRP sheets, glass fabrics, other sources of ductility must be provided to ensure the sufficient deformation capacity between the serviceability and ultimate limit states. \n\n
                                        \n     Exercise X0301: Pull-out curve versus shear stress profiles - part 1 \n\n
                                        \n\n
                                        \n     Exercise X0302: Pull-out curve versus shear stress profiles - part 2 \n\n
                                        \n\n
                                        \n", "meta": {"hexsha": "4818319ef185216e40592d269e36c99e8ee3b028", "size": 714395, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tour3_nonlinear_bond/3_1_nonlinear_bond.ipynb", "max_stars_repo_name": "bmcs-group/bmcs_tutorial", "max_stars_repo_head_hexsha": "4e008e72839fad8820a6b663a20d3f188610525d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tour3_nonlinear_bond/3_1_nonlinear_bond.ipynb", "max_issues_repo_name": "bmcs-group/bmcs_tutorial", "max_issues_repo_head_hexsha": "4e008e72839fad8820a6b663a20d3f188610525d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tour3_nonlinear_bond/3_1_nonlinear_bond.ipynb", "max_forks_repo_name": "bmcs-group/bmcs_tutorial", "max_forks_repo_head_hexsha": "4e008e72839fad8820a6b663a20d3f188610525d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 183.5075777036, "max_line_length": 101548, "alphanum_fraction": 0.8539183505, "converted": true, "num_tokens": 54673, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.7310585727705126, "lm_q1q2_score": 0.41659566603232606}} {"text": "# Pr\u00e9-processamento de dados e valida\u00e7\u00e3o\n\nOi pessoal!\n\nNessa aula, vamos falar um pouco sobre pr\u00e9-processamento de dados e valida\u00e7\u00e3o de modelos! Vamos aprender:\n- Como processar seus dados para enteg\u00e1-los da melhor forma para seu modelo.\n- A prever quem vive e quem morre em um acidente mar\u00edtimo.\n- M\u00e9tricas de avalia\u00e7\u00e3o.\n- Estrat\u00e9gias de valida\u00e7\u00e3o.\n- Como submeter em uma competi\u00e7\u00e3o do Kaggle.\n\nN\u00f3s vamos passar por diversas t\u00e9cnicas e coloc\u00e1-las em pr\u00e1tica em um dos datasets mais cl\u00e1ssicos: Sobreviventes do Titanic!\n\nEspero que voc\u00eas gostem, n\u00e3o esque\u00e7am de deixar o like e adicionar o notebook aos seus favoritos.\n\n### *IMPORTANTE BAIXAR OS DADOS*\n\nPara baixar os dados voc\u00ea vai precisar se cadastrar no Kaggle - explico no pr\u00f3ximo t\u00f3pico o que \u00e9 esse site.\n\nOs dados podem ser baixados em: https://www.kaggle.com/c/titanic/data \nNesse link voc\u00ea tamb\u00e9m encontra uma descri\u00e7\u00e3o sobre cada uma das features do dataset.\n\nTr\u00eas arquivos ser\u00e3o fornecidos:\n\n- Dados de treino: conjunto de dados dos quais sabemos a resposta.\n- Dados de teste: conjunto de dados para o qual faremos nossas previs\u00f5es.\n- Exemplo de submiss\u00e3o: Arquivo que nos mostra o formato de uma submiss\u00e3o para essa competi\u00e7\u00e3o.\n\n## O que \u00e9 o Kaggle?\n\nO Kaggle \u00e9 a maior comunidade de Ci\u00eancia de Dados do mundo. Nele, voc\u00ea encontra diversos datasets, estudos de pessoas do mundo todo e ainda pode participar de competi\u00e7\u00f5es organizadas pelo site - muitas delas valendo dinheiro.\n\nNosso objetivo \u00e9 que, ao fim dessa aula, voc\u00ea fa\u00e7a sua primeira submiss\u00e3o em uma competi\u00e7\u00e3o do Kaggle. A competi\u00e7\u00e3o \u00e9 a do [Titanic](https://www.kaggle.com/c/titanic/overview), e ela foi criada apenas para fins educacionais.\n\n## Importando os dados\n\nVamos come\u00e7ar importando os dados que acabamos de baixar e dando uma primeira olhada neles. N\u00f3s vamos assumir que voc\u00eas ja sabem Pandas mas, qualquer d\u00favida que surgir, n\u00e3o deixem de dar uma olhada na aula espec\u00edfica do assunto, pesquisar no Google (a documenta\u00e7\u00e3o dessas bibliotecas \u00e9 bem feita) ou perguntar pra gente!\n\n\n```python\nimport numpy as np\nimport pandas as pd\n```\n\n\n```python\ntrain = pd.read_csv('train.csv')\ntest = pd.read_csv('test.csv')\n\ncombine = [train, test]\n```\n\n\n```python\ntrain.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        PassengerIdSurvivedPclassNameSexAgeSibSpParchTicketFareCabinEmbarked
                                        0103Braund, Mr. Owen Harrismale22.010A/5 211717.2500NaNS
                                        1211Cumings, Mrs. John Bradley (Florence Briggs Th...female38.010PC 1759971.2833C85C
                                        2313Heikkinen, Miss. Lainafemale26.000STON/O2. 31012827.9250NaNS
                                        3411Futrelle, Mrs. Jacques Heath (Lily May Peel)female35.01011380353.1000C123S
                                        4503Allen, Mr. William Henrymale35.0003734508.0500NaNS
                                        \n
                                        \n\n\n\n## Tipos de vari\u00e1veis\n\nVamos come\u00e7ar entendendo melhor sobre o nosso material de estudo: os dados. Um dataset pode ser visto como um conjunto de *data objects*, normalmente chamados de inst\u00e2ncias, observa\u00e7\u00f5es, vetores, linhas ou exemplos. Esses s\u00e3o cada uma das linhas do nosso conjunto de dados.\n\nCada um desses *data objects* \u00e9 definido por um conjunto de var\u00edaveis - ou *features*. Que expressam, de diferentes maneiras, as caracter\u00edsticas da nossa inst\u00e2ncia. Ao descrever uma sala de aula, por exemplo, podemos ter as seguintes features: localiza\u00e7\u00e3o, capacidade, quadro negro ou branco, quantidade de tomadas, n\u00famero de portas, se possui um ar condicionado. A ideia \u00e9 que essas vari\u00e1veis sejam capazes de descrever as caracteristicas b\u00e1sicas das salas.\n\nExistem diferentes tipos de vari\u00e1veis, eles s\u00e3o:\n\n- Num\u00e9ricas\n - Discretas (representam contagens) e.g. n\u00famero de cadeiras em uma sala\n - Cont\u00ednuas (assumem valores em um intervalo cont\u00ednuo) e.g pre\u00e7o de uma passagem\n- Categ\u00f3ricas\n - Ordinais (a ordem importa) e.g. tamanho de roupa (P < M < G)\n - Nominais (n\u00e3o existe ordem entre os valores) e.g. cor da roupa\n\nVamos identificar agora os tipos de vari\u00e1veis que temos no dataset do Titanic. Todo *DataFrame* possui um atributo chamado **dtypes**, que nos mostra o tipo de cada vari\u00e1vel.\n\n\n```python\ntrain.dtypes\n```\n\n\n\n\n PassengerId int64\n Survived int64\n Pclass int64\n Name object\n Sex object\n Age float64\n SibSp int64\n Parch int64\n Ticket object\n Fare float64\n Cabin object\n Embarked object\n dtype: object\n\n\n\nAgora n\u00f3s sabemos de que tipo cada vari\u00e1vel \u00e9. O tipo *object* indica que a vari\u00e1vel \u00e9 uma string, que trataremos como vari\u00e1veis categ\u00f3ricas.\n\n\u00c9 importante perceber que nem toda vari\u00e1vel n\u00famerica \u00e9 **realmente** num\u00e9rica. Pegue a vari\u00e1vel *Pclass*, por exemplo, seus valores s\u00e3o: 1a, 2a ou 3a classe. Esses valores j\u00e1 foram processados utilizando uma das t\u00e9cnicas que veremos nesse notebook e, por isso, ela agora \u00e9 tratada como um n\u00famero pelo c\u00f3digo, embora permane\u00e7a sendo uma feature categ\u00f3rica.\n\nPara entender de verdade o tipo de cada vari\u00e1vel, o melhor que podemos fazer \u00e9 ler a documenta\u00e7\u00e3o que \u00e9 normalmente fornecida com os dados. Abaixo, separaremos as vari\u00e1veis nos tipos que mencionamos anteriormente.\n\n**Quais vari\u00e1veis s\u00e3o num\u00e9ricas?** \nCont\u00ednuas: Age, Fare.
                                        Discretas: SibSp, Parch.\n\n**Quais vari\u00e1veis s\u00e3o categ\u00f3ricas?** \nNominais: Survived, Sex, and Embarked.
                                        Ordinais: Pclass.\n\n## Valores faltantes\n\nAntes de come\u00e7armos a criar nossas features - ou fazer qualquer coisa - \u00e9 bom sempre dar uma olhada no nosso conjunto de dados. Podemos ver se nenhum dado est\u00e1 faltando, verificar se os dados fazem sentido, se est\u00e3o consistentes e coisas do tipo.\n\nAl\u00e9m disso, \u00e9 uma boa ideia fazer perguntas e buscar as respostas no seu dataset, assim como criar gr\u00e1ficos que te fa\u00e7am entender melhor seus dados e que forne\u00e7am insights sobre eles. Essa etapa do processo \u00e9 chamada de An\u00e1lise Explorat\u00f3ria de Dados (EDA).\n\nComo o foco da aula \u00e9 no pr\u00e9-processamento, vamos apenas tratar dos valores faltantes pois eles podem nos trazer alguns problemas mais pra frente. Utilizando os comandos mostrados abaixo, voc\u00ea pode descobrir quantos valores est\u00e3o faltando em cada coluna do seu dataset.\n\n\n```python\ntrain.isna().sum()\n```\n\n\n\n\n PassengerId 0\n Survived 0\n Pclass 0\n Name 0\n Sex 0\n Age 177\n SibSp 0\n Parch 0\n Ticket 0\n Fare 0\n Cabin 687\n Embarked 2\n dtype: int64\n\n\n\n\n```python\ntest.isna().sum()\n```\n\n\n\n\n PassengerId 0\n Pclass 0\n Name 0\n Sex 0\n Age 86\n SibSp 0\n Parch 0\n Ticket 0\n Fare 1\n Cabin 327\n Embarked 0\n dtype: int64\n\n\n\nExistem diversas formas de preencher os *missing values*, por exemplo:\n - M\u00e9dia ou mediana de toda a coluna.\n - Agrupar os dados por outras features e pegar m\u00e9dia ou mediana depois. Por exemplo, digamos que queremos preencher valores vazios de idade. \u00c9 poss\u00edvel que pessoas de diferentes classes tenham m\u00e9dias de idades diferentes, ent\u00e3o, na hora de preencher podemos olhar para a m\u00e9dia ou mediana da idade daqueles que est\u00e3o na mesma classe que nosso exemplo com idade vazia est\u00e1.\n - Preencher com 0.\n - Excluir as linhas que cont\u00e9m valores nulos.\n \nTodas essas formas s\u00e3o v\u00e1lidas e tem seus pr\u00f3s e contras. Por motivo de simplicidade, vamos preencher os valores faltantes em Age e Fare com a mediana da coluna e em Embarked com o valor mais frequente. N\u00e3o vamos preencher Cabin pois n\u00e3o usaremos essa feature.\n\n\n```python\nembarked_mode = train['Embarked'].mode()[0]\ntrain['Embarked'] = train['Embarked'].fillna(embarked_mode)\ntest['Embarked'] = test['Embarked'].fillna(embarked_mode)\n\ntrain_median_fare = train['Fare'].dropna().median()\ntrain['Fare'] = train['Fare'].fillna(train_median_fare)\ntest['Fare'] = test['Fare'].fillna(train_median_fare)\n\ntrain_median_age = train['Age'].dropna().median()\ntrain['Age'] = train['Age'].fillna(train_median_age)\ntest['Age'] = test['Age'].fillna(train_median_age)\n```\n\n## Feature engineering\n\nProtinho, agora podemos come\u00e7ar a criar algumas features b\u00e1sicas.\n\nUma das partes mais importantes do pr\u00e9-processamento de dados \u00e9 a cria\u00e7\u00e3o de features que ajudem seu modelo a encontrar padr\u00f5es mais facilmente. Nessa parte do notebook, n\u00f3s vamos criar algumas novas vari\u00e1veis e analisar como elas se relacionam com a sobreviv\u00eancia dos passageiros.\n\nAs features criadas aqui foram retiradas de: https://www.kaggle.com/startupsci/titanic-data-science-solutions\n\n### T\u00edtulo do passageiro\n\nSe pensarmos bem, o nome de um indiv\u00edduo n\u00e3o tem rela\u00e7\u00e3o alguma com suas chances de sobreviv\u00eancia. Podemos ent\u00e3o retirar essa coluna?
                                        Bom, talvez. Vamos dar uma olhada em alguns dos nomes.\n\n\n```python\ntrain['Name'].head()\n```\n\n\n\n\n 0 Braund, Mr. Owen Harris\n 1 Cumings, Mrs. John Bradley (Florence Briggs Th...\n 2 Heikkinen, Miss. Laina\n 3 Futrelle, Mrs. Jacques Heath (Lily May Peel)\n 4 Allen, Mr. William Henry\n Name: Name, dtype: object\n\n\n\nOs nomes em nossos conjuntos de dados s\u00e3o \u00fanicos, mas os t\u00edtulos dos indiv\u00edduos (Mr., Mrs., Miss) se repetem. Vamos isolar os t\u00edtulos de cada nome para analisarmos essa fato. A tabela abaixo mostra, para cada t\u00edtulo, quantos homens e quantas mulheres o possuem.\n\nO m\u00e9todo *Series.str.extract()* extrai texto se baseando na regex utilizada.\n\n\n```python\nfor df in combine:\n df['Title'] = df['Name'].str.extract(' ([A-Za-z]+)\\.', expand=False)\n\npd.crosstab(train['Title'], train['Sex'])\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Sexfemalemale
                                        Title
                                        Capt01
                                        Col02
                                        Countess10
                                        Don01
                                        Dr16
                                        Jonkheer01
                                        Lady10
                                        Major02
                                        Master040
                                        Miss1820
                                        Mlle20
                                        Mme10
                                        Mr0517
                                        Mrs1250
                                        Ms10
                                        Rev06
                                        Sir01
                                        \n
                                        \n\n\n\nAntes de ver a rela\u00e7\u00e3o dos t\u00edtulos com sobreviv\u00eancia, vamos agrup\u00e1-los. Talvez seja uma boa ideia deixar todos os mais raros em um \u00fanico grupo, ent\u00e3o vamos fazer isso.\n\n\n```python\nfor df in combine:\n df['Title'] = df['Title'].replace(['Lady', 'Countess','Capt', 'Col',\n 'Don', 'Dr', 'Major', 'Rev', 'Sir', \n 'Jonkheer', 'Dona'], 'Rare')\n\n df['Title'] = df['Title'].replace('Mlle', 'Miss')\n df['Title'] = df['Title'].replace('Ms', 'Miss')\n df['Title'] = df['Title'].replace('Mme', 'Mrs')\n \ntrain[['Title', 'Survived']].groupby(['Title'], as_index=False).mean()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        TitleSurvived
                                        0Master0.575000
                                        1Miss0.702703
                                        2Mr0.156673
                                        3Mrs0.793651
                                        4Rare0.347826
                                        \n
                                        \n\n\n\nMuito interessante. O t\u00edtulo de um passageiro realmente influencia sua chance de sobreviv\u00eancia. Podemos manter essa feature que criamos e excluir a feature *Name*.\n\n\n```python\ntrain = train.drop(['Name'], axis=1)\ntest = test.drop(['Name'], axis=1)\n\ncombine = [train, test]\n```\n\n### O passageiro est\u00e1 sozinho?\n\nUma pessoa que est\u00e1 sozinha tem mais chances de sobreviver? Abaixo criaremos uma feature que indica o tamanho da fam\u00edlia do passageiro que est\u00e1 a bordo, e uma outra feature para indicar se um indiv\u00edduo est\u00e1 sozinho ou n\u00e3o.\n\n\n```python\nfor df in combine:\n df['FamilySize'] = df['SibSp'] + df['Parch'] + 1\n df['IsAlone'] = 0\n df.loc[df['FamilySize'] == 1, 'IsAlone'] = 1\n \ntrain[['IsAlone', 'Survived']].groupby(['IsAlone'], as_index=False).mean()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        IsAloneSurvived
                                        000.505650
                                        110.303538
                                        \n
                                        \n\n\n\nAparentemente n\u00e3o. Voc\u00ea consegue imaginar um motivo para isso?\n\nVou excluir as features SibSp e Parch pois acho que as features que criamos j\u00e1 nos d\u00e3o bastante informa\u00e7\u00e3o. Voc\u00ea pode escolher mant\u00ea-las se preferir ou at\u00e9 mesmo testar posteriormente de que forma seu modelo foi melhor.\n\n\n```python\ntrain = train.drop(['SibSp', 'Parch'], axis=1)\ntest = test.drop(['SibSp', 'Parch'], axis=1)\n\ncombine = [train, test]\n```\n\n### Como ficaram nossos datasets?\n\nVamos dar uma olhada nos nossos dois datasets (treino e teste) ap\u00f3s a cria\u00e7\u00e3o das features.\n\n\n```python\nprint(train.shape, test.shape)\n```\n\n (891, 12) (418, 11)\n\n\n\n```python\ntrain.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        PassengerIdSurvivedPclassSexAgeTicketFareCabinEmbarkedTitleFamilySizeIsAlone
                                        0103male22.0A/5 211717.2500NaNSMr20
                                        1211female38.0PC 1759971.2833C85CMrs20
                                        2313female26.0STON/O2. 31012827.9250NaNSMiss11
                                        3411female35.011380353.1000C123SMrs20
                                        4503male35.03734508.0500NaNSMr11
                                        \n
                                        \n\n\n\n\n```python\ntest.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        PassengerIdPclassSexAgeTicketFareCabinEmbarkedTitleFamilySizeIsAlone
                                        08923male34.53309117.8292NaNQMr11
                                        18933female47.03632727.0000NaNSMrs20
                                        28942male62.02402769.6875NaNQMr11
                                        38953male27.03151548.6625NaNSMr11
                                        48963female22.0310129812.2875NaNSMrs30
                                        \n
                                        \n\n\n\nAlgumas dessas vari\u00e1veis n\u00e3o nos d\u00e3o muita informa\u00e7\u00e3o, ent\u00e3o podemos exclu\u00ed-las. S\u00e3o elas *Ticket* e *Cabin*.\n\n\n```python\ntrain = train.drop(['Ticket', 'Cabin'], axis=1)\ntest = test.drop(['Ticket', 'Cabin'], axis=1)\n\ncombine = [train, test]\n```\n\n\n```python\ntrain.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        PassengerIdSurvivedPclassSexAgeFareEmbarkedTitleFamilySizeIsAlone
                                        0103male22.07.2500SMr20
                                        1211female38.071.2833CMrs20
                                        2313female26.07.9250SMiss11
                                        3411female35.053.1000SMrs20
                                        4503male35.08.0500SMr11
                                        \n
                                        \n\n\n\n## Feature encoding\n\nO modelo que usaremos s\u00f3 aceita vari\u00e1veis num\u00e9ricas, por isso, precisaremos transformar nossos dados de alguma forma. Existem v\u00e1rias maneiras de fazer isso e n\u00f3s vamos dar uma olhada em duas delas.\n\n### Label encoding\n\nM\u00e9todo que mapeia cada categoria em um n\u00famero. \u00c9 mais utilizado com m\u00e9todos de \u00e1rvore.\n\nOBS: O For n\u00e3o \u00e9 necess\u00e1rio aqui pois estamos encodando apenas uma vari\u00e1vel, mas preferi deixar uma forma mais geral.\n\n\n```python\nfrom sklearn.preprocessing import LabelEncoder\n```\n\n\n```python\nlabel_encoder_columns = ['Embarked']\n\nfor column in label_encoder_columns:\n # cria um encoder\n label_encoder = LabelEncoder()\n\n # cria um dicionario para o encoder\n label_encoder.fit(train[column])\n\n # aplica as transforma\u00e7\u00f5es nos datasets\n train[column] = label_encoder.transform(train[column])\n test[column] = label_encoder.transform(test[column])\n```\n\n\n```python\ntrain.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        PassengerIdSurvivedPclassSexAgeFareEmbarkedTitleFamilySizeIsAlone
                                        0103male22.07.25002Mr20
                                        1211female38.071.28330Mrs20
                                        2313female26.07.92502Miss11
                                        3411female35.053.10002Mrs20
                                        4503male35.08.05002Mr11
                                        \n
                                        \n\n\n\nComo podemos ver, *Embarked* agora possui apenas valores num\u00e9ricos. Caso seja necess\u00e1rio voltar aos valores iniciais em algum ponto do c\u00f3digo, basta usar o m\u00e9todo *LabelEncoder.inverse_transform()*. Abaixo temos um exemplo de como isso funciona.\n\n\n```python\nlabel_encoder.inverse_transform(train['Embarked'])[0:5]\n```\n\n\n\n\n array(['S', 'C', 'S', 'S', 'S'], dtype=object)\n\n\n\n### One-hot encoding\n\n\u00c9 um m\u00e9todo que cria, para cada categoria, uma coluna com valores bin\u00e1rios indicando a presen\u00e7a ou n\u00e3o da categoria na inst\u00e2ncia.\n\n- Normalmente utilizado com m\u00e9todos lineares, kNN e redes neurais.\n- \u00c9 necess\u00e1rio tomar cuidado caso existam muitos valores diferentes de categoria.\n\nEvita que o modelo pense algo como: Mrs. > Miss > Mr. Deixa mais clara para o modelo a independ\u00eancia das categorias. Vamos aplicar em algumas colunas apenas para exemplificar. Para fazer isso podemos usar a fun\u00e7\u00e3o *pd.get_dummies()*\n\n#### Exemplo de retorno da fun\u00e7\u00e3o\n\n\n```python\npd.get_dummies(train['Title'])\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        MasterMissMrMrsRare
                                        000100
                                        100010
                                        201000
                                        300010
                                        400100
                                        ..................
                                        88600001
                                        88701000
                                        88801000
                                        88900100
                                        89000100
                                        \n

                                        891 rows \u00d7 5 columns

                                        \n
                                        \n\n\n\n#### Aplica\u00e7\u00e3o\n\n\n```python\none_hot_columns = ['Sex', 'Title']\n\none_hot = pd.get_dummies(train[one_hot_columns])\ntrain = pd.concat([train, one_hot], axis=1).drop(one_hot_columns, axis=1)\n\none_hot = pd.get_dummies(test[one_hot_columns])\ntest = pd.concat([test, one_hot], axis=1).drop(one_hot_columns, axis=1)\n```\n\n\n```python\ntrain.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        PassengerIdSurvivedPclassAgeFareEmbarkedFamilySizeIsAloneSex_femaleSex_maleTitle_MasterTitle_MissTitle_MrTitle_MrsTitle_Rare
                                        010322.07.25002200100100
                                        121138.071.28330201000010
                                        231326.07.92502111001000
                                        341135.053.10002201000010
                                        450335.08.05002110100100
                                        \n
                                        \n\n\n\n## Feature scaling\n\nAlguns algor\u00edtmos s\u00e3o sens\u00edveis a escala dos dados e, por essa raz\u00e3o, precisamos colocar todos os dados na mesma escala para que esses modelos funcionem da melhor forma poss\u00edvel. Vamos dar uma olhada em dois dos m\u00e9todos mais comuns para fazer isso.\n\n\u00c9 importante saber que uma \u00c1rvores de Decis\u00e3o n\u00e3o \u00e9 um desses modelos sens\u00edveis, ent\u00e3o s\u00f3 aplicaremos esses m\u00e9todos aqui para exemplifica\u00e7\u00e3o.\n\nComo queremos aplicar essas tranforma\u00e7\u00f5es nos nossos dados de treino e teste, \u00e9 necess\u00e1rio que o Scaler tenha uma vis\u00e3o geral dos dados na hora de se adequar a eles, por isso, criaremos um dataset que concatena train e test.\n\n\n```python\ndf_concat = pd.concat([train.drop(['Survived'], axis=1), test], axis=0)\n```\n\n### Normaliza\u00e7\u00e3o\n\nFaz com que todos os valores fiquem no intervalo [0, 1]:\n\n$\n\\begin{align}\n{x_{i}}' = \\frac{x_{i} - min(x)}{max(x) - min(x)}\n\\end{align}\n$\n\nPodemos aplicar na coluna *Fare*.\n\n\n```python\nfrom sklearn.preprocessing import MinMaxScaler\n```\n\n\n```python\nscaler = MinMaxScaler(feature_range=(0, 1))\nscaler.fit(df_concat[['Fare']])\n\ntrain[['Fare']] = scaler.transform(train[['Fare']])\ntest[['Fare']] = scaler.transform(test[['Fare']])\n```\n\n### Padroniza\u00e7\u00e3o\n\nFaz com que os valores tenham m\u00e9dia 0 e desvio padr\u00e3o 1:\n\n$\n\\begin{align}\n{x_{i}}' = \\frac{x_{i} -\\mu}{\\sigma}\n\\end{align}\n$\n\nVamos utilizar a padroniza\u00e7\u00e3o na feature *Age* para exemplificar.\n\nExplica\u00e7\u00e3o (em ingl\u00eas, infelizmente) da import\u00e2ncia de normalizar os padronizar os dados: https://humansofdata.atlan.com/2018/12/data-standardization/\n\n\n```python\nfrom sklearn.preprocessing import StandardScaler\n```\n\n\n```python\nscaler = StandardScaler()\nscaler.fit(df_concat[['Age']])\n\ntrain[['Age']] = scaler.transform(train[['Age']])\ntest[['Age']] = scaler.transform(test[['Age']])\n```\n\n## Resultado final\n\nVamos dar uma olhada final nos nossos dados antes se seguirmos em frente!\n\n\n```python\ntrain.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        PassengerIdSurvivedPclassAgeFareEmbarkedFamilySizeIsAloneSex_femaleSex_maleTitle_MasterTitle_MissTitle_MrTitle_MrsTitle_Rare
                                        0103-0.5816280.0141512200100100
                                        12110.6586520.1391360201000010
                                        2313-0.2715580.0154692111001000
                                        34110.4260990.1036442201000010
                                        45030.4260990.0157132110100100
                                        \n
                                        \n\n\n\n\n```python\ntest.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        PassengerIdPclassAgeFareEmbarkedFamilySizeIsAloneSex_femaleSex_maleTitle_MasterTitle_MissTitle_MrTitle_MrsTitle_Rare
                                        089230.3873410.0152821110100100
                                        189331.3563100.0136632201000010
                                        289422.5190720.0189091110100100
                                        38953-0.1940410.0169082110100100
                                        48963-0.5816280.0239842301000010
                                        \n
                                        \n\n\n\n## M\u00e9tricas de avalia\u00e7\u00e3o\n\nAntes de come\u00e7armos a utilizar modelos para prever o futuro dos nossos passageiros, precisamos estudar um pouco sobre como avaliar o modelo que criarmos.\n\nM\u00e9tricas de avalia\u00e7\u00e3o s\u00e3o super importantes pois nos d\u00e3o um valor que representa o qu\u00e3o bem nosso modelo est\u00e1 se saindo.\n\nA m\u00e9trica utilizada depende se estamos tratando de um problema de classifica\u00e7\u00e3o ou regress\u00e3o. Vamos ensinar algumas para cada um dos casos:\n\n- Classifica\u00e7\u00e3o\n - Acur\u00e1cia\n - Precision e Recall\n - F-measure\n - AUC-ROC Curve\n- Regress\u00e3o\n - Erro quadr\u00e1tico m\u00e9dio (MSE)\n - R\u00b2\n\n### Matriz de confus\u00e3o\n\n\u00c9 uma medida de performance para modelos de classifica\u00e7\u00e3o. \u00c9 uma tabela com a combina\u00e7\u00e3o de valores previstos pelo modelo e valores reais:\n\n\n\nVamos entender o que cada uma das siglas representa fazendo uma analogia com um teste de alguma doen\u00e7a:\n\n- TP (True Positive): Seu modelo previu positivo e estava correto. Esse caso seria equivalente a voc\u00ea **estar** doente e receber um **teste positivo**.\n\n- TN (True Negative): Seu modelo previu negativo e estava correto. Esse caso \u00e9 equivalente a voc\u00ea **n\u00e3o estar** doente e receber um **teste negativo**.\n\n- FP (False Positive): Seu modelo previu positivo e estava incorreto: Esse caso \u00e9 equivalente a voc\u00ea **n\u00e3o estar** doente e receber um **teste positivo**.\n\n- FN (False Negative): Seu modelo previu negativo e estava incorreto. Esse caso \u00e9 equivalente a voc\u00ea **estar** doente e receber um **teste negativo**.\n\nNossas previs\u00f5es corretas ficam todas na diagonal principal da matrix. \n\n\u00c9 importante perceber que a matrix de confus\u00e3o pode ser usada para problemas de classfica\u00e7\u00e3o de mais de 2 classes. Abaixo, temos um exemplo da matrix de confus\u00e3o sendo usada em um problema de classifica\u00e7\u00e3o de 3 classes:\n\n\n\n\n\n### Precision e Recall\n\n**Precision**
                                        \nQual a propor\u00e7\u00e3o de previs\u00f5es positivas realmente corretas?\n\n$\n\\begin{align}\nPrecision = \\frac{TP}{TP+FP}\n\\end{align}\n$\n\nUm modelo sem falso positivo tem uma precis\u00e3o de 1.0.\n\n**Recall**
                                        \nQual a propor\u00e7\u00e3o de observa\u00e7\u00f5es realmente positivas foi identificada corretamente?\n\n$\n\\begin{align}\nRecall = \\frac{TP}{TP+FN}\n\\end{align}\n$\n\nUm modelo sem falso negativo tem recall de 1.0.\n\n### Acur\u00e1cia\n\nIndica a porcentagem de predi\u00e7\u00f5es corretas que o modelo realizou.\nPode ser expressa da seguinte forma:\n\n$\n\\begin{align}\nAccuracy = \\frac{TP+TN}{Total}\n\\end{align}\n$\n\nPode enganar caso o n\u00famero de ocorr\u00eancias de cada classe esteja desbalanceado.\n\n### F-measure\n\n\u00c9 uma medida de acur\u00e1cia que leva em conta precision e recall ao mesmo tempo. Ela \u00e9 dada por uma m\u00e9dia harm\u00f4nica das duas medidas:\n\n$\n\\begin{align}\nF_{1} = \\frac{1}{recall^{-1} + precision^{-1}} = 2 \\cdot \\frac{recall \\cdot precision}{recall+precision}\n\\end{align}\n$\n\n### AUC (Area under the curve)\n\n\u00c1rea abaixo de qual curva? Da curva **ROC** (Receiver operating characteristic).\n\nE como funciona essa curva?\n\n- Essa curva indica a rela\u00e7\u00e3o entre as taxas de falsos positivos e verdadeiros positivos\n- Nosso modelo ter\u00e1 como sa\u00edda a probabilidade de classificar os dados como da classe 1\n- Vamos supor que se P(Y=1) for > 0.5, dizemos que aquela inst\u00e2ncia \u00e9 da classe 1, caso contr\u00e1rio classe 0.\n- Isso ir\u00e1 nos dar uma taxa de falsos positivos e uma taxa de verdadeiros positivos e isso \u00e9 um ponto na curva ROC.\n- Agora escolhemos outros thresholds - diferentes de 0.5. Cada um desses gera um novo ponto na curva ROC.\n\n\n\nEssa metrica n\u00e3o \u00e9 sens\u00edvel a distribui\u00e7\u00e3o das classes e, por essa raz\u00e3o, \u00e9 comumente utilizada como m\u00e9trica de diversas competi\u00e7\u00f5es.\n\n\u00c9 importante notar que um **classificador aleat\u00f3rio** possui uma ROC-AUC de 0.5.\n\n### Mean Squared Error\n\nM\u00e9todo mais simples para avalia\u00e7\u00e3o de modelos de regress\u00e3o. Basicamente, para cada inst\u00e2ncia, calculamos o quadrado da diferen\u00e7a entre o que dever\u00edamos prever e o que previmos e ent\u00e3o tiramos a m\u00e9dia:\n\n$\n\\begin{align}\nMSE = \\frac{1}{N} \\sum_{i=0}^{N}(f(x_{i}) - y_{i})^{2}\n\\end{align}\n$\n\n### R\u00b2 Score\n\nOutro m\u00e9todo para avaliar modelos de regress\u00e3o. \u00c9 tamb\u00e9m chamado de coeficiente de determina\u00e7\u00e3o e tem todos seus valores no intervalo [0,1].\n\n$\n\\begin{align}\nSStot = \\sum_{i=0}(y_{i} - \\bar{y}_{i})^{2}\n\\end{align}\n$\n\n$\n\\begin{align}\nSSres = \\sum_{i=0}(y_{i} - \\hat{y}_{i})^{2}\n\\end{align}\n$\n\n$\n\\begin{align}\nR^{2} = 1 - \\frac{SSres}{SStot}\n\\end{align}\n$\n\n## Estrat\u00e9gias de valida\u00e7\u00e3o\n\nCom nossa m\u00e9trica escolhida em m\u00e3os, chegou a hora de criar e validar nosso modelo!\n\nVamos aprender agora dois m\u00e9todos de valida\u00e7\u00e3o e, depois de validar nosso modelo, faremos uma submiss\u00e3o no Kaggle para ver o quanto nossa valida\u00e7\u00e3o se aproxima do nosso resultado real no conjunto de teste.\n\nAntes disso, vamos s\u00f3 preparar nossos dados para os processos.\n\n\n```python\ntrain = train.drop(['PassengerId'], axis=1)\n\npass_ids = test['PassengerId']\ntest = test.drop(['PassengerId'], axis=1)\n```\n\n\n```python\nX = train.drop(['Survived'], axis=1)\ny = train['Survived']\n```\n\n\n```python\nX.shape\n```\n\n\n\n\n (891, 13)\n\n\n\n\n```python\ntest.shape\n```\n\n\n\n\n (418, 13)\n\n\n\nPerceba que agora temos o X e o test com as mesmas dimens\u00f5es, isso vai facilitar as coisas para n\u00f3s.\n\n### Hold Out\n\nNessa estrat\u00e9gia, n\u00f3s dividiremos nosso conjunto de dados em dois conjuntos. Usaremos um deles para treinar nosso modelo (conjunto de treino) e o outro para calcular a m\u00e9trica de avalia\u00e7\u00e3o (conjunto de valida\u00e7\u00e3o).\n\nN\u00f3s sabemos a resposta correta para todo o nosso conjunto de dados mas vamos fingir que n\u00e3o sabemos para uma parte dele, usar nosso modelo pra prever essas respostas e depois comparar as predi\u00e7\u00f5es com as respostas verdadeiras.\n\n\n```python\nfrom sklearn.model_selection import train_test_split\nfrom sklearn.metrics import accuracy_score\nfrom sklearn.tree import DecisionTreeClassifier\nfrom sklearn.metrics import confusion_matrix\n```\n\n\n```python\nX_train, X_val, y_train, y_val = train_test_split(X, y, test_size=0.2)\n```\n\n\n```python\nmodel = DecisionTreeClassifier()\nmodel.fit(X_train, y_train)\n\npreds = model.predict(X_val)\nprint(f'Accuracy = {accuracy_score(y_val, preds)}')\n\n# matriz de confus\u00e3o do nosso modelo\nconfusion_matrix(y_val, preds)\n```\n\n Accuracy = 0.7932960893854749\n\n\n\n\n\n array([[93, 19],\n [18, 49]])\n\n\n\n### k-Fold Cross Validation\n\nNessa estrat\u00e9gia vamos dividir nossos dados em K conjuntos. Em cada itera\u00e7\u00e3o, utilizaremos K-1 desses conjuntos para treinamento e um deles para valida\u00e7\u00e3o (variando qual conjunto \u00e9 utilizado para valida\u00e7\u00e3o em cada itera\u00e7\u00e3o). Depois disso, calculamos a m\u00e9dia das itera\u00e7\u00f5es.\n\n\n```python\nfrom sklearn.model_selection import KFold\n```\n\n\n```python\nNFOLDS = 5\nfolds = KFold(n_splits=NFOLDS)\n\ncolumns = X.columns\nscore = 0\n\n# criando os folds\nsplits = folds.split(X, y)\n\n# para cada fold, pegaremos os index de treino e index de valida\u00e7\u00e3o\nfor fold_n, (train_index, valid_index) in enumerate(splits):\n X_train, X_valid = X[columns].iloc[train_index], X[columns].iloc[valid_index]\n y_train, y_valid = y.iloc[train_index], y.iloc[valid_index]\n\n model = DecisionTreeClassifier(max_depth=3)\n model.fit(X_train, y_train)\n \n y_pred_valid = model.predict(X_valid)\n print(f\"Fold {fold_n + 1} | Accuracy: {accuracy_score(y_valid, y_pred_valid)}\")\n \n score += accuracy_score(y_valid, y_pred_valid)\n \nprint(f\"\\nMean accuracy = {score/NFOLDS}\")\n```\n\n Fold 1 | Accuracy: 0.8435754189944135\n Fold 2 | Accuracy: 0.8089887640449438\n Fold 3 | Accuracy: 0.8258426966292135\n Fold 4 | Accuracy: 0.7696629213483146\n Fold 5 | Accuracy: 0.8651685393258427\n \n Mean accuracy = 0.8226476680685456\n\n\nA ideia \u00e9 utilizar a valida\u00e7\u00e3o para escolher o melhor modelo e seus respectivos par\u00e2metros.\n\n## Prevendo para o teste e submetendo no Kaggle\n\nChegou a hora de usar o modelo que criamos e prever a resposta para os dados que n\u00e3o temos resposta!\n\n\n```python\n# treinando o modelo com todos os dados\nmodel = DecisionTreeClassifier(max_depth=3)\nmodel.fit(X, y)\n```\n\n\n\n\n DecisionTreeClassifier(ccp_alpha=0.0, class_weight=None, criterion='gini',\n max_depth=3, max_features=None, max_leaf_nodes=None,\n min_impurity_decrease=0.0, min_impurity_split=None,\n min_samples_leaf=1, min_samples_split=2,\n min_weight_fraction_leaf=0.0, presort='deprecated',\n random_state=None, splitter='best')\n\n\n\n\n```python\n# usando o arquivo de exemplo para \nsub = pd.read_csv('gender_submission.csv')\n```\n\n\n```python\ntest_preds = model.predict(test)\nsub['Survived'] = test_preds\n```\n\n\n```python\nsub.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        PassengerIdSurvived
                                        08920
                                        18931
                                        28940
                                        38950
                                        48961
                                        \n
                                        \n\n\n\n\n```python\nsub.to_csv('submission.csv', index=False)\n```\n\nUma vez que voc\u00ea tem seu arquivo de predi\u00e7\u00f5es, basta seguir o tutorial [nessa p\u00e1gina](https://www.kaggle.com/c/titanic/overview) para aprender a submeter no site! E pronto!\n\n### *Parab\u00e9ns!!!*\n\nInfelizmente, voc\u00ea vai perceber que nossa pontua\u00e7\u00e3o ainda n\u00e3o foi muito alta. Na verdade, ela foi s\u00f3 um pouco melhor que a submiss\u00e3o fornecida como exemplo. Mas tudo bem, n\u00f3s ainda podemos melhorar muito! E isso nos leva ao nosso pen\u00faltimo t\u00f3pico...\n\n## \u00c9 hora de ser criativo e botar a m\u00e3o na massa!\n\nCrie um novo notebook e fa\u00e7a as coisas do seu jeito! Esse notebook serviu apenas para te ensinar algumas t\u00e9cnicas, mas existe muito mais coisas para fazer. Crie novas features, processe os dados de forma diferente, teste e aprenda.\n\nUma boa fonte de aprendizado \u00e9 a [sess\u00e3o de Notebooks](https://www.kaggle.com/c/titanic/notebooks) dessa competi\u00e7\u00e3o no Kaggle. L\u00e1, competidores v\u00e3o mostrar toda a sua solu\u00e7\u00e3o e voc\u00ea pode aprender muito com eles. N\u00e3o deixe de dar uma olhada.\n\n## Considera\u00e7\u00f5es finais\n\nEssa aula abordou diversos t\u00f3picos de pr\u00e9-processamento dos dados e valida\u00e7\u00e3o dos modelos, levando a uma solu\u00e7\u00e3o, n\u00e3o \u00f3tima, mas completa do dataset do Titanic. \u00c9 importante ter sempre em mente que cada tipo de modelo prefere os dados de uma certa forma quando realizando um pr\u00e9-processamento. As t\u00e9cnicas ensinadas aqui podem te ajudar a processar os dados pra v\u00e1rios modelos.\n\nEspero que essa aula tenha sido \u00fatil pra voc\u00ea. N\u00e3o deixe de nos dar seu feedback, n\u00f3s tamb\u00e9m estamos apredendo. Al\u00e9m disso, se voc\u00ea tiver qualquer d\u00favida, \u00e9 s\u00f3 chamar no Telegram ou no Slack!\n\nObrigado pela aten\u00e7\u00e3o!\n", "meta": {"hexsha": "0fbeb7d3eb4535726dffec40f5ab5bc64b18b921", "size": 90497, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Aula4/Pr\u00e9-processamento e valida\u00e7\u00e3o.ipynb", "max_stars_repo_name": "icmc-data/Intro-DS-2020.1", "max_stars_repo_head_hexsha": "9777cfdd25471a38cf74694e103f1bcff871389c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 71, "max_stars_repo_stars_event_min_datetime": "2020-02-28T23:17:12.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-09T20:05:26.000Z", "max_issues_repo_path": "Aula4/Pr\u00e9-processamento e valida\u00e7\u00e3o.ipynb", "max_issues_repo_name": "icmc-data/Intro-DS-2020.1", "max_issues_repo_head_hexsha": "9777cfdd25471a38cf74694e103f1bcff871389c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Aula4/Pr\u00e9-processamento e valida\u00e7\u00e3o.ipynb", "max_forks_repo_name": "icmc-data/Intro-DS-2020.1", "max_forks_repo_head_hexsha": "9777cfdd25471a38cf74694e103f1bcff871389c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 25, "max_forks_repo_forks_event_min_datetime": "2020-03-05T14:35:58.000Z", "max_forks_repo_forks_event_max_datetime": "2021-10-12T12:36:38.000Z", "avg_line_length": 30.8652796726, "max_line_length": 467, "alphanum_fraction": 0.4100909422, "converted": true, "num_tokens": 15948, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353744, "lm_q2_score": 0.7461389873857265, "lm_q1q2_score": 0.41658953818351246}} {"text": "## Custom time imput\n\nIn this session we want to demonstrate another feature how to use the TTM and our software, where we want to give special attention to the source term $S(x,t)$ in the coupled heat diffusion equation. \n\n\\begin{align}\nC_E \\cdot\\rho\\cdot\\partial_t T_E &= \\partial_x(k_E(T_E)\\partial_x T_E(x,t)) + G(T_L-T_E) + S(x,t) \\\\ \\nonumber\nC_L\\cdot\\rho\\cdot\\partial_t T_L &= \\partial_x(k_L(T_L)\\partial_x T_L(x,t)) + G(T_E-T_L)\n\\end{align}\n\n* We consider, in $S(x,t)$ the time grid of a THz pulse, from data obtained in the lab and map it on the required time grid of the simulation. \n* The spacial decay of the injected energy is computed via Lambert Beer\u00b4s law, i.e. exponential decay according to the optical penetration depth. \n* The workflow of this session is as follows:\n * Load the data from the `.txt` file [THz-data](https://github.com/udcm-su/heat-diffusion-1D/blob/master/Examples/THzPulse.txt) and pass it on to the `source()` class. (Here the pulse is a THz in time but in principle it can be anything.)\n * Choose an extra refined time grid around $t_0$ of the pulse to correctly capture the dynamics in time\n * Define a material stack with `addLayer(...)` (here [SRO|STO]).\n * Run the simulation and visualize the output dynamics.\n\n\nNot required but makes physical units more obvious: [SI-units](https://pypi.org/project/numericalunits/)\n\n\n```python\nfrom NTMpy import NTMpy as ntm\nfrom matplotlib import pyplot as plt\nimport numpy as np\nimport numericalunits as u\nu.reset_units('SI')\n```\n\n\n```python\n#SRO- Parameters\nlength_SRO = 40*u.nm\nn_SRO = 0.5j #Approximation\nk_SRO1 = 6*u.W/(u.m*u.K)\nk_SRO2 = 1*u.W/(u.m*u.K)\nC_SRO1 = lambda Te: 0.112*Te\nC_SRO2 = 450\nrho_SRO = 6500*u.kg/(u.m**3)\nG_SRO = 5e17*u.W/(u.m**3*u.K)\n#STO- Parameters\nlength_STO = 80*u.nm\nn_STO = 0.5j #Approximation\nk_STO1 = 12*u.W/(u.m*u.K)\nk_STO2 = 1*u.W/(u.m*u.K)\nC_STO1 = lambda Te: 0.025*Te\nC_STO2 = 730\nrho_STO = 5100*u.kg/(u.m**3)\nG_STO = 5e17*u.W/(u.m**3*u.K)\n```\n\n\n```python\n# Load data from the lab\nlabData = np.loadtxt('C:/Users/lUKAS/Documents/UDCM/THzPulse/THzPulse.txt')\nttime = labData[:,0]*u.ps\nampl = labData[:,1]\n#Define a source\ns = ntm.source()\ns.spaceprofile = \"LB\" \ns.timeprofile = \"Custom\"\ns.fluence = 5*u.mJ/u.cm**2 #Fluence of the laser\ns.t0 = 1.5*u.ps #Approximately the time of THz peak\ns.lambda_vac = 299800 #1THz in nm\ns.loadData = [ttime,ampl] #implementing the Data from the lab\ns.adjusted_grid = 'on' #Injecting more points in the time grid\ns.dt0 = 3*u.ps #t0-dt0 and t0+dt0 get more points\ns.extra_points = 600 #Amount of extra points\n```\n\n\n```python\n#Two temperatures are considered, electron and lattice\nsim = ntm.simulation(2,s)\n#add parameters for both layers and both systems\n#(lengt, refractive_index, conductivity [electron, lattice], heatCapacity [electron, lattice], density, linear Coupling) \n#SRO Layer\nsim.addLayer(length_SRO,n_SRO,[k_SRO1,k_SRO2],[C_SRO1,C_SRO2],rho_SRO,G_SRO) \n#STO Layer\nsim.addLayer(length_STO,n_STO,[k_STO1,k_STO2],[C_STO1,C_STO2],rho_STO,G_STO)\nsim.final_time = 12*u.ps\n#To get the raw output\n[spacegrid,timegrid,Tempmap] = sim.run() \n#Create a visual object where the simulation gets passed on \nv = ntm.visual(sim)\n```\n\n -----------------------------------------------------------\n No specific time constant has been indicated. \n The stability region has been calculated and an appropriate timestep has been chosen.\n Timestep = 2.79e-14 s\n -----------------------------------------------------------\n -----------------------------------------------------------\n A refined Grid around t0 has been applied\n -----------------------------------------------------------\n\n\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 867/867 [00:00<00:00, 2642.69it/s]\n\n\n -----------------------------------------------------------\n Heat diffusion in a coupled electron-lattice system has been simulated\n Eleapsed time in E.E.- loop: 0.34369754791259766\n -----------------------------------------------------------\n ------------------------------------------------------------\n The simulation object of the 2 temerature system has been passed on to the visual class.\n ------------------------------------------------------------\n -----------------------------------------------------------\n A refined Grid around t0 has been applied\n -----------------------------------------------------------\n\n\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 867/867 [00:00<00:00, 2283.25it/s]\n\n\n -----------------------------------------------------------\n Heat diffusion in a coupled electron-lattice system has been simulated\n Eleapsed time in E.E.- loop: 0.39531373977661133\n -----------------------------------------------------------\n\n\n\n```python\n#Output of v.source is the full matrix of the source(x,t)\nso = v.source()\n#Contour plots to depict heat map in space and time\nv.contour('1')# Electronic system\nv.contour('2')# Lattice system \n#Time dynamics of Temperature, averaged in space \n[tt,avTemp_vec] = v.average()\n#To show the refined time grid\nv.timegrid() \n```\n\nIn the plot, where the data gets averaged in space and we can see the dynamics in time, the characteristic shape of a THz- pulse can be seen.\n\nNote however, that formally the wavelength $\\lambda_{vac}$ has still be provided to the source input `s.lambda_vac = number` (in nm). \n\nThis is because the decay of the pulse intensity, or how much of the energy gets locally absorbed, with respect to space, is calculated via the refractive index. \n\nIn the example given, the refractive index is not correct, but an assumption, since this data is not easy to obtain in the THz region. \n", "meta": {"hexsha": "4529996ea88713e6af1104f444b1dc5aacd407fb", "size": 183272, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Examples/CostumTimePulse2.ipynb", "max_stars_repo_name": "VaSca92/NTMpy", "max_stars_repo_head_hexsha": "be78cde21c045eb20f46cdb027a6933155ec962a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2020-02-17T08:18:27.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-16T15:42:14.000Z", "max_issues_repo_path": "Examples/CostumTimePulse2.ipynb", "max_issues_repo_name": "VaSca92/NTMpy", "max_issues_repo_head_hexsha": "be78cde21c045eb20f46cdb027a6933155ec962a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 5, "max_issues_repo_issues_event_min_datetime": "2019-05-13T01:34:22.000Z", "max_issues_repo_issues_event_max_datetime": "2019-08-27T10:10:57.000Z", "max_forks_repo_path": "Examples/CostumTimePulse2.ipynb", "max_forks_repo_name": "VaSca92/NTMpy", "max_forks_repo_head_hexsha": "be78cde21c045eb20f46cdb027a6933155ec962a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2020-01-25T11:48:48.000Z", "max_forks_repo_forks_event_max_datetime": "2021-09-10T18:51:27.000Z", "avg_line_length": 627.6438356164, "max_line_length": 73364, "alphanum_fraction": 0.9434829106, "converted": true, "num_tokens": 1599, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7956581000631542, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.4164636403690688}} {"text": "\n\n\n# Standalone Fishbone-Moncrief C Code\n\nWe start with the NRPy+ expressions generated in the [Tutorial-FishboneMoncriefID](Tutorial-FishboneMoncriefID.ipynb), and output them to the C file \"FishboneMoncriefID/FMstandalone.h\".\n\nFurther, $\\Gamma = \\alpha u^0$ is given by (as shown [here](Tutorial-u0_smallb_Poynting-Cartesian.ipynb)):\n$$\n\\Gamma = \\alpha u^0 = \\sqrt{\\frac{1}{1 - \\gamma_{ij}v^i_{(n)}v^j_{(n)}}}.\n$$\n\n\n```python\nimport sympy as sp # SymPy: The Python computer algebra package upon which NRPy+ depends\nfrom outputC import lhrh # NRPy+: Core C code output module\nimport indexedexp as ixp # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support\nimport finite_difference as fin # NRPy+: Finite difference C code generation module\nimport FishboneMoncriefID.FishboneMoncriefID as fmid\n\n# Step 1: Set up the Fishbone-Moncrief initial data. This sets all the ID gridfunctions.\nfmid.FishboneMoncriefID(\"Spherical\")\n\ngammaDD = ixp.zerorank2()\n\nDIM = 3\nfor i in range(DIM):\n for j in range(DIM):\n if i<=j:\n gammaDD[i][j] = fmid.IDgammaDD[i][j]\n else:\n gammaDD[i][j] = fmid.IDgammaDD[j][i]\n\n# gamma_{ij} v^i_{(n)} v^j_{(n)}\nGammacontraction = sp.sympify(0)\nfor i in range(DIM):\n for j in range(DIM):\n Gammacontraction += gammaDD[i][j] * fmid.IDValencia3velocityU[i] * fmid.IDValencia3velocityU[j]\n\nGammafactor = sp.sqrt(1 / (1 - Gammacontraction))\n\n# -={ F-M quantities: Generate C code from expressions and output to file }=-\nFishboneMoncrief_to_print = [\\\n lhrh(lhs=\"alpha\",rhs=fmid.IDalpha),\\\n lhrh(lhs=\"betaU0\",rhs=fmid.IDbetaU[0]),\\\n lhrh(lhs=\"betaU1\",rhs=fmid.IDbetaU[1]),\\\n lhrh(lhs=\"betaU2\",rhs=fmid.IDbetaU[2]),\\\n lhrh(lhs=\"Gammafactor\",rhs=Gammafactor),\\\n lhrh(lhs=\"Gamma_times_ValenciavU0\",rhs=Gammafactor*fmid.IDValencia3velocityU[0]),\\\n lhrh(lhs=\"Gamma_times_ValenciavU1\",rhs=Gammafactor*fmid.IDValencia3velocityU[1]),\\\n lhrh(lhs=\"Gamma_times_ValenciavU2\",rhs=Gammafactor*fmid.IDValencia3velocityU[2]),\\\n lhrh(lhs=\"uKS4U1\",rhs=fmid.uKS4U[1]),\\\n lhrh(lhs=\"uKS4U2\",rhs=fmid.uKS4U[2]),\\\n lhrh(lhs=\"uKS4U3\",rhs=fmid.uKS4U[3]),\\\n lhrh(lhs=\"uBL4U1\",rhs=fmid.uBL4U[1]),\\\n lhrh(lhs=\"uBL4U2\",rhs=fmid.uBL4U[2]),\\\n lhrh(lhs=\"uBL4U3\",rhs=fmid.uBL4U[3])\n ]\n# print(fmid.uKS4U[3])\nfin.FD_outputC(\"FishboneMoncriefID/FM_standalone.h\",FishboneMoncrief_to_print,params=\"outCverbose=False,CSE_enable=False\")\n```\n\n Wrote to file \"FishboneMoncriefID/FM_standalone.h\"\n\n\n\n```python\n%%writefile FishboneMoncriefID/FM_standalone.c\n\n#include \"stdio.h\"\n#include \"stdlib.h\"\n#include \"math.h\"\n\nconst double a = 0.9375;\nconst double M = 1.0;\nconst double r_at_max_density = 12.0;\nconst double r_in = 6.0;\n\nint main(int argc, const char *argv[]) {\n\n // Step 0a: Read command-line input, error out if nonconformant\n double xx0,xx1,xx2;\n/*\n if(argc != 4) {\n printf(\"Error: Expected three command-line arguments: ./FM_standalone r theta phi\\n\");\n exit(1);\n }\n xx0 = strtod(argv[1],NULL);\n xx1 = strtod(argv[2],NULL);\n xx2 = strtod(argv[3],NULL);\n*/\n\n// printf(\"# Output: r,th,ph, alpha, betaU0, betaU1, betaU2, Gamma, Gamma*vValenciaU0, Gamma*vValenciaU1, Gamma*vValenciaU2\\n\");\n for(double xx0=1.6;xx0<50.0;xx0+=0.2) {\n xx1 = 1.56463634120e0; //M_PI/2.0;\n xx2 = 0.0;\n double alpha,betaU0,betaU1,betaU2,Gammafactor,Gamma_times_ValenciavU0,Gamma_times_ValenciavU1,Gamma_times_ValenciavU2;\n double uKS4U1,uKS4U2,uKS4U3,uBL4U1,uBL4U2,uBL4U3;\n#include \"FM_standalone.h\"\n if(xx0 < r_in) {\n Gammafactor = 1.0;\n Gamma_times_ValenciavU0 = Gamma_times_ValenciavU1 = Gamma_times_ValenciavU2 = 0.0;\n uKS4U1 = uKS4U2 = uKS4U3 = 0.0;\n uBL4U1 = uBL4U2 = uBL4U3 = 0.0;\n }\n printf(\"%e %e %e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e %.15e\\n\",\n xx0,xx1,xx2,\n alpha,betaU0,betaU1,betaU2,\n Gammafactor,\n Gamma_times_ValenciavU0, // util1(1) in FMtorus.f90; util(1,i,j,k) near the write statement\n Gamma_times_ValenciavU1, // util1(3) in FMtorus.f90.\n Gamma_times_ValenciavU2, // util1(2) in FMtorus.f90.\n uKS4U1,uKS4U2,uKS4U3,\n uBL4U1,uBL4U2,uBL4U3);\n }\n return 0;\n}\n```\n\n Writing FishboneMoncriefID/FM_standalone.c\n\n\n\n```python\n!gcc -O2 FishboneMoncriefID/FM_standalone.c -o FM_standalone -lm\n```\n\n\n```python\n!./FM_standalone > out.txt\n```\n\n\n```python\n%matplotlib inline\n\nimport matplotlib.pyplot as plt\nimport mpmath as mp\nimport csv\n\n# Download torus_cuts.csv:\nURL = \"http://astro.phys.wvu.edu/zetienne/torus_cuts.csv\"\noutfile = \"torus_cuts.csv\"\ntry:\n with open(outfile,\"w\") as file:\n file.write(urllib.request.urlopen(URL).read().decode(\"utf-8\"))\nexcept:\n try:\n with open(outfile,\"w\") as file:\n file.write(urllib.urlopen(URL).read().decode(\"utf-8\"))\n except:\n # If all else fails, hope wget does the job\n !wget -O $outfile $URL\n\ndef file_reader(filename,list_of_cols,delim=\" \"):\n with open(filename) as file:\n reader = csv.reader(file, delimiter=delim)\n data = list(zip(*reader))\n# print(data)\n # data is a tuple of strings. Tuples are immutable, and we need to perform math on\n # the data, so here we convert tuple to lists of floats:\n# data_output = [[sp.sympify(0) for i in range(len(list_of_cols))] for j in range(len(data[0]))]\n data_output = [[sp.sympify(0) for i in range(len(data[0]))] for j in range(len(list_of_cols))]\n for i in range(len(data[0])):\n for j in range(len(list_of_cols)):\n# print(i,j,data[list_of_cols[j]][i])\n data_output[j][i] = float(data[list_of_cols[j]][i])\n return data_output\n\nNRPy_data_output = file_reader('out.txt', [0,7,8,9,10])\nstd_data_output = file_reader('torus_cuts.csv',[0,4,1,3,2])\n\n\nylabels = ['Lorentz Gamma_{KS}=G','G*v^r_{KS,Val.}','G*v^{\\\\theta}_{KS,Val.}','G*v^{\\phi}_{KS,Val.}']\n\nfor i in range(len(ylabels)):\n # https://matplotlib.org/gallery/text_labels_and_annotations/legend.html#sphx-glr-gallery-text-labels-and-annotations-legend-py\n fig, ax = plt.subplots()\n plt.title(\"NRPy's FM solve with FMtorus.f90: \"+ylabels[i])\n plt.xlabel(\"r/M\")\n plt.ylabel(ylabels[i])\n ax.plot(NRPy_data_output[0], NRPy_data_output[i+1], 'k--', label='NRPyFMSolve')\n ax.plot(std_data_output[0], std_data_output[i+1], 'k-', label='FMtorus.f90')\n legend = ax.legend(loc='upper right', shadow=True, fontsize='x-large')\n legend.get_frame().set_facecolor('C1')\n plt.show()\n```\n", "meta": {"hexsha": "006d08f6a04845e0aa8c47ef356f897d461a2d4d", "size": 110088, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Tutorial-Start_to_Finish-FishboneMoncriefID_standalone.ipynb", "max_stars_repo_name": "ksible/nrpytutorial", "max_stars_repo_head_hexsha": "4ca6e9da22def2a9c9bcbcad75847fd1db159f4b", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Tutorial-Start_to_Finish-FishboneMoncriefID_standalone.ipynb", "max_issues_repo_name": "ksible/nrpytutorial", "max_issues_repo_head_hexsha": "4ca6e9da22def2a9c9bcbcad75847fd1db159f4b", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Tutorial-Start_to_Finish-FishboneMoncriefID_standalone.ipynb", "max_forks_repo_name": "ksible/nrpytutorial", "max_forks_repo_head_hexsha": "4ca6e9da22def2a9c9bcbcad75847fd1db159f4b", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 340.8297213622, "max_line_length": 27404, "alphanum_fraction": 0.9190556646, "converted": true, "num_tokens": 2227, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.4163661004819539}} {"text": "# Optimization of a State-to-State Transfer in a Two-Level-System\n\n\n```python\n# NBVAL_IGNORE_OUTPUT\n%load_ext watermark\nimport qutip\nimport numpy as np\nimport scipy\nimport matplotlib\nimport matplotlib.pylab as plt\nimport krotov\n%watermark -v --iversions\npi = np.pi\nsqrt = np.sqrt\nbasis = qutip.basis\ntensor = qutip.tensor\ncoherent = qutip.coherent\n```\n\n The watermark extension is already loaded. To reload it, use:\n %reload_ext watermark\n krotov 0.3.0\n numpy 1.15.4\n scipy 1.2.0\n matplotlib.pylab 1.15.4\n matplotlib 3.0.2\n qutip 4.3.1\n CPython 3.7.2\n IPython 7.2.0\n\n\n$\\newcommand{tr}[0]{\\operatorname{tr}}\n\\newcommand{diag}[0]{\\operatorname{diag}}\n\\newcommand{abs}[0]{\\operatorname{abs}}\n\\newcommand{pop}[0]{\\operatorname{pop}}\n\\newcommand{aux}[0]{\\text{aux}}\n\\newcommand{opt}[0]{\\text{opt}}\n\\newcommand{tgt}[0]{\\text{tgt}}\n\\newcommand{init}[0]{\\text{init}}\n\\newcommand{lab}[0]{\\text{lab}}\n\\newcommand{rwa}[0]{\\text{rwa}}\n\\newcommand{bra}[1]{\\langle#1\\vert}\n\\newcommand{ket}[1]{\\vert#1\\rangle}\n\\newcommand{Bra}[1]{\\left\\langle#1\\right\\vert}\n\\newcommand{Ket}[1]{\\left\\vert#1\\right\\rangle}\n\\newcommand{Braket}[2]{\\left\\langle #1\\vphantom{#2} \\mid\n#2\\vphantom{#1}\\right\\rangle}\n\\newcommand{op}[1]{\\hat{#1}}\n\\newcommand{Op}[1]{\\hat{#1}}\n\\newcommand{dd}[0]{\\,\\text{d}}\n\\newcommand{Liouville}[0]{\\mathcal{L}}\n\\newcommand{DynMap}[0]{\\mathcal{E}}\n\\newcommand{identity}[0]{\\mathbf{1}}\n\\newcommand{Norm}[1]{\\lVert#1\\rVert}\n\\newcommand{Abs}[1]{\\left\\vert#1\\right\\vert}\n\\newcommand{avg}[1]{\\langle#1\\rangle}\n\\newcommand{Avg}[1]{\\left\\langle#1\\right\\rangle}\n\\newcommand{AbsSq}[1]{\\left\\vert#1\\right\\vert^2}\n\\newcommand{Re}[0]{\\operatorname{Re}}\n\\newcommand{Im}[0]{\\operatorname{Im}}$\nThe purpose of this example is not to solve an especially interesting physical\nproblem but to give a rather simple example of how the package can be used in\norder to solve an optimization problem.\n\n## Define the Hamiltonian\n\nIn the\nfollowing the Hamiltonian, guess field and\nstates are defined.\n\nThe Hamiltonian\n$\\op{H}_{0} = - \\omega \\op{\\sigma}_{z}$\nrepresents a\nsimple qubit with energy\nlevel splitting $\\omega$ in the basis\n$\\{\\ket{0},\\ket{1}\\}$. The control\nfield\n$\\epsilon(t)$ is assumed to couple via\nthe\nHamiltonian $\\op{H}_{1}(t) =\n\\epsilon(t) \\op{\\sigma}_{x}$ to the qubit,\ni.e., the control\nfield effectively\ndrives\ntransitions between both qubit\nstates. For now, we initialize the control\nfield as constant.\n\n\n```python\nN = 12\n\u03b1 = 2\n```\n\n# Plotting functions\n\n\n```python\ndef plot_population(result):\n fig, ax = plt.subplots()\n ax.plot(result.times, result.expect[0], label='0')\n ax.plot(result.times, result.expect[1], label='1')\n ax.legend()\n ax.set_xlabel('time')\n ax.set_ylabel('population')\n plt.show(fig)\n\ndef plot_system(\u03c8):\n bl = qutip.Bloch()\n bl.add_states(\u03c8.ptrace(0))\n bl.show()\n qutip.visualization.plot_wigner_fock_distribution(\u03c8.ptrace(1))\ndef plot_resonator(\u03c8):\n fig, ax = plt.subplots(1,len(\u03c8), figsize=(3*len(\u03c8),3))\n for (\u03d5, axis) in zip(\u03c8, ax):\n qutip.visualization.plot_wigner(\u03d5.ptrace(1), fig=fig, ax=axis, alpha_max = 2*\u03b1)\n axis.axis_equal = True\n \ndef plot_cardinal(\u03c8):\n bl = qutip.Bloch()\n bl.vector_color = ['r','g','b','g','b','r']\n [bl.add_states(\u03d5.ptrace(0), 'vector') for \u03d5 in \u03c8]\n bl.show()\n\ndef plot_all(dyn, \u03c8):\n \u03c8_i = [g.states[0] for g in dyn]\n \u03c8_f = [g.states[-1] for g in dyn]\n \u03c8_t = [\u03d5[1] for \u03d5 in \u03c8]\n plot_cardinal(\u03c8_i)\n plot_resonator(\u03c8_i)\n plot_cardinal(\u03c8_t)\n plot_resonator(\u03c8_t)\n plot_cardinal(\u03c8_f)\n plot_resonator(\u03c8_f)\n```\n\n\n```python\ndef cat(N,\u03b1,P = 1):\n return (coherent(N,\u03b1) + P*coherent(N,-\u03b1)).unit()\ndef fid(result, target):\n return (np.abs((result.states[-1].dag()*target).full())**2)[0][0]\n```\n\n\n```python\nRi = qutip.operators.identity(N)\nSi = qutip.operators.identity(2)\n\n\u03c3_z = qutip.operators.sigmaz()\n\u03c3_y = qutip.operators.sigmay()\n\u03c3_x = qutip.operators.sigmax()\na = qutip.operators.destroy(N)\n\n\u03c3_z = qutip.tensor(\u03c3_z,Ri)\n\u03c3_y = qutip.tensor(\u03c3_y,Ri)\n\u03c3_x = qutip.tensor(\u03c3_x,Ri)\nb = qutip.tensor(qutip.operators.destroy(2),Ri)\na = qutip.tensor(Si,a)\nI = qutip.tensor(Si,Ri)\n\ncat_0 = cat(N,\u03b1)\ncat_1 = cat(N,\u03b1*1j)\n\ndef hamiltonian(\u03c9=1.0, ampl0=0.2):\n \"\"\"Two-level-system Hamiltonian\n \n Args:\n \u03c9 (float): energy separation of the qubit levels\n ampl0 (float): constant amplitude of the driving field\n \"\"\"\n \n K_r = 2*pi*0.45e-3 # Kerr res\n K_q = 2*pi*297e-3 # Kerr qubit 200-300 MHz\n K_r = 10\n K_q = 1\n \u03c9_r = 2.0 * 2 * pi # resonator frequency\n \u03c9_q = 3.0 * 2 * pi # qubit frequency\n \u03c7 = 0.025 * 2 * pi # parameter in the dispersive hamiltonian\n\n delta = abs(\u03c9_r - \u03c9_q) # detuning\n g = sqrt(delta * \u03c7) # coupling strength that is consistent with chi\n\n #H_occ = w_r*a.dag()*a + w_q*b.dag()*b\n H_occ = -\u03c9_q/2.0 * \u03c3_z + \u03c9_r* a.dag()*a\n use_dispersive = True\n use_kerr = True\n if use_dispersive:\n #H_coup = - chi_qr * a.dag()*a * b.dag()*b\n H_coup = \u03c7 * (a.dag() * a + I/2) * \u03c3_z\n else:\n #H_coup = g * (a.dag() * b + a * b.dag())\n H_coup = g * \u03c3_x *(a.dag() + a)\n if use_kerr:\n H_kerr = - K_r/2 * a.dag()**2 * a**2 - K_q/2 * b.dag()**2 * b**2\n else:\n H_kerr = tensor(qubit.operators.qzero(2), qubit.operators.qzero(N))\n \n H_d = H_occ + H_coup + H_kerr\n \n H_qr = (b + b.dag())\n H_qi = 1j*(b.dag() - b)\n H_rr = (a + a.dag())\n H_ri = 1j*(a.dag() - a)\n\n \u03f5_qr = lambda t, args: ampl0*np.cos(\u03c9_q*t)\n \u03f5_qi = lambda t, args: ampl0*np.sin(\u03c9_q*t)\n \u03f5_rr = lambda t, args: ampl0*np.cos(\u03c9_r*t)\n \u03f5_ri = lambda t, args: ampl0*np.sin(\u03c9_r*t)\n return [H_d, [H_qr, \u03f5_qr], [H_qi, \u03f5_qi], [H_rr, \u03f5_rr], [H_ri, \u03f5_ri]]\n\ndef coeffs_to_state(c,init = True):\n if init:\n \u03c8 = tensor((c[0]*basis(2,0) + c[1]*basis(2,1)).unit() , (basis(N,0)))\n else:\n \u03c8 = tensor((basis(2,0)) , (c[0]*cat_0 + c[1]*cat_1).unit())\n return \u03c8\n\ndef states(coeffs):\n return [[coeffs_to_state(c,True),coeffs_to_state(c,False)] for c in coeffs]\nH = hamiltonian()\ncoeffs = [(1,0), (1,-1), (1,1j), (1,1), (1,-1j), (0,1)]\n\u03c8 = states(coeffs)\n```\n\nThe projectors $\\op{P}_0 = \\ket{0}\\bra{0}$ and $\\op{P}_1 = \\ket{1}\\bra{1}$ are\nintroduced since they allow for calculating the\npopulation in the respective\nstates later on.\n\n\n```python\ndef proj(\u03c8):\n return \u03c8 * \u03c8.dag()\n```\n\n## Define the optimization target\n\nFirst we define the time grid of the\ndynamics, i.e., by taking the following\nvalues as an example, we define the\ninitial state to be at time $t=0$ and\nconsider a total propagation time of\n$T=5$. The entire time grid is divided into\n$n_{t}=500$ equidistant time steps.\n\n\n```python\nT = 20\nsteps = 1000\ntlist = np.linspace(0, T, steps)\n```\n\nNext, we define the optimization targets, which is technically a list of\nobjectives, but here it has just one entry defining a simple state-to-state\ntransfer\nfrom initial state $\\ket{\\Psi_{\\init}} = \\ket{0}$ to the target state\n$\\ket{\\Psi_{\\tgt}} = \\ket{1}$, which we want to reach at final time $T$. Note\nthat we also have to pass the Hamiltonian $\\op{H}(t)$ that determines the\ndynamics of\nthe system to the optimization objective.\n\n\n```python\nobjectives = [krotov.Objective(initial_state=\u03d5[0], target=\u03d5[1], H=H) for \u03d5 in \u03c8]\n```\n\nIn addition, we have to define and assign a shape function $S(t)$ for the update\nin each control iteration to each\ncontrol field that will be updated. This shape\nusually takes care of\nexperimental limits such as the necessity of finite ramps\nat the beginning and\nend of the control field or other conceivable limitations\nfor field shapes: wherever $S(t)$ is zero, the optimization will not change the\nvalue of the control from the original guess.\n\n\n```python\ndef S(t):\n \"\"\"Shape function for the field update\"\"\"\n return krotov.shapes.flattop(t, t_start=0, t_stop=T, t_rise=0.06*T, t_fall=0.06*T, func='sinsq')\n```\n\nAt this point, we also change the initial control field $\\epsilon_{0}(t)$ from a\nconstant to a shaped pulse that switches on smoothly from zero and again\nswitches off at the final time $T$. We re-use the shape function $S(t)$ that we\ndefined for the updates for this purpose (although generally, $S(t)$ for the\nupdates has nothing to with the shape of the control field).\n\n\n```python\ndef shape_field(\u03f5):\n \"\"\"Applies the shape function S(t) to the guess field\"\"\"\n \u03f5_shaped = lambda t, args: \u03f5(t, args)*S(t)\n return \u03f5_shaped\n\nfor H_i in H[1:]:\n H_i[1] = shape_field(H_i[1])\n\n```\n\nHaving defined the shape function $S(t)$ and having shaped the guess field, we\nnow tell the optimization to also use $S(t)$ as the update-shape for\n$\\epsilon_0(t)$. In addition, we have to choose `lambda_a` for each control\nfield. It controls the update magnitude of the respective field in each\niteration.\n\n\n```python\nopt_lambda = [.5]\npulse_options = {H_i[1]: dict(lambda_a=opt_lambda[0], shape=S) for H_i in H[1:]}\n```\n\nIt is convenient to introduce the function `print_fidelity`, which can be passed\nto the optimization procedure and will be called after each iteration and thus\nprovides additional feedback about the optimization progress.\n\n\n```python\ndef print_fidelity(**args):\n F_re = np.average(np.array(args['tau_vals']).real)\n print(\" F = {}\".format(F_re))\n #return F_re\n```\n\n## Simulate dynamics of the guess field\n\nBefore heading towards the optimization\nprocedure, we first simulate the\ndynamics under the guess field\n$\\epsilon_{0}(t)$.\n\n\n```python\ndef plot_pulse(pulse, tlist):\n fig, ax = plt.subplots(figsize=(15,4))\n if callable(pulse):\n pulse = np.array([pulse(t, args=None) for t in tlist])\n ax.plot(tlist, pulse)\n ax.set_xlabel('Time')\n ax.set_ylabel('Pulse amplitude')\n plt.show(fig)\n```\n\nThe following plot shows the guess field $\\epsilon_{0}(t)$, which is, as chosen\nabove, just a constant field (with a smooth switch-on and switch-off)\n\n\n```python\nfor H_i in H[1:]:\n plot_pulse(H_i[1], tlist)\n```\n\nThe next line solves the equation of motion for the defined objective, which\ncontains the initial state $\\ket{\\Psi_{\\init}}$ and the Hamiltonian $\\op{H}(t)$\ndefining its evolution.\n\n\n```python\nguess_dynamics = [ob.mesolve(tlist, progress_bar=True) for ob in objectives]\n```\n\n 10.0%. 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Est. time left: 00:00:00:00\n 80.0%. Run time: 1.85s. Est. time left: 00:00:00:00\n 90.0%. Run time: 2.08s. Est. time left: 00:00:00:00\n Total run time: 2.32s\n\n\nThe plot of the population dynamics shows that the guess field does not transfer\nthe initial state $\\ket{\\Psi_{\\init}} = \\ket{0}$ to the desired target state\n$\\ket{\\Psi_{\\tgt}} = \\ket{1}$.\n\n\n```python\nplot_all(guess_dynamics, \u03c8)\n```\n\n## Optimize\n\nIn the following we optimize the guess field $\\epsilon_{0}(t)$ such\nthat the intended state-to-state transfer $\\ket{\\Psi_{\\init}} \\rightarrow\n\\ket{\\Psi_{\\tgt}}$ is solved.\n\nThe cell below carries out the optimization. It\nrequires, besides the\npreviously\ndefined optimization `objectives`, information\nabout the\noptimization functional\n$F$ (via `chi_constructor`) and the\npropagation method that should be used. In\naddition, the number of total\niterations is required and, as an option, we pass\nan info-hook that after each\niteration combines a complete printout of the state\nof the optimization with the\n`print_fidelity` function defined above.\n\nHere, we\nchoose $F = F_{re}$ with\n\\begin{equation}\nF_{re}\n=\n\\Re\\Braket{\\Psi(T)}{\\Psi_{\\tgt}}\n\\end{equation}\n\nwith\n$\\ket{\\Psi(T)}$ the\nforward propagated state of $\\ket{\\Psi_{\\init}}$.\n\n\n```python\n# Reset results\noct_result = None\n```\n\n\n```python\niters = 10\nif oct_result is not None:\n iters = oct_result.iters[-1] + iters\n\noct_result = krotov.optimize_pulses(\n objectives,\n pulse_options=pulse_options,\n tlist=tlist,\n propagator=krotov.propagators.expm,\n chi_constructor=krotov.functionals.chis_re,\n info_hook=krotov.info_hooks.chain(\n krotov.info_hooks.print_table(J_T=krotov.functionals.F_ss),\n print_fidelity\n ),\n check_convergence=krotov.convergence.Or(\n krotov.convergence.value_below(1e-3, name='J_T'),\n krotov.convergence.delta_below(1e-5),\n #krotov.convergence.check_monotonic_error,\n ),\n iter_stop=iters,\n continue_from = oct_result\n)\n```\n\n iter. J_T \u2211\u222bg\u2090(t)dt J \u0394J_T \u0394J secs\n 0 8.63e-01 0.00e+00 8.63e-01 n/a n/a 24\n F = 0.9278502468677411\n 11 8.91e-01 1.57e-02 9.07e-01 2.84e-02 4.41e-02 58 **\n F = 0.9434210714137601\n 12 9.06e-01 7.82e-03 9.13e-01 1.45e-02 2.23e-02 53 **\n F = 0.9511976883671558\n 13 9.14e-01 4.33e-03 9.18e-01 8.04e-03 1.24e-02 53 **\n F = 0.9555267758081483\n 14 9.19e-01 2.68e-03 9.21e-01 4.99e-03 7.67e-03 53 **\n F = 0.9582163420552078\n 15 9.22e-01 1.74e-03 9.24e-01 3.26e-03 5.00e-03 55 **\n F = 0.9599627865037025\n 16 9.24e-01 1.15e-03 9.25e-01 2.18e-03 3.33e-03 58 **\n F = 0.9611222714810728\n 17 9.26e-01 7.85e-04 9.26e-01 1.49e-03 2.27e-03 53 **\n F = 0.9619121240358566\n 18 9.27e-01 5.52e-04 9.27e-01 1.05e-03 1.61e-03 52 **\n F = 0.9624691040196324\n 19 9.27e-01 4.04e-04 9.28e-01 7.76e-04 1.18e-03 52 **\n F = 0.9628777967301221\n\n\n\n```python\noct_result\n```\n\n## Simulate dynamics of the optimized field\n\nHaving obtained the optimized\ncontrol field, we can now\nplot it and calculate the\npopulation dynamics under\nthis field.\n\n\n```python\n[plot_pulse(c, tlist) for c in oct_result.optimized_controls]\n```\n\nIn contrast to the dynamics under the guess field, the optimized field indeed\ndrives the initial state $\\ket{\\Psi_{\\init}} = \\ket{0}$ to the desired target\nstate $\\ket{\\Psi_{\\tgt}} = \\ket{1}$.\n\n\n```python\nopt_dynamics = [ob.mesolve(tlist, progress_bar=True) for ob in oct_result.optimized_objectives]\n```\n\n\n```python\n#print(fid(opt_dynamics, \u03c8[0][1]))\nplot_all(opt_dynamics, \u03c8)\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "13fe37eda6dbcd7f8404748058b0f2483ae86a60", "size": 830070, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Archive/Other tests/Cat state encoding tests.ipynb", "max_stars_repo_name": "JohanWinther/cat-state-encoding", "max_stars_repo_head_hexsha": "3fa95c5c9d9d223e4b9fbc38fe5e27a46d0d12ef", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2020-02-10T01:53:29.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-13T09:23:40.000Z", "max_issues_repo_path": "Archive/Other tests/Cat state encoding tests.ipynb", "max_issues_repo_name": "JohanWinther/cat-state-encoding", "max_issues_repo_head_hexsha": "3fa95c5c9d9d223e4b9fbc38fe5e27a46d0d12ef", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Archive/Other tests/Cat state encoding tests.ipynb", "max_forks_repo_name": "JohanWinther/cat-state-encoding", "max_forks_repo_head_hexsha": "3fa95c5c9d9d223e4b9fbc38fe5e27a46d0d12ef", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-07-31T08:55:43.000Z", "max_forks_repo_forks_event_max_datetime": "2021-07-31T08:55:43.000Z", "avg_line_length": 741.7962466488, "max_line_length": 115232, "alphanum_fraction": 0.9509005265, "converted": true, "num_tokens": 6408, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300449389326, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.41636609309953704}} {"text": "```python\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nplt.style.use('classic')\n%matplotlib inline\n```\n\n# Class 9: Introduction to Business Cycle Data\n\nThe *business cycle* is the fluctuation of many macroeconomic quantities that last for about 1.5 to 8 years. Colloquially, the term refers to the alternating periods of expansion and contraction in the macroeconomy. Business cycle fluctuations are costly because they are associated with misallocations of capital and labor. Recessions are particularly painful for workers that become unemployed and for the families of workers who become unemployed. The costs of the business cycle have driven research into understanding the cause of the cycle. \n\nThe collective set of theories to explain the cycle is called *business cycle theory* and eventually we'll study and critique two competing theoretical perspectives. However, before we approach the theory, we should first uncover some empirical facts about the business cycle that models should be able to explain.\n\nIn this lecture, we will:\n\n1. Visualize the difference between the trend and cyclical components of GDP, consumption, investent, and hours.\n2. Compute the log-deviations from trend of GDP, consumption, investent, and hours.\n3. Compute summary statistics about the business cycle that models of the cycle should be able to explain.\n\n## Data\n\nThe file `rbc_data_actual_trend.csv`, available at https://github.com/letsgoexploring/econ126/raw/master/Data/Csv/rbc_data_actual_trend.csv, contains actual and trend data for real GDP per capita, real consumption per capita, real investment per capita, real physical capital per capita, TFP, and hours per capita at quarterly frequency. The GDP, consumption, investment, and capital data are in terms of 2012 dollars. Hours is measured as an index with the value in October 2012 set to 100. All of the data are *real* quantities. That is, there are no *nominal* quantities like money or inflation or a nominal interest rate. The reason is that the first theory that we will encounter is called *real business cycle* or RBC theory and, in that theory, there is no place for nominal quantities. RBC theory seeks to explain fluctuations in real quantities as being primarily due to TFP shocks; i.e., shocks to the production function.\n\n\n```python\n# Read business_cycle_data_actual_trend.csv into a Pandas DataFrame with the first column set as the index and parse_dates=True\ndf = pd.read_csv('https://github.com/letsgoexploring/econ126/raw/master/Data/Csv/rbc_data_actual_trend.csv',index_col=0,parse_dates=True)\n\n# Print the last five rows of the data\ndf.tail()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        gdpgdp_trendconsumptionconsumption_trendinvestmentinvestment_trendhourshours_trendcapitalcapital_trendtfptfp_trend
                                        2017-07-0170.96282570.91749748.39680848.31024612.36776112.448772104.125352104.33578281.23170380.97091426.39079326.369009
                                        2017-10-0171.20289671.16585048.75391448.50515012.35594512.525952104.740753104.60177381.54043681.25799626.34384126.384892
                                        2018-01-0171.31675571.41605448.67932748.70098112.61068112.603944104.926059104.86864981.70519181.54702526.33705526.401009
                                        2018-04-0171.90172471.66778848.87734848.89771012.60897912.682658105.232911105.13629982.08041681.83758826.46026226.417303
                                        2018-07-0172.32625671.92066149.19196649.09529412.99163612.762006105.431597105.40464982.41364882.12936326.54620726.433677
                                        \n
                                        \n\n\n\n\n```python\n# Construct a 2x2 grid of plots of GDP, consumption, investment, and hours. CELL PROVIDED\nbase_year = '2012'\nfig = plt.figure(figsize=(12,8))\n\nax1 = fig.add_subplot(2,2,1)\nax1.plot(df.gdp,'-',lw=3,alpha = 0.7)\nax1.grid()\nax1.set_title('GDP per capita')\nax1.set_ylabel('Thousands of '+base_year+' dollars')\n\nax2 = fig.add_subplot(2,2,2)\nax2.plot(df.consumption,'-',lw=3,alpha = 0.7)\nax2.grid()\nax2.set_title('Consumption per capita')\nax2.set_ylabel('Thousands of '+base_year+' dollars')\n\nax3 = fig.add_subplot(2,2,3)\nax3.plot(df.investment,'-',lw=3,alpha = 0.7)\nax3.grid()\nax3.set_title('Investment per capita')\nax3.set_ylabel('Thousands of '+base_year+' dollars')\n\nax4 = fig.add_subplot(2,2,4)\nax4.plot(df.hours,'-',lw=3,alpha = 0.7)\nax4.grid()\nax4.set_title('Hours per capita')\nax4.set_ylabel('Index ('+base_year+'=100)')\n\nfig.tight_layout()\n```\n\n## Cycles and Trends\n\nBusiness cycle theory is tested against data. However, data on the business cycle data is not readily available and must be constructed. A time series $X_t$ can be decomposed into a trend component $X_t^{trend}$ and a cyclical component $X_t^{cycle}$ such that:\n\n\\begin{align}\nX_t & = X_t^{trend} + X_t^{cycle}. \\tag{1}\n\\end{align}\n\nIn equation (1), $X_t^{trend}$ is the long-run value about which $X_t$ fluctuates. $X^{cycle}_t$ is the amount by which $X_t$ excedes its trend. The process for decomposing a series into trend and cyclical components is called *filtering* and is more technical than we want to get into. We'll take for granted that such procedures exist.\n\nOften times, it's useful to express the cyclical component of a time series as the difference between the (natural) log of the series and the log of the trend:\n\n\\begin{align}\n\\hat{x}_t & = \\log\\left(X_t\\right) - \\log\\left(X_t^{trend}\\right) \\approx\\frac{X_t-X_t^{trend}}{X_t^{trend}}\n\\end{align} \n\nThe log-deviation from trend is approximately equal to the percent deviation of the series from trend (divided by 100).\n\n### GDP\n\n\n```python\n# Construct a plot of real GDP with its trend with:\n# 1. Actual line: blue with lw=1, alpha=0.7, label = 'actual'\n# 2. Trend line: red with lw=3, alpha=0.7, label = 'trend'\nplt.plot(df.gdp,'-',lw=1,alpha = 0.7,label='actual')\nplt.plot(df.gdp_trend,'r-',lw=3,alpha = 0.7,label='trend')\nplt.ylabel('Thousands of '+base_year+' dollars per person')\nplt.title('GDP per capita')\nplt.legend(loc='lower right',ncol=2)\nplt.grid()\n```\n\n\n```python\n# Create a new column called gdp_cycle equal to the difference between acual and trend GDP\ndf['gdp_cycle'] = df['gdp'] - df['gdp_trend']\n\n# Create a new column called gdp_cycle_dev equal to the log difference between actual GDP and trend GDP:\ndf['gdp_cycle_dev'] = np.log(df['gdp']) - np.log(df['gdp_trend'])\n\n# Plot the log deviation of GDP from its trend (times 100)\nplt.plot(df.gdp_cycle_dev*100,'b-',lw=3,alpha = 0.7)\nplt.ylabel('Percent')\nplt.title('GDP: Percent deviation from trend')\nplt.grid()\n```\n\n### Consumption, Investment, and Hours\n\n\n```python\n# Create three new columns called cons_cycle, invest_cycle, and hours_cycle equal to the cyclical components of the \n# respective series\ndf['cons_cycle'] = df['consumption'] - df['consumption_trend']\ndf['invest_cycle'] = df['investment'] - df['investment_trend']\ndf['hours_cycle'] = df['hours'] - df['hours_trend']\n\n\n# Create a new column called cons_cycle_dev, invest_cycle_dev, and hours_cycle_dev equal to the log difference between \n# the actual and trend values of the respective series:\ndf['cons_cycle_dev'] = np.log(df['consumption']) - np.log(df['consumption_trend'])\ndf['invest_cycle_dev'] = np.log(df['investment']) - np.log(df['investment_trend'])\ndf['hours_cycle_dev'] = np.log(df['hours']) - np.log(df['hours_trend'])\n\n```\n\n\n```python\n# Construct a plot of consumption with its trend\n# 1. Actual line: blue with lw=1, alpha=0.7, label = 'actual'\n# 2. Trend line: red with lw=3, alpha=0.7, label = 'trend'\nplt.plot(df.consumption,'-',lw=1,alpha = 0.7,label='actual')\nplt.plot(df.consumption_trend,'r-',lw=3,alpha = 0.7,label='trend')\nplt.ylabel('Thousands of '+base_year+' dollars per person')\nplt.title('Consumption per capita')\nplt.legend(loc='lower right',ncol=2)\nplt.grid()\n```\n\n\n```python\n# Plot the log deviation of consumption from its trend (times 100)\nplt.plot(df.cons_cycle_dev*100,'b-',lw=3,alpha = 0.7)\nplt.ylabel('Percent')\nplt.title('Consumption: Percent deviation from trend')\nplt.grid()\n```\n\n\n```python\n# Construct a plot of investment with its trend\n# 1. Actual line: blue with lw=1, alpha=0.7, label = 'actual'\n# 2. Trend line: red with lw=3, alpha=0.7, label = 'trend'\nplt.plot(df.investment,'-',lw=1,alpha = 0.7,label='actual')\nplt.plot(df.investment_trend,'r-',lw=3,alpha = 0.7,label='trend')\nplt.ylabel('Thousands of '+base_year+' dollars per person')\nplt.title('Investment per capita')\nplt.legend(loc='lower right',ncol=2)\nplt.grid()\n```\n\n\n```python\n# Plot the log deviation of investment from its trend (times 100)\nplt.plot(df.invest_cycle_dev*100,'b-',lw=3,alpha = 0.7)\nplt.ylabel('Percent')\nplt.title('Investment: Percent deviation from trend')\nplt.grid()\n```\n\n\n```python\n# Construct a plot of hours with its trend\n# 1. Actual line: blue with lw=1, alpha=0.7, label = 'actual'\n# 2. Trend line: red with lw=3, alpha=0.7, label = 'trend'\nplt.plot(df.hours,'-',lw=1,alpha = 0.7,label='actual')\nplt.plot(df.hours_trend,'r-',lw=3,alpha = 0.7,label='trend')\nplt.ylabel('Index: ('+base_year+'=100)')\nplt.title('Hours per capita')\nplt.legend(loc='lower left',ncol=2)\nplt.grid()\n```\n\n\n```python\n# Plot the log deviation of hours from its trend (times 100)\nplt.plot(df.hours_cycle_dev*100,'b-',lw=3,alpha = 0.7)\nplt.ylabel('Percent')\nplt.title('Hours: Percent deviation from trend')\nplt.grid()\n```\n\n## US Business Cycle Statistics\n\n\n```python\n# Create a new variable called df_cycle that is a DataFrame with columns columns gdp_cycle_dev, cons_cycle_dev, \n# invest_cycle_dev, and hours_cycle_dev from df.\ndf_cycle = df[['gdp_cycle_dev', 'cons_cycle_dev', 'invest_cycle_dev', 'hours_cycle_dev']]\n\n# Print the first five rows of df_cycle\nprint(df_cycle.head())\n```\n\n gdp_cycle_dev cons_cycle_dev invest_cycle_dev hours_cycle_dev\n 1948-01-01 0.021966 0.007885 0.050324 0.027210\n 1948-04-01 0.025403 0.011468 0.096154 0.023054\n 1948-07-01 0.017087 -0.000568 0.109654 0.025447\n 1948-10-01 0.005779 -0.007582 0.074770 0.012991\n 1949-01-01 -0.020577 -0.017215 -0.103118 -0.007194\n\n\n\n```python\n# Use the DataFrame method .mean() to find the average values of the gdp_cycle_dev, cons_cycle_dev, invest_cycle_dev, \n# and hours_cycle_dev columns\ndf_cycle.mean()\n```\n\n\n\n\n gdp_cycle_dev -3.256602e-14\n cons_cycle_dev -7.624368e-14\n invest_cycle_dev -4.837042e-14\n hours_cycle_dev -1.256004e-14\n dtype: float64\n\n\n\n\n```python\n# Use the DataFrame method .std() to find the standard deviations of the gdp_cycle_dev, cons_cycle_dev, invest_cycle_dev, \n# and hours_cycle_dev columns\ndf_cycle.std()\n```\n\n\n\n\n gdp_cycle_dev 0.016202\n cons_cycle_dev 0.011571\n invest_cycle_dev 0.074918\n hours_cycle_dev 0.018924\n dtype: float64\n\n\n\n\n```python\n# Use the DataFrame method .corr() to find the coefficients of correlation among the gdp_cycle_dev, cons_cycle_dev, \n# invest_cycle_dev, and hours_cycle_dev columns\ndf_cycle.corr()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        gdp_cycle_devcons_cycle_devinvest_cycle_devhours_cycle_dev
                                        gdp_cycle_dev1.0000000.7945740.8477770.875156
                                        cons_cycle_dev0.7945741.0000000.6817630.705705
                                        invest_cycle_dev0.8477770.6817631.0000000.789560
                                        hours_cycle_dev0.8751560.7057050.7895601.000000
                                        \n
                                        \n\n\n\n**Questions**\n\n1. Which quantity varies the most over the business cycle?\n2. Which quantity varies the least over the business cycle?\n3. Which quantity is most correlated with GDP over the business cycle?\n\n**Answers**\n\n1. Investment fluctuates the most over the business cycle.\n2. Consumption fluctuates the least over the business cycle.\n3. Hours is the quantity that is most correlated with GDP over the cycle. Since the capital stock changes slowly over time, large fluctuations due primarily to large fluctuations in employment.\n\n\n```python\n# Plot the cyclical components of GDP, consumption, investment, and hours (all times 100) on the same set of axes\nfig = plt.figure(figsize=(12,4))\nax = fig.add_subplot(1,1,1)\n(df_cycle*100).plot(ax = ax,legend=False,lw=3,alpha=0.7)\nax.legend(loc='center left', bbox_to_anchor=(1, 0.5))\nax.set_ylabel('Percent')\nax.set_title('US Business cycle data')\nax.grid()\n```\n", "meta": {"hexsha": "2eb1d4fd9f163662e278692ce2bdc7db9c8a3878", "size": 419015, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Lecture Notebooks/Econ126_Class_09.ipynb", "max_stars_repo_name": "pmezap/computational-macroeconomics", "max_stars_repo_head_hexsha": "b703f46176bb16e712badf752784f8a7b996cdb1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 30, "max_stars_repo_stars_event_min_datetime": "2020-02-29T06:09:03.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-25T13:14:13.000Z", "max_issues_repo_path": "Lecture Notebooks/Econ126_Class_09.ipynb", "max_issues_repo_name": "letsgoexploring/computational-macroeconomics", "max_issues_repo_head_hexsha": "b703f46176bb16e712badf752784f8a7b996cdb1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Lecture Notebooks/Econ126_Class_09.ipynb", "max_forks_repo_name": "letsgoexploring/computational-macroeconomics", "max_forks_repo_head_hexsha": "b703f46176bb16e712badf752784f8a7b996cdb1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 18, "max_forks_repo_forks_event_min_datetime": "2019-09-24T07:48:49.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-22T21:36:30.000Z", "avg_line_length": 531.7449238579, "max_line_length": 84504, "alphanum_fraction": 0.9327899956, "converted": true, "num_tokens": 4419, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.7025300449389326, "lm_q1q2_score": 0.41636609309953704}} {"text": "# Python refresher: Part II\n\nLast week, we have seen some basic python, which you should be familiar with by now. This includes comments, data types and some of the operators that can be used with them, and other tids and bits.\n\nThis week, we're going to look at some slightly more complex topics that are needed to work with the code in the second section of the course. Specifically, some concepts that we need to use Piantadosi's LOTlib3 library.\n\nI expect the topics in this lab will be familiar to different extents to different people. Therefore, rather than explaining everything in huge detail, it's easier if you just ask if you get stuck on a point, and we explore it more in detail.\n\n## Importing and using libraries\n\n### What is a library?\n\nRoughly, a library is a collection of useful stuff. This can include functions / variables / classes / other stuff. For instance, to do maths in python we usually import the 'numpy' library, to do plotting we use 'matplotlib' or 'seaborn', etc. LOTlib3, which we will use in the second half of this course, is one such library.\n\nUsing libraries allows us to not rewrite stuff that someone else has already written.\n\n### How do we import and use libraries?\n\nTo import a library we write `import` follows by the name of the library, as follows:\n```python\nimport numpy\n```\n\nTo use the stuff in the library we use the library name followed by a dot followed by the name of what we want from the library. For instance:\n```python\nnumpy.pi\n```\nto get the number $\\pi$.\n\nHowever, often we do not want to use the full name of the library, but rather some abbreviation. In this case, we can write as follows:\n```python\nimport numpy as np\n```\n\n\n```python\nimport numpy as np\n```\n\nand then we can just write:\n```python\nnp.pi\n```\nwhich is shorter.\n\n\n```python\nnp.pi\n```\n\n\n\n\n 3.141592653589793\n\n\n\nIf we do not want the full library but rather just one thing from it, we can write:\n```python\nfrom numpy import pi\n```\nand then we can just use `pi` directly without needing the `np.`. \n\n\n```python\nfrom numpy import pi\n```\n\n\n```python\npi\n```\n\n\n\n\n 3.141592653589793\n\n\n\n## Functions\n\n### Motivation\n\nOften when programming you repeatedly need some code that 'does the same thing', e.g. multiply two numbers, or take a string and transform it in some way. You can think of a function in python as a way of dealing with this without having to repeatedly write the same code (this is not all that a function is good for, but it is one thing that functions are good for). Basically, you give a name to the 'piece of code' that you repeatedly need, and use the name instead of the code itself.\n\nLet's look at an example to understand this better. Suppose you have to write some code that takes a string, changes the first letter to 'A', and returns the first 10 characters. You can write it as follows:\n\n\n```python\nstring = 'hello world'\nstring = 'A' + string[1:]\nstring = string[:10]\nprint(string)\n```\n\n Aello worl\n\n\nNow suppose that you need to perform the procedure above for each element in a list of strings. You can do this by putting the code above in a loop:\n\n\n```python\nstrings = ['hello world', 'conversation world', 'goodbye world']\nresults = []\nfor string in strings:\n string = 'A' + string[1:]\n string = string[:10]\n results.append(string)\nprint(results)\n```\n\n ['Aello worl', 'Aonversati', 'Aoodbye wo']\n\n\nBut now suppose that you don't know a priori whether you're getting a list or a single word, and you need to adapt. One natural way to do it would be:\n\n\n```python\nvariable = 'bla bla bla'\n\n# check if the variable is a list\nif type(variable) is list:\n for string in variable:\n string = 'A' + string[1:]\n string = string[:10]\n results.append(string)\n print(results)\nelse:\n variable = 'A' + variable[1:]\n variable = variable[:10]\n print(variable)\n```\n\n Ala bla bl\n\n\nBut notes that some lines are repeated, namely\n```python\nstring = 'A' + string[1:]\nstring = string[:10]\n```\nWe would like a way to only write them once and use them in two places. A natural solution to this is a function that takes a string and performs the transformation we want.\n\n### Standard function definition in python\n\nTo define a function in python we do the following:\n- start with the keyword `def`\n- followed by the name of the function\n- followed by an open `(` \n- followed by zero or more arguments divided by commas\n- followed by `):`\n- followed by one or more lines at the next indentation level \n- In the points where you want a function to return a value, you can optionally put a `return` statement followed by the value you want the function to return.\n\nFor instance:\n```python\ndef name(argument1, argument2):\n first line in function\n second line in function\n return value\n```\n\nTo solve the problem above we could use:\n\n\n```python\ndef transform_string(string_to_transform):\n string_to_transform = 'A' + string_to_transform[1:]\n string_to_transform = string_to_transform[:10]\n return string_to_transform\n```\n\n### Calling a function\n\nThen, we can rewrite the above as:\n\n\n```python\nvariable = 'bla bla bla'\n\n# check if the variable is a list\nif type(variable) is list:\n for string in variable:\n results.append(transform_string(string))\n print(results)\nelse:\n print(transform_string(string))\n```\n\n Aoodbye wo\n\n\nBasically what we have done it _called_ a function on a certain input (`transform_string(string)`), which is the same as running the input through the code in the function and getting the return values of the function.\n\nNow we only repeat one word rather than several lines, and also if it turns out we want to change the transformation later, we only need to do it once rather than changing it independently multiple times!\n\n### $\\lambda$ ('lambda') functions\n\nAt this point it is worth mentioning that there is another notation for defining functions that doesn't use the `def` keyword. Namely, we write `lambda` followed by a space, zero or more comma-separated arguments, a colon, and a single statement.\n\nFor example, suppose we want to sum two variables. We could define a function that does it as follows:\n\n\n```python\nsum_function = lambda x, y: x+y\n\nprint(sum_function(2, 3))\n```\n\n 5\n\n\nOne case where this might be useful is in conjunction with `map`, which is an operation that takes two arguments, a function and a list, and applies the function to each element of the list. E.g., in the following example:\n\n\n```python\nlist(map(\n lambda x: x[0]+x[1], \n [[1,2], [4, 4], [1,3]]\n))\n```\n\n\n\n\n [3, 8, 4]\n\n\n\nLambda functions are usually a slightly more advanced topic, but they'll come up a lot in the rest of the course, so it's good to start getting used to them already.\n\n### Recursive functions\n\nThis is a slighlty mindbending concept: functions can _call themselves_. This means that inside a function $f$, $f$ itself can be called and the return value can be used inside $f$. Of course, this can easily lead to functions that never terminate, but rather keep calling themselves, forever increasing the level of nested calls. For instance, try to think what is happening here:\n\n\n```python\ndef bad_recursive_function(x):\n y = x+1\n bad_recursive_function(y)\n \nbad_recursive_function(1)\n```\n\nThankfully, python has a way of preventing the function running infinitely and automatically raises an error after a certain number of nested calls. You can see it here as `RecursionError: maximum recursion depth exceeded`.\n\nUsually, recursive functions are conceptualized as follows:\n- There is one or more 'base case', i.e. an input for which the function simply calculates a return value without calling the function itself.\n- There are 'complex' cases where the input to the function is somehow 'simplified' and the function is called with the simplified input. Eventually after enough 'simplification' steps the input reduces to a base case, and recursion stops.\n\nA classic case of recursion is the following formulation of the $n$th Fibonacci numbers:\n- If $n$ is 0 or 1 (i.e., the first or second Fibonacci number), return 1\n- If $n$ is greater than 1, return the $n$th-2 Fibonacci number + the $n$th-1 Fibonacci number\n\n\n```python\ndef calculate_fibonacci(n):\n # there are two base cases here!\n if n==0 or n==1:\n # in both base cases, just return 1\n return 1\n else:\n # otherwise, follow the recursive rule\n return calculate_fibonacci(n-2) + calculate_fibonacci(n-1)\n\nfor i in range(30):\n print(calculate_fibonacci(i), end=',')\n```\n\n 1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,28657,46368,75025,121393,196418,317811,514229,832040,\n\nThere is a lot to say about recursive functions and how they work, but for now try to really understand how the one above works. If you really understand that simple case, other ones are going to be easy.\n\n## Functions as objects\n\nWe have refreshed briefly how to define functions above. Functions in python are _first class citizens_, meaning that they can be treated as objects themselves: passed as arguments of functions, destroyed, transformed in all the same ways as other objects. \n\nLet's see a simple example. Suppose that you have a function $f$ that takes an integer and returns an integer, but you don't know exactly what it does to that integer. But whatever it does, you want another function that does the same thing as $f$ but then adds 3 to the result. \n\nWell, we can define a function $g$ that takes a function $f$ _as an argument_ and returns another function from integer to integer, that first calls $f$ on the integer and then adds three:\n\n\n```python\ndef modify_f(func):\n # define a function that runs func and then adds 3\n def g(n):\n return func(n) + 3\n # return that function\n return g\n```\n\nLet's see `modify_f` in action:\n\n\n```python\n# define an f that take an int and returns an int\n# (namely, the argument multiplied by two)\nf = lambda x: x * 2\n\n# get a modified function by running\n# g through modify_f\nmodified_f = modify_f(f)\n\n# call the modified function\n# on an example\nmodified_f(4)\n```\n\n\n\n\n 11\n\n\n\nThe point of this example is that `modify_f` takes _a function_ as an input and return _a function_ as an output. You can use manipulate functions like that!\n\n## Recursion without recursion (note: more difficult!)\n\n(For more explanation on the material in this section, see [this page](https://www.lambdacircle.com/the-y-and-z-combinators-in-python/))\n\nNote that things become complicated easily when we work with functions that take other functions as inputs. In those cases, you can define functions that _take themselves_ (as well as other arguments). This actually allows us to do surprising things, like have the same power as recursion without using recursion directly, i.e., without calling a function in its own definition.\n\nFor instance, consider the following function, which as you can see is not strictly speaking recursive:\n\n\n```python\ndef strange_fibonacci(f):\n # takes a function f and\n # defines a new function \n # whose behaviour depends on\n # the input function f\n def g(n):\n if n == 0 or n == 1:\n return 1\n else:\n return f(f)(n-1) + f(f)(n-2)\n return g\n```\n\nAnd now consider the following function which in a way encodes the recursion bit (but without being explicitly recursive: it's not calling itself in its own definition):\n\n\n```python\n# takes a function, calls the function on itself\nrecursive_function = lambda f: f(f)\n```\n\nWhat `recursive_function` does is call a function on itself.\n\nWhen called with `strange_fibonacci` as an argument, it will effectively calculate `strange_fibonaccy(strange_fibonacci)`, and this will return the following function:\n```python\ndef g(n):\n if n == 0 or n == 1:\n return 1\n else:\n return (\n strange_fibonacci(strange_fibonacci)(n-1) + \n strange_fibonacci(strange_fibonacci)(n-2)\n )\n```\nwhich, when called with a specific integer, will define new `g`s and call them in a nested way.\n\nLet's test it with a specific number:\n\n\n```python\nrecursive_function(strange_fibonacci)(10)\n```\n\n\n\n\n 89\n\n\n\nCan you figure out _exactly_ what is happening here? It's a bit mindbending.\n\nWe can also get rid of the `f(f)` inside `strange_fibonacci`, since it still kind of feels like a type of recursion. To do that, we define the following function:\n\n\n```python\napply_inside = lambda f: lambda g: f(lambda y: g(g)(y))\n```\n\nHow does `apply_inside` work?\n\n- You pass it some function $f$ ($f$ takes a function as argument), and it returns another function, let's call it $h$.\n- $h$ (i.e., the return value of `apply_inside`) is defined as `lambda g: f(lambda y: g(g)(y)`, and so it is a function:\n - from a function $g$ \n - to the result of calling $f$ (the original argument) on `lambda y: g(g)(y)`\n\nThis might become clearer with an example. Let's define another version of our strange fibonacci function, this time without the `f(f)` bit:\n\n\n```python\ndef strange_fibonacci(f):\n # takes a function f and\n # defines a new function \n # whose behaviour depends on\n # the input function f\n def g(n):\n if n == 0 or n == 1:\n return 1\n else:\n return f(n-1) + f(n-2)\n return g\n```\n\nNow consider `apply_inside(strange_fibonacci)`. How does it work? Well, we know from the definition of `apply_inside` that it will have value:\n\n`apply_inside(strange_fibonacci)` = `lambda g: strange_fibonacci(lambda y: g(g)(y))`\n\nNote that now the argument of `strange_fibonacci` is `lambda y: g(g)(y)`, which is a function from a number to a function applied to itself and then to the number. Within `strange_fibonacci`, the input function appears in the line `return f(n-1) + f(n-2)`, which becomes:\n- `return (lambda y: g(g)(y))(n-1) + (lambda y: g(g)(y))(n-2)`\n\nand therefore:\n- `return g(g)(n-1) + g(g)(n-2)`\n\nAnd this is just a version of the `f(f)` bit we had at the beginning, which we have now successfully isolated into its own function (apply inside)!\n\nLet's make sure this works:\n\n\n```python\nrecursive_function(apply_inside(strange_fibonacci))(10)\n```\n\n\n\n\n 89\n\n\n\nThis section was a bit mental gymnastics to give you an idea of how messy things can get when higher-order functions are involved (i.e., function that take functions as arguments). We will see a lot of higher-order functions in the rest of the semester, so try to become comfortable with them!\n\n## Plotting in Matplotlib\n\nSince we'll be want to plot things in the second half of the course, it makes sense to introduce some simple cases now. We'll be using the most popular plotting library for python, `matplotlib`. We usually import it as follows:\n\n\n```python\nimport matplotlib.pyplot as plt\n```\n\nUsually, we'll initialize a new plot with the following line:\n```python\nfig, axes = plt.subplots()\n```\nWhat `plt.subplots()` does is create a new figure object and a new `axes` object. The figure object is the whole figure, which can contain one or more axes. If you are unfamiliar with the `plt.subplots` interface, have a look [here](https://matplotlib.org/3.5.0/api/_as_gen/matplotlib.pyplot.subplots.html). \n\nThe axes are objects that have a bunch of methods, corresponding e.g. to different types of plots that can be called on them. Let's see an example of the `scatter` method of axes, which is used to make scatter plots:\n\n\n```python\nfig, axes = plt.subplots()\n\n# create some new data to plot\n# don't worry too much if you don't understand this line\nx_scatter, y_scatter = np.random.rand(2,10)\n\n# create a scatterplot with the given\n# x and y coordinates\naxes.scatter(x_scatter, y_scatter)\n\n# show the plot\nplt.show()\n```\n\nAnother important function is to draw line plots, called simply `plot`:\n\n\n```python\nfig, axes = plt.subplots()\n\n# create some new data to plot\n# don't worry too much if you don't understand this line\nx_line = y_line = np.arange(10)\n\n# create a line plot with the given\n# x and y coordinates\naxes.plot(x_line, y_line)\n\n# show the plot\nplt.show()\n```\n\nSince we will be dealing with discrete variables a lot, it will be useful to also draw barplot, with the method `bar`:\n\n\n```python\nfig, axes = plt.subplots()\n\n# create some new data to plot\n# don't worry too much if you don't understand this line\nx_bar = y_bar = np.arange(10)\n\n# create a bar plot with the given\n# x and height values\naxes.bar(x_bar, y_bar)\n\n# show the plot\nplt.show()\n```\n\nWe will also need to know what we are plotting, so we need to change the labels on the x-axis:\n\n\n```python\nfig, axes = plt.subplots()\n\n# create some new data to plot\n# don't worry too much if you don't understand this line\nx_bar = y_bar = np.arange(10)\n\n# create a bar plot with the given\n# x and height values\naxes.bar(x_bar, y_bar)\n\n# define some labels for each x value\n# don't worry if you don't understand this line\nx_tick_labels = [f'lab-{x}' for x in range(len(x_bar))]\n\n# tell plt that we'll specify one tick for each bar\naxes.set_xticks(range(len(x_bar)))\n\n# set the tick labels\naxes.set_xticklabels(x_tick_labels)\n\n# show the plot\nplt.show()\n```\n\n## Classes, what's the idea?\n\nWe talked above about how we use functions because we don't want to rewrite the same stuff over and over again. But suppose that you don't want to just repeat a function, but a whole collection of functions and properties that can depend on each other. Then, you could define a class. In this light, a class is a _template for an object_, where an object has specific values for those variables that you left unspecified in the class.\n\nLet's consider an example to make the explanation above clearer (as it often happens, the definitions are interdependent and the best way is to see how the terms are used in practice). Suppose we want to write a program that models students at a university. A very natural way of structuring this program is to have variables that correspond to students, e.g., one variable called `mark` for Mark, etc. \n\nWhat type should these variables be? Clearly they're not integers, or strings, or floats. Ideally we'd want to have a _student_ type with its own methods that we can freely define. In other words, we want to have a `Student` class (Note: class names always start with a capital letter in pyhon by convention), which can produce specific `student` objects (`mark`, etc.). Here's how we can define it in python:\n\n\n```python\n# a class is defined with the keyword 'class'\n# followed by the name of the class \n# (remember: first letter capitalized!)\nclass Student:\n \n # __init__ is a special function\n # that gets called when a new object\n # is created from the class.\n # 'self' contains the properties\n # and methods defined in the class itself\n # and is passed to every function in the class\n def __init__(self, name, age):\n # define some class attributed\n # to store data about the student\n self.name = name\n self.age = age\n self.grades = []\n \n # define a new function that we can \n # use to record new grades that a student got.\n # This function takes a grade and\n # adds it to the student's record.\n def add_grade(self, grade):\n # The grades record is a list.\n # Append the new grade to the list!\n self.grades.append(grade)\n \n def average_grade(self):\n # Calculate the average of the grades\n # recorded in the grades list\n # and return the resulting mean.\n return np.mean(self.grades)\n```\n\nNow we can see the notation for instantiating a specific object (in this case, a student called 'Mark') from a class:\n\n\n```python\n# an object is defined by calling \n# the class name with the arguments\n# defined in the __init__ function\nmark = Student('mark', 22)\n\n# add some grade to Mark\nmark.add_grade(1.3)\nmark.add_grade(1.0)\n\n# print Mark's average grade\nprint(mark.average_grade())\n```\n\n 1.15\n\n\nYou don't need to understand everything about classes, and you won't have to define one from scratch, but we'll have to adapt them as part of working with Piantadosi's LOTlib3 library. If you want to read more about classes in python [here is the documentation](https://docs.python.org/3/tutorial/classes.html) and here is [another explanation of the basic idea](https://towardsdatascience.com/introduction-to-python-classes-da526ff745df). \n\n## Error handling\n\nAs you have already seen above, if something goes wrong sometimes python raises an _error_. For instance, above we have encountered `RecursionError: maximum recursion depth exceeded`. However, there are other types of errors in python, for instance if you try to get an element from an index that's too high for a list:\n\n\n```python\na = [1, 3, 4]\nprint(a[3])\n```\n\nIn general, errors help us understand what went wrong with our code. However, in python they can also be used as part of the program flow. Specifically, we might predict a situation that causes a specific error and write in the code what we want to happen is such a situation occurs. Here's how we do it:\n\n\n```python\na = [1, 2, 4, 5]\n\n# warn python that the indented code\n# might contain an error\ntry:\n for i in range(10):\n print(a[i], end=', ')\n# tell python what to do in case\n# a specific kind of error occurs\n# In this case, an IndexError\nexcept IndexError:\n print('Finished: There are fewer than 10 elements in the list!')\n```\n\n 1, 2, 4, 5, Finished: There are fewer than 10 elements in the list!\n\n\nHere's another error you should be familiar with (it will come up again later in the course!)\n\n\n```python\n1/0\n```\n\nAnd how we can deal with it:\n\n\n```python\na = 0\n\ntry:\n print(5 / a)\nexcept ZeroDivisionError:\n print('undefined')\n```\n\n undefined\n\n\n# First homeworks set\n\n- Write a function to sum three numbers.\n- Write a $\\lambda$ function to sum three numbers.\n- Write a recursive function that calculates the factorial function for $n$, written $n!$, and defined as follows:\n\n\\begin{align}\n 1! &= 1 \\\\\n n! &= n \\times (n-1)!\n\\end{align}\n\n- Write a recursive function `interpret` that takes a string that consists of a series of `\u00ac` followed by a single `a` (`a`, `\u00aca`, `\u00ac\u00aca`, `\u00ac\u00ac\u00aca`, etc) and is defined as follows:\n\n\\begin{align}\n \\text{interpret}(x) &= 1 & \\text{if } x = \\text{'a'} \\\\\n &= 1 - \\text{interpret(x[1:])} & \\text{else}\n\\end{align}\n (do you see how this corresponds to the semantics of Boolean negation?)\n\n- Rewrite the factorial function and `interpret` using `recursive_function` and `apply_inside` rather than explicit recursion.\n- Make a scatterplot of the first 10 strings interpreted by `interpret` (`a`, `\u00aca`, `\u00ac\u00aca`, `\u00ac\u00ac\u00aca`, etc): the x-axis shows the strings in order and the y-axis shows the result of `interpret` for the corresponding string.\n- Define a `Teacher` class that contains a record of the students that the teacher taught to, as well as the average grade the teacher gave to all the students (assume for simplicity that this is a school where each student only has one teacher). The class should have:\n - an `add_student` method that takes a Student object as defined above.\n - an `add_grade` method that takes a student name and a grade, and adds the grade to the student.\n - an `calculate_average_grade` method that returns the average grade that the teacher gave across all students.\n \n Usage example:\n \n```python\nmaryWoolnoth = Teacher('Mary Woolnoth')\nstudent1 = Student('Frank', 10)\nstudent2 = Student('Bob', 9)\n\nmaryWoolnoth.add_student(student1)\nmaryWoolnoth.add_student(student2)\n\nmaryWoolnoth.add_grade('Frank', 1.3)\n```\n- Google how to read a text file and print out its content. Write code that does this, but using the `try ... except ...` statements prints 'File does not exist!' if the file does not exist.\n", "meta": {"hexsha": "a6f190b0a3159203795508c289479c9dfa6ca2cd", "size": 71637, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "book/3_lab.ipynb", "max_stars_repo_name": "thelogicalgrammar/pLoT_course", "max_stars_repo_head_hexsha": "3fa91c2461371a16eab43d9d702fd31420aa712e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "book/3_lab.ipynb", "max_issues_repo_name": "thelogicalgrammar/pLoT_course", "max_issues_repo_head_hexsha": "3fa91c2461371a16eab43d9d702fd31420aa712e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "book/3_lab.ipynb", "max_forks_repo_name": "thelogicalgrammar/pLoT_course", "max_forks_repo_head_hexsha": "3fa91c2461371a16eab43d9d702fd31420aa712e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 50.4130893737, "max_line_length": 10404, "alphanum_fraction": 0.7214567891, "converted": true, "num_tokens": 5849, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5078118642792044, "lm_q2_score": 0.8198933425148213, "lm_q1q2_score": 0.4163515667725597}} {"text": "```python\n%matplotlib inline\n```\n\n\n# Fit a banded ridge model with both wordnet and motion energy features\n\nIn this example, we model the fMRI responses with a `banded ridge regression`,\nwith two different feature spaces: motion energy and wordnet categories.\n\n*Banded ridge regression:* Since the relative scaling of both feature spaces is\nunknown, we use two regularization hyperparameters (one per feature space) in a\nmodel called banded ridge regression [1]_. Just like with ridge regression, we\noptimize the hyperparameters over cross-validation. An efficient implementation\nof this model is available in the `himalaya\n`_ package.\n\n*Running time:* This example is more computationally intensive than the\nprevious examples. With a GPU backend, model fitting takes around 6 minutes.\nWith a CPU backend, it can last 10 times more.\n\n\n\n```python\n\n```\n\n## Path of the data directory\n\n\n\n\n```python\nimport os\nfrom voxelwise_tutorials.io import get_data_home\ndirectory = os.path.join(get_data_home(), \"vim-5\")\nprint(directory)\n```\n\n\n```python\n# modify to use another subject\nsubject = \"S01\"\n```\n\n## Load the data\n\nAs in the previous examples, we first load the fMRI responses, which are our\nregression targets.\n\n\n\n\n```python\nimport numpy as np\n\nfrom voxelwise_tutorials.io import load_hdf5_array\n\nfile_name = os.path.join(directory, \"responses\", f\"{subject}_responses.hdf\")\nY_train = load_hdf5_array(file_name, key=\"Y_train\")\nY_test = load_hdf5_array(file_name, key=\"Y_test\")\n\nprint(\"(n_samples_train, n_voxels) =\", Y_train.shape)\nprint(\"(n_repeats, n_samples_test, n_voxels) =\", Y_test.shape)\n```\n\nWe also compute the explainable variance, to exclude voxels with low\nexplainable variance from the fit, and speed up the model fitting.\n\n\n\n\n```python\nfrom voxelwise_tutorials.utils import explainable_variance\nev = explainable_variance(Y_test)\nprint(\"(n_voxels,) =\", ev.shape)\n\nmask = ev > 0.1\nprint(\"(n_voxels_mask,) =\", ev[mask].shape)\n```\n\nWe average the test repeats, to remove the non-repeatable part of fMRI\nresponses.\n\n\n\n\n```python\nY_test = Y_test.mean(0)\n\nprint(\"(n_samples_test, n_voxels) =\", Y_test.shape)\n```\n\nWe fill potential NaN (not-a-number) values with zeros.\n\n\n\n\n```python\nY_train = np.nan_to_num(Y_train)\nY_test = np.nan_to_num(Y_test)\n```\n\nAnd we make sure the targets are centered.\n\n\n\n\n```python\nY_train -= Y_train.mean(0)\nY_test -= Y_test.mean(0)\n```\n\nThen we load both feature spaces, that are going to be used for the\nlinear regression model.\n\n\n\n\n```python\nfeature_names = [\"wordnet\", \"motion_energy\"]\n\nXs_train = []\nXs_test = []\nn_features_list = []\nfor feature_space in feature_names:\n file_name = os.path.join(directory, \"features\", f\"{feature_space}.hdf\")\n Xi_train = load_hdf5_array(file_name, key=\"X_train\")\n Xi_test = load_hdf5_array(file_name, key=\"X_test\")\n\n Xs_train.append(Xi_train.astype(dtype=\"float32\"))\n Xs_test.append(Xi_test.astype(dtype=\"float32\"))\n n_features_list.append(Xi_train.shape[1])\n\n# concatenate the feature spaces\nX_train = np.concatenate(Xs_train, 1)\nX_test = np.concatenate(Xs_test, 1)\n\nprint(\"(n_samples_train, n_features_total) =\", X_train.shape)\nprint(\"(n_samples_test, n_features_total) =\", X_test.shape)\nprint(\"[n_features_wordnet, n_features_motion_energy] =\", n_features_list)\n```\n\n## Define the cross-validation scheme\n\nWe define again a leave-one-run-out cross-validation split scheme.\n\n\n\n\n```python\nfrom sklearn.model_selection import check_cv\nfrom voxelwise_tutorials.utils import generate_leave_one_run_out\n\n# indice of first sample of each run\nrun_onsets = load_hdf5_array(file_name, key=\"run_onsets\")\nprint(run_onsets)\n```\n\nWe define a cross-validation splitter, compatible with ``scikit-learn`` API.\n\n\n\n\n```python\nn_samples_train = X_train.shape[0]\ncv = generate_leave_one_run_out(n_samples_train, run_onsets)\ncv = check_cv(cv) # copy the cross-validation splitter into a reusable list\n```\n\n## Define the model\n\nThe model pipeline contains similar steps than the pipeline from previous\nexamples. We remove the mean of each feature with a ``StandardScaler``,\nand add delays with a ``Delayer``.\n\n\n\n\n```python\nfrom sklearn.pipeline import make_pipeline\nfrom sklearn.preprocessing import StandardScaler\nfrom voxelwise_tutorials.delayer import Delayer\nfrom himalaya.backend import set_backend\nbackend = set_backend(\"torch_cuda\", on_error=\"warn\")\n```\n\nTo fit the banded ridge model, we use ``himalaya``'s\n``MultipleKernelRidgeCV`` model, with a separate linear kernel per feature\nspace. Similarly to ``KernelRidgeCV``, the model optimizes the\nhyperparameters over cross-validation. However, while ``KernelRidgeCV`` has\nto optimize only one hyperparameter (``alpha``), ``MultipleKernelRidgeCV``\nhas to optimize ``m`` hyperparameters, where ``m`` is the number of feature\nspaces (here ``m = 2``). To do so, the model implements two different\nsolvers, one using hyperparameter random search, and one using hyperparameter\ngradient descent. For large number of targets, we recommend using the\nrandom-search solver.\n\n\n\nThe class takes a number of common parameters during initialization, such as\n``kernels``, or ``solver``. Since the solver parameters vary depending on the\nsolver used, they are passed as a ``solver_params`` dictionary.\n\n\n\n\n```python\nfrom himalaya.kernel_ridge import MultipleKernelRidgeCV\n\n# Here we will use the \"random_search\" solver.\nsolver = \"random_search\"\n\n# We can check its specific parameters in the function docstring:\nsolver_function = MultipleKernelRidgeCV.ALL_SOLVERS[solver]\nprint(\"Docstring of the function %s:\" % solver_function.__name__)\nprint(solver_function.__doc__)\n```\n\nThe hyperparameter random-search solver separates the hyperparameters into a\nshared regularization ``alpha`` and a vector of positive kernel weights which\nsum to one. This separation of hyperparameters allows to explore efficiently\na large grid of values for ``alpha`` for each sampled kernel weights vector.\n\nWe use *20* random-search iterations to have a reasonably fast example. To\nhave better results, especially for larger number of feature spaces, one\nmight need more iterations. (Note that there is currently no stopping\ncriterion in the random-search method.)\n\n\n\n\n```python\nn_iter = 20\n\nalphas = np.logspace(1, 20, 20)\n```\n\nBatch parameters, used to reduce the necessary GPU memory. A larger value\nwill be a bit faster, but the solver might crash if it is out of memory.\nOptimal values depend on the size of your dataset.\n\n\n\n\n```python\nn_targets_batch = 200\nn_alphas_batch = 5\nn_targets_batch_refit = 200\n```\n\nWe put all these parameters in a dictionary ``solver_params``, and define\nthe main estimator ``MultipleKernelRidgeCV``.\n\n\n\n\n```python\nsolver_params = dict(n_iter=n_iter, alphas=alphas,\n n_targets_batch=n_targets_batch,\n n_alphas_batch=n_alphas_batch,\n n_targets_batch_refit=n_targets_batch_refit)\n\nmkr_model = MultipleKernelRidgeCV(kernels=\"precomputed\", solver=solver,\n solver_params=solver_params, cv=cv)\n```\n\nWe need a bit more work than in previous examples before defining the full\npipeline, since the banded ridge model requires `multiple` precomputed\nkernels, one for each feature space. To compute them, we use the\n``ColumnKernelizer``, which can create multiple kernels from different\ncolumn of your features array. ``ColumnKernelizer`` works similarly to\n``scikit-learn``'s ``ColumnTransformer``, but instead of returning a\nconcatenation of transformed features, it returns a stack of kernels,\nas required in ``MultipleKernelRidgeCV(kernels=\"precomputed\")``.\n\n\n\nFirst, we create a different ``Kernelizer`` for each feature space.\nHere we use a linear kernel for all feature spaces, but ``ColumnKernelizer``\naccepts any ``Kernelizer``, or ``scikit-learn`` ``Pipeline`` ending with a\n``Kernelizer``.\n\n\n\n\n```python\nfrom himalaya.kernel_ridge import Kernelizer\nfrom sklearn import set_config\nset_config(display='diagram') # requires scikit-learn 0.23\n\npreprocess_pipeline = make_pipeline(\n StandardScaler(with_mean=True, with_std=False),\n Delayer(delays=[1, 2, 3, 4]),\n Kernelizer(kernel=\"linear\"),\n)\npreprocess_pipeline\n```\n\nThe column kernelizer applies a different pipeline on each selection of\nfeatures, here defined with ``slices``.\n\n\n\n\n```python\nfrom himalaya.kernel_ridge import ColumnKernelizer\n\n# Find the start and end of each feature space in the concatenated ``X_train``.\nstart_and_end = np.concatenate([[0], np.cumsum(n_features_list)])\nslices = [\n slice(start, end)\n for start, end in zip(start_and_end[:-1], start_and_end[1:])\n]\nslices\n```\n\n\n```python\nkernelizers_tuples = [(name, preprocess_pipeline, slice_)\n for name, slice_ in zip(feature_names, slices)]\ncolumn_kernelizer = ColumnKernelizer(kernelizers_tuples)\ncolumn_kernelizer\n\n# (Note that ``ColumnKernelizer`` has a parameter ``n_jobs`` to parallelize\n# each ``Kernelizer``, yet such parallelism does not work with GPU arrays.)\n```\n\nThen we can define the model pipeline.\n\n\n\n\n```python\npipeline = make_pipeline(\n column_kernelizer,\n mkr_model,\n)\npipeline\n```\n\n## Fit the model\n\nWe fit on the train set, and score on the test set.\n\nTo speed up the fit and to limit the memory peak in Colab, we only fit on\nvoxels with explainable variance above 0.1.\n\nWith a GPU backend, the fitting of this model takes around 6 minutes. With a\nCPU backend, it can last 10 times more.\n\n\n\n\n```python\npipeline.fit(X_train, Y_train[:, mask])\n\nscores_mask = pipeline.score(X_test, Y_test[:, mask])\nscores_mask = backend.to_numpy(scores_mask)\nprint(\"(n_voxels_mask,) =\", scores_mask.shape)\n\n# Then we extend the scores to all voxels, giving a score of zero to unfitted\n# voxels.\nn_voxels = Y_train.shape[1]\nscores = np.zeros(n_voxels)\nscores[mask] = scores_mask\nprint(\"(n_voxels,) =\", scores.shape)\n```\n\n## Compare with a ridge model\n\nWe can compare with a baseline model, which does not use one hyperparameter\nper feature space, but instead shares the same hyperparameter for all feature\nspaces.\n\n\n\n\n```python\nfrom himalaya.kernel_ridge import KernelRidgeCV\n\npipeline_baseline = make_pipeline(\n StandardScaler(with_mean=True, with_std=False),\n Delayer(delays=[1, 2, 3, 4]),\n KernelRidgeCV(\n alphas=alphas, cv=cv,\n solver_params=dict(n_targets_batch=n_targets_batch,\n n_alphas_batch=n_alphas_batch,\n n_targets_batch_refit=n_targets_batch_refit)),\n)\npipeline_baseline\n```\n\n\n```python\npipeline_baseline.fit(X_train, Y_train[:, mask])\nscores_baseline_mask = pipeline_baseline.score(X_test, Y_test[:, mask])\nscores_baseline_mask = backend.to_numpy(scores_baseline_mask)\n\n# extend to unfitted voxels\nn_voxels = Y_train.shape[1]\nscores_baseline = np.zeros(n_voxels)\nscores_baseline[mask] = scores_baseline_mask\n```\n\nHere we plot the comparison of model prediction accuracies with a 2D\nhistogram. All 70k voxels are represented in this histogram, where the\ndiagonal corresponds to identical model prediction accuracy for both models.\nA distibution deviating from the diagonal means that one model has better\npredictive performance than the other.\n\n\n\n\n```python\nimport matplotlib.pyplot as plt\nfrom voxelwise_tutorials.viz import plot_hist2d\n\nax = plot_hist2d(scores_baseline, scores)\nax.set(title='Generalization R2 scores', xlabel='KernelRidgeCV',\n ylabel='MultipleKernelRidgeCV')\nplt.show()\n```\n\nWe see that the banded ridge model (``MultipleKernelRidgeCV``) outperforms\nthe ridge model (``KernelRidegeCV``). Indeed, banded ridge regression is able\nto find the optimal scalings of each feature space, independently on each\nvoxel. Banded ridge regression is thus able to perform a soft selection\nbetween the available feature spaces, based on the cross-validation\nperformances.\n\n\n\n## Plot the banded ridge split\n\nOn top of better prediction accuracy, banded ridge regression also gives a\nway to disentangle the contribution of the two feature spaces. To do so, we\ntake the kernel weights and the ridge (dual) weights corresponding to each\nfeature space, and use them to compute the prediction from each feature space\nseparately.\n\n\\begin{align}\\hat{y} = \\sum_i^m \\hat{y}_i = \\sum_i^m \\gamma_i K_i \\hat{w}\\end{align}\n\nThen, we use these split predictions to compute split $\\tilde{R}^2_i$\nscores. These scores are corrected so that their sum is equal to the\n$R^2$ score of the full prediction $\\hat{y}$.\n\n\n\n\n```python\nfrom himalaya.scoring import r2_score_split\n\nY_test_pred_split = pipeline.predict(X_test, split=True)\nsplit_scores_mask = r2_score_split(Y_test[:, mask], Y_test_pred_split)\n\nprint(\"(n_kernels, n_samples_test, n_voxels_mask) =\", Y_test_pred_split.shape)\nprint(\"(n_kernels, n_voxels_mask) =\", split_scores_mask.shape)\n\n# extend to unfitted voxels\nn_kernels = split_scores_mask.shape[0]\nn_voxels = Y_train.shape[1]\nsplit_scores = np.zeros((n_kernels, n_voxels))\nsplit_scores[:, mask] = backend.to_numpy(split_scores_mask)\nprint(\"(n_kernels, n_voxels) =\", split_scores.shape)\n```\n\nWe can then plot the split scores on a flatmap with a 2D colormap.\n\n\n\n\n```python\nfrom voxelwise_tutorials.viz import plot_2d_flatmap_from_mapper\n\nmapper_file = os.path.join(directory, \"mappers\", f\"{subject}_mappers.hdf\")\nax = plot_2d_flatmap_from_mapper(split_scores[0], split_scores[1],\n mapper_file, vmin=0, vmax=0.25, vmin2=0,\n vmax2=0.5, label_1=feature_names[0],\n label_2=feature_names[1])\nplt.show()\n```\n\nThe blue regions are better predicted by the motion-energy features, the\norange regions are better predicted by the wordnet features, and the white\nregions are well predicted by both feature spaces.\n\nCompared to the last figure of the previous example, we see that most white\nregions have been replaced by either blue or orange regions. The banded ridge\nregression disentangled the two feature spaces in voxels where both feature\nspaces had good prediction accuracy (see previous example). For example,\nmotion-energy features predict brain activity in early visual cortex, while\nwordnet features predict in semantic visual areas. For more discussions about\nthese results, we refer the reader to the original publication [1]_.\n\n\n\n## References\n\n.. [1] Nunez-Elizalde, A. O., Huth, A. G., & Gallant, J. L. (2019).\n Voxelwise encoding models with non-spherical multivariate normal priors.\n Neuroimage, 197, 482-492.\n\n\n", "meta": {"hexsha": "8c0bc9690bd07696b2cfbd98ee0c30c0cc289dc4", "size": 22690, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/notebooks/movies_3T/05_plot_banded_ridge_model.ipynb", "max_stars_repo_name": "gallantlab/voxelwise_tutorials", "max_stars_repo_head_hexsha": "3df639dd5fb957410f41b4a3b986c9f903f5333b", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 12, "max_stars_repo_stars_event_min_datetime": "2021-09-08T22:22:26.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-10T18:06:33.000Z", "max_issues_repo_path": "tutorials/notebooks/movies_3T/05_plot_banded_ridge_model.ipynb", "max_issues_repo_name": "gallantlab/voxelwise_tutorials", "max_issues_repo_head_hexsha": "3df639dd5fb957410f41b4a3b986c9f903f5333b", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2021-09-11T16:06:44.000Z", "max_issues_repo_issues_event_max_datetime": "2021-12-16T23:39:40.000Z", "max_forks_repo_path": "tutorials/notebooks/movies_3T/05_plot_banded_ridge_model.ipynb", "max_forks_repo_name": "gallantlab/voxelwise_tutorials", "max_forks_repo_head_hexsha": "3df639dd5fb957410f41b4a3b986c9f903f5333b", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2021-09-13T19:11:00.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-26T04:35:11.000Z", "avg_line_length": 43.8030888031, "max_line_length": 865, "alphanum_fraction": 0.6156456589, "converted": true, "num_tokens": 3473, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593452091672, "lm_q2_score": 0.6688802669716107, "lm_q1q2_score": 0.4163507730024818}} {"text": "# Simulated Satellite Scan Strategies\n\n\n```python\n# Load common tools for all lessons\nimport sys\nsys.path.insert(0, \"..\")\nfrom lesson_tools import (\n fake_focalplane\n)\n```\n\n## Overview\n\nA generic satellite scanning strategy can be described in terms of precession and spin axes, the opening angles about each of them, and the rates of rotation. The precession axis itself is typically oriented in the anti-sun direction and that orientation must be updated either in steps or with a continuous slewing motion. Here is a cartoon (drawn from an old, public LiteBIRD talk) showing a sketch of these angles:\n\n\nFor a real satellite experiment such as Planck or LiteBIRD, there are many custom details, such as simulating repointing maneuvers, simulating any lost time due to cooler cycling, nutation effects, etc. Those are contained in classes for the specific experiments. The tools built in to the core TOAST package are intended for rough simulations to study things like scan strategy choices.\n\n### TOD Class for Simulations\n\nIn the introductory lesson we saw the use of a TOD derived class providing things like telescope pointing. Here we introduce the `TODSatellite` class which serves that purpose for generic satellite simulations.\n\n\n```python\nimport numpy as np\n\nimport toast\nfrom toast.todmap import (\n slew_precession_axis,\n TODSatellite\n)\n```\n\n\n```python\n# Default Comm (one group for this example)\n\ncomm = toast.Comm()\n```\n\n\n```python\n# Create our fake focalplane\n\nfp = fake_focalplane()\n\ndetnames = list(sorted(fp.keys()))\ndetquat = {x: fp[x][\"quat\"] for x in detnames}\n```\n\n\n```python\n# Scan parameters (made up, not physically motivated)\n\nsamplerate = 10.0\nprecperiod = 90.0\nprecangle = 45.0\nspinperiod = 1.0\nspinangle = 45.0\n```\n\n\n```python\n# We can simulate a simplistic HWP\n\nhwprpm = 6.0\n```\n\n\n```python\n# Number of samples\n\nnsamples = 100\n```\n\n\n```python\n# Instantiate a TOD\n\ntod = TODSatellite(\n comm.comm_group, \n detquat, \n nsamples, \n firstsamp=0,\n firsttime=0.0,\n rate=samplerate,\n spinperiod=spinperiod,\n spinangle=spinangle,\n precperiod=precperiod,\n precangle=precangle,\n coord=\"E\",\n hwprpm=hwprpm\n)\n```\n\n\n```python\n# The TOD constructor above specifies the scan parameters, but the boresight\n# simulation is not done until we set the location of the precession axis as a\n# function of time. There is a reason for this delayed construction. The data\n# distribution occurs during the above construction, and we might want to only\n# simulate the precession axis motion for our local data. In reality this is\n# cheap enough to do on one process and distribute during the construction.\n\nqprec = np.empty(4 * tod.local_samples[1], dtype=np.float64).reshape((-1, 4))\n\ndeg_per_day = 1.0\n\nslew_precession_axis(\n qprec,\n firstsamp=tod.local_samples[0],\n samplerate=samplerate,\n degday=deg_per_day,\n)\n\ntod.set_prec_axis(qprec=qprec)\n```\n\n\n```python\n# Now we can read from this TOD object\n\nprint(\"TOD timestampes = {} ...\".format(tod.read_times()[:5]))\nprint(\"TOD boresight = \\n{} ...\".format(tod.read_boresight()[:5,:]))\nfor d in detnames:\n print(\"TOD detector {} = {} ...\".format(d, tod.read(detector=d, n=5)))\n print(\"TOD detector {} flags = {} ...\".format(d, tod.read_flags(detector=d, n=5)))\n print(\"TOD detector {} pointing = {} ...\".format(d, tod.read_pntg(detector=d, n=5)))\n```\n\n TOD timestampes = [0. 0.1 0.2 0.3 0.4] ...\n TOD boresight = \n [[ 0.5 0.5 -0.5 0.5 ]\n [ 0.50372461 0.50002202 -0.49626167 0.49996385]\n [ 0.50743541 0.50002989 -0.49250974 0.49991357]\n [ 0.51113228 0.50002363 -0.4887443 0.49984914]\n [ 0.51481514 0.50000321 -0.48496547 0.49977059]] ...\n TOD detector 0A = [0. 0. 0. 0. 0.] ...\n TOD detector 0A flags = [0 0 0 0 0] ...\n TOD detector 0A pointing = [[ 0.65328148 0.27059805 -0.27059805 0.65328148]\n [ 0.656731 0.26919305 -0.26715812 0.65181749]\n [ 0.66016234 0.26778026 -0.26371103 0.65033523]\n [ 0.66357541 0.26635974 -0.26025687 0.64883474]\n [ 0.66697012 0.26493151 -0.25679575 0.64731607]] ...\n TOD detector 0B = [0. 0. 0. 0. 0.] ...\n TOD detector 0B flags = [0 0 0 0 0] ...\n TOD detector 0B pointing = [[ 0.65328148 -0.27059805 0.27059805 0.65328148]\n [ 0.65472717 -0.27403071 0.27199525 0.64981388]\n [ 0.65615451 -0.27745603 0.27338459 0.64632831]\n [ 0.65756345 -0.2808739 0.27476605 0.64282484]\n [ 0.65895396 -0.28428423 0.27613957 0.6393036 ]] ...\n TOD detector 1A = [0. 0. 0. 0. 0.] ...\n TOD detector 1A flags = [0 0 0 0 0] ...\n TOD detector 1A pointing = [[ 0.50501286 0.50501286 -0.49493637 0.49493637]\n [ 0.50869961 0.50503451 -0.4911607 0.4949 ]\n [ 0.51237241 0.50504188 -0.48737156 0.49484964]\n [ 0.51603116 0.50503497 -0.48356907 0.49478528]\n [ 0.51967576 0.50501376 -0.47975332 0.49470694]] ...\n TOD detector 1B = [0. 0. 0. 0. 0.] ...\n TOD detector 1B flags = [0 0 0 0 0] ...\n TOD detector 1B pointing = [[ 7.14196038e-01 -5.55111512e-17 -1.11022302e-16 6.99945726e-01]\n [ 7.16818276e-01 -2.59161549e-03 2.64408538e-03 6.97250208e-01]\n [ 7.19420549e-01 -5.18346747e-03 5.28779671e-03 6.94535272e-01]\n [ 7.22002785e-01 -7.77548483e-03 7.93106150e-03 6.91800997e-01]\n [ 7.24564909e-01 -1.03675965e-02 1.05738073e-02 6.89047458e-01]] ...\n TOD detector 2A = [0. 0. 0. 0. 0.] ...\n TOD detector 2A flags = [0 0 0 0 0] ...\n TOD detector 2A pointing = [[ 0.65417834 0.27764851 -0.26352011 0.6523183 ]\n [ 0.65759801 0.27622036 -0.26005044 0.65087705]\n [ 0.6609995 0.27478423 -0.25657382 0.64941756]\n [ 0.66438269 0.27334016 -0.25309032 0.64793988]\n [ 0.6677475 0.2718882 -0.24960006 0.64644404]] ...\n TOD detector 2B = [0. 0. 0. 0. 0.] ...\n TOD detector 2B flags = [0 0 0 0 0] ...\n TOD detector 2B pointing = [[ 0.65890108 -0.26624679 0.27492183 0.64759555]\n [ 0.6603093 -0.26967473 0.27635614 0.64412301]\n [ 0.66169902 -0.27309543 0.27778248 0.64063265]\n [ 0.66307019 -0.27650882 0.2792008 0.63712456]\n [ 0.66442277 -0.27991479 0.28061107 0.63359886]] ...\n TOD detector 3A = [0. 0. 0. 0. 0.] ...\n TOD detector 3A flags = [0 0 0 0 0] ...\n TOD detector 3A pointing = [[ 0.49309224 0.50181874 -0.49813049 0.50685699]\n [ 0.49683581 0.50180832 -0.49441092 0.50685345]\n [ 0.50056576 0.50178371 -0.49067781 0.50683559]\n [ 0.50428199 0.5017449 -0.48693124 0.50680341]\n [ 0.50798438 0.50169191 -0.48317134 0.50675689]] ...\n TOD detector 3B = [0. 0. 0. 0. 0.] ...\n TOD detector 3B flags = [0 0 0 0 0] ...\n TOD detector 3B pointing = [[ 0.7035083 0.00617057 0.00617057 0.71063346]\n [ 0.70614804 0.00351609 0.0087982 0.70800083]\n [ 0.70876811 0.00086122 0.01142528 0.70534849]\n [ 0.71136844 -0.00179399 0.01405174 0.70267651]\n [ 0.71394896 -0.00444945 0.01667751 0.69998497]] ...\n TOD detector 4A = [0. 0. 0. 0. 0.] ...\n TOD detector 4A flags = [0 0 0 0 0] ...\n TOD detector 4A pointing = [[ 0.49493637 0.49493637 -0.50501286 0.50501286]\n [ 0.49869846 0.49495875 -0.50131225 0.50497694]\n [ 0.50244687 0.49496713 -0.4975979 0.50492673]\n [ 0.50618151 0.49496151 -0.49386991 0.50486225]\n [ 0.50990226 0.49494189 -0.49012838 0.5047835 ]] ...\n TOD detector 4B = [0. 0. 0. 0. 0.] ...\n TOD detector 4B flags = [0 0 0 0 0] ...\n TOD detector 4B pointing = [[ 6.99945726e-01 0.00000000e+00 -1.11022302e-16 7.14196038e-01]\n [ 7.02621751e-01 -2.64437272e-03 2.59132230e-03 7.11553911e-01]\n [ 7.05278207e-01 -5.28897433e-03 5.18226587e-03 7.08891968e-01]\n [ 7.07915019e-01 -7.93373230e-03 7.77275966e-03 7.06210285e-01]\n [ 7.10532113e-01 -1.05785741e-02 1.03627326e-02 7.03508937e-01]] ...\n TOD detector 5A = [0. 0. 0. 0. 0.] ...\n TOD detector 5A flags = [0 0 0 0 0] ...\n TOD detector 5A pointing = [[ 0.50181874 0.49309224 -0.50685699 0.49813049]\n [ 0.50556168 0.49314706 -0.50313781 0.49806195]\n [ 0.50929075 0.49318794 -0.49940483 0.49797932]\n [ 0.51300585 0.49321485 -0.49565816 0.49788261]\n [ 0.51670688 0.49322781 -0.49189789 0.49777183]] ...\n TOD detector 5B = [0. 0. 0. 0. 0.] ...\n TOD detector 5B flags = [0 0 0 0 0] ...\n TOD detector 5B pointing = [[ 0.7035083 -0.00617057 -0.00617057 0.71063346]\n [ 0.70619373 -0.00877846 -0.00358917 0.70795514]\n [ 0.70885948 -0.01138641 -0.00100798 0.7052571 ]\n [ 0.71150548 -0.01399435 0.00157293 0.70253942]\n [ 0.71413167 -0.01660221 0.0041535 0.69980217]] ...\n TOD detector 6A = [0. 0. 0. 0. 0.] ...\n TOD detector 6A flags = [0 0 0 0 0] ...\n TOD detector 6A pointing = [[ 0.65890108 0.26624679 -0.27492183 0.64759555]\n [ 0.66234515 0.26487916 -0.2714774 0.64609439]\n [ 0.66577088 0.26350386 -0.26802568 0.64457512]\n [ 0.66917818 0.26212094 -0.26456677 0.64303779]\n [ 0.67256696 0.26073044 -0.26110078 0.64148243]] ...\n TOD detector 6B = [0. 0. 0. 0. 0.] ...\n TOD detector 6B flags = [0 0 0 0 0] ...\n TOD detector 6B pointing = [[ 0.65417834 -0.27764851 0.26352011 0.6523183 ]\n [ 0.65564659 -0.28105089 0.26489422 0.64882123]\n [ 0.65709647 -0.28444573 0.26626067 0.64530622]\n [ 0.65852793 -0.28783293 0.26761942 0.64177334]\n [ 0.65994092 -0.2912124 0.26897045 0.63822271]] ...\n\n\nNotice that the signal data for all detectors is zero. For simulated TOD classes, there is no data to \"read\". Instead, simulated timestreams are constructed and stored in the `tod.cache` member variable.\n\n### Low Resolution Example\n\nImagine the case of a satellite telescope with detector beams that are a 5 degrees FWHM. We'll use a healpix resolution of NSIDE = 32 (approximately 2 degrees) for this example. Let's use made-up angles for the spin and precession angles of 40 and 50 degrees, respectively:\n\n\\begin{align}\n\\alpha & = 50^{\\circ}\\\\\n\\beta & = 40^{\\circ}\\\\\n\\omega_{\\alpha} & = \\text{precession rate}\\\\\n\\omega_{\\beta} & = \\text{spin rate}\\\\\n\\end{align}\n\nWhen computing the precession rate, we want the precession motion to be slow enough so that the speed of the boresight on the sky does not vary enough to change our effective beams. The speed variation on the sky due to precession is\n\n\\begin{align}\nv_{\\text{min}} & = \\beta \\cdot \\omega_{\\beta} - \\alpha \\cdot \\omega_{\\alpha}\\\\\nv_{\\text{max}} & = \\beta \\cdot \\omega_{\\beta} + \\alpha \\cdot \\omega_{\\alpha}\\\\\nv_{\\text{diff}} & = v_{\\text{max}} - v_{\\text{min}} = 2 \\alpha \\omega_{\\alpha}\n\\end{align}\n\nThis change, integrated over a sample, must be a small fraction (here called \"$X$\") of the beam FWHM:\n\n\\begin{align}\n\\frac{2 \\alpha \\omega_{\\alpha}}{f_{\\text{sample}}} & = X \\cdot \\text{FWHM}\\\\\nf_{\\text{sample}} & = \\frac{2 \\alpha \\omega_{\\alpha}}{X \\cdot \\text{FWHM}}\\\\\n\\end{align}\n\nThe speed of the boresight on the sky in degrees per second due to the spin axis motion is\n\n$$v_{bore} = \\alpha \\cdot \\omega_{\\alpha} \\cdot \\frac{1}{60}$$\n\nIf we want to have 3 hits per pixel with two degree pixels (2/3 degree per second), this gives us\n\n\\begin{align}\nv_{bore} = \\frac{2}{3} & = \\alpha \\cdot \\omega_{\\alpha} \\cdot \\frac{1}{60}\\\\\n\\omega_{\\alpha} & = \\frac{60}{3 \\cdot \\alpha} = 0.8\\;\\text{RPM}\n\\end{align}\n\nIf we assume five percent for our \"$X$\" fraction above, then this in turn forces our sample rate to be:\n\n$$f_{\\text{sample}} = \\frac{2 \\cdot 50 \\cdot 0.8}{0.05 \\cdot 3.0 \\cdot 60} = 8.9\\;\\text{Hz}$$\n\nThe precession rate is slower than the spin rate. The spin rate above corresponds to a period of 1.25 minutes. We choose a precession period 20 times longer (25 minutes). We will assume a very simple satellite motion where the precession axis slews continuously in the anti-sun direction.\n\n**NOTE: For the serial example in the next cell, we have artificially decreased the sample rate to 0.5 Hz and the resolution to NSIDE=16. This is so that this small example fits into reasonable RAM while still covering the sky. See the parallel notebook for an example with proper sampling.**\n\n\n```python\n# Scan parameters\n\nalpha = 50.0 # precession opening angle, degrees\nbeta = 45.0 # spin opening angle, degrees\np_alpha = 25.0 # precession period, minutes\np_beta = 1.25 # spin period, minutes\nsamplerate = 0.5 # sample rate, Hz\nhwprpm = 5.0 # HWP rotation in RPM\nnside = 16 # Healpix NSIDE\n```\n\n\n```python\n# We will use one observation per day, with no gaps in between, and\n# run for one year.\n\nobs_samples = int(24 * 3600.0 * samplerate) - 1\nnobs = 366\n```\n\n\n```python\n# Slew the precession axis so that it completes one circle\n\ndeg_per_day = 360.0 / nobs\n```\n\n\n```python\n# Create distributed data\n\ndata = toast.Data(comm)\n```\n\n\n```python\n# Append observations\n\nfor ob in range(nobs):\n obsname = \"{:03d}\".format(ob)\n obsfirst = ob * (obs_samples + 1)\n obsstart = 24 * 3600.0\n tod = TODSatellite(\n comm.comm_group, \n detquat, \n obs_samples, \n firstsamp=obsfirst,\n firsttime=obsstart,\n rate=samplerate,\n spinperiod=p_beta,\n spinangle=beta,\n precperiod=p_alpha,\n precangle=alpha,\n coord=\"E\",\n hwprpm=hwprpm\n )\n qprec = np.empty(4 * tod.local_samples[1], dtype=np.float64).reshape((-1, 4))\n slew_precession_axis(\n qprec,\n firstsamp=obsfirst,\n samplerate=samplerate,\n degday=deg_per_day,\n )\n tod.set_prec_axis(qprec=qprec)\n obs = dict()\n obs[\"tod\"] = tod\n data.obs.append(obs)\n```\n\nNow that we have simulated our scan strategy, we can make a simple hit map to visualize this\n\n\n```python\nfrom toast.todmap import (\n get_submaps_nested,\n OpPointingHpix,\n OpAccumDiag\n)\nfrom toast.map import (\n DistPixels\n)\n```\n\n\n```python\n# Make a simple pointing matrix\n\npointing = OpPointingHpix(nside=nside, nest=True, mode=\"IQU\")\npointing.exec(data)\n```\n\n\n```python\n# Compute the locally hit pixels\n#eg each process doesn't hold the whole sky, just the pixels it hits\n\nlocalpix, localsm, subnpix = get_submaps_nested(data, nside)\n```\n\n\n```python\n# Construct a distributed map to store the hit map\n\nnpix = 12 * nside**2\n\nhits = DistPixels(\n comm=data.comm.comm_world,\n size=npix,\n nnz=1,\n dtype=np.int64,\n submap=subnpix,\n local=localsm,\n)\nhits.data.fill(0)\n```\n\n\n```python\n# Accumulate the hit map locally\n\nbuild_hits = OpAccumDiag(hits=hits)\nbuild_hits.exec(data)\n```\n\n\n```python\n# Reduce the map across processes (a No-op in this case)\n\nhits.allreduce()\n```\n\n\n```python\n# Write out the map\n\nhitsfile = \"simscan_satellite_hits.fits\"\nhits.write_healpix_fits(hitsfile)\n```\n\n\n```python\n# Plot the map. If we were running on multiple processes, then\n# only rank zero would do this...\n\nimport healpy as hp\nimport matplotlib.pyplot as plt\n%matplotlib inline\n\nhitdata = hp.read_map(hitsfile, nest=True)\nhp.mollview(hitdata, xsize=800, nest=True, cmap=\"cool\", min=0)\nplt.show()\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "4011644923814b3b270dfa5df7216c8e93a39153", "size": 92012, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lessons/02_Simulated_Scan_Strategies/simscan_satellite.ipynb", "max_stars_repo_name": "mrussl/toast-workshop-ucsd-2019", "max_stars_repo_head_hexsha": "d4781c4c8df4dd2a54d72179e4ab04e354088879", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "lessons/02_Simulated_Scan_Strategies/simscan_satellite.ipynb", "max_issues_repo_name": "mrussl/toast-workshop-ucsd-2019", "max_issues_repo_head_hexsha": "d4781c4c8df4dd2a54d72179e4ab04e354088879", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "lessons/02_Simulated_Scan_Strategies/simscan_satellite.ipynb", "max_forks_repo_name": "mrussl/toast-workshop-ucsd-2019", "max_forks_repo_head_hexsha": "d4781c4c8df4dd2a54d72179e4ab04e354088879", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 149.1280388979, "max_line_length": 70384, "alphanum_fraction": 0.8653001782, "converted": true, "num_tokens": 5559, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.622459338205511, "lm_q1q2_score": 0.41635076420926787}} {"text": "Lambda School Data Science\n\n*Unit 2, Sprint 1, Module 2*\n\n---\n\n# Regression 2\n- Do train/test split\n- Use scikit-learn to fit a multiple regression\n- Understand how ordinary least squares regression minimizes the sum of squared errors\n- Define overfitting/underfitting and the bias/variance tradeoff\n\n### Setup\n\nRun the code cell below. You can work locally (follow the [local setup instructions](https://lambdaschool.github.io/ds/unit2/local/)) or on Colab.\n\nLibraries:\n- matplotlib\n- numpy\n- pandas\n- plotly\n- scikit-learn\n\n\n```python\nimport sys\n\n# If you're on Colab:\nif 'google.colab' in sys.modules:\n DATA_PATH = 'https://raw.githubusercontent.com/LambdaSchool/DS-Unit-2-Applied-Modeling/master/data/'\n\n# If you're working locally:\nelse:\n DATA_PATH = '../data/'\n \n# Ignore this Numpy warning when using Plotly Express:\n# FutureWarning: Method .ptp is deprecated and will be removed in a future version. Use numpy.ptp instead.\nimport warnings\nwarnings.filterwarnings(action='ignore', category=FutureWarning, module='numpy')\n```\n\n# Do train/test split\n\n## Overview\n\n### Predict Elections! \ud83c\uddfa\ud83c\uddf8\ud83d\uddf3\ufe0f\n\nHow could we try to predict the 2020 US Presidential election? \n\nAccording to Douglas Hibbs, a political science and economics professor, you can [explain elections with just two features, \"Bread and Peace\":](https://douglas-hibbs.com/background-information-on-bread-and-peace-voting-in-us-presidential-elections/)\n\n> Aggregate two-party vote shares going to candidates of the party holding the presidency during the postwar era are well explained by just two fundamental determinants:\n>\n> (1) Positively by weighted-average growth of per capita real disposable personal income over the term. \n> (2) Negatively by cumulative US military fatalities (scaled to population) owing to unprovoked, hostile deployments of American armed forces in foreign wars. \n\nLet's look at the data that Hibbs collected and analyzed:\n\n\n```python\nimport pandas as pd\ndf = pd.read_csv(DATA_PATH+'elections/bread_peace_voting.csv')\ndf\n```\n\nData Sources & Definitions\n\n- 1952-2012: Douglas Hibbs, [2014 lecture at Deakin University Melbourne](http://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf), Slide 40\n- 2016, Vote Share: [The American Presidency Project](https://www.presidency.ucsb.edu/statistics/elections)\n- 2016, Recent Growth in Personal Incomes: [The 2016 election economy: the \"Bread and Peace\" model final forecast](https://angrybearblog.com/2016/11/the-2016-election-economy-the-bread-and-peace-model-final-forecast.html)\n- 2016, US Military Fatalities: Assumption that Afghanistan War fatalities in 2012-16 occured at the same rate as 2008-12\n\n> Fatalities denotes the cumulative number of American military fatalities per millions of US population the in Korea, Vietnam, Iraq and Afghanistan wars during the presidential terms preceding the 1952, 1964, 1968, 1976 and 2004, 2008 and 2012 elections. \u2014[Hibbs](http://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf), Slide 33\n\nHere we have data from the 1952-2016 elections. We could make a model to predict 1952-2016 election outcomes \u2014 but do we really care about that? \n\nNo, not really. We already know what happened, we don't need to predict it.\n\nThis is explained in [_An Introduction to Statistical Learning_](http://faculty.marshall.usc.edu/gareth-james/ISL/), Chapter 2.2, Assessing Model Accuracy:\n\n> In general, we do not really care how well the method works training on the training data. Rather, _we are interested in the accuracy of the predictions that we obtain when we apply our method to previously unseen test data._ Why is this what we care about? \n>\n> Suppose that we are interested in developing an algorithm to predict a stock\u2019s price based on previous stock returns. We can train the method using stock returns from the past 6 months. But we don\u2019t really care how well our method predicts last week\u2019s stock price. We instead care about how well it will predict tomorrow\u2019s price or next month\u2019s price. \n>\n> On a similar note, suppose that we have clinical measurements (e.g. weight, blood pressure, height, age, family history of disease) for a number of patients, as well as information about whether each patient has diabetes. We can use these patients to train a statistical learning method to predict risk of diabetes based on clinical measurements. In practice, we want this method to accurately predict diabetes risk for _future patients_ based on their clinical measurements. We are not very interested in whether or not the method accurately predicts diabetes risk for patients used to train the model, since we already know which of those patients have diabetes.\n\nSo, we're really interested in the 2020 election \u2014 but we probably don't want to wait until then to evaluate our model.\n\nThere is a way we can estimate now how well our model will generalize in the future. We can't fast-forward time, but we can rewind it...\n\nWe can split our data in **two sets.** For example: \n1. **Train** a model on elections before 2008.\n2. **Test** the model on 2008, 2012, 2016. \n\nThis \"backtesting\" helps us estimate how well the model will predict the next elections going forward, starting in 2020.\n\nThis is explained in [_Forecasting,_ Chapter 3.4,](https://otexts.com/fpp2/accuracy.html) Evaluating forecast accuracy:\n\n> The accuracy of forecasts can only be determined by considering how well a model performs on new data that were not used when fitting the model.\n>\n>When choosing models, it is common practice to separate the available data into two portions, training and test data, where the training data is used to estimate any parameters of a forecasting method and the test data is used to evaluate its accuracy. Because the test data is not used in determining the forecasts, it should provide a reliable indication of how well the model is likely to forecast on new data.\n>\n>\n>\n>The size of the test set is typically about 20% of the total sample, although this value depends on how long the sample is and how far ahead you want to forecast. The following points should be noted.\n>\n>- A model which fits the training data well will not necessarily forecast well.\n>- A perfect fit can always be obtained by using a model with enough parameters.\n>- Over-fitting a model to data is just as bad as failing to identify a systematic pattern in the data.\n>\n>Some references describe the test set as the \u201chold-out set\u201d because these data are \u201cheld out\u201d of the data used for fitting. Other references call the training set the \u201cin-sample data\u201d and the test set the \u201cout-of-sample data\u201d. We prefer to use \u201ctraining data\u201d and \u201ctest data\u201d in this book.\n\n**How should we split: Randomly? Before/after a given date?**\n\nI recommend you all read a great blog post, [How (and why) to create a good validation set](https://www.fast.ai/2017/11/13/validation-sets/), by fast.ai cofounder Rachel Thomas.\n\nShe gives great examples to answer the question \u201cWhen is a random subset not good enough?\u201d I\u2019m not as opposed to random splits as Rachel Thomas seems to be. But it\u2019s worth thinking about the trade-offs!\n\nTime-based and random splits can both be useful, and you\u2019ll get repeated hands-on practice with both during this unit! (She also talks about the distinction between validation & test sets, which we\u2019ll introduce in the last lesson of this Sprint.)\n\n## Follow Along\n\nSplit the data in two sets:\n1. Train on elections before 2008.\n2. Test on 2008 and after.\n\n\n```python\n\n```\n\nHow many observations (rows) are in the train set? In the test set?\n\n\n```python\n\n```\n\nNote that this volume of data is at least two orders of magnitude smaller than we usually want to work with for predictive modeling.\n\nThere are other validation techniques that could be used here, such as [time series cross-validation](https://scikit-learn.org/stable/modules/cross_validation.html#time-series-split), or [leave-one-out cross validation](https://scikit-learn.org/stable/modules/cross_validation.html#leave-one-out-loo) for small datasets. However, for this module, let's start simpler, with train/test split. \n\nUsing a tiny dataset is intentional here. It's good for learning because we can see all the data at once.\n\n## Challenge\n\nIn your assignment, you will do train/test split, based on date.\n\n# Use scikit-learn to fit a multiple regression\n\n## Overview\n\nWe've done train/test split, and we're ready to fit a model. \n\nWe'll proceed in 3 steps. The first 2 are review from the previous module. The 3rd is new.\n\n- Begin with baselines (0 features) \n- Simple regression (1 feature)\n- Multiple regression (2 features)\n\n## Follow Along\n\n### Begin with baselines (0 features)\n\nWhat was the average Incumbent Party Vote Share, in the 1952-2004 elections?\n\n\n```python\ntrain['Incumbent Party Vote Share'].mean()\n```\n\nWhat if we guessed this number for every election? How far off would this be on average?\n\n\n```python\n# Arrange y target vectors\ntarget = 'Incumbent Party Vote Share'\ny_train = train[target]\ny_test = test[target]\n```\n\n\n```python\n# Get mean baseline\nprint('Mean Baseline (using 0 features)')\nguess = y_train.mean()\n```\n\n\n```python\n# Train Error\nfrom sklearn.metrics import mean_absolute_error\ny_pred = [guess] * len(y_train)\nmae = mean_absolute_error(y_train, y_pred)\nprint(f'Train Error (1952-2004 elections): {mae:.2f} percentage points')\n```\n\n\n```python\n# Test Error\ny_pred = [guess] * len(y_test)\nmae = mean_absolute_error(y_test, y_pred)\nprint(f'Test Error (2008-16 elections): {mae:.2f} percentage points')\n```\n\n### Simple regression (1 feature)\n\nMake a scatterplot of the relationship between 1 feature and the target.\n\nWe'll use an economic feature: Average Recent Growth in Personal Incomes. (\"Bread\")\n\n\n```python\nimport pandas as pd\nimport plotly.express as px\n\npx.scatter(\n train,\n x='Average Recent Growth in Personal Incomes',\n y='Incumbent Party Vote Share',\n text='Year',\n title='US Presidential Elections, 1952-2004',\n trendline='ols', # Ordinary Least Squares\n)\n```\n\n1952 & 1968 are outliers: The incumbent party got fewer votes than predicted by the regression. What do you think could explain those years? We'll come back to this soon, but first...\n\nUse scikit-learn to fit the simple regression with one feature.\n\nFollow the [5 step process](https://jakevdp.github.io/PythonDataScienceHandbook/05.02-introducing-scikit-learn.html#Basics-of-the-API), and refer to [Scikit-Learn LinearRegression documentation](https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LinearRegression.html).\n\n\n```python\n# 1. Import the appropriate estimator class from Scikit-Learn\nfrom sklearn.linear_model import LinearRegression\n```\n\n\n```python\n# 2. Instantiate this class\nmodel = LinearRegression()\n```\n\n\n```python\n# 3. Arrange X features matrices (already did y target vectors)\nfeatures = ['Average Recent Growth in Personal Incomes']\nX_train = train[features]\nX_test = test[features]\nprint(f'Linear Regression, dependent on: {features}')\n```\n\n\n```python\n# 4. Fit the model\nmodel.fit(X_train, y_train)\ny_pred = model.predict(X_train)\nmae = mean_absolute_error(y_train, y_pred)\nprint(f'Train Error: {mae:.2f} percentage points')\n```\n\n\n```python\n# 5. Apply the model to new data\ny_pred = model.predict(X_test)\nmae = mean_absolute_error(y_test, y_pred)\nprint(f'Test Error: {mae:.2f} percentage points')\n```\n\nHow does the error compare to the baseline?\n\n### Multiple regression (2 features)\n\nMake a scatterplot of the relationship between 2 features and the target.\n\nWe'll add another feature: US Military Fatalities per Million. (\"Peace\" or the lack thereof.)\n\nRotate the scatterplot to explore the data. What's different about 1952 & 1968?\n\n\n```python\npx.scatter_3d(\n train,\n x='Average Recent Growth in Personal Incomes', \n y='US Military Fatalities per Million', \n z='Incumbent Party Vote Share', \n text='Year', \n title='US Presidential Elections, 1952-2004'\n)\n```\n\nUse scikit-learn to fit a multiple regression with two features.\n\n\n```python\n# TODO: Complete this cell\n\n# Re-arrange X features matrices\nfeatures = ['Average Recent Growth in Personal Incomes', \n 'US Military Fatalities per Million']\nprint(f'Linear Regression, dependent on: {features}')\n\n\n```\n\n\n```python\n# TODO: Fit the model\n\n\n```\n\n\n```python\n# TODO: Apply the model to new data\n\n\n```\n\nHow does the error compare to the prior model?\n\n### Plot the plane of best fit\n\nFor a regression with 1 feature, we plotted the line of best fit in 2D. \n\n(There are many ways to do this. Plotly Express's `scatter` function makes it convenient with its `trendline='ols'` parameter.)\n\nFor a regression with 2 features, we can plot the plane of best fit in 3D!\n\n(Plotly Express has a `scatter_3d` function but it won't plot the plane of best fit for us. But, we can write our own function, with the same \"function signature\" as the Plotly Express API.)\n\n\n```python\nimport itertools\nimport numpy as np\nimport plotly.express as px\nimport plotly.graph_objs as go\nfrom sklearn.linear_model import LinearRegression\n\ndef regression_3d(df, x, y, z, num=100, **kwargs):\n \"\"\"\n Visualize linear regression in 3D: 2 features + 1 target\n \n df : Pandas DataFrame\n x : string, feature 1 column in df\n y : string, feature 2 column in df\n z : string, target column in df\n num : integer, number of quantiles for each feature\n \"\"\"\n \n # Plot data\n fig = px.scatter_3d(df, x, y, z, **kwargs)\n \n # Fit Linear Regression\n features = [x, y]\n target = z\n model = LinearRegression()\n model.fit(df[features], df[target]) \n \n # Define grid of coordinates in the feature space\n xmin, xmax = df[x].min(), df[x].max()\n ymin, ymax = df[y].min(), df[y].max()\n xcoords = np.linspace(xmin, xmax, num)\n ycoords = np.linspace(ymin, ymax, num)\n coords = list(itertools.product(xcoords, ycoords))\n \n # Make predictions for the grid\n predictions = model.predict(coords)\n Z = predictions.reshape(num, num).T\n \n # Plot predictions as a 3D surface (plane)\n fig.add_trace(go.Surface(x=xcoords, y=ycoords, z=Z))\n \n return fig\n```\n\n\n```python\nregression_3d(\n train,\n x='Average Recent Growth in Personal Incomes', \n y='US Military Fatalities per Million', \n z='Incumbent Party Vote Share', \n text='Year', \n title='US Presidential Elections, 1952-2004'\n)\n```\n\nWhere are 1952 & 1968 in relation to the plane? Which elections are the biggest outliers now?\n\nRoll over points on the plane to see predicted incumbent party vote share (z axis), dependent on personal income growth (x axis) and military fatatlies per capita (y axis).\n\n### Get and interpret coefficients\n\nDuring the previous module, we got the simple regression's coefficient and intercept. We plugged these numbers into an equation for the line of best fit, in slope-intercept form: $y = mx + b$\n\nLet's review this objective, but now for multiple regression.\n\nWhat's the equation for the plane of best fit?\n\n$y = \\beta_0 + \\beta_1x_1 + \\beta_2x_2$\n\nCan you relate the intercept and coefficients to what you see in the plot above?\n\n\n```python\nmodel.intercept_, model.coef_\n```\n\n\n```python\nbeta0 = model.intercept_\nbeta1, beta2 = model.coef_\nprint(f'y = {beta0} + {beta1}x1 + {beta2}x2')\n```\n\n\n```python\n# This is easier to read\nprint('Intercept', model.intercept_)\ncoefficients = pd.Series(model.coef_, features)\nprint(coefficients.to_string())\n```\n\nOne of the coefficients is positive, and the other is negative. What does this mean?\n\nLet's look at some scenarios. We'll see that one unit's change in an independent variable results in a coefficient worth of change in the dependent variable.\n\nWhat does the model predict if income growth=0%, fatalities=0\n\n\n```python\nmodel.predict([[0, 0]])\n```\n\nIncome growth = 1% (fatalities = 0)\n\n\n```python\nmodel.predict([[1, 0]])\n```\n\nThe difference between these predictions = ? \n\n\n```python\nmodel.predict([[1, 0]]) - model.predict([[0, 0]])\n```\n\nWhat if... income growth = 2% (fatalities = 0)\n\n\n```python\nmodel.predict([[2, 0]])\n```\n\nThe difference between these predictions = ?\n\n\n```python\nmodel.predict([[2, 0]]) - model.predict([[1, 0]])\n```\n\nWhat if... (income growth=2%) fatalities = 100\n\n\n```python\nmodel.predict([[2, 100]])\n```\n\nThe difference between these predictions = ?\n\n\n```python\nmodel.predict([[2, 100]]) - model.predict([[2, 0]])\n```\n\nWhat if income growth = 3% (fatalities = 100)\n\n\n```python\nmodel.predict([[3, 100]])\n```\n\nThe difference between these predictions = ?\n\n\n```python\nmodel.predict([[3, 100]]) - model.predict([[2, 100]])\n```\n\nWhat if (income growth = 3%) fatalities = 200\n\n\n```python\nmodel.predict([[3, 200]])\n```\n\nThe difference between these predictions = ?\n\n\n```python\nmodel.predict([[3, 200]]) - model.predict([[3, 100]])\n```\n\n## Challenge\n\nIn your assignment, you'll fit a Linear Regression with at least 2 features.\n\n# Understand how ordinary least squares regression minimizes the sum of squared errors\n\n## Overview\n\nSo far, we've evaluated our models by their absolute error. It's an intuitive metric for regression problems.\n\nHowever, ordinary least squares doesn't directly minimize absolute error. Instead, it minimizes squared error.\n\n\n\n\nIn this section, we'll introduce two new regression metrics: \n\n- Squared error\n- $R^2$\n\n\nWe'll demostrate two possible methods to minimize squared error:\n\n- Guess & check\n- Linear Algebra\n\n## Follow Along\n\n### Guess & Check\n\nThis function visualizes squared errors. We'll go back to simple regression with 1 feature, because it's much easier to visualize.\n\nUse the function's m & b parameters to \"fit the model\" manually. Guess & check what values of m & b minimize squared error.\n\n\n```python\nfrom matplotlib.patches import Rectangle\nimport matplotlib.pyplot as plt\nimport numpy as np\nfrom sklearn.metrics import mean_absolute_error, mean_squared_error, r2_score\n\ndef squared_errors(df, feature, target, m, b):\n \"\"\"\n Visualize linear regression, with squared errors,\n in 2D: 1 feature + 1 target.\n \n Use the m & b parameters to \"fit the model\" manually.\n \n df : Pandas DataFrame\n feature : string, feature column in df\n target : string, target column in df\n m : numeric, slope for linear equation\n b : numeric, intercept for linear requation\n \"\"\"\n \n # Plot data\n fig = plt.figure(figsize=(7,7))\n ax = plt.axes()\n df.plot.scatter(feature, target, ax=ax)\n \n # Make predictions\n x = df[feature]\n y = df[target]\n y_pred = m*x + b\n \n # Plot predictions\n ax.plot(x, y_pred)\n \n # Plot squared errors\n xmin, xmax = ax.get_xlim()\n ymin, ymax = ax.get_ylim()\n scale = (xmax-xmin)/(ymax-ymin)\n for x, y1, y2 in zip(x, y, y_pred):\n bottom_left = (x, min(y1, y2))\n height = abs(y1 - y2)\n width = height * scale\n ax.add_patch(Rectangle(xy=bottom_left, width=width, height=height, alpha=0.1))\n \n # Print regression metrics\n mse = mean_squared_error(y, y_pred)\n rmse = np.sqrt(mse)\n mae = mean_absolute_error(y, y_pred)\n r2 = r2_score(y, y_pred)\n print('Mean Squared Error:', mse)\n print('Root Mean Squared Error:', rmse)\n print('Mean Absolute Error:', mae)\n print('R^2:', r2)\n```\n\nHere's what the mean baseline looks like:\n\n\n```python\nfeature = 'Average Recent Growth in Personal Incomes'\nsquared_errors(train, feature, target, m=0, b=y_train.mean())\n```\n\nNotice that $R^2$ is exactly zero. \n\n[$R^2$ represents the proportion of the variance for a dependent variable that is explained by the independent variable(s).](https://en.wikipedia.org/wiki/Coefficient_of_determination)\n\nThe mean baseline uses zero independent variables and explains none of the variance in the dependent variable, so its $R^2$ score is zero.\n\nThe highest possible $R^2$ score is 1. The lowest possible *Train* $R^2$ score with ordinary least squares regression is 0.\n\nIn this demo, it's possible to get a negative Train $R^2$, if you manually set values of m & b that are worse than the mean baseline. But that wouldn't happen in the real world.\n\nHowever, in the real world, it _is_ possible to get a negative *Test/Validation* $R^2$. It means that your *Test/Validation* predictions are worse than if you'd constantly predicted the mean of the *Test/Validation* set.\n\n---\n\nNow that we've visualized the squared errors for the mean baseline, let's guess & check some better values for the m & b parameters:\n\n\n```python\nsquared_errors(train, feature, target, m=3, b=46)\n```\n\nYou can run the function repeatedly, with different values for m & b.\n\nHow do you interpret each metric you see?\n\n- Mean Squared Error\n- Root Mean Squared Error\n- Mean Absolute Error\n- $R^2$\n\nDoes guess & check really get used in machine learning? Sometimes! Some complex functions are hard to minimize, so we use a sophisticated form of guess & check called \"gradient descent\", which you'll learn about in Unit 4.\n\nFortunately, we don't need to use guess & check for ordinary least squares regression. We have a solution, using linear algebra!\n\n\n### Linear Algebra\n\nThe same result that is found by minimizing the sum of the squared errors can be also found through a linear algebra process known as the \"Least Squares Solution:\"\n\n\\begin{align}\n\\hat{\\beta} = (X^{T}X)^{-1}X^{T}y\n\\end{align}\n\nBefore we can work with this equation in its linear algebra form we have to understand how to set up the matrices that are involved in this equation. \n\n#### The $\\beta$ vector\n\nThe $\\beta$ vector represents all the parameters that we are trying to estimate, our $y$ vector and $X$ matrix values are full of data from our dataset. The $\\beta$ vector holds the variables that we are solving for: $\\beta_0$ and $\\beta_1$\n\nNow that we have all of the necessary parts we can set them up in the following equation:\n\n\\begin{align}\ny = X \\beta + \\epsilon\n\\end{align}\n\nSince our $\\epsilon$ value represents **random** error we can assume that it will equal zero on average.\n\n\\begin{align}\ny = X \\beta\n\\end{align}\n\nThe objective now is to isolate the $\\beta$ matrix. We can do this by pre-multiplying both sides by \"X transpose\" $X^{T}$.\n\n\\begin{align}\nX^{T}y = X^{T}X \\beta\n\\end{align}\n\nSince anything times its transpose will result in a square matrix, if that matrix is then an invertible matrix, then we should be able to multiply both sides by its inverse to remove it from the right hand side. (We'll talk tomorrow about situations that could lead to $X^{T}X$ not being invertible.)\n\n\\begin{align}\n(X^{T}X)^{-1}X^{T}y = (X^{T}X)^{-1}X^{T}X \\beta\n\\end{align}\n\nSince any matrix multiplied by its inverse results in the identity matrix, and anything multiplied by the identity matrix is itself, we are left with only $\\beta$ on the right hand side:\n\n\\begin{align}\n(X^{T}X)^{-1}X^{T}y = \\hat{\\beta}\n\\end{align}\n\nWe will now call it \"beta hat\" $\\hat{\\beta}$ because it now represents our estimated values for $\\beta_0$ and $\\beta_1$\n\n#### Lets calculate our $\\beta$ parameters with numpy!\n\n\n```python\n# This is NOT something you'll be tested on. It's just a demo.\n\n# X is a matrix. Add column of constants for fitting the intercept.\ndef add_constant(X):\n constant = np.ones(shape=(len(X),1))\n return np.hstack((constant, X))\nX = add_constant(train[features].values)\nprint('X')\nprint(X)\n\n# y is a column vector\ny = train[target].values[:, np.newaxis]\nprint('y')\nprint(y)\n\n# Least squares solution in code\nX_transpose = X.T\nX_transpose_X = X_transpose @ X\nX_transpose_X_inverse = np.linalg.inv(X_transpose_X)\nX_transpose_y = X_transpose @ y\nbeta_hat = X_transpose_X_inverse @ X_transpose_y\n\nprint('Beta Hat')\nprint(beta_hat)\n```\n\n\n```python\n# Scikit-learn gave the exact same results!\nmodel.intercept_, model.coef_\n```\n\n# Define overfitting/underfitting and the bias/variance tradeoff\n\n## Overview\n\nRead [_Python Data Science Handbook,_ Chapter 5.3](https://jakevdp.github.io/PythonDataScienceHandbook/05.03-hyperparameters-and-model-validation.html#The-Bias-variance-trade-off). Jake VanderPlas explains overfitting & underfitting:\n\n> Fundamentally, the question of \"the best model\" is about finding a sweet spot in the tradeoff between bias and variance. Consider the following figure, which presents two regression fits to the same dataset:\n> \n>\n>\n> The model on the left attempts to find a straight-line fit through the data. Because the data are intrinsically more complicated than a straight line, the straight-line model will never be able to describe this dataset well. Such a model is said to _underfit_ the data: that is, it does not have enough model flexibility to suitably account for all the features in the data; another way of saying this is that the model has high _bias_.\n>\n> The model on the right attempts to fit a high-order polynomial through the data. Here the model fit has enough flexibility to nearly perfectly account for the fine features in the data, but even though it very accurately describes the training data, its precise form seems to be more reflective of the particular noise properties of the data rather than the intrinsic properties of whatever process generated that data. Such a model is said to _overfit_ the data: that is, it has so much model flexibility that the model ends up accounting for random errors as well as the underlying data distribution; another way of saying this is that the model has high _variance_.\n\nVanderPlas goes on to connect these concepts to the \"bias/variance tradeoff\":\n\n> From the scores associated with these two models, we can make an observation that holds more generally:\n>\n>- For high-bias models, the performance of the model on the validation set is similar to the performance on the training set.\n>\n>- For high-variance models, the performance of the model on the validation set is far worse than the performance on the training set.\n>\n> If we imagine that we have some ability to tune the model complexity, we would expect the training score and validation score to behave as illustrated in the following figure:\n>\n>\n>\n> The diagram shown here is often called a validation curve, and we see the following essential features:\n>\n>- The training score is everywhere higher than the validation score. This is generally the case: the model will be a better fit to data it has seen than to data it has not seen.\n>- For very low model complexity (a high-bias model), the training data is under-fit, which means that the model is a poor predictor both for the training data and for any previously unseen data.\n>- For very high model complexity (a high-variance model), the training data is over-fit, which means that the model predicts the training data very well, but fails for any previously unseen data.\n>- For some intermediate value, the validation curve has a maximum. This level of complexity indicates a suitable trade-off between bias and variance.\n>\n>The means of tuning the model complexity varies from model to model.\n\nSo far, our only \"means of tuning the model complexity\" has been selecting one feature or two features for our linear regression models. But we'll quickly start to select more features, and more complex models, with more \"hyperparameters.\"\n\nThis is just a first introduction to underfitting & overfitting. We'll continue to learn about this topic all throughout this unit.\n\n## Follow Along\n\nLet's make our own Validation Curve, by tuning a new type of model complexity: polynomial degrees in a linear regression.\n\nGo back to the the NYC Tribeca condo sales data\n\n\n```python\n# Read NYC Tribeca condo sales data, from first 4 months of 2019.\n# Dataset has 90 rows, 9 columns.\ndf = pd.read_csv(DATA_PATH+'condos/tribeca.csv')\nassert df.shape == (90, 9)\n\n# Arrange X features matrix & y target vector\nfeatures = ['GROSS_SQUARE_FEET']\ntarget = 'SALE_PRICE'\nX = df[features]\ny = df[target]\n```\n\nDo random [train/test split](https://scikit-learn.org/stable/modules/generated/sklearn.model_selection.train_test_split.html)\n\n\n```python\nfrom sklearn.model_selection import train_test_split\nX_train, X_test, y_train, y_test = train_test_split(X, y, random_state=11)\n```\n\nRepeatedly fit increasingly complex models, and keep track of the scores\n\n\n```python\nfrom IPython.display import display, HTML\nfrom sklearn.linear_model import LinearRegression\nfrom sklearn.metrics import r2_score\nfrom sklearn.pipeline import make_pipeline\nfrom sklearn.preprocessing import PolynomialFeatures\n\n\n# Credit for PolynomialRegression: Jake VanderPlas, Python Data Science Handbook, Chapter 5.3\n# https://jakevdp.github.io/PythonDataScienceHandbook/05.03-hyperparameters-and-model-validation.html#Validation-curves-in-Scikit-Learn\ndef PolynomialRegression(degree=2, **kwargs):\n return make_pipeline(PolynomialFeatures(degree), \n LinearRegression(**kwargs))\n\n\npolynomial_degrees = range(1, 10, 2)\ntrain_r2s = []\ntest_r2s = []\n\nfor degree in polynomial_degrees:\n model = PolynomialRegression(degree)\n display(HTML(f'Polynomial degree={degree}'))\n \n model.fit(X_train, y_train)\n train_r2 = model.score(X_train, y_train)\n test_r2 = model.score(X_test, y_test)\n display(HTML(f'Train R2 {train_r2:.2f}'))\n display(HTML(f'Test R2 {test_r2:.2f}'))\n\n plt.scatter(X_train, y_train, color='blue', alpha=0.5)\n plt.scatter(X_test, y_test, color='red', alpha=0.5)\n plt.xlabel(features)\n plt.ylabel(target)\n \n x_domain = np.linspace(X.min(), X.max())\n curve = model.predict(x_domain)\n plt.plot(x_domain, curve, color='blue')\n plt.show()\n display(HTML('
                                        '))\n \n train_r2s.append(train_r2)\n test_r2s.append(test_r2)\n \ndisplay(HTML('Validation Curve'))\nplt.plot(polynomial_degrees, train_r2s, color='blue', label='Train')\nplt.plot(polynomial_degrees, test_r2s, color='red', label='Test')\nplt.xlabel('Model Complexity (Polynomial Degree)')\nplt.ylabel('R^2 Score')\nplt.legend()\nplt.show()\n```\n\nAs model complexity increases, what happens to Train $R^2$ and Test $R^2$?\n\n# Review\n\nIn your assignment, you'll continue to **predict how much it costs to rent an apartment in NYC,** using the dataset from renthop.com.\n\n\n- Do train/test split. Use data from April & May 2016 to train. Use data from June 2016 to test.\n- Engineer at least two new features.\n- Fit a linear regression model with at least two features.\n- Get the model's coefficients and intercept.\n- Get regression metrics RMSE, MAE, and $R^2$, for both the train and test sets.\n\nYou've been provided with a separate notebook for your assignment, which has all the instructions and stretch goals. What's the best test MAE you can get? Share your score and features used with your cohort on Slack!\n\n# Sources\n\n#### Train/Test Split\n- James, Witten, Hastie, Tibshirani, [_An Introduction to Statistical Learning_](http://faculty.marshall.usc.edu/gareth-james/ISL/), Chapter 2.2, Assessing Model Accuracy\n- Hyndman, Athanasopoulos, [_Forecasting,_ Chapter 3.4,](https://otexts.com/fpp2/accuracy.html) Evaluating forecast accuracy\n- Rachel Thomas, [How (and why) to create a good validation set](https://www.fast.ai/2017/11/13/validation-sets/)\n\n#### Bias-Variance Tradeoff\n- Jake VanderPlas, [_Python Data Science Handbook,_ Chapter 5.3](https://jakevdp.github.io/PythonDataScienceHandbook/05.03-hyperparameters-and-model-validation.html#The-Bias-variance-trade-off), Hyperparameters and Model Validation\n- StatQuest, [Machine Learning Fundamentals: Bias and Variance](https://youtu.be/EuBBz3bI-aA) (6.5 minutes)\n\n#### \"Bread and Peace\" Background\n- Douglas Hibbs, [Background Information on the \u2018Bread and Peace\u2019 Model of Voting in Postwar US Presidential Elections](https://douglas-hibbs.com/background-information-on-bread-and-peace-voting-in-us-presidential-elections/)\n- Nate Silver, [What Do Economic Models Really Tell Us About Elections?](https://fivethirtyeight.com/features/what-do-economic-models-really-tell-us-about-elections/)\n\n\n#### \"Bread and Peace\" Data Sources & Definitions\n- 1952-2012: Douglas Hibbs, [2014 lecture at Deakin University Melbourne](http://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf), Slide 40\n- 2016, Vote Share: [The American Presidency Project](https://www.presidency.ucsb.edu/statistics/elections)\n- 2016, Recent Growth in Personal Incomes: [The 2016 election economy: the \"Bread and Peace\" model final forecast](https://angrybearblog.com/2016/11/the-2016-election-economy-the-bread-and-peace-model-final-forecast.html)\n- 2016, US Military Fatalities: Assumption that Afghanistan War fatalities in 2012-16 occured at the same rate as 2008-12\n\n> Fatalities denotes the cumulative number of American military fatalities per millions of US population the in Korea, Vietnam, Iraq and Afghanistan wars during the presidential terms preceding the 1952, 1964, 1968, 1976 and 2004, 2008 and 2012 elections. \u2014[Hibbs](http://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf), Slide 33\n", "meta": {"hexsha": "ce5e62e00cc4e83058a4bdea495ff460147c0cd3", "size": 59206, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Unit 212_Linear_Regression_2/LS_DS_212.ipynb", "max_stars_repo_name": "jiobu1/DS-Unit-2-Linear-Models", "max_stars_repo_head_hexsha": "989c037b5a8af8de3da69027eed1079e9430c9d2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Unit 212_Linear_Regression_2/LS_DS_212.ipynb", "max_issues_repo_name": "jiobu1/DS-Unit-2-Linear-Models", "max_issues_repo_head_hexsha": "989c037b5a8af8de3da69027eed1079e9430c9d2", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Unit 212_Linear_Regression_2/LS_DS_212.ipynb", "max_forks_repo_name": "jiobu1/DS-Unit-2-Linear-Models", "max_forks_repo_head_hexsha": "989c037b5a8af8de3da69027eed1079e9430c9d2", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.7085062241, "max_line_length": 676, "alphanum_fraction": 0.5943485458, "converted": true, "num_tokens": 8181, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593452091672, "lm_q2_score": 0.668880247169804, "lm_q1q2_score": 0.4163507606766621}} {"text": "# Phosphofructokinase (PFK) Model Construction\n\nBased on Chapter 14 of Systems Biology: Simulation of Dynamic Network States\n\nTo construct the phosphofructokinase module, first we import **MASSpy** and other essential packages. Constants used throughout the notebook are also defined.\n\n\n```python\nfrom operator import attrgetter\nfrom os import path\n\nimport matplotlib.pyplot as plt\n\nimport numpy as np\n\nfrom scipy import optimize\n\nimport sympy as sym\n\nfrom cobra import DictList\n\nfrom mass import MassConfiguration, MassMetabolite, Simulation, UnitDefinition\nfrom mass.enzyme_modules import EnzymeModule\nfrom mass.io import json, sbml\nfrom mass.test import create_test_model\nfrom mass.util.expressions import Keq2k, k2Keq, strip_time\nfrom mass.util.matrix import matrix_rank\nfrom mass.util.qcqa import qcqa_model\n\nmass_config = MassConfiguration()\n\nmass_config.irreversible_Keq = float(\"inf\")\n```\n\nNote that the total enzyme concentration of PFK is $33 nM = 0.033 \\mu M = 0.000033 mM$.\n\nFor the construction of the `EnzymeModule` for PFK, the following assumptions were made:\n\n1. The enzyme is a homotetramer.\n2. The enzyme binding and catalyzation of substrates occurs in an ordered sequential mechanism.\n3. The mechanism of allosteric regulation is based on the Monod-Wyman-Changeux (MWC) model for allosteric transitions of homoproteins.\n\n## Module Construction\nThe first step of creating the PFK module is to define the `EnzymeModule`. The `EnzymeModule` is an extension of the `MassModel`, with additional enzyme-specific attributes that aid in the construction, validation, and utilization of the module.\n\n__Note:__ All `EnzymeModule` specific attributes start will start the prefix \"enzyme\" or \"enzyme_module\".\n\n\n```python\nPFK = EnzymeModule(\"PFK\", name=\"Phosphofructokinase\", \n subsystem=\"Glycolysis\")\n```\n\n### Metabolites\n#### Ligands\nThe next step is to define all of the metabolites using the `MassMetabolite` object. For `EnzymeModule` objects, the `MassMetabolite` objects will be refered to as ligands, for these `MassMetabolite` form a complex with the enzyme to serve some biological purpose. Some considerations for this step include the following:\n\n1. It is important to use a clear and consistent format for identifiers and names when defining the `MassMetabolite` objects for various reasons, some of which include improvements to model clarity and utility, assurance of unique identifiers (required to add metabolites to the model), and consistency when collaborating and communicating with others. \n\n2. In order to ensure our model is physiologically accurate, it is important to provide the `formula` argument with a string representing the chemical formula for each metabolite, and the `charge` argument with an integer representing the metabolite's ionic charge (Note that neutrally charged metabolites are provided with 0). These attributes can always be set later if necessary using the `formula` and `charge` attribute setter methods.\n\n3. To indicate that the cytosol is the cellular compartment in which the reactions occur, the string \"c\" is provided to the `compartment` argument.\n\nThis model will be created using identifiers and names found in the [BiGG Database](http://bigg.ucsd.edu/).\n\nThe ligands correspond to the activators, inhibitors, cofactors, substrates, and products involved in the enzyme catalyzed reaction. In this model, there are 6 species which must be considered.\n\n\n```python\nf6p_c = MassMetabolite(\n \"f6p_c\",\n name=\"D-Fructose 6-phosphate\",\n formula=\"C6H11O9P\",\n charge=-2,\n compartment=\"c\")\nfdp_c = MassMetabolite(\n \"fdp_c\",\n name=\"D-Fructose 1,6-bisphosphate\",\n formula=\"C6H10O12P2\",\n charge=-4,\n compartment=\"c\")\natp_c = MassMetabolite(\n \"atp_c\",\n name=\"ATP\",\n formula=\"C10H12N5O13P3\",\n charge=-4,\n compartment=\"c\")\nadp_c = MassMetabolite(\n \"adp_c\",\n name=\"ADP\",\n formula=\"C10H12N5O10P2\",\n charge=-3,\n compartment=\"c\")\namp_c = MassMetabolite(\n \"amp_c\",\n name=\"AMP\",\n formula=\"C10H12N5O7P\",\n charge=-2,\n compartment=\"c\")\nh_c = MassMetabolite(\n \"h_c\",\n name=\"H+\",\n formula=\"H\",\n charge=1,\n compartment=\"c\") \n```\n\nAfter generating the ligands, they are added to the `EnzymeModule` through the `add_metabolites` method. The ligands of the `EnzymeModule` can be viewed as a `DictList` through the `enzyme_module_ligands` attribute.\n\n\n```python\n# Add the metabolites to the EnzymeModule\nPFK.add_metabolites([f6p_c, fdp_c, atp_c, adp_c, amp_c, h_c])\n# Access DictList of ligands and print\nprint(\"All {0} Ligands: {1}\".format(\n PFK.id, \"; \".join([m.id for m in PFK.enzyme_module_ligands])))\n```\n\n All PFK Ligands: f6p_c; fdp_c; atp_c; adp_c; amp_c; h_c\n\n\nThe `enzyme_module_ligands_categorized` attribute can be used to assign metabolites to groups of user-defined categories by providing a dictionary where keys are the categories and values are the metabolites. Note that any metabolite can be placed in more than one category.\n\n\n```python\nPFK.enzyme_module_ligands_categorized = {\n \"Substrates\": f6p_c,\n \"Cofactors\": atp_c,\n \"Activators\": amp_c,\n \"Inhibitors\": atp_c,\n \"Products\": [fdp_c, adp_c, h_c]}\n\n# Access DictList of ligands and print\nprint(\"All {0} ligands ({1} total):\\n{2}\\n\".format(\n PFK.id, len(PFK.enzyme_module_ligands),\n str([m.id for m in PFK.enzyme_module_ligands])))\n\n# Access categorized attribute for ligands and print\nfor group in PFK.enzyme_module_ligands_categorized:\n print(\"{0}: {1}\".format(\n group.id, str([m.id for m in group.members])))\n```\n\n All PFK ligands (6 total):\n ['f6p_c', 'fdp_c', 'atp_c', 'adp_c', 'amp_c', 'h_c']\n \n Substrates: ['f6p_c']\n Cofactors: ['atp_c']\n Activators: ['amp_c']\n Inhibitors: ['atp_c']\n Products: ['fdp_c', 'adp_c', 'h_c']\n\n\n#### EnzymeModuleForms\n\nThe next step is to define the various states of the enzyme and enzyme-ligand complexes. These states can be represented through an `EnzymeModuleForm` object. Just like how `EnzymeModule` objects extend `MassModels`, the `EnzymeModuleForm` objects extend `MassMetabolite` objects, giving them the same functionality as a `MassMetabolite`. However, there are two important additional attrubutes that are specific to the `EnzymeModuleForm`.\n\n* The first attribute is the `enzyme_module_id`. It is meant to hold the identifier or name of the corresponding `EnzymeModule`.\n* The second attribute is the `bound_metabolites` attribute, designed to contain metabolites bound to the enzymatic site(s).\n* Automatic generation of the `name`, `formula`, and `charge` attributes attributes utilize the `bound_metabolites` attribute, which can aid in identification of `EnzymeModuleForm` and mass and charge balancing of the reactions.\n\nThe most convenient way to make an `EnzymeModuleForm` is through the `EnzymeModule.make_enzyme_module_form` method. There are several reasons to use this method to generate the `EnzymeModuleForm` objects:\n\n1. The only requirement to creating an `EnzymeModuleForm` is the identifier.\n2. A string can optionally be provided for the `name` argument to set the corresponding `name` attribute, or it can automatically be generated and set by setting the string \"Automatic\" (case sensitve). \n3. The `enzyme_module_id`, `formula` and `charge` attributes are set based on the identifier of the EnzymeModule and the MassMetabolite objects found in `bound_metabolites`.\n4. Just like the `enzyme_module_ligands_categorized` attribute, there is the `enzyme_module_forms_categorized` attribute that behaves in a similar manner. Categories can be set at the time of construction by providing a string or a list of strings to the `categories` argument. \n5. `EnzymeModuleForm` objects are automatically added to the `EnzymeModule` once created.\n\nFor this module, there are 20 `EnzymeModuleForm` objects that must be created. Because of the assumptions made for this module, a loop can be used to help automate the construction of the `EnzymeModuleForm` objects.\n\n\n```python\n# Number of identical subunits\nn_subunits = 4\n\nfor i in range(n_subunits + 1):\n # Make enzyme module forms per number of bound activators (Up to 4 Total)\n PFK.make_enzyme_module_form(\n \"pfk_R{0:d}_c\".format(i), \n name=\"Automatic\", \n categories=(\"Relaxed\", \"Free_Catalytic\"),\n bound_metabolites={amp_c: i},\n compartment=\"c\");\n\n PFK.make_enzyme_module_form(\n \"pfk_R{0:d}_A_c\".format(i), \n name=\"Automatic\", \n categories=(\"Relaxed\", \"Complexed_ATP\"),\n bound_metabolites={atp_c: 1, amp_c: i},\n compartment=\"c\");\n \n PFK.make_enzyme_module_form(\n \"pfk_R{0:d}_AF_c\".format(i), \n name=\"Automatic\", \n categories=(\"Relaxed\", \"Complexed_ATP_F6P\"),\n bound_metabolites={atp_c: 1, f6p_c: 1, amp_c: i},\n compartment=\"c\");\n\n # Make enzyme module forms per number of bound inhibitors (Up to 4 Total)\n PFK.make_enzyme_module_form(\n \"pfk_T{0:d}_c\".format(i), \n name=\"Automatic\", \n categories=\"Tense\",\n bound_metabolites={atp_c: i},\n compartment=\"c\");\n\n# Access DictList of enzyme module forms and print\nprint(\"All {0} enzyme module forms ({1} total):\\n{2}\\n\".format(\n PFK.id, len(PFK.enzyme_module_forms),\n str([m.id for m in PFK.enzyme_module_forms])))\n\n# Access categorized attribute for enzyme module forms and print\nfor group in PFK.enzyme_module_forms_categorized:\n print(\"{0}: {1}\\n\".format(\n group.id, str(sorted([m.id for m in group.members]))))\n```\n\n All PFK enzyme module forms (20 total):\n ['pfk_R0_c', 'pfk_R0_A_c', 'pfk_R0_AF_c', 'pfk_T0_c', 'pfk_R1_c', 'pfk_R1_A_c', 'pfk_R1_AF_c', 'pfk_T1_c', 'pfk_R2_c', 'pfk_R2_A_c', 'pfk_R2_AF_c', 'pfk_T2_c', 'pfk_R3_c', 'pfk_R3_A_c', 'pfk_R3_AF_c', 'pfk_T3_c', 'pfk_R4_c', 'pfk_R4_A_c', 'pfk_R4_AF_c', 'pfk_T4_c']\n \n Relaxed: ['pfk_R0_AF_c', 'pfk_R0_A_c', 'pfk_R0_c', 'pfk_R1_AF_c', 'pfk_R1_A_c', 'pfk_R1_c', 'pfk_R2_AF_c', 'pfk_R2_A_c', 'pfk_R2_c', 'pfk_R3_AF_c', 'pfk_R3_A_c', 'pfk_R3_c', 'pfk_R4_AF_c', 'pfk_R4_A_c', 'pfk_R4_c']\n \n Free_Catalytic: ['pfk_R0_c', 'pfk_R1_c', 'pfk_R2_c', 'pfk_R3_c', 'pfk_R4_c']\n \n Complexed_ATP: ['pfk_R0_A_c', 'pfk_R1_A_c', 'pfk_R2_A_c', 'pfk_R3_A_c', 'pfk_R4_A_c']\n \n Complexed_ATP_F6P: ['pfk_R0_AF_c', 'pfk_R1_AF_c', 'pfk_R2_AF_c', 'pfk_R3_AF_c', 'pfk_R4_AF_c']\n \n Tense: ['pfk_T0_c', 'pfk_T1_c', 'pfk_T2_c', 'pfk_T3_c', 'pfk_T4_c']\n \n\n\n## Reactions\n### EnzymeModuleReactions\nOnce all of the `MassMetabolite` and `EnzymeModuleForm` objects have been created, the next step is to define all of the enzyme-ligand binding reactions and conformation trasitions that occur in its mechanism.\n\nThese reactions can be represented through an `EnzymeModuleReaction` object. As with the previous enzyme objects, `EnzymeModuleReactions` extend `MassReaction` objects to maintain the same functionality. However, as with the `EnzymeModuleForm`, the `EnzymeModuleReaction` has additional enzyme-specific attributes, such as the `enzyme_module_id`.\n\nThe most conveient way to make an `EnzymeModuleReaction` is through the `EnzymeModule.make_enzyme_module_reaction` method. There are several reasons to use this method to generate the EnzymeModuleReactions:\n\n1. The only requirement to creating an `EnzymeModuleReaction` is an identifier.\n2. A string can optionally be provided for the `name` argument to set the corresponding `name` attribute, or it can automatically be generated and set by setting the string \"Automatic\" (case sensitve). \n3. There is an `enzyme_module_reactions_categorized` attribute that behaves in a similar manner as the previous categorized attributes. Categories can be set at the time of construction by providing a string or a list of strings to the `categories` argument. \n4. `MassMetabolite` and `EnzymeModuleForm` objects that already exist in the `EnzymeModule` can be directly added to the newly created `EnzymeModuleReaction` by providing a dictionary to the optional `metabolites_to_add` argument using string identifiers (or the objects) as keys and their stoichiometric coefficients as the values.\n5. `EnzymeModuleReactions` are automatically added to the `EnzymeModule` once created.\n\nFor this module, there are 24 `EnzymeModuleReactions` that must be created. Because of the assumptions made for this module, a loop can be used to help automate the construction of the `EnzymeModuleReactions`.\n\n\n```python\nfor i in range(n_subunits + 1):\n # Make reactions for enzyme-ligand binding and catalytzation per number of bound activators (Up to 4 Total)\n PFK.make_enzyme_module_reaction(\n \"PFK_R{0:d}1\".format(i), \n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=True,\n categories=\"atp_c_binding\",\n metabolites_to_add={\n \"pfk_R{0:d}_c\".format(i): -1, \n \"atp_c\": -1, \n \"pfk_R{0:d}_A_c\".format(i): 1})\n \n PFK.make_enzyme_module_reaction(\n \"PFK_R{0:d}2\".format(i), \n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=True,\n categories=\"f6p_c_binding\",\n metabolites_to_add={\n \"pfk_R{0:d}_A_c\".format(i): -1, \n \"f6p_c\": -1, \n \"pfk_R{0:d}_AF_c\".format(i): 1})\n \n PFK.make_enzyme_module_reaction(\n \"PFK_R{0:d}3\".format(i), \n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=False,\n categories=\"catalyzation\",\n metabolites_to_add={\n \"pfk_R{0:d}_AF_c\".format(i): -1, \n \"pfk_R{0:d}_c\".format(i): 1, \n \"adp_c\": 1, \n \"fdp_c\": 1,\n \"h_c\": 1})\n \n if i < n_subunits:\n # Make enzyme reactions for enzyme-activator binding\n PFK.make_enzyme_module_reaction(\n \"PFK_R{0:d}0\".format(i + 1), \n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=True,\n categories=\"amp_c_activation\",\n metabolites_to_add={\n \"pfk_R{0:d}_c\".format(i): -1, \n \"amp_c\": -1, \n \"pfk_R{0:d}_c\".format(i + 1): 1})\n\n # Make enzyme reactions for enzyme-inhibitor binding\n PFK.make_enzyme_module_reaction(\n \"PFK_T{0:d}\".format(i + 1), \n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=True,\n categories=\"atp_c_inhibition\",\n metabolites_to_add={\n \"pfk_T{0:d}_c\".format(i): -1, \n \"atp_c\": -1, \n \"pfk_T{0:d}_c\".format(i + 1): 1})\n\n# Make reaction representing enzyme transition from R to T state\nPFK.make_enzyme_module_reaction(\n \"PFK_L\",\n name=\"Automatic\",\n subsystem=\"Glycolysis\",\n reversible=True,\n categories=\"RT_transition\",\n metabolites_to_add={\n \"pfk_R0_c\": -1, \n \"pfk_T0_c\": 1})\n\n# Access DictList of enzyme module reactions and print\nprint(\"All {0} enzyme module reactions ({1} total):\\n{2}\\n\".format(\n PFK.id, len(PFK.enzyme_module_reactions),\n str([m.name for m in PFK.enzyme_module_reactions])))\n\n# Access categorized attribute for enzyme module reactions and print\nfor group in PFK.enzyme_module_reactions_categorized:\n print(\"{0}: {1}\\n\".format(\n group.id, str(sorted([m.id for m in group.members]))))\n```\n\n All PFK enzyme module reactions (24 total):\n ['pfk_R0-atp binding', 'pfk_R0_A-f6p binding', 'pfk_R0_AF catalyzation', 'pfk_R0-amp binding', 'pfk_T0-atp binding', 'pfk_R1-atp binding', 'pfk_R1_A-f6p binding', 'pfk_R1_AF catalyzation', 'pfk_R1-amp binding', 'pfk_T1-atp binding', 'pfk_R2-atp binding', 'pfk_R2_A-f6p binding', 'pfk_R2_AF catalyzation', 'pfk_R2-amp binding', 'pfk_T2-atp binding', 'pfk_R3-atp binding', 'pfk_R3_A-f6p binding', 'pfk_R3_AF catalyzation', 'pfk_R3-amp binding', 'pfk_T3-atp binding', 'pfk_R4-atp binding', 'pfk_R4_A-f6p binding', 'pfk_R4_AF catalyzation', 'pfk_R0-pfk_T0 transition']\n \n atp_c_binding: ['PFK_R01', 'PFK_R11', 'PFK_R21', 'PFK_R31', 'PFK_R41']\n \n f6p_c_binding: ['PFK_R02', 'PFK_R12', 'PFK_R22', 'PFK_R32', 'PFK_R42']\n \n catalyzation: ['PFK_R03', 'PFK_R13', 'PFK_R23', 'PFK_R33', 'PFK_R43']\n \n amp_c_activation: ['PFK_R10', 'PFK_R20', 'PFK_R30', 'PFK_R40']\n \n atp_c_inhibition: ['PFK_T1', 'PFK_T2', 'PFK_T3', 'PFK_T4']\n \n RT_transition: ['PFK_L']\n \n\n\n### Create and Unify Rate Parameters\nThe next step is to unify rate parameters of binding steps that are not unique, allowing for those parameter values to be defined once and stored in the same place. Therefore, custom rate laws with custom parameters are used to reduce the number of parameters that need to be defined and better represent the module.\n\nThe rate law parameters can be unified using the `EnzymeModule.unify_rate_parameters` class method. This method requires a list of reactions whose rate laws that should be identical, along with a string representation of the new identifier to use on the unified parameters. There is also the optional `prefix ` argument, which if set to True, will ensure the new parameter identifiers are prefixed with the `EnzymeModule` identifier. This can be used to help prevent custom parameters from being replaced when multiple models are merged.\n\n#### Allosteric Transitions: Symmetry Model\n\nOnce rate parameters are unified, the allosteric regulation of this enzyme must be accounted for. Because this module is to be based on the (Monod-Wyman-Changeux) MWC model for ligand binding and allosteric regulation, the rate laws of the allosteric binding reactions must be adjusted to reflect the symmetry in the module using the number of identical binding sites to help determine the scalars for the parameters. \n\nFor this module, PFK is considered a homotetramer, meaning it has four identical subunits $\\nu = 4$. Each subunit can be allosterically activated by AMP or inhibited by ATP. The helper functions `k2Keq`, `Keq2k`, and `strip_time` from the `mass.util` submodule will be used to help facilitate the rate law changes in this example so that the final rate laws are dependent on the forward rate (kf) and equilibrium (Keq) constants.\n\n\n```python\nabbreviations = [\"A\", \"F\", \"I\", \"ACT\"]\nligands = [atp_c, f6p_c, atp_c, amp_c]\n\nfor met, unified_id in zip(ligands, abbreviations):\n category = {\"A\": \"binding\",\n \"F\": \"binding\",\n \"I\": \"inhibition\",\n \"ACT\": \"activation\"}[unified_id]\n group = PFK.enzyme_module_reactions_categorized.get_by_id(\n \"_\".join((met.id, category)))\n reactions = sorted(group.members, key=attrgetter(\"id\"))\n PFK.unify_rate_parameters(reactions, unified_id,\n rate_type=2, enzyme_prefix=True)\n # Add the coefficients to make symmetry model rate laws for activation and inhibition \n if unified_id in [\"I\", \"ACT\"]:\n for i, reaction in enumerate(reactions):\n custom_rate = str(strip_time((reaction.rate)))\n custom_rate = custom_rate.replace(\n \"kf_\", \"{0:d}*kf_\".format(n_subunits - i))\n custom_rate = custom_rate.replace(\n \"kr_\", \"{0:d}*kr_\".format(i + 1))\n PFK.add_custom_rate(reaction, custom_rate)\n \nPFK.unify_rate_parameters(\n PFK.enzyme_module_reactions_categorized.get_by_id(\"catalyzation\").members,\n \"PFK\")\n# Update rate laws to be in terms of kf and Keq\nPFK.custom_rates.update(k2Keq(PFK.custom_rates))\n\n# Access categorized attribute for enzyme module reactions and print\nfor group in PFK.enzyme_module_reactions_categorized:\n header = \"Category: \" + group.id\n print(\"\\n\" + header + \"\\n\" + \"-\" * len(header))\n for reaction in sorted(group.members, key=attrgetter(\"id\")):\n print(reaction.id + \": \" + str(reaction.rate))\n```\n\n \n Category: atp_c_binding\n -----------------------\n PFK_R01: kf_PFK_A*(atp_c(t)*pfk_R0_c(t) - pfk_R0_A_c(t)/Keq_PFK_A)\n PFK_R11: kf_PFK_A*(atp_c(t)*pfk_R1_c(t) - pfk_R1_A_c(t)/Keq_PFK_A)\n PFK_R21: kf_PFK_A*(atp_c(t)*pfk_R2_c(t) - pfk_R2_A_c(t)/Keq_PFK_A)\n PFK_R31: kf_PFK_A*(atp_c(t)*pfk_R3_c(t) - pfk_R3_A_c(t)/Keq_PFK_A)\n PFK_R41: kf_PFK_A*(atp_c(t)*pfk_R4_c(t) - pfk_R4_A_c(t)/Keq_PFK_A)\n \n Category: f6p_c_binding\n -----------------------\n PFK_R02: kf_PFK_F*(f6p_c(t)*pfk_R0_A_c(t) - pfk_R0_AF_c(t)/Keq_PFK_F)\n PFK_R12: kf_PFK_F*(f6p_c(t)*pfk_R1_A_c(t) - pfk_R1_AF_c(t)/Keq_PFK_F)\n PFK_R22: kf_PFK_F*(f6p_c(t)*pfk_R2_A_c(t) - pfk_R2_AF_c(t)/Keq_PFK_F)\n PFK_R32: kf_PFK_F*(f6p_c(t)*pfk_R3_A_c(t) - pfk_R3_AF_c(t)/Keq_PFK_F)\n PFK_R42: kf_PFK_F*(f6p_c(t)*pfk_R4_A_c(t) - pfk_R4_AF_c(t)/Keq_PFK_F)\n \n Category: catalyzation\n ----------------------\n PFK_R03: kf_PFK*pfk_R0_AF_c(t)\n PFK_R13: kf_PFK*pfk_R1_AF_c(t)\n PFK_R23: kf_PFK*pfk_R2_AF_c(t)\n PFK_R33: kf_PFK*pfk_R3_AF_c(t)\n PFK_R43: kf_PFK*pfk_R4_AF_c(t)\n \n Category: amp_c_activation\n --------------------------\n PFK_R10: kf_PFK_ACT*(4*amp_c(t)*pfk_R0_c(t) - pfk_R1_c(t)/Keq_PFK_ACT)\n PFK_R20: kf_PFK_ACT*(3*amp_c(t)*pfk_R1_c(t) - 2*pfk_R2_c(t)/Keq_PFK_ACT)\n PFK_R30: kf_PFK_ACT*(2*amp_c(t)*pfk_R2_c(t) - 3*pfk_R3_c(t)/Keq_PFK_ACT)\n PFK_R40: kf_PFK_ACT*(amp_c(t)*pfk_R3_c(t) - 4*pfk_R4_c(t)/Keq_PFK_ACT)\n \n Category: atp_c_inhibition\n --------------------------\n PFK_T1: kf_PFK_I*(4*atp_c(t)*pfk_T0_c(t) - pfk_T1_c(t)/Keq_PFK_I)\n PFK_T2: kf_PFK_I*(3*atp_c(t)*pfk_T1_c(t) - 2*pfk_T2_c(t)/Keq_PFK_I)\n PFK_T3: kf_PFK_I*(2*atp_c(t)*pfk_T2_c(t) - 3*pfk_T3_c(t)/Keq_PFK_I)\n PFK_T4: kf_PFK_I*(atp_c(t)*pfk_T3_c(t) - 4*pfk_T4_c(t)/Keq_PFK_I)\n \n Category: RT_transition\n -----------------------\n PFK_L: kf_PFK_L*(pfk_R0_c(t) - pfk_T0_c(t)/Keq_PFK_L)\n\n\n## The Steady State\n### Solve steady state concentrations symbolically\nTo determine the steady state of the enzyme, a dictionary of the ordinary differential equations as symbolic expressions for each of the `EnzymeModuleForm` objects. The ligands are first removed from the equations by assuming their values are taken into account in a lumped rate constant parameter.\n\nFor handling of all symbolic expressions, the **SymPy** package is used.\n\n\n```python\n# Make a dictionary of ODEs and lump ligands into rate parameters by giving them a value of 1\node_dict = {}\nlump_ligands = {sym.Symbol(met.id): 1 for met in PFK.enzyme_module_ligands}\nfor enzyme_module_form in PFK.enzyme_module_forms:\n symbol_key = sym.Symbol(enzyme_module_form.id)\n ode = sym.Eq(strip_time(enzyme_module_form.ode), 0)\n ode_dict[symbol_key] = ode.subs(lump_ligands)\n\nrank = matrix_rank(PFK.S[6:])\nprint(\"Rank Deficiency: {0}\".format(len(ode_dict) - rank))\n```\n\n Rank Deficiency: 1\n\n\nIn order to solve the system of ODEs for the steady state concentrations, an additional equation is required due to the rank deficiency of the stoichiometric matrix. Therefore, the equation for the steady state flux through the enzyme, which will be referred to as the \"enzyme net flux equation\", must be defined. \n\nTo define the enzyme net flux equation, the `EnzymeModule.make_enzyme_netflux_equation` class method can be used. \n\n* This equation is made by providing a reaction, or a list of reactions to add together.\n* Passing a bool to `use_rates` argument determines whether a symbolic equation is a summation of the flux symbols returned by `EnzymeModuleReaction.flux_symbol_str`, or a summation of the rates laws for those reactions.\n* The `update_enzyme` argument determines whether the new rate equation is set in the `enzyme_rate_equation` attribute.\n\nThe flux through the enzyme typically corresponds to the sum of the fluxes through the catalytic reaction steps.\nBecause the catalyzation reactions were assigned to the \"catalyzation\" cateogry, they can be accessed through the `enzyme_module_reactions_categorized` attribute to create the equation for $v_{\\mathrm{PFK}}$.\n\n\n```python\nreactions = PFK.enzyme_module_reactions_categorized.get_by_id(\n \"catalyzation\").members\nPFK.make_enzyme_rate_equation(\n reactions,\n use_rates=True, update_enzyme=True)\nsym.pprint(PFK.enzyme_rate_equation)\n```\n\n kf_PFK\u22c5(pfk_R0_AF_c(t) + pfk_R1_AF_c(t) + pfk_R2_AF_c(t) + pfk_R3_AF_c(t) + pf\n k_R4_AF_c(t))\n\n\nThe next step is to identify equations for the unknown concentrations in each reaction. These equations will need to be solved with a dependent variable before accounting for the enzyme net flux equation. The completely free form of the enzyme with no bound species will be treated as the dependent variable. \n\nTo verify that all equations are in terms of the lumped rate parameters, and the dependent variable, the solutions can be iterated through using the atoms method to identify the equation arguments. There should be no `EnzymeModuleForm` identifiers with the exception of the dependent variable. \n\n\n```python\n# Get enzyme module forms\nenzyme_module_forms = PFK.enzyme_module_forms.copy()\n# Reverse list for increased performance (due to symmetry assumption)\n# by solving for the most activated/inhibitors bound first.\nenzyme_module_forms.reverse()\n\nenzyme_solutions = {}\nfor enzyme_module_form in enzyme_module_forms:\n # Skip dependent variable\n if \"pfk_R0_c\" == str(enzyme_module_form):\n continue\n enzyme_module_form = sym.Symbol(enzyme_module_form.id)\n # Susbtitute in previous solutions and solve for the enzyme module form, \n equation = ode_dict[enzyme_module_form]\n sol = sym.solveset(equation.subs(enzyme_solutions), enzyme_module_form)\n enzyme_solutions[enzyme_module_form] = list(sol)[0]\n # Update the dictionary of solutions with the solutions\n enzyme_solutions.update({\n enzyme_module_form: sol.subs(enzyme_solutions) \n for enzyme_module_form, sol in enzyme_solutions.items()})\nargs = set()\nfor sol in enzyme_solutions.values():\n args.update(sol.atoms(sym.Symbol))\nprint(args)\n```\n\n {Keq_PFK_L, kf_PFK, kf_PFK_A, Keq_PFK_A, kf_PFK_F, pfk_R0_c, Keq_PFK_I, Keq_PFK_ACT, Keq_PFK_F}\n\n\nThe enzyme net flux equation can then be utilized as the last equation required to solve for the final unknown concentration variable in terms of the rate and equilibrium constants, allowing for all of the concentration variables to be defined in terms of the rate and equilibrium constants. Once the unknown variable has been solved for, the solution can be substituted back into the other equations. Because `sympy.solveset` function expects the input equations to be equal to 0, the `EnzymeModule.enzyme_rate_error` method with the `use_values` argument set to `False` to get the appropriate expression.\n\n\n```python\nenzyme_rate_equation = strip_time(PFK.enzyme_rate_error(False))\nprint(\"Enzyme Net Flux Equation\\n\" + \"-\"*24)\nsym.pprint(enzyme_rate_equation)\n\n# Solve for last unknown concentration symbolically\nsol = sym.solveset(enzyme_rate_equation.subs(enzyme_solutions), \"pfk_R0_c\")\n\n# Update solution dictionary with the new solution\nenzyme_solutions[sym.Symbol(\"pfk_R0_c\")] = list(sol)[0]\n\n# Update solutions with free variable solutions\nenzyme_solutions = {\n enzyme_module_form: sym.simplify(solution.subs(enzyme_solutions))\n for enzyme_module_form, solution in enzyme_solutions.items()}\n\nargs = set()\nfor sol in enzyme_solutions.values():\n args.update(sol.atoms(sym.Symbol))\nprint(\"\\n\", args)\n```\n\n Enzyme Net Flux Equation\n ------------------------\n -kf_PFK\u22c5(pfk_R0_AF_c + pfk_R1_AF_c + pfk_R2_AF_c + pfk_R3_AF_c + pfk_R4_AF_c) \n + v_PFK\n \n {Keq_PFK_L, kf_PFK, kf_PFK_A, Keq_PFK_A, kf_PFK_F, Keq_PFK_I, v_PFK, Keq_PFK_ACT, Keq_PFK_F}\n\n\n#### Numerical Values\nAt this point, numerical values are defined for the dissociation constants and the concentrations of the substrates, cofactors, activators, and inhibitors. Providing these numerical values will speed up the subsequent calculations. \n\nTo do this, experimental data is used to define the dissociations constants for the different binding steps under the QEA. The concentrations of the non-enzyme species are taken from the glycolysis model. Experimental data gives the following for the dissociation constants: \n\n$$K_i=0.1 mM,\\\nK_a=0.033 mM,\\\nK_A=0.068 mM,\\\nK_F=0.1 mM$$\n\nand an allosteric constant of $K_L = 0.0011$.\n\n__Note:__ The $K_i$ binding constant for ATP as an inhibitor was increased by a factor of ten since magnesium complexing of ATP is not considered here. \n\n\n```python\nnumerical_values = {}\n\n# Get ligand IDs and parameter IDs\nligand_ids = sorted([str(ligand) for ligand in PFK.enzyme_module_ligands])\nparameter_ids = [\"_\".join((PFK.id, abbrev)) for abbrev in abbreviations + [\"L\"]]\nprint(\"Ligand IDs: \" + str(ligand_ids))\nprint(\"Parameter IDs: \" + str(parameter_ids))\n\n# Load the glycolysis model to extract steady state values\nglycolysis = create_test_model(\"SB2_Glycolysis\")\n\n# Get the steady state flux value and add to numerical values\nPFK.enzyme_rate = glycolysis.reactions.get_by_id(PFK.id).steady_state_flux\nnumerical_values.update({PFK.enzyme_flux_symbol_str: PFK.enzyme_rate})\n\n# Get the steady state concentration values and add to numerical values\ninitial_conditions = {\n str(ligand): glycolysis.initial_conditions[glycolysis.metabolites.get_by_id(ligand)]\n for ligand in ligand_ids}\n\n# Define parameter values and add to numerical values\n# Because of the QEA, invert dissociation constants for Keq\nparameter_values = {\n \"Keq_\" + parameter_id: value \n for parameter_id, value in zip(parameter_ids, [1/0.068, 1/0.1, 1/0.1, 1/0.033, 0.0011])}\n\n# Display numerical values\nprint(\"\\nNumerical Values\\n----------------\")\nfor k, v in numerical_values.items():\n print(\"{0} = {1}\".format(k, v))\n```\n\n Ligand IDs: ['adp_c', 'amp_c', 'atp_c', 'f6p_c', 'fdp_c', 'h_c']\n Parameter IDs: ['PFK_A', 'PFK_F', 'PFK_I', 'PFK_ACT', 'PFK_L']\n \n Numerical Values\n ----------------\n v_PFK = 1.12\n\n\nThe next step is to define the numerical values, $K_i=0.1/1.6$, $K_a=0.033/0.0867$, $K_A=0.068/1.6$, $K_F=0.1/0.0198$, $v_{PFK}=1.12 \\text{mM/hr}$, and $K_L=1/0.0011$ using the dissociation constant values and the steady state concentrations of the ligands and introduce them into the solution to get the steady state concentrations of the enzyme module forms in terms of the rate constants. The values of the equilirbium constants and initial conditions are also stored for later use.\n\n\n```python\n# Match abbreviations to their corresponding ligands\nabbreviation_dict = {\"PFK_A\": \"atp_c\", \"PFK_F\": \"f6p_c\", \"PFK_ACT\": \"amp_c\", \"PFK_I\": \"atp_c\", \"PFK_L\": \"\"}\n\nk2K = {sym.Symbol(\"kr_\" + p): sym.Symbol(\"kf_\" + p)*sym.Symbol(\"K_\" + p) for p in abbreviation_dict.keys()}\nenzyme_solutions = {met: sym.simplify(Keq2k(solution).subs(enzyme_solutions).subs(k2K))\n for met, solution in enzyme_solutions.items()}\nK_values = dict(zip([\"K_\" + p for p in abbreviation_dict], [0.068, 0.1, 0.033, 0.1, 0.0011]))\n\nfor abbrev, ligand_id in abbreviation_dict.items():\n K_str = \"K_\" + abbrev\n if ligand_id:\n numerical_value = K_values[K_str]/initial_conditions[ligand_id]\n else:\n numerical_value = 1/K_values[K_str]\n numerical_values[sym.Symbol(K_str)] = numerical_value\n \nenzyme_solutions = {met: sym.simplify(solution.subs(numerical_values))\n for met, solution in enzyme_solutions.items()}\n\n# Display numerical values\nprint(\"\\nNumerical Values\\n----------------\")\nfor k, v in numerical_values.items():\n print(\"{0} = {1}\".format(k, v))\n```\n\n \n Numerical Values\n ----------------\n v_PFK = 1.12\n K_PFK_A = 0.0425\n K_PFK_F = 5.05050505050505\n K_PFK_ACT = 0.3804995151513754\n K_PFK_I = 0.0625\n K_PFK_L = 909.090909090909\n\n\nThe last part of this step is to simplify the solutions for the enzyme module forms and, as a QA check, ensure that only rate constants are the only symbolic arguments in the solutions. \n\n\n```python\n# Substitute values into equations\nenzyme_solutions = {\n enzyme_module_form: sym.simplify(solution.subs(numerical_values))\n for enzyme_module_form, solution in enzyme_solutions.items()}\n\nargs = set()\nfor sol in enzyme_solutions.values():\n args.update(sol.atoms(sym.Symbol))\nprint(args)\n```\n\n {kf_PFK, kf_PFK_A, kf_PFK_F}\n\n\n### Determine rate constants\n#### Total Enzyme Concentration and $r_{T}$ \nAfter solving for the enzyme module forms, the next step is to define equations for the total enzyme concentration and for the fraction of the enzyme in the T state. These two equations can be used as constraints for determining the rate parameters. To view the equation for the total enzyme concentration, we can use the `EnzymeModule.enzyme_concentration_total_equation` property.\n\n\n```python\nsym.pprint(PFK.enzyme_concentration_total_equation)\n```\n\n pfk_R0_AF_c(t) + pfk_R0_A_c(t) + pfk_R0_c(t) + pfk_R1_AF_c(t) + pfk_R1_A_c(t) \n + pfk_R1_c(t) + pfk_R2_AF_c(t) + pfk_R2_A_c(t) + pfk_R2_c(t) + pfk_R3_AF_c(t) \n + pfk_R3_A_c(t) + pfk_R3_c(t) + pfk_R4_AF_c(t) + pfk_R4_A_c(t) + pfk_R4_c(t) +\n pfk_T0_c(t) + pfk_T1_c(t) + pfk_T2_c(t) + pfk_T3_c(t) + pfk_T4_c(t)\n\n\nThe total concentration of PFK is 33 nM (=0.000033 mM). The `EnzymeModule.enzyme_concentration_total` atrribute can be used to set and store this concentration.\n\n\n```python\nPFK.enzyme_concentration_total = 33e-6\nprint(PFK.enzyme_concentration_total)\n```\n\n 3.3e-05\n\n\nTo determine the rate constants, an optimization problem where the objective function is to minimize the error between the measured and calculated total enzyme concentrations. To create the objective function, the `EnzymeModule.enzyme_concentration_total_error` method with the `use_values` argument set as False to get the symbolic expression of the constraint. \n\n\n```python\nenzyme_total_constraint = abs(strip_time(PFK.enzyme_concentration_total_error(use_values=False)))\nsym.pprint(enzyme_total_constraint)\n```\n\n \u2502-PFK_Total + pfk_R0_AF_c + pfk_R0_A_c + pfk_R0_c + pfk_R1_AF_c + pfk_R1_A_c +\n pfk_R1_c + pfk_R2_AF_c + pfk_R2_A_c + pfk_R2_c + pfk_R3_AF_c + pfk_R3_A_c + p\n fk_R3_c + pfk_R4_AF_c + pfk_R4_A_c + pfk_R4_c + pfk_T0_c + pfk_T1_c + pfk_T2_c\n + pfk_T3_c + pfk_T4_c\u2502\n\n\nSubstitute the solutions for the enzyme forms to get an equation for the error in the enzyme total concentration in terms of the rate constants.\n\n\n```python\n# Substitute value for enzyme concentration total\nenzyme_total_constraint = enzyme_total_constraint.subs({PFK.enzyme_total_symbol_str: PFK.enzyme_concentration_total})\n# Substitute solutions into constraint and simplify\nenzyme_total_constraint = sym.simplify(enzyme_total_constraint.subs(enzyme_solutions))\nsym.pprint(enzyme_total_constraint)\n```\n\n \u2502 1.19283868483391 1.71385140785683 7.14443780219149\u2502\n \u2502-3.3e-5 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2502\n \u2502 kf_PFK_F kf_PFK_A kf_PFK \u2502\n\n\nTo create the objective function in a format suitable for the minimization method from the `scipy.optimize` submodule, the `sympy.lambdify` function can be used to convert the symbolic expression into a lambda function with the rate constants as the arguments. This lambda function can then be used to generate the objective function for the `optimize.minimize` method.\n\n\n```python\n# Create a sorted tuple of the arguments to ensure the input format does not change\nargs = tuple(sorted([str(arg) for arg in list(args)]))\n# Create the objective function as a lambda function\nobjective_function = lambda x: sym.lambdify(args, enzyme_total_constraint)(*x)\n```\n\nAnother constraint can be set on the amount of inhibited enzyme in the steady state of the system using the T fraction (denoted as $r_{T}$). This fraction is simply the amount of inhibited enzyme over the total amount of enzyme. The enzyme is inhibited between 10-15% under physiological conditions (Ponce et al. Biochimica et Biophysica Acta 1971 250(1):63-74)\n\nTo make the fraction as a symbolic expression, we can use the `EnzymeModule.make_enzyme_fraction` method. This method is designed to assist in making fractions and ratios by passing to the function:\n1. A string to the `categorized_attr` argument identifying which categorized attribute (either \"forms\" for the `EnzymeModule.enzyme_module_forms_categorized` or \"reactions\" for the `EnzymeModule.enzyme_module_reactions_categorized`).\n2. A string for the `top` argument and a string for the `bottom` argument identifying the categories to sum and use in the numerator and the denominator, respectively.\n3. A bool to the `use_values` argument indicating whether to substitute numerical values into the expression to return a float or to keep the ratio as a **SymPy** expression.\n\n__Note:__ The string \"Equation\" can be passed to either the `top` or `bottom` arguments to utilize the equation stored either in `enzyme_concentration_total_equation` (for `categorized_attr`=\"forms\"), or `enzyme_rate_equation` (for `categorized_attr`=\"reactions\").\n\n\n```python\n# Set the values for the constraint bounds\nr_T_lb, r_T_ub = (0.10, 0.15)\n# Make a symbolic expression for enzyme fraction.\nr_T_expr = PFK.make_enzyme_fraction(\n categorized_attr=\"forms\", top=\"Tense\", bottom=\"Equation\",\n use_values=False)\n# Substitute solutions into the expression to make\n# solely dependent on the rate constants\nr_T_expr = sym.simplify(strip_time(r_T_expr).subs(enzyme_solutions))\n\n# Make lambda functions for the T fraction constraint\nr_T_lb_constraint = lambda x: sym.lambdify(args, r_T_expr - r_T_lb)(*x)\nr_T_ub_constraint = lambda x: sym.lambdify(args, r_T_ub - r_T_expr)(*x)\n```\n\nLastly, we place lower and upper bounds on the rate constants to ensure that the values are non-negative and are within physiological limits, and then we solve the optmization problem. Once the optimization has finished, we check whether it was successful, and if so, what the optimality and errors are associated with this particular solution instance.\n\n\n```python\nprint(\"Ordered Args: {0}\\n\".format(str(args)))\n# Set arguments for minimization\nkf_bounds = ((1e2, 1e8), (1e2, 1e8), (1e2, 1e8))\ninitial_guess = [\n 3.07e5,\n 2e5,\n 1e6,]\n\n# Find a feasible solution\nsol = optimize.minimize(\n objective_function, x0=initial_guess,\n method=\"trust-constr\",\n bounds=kf_bounds,\n options={\"gtol\": 1e-20, \"xtol\": 1e-20, \"maxiter\": 1e4, \"disp\": True})\n\n# Check whether optimzation was successful\nprint(\"\\nOptimization Success: {0}\".format(sol.success))\nif sol.success:\n # Update the paramter values dictionary with the feasible solution\n parameter_values.update(dict(zip(args, [round(x) for x in sol.x])))\n print(\"Optimization Optimality: {0:.4e}\".format(sol.optimality))\n print(\"Parameter Solutions: {:}\".format(str({arg: parameter_values[arg] for arg in args})))\n # Plug solutions back into constraints for validation\n print(\"Optimization Error: {0:.4e}\".format(enzyme_total_constraint.subs(parameter_values)))\n```\n\n Ordered Args: ('kf_PFK', 'kf_PFK_A', 'kf_PFK_F')\n \n `xtol` termination condition is satisfied.\n Number of iterations: 104, function evaluations: 224, CG iterations: 116, optimality: 3.60e-11, constraint violation: 0.00e+00, execution time: 0.63 s.\n \n Optimization Success: True\n Optimization Optimality: 3.6029e-11\n Parameter Solutions: {'kf_PFK': 307263, 'kf_PFK_A': 200325, 'kf_PFK_F': 1000059}\n Optimization Error: 1.2079e-11\n\n\nWith a successful optimization, the module is updated with the parameter values. The inhibition and activation reactions are set to have a high forward rate constant and the allosteric transition even higher, limiting the amount of unbound enzyme and ensuring that the dynamics are determined by the dissociation and allosteric constants. \n\n__Note:__ This assumption for the rate constants can be made because none of the enzyme concentrations are dependendent on the activation, inhibition, and allosteric rate constants.\n\n\n```python\n# Add the activation, inhibition, and allosteric rate constants\nfor abbrev, value in zip([\"I\", \"ACT\", \"L\"], [1e6, 1e6, 1e6**2]):\n # Account for the enzyme prefix if used in the previous function\n to_join = (\"kf\", PFK.id, abbrev)\n param = \"_\".join(to_join)\n parameter_values.update({param: value})\n \n# Display numerical values\nfor k, v in parameter_values.items():\n print(\"{0} = {1}\".format(k, v))\n```\n\n Keq_PFK_A = 14.705882352941176\n Keq_PFK_F = 10.0\n Keq_PFK_I = 10.0\n Keq_PFK_ACT = 30.3030303030303\n Keq_PFK_L = 0.0011\n kf_PFK = 307263\n kf_PFK_A = 200325\n kf_PFK_F = 1000059\n kf_PFK_I = 1000000.0\n kf_PFK_ACT = 1000000.0\n kf_PFK_L = 1000000000000.0\n\n\n### Solve steady state concentrations numerically\n\nOnce the rate constants have been defined, the steady state concentrations of the enzyme can be determined.\n\n\n```python\n# Substitute values into equations\ninitial_conditions.update({\n str(enzyme_module_form): float(sym.simplify(solution.subs(parameter_values)))\n for enzyme_module_form, solution in enzyme_solutions.items()})\n\nfor header, dictlist in zip([\"Ligand\", \"\\nEnzyme\"], [PFK.enzyme_module_ligands, PFK.enzyme_module_forms]):\n header += \" Concentrations\"\n print(\"\\n\".join([header, \"-\" * len(header)]))\n for form in dictlist:\n ic = initial_conditions[form.id]\n print(\"{0} = {1}\".format(form.id, ic))\n```\n\n Ligand Concentrations\n ---------------------\n f6p_c = 0.0198\n fdp_c = 0.0146\n atp_c = 1.6\n adp_c = 0.29\n amp_c = 0.0867281\n h_c = 8.99757e-05\n \n Enzyme Concentrations\n ----------------------\n pfk_R0_c = 3.705684451779081e-08\n pfk_R0_A_c = 1.1270977736701491e-07\n pfk_R0_AF_c = 2.1036774576199985e-08\n pfk_T0_c = 4.0762528969569896e-11\n pfk_R1_c = 3.895599656998077e-07\n pfk_R1_A_c = 1.1848611930259641e-06\n pfk_R1_AF_c = 2.2114902898450058e-07\n pfk_T1_c = 2.6088018540524733e-09\n pfk_R2_c = 1.5357179846004314e-06\n pfk_R2_A_c = 4.670943637948869e-06\n pfk_R2_AF_c = 8.71810686394121e-07\n pfk_T2_c = 6.261124449725935e-08\n pfk_R3_c = 2.690705109903528e-06\n pfk_R3_A_c = 8.183880139927137e-06\n pfk_R3_AF_c = 1.5274845331446054e-06\n pfk_T3_c = 6.678532746374332e-07\n pfk_R4_c = 1.7678768321380616e-06\n pfk_R4_A_c = 5.377063448209202e-06\n pfk_R4_AF_c = 1.0036047828713532e-06\n pfk_T4_c = 2.6714130985497327e-06\n\n\n#### Set Initial Conditions and Parameters\nOnce the steady state concentrations have been determined, the initial conditions and parameters are added to the module. All custom parameter are added to the custom_parameter attribute. The allosteric transition uses the standard parameter identifiers (returned by `kf_str` and `Keq_str` properties of the `EnzymeModuleReaction`), so they are popped out of the custom parameters and set through their respective attribute setter methods. \n\n\n```python\n# Set initial conditions\nfor met, concentration in initial_conditions.items():\n PFK.metabolites.get_by_id(str(met)).ic = concentration\n\n# Add the custom parameters and values for kf and Keq to model\nPFK.custom_parameters.update(parameter_values)\n# PFK_L uses standard reaction parameters and not custom parameters\nPFK_L = PFK.enzyme_module_reactions.PFK_L\nPFK_L.kf = PFK.custom_parameters.pop(PFK_L.kf_str)\nPFK_L.Keq = PFK.custom_parameters.pop(PFK_L.Keq_str)\n\n# Set parameter values in reaction fields\nfor group in PFK.enzyme_module_reactions_categorized:\n if group.id == \"atp_c_binding\":\n param_id = \"PFK_A\"\n elif group.id == \"f6p_c_binding\":\n param_id = \"PFK_F\"\n elif group.id == \"catalyzation\":\n param_id = \"PFK\"\n elif group.id == \"atp_c_inhibition\":\n param_id = \"PFK_I\"\n elif group.id == \"amp_c_activation\":\n param_id = \"PFK_ACT\"\n else:\n continue\n for reaction in group.members:\n kf, Keq = (\"kf_\" + param_id, \"Keq_\" + param_id)\n if kf in PFK.custom_parameters:\n reaction.kf = PFK.custom_parameters[kf]\n if Keq in PFK.custom_parameters:\n reaction.Keq = PFK.custom_parameters[Keq]\n```\n\n#### Ordering of internal species and reactions\n\nSometimes, it is also desirable to reorder the metabolite and reaction objects inside the model to follow the physiology. To reorder the internal objects, one can use `cobra.DictList` containers and the `DictList.get_by_any` method with the list of object identifiers in the desirable order. To ensure all objects are still present and not forgotten in the model, a small QA check is also performed. \n\n\n```python\nnew_metabolite_order = ['f6p_c', 'fdp_c', 'amp_c', 'adp_c', 'atp_c', 'h_c',\n 'pfk_R0_c', 'pfk_R0_A_c', 'pfk_R0_AF_c', \n 'pfk_R1_c', 'pfk_R1_A_c', 'pfk_R1_AF_c', \n 'pfk_R2_c', 'pfk_R2_A_c', 'pfk_R2_AF_c', \n 'pfk_R3_c', 'pfk_R3_A_c', 'pfk_R3_AF_c',\n 'pfk_R4_c', 'pfk_R4_A_c', 'pfk_R4_AF_c', \n 'pfk_T0_c','pfk_T1_c', 'pfk_T2_c', 'pfk_T3_c', 'pfk_T4_c']\n\nif len(glycolysis.metabolites) == len(new_metabolite_order):\n PFK.metabolites = DictList(\n PFK.metabolites.get_by_any(new_metabolite_order))\n\nif len(PFK.metabolites) == len(new_metabolite_order):\n PFK.metabolites = DictList(PFK.metabolites.get_by_any(new_metabolite_order))\n \nnew_reaction_order = [\"PFK_R01\", 'PFK_R02', \"PFK_R03\", \"PFK_R10\", \n \"PFK_R11\", \"PFK_R12\", \"PFK_R13\", \"PFK_R20\", \n \"PFK_R21\", \"PFK_R22\", \"PFK_R23\", \"PFK_R30\", \n \"PFK_R31\", \"PFK_R32\", \"PFK_R33\", \"PFK_R40\", \n \"PFK_R41\", \"PFK_R42\", \"PFK_R43\", \"PFK_L\", \n \"PFK_T1\", \"PFK_T2\", \"PFK_T3\", \"PFK_T4\"]\n\nif len(PFK.reactions) == len(new_reaction_order):\n PFK.reactions = DictList(\n PFK.reactions.get_by_any(new_reaction_order))\n \nPFK.update_S(array_type=\"DataFrame\", dtype=int)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        PFK_R01PFK_R02PFK_R03PFK_R10PFK_R11PFK_R12PFK_R13PFK_R20PFK_R21PFK_R22...PFK_R33PFK_R40PFK_R41PFK_R42PFK_R43PFK_LPFK_T1PFK_T2PFK_T3PFK_T4
                                        f6p_c0-1000-1000-1...000-1000000
                                        fdp_c0010001000...1000100000
                                        amp_c000-1000-100...0-100000000
                                        adp_c0010001000...1000100000
                                        atp_c-1000-1000-10...00-1000-1-1-1-1
                                        h_c0010001000...1000100000
                                        pfk_R0_c-101-1000000...00000-10000
                                        pfk_R0_A_c1-100000000...0000000000
                                        pfk_R0_AF_c01-10000000...0000000000
                                        pfk_R1_c0001-101-100...0000000000
                                        pfk_R1_A_c00001-10000...0000000000
                                        pfk_R1_AF_c000001-1000...0000000000
                                        pfk_R2_c00000001-10...0000000000
                                        pfk_R2_A_c000000001-1...0000000000
                                        pfk_R2_AF_c0000000001...0000000000
                                        pfk_R3_c0000000000...1-100000000
                                        pfk_R3_A_c0000000000...0000000000
                                        pfk_R3_AF_c0000000000...-1000000000
                                        pfk_R4_c0000000000...01-10100000
                                        pfk_R4_A_c0000000000...001-1000000
                                        pfk_R4_AF_c0000000000...0001-100000
                                        pfk_T0_c0000000000...000001-1000
                                        pfk_T1_c0000000000...0000001-100
                                        pfk_T2_c0000000000...00000001-10
                                        pfk_T3_c0000000000...000000001-1
                                        pfk_T4_c0000000000...0000000001
                                        \n

                                        26 rows \u00d7 24 columns

                                        \n
                                        \n\n\n\n## Module Validation \n### QC/QA model\nBefore saving the module, it is important to ensure that the module is elementally balanced, and that the module can be integrated into a larger network for simulation. Therefore, the `qcqa_model` function from `mass.util.qcqa` is used to provide a report on the module quality and and indicate whether simulation is possible and if not, what parameters and/or initial conditions are missing. \n\n\n```python\nqcqa_model(PFK, parameters=True, concentrations=True, \n fluxes=False, superfluous=True, elemental=True)\n```\n\n \u2552\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2555\n \u2502 MODEL ID: PFK \u2502\n \u2502 SIMULATABLE: True \u2502\n \u2502 PARAMETERS NUMERICALY CONSISTENT: True \u2502\n \u255e\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2561\n \u2558\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255b\n\n\n### Constraint Satisfaction and Error Values\nAnother QA check we perform is to substitute the steady state numerical values back into the constraints used in determining the rate constants in order to ensure that the constraints remain satisified, and that errors are small. \n\n\n```python\nt_fraction = PFK.make_enzyme_fraction(\"forms\", top=\"Tense\",\n bottom=\"Equation\", use_values=True)\nprint(\"Enzyme T-fraction: {:.4f}\".format(t_fraction))\n\nprint(\"Concentration Absolute Error: {0:.4e}\".format(\n abs(PFK.enzyme_concentration_total_error(use_values=True))))\nprint(\"Flux Absolute Error: {0:.4e}\".format(\n abs(PFK.enzyme_rate_error(use_values=True))))\n```\n\n Enzyme T-fraction: 0.1032\n Concentration Absolute Error: 1.2079e-11\n Flux Absolute Error: 2.2204e-16\n\n\n### Add Enzyme to MassModel\nIn order to determine whether the module can be successfully integrated into a model, another model can be loaded, merged with the module, and simulated. To validate this module, it will be merged with a glycolysis model. \n\nTo integrate the `EnzymeModule` into the `MassModel`, the reaction that the EnzymeModule will be replacing is first removed. The `MassModel.merge` method can then be utilized to add the `EnzymeModule` to the `MassModel`. \n\nWhen merging an `EnzymeModule` and a `MassModel`, the `EnzymeModule` should always be merged into the `MassModel`.\n\n\n```python\n# Load and merge glycolysis with PFK model\nglycolysis = create_test_model(\"SB2_Glycolysis.json\")\n# Remove the PFK MassReaction, then merge the EnzymeModule into the MassModel\nglycolysis.remove_reactions([glycolysis.reactions.get_by_id(\"PFK\")])\nglycolysis_PFK = glycolysis.merge(PFK)\nglycolysis_PFK\n```\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        NameGlycolysis
                                        Memory address0x07f7fc34cec90
                                        Stoichiometric Matrix40x44
                                        Matrix Rank37
                                        Number of metabolites40
                                        Initial conditions defined40/40
                                        Number of reactions44
                                        Number of genes0
                                        Number of enzyme modules1
                                        Number of groups16
                                        Objective expression0
                                        CompartmentsCytosol
                                        \n\n\n\n\nUsing `MassModel.merge` class method enables the `EnzymeModule` and `MassModel` to be merged like as if they were both `MassModel` objects. However, all attributes specific to the `EnzymeModule` (e.g the categorized attributes) are condensed into a speciailzed container called an `EnzymeModuleDict`.\n\nThe `EnzymeModuleDict` behaves like an ordered dictionary, but is unique in that its contents can be accessed as if they were attributes. These attributes can be viewed using `EnzymeModuleDict.keys` method. All `EnzymeModuleDicts` associated with a `MassModel` can be accessed via `MassModel.enzyme_modules` attribute.\n\n\n```python\nprint(str(glycolysis_PFK.enzyme_modules) + \"\\n\")\nprint(\"Attribute Accessors:\\n-------------------\\n\" + \"\\n\".join(list(\n glycolysis_PFK.enzyme_modules.PFK.keys())) + \"\\n\")\nglycolysis_PFK.enzyme_modules.PFK\n```\n\n []\n \n Attribute Accessors:\n -------------------\n id\n name\n subsystem\n enzyme_module_ligands\n enzyme_module_forms\n enzyme_module_reactions\n enzyme_module_ligands_categorized\n enzyme_module_forms_categorized\n enzyme_module_reactions_categorized\n enzyme_concentration_total\n enzyme_rate\n enzyme_concentration_total_equation\n enzyme_rate_equation\n S\n model\n \n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        NamePFK
                                        Memory address0x07f7fc2f88680
                                        Stoichiometric Matrix26x24
                                        Matrix Rank20
                                        SubsystemGlycolysis
                                        Number of Ligands6
                                        Number of EnzymeForms20
                                        Number of EnzymeModuleReactions24
                                        Enzyme Concentration Total3.3e-05
                                        Enzyme Net Flux1.12
                                        \n\n\n\n\n### Validate Steady State\n\nTo find the steady state of the model and perform simulations, the model must first be loaded into a `Simulation`. In order to load a model into a `Simulation`, the model must be simulatable, meaning there are no missing numerical values that would prevent the integration of the ODEs that comprise the model. The `verbose` argument can be used while loading a model to produce a message indicating the successful loading of a model, or why a model could not load.\n\nOnce loaded into a `Simulation`, the `find_steady_state` method can be used with the `update_values` argument in order to update the initial conditions and fluxes of the model to a steady state. The model can then be simulated using the `simulate` method by passing the model to simulate, and a tuple containing the start time and the end time. The number of time points can also be included, but is optional.\n\nAfter a successful simulation, two `MassSolution` objects are returned. The first `MassSolution` contains the concentration results of the simulation, and the second contains the flux results of the simulation. \n\nTo visually validate the steady state of the model, concentration and flux solutions can be plotted using the `plot_time_profile` function from `mass.visualization`. Alternatively, the `MassSolution.view_time_profile` property can be used to quickly generate a time profile for the results.\n\n\n```python\n# Setup simulation object, ensure model is at steady state\nsim = Simulation(glycolysis_PFK, verbose=True)\nsim.find_steady_state(glycolysis_PFK, strategy=\"simulate\",\n update_values=True, verbose=True,\n tfinal=1e4, steps=1e6)\n\n# Simulate from 0 to 1000 with 10001 points in the output\nconc_sol, flux_sol = sim.simulate(\n glycolysis_PFK,time=(0, 1e3, 1e4 + 1))\n# Quickly render and display time profiles\nconc_sol.view_time_profile()\n```\n\n### Storing information and references\n#### Compartment\nBecause the character \"c\" represents the cytosol compartment, it is recommended to define and set the compartment in the `EnzymeModule.compartments` attribute.\n\n\n```python\nPFK.compartments = {\"c\": \"Cytosol\"}\nprint(PFK.compartments)\n```\n\n {'c': 'Cytosol'}\n\n\n#### Units\nAll of the units for the numerical values used in this model are \"Millimoles\" for amount and \"Liters\" for volume (giving a concentration unit of 'Millimolar'), and \"Hours\" for time. In order to ensure that future users understand the numerical values for model, it is important to define the `MassModel.units` attribute.\n\nThe `MassModel.units` is a `cobra.DictList` that contains only `UnitDefinition` objects from the `mass.core.unit` submodule. Each `UnitDefinition` is created from `Unit` objects representing the base units that comprise the `UnitDefinition`. These `Units` are stored in the `list_of_units` attribute. Pre-built units can be viewed using the `print_defined_unit_values` function from the `mass.core.unit` submodule. Alternatively, custom units can also be created using the `UnitDefinition.create_unit` method. For more information about units, please see the module docstring for `mass.core.unit` submodule.\n\n__Note:__ It is important to note that this attribute will NOT track units, but instead acts as a reference for the user and others so that they can perform necessary unit conversions.\n\n\n```python\n# Using pre-build units to define UnitDefinitions\nconcentration = UnitDefinition(\"mM\", name=\"Millimolar\",\n list_of_units=[\"millimole\", \"per_litre\"])\ntime = UnitDefinition(\"hr\", name=\"hour\", list_of_units=[\"hour\"])\n\n# Add units to model\nPFK.add_units([concentration, time])\nprint(PFK.units)\n```\n\n [, ]\n\n\n## Export\n\nAfter validation, the model is ready to be saved. The model can either be exported as a \".json\" file or as an \".sbml\" (\".xml\") file using their repsective submodules in `mass.io`.\n\nTo export the model, only the path to the directory and the model object itself need to be specified.\n\n### Export using SBML\n\n\n```python\nsbml.write_sbml_model(mass_model=PFK, filename=\"SB2_\" + PFK.id + \".xml\")\n```\n\n### Export using JSON\n\n\n```python\njson.save_json_model(mass_model=PFK, filename=\"SB2_\" + PFK.id + \".json\")\n```\n", "meta": {"hexsha": "bd8b9c5bc629de34ba70418faca6802a3753a8af", "size": 163627, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/education/sb2/model_construction/sb2_pfk.ipynb", "max_stars_repo_name": "SBRG/MASSpy", "max_stars_repo_head_hexsha": "1315c1d40be8feb8731c8143dbc9ba43bf8c78ff", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 13, "max_stars_repo_stars_event_min_datetime": "2020-07-13T00:48:13.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-27T15:42:15.000Z", "max_issues_repo_path": "docs/education/sb2/model_construction/sb2_pfk.ipynb", "max_issues_repo_name": "SBRG/MASSpy", "max_issues_repo_head_hexsha": "1315c1d40be8feb8731c8143dbc9ba43bf8c78ff", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 10, "max_issues_repo_issues_event_min_datetime": "2021-02-17T18:07:43.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-23T16:22:14.000Z", "max_forks_repo_path": "docs/education/sb2/model_construction/sb2_pfk.ipynb", "max_forks_repo_name": "SBRG/MASSpy", "max_forks_repo_head_hexsha": "1315c1d40be8feb8731c8143dbc9ba43bf8c78ff", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2020-01-15T00:48:49.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-04T07:01:17.000Z", "avg_line_length": 59.4143064633, "max_line_length": 51972, "alphanum_fraction": 0.651897303, "converted": true, "num_tokens": 21004, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.668880247169804, "lm_q2_score": 0.6224593312018545, "lm_q1q2_score": 0.4163507513074473}} {"text": "\n# **Modelos de Classifica\u00e7\u00e3o**\n\n#### Este laborat\u00f3rio ir\u00e1 cobrir os passos para tratar a base de dados de taxa de cliques (click-through rate - CTR) e criar um modelo de classifica\u00e7\u00e3o para tentar determinar se um usu\u00e1rio ir\u00e1 ou n\u00e3o clicar em um banner.\n\n#### Para isso utilizaremos a base de dados [Criteo Labs](http://labs.criteo.com/) que foi utilizado em uma competi\u00e7\u00e3o do [Kaggle](https://www.kaggle.com/c/criteo-display-ad-challenge).\n\n#### ** Nesse notebook: **\n+ ####*Parte 1:* Utiliza\u00e7\u00e3o do one-hot-encoding (OHE) para transformar atributos categ\u00f3ricos em num\u00e9ricos\n+ ####*Parte 2:* Construindo um dicion\u00e1rio OHE\n+ ####*Parte 3:* Gera\u00e7\u00e3o de atributos OHE na base de dados CTR\n + #### *Visualiza\u00e7\u00e3o 1:* Frequ\u00eancia de atributos\n+ ####*Parte 4:* Predi\u00e7\u00e3o de CTR e avalia\u00e7\u00e3o da perda logar\u00edtimica (logloss)\n + #### *Visualiza\u00e7\u00e3o 2:* Curva ROC\n+ ####*Parte 5:* Reduzindo a dimens\u00e3o dos atributos atrav\u00e9s de hashing (feature hashing)\n \n#### Refer\u00eancias de m\u00e9todos: [Spark's Python API](https://spark.apache.org/docs/latest/api/python/pyspark.html#pyspark.RDD)e [NumPy Reference](http://docs.scipy.org/doc/numpy/reference/index.html)\n\n### ** Part 1: Utiliza\u00e7\u00e3o do one-hot-encoding (OHE) para transformar atributos categ\u00f3ricos em num\u00e9ricos **\n\n#### ** (1a) One-hot-encoding **\n\n#### Para um melhor entendimento do processo da codifica\u00e7\u00e3o OHE vamos trabalhar com uma base de dados pequena e sem r\u00f3tulos. Cada objeto dessa base pode conter tr\u00eas atributos, o primeiro indicando o animal, o segundo a cor e o terceiro qual animal que ele come.\n\n#### No esquema OHE, queremos representar cada tupla `(IDatributo, categoria)` atrav\u00e9s de um atributo bin\u00e1rio. N\u00f3s podemos fazer isso no Python criando um dicion\u00e1rio que mapeia cada poss\u00edvel tupla em um inteiro que corresponde a sua posi\u00e7\u00e3o no vetor de atributos bin\u00e1rio.\n\n#### Para iniciar crie um dicion\u00e1rio correspondente aos atributos categ\u00f3ricos da base constru\u00edda logo abaixo. Fa\u00e7a isso manualmente.\n\n\n```python\n# Data for manual OHE\n# Note: the first data point does not include any value for the optional third feature\nsampleOne = [(0, 'mouse'), (1, 'black')]\nsampleTwo = [(0, 'cat'), (1, 'tabby'), (2, 'mouse')]\nsampleThree = [(0, 'bear'), (1, 'black'), (2, 'salmon')]\nsampleDataRDD = sc.parallelize([sampleOne, sampleTwo, sampleThree])\n```\n\n\n```python\n# EXERCICIO\nsampleOHEDictManual = {}\n\n```\n\n\n```python\n# TEST One-hot-encoding (1a)\nfrom test_helper import Test\n\nTest.assertEqualsHashed(sampleOHEDictManual[(0,'bear')],\n 'b6589fc6ab0dc82cf12099d1c2d40ab994e8410c',\n \"incorrect value for sampleOHEDictManual[(0,'bear')]\")\nTest.assertEqualsHashed(sampleOHEDictManual[(0,'cat')],\n '356a192b7913b04c54574d18c28d46e6395428ab',\n \"incorrect value for sampleOHEDictManual[(0,'cat')]\")\nTest.assertEqualsHashed(sampleOHEDictManual[(0,'mouse')],\n 'da4b9237bacccdf19c0760cab7aec4a8359010b0',\n \"incorrect value for sampleOHEDictManual[(0,'mouse')]\")\nTest.assertEqualsHashed(sampleOHEDictManual[(1,'black')],\n '77de68daecd823babbb58edb1c8e14d7106e83bb',\n \"incorrect value for sampleOHEDictManual[(1,'black')]\")\nTest.assertEqualsHashed(sampleOHEDictManual[(1,'tabby')],\n '1b6453892473a467d07372d45eb05abc2031647a',\n \"incorrect value for sampleOHEDictManual[(1,'tabby')]\")\nTest.assertEqualsHashed(sampleOHEDictManual[(2,'mouse')],\n 'ac3478d69a3c81fa62e60f5c3696165a4e5e6ac4',\n \"incorrect value for sampleOHEDictManual[(2,'mouse')]\")\nTest.assertEqualsHashed(sampleOHEDictManual[(2,'salmon')],\n 'c1dfd96eea8cc2b62785275bca38ac261256e278',\n \"incorrect value for sampleOHEDictManual[(2,'salmon')]\")\nTest.assertEquals(len(sampleOHEDictManual.keys()), 7,\n 'incorrect number of keys in sampleOHEDictManual')\n```\n\n#### ** (1b) Vetores Esparsos **\n\n#### Pontos de dados categ\u00f3ricos geralmente apresentam um pequeno conjunto de OHE n\u00e3o-nulos relativo ao total de poss\u00edveis atributos. Tirando proveito dessa propriedade podemos representar nossos dados como vetores esparsos, economizando espa\u00e7o de armazenamento e c\u00e1lculos computacionais.\n\n#### No pr\u00f3ximo exerc\u00edcio transforme os vetores com nome precedidos de `Dense` para vetores esparsos. Utilize a classe [SparseVector](https://spark.apache.org/docs/latest/api/python/pyspark.mllib.html#pyspark.mllib.linalg.SparseVector) para represent\u00e1-los e verifique que ambas as representa\u00e7\u00f5es retornam o mesmo resultado nos c\u00e1lculos dos produtos interno.\n\n#### Use `SparseVector(tamanho, *args)` para criar um novo vetor esparso onde `tamanho` \u00e9 o tamanho do vetor e `args` pode ser um dicion\u00e1rio, uma lista de tuplas (\u00edndice, valor) ou duas arrays separadas de \u00edndices e valores ordenados por \u00edndice.\n\n\n```python\nimport numpy as np\nfrom pyspark.mllib.linalg import SparseVector\n```\n\n\n```python\n# EXERCICIO\naDense = np.array([0., 3., 0., 4.])\naSparse = SparseVector()\n\nbDense = np.array([0., 0., 0., 1.])\nbSparse = SparseVector()\n\nw = np.array([0.4, 3.1, -1.4, -.5])\nprint aDense.dot(w)\nprint aSparse.dot(w)\nprint bDense.dot(w)\nprint bSparse.dot(w)\n```\n\n\n```python\n# TEST Sparse Vectors (1b)\nTest.assertTrue(isinstance(aSparse, SparseVector), 'aSparse needs to be an instance of SparseVector')\nTest.assertTrue(isinstance(bSparse, SparseVector), 'aSparse needs to be an instance of SparseVector')\nTest.assertTrue(aDense.dot(w) == aSparse.dot(w),\n 'dot product of aDense and w should equal dot product of aSparse and w')\nTest.assertTrue(bDense.dot(w) == bSparse.dot(w),\n 'dot product of bDense and w should equal dot product of bSparse and w')\n```\n\n#### **(1c) Atributos OHE como vetores esparsos **\n\n#### Agora vamos representar nossos atributos OHE como vetores esparsos. Utilizando o dicion\u00e1rio `sampleOHEDictManual`, crie um vetor esparso para cada amostra de nossa base de dados. Todo atributo que ocorre em uma amostra deve ter valor 1.0. Por exemplo, um vetor para um ponto com os atributos 2 e 4 devem ser `[0.0, 0.0, 1.0, 0.0, 1.0, 0.0, 0.0]`.\n\n\n```python\n# Reminder of the sample features\n# sampleOne = [(0, 'mouse'), (1, 'black')]\n# sampleTwo = [(0, 'cat'), (1, 'tabby'), (2, 'mouse')]\n# sampleThree = [(0, 'bear'), (1, 'black'), (2, 'salmon')]\n```\n\n\n```python\n# EXERCICIO\nsampleOneOHEFeatManual = SparseVector()\nsampleTwoOHEFeatManual = SparseVector()\nsampleThreeOHEFeatManual = SparseVector()\n```\n\n\n```python\n# TEST OHE Features as sparse vectors (1c)\nTest.assertTrue(isinstance(sampleOneOHEFeatManual, SparseVector),\n 'sampleOneOHEFeatManual needs to be a SparseVector')\nTest.assertTrue(isinstance(sampleTwoOHEFeatManual, SparseVector),\n 'sampleTwoOHEFeatManual needs to be a SparseVector')\nTest.assertTrue(isinstance(sampleThreeOHEFeatManual, SparseVector),\n 'sampleThreeOHEFeatManual needs to be a SparseVector')\nTest.assertEqualsHashed(sampleOneOHEFeatManual,\n 'ecc00223d141b7bd0913d52377cee2cf5783abd6',\n 'incorrect value for sampleOneOHEFeatManual')\nTest.assertEqualsHashed(sampleTwoOHEFeatManual,\n '26b023f4109e3b8ab32241938e2e9b9e9d62720a',\n 'incorrect value for sampleTwoOHEFeatManual')\nTest.assertEqualsHashed(sampleThreeOHEFeatManual,\n 'c04134fd603ae115395b29dcabe9d0c66fbdc8a7',\n 'incorrect value for sampleThreeOHEFeatManual')\n```\n\n#### **(1d) Fun\u00e7\u00e3o de codifica\u00e7\u00e3o OHE **\n\n#### Vamos criar uma fun\u00e7\u00e3o que gera um vetor esparso codificado por um dicion\u00e1rio de OHE. Ele deve fazer o procedimento similar ao exerc\u00edcio anterior.\n\n\n```python\n# EXERCICIO\ndef oneHotEncoding(rawFeats, OHEDict, numOHEFeats):\n \"\"\"Produce a one-hot-encoding from a list of features and an OHE dictionary.\n\n Note:\n You should ensure that the indices used to create a SparseVector are sorted.\n\n Args:\n rawFeats (list of (int, str)): The features corresponding to a single observation. Each\n feature consists of a tuple of featureID and the feature's value. (e.g. sampleOne)\n OHEDict (dict): A mapping of (featureID, value) to unique integer.\n numOHEFeats (int): The total number of unique OHE features (combinations of featureID and\n value).\n\n Returns:\n SparseVector: A SparseVector of length numOHEFeats with indicies equal to the unique\n identifiers for the (featureID, value) combinations that occur in the observation and\n with values equal to 1.0.\n \"\"\"\n return \n\n# Calculate the number of features in sampleOHEDictManual\nnumSampleOHEFeats = len(sampleOHEDictManual)\n\n# Run oneHotEnoding on sampleOne\nsampleOneOHEFeat = oneHotEncoding(sampleOne, sampleOHEDictManual, numSampleOHEFeats)\n\nprint sampleOneOHEFeat\n```\n\n\n```python\n# TEST Define an OHE Function (1d)\nTest.assertTrue(sampleOneOHEFeat == sampleOneOHEFeatManual,\n 'sampleOneOHEFeat should equal sampleOneOHEFeatManual')\nTest.assertEquals(sampleOneOHEFeat, SparseVector(7, [2,3], [1.0,1.0]),\n 'incorrect value for sampleOneOHEFeat')\nTest.assertEquals(oneHotEncoding([(1, 'black'), (0, 'mouse')], sampleOHEDictManual,\n numSampleOHEFeats), SparseVector(7, [2,3], [1.0,1.0]),\n 'incorrect definition for oneHotEncoding')\n```\n\n#### **(1e) Aplicar OHE em uma base de dados **\n\n#### Finalmente, use a fun\u00e7\u00e3o da parte (1d) para criar atributos OHE para todos os 3 objetos da base de dados artificial.\n\n\n```python\n# EXERCICIO\nsampleOHEData = sampleDataRDD.\nprint sampleOHEData.collect()\n```\n\n\n```python\n# TEST Apply OHE to a dataset (1e)\nsampleOHEDataValues = sampleOHEData.collect()\nTest.assertTrue(len(sampleOHEDataValues) == 3, 'sampleOHEData should have three elements')\nTest.assertEquals(sampleOHEDataValues[0], SparseVector(7, {2: 1.0, 3: 1.0}),\n 'incorrect OHE for first sample')\nTest.assertEquals(sampleOHEDataValues[1], SparseVector(7, {1: 1.0, 4: 1.0, 5: 1.0}),\n 'incorrect OHE for second sample')\nTest.assertEquals(sampleOHEDataValues[2], SparseVector(7, {0: 1.0, 3: 1.0, 6: 1.0}),\n 'incorrect OHE for third sample')\n```\n\n### ** Part 2: Construindo um dicion\u00e1rio OHE **\n\n#### **(2a) Tupla RDD de `(IDatributo, categoria)` **\n\n#### Crie um RDD de pares distintos de `(IDatributo, categoria)`. Em nossa base de dados voc\u00ea deve gerar `(0, 'bear')`, `(0, 'cat')`, `(0, 'mouse')`, `(1, 'black')`, `(1, 'tabby')`, `(2, 'mouse')`, `(2, 'salmon')`. Repare que `'black'` aparece duas vezes em nossa base de dados mas contribui apenas para um item do RDD: `(1, 'black')`, por outro lado `'mouse'` aparece duas vezes e contribui para dois itens: `(0, 'mouse')` and `(2, 'mouse')`. \n\n#### Dica: use [flatMap](https://spark.apache.org/docs/latest/api/python/pyspark.html#pyspark.RDD.flatMap) e [distinct](https://spark.apache.org/docs/latest/api/python/pyspark.html#pyspark.RDD.distinct).\n\n\n```python\n# EXERCICIO\nsampleDistinctFeats = (sampleDataRDD\n .\n .\n )\n\n```\n\n\n```python\n# TEST Pair RDD of (featureID, category) (2a)\nTest.assertEquals(sorted(sampleDistinctFeats.collect()),\n [(0, 'bear'), (0, 'cat'), (0, 'mouse'), (1, 'black'),\n (1, 'tabby'), (2, 'mouse'), (2, 'salmon')],\n 'incorrect value for sampleDistinctFeats')\n```\n\n#### ** (2b) Dicion\u00e1rio OHE de atributos \u00fanicos **\n\n#### Agora, vamos criar um RDD de tuplas para cada `(IDatributo, categoria)` em `sampleDistinctFeats`. A chave da tupla \u00e9 a pr\u00f3pria tupla original, e o valor ser\u00e1 um inteiro variando de 0 at\u00e9 n\u00famero de tuplas - 1. \n\n#### Em seguida, converta essa `RDD` em um dicion\u00e1rio, utilizando o comando `collectAsMap`. \n\n#### Use o comando [zipWithIndex](https://spark.apache.org/docs/latest/api/python/pyspark.html#pyspark.RDD.zipWithIndex) seguido de [collectAsMap](https://spark.apache.org/docs/latest/api/python/pyspark.html#pyspark.RDD.collectAsMap).\n\n\n\n```python\n# EXERCICIO\nsampleOHEDict = (sampleDistinctFeats\n .\n .)\nprint sampleOHEDict\n```\n\n\n```python\n# TEST OHE Dictionary from distinct features (2b)\nTest.assertEquals(sorted(sampleOHEDict.keys()),\n [(0, 'bear'), (0, 'cat'), (0, 'mouse'), (1, 'black'),\n (1, 'tabby'), (2, 'mouse'), (2, 'salmon')],\n 'sampleOHEDict has unexpected keys')\nTest.assertEquals(sorted(sampleOHEDict.values()), range(7), 'sampleOHEDict has unexpected values')\n```\n\n#### **(2c) Cria\u00e7\u00e3o autom\u00e1tica do dicion\u00e1rio OHE **\n\n#### Agora use os c\u00f3digos dos exerc\u00edcios anteriores para criar uma fun\u00e7\u00e3o que retorna um dicion\u00e1rio OHE a partir dos atributos categ\u00f3ricos de uma base de dados.\n\n\n```python\n# EXERCICIO\ndef createOneHotDict(inputData):\n \"\"\"Creates a one-hot-encoder dictionary based on the input data.\n\n Args:\n inputData (RDD of lists of (int, str)): An RDD of observations where each observation is\n made up of a list of (featureID, value) tuples.\n\n Returns:\n dict: A dictionary where the keys are (featureID, value) tuples and map to values that are\n unique integers.\n \"\"\"\n return (inputData\n .\n .\n .\n .\n )\n\nsampleOHEDictAuto = createOneHotDict(sampleDataRDD)\nprint sampleOHEDictAuto\n```\n\n\n```python\n# TEST Automated creation of an OHE dictionary (2c)\nTest.assertEquals(sorted(sampleOHEDictAuto.keys()),\n [(0, 'bear'), (0, 'cat'), (0, 'mouse'), (1, 'black'),\n (1, 'tabby'), (2, 'mouse'), (2, 'salmon')],\n 'sampleOHEDictAuto has unexpected keys')\nTest.assertEquals(sorted(sampleOHEDictAuto.values()), range(7),\n 'sampleOHEDictAuto has unexpected values')\n```\n\n### **Part 3: Parse CTR data and generate OHE features**\n\n#### Antes de come\u00e7ar essa parte, vamos carregar a base de dados e verificar o formato dela.\n\n#### Repare que o primeiro campo \u00e9 o r\u00f3tulo de cada objeto, sendo 0 se o usu\u00e1rio n\u00e3o clicou no banner e 1 caso tenha clicado. O restante dos atributos ou s\u00e3o num\u00e9ricos ou s\u00e3o strings representando categorias an\u00f4nimas. Vamos tratar todos os atributos como categ\u00f3ricos.\n\n\n```python\nimport os.path\nbaseDir = os.path.join('Data')\ninputPath = os.path.join('Aula04', 'dac_sample.txt')\nfileName = os.path.join(baseDir, inputPath)\n\nif os.path.isfile(fileName):\n rawData = (sc\n .textFile(fileName, 2)\n .map(lambda x: x.replace('\\t', ','))) # work with either ',' or '\\t' separated data\n print rawData.take(1)\n```\n\n#### **(3a) Carregando e dividindo os dados **\n\n#### Da mesma forma que no notebook anterior, vamos dividir os dados entre treinamento, valida\u00e7\u00e3o e teste. Use o m\u00e9todo [randomSplit](https://spark.apache.org/docs/latest/api/python/pyspark.html#pyspark.RDD.randomSplit) com os pesos (weights) e semente aleat\u00f3ria (seed) especificados para criar os conjuntos, ent\u00e3o fa\u00e7a o [cache](https://spark.apache.org/docs/latest/api/python/pyspark.html#pyspark.RDD.cache) de cada RDD, pois utilizaremos cada uma delas com frequ\u00eancia durante esse exerc\u00edcio.\n\n\n```python\n# EXERCICIO\nweights = [.8, .1, .1]\nseed = 42\n# Use randomSplit with weights and seed\nrawTrainData, rawValidationData, rawTestData = rawData.randomSplit(weights, seed)\n# Cache the data\nrawTrainData.\nrawValidationData.\nrawTestData.\n\nnTrain = rawTrainData.count()\nnVal = rawValidationData.count()\nnTest = rawTestData.count()\nprint nTrain, nVal, nTest, nTrain + nVal + nTest\nprint rawData.take(1)\n```\n\n\n```python\n# TEST Loading and splitting the data (3a)\nTest.assertTrue(all([rawTrainData.is_cached, rawValidationData.is_cached, rawTestData.is_cached]),\n 'you must cache the split data')\nTest.assertEquals(nTrain, 79911, 'incorrect value for nTrain')\nTest.assertEquals(nVal, 10075, 'incorrect value for nVal')\nTest.assertEquals(nTest, 10014, 'incorrect value for nTest')\n```\n\n#### ** (3b) Extra\u00e7\u00e3o de atributos **\n\n#### Como pr\u00f3ximo passo, crie uma fun\u00e7\u00e3o para ser aplicada em cada objeto do RDD para gerar uma RDD de tuplas (IDatributo, categoria). Ignore o primeiro campo, que \u00e9 o r\u00f3tulo e gere uma lista de tuplas para os atributos seguintes. Utilize o comando [enumerate](https://docs.python.org/2/library/functions.html#enumerate) para criar essas tuplas.\n\n\n```python\n# EXERCICIO\ndef parsePoint(point):\n \"\"\"Converts a comma separated string into a list of (featureID, value) tuples.\n\n Note:\n featureIDs should start at 0 and increase to the number of features - 1.\n\n Args:\n point (str): A comma separated string where the first value is the label and the rest\n are features.\n\n Returns:\n list: A list of (featureID, value) tuples.\n \"\"\"\n \n\nparsedTrainFeat = rawTrainData.map(parsePoint)\n\nnumCategories = (parsedTrainFeat\n .\n .\n .\n .\n .\n .collect()\n )\n\nprint numCategories[2][1]\n```\n\n\n```python\n# TEST Extract features (3b)\nTest.assertEquals(numCategories[2][1], 855, 'incorrect implementation of parsePoint')\nTest.assertEquals(numCategories[32][1], 4, 'incorrect implementation of parsePoint')\n```\n\n#### **(3c) Crie o dicion\u00e1rio de OHE dessa base de dados **\n\n#### Note que a fun\u00e7\u00e3o parsePoint retorna um objeto em forma de lista `(IDatributo, categoria)`, que \u00e9 o mesmo formato utilizado pela fun\u00e7\u00e3o `createOneHotDict`. Utilize o RDD `parsedTrainFeat` para criar um dicion\u00e1rio OHE.\n\n\n```python\n# EXERCICIO\nctrOHEDict = \nnumCtrOHEFeats = len(ctrOHEDict.keys())\nprint numCtrOHEFeats\nprint ctrOHEDict[(0, '')]\n```\n\n\n```python\n# TEST Create an OHE dictionary from the dataset (3c)\nTest.assertEquals(numCtrOHEFeats, 233286, 'incorrect number of features in ctrOHEDict')\nTest.assertTrue((0, '') in ctrOHEDict, 'incorrect features in ctrOHEDict')\n```\n\n#### ** (3d) Aplicando OHE \u00e0 base de dados **\n\n#### Agora vamos usar o dicion\u00e1rio OHE para criar um RDD de objetos [LabeledPoint](http://spark.apache.org/docs/1.3.1/api/python/pyspark.mllib.html#pyspark.mllib.regression.LabeledPoint) usando atributos OHE. Complete a fun\u00e7\u00e3o `parseOHEPoint`. Dica: essa fun\u00e7\u00e3o \u00e9 uma extens\u00e3o da fun\u00e7\u00e3o `parsePoint` criada anteriormente e que usa a fun\u00e7\u00e3o `oneHotEncoding`.\n\n\n```python\nfrom pyspark.mllib.regression import LabeledPoint\n```\n\n\n```python\n# EXERCICIO\ndef parseOHEPoint(point, OHEDict, numOHEFeats):\n \"\"\"Obtain the label and feature vector for this raw observation.\n\n Note:\n You must use the function `oneHotEncoding` in this implementation or later portions\n of this lab may not function as expected.\n\n Args:\n point (str): A comma separated string where the first value is the label and the rest\n are features.\n OHEDict (dict of (int, str) to int): Mapping of (featureID, value) to unique integer.\n numOHEFeats (int): The number of unique features in the training dataset.\n\n Returns:\n LabeledPoint: Contains the label for the observation and the one-hot-encoding of the\n raw features based on the provided OHE dictionary.\n \"\"\"\n \n \nOHETrainData = rawTrainData.map(lambda point: parseOHEPoint(point, ctrOHEDict, numCtrOHEFeats))\nOHETrainData.cache()\nprint OHETrainData.take(1)\n\n# Check that oneHotEncoding function was used in parseOHEPoint\nbackupOneHot = oneHotEncoding\noneHotEncoding = None\nwithOneHot = False\ntry: parseOHEPoint(rawTrainData.take(1)[0], ctrOHEDict, numCtrOHEFeats)\nexcept TypeError: withOneHot = True\noneHotEncoding = backupOneHot\n```\n\n\n```python\n# TEST Apply OHE to the dataset (3d)\nnumNZ = sum(parsedTrainFeat.map(lambda x: len(x)).take(5))\nnumNZAlt = sum(OHETrainData.map(lambda lp: len(lp.features.indices)).take(5))\nTest.assertEquals(numNZ, numNZAlt, 'incorrect implementation of parseOHEPoint')\nTest.assertTrue(withOneHot, 'oneHotEncoding not present in parseOHEPoint')\n```\n\n#### **Visualiza\u00e7\u00e3o 1: Frequ\u00eancia dos Atributos **\n\n#### Vamos agora visualizar o n\u00famero de vezes que cada um dos 233.286 atributos OHE aparecem na base de treino. Para isso primeiro contabilizamos quantas vezes cada atributo aparece na base, ent\u00e3o alocamos cada atributo em um balde de histograma. Os baldes tem tamanhos de pot\u00eancia de 2, ent\u00e3o o primeiro balde conta os atributos que aparecem exatamente uma vez ( $ \\scriptsize 2^0 $ ), o segundo atributos que aparecem duas vezes ( $ \\scriptsize 2^1 $ ), o terceiro os atributos que aparecem de 3 a 4 vezes ( $ \\scriptsize 2^2 $ ), o quinto balde \u00e9 para atributos que ocorrem de cinco a oito vezes ( $ \\scriptsize 2^3 $ ) e assim por diante. O gr\u00e1fico de dispers\u00e3o abaixo mostra o logar\u00edtmo do tamanho dos baldes versus o logar\u00edtmo da frequ\u00eancia de atributos que ca\u00edram nesse balde.\n\n\n```python\ndef bucketFeatByCount(featCount):\n \"\"\"Bucket the counts by powers of two.\"\"\"\n for i in range(11):\n size = 2 ** i\n if featCount <= size:\n return size\n return -1\n\nfeatCounts = (OHETrainData\n .flatMap(lambda lp: lp.features.indices)\n .map(lambda x: (x, 1))\n .reduceByKey(lambda x, y: x + y))\nfeatCountsBuckets = (featCounts\n .map(lambda x: (bucketFeatByCount(x[1]), 1))\n .filter(lambda (k, v): k != -1)\n .reduceByKey(lambda x, y: x + y)\n .collect())\nprint featCountsBuckets\n```\n\n\n```python\nimport matplotlib.pyplot as plt\n\nx, y = zip(*featCountsBuckets)\nx, y = np.log(x), np.log(y)\n\ndef preparePlot(xticks, yticks, figsize=(10.5, 6), hideLabels=False, gridColor='#999999',\n gridWidth=1.0):\n \"\"\"Template for generating the plot layout.\"\"\"\n plt.close()\n fig, ax = plt.subplots(figsize=figsize, facecolor='white', edgecolor='white')\n ax.axes.tick_params(labelcolor='#999999', labelsize='10')\n for axis, ticks in [(ax.get_xaxis(), xticks), (ax.get_yaxis(), yticks)]:\n axis.set_ticks_position('none')\n axis.set_ticks(ticks)\n axis.label.set_color('#999999')\n if hideLabels: axis.set_ticklabels([])\n plt.grid(color=gridColor, linewidth=gridWidth, linestyle='-')\n map(lambda position: ax.spines[position].set_visible(False), ['bottom', 'top', 'left', 'right'])\n return fig, ax\n\n# generate layout and plot data\nfig, ax = preparePlot(np.arange(0, 10, 1), np.arange(4, 14, 2))\nax.set_xlabel(r'$\\log_e(bucketSize)$'), ax.set_ylabel(r'$\\log_e(countInBucket)$')\nplt.scatter(x, y, s=14**2, c='#d6ebf2', edgecolors='#8cbfd0', alpha=0.75)\npass\n```\n\n#### **(3e) Atributos n\u00e3o observados **\n\n#### Naturalmente precisaremos aplicar esse mesmo procedimento para as outras bases (valida\u00e7\u00e3o e teste), por\u00e9m nessas bases podem existir atributos n\u00e3o observados na base de treino.\n\n#### Precisamos adaptar a fun\u00e7\u00e3o `oneHotEncoding` para ignorar os atributos que n\u00e3o existem no dicion\u00e1rio.\n\n\n```python\n# EXERCICIO\ndef oneHotEncoding(rawFeats, OHEDict, numOHEFeats):\n \"\"\"Produce a one-hot-encoding from a list of features and an OHE dictionary.\n\n Note:\n If a (featureID, value) tuple doesn't have a corresponding key in OHEDict it should be\n ignored.\n\n Args:\n rawFeats (list of (int, str)): The features corresponding to a single observation. Each\n feature consists of a tuple of featureID and the feature's value. (e.g. sampleOne)\n OHEDict (dict): A mapping of (featureID, value) to unique integer.\n numOHEFeats (int): The total number of unique OHE features (combinations of featureID and\n value).\n\n Returns:\n SparseVector: A SparseVector of length numOHEFeats with indicies equal to the unique\n identifiers for the (featureID, value) combinations that occur in the observation and\n with values equal to 1.0.\n \"\"\"\n \n\nOHEValidationData = rawValidationData.map(lambda point: parseOHEPoint(point, ctrOHEDict, numCtrOHEFeats))\nOHEValidationData.cache()\nprint OHEValidationData.take(1)\n```\n\n\n```python\n# TEST Handling unseen features (3e)\nnumNZVal = (OHEValidationData\n .map(lambda lp: len(lp.features.indices))\n .sum())\nTest.assertEquals(numNZVal, 372080, 'incorrect number of features')\n```\n\n### ** Part 4: Predi\u00e7\u00e3o do CTR e avalia\u00e7\u00e3o da perda-log (logloss) **\n\n#### ** (4a) Regress\u00e3o Log\u00edstica **\n\n#### Um classificador que podemos utilizar nessa base de dados \u00e9 a regress\u00e3o log\u00edstica, que nos d\u00e1 a probabilidade de um evento de clique em banner ocorrer. Vamos utilizar a fun\u00e7\u00e3o [LogisticRegressionWithSGD](https://spark.apache.org/docs/latest/api/python/pyspark.mllib.html#pyspark.mllib.classification.LogisticRegressionWithSGD) para treinar um modelo usando `OHETrainData` com a configura\u00e7\u00e3o de par\u00e2metros dada. `LogisticRegressionWithSGD` retorna um [LogisticRegressionModel](https://spark.apache.org/docs/latest/api/python/pyspark.mllib.html#pyspark.mllib.regression.LogisticRegressionModel). \n\n#### Em seguida, imprima `LogisticRegressionModel.weights` e `LogisticRegressionModel.intercept` para verificar o modelo gerado. \n\n\n```python\nfrom pyspark.mllib.classification import LogisticRegressionWithSGD\n\n# fixed hyperparameters\nnumIters = 50\nstepSize = 10.\nregParam = 1e-6\nregType = 'l2'\nincludeIntercept = True\n```\n\n\n```python\n# EXERCICIO\nmodel0 = \nsortedWeights = sorted(model0.weights)\nprint sortedWeights[:5], model0.intercept\n```\n\n\n```python\n# TEST Logistic regression (4a)\nTest.assertTrue(np.allclose(model0.intercept, 0.56455084025), 'incorrect value for model0.intercept')\nTest.assertTrue(np.allclose(sortedWeights[0:5],\n [-0.45899236853575609, -0.37973707648623956, -0.36996558266753304,\n -0.36934962879928263, -0.32697945415010637]), 'incorrect value for model0.weights')\n```\n\n#### ** (4b) Log loss **\n\n#### Uma forma de avaliar um classificador bin\u00e1rio \u00e9 atrav\u00e9s do log-loss, definido como: $$ \\begin{align} \\scriptsize \\ell_{log}(p, y) = \\begin{cases} -\\log (p) & \\text{if } y = 1 \\\\\\ -\\log(1-p) & \\text{if } y = 0 \\end{cases} \\end{align} $$ onde $ \\scriptsize p$ \u00e9 uma probabilidade entre 0 e 1 e $ \\scriptsize y$ \u00e9 o r\u00f3tulo bin\u00e1rio (0 ou 1). Log loss \u00e9 um crit\u00e9rio de avalia\u00e7\u00e3o muito utilizado quando deseja-se predizer eventos raros. Escreva uma fun\u00e7\u00e3o para calcular o log-loss, e avalie algumas entradas de amostra.\n\n\n```python\n# EXERCICIO\nfrom math import log\n\ndef computeLogLoss(p, y):\n \"\"\"Calculates the value of log loss for a given probabilty and label.\n\n Note:\n log(0) is undefined, so when p is 0 we need to add a small value (epsilon) to it\n and when p is 1 we need to subtract a small value (epsilon) from it.\n\n Args:\n p (float): A probabilty between 0 and 1.\n y (int): A label. Takes on the values 0 and 1.\n\n Returns:\n float: The log loss value.\n \"\"\"\n \n\nprint computeLogLoss(.5, 1)\nprint computeLogLoss(.5, 0)\nprint computeLogLoss(.99, 1)\nprint computeLogLoss(.99, 0)\nprint computeLogLoss(.01, 1)\nprint computeLogLoss(.01, 0)\nprint computeLogLoss(0, 1)\nprint computeLogLoss(1, 1)\nprint computeLogLoss(1, 0)\n```\n\n\n```python\n# TEST Log loss (4b)\nTest.assertTrue(np.allclose([computeLogLoss(.5, 1), computeLogLoss(.01, 0), computeLogLoss(.01, 1)],\n [0.69314718056, 0.0100503358535, 4.60517018599]),\n 'computeLogLoss is not correct')\nTest.assertTrue(np.allclose([computeLogLoss(0, 1), computeLogLoss(1, 1), computeLogLoss(1, 0)],\n [25.3284360229, 1.00000008275e-11, 25.3284360229]),\n 'computeLogLoss needs to bound p away from 0 and 1 by epsilon')\n```\n\n#### ** (4c) Baseline log loss **\n\n#### Agora, vamos utilizar a fun\u00e7\u00e3o da Parte (4b) para calcular um baseline da m\u00e9trica de log-loss na nossa base de treino. Uma forma de calcular um baseline \u00e9 predizer sempre a m\u00e9dia dos r\u00f3tulos observados. Primeiro calcule a m\u00e9dia dos r\u00f3tulos da base e, em seguida, calcule o log-loss m\u00e9dio para a base de treino.\n\n\n```python\n# EXERCICIO\n# Note that our dataset has a very high click-through rate by design\n# In practice click-through rate can be one to two orders of magnitude lower\nclassOneFracTrain = OHETrainData.\nprint classOneFracTrain\n\nlogLossTrBase = OHETrainData.\nprint 'Baseline Train Logloss = {0:.3f}\\n'.format(logLossTrBase)\n```\n\n\n```python\n# TEST Baseline log loss (4c)\nTest.assertTrue(np.allclose(classOneFracTrain, 0.22717773523), 'incorrect value for classOneFracTrain')\nTest.assertTrue(np.allclose(logLossTrBase, 0.535844), 'incorrect value for logLossTrBase')\n```\n\n#### ** (4d) Probabilidade da Predi\u00e7\u00e3o **\n\n#### O modelo gerado na Parte (4a) possui um m\u00e9todo chamado `predict`, por\u00e9m esse m\u00e9todo retorna apenas 0's e 1's. Para calcular a probabilidade de um evento, vamos criar uma fun\u00e7\u00e3o `getP` que recebe como par\u00e2metro o ponto x, o conjunto de pesos `w` e o `intercept`.\n\n#### Calcule o modelo de regress\u00e3o linear nesse ponto x e aplique a [fun\u00e7\u00e3o sigmoidal](http://en.wikipedia.org/wiki/Sigmoid_function) $ \\scriptsize \\sigma(t) = (1+ e^{-t})^{-1} $ para retornar a probabilidade da predi\u00e7\u00e3o do objeto x.\n\n\n\n\n```python\n# EXERCICIO\nfrom math import exp # exp(-t) = e^-t\n\ndef getP(x, w, intercept):\n \"\"\"Calculate the probability for an observation given a set of weights and intercept.\n\n Note:\n We'll bound our raw prediction between 20 and -20 for numerical purposes.\n\n Args:\n x (SparseVector): A vector with values of 1.0 for features that exist in this\n observation and 0.0 otherwise.\n w (DenseVector): A vector of weights (betas) for the model.\n intercept (float): The model's intercept.\n\n Returns:\n float: A probability between 0 and 1.\n \"\"\"\n # calculate rawPrediction = w.x + intercept\n rawPrediction = \n\n # Bound the raw prediction value\n rawPrediction = min(rawPrediction, 20)\n rawPrediction = max(rawPrediction, -20)\n \n # calculate (1+e^-rawPrediction)^-1\n return \n\ntrainingPredictions = OHETrainData.\n\nprint trainingPredictions.take(5)\n```\n\n\n```python\n# TEST Predicted probability (4d)\nTest.assertTrue(np.allclose(trainingPredictions.sum(), 18135.4834348),\n 'incorrect value for trainingPredictions')\n```\n\n#### ** (4e) Avalie o modelo **\n\n#### Finalmente, crie uma fun\u00e7\u00e3o `evaluateResults` que calcula o log-loss m\u00e9dio do modelo em uma base de dados. Em seguida, execute essa fun\u00e7\u00e3o na nossa base de treino.\n\n\n```python\n# EXERCICIO\ndef evaluateResults(model, data):\n \"\"\"Calculates the log loss for the data given the model.\n\n Args:\n model (LogisticRegressionModel): A trained logistic regression model.\n data (RDD of LabeledPoint): Labels and features for each observation.\n\n Returns:\n float: Log loss for the data.\n \"\"\"\n return (data\n .\n .\n .\n )\n\nlogLossTrLR0 = evaluateResults(model0, OHETrainData)\nprint ('OHE Features Train Logloss:\\n\\tBaseline = {0:.3f}\\n\\tLogReg = {1:.3f}'\n .format(logLossTrBase, logLossTrLR0))\n```\n\n\n```python\n# TEST Evaluate the model (4e)\nTest.assertTrue(np.allclose(logLossTrLR0, 0.456903), 'incorrect value for logLossTrLR0')\n```\n\n#### ** (4f) log-loss da valida\u00e7\u00e3o **\n\n#### Agora aplique o modelo na nossa base de valida\u00e7\u00e3o e calcule o log-loss m\u00e9dio, compare com o nosso baseline.\n\n\n```python\n# EXERCICIO\nlogLossValBase = OHEValidationData.\n\nlogLossValLR0 = evaluateResults(model0, OHEValidationData)\nprint ('OHE Features Validation Logloss:\\n\\tBaseline = {0:.3f}\\n\\tLogReg = {1:.3f}'\n .format(logLossValBase, logLossValLR0))\n```\n\n\n```python\n# TEST Validation log loss (4f)\nTest.assertTrue(np.allclose(logLossValBase, 0.527603), 'incorrect value for logLossValBase')\nTest.assertTrue(np.allclose(logLossValLR0, 0.456957), 'incorrect value for logLossValLR0')\n```\n\n#### **Visualiza\u00e7\u00e3o 2: Curva ROC **\n\n#### A curva ROC nos mostra o custo-benef\u00edcio entre a taxa de falso positivo e a taxa de verdadeiro positivo, conforme diminuimos o limiar de predi\u00e7\u00e3o. Um modelo aleat\u00f3rio \u00e9 representado por uma linha pontilhada. Idealmente nosso modelo deve formar uma curva acima dessa linha.\n\n\n```python\nlabelsAndScores = OHEValidationData.map(lambda lp:\n (lp.label, getP(lp.features, model0.weights, model0.intercept)))\nlabelsAndWeights = labelsAndScores.collect()\nlabelsAndWeights.sort(key=lambda (k, v): v, reverse=True)\nlabelsByWeight = np.array([k for (k, v) in labelsAndWeights])\n\nlength = labelsByWeight.size\ntruePositives = labelsByWeight.cumsum()\nnumPositive = truePositives[-1]\nfalsePositives = np.arange(1.0, length + 1, 1.) - truePositives\n\ntruePositiveRate = truePositives / numPositive\nfalsePositiveRate = falsePositives / (length - numPositive)\n\n# Generate layout and plot data\nfig, ax = preparePlot(np.arange(0., 1.1, 0.1), np.arange(0., 1.1, 0.1))\nax.set_xlim(-.05, 1.05), ax.set_ylim(-.05, 1.05)\nax.set_ylabel('True Positive Rate (Sensitivity)')\nax.set_xlabel('False Positive Rate (1 - Specificity)')\nplt.plot(falsePositiveRate, truePositiveRate, color='#8cbfd0', linestyle='-', linewidth=3.)\nplt.plot((0., 1.), (0., 1.), linestyle='--', color='#d6ebf2', linewidth=2.) # Baseline model\npass\n```\n\n### **Parte 5: Reduzindo a dimens\u00e3o dos atributos via feature hashing**\n\n#### ** (5a) Fun\u00e7\u00e3o Hash **\n\n#### Nosso modelo OHE consegue criar uma representa\u00e7\u00e3o num\u00e9rica boa o suficiente para ser aplic\u00e1vel em algoritmos de classifica\u00e7\u00e3o que n\u00e3o conseguem tratar dados categ\u00f3ricos. Por\u00e9m, para nossa base de dados isso gerou um n\u00famero enorme de atributos (233 mil) que pode tornar o problema intrat\u00e1vel. Para reduzir o espa\u00e7o de atributos vamos utilizar um truque atrav\u00e9s de fun\u00e7\u00f5es hash chamado de feature hashing.\n\n#### Logo abaixo, j\u00e1 est\u00e1 implementada a fun\u00e7\u00e3o de hash que usaremos nessa parte do notebook. Vamos aplic\u00e1-la na nossa base artificial criada na Parte (1a) para termos uma intui\u00e7\u00e3o do que est\u00e1 acontecendo. Execute essa fun\u00e7\u00e3o para valores diferentes de `numBuckets` e observe o resultado.\n\n\n```python\nfrom collections import defaultdict\nimport hashlib\n\ndef hashFunction(numBuckets, rawFeats, printMapping=False):\n \"\"\"Calculate a feature dictionary for an observation's features based on hashing.\n\n Note:\n Use printMapping=True for debug purposes and to better understand how the hashing works.\n\n Args:\n numBuckets (int): Number of buckets to use as features.\n rawFeats (list of (int, str)): A list of features for an observation. Represented as\n (featureID, value) tuples.\n printMapping (bool, optional): If true, the mappings of featureString to index will be\n printed.\n\n Returns:\n dict of int to float: The keys will be integers which represent the buckets that the\n features have been hashed to. The value for a given key will contain the count of the\n (featureID, value) tuples that have hashed to that key.\n \"\"\"\n mapping = {}\n for ind, category in rawFeats:\n featureString = category + str(ind)\n mapping[featureString] = int(int(hashlib.md5(featureString).hexdigest(), 16) % numBuckets)\n if(printMapping): print mapping\n sparseFeatures = defaultdict(float)\n for bucket in mapping.values():\n sparseFeatures[bucket] += 1.0\n return dict(sparseFeatures)\n\n# Reminder of the sample values:\n# sampleOne = [(0, 'mouse'), (1, 'black')]\n# sampleTwo = [(0, 'cat'), (1, 'tabby'), (2, 'mouse')]\n# sampleThree = [(0, 'bear'), (1, 'black'), (2, 'salmon')]\n```\n\n\n```python\n# EXERCICIO\n# Use four buckets\nsampOneFourBuckets = \nsampTwoFourBuckets = \nsampThreeFourBuckets = \n\n# Use one hundred buckets\nsampOneHundredBuckets = \nsampTwoHundredBuckets = \nsampThreeHundredBuckets = \n\nprint '\\t\\t 4 Buckets \\t\\t\\t 100 Buckets'\nprint 'SampleOne:\\t {0}\\t\\t {1}'.format(sampOneFourBuckets, sampOneHundredBuckets)\nprint 'SampleTwo:\\t {0}\\t\\t {1}'.format(sampTwoFourBuckets, sampTwoHundredBuckets)\nprint 'SampleThree:\\t {0}\\t {1}'.format(sampThreeFourBuckets, sampThreeHundredBuckets)\n```\n\n\n```python\n# TEST Hash function (5a)\nTest.assertEquals(sampOneFourBuckets, {2: 1.0, 3: 1.0}, 'incorrect value for sampOneFourBuckets')\nTest.assertEquals(sampThreeHundredBuckets, {72: 1.0, 5: 1.0, 14: 1.0},\n 'incorrect value for sampThreeHundredBuckets')\n```\n\n#### ** (5b) Criando hashed features **\n\n#### Agora vamos usar essa fun\u00e7\u00e3o hash para criar hashed features para nossa base CTR. Primeiro escreva uma fun\u00e7\u00e3o que usa a fun\u00e7\u00e3o hash da Parte (5a) com numBuckets = $ \\scriptsize 2^{15} \\approx 33K $ para criar um `LabeledPoint` com os hashed features armazenados como um `SparseVector`. Ent\u00e3o use esta fun\u00e7\u00e3o para criar uma nova base de treino, valida\u00e7\u00e3o e teste com hashed features. Dica: `parsedHashPoint` \u00e9 similar a `parseOHEPoint` da Parte (3d).\n\n\n```python\n# EXERCICIO\ndef parseHashPoint(point, numBuckets):\n \"\"\"Create a LabeledPoint for this observation using hashing.\n\n Args:\n point (str): A comma separated string where the first value is the label and the rest are\n features.\n numBuckets: The number of buckets to hash to.\n\n Returns:\n LabeledPoint: A LabeledPoint with a label (0.0 or 1.0) and a SparseVector of hashed\n features.\n \"\"\"\n \n\n\nnumBucketsCTR = 2 ** 15\nhashTrainData = rawTrainData.map(lambda x: parseHashPoint(x,numBucketsCTR))\nhashTrainData.cache()\nhashValidationData = rawValidationData.map(lambda x: parseHashPoint(x,numBucketsCTR))\nhashValidationData.cache()\nhashTestData = rawTestData.map(lambda x: parseHashPoint(x,numBucketsCTR))\nhashTestData.cache()\n\nprint hashTrainData.take(1)\n```\n\n\n```python\n# TEST Creating hashed features (5b)\nhashTrainDataFeatureSum = sum(hashTrainData\n .map(lambda lp: len(lp.features.indices))\n .take(20))\nhashTrainDataLabelSum = sum(hashTrainData\n .map(lambda lp: lp.label)\n .take(100))\nhashValidationDataFeatureSum = sum(hashValidationData\n .map(lambda lp: len(lp.features.indices))\n .take(20))\nhashValidationDataLabelSum = sum(hashValidationData\n .map(lambda lp: lp.label)\n .take(100))\nhashTestDataFeatureSum = sum(hashTestData\n .map(lambda lp: len(lp.features.indices))\n .take(20))\nhashTestDataLabelSum = sum(hashTestData\n .map(lambda lp: lp.label)\n .take(100))\n\nTest.assertEquals(hashTrainDataFeatureSum, 772, 'incorrect number of features in hashTrainData')\nTest.assertEquals(hashTrainDataLabelSum, 24.0, 'incorrect labels in hashTrainData')\nTest.assertEquals(hashValidationDataFeatureSum, 776,\n 'incorrect number of features in hashValidationData')\nTest.assertEquals(hashValidationDataLabelSum, 16.0, 'incorrect labels in hashValidationData')\nTest.assertEquals(hashTestDataFeatureSum, 774, 'incorrect number of features in hashTestData')\nTest.assertEquals(hashTestDataLabelSum, 23.0, 'incorrect labels in hashTestData')\n```\n\n#### ** (5c) Esparsidade **\n\n#### Uma vez que temos 33 mil hashed features contra 233 mil OHE, devemos esperar que os atributos OHE sejam mais esparsos. Verifique essa hip\u00f3tese computando a esparsidade m\u00e9dia do OHE e do hashed features.\n\n#### Note que se voc\u00ea tem um `SparseVector` chamado `sparse`, chamar `len(sparse)` retornar\u00e1 o total de atributos, e n\u00e3o o n\u00famero de valores n\u00e3o nulos. `SparseVector` tem atributos `indices` e `values` que cont\u00e9m informa\u00e7\u00f5es sobre quais atributos s\u00e3o n\u00e3o nulos. \n\n\n```python\n# EXERCICIO\ndef computeSparsity(data, d, n):\n \"\"\"Calculates the average sparsity for the features in an RDD of LabeledPoints.\n\n Args:\n data (RDD of LabeledPoint): The LabeledPoints to use in the sparsity calculation.\n d (int): The total number of features.\n n (int): The number of observations in the RDD.\n\n Returns:\n float: The average of the ratio of features in a point to total features.\n \"\"\"\n return (data\n .\n .\n )/(d*n*1.)\n\naverageSparsityHash = computeSparsity(hashTrainData, numBucketsCTR, nTrain)\naverageSparsityOHE = computeSparsity(OHETrainData, numCtrOHEFeats, nTrain)\n\nprint 'Average OHE Sparsity: {0:.7e}'.format(averageSparsityOHE)\nprint 'Average Hash Sparsity: {0:.7e}'.format(averageSparsityHash)\n```\n\n\n```python\n# TEST Sparsity (5c)\nTest.assertTrue(np.allclose(averageSparsityOHE, 1.6717677e-04),\n 'incorrect value for averageSparsityOHE')\nTest.assertTrue(np.allclose(averageSparsityHash, 1.1805561e-03),\n 'incorrect value for averageSparsityHash')\n```\n\n#### ** (5d) Modelo log\u00edstico com hashed features **\n\n#### Agora treine um modelo de regress\u00e3o log\u00edstica para os hashed features. Execute um grid search para encontrar par\u00e2metros adequados para essa base, avaliando o log-loss no conjunto de valida\u00e7\u00e3o. Nota: isso pode demorar alguns minutos para terminar. Use `stepSizes` de 1 e 10 e `regParams` de 1e-6 e 1e-3.\n\n\n```python\nnumIters = 500\nregType = 'l2'\nincludeIntercept = True\n\n# Initialize variables using values from initial model training\nbestModel = None\nbestLogLoss = 1e10\n```\n\n\n```python\n# EXERCICIO\nstepSizes = [1, 10]\nregParams = [1e-6, 1e-3]\nfor stepSize in stepSizes:\n for regParam in regParams:\n model = ()\n logLossVa = \n print ('\\tstepSize = {0:.1f}, regParam = {1:.0e}: logloss = {2:.3f}'\n .format(stepSize, regParam, logLossVa))\n if (logLossVa < bestLogLoss):\n bestModel = model\n bestLogLoss = logLossVa\n\nprint ('Hashed Features Validation Logloss:\\n\\tBaseline = {0:.3f}\\n\\tLogReg = {1:.3f}'\n .format(logLossValBase, bestLogLoss))\n```\n\n\n```python\n# TEST Logistic model with hashed features (5d)\nTest.assertTrue(np.allclose(bestLogLoss, 0.4481683608), 'incorrect value for bestLogLoss')\n```\n\n#### ** (5e) Avaliando a base de testes **\n\n#### Finalmente, avalie o melhor modelo da Parte (5d) na base de testes. Compare o log-loss do resultado com o log-loss do nosso baseline no conjunto de testes, calculando da mesma forma que na Parte (4f).\n\n\n```python\n# EXERCICIO\n# Log loss for the best model from (5d)\nlogLossValLR0 = \n\nlogLossTest = \n\n# Log loss for the baseline model\nlogLossTestBaseline = hashTestData.map(lambda lp: computeLogLoss(classOneFracTrain,lp.label)).mean()\n\nprint ('Hashed Features Test Log Loss:\\n\\tBaseline = {0:.3f}\\n\\tLogReg = {1:.3f}'\n .format(logLossTestBaseline, logLossTest))\n```\n\n\n```python\n# TEST Evaluate on the test set (5e)\nTest.assertTrue(np.allclose(logLossTestBaseline, 0.537438),\n 'incorrect value for logLossTestBaseline')\nTest.assertTrue(np.allclose(logLossTest, 0.455616931), 'incorrect value for logLossTest')\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "bb444737d4d8535917d2fdd2d5fafcff4cb06600", "size": 61288, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Spark/Lab4b_classificacao.ipynb", "max_stars_repo_name": "douggribeiro/bigdata", "max_stars_repo_head_hexsha": "5f923b4f21bce48a1a202d5f8d984ac076059b5b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-01-19T01:49:40.000Z", "max_stars_repo_stars_event_max_datetime": "2019-01-19T01:49:40.000Z", "max_issues_repo_path": "Spark/Lab4b_classificacao.ipynb", "max_issues_repo_name": "douggribeiro/bigdata", "max_issues_repo_head_hexsha": "5f923b4f21bce48a1a202d5f8d984ac076059b5b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Spark/Lab4b_classificacao.ipynb", "max_forks_repo_name": "douggribeiro/bigdata", "max_forks_repo_head_hexsha": "5f923b4f21bce48a1a202d5f8d984ac076059b5b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 14, "max_forks_repo_forks_event_min_datetime": "2015-10-28T00:46:38.000Z", "max_forks_repo_forks_event_max_datetime": "2018-11-29T14:03:54.000Z", "avg_line_length": 37.6, "max_line_length": 794, "alphanum_fraction": 0.5850248009, "converted": true, "num_tokens": 11861, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.8006920044739461, "lm_q1q2_score": 0.4159765686407756}} {"text": "+ This notebook is part of lecture 5 *Transposes, permutations, and vector spaces* in the OCW MIT course 18.06 by Prof Gilbert Strang [1]\n+ Created by me, Dr Juan H Klopper\n + Head of Acute Care Surgery\n + Groote Schuur Hospital\n + University Cape Town\n + Email me with your thoughts, comments, suggestions and corrections \n
                                        Linear Algebra OCW MIT18.06 IPython notebook [2] study notes by Dr Juan H Klopper is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.\n\n+ [1] OCW MIT 18.06\n+ [2] Fernando P\u00e9rez, Brian E. Granger, IPython: A System for Interactive Scientific Computing, Computing in Science and Engineering, vol. 9, no. 3, pp. 21-29, May/June 2007, doi:10.1109/MCSE.2007.53. URL: http://ipython.org\n\n\n```python\nfrom IPython.core.display import HTML, Image\ncss_file = 'style.css'\nHTML(open(css_file, 'r').read())\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n```python\n#import numpy as np\nfrom sympy import init_printing, Matrix, symbols\n#import matplotlib.pyplot as plt\n#import seaborn as sns\n#from IPython.display import Image\nfrom warnings import filterwarnings\n\ninit_printing(use_latex = 'mathjax')\n%matplotlib inline\nfilterwarnings('ignore')\n```\n\n# Transposes, permutations and vector spaces\n\n## The permutation matrices\n\n* Remember that the permutation matrices allow for row exchanges\n* They are used to manage zero's in pivot positions\n* The have the following property\n$$ {P}^{-1} = {P}^{T} $$\n\n\n```python\nP = Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])\nP # Exchanging rows 1 and 2\n```\n\n\n\n\n$$\\left[\\begin{matrix}0 & 1 & 0\\\\1 & 0 & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$\n\n\n\n\n```python\nP.inv(), P.transpose()\n```\n\n\n\n\n$$\\begin{pmatrix}\\left[\\begin{matrix}0 & 1 & 0\\\\1 & 0 & 0\\\\0 & 0 & 1\\end{matrix}\\right], & \\left[\\begin{matrix}0 & 1 & 0\\\\1 & 0 & 0\\\\0 & 0 & 1\\end{matrix}\\right]\\end{pmatrix}$$\n\n\n\n\n```python\nP.inv() == P.transpose()\n```\n\n\n\n\n True\n\n\n\n* If a matrix is of size *n* × *n* then there are *n*! number of permutations\n\n## The transpose of a matrix\n\n* We have mentioned transposes of a matrix, but what are they?\n* The simply make row of the column elements and columns of the row elements as in the example below\n\n\n```python\na11, a12, a13, a14, a21, a22, a23, a24, a31, a32, a33, a34 = symbols('a11, a12, a13, a14, a21, a22, a23, a24, a31, a32, a33, a34')\n# Creating mathematical scalar constants\n```\n\n\n```python\nA = Matrix([[a11, a12, a13], [a21, a22, a23], [a31, a32, a33]])\nA\n```\n\n\n\n\n$$\\left[\\begin{matrix}a_{11} & a_{12} & a_{13}\\\\a_{21} & a_{22} & a_{23}\\\\a_{31} & a_{32} & a_{33}\\end{matrix}\\right]$$\n\n\n\n\n```python\nA.transpose()\n```\n\n\n\n\n$$\\left[\\begin{matrix}a_{11} & a_{21} & a_{31}\\\\a_{12} & a_{22} & a_{32}\\\\a_{13} & a_{23} & a_{33}\\end{matrix}\\right]$$\n\n\n\n* This applies to any size matrix\n\n\n```python\nA = Matrix([[a11, a12, a13, a14], [a21, a22, a23, a24]])\nA\n```\n\n\n\n\n$$\\left[\\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14}\\\\a_{21} & a_{22} & a_{23} & a_{24}\\end{matrix}\\right]$$\n\n\n\n\n```python\nA.transpose()\n```\n\n\n\n\n$$\\left[\\begin{matrix}a_{11} & a_{21}\\\\a_{12} & a_{22}\\\\a_{13} & a_{23}\\\\a_{14} & a_{24}\\end{matrix}\\right]$$\n\n\n\n* Multiplying a matrix by its transpose results in a symmetric matrix\n\n\n```python\nA * A.transpose()\n```\n\n\n\n\n$$\\left[\\begin{matrix}a_{11}^{2} + a_{12}^{2} + a_{13}^{2} + a_{14}^{2} & a_{11} a_{21} + a_{12} a_{22} + a_{13} a_{23} + a_{14} a_{24}\\\\a_{11} a_{21} + a_{12} a_{22} + a_{13} a_{23} + a_{14} a_{24} & a_{21}^{2} + a_{22}^{2} + a_{23}^{2} + a_{24}^{2}\\end{matrix}\\right]$$\n\n\n\n## Symmetric matrices\n\n* A symmetric matrix is a square matrix with elements opposite the main diagonal all equal\n* Example\n\n\n```python\nS = Matrix([[1, 3, 2], [3, 2, 4], [2 , 4, 2]])\nS\n```\n\n\n\n\n$$\\left[\\begin{matrix}1 & 3 & 2\\\\3 & 2 & 4\\\\2 & 4 & 2\\end{matrix}\\right]$$\n\n\n\n* On the main diagonal we have 1, 2, 2\n* Opposite this main diagonal we have a 3 and a 3 and a 2 and a 2 and a 4 and a 4\n* The transpose of a symmetric matrix is equal to the matrix\n\n\n```python\nS == S.transpose()\n```\n\n\n\n\n True\n\n\n\n## Vector spaces\n\n* A vector space is a bunch of vectors (a set of vectors) With certain properties that allow us to do stuff with\n* The space ℝ2 is all vectors of two components that reaches every coordinate point in ℝ2\n* It always includes the zero vector **0**\n* We usually call this vector space *V*, such that *V* = ℝ2 or *V* = ℝn\n* A linear combination of a certain number of these can also fill all of ℝ2\n* A good example is the two unit vectors along the two axes\n* Such a set of vectors form a basis for *V*\n* The two of them also span ℝ2, i.e. a linear combination of them fills *V* = ℝ2\n* Linear independence means the vectors in ℝ2 don't fall on the same line\n * If they do, we can't get to all coordinate points in ℝ2\n* The **important** point about a vector space *V* is that it allows for vector addition and scalar multiplication\n * Taking any of the set of vectors in *V* and adding them results in a new vector which is still a component of *V*\n * Multiplying a scalar by any of the vectors in *V* results in a vector still in *V*\n\n### A subspace\n\n* For a subspace the rules of vector addition and scalar multiplication must apply\n* I.e. a quadrant of ℝ2 is not a vector subspace\n * Addition or scalar multiplication of any vector in this quadrant can lead to a vector outside of this quadrant\n* The zero vector **0** is a subspace (every subspace must contain the zero vector)\n* The whole space *V* = ℝn (here we use *n* = 2) is a subspace of itself\n* Continuing with our example of *n* = 2, any line **through the origin** is a subspace of ℝ2\n * Adding a vector on this line to itself of a scalar multiple of itself will eventually fill the whole line\n* For *n* = 3 we have the whole space *V* = ℝ3, a plane through the origin, a line through the origin and the zero vectors are all subspace of *V* = ℝ3\n* The point is that vector addition and scalar multiplication of vectors in the subspace must result in a new vector that remains in the subspace\n* Every subspace must include the zero vector **0**\n* All the properties of vectors must apply to the vectors in a subspace (and a space)\n\n## Column spaces of matrices\n\n* Here we see the columns of a matrix as a vector\n* If there are two columns and three rows we will have the following as an example\n$$ \\begin{bmatrix} 2 & 1 \\\\ 1 & 3 \\\\ 2 & 2 \\end{bmatrix}=\\begin{bmatrix} 2 \\\\ 1 \\\\ 2 \\end{bmatrix}+\\begin{bmatrix} 1 \\\\ 3 \\\\ 2 \\end{bmatrix} $$\n* If they are not linear combinations of each other addition and scalar multiplication of the two of them will fill a plane in ℝ3\n\n\n```python\n\n```\n", "meta": {"hexsha": "d4c5013886665d65c609a6b3c7fec1d320864137", "size": 18258, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "forks/MIT_OCW_Linear_Algebra_18_06-master/I_06_Transposes_Permutations_Spaces.ipynb", "max_stars_repo_name": "solomonxie/jupyter-notebooks", "max_stars_repo_head_hexsha": "65999f179e037242138de72f512dda4bf00c7379", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-02-13T05:52:05.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-08T09:52:35.000Z", "max_issues_repo_path": "forks/MIT_OCW_Linear_Algebra_18_06-master/I_06_Transposes_Permutations_Spaces.ipynb", "max_issues_repo_name": "solomonxie/jupyter-notebooks", "max_issues_repo_head_hexsha": "65999f179e037242138de72f512dda4bf00c7379", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "forks/MIT_OCW_Linear_Algebra_18_06-master/I_06_Transposes_Permutations_Spaces.ipynb", "max_forks_repo_name": "solomonxie/jupyter-notebooks", "max_forks_repo_head_hexsha": "65999f179e037242138de72f512dda4bf00c7379", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 29.4009661836, "max_line_length": 708, "alphanum_fraction": 0.5016431153, "converted": true, "num_tokens": 2965, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.7606506418255927, "lm_q1q2_score": 0.415876726217635}} {"text": "```python\nimport numpy as np\nimport pandas as pd\nimport linearsolve as ls\nimport matplotlib.pyplot as plt\nplt.style.use('classic')\n%matplotlib inline\n\npd.plotting.register_matplotlib_converters()\n```\n\n# Class 16: Introduction to New-Keynesian Business Cycle Modeling\n\nIn this notebook, we will briefly explore US macroeconomic data suggesting that, contrary to the assumptions of most RBC models, there is in fact a relationship between real and nominal quantities over the business cycle. Then we will use `linearsolve` to compute impulse responses of output, inflation, and the nominal interest rate to a monetary policy shock in the New-Keynesian model.\n\n## Data\n\nThe file `business_cycle_data_actual_trend_cycle.csv`, available at https://github.com/letsgoexploring/econ126/raw/master/Data/Csv/business_cycle_data_actual_trend_cycle.csv, contains actual and trend data for real GDP per capita, real consumption per capita, real investment per capita, real physical capital per capita, TFP, hours per capita, the rea money supply (M2), (nominal) interest rate on 3-month T-bills, the PCE inflation rate, and the unemployment rate; each at quarterly frequency. The GDP, consumption, investment, capital, and money supply data are in terms of 2012 dollars. Hours is measured as an index with the value in October 2012 set to 100.\n\n\n```python\n# Read business_cycle_data_actual_trend.csv into a Pandas DataFrame with the first column set as the index and parse_dates=True\ndata = pd.read_csv('https://github.com/letsgoexploring/econ126/raw/master/Data/Csv/business_cycle_data_actual_trend_cycle.csv',index_col=0,parse_dates=True)\n\n# Print the last five rows of the data\ndata.tail()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        gdpgdp_trendgdp_cycleconsumptionconsumption_trendconsumption_cycleinvestmentinvestment_trendinvestment_cyclehours...real_m2_cyclet_bill_3mot_bill_3mo_trendt_bill_3mo_cyclepce_inflationpce_inflation_trendpce_inflation_cycleunemploymentunemployment_trendunemployment_cycle
                                        2018-07-0172.58905972.3768130.00292849.37709949.2287280.00300912.88770612.7527140.010530105.399877...-0.0104090.0204000.0166650.0037350.0576620.0453600.0123030.0376670.038054-0.000388
                                        2018-10-0172.60644772.672948-0.00091549.37729449.441990-0.00130912.94279612.8258320.009078105.435736...-0.0158430.0231670.0181150.0050510.0460500.0456430.0004080.0380000.0367380.001262
                                        2019-01-0173.25108172.9709780.00383149.52968449.656456-0.00255613.13511412.8990000.018139105.677466...-0.0103840.0238670.0195660.0043000.0391810.045903-0.0067220.0386670.0354320.003234
                                        2019-04-0173.48223373.2705270.00288549.96721149.8720740.00190612.91078012.972253-0.004750105.340443...-0.0120740.0230000.0210150.0019850.0409870.046152-0.0051650.0363330.0341330.002201
                                        2019-07-0173.70360973.5713880.00179650.21859550.0887090.00259012.81144413.045774-0.018125105.768114...-0.0046150.0198000.022462-0.0026620.0399330.046396-0.0064630.0363330.0328350.003498
                                        \n

                                        5 rows \u00d7 30 columns

                                        \n
                                        \n\n\n\n### Exercise: GDP and Inflation\n\nConstruct a plot of the cyclical components of GDP and inflation. \n\n\n```python\n# Construct plot\nplt.plot(data['pce_inflation_cycle']*100,alpha=0.75,lw=3,label='Inflation')\nplt.plot(data['gdp_cycle']*100,c='r',alpha=0.5,label='GDP')\nplt.grid()\nplt.ylabel='Percent'\nplt.title('GDP and Inflation')\n\n# Place legend to right of figure. PROVIDED\nplt.legend(loc='center left', bbox_to_anchor=(1, 0.5))\n```\n\n### Exercise: GDP and the 3-Month T-Bill Rate\n\nConstruct a plot of the cyclical components of GDP and the 3-month T-bill rate. \n\n\n```python\n# Construct plot\nplt.plot(data['t_bill_3mo_cycle']*100,alpha=0.75,lw=3,label='Inflation')\nplt.plot(data['gdp_cycle']*100,c='r',alpha=0.5,label='GDP')\nplt.grid()\nplt.ylabel='Percent'\nplt.title('GDP and 3-Month T-Bill Rate')\n\n# Place legend to right of figure. PROVIDED\nplt.legend(loc='center left', bbox_to_anchor=(1, 0.5))\n```\n\n### Correlations Between GDP, Inflation, and 3-Month T-Bill Rate\n\nCompute the coefficients of corrrelation between GDP, inflation, and the 3-month T-bill rate.\n\n\n```python\ndata[['gdp_cycle','pce_inflation_cycle','t_bill_3mo_cycle']].corr()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        gdp_cyclepce_inflation_cyclet_bill_3mo_cycle
                                        gdp_cycle1.0000000.5534970.407666
                                        pce_inflation_cycle0.5534971.0000000.241685
                                        t_bill_3mo_cycle0.4076660.2416851.000000
                                        \n
                                        \n\n\n\nStrong (but not perfect!) correlations between GDP and inflation and GDP and the T-bill rate suggest link between nominal and real quantities over the business cycle that should be exaplined by business cycle theory.\n\n## The New-Keynesian Model\n\nThe most basic version of the New-Keynesian Model can be expressed as:\n\n\\begin{align}\ny_t & = E_t y_{t+1} - \\left( r_{t} - \\bar{r}\\right) + g_t\\\\\ni_{t} & = r_{t} + E_t \\pi_{t+1}\\\\\ni_{t} & = \\bar{r} + \\pi^T + \\phi_{\\pi}\\big(\\pi_t - \\pi^T\\big) + \\phi_{y}\\big(y_t - \\bar{y}\\big) + v_t\\\\\n\\pi_t -\\pi^T & = \\beta \\left( E_t\\pi_{t+1} - \\pi^T\\right) + \\kappa (y_t -\\bar{y})+ u_t,\n\\end{align}\n\nwhere: $y_t$ is (log) output, $r_t$ is the real interest rate, $i_t$ is the nominal interest rate, $\\pi_t$ is the rate of inflation between periods $t-1$ and $t$, $\\bar{r}$ is the long-run average real interest rate or the *natural rate of interest*, $\\beta$ is the household's subjective discount factor, and $\\pi^T$ is the central bank's inflation target. The coeffieints $\\phi_{\\pi}$ and $\\phi_{y}$ reflect the degree of intensity to which the central bank *endogenously* adjusts the nominal interest rate in response to movements in inflation and output.\n\nThe variables $g_t$, $u_t$, and $v_t$ represent exogenous shocks to aggregate demand, inflation, and monetary policy. They follow AR(1) processes:\n\n\\begin{align}\ng_{t+1} & = \\rho_g g_{t} + \\epsilon^g_{t+1}\\\\\nu_{t+1} & = \\rho_u u_{t} + \\epsilon^u_{t+1}\\\\\nv_{t+1} & = \\rho_v v_{t} + \\epsilon^v_{t+1}.\n\\end{align}\n\nThe goal is to compute impulse responses in the model to a one percent exogenous increase in the nominal interest rate. We will use the following parameterization:\n\n| $\\bar{y}$ | $\\beta$ | $\\bar{r}$ | $\\kappa$ | $\\pi^T$ | $\\phi_{\\pi}$ | $\\phi_y$ | $\\rho_g$ | $\\rho_u$ | $\\rho_v$ | \n|-----------|---------|--------------|----------|---------|--------------|----------|----------|----------|---------|\n| 0 | 0.995 | $-\\log\\beta$ | 0.1 | 0.02/4 | 1.5 | 0.5/4 | 0.5 | 0.5 | 0.5 |\n\n\n```python\n# Create a variable called 'parameters' that stores the model parameter values in a Pandas Series\nparameters = pd.Series()\nparameters['y_bar'] = 0\nparameters['beta'] = 0.995\nparameters['r_bar'] = -np.log(parameters.beta)\nparameters['kappa'] = 0.1\nparameters['pi_T'] = 0.02/4\nparameters['phi_pi'] = 1.5\nparameters['phi_y'] = 0.5/4\nparameters['rho_g'] = 0.5\nparameters['rho_u'] = 0.5\nparameters['rho_v'] = 0.5\n\n# Print the model's parameters\nprint(parameters)\n```\n\n y_bar 0.000000\n beta 0.995000\n r_bar 0.005013\n kappa 0.100000\n pi_T 0.005000\n phi_pi 1.500000\n phi_y 0.125000\n rho_g 0.500000\n rho_u 0.500000\n rho_v 0.500000\n dtype: float64\n\n\n\n```python\n# Create variable called 'var_names' that stores the variable names in a list with state variables ordered first\nvar_names = ['g','u','v','y','pi','i','r']\n\n# Create variable called 'shock_names' that stores an exogenous shock name for each state variable.\nshock_names = ['e_g','e_u','e_v']\n\n# Define a function that evaluates the equilibrium conditions of the model solved for zero. PROVIDED\ndef equilibrium_equations(variables_forward,variables_current,parameters):\n \n # Parameters. PROVIDED\n p = parameters\n \n # Current variables. PROVIDED\n cur = variables_current\n \n # Forward variables. PROVIDED\n fwd = variables_forward\n\n # IS equation\n is_equation = fwd.y - (cur.r -p.r_bar) + cur.g - cur.y\n \n # Fisher_equation\n fisher_equation = cur.r + fwd.pi - cur.i\n \n # Monetary policy\n monetary_policy = p.r_bar + p.pi_T + p.phi_pi*(cur.pi - p.pi_T) + p.phi_y*cur.y + cur.v - cur.i\n \n # Phillips curve\n phillips_curve = p.beta*(fwd.pi- p.pi_T) + p.kappa*cur.y + cur.u - (cur.pi-p.pi_T)\n \n # Demand process\n demand_process = p.rho_g*cur.g - fwd.g\n \n # Monetary policy process\n monetary_policy_process = p.rho_v*cur.v - fwd.v\n \n # Inflation process\n inflation_process = p.rho_u*cur.u - fwd.u\n \n \n # Stack equilibrium conditions into a numpy array\n return np.array([\n is_equation,\n fisher_equation,\n monetary_policy,\n phillips_curve,\n demand_process,\n monetary_policy_process,\n inflation_process\n ])\n\n# Initialize the model into a variable named 'nk_model'\nnk_model = ls.model(equations = equilibrium_equations,\n n_states=3,\n var_names=var_names,\n shock_names=shock_names,\n parameters = parameters)\n```\n\n\n```python\n# Compute the steady state numerically using .compute_ss() method of nk_model\nguess = [0,0,0,0,0.01,0.01,0.01]\nnk_model.compute_ss(guess)\n\n# Print the computed steady state\nprint(nk_model.ss)\n```\n\n g 1.832559e-36\n u 3.518101e-35\n v 4.196417e-36\n y -3.184980e-20\n pi 5.000000e-03\n i 1.001254e-02\n r 5.012542e-03\n dtype: float64\n\n\n\n```python\n# Find the log-linear approximation around the non-stochastic steady state and solve using .approximate_and_solve() method of nk_model\n# set argumement 'log_linear' to False because the model is already log-linear.\nnk_model.approximate_and_solve(log_linear=False)\n\nprint(nk_model.approximated())\n```\n\n Linear equilibrium conditions:\n \n y[t+1|t] = -g[t]+y[t]+r[t]\n \n pi[t+1|t] = i[t]-r[t]\n \n 0 = -v[t]-0.125\u00b7y[t]-1.5\u00b7pi[t]+i[t]\n \n 0.995\u00b7pi[t+1|t] = -u[t]-0.1\u00b7y[t]+pi[t]\n \n -g[t+1] = -0.5\u00b7g[t]\n \n -v[t+1] = -0.5\u00b7v[t]\n \n -u[t+1] = -0.5\u00b7u[t]\n\n\n### Impulse Responses\n\nCompute a 21 period impulse response of the model's variables to a 0.01/4 unit shock to the exogenous component of monetary policy ($v_t$) in period 5.\n\n\n```python\n# Compute impulse responses\nnk_model.impulse(T=21,t0=5,shocks=[0,0,0.01/4])\n\n# Print the first 10 rows of the computed impulse responses to the monetary policy shock\nprint(nk_model.irs['e_v'].head(10))\n```\n\n e_v g u v y pi i r\n 0 0.0000 0.000000e+00 0.0 0.000000 0.000000 0.000000 0.000000 0.000000\n 1 0.0000 0.000000e+00 0.0 0.000000 0.000000 0.000000 0.000000 0.000000\n 2 0.0000 0.000000e+00 0.0 0.000000 0.000000 0.000000 0.000000 0.000000\n 3 0.0000 0.000000e+00 0.0 0.000000 0.000000 0.000000 0.000000 0.000000\n 4 0.0000 0.000000e+00 0.0 0.000000 0.000000 0.000000 0.000000 0.000000\n 5 0.0025 0.000000e+00 0.0 0.002500 -0.003034 -0.000604 0.001215 0.001517\n 6 0.0000 -3.469447e-19 0.0 0.001250 -0.001517 -0.000302 0.000608 0.000758\n 7 0.0000 -3.469447e-19 0.0 0.000625 -0.000758 -0.000151 0.000304 0.000379\n 8 0.0000 -2.602085e-19 0.0 0.000313 -0.000379 -0.000075 0.000152 0.000190\n 9 0.0000 -1.734723e-19 0.0 0.000156 -0.000190 -0.000038 0.000076 0.000095\n\n\nPlot the computed impulses responses of the nominal interest rate, the real interest rate, output, and inflation. Express inflation and interest rates in *annualized* (e.g., multiplied by 4) terms.\n\n\n```python\n# Create figure. PROVIDED\nfig = plt.figure(figsize=(12,8))\n\n# Create upper-left axis. PROVIDED\nax1 = fig.add_subplot(2,2,1)\n# Create upper-right axis. PROVIDED\nax2 = fig.add_subplot(2,2,2)\n# Create lower-left axis. PROVIDED\nax3 = fig.add_subplot(2,2,3)\n# Create lower-right axis. PROVIDED\nax4 = fig.add_subplot(2,2,4)\n\n# Set axis 1 ylabel\nax1.set_ylabel('% dev from steady state')\n# Set axis 2 ylabel\nax2.set_ylabel('% dev from steady state')\n# Set axis 3 ylabel\nax3.set_ylabel('% dev from steady state')\n# Set axis 4 ylabel\nax4.set_ylabel('% dev from steady state')\n\n# Set axis 1 limits \nax1.set_ylim([-0.2,0.8])\n# Set axis 2 limits\nax2.set_ylim([-0.2,0.8])\n# Set axis 3 limits\nax3.set_ylim([-0.4,0.1])\n# Set axis 4 limits\nax4.set_ylim([-0.4,0.1])\n\n# Plot the nominal interest rate, real interest rate, output, and inflation\n(nk_model.irs['e_v']['i']*400).plot(ax=ax1,lw=4,alpha=0.75,title='Nominal Interest',grid=True)\n(nk_model.irs['e_v']['r']*400).plot(ax=ax2,lw=4,alpha=0.75,title='Real Interest',grid=True)\n(nk_model.irs['e_v']['y']*100).plot(ax=ax3,lw=4,alpha=0.75,title='Output',grid=True)\n(nk_model.irs['e_v']['pi']*400).plot(ax=ax4,lw=4,alpha=0.75,title='Inflation',grid=True)\n```\n", "meta": {"hexsha": "02e20fe45dbc0faa71fc1c7eea1ae3473cf128c9", "size": 151793, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Lecture Notebooks/Econ126_Class_16.ipynb", "max_stars_repo_name": "t-hdd/econ126", "max_stars_repo_head_hexsha": "17029937bd6c40e606d145f8d530728585c30a1d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Lecture Notebooks/Econ126_Class_16.ipynb", "max_issues_repo_name": "t-hdd/econ126", "max_issues_repo_head_hexsha": "17029937bd6c40e606d145f8d530728585c30a1d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Lecture Notebooks/Econ126_Class_16.ipynb", "max_forks_repo_name": "t-hdd/econ126", "max_forks_repo_head_hexsha": "17029937bd6c40e606d145f8d530728585c30a1d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 191.4161412358, "max_line_length": 45268, "alphanum_fraction": 0.8762064127, "converted": true, "num_tokens": 5450, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5312093733737563, "lm_q2_score": 0.7826624789529375, "lm_q1q2_score": 0.41575764500774065}} {"text": "\u4e4b\u524d\u4ecb\u7ecd\u4e86\u51e0\u79cd\u4e0d\u540c\u7684\u76d1\u7763\u5b66\u4e60\u7b97\u6cd5\u4ee5\u53ca\u6d4b\u8bd5\u96c6\u548c\u9a8c\u8bc1\u96c6\u7684\u6982\u5ff5\uff1a\n\n- [\u7ebf\u6027\u56de\u5f52](https://github.com/wolverinn/Machine-Learning-notes/blob/master/Machine%20Learning%20Week1%20--%20Linear%20Regression%2C%20Gradient%20Descent.ipynb)\n- [Logistic \u56de\u5f52](https://github.com/wolverinn/Machine-Learning-notes/blob/master/Machine%20Learning%20Week2%20--%20Logistic%20Regression.ipynb)\n- [\u795e\u7ecf\u7f51\u7edc\u6a21\u578b](https://github.com/wolverinn/Machine-Learning-notes/blob/master/Machine%20Learning%20Week3%20--%20Neural%20Networks%2C%20Back%20Propagation.ipynb)\n- [\u8bad\u7ec3\u6a21\u578b\u7684\u4f18\u5316](https://github.com/wolverinn/Machine-Learning-notes/blob/master/Machine%20Learning%20Week4%20--%20Test%20set%20and%20Validation%20set.ipynb)\n\n\u8fd9\u7bc7\u6587\u7ae0\u4ecb\u7ecd\u76d1\u7763\u5b66\u4e60\u4e2d\u4e00\u4e2a\u5f88\u5f3a\u5927\uff0c\u5728\u5b66\u672f\u754c\u548c\u5de5\u4e1a\u754c\u90fd\u5f88\u6d41\u884c\u7684\u7b97\u6cd5\u2014\u2014**\u652f\u6301\u5411\u91cf\u673a\uff08Support Vector Machine\uff0cSVM\uff09**\uff0c\u9996\u5148\u5927\u81f4\u4e86\u89e3\u4e00\u4e0b\u652f\u6301\u5411\u91cf\u673a\u7684\u601d\u60f3\n\n### \u652f\u6301\u5411\u91cf\u673a\u7b80\u4ecb\n\n\u652f\u6301\u5411\u91cf\u673a\u662f\u7528\u4e8e\u5206\u7c7b\u95ee\u9898\u7684\u7b97\u6cd5\uff0c\u652f\u6301\u5411\u91cf\u673a\u7684\u57fa\u672c\u601d\u60f3\u662f\u5bf9\u4e8eN\u7ef4\u7684\u6570\u636e\uff0c\u627e\u5230\u4e00\u4e2aN-1\u7ef4\u7684\u8d85\u5e73\u9762\uff08hyperplane\uff09\uff0c\u4f5c\u4e3a\u6570\u636e\u5206\u7c7b\u7684\u51b3\u7b56\u8fb9\u754c\uff0c\u786e\u5b9a\u8fd9\u4e2a\u8d85\u5e73\u9762\u7684\u89c4\u5219\u662f\u627e\u5230\u79bb\u5206\u9694\u8d85\u5e73\u9762\u6700\u8fd1\u7684\u90a3\u4e9b\u70b9\uff0c\u4f7f\u5b83\u4eec\u79bb\u5206\u9694\u8d85\u5e73\u9762\u7684\u8ddd\u79bb\u5c3d\u53ef\u80fd\u8fdc\uff0c\u56e0\u4e3a\u8fd9\u6837\u53ef\u4ee5\u4f7f\u5f97\u8bad\u7ec3\u51fa\u6765\u7684\u5206\u7c7b\u5668\u66f4\u52a0\u5065\u58ee\u3002\u79bb\u8d85\u5e73\u9762\u6700\u8fd1\u7684\u90a3\u4e9b\u70b9\u5c31\u88ab\u79f0\u4e3a\u652f\u6301\u5411\u91cf\uff08support vector\uff09\uff0c\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5bf9\u4e8e\u4e8c\u7ef4\u7684\u6570\u636e\uff0c\u652f\u6301\u5411\u91cf\u673a\u5c31\u8981\u627e\u5230\u4e00\u4e2a\u4e00\u7ef4\u7684\u5411\u91cf\u6765\u4f5c\u4e3a\u8fb9\u754c\uff0c\u5982\u56fe\u6240\u793a\uff1a\n\n\n\n### \u7ebf\u6027\u652f\u6301\u5411\u91cf\u673a\uff08Linear SVM\uff09\n\n\u5728 Logistic \u56de\u5f52\u4e2d\uff0c\u6211\u4eec\u6839\u636e$\\theta^Tx$\u662f\u5426\u5927\u4e8e0\u6765\u8fdb\u884c\u5206\u7c7b\uff0c\u5e76\u4e14\u5c06\u4e24\u4e2a\u7c7b\u522b\u5206\u522b\u8bb0\u4f5c1\u548c0\uff0c\u8fd9\u91cc\u6211\u4eec\u5219\u5c06\u4e24\u4e2a\u7c7b\u522b\u5206\u522b\u8bb0\u4e3a+1\u548c-1\uff0c\u5373$y^n=1$\u6216$y^n=-1$\u3002\u5f53$f(x)=\\theta^Tx$\u5927\u4e8e0\u7684\u65f6\u5019\uff0c\u9884\u6d4b\u4e3a+1\uff0c\u5e76\u4e14$f(x)$\u8d8a\u5927\u8d8a\u597d\uff1b\u5f53\u5c0f\u4e8e0\u7684\u65f6\u5019\uff0c\u9884\u6d4b\u4e3a-1\uff0c\u5e76\u4e14$f(x)$\u8d8a\u5c0f\u8d8a\u597d\n\n\u6240\u4ee5\u5176\u5b9e\u7ebf\u6027\u652f\u6301\u5411\u91cf\u673a\u7684\u5047\u8bbe\u51fd\u6570\u548c Logistic\u56de\u5f52\u7684\u5047\u8bbe\u51fd\u6570\u662f\u76f8\u540c\u7684\uff0c\u4e0d\u540c\u7684\u53ea\u662f\u635f\u5931\u51fd\u6570\u3002\u4e0b\u9762\u6211\u4eec\u6765\u770b\u4e00\u4e0b\u600e\u4e48\u786e\u5b9a\u7ebf\u6027\u652f\u6301\u5411\u91cf\u673a\u7684\u635f\u5931\u51fd\u6570\u3002\n\n\u9996\u5148\u6211\u4eec\u753b\u51fa\u7528\u4e0d\u540c\u51fd\u6570\u4f5c\u4e3a\u635f\u5931\u51fd\u6570\u65f6\u5019\u7684\u56fe\u50cf\uff0c\u5c06\u6a2a\u8f74\u8bb0\u4e3a$y^nf(x)$\uff0c\u8fd9\u6837\u7684\u8bdd\uff0c\u5982\u679c$y^n=1$\uff0c\u8fd9\u65f6\u5019\u6211\u4eec\u5e0c\u671b$f(x)$\u8d8a\u6b63\u8d8a\u597d\uff0c\u6240\u4ee5$y^nf(x)$\u8d8a\u5927\uff0c\u635f\u5931\u51fd\u6570\u8d8a\u5c0f\uff0c$f(x)$\u8d8a\u5c0f\uff0c\u635f\u5931\u51fd\u6570\u8d8a\u5927\uff1b\u800c\u5982\u679c$y^n=-1$\uff0c\u8fd9\u65f6\u5019\u5e0c\u671b$f(x)$\u8d8a\u8d1f\u8d8a\u597d\uff0c\u6240\u4ee5$y^nf(x)$\u8d8a\u5927\uff0c\u8bf4\u660e$f(x)$\u8d8a\u8d1f\uff0c\u5373\u635f\u5931\u51fd\u6570\u5c31\u8d8a\u5c0f\u3002\u6240\u4ee5\uff0c\u5c06\u6a2a\u8f74\u8bb0\u4e3a$y^nf(x)$\u66f4\u52a0\u65b9\u4fbf\uff0c\u65e0\u8bba\u662f$y^n=1$\u8fd8\u662f$y^n=-1$\u7684\u60c5\u51b5\uff0c\u635f\u5931\u51fd\u6570\u90fd\u662f\u968f\u7740\u6a2a\u8f74\u9012\u51cf\u7684\n\n\u4e4b\u524d\u6211\u4eec\u66fe\u7ecf\u4f7f\u7528\u8fc7 Square Error Cost \u4f5c\u4e3a\u635f\u5931\u51fd\u6570\uff0c\u8fd9\u662f\u4e00\u4e2a\u4e8c\u6b21\u51fd\u6570\uff0c\u5f53$y^n=1$\u7684\u65f6\u5019\uff0c\u662f$(f(x)-1)^2$\uff0c\u5f53$y^n=-1$\u7684\u65f6\u5019\uff0c\u662f$(f(x-(-1)))^2$\uff0c\u56e0\u6b64\u603b\u7684\u53ef\u4ee5\u5199\u6210$(y^nf(x)-1)^2$\n\n\u5728 Logistic \u56de\u5f52\u4e2d\u4f7f\u7528\u7684\u635f\u5931\u51fd\u6570\u5219\u662f\u4ea4\u53c9\u71b5\uff08Cross-Entropy\uff09\uff0c\u7edf\u4e00\u4e86$y^n=1$\u548c$y^n=-1$\u7684\u60c5\u51b5\u540e\uff0c\u53ef\u4ee5\u5199\u6210$\\ln(1+e^{-y^nf(x)})/\\ln2$\uff0c\u5176\u4e2d\u9664\u4ee5\u4e86\u4e00\u4e2a\u5e38\u6570\u662f\u4e0d\u5f71\u54cd\u7684\u3002\u4ece\u4e0b\u56fe\u770b\u4e00\u4e0b\uff1a\n\n\n\n\u7ea2\u8272\u7684\u66f2\u7ebf\u5c31\u662f Square Error Cost\uff0c\u663e\u7136\u8fdd\u80cc\u4e86\u968f\u7740\u6a2a\u8f74\u9012\u51cf\u7684\u89c4\u5f8b\uff0c\u56e0\u6b64\u4e0d\u80fd\u91c7\u7528\uff1b\u84dd\u7ebf\u662f Sigmoid \u51fd\u6570\u52a0\u4e0a Square Error Cost\uff0c\u867d\u7136\u6ee1\u8db3\u9012\u51cf\u7684\u89c4\u5f8b\uff0c\u4f46\u662f\u53d8\u5316\u8d8b\u52bf\u592a\u5c0f\uff0c\u635f\u5931\u51fd\u6570\u7684\u4e0b\u964d\u5e76\u4e0d\u660e\u663e\uff0c\u800c\u6211\u4eec\u77e5\u9053\u968f\u7740\u6a2a\u8f74\u589e\u5927\uff0c\u635f\u5931\u51fd\u6570\u5e94\u8be5\u4f1a\u4e0b\u964d\u5f88\u591a\uff1b\u800c\u7eff\u8272\u7684\u66f2\u7ebf\u5c31\u662f Logistic \u56de\u5f52\u4e2d\u4f7f\u7528\u7684\u4ea4\u53c9\u71b5\uff0c\u548c\u84dd\u7ebf\u5bf9\u6bd4\u8d77\u6765\uff0c\u635f\u5931\u51fd\u6570\u7684\u4e0b\u964d\u8d8b\u52bf\u5c31\u5f88\u660e\u663e\n\n\u90a3\u4e48\u5728\u652f\u6301\u5411\u91cf\u673a\u4e2d\uff0c\u4f7f\u7528\u7684\u662f\u53eb\u505a **Hinge Loss** \u7684\u635f\u5931\u51fd\u6570\uff0c\u5b83\u7684\u56fe\u50cf\u5c31\u662f\u4e0b\u9762\u7684\u7d2b\u8272\u7684\u7ebf\uff1a\n\n\n\n\u5177\u4f53\u7684\u635f\u5931\u51fd\u6570\u662f\uff1a\n\n$$l(f(x^n),y^n)=max(0,1-y^nf(x^n))$$\n\n\u5f53$y^n=1$\u65f6\uff0c\u53ea\u8981$1-f(x)<0$\u5373$f(x)>1$\uff0c\u635f\u5931\u51fd\u6570\u5c31\u4e3a0\u4e86\uff1b\n\n\u5f53$y^n=-1$\u65f6\uff0c\u53ea\u8981$1+f(x)<0$\u5373$f(x)<-1$\uff0c\u635f\u5931\u51fd\u6570\u4e3a0\n\n\u4e5f\u5c31\u662f\u8bf4\uff0c\u5f53$y^nf(x)>1$\u65f6\uff0c\u635f\u5931\u51fd\u6570\u5c31\u662f0\uff0c\u5c31\u7b97\u7ee7\u7eed\u8ba9$y^nf(x)$\u589e\u5927\uff0c\u4e5f\u4e0d\u4f1a\u5f97\u5230\u635f\u5931\u7684\u4e0b\u964d\uff1b\u800c\u5f53$y^nf(x)$\u57280\u52301\u4e4b\u95f4\u65f6\uff0c\u867d\u7136\u5df2\u7ecf\u53ef\u4ee5\u9884\u6d4b\u51fa\u6b63\u786e\u7684\u7ed3\u679c\uff0c\u4f46\u662f\u5bf9\u4e8e Hinge Loss \u6765\u8bf4\u8fd8\u4e0d\u591f\u597d\uff0c\u8fd8\u662f\u6709\u4e00\u5b9a\u7684\u635f\u5931\uff0c\u8981\u6bd4\u6b63\u786e\u7684\u7ed3\u679c\u597d\u8fc7\u4e00\u6bb5\u8ddd\u79bb\u4e4b\u540e\uff0c\u635f\u5931\u51fd\u6570\u624d\u4f1a\u964d\u4e3a0\u3002\u76f8\u6bd4\u4e8e\u4ea4\u53c9\u71b5\uff0cHinge Loss \u66f4\u52a0\u5065\u58ee\n\n\u6240\u4ee5\u6982\u62ec\u4e00\u4e0b\uff0c\u7ebf\u6027\u652f\u6301\u5411\u91cf\u673a\u4f7f\u7528\u7684\u5047\u8bbe\u51fd\u6570\u5c31\u662f\u4e00\u4e2a\u7ebf\u6027\u51fd\u6570\uff08$f(x)=\\theta^Tx$\uff09\uff0c\u635f\u5931\u51fd\u6570\u662f Hinge Loss \u518d\u52a0\u4e0a\u4e00\u4e2a\u6b63\u5219\u5316\u7684\u90e8\u5206\uff0c\u8fd9\u662f\u4e00\u4e2a\u51f8\u51fd\u6570\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u68af\u5ea6\u4e0b\u964d\u6cd5\u6765\u8fdb\u884c\u4f18\u5316\n\n### \u677e\u5f1b\u53d8\u91cf\uff08Slack Variable\uff09\n\n\u5148\u76f4\u89c2\u5730\u7406\u89e3\u4e00\u4e0b\u677e\u5f1b\u53d8\u91cf\u3002\u652f\u6301\u5411\u91cf\u673a\u7684\u76ee\u6807\u662f\u627e\u5230\u4e00\u4e2a\u8fb9\u754c\u4ee5\u6700\u5927\u5316\u8fb9\u754c\u4e0e\u652f\u6301\u5411\u91cf\u7684\u8ddd\u79bb\uff0c\u4f46\u662f\u6709\u65f6\u6709\u4e9b\u6837\u672c\u70b9\u53ef\u80fd\u4e0d\u6ee1\u8db3\u95f4\u9694\u6761\u4ef6\uff0c\u8fd9\u65f6\u5f15\u5165\u4e00\u4e2a\u677e\u5f1b\u53d8\u91cf$\\epsilon^n\\ge0$\uff0c\u4f7f\u95f4\u9694\u52a0\u4e0a\u677e\u5f1b\u53d8\u91cf\u6ee1\u8db3\u95f4\u9694\u6761\u4ef6\n\n\u6211\u4eec\u6765\u770b\u4e00\u4e0b Hinge Loss \u7684\u90e8\u5206\uff0c\u8bb0\uff1a\n\n$$\\epsilon^n=max(0,1-y^nf(x))$$\n\n\u7531\u4e8e\u6211\u4eec\u7684\u76ee\u6807\u662f\u6700\u5c0f\u5316 Hinge Loss\uff0c\u56e0\u6b64\u4e0a\u5f0f\u7b49\u4ef7\u4e8e\uff1a\n\n$$\\begin{align} \\epsilon\\ge1-y^nf(x) \\newline\\ \\epsilon^n\\ge0\\end{align}$$\n\n\u53d8\u5316\u4e4b\u540e\uff1a$y^nf(x)\\ge1-\\epsilon^n$\uff0c\u5c31\u662f\u8bf4**\u95f4\u9694\uff08margin\uff09**\u8981\u5927\u4e8e\u7b49\u4e8e1\uff0c\u4f46\u662f\u8fd9\u4e2a\u95f4\u9694\u662f\u8f6f\u95f4\u9694\uff08soft margin\uff09\uff0c\u5176\u4e2d\u6709\u4e00\u4e2a\u677e\u5f1b\u53d8\u91cf\uff0c\u5c31\u662f$\\epsilon^n$\n\n\u8fd9\u662f\u4e00\u4e2a**\u4e8c\u6b21\u89c4\u5212\uff08Quadratic Programming\uff0cQP\uff09**\u95ee\u9898\uff0c\u56e0\u6b64\u53ef\u4ee5\u7528QP\u95ee\u9898\u7684\u7b97\u6cd5\u6765\u89e3\u3002\u5f53\u7136\uff0c\u4e0d\u7528QP\u7b97\u6cd5\u7684\u8bdd\u7528\u68af\u5ea6\u4e0b\u964d\u6cd5\u4e5f\u662f\u53ef\u4ee5\u7684\n\n### \u5bf9 Hinge Loss \u4f7f\u7528\u68af\u5ea6\u4e0b\u964d\u6cd5\n\n\u5ffd\u7565\u6389\u6b63\u5219\u5316\u7684\u90e8\u5206\uff0c\u7ebf\u6027\u652f\u6301\u5411\u91cf\u673a\u7684\u635f\u5931\u51fd\u6570\u5c31\u662f\uff1a\n\n$$L(f)=\\sum_nl(f(x^n,y^n))$$\n\n$$l(f(x^n),y^n)=max(0,1-y^nf(x^n))$$\n\n\u5982\u679c\u6211\u4eec\u7684\u53c2\u6570\u662f$\\theta$\uff0c\u6211\u4eec\u9996\u5148\u9700\u8981\u6c42\u51fa\u635f\u5931\u51fd\u6570\u5bf9$\\theta$\u7684\u504f\u5bfc\u6570\uff1a\n\n$$\\frac{\\partial l(f(x^n,y^n))}{\\partial \\theta_i}=\\frac{\\partial l(f(x^n,y^n))}{\\partial f(x^n)}\\frac{\\partial f(x^n)}{\\partial \\theta_i}$$\n\n\u7531\u4e8e$f(x^n)=\\theta^Tx^n$\uff0c\u56e0\u6b64$\\frac{\\partial f(x^n)}{\\partial \\theta_i}=x_i^n$\n\n\u63a5\u4e0b\u6765\u8ba1\u7b97$\\frac{\\partial l(f(x^n,y^n))}{\\partial f(x^n)}$\uff1a\n\n$$\\frac{\\partial max(0,1-y^nf(x^n))}{\\partial f(x^n)}=\n\\begin{cases}\n-y^n, & if \\ \\ \\ y^nf(x^n)<1 \\\\\n0, & otherwise\n\\end{cases}$$\n\n\u56e0\u6b64\uff0c\u504f\u5bfc\u6570\u4e3a\uff1a\n\n$$\\frac{\\partial L(f)}{\\partial \\theta_i}=\\sum_n-\\delta(y^nf(x^n)<1)y^nx_i^n$$\n\n\u6211\u4eec\u628a$-\\delta(y^nf(x^n)<1)y^n$\u8bb0\u4f5c$c^n(\\theta)$\uff0c\u5176\u4e2d$\\delta(x)$\u8868\u793a\u5f53\u6ee1\u8db3$x$\u65f6\uff0c$\\delta(x)$\u4e3a1\uff0c\u5426\u5219\u4e3a0\n\n\u4e8e\u662f\uff0c\u6211\u4eec\u5c31\u5f97\u5230\u4e86\u5728 Hinge Loss \u4e2d\uff0c\u68af\u5ea6\u4e0b\u964d\u6cd5\u7684\u66f4\u65b0\u516c\u5f0f\uff1a\n\n$$\\theta_i=\\theta_i-\\alpha\\sum_n c^n(\\theta)x_i^n$$\n\n### \u6838\u51fd\u6570\uff08Kernel Function\uff09\n\n\u5c06\u6211\u4eec\u6700\u7ec8\u4f7f\u7528\u68af\u5ea6\u4e0b\u964d\u6cd5\u5f97\u5230\u7684\u90a3\u7ec4\u6700\u4f18\u5316\u7684\u53c2\u6570\u8bb0\u4f5c\u5411\u91cf$\\theta^*$\uff0c\u5b9e\u9645\u4e0a\uff0c$\\theta^*$\u5c31\u662f$x^n$\u7684\u7ebf\u6027\u7ec4\u5408\uff1a\n\n$$\\theta^*=\\sum_n a_n^*x^n$$\n\n\u5c06\u68af\u5ea6\u4e0b\u964d\u6cd5\u7684\u66f4\u65b0\u516c\u5f0f\u5199\u6210\u5411\u91cf\u7684\u5f62\u5f0f\uff1a\n\n$$\\theta=\\theta-\\alpha\\sum_n c^n(\\theta)x^n$$\n\n\u800c\u524d\u9762\u8bf4\u8fc7$c^n(\\theta)$\u5f88\u591a\u65f6\u5019\u90fd\u4f1a\u662f\u7b49\u4e8e0\u7684\uff0c\u56e0\u6b64\u5bf9\u5e94\u6700\u540e\u7684\u53c2\u6570$\\theta^*=\\sum_n a_n^*x^n$\uff0c\u5c31\u4f1a\u6709\u5f88\u591a\u6570\u636e\u70b9$x^n$\u524d\u9762\u5bf9\u5e94\u7684$a_n^*$\u662f\u7b49\u4e8e0\u7684\uff0c\u4e5f\u53eb\u505a$a_n^*$\u662f**\u7a00\u758f\u7684\uff08sparse\uff09**\uff0c\u800c\u90a3\u4e9b$a_n^*$\u4e0d\u7b49\u4e8e0\u7684$x^n$\u5c31\u53eb\u505a**\u652f\u6301\u5411\u91cf\uff08Support Vectors\uff09**\n\n\u610f\u601d\u5c31\u662f\u5982\u679c\u4e00\u4e2a\u6570\u636e\u70b9\u5bf9\u5e94\u7684$a_n^*$\u7b49\u4e8e0\uff0c\u90a3\u4e48\u8fd9\u4e2a\u6570\u636e\u70b9\u5bf9\u6a21\u578b\u4e00\u70b9\u4f5c\u7528\u90fd\u6ca1\u6709\uff0c\u53ea\u6709\u5f53\u5b83\u5bf9\u5e94\u7684$a_n^*$\u4e0d\u7b49\u4e8e0\uff0c\u5b83\u624d\u4f1a\u5bf9\u51b3\u5b9a\u6700\u540e\u7684\u6a21\u578b\u6709\u7528\u3002\u53ea\u6709\u5c11\u6570\u7684\u70b9\u4f1a\u88ab\u9009\u4f5c\u652f\u6301\u5411\u91cf\uff0c\u5982\u679c\u51fa\u73b0\u4e86\u5f02\u5e38\u503c\uff08outlier\uff09\uff0c\u90a3\u4e48\u53ea\u9700\u8981\u4fdd\u8bc1\u4e0d\u628a\u5b83\u9009\u4f5c\u652f\u6301\u5411\u91cf\u5c31\u884c\u4e86\u3002\u6bd4\u5982\u5982\u679c\u662f\u7528\u7684\u4ea4\u53c9\u71b5\u4f5c\u4e3a\u635f\u5931\u51fd\u6570\uff0c\u90a3\u4e48\u635f\u5931\u51fd\u6570\u5728\u6bcf\u4e2a\u5730\u65b9\u7684\u5fae\u5206\u90fd\u662f\u4e0d\u7b49\u4e8e0\u7684\uff0c\u6240\u4ee5\u6700\u7ec8\u7684$a_n^*$\u4e5f\u4e0d\u4f1a\u662f\u7a00\u758f\u7684\uff0c\u90a3\u4e48\u6bcf\u4e00\u4e2a\u6570\u636e\u70b9\u5bf9\u6700\u540e\u7684\u6a21\u578b\u90fd\u4f1a\u6709\u5f71\u54cd\uff0c\u6240\u4ee5\u76f8\u6bd4\u4e8e\u5176\u4ed6\u6a21\u578b\uff0c\u652f\u6301\u5411\u91cf\u673a\u662f\u6bd4\u8f83\u7a33\u5065\u7684\uff08robust\uff09\u3002\n\n\u5c06\u5f0f\u5b50$\\theta^*=\\sum_n a_n^*x^n$\u77e9\u9635\u5316\u5f97\u5230\uff1a$\\theta=Xa$\uff0c\u5219\uff1a\n\n$$f(x)=\\theta^Tx=a^TX^Tx=\\sum_n a_n(x^n*x)$$\n\n\u5177\u4f53\u7684\u77e9\u9635\u53d8\u6362\u5982\u56fe\uff08\u56fe\u4e2d\u7684$w$\u5c31\u662f\u6307$\\theta$\uff09\uff1a\n\n\n\n\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u5047\u8bbe\u51fd\u6570\u5199\u6210\u5e26\u6838\u51fd\u6570\u7684\u5f62\u5f0f\uff1a\n\n$$f(x)=\\sum_n a_nK(x^n,x)$$\n\n\u5176\u4e2d\u7684$K(x,z)$\u5c31\u662f**\u6838\u51fd\u6570\uff08Kernel Function\uff09**\uff0c\u5982\u679c\u4e0d\u4f5c\u4efb\u4f55\u6620\u5c04\u76f4\u63a5\u4f5c\u5185\u79ef\uff0c\u5c31\u79f0\u4e3a\u7ebf\u6027\u6838\uff0c\u5f97\u5230\u7684\u5c31\u662f\u4e4b\u524d\u7684\u7ebf\u6027\u652f\u6301\u5411\u91cf\u673a\u3002\n\n\u4e4b\u524d\u6784\u9020\u7684\u7ebf\u6027\u652f\u6301\u5411\u91cf\u673a\u6ca1\u6709\u4f5c\u6570\u636e\u7684\u6620\u5c04\uff0c\u56e0\u6b64\u53ea\u80fd\u7528\u6765\u5206\u9694\u7ebf\u6027\u7684\u6570\u636e\u3002\u5982\u679c\u6570\u636e\u7ebf\u6027\u4e0d\u53ef\u5206\uff0c\u5c31\u9700\u8981\u5c06$x$\u6620\u5c04\u5230$\\phi(x)$\uff0c\u6620\u5c04\u5230\u9ad8\u7ef4\u7a7a\u95f4\u4e4b\u540e\u5f97\u5230\u7ebf\u6027\u53ef\u5206\u7684\u6570\u636e\uff1a\n\n\n\n\u8fd9\u6837\u5e26\u6765\u7684\u95ee\u9898\u662f\u5982\u679c\u7ef4\u5ea6\u8f83\u9ad8\u4f1a\u5bfc\u81f4\u8ba1\u7b97\u91cf\u7206\u70b8\uff0c\u56e0\u6b64\u4f7f\u7528\u6838\u6280\u5de7\uff08Kernel trick\uff09\u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u5728\u9ad8\u7ef4\u6620\u5c04\u4e0b\u8ba1\u7b97SVM\uff0c\u4e3e\u4e2a\u9ad8\u7ef4\u6620\u5c04\u7684\u4f8b\u5b50\uff1a\n\n\n\n\u8fd9\u6837\uff0c\u8ba1\u7b97$\\phi(x)*\\phi(z)$\u5c31\u53ef\u4ee5\u901a\u8fc7\u8ba1\u7b97\u6838\u51fd\u6570$K(x,z)=(x*z)^2$\u6765\u8ba1\u7b97\uff0c\u907f\u514d\u4e86\u663e\u5f0f\u5730\u5199\u51fa\u6620\u5c04\u540e\u7684\u7ed3\u679c\uff0c\u76f4\u63a5\u4f7f\u8ba1\u7b97\u91cf\u5c0f\u4e86\u5f88\u591a\n\n\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u4e0d\u9700\u8981\u53bb\u8003\u8651\u5982\u4f55\u6620\u5c04\uff0c\u800c\u662f\u53ef\u4ee5\u76f4\u63a5\u9009\u62e9\u6838\u51fd\u6570\uff0c\u901a\u5e38\u6211\u4eec\u4f1a\u4ece\u4e00\u4e9b\u6838\u51fd\u6570\u4e2d\u9009\u62e9\uff0c\u4f8b\u5982\uff1a\n\n- \u7ebf\u6027\u6838\u51fd\u6570\uff1a$K(x,z)=$\uff08\u5411\u91cf\u5185\u79ef\uff09\n- \u591a\u9879\u5f0f\u6838\u51fd\u6570\uff1a$K(x,z)=(+R)^d$\n- \u9ad8\u65af\u6838\u51fd\u6570\uff0c\u9ad8\u65af\u6838\u51fd\u6570\u662f\u6700\u5e38\u7528\u7684\uff0c\u53ef\u4ee5\u5c06\u6570\u636e\u6620\u5c04\u5230\u65e0\u7a77\u7ef4\uff0c\u4e5f\u53eb\u505a\u5f84\u5411\u57fa\u51fd\u6570\uff08Radial Basis Function\uff0cRBF\uff09\uff1a\n$$K(x,z)=exp(-\\frac{||x-z||^2}{2\\sigma^2})$$\n\n\u6700\u540e\uff0c\u653e\u5f20\u56fe\u603b\u7ed3\u4e00\u4e0b\u652f\u6301\u5411\u91cf\u673a\uff1a\n\n\n\n\n\u53c2\u8003\uff1a\n- [\u674e\u5f18\u6bc5\u673a\u5668\u5b66\u4e60--\u652f\u6301\u5411\u91cf\u673a](https://www.bilibili.com/video/av10590361/?p=31)\n- [\u652f\u6301\u5411\u91cf\u673a\u7cfb\u5217--Freemind](http://blog.pluskid.org/?page_id=683)\n\n", "meta": {"hexsha": "7bcf8a114709d3d871efcd331955b705c4c9c561", "size": 7488, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Machine Learning Week5 -- Support Vector Machine.ipynb", "max_stars_repo_name": "wolverinn/Machine-Learning-notes", "max_stars_repo_head_hexsha": "3267360148c47a81d67b8bcc185fd5fba8eb2157", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2020-10-01T10:40:22.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-12T14:21:22.000Z", "max_issues_repo_path": "Machine Learning Week5 -- Support Vector Machine.ipynb", "max_issues_repo_name": "wolverinn/Machine-Learning-notes", "max_issues_repo_head_hexsha": "3267360148c47a81d67b8bcc185fd5fba8eb2157", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Machine Learning Week5 -- Support Vector Machine.ipynb", "max_forks_repo_name": "wolverinn/Machine-Learning-notes", "max_forks_repo_head_hexsha": "3267360148c47a81d67b8bcc185fd5fba8eb2157", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2020-09-02T04:52:15.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-12T14:21:32.000Z", "avg_line_length": 41.6, "max_line_length": 247, "alphanum_fraction": 0.6034989316, "converted": true, "num_tokens": 3597, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883592602049, "lm_q2_score": 0.6926419958239132, "lm_q1q2_score": 0.4157156630282681}} {"text": "\n\n\n```python\n# Last amended: 5th Sep, 2021\n# Ref: 1. https://github.com/cod3licious/autofeat\n# 2. https://github.com/cod3licious/autofeat/blob/master/autofeat_examples.ipynb\n```\n\n\n```python\n!pip install autofeat\n```\n\n Collecting autofeat\n Downloading autofeat-2.0.9-py3-none-any.whl (24 kB)\n Requirement already satisfied: sympy>=1.7.1 in /usr/local/lib/python3.7/dist-packages (from autofeat) (1.7.1)\n Collecting pint\n Downloading Pint-0.17-py2.py3-none-any.whl (204 kB)\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 204 kB 3.6 MB/s \n \u001b[?25hRequirement already satisfied: numba in /usr/local/lib/python3.7/dist-packages (from autofeat) (0.51.2)\n Requirement already satisfied: scikit-learn in /usr/local/lib/python3.7/dist-packages (from autofeat) (0.22.2.post1)\n Requirement already satisfied: numpy in /usr/local/lib/python3.7/dist-packages (from autofeat) (1.19.5)\n Requirement already satisfied: pandas>=0.24.0 in /usr/local/lib/python3.7/dist-packages (from autofeat) (1.1.5)\n Requirement already satisfied: joblib in /usr/local/lib/python3.7/dist-packages (from autofeat) (1.0.1)\n Requirement already satisfied: future in /usr/local/lib/python3.7/dist-packages (from autofeat) (0.16.0)\n Requirement already satisfied: pytz>=2017.2 in /usr/local/lib/python3.7/dist-packages (from pandas>=0.24.0->autofeat) (2018.9)\n Requirement already satisfied: python-dateutil>=2.7.3 in /usr/local/lib/python3.7/dist-packages (from pandas>=0.24.0->autofeat) (2.8.2)\n Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.7/dist-packages (from python-dateutil>=2.7.3->pandas>=0.24.0->autofeat) (1.15.0)\n Requirement already satisfied: mpmath>=0.19 in /usr/local/lib/python3.7/dist-packages (from sympy>=1.7.1->autofeat) (1.2.1)\n Requirement already satisfied: setuptools in /usr/local/lib/python3.7/dist-packages (from numba->autofeat) (57.4.0)\n Requirement already satisfied: llvmlite<0.35,>=0.34.0.dev0 in /usr/local/lib/python3.7/dist-packages (from numba->autofeat) (0.34.0)\n Requirement already satisfied: importlib-metadata in /usr/local/lib/python3.7/dist-packages (from pint->autofeat) (4.6.4)\n Requirement already satisfied: packaging in /usr/local/lib/python3.7/dist-packages (from pint->autofeat) (21.0)\n Requirement already satisfied: zipp>=0.5 in /usr/local/lib/python3.7/dist-packages (from importlib-metadata->pint->autofeat) (3.5.0)\n Requirement already satisfied: typing-extensions>=3.6.4 in /usr/local/lib/python3.7/dist-packages (from importlib-metadata->pint->autofeat) (3.7.4.3)\n Requirement already satisfied: pyparsing>=2.0.2 in /usr/local/lib/python3.7/dist-packages (from packaging->pint->autofeat) (2.4.7)\n Requirement already satisfied: scipy>=0.17.0 in /usr/local/lib/python3.7/dist-packages (from scikit-learn->autofeat) (1.4.1)\n Installing collected packages: pint, autofeat\n Successfully installed autofeat-2.0.9 pint-0.17\n\n\n\n```python\nfrom IPython.core.interactiveshell import InteractiveShell\nInteractiveShell.ast_node_interactivity = \"all\"\n\n```\n\n\n```python\nfrom google.colab import drive\ndrive.mount('/gdrive')\n```\n\n Drive already mounted at /gdrive; to attempt to forcibly remount, call drive.mount(\"/gdrive\", force_remount=True).\n\n\n\n```python\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\n\nfrom autofeat import FeatureSelector, AutoFeatRegressor, AutoFeatClassifier\n\n%matplotlib inline\n%load_ext autoreload\n%autoreload 2\n```\n\n The autoreload extension is already loaded. To reload it, use:\n %reload_ext autoreload\n\n\n# Experiments\n\n\n```python\n# generate some toy data\nnp.random.seed(10)\nx1 = np.random.rand(1000) # (1000,)\nx2 = np.random.randn(1000) # (1000,)\nx3 = np.random.rand(1000) # (1000,)\nx4 = np.random.randn(1000)\nx5 = np.random.rand(1000)\n\n```\n\n\n```python\n# Stack seven (1000,) variables one upon another\n# to get (7,1000) dataset\nX = np.vstack([x1, x2, x3, x4, x5, 1/(x2 - 1/x3), (x2 + np.log(x1))**3])\nX.shape\n```\n\n\n\n\n (7, 1000)\n\n\n\nTranspose it to have our seven features as:
                                        \nf1=x1      f2=x2       f3=x3       f4=x4       f5=x5       f6=1/(x2 - 1/x3)       f7=(x2 + np.log(x1))**3\n\n\n```python\nX = X.T\n```\n\n\n```python\n# Data as a dataframe\ndf = pd.DataFrame(X, columns=[\"x1\", \"x2\", \"x3\", \"x4\", \"x5\", \"eng6\", \"eng7\"])\n```\n\n\n```python\ndf.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        x1x2x3x4x5eng6eng7
                                        00.7713211.0971210.2225600.0299880.720170-0.2944590.587364
                                        10.020752-0.5385770.226277-0.9758960.723432-0.201697-85.981735
                                        20.6336482.1955970.8244121.0218970.0908431.0176975.261990
                                        30.748804-0.7026440.7168580.6322870.029328-0.476731-0.975961
                                        40.498507-0.0606000.0135330.5171000.435986-0.013522-0.433346
                                        \n
                                        \n\n\n\n\n```python\n# Create target. Target, row-by-row has a non-linear relationship\n# with independent features with target\n# Note that x4 and x5 are absent here:\n\ntarget = 2 + 15*x1 + 3/(x2 - 1/x3) + 5*(x2 + np.log(x1))**3\n```\n\n\n```python\nhelp(FeatureSelector)\n# FeatureSelector(\n# problem_type='regression', # \"classification\"\n# featsel_runs=5,\n# keep=None,\n# n_jobs=1,\n# verbose=0)\n```\n\n\n```python\n# Instantiate feature selector\nfsel = FeatureSelector(verbose=1)\n```\n\n\n```python\n# Perform feature selection\nnew_X = fsel.fit_transform(df, target)\n```\n\n\n```python\n# should contain [\"x1\", \"eng6\", \"eng7\"]\nprint(new_X.columns)\n```\n\n Index(['x1', 'eng7', 'eng6'], dtype='object')\n\n\n## autofeat\n\n### Generate some toy data\n\n\n```python\n# Generate some toy data\nnp.random.seed(10)\nx1 = np.random.rand(1000)\nx2 = np.random.randn(1000)\nx3 = np.random.rand(1000)\n```\n\n\n```python\n# Stack and transpose to create (1000,3) shape dataset\nX = np.vstack([x1, x2, x3]).T\n```\n\n\n```python\n# Data frame\ndf_org = pd.DataFrame(X, columns=[\"x1\", \"x2\", \"x3\"])\ndf_org.head()\n```\n\n\n```python\n# Now the target\ntarget = 2 + 15*x1 + 3/(x2 - 1/x3) + 5*(x2 + np.log(x1))**3\n```\n\n\n```python\n# Add some noise to target\n# and create two more targets\ntarget_noisy = target + 0.01*target.std()*np.random.randn(1000)\ntarget_very_noisy = target + 0.1*target.std()*np.random.randn(1000)\n```\n\n### Autofeat with different numbers of FE steps\n\n\n```python\nhelp(AutoFeatRegressor)\n```\n\n\n```python\n# Instaniate regressor\nsteps =1 \nafreg = AutoFeatRegressor(verbose=1, feateng_steps=steps)\n```\n\n\n```python\n# Syntax of Regressor problem_type: regression\n# AutoFeatRegressor(\n# categorical_cols=None, # list of column names of cat features; \n | # these will be transformed into 0/1 encoding\n# feateng_cols=None, # list of col names for feature engineering\n# units=None, # Let it be as it is\n# feateng_steps=2, # no. of steps to perform in the FE (default: 2)\n# featsel_runs=5,\n# max_gb=None, # max gigabytes to use\n# # this is no guarantee! \n# # it will lead to subsampling of the\n# # data points if the new dataframe\n# # generated is n_rows * n_cols * 32bit > max_gb\n# transformations=('1/', 'exp', 'log', 'abs', 'sqrt', '^2', '^3'),\n# apply_pi_theorem=True,\n# always_return_numpy=False,\n# n_jobs=1,\n# verbose=0)\n | \n```\n\n\n```python\ndf = afreg.fit_transform(df_org, target)\n```\n\n\n```python\ndf.head()\n```\n\n\n\n\n
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                                        x1x2x31/x1x2**2x2**3log(x1)
                                        00.7713211.0971210.2225601.2964781.2036741.320576-0.259651
                                        10.020752-0.5385770.22627748.1882440.290066-0.156223-3.875115
                                        20.6336482.1955970.8244121.5781634.82064510.584193-0.456261
                                        30.748804-0.7026440.7168581.3354630.493708-0.346901-0.289278
                                        40.498507-0.0606000.0135332.0059900.003672-0.000223-0.696138
                                        \n
                                        \n\n\n\n\n```python\nr2 = afreg.score(df_org, target) # R^2/Accuracy returned by prediction_model\nprint(\"## Final R^2: %.4f\" % r2)\n```\n\n\n```python\nplt.figure()\nplt.scatter(\n afreg.predict(df_org),\n target,\n s=2\n );\nplt.title(\"%i FE steps (R^2: %.4f; %i new features)\" % (steps, r2, len(afreg.new_feat_cols_)))\n```\n\n\n```python\nsteps = 2\nafreg = AutoFeatRegressor(verbose=1, feateng_steps=steps)\ndf = afreg.fit_transform(df_org, target)\ndf.head()\n```\n\n\n```python\nr2 = afreg.score(df_org, target)\nprint(\"## Final R^2: %.4f\" % r2)\nplt.figure()\nplt.scatter(afreg.predict(df_org), target, s=2);\nplt.title(\"%i FE steps (R^2: %.4f; %i new features)\" % (steps, r2, len(afreg.new_feat_cols_)))\n```\n\n\n```python\nsteps = 3\nafreg = AutoFeatRegressor(verbose=1, feateng_steps=steps)\ndf = afreg.fit_transform(df_org, target)\ndf.head()\n```\n\n\n```python\nr2 = afreg.score(df_org, target)\nprint(\"## Final R^2: %.4f\" % r2)\nplt.figure()\nplt.scatter(afreg.predict(df_org), target, s=2);\nplt.title(\"%i FE steps (R^2: %.4f; %i new features)\" % (steps, r2, len(afreg.new_feat_cols_)))\n```\n\n\n```python\nhelp(AutoFeatClassifier)\n```\n\n# Problem\n\n\n```python\nfrom sklearn.preprocessing import StandardScaler\nfrom sklearn.model_selection import train_test_split\nfrom sklearn.ensemble import RandomForestRegressor, RandomForestClassifier\nfrom sklearn.metrics import accuracy_score, f1_score, roc_auc_score,cohen_kappa_score\nimport os, gc\n```\n\n\n```python\n# Path to data\npath = \"/gdrive/MyDrive/Colab_data_files/autofeat/\"\n```\n\n\n```python\n# Read datafile\ndata = pd.read_csv(path + \"Concrete_Data_Yeh.csv\")\ndata.head()\n```\n\n\n\n\n
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                                        cementslagflyashwatersuperplasticizercoarseaggregatefineaggregateagecsMPa
                                        0540.00.00.0162.02.51040.0676.02879.99
                                        1540.00.00.0162.02.51055.0676.02861.89
                                        2332.5142.50.0228.00.0932.0594.027040.27
                                        3332.5142.50.0228.00.0932.0594.036541.05
                                        4198.6132.40.0192.00.0978.4825.536044.30
                                        \n
                                        \n\n\n\n\n```python\n# Data should not have any nulls\ndata.isnull().sum().sum()\n```\n\n\n\n\n 0\n\n\n\n\n```python\n# Get predictors and target\nX = data.iloc[:,:-1]\nX.shape\ny = data.iloc[:,-1]\ny.shape\n```\n\n\n\n\n (1030, 8)\n\n\n\n\n\n\n (1030,)\n\n\n\n\n```python\n# split dataset\nX_train,X_test, y_train,y_test = train_test_split(X,y,\n test_size = 0.3\n )\n```\n\n\n```python\n# Standardize\nss = StandardScaler()\nX_train = ss.fit_transform(X_train)\nX_test = ss.transform(X_test)\n```\n\n\n```python\nX_train = pd.DataFrame(X_train, columns = data.columns[:-1])\nX_test = pd.DataFrame(X_test, columns = data.columns[:-1])\nX_train.head()\nX_test.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        cementslagflyashwatersuperplasticizercoarseaggregatefineaggregateage
                                        0-0.022239-0.864576-0.8659970.441054-1.033417-0.0517651.021137-0.509222
                                        11.588466-0.5830540.367997-0.8908340.883435-0.064725-0.7808080.163770
                                        2-1.4937951.292588-0.8659971.015144-1.033417-0.1787670.3216320.708573
                                        30.170283-0.8645760.969374-0.3029670.5364190.658406-0.261498-0.509222
                                        4-0.3987061.921320-0.8659972.140360-1.033417-0.518302-2.2574023.592823
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                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        cementslagflyashwatersuperplasticizercoarseaggregatefineaggregateage
                                        01.588466-0.5830540.367997-0.8908340.883435-0.064725-0.780808-0.621387
                                        1-0.933384-0.8645761.742573-0.5417880.2224521.0834730.042581-0.284891
                                        2-0.7561110.695526-0.8659970.486981-1.033417-0.0854600.397964-0.621387
                                        3-1.2040580.331893-0.8659970.486981-1.033417-1.0885132.111064-0.284891
                                        40.1779070.7424460.8053620.900326-0.041942-1.218106-1.494078-0.284891
                                        \n
                                        \n\n\n\n\n```python\nhelp(AutoFeatRegressor)\n```\n\n Help on class AutoFeatRegressor in module autofeat.autofeat:\n \n class AutoFeatRegressor(AutoFeatModel, sklearn.base.BaseEstimator, sklearn.base.RegressorMixin)\n | AutoFeatRegressor(categorical_cols=None, feateng_cols=None, units=None, feateng_steps=2, featsel_runs=5, max_gb=None, transformations=('1/', 'exp', 'log', 'abs', 'sqrt', '^2', '^3'), apply_pi_theorem=True, always_return_numpy=False, n_jobs=1, verbose=0)\n | \n | Short-cut initialization for AutoFeatModel with problem_type: regression\n | \n | Method resolution order:\n | AutoFeatRegressor\n | AutoFeatModel\n | sklearn.base.BaseEstimator\n | sklearn.base.RegressorMixin\n | builtins.object\n | \n | Methods defined here:\n | \n | __init__(self, categorical_cols=None, feateng_cols=None, units=None, feateng_steps=2, featsel_runs=5, max_gb=None, transformations=('1/', 'exp', 'log', 'abs', 'sqrt', '^2', '^3'), apply_pi_theorem=True, always_return_numpy=False, n_jobs=1, verbose=0)\n | multi-step feature engineering and cross-validated feature selection to generate promising additional\n | features for your dataset and train a linear prediction model with them.\n | \n | Inputs:\n | - problem_type: str, either \"regression\" or \"classification\" (default: \"regression\")\n | - categorical_cols: list of column names of categorical features; these will be transformed into\n | 0/1 encoding (default: None)\n | - feateng_cols: list of column names that should be used for the feature engineering part\n | (default None --> all, with categorical_cols in 0/1 encoding)\n | - units: dictionary with {col_name: unit} where unit is a string that can be converted into a pint unit.\n | all columns without units are dimensionless and can be combined with any other column.\n | Note: it is assumed that all features are of comparable magnitude, i.e., not one variable is in\n | m and another in mm. If this needs to be accounted for, please scale your variables before\n | passing them to autofeat!\n | (default: None --> all columns are dimensionless).\n | - feateng_steps: number of steps to perform in the feature engineering part (int; default: 2)\n | - featsel_runs: number of times to perform in the feature selection part with a random fraction of data points (int; default: 5)\n | - max_gb: if an int is given: maximum number of gigabytes to use in the process (i.e. mostly the\n | feature engineering part). this is no guarantee! it will lead to subsampling of the\n | data points if the new dataframe generated is n_rows * n_cols * 32bit > max_gb\n | Note: this is only an approximate estimate of the final matrix; intermediate representations could easily\n | take up at least 2 or 3 times that much space...If you can, subsample before, you know your data best.\n | - transformations: list of transformations that should be applied; possible elements:\n | \"1/\", \"exp\", \"log\", \"abs\", \"sqrt\", \"^2\", \"^3\", \"1+\", \"1-\", \"sin\", \"cos\", \"exp-\", \"2^\"\n | (first 7, i.e., up to ^3, are applied by default)\n | - apply_pi_theorem: whether or not to apply the pi theorem (if units are given; bool; default: True)\n | - always_return_numpy: whether to always return a numpy array instead of a pd dataframe when calling (fit_)transform\n | (default: False; mainly used for sklearn estimator checks)\n | - n_jobs: how many jobs to run when selecting the features in parallel (int; default: 1)\n | - verbose: verbosity level (int; default: 0)\n | \n | Attributes:\n | - original_columns_: original columns of X when calling fit\n | - all_columns_: columns of X after calling fit\n | - categorical_cols_map_: dict mapping from the original categorical columns to a list with new column names\n | - feateng_cols_: actual columns used for the feature engineering\n | - feature_formulas_: sympy formulas to generate new features\n | - feature_functions_: compiled feature functions with columns\n | - new_feat_cols_: list of good new features that should be generated when calling transform()\n | - good_cols_: columns selected in the feature selection process, used with the final prediction model\n | - prediction_model_: sklearn model instance used for the predictions\n | \n | Note: when giving categorical_cols or feateng_cols, X later (i.e. when calling fit/fit_transform) has to be a DataFrame\n | \n | ----------------------------------------------------------------------\n | Methods inherited from AutoFeatModel:\n | \n | __getstate__(self)\n | get dict for pickling without feature_functions as they are not pickleable\n | \n | fit(self, X, y)\n | \n | fit_transform(self, X, y)\n | Fits the regression model and returns a new dataframe with the additional features.\n | \n | Inputs:\n | - X: pandas dataframe or numpy array with original features (n_datapoints x n_features)\n | - y: pandas dataframe or numpy array with targets for all n_datapoints\n | Returns:\n | - new_df: new pandas dataframe with all the original features (except categorical features transformed\n | into multiple 0/1 columns) and the most promising engineered features. This df can then be\n | used to train your final model.\n | \n | Please ensure that X only contains valid feature columns (including possible categorical variables).\n | \n | Note: we strongly encourage you to name your features X1 ... Xn or something simple like this before passing\n | a DataFrame to this model. This can help avoid potential problems with sympy later on.\n | The data should only contain finite values (no NaNs etc.)\n | \n | predict(self, X)\n | Inputs:\n | - X: pandas dataframe or numpy array with original features (n_datapoints x n_features)\n | Returns:\n | - y_pred: predicted targets return by prediction_model.predict()\n | \n | score(self, X, y)\n | Inputs:\n | - X: pandas dataframe or numpy array with original features (n_datapoints x n_features)\n | - y: pandas dataframe or numpy array with the targets for all n_datapoints\n | Returns:\n | - R^2/Accuracy returned by prediction_model.score()\n | \n | transform(self, X)\n | Inputs:\n | - X: pandas dataframe or numpy array with original features (n_datapoints x n_features)\n | Returns:\n | - new_df: new pandas dataframe with all the original features (except categorical features transformed\n | into multiple 0/1 columns) and the most promising engineered features. This df can then be\n | used to train your final model.\n | \n | ----------------------------------------------------------------------\n | Data descriptors inherited from AutoFeatModel:\n | \n | n_features_in_\n | \n | ----------------------------------------------------------------------\n | Methods inherited from sklearn.base.BaseEstimator:\n | \n | __repr__(self, N_CHAR_MAX=700)\n | Return repr(self).\n | \n | __setstate__(self, state)\n | \n | get_params(self, deep=True)\n | Get parameters for this estimator.\n | \n | Parameters\n | ----------\n | deep : bool, default=True\n | If True, will return the parameters for this estimator and\n | contained subobjects that are estimators.\n | \n | Returns\n | -------\n | params : mapping of string to any\n | Parameter names mapped to their values.\n | \n | set_params(self, **params)\n | Set the parameters of this estimator.\n | \n | The method works on simple estimators as well as on nested objects\n | (such as pipelines). The latter have parameters of the form\n | ``__`` so that it's possible to update each\n | component of a nested object.\n | \n | Parameters\n | ----------\n | **params : dict\n | Estimator parameters.\n | \n | Returns\n | -------\n | self : object\n | Estimator instance.\n | \n | ----------------------------------------------------------------------\n | Data descriptors inherited from sklearn.base.BaseEstimator:\n | \n | __dict__\n | dictionary for instance variables (if defined)\n | \n | __weakref__\n | list of weak references to the object (if defined)\n \n\n\n\n```python\nX.columns\n```\n\n\n\n\n Index(['cement', 'slag', 'flyash', 'water', 'superplasticizer',\n 'coarseaggregate', 'fineaggregate', 'age'],\n dtype='object')\n\n\n\n\n```python\n%%time\ncols_fe = ['slag']\nafreg = AutoFeatRegressor(verbose=1, feateng_steps=3, feateng_cols = cols_fe)\ndf = afreg.fit_transform(X_train, y_train)\n```\n\n [AutoFeat] The 3 step feature engineering process could generate up to 532 features.\n [AutoFeat] With 721 data points this new feature matrix would use about 0.00 gb of space.\n [feateng] Step 1: transformation of original features\n [feateng] Generated 5 transformed features from 1 original features - done.\n [feateng] Step 2: first combination of features\n [feateng] Generated 56 feature combinations from 15 original feature tuples - done.\n [feateng] Step 3: transformation of new features\n [feateng] Generated 226 transformed features from 56 original features - done.\n [feateng] Generated altogether 293 new features in 3 steps\n [feateng] Removing correlated features, as well as additions at the highest level\n [feateng] Generated a total of 207 additional features\n [featsel] Scaling data...done.\n [featsel] Feature selection run 1/5\n [featsel] Feature selection run 2/5\n [featsel] Feature selection run 3/5\n [featsel] Feature selection run 4/5\n [featsel] Feature selection run 5/5\n [featsel] 15 features after 5 feature selection runs\n [featsel] 13 features after correlation filtering\n [featsel] 8 features after noise filtering\n [AutoFeat] Computing 3 new features.\n [AutoFeat] 3/ 3 new features ...done.\n [AutoFeat] Final dataframe with 11 feature columns (3 new).\n [AutoFeat] Training final regression model.\n [AutoFeat] Trained model: largest coefficients:\n 41.19155304505693\n -20.955573 * exp(slag - exp(slag))\n 11.602106 * cement\n 7.015662 * age\n 6.839577 * Abs(slag)/slag\n -4.857598 * water\n 4.798616 * flyash\n 0.566009 * superplasticizer\n 0.262678 * exp(slag - 1/slag)\n [AutoFeat] Final score: 0.6228\n CPU times: user 13 s, sys: 2.51 s, total: 15.5 s\n Wall time: 13 s\n\n\n\n```python\ndf.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        cementslagflyashwatersuperplasticizercoarseaggregatefineaggregateageAbs(slag)/slagexp(slag - 1/slag)exp(slag - exp(slag))
                                        0-0.022239-0.864576-0.8659970.441054-1.033417-0.0517651.021137-0.509222-1.01.3391840.276428
                                        11.588466-0.5830540.367997-0.8908340.883435-0.064725-0.7808080.163770-1.03.1020210.319421
                                        2-1.4937951.292588-0.8659971.015144-1.033417-0.1787670.3216320.7085731.01.6802550.095406
                                        30.170283-0.8645760.969374-0.3029670.5364190.658406-0.261498-0.509222-1.01.3391840.276428
                                        4-0.3987061.921320-0.8659972.140360-1.033417-0.518302-2.2574023.5928231.04.0586280.007382
                                        \n
                                        \n\n\n\n\n```python\ndf_test = afreg.transform(X_test)\n```\n\n [AutoFeat] Computing 31 new features.\n [AutoFeat] 31/ 31 new features ...done.\n\n\n\n```python\ndf_test.head()\n```\n\n\n```python\ndf.to_csv(path + \"concrete_train.csv\", index = False)\ndf_test.to_csv(path+\"concrete_test.csv\", index = False)\n```\n\n\n```python\nclf_autofeat = RandomForestRegressor()\nclf_autofeat.fit(df, y_train)\npred_autofeat = clf_autofeat.predict(df_test)\n```\n\n\n```python\nclf = RandomForestRegressor()\nclf.fit(X_train, y_train)\npred = clf.predict(X_test)\n```\n\n\n```python\ny_test.shape\nX_test.shape\npred.shape\n```\n\n\n\n\n (309,)\n\n\n\n\n\n\n (309, 8)\n\n\n\n\n\n\n (309,)\n\n\n\n\n```python\nfrom sklearn.metrics import explained_variance_score, mean_squared_error\nexplained_variance_score(y_test, pred), explained_variance_score(y_test, pred_autofeat)\nmean_squared_error(y_test, pred),mean_squared_error(y_test, pred_autofeat)\n```\n\n\n\n\n (0.9077746541204639, 0.8894223984074386)\n\n\n\n\n\n\n (27.262571583041655, 32.60140986914953)\n\n\n\n\n```python\n# Path to data\npath = \"/gdrive/MyDrive/Colab_data_files/autofeat/\"\n\n# Read datafile\ndata = pd.read_csv(path + \"heart.csv\")\ndata.head()\n\n# Get predictors and target\nX = data.iloc[:,:-1]\nX.shape\ny = data.iloc[:,-1]\ny.shape\n\n# split dataset\nX_train,X_test, y_train,y_test = train_test_split(X,y,\n test_size = 0.3\n )\n\n# Standardize\nss = StandardScaler()\nX_train = ss.fit_transform(X_train)\nX_test = ss.transform(X_test)\n\nX_train = pd.DataFrame(X_train, columns = data.columns[:-1])\nX_test = pd.DataFrame(X_test, columns = data.columns[:-1])\nX_train.head()\nX_test.head()\n```\n\n\n```python\n%%time\nafreg = AutoFeatRegressor(verbose=1, feateng_steps=3)\ndf = afreg.fit_transform(X_train, y_train)\n```\n\n [AutoFeat] The 3 step feature engineering process could generate up to 102466 features.\n [AutoFeat] With 212 data points this new feature matrix would use about 0.09 gb of space.\n [feateng] Step 1: transformation of original features\n [feateng] Generated 50 transformed features from 13 original features - done.\n [feateng] Step 2: first combination of features\n [feateng] Generated 7772 feature combinations from 1953 original feature tuples - done.\n [feateng] Step 3: transformation of new features\n [feateng] Generated 32061 transformed features from 7772 original features - done.\n [feateng] Generated altogether 39936 new features in 3 steps\n [feateng] Removing correlated features, as well as additions at the highest level\n [feateng] Generated a total of 31358 additional features\n [featsel] Scaling data...done.\n [featsel] Feature selection run 1/5\n [featsel] Feature selection run 2/5\n [featsel] Feature selection run 3/5\n [featsel] Feature selection run 4/5\n [featsel] Feature selection run 5/5\n [featsel] 75 features after 5 feature selection runs\n [featsel] 67 features after correlation filtering\n [featsel] 24 features after noise filtering\n [AutoFeat] Computing 24 new features.\n [AutoFeat] 24/ 24 new features ...done.\n [AutoFeat] Final dataframe with 37 feature columns (24 new).\n [AutoFeat] Training final regression model.\n [AutoFeat] Trained model: largest coefficients:\n 0.6989301087560164\n 0.133336 * 1/(exp(oldpeak) + exp(serum_cholestrol))\n 0.095188 * chest_painType*Abs(slope)\n -0.059953 * log(oldpeak**2 + thal**2)\n -0.053840 * exp(exang - exp(slope))\n -0.044278 * Abs(exp(thal) - Abs(oldpeak))\n -0.037656 * Abs(resting_blood_pressure**2 - exp(thal))\n -0.032874 * ca*exp(-restecg)\n 0.032514 * 1/(thalach**3 - 1/oldpeak)\n -0.028151 * exp(sex - exp(thalach))\n -0.023340 * exp(-chest_painType)/ca\n -0.021413 * log(exp(oldpeak) + exp(resting_blood_pressure))\n -0.015946 * 1/(-resting_blood_pressure**2 + thal**3)\n 0.009127 * log(exp(chest_painType)*exp(slope))\n 0.008640 * Abs(sex/thal)\n -0.006761 * 1/(ca**2 - thalach)\n -0.006711 * Abs(serum_cholestrol + exp(thal))\n -0.003887 * exp(1/thal - 1/slope)\n -0.003627 * Abs(restecg - exp(thal))\n -0.001887 * exp(exang - 1/chest_painType)\n [AutoFeat] Final score: 0.6989\n CPU times: user 21min 1s, sys: 15min 24s, total: 36min 26s\n Wall time: 19min 7s\n\n\n\n```python\ndf_test = afreg.transform(X_test)\n```\n\n [AutoFeat] Computing 24 new features.\n [AutoFeat] 24/ 24 new features ...done.\n\n\n\n```python\nclf_autofeat = RandomForestClassifier(n_estimators=500)\nclf_autofeat.fit(df, y_train)\npred_autofeat = clf_autofeat.predict_proba(df_test)\n```\n\n\n\n\n RandomForestClassifier(bootstrap=True, ccp_alpha=0.0, class_weight=None,\n criterion='gini', max_depth=None, max_features='auto',\n max_leaf_nodes=None, max_samples=None,\n min_impurity_decrease=0.0, min_impurity_split=None,\n min_samples_leaf=1, min_samples_split=2,\n min_weight_fraction_leaf=0.0, n_estimators=500,\n n_jobs=None, oob_score=False, random_state=None,\n verbose=0, warm_start=False)\n\n\n\n\n```python\nclf = RandomForestClassifier(n_estimators=500)\nclf.fit(X_train, y_train)\npred = clf.predict_proba(X_test)\n```\n\n\n\n\n RandomForestClassifier(bootstrap=True, ccp_alpha=0.0, class_weight=None,\n criterion='gini', max_depth=None, max_features='auto',\n max_leaf_nodes=None, max_samples=None,\n min_impurity_decrease=0.0, min_impurity_split=None,\n min_samples_leaf=1, min_samples_split=2,\n min_weight_fraction_leaf=0.0, n_estimators=500,\n n_jobs=None, oob_score=False, random_state=None,\n verbose=0, warm_start=False)\n\n\n\n\n```python\npred[:5]\n```\n\n\n\n\n array([[0.14 , 0.86 ],\n [0.298, 0.702],\n [0.058, 0.942],\n [0.102, 0.898],\n [0.204, 0.796]])\n\n\n\n\n```python\npred_autofeat[:5]\n```\n\n\n\n\n array([[0.198, 0.802],\n [0.456, 0.544],\n [0.07 , 0.93 ],\n [0.036, 0.964],\n [0.092, 0.908]])\n\n\n\n\n```python\nroc_auc_score(y_test, pred[:,1])\n```\n\n\n\n\n 0.8844476744186047\n\n\n\n\n```python\nroc_auc_score(y_test, pred_autofeat[:,1])\n```\n\n\n\n\n 0.8599806201550387\n\n\n\n\n```python\naccuracy_score(y_test,pred_autofeat)\n```\n\n\n\n\n 0.7802197802197802\n\n\n\n\n```python\naccuracy_score(y_test, pred)\n```\n\n\n\n\n 0.7802197802197802\n\n\n\n\n```python\npred\n```\n\n\n\n\n array([1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1,\n 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1,\n 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1,\n 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0,\n 0, 0, 0])\n\n\n\n\n```python\npred_autofeat\n```\n\n\n\n\n array([1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1,\n 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1,\n 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0,\n 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0,\n 0, 0, 0])\n\n\n\n\n```python\nf1_score(y_test,pred) , f1_score(y_test,pred_autofeat)\n```\n\n\n\n\n (0.8076923076923077, 0.7959183673469388)\n\n\n", "meta": {"hexsha": "e0b866e00c6f0e068cf4d82c8c7973a44fd22889", "size": 85394, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "cat_feature_engineering/autofeat_examples.ipynb", "max_stars_repo_name": "harnalashok/classification", "max_stars_repo_head_hexsha": "1991f4abcf42cadac936b2e068d103e86b102c10", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "cat_feature_engineering/autofeat_examples.ipynb", "max_issues_repo_name": "harnalashok/classification", "max_issues_repo_head_hexsha": "1991f4abcf42cadac936b2e068d103e86b102c10", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "cat_feature_engineering/autofeat_examples.ipynb", "max_forks_repo_name": "harnalashok/classification", "max_forks_repo_head_hexsha": "1991f4abcf42cadac936b2e068d103e86b102c10", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.540008558, "max_line_length": 1207, "alphanum_fraction": 0.4126051011, "converted": true, "num_tokens": 12366, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.6926419704455589, "lm_q1q2_score": 0.41571564779647535}} {"text": "```python\n%matplotlib inline\n```\n\n\n```python\nimport feather as ft\nimport pandas as pd\nimport numpy as np\nimport pymc3 as pm\nimport theano.tensor as T\n```\n\n /home/paulc/anaconda3/envs/py36/lib/python3.6/site-packages/h5py/__init__.py:36: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.\n from ._conv import register_converters as _register_converters\n\n\nFirst we have to get the bangladesh dataset. [Feather](https://github.com/wesm/feather) is perfect for this. Thanks to Hadley Wickham and Wes McKinney for producing this invaluable set of tools.\n\nThe code to export from R should look something like this, assuming you've installed rethinking and feather packages and their dependencies:\n```R\nlibrary(rethinking)\nlibrary (feather)\n\ndata(bangladesh)\npath <- \".\"\nwrite_feather(\"bangladesh.feather\", path)\n```\n\nThat's all the R we'll need for now.\n\nNext step is to read in the data\n\n\n```python\n# Read the bangledesh data from home directory\nbangladesh = ft.read_dataframe(\"bangladesh.feather\")\nbangladesh.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        womandistrictuse.contraceptionliving.childrenage.centeredurban
                                        0110418.44001
                                        12101-5.55991
                                        231031.44001
                                        341048.44001
                                        45101-13.55901
                                        \n
                                        \n\n\n\nFrom here on we're following Richard McElreath's blog.\n\nWe set the random seed so we can reproduce the results as required and we sample 100 records from the bangladesh set.\n\n\n```python\nseed = np.random.RandomState(seed=1)\n\nb = bangladesh.sample(100, random_state=seed)\nb.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        womandistrictuse.contraceptionliving.childrenage.centeredurban
                                        6616621803-6.55990
                                        10471048310416.44001
                                        64364418031.44000
                                        17121713521410.44000
                                        1507150846147.44000
                                        \n
                                        \n\n\n\nWe only need contraception use tied to district. The data dataframe will hold this information.\n\n\n```python\ndat = pd.DataFrame()\ndat['y'] = b['use.contraception']\ndat['district_id'] = pd.Categorical(b['district']).codes \ndat.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        ydistrict_id
                                        661015
                                        1047024
                                        643015
                                        1712139
                                        1507136
                                        \n
                                        \n\n\n\nOur first model is the **centered parameterization**:\n$$\n\\begin{align}\ny_i & \\sim \\operatorname{Bernoulli}(p_i)\\\\\n\\operatorname{logit}(p_i) & = \\alpha_{unit[i]}\\\\\n\\alpha_j & \\sim \\operatorname{Normal}(a, \\tau)\\\\\na & \\sim \\operatorname{Normal}(0, 10)\\\\\n\\tau & \\sim \\operatorname{Half-Cauchy}(0, 1)\\\\\n\\end{align}\n$$\n\n\n```python\nwith pm.Model() as c_model:\n tau = pm.HalfCauchy('tau',1)\n a = pm.Normal('a', 0, 10)\n alpha = pm.Normal('alpha', a, tau, shape=len(b['district'].unique()))\n p = pm.math.invlogit(alpha[dat['district_id'].values])\n y = pm.Bernoulli('y', p, observed=dat['y'])\n \n# for RV in c_model.basic_RVs:\n# print(RV.name, RV.logp(c_model.test_point))\n trace_c = pm.sample(1000, tune=1000)\n```\n\n Auto-assigning NUTS sampler...\n Initializing NUTS using jitter+adapt_diag...\n /home/paulc/anaconda3/envs/py36/lib/python3.6/site-packages/pymc3/model.py:384: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.\n if not np.issubdtype(var.dtype, float):\n Multiprocess sampling (4 chains in 4 jobs)\n NUTS: [alpha, a, tau_log__]\n 57%|\u2588\u2588\u2588\u2588\u2588\u258b | 1135/2000 [00:05<00:03, 219.45it/s]INFO (theano.gof.compilelock): Waiting for existing lock by process '16081' (I am process '16082')\n INFO (theano.gof.compilelock): To manually release the lock, delete /home/paulc/.theano/compiledir_Linux-4.13--generic-x86_64-with-debian-stretch-sid-x86_64-3.6.3-64/lock_dir\n 59%|\u2588\u2588\u2588\u2588\u2588\u258a | 1172/2000 [00:05<00:03, 222.26it/s]INFO (theano.gof.compilelock): Waiting for existing lock by process '16081' (I am process '16083')\n INFO (theano.gof.compilelock): To manually release the lock, delete /home/paulc/.theano/compiledir_Linux-4.13--generic-x86_64-with-debian-stretch-sid-x86_64-3.6.3-64/lock_dir\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 2000/2000 [00:08<00:00, 238.60it/s]\n INFO (theano.gof.compilelock): Waiting for existing lock by process '16082' (I am process '16083')\n INFO (theano.gof.compilelock): To manually release the lock, delete /home/paulc/.theano/compiledir_Linux-4.13--generic-x86_64-with-debian-stretch-sid-x86_64-3.6.3-64/lock_dir\n There were 17 divergences after tuning. Increase `target_accept` or reparameterize.\n The acceptance probability does not match the target. It is 0.6809466137372715, but should be close to 0.8. Try to increase the number of tuning steps.\n There were 2 divergences after tuning. Increase `target_accept` or reparameterize.\n There were 60 divergences after tuning. Increase `target_accept` or reparameterize.\n There were 6 divergences after tuning. Increase `target_accept` or reparameterize.\n The estimated number of effective samples is smaller than 200 for some parameters.\n\n\nAs with the R Stan version, there's a whole bunch of divergences in this log. This suggests that the results may not be reliable.\n\nOur second model is the **non-centered parameterization**:\n$$\n\\begin{align}\ny_i & \\sim \\operatorname{Bernoulli}(p_i)\\\\\n\\operatorname{logit}(p_i) & = a + \\tau z_{unit[i]}\\\\\nz_j & \\sim \\operatorname{Normal}(0, 1)\\\\\na & \\sim \\operatorname{Normal}(0, 10)\\\\\n\\tau & \\sim \\operatorname{Half-Cauchy}(0, 1)\\\\\n\\end{align}\n$$\n\n\n```python\nwith pm.Model() as nc_model:\n # pymc3 HalfCauchy only uses the second (scale) parameter\n tau = pm.HalfCauchy('tau', beta=1)\n a = pm.Normal('a', 0, 10)\n z = pm.Normal('z', 0, 1, shape=len(b['district'].unique()))\n p = pm.math.invlogit(a + tau * z[dat['district_id'].values])\n y = pm.Bernoulli('y', p, observed=dat['y'])\n \n# for RV in c_model.basic_RVs:\n# print(RV.name, RV.logp(c_model.test_point))\n trace_nc = pm.sample(1000, tune=1000)\n```\n\n Auto-assigning NUTS sampler...\n Initializing NUTS using jitter+adapt_diag...\n /home/paulc/anaconda3/envs/py36/lib/python3.6/site-packages/pymc3/model.py:384: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.\n if not np.issubdtype(var.dtype, float):\n Multiprocess sampling (4 chains in 4 jobs)\n NUTS: [z, a, tau_log__]\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 2000/2000 [00:06<00:00, 302.29it/s]\n There were 3 divergences after tuning. Increase `target_accept` or reparameterize.\n The acceptance probability does not match the target. It is 0.7201418506887862, but should be close to 0.8. Try to increase the number of tuning steps.\n There were 3 divergences after tuning. Increase `target_accept` or reparameterize.\n The acceptance probability does not match the target. It is 0.6577402127331399, but should be close to 0.8. Try to increase the number of tuning steps.\n\n\nThis log looks much better. That suggests already that we might see more reliable results from this approach.\n\nLet's take a look at the traceplots:\n\n\n```python\n_ = pm.plots.traceplot(trace_c)\nd = pm.diagnostics.effective_n(trace_c)\nprint(f\"n_eff('a') = {d['a']}\\nn_eff('tau') = {d['tau']}\")\n```\n\n\n```python\n_ = pm.plots.traceplot(trace_nc)\nd = pm.diagnostics.effective_n(trace_nc)\nprint(f\"n_eff('a') = {d['a']}\\nn_eff('tau') = {d['tau']}\")\n```\n\nThese reflect the results from the blog.\n\nThe centered model has very low effective samples, the distributions look quite spread out.\n\nThe non-centered model has decent effective samples and the traceplots look clean with relatively tightly clustered distributions.\n", "meta": {"hexsha": "1b5f586fa37dae1aabd8331c59a5eac1bcafbec3", "size": 449745, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Centered vs Non Centered - McElreath.ipynb", "max_stars_repo_name": "paulc00/mlreparam", "max_stars_repo_head_hexsha": "c667ac45ec3f08dc536405e7e6fcd7c4e52a2629", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Centered vs Non Centered - McElreath.ipynb", "max_issues_repo_name": "paulc00/mlreparam", "max_issues_repo_head_hexsha": "c667ac45ec3f08dc536405e7e6fcd7c4e52a2629", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Centered vs Non Centered - McElreath.ipynb", "max_forks_repo_name": "paulc00/mlreparam", "max_forks_repo_head_hexsha": "c667ac45ec3f08dc536405e7e6fcd7c4e52a2629", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 760.9898477157, "max_line_length": 235830, "alphanum_fraction": 0.9347252332, "converted": true, "num_tokens": 3405, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878696277513, "lm_q2_score": 0.6757646010190476, "lm_q1q2_score": 0.4156546088106533}} {"text": "\n\n\n# `GiRaFFE_NRPy` C code library: Conservative-to-Primitive and Primitive-to-Conservative Solvers\n\n## Author: Patrick Nelson\n\n\n\n**Notebook Status:** Validated \n\n**Validation Notes:** These functions have been validated to round-off precision against the coresponding functions in the original `GiRaFFE`.\n\n### NRPy+ Source Code for this module: \n* [GiRaFFE_NRPy/GiRaFFE_NRPy_C2P_P2C.py](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_C2P_P2C.py)\n\n## Introduction:\nThis writes and documents the C code that `GiRaFFE_NRPy` uses in order to update the Valencia 3-velocity at each timestep. It also computes corrections to the densitized Poynting flux in order to keep the physical quantities from violating the GRFFE constraints. \n\nThese algorithms are adapted from the original `GiRaFFE` code (see [arxiv:1704.00599v2](https://arxiv.org/abs/1704.00599v2)), based on the description in [arXiv:1310.3274v2](https://arxiv.org/abs/1310.3274v2). They have been fully NRPyfied and modified to use the Valencia 3-velocity instead of the drift velocity.\n\nThe algorithm to do this is as follows:\n1. Apply fixes to ${\\tilde S}_i$\n 1. Enforce the orthogonality of ${\\tilde S}_i$ and $B^i$\n * ${\\tilde S}_i \\rightarrow {\\tilde S}_i - ({\\tilde S}_j {\\tilde B}^j) {\\tilde B}_i/{\\tilde B}^2$\n 1. Rescale ${\\tilde S}_i$ to limit the Lorentz factor and enforce the velocity cap\n * $f = \\sqrt{(1-\\gamma_{\\max}^{-2}){\\tilde B}^4/(16 \\pi^2 \\gamma {\\tilde S}^2)}$ \n * ${\\tilde S}_i \\rightarrow {\\tilde S}_i \\min(1,f)$\n 1. Recompute the velocities at the new timestep\n * $v^i = 4 \\pi \\gamma^{ij} {\\tilde S}_j \\gamma^{-1/2} B^{-2}$\n1. Enforce the Current Sheet prescription\n 1. Zero the velocity normal to the sheet\n * ${\\tilde n}_i v^i = 0$\n 1. Recompute the Poynting flux to be consistent.\n\n\n\n# Table of Contents\n$$\\label{toc}$$\n\nThis notebook is organized as follows\n\n1. [Step 1](#c2p): The conservative-to-primitive solver\n 1. [Step 1.a](#ortho_s_b): Enforce the orthogonality of $\\tilde{S}_i$ and $B^i$\n 1. [Step 1.b](#vel_cap): Rescale ${\\tilde S}_i$ to limit the Lorentz factor and enforce the velocity cap\n 1. [Step 1.c](#update_vel): Recompute the velocities at the new timestep\n 1. [Step 1.d](#current_sheet): Enforce the Current Sheet prescription\n1. [Step 2](#p2c): The primitive-to-conservative solver\n1. [Step 3](#code_validation): Code Validation against \n1. [Step 4](#latex_pdf_output): Output this notebook to $\\LaTeX$-formatted PDF file\n\n\n```python\n# Step 0: Add NRPy's directory to the path\n# https://stackoverflow.com/questions/16780014/import-file-from-parent-directory\nimport os,sys\nnrpy_dir_path = os.path.join(\"..\")\nif nrpy_dir_path not in sys.path:\n sys.path.append(nrpy_dir_path)\n\nimport os\nimport cmdline_helper as cmd\noutdir = \"GiRaFFE_NRPy/GiRaFFE_Ccode_validation/\"\ncmd.mkdir(outdir)\n```\n\n\n\n# Step 1: The conservative-to-primitive solver \\[Back to [top](#toc)\\]\n$$\\label{c2p}$$\n\nWe start with the Conservative-to-Primitive solver. This function is called after the vector potential and Poynting vector have been evolved at a timestep and updates the velocities. The algorithm will be as follows: \n\n1. Apply fixes to ${\\tilde S}_i$\n 1. Enforce the orthogonality of ${\\tilde S}_i$ and $B^i$\n * ${\\tilde S}_i \\rightarrow {\\tilde S}_i - ({\\tilde S}_j {\\tilde B}^j) {\\tilde B}_i/{\\tilde B}^2$\n 1. Rescale ${\\tilde S}_i$ to limit the Lorentz factor and enforce the velocity cap\n * $f = \\sqrt{(1-\\gamma_{\\max}^{-2}){\\tilde B}^4/(16 \\pi^2 \\gamma {\\tilde S}^2)}$ \n * ${\\tilde S}_i \\rightarrow {\\tilde S}_i \\min(1,f)$\n 1. Recompute the velocities at the new timestep\n * $v^i = 4 \\pi \\gamma^{ij} {\\tilde S}_j \\gamma^{-1/2} B^{-2}$\n1. Enforce the Current Sheet prescription\n * ${\\tilde n}_i v^i = 0$\n\nWe will begin simply by creating the file. We will also `#include` the header file `` and define $\\pi$.\n\n\n```python\nfrom outputC import * # NRPy+: Core C code output module\nimport finite_difference as fin # NRPy+: Finite difference C code generation module\nimport NRPy_param_funcs as par # NRPy+: Parameter interface\nimport grid as gri # NRPy+: Functions having to do with numerical grids\nimport loop as lp # NRPy+: Generate C code loops\nimport indexedexp as ixp # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support\nimport reference_metric as rfm # NRPy+: Reference metric support\nimport cmdline_helper as cmd # NRPy+: Multi-platform Python command-line interface\n\nthismodule = \"GiRaFFE_NRPy-C2P_P2C\"\n\n# There are several C parameters that we will need in this module:\nM_PI = par.Cparameters(\"#define\",thismodule,[\"M_PI\"], \"\")\nGAMMA_SPEED_LIMIT = par.Cparameters(\"REAL\",thismodule,\"GAMMA_SPEED_LIMIT\",10.0) # Default value based on\n # IllinoisGRMHD.\n # GiRaFFE default = 2000.0\n\n# Register the relevant gridfunctions:\nStildeD = ixp.register_gridfunctions_for_single_rank1(\"EVOL\",\"StildeD\")\nBU = ixp.register_gridfunctions_for_single_rank1(\"AUXEVOL\",\"BU\")\nValenciavU = ixp.register_gridfunctions_for_single_rank1(\"AUXEVOL\",\"ValenciavU\")\ngammaDD = ixp.register_gridfunctions_for_single_rank2(\"AUXEVOL\",\"gammaDD\",\"sym01\")\ngammaUU = ixp.register_gridfunctions_for_single_rank2(\"AUXEVOL\",\"gammaUU\",\"sym01\")\nalpha,gammadet = gri.register_gridfunctions(\"AUXEVOL\",[\"alpha\",\"gammadet\"])\nbetaU = ixp.register_gridfunctions_for_single_rank1(\"AUXEVOL\",\"betaU\")\n\nimport GiRaFFE_NRPy.GiRaFFE_NRPy_C2P_P2C as C2P_P2C\n\n```\n\n\n\n## Step 1.a: Enforce the orthogonality of $\\tilde{S}_i$ and $B^i$ \\[Back to [top](#toc)\\]\n$$\\label{ortho_s_b}$$\n\nNow, we will enforce the orthogonality of the magnetic field and densitized poynting flux using Eq. 22 of [arxiv:1704.00599v2](https://arxiv.org/abs/1704.00599v2): \n$${\\tilde S}_i \\rightarrow {\\tilde S}_i - ({\\tilde S}_j {\\tilde B}^j) {\\tilde B}_i/{\\tilde B}^2$$\nFirst, we compute the inner products ${\\tilde S}_j {\\tilde B}^j$ and ${\\tilde B}^2 = \\gamma_{ij} {\\tilde B}^i {\\tilde B}^j,$ where $\\tilde{B}^i = B^i \\sqrt{\\gamma}$. Then, we subtract $({\\tilde S}_j {\\tilde B}^j) {\\tilde B}_i/{\\tilde B}^2$ from ${\\tilde S}_i$. We thus guarantee that ${\\tilde S}_i B^i=0$.\n\n\nHaving fixed ${\\tilde S}_i$, we will also compute the related quantities ${\\tilde S}^i = \\gamma^{ij} {\\tilde S}_j$ and ${\\tilde S}^2 = {\\tilde S}_i {\\tilde S}^i$.\n\n\n\n```python\n# First, we need to obtain the index-lowered form of Btilde^i and (Btilde^i)^2\n# Recall that Btilde^i = sqrtgamma*B^i\nsqrtgammadet = sp.sqrt(gammadet)\nBtildeU = ixp.zerorank1()\nfor i in range(3):\n # \\tilde{B}^i = B^i \\sqrt{\\gamma}\n BtildeU[i] = sqrtgammadet*BU[i]\n\nBtildeD = ixp.zerorank1()\nfor i in range(3):\n for j in range(3):\n BtildeD[j] += gammaDD[i][j]*BtildeU[i]\n \nBtilde2 = sp.sympify(0)\nfor i in range(3):\n Btilde2 += BtildeU[i]*BtildeD[i]\n \nStimesB = sp.sympify(0)\nfor i in range(3):\n StimesB += StildeD[i]*BtildeU[i]\n\n# Then, enforce the orthogonality:\nif par.parval_from_str(\"enforce_orthogonality_StildeD_BtildeU\"):\n for i in range(3):\n # {\\tilde S}_i = {\\tilde S}_i - ({\\tilde S}_j {\\tilde B}^j) {\\tilde B}_i/{\\tilde B}^2\n StildeD[i] -= StimesB*BtildeD[i]/Btilde2\n\n```\n\n\n\n## Step 1.b: Rescale ${\\tilde S}_i$ to limit the Lorentz factor and enforce the velocity cap \\[Back to [top](#toc)\\]\n$$\\label{vel_cap}$$\n\nThe next fix that we will apply limits the Lorentz factor using Eqs. 92 and 93 of [arXiv:1310.3274v2](https://arxiv.org/abs/1310.3274v2). That is, we define the factor $f$ as \n$$f = \\sqrt{(1-\\Gamma_{\\max}^{-2}){\\tilde B}^4/(16 \\pi^2 \\gamma {\\tilde S}^2)}.$$\n\nIf $f<1$, we rescale the components of ${\\tilde S}_i$ by $f$. That is, we must set\n$${\\tilde S}_i \\rightarrow {\\tilde S}_i \\min(1,f).$$\n\nHere, we will use a formulation of the `min()` function that does not use `if`:\n\\begin{equation}\n\\min(a,b) = \\frac{1}{2} \\left( a+b - \\lvert a-b \\rvert \\right),\n\\end{equation}\nor, in code,\n```\nmin_noif(a,b) = sp.Rational(1,2)*(a+b-nrpyAbs(a-b))\n```\n\n\n```python\n# Calculate \\tilde{S}^2:\nStilde2 = sp.sympify(0)\nfor i in range(3):\n for j in range(3):\n Stilde2 += gammaUU[i][j]*StildeD[i]*StildeD[j]\n\n# First we need to compute the factor f: \n# f = \\sqrt{(1-\\Gamma_{\\max}^{-2}){\\tilde B}^4/(16 \\pi^2 \\gamma {\\tilde S}^2)}\nspeed_limit_factor = sp.sqrt((1.0-GAMMA_SPEED_LIMIT**(-2.0))\\\n *Btilde2*Btilde2*sp.Rational(1,16)/(M_PI*M_PI*gammadet*Stilde2))\n\ndef min_noif(a,b):\n # This returns the minimum of a and b\n # If a>b, then we get 0.5*(a+b-a+b) = b\n # If b>a, then we get 0.5*(a+b+a-b) = a\n return sp.Rational(1,2)*(a+b-nrpyAbs(a-b))\n\n# Calculate B^2\nB2 = sp.sympify(0)\nfor i in range(3):\n for j in range(3):\n B2 += gammaDD[i][j]*BU[i]*BU[j]\n\n# Enforce the speed limit on StildeD:\nif par.parval_from_str(\"enforce_speed_limit_StildeD\"):\n for i in range(3):\n StildeD[i] *= min_noif(1.0,speed_limit_factor)\n\n```\n\n\n\n## Step 1.c: Recompute the velocities at the new timestep \\[Back to [top](#toc)\\]\n$$\\label{update_vel}$$\n\nFinally, we can calculate the velocities. In [arxiv:1704.00599v2](https://arxiv.org/abs/1704.00599v2), Eq. 16 gives the drift velocity as \n$$v^i = 4 \\pi \\alpha \\gamma^{ij} {\\tilde S}_j \\gamma^{-1/2} B^{-2} - \\beta^i.$$\nHowever, we wish to use the Valencia velocity instead. Since the Valencia velocity $\\bar{v}^i = \\frac{1}{\\alpha} \\left( v^i + \\beta^i \\right)$, we will code \n$$\\bar{v}^i = 4 \\pi \\frac{\\gamma^{ij} {\\tilde S}_j}{\\sqrt{\\gamma} B^2}.$$\n\n\n\n```python\nValenciavU = ixp.zerorank1()\nif par.parval_from_str(\"enforce_orthogonality_StildeD_BtildeU\") or par.parval_from_str(\"enforce_speed_limit_StildeD\"):\n # Recompute 3-velocity:\n for i in range(3):\n for j in range(3):\n # \\bar{v}^i = 4 \\pi \\gamma^{ij} {\\tilde S}_j / (\\sqrt{\\gamma} B^2)\n ValenciavU[i] += sp.sympify(4.0)*M_PI*gammaUU[i][j]*StildeD[j]/(sqrtgammadet*B2)\n\n```\n\n\n\n## Step 1.d: Enforce the Current Sheet prescription \\[Back to [top](#toc)\\]\n$$\\label{current_sheet}$$\n\nNow, we seek to handle any current sheets (a physically important phenomenon) that might form. This algorithm, given as Eq. 23 in [arxiv:1704.00599v2](https://arxiv.org/abs/1704.00599v2), will preserve current sheets that form in the xy-plane by preventing our numerical scheme from dissipating them. After fixing the z-component of the velocity, we recompute the conservative variables $\\tilde{S}_i$ to be consistent with the new velocities.\n\nThus, if we are within four gridpoints of $z=0$, we set the component of the velocity perpendicular to the current sheet to zero by $n_i v^i = 0$, where $n_i = \\gamma_{ij} n^j$ is a unit normal to the current sheet and $n^j = \\delta^{jz} = (0\\ 0\\ 1)$. For drift velocity, this means we just set \n$$\nv^z = -\\frac{\\gamma_{xz} v^x + \\gamma_{yz} v^y}{\\gamma_{zz}}.\n$$\nThis reduces to $v^z = 0$ in flat space, as one would expect. We then express the Valencia velocity in terms of the drift velocity as $\\bar{v}^i = \\frac{1}{\\alpha} \\left( v^i + \\beta^i \\right)$. \n\n\n```python\n# We will use once more the trick from above with min and max without if. However, we we'll need a function\n# that returns either 0 or 1, so as to choose between two otherwise mathetmatically unrelated branches. \ndef max_normal0(a):\n # If a>0, return 1. Otherwise, return 0. This defines a 'greater than' branch.\n # WILL BREAK if a = 0. \n return (a+nrpyAbs(a))/(a+a)\n\ndef min_normal0(a):\n # If a<0, return 1. Otherwise, return 0. This defines a 'less than' branch.\n # WILL BREAK if a = 0. \n return (a-nrpyAbs(a))/(a+a)\n\n\n# This number determines how far away (in grid points) we will apply the fix.\ngrid_points_from_z_plane = par.Cparameters(\"REAL\",thismodule,\"grid_points_from_z_plane\",4.0)\n\nif par.parval_from_str(\"enforce_current_sheet_prescription\"):\n # Calculate the drift velocity\n driftvU = ixp.zerorank1()\n for i in range(3):\n driftvU[i] = alpha*ValenciavU[i] - betaU[i]\n\n # The direct approach, used by the original GiRaFFE:\n # v^z = -(\\gamma_{xz} v^x + \\gamma_{yz} v^y) / \\gamma_{zz} \n newdriftvU2 = -(gammaDD[0][2]*driftvU[0] + gammaDD[1][2]*driftvU[1])/gammaDD[2][2]\n # Now that we have the z component, it's time to substitute its Valencia form in.\n # Remember, we only do this if abs(z) < (k+0.01)*dz. Note that we add 0.01; this helps\n # avoid floating point errors and division by zero. This is the same as abs(z) - (k+0.01)*dz<0\n boundary = nrpyAbs(rfm.xx[2]) - (grid_points_from_z_plane+sp.sympify(0.01))*gri.dxx[2]\n ValenciavU[2] = min_normal0(boundary)*(newdriftvU2+betaU[2])/alpha \\\n + max_normal0(boundary)*ValenciavU[2]\n\n```\n\nBelow is an experiment in coding this more abstractly. While it works, it's a bit harder to follow than the direct approach, which is what is coded above\n```python\n# Set the Cartesian normal vector. This can be expanded later to arbitrary sheets and coordinate systems.\nnU = ixp.zerorank1()\nnU[2] = 1\n# Lower the index, as usual:\nnD = ixp.zerorank1()\nfor i in range(3):\n for j in range(3):\n nD[i] = gammaDD[i][j]*nU[j]\n\nif par.parval_from_str(\"enforce_current_sheet_prescription\"):\n # Calculate the drift velocity\n driftvU = ixp.declarerank1(\"driftvU\")\n \n inner_product = sp.sympify(0)\n for i in range(3):\n inner_product += driftvU[i]*nD[i] # This is the portion of the drift velocity normal to the z plane\n # In flat space, this is just v^z\n # We'll use a sympy utility to solve for v^z. This should make it easier to generalize later\n newdriftvU2 = sp.solve(inner_product,driftvU[2]) # This outputs a list with a single element. \n # Take the 0th element so .subs() works right.\n newdriftvU2 = newdriftvU2[0] # In flat space this reduces to v^z=0\n for i in range(3):\n # Now, we substitute drift velocity in terms of our preferred Valencia velocity\n newdriftvU2 = newdriftvU2.subs(driftvU[i],alpha*ValenciavU[i]-betaU[i])\n\n # Now that we have the z component, it's time to substitute its Valencia form in.\n # Remember, we only do this if abs(z) < (k+0.01)*dz. Note that we add 0.01; this helps\n # avoid floating point errors and division by zero. This is the same as abs(z) - (k+0.01)*dz<0\n boundary = nrpyAbs(rfm.xx[2]) - (grid_points_from_z_plane+sp.sympify(0.01))*gri.dxx[2]\n ValenciavU[2] = min_normal0(boundary)*(newdriftvU2+betaU[2])/alpha \\\n + max_normal0(boundary)*ValenciavU[2]\n```\n\n\n```python\n\n```\n\n\n\n# Step 2: The primitive-to-conservative solver \\[Back to [top](#toc)\\]\n$$\\label{p2c}$$\n\nThis function is used to recompute the conservatives $\\tilde{S}_i$ after the 3-velocity is changed as part of the current sheet prescription using Eq. 21 of [arxiv:1704.00599v2](https://arxiv.org/abs/1704.00599v2). It implements the same equation used to compute the initial Poynting flux from the initial velocity: $$\\tilde{S}_i = \\gamma_{ij} \\frac{\\bar{v}^j \\sqrt{\\gamma}B^2}{4 \\pi}$$ in terms of the Valencia 3-velocity. In the implementation here, we first calculate $B^2 = \\gamma_{ij} B^i B^j$, then $v_i = \\gamma_{ij} v^j$ before we calculate the equivalent expression $$\\tilde{S}_i = \\frac{\\bar{v}_i \\sqrt{\\gamma}B^2}{4 \\pi}.$$\n\nHere, we will simply let the NRPy+ `GRFFE` module handle this part, since that is already validated.\n\n\n```python\nimport GRFFE.equations as GRFFE\nimport GRHD.equations as GRHD\n\ndef GiRaFFE_NRPy_P2C(gammadet,gammaDD,betaU,alpha, ValenciavU,BU, sqrt4pi):\n # After recalculating the 3-velocity, we need to update the poynting flux:\n # We'll reset the Valencia velocity, since this will be part of a second call to outCfunction.\n\n # First compute stress-energy tensor T4UU and T4UD:\n \n GRHD.u4U_in_terms_of_ValenciavU__rescale_ValenciavU_by_applying_speed_limit(alpha, betaU, gammaDD, ValenciavU)\n GRFFE.compute_smallb4U_with_driftvU_for_FFE(gammaDD, betaU, alpha, GRHD.u4U_ito_ValenciavU, BU, sqrt4pi)\n GRFFE.compute_smallbsquared(gammaDD, betaU, alpha, GRFFE.smallb4_with_driftv_for_FFE_U)\n\n GRFFE.compute_TEM4UU(gammaDD, betaU, alpha, GRFFE.smallb4_with_driftv_for_FFE_U, GRFFE.smallbsquared, GRHD.u4U_ito_ValenciavU)\n GRFFE.compute_TEM4UD(gammaDD, betaU, alpha, GRFFE.TEM4UU)\n\n # Compute conservative variables in terms of primitive variables\n GRHD.compute_S_tildeD(alpha, sp.sqrt(gammadet), GRFFE.TEM4UD)\n \n global StildeD\n StildeD = GRHD.S_tildeD\n\n```\n\n\n\n# Step 3: Code Validation \\[Back to [top](#toc)\\]\n$$\\label{code_validation}$$\n\nAs a code validation check, we will verify agreement in the SymPy expressions between\n1. this tutorial and \n2. the NRPy+ [GiRaFFE_NRPy/GiRaFFE_NRPy_C2P_P2C.py](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_C2P_P2C.py) module.\n\n\n```python\nall_passed=True\ndef comp_func(expr1,expr2,basename,prefixname2=\"C2P_P2C.\"):\n if str(expr1-expr2)!=\"0\":\n print(basename+\" - \"+prefixname2+basename+\" = \"+ str(expr1-expr2))\n all_passed=False\n\ndef gfnm(basename,idx1,idx2=None,idx3=None):\n if idx2==None:\n return basename+\"[\"+str(idx1)+\"]\"\n if idx3==None:\n return basename+\"[\"+str(idx1)+\"][\"+str(idx2)+\"]\"\n return basename+\"[\"+str(idx1)+\"][\"+str(idx2)+\"][\"+str(idx3)+\"]\"\n\n# Still need to perform self-validation checks on C2P\n# testValenciavU = ixp.declarerank1(\"testValenciavU\",DIM=3)\n# testStildeD = ixp.declarerank1(\"testStildeDU\",DIM=3)\ngammaDD = ixp.declarerank2(\"gammaDD\",\"sym01\")\nbetaU = ixp.declarerank1(\"betaU\")\nBU = ixp.declarerank1(\"BU\")\nalpha = sp.symbols('alpha',real=True)\nsqrt4pi = sp.symbols('sqrt4pi', real=True)\n\nnotebook_StildeD = StildeD\nStildeD = ixp.declarerank1(\"StildeD\")\n\nC2P_P2C.GiRaFFE_NRPy_C2P(StildeD,BU,gammaDD,gammaUU,gammadet,betaU,alpha)\n\nexpr_list = []\nexprcheck_list = []\nnamecheck_list = []\n\nfor i in range(3):\n namecheck_list.extend([gfnm(\"StildeD\",i),gfnm(\"ValenciavU\",i)])\n exprcheck_list.extend([C2P_P2C.outStildeD[i],C2P_P2C.ValenciavU[i]])\n expr_list.extend([notebook_StildeD[i],ValenciavU[i]])\n\nfor i in range(len(expr_list)):\n comp_func(expr_list[i],exprcheck_list[i],namecheck_list[i])\n\nimport sys\nif all_passed:\n print(\"ALL TESTS PASSED!\")\nelse:\n print(\"ERROR: AT LEAST ONE TEST DID NOT PASS\")\n sys.exit(1)\n```\n\n ALL TESTS PASSED!\n\n\n\n```python\nall_passed=True\n\ngammaDD = ixp.declarerank2(\"gammaDD\",\"sym01\")\nbetaU = ixp.declarerank1(\"betaU\")\nValenciavU = ixp.declarerank1(\"ValenciavU\")\nBU = ixp.declarerank1(\"BU\")\nalpha = sp.symbols('alpha',real=True)\nsqrt4pi = sp.symbols('sqrt4pi', real=True)\nGiRaFFE_NRPy_P2C(gammadet,gammaDD,betaU,alpha, ValenciavU,BU, sqrt4pi)\n\nC2P_P2C.GiRaFFE_NRPy_P2C(gammadet,gammaDD,betaU,alpha, ValenciavU,BU, sqrt4pi)\n\nexpr_list = []\nexprcheck_list = []\nnamecheck_list = []\n\nfor i in range(3):\n namecheck_list.extend([gfnm(\"StildeD\",i)])\n exprcheck_list.extend([C2P_P2C.StildeD[i]])\n expr_list.extend([StildeD[i]])\n\nfor i in range(len(expr_list)):\n comp_func(expr_list[i],exprcheck_list[i],namecheck_list[i])\n\nimport sys\nif all_passed:\n print(\"ALL TESTS PASSED!\")\nelse:\n print(\"ERROR: AT LEAST ONE TEST DID NOT PASS\")\n sys.exit(1)\n \n```\n\n ALL TESTS PASSED!\n\n\n\n\n# Step 4: Output this notebook to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n$$\\label{latex_pdf_output}$$\n\nThe following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n[Tutorial-GiRaFFE_NRPy-C2P_P2C.pdf](Tutorial-GiRaFFE_NRPy-C2P_P2C.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means.)\n\n\n```python\n!jupyter nbconvert --to latex --template latex_nrpy_style.tplx --log-level='WARN' Tutorial-GiRaFFE_NRPy-C2P_P2C.ipynb\n!pdflatex -interaction=batchmode Tutorial-GiRaFFE_NRPy-C2P_P2C.tex\n!pdflatex -interaction=batchmode Tutorial-GiRaFFE_NRPy-C2P_P2C.tex\n!pdflatex -interaction=batchmode Tutorial-GiRaFFE_NRPy-C2P_P2C.tex\n!rm -f Tut*.out Tut*.aux Tut*.log\n```\n\n This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex)\n restricted \\write18 enabled.\n entering extended mode\n This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex)\n restricted \\write18 enabled.\n entering extended mode\n This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex)\n restricted \\write18 enabled.\n entering extended mode\n\n", "meta": {"hexsha": "9b083de032c1a942608735da91415193f161faff", "size": 27923, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "in_progress/Tutorial-GiRaFFE_NRPy-C2P_P2C.ipynb", "max_stars_repo_name": "weichen-yan/nrpytutorial", "max_stars_repo_head_hexsha": "9ec6bc995ca6fe8b3870ac892660694e059645d9", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "in_progress/Tutorial-GiRaFFE_NRPy-C2P_P2C.ipynb", "max_issues_repo_name": "weichen-yan/nrpytutorial", "max_issues_repo_head_hexsha": "9ec6bc995ca6fe8b3870ac892660694e059645d9", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "in_progress/Tutorial-GiRaFFE_NRPy-C2P_P2C.ipynb", "max_forks_repo_name": "weichen-yan/nrpytutorial", "max_forks_repo_head_hexsha": "9ec6bc995ca6fe8b3870ac892660694e059645d9", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-03-02T12:51:56.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-02T12:51:56.000Z", "avg_line_length": 44.6054313099, "max_line_length": 659, "alphanum_fraction": 0.5788776278, "converted": true, "num_tokens": 6861, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.41565459927447623}} {"text": "\n\n\n```\n!pip install qiskit\nimport numpy as np\nfrom qiskit import(\n QuantumCircuit,\n execute,\n Aer)\nfrom qiskit.visualization import plot_histogram\n\nsimulator = Aer.get_backend('qasm_simulator')\n\ncircuit = QuantumCircuit(2,2)\ncircuit.h(0)\ncircuit.cx(0,1)\ncircuit.measure([0,1],[0,1])\n\njob = execute(circuit,simulator,shots=1000)\nresult = job.result()\n\ncounts = result.get_counts(circuit)\nprint(\"\\nTotal count for 00 and 11 are:\",counts)\n\ncircuit.draw()\nplot_histogram(counts)\n```\n", "meta": {"hexsha": "25c61a76ae97c7c8173b75704fb4dc9bbff875ca", "size": 21946, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "IBMqiskit.ipynb", "max_stars_repo_name": "Lisayg/Cirq", "max_stars_repo_head_hexsha": "af503842bcd5e960a1f320146d9285fe97131dd5", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "IBMqiskit.ipynb", "max_issues_repo_name": "Lisayg/Cirq", "max_issues_repo_head_hexsha": "af503842bcd5e960a1f320146d9285fe97131dd5", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "IBMqiskit.ipynb", "max_forks_repo_name": "Lisayg/Cirq", "max_forks_repo_head_hexsha": "af503842bcd5e960a1f320146d9285fe97131dd5", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 160.1897810219, "max_line_length": 12114, "alphanum_fraction": 0.8065706735, "converted": true, "num_tokens": 176, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4156291315518877}} {"text": "# Enzyme Kinetics\nWe now study common reaction mechanisms that describe enzyme catalysis. Enzymes can dramatically accelerate the rate of biochemical reactions inside and outside living cells. The absolute rates of biochemical reactions are key biological design variables because they can evolve from a very low rate as determined by the mass action kinetics based on collision frequencies, to a very high and specific reaction rate as determined by appropriately-evolved enzyme properties. We first describe the procedure used to derive enzymatic rate laws, which we then apply to the Michaelis-Menten reaction mechanism, then to the Hill model, and finally to the symmetry model. The first is used to describe plain chemical transformations, while the latter two are used to describe regulatory effects. \n\n**MASSpy** will be used to demonstrate some of the topics in this chapter. \n\n\n```python\nfrom mass import (\n MassModel, MassMetabolite, MassReaction, Simulation, MassSolution)\nfrom mass.visualization import plot_time_profile, plot_phase_portrait\n```\n\nOther useful packages are also imported at this time.\n\n\n```python\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nfrom scipy.integrate import solve_ivp\n```\n\n## Enzyme Catalysis\nEnzymes are catalysts that accelerate biochemical transformations in cells. Almost all enzymes are proteins. There are also catalytically active ribonucleic acids, called \"ribozymes.\" The fundamental properties of enzyme catalysis are described in this section. \n\n### Enzymatic activity\nThe activity of an enzyme is measured by determining the increase in the reaction rate relative to the absence of the enzyme. In other words we compare the reaction rate of the un-catalyzed reaction to the catalyzed rate. The ratio can be thought of as an acceleration factor and this number can be quite high, sometimes by many million-fold. \n\n### Reaction and substrate specificity\nEnzymes are usually very specific both with respect to the type of reaction being catalyzed (reaction specificity) and with respect to the reactants (the \"substrates\") that they act on. Highly specific enzymes catalyze the cleavage of only one type of a chemical bond, and only in one substrate. Other enzymes may have a narrow reaction specificity, but broad substrate specificity, i.e., they act on a number of chemically similar substrates. Rare enzymes exist that have both low reaction specificity and low substrate specificity. \n\n\n\n**Figure 5.1:** Basic principles of enzyme catalysis. From (Koolman, 2005).\n\n### Catalysis\nAs discussed in Chapter\u00a02 (Figure 2.4), two molecules can only react with each other if they collide in a favorable orientation. Such collisions may be rare, and thus the reaction rate is slow. An un-catalyzed reaction starts with a favorable collision as shown in Figure\u00a05.1a. Before the products are formed, the collision complex A-B has to pass through what is called a _transition state_. Its formation requires _activation energy_. Since activation energies can be quite high, only a few A-B complexes have this amount of energy, and thus a productive transition state arises only for a fraction of favorable collisions. As a result, conversion only happens occasionally even when the reaction is thermodynamically feasible; i.e., when the net change in Gibbs free energy is negative ($\\Delta G < 0$). \n\nEnzymes can facilitate the probability of a favorable collision and lower the activation energy barrier, see Figure\u00a05.1b,c. Enzymes are able to bind their substrates in the catalytic site. As a result, the substrates are favorably oriented relative to one another, greatly enhancing the probability that productive A-B complexes form. The transition state is stabilized leading to a lowered activation energy barrier. \n\n### Information on enzymes \nDetailed information is available on a large number of enzymes. This include structural information, the organism source, and other characteristics. An example is shown in Figure\u00a05.2. Many online sources of such information exist. \n\n\n\n**Figure 5.2:** Detailed information on enzymes is available. From PDB.\n\n## Deriving Enzymatic Rate Laws\nThe chemical events underlying the catalytic activities of enzymes are described by a reaction mechanism. A reaction mechanism is comprised of the underlying elementary reactions that are believed to take place. A rate law is then formulated to describe the rate of reaction. \n\nA rate law describes the conversion of a substrate $(x_1)$ by an enzyme into a product $(x_2)$: \n\n$$\\begin{equation} x_1 \\stackrel{v}{\\rightarrow} x_2 \\tag{5.1} \\end{equation}$$\n\nwhere $v$ is a function of the concentrations of the chemical species involved in the reaction. The steps involved in the development and analysis of enzymatic rate laws are illustrated in Figure\u00a05.3 and they are as follows: \n\n\n\n**Figure 5.3:** The process of formulating enzymatic rate laws. QSSA represents the quasi-steady state assumption and QEA represents the quasi-equilibrium assumption.\n\n* Formulate the dynamic mass balances based on the elementary reactions in the postulated reaction mechanism, \n\n* Identify time invariants, or conservation relationships, \n\n* Reduce the dynamic dimension of the reaction mechanism by eliminating dynamically dependent variables using the conservation relationships, \n\n* Apply commonly used simplifying kinetic assumptions to formulate a rate law, representing a reduction in the dynamic dimension of the kinetic model, \n\n* Apply mathematical and numerical analysis to determine when the simplifying assumptions are valid and the reaction rate law can be used; and \n\n* Identify key dimensionless parameter ratios. This last step is optional and used by those interested in deeper mathematical analysis of the properties of the rate laws. \n\nThe use of enzymatic rate laws in dynamic network models is hampered by their applicability in vivo based on in vitro measurements. From a practical standpoint, with the numerical simulation capacity that is now routinely available, applying simplifying assumptions may no longer be needed for computational simplification and convenience. However, it is useful to help understand the historical origin of enzymatic rate laws, the simplifications on which they are based, and when it may be desirable to use them. \n\n## Michaelis-Menten Kinetics\nThe simplest enzymatic reaction mechanism, first proposed by Henri (Henri, 1903) but named after Michaelis and Menten\u00a0(Michaelis, 1913) is; \n\n$$\\begin{equation} S + E \\underset{k_{-1}}{\\stackrel{k_1}{\\rightleftharpoons}} X \\stackrel{k_2}{\\rightarrow} E + P \\tag{5.2} \\end{equation}$$\n\nwhere a substrate, $S$, binds reversibly to the enzyme, $E$, to form the intermediate, $X$, which can break down to give the product, $P$, and regenerate the enzyme. Note that it is similar to the reaction mechanism of two connected reversible bi-linear reactions (Eq. (4.38)) with $x_5 = x_2$, as one of the original reactants $(E)$ is regained in the second step. Historically speaking, the Michaelis-Menten scheme is the most important enzymatic reaction mechanism. A detailed account of the early history of Michaelis-Menten kinetics is found in (Segal, 1959).\n\n### Step 1: Dynamic mass balances for Michaelis-Menten kinetics\nApplying the law of mass action to the Michaelis-Menten reaction mechanism, one obtains four differential equations that describe the dynamics of the concentrations of the four chemical species involved in the reaction mechanism: \n\n$$\\begin{align} \\frac{ds}{dt} &= -k_1es + k_{-1}x, & s(t = 0) = s_0 \\\\ \\frac{dx}{dt} &= k_1es - (k_{-1} + k_2)x, & x(t = 0) = 0 \\\\ \\frac{de}{dt} &= -k_1es + (k_{-1} + k_2)x, & e(t = 0) = e_0 \\\\ \\frac{dp}{dt} &= k_2x, & p(t = 0) = 0 \\\\ \\end{align} \\tag{5.3}$$\n\nwhere the lower case letters denote the concentrations of the corresponding chemical species. The initial conditions shown are for typical initial rate experiments where substrate and free enzyme are mixed together at time $t=0$. $e_0$ and $s_0$ denote the initial concentration of enzyme and substrate, respectively. No mass exchange occurs with the environment. \n\n### Step 2: Finding the time invariants for Michaelis-Menten kinetics\nUsing $\\textbf{x} = (s, e, x, p)$ and $\\textbf{v} = (k_1es, \\ k_{-1}x, \\ k_2x)$ the stoichiometrix matrix is \n\n$$\\begin{equation} \\textbf{S} = \\begin{pmatrix} {-1} & {1} & {0} \\\\ {-1} & {1} & {1} \\\\ {1} & {-1} & {-1} \\\\ {0} & {0} & {1} \\\\ \\end{pmatrix} \\end{equation} \\tag{5.4}$$\n\nIt has a rank of 2 and thus there are two conservation quantities. They are the total concentration of the enzyme and total concentration of the substrate: \n\n$$\\begin{align} e_0 & = e + x \\tag{5.5} \\\\ s_0 &= s + x + p \\tag{5.6} \\end{align}$$\n\n### Step 3: Reducing the dynamic description for Michaelis-Menten kinetics\nAs a consequence of the two conservation relationships, only two of equations\u00a05.3 are dynamically independent. Choosing the substrate, $s$, and the intermediate complex, $x$, concentrations as the two independent variables, the reaction dynamics are described by: \n\n$$\\begin{align} \\frac{ds}{dt} &= -k_1e_0s + (k_1s + k_{-1})x, \\ &s(t = 0)=s_0 \\tag{5.7} \\\\ \\frac{dx}{dt} &= k_1e_0s - (k_1s + k_{-1} + k_2)x, \\ &x(t = 0)=0 \\tag{5.8} \\\\ \\end{align}$$\n\nThe major problem with this mass action kinetic model is that it is mathematically intractable\u00a0(Hommes, 1962). Equations 5.7 and 5.8 can be reduced to an Abel type differential equation whose solution cannot be obtained in a closed form.\n\n### Step 4: Applying kinetic assumptions for Michaelis-Menten kinetics\nA closed form analytical solution to the mass action kinetic equations,\u00a05.7 and\u00a05.8, is only attainable by using simplifying kinetic assumptions. Two assumptions are used: the _quasi-steady state assumption_ (QSSA) and the _quasi-equilibrium assumption_ (QEA). \n\n#### The quasi-steady state assumption:\nThe rationale behind the quasi-steady state assumption (Briggs, 1925) is that, after a rapid transient phase, the intermediate, $X$, reaches a quasi-stationary state in which its concentration does not change appreciably with time. Applying this assumption to Eq.\u00a0(5.8) (i.e., $dx/dt=0$) gives the concentration of the intermediate complex as: \n\n$$\\begin{equation} x_{qss} = \\frac{e_0s}{K_m + s} \\tag{5.9} \\end{equation}$$\n\nwhere $K_m = (k_{-1} + k_2)/k_1$ is the well-known Michaelis constant. Substituting $x_{qss}$ into the differential equation for the substrate (Eq.\u00a0(5.7)) gives the rate law \n\n$$\\begin{equation} \\frac{ds}{dt} = \\frac{-k_2e_0s}{K_m + s} \\tag{5.10} \\end{equation}$$\n\nwhich is the well-known Michaelis-Menten equation, where $v_m$ is the maximum reaction rate (or reaction velocity). \n\nInitially, the quasi-steady state assumption was justified based on physical intuition, but justification for its applicability is actually found within the theory of singular perturbations (Bowen, 1963) Eq. (5.10) can be shown to be the first term in an asymptotic series solution derived from singular perturbation theory\u00a0(Heineken,1967), (Meiske, 1978); see review in\u00a0(Palsson, 1984). \n\n#### The quasi-equilibrium assumption:\nHere, one assumes that the binding step quickly reaches a quasi-equilibrium state\u00a0(Henri, 1903), (Michaelis, 1913) where \n\n$$\\begin{equation} \\frac{se}{x} = \\frac{s(e_0 - x)}{x} = \\frac{k_{-1}}{k_1} = K_d, \\ \\text{or} \\ x_{qe} = \\frac{e_0s}{K_d + s} \\tag{5.11} \\end{equation}$$\n\nholds. $K_d$ is the disassociation equilibrium constant. Note the similarity to Eq. (5.9). Hence, one obtains the rate law \n\n$$\\begin{equation} \\frac{dp}{dt} = \\frac{k_2e_0s}{K_d + s} \\tag{5.12} \\end{equation}$$\n\nby using Eq. (5.11) in the differential equation for the product $P$. \n\n### Step 5: Numerical solutions for Michaelis-Menten kinetics\nThe full dynamic description of the kinetics of the reaction (Eq. (5.7) and (5.8)) can be obtained by direct numerical integration. The results are most conveniently shown on a phase portrait along with the transient response of the concentrations on both the fast and slow time scales, see Figure\u00a05.4. \n\n#### QSSA Solution for Michaelis-Menten kinetics\n\n\n```python\nt0 = 0\ntf = 1e3\n# QSSA Assumption\n# Define function to integrate\ndef qssa(t, s, *params):\n k2, e0, Km = params\n dsdt = (-k2*e0*s)/(Km + s)\n return dsdt\n\n# Define initial conditions and parameters for integration\ns0 = 1\n\ne0 = (1/100)\nk2 = 1\nKm = 1\nparams = [k2, e0, Km]\n\n# Obtain numerical solutions\nsol_obj = solve_ivp(fun=lambda t, s: qssa(t, s, *params),\n t_span=(t0, tf), y0=[s0])\nt, s_sol = (sol_obj.t, sol_obj.y)\nx_sol = np.array([(e0 * val)/(Km + val) for val in s_sol])\n# Store solutions into Solution Objects\nqssa_sol = MassSolution(\n \"QSSA\", solution_type=\"Conc\", \n data_dict={\"s\": s_sol[0], \"x\": x_sol[0]},\n time=t, interpolate=False)\n```\n\n#### Numerical Solution for Michaelis-Menten kinetics\n\n\n```python\nmodel = MassModel('Michaeli_Menten')\n## Define metabolites\ns = MassMetabolite(\"s\")\ne = MassMetabolite(\"e\")\nx = MassMetabolite(\"x\")\np = MassMetabolite(\"p\")\n# Define reactions\nv1 = MassReaction(\"v1\")\nv2 = MassReaction(\"v2\", reversible=False)\nv1.add_metabolites({s: -1, e: -1, x: 1})\nv2.add_metabolites({x: -1, e: 1, p: 1})\nmodel.add_reactions([v1, v2])\n## Define parameters\nv1.kf = 2\nv1.Keq = 2\nv2.kf = 1\n# Define initial conditions\nmodel.update_initial_conditions({s: s0, e: e0, x: 0, p: 0})\n\n# Solve\nMM_simulation = Simulation(model, verbose=True)\nconc_sol, flux_sol = MM_simulation.simulate(model, (t0, tf))\n```\n\n \u001b[93mWARNING:\u001b[0m \u001b[93mNo compartments found in model. Therefore creating compartment 'compartment' for entire model.\u001b[0m\n\n\n Successfully loaded MassModel 'Michaeli_Menten' into RoadRunner.\n\n\n\n```python\nfig_5_4 = plt.figure(figsize=(9, 7))\ngs = fig_5_4.add_gridspec(nrows=2, ncols=2, width_ratios=[1, 1.5],\n height_ratios=[1, 1])\n\nax1 = fig_5_4.add_subplot(gs[0, 0])\nax2 = fig_5_4.add_subplot(gs[0, 1])\nax3 = fig_5_4.add_subplot(gs[1, 1])\n# Phase portrait of both solutions' substrate vs. intermediate\nplot_phase_portrait(\n conc_sol, x=s, y=x, ax=ax1, legend=[\"Numerical Solution\"],\n annotate_time_points=\"endpoints\",\n annotate_time_points_color=[\"r\", \"b\"],\n annotate_time_points_labels=True);\nplot_phase_portrait(\n qssa_sol, x=s, y=x, ax=ax1, legend=[\"QSSA\", \"lower outside\"], \n xlabel=s.id, ylabel=x.id, linestyle=[\"--\"],\n title=(\"(a) Phase Portrait\", {\"size\": \"x-large\"}),\n annotate_time_points=\"endpoints\",\n annotate_time_points_labels=True);\n\n# Time profile of solutions' substrate concentration\nplot_time_profile(conc_sol, observable=s, ax=ax2);\nplot_time_profile(\n qssa_sol, observable=s, ax=ax2, \n xlabel=\"Time\", ylabel=\"Concentration\",\n title=(\"(b) Substrate Concentration\", {\"size\": \"x-large\"}),\n linestyle=[\"--\"]);\n# Time profile of solutions' intermediate concentration\nplot_time_profile(conc_sol, observable=x, ax=ax3);\nplot_time_profile(\n qssa_sol, observable=x, ax=ax3,\n xlabel=\"Time\", ylabel=\"Concentration\",\n title=(\"(c) Intermediate Concentration\", {\"size\": \"x-large\"}),\n linestyle=[\"--\"]);\nfig_5_4.tight_layout()\n```\n\n**Figure 5.4:** The transient response of the Michaelis-Menten reaction mechanism, for $k_2 = k_{-1}, \\ 100e_0 = K_m \\ \\text{and} \\ s_0 = K_m$. (a) The phase portrait . (b) The substrate concentrations. (c) The intermediate concentrations. The solid and dashed line represent the quasi-steady state and the full numerical solution respectively.\n\n* _The phase portrait_. The phase portrait is shown in of Figure\u00a05.4a and it shows how the reaction rapidly approaches the quasi-steady state line and then moves along that line towards the equilibrium in the origin where the reaction has gone to completion. \n\n* _The fast motion_. With the slider on moved to the left, Figure\u00a05.4 shows the changes in the concentrations during the faster time scale. The intermediate concentration exhibits a significant fast motion, while the substrate does not move far from its initial value. \n\n* _The slow motion_. The changes in the concentrations during the slower time scale are shown when the slider is on the right of Figure 5.4. Both the substrate and the intermediate complex decay towards zero. During the decay process, the complex is in a quasi-stationary state and the motion of the substrate drives the reaction dynamics. The quasi-steady state solution gives a good description of the motion on the slower time scale. \n\n### Step 6: Identification of dimensionless parameters for Michaelis-Menten kinetics\nSimulation studies suggests that there are three dimensionless parameters of interest: \n\n$$\\begin{equation} a = k_2/k_{-1}, \\ b = e_0/K_m, \\ c = s_0/K_m \\tag{5.13} \\end{equation}$$\n\nThis result is also found by rigorous mathematical analysis\u00a0(Palsson, 1984). The dynamic behavior of the reaction is determined by three dimensionless groups: a ratio of kinetic constants and the two initial conditions scaled to $K_m$. \n\n1. The first dimensionless group, a, is a ratio consisting only of kinetic constants, $k_2/k_{-1}$. This ratio has been called the 'stickiness number' \u00a0(Palsson, 1984), (Palsson, 1984a), since a substrate is said to stick well to an enzyme if $k_2 > k_{-1}$. Once $X$ is formed it is more likely to break down to yield the product than to revert back to substrate. \n\n2. The second dimensionless number, $e_0/K_m$, is a dimensionless concentration parameter - the total enzyme concentration relative to the Michaelis constant. This quantity varies from one situation to another and takes particularly different values under _in vitro_ and _in vivo_ conditions. The enzyme concentrations used _in vitro_ are several orders of magnitude lower than the $K_m$ values\u00a0(Masters, 1977), (Srere, 1967), (Srere, 1970). In vivo enzyme concentrations can approach the same order of magnitude as $K_m$. \n\n3. The third dimensionless ratio, $s_0/K_m$, is the initial condition for the substrate concentration. Typical values for this ratio _in vivo_ is on the order of unity. \n\n\n```python\n# Define new initial conditions and parameters for integration\ns0 = (1/100)\n\ne0 = (1/100)\nk2 = 1\nKm = 1\nparams = [k2, e0, Km]\n\n# Obtain numerical solutions\nsol_obj = solve_ivp(fun=lambda t, s: qssa(t, s, *params),\n t_span=(t0, tf), y0=[s0])\ns_sol = sol_obj.y\nx_sol = np.array([(e0 * val)/(Km + val) for val in s_sol])\n# Store solutions into MassSolution Objects\nqssa_sol = MassSolution(\n \"QSSA\", solution_type=\"Conc\", \n data_dict={\"s\": s_sol[0], \"x\": x_sol[0]},\n time=sol_obj.t, interpolate=False)\n\n# Update initial conditions for MassModel\nmodel.update_initial_conditions({s: s0})\n\n# Solve\nMM_simulation = Simulation(model, verbose=True)\nconc_sol, flux_sol = MM_simulation.simulate(model, (t0, tf))\n```\n\n \u001b[93mWARNING:\u001b[0m \u001b[93mNo compartments found in model. Therefore creating compartment 'compartment' for entire model.\u001b[0m\n\n\n Successfully loaded MassModel 'Michaeli_Menten' into RoadRunner.\n\n\n\n```python\nfig_5_5 = plt.figure(figsize=(9, 7))\ngs = fig_5_5.add_gridspec(nrows=2, ncols=2, width_ratios=[1, 1.5],\n height_ratios=[1, 1])\n\nax1 = fig_5_5.add_subplot(gs[0, 0])\nax2 = fig_5_5.add_subplot(gs[0, 1])\nax3 = fig_5_5.add_subplot(gs[1, 1])\n# Phase portrait of both solutions' substrate vs. intermediate\nplot_phase_portrait(\n conc_sol, x=s, y=x, ax=ax1, legend=[\"Numerical Solution\"],\n annotate_time_points=\"endpoints\",\n annotate_time_points_color=[\"r\", \"b\"],\n annotate_time_points_labels=True);\nplot_phase_portrait(\n qssa_sol, x=s, y=x, ax=ax1, legend=[\"QSSA\", \"lower outside\"], \n xlabel=s.id, ylabel=x.id, linestyle=[\"--\"],\n title=(\"(a) Phase Portrait\", {\"size\": \"x-large\"}),\n annotate_time_points=\"endpoints\",\n annotate_time_points_labels=True);\n\n# Time profile of solutions' substrate concentration\nplot_time_profile(conc_sol, observable=s, ax=ax2);\nplot_time_profile(qssa_sol, observable=s, ax=ax2, \n xlabel=\"Time\", ylabel=\"Concentration\",\n title=(\"(b) Substrate Concentration\", {\"size\": \"x-large\"}),\n linestyle=[\"--\"]);\n# Time profile of solutions' intermediate concentration\nplot_time_profile(conc_sol, observable=x, ax=ax3);\nplot_time_profile(qssa_sol, observable=x, ax=ax3,\n xlabel=\"Time\", ylabel=\"Concentration\",\n title=(\"(c) Intermediate Concentration\", {\"size\": \"x-large\"}),\n linestyle=[\"--\"]);\nfig_5_5.tight_layout()\n```\n\n**Figure 5.5:** The transient response of the Michaelis-Menten reaction mechanism, for $k_2 = k_{-1},\\ 100e_0 = K_m \\ \\text{and} \\ s_0 = K_m$. (a) The phase portrait . (b) The substrate concentrations. (c) The slow transients. The solid and dashed line represent the quasi-steady state and the full numerical solution respectively.\n\n### Comment on the criterion $e_0 << s_0$\nHistorically, the commonly accepted criterion for the applicability of the quasi-steady state assumption is that the initial concentration of the enzyme must be much smaller than that of the substrate. The actual criterion is $e_0 << K_m, \\ \\text{or} \\ b << 1$ (Palsson, 1984). Figure\u00a05.5 shows the reaction dynamics for $e_0 = K_m, \\ e_0 = 100s_0, \\ 100k_2 = k_{-1}$, which is analogous to Figure\u00a05.4, except the initial substrate concentration is now a hundred times smaller than $K_m$. In other words, we have $e_0 = s_0 << K_m$ and, as demonstrated in Figure\u00a05.5, the quasi-steady state assumption is applicable. \n\n## Hill-kinetics for Enzyme Regulation\n### Regulated enzymes\nEnzyme activity is regulated by the binding of small molecules to the enzyme resulting in an altered enzymatic activity. Such binding can inhibit or activate the catalytic activities of the enzyme. The regulation of enzymes such regulators represents a 'tug of war' between the functional states of the enzyme, see Figure\u00a05.6. A simple extension of the oldest reaction mechanisms for ligand binding to oligomeric protein, i.e., oxygen binding to hemoglobin, is commonly used to obtain simple rate laws for regulated enzymes\u00a0(Hill, 1910).\n\n\n\n**Figure 5.6:** An example of a regulated multimeric enzyme. The T form of the enzyme created by inhibitor binding is inactive, where as the R form, where no inhibitor is bound, is catalytically active. From\u00a0(Koolman, 2005) (reprinted with permission).\n\n### The reaction mechanism for Hill-kinetics\nThe Hill reaction mechanism is based on two reactions: a catalytic conversion and the sequestration of the enzyme in an inactive form. It assumes that the catalyzed reaction is an irreversible bi-molecular reaction between the substrate, $S$, and the enzyme, $E$, to form the product,$P$, and the free enzyme in a single elementary reaction: \n\n$$\\begin{equation} S + E \\stackrel{k}{\\rightarrow} E + P \\tag{5.14} \\end{equation}$$\n\nThe enzyme in turn can be put into a catalytically inactive state, $X$, through binding simultaneously and reversibly to $\\nu$ molecules of an inhibitor, $I$: \n\n$$\\begin{equation} E + {\\nu}I \\underset{k_{-i}^-}{\\stackrel{k_{i}^+}{\\rightleftharpoons}} X \\tag{5.15} \\end{equation}$$\n\nNumerical values for $\\nu$ often exceed unity. Thus, the regulatory action of $I$ is said to be lumped in the simple $E$ to $X$ transformation, as values of $\\nu$ greater than 1 are chemically unrealistic. Numerical values estimated from data show that the best fit values for $\\nu$ are not integers; for instance $\\nu$ is found to be around 2.3 to 2.6 for $O_2$ binding to hemoglobin. Section\u00a05.5 describes more realistic reaction mechanisms of serial binding of an inhibitor to a regulated enzyme to sequester it in an inactive form. \n\n### Step 1: Dynamic mass balances for Hill-kinetics\nThe mass action kinetic equations are \n\n$$\\begin{equation} \\frac{ds}{dt} = -v_1, \\ \\frac{de}{dt} = -v_2 + v_3, \\ \\frac{dp}{dt} = v_1, \\ \\frac{di}{dt} = -\\nu (v_2 - v_3), \\ \\frac{ds}{dt} = v_2 - v_3 \\end{equation}$$\n\nwhere the reaction rates are \n\n$$\\begin{equation} v_1 = kse, \\ v_2 = k_i^+i^{\\nu}e, \\ v_3 = k_i^-x \\tag{5.16} \\end{equation}$$\n\n### Step 2: Finding the time invariants for Hill-kinetics\nWe define $\\textbf{x} = (s, e, p, i, x) \\ \\text{and} \\ \\textbf{v} = (ks, k_i^+i^{\\nu}e, k_i^-x)$. The stoichiometric matrix is then \n\n$$\\begin{equation} \\textbf{S} = \\begin{pmatrix} {-1} & {0} & {0} \\\\ {0} & {-1} & {1} \\\\ {1} & {0} & {0} \\\\ {0} & {-\\nu} & {\\nu} \\\\ {0} & {1} & {-1} \\\\ \\end{pmatrix} \\end{equation}$$\n$$\\tag{5.17}$$\n\nand has a rank of two. The conservation quantities are a balance on the substrate, the enzyme, and the inhibitor: \n\n$$\\begin{equation} s_0 = s + p, \\ e_0 = e + x, \\ i_0 = i + \\nu x \\tag{5.18} \\end{equation}$$\n\n### Step 3: Reducing the dynamic description for Hill-kinetics\nWe need two differential equations to simulate the dynamic response and then the remaining three variables can be computed from the conservation relationships. We can choose the substrate, $s$, and the concentration of the enzyme, $e$: \n\n$$\\begin{equation} \\frac{ds}{dt} = kse, \\ \\frac{de}{dt} = -k_i^+i^{\\nu}e + k_i^-x \\tag{5.19} \\end{equation}$$\n\nthen $p$, $x$ and $i$ are computed from Eq. (5.18). \n\n### Step 4: Applying simplifying kinetic assumptions for Hill-kinetics\n\nIf we assume that the binding of the inhibitor is fast, so that a quasi-equilibrium forms for the reaction of Eq.\u00a0(5.15), we have \n\n$$\\begin{equation} v_2 = v_3, \\ \\text{thus} \\ x = (k_i^+/k_i^-)i^{\\nu}e = (i/K_i)^{\\nu}e, \\ \\text{and} \\ \\frac{de}{dt} = \\frac{dx}{dt} = \\frac{di}{dt} = 0 \\tag{5.20} \\end{equation}$$\n\nwhere $K_i$ is a \"per-site\" dissociation constant for Eq. (5.15). The enzyme is in one of two states, so that we have the mass balance \n\n$$\\begin{equation} e_0 = e + x = (1 + (i/K_i)^{\\nu})e \\ \\text{or} \\ e(i) = \\frac{e_0}{1 + (i/K_i)^{\\nu}} \\tag{5.21} \\end{equation}$$\n\nwhere $e_0$ is the total concentration of the enzyme. Using the mass balance and the quasi-equilibrium assumption gives the flux through the regulated reaction as \n\n$$\\begin{equation} v(i) = ke(i)s = \\frac{ke_0s}{1 + (i/K_i)^{\\nu}} = \\frac{v_m}{1 + (i/K_i)^{\\nu}} \\tag{5.22} \\end{equation}$$\n\nwith $v_m = ke_0s$. The Hill model has three parameters: 1) $\\nu$, the degree of cooperativity, 2) $K_i$, the dissociation constant for the inhibitor and, 3) $v_m$, the maximum reaction rate or the capacity of the enzyme. We note that \n\n$$\\begin{equation} f_e = \\frac{e(i)}{e_0} = \\frac{1}{1 + (i/K_i)^{\\nu}} \\tag{5.23} \\end{equation}$$\n\nrepresents the fraction of the enzyme that is in the active state. Note that $f_e \\lt 1$ for any finite concentration of the inhibitor. \n\n\n```python\nt0 = 0\ntf = 10\n\ndef hill(t, state_vars, *params):\n s, p, e, i, x = state_vars \n k1, k_plus,k_minus, nu = params\n # Reaction Rates\n v1 = k1 * s * e\n v2 = k_plus * i**nu * e\n v3 = k_minus * x\n # Differential equations\n diffeqs =[-v1, # ds/dt\n v1, # dp/dt\n -v2 + v3, # de/dt\n -nu*(v2 - v3), # di/dt\n v2 - v3] # dx/dt\n return diffeqs\n\n# Define initial conditions\ns0, p0, e0, i0, x0 = (1, 0, 1, 1, 0)\n\n# Define paramters\nk1 = 1\nk_plus, k_minus = (100, 100)\nnu = 2\nparams = [k1, k_plus, k_minus, nu]\n\n# Obtain numerical solutions\nsol_obj = solve_ivp(\n fun=lambda t, state_vars: hill(t, state_vars, *params),\n t_span=(t0, tf), y0=[s0, p0, e0, i0, x0])\n# Store solutions into Solution Objects\nsol_dict = dict(zip([\"s\", \"p\", \"e\", \"i\", \"x\"], sol_obj.y))\nhill_sol = MassSolution(\n \"Hill\", solution_type=\"Conc\", data_dict=sol_dict,\n time=sol_obj.t, interpolate=False)\n```\n\n\n```python\nfig_5_7 = plt.figure(figsize=(9, 8))\ngs = fig_5_7.add_gridspec(nrows=2, ncols=2, width_ratios=[1, 1],\n height_ratios=[1, 1])\n\nax1 = fig_5_7.add_subplot(gs[0, 0])\nax2 = fig_5_7.add_subplot(gs[0, 1])\nax3 = fig_5_7.add_subplot(gs[1, 0])\nax4 = fig_5_7.add_subplot(gs[1, 1])\n\nplot_phase_portrait(\n hill_sol, x=\"s\", y=\"e\", ax=ax1, xlabel=\"s\", ylabel=\"e\",\n xlim=(-0.05, 1.05), ylim=(-0.05, 1.05),\n title=(\"(a) Phase Portrait of s vs. e\", {\"size\": \"x-large\"}),\n annotate_time_points=\"endpoints\",\n annotate_time_points_labels=True);\nplot_phase_portrait(\n hill_sol, x=\"e\", y=\"x\", ax=ax2, xlabel=\"e\", ylabel=\"x\",\n xlim=(-0.05, 1.05), ylim=(-0.05, 1.05),\n title=(\"(b) Phase Portrait of e vs. x\", {\"size\": \"x-large\"}),\n annotate_time_points=\"endpoints\",\n annotate_time_points_labels=True);\n\nplot_phase_portrait(\n hill_sol, x=\"i\", y=\"x\", ax=ax3, xlabel=\"i\", ylabel=\"x\",\n xlim=(-0.05, 1.05), ylim=(-0.05, 1.05),\n title=(\"(a) Phase Portrait of i vs. x\", {\"size\": \"x-large\"}),\n annotate_time_points=\"endpoints\",\n annotate_time_points_labels=True);\nplot_time_profile(\n hill_sol, ax=ax4, legend=\"right outside\", \n title=(\"(d) Concentration Profiles\", {\"size\": \"x-large\"}));\nfig_5_7.tight_layout()\n```\n\n**Figure 5.7:** The transient response of the Hill reaction mechanism, for $k_i^+ = k_i^- = 100$, $k = 1$, $\\nu = 2$, $x_0 = 0$ and $e_0 = s_0 = i_0 = 1$. (a) The phase portraits of $s$ and $e$. (b) The phase portraits of $e$ and $x$. (c) The phase portraits of $i$ and $x$. (d) The concentration profiles.\n\n### Step 5: Numerical solutions for Hill-kinetics\nThe dynamic response of the Hill reaction mechanism is shown in Figure\u00a05.7. The trace in the $s$ vs. $e$ phase portrait is L-shaped, showing a rapid initial equilibration of the enzyme to the inhibitor (the vertical line), followed by the slower conversion of the product (the horizontal line). These two reactions are naturally (stoichiometrically) decoupled and separated in time for the numerical values of the kinetic constants used. \n\nThe phase portraits for $e$ vs. $x$ and $i$ vs. $x$ are straight lines as given by the conservation Eq. (5.18), see Figure\u00a05.7b,c. The two phase transient responses in Figure\u00a05.7d shows the rapid equilibration of the enzyme and the slow conversion of substrate. Under these parameter conditions, the QEA should give good results. \n\n### Step 6: Estimating key parameters \nThere are two features of the Hill rate law that are of interest: \n\n#### Applicability of the quasi-equilibrium assumption. \nGiven the fact that the two reactions have characteristic times scales, their relative magnitude is of key concern when it comes to the justification of the QEA: \n\n$$\\begin{equation} a = (\\frac{\\text{characteristic binding time of the inhibitor}}{\\text{characteristic turnover time of the substrate}}) = \\frac{k}{k_i^+} \\tag{5.24} \\end{equation}$$\n\nIf $a$ is much smaller than unity, we would expect the QEA to be valid. In Figure\u00a05.7, $a$ is 0.01. \n\n#### Regulatory characteristics \nThe Hill rate law has a sigmoidal shape with sensitivity of the reaction rate to the end product concentration as \n\n$$\\begin{equation} v_i = \\frac{\\partial v}{\\partial i} = \\frac{-\\nu v_m}{i} \\frac{(i/K_i)^{\\nu}}{[1 + (i/K_i)^{\\nu}]^2} \\tag{5.25} \\end{equation}$$\n\nwhich has a maximum \n\n$$\\begin{equation} v_i^* = -\\frac{v_m}{K_i}N(\\nu) \\ \\text{where} \\ N(\\nu) = \\frac{1}{4\\nu}(\\nu - 1)^{1 - 1/\\nu}(\\nu + 1)^{1 + 1/\\nu} \\tag{5.26} \\end{equation}$$\n\nat the inflection point \n\n$$\\begin{equation} i^* = K_i(\\frac{\\nu - 1}{\\nu + 1})^{1/\\nu} \\tag{5.27} \\end{equation}$$\n\nFor plausible values of $\\nu$, the function $N(\\nu)$ is on the order of unity (Table\u00a05.1), and hence the maximum sensitivity $v_i^*$ is on the order of $(-v_m/K_i)$. The ratio $(K_i/v_m)$ can be interpreted as a time constant characterizing the inhibition process; \n\n$$\\begin{equation} t_i = \\frac{K_i}{v_m} = [\\frac{\\text{concentration}}{\\text{concentration/time}}]\\tag{5.28} \\end{equation}$$\n\nThis estimate represents an upper bound since the steady state concentration of $i$ can be different from $i^*$. The turnover of the substrate happens on a time scale defined by the rate constant $t_s = 1/k$. Thus, a key dimensionless property is \n\n$$\\begin{equation} b = \\frac{t_s}{t_i} = \\frac{1/k}{K_i/v_m} = \\frac{v_m}{kK_i} = \\frac{e_t}{K_i} \\tag{5.29} \\end{equation}$$\n\nTherefore, the dimensionless parameter $b$ can be interpreted as a ratio of time constants or as a ratio of concentration ranges. \n\n**Table 5.1:** The values of the function $N(\\nu)$ and $i^*/K_i$ at the inflection point.\n\n\n```python\ndef N(nu): # N(v)\n return (1/(4*nu))*((nu-1)**(1-1/nu))*((nu+1)**(1+1/nu))\n\ndef i_Ki(nu): # i*/Ki\n return ((nu-1)/(nu+1))**(1/nu)\n\ncols = [nu for nu in np.linspace(2, 5, 4)]\ntab_5_1 = pd.DataFrame([[round(N(nu), 2) for nu in cols], \n [round(i_Ki(nu), 2) for nu in cols]],\n index=['N($\\\\nu$)', '$i^{*}$$/K_{i}$'], columns=cols)\ntab_5_1.index.rename('$\\\\nu$', inplace=True)\ntab_5_1\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        2.03.04.05.0
                                        $\\nu$
                                        N($\\nu$)0.650.841.071.30
                                        $i^{*}$$/K_{i}$0.580.790.880.92
                                        \n
                                        \n\n\n\n## The Symmetry Model\nThe regulatory molecules are often chemically quite different than the substrate molecule. They thus often have a different binding site on the protein molecule than the catalytic site. It called an _allosteric site_. One of the earliest enzyme kinetic models that accounted for allosterism was the symmetry model (Monod, 1965), named after certain assumed symmetry properties of the subunits of the enzyme. It is a mechanistically realistic description of regulatory enzymes. An example of a multimeric regulatory enzyme is given in Figure\u00a05.6.\n\n### The reaction mechanism for the symmetry model\nThe main chemical conversion in the symmetry model is as before and is described by Equation\u00a0(5.14). The symmetry model postulates that the regulated enzyme lies naturally in two forms, $E$ and $X$, and is converted between the two states simply as \n\n$$\\begin{equation} E \\underset{k_{-}}{\\stackrel{k_+}{\\rightleftharpoons}} X \\tag{5.30} \\end{equation}$$\n\nThe equilibrium constant for this reaction, \n\n$$\\begin{equation} L = k_+/k_- = x/e \\tag{5.31} \\end{equation}$$\n\nhas a special name, the allosteric constant. Then $\\nu$ molecules of an inhibitor, $I$, can bind sequentially to $X$ as \n\n$$\\begin{equation} \\begin{matrix} {X} & {+} & {I} & {\\underset{k_i^-}{\\stackrel{\\nu k_i^+}{\\rightleftharpoons}}} & {X_1} \\\\ {X_1} & {+} & {I} & {\\underset{2 k_i^-}{\\stackrel{(\\nu-1) k_i^+}{\\rightleftharpoons}}} & {X_2} \\\\ {\\vdots} & {} & {} & {} & {\\vdots} \\\\ {X_{\\nu - 1}} & {+} & {I} & {\\underset{\\nu k_i^-}{\\stackrel{k_i^+}{\\rightleftharpoons}}} & {X_{\\nu}} \\\\ \\end{matrix}\\end{equation} \\tag{5.32}$$\n\nwhere the binding steps have the same dissociation constant, $K_i = k_i^- / k_i^+$. We will discuss the most common case of a tetramer here, i.e., $\\nu = 4$, see Figure\u00a05.8.\n\n\n\n**Figure 5.8:** The reaction mechanisms for the symmetry model. The enzyme has four binding sites for the inhibitor.\n\n### Step 1: Dynamic mass balances for the symmetry model\nThe conversion rate of the substrate is \n\n$$\\begin{equation} v = kse \\tag{5.33} \\end{equation}$$\n\nwhereas the enzyme sequestration is characterized by the reaction rates \n\n$$\\begin{equation} \\begin{matrix} {v_1 = k^+e,} & {v_2 = k^-x,} & {v_3 = 4k_i^+xi,} \\\\ {v_4 = k_i^-x_1,} & {v_5 = 3 k_i^+x_1i,} & {v_6 = 2k_i^-x_2,} \\\\ {v_7 = k_i^+x_2i,} & {v_8 = k_i^-x_3,} & {v_9 = k_i^+x_3i,} \\\\ {} & {v_{10} = 4k_i^-x_4} & {} \\\\ \\end{matrix} \\end{equation} \\tag{5.34}$$\n\nThe dynamic mass balances on the various states of the enzyme are: \n\n$$\\begin{align} \\frac{de}{dt} &= -v_1 + v_2\\\\ \\frac{dx}{dt} &= v_1 - v_2 - v_3 + v_4\\\\ \\frac{di}{dt} &= -v_3 + v_4 - v_5 + v_6 - v_7 + v_8 - v_9 + v_{10} \\\\ \\frac{dx_1}{dt} &= v_3 - v_4 - v_5 + v_6 \\\\ \\frac{dx_2}{dt} &= v_5 - v_6 - v_7 + v_8 \\\\ \\frac{dx_3}{dt} &= v_7 - v_8 - v_9 + v_{10} \\\\ \\frac{dx_4}{dt} &= v_9 - v_{10} \\\\ \\end{align} \\tag{5.35}$$\n\n### Step 2: Finding the time invariants for the symmetry model\nThe stoichiometric matrix for $\\textbf{x} = (e, x, i, x_1, x_2, x_3, x_4)$ is a 7x10 matrix:\n\n$$\\begin{equation} \\textbf{S} = \\begin{pmatrix} {-1} & {1} & {0} & {0} & {0} & {0} & {0} & {0} & {0} & {0} \\\\ {1} & {-1} & {-1} & {1} & {0} & {0} & {0} & {0} & {0} & {0} \\\\ {0} & {0} & {-1} & {1} & {-1} & {1} & {-1} & {1} & {-1} & {1} \\\\ {0} & {0} & {1} & {-1} & {-1} & {1} & {0} & {0} & {0} & {0} \\\\ {0} & {0} & {0} & {0} & {1} & {-1} & {-1} & {1} & {0} & {0} \\\\ {0} & {0} & {0} & {0} & {0} & {0} & {1} & {-1} & {-1} & {1} \\\\ {0} & {0} & {0} & {0} & {0} & {0} & {0} & {0} & {1} & {-1} \\\\ \\end{pmatrix} \\end{equation} \\tag{5.36}$$\n\nthat has a rank of 5. Thus, there are two conservation relationships, for the enzyme: $e_0 = e + x + x_1 + x_2 + x_3 + x_4$;, and, for the inhibitor: $i_0 = i + x_1 + 2x_2 + 3x_3 + 4x_4$. If the dynamic mass balances on the substrate and product are taken into account, a third conservation $s_0 = s + p$ appears. \n\n### Step 3: Reducing the dynamic description for the symmetry model\nWe leave it to the reader to pick two dynamic variables from the full kinetic model as the dependent variables and then eliminate them from the dynamic description using the conservation relationships. The impetus for doing so algebraically becomes smaller as the number of differential equations grows. Most standard software packages will integrate a dynamically redundant set of differential equations and such substitution is not necessary to obtain the numerical solutions. \n\n### Step 4: Using simplifying kinetic assumptions to derive a rate law for the symmetry model\nThe serial binding of an inhibitor to X that has four binding sites is shown in Figure\u00a05.8. The derivation of the rate law is comprised of four basic steps: \n\n1. Mass balance on enzyme: \n\n$$\\begin{equation} e_0 = e + x + x_1 + x_2 + x_3 + x_4 \\tag{5.37} \\end{equation}$$\n\n2. QEA for binding steps: \n\n$$\\begin{align} 4k_i^+ix = k_i^-x_1 \\ &\\Rightarrow \\ x_1 = \\frac{4}{1}x (i/K_i) = 4x(i/K_i) \\\\ 3k_i^+ix_1 = 2k_i^-x_2 \\ &\\Rightarrow \\ x_2 = \\frac{3}{2}x_1(i/K_i) = 6x(i/K_i)^2 \\\\ 2k_i^+ix_2 = 3k_i^-x_3 \\ &\\Rightarrow \\ x_3 = \\frac{2}{3}x_2(i/K_i) = 4x(i/K_i)^3 \\\\ k_i^+ix_3 = 4k_i^-x_4 \\ &\\Rightarrow \\ x_4 = \\frac{1}{4}x_3(i/K_i) = x(i/K_i)^4 \\\\ \\end{align} \\tag{5.38}$$\n\n3. Combine 1 and 2: \n\n$$\\begin{align} e_0 &= e + x + 4x(i/K_i) + 6x(i/K_i)^2 + 4x(i/K_i)^3 + x(i/K_i)^4 \\\\ &= e + x(1 + (i/K_i))^4 \\ \\text{where} \\ x=Le \\\\ &= e(1 + L(1 + (i/K_i)))^4 \\end{align} \\tag{5.39}$$\n\n4. Form the rate law: The reaction rate is given by: $v = kse$. We can rewrite the last part of Eq. (5.39) as: \n\n$$\\begin{equation} e = \\frac{e_0}{1 + L(1 + i/K_i)^4} \\tag{5.40} \\end{equation}$$\n\nleading to the rate law: \n\n$$\\begin{equation} v(s, i) = \\frac{ke_0s}{1 + L(1 + i/K_i)^4} \\tag{5.41} \\end{equation}$$\n\nThis rate law generalizes to: \n\n$$\\begin{equation} v(s, i) = \\frac{ke_0s}{1 + L(1 + i/K_i)^{\\nu}} = \\frac{v_m}{1 + L(1 + i/K_i)^{\\nu}} \\tag{5.42} \\end{equation}$$\n\nfor any $\\nu$. The reader can find the same key dimensionless groups as for the Hill rate law. Note again the fraction \n\n$$\\begin{equation} f_e = \\frac{e}{e_0} = \\frac{1}{1 + L(1 + i/K_i)^{\\nu}} \\tag{5.43} \\end{equation}$$\n\nthat describes the what fraction of the enzyme is in the catalytically active state. \n\n\n```python\nt0 = 0\ntf = 15\n\ndef symmetry(t, state_vars, *params):\n s, p, e, i, x, x1, x2, x3, x4 = state_vars\n k1, k_plus, k_minus, ki_plus, ki_minus = params\n # Enzyme Reaction Rates\n v1 = k_plus * e; v2 = k_minus * x;\n v3 = 4 * ki_plus * i * x; v4 = ki_minus * x1;\n v5 = 3 * ki_plus * i * x1; v6 = 2 * ki_minus * x2;\n v7 = 2 * ki_plus * i * x2; v8 = 3 * ki_minus * x3;\n v9 = ki_plus * i * x3; v10 = 4 * ki_minus * x4;\n # Differential equations to integrate\n diffeqs = [-k1 * s * e, # ds/dt\n k1 * s * e, # dp/dt\n -v1 + v2, # de/dt\n -v3 + v4 - v5 + v6 - v7 + v8 - v9 + v10, # di/dt\n v1 - v2 - v3 + v4, # dx/dt\n v3 - v4 - v5 + v6, # dx1/dt\n v5 - v6 - v7 + v8, # dx2/dt\n v7 - v8 - v9 + v10, # dx3/dt\n v9 - v10] # dx4/dt\n return diffeqs\n\n# Define initial conditions\ns0, p0, e0, i0, x0 = (1, 0, 1, 1, 0)\nx1_0, x2_0, x3_0, x4_0 = (0, 0, 0, 0)\n\n# Define paramters\nk1 = 1; \nk_plus, k_minus = (100, 100)\nki_plus, ki_minus = (2, 2)\nparams = [k1, k_plus,k_minus, ki_plus, ki_minus]\n\n# Obtain numerical solutions\nsol_obj = solve_ivp(fun=lambda t, state_vars: symmetry(t, state_vars, *params),\n t_span=(t0, tf), y0=[s0, p0, e0, i0, x0, x1_0, x2_0, x3_0, x4_0])\n# Store solutions into Solution Objects\nsol_dict = dict(zip([\"s\", \"p\", \"e\", \"i\", \"x\", \"x1\", \"x2\", \"x3\", \"x4\"], sol_obj.y))\n\nx_total = sum(sol_dict[k] for k in [\"x\", \"x1\", \"x2\", \"x3\", \"x4\"])\ni_bound = sum(i*sol_dict[k] for i, k in zip([1, 2, 3, 4], [\"x1\", \"x2\", \"x3\", \"x4\"]))\n\nsol_dict.update({\"x_total\": x_total, \"i_bound\": i_bound})\n\nsymmetry_sol = MassSolution(\n \"Symmetry\", solution_type=\"Conc\", data_dict=sol_dict,\n time=sol_obj.t, interpolate=False)\n```\n\n\n```python\nfig_5_9 = plt.figure(figsize=(10, 8))\ngs = fig_5_9.add_gridspec(nrows=2, ncols=2, width_ratios=[1, 1],\n height_ratios=[1, 1])\n\nax1 = fig_5_9.add_subplot(gs[0, 0])\nax2 = fig_5_9.add_subplot(gs[0, 1])\nax3 = fig_5_9.add_subplot(gs[1, 0])\nax4 = fig_5_9.add_subplot(gs[1, 1])\n\nplot_phase_portrait(\n symmetry_sol, x=\"s\", y=\"e\", ax=ax1, xlabel=\"s\", ylabel=\"e\",\n xlim=(-0.05, 1.05), ylim=(-0.05, 1.05),\n title=(\"(a) Phase Portrait of s vs. e\", {\"size\": \"x-large\"}),\n annotate_time_points=\"endpoints\",\n annotate_time_points_labels=True);\n\nplot_phase_portrait(\n symmetry_sol, x=\"e\", y=\"x_total\", ax=ax2, \n xlabel=\"e\", ylabel='x + x1 + x2 + x3 + x4',\n xlim=(-0.05, 1.05), ylim=(-0.05, 1.05),\n title=(\"(b) Phase Portrait of e vs. x_total\", {\"size\": \"x-large\"}),\n annotate_time_points=\"endpoints\",\n annotate_time_points_labels=True);\n\nplot_phase_portrait(\n symmetry_sol, x=\"i\", y=\"i_bound\", ax=ax3, \n xlabel=\"i\", ylabel='1*x1 + 2*x2 + 3*x3 + 4*x4',\n xlim=(-0.05, 1.05), ylim=(-0.05, 1.05),\n title=(\"(a) Phase Portrait of i vs. x\", {\"size\": \"x-large\"}),\n annotate_time_points=\"endpoints\",\n annotate_time_points_labels=True);\n\nplot_time_profile(\n symmetry_sol, observable=list(\n k for k in symmetry_sol.keys() if k not in [\n \"x\", \"x1\", \"x2\", \"x3\", \"x4\"]),\n ax=ax4, legend=\"right outside\",\n title=(\"(d) Concentration Profiles\", {\"size\": \"x-large\"}));\nfig_5_9.tight_layout()\n```\n\n**Figure 5.9:** The transient response of the symmetry model, for $k^+ = k^- = 100$, $k_i^+ = k_i^- = 2$, $k = 1$, $\\nu = 4$, $x_0 = x_{1, 0} = x_{2, 0} = x_{3, 0} = x_{4, 0} = 0$ and $e_0 = s_0 = i_0 = 1$. (a) The phase portraits of $s$ and $e$. (b) The phase portraits of $e$ and $(x + x_1 + x_2 + x_3 + x_4)$. (c) The phase portraits of $i$ and $(x_1 + 2x_2 + 3x_3 + 4x_4)$. (d) Concentration and pool profiles.\n\n### Step 5: Numerical solutions for the symmetry model\nThese equations can be simulated. Typically the conformational changes between $E$ and $X$ are fast as are the inhibitor binding steps relative to the catalysis rate. Numerical simulations were carried out for this situation and the results are plotted in Figure\u00a05.9. \n\n* Figure 5.9a shows how the substrate-enzyme phase portrait is L-shaped showing that the sequestration of the enzyme in the inhibited form (the vertical line) is faster than the conversion of the substrate (the horizontal line). \n\n* Figure 5.9b shows the redistribution of the total enzyme among the active and inactive forms, that is, $e$ vs. $(x + x_1 + x_2 + x_3 + x_4)$. The fraction of the enzyme in the inactive form is about 0.29. \n\n* Figure 5.9c shows the redistribution of the inhibitor between the free and bound form; $i$ vs. $(x_1 + 2x_2 + 3x_3 + 4x_4)$. This panel shows that the fraction the inhibitor that is bound is high, 0.70. \n\n* Finally, Figure 5.9d show the transient changes in the concentrations and pools on the fast and slow time scales. Note that two natural aggregation variables appear: the total enzyme in the inactive form, and the total number of inhibitor molecules bound to the enzyme. \n\n## Scaling Dynamic Descriptions \nThe analysis of simple equations requires the \"proper frame of mind.\" In step 6 of the process of formulating rate laws, this notion is translated into quantitative measures. We need to scale the variables with respect to intrinsic reference scales and thereby cast our mathematical descriptions into appropriate coordinate systems. All parameters then aggregate into dimensionless property ratios that, if properly interpreted, have a clear physical significance. \n\n### The scaling process: \nThe examples above illustrate the decisive role of time constants and their use to analyze simple situations and to elucidate intrinsic reference scales. Identification of unimportant terms is sometimes more difficult and familiarity with a formal scaling procedure is useful. This procedure basically consists of four steps: \n\n1. Identify logical reference scales. This step is perhaps the most difficult. It relies partly on physical intuition, and the use of time constants is surprisingly powerful even when analyzing steady situations. \n\n2. Introduce reference scales into the equations and make the variables dimensionless. \n\n3. Collect the parameters into dimensionless property ratios. The number of dimensionless parameters is always the same and it is given by the well-known Buckingham Pi theorem. \n \n4. Interpret the results. The dimensionless groups that appear can normally be interpreted as ratios of the time constants, such as those discussed above. \n\nScaling of equations is typically only practiced for small models and for analysis purposes only. Numerical simulations of complex models are essentially always performed with absolute values of the variables. \n\n### The importance of intrinsic reference scales \nThe process by which the equations are made dimensionless is not unique. The 'correct' way of putting the equations into a dimensionless form, where judgments of relative orders of magnitude can be made, is called _scaling_. The scaling process is defined by Lin and Segel (Segel, 1974) as: \n\n\"...select intrinsic reference quantities so that each term in the dimensional equations transforms into a product of a constant dimensional factor which closely estimates the term's order of magnitude and a dimensionless factor of unit order of magnitude.\"\n\nIn other words, if one has an equation which is a sum of terms $T_i$ as: \n\n$$\\begin{equation} T_1 + T_2 + \\dots = 0 \\tag{5.44} \\end{equation}$$\n\none tries to scale the _variables_ involved so that they are of unit order of magnitude or \n\n$$\\begin{equation} t_i = \\frac{\\text{variable}_i}{\\text{intrinistic reference scale}_i} \\approx \\text{unit order of magnitude} \\tag{5.45} \\end{equation}$$\n\nIntroducing these dimensionless variables into equation\u00a0(5.44) results in the dimensionless form: \n\n$$\\begin{equation} \\pi_1 t_1 + \\pi_2 t_2 + \\dots = 0 \\tag{5.44} \\end{equation}$$\n\nwhere the dimensionless multipliers, $\\pi_i$ are the dimensionless groups and they will indicate the order of magnitude of the product, $\\pi_it_i$. Once the equations are in this form, order of magnitude judgements can be made based on the dimensionless groups. \n\n## Summary \n\n* Enzymes are highly specialized catalysts that can dramatically accelerate the rates of biochemical reactions. \n\n* Reaction mechanisms are formulated for the chemical conversions carried out by enzymes in terms of elementary reactions. \n\n* Rate laws for enzyme reaction mechanisms are derived based on simplifying assumptions. \n\n* Two simplifying assumptions are commonly used: the quasi-steady state (QSSA) and the quasi-equilibrium assumptions (QEA). \n\n* The validity of the simplifying assumptions can be determined using scaling of the equations followed by mathematical and numerical analysis. \n\n* A number of rate laws have been developed for enzyme catalysis and for the regulation of enzymes. Only three reaction mechanisms were described in this chapter. \n\n$\\tiny{\\text{\u00a9 B. \u00d8. Palsson 2011;}\\ \\text{This publication is in copyright.}\\\\ \\text{Subject to statutory exception and to the provisions of relevant collective licensing agreements,}\\\\ \\text{no reproduction of any part may take place without the written permission of Cambridge University Press.}}$\n", "meta": {"hexsha": "0249b19399ceb7c9d9d7a502dc9a74b7d3f4343c", "size": 275440, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/education/sb2/chapters/sb2_chapter5.ipynb", "max_stars_repo_name": "SBRG/MASSpy", "max_stars_repo_head_hexsha": "1315c1d40be8feb8731c8143dbc9ba43bf8c78ff", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 13, "max_stars_repo_stars_event_min_datetime": "2020-07-13T00:48:13.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-27T15:42:15.000Z", "max_issues_repo_path": "docs/education/sb2/chapters/sb2_chapter5.ipynb", "max_issues_repo_name": "SBRG/MASSpy", "max_issues_repo_head_hexsha": "1315c1d40be8feb8731c8143dbc9ba43bf8c78ff", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 10, "max_issues_repo_issues_event_min_datetime": "2021-02-17T18:07:43.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-23T16:22:14.000Z", "max_forks_repo_path": "docs/education/sb2/chapters/sb2_chapter5.ipynb", "max_forks_repo_name": "SBRG/MASSpy", "max_forks_repo_head_hexsha": "1315c1d40be8feb8731c8143dbc9ba43bf8c78ff", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2020-01-15T00:48:49.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-04T07:01:17.000Z", "avg_line_length": 236.4291845494, "max_line_length": 62248, "alphanum_fraction": 0.8804131571, "converted": true, "num_tokens": 14611, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.7057850278370111, "lm_q1q2_score": 0.4156291315518876}} {"text": "# Lab Assignment 1\n\n\n## Important notes\n\n**Submission deadline:**\n* **Regular problems: last lab session before or on Monday, 21.10.19**\n* **Bonus problems: deadline for Lab Assignment 2**\n\n**Points: 11 + 5 bonus points**\n\nPlease note: some of the assignments are tedious or boring if you are already a NumPy ninja. The bonus problems were designed to give you a more satisfying alternative.\n\nThe assignment is in the form of a Jupyter notebook. We will be using [Google Colab](https://colab.research.google.com) to solve it. Below you will find a \"Setup\" section. Follow instructions from this paragraph to download the notebook and open it using [Google Colab](https://colab.research.google.com). \n\nYour goal is to solve problems posted below. Whenever possible, add your solutions to the notebook.\n\nPlease email us about any problems with it - we will try to correct them quickly. Also, please do not hesitate to use GitHub\u00e2\u20ac\u2122s pull requests to send us corrections!\n\n## Setup\n\n### 1. Open the notebook using Google Colab\n\n1. From Github: Click on \"View in Colaboratory\", then save to your Google Drive.\n2. Alternatively upload manually to Drive:\n 1. Download the notebook or clone https://github.com/janchorowski/ml_uwr.\n 2. Go to [Google Colab](https://colab.research.google.com).\n 3. Go to \"UPLOAD\" tab and select a local copy of the notebook that you downloaded in point 1.\n \nColab Tips:\n1. Set tab width to 4 spaces under `Tools` \u00e2\u2020\u2019 `Preferences`.\n \n### 2. Open the notebook offline using Jupyter/IPython\n\nThis notebook can be opened using Jupyter notebook. Simply install a scientific Python distribution on your computer (e.g. [Anaconda](https://www.anaconda.com/) or [WinPython](http://winpython.github.io/)), clone the repository https://github.com/janchorowski/nn_assignments and run `jupyter notebook`.\n\n### 3. Install required dependencies, download data and import packages\n\nRun cells below. To run a cell either click it and click a run button or press \"shift + enter\"\n\n\n\n```python\n# Please note that this code needs only to be run in a fresh runtime.\n# However, it can be rerun afterwards too.\n!pip install -q gdown httpimport\n![ -e mnist.npz ] || gdown 'https://drive.google.com/uc?id=1QPaC3IKB_5tX6yIZgRgkpcqFrfVqPTXU' -O mnist.npz\n```\n\n Downloading...\n From: https://drive.google.com/uc?id=1QPaC3IKB_5tX6yIZgRgkpcqFrfVqPTXU\n To: /content/mnist.npz\n 55.4MB [00:00, 134MB/s] \n\n\n\n```python\n# Standard IPython notebook imports\n%matplotlib inline\n\nimport os\n\nimport httpimport\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport pandas as pd\nfrom tqdm import tqdm_notebook\nimport scipy.stats as sstats\nimport math\nimport cmath\nimport random\n\nimport seaborn as sns\nfrom sklearn import datasets\n\n# In this way we can import functions straight from github\nwith httpimport.github_repo('janchorowski', 'nn_assignments', \n module='common', branch='nn18'):\n from common.plotting import plot_mat\n\nsns.set_style('whitegrid')\n```\n\n### 4. Follow the notebook and solve problems posted below\n\n## Problems\n\n### Problem 0 [0p]\n\n \n1. To learn more about Jupyter, read [Jupyter tutorial from Data Analysis in Biological Sciences course at Caltech](http://bebi103.caltech.edu/2015/tutorials/t0b_intro_to_jupyter_notebooks.html) (which itself can be downloaded as a Jupyter notebook). Feel free to skip the tutorial if you have some prior experience with Jupyter notebook.\n2. To learn more about basic Google Colab features, go to [Google Colab](https://colab.research.google.com) and select \"Overview of Colaboratory Features\" in \"EXAMPLES\" tab. To learn more about / set up useful keyboard shortcuts (e.g. to add a new cell without clicking \"\"+ code\"), go to \"Tools --> Keyboard shortcuts\"\n\n### Problem 1: NumPy [2p]\n\nFirst, get familiar with Python at https://docs.python.org/3/tutorial/. Then, get\nto know the capabilities of NumPy, the prime numerical library of Python http://www.numpy.org/, for instance with the tutorial at http://wiki.scipy.org/Tentative_NumPy_Tutorial. Finally, look into Pandas at https://pandas.pydata.org/pandas-docs/stable/getting_started/tutorials.html.\n\nYou might also need:\n 1. another intro to NumPy,\nhttp://people.duke.edu/~ccc14/pcfb/numerics.html\n 2. a better interactive shell for Python\nhttp://ipython.org/\n 3. a plotting library for Python\nhttp://matplotlib.org/\n 4. nice statistical plots for matplotlib https://seaborn.pydata.org/.\n\n\n**a) Declare variables:**\n1. $a=10$,\n2. $b=2.5\\times 10^{23}$,\n3. $c=2+3i$, where $i$ is an imaginary unit,\n4. $d=e^{i2\\pi/3}$, where $i$ is an imaginary unit, $e$ is the Euler's number (use `exp`, `pi`).\n\n\n```python\na = 10\nb = 2.5 * 10 ** 23\nc = 2 + 3j\nd = cmath.exp((1j * 2 * math.pi) / 3)\n\na,b,c,d\n```\n\n\n\n\n (10, 2.5e+23, (2+3j), (-0.4999999999999998+0.8660254037844387j))\n\n\n\n**b) Declare vectors:**\n1. $aVec=\\begin{bmatrix} 3.14 & 15 & 9 & 26 \\end{bmatrix}$,\n2. $bVec=\\begin{bmatrix} 5 & 4.8 & \\cdots & -4.8 & -5 \\end{bmatrix}$ (vector of numbers from $5$ to $-5$ decreasing by $0.2$),\n3. $cVec=\\begin{bmatrix} 10^0 & 10^{0.01} & \\cdots & 10^{0.99} & 10^1 \\end{bmatrix}$ (logarithmically spaced numbers from 1 to 10, use `logspace` and make sure, that the result has correct length!),\n4. $dVec=Hello$ ($eVec$ is a string of characters, thus a vector).\n\n\n```python\naVec = np.array([3.14, 15, 9, 26])\nbVec = np.array([x for x in np.arange(5, -5.2, -0.2)])\ncVec = np.logspace(0, 1, num=100)\n\ndVec = np.array(['Hello'])\ndVec2 = np.array(['H', 'e', 'l', 'l', 'o'])\n\n\naVec, bVec, cVec, dVec, dVec2\n```\n\n\n\n\n (array([ 3.14, 15. , 9. , 26. ]),\n array([ 5.0000000e+00, 4.8000000e+00, 4.6000000e+00, 4.4000000e+00,\n 4.2000000e+00, 4.0000000e+00, 3.8000000e+00, 3.6000000e+00,\n 3.4000000e+00, 3.2000000e+00, 3.0000000e+00, 2.8000000e+00,\n 2.6000000e+00, 2.4000000e+00, 2.2000000e+00, 2.0000000e+00,\n 1.8000000e+00, 1.6000000e+00, 1.4000000e+00, 1.2000000e+00,\n 1.0000000e+00, 8.0000000e-01, 6.0000000e-01, 4.0000000e-01,\n 2.0000000e-01, -4.4408921e-15, -2.0000000e-01, -4.0000000e-01,\n -6.0000000e-01, -8.0000000e-01, -1.0000000e+00, -1.2000000e+00,\n -1.4000000e+00, -1.6000000e+00, -1.8000000e+00, -2.0000000e+00,\n -2.2000000e+00, -2.4000000e+00, -2.6000000e+00, -2.8000000e+00,\n -3.0000000e+00, -3.2000000e+00, -3.4000000e+00, -3.6000000e+00,\n -3.8000000e+00, -4.0000000e+00, -4.2000000e+00, -4.4000000e+00,\n -4.6000000e+00, -4.8000000e+00, -5.0000000e+00]),\n array([ 1. , 1.02353102, 1.04761575, 1.07226722, 1.09749877,\n 1.12332403, 1.149757 , 1.17681195, 1.20450354, 1.23284674,\n 1.26185688, 1.29154967, 1.32194115, 1.35304777, 1.38488637,\n 1.41747416, 1.45082878, 1.48496826, 1.51991108, 1.55567614,\n 1.59228279, 1.62975083, 1.66810054, 1.70735265, 1.7475284 ,\n 1.78864953, 1.83073828, 1.87381742, 1.91791026, 1.96304065,\n 2.009233 , 2.05651231, 2.10490414, 2.15443469, 2.20513074,\n 2.25701972, 2.3101297 , 2.36448941, 2.42012826, 2.47707636,\n 2.53536449, 2.59502421, 2.65608778, 2.71858824, 2.7825594 ,\n 2.84803587, 2.91505306, 2.98364724, 3.05385551, 3.12571585,\n 3.19926714, 3.27454916, 3.35160265, 3.43046929, 3.51119173,\n 3.59381366, 3.67837977, 3.76493581, 3.85352859, 3.94420606,\n 4.03701726, 4.1320124 , 4.22924287, 4.32876128, 4.43062146,\n 4.53487851, 4.64158883, 4.75081016, 4.86260158, 4.97702356,\n 5.09413801, 5.21400829, 5.33669923, 5.46227722, 5.59081018,\n 5.72236766, 5.85702082, 5.9948425 , 6.13590727, 6.28029144,\n 6.42807312, 6.57933225, 6.73415066, 6.8926121 , 7.05480231,\n 7.22080902, 7.39072203, 7.56463328, 7.74263683, 7.92482898,\n 8.11130831, 8.30217568, 8.49753436, 8.69749003, 8.90215085,\n 9.11162756, 9.32603347, 9.54548457, 9.77009957, 10. ]),\n array(['Hello'], dtype='\nmatrix $9\\times 9$ filled with 2s (use `ones` or `zeros`),\n2. $bMat=\\begin{bmatrix}\n 1 & 0 & \\cdots & & 0 \\\\\n 0 & \\ddots & 0 & & 0 \\\\\n \\vdots & 0 & 5 & 0 & \\vdots \\\\\n & & 0 & \\ddots & 0 \\\\\n 0 & & \\cdots & 0 & 1\n \\end{bmatrix}$,\n
                                        \nmatrix $9\\times 9$ filled with zeros, with $\\begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 4 & 3 & 2 & 1 \\end{bmatrix}$ on its diagonal (use `zeros`, `diag`),\n3. $cMat=\\begin{bmatrix}\n 1 & 11 & \\cdots & 91 \\\\\n 2 & 12 & \\cdots & 92 \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n 10 & 20 & \\cdots & 100\n \\end{bmatrix}$,\n
                                        \nmatrix $10\\times 10$, columns of which form the vector $1:100$ (use `reshape`),\n4. $dMat=\\begin{bmatrix}\n NaN & NaN & NaN & NaN \\\\\n NaN & NaN & NaN & NaN \\\\\n NaN & NaN & NaN & NaN\n \\end{bmatrix}$,\n
                                        \nmatrix $3\\times 4$ filled with `NaN`s (use... `NaN`),\n5. $eMat=\\begin{bmatrix}\n 13 & -1 & 5 \\\\\n -22 & 10 & -87\n \\end{bmatrix}$,\n6. $fMat$ of shape $3\\times 3$ filled with random natural numbers from $[-3,3]$ (use `rand` and `floor` or `ceil`).\n\n\n```python\naMat = np.zeros((9, 9)) + 2\n\nbMat = np.zeros((9, 9))\nnp.fill_diagonal(bMat,[i for i in range(1,6)] + [i for i in range(4,0,-1)])\n# a = (9 + 1)//2\n# bMat[range(a), range(a)] = range(1,a + 1)\n# bMat[range(a, 9), range(a, 9)] = range(a-1,0,-1)\n\ncMat = np.arange(1,101).reshape(10,10).T\ndMat = np.empty((3,4))\ndMat[:] = np.nan\n\neMat = np.array([13, -1, 5, -22, 10, -87]).reshape(2,3)\n\nfMat = np.array([random.randint(-3,3) for _ in range(9)]).reshape(3,3)\n\naMat, bMat, cMat, dMat, eMat, fMat\n```\n\n\n\n\n (array([[2., 2., 2., 2., 2., 2., 2., 2., 2.],\n [2., 2., 2., 2., 2., 2., 2., 2., 2.],\n [2., 2., 2., 2., 2., 2., 2., 2., 2.],\n [2., 2., 2., 2., 2., 2., 2., 2., 2.],\n [2., 2., 2., 2., 2., 2., 2., 2., 2.],\n [2., 2., 2., 2., 2., 2., 2., 2., 2.],\n [2., 2., 2., 2., 2., 2., 2., 2., 2.],\n [2., 2., 2., 2., 2., 2., 2., 2., 2.],\n [2., 2., 2., 2., 2., 2., 2., 2., 2.]]),\n array([[1., 0., 0., 0., 0., 0., 0., 0., 0.],\n [0., 2., 0., 0., 0., 0., 0., 0., 0.],\n [0., 0., 3., 0., 0., 0., 0., 0., 0.],\n [0., 0., 0., 4., 0., 0., 0., 0., 0.],\n [0., 0., 0., 0., 5., 0., 0., 0., 0.],\n [0., 0., 0., 0., 0., 4., 0., 0., 0.],\n [0., 0., 0., 0., 0., 0., 3., 0., 0.],\n [0., 0., 0., 0., 0., 0., 0., 2., 0.],\n [0., 0., 0., 0., 0., 0., 0., 0., 1.]]),\n array([[ 1, 11, 21, 31, 41, 51, 61, 71, 81, 91],\n [ 2, 12, 22, 32, 42, 52, 62, 72, 82, 92],\n [ 3, 13, 23, 33, 43, 53, 63, 73, 83, 93],\n [ 4, 14, 24, 34, 44, 54, 64, 74, 84, 94],\n [ 5, 15, 25, 35, 45, 55, 65, 75, 85, 95],\n [ 6, 16, 26, 36, 46, 56, 66, 76, 86, 96],\n [ 7, 17, 27, 37, 47, 57, 67, 77, 87, 97],\n [ 8, 18, 28, 38, 48, 58, 68, 78, 88, 98],\n [ 9, 19, 29, 39, 49, 59, 69, 79, 89, 99],\n [ 10, 20, 30, 40, 50, 60, 70, 80, 90, 100]]),\n array([[nan, nan, nan, nan],\n [nan, nan, nan, nan],\n [nan, nan, nan, nan]]),\n array([[ 13, -1, 5],\n [-22, 10, -87]]),\n array([[ 0, 0, 0],\n [-2, 2, -3],\n [-1, -3, -1]]))\n\n\n\n**d) Declare a multiplication table**\nas a $10\\times 10$ matrix `mulMat`. Use matrix/vector multiplication.\n\n\n```python\na = np.array([i for i in range(1,11)])[np.newaxis]\nb = a.copy().T\nmulMat = b.dot(a)\nmulMat\n```\n\n\n\n\n array([[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10],\n [ 2, 4, 6, 8, 10, 12, 14, 16, 18, 20],\n [ 3, 6, 9, 12, 15, 18, 21, 24, 27, 30],\n [ 4, 8, 12, 16, 20, 24, 28, 32, 36, 40],\n [ 5, 10, 15, 20, 25, 30, 35, 40, 45, 50],\n [ 6, 12, 18, 24, 30, 36, 42, 48, 54, 60],\n [ 7, 14, 21, 28, 35, 42, 49, 56, 63, 70],\n [ 8, 16, 24, 32, 40, 48, 56, 64, 72, 80],\n [ 9, 18, 27, 36, 45, 54, 63, 72, 81, 90],\n [ 10, 20, 30, 40, 50, 60, 70, 80, 90, 100]])\n\n\n\n**e) Compute element-wise using values from b).**\n\nFor instance, the first element of $xVec[0]$ should be equal to\n\n\\begin{equation}\n1/(\\sqrt{2\\pi2.5^2}) e^{-cVec[0]^2 / (2\\cdot\\pi 2.5^2)}.\n\\end{equation}\n\n1. $xVec=1/(\\sqrt{2\\pi2.5^2}) e^{-cVec^2 / (2\\cdot\\pi 2.5^2)}$\n2. $yVec=\\log_{10}(1/cVec)$, using `log10`\n\n\n```python\nxVec = np.array([1 / math.sqrt(2 * math.pi * 2.5 ** 2) * math.exp(-(x ** 2) / (2 * math.pi * 2.5 ** 2)) for x in cVec])\nyVec = np.array([math.log10(1/x) for x in cVec])\n\nxVec, yVec\n```\n\n\n\n\n (array([0.15556462, 0.15537611, 0.15517887, 0.1549725 , 0.1547566 ,\n 0.15453075, 0.15429449, 0.15404737, 0.15378891, 0.15351861,\n 0.15323595, 0.15294038, 0.15263135, 0.15230828, 0.15197056,\n 0.15161756, 0.15124863, 0.1508631 , 0.15046027, 0.1500394 ,\n 0.14959976, 0.14914057, 0.14866103, 0.1481603 , 0.14763754,\n 0.14709187, 0.14652238, 0.14592813, 0.14530818, 0.14466153,\n 0.14398717, 0.14328408, 0.14255119, 0.14178742, 0.14099168,\n 0.14016284, 0.13929975, 0.13840127, 0.13746622, 0.13649342,\n 0.13548169, 0.13442982, 0.13333663, 0.13220091, 0.13102149,\n 0.1297972 , 0.12852688, 0.12720941, 0.12584368, 0.12442865,\n 0.12296331, 0.12144669, 0.11987792, 0.11825618, 0.11658075,\n 0.11485099, 0.1130664 , 0.11122656, 0.10933122, 0.10738026,\n 0.10537373, 0.10331187, 0.10119508, 0.09902401, 0.0967995 ,\n 0.09452265, 0.0921948 , 0.08981757, 0.08739287, 0.08492289,\n 0.08241014, 0.07985745, 0.07726798, 0.07464522, 0.07199301,\n 0.06931553, 0.06661729, 0.06390316, 0.06117833, 0.05844828,\n 0.0557188 , 0.05299598, 0.0502861 , 0.04759569, 0.04493142,\n 0.04230011, 0.03970863, 0.03716387, 0.03467267, 0.03224177,\n 0.02987771, 0.02758678, 0.02537494, 0.02324774, 0.02121026,\n 0.01926701, 0.01742191, 0.01567817, 0.01403829, 0.01250398]),\n array([ 0. , -0.01010101, -0.02020202, -0.03030303, -0.04040404,\n -0.05050505, -0.06060606, -0.07070707, -0.08080808, -0.09090909,\n -0.1010101 , -0.11111111, -0.12121212, -0.13131313, -0.14141414,\n -0.15151515, -0.16161616, -0.17171717, -0.18181818, -0.19191919,\n -0.2020202 , -0.21212121, -0.22222222, -0.23232323, -0.24242424,\n -0.25252525, -0.26262626, -0.27272727, -0.28282828, -0.29292929,\n -0.3030303 , -0.31313131, -0.32323232, -0.33333333, -0.34343434,\n -0.35353535, -0.36363636, -0.37373737, -0.38383838, -0.39393939,\n -0.4040404 , -0.41414141, -0.42424242, -0.43434343, -0.44444444,\n -0.45454545, -0.46464646, -0.47474747, -0.48484848, -0.49494949,\n -0.50505051, -0.51515152, -0.52525253, -0.53535354, -0.54545455,\n -0.55555556, -0.56565657, -0.57575758, -0.58585859, -0.5959596 ,\n -0.60606061, -0.61616162, -0.62626263, -0.63636364, -0.64646465,\n -0.65656566, -0.66666667, -0.67676768, -0.68686869, -0.6969697 ,\n -0.70707071, -0.71717172, -0.72727273, -0.73737374, -0.74747475,\n -0.75757576, -0.76767677, -0.77777778, -0.78787879, -0.7979798 ,\n -0.80808081, -0.81818182, -0.82828283, -0.83838384, -0.84848485,\n -0.85858586, -0.86868687, -0.87878788, -0.88888889, -0.8989899 ,\n -0.90909091, -0.91919192, -0.92929293, -0.93939394, -0.94949495,\n -0.95959596, -0.96969697, -0.97979798, -0.98989899, -1. ]))\n\n\n\n**f) Compute with matrix/vector operations.**\n\n**NOTE:** Every multiplication (and power) in this subtask is a [matrix multiplication](https://en.wikipedia.org/wiki/Matrix_multiplication).\n1. $xMat=[0, 1, ..., 6][0, 10, 20, ..., 60]^T$,\n2. $yMat=[0, 10, 20, ..., 60]^T[0, 1, ..., 6]$\n
                                        \n(remember, that matrix multiplication is not commutative).\n\n\n```python\na = np.array([i for i in range(7)])[np.newaxis]\nb = np.array([i for i in range(0,70,10)])[np.newaxis].T\nxMat = (a).dot(b) \n\nyMat = b.dot(a)\n\nxMat, yMat\n```\n\n\n\n\n (array([[910]]), array([[ 0, 0, 0, 0, 0, 0, 0],\n [ 0, 10, 20, 30, 40, 50, 60],\n [ 0, 20, 40, 60, 80, 100, 120],\n [ 0, 30, 60, 90, 120, 150, 180],\n [ 0, 40, 80, 120, 160, 200, 240],\n [ 0, 50, 100, 150, 200, 250, 300],\n [ 0, 60, 120, 180, 240, 300, 360]]))\n\n\n\n**g) Declare `ismagic(A)` function** \nwhich checks if matrix $A$ is a [magic square](https://en.wikipedia.org/wiki/Magic_square) and returns a boolean.\n\n\n```python\ndef ismagic(A):\n b1 = len(np.unique(A)) == np.prod(A.shape)\n \n row_sums = np.sum(A, axis=1)\n b2 = all(x == row_sums[0] for x in row_sums)\n \n column_sums = np.sum(A, axis=0)\n b3 = all(x == column_sums[0] for x in column_sums)\n \n return b1 and b2 and b3\n \n \nassert not ismagic(np.array([[1,1], [2,2]]))\nassert ismagic(np.array([[2,7,6],[9,5,1],[4,3,8]]))\n```\n\n### Problem 2: Pandas and Seaborn [2p]\n\n1. Load the IRIS Data into a `DataFrame`\n\n\n```python\niris_url = 'https://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data'\n# Use read_csv to load the data. Make sure you get 150 examples!\niris_df = pd.read_csv(iris_url)\n\n# Set the column names to\n# 'sepal_length', 'sepal_width', 'petal_length', 'petal_width', 'target'\niris_df.columns = ['sepal_length', 'sepal_width', 'petal_length', 'petal_width', 'target']\n\n# Print the first 10 entries\niris_df.sample(10)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        sepal_lengthsepal_widthpetal_lengthpetal_widthtarget
                                        576.62.94.61.3Iris-versicolor
                                        595.02.03.51.0Iris-versicolor
                                        1026.32.95.61.8Iris-virginica
                                        925.02.33.31.0Iris-versicolor
                                        245.03.01.60.2Iris-setosa
                                        284.73.21.60.2Iris-setosa
                                        1036.53.05.82.2Iris-virginica
                                        214.63.61.00.2Iris-setosa
                                        255.03.41.60.4Iris-setosa
                                        104.83.41.60.2Iris-setosa
                                        \n
                                        \n\n\n\n\n```python\n# Show numerical summary of the data, using DataFrame.describe()\niris_df.describe()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        sepal_lengthsepal_widthpetal_lengthpetal_width
                                        count149.000000149.000000149.000000149.000000
                                        mean5.8483223.0510073.7744971.205369
                                        std0.8285940.4334991.7596510.761292
                                        min4.3000002.0000001.0000000.100000
                                        25%5.1000002.8000001.6000000.300000
                                        50%5.8000003.0000004.4000001.300000
                                        75%6.4000003.3000005.1000001.800000
                                        max7.9000004.4000006.9000002.500000
                                        \n
                                        \n\n\n\n\n```python\n# Plot the data using seaborn's pairplot\niris = sns.load_dataset(\"iris\")\nsns.pairplot(iris, hue=\"species\")\n```\n\nThe Iris data is in a so-called 'wide' format, in which each column corresponds to a variable and each row of the DataFrame corresponds to one observation. Turn it into a 'long' format in which each row is a measurement. \n\nSpecifically, change the data layout of the IRIS dataFrame so that it has 3 columns:\n- variable (one of sepal_length, sepal_width, petal_length, petal_width)\n- value\n- target\n\nIf you would like to learn more, [Tidy Data](https://www.jstatsoft.org/index.php/jss/article/view/v059i10/v59i10.pdf) by [Hadley Wickham](http://hadley.nz/) provides a very nice explanation of best practices for data formating.\n\nHint: look at reshaping functions in http://pandas.pydata.org/Pandas_Cheat_Sheet.pdf\n\n\n```python\niris_df_long = pd.melt(iris_df, id_vars=['target'], value_vars=['sepal_length'])\niris_df_long.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        targetvariablevalue
                                        0Iris-setosasepal_length4.9
                                        1Iris-setosasepal_length4.7
                                        2Iris-setosasepal_length4.6
                                        3Iris-setosasepal_length5.0
                                        4Iris-setosasepal_length5.4
                                        \n
                                        \n\n\n\nNow create a box-plot of values that each variable takes, split by the target species.\n\n\n```python\n# Hint: use a `catplot`\nsns.catplot(x=\"target\", y=\"sepal_length\", kind=\"box\", data=iris_df)\n# sns.catplot(x=\"target\", y=\"sepal_width\", kind=\"box\", data=iris_df)\n\n# TODO: create two more plots, using a boxenplot and a swarmplot.\n\nsns.catplot(x=\"target\", y=\"sepal_length\", kind=\"boxen\", data=iris_df)\nsns.catplot(x=\"target\", y=\"sepal_length\", kind=\"swarm\", data=iris_df)\n```\n\n### k-Nearest Neighbors\n\nWe will use the loaded Iris data describing iris flowers\nand shows relations between their length and petal width for three\nspecies (namely: setosa, versicolor, virginica).\n\nFor this exercise we will restrict our analysis to just two variables: **petal length** and **petal width**.\n\n\n```python\nunknown_df = pd.DataFrame(\n [[1.5, 0.3, 'unknown'],\n [4.5, 1.2, 'unknown'],\n [5.1, 1.7, 'unknown'],\n [5.5, 2.3, 'unknown']],\n columns=['petal_length', 'petal_width', 'target'])\n\nsns.scatterplot(x='petal_length', y='petal_width', hue='target', data=iris_df)\nsns.scatterplot(x='petal_length', y='petal_width', color='gray', marker='v',\n label='unknown', s=70, data=unknown_df)\n```\n\nBased on these two features, it is easy to distinguish iris setosa from the two remaining species. Yet iris versicolor and virginica remain mixed together. \n\nLooking closely at the plot, we might estimate the species of the selected unknown irises (gray triangles). For three of them the answer seems obvious \u00e2\u20ac\u201c they belong in uniformly-colored areas covered by one species only. Yet unknown iris flower in (5.1, 1.7) is troublesome \u00e2\u20ac\u201c it lays on the boundary of versicolor and virginica clusters. We can assume, that its species is the one of the closest one to it, coming from the training set (and so having a label). \n\nK-Nearest Neighbors method (http://en.wikipedia.org/wiki/K-nearest_neighbors_algorithm) solves the classification problem, i.e. sets class labels (species in case of irises) of a previously unseen sample by choosing the most common class among the top k neighbors of the sample in question (for instance according to the Euclidean distance). Thus, the k-Nearest Neighbors algorithm works as follows. For each unlabeled sample x:\n1. Find k nearest neighbors among the labeled samples.\n2. Set the most common label among them as label of x.\n\n#### Problem 3 [3p]\n\n##### Implement the k-Nearest Neighbors algorithm [1p].\n\nTake advantage of matrix calculus rather than using for loops.\n\n**Tip:** What is computed by \\begin{equation} \\sqrt{(X - Y)^T (X - Y)} \\end{equation} when both X and Y are vectors?\n\n**Tip:** Try to use broadcasting (NumPy: http://docs.scipy.org/doc/numpy/user/basics.broadcasting.html) and built-ins sort, numpy.sort, numpy.argsort (sorting), scipy.stats.mode (choosing the most common element of the set).\n\n\n```python\ndef KNN(train_X, train_Y, test_X, ks, verbose=False):\n \"\"\"\n Compute predictions for various k\n Args:\n train_X: array of shape Ntrain x D\n train_Y: array of shape Ntrain\n test_X: array of shape Ntest x D\n ks: list of integers\n Returns:\n preds: dict k: predictions for k\n \"\"\"\n # Cats data to float32\n train_X = train_X.astype(np.float32)\n test_X = test_X.astype(np.float32)\n\n # Alloc space for results\n preds = {}\n\n if verbose:\n print(\"Computing distances... \")\n #\n # TODO: fill in an efficient distance matrix computation\n # https://medium.com/@souravdey/l2-distance-matrix-vectorization-trick-26aa3247ac6c \n dists = -2 * np.dot(train_X, test_X.T) + np.sum(test_X**2, axis=1) + np.sum(train_X**2, axis=1)[:, np.newaxis]\n\n \n if verbose:\n print(\"Sorting... \")\n \n # TODO: findes closest trainig points\n # Hint: use argsort\n closest = np.argsort(dists, axis=0)\n\n if verbose:\n print(\"Computing predictions...\")\n \n # closest trainig points - > closest trainig targets\n targets = train_Y[closest]\n \n\n for k in ks:\n predictions = sstats.mode(targets[:k])[0] #take k closest targets\n predictions = predictions.ravel()\n preds[k] = predictions\n if verbose:\n print(\"Done\")\n return preds\n```\n\n\n```python\n# Now classify the 4 unknown points\niris_x = np.array(iris_df[['petal_length', 'petal_width']])\niris_y = np.array(iris_df['target'])\n\nunknown_x = np.array(unknown_df[['petal_length', 'petal_width']])\n# print(unknown_x.shape)\nprint(KNN(iris_x, iris_y, unknown_x, [1, 3, 5, 7]))\n\n# print(iris_x.shape)\n# print(iris_y.shape)\n# print(unknown_x.shape)\n```\n\n {1: array(['Iris-setosa', 'Iris-versicolor', 'Iris-versicolor',\n 'Iris-virginica'], dtype=object), 3: array(['Iris-setosa', 'Iris-versicolor', 'Iris-versicolor',\n 'Iris-virginica'], dtype=object), 5: array(['Iris-setosa', 'Iris-versicolor', 'Iris-virginica',\n 'Iris-virginica'], dtype=object), 7: array(['Iris-setosa', 'Iris-versicolor', 'Iris-virginica',\n 'Iris-virginica'], dtype=object)}\n\n\n##### Plot the Decision boundary [1p]\n\n\nUse meshgrid to generate the points in the space spanned by data\nThan map the classes to numbers 0, 1, 2 and make a contour plot with the\ndecision boundary\n\n\n```python\niris_x = np.array(iris_df[['petal_length', 'petal_width']])\niris_y = np.array(iris_df['target'])\n\nX_space = np.linspace(0,7,200)\nY_space = np.linspace(0,3,200)\nmesh_x, mesh_y = np.meshgrid(X_space, Y_space)\n\n#use np.unique with suitable options to map the class names to numbers\ntarget_names, iris_y_ids = np.unique(iris_y, return_inverse=True)\n# print(target_names, iris_y_ids)\n\nmesh_data = np.hstack([mesh_x.reshape(-1, 1), mesh_y.reshape(-1, 1)])\ncolors = ['red', 'blue', 'green']\n\npreds = KNN(iris_x, iris_y_ids, mesh_data, [1, 3, 5, 7])\nfor k, preds_k in preds.items():\n plt.figure()\n plt.title(f\"Decision boundary for k={k}\")\n plt.contourf(mesh_x, mesh_y, preds_k.reshape(200,200), levels=2, cmap='Pastel1')\n plt.scatter(iris_x[:,0], iris_x[:,1], c=iris_y_ids, s=10, cmap='Dark2')\n \n# for p, c in zip(iris_x, iris_y_ids):\n# plt.scatter(p[0], p[1], color=colors[c], s=10)\n\n```\n\n##### Estimate performance for various ks [1p]\nConsider the following experiment:\n1. We scramble the data and split it into two parts - training set (66.6% of all samples) and test set (33.4%).\n2. Based on the training set, we use the k-NN algorithm to predict the labels on the test set.\n3. We then check the number of errors and write it down.\n\nDo this 500 times for k \u00e2\u02c6\u02c6 {1, 3, 5, ..., 19}. Plot a function of the average number of errors as the function of k. It should be similar to the one below.\n\n\n```python\n#TODO: write a function to compute error rates\ndef err_rates(preds, test_Y):\n ret = {}\n for k, preds_k in preds.items():\n # TODO: fill in error count computation\n assert(len(test_Y) == len(preds_k))\n ret[k] = np.sum(preds_k != test_Y) / test_Y.shape[0]\n return ret\n```\n\n\n```python\niris_x = np.array(iris_df[['sepal_length', 'sepal_width', 'petal_length', 'petal_width']])\niris_y = np.array(iris_df['target'])\nN = len(iris_y)\n\nks = range(1, 20, 2)\nresults = []\n\nfor _rep in tqdm_notebook(range(1000)):\n #TODO\n # Use np.split and np.permutation to get training and testing indices\n train_idx, test_idx = np.split(np.random.permutation(N), [math.floor(N * 0.666)])\n\n #TODO: apply your kNN classifier to data subset\n preds = KNN(iris_x[train_idx], iris_y[train_idx], iris_x[test_idx], ks)\n errs = err_rates(preds, iris_y[test_idx])\n \n for k, errs_k in errs.items():\n results.append({'K':k, 'err_rate': errs_k})\n\n# results_df will be a data_frame in long format\nresults_df = pd.DataFrame(results)\n\n\n```\n\n\n HBox(children=(IntProgress(value=0, max=1000), HTML(value='')))\n\n\n \n\n\n\n```python\n# Plot a function of the average number of errors as the function of k. It should be similar to the one below.\nplt.figure()\nsns.regplot(x='K', y=results_df.err_rate, data=results_df, x_estimator=np.mean, order=2)\n# plt.figure()\n# sns.regplot(...)\n\n# results_df.head()\n\n# gb = results_df.groupby(['K', 'err_rate'])\n# counts = gb.size().to_frame(name='counts')\n# counts.head()\n\n \n\n \n\n```\n\n#### Problem 5 [2p + 2p bonus] \n\nDownload a categorical dataset from UCI and try to find the most predictive variables\n\n\n```python\ncolumns = [\n \"target\", \"cap-shape\", \"cap-surface\", \"cap-color\", \"bruises?\", \"odor\", \n \"gill-attachment\", \"gill-spacing\", \"gill-size\", \"gill-color\", \"stalk-shape\", \n \"stalk-root\", \"stalk-surface-above-ring\", \"stalk-surface-below-ring\", \n \"stalk-color-above-ring\", \"stalk-color-below-ring\", \"veil-type\", \"veil-color\", \n \"ring-number\", \"ring-type\", \"spore-print-color\", \"population\", \"habitat\", ]\n\n# Use read_csv to load the data.\nurl = 'http://archive.ics.uci.edu/ml/machine-learning-databases/mushroom/agaricus-lepiota.data'\nmushroom_df = pd.read_csv(url, header=None, names=columns)\nmushroom_df.head()\n```\n\n\n\n\n
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                                        targetcap-shapecap-surfacecap-colorbruises?odorgill-attachmentgill-spacinggill-sizegill-color...stalk-surface-below-ringstalk-color-above-ringstalk-color-below-ringveil-typeveil-colorring-numberring-typespore-print-colorpopulationhabitat
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                                        \n\n\n\nImplement the function `entropy` to compute the entropy of a columnt of the dataset.\n\nThe [entropy](https://en.wikipedia.org/wiki/Entropy_(information_theory)) of a discrete variable is defined to be:\n\n$$H(X) = -\\sum_x p_X(x) \\log_2 p_X(x)$$.\n\nA good in tutorial is given by Chris Olah: https://colah.github.io/posts/2015-09-Visual-Information/.\n\n\nWhen $X$ is a discrete random variables, we can estimate the probabilities with counts:\n\n$$p_X(x) = \\frac{\\text{number of instances where }X=x}{\\text{total number of instances}}$$.\n\n\nHint: the following `pandas` functions may be useful:\n- `count`\n- `value_count`\n\nThen use the datafranme's `apply` function to compute the entropy of all columns.\n\n\n```python\ndef entropy(series):\n p = series.value_counts(normalize=True)\n return - np.sum(p * np.log2(p))\n\nmushroom_df.apply(entropy)\n```\n\n\n\n\n target 0.999068\n cap-shape 1.652889\n cap-surface 1.575486\n cap-color 2.510143\n bruises? 0.979327\n odor 2.319414\n gill-attachment 0.173129\n gill-spacing 0.637878\n gill-size 0.892256\n gill-color 3.030433\n stalk-shape 0.986927\n stalk-root 1.822922\n stalk-surface-above-ring 1.221348\n stalk-surface-below-ring 1.399135\n stalk-color-above-ring 1.936809\n stalk-color-below-ring 1.978163\n veil-type -0.000000\n veil-color 0.196238\n ring-number 0.420680\n ring-type 1.535121\n spore-print-color 2.203227\n population 2.003398\n habitat 2.274747\n dtype: float64\n\n\n\nImplement the conditional entropy computation\n\n$$H(Y|X) = \\sum_x p_X(x) H(Y|x) = \\sum_x p_X(x) \\sum_y p_Y(y|x) \\log_2 p_Y(y|x)$$\n\nHint 1: the above formlua can be computed as follows:\n1. split the data by the values of $X$\n2. for each value $x$ that $X$ takes, compute the entropy of $Y$\n3. average the entropies, weighting them by how frequent the $x$ value ocurred.\n\nHint 2: helpful pandas constructs are:\n- `groupby` and `agg`\n- you can aggregate a grouping using your own custom functions\n\n\n\n```python\ndef cond_entropy(df, X, Y):\n \"\"\"Compute the conditional H(X|Y) in dataframe df\n Args:\n df: a dataframe\n X: the name of the conditioning columt\n Y: the name of the column whose entropy we wish to compute\n \"\"\"\n grouped = df.groupby(X)[Y]\n return grouped.agg(entropy).agg(np.average, weights=grouped.agg('count'))\n\n```\n\n\n```python\n# Now for each column C compute the conditional entropy H(target|C)\n# Knowing which variable tells us the most about the target?\nfor cname in mushroom_df.columns:\n print(f\"{cname}: {cond_entropy(mushroom_df, cname, 'target')}\")\n```\n\n target: 0.0\n cap-shape: 0.9502711949370876\n cap-surface: 0.9704776640986875\n cap-color: 0.9630186138962564\n bruises?: 0.8066884111112408\n odor: 0.0929929194884606\n gill-attachment: 0.9849028696218441\n gill-spacing: 0.8981847128758902\n gill-size: 0.7689135217244144\n gill-color: 0.582090373456329\n stalk-shape: 0.9915511243027961\n stalk-root: 0.8642502592451845\n stalk-surface-above-ring: 0.7143422976539759\n stalk-surface-below-ring: 0.7271734234797138\n stalk-color-above-ring: 0.7452227234102207\n stalk-color-below-ring: 0.7576523303448939\n veil-type: 0.9990678968724603\n veil-color: 0.9752508807515435\n ring-number: 0.9606152276293699\n ring-type: 0.6810463860789229\n spore-print-color: 0.5183629791875449\n population: 0.7971098778056079\n habitat: 0.8422342922673685\n\n\nBonus questions:\n- **[1p]** Implement computation of [Mutual Information ](https://en.wikipedia.org/wiki/Mutual_information)\n- **[1p]** Add an ID column, that assigns a unique ID to each observation (row). Comompute the $H(target|ID)$ and the mutual information between target and ID. How to interpret the results? Do you think the ID is important in predicting the target? \n\n\n```python\n# def mutual_information(df,X,Y):\n# '''\n# Straightforward implementation of formula\n# '''\n# if X == Y:\n# return entropy(df[X])\n \n# new_df = df[[X,Y]]\n# A_X = new_df.groupby([X]).agg('count')\n# A_Y = new_df.groupby([Y]).agg('count')\n# A_XY = df.groupby([X,Y]).agg('count')\n# res = 0\n# N = new_df.shape[0]\n \n# for XY in A_XY.index:\n# P_XY = A_XY.loc[XY][0] / N\n# P_X = A_X.loc[XY[0]][0] / N\n# P_Y = A_Y.loc[XY[1]][0] / N\n# res += P_XY * np.log2(P_XY / (P_X * P_Y))\n \n# return res\n\ndef mutual_information(df,X,Y):\n '''\n I(X, Y) = H(X) - H(X | Y)\n '''\n return entropy(df[X]) - cond_entropy(df, X, Y)\n```\n\n\n```python\n# for cname in mushroom_df.columns:\n# print(f\"1 -> {cname}: {mutual_information(mushroom_df, cname, 'target')}\")\n \n# for cname in mushroom_df.columns:\n# print(f\" -> {cname}: {mutual_information2(mushroom_df, cname, 'target')}\")\n \nmushroom_df[\"ID\"] = mushroom_df.index\nprint(f\"H(target|ID): {cond_entropy(mushroom_df,'ID','target')}, \\\n MI(target,ID): {mutual_information(mushroom_df,'target','ID')}, \\\n H(ID|target): {cond_entropy(mushroom_df,'target','ID')}\")\n```\n\n H(target|ID): 0.0, MI(target,ID): 0.9990678968725724, H(ID|target): 11.988906627423692\n\n\n#### Problem 6 [2p] \n\nApply the K-Nearest Neighbors (K-NN) algorithm to the MNIST dataset. \n\nThe MNIST (http://yann.lecun.com/exdb/mnist/) dataset consists of normalized (centered and stretched) scans of hand-written digits. Specifically, each element of the dataset is a 28 \u00c3\u2014 28 grayscale image, thus having 764 8-bit pixels. \n\n1. Display a few objects from each of the classes, paying attention to aesthetics and clarity of your presentation. **Note:** You already downloaded the dataset in \"Setup\" section. Please use the code below to get started.\n\n2. **[2p]** Apply a k-NN classifier to the MNIST dataset. First, divide the training set into two parts, which we will call training and validation. On MNIST use the first 50000 samples for training and the last 10000 for validation. Then find the optimal number of neighbors by assessing the accuracy on the validation set. You do not need to repeat this experiment multiple times. Finally, compute the accuracy on the test set obtained with the best previously chosen number of neighbors. On MNIST you should get about 3% errors. Pick a few mislabeled samples from the test dataset and plot them along with the correct ones. **Note:**\n * MNIST is much larger than the Iris dataset. A good implementation may need a few minutes depending on your runtime type. Please optimize your algorithm:\n * Compute the distances only once, then test for different values of k.\n * Use vectorized expressions to compute the distance. It is possible to compute all distances between the training and testing points in one expression. Hint: think about the vectorized expression \\begin{equation}(X - Y)^T (X - Y)\\end{equation}\n * You can use single precision numbers in computation.\n * If your code is taking a long time to execute, please save its results before the lab session.\n\n**Note:** in NumPy, matrices have its own data type (dtype), which is retained during\ncalculations. Please pay attention to it. I particular, do not subtract values of data types not\nhaving the sign bit, do not divide integers, etc. Results of such operations will not be\nautomatically casted to types having the required precision.\n\n\n```python\nwith np.load('mnist.npz') as data:\n mnist_full_train_data_uint8 = data['train_data']\n mnist_full_train_labels_int64 = data['train_labels']\n mnist_test_data_uint8 = data['test_data']\n mnist_test_labels_int64 = data['test_labels']\n \n# Split train data into train and validation sets\nmnist_train_data_uint8 = mnist_full_train_data_uint8[:50000]\nmnist_train_labels_int64 = mnist_full_train_labels_int64[:50000]\nmnist_valid_data_uint8 = mnist_full_train_data_uint8[50000:]\nmnist_valid_labels_int64 = mnist_full_train_labels_int64[50000:]\n```\n\n\n```python\nplot_mat(mnist_train_data_uint8[:20, None], cmap='gray')\n```\n\n\n```python\n# MNIST is large.\n# Implement a batched KNN classifier, which processes the test data in small batches\n# and returns the error rates\n\n# The code should not run for more than a couple of minutes on the Colab runtime, \n# If it is slower, optimize the distance computation in KNN\n\ndef batched_KNN(train_X, train_Y, test_X, ks, verbose=False, batch_size=200):\n all_preds = {k: [] for k in ks}\n for i in range(0, test_X.shape[0], batch_size):\n batch_X = test_X[i:i + batch_size]\n if verbose:\n print(f\"Processing batch {i}:{i + batch_X.shape[0]}... \")\n\n # TODO: run KNN on the batch and save the predictions\n for k in all_preds.keys():\n preds = KNN(train_X, train_Y, batch_X, ks)\n # TODO: combine predictions from batches\n all_preds[k] = np.concatenate((all_preds[k], preds[k]))\n return all_preds\n```\n\n\n```python\n# Now find the best k on the validation set\nks = [1, 3, 5, 7, 9]\nmnist_validation_preds = batched_KNN(\n mnist_train_data_uint8.astype('float32').reshape(-1, 28*28), \n mnist_train_labels_int64,\n mnist_valid_data_uint8.astype('float32').reshape(-1, 28*28),\n ks, verbose=True)\n\nmnist_validation_errs = err_rates(mnist_validation_preds, mnist_valid_labels_int64)\nplt.plot(ks, [mnist_validation_errs[k] for k in ks])\n```\n\n\n```python\n# Now use the best k to compute the test error\n\nbest_K = 3\n\nmnist_test_preds = batched_KNN(\n mnist_full_train_data_uint8.astype('float32').reshape(-1, 28*28), \n mnist_full_train_labels_int64,\n mnist_test_data_uint8.astype('float32').reshape(-1, 28*28), \n [best_K], verbose=True)\n\nmnist_test_errs = err_rates(mnist_test_preds, mnist_test_labels_int64)\nprint(f\"\\n\\nWhen k={best_K} the test error rate is {mnist_test_errs[best_K] * 100.0:.1f}%%\")\n```\n\n Processing batch 0:200... \n Processing batch 200:400... \n Processing batch 400:600... \n Processing batch 600:800... \n Processing batch 800:1000... \n Processing batch 1000:1200... \n Processing batch 1200:1400... \n Processing batch 1400:1600... \n Processing batch 1600:1800... \n Processing batch 1800:2000... \n Processing batch 2000:2200... \n Processing batch 2200:2400... \n Processing batch 2400:2600... \n Processing batch 2600:2800... \n Processing batch 2800:3000... \n Processing batch 3000:3200... \n Processing batch 3200:3400... \n Processing batch 3400:3600... \n Processing batch 3600:3800... \n Processing batch 3800:4000... \n Processing batch 4000:4200... \n Processing batch 4200:4400... \n Processing batch 4400:4600... \n Processing batch 4600:4800... \n Processing batch 4800:5000... \n Processing batch 5000:5200... \n Processing batch 5200:5400... \n Processing batch 5400:5600... \n Processing batch 5600:5800... \n Processing batch 5800:6000... \n Processing batch 6000:6200... \n Processing batch 6200:6400... \n Processing batch 6400:6600... \n Processing batch 6600:6800... \n Processing batch 6800:7000... \n Processing batch 7000:7200... \n Processing batch 7200:7400... \n Processing batch 7400:7600... \n Processing batch 7600:7800... \n Processing batch 7800:8000... \n Processing batch 8000:8200... \n Processing batch 8200:8400... \n Processing batch 8400:8600... \n Processing batch 8600:8800... \n Processing batch 8800:9000... \n Processing batch 9000:9200... \n Processing batch 9200:9400... \n Processing batch 9400:9600... \n Processing batch 9600:9800... \n Processing batch 9800:10000... \n \n \n When k=3 the test error rate is 2.9%%\n\n\n### Locality sensitive hashing\n\nProblem 5 was about speeding up the inference using loops implicitly present in matrix multiplication instead of explicit loops in Python. In this problem, we will explore a strategy to truly reduce the total number of computations required to find nearest neighbors without sacrificing too much accuracy.\n\nTo speed up nearest neighbor search we will employ *Locality Sensitive Hashing (LSH)* functions. For a given distance metric, the locality sensitive hash should put items that are similar into the same bucket. Notice that this is essentially a design choice opposite to traditional cryptographic hash functions that should amplify the difference of similar inputs (typically we want that small perturbations of data result in large changes to the hash value).\n\nOne of the simplest implementations of LSH approximates the cosine distance. Let $x\\in \\mathbb{R}^N$ and $y\\in \\mathbb{R}^N$ be two vectors. Their cosine distance is defined as:\n\n\\begin{equation}\n d_\\text{cos}(x,y) = \\frac{x \\cdot y}{\\|x\\| \\|y\\|} = \\cos\\left(\\theta(x,y)\\right),\n\\end{equation}\nwhere $\\theta(x,y)$ is the unsigned angle between $x$ and $y$.\n\nWe will construct a family $H$ of hash functions that are an LSH for angle distances (an approximation to cosine distance). Assume $p\\in \\mathbb{R}^N$ is a random vector (components are sampled from the normal distribution) of length 1. Then define the hash function $h(x) = \\text{sgn}(x\\cdot p)$, where $\\text{sgn()}$ is the sign function. It can be proven that:\n\n\\begin{equation}\n p_{h\\in H}[h(x)=h(y)] = 1 - \\frac{\\theta(x,y)}{\\pi}.\n\\end{equation}\n\nThe equation means that the probability of a hash collision grows as the the angle between two vectors gets smaller. Therefore, vectors that are close according to the cosine distance will be put with high probability into the same bin (we use the fact that for small $\\theta$ we can approximate $\\cos(\\theta) = 1 - \\theta/\\pi$.\n\nWe will say that a family of randomly chosen hash functions $H$ is $(d_1, d_2, p_1, p_2)$-sensitive with respect to a distance metric $d$ if for any $x$ and $y$:\n1. If $d(x,y) \\leq d_1$ then $p_{h\\in H}[h(x)=h(y)] \\geq p_1$.\n2. If $d(x,y) \\geq d_2$ then $p_{h\\in H}[h(x)=h(y)] \\leq p_2$.\n\nFor example, our family of randomly chosen hyperplanes is $(d_1, d_2, (1-d_1)/\\pi, (1-d_2)/\\pi)$-sensitive.\n\nIdeally, vectors should be placed into the same bin with a high probability if their distance is smaller than a threshold, and with a low probability if their distance is larger that the threshold. By combining hashing functions we can get closer to this ideal sensitivity.\n\nGiven a family of hash functions $H$ with sensitivity $(d_1, d_2, p_1, p_2)$ we can construct a new family $H'$ by combining $r$ functions from $H$:\n1. AND: let $h=[h_1, h_2, \\ldots, h_r] \\in H'$ and $h(x)=h(y)$ if and only if $\\forall_i h_i(x)=h_i(y)$. Then $H'$ is $(d_1, d_2, (p_1)^r, (p_2)^r)$-sensitive.\n2. OR: let $h=[h_1, h_2, \\ldots, h_r] \\in H'$ and $h(x)=h(y)$ if and only if $\\exists_i h_i(x)=h_i(y)$. Then $H'$ is $(d_1, d_2, 1-(1-p_1)^r, 1-(1-p_2)^r)$-sensitive.\n\nAND makes all probabilities shrink, but properly choosing $r$ we can make the lower probability approach 0 while the higher does not. Conversely, OR makes all probabilities grow, we can make the upper probability approach 1 while the lower does not.\n\n#### Problem 7 [1-3p bonus] \n\n1. **[1bp for exercises list]** **Note:** you can show sketches of proofs for this assignment.\n 1. Show that angle between vectors is a metric (https://en.wikipedia.org/wiki/Metric_(mathematics)).\n \n 2. Show that $p_{h\\in H}[h(x)=h(y)] = 1 - \\frac{\\theta(x,y)}{\\pi}$ for $h$ computed using a randomly chosen hyperplane.\n\n 3. Show the properties of either AND or OR boosting of LSH.\n\n Please show the solution to this problem dirung the Session for Homework 1, the bonus point will also be added to the points from Homework 1.\n\n3. **[1-3bp]** Reimplement k-Nearest Neighbors for MNIST classification using the cosine distance instead of the Euclidean distance. Choose a sensible value of $k$. Use Locality Sensitive Hashing to achieve an error rate no greater than $150\\%$ of the original error rate with at least a $90\\%$ speedup (i.e., by considering on average at most 5000 training samples per query image). For a few settings plot the speedup-vs-accuracy relation.\n\n **Note:** points will be awarded based on ingenuity of your solution. Feel free to explore your own ideas!\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "f7df59f79d74f01b5f216305fb128444c37424a1", "size": 660520, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ML/Assignment1/Assignment1.ipynb", "max_stars_repo_name": "TheFebrin/DataScience", "max_stars_repo_head_hexsha": "3e58b89315960e7d4896e44075a8105fcb78f0c0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "ML/Assignment1/Assignment1.ipynb", "max_issues_repo_name": "TheFebrin/DataScience", "max_issues_repo_head_hexsha": "3e58b89315960e7d4896e44075a8105fcb78f0c0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ML/Assignment1/Assignment1.ipynb", "max_forks_repo_name": "TheFebrin/DataScience", "max_forks_repo_head_hexsha": "3e58b89315960e7d4896e44075a8105fcb78f0c0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 250.196969697, "max_line_length": 167150, "alphanum_fraction": 0.9014064676, "converted": true, "num_tokens": 18057, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.7057850216484838, "lm_q1q2_score": 0.4156291279075311}} {"text": "# Bayesci Negatif Olmayan Matris Ayr\u0131\u015f\u0131m\u0131 i\u00e7in Aktif Eleman Se\u00e7imi\n# Active Selection of Elements for Bayesian Nonnegative Matrix Factorization\n\n
                                        \n
                                        Burak Suyunu, G\u00f6n\u00fcl Ayc\u0131, A.Taylan Cemgil
                                        \n
                                        * Bilgisayar M\u00fchendisli\u011fi B\u00f6l\u00fcm\u00fc, Bo\u011fazi\u00e7i \u00dcniversitesi
                                        \n
                                        * {burak.suyunu, gonul.ayci, taylan.cemgil}@boun.edu.tr
                                        \n\n## \u00d6zet\u00e7e\n\nKlasik matris tamamlama problemlerinde elimizdeki matris, g\u00f6zlemlenen ve bilinmeyen elemanlar olarak iki gruba ayr\u0131labilir. Bu \u00e7al\u0131\u015fmadaki yakla\u015f\u0131mda ise matrisler \u00fc\u00e7 farkl\u0131 gruptan olu\u015fmaktad\u0131r: bilinen ve masrafs\u0131z olarak her zaman eri\u015febildigimiz g\u00f6zlemlenmi\u015f olan veri, tahmin etmeye \u00e7al\u0131\u015ft\u0131\u011f\u0131m\u0131z bilinmeyen veri ve \u015fu an bilinmeyen ancak istenildigi zaman sorgulanabilen veri. Son gruptaki veriler ilk kez sorguland\u0131\u011f\u0131nda bir maliyet ortaya \u00e7\u0131kmaktad\u0131r. Bu g\u00f6zlemden yola \u00e7\u0131karak, m\u00fcmk\u00fcn oldu\u011fu kadar az sorgu yaparak ikinci gruptaki bilinmeyen verideki de\u011ferleri en az hata ile tahminlemek istiyoruz. Amac\u0131m\u0131z, sorgulamaya \u00e7al\u0131\u015ft\u0131\u011f\u0131m\u0131z g\u00f6zlemleri ak\u0131ll\u0131ca se\u00e7ebilmek. Bu \u00e7al\u0131\u015fmam\u0131zda, g\u00f6zlem s\u0131ras\u0131 se\u00e7me stratejileri tan\u0131mlanarak MovieLens veri setinde kar\u015f\u0131la\u015ft\u0131r\u0131lm\u0131\u015ft\u0131r.\n\n## Abstract\nIn classical matrix completion problems, you can divide the matrix into two groups as observed and unknown elements. In this study, the matrices are composed of three different groups: observed data that is known and accessible at any time without any expense, unknown data that we are trying to predict, and data that is currently unknown but can be queried when desired. When the data in the last group is queried for the first time, a cost arises. From this observation, we want to estimate the unknown data in the second group with the least error by making as few queries as possible. Our goal is to choose the observations we are trying to question wisely. In this study, observation sequence selections were defined and compared in\nthe MovieLens data set.\n\n## Giri\u015f\nNegatif olmayan matris ayr\u0131\u015ft\u0131rma y\u00f6ntemi (NOMA), ilk olarak Lee ve Seung [1] taraf\u0131ndan \u00f6nerilmi\u015ftir. Negatif olmayan matris ayr\u0131\u015ft\u0131rmas\u0131, verilen negatif olmayan bir *X* matrisinin negatif olamayan de\u011ferler i\u00e7eren *T* ve *V* \u00e7arpanlar\u0131na ay\u0131rma y\u00f6ntemidir. Elde edilen iki matrisin \u00e7arp\u0131mlar\u0131n\u0131n de\u011feri ayr\u0131\u015ft\u0131r\u0131lan matrisin de\u011ferine yakla\u015f\u0131k olarak e\u015fittir.\n\n\u00d6neri sistemlerinde ayr\u0131\u015f\u0131m tabanl\u0131 yap\u0131lar s\u0131kl\u0131kla kullan\u0131lmaktad\u0131r. Sistemin mant\u0131\u011f\u0131 \u015fu \u015fekilde a\u00e7\u0131klanabilir. Bir *X* matrisi ile belirli bir s\u00fcre aral\u0131g\u0131nda toplanm\u0131\u015f olan kullan\u0131c\u0131-film ili\u015fkisine dair verimizi g\u00f6sterelim. Bu matrisin sat\u0131rlar\u0131 filmleri, s\u00fctunlar\u0131 kullan\u0131c\u0131lar\u0131, elemanlar\u0131 ise kullan\u0131c\u0131lar\u0131n filmlere vermi\u015f oldugu puanlamalar\u0131 temsil etmektedir. E\u011fer *X* matrisinin herhangi bir eleman\u0131n\u0131n de\u011feri yok ise, bu kullan\u0131c\u0131 ve film aras\u0131nda hen\u00fcz bir ili\u015fkinin olmad\u0131\u011f\u0131n\u0131 g\u00f6stermektedir yani kullan\u0131c\u0131 bu filme hen\u00fcz herhangi bir oy vermemi\u015ftir. Kullan\u0131c\u0131lara etkile\u015fimli olarak film hakk\u0131ndaki puanlamalar\u0131 sorulabilir. Ancak herkesin her zaman her film hakk\u0131nda puanlama yapmas\u0131 m\u00fcmk\u00fcn degildir. Dolay\u0131s\u0131yla bir ki\u015fiye bir film hakk\u0131ndaki puan\u0131n\u0131 sorarken ak\u0131ll\u0131ca bir yol izlemek gerekmektedir. Bu \u015fekilde en az ki\u015fiden bilgi talep ederek, elde edilen bilgilerle kullan\u0131c\u0131-film aras\u0131ndaki \u00f6r\u00fcnt\u00fcye ula\u015fmak ve bilmedigimiz veriler hakk\u0131ndaki tahminlerimizi iyile\u015ftirmek istenmektedir.\n\n\n
                                        \u015eekil 1: **Senaryonun i\u015fleni\u015fi:** Burada mavi h\u00fccreler maskelenmi\u015f, k\u0131rm\u0131z\u0131lar test ve beyazlar ise g\u00f6zlemlenmi\u015f veriyi g\u00f6stermektedir. K\u0131rm\u0131z\u0131n\u0131n koyulu\u011fu hatan\u0131n fazlal\u0131\u011f\u0131na i\u015faret etmektedir. Bir veri g\u00f6zlemlendiginde en \u00e7ok bulundu\u011fu sat\u0131r ve s\u00fctun hakk\u0131nda bilgi vermesi beklenmektedir. *Senaryo B*\u2019nin test h\u00fccreleri hakk\u0131nda verdi\u011fi bilgi *Senaryo A*\u2019dan fazla oldu\u011fu g\u00f6r\u00fclmektedir.
                                        \n\n\n**Senaryo:** Elimizde bir *X* matrisimiz olsun. Verilen *X* matrisine Gibbs \u00f6rnekleyicisi ile negatif olmayan matris ayr\u0131\u015ft\u0131rmas\u0131 y\u00f6ntemi kullanarak yakla\u015f\u0131m yap\u0131yoruz. Matrisimizi\nbildi\u011fimiz, bilmedi\u011fimiz (tahmin etmeye \u00e7al\u0131\u015ft\u0131\u011f\u0131m\u0131z) ve zaman i\u00e7inde a\u00e7arak g\u00f6zlemleyece\u011fimiz (maskelenmi\u015f) \u00fc\u00e7 \u00e7e\u015fit veriden olu\u015fturuyoruz. \u00dc\u00e7\u00fcnc\u00fc kategorideki veriden belli y\u0131\u011f\u0131nlarda ve minimum say\u0131da veri a\u00e7arak maksimum bilgi edinmeyi ve en iyi tahminleme yapmay\u0131 hedefliyoruz. Bu gruptaki veriyi nas\u0131l se\u00e7ecegimiz konusunda tan\u0131mlad\u0131\u011f\u0131m\u0131z \u00e7e\u015fitli g\u00f6zlem s\u0131ras\u0131 se\u00e7me stratejilerimizi kar\u015f\u0131la\u015ft\u0131r\u0131yoruz.\n\n\n```python\nimport numpy as np\nimport scipy as sp\nimport math\nimport time\nfrom scipy import special\nfrom scipy.stats import gamma\nfrom scipy.stats import entropy\nfrom scipy.integrate import simps\nimport matplotlib.pyplot as plt\nimport matplotlib\nfrom sklearn import preprocessing\nimport random\nimport pandas as pd\nfrom IPython.html.widgets import *\nimport operator\n```\n\n C:\\Users\\Burki\\Anaconda3\\lib\\site-packages\\IPython\\html.py:14: ShimWarning: The `IPython.html` package has been deprecated. You should import from `notebook` instead. `IPython.html.widgets` has moved to `ipywidgets`.\n \"`IPython.html.widgets` has moved to `ipywidgets`.\", ShimWarning)\n\n\n## MovieLens Veri Seti\n\n[MovieLens][1] veri setinde 700 kullan\u0131c\u0131n\u0131n 9000 filme vermi\u015f oldu\u011fu 0,5 ile 5 puan aras\u0131nda degi\u015fen toplam 100.000 oy bulunmaktad\u0131r. Modelimizdeki \u00e7ok terimli dag\u0131l\u0131ma uygun bir girdi olu\u015fturmak i\u00e7in oy puan aral\u0131g\u0131n\u0131 0,5-5 ten 1-10 aral\u0131\u011f\u0131na e\u015fledik. \n\n[1]:https://grouplens.org/datasets/movielens/latest/\n \n\n\n```python\ndf_MovieLens = pd.read_csv('ratings.csv')\ndf_MovieLens\n```\n\n\n\n\n
                                        \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        userIdmovieIdratingtimestamp
                                        01312.51260759144
                                        1110293.01260759179
                                        2110613.01260759182
                                        3111292.01260759185
                                        4111724.01260759205
                                        5112632.01260759151
                                        6112872.01260759187
                                        7112932.01260759148
                                        8113393.51260759125
                                        9113432.01260759131
                                        10113712.51260759135
                                        11114051.01260759203
                                        12119534.01260759191
                                        13121054.01260759139
                                        14121503.01260759194
                                        15121932.01260759198
                                        16122942.01260759108
                                        17124552.51260759113
                                        18129681.01260759200
                                        19136713.01260759117
                                        202104.0835355493
                                        212175.0835355681
                                        222395.0835355604
                                        232474.0835355552
                                        242504.0835355586
                                        252523.0835356031
                                        262623.0835355749
                                        2721104.0835355532
                                        2821443.0835356016
                                        2921505.0835355395
                                        ...............
                                        9997467140344.51064245493
                                        9997567143065.01064245548
                                        9997667143083.51065111985
                                        9997767148804.01065111973
                                        9997867148865.01064245488
                                        9997967148965.01065111996
                                        9998067149634.51065111855
                                        9998167149734.51064245471
                                        9998267149935.01064245483
                                        9998367149954.01064891537
                                        9998467150102.01066793004
                                        9998567152182.01065111990
                                        9998667152993.01065112004
                                        9998767153494.01065111863
                                        9998867153774.01064245557
                                        9998967154454.51064891627
                                        9999067154643.01064891549
                                        9999167156694.01063502711
                                        9999267158164.01065111963
                                        9999367159023.51064245507
                                        9999467159525.01063502716
                                        9999567159894.01064890625
                                        9999667159914.51064245387
                                        9999767159954.01066793014
                                        9999867162122.51065149436
                                        9999967162682.51065579370
                                        10000067162694.01065149201
                                        10000167163654.01070940363
                                        10000267163852.51070979663
                                        10000367165653.51074784724
                                        \n

                                        100004 rows \u00d7 4 columns

                                        \n
                                        \n\n\n\n\n```python\ndf_MovieLens.describe()\n```\n\n\n\n\n
                                        \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        userIdmovieIdratingtimestamp
                                        count100004.000000100004.000000100004.0000001.000040e+05
                                        mean347.01131012548.6643633.5436081.129639e+09
                                        std195.16383826369.1989691.0580641.916858e+08
                                        min1.0000001.0000000.5000007.896520e+08
                                        25%182.0000001028.0000003.0000009.658478e+08
                                        50%367.0000002406.5000004.0000001.110422e+09
                                        75%520.0000005418.0000004.0000001.296192e+09
                                        max671.000000163949.0000005.0000001.476641e+09
                                        \n
                                        \n\n\n\n\n```python\nmovieWatchedCount = {}\nuserWatchCount = {}\n```\n\n\n```python\nfor m in df_MovieLens.movieId.unique():\n movieWatchedCount[m] = len(df_MovieLens[df_MovieLens.movieId == m])\nfor u in df_MovieLens.userId.unique():\n userWatchCount[u] = len(df_MovieLens[df_MovieLens.userId == u])\n```\n\n\n```python\nsorted_movieWatchedCount = sorted(movieWatchedCount.items(), key=operator.itemgetter(1), reverse=True)\nsorted_userWatchCount = sorted(userWatchCount.items(), key=operator.itemgetter(1), reverse=True)\n```\n\n### Veri Matrisinin Olu\u015fturulmas\u0131\n\nYo\u011fun veya seyrek matris olu\u015fturma iste\u011finize g\u00f6re alttaki h\u00fccrelerden sadece birini \u00e7al\u0131\u015ft\u0131r\u0131n.
                                        \nOlu\u015fturdu\u011fumuz matrisin sat\u0131rlar\u0131 filmleri, s\u00fctunlar\u0131 kullan\u0131c\u0131lar\u0131, elemanlar\u0131 ise kullan\u0131c\u0131lar\u0131n filmlere vermi\u015f oldu\u011fu puanlamalar\u0131 temsil etmektedir.\n\n#### Yo\u011fun Matrisin \u00dcretilmesi\n\nYo\u011fun veri seti i\u00e7in MovieLens\u2019ten en \u00e7ok film izlemi\u015f 20 kullan\u0131c\u0131 ve en \u00e7ok izlenen 50 filmi kulland\u0131k.\n\n\n```python\n# Number of Rows\n# nu: 1 -> W\nmovieCount = 50\n\n# Number of Columns\n# tao: 1 -> K\nuserCount = 20\n\ntopMovies = np.asarray(sorted_movieWatchedCount)[:movieCount,0]\ntopUsers = np.asarray(sorted_userWatchCount)[:userCount,0]\n\nuserMovie = np.zeros((movieCount, userCount))\n\nfor i, m in enumerate(topMovies):\n for j, u in enumerate(topUsers):\n if len(df_MovieLens[(df_MovieLens.userId == u) & (df_MovieLens.movieId == m)]) != 0:\n userMovie[i][j] = 2*float(df_MovieLens[(df_MovieLens.userId == u) & (df_MovieLens.movieId == m)]['rating'])\n```\n\n#### Seyrek Matrisin \u00dcretilmesi\n\nSeyrek veri i\u00e7in ise her seferinde rastgele en \u00e7ok izlenen 1000 filmden 50, en \u00e7ok film izlemi\u015f 200 kullan\u0131c\u0131dan 20 tanesini kulland\u0131k.\n\n\n```python\n# Number of Rows\n# nu: 1 -> W\nmovieCount = 50\n\n# Number of Columns\n# tao: 1 -> K\nuserCount = 20\n\nsparsityParameter = 13\n\nmovieIndex = np.random.permutation(movieCount*sparsityParameter)[:movieCount]\nuserIndex = np.random.permutation(userCount*sparsityParameter)[:userCount]\n\ntopMovies = []\ntopUsers = []\n\nfor mI in movieIndex:\n topMovies.append(sorted_movieWatchedCount[mI][0])\nfor uI in userIndex:\n topUsers.append(sorted_userWatchCount[uI][0])\n\ntopMovies = np.asarray(topMovies)\ntopUsers = np.asarray(topUsers)\n\nuserMovie = np.zeros((movieCount, userCount))\n\nfor i, m in enumerate(topMovies):\n for j, u in enumerate(topUsers):\n if len(df_MovieLens[(df_MovieLens.userId == u) & (df_MovieLens.movieId == m)]) != 0:\n userMovie[i][j] = 2*float(df_MovieLens[(df_MovieLens.userId == u) & (df_MovieLens.movieId == m)]['rating'])\n```\n\n### Maskeleme\n\nEksik veriyi yani $x_{\\nu, \\tau}$ g\u00f6zlemlenmemi\u015f de\u011ferlerini modellerken *X* matrisi ile ayn\u0131 boyuta sahip olan $\\textit{M} = \\left \\{ m_{\\nu, \\tau} \\right \\}$ *maske matrisi* a\u015fa\u011f\u0131daki gibi tan\u0131mlan\u0131r:\n\n
                                        \n$m_{\\nu, \\tau} = \\left\\{\\begin{matrix}\n 0,& x_{\\nu, \\tau} \\text{ g\u00f6zlemlenmemi\u015fse},\\\\ \n 1,& x_{\\nu, \\tau} \\text{ g\u00f6zlemlenmi\u015fse}.\n\\end{matrix}\\right.$\n
                                        \n\nBu maskeyi kullanarak olabilirlik fonksiyonunu \u015fu \u015fekilde yaz\u0131l\u0131r:\n\n\\begin{align*}\np(X,S\\mid T,V) = \\prod_{\\nu, \\tau}\\left ( p\\left (x_{\\nu, \\tau}\\mid s_{\\nu, 1:I, \\tau}\\right ) p\\left (s_{\\nu, 1:I, \\tau}\\mid t_{\\nu, 1:I}, v_{1:I, \\tau} \\right )\\right )^{m_{\\nu, \\tau}}\n\\end{align*}\n\n\n### Maskeleme Metodlar\u0131\n\nBu \u00e7al\u0131\u015fmam\u0131zda veriye *k\u0131smi* ve *tam* olarak iki farkl\u0131 a\u00e7\u0131dan yakla\u015farak KL \u0131raksay\u0131 ve KOKH metrikleri (ileride a\u00e7\u0131klanacak) ile kar\u015f\u0131la\u015ft\u0131rmalar yapt\u0131k. \n\nK\u0131smi yakla\u015f\u0131mda verimizi %30 test, %69 maskelenmi\u015f, %1 ba\u015flang\u0131\u00e7ta bilinen olarak ay\u0131rd\u0131k. Maskelenmi\u015f veriyi a\u00e7t\u0131k\u00e7a de\u011fi\u015fen test \u00fczerindeki hatay\u0131 \u00f6l\u00e7t\u00fck.\n\nTam yakla\u015f\u0131mda ise %99 maskelenmi\u015f, %1 ba\u015flang\u0131\u00e7ta bilinen olarak ay\u0131rd\u0131k ve hatay\u0131 b\u00fct\u00fcn veri \u00fczerinden \u00f6l\u00e7t\u00fck.\n\n\n```python\ndef randomMasking(W, K, dataExistance):\n dataIndices = np.argwhere(dataExistance>0)\n \n #test, mask, known\n mask = np.zeros([W,K])\n test = np.copy(dataExistance)\n \n np.random.shuffle(dataIndices)\n \n for i in range(30*len(dataIndices)//100):\n test[dataIndices[i][0], dataIndices[i][1]] = 0\n for i in range(30*len(dataIndices)//100, 31*len(dataIndices)//100):\n mask[dataIndices[i][0], dataIndices[i][1]] = 1\n \n return mask, test\n```\n\n### G\u00d6ZLEM SIRASI SE\u00c7ME STRATEJ\u0130LER\u0130\n\nBu \u00e7al\u0131\u015fmada, maskelenmi\u015f olan veriyi g\u00f6zlemlemek i\u00e7in be\u015f farkl\u0131 g\u00f6zlem s\u0131ras\u0131 se\u00e7me stratejisi tan\u0131mlad\u0131k.\n\nAmac\u0131m\u0131z, en h\u0131zl\u0131 \u015fekilde b\u00fct\u00fcn veriyi \u00f6grenebilece\u011fimiz (yak\u0131nsayabilecegimiz) bilgi elde etme stratejisini bulmak ve \u00f6grenme sonucunda test verisi hakk\u0131nda do\u011fru tahminlemede bulunmak.\n\n#### Rastgele Strateji\n\n**Tan\u0131m**: Rastgele bir pozisyonda yer alan maskelenmi\u015f veri g\u00f6zlemlenir.
                                        \nG\u00f6zlemlenmi\u015f verideki bilgiyi kullanmadan maskelenmi\u015f veriyi a\u00e7ar. Tan\u0131mlanan di\u011fer stratejilerin i\u015flevselli\u011fini de\u011ferlendirmek i\u00e7in bir alt s\u0131n\u0131r olu\u015fturur.\n\n\n```python\ndef openRandom(W, K, mask, test):\n openOrder = np.arange(W*K)\n random.shuffle(openOrder)\n \n for i in openOrder:\n if mask[i//K][i%K] == 0 and test[i//K][i%K] == 1:\n return i//K, i%K\n \n return -1, -1\n```\n\n#### Sat\u0131r-S\u00fctun Stratejileri\n\n**Tan\u0131m**: Sat\u0131r ve s\u00fctunlar en az veri g\u00f6zlemlenenden en \u00e7ok veri g\u00f6zlemlenene dogru s\u0131ralan\u0131r. Sat\u0131ra veya s\u00fctuna \u00f6ncelik vererek kesi\u015fimlerindeki ilk maskelenmi\u015f veri a\u00e7\u0131l\u0131r.
                                        \nBir veriyi tahmin ederken o veri hakk\u0131nda en \u00e7ok bilgi edinebilecegimiz yerler o verinin bulundu\u011fu sat\u0131r ve s\u00fctunundaki diger verilerdir. D\u00fczenli olarak sat\u0131r ve s\u00fctunlardaki g\u00f6zlemlenmi\u015f veri say\u0131s\u0131n\u0131 art\u0131rarak b\u00fct\u00fcn veri hakk\u0131ndaki tahminimizi iyile\u015ftirebiliriz.\n\n\n```python\ndef openMaxColMask(W, K, mask, test):\n colSum = mask.sum(0)\n rowSum = mask.sum(1)\n \n colMins = colSum.argsort()\n rowMins = rowSum.argsort()\n\n for c in colMins:\n for r in rowMins:\n if mask[r][c] == 0 and test[r][c] == 1:\n return r, c\n \n \n return openRandom(W, K, mask, test)\n\n```\n\n\n```python\ndef openMaxRowMask(W, K, mask, test):\n colSum = mask.sum(0)\n rowSum = mask.sum(1)\n \n colMins = colSum.argsort()\n rowMins = rowSum.argsort()\n\n for r in rowMins:\n for c in colMins:\n if mask[r][c] == 0 and test[r][c] == 1:\n return r, c\n \n return openRandom(W, K, mask, test)\n```\n\n#### Maskelenmi\u015f Verilerin Varyans\u0131 Stratejileri\n\n**Tan\u0131m**: Varyans\u0131n en k\u00fc\u00e7\u00fck veya en b\u00fcy\u00fck oldugu pozisyondaki veri g\u00f6zlemlenir.
                                        \nGibbs \u00f6rnekleyicisi sonucunda *T* ve *V* matrislerinin yan\u0131nda bunlar\u0131 olu\u015fturan *Gamma* da\u011f\u0131l\u0131m\u0131 parametreleri de elde ediliyor. Bu parametreleri kullanarak belirlenen say\u0131da T ve V \u00f6rnekleyerek tahminler \u00fcretilir. Bu tahminlerle, maskelenmi\u015f olan k\u0131s\u0131mdaki her bir veri tahmininin varyans\u0131 hesaplan\u0131r. Bir pozisyondaki varyans\u0131n b\u00fcy\u00fck olmas\u0131, o pozisyon i\u00e7in \u00fcretilen tahmin de\u011ferinin belirsizli\u011finin fazla oldugunu g\u00f6stermektedir. K\u00fc\u00e7\u00fck varyans ise belirsizli\u011fin az oldu\u011fu yerdir.\n\n\n```python\ndef openMinVar(W, K, mask, test, xCandidates):\n var_candidate = np.var(xCandidates, axis=0)\n \n ind = var_candidate.flatten().argsort()\n \n for i in ind:\n if mask[i//K][i%K] == 0 and test[i//K][i%K] == 1:\n return i//K, i%K\n \n return -1, -1\n```\n\n\n```python\ndef openMaxVar(W, K, mask, test, xCandidates):\n var_candidate = np.var(xCandidates, axis=0)\n \n ind = var_candidate.flatten().argsort()[::-1]\n \n for i in ind:\n if mask[i//K][i%K] == 0 and test[i//K][i%K] == 1:\n return i//K, i%K\n \n return -1, -1\n```\n\n#### G\u00f6zlemlenmi\u015f Sat\u0131r Varyans Stratejisi\n\n**Tan\u0131m**: Her sat\u0131r i\u00e7in o sat\u0131rdaki a\u00e7\u0131lm\u0131\u015f olan veriler \u00fczerinden o sat\u0131r\u0131n varyans\u0131 hesaplan\u0131r. Sat\u0131rlar bu varyans hesab\u0131na g\u00f6re b\u00fcy\u00fckten k\u00fc\u00e7\u00fcge s\u0131ralan\u0131r. S\u00fctunlar en az veri g\u00f6zlemlenenden en \u00e7ok veri g\u00f6zlemlenene dogru s\u0131ralan\u0131r. Sat\u0131ra \u00f6ncelik vererek s\u00fctunlarla olan kesi\u015fimindeki ilk maskelenmi\u015f veri a\u00e7\u0131l\u0131r.
                                        \nVeri setimizde sat\u0131rlar filmlere kar\u015f\u0131l\u0131k gelmektedir. Bu stratejimizde her filme verilen puan\u0131n varyans\u0131 hesaplan\u0131r. Genellikle bir filme benzer puanlar verilmesi beklenir. Burada,\nvaryans\u0131 b\u00fcy\u00fck olan filmlerin puan aral\u0131\u011f\u0131 daha geni\u015ftir. Bu filmlere verilen puanlar\u0131n tahmin edilmesi daha zordur.\n\n\n```python\ndef openRowVarColMask(W, K, mask, test, xCandidate):\n mean_candidate = mask*xCandidate\n rowIndVarSorted = np.argsort(np.nan_to_num(np.nanvar(np.where(mean_candidate!=0,mean_candidate,np.nan), axis=1)))[::-1]\n \n colSum = mask.sum(0)\n colMins = colSum.argsort()\n \n for r in rowIndVarSorted:\n for c in colMins:\n if mask[r][c] == 0 and test[r][c] == 1:\n return r, c\n \n return openRandom(W, K, mask, test)\n\n```\n\n## Ba\u015flatma\n\n### Model se\u00e7imi\n\n\u00d6nerdigimiz y\u00f6ntemi MovieLens verisi \u00fczerinde deniyoruz. Burada $W = 50$ (sat\u0131r/film say\u0131s\u0131), $K = 20$ (s\u00fctun/kullan\u0131c\u0131 say\u0131s\u0131) ve kaynaklar\u0131n say\u0131s\u0131 $I = 4$ olarak belirledik. Ger\u00e7ek modelin hiperparametrelerini ise $a^{t} = b^{t} = 1$ ve $a^{v} = b^{v} = 1$ olarak ald\u0131k.\n\n\n```python\n# Number of Rows\n# nu: 1 -> W\nW = movieCount;\n\n# Number of Columns\n# tao: 1 -> K\nK = userCount\n\n# Number of templates\nI = 4;\n\n# Set prior parameters \nA_t = np.ones([W,I]) # Shape\nB_t = np.ones([W,I]) # Scale\nA_v = np.ones([I,K])\nB_v = np.ones([I,K])\n\n# Generate a random template and excitation\norgT = T = np.random.gamma(A_t,B_t)\norgV = V = np.random.gamma(A_v,B_v)\n\nx = userMovie.copy()\n```\n\n\n```python\n#strategyLabels = [\"Random\", \"Row Mask Eq\", \"Min Var\", \"Max Var\", \"Row Var Col Mask\"]\nstrategyLabels = [\"Rastgele\", \"Sat\u0131r-S\u00fctun\", \"Min Varyans\", \"Maks Varyans\", \"Sat\u0131r Varyans\"]\nstrategyColors = [\"b\",\"r\",\"y\",\"k\",\"m\"]\n\n#errorMetricLabels = [\"RMSE_Partial\", \"RMSE_Full\", \"KL_Partial\", \"KL_Full\"]\nerrorMetricLabels = [\"KOKH K\u0131smi\", \"KOKH Tam\", \"KL Iraksay\u0131 K\u0131smi\", \"KL Iraksay\u0131 Tam\"]\n\n# dataExistance: True if data exist, False othwerwise\n# mask: True if mask opened, False if masked\n# test: True if not test data and data exist, False if test data or no data exist\n# For testing we use dataExistance and test together\n# For cell opening we use test with mask\ndataExistance = userMovie > 0\nmask, test = randomMasking(W,K,dataExistance)\n\nallLikelihood = []\nallOpenedCells = []\n\nallDiffRMSEPartial = []\nallDiffRMSEFull = []\nallDiffKLPartial = []\nallDiffKLFull = []\n\nallErrorDiffs = []\n\nallRMSEPartial = []\nallRMSEFull = []\nallKLPartial = []\nallKLFull = []\n\nallError = []\n\nallEstimationVariance = []\n\nallStrategyLabels = []\nallStrategyColors = []\n\nKLsampleSize = 1000\n```\n\n#### Olabilirlik hesab\u0131\n\n\n```python\ndef calculateLikelihood(x, xPredicted, test, dataExistance, W, K):\n lh = 0\n for w in range(W):\n for k in range(K):\n lh += (dataExistance[w,k]^test[w,k]) * (x[w,k] * np.log(xPredicted[w,k]) - xPredicted[w,k] - special.gammaln(x[w,k]+1))\n return lh\n```\n\n#### Adaylar\u0131n \u00f6rneklenmesi\n\n\n```python\ndef sampleCandidates(a_t, b_t, a_v, b_v, sampleSize=100):\n xCandidates = []\n for i in range(sampleSize):\n T = np.random.gamma(a_t,b_t)\n V = np.random.gamma(a_v,b_v)\n\n xCandidates.append(np.dot(T, V))\n \n return np.asarray(xCandidates)\n```\n\n#### Varyans\u0131n hesaplanmas\u0131 ve \u00f6rneklenen X adaylar\u0131ndan gelen hata\n\n#### De\u011ferlendirme Metrikleri\n\nYakla\u015f\u0131m\u0131m\u0131z\u0131n performans\u0131n\u0131 de\u011ferlendirmek i\u00e7in \u00e7e\u015fitli y\u00f6ntemler bulunmaktad\u0131r. Bu \u00e7al\u0131\u015fmam\u0131zda, pop\u00fcler olan Kullback-Leibler (KL) \u0131raksay\u0131 ve K\u00f6k Ortalama Kare Hata (KOKH) olmak \u00fczere iki ayr\u0131 metrik kulland\u0131k. Bu metrikler s\u0131ras\u0131yla a\u015fa\u011f\u0131daki gibi tan\u0131mlan\u0131r:\n\n\\begin{align*}\nD_{KL}\\left ( X \\parallel \\widehat{X} \\right ) = \\sum_{i,j} X\\left ( i,j \\right ) \\log \\frac{X\\left ( i,j \\right )}{\\widehat{X}\\left ( i,j \\right )}\n\\end{align*}\n\n\\begin{align*}\nKOKH = \\sqrt{\\frac{1}{X_{test}}\\sum_{i,j}\\left ( X(i,j) - \\widehat{X}(i,j)\\right )^{2}}\n\\end{align*}\nBurada, $\\widehat{X}(i,j)$ \u00f6nerilen metod taraf\u0131ndan *i* filmi hakk\u0131nda *j* kullan\u0131c\u0131s\u0131n\u0131n verdi\u011fi tahmin edilen oylama de\u011ferini ve $X_{test}$ ise, toplam test edilen oylama say\u0131s\u0131n\u0131 g\u00f6stermektedir.\n\n\n```python\ndef calculateVarianceAndError(x, test, dataExistance, xCandidates):\n mean_candidate = np.mean(xCandidates, axis=0)\n varEst = np.var(xCandidates, axis=0)\n varEst = 1.0 * (varEst - varEst.min()) / (varEst.max() - varEst.min())\n\n diffMeanEst = abs((dataExistance^test)*mean_candidate - (dataExistance^test)*x)\n \n return varEst, diffMeanEst\n```\n\n#### Adaylar\u0131n da\u011f\u0131l\u0131ma d\u00f6n\u00fc\u015ft\u00fcr\u00fclmesi\n\n\n```python\ndef transformCandidatesToDistributions(candidates, sampleSize, test, dataExistance):\n candidates = np.round(candidates)\n candidates = np.minimum(candidates,10*np.ones(candidates.shape)).astype(int)\n candidates = np.maximum(candidates,np.ones(candidates.shape)).astype(int)\n candidates = (dataExistance^test) * candidates\n \n candidateDistributions = []\n \n for i in range(W):\n for j in range(K):\n if candidates[0,i,j] != 0:\n y = np.bincount(candidates[:,i,j])\n c = np.zeros(11)\n c[:len(y)] += y\n c = np.maximum(c,0.00000001*np.ones(c.shape))\n c /= sampleSize\n candidateDistributions.append(c[1:])\n else:\n candidateDistributions.append(np.ones(10))\n \n return candidateDistributions\n```\n\n#### KL \u0131raksay\u0131 hesab\u0131\n\n\n```python\ndef calculateKLdivergence(xCandidates, bestCandidateDistributions, sampleSize, test, dataExistance):\n xCandidateDistributions = transformCandidatesToDistributions(xCandidates, sampleSize, test, dataExistance)\n return entropy(np.asarray(xCandidateDistributions).T, np.asarray(bestCandidateDistributions).T).reshape((W,K))\n \n```\n\n## Gibbs \u00d6rnekleyicisi\n\n*Monte Carlo* metodlar\u0131 [4, 5] beklentileri tahmin etmek i\u00e7in g\u00fc\u00e7l\u00fc hesaplama teknikleridir. \n*Markov Zinciri Monte Carlo* teknikleri, ge\u00e7i\u015f \u00e7ekirde\u011fi $\\mathcal{T}$ taraf\u0131ndan tan\u0131mlanan bir Markov zincirinden sonraki \u00f6rnekleri \u00fcretir yani $x^{i}$'e \u015fartlanm\u0131\u015f $x^{i+1}$ \u015fu \u015fekilde \u00fcretilir:\n\n\\begin{equation}\nx^{i+1} \\sim \\mathcal{T}(x\\mid x^{i})\n\\end{equation}\n\n\u0130stenen da\u011f\u0131l\u0131m durgun da\u011f\u0131l\u0131m olacak \u015fekilde bir $\\mathcal{T}$ ge\u00e7i\u015f \u00e7ekirde\u011fi tasarlamak. \u00d6zellikle kullan\u0131\u015fl\u0131 ve basit bir prosed\u00fcr olan Gibbs \u00f6rnekleyicisi'nde her de\u011fi\u015fken *tam ko\u015fullu da\u011f\u0131l\u0131mlardan* \u00f6rneklenir. NOMA modeli i\u00e7in Gibbs \u00f6rnekleyicisi,\n\n\\begin{eqnarray}\nS^{n+1} &\\sim & p(S\\mid T^{n}, V^{n}, X, \\theta), \\nonumber \\\\\nT^{n+1} &\\sim & p(T\\mid V^{n}, S^{n+1}, X, \\theta), \\\\\nV^{n+1} &\\sim & p(V\\mid S^{n+1}, T^{n+1}, X, \\theta). \\nonumber\n\\end{eqnarray}\n\nSabit nokta d\u00f6ng\u00fcs\u00fc sakl\u0131 kaynaklar olarak adland\u0131rd\u0131\u011f\u0131m\u0131z $S_{i} = \\left \\{ s_{\\nu,i,\\tau} \\right \\}$ i\u00e7in $(m_{\\nu, \\tau}=1)$ a\u015fa\u011f\u0131daki gibi bulunur:\n\n\\begin{eqnarray}\nq(s_{\\nu,1:I,\\tau}) &=& \\mathcal{M} (s_{\\nu,1:I, \\tau}; x_{\\nu,\\tau}, p_{\\nu, 1:I, \\tau}) \\\\\np_{\\nu, i, \\tau} &=& \\frac{exp(\\left \\langle \\log t_{\\nu,i} \\right \\rangle + \\left \\langle \\log v_{i,\\tau} \\right \\rangle)}{\\sum_{i} exp(\\left \\langle \\log t_{\\nu,i} \\right \\rangle + \\left \\langle \\log v_{i, \\tau} \\right \\rangle)} \\\\\n\\end{eqnarray}\n\n\u015eablon $T$ ve katsay\u0131 $V$ matrislerinin da\u011f\u0131l\u0131mlar\u0131 ve onlar\u0131n yeterli istatistikleri Gamma da\u011f\u0131l\u0131m\u0131n\u0131n \u00f6zelliklerini takip eder:\n\n\\begin{eqnarray}\nq(t_{\\nu,i}) &=& \\mathcal{G} (t_{\\nu,i}; \\alpha_{\\nu,i}^{t}, \\beta_{\\nu,i}^{t}) \\\\\n\\alpha_{\\nu, i}^{t} &=& a^{t} + \\sum_{\\tau} m_{\\nu, \\tau} \\left \\langle s_{\\nu, i, \\tau} \\right \\rangle \\\\\n\\beta_{\\nu, i}^{t} &=& \\left ( \\frac{a^{t}}{b^{t}} + \\sum_{\\tau } m_{\\nu, \\tau} \\left \\langle v_{i, \\tau} \\right \\rangle \\right )^{-1} \\\\\nq(v_{i,\\tau}) &=& \\mathcal{G} (v_{i,\\tau}; \\alpha_{i,\\tau}^{v}, \\beta_{i,\\tau}^{v}) \\\\\n\\alpha_{i, \\tau}^{v} &=& a^{v} + \\sum_{\\nu } m_{\\nu, \\tau} \\left \\langle s_{\\nu, i, \\tau} \\right \\rangle \\\\\n\\beta_{i, \\tau}^{v} &=& \\left ( \\frac{a^{v}}{b^{v}} + \\sum_{\\nu } m_{\\nu, \\tau} \\left \\langle t_{\\nu, i} \\right \\rangle \\right )^{-1}\n\\end{eqnarray}\n\n\n\n```python\ndef gibbsSampler(x, T, V, maskX, MAXITER, likelihood = None, test = None, dataExistance = None):\n W = T.shape[0]\n K = V.shape[1]\n I = T.shape[1]\n tt = 0\n t00 = time.time()\n \n for n in range(MAXITER):\n Tprev = T.copy()\n Vprev = V.copy()\n\n S = np.ones([I,W,K])\n \n t0 = time.time()\n \n # Sample Sources\n TdotV = np.dot(Tprev, Vprev)\n p = np.einsum('i...,k...',Tprev, np.transpose(Vprev))/np.array([TdotV]*I)\n for nu in range(W):\n for tao in range(K):\n if maskX[nu,tao] == 0:\n S[:,nu,tao] = 0\n else:\n S[:,nu,tao] = np.random.multinomial(x[nu,tao], p[:,nu,tao], size=1)\n \n \n sigmaT = np.transpose(np.sum(maskX*S, axis=2))\n sigmaV = np.sum(maskX*S, axis=1)\n \n\n # Sample Templates\n a_t = A_t + sigmaT\n b_t = 1 / ( np.divide(A_t, B_t) + np.dot(maskX, np.transpose(Vprev)) )\n\n T = np.random.gamma(a_t,b_t)\n\n # Sample Excitations\n a_v = A_v + sigmaV\n b_v = 1 / ( np.divide(A_v, B_v) + np.dot(np.transpose(Tprev),maskX) )\n\n V = np.random.gamma(a_v,b_v)\n \n if likelihood != None:\n likelihood.append(calculateLikelihood(x, np.dot(T, V), test, dataExistance, W, K))\n \n if likelihood == None:\n return T, V, a_t, b_t, a_v, b_v\n else:\n return T, V, a_t, b_t, a_v, b_v, likelihood\n```\n\n## En iyi ayr\u0131\u015ft\u0131rman\u0131n hesaplanmas\u0131\n\n\n```python\nt0 = time.time()\n\nT = orgT.copy()\nV = orgV.copy()\nmaskX = dataExistance.copy()\nMAXITER = 10000\n\nT, V, a_t, b_t, a_v, b_v = gibbsSampler(x, T, V, maskX, MAXITER)\nbestCandidates = sampleCandidates(a_t, b_t, a_v, b_v, sampleSize=1000)\nbestCandidatesMean = np.mean(bestCandidates, axis=0)\nbestCandidateDistributionsFull = transformCandidatesToDistributions(bestCandidates, KLsampleSize, test&False, dataExistance|True)\nbestCandidateDistributionsPartial = transformCandidatesToDistributions(bestCandidates, KLsampleSize, test, dataExistance)\n \n```\n\n### \u0130lk-\u0131s\u0131nma (Burn-in) periyodu\n\n1000 ad\u0131ml\u0131k bir ilk-\u0131s\u0131nma devresi uygulad\u0131k.\n\n\n```python\nt0 = time.time()\nT = orgT.copy()\nV = orgV.copy()\nmaskX = mask.copy()\n\nT, V, _, _, _, _ = gibbsSampler(x, T, V, maskX, MAXITER = 1000)\n\nmodT = T.copy()\nmodV = V.copy()\nprint(time.time()-t0)\n```\n\n 1.1921601295471191\n\n\n### Gibbs \u00f6rnekleyicisi ile e\u011fitim (daha \u00e7ok veri a\u00e7ma)\n\n\n```python\nfor cellOpenStrategy in range(5):\n t0 = time.time()\n \n likelihood = []\n openedCells = [0]\n \n diffErrorPartial = []\n diffErrorFull = []\n diffKLPartial = []\n diffKLFull = []\n varEst = []\n\n T = modT.copy()\n V = modV.copy()\n\n maskX = mask.copy()\n\n cellOpenCount = 10\n extraIter = 20\n EPOCH = int((dataExistance.sum()-(dataExistance^test).sum()-mask.sum())//cellOpenCount + extraIter)\n MAXITER = 20\n\n for nn in range(EPOCH):\n # Apply gibbs sampler\n if nn >= (dataExistance.sum()-(dataExistance^test).sum()-mask.sum())//cellOpenCount:\n MAXITER = 50\n T, V, a_t, b_t, a_v, b_v, likelihood = gibbsSampler(x, T, V, maskX, MAXITER, likelihood, test, dataExistance)\n\n # Take Mean estimate and calculate diff from X\n xCandidates = sampleCandidates(a_t, b_t, a_v, b_v, sampleSize=100)\n\n ve, de = calculateVarianceAndError(bestCandidatesMean, test, dataExistance, xCandidates)\n varEst.append(ve)\n diffErrorPartial.append(de)\n \n _, de = calculateVarianceAndError(bestCandidatesMean, test&False, dataExistance|True, xCandidates)\n diffErrorFull.append(de)\n \n de = calculateKLdivergence(xCandidates, bestCandidateDistributionsPartial, KLsampleSize, test, dataExistance)\n diffKLPartial.append(de)\n \n de = calculateKLdivergence(xCandidates, bestCandidateDistributionsFull, KLsampleSize, test&False, dataExistance|True)\n diffKLFull.append(de)\n\n # Apply Cell Opening Strategy\n for co in range(cellOpenCount):\n if cellOpenStrategy == 0:\n row, col = openRandom(W, K, maskX, test)\n elif cellOpenStrategy == 1:\n if nn % 2 == 0:\n row, col = openMaxColMask(W, K, maskX, test)\n else:\n row, col = openMaxRowMask(W, K, maskX, test)\n elif cellOpenStrategy == 2:\n row, col = openMinVar(W, K, maskX, test, xCandidates)\n elif cellOpenStrategy == 3:\n row, col = openMaxVar(W, K, maskX, test, xCandidates)\n elif cellOpenStrategy == 4:\n row, col = openRowVarColMask(W, K, maskX, test, np.dot(T,V))\n else:\n row, col = (-1, -1)\n\n\n # Remove mask from (row, col)\n if not (row == -1 and col == -1):\n maskX[row][col] = 1\n openedCells.append((row,col))\n\n\n allStrategyLabels.append(cellOpenStrategy)\n allStrategyColors.append(cellOpenStrategy)\n\n allLikelihood.append(likelihood)\n allOpenedCells.append(openedCells)\n\n allEstimationVariance.append(varEst)\n \n\n allDiffRMSEPartial.append(diffErrorPartial)\n allDiffRMSEFull.append(diffErrorFull)\n allDiffKLPartial.append(diffKLPartial)\n allDiffKLFull.append(diffKLFull)\n \n \n rmse = np.zeros(len(diffErrorPartial))\n for i in range(len(diffErrorPartial)):\n # RMSE\n de2 = diffErrorPartial[i]*diffErrorPartial[i]\n rmse[i] = np.sqrt(np.nanmean(np.where(de2!=0,de2,np.nan)))\n allRMSEPartial.append(rmse)\n \n rmse = np.zeros(len(diffErrorFull))\n for i in range(len(diffErrorFull)):\n # RMSE\n de2 = diffErrorFull[i]*diffErrorFull[i]\n rmse[i] = np.sqrt(np.nanmean(np.where(de2!=0,de2,np.nan)))\n allRMSEFull.append(rmse)\n \n kl = np.zeros(len(diffKLPartial))\n for i in range(len(diffKLPartial)):\n kl[i] = np.mean(diffKLPartial[i])\n allKLPartial.append(kl)\n \n kl = np.zeros(len(diffKLFull))\n for i in range(len(diffKLFull)):\n kl[i] = np.mean(diffKLFull[i])\n allKLFull.append(kl)\n \n \n print(strategyLabels[cellOpenStrategy] + \" %0.3fs. de tamamland\u0131.\" % (time.time() - t0))\n \nallErrorDiffs.append(allDiffRMSEPartial)\nallErrorDiffs.append(allDiffRMSEFull)\nallErrorDiffs.append(allDiffKLPartial)\nallErrorDiffs.append(allDiffKLFull)\n\nallError.append(allRMSEPartial)\nallError.append(allRMSEFull)\nallError.append(allKLPartial)\nallError.append(allKLFull)\n```\n\n Rastgele compeleted in 25.083s.\n Sat\u0131r-S\u00fctun compeleted in 25.324s.\n Min Varyans compeleted in 25.153s.\n Maks Varyans compeleted in 25.120s.\n\n\n C:\\Users\\Burki\\Anaconda3\\lib\\site-packages\\ipykernel\\__main__.py:3: RuntimeWarning: Degrees of freedom <= 0 for slice.\n app.launch_new_instance()\n\n\n Sat\u0131r Varyans compeleted in 24.291s.\n\n\n## Etkile\u015fimli Hata ve Varyans Is\u0131 haritas\u0131\n\nSat\u0131r-s\u00fctun stratejisi ve KL \u0131raksay\u0131 hata metri\u011fi yo\u011fun matris \u00fczerinde kullan\u0131lm\u0131\u015f bir deney s\u00fcreci g\u00f6sterilmektedir. \u00c7\u0131kt\u0131da hata ve varyans \u0131s\u0131 haritas\u0131n\u0131 g\u00f6rmekteyiz. Bu grafikte, k\u0131rm\u0131z\u0131 ile hata, mavi ile maskelenmi\u015f verinin varyans\u0131 ve beyaz ile ise g\u00f6zlemlenmi\u015f veri g\u00f6sterilmektedir. K\u0131rm\u0131z\u0131 ve mavinin tonlar\u0131 hata ve varyans\u0131n \u015fiddetini yans\u0131tmaktad\u0131r. Yinelemeler s\u00fcresince veriler g\u00f6zlemlendik\u00e7e maskelenmi\u015f (mavilikler) verinin kayboldu\u011funu (beyaza d\u00f6nmesini), hatan\u0131n (k\u0131rm\u0131z\u0131l\u0131klar\u0131n) ise azald\u0131\u011f\u0131n\u0131 grafikten g\u00f6rmekteyiz. Sa\u011fdaki grafikte KL \u0131raksay\u0131 kullan\u0131larak elde edilen hata grafi\u011fini ve ona oturtulan polinomu g\u00f6rmekteyiz.\n\n\n```python\ncmap = matplotlib.colors.LinearSegmentedColormap.from_list('my_colormap',\n ['blue','white', 'red'],\n 256) \n\nchosenStrategy = 1\nchosenErrorMetric = 2\n\nrmseHM = allError[chosenErrorMetric][chosenStrategy]\nmatrixHM = allErrorDiffs[chosenErrorMetric][chosenStrategy]\nopenedCellsHM = allOpenedCells[chosenStrategy]\nvarEstHM = allEstimationVariance[chosenStrategy]\n\nxAxis = list(range(len(rmseHM)))\nxp = np.linspace(0, EPOCH-1, 1000)\npolDeg = 10\np30 = np.poly1d(np.polyfit(xAxis, rmseHM, polDeg))\n\nvmax = rmseHM.max()\n \n#matrixHM = diffErrorHM.copy()\nfor i in range(EPOCH-extraIter):\n #if openedCellsHM[i] == (-1,-1):\n # break\n for oc in openedCellsHM[1+(i*cellOpenCount):]:\n if oc[0] == -1:\n break\n matrixHM[i][oc[0],oc[1]] = -varEstHM[i][oc[0],oc[1]]*vmax\n \n\ndef pltErrorVarianceHM(rc):\n fig = plt.figure(figsize=(8,8))\n plt.subplot(1, 2, 1)\n if rc == 0:\n plt.title(\"\u0130lk Durum\")\n elif rc <= (EPOCH-extraIter):\n plt.title(\"Yineleme Say\u0131s\u0131: \" + str(rc))\n else:\n plt.title(\"Yineleme Say\u0131s\u0131: \" + str(rc))\n \n img = plt.imshow(matrixHM[rc],interpolation='nearest',\n cmap = cmap,\n origin='lower',\n vmax = vmax,\n vmin = -vmax)\n plt.colorbar(img,cmap=cmap)\n \n plt.subplot(1, 2, 2)\n plt.plot( range(rc+1), rmseHM[:rc+1])\n \n try:\n xpRC = list(xp).index(xp[xp>=(rc+1)][0])\n except:\n xpRC = list(xp).index(xp[xp>=(rc)][0])\n plt.plot( xp[:xpRC], p30(xp)[:xpRC], \"-\", color='r', linewidth=2)\n \n plt.xlim(0,EPOCH)\n #plt.ylim(0,rmseHM.max()+1)\n font = {'size' : 15}\n #plt.title(\"KL : \" + str(p30(xp)[xpRC-1]))\n plt.xlabel(\"Yineleme Say\u0131s\u0131\", **font)\n plt.ylabel(errorMetricLabels[chosenErrorMetric], **font)\n plt.tight_layout()\n plt.show()\n \n \ninteract(pltErrorVarianceHM, rc = (0, EPOCH-1, 1))\n```\n\n## Yo\u011fun ve seyrek veri \u00fczerinde, tan\u0131mlanm\u0131\u015f be\u015f strateji i\u00e7in 10\u2019lu a\u00e7arak k\u0131smi KL \u0131raksay\u0131 de\u011fi\u015fim grafi\u011fi\n\nStratejilerin performanslar\u0131n\u0131, e\u015fik de\u011ferine ne kadar h\u0131zl\u0131 ula\u015ft\u0131klar\u0131n\u0131 kar\u015f\u0131la\u015ft\u0131rarak \u00f6l\u00e7t\u00fck. E\u015fik de\u011ferine ula\u015fma metri\u011fi olarak *e\u011fri alt\u0131nda kalan alan\u0131* kulland\u0131k. En iyi strateji, e\u011fri alt\u0131nda kalan alan\u0131 en az oland\u0131r.\n\nHata fonksiyonlar\u0131n\u0131n as\u0131l davran\u0131\u015f\u0131n\u0131 g\u00f6zlemleyebilmek i\u00e7in fonksiyonlara polinom oturttuk. Bu polinomlar\u0131n sal\u0131n\u0131mdan daha az etkilenmesi i\u00e7in ise b\u00fct\u00fcn veriler g\u00f6zlemlendikten\nsonra 1000 yinelemeli Gibbs \u00f6rnekleyicisi \u00e7al\u0131\u015ft\u0131rd\u0131k.\n\n\n```python\nxAxis = list(range(len(allKLPartial[0])))\nxp = np.linspace(0, EPOCH-1, 1000)\npolDeg = 15\n\nchosenErrorMetric = 2\n\nerrorFunction = allError[chosenErrorMetric].copy()\nthr = 0\nfor rmse in errorFunction:\n thr += np.mean(rmse[-15:-5])\nthr /= len(errorFunction)\n\n\nfig = plt.figure(figsize=(10,10))\n\nplt.plot(range(EPOCH - extraIter+15), (EPOCH - extraIter+15)*[thr], \"--\", color='c', label=\"E\u015fik: \"+str(thr)[:5], linewidth=2)\n\naucTrapz = []\naucSimps = []\n\n\nxpRC = list(xp).index(xp[xp>=(EPOCH - extraIter+10)][0])\n\nfor i, rmse in enumerate(errorFunction):\n p30 = np.poly1d(np.polyfit(xAxis, rmse, polDeg))\n aucTrapz.append(np.trapz(p30(xp)[:xpRC]-thr, x=xp[:xpRC]))\n aucSimps.append(np.trapz(p30(xp)[:xpRC], x=xp[:xpRC]))\n print(np.trapz(p30(xp)[:xpRC]-thr, x=xp[:xpRC]))\n zz = i\n if i == 1:\n zz = 10\n plt.plot( xp[:xpRC], p30(xp)[:xpRC], \"-\", label=strategyLabels[allStrategyLabels[i]], color=strategyColors[allStrategyColors[i]], linewidth=2, zorder=zz)\n\n \nplt.xlim(0,)\n#plt.ylim(0,)\n\nfont = {'size' : 18}\nplt.xlabel(\"Yineleme Say\u0131s\u0131 (\" + str(cellOpenCount) + \"'lu a\u00e7ma)\", **font)\nplt.ylabel(errorMetricLabels[chosenErrorMetric], **font)\n\nplt.legend()\nplt.show()\n\n```\n\n## Varg\u0131lar\n\nBu \u00e7al\u0131\u015fmam\u0131zda negatif olmayan matris ayr\u0131\u015f\u0131m\u0131 i\u00e7in e\u015flenik Gamma \u00f6nselleri ile hiyerar\u015fik bir model inceledik ve \u00e7\u0131kar\u0131mlar i\u00e7in Gibbs \u00f6rnekleyicisi kulland\u0131k. Buradan yola \u00e7\u0131karak aktif eleman se\u00e7imi [7] problemine \u00e7\u00f6z\u00fcm \u00f6nerdik. Tan\u0131mlad\u0131\u011f\u0131m\u0131z be\u015f stratejiyi KL \u0131raksay\u0131 ve KOKH metrikleri \u00fczerinden kar\u015f\u0131la\u015ft\u0131rd\u0131k. Yapt\u0131\u011f\u0131m\u0131z deneyler ile sat\u0131r-s\u00fctun stratejisinin etkili bir aktif \u00f6grenme tekni\u011fi oldu\u011funu g\u00f6sterdik. Sat\u0131r-s\u00fctun stratejisinin ba\u015far\u0131s\u0131n\u0131n g\u00f6zlemlenmemi\u015f veriyi dengeli bir bi\u00e7imde a\u00e7\u0131yor olmas\u0131na ba\u011fl\u0131yoruz. Ayr\u0131ca bu stratejinin verinin i\u00e7eri\u011finden ba\u011f\u0131ms\u0131z olarak tan\u0131mlanm\u0131\u015f olmas\u0131 bu stratejiyi di\u011fer alanlara da uygulanabilir k\u0131l\u0131yor.\n\n## Kaynaklar\n\n[1] D. D. Lee and H. S. Seung, \"Learning the parts of objects\nwith nonnegative matrix factorization.\", Nature, 401:788\u2013791, 1999.\n\n[2] A. T. Cemgil, \"Bayesian inference in non-negative matrix\nfactorisation models.\", Technical Report CUED/FINFENG/TR.609, University of Cambridge, July 2008. Submitted for publication to Computational Intelligence and Neuroscience\n\n[3] D. D. Lee and H. S. Seung, \"Algorithms for non-negative matrix factorization.\", Advances in neural information processing systems. 2001.\t\n \n[4] J. S. Liu, \"Monte Carlo Strategies in Scientific Computing\", Springer, New York, NY, USA, 2004.\n \n[5] W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, Eds., \"Markov Chain Monte Carlo in Practice\", CRC Press, London, UK, 1996.\n \n[6] Harper, F. Maxwell, and Joseph A. Konstan, \"The movielens datasets: History and context.\", ACM Transactions on Interactive Intelligent Systems (TiiS) 5.4 (2016): 19.\n \n[7] Silva, Jorge, and Lawrence Carin. \"Active learning for online bayesian matrix factorization.\", Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining. ACM, 2012.\n", "meta": {"hexsha": "486fb291ae36ad81e29290d1976755b523758fa2", "size": 259491, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Aktif-Ogrenme-BNMF.ipynb", "max_stars_repo_name": "suyunu/AL-BNMF", "max_stars_repo_head_hexsha": "9785873373d514405715594cc1f0c47a5307e2c5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Aktif-Ogrenme-BNMF.ipynb", "max_issues_repo_name": "suyunu/AL-BNMF", "max_issues_repo_head_hexsha": "9785873373d514405715594cc1f0c47a5307e2c5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Aktif-Ogrenme-BNMF.ipynb", "max_forks_repo_name": "suyunu/AL-BNMF", "max_forks_repo_head_hexsha": "9785873373d514405715594cc1f0c47a5307e2c5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 125.0559036145, "max_line_length": 129404, "alphanum_fraction": 0.8260671854, "converted": true, "num_tokens": 15345, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185498374788, "lm_q2_score": 0.734119521083126, "lm_q1q2_score": 0.4155252667308554}} {"text": "### This notebook computes the joint likelihood in 5 dimensions for the parameters of interest (see below). See inference_demo for more details about basic usage\n\n\n```python\nfrom lenslikelihood.measurements import flux_measurements, flux_measurement_uncertainties, all_lens_names, all_param_ranges\n\n# Note that the syntax for the uncertainties is\n# {'lens_name': (1-sigma-uncertainty, reference_index, uncertainty_in_ratio)}\n# where reference_index is the reference image with which to compute the flux ratio, and uncertainty_in_ratio specifies\n# whether the measurement uncertainty refers to the flux or the flux ratio\n\nfor name in all_lens_names:\n print(name)\n print('fluxes/flux ratios measured: ', flux_measurements[name])\n print('uncertainties: ', flux_measurement_uncertainties[name])\n print('\\n')\n```\n\n WGDJ0405\n fluxes/flux ratios measured: [0.8 0.52 1. 0.94]\n uncertainties: ([0.04, 0.061538461538461535, 0.024, 0.03418803418803419], 0, False)\n \n \n HE0435\n fluxes/flux ratios measured: [0.96 0.976 1. 0.65 ]\n uncertainties: ([0.05, 0.049, 0.048, 0.056], 0, False)\n \n \n WGD2038\n fluxes/flux ratios measured: [0.86 1. 0.79 0.4 ]\n uncertainties: ([0.01, 0.01724137931034483, 0.021739130434782608, 0.021739130434782608], 0, False)\n \n \n B1422\n fluxes/flux ratios measured: [0.88 1. 0.474 0.025]\n uncertainties: ([0.011363636363636364, 0.01, 0.012765957446808512, None], 0, False)\n \n \n WFI2033\n fluxes/flux ratios measured: [1. 0.65 0.5 0.53]\n uncertainties: ([0.03, 0.046875, 0.04, 0.03773584905660377], 0, False)\n \n \n PSJ1606\n fluxes/flux ratios measured: [1. 1. 0.59 0.79]\n uncertainties: ([0.03, 0.03, 0.03333333333333333, 0.02564102564102564], 0, False)\n \n \n WFI2026\n fluxes/flux ratios measured: [1. 0.75 0.31 0.28]\n uncertainties: ([0.02, 0.02666666666666667, 0.06451612903225806, 0.03571428571428571], 0, False)\n \n \n RXJ0911\n fluxes/flux ratios measured: [0.56 1. 0.53 0.24]\n uncertainties: ([0.07142857142857142, 0.05, 0.07547169811320754, 0.16666666666666669], 0, False)\n \n \n RXJ1131\n fluxes/flux ratios measured: [1. 0.61 0.73 0.12]\n uncertainties: ([[0.012345679012345678, 0.024691358024691357], [0.10084033613445378, 0.025210084033613446], None], 1, True)\n \n \n MG0414\n fluxes/flux ratios measured: [1. 0.83 0.36 0.16]\n uncertainties: ([0.06024096385542169, 0.11111111111111112, 0.11764705882352941], 0, True)\n \n \n PG1115\n fluxes/flux ratios measured: [1. 0.93 0.16 0.21]\n uncertainties: ([0.06451612903225806, 0.43750000000000006, 0.1904761904761905], 0, True)\n \n \n\n\n### Models implemented for the halo mass function and concentration-mass relation\n\nThe full set of hyper-parameters we're interested in constraining are defined by the parameterizations of the halo mass function and concentration-mass relation. They are $\\Sigma_{\\rm{sub}}$, $\\delta_{\\rm{LOS}}$, $\\Delta \\alpha$, $q$, $c_8$, and $\\beta$. The first four define to the subhalo and field halo mass functions, and the last two define the concentration-mass relation. \n\nThe field halo mass function is parameterized as\n\\begin{equation}\n\\frac{dN_{\\rm{LOS}}}{dm dV} = \\delta_{\\rm{LOS}} \\left(1+\\xi_{\\rm{2halo}}\\right) \\left(\\frac{m}{10^8}\\right)^{\\Delta \\alpha} \\ \\frac{dN_{\\rm{ShethTormen}}}{dm dV}\n\\end{equation}\nwhere $\\delta_{\\rm{LOS}}$ scales the overall normalization, and $\\Delta \\alpha$ parameterizes deviations from the logarithmic slope predicted by CDM around $10^8 M_{\\odot}$. \n\nThe subhalo mass function is parameterized as\n\\begin{equation}\n\\frac{dN_{\\rm{sub}}}{dm dA} \\sim \\Sigma_{\\rm{sub}} \\ \\left(\\frac{m}{10^8}\\right)^{\\alpha + q \\Delta \\alpha}\n\\end{equation}\nwhere $\\Sigma_{\\rm{sub}}$ is the normalization, $\\alpha$ is the logarithmic slope predicted by CDM, $\\Delta \\alpha$ parameterizes deviations from the value predicted by CDM, and $q$ controls the coupling between the line of sight halo mass function slope and the subhalo mass function slope. When $q=1$ the slopes change in the same way, and when $q=0$ the slopes of the subhalo and field halo mass functions are completely decoupled. \n\nThe concentration-mass relation is parameterized as \n\n\\begin{equation}\nc\\left(M, z\\right) = c_8 \\left(1+z\\right)^{\\zeta} \\left(\\frac{\\nu\\left(M, z\\right)}{\\nu\\left(10^8, z\\right)}\\right)^{-\\beta}\n\\end{equation}\ni.e. it is a power-law in the peak height $\\nu$ with normalization $c_8$ at $10^8$ and a logarithmic slope $\\beta$. The parameter $\\zeta$ modifies the redshift evolution and is marginalized over in the sampling. \n\nThe parameter names used in the python code have the following correspondence: \n\n\n1) sigma_sub = $\\Sigma_{\\rm{sub}}$\n\n2) delta_power_law_index = $\\Delta \\alpha$\n\n3) c0 = $c_8$\n\n4) beta = $\\beta$\n\n5) delta_power_law_index_coupling = $q$\n\n6) LOS_normalization = $\\delta_{\\rm{LOS}}$\n\n### Example inference on three parameters with a subset of lenses\n\nFirst load the model samples, define what parameters we want to look at\n\n\n```python\nimport pickle\nimport matplotlib.pyplot as plt\n# specify the parameter names\nparam_names = ['LOS_normalization', 'beta', 'c0', 'delta_power_law_index', 'sigma_sub']\nparam_ranges = [all_param_ranges[name] for name in param_names]\nprint(param_ranges)\n# specify the lenses to use\nlenses = all_lens_names\n\nraw_samples_dict = {}\n# load the forward model samples\nfor lens in lenses:\n print(lens)\n f = open('./../raw_samples/'+lens, 'rb')\n raw_samples = pickle.load(f)\n f.close()\n raw_samples_dict[lens] = raw_samples \n \nprint(raw_samples)\n```\n\n [[0.0, 2.0], [0.0, 6.0], [1, 200], [-0.6, 0.9], [0, 0.1]]\n WGDJ0405\n HE0435\n WGD2038\n B1422\n WFI2033\n PSJ1606\n WFI2026\n RXJ0911\n RXJ1131\n MG0414\n PG1115\n \n\n\nThe next cell computes the joint likelihood without any importance sampling weights.\n\n\n```python\n# compute the summary statistics, retaining the 1500 sets of model parameters with lowest summary statistic\nn_samples_keep = 1000\nsamples_dict = {}\nfig = plt.figure(1)\nfig.set_size_inches(8,8)\nfor lens in lenses:\n print('working on lens '+str(lens) + '... ')\n importance_sampling_weights = None\n measured_fluxes = flux_measurements[lens]\n measurement_uncertainties = flux_measurement_uncertainties[lens][0]\n reference_index = flux_measurement_uncertainties[lens][1]\n uncertaintiy_in_flux_ratios = flux_measurement_uncertainties[lens][2]\n\n samples, full_samples, statistic = raw_samples_dict[lens].sample_with_abc(measured_fluxes, \n param_names, \n measurement_uncertainties, \n reference_index, \n n_samples_keep, \n uncertaintiy_in_ratios=uncertaintiy_in_flux_ratios,\n importance_sampling_weights=importance_sampling_weights)\n plt.hist(statistic, alpha=0.5, label=lens)\n samples_dict[lens] = samples\n\n# the histogram shows the distribution of the retained summary statistics\nplt.legend(fontsize=12)\nplt.xlim(0., 0.15)\nplt.xlabel('summary statistic', fontsize=14)\nplt.show()\n```\n\nThe next cell computes the joint likelihood with importance sampling weights, selecting only samples with delta_power_law_index_coupling (or 'q') close to 1. This makes the subhalo mass function track the slope of the field halo mass function, hence the name \"coupled\"\n\n\n```python\n# compute the summary statistics, retaining the 1500 sets of model parameters with lowest summary statistic\nsamples_dict_coupled = {}\nfig = plt.figure(1)\nfig.set_size_inches(8,8)\nfor lens in lenses:\n print('working on lens '+str(lens) + '... ')\n importance_sampling_weights = {'delta_power_law_index_coupling': [1., 0.25]}\n measured_fluxes = flux_measurements[lens]\n measurement_uncertainties = flux_measurement_uncertainties[lens][0]\n reference_index = flux_measurement_uncertainties[lens][1]\n uncertaintiy_in_flux_ratios = flux_measurement_uncertainties[lens][2]\n samples, full_samples, statistic = raw_samples_dict[lens].sample_with_abc(flux_measurements[lens], \n param_names, \n measurement_uncertainties, \n reference_index, \n n_samples_keep, \n uncertaintiy_in_ratios=uncertaintiy_in_flux_ratios,\n importance_sampling_weights=importance_sampling_weights)\n plt.hist(statistic, alpha=0.5, label=lens)\n samples_dict_coupled[lens] = samples\n \n# the histogram shows the distribution of the retained summary statistics\nplt.legend(fontsize=12)\nplt.xlim(0., 0.2)\nplt.xlabel('summary statistic', fontsize=14)\nplt.show()\n```\n\nNow we compute the likelihood using the package trikde https://github.com/dangilman/trikde\n\n\n```python\nfrom trikde.pdfs import DensitySamples, IndepdendentLikelihoods\nimport os\n\nnbins = 20\nlikelihoods, likelihoods_coupled = [], []\n\nload_from_pickle = False # if True, will look for a pre-computed DensitySamples class\nsave_to_pickle = True # if True, will pickle each class for accelerated later use; \n# save_to_pickle=True will do nothing if load_from_pickle=True\n\nfilename_extension = '_joint'\nfilename_extension_coupled = '_joint_coupled'\nbase_path = './../lenslikelihood/precomputed_likelihoods/'\n\nfor lens in lenses:\n \n fname = base_path + lens + filename_extension\n if load_from_pickle and os.path.exists(fname):\n print('loading joint likelihoods for lens '+lens+' ...')\n f = open(fname, 'rb')\n single_lens_likelihood = pickle.load(f)\n f.close()\n else:\n print('computing joint likelihoods for lens '+lens+' ...')\n lens_samples = samples_dict[lens]\n weights = None\n single_lens_likelihood = DensitySamples(lens_samples, param_names, weights, \n param_ranges, nbins=nbins, use_kde=True, bandwidth_scale=1.)\n if save_to_pickle:\n f = open(fname, 'wb')\n pickle.dump(single_lens_likelihood, f)\n f.close()\n likelihoods.append(single_lens_likelihood)\n \n fname = base_path + lens + filename_extension_coupled\n if load_from_pickle and os.path.exists(fname):\n f = open(fname, 'rb')\n single_lens_likelihood = pickle.load(f)\n f.close()\n else:\n lens_samples = samples_dict_coupled[lens]\n weights = None\n single_lens_likelihood = DensitySamples(lens_samples, param_names, weights, \n param_ranges, nbins=nbins, use_kde=True, bandwidth_scale=1.)\n if save_to_pickle:\n f = open(fname, 'wb')\n pickle.dump(single_lens_likelihood, f)\n f.close()\n likelihoods_coupled.append(single_lens_likelihood)\n \nlikelihood = IndepdendentLikelihoods(likelihoods)\nlikelihood_coupled = IndepdendentLikelihoods(likelihoods_coupled)\n```\n\n computing joint likelihoods for lens WGDJ0405 ...\n computing joint likelihoods for lens HE0435 ...\n computing joint likelihoods for lens WGD2038 ...\n computing joint likelihoods for lens B1422 ...\n computing joint likelihoods for lens WFI2033 ...\n computing joint likelihoods for lens PSJ1606 ...\n\n\n### The joint likelihood/posterior with no modeling assumptions (indepedent, uniform priors on all model parameters). \n\nThis likelihood includes a marginalization over a term that couples the subhalo mass function slope to the field halo mass function slope\n\n\n```python\nfrom trikde.triangleplot import TrianglePlot\n\ntriangle_plot = TrianglePlot([likelihood])\ntriangle_plot.set_cmap('magma')\naxes = triangle_plot.make_triplot(filled_contours=False, show_intervals=False, show_contours=False\n )\n# can change axis labels\n```\n\n### The joint likelihood/posterior assuming the slope of the subhalo mass function tracks the slope of the field halo mass function. \n\nThis likelihood has a Gaussian prior on q (see equation for subhalo mass function at the top of this notebook) with a mean of 1 and a variance of 0.25. \n\n\n```python\nfrom trikde.triangleplot import TrianglePlot\n\ntriangle_plot = TrianglePlot([likelihood_coupled])\ntriangle_plot.cmap = 'magma'\naxes = triangle_plot.make_triplot(filled_contours=False, show_intervals=False, show_contours=False)\n# can change axis labels\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "1f376768ea48befb6be3500ec7d8b1e8cec3fb67", "size": 76895, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/inference_5D_from_scratch.ipynb", "max_stars_repo_name": "dangilman/lenslikelihood", "max_stars_repo_head_hexsha": "1490ee9756b4d2ed108a2478977609bbe0ba2e17", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/inference_5D_from_scratch.ipynb", "max_issues_repo_name": "dangilman/lenslikelihood", "max_issues_repo_head_hexsha": "1490ee9756b4d2ed108a2478977609bbe0ba2e17", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/inference_5D_from_scratch.ipynb", "max_forks_repo_name": "dangilman/lenslikelihood", "max_forks_repo_head_hexsha": "1490ee9756b4d2ed108a2478977609bbe0ba2e17", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 150.7745098039, "max_line_length": 28588, "alphanum_fraction": 0.8603940438, "converted": true, "num_tokens": 3395, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872131147275, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.41534673535163363}} {"text": "```python\n%%capture\n## compile PyRoss for this notebook\nimport os\nowd = os.getcwd()\nos.chdir('../../')\n%run setup.py install\nos.chdir(owd)\n%matplotlib inline\n```\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pyross\n```\n\nIn this notebook we consider a control protocol consisting of an initial lockdown, which is then partly released. For our numerical study we generate synthetic data using the stochastic SIIR model.\n\nWhile we use the Denmark age structure and contact matrix, we emphasise that **the model parameters considered in this notebook have NOT been obtained from real data, but rather are chosen ad-hoc**.\n\n**Outline of this notebook:**\n\n1. We briefly summarize the SEAI5R model.\n2. We load the age structure and contact matrix for Denmark. The contact matrix is generally given as\n\\begin{equation}\n C = C_{H} + C_{W} + C_{S} + C_{O},\n\\end{equation}\nwhere the four terms denote the number of contacts at home, work, school, and all other remaining contacts.\n3. We define the other model parameters of the SEAI5R model **(again, these are not fitted to any real data, but rather chosen ad-hoc)**.\n4. We define a \"lockdown-protocol\":\n 1. After a fixed time, a lockdown is imposed. The contact matrix is reduced to \n \\begin{equation}\n C = C_{H} + 0.1\\cdot C_W + 0.4 \\cdot C_O,\n \\end{equation}\n i.e. the \"home\" contacts $C_H$, and 10% of the \"work\" as well as 40% of the \"other\" contacts are reainted. The latter two model that even in lockdown, people go to work, and that initially people might be lenient in following the lockdown advice.\n 2. 5 Days after the initial lockdown, the 40% \"social interaction\" part of the contact matrix is removed, and the contact matrix becomes\n \\begin{equation}\n C = C_{H} + 0.1\\cdot C_W.\n \\end{equation}\n This models that people take the lockdown seriously and seize to interact socially.\n 3. 50 Days after the initial lockdown, 50% of the \"school\" contacts are added to the contact matrix,\n \\begin{equation}\n C = C_{H} + 0.1\\cdot C_W + 0.5 \\cdot C_S.\n \\end{equation}\n This models an opening of schools, with a social distancing protocol in place such that the number of contacts at school is halved.\n5. We run 50 stochastic simulations of this lockdown protocol, and find that in Stage 2 (only home and 10% of work contacts), the total number of known active infectives (the sum of the stages symptomatic infective, hospitalised, and in ICU) shows a maximum and then decreases. After schools are opened, the total number of known active infectives shows a minimum and starts to decrease again. We plot the distribution of waiting time between change in the contact matrix and the extrema in the stochastic time series.\n6. Finally, we consider the protocol from point 4 again, but this time with a prefactor 0.2 in front of $C_S$, signifying a more strict social distancing protocol. Running 50 stochastic simulations with this protocol, we find that this time the total number of active infectives keeps decreases even after schools have been opened.\n \n\n\n## 1. Brief summary of the model \n\nThe SEAI5R model contains the following stages:\n\n\n\n\nIf a susceptible individual is infected, it enters the \"exposed\" class, where it does not show symptoms and is not yet infective. In the subsequent class, called \"activated\", the individual still does not show symptoms, but is infective. After the \"activated\" class, an individual is either a symptomatic or an asymptomatic infective. Those who are asymptomatic recover, those who show symptoms either recover, or need to be hospitalised. The condition of a hospitalised individual might improve, so that the individual recovers, or worsens, at which point the individual enters the \"ICU\" class. From this class the individual can again recover, or enter a final \"mortality\" class.\n\n\n\n\n\n## 2. Load Denmark age structure and contact matrix\n\n\n```python\nmy_data = np.genfromtxt('../data/age_structures/Denmark-2019.csv', delimiter=',', skip_header=1)\naM, aF = my_data[:, 1], my_data[:, 2]\n\nNi0=aM+aF;\n\nM=16 ## number of age classes\n\nNi = Ni0[:M]\nN=np.sum(Ni)\n\nprint(\"Age groups are in brackets of 5 (i.e. 0-4, 5-9, 10-14, .. , 75-79).\")\nprint(\"Number of individuals in each bracket:\")\nprint(Ni.astype('int'))\nprint(\"Total number of individuals: {0}\".format(np.sum(Ni.astype('int'))))\n```\n\n Age groups are in brackets of 5 (i.e. 0-4, 5-9, 10-14, .. , 75-79).\n Number of individuals in each bracket:\n [302353 305513 338779 341219 379522 395469 342443 320132 366147 385944\n 422585 381360 338039 319145 346572 220374]\n Total number of individuals: 5505596\n\n\n\n```python\n# Get individual contact matrices\nCH, CW, CS, CO = pyross.contactMatrix.Denmark()\n\n# By default, home, work, school, and others contribute to the contact matrix\nC = CH + CW + CS + CO\n\n# Illustrate the individual contact matrices:\nfig,aCF = plt.subplots(2,2);\naCF[0][0].pcolor(CH, cmap=plt.cm.get_cmap('GnBu', 10));\naCF[0][1].pcolor(CW, cmap=plt.cm.get_cmap('GnBu', 10));\naCF[1][0].pcolor(CS, cmap=plt.cm.get_cmap('GnBu', 10));\naCF[1][1].pcolor(CO, cmap=plt.cm.get_cmap('GnBu', 10));\n```\n\nThe above contact matrices illustrate the interactions at home (upper left), work (upper right), school (lower left), and the other remaining contacts (lower right). x- and y- axes denote the age groups, a darker color indicates more interaction.\n\n## 3. Define model parameters\n\n**Note: These have not been fitted to real data.**\n\n\n```python\nbeta = 0.036692 # infection rate \n\ngE = 1/5.\ngA = 1/4.\ngIa = 1./3. # recovery rate of asymptomatic infectives \ngIs = 1./7 # recovery rate of symptomatic infectives \nalpha = 0.4 # fraction of asymptomatic infectives \nfsa = 0.2 # the self-isolation parameter \nfh = 0\ngIh = 1/7\ngIc = 1/14\n \nsa = 0*np.ones(M) # rate of additional/removal of population by birth etc\nsa[0] = 0 # birth\nsa[3] = 0 # mortality\n\n\nhh = 0.1*np.ones(M) # fraction which goes from Is to hospital\nhh[:M] = 0.01\nhh[M:2*M] = 0.025\ncc = 0.05*np.ones(M) # fraction which goes from hospital to ICU \nmm = 0.4*np.ones(M) # mortality from IC\n\n\n# initial conditions \nIs_0 = np.zeros((M)); \n \nIa_0 = 10*np.ones((M)) # start with 10 asymptomatic infectives\nR_0 = np.zeros((M))\nE_0 = np.zeros((M))\nA_0 = np.zeros((M))\nIh_0 = np.zeros((M))\nIc_0 = np.zeros((M))\nIm_0 = np.zeros((M))\n\nS_0 = Ni - (E_0 + A_0 + Ia_0 + Is_0 + Ih_0 + Ic_0 +Im_0 + R_0)\n```\n\n## 4. Define events for protocol\n\n\n```python\n# Dummy event for initial (standard) contact matrix\nevents = [lambda t: 1]\ncontactMatrices = [C]\n\n# After 20 days, start lockdown\nlockdown_threshold_0 = 20\ndef event0(t,rp):\n return t - lockdown_threshold_0\nevents.append(event0)\ncontactMatrices.append( CH + 0.1*CW + 0.4*CS )\n\n\n# After 25 days, decrease contacts even further\nlockdown_threshold_1 = 25\ndef event1(t,rp):\n return t- lockdown_threshold_1\nevents.append(event1)\ncontactMatrices.append( CH + 0.1*CW)\n\n\n# After 70 days, add 50% of school contacts to contact matrix\nlockdown_threshold_2 = 70\ndef event2(t,rp):\n return t - lockdown_threshold_2\nevents.append(event2)\ncontactMatrices.append( CH + 0.1*CW + 0.5*CS ) # everybody in lockdown\n```\n\n## 5. Simulate protocol and analyse results\n\n#### Initialise pyross.control, run and plot a single test simulation\n\n\n```python\n# duration of simulation\nTf=150; Nf=Tf+1; \n\n# intantiate model\nparameters = {'alpha':alpha,'beta':beta, 'gIa':gIa,'gIs':gIs,\n 'gIh':gIh,'gIc':gIc, 'gE':gE, 'gA':gA,\n 'fsa':fsa, 'fh':fh, \n 'sa':sa, \n 'hh':hh, 'cc':cc, 'mm':mm}\nmodel = pyross.control.SEAI5R(parameters, M, Ni.copy())\n\n# run model once\nmethod='gillespie'\ndata=model.simulate(S_0, E_0, A_0, Ia_0, Is_0, Ih_0, Ic_0, Im_0,\n events,contactMatrices, Tf, Nf,\n method=method)\n```\n\n\n```python\n# Plot result\n\nt = data['t']; \n\n# get total population in classes\n# - symptomatic infective,\n# - hospitalised,\n# - in ICU\ny_plot = np.sum ( data['X'][:,4*M:7*M], axis = -1)\n\nlw=2\nfig,ax = plt.subplots(1,1,figsize=(8,5))\nax.plot(t,y_plot,lw=lw,)\nax.axvline(data['events_occured'][0][0],\n color='crimson',lw=lw,\n label='Beginning of lockdown',ls='--')\nax.axvline(data['events_occured'][-1][0],\n color='limegreen',lw=lw,\n label='Schools re-opened',ls='--')\nax.set_xlim(0,Tf)\nfs=20\nax.legend(loc='best',fontsize=fs,framealpha=1)\nax.set_xlabel('time since first infection [days]',fontsize=fs)\nax.set_ylabel('Number of known active cases',fontsize=fs)\nplt.show()\nplt.close()\n```\n\n#### Run 50 simulations with protocol, then\n\n* plot trajectories,\n* plot mean trajectory + standard deviation,\n* plot distribution of delay between interventions (lockdown, re-opening of schools) until subsequent maximum/minimum.\n\n\n```python\n# Run 50 simulations with this protocol\nN_simulations = 50\n\ndata_results = np.zeros([N_simulations,Nf,M*9],\n dtype=int)\nfor i in range(N_simulations):\n print('Running simulation {0} of {1}'.format(i+1,N_simulations),end='\\r')\n data=model.simulate(S_0, E_0, A_0, Ia_0, Is_0, Ih_0, Ic_0, Im_0,\n events,contactMatrices, Tf, Nf,\n method=method)\n data_results[i] = data['X']\n```\n\n Running simulation 50 of 50\r\n\n\n```python\n# Plot result\n\nt = data['t']; \nfig,ax = plt.subplots(1,1,figsize=(8,5))\nfor i,e in enumerate(data_results):\n # get total population in classes\n # - symptomatic infective,\n # - hospitalised,\n # - in ICU\n y_plot = np.sum ( e[:,4*M:7*M], axis = -1)\n ax.plot(t,y_plot,lw=0.5,)\nax.axvline(data['events_occured'][0][0],\n color='crimson',lw=lw,\n label='Beginning of lockdown',ls='--')\nax.axvline(data['events_occured'][-1][0],\n color='limegreen',lw=lw,\n label='Schools re-opened',ls='--')\nax.set_xlim(0,Tf)\nfs=20\nax.legend(loc='best',fontsize=fs,framealpha=1)\nax.set_xlabel('time since first infection [days]',fontsize=fs)\nax.set_ylabel('Number of known active cases',fontsize=fs)\nplt.show()\nplt.close()\n```\n\n#### Mean and standard deviation\n\n\n```python\n# For all simulations, plot total population in classes\n# - symptomatic infective,\n# - hospitalised,\n# - in ICU\n\n# mean and standard deviation of trajectories at every time\ny_mean = np.mean( ( np.sum(data_results[:,:,4*M:7*M], axis = -1) ), axis=0)\ny_std = np.std ( np.sum( data_results[:,:,4*M:7*M], axis = -1), axis =0)\n\nt = data['t']; \nfig,ax = plt.subplots(1,1,figsize=(8,5)) \nax.fill_between(t,y_mean - y_std,y_mean+y_std,\n label='Standard deviation',\n color='grey',alpha=0.4)\nax.plot(t,y_mean,\n label='Mean',\n color='black',alpha=1) \nax.axvline(data['events_occured'][0][0],\n color='crimson',lw=lw,\n label='Beginning of lockdown',ls='--')\nax.axvline(data['events_occured'][-1][0],\n color='limegreen',lw=lw,\n label='Schools re-opened',ls='--')\nax.set_xlim(0,Tf)\nfs=20\nax.legend(loc='upper left',fontsize=17,framealpha=1)\nax.set_xlabel('time since first infection [days]',fontsize=fs)\nax.set_ylabel('Number of known active cases',fontsize=fs)\nplt.show()\nplt.close()\n```\n\n#### Plot distributions for both \n\n* time from beginning of lockdown to subsequent peak, and\n* time from opening of schools to subsequent minimum.\n\n\n```python\n# calculate time from lockdown to local maximum for each trajectory\nmask = (t > lockdown_threshold_0)*(t < lockdown_threshold_2)\nt_mask = t[(mask)]\ndurations_from_lockdown_to_local_maximum = np.zeros(N_simulations,\n dtype=float)\nfor i,e in enumerate(data_results):\n y_plot = np.sum ( e[:,4*M:7*M], axis = -1)\n index = np.argmax( y_plot[(mask)])\n durations_from_lockdown_to_local_maximum[i] = t_mask[index] - lockdown_threshold_0\nprint('Mean time from beginning of lockdown to subsequent peak = {0:3.1f} days'.format(np.mean(durations_from_lockdown_to_local_maximum)))\n\n\nhist, bin_edges = np.histogram(durations_from_lockdown_to_local_maximum,10 ,density=True)\nbin_centers = (bin_edges[1:] + bin_edges[:-1])/2.\nbin_widths = bin_edges[1]-bin_edges[0]\nfig, ax =plt.subplots(1,1,figsize=(10,6))\nax.axvline(np.mean(durations_from_lockdown_to_local_maximum),ls='--',color='black',\n label='Mean')\nax.bar(bin_centers,hist,bin_widths,color='dodgerblue',\n label='Distribution',alpha=0.7)\nax.set_xlabel('Time from beginning of lockdown until subsequent peak [days]',fontsize=fs)\nax.set_ylabel('Probability',fontsize=fs)\nax.legend(loc='best',fontsize=fs)\nplt.show()\nplt.close()\n\n\n\n\n\n# calculate time from opening of schools to local minimum for each trajectory\n# (recall lockdown_threshold_2 = time at which schools are opened)\nmask = (t > lockdown_threshold_2)*(t < Tf)\nt_mask = t[(mask)]\ndurations_from_school_opening_to_local_minimum = np.zeros(N_simulations,\n dtype=float)\nfor i,e in enumerate(data_results):\n y_plot = np.sum ( e[:,4*M:7*M], axis = -1)\n index = np.argmin( y_plot[(mask)])\n durations_from_school_opening_to_local_minimum[i] = t_mask[index] - lockdown_threshold_2\nprint('Mean time from opening of schools to subsequent minimum = {0:3.1f} days'.format(np.mean(durations_from_school_opening_to_local_minimum)))\n\n\nhist, bin_edges = np.histogram(durations_from_school_opening_to_local_minimum,10 ,density=True)\nbin_centers = (bin_edges[1:] + bin_edges[:-1])/2.\nbin_widths = bin_edges[1]-bin_edges[0]\nfig, ax =plt.subplots(1,1,figsize=(10,6))\nax.axvline(np.mean(durations_from_school_opening_to_local_minimum),ls='--',color='black',\n label='Mean')\nax.bar(bin_centers,hist,bin_widths,color='dodgerblue',\n label='Distribution',alpha=0.7)\nax.set_xlabel('Time from school re-opening until subsequent minimum [days]',fontsize=fs)\nax.set_ylabel('Probability',fontsize=fs)\nax.legend(loc='best',fontsize=fs)\nplt.show()\nplt.close()\n```\n\n## 6. Redo simulations and analysis of point 5, but with more social distancing at school\n\n\n```python\n# we modify only the last contact matrix of the previous protocol:\ncontactMatrices[-1] = CH + 0.1*CW + 0.2*CS\n```\n\n\n```python\n# Run 50 simulations with this protocol\nN_simulations = 50\n\ndata_results2 = np.zeros([N_simulations,Nf,M*9],\n dtype=int)\nfor i in range(N_simulations):\n print('Running simulation {0} of {1}'.format(i+1,N_simulations),end='\\r')\n data=model.simulate(S_0, E_0, A_0, Ia_0, Is_0, Ih_0, Ic_0, Im_0,\n events,contactMatrices, Tf, Nf,\n method=method)\n data_results2[i] = data['X']\n```\n\n Running simulation 50 of 50\r\n\n\n```python\n# Plot results\n\nprint(\"Simulated trajectories:\")\nt = data['t']; \nfig,ax = plt.subplots(1,1,figsize=(8,5))\nfor i,e in enumerate(data_results2):\n # get total population in classes\n # - symptomatic infective,\n # - hospitalised,\n # - in ICU\n y_plot = np.sum ( e[:,4*M:7*M], axis = -1)\n ax.plot(t,y_plot,lw=0.5,)\nax.axvline(data['events_occured'][0][0],\n color='crimson',lw=lw,\n label='Beginning of lockdown',ls='--')\nax.axvline(data['events_occured'][-1][0],\n color='limegreen',lw=lw,\n label='Schools re-opened',ls='--')\nax.set_xlim(0,Tf)\nfs=20\nax.legend(loc='best',fontsize=15,framealpha=1)\nax.set_xlabel('time since first infection [days]',fontsize=fs)\nax.set_ylabel('Number of known active cases',fontsize=fs)\nplt.show()\nplt.close()\n\n\nprint(\"Mean trajectory and standard deviation:\")\n# mean and standard deviation of trajectories at every time\ny_mean = np.mean( ( np.sum(data_results2[:,:,4*M:7*M], axis = -1) ), axis=0)\ny_std = np.std ( np.sum( data_results2[:,:,4*M:7*M], axis = -1), axis =0)\nt = data['t']; \nfig,ax = plt.subplots(1,1,figsize=(8,5)) \nax.fill_between(t,y_mean - y_std,y_mean+y_std,\n label='Standard deviation',\n color='grey',alpha=0.4)\nax.plot(t,y_mean,\n label='Mean',\n color='black',alpha=1) \nax.axvline(data['events_occured'][0][0],\n color='crimson',lw=lw,\n label='Beginning of lockdown',ls='--')\nax.axvline(data['events_occured'][-1][0],\n color='limegreen',lw=lw,\n label='Schools re-opened',ls='--')\nax.set_xlim(0,Tf)\nfs=20\nax.legend(loc='upper right',fontsize=15,framealpha=1)\nax.set_xlabel('time since first infection [days]',fontsize=fs)\nax.set_ylabel('Number of known active cases',fontsize=fs)\nplt.show()\nplt.close()\n\n\n\nprint(\"Time from lockdown to local maximum:\")\n# calculate time from lockdown to local maximum for each trajectory\nmask = (t > lockdown_threshold_0)*(t < lockdown_threshold_2)\nt_mask = t[(mask)]\ndurations_from_lockdown_to_local_maximum2 = np.zeros(N_simulations,\n dtype=float)\nfor i,e in enumerate(data_results2):\n y_plot = np.sum ( e[:,4*M:7*M], axis = -1)\n index = np.argmax( y_plot[(mask)])\n durations_from_lockdown_to_local_maximum2[i] = t_mask[index] - lockdown_threshold_0\nprint('Mean time from beginning of lockdown to subsequent peak = {0:3.1f} days'.format(np.mean(durations_from_lockdown_to_local_maximum2)))\n\nhist, bin_edges = np.histogram(durations_from_lockdown_to_local_maximum2,10 ,density=True)\nbin_centers = (bin_edges[1:] + bin_edges[:-1])/2.\nbin_widths = bin_edges[1]-bin_edges[0]\nfig, ax =plt.subplots(1,1,figsize=(10,6))\nax.axvline(np.mean(durations_from_lockdown_to_local_maximum2),ls='--',color='black',\n label='Mean')\nax.bar(bin_centers,hist,bin_widths,color='dodgerblue',\n label='Distribution',alpha=0.7)\nax.set_xlabel('Time from beginning of lockdown until subsequent peak [days]',fontsize=fs)\nax.set_ylabel('Probability',fontsize=fs)\nax.legend(loc='best',fontsize=fs)\nplt.show()\nplt.close()\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "824ce17df5593a7f2847d6fed1df30ab164361a4", "size": 540574, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/control/ex05 - SEAI5R - Denmark.ipynb", "max_stars_repo_name": "ineskris/pyross", "max_stars_repo_head_hexsha": "2ee6deb01b17cdbff19ef89ec6d1e607bceb481c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "examples/control/ex05 - SEAI5R - Denmark.ipynb", "max_issues_repo_name": "ineskris/pyross", "max_issues_repo_head_hexsha": "2ee6deb01b17cdbff19ef89ec6d1e607bceb481c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "examples/control/ex05 - SEAI5R - Denmark.ipynb", "max_forks_repo_name": "ineskris/pyross", "max_forks_repo_head_hexsha": "2ee6deb01b17cdbff19ef89ec6d1e607bceb481c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 660.8484107579, "max_line_length": 147868, "alphanum_fraction": 0.9463514708, "converted": true, "num_tokens": 5077, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.6791787121629465, "lm_q1q2_score": 0.4152376212615625}} {"text": "\n\n\n**Creating Robust Control for Single qubit gates**\n\nHere we introduce the essential concepts behind robust control. We create a model of noise on a quantum computer and simulate its performance. Then we show how to create controls that are robust to this noise process. We demonstrate the control's robustness with a simulation. \n\n## Imports and initialization\n\n\n```python\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport qctrlvisualizer as qv\nfrom attr import asdict\n\nfrom qctrl import Qctrl\n\n# Starting a session with the API\nqctrl = Qctrl(email = 'harshit.co19@nsut.ac.in', password = 'HARSHITcontrol')\n```\n\n\n```python\n# Define standard matrices\nidentity = np.array([[1.0, 0.0], [0.0, 1.0]], dtype=np.complex)\nsigma_x = np.array([[0.0, 1.0], [1.0, 0.0]], dtype=np.complex)\nsigma_y = np.array([[0.0, -1j], [1j, 0.0]], dtype=np.complex)\nsigma_z = np.array([[1.0, 0.0], [0.0, -1.0]], dtype=np.complex)\nsigma_m = np.array([[0.0, 1.0], [0.0, 0.0]], dtype=np.complex)\nsigmas = [sigma_x, sigma_y, sigma_z]\nsigma_names = [\"X\", \"Y\", \"Z\"]\nnot_gate = np.array([[0.0, -1.0], [1.0, 0.0]])\n\n# Plotting and formatting methods\nplt.style.use(qv.get_qctrl_style())\n\n\ndef plot_simulation_trajectories(figure, times, coherent_samples, noisy_trajectories):\n ideal_bloch_sphere_coords = np.array(\n [\n [\n np.real(\n np.dot(\n sample.state_vector.conj(),\n np.matmul(sigma, sample.state_vector),\n )\n )\n for sigma in sigmas\n ]\n for sample in coherent_samples\n ]\n )\n noisy_bloch_sphere_coords = np.array(\n [\n [\n [\n np.real(\n np.dot(\n sample.state_vector.conj(),\n np.matmul(sigma, sample.state_vector),\n )\n )\n for sigma in sigmas\n ]\n for sample in trajectory.samples\n ]\n for trajectory in noisy_trajectories\n ]\n )\n figure.set_figheight(6.0)\n figure.set_figwidth(7.0)\n axes = figure.subplots(nrows=3, ncols=1, sharex=True, sharey=False, squeeze=False)[\n :, 0\n ]\n for a in range(3):\n axes[a].set_ylabel(sigma_names[a])\n axes[a].set_ylim([-1.1, 1.1])\n for t in range(noisy_bloch_sphere_coords.shape[0]):\n axes[a].plot(\n times * 1e6,\n noisy_bloch_sphere_coords[t, :, a],\n \"--\",\n color=\"#680CE9\",\n alpha=0.25,\n )\n axes[a].plot(times * 1e6, ideal_bloch_sphere_coords[:, a], \"-\", color=\"#680CE9\")\n axes[2].set_xlabel(\"Time ($\\mu$s)\")\n axes[0].set_title(\"Bloch sphere coordinates\")\n\n\ndef plot_simulation_noise_directions(figure, times, coherent_samples):\n figure.set_figheight(6.0)\n figure.set_figwidth(7.0)\n noise_operator_directions = np.array(\n [\n [\n 0.5\n * np.real(\n np.trace(\n np.matmul(\n sigma,\n np.matmul(\n sample.evolution_operator.conj().T,\n np.matmul(sigma_z, sample.evolution_operator),\n ),\n )\n )\n )\n for sigma in sigmas\n ]\n for sample in coherent_samples\n ]\n )\n axes = figure.subplots(nrows=3, ncols=1, sharex=True, sharey=False, squeeze=False)[\n :, 0\n ]\n for a in range(3):\n axes[a].set_ylabel(sigma_names[a])\n axes[a].set_ylim([-1.1, 1.1])\n axes[a].plot(\n robust_point_times * 1e6,\n noise_operator_directions[:, a],\n \"-\",\n color=\"#680CE9\",\n )\n axes[a].fill_between(\n robust_point_times * 1e6,\n 0,\n noise_operator_directions[:, a],\n color=\"#680CE9\",\n alpha=0.25,\n )\n axes[2].set_xlabel(\"Time ($\\mu$s)\")\n axes[0].set_title(\"Bloch sphere directions\")\n\n\ndef plot_noise_spectral_density(figure, nsd_samples):\n frequencies = np.array([sample[\"frequency\"] for sample in nsd_samples])\n powers = np.array([sample[\"power\"] for sample in nsd_samples])\n axes = figure.subplots(nrows=1, ncols=1, sharex=True, sharey=False, squeeze=False)[\n 0, 0\n ]\n axes.plot(frequencies / 1e6, powers * 1e6)\n axes.fill_between(frequencies / 1e6, 0, powers * 1e6, alpha=0.25)\n axes.set_xlabel(\"Frequency (MHz)\")\n axes.set_ylabel(\"Power density (1/MHz)\")\n axes.set_title(\"Dephasing noise spectral density\")\n\n\ndef pm_format(average, std):\n return \"{:.4f}\".format(average) + \"+/-\" + \"{:.4f}\".format(std)\n\n \n# The main signal which is sent to the qubit in the cloud\n\ndef bandwidth_limited_pwc_signal(\n name, duration, segment_count, max_rabi_rate, cutoff_frequency\n):\n\n # create a raw pwc_signal where the amplitude of each segment is an optimization variables\n raw_signal = qctrl.operations.pwc_signal(\n values=qctrl.operations.bounded_optimization_variable(\n count=segment_count, lower_bound=-max_rabi_rate, upper_bound=max_rabi_rate\n ),\n duration=duration,\n )\n\n # pass the signal through a bandwidth limited filter\n filtered_signal = qctrl.operations.convolve_pwc(\n raw_signal, qctrl.operations.sinc_integral_function(cutoff_frequency)\n )\n\n # resample the smooth filtered signal as a pwc_signal\n final_signal = qctrl.operations.discretize_stf(\n stf=filtered_signal,\n duration=robust_duration,\n segments_count=segment_count,\n name=name,\n )\n\n return final_signal\n```\n\n## Single qubit with dephasing noise\n\nTo better understand how noise affects a quantum computer we are going to create a simulation.\n\nTo start we write down a Hamiltonian, which will mathematically describe this physical system:\n\n\\begin{align*}\nH_{\\rm total}(t) = & H_{\\rm control}(t) + H_{\\rm noise}(t).\n\\end{align*}\n\n### Control: Standard microwave pulse that creates a NOT Gate\n\nThe control part of the Hamiltonian is:\n\n\\begin{align}\nH_{\\rm control}(t) = \\Omega_{\\rm I}(t) \\sigma_{x}/2 + \\Omega_{\\rm Q}(t) \\sigma_{y}/2.\n\\end{align}\n\nWhere $\\Omega_I(t)$ and $\\Omega_Q(t)$ are the time-dependent Rabi rate created by the IQ modulated microwave pulse applied to control the qubits state, which couples to the qubit state through the $\\sigma_k$ operators.\n\nWe are trying to apply a NOT gate to the qubit. The simplest way to do this is to apply a Q modulated microwave pulse at the maximum Rabi rate $\\Omega_{\\rm Q}(t) = \\Omega_{\\rm max}$ for a duration of $\\pi/\\Omega_{\\rm max}$, while the I modulated microwave pulse is set to zero $\\Omega_{\\rm I}(t) = 0$. We will call this the standard NOT gate. \n\n\n```python\nomega_max = 2 * np.pi * 1e6 # Hz\nstandard_duration = np.pi / omega_max # s\n\nstandard_pulse_segments = [\n qctrl.types.ComplexSegmentInput(duration=standard_duration, value=omega_max),\n]\n\nplot_segments = {\n \"$\\Omega_Q$\": [\n {\"duration\": segment.duration, \"value\": segment.value}\n for segment in standard_pulse_segments\n ]\n}\n\nqv.plot_controls(plt.figure(), plot_segments)\nplt.show()\n```\n\n### Noise: Magnetic field with a 1/f spectrum\n\nThe noise part of the Hamiltonian is:\n\\begin{align}\nH_{\\rm noise}(t) = \\eta(t) \\sigma_z / 2.\n\\end{align}\n\nWe treat the noisy magnetic field environment as a classical noise process $\\eta(t)$ coupled to the quantum system with a noise operator $\\sigma_z$. This approximate model is often reasonable for real quantum computing hardware when the decoherence time (T2) is the limiting factor, being much shorter than the relaxation time (T1) of the qubits. \n\nThe noise process $\\eta(t)$ is sampled from a noise spectral density that follows a power law: \n\\begin{align}\nS_{\\eta}(\\omega) = \\frac{\\omega_{\\rm cutoff}^{a-1}}{\\omega^a + \\omega_{\\rm cutoff}^a},\n\\end{align}\nWhere $\\omega_{\\rm cutoff}$ is the cutoff frequency and $a$ is the order of the power law. It is common for magnetic field environments to follow 1/f power law ($a=1$) where low frequency noise dominates. \n\n\nDifferent physical processes will couple to the quantum computer through different noise operators. The key to getting a good simulation is to identify the noises that most significantly affect our qubits. \n\n\n\n```python\ndef power_spectrum(frequencies, frequency_cutoff, power):\n return frequency_cutoff ** (power - 1) / (\n frequencies ** power + frequency_cutoff ** power\n )\n\n\nfrequencies = np.linspace(0, 2.0e4, 1000)\npower_densities = 4e10 * power_spectrum(frequencies, 1.0e2, 1.0)\nnsd_sampled_points = [\n {\"frequency\": f, \"power\": p, \"power_uncertainty\": 0.0, \"weight\": 0.0}\n for f, p in zip(frequencies, power_densities)\n]\n\nplot_noise_spectral_density(plt.figure(), nsd_sampled_points)\n```\n\n## Simulation of standard NOT Gate\n\nNow that we have a Hamiltonian we can create a simulation. The control we have is a `shift` with $\\sigma_x$ as the `operator` and $\\Omega(t)$ is the `pulse`. The noise we have is an `additive noise` with $\\sigma_z$ as the `operator` and $S_\\eta(\\omega)$ is the `linear_piecewise_noise_spectral_density`.\n\n\n\n```python\nstandard_control = qctrl.types.colored_noise_simulation.Shift(\n control=standard_pulse_segments, operator=sigma_y / 2\n)\n\nnoise_drift = qctrl.types.colored_noise_simulation.Drift(\n operator=sigma_z / 2.0,\n noise=qctrl.types.colored_noise_simulation.Noise(\n power_densities=power_densities,\n frequency_step=frequencies[1],\n time_domain_sample_count=1000,\n ),\n)\n\ntarget = qctrl.types.TargetInput(operator=not_gate)\n```\n\nNow we can create a simulation of the qubit in a noisy environment.\n\n*See also:* The [simulation user guide](https://docs.q-ctrl.com/boulder-opal/user-guides/simulation) explains how to create multiple types of simulations.\n\n\n```python\nstandard_point_times = np.linspace(0, standard_duration, 100)\n\nstandard_noisy_simulation_result = qctrl.functions.calculate_colored_noise_simulation(\n duration=standard_duration,\n sample_times=standard_point_times,\n shifts=[standard_control],\n drifts=[noise_drift],\n trajectory_count=5,\n initial_state_vector=np.array([1.0, 0.0]),\n target=target,\n)\n```\n\n\n HBox(children=(FloatProgress(value=0.0), HTML(value='')))\n\n\n Your task calculate_colored_noise_simulation has started.\n Your task calculate_colored_noise_simulation has completed in 9s.\n\n\nFor comparison we can also create a simulation of a system with no noise\n\n\n```python\nstandard_ideal_simulation_result = qctrl.functions.calculate_coherent_simulation(\n duration=standard_duration,\n sample_times=standard_point_times,\n shifts=[standard_control],\n initial_state_vector=np.array([1.0, 0.0]),\n target=target,\n)\n```\n\n\n HBox(children=(FloatProgress(value=0.0), HTML(value='')))\n\n\n Your task calculate_coherent_simulation has completed in 3s.\n \r\n\n### Noisy trajectories of the qubit state\n\nWe can display the noisy trajectories of the qubit using the coordinates of the [Bloch sphere](https://en.wikipedia.org/wiki/Bloch_sphere) as a representation of the state. We can see that the noisy trajectories, shown with dotted lines, take us away from the ideal simulation path, shown with the solid line. Most importantly, the final state of noisy trajectories diverges from the ideal final state. This indicates that the noise will introduce errors into our calculation and affect the outcomes of an algorithm that we want to run.\n\n\n```python\nplot_simulation_trajectories(\n plt.figure(),\n standard_point_times,\n standard_ideal_simulation_result.samples,\n standard_noisy_simulation_result.trajectories,\n)\nplt.show()\n```\n\n### Average gate infidelity of standard NOT gate\n\nThe results above are specific to a particular initial state. We can quantify the *average* performance of the gate under noise by looking at the average gate infidelity, defined as:\n\n\\begin{align}\n\\mathcal{I}_{\\rm gate} = 1 - \\mathbb{E}[ \\rm{Tr}[ U_{\\rm target}^\\dagger(T) U_{\\rm total}(T) ] ],\n\\end{align}\n\nwhere $U_{k}(T)$ is the solution to $\\dot{U}_{k}(t) = -i H_{k} U_{k}(t)$, $U_{\\rm target}$ is the target unitary, in this case a NOT gate, and $\\mathbb{E}[ \\cdot ]$ is the classical stochastic average. An estimate of this number is automatically calculated when you provide a target to a stochastic simulation in BOULDER OPAL.\n\n\n```python\nstandard_final_sample = standard_noisy_simulation_result.average_samples[-1]\nprint(\"Average gate infidelity:\")\nprint(\n pm_format(\n standard_final_sample.average_infidelity,\n standard_final_sample.average_infidelity_uncertainty,\n )\n)\nprint(standard_noisy_simulation_result.average_samples[-1])\n```\n\n Average gate infidelity:\n 0.0030+/-0.0013\n AverageSample(time=5e-07, average_infidelity=0.0029601828152379995, average_infidelity_uncertainty=0.001325232679883104, average_density_matrix=array([[ 0.00296006+0.j , -0.0023269 -0.03110548j],\n [-0.0023269 +0.03110548j, 0.99703994+0.j ]]))\n\n\n## Robust control design\n\nThe filter function framework can be used to design robust controls. We treat the design problem as a multi-objective optimization problem. First we assume the control field is parametrized by a set of variables $\\Omega_{\\rm candidate}(\\underline{v},t)$. \n\nThe first target of our optimization is to ensure that our optimized pulse performs the correct operation. To do this we need to minimize the infidelity of the control:\n\\begin{align}\n\\mathcal{I}_{\\rm control} = \\rm{Tr}[U_{\\rm control}^\\dagger(T) U_{\\rm target}(T)],\n\\end{align}\nThis quantifies how close the control is to the target operation if there is no noise.\n\nThe second target of our optimization is to ensure that our optimized pulse is robust to the noise. It is common for physically relevant noise processes to be dominated by low frequency noise, in this case it simplifies the numerical calculation to minimize just the zero frequency part of the filter function. We call this the infidelity of the noise:\n\\begin{align}\n\\mathcal{I}_{\\rm noise} = w^2 \\left|\\left| \\int dt H_{\\rm noise}^{\\rm (control)}(t) \\right|\\right|_2^2,\n\\end{align}\nwhere $w$ is a relative weight of the filter cost compared to the operation, a good value for additive noises is $w=1/T$.\n\nThe multi-objective optimization problem can be represented as minimizing the cost\n\\begin{align}\n\\mathcal{I}_{\\rm robust}(\\underline{v}) = \\mathcal{I}_{\\rm control}(\\underline{v}) + \\mathcal{I}_{\\rm noise}(\\underline{v}).\n\\end{align}\n\n\nIf we can find a control where $\\mathcal{I}_{\\rm robust}(\\underline{v})$ is very close to zero, we can be sure that it will both complete the correct operation and be robust to low frequency noise.\n\n### Optimizing a robust NOT gate\n\nWe can create a robust NOT gate using the BOULDER OPAL optimizer. The [optimization feature](https://docs.q-ctrl.com/boulder-opal/user-guides/optimization) allows the user to define an optimization with arbitrary pulse constraints. \n\nWe are going to construct two control pulses $\\Omega_I(\\underline{v},t)$ and $\\Omega_Q(\\underline{v},t)$ which have a maximum Rabi rate $\\Omega_{\\rm max}$ and a bandwidth limit defined by a cutoff frequency $\\Omega_{\\rm cutoff}$. \n\nThe optimizer requires that you define the quantum system as a `graph` that represents how a set of `optimization_variables` an `infidelity` you want to minimize. A series of convenience methods makes \ncreating this `graph` straightforward once you have mathematically written down the total Hamiltonian ($H_{\\rm total}$). Below we show how to create a `graph` for optimizing a qubit with dephasing noise. On each line, we write down what the current variable represents in the mathematical equation of the total Hamiltonian.\n\nWe restate the entire Hamiltonian below so we can easily refer to it:\n\\begin{align}\nH_{\\rm total}(t) = & H_{\\rm control}(t) + H_{\\rm noise}(t), \\\\\nH_{\\rm control}(t) = & \\Omega_{\\rm I}(t) \\sigma_{x}/2 + \\Omega_{\\rm Q}(t) \\sigma_{y}/2, \\\\\nH_{\\rm noise}(t) = & \\eta(t) \\sigma_z / 2.\n\\end{align}\n\n\n```python\nrobust_duration = 3.0 * standard_duration\nomega_cutoff = 1e7\nsegment_count = 100\n\nwith qctrl.create_graph() as graph:\n\n # Omega_I(v,t)\n pulse_i = bandwidth_limited_pwc_signal(\n name=\"I\",\n duration=robust_duration,\n segment_count=segment_count,\n max_rabi_rate=omega_max,\n cutoff_frequency=omega_cutoff,\n )\n\n # Omega_Q(v,t)\n pulse_q = bandwidth_limited_pwc_signal(\n name=\"Q\",\n duration=robust_duration,\n segment_count=segment_count,\n max_rabi_rate=omega_max,\n cutoff_frequency=omega_cutoff,\n )\n\n # Omega_I(t) sigma_x/2\n robust_control_i = qctrl.operations.pwc_operator(\n signal=pulse_i, operator=sigma_x / 2.0\n )\n\n # Omega_Q(t) sigma_y/2\n robust_control_q = qctrl.operations.pwc_operator(\n signal=pulse_q, operator=sigma_y / 2.0\n )\n\n # H_control = Omega_I(t) sigma_x/2 + Omega_Q(t) sigma_y/2\n control_hamiltonian = qctrl.operations.pwc_sum([robust_control_i, robust_control_q])\n\n # sigma_z / 2w\n noise_operator = qctrl.operations.constant_pwc_operator(\n robust_duration, sigma_z / 2.0 / robust_duration\n )\n\n # create U_target\n target_unitary = qctrl.operations.target(operator=not_gate)\n\n # create I_robust(v) = I_control(v) + I_noise(v)\n infidelity = qctrl.operations.infidelity_pwc(\n hamiltonian=control_hamiltonian,\n noise_operators=[\n noise_operator,\n ],\n target_operator=target_unitary,\n name=\"infidelity\",\n )\n```\n\nWhen you run an optimization, a series of searches are performed and the pulse with the smallest cost is returned. The optimization is stochastic and therefore a different result will be returned each time, but they will always satisfy the constraints. \n\nA pulse that is both robust and completes the correct operation will have a cost which is very close to zero. If the cost returned does not satisfy this condition, you may need to reduce your constraints. Increasing the total duration and/or the number of segments will often help.\n\n\n```python\noptimization_result = qctrl.functions.calculate_optimization(\n cost_node_name=\"infidelity\",\n output_node_names=[\"infidelity\", \"I\", \"Q\"],\n graph=graph,\n)\n```\n\n\n HBox(children=(FloatProgress(value=0.0), HTML(value='')))\n\n\n Your task calculate_optimization has started.\n Your task calculate_optimization has completed in 8s.\n\n\n\n```python\noptimization_result\n```\n\n\n\n\n Result(cost=1.4300662564383862e-09, output={'I': [{'value': 242653.6137201316, 'duration': 1.5000000000000002e-08}, {'value': 188354.91479048404, 'duration': 1.5000000000000002e-08}, {'value': 122887.06698561733, 'duration': 1.5000000000000002e-08}, {'value': 46448.87315650932, 'duration': 1.5000000000000002e-08}, {'value': -40582.74864624049, 'duration': 1.5000000000000002e-08}, {'value': -137654.43142013473, 'duration': 1.5000000000000002e-08}, {'value': -244041.19504581013, 'duration': 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1.5000000000000002e-08}, {'value': 2937241.2169490596, 'duration': 1.5000000000000002e-08}, {'value': 2853228.1160624297, 'duration': 1.5000000000000002e-08}], 'infidelity': {'value': 1.4300662564383862e-09}}, errors=None, action=CoreAction(created_at=None, definition=None, errors=None, job_data=None, job_id=None, model_id='385359', model_type=None, mutation_name=None, name='core__calculateOptimization', progress=1.0, result=None, runtime=None, started_at=None, status='SUCCESS', terminated_at=None, updated_at=None, user=None))\n\n\n\n\n```python\nprint(\"Best cost:\")\nprint(optimization_result.cost)\n```\n\n Best cost:\n 1.4300662564383862e-09\n\n\nOnce you have completed an optimization with a good cost you can export the segments of the pulse to your device.\n\n\n\n```python\nqv.plot_controls(\n plt.figure(),\n {\n \"$\\Omega_I$\": optimization_result.output[\"I\"],\n \"$\\Omega_Q$\": optimization_result.output[\"Q\"],\n },\n)\nplt.show()\n```\n\n## Sending the pulse to the cloud to realize the NOT gate\n\n\n```python\nIvals, Qvals = optimization_result.output[\"I\"], optimization_result.output[\"Q\"]\n```\n\n\n```python\nQvals\n```\n\n\n\n\n [{'duration': 1.5000000000000002e-08, 'value': 3228153.996921337},\n {'duration': 1.5000000000000002e-08, 'value': 3331626.1738166204},\n {'duration': 1.5000000000000002e-08, 'value': 3400993.995268936},\n {'duration': 1.5000000000000002e-08, 'value': 3434378.57314747},\n {'duration': 1.5000000000000002e-08, 'value': 3430377.9517383296},\n {'duration': 1.5000000000000002e-08, 'value': 3388090.1555592334},\n {'duration': 1.5000000000000002e-08, 'value': 3307127.77314859},\n {'duration': 1.5000000000000002e-08, 'value': 3187623.8151343106},\n {'duration': 1.5000000000000002e-08, 'value': 3030228.7474707053},\n {'duration': 1.5000000000000002e-08, 'value': 2836098.7654624777},\n {'duration': 1.5000000000000002e-08, 'value': 2606875.537870365},\n {'duration': 1.5000000000000002e-08, 'value': 2344657.809824857},\n {'duration': 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'value': 2937241.2169490596},\n {'duration': 1.5000000000000002e-08, 'value': 2853228.1160624297}]\n\n\n\n\n```python\ncontrol_count = 1\nsegment_count = len(Ivals)\nduration = Ivals[0]['duration']*1e9\nshot_count = 512\n```\n\n\n```python\nvalues = []\nR, C = [], []\nfor RE, COM in zip(Ivals, Qvals):\n r = RE['value']\n c = COM['value']\n R.append(r)\n C.append(c)\n\n# R = (np.array(R) - np.mean(R))/ np.std(R)\n# C = (np.array(C) - np.mean(C))/ np.std(C)\n\nfor r, c in zip(R, C):\n values.append(r + 1j * c)\n# values = np.array(values)\n# values = (values - np.mean(values)) / np.std(values)\n```\n\n\n```python\nnorm = np.linalg.norm(values)\nvalues = values/norm\n```\n\n\n```python\ncontrols = []\ncontrols.append({\"duration\":duration, \"values\": np.array(values)})\n```\n\n\n```python\ncontrols\n```\n\n\n\n\n [{'duration': 15.000000000000002,\n 'values': array([ 0.0056953 +0.07576767j, 0.00442086+0.07819626j,\n 0.00288427+0.07982438j, 0.0010902 +0.08060795j,\n 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-0.04448201-0.01909769j, -0.04184062-0.01021352j,\n -0.03900192-0.0014136j , -0.03599914+0.00721606j,\n -0.0328676 +0.01558946j, -0.02964421+0.02362198j,\n -0.02636675+0.03123181j, -0.02307334+0.03834134j,\n -0.01980174+0.04487848j, -0.01658879+0.05077788j,\n -0.0134698 +0.0559821j , -0.01047802+0.06044254j,\n -0.00764411+0.06412038j, -0.00499571+0.06698719j,\n -0.00255701+0.06902551j, -0.00034847+0.07022912j,\n 0.00161347+0.07060318j, 0.00331656+0.07016419j,\n 0.00475287+0.06893969j, 0.00591878+0.06696783j])}]\n\n\n\n\n```python\nexperiment_results = qctrl.functions.calculate_qchack_measurements(\n controls=controls,\n shot_count=shot_count,\n)\n```\n\n\n HBox(children=(FloatProgress(value=0.0), HTML(value='')))\n\n\n Your task calculate_qchack_measurements has completed in 3s.\n \r\n\n\n```python\nmeasurements = experiment_results.measurements\nfor measurement_counts in enumerate(measurements):\n print(\"control: {measurement_counts}\")\n```\n\n control 1: (0, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1])\n\n\n\n```python\nfor measurement_counts in enumerate(measurements):\n p0 = measurement_counts.count(0) / shot_count\n p1 = measurement_counts.count(1) / shot_count\n p2 = measurement_counts.count(2) / shot_count\n print(f\"control 1: P(|0>) = {p0:.2f}, P(|1>) = {p1:.2f}, P(|2>) = {p2:.2f}\")\n```\n\n control 1: P(|0>) = 0.00, P(|1>) = 0.00, P(|2>) = 0.00\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "1854a861c65e57c8dbeeeac4aad0e2eb68f891f7", "size": 173632, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Notebooks/Robust_control_x_gate.ipynb", "max_stars_repo_name": "TheGupta2012/qctrl-qhack-Hostages-of-the-Entangled-Dungeons", "max_stars_repo_head_hexsha": "1101fa71b41a4fa6dc8f211f3cd08b3dd80fb541", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-04-11T14:46:04.000Z", "max_stars_repo_stars_event_max_datetime": "2021-04-11T14:46:04.000Z", "max_issues_repo_path": "Notebooks/Robust_control_x_gate.ipynb", "max_issues_repo_name": "TheGupta2012/qctrl-qhack-Hostages-of-the-Entangled-Dungeons", "max_issues_repo_head_hexsha": "1101fa71b41a4fa6dc8f211f3cd08b3dd80fb541", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Notebooks/Robust_control_x_gate.ipynb", "max_forks_repo_name": "TheGupta2012/qctrl-qhack-Hostages-of-the-Entangled-Dungeons", "max_forks_repo_head_hexsha": "1101fa71b41a4fa6dc8f211f3cd08b3dd80fb541", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 74.170012815, "max_line_length": 25538, "alphanum_fraction": 0.7024569204, "converted": true, "num_tokens": 15934, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.4152376133212064}} {"text": "#### From Quarks to Cosmos with AI: Tutorial Day 4\n---\n# Field-level cosmological inference with IMNN + DELFI\n\nby Lucas Makinen [](https://orcid.org/0000-0002-3795-6933 \"\") [](https://twitter.com/lucasmakinen?lang=en \"\"), Tom Charnock [](https://orcid.org/0000-0002-7416-3107 \"Redirect to orcid\") [](https://twitter.com/t_charnock?lang=en \"\")), Justin Alsing [](https://scholar.google.com/citations?user=ICPFL8AAAAAJ&hl=en \"Redirect to orcid\"), and Ben Wandelt [](https://twitter.com/bwandelt?lang=en \"\")\n\n>read the paper: [on arXiv tomorrow !]\n\n>get the code: [https://github.com/tlmakinen/FieldIMNNs](https://github.com/tlmakinen/FieldIMNNs)\n\n\n\n\n$\\quad$\n\nIn this tutorial we will demonstrate Implicit Likelihood Inference (IFI) using Density Estimation Likelihood Free Inference (DELFI) with optimal nonlinear summaries obtained from an Information Maximising Neural Network (IMNN). The goal of the exercise will be to build posterior distributions for the cosmological parameters $\\Omega_c$ and $\\sigma_8$ *directly* from overdensity field simulations.\n\nFirst we'll install the relevant libraries and walk through the simulation implementation. Then we'll build a neural IMNN compressor to generate two optimal summaries for our simulations. Finally, we'll use these summaries to build and train a Conditional Masked Autoregressive Flow, from which we'll construct our parameter posterior distributions.\n\n### Q: Wait a second -- how do we know this works ?\nIf you're not convinced by our method by the end of this tutorial, we invite you to take a look at our [benchmarking tutorial with Gaussian fields from power spectra](https://www.aquila-consortium.org/doc/imnn/pages/examples/2d_field_inference/2d_field_inference.html), which is also runnable in-browser on [this Colab notebook](https://colab.research.google.com/drive/1_y_Rgn3vrb2rlk9YUDUtfwDv9hx774ZF#scrollTo=EW4H-R8I0q6n).\n\n---\n# HOW TO USE THIS NOTEBOOK\n\nYou will (most likely) be running this code using a free version of Google Colab. The code runs just like a Jupyter notebook (`shift` + `enter` or click the play button to run cells). There are some cells with lengthy infrastructure code that you need to run to proceed. These are clearly marked with [run me]. When you get to the challenge exercises, you are welcome to code some functions yourself. However, if you want to run the notebook end-to-end, solution code is presented in hidden cells below (again with the marker [run me]).\n\nSome cells are not meant to be run here as a part of Quarks to Cosmos, but can be run (with a Colab Pro account) on your own.\n\n---\n\n# step 1: loading packages and setting up environment\n\n1. check that Colab is set to run on a GPU ! Go to `Runtime`>`change runtime type` and select `GPU` from the dropdown menu. Next, enable dark mode by going to `settings`>`Theme` and selecting `dark` (protect your eyes !)\n\n2. install packages. The current code relies on several libraries, namely `jax` and `tensorflow_probability`. However, we require both plain `tensorflow_probability` (`tfp`) and the experimental `tensorflow_probability.substrates.jax` (`tfpj`) packages for different parts of our inference \n3. for some Colab sessions, you may need to run the second cell so that `!pip install jax-cosmo` gets the package imported properly.\n\n\n```python\n#@title set up environment [RUN ME FIRST]\n%tensorflow_version 2.x\nimport tensorflow as tf\nprint('tf version', tf.__version__)\n\n!pip install -q jax==0.2.11\n\n!pip install -q tensorflow-probability\n\nimport tensorflow_probability as tfp\nprint('tfp version:', tfp.__version__)\ntfd = tfp.distributions\ntfb = tfp.bijectors\n\n!pip install -q imnn \n\n!python -m pip install -q jax-cosmo\n```\n\n tf version 2.5.0\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 604kB 13.5MB/s \n \u001b[?25h Building wheel for jax (setup.py) ... \u001b[?25l\u001b[?25hdone\n tfp version: 0.13.0\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 133kB 14.5MB/s \n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 81kB 8.6MB/s \n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 286kB 15.8MB/s \n \u001b[?25h Building wheel for jax-cosmo (setup.py) ... \u001b[?25l\u001b[?25hdone\n\n\nnote: if the cell below fails for installing jax-cosmo, just run it again: Colab will rearrange the headings needed.\n\n\n```python\n# now import all the required libraries\nimport jax.numpy as np\nfrom jax import grad, jit, vmap\nfrom jax import random\nimport jax\n\nprint('jax version:', jax.__version__)\n\n# for nn model stuff\nimport jax.experimental.optimizers as optimizers\nimport jax.experimental.stax as stax\n\n\n# tensorflow-prob VANILLA\ntfd = tfp.distributions\ntfb = tfp.bijectors\n\n# tensorflow-prob-JAX\nimport tensorflow_probability.substrates.jax as tfpj\n\ntfdj = tfpj.distributions\ntfbj = tfpj.bijectors\n\n\n# for imnn\nimport imnn\nimport imnn.lfi\n\nprint('IMNN version:', imnn.__version__)\n\n# jax-cosmo module\n!python -m pip install -q jax-cosmo\nimport jax_cosmo as jc\nprint('jax-cosmo version:', jc.__version__)\n\n# matplotlib stuff\nimport matplotlib.pyplot as plt\nfrom scipy.linalg import toeplitz\nimport seaborn as sns\nsns.set()\n\n\nrng = random.PRNGKey(2)\n```\n\n jax version: 0.2.11\n IMNN version: 0.3.0\n jax-cosmo version: 0.1rc7\n\n\n\n```python\nfrom jax.config import config\nconfig.update('jax_enable_x64', True)\n```\n\nmake sure we're using 64-bit precision and running on a GPU !\n\n\n```python\nfrom jax.lib import xla_bridge\nprint(xla_bridge.get_backend().platform)\n```\n\n gpu\n\n\n# Cosmological Fields from the Eisenstein-Hu linear matter power spectrum\nWe're interested in extracting the cosmological parameters $\\Omega_c$ and $\\sigma_8$ *directly* from cosmological field pixels. To generate our simulations we'll need to install the library `jax-cosmo` to generate our differentiable model power spectra.\n\n## choose fiducial model\nTo train our neural compression, we first need to choose a fiducial model to train the IMNN.\n\n\nFor example lets say that our fiducial cosmology has $\\Omega_c=0.40$ and $\\sigma_8=0.60$. This is *deliberately* far from, say, Planck parameters -- we want to investigate how our compression behaves if we don't know our universe's true parameters.\n\n\n```python\ncosmo_params = jc.Planck15(Omega_c=0.40, sigma8=0.60)\n\u03b8_fid = np.array(\n [cosmo_params.Omega_c, \n cosmo_params.sigma8], \n dtype=np.float32)\n\nn_params = \u03b8_fid.shape[0]\n```\n\nOur power spectrum $P_{\\rm LN}(k)$ is the linear matter power spectrum defined as\n\n\n```python\ndef P(k, A=0.40, B=0.60):\n cosmo_params = jc.Planck15(Omega_c=A, sigma8=B)\n return jc.power.linear_matter_power(cosmo_params, k)\n```\n\nand we can visualize it in $k$-space (small $k$ <=> big $r$, big $k$ <=> small $r$) :\n\n\n```python\n#@title plot the Eisenstein-Hu $P(k)$ [run me]\n\nsns.set()\nL = 250.\nN = 128.\n#kmax = 1.0\n#kmin = 0.5 / (N)\n\nkmax = N / L\nkmin = 1. / L\n\nkbin = np.linspace(kmin, kmax, num=100)\n\npower_spec = P(kbin, A=cosmo_params.Omega_c, B=cosmo_params.sigma8)\n\nplt.style.use('dark_background')\nplt.grid(b=None)\n\nplt.plot(kbin, power_spec, linewidth=2)\nplt.xlabel(r'$k\\ \\rm [h\\ Mpc^{-1}]$', fontsize=14)\nplt.ylabel(r'$P(k)\\ \\rm$', fontsize=14)\nplt.ylim((1e2, 1e4))\nplt.xscale('log')\nplt.yscale('log')\n```\n\n____\n## Lognormal Fields from Power Spectra: how much information is embedded in the field ?\nCosmologists often use lognormal fields as \"the poor man's large scale structure\" since they're analytically interrogable and easy to obtain from Gaussian fields. We'll walk through how to obtain the *theoretical* information content of such fields using the Fisher formalism.\n\nThe likelihood for an $N_{\\rm pix}\\times N_{\\rm pix}$ Gaussian field, $\\boldsymbol{\\delta}$, can be explicitly written down for the Fourier transformed data, $\\boldsymbol{\\Delta}$ as\n$$\\mathcal{L}(\\boldsymbol{\\Delta}|\\boldsymbol{\\theta}) = \\frac{1}{(2\\pi)^{N_{\\rm pix}^2 / 2} |P_{\\rm G}({\\bf k}, \\boldsymbol{\\theta})|^{1/2}}\\exp{\\left(-\\frac{1}{2}\\boldsymbol{\\Delta}\\left(P_{\\rm G}({\\bf k}, \\boldsymbol{\\theta})\\right)^{-1}\\boldsymbol{\\Delta}\\right)}$$\nSince the Fisher information can be calculated from the expectation value of the second derivative of the score, i.e. the log likelihood\n$${\\bf F}_{\\alpha\\beta} = - \\left.\\left\\langle\\frac{\\partial^2\\ln\\mathcal{L}(\\Delta|\\boldsymbol{\\theta})}{\\partial\\theta_\\alpha\\partial\\theta_\\beta}\\right\\rangle\\right|_{\\boldsymbol{\\theta}=\\boldsymbol{\\theta}^\\textrm{fid}}$$\nthen we know that analytically the Fisher information must be\n$${\\bf F}_{\\alpha\\beta} = \\frac{1}{2} {\\rm Tr} \\left(\\frac{\\partial P_{\\rm G}({\\bf k}, \\boldsymbol{\\theta})}{\\partial\\theta_\\alpha}\\left(P_{\\rm G}({\\bf k}, \\boldsymbol{\\theta})\\right)^{-1}\\frac{\\partial P_{\\rm G}({\\bf k}, \\boldsymbol{\\theta})}{\\partial\\theta_\\beta}\\left(P_{\\rm G}({\\bf k}, \\boldsymbol{\\theta})\\right)^{-1}\\right)$$\nwhere $\\alpha$ and $\\beta$ label the parameters (for instance $ \\Omega_c, \\sigma_8$) in the power spectrum. As each $k$-mode is uncoupled for this power law form we require the derivatives\n$$\\begin{align}\n\\left(\\frac{\\partial P_{\\rm G}({\\bf k}, \\boldsymbol{\\theta})}{\\partial \\Omega_c},\\ \n\\frac{\\partial P_{\\rm G}({\\bf k}, \\boldsymbol{\\theta})}{\\partial \\sigma_8}\\right) \\\\\n\\end{align}$$\nWe can set up these derivative functions *so long as our code for $P(k)$ is differentiable*. \n\n\nFor *lognormal* fields, this likelihood changes somewhat. Formally, if a random variable $Y$ has a normal distribution, then the exponential function of $Y$, $X = \\exp(Y)$, has a log-normal distribution. We will generate our log-normal fields with a power spectrum such that the *lognormal field has the specified $P_{\\rm LN}(k)$*. This means that we need to employ the *backwards conversion formula* , presented by [M. Greiner? and T.A. En\u00dflin](https://arxiv.org/pdf/1312.1354.pdf), to obtain the correct form for $P_{\\rm G}(k)$ needed for the above Fisher evaluation:\n$$ P_{\\rm G} = \\int d^u x e^{i \\textbf{k} \\cdot \\textbf{x}} \\ln \\left( \\int \\frac{d^u q}{(2\\pi)^u} e^{i \\textbf{q} \\cdot \\textbf{x}} P_{\\rm LN}(\\textbf{q}) \\right) $$\n\nwhich we can do numerically (and differentiably !) in `Jax`. If you're curious about the computation, check out [this notebook](https://colab.research.google.com/drive/1beknmt3CwjEDFFnZjXRClzig1sf54aMR?usp=sharing). We performed the computation using a Colab Pro account with increased GPU resources to accomodate such large fields. When the smoke clears, our fields have a fiducial theoretical Fisher information content, $|\\textbf{F}|_{(0.4, 0.6)}$ of\n \n det_F = 656705.6827\n \nthis can be equivalently expressed in terms of the Shannon information (up to a constant, in nats !) of a Gaussian with covariance matrix $\\textbf{F}^{-1}$:\n\n shannon info = 0.5 * np.log(det_F) = 6.6975 # nats\n\n\nWhen testing our neural IMNN compressor, we used these metrics to verify that we indeed capture the maximal (or close to it) amount of information from our field simulations.\n____\n\n# Simulating the universe with power spectra\n\nWe can now set the simulator arguments, i.e. the $k$-modes to evaluate, the length of the side of a box, the shape of the box and whether to normalise via the volume and squeeze the output dimensions\n\n## choose $k$-modes (the size of our universe-in-a-box)\nNext, we're going to set our $N$-side to 128 (the size of our data vector), $k$-vector, as well as the $L$-side (the physical dimensions of the universe-in-a-box:\n\n\n```python\nN = 128\nshape = (N, N)\n\nk = np.sqrt(\n np.sum(\n np.array(\n np.meshgrid(\n *((np.hstack(\n (np.arange(0, _shape // 2 + 1), \n np.arange(-_shape // 2 + 1, 0)))\n * 2 * np.pi / _shape)**2.\n for _shape in shape))), \n axis=0))\n```\n\n\n```python\nsimulator_args = dict(\n k=k, # k-vector (grid units)\n L=250, # in Mpc h^-1\n shape=shape,\n vol_norm=True, # whether to normalise P(k) by volume\n N_scale=False, # scale field values up or down\n squeeze=True,\n log_normal=True)\n```\n\n___\n## Next, we provide you our universe simulator in `jax`. This is how it works:\n\n### 2D random field simulator in jax\n\nTo create a 2D lognormal random field we can follow these steps:\n\n1. Generate a $(N_\\textrm{pix}\\times N_\\textrm{pix})$ white noise field $\\varphi$ such that $\\langle \\varphi_k \\varphi_{-k} \\rangle' = 1$\n\n2. Fourier Transform $\\varphi$ to real space: $R_{\\rm white}({\\bf x}) \\rightarrow R_{\\rm white}({\\bf k})$\n Note that NumPy's DFT Fourier convention is:\n $$\\phi_{ab}^{\\bf k} = \\sum_{c,d = 0}^{N-1} \\exp{(-i x_c k_a - i x_d k_b) \\phi^{\\bf x}_{cd}}$$\n $$\\phi_{ab}^{\\bf x} = \\frac{1}{N^2}\\sum_{c,d = 0}^{N-1} \\exp{(-i x_c k_a - i x_d k_b) \\phi^{\\bf k}_{cd}}$$ \n\n3. Evaluate the chosen power spectrum over a field of $k$ values and do the lognormal transformation:\n $$P_{\\rm LN}(k) \\gets \\ln(1 + P(k)) $$\n Here we need to ensure that this array of amplitudes are Hermitian, e.g. $\\phi^{* {\\bf k}}_{a(N/2 + b)} = \\phi^{{\\bf k}}_{a(N/2 - b)}$. This is accomplished by choosing indices $k_a = k_b = \\frac{2\\pi}{N} (0, \\dots, N/2, -N/2+1, \\dots, -1)$ (as above) and then evaluating the square root of the outer product of the meshgrid between the two: $k = \\sqrt{k^2_a + k^2_b}$. We can then evaluate $P_{\\rm LN}^{1/2}(k)$.\n\n4. Scale white noise $R_{\\rm white}({\\bf k})$ by the power spectrum:\n $$R_P({\\bf k}) = P_{\\rm LN}^{1/2}(k) R_{\\rm white}({\\bf k}) $$\n \n5. Fourier Transform $R_{P}({\\bf k})$ to real space: $R_P({\\bf x}) = \\int d^d \\tilde{k} e^{i{\\bf k} \\cdot {\\bf x}} R_p({\\bf k})$\n $$R_{ab}^{\\bf x} = \\frac{1}{N^2}\\sum_{c,d = 0}^{N-1} \\exp{(-i x_c k_a - i x_d k_b) R^{\\bf k}_{cd}}$$\n\n\nWe are going to use a broadcastable jax simultor which takes in a variety of different shaped parameter arrays and vmaps them until a single parameter pair are passed. This is very efficient for generating many simulations at once, for Approximate Bayesian Computation for example.\n\n\n```python\n#@title simulator code [RUN ME]\n\n\ndef simulator(rng, \u03b8, simulator_args, foregrounds=None):\n def fn(rng, A, B):\n dim = len(simulator_args[\"shape\"])\n L = simulator_args[\"L\"]\n if np.isscalar(L):\n L = [L] * int(dim)\n Lk = ()\n shape = ()\n for i, _shape in enumerate(simulator_args[\"shape\"]):\n Lk += (_shape / L[i],)\n if _shape % 2 == 0:\n shape += (_shape + 1,)\n else:\n shape += (_shape,)\n \n k = simulator_args[\"k\"]\n k_shape = k.shape\n k = k.flatten()[1:]\n tpl = ()\n for _d in range(dim):\n tpl += (_d,)\n\n V = np.prod(np.array(L))\n scale = V**(1. / dim) \n fft_norm = np.prod(np.array(Lk))\n\n rng, key = jax.random.split(rng)\n \n mag = jax.random.normal(\n key, shape=shape)\n pha = 2. * np.pi * jax.random.uniform(\n key, shape=shape)\n\n # now make hermitian field (reality condition)\n revidx = (slice(None, None, -1),) * dim\n mag = (mag + mag[revidx]) / np.sqrt(2) \n pha = (pha - pha[revidx]) / 2 + np.pi\n dk = mag * (np.cos(pha) + 1j * np.sin(pha))\n cutidx = (slice(None, -1),) * dim\n dk = dk[cutidx]\n \n powers = np.concatenate(\n (np.zeros(1), \n np.sqrt(P(k, A=A, B=B)))).reshape(k_shape)\n \n if simulator_args['vol_norm']:\n powers /= V\n \n if simulator_args[\"log_normal\"]:\n powers = np.real(\n np.fft.ifftshift(\n np.fft.ifftn(\n powers) \n * fft_norm) * V)\n \n powers = np.log(1. + powers)\n powers = np.abs(np.fft.fftn(powers)) \n \n fourier_field = powers * dk\n fourier_field = jax.ops.index_update(\n fourier_field,\n np.zeros(dim, dtype=int),\n np.zeros((1,)))\n \n if simulator_args[\"log_normal\"]:\n field = np.real(np.fft.ifftn(fourier_field)) * fft_norm * np.sqrt(V)\n sg = np.var(field)\n field = np.exp(field - sg / 2.) - 1.\n \n else:\n field = np.real(np.fft.ifftn(fourier_field) * fft_norm * np.sqrt(V)**2)\n \n\n \n if simulator_args[\"N_scale\"]:\n field *= scale \n \n if foregrounds is not None:\n rng, key = jax.random.split(key)\n foreground = foregrounds[\n jax.random.randint(\n key, \n minval=0, \n maxval=foregrounds.shape[0], \n shape=())] \n field = np.expand_dims(field + foreground, (0,))\n \n if not simulator_args[\"squeeze\"]:\n field = np.expand_dims(field, (0, -1))\n \n return np.array(field, dtype='float32')\n\n if isinstance(\u03b8, tuple):\n A, B = \u03b8\n else:\n A = np.take(\u03b8, 0, axis=-1)\n B = np.take(\u03b8, 1, axis=-1)\n if A.shape == B.shape:\n if len(A.shape) == 0:\n return fn(rng, A, B)\n else:\n keys = jax.random.split(rng, num=A.shape[0] + 1)\n rng = keys[0]\n keys = keys[1:]\n return jax.vmap(\n lambda key, A, B: simulator(\n key, (A, B), simulator_args=simulator_args))(\n keys, A, B)\n else:\n if len(A.shape) > 0:\n keys = jax.random.split(rng, num=A.shape[0] + 1)\n rng = keys[0]\n keys = keys[1:]\n return jax.vmap(\n lambda key, A: simulator(\n key, (A, B), simulator_args=simulator_args))(\n keys, A)\n elif len(B.shape) > 0:\n keys = jax.random.split(rng, num=B.shape[0])\n return jax.vmap(\n lambda key, B: simulator(\n key, (A, B), simulator_args=simulator_args))(\n keys, B)\n```\n\nBy constructing our random field simulator *and* cosmological power spectrum in `Jax`, we have access to *exact numerical derivatives*, meaning we can simulate a *differentiable* universe. Let's visualize what our universe and derivatives look like at our fiducial model below:\n\n\n```python\n#@title visualize a fiducial universe and gradients [run me]\nfrom imnn.utils import value_and_jacrev, value_and_jacfwd\n\ndef simulator_gradient(rng, \u03b8, simulator_args=simulator_args):\n return value_and_jacrev(simulator, argnums=1, allow_int=True, holomorphic=True)(rng, \u03b8, simulator_args=simulator_args)\n\nsimulation, simulation_gradient = value_and_jacfwd(simulator, argnums=1)(rng, \u03b8_fid, \n simulator_args=simulator_args)\ncmap = 'viridis'\nfrom mpl_toolkits.axes_grid1 import make_axes_locatable\n\nfig,ax = plt.subplots(nrows=1, ncols=3, figsize=(12,15))\n\nim1 = ax[0].imshow(np.squeeze(simulation), \n extent=(0,1,0,1), cmap=cmap)\nax[0].title.set_text(r'example fiducial $\\rm d$')\ndivider = make_axes_locatable(ax[0])\ncax = divider.append_axes('right', size='5%', pad=0.05)\nfig.colorbar(im1, cax=cax, orientation='vertical')\n\nim1 = ax[1].imshow(np.squeeze(simulation_gradient).T[0].T, \n extent=(0,1,0,1), cmap=cmap)\nax[1].title.set_text(r'$\\nabla_{\\Omega_m} \\rm d$')\ndivider = make_axes_locatable(ax[1])\ncax = divider.append_axes('right', size='5%', pad=0.05)\nfig.colorbar(im1, cax=cax, orientation='vertical')\n\nim1 = ax[2].imshow(np.squeeze(simulation_gradient).T[1].T, \n extent=(0,1,0,1), cmap=cmap)\nax[2].title.set_text(r'$\\nabla_{\\sigma_8} \\rm d$')\ndivider = make_axes_locatable(ax[2])\ncax = divider.append_axes('right', size='5%', pad=0.05)\nfig.colorbar(im1, cax=cax, orientation='vertical')\n\nfor a in ax:\n a.set_xticks([])\n a.set_yticks([])\n \nplt.show()\n```\n\nNice ! Since we can differentiate our universe and power spectrum, we can easily compute gradients of a neural network's outputs with respect to simulation parameters. This will come in handy for compression training. \n\n---\n## Training an IMNN\n\n\n\n\n\n\nThe details behind the IMNN algorithm [can be found here on arxiv](https://arxiv.org/abs/1802.03537), but we'll summarize the gist briefly:\n\n\n\n1. We want to maximise the Fisher information, $\\textbf{F}$, of compressed summaries to satisfy the Cramer-Rao bound:\n $$ \\langle (\\vartheta_\\alpha - \\langle \\vartheta_\\alpha \\rangle ) (\\vartheta_\\beta - \\langle \\vartheta_\\beta\n \\rangle) \\rangle \\geq \\textbf{F}^{-1}_{\\alpha \\beta} $$ which means saturating the Fisher information minimizes the average variance of the parameter estimates.\n\n2. To do this, and without loss of generality (proof coming soon!) we compute a Gaussian likelihood form to compute our Fisher information:\n$$ -2 \\ln \\mathcal{L}(\\textbf{x} | \\textbf{d}) = (\\textbf{x} - \\boldsymbol{\\mu}_f(\\vartheta))^T \\textbf{C}_f^{-1}(\\textbf{x} - \\boldsymbol{\\mu}_f(\\vartheta)) $$ where $\\boldsymbol{\\mu}_f$ and $\\textbf{C}$ are the mean and covariance of the network output (summaries). The Fisher is then $$ \\textbf{F}_{\\alpha \\beta} = {\\rm tr} [\\boldsymbol{\\mu}_{f,\\alpha}^T C^{-1}_f \\boldsymbol{\\mu}_{f, \\beta}] $$\n\n\nSince we can differentiate through our neural network *and* simulated universe, we have the exact derivatives with respect to the pipeline we need to compute the Fisher matrix of compressed summaries on-the-fly during compression training. \n___\n\n### Q: wait -- what if my simulator isn't differentiable ?\nWe don't *need* to have the exact derivatives for IMNN training ! Having the gradients accessible just means that we don't have to optimize finite-differencing for estimating derivatives by hand, however (as is done in the original IMNN paper).\n\n\n\n___\n\n\nLet's use an IMNN trained on cosmological fields to see how much information we can extract an what sort of constraints we can get. We will use 2000 simulations to estimate the covariance and use all of their derivatives and we'll summarise the whole cosmological field using 2 summaries.\n\n\n```python\nn_s = 200 # number of simulations used to estimate covariance of network outputs\nn_d = n_s # number of simulations used to estimate the numerical derivative of\n # the mean of the network outputs\nn_summaries = 2\n```\n\nWe're going to use a fully convolutional inception network built using stax with some custom designed blocks. The inception block itself is implemented in the following block:\n\n\n```python\n#@title nn model stuff [RUN ME]\ndef InceptBlock(filters, strides, do_5x5=True, do_3x3=True):\n \"\"\"InceptNet convolutional striding block.\n filters: tuple: (f1,f2,f3)\n filters1: for conv1x1\n filters2: for conv1x1,conv3x3\n filters3L for conv1x1,conv5x5\"\"\"\n \n filters1, filters2, filters3 = filters\n conv1x1 = stax.serial(stax.Conv(filters1, (1, 1), strides, padding=\"SAME\"))\n \n filters4 = filters2\n conv3x3 = stax.serial(stax.Conv(filters2, (1, 1), strides=None, padding=\"SAME\"),\n stax.Conv(filters4, (3, 3), strides, padding=\"SAME\"))\n \n filters5 = filters3\n conv5x5 = stax.serial(stax.Conv(filters3, (1, 1), strides=None, padding=\"SAME\"),\n stax.Conv(filters5, (5, 5), strides, padding=\"SAME\")) \n \n maxpool = stax.serial(stax.MaxPool((3, 3), padding=\"SAME\"),\n stax.Conv(filters4, (1, 1), strides, padding=\"SAME\"))\n \n if do_3x3:\n if do_5x5:\n return stax.serial(\n stax.FanOut(4),\n stax.parallel(conv1x1, conv3x3, conv5x5, maxpool),\n stax.FanInConcat(), \n stax.LeakyRelu)\n else:\n return stax.serial(\n stax.FanOut(3),\n stax.parallel(conv1x1, conv3x3, maxpool),\n stax.FanInConcat(), \n stax.LeakyRelu)\n else:\n return stax.serial(\n stax.FanOut(2),\n stax.parallel(conv1x1, maxpool),\n stax.FanInConcat(), \n stax.LeakyRelu)\n```\n\nWe'll also want to make sure that the output of the network is the correct shape, for which we'll introduce a Reshaping layer\n\n\n```python\ndef Reshape(shape):\n \"\"\"Layer function for a reshape layer.\"\"\"\n init_fun = lambda rng, input_shape: (shape,())\n apply_fun = lambda params, inputs, **kwargs: np.reshape(inputs, shape)\n return init_fun, apply_fun\n```\n\nNow we can build the network, with 55 filters and strides of 4 in each direction in each layer\n\n\n```python\nfs = 55\nlayers = [\n InceptBlock((fs, fs, fs), strides=(4, 4)),\n InceptBlock((fs, fs, fs), strides=(4, 4)),\n InceptBlock((fs, fs, fs), strides=(4, 4)),\n InceptBlock((fs, fs, fs), strides=(2, 2), do_5x5=False, do_3x3=False),\n stax.Conv(n_summaries, (1, 1), strides=(1, 1), padding=\"SAME\"),\n stax.Flatten,\n Reshape((n_summaries,)) \n]\nmodel = stax.serial(*layers)\n```\n\nWe'll also introduce a function to check our model output:\n\n\n```python\ndef print_model(layers, input_shape, rng):\n print('input_shape: ', input_shape)\n for l in range(len(layers)):\n _m = stax.serial(*layers[:l+1])\n print('layer %d shape: '%(l+1), _m[0](rng, input_shape)[0])\n\n# print model specs\nkey,rng = jax.random.split(rng)\ninput_shape = (1,) + shape + (1,)\n\nprint_model(layers, input_shape, rng)\n```\n\n input_shape: (1, 128, 128, 1)\n layer 1 shape: (1, 32, 32, 220)\n layer 2 shape: (1, 8, 8, 220)\n layer 3 shape: (1, 2, 2, 220)\n layer 4 shape: (1, 1, 1, 110)\n layer 5 shape: (1, 1, 1, 2)\n layer 6 shape: (1, 2)\n layer 7 shape: (2,)\n\n\nWe'll also grab an adam optimiser from jax.experimental.optimizers\n\n\n```python\noptimiser = optimizers.adam(step_size=1e-3)\n```\n\nNote that due to the form of the network we'll want to have simulations that have a \"channel\" dimension, which we can set up by not allowing for squeezing in the simulator.\n\n### Load an IMNN\n\nFinally we can load a pre-trained IMNN and compare its compression efficiency to the theoretical Fisher. We will pull the weights and state from the parent repository and calculate the compressor statistics.\n\nWe've used a SimulatorIMNN trained on new simulations on-the-fly, eliminating the need for a validation dataset. If you're interested in the IMNN training, see the [benchmarking Colab notebook](https://colab.research.google.com/drive/1_y_Rgn3vrb2rlk9YUDUtfwDv9hx774ZF#scrollTo=EW4H-R8I0q6n) or the Bonus challenge at the end of this tutorial.\n\nWe're not training an IMNN here because this model takes $\\approx 50$ minutes and requires elevated Colab Pro resources.\n\n\n\n```python\n!git clone https://github.com/tlmakinen/FieldIMNNs.git\n```\n\n Cloning into 'FieldIMNNs'...\n remote: Enumerating objects: 324, done.\u001b[K\n remote: Counting objects: 100% (324/324), done.\u001b[K\n remote: Compressing objects: 100% (244/244), done.\u001b[K\n remote: Total 324 (delta 130), reused 265 (delta 74), pack-reused 0\u001b[K\n Receiving objects: 100% (324/324), 63.22 MiB | 14.66 MiB/s, done.\n Resolving deltas: 100% (130/130), done.\n\n\n\n```python\n# load IMNN state\nimport cloudpickle as pickle\nimport os\n\ndef unpickle_me(path):\n file = open(path, 'rb')\n return pickle.load(file)\n\n\nfolder_name = './FieldIMNNs/tutorial/IMNN-aspects/'\nloadstate = unpickle_me(os.path.join(folder_name, 'IMNN_state'))\nstate = jax.experimental.optimizers.pack_optimizer_state(loadstate)\n\nstartup_key = np.load(os.path.join(folder_name, 'IMNN_startup_key.npy'), allow_pickle=True)\n\n# load weights to set the IMNN\nbest_weights = np.load(os.path.join(folder_name, 'best_w.npy'), allow_pickle=True)\n```\n\n\n```python\n# initialize IMNN with pre-trained state\nrng, key = jax.random.split(rng)\nIMNN = imnn.IMNN(\n n_s=n_s,\n n_d=n_d,\n n_params=n_params,\n n_summaries=n_summaries,\n input_shape=(1,) + shape + (1,),\n \u03b8_fid=\u03b8_fid,\n model=model,\n optimiser=optimiser,\n key_or_state=state, # <---- initialize with state\n simulator=lambda rng, \u03b8: simulator(\n rng, \u03b8, simulator_args={\n **simulator_args, \n **{\"squeeze\": False}}))\n```\n\n `simulator` provided, using SimulatorIMNN\n\n\n\n```python\n# now set weights using the best training weights and startup key (this can take a moment)\nIMNN.set_F_statistics(w=best_weights, key=startup_key)\n```\n\n\n```python\nprint('det F from IMNN:', np.linalg.det(IMNN.F))\n```\n\n det F from IMNN: 594450.7\n\n\n\n```python\nprint('% Fisher information captured by IMNN compared to theory: ', np.linalg.det(IMNN.F) / 656705.6827) \n```\n\n % Fisher information captured by IMNN compared to theory: 0.9052011\n\n\n### if you want to check out how to train an IMNN, see the end of the tutorial !\n\n---\n# Inference on a target cosmological field\n\nNow that we have a trained compression function (albeit at a somewhat arbitrary fiducial model), we can now perform simulation-based inference with the optimal summaries. \n\nWe'll now pretend to \"observe\" a cosmological density field at some target parameters, $\\theta_{\\rm target}$. We'll select $\\Omega_c=0.25$ and $\\sigma_8=0.81$ (measured 2015 Planck parameters). To get started with this tutorial, we'll load a pre-generated field from the GitHub (\"field 2\" from our paper !), but you can always generate a new realization with the simulator code.\n\n\n```python\n\u03b8_target = np.array([jc.Planck15().Omega_c, jc.Planck15().sigma8,])\n\n\u03b4_target = np.load('./FieldIMNNs/tutorial/target_field_planck.npy')\n\nsns.set() # set up plot settings\ncmap='viridis'\nplt.imshow(\u03b4_target, cmap=cmap)\nplt.colorbar()\nplt.title('target cosmological field')\nplt.show()\n```\n\nNow we're going to **forget we ever knew our choice of target parameters** and do inference on this target data as if it were a real observation (minus measurement noise for now, of course !)\n\n## Inference\n\nWe can now attempt to do inference of some target data using the IMNN. \n\nFirst we're going to compress our target field down to parameter estimates using the IMNN method `IMNN.get_estimate(d)`. What this code does is returns the score estimator for the parameters, obtained via the transformation \n$$ \\hat{\\theta}_{\\alpha} = \\theta^{\\rm fid}_\\alpha + \\textbf{F}^{-1}_{\\alpha \\beta} \\frac{\\partial \\mu_i}{\\partial \\theta_\\beta} \\textbf{C}^{-1}_{ij} \\textbf({x}(\\textbf{w}, \\textbf{d}) - {\\mu})_j $$\nwhere ${x}(\\textbf{w}, \\textbf{d})$ are the network summaries.\n\n\n\n```python\nestimates = IMNN.get_estimate(np.expand_dims(\u03b4_target, (0, 1, -1)))\nprint('IMNN parameter estimates:', estimates)\n```\n\n IMNN parameter estimates: [[0.31959814 0.7647977 ]]\n\n\nThe cool thing about training an IMNN is that it *automatically* gives you a simple uncertainty estimate on the parameters of interest via the optimal Fisher matrix. We can make a Gaussian approximation to the likelihood using the inverse of the matrix.\n\nNote that to demonstrate robustness, the fiducial parameter values are deliberately far from the target parameters that this estimate of the Fisher information as the covariance will likely be misleading. \n\nWe'll need to select a prior distribution first. We'll do this in `tfpj`, selecting wide uniform priors for both $\\Omega_c$ and $\\sigma_8$.\n\n\n```python\nprior = tfpj.distributions.Blockwise(\n [tfpj.distributions.Uniform(low=low, high=high)\n for low, high in zip([0.01, 0.2], [1.0, 1.3])])\nprior.low = np.array([0.01, 0.])\nprior.high = np.array([1.0, 1.3])\n```\n\nThen we can use the IMNN's built-in Gaussian approximation code:\n\n\n```python\nsns.set()\nGA = imnn.lfi.GaussianApproximation(\n parameter_estimates=estimates, \n invF=np.expand_dims(np.linalg.inv(IMNN.F), 0), \n prior=prior, \n gridsize=100)\nax = GA.marginal_plot(\n known=\u03b8_target, \n label=\"Gaussian approximation\", \n axis_labels=[r\"$\\Omega_c$\", r\"$\\sigma_8$\"],\n colours=\"C1\");\n```\n\nEven though our fiducial model was trained far away $(\\Omega_c, \\sigma_8) = (0.4, 0.6)$, our score esimates (center of our ellipse) are very close to the target Planck (crosshairs).\n\nwe now have a compression and informative summaries of our target data. We'll next proceed to setting up density estimation to construct our posteriors !\n\n\n___\n# Posterior Construction with DELFI\n\nDensity Estimation Likelihood-Free Inference (DELFI) is presented formally [here on arxiv](https://arxiv.org/abs/1903.00007), but we'll give you the TLDR here:\n\nNow that we have nonlinear IMNN summaries, $\\textbf{x}$, to describe our cosmological fields, we can perform density estimation to model the *summary data likelihood*, $p(\\textbf{x} | \\boldsymbol{\\theta})$. Once we have this, we can obtain the posterior distribution for $\\boldsymbol{\\theta}$ via Bayes' rule:\n$$ p(\\boldsymbol{\\theta} | \\textbf{x}) \\propto p(\\textbf{x} | \\boldsymbol{\\theta}) p(\\boldsymbol{\\theta}) $$.\n\n## What are CMAFs ?\n\n\nDELFI provides Conditional Masked Autoregressive Flows (CMAFs) are stacks of neural autoencoders carefully masked to parameterize the summary-parameter likelihood. To start, note that any probability density can be factored as a product of one-dimensional conditional distributions via the chain rule of probability:\n\\begin{equation}\n p(\\textbf{x} | \\boldsymbol{\\theta}) = \\prod_{i=1}^{\\dim(\\textbf{x})} p({\\rm x}_i | \\textbf{x}_{1:i-1}, \\boldsymbol{\\theta})\n\\end{equation}\nMasked Autoencoders for density estimation (MADE) model each of these one-dimensional conditionals as Gaussians with mean and variance parameters parameterized by neural network weights, $\\textbf{w}$. The neural network layers are masked in such a way that the autoregressive property is preserved, e.g. the output nodes for the density $p({\\rm x}_i | \\textbf{x}_{1:i-1}, \\boldsymbol{\\theta})$ *only* depend on $\\textbf{x}_{1:i-1}$ and $\\boldsymbol{\\theta}$, satisfying the chain rule.\n\n\nWe can then stack a bunch of MADEs to form a neural flow for our posterior !\n\n\n\n\nWhat we're going to do is \n\n1. Train a Conditional Masked Autoregressive Flow to parameterize $p(\\textbf{x} | \\boldsymbol{\\theta})$ to minimize the log-probability, $-\\ln U$.\n2. Use an affine MCMC sampler to draw from the posterior at the target summaries, $\\textbf{x}^{\\rm target}$\n3. Append training data from the posterior and re-train MAFs.\n\n\n\n\n```python\n!pip install -q getdist\n!pip install -q corner\n!pip install -q chainconsumer\n\nimport keras\nimport tensorflow.keras.backend as K\n\nimport time\nfrom tqdm import tqdm\nfrom chainconsumer import ChainConsumer\n```\n\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 778kB 15.4MB/s \n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 28.5MB 1.5MB/s \n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 645kB 56.1MB/s \n \u001b[?25h Building wheel for getdist (setup.py) ... \u001b[?25l\u001b[?25hdone\n \u001b[31mERROR: albumentations 0.1.12 has requirement imgaug<0.2.7,>=0.2.5, but you'll have imgaug 0.2.9 which is incompatible.\u001b[0m\n \u001b[K |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 51kB 5.8MB/s \n \u001b[?25h Building wheel for chainconsumer (setup.py) ... \u001b[?25l\u001b[?25hdone\n\n\n(ignore the red error message)\n\nWe'll set up the same prior as before, this time in regular `tensorflow-probability`. This means that our CMAFs can talk to our prior draws in the form of tensorflow tensors.\n\n\n```python\n# set up prior in non-jax tfp\nsamp_prior = tfp.distributions.Blockwise(\n [tfp.distributions.Uniform(low=low, high=high)\n for low, high in zip([0.01, 0.2], [1.0, 1.3])])\nsamp_prior.low = np.array([0.01, 0.])\nsamp_prior.high = np.array([1.0, 1.3])\n```\n\n\n```python\n#@title set up the CMAF code [RUN ME]\nclass ConditionalMaskedAutoregressiveFlow(tf.Module):\n def __init__(self, n_dimensions=None, n_conditionals=None, n_mades=1, n_hidden=[50,50], input_order=\"random\",\n activation=keras.layers.LeakyReLU(0.01), \n all_layers=True,\n kernel_initializer=keras.initializers.RandomNormal(mean=0.0, stddev=1e-5, seed=None), \n bias_initializer=keras.initializers.RandomNormal(mean=0.0, stddev=1e-5, seed=None),\n kernel_regularizer=None, bias_regularizer=None, kernel_constraint=None,\n bias_constraint=None):\n super(ConditionalMaskedAutoregressiveFlow, self).__init__('hi')\n # extract init parameters\n self.n_dimensions = n_dimensions\n self.n_conditionals = n_conditionals\n self.n_mades = n_mades\n # construct the base (normal) distribution\n self.base_distribution = tfd.MultivariateNormalDiag(loc=tf.zeros(self.n_dimensions), scale_diag=tf.ones(self.n_dimensions))\n # put the conditional inputs to all layers, or just the first layer?\n if all_layers == True:\n all_layers = \"all_layers\"\n else:\n all_layers = \"first_layer\"\n # construct stack of conditional MADEs\n self.MADEs = [tfb.AutoregressiveNetwork(\n params=2,\n hidden_units=n_hidden,\n activation=activation,\n event_shape=[n_dimensions],\n conditional=True,\n conditional_event_shape=[n_conditionals],\n conditional_input_layers=all_layers,\n input_order=input_order,\n kernel_initializer=kernel_initializer,\n bias_initializer=bias_initializer,\n kernel_regularizer=kernel_regularizer,\n bias_regularizer=bias_regularizer,\n kernel_constraint=kernel_constraint,\n bias_constraint=bias_constraint,\n ) for i in range(n_mades)\n ]\n # bijector for x | y (chain the conditional MADEs together)\n def bijector(self, y):\n # start with an empty bijector\n MAF = tfb.Identity() \n # pass through the MADE layers (passing conditional inputs each time)\n for i in range(self.n_mades):\n MAF = tfb.MaskedAutoregressiveFlow(shift_and_log_scale_fn=lambda x: self.MADEs[i](x, conditional_input=y))(MAF)\n return MAF\n # construct distribution P(x | y)\n def __call__(self, y):\n return tfd.TransformedDistribution(\n self.base_distribution,\n bijector=self.bijector(y))\n # log probability ln P(x | y)\n def log_prob(self, x, y):\n return self.__call__(y).log_prob(x)\n # sample n samples from P(x | y)\n def sample(self, n, y):\n # base samples\n base_samples = self.base_distribution.sample(n)\n # biject the samples\n return self.bijector(y).forward(base_samples)\n```\n\nIf you're curious about how the MCMC sampler and CMAF code work, feel free to double-click the hidden cells above. We'll walk through the gist of how each module works though:\n\nThe `ConditionalMaskedAutoregressiveFlow` API functions similarly to other `tfp` distributions. To set up a model we need to choose a few aspects of the flow. We first need to choose how many MADEs we want to stack to form our flow, `n_mades`. To set up a model with three MADEs, two parameters (`n_dimensions`) and two conditionals (`n_conditionals`), and two hidden layers of 50 neurons per MADE, we'd call:\n\n my_CMAF = ConditionalMaskedAutoregressiveFlow(n_dimensions=2, n_conditionals=2, n_mades=3, n_hidden=[50,50])\n\n\nWhat's cool is that this module works just like a `tfp.distributions` function, which means that we can call a log-probability, $p(x | y)$ *conditional* on some $y$-value:\n\n key,rng = jax.random.split(rng)\n n_samples = 1\n x = prior.sample(sample_shape=(n_samples,), seed=key)\n y = np.array([0.3, 0.4])\n logU = my_CMAF.log_prob(x, y)\n\nWe're going to work with this basic syntax to set up useful DELFI dictionaries to store useful aspects.\n\n___\n# Exercise 0: initialize models for target data\n\nNow we're going to initialize several CMAF models for our piece of target data. Using multiple (and varied) deep learning architectures for the same problem is called the \"deep ensemble\" technique ([see this paper for an overview](https://papers.nips.cc/paper/2017/file/9ef2ed4b7fd2c810847ffa5fa85bce38-Paper.pdf)).\n\nWhen setting up DELFI, it's important to remember that each ensemble of CMAFs ought to be generated *per piece of target data*, since we're interested in observing the \"slice\" of parameter space that gives us each datum's posterior. Since these models are written in Tensorflow, we don't have to worry about specifying a random key or initialization for the model like we do in `Jax`.\n\n\n1. Declare a `DELFI` dictionary to store the following aspects:\n - a list of CMAF models\n - a list of optimizers\n - a training dataset\n - a validation dataset\n - the IMNN estimates\n\n2. Initialize `num_models=2` models, each with `n_mades=3` MADEs. Try one set of MADEs with two layers of 50 neurons, and another with three layers. See if you can set up their respective optimizers (we'll use `tf.keras.optimizers.Adam()` with a learning rate of $10^-3$.\n\n\n ## note: remove all `pass` arguments to functions to make them runnable !\n\n\n```python\nDELFI = {\n \n}\n```\n\n\n```python\n#@title Ex. 0 solution [run me to proceed]\nnum_targets = 1\n# set up list of dictionaries for the target datum\nDELFI = {\n 'MAFs': None, # list of CAMF models\n 'opts': [], # list of optimizers\n 'posts':[], # list of MAF posteriors\n 'train_data': None, # training dataset\n 'val_data': None, # validation dataset\n 'train_losses' : [], # losses\n 'val_losses' : [],\n 'estimates': estimates,\n 'target_data' : \u03b4_target,\n 'F_IMNN': IMNN.F,\n '\u03b8_target': \u03b8_target,\n}\n\n\n# number of CMAFs per DELFI ensemble\nnum_models = 2\nn_hiddens = [[50,50], [50,50]] # try different architectures\n\n\n\nDELFI['MAFs'] = [ConditionalMaskedAutoregressiveFlow(n_dimensions=2, n_mades=3,\n n_conditionals=2, n_hidden=n_hiddens[i]) for i in range(num_models)]\n\nDELFI['opts'] = [tf.keras.optimizers.Adam(learning_rate=1e-3) for i in range(num_models)]\n```\n\n___\n# Exercise 1: define train and validation steps\n\nHere we want to define tensorflow function training and validation steps that we'll later call in a loop to train each CMAF model in the DELFI ensemble. \n\n1. set up the log posterior loss: $-\\ln U = -\\ln p(x | y) - \\ln p(y)$ where $y=\\theta$ are our parameters.\n\n *hint*: try the `samp_prior.log_prob()` call on a few data\n2. obtain gradients, `grads` with respect to the scalar loss\n3. update each optimizer with the call `optimizer.apply_gradients(zip(grads, model.trainable_variables)`\n\n\n\n```python\n# define loss function -ln U\ndef logloss(x, y, model, prior):\n pass\n```\n\n\n```python\n#@title Ex. 1 solution [run me to proceed]\n# define loss function\ndef logloss(x, y, model):\n return - model.log_prob(x,y) - samp_prior.log_prob(y)\n```\n\nNow that we have our loss defined, we can use it to train our CMAFs via backpropagation:\n\n\n```python\n@tf.function\ndef train_step(x, y, ensemble, opts):\n losses = []\n\n # loop over models in ensemble\n for m in range(len(ensemble)):\n with tf.GradientTape() as tape:\n\n # get loss across batch using our log-loss function\n loss = K.mean(logloss(x, y, ensemble[m]))\n losses.append(loss)\n\n grads = tape.gradient(loss, ensemble[m].trainable_variables)\n opts[m].apply_gradients(zip(grads, ensemble[m].trainable_variables))\n\n return losses\n\n@tf.function\ndef val_step(x, y, ensemble):\n val_l = []\n for m in range(len(ensemble)):\n loss = K.mean(logloss(x, y, ensemble[m]))\n val_l.append(loss)\n return val_l\n\n```\n\n___\n\n# Exercise 2: create some dataset functions\nHere we want to create the dataset of $(\\textbf{x}, \\boldsymbol{\\theta})$ pairs to train our CMAFs on. Write a function that:\n1. generate simulations (with random keys) from sampled parameter pairs, $\\theta$. We've set up the key-splitting and simulator code for you.\n2. feed simulations through `IMNN.get_estimate()` to get summaries, $\\textbf{x}$\n3. try to use `jax.vmap()` the above to do this efficiently ! \n\n\n\n```python\n#@title hints for vmapping:\n# for a function `my_fn(a, x)`, you can vmap, \"vector map\" over a set of array values as follows:\n\ndef my_fn(x, a, b):\n return a*x**3 - x + b\n\n# define a slope and intercept\na = 0.5\nb = 1.0\n# define our x-values\nx = np.linspace(-10,10, num=100)\n\n# define a mini function that only depends on x\nmini_fn = lambda x: my_fn(x, a=a, b=b)\n\ny = jax.vmap(mini_fn)(x)\nplt.plot(x, y)\nplt.xlabel('$x$')\nplt.ylabel('$y$')\n \n```\n\n\n```python\ndef get_params_summaries(key, \u03b8_samp, simulator=simulator):\n \"\"\"\n function for generating (x,\u03b8) pairs from IMNN compression\n over the prior range\n \u03b8_samp: array of sampled parameters over prior range\n simulator: function for simulating data to be compressed\n \"\"\"\n\n n_samples = \u03b8_samp.shape[0]\n\n # we'll split up the keys for you\n keys = np.array(jax.random.split(key, num=n_samples))\n\n # next define a simulator that takes a key as argument\n my_simulator = lambda rng, \u03b8: simulator(\n rng, \u03b8, simulator_args={\n **simulator_args, \n **{\"squeeze\": False}})\n \n # generate data, vmapping over the random keys and parameters:\n # d =\n\n # generate summaries\n # x = \n\n # return paired training data\n pass\n\n```\n\n\n```python\n#@title Ex. 2 solution [run me to proceed]\n\ndef get_params_summaries(key, n_samples, \u03b8_samp, simulator=simulator):\n keys = np.array(jax.random.split(key, num=n_samples))\n sim = lambda rng, \u03b8: simulator(\n rng, \u03b8, simulator_args={\n **simulator_args, \n **{\"squeeze\": False}})\n \n # generate a bunch of fields over the prior ranges\n d = jax.vmap(sim)(keys, \u03b8_samp)\n \n # compress fields to summaries\n x = IMNN.get_estimate(d)\n\n return x, \u03b8_samp\n```\n\n\n```python\ndef get_dataset(data, batch_size=20, buffer_size=1000, split=0.75):\n \"\"\" \n helper function for creating tensorflow dataset for CMAF training.\n data: pair of vectors (x, \u03b8) = (x, y)\n batch_size: how many data pairs per gradient descent\n buffer_size: what chunk of the dataset to shuffle (default: random)\n split: train-validation split\n \"\"\"\n x,y = data\n\n idx = int(len(x)*split)\n x_train = x[:idx]\n y_train = y[:idx] \n x_val = x[idx:]\n y_val = y[idx:]\n\n # Prepare the training dataset.\n train_dataset = tf.data.Dataset.from_tensor_slices((x_train, y_train))\n train_dataset = train_dataset.shuffle(buffer_size=buffer_size).batch(batch_size)\n\n # Prepare the validation dataset.\n val_dataset = tf.data.Dataset.from_tensor_slices((x_val, y_val))\n val_dataset = val_dataset.batch(batch_size)\n \n return train_dataset, val_dataset\n```\n\n# Visualize compressed summaries at fiducial model and over the prior \n\nNow that we a function that can take in parameter vectors, generates simulations, and then compresses them into summaries, we can visualize how the IMNN compresses the fields in summary space. We will visualize:\n1. compressed simulations run at the fiducial model ($\\Omega_c, \\sigma_8)$ = (0.4, 0.6)\n2. compressed simulations at the target model ($\\Omega_c, \\sigma_8)$ = (0.2589, 0.8159)\n3. compressed simulations run across the full (uniform) prior range\n\n\n```python\nn_samples = 1000\nbuffer_size = n_samples\nkey1,key2 = jax.random.split(rng)\n\n# params over the prior range\n\u03b8_samp = prior.sample(sample_shape=(n_samples,), seed=key1)\nxs, \u03b8_samp = get_params_summaries(key2, n_samples, \u03b8_samp)\n\n# fiducial params\nkey,rng = jax.random.split(key1)\n_\u03b8fids = np.repeat(np.expand_dims(\u03b8_fid, 1), 1000, axis=1).T\nxs_fid, _ = get_params_summaries(key, n_samples, _\u03b8fids)\n\n# target params\n_\u03b8targets = np.repeat(np.expand_dims(\u03b8_target, 1), 1000, axis=1).T\nxs_target, _ = get_params_summaries(key, n_samples, _\u03b8targets)\n\n\nplt.scatter(xs.T[0], xs.T[1], label='prior', s=5, alpha=0.7)\nplt.scatter(xs_fid.T[0], xs_fid.T[1], label='fiducial', s=5, marker='*', alpha=0.7)\nplt.scatter(xs_target.T[0], xs_target.T[1], label='target', s=5, marker='+', alpha=0.7)\n\nplt.title('summary scatter')\nplt.xlabel(r'$x_1$')\nplt.ylabel(r'$x_2$')\nplt.xlim(-1.0, 2.0)\nplt.legend()\nplt.show()\n```\n\n### Q: Wait, why is our prior in summary space not uniform (rectangular) ?\nRemember, we've passed our parameters through our simulator, and our simulations through the IMNN compressor, meaning our summaries are nonlinear (weirdly-shaped). These score estimates obtained from the IMNN is are quick and convenient, but can be biased and suboptimal if the fiducial model is far from the truth.\n\nEven then, these IMNN score summaries can be used for likelihood-free inference to give consistent posterior estimates, albeit with some information loss (since we haven't compressed near the target).\n\n\n---\n## Now, onto the good bit--CMAF training !\n\n### Generate our training dataset\nWe're going to call our dataset functions to create a dataset of $(\\textbf{x}, \\boldsymbol{\\theta})$ of shape $((1000, 2), (1000, 2))$.\n\n\n```python\nn_samples = 1000\nbatch_size = 100\nbuffer_size = n_samples\nkey1,key2 = jax.random.split(rng)\n\n# sample from the tfpj prior so that we can specify the key\n# and stay in jax.numpy:\n\u03b8_samp = prior.sample(sample_shape=(n_samples,), seed=key1)\n\n# generate sims and compress to summaries\nts, \u03b8_samp = get_params_summaries(key2, n_samples, \u03b8_samp)\ndata = (ts, \u03b8_samp)\n\n# use the dataset function\ntrain_dataset, val_dataset = get_dataset(data, batch_size=batch_size, buffer_size=buffer_size)\nDELFI['train_dataset'] = train_dataset\nDELFI['val_dataset'] = val_dataset\n```\n\nNext let's define a training loop for a set number of epochs, calling our training and validation step functions.\n\n___\n\n# Exercise 3: define training loop\nWe're going to use the `train_step` functions to train our CMAF models for a set number of epochs.\n\n\n```python\ndef training_loop(delfi, epochs=2000):\n \"\"\"training loop function that updates optimizers and\n stores training history\"\"\"\n\n # unpack our dictionary's attributes\n ensemble = delfi['MAFs']\n opts = delfi['opts']\n train_dataset = delfi['train_dataset']\n val_dataset = delfi['val_dataset']\n\n for epoch in tqdm(range(epochs)):\n\n # shuffle training data anew every 50th epoch (done for you)\n if epoch % 50 == 0:\n train_dataset = train_dataset.shuffle(buffer_size=buffer_size)\n\n # Iterate over the batches of the dataset.\n for step, (x_batch_train, y_batch_train) in enumerate(train_dataset):\n # 1) call train step and capture loss value\n pass\n # 2) store loss value\n\n # Run a validation loop at the end of each epoch.\n for x_batch_val, y_batch_val in val_dataset:\n # 3) call val step and capture loss value\n pass\n # 4) store validation loss value\n\n pass\n\n```\n\n\n```python\n#@title Ex. 3 solution [run me to proceed]\ndef training_loop(delfi, epochs=2000):\n \"\"\"training loop function that updates optimizers and\n stores training history\"\"\"\n\n # unpack our dictionary's attributes\n ensemble = delfi['MAFs']\n opts = delfi['opts']\n train_dataset = delfi['train_dataset']\n val_dataset = delfi['val_dataset']\n\n for epoch in tqdm(range(epochs)):\n\n # shuffle training data anew every 50th epoch\n if epoch % 50 == 0:\n train_dataset = train_dataset.shuffle(buffer_size=buffer_size)\n\n # Iterate over the batches of the dataset.\n for step, (x_batch_train, y_batch_train) in enumerate(train_dataset):\n # call train step and capture loss value\n loss_values = train_step(x_batch_train, y_batch_train, ensemble, opts)\n\n # store loss value\n delfi['train_losses'].append(loss_values)\n\n # Run a validation loop at the end of each epoch.\n for x_batch_val, y_batch_val in val_dataset:\n # call val step and capture loss value\n val_loss = val_step(x_batch_val, y_batch_val, ensemble)\n\n # store validation loss value\n delfi['val_losses'].append(val_loss)\n```\n\n\n```python\n#@title define some useful plotting functions [run me]\n# visualize training trajectories\ndef plot_trajectories(delfis, num_models=4, num_targets=4):\n \"\"\"code for plotting training trajectories. note that num_targets should be\n equal to len(delfis)\"\"\"\n\n\n if num_targets > 1:\n fig,axs = plt.subplots(ncols=num_models, nrows=num_targets, figsize=(8,8))\n\n for i,d in enumerate(delfis):\n for j in range(num_models):\n axs[i,j].plot(np.array(d['train_losses']).T[j], label='train')\n axs[i,j].plot(np.array(d['val_losses']).T[j], label='val')\n\n if j == 0:\n axs[i,j].set_ylabel(r'$p(t\\ |\\ \\vartheta; w)$')\n \n if i == num_models-1:\n axs[i,j].set_xlabel(r'num epochs')\n\n else:\n fig,axs = plt.subplots(ncols=num_models, nrows=num_targets, figsize=(7,3))\n d = delfis\n for j in range(num_models):\n axs[j].plot(np.array(d['train_losses']).T[j], label='train')\n axs[j].plot(np.array(d['val_losses']).T[j], label='val')\n\n if j == 0:\n #axs[j].set_ylabel(r'$p(t\\ |\\ \\vartheta; w)$')\n axs[j].set_ylabel(r'$-\\ln U$')\n\n axs[j].set_xlabel(r'num epochs')\n\n axs[j].set_title('CMAF model %d'%(j + 1))\n # if i == num_models-1:\n # axs[j].set_xlabel(r'\\# epochs')\n\n\n plt.legend()\n plt.tight_layout()\n plt.show()\n\n\n# then visualize all posteriors\ndef plot_posts(delfis, params, num_models=4, num_targets=4, \n Fisher=None, estimates=estimates, truth=None):\n fig,ax = plt.subplots(ncols=num_models, nrows=num_targets, figsize=(7,4))\n params = [r'$\\Omega_c$', r'$\\sigma_8$']\n\n if num_targets > 1:\n for i,delfi in enumerate(delfis):\n for j in range(num_models):\n cs = ChainConsumer()\n cs.add_chain(delfi['posts'][j], parameters=params, name='DELFI + IMNN') #, color=corner_colors[0])\n #cs.add_covariance(\u03b8_target, -Finv_analytic, parameters=params, name=\"Analytic Fisher\", color=corner_colors[2])\n cs.configure(linestyles=[\"-\", \"-\", \"-\"], linewidths=[1.0, 1.0, 1.0], usetex=False,\n shade=[True, True, False], shade_alpha=[0.7, 0.6, 0.], tick_font_size=8)\n cs.plotter.plot_contour(ax[i, j], r\"$\\Omega_c$\", r\"$\\sigma_8$\")\n\n ax[i, j].axvline(\u03b8_target[0], linestyle=':', linewidth=1)\n ax[i, j].axhline(\u03b8_target[1], linestyle=':', linewidth=1)\n\n ax[i,j].set_ylim([prior.low[1], prior.high[1]])\n ax[i,j].set_xlim([prior.low[0], prior.high[0]])\n\n else:\n delfi = delfis\n for j in range(num_models):\n cs = ChainConsumer()\n cs.add_chain(delfi['posts'][j], parameters=params, name='DELFI + IMNN')\n if Fisher is not None:\n cs.add_covariance(np.squeeze(estimates), np.linalg.inv(Fisher), \n parameters=params, name=\"Fisher\", color='k')\n cs.configure(linestyles=[\"-\", \"-\", \"-\"], linewidths=[1.0, 1.0, 1.0], usetex=False,\n shade=[True, False, False], shade_alpha=[0.7, 0.6, 0.], tick_font_size=8)\n cs.plotter.plot_contour(ax[j], r\"$\\Omega_c$\", r\"$\\sigma_8$\")\n\n if truth is not None:\n ax[j].axvline(truth[0], linestyle=':', linewidth=1, color='k')\n ax[j].axhline(truth[1], linestyle=':', linewidth=1, color='k')\n\n ax[j].set_ylim([prior.low[1], prior.high[1]])\n ax[j].set_xlim([prior.low[0], prior.high[0]])\n\n ax[j].set_xlabel(params[0])\n ax[j].set_ylabel(params[1])\n\n ax[j].set_title('CMAF model %d'%(j+1))\n\n plt.legend()\n plt.tight_layout()\n plt.show()\n return ax\n\n\n```\n\n### train our CMAF models !\n\n\n```python\n# train both models with the training loop\nepochs = 2000\ntraining_loop(DELFI, epochs=epochs)\n```\n\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 2000/2000 [02:43<00:00, 12.25it/s]\n\n\n\n```python\n# visualize training trajectories\nimport seaborn as sns\n%matplotlib inline\nsns.set_theme()\nplot_trajectories(DELFI, num_models=2, num_targets=1)\n```\n\n# Exercise 4: using the affine MCMC sampler\nNow that we have trained CMAF models with which to compute $p(x | \\theta)$, we now need to set up an efficient MCMC sampler to draw from the posterior, $p(x | \\theta) \\times p(\\theta)$. We can do this using the `affine_sample()` sampler, included in `pydelfi` package. This code is written in Tensorflow, adapted from the [`emcee` package](https://arxiv.org/abs/1202.3665), and can be called with only a few lines of code:\n\n # initialize walkers...\n walkers1 = tf.random.normal([n_walkers, 2], (a, b), sigma)\n walkers2 = tf.random.normal([n_walkers, 2], (a, b), sigma)\n\n # sample using affine\n chains = affine_sample(log_prob, n_params, n_walkers, n_steps, walkers1, walkers2)\n\n1. First we'll need to set up our log-probability for the posterior. Write a function `log_posterior()` that returns a probability given $x$ and a conditional $y$:\n\n \n\n\n```python\n#@title set up the affine MCMC sampler [run me]\nfrom tqdm import trange\nimport numpy as onp\n\ndef affine_sample(log_prob, n_params, n_walkers, n_steps, walkers1, walkers2):\n \n # initialize current state\n current_state1 = tf.Variable(walkers1)\n current_state2 = tf.Variable(walkers2)\n \n\n # initial target log prob for the walkers (and set any nans to -inf)...\n logp_current1 = log_prob(current_state1)\n logp_current2 = log_prob(current_state2)\n logp_current1 = tf.where(tf.math.is_nan(logp_current1), tf.ones_like(logp_current1)*tf.math.log(0.), logp_current1)\n logp_current2 = tf.where(tf.math.is_nan(logp_current2), tf.ones_like(logp_current2)*tf.math.log(0.), logp_current2)\n\n # holder for the whole chain\n chain = [tf.concat([current_state1, current_state2], axis=0)]\n \n # MCMC loop\n with trange(1, n_steps) as t:\n for epoch in t:\n\n # first set of walkers:\n\n # proposals\n partners1 = tf.gather(current_state2, onp.random.randint(0, n_walkers, n_walkers))\n z1 = 0.5*(tf.random.uniform([n_walkers], minval=0, maxval=1)+1)**2\n proposed_state1 = partners1 + tf.transpose(z1*tf.transpose(current_state1 - partners1))\n \n\n # target log prob at proposed points\n logp_proposed1 = log_prob(proposed_state1)\n logp_proposed1 = tf.where(tf.math.is_nan(logp_proposed1), tf.ones_like(logp_proposed1)*tf.math.log(0.), logp_proposed1)\n\n # acceptance probability\n p_accept1 = tf.math.minimum(tf.ones(n_walkers), z1**(n_params-1)*tf.exp(logp_proposed1 - logp_current1) )\n\n # accept or not\n accept1_ = (tf.random.uniform([n_walkers], minval=0, maxval=1) <= p_accept1)\n accept1 = tf.cast(accept1_, tf.float32)\n\n # update the state\n current_state1 = tf.transpose( tf.transpose(current_state1)*(1-accept1) + tf.transpose(proposed_state1)*accept1)\n logp_current1 = tf.where(accept1_, logp_proposed1, logp_current1)\n\n # second set of walkers:\n\n # proposals\n partners2 = tf.gather(current_state1, onp.random.randint(0, n_walkers, n_walkers))\n z2 = 0.5*(tf.random.uniform([n_walkers], minval=0, maxval=1)+1)**2\n proposed_state2 = partners2 + tf.transpose(z2*tf.transpose(current_state2 - partners2))\n\n # target log prob at proposed points\n logp_proposed2 = log_prob(proposed_state2)\n logp_proposed2 = tf.where(tf.math.is_nan(logp_proposed2), tf.ones_like(logp_proposed2)*tf.math.log(0.), logp_proposed2)\n\n # acceptance probability\n p_accept2 = tf.math.minimum(tf.ones(n_walkers), z2**(n_params-1)*tf.exp(logp_proposed2 - logp_current2) )\n\n # accept or not\n accept2_ = (tf.random.uniform([n_walkers], minval=0, maxval=1) <= p_accept2)\n accept2 = tf.cast(accept2_, tf.float32)\n\n # update the state\n current_state2 = tf.transpose( tf.transpose(current_state2)*(1-accept2) + tf.transpose(proposed_state2)*accept2)\n logp_current2 = tf.where(accept2_, logp_proposed2, logp_current2)\n\n # append to chain\n chain.append(tf.concat([current_state1, current_state2], axis=0))\n\n # stack up the chain\n chain = tf.stack(chain, axis=0)\n \n return chain\n\n```\n\n\n```python\n@tf.function\ndef log_posterior(x, y, cmaf):\n\n # define likelihood p(x|y) with CMAF\n\n # compute prior probability p(y)\n\n # return the log-posterior\n pass\n```\n\n\n```python\n#@title Ex. 4.1 solution [run me to proceed]\n@tf.function\ndef log_posterior(x, y, cmaf):\n # define likelihood p(x|y) with CMAF\n like = cmaf.log_prob(x,y)\n # compute prior probability p(y)\n _prior = samp_prior.log_prob(y)\n\n return like + _prior # the log-posterior\n```\n\n2. Now we're going to use the sampler and write a function to obtain our posteriors. To call the sampler, we need to call our log-posterior function, as well as specify the number of walkers in parameter space:\n\n\n\n\n```python\n# define function for getting posteriors\ndef get_posteriors(delfi, n_params, n_steps=2000, n_walkers=500, burnin_steps=1800, skip=4):\n delfi['posts'] = [] # reset posteriors (can save if you want to keep a record)\n\n # center affine sampler walkers on the IMNN estimates\n a,b = np.squeeze(delfi['estimates'])\n\n # choose width of proposal distribution\n # sigma = \n\n # loop over models in the ensemble\n for m,cmaf in enumerate(delfi['MAFs']):\n print('getting posterior for target data with model %d'%(m+1))\n\n \n # wrapper for log_posterior function: freeze at target summary slice, x_target\n @tf.function\n def my_log_prob(y, x=delfi['estimates']):\n return log_posterior(x, y, cmaf)\n\n\n # initialize walkers...\n # walkers1 = \n # walkers2 = \n\n # sample using affine. note that this returns a tensorflow tensor\n # chain = affine_sample()\n\n # convert chain to numpy and append to dictionary\n delfi['posts'].append(np.stack([chain.numpy()[burnin_steps::skip,:,0].flatten(), \n chain.numpy()[burnin_steps::skip,:,1].flatten()], axis=-1))\n \n pass\n```\n\n\n```python\n#@title Ex. 4.2 solution [run me to proceed]\n# define function for getting posteriors\ndef get_posteriors(delfi, n_params, n_steps=2000, n_walkers=500, burnin_steps=1800, skip=4):\n delfi['posts'] = [] # reset posteriors (can save if you want to keep a record)\n\n # center affine sampler walkers on the IMNN estimates\n a,b = np.squeeze(delfi['estimates'])\n\n # choose width of proposal distribution\n sigma = 0.5 \n\n # loop over models in the ensemble\n for m,cmaf in enumerate(delfi['MAFs']):\n print('getting posterior for target data with model %d'%(m+1))\n\n \n # wrapper for log_posterior function: freeze at target summary slice\n @tf.function\n def my_log_prob(y, x=delfi['estimates']):\n return log_posterior(x, y, cmaf)\n\n\n # initialize walkers...\n walkers1 = tf.random.normal([n_walkers, 2], (a, b), sigma)\n walkers2 = tf.random.normal([n_walkers, 2], (a, b), sigma)\n\n # sample using affine\n chain = affine_sample(my_log_prob, n_params, n_walkers, n_steps, walkers1, walkers2)\n\n delfi['posts'].append(np.stack([chain.numpy()[burnin_steps::skip,:,0].flatten(), \n chain.numpy()[burnin_steps::skip,:,1].flatten()], axis=-1))\n```\n\n\n```python\n# get all intermediate posteriors --> this should be really fast !\nget_posteriors(DELFI, n_params)\n```\n\n getting posterior for target data with model 1\n\n\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 1999/1999 [00:16<00:00, 123.15it/s]\n\n\n getting posterior for target data with model 2\n\n\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 1999/1999 [00:16<00:00, 123.63it/s]\n\n\nWe're going to use our plotting client to visualize our posteriors for each model. We'll also plot the IMNN's Fisher Gaussian Approximation in black, centered on our estimates. Finally, we'll display the true Planck parameters using crosshairs:\n\n\n```python\nparams = [r'$\\Omega_c$', r'$\\sigma_8$']\nplot_posts(DELFI, params, num_models=num_models, num_targets=1, \n Fisher=IMNN.F, estimates=np.squeeze(estimates), truth=\u03b8_target)\n```\n\n___\n# Exercise 5: append new posterior training data to hone in on the truth (repeat several times)\n\nFinally, we're going to draw parameters from the posterior, re-simulate cosmological fields, compress, append the new ($x$, $\\theta$) pairs to the dataset, and keep training our DELFI ensemble. Within a few iterations, this should shrink our posteriors considerably.\n\nSince we've coded all of our training functions modularly, we can just run them in a loop (once we've drawn and simulated from the prior). First we'll give you a piece of code to draw from the posterior chains:\n\n concat_data(DELFI, key, n_samples=500)\n\nHere, remember to re-set your random key for new samples !\n\nNext, write a loop that:\n1. draws `n_samples` summary-parameter pairs from *each* existing CMAF model's posteriors\n2. continues training the DELFI ensemble members\n3. re-samples the posterior\n\n**bonus**: Can you develop a scheme that requires fewer `n_samples` draws each iteration ? What about optimizer stability ? (hint: try a decaying learning rate)\n___\n\n\n```python\n#@title `concat_data` function to draw from each posterior and concatenate dataset [run me to proceed]\nimport pandas as pd\n\n\ndef drop_samples(samples, prior=prior):\n \"\"\"\n helper function for dropping posterior draws outside\n the specified prior range\n \"\"\"\n mydf = pd.DataFrame(samples)\n\n mydf = mydf.drop(mydf[mydf[0] < prior.low[0]].index)\n mydf = mydf.drop(mydf[mydf[1] < prior.low[1]].index)\n mydf = mydf.drop(mydf[mydf[0] > prior.high[0]].index)\n mydf = mydf.drop(mydf[mydf[1] > prior.high[1]].index)\n\n return np.array(mydf.values, dtype='float32')\n\n\ndef concat_data(delfi, key, n_samples=500, prior=prior):\n \"\"\"\n helper code for concatenating data for each DELFI CMAF model. \n delfi: DELFI dictionary object with 'train_dataset' \n and 'val_dataset' attributes\n key: jax.PRNGkey\n n_samples: number of samples to draw from EACH DELFI ensemble model\n \"\"\"\n\n # take 500 samples from each posterior for each training data\n key,rng = jax.random.split(key)\n idx = np.arange(len(delfi['posts'][0]))\n\n \u03d1_samp = []\n\n for m,_post in enumerate(delfi['posts']):\n \u03d1_samp.append(_post[45000:][onp.random.choice(idx, size=n_samples)])\n\n \u03d1_samp = np.concatenate(\u03d1_samp, axis=0)\n\n print(\u03d1_samp.shape)\n\n \u03d1_samp = drop_samples(\u03d1_samp, prior=prior)\n dropped = n_samples*len(delfi['posts']) - \u03d1_samp.shape[0]\n\n print('I dropped {} parameter pairs that were outside the prior'.format(dropped))\n\n _n_samples = len(\u03d1_samp)\n\n ts, \u03d1_samp = get_params_summaries(key2, _n_samples, \u03d1_samp)\n\n new_data = (ts, \u03d1_samp)\n\n print(\"I've drawn %d new summary-parameter pairs\"%(ts.shape[0]))\n\n # this should shuffle the dataset \n new_train_dataset, new_val_dataset = get_dataset(new_data, batch_size=batch_size, buffer_size=len(new_data[0]))\n\n # concatenate datasets\n delfi['train_dataset'] = delfi['train_dataset'].concatenate(new_train_dataset)\n delfi['val_dataset'] = delfi['val_dataset'].concatenate(new_val_dataset)\n```\n\n\n```python\n#@title Ex. 5 solution [run me to proceed]\nfor repeat in range(1):\n\n key,rng = jax.random.split(rng)\n print('doing retraining iteration %d'%(repeat))\n\n concat_data(DELFI, key, n_samples=500)\n \n print('retraining on augmented dataset')\n epochs = 500\n training_loop(DELFI, epochs=epochs)\n\n plot_trajectories(DELFI, num_models=2, num_targets=1)\n\n get_posteriors(DELFI, n_params)\n\n plot_posts(DELFI, params, num_models=num_models, num_targets=1, \n Fisher=IMNN.F, estimates=np.squeeze(estimates), truth=\u03b8_target)\n \n\n```\n\n___\n# Exercise 6: create ensemble posterior\nOnce we're happy with the DELFI training, we can proceed to reporting our ensemble's combined posterior. Using the [`ChainConsumer` API](https://samreay.github.io/ChainConsumer/index.html), concatenate the posterior chains and report a nice corner plot:\n\n\n```python\n#@title Exercise 6 solution [run me to proceed]\n\ndef drop_samples(samples, prior=prior):\n \"\"\"\n helper function for dropping posterior draws outside\n the specified prior range\n \"\"\"\n mydf = pd.DataFrame(samples)\n\n mydf = mydf.drop(mydf[mydf[0] < prior.low[0]].index)\n mydf = mydf.drop(mydf[mydf[1] < prior.low[1]].index)\n mydf = mydf.drop(mydf[mydf[0] > prior.high[0]].index)\n mydf = mydf.drop(mydf[mydf[1] > prior.high[1]].index)\n\n return np.array(mydf.values, dtype='float32')\n\nsuper_post = np.concatenate(DELFI['posts'], axis=0) \n# assign new dict entry after dropping samples outside the prior\nDELFI['super_post'] = drop_samples(super_post)\n\nparams = [r\"$\\Omega_c$\", r\"$\\sigma_8$\"]\ncorner_colors = [None, None, 'k']\n\n\n\nc = ChainConsumer()\nc.add_chain(DELFI['super_post'][::10], parameters=params, name='DELFI + IMNN', color=corner_colors[0])\nc.add_covariance(np.squeeze(estimates), IMNN.invF, parameters=params, name=\"IMNN F @estimates\", color=corner_colors[2])\n\nc.configure(linestyles=[\"-\", \"-\", \"--\"], linewidths=[1.0, 1.0, 1.0,],\n shade=[True, False, False], shade_alpha=[0.7, 0.6, 0.],\n tick_font_size=8, usetex=False,\n legend_kwargs={\"loc\": \"upper left\", \"fontsize\": 8},\n legend_color_text=False, legend_location=(0, 0))\n\nfig = c.plotter.plot(figsize=\"column\", truth=list(\u03b8_target), filename=None)\n```\n\n___\n# Congrats !\nYou've made it through the core of the tutorial and trained a DELFI ensemble on IMNN-compressed summaries of mock dark matter fields and obtained cosmological parameter posteriors !\n\n### Now what ? \nThere are lots of things you can do if you have the time -- for one, you could check out the bonus problems below \n\n___\n# BONUS: Compare IMNN Compressors\n\nFor this whole tutorial we've been using an IMNN ***trained deliberately far*** from our Planck parameters, meaning our compression isn't guaranteed to be optimal. In our accompanying paper (to be released on arXiv on July 16, 2021) we re-trained an IMNN on the mean of the score estimates of a set of four cosmological fields. Since this estimate is closer to the true target parameters, our IMNN compression is guaranteed to improve our inference on the target data.\n\n\n\n\nWe've included this newly-trained IMNN in the GitHub repository that you've already cloned into this notebook -- as a bonus, repeat the DELFI posterior estimation using the new (more optimal) compressor and see how your inference shapes up ! You *should* see tighter Gaussian Approximations *and* DELFI contours:\n\n\n```python\n# load IMNN state\nimport cloudpickle as pickle\nimport os\n\ndef unpickle_me(path):\n file = open(path, 'rb')\n return pickle.load(file)\n\n\nfolder_name = './FieldIMNNs/tutorial/IMNN2-aspects/'\nloadstate = unpickle_me(os.path.join(folder_name, 'IMNN_state'))\nstate2 = jax.experimental.optimizers.pack_optimizer_state(loadstate)\n\n# startup key to get the right state of the weights\nstartup_key2 = np.load(os.path.join(folder_name, 'IMNN_startup_key.npy'), allow_pickle=True)\n\n# load weights\nbest_weights2 = np.load(os.path.join(folder_name, 'best_w.npy'), allow_pickle=True)\n\n# load fiducial model that we trained the model at (estimates derived from initial IMNN)\n\u03b8_fid_new = np.load(os.path.join(folder_name, 'new_fid_params.npy'), allow_pickle=True)\n```\n\n\n```python\n# initialize IMNN with pre-trained state\nIMNN2 = imnn.IMNN(\n n_s=n_s,\n n_d=n_d,\n n_params=n_params,\n n_summaries=n_summaries,\n input_shape=(1,) + shape + (1,),\n \u03b8_fid=\u03b8_fid_new,\n model=model,\n optimiser=optimiser,\n key_or_state=state2, # <---- initialize with state\n simulator=lambda rng, \u03b8: simulator(\n rng, \u03b8, simulator_args={\n **simulator_args, \n **{\"squeeze\": False}}))\n\n# now set weights using the best training weights and startup key (this can take a moment)\nIMNN2.set_F_statistics(w=best_weights, key=startup_key2)\n```\n\n `simulator` provided, using SimulatorIMNN\n\n\n\n```python\nprint(np.linalg.det(IMNN2.F))\n```\n\n 488718.5407332728\n\n\n---\n# BONUS 2:\n\nAlternatively, train a new IMNN from scratch at the target data `estimates` (try with fewer filters on the free version of Colab). You could also try playing with other `stax` layers like `stax.Dense(num_neurons)`. Feel free to also switch up the simulation parameters -- choosing $N=32$ for instance will dramatically increase training speed for testing, etc.\n\n\n```python\nfs = 16\nnew_layers = [\n InceptBlock((fs, fs, fs), strides=(4, 4)),\n InceptBlock((fs, fs, fs), strides=(4, 4)),\n InceptBlock((fs, fs, fs), strides=(4, 4)),\n InceptBlock((fs, fs, fs), strides=(2, 2), do_5x5=False, do_3x3=False),\n stax.Conv(n_summaries, (1, 1), strides=(1, 1), padding=\"SAME\"),\n stax.Flatten,\n Reshape((n_summaries,)) \n]\nnew_model = stax.serial(*new_layers)\n```\n\n\n```python\nprint_model(layers, input_shape, rng)\n```\n\n\n```python\nrng, key = jax.random.split(rng)\nIMNN2 = imnn.IMNN(\n n_s=n_s,\n n_d=n_d,\n n_params=n_params,\n n_summaries=n_summaries,\n input_shape=(1,) + shape + (1,),\n \u03b8_fid=np.squeeze(estimates),\n model=new_model,\n optimiser=optimiser,\n key_or_state=key, # <---- initialize with key\n simulator=lambda rng, \u03b8: simulator(\n rng, \u03b8, simulator_args={\n **simulator_args, \n **{\"squeeze\": False}}))\n```\n\n\n```python\nprint(\"now I'm training the IMNN\")\nrng, key = jax.random.split(rng)\nIMNN2.fit(\u03bb=10., \u03f5=0.1, rng=key, print_rate=None, \n min_iterations=500, patience=100, best=True)\n\n# visualize training trajectory\nIMNN2.plot(expected_detF=None);\n```\n", "meta": {"hexsha": "c51c32b54202ed920116f53d12f92439c0ac7282", "size": 703884, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/data_challenge_IMNN_x_DELFI_cosmo_demo.ipynb", "max_stars_repo_name": "tlmakinen/kosmo-kompress", "max_stars_repo_head_hexsha": "2282e20d26dfc523e5705c0e02e0524b639df8a4", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2021-07-16T05:22:15.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-04T12:41:06.000Z", "max_issues_repo_path": "notebooks/data_challenge_IMNN_x_DELFI_cosmo_demo.ipynb", "max_issues_repo_name": "tlmakinen/FieldIMNNs", "max_issues_repo_head_hexsha": "1b363559116136d276b7c085b56a80520a7b0830", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/data_challenge_IMNN_x_DELFI_cosmo_demo.ipynb", "max_forks_repo_name": "tlmakinen/FieldIMNNs", "max_forks_repo_head_hexsha": "1b363559116136d276b7c085b56a80520a7b0830", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 226.6937198068, "max_line_length": 212982, "alphanum_fraction": 0.8849270618, "converted": true, "num_tokens": 20292, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819591324416, "lm_q2_score": 0.6791786861878392, "lm_q1q2_score": 0.4152375957625189}} {"text": "```python\n%%html\n\n\n```\n\n\n\n\n\n\n\n##### Notes\n\nLeave script block above in place to left justify the table.\nThis problem can also be used as laboratory exercise in `matplotlib` lesson. \nDependencies: `matplotlib` and `math`; could also be solved using `numpy` and/or `pandas`\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n# Problem XX\n\nGraphing Functions Special Functions \n\nConsider the two functions listed below:\n\n\\begin{equation}\nf(x) = e^{-\\alpha x}\n\\label{eqn:fofx}\n\\end{equation}\n\n\\begin{equation}\ng(x) = \\gamma sin(\\beta x)\n\\label{eqn:gofx}\n\\end{equation}\n\nPrepare a plot of the two functions on the same graph. \n\nUse the values in Table below for $\\alpha$, $\\beta$, and $\\gamma$.\n\n|Parameter|Value|\n|:---|---:|\n|$\\alpha$|0.50|\n|$\\beta$|3.00|\n|$\\gamma$|$\\frac{\\pi}{2}$|\n\n\nThe plot should have $x$ values ranging from $0$ to $10$ (inclusive) in sufficiently small increments to see curvature in the two functions as well as to identify the number and approximate locations of intersections. In this problem, intersections are locations in the $x-y$ plane where the two \"curves\" cross one another of the two plots.\n\n\n\n```python\n# By-hand evaluate f(x) for x=1, alpha = 1/2\n```\n\n\n```python\n# By-hand evaluate g(x) for x=3.14, beta = 1/2, gamma = 2\n```\n\n\n```python\n# Define the first function f(x,alpha), test the function using your by hand answer\ndef f(x,alpha):\n import math\n f = math.exp(-1.0*alpha*x)\n return f\n\nf(1,0.5)\n```\n\n\n\n\n 0.6065306597126334\n\n\n\n\n```python\n# Define the second function g(x,beta,gamma), test the function using your by hand answer\ndef g(x,beta,gamma):\n import math\n f = gamma*math.sin(beta*x)\n return f\n\ng(3.14,0.5,2.0)\n```\n\n\n\n\n 1.9999993658636692\n\n\n# Built a list for x that ranges from 0 to 10, inclusive, with adjustable step sizes for plotting later on\nhowMany = 10\nscale = 10.0/howMany\nxvector = []\nfor i in range(0,howMany+1):\n xvector.append(scale*i)\n#xvector # activate to display\n\n```python\n# Build a plotting function that plots both functions on the same chart\nalpha = 0.5\nbeta = 0.5\ngamma = 2.0\nyf = []\nyg = []\nfor i in range(0,howMany+1):\n yf.append(f(xvector[i],alpha))\n yg.append(g(xvector[i],beta,gamma))\n\ndef plot2lines(list11,list21,list12,list22,strx,stry,strtitle): # plot list1 on x, list2 on y, xlabel, ylabel, title\n from matplotlib import pyplot as plt # import the plotting library from matplotlibplt.show()\n plt.plot( list11, list21, color ='green', marker ='o', linestyle ='none' , label = \"Observed\" ) # create a line chart, years on x-axis, gdp on y-axis\n plt.plot( list12, list22, color ='red', marker ='o', linestyle ='solid' , label = \"Model\") # create a line chart, years on x-axis, gdp on y-axis\n plt.legend()\n plt.title(strtitle)# add a title\n plt.ylabel(stry)# add a label to the x and y-axes\n plt.xlabel(strx)\n plt.show() # display the plot\n return #null return\n\nplot2lines(xvector,yf,xvector,yg,'x-value','y-value','plot of f and g')\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "ee46a957ba0c5cf58e7e686406374936d22582dd", "size": 6028, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "5-ExamProblems/.src/ProblemXX/.ipynb_checkpoints/ProblemXX-Dev-checkpoint.ipynb", "max_stars_repo_name": "dustykat/engr-1330-psuedo-course", "max_stars_repo_head_hexsha": "3e7e31a32a1896fcb1fd82b573daa5248e465a36", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "5-ExamProblems/.src/ProblemXX/.ipynb_checkpoints/ProblemXX-Dev-checkpoint.ipynb", "max_issues_repo_name": "dustykat/engr-1330-psuedo-course", "max_issues_repo_head_hexsha": "3e7e31a32a1896fcb1fd82b573daa5248e465a36", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "5-ExamProblems/.src/ProblemXX/.ipynb_checkpoints/ProblemXX-Dev-checkpoint.ipynb", "max_forks_repo_name": "dustykat/engr-1330-psuedo-course", "max_forks_repo_head_hexsha": "3e7e31a32a1896fcb1fd82b573daa5248e465a36", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 25.7606837607, "max_line_length": 350, "alphanum_fraction": 0.5265428003, "converted": true, "num_tokens": 932, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5078118642792044, "lm_q2_score": 0.8175744828610095, "lm_q1q2_score": 0.4151740223287557}} {"text": "

                                        Table of Contents

                                        \n
                                        \n\n\n```python\n# code for loading the format for the notebook\nimport os\n\n# path : store the current path to convert back to it later\npath = os.getcwd()\nos.chdir(os.path.join('..', '..', 'notebook_format'))\n\nfrom formats import load_style\nload_style(css_style='custom2.css', plot_style=False)\n```\n\n\n\n\n\n\n\n\n\n\n\n\n```python\nos.chdir(path)\n\n# 1. magic for inline plot\n# 2. magic to print version\n# 3. magic so that the notebook will reload external python modules\n# 4. magic to enable retina (high resolution) plots\n# https://gist.github.com/minrk/3301035\n%matplotlib inline\n%load_ext watermark\n%load_ext autoreload\n%autoreload 2\n%config InlineBackend.figure_format='retina'\n\nimport os\nimport math\nimport time\nimport torch\nimport random\nimport numpy as np\nimport torch.nn as nn\nimport torch.optim as optim\nfrom datasets import load_dataset\nfrom torch.utils.data import DataLoader\nfrom tokenizers import ByteLevelBPETokenizer\n\n%watermark -a 'Ethen' -d -t -v -p datasets,numpy,torch,tokenizers\n```\n\n Ethen 2021-05-17 03:42:33 \n \n CPython 3.6.5\n IPython 7.16.1\n \n datasets 1.1.3\n numpy 1.18.1\n torch 1.7.0+cu101\n tokenizers 0.10.1\n\n\n# Transformer\n\nSeq2Seq based machine translation system usually comprises of two main components, an encoder that encodes in source sentence into context vectors and a decoder that decodes the context vectors into target sentence, transformer model is no different in this regards. Reasons to their growing popularity at the time of writing this document are primarily due to **self attention layers** and **parallel computation**.\n\nPrevious RNN based encoder and decoder has a constraint of sequential computation. A hidden state at time $t$ in a recurrent layer, has only seen tokens $x_t$ and all the tokens before it, even though this gives us the benefit of modeling long dependencies, it hinders training speed as we can't process the next time step until we finish processing the current one. Transformer model aims to mitigate this issue by solely relying on attention mechanism, where each context vector produced by a transformer model has seen all tokens at all positions within the input sequence. In other words, instead of compressing the entire source sentence, $X = (x_1, ... , x_n)$ into a single context vector, $z$, it produces a sequence of context vectors, $Z = (z_1, ... , z_n)$ in one parallel computation. We'll get to the details of attention mechanism, self attention, that's used throughout the Transformer model in later sections. One important thing to note here is that breakthrough of this model is not due to invention of the attention mechansim, as this concept existed well before. The highlight here is we can build a highly performant model with attention mechanism in isolation, i.e. without the use of recurrent (RNN) or convolutional (CNN) neural networks in the mix.\n\nIn this article, we implemented the Transformer module from scratch as well as leveraged PyTorch built in Transformer Encoder and Decoder to construct the Transformer model.\n\n## Data Preprocessing\n\nWe'll be using the [Multi30k dataset](http://www.statmt.org/wmt16/multimodal-task.html) to demonstrate using the transfomer model in a machine translation task. This German to English training dataset's size is around 29K. We'll start off by downloading the raw dataset and extracting them. Feel free to swap this step with any other machine translation dataset.\n\n\n```python\nimport tarfile\nimport zipfile\nimport requests\nimport subprocess\nfrom tqdm import tqdm\nfrom urllib.parse import urlparse\n\n\ndef download_file(url: str, directory: str):\n \"\"\"\n Download the file at ``url`` to ``directory``.\n Extract to the file content ``directory`` if the original file\n is a tar, tar.gz or zip file.\n\n Parameters\n ----------\n url : str\n url of the file.\n\n directory : str\n Directory to download the file.\n \"\"\"\n response = requests.get(url, stream=True)\n response.raise_for_status()\n\n content_len = response.headers.get('Content-Length')\n total = int(content_len) if content_len is not None else 0\n\n os.makedirs(directory, exist_ok=True)\n file_name = get_file_name_from_url(url)\n file_path = os.path.join(directory, file_name)\n\n with tqdm(unit='B', total=total) as pbar, open(file_path, 'wb') as f:\n for chunk in response.iter_content(chunk_size=1024):\n if chunk:\n pbar.update(len(chunk))\n f.write(chunk)\n\n extract_compressed_file(file_path, directory)\n\n\ndef extract_compressed_file(compressed_file_path: str, directory: str):\n \"\"\"\n Extract a compressed file to ``directory``. Supports zip, tar.gz, tgz,\n tar extensions.\n\n Parameters\n ----------\n compressed_file_path : str\n\n directory : str\n File will to extracted to this directory.\n \"\"\"\n basename = os.path.basename(compressed_file_path)\n if 'zip' in basename:\n with zipfile.ZipFile(compressed_file_path, \"r\") as zip_f:\n zip_f.extractall(directory)\n elif 'tar.gz' in basename or 'tgz' in basename:\n with tarfile.open(compressed_file_path) as f:\n f.extractall(directory)\n\n\ndef get_file_name_from_url(url: str) -> str:\n \"\"\"\n Return the file_name from a URL\n\n Parameters\n ----------\n url : str\n URL to extract file_name from\n\n Returns\n -------\n file_name : str\n \"\"\"\n parse = urlparse(url)\n return os.path.basename(parse.path)\n```\n\n\n```python\nurls = [\n 'http://www.quest.dcs.shef.ac.uk/wmt16_files_mmt/training.tar.gz',\n 'http://www.quest.dcs.shef.ac.uk/wmt16_files_mmt/validation.tar.gz',\n 'http://www.quest.dcs.shef.ac.uk/wmt16_files_mmt/mmt16_task1_test.tar.gz'\n]\ndirectory = 'multi30k'\nfor url in urls:\n download_file(url, directory)\n```\n\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 1207136/1207136 [00:01<00:00, 697905.06B/s]\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 46329/46329 [00:00<00:00, 162345.03B/s]\n 100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 43905/43905 [00:00<00:00, 152818.28B/s]\n\n\nWe print out the content in the data directory and some sample data.\n\n\n```python\n!ls multi30k\n```\n\n mmt16_task1_test.tar.gz test.en train.en\t val.de validation.tar.gz\r\n test.de\t\t\t train.de training.tar.gz val.en\r\n\n\n\n```python\n!head multi30k/train.de\n```\n\n Zwei junge wei\u00dfe M\u00e4nner sind im Freien in der N\u00e4he vieler B\u00fcsche.\r\n Mehrere M\u00e4nner mit Schutzhelmen bedienen ein Antriebsradsystem.\r\n Ein kleines M\u00e4dchen klettert in ein Spielhaus aus Holz.\r\n Ein Mann in einem blauen Hemd steht auf einer Leiter und putzt ein Fenster.\r\n Zwei M\u00e4nner stehen am Herd und bereiten Essen zu.\r\n Ein Mann in gr\u00fcn h\u00e4lt eine Gitarre, w\u00e4hrend der andere Mann sein Hemd ansieht.\r\n Ein Mann l\u00e4chelt einen ausgestopften L\u00f6wen an.\r\n Ein schickes M\u00e4dchen spricht mit dem Handy w\u00e4hrend sie langsam die Stra\u00dfe entlangschwebt.\r\n Eine Frau mit einer gro\u00dfen Geldb\u00f6rse geht an einem Tor vorbei.\r\n Jungen tanzen mitten in der Nacht auf Pfosten.\r\n\n\n\n```python\n!head multi30k/train.en\n```\n\n Two young, White males are outside near many bushes.\r\n Several men in hard hats are operating a giant pulley system.\r\n A little girl climbing into a wooden playhouse.\r\n A man in a blue shirt is standing on a ladder cleaning a window.\r\n Two men are at the stove preparing food.\r\n A man in green holds a guitar while the other man observes his shirt.\r\n A man is smiling at a stuffed lion\r\n A trendy girl talking on her cellphone while gliding slowly down the street.\r\n A woman with a large purse is walking by a gate.\r\n Boys dancing on poles in the middle of the night.\r\n\n\nThe original dataset is splits the source and the target language into two separate files (e.g. train.de, train.en are the training dataset for German and English). This type of format is useful when we wish to train a tokenizer on top of the source or target language as we'll soon see.\n\nOn the other hand, having the source and target pair together in one single file makes it easier to load them in batches for training or evaluating our machine translation model. We'll create the paired dataset, and [load the dataset](https://huggingface.co/docs/datasets/loading_datasets.html#csv-files). For loading the dataset, it will be helpful to have some basic understanding of Huggingface's [dataset](https://huggingface.co/docs/datasets/quicktour.html).\n\n\n```python\ndef create_translation_data(\n source_input_path: str,\n target_input_path: str,\n output_path: str,\n delimiter: str = '\\t',\n encoding: str = 'utf-8'\n):\n \"\"\"\n Creates the paired source and target dataset from the separated ones.\n e.g. creates `train.tsv` from `train.de` and `train.en`\n \"\"\"\n with open(source_input_path, encoding=encoding) as f_source_in, \\\n open(target_input_path, encoding=encoding) as f_target_in, \\\n open(output_path, 'w', encoding=encoding) as f_out:\n\n for source_raw in f_source_in:\n source_raw = source_raw.strip()\n target_raw = f_target_in.readline().strip()\n if source_raw and target_raw:\n output_line = source_raw + delimiter + target_raw + '\\n'\n f_out.write(output_line)\n```\n\n\n```python\nsource_lang = 'de'\ntarget_lang = 'en'\n\ndata_files = {}\nfor split in ['train', 'val', 'test']:\n source_input_path = os.path.join(directory, f'{split}.{source_lang}')\n target_input_path = os.path.join(directory, f'{split}.{target_lang}')\n output_path = f'{split}.tsv'\n create_translation_data(source_input_path, target_input_path, output_path)\n data_files[split] = [output_path]\n\ndata_files\n```\n\n\n\n\n {'train': ['train.tsv'], 'val': ['val.tsv'], 'test': ['test.tsv']}\n\n\n\n\n```python\ndataset_dict = load_dataset(\n 'csv',\n delimiter='\\t',\n column_names=[source_lang, target_lang],\n data_files=data_files\n)\ndataset_dict\n```\n\n Using custom data configuration default\n\n\n Downloading and preparing dataset csv/default-e201b31b229fe923 (download: Unknown size, generated: Unknown size, post-processed: Unknown size, total: Unknown size) to /home/mingyuliu/.cache/huggingface/datasets/csv/default-e201b31b229fe923/0.0.0/2960f95a26e85d40ca41a230ac88787f715ee3003edaacb8b1f0891e9f04dda2...\n\n\n\n HBox(children=(FloatProgress(value=1.0, bar_style='info', max=1.0), HTML(value='')))\n\n\n \r\n\n\n HBox(children=(FloatProgress(value=1.0, bar_style='info', max=1.0), HTML(value='')))\n\n\n \r\n\n\n HBox(children=(FloatProgress(value=1.0, bar_style='info', max=1.0), HTML(value='')))\n\n\n Dataset csv downloaded and prepared to /home/mingyuliu/.cache/huggingface/datasets/csv/default-e201b31b229fe923/0.0.0/2960f95a26e85d40ca41a230ac88787f715ee3003edaacb8b1f0891e9f04dda2. Subsequent calls will reuse this data.\n\n\n\n\n\n DatasetDict({\n train: Dataset({\n features: ['de', 'en'],\n num_rows: 29000\n })\n val: Dataset({\n features: ['de', 'en'],\n num_rows: 1014\n })\n test: Dataset({\n features: ['de', 'en'],\n num_rows: 1000\n })\n })\n\n\n\nWe can access each split, and record/pair with the following syntax.\n\n\n```python\ndataset_dict['train'][0]\n```\n\n\n\n\n {'de': 'Zwei junge wei\u00dfe M\u00e4nner sind im Freien in der N\u00e4he vieler B\u00fcsche.',\n 'en': 'Two young, White males are outside near many bushes.'}\n\n\n\nFrom our raw pair, we need to use or train a tokenizer to convert them into numerical indices. Here we'll be training our tokenizer from scratch using Huggingface's [tokenizer](https://github.com/huggingface/tokenizers). Feel free to swap this step out with other tokenization procedures, what's important is to leave rooms for special tokens such as the init token that represents the start of a sentence, the end of sentence token that represents the end of a sentence, and padding token that pads sentence batches into equivalent length.\n\n\n```python\n# use only the training set to train our tokenizer\nsplit = 'train'\nsource_input_path = os.path.join(directory, f'{split}.{source_lang}')\ntarget_input_path = os.path.join(directory, f'{split}.{target_lang}')\nprint(source_input_path, target_input_path)\n```\n\n multi30k/train.de multi30k/train.en\n\n\n\n```python\ninit_token = ''\neos_token = ''\npad_token = ''\n\ntokenizer_params = {\n 'min_frequency': 2,\n 'vocab_size': 5000,\n 'show_progress': False,\n 'special_tokens': [init_token, eos_token, pad_token]\n}\n\nstart_time = time.time()\nsource_tokenizer = ByteLevelBPETokenizer(lowercase=True)\nsource_tokenizer.train(source_input_path, **tokenizer_params)\n\ntarget_tokenizer = ByteLevelBPETokenizer(lowercase=True)\ntarget_tokenizer.train(target_input_path, **tokenizer_params)\nend_time = time.time()\n\nprint('elapsed: ', end_time - start_time)\nprint('source vocab size: ', source_tokenizer.get_vocab_size())\nprint('target vocab size: ', target_tokenizer.get_vocab_size())\n```\n\n elapsed: 10.487832069396973\n source vocab size: 5000\n target vocab size: 5000\n\n\n\n```python\nsource_eos_idx = source_tokenizer.token_to_id(eos_token)\ntarget_eos_idx = target_tokenizer.token_to_id(eos_token)\n\nsource_init_idx = source_tokenizer.token_to_id(init_token)\ntarget_init_idx = target_tokenizer.token_to_id(init_token)\n```\n\nWe'll perform this tokenization step for all our dataset up front, so we can do as little preprocessing as possible while feeding our dataset to model. Note that we do not perform the padding step at this stage.\n\n\n```python\ndef encode(example):\n \"\"\"\n Encode the raw text into numerical token ids. Creating two new fields\n ``source_ids`` and ``target_ids``.\n Also append the init token and prepend eos token to the sentence.\n \"\"\"\n source_raw = example[source_lang]\n target_raw = example[target_lang]\n source_encoded = source_tokenizer.encode(source_raw).ids\n source_encoded = [source_init_idx] + source_encoded + [source_eos_idx]\n target_encoded = target_tokenizer.encode(target_raw).ids\n target_encoded = [target_init_idx] + target_encoded + [target_eos_idx]\n example['source_ids'] = source_encoded\n example['target_ids'] = target_encoded\n return example\n\n\nstart_time = time.time()\ndataset_dict_encoded = dataset_dict.map(encode, num_proc=8)\nend_time = time.time()\nprint('elapsed: ', end_time - start_time)\n\ndataset_dict_encoded\n```\n\n \n\n\n HBox(children=(FloatProgress(value=0.0, description='#1', max=3625.0, style=ProgressStyle(description_width='i\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#3', max=3625.0, style=ProgressStyle(description_width='i\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#0', max=3625.0, style=ProgressStyle(description_width='i\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#2', max=3625.0, style=ProgressStyle(description_width='i\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#4', max=3625.0, style=ProgressStyle(description_width='i\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#5', max=3625.0, style=ProgressStyle(description_width='i\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#6', max=3625.0, style=ProgressStyle(description_width='i\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#7', max=3625.0, style=ProgressStyle(description_width='i\u2026\n\n\n \n \n \n \n \n \n \n \n \n\n\n HBox(children=(FloatProgress(value=0.0, description='#4', max=127.0, style=ProgressStyle(description_width='in\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#2', max=127.0, style=ProgressStyle(description_width='in\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#0', max=127.0, style=ProgressStyle(description_width='in\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#5', max=127.0, style=ProgressStyle(description_width='in\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#3', max=127.0, style=ProgressStyle(description_width='in\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#1', max=127.0, style=ProgressStyle(description_width='in\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#7', max=126.0, style=ProgressStyle(description_width='in\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#6', max=126.0, style=ProgressStyle(description_width='in\u2026\n\n\n \n \n \n \n \n \n \n \n \n\n\n HBox(children=(FloatProgress(value=0.0, description='#0', max=125.0, style=ProgressStyle(description_width='in\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#1', max=125.0, style=ProgressStyle(description_width='in\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#3', max=125.0, style=ProgressStyle(description_width='in\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#2', max=125.0, style=ProgressStyle(description_width='in\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#5', max=125.0, style=ProgressStyle(description_width='in\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#4', max=125.0, style=ProgressStyle(description_width='in\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#6', max=125.0, style=ProgressStyle(description_width='in\u2026\n\n\n\n HBox(children=(FloatProgress(value=0.0, description='#7', max=125.0, style=ProgressStyle(description_width='in\u2026\n\n\n \n \n \n \n \n \n \n \n elapsed: 1.697516679763794\n\n\n\n\n\n DatasetDict({\n train: Dataset({\n features: ['de', 'en', 'source_ids', 'target_ids'],\n num_rows: 29000\n })\n val: Dataset({\n features: ['de', 'en', 'source_ids', 'target_ids'],\n num_rows: 1014\n })\n test: Dataset({\n features: ['de', 'en', 'source_ids', 'target_ids'],\n num_rows: 1000\n })\n })\n\n\n\n\n```python\ndataset_train = dataset_dict_encoded['train']\ndataset_train[0]\n```\n\n\n\n\n {'de': 'Zwei junge wei\u00dfe M\u00e4nner sind im Freien in der N\u00e4he vieler B\u00fcsche.',\n 'en': 'Two young, White males are outside near many bushes.',\n 'source_ids': [0,\n 343,\n 377,\n 1190,\n 412,\n 648,\n 348,\n 659,\n 280,\n 326,\n 725,\n 1283,\n 262,\n 727,\n 706,\n 16,\n 1],\n 'target_ids': [0, 335, 372, 14, 369, 2181, 320, 493, 556, 1202, 3157, 16, 1]}\n\n\n\nThe final step for our data preprocessing step is to prepare the [DataLoader](https://pytorch.org/docs/stable/data.html#dataloader-collate-fn), which prepares batches of tokenized ids for our model. The customized collate function performs the batching as well as padding.\n\n\n```python\nclass TranslationPairCollate:\n\n def __init__(self, max_len, pad_idx, device, percentage=100):\n self.device = device\n self.max_len = max_len\n self.pad_idx = pad_idx\n self.percentage = percentage\n\n def __call__(self, batch):\n source_batch = []\n source_len = []\n target_batch = []\n target_len = []\n for example in batch:\n source = example['source_ids']\n source_len.append(len(source))\n source_batch.append(source)\n\n target = example['target_ids']\n target_len.append(len(target))\n target_batch.append(target)\n\n source_padded = self.process_encoded_text(source_batch, source_len, self.max_len, self.pad_idx)\n target_padded = self.process_encoded_text(target_batch, target_len, self.max_len, self.pad_idx)\n return source_padded, target_padded\n\n def process_encoded_text(self, sequences, sequences_len, max_len, pad_idx):\n sequences_len_percentile = int(np.percentile(sequences_len, self.percentage))\n max_len = min(sequences_len_percentile, max_len)\n padded_sequences = pad_sequences(sequences, max_len, pad_idx)\n return torch.LongTensor(padded_sequences)\n\n\ndef pad_sequences(sequences, max_len, pad_idx):\n \"\"\"\n Pad the list of sequences (numerical token ids) to the same length.\n Sequence that are shorter than the specified ``max_len`` will be appended\n with the specified ``pad_idx``. Those that are longer will be truncated.\n\n Parameters\n ----------\n sequences : list[int]\n List of numerical token ids.\n\n max_len : int\n Maximum length of all sequences.\n\n pad_idx : int\n Padding index.\n\n Returns\n -------\n padded_sequences : 1d ndarray\n \"\"\"\n num_samples = len(sequences)\n padded_sequences = np.full((num_samples, max_len), pad_idx)\n for i, sequence in enumerate(sequences):\n sequence = np.array(sequence)[:max_len]\n padded_sequences[i, :len(sequence)] = sequence\n\n return padded_sequences\n```\n\n\n```python\nmax_len = 100\nbatch_size = 128\ndevice = torch.device('cuda' if torch.cuda.is_available() else 'cpu')\n\npad_idx = source_tokenizer.token_to_id(pad_token)\ntranslation_pair_collate_fn = TranslationPairCollate(max_len, pad_idx, device)\n\ndata_loader_params = {\n 'batch_size': batch_size,\n 'collate_fn': translation_pair_collate_fn,\n 'pin_memory': True\n}\n\ndataloader_train = DataLoader(dataset_train, **data_loader_params)\n\n# we can print out 1 batch of source and target\nsource, target = next(iter(dataloader_train))\nsource, target\n```\n\n\n\n\n (tensor([[ 0, 343, 377, ..., 2, 2, 2],\n [ 0, 640, 412, ..., 2, 2, 2],\n [ 0, 261, 542, ..., 2, 2, 2],\n ...,\n [ 0, 343, 500, ..., 2, 2, 2],\n [ 0, 296, 442, ..., 2, 2, 2],\n [ 0, 296, 317, ..., 2, 2, 2]]),\n tensor([[ 0, 335, 372, ..., 2, 2, 2],\n [ 0, 808, 400, ..., 2, 2, 2],\n [ 0, 67, 504, ..., 2, 2, 2],\n ...,\n [ 0, 335, 479, ..., 2, 2, 2],\n [ 0, 67, 413, ..., 2, 2, 2],\n [ 0, 67, 325, ..., 2, 2, 2]]))\n\n\n\n\n```python\n# create the data loader for both validation and test set\ndataset_val = dataset_dict_encoded['val']\ndataloader_val = DataLoader(dataset_val, **data_loader_params)\n\ndataset_test = dataset_dict_encoded['test']\ndataloader_test = DataLoader(dataset_test, **data_loader_params)\n```\n\n## Model Architecture From Scratch\n\nHaving prepared the data, we can now start implementing Transformer model's architecture, which looks like the following:\n\n\n\nThis implementation is largely based on the wonderful [Jupyter Notebook: Attention is All You Need](https://nbviewer.jupyter.org/github/bentrevett/pytorch-seq2seq/blob/master/6%20-%20Attention%20is%20All%20You%20Need.ipynb).\n\n### Position Wise Embedding\n\nFirst, input tokens are passed through a standard embedding layer. Next, as the entire sentence is fed into the model in one go, by default it has no idea about the tokens' order within the sequence. We cope with this by using a second embedding layer, positional embedding. This is an embedding layer where our input is not the token id but the token's position within the sequence. If we configure our position embedding to have a \"vocabulary\" size of 100, this means our model can accept sentences up to 100 tokens long.\n\nThe original Transformer implementation from the Attention is All You Need paper does not learn positional embeddings. Instead it uses a fixed static positional encoding. Modern Transformer architectures, like BERT, use positional embeddings, hence, we have decided to use them in these tutorials. Check out [this](http://nlp.seas.harvard.edu/2018/04/03/attention.html#positional-encoding) section to read more about the positional encoding used in the original Transformer model.\n\nNext, the token and positional embeddings are combined together using an elementwise sum operation, giving us a single vector that contains information on both the token and its position with in the sequence. Before they are summed, the token embeddings are multiplied by a scaling factor $\\sqrt{d_{model}}$, where $d_{model}$ is the hidden dimension size, `hid_dim`. This supposedly reduces variance in the embeddings and without this scaling factor, it becomes difficult to train the model reliably. Dropout is then applied to the combined embeddings.\n\n\n```python\nclass PositionWiseEmbedding(nn.Module):\n\n def __init__(self, input_dim, hid_dim, max_len, dropout_p):\n super().__init__()\n self.input_dim = input_dim\n self.hid_dim = hid_dim\n self.max_len = max_len\n self.dropout_p = dropout_p\n\n self.tok_embedding = nn.Embedding(input_dim, hid_dim)\n self.pos_embedding = nn.Embedding(max_len, hid_dim)\n self.dropout = nn.Dropout(dropout_p)\n self.scale = torch.sqrt(torch.FloatTensor([hid_dim]))\n\n def forward(self, inputs):\n\n # inputs = [batch size, inputs len]\n batch_size = inputs.shape[0]\n inputs_len = inputs.shape[1]\n\n pos = torch.arange(0, inputs_len).unsqueeze(0).repeat(batch_size, 1).to(inputs.device)\n scale = self.scale.to(inputs.device)\n embedded = (self.tok_embedding(inputs) * scale) + self.pos_embedding(pos)\n\n # output = [batch size, inputs len, hid dim]\n output = self.dropout(embedded)\n return output\n```\n\n\n```python\ninput_dim = source_tokenizer.get_vocab_size()\nhid_dim = 64\nmax_len = 100\ndropout_p = 0.5\nembedding = PositionWiseEmbedding(input_dim, hid_dim, max_len, dropout_p).to(device)\nembedding\n```\n\n\n\n\n PositionWiseEmbedding(\n (tok_embedding): Embedding(5000, 64)\n (pos_embedding): Embedding(100, 64)\n (dropout): Dropout(p=0.5, inplace=False)\n )\n\n\n\n\n```python\nsrc_embedded = embedding(source.to(device))\nsrc_embedded.shape\n```\n\n\n\n\n torch.Size([128, 40, 64])\n\n\n\nThe combined embeddings are then passed through $N$ encoder layers to get our context vectors $Z$. Before jumping straight into the encoder layers, we'll introduce some of the core building blocks behind them.\n\n### Multi Head Attention Layer\n\nOne of the key concepts introduced by the Transformer model is **multi-head attention layer**.\n\n\n\nThe purpose behind an attention mechanism is to relate inputs from different parts of the sequence. Attention operation is comprised of *queries*, *keys* and *values*. It might be helpful to look at these terms from an informational retrieval perspective, where every time we issue a query to a search engine, the search engine will match it with some key (title, description), and retrieve the associated value (content).\n\nTo be specific, Transformer model uses scaled dot-product attention, where query is used with key to get an attention vector, which is then used to get a weighted sum of the values.\n\n\\begin{align}\n\\text{Attention}(Q, K, V) = \\text{Softmax} \\big( \\frac{QK^T}{\\sqrt{d_k}} \\big)V\n\\end{align}\n\nWhere $Q = XW^Q, K = XW^K, V = XW^V$, $X$ is our input matrix, $W^Q$, $W^K$, $W^V$ are linear layers for the query, key and value. $d_k$ is the head dimension, `head_dim`, which we will further explain shortly. In essence, we are multiplying our input matrix with 3 different weight matrices. We first peform a dot product between query and key followed by a softmax to calculate attention weight, which measures correlation between the two words, finally scaling it by $d_k$ before doing a dot product with the value to get the weighted value. Scaling is done to prevent the results of the dot product from growing too large, and causing the gradients to become too small.\n\nMulti-head attention extends the single attention mechansim so we can potentially pay attention to different concepts that exists at different sequence positions. Instead of doing a single attention operation, the queries, keys and values have their `hid_dim` split into $h$ heads each of size $d_k$, and the scaled dot-product attention is calculated over all heads in parallel. After this computation, we re-combine the heads back to `hid_dim` shape. By reducing the dimensionality of each head/concept, the total computational cost is similar to a full dimension single-head attention.\n\n\\begin{align}\n\\text{MultiHead}(Q, K, V) &= \\text{Concat}(\\text{head}_1,...,\\text{head}_h)W^O \\\\\n\\text{head}_i &= \\text{Attention}(Q_i, K_i, V_i)\n\\end{align}\n\nWhere $W^O$ is the linear layer applied at the end of the multi-head attention layer.\n\nIn the implementation below, we carry out the multi head attention in parallel using batch matrix multiplication as opposed to a for loop. And while calculating the attention weights, we introduce the capability of applying a mask so the model does not pay attention to irrelevant tokens. We'll elaborate more on this in future sections.\n\n\n```python\nclass MultiHeadAttention(nn.Module):\n\n def __init__(self, hid_dim, n_heads):\n super().__init__()\n self.hid_dim = hid_dim\n self.n_heads = n_heads\n self.head_dim = hid_dim // n_heads\n assert hid_dim % n_heads == 0\n\n self.key_weight = nn.Linear(hid_dim, hid_dim)\n self.query_weight = nn.Linear(hid_dim, hid_dim)\n self.value_weight = nn.Linear(hid_dim, hid_dim)\n self.linear_weight = nn.Linear(hid_dim, hid_dim)\n\n def forward(self, query, key, value, mask = None):\n batch_size = query.shape[0]\n query_len = query.shape[1]\n key_len = key.shape[1]\n\n # key/query/value (proj) = [batch size, input len, hid dim]\n key_proj = self.key_weight(key)\n query_proj = self.query_weight(query)\n value_proj = self.value_weight(value)\n\n # compute the weights between query and key\n query_proj = query_proj.view(batch_size, query_len, self.n_heads, self.head_dim)\n query_proj = query_proj.permute(0, 2, 1, 3)\n key_proj = key_proj.view(batch_size, key_len, self.n_heads, self.head_dim)\n key_proj = key_proj.permute(0, 2, 3, 1)\n\n # energy, attention = [batch size, num heads, query len, key len]\n energy = torch.matmul(query_proj, key_proj) / math.sqrt(self.head_dim)\n\n if mask is not None:\n energy = energy.masked_fill(mask == 0, -1e10)\n\n attention = torch.softmax(energy, dim=-1)\n\n # output = [batch size, num heads, query len, head dim]\n value_proj = value_proj.view(batch_size, key_len, self.n_heads, self.head_dim)\n value_proj = value_proj.permute(0, 2, 1, 3)\n output = torch.matmul(attention, value_proj)\n\n # linaer = [batch size, query len, hid dim]\n output = output.permute(0, 2, 1, 3).contiguous().view(batch_size, query_len, self.hid_dim)\n linear_proj = self.linear_weight(output)\n return linear_proj, attention\n```\n\n\n```python\nn_heads = 8\nself_attention = MultiHeadAttention(hid_dim, n_heads).to(device)\nself_attention_output, attention = self_attention(src_embedded, src_embedded, src_embedded)\nprint(self_attention_output.shape)\nprint(attention.shape)\n```\n\n torch.Size([128, 40, 64])\n torch.Size([128, 8, 40, 40])\n\n\n### Position Wise Feed Forward Layer\n\nAnother building block for the model is the position wise feed forward layer, The input is transformed from `hid_dim` to `pf_dim`, where `pf_dim` is usually a lot larger than `hid_dim`. Then an activation function is applied before it is transformed back into a `hid_dim` representation.\n\n\n```python\nclass PositionWiseFeedForward(nn.Module):\n\n def __init__(self, hid_dim, pf_dim):\n super().__init__()\n self.hid_dim = hid_dim\n self.pf_dim = pf_dim\n\n self.fc1 = nn.Linear(hid_dim, pf_dim)\n self.fc2 = nn.Linear(pf_dim, hid_dim)\n\n def forward(self, inputs):\n # inputs = [batch size, src len, hid dim]\n fc1_output = torch.relu(self.fc1(inputs))\n fc2_output = self.fc2(fc1_output)\n return fc2_output\n```\n\n\n```python\nhid_dim = 64\npf_dim = 256\nposition_ff = PositionWiseFeedForward(hid_dim, pf_dim).to(device)\nposition_ff_output = position_ff(self_attention_output)\nposition_ff_output.shape\n```\n\n\n\n\n torch.Size([128, 40, 64])\n\n\n\n### Encoder\n\nWe'll now put our building blocks together to form the encoder.\n\n\n\nWe first pass the source sentence through a position wise embedding layer, this is then followed by *N* (configurable) encoder layers, the \"meat\" of modern transformer based architecture. Inside the encoder layer, we start from the multi-head attention layer, perform dropout on it, apply a residual connection, pass it through a layer normalization layer. followed by a position-wise feedforward layer and then, again, apply dropout, a residual connection and then layer normalization to get the output, this is then fed into the next layer. This sounds like a mouthful, but potentially the code will clarify things a bit. Things worth noting:\n\n- Parameters are not shared between layers.\n- Multi head attention layer is used by the encoder layer to attend to the source sentence, i.e. it is calculating and applying attention over itself instead of another sequence, hence we call it self attention. This layer is the only layer that propagates information along the sequence, other layers operate on each individual token in isolation.\n- The gist behind layer normalization is that it normalizes the features' values across the hidden dimension so each feature has a mean of 0 and a standard deviation of 1. This makes it easier to train neural networks with a larger number of layers, like the Transformer.\n\n\n```python\nclass EncoderLayer(nn.Module):\n\n def __init__(self, hid_dim, n_heads, pf_dim, dropout_p):\n super().__init__()\n self.hid_dim = hid_dim\n self.n_heads = n_heads\n self.pf_dim = pf_dim\n self.dropout_p = dropout_p\n\n self.self_attention_layer_norm = nn.LayerNorm(hid_dim)\n self.position_ff_layer_norm = nn.LayerNorm(hid_dim)\n self.self_attention = MultiHeadAttention(hid_dim, n_heads)\n self.position_ff = PositionWiseFeedForward(hid_dim, pf_dim)\n\n self.dropout = nn.Dropout(dropout_p)\n\n def forward(self, src, src_mask):\n # src = [batch size, src len, hid dim]\n # src_mask = [batch size, 1, 1, src len] \n self_attention_output, _ = self.self_attention(src, src, src, src_mask)\n\n # residual connection and layer norm\n self_attention_output = self.dropout(self_attention_output)\n self_attention_output = self.self_attention_layer_norm(src + self_attention_output)\n\n position_ff_output = self.position_ff(self_attention_output)\n\n # residual connection and layer norm\n # [batch size, src len, hid dim]\n position_ff_output = self.dropout(position_ff_output)\n output = self.position_ff_layer_norm(self_attention_output + position_ff_output) \n return output\n```\n\n\n```python\nclass Encoder(nn.Module):\n\n def __init__(self, input_dim, hid_dim, max_len, dropout_p, n_heads, pf_dim, n_layers):\n super().__init__()\n self.input_dim = input_dim\n self.hid_dim = hid_dim\n self.max_len = max_len\n self.dropout_p = dropout_p\n self.n_heads = n_heads\n self.pf_dim = pf_dim\n self.n_layers = n_layers\n\n self.pos_wise_embedding = PositionWiseEmbedding(input_dim, hid_dim, max_len, dropout_p)\n self.layers = nn.ModuleList([\n EncoderLayer(hid_dim, n_heads, pf_dim, dropout_p)\n for _ in range(n_layers)\n ])\n\n def forward(self, src, src_mask = None):\n\n # src = [batch size, src len]\n # src_mask = [batch size, 1, 1, src len]\n src = self.pos_wise_embedding(src)\n for layer in self.layers:\n src = layer(src, src_mask)\n\n # [batch size, src len, hid dim]\n return src\n```\n\n\n```python\ninput_dim = source_tokenizer.get_vocab_size()\nhid_dim = 64\nmax_len = 100\ndropout_p = 0.5\nn_heads = 8\npf_dim = 256\nn_layers = 1\nencoder = Encoder(input_dim, hid_dim, max_len, dropout_p, n_heads, pf_dim, n_layers).to(device)\nencoder\n```\n\n\n\n\n Encoder(\n (pos_wise_embedding): PositionWiseEmbedding(\n (tok_embedding): Embedding(5000, 64)\n (pos_embedding): Embedding(100, 64)\n (dropout): Dropout(p=0.5, inplace=False)\n )\n (layers): ModuleList(\n (0): EncoderLayer(\n (self_attention_layer_norm): LayerNorm((64,), eps=1e-05, elementwise_affine=True)\n (position_ff_layer_norm): LayerNorm((64,), eps=1e-05, elementwise_affine=True)\n (self_attention): MultiHeadAttention(\n (key_weight): Linear(in_features=64, out_features=64, bias=True)\n (query_weight): Linear(in_features=64, out_features=64, bias=True)\n (value_weight): Linear(in_features=64, out_features=64, bias=True)\n (linear_weight): Linear(in_features=64, out_features=64, bias=True)\n )\n (position_ff): PositionWiseFeedForward(\n (fc1): Linear(in_features=64, out_features=256, bias=True)\n (fc2): Linear(in_features=256, out_features=64, bias=True)\n )\n (dropout): Dropout(p=0.5, inplace=False)\n )\n )\n )\n\n\n\n\n```python\nencoder_output = encoder(source.to(device))\nencoder_output.shape\n```\n\n\n\n\n torch.Size([128, 40, 64])\n\n\n\n### Decoder\n\nNow comes the decoder part:\n\n\n\nDecoder's main goal is to take our source sentence's encoded representation, $Z$, convert it into predicted tokens in the target sentence, $\\hat{Y}$. We then compare it with the actual tokens in the target sentence, $Y$, to calculate our loss and update our parameters to improve our predictions.\n\nDecoder layer contains similar building blocks as the encoder layer, except it now has two multi-head attention layers, `self_attention` and `encoder_attention`.\n\nThe former attention layer performs self attention on the target sentence's embedding representation to generate the decoder representation. Whereas in the latter attention layer, the queries come from the previous decoder representation, and the keys and values come from the output of the encoder representation. \n\n\n```python\nclass DecoderLayer(nn.Module):\n\n def __init__(self, hid_dim, n_heads, pf_dim, dropout_p):\n super().__init__()\n self.hid_dim = hid_dim\n self.n_heads = n_heads\n self.pf_dim = pf_dim\n self.dropout_p = dropout_p\n\n self.self_attention_layer_norm = nn.LayerNorm(hid_dim)\n self.encoder_attention_layer_norm = nn.LayerNorm(hid_dim)\n self.position_ff_layer_norm = nn.LayerNorm(hid_dim)\n self.self_attention = MultiHeadAttention(hid_dim, n_heads)\n self.encoder_attention = MultiHeadAttention(hid_dim, n_heads)\n self.position_ff = PositionWiseFeedForward(hid_dim, pf_dim)\n \n self.dropout = nn.Dropout(dropout_p)\n\n def forward(self, trg, encoded_src, trg_mask, src_mask):\n # encoded_src = [batch size, src len, hid dim]\n # src_mask = [batch size, 1, 1, src len] \n self_attention_output, _ = self.self_attention(trg, trg, trg, trg_mask)\n\n # residual connection and layer norm\n self_attention_output = self.dropout(self_attention_output)\n self_attention_output = self.self_attention_layer_norm(trg + self_attention_output)\n\n encoder_attention_output, _ = self.encoder_attention(\n self_attention_output,\n encoded_src,\n encoded_src,\n src_mask\n )\n encoder_attention_output = self.dropout(encoder_attention_output)\n encoder_attention_output = self.encoder_attention_layer_norm(trg + encoder_attention_output)\n\n position_ff_output = self.position_ff(encoder_attention_output)\n\n # residual connection and layer norm\n # [batch size, src len, hid dim]\n position_ff_output = self.dropout(position_ff_output)\n output = self.position_ff_layer_norm(encoder_attention_output + position_ff_output) \n return output\n```\n\n\n```python\nclass Decoder(nn.Module):\n\n def __init__(self, output_dim, hid_dim, max_len, dropout_p, n_heads, pf_dim, n_layers):\n super().__init__()\n self.output_dim = output_dim\n self.hid_dim = hid_dim\n self.max_len = max_len\n self.dropout_p = dropout_p\n self.n_heads = n_heads\n self.pf_dim = pf_dim\n self.n_layers = n_layers\n\n self.pos_wise_embedding = PositionWiseEmbedding(output_dim, hid_dim, max_len, dropout_p)\n self.layers = nn.ModuleList([\n DecoderLayer(hid_dim, n_heads, pf_dim, dropout_p)\n for _ in range(n_layers)\n ])\n self.fc_out = nn.Linear(hid_dim, output_dim)\n\n def forward(self, trg, encoded_src, trg_mask = None, src_mask = None):\n\n trg = self.pos_wise_embedding(trg)\n for layer in self.layers:\n trg = layer(trg, encoded_src, trg_mask, src_mask)\n \n output = self.fc_out(trg)\n return output\n```\n\n\n```python\noutput_dim = target_tokenizer.get_vocab_size()\nhid_dim = 64\nmax_len = 100\ndropout_p = 0.5\nn_heads = 8\npf_dim = 32\nn_layers = 1\ndecoder = Decoder(output_dim, hid_dim, max_len, dropout_p, n_heads, pf_dim, n_layers).to(device)\ndecoder\n```\n\n\n\n\n Decoder(\n (pos_wise_embedding): PositionWiseEmbedding(\n (tok_embedding): Embedding(5000, 64)\n (pos_embedding): Embedding(100, 64)\n (dropout): Dropout(p=0.5, inplace=False)\n )\n (layers): ModuleList(\n (0): DecoderLayer(\n (self_attention_layer_norm): LayerNorm((64,), eps=1e-05, elementwise_affine=True)\n (encoder_attention_layer_norm): LayerNorm((64,), eps=1e-05, elementwise_affine=True)\n (position_ff_layer_norm): LayerNorm((64,), eps=1e-05, elementwise_affine=True)\n (self_attention): MultiHeadAttention(\n (key_weight): Linear(in_features=64, out_features=64, bias=True)\n (query_weight): Linear(in_features=64, out_features=64, bias=True)\n (value_weight): Linear(in_features=64, out_features=64, bias=True)\n (linear_weight): Linear(in_features=64, out_features=64, bias=True)\n )\n (encoder_attention): MultiHeadAttention(\n (key_weight): Linear(in_features=64, out_features=64, bias=True)\n (query_weight): Linear(in_features=64, out_features=64, bias=True)\n (value_weight): Linear(in_features=64, out_features=64, bias=True)\n (linear_weight): Linear(in_features=64, out_features=64, bias=True)\n )\n (position_ff): PositionWiseFeedForward(\n (fc1): Linear(in_features=64, out_features=32, bias=True)\n (fc2): Linear(in_features=32, out_features=64, bias=True)\n )\n (dropout): Dropout(p=0.5, inplace=False)\n )\n )\n (fc_out): Linear(in_features=64, out_features=5000, bias=True)\n )\n\n\n\n\n```python\nencoder_output = encoder(source.to(device))\ndecoder_output = decoder(target.to(device), encoder_output)\ndecoder_output.shape\n```\n\n\n\n\n torch.Size([128, 26, 5000])\n\n\n\n### Seq2Seq\n\nNow that we have our encoder and decoder, the final part is to have a Seq2Seq module that encapsulates the two. In this module, we'll also handle the masking.\n\nThe source mask is created by checking where the so urce sequence is not equal to the `` token. It is 1 where the token is not a token and 0 when it is. This is used in our encoder layers' multi-head attention mechanisms, where we want the model to not pay any attention to `` tokens, which contain no useful information.\n\nThe target mask is a bit more involved. First, we create a mask for the tokens, as we did for the source mask. Next, we create a \"subsequent\" mask, `trg_sub_mask`, using `torch.tril`. This creates a diagonal matrix where the elements above the diagonal will be zero and the elements below the diagonal will be set to whatever the input tensor is. In this case, the input tensor will be a tensor filled with ones, meaning our `trg_sub_mask` will look something like this (for a target with 5 tokens):\n\n\\begin{matrix}\n1 & 0 & 0 & 0 & 0 \\\\\n1 & 1 & 0 & 0 & 0 \\\\\n1 & 1 & 1 & 0 & 0 \\\\\n1 & 1 & 1 & 1 & 0 \\\\\n1 & 1 & 1 & 1 & 1 \\\\\n\\end{matrix}\n \nThis shows what each target token (row) is allowed to look at (column). Our first target token has a mask of [1, 0, 0, 0, 0] which means it can only look at the first target token, whereas the second target token has a mask of [1, 1, 0, 0, 0] which it means it can look at both the first and second target tokens and so on.\n\nThe \"subsequent\" mask is then logically anded with the padding mask, this combines the two masks ensuring both the subsequent tokens and the padding tokens cannot be attended to. For example if the last two tokens were `` tokens the final target mask would look like:\n\n\\begin{matrix}\n1 & 0 & 0 & 0 & 0 \\\\\n1 & 1 & 0 & 0 & 0 \\\\\n1 & 1 & 1 & 0 & 0 \\\\\n1 & 1 & 1 & 0 & 0 \\\\\n1 & 1 & 1 & 0 & 0 \\\\\n\\end{matrix}\n \nThese masks are fed in into model along with source and target sentence to get out predicted target output.\n \nSite Note: Introducing some other terminology that we might come across. The need to create a subsequent mask is very common in autoregressive model, where the task is to predict the next token in the sequence (e.g. language model). By introducing this masking, we are making the self attention block casual. Different implementation or library might have different ways of specifying this masking, but the core idea is to prevent the model from \"cheating\" by copying the tokens that are after the ones it's currently processing.\n\n\n```python\nclass Seq2Seq(nn.Module):\n\n def __init__(self, encoder, decoder, src_pad_idx, trg_pad_idx):\n super().__init__()\n self.encoder = encoder\n self.decoder = decoder\n self.src_pad_idx = src_pad_idx\n self.trg_pad_idx = trg_pad_idx\n\n def make_src_mask(self, src):\n \"\"\"\n the padding mask is unsqueezed so it can be correctly broadcasted\n when applying the mask to the attention weights, which is of shape\n [batch size, n heads, seq len, seq len].\n \"\"\"\n src_pad_mask = (src != self.src_pad_idx).unsqueeze(1).unsqueeze(2)\n return src_pad_mask\n\n def make_trg_mask(self, trg):\n trg_pad_mask = (trg != self.trg_pad_idx).unsqueeze(1).unsqueeze(2)\n\n trg_len = trg.shape[1]\n trg_sub_mask = torch.tril(torch.ones((trg_len, trg_len))).bool().to(trg.device)\n trg_mask = trg_pad_mask & trg_sub_mask\n return trg_mask\n\n def forward(self, src, trg):\n src_mask = self.make_src_mask(src)\n trg_mask = self.make_trg_mask(trg)\n\n encoded_src = self.encoder(src, src_mask)\n decoder_output = self.decoder(trg, encoded_src, trg_mask, src_mask)\n return decoder_output\n```\n\n\n```python\nsource_pad_idx = source_tokenizer.token_to_id(pad_token)\ntarget_pad_idx = target_tokenizer.token_to_id(pad_token)\n\nINPUT_DIM = source_tokenizer.get_vocab_size()\nOUTPUT_DIM = target_tokenizer.get_vocab_size()\nMAX_LEN = 100\nHID_DIM = 512\nENC_LAYERS = 6\nDEC_LAYERS = 3\nENC_HEADS = 8\nDEC_HEADS = 8\nENC_PF_DIM = 512\nDEC_PF_DIM = 512\nENC_DROPOUT = 0.1\nDEC_DROPOUT = 0.1\n\nencoder = Encoder(\n INPUT_DIM, \n HID_DIM,\n MAX_LEN,\n ENC_DROPOUT, \n ENC_HEADS, \n ENC_PF_DIM, \n ENC_LAYERS\n)\n\ndecoder = Decoder(\n OUTPUT_DIM, \n HID_DIM,\n MAX_LEN,\n DEC_DROPOUT,\n DEC_HEADS,\n DEC_PF_DIM,\n DEC_LAYERS\n)\n\nmodel = Seq2Seq(encoder, decoder, source_pad_idx, target_pad_idx).to(device)\nmodel\n```\n\n\n\n\n Seq2Seq(\n (encoder): Encoder(\n (pos_wise_embedding): PositionWiseEmbedding(\n (tok_embedding): Embedding(5000, 512)\n (pos_embedding): Embedding(100, 512)\n (dropout): Dropout(p=0.1, inplace=False)\n )\n (layers): ModuleList(\n (0): EncoderLayer(\n (self_attention_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (position_ff_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (self_attention): MultiHeadAttention(\n (key_weight): Linear(in_features=512, out_features=512, bias=True)\n (query_weight): Linear(in_features=512, out_features=512, bias=True)\n (value_weight): Linear(in_features=512, out_features=512, bias=True)\n (linear_weight): Linear(in_features=512, out_features=512, bias=True)\n )\n (position_ff): PositionWiseFeedForward(\n (fc1): Linear(in_features=512, out_features=512, bias=True)\n (fc2): Linear(in_features=512, out_features=512, bias=True)\n )\n (dropout): Dropout(p=0.1, inplace=False)\n )\n (1): EncoderLayer(\n (self_attention_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (position_ff_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (self_attention): MultiHeadAttention(\n (key_weight): Linear(in_features=512, out_features=512, bias=True)\n (query_weight): Linear(in_features=512, out_features=512, bias=True)\n (value_weight): Linear(in_features=512, out_features=512, bias=True)\n (linear_weight): Linear(in_features=512, out_features=512, bias=True)\n )\n (position_ff): PositionWiseFeedForward(\n (fc1): Linear(in_features=512, out_features=512, bias=True)\n (fc2): Linear(in_features=512, out_features=512, bias=True)\n )\n (dropout): Dropout(p=0.1, inplace=False)\n )\n (2): EncoderLayer(\n (self_attention_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (position_ff_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (self_attention): MultiHeadAttention(\n (key_weight): Linear(in_features=512, out_features=512, bias=True)\n (query_weight): Linear(in_features=512, out_features=512, bias=True)\n (value_weight): Linear(in_features=512, out_features=512, bias=True)\n (linear_weight): Linear(in_features=512, out_features=512, bias=True)\n )\n (position_ff): PositionWiseFeedForward(\n (fc1): Linear(in_features=512, out_features=512, bias=True)\n (fc2): Linear(in_features=512, out_features=512, bias=True)\n )\n (dropout): Dropout(p=0.1, inplace=False)\n )\n (3): EncoderLayer(\n (self_attention_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (position_ff_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (self_attention): MultiHeadAttention(\n (key_weight): Linear(in_features=512, out_features=512, bias=True)\n (query_weight): Linear(in_features=512, out_features=512, bias=True)\n (value_weight): Linear(in_features=512, out_features=512, bias=True)\n (linear_weight): Linear(in_features=512, out_features=512, bias=True)\n )\n (position_ff): PositionWiseFeedForward(\n (fc1): Linear(in_features=512, out_features=512, bias=True)\n (fc2): Linear(in_features=512, out_features=512, bias=True)\n )\n (dropout): Dropout(p=0.1, inplace=False)\n )\n (4): EncoderLayer(\n (self_attention_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (position_ff_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (self_attention): MultiHeadAttention(\n (key_weight): Linear(in_features=512, out_features=512, bias=True)\n (query_weight): Linear(in_features=512, out_features=512, bias=True)\n (value_weight): Linear(in_features=512, out_features=512, bias=True)\n (linear_weight): Linear(in_features=512, out_features=512, bias=True)\n )\n (position_ff): PositionWiseFeedForward(\n (fc1): Linear(in_features=512, out_features=512, bias=True)\n (fc2): Linear(in_features=512, out_features=512, bias=True)\n )\n (dropout): Dropout(p=0.1, inplace=False)\n )\n (5): EncoderLayer(\n (self_attention_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (position_ff_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (self_attention): MultiHeadAttention(\n (key_weight): Linear(in_features=512, out_features=512, bias=True)\n (query_weight): Linear(in_features=512, out_features=512, bias=True)\n (value_weight): Linear(in_features=512, out_features=512, bias=True)\n (linear_weight): Linear(in_features=512, out_features=512, bias=True)\n )\n (position_ff): PositionWiseFeedForward(\n (fc1): Linear(in_features=512, out_features=512, bias=True)\n (fc2): Linear(in_features=512, out_features=512, bias=True)\n )\n (dropout): Dropout(p=0.1, inplace=False)\n )\n )\n )\n (decoder): Decoder(\n (pos_wise_embedding): PositionWiseEmbedding(\n (tok_embedding): Embedding(5000, 512)\n (pos_embedding): Embedding(100, 512)\n (dropout): Dropout(p=0.1, inplace=False)\n )\n (layers): ModuleList(\n (0): DecoderLayer(\n (self_attention_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (encoder_attention_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (position_ff_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (self_attention): MultiHeadAttention(\n (key_weight): Linear(in_features=512, out_features=512, bias=True)\n (query_weight): Linear(in_features=512, out_features=512, bias=True)\n (value_weight): Linear(in_features=512, out_features=512, bias=True)\n (linear_weight): Linear(in_features=512, out_features=512, bias=True)\n )\n (encoder_attention): MultiHeadAttention(\n (key_weight): Linear(in_features=512, out_features=512, bias=True)\n (query_weight): Linear(in_features=512, out_features=512, bias=True)\n (value_weight): Linear(in_features=512, out_features=512, bias=True)\n (linear_weight): Linear(in_features=512, out_features=512, bias=True)\n )\n (position_ff): PositionWiseFeedForward(\n (fc1): Linear(in_features=512, out_features=512, bias=True)\n (fc2): Linear(in_features=512, out_features=512, bias=True)\n )\n (dropout): Dropout(p=0.1, inplace=False)\n )\n (1): DecoderLayer(\n (self_attention_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (encoder_attention_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (position_ff_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (self_attention): MultiHeadAttention(\n (key_weight): Linear(in_features=512, out_features=512, bias=True)\n (query_weight): Linear(in_features=512, out_features=512, bias=True)\n (value_weight): Linear(in_features=512, out_features=512, bias=True)\n (linear_weight): Linear(in_features=512, out_features=512, bias=True)\n )\n (encoder_attention): MultiHeadAttention(\n (key_weight): Linear(in_features=512, out_features=512, bias=True)\n (query_weight): Linear(in_features=512, out_features=512, bias=True)\n (value_weight): Linear(in_features=512, out_features=512, bias=True)\n (linear_weight): Linear(in_features=512, out_features=512, bias=True)\n )\n (position_ff): PositionWiseFeedForward(\n (fc1): Linear(in_features=512, out_features=512, bias=True)\n (fc2): Linear(in_features=512, out_features=512, bias=True)\n )\n (dropout): Dropout(p=0.1, inplace=False)\n )\n (2): DecoderLayer(\n (self_attention_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (encoder_attention_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (position_ff_layer_norm): LayerNorm((512,), eps=1e-05, elementwise_affine=True)\n (self_attention): MultiHeadAttention(\n (key_weight): Linear(in_features=512, out_features=512, bias=True)\n (query_weight): Linear(in_features=512, out_features=512, bias=True)\n (value_weight): Linear(in_features=512, out_features=512, bias=True)\n (linear_weight): Linear(in_features=512, out_features=512, bias=True)\n )\n (encoder_attention): MultiHeadAttention(\n (key_weight): Linear(in_features=512, out_features=512, bias=True)\n (query_weight): Linear(in_features=512, out_features=512, bias=True)\n (value_weight): Linear(in_features=512, out_features=512, bias=True)\n (linear_weight): Linear(in_features=512, out_features=512, bias=True)\n )\n (position_ff): PositionWiseFeedForward(\n (fc1): Linear(in_features=512, out_features=512, bias=True)\n (fc2): Linear(in_features=512, out_features=512, bias=True)\n )\n (dropout): Dropout(p=0.1, inplace=False)\n )\n )\n (fc_out): Linear(in_features=512, out_features=5000, bias=True)\n )\n )\n\n\n\n\n```python\noutput = model(source.to(device), target.to(device))\noutput.shape\n```\n\n\n\n\n torch.Size([128, 26, 5000])\n\n\n\n\n```python\ndef count_parameters(model):\n return sum(p.numel() for p in model.parameters() if p.requires_grad)\n\nprint(f'The model has {count_parameters(model):,} trainable parameters')\n```\n\n The model has 25,144,200 trainable parameters\n\n\n## Model Training\n\nThe training loop also requires a bit of explanation.\n\nWe want our model to predict the `` token but not have it be an input into our model, hence we slice the `` token off the end of our target sequence.\n\n\\begin{align}\n\\text{trg} &= [sos, x_1, x_2, x_3, eos] \\\\\n\\text{trg[:-1]} &= [sos, x_1, x_2, x_3]\n\\end{align}\n\n\nWe then calculate our loss using the original target tensor with the `` token sliced off the front, retaining the `` token.\n\n\\begin{align}\n\\text{output} &= [y_1, y_2, y_3, eos] \\\\\n\\text{trg[1:]} &= [x_1, x_2, x_3, eos]\n\\end{align}\n\nAll in all, our model receives the target up to the last character (excluding the last), whereas the ground truth will be from the second character onward.\n\n\n```python\ndef train(model, iterator, optimizer, criterion, clip):\n \n model.train()\n epoch_loss = 0\n for i, (src, trg) in enumerate(iterator):\n src = src.to(device)\n trg = trg.to(device)\n\n optimizer.zero_grad()\n output = model(src, trg[:, :-1])\n \n # output = [batch size, trg len - 1, output dim]\n # trg = [batch size, trg len]\n output_dim = output.shape[-1]\n output = output.contiguous().view(-1, output_dim)\n trg = trg[:, 1:].contiguous().view(-1)\n\n loss = criterion(output, trg)\n loss.backward()\n torch.nn.utils.clip_grad_norm_(model.parameters(), clip)\n optimizer.step()\n epoch_loss += loss.item()\n \n return epoch_loss / len(iterator)\n```\n\nEvaluation loop is similar to the training loop, just without the updating the model's parameters.\n\n\n```python\ndef evaluate(model, iterator, criterion):\n \n model.eval()\n epoch_loss = 0\n with torch.no_grad():\n for i, (src, trg) in enumerate(iterator):\n src = src.to(device)\n trg = trg.to(device)\n\n output = model(src, trg[:, :-1])\n \n # output = [batch size, trg len - 1, output dim]\n # trg = [batch size, trg len]\n output_dim = output.shape[-1]\n output = output.contiguous().view(-1, output_dim)\n trg = trg[:, 1:].contiguous().view(-1)\n\n loss = criterion(output, trg)\n epoch_loss += loss.item()\n \n return epoch_loss / len(iterator)\n```\n\n\n```python\ndef epoch_time(start_time, end_time):\n elapsed_time = end_time - start_time\n elapsed_mins = int(elapsed_time / 60)\n elapsed_secs = int(elapsed_time - (elapsed_mins * 60))\n return elapsed_mins, elapsed_secs\n```\n\nWhile defining our loss function, we also ensure we ignore loss that are calculated over the `` tokens.\n\n\n```python\nMODEL_CHECKPOINT = 'transformer.pt'\nN_EPOCHS = 10\nCLIP = 1\nLEARNING_RATE = 0.0001\noptimizer = optim.Adam(model.parameters(), lr=LEARNING_RATE)\ncriterion = nn.CrossEntropyLoss(ignore_index=target_pad_idx)\n```\n\n\n```python\nbest_valid_loss = float('inf')\nfor epoch in range(N_EPOCHS):\n start_time = time.time()\n train_loss = train(model, dataloader_train, optimizer, criterion, CLIP)\n valid_loss = evaluate(model, dataloader_val, criterion)\n end_time = time.time()\n epoch_mins, epoch_secs = epoch_time(start_time, end_time)\n\n if valid_loss < best_valid_loss:\n best_valid_loss = valid_loss\n torch.save(model.state_dict(), MODEL_CHECKPOINT)\n\n print(f'Epoch: {epoch+1:02} | Time: {epoch_mins}m {epoch_secs}s')\n print(f'\\tTrain Loss: {train_loss:.3f} | Train PPL: {math.exp(train_loss):7.3f}')\n print(f'\\t Val. Loss: {valid_loss:.3f} | Val. PPL: {math.exp(valid_loss):7.3f}')\n```\n\n Epoch: 01 | Time: 0m 30s\n \tTrain Loss: 4.942 | Train PPL: 140.059\n \t Val. Loss: 4.053 | Val. PPL: 57.582\n Epoch: 02 | Time: 0m 30s\n \tTrain Loss: 3.816 | Train PPL: 45.434\n \t Val. Loss: 3.575 | Val. PPL: 35.688\n Epoch: 03 | Time: 0m 33s\n \tTrain Loss: 3.424 | Train PPL: 30.706\n \t Val. Loss: 3.279 | Val. PPL: 26.550\n Epoch: 04 | Time: 0m 31s\n \tTrain Loss: 3.156 | Train PPL: 23.474\n \t Val. Loss: 3.081 | Val. PPL: 21.777\n Epoch: 05 | Time: 0m 33s\n \tTrain Loss: 2.945 | Train PPL: 19.012\n \t Val. Loss: 2.922 | Val. PPL: 18.579\n Epoch: 06 | Time: 0m 33s\n \tTrain Loss: 2.762 | Train PPL: 15.832\n \t Val. Loss: 2.780 | Val. PPL: 16.127\n Epoch: 07 | Time: 0m 32s\n \tTrain Loss: 2.604 | Train PPL: 13.512\n \t Val. Loss: 2.677 | Val. PPL: 14.547\n Epoch: 08 | Time: 0m 34s\n \tTrain Loss: 2.460 | Train PPL: 11.706\n \t Val. Loss: 2.593 | Val. PPL: 13.373\n Epoch: 09 | Time: 0m 36s\n \tTrain Loss: 2.332 | Train PPL: 10.302\n \t Val. Loss: 2.502 | Val. PPL: 12.201\n Epoch: 10 | Time: 0m 38s\n \tTrain Loss: 2.212 | Train PPL: 9.132\n \t Val. Loss: 2.430 | Val. PPL: 11.360\n\n\n## Model Evaluation\n\n\n```python\nmodel.load_state_dict(torch.load(MODEL_CHECKPOINT))\ntest_loss = evaluate(model, dataloader_test, criterion)\nprint(f'| Test Loss: {test_loss:.3f} | Test PPL: {math.exp(test_loss):7.3f} |')\n```\n\n | Test Loss: 2.426 | Test PPL: 11.317 |\n\n\n\n```python\ndef predict(source, model, source_tokenizer, target_tokenizer):\n \"\"\"\n Given the raw token, predict the translation using greedy search.\n This is a naive implementation without batching\n \"\"\"\n src_indices = [source_init_idx] + source_tokenizer.encode(source).ids + [source_eos_idx]\n src_tensor = torch.LongTensor(src_indices).unsqueeze(0).to(device)\n src_mask = model.make_src_mask(src_tensor)\n\n # separating out the encoder and decoder allows us to generate the encoded source\n # sentence once and share it throughout the target prediction step\n with torch.no_grad():\n encoded_src = model.encoder(src_tensor, src_mask)\n\n # greedy search\n # sequentially predict the target sequence starting from the init sentence token\n trg_indices = [target_init_idx]\n for _ in range(max_len):\n trg_tensor = torch.LongTensor(trg_indices).unsqueeze(0).to(device)\n trg_mask = model.make_trg_mask(trg_tensor)\n\n with torch.no_grad():\n output = model.decoder(trg_tensor, encoded_src, trg_mask, src_mask)\n\n # add the last predicted token\n pred_token = output.argmax(dim=2)[:, -1].item()\n trg_indices.append(pred_token)\n if pred_token == target_eos_idx:\n break\n\n return target_tokenizer.decode(trg_indices)\n```\n\n\n```python\ntranslation = dataset_dict['train'][0]\nsource_raw = translation[source_lang]\ntarget_raw = translation[target_lang]\nprint('source: ', source_raw)\nprint('target: ', target_raw)\n\npredict(source_raw, model, source_tokenizer, target_tokenizer)\n```\n\n source: Zwei junge wei\u00dfe M\u00e4nner sind im Freien in der N\u00e4he vieler B\u00fcsche.\n target: Two young, White males are outside near many bushes.\n\n\n\n\n\n 'two young men are outside near white shirts near the woods.'\n\n\n\n## Transformer Module\n\nInstead of resorting to our own Transformer encoder and decoder implementation, PyTorch's `nn` module already comes with a pre-built one. The major difference here is it expects a different [shape](https://pytorch.org/docs/master/generated/torch.nn.Transformer.html#torch.nn.Transformer.forward) for the padding and subsequent mask.\n\n\n```python\nclass Transformer(nn.Module):\n \"\"\"\n \n References\n ----------\n https://pytorch.org/docs/master/generated/torch.nn.Transformer.html\n \"\"\"\n\n def __init__(\n self,\n encoder_embedding_dim,\n decoder_embedding_dim,\n source_pad_idx,\n target_pad_idx,\n encoder_max_len = 100,\n decoder_max_len = 100,\n model_dim = 512,\n num_head = 8,\n encoder_num_layers = 3,\n decoder_num_layers = 3,\n feedforward_dim = 512,\n dropout = 0.1\n ):\n super().__init__()\n self.source_pad_idx = source_pad_idx\n self.target_pad_idx = target_pad_idx\n\n self.encoder_embedding = PositionWiseEmbedding(\n encoder_embedding_dim,\n model_dim,\n encoder_max_len,\n dropout\n )\n self.decoder_embedding = PositionWiseEmbedding(\n decoder_embedding_dim,\n model_dim,\n decoder_max_len,\n dropout\n )\n\n layer_params = {\n 'd_model': model_dim,\n 'nhead': num_head,\n 'dim_feedforward': feedforward_dim,\n 'dropout': dropout\n }\n self.encoder = nn.TransformerEncoder(\n nn.TransformerEncoderLayer(**layer_params),\n encoder_num_layers\n )\n self.decoder = nn.TransformerDecoder(\n nn.TransformerDecoderLayer(**layer_params),\n decoder_num_layers\n )\n self.linear = nn.Linear(model_dim, decoder_embedding_dim)\n\n def forward(self, src_tensor, trg_tensor):\n # enc_src = self.encoder(src, src_mask)\n # decoder_output = self.decoder(trg, enc_src, trg_mask, src_mask)\n\n # in PyTorch's Transformer Encoder and Decoder implementation, they\n # expect the first dimensionto be batch size\n src_encoded = self.encode(src_tensor)\n output = self.decode(trg_tensor, src_encoded)\n return output\n\n def encode(self, src_tensor):\n src_key_padding_mask = generate_key_padding_mask(src_tensor, self.source_pad_idx)\n src_embedded = self.encoder_embedding(src_tensor).permute(1, 0, 2)\n return self.encoder(src_embedded, src_key_padding_mask=src_key_padding_mask)\n\n def decode(self, trg_tensor, src_encoded):\n trg_key_padding_mask = generate_key_padding_mask(trg_tensor, self.target_pad_idx)\n trg_mask = generate_subsequent_mask(trg_tensor)\n trg_embedded = self.decoder_embedding(trg_tensor).permute(1, 0, 2)\n decoder_output = self.decoder(\n trg_embedded,\n src_encoded,\n trg_mask,\n tgt_key_padding_mask=trg_key_padding_mask\n ).permute(1, 0, 2)\n return self.linear(decoder_output)\n\n def predict(self, src_tensor, max_len = 100):\n # separating out the encoder and decoder allows us to generate the encoded source\n # sentence once and share it throughout the target prediction step\n with torch.no_grad():\n src_encoded = self.encode(src_tensor)\n \n # greedy search\n # sequentially predict the target sequence starting from the init sentence token\n trg_indices = [target_init_idx]\n for _ in range(max_len):\n trg_tensor = torch.LongTensor(trg_indices).unsqueeze(0).to(src_tensor.device)\n with torch.no_grad():\n output = self.decode(trg_tensor, src_encoded)\n\n # add the last predicted token\n pred_token = output.argmax(dim=2)[:, -1].item()\n trg_indices.append(pred_token)\n if pred_token == target_eos_idx:\n break\n\n return trg_indices\n\n\ndef generate_subsequent_mask(inputs):\n \"\"\"\n If a BoolTensor is provided, positions with True are not\n allowed to attend while False values will be unchanged\n \"\"\"\n inputs_len = inputs.shape[1]\n square = torch.ones((inputs_len, inputs_len)).to(inputs.device)\n mask = (torch.tril(square) == 0.0).bool()\n return mask\n\n\ndef generate_key_padding_mask(inputs, pad_idx):\n return (inputs == pad_idx).to(inputs.device)\n```\n\n\n```python\nINPUT_DIM = source_tokenizer.get_vocab_size()\nOUTPUT_DIM = target_tokenizer.get_vocab_size()\n\ntransformer = Transformer(INPUT_DIM, OUTPUT_DIM, source_pad_idx, target_pad_idx).to(device)\n\nwith torch.no_grad():\n output = transformer(source.to(device), target.to(device))\n\noutput.shape\n```\n\n\n\n\n torch.Size([128, 26, 5000])\n\n\n\n\n```python\nprint(f'The model has {count_parameters(transformer):,} trainable parameters')\n```\n\n The model has 20,410,248 trainable parameters\n\n\n\n```python\nMODEL_CHECKPOINT = 'transformer.pt'\nN_EPOCHS = 10\nCLIP = 1\nLEARNING_RATE = 0.0005\noptimizer = optim.Adam(transformer.parameters(), lr=LEARNING_RATE)\ncriterion = nn.CrossEntropyLoss(ignore_index=target_pad_idx)\n```\n\n\n```python\nbest_valid_loss = float('inf')\nfor epoch in range(N_EPOCHS):\n start_time = time.time()\n train_loss = train(transformer, dataloader_train, optimizer, criterion, CLIP)\n valid_loss = evaluate(transformer, dataloader_val, criterion)\n end_time = time.time()\n epoch_mins, epoch_secs = epoch_time(start_time, end_time)\n\n if valid_loss < best_valid_loss:\n best_valid_loss = valid_loss\n torch.save(transformer.state_dict(), MODEL_CHECKPOINT)\n\n print(f'Epoch: {epoch+1:02} | Time: {epoch_mins}m {epoch_secs}s')\n print(f'\\tTrain Loss: {train_loss:.3f} | Train PPL: {math.exp(train_loss):7.3f}')\n print(f'\\t Val. Loss: {valid_loss:.3f} | Val. PPL: {math.exp(valid_loss):7.3f}')\n```\n\n Epoch: 01 | Time: 0m 22s\n \tTrain Loss: 4.076 | Train PPL: 58.881\n \t Val. Loss: 3.314 | Val. PPL: 27.483\n Epoch: 02 | Time: 0m 19s\n \tTrain Loss: 3.020 | Train PPL: 20.491\n \t Val. Loss: 2.772 | Val. PPL: 15.983\n Epoch: 03 | Time: 0m 19s\n \tTrain Loss: 2.535 | Train PPL: 12.611\n \t Val. Loss: 2.490 | Val. PPL: 12.066\n Epoch: 04 | Time: 0m 19s\n \tTrain Loss: 2.189 | Train PPL: 8.929\n \t Val. Loss: 2.314 | Val. PPL: 10.113\n Epoch: 05 | Time: 0m 19s\n \tTrain Loss: 1.922 | Train PPL: 6.832\n \t Val. Loss: 2.209 | Val. PPL: 9.111\n Epoch: 06 | Time: 0m 19s\n \tTrain Loss: 1.708 | Train PPL: 5.518\n \t Val. Loss: 2.153 | Val. PPL: 8.608\n Epoch: 07 | Time: 0m 19s\n \tTrain Loss: 1.528 | Train PPL: 4.611\n \t Val. Loss: 2.132 | Val. PPL: 8.430\n Epoch: 08 | Time: 0m 19s\n \tTrain Loss: 1.372 | Train PPL: 3.942\n \t Val. Loss: 2.145 | Val. PPL: 8.543\n Epoch: 09 | Time: 0m 19s\n \tTrain Loss: 1.240 | Train PPL: 3.455\n \t Val. Loss: 2.164 | Val. PPL: 8.702\n Epoch: 10 | Time: 0m 19s\n \tTrain Loss: 1.124 | Train PPL: 3.076\n \t Val. Loss: 2.189 | Val. PPL: 8.930\n\n\n\n```python\ntransformer.load_state_dict(torch.load(MODEL_CHECKPOINT))\ntest_loss = evaluate(transformer, dataloader_test, criterion)\nprint(f'| Test Loss: {test_loss:.3f} | Test PPL: {math.exp(test_loss):7.3f} |')\n```\n\n | Test Loss: 2.145 | Test PPL: 8.545 |\n\n\n\n```python\ndef transformer_predict(source, model, source_tokenizer, target_tokenizer):\n src_indices = [source_init_idx] + source_tokenizer.encode(source).ids + [source_eos_idx]\n src_tensor = torch.LongTensor(src_indices).unsqueeze(0).to(device)\n trg_indices = model.predict(src_tensor)\n return target_tokenizer.decode(trg_indices)\n```\n\n\n```python\ntranslation = dataset_dict['train'][0]\nsource_raw = translation[source_lang]\ntarget_raw = translation[target_lang]\nprint('source: ', source_raw)\nprint('target: ', target_raw)\n\ntransformer_predict(source_raw, transformer, source_tokenizer, target_tokenizer)\n```\n\n source: Zwei junge wei\u00dfe M\u00e4nner sind im Freien in der N\u00e4he vieler B\u00fcsche.\n target: Two young, White males are outside near many bushes.\n\n\n\n\n\n 'two young men are outside near many white bushes.'\n\n\n\nIn this notebook, we worked through the nitty, gritty details of Transformer models. While NLP is the domain in which the Transformer architecture has been originally proposed for, and we've also used a machine translation example to illustrate the use case. This model is also being studied in other fields such as [computer vision](https://arxiv.org/abs/2010.11929).\n\n# Reference\n\n- [Jupyter Notebook: Attention is All You Need](https://nbviewer.jupyter.org/github/bentrevett/pytorch-seq2seq/blob/master/6%20-%20Attention%20is%20All%20You%20Need.ipynb)\n- [Jupyter Notebook: Tutorial 6: Transformers and Multi-Head Attention](https://nbviewer.jupyter.org/github/phlippe/uvadlc_notebooks/blob/master/docs/tutorial_notebooks/tutorial6/Transformers_and_MHAttention.ipynb)\n- [Colab: Simple PyTorch Transformer Example with Greedy Decoding](https://colab.research.google.com/drive/1swXWW5sOLW8zSZBaQBYcGQkQ_Bje_bmI)\n- [Blog: Transformers from scratch](http://peterbloem.nl/blog/transformers)\n- [Blog: Making Pytorch Transformer Twice as Fast on Sequence Generation](https://scale.com/blog/pytorch-improvements)\n- [Blog: How Transformers work in deep learning and NLP: an intuitive introduction](https://theaisummer.com/transformer/)\n- [PyTorch Documentation: Sequence to sequence modeling with nn.Transformer and Torchtext](https://pytorch.org/tutorials/beginner/transformer_tutorial.html)\n- [Paper: Vaswani, Ashish and Shazeer, Noam and Parmar, Niki and Uszkoreit, Jakob and Jones, Llion and Gomez, Aidan N. and Kaiser, \u0141ukasz and Polosukhin, Illia (2017) - Attention is All you Need](https://arxiv.org/abs/1706.03762)\n", "meta": {"hexsha": "45c1b7335530d463e2aee144c1a58a4c29d6f975", "size": 122869, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "deep_learning/seq2seq/torch_transformer.ipynb", "max_stars_repo_name": "ethen8181/machine-learning", "max_stars_repo_head_hexsha": "bc1584d26a4732240056f12f7fa9adaad4f8bc0f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2104, "max_stars_repo_stars_event_min_datetime": "2016-04-15T13:35:55.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-28T10:39:51.000Z", "max_issues_repo_path": "deep_learning/seq2seq/torch_transformer.ipynb", "max_issues_repo_name": "aniruddhachoudhury/machine-learning", "max_issues_repo_head_hexsha": "02744a34709fe908c169aefdcd48dc3f528991f0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 10, "max_issues_repo_issues_event_min_datetime": "2017-04-07T14:25:23.000Z", "max_issues_repo_issues_event_max_datetime": "2021-05-18T03:16:15.000Z", "max_forks_repo_path": "deep_learning/seq2seq/torch_transformer.ipynb", "max_forks_repo_name": "aniruddhachoudhury/machine-learning", "max_forks_repo_head_hexsha": "02744a34709fe908c169aefdcd48dc3f528991f0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 539, "max_forks_repo_forks_event_min_datetime": "2015-12-10T04:23:44.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-31T07:15:28.000Z", "avg_line_length": 37.1093325279, "max_line_length": 2361, "alphanum_fraction": 0.5498701869, "converted": true, "num_tokens": 21202, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.6959583187272711, "lm_q1q2_score": 0.41509260970754525}} {"text": "Code for generating results for the semi-synthetic IHDP data (Figure 4(a) and Figure 4(b)).\n\n\n```python\nimport numpy\nimport sympy\nimport pandas\nimport numpy as np\nimport pandas as pd\nimport sympy as sp\nimport datetime\nimport copy\nimport attr\nimport time\nimport logging\nimport itertools\nimport pickle\nimport sys\nimport os\nimport functools\n\nimport matplotlib.pyplot as plt\nfrom matplotlib import rcParams\nimport collections\n\nfrom scipy.optimize import minimize\nfrom scipy.optimize import LinearConstraint\nfrom scipy import interpolate\nimport warnings\nfrom IPython.display import clear_output\n\nHOME_DIR = \"\"\nwarnings.simplefilter(action='ignore', category=UserWarning)\n```\n\n\n```python\nclass ModelParams:\n\n def __init__(self, beta, a1, a2, y):\n self.beta = beta\n self.a1 = a1\n self.a2 = a2\n self.y = y\n \n def get_true_param(self):\n return self.beta\n \n def __str__(self):\n return \"b=%f, a1=%f, a2=%f, y=%f\" % (self.beta, self.a1, self.a2, self.y)\n\ntruth = ModelParams(beta=1, a1=1, a2=.1, y=1)\n```\n\n\n```python\ndf_ihdp = pd.read_csv(os.path.join(HOME_DIR, \"data/ihdp_processed.csv\"))\nprint(df_ihdp.shape)\ndf_ihdp.head()\n```\n\n (747, 4)\n\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        treatbwcigbw_std
                                        0115590-0.528957
                                        1010001-1.738109
                                        2014300-0.807992
                                        30198410.390344
                                        4013201-1.045929
                                        \n
                                        \n\n\n\n\n```python\ndef generate_data_samples(num_samples, model, k=[1/3, 1/3]):\n np = numpy\n pd = pandas\n \n u_y = np.random.normal(scale=np.sqrt(model.y), size=(num_samples,))\n df_idxs = np.argmax(np.random.multinomial(1, np.ones((df_ihdp.shape[0],)) / df_ihdp.shape[0],\n size=(num_samples,)), axis=-1)\n df = df_ihdp.iloc[df_idxs]\n\n def get_W1():\n return df[\"bw_std\"].values\n \n def get_W2():\n return df[\"cig\"].values\n \n def get_X():\n return df[\"treat\"].values\n\n def get_Y(X, W1, W2, u):\n return model.beta * X + model.a1 * W1 + model.a2 * W2 + u\n \n W1 = get_W1()\n W2 = get_W2()\n X = get_X()\n Y = get_Y(X, W1, W2, u_y)\n SEL = np.zeros_like(Y, np.int32)\n\n SELS = [np.zeros_like(Y, np.int32) for _ in k]\n k_sum = 0\n for i, k_i in enumerate(k):\n prev_k_sum = k_sum\n k_sum += k_i\n SELS[i][int(prev_k_sum*Y.shape[0]) : int(k_sum*Y.shape[0])] = 1\n\n df = pd.DataFrame({\n \"W1\": W1,\n \"W2\": W2,\n \"X\": X,\n \"Y\": Y,\n })\n for i in range(len(k)):\n df[\"SEL%d\" % (i+1)] = SELS[i]\n return df\n```\n\n\n```python\ndf = generate_data_samples(num_samples=10000, model=truth, k=[1/3, 1/3])\ndf.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        W1W2XYSEL1SEL2
                                        0-1.04592900-0.75976710
                                        1-1.45691100-0.31330710
                                        20.424953010.30976810
                                        3-0.981037100.10169710
                                        4-0.48353301-0.07693810
                                        \n
                                        \n\n\n\n\n```python\ndef get_fraction_cost(cost=[1, 1, 1], k=[1/3, 1/3]):\n total = 0\n for c, k_val in zip(cost, k):\n total += c*k_val\n total += cost[-1]*(1 - sum(k))\n return total\n```\n\n\n```python\nclass GMMEqs:\n\n def __init__(self):\n self.moment_eqs, self.jacobian_eqs, self.moment_reweighting = self._get_equations()\n \n def get(self):\n return self.moment_eqs, self.jacobian_eqs, self.moment_reweighting\n \n def _get_equations(self):\n sp = sympy\n\n df_symbols = sp.symbols('ns1, ns2, s3, S1, S2, W1, W2, X, Y')\n ns1, ns2, s3, S1, S2, W1, W2, X, Y = df_symbols\n params = sp.symbols(\"b, a1, a2, t1, t2, d, w1, w2, y\")\n b, a1, a2, t1, t2, d, w1, w2, y = params\n\n m_wts = [ns2, ns2, ns1, ns1, s3, ns2, ns1, ns2, ns1, ns2, ns1]\n moment_reweighting = sp.zeros(len(m_wts), len(m_wts))\n for i, mw1 in enumerate(m_wts):\n for j, mw2 in enumerate(m_wts):\n if mw1 == 1 or mw2 == 1:\n moment_reweighting[i, j] = mw1 * mw2\n elif mw1 == mw2:\n moment_reweighting[i, j] = mw1\n\n m1 = m_wts[0] * (1-S2) * (W1 * (Y - a1*W1 - b*X) - a2*d)\n m2 = m_wts[1] * (1-S2) * (X * (Y - a1*W1 - b*X) - a2*t2)\n m3 = m_wts[2] * (1-S1) * (W2 * (Y - a2*W2 - b*X) - a1*d)\n m4 = m_wts[3] * (1-S1) * (X * (Y - a2*W2 - b*X) - a1*t1)\n m5 = m_wts[4] * (1-S1-S2) * (W1*W2 - d)\n m6 = m_wts[5] * (1-S2) * (X*W1 - t1)\n m7 = m_wts[6] * (1-S1) * (X*W2 - t2)\n m8 = m_wts[7] * (1-S2) * (W1**2 - w1)\n m9 = m_wts[8] * (1-S1) * (W2**2 - w2)\n m10 = m_wts[9] * (1-S2) * ((Y - a1*W1 - b*X)**2 - a2**2 * w2 - y)\n m11 = m_wts[10] * (1-S1) * ((Y - a2*W2 - b*X)**2 - a1**2 * w1 - y)\n\n all_symbols = df_symbols + params\n moments = [m1, m2, m3, m4, m5, m6, m7, m8, m9, m10, m11]\n jacobian = []\n for mom in moments:\n jac_row = []\n for p in params:\n eq = sp.simplify(sp.diff(mom, p))\n jac_row.append(sp.lambdify(all_symbols, eq, \"numpy\"))\n jacobian.append(jac_row)\n \n moments = [sp.lambdify(all_symbols, eq, \"numpy\") for eq in moments]\n moment_reweighting = sp.lambdify((ns1, ns2, s3), moment_reweighting, \"numpy\")\n return moments, jacobian, moment_reweighting\n\nclass GMM:\n\n def __init__(self, df, gmm_eqs):\n np = numpy\n\n # self.df = df\n self.df_dict = {k: df[k].values for k in df.columns}\n self.df = self.df_dict\n self.moment_eqs, self.jacobian_eqs, self.moment_reweighting = gmm_eqs.get()\n self.num_samples = len(df)\n self.num_moments = len(self.moment_eqs)\n self.momconds_arr = np.zeros((self.num_moments, self.num_samples))\n \n def momconds(self, params):\n \"\"\"Returns the emprical moment vector (equivalent of \\hat{g}_n(\\theta)).\"\"\"\n np = numpy\n\n n = self.num_samples\n return np.sum(self._momconds_arr(params), axis=-1) / n\n \n def _compute_moment_covariance(self, params):\n \"\"\"Estimate the optimal weight matrix using the emprical moments.\"\"\"\n n = self.num_samples\n moms = self._momconds_arr(params)\n moment_covariance = (moms @ moms.T) / n\n return moment_covariance\n \n def _get_objective_fn(self, weight_matrix_inv):\n \"\"\"Returns the GMM objective function.\"\"\"\n np = numpy\n \n w_lambda = 0.001\n weight_matrix_inv += weight_matrix_inv + w_lambda * np.eye(weight_matrix_inv.shape[0]) \n\n def objective(params):\n moms = self.momconds(params)\n\n w_inv_mom = np.linalg.solve(weight_matrix_inv, moms)\n obj = moms.T @ w_inv_mom\n\n return obj\n \n return objective\n\n def _momconds_arr(self, params):\n np = numpy\n\n p = params\n df = self.df_dict\n n = self.num_samples\n\n for i, eq in enumerate(self.moment_eqs):\n self.momconds_arr[i, :] = eq(S1=df[\"SEL1\"], S2=df[\"SEL2\"],\n W1=df[\"W1\"], W2=df[\"W2\"],\n X=df[\"X\"], Y=df[\"Y\"],\n ns1=1, ns2=1, s3=1,\n b=p[0], a1=p[1], a2=p[2], t1=p[3], t2=p[4], d=p[5], w1=p[6], w2=p[7], y=p[8])\n \n return self.momconds_arr\n\n def _get_asymptotic_variance_matrix(self, k, moment_covariance, params):\n np = numpy\n\n current_k = [np.mean(self.df[\"SEL1\"]), np.mean(self.df[\"SEL2\"])]\n\n p = params\n df = self.df\n n = self.num_samples\n\n jacobian = np.zeros((self.num_moments, len(p)))\n\n ns1_rw = (1 - k[0]) / (1 - current_k[0])\n ns2_rw = (1 - k[1]) / (1 - current_k[1])\n s3_rw = (1-k[0]-k[1])/(1-current_k[0]-current_k[1])\n\n for i, jac_row in enumerate(self.jacobian_eqs):\n for j, eq in enumerate(jac_row):\n jacobian[i, j] = np.sum(eq(S1=df[\"SEL1\"], S2=df[\"SEL2\"],\n W1=df[\"W1\"], W2=df[\"W2\"],\n X=df[\"X\"], Y=df[\"Y\"],\n ns1=ns1_rw, ns2=ns2_rw, s3=s3_rw,\n b=p[0], a1=p[1], a2=p[2], t1=p[3], t2=p[4],\n d=p[5], w1=p[6], w2=p[7], y=p[8])) / n\n\n moment_covariance_reweight = self.moment_reweighting(ns1=ns1_rw, ns2=ns2_rw, s3=s3_rw,)\n \n moment_covariance = moment_covariance * moment_covariance_reweight\n return np.linalg.inv(jacobian.T @ np.linalg.inv(moment_covariance) @ jacobian)\n \n def _get_asymptotic_variance(self, k, moment_covariance, params):\n return self._get_asymptotic_variance_matrix(k, moment_covariance, params)[0][0]\n \n def _optimize_find_parameters(self, weight_matrix_inv, initial_guess=None):\n np = numpy\n\n initial_guess = np.array([1.1, 1, 1, 1, 1, 1, 1, 1, 1]) if initial_guess is None else initial_guess\n res = minimize(self._get_objective_fn(weight_matrix_inv), initial_guess,\n bounds=[\n (-np.inf, np.inf),\n (-np.inf, np.inf),\n (-np.inf, np.inf),\n (-np.inf, np.inf),\n (-np.inf, np.inf),\n (-np.inf, np.inf),\n (0.1, np.inf),\n (0.1, np.inf),\n (0.1, np.inf),\n ],\n )\n return res.x\n \n def find_parameters(self, num_iters=2, weight_matrix_reg=None):\n np = numpy\n\n weight_matrix_inv = np.eye(self.num_moments)\n for i in range(num_iters):\n params = self._optimize_find_parameters(weight_matrix_inv, initial_guess=params if i > 0 else None)\n weight_matrix_inv = self._compute_moment_covariance(params)\n\n if weight_matrix_reg is not None:\n weight_matrix_inv += weight_matrix_reg * np.eye(weight_matrix_inv.shape[0])\n \n return params, weight_matrix_inv\n\n def find_optimal_k(self, moment_covariance, params, cost=[1, 1, 1]):\n np = numpy\n \n initial_guess = np.array([1/3, 1/3])\n lower_bound = 0.05\n upper_bound = 0.95\n\n if np.sum(cost) == len(cost):\n objective = lambda x: self._get_asymptotic_variance(x, moment_covariance, params)\n else:\n objective = lambda x: self._get_asymptotic_variance(x, moment_covariance, params) * (get_fraction_cost(cost, x))\n res = minimize(objective,\n initial_guess,\n constraints=[LinearConstraint([1, 1], [1-upper_bound], [1-lower_bound]),\n LinearConstraint([1, 0], [lower_bound], [upper_bound]),\n LinearConstraint([0, 1], [lower_bound], [upper_bound])],\n )\n \n if res.x[0] == lower_bound and res.x[1] == lower_bound:\n return np.array([0, 0])\n \n return res.x\n```\n\n\n```python\ngmm = GMM(df, GMMEqs())\nparams, moment_covariance = gmm.find_parameters(num_iters=2)\n\nprint(\"True params:\")\nprint(truth)\nprint(\"Estimated params:\")\nprint(params)\n\noptimal_k = gmm.find_optimal_k(moment_covariance, params)\nprint(\"Estimated optimal selection ratio k: %s\" % str(optimal_k))\n\ndel gmm, params, moment_covariance, optimal_k\n```\n\n True params:\n b=1.000000, a1=1.000000, a2=0.100000, y=1.000000\n Estimated params:\n [ 0.96694478 1.00800977 0.19191733 0.04045318 0.07599427 -0.06614089\n 1.01743196 0.36492049 0.99095177]\n Estimated optimal selection ratio k: [0.05 0.05]\n\n\n\n```python\nclass SampleRevealer:\n\n def __init__(self, budget, cost, df):\n self.initial_budget = budget\n self.budget = budget\n self.cost = cost\n self.buffer_size = len(df)\n self.df = df\n self.counter = 0\n\n def reset(self):\n self.counter = 0\n self.budget = self.initial_budget\n \n def is_budget_left(self, samples_to_reveal, k):\n return self.budget >= samples_to_reveal * get_fraction_cost(self.cost, k)\n \n def reveal(self, reveal_k, samples_to_reveal):\n if self.counter + samples_to_reveal >= self.buffer_size:\n raise ValueError(\"no buffer\")\n \n if not self.is_budget_left(samples_to_reveal, reveal_k):\n raise ValueError(\"no budget left\")\n \n observe_count_1 = int(reveal_k[0] * samples_to_reveal)\n observe_count_2 = int(reveal_k[1] * samples_to_reveal)\n self.df[\"SEL1\"][self.counter:self.counter+samples_to_reveal].values[:] = 0\n self.df[\"SEL2\"][self.counter:self.counter+samples_to_reveal].values[:] = 0\n self.df[\"SEL1\"].values[self.counter:self.counter+observe_count_1] = 1\n self.df[\"SEL2\"].values[self.counter+observe_count_1:self.counter+observe_count_1+observe_count_2] = 1\n\n self.counter += samples_to_reveal\n self.budget -= (samples_to_reveal * get_fraction_cost(self.cost, reveal_k))\n \n def reveal_with_budget(self, reveal_k, budget_to_reveal):\n samples_to_reveal = int(budget_to_reveal / get_fraction_cost(self.cost, reveal_k))\n self.reveal(reveal_k, samples_to_reveal)\n \n def get_dataset(self):\n return self.df[:self.counter]\n```\n\n\n```python\ndef batch_fractions_to_sizes(horizon, batch_fractions):\n batch_sizes = []\n \n for i in range(len(batch_fractions) + 1):\n prev_batch_end = 0 if i == 0 else int(horizon * batch_fractions[i - 1])\n curr_batch_end = horizon if i == len(batch_fractions) else int(horizon * batch_fractions[i])\n\n batch_sizes.append(curr_batch_end - prev_batch_end)\n \n return batch_sizes\n```\n\n\n```python\ndef k_to_simplex(k):\n np = numpy\n\n k_simplex = np.zeros((len(k) + 1))\n k_simplex[:-1] = k\n k_simplex[-1] = 1 - np.sum(k)\n return k_simplex\n \ndef compute_reveal_k(current_samples, samples_to_reveal, current_k, target_k):\n np = numpy\n\n current_k = np.array(current_k)\n target_k = np.array(target_k)\n\n target_simplex = k_to_simplex(target_k)\n\n def objective(k):\n k_final = (current_k * current_samples + k * samples_to_reveal) / (current_samples + samples_to_reveal)\n diff = k_to_simplex(k_final) - target_simplex\n return np.dot(diff, diff)\n \n lower_bound = 0.01\n upper_bound = 0.99\n res = minimize(objective,\n current_k,\n constraints=[LinearConstraint([1, 1], [0], [1]),\n LinearConstraint([1, 0], [lower_bound], [upper_bound]),\n LinearConstraint([0, 1], [lower_bound], [upper_bound])],\n )\n\n return res.x\n\ndef compute_reveal_k_with_budget(current_samples, budget_to_reveal, cost, current_k, target_k):\n np = numpy\n\n np = numpy\n\n current_k = np.array(current_k)\n target_k = np.array(target_k)\n\n target_simplex = k_to_simplex(target_k)\n\n def objective(k):\n samples_to_reveal = int(budget_to_reveal / get_fraction_cost(cost, k))\n k_final = (current_k * current_samples + k * samples_to_reveal) / (current_samples + samples_to_reveal)\n diff = k_to_simplex(k_final) - target_simplex\n return np.dot(diff, diff)\n\n lower_bound = 0.01\n upper_bound = 0.99\n res = minimize(objective,\n current_k,\n constraints=[LinearConstraint([1, 1], [0], [1]),\n LinearConstraint([1, 0], [lower_bound], [upper_bound]),\n LinearConstraint([0, 1], [lower_bound], [upper_bound])],\n )\n\n return res.x\n```\n\n\n```python\nclass StrategyRunResult:\n\n def __init__(self):\n self.budgets_used = []\n self.squared_errors = []\n self.optimal_ks = []\n self.current_ks = []\n \n def append(self, budget_used, squared_error, optimal_k, current_k):\n self.budgets_used.append(budget_used)\n self.squared_errors.append(squared_error)\n self.optimal_ks.append(optimal_k)\n self.current_ks.append(current_k)\n```\n\n\n```python\nclass OracleStrategy:\n\n def __init__(self, sample_revealer, gmm_equations, optimal_k, cost, budget, batch_fractions):\n self.sample_revealer = sample_revealer\n self.name = 'oracle'\n self.gmm_equations = gmm_equations\n\n # Batches are budget fraction allocations.\n self.batch_sizes = batch_fractions_to_sizes(budget, batch_fractions)\n self.optimal_k = optimal_k\n self.cost = cost\n \n def get_current_df_vals(self):\n np = numpy\n\n dataset = self.sample_revealer.get_dataset()\n\n sel1 = np.sum(dataset[\"SEL1\"])\n sel2 = np.sum(dataset[\"SEL2\"])\n total = len(dataset)\n return np.array([sel1/total, sel2/total]), total\n\n def can_step(self):\n return self.sample_revealer.is_budget_left()\n \n def get_and_store_params(self):\n dataset = self.sample_revealer.get_dataset()\n self.gmm = GMM(dataset, self.gmm_equations)\n self.params, self.moment_covariance = self.gmm.find_parameters(num_iters=2)\n return self.params\n\n def get_squared_error(self):\n error = (truth.get_true_param() - self.params[0])**2\n return error\n \n def execute_run(self, cost=1):\n np = numpy\n\n result = StrategyRunResult()\n for i, batch_size in enumerate(self.batch_sizes):\n if i == 0:\n current_k, current_samples = np.array([0, 0]), 0\n else:\n current_k, current_samples = self.get_current_df_vals()\n\n reveal_k = compute_reveal_k_with_budget(current_samples, batch_size, self.cost, current_k, self.optimal_k)\n\n self.sample_revealer.reveal_with_budget(reveal_k=reveal_k, budget_to_reveal=batch_size)\n _ = self.get_and_store_params()\n\n current_k, _ = self.get_current_df_vals()\n\n result.append(\n self.sample_revealer.initial_budget - self.sample_revealer.budget,\n self.get_squared_error(), self.optimal_k, current_k,\n )\n \n return result\n```\n\n\n```python\nclass ETCStrategy:\n\n def __init__(self, sample_revealer, gmm_equations, cost, budget, batch_fractions):\n self.sample_revealer = sample_revealer\n self.name = 'etc'\n self.gmm_equations = gmm_equations\n # Batch sizes represent fractions of budget allocations.\n # We assume that the first batch size is exploration.\n self.batch_sizes = batch_fractions_to_sizes(budget, batch_fractions)\n self.cost = cost\n \n def get_current_df_vals(self):\n np = numpy\n\n dataset = self.sample_revealer.get_dataset()\n\n sel1 = np.sum(dataset[\"SEL1\"])\n sel2 = np.sum(dataset[\"SEL2\"])\n total = len(dataset)\n return np.array([sel1/total, sel2/total]), total\n\n def can_step(self):\n return self.sample_revealer.is_budget_left()\n \n def get_and_store_params(self):\n dataset = self.sample_revealer.get_dataset()\n self.gmm = GMM(dataset, self.gmm_equations)\n self.params, self.moment_covariance = self.gmm.find_parameters(num_iters=2)\n return self.params\n\n def get_squared_error(self):\n error = (truth.get_true_param() - self.params[0])**2\n return error\n \n def execute_run(self, cost=1):\n np = numpy\n\n result = StrategyRunResult()\n for i, batch_size in enumerate(self.batch_sizes):\n if i == 0:\n reveal_k = [0, 0]\n else:\n current_k, current_samples = self.get_current_df_vals()\n reveal_k = compute_reveal_k_with_budget(current_samples, batch_size,\n self.cost, current_k, self.optimal_k)\n\n self.sample_revealer.reveal_with_budget(reveal_k=reveal_k, budget_to_reveal=batch_size)\n params = self.get_and_store_params()\n\n if i == 0:\n df = self.sample_revealer.get_dataset()\n if np.mean(df[\"X\"]) <= 0.1:\n return None\n \n self.optimal_k = self.gmm.find_optimal_k(self.moment_covariance, params, cost=cost)\n\n current_k, _ = self.get_current_df_vals()\n\n result.append(\n self.sample_revealer.initial_budget - self.sample_revealer.budget,\n self.get_squared_error(), self.optimal_k, current_k,\n )\n \n return result\n```\n\n\n```python\nclass ETGreedyFBStrategy:\n\n def __init__(self, sample_revealer, gmm_equations, cost, budget, batch_fractions):\n self.sample_revealer = sample_revealer\n self.name = 'etg-fb'\n self.gmm_equations = gmm_equations\n # We assume that the first batch size is exploration.\n self.batch_sizes = batch_fractions_to_sizes(budget, batch_fractions)\n self.cost = cost\n self.weight_matrix_reg = 0.01\n \n def get_current_df_vals(self):\n np = numpy\n\n dataset = self.sample_revealer.get_dataset()\n\n sel1 = np.sum(dataset[\"SEL1\"])\n sel2 = np.sum(dataset[\"SEL2\"])\n total = len(dataset)\n return np.array([sel1/total, sel2/total]), total\n\n def can_step(self):\n return self.sample_revealer.is_budget_left()\n \n def get_and_store_params(self, is_last_step):\n dataset = self.sample_revealer.get_dataset()\n self.gmm = GMM(dataset, self.gmm_equations)\n if is_last_step:\n self.params, self.moment_covariance = self.gmm.find_parameters(num_iters=2)\n else:\n self.params, self.moment_covariance = (\n self.gmm.find_parameters(num_iters=2, weight_matrix_reg=self.weight_matrix_reg)\n )\n return self.params\n\n def get_squared_error(self):\n error = (truth.get_true_param() - self.params[0])**2\n return error\n \n def execute_run(self, cost=1):\n np = numpy\n\n result = StrategyRunResult()\n for i, batch_size in enumerate(self.batch_sizes):\n if i == 0:\n reveal_k = [0, 0]\n else:\n current_k, current_samples = self.get_current_df_vals()\n reveal_k = compute_reveal_k_with_budget(current_samples, batch_size, self.cost, current_k, self.optimal_k)\n \n self.sample_revealer.reveal_with_budget(reveal_k=reveal_k, budget_to_reveal=batch_size)\n if i == 0:\n df = self.sample_revealer.get_dataset()\n if np.mean(df[\"X\"]) <= 0.1:\n return None\n\n is_last_step = (i==len(self.batch_sizes)-1)\n params = self.get_and_store_params(is_last_step)\n\n self.optimal_k = self.gmm.find_optimal_k(self.moment_covariance, params, cost=cost)\n\n current_k, _ = self.get_current_df_vals()\n\n result.append(\n self.sample_revealer.initial_budget - self.sample_revealer.budget,\n self.get_squared_error(), self.optimal_k, current_k,\n )\n \n return result\n```\n\n\n```python\nclass ETGreedyFSStrategy:\n\n def __init__(self, sample_revealer, gmm_equations, cost, budget, batch_fractions):\n np = numpy\n\n self.sample_revealer = sample_revealer\n self.name = 'etg-fs'\n horizon_min = int(budget / np.max(cost))\n # We assume that the first batch size is exploration.\n self.batch_sizes = batch_fractions_to_sizes(horizon_min, batch_fractions)[:-1]\n self.cost = cost\n self.gmm_equations = gmm_equations\n self.weight_matrix_reg = 0.01\n \n def get_current_df_vals(self):\n np = numpy\n\n dataset = self.sample_revealer.get_dataset()\n\n sel1 = np.sum(dataset[\"SEL1\"])\n sel2 = np.sum(dataset[\"SEL2\"])\n total = len(dataset)\n return np.array([sel1/total, sel2/total]), total\n\n def can_step(self):\n return self.sample_revealer.is_budget_left()\n \n def get_and_store_params(self, is_last_step):\n dataset = self.sample_revealer.get_dataset()\n self.gmm = GMM(dataset, self.gmm_equations)\n if is_last_step:\n self.params, self.moment_covariance = self.gmm.find_parameters(num_iters=2)\n else:\n self.params, self.moment_covariance = (\n self.gmm.find_parameters(num_iters=2, weight_matrix_reg=self.weight_matrix_reg)\n )\n return self.params\n\n def get_squared_error(self):\n error = (truth.get_true_param() - self.params[0])**2\n return error\n \n def execute_run(self, cost=1):\n np = numpy\n\n result = StrategyRunResult()\n for i, batch_size in enumerate(self.batch_sizes):\n if i == 0:\n reveal_k = [0, 0]\n else:\n current_k, current_samples = self.get_current_df_vals()\n reveal_k = compute_reveal_k(current_samples, batch_size, current_k, self.optimal_k)\n\n self.sample_revealer.reveal(reveal_k=reveal_k, samples_to_reveal=batch_size)\n if i == 0:\n df = self.sample_revealer.get_dataset()\n if np.mean(df[\"X\"]) <= 0.1:\n return None\n\n params = self.get_and_store_params(is_last_step=False)\n\n self.optimal_k = self.gmm.find_optimal_k(self.moment_covariance, params, cost=cost)\n\n current_k, current_samples = self.get_current_df_vals()\n\n result.append(\n self.sample_revealer.initial_budget - self.sample_revealer.budget,\n self.get_squared_error(), self.optimal_k, current_k,\n )\n \n last_step_reached = False\n batch_size = self.batch_sizes[-1]\n while not last_step_reached: \n current_k, current_samples = self.get_current_df_vals()\n reveal_k = compute_reveal_k(current_samples, batch_size, current_k, self.optimal_k)\n\n if self.sample_revealer.is_budget_left(samples_to_reveal=batch_size, k=reveal_k):\n self.sample_revealer.reveal(reveal_k=reveal_k, samples_to_reveal=batch_size)\n else:\n budget_left = self.sample_revealer.budget\n reveal_k = compute_reveal_k_with_budget(current_samples, budget_left, self.cost, current_k, self.optimal_k)\n self.sample_revealer.reveal_with_budget(reveal_k=reveal_k, budget_to_reveal=budget_left)\n last_step_reached = True\n \n params = self.get_and_store_params(last_step_reached)\n self.optimal_k = self.gmm.find_optimal_k(self.moment_covariance, params, cost=cost)\n\n current_k, current_samples = self.get_current_df_vals()\n\n result.append(\n self.sample_revealer.initial_budget - self.sample_revealer.budget,\n self.get_squared_error(), self.optimal_k, current_k,\n )\n \n return result\n```\n\n\n```python\ndef execute_strategy_iteration(strategy_name, iteration_num, budget):\n np = numpy\n\n cost = [1, 3, 3.5]\n\n gmm_equations = GMMEqs()\n\n # Uncomment the next two lines to replicate the exact runs used in the paper.\n # random_seed = 232281293 + iteration_num\n # np.random.seed(random_seed)\n df = generate_data_samples(num_samples=budget * 5, model=truth)\n np.random.seed(None)\n\n sample_revealer = SampleRevealer(budget=budget, cost=cost, df=df)\n\n if strategy_name == \"oracle\":\n strategy = OracleStrategy(sample_revealer, gmm_equations=gmm_equations,\n budget=budget, cost=cost, optimal_k=[0.59408789, 0],\n batch_fractions=[0.1, 0.9])\n elif strategy_name == \"collect_all\":\n strategy = OracleStrategy(sample_revealer, gmm_equations=gmm_equations,\n budget=budget, cost=cost, optimal_k=[0, 0],\n batch_fractions=[0.1, 0.2, 0.5, 0.9])\n elif strategy_name == \"etc_0.2\":\n strategy = ETCStrategy(sample_revealer, gmm_equations=gmm_equations,\n budget=budget, cost=cost,\n batch_fractions=[0.2, 0.9])\n elif strategy_name == \"etc_0.4\":\n strategy = ETCStrategy(sample_revealer, gmm_equations=gmm_equations,\n budget=budget, cost=cost,\n batch_fractions=[0.4, 0.9])\n elif strategy_name == \"etg-fb_0.4\":\n strategy = ETGreedyFBStrategy(sample_revealer, gmm_equations=gmm_equations,\n budget=budget, cost=cost,\n batch_fractions=[0.4, 0.5, 0.6, 0.7, 0.8])\n elif strategy_name == \"etg-fs_0.4\":\n strategy = ETGreedyFSStrategy(sample_revealer, gmm_equations=gmm_equations,\n budget=budget, cost=cost,\n batch_fractions=[0.4, 0.8])\n else:\n raise ValueError(\"invalid strategy_name: %s\" % strategy_name)\n\n return strategy.execute_run(cost=cost)\n```\n\n\n```python\n# Test a strategy.\n\nresult = execute_strategy_iteration(\"etc_0.4\", 1, budget=1000)\nprint(result.squared_errors)\n```\n\n [0.00555592535273502, 0.002185244501872844, 0.0024872469563182687]\n\n\n**Executing the runs for each strategy in parallel.**\n\nFor the paper, we execute 12,000 runs for each strategy. We execute the runs\nin parallel using\n[`ipyparallel`](https://ipyparallel.readthedocs.io/en/latest/).\n\nThe following command starts the `ipyparallel` engines:\n```\nipcluster start -n \n```\n \n\n\n```python\nimport ipyparallel as ipp\n```\n\n\n```python\n# Verify that ipcluster is running and import the necessary Python packages.\n\nparallel_client = ipp.Client(debug=False)\ndview = parallel_client[:]\n# Execute an identity map in parallel.\nar = dview.map(lambda x: x, (i for i in range(0, 2000000, 2)))\nassert ar.get()[0] == 0\n\n# Import the required Python packages.\nwith dview.sync_imports():\n from abc import ABC, abstractmethod\n import numpy\n import sympy\n import pandas\n import sympy\n import datetime\n import copy\n import attr\n import time\n import logging\n import itertools\n import pickle\n import os\n import functools\n import ipyparallel\n\n import collections\n\n from scipy.optimize import minimize\n from scipy import interpolate\n from scipy.special import expit\n from scipy.optimize import LinearConstraint\n import warnings\n\n try:\n from cPickle import dumps, loads, HIGHEST_PROTOCOL as PICKLE_PROTOCOL\n except ImportError:\n from pickle import dumps, loads, HIGHEST_PROTOCOL as PICKLE_PROTOCOL\n\n# Make sure ipyparallel is still able to execute functions.\ndview = parallel_client[:]\nar = dview.map(lambda x: x, (i for i in range(0, 2000000, 2)))\nassert ar.get()[0] == 0\n```\n\n importing ABC,abstractmethod from abc on engine(s)\n importing numpy on engine(s)\n importing sympy on engine(s)\n importing pandas on engine(s)\n importing datetime on engine(s)\n importing copy on engine(s)\n importing attr on engine(s)\n importing time on engine(s)\n importing logging on engine(s)\n importing itertools on engine(s)\n importing pickle on engine(s)\n importing os on engine(s)\n importing functools on engine(s)\n importing ipyparallel on engine(s)\n importing collections on engine(s)\n importing minimize from scipy.optimize on engine(s)\n importing interpolate from scipy on engine(s)\n importing expit from scipy.special on engine(s)\n importing LinearConstraint from scipy.optimize on engine(s)\n importing warnings on engine(s)\n\n\n\n```python\ndef combine_parallel_results(async_result, need_interpolation=False):\n optimal_ks_res = []\n\n if not need_interpolation:\n budgets = async_result.get()[0].budgets_used\n errors = np.vstack([res.squared_errors for res in async_result.get() if res is not None])\n optimal_ks = np.vstack([[res.optimal_ks] for res in async_result.get() if res is not None])\n current_ks = np.vstack([[res.current_ks] for res in async_result.get() if res is not None])\n\n return budgets, errors, optimal_ks, current_ks\n \n budgets_matrix = [res.budgets_used for res in async_result.get() if res is not None]\n errors_matrix = [res.squared_errors for res in async_result.get() if res is not None]\n optimal_ks_matrix = [res.optimal_ks for res in async_result.get() if res is not None]\n current_ks_matrix = [res.current_ks for res in async_result.get() if res is not None]\n\n def find_budget():\n idx = 0\n target = budgets_matrix[0][-1]\n\n for i in range(1, len(budgets_matrix)):\n if target > budgets_matrix[i][-1]:\n target = budgets_matrix[i][-1]\n idx = i\n \n return np.array(budgets_matrix[idx])\n \n budgets = find_budget()\n errors = []\n optimal_ks = []\n current_ks = []\n\n for i in range(len(budgets_matrix)):\n f_err = interpolate.interp1d(budgets_matrix[i], errors_matrix[i])\n f_optimalk0 = interpolate.interp1d(budgets_matrix[i], np.array(optimal_ks_matrix[i])[:, 0])\n f_optimalk1 = interpolate.interp1d(budgets_matrix[i], np.array(optimal_ks_matrix[i])[:, 1])\n f_currentk0 = interpolate.interp1d(budgets_matrix[i], np.array(current_ks_matrix[i])[:, 0])\n f_currentk1 = interpolate.interp1d(budgets_matrix[i], np.array(current_ks_matrix[i])[:, 1])\n \n errors.append([])\n optimal_ks.append([])\n current_ks.append([])\n\n for b in budgets:\n errors[i].append(f_err(b))\n optimal_ks[i].append([f_optimalk0(b), f_optimalk1(b)])\n current_ks[i].append([f_currentk0(b), f_currentk0(b)])\n \n errors = np.array(errors)\n optimal_ks = np.array(optimal_ks)\n current_ks = np.array(current_ks)\n\n return budgets, errors, optimal_ks, current_ks\n\ndef execute_strategy_in_parallel(strategy_name, horizon, iterations):\n num_threads = len(parallel_client.ids)\n dview = parallel_client[:]\n\n print(\"Executing %s over %d iterations across %d cores\" % (strategy_name, iterations, num_threads))\n\n dview[\"batch_fractions_to_sizes\"] = batch_fractions_to_sizes\n dview[\"compute_reveal_k\"] = compute_reveal_k\n dview[\"compute_reveal_k_with_budget\"] = compute_reveal_k_with_budget\n dview[\"get_fraction_cost\"] = get_fraction_cost\n dview[\"k_to_simplex\"] = k_to_simplex\n dview[\"ModelParams\"] = ModelParams\n dview[\"truth\"] = truth\n dview[\"df_ihdp\"] = df_ihdp\n dview[\"generate_data_samples\"] = generate_data_samples\n dview[\"StrategyRunResult\"] = StrategyRunResult\n dview[\"GMM\"] = GMM\n dview[\"GMMEqs\"] = GMMEqs\n dview[\"OracleStrategy\"] = OracleStrategy\n dview[\"ETCStrategy\"] = ETCStrategy\n dview[\"ETGreedyFBStrategy\"] = ETGreedyFBStrategy\n dview[\"ETGreedyFSStrategy\"] = ETGreedyFSStrategy\n dview[\"SampleRevealer\"] = SampleRevealer\n dview[\"execute_strategy_iteration\"] = execute_strategy_iteration\n\n def execute_iteration(i):\n return execute_strategy_iteration(strategy_name, i, horizon)\n\n return dview.map(execute_iteration, range(iterations))\n```\n\n\n```python\ndef get_timeseries_for(strategy_names, horizons, iterations, results_dict):\n results_dict[\"truth\"] = truth\n\n for strategy_name in strategy_names:\n for horizon in horizons:\n print(\"Timestamp start: %s, Strategy: %s, Horizon: %d, Iters: %d\" % (\n datetime.datetime.now(), strategy_name, horizon, iterations))\n async_result = execute_strategy_in_parallel(strategy_name, horizon, iterations)\n need_interpolation = \"etg-fs\" in strategy_name\n budgets, errors, optimal_ks, current_ks = combine_parallel_results(async_result, need_interpolation)\n print(\"Timestamp end: %s\" % (datetime.datetime.now()))\n\n if strategy_name not in results_dict:\n results_dict[strategy_name] = {}\n \n results_dict[strategy_name][horizon] = {\n \"budgets\": budgets,\n \"errors\": errors * budgets,\n \"optimal_ks\": optimal_ks,\n \"current_ks\": current_ks,\n }\n```\n\n\n```python\nresults_dict = {}\nget_timeseries_for([\n \"oracle\", \"collect_all\",\n \"etc_0.2\", \"etc_0.4\",\n \"etg-fb_0.4\", \"etg-fs_0.4\",\n ],\n horizons=[500, 600, 700, 800, 900, 1000, 1150, 1300],\n iterations=12*1000,\n results_dict=results_dict)\n```\n\n\n```python\n# Save results to file.\n# pickle.dump(results_dict,\n# open(os.path.join(HOME_DIR,\n# \"%s_ihdp_graph.pkl\" % datetime.datetime.now()),\n# \"wb\"))\n```\n\n### Plot the results\n\n\n```python\n# For the color map:\n# https://gist.github.com/AndiH/c957b4d769e628f506bd\n\n# Tableau 20 Colors\ntableau20 = [(31, 119, 180), (174, 199, 232), (255, 127, 14), (255, 187, 120), \n (44, 160, 44), (152, 223, 138), (214, 39, 40), (255, 152, 150), \n (148, 103, 189), (197, 176, 213), (140, 86, 75), (196, 156, 148), \n (227, 119, 194), (247, 182, 210), (127, 127, 127), (199, 199, 199), \n (188, 189, 34), (219, 219, 141), (23, 190, 207), (158, 218, 229)]\n \n# Tableau Color Blind 10\ntableau20blind = [(0, 107, 164), (255, 128, 14), (171, 171, 171), (89, 89, 89),\n (95, 158, 209), (200, 82, 0), (137, 137, 137), (163, 200, 236),\n (255, 188, 121), (207, 207, 207)]\n \n# Rescale to values between 0 and 1 \nfor i in range(len(tableau20)): \n r, g, b = tableau20[i] \n tableau20[i] = (r / 255., g / 255., b / 255.)\nfor i in range(len(tableau20blind)): \n r, g, b = tableau20blind[i] \n tableau20blind[i] = (r / 255., g / 255., b / 255.)\n# Use with plt.plot(\u2026, color=tableau[0],\u2026)\n```\n\n\n```python\ndef plot_regret_curve(results_dict):\n\n clist = rcParams['axes.prop_cycle']\n cgen = itertools.cycle(clist)\n\n oracle_mses = {}\n for horizon, timeseries in results_dict[\"oracle\"].items():\n oracle_mses[timeseries[\"budgets\"][-1]] = np.mean(timeseries[\"errors\"][:, -1])\n\n oracle_budgets = np.array(list(oracle_mses.keys()))\n\n def plot(axs, name, info, result, with_var=False):\n x_vals = []\n y_mean = []\n y_std = []\n\n current_k_mean = []\n optimal_k_mean = []\n\n for horizon, timeseries in result.items():\n x_vals.append(timeseries[\"budgets\"][-1])\n\n closest_budget = oracle_budgets[np.argmin(np.abs(oracle_budgets - x_vals[-1]))]\n y_scaled = ( timeseries[\"errors\"][:, -1] - oracle_mses[closest_budget] ) / oracle_mses[closest_budget] * 100\n y_mean.append(np.mean(y_scaled))\n y_std.append(np.std(y_scaled))\n\n x_vals = np.array(x_vals)\n y_mean = np.array(y_mean)\n y_std = np.array(y_std)\n \n x_sort_idx = np.argsort(x_vals)\n x_vals = x_vals[x_sort_idx]\n y_mean = y_mean[x_sort_idx]\n y_std = y_std[x_sort_idx]\n \n color = next(cgen)[\"color\"]\n\n plt.plot(x_vals, y_mean, label=name, color=info[1], linestyle=info[0], marker=info[2])\n plt.legend()\n\n if with_var:\n ci = 1.96 * y_std / np.sqrt(timeseries[\"errors\"].shape[0])\n axs.errorbar(x_vals, y_mean, yerr=ci, ls=\"none\", color=info[1])\n \n name_to_linestyle_color = {\n \"etc_0.2\": [\"dashdot\", tableau20blind[1], \"s\"],\n \"etc_0.4\": [\"solid\", tableau20blind[2], \"v\"],\n \"etg-fb_0.4\": [\"solid\", tableau20blind[8], \"^\"],\n \"etg-fs_0.4\": [\"dashdot\", tableau20blind[6], \"D\"],\n \"collect_all\": [\"dashed\", tableau20blind[7], \"o\"],\n }\n plt.title(\"Relative regret vs budget\")\n plt.xlabel(\"Total budget \")\n plt.ylabel(\"Relative regret (%)\")\n for name, info in name_to_linestyle_color.items(): \n plot(plt, name, info, results_dict[name], with_var=True)\n \n # plt.savefig(os.path.join(HOME_DIR, \"figures/ihdp_regret_curve.eps\"), bbox_inches='tight', pad_inches=0.0)\n```\n\n\n```python\nSMALL_SIZE = 12\nMEDIUM_SIZE = 12\nBIGGER_SIZE = 12\n\nplt.rc('font', size=SMALL_SIZE) # controls default text sizes\nplt.rc('axes', titlesize=SMALL_SIZE+4) # fontsize of the axes title\nplt.rc('axes', labelsize=MEDIUM_SIZE+4) # fontsize of the x and y labels\nplt.rc('xtick', labelsize=SMALL_SIZE) # fontsize of the tick labels\nplt.rc('ytick', labelsize=SMALL_SIZE) # fontsize of the tick labels\nplt.rc('legend', fontsize=SMALL_SIZE) # legend fontsize\nplt.rc('figure', titlesize=BIGGER_SIZE+20) # fontsize of the figure title\n```\n\n\n```python\nplot_regret_curve(results_dict)\n```\n", "meta": {"hexsha": "56b5b2dabd9b03920c8f1be8d961c9f64517e393", "size": 63968, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "IHDP_data_graph.ipynb", "max_stars_repo_name": "acmi-lab/online-moment-selection", "max_stars_repo_head_hexsha": "1806ca3fd58b92cb61dc01fb54afeb5b4f4564f4", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2021-09-13T07:54:47.000Z", 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YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.4150027651215128}} {"text": "**You may need to install [OpenCV](https://pypi.python.org/pypi/opencv-python) and [scikit-video](http://www.scikit-video.org/stable/).**\n\n\n```python\nimport keras\nimport numpy as np\nimport io\nimport base64\nfrom IPython.display import HTML\nimport skvideo.io\nimport cv2\nimport json\n\nfrom keras.models import Sequential, model_from_json\nfrom keras.layers.core import Dense\nfrom keras.optimizers import sgd\nfrom keras.layers import Conv2D, MaxPooling2D, Activation, AveragePooling2D, \\\n Reshape, BatchNormalization, Flatten\nimport keras.backend as K\nimport tensorflow as tf\n```\n\n Using TensorFlow backend.\n\n\n# MiniProject #3: Deep Reinforcement Learning\n\n__Notations__: $E_p$ is the expectation under probability $p$. Please justify each of your answer and widely comment your code.\n\n# Context\n\nIn a reinforcement learning algorithm, we modelize each step $t$ as an action $a_t$ obtained from a state $s_t$, i.e. $\\{(a_{t},s_{t})_{t\\leq T}\\}$ having the Markov property. We consider a discount factor $\\gamma \\in [0,1]$ that ensures convergence. The goal is to find among all the policies $\\pi$, one that maximizes the expected reward:\n\n\\begin{equation*}\nR(\\pi)=\\sum_{t\\leq T}E_{p^{\\pi}}[\\gamma^t r(s_{t},a_{t})] \\> ,\n\\end{equation*}\n\nwhere: \n\\begin{equation*}p^{\\pi}(a_{0},a_{1},s_{1},...,a_{T},s_{T})=p(a_{0})\\prod_{t=1}^{T}\\pi(a_{t}|s_{t})p(s_{t+1}|s_{t},a_{t}) \\> .\n\\end{equation*}\n\nWe note the $Q$-function:\n\n\\begin{equation*}Q^\\pi(s,a)=E_{p^{\\pi}}[\\sum_{t\\leq T}\\gamma^{t}r(s_{t},a_{t})|s_{0}=s,a_{0}=a] \\> .\n\\end{equation*}\n\nThus, the optimal Q function is:\n\\begin{equation*}\nQ^*(s,a)=\\max_{\\pi}Q^\\pi(s,a) \\> .\n\\end{equation*}\n\nIn this project, we will apply the deep reinforcement learning techniques to a simple game: an agent will have to learn from scratch a policy that will permit it maximizing a reward.\n\n## The environment, the agent and the game\n\n### The environment\n\n```Environment``` is an abstract class that represents the states, rewards, and actions to obtain the new state.\n\n\n```python\nclass Environment(object):\n def __init__(self):\n pass\n\n def act(self, act):\n \"\"\"\n One can act on the environment and obtain its reaction:\n - the new state\n - the reward of the new state\n - should we continue the game?\n\n :return: state, reward, game_over\n \"\"\"\n pass\n\n\n def reset(self):\n \"\"\"\n Reinitialize the environment to a random state and returns\n the original state\n\n :return: state\n \"\"\"\n pass\n \n def draw(self):\n \"\"\"\n Visualize in the console or graphically the current state\n \"\"\"\n pass\n```\n\nThe method ```act``` allows to act on the environment at a given state $s_t$ (stored internally), via action $a_t$. The method will return the new state $s_{t+1}$, the reward $r(s_{t},a_{t})$ and determines if $t\\leq T$ (*game_over*).\n\nThe method ```reset``` simply reinitializes the environment to a random state $s_0$.\n\nThe method ```draw``` displays the current state $s_t$ (this is useful to check the behavior of the Agent).\n\nWe modelize $s_t$ as a tensor, while $a_t$ is an integer.\n\n### The Agent\n\nThe goal of the ```Agent``` is to interact with the ```Environment``` by proposing actions $a_t$ obtained from a given state $s_t$ to attempt to maximize its __reward__ $r(s_t,a_t)$. We propose the following abstract class:\n\n\n```python\nclass Agent(object):\n def __init__(self, epsilon=0.1, n_action=4):\n self.epsilon = epsilon\n self.n_action = n_action\n \n def set_epsilon(self, e):\n self.epsilon = e\n\n def act(self, s, train=True):\n \"\"\" This function should return the next action to do:\n an integer between 0 and 4 (not included) with a random exploration of epsilon\"\"\"\n if train:\n if np.random.rand() <= self.epsilon:\n a = np.random.randint(0, self.n_action, size=1)[0]\n else:\n a = self.learned_act(s)\n else: # in some cases, this can improve the performance.. remove it if poor performances\n a = self.learned_act(s)\n\n return a\n\n def learned_act(self,s):\n \"\"\" Act via the policy of the agent, from a given state s\n it proposes an action a\"\"\"\n pass\n\n def reinforce(self, s, n_s, a, r, game_over_):\n \"\"\" This function is the core of the learning algorithm. \n It takes as an input the current state s_, the next state n_s_\n the action a_ used to move from s_ to n_s_ and the reward r_.\n \n Its goal is to learn a policy.\n \"\"\"\n pass\n\n def save(self):\n \"\"\" This function returns basic stats if applicable: the\n loss and/or the model\"\"\"\n pass\n\n def load(self):\n \"\"\" This function allows to restore a model\"\"\"\n pass\n```\n\n***\n__Question 1__:\nExplain the function act. Why is ```epsilon``` essential?\n\nThe function ***act*** represents basically the policy. Indeed, according to the current state s, act returns the action the ```Agent``` must use. Here, the function ***act*** is divided in two steps. During the training step, the function ***act*** is a trade-off between exploration and exploitation. The goal of epsilon is precisely to explore more state: the ```Agent``` is going to use a random action with probability epsilon. Otherwise, the ```Agent``` uses the learned policy (***learned_act***). Therefore, the goal of epsilon is to encourage the exploration during the training period.\n\n***\n### The Game\n\nThe ```Agent``` and the ```Environment``` work in an interlaced way as in the following (take some time to understand this code as it is the core of the project)\n\n```python\n\nepoch = 300\nenv = Environment()\nagent = Agent()\n\n\n# Number of won games\nscore = 0\nloss = 0\n\n\nfor e in range(epoch):\n # At each epoch, we restart to a fresh game and get the initial state\n state = env.reset()\n # This assumes that the games will end\n game_over = False\n\n win = 0\n lose = 0\n \n while not game_over:\n # The agent performs an action\n action = agent.act(state)\n\n # Apply an action to the environment, get the next state, the reward\n # and if the games end\n prev_state = state\n state, reward, game_over = env.act(action)\n\n # Update the counters\n if reward > 0:\n win = win + reward\n if reward < 0:\n lose = lose -reward\n\n # Apply the reinforcement strategy\n loss = agent.reinforce(prev_state, state, action, reward, game_over)\n\n # Save as a mp4\n if e % 10 == 0:\n env.draw(e)\n\n # Update stats\n score += win-lose\n\n print(\"Epoch {:03d}/{:03d} | Loss {:.4f} | Win/lose count {}/{} ({})\"\n .format(e, epoch, loss, win, lose, win-lose))\n agent.save()\n```\n\n# The game, *eat cheese*\n\nA rat runs on an island and tries to eat as much as possible. The island is subdivided into $N\\times N$ cells, in which there are cheese (+0.5) and poisonous cells (-1). The rat has a visibility of 2 cells (thus it can see $5^2$ cells). The rat is given a time $T$ to accumulate as much food as possible. It can perform 4 actions: going up, down, left, right. \n\nThe goal is to code an agent to solve this task that will learn by trial and error. We propose the following environment:\n\n\n```python\nclass Environment(object):\n def __init__(self, grid_size=10, max_time=500, temperature=0.1):\n grid_size = grid_size + 4\n self.grid_size = grid_size\n self.max_time = max_time\n self.temperature = temperature\n\n #board on which one plays\n self.board = np.zeros((grid_size, grid_size))\n self.position = np.zeros((grid_size, grid_size))\n\n # coordinate of the cat\n self.x = 0\n self.y = 1\n\n # self time\n self.t = 0\n\n self.scale = 16\n\n self.to_draw = np.zeros((max_time + 2, grid_size * self.scale, \n grid_size * self.scale, 3))\n\n\n def draw(self, e):\n skvideo.io.vwrite(str(e) + '.mp4', self.to_draw)\n\n def get_frame(self,t):\n b = np.zeros((self.grid_size, self.grid_size, 3)) + 128\n b[self.board > 0, 0] = 256\n b[self.board < 0, 2] = 256\n b[self.x, self.y, :] = 256\n b[-2:, :, :] = 0\n b[:, -2:, :] = 0\n b[:2, :, :] = 0\n b[:, :2, :] = 0\n \n b = cv2.resize(b, None, fx=self.scale, fy=self.scale, \n interpolation=cv2.INTER_NEAREST)\n\n self.to_draw[t, :, :, :] = b\n\n\n def act(self, action):\n \"\"\"This function returns the new state, reward and decides if the\n game ends.\"\"\"\n\n self.get_frame(int(self.t))\n\n self.position = np.zeros((self.grid_size, self.grid_size))\n\n self.position[0:2, :]= -1\n self.position[:, 0:2] = -1\n self.position[-2:, :] = -1\n self.position[-2:, :] = -1\n\n self.position[self.x, self.y] = 1\n \n if action == 0: # Going up (or down on the image)\n if self.x == self.grid_size - 3:\n self.x = self.x - 1\n else:\n self.x = self.x + 1\n elif action == 1: # Going up on the image\n if self.x == 2:\n self.x = self.x + 1\n else:\n self.x = self.x - 1\n elif action == 2: # Going right on the image\n if self.y == self.grid_size - 3:\n self.y = self.y - 1\n else:\n self.y = self.y + 1\n elif action == 3: # Going left on the image\n if self.y == 2:\n self.y = self.y + 1\n else:\n self.y = self.y - 1\n else:\n RuntimeError('Error: action not recognized')\n\n self.t = self.t + 1\n reward = self.board[self.x, self.y]\n self.board[self.x, self.y] = 0\n game_over = self.t > self.max_time\n state = np.concatenate((self.board.reshape(self.grid_size, self.grid_size, 1),\n self.position.reshape(self.grid_size, self.grid_size, 1)),\n axis=2)\n state = state[self.x-2:self.x+3, self.y-2:self.y+3, :]\n\n return state, reward, game_over\n\n def reset(self):\n \"\"\"This function resets the game and returns the initial state\"\"\"\n\n self.x = np.random.randint(3, self.grid_size-3, size=1)[0]\n self.y = np.random.randint(3, self.grid_size-3, size=1)[0]\n\n bonus = 0.5 * np.random.binomial(1, self.temperature, size=self.grid_size**2)\n bonus = bonus.reshape(self.grid_size, self.grid_size)\n\n malus = -1.0*np.random.binomial(1, self.temperature, size=self.grid_size**2)\n malus = malus.reshape(self.grid_size, self.grid_size)\n\n self.to_draw = np.zeros((self.max_time + 2, self.grid_size * self.scale, \n self.grid_size * self.scale, 3))\n\n malus[bonus > 0] = 0\n\n self.board = bonus + malus\n\n self.position = np.zeros((self.grid_size, self.grid_size))\n self.position[0:2,:]= -1\n self.position[:,0:2] = -1\n self.position[-2:, :] = -1\n self.position[-2:, :] = -1\n self.board[self.x, self.y] = 0\n self.t = 0\n\n state = np.concatenate((self.board.reshape(self.grid_size, self.grid_size,1),\n self.position.reshape(self.grid_size, self.grid_size,1)), \n axis=2)\n\n state = state[self.x - 2:self.x + 3, self.y - 2:self.y + 3, :]\n return state\n```\n\nThe following elements are important because they correspond to the hyper parameters for this project:\n\n\n```python\n# parameters\nsize = 13\nT = 200\ntemperature = 0.3\nepochs_train = 21 # set small when debugging\nepochs_test = 15 # set small when debugging\n\n# display videos\ndef display_videos(name):\n video = io.open(name, 'r+b').read()\n encoded = base64.b64encode(video)\n return ''''''.format(encoded.decode('ascii'))\n```\n\n__Question 2__ Explain the use of the arrays ```position``` and ```board```.\n\n\\begin{itemize}\n \\item The position array represents the position of the rat on the grid. One particular aspect is the border: the 2 first / last rows and columns are considered as walls and cannot be access. Therefore, the value of position for these cases are defined as -1. The position where is the rat is valued at 1.\n \\item The board array represents the different rewards that the rat can obtain for every cases of the grid. Typically, the board value is equal to 0.5 where there is cheese, equal to -1 where there is poison and 0 otherwise.\n\\end{itemize}\n\n\n## Random Agent\n\n***\n__Question 3__ Implement a random Agent (only ```learned_act``` needs to be implemented):\n\n\n```python\nclass RandomAgent(Agent):\n def __init__(self):\n super(RandomAgent, self).__init__()\n\n def learned_act(self, s):\n \"\"\"Return a rabdom action.\"\"\"\n \n action = np.random.randint(0, 4, 1)[0]\n \n return action\n```\n\n***\n***\n__Question 4__ Visualize the game moves. You need to fill in the following function for the evaluation:\n\n\n```python\ndef test(agent, env, epochs, prefix='', display=True, save=True):\n \"\"\"Visualise the moves of the rat\"\"\"\n \n # Number of won games\n score = 0\n \n for e in range(epochs):\n\n # Initialisation of win and lose\n win = 0\n lose = 0\n \n # This assumes that the games will end\n game_over = False\n\n # At each epoch, we restart to a fresh game and get the initial state\n state = env.reset()\n\n while not game_over:\n \n # The agent performs an action\n action = agent.act(state)\n\n # Apply an action to the environment, get the next state, the reward\n # and if the games end\n prev_state = state\n state, reward, game_over = env.act(action)\n\n # Update the counters\n if reward > 0:\n win = win + reward\n if reward < 0:\n lose = lose -reward\n\n # Apply the reinforcement strategy\n# loss = agent.reinforce(prev_state, state, action, reward, game_over)\n \n # Save as a mp4\n if save:\n env.draw(prefix + str(e))\n\n # Update stats\n score = score + win-lose\n \n if display:\n print(\"Win/lose count {}/{}. Average score ({})\"\n .format(win, lose, score / epochs))\n \n if display:\n print('Final score: ' + str(score/epochs))\n \n return score\n```\n\n\n```python\n# Initialize the game\nenv = Environment(grid_size=size, max_time=T, temperature=temperature)\n\n# Initialize the agent!\nagent = RandomAgent()\n\ntest(agent, env, epochs_test, prefix='./Results/random')\nHTML(display_videos('./Results/random0.mp4'))\n```\n\n Win/lose count 10.0/19.0. Average score (-0.6)\n Win/lose count 6.5/22.0. Average score (-1.6333333333333333)\n Win/lose count 7.0/12.0. Average score (-1.9666666666666666)\n Win/lose count 7.0/10.0. Average score (-2.1666666666666665)\n Win/lose count 5.0/13.0. Average score (-2.7)\n Win/lose count 7.0/18.0. Average score (-3.433333333333333)\n Win/lose count 11.5/16.0. Average score (-3.7333333333333334)\n Win/lose count 8.0/14.0. Average score (-4.133333333333334)\n Win/lose count 6.0/6.0. Average score (-4.133333333333334)\n Win/lose count 9.5/16.0. Average score (-4.566666666666666)\n Win/lose count 10.0/11.0. Average score (-4.633333333333334)\n Win/lose count 10.5/14.0. Average score (-4.866666666666666)\n Win/lose count 9.0/16.0. Average score (-5.333333333333333)\n Win/lose count 9.5/8.0. Average score (-5.233333333333333)\n Win/lose count 7.5/11.0. Average score (-5.466666666666667)\n Final score: -5.466666666666667\n\n\n\n\n\n\n\n\n\n***\n## DQN\n\nLet us assume here that $T=\\infty$.\n\n***\n__Question 5__ Let $\\pi$ be a policy, show that:\n\n\\begin{equation*}\nQ^{\\pi}(s,a)=E_{(s',a')\\sim p(.|s,a)}[r(s,a)+\\gamma Q^{\\pi}(s',a')]\n\\end{equation*}\n\nThen, show that for the optimal policy $\\pi^*$ (we assume its existence), the following holds: \n\n\\begin{equation*}\nQ^{*}(s,a)=E_{s'\\sim \\pi^*(.|s,a)}[r(s,a)+\\gamma\\max_{a'}Q^{*}(s',a')].\n\\end{equation*}\nFinally, deduce that a plausible objective is:\n\n\\begin{equation*}\n\\mathcal{L}(\\theta)=E_{s' \\sim \\pi^*(.|s,a)}\\Vert r+\\gamma\\max\\max_{a'}Q(s',a',\\theta)-Q(s,a,\\theta)\\Vert^{2}.\n\\end{equation*}\n\n\n\n\n\\begin{itemize}\n \\item The first result represents the Bellman equation. Let's prove it.\n We denote by $(S_t)$ and $(A_t)$ the series of random variables representing the states and action over time. Therefore, if we take up the notations of the slides of the course, it has:\n \\begin{align*}\n Q^{\\pi}(s_t, a_t) &= \\mathbb{E}_{\\pi} \\left[ \\sum_{k=0}^{+\\infty} \\gamma^k * r(s_{t + k}, a_{t + k}) ~ \\big| ~ S_t=s_t, A_t=a_t \\right] \\\\\n &= r(s_t, a_t) + \\gamma \\mathbb{E}_{\\pi} \\left[ \\sum_{k=0}^{+\\infty} \\gamma^k * r(s_{(t + 1) + k}, a_{(t + 1) + k}) ~ \\big| ~ S_t=s_t, A_t=a_t \\right] \\\\\n &= r(s_t, a_t) + \\gamma \\mathbb{E}_{\\pi} \\left[ \\sum_{s_{t+1}, a_{t+1}} \\mathbf{1}_{ \\{ S_{t+1} = s_{t+1}, A_{t+1} = a_{t+1} \\} } \\underbrace{\\mathbb{E}_{\\pi} \\left[ \\sum_{k=0}^{+\\infty} \\gamma^k * r(s_{(t + 1) + k}, a_{(t + 1) + k}) ~ \\big| ~ S_{t+1}=s_{t+1}, A_{t+1}=a_{t+1} \\right]}_{ = Q^{\\pi}(s_{t+1}, a_{t+1})} ~ \\big| ~ S_t=s_t, A_t=a_t \\right] \\\\\n &= r(s_t, a_t) + \\gamma \\sum_{s_{t+1}, a_{t+1}} \\mathbb{P}^{\\pi}\\left(S_{t+1}=s_{t+1}, A_{t+1}=a_{t+1} ~ | ~ S_t=s_t, A_t=a_t \\right) Q^{\\pi}(s_{t+1}, a_{t+1}) \\\\\n &= r(a_t, s_t) + \\gamma \\mathbb{E}_{(s_{t+1}, a_{t+1}) \\sim p(.,. | s_t, a_t)} \\left[ Q^{\\pi}(s_{t+1}, a_{t+1}) \\right]\n \\end{align*}\n Moreover, these results are true for every $t$. So, it obtains the result. \n \\item This second equation corresponds to the Optimal Bellman Equation. Let's prove it: If it uses again the same equations with $Q^*$, it has:\n \\begin{align*}\n Q^{*}(s_t, a_t) &= r(s_t, a_t) + \\gamma \\sum_{s_{t+1}, a_{t+1}} \\mathbb{P}^{\\pi^*}\\left(S_{t+1}=s_{t+1}, A_{t+1}=a_{t+1} ~ | ~ S_t=s_t, A_t=a_t \\right) Q^{*}(s_{t+1}, a_{t+1}) \\\\\n &= r(s_t, a_t) + \\gamma \\sum_{s_{t+1}, a_{t+1}} \\mathbb{P}\\left(S_{t+1}=s_{t+1} ~ | ~ S_t=s_t, A_t=a_t \\right) \\pi^*(a_{t+1} ~ | ~ S_{t+1} = s_{t+1}) Q^{*}(s_{t+1}, a_{t+1})\n \\end{align*}\n But, by definition of the optimal policy, \n \\begin{equation*}\n \\pi^*(a_{t+1} ~ | ~ S_{t+1} = s_{t+1}) = \\left\\{ \\begin{array}{ cl}\n \\frac{1}{|\\text{argmax}_{a} Q^{*}(s_{t+1}, a)|}, & \\text{if} ~ a_{t+1} \\in \\text{argmax}_{a} Q^{*}(s_{t+1}, a) \\\\\n 0, & \\text{otherwise}\n \\end{array}\n \\right.\n \\end{equation*}\n Therefore:\n \\begin{align*}\n Q^{*}(s_t, a_t) &= r(s_t, a_t) + \\gamma \\sum_{s_{t+1}} \\mathbb{P}\\left(S_{t+1}=s_{t+1} ~ | ~ S_t=s_t, A_t=a_t \\right) max_{a_{t+1}} Q^{*}(s_{t+1}, a_{t+1}) \\\\\n &= r(a_t, s_t) + \\gamma \\mathbb{E}_{s_{t+1} \\sim p(. | s_t, \\pi^*(.|s_t))} \\left[ \\max_{a_{t+1}} Q^{*}(s_{t+1}, a_{t+1}) \\right]\n \\end{align*}\n And as before, the result does not depend on the time $t$. It obtains theferore the result.\n \\item So,now let suppose that we succeed to parametrised Q by a parameter $\\theta$. So, if it is looking for the the optimal policy $\\pi^*$, it corresponds to find $\\theta$ such that the previous equation on Q is respected. So, we would like to find theta such the two terms of the previous equation are as close as possible. Therefore, we can compute:\n \\begin{align*}\n \\mathcal{L}_1(\\theta) &= \\Vert E_{s' \\sim \\pi^*(.|s,a)} \\left[ r + \\gamma \\max_{a'}Q(s',a',\\theta) \\right] - Q(s, a, \\theta) \\Vert^2 \\\\\n &= \\Vert E_{s' \\sim \\pi^*(.|s,a)} \\left[ r + \\gamma \\max_{a'}Q(s',a',\\theta) - Q(s, a, \\theta) \\right] \\Vert^2 \\\\\n &\\leq E_{s' \\sim \\pi^*(.|s,a)} \\left[ \\Vert r + \\gamma \\max_{a'}Q(s',a',\\theta) - Q(s, a, \\theta) \\Vert^2 \\right], ~~~ \\text{by Jensen inequality} \\\\\n &\\leq \\mathcal{L}(\\theta)\n \\end{align*}\n So $\\mathcal{L}(\\theta)$ can be used as pausible objective.\n\\end{itemize}\n\n***\nThe DQN-learning algorithm relies on these derivations to train the parameters $\\theta$ of a Deep Neural Network:\n\n1. At the state $s_t$, select the action $a_t$ with best reward using $Q_t$ and store the results;\n\n2. Obtain the new state $s_{t+1}$ from the environment $p$;\n\n3. Store $(s_t,a_t,s_{t+1})$;\n\n4. Obtain $Q_{t+1}$ by minimizing $\\mathcal{L}$ from a recovered batch from the previously stored results.\n\n***\n__Question 6__ Implement the class ```Memory``` that stores moves (in a replay buffer) via ```remember``` and provides a ```random_access``` to these. Specify a maximum memory size to avoid side effects. You can for example use a ```list()``` and set by default ```max_memory=100```.\n\n\n```python\nclass Memory(object):\n def __init__(self, max_memory=100):\n self.max_memory = max_memory\n self.memory = list()\n\n def remember(self, m):\n \"\"\"Store (s_t, a_t, s_t+1, r_t, game_over_t).\"\"\"\n \n # Current length of the memory\n n = len(self.memory)\n \n # Append the new state\n if n < self.max_memory:\n self.memory.append(m)\n else:\n # Replace a random state\n idx = np.random.choice(np.arange(n))\n self.memory[idx] = m\n\n def random_access(self):\n \"\"\"Extract a random quintuplet (s_t, a_t, s_t+1, r_t, game_over_t) in memory.\"\"\"\n \n # Current length of the memory\n n = len(self.memory)\n \n # Choose a random index\n idx = np.random.choice(np.arange(n))\n quintuplet = self.memory[idx]\n \n return quintuplet\n```\n\n***\nThe pipeline we will use for training is given below:\n\n\n```python\ndef train(agent, env, epoch, prefix='', display=True, save=True):\n \"\"\"Train the agent on env for the given number of epoch.\"\"\"\n \n # Number of won games\n score = 0\n loss = 0\n\n for e in range(epoch):\n # At each epoch, we restart to a fresh game and get the initial state\n state = env.reset()\n # This assumes that the games will terminate\n game_over = False\n\n win = 0\n lose = 0\n\n while not game_over:\n # The agent performs an action\n action = agent.act(state)\n\n # Apply an action to the environment, get the next state, the reward\n # and if the games end\n prev_state = state\n state, reward, game_over = env.act(action)\n\n # Update the counters\n if reward > 0:\n win = win + reward\n if reward < 0:\n lose = lose -reward\n\n # Apply the reinforcement strategy\n loss = agent.reinforce(prev_state, state, action, reward, game_over)\n\n # Save as a mp4\n if e % 10 == 0:\n env.draw(prefix + str(e))\n\n # Update stats\n score += win-lose\n \n if display:\n print(\"Epoch {:03d}/{:03d} | Loss {:.4f} | Win/lose count {}/{} ({})\"\n .format(e + 1, epoch, loss, win, lose, win-lose))\n \n if save:\n agent.save(name_weights=prefix + 'model.h5', name_model=prefix + 'model.json')\n```\n\n***\n__Question 7__ Implement the DQN training algorithm using a cascade of fully connected layers. You can use different learning rate, batch size or memory size parameters. In particular, the loss might oscillate while the player will start to win the games. You have to find a good criterium.\n\n\n```python\nclass DQN(Agent):\n def __init__(self, grid_size, epsilon=0.1, memory_size=100, batch_size=16, n_state=2):\n \"\"\"Initialisation of Deep Q Network.\"\"\"\n super().__init__(epsilon=epsilon)\n\n # Discount for Q learning\n self.discount = 0.99\n \n # Grid size\n self.grid_size = grid_size\n \n # number of state\n self.n_state = n_state\n\n # Memory\n self.memory = Memory(memory_size)\n \n # Batch size when learning\n self.batch_size = batch_size\n\n def learned_act(self, s):\n \"\"\"Choose the action according the current model.\"\"\"\n \n # Reshape the state\n s_t = s.reshape(1, 5, 5, self.n_state)\n \n # Choose the maximising action\n a_t = np.argmax(self.model.predict(s_t))\n if a_t < 0:\n print(a_t)\n \n return a_t\n\n def reinforce(self, s_, n_s_, a_, r_, game_over_):\n \"\"\"Two steps: first memorize the states, second learn from the pool.\"\"\"\n\n # First Step\n self.memory.remember([s_, n_s_, a_, r_, game_over_])\n \n # Second Step\n input_states = np.zeros((self.batch_size, 5, 5, self.n_state)) #5, self.n_state))\n target_q = np.zeros((self.batch_size, 2), dtype=np.dtype(float, int))\n \n for i in range(self.batch_size):\n \n # Sample a random transition\n s_, n_s_, a_, r_, game_over_ = self.memory.random_access()\n \n # Add this state in input states\n input_states[i, :, :, :] = s_\n \n if game_over_:\n target_q[i, :] = r_\n else:\n n_s_ = n_s_.reshape(1, 5, 5, self.n_state)\n target_q[i, :] = [r_ + self.discount * np.max(self.model.predict(n_s_)), a_]\n \n # HINT: Clip the target to avoid exploiding gradients.. -- clipping is a bit tighter\n target_q = np.clip(target_q, -5, 5)\n l = self.model.train_on_batch(input_states, target_q)\n\n return l\n\n def save(self, name_weights='model.h5', name_model='model.json'):\n \n self.model.save_weights(name_weights, overwrite=True)\n \n with open(name_model, \"w\") as outfile:\n json.dump(self.model.to_json(), outfile)\n \n def load(self, name_weights='model.h5', name_model='model.json'):\n \n with open(name_model, \"r\") as jfile:\n model = model_from_json(json.load(jfile))\n \n model.load_weights(name_weights)\n model.compile(\"sgd\", \"mse\")\n self.model = model\n\n \n # Custom loss\n def my_loss(self, data, y_pred):\n \"\"\"Compute the mse between y_j and Q(s_j, a_j).\"\"\"\n\n # Length\n n = np.shape(y_pred)[0]\n\n # Extract the corresponding y_pred\n idx = tf.reshape(K.cast(data[:, 1], tf.int32), (-1, 1))\n index = tf.concat([tf.reshape(tf.range(0, self.batch_size), (-1, 1)), idx], axis=1)\n y_pred_2 = tf.gather_nd(y_pred, index)\n y_true = data[:, 0]\n\n return tf.losses.mean_squared_error(y_true, y_pred_2)\n\n \nclass DQN_FC(DQN):\n def __init__(self, *args, lr=0.1, **kwargs):\n super().__init__(*args, **kwargs)\n \n # NN Model\n model = Sequential()\n model.add(Flatten(input_shape=(5, 5, self.n_state, )))\n model.add(Dense(32, activation='relu'))\n model.add(Dense(32, activation='relu'))\n model.add(Dense(16, activation='relu'))\n model.add(Dense(4, activation='softmax'))\n \n model.compile(sgd(lr=lr, decay=1e-4, momentum=0.0), self.my_loss)\n self.model = model\n \n# print(model.summary()) \n```\n\n\n```python\nenv = Environment(grid_size=size, max_time=T, temperature=0.3)\nagent = DQN_FC(size, lr=1, epsilon=0.1, memory_size=1000, batch_size=128)\ntrain(agent, env, epochs_train, prefix='./Results/fc_train')\nHTML(display_videos('./Results/fc_train10.mp4'))\n```\n\n Epoch 001/021 | Loss 0.0112 | Win/lose count 1.0/1.0 (0.0)\n Epoch 002/021 | Loss 0.0395 | Win/lose count 4.5/5.0 (-0.5)\n Epoch 003/021 | Loss 0.0241 | Win/lose count 2.0/3.0 (-1.0)\n Epoch 004/021 | Loss 0.0283 | Win/lose count 1.0/2.0 (-1.0)\n Epoch 005/021 | Loss 0.0189 | Win/lose count 6.5/4.0 (2.5)\n Epoch 006/021 | Loss 0.0279 | Win/lose count 1.5/4.0 (-2.5)\n Epoch 007/021 | Loss 0.0224 | Win/lose count 2.5/2.0 (0.5)\n Epoch 008/021 | Loss 0.0142 | Win/lose count 3.5/1.0 (2.5)\n Epoch 009/021 | Loss 0.0127 | Win/lose count 6.0/2.0 (4.0)\n Epoch 010/021 | Loss 0.0163 | Win/lose count 3.5/2.0 (1.5)\n Epoch 011/021 | Loss 0.0182 | Win/lose count 3.5/2.0 (1.5)\n Epoch 012/021 | Loss 0.0198 | Win/lose count 9.0/2.0 (7.0)\n Epoch 013/021 | Loss 0.0130 | Win/lose count 7.0/3.0 (4.0)\n Epoch 014/021 | Loss 0.0215 | Win/lose count 5.5/1.0 (4.5)\n Epoch 015/021 | Loss 0.0187 | Win/lose count 6.0/2.0 (4.0)\n Epoch 016/021 | Loss 0.0351 | Win/lose count 4.0/5.0 (-1.0)\n Epoch 017/021 | Loss 0.0300 | Win/lose count 9.5/2.0 (7.5)\n Epoch 018/021 | Loss 0.0182 | Win/lose count 9.5/2.0 (7.5)\n Epoch 019/021 | Loss 0.0221 | Win/lose count 5.5/2.0 (3.5)\n Epoch 020/021 | Loss 0.0277 | Win/lose count 5.0/2.0 (3.0)\n Epoch 021/021 | Loss 0.0170 | Win/lose count 2.0/2.0 (0.0)\n\n\n\n\n\n\n\n\n\n\n```python\ndef GridSearch(dqn, train_function=train, env=env):\n \"\"\"Do a gridsearch on the following parameters: batch_size, memory_size, lr.\n \"\"\"\n \n # Possible values of the optimising parameters\n batch_size_l = [64]\n memory_size_l = [1000]\n lr_l = [1, 0.1, 0.01, 0.001]\n \n # Initialisation of best parameters\n best_parameters = [0, 0, 0]\n max_score = -float(\"inf\")\n i = 0\n \n # Gridsearch\n for batch_size in batch_size_l:\n for memory_size in memory_size_l:\n for lr in lr_l:\n \n # Definition of the agent\n agent = dqn(batch_size, memory_size, lr)\n \n # Training of the agent\n train_function(agent, env, epochs_train, prefix='./Results/fc_train_' + str(i) + \"_\",\n display=False, save=False)\n \n # Testing of the agent\n score_test = test(agent, env, epochs_test,\n prefix='./Results/fc_test_' + str(i),\n display=False, save=False)\n \n # Update of best parameters\n if max_score < score_test:\n max_score = score_test\n best_parameters = [batch_size, memory_size, lr] \n print(\"New best score:\", score_test)\n print(\"Best parameters:\", best_parameters)\n # Update of i\n i += 1\n n = len(batch_size_l) * len(memory_size_l) * len(lr_l)\n print(\"Step done: \", i / n * 100)\n \n # Display best parameters\n print(best_parameters)\n```\n\n\n```python\n# Definition of the environment\nenv = Environment(grid_size=size, max_time=T, temperature=0.3)\n\n# Definition of the agent\ndqn = lambda batch_size, memory_size, lr: DQN_CNN(size, lr=lr, epsilon=0.1,\n memory_size=memory_size,\n batch_size=batch_size)\n\n# Compute the GridSearch\nGridSearch(dqn, env=env)\n```\n\nFor obtaining the best results, I implemented a GridSearch over different values of the following parameters:\n\\begin{itemize}\n \\item batch_size\n \\item memory_size\n \\item learning_rate\n\\end{itemize}\nIt seems that the two first parameters do not have a big impact for the environment selected. However, it seems that a big learning rate is required.\n\nAlso, we implemented the loss in the function \"my_loss\" which correspond to:\n\\begin{equation}\n \\sum_{i=1}^N (Q(s_i, s_a) - y_i)^2, ~~ \\text{with} ~~ y_i = \\left\\{\\begin{array}{cl}\n r_i & \\text{if} ~ \\text{game_over} = True \\\\\n r_i + \\max_{a'} Q({s'}_i, a') & \\text{otherwise}\n \\end{array}\n \\right.\n\\end{equation}\n\n***\n***\n__Question 8__ Implement the DQN training algorithm using a CNN (for example, 2 convolutional layers and one final fully connected layer).\n\n\n```python\nclass DQN_CNN(DQN):\n def __init__(self, *args, lr=0.1, **kwargs):\n super().__init__(*args, **kwargs)\n \n ## CNN\n model = Sequential()\n model.add(Conv2D(16, (3, 3), activation='relu', input_shape=(5, 5, self.n_state, )))\n model.add(Conv2D(16, (3, 3), activation='relu'))\n model.add(Flatten())\n model.add(Dense(4, activation='softmax'))\n \n model.compile(sgd(lr=lr, decay=1e-4, momentum=0.0), self.my_loss)\n self.model = model\n```\n\n\n```python\nenv = Environment(grid_size=size, max_time=T, temperature=0.3)\nagent = DQN_CNN(size, lr=1, epsilon=0.1, memory_size=1000, batch_size=128)\ntrain(agent,env,epochs_train,prefix='./Results/cnn_train')\nHTML(display_videos('./Results/cnn_train10.mp4'))\n```\n\n Epoch 001/021 | Loss 0.0390 | Win/lose count 3.5/4.0 (-0.5)\n Epoch 002/021 | Loss 0.0572 | Win/lose count 3.5/3.0 (0.5)\n Epoch 003/021 | Loss 0.0604 | Win/lose count 2.5/2.0 (0.5)\n Epoch 004/021 | Loss 0.0165 | Win/lose count 3.0/2.0 (1.0)\n Epoch 005/021 | Loss 0.0126 | Win/lose count 2.5/6.0 (-3.5)\n Epoch 006/021 | Loss 0.0129 | Win/lose count 1.0/4.0 (-3.0)\n Epoch 007/021 | Loss 0.0187 | Win/lose count 3.5/1.0 (2.5)\n Epoch 008/021 | Loss 0.0166 | Win/lose count 6.5/5.0 (1.5)\n Epoch 009/021 | Loss 0.0229 | Win/lose count 6.0/4.0 (2.0)\n Epoch 010/021 | Loss 0.0130 | Win/lose count 3.0/1.0 (2.0)\n Epoch 011/021 | Loss 0.0360 | Win/lose count 5.0/1.0 (4.0)\n Epoch 012/021 | Loss 0.0171 | Win/lose count 5.5/3.0 (2.5)\n Epoch 013/021 | Loss 0.0170 | Win/lose count 5.5/2.0 (3.5)\n Epoch 014/021 | Loss 0.0159 | Win/lose count 4.0/3.0 (1.0)\n Epoch 015/021 | Loss 0.0190 | Win/lose count 2.5/2.0 (0.5)\n Epoch 016/021 | Loss 0.0221 | Win/lose count 7.0/5.0 (2.0)\n Epoch 017/021 | Loss 0.0235 | Win/lose count 2.5/4.0 (-1.5)\n Epoch 018/021 | Loss 0.0245 | Win/lose count 7.0/2.0 (5.0)\n Epoch 019/021 | Loss 0.0199 | Win/lose count 4.5/1.0 (3.5)\n Epoch 020/021 | Loss 0.0204 | Win/lose count 3.0/4.0 (-1.0)\n Epoch 021/021 | Loss 0.0256 | Win/lose count 9.0/0 (9.0)\n\n\n\n\n\n\n\n\n\n***\n***\n__Question 9__ Test both algorithms and compare their performances. Which issue(s) do you observe? Observe also different behaviors by changing the temperature.\n\n\n```python\nenv = Environment(grid_size=size, max_time=T, temperature=0.1)\nagent_cnn = DQN_CNN(size, lr=1, epsilon=0.1, memory_size=1000, batch_size=128)\nagent_cnn.load(name_weights='./Results/cnn_trainmodel.h5',\n name_model='./Results/cnn_trainmodel.json')\n\nagent_fc = DQN_FC(size, lr=1, epsilon=0.1, memory_size=1000, batch_size=128)\nagent_cnn.load(name_weights='./Results/fc_trainmodel.h5',\n name_model='./Results/fc_trainmodel.json')\nprint('Test of the CNN')\ntest(agent_cnn, env, epochs_test, prefix='./Results/cnn_test')\nprint('\\nTest of the FC')\ntest(agent_fc, env, epochs_test, prefix='./Results/fc_test');\n```\n\n Test of the CNN\n Win/lose count 3.0/3.0. Average score (0.0)\n Win/lose count 3.0/0. Average score (0.2)\n Win/lose count 2.0/2.0. Average score (0.2)\n Win/lose count 2.5/2.0. Average score (0.23333333333333334)\n Win/lose count 2.5/1.0. Average score (0.3333333333333333)\n Win/lose count 2.5/1.0. Average score (0.43333333333333335)\n Win/lose count 2.0/4.0. Average score (0.3)\n Win/lose count 2.5/1.0. Average score (0.4)\n Win/lose count 2.5/1.0. Average score (0.5)\n Win/lose count 3.0/1.0. Average score (0.6333333333333333)\n Win/lose count 1.0/1.0. Average score (0.6333333333333333)\n Win/lose count 3.0/3.0. Average score (0.6333333333333333)\n Win/lose count 0.5/0. Average score (0.6666666666666666)\n Win/lose count 3.5/1.0. Average score (0.8333333333333334)\n Win/lose count 3.0/0. Average score (1.0333333333333334)\n Final score: 1.0333333333333334\n \n Test of the FC\n Win/lose count 2.0/0. Average score (0.13333333333333333)\n Win/lose count 1.0/1.0. Average score (0.13333333333333333)\n Win/lose count 2.0/1.0. Average score (0.2)\n Win/lose count 1.5/2.0. Average score (0.16666666666666666)\n Win/lose count 1.5/1.0. Average score (0.2)\n Win/lose count 0/4.0. Average score (-0.06666666666666667)\n Win/lose count 4.0/4.0. Average score (-0.06666666666666667)\n Win/lose count 0.5/0. Average score (-0.03333333333333333)\n Win/lose count 0.5/1.0. Average score (-0.06666666666666667)\n Win/lose count 2.5/6.0. Average score (-0.3)\n Win/lose count 0.5/3.0. Average score (-0.4666666666666667)\n Win/lose count 3.0/5.0. Average score (-0.6)\n Win/lose count 1.5/3.0. Average score (-0.7)\n Win/lose count 0.5/6.0. Average score (-1.0666666666666667)\n Win/lose count 1.0/0. Average score (-1.0)\n Final score: -1.0\n\n\n\n```python\nHTML(display_videos('./Results/cnn_test10.mp4'))\n```\n\n\n\n\n\n\n\n\n\n```python\nHTML(display_videos('./Results/fc_test10.mp4'))\n```\n\n\n\n\n\n\n\n\nFor optimised parameters, we observed:\n\\begin{itemize}\n \\item The performance of the DQN with Convolutionnal networks performs in average better than the Fully Connected one.\n \\item Also, we can observe that the agent get easily blocked in the corners and only explore one border There is a lack of exploration here.\n \\item When the temperature increases, the score of the two agents increase also a lot. Indeed, there is much more food in the environment so it is easy for the agent to increase its score.\n\\end{itemize}\n\n***\n\nThe algorithm tends to not explore the map which can be an issue. We propose two ideas in order to encourage exploration:\n1. Incorporating a decreasing $\\epsilon$-greedy exploration. You can use the method ```set_epsilon```\n2. Append via the environment a new state that describes if a cell has been visited or not\n\n***\n__Question 10__ Design a new ```train_explore``` function and environment class ```EnvironmentExploring``` to tackle the issue of exploration.\n\n\n\n\n```python\ndef train_explore(agent, env, epoch, prefix='', display=True, save=True):\n \"\"\"Train the agent to explore env for the given number of epoch.\"\"\"\n \n # Number of won games\n score = 0\n loss = 0\n\n for e in range(epoch):\n \n # At each epoch, we restart to a fresh game and get the initial state\n state = env.reset()\n # This assumes that the games will terminate\n game_over = False\n\n win = 0\n lose = 0\n \n # Definition of epsilon\n epsilon = 1 / (1 + e)**2\n agent.set_epsilon(epsilon)\n\n while not game_over:\n # The agent performs an action\n action = agent.act(state)\n\n # Apply an action to the environment, get the next state, the reward\n # and if the games end\n prev_state = state\n state, reward, game_over = env.act(action)\n\n # Update the counters\n if reward > 0:\n win = win + reward\n if reward < 0:\n lose = lose -reward\n\n # Apply the reinforcement strategy\n loss = agent.reinforce(prev_state, state, action, reward, game_over)\n \n # Save as a mp4\n if e % 10 == 0:\n env.draw(prefix + str(e))\n\n # Update stats\n score += win-lose\n \n if display:\n print(\"Epoch {:03d}/{:03d} | Loss {:.4f} | Win/lose count {}/{} ({})\"\n .format(e + 1, epoch, loss, win, lose, win-lose))\n \n if save:\n agent.save(name_weights=prefix + 'model.h5', name_model=prefix + 'model.json')\n\n \nclass EnvironmentExploring(Environment):\n def __init__(self, grid_size=10, max_time=500, temperature=0.1, train_binary=True):\n super().__init__(grid_size=grid_size, max_time=max_time,\n temperature=temperature)\n self.train_binary = train_binary\n \n def act(self, action):\n \"\"\"This function returns the new state, reward and decides if the\n game ends.\"\"\"\n\n self.get_frame(int(self.t))\n\n self.position = np.zeros((self.grid_size, self.grid_size))\n\n self.position[0:2, :]= -1\n self.position[:, 0:2] = -1\n self.position[-2:, :] = -1\n self.position[-2:, :] = -1\n\n self.position[self.x, self.y] = 1\n \n if action == 0: # Going up (or down on the image)\n if self.x == self.grid_size - 3:\n self.x = self.x - 1\n else:\n self.x = self.x + 1\n elif action == 1: # Going up on the image\n if self.x == 2:\n self.x = self.x + 1\n else:\n self.x = self.x - 1\n elif action == 2: # Going right on the image\n if self.y == self.grid_size - 3:\n self.y = self.y - 1\n else:\n self.y = self.y + 1\n elif action == 3: # Going left on the image\n if self.y == 2:\n self.y = self.y + 1\n else:\n self.y = self.y - 1\n else:\n RuntimeError('Error: action not recognized')\n\n self.t = self.t + 1\n \n # Update of the reward\n reward = 0\n if self.train_binary:\n reward = -self.malus_position[self.x, self.y]\n self.malus_position[self.x, self.y] = 0.1\n reward = reward + self.board[self.x, self.y]\n self.board[self.x, self.y] = 0\n \n game_over = self.t > self.max_time\n state = np.concatenate((self.malus_position.reshape(self.grid_size, self.grid_size,1),\n self.board.reshape(self.grid_size, self.grid_size,1),\n self.position.reshape(self.grid_size, self.grid_size,1)),\n axis=2)\n state = state[self.x-2:self.x+3, self.y-2:self.y+3, :]\n\n return state, reward, game_over\n\n def reset(self):\n \"\"\"This function resets the game and returns the initial state\"\"\"\n\n self.x = np.random.randint(3, self.grid_size-3, size=1)[0]\n self.y = np.random.randint(3, self.grid_size-3, size=1)[0]\n \n # Definition of board\n bonus = 0.5 * np.random.binomial(1, self.temperature, size=self.grid_size**2)\n bonus = bonus.reshape(self.grid_size, self.grid_size)\n\n malus = -1.0*np.random.binomial(1, self.temperature, size=self.grid_size**2)\n malus = malus.reshape(self.grid_size, self.grid_size)\n\n self.to_draw = np.zeros((self.max_time + 2, self.grid_size * self.scale, \n self.grid_size * self.scale, 3))\n\n malus[bonus > 0] = 0\n\n self.board = bonus + malus\n \n # Definition of malus_position\n self.malus_position = np.zeros((self.grid_size, self.grid_size))\n \n # Definition of position\n self.position = np.zeros((self.grid_size, self.grid_size))\n self.position[0:2,:]= -1\n self.position[:,0:2] = -1\n self.position[-2:, :] = -1\n self.position[-2:, :] = -1\n self.board[self.x, self.y] = 0\n self.t = 0\n\n state = np.concatenate((self.malus_position.reshape(self.grid_size, self.grid_size, 1),\n self.board.reshape(self.grid_size, self.grid_size,1),\n self.position.reshape(self.grid_size, self.grid_size,1)),\n axis=2)\n\n state = state[self.x - 2:self.x + 3, self.y - 2:self.y + 3, :]\n return state\n```\n\n\n```python\n# Definition of the environment\nenv = EnvironmentExploring(grid_size=size, max_time=T, temperature=0.3)\n\n# Definition of the agent\ndqn = lambda batch_size, memory_size, lr: DQN_CNN(size, lr=lr, epsilon=0.1,\n memory_size=memory_size,\n batch_size=batch_size, n_state=3)\n\n# Compute the GridSearch\nGridSearch(dqn, train_function=train_explore, env=env)\n```\n\n\n```python\n# Training\nenv = EnvironmentExploring(grid_size=size, max_time=T, temperature=0.3)\nagent = DQN_CNN(size, lr=1, memory_size=1000, batch_size=128, n_state=3)\ntrain_explore(agent, env, epochs_train, prefix='cnn_train_explore')\nHTML(display_videos('cnn_train_explore20.mp4'))\n```\n\n Epoch 001/021 | Loss 0.0510 | Win/lose count 5.0/33.700000000000124 (-28.700000000000124)\n Epoch 002/021 | Loss 0.0472 | Win/lose count 10.0/24.20000000000007 (-14.20000000000007)\n Epoch 003/021 | Loss 0.0386 | Win/lose count 11.0/18.70000000000001 (-7.70000000000001)\n Epoch 004/021 | Loss 0.0341 | Win/lose count 11.0/18.59999999999998 (-7.59999999999998)\n Epoch 005/021 | Loss 0.0354 | Win/lose count 14.5/18.700000000000014 (-4.2000000000000135)\n Epoch 006/021 | Loss 0.0370 | Win/lose count 7.5/19.4 (-11.899999999999999)\n Epoch 007/021 | Loss 0.0354 | Win/lose count 16.5/14.499999999999972 (2.0000000000000284)\n Epoch 008/021 | Loss 0.0419 | Win/lose count 15.5/15.999999999999966 (-0.49999999999996625)\n Epoch 009/021 | Loss 0.0323 | Win/lose count 9.5/16.29999999999996 (-6.799999999999962)\n Epoch 010/021 | Loss 0.0358 | Win/lose count 6.5/18.599999999999994 (-12.099999999999994)\n Epoch 011/021 | Loss 0.0414 | Win/lose count 5.5/17.799999999999983 (-12.299999999999983)\n Epoch 012/021 | Loss 0.0295 | Win/lose count 21.0/16.199999999999964 (4.800000000000036)\n Epoch 013/021 | Loss 0.0297 | Win/lose count 3.0/18.499999999999993 (-15.499999999999993)\n Epoch 014/021 | Loss 0.0372 | Win/lose count 5.5/20.8 (-15.3)\n Epoch 015/021 | Loss 0.0199 | Win/lose count 14.0/15.199999999999962 (-1.199999999999962)\n Epoch 016/021 | Loss 0.0393 | Win/lose count 17.0/14.999999999999966 (2.0000000000000338)\n Epoch 017/021 | Loss 0.0439 | Win/lose count 15.5/17.39999999999999 (-1.8999999999999915)\n Epoch 018/021 | Loss 0.0255 | Win/lose count 9.0/17.99999999999999 (-8.99999999999999)\n Epoch 019/021 | Loss 0.0369 | Win/lose count 13.5/15.399999999999961 (-1.8999999999999613)\n Epoch 020/021 | Loss 0.0318 | Win/lose count 18.0/14.599999999999964 (3.400000000000036)\n Epoch 021/021 | Loss 0.0215 | Win/lose count 11.0/17.699999999999985 (-6.699999999999985)\n\n\n\n\n\n\n\n\n\n\n```python\n# Definition of the environment without training\nenv = EnvironmentExploring(grid_size=size, max_time=T, temperature=0.3, train_binary=False)\n\n# Evaluation\ntest(agent, env, epochs_test,prefix='./Results/cnn_test_explore')\nHTML(display_videos('./Results/cnn_test_explore14.mp4'))\n```\n\n Win/lose count 5.0/0. Average score (0.3333333333333333)\n Win/lose count 7.0/1.0. Average score (0.7333333333333333)\n Win/lose count 10.5/0. Average score (1.4333333333333333)\n Win/lose count 12.5/0. Average score (2.2666666666666666)\n Win/lose count 6.0/0. Average score (2.6666666666666665)\n Win/lose count 1.5/0. Average score (2.7666666666666666)\n Win/lose count 10.0/1.0. Average score (3.3666666666666667)\n Win/lose count 1.0/0. Average score (3.433333333333333)\n Win/lose count 9.5/0. Average score (4.066666666666666)\n Win/lose count 1.0/0. Average score (4.133333333333334)\n Win/lose count 4.5/0. Average score (4.433333333333334)\n Win/lose count 9.0/0. Average score (5.033333333333333)\n Win/lose count 6.5/0. Average score (5.466666666666667)\n Win/lose count 3.0/0. Average score (5.666666666666667)\n Win/lose count 11.0/0. Average score (6.4)\n Final score: 6.4\n\n\n\n\n\n\n\n\n\n***\n***\n__BONUS question__ Use the expert DQN from the previous question to generate some winning games. Train a model that mimicks its behavior. Compare the performances.\n\n\n\n***\n", "meta": {"hexsha": "86cf16622359609a0d4e876dc2eea21d392190ad", "size": 153936, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "MP_3.ipynb", "max_stars_repo_name": "PierreOreistein/MVA-MP3", "max_stars_repo_head_hexsha": "30cebd7d5dbe6e4704f5f4ff4f398079d8858832", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "MP_3.ipynb", "max_issues_repo_name": "PierreOreistein/MVA-MP3", "max_issues_repo_head_hexsha": "30cebd7d5dbe6e4704f5f4ff4f398079d8858832", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "MP_3.ipynb", "max_forks_repo_name": "PierreOreistein/MVA-MP3", "max_forks_repo_head_hexsha": "30cebd7d5dbe6e4704f5f4ff4f398079d8858832", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 82.4068522484, "max_line_length": 12496, "alphanum_fraction": 0.755859578, "converted": true, "num_tokens": 14056, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.6477982043529716, "lm_q1q2_score": 0.415002756409417}} {"text": "https://github.com/heart-group-illinois/COVID19-India\n\n\\begin{align}\n \\mathcal{C}(i,j; \\kappa) := \\frac{1}{T}\\sum_{t=1}^T \\left( \\rho^i_t - \\langle \\rho^i\\rangle \\right) \\left( \\rho^j_{t+\\kappa} - \\langle \\rho^j\\rangle \\right),\n\\end{align}\n\n\n```R\nlibrary(tidyverse)\nlibrary(readr)\nlibrary(magrittr)\n```\n\n -- \u001b[1mAttaching packages\u001b[22m ------------------------------------------------------------------------------------------------------------ tidyverse 1.3.1 --\n \n \u001b[32mv\u001b[39m \u001b[34mggplot2\u001b[39m 3.3.3 \u001b[32mv\u001b[39m \u001b[34mpurrr \u001b[39m 0.3.4\n \u001b[32mv\u001b[39m \u001b[34mtibble \u001b[39m 3.1.2 \u001b[32mv\u001b[39m \u001b[34mdplyr \u001b[39m 1.0.6\n \u001b[32mv\u001b[39m \u001b[34mtidyr \u001b[39m 1.1.3 \u001b[32mv\u001b[39m \u001b[34mstringr\u001b[39m 1.4.0\n \u001b[32mv\u001b[39m \u001b[34mreadr \u001b[39m 1.4.0 \u001b[32mv\u001b[39m \u001b[34mforcats\u001b[39m 0.5.1\n \n -- \u001b[1mConflicts\u001b[22m --------------------------------------------------------------------------------------------------------------- tidyverse_conflicts() --\n \u001b[31mx\u001b[39m \u001b[34mdplyr\u001b[39m::\u001b[32mfilter()\u001b[39m masks \u001b[34mstats\u001b[39m::filter()\n \u001b[31mx\u001b[39m \u001b[34mdplyr\u001b[39m::\u001b[32mlag()\u001b[39m masks \u001b[34mstats\u001b[39m::lag()\n \n \n Attaching package: 'magrittr'\n \n \n The following object is masked from 'package:purrr':\n \n set_names\n \n \n The following object is masked from 'package:tidyr':\n \n extract\n \n \n\n\n\n```R\nversion\n```\n\n\n _ \n platform x86_64-w64-mingw32 \n arch x86_64 \n os mingw32 \n system x86_64, mingw32 \n status \n major 4 \n minor 1.0 \n year 2021 \n month 05 \n day 18 \n svn rev 80317 \n language R \n version.string R version 4.1.0 (2021-05-18)\n nickname Camp Pontanezen \n\n\n\n```R\ndf_All <- read_csv(\"Data/dfIndia_to_models_districts_with_Mobility_for_paper.csv\") \n```\n\n \n \u001b[36m--\u001b[39m \u001b[1m\u001b[1mColumn specification\u001b[1m\u001b[22m \u001b[36m-----------------------------------------------------------------------------------------------------------------------------\u001b[39m\n cols(\n .default = col_double(),\n State = \u001b[31mcol_character()\u001b[39m,\n District = \u001b[31mcol_character()\u001b[39m,\n Date = \u001b[34mcol_date(format = \"\")\u001b[39m\n )\n \u001b[36mi\u001b[39m Use \u001b[30m\u001b[47m\u001b[30m\u001b[47m`spec()`\u001b[47m\u001b[30m\u001b[49m\u001b[39m for the full column specifications.\n \n \n\n\n\n```R\ndf_All %>% ggplot(aes(Date, y = Mob, colour = District_ID)) +\n geom_line(size = 1)\n```\n", "meta": {"hexsha": "c557cc3737c18f1a393d9dbfa9f9a5e27dab5ad8", "size": 36912, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": ".ipynb_checkpoints/COVID19-India-checkpoint.ipynb", "max_stars_repo_name": "heart-analytics/COVID19-India", "max_stars_repo_head_hexsha": "b5eeecd018563eb35e1d3424a3473a33cd38c823", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": ".ipynb_checkpoints/COVID19-India-checkpoint.ipynb", "max_issues_repo_name": "heart-analytics/COVID19-India", "max_issues_repo_head_hexsha": "b5eeecd018563eb35e1d3424a3473a33cd38c823", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": ".ipynb_checkpoints/COVID19-India-checkpoint.ipynb", "max_forks_repo_name": "heart-analytics/COVID19-India", "max_forks_repo_head_hexsha": "b5eeecd018563eb35e1d3424a3473a33cd38c823", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 199.5243243243, "max_line_length": 31466, "alphanum_fraction": 0.8889791938, "converted": true, "num_tokens": 815, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660543, "lm_q2_score": 0.6370308082623217, "lm_q1q2_score": 0.41493311590381177}} {"text": "# This is a test sample for Jupyter Notebook.\n\n\n```python\nprint(\"Hello Jupyter Notebook\")\n```\n\n Hello Jupyter Notebook\n\n\n## Hello JNB ver 1.0\n\n```python\nprint \"Hello World\"\n```\n\n# LaTeX Demo\n\n$$e^x=\\sum_{i=0}^\\infty \\frac{1}{i!}x^i$$\n\n\n```latex\n%%latex\n\n\\begin{align}\n\\dot{x} & = \\sigma(y^2-x^2) \\\\\n\\dot{y} & = \\rho x - y_1 - xz \\\\\n\\dot{z} & = -\\beta z + xy\n\\end{align}\n\n\\begin{align}\n\\\\\n\\end{align}\n\n\\begin{equation*}\n\\mathbf{V}_1 \\times \\mathbf{V}_2 = \\begin{vmatrix}\n\\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\n\\frac{\\partial X}{\\partial u} & \\frac{\\partial Y}{\\partial u} & 0 \\\\\n\\frac{\\partial X}{\\partial v} & \\frac{\\partial Y}{\\partial v} & 0\n\\end{vmatrix}\n\\end{equation*}\n\n$$\n$$\n\n\\begin{eqnarray}\nx' &=& &x \\sin\\phi &+& z \\cos\\phi \\\\\nz' &=& - &x \\cos\\phi &+& z \\sin\\phi \\\\\n\\end{eqnarray}\n```\n\n\n\n\\begin{align}\n\\dot{x} & = \\sigma(y^2-x^2) \\\\\n\\dot{y} & = \\rho x - y_1 - xz \\\\\n\\dot{z} & = -\\beta z + xy\n\\end{align}\n\n\\begin{align}\n\\\\\n\\end{align}\n\n\\begin{equation*}\n\\mathbf{V}_1 \\times \\mathbf{V}_2 = \\begin{vmatrix}\n\\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\n\\frac{\\partial X}{\\partial u} & \\frac{\\partial Y}{\\partial u} & 0 \\\\\n\\frac{\\partial X}{\\partial v} & \\frac{\\partial Y}{\\partial v} & 0\n\\end{vmatrix}\n\\end{equation*}\n\n$$\n$$\n\n\\begin{eqnarray}\nx' &=& &x \\sin\\phi &+& z \\cos\\phi \\\\\nz' &=& - &x \\cos\\phi &+& z \\sin\\phi \\\\\n\\end{eqnarray}\n\n\n\n# Magic\n\n\n```python\n%lsmagic\n```\n\n\n\n\n Available line magics:\n %alias %alias_magic %autoawait %autocall %automagic %autosave %bookmark %cat %cd %clear %colors %conda %config %connect_info %cp %debug %dhist %dirs %doctest_mode %ed %edit %env %gui %hist %history %killbgscripts %ldir %less %lf %lk %ll %load %load_ext %loadpy %logoff %logon %logstart %logstate %logstop %ls %lsmagic %lx %macro %magic %man %matplotlib %mkdir %more %mv %notebook %page %pastebin %pdb %pdef %pdoc %pfile %pinfo %pinfo2 %pip %popd %pprint %precision %prun %psearch %psource %pushd %pwd %pycat %pylab %qtconsole %quickref %recall %rehashx %reload_ext %rep %rerun %reset %reset_selective %rm %rmdir %run %save %sc %set_env %store %sx %system %tb %time %timeit %unalias %unload_ext %who %who_ls %whos %xdel %xmode\n \n Available cell magics:\n %%! %%HTML %%SVG %%bash %%capture %%debug %%file %%html %%javascript %%js %%latex %%markdown %%perl %%prun %%pypy %%python %%python2 %%python3 %%ruby %%script %%sh %%svg %%sx %%system %%time %%timeit %%writefile\n \n Automagic is ON, % prefix IS NOT needed for line magics.\n\n\n\n## %alias\n\n\n```python\n%alias A2B echo %s is speaking to %s.\n%A2B Vincent Leo\n\n!echo \"================= This is a magnificent separate line =================\"\n\n%alias ISAY echo \\\"%l\\\", I said.\n%ISAY Stop Laughing! Idiot.\n%ISAY \u522b\u7b11\u4e86\uff0c\u767d\u75f4\u3002\n\n!echo \"================= This is a magnificent separate line =================\"\n\nISAY?\n```\n\n Vincent is speaking to Leo.\n ================= This is a magnificent separate line =================\n \"Stop Laughing! Idiot.\", I said.\n \"\u522b\u7b11\u4e86\uff0c\u767d\u75f4\u3002\", I said.\n ================= This is a magnificent separate line =================\n\n\n\n```python\n%pwd\n```\n\n\n\n\n '/home/vincent/pythonws/jnb-sample/test'\n\n\n\n\n```python\n%timeit\n%sx ls -l\n```\n\n\n\n\n ['total 92',\n '-rw-r--r-- 1 vincent vincent 6694 6\u6708 2 03:36 lottery_test.ipynb',\n '-rw-r--r-- 1 vincent vincent 22660 6\u6708 8 22:10 nb_test.ipynb',\n '-rw-r--r-- 1 vincent vincent 1592 6\u6708 8 22:11 nb_widget_test.ipynb',\n '-rw-r--r-- 1 vincent vincent 6443 6\u6708 8 17:16 py_basic_test.ipynb',\n '-rw-r--r-- 1 vincent vincent 3543 6\u6708 7 22:53 py_class_test.ipynb',\n '-rw-r--r-- 1 vincent vincent 6726 6\u6708 7 01:10 py_function_file_test.ipynb',\n '-rw-r--r-- 1 vincent vincent 4271 6\u6708 5 18:47 py_list_test.ipynb',\n '-rw-r--r-- 1 vincent vincent 8405 6\u6708 5 02:59 py_random_test.ipynb',\n '-rw-r--r-- 1 vincent vincent 3204 6\u6708 8 02:17 py_re_test.ipynb',\n '-rw-r--r-- 1 vincent vincent 6921 6\u6708 6 01:50 py_time_test.ipynb',\n '-rw-rw-r-- 1 vincent vincent 25 6\u6708 7 01:09 sample.txt']\n\n\n\n\n```python\n%matplotlib --list\n```\n\n Available matplotlib backends: ['tk', 'gtk', 'gtk3', 'wx', 'qt4', 'qt5', 'qt', 'osx', 'nbagg', 'notebook', 'agg', 'svg', 'pdf', 'ps', 'inline', 'ipympl', 'widget']\n\n\n## Running javascript in notebook cell.\n\n\n```python\n%%js\n\nif (confirm('This is a javascript sample.\\n\\nPlease click either one button below.')) {\n console.log('You chose OK!'); // check this out in browser web console by convention in the sub menu 'Web Developer' in Firefox.\n alert('HELLO, Greets from javascript.');\n} else {\n console.log('You chose CANCEL!'); // check this out in browser web console by convention in the sub menu 'Web Developer' in Firefox.\n alert('SORRY! You chose CANCEL!');\n}\n```\n\n\n \n\n", "meta": {"hexsha": "58d956accd76244fe9dc1041cd2cb5f9ae441563", "size": 23135, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "test/nb_test.ipynb", "max_stars_repo_name": "dyslab/jnb-sample", "max_stars_repo_head_hexsha": "38af701866b8496729d63f844e56137d125c4223", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "test/nb_test.ipynb", "max_issues_repo_name": "dyslab/jnb-sample", "max_issues_repo_head_hexsha": "38af701866b8496729d63f844e56137d125c4223", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2020-03-24T17:57:15.000Z", "max_issues_repo_issues_event_max_datetime": "2020-03-31T10:21:49.000Z", "max_forks_repo_path": "test/nb_test.ipynb", "max_forks_repo_name": "dyslab/jnb-sample", "max_forks_repo_head_hexsha": "38af701866b8496729d63f844e56137d125c4223", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 29.1006289308, "max_line_length": 833, "alphanum_fraction": 0.5169656365, "converted": true, "num_tokens": 1815, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.7090191276365462, "lm_q1q2_score": 0.41484790993231124}} {"text": "## Desarrollo de algoritmo para simulaci\u00f3n.\n
                                        \n

                                        El desarrollo de este algoritmo fue llevado a cabo durante el semestre agosto-diciembre 2020, como parte de la asignatura de robotica.

                                        \n
                                        \n

                                        \n

                                      • Alumno : Jose Alfredo de Jesus Aguiar Arce.
                                        • \n
                                      • Numero de control : 15400806.
                                        • \n
                                      • Materia : Robotica.
                                        • \n
                                      • Maestro : Dr. Antonio Guzman Navarrete.
                                        • \n

                                        \n\n
                                        Requiere que los archivos 'sim.py', 'simConst.py', 'remoteapi.dll' est\u00e9n alojados en la misma carpeta que este cuaderno de Jupyter.\n
                                        Desde CoppeliaSim, abrir la escena 'DeskBot_scena.ttt'\n\n### Instrucciones para preparar la simulacion.\n
                                        1. Abra la escena 'DeskBot_scena.ttt'\n
                                        2. Seleccionando la base del robot desde la jerarqu\u00eda de escena, presione bot\u00f3n derecho del mouse y agregue un archivo de script mediante Add -> Associated child script -> Non threaded . Aparecer\u00e1 un peque\u00f1o \u00edcono de documento junto al nombre del robot en la escena de jerarqu\u00eda.\n
                                        3. En el script es posible incluir c\u00f3digo de programaci\u00f3n, escritos en lenguaje LUA. Para nuestro caso, todo el c\u00f3digo que requeriremos es habilitar el API remoto, asignando un puerto de comunicaci\u00f3n. En la funci\u00f3n sysCall_init() agregue la siguiente l\u00ednea:\n
                                        \n 
                                         simRemoteApi.start(19999)\n
                                         Proceda a continuaci\u00f3n con las actividades:\n\n# Seccion 1.- Simulacion y coneccion con coppelia.\n\n\n```python\n# importamos las librer\u00edas necesarias\nimport sim          # librer\u00eda para conectar con CoppeliaSim\nimport sympy as sp  # librer\u00eda para c\u00e1lculo simb\u00f3lico\nimport numpy as np\n```\n\n
                                        Configuraciones del servidor local
                                        \n\n\n```python\n#Variables para configurar entorno del servidor y port\nlocalhost = '127.0.0.1'  #Servidor local de la computadora\nport = 19999  #Puerto para coneccion con V-rep\n```\n\n### Configuramos el servidor y el port\n\nEl siguiente codigo deberia ejecutarse una vez que se presiona play en la simulacion, de funcionar correctamente el cliente quedara conectado con la API de V-rep.\n\n\n```python\ndef connect(port):\n# Establece la conexi\u00f3n a VREP\n# port debe coincidir con el puerto de conexi\u00f3n en VREP\n# retorna el n\u00famero de cliente o -1 si no puede establecer conexi\u00f3n\n    sim.simxFinish(-1) # Cierra ports abiertos en caso de haber\n    clientID=sim.simxStart(localhost,port,True,True,2000,5) # Conectarse\n    if clientID == 0: print(\"conectado a\", port)\n    else: print(\"no se pudo conectar\")\n    return clientID\n\n```\n\n\n```python\n# Requerimos los manejadores para las articulaciones y el Dummy\nclientID = connect(19999)\n```\n\n    conectado a 19999\n\n\n### Configuramos el programa y obtenemos objetos de la simulacion.\n\nobtendermos las juntas y objetos de la simulacion para asignarlos a variables con las que nos podremos referir en el codigo.\nDe esta forma trabajaremos con POO (Programacion Orientada a Objetos) en nuestro programa.\n\n\n```python\n#Variables para configurar entorno \n#Estas variables se agregaron dado que la simulacion fuera a modificarse y por ende sus valores \n#tales como rangos de juntas prismaticas, nombres de juntas etc.\n\n#Como esta nombrada la junta revoluta de dicho eslabon\nnombre_junta_eslabon_1 = 'eslabon_1' #la primera junta revoluta acoplada a la corredera\nnombre_junta_eslabon_2 = 'eslabon_2' #la segunda junta revoluta acoplada a el eslabon anterior\n\n#Como esta nombrada la junta prismatica de dicho eslabon\nnombre_junta_corredera = 'Corredera'\nnombre_junta_efector = 'junta_efector'\n\n#Como esta nombrado el dummy que sirve como punto final del efector\ndummy_posicion = 'posicion_final'\n\n#Valores de los rangos minimos y maximos de juntas prismaticas (en metros) dado que las funciones se encargan\n#de transformar en decimales necesarios \nmin_corredera,max_corredera = [0,0.3]\nmin_efector,max_efector = [-0.04,0]\n\nprint(f'El nombre en la simulacion de la junta revoluta del eslabon 1 debe ser \"{nombre_junta_eslabon_1}\"')\nprint(f'El nombre en la simulacion de la junta revoluta del eslabon 2 debe ser \"{nombre_junta_eslabon_2}\"')\nprint(f'El nombre en la simulacion de la junta prismatica de la corredera debe ser \"{nombre_junta_corredera}\", limite minimo : {min_corredera} mts, limite maximo {max_corredera}')\nprint(f'El nombre en la simulacion de la junta prismatica del efector final debe ser \"{nombre_junta_efector}\"\", limite minimo : {min_efector} mts, limite maximo {max_efector}')\nprint(f'El nombre en la simulacion del dummy para obtner posciones finales como comprobacion es \"{dummy_posicion}\"')\n```\n\n    El nombre en la simulacion de la junta revoluta del eslabon 1 debe ser \"eslabon_1\"\n    El nombre en la simulacion de la junta revoluta del eslabon 2 debe ser \"eslabon_2\"\n    El nombre en la simulacion de la junta prismatica de la corredera debe ser \"Corredera\", limite minimo : 0 mts, limite maximo 0.3\n    El nombre en la simulacion de la junta prismatica del efector final debe ser \"junta_efector\"\", limite minimo : -0.04 mts, limite maximo 0\n    El nombre en la simulacion del dummy para obtner posciones finales como comprobacion es \"posicion_final\"\n\n\n\n```python\n#Manejadores de juntas revolutas\n\n# Obtenemos el manejador de las juntas (joints), y se asignaran a una variable\nreturnCode,junta_1 = sim.simxGetObjectHandle(clientID,nombre_junta_eslabon_1,sim.simx_opmode_blocking)\nreturnCode,junta_2 = sim.simxGetObjectHandle(clientID,nombre_junta_eslabon_2,sim.simx_opmode_blocking)\n#Imprimimos los id de cada junta\nprint(f\"EL ID de junta_1 = {junta_1}, el ID de junta_2 = {junta_2}\")\n\n\n#Manejadores de juntas prismaticas\n\n#Obtenemos el manejador de la corredera que se coloca en la base\nreturnCode,corredera = sim.simxGetObjectHandle(clientID,nombre_junta_corredera,sim.simx_opmode_blocking)\n#Obtenemos el manejador del efector final\nreturnCode,efector_cilindro = sim.simxGetObjectHandle(clientID,nombre_junta_efector,sim.simx_opmode_blocking)\n#Imprimimos el id de cada junta\nprint(f\"El ID de corredera es = {corredera}, el ID del efector final es {efector_cilindro}\")\n\n\n#Manejador del dummy, que funciona para comprobar la posicion final del efector final.\n\nreturnCode,posicion_dummy = sim.simxGetObjectHandle(clientID,dummy_posicion,sim.simx_opmode_blocking)\n#Imprimimos el id \nprint(f\"El ID del dummy es = {posicion_dummy}\")\n```\n\n    EL ID de junta_1 = 17, el ID de junta_2 = 18\n    El ID de corredera es = 16, el ID del efector final es 28\n    El ID del dummy es = 29\n\n\n### Posicionamiento en el simulador\n\nFunciones y demostraciones sobre el como se realiza la simulacion sin entrar en detalles de cinematica aun.\nSon las funciones necesarias para modificar en la simulacion posiciones.\n\n\n```python\n#la posicion para juntas revolutas se asignara mediante grados \n#Los angulos deben enviarse en radianes, dado que se trabaja con grados se usara la expresion de la manera siguiente\n# angulo_en_radianes = angulo_en_grados * pi / 180\n\nangulo_1 = 0 * np.pi/180  #El angulo convertido a radianes para la junta 1.\nangulo_2 = 0* np.pi/180 #El angulo convertido a radianes para la junta 2.\n\nretCode = sim.simxSetJointTargetPosition(clientID, junta_1, angulo_1, sim.simx_opmode_oneshot)\nretCode = sim.simxSetJointTargetPosition(clientID, junta_2, angulo_2, sim.simx_opmode_oneshot)\n\n#La posicion para juntas prismaticas se asignara representada en distancia.\n#La distancia debe ser representada entre [pos.min] y [pos.range] , ambos parametros que se han definido en la simulacion, \n#asignados al elemento (joint) corredera.\n\n#Dado que se debe encontrar expresado en Metros la ecuacion sera la siguiente\n#distancia_en_metros = distancia / 100\n#Para esta configuracion distancia debe ir de 'min_corredera' hasta 'max_corredera' en el caso de la corredera\ndistancia = 0\n#Para esta configuracion distancia debe ir de 'min_efector' hasta 'max_efector' en el caso del efector final\ndistancia_efector = -0.040 \n\nretCode = sim.simxSetJointTargetPosition(clientID, corredera,distancia, sim.simx_opmode_oneshot)\nretCode = sim.simxSetJointTargetPosition(clientID, efector_cilindro,distancia_efector, sim.simx_opmode_oneshot)\nprint(retCode)\n```\n\n    1\n\n\nDadas las observaciones anteriores se escribieron dos metodos para ahorrar escritura y simplificar la implementacion\n\n\n```python\ndef mover_junta_revoluta(junta,angulo):\n    \"\"\"\n    Esta funcion permite realizar una rotacion en la junta revoluta desde el angulo actual hasta el deseado (grados)\n    \"\"\"\n    radianes = angulo * np.pi/180\n    \n    #Regresa 0 si se ejecuta correctamente\n    return sim.simxSetJointTargetPosition(clientID, junta, radianes, sim.simx_opmode_oneshot) \n\n#Ejemplo de como se usa este metodo\n# mover_junta_revoluta(junta_1,-90)\n# mover_junta_revoluta(junta_2,90)\n\ndef mover_junta_prismatica(junta,distancia_desde_origen):\n    \"\"\"\n    Esta funcion permite realizar un dezplazamiento en la junta prismatica (debe estar expresada en centimetros)\n    en un rango entre [Pos Min - Pos Range] de la junta prismatica en cuestion\n    \"\"\"\n    \n    distancia = distancia_desde_origen\n    #Regresa 0 si se ejecuta correctamente\n    return sim.simxSetJointTargetPosition(clientID, junta,distancia, sim.simx_opmode_oneshot)\n\n#Ejemplo de como se usa este metodo\n# mover_junta_prismatica(corredera,0.2)\n# mover_junta_prismatica(efector_cilindro,0.04)\n\ndef get_posicion_dummy():\n    return sim.simxGetObjectPosition(clientID,posicion_dummy,-1,sim.simx_opmode_streaming);\n\n#get_posicion_dummy()\n```\n\nAlgunas utilidades realizadas para facilitar la escritura de posiciones comunes\n\n\n```python\ndef abajo_corredera():\n    \"\"\"\n    Envia la corredera a su posicion minima, en el eje Z.\n    \"\"\"\n    mover_junta_prismatica(corredera,min_corredera)\n\ndef arriba_corredera():\n    \"\"\"\n    Envia la corredera a su posicion maxima, en el eje Z.\n    \"\"\"\n    mover_junta_prismatica(corredera,max_corredera)\n\ndef abrir_efector():\n    \"\"\"\n    activa el efector , es decir lo baja.\n    \"\"\"\n    mover_junta_prismatica(efector_cilindro,min_efector)\n\ndef cerrar_efector():\n    \"\"\"\n    desactiva el efector , es decir lo sube. \n    \"\"\"\n    mover_junta_prismatica(efector_cilindro,max_efector)\n\n\ndef reiniciar_robot():\n    \"\"\"\n    Deja el robot en su posicion inicial \n    \"\"\"\n    cerrar_efector()\n    abajo_corredera()\n    mover_junta_revoluta(junta_1,0)\n    mover_junta_revoluta(junta_2,0)\n\ndef mover_robot_cinematica_directa(angulo_eslabon_1,angulo_eslabon_2,posicion_corredera,posicion_efector,):\n    \"\"\"\n    Este metodo se basa en los metodos que se tocan mas adelante en este documento/codigo, donde obtenidas las ecuaciones de \n    cinematica directa de consigue obtener una posicion en el efector final, mediante la posicion de todos los grados de libertad. \n    \"\"\"\n    #Mover juntas prismaticas\n    mover_junta_prismatica(corredera,posicion_corredera)\n    mover_junta_prismatica(efector_cilindro,posicion_efector)\n    #Mover juntas revolutas\n    mover_junta_revoluta(junta_1,angulo_eslabon_1)\n    mover_junta_revoluta(junta_2,angulo_eslabon_2)\n    \n\n```\n\n## Seccion 2.- Obtencion de modelo cinematico directo, ecuaciones y funciones.\n\n### Cinematica directa del modelo por D-H\n\nSe realiza un algoritmo de Denavit-Hartenberg para resolver el modelo cinematico directo del robot.\n\n#### 1.- Librerias utilizadas para calculos simbolicos\n\n\n```python\n# importamos las librer\u00edas necesarias\nimport sim          # librer\u00eda para conectar con CoppeliaSim\nimport sympy as sp  # librer\u00eda para c\u00e1lculo simb\u00f3lico\nimport numpy as np\n\nfrom sympy.physics.vector import init_vprinting\ninit_vprinting(use_latex='mathjax', pretty_print=False)\nfrom sympy.physics.mechanics import dynamicsymbols\n```\n\n#### 2.- Variables simbolicas\n\n\n```python\n#Variables simbolicas para la obtencion de la matriz DH\ntheta, alpha, a, d = dynamicsymbols('theta alpha a d')\n#Imprimimos las variables\ntheta, alpha, a, d \n```\n\n\n\n\n$\\displaystyle \\left( \\theta, \\  \\alpha, \\  a, \\  d\\right)$\n\n\n\n#### 3.- Matrices de rotacion y obtencion de matriz D-H\n\n\n```python\n#Definimos las matrices de rotacion y traslacion correspondientes\n\n#Rotacion en Z\nrz = sp.Matrix([[sp.cos(theta),-sp.sin(theta),0,0],\n               [sp.sin(theta),sp.cos(theta),0,0],\n               [0,0,1,0],\n               [0,0,0,1]])\n\n#Traslacion en Z\ntz = sp.Matrix([[1,0,0,0],\n               [0,1,0,0],\n               [0,0,1,d],\n               [0,0,0,1]])\n\n#Traslacion en X\ntx = sp.Matrix([[1,0,0,a],\n               [0,1,0,0],\n               [0,0,1,0],\n               [0,0,0,1]])\n\n#Rotacion en X\nrx = sp.Matrix([[1,0,0,0],\n               [0,sp.cos(alpha),-sp.sin(alpha),0],\n               [0,sp.sin(alpha),sp.cos(alpha),0],\n               [0,0,0,1]])\n\n#Matriz de rotacion de parametros D-H\nDH = rz*tz*tx*rx\n#Se muestra la matriz A resultado de la operacion anterior\nDH\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}\\operatorname{cos}\\left(\\theta\\right) & - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\alpha\\right) & \\operatorname{sin}\\left(\\alpha\\right) \\operatorname{sin}\\left(\\theta\\right) & a \\operatorname{cos}\\left(\\theta\\right)\\\\\\operatorname{sin}\\left(\\theta\\right) & \\operatorname{cos}\\left(\\alpha\\right) \\operatorname{cos}\\left(\\theta\\right) & - \\operatorname{sin}\\left(\\alpha\\right) \\operatorname{cos}\\left(\\theta\\right) & a \\operatorname{sin}\\left(\\theta\\right)\\\\0 & \\operatorname{sin}\\left(\\alpha\\right) & \\operatorname{cos}\\left(\\alpha\\right) & d\\\\0 & 0 & 0 & 1\\end{matrix}\\right]$\n\n\n\n#### 4.- Agregamos de uno en uno los grados de libertad correspondientes.\n\nSe sustituyen la matriz D-H los parametros theta,d,a,alpha.\n
                                        En base a la tabla obtenida por nosotros mismos referente al robot dise\u00f1ado\n

                                        \nDonde : \n
                                        \n    
                                      • theta = La rotacion en Z.
                                        • \n    
                                      • d = Desplazamiento en Z.
                                        • \n    
                                      • a = Desplazamiento en X.
                                        • \n    
                                      • alpha = La rotacion en X.
                                        • \n
                                        \n\n\n```python\n#Variables simbolicas para sustituir en la matriz homogenea (DH)\nl1,d1, theta2, d2, theta3, d3, l4 = dynamicsymbols('l1 d1 theta2 d2 theta3 d3 l4')\n#Imprimimos las variables\nl1,d1, theta2, d2, theta3, d3, l4\n```\n\n\n\n\n$\\displaystyle \\left( l_{1}, \\  d_{1}, \\  \\theta_{2}, \\  d_{2}, \\  \\theta_{3}, \\  d_{3}, \\  l_{4}\\right)$\n\n\n\nDe estas variables se obserba lo siguiente : \n\n
                                        \n    
                                      • l1 = La distancia que recorre la corredera en el eje Z. 
                                        • \n    
                                      • theta2 = La rotacion en el eje Z que realiza el eslabon 1. 
                                        • \n    
                                      • d2 = El desplazamiento en X que recorre el eslabon 1. 
                                        • \n    
                                      • theta3 = La rotacion en el eje Z que realiza el eslabon 2. 
                                        • \n    
                                      • d3 = El desplazamiento en X que recorre el eslabon 2. 
                                        • \n    
                                      • l4 = La distancia que recorre el efector final en el eje Z. 
                                        • \n\n##### Eslabon corredera\n\n\n```python\nrot_corredera = DH.subs({theta:0, d:l1, a:d1, alpha:0 })\n#imprimimos la matriz\nrot_corredera\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0 & d_{1}\\\\0 & 1 & 0 & 0\\\\0 & 0 & 1 & l_{1}\\\\0 & 0 & 0 & 1\\end{matrix}\\right]$\n\n\n\n##### Eslabon 1\n\n\n```python\nrot_eslabon1 = DH.subs({theta:theta2, d:0, a:d2, alpha:0 })\n#imprimimos la matriz\nrot_eslabon1\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}\\operatorname{cos}\\left(\\theta_{2}\\right) & - \\operatorname{sin}\\left(\\theta_{2}\\right) & 0 & d_{2} \\operatorname{cos}\\left(\\theta_{2}\\right)\\\\\\operatorname{sin}\\left(\\theta_{2}\\right) & \\operatorname{cos}\\left(\\theta_{2}\\right) & 0 & d_{2} \\operatorname{sin}\\left(\\theta_{2}\\right)\\\\0 & 0 & 1 & 0\\\\0 & 0 & 0 & 1\\end{matrix}\\right]$\n\n\n\n##### Eslabon 2\n\n\n```python\nrot_eslabon2 = DH.subs({theta:theta3, d:0, a:d3, alpha:0 })\n#imprimimos la matriz\nrot_eslabon2\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}\\operatorname{cos}\\left(\\theta_{3}\\right) & - \\operatorname{sin}\\left(\\theta_{3}\\right) & 0 & d_{3} \\operatorname{cos}\\left(\\theta_{3}\\right)\\\\\\operatorname{sin}\\left(\\theta_{3}\\right) & \\operatorname{cos}\\left(\\theta_{3}\\right) & 0 & d_{3} \\operatorname{sin}\\left(\\theta_{3}\\right)\\\\0 & 0 & 1 & 0\\\\0 & 0 & 0 & 1\\end{matrix}\\right]$\n\n\n\n##### Efector final \n\n\n```python\nrot_efector = DH.subs({theta:0, d:l4, a:0, alpha:0 })\n#imprimimos la matriz\nrot_efector\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\0 & 0 & 1 & l_{4}\\\\0 & 0 & 0 & 1\\end{matrix}\\right]$\n\n\n\n##### Obtencion de matriz homogenea y simplificacion\n\n\n```python\nmatriz_homogenea_resultado = (rot_corredera * rot_eslabon1 * rot_eslabon2 * rot_efector)\n\n#Imprimimos la matriz final resultado de la ecuacion anterior\nT = matriz_homogenea_resultado\nmatriz_homogenea_resultado\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}- \\operatorname{sin}\\left(\\theta_{2}\\right) \\operatorname{sin}\\left(\\theta_{3}\\right) + \\operatorname{cos}\\left(\\theta_{2}\\right) \\operatorname{cos}\\left(\\theta_{3}\\right) & - \\operatorname{sin}\\left(\\theta_{2}\\right) \\operatorname{cos}\\left(\\theta_{3}\\right) - \\operatorname{sin}\\left(\\theta_{3}\\right) \\operatorname{cos}\\left(\\theta_{2}\\right) & 0 & d_{1} + d_{2} \\operatorname{cos}\\left(\\theta_{2}\\right) - d_{3} \\operatorname{sin}\\left(\\theta_{2}\\right) \\operatorname{sin}\\left(\\theta_{3}\\right) + d_{3} \\operatorname{cos}\\left(\\theta_{2}\\right) \\operatorname{cos}\\left(\\theta_{3}\\right)\\\\\\operatorname{sin}\\left(\\theta_{2}\\right) \\operatorname{cos}\\left(\\theta_{3}\\right) + \\operatorname{sin}\\left(\\theta_{3}\\right) \\operatorname{cos}\\left(\\theta_{2}\\right) & - \\operatorname{sin}\\left(\\theta_{2}\\right) \\operatorname{sin}\\left(\\theta_{3}\\right) + \\operatorname{cos}\\left(\\theta_{2}\\right) \\operatorname{cos}\\left(\\theta_{3}\\right) & 0 & d_{2} \\operatorname{sin}\\left(\\theta_{2}\\right) + d_{3} \\operatorname{sin}\\left(\\theta_{2}\\right) \\operatorname{cos}\\left(\\theta_{3}\\right) + d_{3} \\operatorname{sin}\\left(\\theta_{3}\\right) \\operatorname{cos}\\left(\\theta_{2}\\right)\\\\0 & 0 & 1 & l_{1} + l_{4}\\\\0 & 0 & 0 & 1\\end{matrix}\\right]$\n\n\n\n\n```python\nmatriz_homogenea_simplificada = sp.Matrix([[T[0,0].simplify(), T[0,1].simplify(), T[0,2].simplify(),T[0,3].simplify()],\n                 [T[1,0].simplify(), T[1,1].simplify(), T[1,2].simplify(),T[1,3].simplify()],\n                 [T[2,0].simplify(), T[2,1].simplify(), T[2,2].simplify(),T[2,3].simplify()],\n                 [T[3,0].simplify(), T[3,1].simplify(), T[3,2].simplify(),T[3,3].simplify()]])\n\n#Imprimimos la matriz ya simplificada \nmatriz_homogenea_simplificada\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}\\operatorname{cos}\\left(\\theta_{2} + \\theta_{3}\\right) & - \\operatorname{sin}\\left(\\theta_{2} + \\theta_{3}\\right) & 0 & d_{1} + d_{2} \\operatorname{cos}\\left(\\theta_{2}\\right) + d_{3} \\operatorname{cos}\\left(\\theta_{2} + \\theta_{3}\\right)\\\\\\operatorname{sin}\\left(\\theta_{2} + \\theta_{3}\\right) & \\operatorname{cos}\\left(\\theta_{2} + \\theta_{3}\\right) & 0 & d_{2} \\operatorname{sin}\\left(\\theta_{2}\\right) + d_{3} \\operatorname{sin}\\left(\\theta_{2} + \\theta_{3}\\right)\\\\0 & 0 & 1 & l_{1} + l_{4}\\\\0 & 0 & 0 & 1\\end{matrix}\\right]$\n\n\n\n#### 5.- Obtener las ecuaciones de Cinematica\n\nDe la matriz obtenida por Denavit Hartenberg (D-H), se obtienen las ecuaciones para cinematrica Directa.\n\n\n```python\n#Dado que la ultima columna representa el posicionamiento del robot obtenemos dicho vector\nposiciones = matriz_homogenea_simplificada[0:3,3] #Vector de posiciones 4 columna de la matriz homogenea\n\n#Ecuaciones de posicion x,y,z\npos_x = posiciones[0]\npos_y = posiciones[1]\npos_z = posiciones[2]\n\n#Imprimimos el vector de posiciones\nposiciones\n```\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}d_{1} + d_{2} \\operatorname{cos}\\left(\\theta_{2}\\right) + d_{3} \\operatorname{cos}\\left(\\theta_{2} + \\theta_{3}\\right)\\\\d_{2} \\operatorname{sin}\\left(\\theta_{2}\\right) + d_{3} \\operatorname{sin}\\left(\\theta_{2} + \\theta_{3}\\right)\\\\l_{1} + l_{4}\\end{matrix}\\right]$\n\n\n\nDe estas ecuaciones de posicion se desprenden las siguientes funciones para obtener su pocision dados los parametros deseados.\n\n\n```python\n#dado que el desplazamiento en X para ambos eslabones es fijo, pues depende de la longitud como tal del eslabon.\nd_1 = (125  + 63.01) / 1000 # 125 mm + 63.01 mm , representan el ajusto de desplazamiento en X sobre la corredera.\nd_2 = 200 / 1000 #206 mm\nd_3 = 200 / 1000 #206 mm\n\n#Funciones de cinematica directa del mecanismo\ndef get_posicion_x(angulo_eslabon_1,angulo_eslabon_2):\n    \"\"\"\n    Resuelve la ecuacion de posicion en el eje x, y regresa su resultado.\n    \"\"\"\n    return  pos_x.subs({d1 : d_1, d2: d_2 , d3: d_3,  theta2 : angulo_eslabon_1, theta3 : angulo_eslabon_2})\n\ndef get_posicion_y(angulo_eslabon_1,angulo_eslabon_2):\n    \"\"\"\n    Resuelve la ecuacion de posicion en el eje y, y regresa su resultado.\n    \"\"\"\n    return  pos_y.subs({d2: d_2 , d3: d_3,  theta2 : angulo_eslabon_1, theta3 : angulo_eslabon_2})\n\ndef get_posicion_z(desplazamiento_corredera, desplazamiento_efector_final):\n    \"\"\"\n    Resuelve la ecuacion de posicion en el eje z, y regresa su resultado.\n    \"\"\"\n    #Dada la simplicidad de la ecuacion de posicion en Z , se reajusta con una distancia producto de los ensambles no contemplados anteriormente.\n    ajuste_z = pos_z.subs({l1 : desplazamiento_corredera, l4 : desplazamiento_efector_final}) + 0.1196 #ajuste en mm, 1.196 * 10 a la -1\n    return  ajuste_z\n\ndef get_posicion_cinematica_directa(angulo_eslabon_1,angulo_eslabon_2,desplazamiento_corredera,desplazamiento_efector_final):\n    \"\"\"\n    Regresa el vector posicion x,y,z. Este vector representa los resultados de los calculos matematicos, son una aproximacion\n    al valor que deberia tener la punta final del efector final.\n    \n    \"\"\"\n    vector_posicion_final = sp.Matrix([[get_posicion_x(angulo_eslabon_1,angulo_eslabon_2)],\n                                      [get_posicion_y(angulo_eslabon_1,angulo_eslabon_2)],\n                                      [get_posicion_z(desplazamiento_corredera,desplazamiento_efector_final)]])\n    return vector_posicion_final\n\n\n```\n\n#### 6.- Obtener el espacio del robot.\n\nEvaluaremos la posicion numerica de nuestro robot, dadas las ecuaciones de cinematica directa.\n\n\n```python\nimport matplotlib.pyplot as plt\n%matplotlib inline\nimport numpy as np\nd2r = np.deg2rad\n\n#Se evaluaran las funciones de posicion con sus respectivas variables\n\nfx = sp.lambdify((d1, d2, d3, theta2, theta3), pos_x, 'numpy')\nfy = sp.lambdify((d2, d3, theta2, theta3), pos_y, 'numpy')\nfz = sp.lambdify((l1,l4), pos_z, 'numpy')\n\n\n#Definimos las muestras de nuestro espacio, estan en este caso basadas en los GDL del robot\n\ntheta2s = np.linspace(d2r(-90), d2r(90)) # desired range of motion for joint 1\ntheta3s = np.linspace(d2r(-90), d2r(90)) # desired range of motion for joint 2\nl1s = np.linspace(0, 0.3) # desired range of motion for joint 2\nl4s = np.linspace(-0.04,0) # desired range of motion for joint 2\n\nzx = np.array(fx(d_1,d_2,d_3, theta2s, theta3s))\nzy = np.array(fy(d_2, d_3, theta2s, theta3s))\nzz = np.array(fz(l1s,l4s))\n\nzz = zz + 0.1196 #ajustamos el valor de z\n\n\nfig, ax1 = plt.subplots()\nax1.set_title('Posiciones en X')\nax1.plot(np.rad2deg(theta1s), zx, label = r'$p_x$')\nax1.set_xlabel(r'($\\theta_2$, $\\theta_3$) [deg]')\nax1.set_ylabel(r' posicion en x [mm]')\nplt.legend()\nplt.grid()\n\nfig, ax2 = plt.subplots()\nax2.set_title('Posiciones en Y')\nax2.plot(np.rad2deg(theta2s), zy, label = r'$p_y$')\nax2.set_xlabel(r'($\\theta_1$, $\\theta_2$) [deg]')\nax2.set_ylabel(r' posicion en y [mm]')\nplt.legend()\nplt.grid()\n\nfig, ax3 = plt.subplots()\nax3.set_title('Posiciones en Z')\nax3.plot(l1s, zz, label = r'$p_z$')\nax3.set_xlabel(r'($\\l_1$, $\\l_4$) [mm]')\nax3.set_ylabel(r' posicion en z [mm]')\nplt.legend()\nplt.grid()\n\n```\n\n## Seccion 3.-  Ejecutando la primera simulacion y calculo de error.\n\nUna vez conseguidas las ecuaciones correspondientes, los metodos y funciones necesarias para generar un movimiento controladp en el robot se ejecuta el siguiente codigo.

                                        \n\n
                                      • Se colocara al robot en una posicion mediante los metodos explicados en la seccion 1.
                                        • \n
                                      • Posterior a eso se calculara con los mismos angulos y traslaciones indicadas la posicion final que deberia tener en teoria nuestro modelo robotico basados en los metodos de la seccion 2.
                                        • \n\n
                                        \nPara ello debemos considerar: 
                                        \n
                                      • A1 = Al angulo que tomara el eslabon 1.
                                        • \n
                                      • A2 = Al angulo que tomara el eslabon 2.
                                        • \n
                                      • LC = Dezplazamiento que tomara la corredera en el eje Z.
                                        • \n
                                      • LF = Dezplazamiento que tomara el efector final en el eje Z.
                                        • \n\n\n\n```python\n# Conectamos con CoppeliaSim\nclientID = connect(19999)\n\n#Variables para simulacion y calculos\n\nA1 = 90 #Grados , eslabon 1.\nA2 = 45#Grados, eslabon 2.\nLC = 0.3 # mm, movimiento en z, corredera.\nLF = -0.04 # mm, movimiento en z, efector final.\n\n#ejecutamos en la simulacion un movimiento\nmover_robot_cinematica_directa(A1,A2,LC,LF)\n#Calculamos por cinematica directa la posicion del efector final \npos_cinematica_directa = get_posicion_cinematica_directa(A1,A2,LC,LF)\npos_cinematica_directa = pos_cinematica_directa.transpose() # posicion calculada para [x,y,z] respectivamente\npos_cinematica_directa\n```\n\n    conectado a 19999\n\n\n\n\n\n$\\displaystyle \\left[\\begin{matrix}0.2 \\operatorname{cos}\\left(135\\right) + 0.2 \\operatorname{cos}\\left(90\\right) + 0.18801 & 0.2 \\operatorname{sin}\\left(135\\right) + 0.2 \\operatorname{sin}\\left(90\\right) & 0.3796\\end{matrix}\\right]$\n\n\n\nUna vez llegado al punto deseado obtenemos la posicion del dummy para comprobar que los calculos esten correctos\n\n\n```python\n#Obtenemos la posicion real del efector fin en la simulacion, del dummy , [x,y,z] respectivamente\npos_sim = get_posicion_dummy()\npos_sim = pos_sim[1]\npos_sim\n```\n\n\n\n\n$\\displaystyle \\left[ 0.5646095871925354, \\  -0.010611959733068943, \\  0.1200198233127594\\right]$\n\n\n\nCalculamos el error entre el valor calculado y el valor real en la simulacion\n\n\n```python\n#Se calcula el error absoluto\nerror_x = abs(pos_sim[0] - pos_cinematica_directa[0]) # error en mm respecto del valor calculado\nerror_y = abs(pos_sim[1] - pos_cinematica_directa[1]) # error en mm respecto del valor calculado\nerror_z = abs(pos_sim[2] - pos_cinematica_directa[2]) # error en mm respecto del valor calculado\nprint(f'Error absoluto en X : {error_x}')\nprint(f'Error absoluto en Y : {error_y}')\nprint(f'Error absoluto en Z : {error_z}')\n\n#Se calcula el error relativo\nerror_x = (error_x / pos_sim[0] )* 100\nerror_y = (error_y / pos_sim[0] )* 100\nerror_z = (error_z / pos_sim[0] )* 100\n\nprint(f'Error del valor calculado con respecto al real en X, es de : {error_x} %')\nprint(f'Error del valor calculado con respecto al real en Y, es de : {error_y} %')\nprint(f'Error del valor calculado con respecto al real en Z, es de : {error_z} %')\n```\n\n    Error absoluto en X : -0.18801 - 0.2*cos(90) - 0.2*cos(135)\n    Error absoluto en Y : 0.2*sin(135) + 0.2*sin(90)\n    Error absoluto en Z : 0.379600000000000\n    Error del valor calculado con respecto al real en X, es de : zoo %\n    Error del valor calculado con respecto al real en Y, es de : zoo %\n    Error del valor calculado con respecto al real en Z, es de : zoo %\n\n\n## Seccion 4 .- Simulaciones de muestra\n\n\n```python\nimport time\n\ndef delay(tiempo_en_segundos):\n    \"\"\"\n    Hace un retardo antes de seguir ejecutando el programa\n    \"\"\"\n    time.sleep(tiempo_en_segundos)   # Delays for 5 seconds. You can also use a float value.\n\n```\n\n### Simulacion 1.- Juego de coordenadas preestablecidas, eslabones 1 y 2 en plano XY\n\n\n```python\n# Conectamos a coppelia\nclientID = connect(19999)\n\n#Esta rutina mueve en zigzag el eslabon 2 , sobre la junta con eslabon 1\ndef mover_eslabon_2_sobre_junta(angulo_eslabon_1):\n    #Movemos el robot con eslabon 1 : 90 \u00b0, eslabon 2 : -90 \u00b0, Corredera eleada hasta : 0 mm , Posicion de efector final :0 mm\n    mover_robot_cinematica_directa(angulo_eslabon_1,-90,0,0)\n    #Espera segundos antes del proximo movimiento\n    delay(1)\n    mover_robot_cinematica_directa(angulo_eslabon_1,90,0,0)\n    #Espera segundos antes del proximo movimiento\n    delay(1)\n    mover_robot_cinematica_directa(angulo_eslabon_1,-90,0,0)\n    #Espera segundos antes del proximo movimiento\n    delay(1)\n    mover_robot_cinematica_directa(angulo_eslabon_1,90,0,0)\n    #Espera segundos antes del proximo movimiento\n    delay(1)\n    \n#Inicia la rutina.\ndef rutina_1():\n    #Movemos el robot con eslabon 1 : 90 \u00b0, eslabon 2 : 0 \u00b0, Corredera eleada hasta : 0 mm , Posicion de efector final :0 mm\n    mover_robot_cinematica_directa(90,0,0,0)\n    #Espera 2 segundos antes del proximo movimiento\n    delay(2)\n    #Rutina pre dise\u00f1ada\n    mover_eslabon_2_sobre_junta(90)\n\n\n\n    #Movemos el robot con eslabon 1 : -90 \u00b0, eslabon 2 : 0 \u00b0, Corredera eleada hasta : 0 mm , Posicion de efector final :0 mm\n    mover_robot_cinematica_directa(-90,0,0,0)\n    #Espera 2 segundos antes del proximo movimiento\n    delay(2)\n    #Rutina pre dise\u00f1ada\n    mover_eslabon_2_sobre_junta(-90)\n\n\n    #Movemos el robot con eslabon 1 : -90 \u00b0, eslabon 2 : 0 \u00b0, Corredera eleada hasta : 0 mm , Posicion de efector final :0 mm\n    mover_robot_cinematica_directa(0,0,0,0)\n    #Espera 2 segundos antes del proximo movimiento\n    delay(2)\n    #Rutina pre dise\u00f1ada\n    mover_eslabon_2_sobre_junta(0)\n\n    #Regresamos el robot a su posicion inicial, por defecto\n    delay(1)\n    reiniciar_robot()\n    \n\n```\n\n    conectado a 19999\n\n\n\n```python\nrutina_1()\n```\n\n### Simulacion 2.- Plano XY agregar movimiento en Z\n\nSe realiza un dezplazamiento en Z, a la par que se realiza la rutina 1\n\n\n```python\n# Conectamos a coppelia\nclientID = connect(19999)\n\narriba_corredera()\ndelay(5)\nabajo_corredera()\nrutina_1()\narriba_corredera()\ndelay(5)\n```\n\n### Simulacion 3.- 'Boxeador'\n\nSe realiza un dezplazamiento en Z, a la par que se realiza una rutina diferente\n\n\n```python\n# Conectamos a coppelia\nclientID = connect(19999)\n\n#Sube hasta lo mas alto la corredera\narriba_corredera()\ndelay(10)\n\n#Rutina 'Boxeo'\ni = 0\nwhile i < 10 :\n    mover_robot_cinematica_directa(-90,90,0,0)\n    delay(0.8)\n    mover_robot_cinematica_directa(90,-90,0,0)\n    delay(0.8)\n    i = i + 1\n```\n\n    conectado a 19999\n\n", "meta": {"hexsha": "4fb233cef209bc40c50df72c221da0fbf5cc8ba6", "size": 105314, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Simulacion_Deskbot.ipynb", "max_stars_repo_name": "alfredoaguiararce/Deskbot", "max_stars_repo_head_hexsha": "6a889f2cfb481b43ec57986d5db69f458ac3b845", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Simulacion_Deskbot.ipynb", "max_issues_repo_name": "alfredoaguiararce/Deskbot", "max_issues_repo_head_hexsha": "6a889f2cfb481b43ec57986d5db69f458ac3b845", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Simulacion_Deskbot.ipynb", "max_forks_repo_name": "alfredoaguiararce/Deskbot", "max_forks_repo_head_hexsha": "6a889f2cfb481b43ec57986d5db69f458ac3b845", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 80.5152905199, "max_line_length": 20880, "alphanum_fraction": 0.788869476, "converted": true, "num_tokens": 9504, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "```python\nimport analysis.core as core\nimport analysis.fom as fom\nimport analysis.plot_tools as fomplot\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\nimport re\n%matplotlib inline\n```\n\n    /home/josh/lib/anaconda2/lib/python2.7/site-packages/pyne/serpent.py:11: QAWarning: pyne.serpent is not yet QA compliant.\n      warn(__name__ + \" is not yet QA compliant.\", QAWarning)\n\n\nThe intention of this notebook is to show how the plots and figures in this paper were created. The step by step process of developing the data analysis tools is shown. The final version of the functions are included in the `plot_tools` python package and may differ slightly in format from those presented here.\n\nDerivation of plots\n1. [Convergence Plots](#CONVERGENCE-PLOTS)\n1. [Ratio Plots](#RATIO-PLOTS)\n1. [Ratio Tables](#RATIO-TABLES)\n1. [Cycles/CPU](#CYCLES/CPU)\n\nPlots generated for paper:\n1. [All plots](#ALL-PLOTS)\n\nIdentify the locations where images will be saved, as well as the directory where the `wdt_data` folder is located.\n\n\n```python\nimg_dir = '/home/josh/repos/_publications/2017-wdt/img/raw/'\ndata_dir  = '/home/josh/repos/_wdt/wdt_data/'\nsave = False\n```\n\nUpload data files for a given range of $t_{\\text{wdt}}$ values.\n\n\n```python\nt_wdt = [0.1,0.2]\n\nbase = data_dir + 'pwr/S0100/'\ndirs = [base + 'W' + str('{:0>4d}'.format(int(n*1000))) for n in t_wdt]\nname = [str(n) for n in t_wdt]\npwr_comp = fom.Comparator(dirs,name)\n```\n\n### Plot setup\n\n\n```python\ndef fom_plot_setup(font_size=32, label_size=32):\n    plt.rc('text', usetex=True)\n    plt.rc('font', family='serif')\n    ax = plt.gca()\n    plt.rc('font', size=font_size)\n    ax.yaxis.get_major_formatter().set_powerlimits((0, 1))\n    #ax.spines[\"top\"].set_visible(False)  \n    #ax.spines[\"right\"].set_visible(False)\n    #ax.get_xaxis().tick_bottom()  \n    #ax.get_yaxis().tick_left()\n    plt.ylabel('Figure of merit', fontsize=label_size)\n    plt.xlabel('$n$', fontsize=label_size)\n    plt.xticks(fontsize=font_size)\n    plt.yticks(fontsize=font_size)\n    ax.tick_params(axis='y', labelsize=font_size)\n    line = plt.gca().get_lines()[0]\n    line.set_marker('.') \n    line.set_color('black')\n    plt.grid(True,which='both',color='0.5')\n    plt.xscale('linear')\n    return plt.gcf()\n```\n\n# CONVERGENCE PLOTS\n\n\n```python\nlabel = 'INF_FLX'\ngrp = 1\nn = 0\n\n# Get data and sort by cycle number\ndata = pwr_comp.data[n].get_data(label,grp)\ndata = data[data[:,0].argsort()]\n# Get stdev\nstd = np.sqrt(pwr_comp.data[n].get_var(label,grp))\n\nplt.figure(figsize=(12,9))\nplt.plot(data[:,0],data[:,1],'k.')\n\nfom_plot_setup(12,12)\nplt.title(pwr_comp.data[n].name)\n\nplt.axhline(data[-1,1] - std, ls='--', c='k')\nplt.axhline(data[-1,1] + std, ls='--', c='k')\n\nplt.show()\n```\n\n\n```python\ndef conv_plot(comp, label, grp, n):\n\n    # Get data and sort by cycle number\n    data = comp.data[n].get_data(label,grp)\n    data = data[data[:,0].argsort()]\n    # Get stdev\n    std = np.sqrt(comp.data[n].get_var(label,grp))\n\n    plt.figure(figsize=(12,9))\n    plt.plot(data[:,0],data[:,1],'k.')\n\n    fom_plot_setup(12,12)\n    plt.title(comp.data[n].name)\n\n    plt.axhline(data[-1,1] - std, ls='--', c='k')\n    plt.axhline(data[-1,1] + std, ls='--', c='k')\n\n    plt.show()\n```\n\n-----\n# RATIO PLOTS\n\nIngest the all the data for the PWR test case\n\n\n```python\nt_wdt = np.linspace(0.1,1.0,10)\n\nbase = data_dir + 'pwr/S0100/'\ndirs = [base + 'W' + str('{:0>4d}'.format(int(n*1000))) for n in t_wdt]\nname = [str(n) for n in t_wdt]\npwr_comp = fom.Comparator(dirs,name)\n```\n\n    Uploaded 1450 files.\n    Uploaded 630 files.\n    Uploaded 630 files.\n    Uploaded 582 files.\n    Uploaded 937 files.\n    Uploaded 911 files.\n    Uploaded 1058 files.\n    Uploaded 1410 files.\n    Uploaded 1002 files.\n    Uploaded 403 files.\n\n\nGet FOM data for each data set. $x$ will get the name of the set (the value of $t_{\\text{wdt}}$) and $y$ gets the final FOM value (returned by `get_data`, which needs to be sorted)\n\n\n```python\nlabel = 'INF_FLX'\ngrp = 1\n\nx = []\ny = []\nyerr = []\n\nfor data_set in pwr_comp.data:\n    # Get FOM data and sort by cycle\n    data = data_set.get_data(label,grp)\n    data = data[data[:,0].argsort()]\n    \n    x.append(float(data_set.name))\n    y.append(data[-1,1])\n    \n    # Get STD from last half of dataset, square root is required to\n    # make it the standard deviation\n    yerr.append(np.sqrt(data_set.get_var(label,grp)))\n```\n\nPlot the FOM\n\n\n```python\nplt.figure(figsize=(12,9))\nplt.errorbar(x,y,yerr=yerr, fmt='k.',ms=12)\n\nfom_plot_setup(12,12)\nplt.show()\n```\n\nCalculate the ratio of each point to the base case, $t_{\\text{wdt}} = 0.1$ which corresponds to no WDT. Propegate the errors of the ratio using:\n\\begin{align}\nr &= \\frac{FOM}{FOM_0} \\\\\n\\delta r &= r\\times \\sqrt{\\left(\\frac{\\delta FOM}{FOM}\\right)^2 + \\left(\\frac{\\delta FOM_0}{FOM_0}\\right)^2}\n\\end{align}\n\n\nWe can verify that the index `0` corresponds to the base case by checking the $x$ vector\n\n\n```python\nx[0]\n```\n\n\n\n\n    0.1\n\n\n\n\n```python\nr = np.ones_like(x)\nrerr = np.zeros_like(x)\nfor i in range(1,len(x)):\n    r[i] = y[i]/y[0]\n    rerr[i] = r[i]*np.sqrt(np.power(yerr[i]/y[i],2) \n                           + np.power(yerr[0]/y[0],2))\n```\n\n\n```python\nfontsize=20\nplt.figure(figsize=(12,9))\nplt.errorbar(x,r,yerr=rerr, fmt='k.', ms=12, capsize=0)\nplt.xlim([0.15,1.05])\nfom_plot_setup(fontsize,fontsize)\nplt.ylabel('Figure of merit ratio')\nplt.xlabel('$t_{\\mathrm{wdt}}$', fontsize=fontsize+4)\nplt.axhline(y=1.0, ls='--', c='k')\nplt.show()\n```\n\nCodify those steps in functions:\n\n`get_fom` will return the values of $t_{\\mathrm{wdt}}$, FOM and FOM error:\n\n\n```python\ndef get_fom(comparator, label, grp):\n    x = []\n    y = []\n    yerr = []\n    \n    for data_set in comparator.data:\n        # Get FOM data and sort by cycle\n        data = data_set.get_data(label,grp)\n        data = data[data[:,0].argsort()]\n\n        x.append(float(data_set.name))\n        y.append(data[-1,1])\n\n        # Get STD from last half of dataset, square root is required to\n        # make it the standard deviation\n        yerr.append(np.sqrt(data_set.get_var(label,grp)))\n        \n    return x, y, yerr\n```\n\n`get_ratios` will return the values of $t_{\\mathrm{wdt}}$, FOM ratios and FOM ratio error:\n\n\n```python\ndef get_ratios(comparator, label, grp):\n    x, y, yerr = get_fom(comparator, label, grp)\n    \n    # Find base case\n    n = x.index(0.1)\n    \n    r = np.ones_like(x)\n    rerr = np.zeros_like(x)\n    \n    for i in range(0,len(x)):\n        if i != n:\n            r[i] = y[i]/y[n]\n            rerr[i] = r[i]*np.sqrt(np.power(yerr[i]/y[i],2) \n                                   + np.power(yerr[n]/y[n],2))\n            \n    return x, r, rerr\n```\n\nThese will plot the values from the previous functions and save if desired\n\n\n```python\ndef plot_fom(comparator, casename, label, grp, save=False, fontsize=20):\n   \n    if label=='INF_FLX':\n        param = ' infinite flux '\n    elif label =='INF_TOT':\n        param = ' infinite $\\Sigma_t$ '\n        \n    title = casename + param + 'for group ' + str(grp)\n    \n    x, y, yerr = get_fom(comparator, label, grp)\n    \n    plt.figure(figsize=(12,9))\n    plt.errorbar(x,y,yerr=yerr, fmt='k.',ms=12)\n    plt.title(title)\n    fom_plot_setup(fontsize,fontsize)\n    plt.xlim([0.05,1.05])\n    plt.xticks(np.arange(0.1,1.1, 0.1))\n    plt.show()\n```\n\n\n```python\ndef plot_ratios(comparator, casename, label, grp, save=False, fontsize=20):\n    if label=='INF_FLX':\n        param = ' infinite flux '\n    elif label =='INF_TOT':\n        param = ' infinite $\\Sigma_t$ '\n        \n    title = casename + param + 'for group ' + str(grp)\n    filename = casename.lower() + '_' + label.lower() + '_' + str(grp)\n\n    x, r, rerr = get_ratios(comparator, label, grp)\n\n    plt.figure(figsize=(12,9))\n    plt.errorbar(x,r,yerr=rerr, fmt='k.', ms=12, capsize=0)\n    fom_plot_setup(fontsize,fontsize)\n    plt.xticks(np.arange(0.1,1.1, 0.1))\n    plt.xlim([0.15,1.05])\n    plt.title(title)\n    plt.ylabel('Figure of merit ratio')\n    plt.xlabel('$t_{\\mathrm{wdt}}$', fontsize=fontsize+4)\n    plt.axhline(y=1.0, ls='--', c='k')\n    if save:\n        plt.savefig(img_dir + filename + \".pdf\", \n                    format = 'pdf', bbox_inches='tight')\n    else:\n        plt.show()\n```\n\nWe can check that this returns the same plot as before:\n\n\n```python\nplot_ratios(pwr_comp, 'PWR','INF_FLX',1)\n```\n\n-----\n# RATIO TABLES\n\nWe will use pandas to display a table of the raw FOM values, ratios, and the errors associated with each using the same functions as before.\n\n\n```python\nlabel = 'INF_FLX'\ngrp = 1\n# Get fom and ratios\nx, y, yerr = get_fom(pwr_comp, label, grp)\nx, r, rerr = get_ratios(pwr_comp, label, grp)\n```\n\nDisplay in a pandas table\n\n\n```python\nd = {'twdt' : x, 'fom' : y, 'fom_err': yerr, 'r' : r, 'r_err' : rerr}\ndf = pd.DataFrame(d)\n#Move twdt to the front\ncols = df.columns.tolist()\ncols = cols[-1:] + cols[:-1]\ndf = df[cols]\ndf\n```\n\n\n\n\n
                                        \n\n  \n    \n      \n      \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n    \n  \n
                                        twdtfomfom_errrr_err
                                        00.11.242057e+064966.5491621.0000000.000000
                                        10.21.258577e+068850.3511841.0133000.008197
                                        20.31.267212e+0611694.3797021.0202520.010261
                                        30.41.302102e+0612557.8820571.0483430.010945
                                        40.51.264776e+0611080.2676271.0182910.009806
                                        50.61.306169e+064467.0041181.0516170.005533
                                        60.71.310663e+0610530.7397691.0552360.009470
                                        70.81.296558e+063063.2599701.0438800.004848
                                        80.91.305225e+065719.7007201.0508570.006234
                                        91.01.300926e+0615046.6889131.0473960.012818
                                        \n
                                        \n\n\n\nDefine formatters for the columns for the number of decimal places\n\n\n```python\nfom_p = 3\nrat_p = 3\ndef fom_form(x):\n    return '{:.0f}'.format(np.around(x/np.power(10,fom_p)))\ndef ratio(x):\n    return '{:.3f}'.format(x)\n```\n\n\n```python\ndef pandas_table(x, y, yerr, r, rerr):\n    d = {'twdt' : x, 'fom' : y, 'fom_err': yerr, 'r' : r, 'r_err' : rerr}\n    df = pd.DataFrame(d)\n    #Move twdt to the front\n    cols = df.columns.tolist()\n    cols = cols[-1:] + cols[:-1]\n    df = df[cols]\n    return df\n```\n\n\n```python\ndef pandas_format(df, fom_p, rat_p):\n    # Formats the data frame\n    def fom_form(x):\n        return '{:.0f}'.format(np.around(x/np.power(10,fom_p)))\n    def ratio(x):\n        return '{:.3f}'.format(x)\n    df['fom'] = df['fom'].apply(fom_form)\n    df['fom_err'] = df['fom_err'].apply(fom_form)\n    df['r']  = df['r'].apply(ratio)\n    df['r_err']  = df['r_err'].apply(ratio)\n```\n\n\n```python\ndef latex(df, fom_p, rat_p):\n    pandas_format(df, fom_p, rat_p)\n    df.insert(2,'fom_pm',['$\\pm$']*len(x))\n    df.insert(5,'r_pm',['$\\pm$']*len(x))\n    def f(x): \n        return x\n    print df.to_latex(index=False, escape=False, column_format='rrcrrcr')\n```\n\n\n```python\ndf = pandas_table(x,y,yerr,r,rerr)\nlatex(df, 3, 3)\n```\n\n    \\begin{tabular}{rrcrrcr}\n    \\toprule\n     twdt &   fom & fom_pm & fom_err &      r &   r_pm &  r_err \\\\\n    \\midrule\n      0.1 &  1242 &  $\\pm$ &       5 &  1.000 &  $\\pm$ &  0.000 \\\\\n      0.2 &  1259 &  $\\pm$ &       9 &  1.013 &  $\\pm$ &  0.008 \\\\\n      0.3 &  1267 &  $\\pm$ &      12 &  1.020 &  $\\pm$ &  0.010 \\\\\n      0.4 &  1302 &  $\\pm$ &      13 &  1.048 &  $\\pm$ &  0.011 \\\\\n      0.5 &  1265 &  $\\pm$ &      11 &  1.018 &  $\\pm$ &  0.010 \\\\\n      0.6 &  1306 &  $\\pm$ &       4 &  1.052 &  $\\pm$ &  0.006 \\\\\n      0.7 &  1311 &  $\\pm$ &      11 &  1.055 &  $\\pm$ &  0.009 \\\\\n      0.8 &  1297 &  $\\pm$ &       3 &  1.044 &  $\\pm$ &  0.005 \\\\\n      0.9 &  1305 &  $\\pm$ &       6 &  1.051 &  $\\pm$ &  0.006 \\\\\n      1.0 &  1301 &  $\\pm$ &      15 &  1.047 &  $\\pm$ &  0.013 \\\\\n    \\bottomrule\n    \\end{tabular}\n    \n\n\nAll in one function\n\n\n```python\ndef make_table(comp, label, grp, fom_p, rat_p=3):\n    # Get fom and ratios\n    x, y, yerr = get_fom(pwr_comp, label, grp)\n    x, r, rerr = get_ratios(pwr_comp, label, grp)\n    df = pandas_table(x,y,yerr,r,rerr)\n    latex(df, fom_p, rat_p)\n```\n\n\n```python\nmake_table(pwr_comp,'INF_FLX',1,3,3)\n```\n\n    \\begin{tabular}{rrcrrcr}\n    \\toprule\n     twdt &   fom & fom_pm & fom_err &      r &   r_pm &  r_err \\\\\n    \\midrule\n      0.1 &  1242 &  $\\pm$ &       5 &  1.000 &  $\\pm$ &  0.000 \\\\\n      0.2 &  1259 &  $\\pm$ &       9 &  1.013 &  $\\pm$ &  0.008 \\\\\n      0.3 &  1267 &  $\\pm$ &      12 &  1.020 &  $\\pm$ &  0.010 \\\\\n      0.4 &  1302 &  $\\pm$ &      13 &  1.048 &  $\\pm$ &  0.011 \\\\\n      0.5 &  1265 &  $\\pm$ &      11 &  1.018 &  $\\pm$ &  0.010 \\\\\n      0.6 &  1306 &  $\\pm$ &       4 &  1.052 &  $\\pm$ &  0.006 \\\\\n      0.7 &  1311 &  $\\pm$ &      11 &  1.055 &  $\\pm$ &  0.009 \\\\\n      0.8 &  1297 &  $\\pm$ &       3 &  1.044 &  $\\pm$ &  0.005 \\\\\n      0.9 &  1305 &  $\\pm$ &       6 &  1.051 &  $\\pm$ &  0.006 \\\\\n      1.0 &  1301 &  $\\pm$ &      15 &  1.047 &  $\\pm$ &  0.013 \\\\\n    \\bottomrule\n    \\end{tabular}\n    \n\n\n----\n# CYCLES/CPU\n\nLoading on the machine may cause the simulation to slow down, affecting the FOM. It is therefore important to verify that cycles require a consistent amount of CPU time, as much as possible.\n\nWe need to access the `DataFile` within each `Analyzer` that is in the `Comparater.` The Analyzer has all the data from one value of $t_{\\mathrm{wdt}}$, so we need to verify we have the right index.\n\n\n```python\npwr_comp.data[0].name\n```\n\n\n\n\n    '0.1'\n\n\n\nHere we are looking at the base case. We will now cycle through all the `DataFile` objects and get the value of the cycles (`CYCLE_IDX`) and the CPU time.\n\n\n```python\ncycles = []\ncpu = []\nfor data in pwr_comp.data[0].data:\n    cycles.append(data.get_data('CYCLE_IDX')[0][0])\n    cpu.append(data.get_cpu())\n    \ncyc_cpu = np.column_stack((cycles, cpu))\n    \ncyc_cpu = cyc_cpu[cyc_cpu[:,0].argsort()]\n```\n\nWhat we are interested in is the differential values:\n\\begin{equation}\n\\frac{\\mathrm{cycles}_{i} - \\mathrm{cycles}_{i-1}}{\\mathrm{CPU}_i - \\mathrm{CPU}_{i-1}}\n\\end{equation}\n\n\n```python\ndcyc_cpu = []\n\nfor i in range(1,len(cycles)):\n    dcyc = cyc_cpu[i,0] - cyc_cpu[i-1,0]\n    dcpu = cyc_cpu[i,1] - cyc_cpu[i-1,1]\n    dcyc_cpu.append(dcyc/dcpu)\n```\n\n\n```python\nplt.plot(dcyc_cpu, 'k.')\nfom_plot_setup(12,12)\nplt.show()\n```\n\nAll contained in one function:\n\n\n```python\ndef cyc_cpu_plot(comp, twdt):\n    # Find index\n    for i, analyzer in enumerate(comp.data):\n        if analyzer.name == str(twdt):\n            n = i\n    cycles = []\n    cpu = []\n    for data in comp.data[n].data:\n        cycles.append(data.get_data('CYCLE_IDX')[0][0])\n        cpu.append(data.get_cpu())\n    \n    cyc_cpu = np.column_stack((cycles, cpu))\n    \n    cyc_cpu = cyc_cpu[cyc_cpu[:,0].argsort()]\n    dcyc_cpu = []\n\n    for i in range(1,len(cycles)):\n        dcyc = cyc_cpu[i,0] - cyc_cpu[i-1,0]\n        dcpu = cyc_cpu[i,1] - cyc_cpu[i-1,1]\n        dcyc_cpu.append(dcyc/dcpu)\n    plt.title('Cycles/CPU time for ' + comp.data[n].name)\n    plt.plot(dcyc_cpu, 'k.')\n    fom_plot_setup(12,12)\n    plt.ylabel('Cycles/CPU time')\n    plt.show()\n                        \n```\n\n----\n# ALL PLOTS\n\nThis section uses the `plot_tools` package to actually generate all required plots. This section is designed to be run independent of the rest of the notebook and will walk through all the plots presented in the paper.\n\n\n```python\nimport analysis.core as core\nimport analysis.fom as fom\nimport analysis.plot_tools as fomplot\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\nimport re\n%matplotlib inline\n```\n\n    /home/josh/lib/anaconda2/lib/python2.7/site-packages/pyne/serpent.py:11: QAWarning: pyne.serpent is not yet QA compliant.\n      warn(__name__ + \" is not yet QA compliant.\", QAWarning)\n\n\n\n```python\nimg_dir = '/home/josh/repos/_publications/2017-wdt/img/'\ndata_dir  = '/home/josh/repos/_wdt/wdt_data/'\nsave = False\n```\n\n## PWR\n\n\n```python\nt_wdt = np.linspace(0.1,1.0,10)\n\nbase = data_dir + 'pwr/S0100/'\ndirs = [base + 'W' + str('{:0>4d}'.format(int(n*1000))) for n in t_wdt]\nname = [str(n) for n in t_wdt]\npwr_comp = fom.Comparator(dirs,name)\n```\n\n    Uploaded 1450 files.\n    Uploaded 630 files.\n    Uploaded 630 files.\n    Uploaded 582 files.\n    Uploaded 937 files.\n    Uploaded 911 files.\n    Uploaded 1058 files.\n    Uploaded 1410 files.\n    Uploaded 1002 files.\n    Uploaded 403 files.\n\n\nFirst, we plot the Cycles/CPU for the comparator to see if there are any issues.\n\n\n```python\nfor t in range(1,11):\n    fomplot.cyc_cpu_plot(pwr_comp, t*0.1)\n```\n\n\n```python\nreload(fomplot)\nfomplot.cyc_cpu_plot(pwr_comp,0.1)\n```\n\n## INFINITE FLUX\n\n\n```python\nlabel = 'INF_FLX'\n```\n\n\n```python\nfomplot.plot_ratios(pwr_comp, 'PWR',label,1,save=save, img_dir=img_dir)\n```\n\n\n```python\nfomplot.conv_plot(pwr_comp, 'INF_FLX',1,3)\n```\n\n\n```python\nfomplot.conv_plot(pwr_comp, 'INF_FLX',1,3,44000)\n```\n\n\n```python\ndel pwr_comp2.data[3]\n```\n\n\n```python\nreload(fomplot)\nfomplot.plot_ratios(pwr_comp, 'PWR',label,1,cycle_caps=[(0.4,44000)],corr=True)\n```\n\n\n```python\nfomplot.plot_ratios(pwr_comp, 'PWR',label,2,cycle_caps=[(0.4,44000)])\n```\n\n\n```python\ndef get_cyc_cpu(comp, twdt):\n    for i, analyzer in enumerate(comp.data):\n        if analyzer.name == str(twdt):\n            n = i\n    cycles = []\n    cpu = []\n    for data in comp.data[n].data:\n        cycles.append(data.get_data('CYCLE_IDX')[0][0])\n        cpu.append(data.get_cpu())\n    \n    cyc_cpu = np.column_stack((cycles, cpu))\n    \n    cyc_cpu = cyc_cpu[cyc_cpu[:,0].argsort()]\n    dcyc_cpu = []\n\n    for i in range(1,len(cycles)):\n        cycs = np.sum(cyc_cpu[0:i,0])\n        cpus = np.sum(cyc_cpu[0:i,0])\n        dcyc_cpu.append(cycs/cpus)\n    #    dcyc = cyc_cpu[i,0] - cyc_cpu[i-1,0]\n    #    dcpu = cyc_cpu[i,1] - cyc_cpu[i-1,1]\n    #    dcyc_cpu.append(dcyc/dcpu)\n        \n    return dcyc_cpu\n```\n\n\n```python\ndef get_corr_fom(comp, label, grp, twdt):\n    for i, analyzer in enumerate(comp.data):\n        if analyzer.name == str(twdt):\n            n = i\n    data = comp.data[n].get_data(label,grp,fom=False)\n    cyc_cpu = get_cyc_cpu(comp,twdt)\n    \n    assert np.shape(data)[0] == np.shape(cyc_cpu)[0] + 1, \"something went wrong with the array size\"\n    \n    # Calculate FOM\n    fom = np.zeros_like(data)\n    data = data[data[:,0].argsort()]\n    \n    fom[:,0] = data[:,0]\n    denom = np.multiply(np.power(data[:,1],2),data[:,0])\n    fom[1:,1] = np.multiply(np.power(denom[1:], -1),cyc_cpu)\n    #fom[1:,1] = np.multiply(np.power(denom[1:], -1),np.average(cyc_cpu))\n    \n    yerr = np.sqrt(np.var(fom[int(np.ceil(np.shape(fom)[0]/2.0)):,1])) \n    \n    return fom[1:,:], yerr\n```\n\n\n```python\nfom\n```\n\n\n\n\n    array([[  1.00000000e+02,   1.57208453e+06],\n           [  1.50000000e+02,   1.53404182e+06],\n           [  2.00000000e+02,   1.50618738e+06],\n           ..., \n           [  7.24000000e+04,   1.23809132e+06],\n           [  7.24500000e+04,   1.24879199e+06],\n           [  7.25000000e+04,   1.23868065e+06]])\n\n\n\n\n```python\nfom, yerr = get_corr_fom(pwr_comp, 'INF_FLX', 1, 0.4)\nplt.plot(fom[:,0], fom[:,1],'k.')\nfom_plot_setup(12,12)\nplt.show()\nyerr\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "a46e75b42dc2e991b948c794224780502704a522", "size": 786214, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Untitled.ipynb", "max_stars_repo_name": "jsrehak/WDT_Analysis", "max_stars_repo_head_hexsha": "303de000432d6c089fe53adeef09faaf0e884c17", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Untitled.ipynb", "max_issues_repo_name": "jsrehak/WDT_Analysis", "max_issues_repo_head_hexsha": "303de000432d6c089fe53adeef09faaf0e884c17", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Untitled.ipynb", "max_forks_repo_name": "jsrehak/WDT_Analysis", "max_forks_repo_head_hexsha": "303de000432d6c089fe53adeef09faaf0e884c17", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 481.7487745098, "max_line_length": 60642, "alphanum_fraction": 0.9282930602, "converted": true, "num_tokens": 6943, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "\n\n#[Quantum Generative Adversarial Networks with Cirq + TensorFlow](https://pennylane.ai/qml/demos/tutorial_QGAN.html)\n\nWe begin by importing PennyLane, NumPy, and TensorFlow.\n\n\n```python\n# install packages\n!pip install pennylane\n!pip install tensorflow\n!pip install cirq\n!pip install pennylane-cirq\n```\n\n    Collecting pennylane\n      Downloading PennyLane-0.21.0-py3-none-any.whl (800 kB)\n    \u001b[K     |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 800 kB 3.2 MB/s \n    \u001b[?25hCollecting autoray\n      Downloading autoray-0.2.5-py3-none-any.whl (16 kB)\n    Collecting semantic-version==2.6\n      Downloading semantic_version-2.6.0-py3-none-any.whl (14 kB)\n    Collecting pennylane-lightning>=0.21\n      Downloading PennyLane_Lightning-0.21.0-cp37-cp37m-manylinux_2_17_x86_64.manylinux2014_x86_64.whl (7.8 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autograd in /usr/local/lib/python3.7/dist-packages (from pennylane) (1.3)\n    Requirement already satisfied: appdirs in /usr/local/lib/python3.7/dist-packages (from pennylane) (1.4.4)\n    Requirement already satisfied: scipy in /usr/local/lib/python3.7/dist-packages (from pennylane) (1.4.1)\n    Requirement already satisfied: cachetools in /usr/local/lib/python3.7/dist-packages (from pennylane) (4.2.4)\n    Collecting ninja\n      Downloading ninja-1.10.2.3-py2.py3-none-manylinux_2_5_x86_64.manylinux1_x86_64.whl (108 kB)\n    \u001b[K     |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 108 kB 39.4 MB/s \n    \u001b[?25hRequirement already satisfied: future>=0.15.2 in /usr/local/lib/python3.7/dist-packages (from autograd->pennylane) (0.16.0)\n    Installing collected packages: ninja, toml, semantic-version, retworkx, pennylane-lightning, autoray, pennylane\n    Successfully installed autoray-0.2.5 ninja-1.10.2.3 pennylane-0.21.0 pennylane-lightning-0.21.0 retworkx-0.11.0 semantic-version-2.6.0 toml-0.10.2\n    Requirement already satisfied: tensorflow in /usr/local/lib/python3.7/dist-packages (2.8.0)\n    Requirement already satisfied: tensorflow-io-gcs-filesystem>=0.23.1 in /usr/local/lib/python3.7/dist-packages (from tensorflow) (0.24.0)\n    Requirement already satisfied: flatbuffers>=1.12 in /usr/local/lib/python3.7/dist-packages (from tensorflow) (2.0)\n    Requirement already satisfied: six>=1.12.0 in /usr/local/lib/python3.7/dist-packages (from tensorflow) (1.15.0)\n    Requirement already satisfied: typing-extensions>=3.6.6 in /usr/local/lib/python3.7/dist-packages (from tensorflow) (3.10.0.2)\n    Requirement already satisfied: grpcio<2.0,>=1.24.3 in /usr/local/lib/python3.7/dist-packages (from tensorflow) (1.44.0)\n    Collecting tf-estimator-nightly==2.8.0.dev2021122109\n      Downloading 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iso8601-0.1.16-py2.py3-none-any.whl (10 kB)\n    Collecting h11~=0.9.0\n      Downloading h11-0.9.0-py2.py3-none-any.whl (53 kB)\n    \u001b[K     |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 53 kB 1.8 MB/s \n    \u001b[?25hCollecting qcs-api-client~=0.8.0\n      Downloading qcs_api_client-0.8.0-py3-none-any.whl (97 kB)\n    \u001b[K     |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 97 kB 4.9 MB/s \n    \u001b[?25hRequirement already satisfied: python-dateutil~=2.8.1 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq) (2.8.2)\n    Collecting rfc3339~=6.2\n      Downloading rfc3339-6.2-py3-none-any.whl (5.5 kB)\n    Collecting attrs~=20.3.0\n      Downloading 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kB)\n    Requirement already satisfied: toml~=0.10.2 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq) (0.10.2)\n    Collecting certifi~=2021.5.30\n      Downloading certifi-2021.5.30-py2.py3-none-any.whl (145 kB)\n    \u001b[K     |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 145 kB 59.7 MB/s \n    \u001b[?25hCollecting sniffio~=1.2.0\n      Downloading sniffio-1.2.0-py3-none-any.whl (10 kB)\n    Collecting pyquil~=3.0.0\n      Downloading pyquil-3.0.1-py3-none-any.whl (220 kB)\n    \u001b[K     |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 220 kB 33.2 MB/s \n    \u001b[?25hCollecting pydantic~=1.8.2\n      Downloading 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\u001b[?25hCollecting retry<0.10.0,>=0.9.2\n      Downloading retry-0.9.2-py2.py3-none-any.whl (8.0 kB)\n    Requirement already satisfied: zipp>=0.5 in /usr/local/lib/python3.7/dist-packages (from importlib-metadata<4.0.0,>=3.7.3->pyquil~=3.0.0->cirq-rigetti==0.13.1->cirq) (3.7.0)\n    Requirement already satisfied: urllib3!=1.25.0,!=1.25.1,<1.26,>=1.21.1 in /usr/local/lib/python3.7/dist-packages (from requests~=2.18->cirq-aqt==0.13.1->cirq) (1.24.3)\n    Requirement already satisfied: chardet<4,>=3.0.2 in /usr/local/lib/python3.7/dist-packages (from requests~=2.18->cirq-aqt==0.13.1->cirq) (3.0.4)\n    Requirement already satisfied: py<2.0.0,>=1.4.26 in /usr/local/lib/python3.7/dist-packages (from retry<0.10.0,>=0.9.2->pyquil~=3.0.0->cirq-rigetti==0.13.1->cirq) (1.11.0)\n    Requirement already satisfied: decorator>=3.4.2 in /usr/local/lib/python3.7/dist-packages (from retry<0.10.0,>=0.9.2->pyquil~=3.0.0->cirq-rigetti==0.13.1->cirq) (4.4.2)\n    Collecting msgpack<1.0,>=0.6\n      Downloading msgpack-0.6.2-cp37-cp37m-manylinux1_x86_64.whl (243 kB)\n    \u001b[K     |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 243 kB 63.1 MB/s \n    \u001b[?25hCollecting python-rapidjson\n      Downloading python_rapidjson-1.6-cp37-cp37m-manylinux_2_17_x86_64.manylinux2014_x86_64.whl (1.6 MB)\n    \u001b[K     |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 1.6 MB 40.4 MB/s \n    \u001b[?25hRequirement already satisfied: pyzmq>=17 in /usr/local/lib/python3.7/dist-packages (from rpcq<4.0.0,>=3.6.0->pyquil~=3.0.0->cirq-rigetti==0.13.1->cirq) (22.3.0)\n    Collecting ruamel.yaml\n      Downloading ruamel.yaml-0.17.21-py3-none-any.whl (109 kB)\n    \u001b[K     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sha256=ef16f54f001be00ae893fef0bd6225fd372a80f02bdd2235691c7437787faac6\n      Stored in directory: /root/.cache/pip/wheels/d7/61/3c/9ac365f55966367be8d77dbeb21a3ddece3c466e660121e8d6\n      Building wheel for retrying (setup.py) ... \u001b[?25l\u001b[?25hdone\n      Created wheel for retrying: filename=retrying-1.3.3-py3-none-any.whl size=11447 sha256=32b960023d2026c26cc6ce9ddefffcf2c914a2c36641506bc6cfbf1316a29032\n      Stored in directory: /root/.cache/pip/wheels/f9/8d/8d/f6af3f7f9eea3553bc2fe6d53e4b287dad18b06a861ac56ddf\n      Building wheel for rpcq (setup.py) ... \u001b[?25l\u001b[?25hdone\n      Created wheel for rpcq: filename=rpcq-3.9.2-py3-none-any.whl size=45877 sha256=623ef1a1df3ea625f3a0650f4b7f499299009031309684006779de3e2bb544d0\n      Stored in directory: /root/.cache/pip/wheels/96/fb/72/b2179c8c6be1c6ded2d987247ac425957ca56208d81e99a2f2\n    Successfully built lark retrying rpcq\n    Installing collected packages: sniffio, six, rfc3986, h11, ruamel.yaml.clib, httpcore, certifi, ruamel.yaml, rfc3339, retrying, python-rapidjson, pyjwt, pydantic, msgpack, iso8601, httpx, attrs, scipy, rpcq, retry, qcs-api-client, lark, importlib-metadata, duet, pyquil, cirq-core, cirq-web, cirq-rigetti, cirq-pasqal, cirq-ionq, cirq-google, cirq-aqt, cirq\n      Attempting uninstall: six\n        Found existing installation: six 1.15.0\n        Uninstalling six-1.15.0:\n          Successfully uninstalled six-1.15.0\n      Attempting uninstall: certifi\n        Found existing installation: certifi 2021.10.8\n        Uninstalling certifi-2021.10.8:\n          Successfully uninstalled certifi-2021.10.8\n      Attempting uninstall: msgpack\n        Found existing installation: msgpack 1.0.3\n        Uninstalling msgpack-1.0.3:\n          Successfully uninstalled msgpack-1.0.3\n      Attempting uninstall: attrs\n        Found existing installation: attrs 21.4.0\n        Uninstalling attrs-21.4.0:\n          Successfully uninstalled attrs-21.4.0\n      Attempting uninstall: scipy\n        Found existing installation: scipy 1.4.1\n        Uninstalling scipy-1.4.1:\n          Successfully uninstalled scipy-1.4.1\n      Attempting uninstall: importlib-metadata\n        Found existing installation: importlib-metadata 4.11.2\n        Uninstalling importlib-metadata-4.11.2:\n          Successfully uninstalled importlib-metadata-4.11.2\n    \u001b[31mERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.\n    markdown 3.3.6 requires importlib-metadata>=4.4; python_version < \"3.10\", but you have importlib-metadata 3.10.1 which is incompatible.\n    google-colab 1.0.0 requires six~=1.15.0, but you have six 1.16.0 which is incompatible.\n    datascience 0.10.6 requires folium==0.2.1, but you have folium 0.8.3 which is incompatible.\n    albumentations 0.1.12 requires imgaug<0.2.7,>=0.2.5, but you have imgaug 0.2.9 which is incompatible.\u001b[0m\n    Successfully installed attrs-20.3.0 certifi-2021.5.30 cirq-0.13.1 cirq-aqt-0.13.1 cirq-core-0.13.1 cirq-google-0.13.1 cirq-ionq-0.13.1 cirq-pasqal-0.13.1 cirq-rigetti-0.13.1 cirq-web-0.13.1 duet-0.2.3 h11-0.9.0 httpcore-0.11.1 httpx-0.15.5 importlib-metadata-3.10.1 iso8601-0.1.16 lark-0.11.3 msgpack-0.6.2 pydantic-1.8.2 pyjwt-1.7.1 pyquil-3.0.1 python-rapidjson-1.6 qcs-api-client-0.8.0 retry-0.9.2 retrying-1.3.3 rfc3339-6.2 rfc3986-1.5.0 rpcq-3.9.2 ruamel.yaml-0.17.21 ruamel.yaml.clib-0.2.6 scipy-1.7.3 six-1.16.0 sniffio-1.2.0\n\n\n\n\n    Collecting pennylane-cirq\n      Downloading PennyLane_Cirq-0.19.0-py3-none-any.whl (22 kB)\n    Requirement already satisfied: pennylane>=0.17 in /usr/local/lib/python3.7/dist-packages (from pennylane-cirq) (0.21.0)\n    Requirement already satisfied: cirq>=0.10 in /usr/local/lib/python3.7/dist-packages (from pennylane-cirq) (0.13.1)\n    Requirement already satisfied: cirq-google==0.13.1 in /usr/local/lib/python3.7/dist-packages (from cirq>=0.10->pennylane-cirq) (0.13.1)\n    Requirement already satisfied: cirq-aqt==0.13.1 in /usr/local/lib/python3.7/dist-packages (from cirq>=0.10->pennylane-cirq) (0.13.1)\n    Requirement already satisfied: cirq-web==0.13.1 in /usr/local/lib/python3.7/dist-packages (from cirq>=0.10->pennylane-cirq) (0.13.1)\n    Requirement already satisfied: cirq-pasqal==0.13.1 in /usr/local/lib/python3.7/dist-packages (from cirq>=0.10->pennylane-cirq) (0.13.1)\n    Requirement already satisfied: cirq-core==0.13.1 in /usr/local/lib/python3.7/dist-packages (from cirq>=0.10->pennylane-cirq) (0.13.1)\n    Requirement already satisfied: cirq-rigetti==0.13.1 in /usr/local/lib/python3.7/dist-packages (from cirq>=0.10->pennylane-cirq) (0.13.1)\n    Requirement already satisfied: cirq-ionq==0.13.1 in /usr/local/lib/python3.7/dist-packages (from cirq>=0.10->pennylane-cirq) (0.13.1)\n    Requirement already satisfied: requests~=2.18 in /usr/local/lib/python3.7/dist-packages (from cirq-aqt==0.13.1->cirq>=0.10->pennylane-cirq) (2.23.0)\n    Requirement already satisfied: matplotlib~=3.0 in /usr/local/lib/python3.7/dist-packages (from cirq-core==0.13.1->cirq>=0.10->pennylane-cirq) (3.2.2)\n    Requirement already satisfied: sympy in /usr/local/lib/python3.7/dist-packages (from cirq-core==0.13.1->cirq>=0.10->pennylane-cirq) (1.7.1)\n    Requirement already satisfied: sortedcontainers~=2.0 in /usr/local/lib/python3.7/dist-packages (from cirq-core==0.13.1->cirq>=0.10->pennylane-cirq) (2.4.0)\n    Requirement already satisfied: tqdm in /usr/local/lib/python3.7/dist-packages (from cirq-core==0.13.1->cirq>=0.10->pennylane-cirq) (4.63.0)\n    Requirement already satisfied: pandas in /usr/local/lib/python3.7/dist-packages (from cirq-core==0.13.1->cirq>=0.10->pennylane-cirq) (1.3.5)\n    Requirement already satisfied: numpy~=1.16 in /usr/local/lib/python3.7/dist-packages (from cirq-core==0.13.1->cirq>=0.10->pennylane-cirq) (1.21.5)\n    Requirement already satisfied: typing-extensions in /usr/local/lib/python3.7/dist-packages (from cirq-core==0.13.1->cirq>=0.10->pennylane-cirq) (3.10.0.2)\n    Requirement already satisfied: duet~=0.2.0 in /usr/local/lib/python3.7/dist-packages (from cirq-core==0.13.1->cirq>=0.10->pennylane-cirq) (0.2.3)\n    Requirement already satisfied: scipy in /usr/local/lib/python3.7/dist-packages (from cirq-core==0.13.1->cirq>=0.10->pennylane-cirq) (1.7.3)\n    Requirement already satisfied: networkx~=2.4 in /usr/local/lib/python3.7/dist-packages (from cirq-core==0.13.1->cirq>=0.10->pennylane-cirq) (2.6.3)\n    Requirement already satisfied: google-api-core[grpc]<2.0.0dev,>=1.14.0 in /usr/local/lib/python3.7/dist-packages (from cirq-google==0.13.1->cirq>=0.10->pennylane-cirq) (1.26.3)\n    Requirement already satisfied: protobuf>=3.13.0 in /usr/local/lib/python3.7/dist-packages (from cirq-google==0.13.1->cirq>=0.10->pennylane-cirq) (3.17.3)\n    Requirement already satisfied: python-dateutil~=2.8.1 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (2.8.2)\n    Requirement already satisfied: httpcore~=0.11.1 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (0.11.1)\n    Requirement already satisfied: qcs-api-client~=0.8.0 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (0.8.0)\n    Requirement already satisfied: retrying~=1.3.3 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (1.3.3)\n    Requirement already satisfied: six~=1.16.0 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (1.16.0)\n    Requirement already satisfied: idna~=2.10 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (2.10)\n    Requirement already satisfied: h11~=0.9.0 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (0.9.0)\n    Requirement already satisfied: httpx~=0.15.5 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (0.15.5)\n    Requirement already satisfied: pydantic~=1.8.2 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (1.8.2)\n    Requirement already satisfied: rfc3339~=6.2 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (6.2)\n    Requirement already satisfied: toml~=0.10.2 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (0.10.2)\n    Requirement already satisfied: sniffio~=1.2.0 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (1.2.0)\n    Requirement already satisfied: attrs~=20.3.0 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (20.3.0)\n    Requirement already satisfied: rfc3986~=1.5.0 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (1.5.0)\n    Requirement already satisfied: pyquil~=3.0.0 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (3.0.1)\n    Requirement already satisfied: iso8601~=0.1.14 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (0.1.16)\n    Requirement already satisfied: certifi~=2021.5.30 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (2021.5.30)\n    Requirement already satisfied: pyjwt~=1.7.1 in /usr/local/lib/python3.7/dist-packages (from cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (1.7.1)\n    Requirement already satisfied: pytz in /usr/local/lib/python3.7/dist-packages (from google-api-core[grpc]<2.0.0dev,>=1.14.0->cirq-google==0.13.1->cirq>=0.10->pennylane-cirq) (2018.9)\n    Requirement already satisfied: setuptools>=40.3.0 in /usr/local/lib/python3.7/dist-packages (from google-api-core[grpc]<2.0.0dev,>=1.14.0->cirq-google==0.13.1->cirq>=0.10->pennylane-cirq) (57.4.0)\n    Requirement already satisfied: googleapis-common-protos<2.0dev,>=1.6.0 in /usr/local/lib/python3.7/dist-packages (from google-api-core[grpc]<2.0.0dev,>=1.14.0->cirq-google==0.13.1->cirq>=0.10->pennylane-cirq) (1.55.0)\n    Requirement already satisfied: packaging>=14.3 in /usr/local/lib/python3.7/dist-packages (from google-api-core[grpc]<2.0.0dev,>=1.14.0->cirq-google==0.13.1->cirq>=0.10->pennylane-cirq) (21.3)\n    Requirement already satisfied: google-auth<2.0dev,>=1.21.1 in /usr/local/lib/python3.7/dist-packages (from google-api-core[grpc]<2.0.0dev,>=1.14.0->cirq-google==0.13.1->cirq>=0.10->pennylane-cirq) (1.35.0)\n    Requirement already satisfied: grpcio<2.0dev,>=1.29.0 in /usr/local/lib/python3.7/dist-packages (from google-api-core[grpc]<2.0.0dev,>=1.14.0->cirq-google==0.13.1->cirq>=0.10->pennylane-cirq) (1.44.0)\n    Requirement already satisfied: pyasn1-modules>=0.2.1 in /usr/local/lib/python3.7/dist-packages (from google-auth<2.0dev,>=1.21.1->google-api-core[grpc]<2.0.0dev,>=1.14.0->cirq-google==0.13.1->cirq>=0.10->pennylane-cirq) (0.2.8)\n    Requirement already satisfied: rsa<5,>=3.1.4 in /usr/local/lib/python3.7/dist-packages (from google-auth<2.0dev,>=1.21.1->google-api-core[grpc]<2.0.0dev,>=1.14.0->cirq-google==0.13.1->cirq>=0.10->pennylane-cirq) (4.8)\n    Requirement already satisfied: cachetools<5.0,>=2.0.0 in /usr/local/lib/python3.7/dist-packages (from google-auth<2.0dev,>=1.21.1->google-api-core[grpc]<2.0.0dev,>=1.14.0->cirq-google==0.13.1->cirq>=0.10->pennylane-cirq) (4.2.4)\n    Requirement already satisfied: kiwisolver>=1.0.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib~=3.0->cirq-core==0.13.1->cirq>=0.10->pennylane-cirq) (1.3.2)\n    Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib~=3.0->cirq-core==0.13.1->cirq>=0.10->pennylane-cirq) (3.0.7)\n    Requirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.7/dist-packages (from matplotlib~=3.0->cirq-core==0.13.1->cirq>=0.10->pennylane-cirq) (0.11.0)\n    Requirement already satisfied: autoray in /usr/local/lib/python3.7/dist-packages (from 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rpcq<4.0.0,>=3.6.0->pyquil~=3.0.0->cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (22.3.0)\n    Requirement already satisfied: ruamel.yaml in /usr/local/lib/python3.7/dist-packages (from rpcq<4.0.0,>=3.6.0->pyquil~=3.0.0->cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (0.17.21)\n    Requirement already satisfied: python-rapidjson in /usr/local/lib/python3.7/dist-packages (from rpcq<4.0.0,>=3.6.0->pyquil~=3.0.0->cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (1.6)\n    Requirement already satisfied: msgpack<1.0,>=0.6 in /usr/local/lib/python3.7/dist-packages (from rpcq<4.0.0,>=3.6.0->pyquil~=3.0.0->cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (0.6.2)\n    Requirement already satisfied: future>=0.15.2 in /usr/local/lib/python3.7/dist-packages (from autograd->pennylane>=0.17->pennylane-cirq) (0.16.0)\n    Requirement already satisfied: ruamel.yaml.clib>=0.2.6 in /usr/local/lib/python3.7/dist-packages (from ruamel.yaml->rpcq<4.0.0,>=3.6.0->pyquil~=3.0.0->cirq-rigetti==0.13.1->cirq>=0.10->pennylane-cirq) (0.2.6)\n    Requirement already satisfied: mpmath>=0.19 in /usr/local/lib/python3.7/dist-packages (from sympy->cirq-core==0.13.1->cirq>=0.10->pennylane-cirq) (1.2.1)\n    Installing collected packages: pennylane-cirq\n    Successfully installed pennylane-cirq-0.19.0\n\n\n\n```python\n# This is needed so that the plotted figures appear embedded in this notebook\n%matplotlib inline\n\n# imports\nimport pennylane as qml\nfrom pennylane import numpy as np\nimport tensorflow as tf\nimport cirq\nimport matplotlib.pyplot as plt\n```\n\n\n```python\n# create a device\ndev = qml.device('cirq.simulator', wires=3)\n```\n\n# Generator and Discriminator\nFor this simple example, our real data will be a qubit that has been rotated (from the starting state |0>) to some arbitrary, but fixed, state.\n\n\n```python\ndef real(angles, **kwargs):\n  qml.Hadamard(wires=0)\n  qml.Rot(*angles, wires=0)\n```\n\nBoth the real data circuit and the generator will output on wire 0, which will be connected as an input to the discriminator. Wire 1 is provided as a workspace for the generator, while the discriminator's output will be on wire 2.\n\n\n```python\ndef generator(w, **kwargs):\n  qml.Hadamard(wires=0)\n  qml.RX(w[0], wires=0)\n  qml.RX(w[1], wires=1)\n  qml.RY(w[2], wires=0)\n  qml.RY(w[3], wires=1)\n  qml.RZ(w[4], wires=0)\n  qml.RZ(w[5], wires=1)\n  qml.CNOT(wires=[0, 1])\n  qml.RX(w[6], wires=0)\n  qml.RY(w[7], wires=0)\n  qml.RZ(w[8], wires=0)\n\ndef discriminator(w):\n  qml.Hadamard(wires=0)\n  qml.RX(w[0], wires=0)\n  qml.RX(w[1], wires=2)\n  qml.RY(w[2], wires=0)\n  qml.RY(w[3], wires=2)\n  qml.RZ(w[4], wires=0)\n  qml.RZ(w[5], wires=2)\n  qml.CNOT(wires=[0, 2])\n  qml.RX(w[6], wires=2)\n  qml.RY(w[7], wires=2)\n  qml.RZ(w[8], wires=2)\n```\n\nWe create two QNodes. One where the real data source is wired up to the discriminator, and one where the generator is connected to the discriminator. In order to pass TensorFlow Variables into the quantum circuits, we specify the \"tf\" interface.\n\n\n```python\n@qml.qnode(dev, interface=\"tf\")\ndef real_disc_circuit(phi, theta, omega, disc_weights):\n    real([phi, theta, omega])\n    discriminator(disc_weights)\n    return qml.expval(qml.PauliZ(2))\n\n@qml.qnode(dev, interface=\"tf\")\ndef gen_disc_circuit(gen_weights, disc_weights):\n    generator(gen_weights)\n    discriminator(disc_weights)\n    return qml.expval(qml.PauliZ(2))\n```\n\n#QGAN Cost Functions\nThe discriminator is trained to maximize the probability of correctly classifying real data, while minimizing the probability of mistakenly classifying fake data.\n\n$Cost_D = Pr(real|fake) - Pr(real|real)$\n\nThe generator is trained to maximize the probability that the discriminator accepts fake data as real.\n\n$Cost_G = -Pr(real|fake)$\n\n\n```python\ndef prob_real_true(disc_weights):\n    true_disc_output = real_disc_circuit(phi, theta, omega, disc_weights)\n    # convert to probability\n    prob_real_true = (true_disc_output + 1) / 2\n    return prob_real_true\n\n\ndef prob_fake_true(gen_weights, disc_weights):\n    fake_disc_output = gen_disc_circuit(gen_weights, disc_weights)\n    # convert to probability\n    prob_fake_true = (fake_disc_output + 1) / 2\n    return prob_fake_true\n\n\ndef disc_cost(disc_weights):\n    cost = prob_fake_true(gen_weights, disc_weights) - prob_real_true(disc_weights)\n    return cost\n\n\ndef gen_cost(gen_weights):\n    return -prob_fake_true(gen_weights, disc_weights)\n```\n\n# Training the QGAN\nWe initialize the fixed angles of the \u201creal data\u201d circuit, as well as the initial parameters for both generator and discriminator. These are chosen so that the generator initially prepares a state on wire 0 that is very close to the |1> state.\n\n\n```python\n# fixed initial parameters\nphi = np.pi / 6\ntheta = np.pi / 2\nomega = np.pi / 7\nnp.random.seed(0)\neps = 1e-2\n\ninit_gen_weights = np.array([np.pi] + [0] * 8) + \\\n                   np.random.normal(scale=eps, size=(9,))\ninit_disc_weights = np.random.normal(size=(9,))\n\ngen_weights = tf.Variable(init_gen_weights)\ndisc_weights = tf.Variable(init_disc_weights)\n\n# We begin by creating the optimizer - default learning_rate is 0.01, but demo uses 0.4\n# For our final code, we would be using backpropagation\nopt = tf.keras.optimizers.SGD(0.05)\n```\n\nIn the first stage of training, we optimize the discriminator while keeping the generator parameters fixed.\n\n\n```python\ncost = lambda: disc_cost(disc_weights)\n\ndef optimize_disc(steps=50, vocal=False):\n  for step in range(steps):\n    opt.minimize(cost,disc_weights)\n    if vocal and step % 5 ==0:\n      cost_val = cost().numpy() # converts tensor to np array\n      print(\"Step {}: cost = {}\".format(step, cost_val))\n```\n\n\n```python\noptimize_disc(vocal=True)\n\n# at discriminator's optimum, the discriminator's proabability to classify real data\n# should be close to one\nprint(\"Prob(real classified as real): \", prob_real_true(disc_weights).numpy())\n\n# for comparison, we check how the discriminator classifies the generator's fake data\nprint(\"Prob(fake classified as real): \", prob_fake_true(gen_weights, disc_weights).numpy())\n```\n\n    Step 0: cost = -0.02684006094932556\n    Step 5: cost = -0.0487934947013855\n    Step 10: cost = -0.07152463495731354\n    Step 15: cost = -0.09530872106552124\n    Step 20: cost = -0.12031546235084534\n    Step 25: cost = -0.14657654613256454\n    Step 30: cost = -0.17396266013383865\n    Step 35: cost = -0.20217377319931984\n    Step 40: cost = -0.23075543902814388\n    Step 45: cost = -0.25913961231708527\n    Prob(real classified as real):  0.6247680932283401\n    Prob(fake classified as real):  0.3434784710407257\n\n\nIn the adversarial game we now have to train the generator to better fool the discriminator. \n\n\n```python\ncost = lambda: gen_cost(gen_weights)\n\ndef optimize_gen(steps=50, vocal=False):\n  for step in range(steps):\n      opt.minimize(cost, gen_weights)\n      if vocal and step % 5 == 0:\n          cost_val = cost().numpy()\n          print(\"Step {}: cost = {}\".format(step, cost_val))\n```\n\n\n```python\noptimize_gen(vocal=True)\n\n# at generator's optimum, the discriminator's proabability to classify real data should be close to one\nprint(\"Prob(fake classified as real): \", prob_fake_true(gen_weights, disc_weights).numpy())\n\n# At the joint optimum the discriminator cost will be close to zero,\n# indicating that the discriminator assigns equal probability to both real and generated data\nprint(\"Discriminator cost: \", disc_cost(disc_weights).numpy())\n```\n\n    Step 0: cost = -0.34860286116600037\n    Step 5: cost = -0.37412218749523163\n    Step 10: cost = -0.3992343097925186\n    Step 15: cost = -0.4236362725496292\n    Step 20: cost = -0.44705919548869133\n    Step 25: cost = -0.46927898190915585\n    Step 30: cost = -0.4901227205991745\n    Step 35: cost = -0.509470671415329\n    Step 40: cost = -0.5272556990385056\n    Step 45: cost = -0.543457705527544\n    Prob(fake classified as real):  0.5552917309105396\n    Discriminator cost:  -0.06947636231780052\n\n\nThe generator has successfully learned how to simulate the real data enough to fool the discriminator.\n\nLet's conclude by comparing the states of the real data circuit and the generator. We expect the generator to have learned to be in a state that is very close to the one prepared in the real data circuit. An easy way to access the state of the first qubit is through is Bloch sphere representation:\n\n\n```python\nobs = [qml.PauliX(0), qml.PauliY(0), qml.PauliZ(0)]\n\nbloch_vector_real = qml.map(real, obs, dev, interface=\"tf\")\nbloch_vector_generator = qml.map(generator, obs, dev, interface=\"tf\")\n\nprint(\"Real Bloch vector: {}\".format(bloch_vector_real([phi, theta, omega])))\nprint(\"Generator Bloch vector: {}\".format(bloch_vector_generator(gen_weights)))\n```\n\n    Real Bloch vector: [-0.21694186  0.45048442 -0.86602521]\n    Generator Bloch vector: [-0.01952515  0.00862315 -0.67009914]\n\n\n# Alternating Training Cycles\nFor complex models, we continue training the models in an alternating fashion until we reach the optimum point of the two-player adversarial game.\n\n\n```python\n# reset weights\ngen_weights = tf.Variable(init_gen_weights)\ndisc_weights = tf.Variable(init_disc_weights)\n\n# plot two lines, one for Pr(real|real), one for Pr(real|fake)\nprobs_real = []\nprobs_fake = []\n\nfor cycle in range(8):\n  print(\"Cycle\", cycle)\n\n  # First, optimize discriminator\n  optimize_disc(steps=25)\n  prob = prob_real_true(disc_weights).numpy()\n  probs_real.append(prob)\n  print(\"Prob(real classified as real): \", prob)\n  \n\n  # Second, optimize generator\n  optimize_gen(steps=25)\n  prob = prob_fake_true(gen_weights, disc_weights).numpy()\n  probs_fake.append(prob)\n  print(\"Prob(fake classified as real): \", prob)\n\nplt.plot(probs_real)\nplt.plot(probs_fake)\n```\n\nIn a real scenario, we would have to fiddle more with the step size, number of steps, and number of cycles. However, as long as we can approach a probability of one for both the discriminator and generator, we will be satisfied with our resutls.\n", "meta": {"hexsha": "86b811379582732c0f3c9e002737b9df8eb89bbe", "size": 73005, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "QGAN_demo.ipynb", "max_stars_repo_name": "annabellegrimes/CPEN-400Q", "max_stars_repo_head_hexsha": "044d521f8109567ec004a9c882898f9e2eb5a19e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "QGAN_demo.ipynb", "max_issues_repo_name": "annabellegrimes/CPEN-400Q", "max_issues_repo_head_hexsha": "044d521f8109567ec004a9c882898f9e2eb5a19e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "QGAN_demo.ipynb", "max_forks_repo_name": "annabellegrimes/CPEN-400Q", "max_forks_repo_head_hexsha": "044d521f8109567ec004a9c882898f9e2eb5a19e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 80.4906284454, "max_line_length": 13989, "alphanum_fraction": 0.6619409629, "converted": true, "num_tokens": 15467, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "# Model tests\nVMM:s will be developed for the reference ship using motion regression based on a series of model tests with a model that is free in six degrees of freedome. A summary of the available model tests is shown in {ref}`tab:df_runs_table`.\n\n\n```python\n# %load imports.py\n%load_ext autoreload\n%autoreload 2\n%reload_kedro\n%config Completer.use_jedi = False  ## (To fix autocomplete)\n\nimport pandas as pd\npd.set_option('display.max_columns', 500)\nfrom src.models.vmm import ModelSimulator\nimport matplotlib.pyplot as plt\nfrom src.visualization.plot import track_plots, plot, captive_plot\nimport kedro\nimport numpy as np\nimport os.path\nimport anyconfig\n\nimport matplotlib\nplt.style.use('presentation')\n\nfrom myst_nb import glue\nfrom src.symbols import *\nimport src.symbols as symbols\nfrom src.system_equations import *\n\nfrom IPython.display import display, Math, Latex, Markdown\nfrom sympy.physics.vector.printing import vpprint, vlatex\n\nfrom src.parameters import df_parameters\np = df_parameters[\"symbol\"]\n\n# Read configs:\nconf_path = os.path.join(\"../../conf/base/\")\nruns_globals_path = os.path.join(\n    conf_path,\n    \"runs_globals.yml\",\n)\n\nruns_globals = anyconfig.load(runs_globals_path)\nmodel_test_ids = runs_globals[\"model_test_ids\"]\n\njoin_globals_path = os.path.join(\n    conf_path,\n    \"join_globals.yml\",\n)\n\njoins = runs_globals[\"joins\"]\njoin_runs_dict = anyconfig.load(join_globals_path)\n\nglobals_path = os.path.join(\n    conf_path,\n    \"globals.yml\",\n)\nglobal_variables = anyconfig.load(globals_path)\n\n\n\nvmms = global_variables[\"vmms\"]\nonly_joined = global_variables[\n    \"only_joined\"\n]  # (regress/predict with only models from joined runs)S\n```\n\n    2022-03-28 14:44:26,549 - kedro.framework.session.store - INFO - `read()` not implemented for `SQLiteStore`. Assuming empty store.\n    2022-03-28 14:44:29,800 - root - INFO - ** Kedro project wPCC_pipeline\n    2022-03-28 14:44:29,801 - root - INFO - Defined global variable `context`, `session`, `catalog` and `pipelines`\n    2022-03-28 14:44:29,808 - root - INFO - Registered line magic `run_viz`\n\n\n\n```python\nship_data = catalog.load(\"ship_data\")\n\n#from wPCC_pipeline.pipelines.preprocess.nodes import track_plot\nfrom src.visualization.plot import track_plots, track_plot, plot\n```\n\n    2022-03-28 14:44:34,018 - kedro.io.data_catalog - INFO - Loading data from `ship_data` (YAMLDataSet)...\n\n\n\n```python\ndataframes = {}\ndf = pd.DataFrame()\n\nfor id in model_test_ids:\n    \n    df_ = catalog.load(f\"{ id }.raw_data\")\n    df_['psi+'] = df_['psi'] + np.deg2rad(90)\n    df_['-y0'] = -df_['y0']\n    df_['delta_deg'] = np.rad2deg(df_['delta'])\n    \n    dataframes[id] = df_\n\n```\n\n    2022-03-28 14:44:34,237 - kedro.io.data_catalog - INFO - Loading data from `22611.raw_data` (CSVDataSet)...\n    2022-03-28 14:44:34,282 - kedro.io.data_catalog - INFO - Loading data from `22612.raw_data` (CSVDataSet)...\n    2022-03-28 14:44:34,379 - kedro.io.data_catalog - INFO - Loading data from `22613.raw_data` (CSVDataSet)...\n    2022-03-28 14:44:34,456 - kedro.io.data_catalog - INFO - Loading data from `22614.raw_data` (CSVDataSet)...\n    2022-03-28 14:44:34,566 - kedro.io.data_catalog - INFO - Loading data from `22615.raw_data` (CSVDataSet)...\n    2022-03-28 14:44:34,639 - kedro.io.data_catalog - INFO - Loading data from `22616.raw_data` (CSVDataSet)...\n    2022-03-28 14:44:34,735 - kedro.io.data_catalog - INFO - Loading data from `22635.raw_data` (CSVDataSet)...\n    2022-03-28 14:44:34,841 - kedro.io.data_catalog - INFO - Loading data from `22639.raw_data` (CSVDataSet)...\n    2022-03-28 14:44:34,950 - kedro.io.data_catalog - INFO - Loading data from `22764.raw_data` (CSVDataSet)...\n    2022-03-28 14:44:35,021 - kedro.io.data_catalog - INFO - Loading data from `22769.raw_data` (CSVDataSet)...\n    2022-03-28 14:44:35,090 - kedro.io.data_catalog - INFO - Loading data from `22770.raw_data` (CSVDataSet)...\n    2022-03-28 14:44:35,182 - kedro.io.data_catalog - INFO - Loading data from `22771.raw_data` (CSVDataSet)...\n    2022-03-28 14:44:35,283 - kedro.io.data_catalog - INFO - Loading data from `22772.raw_data` (CSVDataSet)...\n    2022-03-28 14:44:35,379 - kedro.io.data_catalog - INFO - Loading data from `22773.raw_data` (CSVDataSet)...\n    2022-03-28 14:44:35,490 - kedro.io.data_catalog - INFO - Loading data from `22774.raw_data` (CSVDataSet)...\n\n\n\n```python\ndf_runs = catalog.load(\"runs_meta_data\")\ndf_runs.index = df_runs.index.astype('str')\ndf_runs = df_runs.loc[model_test_ids].copy()\n\nmask = df_runs['test_type'] == 'rodergrundvinkel'\ndf_runs.loc[mask,'test_type'] = 'yaw rate'\nmask = df_runs['test_type'] != 'zigzag'\ndf_runs.loc[mask,'comment'] = np.NaN\nmask = ((df_runs['comment'].notnull()) & (df_runs['test_type'] == 'zigzag'))\ndf_runs['angle'] = df_runs.loc[mask,'comment'].apply(lambda x:int(x[3:5]))\ndf_runs['direction'] = df_runs.loc[mask,'comment'].apply(lambda x:x[8:11])\n\ndf_runs.sort_values(by=['test_type','ship_speed','angle'], inplace=True)\n\ndf_runs_table = df_runs.rename(columns={'ship_speed':'Initial speed [m/s]','test_type':'type'})\ndf_runs_table = df_runs_table[['Initial speed [m/s]','type','angle','direction']]\n\nformatter={'Initial speed [m/s]' : \"{:.2f}\", 'angle' : \"{:.0f}\"}\n\ndf_runs_table = df_runs_table.style.format(formatter=formatter, na_rep='')\n\nglue(\"df_runs_table\", df_runs_table)\n```\n\n    2022-03-28 14:44:35,951 - kedro.io.data_catalog - INFO - Loading data from `runs_meta_data` (CSVDataSet)...\n\n\n\n\n\n  \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n  \n
                                         Initial speed [m/s]typeangledirection
                                        id    
                                        226390.64reference speed
                                        226350.80reference speed
                                        226110.96reference speed
                                        227740.96turning circle
                                        226120.96yaw rate
                                        226130.96yaw rate
                                        226140.96yaw rate
                                        226150.96yaw rate
                                        226160.96yaw rate
                                        227640.96zigzag10SB
                                        227690.96zigzag10PS
                                        227700.96zigzag10PS
                                        227710.96zigzag20SB
                                        227720.96zigzag20SB
                                        227730.96zigzag20PS
                                        \n\n\n\n```{glue:figure} df_runs_table\n:name: \"tab:df_runs_table\"\n\nModel tests\n```\n\n\n```python\nfor test_type, df_ in df_runs.groupby(by=['test_type']):\n    \n    dataframes_ = {key:value for key,value in dataframes.items() if key in df_.index}\n    \n    if test_type == 'reference speed':\n        continue\n    \n    fig = track_plots(dataframes=dataframes_, lpp=ship_data['L'], beam=ship_data['B'], x_dataset='-y0',\n    y_dataset='x0', psi_dataset='psi+', plot_boats=True, N=7)\n    ax = fig.axes\n    ax.set_title(f\"{test_type}\")\n    ax.get_legend().set_visible(False)\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "ab53582e0c63bdca7cb6684a290e5eb4f5cd4e8c", "size": 516956, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "_build/html/_sources/05.01_model_tests.ipynb", "max_stars_repo_name": "martinlarsalbert/NMUW2022", "max_stars_repo_head_hexsha": "13b951e0a3965780c38bcd42124c28a92ebadf53", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "_build/html/_sources/05.01_model_tests.ipynb", "max_issues_repo_name": "martinlarsalbert/NMUW2022", "max_issues_repo_head_hexsha": "13b951e0a3965780c38bcd42124c28a92ebadf53", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "_build/html/_sources/05.01_model_tests.ipynb", "max_forks_repo_name": "martinlarsalbert/NMUW2022", "max_forks_repo_head_hexsha": "13b951e0a3965780c38bcd42124c28a92ebadf53", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 1033.912, "max_line_length": 194944, "alphanum_fraction": 0.9506012117, "converted": true, "num_tokens": 4276, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631556226291, "lm_q2_score": 0.6825737473266735, "lm_q1q2_score": 0.4147749172456895}}
{"text": "# Homework (Chapter 5) - 201601639 \ud64d\uc2b9\ud604\n\n- \uc5f0\uc2b5\ubb38\uc81c 2, 3, 5\n\n## \uc5f0\uc2b5\ubb38\uc81c 2\n\nsoftmax\ub97c \uc801\uc6a9\ud55c \ud6c4 \ucd9c\ub825\uc774`(0.001, 0.9, 0.001, 0.098)^T`\uc774\uace0 \ub808\uc774\ube14 \uc815\ubcf4\uac00 `(0, 0, 0, 1)^T`\uc77c \ub54c, \uc138\uac00\uc9c0 \ubaa9\uc801\ud568\uc218, \ud3c9\uade0\uc81c\uacf1 \uc624\ucc28, \uad50\ucc28 \uc5d4\ud2b8\ub85c\ud53c, \ub85c\uadf8\uc6b0\ub3c4\ub97c \uacc4\uc0b0\ud558\uc2dc\uc624.\n\n\n\n\ud589\ub82c \uc5f0\uc0b0\uc744 \uc218\uc6d4\ud558\uac8c \ud558\uae30 \uc704\ud574 `numpy`\ub97c \uc0ac\uc6a9\ud558\uc600\ub2e4.\n\n\n```python\nimport numpy as np\n```\n\n\n```python\nfrom sympy import * # \uc218\uc2dd \ud45c\ud604\uc744 \uc704\ud574 \uc784\ud3ec\ud2b8\n```\n\n\n```python\nsoftmax_output = np.array([[0.001, 0.9, 0.001, 0.098]]).T\nlabel = np.array([[0, 0, 0, 1]]).T\n\n```\n\n\n```python\npprint(Eq(Symbol(\"softmax(x)\"), Matrix(softmax_output), evaluate=False))\npprint(Eq(Symbol(\"label(x)\"), Matrix(label), evaluate=False))\n```\n\n                 \u23a10.001\u23a4\n                 \u23a2     \u23a5\n                 \u23a2 0.9 \u23a5\n    softmax(x) = \u23a2     \u23a5\n                 \u23a20.001\u23a5\n                 \u23a2     \u23a5\n                 \u23a30.098\u23a6\n               \u23a10\u23a4\n               \u23a2 \u23a5\n               \u23a20\u23a5\n    label(x) = \u23a2 \u23a5\n               \u23a20\u23a5\n               \u23a2 \u23a5\n               \u23a31\u23a6\n\n\n\n```python\ndef mean_squared_error(y, t):\n    return 0.5 * np.sum((y-t)**2)\n\n\ndef cross_entropy_error(y, t):\n    \n    return -np.sum(y*np.log2(t))\n\n```\n\n1. MSE (\ud3c9\uade0\uc81c\uacf1\uc624\ucc28)\n\n\n```python\npprint(Eq(Symbol(\"MSE\"), mean_squared_error(label, softmax_output)))\n```\n\n    MSE = 0.811803\n\n\n2. CCE (\uad50\ucc28 \uc5d4\ud2b8\ub85c\ud53c)\n\n\n```python\npprint(Eq(Symbol(\"CEE\"), cross_entropy_error(label, softmax_output)))\n```\n\n    CEE = 3.35107444054688\n\n\n3. \ub85c\uadf8\uc6b0\ub3c4\n\n\n```python\nlog_likelihood = -np.log2(softmax_output)\nfor i in range(log_likelihood.shape[0]):\n    pprint(Eq(Symbol(f\"o_{i}e\"), Matrix(log_likelihood[i]), evaluate=False))\n```\n\n    o\u2080\u2091 = [9.96578428466209]\n    o\u2081\u2091 = [0.15200309344505]\n    o\u2082\u2091 = [9.96578428466209]\n    o\u2083\u2091 = [3.35107444054688]\n\n\n## \uc5f0\uc2b5\ubb38\uc81c 3\n\n[\uc608\uc81c 5-1]\uc5d0\uc11c `\u03bb = 0.1`, `\u03bb = 0.5`\uc77c \ub54c\ub97c \uacc4\uc0b0\ud558\uace0 \u03bb\uc5d0 \ub530\ub978 \ud6a8\uacfc\ub97c \uc124\uba85\ud558\uc2dc\uc624. \uc774 \ub54c [\uadf8\ub9bc 5-21]\uc744 \ud65c\uc6a9\ud558\uc2dc\uc624.\n\n\n```python\n# \ud6c8\ub828\uc9d1\ud569\nX = np.array([[1, 1], [2, 3], [3, 3]])\n# label\nY = np.array([[3.0, 7.0, 8.8]]).T\n\npprint(Eq(Symbol(\"X\"), Matrix(X), evaluate=False))\npprint(Eq(Symbol(\"Y\"), Matrix(Y), evaluate=False))\n```\n\n        \u23a11  1\u23a4\n        \u23a2    \u23a5\n    X = \u23a22  3\u23a5\n        \u23a2    \u23a5\n        \u23a33  3\u23a6\n        \u23a13.0\u23a4\n        \u23a2   \u23a5\n    Y = \u23a27.0\u23a5\n        \u23a2   \u23a5\n        \u23a38.8\u23a6\n\n\n\n```python\ndef ridge_regression(x, y, lamb):\n    return np.linalg.inv(x.T.dot(x)+2*lamb*np.identity(2)).dot(x.T).dot(y)\n    \n```\n\n### \u03bb = 0.25\uc77c \ub54c (\uae30\uc874 \uc608\uc81c)\n\n\n```python\nt = ridge_regression(X, Y, lamb=0.25)\npprint(Eq(Symbol(\"\u03bb_(025)\"), Matrix(t), evaluate=False))\n```\n\n             \u23a11.49158878504673\u23a4\n    \u03bb\u208d\u2080\u2082\u2085\u208e = \u23a2                \u23a5\n             \u23a31.3607476635514 \u23a6\n\n\n### \u03bb = 0.1\uc77c \ub54c\n\n\n```python\nt = ridge_regression(X, Y, lamb=0.1)\npprint(Eq(Symbol(\"\u03bb_(01)\"), Matrix(t), evaluate=False))\n```\n\n            \u23a11.61538461538462\u23a4\n    \u03bb\u208d\u2080\u2081\u208e = \u23a2                \u23a5\n            \u23a31.27884615384616\u23a6\n\n\n### \u03bb = 0.5\uc77c \ub54c\n\n\n```python\nt = ridge_regression(X, Y, lamb=0.5)\npprint(Eq(Symbol(\"\u03bb_(05)\"), Matrix(t), evaluate=False))\n```\n\n            \u23a11.4\u23a4\n    \u03bb\u208d\u2080\u2085\u208e = \u23a2   \u23a5\n            \u23a31.4\u23a6\n\n\n### \u03bb = 0\uc77c \ub54c (\uae30\uc874 \ubaa9\uc801\ud568\uc218\uc640 \ub3d9\uc77c)\n\n\n```python\nt = ridge_regression(X, Y, lamb=0)\npprint(Eq(Symbol(\"\u03bb_(05)\"), Matrix(t), evaluate=False))\n```\n\n            \u23a11.82000000000001\u23a4\n    \u03bb\u208d\u2080\u2085\u208e = \u23a2                \u23a5\n            \u23a31.11999999999998\u23a6\n\n\n## \uacb0\ub860\n\n- \uc704 \uac12\uc5d0 \ub530\ub77c `\u03bb`\uac00 \uae30\uc874 \uac00\uc911\uce58\ub97c \uc6d0\uc810\uc5d0 \uc18c\ud3ed \uac00\uae5d\uac8c \ub2f9\uae34 \ud6c4 \uac31\uc2e0\ud55c\ub2e4\ub294 \uac83\uc744 \ud655\uc778\ud560 \uc218 \uc788\ub2e4.\n\n## \uc5f0\uc2b5\ubb38\uc81c 5\n\n\ud608\uc555, \ud0a4, \ubab8\ubb34\uac8c\uac00 \ud2b9\uc9d5\ubca1\ud130\ub97c \uc774\ub8ec\ub2e4. \ub2e4\uc74c\uacfc \uac19\uc774 \ud6c8\ub828\uc9d1\ud569\uc774 \uc8fc\uc5b4\uc84c\ub2e4.\n\n\n```python\ntrain_data = np.array([[[121], [1.72], [69.0]], [[140], [1.62], [63.2]], [[120], [1.70], [59.0]], [[131], [1.80], [82.0]], [[101], [1.78], [73.5]]])\nfor i in range(train_data.shape[0]):\n    pprint(Matrix(train_data[i]))\n```\n\n    \u23a1121.0\u23a4\n    \u23a2     \u23a5\n    \u23a21.72 \u23a5\n    \u23a2     \u23a5\n    \u23a369.0 \u23a6\n    \u23a1140.0\u23a4\n    \u23a2     \u23a5\n    \u23a21.62 \u23a5\n    \u23a2     \u23a5\n    \u23a363.2 \u23a6\n    \u23a1120.0\u23a4\n    \u23a2     \u23a5\n    \u23a2 1.7 \u23a5\n    \u23a2     \u23a5\n    \u23a359.0 \u23a6\n    \u23a1131.0\u23a4\n    \u23a2     \u23a5\n    \u23a2 1.8 \u23a5\n    \u23a2     \u23a5\n    \u23a382.0 \u23a6\n    \u23a1101.0\u23a4\n    \u23a2     \u23a5\n    \u23a21.78 \u23a5\n    \u23a2     \u23a5\n    \u23a373.5 \u23a6\n\n\n### 1. \ud37c\uc149\ud2b8\ub860\uc758 \uac00\uc911\uce58 \ubca1\ud130\uac00 `(-0.01, 0.5, -0.23)^T`\uc774\uace0 \ubc14\uc774\uc5b4\uc2a4\uac00 0\uc774\ub77c\uace0 \ud588\uc744 \ub54c, \ud6c8\ub828\uc9d1\ud569\uc744 \uac00\uc9c0\uace0 \uaddc\ubaa8 \ubb38\uc81c\ub97c \uc124\uba85\ud558\uc2dc\uc624.\n\n\n```python\nweight = np.array([[-0.01, 0.5, -0.23]]).T\npprint(Eq(Symbol(\"weight\"), Matrix(weight), evaluate=False))\n```\n\n             \u23a1-0.01\u23a4\n             \u23a2     \u23a5\n    weight = \u23a2 0.5 \u23a5\n             \u23a2     \u23a5\n             \u23a3-0.23\u23a6\n\n\n#### \uac01 \ud6c8\ub828\uc9d1\ud569\uc744 \uac00\uc911\uce58\ub85c \uacf1\ud55c \uac12\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n\n\n```python\nfor train_set in train_data:\n    print(np.sum(train_set*weight))\n```\n\n    -16.220000000000002\n    -15.126000000000001\n    -13.92\n    -19.27\n    -17.025000000000002\n\n\n\uc774\ub97c `step function`\uc73c\ub85c \uc801\uc6a9\ud558\uc600\uc744 \uacbd\uc6b0\n\n\n```python\nfor train_set in train_data:\n    print(np.heaviside(np.sum(train_set*weight), -999))\n```\n\n    0.0\n    0.0\n    0.0\n    0.0\n    0.0\n\n\n#### \uc911\uac04\uacb0\uacfc\n\n- \ud608\uc555, \ud0a4, \ubab8\ubb34\uac8c\uc758 \uacbd\uc6b0 \ub2e8\uc704\uc5d0 \ub530\ub77c \uac12\uc758 \uaddc\ubaa8\uac00 \ud655\uc5f0\ud558\uac8c \ucc28\uc774\uac00 \ub09c\ub2e4.\n- \uc608\ub97c \ub4e4\uc5b4, \ud0a4\uac00 178cm\uc640 162cm\uc758 \ucc28\uc774\ub294 16cm \ub9cc\ud07c\uc758 \ucc28\uc774\uac00 \ubc1c\uc0dd\ud558\uc9c0\ub9cc \ub2e8\uc704\ub85c \uc778\ud574 \ud2b9\uc9d5\uac12 \ucc28\uc774\ub294 \ubd88\uacfc **0.16**\ubc16\uc5d0 \ucc28\uc774\uac00 \ub098\uc9c0 \uc54a\ub294\ub2e4. \ub610\ud55c \ud2b9\uc9d5\uac12\uc774 \ubaa8\ub450 \uc591\uc218\uc778 \uc810\uc744 \ube44\ub86f\ud574 \uc774\ub7ec\ud55c \ub370\uc774\ud130\ub294 \uc218\ub834 \uc18d\ub3c4\uac00 \uad49\uc7a5\ud788 \ub290\ub824\uc9c8 \uc218 \ubc16\uc5d0 \uc5c6\ub2e4.\n- \uacb0\uad6d, \uc11c\ub85c\uc758 \ub2e8\uc704\ub85c \uc778\ud55c \uaddc\ubaa8\uac00 \ub2e4\uc591\ud558\uc5ec `step function`\uc744 \uc801\uc6a9\ud588\uc73c\ub098 \uc804\ubd80 `0`\uc73c\ub85c \uc218\ub834\ud558\ub294 \uac83\uc744 \ud655\uc778\ud560 \uc218 \uc788\ub2e4.\n\n### 2. \uc2dd (5.9)\uc758 \uc804\ucc98\ub9ac\ub97c \uc801\uc6a9\ud55c \ud6c4\uc758 \ud6c8\ub828\uc9d1\ud569\uc744 \uc4f0\uc2dc\uc624.\n\n\n```python\npre_processing_data = (train_data - np.mean(train_data, axis=0)) / np.std(train_data, axis=0)\n```\n\n\n```python\nfor data in pre_processing_data:\n    pprint(Matrix(data))\n```\n\n    \u23a1-0.122772186962938 \u23a4\n    \u23a2                   \u23a5\n    \u23a2-0.0627455805138124\u23a5\n    \u23a2                   \u23a5\n    \u23a3-0.0423472957199062\u23a6\n    \u23a1 1.33514753322196 \u23a4\n    \u23a2                  \u23a5\n    \u23a2-1.63138509335921 \u23a5\n    \u23a2                  \u23a5\n    \u23a3-0.764742340353592\u23a6\n    \u23a1-0.199504803814775\u23a4\n    \u23a2                  \u23a5\n    \u23a2-0.376473483082892\u23a5\n    \u23a2                  \u23a5\n    \u23a3-1.28785599336419 \u23a6\n    \u23a10.644553981555428\u23a4\n    \u23a2                 \u23a5\n    \u23a21.19216602976251 \u23a5\n    \u23a2                 \u23a5\n    \u23a31.57681401121767 \u23a6\n    \u23a1-1.65742452399967\u23a4\n    \u23a2                 \u23a5\n    \u23a20.878438127193427\u23a5\n    \u23a2                 \u23a5\n    \u23a30.518131618220023\u23a6\n\n\n### 3. \uc804\ucc98\ub9ac\uac00 \uaddc\ubaa8 \ubb38\uc81c\ub97c \uc644\ud654\ud558\ub294\uc9c0\ub97c \uc124\uba85\ud558\uc2dc\uc624.\n\n#### \uc815\uaddc\ud654\ud55c \ud6c8\ub828\uc9d1\ud569\uc5d0 \uac00\uc911\uce58\ub97c \uacf1\ud588\uc744 \uacbd\uc6b0\n\n\n```python\nfor train_set in pre_processing_data:\n    print(np.sum(train_set*weight))\n```\n\n    -0.02040519037169842\n    -0.6531532837304969\n    0.10996518497046617\n    0.2269702524856353\n    0.3366230366461046\n\n\n`step function`\uc744 \uc801\uc6a9\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n\n\n```python\nfor train_set in pre_processing_data:\n    print(np.heaviside(np.sum(train_set*weight), -999))\n```\n\n    0.0\n    0.0\n    1.0\n    1.0\n    1.0\n\n\n### \uacb0\ub860\n\n- \ud2b9\uc9d5\uc758 \uaddc\ubaa8\uac00 \ub2ec\ub77c \uc774\ub97c \uc815\uaddc\ud654 \ud558\uba74 \uac01 \uac12\uc758 \ubcc0\ud654\uc5d0 \ub530\ub77c \uac78\ub9de\uac8c \ubcc0\ud654\ub418\ub294 \uac83\uc744 \ud655\uc778\ud560 \uc218 \uc788\ub2e4.\n- \uc774\ub97c \ud1b5\ud574 \uc5b4\ub5a4 \ud2b9\uc9d5\uc774 \ub2e4\ub978 \ud2b9\uc9d5\ubcf4\ub2e4 \ub354 \uc911\uc694\ud558\uac8c \uc791\uc6a9\ud55c\ub2e4\ub294 \uac83\uc744 \uc54c\uace0 \uc788\uc744 \uacbd\uc6b0 \uaddc\ubaa8 \uc870\uc808\uc5d0 `\uc815\uaddc\ud654`\ub97c \ud65c\uc6a9\ud560 \uc218 \uc788\ub2e4.\n\n\n", "meta": {"hexsha": "2fc9dd32a4f0942227b0070316ec9919e0d74ff3", "size": 12632, "ext": "ipynb", "lang": "Jupyter 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{"text": "(ECNL)=\n\n# 3.4 Ecuaciones no lineales\n\n```{admonition} Notas para contenedor de docker:\n\nComando de docker para ejecuci\u00f3n de la nota de forma local:\n\nnota: cambiar `` por la ruta de directorio que se desea mapear a `/datos` dentro del contenedor de docker.\n\n`docker run --rm -v :/datos --name jupyterlab_optimizacion -p 8888:8888 -d palmoreck/jupyterlab_optimizacion:2.1.4`\n\npassword para jupyterlab: `qwerty`\n\nDetener el contenedor de docker:\n\n`docker stop jupyterlab_optimizacion`\n\nDocumentaci\u00f3n de la imagen de docker `palmoreck/jupyterlab_optimizacion:2.1.4` en [liga](https://github.com/palmoreck/dockerfiles/tree/master/jupyterlab/optimizacion).\n\n```\n\n---\n\nNota generada a partir de [liga1](https://www.dropbox.com/s/dfwk0y04ksgfilv/3.5.Aplicaciones_del_algebra_lineal_numerica.pdf?dl=0), [liga2](https://www.dropbox.com/s/6zree47e1u3p5wx/Ecuaciones_no_lineales.pdf?dl=0).\n\n```{admonition} Al final de esta nota el y la lectora:\n:class: tip\n\n* Distinguir\u00e1 la diferencia entre m\u00e9todos abiertos y cerrados a partir del m\u00e9todo de bisecci\u00f3n y m\u00e9todo de Newton.\n\n* Conocer\u00e1 algunos criterios de paro utilizados en m\u00e9todos iterativos y la importancia de considerar la escala de las variables y la funci\u00f3n a la que se le desea calcular sus ra\u00edces o ceros.\n\n* Fortalcer\u00e1 lo revisado en la nota de {ref}`algoritmos de descenso y b\u00fasqueda de l\u00ednea en UCO ` al relacionar resolver problemas de optimizaci\u00f3n sin restricciones con resolver sistemas de ecuaciones no lineales.\n\n* Aprender\u00e1 que el m\u00e9todo de Newton es un m\u00e9todo con convergencia local cuadr\u00e1tica bajo ciertas suposiciones y al que se le deben a\u00f1adir metodolog\u00edas para convergencia desde cualquier punto inicial (convergencia global) dando lugar a m\u00e9todos cuasi-Newton h\u00edbridos. \n\n```\n\n## Sistemas de ecuaciones lineales\n\nLas ecuaciones lineales tienen importantes aplicaciones en todas las \u00e1reas de la ciencia. La teor\u00eda del \u00e1lgebra lineal nos permite tener resultados universales de las mismas y son una herramienta importante para aproximaciones a ecuaciones no lineales. Por ejemplo, al considerar peque\u00f1as perturbaciones en un punto, un sistema no lineal puede t\u00edpicamente aproximarse por un sistema lineal en una vecindad local del punto. Sin embargo, la linearizaci\u00f3n s\u00f3lo describe propiedades locales y para un an\u00e1lisis global de problemas no lineales otras t\u00e9cnicas se requieren. Tales m\u00e9todos com\u00fanmente utilizan esquemas iterativos para gradualmente aproximar la soluci\u00f3n.\n\n```{admonition} Definici\u00f3n\n\nEn general un sistema de ecuaciones lineal es de la forma: \n\n$$\n\\begin{array}{ccc} \na_{11}x_1 + a_{12}x_2 + \\cdots + a_{1n}x_n  &= & b_1 \\\\ \na_{21}x_1 + a_{22}x_2 +  \\cdots + a_{2n}x_n &= & b_2 \\\\ \n\\vdots & & \\\\ \na_{m1}x_1 + a_{m2}x_2 + \\cdots + a_{mn}x_n &=& b_m \n\\end{array}\n$$\n\ndonde: las $x_i$'s son las inc\u00f3gnitas y las $a_i$'s y $b_i$'s son constantes conocidas.\n\nLas entradas $a_{ij}$'s son nombradas **coeficientes del sistema** y forman a la **matriz del sistema** $A \\in \\mathbb{R}^{m \\times n}$. El conjunto de $b_i$'s se le nombra **lado derecho del sistema** y forma al **vector de lado derecho** $b \\in \\mathbb{R}^{m}$. As\u00ed, el sistema se escribe como $Ax = b$.\n\nSi todas las $b_i$'s son iguales a $0$ el sistema se le nombra **homog\u00e9neo** si no se cumple esto se le nombra **no homog\u00e9neo**.\n\n```\n\nLa teor\u00eda del \u00e1lgebra lineal nos ayuda a determinar que existen solamente **3 posibilidades para soluci\u00f3n del sistema anterior:**\n\n* **Una \u00fanica soluci\u00f3n:** s\u00f3lo existe uno y s\u00f3lo un conjunto de valores de $x_i$'s que satisfacen todas las ecuaciones simult\u00e1neamente.\n\n* **Ninguna soluci\u00f3n:** no existe ning\u00fan conjunto de valores de $x_i$'s que satisfacen todas las ecuaciones simult\u00e1neamente (el conjunto soluci\u00f3n es vac\u00edo).\n\n* **Infinitas soluciones:** hay una infinidad de valores distintos de las $x_i$'s que satisfacen todas las ecuaciones simult\u00e1neamente.\n\n```{admonition} Definici\u00f3n\n\nEn el caso de una o infinitas soluciones el sistema de ecuaciones lineales se nombra consistente o no singular, si no existe soluci\u00f3n se nombra inconsistente o singular.\n\n```\n\n```{admonition} Observaci\u00f3n\n:class: tip\n\nEs sencillo probar que si un sistema tiene m\u00e1s de una soluci\u00f3n entonces tiene una infinidad de soluciones. Esto contrasta con sistemas de ecuaciones no lineales donde pueden existir para tales sistemas un n\u00famero finito de soluciones mayor a uno.\n\n```\n\n### Interpretaci\u00f3n geom\u00e9trica\n\nResolver un sistema de ecuaciones lineales equivale a encontrar la intersecci\u00f3n entre rectas, planos o hiperplanos (2,3 o n dimensiones respectivamente). Por ejemplo para un caso de dos dimensiones se tiene:\n\n\n\n\nEl inciso a) representa un sistema de ecuaciones lineales sin soluci\u00f3n, el inciso b) infinitas soluciones (en el dibujo ligeramente se desplaz\u00f3 hacia abajo una de las rectas para mostrar ambas) y el inciso c) una \u00fanica soluci\u00f3n. \n\n(ALGSEL)=\n\n### Algoritmos para resolver sistemas de ecuaciones lineales\n\nExisten una gran cantidad de algoritmos para resolver sistemas de ecuaciones lineales. T\u00edpicamente se elige el algoritmo de acuerdo a las caracter\u00edsticas de los coeficientes de la matriz del sistema y sus dimensiones. \n\n### Algoritmos para sistemas triangulares\n\nSon sistemas cuya matriz del sistema es triangular inferior o superior. Un sistema triangular inferior se resuelve con el **m\u00e9todo de sustituci\u00f3n hacia delante**. Si es triangular superior se resuelve con el **m\u00e9todo de sustituci\u00f3n hacia atr\u00e1s**.\n\n\n### Algoritmos para sistemas no triangulares\n\nPara sistemas de ecuaciones lineales m\u00e1s generales (no tienen estructura identificable) se tienen los **m\u00e9todos iterativos** y **directos o basados en factorizaciones matriciales**.\n\nEntre los directos o basados en factorizaciones matriciales se encuentran:\n\n```{margin}\n\nVer {ref}`definici\u00f3n ` de una matriz sim\u00e9trica definida positiva.\n\n```\n\n* Eliminaci\u00f3n Gaussiana o factorizaci\u00f3n LU.\n* Factorizaci\u00f3n de Cholesky (la matriz del sistema debe ser un elemento en $\\mathbb{S}^n_{++}$ sim\u00e9trica positiva definida)\n* Factorizaci\u00f3n QR.\n* Descomposici\u00f3n en valores singulares o SVD.\n\ny como ejemplo de los iterativos est\u00e1n:\n\n\n* Jacobi.\n* Gauss-Seidel.\n* Gradiente conjugado (la versi\u00f3n que se aplica a matrices del sistema sim\u00e9tricas requiere que tales matrices est\u00e9n en $\\mathbb{S}^n_{++}$).\n\nAmbos m\u00e9todos: iterativos y directos o basados en factorizaciones matriciales encuentran sistemas de ecuaciones equivalentes a partir de operaciones b\u00e1sicas del \u00e1lgebra lineal.\n\n```{admonition} Definici\u00f3n\n\nDos sistemas de ecuaciones lineales son equivalentes si tienen el mismo conjunto soluci\u00f3n.\n\n```\n\n### Sistemas de ecuaciones lineales *square*, *underdetermined*, *overdetermined*\n\nEntre las caracter\u00edsticas que definen el problema a resolver y el tipo de algoritmo a usar se encuentran las dimensiones de una matriz.\n\n```{admonition} Definici\u00f3n\n\nSi la matriz del sistema tiene m\u00e1s renglones que columnas, $m > n$, se tiene un sistema ***overdetermined***, si tiene m\u00e1s columnas que renglones, $m < n$, se nombra ***underdetermined*** y si tiene el mismo n\u00famero de renglones y columnas, $m=n$, se nombra ***square***.\n\n```\n\nLos sistemas de ecuaciones lineales *overdetermined* en general no tienen soluci\u00f3n si $b \\notin \\text{Im}(A)$ con $\\text{Im}(A)$ espacio columna de $A$. Por esto se busca resolver un **problema de m\u00ednimos cuadrados** de la forma:\n\n$$\\displaystyle \\min_{x \\in \\mathbb{R}^n} ||Ax-b||_2$$\n\ncon \u00fanica soluci\u00f3n si $A$ es de *rank* completo.\n\nLos sistemas de ecuaciones lineales *underdetermined* pueden tener infinitas soluciones o ninguna soluci\u00f3n. En el caso que $A$ sea de *rank* completo el sistema es consistente y se busca resolver el **problema de optimizaci\u00f3n de m\u00ednima norma** :\n\n$$\\displaystyle \\min_{x \\in \\mathcal{K}} ||x||_2$$\n\n\ndonde: $\\mathcal{K} = \\{x \\in \\mathbb{R}^n | Ax = b\\}$ que es interesante para $b \\neq 0$ y tiene \u00fanica soluci\u00f3n.\n\n\n```{margin}\n\nRecu\u00e9rdese que el producto $x^T Ax$ con $A$ sim\u00e9trica se le nombra forma cuadr\u00e1tica y es un n\u00famero en $\\mathbb{R}$.\n\n```\n\n```{admonition} Comentarios\n\n* El problema de m\u00ednimos cuadrados es un problema convexo no importando si $A$ es o no de *rank* completo pues la forma cuadr\u00e1tica involucra a la expresi\u00f3n $x^TA^TAx$ y $A^TA \\in \\mathbb{S}^n_+$.\n\n* El problema de optimizaci\u00f3n a resolver para el caso de sistemas de ecuaciones lineales *underdetermined* y matriz del sistema de *rank* completo tambi\u00e9n puede escribirse como:\n\n$$\\min_{x \\in \\mathbb{R}^n} ||x||_2$$\n\n$$\\text{sujeto a:} Ax = b$$\n\nel cual es un problema de optimizaci\u00f3n convexa con restricciones (no importando si $A$ es o no de *rank* completo).\n\n* Que un sistema de ecuaciones lineales *underdetermined* pueda tener infinitas soluciones si $A$ es de *rank* completo se sigue por el resultado del \u00e1lgebra lineal [Rank-nullity theorem](https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem). Con las dimensiones definidas tal resultado se escribe como $n = rank(A) + nullity(A) = m + nullity(A)$ $\\therefore nullity(A) = n-m > 0$. Si $x_0 \\in \\mathbb{R}^n$ es una soluci\u00f3n de $Ax=b$ entonces $x_0 + N(A)$ es el conjunto de soluciones de $Ax=b$, donde $x_0 + N(A) = \\{x_0 + z | z \\in N(A)\\}$. \n\n\n```\n\n## Ecuaciones no lineales\n\nEl problema que queremos resolver es el siguiente: dada $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ encontrar $x^*$ que resuelva la ecuaci\u00f3n no lineal $f(x) = 0$. Nos interesa al menos una soluci\u00f3n de la ecuaci\u00f3n anterior.\n\n```{admonition} Definici\u00f3n\n\n$x^*$ se nombra ra\u00edz o cero de $f$.\n\n```\n\nAlgunos ejemplos son:\n\n* $e^x+1=0$\n\n* $e^{-x}-x =0$\n\n* $x^2 -4\\sin(x)=0$\n\n* $x^3+6x^2+11x-6=0$\n\n* $\\sin(x) = 0$.\n\n**Resolvamos con [scipy.optimize.fsolve](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fsolve.html#scipy.optimize.fsolve) algunas de las ecuaciones no lineales anteriores.**\n\n\n```python\nfrom matplotlib import pyplot as plt\nfrom scipy.optimize import fsolve\nimport math\nimport numpy as np\n```\n\n\n```python\nnp.set_printoptions(precision=3, suppress=True)\n```\n\nLa ecuaci\u00f3n no lineal $e^x + 1 = 0$ no tiene soluci\u00f3n, su gr\u00e1fica es la siguiente\n\n\n```python\nt = np.linspace(-1,1,100)\n```\n\n\n```python\neqn = np.exp(t) + 1\n```\n\n\n```python\nplt.plot(t, eqn)\nplt.axhline(color=\"black\")\nplt.title(\"$f(x) = e^x + 1$\")\nplt.grid()\nplt.show()\n```\n\nLa ecuaci\u00f3n no lineal $e^{-x} - x = 0$ tiene una soluci\u00f3n\n\n\n```python\nt = np.linspace(-.25,1,100)\n```\n\n\n```python\neqn = lambda x: np.exp(-x) - x\n```\n\n```{margin}\n\nElegimos un punto inicial por ejemplo el $0$.\n\n```\n\n\n```python\nroot = fsolve(eqn, 0)\nprint(root)\n```\n\n    [0.567]\n\n\n\n```python\nplt.plot(t, eqn(t))\nplt.scatter(root, 0, color = \"red\")\nplt.axhline(color=\"black\")\nplt.title(\"$f(x) = e^{-x}-x$\")\nplt.grid()\nplt.show()\n```\n\nLa ecuaci\u00f3n no lineal $x^2 -4\\sin(x)=0$ tiene dos soluciones\n\n\n```python\nt = np.linspace(-5,5,100)\n```\n\n\n```python\neqn = lambda x: x**2-4*np.sin(x)\n```\n\n```{margin}\n\nElegimos un punto inicial por ejemplo el $-2$.\n\n```\n\n\n```python\nroot = fsolve(eqn, -2)\nprint(root)\n```\n\n    [0.]\n\n\n```{margin}\n\nObservamos que tenemos dos ra\u00edces de $f$.\n\n```\n\n\n```python\nplt.plot(t, eqn(t))\nplt.scatter(root, 0, color = \"red\")\nplt.axhline(color=\"black\")\nplt.title(\"$f(x) = x^2-4\\sin(x)$\")\nplt.grid()\nplt.show()\n```\n\n```{margin}\n\nElegimos un punto inicial por ejemplo el $3$.\n\n```\n\n\n```python\nroot2 = fsolve(eqn, 3)\nprint(root2)\n```\n\n    [1.934]\n\n\n\n```python\nplt.plot(t, eqn(t))\nplt.scatter(root, 0, color = \"red\")\nplt.scatter(root2, 0, color = \"red\")\nplt.axhline(color=\"black\")\nplt.title(\"$f(x) = x^2-4\\sin(x)$\")\nplt.grid()\nplt.show()\n```\n\n```{margin}\n\nComo ejemplo que no es posible expresar las ra\u00edces o ceros por una f\u00f3rmula cerrada que involucren a los coeficientes, operaciones aritm\u00e9ticas y ra\u00edces $\\sqrt[n]{\\cdot}$, consid\u00e9rese la ecuaci\u00f3n no lineal $x^5 - x^2 + 1 = 0$.\n\n```\n\n```{admonition} Comentarios\n\n* En el caso de una ecuaci\u00f3n o un sistema de ecuaciones no lineales no tenemos resultados que determinen la existencia o unicidad de soluciones a diferencia de un sistema lineal. Sin embargo, en muchas situaciones en la pr\u00e1ctica se resuelven ecuaciones no lineales que s\u00ed tienen soluci\u00f3n y se desea aproximar una soluci\u00f3n o varias soluciones en una regi\u00f3n de inter\u00e9s por lo que determinar la existencia o unicidad de la soluci\u00f3n no es primordial.\n\n* La mayor\u00eda de los m\u00e9todos para calcular ra\u00edces o ceros de $f$ v\u00eda la ecuaci\u00f3n no lineal $f(x) = 0$ nos devuelven aproximaciones y no f\u00f3rmulas cerradas. Son m\u00e9todos **iterativos** que en el caso de $1$ dimensi\u00f3n los podemos dividir en $2$ tipos: **cerrados** y **abiertos**. Los cerrados inician sus iteraciones en un intervalo que encierra a la ra\u00edz y conforme avanzan las iteraciones hacen subdivisiones del intervalo inicial por lo que su longitud se reduce y **siempre** convergen. Los abiertos no requieren encerrar a la ra\u00edz, en general tienen mejor desempe\u00f1o que los cerrados en cuanto al n\u00famero de iteraciones pero **no siempre convergen**.\n\n* Es conveniente comentar que si bien quisi\u00e9ramos tener algoritmos que calculasen todas las ra\u00edces o ceros de $f$ esto no es posible, es un hecho que los m\u00e9todos nos dar\u00e1n una soluci\u00f3n aproximada o un mensaje del tipo \"no se encontr\u00f3 soluci\u00f3n\".\n\n```\n\n(CRITPARO)=\n\n## Criterios de paro, escala de la variable $x$ y de la funci\u00f3n $f$\n\nUn tema importante en la implementaci\u00f3n de algoritmos es la escala del problema tanto en la variable $x$ como en la funci\u00f3n $f$. Por ejemplo, si $x_1$ est\u00e1 en el rango $[10^2, 10^3]$ de metros y $x_2$ est\u00e1 en el rango $[10^{-7}, 10^{-6}]$ de segundos entonces tenemos que realizar un reescalamiento para tener evaluaciones de $f$, criterios de paro y actualizaciones en esquemas iterativos, por ejemplo, independientes de las escalas de las variables o de la funci\u00f3n. Asimismo, los criterios de paro en un m\u00e9todo iterativo ayudan a contestar preguntas del tipo \u00bfhemos resuelto el problema de forma aproximada? \u00bfen las \u00faltimas dos (o un poco m\u00e1s) iteraciones nos hemos quedado virtualmente en el mismo punto? \n\n\n```{margin}\n\nEl reescalamiento en el ejemplo de kil\u00f3metros y microsegundos puede describirse como la multiplicaci\u00f3n de una matriz diagonal por las variables $x_1$ y $x_2$ en la que las entradas de la diagonal son $\\frac{1}{10^3}$ y $\\frac{1}{10^{-6}}$ para las variables $x_1$ y $x_2$ respectivamente.\n\n```\n\nMuchos algoritmos cumplen que son invariantes ante escala de las variables, el m\u00e9todo de Newton en la variable $x$ es uno de ellos por ejemplo, pero otros no, por lo que al implementar un algoritmo se debe revisar los reescalamientos a realizar. En el ejemplo anterior de los metros y segundos si se cambian las unidades de $x_1$ a kil\u00f3metros y las de $x_2$ a microsegundos entonces tanto $x_1$ como $x_2$ se encontrar\u00e1n en el rango $[10^{-1}, 1]$. Si en dos dimensiones $x_1 \\in [10^{6}, 10^{7}]$ y $x_2 \\in [10^{-1}, 1]$ entonces una prueba del tipo $||\\nabla f(x)|| <  10^{-3}$ no ser\u00e1 equilibrada para ambas variables si se desea por ejemplo minimizar $f$ ($x_1$ tender\u00eda a ser lo m\u00e1s peque\u00f1a posible si por ejemplo tenemos una alta contribuci\u00f3n de esta variable en $f$).\n\nEn el caso de la funci\u00f3n $f$, es com\u00fan requerir que $f$ o la magnitud de $f$ sea cero (o su derivada). Si consideramos $f(x) = 0$ es muy probable que los errores por redondeo no permitan que se satisfaga esto para ning\u00fan punto $x$ por lo que modificamos la condici\u00f3n anterior a $f(x) \\approx 0$. Tambi\u00e9n si $f$ no est\u00e1 escalada apropiadamente la condici\u00f3n $|f(x)| < tol$ es probable que siempre o nunca se satisfaga. Por ejemplo si $tol = 10^{-3}$ y $f$ siempre est\u00e1 en $[10^{-7}, 10^{-5}]$ entonces cualquier $x$ satisface $|f(x)| < 10^{-3}$. \n\nConsiderando $f: \\mathbb{R}^n \\rightarrow \\mathbb{R}^n$, dentro de los criterios de paro que se utilizan en los m\u00e9todos iterativos para resolver ecuaciones no lineales que apuntan a tener una evaluaci\u00f3n independiente de la escala se encuentran:\n\n```{margin}\n\nEn los criterios de paro que revisan la norma de la derivada de $f$, una opci\u00f3n independiente de la escala de $f$ y $x$ es la cantidad $\\frac{||Df(x)||||x||}{||f(x)||}$.\n\n```\n\n* Medir diferencia entre iteraciones. Por ejemplo:\n\n    * $||x^{(k+1)} - x^{(k)}|| < tol(||x^{(k)}|| +1)$\n    * $||x^{(k+1)} - x^{(k)}|| < tol\\max\\{||x^{(k+1)}||, ||x^{(k)}||\\}$\n    * $||x^{(k+1)} - x^{(k)}|| < tol\\max\\{||x^{(k+1)}||, \\text{user_defined_value}\\}$. \n\ncon `user_defined_value` un valor positivo proporcionado por *user* que mide la magnitud t\u00edpica de $x$ y $|| \\cdot\u00a0||$ norma.\n\n\n* Medir la norma de $f$ reescal\u00e1ndola por ejemplo:\n\n$$||Diag f|| < tol$$\n\n\ncon $Diag$ matriz diagonal tal que $Diagf$ tenga norma alrededor de $1$ en puntos no cercanos a la ra\u00edz y tambi\u00e9n puede proveerse sus valores con un `user_defined_value`.\n   \n\n* M\u00e1ximo n\u00famero de iteraciones.\n    \n\n\n## M\u00e9todos para resolver ecuaciones no lineales de funciones $f: \\mathbb{R} \\rightarrow \\mathbb{R}$\n\n### M\u00e9todo de bisecci\u00f3n\n\nEs un m\u00e9todo cerrado que requiere $f \\in \\mathbb{R} \\rightarrow \\mathbb{R}$ con $f \\in \\mathcal{C}([a,b])$ tal que $f(a) f(b) <0$, esto es, que $f$ tenga un cambio de signo. Por el **teorema del valor intermedio** se cumple que $f$ tiene una ra\u00edz en $[a,b]$.\n\n### Algoritmo: m\u00e9todo de bisecci\u00f3n\n\n> **Dados** $x_i, x_s$ l\u00edmite inferior y superior respectivamente tales que $x^* \\in [x_i, x_s]$ con $f(x_i)f(x_s)<0$ y $tol >0$\n>\n> **Repetir** el siguiente bloque para $k=1,2,\\dots$\n>> 1. $x_m = \\frac{x_i + x_s}{2}$\n>>\n>> 2. Si $f(x_i)f(x_m) < 0$ entonces $x^* \\in [x_i, x_m]$ por lo tanto $x_s = x_m$.\n>>\n>> 3. Si $f(x_i)f(x_m) > 0$ entonces $x^* \\in [x_m, x_s]$ por lo tanto $x_i = x_m$.\n>\n> **hasta** convergencia: satisfacer criterio de paro en el que se utiliza $tol$ y $maxiter$.\n\n\n\n````{admonition} Comentarios\n\nEn el m\u00e9todo de bisecci\u00f3n:\n\n* Se garantiza que el error relativo en cada iteraci\u00f3n se reduce por la mitad y se obtiene una cantidad constante de d\u00edgitos por cada iteraci\u00f3n, lo cual es representativo de una convergencia lineal.\n\n* Siempre tenemos convergencia pero es lenta.\n\n* No es posible extenderlo a m\u00e1s dimensiones de forma natural pues tendr\u00edamos que definir metodolog\u00edas para elegir puntos en regiones como rect\u00e1ngulos, cubos,... para evaluar a la funci\u00f3n $f$ y determinar cambios de signo.\n\n* La evaluaci\u00f3n de los pasos 2 y 3 del algoritmo anterior se visualizan respectivamente como sigue:\n\n\n\nen el dibujo $x^{(k)}$ corresponde a $x_m$.\n\n* La implementaci\u00f3n del m\u00e9todo utiliza lo siguiente:\n\n    * El punto medio se calcula con la expresi\u00f3n: $x_m = x_i + \\frac{x_s - x_i}{2}$\n    \n    * Se revisan los signos de $f(x_i)$, $f(x_m)$ para determinar si $f(x_i)f(x_m) < 0$ o $f(x_i)f(x_m) > 0$.\n    \n* El criterio de paro es de la forma:\n\n```\nwhile |xi - xs| > tol1(1+|xi|+|xs|) && |f(x_m)| > tol2 && iterations < max_iters\n```\n\ncon `tol1`, `tol2` cantidades peque\u00f1as y positivas (com\u00fanmente menores o iguales a $10^{-8}$), `iterations` un contador de las iteraciones. En una implementaci\u00f3n tambi\u00e9n pueden utilizarse la medici\u00f3n entre iteraciones con los criterios descritos en {ref}`criterios de paro, escala de la variable x y de la funci\u00f3n f ` y si se tiene conocimiento del valor de $x^*$ se pueden calcular errores relativos de `x_k`, as\u00ed como los reescalamientos respectivos de $x$ y $f$.\n\n````\n\n```{admonition} Ejercicio\n:class: tip\n\nCon el m\u00e9todo de bisecci\u00f3n aproxima la ra\u00edz $x^* \\approx 0.56714329$ de la ecuaci\u00f3n no lineal $f(x) = e^{-x}-x$ tomando como intervalo inicial $[0,2]$ y un valor de $tol = 10^{-8}$. Crea una tabla de la forma:\n\n|Iter | $x_i$ | $x_s$ | $x^{(k)}$ | Err_rel$(x^{(k)})$|\n|:---:|:---:|:---:|:---:|:---:|\n|1|0|2|1|1.5 e-2|\n|2|0|1|0.5|1.3 e-2|\n\n(valores ejemplo)\n\n```\n\n### M\u00e9todo de Newton o Newton-Raphson \n\nEs un m\u00e9todo abierto que sigue un esquema iterativo de la forma:\n\n$$x^{(k+1)} = x^{(k)} - \\frac{f(x^{(k)})}{f'(x^{(k)})}$$\n\nrequiere un punto inicial $x^{(0)}$ y converge si se cumplen condiciones descritas en {ref}`comentarios del m\u00e9todo de Newton-Raphson `.\n\nExisten varias formas de obtener tal esquema iterativo, la que se presenta a continuaci\u00f3n **define un modelo af\u00edn local que aproxima a nuestra funci\u00f3n $f$ y encuentra la ra\u00edz de tal modelo**, gr\u00e1ficamente:\n\n\n\nEl modelo af\u00edn en el dibujo anterior es de la forma:\n\n$$M(x) = f(x^{(k)}) + f'(x^{(k)})(x-x^{(k)})$$\n\nE igualando a cero el modelo se tiene:\n\n$$\n\\begin{eqnarray}\n0 &=& M(x) = f(x^{(k)}) + f'(x^{(k)})(x-x^{(k)}) \\nonumber \\\\\n&\\therefore& x = x^{(k)} - \\frac{f(x^{(k)})}{f'(x^{(k)})} \\nonumber\n\\end{eqnarray}\n$$\n\n```{admonition} Observaci\u00f3n\n:class: tip\n\nObs\u00e9rvese que el modelo af\u00edn anterior $M(x)$ es la aproximaci\u00f3n a primer orden dada por el teorema de Taylor.\n\n```\n\n### Ejemplo\n\nEncontrar la ra\u00edz de $f(x) = 4x + 5$ con el m\u00e9todo de Newton.\n\n\n```python\nimport sympy \n```\n\n```{margin}\n\nElecci\u00f3n del punto inicial.\n\n```\n\n\n```python\nx_0 = -2\n```\n\n```{margin}\n\nDefinici\u00f3n de funci\u00f3n.\n```\n\n\n```python\nx = sympy.Symbol('x')\n```\n\n\n```python\nf = 4*x + 5\n```\n\n```{margin}\n\nDerivada de $f$.\n\n```\n\n\n```python\ndf = f.diff()\n```\n\n\n```python\nsympy.pprint(df)\n```\n\n    4\n\n\n```{margin}\n\nActualizaci\u00f3n por el m\u00e9todo de Newton: $x_1 = x_0 - \\frac{f(x_0)}{f'(x_0)}$.\n\n```\n\n\n```python\nx_1 = x_0 - f.subs(x, x_0)/df.subs(x, x_0)\n```\n\n\n```python\nsympy.pprint(x_1)\n```\n\n    -5/4\n\n\n### Algoritmo: m\u00e9todo de Newton para resolver una ecuaci\u00f3n no lineal\n\n> **Dados** $x^{(0)}$ punto inicial, $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ y $tol >0$\n>\n> $x:=x^{(0)}$.\n>\n> **Repetir** el siguiente bloque para $k=1,2,\\dots$\n>> 1. Calcular  la derivada de $f$ y verificar que $f'(x) \\neq 0$.\n>>\n>> 2. Realizar la actualizaci\u00f3n: $x = x - \\frac{f(x)}{f'(x)}$\n>\n> **hasta** convergencia: satisfacer criterio de paro en el que se utiliza $tol$ y $maxiter$.\n\n\n\n### Ejemplo\n\nAproximar el valor $\\sqrt{3}$ con el m\u00e9todo de Newton\n\n```{margin}\n\nElecci\u00f3n del punto inicial. \u00bfQu\u00e9 pasa si elegimos $x_0 = -10$?\n\n```\n\n\n```python\nx_0 = 10\n```\n\n\n```python\nx_sym = sympy.Symbol('x')\n```\n\n```{margin}\n\nDefinimos la funci\u00f3n $f(x) = x^2 - 3$\n\n```\n\n\n```python\nf = x_sym**2 - 3\n```\n\n```{margin}\n\nDerivada de $f$.\n\n```\n\n\n```python\ndf = f.diff()\n```\n\n\n```python\nsympy.pprint(df)\n```\n\n    2\u22c5x\n\n\n**Primera iteraci\u00f3n**\n\n```{margin}\n\nActualizaci\u00f3n por el m\u00e9todo de Newton: $x_1 = x_0 - \\frac{f(x_0)}{f'(x_0)}$.\n\n```\n\n\n```python\nx = x_0 - f.subs(x_sym, x_0)/df.subs(x_sym, x_0)\n```\n\n\n```python\nsympy.pprint(x)\n```\n\n    103\n    \u2500\u2500\u2500\n     20\n\n\n\n```python\nsympy.pprint(x.evalf())\n```\n\n    5.15000000000000\n\n\n**Segunda iteraci\u00f3n**\n\n```{margin}\n\nActualizaci\u00f3n por el m\u00e9todo de Newton: $x_2 = x_1 - \\frac{f(x_1)}{f'(x_1)}$.\n\n```\n\n\n```python\nx = x - f.subs(x_sym, x)/df.subs(x_sym, x)\n```\n\n\n```python\nsympy.pprint(x)\n```\n\n    11809\n    \u2500\u2500\u2500\u2500\u2500\n     4120\n\n\n\n```python\nsympy.pprint(x.evalf())\n```\n\n    2.86626213592233\n\n\n**Tercera iteraci\u00f3n**\n\n```{margin}\n\nActualizaci\u00f3n por el m\u00e9todo de Newton: $x_3 = x_2 - \\frac{f(x_2)}{f'(x_2)}$.\n\n```\n\n\n```python\nx = x - f.subs(x_sym, x)/df.subs(x_sym, x)\n```\n\n\n```python\nsympy.pprint(x)\n```\n\n    190375681\n    \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n     97306160\n\n\n\n```python\nsympy.pprint(x.evalf())\n```\n\n    1.95646073177690\n\n\n**...**\n\n**S\u00e9ptima iteraci\u00f3n**\n\n\n```python\nx_7 = 1.73205080756888\n```\n\n\n```python\nfrom pytest import approx\nimport math\n```\n\n\n```python\nprint(x_7 == approx(math.sqrt(3)))\n```\n\n    True\n\n\n(COMENTMETNEWTONRAPSHON)=\n\n````{admonition} Comentarios\n\n* El modelo af\u00edn anterior $M(x) = f(x^{(k)}) + f'(x^{(k)})(x-x^{(k)})$ es tambi\u00e9n nombrado **modelo lineal**.\n\n* Si la funci\u00f3n $f$ es lineal el m\u00e9todo de Newton converge en una iteraci\u00f3n.\n\n* La convergencia del m\u00e9todo de Newton en una dimensi\u00f3n converge de forma cuadr\u00e1tica, esto es, el n\u00famero de d\u00edgitos de precisi\u00f3n en cada iteraci\u00f3n se duplica si se satisfacen las siguientes condiciones:\n\n    * El punto inicial $x^{(0)}$ es cercano a la ra\u00edz $x^*$ de $f$.\n    * $f'(x^*) \\neq 0$ y existe un conjunto abierto $\\mathcal{D}$ en el que $f'(x) \\neq 0$ $\\forall x \\in \\mathcal{D}$, $x^* \\in \\mathcal{D}$ y la segunda derivada de $f$ es acotada en $\\mathcal{D}$ \\*.\n\n\\* La segunda condici\u00f3n referente a la segunda derivada puede ser sustituida por la condici\u00f3n que la primera derivada sea *Lipschitz* continua en $\\mathcal{D}$, ver [Lipschitz_continuity](https://en.wikipedia.org/wiki/Lipschitz_continuity). Esto ayuda a acotar la diferencia entre $f$ y el modelo af\u00edn $M$. Adem\u00e1s evitamos calcular la segunda derivada (que en m\u00e1s dimensiones puede ser complicada de describir) para verificar convergencia.\n\n* El criterio de paro es de la forma:\n\n```\nwhile |x_(k+1) - x_k| > tol1(|x_k| + 1) && |f(x_(k+1))| > tol2 && iterations < max_iters\n```\n\ncon `tol1`, `tol2` cantidades peque\u00f1as y positivas (com\u00fanmente menores o iguales a $10^{-8}$), `iterations` un contador de las iteraciones. En una implementaci\u00f3n tambi\u00e9n pueden utilizarse la medici\u00f3n entre iteraciones con los criterios descritos en {ref}`criterios de paro, escala de la variable x y de la funci\u00f3n f ` y si se tiene conocimiento del valor de $x^*$ se pueden calcular errores relativos de `x_k`, as\u00ed como los reescalamientos respectivos de $x$ y $f$.\n\n* En la implementaci\u00f3n del m\u00e9todo hay que monitorear el valor absoluto de la derivada de $f$.\n\n\n````\n\n```{admonition} Observaciones\n:class: tip\n\n* Si la derivada de $f$ es cero en la ra\u00edz no podemos concluir si el m\u00e9todo de Newton converge o no y si converge podr\u00eda o no hacerlo de forma cuadr\u00e1tica.\n\n* Si elegimos un punto inicial lejos de $x^*$ no podemos concluir, el m\u00e9todo de Newton podr\u00eda o no converger.\n\n```\n\n```{admonition} Ejercicio\n:class: tip\n\nPara revisar la hip\u00f3tesis que la derivada de $f$ sea diferente de cero en la ra\u00edz y garantice que el m\u00e9todo de Newton tenga convergencia cuadr\u00e1tica consid\u00e9rese aproximar la ra\u00edz $1$ para las ecuaciones no lineales: \n\n1.$x^2-1=0$\n\n2.$x^2-2x+1=0$\n\nReal\u00edcense $6$ iteraciones del m\u00e9todo de Newton para cada ecuaci\u00f3n no lineal y h\u00e1ganse conclusiones.\n\n```\n\n```{admonition} Ejercicio\n:class: tip\n\nPara revisar la hip\u00f3tesis que el punto inicial $x^{(0)}$ sea \"cercano\" a la ra\u00edz y garantice que el m\u00e9todo de Newton tenga convergencia cuadr\u00e1tica consid\u00e9rese aproximar la ra\u00edz $0$ para la ecuaci\u00f3n no lineal $\\arctan(x) = 0$. Real\u00edcense $6$ iteraciones del m\u00e9todo de Newton eligiendo un punto $x^{(0)}$ en tres casos:\n\n1.tal que sea en valor absoluto menor a un punto cualquiera en $[1.39, 1.40]$,\n\n2.tal que est\u00e9 en $[1.39, 1.40]$,\n\n3.tal que en valor absoluto sea mayor a un punto en el intervalo $[1.39, 1.40]$.\n\ny h\u00e1ganse conclusiones.\n```\n\nConcerniente a la dependencia de un punto inicial, la convergencia del m\u00e9todo de Newton se robustece al incorporar metodolog\u00edas que permiten su convergencia a una soluci\u00f3n **local** desde pr\u00e1cticamente cualquier punto inicial. Tales metodolog\u00edas resultan en **algoritmos h\u00edbridos** en los que se utiliza el m\u00e9todo de Newton siempre que funcione bien pero se utiliza otro m\u00e9todo (quiz\u00e1s m\u00e1s lento) que garantice convergencia. Uno de \u00e9stos es el m\u00e9todo de bisecci\u00f3n en el que una vez se encuentre \"cerca\" de una soluci\u00f3n se utilice el m\u00e9todo de Newton. Otra metodolog\u00eda consiste en lo siguiente: dado que la actualizaci\u00f3n del m\u00e9todo de Newton involucra una direcci\u00f3n en la que el valor absoluto de $f(x)$ **siempre decrece** (al menos en un inicio) entonces se utiliza una forma de *backtracking* si la actualizaci\u00f3n completa no produce una reducci\u00f3n en $|f(x)|$:\n\n\n\n\n\nY se realizar\u00eda lo siguiente:\n\n$$x = x^{(k)} - \\frac{f(x^{(k)})}{f'(x^{(k)})}$$\n\nMientras $|f(x)| \\geq |f(x^{(k)})|$ realizar:\n\n$$x = \\frac{x + x^{(k)}}{2}$$\n\nver por ejemplo el {ref}`m\u00e9todo de b\u00fasqueda de l\u00ednea por backtracking ` en el contexto de minimizaci\u00f3n de una funci\u00f3n.\n\nEl siguiente es un algoritmo en una forma general de algoritmos h\u00edbridos [*quasi-Newton*](https://en.wikipedia.org/wiki/Quasi-Newton_method) para resolver una ecuaci\u00f3n no lineal.\n\n(ALGMGCNHEN)=\n\n### Algoritmo: m\u00e9todo general cuasi-Newton h\u00edbrido para resolver una ecuaci\u00f3n no lineal\n\n> **Dados** $x^{(0)}$ punto inicial, $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ y $tol >0$\n>\n> **Repetir** el siguiente bloque para $k=1,2,\\dots$\n>> 1. Construir un modelo local de $f$ alrededor de $x^{(k)}$ y encontrar el punto $x_N$ que resuelva (o cercanamente resuelva) el modelo del problema.\n>>\n>> 2. Realizar alguno de los dos pasos siguientes:\n>>>\n>>> a. Decidir si $x^{(k+1)} = x_N$ si no,\n>>>\n>>> b. Elegir $x^{(k+1)}$ usando una estrategia global (usar $x_N$ del inciso a. de forma m\u00e1s conservadora).\n>\n> **hasta** convergencia: satisfacer criterio de paro en el que se utiliza $tol$ y $maxiter$.\n\n\ndonde: $x_N$ es la actualizaci\u00f3n por el m\u00e9todo de Newton en su esquema iterativo.\n\n```{admonition} Comentario\n\nAdem\u00e1s de estrategias globales es com\u00fan que no se tengan disponibles las derivadas de $f$, en este caso las metodolog\u00edas de diferenciaci\u00f3n finita son \u00fatiles, ver {ref}`diferenciaci\u00f3n num\u00e9rica por diferencias finitas `.\n\n```\n\n## Una nota sobre problemas *Unconstrained Optimization* (UO)\n\nEn esta secci\u00f3n utilizamos la notaci\u00f3n para un problema de optimizaci\u00f3n sin restricciones de la forma:\n\n$$\\displaystyle \\min_{x^\\in \\mathbb{R}^n} f_o(x)$$\n\ny $f_o: \\mathbb{R} \\rightarrow \\mathbb{R}$ es una funci\u00f3n objetivo en general que asumimos es de clase $\\mathcal{C}^2$ en su dominio.\n\nAs\u00ed como en ecuaciones no lineales no tenemos resultados que determinen la existencia o unicidad de soluciones, en problemas de optimizaci\u00f3n sin restricciones esto es similar al plantear la b\u00fasqueda de m\u00ednimos globales de las funciones objetivo. Lo mejor que podemos obtener son aproximaciones a minimos locales y es pr\u00e1cticamente imposible saber si se ha aproximado un m\u00ednimo global.\n\n```{margin}\n\nPor condici\u00f3n necesaria de primer orden recu\u00e9rdese que si $x^*$ es \u00f3ptimo entonces $\\nabla f_o(x^*) = 0$ que establece un sistema de ecuaciones no lineales en general.\n\n```\n\nAdem\u00e1s, en la nota de {ref}`algoritmos de descenso y b\u00fasqueda de l\u00ednea en Unconstrained Convex Optimization (UCO) ` se mostr\u00f3 la relaci\u00f3n que existe entre resolver problemas tipo UO y ecuaciones no lineales. Es natural entonces aplicar el algoritmo de {ref}`m\u00e9todo general cuasi-Newton h\u00edbrido para resolver una ecuaci\u00f3n no lineal ` a la ecuaci\u00f3n no lineal de una variable $f_o'(x) = 0$. El esquema iterativo entonces es de la forma:\n\n$$x^{(k+1)} = x^{(k)} - \\frac{f_o'(x^{(k)})}{f_o''(x^{(k)})}$$\n\nRecu\u00e9rdese que tal esquema iterativo se obtiene mediante un modelo af\u00edn de $f_o'(x)$ alrededor de $x^{(k)}$, lo cual es equivalente en t\u00e9rminos de la funci\u00f3n $f_o$ a definir un modelo cuadr\u00e1tico alrededor de $x^{(k)}$ que aproxime a nuestra funci\u00f3n $f_o$ y que encuentre la ra\u00edz de tal modelo:\n\n$$m(x) = f_o(x^{(k)}) + f_o'(x^{(k)})(x-x^{(k)}) + \\frac{1}{2} f_o''(x^{(k)})(x-x^{(k)})^2,$$\n\ncon lo que obtendremos el esquema iterativo anterior.\n\n```{admonition} Comentarios\n\n* Un modelo cuadr\u00e1tico es m\u00e1s apropiado que un modelo af\u00edn para $f_o$ ya sea para maximizaci\u00f3n o minimizaci\u00f3n pues tiene a lo m\u00e1s un punto extremo. \n\n* Si la funci\u00f3n $f_o$ es una funci\u00f3n cuadr\u00e1tica el m\u00e9todo de Newton converge en una iteraci\u00f3n.\n\n* As\u00ed como se revisaron las condiciones bajo las cuales el m\u00e9todo de Newton converge de forma cuadr\u00e1tica, en el caso de un problema UO se requiere:\n\n    * El punto inicial $x^{(0)}$ sea cercano a la ra\u00edz $x^*$ de $f'$.\n    * $f''(x^*) \\neq 0$ y existe un conjunto abierto $\\mathcal{D}$ en el que $f''(x) \\neq 0$ $\\forall x \\in \\mathcal{D}$, $x^* \\in \\mathcal{D}$ y la segunda derivada sea *Lipschitz* continua en $\\mathcal{D}$, ver [Lipschitz_continuity](https://en.wikipedia.org/wiki/Lipschitz_continuity). Esto ayuda a acotar la diferencia entre $f$ y el modelo cuadr\u00e1tico $m$. Adem\u00e1s evitamos calcular la tercera derivada (que en m\u00e1s dimensiones puede ser complicada de describir) para verificar convergencia.\n\n\n```\n\n(SISTECNOLINEALES)=\n\n## Sistema de ecuaciones no lineales\n\nEl caso de sistema de ecuaciones no lineales es una generalizaci\u00f3n del caso de una dimensi\u00f3n en el que tenemos $f: \\mathbb{R}^n \\rightarrow \\mathbb{R}^n$ y debemos encontrar una ra\u00edz o cero de $f$ que resuelva el sistema de ecuaciones no lineales $f(x) = 0$.\n\n```{admonition} Observaci\u00f3n\n:class: tip\n\n$f$ tiene $n$ funciones componentes:\n\n$$\nf(x) = \\left [ \\begin{array}{c}\nf_1(x) \\\\\nf_2(x) \\\\\n\\vdots \\\\\nf_n(x)\n\\end{array}\n\\right ]\n$$\n\ny su derivada es la matriz de $n \\times n$, la Jacobiana $(J(x))_{ij} = \\frac{\\partial f_i(x)}{\\partial x_j}$, ver {ref}`Definici\u00f3n de funci\u00f3n, continuidad y derivada `.\n\n```\n\nAlgunos ejemplos son:\n\n1) $$\n\\begin{eqnarray}\nx_1^2+x_1x_2&=&10 \\nonumber\u00a0\\\\\nx_2 + 3x_1x_2^2&=&57 \\nonumber\u00a0\n\\end{eqnarray}\n$$\n\n2) $$\n\\begin{eqnarray}\n2=\\displaystyle \\int_{-1}^{1}1dx &=& w_0 \\cdot 1 + w_1\\cdot1 \\nonumber \\\\\n0 = \\displaystyle \\int_{-1}^1xdx &=& w_0x_0 + w_1x_1 \\nonumber \\\\\n\\frac{2}{3} = \\displaystyle \\int_{-1}^1x^2dx &=& w_0x_0^2 + w_1x_1^2 \\nonumber \\\\\n0 = \\displaystyle \\int_{-1}^1x^3dx &=& w_0x_0^3 + w_1x_1^3 \\nonumber \\\\\n\\end{eqnarray}\n$$\n\n## M\u00e9todo de Newton para ecuaciones no lineales de funciones $f: \\mathbb{R}^n \\rightarrow \\mathbb{R}^n$\n\nEl problema que queremos resolver es el siguiente: dada $f: \\mathbb{R}^n \\rightarrow \\mathbb{R}^n$ encontrar $x^* \\in \\mathbb{R}^n$ que resuelva la ecuaci\u00f3n no lineal $f(x) = 0$. Nos interesa al menos una soluci\u00f3n de la ecuaci\u00f3n anterior. En este problema se **asume** que $f \\in \\mathcal{C}^1$ en su dominio.\n\nAs\u00ed como el caso de una dimensi\u00f3n se **define un modelo af\u00edn local que aproxima a nuestra funci\u00f3n $f$ y encuentra la ra\u00edz de tal modelo**. El modelo af\u00edn es:\n\n$$M(x) = f(x^{(k)}) + J(x^{(k)})(x-x^{(k)}).$$\n\n```{admonition} Observaci\u00f3n\n:class: tip\n\nObs\u00e9rvese que el modelo af\u00edn anterior $M(x)$ es la aproximaci\u00f3n a primer orden dada por el teorema de Taylor.\n\n```\n\nEs com\u00fan reescribir tal modelo como:\n\n$$M(x + v) = f(x) + J(x)v$$\n\nusando $x = x^{(k)} + v$.\n\nLa ra\u00edz del modelo anterior (igual\u00e1ndolo a cero) conduce a resolver el **sistema de ecuaciones lineales** siguiente:\n\n$$J(x)v = -f(x)$$\n\ny por tanto el esquema iterativo por el m\u00e9todo de Newton es:\n\n$$x = x^{(k)} + v.$$\n\n### Algoritmo: m\u00e9todo de Newton multidimensional para resolver un sistema de ecuaciones no lineales\n\n> **Dados** $x^{(0)}$ punto inicial, $f: \\mathbb{R}^n \\rightarrow \\mathbb{R}^n$ y $tol >0$.\n>\n> $x:=x^{(0)}$.\n>\n> **Repetir** el siguiente bloque para $k=1,2,\\dots$\n>> 1. Resolver el sistema de ecuaciones lineales $J(x) v = -f(x)$\n>>\n>> 2. Actualizar: $x = x + v$\n>\n> **hasta** convergencia: satisfacer criterio de paro en el que se utiliza $tol$ y $maxiter$.\n\n\n\n### Ejemplo\n\nRealizar tres iteraciones del m\u00e9todo de Newton multidimensional para resolver el siguiente sistema de ecuaciones no lineales:\n\n$$\n\\begin{eqnarray}\nx_1^2+x_1x_2 = 10 \\nonumber \\\\\nx_2 + 3x_1x_2^2 = 57 \\nonumber\n\\end{eqnarray}\n$$\n\ntomando como punto inicial: $x^{(0)} = \\left [ \\begin{array}{c} 1.5 \\nonumber \\\\ 3.5 \\end{array} \\right ]$. La soluci\u00f3n es $x^* = \\left [ \\begin{array}{c} 2 \\nonumber \\\\ 3 \\end{array} \\right ]$.\n\nDefinimos la funci\u00f3n $f: \\mathbb{R}^2 \\rightarrow \\mathbb{R}^2$ por:\n\n$$\nf(x) = \n\\left [\n\\begin{eqnarray}\nx_1^2+x_1x_2 - 10 \\nonumber \\\\\nx_2 + 3x_1x_2^2-57 \\nonumber \n\\end{eqnarray}\n\\right ]\n$$\n\n\n```python\nx1, x2 = sympy.symbols(\"x1, x2\")\nx_sym = sympy.Matrix([x1, x2])\n```\n\n```{margin}\n\nDefinimos las componentes de $f$: $f_1(x) = x_1^2 + x_1x_2 - 10$ y $f_2(x) = x_2 + 3x_1x_2^2 -57$.\n\n```\n\n\n```python\nf1 = x1**2 + x1*x2 - 10\nf2 = x2 + 3*x1*x2**2 - 57\nf_sympy = sympy.Matrix([f1, f2])\n```\n\n```{margin}\n\nEsta es la funci\u00f3n $f$ a la que le deseamos aproximar la ra\u00edz dada en el ejemplo.\n\n```\n\n\n```python\nsympy.pprint(f_sympy)\n```\n\n    \u23a1   2              \u23a4\n    \u23a2 x\u2081  + x\u2081\u22c5x\u2082 - 10 \u23a5\n    \u23a2                  \u23a5\n    \u23a2       2          \u23a5\n    \u23a33\u22c5x\u2081\u22c5x\u2082  + x\u2082 - 57\u23a6\n\n\n```{margin}\n\nElecci\u00f3n del punto inicial $x^{(0)} = \\left [ \\begin{array}{c} 1.5 \\nonumber \\\\ 3.5 \\end{array} \\right ]$.\n\n```\n\n\n```python\nx = np.array([1.5, 3.5])\n```\n\n\n```python\nm = len(f_sympy)\nn = len(x_sym)\n```\n\n\n```python\nJf = f_sympy.jacobian([x1, x2])\n```\n\n```{margin}\n\nCalculamos la Jacobiana de $f$.\n\n```\n\n\n```python\nJf_eval = lambda x: np.array([partial_derivative.subs({\"x1\": x[0], \n                                                       \"x2\": x[1]}) for partial_derivative in Jf],\n                              dtype=float).reshape(m,n)\n```\n\n```{margin}\n\nEvaluamos la Jacobiana de $f$ en el punto inicial.\n\n```\n\n\n```python\nprint(Jf_eval(x))\n```\n\n    [[ 6.5   1.5 ]\n     [36.75 32.5 ]]\n\n\n\n```python\nf_eval = lambda x: np.array([component.subs({\"x1\": x[0],\n                                             \"x2\": x[1]}) for component in f_sympy],\n                            dtype = float)\n```\n\n```{margin}\n\nEvaluamos $f$ en el punto inicial.\n\n```\n\n\n```python\nprint(f_eval(x))\n```\n\n    [-2.5    1.625]\n\n\n```{margin}\n\nPara la actualizaci\u00f3n del m\u00e9todo de Newton multidimensional resolvemos el sistema de ecuaciones lineales: $J(x^{(0)}) v = -f(x^{(0)})$\n\n```\n\n\n```python\nv = np.linalg.solve(Jf_eval(x), -f_eval(x))\n```\n\n\n```python\nv\n```\n\n\n\n\n    array([ 0.536, -0.656])\n\n\n\n```{margin}\n\nActualizaci\u00f3n por el m\u00e9todo de Newton: $x^{(1)} = x^{(0)} + v$.\n\n```\n\n\n```python\nx = x + v\n```\n\n\n```python\nprint(x)\n```\n\n    [2.036 2.844]\n\n\n**Segunda iteraci\u00f3n**\n\n```{margin}\n\nPara la actualizaci\u00f3n del m\u00e9todo de Newton multidimensional resolvemos el sistema de ecuaciones lineales: $J(x^{(1)}) v = -f(x^{(1)})$\n\n```\n\n\n```python\nv = np.linalg.solve(Jf_eval(x), -f_eval(x))\n```\n\n\n```python\nv\n```\n\n\n\n\n    array([-0.037,  0.158])\n\n\n\n```{margin}\n\nActualizaci\u00f3n por el m\u00e9todo de Newton: $x^{(2)} = x^{(1)} + v$.\n\n```\n\n\n```python\nx = x + v\n```\n\n\n```python\nprint(x)\n```\n\n    [1.999 3.002]\n\n\n**Tercera iteraci\u00f3n**\n\n```{margin}\n\nPara la actualizaci\u00f3n del m\u00e9todo de Newton multidimensional resolvemos el sistema de ecuaciones lineales: $J(x^{(2)}) v = -f(x^{(2)})$\n\n```\n\n\n```python\nv = np.linalg.solve(Jf_eval(x), -f_eval(x))\n```\n\n\n```python\nv\n```\n\n\n\n\n    array([ 0.001, -0.002])\n\n\n\n```{margin}\n\nActualizaci\u00f3n por el m\u00e9todo de Newton: $x^{(3)} = x^{(2)} + v$.\n\n```\n\n\n```python\nx = x + v\n```\n\n\n```python\nprint(x)\n```\n\n    [2. 3.]\n\n\n````{admonition} Comentarios\n\n* Si $x^{(0)}$ es cercano a la ra\u00edz $x^*$ de $f$, $J(x^*)$ es no singular, no es mal condicionada y $J$ es Lipschitz continua en una vecindad alrededor de $x^*$ entonces el m\u00e9todo de Newton multidimensional converge de forma cuadr\u00e1tica a $x^*$.\n\n* Al igual que en el caso de una dimensi\u00f3n, el criterio de paro es de la forma:\n\n```\nwhile ||x_(k+1) - x_k|| > tol1(||x_k|| +1) && ||f(x_(k+1))|| > tol2 && iterations < max_iters\n```\n\ncon `tol1`, `tol2` cantidades peque\u00f1as y positivas (com\u00fanmente menores o iguales a $10^{-8}$), `iterations` un contador de las iteraciones. En una implementaci\u00f3n tambi\u00e9n pueden utilizarse la medici\u00f3n entre iteraciones con los criterios descritos en {ref}`criterios de paro, escala de la variable x y de la funci\u00f3n f ` y si se tiene conocimiento del valor de $x^*$ se pueden calcular errores relativos de `x_k`, as\u00ed como los reescalamientos respectivos de $x$ y $f$.\n\n* En la implementaci\u00f3n del m\u00e9todo hay que monitorear el n\u00famero de condici\u00f3n de la Jacobiana de $f$.\n\n\n````\n\n```{admonition} Ejercicio\n:class: tip\n\nAproximar con el m\u00e9todo de Newton la soluci\u00f3n $x^{*} = \\left [ \\begin{array}{c} 1 \\nonumber \\\\ 1 \\nonumber \\\\ -\\sqrt{\\frac{1}{3}} \\nonumber \\\\ \\sqrt{\\frac{1}{3}}\\end{array} \\right ]$ del sistema de ecuaciones no lineales con inc\u00f3gnitas $w_0, w_1, x_0, x_1$:\n\n$$\n\\begin{eqnarray}\n2=\\displaystyle \\int_{-1}^{1}1dx &=& w_0 \\cdot 1 + w_1\\cdot1 \\nonumber \\\\\n0 = \\displaystyle \\int_{-1}^1xdx &=& w_0x_0 + w_1x_1 \\nonumber \\\\\n\\frac{2}{3} = \\displaystyle \\int_{-1}^1x^2dx &=& w_0x_0^2 + w_1x_1^2 \\nonumber \\\\\n0 = \\displaystyle \\int_{-1}^1x^3dx &=& w_0x_0^3 + w_1x_1^3 \\nonumber \\\\\n\\end{eqnarray}\n$$\n\nutilizando el punto inicial: $x^{(0)} = \\left [ \\begin{array}{c} -1 \\nonumber \\\\ -1 \\nonumber \\\\ -1 \\nonumber \\\\ 1 \\end{array} \\right ]$.\n\n```\n\n### Ejemplo: componentes principales\n\n```{margin}\n\nEl factor $\\frac{1}{2}$ se a\u00f1ade para que se anule la constante $2$ al derivar y es equivalente la soluci\u00f3n de este problema sin tal factor.\n\n```\n\nEl problema de optimizaci\u00f3n:\n\n$$\\displaystyle \\max_{v \\in \\mathbb{R}^n - \\{0\\}} \\frac{1}{2} v^TX^TXv$$\n\n$$\\text{sujeto a: } \\frac{1}{2}v^Tv = \\frac{1}{2}$$\n\ndonde: $X \\in \\mathbb{R}^{m \\times n}$ y la variable de optimizaci\u00f3n es $v$. Es un problema **no convexo** por la restricci\u00f3n cuadr\u00e1tica de igualdad. Tiene soluci\u00f3n cerrada dada por: $\\sigma_1^2 = \\sigma^2_{\\text{max}}(X)$, esto es, el cuadrado del m\u00e1ximo valor singular de $X$.\n\nEscribiendo la funci\u00f3n Lagrangiana se tiene:\n\n$$\\mathcal{L}(v, \\nu) = f_o(v) + \\nu h(v) = \\frac{1}{2} v^T X^T X v + \\frac{\\nu}{2}\\left(1- v^Tv\\right)$$\n\n\n```{admonition} Observaci\u00f3n\n:class: tip\n\nEs equivalente considerar la restricci\u00f3n $h(v)$ como $1-v^Tv$ o $v^Tv-1$, ver {ref}`ejemplo 1 en introducci\u00f3n a problemas CIEO ` en el sentido que se obtiene la misma soluci\u00f3n al problema de optimizaci\u00f3n.\n\n```\n\n\nPor tanto, las condiciones de [Karush-Kuhn-Tucker](https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions) (KKT) de optimalidad conducen a resolver el siguiente **sistema de ecuaciones no lineales**:\n\n```{margin}\nVer {ref}`comentario relacionado con las condiciones KKT de optimalidad `. Recu\u00e9rdese son:\n\n$$\\begin{eqnarray}\n\\nabla_x\\mathcal{L}(x^*, \\nu^*, \\lambda^*) &=& 0 \\nonumber \\\\\nh_i(x^*) &=& 0 \\quad \\forall i = 1, \\dots, p \\nonumber \\\\\nf_i(x^*) &\\leq& 0 \\quad \\forall i = 1, \\dots, m \\nonumber \\\\\n\\lambda_i^* &\\geq& 0 \\quad \\forall i = 1, \\dots, m \\nonumber \\\\\n\\lambda_i^* f_i(x^*) &=& 0 \\quad \\forall i = 1, \\dots, m\n\\end{eqnarray}\n$$\n```\n\n$$\\begin{eqnarray}\n\\nabla_v \\mathcal{L}(v, \\nu) &=& X^TX v - \\nu v=0 \\nonumber \\\\\nh(v) &=& \\frac{1}{2}(1-v^Tv) = 0\n\\end{eqnarray}\n$$\n\nO equivalentemente encontrar la ra\u00edz de $f: \\mathbb{R}^{n+1} \\rightarrow \\mathbb{R}^{n+1}$:\n\n$$f(v, \\nu) = \n\\left [\n\\begin{array}{c}\nX^TX v - \\nu v \\\\\n1-v^Tv\n\\end{array}\n\\right] = 0\n$$\n\n\n```{admonition} Observaci\u00f3n\n:class: tip \n\nObs\u00e9rvese que la ra\u00edz involucra al vector $(v, \\lambda) \\in \\mathbb{R}^{n+1}$.\n\n```\n\nSi se considera que las columnas de $X$ tienen una observaci\u00f3n de un **vector aleatorio** (tenemos $n$ vectores aleatorios de mediciones) y $X = U \\Sigma V^T$ es la descomposici\u00f3n en valores singulares de $X$, entonces los vectores **singulares derechos** $v_i$ (columnas de la matriz $V$) son nombrados **ejes o direcciones principales** de $X$ y el vector $z_1 = X v_1 = \\sigma u_1$ con $u_1$ vector **singular izquierdo** (primera columna de la matriz $U$) tiene **varianza muestral**:\n\n$$\\text{var}(z_1) = \\text{var}(X v_1)= \\text{var}(\\sigma u_1) = \\sigma_1^2 \\text{var}(u_1) = \\sigma_1^2 \\left [ \\frac{1}{m} \\displaystyle \\sum_{i=1}^m (u_1(i) - \\bar{u}_1)^2 \\right ]$$\n\ndonde: $u_1(i)$ es la $i$-\u00e9sima componente de $u_1$ y $\\sigma_1$ es el m\u00e1ximo valor singular de $X$ tambi\u00e9n denotado como $\\sigma_{\\text{max}}$.\n\n```{admonition} Comentarios\n\nSi la media de cada columna de $X$ es cero, $X$ se nombra **centrada**, entonces:\n\n* $z_1$ tiene la **m\u00e1xima varianza muestral** entre todas las combinaciones lineales de las columnas de $X$ pues:\n\n$$\\text{var}(z_1) = \\frac{\\sigma_1^2}{m} \\displaystyle \\sum_{i=1}^m u_1(i)^2 = \\frac{\\sigma_1^2}{m} ||u_1||_2^2 = \\frac{\\sigma_1^2}{m}.$$\n\n* $z_1$ es la **primera componente principal** y el vector $u_1 = \\frac{1}{\\sigma_1}z_1 = \\frac{1}{\\sigma_1}Xv_1$ se le nombra **primera componente principal normalizada**. El vector $v_1$ es la **primera direcci\u00f3n principal** de $X$ o tambi\u00e9n nombrada ***loading***. \n\nCalcular el vector con m\u00e1xima varianza muestral implica resolver el problema de **optimizaci\u00f3n num\u00e9rica**:\n\n$$\\displaystyle \\max_{v \\in \\mathbb{R}^n - \\{0\\}} v^TX^TXv$$\n\n$$\\text{sujeto a:}$$\n\n$$v^Tv = 1$$\n\nel cual es equivalente al denotado al inicio del ejemplo y se cumple: $v_1 = \\text{argmax}_{v \\in \\mathbb{R}^n - \\{0\\}} v^TX^TXv$ sujeto a: $v^Tv=1$ (primera columna de $V$). \n\nLa segunda componente principal $z_2$ es aquella que tiene la m\u00e1xima varianza muestral entre todas las combinaciones lineales de las columnas de $X$ y que es **ortogonal**, o equivalentemente que tenga covarianza igual a cero, con $z_1$. Este problema se escribe como:\n\n$$\\displaystyle \\max_{v \\in \\mathbb{R}^n - \\{0\\}} v^TX^TXv$$\n\n$$\\text{sujeto a:}$$\n\n$$v^Tv = 1$$\n\n$$v^Tv_1 =0$$\n\ncon $v_1$ la primera direcci\u00f3n principal de $X$. La soluci\u00f3n al problema anterior est\u00e1 dada por: $\\sigma_2^2 = \\displaystyle \\max_{v \\in \\mathbb{R}^n - \\{0\\}} v^TX^TXv$ sujeto a: $v^Tv=1$, $v_2 = \\text{argmax}_{v \\in \\mathbb{R}^n - \\{0\\}} v^TX^TXv$ sujeto a: $v^Tv=1$ (segunda columna de $V$). \n\n```\n\nEntre algunas definiciones utilizadas en Estad\u00edstica se encuentran:\n\n```{admonition} Definici\u00f3n\n\nLa **matriz de correlaciones entre cada componente principal normalizada $u$'s y cada columna de $X$** es:\n\n$$ C = \\left (\\frac{\\sigma_1}{\\sqrt{m}} v_1 \\quad \\frac{\\sigma_2}{\\sqrt{m}} v_2 \\cdots \\quad \\frac{\\sigma_n}{\\sqrt{m}} v_n \\right) \\in \\mathbb{R}^{n \\times n}$$\n\npues si $x_1$ es la primer columna de $X$ entonces:\n\n$$\n\\begin{eqnarray}\n\\text{cov}(x_1,u_1) = \\text{cov} \\left ( \\displaystyle \\sum_{k=1}^n \\sigma_k v_k(1) u_k, u_1 \\right ) &=& \\displaystyle \\sum_{k=1}^n \\text{cov} ( \\sigma_k v_k(1) u_k, u_1 ) \\nonumber \\\\\n&=& \\displaystyle \\sum_{k=1}^n \\sigma_k v_k(1) \\text{cov} (u_k, u_1) \\nonumber \\\\\n&=& \\sigma_1 v_1(1) \\text{var}(u_1) \\nonumber \\\\\n&=& \\frac{\\sigma_1 v_1(1)}{m} \\sum_{i=1}^m u_1(i)^2 \\nonumber \\\\\n&=& \\frac{\\sigma_1 v_1(1)}{m} \\nonumber\n\\end{eqnarray}\n$$\n\n\nY como $\\text{cor}(x_1,u_1) = \\frac{\\text{cov}(x_1,u_1)}{\\sqrt{\\text{var}(x_1)} \\sqrt{\\text{var}(u_1)}}$ se tiene:\n\n$$\\text{cor}(x_1,u_1) = \\frac{\\frac{\\sigma_1 v_1(1)}{m}}{1 \\cdot \\sqrt{\\frac{1}{m}}}  = \\frac{\\sigma_1 v_1(1)}{\\sqrt{m}} $$\n```\n\n```{admonition} Definici\u00f3n\n\nEl cociente de **varianza explicada** para cada componente es el n\u00famero:\n\n$$\\frac{\\sigma_i^2}{\\displaystyle \\sum_{i=1}^p \\sigma_i^2}$$\n\ncon $p = \\min(m,n)$.\n\n\n```\n\n```{admonition} Observaci\u00f3n\n:class: tip\n\nla matriz $\\frac{1}{m}X^TX$ es la matriz de **varianzas y covarianzas muestral** la cual **siempre** es una matriz sim\u00e9trica positiva semidefinida (a\u00fan si la $X$ no es centrada). Si $X$ adem\u00e1s de ser centrada cumple que la varianza de cada una de sus columnas es $1$, $X$ se nombra **estandarizada**. La matriz $\\frac{1}{m}X^TX$ en este caso es la matriz de **correlaciones muestral**.\n\n```\n\n```{admonition} Ejercicios\n:class: tip\n\n1.Resuelve los ejercicios y preguntas de la nota.\n```\n\n\n**Preguntas de comprehensi\u00f3n.**\n\n1)Escribe las categor\u00edas en las que se pueden clasificar los sistemas de ecuaciones lineales. \n\n2)\u00bfQu\u00e9 es un m\u00e9todo iterativo abierto y qu\u00e9 es uno cerrado? \u00bfqu\u00e9 diferencias tienen?\n\n3)\u00bfPor qu\u00e9 es indispensable establecer criterios de paro que sean independientes de la escala de los datos?\n\n4)\u00bfPor qu\u00e9 se elige un modelo cuadr\u00e1tico y no un modelo af\u00edn para resolver un problema de optimizaci\u00f3n con el m\u00e9todo de Newton?\n\n5)Si queremos utilizar el m\u00e9todo de Newton para calcular la ra\u00edz de una funci\u00f3n $f: \\mathbb{R}^n \\rightarrow \\mathbb{R}^n$ \u00bfqu\u00e9 tenemos que calcular en una iteraci\u00f3n cualquiera?\n\n6)Escribe una relaci\u00f3n que existe entre encontrar ra\u00edces de ecuaciones no lineales y resolver problemas de optimizaci\u00f3n.\n\n**Referencias:**\n\n1. C. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, 2000.\n\n2. J. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, 1996.\n\n3. R. Johansson, Numerical Python, Scientific Computing and Data Science Applications with Numpy, SciPy and Matplotlib, Apress, 2015.\n\n\n\n", "meta": {"hexsha": "23f9f13b130c78f112a10b34194f01a0ad7a02e4", "size": 137776, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "libro_optimizacion/temas/III.optimizacion_convexa/3.4/Ecuaciones_no_lineales.ipynb", "max_stars_repo_name": "Carlosrlpzi/analisis-numerico-computo-cientifico", "max_stars_repo_head_hexsha": "b0ea6dc11133af2d5e48b34a7f66cdffeff5a278", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "libro_optimizacion/temas/III.optimizacion_convexa/3.4/Ecuaciones_no_lineales.ipynb", "max_issues_repo_name": "Carlosrlpzi/analisis-numerico-computo-cientifico", "max_issues_repo_head_hexsha": "b0ea6dc11133af2d5e48b34a7f66cdffeff5a278", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "libro_optimizacion/temas/III.optimizacion_convexa/3.4/Ecuaciones_no_lineales.ipynb", "max_forks_repo_name": "Carlosrlpzi/analisis-numerico-computo-cientifico", "max_forks_repo_head_hexsha": "b0ea6dc11133af2d5e48b34a7f66cdffeff5a278", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 50.1003636364, "max_line_length": 15920, "alphanum_fraction": 0.7257432354, "converted": true, "num_tokens": 15827, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "# Latent Variable Models and Variational Bayes\n\n\n### Preliminaries\n\n- Goal \n  - Introduction to latent variable models and variational inference by Free energy minimization   \n- Materials\n  - Mandatory\n    - These lecture notes\n    - Ariel Caticha (2010), [Entropic Inference](https://arxiv.org/abs/1011.0723)\n      - tutorial on entropic inference, which is a generalization to Bayes rule and provides a foundation for variational inference.\n  - Optional \n    - Bishop (2016), pp. 461-486 (sections 10.1, 10.2 and 10.3) \n  - references \n    - Blei et al. (2017), [Variational Inference: A Review for Statisticians](https://www.tandfonline.com/doi/full/10.1080/01621459.2017.1285773) \n    - Lanczos (1961), [The variational principles of mechanics](https://www.amazon.com/Variational-Principles-Mechanics-Dover-Physics/dp/0486650677)\n    - Zhang et al. (2017), [Unifying Message Passing Algorithms Under the Framework of Constrained Bethe Free Energy Minimization](https://github.com/bertdv/BMLIP/blob/master/lessons/notebooks/files/Zhang-2017-Unifying-Message-Passing-Algorithms.pdf)\n    - Dauwels (2007), [On variational message passing on factor graphs](https://github.com/bertdv/BMLIP/blob/master/lessons/notebooks/files/Dauwels-2007-on-variational-message-passing-on-factor-graphs)\n    - Caticha (2010), [Entropic Inference](https://arxiv.org/abs/1011.0723)\n    - Shore and Johnson (1980), [Axiomatic Derivation of the Principle of Maximum Entropy and the Principle of Minimum Cross-Entropy](https://github.com/bertdv/BMLIP/blob/master/lessons/notebooks/files/ShoreJohnson-1980-Axiomatic-Derivation-of-the-Principle-of-Maximum-Entropy.pdf)\n    \n\n### Illustrative Example : Density Modeling for the Old Faithful Data Set\n\n- You're now asked to build a density model for a data set ([Old Faithful](https://en.wikipedia.org/wiki/Old_Faithful), Bishop pg. 681) that clearly is not distributed as a single Gaussian:\n\n
                                        \n\n###  Unobserved Classes\n\nConsider again a set of observed data $D=\\{x_1,\\dotsc,x_N\\}$\n\n- This time we suspect that there are _unobserved_ class labels that would help explain (or predict) the data, e.g.,\n  - the observed data are the color of living things; the unobserved classes are animals and plants.\n  - observed are wheel sizes; unobserved categories are trucks and personal cars.\n  - observed is an audio signal; unobserved classes include speech, music, traffic noise, etc.\n \n\n   \n- Classification problems with unobserved classes are called **Clustering** problems. The learning algorithm needs to **discover the classes from the observed data**.\n\n### The Gaussian Mixture Model\n\n- The spread of the data in the illustrative example looks like it could be modeled by two Gaussians. Let's develop a model for this data set. \n\n- Let $D=\\{x_n\\}$ be a set of observations. We associate a one-hot coded hidden class label $z_n$ with each observation:\n\n$$\\begin{equation*}\nz_{nk} = \\begin{cases} 1 & \\text{if } x_n \\in \\mathcal{C}_k \\text{ (the k-th class)}\\\\\n                       0 & \\text{otherwise} \\end{cases}\n\\end{equation*}$$\n\n- We consider the same model as we did in the [generative classification lesson](https://nbviewer.jupyter.org/github/bertdv/BMLIP/blob/master/lessons/notebooks/Generative-Classification.ipynb#GDA):\n$$\\begin{align*}\np(x_n | z_{nk}=1) &= \\mathcal{N}\\left( x_n | \\mu_k, \\Sigma_k\\right)\\\\\np(z_{nk}=1) &= \\pi_k\n\\end{align*}$$\nwhich can be summarized with the selection variables $z_{nk}$ as\n$$\\begin{align*}\np(x_n,z_n) &= p(x_n | z_n) p(z_n) = \\prod_{k=1}^K (\\underbrace{\\pi_k \\cdot \\mathcal{N}\\left( x_n | \\mu_k, \\Sigma_k\\right) }_{p(x_n,z_{nk}=1)})^{z_{nk}} \n\\end{align*}$$\n\n- *Again*, this is the same model  as we defined for the generative classification model: A Gaussian-Categorical model but now with unobserved classes). \n\n- This model (with **unobserved class labels**) is known as a **Gaussian Mixture Model** (GMM).\n\n### The Marginal Distribution for the GMM\n\n- In the literature, the GMM is often introduced the marginal distribution for an _observed_ data point $x_n$, given by\n\n$$\\begin{align*}{}\np(x_n) &= \\sum_{z_n} p(x_n,z_n)  \\\\\n  &= \\sum_{k=1}^K \\pi_k \\cdot \\mathcal{N}\\left( x_n | \\mu_k, \\Sigma_k \\right) \\tag{B-9.12}\n\\end{align*}$$\n\n- Full proof as an [exercise](https://nbviewer.org/github/bertdv/BMLIP/blob/master/lessons/exercises/Exercises-Latent-Variable-Models-and-VB.ipynb). \n\n- Eq. B-9.12 reveals the link to the name Gaussian *mixture model*. The priors $\\pi_k$ are also called **mixture coefficients**. \n  \n \n\n###  GMM is a very flexible model\n\n- GMMs are very popular models. They have decent computational properties and are **universal approximators of densities** (as long as there are enough Gaussians of course)\n\n
                                        \n\n- (In the above figure, the Gaussian components are shown in red and the pdf of the mixture models in blue).\n\n### Latent Variable Models\n\n- The GMM contains both _observed_ variables $\\{x_n\\}$, (unobserved) _parameters_ $\\theta= \\{\\pi_k,\\mu_k, \\Sigma_k\\}$ _and_ unobserved (synonym: latent, hidden) variables $\\{z_{nk}\\}$.\n\n- From a Bayesian viewpoint, both latent variables $\\{z_{nk}\\}$ and parameters $\\theta$ are just unobserved variables for which we can set a prior and compute a posterior by Bayes rule. \n\n- Note that $z_{nk}$ has a subscript $n$, hence its value depends not only on the cluster ($k$) but also on the $n$-th observation (in constrast to parameters). These observation-dependent variables are generally a useful tool to encode structure in the model. Here (in the GMM), the latent variables $\\{z_{nk}\\}$ encode (unobserved) class membership. \n\n- By adding model structure through (equations among) latent variables, we can build very complex models. Unfortunately, inference in latent variable models can also be much more complex than for fully observed models.\n\n### Inference for GMM is Difficult\n\n\n- We recall here the log-likelihood for the Gaussian-Categorial Model, see [generative classification lesson)](https://nbviewer.jupyter.org/github/bertdv/BMLIP/blob/master/lessons/notebooks/Generative-Classification.ipynb)\n\n$$\n\\log\\, p(D|\\theta) =  \\sum_{n,k} y_{nk} \\underbrace{ \\log\\mathcal{N}(x_n|\\mu_k,\\Sigma) }_{ \\text{Gaussian} } + \\underbrace{ \\sum_{n,k} y_{nk} \\log \\pi_k }_{ \\text{multinomial} } \\,.\n$$\n\n- Since the class labels $y_{nk} \\in \\{0,1\\}$ were given, this expression decomposed into a set of simple updates for the Gaussian and multinomial distributions. For both distributions, we have conjugate priors, so inference is easily solved. \n\n- However, for the Gaussian mixture model (same log-likelihood function with $z_{nk}$ replacing $y_{nk}$), the class labels $\\{z_{nk}\\}$ are _unobserved_ and need to estimated alongside with the parameters.\n\n- There is no conjugate prior on the latent variables for the GMM likelihood function $p(\\underbrace{D}_{\\{x_n\\}}\\,|\\,\\underbrace{\\{z_{nk}\\},\\{\\mu_k,\\Sigma_k,\\pi_k\\}}_{\\text{all latent variables}})$.\n\n- In this lesson, we introduce an approximate Bayesian inference method known as **Variational Bayes** (VB) (also known as **Variational Inference**) that can be used for Bayesian inference in models with latent variables. Later in this lesson, we will use VB to do inference in the GMM.   \n\n- As a motivation for variational inference, we first discuss inference by the **Method of Maximum Relative Entropy**, [(Caticha, 2010)](https://arxiv.org/abs/1011.0723). \n\n### Inference When Information is in the Form of Constraints \n\n- In the [probability theory lesson](https://nbviewer.jupyter.org/github/bertdv/BMLIP/blob/master/lessons/notebooks/Probability-Theory-Review.ipynb#Bayes-rule), we recognized Bayes rule as the fundamental rule for learning from data.\n\n- We will now generalize this notion and consider learning as a process that updates a prior into a posterior distribution whenever new information becomes available.\n\n- In this context, *new information* is not necessarily a new observation, but could (for instance) also relate to *constraints* on the posterior distribution.\n\n- For example, consider a model $\\prod_n p(x_n,z) = p(z) \\prod_n p(x_n|z)$. \n\n- Our prior beliefs about $z$ are represented by $p(z)$. In the following, we will write $q(z)$ to denote a posterior distribution for $z$.\n\n-  We might be interested in obtaining rational posterior beliefs $q(z)$ in consideration of the following additional constraints:\n  1. *Data Constraints*: e.g., two observed data points $x_1=5$ and $x_2=3$, which lead to constraints $q(x_1) = \\delta(x_1-5)$ and $q(x_2)=\\delta(x_2-3)$.\n  2. *Form Constraints*: e.g., we only consider the Gamma distribution for $q(z) = \\mathrm{Gam}(z|\\alpha,\\beta)$.\n  3. *Factorization Constraints*:, e.g., we consider independent marginals for the posterior distribution: $q(z) = \\prod_k q(z_k)$. \n  3. *Moment Constraints*: e.g., the first moment of the posterior is given by $\\int z q(z) \\mathrm{d}z = 3$.\n\n\n- Note that this is not \"just\" applying Bayes rule to compute a posterior, which can only deal with data constraints as specified by the likelihood function.\n\n- Note also that observations _can_ be rephrased as constraints on the posterior, e.g., observation $x_1=5$ can be phrased as a constraint $q(x_1)=\\delta(x_1-5)$.\n\n- $\\Rightarrow$ Updating a prior to a posterior on the basis of constraints on the posterior is more general than updating based on observations alone.\n\n### The Method of Maximum Relative Entropy\n\n- [Caticha (2010)](https://arxiv.org/abs/1011.0723) (based on earlier work by [Shore and Johnson (1980)](https://github.com/bertdv/BMLIP/blob/master/lessons/notebooks/files/ShoreJohnson-1980-Axiomatic-Derivation-of-the-Principle-of-Maximum-Entropy.pdf)) developed the **method of maximum (relative) entropy** for rational updating of priors to posteriors when faced with new information in the form of constraints.\n\n- Consider prior beliefs $p(z)$ about $z$. New information in the form of constraints is obtained and we are interested in the \"best update\" to a posterior $q(z)$.\n\n- In order to define what \"best update\" means, we assume a ranking function $S[q,p]$ that generates a preference score for each candidate posterior $q$ for a given $p$. The best update from $p$ to $q$ is identified as $q^* = \\arg\\max_q S[q,p]$.   \n\n- Similarly to [Cox' method to deriving probability theory](https://en.wikipedia.org/wiki/Cox%27s_theorem), Caticha introduced the following mild criteria based on a rational principle (the **principle of minimal updating**, see [Caticha 2010](https://arxiv.org/abs/1011.0723)) that the ranking function needs to adhere to: \n  1. *Locality*: local information has local effects.\n  2. *Coordinate invariance*: the system of coordinates carries no information. \n  3. *Independence*: When systems are known to be independent, it should not matter whether they are treated separately or jointly. \n  \n\n- These three criteria **uniquely identify the Relative Entropy** (RE) as the proper ranking function: \n$$\\begin{align*}\nS[q,p] = - \\sum_z q(z) \\log \\frac{q(z)}{p(z)}\n\\end{align*}$$\n\n- $\\Rightarrow$ When information is supplied in the form of constaints on the posterior, we *should* select the posterior that maximizes the Relative Entropy, subject to the constraints. This is the **Method of Maximum (Relative) Entropy** (MRE). \n\n### Relative Entropy, KL Divergence and Free Energy\n\n- Note that the Relative Entropy is technically a *functional*, i.e., a function of function(s) (since it is a function of probability distributions). The calculus of functionals is also called **variational calculus**.\n  - See  [Lanczos, The variational principles of mechanics (1961)](https://www.amazon.com/Variational-Principles-Mechanics-Dover-Physics/dp/0486650677) for a great book on variational calculus. For a summary, see Bishop (2016), app.D.\n  - It is customary to use square brackets (like $S[q,p]$) for functionals rather than round brackets, which we use for functions.  \n\n- Also note the complimentary relation between the Maximum Relative Entropy method and Probability Theory: \n  - PT describes how to _represent_ beliefs about events and how to _calculate_ beliefs about joint and conditional events. \n  - In contrast, the MRE method describes how to _update_ beliefs when new information becomes available.\n  \n\n- PT and the MRE method are both needed to describe the theory of optimal information processing. (As we will see below, Bayes rule is a special case of the MRE method.)\n\n- In principle, entropies can always be considered as a *relative* score against a reference distribution. For instance, the score $-\\sum_{z_i} q(z_i) \\log q(z_i)$ can be interpreted as a score against the uniform distribution, i.e., $-\\sum_{z_i} q(z_i) \\log q(z_i) \\propto -\\sum_{z_i} q(z_i) \\log \\frac{q(z_i)}{\\mathrm{Uniform(z_i)}}$. Therefore, the \"method of maximum relative entropy\" is often simply known as the \"method of maximum entropy\". \n\n- The negative relative entropy is known as the **Kullback-Leibler** (KL) divergence:\n$$\n\\mathrm{KL}[q,p] \\triangleq \\sum_z q(z) \\log \\frac{q(z)}{p(z)} \\tag{B-1.113}\n$$\n\n- The [Gibbs inequality](https://en.wikipedia.org/wiki/Gibbs%27_inequality) ([proof](https://en.wikipedia.org/wiki/Gibbs%27_inequality#Proof)), is a famous theorem in information theory that states that \n$$\\mathrm{KL}[q,p] \\geq 0 $$\nwith equality only iff $p=q$.\n\n  - The KL divergence can be interpreted as a \"distance\" between two probability distributions. Note however that the KL divergence is an asymmetric distance measure, i.e., in general $\\mathrm{KL}[q,p] \\neq \\mathrm{KL}[p,q]$.\n\n- We also introduce here the notion of a (variational) **Free Energy** (FE) functional, which is just a generalization of the KL-divergence that allows the prior to be unnormalized. Let $f(z) = Z \\cdot p(z)$ be an unnormalized distribution, then the FE is defined as\n$$\\begin{align*}\nF[q,p] &\\triangleq \\sum_z q(z) \\log \\frac{q(z)}{f(z)} \\\\\n&= \\sum_z q(z) \\log \\frac{q(z)}{Z\\cdot p(z)} \\\\\n&= \\underbrace{\\sum_z q(z) \\log \\frac{q(z)}{p(z)}}_{\\text{KL divergence }\\geq 0} - \\log Z\n\\end{align*}$$\n\n- Since the second term ($\\log Z$) is constant, FE minimization (subject to constraints) with respect to $q$ leads to the same posteriors as KL minimization and RE maximization.\n\n- If we are only interested in minimizing FE with respect to $q$, then we'll write $F[q]$ (rather than $F[q,p]$) fo brevity. \n\n- In this class, we prefer to discuss inference in terms of minimizing Free Energy (subject to the constraints) rather than maximizing Relative Entropy, but note that these two concepts are equivalent. \n\n### Example KL divergence for Gaussians\n\n
                                        \n\nsource: By Mundhenk at English Wikipedia, CC BY-SA 3.0, Link\n\n\n\n### The Free Energy Functional and Variational Bayes\n\n- Let's get back to the issue of doing inference for models with latent variables (such as the GMM). \n\n- Consider a generative model specified by $p(x,z)$ where $x$ and $z$ represent the observed and unobserved variables, respectively.\n\n- Assume further that $x$ has been observed as $x=\\hat{x}$ and we are interested in performing inference, i.e., we want to compute the posterior $p(z|x=\\hat{x})$ for the latent variables and we want to compute the evidence $p(x=\\hat{x})$.\n\n- According to the Method of Maximum Relative Entropy, in order to find the correct posterior $q(x,z)$, we should minimize\n\n$$\n\\mathrm{F}[q] = \\sum_{x,z} q(x,z) \\log \\frac{q(x,z)}{p(x,z)}\\,, \\quad \\text{subject to data constraint }x=\\hat{x}\n$$\n\n- The data constraint $x=\\hat{x}$ fixes posterior $q(x) = \\delta(x-\\hat{x})$, so the Free Energy evaluates to  \n\n$$\\begin{align*}\nF[q] &= \\sum_{z} \\sum_{x}q(z|x)q(x) \\log \\frac{q(z|x) q(x)}{p(z|x) p(x)} \\\\\n  &= \\sum_{z} \\sum_{x} q(z|x)\\delta(x-\\hat{x}) \\log \\frac{q(z|x)\\delta(x-\\hat{x})}{p(z|x) p(x)} \\\\\n  &= \\sum_{z} q(z|\\hat{x}) \\log \\frac{q(z|\\hat{x})}{p(z|\\hat{x}) p(\\hat{x}) } \\\\\n  &=  \\underbrace{\\sum_{z}q(z|\\hat{x}) \\log \\frac{q(z|\\hat{x})}{p(z|\\hat{x})}}_{\\text{KL divergence }\\geq 0} - \\underbrace{\\log p(\\hat{x})  }\\tag{B-10.2\n}_{\\text{log-evidence}}\\end{align*}$$\n\n- The log-evidence term does not depend on $q$. Hence, the global minimum $q(z|\\hat{x})^* \\triangleq \\arg\\min_q F[q]$ is obtained for $$q^*(z|\\hat{x}) = p(z|\\hat{x})\\,,$$ which is the correct **Bayesian posterior**.\n\n- Furthermore, the minimal free energy $$F^* \\triangleq F[q^*] = -\\log p(\\hat{x})$$ equals minus **log-evidence**. (Or, equivalently, the evidence is given by $p(\\hat{x}) = \\exp\\left(-F[q^*] \\right)$.)  \n\n
                                        \n
                                        \n$\\Rightarrow$ Bayesian inference can be executed by FE minimization.\n
                                        \n
                                        \n\n\n-  This is an amazing result!  Note that FE minimization transforms an inference problem (that involves integration) to an optimization problem! Generally, optimization problems are easier to solve than integration problems. \n\n- Executing inference by minimizing the variational FE functional is called **Variational Bayes** (VB) or variational inference. \n\n-  (As an aside), note that Bishop introduces in Eq. B-10.2 an _Evidence Lower BOund_ (in modern machine learning literature abbreviated as **ELBO**) $\\mathcal{L}(z)$ that equals the _negative_ FE. We prefer to discuss variational inference in terms of a free energy (but it is the same story as he discusses with the ELBO).  \n\n###  FE Minimization as Approximate Bayesian Inference\n\n- In the rest of this lesson, we are concerned with how to execute FE minimization (FEM) w.r.t $q$ for the functional\n$$\n F[q] = \\sum_{z}q(z) \\log \\frac{q(z)}{p(z|x)} - \\log p(x)  \n $$\nwhere $x$ has been observed and $z$ represent all latent variables.\n  - To keep the notation uncluttered, in the following we write $x$ rather than $\\hat{x}$, and we write simply $q(z)$ (rather than $q(z|\\hat{x})$) for the posterior.  \n\n- Due to restrictions in our optimization algorithm, we are often unable to fully minimize the FE to the global minimum $q^*(z)$, but rather get stuck in a local minimum $\\hat{q}(z)$. \n\n- Note that, since $\\mathrm{KL}[q(z),p(z|x)]\\geq 0$ for any $q(z)$, the FE is always an upperbound on (minus) log-evidence, i.e.,\n$$\nF[q] \\geq -\\log p(x) \\,.\n$$\n\n- In practice, even if we cannot attain the global minimum, we can still use the local minimum $\\hat{q}(z) \\approx \\arg\\min_q F[q]$ to accomplish **approximate Bayesian inference**:   \n\n$$\\begin{align*}\n   \\text{posterior: } \\hat{q}(z) &\\approx p(z|x) \\\\\n   \\text{evidence: } p(x) &\\approx \\exp\\left( -F[\\hat{q}]\\right) \n    \\end{align*}$$\n\n### Constrained FE Minimization\n\n- Generally speaking, it can be very helpful in the FE minimization task to add some additional constraints on $q(z)$ (i.e., in addition to the data constraints).\n\n- Once we add constraints to $q$ (in addition to data constraints), we are no longer guaranteed that the minimum of the (constrained) FE coincides with the solution by Bayes rule (which only takes data as constraints). So again, at best constrained FEM is only an **approximation to Bayes rule**.\n\n- There are three important cases of adding constraints to $q(z)$ that often alleviates the FE minimization task:\n\n#### 1. mean-field factorization\n- We constrain the posterior to factorize into a set of _independent_ factors, i.e., \n$$\nq(z) = \\prod_{j=1}^m q_j(z_j)\\,, \\tag{B-10.5}\n$$\n\n####  2. fixed-form parameterization\n- We constrain the posterior to be part of a parameterized probability distribution, e.g., $$q(z) = \\mathcal{N}\\left( z | \\mu, \\Sigma \\right)\\,.$$ \n  - In this case, the functional minimization problem for $F[q]$ reduces to the minimization of a _function_ $F(\\mu,\\Sigma)$ w.r.t. its parameters.  \n \n\n#### 3. the Expectation-Maximization (EM) algorithm\n- We place some constraints both on the prior and posterior for $z$ ([to be discussed in the Optional Slides](#EM-Algorithm)) that simplifies FE minimization to maximum-likelihood estimation. \n\n### FE Minimization with Mean-field Factorization Constraints: the CAVI Approach \n\n- Let's work out FE minimization with additional mean-field constraints (=full factorization) constraints:  $$q(z) = \\prod_{j=1}^m q_j(z_j)\\,.$$\n  - In other words, the posteriors for $z_j$ are all considered independent. This is a strong constraint but leads often to good solutions.\n\n- Given the mean-field constraints, it is possible to derive the following expression for the optimal solutions $q_j^*(z_j)$, for $j=1,\\ldots,m$: \n\n\\begin{equation*} \\tag{B-10.9}\n\\boxed{\n\\begin{aligned}\n\\log q_j^*(z_j) &\\propto \\mathrm{E}_{q_{-j}^*}\\left[ \\log p(x,z) \\right]  \\\\\n  &= \\underbrace{\\sum_{z_{-j}} q_{-j}^*(z_{-j}) \\underbrace{\\log p(x,z)}_{\\text{\"field\"}}}_{\\text{\"mean field\"}} \n\\end{aligned}}\n\\end{equation*}\n\nwhere we defined $q_{-j}^*(z_{-j}) \\triangleq q_1^*(z_1)q_2^*(z_2)\\cdots q_{j-1}^*(z_{j-1})q_{j+1}^*(z_{j+1})\\cdots q_m^*(z_m)$.\n\n- **Proof** (from [Blei, 2017](https://www.tandfonline.com/doi/full/10.1080/01621459.2017.1285773)): We first rewrite the FE as a function of $q_j(z_j)$ only: \n  $$ F[q_j] = \\mathbb{E}_{q_{j}}\\left[ \\mathbb{E}_{q_{-j}}\\left[ \\log p(x,z_j,z_{-j})\\right]\\right] - \\mathbb{E}_{q_j}\\left[ \\log q_j(z_j)\\right] + \\mathtt{const.}\\,,$$\n  where the constant holds all terms that do not depend on $z_j$. This expression can be written as \n  $$ F[q_j] = \\sum_{z_j} q_j(z_j) \\log \\frac{q_j(z_j)}{\\exp\\left( \\mathbb{E}_{q_{-j}}\\left[ \\log p(x,z_j,z_{-j})\\right]\\right)}$$\n  which is a KL-divergence that is minimized by Eq. B-10.9.  (end proof)\n\n  \n- This is not yet a full solution to the FE minimization task since the solution $q_j^*(z_j)$ depends on expectations that involve other solutions $q_{i\\neq j}^*(z_{i \\neq j})$, and each of these other solutions $q_{i\\neq j}^*(z_{i \\neq j})$ depends on an expection that involves $q_j^*(z_j)$. \n\n- In practice, we solve this chicken-and-egg problem by an iterative approach: we first initialize all $q_j(z_j)$ (for $j=1,\\ldots,m$) to an appropriate initial distribution and then cycle through the factors in turn by solving eq.B-10.9 and update $q_{-j}^*(z_{-j})$ with the latest estimates. (See [Blei, 2017](https://www.tandfonline.com/doi/full/10.1080/01621459.2017.1285773), Algorithm 1, p864).  \n\n- This algorithm for approximating Bayesian inference is known **Coordinate Ascent Variational Inference** (CAVI).  \n\n### Example: FEM for the Gaussian Mixture Model (CAVI Approach)\n\n- Let's get back to the illustrative example at the beginning of this lesson: we want to do [density modeling for the Old Faithful data set](#illustrative-example).\n\n#### model specification\n\n\n- We consider a Gaussian Mixture Model, specified by \n$$\\begin{align*}\np(x,z|\\theta) &= p(x|z,\\mu,\\Lambda)p(z|\\pi) \\\\\n  &= \\prod_{n=1}^N \\prod_{k=1}^K \\left(\\pi_k \\cdot \\mathcal{N}\\left( x_n | \\mu_k, \\Lambda_k^{-1}\\right)\\right)^{z_{nk}} \\tag{B-10.37,38}\n\\end{align*}$$\nwith tuning parameters $\\theta=\\{\\pi_k, \\mu_k,\\Lambda_k\\}$.\n\n- Let us introduce some priors for the parameters. We factorize the prior and choose conjugate distributions by\n$$\np(\\theta) = p(\\pi,\\mu,\\Lambda) = p(\\pi) p(\\mu|\\Lambda) p(\\Lambda)\n$$\nwith \n$$\\begin{align*}\np(\\pi) &= \\mathrm{Dir}(\\pi|\\alpha_0) = C(\\alpha_0) \\prod_k \\pi_k^{\\alpha_0-1} \\tag{B-10.39}\\\\\np(\\mu|\\Lambda) &= \\prod_k \\mathcal{N}\\left(\\mu_k | m_0, \\left( \\beta_0 \\Lambda_k\\right)^{-1} \\right) \\tag{B-10.40}\\\\\np(\\Lambda) &= \\prod_k \\mathcal{W}\\left( \\Lambda_k | W_0, \\nu_0 \\right) \\tag{B-10.40}\n\\end{align*}$$\nwhere $\\mathcal{W}\\left( \\cdot \\right)$ is a [Wishart distribution](https://en.wikipedia.org/wiki/Wishart_distribution) (i.e., a multi-dimensional Gamma distribution).\n\n- The full generative model is now specified by\n$$\np(x,z,\\pi,\\mu,\\Lambda) = p(x|z,\\mu,\\Lambda) p(z|\\pi) p(\\pi) p(\\mu|\\Lambda) p(\\Lambda) \\tag{B-10.41}\n$$\nwith hyperparameters $\\{ \\alpha_0, m_0, \\beta_0, W_0, \\nu_0\\}$.\n\n#### inference\n\n- Assume that we have observed $D = \\left\\{x_1, x_2, \\ldots, x_N\\right\\}$ and are interested to infer the posterior distribution for the tuning parameters: \n$$\np(\\theta|D) = p(\\pi,\\mu,\\Lambda|D)\n$$\n\n- The (perfect) Bayesian solution is \n$$\np(\\theta|D) = \\frac{p(x=D,\\theta)}{p(x=D)} = \\frac{\\sum_z p(x=D,z,\\pi,\\mu,\\Lambda)}{\\sum_z \\sum_{\\pi} \\iint p(x=D,z,\\pi,\\mu,\\Lambda) \\,\\mathrm{d}\\mu\\mathrm{d}\\Lambda}\n$$\nbut this is intractable (See [Blei (2017), p861, eqs. 8 and 9](https://www.tandfonline.com/doi/full/10.1080/01621459.2017.1285773)).\n\n- Alternatively, we can use **FE minimization with factorization constraint** \n$$\\begin{equation}\nq(\\theta) = q(z) \\cdot q(\\pi,\\mu,\\Lambda) \\tag{B-10.42}\n\\end{equation}$$ \non the posterior. For the specified model, this leads to FE minimization wrt the hyperparameters, i.e., we need to minimize the function $F(\\alpha_0, m_0, \\beta_0, W_0, \\nu_0)$.\n\n- Bishop shows that the equations for the [optimal solutions (Eq. B-10.9)](#optimal-solutions) are analytically solvable for the GMM as specified above, leading to the following variational update equations (for $k=1,\\ldots,K$): \n$$\n\\begin{align*}\n\\alpha_k &= \\alpha_0 + N_k  \\tag{B-10.58} \\\\\n\\beta_k &= \\beta_0 + N_k  \\tag{B-10.60} \\\\\nm_k &= \\frac{1}{\\beta_k} \\left( \\beta_0 m_0 + N_k \\bar{x}_k \\right) \\tag{B-10.61} \\\\\nW_k^{-1} &= W_0^{-1} + N_k S_k + \\frac{\\beta_0 N_k}{\\beta_0 + N_k}\\left( \\bar{x}_k - m_0\\right) \\left( \\bar{x}_k - m_0\\right)^T \\tag{B-10.62} \\\\\n\\nu_k &= \\nu_0 + N_k \\tag{B-10.63}\n\\end{align*}\n$$\nwhere we used\n$$\n\\begin{align*}\n\\log \\rho_{nk} &= \\mathbb{E}\\left[ \\log \\pi_k\\right] + \\frac{1}{2}\\mathbb{E}\\left[ \\log | \\Lambda_k | \\right] - \\frac{D}{2} \\log(2\\pi) \\\\ \n & \\qquad - \\frac{1}{2}\\mathbb{E}\\left[(x_k - \\mu_k)^T \\Lambda_k(x_k - \\mu_k)  \\right]  \\tag{B-10.46} \\\\\nr_{nk} &= \\frac{\\rho_{nk}}{\\sum_{j=1}^K \\rho_{nj}} \\tag{B-10.49} \\\\\nN_k &= \\sum_{n=1}^N r_{nk} x_n \\tag{B-10.51} \\\\\n\\bar{x}_k &= \\frac{1}{N_k} \\sum_{n=1}^N r_{nk} x_n \\tag{B-10.52} \\\\\nS_k &= \\frac{1}{N_k} \\sum_{n=1}^N r_{nk} \\left( x_n - \\bar{x}_k\\right) \\left( x_n - \\bar{x}_k\\right)^T \\tag{B-10.53}\n\\end{align*}\n$$\n\n- Exam guide: Working out FE minimization for the GMM to these update equations (eqs B-10.58 through B-10.63) is not something that you need to reproduce without assistance at the exam. Rather, the essence is that *it is possible* to arrive at closed-form variational update equations for the GMM. You should understand though how FEM works conceptually and in principle be able to derive variational update equations for very simple models that do not involve clever mathematical tricks.\n\n### Code Example: FEM for GMM on Old Faithfull data set\n\n- Below we exemplify training of a Gaussian Mixture Model on the Old Faithful data set by Free Energy Minimization, using the constraints as specified above, e.g., [(B-10.42)](#mf-constraint). \n\n\n```julia\nusing Pkg;Pkg.activate(\"probprog/workspace\");Pkg.instantiate()\nIJulia.clear_output();\n```\n\n\n```julia\nusing DataFrames, CSV, LinearAlgebra, PDMats, SpecialFunctions\ninclude(\"scripts/gmm_plot.jl\") # Holds plotting function \nold_faithful = CSV.read(\"datasets/old_faithful.csv\",DataFrame);\nX = convert(Matrix{Float64}, [old_faithful[!,1] old_faithful[!,2]]');#data matrix\nN = size(X, 2) #number of observations\nK = 6\n\nfunction sufficientStatistics(X,r,k::Int) #function to compute sufficient statistics\n    N_k = sum(r[k,:])\n    hat_x_k = sum([r[k,n]*X[:,n] for n in 1:N]) ./ N_k\n    S_k = sum([r[k,n]*(X[:,n]-hat_x_k)*(X[:,n]-hat_x_k)' for n in 1:N]) ./ N_k\n    return N_k, hat_x_k, S_k\nend\n\nfunction updateMeanPrecisionPi(m_0,\u03b2_0,W_0,\u03bd_0,\u03b1_0,r) #variational maximisation function\n    m = Array{Float64}(undef,2,K) #mean of the clusters \n    \u03b2 = Array{Float64}(undef,K) #precision scaling for Gausian distribution\n    W = Array{Float64}(undef,2,2,K) #precision prior for Wishart distributions\n    \u03bd = Array{Float64}(undef,K) #degrees of freedom parameter for Wishart distribution\n    \u03b1 = Array{Float64}(undef,K) #Dirichlet distribution parameter \n    for k=1:K\n        sst = sufficientStatistics(X,r,k)\n        \u03b1[k] = \u03b1_0[k] + sst[1]; \u03b2[k] = \u03b2_0[k] + sst[1]; \u03bd[k] = \u03bd_0[k] .+ sst[1]\n        m[:,k] = (1/\u03b2[k])*(\u03b2_0[k].*m_0[:,k] + sst[1].*sst[2])\n        W[:,:,k] = inv(inv(W_0[:,:,k])+sst[3]*sst[1] + ((\u03b2_0[k]*sst[1])/(\u03b2_0[k]+sst[1])).*(sst[2]-m_0[:,k])*(sst[2]-m_0[:,k])')\n    end\n    return m,\u03b2,W,\u03bd,\u03b1\nend\n\nfunction updateR(\u039b,m,\u03b1,\u03bd,\u03b2) #variational expectation function\n    r = Array{Float64}(undef,K,N) #responsibilities \n    hat_\u03c0 = Array{Float64}(undef,K) \n    hat_\u039b = Array{Float64}(undef,K)\n    for k=1:K\n        hat_\u039b[k] = 1/2*(2*log(2)+logdet(\u039b[:,:,k])+digamma(\u03bd[k]/2)+digamma((\u03bd[k]-1)/2))\n        hat_\u03c0[k] = exp(digamma(\u03b1[k])-digamma(sum(\u03b1)))\n        for n=1:N\n           r[k,n] = hat_\u03c0[k]*exp(-hat_\u039b[k]-1/\u03b2[k] - (\u03bd[k]/2)*(X[:,n]-m[:,k])'*\u039b[:,:,k]*(X[:,n]-m[:,k]))\n        end\n    end\n    for n=1:N\n        r[:,n] = r[:,n]./ sum(r[:,n]) #normalize to ensure r represents probabilities \n    end\n    return r\nend\n\nmax_iter = 15\n#store the inference results in these vectors\n\u03bd = fill!(Array{Float64}(undef,K,max_iter),3)\n\u03b2 = fill!(Array{Float64}(undef,K,max_iter),1.0)\n\u03b1 = fill!(Array{Float64}(undef,K,max_iter),0.01)\nR = Array{Float64}(undef,K,N,max_iter)\nM = Array{Float64}(undef,2,K,max_iter)\n\u039b = Array{Float64}(undef,2,2,K,max_iter)\nclusters_vb = Array{Distribution}(undef,K,max_iter) #clusters to be plotted\n#initialize prior distribution parameters\nM[:,:,1] = [[3.0; 60.0];[4.0; 70.0];[2.0; 50.0];[2.0; 60.0];[3.0; 100.0];[4.0; 80.0]]\nfor k=1:K\n    \u039b[:,:,k,1] = [1.0 0;0 0.01]\n    R[k,:,1] = 1/(K)*ones(N)\n    clusters_vb[k,1] = MvNormal(M[:,k,1],PDMats.PDMat(convert(Matrix,Hermitian(inv(\u03bd[1,1].*\u039b[:,:,k,1])))))\nend\n#variational inference\nfor i=1:max_iter-1\n    #variational expectation \n    R[:,:,i+1] = updateR(\u039b[:,:,:,i],M[:,:,i],\u03b1[:,i],\u03bd[:,i],\u03b2[:,i]) \n    #variational minimisation\n    M[:,:,i+1],\u03b2[:,i+1],\u039b[:,:,:,i+1],\u03bd[:,i+1],\u03b1[:,i+1] = updateMeanPrecisionPi(M[:,:,i],\u03b2[:,i],\u039b[:,:,:,i],\u03bd[:,i],\u03b1[:,i],R[:,:,i+1])\n    for k=1:K\n        clusters_vb[k,i+1] = MvNormal(M[:,k,i+1],PDMats.PDMat(convert(Matrix,Hermitian(inv(\u03bd[k,i+1].*\u039b[:,:,k,i+1])))))\n    end\nend\n\n\nsubplot(2,3,1); plotGMM(X, clusters_vb[:,1], R[:,:,1]); title(\"Initial situation\")\nfor i=2:6\n    subplot(2,3,i)\n    plotGMM(X, clusters_vb[:,i*2], R[:,:,i*2]); title(\"After $(i*2) iterations\")\nend\nPyPlot.tight_layout();\n```\n\nThe generated figure looks much like Figure 10.6 in Bishop. The plots show results for Variational Bayesian\nmixture of K = 6 Gaussians applied to the Old Faithful data set, in\nwhich the ellipses denote the one\nstandard-deviation density contours\nfor each of the components, and the\ncolor coding of the data points reflects the \"soft\" class label assignments. Components whose expected\nmixing coefficient are numerically indistinguishable from zero are not\nplotted. Note that this procedure learns the number of classes (two), learns the class label for each observation, and learns the mean and variance for the Gaussian data distributions.\n\n###  Free Energy Decompositions and Fixed-form Constraints\n\n\n-  Making use of $p(x,z) = p(z|x)p(x) = p(x|z)p(z)$, the FE functional can be rewritten as \n\n$$\\begin{align*}\n\\mathrm{F}[q] &= \\underbrace{-\\sum_z q(z) \\log p(x,z)}_{\\text{energy}} - \\underbrace{\\sum_z q(z) \\log \\frac{1}{q(z)}}_{\\text{entropy}} \\tag{EE} \\\\\n&= \\underbrace{\\sum_z q(z) \\log \\frac{q(z)}{p(z|x)}}_{\\text{KL divergence}\\geq 0} - \\underbrace{\\log p(x)}_{\\text{log-evidence}} \\tag{DE}\\\\\n&= \\underbrace{\\sum_z q(z)\\log\\frac{q(z)}{p(z)}}_{\\text{complexity}} - \\underbrace{\\sum_z q(z) \\log p(x|z)}_{\\text{accuracy}}  \\tag{CA}\n\\end{align*}$$\n\n- These decompositions are very insightful (we will revisit them later) and we will label them respectively as _energy-entropy_ (EE), _divergence-evidence_ (DE), and _complexity-accuracy_ (CA) decompositions. \n\n\n\n- In the [Bayesian Machine Learning](https://nbviewer.org/github/bertdv/BMLIP/blob/master/lessons/notebooks/Bayesian-Machine-Learning.ipynb) lecture, we discussed the CA decomposition of Bayesian model evidence to support the interpretation of evidence as a model performance criterion. Here, we recognize that FE allows a similar CA decomposition: minimizing FE increases data accuracy and decreases model complexity. \n\n- The CA decomposition makes use of the prior $p(z)$ and likelihood $p(x|z)$, both of which are selected by the engineer, so the FE can be evaluated with this decomposition!\n\n- If we now assume some parametric form for $q(z)$, e.g. $q(z) = \\mathcal{N}(z\\mid \\mu, \\Sigma)$, then the FE functional degenerates to a **regular function** $F(\\mu,\\Sigma)$. In principle, this function can be evaluated by the CA decomposition and minimized by standard (function) minimization methods. \n\n- The EE decomposition provides a link to a second law of thermodynamics: Minimizing FE leads to entropy maximization. \n\n\n\n\n###  Summary\n\n- Latent variable models (LVM) contain a set of unobserved variables whose size grows with the number of observations.\n\n- LVMs can model more complex phenomena than fully observed models, but inference in LVM models is usually not analytically solvable.\n\n- The Free Energy (FE) functional transforms Bayesian inference computations (very large summations or integrals) to an optimization problem. \n\n- Inference by minimizing FE, also known as variational inference, is fully consistent with the \"Method of Maximum Relative Entropy\", which is by design the rational way to update beliefs from priors to posteriors when new information becomes available. Thus, FE mimimization is a \"correct\" inference procedure that generalizes Bayes rule. \n\n- In general, global FE minimization is an unsolved problem and finding good local minima is at the heart of current Bayesian technology research. Three simplifying constraints on the posterior $q(z)$ in the FE functional are currently popular in practical settings:\n    - mean-field assumptions\n    - assuming a parametric from for $q$\n    - EM algorithm\n \n\n   \n- These constraints often makes FE minimization implementable at the price of obtaining approximately Bayesian inference results. \n\n- The back-ends of Probabilistic Programming languages often contain lots of methods for constrained FE minimization. \n\n\n## 
                                        OPTIONAL SLIDES
                                        \n\n### FE Minimization by the Expectation-Maximization (EM) Algorithm\n\n- The EM algorithm is a special case of FE minimization that focusses on Maximum-Likelihood estimation for models with latent variables. \n- Consider a model $$p(x,z,\\theta)$$ with observations $x = \\{x_n\\}$, latent variables $z=\\{z_n\\}$ and parameters $\\theta$.\n\n- We can write the following FE functional for this model:\n$$\\begin{align*}\nF[q] =  \\sum_z \\sum_\\theta q(z,\\theta) \\log \\frac{q(z,\\theta)}{p(x,z,\\theta)} \n\\end{align*}$$\n\n- The EM algorithm makes the following simplifying assumptions:\n  1. The prior for the parameters is uninformative (uniform). This implies that \n  $$p(x,z,\\theta) = p(x,z|\\theta) p(\\theta) \\propto p(x,z|\\theta)$$\n  2. A factorization constraint $$q(z,\\theta) = q(z) q(\\theta)$$\n  3. The posterior for the parameters is a delta function:\n  $$q(\\theta) = \\delta(\\theta - \\hat{\\theta})$$\n \n\n \n- Basically, these three assumptions turn FE minimization into maximum likelihood estimation for the parameters $\\theta$ and the FE simplifies to \n$$\\begin{align*}\nF[q,\\theta] =  \\sum_z q(z) \\log \\frac{q(z)}{p(x,z|\\theta)} \n\\end{align*}$$\n\n- The EM algorithm minimizes this FE by iterating (iteration counter: $i$) over \n\n\\begin{equation*}\n\\boxed{\n\\begin{aligned}\n\\mathcal{L}^{(i)}(\\theta) &= \\sum_z \\overbrace{p(z|x,\\theta^{(i-1)})}^{q^{(i)}(z)}  \\log p(x,z|\\theta) \\quad &&\\text{the E-step} \\\\\n\\theta^{(i)} &= \\arg\\max_\\theta \\mathcal{L}^{(i)}(\\theta) &&\\text{the M-step}\n\\end{aligned}}\n\\end{equation*}\n\n- These choices are optimal for the given FE functional. In order to see this, consider the two decompositions\n$$\\begin{align}\nF[q,\\theta] &= \\underbrace{-\\sum_z q(z) \\log p(x,z|\\theta)}_{\\text{energy}} - \\underbrace{\\sum_z q(z) \\log \\frac{1}{q(z)}}_{\\text{entropy}} \\tag{EE}\\\\\n  &= \\underbrace{\\sum_z q(z) \\log \\frac{q(z)}{p(z|x,\\theta)}}_{\\text{divergence}} - \\underbrace{\\log p(x|\\theta)}_{\\text{log-likelihood}}  \\tag{DE}\n\\end{align}$$\n\n- The DE decomposition shows that the FE is minimized for the choice $q(z) := p(z|x,\\theta)$. Also, for this choice, the FE equals the (negative) log-evidence (, which is this case simplifies to the log-likelihood). \n\n- The EE decomposition shows that the FE is minimized wrt $\\theta$ by minimizing the energy term. The energy term is computed in the E-step and optimized in the M-step.\n  - Note that in the EM literature, the energy term is often called the _expected complete-data log-likelihood_.)\n\n- In order to execute the EM algorithm, it is assumed that we can analytically execute the E- and M-steps. For a large set of models (including models whose distributions belong to the exponential family of distributions), this is indeed the case and hence the large popularity of the EM algorithm. \n\n- The EM algorihm imposes rather severe assumptions on the FE (basically approximating Bayesian inference by maximum likelihood estimation). Over the past few years, the rise of Probabilistic Programming languages has dramatically increased the range of models for which the parameters can by estimated autmatically by (approximate) Bayesian inference, so the popularity of EM is slowly waning. (More on this in the Probabilistic Programming lessons). \n\n- Bishop (2006) works out EM for the GMM in section 9.2.\n\n### Code Example: EM-algorithm for the GMM on the Old-Faithful data set\n\nWe'll perform clustering on the data set from the [illustrative example](#illustrative-example) by fitting a GMM consisting of two Gaussians using the EM algorithm. \n\n\n```julia\nusing Pkg; Pkg.activate(\"probprog/workspace\");Pkg.instantiate();\nIJulia.clear_output();\n\nusing DataFrames, CSV, LinearAlgebra\ninclude(\"scripts/gmm_plot.jl\") # Holds plotting function \nold_faithful = CSV.read(\"datasets/old_faithful.csv\", DataFrame);\n\nX =  Array(Matrix{Float64}(old_faithful)')\nN = size(X, 2)\n\n# Initialize the GMM. We assume 2 clusters.\nclusters = [MvNormal([4.;60.], [.5 0;0 10^2]); \n            MvNormal([2.;80.], [.5 0;0 10^2])];\n\u03c0_hat = [0.5; 0.5]                    # Mixing weights\n\u03b3 = fill!(Matrix{Float64}(undef,2,N), NaN)  # Responsibilities (row per cluster)\n\n# Define functions for updating the parameters and responsibilities\nfunction updateResponsibilities!(X, clusters, \u03c0_hat, \u03b3)\n    # Expectation step: update \u03b3\n    norm = [pdf(clusters[1], X) pdf(clusters[2], X)] * \u03c0_hat\n    \u03b3[1,:] = (\u03c0_hat[1] * pdf(clusters[1],X) ./ norm)'\n    \u03b3[2,:] = 1 .- \u03b3[1,:]\nend\nfunction updateParameters!(X, clusters, \u03c0_hat, \u03b3)\n    # Maximization step: update \u03c0_hat and clusters using ML estimation\n    m = sum(\u03b3, dims=2)\n    \u03c0_hat = m / N\n    \u03bc_hat = (X * \u03b3') ./ m'\n    for k=1:2\n        Z = (X .- \u03bc_hat[:,k])\n        \u03a3_k = Symmetric(((Z .* (\u03b3[k,:])') * Z') / m[k])\n        clusters[k] = MvNormal(\u03bc_hat[:,k], convert(Matrix, \u03a3_k))\n    end\nend\n\n# Execute the algorithm: iteratively update parameters and responsibilities\nsubplot(2,3,1); plotGMM(X, clusters, \u03b3); title(\"Initial situation\")\nupdateResponsibilities!(X, clusters, \u03c0_hat, \u03b3)\nsubplot(2,3,2); plotGMM(X, clusters, \u03b3); title(\"After first E-step\")\nupdateParameters!(X, clusters, \u03c0_hat, \u03b3)\nsubplot(2,3,3); plotGMM(X, clusters, \u03b3); title(\"After first M-step\")\niter_counter = 1\nfor i=1:3\n    for j=1:i+1\n        updateResponsibilities!(X, clusters, \u03c0_hat, \u03b3)\n        updateParameters!(X, clusters, \u03c0_hat, \u03b3)\n        iter_counter += 1\n    end\n    subplot(2,3,3+i); \n    plotGMM(X, clusters, \u03b3); \n    title(\"After $(iter_counter) iterations\")\nend\nPyPlot.tight_layout()\n```\n\nNote that you can step through the interactive demo yourself by running [this script](https://github.com/bertdv/AIP-5SSB0/blob/master/lessons/notebooks/scripts/interactive_em_demo.jl) in julia. You can run a script in julia by    \n`julia> include(\"path/to/script-name.jl\")`\n\n### Message Passing for Free Energy Minimization\n\n- The Sum-Product (SP) update rule implements perfect Bayesian inference. \n- Sometimes, the SP update rule is not analytically solvable. \n- Fortunately, for many well-known Bayesian approximation methods, a message passing update rule can be created, e.g. [Variational Message Passing](https://en.wikipedia.org/wiki/Variational_message_passing) (VMP) for variational inference. \n- In general, all of these message passing algorithms can be interpreted as minimization of a constrained free energy (e.g., see [Zhang et al. (2017)](https://github.com/bertdv/BMLIP/blob/master/lessons/notebooks/files/Zhang-2017-Unifying-Message-Passing-Algorithms.pdf), and hence these message passing schemes comply with [Caticha's Method of Maximum Relative Entropy](https://arxiv.org/abs/1011.0723), which, as discussed in the [variational Bayes lesson](https://nbviewer.jupyter.org/github/bertdv/BMLIP/blob/master/lessons/notebooks/Latent-Variable-Models-and-VB.ipynb) is the proper way for updating beliefs. \n- Different message passing updates rules can be combined to get a hybrid inference method in one model. \n\n### The Local Free Energy in a Factor Graph \n\n- Consider an edge $x_j$ in a Forney-style factor graph for a generative model $p(x) = p(x_1,x_2,\\ldots,x_N)$.\n\n\n\n- Assume that the graph structure (factorization) is specified by\n$$\np(x) = \\prod_{a=1}^M p_a(x_a)\n$$\nwhere $a$ is a set of indices.\n- Also, we assume a mean-field approximation for the posterior:\n$$\nq(x) = \\prod_{i=1}^N q_i(x_i)\n$$\nand consequently a corresponding free energy functional  \n$$\\begin{align*}\nF[q] &= \\sum_x q(x) \\log \\frac{q(x)}{p(x)} \\\\\n  &= \\sum_i \\sum_{x_i} \\left(\\prod_{i=1}^N q_i(x_i)\\right) \\log \\frac{\\prod_{i=1}^N q_i(x_i)}{\\prod_{a=1}^M p_a(x_a)}\n\\end{align*}$$\n\n- With these assumptions, it can be shown that the FE evaluates to (exercise)\n$$\nF[q] = \\sum_{a=1}^M \\underbrace{\\sum_{x_a} \\left( \\prod_{j\\in N(a)} q_j(x_j)\\cdot \\left(-\\log p_a(x_a)\\right) \\right) }_{\\text{node energy }U[p_a]} - \\sum_{i=1}^N \\underbrace{\\sum_{x_i} q_i(x_i) \\log \\frac{1}{q_i(x_i)}}_{\\text{edge entropy }H[q_i]}\n$$\n\n- In words, the FE decomposes into a sum of (expected) energies for the nodes minus the entropies on the edges. \n \n\n### Variational Message Passing\n\n- Let us now consider the local free energy that is associated with edge corresponding to $x_j$. \n  \n  
                                        \n  \n  \n- Apparently (see previous slide), there are three contributions to the free energy for $x_j$:\n    - one entropy term for the edge $x_j$\n    - two energy terms: one for each node that attaches to $x_j$ (in the figure: nodes $p_a$ and $p_b$)\n    \n- The local free energy for $x_j$ can be written as (exercise)\n  $$\n  F[q_j] \\propto \\sum_{x_j} q(x_j) \\log \\frac{q_j(x_j)}{\\nu_a(x_j)\\cdot \\nu_b(x_j)}\n  $$\n  where\n  $$\\begin{align*} \n  \\nu_a(x_j) &\\propto \\exp\\left( \\mathbb{E}_{q_{k}}\\left[ \\log p_a(x_a)\\right]\\right) \\\\\n  \\nu_b(x_j) &\\propto \\exp\\left( \\mathbb{E}_{q_{l}}\\left[ \\log p_b(x_b)\\right]\\right) \n  \\end{align*}$$\n  and $\\mathbb{E}_{q_{k}}\\left[\\cdot\\right]$ is an expectation w.r.t. all $q(x_k)$ with $k \\in N(a)\\setminus {j}$.\n  \n- $\\nu_a(x_j)$ and $\\nu_b(x_j)$  can be locally computed in nodes $a$ and $b$ respectively and can be interpreted as colliding messages over edge $x_j$. \n  \n- Local free energy minimization is achieved by setting\n  $$\n  q_j(x_j) \\propto \\nu_a(x_j) \\cdot \\nu_b(x_j)\n  $$\n  \n- Note that message $\\nu_a(x_j)$ depends on posterior beliefs over incoming edges ($k$) for node $a$, and in turn, the message from node $a$ towards edge $x_k$ depends on the belief $q_j(x_j)$. I.o.w., direct mutual dependencies exist between posterior beliefs over edges that attach to the same node. \n  \n- These considerations lead to the [Variational Message Passing](https://en.wikipedia.org/wiki/Variational_message_passing) procedure, which is an iterative free energy minimization procedure that can be executed completely through locally computable messages.  \n\n- Procedure VMP, see [Dauwels (2007), section 3](https://github.com/bertdv/BMLIP/blob/master/lessons/notebooks/files/Dauwels-2007-on-variational-message-passing-on-factor-graphs)\n  > 1. Initialize all messages $q$ and $\u03bd$, e.g., $q(\\cdot) \\propto 1$ and $\\nu(\\cdot) \\propto 1$. 
                                        \n  > 2. Select an edge $z_k$ in the factor graph of $f(z_1,\\ldots,z_m)$.
                                        \n  > 3. Compute the two messages $\\overrightarrow{\\nu}(z_k)$ and $\\overleftarrow{\\nu}(z_k)$ by applying the following generic rule:\n  $$\n  \\overrightarrow{\\nu}(y) \\propto \\exp\\left( \\mathbb{E}_{q}\\left[ \\log g(x_1,\\dots,x_n,y)\\right] \\right) \n  $$\n  > 4. Compute the marginal $q(z_k)$\n  $$\n  q(z_k) \\propto \\overrightarrow{\\nu}(z_k) \\overleftarrow{\\nu}(z_k)\n  $$\n  and send it to the two nodes connected to the edge $x_k$.
                                        \n  > 5. Iterate 2\u20134 until convergence. \n\n### The Bethe Free Energy and Belief Propagation\n\n- We showed that, under mean field assumptions, the FE can be decomposed into a sum of local FE contributions for the nodes ($a$) and edges ($i$):\n$$\nF[q] = \\sum_{a=1}^M \\underbrace{\\sum_{x_a} \\left( \\prod_{j\\in N(a)} q_j(x_j)\\cdot \\left(-\\log p_a(x_a)\\right) \\right) }_{\\text{node energy }U[p_a]} - \\sum_{i=1}^N \\underbrace{\\sum_{x_i} q_i(x_i) \\log \\frac{1}{q_i(x_i)}}_{\\text{edge entropy }H[q_i]}\n$$\n\n- The mean field assumption is very strong and may lead to large inference costs ($\\mathrm{KL}(q(x),p(x|\\text{data}))$). A more relaxed assumption is to allow joint posterior beliefs over the variables that attach to a node. This idea is expressed by the Bethe Free Energy:\n$$\nF_B[q] = \\sum_{a=1}^M \\left( \\sum_{x_a} q_a(x_a) \\log \\frac{q_a(x_a)}{p_a(x_a)} \\right)  - \\sum_{i=1}^N (d_i - 1) \\sum_{x_i} q_i(x_i) \\log {q_i(x_i)}\n$$\nwhere $q_a(x_a)$ is the posterior joint belief over the variables $x_a$ (i.e., the set of variables that attach to node $a$), $q_i(x_i)$ is the posterior marginal belief over the variable $x_i$ and $d_i$ is the number of factor nodes that link to edge $i$. Moreover, $q_a(x_a)$ and $q_i(x_i)$ are constrained to obey the following equalities:\n$$\n  \\sum_{x_a \\backslash x_i} q_a(x_a) = q_i(x_i), ~~~ \\forall i, \\forall a \\\\\n  \\sum_{x_i} q_i(x_i) = 1, ~~~ \\forall i \\\\\n  \\sum_{x_a} q_a(x_a) = 1, ~~~ \\forall a \\\\\n$$\n\n- We form the Lagrangian by augmenting the Bethe Free Energy functional with the constraints:\n$$\nL[q] = F_B[q] + \\sum_i\\sum_{a \\in N(i)} \\lambda_{ai}(x_i) \\left(q_i(x_i) - \\sum_{x_a\\backslash x_i} q(x_a) \\right) + \\sum_{i} \\gamma_i \\left(  \\sum_{x_i}q_i(x_i) - 1\\right) + \\sum_{a}\\gamma_a \\left(  \\sum_{x_a}q_a(x_a) -1\\right)\n$$\n\n- The stationary solutions for this Lagrangian are given by\n$$\nq_a(x_a) = f_a(x_a) \\exp\\left(\\gamma_a -1 + \\sum_{i \\in N(a)} \\lambda_{ai}(x_i)\\right) \\\\ \nq_i(x_i) = \\exp\\left(1- \\gamma_i + \\sum_{a \\in N(i)} \\lambda_{ai}(x_i)\\right) ^{\\frac{1}{d_i - 1}}\n$$\nwhere $N(i)$ denotes the factor nodes that have $x_i$ in their arguments and $N(a)$ denotes the set of variables in the argument of $f_a$.\n\n- Stationary solutions are functions of Lagrange multipliers. This means that Lagrange multipliers need to be determined. Lagrange multipliers can be determined by plugging the stationary solutions back into the constraint specification and solving for the multipliers which ensure that the constraint is satisfied. The first constraint we consider is normalization, which yields the following identification:\n$$\n\\gamma_a = 1 - \\log \\Bigg(\\sum_{x_a}f_a(x_a)\\exp\\left(\\sum_{i \\in N(a)}\\lambda_{ai}(x_i)\\right)\\Bigg)\\\\\n\\gamma_i = 1 + (d_i-1) \\log\\Bigg(\\sum_{x_i}\\exp\\left( \\frac{1}{d_i-1}\\sum_{a \\in N(i)} \\lambda_{ai}(x_i)\\right)\\Bigg).\n$$\n\n- The functional form of the Lagrange multipliers that corresponds to the normalization constraint enforces us to obtain the Lagrange multipliers that correspond to the marginalization constraint. To do so we solve for \n$$ \\sum_{x_a \\backslash x_i} f_a(x_a) \\exp\\left(\\sum_{i \\in N(a)} \\lambda_{ai}(x_i)\\right) = \\exp\\left(\\sum_{a \\in N(i)} \\lambda_{ai}(x_i)\\right) ^{\\frac{1}{d_i - 1}} \\\\ \\exp\\left(\\lambda_{ai}(x_i)\\right)\\sum_{x_a \\backslash x_i} f_a(x_a) \\exp\\Bigg(\\sum_{\\substack{{j \\in N(a)} \\\\ j \\neq i}}\\lambda_{aj}(x_j)\\Bigg) = \\exp\\left(\\sum_{a \\in N(i)} \\lambda_{ai}(x_i)\\right) ^{\\frac{1}{d_i - 1}}\\\\ \\exp\\left(\\lambda_{ai}(x_i) + \\lambda_{ia}(x_i)\\right) = \\exp\\left(\\sum_{a \\in N(i)} \\lambda_{ai}(x_i)\\right) ^{\\frac{1}{d_i - 1}}\\,, $$ \nwhere we defined an auxilary function\n$$\n\\exp(\\lambda_{ia}(x_i)) \\triangleq \\sum_{x_a \\backslash x_i} f_a(x_a) \\exp\\Bigg(\\sum_{\\substack{{j \\in N(a)} \\\\ j \\neq i}}\\lambda_{aj}(x_j)\\Bigg) \\,.\n$$\nThis definition is valid since it can be inverted by the relation\n$$\n\\lambda_{ia}(x_i) = \\frac{2-d_i}{d_i - 1}\\lambda_{ai}(x_i) + \\frac{1}{d_i -1}\\sum_{\\substack{c \\in N(i)\\\\c \\neq a}}\\lambda_{ci}(x_i)\n$$\n\n- In general it is not possible to solve for the Lagrange multipliers analytically and we resort to iteratively obtaining the solutions. This leads to the **Belief Propagation algorithm** where the exponentiated Lagrange multipliers (messages) are updated iteratively via \n$$ \\mu_{ia}^{(k+1)}(x_i) = \\sum_{x_a \\backslash x_i} f_a(x_a) \\prod_{\\substack{{j \\in N(a)} \\\\ j \\neq i}}\\mu^{(k)}_{aj}(x_j) \\\\ \\mu_{ai}^{(k)}(x_i) = \\prod_{\\substack{c \\in N(i) \\\\ c \\neq a}}\\mu^{(k)}_{ic}(x_i)\\,, $$ \nwhere $k$ denotes iteration number and the messages are defined as\n$$\n\\mu_{ia}(x_i) \\triangleq \\exp(\\lambda_{ia}(x_i))\\\\\n\\mu_{ai}(x_i) \\triangleq \\exp(\\lambda_{ai}(x_i))\\,.\n$$\n\n- For a more complete overview of message passing as Bethe Free Energy minimization, see Zhang (2017).\n\n\n```julia\nopen(\"../../styles/aipstyle.html\") do f\n    display(\"text/html\", read(f, String))\nend\n```\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n```julia\n\n```\n", "meta": {"hexsha": "e0d1abd933be9233924c21bd624782e7fc412c6a", "size": 404396, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lessons/notebooks/Latent-Variable-Models-and-VB.ipynb", "max_stars_repo_name": "Yikeru/BMLIP", "max_stars_repo_head_hexsha": "296f5330210d387809b2c3ce7a6847f2bd69b24c", "max_stars_repo_licenses": ["CC-BY-3.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "lessons/notebooks/Latent-Variable-Models-and-VB.ipynb", "max_issues_repo_name": "Yikeru/BMLIP", "max_issues_repo_head_hexsha": "296f5330210d387809b2c3ce7a6847f2bd69b24c", "max_issues_repo_licenses": ["CC-BY-3.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "lessons/notebooks/Latent-Variable-Models-and-VB.ipynb", "max_forks_repo_name": "Yikeru/BMLIP", "max_forks_repo_head_hexsha": "296f5330210d387809b2c3ce7a6847f2bd69b24c", "max_forks_repo_licenses": ["CC-BY-3.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 204.7574683544, "max_line_length": 165946, "alphanum_fraction": 0.8883767396, "converted": true, "num_tokens": 16551, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "# Spectrum\n\n[Colour](http://en.wikipedia.org/wiki/Colour) is defined as the characteristic of visual perception that can be described by attributes of hue, brightness (or lightness) and colourfulness (or saturation or chroma).\n\nWhen necessary, to avoid confusion between other meanings of the word, the term \"perceived colour\" may be used.\n\nPerceived colour depends on the spectral distribution of the colour stimulus, on the size, shape, structure and surround of the stimulus area, on the state of adaptation of the observer's visual system, and on the observer's experience of the prevailing and similar situations of observation. [1]\n\n[Light](http://en.wikipedia.org/wiki/Light) is the electromagnetic radiation that is considered from the point of view of its ability to excite the human visual system. [2]\n\nThe portion of the electromatic radiation frequencies perceived in the approximate wavelength range 360-780 nanometres (nm) is called the [visible spectrum](http://en.wikipedia.org/wiki/Visible_spectrum).\n\n\n```python\n%matplotlib inline\n```\n\n\n```python\nimport colour\nfrom colour.plotting import *\n\ncolour.filter_warnings(True, False)\n\ncolour_plotting_defaults()\n\n# Plotting the visible spectrum.\nvisible_spectrum_plot()\n```\n\nThe spectrum is defined as the display or specification of the monochromatic components of the radiation considered. [3]\n\nAt the core of [Colour](https://github.com/colour-science/colour/) is the *colour.colorimetry* sub-package, it defines the objects needed for spectral related computations and many others:\n\n\n```python\nfrom pprint import pprint\n\nimport colour.colorimetry as colorimetry\n\npprint(colorimetry.__all__)\n```\n\n    ['SpectralShape',\n     'SpectralPowerDistribution',\n     'MultiSpectralPowerDistribution',\n     'DEFAULT_SPECTRAL_SHAPE',\n     'constant_spd',\n     'zeros_spd',\n     'ones_spd',\n     'blackbody_spd',\n     'blackbody_spectral_radiance',\n     'planck_law',\n     'LMS_ConeFundamentals',\n     'RGB_ColourMatchingFunctions',\n     'XYZ_ColourMatchingFunctions',\n     'CMFS',\n     'LMS_CMFS',\n     'RGB_CMFS',\n     'STANDARD_OBSERVERS_CMFS',\n     'ILLUMINANTS',\n     'D_ILLUMINANTS_S_SPDS',\n     'HUNTERLAB_ILLUMINANTS',\n     'ILLUMINANTS_RELATIVE_SPDS',\n     'LIGHT_SOURCES',\n     'LIGHT_SOURCES_RELATIVE_SPDS',\n     'LEFS',\n     'PHOTOPIC_LEFS',\n     'SCOTOPIC_LEFS',\n     'BANDPASS_CORRECTION_METHODS',\n     'bandpass_correction',\n     'bandpass_correction_Stearns1988',\n     'D_illuminant_relative_spd',\n     'CIE_standard_illuminant_A_function',\n     'mesopic_luminous_efficiency_function',\n     'mesopic_weighting_function',\n     'LIGHTNESS_METHODS',\n     'lightness',\n     'lightness_Glasser1958',\n     'lightness_Wyszecki1963',\n     'lightness_CIE1976',\n     'lightness_Fairchild2010',\n     'lightness_Fairchild2011',\n     'LUMINANCE_METHODS',\n     'luminance',\n     'luminance_Newhall1943',\n     'luminance_ASTMD153508',\n     'luminance_CIE1976',\n     'luminance_Fairchild2010',\n     'luminance_Fairchild2011',\n     'dominant_wavelength',\n     'complementary_wavelength',\n     'excitation_purity',\n     'colorimetric_purity',\n     'luminous_flux',\n     'luminous_efficiency',\n     'luminous_efficacy',\n     'RGB_10_degree_cmfs_to_LMS_10_degree_cmfs',\n     'RGB_2_degree_cmfs_to_XYZ_2_degree_cmfs',\n     'RGB_10_degree_cmfs_to_XYZ_10_degree_cmfs',\n     'LMS_2_degree_cmfs_to_XYZ_2_degree_cmfs',\n     'LMS_10_degree_cmfs_to_XYZ_10_degree_cmfs',\n     'SPECTRAL_TO_XYZ_METHODS',\n     'spectral_to_XYZ',\n     'ASTME30815_PRACTISE_SHAPE',\n     'lagrange_coefficients_ASTME202211',\n     'tristimulus_weighting_factors_ASTME202211',\n     'adjust_tristimulus_weighting_factors_ASTME30815',\n     'spectral_to_XYZ_integration',\n     'spectral_to_XYZ_tristimulus_weighting_factors_ASTME30815',\n     'spectral_to_XYZ_ASTME30815',\n     'wavelength_to_XYZ',\n     'WHITENESS_METHODS',\n     'whiteness',\n     'whiteness_Berger1959',\n     'whiteness_Taube1960',\n     'whiteness_Stensby1968',\n     'whiteness_ASTME313',\n     'whiteness_Ganz1979',\n     'whiteness_CIE2004',\n     'YELLOWNESS_METHODS',\n     'yellowness',\n     'yellowness_ASTMD1925',\n     'yellowness_ASTME313']\n\n\n> Note: *colour.colorimetry* sub-package public API is directly available from *colour* namespace.\n\n[Colour](https://github.com/colour-science/colour/) computations are based on a comprehensive dataset available in pretty much each sub-packages, for example *colour.colorimetry.dataset* defines the following data:\n\n\n```python\nimport colour.colorimetry.dataset as dataset\n\npprint(dataset.__all__)\n```\n\n    ['CMFS',\n     'LMS_CMFS',\n     'RGB_CMFS',\n     'STANDARD_OBSERVERS_CMFS',\n     'ILLUMINANTS',\n     'D_ILLUMINANTS_S_SPDS',\n     'HUNTERLAB_ILLUMINANTS',\n     'ILLUMINANTS_RELATIVE_SPDS',\n     'LIGHT_SOURCES',\n     'LIGHT_SOURCES_RELATIVE_SPDS',\n     'LEFS',\n     'PHOTOPIC_LEFS',\n     'SCOTOPIC_LEFS']\n\n\n> Note: *colour.colorimetry.dataset* sub-package public API is directly available from *colour* namespace.\n\n## Spectral Power Distribution\n\nWhether it be a sample spectral power distribution, colour matching functions or illuminants, spectral data is manipulated using an object built with the *colour.SpectralPowerDistribution* class or based on it:\n\n\n```python\nimport colour\n\n# Defining a sample spectral power distribution data.\nsample_spd_data = {\n    380: 0.048,\n    385: 0.051,\n    390: 0.055,\n    395: 0.06,\n    400: 0.065,\n    405: 0.068,\n    410: 0.068,\n    415: 0.067,\n    420: 0.064,\n    425: 0.062,\n    430: 0.059,\n    435: 0.057,\n    440: 0.055,\n    445: 0.054,\n    450: 0.053,\n    455: 0.053,\n    460: 0.052,\n    465: 0.052,\n    470: 0.052,\n    475: 0.053,\n    480: 0.054,\n    485: 0.055,\n    490: 0.057,\n    495: 0.059,\n    500: 0.061,\n    505: 0.062,\n    510: 0.065,\n    515: 0.067,\n    520: 0.070,\n    525: 0.072,\n    530: 0.074,\n    535: 0.075,\n    540: 0.076,\n    545: 0.078,\n    550: 0.079,\n    555: 0.082,\n    560: 0.087,\n    565: 0.092,\n    570: 0.100,\n    575: 0.107,\n    580: 0.115,\n    585: 0.122,\n    590: 0.129,\n    595: 0.134,\n    600: 0.138,\n    605: 0.142,\n    610: 0.146,\n    615: 0.150,\n    620: 0.154,\n    625: 0.158,\n    630: 0.163,\n    635: 0.167,\n    640: 0.173,\n    645: 0.180,\n    650: 0.188,\n    655: 0.196,\n    660: 0.204,\n    665: 0.213,\n    670: 0.222,\n    675: 0.231,\n    680: 0.242,\n    685: 0.251,\n    690: 0.261,\n    695: 0.271,\n    700: 0.282,\n    705: 0.294,\n    710: 0.305,\n    715: 0.318,\n    720: 0.334,\n    725: 0.354,\n    730: 0.372,\n    735: 0.392,\n    740: 0.409,\n    745: 0.420,\n    750: 0.436,\n    755: 0.450,\n    760: 0.462,\n    765: 0.465,\n    770: 0.448,\n    775: 0.432,\n    780: 0.421}\n\nspd = colour.SpectralPowerDistribution(sample_spd_data, name='Sample')\nprint(spd)\n```\n\n    [[  3.80000000e+02   4.80000000e-02]\n     [  3.85000000e+02   5.10000000e-02]\n     [  3.90000000e+02   5.50000000e-02]\n     [  3.95000000e+02   6.00000000e-02]\n     [  4.00000000e+02   6.50000000e-02]\n     [  4.05000000e+02   6.80000000e-02]\n     [  4.10000000e+02   6.80000000e-02]\n     [  4.15000000e+02   6.70000000e-02]\n     [  4.20000000e+02   6.40000000e-02]\n     [  4.25000000e+02   6.20000000e-02]\n     [  4.30000000e+02   5.90000000e-02]\n     [  4.35000000e+02   5.70000000e-02]\n     [  4.40000000e+02   5.50000000e-02]\n     [  4.45000000e+02   5.40000000e-02]\n     [  4.50000000e+02   5.30000000e-02]\n     [  4.55000000e+02   5.30000000e-02]\n     [  4.60000000e+02   5.20000000e-02]\n     [  4.65000000e+02   5.20000000e-02]\n     [  4.70000000e+02   5.20000000e-02]\n     [  4.75000000e+02   5.30000000e-02]\n     [  4.80000000e+02   5.40000000e-02]\n     [  4.85000000e+02   5.50000000e-02]\n     [  4.90000000e+02   5.70000000e-02]\n     [  4.95000000e+02   5.90000000e-02]\n     [  5.00000000e+02   6.10000000e-02]\n     [  5.05000000e+02   6.20000000e-02]\n     [  5.10000000e+02   6.50000000e-02]\n     [  5.15000000e+02   6.70000000e-02]\n     [  5.20000000e+02   7.00000000e-02]\n     [  5.25000000e+02   7.20000000e-02]\n     [  5.30000000e+02   7.40000000e-02]\n     [  5.35000000e+02   7.50000000e-02]\n     [  5.40000000e+02   7.60000000e-02]\n     [  5.45000000e+02   7.80000000e-02]\n     [  5.50000000e+02   7.90000000e-02]\n     [  5.55000000e+02   8.20000000e-02]\n     [  5.60000000e+02   8.70000000e-02]\n     [  5.65000000e+02   9.20000000e-02]\n     [  5.70000000e+02   1.00000000e-01]\n     [  5.75000000e+02   1.07000000e-01]\n     [  5.80000000e+02   1.15000000e-01]\n     [  5.85000000e+02   1.22000000e-01]\n     [  5.90000000e+02   1.29000000e-01]\n     [  5.95000000e+02   1.34000000e-01]\n     [  6.00000000e+02   1.38000000e-01]\n     [  6.05000000e+02   1.42000000e-01]\n     [  6.10000000e+02   1.46000000e-01]\n     [  6.15000000e+02   1.50000000e-01]\n     [  6.20000000e+02   1.54000000e-01]\n     [  6.25000000e+02   1.58000000e-01]\n     [  6.30000000e+02   1.63000000e-01]\n     [  6.35000000e+02   1.67000000e-01]\n     [  6.40000000e+02   1.73000000e-01]\n     [  6.45000000e+02   1.80000000e-01]\n     [  6.50000000e+02   1.88000000e-01]\n     [  6.55000000e+02   1.96000000e-01]\n     [  6.60000000e+02   2.04000000e-01]\n     [  6.65000000e+02   2.13000000e-01]\n     [  6.70000000e+02   2.22000000e-01]\n     [  6.75000000e+02   2.31000000e-01]\n     [  6.80000000e+02   2.42000000e-01]\n     [  6.85000000e+02   2.51000000e-01]\n     [  6.90000000e+02   2.61000000e-01]\n     [  6.95000000e+02   2.71000000e-01]\n     [  7.00000000e+02   2.82000000e-01]\n     [  7.05000000e+02   2.94000000e-01]\n     [  7.10000000e+02   3.05000000e-01]\n     [  7.15000000e+02   3.18000000e-01]\n     [  7.20000000e+02   3.34000000e-01]\n     [  7.25000000e+02   3.54000000e-01]\n     [  7.30000000e+02   3.72000000e-01]\n     [  7.35000000e+02   3.92000000e-01]\n     [  7.40000000e+02   4.09000000e-01]\n     [  7.45000000e+02   4.20000000e-01]\n     [  7.50000000e+02   4.36000000e-01]\n     [  7.55000000e+02   4.50000000e-01]\n     [  7.60000000e+02   4.62000000e-01]\n     [  7.65000000e+02   4.65000000e-01]\n     [  7.70000000e+02   4.48000000e-01]\n     [  7.75000000e+02   4.32000000e-01]\n     [  7.80000000e+02   4.21000000e-01]]\n\n\nThe sample spectral power distribution can be easily plotted against the visible spectrum:\n\n\n```python\n# Plotting the sample spectral power distribution.\nsingle_spd_plot(spd)\n```\n\nWith the sample spectral power distribution defined, we can retrieve its shape: \n\n\n```python\n# Displaying the sample spectral power distribution shape.\nprint(spd.shape)\n```\n\n    (380.0, 780.0, 5.0)\n\n\nThe shape returned is an instance of *colour.SpectralShape* class:\n\n\n```python\nrepr(spd.shape)\n```\n\n\n\n\n    'SpectralShape(380.0, 780.0, 5.0)'\n\n\n\n*colour.SpectralShape* is used throughout [Colour](https://github.com/colour-science/colour/) to define spectral dimensions and is instantiated as follows:\n\n\n```python\n# Using *colour.SpectralShape* with iteration.\nshape = colour.SpectralShape(start=0, end=10, interval=1)\nfor wavelength in shape:\n    print(wavelength)\n\n# *colour.SpectralShape.range* method is providing the complete range of values. \nshape = colour.SpectralShape(0, 10, 0.5)\nshape.range()\n```\n\n    0.0\n    1.0\n    2.0\n    3.0\n    4.0\n    5.0\n    6.0\n    7.0\n    8.0\n    9.0\n    10.0\n\n\n\n\n\n    array([  0. ,   0.5,   1. ,   1.5,   2. ,   2.5,   3. ,   3.5,   4. ,\n             4.5,   5. ,   5.5,   6. ,   6.5,   7. ,   7.5,   8. ,   8.5,\n             9. ,   9.5,  10. ])\n\n\n\n[Colour](https://github.com/colour-science/colour/) defines three convenient objects to create constant spectral power distributions:\n\n*  *colour.constant_spd*\n*  *colour.zeros_spd*\n*  *colour.ones_spd*\n\n\n```python\n# Defining a constant spectral power distribution.\nconstant_spd = colour.constant_spd(100)\nprint('\"Constant Spectral Power Distribution\"')\nprint(constant_spd.shape)\nprint(constant_spd[400])\n\n# Defining a zeros filled spectral power distribution.\nprint('\\n\"Zeros Filled Spectral Power Distribution\"')\nzeros_spd = colour.zeros_spd()\nprint(zeros_spd.shape)\nprint(zeros_spd[400])\n\n# Defining a ones filled spectral power distribution.\nprint('\\n\"Ones Filled Spectral Power Distribution\"')\nones_spd = colour.ones_spd()\nprint(ones_spd.shape)\nprint(ones_spd[400])\n```\n\n    \"Constant Spectral Power Distribution\"\n    (360.0, 780.0, 1.0)\n    100.0\n    \n    \"Zeros Filled Spectral Power Distribution\"\n    (360.0, 780.0, 1.0)\n    0.0\n    \n    \"Ones Filled Spectral Power Distribution\"\n    (360.0, 780.0, 1.0)\n    1.0\n\n\nBy default the shape used by *colour.constant_spd*, *colour.zeros_spd* and *colour.ones_spd* is the one defined by *colour.DEFAULT_SPECTRAL_SHAPE* attribute using the *CIE 1931 2\u00b0 Standard Observer* shape.\n\n\n```python\nprint(repr(colour.DEFAULT_SPECTRAL_SHAPE))\n```\n\n    SpectralShape(360, 780, 1)\n\n\nA custom shape can be passed to construct a constant spectral power distribution with tailored dimensions:\n\n\n```python\ncolour.ones_spd(colour.SpectralShape(400, 700, 5))[450]\n```\n\n\n\n\n    1.0\n\n\n\nOften interpolation of the spectral power distribution is needed, this is achieved with the *colour.SpectralPowerDistribution.interpolate* method. Depending on the wavelengths uniformity, the default interpolation method will differ. Following *CIE 167:2005* recommendation: The method developed by Sprague (1880) should be used for interpolating functions having a uniformly spaced independent variable and a *Cubic Spline* method for non-uniformly spaced independent variable. [4]\n\nWe can check the uniformity of the sample spectral power distribution:\n\n\n```python\n# Checking the sample spectral power distribution uniformity.\nprint(spd.is_uniform())\n```\n\n    True\n\n\nSince the sample spectral power distribution is uniform the interpolation will be using the *colour.SpragueInterpolator* interpolator.\n\n> Note: Interpolation happens in place and may alter your original data, use the *colour.SpectralPowerDistribution.clone* method to produce a copy of your spectral power distribution before interpolation.\n\n\n```python\n# Copying the sample spectral power distribution.\nspd_copy = spd.copy()\n\n# Interpolating the copied sample spectral power distribution.\nspd_copy.interpolate(colour.SpectralShape(400, 770, 1))\nspd_copy[401]\n```\n\n\n\n\n    0.065809599999999996\n\n\n\n\n```python\n# Comparing the interpolated spectral power distribution with the original one.\nmulti_spd_plot([spd, spd_copy], bounding_box=[730,780, 0.1, 0.5])\n```\n\nExtrapolation although dangerous can be used to help aligning two spectral power distributions together. *CIE 015:2004 Colorimetry, 3rd Edition* recommends that unmeasured values may be set equal to the nearest measured value of the appropriate quantity in truncation: [5]\n\n\n```python\n# Extrapolating the copied sample spectral power distribution.\nspd_copy.extrapolate(colour.SpectralShape(340, 830))\nspd_copy[340], spd_copy[830]\n```\n\n\n\n\n    (0.065000000000000002, 0.44800000000000018)\n\n\n\nThe underlying interpolator can be swapped for any of the *Colour* interpolators.\n\n\n```python\npprint([\n    export for export in colour.algebra.interpolation.__all__\n    if 'Interpolator' in export\n])\n```\n\n    [u'KernelInterpolator',\n     u'LinearInterpolator',\n     u'SpragueInterpolator',\n     u'CubicSplineInterpolator',\n     u'PchipInterpolator',\n     u'NullInterpolator']\n\n\n\n```python\n# Changing interpolator while trimming the copied spectral power distribution.\nspd_copy.interpolate(\n    colour.SpectralShape(400, 700, 10), interpolator=colour.LinearInterpolator)\n```\n\n\n\n\n    SpectralPowerDistribution([[  4.00000000e+02,   6.50000000e-02],\n                               [  4.10000000e+02,   6.80000000e-02],\n                               [  4.20000000e+02,   6.40000000e-02],\n                               [  4.30000000e+02,   5.90000000e-02],\n                               [  4.40000000e+02,   5.50000000e-02],\n                               [  4.50000000e+02,   5.30000000e-02],\n                               [  4.60000000e+02,   5.20000000e-02],\n                               [  4.70000000e+02,   5.20000000e-02],\n                               [  4.80000000e+02,   5.40000000e-02],\n                               [  4.90000000e+02,   5.70000000e-02],\n                               [  5.00000000e+02,   6.10000000e-02],\n                               [  5.10000000e+02,   6.50000000e-02],\n                               [  5.20000000e+02,   7.00000000e-02],\n                               [  5.30000000e+02,   7.40000000e-02],\n                               [  5.40000000e+02,   7.60000000e-02],\n                               [  5.50000000e+02,   7.90000000e-02],\n                               [  5.60000000e+02,   8.70000000e-02],\n                               [  5.70000000e+02,   1.00000000e-01],\n                               [  5.80000000e+02,   1.15000000e-01],\n                               [  5.90000000e+02,   1.29000000e-01],\n                               [  6.00000000e+02,   1.38000000e-01],\n                               [  6.10000000e+02,   1.46000000e-01],\n                               [  6.20000000e+02,   1.54000000e-01],\n                               [  6.30000000e+02,   1.63000000e-01],\n                               [  6.40000000e+02,   1.73000000e-01],\n                               [  6.50000000e+02,   1.88000000e-01],\n                               [  6.60000000e+02,   2.04000000e-01],\n                               [  6.70000000e+02,   2.22000000e-01],\n                               [  6.80000000e+02,   2.42000000e-01],\n                               [  6.90000000e+02,   2.61000000e-01],\n                               [  7.00000000e+02,   2.82000000e-01]],\n                              interpolator=SpragueInterpolator,\n                              interpolator_args={},\n                              extrapolator=Extrapolator,\n                              extrapolator_args={u'right': None, u'method': u'Constant', u'left': None})\n\n\n\nThe extrapolation behaviour can be changed for *Linear* method instead of the *Constant* default method or even use arbitrary constant *left* and *right* values:\n\n\n```python\n# Extrapolating the copied sample spectral power distribution with *Linear* method.\nspd_copy.extrapolate(\n    colour.SpectralShape(340, 830),\n    extrapolator_args={'method': 'Linear',\n                       'right': 0})\nspd_copy[340], spd_copy[830]\n```\n\n\n\n\n    (0.046999999999999348, 0.0)\n\n\n\nAligning a spectral power distribution is a convenient way to first interpolate the current data within its original bounds then if needed extrapolates any missing values to match the requested shape:\n\n\n```python\n# Aligning the cloned sample spectral power distribution.\n# We first trim the spectral power distribution as above.\nspd_copy.interpolate(colour.SpectralShape(400, 700))  \nspd_copy.align(colour.SpectralShape(340, 830, 5))\nspd_copy[340], spd_copy[830]\n```\n\n\n\n\n    (0.065000000000000002, 0.28199999999999975)\n\n\n\nThe *colour.SpectralPowerDistribution* class also supports various arithmetic operations like *addition*, *subtraction*, *multiplication*, *division* or *exponentiation* with *numeric* and *array_like* variables or other *colour.SpectralPowerDistribution* class instances:\n\n\n```python\nspd = colour.SpectralPowerDistribution({\n    410: 0.25,\n    420: 0.50,\n    430: 0.75,\n    440: 1.0,\n    450: 0.75,\n    460: 0.50,\n    480: 0.25\n})\n\nprint((spd.copy() + 1).values)\nprint((spd.copy() * 2).values)\nprint((spd * [0.35, 1.55, 0.75, 2.55, 0.95, 0.65, 0.15]).values)\nprint((spd * colour.constant_spd(2, spd.shape) * colour.constant_spd(3, spd.shape)).values)\n```\n\n    [ 1.25  1.5   1.75  2.    1.75  1.5   1.25]\n    [ 0.5  1.   1.5  2.   1.5  1.   0.5]\n    [ 0.0875  0.775   0.5625  2.55    0.7125  0.325   0.0375]\n    [ 1.5  3.   4.5  6.   4.5  3.   nan  1.5]\n\n\nThe spectral power distribution can be normalised with an arbitrary factor:\n\n\n```python\nprint(spd.normalise().values)\nprint(spd.normalise(100).values)\n```\n\n    [ 0.25  0.5   0.75  1.    0.75  0.5   0.25]\n    [  25.   50.   75.  100.   75.   50.   25.]\n\n\n## Colour Matching Functions\n\nIn the late 1920's, Wright (1928) and Guild (1931) independently conducted a series of colour matching experiments to quantify the colour ability of an average human observer which laid the foundation for the specification of the [CIE XYZ colourspace](http://en.wikipedia.org/wiki/CIE_color_space#Definition_of_the_CIE_XYZ_color_space). The results obtained were summarized by the *Wright & Guild 1931 2\u00b0 RGB CMFs* $\\bar{r}(\\lambda)$,$\\bar{g}(\\lambda)$,$\\bar{b}(\\lambda)$ colour matching functions: they represent the amounts of three monochromatic primary colours $\\textbf{R}$,$\\textbf{G}$,$\\textbf{B}$ needed to match the test colour at a single wavelength of light.\n\n> See Also: The [Colour Matching Functions](cmfs.ipynb) notebook for in-depth informations about the colour matching functions.\n\n\n```python\n# Plotting *Wright & Guild 1931 2 Degree RGB CMFs* colour matching functions.\nsingle_cmfs_plot('Wright & Guild 1931 2 Degree RGB CMFs')\n```\n\nWith an RGB model of human vision based on *Wright & Guild 1931 2\u00b0 RGB CMFs* $\\bar{r}(\\lambda)$,$\\bar{g}(\\lambda)$,$\\bar{b}(\\lambda)$ colour matching functions and for pragmatic reasons the *CIE* members developed a new colour space that would relate to the *CIE RGB* colourspace but for which all tristimulus values would be positive for real colours: *CIE XYZ* described with $\\bar{x}(\\lambda)$,$\\bar{y}(\\lambda)$,$\\bar{z}(\\lambda)$ colour matching functions.\n\n\n```python\n# Plotting *CIE XYZ 1931 2 Degree Standard Observer* colour matching functions.\nsingle_cmfs_plot('CIE 1931 2 Degree Standard Observer')\n```\n\nIn the 1960's it appeared that cones were present in a larger region of eye than the one initially covered by the experiments that lead to the *CIE 1931 2\u00b0 Standard Observer* specification.\n\nAs a result, colour computations done with the *CIE 1931 2\u00b0 Standard Observer* do not always correlate to the visual observation.\n\nIn 1964, the *CIE* defined an additional standard observer: the *CIE 1964 10\u00b0 Standard Observer* derived from the work of Stiles and Burch (1959), and Speranskaya (1959). The *CIE 1964 10\u00b0 Standard Observer* is believed to be a better representation of the human vision spectral response and recommended when dealing with a field of view of more than 4\u00b0.\n\nFor example and as per *CIE* recommendation, the *CIE 1964 10\u00b0 Standard Observer* is commonly used with spectrophotometers for colour measurements whereas colorimeters generally use the *CIE 1931 2\u00b0 Standard Observer* for quality control and other colour evaluation applications.\n\n## CIE XYZ Tristimulus Values\n\nThe *CIE XYZ* tristimulus values specify a colour stimulus in terms of the visual system. Their values for colour of a surface with spectral reflectance $\\beta(\\lambda)$ under an illuminant of relative spectral power $S(\\lambda)$ are calculated using the following equations: [6]\n\n$$\n\\begin{equation}\nX=k\\int_{\\lambda}\\beta(\\lambda)S(\\lambda)\\bar{x}(\\lambda)d\\lambda\\\\\nY=k\\int_{\\lambda}\\beta(\\lambda)S(\\lambda)\\bar{y}(\\lambda)d\\lambda\\\\\nZ=k\\int_{\\lambda}\\beta(\\lambda)S(\\lambda)\\bar{z}(\\lambda)d\\lambda\n\\end{equation}\n$$\nwhere\n$$\n\\begin{equation}\nk=\\cfrac{100}{\\int_{\\lambda}S(\\lambda)\\bar{y}(\\lambda)d\\lambda}\n\\end{equation}\n$$\n\nHowever in virtually all practical computations of *CIE XYZ* tristimulus values, the integrals are replaced by summations:\n\n$$\n\\begin{equation}\nX=k\\sum\\limits_{\\lambda=\\lambda_a}^{\\lambda_b}\\beta(\\lambda)S(\\lambda)\\bar{x}(\\lambda)\\Delta\\lambda\\\\\nY=k\\sum\\limits_{\\lambda=\\lambda_a}^{\\lambda_b}\\beta(\\lambda)S(\\lambda)\\bar{y}(\\lambda)\\Delta\\lambda\\\\\nZ=k\\sum\\limits_{\\lambda=\\lambda_a}^{\\lambda_b}\\beta(\\lambda)S(\\lambda)\\bar{z}(\\lambda)\\Delta\\lambda\\\\\n\\end{equation}\n$$\nwhere\n$$\n\\begin{equation}\nk=\\cfrac{100}{\\sum\\limits_{\\lambda=\\lambda_a}^{\\lambda_b}S(\\lambda)\\bar{y}(\\lambda)\\Delta\\lambda}\n\\end{equation}\n$$\n\nCalculating the *CIE XYZ* tristimulus values of a colour stimulus is done using the *colour.spectral_to_XYZ* definition which follows *ASTM E2022\u201311* and *ASTM E308\u201315* practises computation method:\n\n\n```python\nspd = colour.SpectralPowerDistribution(sample_spd_data, name='Sample')\ncmfs = colour.STANDARD_OBSERVERS_CMFS['CIE 1931 2 Degree Standard Observer']\nilluminant = colour.ILLUMINANTS_RELATIVE_SPDS['A']\n\n# Calculating the sample spectral power distribution *CIE XYZ* tristimulus values.\ncolour.spectral_to_XYZ(spd, cmfs, illuminant)\n```\n\n\n\n\n    array([ 14.78676625,  10.97815867,   1.99023459])\n\n\n\n> Note: Output *CIE XYZ* colourspace matrix is in domain [0, 100].\n\n*CIE XYZ* tristimulus values can be plotted into the *CIE 1931 Chromaticity Diagram*:\n\n\n```python\nimport pylab\n\n# Plotting the *CIE 1931 Chromaticity Diagram*.\n# The argument *standalone=False* is passed so that the plot doesn't get displayed\n# and can be used as a basis for other plots.\nchromaticity_diagram_plot_CIE1931(standalone=False)\n\n# Calculating the *xy* chromaticity coordinates.\n# The output domain of *colour.spectral_to_XYZ* is [0, 100] and \n# the input domain of *colour.XYZ_to_sRGB* is [0, 1].\n# We need to take it in account and rescale the input *CIE XYZ* colourspace matrix.\nx, y = colour.XYZ_to_xy(colour.spectral_to_XYZ(spd, cmfs, illuminant) / 100)\n\n# Plotting the *xy* chromaticity coordinates.\npylab.plot(x, y, 'o-', color='white')\n\n# Annotating the plot.\npylab.annotate(spd.name,\n               xy=(x, y),\n               xytext=(-50, 30),\n               textcoords='offset points',\n               arrowprops=dict(arrowstyle='->', connectionstyle='arc3, rad=-0.2'))\n\n# Displaying the plot.\nrender(standalone=True)\n```\n\nRetrieving the *CIE XYZ* tristimulus values of any wavelength from colour matching functions is done using the *colour.wavelength_to_XYZ* definition, if the value requested is not available, the colour matching functions will be interpolated following *CIE 167:2005* recommendation:\n\n\n```python\ncolour.wavelength_to_XYZ(546.1, colour.STANDARD_OBSERVERS_CMFS['CIE 1931 2 Degree Standard Observer'])\n```\n\n\n\n\n    array([ 0.3755316 ,  0.98444552,  0.01220285])\n\n\n\n## Bibliography\n\n1. ^ CIE. (n.d.). 17-198 colour (perceived). Retrieved June 26, 2014, from http://eilv.cie.co.at/term/198\n2. ^ CIE. (n.d.). 17-659 light. Retrieved June 26, 2014, from http://eilv.cie.co.at/term/659\n3. ^ CIE. (n.d.). 17-1238 spectrum. Retrieved June 27, 2014, from http://eilv.cie.co.at/term/1238\n4. ^ CIE TC 1-38. (2005). 9. INTERPOLATION. In CIE 167:2005 Recommended Practice for Tabulating Spectral Data for Use in Colour Computations (pp. 14\u201319). ISBN:978-3-901-90641-1\n5. ^ CIE TC 1-48. (2004). CIE 015:2004 Colorimetry, 3rd Edition. CIE 015:2004 Colorimetry, 3rd Edition (pp. 1\u201382). ISBN:978-3-901-90633-6\n6. ^ Wyszecki, G., & Stiles, W. S. (2000). Integration Replace by Summation. In *Color Science: Concepts and Methods, Quantitative Data and Formulae* (pp. 158\u2013163). Wiley. 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{"text": "\n\n## First Assignment\n\n#### 1) Apply the appropriate string methods to the **x** variable (as '.upper') to change it exactly to: \"$Dichlorodiphenyltrichloroethane$\".\n\n\n```python\nx = \"DiClOrod  IFeNi  lTRicLOr oETaNo  DiChlorod iPHeny lTrichL oroEThaNe\"\n\n```\n\n\n```python\nprint(x)\n\n```\n\n    DiClOrod  IFeNi  lTRicLOr oETaNo  DiChlorod iPHeny lTrichL oroEThaNe\n\n\n\n```python\nx = x.replace(' ', '').title()\n\n\n```\n\n\n```python\nx = x[x.find('Dichlorodipheny') :]\n```\n\n\n```python\nprint(x)\n```\n\n    e\n\n\n\n```python\nx = x.split(',',31)\n```\n\n\n```python\nprint(x)\n```\n\n    e\n\n\n\n```python\nuppercase_string=x.upper()\n```\n\n\n```python\nprint(uppercase_string)\n```\n\n    DICLOROD  IFENI  LTRICLOR OETANO  DICHLOROD IPHENY LTRICHL OROETHANE\n\n\n#### 2) Assign respectively the values: 'word', 15, 3.14 and 'list' to variables A, B, C and D in a single line of code. Then, print them in that same order on a single line separated by a space, using only one print statement.\n\n\n```python\n# 1\nA,B,C,D = ['word', 15, 3.14, 'list']\n```\n\n\n```python\nD\n```\n\n\n\n\n    'list'\n\n\n\n#### 3) Use the **input()** function to receive an input in the form **'68.4 1.71'**, that is, two floating point numbers in a line separated by space. Then, assign these numbers to the variables **w** and **h** respectively, which represent an individual's weight and height (hint: take a look at the '.split()' method). With this data, calculate the individual's Body Mass Index (BMI) from the following relationship: \n    \n\\begin{equation}\nBMI = \\dfrac{weight}{height^2}\n\\end{equation}\n\n\n```python\n#1\ndef bmi (weight = 0, height = 0):\n    if bmi== 0:\n        bmi = input('68.4' '1.71')    \n        bmi.split('','')\n        bmi = float(weight)/((float(height)/100)**2)\n        print(bmi)\n  \n```\n\n\n```python\nprint(bmi)\n```\n\n    \n\n\n\n```python\nw, h = 68.4, 1.71\nprint(w)\nprint(h)\n```\n\n    68.4\n    1.71\n\n\n\n```python\n#2\n```\n\n\n```python\nx = input('enter height space weight : ')\n```\n\n    enter height and weight : 168 68\n\n\n\n```python\nx\n```\n\n\n\n\n    '168 68'\n\n\n\n\n```python\ny = x.split(' ')\n```\n\n\n```python\ny\n```\n\n\n\n\n    ['168', '68']\n\n\n\n\n```python\nheight = float(y[0])\n```\n\n\n```python\ntype(height)\n```\n\n\n\n\n    float\n\n\n\n#### This value can also be classified according to ranges of values, following to the table below. Use conditional structures to classify and print the classification assigned to the individual. \n\n
                                        <\\center>  \n\n\n(source: https://healthtravelguide.com/bmi-calculator/)\n\n\n```python\n#1\nbmi = int(input())\ndef bmi(weight= 0, height= 0):\n    if weight == 0:\n        weight = input(w)\n    if height == 0:\n        height = input(h)\n    if bmi  < 18.5 and bmi == 18.5:\n       print('Underweight')\n    if bmi  > 18.5 and bmi < 24.9 :\n       print('Normal weight')\n    if bmi > 25.0 and bmi ==25.0:\n       print('Pre-obesity or obesity')\n```\n\n    2\n\n\n\n```python\n#2\nbmi = int(input('BMI: '))\n    if bmi < 18.5:\n        print('Underweight')\n    elif bmi > 18.5 and bmi < 24.9:\n        print('Normal weight')\n    elif bmi > 25.0 and bmi ==25.0:\n        print('Pre-obesity or obesity')\n```\n\n\n```python\nprint(bmi)\n```\n\n#### 4) Receive an integer as an input and, using a loop, calculate the factorial of this number, that is, the product of all the integers from one to the number provided. \n\n\n```python\nfor integer in range (1,10):\n    print('   ')\n    print(f'{integer} ')\n    print()\n\n```\n\n       \n    1 \n    \n       \n    2 \n    \n       \n    3 \n    \n       \n    4 \n    \n       \n    5 \n    \n       \n    6 \n    \n       \n    7 \n    \n       \n    8 \n    \n       \n    9 \n    \n\n\n#### 5) Using a while loop and the input function, read an indefinite number of integers until the number read is -1. Present the sum of all these numbers in the form of a print, excluding the -1 read at the end. \n\n\n```python\nx = 1\nwhile x < 1:\n    print(x)\n    x += -1\n    print(do_sum(x)).pop\n```\n\n\n```python\nprint(x)\n```\n\n    1\n\n\n#### 6) Read the **first name** of an employee, his **amount of hours worked** and the **salary per hour** in a single line separated by commas. Next, calculate the **total salary** for this employee and show it to two decimal places.\n\n\n```python\n# which data, should I invent name and salary?\nrecord = {'first name': 'Susan', 'amount hours worked':150, 'Salary per hour':'12'}\nprint(record)\ndef total_salary(hour_worked, wage):\n    hours_worked = 150\n    wage = 12.00\n    total_salary = [hours_worked]*[wage]\nprint(total_salary)\n```\n\n    {'first name': 'Susan', 'amount hours worked': 150, 'Salary per hour': '12'}\n    \n\n\n#### 7) Read three floating point values **A**, **B** and **C** respectively. Then calculate itens a, b, c, d and e: \n\n\n```python\nimport math\n```\n\n    a) the area of the triangle rectangle with A as the base and C as the height.\n\n\n```python\nA = 2\nC = 1.5\nfrom math import area\narea = float(input('area of trinagle'('1/2' *A * C)\nprint(('area of trinagle ') + str(A) + 'is: ' str(1/2 *A*C))\n```\n\n    b) the area of the circle of radius C. (pi = 3.14159) \n\n\n```python\nfrom math import pi\nr = float(input('area of the circle radius C : '))\nprint ('area of the circle radius C ' + str(r) + \" is: \" + str(pi * r**2))\n```\n\n    area of the circle radius C : 1\n    area of the circle radius C 1.0 is: 3.141592653589793\n\n\n    c) the area of the trapezoid that has A and B for bases and C for height. \n\n\n```python\nA = b1 \nB = b2 \nC = h \ndef area(b1, b2, h):\n    return ((b1 + b2) / 2) * h\narea = area(b1, b2, h)\nprint('area is:', area)\n\n```\n\n    area is: 4.275\n\n\n    d) the area of the square that has side B. \n\n\n```python\nB = side\ndef areaSquare( side ):\n    area = side * side\n    return area\n   \n```\n\n    e) the area of the rectangle that has sides A and B. \n\n\n```python\nA = l\nb = b\nl = float(input('Enter the length of a Rectangle: '))\nb = float(input('Enter the breadth of a Rectangle: '))\nArea = l * b\nprint(\"Area of a Rectangle is: %.2f\" %Area)\nl = A\nb = B\n\n\n```\n\n    Enter the length of a Rectangle: 2\n    Enter the breadth of a Rectangle: 5\n    Area of a Rectangle is: 10.00\n\n\n#### 8) Read **the values a, b and c** and calculate the **roots of the second degree equation** $ax^{2}+bx+c=0$ using [this formula](https://en.wikipedia.org/wiki/Quadratic_equation). If it is not possible to calculate the roots, display the message **\u201cThere are no real roots\u201d**. \n\n\n```python\nimport math\ndef equationroots (a, b, c):\n    dis = (b * b - 4 * a * c)\n    sqrt_val = math.sqrt(abs(dis))\n    if dis > 0: \n        print(' There are no real roots ') \n        print((-b + sqrt_val)/(2 * a)) \n        print((-b - sqrt_val)/(2 * a))\n   \n\n    \n\n```\n\n#### 9) Read four floating point numerical values corresponding to the coordinates of two geographical coordinates in the cartesian plane. Each point will come in a line with its coordinates separated by space. Then calculate and show the distance between these two points. \n\n(obs: $d=\\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$)\n\n\n```python\npoint1 = [30.0443879, 31.2357257]\npoint2=  [41.8933203, 12.4829321]\npoint1 = (x1, x2)\npoint2 = (y1, y2)\nfloat(input('Point 1: ', (point1))\nfloat(input('Point 2: ', (point2))\ndistance = equationroots(x,y)\nprint((\"Distance between two points is: \") %.2 f %2)\n```\n\n#### 10) Read **two floating point numbers** on a line that represent **coordinates of a cartesian point**. With this, use **conditional structures** to determine if you are at the origin, printing the message **'origin'**; in one of the axes, printing **'x axis'** or **'y axis'**; or in one of the four quadrants, printing **'q1'**, **'q2**', **'q3'** or **'q4'**. \n\n\n```python\nmy_origin = [30.0443879, 31.2357257]\n    if i == my_origin:\n        print('origin')\n\n```\n\n#### 11) Read an integer that represents a phone code for international dialing.  \n#### Then, inform to which country the code belongs to, considering the generated table below:\n(You just need to consider the first 10 entries)  \n\n\n```python\nimport pandas as pd\ndf = pd.read_html('https://en.wikipedia.org/wiki/Telephone_numbers_in_Europe')[1]\ndf = df.iloc[:,:2]\ndf.head(20)\n\n```\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n  \n
                                        CountryCountry calling code
                                        0Austria43
                                        1Belgium32
                                        2Bulgaria359
                                        3Croatia385
                                        4Cyprus357
                                        5Czech Republic420
                                        6Denmark45
                                        7Estonia372
                                        8Finland358
                                        9France33
                                        10Germany49
                                        11Greece30
                                        12Hungary36
                                        13Iceland354
                                        14Ireland353
                                        15Italy39
                                        16Latvia371
                                        17Liechtenstein423
                                        18Lithuania370
                                        19Luxembourg352
                                        \n
                                        \n\n\n\n\n```python\ncodes = {'43': 'Austria','32': 'Belgium','358': 'Finland','33': 'France','49': 'Germany','30': 'Greece', '36' : 'Hungary','354' : 'Iceland','353' : 'Ireland','39': 'Italy'}\n```\n\n\n```python\ncodes\n```\n\n\n\n\n    {'30': 'Greece',\n     '32': 'Belgium',\n     '33': 'France',\n     '353': 'Ireland',\n     '354': 'Iceland',\n     '358': 'Finland',\n     '36': 'Hungary',\n     '39': 'Italy',\n     '43': 'Austria',\n     '49': 'Germany'}\n\n\n\n\n```python\nmy_code = input('What is your country code? ')\n```\n\n    What is your country code? 33\n\n\n\n```python\nmy_code\n```\n\n\n\n\n    '33'\n\n\n\n\n```python\ncodes[my_code]\n```\n\n\n\n\n    'France'\n\n\n\n#### 12) Write a piece of code that reads 6 numbers in a row. Next, show the number of positive values entered. On the next line, print the average of the values to one decimal place. \n\n\n```python\nmy_new_list= ('5','7','15','1','12','2')\nprint(my_new_list)\n```\n\n    ('5', '7', '15', '1', '12', '2')\n\n\n\n```python\nmy_new_list=int\nsum(my_new_list)/6\n```\n\n#### 13) Read an integer **N**. Then print the **square of each of the even values**, from 1 to N, including N, if applicable, arranged one per line. \n\n\n```python\nN = i\nint(i) for i in range(1,21)*2:\n    print(i)\n\n```\n\n#### 14) Using **input()**, read an integer and print its classification as **'even / odd'** and **'positive / negative'** . The two classes for the number must be printed on the same line separated by a space. In the case of zero, print only **'null'**. \n\n\n```python\nmy_new_integer = int(input())\nif my_new_integer %2 and my_new_integer > 0:\n    print('even positive')\nelif my_new_integer == 0:\n    print('null')\nelif my_new_integer < 0 and my_new_integer %2 == 0:\n    print('even negative')\nelif my_new_integer > 0 and my_new_integer %2 != 0:\n    print('odd positive')\nelse:\n    print('odd negative')\n          \n```\n\n    0\n    null\n\n\n## Challenge\n#### 15) Ordering problems are recurrent in the history of programming. Over time, several algorithms have been developed to fulfill this function. The simplest of these algorithms is the [**Bubble Sort**](https://en.wikipedia.org/wiki/Bubble_sort), which is based on comparisons of elements two by two in a loop of passes through the elements. Your mission, if you decide to accept it, will be to input six whole numbers ramdonly ordered. Then implement the **Bubble Sort** principle to order these six numbers **using only loops and conditionals**.  \n#### At the end, print the six numbers in ascending order on a single line separated by spaces. \n\n\n```python\n\n```\n", "meta": {"hexsha": "f8b611fc44a0943f61f016c746235b6109b2d511", "size": 43736, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Borowska_Anna_Maria_Assignment_1.ipynb", "max_stars_repo_name": "ambo2020/Anna-Borowska", "max_stars_repo_head_hexsha": "607ee5c60f5042a6aa7a3aa518b13da3c8cf7967", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Borowska_Anna_Maria_Assignment_1.ipynb", "max_issues_repo_name": "ambo2020/Anna-Borowska", "max_issues_repo_head_hexsha": "607ee5c60f5042a6aa7a3aa518b13da3c8cf7967", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Borowska_Anna_Maria_Assignment_1.ipynb", "max_forks_repo_name": "ambo2020/Anna-Borowska", "max_forks_repo_head_hexsha": "607ee5c60f5042a6aa7a3aa518b13da3c8cf7967", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 28.7736842105, "max_line_length": 565, "alphanum_fraction": 0.4162932138, "converted": true, "num_tokens": 3971, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.7690802476562641, "lm_q1q2_score": 0.41452134884905495}}
{"text": "\n\n\n# Tutorial-IllinoisGRMHD: Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.C\n\n## Authors: Leo Werneck & Zach Etienne\n\n**This module is currently under development**\n\n## In this tutorial module we explain the construction of the right-hand side of the evolution equations of $\\left[\\sqrt{\\gamma}\\Phi\\right]$ and add gauge terms to the right-hand side of the evolution equations of $A_{i}$\n\n### Required and recommended citations:\n\n* **(Required)** Etienne, Z. B., Paschalidis, V., Haas R., M\u00f6sta P., and Shapiro, S. L. IllinoisGRMHD: an open-source, user-friendly GRMHD code for dynamical spacetimes. Class. Quantum Grav. 32 (2015) 175009. ([arxiv:1501.07276](http://arxiv.org/abs/1501.07276)).\n* **(Required)** Noble, S. C., Gammie, C. F., McKinney, J. C., Del Zanna, L. Primitive Variable Solvers for Conservative General Relativistic Magnetohydrodynamics. Astrophysical Journal, 641, 626 (2006) ([astro-ph/0512420](https://arxiv.org/abs/astro-ph/0512420)).\n* **(Recommended)** Del Zanna, L., Bucciantini N., Londrillo, P. An efficient shock-capturing central-type scheme for multidimensional relativistic flows - II. Magnetohydrodynamics. A&A 400 (2) 397-413 (2003). DOI: 10.1051/0004-6361:20021641 ([astro-ph/0210618](https://arxiv.org/abs/astro-ph/0210618)).\n\nIf using the version of `IllinoisGRMHD` with piecewise polytropic *or* tabulated (coming soon!) EOS support, then the following citation is also required:\n\n* **(Required)** Etienne, Z. B., Werneck, L., Paschalidis, V., Haas R., M\u00f6sta P., and Shapiro, S. L., *IllinoisGRMHD github repository* (2019). Source Code URL: https://github.com/zachetienne/nrpytutorial/tree/master/IllinoisGRMHD/.\n\n\n\n# Table of Contents\n$$\\label{toc}$$\n\nThis module is organized as follows\n\n0. [Step 0](#src_dir): **Source directory creation**\n1. [Step 1](#lorenz_psi6phi_rhs__add_gauge_terms_to_a_i_rhs__c): **`Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.C`**\n    1. [Step 1.a](#interpolations): *Performing all necessary interpolations*\n        1. [Step 1.a.i](#interpolation_algorithm): The interpolation algorithm\n        1. [Step 1.a.ii](#interp_and_alpha_phi_minus_betaj_A_j): Interpolating gridfunctions and computing $\\left(\\alpha\\Phi-\\beta^{j}A_{j}\\right)$\n    1. [Step 1.b](#partial_t_a_i_gauge): *Computing $\\partial_{t}A_{i}^{\\rm gauge}$*\n    1. [Step 1.c](#shift_advection_terms): *Computing $\\partial_{j}\\beta^{j}\\left[\\sqrt{\\gamma}\\Phi\\right]$*\n    1. [Step 1.d](#partial_j_alpha_psi6_aj): *Computing $-\\partial_{j}\\left(\\alpha\\sqrt{\\gamma}A^{j}\\right)-\\xi\\alpha\\left[\\sqrt{\\gamma}\\Phi\\right]$*\n    1. [Step 1.e](#fct_avg): *The `avg()` function*\n1. [Step 2](#code_validation): **Code validation**\n1. [Step 3](#latex_pdf_output): **Output this notebook to $\\LaTeX$-formatted PDF file**\n\n\n\n# Step 0: Source directory creation \\[Back to [top](#toc)\\]\n$$\\label{src_dir}$$\n\nWe will now use the [cmdline_helper.py NRPy+ module](Tutorial-Tutorial-cmdline_helper.ipynb) to create the source directory within the `IllinoisGRMHD` NRPy+ directory, if it does not exist yet.\n\n\n```python\n# Step 0: Creation of the IllinoisGRMHD source directory\n# Step 0a: Add NRPy's directory to the path\n# https://stackoverflow.com/questions/16780014/import-file-from-parent-directory\nimport os,sys\nnrpy_dir_path = os.path.join(\"..\",\"..\")\nif nrpy_dir_path not in sys.path:\n    sys.path.append(nrpy_dir_path)\n\n# Step 0b: Load up cmdline_helper and create the directory\nimport cmdline_helper as cmd\noutdir = os.path.join(\"..\",\"src\")\ncmd.mkdir(outdir)\n```\n\n\n\n# Step 1: The `Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.C` file \\[Back to [top](#toc)\\]\n$$\\label{lorenz_psi6phi_rhs__add_gauge_terms_to_a_i_rhs__c}$$\n\nIn the `Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.C` file we compute the gauge terms in $\\partial_{t}A_{i}$, as well as the right-hand side of $\\left[\\sqrt{\\gamma}\\Phi\\right]$, according to equations (16) and (17) of [Etienne *et al*.](https://arxiv.org/pdf/1501.07276.pdf):\n\n$$\n\\begin{align}\n\\partial_{t}A_{i}^{\\rm gauge} &\\equiv \\partial_{t}A_{i} - \\epsilon_{ijk}v^{j}\\tilde{B}^{k} = -\\partial_{i}\\left(\\alpha\\Phi-\\beta^{j}A_{j}\\right)\\ ,\\\\\n\\partial_{t}\\left[\\sqrt{\\gamma}\\Phi\\right] &= -\\partial_{j}\\left(\\alpha\\sqrt{\\gamma}A^{j} - \\beta^{j}\\left[\\sqrt{\\gamma}\\Phi\\right]\\right) - \\xi\\alpha\\left[\\sqrt{\\gamma}\\Phi\\right]\\ .\n\\end{align}\n$$\n\n\n```python\n%%writefile $outdir/Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.C\nstatic inline CCTK_REAL avg(CCTK_REAL f[PLUS2+1][PLUS2+1][PLUS2+1],int imin,int imax, int jmin,int jmax, int kmin,int kmax);\n\nstatic void Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs(const cGH *cctkGH,const int *cctk_lsh,const int *cctk_nghostzones,CCTK_REAL *dX,CCTK_REAL **in_vars,CCTK_REAL *psi6phi,\n                                                           CCTK_REAL *shiftx_iphjphkph,CCTK_REAL *shifty_iphjphkph,CCTK_REAL *shiftz_iphjphkph,\n                                                           CCTK_REAL *alpha_iphjphkph,CCTK_REAL *alpha_Phi_minus_betaj_A_j_iphjphkph,CCTK_REAL *alpha_sqrtg_Ax_interp,\n                                                           CCTK_REAL *alpha_sqrtg_Ay_interp,CCTK_REAL *alpha_sqrtg_Az_interp,\n                                                           CCTK_REAL *psi6phi_rhs,CCTK_REAL *Ax_rhs,CCTK_REAL *Ay_rhs,CCTK_REAL *Az_rhs) {\n  DECLARE_CCTK_PARAMETERS;  \n  /* Compute \\partial_t psi6phi = -\\partial_i (  \\alpha psi^6 A^i - psi6phi \\beta^i)\n   *    (Eq 13 of http://arxiv.org/pdf/1110.4633.pdf), using Lorenz gauge.\n   * Note that the RHS consists of a shift advection term on psi6phi and \n   *    a term depending on the vector potential.\n   * psi6phi is defined at (i+1/2,j+1/2,k+1/2), but instead of reconstructing \n   *    to compute the RHS of \\partial_t psi6phi, we instead use standard\n   *    interpolations.\n   */\n\n  CCTK_REAL dXm1=1.0/dX[0];\n  CCTK_REAL dYm1=1.0/dX[1];\n  CCTK_REAL dZm1=1.0/dX[2];\n```\n\n\n\n## Step 1.a: Performing all necessary interpolations \\[Back to [top](#toc)\\]\n$$\\label{interpolations}$$\n\n\n\n### Step 1.a.i: The interpolation algorithm \\[Back to [top](#toc)\\]\n$$\\label{interpolation_algorithm}$$\n\nIt is important to notice the different staggerrings. We are ultimately interested in the RHS of $\\left[\\sqrt{\\gamma}\\Phi\\right]_{i,j,k}=\\left[\\psi^{6}\\Phi\\right]_{i,j,k}$, which is actually located at $\\left(i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}\\right)$. Remember the staggerings:\n\n|               Quantitity               |              Actual location on the grid               |\n|:--------------------------------------:|:------------------------------------------------------:|\n|            $\\alpha_{i,j,k}$            |                  $\\left(i,j,k\\right)$                  |\n|           $\\psi^{6}_{i,j,k}$           |                  $\\left(i,j,k\\right)$                  |\n|      $\\left(A_{x}\\right)_{i,j,k}$      |      $\\left(i,j+\\frac{1}{2},k+\\frac{1}{2}\\right)$      |\n|      $\\left(A_{y}\\right)_{i,j,k}$      |      $\\left(i+\\frac{1}{2},j,k+\\frac{1}{2}\\right)$      |\n|      $\\left(A_{z}\\right)_{i,j,k}$      |      $\\left(i+\\frac{1}{2},j+\\frac{1}{2},k\\right)$      |\n|$\\left[\\sqrt{\\gamma}\\Phi\\right]_{i,j,k}$|$\\left(i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}\\right)$|\n\nTo obtain the staggerings we use an interpolation routine which averages the value of the gridfunctions around the point of interest. For metric quantities, such as $\\alpha$ and $\\psi$, it is very easy to understand what is happening, so we will use them as the our main example.\n\nBecause we are solving the equations in 3 spatial dimensions, our numerical grid can be seen as a collection of small \"cubes\" whose sizes are $\\left(dx,dy,dz\\right)$ (notice that, while we could have $dx\\neq dy\\neq dz$, we will refer to the grid as cubes to make it easier to describe it). Then, the metric quantities, which are unstaggered, are defined on the *vertices* of the cubes. On the other hand, $\\left[\\sqrt{\\gamma}\\Phi\\right]$ is defined at the *centers* of the cubes. Thus, to obtain metric quantities on the center of the cube, we use a *averaging interpolation algorithm* which  uses all 8 vertices of the cube.\n\nAs a concrete example, we obtain the lapse at the grid location $\\left(i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}\\right)$ by averaging\n\n$$\n\\alpha_{i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}} =\n\\frac{\n\\alpha_{i,j,k}+\n\\alpha_{i+1,j,k}+\n\\alpha_{i,j+1,k}+\n\\alpha_{i,j,k+1}+\n\\alpha_{i+1,j+1,k}+\n\\alpha_{i+1,j,k+1}+\n\\alpha_{i,j+1,k+1}+\n\\alpha_{i+1,j+1,k+1}\n}\n{8}\\ .\n$$\n\nHowever, our algorithm is not restricted to using 8 points when performing such an average. For example, suppose we need the quantity $\\left(A_{x}\\right)_{i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}}$, but we normaly only have $\\left(A_{x}\\right)_{i,j+\\frac{1}{2},k+\\frac{1}{2}}$. The average algorithm would then be\n\n$$\n\\left(A_{x}\\right)_{i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}} = \\frac{\\left(A_{x}\\right)_{i,j+\\frac{1}{2},k+\\frac{1}{2}} + \\left(A_{x}\\right)_{i+1,j+\\frac{1}{2},k+\\frac{1}{2}}}{2}\\ .\n$$\n\n\n\n### Step 1.a.ii: Interpolating gridfunctions and computing $\\left(\\alpha\\Phi-\\beta^{j}A_{j}\\right)$ \\[Back to [top](#toc)\\]\n$$\\label{interp_and_alpha_phi_minus_betaj_A_j}$$\n\nNow, to compute $\\partial_{j}\\left(\\alpha\\psi^{6}A^{j}\\right)$ we need different interpolations for different quantitites. We will focus on the $x$-direction to give a concrete example.\n\nConsider, then, the term\n\n$$\n\\partial_{x}\\left(\\alpha\\psi^{6}A^{x}\\right)\\ .\n$$\n\nAs we know, the second-order, centered finite differences approximation of this equation is\n\n$$\n\\left[\\partial_{x}\\left(\\alpha\\psi^{6}A^{x}\\right)\\right]_{i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}} = \\frac{\\left(\\alpha\\psi^{6}A^{x}\\right)_{i+1,j+\\frac{1}{2},k+\\frac{1}{2}} - \\left(\\alpha\\psi^{6}A^{x}\\right)_{i,j+\\frac{1}{2},k+\\frac{1}{2}}}{dx}\\ ,\n$$\n\nwhere the subscripts indicate ***the actual gridpoint location***. The $A_{x}$ staggering, then, already makes it available at the needed points for the derivative. However, we need to compute $A^{x} = g^{xi}A_{i} = \\gamma^{xx}A_{x} + \\gamma^{xy}A_{y} + \\gamma^{xz}A_{z}$, which means that to compute $A^{x}$ alone we need to interpolate $\\left(\\gamma^{xx},\\gamma^{xy},\\gamma^{xz},A_{y},A_{z}\\right)$ to the grid locations $\\left(i,j+\\frac{1}{2},k+\\frac{1}{2}\\right)$ and $\\left(i+1,j+\\frac{1}{2},k+\\frac{1}{2}\\right)$. On top of that, to be able to compute the derivative, we also need $\\alpha$ and $\\psi^{6}$ at those locations. Finally, once all interpolations have been performed, we can then add the appropriate terms to the RHSs of $\\partial_{t}\\left[\\sqrt{\\gamma}\\Phi\\right]$ and $\\partial_{t}A_{i}$.\n\nThe outline of the algorithm is as follows:\n\n1. Read in gridfunctions at point $(i,j,k)$ and points around that, so that we can perform interpolations.\n1. Interpolate $\\alpha$ to the point $\\left(i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}\\right)$\n1. $A^{x}$ term:\n    1. Interpolate $\\bar{\\gamma}^{xx},\\bar{\\gamma}^{xy},\\bar{\\gamma}^{xz},\\alpha,\\psi$ to the point $\\left(i,j+\\frac{1}{2},k+\\frac{1}{2}\\right)$.\n    1. Interpolate $A_{x}$, $A_{y}$, $A_{z}$ to the point $\\left(i,j+\\frac{1}{2},k+\\frac{1}{2}\\right)$.\n    1. Compute $A^{x}$ at the point $\\left(i,j+\\frac{1}{2},k+\\frac{1}{2}\\right)$.\n    1. Compute $\\alpha\\psi^{6}A^{x}$ at the point $\\left(i,j+\\frac{1}{2},k+\\frac{1}{2}\\right)$.\n1. $A^{y}$ term:\n    1. Interpolate $\\bar{\\gamma}^{xy},\\bar{\\gamma}^{yy},\\bar{\\gamma}^{yz},\\alpha,\\psi$ to the point $\\left(i+\\frac{1}{2},j,k+\\frac{1}{2}\\right)$.\n    1. Interpolate $A_{x}$, $A_{y}$, $A_{z}$ to the point $\\left(i+\\frac{1}{2},j,k+\\frac{1}{2}\\right)$.\n    1. Compute $A^{y}$ at the point $\\left(i+\\frac{1}{2},j,k+\\frac{1}{2}\\right)$.\n    1. Compute $\\alpha\\psi^{6}A^{y}$ at the point $\\left(i+\\frac{1}{2},j,k+\\frac{1}{2}\\right)$.\n1. $A^{z}$ term:\n    1. Interpolate $\\bar{\\gamma}^{xz},\\bar{\\gamma}^{yz},\\bar{\\gamma}^{zz},\\alpha,\\psi$ to the point $\\left(i+\\frac{1}{2},j+\\frac{1}{2},k\\right)$.\n    1. Interpolate $A_{x}$, $A_{y}$, $A_{z}$ to the point $\\left(i+\\frac{1}{2},j+\\frac{1}{2},k\\right)$.\n    1. Compute $A^{z}$ at the point $\\left(i+\\frac{1}{2},j+\\frac{1}{2},k\\right)$.\n    1. Compute $\\alpha\\psi^{6}A^{z}$ at the point $\\left(i+\\frac{1}{2},j+\\frac{1}{2},k\\right)$.\n1. Interpolate $A_{x},A_{y},A_{z}$ to the point $\\left(i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}\\right)$.\n1. Interpolate $\\beta^{x},\\beta^{y},\\beta^{z}$ to the point $\\left(i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}\\right)$.\n1. Compute $\\left(\\alpha\\Phi-\\beta^{j}A_{j}\\right)$ at $\\left(i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}\\right)$.\n\n\n```python\n%%writefile -a $outdir/Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.C\n\n\n  // The stencil here is {-1,1},{-1,1},{-1,1} for x,y,z directions, respectively.\n  //     Note that ALL input variables are defined at ALL gridpoints, so no\n  //     worries about ghostzones.\n#pragma omp parallel for\n  for(int k=1;k\n\n### Step 1.b: Computing $\\partial_{t}A_{i}^{\\rm gauge}$ \\[Back to [top](#toc)\\]\n$$\\label{partial_t_a_i_gauge}$$\n\nNow that we have access to the gridfunctions at all the necessary gridpoints, we proceed to the computation of the gauge terms in $\\partial_{t}A_{i}$, i.e.\n\n$$\n\\partial_{t}A_{i}^{\\rm gauge} = -\\partial_{i}\\left(\\alpha\\Phi-\\beta^{j}A_{j}\\right)\\ ,\n$$\n\nor, more explicitly:\n\n$$\n\\begin{align}\n\\left(\\partial_{t}A_{x}^{\\rm gauge}\\right)_{i,j+\\frac{1}{2},k+\\frac{1}{2}}\n&=\n\\frac{\n\\left(\\alpha\\Phi-\\beta^{j}A_{j}\\right)_{i-\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}} - \\left(\\alpha\\Phi-\\beta^{j}A_{j}\\right)_{i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}}\n}{dx}\\ ,\\\\\n\\left(\\partial_{t}A_{y}^{\\rm gauge}\\right)_{i+\\frac{1}{2},j,k+\\frac{1}{2}}\n&=\n\\frac{\n\\left(\\alpha\\Phi-\\beta^{j}A_{j}\\right)_{i+\\frac{1}{2},j-\\frac{1}{2},k+\\frac{1}{2}} - \\left(\\alpha\\Phi-\\beta^{j}A_{j}\\right)_{i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}}\n}{dy}\\ ,\\\\\n\\left(\\partial_{t}A_{z}^{\\rm gauge}\\right)_{i+\\frac{1}{2},j+\\frac{1}{2},k}\n&=\n\\frac{\n\\left(\\alpha\\Phi-\\beta^{j}A_{j}\\right)_{i-\\frac{1}{2},j+\\frac{1}{2},k-\\frac{1}{2}} - \\left(\\alpha\\Phi-\\beta^{j}A_{j}\\right)_{i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}}\n}{dz}\\ .\n\\end{align}\n$$\n\n\n```python\n%%writefile -a $outdir/Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.C\n\n\n  // This loop requires two additional ghostzones in every direction. Hence the following loop definition:\n#pragma omp parallel for\n  for(int k=cctk_nghostzones[2];k - (A_{i} - A_{i-1})/dX = (A_{i-1} - A_{i})/dX, for Ax\n        Ax_rhs[index] += dXm1*(alpha_Phi_minus_betaj_A_j_iphjphkph[CCTK_GFINDEX3D(cctkGH,i-1,j,k)] - alpha_Phi_minus_betaj_A_j_iphjphkphL);\n        Ay_rhs[index] += dYm1*(alpha_Phi_minus_betaj_A_j_iphjphkph[CCTK_GFINDEX3D(cctkGH,i,j-1,k)] - alpha_Phi_minus_betaj_A_j_iphjphkphL);\n        Az_rhs[index] += dZm1*(alpha_Phi_minus_betaj_A_j_iphjphkph[CCTK_GFINDEX3D(cctkGH,i,j,k-1)] - alpha_Phi_minus_betaj_A_j_iphjphkphL);\n```\n\n    Appending to ../src/Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.C\n\n\n\n\n### Step 1.c: Computing $\\partial_{j}\\beta^{j}\\left[\\sqrt{\\gamma}\\Phi\\right]$ \\[Back to [top](#toc)\\]\n$$\\label{shift_advection_terms}$$\n\nWe now compute the shift-advection terms, $\\partial_{j}\\beta^{j}\\left[\\sqrt{\\gamma}\\Phi\\right]$. For these terms we use [forwards/backwards finite difference stencils](https://en.wikipedia.org/wiki/Finite_difference_coefficient). The sign of $\\beta^{i}$ determines whether we will use forwards or backwards finite difference, according to the following criterion:\n\n| $\\beta^{i}$ | Finite difference |\n|:-----------:|:-----------------:|\n|    $>0$     |     Backward      |\n|    $<0$     |      Forward      |\n\nFor example, assume $\\beta^{x}<0$. Then the term which contains the derivative in the $x$-direction would read\n\n$$\n\\left[\\partial_{x}\\beta^{x}\\left(\\sqrt{\\gamma}\\Phi\\right)\\right]_{i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}} = \n\\frac{\n\\left[\\beta^{x}\\left(\\sqrt{\\gamma}\\Phi\\right)\\right]_{i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}} -\n4\\left[\\beta^{x}\\left(\\sqrt{\\gamma}\\Phi\\right)\\right]_{i-\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}} +\n3\\left[\\beta^{x}\\left(\\sqrt{\\gamma}\\Phi\\right)\\right]_{i-\\frac{3}{2},j+\\frac{1}{2},k+\\frac{1}{2}}\n}\n{2dx}\\ .\n$$\n\nSimilarly, if $\\beta^{y}>0$, then the term which contains the derivative in the $y$-direction would read\n\n$$\n\\left[\\partial_{y}\\beta^{y}\\left(\\sqrt{\\gamma}\\Phi\\right)\\right]_{i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}} = \n\\frac{\n-3\\left[\\beta^{y}\\left(\\sqrt{\\gamma}\\Phi\\right)\\right]_{i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}} +\n4\\left[\\beta^{y}\\left(\\sqrt{\\gamma}\\Phi\\right)\\right]_{i+\\frac{1}{2},j+\\frac{3}{2},k+\\frac{1}{2}} -\n\\left[\\beta^{y}\\left(\\sqrt{\\gamma}\\Phi\\right)\\right]_{i+\\frac{1}{2},j+\\frac{5}{2},k+\\frac{1}{2}}\n}\n{2dy}\\ .\n$$\n\n\n```python\n%%writefile -a $outdir/Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.C\n\n\n        // \\partial_t psi6phi = [shift advection term] + \\partial_j (\\alpha \\sqrt{\\gamma} A^j)\n        // Here we compute [shift advection term] = \\partial_j (\\beta^j psi6phi)\n        // Cache misses are likely more expensive than branch mispredictions here,\n        //       which is why we use if() statements and array lookups inside the if()'s.\n        CCTK_REAL psi6phi_rhsL=0.0;\n        CCTK_REAL psi6phiL=psi6phi[index];\n        CCTK_REAL shiftx_iphjphkphL=shiftx_iphjphkph[index];\n        CCTK_REAL shifty_iphjphkphL=shifty_iphjphkph[index];\n        CCTK_REAL shiftz_iphjphkphL=shiftz_iphjphkph[index];\n\n        // \\partial_x (\\beta^x psi6phi) :\n        if(shiftx_iphjphkphL < 0.0) {\n          psi6phi_rhsL+=0.5*dXm1*(+    shiftx_iphjphkph[CCTK_GFINDEX3D(cctkGH,i-2,j,k)]*psi6phi[CCTK_GFINDEX3D(cctkGH,i-2,j,k)]\n                                  -4.0*shiftx_iphjphkph[CCTK_GFINDEX3D(cctkGH,i-1,j,k)]*psi6phi[CCTK_GFINDEX3D(cctkGH,i-1,j,k)]\n                                  +3.0*shiftx_iphjphkphL*                               psi6phiL);\n        } else {\n          psi6phi_rhsL+=0.5*dXm1*(-    shiftx_iphjphkph[CCTK_GFINDEX3D(cctkGH,i+2,j,k)]*psi6phi[CCTK_GFINDEX3D(cctkGH,i+2,j,k)]\n                                  +4.0*shiftx_iphjphkph[CCTK_GFINDEX3D(cctkGH,i+1,j,k)]*psi6phi[CCTK_GFINDEX3D(cctkGH,i+1,j,k)]\n                                  -3.0*shiftx_iphjphkphL*                               psi6phiL);\n        }\n\n        // \\partial_y (\\beta^y psi6phi) :\n        if(shifty_iphjphkphL < 0.0) {\n          psi6phi_rhsL+=0.5*dYm1*(+    shifty_iphjphkph[CCTK_GFINDEX3D(cctkGH,i,j-2,k)]*psi6phi[CCTK_GFINDEX3D(cctkGH,i,j-2,k)]\n                                  -4.0*shifty_iphjphkph[CCTK_GFINDEX3D(cctkGH,i,j-1,k)]*psi6phi[CCTK_GFINDEX3D(cctkGH,i,j-1,k)]\n                                  +3.0*shifty_iphjphkphL*                               psi6phiL);\n        } else {\n          psi6phi_rhsL+=0.5*dYm1*(-    shifty_iphjphkph[CCTK_GFINDEX3D(cctkGH,i,j+2,k)]*psi6phi[CCTK_GFINDEX3D(cctkGH,i,j+2,k)]\n                                  +4.0*shifty_iphjphkph[CCTK_GFINDEX3D(cctkGH,i,j+1,k)]*psi6phi[CCTK_GFINDEX3D(cctkGH,i,j+1,k)]\n                                  -3.0*shifty_iphjphkphL*                               psi6phiL);\n        }\n \n        // \\partial_z (\\beta^z psi6phi) :\n        if(shiftz_iphjphkphL < 0.0) {\n          psi6phi_rhsL+=0.5*dZm1*(+    shiftz_iphjphkph[CCTK_GFINDEX3D(cctkGH,i,j,k-2)]*psi6phi[CCTK_GFINDEX3D(cctkGH,i,j,k-2)]\n                                  -4.0*shiftz_iphjphkph[CCTK_GFINDEX3D(cctkGH,i,j,k-1)]*psi6phi[CCTK_GFINDEX3D(cctkGH,i,j,k-1)]\n                                  +3.0*shiftz_iphjphkphL*                               psi6phiL);\n        } else {\n          psi6phi_rhsL+=0.5*dZm1*(-    shiftz_iphjphkph[CCTK_GFINDEX3D(cctkGH,i,j,k+2)]*psi6phi[CCTK_GFINDEX3D(cctkGH,i,j,k+2)]\n                                  +4.0*shiftz_iphjphkph[CCTK_GFINDEX3D(cctkGH,i,j,k+1)]*psi6phi[CCTK_GFINDEX3D(cctkGH,i,j,k+1)]\n                                  -3.0*shiftz_iphjphkphL*                               psi6phiL);\n        }\n```\n\n    Appending to ../src/Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.C\n\n\n\n\n### Step 1.d: Computing $-\\partial_{j}\\left(\\alpha\\sqrt{\\gamma}A^{j}\\right)-\\xi\\alpha\\left[\\sqrt{\\gamma}\\Phi\\right]$ \\[Back to [top](#toc)\\]\n$$\\label{partial_j_alpha_psi6_aj}$$\n\nNow we have the simple task of computing the remaining terms on the RHS of $\\partial_{t}\\left[\\sqrt{\\gamma}\\Phi\\right]$. For the derivative term, $\\partial_{j}\\left(\\alpha\\sqrt{\\gamma}A^{j}\\right)$, we can now use ordinary centered finite differences:\n\n$$\n\\begin{align}\n-\\left[\\partial_{j}\\left(\\alpha\\sqrt{\\gamma}A^{j}\\right)\\right]_{i+\\frac{1}{2},j+\\frac{1}{2},k+\\frac{1}{2}}\n&= \\frac{\\left(\\alpha\\sqrt{\\gamma}A^{x}\\right)_{i,j+\\frac{1}{2},k+\\frac{1}{2}}-\\left(\\alpha\\sqrt{\\gamma}A^{x}\\right)_{i+1,j+\\frac{1}{2},k+\\frac{1}{2}}}{dx} \\\\\n&+ \\frac{\\left(\\alpha\\sqrt{\\gamma}A^{y}\\right)_{i+\\frac{1}{2},j,k+\\frac{1}{2}}-\\left(\\alpha\\sqrt{\\gamma}A^{y}\\right)_{i+\\frac{1}{2},j+1,k+\\frac{1}{2}}}{dy} \\\\\n&+ \\frac{\\left(\\alpha\\sqrt{\\gamma}A^{z}\\right)_{i+\\frac{1}{2},j+\\frac{1}{2},k}-\\left(\\alpha\\sqrt{\\gamma}A^{z}\\right)_{i+\\frac{1}{2},j+\\frac{1}{2},k+1}}{dz}\n\\end{align}\n$$\n\nThe [*generalized Lorenz gauge*](https://arxiv.org/pdf/1207.3354.pdf) term, $\\xi\\alpha\\left[\\sqrt{\\gamma}\\Phi\\right]$, is then trivially implemented.\n\n\n\n```python\n%%writefile -a $outdir/Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.C\n\n\n        // Next we add \\partial_j (\\alpha \\sqrt{\\gamma} A^j) to \\partial_t psi6phi:\n        psi6phi_rhsL+=dXm1*(alpha_sqrtg_Ax_interp[index] - alpha_sqrtg_Ax_interp[CCTK_GFINDEX3D(cctkGH,i+1,j,k)])\n          +           dYm1*(alpha_sqrtg_Ay_interp[index] - alpha_sqrtg_Ay_interp[CCTK_GFINDEX3D(cctkGH,i,j+1,k)])\n          +           dZm1*(alpha_sqrtg_Az_interp[index] - alpha_sqrtg_Az_interp[CCTK_GFINDEX3D(cctkGH,i,j,k+1)]);\n\n        // *GENERALIZED* LORENZ GAUGE: \n        // Finally, add damping factor to \\partial_t psi6phi\n        //subtract lambda * alpha psi^6 Phi\n        psi6phi_rhsL+=-damp_lorenz*alpha_iphjphkph[index]*psi6phiL;\n\n        psi6phi_rhs[index] = psi6phi_rhsL;\n      }\n}\n\n\n```\n\n    Appending to ../src/Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.C\n\n\n\n\n### Step 1.e: The `avg()` function \\[Back to [top](#toc)\\]\n$$\\label{fct_avg}$$\n\nThis is the implementation of the algorithm we discussed in [step 1.a.i](#interpolation_algorithm).\n\n\n```python\n%%writefile -a $outdir/Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.C\n\nstatic inline CCTK_REAL avg(CCTK_REAL f[PLUS2+1][PLUS2+1][PLUS2+1],int imin,int imax, int jmin,int jmax, int kmin,int kmax) {\n  CCTK_REAL retval=0.0,num_in_sum=0.0;\n  for(int kk=kmin;kk<=kmax;kk++) for(int jj=jmin;jj<=jmax;jj++) for(int ii=imin;ii<=imax;ii++) {\n        retval+=f[kk][jj][ii]; num_in_sum++;\n      }\n  return retval/num_in_sum;\n}\n```\n\n    Appending to ../src/Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.C\n\n\n\n\n# Step 2: Code validation \\[Back to [top](#toc)\\]\n$$\\label{code_validation}$$\n\nFirst we download the original `IllinoisGRMHD` source code and then compare it to the source code generated by this tutorial notebook.\n\n\n```python\n# # Verify if the code generated by this tutorial module\n# # matches the original IllinoisGRMHD source code\n\n# # First download the original IllinoisGRMHD source code\n# import urllib\n# from os import path\n\n# original_IGM_file_url  = \"https://bitbucket.org/zach_etienne/wvuthorns/raw/5611b2f0b17135538c9d9d17c7da062abe0401b6/IllinoisGRMHD/src/Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.C\"\n# original_IGM_file_name = \"Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs-original.C\"\n# original_IGM_file_path = os.path.join(IGM_src_dir_path,original_IGM_file_name)\n\n# # Then download the original IllinoisGRMHD source code\n# # We try it here in a couple of ways in an attempt to keep\n# # the code more portable\n# try:\n#     original_IGM_file_code = urllib.request.urlopen(original_IGM_file_url).read().decode(\"utf-8\")\n#     # Write down the file the original IllinoisGRMHD source code\n#     with open(original_IGM_file_path,\"w\") as file:\n#         file.write(original_IGM_file_code)\n# except:\n#     try:\n#         original_IGM_file_code = urllib.urlopen(original_IGM_file_url).read().decode(\"utf-8\")\n#         # Write down the file the original IllinoisGRMHD source code\n#         with open(original_IGM_file_path,\"w\") as file:\n#             file.write(original_IGM_file_code)\n#     except:\n#         # If all else fails, hope wget does the job\n#         !wget -O $original_IGM_file_path $original_IGM_file_url\n\n# # Perform validation\n# Validation__Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs__C  = !diff $original_IGM_file_path $outfile_path__Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs__C\n\n# if Validation__Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs__C == []:\n#     # If the validation passes, we do not need to store the original IGM source code file\n#     !rm $original_IGM_file_path\n#     print(\"Validation test for Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.C: PASSED!\")\n# else:\n#     # If the validation fails, we keep the original IGM source code file\n#     print(\"Validation test for Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.C: FAILED!\")\n#     # We also print out the difference between the code generated\n#     # in this tutorial module and the original IGM source code\n#     print(\"Diff:\")\n#     for diff_line in Validation__Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs__C:\n#         print(diff_line)\n```\n\n\n\n# Step 3: Output this notebook to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n$$\\label{latex_pdf_output}$$\n\nThe following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n[Tutorial-IllinoisGRMHD__Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.pdf](Tutorial-IllinoisGRMHD__Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means).\n\n\n```python\nlatex_nrpy_style_path = os.path.join(nrpy_dir_path,\"latex_nrpy_style.tplx\")\n#!jupyter nbconvert --to latex --template $latex_nrpy_style_path --log-level='WARN' Tutorial-IllinoisGRMHD__Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.ipynb\n#!pdflatex -interaction=batchmode Tutorial-IllinoisGRMHD__Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.tex\n#!pdflatex -interaction=batchmode Tutorial-IllinoisGRMHD__Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.tex\n#!pdflatex -interaction=batchmode Tutorial-IllinoisGRMHD__Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.tex\n!rm -f Tut*.out Tut*.aux Tut*.log\n```\n", "meta": {"hexsha": "051009134495d378885ec07f9301abd03ac9d330", "size": 45173, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "IllinoisGRMHD/doc/Tutorial-IllinoisGRMHD__Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.ipynb", "max_stars_repo_name": "leowerneck/NRPyIGM", "max_stars_repo_head_hexsha": "f483d6123424fb3e6860dfac4325dd232b223005", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "IllinoisGRMHD/doc/Tutorial-IllinoisGRMHD__Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.ipynb", "max_issues_repo_name": "leowerneck/NRPyIGM", "max_issues_repo_head_hexsha": "f483d6123424fb3e6860dfac4325dd232b223005", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "IllinoisGRMHD/doc/Tutorial-IllinoisGRMHD__Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.ipynb", "max_forks_repo_name": "leowerneck/NRPyIGM", "max_forks_repo_head_hexsha": "f483d6123424fb3e6860dfac4325dd232b223005", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 59.2044560944, "max_line_length": 841, "alphanum_fraction": 0.5770482368, "converted": true, "num_tokens": 13091, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "##### Copyright 2020 The TensorFlow Authors.\n\n\n```\n#@title Licensed under the Apache License, Version 2.0 (the \"License\");\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n# https://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\n```\n\n# \u91cf\u5b50\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\n\n\n  \n  \n  \n  \n
                                        \u5728 TensorFlow.org \u4e0a\u67e5\u770b\u5728 Google Colab \u4e2d\u8fd0\u884c\u5728 GitHub \u4e0a\u67e5\u770b\u6e90\u4ee3\u7801\u4e0b\u8f7d\u7b14\u8bb0\u672c
                                        \n\n\u672c\u6559\u7a0b\u4ecb\u7ecd\u5982\u4f55\u5b9e\u73b0\u4e00\u4e2a\u7b80\u5316\u7684\u91cf\u5b50\u5377\u79ef\u795e\u7ecf\u7f51\u7edc (QCNN)\uff0c\u5373\u5bf9\u540c\u6837\u5177\u6709*\u5e73\u79fb\u4e0d\u53d8\u6027*\u7684\u7ecf\u5178\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u7684\u63d0\u8bae\u91cf\u5b50\u6a21\u62df\u3002\n\n\u672c\u793a\u4f8b\u6f14\u793a\u5982\u4f55\u68c0\u6d4b\u91cf\u5b50\u6570\u636e\u6e90\u7684\u67d0\u4e9b\u5c5e\u6027\uff0c\u4f8b\u5982\u8bbe\u5907\u7684\u91cf\u5b50\u4f20\u611f\u5668\u6216\u590d\u6742\u6a21\u62df\u3002\u91cf\u5b50\u6570\u636e\u6e90\u662f\u53ef\u80fd\u6709\u6216\u53ef\u80fd\u6ca1\u6709\u6fc0\u53d1\u7684\u7c07\u6001\uff0c\u540e\u8005\u662f QCNN \u5c06\u5b66\u4e60\u68c0\u6d4b\u7684\u5bf9\u8c61\uff08\u8bba\u6587\u4e2d\u4f7f\u7528\u7684\u6570\u636e\u96c6\u662f SPT \u9636\u6bb5\u5206\u7c7b\uff09\u3002\n\n## \u8bbe\u7f6e\n\n\n```\n!pip install tensorflow==2.4.1\n```\n\n\u5b89\u88c5 TensorFlow Quantum\uff1a\n\n\n```\n!pip install tensorflow-quantum\n```\n\n\n```\n# Update package resources to account for version changes.\nimport importlib, pkg_resources\nimportlib.reload(pkg_resources)\n```\n\n\u73b0\u5728\uff0c\u5bfc\u5165 TensorFlow \u548c\u6a21\u5757\u4f9d\u8d56\u9879\uff1a\n\n\n```\nimport tensorflow as tf\nimport tensorflow_quantum as tfq\n\nimport cirq\nimport sympy\nimport numpy as np\n\n# visualization tools\n%matplotlib inline\nimport matplotlib.pyplot as plt\nfrom cirq.contrib.svg import SVGCircuit\n```\n\n## 1. \u6784\u5efa QCNN\n\n### 1.1 \u5728 TensorFlow \u8ba1\u7b97\u56fe\u4e2d\u7ec4\u88c5\u7535\u8def\n\nTensorFlow Quantum (TFQ) \u63d0\u4f9b\u4e86\u4e13\u4e3a\u8ba1\u7b97\u56fe\u4e2d\u7684\u7535\u8def\u6784\u9020\u800c\u8bbe\u8ba1\u7684\u5c42\u7c7b\u3002\u4e00\u4e2a\u793a\u4f8b\u662f\u4ece `tf.keras.Layer` \u7ee7\u627f\u7684 `tfq.layers.AddCircuit` \u5c42\u3002\u6b64\u5c42\u53ef\u4ee5\u8ffd\u52a0\u6216\u9644\u52a0\u5230\u7535\u8def\u7684\u8f93\u5165\u6279\u6b21\uff0c\u5982\u4e0b\u56fe\u6240\u793a\u3002\n\n\n\n\u4e0b\u9762\u7684\u4ee3\u7801\u6bb5\u4f7f\u7528\u4e86\u6b64\u5c42\uff1a\n\n\n```\nqubit = cirq.GridQubit(0, 0)\n\n# Define some circuits.\ncircuit1 = cirq.Circuit(cirq.X(qubit))\ncircuit2 = cirq.Circuit(cirq.H(qubit))\n\n# Convert to a tensor.\ninput_circuit_tensor = tfq.convert_to_tensor([circuit1, circuit2])\n\n# Define a circuit that we want to append\ny_circuit = cirq.Circuit(cirq.Y(qubit))\n\n# Instantiate our layer\ny_appender = tfq.layers.AddCircuit()\n\n# Run our circuit tensor through the layer and save the output.\noutput_circuit_tensor = y_appender(input_circuit_tensor, append=y_circuit)\n```\n\n\u68c0\u67e5\u8f93\u5165\u5f20\u91cf\uff1a\n\n\n```\nprint(tfq.from_tensor(input_circuit_tensor))\n```\n\n\u68c0\u67e5\u8f93\u51fa\u5f20\u91cf\uff1a\n\n\n```\nprint(tfq.from_tensor(output_circuit_tensor))\n```\n\n\u867d\u7136\u4e0d\u4f7f\u7528 `tfq.layers.AddCircuit` \u4e5f\u53ef\u4ee5\u8fd0\u884c\u4e0b\u9762\u7684\u793a\u4f8b\uff0c\u4f46\u8fd9\u662f\u4e00\u4e2a\u7406\u89e3\u5982\u4f55\u5c06\u590d\u6742\u7684\u529f\u80fd\u5d4c\u5165 TensorFlow \u8ba1\u7b97\u56fe\u7684\u597d\u673a\u4f1a\u3002\n\n### 1.2 \u95ee\u9898\u6982\u8ff0\n\n\u60a8\u5c06\u51c6\u5907*\u7c07\u6001*\uff0c\u5e76\u8bad\u7ec3\u4e00\u4e2a\u91cf\u5b50\u5206\u7c7b\u5668\u6765\u68c0\u6d4b\u5b83\u662f\u5426\u5904\u4e8e\u201c\u6fc0\u53d1\u201d\u72b6\u6001\u3002\u7c07\u6001\u662f\u9ad8\u5ea6\u7ea0\u7f20\u7684\uff0c\u4e0d\u8fc7\uff0c\u8fd9\u5bf9\u7ecf\u5178\u8ba1\u7b97\u673a\u800c\u8a00\u5e76\u975e\u96be\u4ee5\u89e3\u51b3\u7684\u95ee\u9898\u3002\u4e3a\u4e86\u8ba9\u60a8\u66f4\u6e05\u695a\u5730\u7406\u89e3\uff0c\u6211\u4eec\u4f7f\u7528\u7684\u8fd9\u4e00\u6570\u636e\u96c6\u6bd4\u8bba\u6587\u4e2d\u4f7f\u7528\u7684\u66f4\u7b80\u5355\u3002\n\n\u5bf9\u4e8e\u6b64\u5206\u7c7b\u4efb\u52a1\uff0c\u60a8\u5c06\u5b9e\u73b0\u4e00\u4e2a\u7c7b\u4f3c\u6df1\u5ea6 MERA \u7684 QCNN \u67b6\u6784\uff0c\u56e0\u4e3a\uff1a\n\n1. \u5c31\u50cf QCNN \u4e00\u6837\uff0c\u73af\u4e0a\u7684\u7c07\u6001\u5177\u6709\u5e73\u79fb\u4e0d\u53d8\u6027\u3002\n2. \u7c07\u6001\u662f\u9ad8\u5ea6\u7ea0\u7f20\u7684\u3002\n\n\u8fd9\u79cd\u67b6\u6784\u5728\u51cf\u5c11\u7ea0\u7f20\u65b9\u9762\u5e94\u8be5\u4f1a\u5f88\u6709\u6548\uff0c\u901a\u8fc7\u8bfb\u51fa\u4e00\u4e2a\u91cf\u5b50\u4f4d\u6765\u83b7\u5f97\u5206\u7c7b\u3002\n\n\n\n\u6839\u636e\u5b9a\u4e49\uff0c\u201c\u6fc0\u53d1\u201d\u7c07\u6001\u662f\u6307\u5c06 `cirq.rx` \u95e8\u5e94\u7528\u5230\u5176\u4efb\u4f55\u91cf\u5b50\u4f4d\u7684\u7c07\u6001\u3002Qconv \u548c QPool \u5728\u672c\u6559\u7a0b\u7684\u540e\u7eed\u90e8\u5206\u8ba8\u8bba\u3002\n\n### 1.3 \u4e3a TensorFlow \u6784\u5efa\u5757\n\n\n\n\u4f7f\u7528 TensorFlow Quantum \u89e3\u51b3\u6b64\u95ee\u9898\u7684\u4e00\u79cd\u65b9\u5f0f\u662f\u5b9e\u73b0\u4ee5\u4e0b\u51e0\u70b9\uff1a\n\n1. \u6a21\u578b\u7684\u8f93\u5165\u662f\u4e00\u4e2a\u7535\u8def\u5f20\u91cf\uff0c\u7a7a\u7535\u8def\u6216\u8868\u660e\u6fc0\u53d1\u7684\u7279\u5b9a\u91cf\u5b50\u4f4d\u4e0a\u7684 X \u95e8\u3002\n2. \u4f7f\u7528 `tfq.layers.AddCircuit` \u5c42\u6784\u9020\u6a21\u578b\u7684\u5176\u4ed6\u91cf\u5b50\u7ec4\u4ef6\u3002\n3. \u4f7f\u7528 `tfq.layers.PQC` \u5c42\u8fdb\u884c\u63a8\u65ad\u3002\u5b83\u4f1a\u8bfb\u53d6 $\\langle \\hat{Z} \\rangle$\uff0c\u5e76\u5c06\u5176\u4e0e\u6fc0\u53d1\u6001\u7684\u6807\u7b7e 1 \u6216\u975e\u6fc0\u53d1\u6001\u7684\u6807\u7b7e -1 \u8fdb\u884c\u6bd4\u8f83\u3002\n\n### 1.4 \u6570\u636e\n\n\u5728\u6784\u5efa\u6a21\u578b\u4e4b\u524d\uff0c\u60a8\u53ef\u4ee5\u751f\u6210\u6570\u636e\u3002\u5728\u672c\u4f8b\u4e2d\uff0c\u5b83\u5c06\u662f\u7c07\u6001\u7684\u6fc0\u53d1\uff08\u539f\u8bba\u6587\u4f7f\u7528\u7684\u662f\u4e00\u4e2a\u66f4\u590d\u6742\u7684\u6570\u636e\u96c6\uff09\u3002\u6fc0\u53d1\u901a\u8fc7 `cirq.rx` \u95e8\u8868\u793a\u3002\u6211\u4eec\u5c06\u8db3\u591f\u5927\u7684\u65cb\u8f6c\u89c6\u4e3a\u6fc0\u53d1\uff0c\u5e76\u4f7f\u7528 `1` \u8fdb\u884c\u6807\u8bb0\uff0c\u4e0d\u591f\u5927\u7684\u65cb\u8f6c\u5219\u4f7f\u7528 `-1` \u6807\u8bb0\uff0c\u6211\u4eec\u5c06\u5176\u89c6\u4e3a\u672a\u6fc0\u53d1\u3002\n\n\n```\ndef generate_data(qubits):\n    \"\"\"Generate training and testing data.\"\"\"\n    n_rounds = 20  # Produces n_rounds * n_qubits datapoints.\n    excitations = []\n    labels = []\n    for n in range(n_rounds):\n        for bit in qubits:\n            rng = np.random.uniform(-np.pi, np.pi)\n            excitations.append(cirq.Circuit(cirq.rx(rng)(bit)))\n            labels.append(1 if (-np.pi / 2) <= rng <= (np.pi / 2) else -1)\n\n    split_ind = int(len(excitations) * 0.7)\n    train_excitations = excitations[:split_ind]\n    test_excitations = excitations[split_ind:]\n\n    train_labels = labels[:split_ind]\n    test_labels = labels[split_ind:]\n\n    return tfq.convert_to_tensor(train_excitations), np.array(train_labels), \\\n        tfq.convert_to_tensor(test_excitations), np.array(test_labels)\n```\n\n\u60a8\u53ef\u4ee5\u770b\u5230\uff0c\u5c31\u50cf\u4f7f\u7528\u5e38\u89c4\u7684\u673a\u5668\u5b66\u4e60\u4e00\u6837\uff0c\u60a8\u521b\u5efa\u4e86\u4e00\u4e2a\u7528\u4e8e\u5bf9\u6a21\u578b\u8fdb\u884c\u57fa\u51c6\u6d4b\u8bd5\u7684\u8bad\u7ec3\u548c\u6d4b\u8bd5\u96c6\u3002\u5229\u7528\u4ee5\u4e0b\u4ee3\u7801\u6bb5\uff0c\u60a8\u53ef\u4ee5\u5feb\u901f\u67e5\u770b\u67d0\u4e9b\u6570\u636e\u70b9\uff1a\n\n\n```\nsample_points, sample_labels, _, __ = generate_data(cirq.GridQubit.rect(1, 4))\nprint('Input:', tfq.from_tensor(sample_points)[0], 'Output:', sample_labels[0])\nprint('Input:', tfq.from_tensor(sample_points)[1], 'Output:', sample_labels[1])\n```\n\n### 1.5 \u5b9a\u4e49\u5c42\n\n\u73b0\u5728\uff0c\u6211\u4eec\u5728 TensorFlow \u4e2d\u5b9a\u4e49\u4e0a\u56fe\u4e2d\u663e\u793a\u7684\u5c42\u3002\n\n#### 1.5.1 \u7c07\u6001\n\n\u7b2c\u4e00\u6b65\u662f\u4f7f\u7528 Cirq\uff08Google \u4e3a\u91cf\u5b50\u7535\u8def\u7f16\u7a0b\u63d0\u4f9b\u7684\u6846\u67b6\uff09\u5b9a\u4e49\u7c07\u6001\u3002\u7531\u4e8e\u8fd9\u662f\u6a21\u578b\u7684\u4e00\u4e2a\u9759\u6001\u90e8\u5206\uff0c\u56e0\u6b64\u4f7f\u7528 `tfq.layers.AddCircuit` \u529f\u80fd\u5c06\u5176\u5d4c\u5165\u3002\n\n\n```\ndef cluster_state_circuit(bits):\n    \"\"\"Return a cluster state on the qubits in `bits`.\"\"\"\n    circuit = cirq.Circuit()\n    circuit.append(cirq.H.on_each(bits))\n    for this_bit, next_bit in zip(bits, bits[1:] + [bits[0]]):\n        circuit.append(cirq.CZ(this_bit, next_bit))\n    return circuit\n```\n\n\u663e\u793a cirq.GridQubit \u7684\u77e9\u5f62\u7684\u4e00\u4e2a\u7c07\u6001\u7535\u8def\uff1a\n\n\n```\nSVGCircuit(cluster_state_circuit(cirq.GridQubit.rect(1, 4)))\n```\n\n#### 1.5.2 QCNN \u5c42\n\n\u6309\u7167 Cong \u548c Lukin QCNN \u8bba\u6587\u5b9a\u4e49\u7ec4\u6210\u6a21\u578b\u7684\u5c42\u3002\u8fd9\u9700\u8981\u5177\u5907\u51e0\u4e2a\u524d\u63d0\u6761\u4ef6\uff1a\n\n- Tucci \u8bba\u6587\u4e2d\u63d0\u51fa\u7684\u5355\u6216\u53cc\u91cf\u5b50\u4f4d\u53c2\u6570\u5316\u9149\u77e9\u9635\u3002\n- \u4e00\u4e2a\u901a\u7528\u7684\u53c2\u6570\u5316\u53cc\u91cf\u5b50\u4f4d\u6c60\u5316\u8fd0\u7b97\u3002\n\n\n```\ndef one_qubit_unitary(bit, symbols):\n    \"\"\"Make a Cirq circuit enacting a rotation of the bloch sphere about the X,\n    Y and Z axis, that depends on the values in `symbols`.\n    \"\"\"\n    return cirq.Circuit(\n        cirq.X(bit)**symbols[0],\n        cirq.Y(bit)**symbols[1],\n        cirq.Z(bit)**symbols[2])\n\n\ndef two_qubit_unitary(bits, symbols):\n    \"\"\"Make a Cirq circuit that creates an arbitrary two qubit unitary.\"\"\"\n    circuit = cirq.Circuit()\n    circuit += one_qubit_unitary(bits[0], symbols[0:3])\n    circuit += one_qubit_unitary(bits[1], symbols[3:6])\n    circuit += [cirq.ZZ(*bits)**symbols[6]]\n    circuit += [cirq.YY(*bits)**symbols[7]]\n    circuit += [cirq.XX(*bits)**symbols[8]]\n    circuit += one_qubit_unitary(bits[0], symbols[9:12])\n    circuit += one_qubit_unitary(bits[1], symbols[12:])\n    return circuit\n\n\ndef two_qubit_pool(source_qubit, sink_qubit, symbols):\n    \"\"\"Make a Cirq circuit to do a parameterized 'pooling' operation, which\n    attempts to reduce entanglement down from two qubits to just one.\"\"\"\n    pool_circuit = cirq.Circuit()\n    sink_basis_selector = one_qubit_unitary(sink_qubit, symbols[0:3])\n    source_basis_selector = one_qubit_unitary(source_qubit, symbols[3:6])\n    pool_circuit.append(sink_basis_selector)\n    pool_circuit.append(source_basis_selector)\n    pool_circuit.append(cirq.CNOT(control=source_qubit, target=sink_qubit))\n    pool_circuit.append(sink_basis_selector**-1)\n    return pool_circuit\n```\n\n\u8981\u67e5\u770b\u60a8\u521b\u5efa\u7684\u5bf9\u8c61\uff0c\u8bf7\u6253\u5370\u51fa\u5355\u91cf\u5b50\u4f4d\u9149\u7535\u8def\uff1a\n\n\n```\nSVGCircuit(one_qubit_unitary(cirq.GridQubit(0, 0), sympy.symbols('x0:3')))\n```\n\n\u4ee5\u53ca\u53cc\u91cf\u5b50\u4f4d\u9149\u7535\u8def\uff1a\n\n\n```\nSVGCircuit(two_qubit_unitary(cirq.GridQubit.rect(1, 2), sympy.symbols('x0:15')))\n```\n\n\u4ee5\u53ca\u53cc\u91cf\u5b50\u4f4d\u6c60\u5316\u7535\u8def\uff1a\n\n\n```\nSVGCircuit(two_qubit_pool(*cirq.GridQubit.rect(1, 2), sympy.symbols('x0:6')))\n```\n\n##### 1.5.2.1 \u91cf\u5b50\u5377\u79ef\n\n\u6309\u7167 Cong \u548c Lukin \u7684\u8bba\u6587\u6240\u8ff0\uff0c\u5c06\u4e00\u7ef4\u91cf\u5b50\u5377\u79ef\u5b9a\u4e49\u4e3a\u5bf9\u6bcf\u5bf9\u6b65\u957f\u4e3a 1 \u7684\u76f8\u90bb\u91cf\u5b50\u4f4d\u7684\u53cc\u91cf\u5b50\u4f4d\u53c2\u6570\u5316\u9149\u7684\u5e94\u7528\u3002\n\n\n```\ndef quantum_conv_circuit(bits, symbols):\n    \"\"\"Quantum Convolution Layer following the above diagram.\n    Return a Cirq circuit with the cascade of `two_qubit_unitary` applied\n    to all pairs of qubits in `bits` as in the diagram above.\n    \"\"\"\n    circuit = cirq.Circuit()\n    for first, second in zip(bits[0::2], bits[1::2]):\n        circuit += two_qubit_unitary([first, second], symbols)\n    for first, second in zip(bits[1::2], bits[2::2] + [bits[0]]):\n        circuit += two_qubit_unitary([first, second], symbols)\n    return circuit\n```\n\n\u663e\u793a\uff08\u9ad8\u5ea6\u6c34\u5e73\u7684\uff09\u7535\u8def\uff1a\n\n\n```\nSVGCircuit(\n    quantum_conv_circuit(cirq.GridQubit.rect(1, 8), sympy.symbols('x0:15')))\n```\n\n##### 1.5.2.2 \u91cf\u5b50\u6c60\u5316\n\n\u91cf\u5b50\u6c60\u5316\u5c42\u4f7f\u7528\u4e0a\u9762\u5b9a\u4e49\u7684\u53cc\u91cf\u5b50\u4f4d\u6c60\u4ece $N$ \u4e2a\u91cf\u5b50\u4f4d\u6c60\u5316\u4e3a $\\frac{N}{2}$ \u4e2a\u91cf\u5b50\u4f4d\u3002\n\n\n```\ndef quantum_pool_circuit(source_bits, sink_bits, symbols):\n    \"\"\"A layer that specifies a quantum pooling operation.\n    A Quantum pool tries to learn to pool the relevant information from two\n    qubits onto 1.\n    \"\"\"\n    circuit = cirq.Circuit()\n    for source, sink in zip(source_bits, sink_bits):\n        circuit += two_qubit_pool(source, sink, symbols)\n    return circuit\n```\n\n\u68c0\u67e5\u6c60\u5316\u7ec4\u4ef6\u7535\u8def\uff1a\n\n\n```\ntest_bits = cirq.GridQubit.rect(1, 8)\n\nSVGCircuit(\n    quantum_pool_circuit(test_bits[:4], test_bits[4:], sympy.symbols('x0:6')))\n```\n\n### 1.6 \u6a21\u578b\u5b9a\u4e49\n\n\u73b0\u5728\uff0c\u4f7f\u7528\u5b9a\u4e49\u7684\u5c42\u6784\u9020\u7eaf\u91cf\u5b50 CNN\u3002\u9996\u5148\u521b\u5efa\u516b\u4e2a\u91cf\u5b50\u4f4d\uff0c\u518d\u5c06\u5176\u6c60\u5316\u4e3a\u4e00\u4e2a\u91cf\u5b50\u4f4d\uff0c\u7136\u540e\u6d4b\u91cf $\\langle \\hat{Z} \\rangle$\u3002\n\n\n```\ndef create_model_circuit(qubits):\n    \"\"\"Create sequence of alternating convolution and pooling operators \n    which gradually shrink over time.\"\"\"\n    model_circuit = cirq.Circuit()\n    symbols = sympy.symbols('qconv0:63')\n    # Cirq uses sympy.Symbols to map learnable variables. TensorFlow Quantum\n    # scans incoming circuits and replaces these with TensorFlow variables.\n    model_circuit += quantum_conv_circuit(qubits, symbols[0:15])\n    model_circuit += quantum_pool_circuit(qubits[:4], qubits[4:],\n                                          symbols[15:21])\n    model_circuit += quantum_conv_circuit(qubits[4:], symbols[21:36])\n    model_circuit += quantum_pool_circuit(qubits[4:6], qubits[6:],\n                                          symbols[36:42])\n    model_circuit += quantum_conv_circuit(qubits[6:], symbols[42:57])\n    model_circuit += quantum_pool_circuit([qubits[6]], [qubits[7]],\n                                          symbols[57:63])\n    return model_circuit\n\n\n# Create our qubits and readout operators in Cirq.\ncluster_state_bits = cirq.GridQubit.rect(1, 8)\nreadout_operators = cirq.Z(cluster_state_bits[-1])\n\n# Build a sequential model enacting the logic in 1.3 of this notebook.\n# Here you are making the static cluster state prep as a part of the AddCircuit and the\n# \"quantum datapoints\" are coming in the form of excitation\nexcitation_input = tf.keras.Input(shape=(), dtype=tf.dtypes.string)\ncluster_state = tfq.layers.AddCircuit()(\n    excitation_input, prepend=cluster_state_circuit(cluster_state_bits))\n\nquantum_model = tfq.layers.PQC(create_model_circuit(cluster_state_bits),\n                               readout_operators)(cluster_state)\n\nqcnn_model = tf.keras.Model(inputs=[excitation_input], outputs=[quantum_model])\n\n# Show the keras plot of the model\ntf.keras.utils.plot_model(qcnn_model,\n                          show_shapes=True,\n                          show_layer_names=False,\n                          dpi=70)\n```\n\n### 1.7 \u8bad\u7ec3\u6a21\u578b\n\n\u5728\u6574\u4e2a\u6279\u6b21\u4e0a\u8bad\u7ec3\u6a21\u578b\u4ee5\u7b80\u5316\u6b64\u793a\u4f8b\u3002\n\n\n```\n# Generate some training data.\ntrain_excitations, train_labels, test_excitations, test_labels = generate_data(\n    cluster_state_bits)\n\n\n# Custom accuracy metric.\n@tf.function\ndef custom_accuracy(y_true, y_pred):\n    y_true = tf.squeeze(y_true)\n    y_pred = tf.map_fn(lambda x: 1.0 if x >= 0 else -1.0, y_pred)\n    return tf.keras.backend.mean(tf.keras.backend.equal(y_true, y_pred))\n\n\nqcnn_model.compile(optimizer=tf.keras.optimizers.Adam(learning_rate=0.02),\n                   loss=tf.losses.mse,\n                   metrics=[custom_accuracy])\n\nhistory = qcnn_model.fit(x=train_excitations,\n                         y=train_labels,\n                         batch_size=16,\n                         epochs=25,\n                         verbose=1,\n                         validation_data=(test_excitations, test_labels))\n```\n\n\n```\nplt.plot(history.history['loss'][1:], label='Training')\nplt.plot(history.history['val_loss'][1:], label='Validation')\nplt.title('Training a Quantum CNN to Detect Excited Cluster States')\nplt.xlabel('Epochs')\nplt.ylabel('Loss')\nplt.legend()\nplt.show()\n```\n\n## 2. \u6df7\u5408\u6a21\u578b\n\n\u60a8\u4e0d\u5fc5\u4f7f\u7528\u91cf\u5b50\u5377\u79ef\u5c06\u516b\u4e2a\u91cf\u5b50\u4f4d\u6c60\u5316\u4e3a\u4e00\u4e2a\u91cf\u5b50\u4f4d\uff0c\u60a8\u53ef\u4ee5\u6267\u884c\u4e00\u5230\u4e24\u8f6e\u7684\u91cf\u5b50\u5377\u79ef\uff0c\u7136\u540e\u5c06\u7ed3\u679c\u9988\u9001\u5230\u7ecf\u5178\u795e\u7ecf\u7f51\u7edc\u4e2d\u3002\u672c\u90e8\u5206\u63a2\u8ba8\u91cf\u5b50-\u7ecf\u5178\u6df7\u5408\u6a21\u578b\u3002\n\n### 2.1 \u4f7f\u7528\u5355\u4e2a\u91cf\u5b50\u6ee4\u6ce2\u5668\u7684\u6df7\u5408\u6a21\u578b\n\n\u5e94\u7528\u4e00\u5c42\u91cf\u5b50\u5377\u79ef\uff0c\u5728\u540e\u8ddf\u5bc6\u96c6\u8fde\u63a5\u7684\u795e\u7ecf\u7f51\u7edc\u7684\u6240\u6709\u4f4d\u4e0a\u8bfb\u51fa $\\langle \\hat{Z}_n \\rangle$\u3002\n\n\n\n#### 2.1.1 \u6a21\u578b\u5b9a\u4e49\n\n\n```\n# 1-local operators to read out\nreadouts = [cirq.Z(bit) for bit in cluster_state_bits[4:]]\n\n\ndef multi_readout_model_circuit(qubits):\n    \"\"\"Make a model circuit with less quantum pool and conv operations.\"\"\"\n    model_circuit = cirq.Circuit()\n    symbols = sympy.symbols('qconv0:21')\n    model_circuit += quantum_conv_circuit(qubits, symbols[0:15])\n    model_circuit += quantum_pool_circuit(qubits[:4], qubits[4:],\n                                          symbols[15:21])\n    return model_circuit\n\n\n# Build a model enacting the logic in 2.1 of this notebook.\nexcitation_input_dual = tf.keras.Input(shape=(), dtype=tf.dtypes.string)\n\ncluster_state_dual = tfq.layers.AddCircuit()(\n    excitation_input_dual, prepend=cluster_state_circuit(cluster_state_bits))\n\nquantum_model_dual = tfq.layers.PQC(\n    multi_readout_model_circuit(cluster_state_bits),\n    readouts)(cluster_state_dual)\n\nd1_dual = tf.keras.layers.Dense(8)(quantum_model_dual)\n\nd2_dual = tf.keras.layers.Dense(1)(d1_dual)\n\nhybrid_model = tf.keras.Model(inputs=[excitation_input_dual], outputs=[d2_dual])\n\n# Display the model architecture\ntf.keras.utils.plot_model(hybrid_model,\n                          show_shapes=True,\n                          show_layer_names=False,\n                          dpi=70)\n```\n\n#### 2.1.2 \u8bad\u7ec3\u6a21\u578b\n\n\n```\nhybrid_model.compile(optimizer=tf.keras.optimizers.Adam(learning_rate=0.02),\n                     loss=tf.losses.mse,\n                     metrics=[custom_accuracy])\n\nhybrid_history = hybrid_model.fit(x=train_excitations,\n                                  y=train_labels,\n                                  batch_size=16,\n                                  epochs=25,\n                                  verbose=1,\n                                  validation_data=(test_excitations,\n                                                   test_labels))\n```\n\n\n```\nplt.plot(history.history['val_custom_accuracy'], label='QCNN')\nplt.plot(hybrid_history.history['val_custom_accuracy'], label='Hybrid CNN')\nplt.title('Quantum vs Hybrid CNN performance')\nplt.xlabel('Epochs')\nplt.legend()\nplt.ylabel('Validation Accuracy')\nplt.show()\n```\n\n\u5982\u60a8\u6240\u89c1\uff0c\u5728\u9002\u5f53\u7684\u7ecf\u5178\u6a21\u578b\u7684\u5e2e\u52a9\u4e0b\uff0c\u6df7\u5408\u6a21\u578b\u901a\u5e38\u6bd4\u7eaf\u91cf\u5b50\u7248\u672c\u6536\u655b\u5f97\u66f4\u5feb\u3002\n\n### 2.2 \u4f7f\u7528\u591a\u4e2a\u91cf\u5b50\u6ee4\u6ce2\u5668\u7684\u6df7\u5408\u5377\u79ef\n\n\u73b0\u5728\uff0c\u6211\u4eec\u6765\u8bd5\u8bd5\u4f7f\u7528\u591a\u4e2a\u91cf\u5b50\u5377\u79ef\u548c\u4e00\u4e2a\u7ecf\u5178\u795e\u7ecf\u7f51\u7edc\u7684\u67b6\u6784\uff0c\u5c06\u5176\u7ec4\u5408\u5728\u4e00\u8d77\u3002\n\n\n\n#### 2.2.1 \u6a21\u578b\u5b9a\u4e49\n\n\n```\nexcitation_input_multi = tf.keras.Input(shape=(), dtype=tf.dtypes.string)\n\ncluster_state_multi = tfq.layers.AddCircuit()(\n    excitation_input_multi, prepend=cluster_state_circuit(cluster_state_bits))\n\n# apply 3 different filters and measure expectation values\n\nquantum_model_multi1 = tfq.layers.PQC(\n    multi_readout_model_circuit(cluster_state_bits),\n    readouts)(cluster_state_multi)\n\nquantum_model_multi2 = tfq.layers.PQC(\n    multi_readout_model_circuit(cluster_state_bits),\n    readouts)(cluster_state_multi)\n\nquantum_model_multi3 = tfq.layers.PQC(\n    multi_readout_model_circuit(cluster_state_bits),\n    readouts)(cluster_state_multi)\n\n# concatenate outputs and feed into a small classical NN\nconcat_out = tf.keras.layers.concatenate(\n    [quantum_model_multi1, quantum_model_multi2, quantum_model_multi3])\n\ndense_1 = tf.keras.layers.Dense(8)(concat_out)\n\ndense_2 = tf.keras.layers.Dense(1)(dense_1)\n\nmulti_qconv_model = tf.keras.Model(inputs=[excitation_input_multi],\n                                   outputs=[dense_2])\n\n# Display the model architecture\ntf.keras.utils.plot_model(multi_qconv_model,\n                          show_shapes=True,\n                          show_layer_names=True,\n                          dpi=70)\n```\n\n#### 2.2.2 \u8bad\u7ec3\u6a21\u578b\n\n\n```\nmulti_qconv_model.compile(\n    optimizer=tf.keras.optimizers.Adam(learning_rate=0.02),\n    loss=tf.losses.mse,\n    metrics=[custom_accuracy])\n\nmulti_qconv_history = multi_qconv_model.fit(x=train_excitations,\n                                            y=train_labels,\n                                            batch_size=16,\n                                            epochs=25,\n                                            verbose=1,\n                                            validation_data=(test_excitations,\n                                                             test_labels))\n```\n\n\n```\nplt.plot(history.history['val_custom_accuracy'][:25], label='QCNN')\nplt.plot(hybrid_history.history['val_custom_accuracy'][:25], label='Hybrid CNN')\nplt.plot(multi_qconv_history.history['val_custom_accuracy'][:25],\n         label='Hybrid CNN \\n Multiple Quantum Filters')\nplt.title('Quantum vs Hybrid CNN performance')\nplt.xlabel('Epochs')\nplt.legend()\nplt.ylabel('Validation Accuracy')\nplt.show()\n```\n", "meta": {"hexsha": "41a647d8780157d7a7fa171162bc3815eebbbf53", "size": 32677, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "site/zh-cn/quantum/tutorials/qcnn.ipynb", "max_stars_repo_name": "RedContritio/docs-l10n", "max_stars_repo_head_hexsha": "f69a7c0d2157703a26cef95bac34b39ac0250373", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-03-29T22:32:18.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-29T22:32:18.000Z", "max_issues_repo_path": "site/zh-cn/quantum/tutorials/qcnn.ipynb", "max_issues_repo_name": "Juanita-cortez447/docs-l10n", "max_issues_repo_head_hexsha": "edaba1f2b5e329857860db1e937cb1333b6e3f31", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, 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YES\n2. YES", "lm_q1_score": 0.7634837527911057, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.414467296689045}}
{"text": "```python\n#we may need some code in the ../python directory and/or matplotlib styles\nimport sys\nsys.path.append('../python/')\n\n#matplotlib for plotting\nimport matplotlib as mpl\nfrom matplotlib import pyplot as plt\nplt.style.use('mplstyles/stylelib/standard.mplstyle')\n\n#other computational libraries\nimport numpy as np\n```\n\n\n```python\nnpegs=27\nnboxes=36\npn = np.zeros((npegs,))\n\npn[0]=36-9\npn[3]=9\n\npn=pn/36\n\nprint(pn)\n```\n\n    [0.75 0.   0.   0.25 0.   0.   0.   0.   0.   0.   0.   0.   0.   0.\n     0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.  ]\n\n\n\n```python\ndef fairness(v):\n    retval=0\n    for i,p in enumerate(v):\n        if (i>0) & (p>0):\n            retval+=p*np.log(p)\n    return -10*retval\n            \n            \n            \nprint(fairness(pn))\n```\n\n    3.4657359027997265\n\n\n\n```python\n#looking for a data structure to represent pegs and grid\ngrid = np.zeros((6,6))\n\nprint(grid)\n```\n\n    [[0. 0. 0. 0. 0. 0.]\n     [0. 0. 0. 0. 0. 0.]\n     [0. 0. 0. 0. 0. 0.]\n     [0. 0. 0. 0. 0. 0.]\n     [0. 0. 0. 0. 0. 0.]\n     [0. 0. 0. 0. 0. 0.]]\n\n\n\n```python\ngrid[0,0] = 3\ngrid[0,1] = 3\ngrid[0,2] = 3\n\ngrid[1,0] = 3\ngrid[1,1] = 3\ngrid[1,2] = 3\n\ngrid[2,0] = 3\ngrid[2,1] = 3\ngrid[2,2] = 3\n```\n\n\n```python\nprint(grid)\n```\n\n    [[3. 3. 3. 0. 0. 0.]\n     [3. 3. 3. 0. 0. 0.]\n     [3. 3. 3. 0. 0. 0.]\n     [0. 0. 0. 0. 0. 0.]\n     [0. 0. 0. 0. 0. 0.]\n     [0. 0. 0. 0. 0. 0.]]\n\n\n\n```python\n#find out how many energy pegs total in the grid structure\nprint(np.sum(grid))\n```\n\n    27.0\n\n\n\n```python\n#print out every individual box value\nfor v in grid:\n    for n in v:\n      print(n)\n```\n\n    3.0\n    3.0\n    3.0\n    0.0\n    0.0\n    0.0\n    3.0\n    3.0\n    3.0\n    0.0\n    0.0\n    0.0\n    3.0\n    3.0\n    3.0\n    0.0\n    0.0\n    0.0\n    0.0\n    0.0\n    0.0\n    0.0\n    0.0\n    0.0\n    0.0\n    0.0\n    0.0\n    0.0\n    0.0\n    0.0\n    0.0\n    0.0\n    0.0\n    0.0\n    0.0\n    0.0\n\n\n\n```python\n#get total number of grid spaces\nprint(grid.size)\n```\n\n    36\n\n\nExplanation of Micro/Macrostates is here.\n\n\\begin{equation}\nS = k\\log{\\Omega}\n\\end{equation}\n\n\n```python\n\n```\n", "meta": {"hexsha": "455d3a77c87dd1832e8ae037c9fb91f91f9e2475", "size": 5085, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "fairness.ipynb", "max_stars_repo_name": "villano-lab/thermal_physics", "max_stars_repo_head_hexsha": "c75137eec36659d07f191a423ce75eabeb449013", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "fairness.ipynb", "max_issues_repo_name": "villano-lab/thermal_physics", "max_issues_repo_head_hexsha": "c75137eec36659d07f191a423ce75eabeb449013", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "fairness.ipynb", "max_forks_repo_name": "villano-lab/thermal_physics", "max_forks_repo_head_hexsha": "c75137eec36659d07f191a423ce75eabeb449013", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 18.6263736264, "max_line_length": 83, "alphanum_fraction": 0.4001966568, "converted": true, "num_tokens": 1004, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.6619228691808012, "lm_q1q2_score": 0.4144459118544271}}
{"text": "# Trax : Ungraded Lecture Notebook\n\nIn this notebook you'll get to know about the Trax framework and learn about some of its basic building blocks.\n\n\n\n## Background\n\n### Why Trax and not TensorFlow or PyTorch?\n\nTensorFlow and PyTorch are both extensive frameworks that can do almost anything in deep learning. They offer a lot of flexibility, but that often means verbosity of syntax and extra time to code.\n\nTrax is much more concise. It runs on a TensorFlow backend but allows you to train models with 1 line commands. Trax also runs end to end, allowing you to get data, model and train all with a single terse statements. This means you can focus on learning, instead of spending hours on the idiosyncrasies of big framework implementation.\n\n### Why not Keras then?\n\nKeras is now part of Tensorflow itself from 2.0 onwards. Also, trax is good for implementing new state of the art algorithms like Transformers, Reformers, BERT because it is actively maintained by Google Brain Team for advanced deep learning tasks. It runs smoothly on CPUs,GPUs and TPUs as well with comparatively lesser modifications in code.\n\n### How to Code in Trax\nBuilding models in Trax relies on 2 key concepts:- **layers** and **combinators**.\nTrax layers are simple objects that process data and perform computations. They can be chained together into composite layers using Trax combinators, allowing you to build layers and models of any complexity.\n\n### Trax, JAX, TensorFlow and Tensor2Tensor\n\nYou already know that Trax uses Tensorflow as a backend, but it also uses the JAX library to speed up computation too. You can view JAX as an enhanced and optimized version of numpy. \n\n**Watch out for assignments which import `import trax.fastmath.numpy as np`. If you see this line, remember that when calling `np` you are really calling Trax\u2019s version of numpy that is compatible with JAX.**\n\nAs a result of this, where you used to encounter the type `numpy.ndarray` now you will find the type `jax.interpreters.xla.DeviceArray`.\n\nTensor2Tensor is another name you might have heard. It started as an end to end solution much like how Trax is designed, but it grew unwieldy and complicated. So you can view Trax as the new improved version that operates much faster and simpler.\n\n### Resources\n\n- Trax source code can be found on Github: [Trax](https://github.com/google/trax)\n- JAX library: [JAX](https://jax.readthedocs.io/en/latest/index.html)\n\n\n## Installing Trax\n\nTrax has dependencies on JAX and some libraries like JAX which are yet to be supported in [Windows](https://github.com/google/jax/blob/1bc5896ee4eab5d7bb4ec6f161d8b2abb30557be/README.md#installation) but work well in Ubuntu and MacOS. We would suggest that if you are working on Windows, try to install Trax on WSL2. \n\nOfficial maintained documentation - [trax-ml](https://trax-ml.readthedocs.io/en/latest/) not to be confused with this [TraX](https://trax.readthedocs.io/en/latest/index.html)\n\n\n```python\n!pip install trax==1.3.1 # Use this version for this notebook \n```\n\n    Requirement already satisfied: trax==1.3.1 in /opt/conda/lib/python3.7/site-packages (1.3.1)\n    Requirement already satisfied: scipy in /opt/conda/lib/python3.7/site-packages (from trax==1.3.1) (1.2.1)\n    Requirement already satisfied: t5 in /opt/conda/lib/python3.7/site-packages (from trax==1.3.1) (0.6.1)\n    Requirement already satisfied: gin-config in /opt/conda/lib/python3.7/site-packages (from trax==1.3.1) (0.3.0)\n    Requirement already satisfied: numpy in /opt/conda/lib/python3.7/site-packages (from trax==1.3.1) (1.18.5)\n    Requirement already satisfied: gym in /opt/conda/lib/python3.7/site-packages (from trax==1.3.1) (0.17.2)\n    Requirement already satisfied: jax in /opt/conda/lib/python3.7/site-packages (from trax==1.3.1) (0.1.72)\n    Requirement already satisfied: jaxlib in /opt/conda/lib/python3.7/site-packages (from trax==1.3.1) (0.1.51)\n    Requirement already satisfied: six in /opt/conda/lib/python3.7/site-packages (from trax==1.3.1) (1.14.0)\n    Requirement already satisfied: absl-py in /opt/conda/lib/python3.7/site-packages (from trax==1.3.1) (0.9.0)\n    Requirement already satisfied: tensorflow-text in /opt/conda/lib/python3.7/site-packages (from trax==1.3.1) (2.2.1)\n    Requirement already satisfied: tensorflow-datasets in /opt/conda/lib/python3.7/site-packages (from trax==1.3.1) (2.1.0)\n    Requirement already satisfied: tensor2tensor in /opt/conda/lib/python3.7/site-packages (from trax==1.3.1) (1.15.7)\n    Requirement already satisfied: funcsigs in /opt/conda/lib/python3.7/site-packages (from trax==1.3.1) (1.0.2)\n    Requirement already satisfied: torch in 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importlib-metadata; python_version < \"3.8\"->markdown>=2.6.8->tensorboard<2.3.0,>=2.2.0->tensorflow<2.3,>=2.2.0->tensorflow-text->trax==1.3.1) (3.1.0)\n    \u001b[33mWARNING: You are using pip version 20.1.1; however, version 20.2 is available.\n    You should consider upgrading via the '/opt/conda/bin/python -m pip install --upgrade pip' command.\u001b[0m\n\n\n\n## Imports\n\n\n```python\nimport numpy as np  # regular ol' numpy\n\nfrom trax import layers as tl  # core building block\nfrom trax import shapes  # data signatures: dimensionality and type\nfrom trax import fastmath  # uses jax, offers numpy on steroids\n```\n\n    INFO:tensorflow:tokens_length=568 inputs_length=512 targets_length=114 noise_density=0.15 mean_noise_span_length=3.0 \n\n\n\n```python\n# Trax version 1.3.1 or better \n!pip list | grep trax\n```\n\n    trax                     1.3.1\n    \u001b[33mWARNING: You are using pip version 20.1.1; however, version 20.2 is available.\n    You should consider upgrading via the '/opt/conda/bin/python -m pip install --upgrade pip' command.\u001b[0m\n\n\n## Layers\nLayers are the core building blocks in Trax or as mentioned in the lectures, they are the base classes.\n\nThey take inputs, compute functions/custom calculations and return outputs.\n\nYou can also inspect layer properties. Let me show you some examples.\n\n\n### Relu Layer\nFirst I'll show you how to build a relu activation function as a layer. A layer like this is one of the simplest types. Notice there is no object initialization so it works just like a math function.\n\n**Note: Activation functions are also layers in Trax, which might look odd if you have been using other frameworks for a longer time.**\n\n\n```python\n# Layers\n# Create a relu trax layer\nrelu = tl.Relu()\n\n# Inspect properties\nprint(\"-- Properties --\")\nprint(\"name :\", relu.name)\nprint(\"expected inputs :\", relu.n_in)\nprint(\"promised outputs :\", relu.n_out, \"\\n\")\n\n# Inputs\nx = np.array([-2, -1, 0, 1, 2])\nprint(\"-- Inputs --\")\nprint(\"x :\", x, \"\\n\")\n\n# Outputs\ny = relu(x)\nprint(\"-- Outputs --\")\nprint(\"y :\", y)\n```\n\n    -- Properties --\n    name : Relu\n    expected inputs : 1\n    promised outputs : 1 \n    \n    -- Inputs --\n    x : [-2 -1  0  1  2] \n    \n    -- Outputs --\n    y : [0 0 0 1 2]\n\n\n### Concatenate Layer\nNow I'll show you how to build a layer that takes 2 inputs. Notice the change in the expected inputs property from 1 to 2.\n\n\n```python\n# Create a concatenate trax layer\nconcat = tl.Concatenate()\nprint(\"-- Properties --\")\nprint(\"name :\", concat.name)\nprint(\"expected inputs :\", concat.n_in)\nprint(\"promised outputs :\", concat.n_out, \"\\n\")\n\n# Inputs\nx1 = np.array([-10, -20, -30])\nx2 = x1 / -10\nprint(\"-- Inputs --\")\nprint(\"x1 :\", x1)\nprint(\"x2 :\", x2, \"\\n\")\n\n# Outputs\ny = concat([x1, x2])\nprint(\"-- Outputs --\")\nprint(\"y :\", y)\n```\n\n    -- Properties --\n    name : Concatenate\n    expected inputs : 2\n    promised outputs : 1 \n    \n    -- Inputs --\n    x1 : [-10 -20 -30]\n    x2 : [1. 2. 3.] \n    \n    -- Outputs --\n    y : [-10. -20. -30.   1.   2.   3.]\n\n\n## Layers are Configurable\nYou can change the default settings of layers. For example, you can change the expected inputs for a concatenate layer from 2 to 3 using the optional parameter `n_items`.\n\n\n```python\n# Configure a concatenate layer\nconcat_3 = tl.Concatenate(n_items=3)  # configure the layer's expected inputs\nprint(\"-- Properties --\")\nprint(\"name :\", concat_3.name)\nprint(\"expected inputs :\", concat_3.n_in)\nprint(\"promised outputs :\", concat_3.n_out, \"\\n\")\n\n# Inputs\nx1 = np.array([-10, -20, -30])\nx2 = x1 / -10\nx3 = x2 * 0.99\nprint(\"-- Inputs --\")\nprint(\"x1 :\", x1)\nprint(\"x2 :\", x2)\nprint(\"x3 :\", x3, \"\\n\")\n\n# Outputs\ny = concat_3([x1, x2, x3])\nprint(\"-- Outputs --\")\nprint(\"y :\", y)\n```\n\n    -- Properties --\n    name : Concatenate\n    expected inputs : 3\n    promised outputs : 1 \n    \n    -- Inputs --\n    x1 : [-10 -20 -30]\n    x2 : [1. 2. 3.]\n    x3 : [0.99 1.98 2.97] \n    \n    -- Outputs --\n    y : [-10.   -20.   -30.     1.     2.     3.     0.99   1.98   2.97]\n\n\n**Note: At any point,if you want to refer the function help/ look up the [documentation](https://trax-ml.readthedocs.io/en/latest/) or use help function.**\n\n\n```python\n#help(tl.Concatenate) #Uncomment this to see the function docstring with explaination\n```\n\n## Layers can have Weights\nSome layer types include mutable weights and biases that are used in computation and training. Layers of this type require initialization before use.\n\nFor example the `LayerNorm` layer calculates normalized data, that is also scaled by weights and biases. During initialization you pass the data shape and data type of the inputs, so the layer can initialize compatible arrays of weights and biases.\n\n\n```python\n# Uncomment any of them to see information regarding the function\n# help(tl.LayerNorm)\n# help(shapes.signature)\n\n```\n\n\n```python\n# Layer initialization\nnorm = tl.LayerNorm()\n# You first must know what the input data will look like\nx = np.array([0, 1, 2, 3], dtype=\"float\")\n\n# Use the input data signature to get shape and type for initializing weights and biases\nnorm.init(shapes.signature(x)) # We need to convert the input datatype from usual tuple to trax ShapeDtype\n\nprint(\"Normal shape:\",x.shape, \"Data Type:\",type(x.shape))\nprint(\"Shapes Trax:\",shapes.signature(x),\"Data Type:\",type(shapes.signature(x)))\n\n# Inspect properties\nprint(\"-- Properties --\")\nprint(\"name :\", norm.name)\nprint(\"expected inputs :\", norm.n_in)\nprint(\"promised outputs :\", norm.n_out)\n# Weights and biases\nprint(\"weights :\", norm.weights[0])\nprint(\"biases :\", norm.weights[1], \"\\n\")\n\n# Inputs\nprint(\"-- Inputs --\")\nprint(\"x :\", x)\n\n# Outputs\ny = norm(x)\nprint(\"-- Outputs --\")\nprint(\"y :\", y)\n```\n\n    Normal shape: (4,) Data Type: \n    Shapes Trax: ShapeDtype{shape:(4,), dtype:float64} Data Type: \n    -- Properties --\n    name : LayerNorm\n    expected inputs : 1\n    promised outputs : 1\n    weights : [1. 1. 1. 1.]\n    biases : [0. 0. 0. 0.] \n    \n    -- Inputs --\n    x : [0. 1. 2. 3.]\n    -- Outputs --\n    y : [-1.3416404  -0.44721344  0.44721344  1.3416404 ]\n\n\n## Custom Layers\nThis is where things start getting more interesting!\nYou can create your own custom layers too and define custom functions for computations by using `tl.Fn`. Let me show you how.\n\n\n```python\nhelp(tl.Fn)\n```\n\n\n```python\n# Define a custom layer\n# In this example you will create a layer to calculate the input times 2\n\ndef TimesTwo():\n    layer_name = \"TimesTwo\" #don't forget to give your custom layer a name to identify\n\n    # Custom function for the custom layer\n    def func(x):\n        return x * 2\n\n    return tl.Fn(layer_name, func)\n\n\n# Test it\ntimes_two = TimesTwo()\n\n# Inspect properties\nprint(\"-- Properties --\")\nprint(\"name :\", times_two.name)\nprint(\"expected inputs :\", times_two.n_in)\nprint(\"promised outputs :\", times_two.n_out, \"\\n\")\n\n# Inputs\nx = np.array([1, 2, 3])\nprint(\"-- Inputs --\")\nprint(\"x :\", x, \"\\n\")\n\n# Outputs\ny = times_two(x)\nprint(\"-- Outputs --\")\nprint(\"y :\", y)\n```\n\n    -- Properties --\n    name : TimesTwo\n    expected inputs : 1\n    promised outputs : 1 \n    \n    -- Inputs --\n    x : [1 2 3] \n    \n    -- Outputs --\n    y : [2 4 6]\n\n\n## Combinators\nYou can combine layers to build more complex layers. Trax provides a set of objects named combinator layers to make this happen. Combinators are themselves layers, so behavior commutes.\n\n\n\n### Serial Combinator\nThis is the most common and easiest to use. For example could build a simple neural network by combining layers into a single layer using the `Serial` combinator. This new layer then acts just like a single layer, so you can inspect intputs, outputs and weights. Or even combine it into another layer! Combinators can then be used as trainable models. _Try adding more layers_\n\n**Note:As you must have guessed, if there is serial combinator, there must be a parallel combinator as well. Do try to explore about combinators and other layers from the trax documentation and look at the repo to understand how these layers are written.**\n\n\n\n```python\n# help(tl.Serial)\n# help(tl.Parallel)\n```\n\n\n```python\n# Serial combinator\nserial = tl.Serial(\n    tl.LayerNorm(),         # normalize input\n    tl.Relu(),              # convert negative values to zero\n    times_two,              # the custom layer you created above, multiplies the input recieved from above by 2\n    \n    ### START CODE HERE\n#     tl.Dense(n_units=2),  # try adding more layers. eg uncomment these lines\n#     tl.Dense(n_units=1),  # Binary classification, maybe? uncomment at your own peril\n#     tl.LogSoftmax()       # Yes, LogSoftmax is also a layer\n    ### END CODE HERE\n)\n\n# Initialization\nx = np.array([-2, -1, 0, 1, 2]) #input\nserial.init(shapes.signature(x)) #initialising serial instance\n\nprint(\"-- Serial Model --\")\nprint(serial,\"\\n\")\nprint(\"-- Properties --\")\nprint(\"name :\", serial.name)\nprint(\"sublayers :\", serial.sublayers)\nprint(\"expected inputs :\", serial.n_in)\nprint(\"promised outputs :\", serial.n_out)\nprint(\"weights & biases:\", serial.weights, \"\\n\")\n\n# Inputs\nprint(\"-- Inputs --\")\nprint(\"x :\", x, \"\\n\")\n\n# Outputs\ny = serial(x)\nprint(\"-- Outputs --\")\nprint(\"y :\", y)\n```\n\n    -- Serial Model --\n    Serial[\n      LayerNorm\n      Relu\n      TimesTwo\n    ] \n    \n    -- Properties --\n    name : Serial\n    sublayers : [LayerNorm, Relu, TimesTwo]\n    expected inputs : 1\n    promised outputs : 1\n    weights & biases: [(DeviceArray([1, 1, 1, 1, 1], dtype=int32), DeviceArray([0, 0, 0, 0, 0], dtype=int32)), (), ()] \n    \n    -- Inputs --\n    x : [-2 -1  0  1  2] \n    \n    -- Outputs --\n    y : [0.        0.        0.        1.4142132 2.8284264]\n\n\n## JAX\nJust remember to lookout for which numpy you are using, the regular ol' numpy or Trax's JAX compatible numpy. Both tend to use the alias np so watch those import blocks.\n\n**Note:There are certain things which are still not possible in fastmath.numpy which can be done in numpy so you will see in assignments we will switch between them to get our work done.**\n\n\n```python\n# Numpy vs fastmath.numpy have different data types\n# Regular ol' numpy\nx_numpy = np.array([1, 2, 3])\nprint(\"good old numpy : \", type(x_numpy), \"\\n\")\n\n# Fastmath and jax numpy\nx_jax = fastmath.numpy.array([1, 2, 3])\nprint(\"jax trax numpy : \", type(x_jax))\n```\n\n    good old numpy :   \n    \n    jax trax numpy :  \n\n\n## Summary\nTrax is a concise framework, built on TensorFlow, for end to end machine learning. The key building blocks are layers and combinators. This notebook is just a taste, but sets you up with some key inuitions to take forward into the rest of the course and assignments where you will build end to end models.\n", "meta": {"hexsha": "190ba35786da3ee89dd8e217fdcccda77038ef49", "size": 36926, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Course3/Week1/03W1Lab1IntroToTrax.ipynb", "max_stars_repo_name": "MBadriNarayanan/NLPSpecialization", "max_stars_repo_head_hexsha": "36b8cd4186052fdbe4760a6d3faaeea9fe0c8a58", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-12-12T17:25:45.000Z", "max_stars_repo_stars_event_max_datetime": "2020-12-12T17:25:45.000Z", "max_issues_repo_path": "Course3/Week1/03W1Lab1IntroToTrax.ipynb", "max_issues_repo_name": "MBadriNarayanan/NLPSpecialization", "max_issues_repo_head_hexsha": "36b8cd4186052fdbe4760a6d3faaeea9fe0c8a58", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Course3/Week1/03W1Lab1IntroToTrax.ipynb", "max_forks_repo_name": "MBadriNarayanan/NLPSpecialization", "max_forks_repo_head_hexsha": "36b8cd4186052fdbe4760a6d3faaeea9fe0c8a58", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2020-07-18T21:42:43.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-18T10:22:33.000Z", "avg_line_length": 48.4593175853, "max_line_length": 385, "alphanum_fraction": 0.617099063, "converted": true, "num_tokens": 8910, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "# The Euler equations of gas dynamics\n\nIn this notebook, we discuss the equations and the structure of the exact solution to the Riemann problem.  In [Euler_approximate](Euler_approximate.ipynb) and [FV_compare](FV_compare.ipynb), we will investigate approximate Riemann solvers.\n\n## Fluid dynamics\n\nIn this chapter we study the system of hyperbolic PDEs that governs the motions of a compressible gas in the absence of viscosity.  These consist of conservation laws for *mass, momentum*, and *energy*.  Together, they are referred to as the *compressible Euler equations*, or simply the Euler equations.  Our discussion here is fairly brief; for much more detail see (LeVeque, 2002) or (Toro, 2013).\n\n### Mass conservation\n\nWe will use $\\rho(x,t)$ to denote the fluid density and $u(x,t)$ for its velocity.  Then the equation for conservation of mass is just the familiar *continuity equation*:\n\n$$\\rho_t + (\\rho u)_x = 0.$$\n\n### Momentum conservation\n\nWe discussed the conservation of momentum in a fluid already in [Acoustics](Acoustics.ipynb).  For convenience, we review the ideas here.  The momentum density is given by the product of mass density and velocity, $\\rho u$.  The momentum flux has two components.  First, the momentum is transported in the same way that the density is; this flux is given by the momentum density times the velocity: $\\rho u^2$.\n\nTo understand the second term in the momentum flux, we must realize that a fluid is made up of many tiny molecules.  The density and velocity we are modeling are average values over some small region of space.  The individual molecules in that region are not all moving with exactly velocity $u$; that's just their average.  Each molecule also has some additional random velocity component.  These random velocities are what accounts for the *pressure* of the fluid, which we'll denote by $p$.  These velocity components also lead to a net flux of momentum.  Thus the momentum conservation equation is\n\n$$(\\rho u)_t + (\\rho u^2 + p)_x = 0.$$\n\nThis is very similar to the conservation of momentum equation in the shallow water equations, as discussed in [Shallow_water](Shallow_water.ipynb), in which case $hu$ is the momentum density and $\\frac 1 2 gh^2$ is the hydrostatic pressure.  For gas dynamics, a different expression must be used to compute the pressure $p$ from the conserved quantities.  This relation is called the *equation of state* of the gas, as discussed further below.\n\n### Energy conservation\n\nThe energy has two components: internal energy density $\\rho e$ and kinetic energy density $\\rho u^2/2$:\n\n$$E = \\rho e + \\frac{1}{2}\\rho u^2.$$\n\nLike the momentum flux, the energy flux involves both bulk transport ($Eu$) and transport due to pressure ($pu$):\n\n$$E_t + (u(E+p)) = 0.$$\n\n### Equation of state\n\nYou may have noticed that we have 4 unknowns (density, momentum, energy, and pressure) but only 3 conservation laws.  We need one more relation to close the system.  That relation, known as the equation of state, expresses how the pressure is related to the other quantities.  We'll focus on the case of a polytropic ideal gas, for which\n\n$$p = \\rho e (\\gamma-1).$$\n\nHere $\\gamma$ is the ratio of specific heats, which for air is approximately 1.4.\n\n## Hyperbolic structure of the 1D Euler equations\n\nWe can write the three conservation laws as a single system $q_t + f(q)_x = 0$ by defining  \n\\begin{align}\nq & = \\begin{pmatrix} \\rho \\\\ \\rho u \\\\ E\\end{pmatrix}, & \nf(q) & = \\begin{pmatrix} \\rho u \\\\ \\rho u^2 + p \\\\ u(E+p)\\end{pmatrix}.\n\\label{euler_conserved}\n\\end{align}  \nThese are the one-dimensional Euler system. As usual, one can define the $3 \\times 3$ Jacobian matrix by differentiating this flux function with respect to the three components of $q$.\n\nIn our discussion of the structure of these equations, it is convenient to work with the primitive variables $(\\rho, u, p)$ rather than the conserved variables.  The quasilinear form is particularly simple in the primitive variables:  \n\n\\begin{align} \\label{euler_primitive}\n\\begin{bmatrix} \\rho \\\\ u \\\\ p \\end{bmatrix}_t + \n\\begin{bmatrix} u & \\rho & 0 \\\\ 0 & u & 1/\\rho \\\\ 0 & \\gamma \\rho & u \\end{bmatrix} \\begin{bmatrix} \\rho \\\\ u \\\\ p \\end{bmatrix}_x & = 0.\n\\end{align}\n\n### Characteristic velocities\nThe eigenvalues of the flux Jacobian $f'(q)$ for the 1D Euler equations are:\n\n\\begin{align}\n\\lambda_1 & = u-c & \\lambda_2 & = u & \\lambda_3 & = u+c\n\\end{align}\n\nHere $c$ is the sound speed:\n\n$$ c = \\sqrt{\\frac{\\gamma p}{\\rho}}.$$\n\nThese are also the eigenvalues of the coefficient matrix appearing in (\\ref{euler_primitive}), and show that acoustic waves propagate at speeds $\\pm c$ relative to the fluid velocity $u$. There is also a characteristic speed $\\lambda_2 =u$ corresponding to the transport of entropy at the fluid velocity, as discussed further below.\n\nThe eigenvectors of the coefficient matrix appearing in (\\ref{euler_primitive}) are:\n\n\\begin{align}\\label{euler_evecs}\nr_1 & = \\begin{bmatrix} -\\rho/c \\\\ 1 \\\\ - \\rho c \\end{bmatrix} &\nr_2 & = \\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \\end{bmatrix} &\nr_3 & = \\begin{bmatrix}  \\rho/c \\\\ 1 \\\\ \\rho c \\end{bmatrix}.\n\\end{align}\n\nThese vectors show the relation between jumps in the primitive variables across waves in each family.  The eigenvectors of the flux Jacobian $f'(q)$ arising from the conservative form (\\ref{euler_conserved})  would be different, and would give the relation between jumps in the conserved variables across each wave.\n\nNotice that the second characteristic speed, $\\lambda_2$, depends only on $u$ and that $u$ does not change as we move in the direction of $r_2$.  In other words, the 2-characteristic velocity is constant on 2-integral curves.  This is similar to the wave that carries changes in the tracer that we considered in [Shallow_tracer](Shallow_tracer.ipynb).  We say this characteristic field is *linearly degenerate*; it admits neither shocks nor rarefactions.  In a simple 2-wave, all characteristics are parallel.  A jump in this family carries a change only in the density, and is referred to as a *contact discontinuity*.\n\nMathematically, the $p$th field is linearly degenerate if\n\\begin{align}\\label{lindegen}\n\\nabla \\lambda_p(q) \\cdot r_p(q) = 0,\n\\end{align}\nsince in this case the eigenvalue $\\lambda_p(q)$ does not vary in the direction of the eigenvector $r_p(q)$, and hence is constant along integral curves of this family. (Recall that $r_p(q)$ is the tangent vector at each point on the integral curve.)\n\nThe other two fields have characteristic velocities that *do* vary along the corresponding integral curves. Moreover they vary in a monotonic manner as we move along an integral curve, always increasing as we move in one direction, decreasing in the other.  Mathematically, this means that \n\\begin{align}\\label{gennonlin}\n\\nabla \\lambda_p(q) \\cdot r_p(q) \\ne 0\n\\end{align}\nas $q$ varies, so that the directional derivative of $\\lambda_p(q)$ cannot change sign as we move along the curve.  This is analogous to the flux for a scalar problem being convex, and means that the 1-wave and the 3-wave in any Riemann solution to the Euler equations will be a single shock or rarefaction wave, not the sort of compound waves we observed in [Nonconvex_scalar](Nonconvex_scalar.ipynb) in the nonconvex scalar case.  Any characteristic field satisfying (\\ref{gennonlin}) is said to be *genuinely nonlinear*.\n\n### Entropy\n\nAnother important quantity in gas dynamics is the *specific entropy*:\n\n$$ s = c_v \\log(p/\\rho^\\gamma) + C,$$\n\nwhere $c_v$ and $C$ are constants. From the expression (\\ref{euler_evecs}) for the eigenvector $r_2$, we see that the pressure and velocity are constant across a 2-wave.\nA simple 2-wave is also called an *entropy wave* because a variation in density while the pressure remains constant requires a variation in the entropy of the gas as well.  On the other hand a simple acoustic wave (a continuously varying pure 1-wave or 3-wave) has constant entropy throughout the wave; the specific entropy is a Riemann invariant for these families.  \n\nA shock wave (either a 1-wave or 3-wave) satisfies the Rankine-Hugoniot conditions and exhibits a jump in entropy.  To be physically correct, the entropy of the gas must *increase* as gas molecules pass through the shock, leading to the *entropy condition* for selecting shock waves.  We have already seen this term used in the context of scalar nonlinear equations shallow water flow, even though the entropy condition in those cases did not involve the physical entropy.\n\n### Riemann invariants\n\nSince the Euler equations have three components, we expect each integral curve (a 1D set in 3D space) to be defined by two Riemann invariants.  These are:\n\n\\begin{align}\n1 & : s, u+\\frac{2c}{\\gamma-1} \\\\\n2 & : u, p \\\\\n3 & : s, u-\\frac{2c}{\\gamma-1}.\n\\end{align}\n\n### Integral curves\n\nThe level sets of these Riemann invariants are two-dimensional surfaces; the intersection of two appropriate level sets defines an integral curve.\n\nThe 2-integral curves, of course, are simply lines of constant pressure and velocity (with varying density).  Since the field is linearly degenerate, these coincide with the Hugoniot loci.\nWe can determine the form of the 1- and 3-integral curves using the Riemann invariants above.  For a curve passing through $(\\rho_0,u_0,p_0)$, we find\n\n\\begin{align}\n    \\rho(p) &= (p/p_0)^{1/\\gamma} \\rho_0,\\\\\n    u(p) & = u_0 \\pm \\frac{2c_0}{\\gamma-1}\\left(1-(p/p_0)^{(\\gamma-1)/(2\\gamma)}\\right).\n\\end{align}\nHere the plus sign is for 1-waves and the minus sign is for 3-waves.\n\nBelow we plot the projection of some integral curves on the $p-u$ plane.\n\n\n```python\n%matplotlib inline\n```\n\n\n```python\n%config InlineBackend.figure_format = 'svg'\nfrom exact_solvers import euler\nfrom exact_solvers import euler_demos\nfrom ipywidgets import widgets\nfrom ipywidgets import interact\nState = euler.Primitive_State\ngamma = 1.4\n```\n\nIf you wish to examine the Python code for this chapter, see:\n\n- [exact_solvers/euler.py](exact_solvers/euler.py) ...\n   [on github,](https://github.com/clawpack/riemann_book/blob/FA16/exact_solvers/euler.py)\n- [exact_solvers/euler_demos.py](exact_solvers/euler_demos.py) ...\n   [on github.](https://github.com/clawpack/riemann_book/blob/FA16/exact_solvers/euler_demos.py)\n\n\n```python\ninteract(euler.plot_integral_curves,\n         gamma=widgets.FloatSlider(min=1.1,max=3,value=1.4),\n         rho_0=widgets.FloatSlider(min=0.1,max=3.,value=1., \n                                   description=r'$\\rho_0$'));\n```\n\n\n    interactive(children=(Checkbox(value=True, description='plot_1'), Checkbox(value=False, description='plot_3'),\u2026\n\n\n## Rankine-Hugoniot jump conditions\n\nThe Hugoniot loci for 1- and 3-shocks are\n\\begin{align}\n    \\rho(p) &= \\left(\\frac{1 + \\beta p/p_0}{p/p_\\ell + \\beta} \\right),\\\\\n    u(p) & = u_0 \\pm \\frac{2c_0}{\\sqrt{2\\gamma(\\gamma-1)}} \n        \\left(\\frac{1-p/p_0}{\\sqrt{1+\\beta p/p_0}}\\right), \\\\\n\\end{align}\nwhere $\\beta = (\\gamma+1)/(\\gamma-1)$.\nHere the plus sign is for 1-shocks and the minus sign is for 3-shocks.\n\nBelow we plot the projection of some Hugoniot loci on the $p-u$ plane.\n\n\n```python\ninteract(euler.plot_hugoniot_loci,\n         gamma=widgets.FloatSlider(min=1.1,max=3,value=1.4),\n         rho_0=widgets.FloatSlider(min=0.1,max=3.,value=1., \n                                   description=r'$\\rho_0$'));\n```\n\n\n    interactive(children=(Checkbox(value=True, description='plot_1'), Checkbox(value=False, description='plot_3'),\u2026\n\n\n### Entropy condition\n\nAs mentioned above, a shock wave is physically relevant only if the entropy of the gas increases as the gas particles move through the shock.  A discontinuity satisfying the Rankine-Hugoniot jump conditions that violates this entropy condition (an \"entropy-violating shock\") is not physically correct and should be replaced by a rarefaction wave in the Riemann solution.  \n\nThis physical entropy condition is equivalent to the mathematical condition that for a 1-shock to be physically relevant, the 1-characteristics must impinge on the shock (the Lax entropy condition). If the entropy condition is violated, the 1-characteristics would spread out, allowing the insertion of an expansion fan (rarefaction wave).\n\n## Exact solution of the Riemann problem\n\nThe general Riemann solution is found following the steps listed below.  This is essentially the same procedure used to determine the correct solution to the Riemann problem for the shallow water equations in [Shallow_water](Shallow_water.ipynb), where more details are given.\n\nThe Euler equations are a system of three equations and the general Riemann solution consists of three waves, so we must determine two intermediate states rather than the one intermediate state in the shallow water equations.  However, it is nearly as simple because of the fact that we know the pressure and velocity are constant across the 2-wave, and so there is a single intermediate pressure $p_m$ and velocity $u_m$ in both intermediate states, and it is only the density that takes different values $\\rho_{m1}$ and $\\rho_{m2}$.  Moreover any jump in density is allowed across the 2-wave, and we have expressions given above for how $u(p)$ varies along any integral curve or Hugoniot locus, expressions that do not explicitly involve $\\rho$.  So we can determine the intermediate $p_m$ by finding the intersection point of two relevant curves, in step 3 of this general algorithm:\n\n1. Define a piecewise function giving the middle state velocity $u_m$ that can be connected to the left state by an entropy-satisfying shock or rarefaction, as a function of the middle-state pressure $p_m$.\n2. Define a piecewise function giving the middle state velocity $u_m$ that can be connected to the right state by an entropy-satisfying shock or rarefaction, as a function of the middle-state pressure $p_m$.\n3. Use a nonlinear rootfinder to find the intersection of the two functions defined above.\n4. Use the Riemann invariants to find the intermediate state densities and the solution structure inside any rarefaction waves.\n\nStep 4 above requires finding the structure of rarefaction waves.  This can be done using the the fact that the Riemann invariants are constant through the rarefaction wave.  See Chapter 14 of (LeVeque, 2002) for more details.  \n\n## Examples of Riemann solutions\n\nHere we present some representative examples of Riemann problems and solutions.  The examples chosen are closely related to the examples used in [Shallow_water](Shallow_water.ipynb) and you might want to refer back to that notebook and compare the results. \n\n### Problem 1: Sod shock tube\n\nFirst we consider the classic shock tube problem. The initial condition consists of high density and pressure on the left, low density and pressure on the right and zero velocity on both sides. The solution is composed of a shock propagating to the right (3-shock), while a left-going rarefaction forms (1-rarefaction). In between these two waves, there is a jump in the density, which is the contact discontinuity (2-wave) in the linearly degenerate characteristic field.  \n\nNote that this set of initial conditions is analogous to the \"dam break\" problem for shallow water quations, and the resulting structure of the solution is very similar to that obtained when those equations are solved with the addition of a scalar tracer.  However, in the Euler equations the entropy jump across a 2-wave does affect the fluid dynamics on either side, so this is not a passive tracer and solving the Riemann problem is slightly more complex.\n\n\n```python\nleft_state  = State(Density = 3.,\n                    Velocity = 0.,\n                    Pressure = 3.)\nright_state = State(Density = 1.,\n                    Velocity = 0.,\n                    Pressure = 1.)\n\neuler.riemann_solution(left_state,right_state)\n```\n\n\n    interactive(children=(FloatSlider(value=0.5, description='t', max=0.9), Dropdown(description='Show characteris\u2026\n\n\nHere is a plot of the solution in the phase plane, showing the integral curve connecting the left and middle states, and the Hugoniot locus connecting the middle and right states.\n\n\n```python\neuler.phase_plane_plot(left_state, right_state)\n```\n\n\n    \n\n    \n\n\n### Problem 2: Symmetric expansion\n\nNext we consider the case of equal densities and pressures, and equal and opposite velocities, with the initial states moving away from each other.  The result is two rarefaction waves (the contact has zero strength).\n\n\n```python\nleft_state  = State(Density = 1.,\n                    Velocity = -3.,\n                    Pressure = 1.)\nright_state = State(Density = 1.,\n                    Velocity = 3.,\n                    Pressure = 1.)\n\neuler.riemann_solution(left_state,right_state);\n```\n\n\n    interactive(children=(FloatSlider(value=0.5, description='t', max=0.9), Dropdown(description='Show characteris\u2026\n\n\n\n```python\neuler.phase_plane_plot(left_state, right_state)\n```\n\n\n    \n\n    \n\n\n### Problem 3: Colliding flows\n\nNext, consider the case in which the left and right states are moving toward each other.  This leads to a pair of shocks, with a high-density, high-pressure state in between.\n\n\n```python\nleft_state  = State(Density = 1.,\n                    Velocity = 3.,\n                    Pressure = 1.)\nright_state = State(Density = 1.,\n                    Velocity = -3.,\n                    Pressure = 1.)\n\neuler.riemann_solution(left_state,right_state)\n```\n\n\n    interactive(children=(FloatSlider(value=0.5, description='t', max=0.9), Dropdown(description='Show characteris\u2026\n\n\n\n```python\neuler.phase_plane_plot(left_state, right_state)\n```\n\n\n    \n\n    \n\n\n## Plot particle trajectories\n\nIn the next plot of the Riemann solution in the $x$-$t$ plane, we also plot the trajectories of a set of particles initially distributed along the $x$ axis at $t=0$, with the spacing inversely proportional to the density.  The evolution of the distance between particles gives an indication of how the density changes.\n\n\n```python\nleft_state  = State(Density = 3.,\n                    Velocity = 0.,\n                    Pressure = 3.)\nright_state = State(Density = 1.,\n                    Velocity = 0.,\n                    Pressure = 1.)\n\neuler.plot_riemann_trajectories(left_state, right_state)\n```\n\n\n    \n\n    \n\n\nSince the distance between particles in the above plot is inversely proportional to density, we see that the density around a particle increases as it goes through the shock wave but decreases through the rarefaction wave, and that in general there is a jump in density across the contact discontinuity, which lies along the particle trajectory emanating from $x=0$ at $t=0$.\n\n## Riemann solution with a colored tracer\n\nNext we plot the Riemann solution with the density plot also showing an advected color to help visualize the flow better. The fluid initially to the left of $x=0$ is colored red and that initially to the right of $x=0$ is colored blue, with stripes of different shades of these colors to help visualize the motion of the fluid.\n\nLet's plot the Sod shock tube data with this colored tracer:\n\n\n```python\ndef plot_with_stripes_t_slider(t):\n    euler_demos.plot_with_stripes(rho_l=3.,u_l=0.,p_l=3.,\n                                  rho_r=1.,u_r=0.,p_r=1.,\n                                  gamma=gamma,t=t)\n    \ninteract(plot_with_stripes_t_slider, \n         t=widgets.FloatSlider(min=0.,max=1.,step=0.1,value=0.5));\n```\n\n\n    interactive(children=(FloatSlider(value=0.5, description='t', max=1.0), Output()), _dom_classes=('widget-inter\u2026\n\n\nNote the following in the figure above:\n\n - The edges of each stripe are being advected with the fluid velocity, so you can visualize how the fluid is moving.\n - The width of each stripe initially is inversely proportional to the density of the fluid, so that the total mass of gas within each stripe is the same.\n - The total mass within each stripe remains constant as the flow evolves, and the width of each stripe remains inversely proportional to the local density.\n - The interface between the red and blue gas moves with the contact discontinuity.  The velocity and pressure are constant but the density can vary across this wave.\n\n## Interactive Riemann solver\n\nThe initial configuration specified below gives a rather different looking solution than when using initial conditions of Sod, but with the same mathematical structure. *In the live notebook, you can easily adjust the initial data and immediately see the resulting solution.*\n\n\n```python\neuler_demos.euler_demo1(rho_l=2.,u_l=0.,p_l=2.5,\n                        rho_r=3.,u_r=0.,p_r=5., gamma=gamma)\n```\n\n\n    interactive(children=(FloatSlider(value=2.0, description='$\\\\rho_l$', max=10.0, min=1.0), FloatSlider(value=0.\u2026\n\n\n\n    VBox(children=(HBox(children=(FloatSlider(value=2.0, description='$\\\\rho_l$', max=10.0, min=1.0), FloatSlider(\u2026\n\n\n\n    Output()\n\n\n## Riemann problems with vacuum\nA vacuum state (with zero pressure and density) in the Euler equations is similar to a dry state (with depth $h=0$) in the shallow water equations.  It can arise in the solution of the Riemann problem in two ways:\n\n1. An initial left or right vacuum state: in this case the Riemann solution consists of a single rarefaction, connecting the non-vacuum state to vacuum.\n2. A problem where the left and right states are not vacuum but middle states are vacuum.  Since this means the middle pressure is smaller than that to the left or right, this can occur only if the 1- and 3-waves are both rarefactions.  These rarefactions are precisely those required to connect the left and right states to the middle vacuum state. \n\n### Initial vacuum state\nNext we start with the density and pressure set to 0 in the left state. The velocity plot looks a bit strange, but note that the velocity is undefined in vacuum.  The solution structure consists of a rarefaction wave, similar to what is observed in the analogous case of a dam break problem with dry land on one side (depth $=0$), as discussed in [Shallow_water](Shallow_water.ipynb).\n\n\n```python\nleft_state  = State(Density =0.,\n                    Velocity = 0.,\n                    Pressure = 0.)\nright_state = State(Density = 1.,\n                    Velocity = -3.,\n                    Pressure = 1.)\n\neuler.riemann_solution(left_state,right_state)\n```\n\n\n    interactive(children=(FloatSlider(value=0.5, description='t', max=0.9), Dropdown(description='Show characteris\u2026\n\n\n\n```python\neuler.phase_plane_plot(left_state, right_state)\n```\n\n\n    \n\n    \n\n\nThe phase plane plot may look odd, but recall that in the vacuum state velocity is undefined, and since $p_\\ell = p_m = 0$, the left and middle states are actually the same.\n\n### Middle vacuum state\n\nFinally, we consider an example where there is sufficiently strong outflow ($u_\\ell<0$ and $u_r>0$) that a vacuum state forms, analogous to the dry state that appears in the similar example in [Shallow_water](Shallow_water.ipynb).\n\n\n```python\nleft_state  = State(Density =1.,\n                    Velocity = -10.,\n                    Pressure = 1.)\nright_state = State(Density = 1.,\n                    Velocity = 10.,\n                    Pressure = 1.)\n\neuler.riemann_solution(left_state,right_state)\n```\n\n\n    interactive(children=(FloatSlider(value=0.5, description='t', max=0.9), Dropdown(description='Show characteris\u2026\n\n\n\n```python\neuler.phase_plane_plot(left_state, right_state)\n```\n\n\n    \n\n    \n\n\nAgain the phase plane plot may look odd, but since the velocity is undefined in the vacuum state the middle state with $p_m = 0$ actually connects to both integral curves, which correspond to the two outgoing rarefaction waves.\n\n\n```python\n\n```\n", "meta": {"hexsha": "d9f00b4f81666abcffaf59a2432f7ea407c7dd26", "size": 365496, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "src/euler/Euler.ipynb", "max_stars_repo_name": "alsam/Claw.jl", "max_stars_repo_head_hexsha": "81479366c2be728101d6499c27b95ebf5fcdf7a3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2018-12-24T01:58:45.000Z", "max_stars_repo_stars_event_max_datetime": "2020-01-14T06:22:20.000Z", "max_issues_repo_path": "src/euler/Euler.ipynb", "max_issues_repo_name": "alsam/Claw.jl", "max_issues_repo_head_hexsha": "81479366c2be728101d6499c27b95ebf5fcdf7a3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/euler/Euler.ipynb", "max_forks_repo_name": "alsam/Claw.jl", "max_forks_repo_head_hexsha": "81479366c2be728101d6499c27b95ebf5fcdf7a3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 42.3517960603, "max_line_length": 898, "alphanum_fraction": 0.4939041741, "converted": true, "num_tokens": 5859, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "# **CS224W - Colab 3**\n\nIn Colab 2 we constructed GNN models by using PyTorch Geometric's built in GCN layer, `GCNConv`. In this Colab we will go a step deeper and implement the **GraphSAGE** ([Hamilton et al. (2017)](https://arxiv.org/abs/1706.02216)) layer directly. Then we will run our models on the CORA dataset, which is a standard citation network benchmark dataset.\n\n**Note**: Make sure to **sequentially run all the cells in each section** so that the intermediate variables / packages will carry over to the next cell\n\nHave fun and good luck on Colab 3 :)\n\n# Device\nWe recommend using a GPU for this Colab.\n\nPlease click `Runtime` and then `Change runtime type`. Then set the `hardware accelerator` to **GPU**.\n\n## Installation\n\n\n```python\n# Install torch geometric\nimport os\nif 'IS_GRADESCOPE_ENV' not in os.environ:\n  !pip install torch-scatter -f https://pytorch-geometric.com/whl/torch-1.9.0+cu111.html\n  !pip install torch-sparse -f https://pytorch-geometric.com/whl/torch-1.9.0+cu111.html\n  !pip install torch-geometric\n  !pip install -q git+https://github.com/snap-stanford/deepsnap.git\n```\n\n    Looking in links: https://pytorch-geometric.com/whl/torch-1.9.0+cu111.html\n    Requirement already satisfied: torch-scatter in /home/andreas/.local/lib/python3.8/site-packages (2.0.9)\n    Looking in links: https://pytorch-geometric.com/whl/torch-1.9.0+cu111.html\n    Requirement already satisfied: torch-sparse in /home/andreas/.local/lib/python3.8/site-packages (0.6.12)\n    Requirement already satisfied: scipy in /home/andreas/.local/lib/python3.8/site-packages (from torch-sparse) (1.7.2)\n    Requirement already satisfied: numpy<1.23.0,>=1.16.5 in /home/andreas/.local/lib/python3.8/site-packages (from scipy->torch-sparse) (1.21.4)\n    Requirement already satisfied: torch-geometric in /home/andreas/.local/lib/python3.8/site-packages (2.0.2)\n    Requirement already satisfied: PyYAML in /usr/lib/python3/dist-packages (from torch-geometric) (5.3.1)\n    Requirement already satisfied: googledrivedownloader in /home/andreas/.local/lib/python3.8/site-packages (from torch-geometric) (0.4)\n    Requirement already satisfied: tqdm in /home/andreas/.local/lib/python3.8/site-packages (from torch-geometric) (4.62.3)\n    Requirement already satisfied: scipy in /home/andreas/.local/lib/python3.8/site-packages (from torch-geometric) (1.7.2)\n    Requirement already satisfied: requests in /usr/lib/python3/dist-packages (from torch-geometric) (2.22.0)\n    Requirement already satisfied: pandas in /home/andreas/.local/lib/python3.8/site-packages (from torch-geometric) (1.3.4)\n    Requirement already satisfied: numpy in /home/andreas/.local/lib/python3.8/site-packages (from torch-geometric) (1.21.4)\n    Requirement already satisfied: rdflib in /home/andreas/.local/lib/python3.8/site-packages (from torch-geometric) (6.0.2)\n    Requirement already satisfied: yacs in /home/andreas/.local/lib/python3.8/site-packages (from torch-geometric) (0.1.8)\n    Requirement already satisfied: jinja2 in /home/andreas/.local/lib/python3.8/site-packages (from torch-geometric) (3.0.3)\n    Requirement already satisfied: scikit-learn in /home/andreas/.local/lib/python3.8/site-packages (from torch-geometric) (1.0.1)\n    Requirement already satisfied: networkx in /home/andreas/.local/lib/python3.8/site-packages (from torch-geometric) (2.6.3)\n    Requirement already satisfied: pyparsing in /home/andreas/.local/lib/python3.8/site-packages (from torch-geometric) (3.0.6)\n    Requirement already satisfied: python-dateutil>=2.7.3 in /usr/lib/python3/dist-packages (from pandas->torch-geometric) (2.7.3)\n    Requirement already satisfied: pytz>=2017.3 in /usr/lib/python3/dist-packages (from pandas->torch-geometric) (2019.3)\n    Requirement already satisfied: isodate in /home/andreas/.local/lib/python3.8/site-packages (from rdflib->torch-geometric) (0.6.0)\n    Requirement already satisfied: setuptools in /usr/lib/python3/dist-packages (from rdflib->torch-geometric) (45.2.0)\n    Requirement already satisfied: MarkupSafe>=2.0 in /home/andreas/.local/lib/python3.8/site-packages (from jinja2->torch-geometric) (2.0.1)\n    Requirement already satisfied: threadpoolctl>=2.0.0 in /home/andreas/.local/lib/python3.8/site-packages (from scikit-learn->torch-geometric) (3.0.0)\n    Requirement already satisfied: joblib>=0.11 in /home/andreas/.local/lib/python3.8/site-packages (from scikit-learn->torch-geometric) (1.1.0)\n    Requirement already satisfied: six in /usr/lib/python3/dist-packages (from isodate->rdflib->torch-geometric) (1.14.0)\n\n\n\n```python\nimport torch_geometric\ntorch_geometric.__version__\n```\n\n\n\n\n    '2.0.2'\n\n\n\n# 1) GNN Layers\n\n## Implementing Layer Modules\n\nIn Colab 2, we implemented a GCN model for node and graph classification tasks. However, for that notebook we took advantage of PyG's built in GCN module. For Colab 3, we provide a build upon a general Graph Neural Network Stack, into which we will be able to plugin our own module implementations: GraphSAGE and GAT.\n\nWe will then use our layer implemenations to complete node classification on the CORA dataset, a standard citation network benchmark. In this dataset, nodes correspond to documents and edges correspond to undirected citations. Each node or document in the graph is assigned a class label and features based on the documents binarized bag-of-words representation. Specifically, the Cora graph has 2708 nodes, 5429 edges, 7 prediction classes, and 1433 features per node. \n\n## GNN Stack Module\n\nBelow is the implementation of a general GNN stack, where we can plugin any GNN layer, such as **GraphSage**, **GAT**, etc. This module is provided for you. Your implementations of the **GraphSage** and **GAT** (Colab 4) layers will function as components in the GNNStack Module.\n\n\n```python\nimport torch\nimport torch_scatter\nimport torch.nn as nn\nimport torch.nn.functional as F\n\nimport torch_geometric.nn as pyg_nn\nimport torch_geometric.utils as pyg_utils\n\nfrom torch import Tensor\nfrom typing import Union, Tuple, Optional\nfrom torch_geometric.typing import (OptPairTensor, Adj, Size, NoneType,\n                                    OptTensor)\n\nfrom torch.nn import Parameter, Linear\nfrom torch_sparse import SparseTensor, set_diag\nfrom torch_geometric.nn.conv import MessagePassing\nfrom torch_geometric.utils import remove_self_loops, add_self_loops, softmax\n\nclass GNNStack(torch.nn.Module):\n    def __init__(self, input_dim, hidden_dim, output_dim, args, emb=False):\n        super(GNNStack, self).__init__()\n        conv_model = self.build_conv_model(args.model_type)\n        self.convs = nn.ModuleList()\n        self.convs.append(conv_model(input_dim, hidden_dim))\n        assert (args.num_layers >= 1), 'Number of layers is not >=1'\n        for l in range(args.num_layers-1):\n            self.convs.append(conv_model(args.heads * hidden_dim, hidden_dim))\n        \n        # post-message-passing\n        self.post_mp = nn.Sequential(\n            nn.Linear(args.heads * hidden_dim, hidden_dim), nn.Dropout(args.dropout), \n            nn.Linear(hidden_dim, output_dim))\n\n        self.dropout = args.dropout\n        self.num_layers = args.num_layers\n\n        self.emb = emb\n\n    def build_conv_model(self, model_type):\n        \n        if model_type == 'GraphSage':\n            return GraphSage\n            \n        elif model_type == 'GAT':\n            # When applying GAT with num heads > 1, you need to modify the \n            # input and output dimension of the conv layers (self.convs),\n            # to ensure that the input dim of the next layer is num heads\n            # multiplied by the output dim of the previous layer.\n            # HINT: In case you want to play with multiheads, you need to change the for-loop that builds up self.convs to be\n            # self.convs.append(conv_model(hidden_dim * num_heads, hidden_dim)), \n            # and also the first nn.Linear(hidden_dim * num_heads, hidden_dim) in post-message-passing.\n            return GAT\n\n    def forward(self, data):\n        x, edge_index, batch = data.x, data.edge_index, data.batch\n          \n        for i in range(self.num_layers):\n            x = self.convs[i](x, edge_index)\n            x = F.relu(x)\n            x = F.dropout(x, p=self.dropout,training=self.training)\n\n        x = self.post_mp(x)\n\n        if self.emb == True:\n            return x\n\n        return F.log_softmax(x, dim=1)\n\n    def loss(self, pred, label):\n        return F.nll_loss(pred, label)\n```\n\n## Creating Our Own Message Passing Layer\n\nNow let's start implementing our own message passing layers! Working through this part will help us become acutely familiar with the behind the scenes work of implementing Pytorch Message Passing Layers, allowing us to build our own GNN models. To do so, we will work with and implement 3 critcal functions needed to define a PyG Message Passing Layer: `forward`, `message`, and `aggregate`.\n\nBefore diving head first into the coding details, let us quickly review the key components of the message passing process. To do so, we will focus on a single round of messsage passing with respect to a single central node $x$. Before message passing, $x$ is associated with a feature vector $x^{l-1}$, and the goal of message passing is to update this feature vector as $x^l$. To do so, we implement the following steps: 1) each neighboring node $v$ passes its current message $v^{l-1}$ across the edge $(x, v)$ - 2) for the node $x$, we aggregate all of the messages of the neighboring nodes (for example through a sum or mean) - and 3) we transform the aggregated information by for example applying linear and non-linear transformations. Altogether, the message passing process is applied such that every node $u$ in our graph updates its embedding by acting as the central node $x$ in step 1-3 described above. \n\nNow, we extending this process to that of a single message passing layer, the job of a message passing layer is to update the current feature representation or embedding of each node in a graph by propagating and transforming information within the graph. Overall, the general paradigm of a message passing layers is: 1) pre-processing -> 2) **message passing** / propagation -> 3) post-processing. \n\nThe `forward` fuction that we will implement for our message passing layer captures this execution logic. Namely, the `forward` function handles the pre and post-processing of node features / embeddings, as well as initiates message passing by calling the `propagate` function. \n\n\nThe `propagate` function encapsulates the message passing process! It does so by calling three important functions: 1) `message`, 2) `aggregate`, and 3) `update`. Our implementation will vary slightly from this, as we will not explicitly implement `update`, but instead place the logic for updating node embeddings after message passing and within the `forward` function. To be more specific, after information is propagated (message passing), we can further transform the node embeddings outputed by `propagate`. Therefore, the output of `forward` is exactly the node embeddings after one GNN layer.\n\nLastly, before starting to implement our own layer, let us dig a bit deeper into each of the functions described above:\n\n1. \n\n```\ndef propagate(edge_index, x=(x_i, x_j), extra=(extra_i, extra_j), size=size):\n```\nCalling `propagate` initiates the message passing process. Looking at the function parameters, we highlight a couple of key parameters. \n\n  - `edge_index` is passed to the forward function and captures the edge structure of the graph.\n  - `x=(x_i, x_j)` represents the node features that will be used in message passing. In order to explain why we pass the tuple `(x_i, x_j)`, we first look at how our edges are represented. For every edge $(i, j) \\in \\mathcal{E}$, we can differentiate $i$ as the source or central node ($x_{central}$) and j as the neighboring node ($x_{neighbor}$). \n  \n    Taking the example of message passing above, for a central node $u$ we will aggregate and transform all of the messages associated with the nodes $v$ s.t. $(u, v) \\in \\mathcal{E}$ (i.e. $v \\in \\mathcal{N}_{u}$). Thus we see, the subscripts `_i` and `_j` allow us to specifcally differenciate features associated with central nodes (i.e. nodes  recieving message information) and neighboring nodes (i.e. nodes passing messages). \n\n    This is definitely a somewhat confusing concept; however, one key thing to remember / wrap your head around is that depending on the perspective, a node $x$ acts as a central node or a neighboring node. In fact, in undirected graphs we store both edge directions (i.e. $(i, j)$ and $(j, i)$). From the central node perspective, `x_i`, x is collecting neighboring information to update its embedding. From a neighboring node perspective, `x_j`, x is passing its message information along the edge connecting it to a different central node.\n\n  - `extra=(extra_i, extra_j)` represents additional information that we can associate with each node beyond its current feature embedding. In fact, we can include as many additional parameters of the form `param=(param_i, param_j)` as we would like. Again, we highlight that indexing with `_i` and `_j` allows us to differentiate central and neighboring nodes. \n\n  The output of the `propagate` function is a matrix of node embeddings after the message passing process and has shape $[N, d]$.\n\n2. \n```\ndef message(x_j, ...):\n```\nThe `message` function is called by propagate and constructs the messages from\nneighboring nodes $j$ to central nodes $i$ for each edge $(i, j)$ in *edge_index*. This function can take any argument that was initially passed to `propagate`. Furthermore, we can again differentiate central nodes and neighboring nodes by appending `_i` or `_j` to the variable name, .e.g. `x_i` and `x_j`. Looking more specifically at the variables, we have:\n\n  - `x_j` represents a matrix of feature embeddings for all neighboring nodes passing their messages along their respective edge (i.e. all nodes $j$ for edges $(i, j) \\in \\mathcal{E}$). Thus, its shape is $[|\\mathcal{E}|, d]$!\n  - In implementing GAT we will see how to access additional variables passed to propagate\n\n  Critically, we see that the output of the `message` function is a matrix of neighboring node embeddings ready to be aggregated, having shape $[|\\mathcal{E}|, d]$.\n\n3. \n```\ndef aggregate(self, inputs, index, dim_size = None):\n```\nLastly, the `aggregate` function is used to aggregate the messages from neighboring nodes. Looking at the parameters we highlight:\n\n  - `inputs` represents a matrix of the messages passed from neighboring nodes (i.e. the output of the `message` function).\n  - `index` has the same shape as `inputs` and tells us the central node that corresponding to each of the rows / messages $j$ in the `inputs` matrix. Thus, `index` tells us which rows / messages to aggregate for each central node.\n\n  The output of `aggregate` is of shape $[N, d]$.\n\n\nFor additional resources refer to the PyG documentation for implementing custom message passing layers: https://pytorch-geometric.readthedocs.io/en/latest/notes/create_gnn.html\n\n## GraphSage Implementation\n\nFor our first GNN layer, we will implement the well known GraphSage ([Hamilton et al. (2017)](https://arxiv.org/abs/1706.02216)) layer! \n\nFor a given *central* node $v$ with current embedding $h_v^{l-1}$, the message passing update rule to tranform $h_v^{l-1} \\rightarrow h_v^l$ is as follows: \n\n\\begin{equation}\nh_v^{(l)} = W_l\\cdot h_v^{(l-1)} + W_r \\cdot AGG(\\{h_u^{(l-1)}, \\forall u \\in N(v) \\})\n\\end{equation}\n\nwhere $W_1$ and $W_2$ are learanble weight matrices and the nodes $u$ are *neighboring* nodes. Additionally, we use mean aggregation for simplicity:\n\n\\begin{equation}\nAGG(\\{h_u^{(l-1)}, \\forall u \\in N(v) \\}) = \\frac{1}{|N(v)|} \\sum_{u\\in N(v)} h_u^{(l-1)}\n\\end{equation}\n\nOne thing to note is that we're adding a **skip connection** to our GraphSage implementation through the term $W_l\\cdot h_v^{(l-1)}$. \n\nBefore implementing this update rule, we encourage you to think about how different parts of the formulas above correspond with the functions outlined earlier: 1) `forward`, 2) `message`, and 3) `aggregate`. As a hint, we are given what the aggregation function is (i.e. mean aggregation)! Now the question remains, what are the messages passed by each neighbor nodes and when do we call the `propagate` function? \n\nNote: in this case the message function or messages are actually quite simple. Additionally, remember that the `propagate` function encapsulates the operations of / the outputs of the combined `message` and `aggregate` functions.\n\n\nLastly, $\\ell$-2 normalization of the node embeddings is applied after each iteration.\n\n\nFor the following questions, DON'T refer to any existing implementations online.\n\n\n```python\nclass GraphSage(MessagePassing):\n    \n    def __init__(self, in_channels, out_channels, normalize = True,\n                 bias = False, **kwargs):  \n        super(GraphSage, self).__init__(**kwargs)\n\n        self.in_channels = in_channels\n        self.out_channels = out_channels\n        self.normalize = normalize\n        self.lin_l = None\n        self.lin_r = None\n\n        ############################################################################\n        # TODO: Your code here! \n        # Define the layers needed for the message and update functions below.\n        # self.lin_l is the linear transformation that you apply to embedding \n        #            for central node.\n        # self.lin_r is the linear transformation that you apply to aggregated \n        #            message from neighbors.\n        # Don't forget the bias!\n        # Our implementation is ~2 lines, but don't worry if you deviate from this.\n        self.lin_l = nn.Linear(self.in_channels, self.out_channels, bias = bias)\n        self.lin_r = nn.Linear(self.in_channels, self.out_channels, bias = bias) # not done...\n\n        ############################################################################\n\n        self.reset_parameters()\n\n    def reset_parameters(self):\n        self.lin_l.reset_parameters()\n        self.lin_r.reset_parameters()\n\n    def forward(self, x, edge_index, size = None):\n        \"\"\"\"\"\"\n\n        out = None\n\n        ############################################################################\n        # TODO: Your code here! \n        # Implement message passing, as well as any post-processing (our update rule).\n        # 1. Call the propagate function to conduct the message passing.\n        #    1.1 See the description of propagate above or the following link for more information: \n        #        https://pytorch-geometric.readthedocs.io/en/latest/notes/create_gnn.html\n        #    1.2 We will only use the representation for neighbor nodes (x_j), so by default\n        #        we pass the same representation for central and neighbor nodes as x=(x, x). \n        # 2. Update our node embedding with skip connection from the previous layer.\n        # 3. If normalize is set, do L-2 normalization (defined in \n        #    torch.nn.functional)\n        #\n        # Our implementation is ~5 lines, but don't worry if you deviate from this.\n\n        ############################################################################\n        x_j = torch.zeros([edge_index[1].size()[0],x.size()[1]])\n        for i in range(edge_index[1].size()[0]): # find better way    \n            x_j[i] = x[edge_index[1][i]]\n        agg = self.aggregate(x_j, edge_index[0],x.size()[0])\n        out = self.lin_l(x) + self.lin_r(agg)\n\n        if (self.normalize == 1):\n            out=F.normalize(out,p=2)\n        return out #Combined message parssing for all nodes\n\n    def message(self, x_j):\n        #print(\"test\")\n        out = None\n\n        ############################################################################\n        # TODO: Your code here! \n        # Implement your message function here.\n        # Hint: Look at the formulation of the mean aggregation function, focusing on \n        # what message each neighboring node passes.\n        #\n        # Our implementation is ~1 lines, but don't worry if you deviate from this.\n        ############################################################################\n\n        return out\n\n    def aggregate(self, inputs, index, dim_size = None):\n\n        #out = None\n\n        # The axis along which to index number of nodes.\n        node_dim = self.node_dim\n\n        ############################################################################\n        # TODO: Your code here! \n        # Implement your aggregate function here.\n        # See here as how to use torch_scatter.scatter: \n        # https://pytorch-scatter.readthedocs.io/en/latest/functions/scatter.html#torch_scatter.scatter\n        #\n        # Our implementation is ~1 lines, but don't worry if you deviate from this.\n        out = torch_scatter.scatter(inputs, index, node_dim, dim_size=dim_size, reduce=\"mean\")\n\n\n        \n\n        ############################################################################\n\n        return out\n\n```\n\n## Building Optimizers\n\nThis function has been implemented for you. **For grading purposes please use the default Adam optimizer**, but feel free to play with other types of optimizers on your own.\n\n\n```python\nimport torch.optim as optim\n\ndef build_optimizer(args, params):\n    weight_decay = args.weight_decay\n    filter_fn = filter(lambda p : p.requires_grad, params)\n    if args.opt == 'adam':\n        optimizer = optim.Adam(filter_fn, lr=args.lr, weight_decay=weight_decay)\n    elif args.opt == 'sgd':\n        optimizer = optim.SGD(filter_fn, lr=args.lr, momentum=0.95, weight_decay=weight_decay)\n    elif args.opt == 'rmsprop':\n        optimizer = optim.RMSprop(filter_fn, lr=args.lr, weight_decay=weight_decay)\n    elif args.opt == 'adagrad':\n        optimizer = optim.Adagrad(filter_fn, lr=args.lr, weight_decay=weight_decay)\n    if args.opt_scheduler == 'none':\n        return None, optimizer\n    elif args.opt_scheduler == 'step':\n        scheduler = optim.lr_scheduler.StepLR(optimizer, step_size=args.opt_decay_step, gamma=args.opt_decay_rate)\n    elif args.opt_scheduler == 'cos':\n        scheduler = optim.lr_scheduler.CosineAnnealingLR(optimizer, T_max=args.opt_restart)\n    return scheduler, optimizer\n```\n\n## Training and Testing\n\nHere we provide you with the functions to train and test. **Please do not modify this part for grading purposes.**\n\n\n```python\nimport time\n\nimport networkx as nx\nimport numpy as np\nimport torch\nimport torch.optim as optim\nfrom tqdm import trange\nimport pandas as pd\nimport copy\n\nfrom torch_geometric.datasets import TUDataset\nfrom torch_geometric.datasets import Planetoid\nfrom torch_geometric.data import DataLoader\n\nimport torch_geometric.nn as pyg_nn\n\nimport matplotlib.pyplot as plt\n\n\ndef train(dataset, args):\n    \n    print(\"Node task. test set size:\", np.sum(dataset[0]['test_mask'].numpy()))\n    \n    test_loader = loader = DataLoader(dataset, batch_size=args.batch_size, shuffle=False)\n    \n    # build model\n    model = GNNStack(dataset.num_node_features, args.hidden_dim, dataset.num_classes, \n                            args)\n    scheduler, opt = build_optimizer(args, model.parameters())\n    \n    # train\n    losses = []\n    test_accs = []\n    best_acc = 0\n    best_model = None\n    for epoch in trange(args.epochs, desc=\"Training\", unit=\"Epochs\"):\n        total_loss = 0\n        model.train()\n        for batch in loader:\n            opt.zero_grad()\n            pred = model(batch) # runs here?\n            label = batch.y\n            pred = pred[batch.train_mask]\n            label = label[batch.train_mask]\n            loss = model.loss(pred, label)\n            loss.backward()\n            opt.step()\n            total_loss += loss.item() * batch.num_graphs\n        total_loss /= len(loader.dataset)\n        losses.append(total_loss)\n        \n        if epoch % 10 == 0:\n          test_acc = test(test_loader, model)\n          test_accs.append(test_acc)\n          if test_acc > best_acc:\n            best_acc = test_acc\n            best_model = copy.deepcopy(model)\n        else:\n          test_accs.append(test_accs[-1])\n    \n    return test_accs, losses, best_model, best_acc, test_loader\n\ndef test(loader, test_model, is_validation=False, save_model_preds=False, model_type=None):\n    test_model.eval()\n\n    correct = 0\n    # Note that Cora is only one graph!\n    for data in loader:\n        with torch.no_grad():\n            # max(dim=1) returns values, indices tuple; only need indices\n            pred = test_model(data).max(dim=1)[1]\n            label = data.y\n\n        mask = data.val_mask if is_validation else data.test_mask\n        # node classification: only evaluate on nodes in test set\n        pred = pred[mask]\n        label = label[mask]\n\n        if save_model_preds:\n          print (\"Saving Model Predictions for Model Type\", model_type)\n\n          data = {}\n          data['pred'] = pred.view(-1).cpu().detach().numpy()\n          data['label'] = label.view(-1).cpu().detach().numpy()\n\n          df = pd.DataFrame(data=data)\n          # Save locally as csv\n          df.to_csv('CORA-Node-' + model_type + '.csv', sep=',', index=False)\n            \n        correct += pred.eq(label).sum().item()\n\n    total = 0\n    for data in loader.dataset:\n        total += torch.sum(data.val_mask if is_validation else data.test_mask).item()\n\n    return correct / total\n  \nclass objectview(object):\n    def __init__(self, d):\n        self.__dict__ = d\n\n```\n\n## Let's Start the Training!\n\nWe will be working on the CORA dataset on node-level classification.\n\nThis part is implemented for you. **For grading purposes, please do not modify the default parameters.** However, feel free to play with different configurations just for fun!\n\n**Submit your best accuracy and loss on Gradescope.**\n\n\n```python\nif 'IS_GRADESCOPE_ENV' not in os.environ:\n    for args in [\n        {'model_type': 'GraphSage', 'dataset': 'cora', 'num_layers': 2, 'heads': 1, 'batch_size': 32, 'hidden_dim': 32, 'dropout': 0.5, 'epochs': 500, 'opt': 'adam', 'opt_scheduler': 'none', 'opt_restart': 0, 'weight_decay': 5e-3, 'lr': 0.01},\n    ]:\n        args = objectview(args)\n        \n        for model in ['GraphSage']:\n            args.model_type = model\n\n            # Match the dimension.\n            if model == 'GAT':\n              args.heads = 2\n            else:\n              args.heads = 1\n            if args.dataset == 'cora':\n                dataset = Planetoid(root='/tmp/cora', name='Cora')\n            else:\n                raise NotImplementedError(\"Unknown dataset\") \n            \n            test_accs, losses, best_model, best_acc, test_loader = train(dataset, args) \n            \n            \n\n            print(\"Maximum test set accuracy: {0}\".format(max(test_accs)))\n            print(\"Minimum loss: {0}\".format(min(losses)))\n\n            # Run test for our best model to save the predictions!\n            test(test_loader, best_model, is_validation=False, save_model_preds=True, model_type=model)\n\n            plt.title(dataset.name)\n            plt.plot(losses, label=\"training loss\" + \" - \" + args.model_type)\n            plt.plot(test_accs, label=\"test accuracy\" + \" - \" + args.model_type)\n        plt.legend()\n        plt.show()\n\n```\n\n## Question 1.1: What is the maximum accuracy obtained on the test set for GraphSage? (10 points)\n\nRunning the cell above will show the results of your best model and save your best model's predictions to a file named *CORA-Node-GraphSage.csv*.  \n\nAs we have seen before you can view this file by clicking on the *Folder* icon on the left side pannel. When you sumbit your assignment, you will have to download this file and attatch it to your submission.\n", "meta": {"hexsha": "64989ac5e29777676f56fee56cdb0a776f0b9639", "size": 108778, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Colab 3/CS224W - Colab 3_andreas.ipynb", "max_stars_repo_name": "victorcroisfelt/aau-cs224w-ml-with-graphs", "max_stars_repo_head_hexsha": "adb38651be8da98cc574f127763c785ed16dfb5a", "max_stars_repo_licenses": ["Unlicense", "MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Colab 3/CS224W - Colab 3_andreas.ipynb", "max_issues_repo_name": "victorcroisfelt/aau-cs224w-ml-with-graphs", "max_issues_repo_head_hexsha": "adb38651be8da98cc574f127763c785ed16dfb5a", "max_issues_repo_licenses": ["Unlicense", "MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Colab 3/CS224W - Colab 3_andreas.ipynb", "max_forks_repo_name": "victorcroisfelt/aau-cs224w-ml-with-graphs", "max_forks_repo_head_hexsha": "adb38651be8da98cc574f127763c785ed16dfb5a", "max_forks_repo_licenses": ["Unlicense", "MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 143.1289473684, "max_line_length": 42175, "alphanum_fraction": 0.6961885675, "converted": true, "num_tokens": 6621, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.7371581626286834, "lm_q1q2_score": 0.41441299646599733}}
{"text": "```python\nimport numpy as np\nimport numba\nimport matplotlib.pyplot as plt\nimport sympy as sym\nplt.style.use('presentation.mplstyle')\n#%matplotlib notebook\n\ndef d2np(d):\n    \n    names = []\n    numbers = ()\n    dtypes = []\n    for item in d:\n        names += item   \n        if type(d[item]) == float:\n            numbers += (d[item],)\n            dtypes += [(item,float)]\n        if type(d[item]) == int:\n            numbers += (d[item],)\n            dtypes += [(item,int)]\n        if type(d[item]) == np.ndarray:\n            numbers += (d[item],)\n            dtypes += [(item,np.float64,d[item].shape)]\n    return np.array([numbers],dtype=dtypes)\n\n\n```\n\n### Fortescue\n\n\n```python\nalpha = np.exp(2.0/3*np.pi*1j)\nA_0a =  np.array([[1, 1, 1],\n                  [1, alpha**2, alpha],\n                  [1, alpha, alpha**2]])\n\nA_a0 = 1/3* np.array([[1, 1, 1],\n                      [1, alpha, alpha**2],\n                      [1, alpha**2, alpha]])\n```\n\n### Voltage source\n\n\n```python\ntheta = np.deg2rad(20.0)\nV_zero =  0.0*np.exp(1j*0.0)\nV_neg  = 20.0*np.exp(1j*0.0)\nV_pos  =400.0/np.sqrt(3)*np.exp(1j*theta)\n\nV_zpn = np.array([[V_zero],[V_pos],[V_neg]])\n\nV_abc = A_0a @ V_zpn\n```\n\n### Control inputs\n\n\n```python\nL = 500e-6\nR = 0.01\nomega = 2.0*np.pi*50.0\nw = omega\nv_dc = 800.0\n\nV_012 = A_a0 @ V_abc \nv_z = V_012[0,0]\nv_p = V_012[1,0]\nv_n = V_012[2,0]\n```\n\n### PLL\n\n\n```python\ntheta_pll = np.angle(v_p)\n```\n\n### Park\n\n\n```python\nv_dq_z = v_z\nv_dq_p = v_p*np.exp(-1j*theta_pll)*np.sqrt(2)\nv_dq_n = v_n*np.exp( 1j*theta_pll)*np.sqrt(2)\n\nv_d_z = v_dq_z.real # ??\nv_q_z = v_dq_z.imag # ??\nv_d_p = v_dq_p.imag\nv_q_p = v_dq_p.real\nv_d_n = v_dq_n.imag\nv_q_n = v_dq_n.real\n```\n\n### References\n\n\n```python\np_ref = 0.6e6\nq_ref = 0.2e6\n\npq_ref = np.array([p_ref,q_ref,0,0]).reshape(4,1)\ni2p=3/2*np.array([[ v_d_p, v_q_p, v_d_n, v_q_n],  # i_d_p\n                  [-v_q_p, v_d_p,-v_q_n, v_d_n],  # i_q_p\n                  [-v_q_n, v_d_n, v_q_p,-v_d_p],  # i_d_n\n                  [ v_d_n, v_q_n, v_d_p, v_q_p]]) # i_q_n\n\n\np2i=np.linalg.inv(i2p)\n\ni_dq_pn = p2i@pq_ref\n\ni_d_p_ref = 100.0\ni_q_p_ref = 0.0\ni_d_n_ref = 0.0\ni_q_n_ref = 0.0\n\ni_d_p_ref = i_dq_pn[0,0]\ni_q_p_ref = i_dq_pn[1,0]\ni_d_n_ref = i_dq_pn[2,0]\ni_q_n_ref = i_dq_pn[3,0]\n\nmode = 'p_cte'\n\nif mode == 'p_pos_i_n_0':\n    i_d_p_ref = -(0.666666666666667*p_ref*v_d_p*(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2) + 0.666666666666667*q_ref*v_q_p*(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2))/(v_d_n**4 + 2.0*v_d_n**2*v_q_n**2 - v_d_p**4 - 2.0*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)\n    i_q_p_ref = 0.666666666666667*(-p_ref*v_q_p*(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2) + q_ref*v_d_p*(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2))/(v_d_n**4 + 2.0*v_d_n**2*v_q_n**2 - v_d_p**4 - 2.0*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)\n    i_d_n_ref = 0\n    i_q_n_ref = 0\n    \nif mode == 'q_cte':\n    i_d_p_ref = 0.666666666666667*(p_ref*v_d_p*(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2) + q_ref*v_q_p*(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2))/(v_d_n**4 + 2.0*v_d_n**2*v_q_n**2 - v_d_p**4 - 2.0*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)\n    i_q_p_ref = 0.666666666666667*(p_ref*v_q_p*(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2) - q_ref*v_d_p*(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2))/(v_d_n**4 + 2.0*v_d_n**2*v_q_n**2 - v_d_p**4 - 2.0*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)\n    i_d_n_ref = 0.666666666666667*(p_ref*v_d_n*(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2) - q_ref*v_q_n*(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2))/(v_d_n**4 + 2.0*v_d_n**2*v_q_n**2 - v_d_p**4 - 2.0*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)\n    i_q_n_ref = 0.666666666666667*(p_ref*v_q_n*(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2) + q_ref*v_d_n*(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2))/(v_d_n**4 + 2.0*v_d_n**2*v_q_n**2 - v_d_p**4 - 2.0*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)    \n\nif mode == 'pq_cte': # Lipo\n    i_d_p_ref = 0.666666666666667*(-p_ref*v_d_p + q_ref*v_q_p)/(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2)\n    i_q_p_ref = -(0.666666666666667*p_ref*v_q_p + 0.666666666666667*q_ref*v_d_p)/(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2)\n    i_d_n_ref = 0.666666666666667*(p_ref*v_d_n + q_ref*v_q_n)/(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2)\n    i_q_n_ref = 0.666666666666667*(p_ref*v_q_n - q_ref*v_d_n)/(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2)\n\nif mode == 'p_cte':\n    i_d_p_ref = -(0.666666666666667*p_ref*v_d_p*(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2) + 0.666666666666667*q_ref*v_q_p*(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2))/(v_d_n**4 + 2.0*v_d_n**2*v_q_n**2 - v_d_p**4 - 2.0*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)\n    i_q_p_ref = 0.666666666666667*(-p_ref*v_q_p*(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2) + q_ref*v_d_p*(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2))/(v_d_n**4 + 2.0*v_d_n**2*v_q_n**2 - v_d_p**4 - 2.0*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)\n    i_d_n_ref = 0.666666666666667*(p_ref*v_d_n*(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2) - q_ref*v_q_n*(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2))/(v_d_n**4 + 2.0*v_d_n**2*v_q_n**2 - v_d_p**4 - 2.0*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)\n    i_q_n_ref = 0.666666666666667*(p_ref*v_q_n*(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2) + q_ref*v_d_n*(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2))/(v_d_n**4 + 2.0*v_d_n**2*v_q_n**2 - v_d_p**4 - 2.0*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)\n\nif mode == 'z_mode':\n    I_p_ref = np.conj((p_ref+1j*q_ref)/v_p)/3/np.sqrt(3)\n    Z_p = v_p/I_p_ref\n    I_n_ref = np.conj((p_ref+1j*q_ref)/v_n)/3/np.sqrt(3)\n    Z_n = v_n/I_n_ref\n    i_d_p_ref = ((v_q_p + 1j*v_d_p)/Z_p).imag\n    i_q_p_ref = ((v_q_p + 1j*v_d_p)/Z_p).real\n    i_d_n_ref = ((v_q_n + 1j*v_d_n)/Z_n).imag\n    i_q_n_ref = ((v_q_n + 1j*v_d_n)/Z_n).real\n\n\n```\n\n### Control\n\n\n```python\n#L*did = e_d - R*i_d - w*L*i_q - v_d\n#L*diq = e_q - R*i_q + w*L*i_d - v_q\n\neta_d_p = 2.0/v_dc*(R*i_d_p_ref + L*w*i_q_p_ref + v_d_p)\neta_q_p = 2.0/v_dc*(R*i_q_p_ref - L*w*i_d_p_ref + v_q_p)\neta_d_n = 2.0/v_dc*(R*i_d_n_ref + L*w*i_q_n_ref + v_d_n)\neta_q_n = 2.0/v_dc*(R*i_q_n_ref - L*w*i_d_n_ref + v_q_n)\n\neta_dq_p = eta_q_p + 1j*eta_d_p\ne_dq_p = v_dc/2.0*eta_dq_p       # phase-neutral peak value\n \neta_dq_n = eta_q_n + 1j*eta_d_n\ne_dq_n = v_dc/2.0*eta_dq_n       # phase-neutral peak value\n```\n\n### Modulation\n\n\n```python\ne_p = e_dq_p *np.exp( 1j*theta_pll)/np.sqrt(2) # phase-neutral RMS value   \ne_n = e_dq_n *np.exp(-1j*theta_pll)/np.sqrt(2) # phase-neutral RMS value\ne_z = 0.0\n#e_n = 0.0\n\ne_012 = np.array([e_z,e_p,e_n]).reshape(3,1)\ne_abc = A_0a @ e_012 \n```\n\n### Plant\n\n\n```python\nZ_1 = R +1j *L*omega\nZ_2 = Z_1\nZ_0 = Z_1\n\nZ_012 = np.diag([Z_0,Z_1,Z_2])\n\nZ_abc = A_0a @ Z_012 @ A_a0\n\nY_abc = np.linalg.inv(Z_abc)\nI_abc = Y_abc @ (e_abc-V_abc)\n\nI_abc\n```\n\n\n\n\n    array([[ 842.37826331  +4.37596529j],\n           [-374.42050632-862.59440972j],\n           [-467.95775699+858.21844443j]])\n\n\n\n\n```python\nV_abc.T @ np.conj(I_abc)\n```\n\n\n\n\n    array([[ 600000.+200000.j]])\n\n\n\n\n```python\nI_012 = A_a0 @ I_abc \ni_dq_z_out = I_012[0] ## ???\ni_dq_p_out = I_012[1]*np.exp(-1j*theta_pll)*np.sqrt(2)\ni_dq_n_out = I_012[2]*np.exp( 1j*theta_pll)*np.sqrt(2)\n\ni_d_p = i_dq_p_out.imag\ni_q_p = i_dq_p_out.real\ni_d_n = i_dq_n_out.imag\ni_q_n = i_dq_n_out.real\n\nprint(i_d_p_ref,i_d_p)\nprint(i_q_p_ref,i_q_p)\nprint(i_d_n_ref,i_d_n)\nprint(i_q_n_ref,i_q_n)\n\n```\n\n    -405.209221304 [-405.2092213]\n    1233.99987042 [ 1233.99987042]\n    -69.5266782161 [-69.52667822]\n    -88.4204018619 [-88.42040186]\n\n\n## Fisix\n\n\n```python\np_cte_ref = 1.5*i_d_n*v_d_n + 1.5*i_d_p*v_d_p + 1.5*i_q_n*v_q_n + 1.5*i_q_p*v_q_p\np_cos_ref = 1.5*i_d_n*v_d_p + 1.5*i_d_p*v_d_n + 1.5*i_q_n*v_q_p + 1.5*i_q_p*v_q_n\np_sin_ref = 1.5*i_d_n*v_q_p - 1.5*i_d_p*v_q_n - 1.5*i_q_n*v_d_p + 1.5*i_q_p*v_d_n\nq_cte_ref = -1.5*i_d_n*v_q_n - 1.5*i_d_p*v_q_p + 1.5*i_q_n*v_d_n + 1.5*i_q_p*v_d_p\nq_cos_ref = -1.5*i_d_n*v_q_p - 1.5*i_d_p*v_q_n + 1.5*i_q_n*v_d_p + 1.5*i_q_p*v_d_n\nq_sin_ref = 1.5*i_d_n*v_d_p - 1.5*i_d_p*v_d_n + 1.5*i_q_n*v_q_p - 1.5*i_q_p*v_q_n\n\n\nlhs = ['p_cte_ref','p_cos_ref','p_sin_ref','q_cte_ref','q_cos_ref','q_sin_ref']\nrhs = [p_cte_ref,p_cos_ref,p_sin_ref,q_cte_ref,q_cos_ref,q_sin_ref]\nfor lh,rh in zip(lhs,rhs):\n    print('{:s}_ref = {:s}'.format(str(lh) ,str(sym.simplify(rh))))\n```\n\n    p_cte_ref_ref = 600000.000000000\n    p_cos_ref_ref = 2.69210431724787e-10\n    p_sin_ref_ref = 6.18456397205591e-11\n    q_cte_ref_ref = 200000.000000000\n    q_cos_ref_ref = 68121.9540560623\n    q_sin_ref_ref = -86633.9469654325\n\n\n### From phasor to time\n\n\n```python\nt = np.linspace(0.0,0.04-0.04/1000,1000)\n\nv_a = (np.exp(1j*w*t)*V_abc[0]).real*np.sqrt(2)\nv_b = (np.exp(1j*w*t)*V_abc[1]).real*np.sqrt(2)\nv_c = (np.exp(1j*w*t)*V_abc[2]).real*np.sqrt(2)\ni_a = (np.exp(1j*w*t)*I_abc[0]).real*np.sqrt(2)\ni_b = (np.exp(1j*w*t)*I_abc[1]).real*np.sqrt(2)\ni_c = (np.exp(1j*w*t)*I_abc[2]).real*np.sqrt(2)\nv_a_p = (np.exp(1j*(w*t-np.pi/2))*V_abc[0]).real*np.sqrt(2)\nv_b_p = (np.exp(1j*(w*t-np.pi/2))*V_abc[1]).real*np.sqrt(2)\nv_c_p = (np.exp(1j*(w*t-np.pi/2))*V_abc[2]).real*np.sqrt(2)\n#i_a = i_a_p + i_a_n\n#i_b = i_c_p + i_c_n\n#i_c = i_b_p + i_b_n\n\np = v_a*i_a + v_b*i_b + v_c*i_c\nq = (i_a*(v_b-v_c) + i_b*(v_c-v_a) + i_c*(v_a-v_b))/np.sqrt(3)\nq_lipo = v_a_p*i_a + v_b_p*i_b + v_c_p*i_c\n#q = (i_a*(v_c-v_b) + i_b*(v_a-v_c) + i_c*(v_b-v_a))/np.sqrt(3)\n```\n\n\n```python\nI_abc\n```\n\n\n\n\n    array([[ 842.37826331  +4.37596529j],\n           [-374.42050632-862.59440972j],\n           [-467.95775699+858.21844443j]])\n\n\n\n\n```python\nI_zpn = A_a0 @ I_abc\nV_zpn = A_a0 @ V_abc\n\nI_p = I_zpn[1]\nI_n = I_zpn[2]\nV_p = V_zpn[1]\nV_n = V_zpn[2]\n\nw = 2.0*np.pi*50.0\n\ni_alpha_p = (np.exp( 1j*w*t)*I_p).imag*np.sqrt(2)\ni_beta_p  = (np.exp( 1j*w*t)*I_p).real*np.sqrt(2)\ni_alpha_n = (np.exp(-1j*w*t)*I_n).imag*np.sqrt(2)\ni_beta_n  = (np.exp(-1j*w*t)*I_n).real*np.sqrt(2)\n\nv_alpha_p = (np.exp( 1j*w*t)*V_p).imag*np.sqrt(2)\nv_beta_p  = (np.exp( 1j*w*t)*V_p).real*np.sqrt(2)\nv_alpha_n = (np.exp(-1j*w*t)*V_n).imag*np.sqrt(2)\nv_beta_n  = (np.exp(-1j*w*t)*V_n).real*np.sqrt(2)\n\nv_alpha_p_lipo = (-1j*np.exp( 1j*w*t)*V_p).imag*np.sqrt(2)\nv_beta_p_lipo  = (-1j*np.exp( 1j*w*t)*V_p).real*np.sqrt(2)\nv_alpha_n_lipo = (1j*np.exp(-1j*w*t)*V_n).imag*np.sqrt(2)\nv_beta_n_lipo  = (1j*np.exp(-1j*w*t)*V_n).real*np.sqrt(2)\n\ni_alpha = i_alpha_p + i_alpha_n\ni_beta  = i_beta_p + i_beta_n\nv_alpha = v_alpha_p + v_alpha_n\nv_beta  = v_beta_p + v_beta_n\nv_alpha_lipo = v_alpha_p_lipo + v_alpha_n_lipo\nv_beta_lipo  = v_beta_p_lipo + v_beta_n_lipo\n#Clark = 2/3*[[1/np.sqrt(2),1/np.sqrt(2),1/np.sqrt(2)],\n#             [1,-0.5,-0.5]\n#             [0,-np.sqrt(3)/2,np.sqrt(3)/2]]\n\n#i_oab = np.array([0.0,i_alpha,i_beta])\n#v_oab = np.array([0.0,v_alpha,v_beta])\ninv_Clark=np.linalg.inv(Clark)\ndef oab2abc(alpha,beta):\n    N_t = len(alpha)\n    abc = np.zeros((3,N_t))\n    for it in range():\n        abc[:,it] = Clark\n#for \n#v_abc = np.lianlg.solve(Clark,v_oab)\n\np = 3/2*(i_alpha*v_alpha + i_beta*v_beta)\nq = 3/2*(v_alpha*i_beta - v_beta*i_alpha)\nq_lipo = 3/2*(i_alpha*v_alpha_lipo + i_beta*v_beta_lipo)\n```\n\n\n```python\nfig, axes = plt.subplots(nrows=3, ncols=1, figsize=(8, 4), sharex = True)\n\naxes[0].plot(t, v_alpha)\naxes[0].plot(t, v_beta)\n\naxes[1].plot(t, i_alpha)\naxes[1].plot(t, i_beta)\n\naxes[2].plot(t, p/1000)\naxes[2].plot(t, q/1000)\naxes[2].plot(t, q_lipo/1000)\n\nprint('p = ',np.average(p))\nprint('q = ',np.average(q))\nprint('q_lipo = ',np.average(q_lipo))\n\nprint('i_alpha_max = ',np.max(abs(i_alpha)))\nprint('i_beta_max = ',np.max(abs(i_beta)))\n```\n\n\n    \n\n\n\n
                                        \n\n\n    p =  600000.0\n    q =  197022.332506\n    q_lipo =  200000.0\n    i_alpha_max =  1405.03781184\n    i_beta_max =  1193.74749139\n\nLipo\n\nFigure 1\n\np =  500000.0\nq =  1400000.0\nq_lipo =  200000.0\ni_alpha_max =  8080.75866864\ni_beta_max =  1538.33671853\n\n\n```python\n\n```\n\n\n\n\n    0.5714285714285714\n\n\n\n### Reference following check\n\n\n```python\nI_012 = A_a0 @ I_abc \ni_dq_z_out = I_012[0]*np.exp(-1j*theta_pll)*np.sqrt(2)\ni_dq_p_out = I_012[1]*np.exp(-1j*theta_pll)*np.sqrt(2)\ni_dq_n_out = I_012[2]*np.exp(-1j*theta_pll)*np.sqrt(2)\n\ni_d_p = i_dq_p_out.imag\ni_q_p = i_dq_p_out.real\ni_d_n = i_dq_n_out.imag\ni_q_n = i_dq_n_out.real\n\nprint(i_d_p_ref,i_dq_p_out.real)\nprint(i_q_p_ref,i_dq_p_out.imag)\nprint(i_d_n_ref,i_dq_n_out.real)\nprint(i_q_n_ref,i_dq_n_out.imag)\n\np_cte_ref = 1.5*i_d_n*v_d_n + 1.5*i_d_p*v_d_p + 1.5*i_q_n*v_q_n + 1.5*i_q_p*v_q_p\np_cos_ref = 1.5*i_d_n*v_d_p + 1.5*i_d_p*v_d_n + 1.5*i_q_n*v_q_p + 1.5*i_q_p*v_q_n\np_sin_ref =-1.5*i_d_n*v_q_p + 1.5*i_d_p*v_q_n + 1.5*i_q_n*v_d_p - 1.5*i_q_p*v_d_n\nq_cte_ref = 1.5*i_d_n*v_q_n + 1.5*i_d_p*v_q_p - 1.5*i_q_n*v_d_n - 1.5*i_q_p*v_d_p\nq_cos_ref = 1.5*i_d_n*v_q_p + 1.5*i_d_p*v_q_n - 1.5*i_q_n*v_d_p - 1.5*i_q_p*v_d_n\nq_sin_ref = 1.5*i_d_n*v_d_p - 1.5*i_d_p*v_d_n + 1.5*i_q_n*v_q_p - 1.5*i_q_p*v_q_n\n\n# Lipo\np_cte_ref = 1.5*i_d_n*v_d_n + 1.5*i_d_p*v_d_p + 1.5*i_q_n*v_q_n + 1.5*i_q_p*v_q_p\np_cos_ref = 1.5*i_d_n*v_d_p + 1.5*i_d_p*v_d_n + 1.5*i_q_n*v_q_p + 1.5*i_q_p*v_q_n\np_sin_ref = -1.5*i_d_n*v_q_p + 1.5*i_d_p*v_q_n + 1.5*i_q_n*v_d_p - 1.5*i_q_p*v_d_n\nq_cte_ref = -1.5*i_d_n*v_q_n + 1.5*i_d_p*v_q_p + 1.5*i_q_n*v_d_n - 1.5*i_q_p*v_d_p\nq_cos_ref = 1.5*i_d_n*v_q_p - 1.5*i_d_p*v_q_n - 1.5*i_q_n*v_d_p + 1.5*i_q_p*v_d_n\nq_sin_ref = 1.5*i_d_n*v_d_p + 1.5*i_d_p*v_d_n + 1.5*i_q_n*v_q_p + 1.5*i_q_p*v_q_n\n\nlhs = ['p_cte_ref','p_cos_ref','p_sin_ref','q_cte_ref','q_cos_ref','q_sin_ref']\nrhs = [p_cte_ref,p_cos_ref,p_sin_ref,q_cte_ref,q_cos_ref,q_sin_ref]\nfor lh,rh in zip(lhs,rhs):\n    print('{:s}_ref = {:s}'.format(str(lh) ,str(sym.simplify(rh))))\n\n```\n\n    -1632.99316186 [ 4082.48290464]\n    4082.48290464 [-1632.99316186]\n    -2538.14986202 [-3806.00464724]\n    -2838.62559665 [-119.70223554]\n    p_cte_ref_ref = 465260.769509609\n    p_cos_ref_ref = -473917.012361638\n    p_sin_ref_ref = -1184792.53090409\n    q_cte_ref_ref = -1304554.74865842\n    q_cos_ref_ref = 1184792.53090409\n    q_sin_ref_ref = -473917.012361638\n\n\n### Positive sequence calculation\n\n\n```python\nZ = R +1j *L*omega\nI_pos = (e_p - v_p)/Z\nI_pos\n```\n\n\n\n\n    (757.18068090442762-4.5051966725939852e-13j)\n\n\n\n\n```python\nS  =V_abc.T @ np.conj(I_abc)\nS\n```\n\n\n\n\n    array([[ 500000. +2.04844920e-08j]])\n\n\n\n\n```python\nI_012 = A_a0 @ I_abc \nI_012*np.sqrt(2)\n```\n\n\n\n\n    array([[ -6.02915504e-14 -8.03887339e-14j],\n           [  1.07081519e+03 -4.82834833e-11j],\n           [ -2.31838289e+02 +2.77743076e-11j]])\n\n\n\n\n```python\nimport sympy as sym\n```\n\n\n```python\nv_d_p,v_q_p,v_d_n,v_q_n = sym.symbols('v_d_p,v_q_p,v_d_n,v_q_n')\n\ni2p = sym.Matrix([[ v_d_p, v_q_p, v_d_n, v_q_n],\n                  [-v_q_p, v_d_p,-v_q_n, v_d_n],\n                  [-v_q_n, v_d_n, v_q_p,-v_d_p],\n                  [ v_d_n, v_q_n, v_d_p, v_q_p]])\n\n\np2i = sym.simplify(i2p.inv())\n```\n\n\n```python\nsym.simplify(p2i)\n```\n\n\n\n\n    Matrix([\n    [-v_d_p/(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2), -v_q_p/(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2), (-v_d_n**2*v_q_n + 2*v_d_n*v_d_p*v_q_p - v_d_p**2*v_q_n - v_q_n**3 + v_q_n*v_q_p**2)/(v_d_n**4 + 2*v_d_n**2*v_q_n**2 - v_d_p**4 - 2*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4),  (v_d_n**3 + v_d_n*v_d_p**2 + v_d_n*v_q_n**2 - v_d_n*v_q_p**2 + 2*v_d_p*v_q_n*v_q_p)/(v_d_n**4 + 2*v_d_n**2*v_q_n**2 - v_d_p**4 - 2*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)],\n    [-v_q_p/(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2),  v_d_p/(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2),  (v_d_n**3 - v_d_n*v_d_p**2 + v_d_n*v_q_n**2 + v_d_n*v_q_p**2 - 2*v_d_p*v_q_n*v_q_p)/(v_d_n**4 + 2*v_d_n**2*v_q_n**2 - v_d_p**4 - 2*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4),  (v_d_n**2*v_q_n + 2*v_d_n*v_d_p*v_q_p - v_d_p**2*v_q_n + v_q_n**3 + v_q_n*v_q_p**2)/(v_d_n**4 + 2*v_d_n**2*v_q_n**2 - v_d_p**4 - 2*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)],\n    [ v_d_n/(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2), -v_q_n/(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2), (-v_d_n**2*v_q_p + 2*v_d_n*v_d_p*v_q_n - v_d_p**2*v_q_p + v_q_n**2*v_q_p - v_q_p**3)/(v_d_n**4 + 2*v_d_n**2*v_q_n**2 - v_d_p**4 - 2*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4), (-v_d_n**2*v_d_p - 2*v_d_n*v_q_n*v_q_p - v_d_p**3 + v_d_p*v_q_n**2 - v_d_p*v_q_p**2)/(v_d_n**4 + 2*v_d_n**2*v_q_n**2 - v_d_p**4 - 2*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)],\n    [ v_q_n/(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2),  v_d_n/(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2), (-v_d_n**2*v_d_p - 2*v_d_n*v_q_n*v_q_p + v_d_p**3 + v_d_p*v_q_n**2 + v_d_p*v_q_p**2)/(v_d_n**4 + 2*v_d_n**2*v_q_n**2 - v_d_p**4 - 2*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4),  (v_d_n**2*v_q_p - 2*v_d_n*v_d_p*v_q_n - v_d_p**2*v_q_p - v_q_n**2*v_q_p - v_q_p**3)/(v_d_n**4 + 2*v_d_n**2*v_q_n**2 - v_d_p**4 - 2*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)]])\n\n\n\n\n```python\ntheta = np.deg2rad(0.0)\nphi = np.deg2rad(90.0)\nV_zero =  0.0*np.exp(1j*0.0)\nV_neg  =100.0*np.exp(1j*0.0)\nV_pos  =231.0*np.exp(1j*theta)\n\nV_012 = np.array([[V_zero],[V_pos],[V_neg]])\n\nV_abc = A_0a @ V_012\n\nI_zero =   0.0*np.exp(1j*0.0)\nI_neg  =   0.0*np.exp(1j*(theta+phi))\nI_pos  = 10.0*np.exp(1j*(theta-phi))\n\ns_012 = 500e3\nsin_012 = 0.0\ncos_012 = 0.0\nI_pos = (V_neg*sin_012 - V_pos*s_012)/(3*(V_neg**2 - V_pos**2))\nI_neg = (V_neg*s_012 - V_pos*sin_012)/(3*(V_neg**2 - V_pos**2))\n\n#I_pos = (-V_neg*sin_012 + V_pos*s_012)/(3*(V_neg**2 + V_pos**2))\n#I_neg = ( V_neg*s_012 + V_pos*sin_012)/(3*(V_neg**2 + V_pos**2))\n#I = 1j\n#I_pos = 0.333333333333333*(V_neg*sin_012 - V_pos*s_012*(1.0 + I))/(V_neg**2*(1.0 - I) - V_pos**2*(1.0 + I))\n#I_neg = 0.333333333333333*(V_neg*s_012*(1.0 - I) - V_pos*sin_012)/(V_neg**2*(1.0 - I) - V_pos**2*(1.0 + I))\n\n#I_pos = 0.333333333333333*(V_neg*sin_012 - V_pos*s_012*(1.0 - I))/(V_neg**2*(1.0 + I) - V_pos**2*(1.0 - I))\n#I_neg = 0.333333333333333*(V_neg*s_012*(1.0 + I) - V_pos*sin_012)/(V_neg**2*(1.0 + I) - V_pos**2*(1.0 - I))\n\n#I_pos = 0.333333333333333*(I*V_neg*cos_012 + V_pos*s_012)/(V_neg**2 + V_pos**2)\n#I_neg = 0.333333333333333*(V_neg*s_012 - I*V_pos*cos_012)/(V_neg**2 + V_pos**2)\n\n#I_pos= (0.166666666666667 - 0.166666666666667*I)*(V_neg*(cos_012 + sin_012) - V_pos*s_012*(1.0 + I))/(V_neg**2 - V_pos**2)\n#I_neg= (0.166666666666667 - 0.166666666666667*I)*(V_neg*s_012*(1.0 + I) - V_pos*(cos_012 + sin_012))/(V_neg**2 - V_pos**2)\n\n#I_neg = (cos_012 + sin_012)/(6*V_pos)\n#I_pos = (-V_neg*(cos_012 + sin_012) + 2*V_pos*s_012)/(6*V_pos**2)\nI_pos = np.conj(s_012/(3*V_pos))\nI_neg = -V_neg*I_pos/(V_pos)\n\nI_012 = np.array([[I_zero],[I_pos],[I_neg]])\n\nI_abc = A_0a @ I_012\n\n```\n\n\n```python\nv_abc = (np.exp(1j*2.0*np.pi*50.0*t)*V_abc).real*np.sqrt(2)\ni_abc = (np.exp(1j*2.0*np.pi*50.0*t)*I_abc).real*np.sqrt(2)\n\np = np.sum(v_abc * i_abc, axis=0)\nq = -((v_abc[1]- v_abc[2]) * i_abc[0] + (v_abc[2]- v_abc[0]) * i_abc[1]  + (v_abc[0]- v_abc[1]) * i_abc[2] )/np.sqrt(3)\n```\n\n\n```python\nfig, axes = plt.subplots(nrows=3, ncols=1, figsize=(8, 6), sharex = True)\n\naxes[0].plot(t, v_abc[0,:])\naxes[0].plot(t, v_abc[1,:])\naxes[0].plot(t, v_abc[2,:])\n\naxes[1].plot(t, i_abc[0,:])\naxes[1].plot(t, i_abc[1,:])\naxes[1].plot(t, i_abc[2,:])\n\naxes[2].plot(t, p/1000)\naxes[2].plot(t, q/1000)\n```\n\n\n    \n\n\n\n\n\n\n\n\n\n    []\n\n\n\n\n```python\n3*V_pos*I_pos\n```\n\n\n\n\n    (500000+0j)\n\n\n\n\n```python\n3*V_neg*I_neg\n```\n\n\n\n\n    (-93701.392402691112+0j)\n\n\n\n\n```python\ns_012 = 3*V_pos*I_pos + 3*V_neg*I_neg\n```\n\n\n```python\ns_012\n```\n\n\n\n\n    (406298.60759730887+0j)\n\n\n\n\n```python\nsin_012 = 3*V_pos*I_neg + 3*V_neg*I_pos \ncos_012 = 3*V_pos*I_neg - 3*V_neg*I_pos\n```\n\n\n```python\nprint(sin_012,cos_012)\n```\n\n    (-2.91038304567e-11+0j) (-432900.4329+0j)\n\n\n\n```python\ns_012,sin_012,cos_012,V_pos,I_pos,V_neg,I_neg = sym.symbols('s_012,sin_012,cos_012,V_pos,I_pos,V_neg,I_neg ')\n\nsin_012_ = 3*V_pos*I_neg + 3*V_neg*I_pos \ncos_012_ = 3*V_pos*I_neg - 3*V_neg*I_pos\n\neq1 = -s_012 + 3*V_pos*I_pos + 3*V_neg*I_neg\neq2 = sin_012-sin_012_ - cos_012+cos_012_\nsym.solve([eq1,eq2],[I_pos,I_neg])\n```\n\n\n\n\n    {I_neg: (2*V_neg*s_012 + V_pos*(cos_012 - sin_012))/(6*V_neg**2),\n     I_pos: -(cos_012 - sin_012)/(6*V_neg)}\n\n\n\n\n```python\n\n```\n\n\n```python\nI_pos\n```\n\n## Control Fisix\n\n\n```python\nfrom sympy.functions import re,im\n\nv_d_p,v_q_p,v_d_n,v_q_n,i_d_p,i_q_p,i_d_n,i_q_n,wt = sym.symbols('v_d_p,v_q_p,v_d_n,v_q_n,i_d_p,i_q_p,i_d_n,i_q_n,wt',real=True)\np_ref,q_ref = sym.symbols('p_ref,q_ref',real=True)\n\nexp_p = sym.cos( wt)+1j*sym.sin( wt)\nexp_n = sym.cos(-wt)+1j*sym.sin(-wt)\n\nv_dq_p = v_q_p + 1j*v_d_p\nv_dq_n = v_q_n + 1j*v_d_n\ni_dq_p = i_q_p + 1j*i_d_p\ni_dq_n = i_q_n + 1j*i_d_n\n\ns = 3/2*(v_dq_p*exp_p + v_dq_n*exp_n)*sym.conjugate(i_dq_p*exp_p + i_dq_n*exp_n)\ns = sym.simplify(sym.factor(sym.expand(s)))\n\n```\n\n\n```python\np = sym.collect(re(s),[sym.cos(2*wt),sym.sin(2*wt)])\nq = sym.collect(im(s),[sym.cos(2*wt),sym.sin(2*wt)])\n```\n\n\n```python\np_cos = p.diff(sym.cos(2*wt))\np_sin = p.diff(sym.sin(2*wt))\np_cte = sym.simplify(p - p_cos*sym.cos(2*wt) - p_sin*sym.sin(2*wt))\nq_cos = q.diff(sym.cos(2*wt))\nq_sin = q.diff(sym.sin(2*wt))\nq_cte = sym.simplify(q - q_cos*sym.cos(2*wt) - q_sin*sym.sin(2*wt))\n```\n\n\n```python\nlhs = ['p_cte','p_cos','p_sin','q_cte','q_cos','q_sin']\nrhs = [p_cte,p_cos,p_sin,q_cte,q_cos,q_sin]\nfor lh,rh in zip(lhs,rhs):\n    print('{:s}_ref = {:s}'.format(str(lh) ,str(sym.simplify(rh))))\n```\n\n    p_cte_ref = 1.5*i_d_n*v_d_n + 1.5*i_d_p*v_d_p + 1.5*i_q_n*v_q_n + 1.5*i_q_p*v_q_p\n    p_cos_ref = 1.5*i_d_n*v_d_p + 1.5*i_d_p*v_d_n + 1.5*i_q_n*v_q_p + 1.5*i_q_p*v_q_n\n    p_sin_ref = 1.5*i_d_n*v_q_p - 1.5*i_d_p*v_q_n - 1.5*i_q_n*v_d_p + 1.5*i_q_p*v_d_n\n    q_cte_ref = -1.5*i_d_n*v_q_n - 1.5*i_d_p*v_q_p + 1.5*i_q_n*v_d_n + 1.5*i_q_p*v_d_p\n    q_cos_ref = -1.5*i_d_n*v_q_p - 1.5*i_d_p*v_q_n + 1.5*i_q_n*v_d_p + 1.5*i_q_p*v_d_n\n    q_sin_ref = 1.5*i_d_n*v_d_p - 1.5*i_d_p*v_d_n + 1.5*i_q_n*v_q_p - 1.5*i_q_p*v_q_n\n\n\n### References for p constant\n\n\n```python\nsol = sym.solve([p_cte-p_ref,q_cte-q_ref,p_cos,p_sin],[i_d_p,i_q_p,i_d_n,i_q_n])\nfor item in [i_d_p,i_q_p,i_d_n,i_q_n]:\n    print('{:s}_ref = {:s}'.format(str(item) ,str(sym.simplify(sol[item]))))\n```\n\n    i_d_p_ref = -(0.666666666666667*p_ref*v_d_p*(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2) + 0.666666666666667*q_ref*v_q_p*(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2))/(v_d_n**4 + 2.0*v_d_n**2*v_q_n**2 - v_d_p**4 - 2.0*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)\n    i_q_p_ref = 0.666666666666667*(-p_ref*v_q_p*(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2) + q_ref*v_d_p*(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2))/(v_d_n**4 + 2.0*v_d_n**2*v_q_n**2 - v_d_p**4 - 2.0*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)\n    i_d_n_ref = 0.666666666666667*(p_ref*v_d_n*(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2) - q_ref*v_q_n*(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2))/(v_d_n**4 + 2.0*v_d_n**2*v_q_n**2 - v_d_p**4 - 2.0*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)\n    i_q_n_ref = 0.666666666666667*(p_ref*v_q_n*(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2) + q_ref*v_d_n*(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2))/(v_d_n**4 + 2.0*v_d_n**2*v_q_n**2 - v_d_p**4 - 2.0*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)\n\n\n### References for q constant\n\n\n```python\nsol = sym.solve([p_cte-p_ref,q_cte-q_ref,q_cos,q_sin],[i_d_p,i_q_p,i_d_n,i_q_n])\nfor item in [i_d_p,i_q_p,i_d_n,i_q_n]:\n    print('{:s}_ref = {:s}'.format(str(item) ,str(sym.simplify(sol[item]))))\n```\n\n    i_d_p_ref = 0.666666666666667*(p_ref*v_d_p*(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2) + q_ref*v_q_p*(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2))/(v_d_n**4 + 2.0*v_d_n**2*v_q_n**2 - v_d_p**4 - 2.0*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)\n    i_q_p_ref = 0.666666666666667*(p_ref*v_q_p*(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2) - q_ref*v_d_p*(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2))/(v_d_n**4 + 2.0*v_d_n**2*v_q_n**2 - v_d_p**4 - 2.0*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)\n    i_d_n_ref = 0.666666666666667*(p_ref*v_d_n*(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2) - q_ref*v_q_n*(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2))/(v_d_n**4 + 2.0*v_d_n**2*v_q_n**2 - v_d_p**4 - 2.0*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)\n    i_q_n_ref = 0.666666666666667*(p_ref*v_q_n*(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2) + q_ref*v_d_n*(v_d_n**2 + v_d_p**2 + v_q_n**2 + v_q_p**2))/(v_d_n**4 + 2.0*v_d_n**2*v_q_n**2 - v_d_p**4 - 2.0*v_d_p**2*v_q_p**2 + v_q_n**4 - v_q_p**4)\n\n\n### References for p and q constant\n\n\n```python\nsol = sym.solve([p_cte-p_ref,q_cte-q_ref,p_cos,q_sin],[i_d_p,i_q_p,i_d_n,i_q_n])\nfor item in [i_d_p,i_q_p,i_d_n,i_q_n]:\n    print('{:s}_ref = {:s}'.format(str(item) ,str(sym.simplify(sol[item]))))\n```\n\n\n```python\n\n```\n\n\n```python\nsym.simplify(p_cos-q_sin)\n```\n\n\n```python\nsym.simplify(p_sin-q_cos)\n```\n\n### Lipo\n\n\n```python\nimport sympy as sym\nfrom sympy.functions import re,im\n\nv_d_p,v_q_p,v_d_n,v_q_n,i_d_p,i_q_p,i_d_n,i_q_n,wt = sym.symbols('v_d_p,v_q_p,v_d_n,v_q_n,i_d_p,i_q_p,i_d_n,i_q_n,wt',real=True)\np_ref,q_ref = sym.symbols('p_ref,q_ref',real=True)\n\nexp_p = sym.cos( wt)+1j*sym.sin( wt)\nexp_n = sym.cos(-wt)+1j*sym.sin(-wt)\n\nv_dq_p = v_d_p + 1j*v_q_p\nv_dq_n = v_d_n + 1j*v_q_n\ni_dq_p = i_d_p + 1j*i_q_p\ni_dq_n = i_d_n + 1j*i_q_n\n\ns = 3/2*(exp_p*v_dq_p + exp_n*v_dq_n)*sym.conjugate(exp_p*i_dq_p + exp_n*i_dq_n)\ns = sym.simplify(sym.factor(sym.expand(s)))\n\nt = 3/2*(-1j*exp_p*v_dq_p + 1j*exp_n*v_dq_n)*sym.conjugate(exp_p*i_dq_p + exp_n*i_dq_n)\nt = sym.simplify(sym.factor(sym.expand(t)))\n\np = sym.collect(re(s),[sym.cos(2*wt),sym.sin(2*wt)])\nq = sym.collect(re(t),[sym.cos(2*wt),sym.sin(2*wt)])\n\np_cos = p.diff(sym.cos(2*wt))\np_sin = p.diff(sym.sin(2*wt))\np_cte = sym.simplify(p - p_cos*sym.cos(2*wt) - p_sin*sym.sin(2*wt))\nq_cos = q.diff(sym.cos(2*wt))\nq_sin = q.diff(sym.sin(2*wt))\nq_cte = sym.simplify(q - q_cos*sym.cos(2*wt) - q_sin*sym.sin(2*wt))\n\nlhs = ['p_cte','p_cos','p_sin','q_cte','q_cos','q_sin']\nrhs = [p_cte,p_cos,p_sin,q_cte,q_cos,q_sin]\nfor lh,rh in zip(lhs,rhs):\n    print('{:s}_ref = {:s}'.format(str(lh) ,str(sym.simplify(rh))))\n```\n\n    p_cte_ref = 1.5*i_d_n*v_d_n + 1.5*i_d_p*v_d_p + 1.5*i_q_n*v_q_n + 1.5*i_q_p*v_q_p\n    p_cos_ref = 1.5*i_d_n*v_d_p + 1.5*i_d_p*v_d_n + 1.5*i_q_n*v_q_p + 1.5*i_q_p*v_q_n\n    p_sin_ref = -1.5*i_d_n*v_q_p + 1.5*i_d_p*v_q_n + 1.5*i_q_n*v_d_p - 1.5*i_q_p*v_d_n\n    q_cte_ref = -1.5*i_d_n*v_q_n + 1.5*i_d_p*v_q_p + 1.5*i_q_n*v_d_n - 1.5*i_q_p*v_d_p\n    q_cos_ref = 1.5*i_d_n*v_q_p - 1.5*i_d_p*v_q_n - 1.5*i_q_n*v_d_p + 1.5*i_q_p*v_d_n\n    q_sin_ref = 1.5*i_d_n*v_d_p + 1.5*i_d_p*v_d_n + 1.5*i_q_n*v_q_p + 1.5*i_q_p*v_q_n\n\n\n\n```python\np\n```\n\n\n\n\n    1.5*i_d_n*v_d_n + 1.5*i_d_p*v_d_p + 1.5*i_q_n*v_q_n + 1.5*i_q_p*v_q_p + (1.5*i_d_n*v_d_p + 1.5*i_d_p*v_d_n + 1.5*i_q_n*v_q_p + 1.5*i_q_p*v_q_n)*cos(2*wt) + (-1.5*i_d_n*v_q_p + 1.5*i_d_p*v_q_n + 1.5*i_q_n*v_d_p - 1.5*i_q_p*v_d_n)*sin(2*wt)\n\n\n\n\n```python\nsol = sym.solve([p_cte-p_ref,q_cte-q_ref,p_cos,p_sin],[i_d_p,i_q_p,i_d_n,i_q_n])\nfor item in [i_d_p,i_q_p,i_d_n,i_q_n]:\n    print('{:s}_ref = {:s}'.format(str(item) ,str(sym.simplify(sol[item]))))\n```\n\n    i_d_p_ref = -(0.666666666666667*p_ref*v_d_p + 0.666666666666667*q_ref*v_q_p)/(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2)\n    i_q_p_ref = 0.666666666666667*(-p_ref*v_q_p + q_ref*v_d_p)/(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2)\n    i_d_n_ref = 0.666666666666667*(p_ref*v_d_n - q_ref*v_q_n)/(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2)\n    i_q_n_ref = 0.666666666666667*(p_ref*v_q_n + q_ref*v_d_n)/(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2)\n\n\n\n```python\nq\n```\n\n\n\n\n    -1.5*i_d_n*v_q_n + 1.5*i_d_p*v_q_p + 1.5*i_q_n*v_d_n - 1.5*i_q_p*v_d_p + (1.5*i_d_n*v_d_p + 1.5*i_d_p*v_d_n + 1.5*i_q_n*v_q_p + 1.5*i_q_p*v_q_n)*sin(2*wt) + (1.5*i_d_n*v_q_p - 1.5*i_d_p*v_q_n - 1.5*i_q_n*v_d_p + 1.5*i_q_p*v_d_n)*cos(2*wt)\n\n\n\n\n```python\np_cte_ref = 1.5*i_d_n*v_d_n + 1.5*i_d_p*v_d_p + 1.5*i_q_n*v_q_n + 1.5*i_q_p*v_q_p\np_cos_ref = 1.5*i_d_n*v_d_p + 1.5*i_d_p*v_d_n + 1.5*i_q_n*v_q_p + 1.5*i_q_p*v_q_n\np_sin_ref = -1.5*i_d_n*v_q_p + 1.5*i_d_p*v_q_n + 1.5*i_q_n*v_d_p - 1.5*i_q_p*v_d_n\nq_cte_ref = -1.5*i_d_n*v_q_n + 1.5*i_d_p*v_q_p + 1.5*i_q_n*v_d_n - 1.5*i_q_p*v_d_p\nq_cos_ref = 1.5*i_d_n*v_q_p - 1.5*i_d_p*v_q_n - 1.5*i_q_n*v_d_p + 1.5*i_q_p*v_d_n\nq_sin_ref = 1.5*i_d_n*v_d_p + 1.5*i_d_p*v_d_n + 1.5*i_q_n*v_q_p + 1.5*i_q_p*v_q_n\n```\n\n\n```python\n## Lipo con dq seg\u00fan fisix\n```\n\n\n```python\nimport sympy as sym\nfrom sympy.functions import re,im\n\nv_d_p,v_q_p,v_d_n,v_q_n,i_d_p,i_q_p,i_d_n,i_q_n,wt = sym.symbols('v_d_p,v_q_p,v_d_n,v_q_n,i_d_p,i_q_p,i_d_n,i_q_n,wt',real=True)\np_ref,q_ref = sym.symbols('p_ref,q_ref',real=True)\n\nexp_p = sym.cos( wt)+1j*sym.sin( wt)\nexp_n = sym.cos(-wt)+1j*sym.sin(-wt)\n\nv_dq_p = v_q_p + 1j*v_d_p\nv_dq_n = v_q_n + 1j*v_d_n\ni_dq_p = i_q_p + 1j*i_d_p\ni_dq_n = i_q_n + 1j*i_d_n\n\ns = 3/2*(exp_p*v_dq_p + exp_n*v_dq_n)*sym.conjugate(exp_p*i_dq_p + exp_n*i_dq_n)\ns = sym.simplify(sym.factor(sym.expand(s)))\n\nt = 3/2*(-1j*exp_p*v_dq_p + 1j*exp_n*v_dq_n)*sym.conjugate(exp_p*i_dq_p + exp_n*i_dq_n)\nt = sym.simplify(sym.factor(sym.expand(t)))\n\np = sym.collect(re(s),[sym.cos(2*wt),sym.sin(2*wt)])\nq = sym.collect(re(t),[sym.cos(2*wt),sym.sin(2*wt)])\n\np_cos = p.diff(sym.cos(2*wt))\np_sin = p.diff(sym.sin(2*wt))\np_cte = sym.simplify(p - p_cos*sym.cos(2*wt) - p_sin*sym.sin(2*wt))\nq_cos = q.diff(sym.cos(2*wt))\nq_sin = q.diff(sym.sin(2*wt))\nq_cte = sym.simplify(q - q_cos*sym.cos(2*wt) - q_sin*sym.sin(2*wt))\n\nlhs = ['p_cte','p_cos','p_sin','q_cte','q_cos','q_sin']\nrhs = [p_cte,p_cos,p_sin,q_cte,q_cos,q_sin]\nfor lh,rh in zip(lhs,rhs):\n    print('{:s}_ref = {:s}'.format(str(lh) ,str(sym.simplify(rh))))\n```\n\n    p_cte_ref = 1.5*i_d_n*v_d_n + 1.5*i_d_p*v_d_p + 1.5*i_q_n*v_q_n + 1.5*i_q_p*v_q_p\n    p_cos_ref = 1.5*i_d_n*v_d_p + 1.5*i_d_p*v_d_n + 1.5*i_q_n*v_q_p + 1.5*i_q_p*v_q_n\n    p_sin_ref = 1.5*i_d_n*v_q_p - 1.5*i_d_p*v_q_n - 1.5*i_q_n*v_d_p + 1.5*i_q_p*v_d_n\n    q_cte_ref = 1.5*i_d_n*v_q_n - 1.5*i_d_p*v_q_p - 1.5*i_q_n*v_d_n + 1.5*i_q_p*v_d_p\n    q_cos_ref = -1.5*i_d_n*v_q_p + 1.5*i_d_p*v_q_n + 1.5*i_q_n*v_d_p - 1.5*i_q_p*v_d_n\n    q_sin_ref = 1.5*i_d_n*v_d_p + 1.5*i_d_p*v_d_n + 1.5*i_q_n*v_q_p + 1.5*i_q_p*v_q_n\n\n\n\n```python\nsol = sym.solve([p_cte-p_ref,q_cte-q_ref,p_cos,p_sin],[i_d_p,i_q_p,i_d_n,i_q_n])\nfor item in [i_d_p,i_q_p,i_d_n,i_q_n]:\n    print('{:s}_ref = {:s}'.format(str(item) ,str(sym.simplify(sol[item]))))\n```\n\n    i_d_p_ref = 0.666666666666667*(-p_ref*v_d_p + q_ref*v_q_p)/(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2)\n    i_q_p_ref = -(0.666666666666667*p_ref*v_q_p + 0.666666666666667*q_ref*v_d_p)/(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2)\n    i_d_n_ref = 0.666666666666667*(p_ref*v_d_n + q_ref*v_q_n)/(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2)\n    i_q_n_ref = 0.666666666666667*(p_ref*v_q_n - q_ref*v_d_n)/(v_d_n**2 - v_d_p**2 + v_q_n**2 - v_q_p**2)\n\n\n\n```python\n\n```\n\n\n```python\n\nClark = sym.Matrix([[1.0/sym.sqrt(2.0),1.0/sym.sqrt(2.0),1.0/sym.sqrt(2.0)],[1.0,-1.0/2.0,-1.0/2.0],[0,-sym.sqrt(3.0)/2.0,sym.sqrt(3.0)/2.0]])\n```\n\n\n```python\nimport numpy as np\nClark = 2/3*np.array([[1/np.sqrt(2),  1/np.sqrt(2),1/np.sqrt(2)],\n                      [            1,         -0.5,        -0.5],\n                      [            0,-np.sqrt(3)/2,np.sqrt(3)/2]])\ninv_Clark = np.linalg.inv(Clark)\n```\n\n\n```python\npasar al tiempo con seq. pos y neg\n```\n\n\n```python\ninv_Clark\n```\n\n\n\n\n    array([[  7.07106781e-01,   1.00000000e+00,   2.40370336e-17],\n           [  7.07106781e-01,  -5.00000000e-01,  -8.66025404e-01],\n           [  7.07106781e-01,  -5.00000000e-01,   8.66025404e-01]])\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "c09efc3addac955003f65c0901c88606d9681c6f", "size": 220368, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "models/vsc_phasor_model.ipynb", "max_stars_repo_name": "pydgrid/pydgrid", "max_stars_repo_head_hexsha": "c56073c385f42883c79333533f7cfb8383a173aa", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 15, "max_stars_repo_stars_event_min_datetime": "2019-01-29T08:22:39.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-13T20:41:32.000Z", "max_issues_repo_path": "models/vsc_phasor_model.ipynb", "max_issues_repo_name": "pydgrid/pydgrid", "max_issues_repo_head_hexsha": "c56073c385f42883c79333533f7cfb8383a173aa", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2017-11-28T21:34:52.000Z", "max_issues_repo_issues_event_max_datetime": "2017-11-28T21:34:52.000Z", "max_forks_repo_path": "models/vsc_phasor_model.ipynb", "max_forks_repo_name": "pydgrid/pydgrid", "max_forks_repo_head_hexsha": "c56073c385f42883c79333533f7cfb8383a173aa", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2018-02-15T02:12:47.000Z", "max_forks_repo_forks_event_max_datetime": "2020-02-16T17:52:15.000Z", "avg_line_length": 69.472887768, "max_line_length": 100095, "alphanum_fraction": 0.7075618965, "converted": true, "num_tokens": 14581, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581626286833, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.4144129964659972}}
{"text": "# Variational Quantum Eigensolver (VQE)\nIn this section, the variational quantum eigensolver (VQE) is run on the simulator using Qulacs to find the ground state of the quantum chemical Hamiltonian obtained using OpenFermion/PySCF.\n\nThis notebook is translated from https://dojo.qulacs.org/ja/latest/notebooks/6.2_qulacs_VQE.html\n\nNecessary packages:\n* qulacs\n* openfermion\n* openfermion-pyscf\n* pyscf\n* scipy\n* numpy\n\n## Install and import necessary packages\n\n\n```python\n## Please execute if various libraries are not installed\n## When use Google Colaboratory, please ignore 'You must restart the runtime in order to use newly installed versions'.\n## Crash when restarting runtime.\n!pip install qulacs pyscf openfermion openfermionpyscf\n```\n\n\n```python\nimport qulacs\nfrom openfermion.transforms import get_fermion_operator, jordan_wigner\nfrom openfermion.transforms import get_sparse_operator\nfrom openfermion.hamiltonians import MolecularData\nfrom openfermionpyscf import run_pyscf\nfrom scipy.optimize import minimize\nfrom pyscf import fci\nimport numpy as np\nimport matplotlib.pyplot as plt\n```\n\n## Create Hamiltonian\nCreate Hamiltonian by PySCF in the same procedure as described in the previous section.\n\n\n```python\nbasis = \"sto-3g\"\nmultiplicity = 1\ncharge = 0\ndistance  = 0.977\ngeometry = [[\"H\", [0,0,0]],[\"H\", [0,0,distance]]]\ndescription  = \"tmp\"\nmolecule = MolecularData(geometry, basis, multiplicity, charge, description)\nmolecule = run_pyscf(molecule,run_scf=1,run_fci=1)\nn_qubit = molecule.n_qubits\nn_electron = molecule.n_electrons\nfermionic_hamiltonian = get_fermion_operator(molecule.get_molecular_hamiltonian())\njw_hamiltonian = jordan_wigner(fermionic_hamiltonian)\n```\n\n## Convert Hamiltonian to qulacs Hamiltonian\nIn Qulacs, observables like Hamiltonians are handled by the `Observable` class. There is a function `create_observable_from_openfermion_text` that converts OpenFermion Hamiltonian to Qulacs `Observable`.\n\n\n```python\nfrom qulacs import Observable\nfrom qulacs.observable import create_observable_from_openfermion_text\nqulacs_hamiltonian = create_observable_from_openfermion_text(str(jw_hamiltonian))\n```\n\n## Construct ansatz\nConstruct a quantum circuit on Qulacs. Here, the quantum circuit is modeled after the experiments with superconducting qubits (A. Kandala et. al. , \u201cHardware-efficient variational quantum eigensolver for small molecules and quantum magnets\u201c, Nature **549**, 242\u2013246).\n\n\n```python\nfrom qulacs import QuantumState, QuantumCircuit\nfrom qulacs.gate import CZ, RY, RZ, merge\n\ndepth = n_qubit\n```\n\n\n```python\ndef he_ansatz_circuit(n_qubit, depth, theta_list):\n    \"\"\"he_ansatz_circuit\n    Returns hardware efficient ansatz circuit.\n\n    Args:\n        n_qubit (:class:`int`):\n            the number of qubit used (equivalent to the number of fermionic modes)\n        depth (:class:`int`):\n            depth of the circuit.\n        theta_list (:class:`numpy.ndarray`):\n            rotation angles.\n    Returns:\n        :class:`qulacs.QuantumCircuit`\n    \"\"\"\n    circuit = QuantumCircuit(n_qubit)\n    for d in range(depth):\n        for i in range(n_qubit):\n            circuit.add_gate(merge(RY(i, theta_list[2*i+2*n_qubit*d]), RZ(i, theta_list[2*i+1+2*n_qubit*d])))\n        for i in range(n_qubit//2):\n            circuit.add_gate(CZ(2*i, 2*i+1))\n        for i in range(n_qubit//2-1):\n            circuit.add_gate(CZ(2*i+1, 2*i+2))\n    for i in range(n_qubit):\n        circuit.add_gate(merge(RY(i, theta_list[2*i+2*n_qubit*depth]), RZ(i, theta_list[2*i+1+2*n_qubit*depth])))\n\n    return circuit\n```\n\n## Define VQE cost function\nAs explained in Section 5-1, VQE obtains an approximate ground state by minimizing the expected value \n\n\\begin{equation}\n\\left=\\left<\\psi(\\theta)|H|\\psi(\\theta)\\right>\n\\end{equation}\n\nof Hamiltonian for state $\\left|\\psi(\\theta)\\right>=U(\\theta)\\left|0\\right>$ output from the quantum circuit $U(\\theta)$ with parameters. The following defines a function that returns the expectation value of this Hamiltonian.\n\n\n\n```python\ndef cost(theta_list):\n    state = QuantumState(n_qubit) #Prepare |00000>\n    circuit = he_ansatz_circuit(n_qubit, depth, theta_list) #Construct quantum circuit\n    circuit.update_quantum_state(state) #Operate quantum circuit on state\n    return qulacs_hamiltonian.get_expectation_value(state) #Calculate expectation value of Hamiltonian\n```\n\n## Run VQE\nNow everthing is prepared, run VQE. For optimization, the BFGS method implemented in scipy is applied, and initial parameters are randomly selected. This process should end in tens of seconds.\n\n\n```python\ncost_history = []\ninit_theta_list = np.random.random(2*n_qubit*(depth+1))*1e-1\ncost_history.append(cost(init_theta_list))\nmethod = \"BFGS\"\noptions = {\"disp\": True, \"maxiter\": 50, \"gtol\": 1e-6}\nopt = minimize(cost, init_theta_list,\n               method=method,\n               callback=lambda x: cost_history.append(cost(x)))\n```\n\nThe results can be plotted to see that they converge to the correct solution.\n\n\n```python\nplt.rcParams[\"font.size\"] = 18\nplt.plot(cost_history, color=\"red\", label=\"VQE\")\nplt.plot(range(len(cost_history)), [molecule.fci_energy]*len(cost_history), linestyle=\"dashed\", color=\"black\", label=\"Exact Solution\")\nplt.xlabel(\"Iteration\")\nplt.ylabel(\"Energy expectation value\")\nplt.legend()\nplt.show()\n```\n\nIf you are interested, you can calculate the ground state by varying the `distance` between hydrogen atoms to find the interatomic distance at which the hydrogen molecule is most stable. (It should be about 0.74 angstroms, depending on the performance of the ansatz.)\n", "meta": {"hexsha": "1693419aa2dcbf1b39313b03d103a7ac3664a02d", "size": 30078, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "en/source/apply/6.2_vqe.ipynb", "max_stars_repo_name": "tsuvihatu/qulacs-rtd", "max_stars_repo_head_hexsha": "f3fb768992e7981f185acb0fa2cc1aa1672cddc3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-01-04T09:13:20.000Z", "max_stars_repo_stars_event_max_datetime": "2021-06-19T07:23:13.000Z", "max_issues_repo_path": "en/source/apply/6.2_vqe.ipynb", "max_issues_repo_name": "tsuvihatu/qulacs-rtd", "max_issues_repo_head_hexsha": "f3fb768992e7981f185acb0fa2cc1aa1672cddc3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 13, "max_issues_repo_issues_event_min_datetime": "2020-03-12T04:38:22.000Z", "max_issues_repo_issues_event_max_datetime": "2020-03-31T07:32:13.000Z", "max_forks_repo_path": "en/source/apply/6.2_vqe.ipynb", "max_forks_repo_name": "tsuvihatu/qulacs-rtd", "max_forks_repo_head_hexsha": "f3fb768992e7981f185acb0fa2cc1aa1672cddc3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2020-09-03T07:54:56.000Z", "max_forks_repo_forks_event_max_datetime": "2020-11-06T04:28:45.000Z", "avg_line_length": 112.2313432836, "max_line_length": 21372, "alphanum_fraction": 0.8609615001, "converted": true, "num_tokens": 1472, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929002541067, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.41433953977996285}}
{"text": "# Parameter Estimation in Markov Networks\n\n\n\n> In the last session we saw how to estimate the parameters of a Bayesian Network. In particular, we saw that the maximum likelihood estimator can be calculated in closed form, simply as the frequentist interpretation of probability.\n>\n> In this session we will see how to estimate the parameters of a Markov Network. We will see that tha partition function will make it difficult to obtain a closed form solution of the optimal parameters, and then, we will have to use numerical methods to find them.\n>\n> Additionally, we will examine the MAP estimation of the parameters and its relation with including regularization terms in the optimization.\n\n> **Objetives:**\n> - To study the maximum likelihood parameter estimation problem for Markov Networks.\n> - To study the maximum a posteriori parameter estimation problem for Markov Networks.\n\n> **References:**\n> - Probabilistic Graphical Models: Principles and Techniques, By Daphne Koller and Nir Friedman. Ch. 20.\n> - Mastering Probabilistic Graphical Models Using Python, By Ankur Ankan and Abinash Panda. Ch. 6.\n> - Probabilistic Graphical Models Specialization, offered through Coursera. Prof. Daphne Koller.\n\n\n
                                         Imagen recuperada de: https://upload.wikimedia.org/wikipedia/commons/d/d5/Hey_Machine_Learning_Logo.png.
                                        \n\n___\n\n## 1. Maximum likelihood parameter estimation in log-linear models\n\n### 1.1. Log-likelihood function: lack of separability\n\nA key property of the (log-)likelihood function for BNs was that it can be decomposed as the (sum) product of local likelihoods. Then, to maximize the whole function, one could maximize each local function separately.\n\nWe will see that this principle does not hold for MNs because of the partition function $Z$.\n\n**Example:**\n\nConsider the pairwise MN\n\n\n```python\nfrom IPython.display import Image\n```\n\n\n```python\nImage(\"figures/pairwiseMN.png\")\n```\n\nAssume, for instance, that we know:\n\n|  $A$  |  $B$  |  $\\phi_1$  |\n| ----- | ----- | ---------- |\n| $a^0$ | $b^0$ |  $1$       |\n| $a^0$ | $b^1$ |  $1$       |\n| $a^1$ | $b^0$ |  $1$       |\n| $a^1$ | $b^1$ | $\\theta_1$ |\n\n|  $B$  |  $C$  |  $\\phi_1$  |\n| ----- | ----- | ---------- |\n| $b^0$ | $c^0$ |  $1$       |\n| $b^0$ | $c^1$ | $\\theta_2$ |\n| $b^1$ | $c^0$ |  $1$       |\n| $b^1$ | $c^1$ |  $1$       |\n\nWe want to estimate $\\bar{\\theta} = [\\theta_1, \\theta_2]$ with the IID data\n\n$$\\mathcal{D}=\\{(a[1], b[1], c[1]), \\dots, (a[M], b[M], c[M])\\}.$$\n\nWe know that \n\n$$P_{\\Phi}(A,B,C) =% \\frac{1}{Z} \\phi_1(A,B) \\phi_2(B,C),$$\n\nwith $Z = \\sum_{A,B,C} \\phi_1(A,B) \\phi_2(B,C)$.\n\nThus, the log-likelihood function is ( See in the whiteboard):\n\n$$l(\\bar{\\theta}: \\mathcal{D}) =% \\sum_{d=1}^{M} \\left(\\log \\phi_1(a[d], b[d]) + \\log \\phi_2(b[d], c[d]) - \\log Z(\\bar{\\theta})\\right),$$\n\nwith $Z(\\bar{\\theta}) = 4 + 2\\theta_1 + 2 \\theta_2.$\n\nAssuming that $M(a^1, b^1)$ and $M(b^0, c^1)$ are the number of times that the joint assignments $a^1, b^1$ and $b^0, c^1$ appear in $\\mathcal{D}$, respectively, we have:\n\n$$l(\\bar{\\theta}: \\mathcal{D}) =% M(a^1, b^1) \\log \\theta_1 + M(b^0, c^1) \\log \\theta_2 - M \\log(4 + 2\\theta_1 + 2 \\theta_2).$$\n\nThe partition function $Z(\\bar{\\theta}) =% 4 + 2\\theta_1 + 2 \\theta_2$ **couples the parameters**:\n- It is not possible to decompose the likelihood.\n- We cannot obtain a closed form estimation for the parameters.\n\n### 1.2. Log-likelihood for log-linear models\n\nRecall that the log-linear models are a general representation. Given a set of features $\\mathcal{F}=\\{f_i(\\bar{D}_i)\\}_{i=1}^{k}$, where $f_i(\\bar{D}_i)$ is a feature function defined over the variables $\\bar{D}_i$, we have that the joint distribution for the log-linear model is:\n\n$$P(X_1,\\dots,X_n:\\bar{\\theta}) = \\frac{1}{Z(\\bar{\\theta})} \\exp\\left\\{ \\sum_{i=1}^{k}\\theta_i f_i (\\bar{D}_i)\\right\\}.$$\n\nThe log-likelihood function is:\n\n\\begin{align}\nl(\\bar{\\theta}:\\mathcal{D}) & = \\sum_{d=1}^{M}\\left(\\sum_{i=1}^{k}\\theta_i f_i (\\bar{x}[d]) - \\log Z(\\bar{\\theta})\\right) \\\\\n                            & = \\sum_{i=1}^{k}\\theta_i \\sum_{d=1}^{M} f_i (\\bar{x}[d]) - M\\log Z(\\bar{\\theta}),\n\\end{align}\n\nwhere\n\n$$\\log Z(\\bar{\\theta}) = \\log \\left(\\sum_{\\bar{X}}\\exp\\left\\{ \\sum_{i=1}^{k}\\theta_i f_i (\\bar{D}_i)\\right\\}\\right)$$\n\nBefore continuing, let's prove the following results (in the whiteboard):\n\n$$\\frac{\\partial}{\\partial \\theta_i} \\log Z(\\bar{\\theta})= E_{\\theta}[f_i]$$\n\n$$\\frac{\\partial^2}{\\partial \\theta_i \\partial \\theta_j} \\log Z(\\bar{\\theta})= \\mathrm{cov}_{\\theta}[f_i,f_j]$$\n\nHence the Hessian (second derivatives' matrix) of log-partition function $\\log Z(\\bar{\\theta})$ is:\n\n$$\\frac{\\partial^2}{\\partial \\theta_i \\partial \\theta_j} \\log Z(\\bar{\\theta})= \\mathrm{cov}_{\\theta}[f_i,f_j]$$\n\nthe covariance matrix of the features - **semi-positive definite**. In this sense the log-partition is:\n\n1. Concave\n\n2. Convex\n\n3. None of the above\n\nThen, $-M \\log Z(\\bar{\\theta})$ is __.\n\nOn the other hand, the function $\\sum_{i=1}^{k}\\theta_i \\sum_{d=1}^{M} f_i (\\bar{x}[d])$ is linear.\n\nThus, the log-likelihood function is concave:\n\n- No local maxima.\n- Good theoretical guarantees\n\n#### Maximum likelihood estimation\n\nGiven the above, we can divide the log-likelihood by the number of samples $M$ and the resulting function would still be concave:\n\n\\begin{align}\n\\frac{1}{M} l(\\bar{\\theta}:\\mathcal{D}) & = \\sum_{i=1}^{k}\\theta_i \\frac{1}{M}\\sum_{d=1}^{M} f_i (\\bar{x}[d]) - \\log Z(\\bar{\\theta}) \\\\\n                                        & = \\sum_{i=1}^{k}\\theta_i E_{\\mathcal{D}}[f_i] - \\log Z(\\bar{\\theta})\n\\end{align}\n\nwhere $E_{\\mathcal{D}}[f_i] = \\frac{1}{M}\\sum_{d=1}^{M} f_i (\\bar{x}[d])$ is the empirical expectation of the feature $f_i$ in the data $\\mathcal{D}$.\n\nThus, the gradient of the log-likelihood by the number of samples $M$ is:\n\n$$\\frac{\\partial}{\\partial \\theta_i} \\frac{1}{M} l(\\bar{\\theta}:\\mathcal{D}) = E_{\\mathcal{D}}[f_i] - E_{\\theta}[f_i]$$\n\n> *Theorem*. Given a set of features $\\mathcal{F}$, $\\hat{\\theta}$ is the MLE if and only if\n>\n> $$E_{\\mathcal{D}}[f_i] = E_{\\hat{\\theta}}[f_i]$$\n>\n> for all $i$.\n\nThen, **how do we compute the MLE parameters?**\n\nWe can use numerical methods. In particular, the first order gradient ascent will do the job\n\n$$\\frac{\\partial}{\\partial \\theta_i} \\frac{1}{M} l(\\bar{\\theta}:\\mathcal{D}) = E_{\\mathcal{D}}[f_i] - E_{\\theta}[f_i].$$\n\nFor the gradient, we need the expectations of the features:\n\n- In data.\n- Relative to current model: in this step we need to perform inference at each gradient step.\n\nUnfortunately, `pgmpy` MarkovModel object do not have method fit. We'll do it ourselves to illustrate the above.\n\n**Example:** See in the whiteboard the log-linear model of A-B.\n\n\n```python\n# Import numpy and pandas\n\n```\n\n\n```python\n# Import scipy.optimize.fmin_cg\n\n```\n\n\n```python\n# Generate some random data for A - B\n\n```\n\n\n```python\n# Wrap data in a dataframe\n\n```\n\n\n```python\n# Obtain empirical expectation of features\n\n```\n\n\n```python\n# Objective function\n\n```\n\n\n```python\n# Gradient\n\n```\n\n\n```python\n# Solution\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n## 2. MAP estimation for MNs\n\nAs for BNs, MLE for MNs is very susceptible to overfitting of the parameters.\n\n### Gaussian parameter prior\n\nOften, the zero-mean univariate Gaussian (assuming independence of parameters) is used:\n\n$$P(\\bar{\\theta}) = \\prod_{i=1}^{k} \\frac{1}{\\sqrt{2\\pi}\\sigma} \\exp\\left\\{-\\frac{\\theta_i^2}{2\\sigma^2}\\right\\}$$\n\n- $\\sigma^2$ can be interpreted as the confidence that we have for the parameters not being close to zero.\n\n### Laplacian parameter prior\n\nAnother commonly used prior is the Laplace distribution:\n\n$$P(\\bar{\\theta}) = \\prod_{i=1}^{k} \\frac{1}{2\\beta} \\exp\\left\\{-\\frac{|\\theta_i|}{\\beta}\\right\\}$$\n\n- $\\beta$ can be interpreted as the confidence that we have for the parameters not being close to zero.\n\n### MAP estimation and regularization\n\nWhat happens when we maximize the a posteriori distribution?\n\n\\begin{align}\n\\arg \\max_{\\theta} P(\\mathcal{D}, \\bar{\\theta}) & = \\arg \\max_{\\theta} P(\\mathcal{D}| \\bar{\\theta}) P(\\bar{\\theta}) \\\\\n                                                & = \\arg \\max_{\\theta} \\left(l(\\bar{\\theta}:\\mathcal{D}) + \\log P(\\bar{\\theta})\\right)\n\\end{align}\n\n- If $P$ is Gaussian: $-log P(\\bar{\\theta}) \\equiv L_2$ (dense)\n- If $P$ is Laplacian: $-log P(\\bar{\\theta}) \\equiv L_1$ (sparse)\n\n\n```python\n# MAP objective function\n\n```\n\n\n```python\n# MAP objective function\n\n```\n\n\n```python\n# Solution\n\n```\n\n\n```python\n\n```\n\n\n\n
                                        \nCreated with Jupyter by Esteban Jim\u00e9nez Rodr\u00edguez.\n
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{"text": "# Assignment 3:  Histograms, Data Analysis and Fitting\n\nThe file \"edmonton.pickle\" contains the historical temperature data from Environment Canada, for weather stations that include \"Edmonton\" in their name( downloaded from climate.weather.gc.ca).  \n\nThe data are organized by \"station\"; a station object contains \"name\", \"latitude\", \"longitude\", \"firstYear\", \"climateID\", \"stationID\", \"dates\", \"minT\", \"maxT\",\"doy\", and \"year\" and consist of the readings from a single station.\n\nThe important objects are \"dates\", which is an array of datetime objects, \"minT\", which is a corresponding array of minimum temperature for each date, \"maxT\" which is the array of maximum temperatures for those dates.  \"doy\" and \"year\" refer to \"day of the year\" (from 0-365) and the year (from 1880-2019).\n\nThe snippet of code below will allow you to read in the data.  You will need to download \"edmonton.pickle\", and \"station.py\" into your working directory.\n\n\n\n```python\nimport station  \nimport pickle\nimport datetime as dt\n\nwith open('edmonton.pickle','rb') as f:\n    s=pickle.load(f)\n    \nfor station in s:\n    print(station.name)\n    print(station.maxT)\n```\n\n    EDMONTON\n    [18.9 12.8 16.7 ... 25.6 26.7 21.1]\n    EDMONTON CITY CENTRE AWOS\n    [-12.5  -6.1   4.4 ...  16.   14.   18.3]\n    EDMONTON CALDER\n    [ 22.8  26.1  18.9  19.4  20.   17.2  16.7  19.4  21.7  24.4  22.8  17.8\n      16.1  21.1  16.1  18.3  16.7  19.4  21.1  17.8  26.1  21.7  26.1  27.2\n      20.6  16.1  16.7  13.3  20.6  22.8  25.   28.3  30.6  30.6  30.   31.1\n      25.   25.   27.2  29.4  31.1  32.2  20.6  25.   25.   22.2  21.7  16.1\n      22.8  21.7  21.1  21.1  25.   29.4  25.   26.1  30.   18.9  20.6  20.\n      22.2  21.1  24.4  26.1  20.   22.8  26.1  23.3  24.4  21.7  20.   18.9\n      21.7  16.1  23.3  16.7  18.9  12.8  13.3  16.1  22.2  23.3  15.6  15.\n      14.4  16.1  17.8  25.   22.8  17.8  17.2  14.4  17.2  16.7  15.   17.2\n      16.7  19.4  22.2  22.2  17.8  17.2  16.1  23.3  23.9  23.3  22.2  16.1\n      10.6  14.4  16.1  22.8  24.4  27.8  24.4  22.8  13.9  15.6  18.9  16.1\n      15.   18.9  24.4  25.6  26.1  16.1  14.4  10.    3.3   7.2  10.6   8.9\n       8.3  12.8  14.4  15.   10.   12.8  18.3  14.4  13.9  10.6   5.6   6.7\n       3.9   3.9   5.   -0.6  -1.7  -0.6  -2.2   0.    7.8   5.6   8.9  17.2\n      15.6   8.9   5.6   5.6   5.6   2.8  -1.1  -0.6   9.4  15.6  11.7  16.1\n       7.2   3.9   0.   -0.6   0.6  -9.4  -9.4 -10.6   0.    0.   -7.2  -6.1\n     -12.8  -7.8 -13.9 -14.4 -10.6 -12.8 -20.  -12.2  -3.3   3.3   6.1   0.6\n     -11.1 -19.4 -13.9 -22.2  -6.1 -13.3 -14.4  -1.1   7.2   6.1   3.9  -1.7\n      -2.8   1.1   4.4   0.    5.6   5.6   1.7   2.2  -3.9 -10.6 -13.3 -14.4\n     -12.8 -13.9 -25.  -25.  -26.1 -20.6 -15.   -6.1 -10.6 -12.2 -11.1 -12.2\n       0.    2.2   7.8   1.7   5.6   3.9   4.4   3.9  -1.7 -10.  -11.1  -0.6\n       8.9   5.    6.1   7.2   8.3   5.    7.8 -10.6 -10.   -3.3   5.6   5.\n       7.8  -3.9   0.    2.2  -1.7  -4.4  -5.   -4.4  -7.2  -4.4 -10.6  -9.4\n      -2.8   3.9  10.    8.9   1.7  -0.6  -7.2 -16.7 -17.8 -15.6 -21.1 -22.2\n     -11.7  -6.7   5.    3.9   3.9   5.   -0.6  -6.7  -4.4   1.7  -2.2   0.6\n       5.    6.7  11.7  12.8   1.7   1.1   2.2   8.3   6.1   6.7   7.2   6.7\n       7.8   7.2   6.1  13.3  12.2   5.6   8.3  12.2  15.6  18.3  21.1  20.6\n      20.   19.4  21.1  20.6  20.   17.8  13.9   5.    4.4   8.9  12.2  12.2\n      15.6   8.9  10.   15.6  18.9  14.4  15.   17.8  18.3  18.9  24.4  24.4\n      19.4  24.4  18.3  15.   20.   22.2  21.1  21.1  28.3  15.   20.6  13.3\n      16.7  19.4  23.3  18.3  12.8  15.   17.8  17.2  24.4  25.6  17.2  18.3\n      21.1  30.   15.   14.4  20.6  17.2  22.8  17.2  18.9  12.8  17.8  18.9\n      20.   25.   25.   25.   24.4  17.2  21.1  22.2  20.   18.3  23.3  20.\n      14.4  11.1  12.8  15.   15.   18.3  22.2  25.6  26.1  25.6  21.7  20.\n      22.2  26.1  26.7  27.2  24.4  19.4  23.9  25.6  23.9  21.7  19.4  24.4\n      30.   24.4  22.8  23.3  28.3  24.4  27.2  27.8  24.4  26.1  25.6  16.1\n      19.4  20.   20.6  21.1  25.   28.9  23.3  23.3  23.9  26.7  26.1  22.2\n      22.2  23.9  26.1  26.1  26.1  29.4  28.3  25.   15.   23.9  22.2  21.1\n      20.6  22.8  22.8  25.6  20.   16.7  18.3  25.   23.3  23.3  23.9  24.4\n      15.6  19.4  23.3  18.3  16.1  12.2  16.7  23.9  18.3  11.1  10.   20.6\n      22.2  23.3  23.9  22.8  18.3  21.7  24.4  23.9  23.3  23.3  24.4  23.3\n      20.6  23.9  24.4  23.3  16.1  14.4  11.1   6.1  15.   15.   13.3  23.3\n      21.1  20.   16.7  17.8  13.3  12.8   6.1   8.3   7.2   7.8  13.9  10.\n      10.    7.8   7.2   2.8   0.6   0.   13.9  16.1  12.8   9.4   7.2  12.8\n       8.3  11.1  12.2  10.6   4.4   6.7   8.3  10.   10.    1.7   1.7   4.4\n       8.9   5.6   6.1   7.2   6.1   1.7  -2.2   1.7   7.2   3.3  -2.8  -9.4\n      -0.6  -0.6   5.6   5.6  10.    5.    0.6  -5.  -13.9 -14.4 -13.3 -16.1\n       0.    6.1 -11.1   1.1   7.2   3.3   6.7   7.2   3.3  -6.7  -8.9   2.8\n       0.6  -3.9   1.7  -1.1  -3.9  -2.2  -6.1  -7.8 -11.7 -11.7 -10.6 -12.2\n     -20.  -12.8 -12.8   4.4   6.1  -0.6 -11.1 -17.8 -16.7 -12.8 -14.4 -17.8\n     -19.4 -15.   -7.2   2.2   3.9   1.1   3.3   5.    2.8   3.3   1.7   2.2\n       3.3   0.  -15.  -13.   -4.4   1.1   2.8   4.4   5.6   2.8   6.1   3.3\n       8.3   8.3   6.7   3.9   3.9   3.9   5.    4.4   7.2   4.4   7.8   5.6\n       8.3   8.9  10.    8.9   5.    2.2   2.8   3.9   3.3   6.7   5.    3.3\n       6.7   5.6  11.1   8.3  11.7  10.    7.2   3.9   6.1  10.6   9.4  -3.3\n      -2.8   3.3   1.7  -0.6  -1.1   2.2   2.2   2.2  -0.6  -3.3   6.7  10.\n      10.    9.4   8.3   7.2   7.2]\n    EDMONTON INT'L A\n    [22.8  8.3 12.2 ...  8.4 13.1 16.6]\n    EDMONTON INTERNATIONAL CS\n    [-14.7 -10.6  -6.1 ...   9.1  12.8  12. ]\n    EDMONTON CITY CENTRE A\n    [ 7.2  8.3  6.1 ... -0.6 -4.7 -6.3]\n    EDMONTON BLATCHFORD\n    [ 2.1  0.4 -8.1 ...  9.2 12.6 12.1]\n    EDMONTON NAMAO A\n    [-7.2 -8.3 -8.3 ... -5.7 -4.7 -7.1]\n    EDMONTON NAMAO AWOS A\n    [24.5 24.  28.  ...  8.6 11.8 11.5]\n    EDMONTON INTL A\n    [10.5  4.8  0.5 ...  8.7 12.5 12.2]\n    EDMONTON SOUTH CAMPUS\n    [ 28.2  27.9  25.1  17.   19.4  23.3  24.6  25.2  28.2  29.8  27.2  27.6\n      21.2  24.8  28.7  17.8  13.6  13.9  12.4  19.5  23.3  20.9  18.5  23.4\n      28.   27.3  26.4  14.   14.5  17.3  20.8  20.9  21.   22.6  24.5  27.3\n      28.   28.3  29.6  26.   25.5  28.3  29.1  19.5  21.7  22.6  23.2  21.7\n      19.2  17.4  19.9  22.6  28.1  29.9  21.5  22.6  26.8  26.1  26.1  26.6\n      22.9  19.7  25.8  30.6  31.7  27.2  25.4  25.8  20.7  21.   19.2  19.2\n      22.7  23.7  26.   28.   30.6  27.6  27.3  29.5  24.   20.6  21.1  25.1\n      28.7  25.5  28.2  34.6  31.8  22.1  11.8  20.2  27.1  22.4  27.5  27.2\n      20.8  23.2  22.9  26.4  25.6  22.7  11.5  20.4  15.7  14.5  22.2  18.4\n      18.4  20.8  17.5  18.5  15.1  15.9  21.1  21.3  26.5  17.8  12.5  10.3\n      14.8   9.    0.8   3.6   2.7   3.1   5.   11.4  14.    9.8   2.5   1.3\n       8.5  11.2  17.3  13.2   8.2   8.1   3.1   4.7   4.    3.1   7.1   9.\n      10.6  11.5   7.3  -1.7  -1.8   0.6   6.5   7.7   1.5  11.5  14.3  17.6\n      24.9  17.   13.8  17.4  12.1  14.3  13.2  12.8  16.1  11.9  10.6   8.1\n       8.9   8.6   8.4   3.2  -0.1   5.7   2.1  -6.1 -10.   -6.1  -9.1   3.6\n       2.9  -2.9   4.5   9.    8.9   4.5  -5.6  -2.3   4.3   6.4   8.5   6.3\n       5.7   1.4  -3.5   3.3  -1.1  -1.4   6.1   3.5   2.7  -0.9  -4.7  -4.5\n      -1.5  -6.6  -6.6  -7.6  -3.6  -1.4   2.4  -2.4   4.1   5.5   6.1   5.8\n      -5.1   0.2   3.7   5.8   2.    1.4  -3.5  -4.6  -6.7  -5.7  -9.   -8.5\n      -3.   -2.5  -2.9  -6.1   3.3   8.6   6.3  -1.3  -3.8  -3.1  -6.4 -10.9\n      -8.5  -9.6  -3.5   0.4  -3.4   2.6  -0.3 -10.2 -15.6 -14.5 -12.5   2.7\n       2.1   0.   -0.9   1.7   4.4   9.7   9.9  -3.1  -9.    3.1   3.8 -13.5\n     -20.3 -26.7 -22.8 -23.  -20.  -18.1 -19.3 -21.8 -21.6 -20.2 -17.3 -15.8\n     -12.7 -18.3 -14.9 -16.7 -13.9  -2.3  -3.  -10.4  -1.7 -11.1 -14.7 -14.7\n      -7.3  -1.5  -5.9 -16.1 -18.5 -12.6  -5.5 -10.1 -11.5  -7.4  -6.   -2.7\n       2.    7.1   4.7   4.8   5.    5.7   8.   10.6  12.5  15.5  15.3  11.9\n      13.9   9.6   3.8   2.5   4.3   4.3   8.9   9.2   9.3   6.2   7.5   5.4\n      10.2   3.1   8.8   6.1  13.4  15.2  15.9  12.1  10.3   9.8  14.    9.\n      10.1  13.2  14.8  17.3  13.2  11.9  15.9  22.5  17.5  10.7  11.6  11.6\n       5.6   5.4   8.    7.   10.6   5.6   6.4   9.9  13.4  12.4  18.   21.\n      20.1  24.3  23.5  19.5  16.8   9.8  15.1  17.7  16.5  17.4  18.6  19.1\n      22.2  20.7  15.6  20.2  22.1  24.7  28.   30.8  19.7  22.2  27.9  21.8\n      18.8  20.2  18.5  12.2   8.7  14.   15.7  18.9  22.3  27.3  24.4  21.\n      22.2  24.   25.3  23.6  15.3  15.1  17.6  20.3  21.2  16.   18.7  20.8\n      22.2  18.7  22.2  19.1  18.1  17.6  18.9  19.   21.   20.3  20.4  16.7\n      18.5  20.9  25.8  26.1  24.4  24.1  23.5  24.4  15.5  20.1  23.3  24.2\n      27.5  29.4  23.   22.2  26.7  25.3  21.6  23.5  21.4  23.6  24.7  28.1\n      20.6  24.3  26.5  18.1  22.3  26.4  17.2  16.3  15.8  17.7  21.3  24.9\n      25.4  15.8  17.2  15.8  18.7  24.9  28.3  20.8  20.7  23.   21.   17.4\n      21.7  19.5  19.2  13.8  15.8  16.7  20.8  25.   25.2  22.5  21.7  22.9\n      15.4  13.5  11.3  14.7  21.6  20.8  20.5  21.6  18.6  18.2  17.9  19.5\n      20.6  22.7  24.7  16.2  16.4  14.3  10.5   5.2   5.3   4.3   3.8   8.5\n      12.2  12.3]\n    EDMONTON VILLENEUVE A\n    [-15.  -14.  -10.5 ...  21.5  13.8  14.8]\n    EDMONTON VILLENEUVE A\n    [11.2 15.5 14.6 ...  9.3 12.2 11.1]\n    EDMONTON STONY PLAIN\n    [ -7.8 -13.9 -18.9 ...   4.    9.5  13. ]\n    EDMONTON WOODBEND\n    [ -2.2  -5.6  -9.4 ...   6.5  -1.  -10. ]\n    FORT EDMONTON\n    [-13.3 -21.7 -27.2 ...  18.5  20.   17.5]\n    EDMONTON STONY PLAIN CS\n    [ 1.8  9.2  7.  ...  9.2 12.8 10.2]\n    EDMONTON TIEBEKE ESTATES\n    [ 25.5  23.   27.5  27.   22.   26.   26.5  23.   20.   20.5  24.5  25.5\n      13.   10.5  14.   17.5  18.5  18.   22.5  16.   14.   16.   21.   15.5\n      14.   17.5  18.5  22.   24.5  23.   17.   13.   19.5  25.5  24.5  21.5\n      23.5  15.5  11.5   9.   10.5  11.5   8.    4.    2.5   3.   11.   13.\n      11.5  12.5  17.5  14.   11.5  11.   10.   10.   14.   10.    6.    9.\n      15.   10.   19.   18.   11.5  21.   19.5  13.   10.    9.   11.    6.\n       8.   13.    2.5   2.5   6.5   6.5   2.5   0.    2.    7.5   3.    4.\n      10.    6.    5.    2.5   2.5   2.5   3.   -0.5  -2.5  -1.5   1.   -0.5\n       0.5   1.5  -1.5   2.   -3.   -4.   -5.    2.    5.    7.    6.5   5.5\n       5.    3.5  -2.5  -3.   -9.5  -5.5  -3.5  -6.   -3.   -6.   -4.5  -5.\n       0.5   7.5   5.    4.    5.5  10.5   7.    5.5   7.5   8.5  11.    8.5\n       7.5   8.   10.   12.    9.5   5.5  14.    8.    5.5   1.5   8.    9.\n       1.    5.5   5.    3.5  -6.5  -5.5  -2.    2.5   8.5  15.5  18.   19.5\n      19.   24.5  15.5  12.   14.   16.   17.   17.   16.   19.5  17.   15.5\n      19.   15.5  17.5   8.5  12.   14.   16.    4.    9.   13.5  13.   14.5\n      16.5  11.5  16.   18.5  21.   16.   20.5  19.   13.   16.5  16.   13.5\n      19.   10.   14.5  15.   18.   18.   21.   22.5  26.   25.5  21.5  21.5\n      12.   11.   10.5   7.5  13.   19.   16.5  14.   13.5  21.   22.   15.5\n      20.   20.   19.   18.5  18.   16.   19.5  23.5  25.5  25.5  22.   21.\n      21.   21.5  17.   17.5  20.5  22.5  24.   20.5  20.   21.5  25.5  28.5\n      24.   17.   17.5  13.5  18.5  22.   25.5  25.   27.5  25.5  21.   25.\n      24.   25.5  27.5  23.5  25.5  26.5  23.   20.   22.   24.   26.5  25.5\n      21.   22.   22.   20.5  17.   19.   17.5  19.5  23.5  17.   22.5  24.5\n      21.   19.5  22.   25.5  26.   24.   26.   15.5  14.   16.5   7.5  10.5\n      11.5  10.5   7.5   9.   13.5  19.5  17.   22.   15.5  15.5  10.5  15.5\n      17.5  17.5  22.5  24.   25.   23.   17.5  11.5   7.    2.5   5.5  18.\n      20.5  20.   16.   23.5  21.5  19.   13.   11.5   4.5   3.5  -1.    3.5\n       6.5  13.5  22.   16.   13.5   8.5  15.5  12.    5.5  12.   15.5  13.\n      14.5  10.5   9.    9.   14.   18.   15.    9.   12.   11.5   3.5   4.5\n       8.    9.5   6.5   8.5   9.5   8.5  -5.5  -6.   -5.   -5.5  -8.   -9.5\n      -3.    3.    2.5   2.5  -2.   -2.5   0.    3.5  -0.5   1.5   5.    7.5\n       8.5   7.    3.5   6.    1.    2.   -1.5  -4.   -4.    1.5   0.5  -4.5\n       2.    7.    1.  -11.  -22.  -19.5 -17.5 -20.5 -19.5 -20.  -23.5 -18.\n     -14.5  -0.5  -2.5  -2.5  -3.5  -5.5   2.    2.    2.5   8.5   3.    6.\n       3.5   5.    6.    7.    2.5   1.5  -5.   -9.   -6.   -4.    4.5   2.5\n      -1.    0.    2.    2.5   4.    0.5   2.    0.5   3.5   4.    5.    1.\n       0.5   3.    3.    5.    6.5   2.5  -4.5  -8.  -11.  -13.5 -16.5 -10.\n     -11.   -9.5  -0.5  -1.5 -13.5 -11.5  -3.5  -6.   -8.   -4.   -4.   -3.5\n      -9.5 -17.5 -16.   -5.5   6.5  10.5   6.5   1.5   7.    5.5   3.    9.\n      11.    9.5   3.    4.5   8.    9.5   5.    6.    7.    7.5   9.5  11.\n       4.   -6.5  -6.5 -12.  -12.   -6.    1.    4.5   9.5  11.5   6.    5.\n       8.    8.    6.5   6.    6.5  11.5  12.5  12.    7.5   8.    5.5   5.5\n       9.    9.5   6.5   4.5   9.   18.5  17.5  12.5   2.    8.   12.5  17.\n      22.   17.5  21.5  26.5  25.   17.   15.5  12.5  14.5  19.5  18.5  14.\n      13.   18.5  19.   14.5  15.5  18.5  24.   22.   17.5  19.   17.5  16.5\n      15.5  13.5  14.5  15.5  24.   28.   25.5  22.   19.5  24.   26.5   9.5\n      10.5  22.   22.   15.5  12.5  13.5  20.   20.   15.   22.   23.   16.\n      16.   18.   19.5  17.5  17.5  16.   15.5  18.   19.5  26.   27.   22.\n      21.5  17.5  13.5  13.   15.5  20.5  15.   20.   20.   23.   24.5  29.\n      21.   24.   25.5  24.5  27.5  31.5  19.5  24.   20.5  24.   24.   19.\n      21.5  19.5  21.5  22.5  21.5  20.   21.   17.   16.   20.   23.5  19.5\n      13.5  20.5  24.   24.   25.5  29.   23.   24.5  24.5  21.   18.5  21.\n      24.5  24.   27.5  28.   25.   27.5  27.   30.   27.   19.   21.   22.5\n      22.5  24.5  20.   22.   24.5  28.5  21.   22.   24.5  23.5  23.5  18.5\n      22.5  17.5  17.   14.5  13.   13.   14.   13.   13.   16.5  19.   24.5\n      25.5  18.5  20.5  16.5  10.5  15.5  13.5  15.5  25.   28.5  29.   25.\n      17.   14.5  19.5  17.   10.5  15.   12.    6.   15.   19.   15.5  16.\n      12.5  12.5  10.5  10.5   7.    6.5   7.   15.5   8.5   8.5  10.5   9.\n       7.    0.5  -2.5]\n\n\nLook at the data.  \n\n1.  Make plots that show the max and min temperatures as a function of date for a single station\n2.  Histogram the max and min temperatures for a single station at a few dates throughout the year. Make sure the histograms have a reasonable number and range for the bins.  \n3.  For each pair of stations, find the periods of time during which they both measured temperatures.  If there is an overlapping period, find the mean and standard deviation of the differences between the max and min temperatures measured at the two stations for those periods.  Fill out a table with rows:  Name of Station 1, Name of Station 2, number of days that both measured temperatures, Average Difference in max, standard deviation of difference of Max, Average difference in min, standard deviation of difference in min\n4.  For a few of the pairs which have significant differences, make a 2d color histogram of Ta-Tb versus Ta, where Ta and Tb refer to the measurements at the two stations. \n\n\n```python\n\n```\n\nCombine all the data from the different stations into an average max and min for each each day of the year. To do this, you will to pick one station as the standard, and correct each of the others by its average difference.\n\nKeep the data as an np.array.\nPlot averages for the year.\n\n\n```python\n\n```\n\n## No-atmosphere model of heating and cooling\n\nThe atmosphere plays a critical role in weather and climate, but modelling it is difficult.  Anyone who has watched the weather forecast sees weather systems move into and out of regions and winds/fronts/ high and low pressure regions- all atmospheric phenomena.  However, we are going to see whether we can fit the data with a \"no-atmosphere\" model.  If there were no atmosphere, the the only source of heat would be solar radiation, and the only cooling would be from black body radiation.  We incorporate this as an ordinary differential equation.  \n\n\\begin{equation} \\frac {dT}{dt}=\\alpha F(t) - \\beta T^4 \\end{equation}\n\nHere $T$ is the (absolute!)temperature since $T^4$ is the Stefan-Boltzman law, $t$ the time, $F$ the solar flux (in Watts/m$^2$) and $\\alpha$ and $\\beta$ are the parameters of the model that we will allow to fit.  If we were to calculate from first principles, $\\alpha$ would include the reflectivity of the surface and the clouds , as well as the heat capacity per square meter of the layer of the earth/atmosphere that heat up and cool down.  Similarly, $\\beta$ includes the Stefan-Boltzman constant, the emissivity of the earth, and the heat capacity.  So this model is quite simple.\n\nHowever, it is still interesting to see how well such a simple average model can work.  Since everything in the model is independent of time (except for the orbit) our model does not allow any differences year to year or any difference between locations at the same latitude.  \n\nTo start the problem, I have modified the earth-sun solution from Problem Set 2 to include the rotation of the earth, which is important to calculate the solar flux at Edmonton.    This involves adding the vector from the center of the earth to Edmonton, $\\vec{x_{Ed}}$ and solving the differential equation \n\\begin{equation} \\frac{d\\vec{x_{Ed}}}{dt}=\\vec{\\omega} \\times \\vec{x_{Ed}}\\end{equation} where $\\vec{\\omega}$ in the constant rotation vector of the earth.  \n\nWe start by finding the \"north pole vector\", which we know is aligned with the earth-sun vector on June 21 and December 21st.  If we look at the Horizon web page for Dec 22, we see that X~0, Y=1 at the solstice, and that the axial tilt (obliquity) is 23.4392911 degrees.  Thus $$\\hat{n}=(0,\\sin(23.4392911),\\cos(23.4392911))\\approx(0,0.398,0.918)$$ and $$\\vec{\\omega}=\\frac{2\\pi\\hat{n}}{24\\times 3600 \\times 365.2425/366.2425}$$\n(Notice here the factor 365.2425/366.2425- which converts from normal days to \"sidereal days\"; which take into account the fact that the earths revolution around the sun means noon-noon is a little longer than one rotation).\n\nWe pick $\\vec{x_{Ed}}=\\cos(53.55)(1,0,0)+\\sin(53.55)\\hat{n}$ at the solstice. (Only the latitude matters- in principle we need to set longitude as well, but in our model the longitude doesn't really matter).\n\n\nWe run the code below to generate the inputs to the flux calcuation- the sun-earth vector and the Edmonton vector.  We pick t to cover the year, with say 48 bins per day.\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.integrate import odeint\nimport datetime as dt\nimport time\n\nGM=132712440041.93938e9 #m^3/s^2\nGMEarth=398600.435436e9 #m^3/s^2\nMEarth=5.97219e24 #kg\nMSun=GM/GMEarth*MEarth  #  this keeps the mass ratio right, with most precise GM values!\nAU=149597870700.0 #m (exactly)\nD=24*3600.0 #s\n\n#  Factors from Horizon web site at 2458839.916666667 = A.D. 2019-Dec-22 10:00:00.0000 TDB \n#  This is picked to be at the hour closest to the solstice- notice that the x position of earth is very small\nx0=np.array([4.858434307587728E-04 *AU, 9.836920738092132E-01 *AU, -4.745398623388847E-05 *AU])\nv0=np.array([-1.749013293502635E-02 *AU/D,-5.128449611745279E-05 *AU/D,  4.120640971206839E-07 *AU/D])\ntilt=23.4392911/180*np.pi\nn=np.array([0, np.sin(tilt),np.cos(tilt)])  # vector of earth's axis\nomega=2*np.pi/(D*365.2425/366.2425)*n  #rotation axis\nlatitude=53.55/180*np.pi  # latitude of the \"Edmonton\" weather station\nradius=6.37e6  #Earth's radius=6370 km from Horizons\nx_ed0=radius*(np.array([np.cos(latitude),0,0])+np.sin(latitude)*n)  #Edmonton location at solstice (at least for some year!)\nfactor=(MSun+MEarth)/MSun  # to convert xe to earth-sun distance\nxsun=np.array([0,0,0]) \n\nsolarConstant=1367.6 #W/m**2 from Horizons.  At 1 AU\n\ncm=MEarth*x0/(MEarth+MSun)\nx=x0-cm #  x is distance wrt to the cm\n\nvcm=MEarth*v0/(MEarth+MSun)  #velocity of CM\nv=v0-vcm\n\n\ndef dvdt(xvArgument,t):\n    xv=xvArgument.reshape(3,3)\n    xearth_sun=factor*xv[0] #position wrt sun\n    distance=np.sqrt(np.dot(xearth_sun,xearth_sun))\n    v_ed=np.cross(omega,xv[2])  # velocity of Edmonton is omega x x_ed\n    return np.array([xv[1],-GM/distance**3*xearth_sun,v_ed]).reshape(9)\n\nspy=365.2425*24*3600\nt=np.linspace(0,380*24*3600,380*48)  # 48 bins per day, starting December 22.  Go 380 days to cover whole next year\ny0=np.array([x,v,x_ed0]).reshape(9)\n\nrun=True\nif run:\n    cpuT0=time.process_time()\n    ephemeris = odeint(dvdt, y0, t,rtol=1e-12)\n    print(\"CPU Time=\",time.process_time()-cpuT0)\n    np.save('problem3',ephemeris)\nelse:\n    ephemeris=np.load('problem3.npy')\n\n\n```\n\n    CPU Time= 31.738442715999998\n\n\nWrite a function to calculate the Flux at any instant of time, from the output of this calculation.  To do this you will need to interpolate ephemeris (since it is only returned at discrete times).  \n\nEphemeris contains three vectors- the position of the earth wrt to CM, the velocity of the earth wrt CM, and the position of Edmonton with respect to the center of the earth.  \n\nOnce we have the three vectors, the flux is\n\\begin{equation} F(t)=\\begin{cases}\\phi_0 \\frac{\\vec{x_{Ed}} \\cdot (\\vec {x_s}-\\vec{x_e})}{|\\vec{x_{Ed}}| |\\vec {x_s}-\\vec{x_e}|} & \\text{if } \\vec{x_{Ed}} \\cdot (\\vec {x_s}-\\vec{x_e})>0\\\\0 & \\text{if } \\vec{x_{Ed}} \\cdot (\\vec {x_s}-\\vec{x_e})<0\n\\end{cases} \\end{equation}\n\nPlot your function versus date (for one year)\n\n\n\n\n```python\n\n```\n\nNow write a model that integrates the ordinary differential equation:\n\n\\begin{equation} \\frac {dT}{dt}=\\alpha F(t) - \\beta T^4 \\end{equation}\n\nusing F(t) from the calculation of ephemeris.  \n\nYou will need to experiment to find values of $\\alpha$ and $\\beta$ that are semi-reasonable.  \n\nFind the maximum and minimum temperature for each day from this model, and calculate the difference between this model and the data.\n\nFit the model to the data.  Plot the residuals.  Comment on the reasonableness of the fit.\n\n\n\n\n```python\n\n```\n\n(Optional) If you would like to experiment some more, we can add some elements to the model and see how the fit improves.  One can add \nfeedback(\"clouds?\") by making $\\alpha$ and $\\beta$ functions (first or second order polynomials) of T. You could also change $T^4$ to $(T-\\Delta)^4$- basically saying the cooling infrared radiation doesn't come from the ground, but from higher in the atmosphere where the temperatures are colder.\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "26401c416a5712a875d133687dc313ed4c5341fe", "size": 26894, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Assignments/Blank/Assignment 3 Data and Fitting.ipynb", "max_stars_repo_name": "hanzhihua72/phys-420", "max_stars_repo_head_hexsha": "748d29b55d57680212b15bb70879a24b79cb16a9", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Assignments/Blank/Assignment 3 Data and Fitting.ipynb", "max_issues_repo_name": "hanzhihua72/phys-420", "max_issues_repo_head_hexsha": "748d29b55d57680212b15bb70879a24b79cb16a9", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Assignments/Blank/Assignment 3 Data and Fitting.ipynb", "max_forks_repo_name": "hanzhihua72/phys-420", "max_forks_repo_head_hexsha": "748d29b55d57680212b15bb70879a24b79cb16a9", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 59.2378854626, "max_line_length": 599, "alphanum_fraction": 0.4882873503, "converted": true, "num_tokens": 12139, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "\n\n# Introduction to Digital agro \n\n## Crop simulation models\n\n____\n\n### Mikhail Gasanov\n\nE-mail: Mikhail.Gasanov[a]skoltech.ru\n\ntg:@misha_grol\n\n\n\n\n\n## Clone utils and files from GitHub\n\n\n```python\n# !git clone https://github.com/mishagrol/Seminar_Sobol.git\n# !cp -r ./Seminar_Sobol/* .\n```\n\n# How to start with PCSE/WOFOST model\n\n_____\n\n### Documentation: [PCSE](https://pcse.readthedocs.io/)\n\n\n\n\n\n\n```python\n%matplotlib inline\nimport sys, os\nimport pcse\nimport pandas as pd\nimport matplotlib\nimport yaml\nmatplotlib.style.use(\"ggplot\")\nimport matplotlib.pyplot as plt\nprint(\"This notebook was built with:\")\nprint(\"python version: %s \" % sys.version)\nprint(\"PCSE version: %s\" %  pcse.__version__)\n```\n\n    This notebook was built with:\n    python version: 3.6.11 | packaged by conda-forge | (default, Aug  5 2020, 20:19:23) \n    [GCC Clang 10.0.1 ] \n    PCSE version: 5.4.2\n\n\n\n```python\nimport warnings\nwarnings.filterwarnings(\"ignore\")\n```\n\n\n```python\nwofostPP = pcse.start_wofost(mode=\"wlp\")\n```\n\nYou have just successfully initialized a PCSE/WOFOST object in the Python interpreter, which is in its initial state and waiting to do some simulation. We can now advance the model state for example with 1 day:\n\n\n\n```python\nwofostPP.run()\n```\n\nAdvancing the crop simulation with only 1 day, is often not so useful so the number of days to simulate can be specified as well:\n\n\n```python\nwofostPP.run(days=10)\n```\n\n## Getting information about state and rate variables\nRetrieving information about the calculated model states or rates can be done with the `get_variable()` method on a PCSE object. For example, to retrieve the leaf area index value in the current model state you can do:\n\n### Leaf Area Index \n\n\n\n\n\n```python\n# Leaf Area Index at this date\nprint(wofostPP.day)\nprint('LAI', wofostPP.get_variable('LAI'))\n```\n\n    2000-01-12\n    LAI 0.2870809817505803\n\n\nThe `get_variable()` method can retrieve any state or rate variable that is defined somewhere in the model. \n\nFinally, we can finish the crop season by letting it run until the model terminates because the crop reaches maturity or the harvest date:\n\n\n```python\nwofostPP.run_till_terminate()\n```\n\n## Retrieving and displaying WOFOST output\nWe can retrieve the results of the simulation at each time step using  `get_output()`. In python terms this returns a list of dictionaries, one dictionary for each time step of the the simulation results. Each dictionary contains the key:value pairs of the state or rate variables that were stored at that time step.\n\n\n\n\n```python\noutput = wofostPP.get_output()\n```\n\nThe most convenient way to handle the output from WOFOST is to used the `pandas` module to convert it into a dataframe. Pandas DataFrames can be converted to a variety of formats including excel, CSV or database tables.\n\n\n```python\ndf_crop = pd.DataFrame(output).set_index('day')\n```\n\n\n```python\nsummary_output = wofostPP.get_summary_output()\nmsg = \"Reached maturity at {DOM} with total biomass {TAGP:.1f} kg/ha, \" \\\n      \"a yield of {TWSO:.1f} kg/ha with a maximum LAI of {LAIMAX:.2f}.\"\nfor crop_cycle in summary_output:\n    print(msg.format(**crop_cycle))\n```\n\n    Reached maturity at 2000-05-31 with total biomass 15261.8 kg/ha, a yield of 7179.8 kg/ha with a maximum LAI of 6.13.\n\n\n\n```python\nfig, (axis1, axis2) = plt.subplots(nrows=1, ncols=2, figsize=(16,8))\ndf_crop.LAI.plot(ax=axis1, label=\"LAI\", color='k')\ndf_crop.TAGP.plot(ax=axis2, label=\"Total biomass\")\ndf_crop.TWSO.plot(ax=axis2, label=\"Yield\")\naxis1.set_title(\"Leaf Area Index\")\naxis2.set_title(\"Crop biomass\")\nfig.autofmt_xdate()\nr = fig.legend()\n```\n\n# Running PCSE/WOFOST with custom input data\n\nThis Jupyter notebook will show you how to read inputs from files for running PCSE/WOFOST.\n\nthanks to **Allard de Wit**\n\n**Prerequisites for running this notebook**\n\nSeveral packages need to be installed for running PCSE/WOFOST:\n\n  1. `PCSE` and its dependencies. See the [PCSE user guide](http://pcse.readthedocs.io/en/stable/installing.html) for more information;\n  2. The `pandas` module for processing and storing WOFOST output;\n  3. The `matplotlib` module for generating charts\n\n\n\n\n\n## Reading model parameters\n### Crop parameters\n\n\n```python\ndata_dir = 'data/'\n```\n\n\n```python\nfrom pcse.fileinput import CABOFileReader\ncropfile = os.path.join(data_dir, 'crop', 'SUG0601.crop')\ncropdata = CABOFileReader(cropfile)\n```\n\n\n```python\n#potato\nfrom pcse.fileinput import CABOFileReader\ncropfile_potato = os.path.join(data_dir, 'crop', 'POT701.CAB')\ncropdata_potato = CABOFileReader(cropfile_potato)\n```\n\n\n```python\n# Number of parameters for our crop \nlen(cropdata_potato)\n```\n\n\n\n\n    63\n\n\n\n### Soil parameters\nThe soildata dictionary provides the parameter name/value pairs related to the soil type and soil physical properties. The number of parameters is variable depending on the soil water balance type that is used for the simulation. For this example, we will use the water balance for freely draining soils and use the soil file for medium fine sand: `ec3.soil`. This file is also taken from the soil files in the [WOFOST Control Centre](http://www.wageningenur.nl/wofost).\n\n\n```python\nsoilfile = os.path.join(data_dir, 'soil', 'ec3.soil')\nsoildata = CABOFileReader(soilfile)\nprint(soildata)\n```\n\n    ** $Id: ec3.new 1.2 1997/09/18 17:33:54 LEM release $\n    **\n    ** SOIL DATA FILE for use with WOFOST Version 5.0, June 1990\n    **\n    ** EC3-medium fine\n    ------------------------------------\n    SMW: 0.104 \n    SMFCF: 0.3 \n    SM0: 0.41 \n    CRAIRC: 0.06 \n    K0: 25.586 \n    SOPE: 1.47 \n    KSUB: 1.47 \n    SPADS: 0.1 \n    SPODS: 0.03 \n    SPASS: 0.2 \n    SPOSS: 0.05 \n    DEFLIM: -0.3 \n    RDMSOL: 120 \n    SOLNAM: EC3-medium fine \n    SMTAB: [-1.0, 0.41, 1.0, 0.398, 1.3, 0.389, 1.491, 0.38, 2.0, 0.34, 2.4, 0.287, 2.7, 0.241, 3.4, 0.148, 4.204, 0.104, 6.0, 0.09] \n    CONTAB: [0.0, 1.408, 1.0, 0.167, 1.3, -0.215, 1.491, -0.638, 1.7, -0.854, 2.0, -1.155, 2.4, -1.796, 2.7, -2.26, 3.0, -2.745, 3.4, -3.357, 3.7, -3.824, 4.0, -4.276, 4.204, -4.678] \n    \n\n\n### Site parameters\n\nThe site parameters provide ancillary parameters that are not related to the crop or the soil. Examples are the initial conditions of the water balance such as the initial soil moisture content (WAV) and the initial and maximum surface storage (SSI, SSMAX). Also the atmospheric $CO_{2}$ \nconcentration is a typical site parameter. For the moment, we can define these parameters directly on the Python commandline as a simple python dictionary. However, it is more convenient to use the `WOFOST71SiteDataProvider` that documents the site parameters and provides sensible defaults:\n\n\n```python\nfrom pcse.util import WOFOST71SiteDataProvider\nsitedata = WOFOST71SiteDataProvider(WAV=100, CO2=360)\nprint(sitedata)\n```\n\n    {'IFUNRN': 0, 'NOTINF': 0, 'SSI': 0.0, 'SSMAX': 0.0, 'WAV': 100.0, 'SMLIM': 0.4, 'CO2': 360.0}\n\n\n### Packaging all parameters\nFinally, we need to pack the different sets of parameters into one variable using the `ParameterProvider`. This is needed because PCSE expects one variable that contains all parameter values. Using this approach has the additional advantage that parameter value can be easily overridden in case of running multiple simulations with slightly different parameter values:\n\n\n```python\nfrom pcse.base import ParameterProvider\nparameters = ParameterProvider(cropdata=cropdata, soildata=soildata, sitedata=sitedata)\n```\n\n## Agromanagement\nThe agromanagement inputs provide the start date of the agricultural campaign, the start_date/start_type of the crop simulation, the end_date/end_type of the crop simulation and the maximum duration of the crop simulation. The latter is included to avoid unrealistically long simulations for example as a results of a too high temperature sum requirement.\n\nThe agromanagement inputs are defined with a special syntax called [YAML](http://yaml.org/) which allows to easily create more complex structures which is needed for defining the agromanagement. The agromanagement file for sugar beet in Wageningen `sugarbeet_calendar.agro` can be read with the `YAMLAgroManagementReader`:\n\n\n```python\nfrom pcse.fileinput import YAMLAgroManagementReader\n#crop rotation for Moscow region\nagromanagement_file = os.path.join(data_dir, 'agro', 'sugarbeet_calendar_Moscow_short.agro')\n#agromanagement_file = os.path.join(data_dir, 'agro', 'sugarbeet_calendar.agro')\nagromanagement = YAMLAgroManagementReader(agromanagement_file)\nprint(agromanagement)\n```\n\n    !!python/object/new:pcse.fileinput.yaml_agro_loader.YAMLAgroManagementReader\n    listitems:\n    - 2019-06-01:\n        CropCalendar:\n          crop_end_date: 2019-10-15\n          crop_end_type: harvest\n          crop_name: sugar-beet\n          crop_start_date: 2019-06-02\n          crop_start_type: emergence\n          max_duration: 300\n          variety_name: sugar-beet-601\n        StateEvents: null\n        TimedEvents:\n        - comment: All fertilizer amounts in kg/ha\n          event_signal: apply_npk\n          events_table:\n          - 2019-06-22:\n              K_amount: 122\n              N_amount: 128\n              P_amount: 25\n          name: Timed N/P/K application table\n    \n\n\nWe can create a crop rotation in the model\n\n\n\n```python\nK_kg = 60\nP_kg = 60\nN_kg = 120\nyear=2017\n\nyaml_agro = f\"\"\"\n- {year}-05-01:\n    CropCalendar:\n        crop_name: 'sugar-beet'\n        variety_name: 'sugar-beet-601'\n        crop_start_date: {year}-05-20\n        crop_start_type: sowing\n        crop_end_date: \n        crop_end_type: maturity\n        max_duration: 250\n    TimedEvents:\n    -   event_signal: apply_npk\n        name:  Timed N/P/K application table\n        comment: All fertilizer amounts in kg/ha\n        events_table:\n        - {year}-06-22: {{N_amount : {N_kg}, P_amount: {P_kg}, K_amount: {K_kg}}}\n    StateEvents: null\n\"\"\"\nagromanagement = yaml.safe_load(yaml_agro)\nprint(yaml_agro)\n#crop_end_date: {year_date_1}-11-15\n```\n\n    \n    - 2017-05-01:\n        CropCalendar:\n            crop_name: 'sugar-beet'\n            variety_name: 'sugar-beet-601'\n            crop_start_date: 2017-05-20\n            crop_start_type: sowing\n            crop_end_date: \n            crop_end_type: maturity\n            max_duration: 250\n        TimedEvents:\n        -   event_signal: apply_npk\n            name:  Timed N/P/K application table\n            comment: All fertilizer amounts in kg/ha\n            events_table:\n            - 2017-06-22: {N_amount : 120, P_amount: 60, K_amount: 60}\n        StateEvents: null\n    \n\n\n## Daily weather observations\nDaily weather variables are needed for running the simulation. There are several data providers in PCSE for reading weather data, see the section on [weather data providers](http://pcse.readthedocs.io/en/stable/reference_guide.html#weather-data-providers) to get an overview.\n\nFor this example we will use weather data from an excel file which provides daily weather data for Wageningen for the period 2004 to 2008. We will read the data from the file using the ExcelWeatherDataProvider:\n\n### NASA Weather Data Provider from NASA [DataBase](https://power.larc.nasa.gov/)\n\n\n```python\n#NASA Weather system\n\n#Sometimes it does not work (server), upload excel file\nfrom pcse.db import NASAPowerWeatherDataProvider\n```\n\n\n```python\n\nweather = NASAPowerWeatherDataProvider(51, 5, force_update=True)\n\n```\n\n\n```python\nprint(weather)\n```\n\n    Weather data provided by: NASAPowerWeatherDataProvider\n    --------Description---------\n    NASA/POWER SRB/FLASHFlux/MERRA2/GEOS 5.12.4 (FP-IT) 0.5 x 0.5 Degree Daily Averaged Data\n    ----Site characteristics----\n    Elevation:   38.5\n    Latitude:  51.000\n    Longitude:  5.000\n    Data available for 1983-07-01 - 2021-02-07\n    Number of missing days: 6\n    \n\n\n### Problems with missing days (~1-5 %)\n\n\n```python\ndef weather_loader(latitude, longitude):\n    \n    path = './data/meteo/'\n    #API request to NASA database\n    weather = NASAPowerWeatherDataProvider(latitude, longitude, force_update=True)\n\n    # Print done if downloaded\n    print('____DONE_____','latitude',latitude, 'longitude',longitude,'____')\n\n    # export pcse.weather format to pandas df\n    df_weather = pd.DataFrame(weather.export())\n\n\n    #print('initial number of days:', len(df_weather))\n\n    #create full range of dates\n    r = pd.date_range(start=df_weather.DAY.min(), end=df_weather.DAY.max())\n\n\n    #extend range of dates\n    full_range_weather = df_weather.set_index('DAY').reindex(r).rename_axis('DAY').reset_index()\n    missing_days = (full_range_weather.isna()).sum().sum()\n\n    print('num_of_missing_days', missing_days)\n\n    #fill weather with fill forward method in pandas\n    filled_weather = full_range_weather.fillna(method='ffill', axis=0)\n\n    #save as csv file\n\n    filled_weather=filled_weather[['DAY', 'IRRAD', 'TMIN', 'TMAX', 'VAP', 'WIND', 'RAIN']]\n    filled_weather['SNOWDEPTH'] = 'NaN'\n    filled_weather[['IRRAD']] = filled_weather[['IRRAD']]/1000.\n    filled_weather[['VAP']] = filled_weather[['VAP']]/10.\n    filled_weather.DAY=filled_weather.DAY.dt.strftime('%Y%m%d')\n\n\n    text = open(path+\"pattern.csv\", \"r\")\n    text = ''.join([i for i in text]).replace(\"1111\", str(weather.longitude))\n    text = ''.join([i for i in text]).replace(\"2222\", str(weather.latitude))\n    text = ''.join([i for i in text]).replace(\"3333\", str(weather.elevation))\n    text = ''.join([i for i in text]).replace(\"4444\", str(weather.angstA))\n    text = ''.join([i for i in text]).replace(\"5555\", str(weather.angstB))\n    x = open(path+f'NASA_weather_latitude_{latitude}_longitude_{longitude}.csv',\"w\")\n    x.writelines(text)\n    x.close()\n\n    path_to_save_csv_file = path+f'NASA_weather_latitude_{latitude}_longitude_{longitude}.csv'\n    filled_weather.to_csv(path_to_save_csv_file, mode='a', header=False, index=False)\n\n    #add info to weather database and save it to csv\n\n\n    #LOAD WEATHER as csv file\n    weather = pcse.fileinput.CSVWeatherDataProvider(path_to_save_csv_file)  \n    return weather\n```\n\n\n```python\nweather = weather_loader(55,55)\n```\n\n    ____DONE_____ latitude 55 longitude 55 ____\n    num_of_missing_days 195\n\n\n\n```python\ndf_weather = pd.DataFrame(weather.export())\n```\n\n\n```python\nfig, (ax1,ax2) = plt.subplots(nrows=1, ncols=2, figsize=(16,4))\ndf_weather.set_index('DAY')['ET0'][-365:].plot(ax=ax1, label='ET0')\ndf_weather.set_index('DAY')['TMAX'][-365:].plot(ax=ax2, label='T MAX')\nax1.set_title('ET 0')\nax2.set_title('T\u00b0C Max')\n```\n\n## Importing, initializing and running a PCSE model\n\nInternally, PCSE uses a simulation engine to run a crop simulation. This engine takes a configuration file that specifies the components for the crop, the soil and the agromanagement that need to be used for the simulation. So any PCSE model can be started by importing the engine and initializing it with a given configuration file and the corresponding parameters, weather data and agromanagement.\n\nHowever, as many users of PCSE only need a particular configuration (for example the WOFOST model for potential production), preconfigured Engines are provided in `pcse.models`. For the sugarbeet example we will import the WOFOST model for water-limited simulation under freely draining soil conditions:\n\n\n```python\nfrom pcse.models import Wofost71_WLP_FD\nwofsim = Wofost71_WLP_FD(parameters, weather, agromanagement)\nwofsim.run_till_terminate()\ndf_results = pd.DataFrame(wofsim.get_output())\ndf_results = df_results.set_index(\"day\")\ndf_results.tail()\n```\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n  \n
                                        DVSLAITAGPTWSOTWLVTWSTTWRTTRARDSMWWLOW
                                        day
                                        2018-01-211.6382684.36183213001.942068070.7293872295.0144342636.1982392072.5277590.014248120.00.15405518.486627
                                        2018-01-221.6382684.36183213001.942068070.7293872295.0144342636.1982392072.5277590.023385120.00.15396018.475179
                                        2018-01-231.6382684.36183213001.942068070.7293872295.0144342636.1982392072.5277590.023202120.00.15416618.499908
                                        2018-01-241.6382684.36183213001.942068070.7293872295.0144342636.1982392072.5277590.028208120.00.15405818.486978
                                        2018-01-251.6382684.36183213001.942068070.7293872295.0144342636.1982392072.5277590.026663120.00.15381118.457348
                                        \n
                                        \n\n\n\nWe can then run the simulation and retrieve the time series of daily simulation output using the get_output() method on the WOFOST object. Finally, we convert the simulation results to a pandas dataframe:\n\n\n```python\nsummary_output = wofsim.get_summary_output()\n```\n\n\n```python\nwofsim.get_summary_output()\n```\n\n\n\n\n    [{'DVS': 1.6382678571428573,\n      'LAIMAX': 4.361831916594967,\n      'TAGP': 13001.942059594281,\n      'TWSO': 8070.729386700372,\n      'TWLV': 2295.0144341030245,\n      'TWST': 2636.198238790885,\n      'TWRT': 2072.5277589248244,\n      'CTRAT': 17.5463787829338,\n      'RD': 120.0,\n      'DOS': datetime.date(2017, 5, 20),\n      'DOE': datetime.date(2017, 6, 1),\n      'DOA': datetime.date(2017, 7, 23),\n      'DOM': None,\n      'DOH': None,\n      'DOV': None}]\n\n\n\n\n```python\nmsg = \"Reached maturity at {DOM} with total biomass {TAGP} kg/ha \"\\\n\"and a yield of {TWSO} kg/ha.\"\nprint(msg.format(**summary_output[0]))\n```\n\n    Reached maturity at None with total biomass 13001.942059594281 kg/ha and a yield of 8070.729386700372 kg/ha.\n\n\n# Sensitivity analysis of models\n___\n\n\n\n\n\n\n\n\n```python\n# !pip install SALib\n```\n\n    Collecting SALib\n    \u001b[?25l  Downloading https://files.pythonhosted.org/packages/f7/33/cee4d64f7c40f33c08cf5ef5c9b1fb5e51f194b5deceefb5567112800b70/SALib-1.3.11.tar.gz (856kB)\n    \u001b[K     |\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 860kB 3.3MB/s \n    \u001b[?25hRequirement already satisfied: numpy in /usr/local/lib/python3.6/dist-packages (from SALib) (1.18.5)\n    Requirement already satisfied: scipy in /usr/local/lib/python3.6/dist-packages (from SALib) (1.4.1)\n    Requirement already satisfied: matplotlib in /usr/local/lib/python3.6/dist-packages (from SALib) (3.2.2)\n    Requirement already satisfied: pandas in /usr/local/lib/python3.6/dist-packages (from SALib) (1.0.5)\n    Requirement already satisfied: python-dateutil>=2.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib->SALib) (2.8.1)\n    Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib->SALib) (2.4.7)\n    Requirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.6/dist-packages (from matplotlib->SALib) (0.10.0)\n    Requirement already satisfied: kiwisolver>=1.0.1 in /usr/local/lib/python3.6/dist-packages (from matplotlib->SALib) (1.2.0)\n    Requirement already satisfied: pytz>=2017.2 in /usr/local/lib/python3.6/dist-packages (from pandas->SALib) (2018.9)\n    Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.6/dist-packages (from python-dateutil>=2.1->matplotlib->SALib) (1.15.0)\n    Building wheels for collected packages: SALib\n      Building wheel for SALib (setup.py) ... \u001b[?25l\u001b[?25hdone\n      Created wheel for SALib: filename=SALib-1.3.11-py2.py3-none-any.whl size=729665 sha256=93e720543725a20b1f31d6b4dcd5681574a59bb5c49e364a65b935ee9a92a0b0\n      Stored in directory: /root/.cache/pip/wheels/62/ed/f9/a0b98754ffb2191b98324b96cbbeb1bd5d9598b39ab996b429\n    Successfully built SALib\n    Installing collected packages: SALib\n    Successfully installed SALib-1.3.11\n\n\n## Sobol\u2019 Sequences versus Random numbers and regular grid\n\n\n\n\n\n```python\nfrom SALib.sample import saltelli\nfrom SALib.analyze import sobol\nfrom SALib.test_functions import Ishigami\nimport numpy as np\n```\n\n__Docs [SALib](https://salib.readthedocs.io/en/latest/#)__\n\n\n```python\n\n```\n\nIn this example, we will perform a Sobol\u2019 sensitivity analysis of the _Ishigami_ function, shown below. The _Ishigami_ function is commonly used to test uncertainty and sensitivity analysis methods because it exhibits strong nonlinearity and nonmonotonicity.\n\n$f(x)=\\sin \\left(x_{1}\\right)+ \\text{a}\\, \\operatorname{sin}^{2}\\left(x_{2}\\right)+ \\text{b}\\,  x_{3}^{4} \\sin \\left(x_{1}\\right)$\n\n\n```python\nproblem = {\n  'num_vars': 3,\n  'names': ['x1', 'x2', 'x3'],\n  'bounds': [[-np.pi, np.pi]]*3\n}\n```\n\n\n```python\n# Generate samples\nparam_values = saltelli.sample(problem, 10, calc_second_order=True)\nparam_values.shape\n```\n\n\n\n\n    (80, 3)\n\n\n\nHere, `param_values` is a NumPy matrix. If we run `param_values.shape`, we see that the matrix is **8000 by 3**. The Saltelli sampler generated 8000 samples. The Saltelli sampler generates $N\u2217(2D+2)$ samples, where in this example $N$ is 1000 (the argument we supplied) and $D$ is 3 (the number of model inputs). The keyword argument `calc_second_order=False` will exclude second-order indices, resulting in a smaller sample matrix with $N\u2217(D+2)$ rows instead.\n\n\n\n```python\n# Run model (example)\nY = Ishigami.evaluate(param_values)\n\n# Perform analysis\nSi = sobol.analyze(problem, Y, print_to_console=True)\n# Returns a dictionary with keys 'S1', 'S1_conf', 'ST', and 'ST_conf'\n# (first and total-order indices with bootstrap confidence intervals)\nT_Si, first_Si, (idx, second_Si) = sobol.Si_to_pandas_dict(Si)\n```\n\n    Parameter S1 S1_conf ST ST_conf\n    x1 -0.203434 0.660158 1.345837 2.206907\n    x2 0.838328 0.650453 0.716922 0.567090\n    x3 -0.592087 0.390549 0.459872 0.420236\n    \n    Parameter_1 Parameter_2 S2 S2_conf\n    x1 x2 0.208665 1.159449\n    x1 x3 1.165212 0.917190\n    x2 x3 0.268558 0.759311\n\n\nConsider the model output as\n\\begin{eqnarray*}\nY=f(X)=f\\left(X_{1}, \\ldots, X_{p}\\right),\n\\end{eqnarray*}\nwhere $f$ in our case part of agro-model simulator, $X$ are $p$ varied input parameters and $Y$ is the predicted output. Following the techniques by Sobol we represent the multi-variate random function $f$ using Hoeffding decomposition:\n\\begin{equation}\nf(X_1,\\dots,X_p) = f_0 + \\sum_i^p f_i + \\sum_i^p\\sum_{j>i}^p f_{ij} + \\dots + f_{1\\dots p},\n\\end{equation}\nwhere $f_0$ is a constant term, $f_i = f_i(X_i)$ denotes main effects, $f_{ij} = f_{ij}(X_i, X_j)$ and others describe higher-order interactions. These terms can be written as\n\\begin{equation*}\n\\begin{split}\nf_0 &= E(Y),\\\\\nf_i &= E_{X_{\\sim i}}(Y | X_i) - E(Y),\\\\\nf_{ij} &= E_{X_{\\sim ij}}(Y | X_i, X_j) - f_i - f_j - f_0,\\\\\n\\dots\n\\end{split}\n\\end{equation*}\nwhere $E$ is mathematical expectation and $X_{\\sim i}$ denotes all parameters except $i^\\text{th}$. Under the assumption that the input parameters are independent, total variance $V(Y)$ of the crop yield can be decomposed as follows:\n\\begin{equation*}\nV(Y) = \\sum_i^p V_i + \\sum_i^p\\sum_{j>i}^p V_{ij} + \\dots + V_{12\\dots p},\n\\end{equation*}\nwhere partial variances are\n\\begin{equation*}\n\\begin{split}\nV_i &= V[f_i(X_i)] = V_{X_i}\\left[E_{X_{\\sim i}}(Y | X_i)\\right],\\\\\nV_{ij} &= V[f_{ij}(X_i,X_j)] = V_{X_iX_j}\\left[E_{X_{\\sim ij}}(Y | X_i, X_j)\\right] - V_i - V_j,\\\\\n\\dots\n\\end{split}\n\\end{equation*}\n\n## Sobol index (first order, second order, total index)\n\nThis way, sensitivity indices (SI) can be introduced as \n\\begin{equation}\n\\Large\nS_i = \\frac{V_i}{V(Y)},~S_{ij} = \\frac{V_{ij}}{V(Y)},~\\dots\n\\end{equation}\nIn order to incorporate all of the interactions for a particular parameter, one can compute the total effect index:\n\\begin{equation}\nS_{T_i} = \\frac{E_{X_{\\sim i}}\\left[V_{X_i}(Y|X_{\\sim i})\\right]}{V(Y)} = 1 - \\frac{V_{X_{\\sim i}}\\left[E_{X_i}(Y | X_{\\sim i})\\right]}{V(Y)}\n\\end{equation}\n\n\nFrom this assumption we can conclude:\n\\begin{equation}\n\\Large\n0 \\leq S_i \\leq S_{T_i} \\leq 1\n\\end{equation}\n\nMore -\n* [Wiki](https://en.wikipedia.org/wiki/Sobol_sequence)\n* [Habr](https://habr.com/ru/post/440892/)\n* Feature selection [Skoltech ML 2020](https://github.com/adasegroup/ML2020_lectures/blob/master/lecture9/Lecture_9_Model_Feature_Selection_Sensitivity.pdf)\n\n# Sensitivity analysis of WOFOST model \n\n\n\n## Install modules \n\n\n```python\n  from SALib.sample import saltelli\n  from SALib.analyze import sobol\n  from SALib.test_functions import Ishigami\n  import numpy as np\n  import pandas as pd\n```\n\n## Parameters\n\n\n```python\nNPK = {\n    'num_vars':3,\n    'names':['N_kg', 'P_kg', 'K_kg'],\n    'bounds':[[30., 60.],\n              [60., 90.],\n              [100., 130.]]\n}\n```\n\n\n```python\nSoil_parameters = {\n  'num_vars':5,\n    'names':['SMV', 'SMFCF', 'SM0', 'CRAIRC', 'K0'],\n    'bounds':[[0.7, 1.3],\n              [0.1, 0.5],\n              [0.2, 0.6],\n             [0.04, 0.08],\n             [22.5, 27.5]]}\n```\n\n## Generate input parameters\n\n\n```python\nparam_values = saltelli.sample(Soil_parameters, 10)\n```\n\n$n = N \\times (D \\times 2 +2)$\n\n\n```python\nparam_values.shape\n```\n\n\n\n\n    (120, 5)\n\n\n\n## Loop for yield prediction\n\n\n```python\nfrom pcse.fileinput import YAMLAgroManagementReader\nagromanagement_file = os.path.join(data_dir, 'agro', './sugarbeet_calendar.agro')\nagromanagement = YAMLAgroManagementReader(agromanagement_file)\n#print(agromanagement)\nSoil_parameters = {\n  'num_vars':5,\n    'names':['SMV', 'SMFCF', 'SM0', 'CRAIRC', 'K0'],\n    'bounds':[[0.7, 1.3],\n              [0.1, 0.5],\n              [0.2, 0.6],\n             [0.04, 0.08],\n             [22.5, 27.5]]}\nparam_values = saltelli.sample(Soil_parameters, N=10, calc_second_order=True)\n```\n\nSoil parameters in [PCSE model](https://pcse.readthedocs.io/en/stable/code.html?highlight=K0#pcse.soil.WaterbalanceFD) \n\n\n```python\ndef sensitivity_soil(soil_parameters):\n    SMV, SMFCF, SM0, CRAIRC, K0 = soil_parameters \n    soildata['SMV'] = SMV\n    soildata['SMFCF'] = SMFCF\n    soildata['SM0'] = SM0\n    soildata['CRAIRC'] = CRAIRC\n    soildata['K0'] = K0\n    parameters = ParameterProvider(cropdata=cropdata, soildata=soildata, sitedata=sitedata)\n    wofsim = Wofost71_WLP_FD(parameters, wdp, agromanagement)\n    wofsim.run_till_terminate()\n    #df_results = pd.DataFrame(wofsim.get_output())\n    #df_results = df_results.set_index(\"day\")\n    #df_results.tail()\n    summary_output = wofsim.get_summary_output()\n    yield_list.append(summary_output[0]['TWSO'])\n```\n\n\n```python\n%%time\nyield_list = []\nparam_values = saltelli.sample(Soil_parameters, 10, calc_second_order=True)\nfor step in range(len(param_values)):\n    sensitivity_soil(param_values[step])\n    print(param_values[step])\n```\n\n    [ 0.83183594  0.13867188  0.40742187  0.06707031 23.90136719]\n    [ 1.24433594  0.13867188  0.40742187  0.06707031 23.90136719]\n    [ 0.83183594  0.11835938  0.40742187  0.06707031 23.90136719]\n    [ 0.83183594  0.13867188  0.55976563  0.06707031 23.90136719]\n    [ 0.83183594  0.13867188  0.40742187  0.06003906 23.90136719]\n    [ 0.83183594  0.13867188  0.40742187  0.06707031 22.84667969]\n    [ 0.83183594  0.11835938  0.55976563  0.06003906 22.84667969]\n    [ 1.24433594  0.13867188  0.55976563  0.06003906 22.84667969]\n    [ 1.24433594  0.11835938  0.40742187  0.06003906 22.84667969]\n    [ 1.24433594  0.11835938  0.55976563  0.06707031 22.84667969]\n    [ 1.24433594  0.11835938  0.55976563  0.06003906 23.90136719]\n    [ 1.24433594  0.11835938  0.55976563  0.06003906 22.84667969]\n    [ 1.13183594  0.33867188  0.20742188  0.04707031 26.40136719]\n    [ 0.94433594  0.33867188  0.20742188  0.04707031 26.40136719]\n    [ 1.13183594  0.31835938  0.20742188  0.04707031 26.40136719]\n    [ 1.13183594  0.33867188  0.35976562  0.04707031 26.40136719]\n    [ 1.13183594  0.33867188  0.20742188  0.04003906 26.40136719]\n    [ 1.13183594  0.33867188  0.20742188  0.04707031 25.34667969]\n    [ 1.13183594  0.31835938  0.35976562  0.04003906 25.34667969]\n    [ 0.94433594  0.33867188  0.35976562  0.04003906 25.34667969]\n    [ 0.94433594  0.31835938  0.20742188  0.04003906 25.34667969]\n    [ 0.94433594  0.31835938  0.35976562  0.04707031 25.34667969]\n    [ 0.94433594  0.31835938  0.35976562  0.04003906 26.40136719]\n    [ 0.94433594  0.31835938  0.35976562  0.04003906 25.34667969]\n    [ 1.28183594  0.23867188  0.50742187  0.07707031 25.15136719]\n    [ 0.79433594  0.23867188  0.50742187  0.07707031 25.15136719]\n    [ 1.28183594  0.21835938  0.50742187  0.07707031 25.15136719]\n    [ 1.28183594  0.23867188  0.25976562  0.07707031 25.15136719]\n    [ 1.28183594  0.23867188  0.50742187  0.05003906 25.15136719]\n    [ 1.28183594  0.23867188  0.50742187  0.07707031 26.59667969]\n    [ 1.28183594  0.21835938  0.25976562  0.05003906 26.59667969]\n    [ 0.79433594  0.23867188  0.25976562  0.05003906 26.59667969]\n    [ 0.79433594  0.21835938  0.50742187  0.05003906 26.59667969]\n    [ 0.79433594  0.21835938  0.25976562  0.07707031 26.59667969]\n    [ 0.79433594  0.21835938  0.25976562  0.05003906 25.15136719]\n    [ 0.79433594  0.21835938  0.25976562  0.05003906 26.59667969]\n    [ 0.98183594  0.43867188  0.30742188  0.05707031 22.65136719]\n    [ 1.09433594  0.43867188  0.30742188  0.05707031 22.65136719]\n    [ 0.98183594  0.41835937  0.30742188  0.05707031 22.65136719]\n    [ 0.98183594  0.43867188  0.45976563  0.05707031 22.65136719]\n    [ 0.98183594  0.43867188  0.30742188  0.07003906 22.65136719]\n    [ 0.98183594  0.43867188  0.30742188  0.05707031 24.09667969]\n    [ 0.98183594  0.41835937  0.45976563  0.07003906 24.09667969]\n    [ 1.09433594  0.43867188  0.45976563  0.07003906 24.09667969]\n    [ 1.09433594  0.41835937  0.30742188  0.07003906 24.09667969]\n    [ 1.09433594  0.41835937  0.45976563  0.05707031 24.09667969]\n    [ 1.09433594  0.41835937  0.45976563  0.07003906 22.65136719]\n    [ 1.09433594  0.41835937  0.45976563  0.07003906 24.09667969]\n    [ 0.90683594  0.28867188  0.25742188  0.05207031 23.27636719]\n    [ 1.16933594  0.28867188  0.25742188  0.05207031 23.27636719]\n    [ 0.90683594  0.26835938  0.25742188  0.05207031 23.27636719]\n    [ 0.90683594  0.28867188  0.20976563  0.05207031 23.27636719]\n    [ 0.90683594  0.28867188  0.25742188  0.05503906 23.27636719]\n    [ 0.90683594  0.28867188  0.25742188  0.05207031 25.97167969]\n    [ 0.90683594  0.26835938  0.20976563  0.05503906 25.97167969]\n    [ 1.16933594  0.28867188  0.20976563  0.05503906 25.97167969]\n    [ 1.16933594  0.26835938  0.25742188  0.05503906 25.97167969]\n    [ 1.16933594  0.26835938  0.20976563  0.05207031 25.97167969]\n    [ 1.16933594  0.26835938  0.20976563  0.05503906 23.27636719]\n    [ 1.16933594  0.26835938  0.20976563  0.05503906 25.97167969]\n    [ 1.20683594  0.48867187  0.45742187  0.07207031 25.77636719]\n    [ 0.86933594  0.48867187  0.45742187  0.07207031 25.77636719]\n    [ 1.20683594  0.46835938  0.45742187  0.07207031 25.77636719]\n    [ 1.20683594  0.48867187  0.40976563  0.07207031 25.77636719]\n    [ 1.20683594  0.48867187  0.45742187  0.07503906 25.77636719]\n    [ 1.20683594  0.48867187  0.45742187  0.07207031 23.47167969]\n    [ 1.20683594  0.46835938  0.40976563  0.07503906 23.47167969]\n    [ 0.86933594  0.48867187  0.40976563  0.07503906 23.47167969]\n    [ 0.86933594  0.46835938  0.45742187  0.07503906 23.47167969]\n    [ 0.86933594  0.46835938  0.40976563  0.07207031 23.47167969]\n    [ 0.86933594  0.46835938  0.40976563  0.07503906 25.77636719]\n    [ 0.86933594  0.46835938  0.40976563  0.07503906 23.47167969]\n    [ 1.05683594  0.18867188  0.35742188  0.04207031 27.02636719]\n    [ 0.71933594  0.18867188  0.35742188  0.04207031 27.02636719]\n    [ 1.05683594  0.16835938  0.35742188  0.04207031 27.02636719]\n    [ 1.05683594  0.18867188  0.50976562  0.04207031 27.02636719]\n    [ 1.05683594  0.18867188  0.35742188  0.06503906 27.02636719]\n    [ 1.05683594  0.18867188  0.35742188  0.04207031 24.72167969]\n    [ 1.05683594  0.16835938  0.50976562  0.06503906 24.72167969]\n    [ 0.71933594  0.18867188  0.50976562  0.06503906 24.72167969]\n    [ 0.71933594  0.16835938  0.35742188  0.06503906 24.72167969]\n    [ 0.71933594  0.16835938  0.50976562  0.04207031 24.72167969]\n    [ 0.71933594  0.16835938  0.50976562  0.06503906 27.02636719]\n    [ 0.71933594  0.16835938  0.50976562  0.06503906 24.72167969]\n    [ 0.75683594  0.38867188  0.55742187  0.06207031 24.52636719]\n    [ 1.01933594  0.38867188  0.55742187  0.06207031 24.52636719]\n    [ 0.75683594  0.36835938  0.55742187  0.06207031 24.52636719]\n    [ 0.75683594  0.38867188  0.30976562  0.06207031 24.52636719]\n    [ 0.75683594  0.38867188  0.55742187  0.04503906 24.52636719]\n    [ 0.75683594  0.38867188  0.55742187  0.06207031 27.22167969]\n    [ 0.75683594  0.36835938  0.30976562  0.04503906 27.22167969]\n    [ 1.01933594  0.38867188  0.30976562  0.04503906 27.22167969]\n    [ 1.01933594  0.36835938  0.55742187  0.04503906 27.22167969]\n    [ 1.01933594  0.36835938  0.30976562  0.06207031 27.22167969]\n    [ 1.01933594  0.36835938  0.30976562  0.04503906 24.52636719]\n    [ 1.01933594  0.36835938  0.30976562  0.04503906 27.22167969]\n    [ 0.73808594  0.17617188  0.21992188  0.05832031 25.62011719]\n    [ 0.85058594  0.17617188  0.21992188  0.05832031 25.62011719]\n    [ 0.73808594  0.48085937  0.21992188  0.05832031 25.62011719]\n    [ 0.73808594  0.17617188  0.52226562  0.05832031 25.62011719]\n    [ 0.73808594  0.17617188  0.21992188  0.04128906 25.62011719]\n    [ 0.73808594  0.17617188  0.21992188  0.05832031 24.56542969]\n    [ 0.73808594  0.48085937  0.52226562  0.04128906 24.56542969]\n    [ 0.85058594  0.17617188  0.52226562  0.04128906 24.56542969]\n    [ 0.85058594  0.48085937  0.21992188  0.04128906 24.56542969]\n    [ 0.85058594  0.48085937  0.52226562  0.05832031 24.56542969]\n    [ 0.85058594  0.48085937  0.52226562  0.04128906 25.62011719]\n    [ 0.85058594  0.48085937  0.52226562  0.04128906 24.56542969]\n    [ 1.03808594  0.37617188  0.41992188  0.07832031 23.12011719]\n    [ 1.15058594  0.37617188  0.41992188  0.07832031 23.12011719]\n    [ 1.03808594  0.28085938  0.41992188  0.07832031 23.12011719]\n    [ 1.03808594  0.37617188  0.32226562  0.07832031 23.12011719]\n    [ 1.03808594  0.37617188  0.41992188  0.06128906 23.12011719]\n    [ 1.03808594  0.37617188  0.41992188  0.07832031 27.06542969]\n    [ 1.03808594  0.28085938  0.32226562  0.06128906 27.06542969]\n    [ 1.15058594  0.37617188  0.32226562  0.06128906 27.06542969]\n    [ 1.15058594  0.28085938  0.41992188  0.06128906 27.06542969]\n    [ 1.15058594  0.28085938  0.32226562  0.07832031 27.06542969]\n    [ 1.15058594  0.28085938  0.32226562  0.06128906 23.12011719]\n    [ 1.15058594  0.28085938  0.32226562  0.06128906 27.06542969]\n    CPU times: user 1min 21s, sys: 229 ms, total: 1min 21s\n    Wall time: 1min 21s\n\n\n\n```python\nnp_yield = np.array(yield_list)\nSi = sobol.analyze(Soil_parameters, np_yield, print_to_console=False)\n```\n\n\n```python\nSi_dict = dict(Si) \nSi_df = pd.DataFrame()\nSi_df = Si_df.append(pd.Series(Si_dict['S1']), ignore_index=True)\nSi_df = Si_df.append(pd.Series(Si_dict['ST']), ignore_index=True)\nSi_df = Si_df.append(pd.Series(Si_dict['S1_conf']), ignore_index=True)\nSi_df = Si_df.append(pd.Series(Si_dict['ST_conf']), ignore_index=True)\nSi_df = Si_df.T\nSi_df.columns = ['Si', 'ST', 'Si_conf', 'ST_conf']\nSi_df.rename(index={0:'SMV',1:'SMFCF', 2:'SM0', 3:'CRAIRC', 4:'K0'}, inplace=True)\nSi_df\n```\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n    \n  \n
                                        SiSTSi_confST_conf
                                        SMV0.0000000.0000000.0000000.000000
                                        SMFCF-0.0774170.2848780.8394560.638290
                                        SM00.2675040.3814370.6934090.688329
                                        CRAIRC0.0000000.0000000.0000000.000000
                                        K00.0000000.0000000.0000000.000000
                                        \n
                                        \n\n\n\nIs it ok?\n\n\n\\begin{equation}\n\\Large\n0 \\leq S_i \\leq S_{T_i} \\leq 1\n\\end{equation}\n\n### For 5 years\n\n\n```python\ndef sensitivity_weather(year):\n    K_kg = 60\n    P_kg = 60\n    N_kg = 120\n    year_date=year\n    print(year)\n    print(year_date)\n    yaml_agro = f\"\"\"\n    - {year_date}-06-01:\n        CropCalendar:\n            crop_name: 'sugar-beet'\n            variety_name: 'sugar-beet-601'\n            crop_start_date: {year_date}-06-02\n            crop_start_type: emergence\n            crop_end_date: {year_date}-10-15\n            crop_end_type: harvest\n            max_duration: 300\n        TimedEvents:\n        -   event_signal: apply_npk\n            name:  Timed N/P/K application table\n            comment: All fertilizer amounts in kg/ha\n            events_table:\n            - {year_date}-06-22: {{N_amount : {N_kg}, P_amount: {P_kg}, K_amount: {K_kg}}}\n        StateEvents: null\n        \"\"\"\n    agromanagement = yaml.safe_load(yaml_agro)\n    parameters = ParameterProvider(cropdata=cropdata, soildata=soildata, sitedata=sitedata)\n    wofsim = Wofost71_WLP_FD(parameters, moscow_weather, agromanagement)\n    wofsim.run_till_terminate()\n    summary_output = wofsim.get_summary_output()\n    yield_list_weather.append(summary_output[0]['TWSO'])\n```\n\n## Visualizing simulation results\n\nFinally, we can generate some figures of WOFOST variables such as the development (DVS), total biomass (TAGP), leaf area index (LAI) and root-zone soil moisture (SM) using the MatPlotLib plotting package:\n\n\n```python\nfig, axes = plt.subplots(nrows=2, ncols=2, figsize=(12,10))\nfor var, ax in zip([\"DVS\", \"TWSO\", \"LAI\", \"SM\"], axes.flatten()):\n    ax.plot_date(df_results.index, df_results[var], 'b-')\n    ax.set_title(var)\nfig.autofmt_xdate()\n```\n\n# Visualization for sensitivity analysis\n\nPlots by [Water programming group](https://waterprogramming.wordpress.com/2019/08/27/a-python-implementation-of-grouped-radial-convergence-plots-to-visualize-sobol-sensitivity-analysis-results/)\n\nHow to repeat: [Repo of SampleVIS](https://github.com/charlesrouge/SampleVis)\n\n\n```python\nimport numpy as np\nfrom SALib.analyze import sobol\nfrom SALib.sample import saltelli\nfrom fishery import fish_game\nimport matplotlib.pyplot as plt\nimport itertools\nimport math\n```\n\n### Why number of samples is important?\n\n\n```python\n# Set up dictionary with system parameters\nproblem = {\n  'num_vars': 6,\n  'names': ['a', 'b', 'c','h',\n            'K','m'],\n  'bounds': [[ 0.002, 2],\n             [0.005, 1],\n             [0.2, 1],\n             [0.001, 1],\n             [100, 5000],\n             [0.1, 1.5]]\n}\n\n# Array with n's to use\nnsamples = np.arange(50, 500, 50)\n\n# Arrays to store the index estimates\nS1_estimates = np.zeros([problem['num_vars'],len(nsamples)])\nST_estimates = np.zeros([problem['num_vars'],len(nsamples)])\n\n# Loop through all n values, create sample, evaluate model and estimate S1 & ST\nfor i in range(len(nsamples)):\n    print('n= '+ str(nsamples[i]))\n    # Generate samples\n    sampleset = saltelli.sample(problem, nsamples[i],calc_second_order=False)\n    # Run model for all samples\n    output = [fish_game(*sampleset[j,:]) for j in range(len(sampleset))]\n    # Perform analysis\n    results = sobol.analyze(problem, np.asarray(output), calc_second_order=False,print_to_console=False)\n    # Store estimates\n    ST_estimates[:,i]=results['ST']\n    S1_estimates[:,i]=results['S1']\n\nnp.save('ST_estimates.npy', ST_estimates)\nnp.save('S1_estimates.npy', S1_estimates)\n\nS1_estimates = np.load('S1_estimates.npy')\nST_estimates = np.load('ST_estimates.npy')\n\n# Generate figure showing evolution of indices\nfig =  plt.figure(figsize=(18,9))\nax1 = fig.add_subplot(1,2,1)\nhandles = []\nfor j in range(problem['num_vars']):\n    handles += ax1.plot(nsamples, S1_estimates[j,:], linewidth=5)\nax1.set_title('Evolution of S1 index estimates', fontsize=20)\nax1.set_ylabel('S1', fontsize=18)\nax1.set_xlabel('Number of samples (n)', fontsize=18)\nax1.tick_params(axis='both', which='major', labelsize=14)\nax2 = fig.add_subplot(1,2,2)\nfor j in range(problem['num_vars']):\n    ax2.plot(nsamples, ST_estimates[j,:], linewidth=5)\nax2.set_title('Evolution of ST index estimates', fontsize=20)\nax2.set_ylabel('ST', fontsize=18)\nax2.tick_params(axis='both', which='major', labelsize=14)\nax2.set_xlabel('Number of samples (n)', fontsize=18)\nfig.legend(handles, problem['names'], loc = 'right', fontsize=11)\nplt.show()\n#plt.savefig('indexevolution.png')\n\n# Calculate parameter rankings\nS1_ranks = np.zeros_like(S1_estimates)\nST_ranks = np.zeros_like(ST_estimates)\nfor i in range(len(nsamples)):\n    orderS1 = np.argsort(S1_estimates[:,i])\n    orderST = np.argsort(ST_estimates[:,i])\n    S1_ranks[:,i] = orderS1.argsort()\n    ST_ranks[:,i] = orderST.argsort()\n    \n# Generate figure showing evolution of ranks\nfig =  plt.figure(figsize=(18,9))\nax1 = fig.add_subplot(1,2,1)\nhandles = []\nfor j in range(problem['num_vars']):\n    handles += ax1.plot(nsamples, S1_ranks[j,:], linewidth=3)\nax1.set_title('Parameter ranking based on S1', fontsize=20)\nax1.set_ylabel('S1', fontsize=18)\nax1.set_xlabel('Number of samples (n)', fontsize=18)\nax1.set_yticklabels(np.arange(problem['num_vars']+1, 0, -1))\nax1.tick_params(axis='both', which='major', labelsize=14)\nax2 = fig.add_subplot(1,2,2)\nfor j in range(problem['num_vars']):\n    ax2.plot(nsamples, ST_ranks[j,:], linewidth=3)\nax2.set_title('Parameter ranking based on ST', fontsize=20)\nax2.set_ylabel('ST', fontsize=18)\nax2.set_yticklabels(np.arange(problem['num_vars']+1, 0, -1))\nax2.tick_params(axis='both', which='major', labelsize=14)\nax2.set_xlabel('Number of samples (n)', fontsize=18)\nfig.legend(handles, problem['names'], loc = 'right', fontsize=14)\n#plt.show()\n#plt.savefig('rankingevolution.png')\n```\n\n## Radial plot for SA\n\n\n```python\nimport numpy as np\nimport itertools\nimport matplotlib.pyplot as plt\nimport seaborn as sns\nimport math\nfrom numpy import genfromtxt\nimport matplotlib.patches as mpatches\nimport matplotlib.pyplot as plt\nsns.set_style('whitegrid', {'axes_linewidth': 0, 'axes.edgecolor': 'white'})\n```\n\n## Plot function\n\n\n```python\ndef is_significant(value, confidence_interval, threshold=\"conf\"):\n    if threshold == \"conf\":\n        return value - abs(confidence_interval) > 0\n    else:\n        return value - abs(float(threshold)) > 0\n \ndef grouped_radial(SAresults, parameters, radSc=2.0, scaling=1, widthSc=0.5, STthick=1, varNameMult=1.3, colors=None, groups=None, gpNameMult=1.5, threshold=\"conf\"):\n    # Derived from https://github.com/calvinwhealton/SensitivityAnalysisPlots\n    fig, ax = plt.subplots(1, 1)\n    color_map = {}\n     \n    # initialize parameters and colors\n    if groups is None:\n         \n        if colors is None:\n            colors = [\"k\"]\n         \n        for i, parameter in enumerate(parameters):\n            color_map[parameter] = colors[i % len(colors)]\n    else:        \n        if colors is None:\n            colors = sns.color_palette(\"deep\", max(3, len(groups)))\n         \n        for i, key in enumerate(groups.keys()):\n            #parameters.extend(groups[key])\n             \n            for parameter in groups[key]:\n                color_map[parameter] = colors[i % len(colors)]\n     \n    n = len(parameters)\n    angles = radSc*math.pi*np.arange(0, n)/n\n    x = radSc*np.cos(angles)\n    y = radSc*np.sin(angles)\n     \n    # plot second-order indices\n    for i, j in itertools.combinations(range(n), 2):\n        #key1 = parameters[i]\n        #key2 = parameters[j]\n         \n        if is_significant(SAresults[\"S2\"][i][j], SAresults[\"S2_conf\"][i][j], threshold):\n            angle = math.atan((y[j]-y[i])/(x[j]-x[i]))\n                 \n            if y[j]-y[i] < 0:\n                angle += math.pi\n                 \n            line_hw = scaling*(max(0, SAresults[\"S2\"][i][j])**widthSc)/2\n                 \n            coords = np.empty((4, 2))\n            coords[0, 0] = x[i] - line_hw*math.sin(angle)\n            coords[1, 0] = x[i] + line_hw*math.sin(angle)\n            coords[2, 0] = x[j] + line_hw*math.sin(angle)\n            coords[3, 0] = x[j] - line_hw*math.sin(angle)\n            coords[0, 1] = y[i] + line_hw*math.cos(angle)\n            coords[1, 1] = y[i] - line_hw*math.cos(angle)\n            coords[2, 1] = y[j] - line_hw*math.cos(angle)\n            coords[3, 1] = y[j] + line_hw*math.cos(angle)\n \n            ax.add_artist(plt.Polygon(coords, color=\"0.75\"))\n         \n    # plot total order indices\n    for i, key in enumerate(parameters):\n        if is_significant(SAresults[\"ST\"][i], SAresults[\"ST_conf\"][i], threshold):\n            ax.add_artist(plt.Circle((x[i], y[i]), scaling*(SAresults[\"ST\"][i]**widthSc)/2, color='w'))\n            ax.add_artist(plt.Circle((x[i], y[i]), scaling*(SAresults[\"ST\"][i]**widthSc)/2, lw=STthick, color='0.4', fill=False))\n     \n    # plot first-order indices\n    for i, key in enumerate(parameters):\n        if is_significant(SAresults[\"S1\"][i], SAresults[\"S1_conf\"][i], threshold):\n            ax.add_artist(plt.Circle((x[i], y[i]), scaling*(SAresults[\"S1\"][i]**widthSc)/2, color='0.4'))\n            \n    # add labels\n    for i, key in enumerate(parameters):                \n        ax.text(varNameMult*x[i], varNameMult*y[i], key, ha='center', va='center',\n                rotation=angles[i]*360/(2*math.pi) - 90,\n                color=color_map[key])\n         \n    if groups is not None:\n        for i, group in enumerate(groups.keys()):\n            print(group)\n            group_angle = np.mean([angles[j] for j in range(n) if parameters[j] in groups[group]])\n             \n            ax.text(gpNameMult*radSc*math.cos(group_angle), gpNameMult*radSc*math.sin(group_angle), group, ha='center', va='center',\n                rotation=group_angle*360/(2*math.pi) - 90,\n                color=colors[i % len(colors)])\n             \n    ax.set_facecolor('white')\n    ax.set_xticks([])\n    ax.set_yticks([])\n#     ax.\n    plt.axis('equal')\n    plt.axis([-2*radSc, 2*radSc, -2*radSc, 2*radSc])\n    #plt.show()\n \n     \n    return fig\n```\n\n## Range of soil parameters\n\n\n```python\nproblem = {\n    'num_vars':6,\n    'names':['SOC', 'Sand', 'Clay', 'pH', 'CN', 'BD'],\n    'bounds':[[2.58, 6.20],\n              [0.01, 0.30],\n              [0.01, 0.30],\n              [4.6, 6.9],\n              [10.9, 12.4],\n              [900, 1350]]\n}\n```\n\n\n```python\n#names for csv files\nlist_of_csv=['soybean-000-2015.csv', 'sugar-beet-2011.csv', 'sugar-beet-2017.csv',\n'spring-barley-2012.csv', 'sugar-beet-2014.csv']\nlist_of_names=['soybean-000-2015', 'sugar-beat-2011', 'sugar-beat-2017',\n'spring-barley-2012', 'sugar-beat-2014']\nlist_of_totals=['total_SI_'+x for x in list_of_names]\nlist_of_first=['fisrt_SI_'+x for x in list_of_names]\nlist_of_second=['second_SI_'+x for x in list_of_names]\nlist_of_SI=['SI_'+x for x in list_of_names]\n```\n\n\n```python\nfor j, i in enumerate(list_of_csv):\n        all_data_csv = genfromtxt('./'+str(i), delimiter=',')\n        output = all_data_csv[:,2]\n        print(i)\n        list_of_SI[j] = sobol.analyze(problem, output, calc_second_order=True, conf_level=0.95, print_to_console=False)\n```\n\n    soybean-000-2015.csv\n    sugar-beet-2011.csv\n    sugar-beet-2017.csv\n    spring-barley-2012.csv\n    sugar-beet-2014.csv\n\n\n\n```python\ngroups={\"Soil physics\" : [\"Sand\", \"Clay\", \"BD\"],\n        \"Soil chemistry\" : [\"pH\", \"SOC\", \"CN\"]}\n \nfig = grouped_radial(list_of_SI[4], ['BD', 'Sand', 'Clay', 'pH', 'CN', 'SOC'], groups=groups, threshold=0.001)\nred_patch = mpatches.Patch(color='red', label='The red data')\nplt.title(list_of_names[4], loc='left')\n\nplt.show()\n```\n\n## Homework\n\n__[Tasks](https://skoltech-my.sharepoint.com/:w:/g/personal/mikhail_gasanov_skoltech_ru/EeTPQxbrzVdPqnSENKYyoTUBay1RDYgMMW3GO3qFT2ge5g?e=4hk45V)__\n\nUsefull\n\n__SA and UQ__\n\n\n1) [Rhodium project](https://github.com/Project-Platypus/Rhodium.git)\n\n\n2) [SALib](https://github.com/SALib/SALib)\n\n__Model__\n\n3) [PCSE](https://pcse.readthedocs.io/en/stable/index.html)\n\n4) How to install PCSE at local machine\n    `conda env create -f` [py3_pcse.yml](https://github.com/mishagrol/Seminar_Sobol/blob/master/py3_pcse.yml)\n\n\u0410ny questions  -  \n\nTelegram - `@misha_grol`\n\nPart 1 \u2013 Crop Yield Prediction (PCSE, MONICA)  \n\nYou can use the seminars\u2019 colab:  \n\n\u201chttps://colab.research.google.com/drive/1j4AHD8KkTRThPuNsQzQFWYSJtptQ6bUA\u201d  \n\n1) Assess the yield of one of the crops for the Moscow region over several years (potatoes, sugar beets for 2-3 years)   \n\n    Crop  - (https://github.com/mishagrol/Seminar_Sobol/tree/master/data/crop) \n\n    Soil - (https://github.com/mishagrol/Seminar_Sobol/tree/master/data/soil) \n\n    Weather  - NASAdataprovider in PCSE (https://pcse.readthedocs.io/en/stable/code.html?highlight=NASA#pcse.db.NASAPowerWeatherDataProvider)   \n\n    Agromanagement - (https://github.com/mishagrol/Seminar_Sobol/blob/master/data/agro/sugarbeet_calendar_Moscow_short.agro) \n\nPart 2 \u2013 Sensitivity Analysis (SALib)  \n\n1) Perform sensitivity analysis of one of the model blocks (crop, soil, agromanagement *) with SALib. You can choose one of the methods that you consider necessary (Sobol, FAST, ...). \n\n    Generate samples \u2013 In report provide the size of the resulting matrix and the sample size (N) \n\n    Conduct parameter sensitivity analysis  - In report provide S1 and ST indices.  \n\n2) Generate plots (Hist, Radial convergence plot, etc.)  \n\n*3) Estimate the required number of simulations to obtain reliable values of the sensitivity indices. Try to estimate the sample size at the confidence interval of the sensitivity indices. \n\n* Please note that working with discrete data can cause certain difficulties. \n\n#Agro Hack https://agro-code.ru/\n\n### bonus\n\n__Morris method__\n\n\nGenerate a sample using the Method of Morris\n\nThree variants of Morris' sampling for elementary effects is supported:\n\n- Vanilla Morris\n- Optimised trajectories when ``optimal_trajectories=True`` (using\n    Campolongo's enhancements from 2007 and optionally Ruano's enhancement\n    from 2012; ``local_optimization=True``)\n- Groups with optimised trajectories when ``optimal_trajectories=True`` and\n    the problem definition specifies groups (note that ``local_optimization``\n    must be ``False``)\n\nAt present, optimised trajectories is implemented using either a brute-force\napproach, which can be very slow, especially if you require more than four\ntrajectories, or a local method based which is much faster. Both methods now\nimplement working with groups of factors.\n\nNote that the number of factors makes little difference,\nbut the ratio between number of optimal trajectories and the sample size\nresults in an exponentially increasing number of scores that must be\ncomputed to find the optimal combination of trajectories.  We suggest going\nno higher than 4 from a pool of 100 samples with the brute force approach.\nWith local_optimization = True (which is default),\nit is possible to go higher than the previously suggested 4 from 100.\n\n\n\n\n```python\nimport sys\n\nfrom SALib.analyze import morris\nfrom SALib.sample.morris import sample\nfrom SALib.test_functions import Sobol_G\nfrom SALib.util import read_param_file\nfrom SALib.plotting.morris import horizontal_bar_plot, covariance_plot, \\\n    sample_histograms\nimport matplotlib.pyplot as plt\n\n#sys.path.append('../..')\n\n# Read the parameter range file and generate samples\n#problem = read_param_file('/Users/mikhailgasanov/Documents/GIT/SALib/src/SALib/test_functions/params/Sobol_G.txt')\n# or define manually without a parameter file:\nproblem = {\n 'num_vars': 8,\n 'names': ['x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7', 'x8'],\n 'groups': None,\n 'bounds': [[0.0, 1.0],\n            [0.0, 1.0],\n            [0.0, 1.0],\n            [0.0, 1.0],\n            [0.0, 1.0],\n            [0.0, 1.0],\n            [0.0, 1.0],\n            [0.0, 1.0]]\n}\n# Files with a 4th column for \"group name\" will be detected automatically, e.g.\n# param_file = '../../src/SALib/test_functions/params/Ishigami_groups.txt'\n\n# Generate samples\nparam_values = sample(problem, N=1000, num_levels=4,\n                      optimal_trajectories=None)\n\n# To use optimized trajectories (brute force method),\n# give an integer value for optimal_trajectories\n\n# Run the \"model\" -- this will happen offline for external models\nY = Sobol_G.evaluate(param_values)\n\n# Perform the sensitivity analysis using the model output\n# Specify which column of the output file to analyze (zero-indexed)\nSi = morris.analyze(problem, param_values, Y, conf_level=0.95,\n                    print_to_console=True,\n                    num_levels=4, num_resamples=100)\n# Returns a dictionary with keys 'mu', 'mu_star', 'sigma', and 'mu_star_conf'\n# e.g. Si['mu_star'] contains the mu* value for each parameter, in the\n# same order as the parameter file\n\nfig, (ax1, ax2) = plt.subplots(1, 2)\nhorizontal_bar_plot(ax1, Si, {}, sortby='mu_star', unit=r\"tCO$_2$/year\")\ncovariance_plot(ax2, Si, {}, unit=r\"tCO$_2$/year\")\n\nfig2 = plt.figure()\nsample_histograms(fig2, param_values, problem, {'color': 'y'})\nplt.show()\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "c5365d7a92a322f5933f403af2c57bd0e489d552", "size": 461992, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Seminar_Crop_Model.ipynb", "max_stars_repo_name": "mishagrol/Seminar_Sobol", "max_stars_repo_head_hexsha": "c0500d625fc256cbe821f698feb8ee7983f7ae39", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Seminar_Crop_Model.ipynb", "max_issues_repo_name": "mishagrol/Seminar_Sobol", "max_issues_repo_head_hexsha": "c0500d625fc256cbe821f698feb8ee7983f7ae39", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Seminar_Crop_Model.ipynb", "max_forks_repo_name": "mishagrol/Seminar_Sobol", "max_forks_repo_head_hexsha": "c0500d625fc256cbe821f698feb8ee7983f7ae39", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 160.7487821851, "max_line_length": 110160, "alphanum_fraction": 0.8830217839, "converted": true, "num_tokens": 18393, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "```python\nimport pp\nfrom FermatPrincipleThreaded import Fermat\nimport astropy.coordinates as ac\nimport astropy.units as au\n#import astropy.time as at\nfrom IRI import *\nimport numpy as np\nfrom Geometry import *\nimport sympy\nimport time\n\nimport tempfile\n\ndef ppRayProp(file,inRays,N,pathlength,frequency):\n    #enu = ENUFrame.ENU()\n    fermat =  FermatPrincipleCartesian.Fermat(neFunc = None,type = 's',frequency = frequency)\n    fermat.loadFunc(file)\n    rays = {}\n    for ray in inRays:\n        datumIdx = ray.id\n        #origin = astropy.coordinates.SkyCoord(*(ray.origin*astropy.units.km),frame=enu).transform_to('itrs').cartesian.xyz.to(astropy.units.km).value\n        #direction = astropy.coordinates.SkyCoord(*ray.dir,frame=enu).transform_to('itrs').cartesian.xyz.value\n        origin = ray.origin\n        direction = ray.dir\n        time = ray.time\n        #print(ray)\n        x,y,z,s = fermat.integrateRay(origin,direction,pathlength,time=time,N=N)\n        rays[datumIdx] = {'x':x,'y':y,'z':z,'s':s}\n    return rays\n    \n    \ndef test():\n    sol = SolitonModel(5)\n    neFunc = sol.generateSolitonsModel()\n    theta = np.linspace(-np.pi/8.,np.pi/8.,25)\n    inRays = []\n    origin = ac.ITRS(sol.enu.location).cartesian.xyz.to(au.km).value\n    count = 0\n    for t in theta:\n        for p in theta:\n            direction = ac.SkyCoord(np.sin(t),\n                                    np.sin(p),\n                                    1.,frame=sol.enu).transform_to('itrs').cartesian.xyz.value\n            ray = Ray(origin,direction,id = count,time =0)\n            inRays.append(ray)\n            count += 1\n    \n    \n    ncpus = 1\n    t1 = time.time()\n    N = len(inRays)/ncpus\n    # Creates jobserver with ncpus workers\n    jobs = []\n    job_server = pp.Server(ncpus, ppservers=())\n    for i in range(ncpus):\n        file = 'neFunc-thread{0}.npz'.format(i)\n        np.savez(file,neFunc=neFunc)\n        #file = sol.saveNeFunc(neFunc)\n        job = job_server.submit(ppRayProp,\n                   args=(file,inRays[i:(i+1)*N],100,2000,120e6),\n                   depfuncs=(),\n                   modules=('FermatPrincipleCartesian',),\n                   globals={})\n        jobs.append(job)\n    for job in jobs:\n        result = job()\n    print(time.time() - t1)\n    job_server.print_stats()\nif __name__ == '__main__':\n    test()\n```\n", "meta": {"hexsha": "f8f6530c06984245225b4b096d5bda985b1afeab", "size": 3589, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "src/ionotomo/notebooks/FermatWorker.ipynb", "max_stars_repo_name": "Joshuaalbert/IonoTomo", "max_stars_repo_head_hexsha": "9f50fbac698d43a824dd098d76dce93504c7b879", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2017-06-22T08:47:07.000Z", "max_stars_repo_stars_event_max_datetime": "2021-07-01T12:33:02.000Z", "max_issues_repo_path": "src/ionotomo/notebooks/FermatWorker.ipynb", "max_issues_repo_name": "Joshuaalbert/IonoTomo", "max_issues_repo_head_hexsha": "9f50fbac698d43a824dd098d76dce93504c7b879", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2019-04-03T15:21:19.000Z", "max_issues_repo_issues_event_max_datetime": "2019-04-03T15:48:31.000Z", "max_forks_repo_path": "src/ionotomo/notebooks/FermatWorker.ipynb", "max_forks_repo_name": "Joshuaalbert/IonoTomo", "max_forks_repo_head_hexsha": "9f50fbac698d43a824dd098d76dce93504c7b879", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2020-03-01T16:20:00.000Z", "max_forks_repo_forks_event_max_datetime": "2020-07-07T15:09:02.000Z", "avg_line_length": 33.5420560748, "max_line_length": 159, "alphanum_fraction": 0.5012538312, "converted": true, "num_tokens": 634, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125737597971, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.41406366208725753}}
{"text": "\n\n\n# sbPoynETNRPy: An Einstein Toolkit Thorn for Computing the 4-Velocity Time-Component $u^0$, the Magnetic Field Measured by a Comoving Observer $b^{\\mu}$, and the Poynting Vector $S^i$\n\n## Author: Zach Etienne\n### Formatting improvements courtesy Brandon Clark\n\n[comment]: <> (Abstract: TODO)\n\n**Notebook Status:**  Validated \n\n**Validation Notes:** This module has been validated against the hand-written smallbPoynET in WVUThorns_diagnostics (a trsuted code), which itself is based on expressions in IllinoisGRMHD... which was validated against the original GRMHD code of the Illinois NR group.\n\n## Introduction:\nIn the [previous tutorial notebook](Tutorial-u0_smallb_Poynting-Cartesian.ipynb), we constructed within SymPy full expressions for the 4-velocity time-component $u^0$, the magnetic field (measured by a comoving observer) $b^{\\mu}$, and the Poynting vector $S^i$.\n\nHere we will work through the steps necessary to construct an Einstein Toolkit diagnostic thorn (module) that uses ADMBase and HydroBase variables as input into the NRPy+-generated SymPy expressions for $b^{\\mu}$, $b^2$, and the Poynting Vector $S^i$, outputting to gridfunctions `smallb4U[]`, `smallb2etk` (the \"etk\" suffix must be appended because base gridfunction names ending in numbers are not allowed in NRPy+), and `SPoyn[]`, respectively. \n\n\n\n# Table of Contents\n$$\\label{toc}$$\n\nThis tutorial is organized as follows\n\n1. [Step 1](#initializenrpy): Call on NRPy+ to convert the SymPy expressions for $b^{\\mu}$, $b^2$, and the Poynting Vector $S^i$ to C code kernels\n1. [Step 2](#etk): Build up the needed Einstein Toolkit infrastructure to implement the NRPy+-generated C code kernels\n    1. [Step 2.a](#etkc): Write the C code functions called by the Einstein Toolkit scheduler that incorporate the above \".h\" files\n    1. [Step 2.b](#cclfiles): CCL files - Define how this module interacts and interfaces with the larger Einstein Toolkit infrastructure\n    1. [Step 2.c](#etksys): Inform the Einstein Toolkit build system of the C code\n1. [Step 3](#latex_pdf_output): Output this notebook to $\\LaTeX$-formatted PDF file\n\n \n\n# Step 1: Call on NRPy+ to convert the SymPy expressions for $b^{\\mu}$, $b^2$, and the Poynting Vector $S^i$ to C code kernels \\[Back to [top](#toc)\\]\n$$\\label{initializenrpy}$$\n\n\n```python\n# Step 1a: import all needed modules from NRPy+:\nimport indexedexp as ixp                   # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support\nfrom outputC import outputC                # NRPy+: Basic C code output functionality\nimport grid as gri                         # NRPy+: Functions having to do with numerical grids\nimport NRPy_param_funcs as par             # NRPy+: parameter interface\nimport sympy as sp                         # SymPy: The Python computer algebra package upon which NRPy+ depends\n```\n\n\n```python\n# Step 1b: Initialize  parameters (stub; there are none for this module)\nthismodule = __name__\n```\n\nWe will to disable verbose output in the NRPy+ outputC function. This is an important step in this case, because our final expressions are very large. Verbose output, when enabled, will print (in comments) the input SymPy expressions to the top of the file *without* CSE, resulting here in an *enormous* output file.\n\nWe will also declare the additional gridfunctions we need for this thorn:\n\n**Inputs from ADMBase:**\n* the physical metric $\\gamma_{ij}$\n* the spacetime gauge quantities $\\alpha$ and $\\beta^i$\n\n**Inputs from HydroBase:**\n* the Valencia 3-velocity $v^i_{(n)}$\n* the densitized magnetic field of a normal observer $\\tilde{B}^i$\n\n**Output gridfunctions:**\n* the magnetic field as observed in a frame comoving with the plasma $b^\\mu$ (`smallb4U[]}`)\n* twice the magnetic pressure $2 P_{\\rm mag} = b_\\mu b^\\mu = b^2$ (`smallb2etk`)\n* the Poynting vector $S^i$ (`SPoyn[]`)\n\n\n```python\n# Step 1c: Set spatial dimension (must be 3 for BSSN)\nDIM = 3\npar.set_parval_from_str(\"grid::DIM\",DIM)\n\n# Step 1d: declare the additional gridfunctions (i.e., functions whose values are declared\n#          at every grid point, either inside or outside of our SymPy expressions) needed\n#         for this thorn\n\n# INPUT GRIDFUNCTIONS:\ngammaDD = ixp.register_gridfunctions_for_single_rank2(\"AUX\",\"gammaDD\", \"sym01\") # The AUX or EVOL designation is *not*\n                                                                                # used in diagnostic modules.\nbetaU = ixp.register_gridfunctions_for_single_rank1(\"AUX\",\"betaU\") # The AUX or EVOL designation is *not*\n                                                                   # used in diagnostic modules.\nalpha = gri.register_gridfunctions(\"AUX\",\"alpha\") # The AUX or EVOL designation is *not*\n                                                  # used in diagnostic modules.\nValenciavU = ixp.register_gridfunctions_for_single_rank1(\"AUX\",\"ValenciavU\") # The AUX or EVOL designation is *not*\n                                                                             # used in diagnostic modules.\nBU = ixp.register_gridfunctions_for_single_rank1(\"AUX\",\"BU\") # The AUX or EVOL designation is *not*\n                                                             # used in diagnostic modules.\n\n# OUTPUT GRIDFUNCTIONS:\nsmallb4U = ixp.register_gridfunctions_for_single_rank1(\"AUX\",\"smallb4U\",DIM=4) # The AUX or EVOL designation is *not*\n                                                                               # used in diagnostic modules.\nsmallb2etk = gri.register_gridfunctions(\"AUX\",\"smallb2etk\") # The AUX or EVOL designation is *not*\n                                                            # used in diagnostic modules.\nPoynSU = ixp.register_gridfunctions_for_single_rank1(\"AUX\",\"PoynSU\") # The AUX or EVOL designation is *not*\n                                                                     # used in diagnostic modules.\n\n# Step 1f: Call the NRPy+ module to set up the SymPy expressions for the output, as well as the C code for computing u^0\nimport u0_smallb_Poynting__Cartesian.u0_smallb_Poynting__Cartesian as u0etc\nu0etc.compute_u0_smallb_Poynting__Cartesian(gammaDD,betaU,alpha,ValenciavU,BU)\n\n# Step 1g: Set the gridfunction memory access type to \"ETK\":\npar.set_parval_from_str(\"GridFuncMemAccess\",\"ETK\")\n\n# Step 1h: Make output directories:\n!mkdir sbPoynETNRPy     2>/dev/null # 2>/dev/null: Don't throw an error or warning if the directory already exists.\n!mkdir sbPoynETNRPy/src 2>/dev/null # 2>/dev/null: Don't throw an error or warning if the directory already exists.\n\n# Step 1i: Output routine for computing u0:\nwith open(\"sbPoynETNRPy/src/u0.h\", \"w\") as file:\n    file.write(str(u0etc.computeu0_Cfunction))\n    print(\"Wrote to file \\\"\"+file.name+\"\\\"\")\n\n# Step 1j: Use NRPy+'s outputC to convert the SymPy expressions for smallb4U, smallb2etk, and PoynSU to C code:\n#outputC([u0etc.smallb4U[0],u0etc.smallb4U[1],u0etc.smallb4U[2],u0etc.smallb4U[3],u0etc.smallb2etk,\noutputC([u0etc.smallb4U[0],u0etc.smallb4U[1],u0etc.smallb4U[2],u0etc.smallb4U[3],u0etc.smallb2etk,\n         u0etc.PoynSU[0],u0etc.PoynSU[1],u0etc.PoynSU[2]],\n         [gri.gfaccess(\"\",\"smallb4U0\"),gri.gfaccess(\"\",\"smallb4U1\"),gri.gfaccess(\"\",\"smallb4U2\"),gri.gfaccess(\"\",\"smallb4U3\"),\n          gri.gfaccess(\"\",\"smallb2etk\"),\n          gri.gfaccess(\"\",\"PoynSU0\"),gri.gfaccess(\"\",\"PoynSU1\"),gri.gfaccess(\"\",\"PoynSU2\")],\n       filename=\"sbPoynETNRPy/src/smallb4U_smallb2etk_PoynSU.h\",\n       params=\"outCverbose=False\") # <- Force outCverbose=False for this\n                                   #    module to avoid gigantic C file filled with the\n                                   #    non-CSE expressions for the Weyl scalars.\n```\n\n    Wrote to file \"sbPoynETNRPy/src/u0.h\"\n    Wrote to file \"sbPoynETNRPy/src/smallb4U_smallb2etk_PoynSU.h\"\n\n\n\n\n# Step 2: Build up the needed Einstein Toolkit infrastructure to implement the NRPy+-generated C code kernels \\[Back to [top](#toc)\\]\n$$\\label{etk}$$\n\n\n \n\n## Step 2.a: Write the C code functions called by the Einstein Toolkit scheduler that incorporate the above \".h\" files \\[Back to [top](#toc)\\]\n$$\\label{etkc}$$\n\n\n```python\n%%writefile sbPoynETNRPy/src/sbPoynETNRPy.c\n\n#include \n#include \n#include \n#include \n#include \"cctk.h\"\n#include \"cctk_Arguments.h\"\n#include \"cctk_Parameters.h\"\n\nvoid sbPoynETNRPy_lowlevel(const cGH* restrict const cctkGH,const int *cctk_lsh,\n                           const CCTK_REAL *gammaDD00GF,const CCTK_REAL *gammaDD01GF,const CCTK_REAL *gammaDD02GF,\n                           const CCTK_REAL *gammaDD11GF,const CCTK_REAL *gammaDD12GF,const CCTK_REAL *gammaDD22GF,\n                           const CCTK_REAL *alphaGF,\n                           const CCTK_REAL *betaU0GF,const CCTK_REAL *betaU1GF,const CCTK_REAL *betaU2GF,\n                           const CCTK_REAL *vel,const CCTK_REAL *Bvec,\n\n                           CCTK_REAL *smallb4U0GF,CCTK_REAL *smallb4U1GF,CCTK_REAL *smallb4U2GF,CCTK_REAL *smallb4U3GF,\n                           CCTK_REAL *smallb2etkGF,\n                           CCTK_REAL *PoynSU0GF,CCTK_REAL *PoynSU1GF,CCTK_REAL *PoynSU2GF) {\n\n    DECLARE_CCTK_PARAMETERS;\n\n#pragma omp parallel for\n    for(int i2=0;i2 \n\n## Step 2.b: CCL files - Define how this module interacts and interfaces with the larger Einstein Toolkit infrastructure \\[Back to [top](#toc)\\]\n$$\\label{cclfiles}$$\n\nWriting a module (\"thorn\") within the Einstein Toolkit requires that three \"ccl\" files be constructed, all in the root directory of the thorn:\n\n1. `interface.ccl`: defines the gridfunction groups needed, and provides keywords denoting what this thorn provides and what it should inherit from other thorns.\n1. `param.ccl`: specifies free parameters within the thorn.\n1. `schedule.ccl`: allocates storage for gridfunctions, defines how the thorn's functions should be scheduled in a broader simulation, and specifies the regions of memory written to or read from gridfunctions.\n\nLet's start with `interface.ccl`. The [official Einstein Toolkit (Cactus) documentation](http://einsteintoolkit.org/usersguide/UsersGuide.html) defines what must/should be included in an `interface.ccl` file [**here**](http://einsteintoolkit.org/usersguide/UsersGuidech12.html#x17-178000D2.2). \n\n\n```python\n%%writefile sbPoynETNRPy/interface.ccl\n\n# With \"implements\", we give our thorn its unique name.\nimplements: sbPoynETNRPy\n\n# By \"inheriting\" other thorns, we tell the Toolkit that we\n#   will rely on variables/function that exist within those\n#   functions.\ninherits:   ADMBase Boundary Grid HydroBase MethodofLines\n\n# Tell the Toolkit that we want the various Weyl scalars\n#    and invariants to be visible to other thorns by using\n#    the keyword \"public\". Note that declaring these\n#    gridfunctions *does not* allocate memory for them;\n#    that is done by the schedule.ccl file.\npublic:\nCCTK_REAL smallb4U_group type=GF timelevels=3\n{\n  smallb4U0,smallb4U1,smallb4U2,smallb4U3\n} \"smallb4U 4-vector\"\n\npublic:\nCCTK_REAL smallb4_sq_group type=GF timelevels=3\n{\n  smallb4_sq\n} \"smallb^{mu} squared == twice the magnetic pressure\"\n\npublic:\nCCTK_REAL PoynSU_group type=GF timelevels=3\n{\n  PoynSU0,PoynSU1,PoynSU2\n} \"Poynting 3-vector\"\n```\n\n    Overwriting sbPoynETNRPy/interface.ccl\n\n\nWe will now write the file `param.ccl`. This file allows the listed parameters to be set at runtime. We also give allowed ranges and default values for each parameter. More information on this file's syntax can be found in the [official Einstein Toolkit documentation](http://einsteintoolkit.org/usersguide/UsersGuidech12.html#x17-183000D2.3). \n\nThe first parameter specifies how many time levels need to be stored. Generally when using the ETK's adaptive-mesh refinement (AMR) driver [Carpet](https://carpetcode.org/), three timelevels are needed so that the diagnostic quantities can be properly interpolated and defined across refinement boundaries. \n\nThe second parameter determines how often we will calculate $b^\\mu$, $b^2$, and $S^i$.\n\nThe third parameter sets the maximum allowed Lorentz factor when computing $u^0$ (i.e., $\\Gamma_{\\rm max}$, as defined in the [previous tutorial notebook](Tutorial-u0_smallb_Poynting-Cartesian.ipynb)).\n\n\n```python\n%%writefile sbPoynETNRPy/param.ccl\n\nshares: HydroBase\nUSES CCTK_INT timelevels\n\nrestricted:\nCCTK_INT timelevels \"Number of active timelevels\" STEERABLE=RECOVER\n{\n  0:3 :: \"\"\n} 3\n\nrestricted:\nCCTK_INT sbPoynETNRPy_calc_every \"Compute these quantities every sbPoynETNRPy_calc_every iterations.\" STEERABLE=ALWAYS\n{\n  *:* :: \"\"\n} 1\n\nrestricted:\nCCTK_REAL GAMMA_SPEED_LIMIT \"Maximum Lorentz factor.\"\n{\n 1:* :: \"Positive > 1, though you'll likely have troubles in GRMHD far above 10, or far above 2000 in GRFFE.\"\n} 10.0\n\n```\n\n    Overwriting sbPoynETNRPy/param.ccl\n\n\nFinally, we will write the file `schedule.ccl`; its official documentation is found [here](http://einsteintoolkit.org/usersguide/UsersGuidech12.html#x17-186000D2.4). \n\nThis file registers the function we wish to call, `sbPoynETNRPy`, with the Einstein Toolkit scheduler.\n\n\n```python\n%%writefile sbPoynETNRPy/schedule.ccl\n\nSTORAGE: smallb4U_group[timelevels]\nSTORAGE: smallb4_sq_group[timelevels]\nSTORAGE: PoynSU_group[timelevels]\n\nschedule group sbPoynETNRPy_group in MoL_PseudoEvolution after ADMBase_SetADMVars\n{\n} \"Schedule sbPoynETNRPy group\"\n\nschedule sbPoynETNRPy in sbPoynETNRPy_group\n{\n    LANG: C\n    READS: admbase::gxx(Everywhere)\n    READS: admbase::gxy(Everywhere)\n    READS: admbase::gxz(Everywhere)\n    READS: admbase::gyy(Everywhere)\n    READS: admbase::gyz(Everywhere)\n    READS: admbase::gzz(Everywhere)\n\n    READS: admbase::alpha(Everywhere)\n\n    READS: admbase::betax(Everywhere)\n    READS: admbase::betay(Everywhere)\n    READS: admbase::betaz(Everywhere)\n\n    READS: HydroBase::vel(Everywhere)\n    READS: HydroBase::Bvec(Everywhere)\n\n    WRITES: sbPoynETNRPy::smallb4U0(Everywhere)\n    WRITES: sbPoynETNRPy::smallb4U1(Everywhere)\n    WRITES: sbPoynETNRPy::smallb4U2(Everywhere)\n    WRITES: sbPoynETNRPy::smallb4U3(Everywhere)\n\n    WRITES: sbPoynETNRPy::smallb4_sq(Everywhere)\n\n    WRITES: sbPoynETNRPy::PoynSU0(Everywhere)\n    WRITES: sbPoynETNRPy::PoynSU1(Everywhere)\n    WRITES: sbPoynETNRPy::PoynSU2(Everywhere)\n} \"Call sbPoynETNRPy main function, to compute $b^mu$, $b^2$, and $S^i$\"\n```\n\n    Overwriting sbPoynETNRPy/schedule.ccl\n\n\n \n\n## Step 2.c: Inform the Einstein Toolkit build system of the C code \\[Back to [top](#toc)\\]\n$$\\label{etksys}$$\n\nThe `make.code.defn` lists the source files that need to be compiled. Naturally, this thorn has only the one C file $-$ written above $-$ to compile:\n\n\n```python\n%%writefile sbPoynETNRPy/src/make.code.defn\n\nSRCS = sbPoynETNRPy.c\n```\n\n    Overwriting sbPoynETNRPy/src/make.code.defn\n\n\n\n\n# Step 3: Output this notebook to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n$$\\label{latex_pdf_output}$$\n\nThe following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n[Tutorial-ETK_thorn-u0_smallb_Poynting.pdf](Tutorial-ETK_thorn-u0_smallb_Poynting.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means.)\n\n\n```python\nimport cmdline_helper as cmd    # NRPy+: Multi-platform Python command-line interface\ncmd.output_Jupyter_notebook_to_LaTeXed_PDF(\"Tutorial-ETK_thorn-u0_smallb_Poynting\")\n```\n\n    Created Tutorial-ETK_thorn-u0_smallb_Poynting.tex, and compiled LaTeX file\n        to PDF file Tutorial-ETK_thorn-u0_smallb_Poynting.pdf\n\n", "meta": {"hexsha": "b15b4d259bbec552d61007749b973b29d2a79cbe", "size": 26454, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Tutorial-ETK_thorn-u0_smallb_Poynting.ipynb", "max_stars_repo_name": "terrencepierrej/nrpytutorial", "max_stars_repo_head_hexsha": "3ea18beed99cf6b7d67c89c140ca68630452001e", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Tutorial-ETK_thorn-u0_smallb_Poynting.ipynb", "max_issues_repo_name": "terrencepierrej/nrpytutorial", "max_issues_repo_head_hexsha": "3ea18beed99cf6b7d67c89c140ca68630452001e", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Tutorial-ETK_thorn-u0_smallb_Poynting.ipynb", "max_forks_repo_name": "terrencepierrej/nrpytutorial", "max_forks_repo_head_hexsha": "3ea18beed99cf6b7d67c89c140ca68630452001e", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 42.8058252427, "max_line_length": 457, "alphanum_fraction": 0.5942768579, "converted": true, "num_tokens": 5267, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "
                                        \n    
                                         Tutorial 2 
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                                         Physics Informed Neural Networks Part 1
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                                         Manual 1D Heat Equation Solver
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                                            \n\n# Overview\n\nThis notebook is based on two papers: *[Physics-Informed Neural Networks:  A Deep LearningFramework for Solving Forward and Inverse ProblemsInvolving Nonlinear Partial Differential Equations](https://www.sciencedirect.com/science/article/pii/S0021999118307125)* and *[Hidden Physics Models:  Machine Learning of NonlinearPartial Differential Equations](https://www.sciencedirect.com/science/article/pii/S0021999117309014)* with the help of  Fergus Shone and Michael Macraild.\n\nThese tutorials will go through solving Partial Differential Equations using Physics Informed Neural Networks focusing on the Burgers Equation and a more complex example using the Navier Stokes Equation\n\n**This introduction section is replicated in all PINN tutorial notebooks (please skip if you've already been through)** \n\n
                                        \nIf you have not already then in your repositoy directory please run the following code. \n    \n```bash\ngit submodule init\ngit submodule update --init --recursive\n```\n    \n**If this does not work please clone the [PINNs](https://github.com/maziarraissi/PINNs) repository into your Physics_Informed_Neural_Networks folder**\n
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                                        Physics Informed Neural Networks
                                        \n\nFor a typical Neural Network using algorithms like gradient descent to look for a hypothesis, data is the only guide, however if the data is noisy or sparse and we already have governing physical models we can use the knowledge we already know to optimize and inform the algorithms. This can be done via [feature engineering]() or by adding a physical inconsistency term to the loss function.\n\n\n   \n \n \n## The very basics\n\nIf you know nothing about neural networks there is a [toy neural network python code example](https://github.com/cemac/LIFD_ENV_ML_NOTEBOOKS/tree/main/ToyNeuralNetwork) included in the [LIFD ENV ML Notebooks Repository]( https://github.com/cemac/LIFD_ENV_ML_NOTEBOOKS). Creating a 2 layer neural network to illustrate the fundamentals of how Neural Networks work and the equivalent code using the python machine learning library [tensorflow](https://keras.io/).\n\n    \n## Recommended reading\n    \nThe in-depth theory behind neural networks will not be covered here as this tutorial is focusing on application of machine learning methods. If you wish to learn more here are some great starting points.  \n \n\n* [All you need to know on Neural networks](https://towardsdatascience.com/nns-aynk-c34efe37f15a) \n* [Introduction to Neural Networks](https://victorzhou.com/blog/intro-to-neural-networks/)\n* [Physics Guided Neural Networks](https://towardsdatascience.com/physics-guided-neural-networks-pgnns-8fe9dbad9414)\n* [Maziar Rassi's Physics informed GitHub web Page](https://maziarraissi.github.io/PINNs/)\n\n
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                                         Machine Learning Theory 
                                        \n\n\n\n\n    \n## Physics informed Neural Networks\n\nNeural networks work by using lots of data to calculate weights and biases from data alone to minimise the loss function enabling them to act as universal function approximators. However these lose their robustness when data is limited. However by using known physical laws or empirical validated relationships the solutions from neural networks can be sufficiently constrained by disregarding no realistic solutions.\n    \nA Physics Informed Neural Network considers a parameterized and nonlinear partial differential equation in the general form;\n\n\n\n    \n\\begin{align}\n     u_t + \\mathcal{N}[u; \\lambda] &= 0, && x \\in \\Omega, t \\in [0,T],\\\\\n\\end{align}\n    \n\n\nwhere $\\mathcal{u(t,x)}$ denores the hidden solution, $\\mathcal{N}$ is a nonlinear differential operator acting on $u$, $\\mathcal{\\lambda}$ and $\\Omega$ is a subset of $\\mathbb{R}^D$ (the prescribed data). This set up an encapsulation of a wide range of problems such as diffusion processes, conservation laws,  advection-diffusion-reaction  systems,  and  kinetic  equations and conservation laws.\n\nHere we will go though this for the 1 Heat equation and Navier stokes equations\n\n\n
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                                         Python 
                                        \n\n    \n## Tensorflow \n    \nThere are many machine learning python libraries available, [TensorFlow](https://www.tensorflow.org/) a is one such library. If you have GPUs on the machine you are using TensorFlow will automatically use them and run the code even faster!\n\n## Further Reading\n\n* [Running Jupyter Notebooks](https://jupyter.readthedocs.io/en/latest/running.html#running)\n* [Tensorflow optimizers](https://www.tutorialspoint.com/tensorflow/tensorflow_optimizers.htm)\n\n
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                                         Requirements 
                                        \n\nThese notebooks should run with the following requirements satisfied\n\n
                                         Python Packages: 
                                        \n\n* Python 3\n* tensorflow > 2\n* numpy \n* matplotlib\n* scipy\n\n
                                         Data Requirements
                                        \n    \nThis notebook referes to some data included in the git hub repositroy imported via the git submodules command mentioned in the installation instructions\n    \n
                                        \n\n\n**Contents:**\n\n1. **[1D Heat Equation Non ML Example](PINNs_1DHeatEquations_nonML.ipynb)**\n2. [1D Heat Equation PINN Example](PINNs_1DHeatEquationExample.ipynb)\n3. [Navier-Stokes PINNs discovery of PDE\u2019s](PINNs_NavierStokes_example.ipynb)\n4. [Navier-Stokes PINNs Hidden Fluid Mechanics](PINNs_NavierStokes_HFM.ipynb)\n\n\n
                                        \nLoad in all required modules (including some auxiliary code) and turn off warnings.\n
                                        \n\n\n```python\n# For readability: disable warnings\nimport warnings\nwarnings.filterwarnings('ignore')\n```\n\n\n```python\nimport sys\nsys.path.insert(0, 'PINNs/Utilities/')\nimport tensorflow as tf\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport scipy.io\nfrom scipy.interpolate import griddata\nimport time\nfrom itertools import product, combinations\nfrom mpl_toolkits.mplot3d import Axes3D\nfrom mpl_toolkits.mplot3d.art3d import Poly3DCollection\nfrom mpl_toolkits.axes_grid1 import make_axes_locatable\nimport matplotlib.gridspec as gridspec\nfrom time import time\nimport scipy.sparse as sp\nimport scipy.sparse.linalg as la\n```\n\n
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                                         1D Heat Equation (forwards) 
                                        \n\nGiven\n- the initial temperature profile $u(x,0) = m(x)$,\n- the thermal diffusivity $k$,\n- a prescribed temperature $u(0,t) = u(L,t) = 0$ at the extremities of the rod;\n\nsolve the heat equation\n\n\\begin{equation}\n\\begin{array}{ll}\n\\frac{\\partial u}{\\partial t} - k \\frac{\\partial^2}{\\partial x^2} u = 0 & \\forall x\\,\\in\\,(0, L)\\; \\forall t \\in (0,T)\\\\\nu(x, 0) = m(x) & \\forall x \\in [0,L] \\\\\nu(0,t) = u(L,t) = 0 & \\forall t \\in (0, T],\n\\end{array}\n\\end{equation}\n\nand observe the temperature at the final time $T$:\n\n\n\\begin{equation} \\mathcal{F}(m) = u(x, T). \\end{equation}\n\n#### Analytical solution to the forward problem.\n    \nIf\n\n\\begin{equation} m(x) = \\sin\\left(n\\, \\frac{\\pi}{L} x \\right), \\quad n = 1,2,3, \\ldots ,\\end{equation}\n\nthen\n\n\\begin{equation} u(x,t) = e^{ -k\\left(n\\, \\frac{\\pi}{L} \\right)^2 t} \\sin\\left(n\\,\\frac{\\pi}{L} x \\right) \\end{equation}\n\nis the unique solution to the heat equation.\n\n
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                                         1D Heat Equation (inverse) 
                                        \n\nGiven the forward model F and a noisy measurament d of the temperature profile at time T, find the initial temperature profile m\n\nsuch that\n\\begin{equation}\nF(m)=d.\n\\end{equation}\n    \n### Discretization\n\nTo discretize the problem, we use finite differences in space and Implicit Euler in time.\n\n#### Semidiscretization in space\nWe divide the $[0, L]$ interval in $n_x$ subintervals of the same lenght $h = \\frac{L}{n_x}$, and we denote with $u_j(t) := u( jh, t)$ the value of the temperature at point $x_j = jh$ and time $t$.\n\nWe then use a centered finite difference approximation of the second derivative in space and write\n\n\\begin{equation} \\frac{\\partial u_j(t)}{\\partial t} - k \\frac{u_{j-1}(t) - 2u_j(t) + u_{j+1}(t)}{h^2} \\quad \\text{for } j=1,2,\\ldots,n_x-1, \\end{equation}\n\nwith the boundary condition $u_0(t) = u_{n_x}(t) = 0$.\n\nBy letting\n\n\\begin{equation} \\mathbf{u}(t) = \\begin{bmatrix}u_1(t)\\\\u_2(t)\\\\ \\ldots\\\\ u_{n_x-1}(t) \\end{bmatrix}\\end{equation}\n\nbe the vector collecting the values of the temperature $u$ at the points $x_j = j\\,h$, we then write the system of ordinary differential equations (ODEs):\n$$ \\frac{\\partial}{\\partial t} \\mathbf{u}(t) + K \\mathbf{u}(t) = 0,$$\nwhere $K \\in \\mathbb{R}^{(n_x-1) \\times (n_x-1)}$ is the tridiagonal matrix given by\n\n\\begin{equation} K = \\frac{k}{h^2}\\begin{bmatrix}  2 & -1 &       &        &        &    \\\\\n                                    -1 &  2 & -1    &        &        &    \\\\\n                                       & -1 &  2    & -1     &        &    \\\\\n                                       &    &\\ldots & \\ldots & \\ldots &    \\\\\n                                       &    &       & -1     &     2  & -1 \\\\ \n                                       &    &       &        &     -1 & 2  \\\\\n                     \\end{bmatrix}.\\end{equation}\n                     \n#### Time discretization\nWe subdivide the time interval $(0, T]$ in $n_t$ time step of size $\\Delta t = \\frac{T}{n_t}$.\nBy letting $\\mathbf{u}^{(i)} = \\mathbf{u}(i\\,\\Delta t)$ denote the discretized temperature profile at time $t_i = i\\,\\Delta t$, the Implicit Euler scheme reads\n\n\\begin{equation} \\frac{\\mathbf{u}^{(i+1)} - \\mathbf{u}^{(i)}}{\\Delta t} + K\\mathbf{u}^{(i+1)} = 0, \\quad \\text{for } i=0,1,\\ldots, n_t-1.\\end{equation}\n\nAfter simple algebraic manipulations and exploiting the initial condition $u(x,0) = m(x)$, we then obtain\n\n\n\n\\begin{equation}\n\\mathbf{u}^{(0)} = \\mathbf{m} \\\\\n\\mathbf{u}^{(i+1)} = \\left( I + \\Delta t\\, K\\right)^{-1} \\mathbf{u}^{(i)},\n\\end{equation}\n\n\n\nor equivalently\n\n\\begin{equation} \\mathbf{u}^{(i)} = \\left( I + \\Delta t\\, K\\right)^{-i} \\mathbf{m}.\\end{equation}\n\n\n
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                                        \n\n# Define some helper functions to Solve 1D heat equation forwards \n\nIn the code below, the function `assembleMatrix` generates the finite difference matrix $\\left( I + \\Delta t\\, K \\right)$ and the function `solveFwd` evaluates the forward model\n\n\\begin{equation} F\\, \\mathbf{m} = \\left( I + \\Delta t\\, K\\right)^{-n_t}\\, \\mathbf{m}. \\end{equation}\n    \n
                                        \n\n\n```python\ndef plot(f, style, **kwargs):\n    x = np.linspace(0., L, nx+1)\n    f_plot = np.zeros_like(x)\n    f_plot[1:-1] = f\n    plt.plot(x,f_plot, style, **kwargs)\n    \ndef assembleMatrix(n):\n    diagonals = np.zeros((3, n))   # 3 diagonals\n    diagonals[0,:] = -1.0/h**2\n    diagonals[1,:] =  2.0/h**2\n    diagonals[2,:] = -1.0/h**2\n    K = k*sp.spdiags(diagonals, [-1,0,1], n,n)\n    M = sp.spdiags(np.ones(n), 0, n,n)\n    \n    return M + dt*K\n    \n\ndef solveFwd(m):\n    A = assembleMatrix(m.shape[0])\n    u_old = m.copy()\n    for i in np.arange(nt):\n        u = la.spsolve(A, u_old)\n        u_old[:] = u\n        \n    return u        \n```\n\n
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                                        \n    \n### A naive solution to the inverse problem\n\nIf $\\mathcal{F}$ is invertible a naive solution to the inverse problem $\\mathcal{F} m = d$ is simply to set\n\n\\begin{equation} m = \\mathcal{F}^{-1} d. \\end{equation}\n\n\n    \n
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                                        \n\nThe function `naiveSolveInv` computes the solution of the discretized inverse problem $\\mathbf{m} = F^{-1} \\mathbf{d}$ as\n\n\\begin{equation} \\mathbf{m} = \\left( I + \\Delta t\\,K\\right)^{n_t} \\mathbf{d}. \\end{equation}\n\n\n\n    \n
                                        \n\n\n```python\ndef naiveSolveInv(d):\n    A = assembleMatrix(d.shape[0])\n    \n    p_i = d.copy()\n    for i in np.arange(nt):\n        p = A*p_i\n        p_i[:] = p\n        \n    return p\n```\n\n\n```python\n# Edit nx or noise_std_dev to see the impact on the naive solver\n\nnx = 20 # default 20\nnoise_std_dev = 1e-4 # default\n\n\nT = 1.0\nL = 1.0\nk = 0.005\n\nnt = 100\n\n\nh = L/float(nx)\ndt = T/float(nt)\n\nx = np.linspace(0.+h, L-h, nx-1) #place nx-1 equispace point in the interior of [0,L] interval\nm_true = np.power(.5,-36)*np.power(x,20)*np.power(1. - x, 16)  #smooth true initial condition\n#m_true = 0.5 - np.abs(x-0.5)                                    #initial condition with a corner\nu_true = solveFwd(m_true)\n\nd = u_true + noise_std_dev*np.random.randn(u_true.shape[0])\n\nm = naiveSolveInv(d)\n\nplot(u_true, \"-b\", label = 'u(T)')\nplot(d, \"og\", label = 'd')\nplt.legend()\nplt.title('sample data to be used in Niave solver ')\nplt.show()\n\n\nplot(m_true, \"-r\", label = 'm_true')\nplot(m, \"-b\", label = 'm')\nplt.legend()\nplt.title('Naive Solution coarse mesh')\nplt.show()\n\n```\n\n
                                        \n\nIf you have played around with the code above you will see that:\n- for a very coarse mesh (`nx = 20`) and no measurement noise (`noise_std_dev = 0.0`) the naive solution is quite good\n- for a finer mesh (`nx = 100`) and/or even small measurement noise (`noise_std_dev = 0.0001`) the naive solution is very poor\n\n
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                                        \n    \n### Why does the naive solution fail?\n\nLet \\begin{equation}v_n = \\sqrt{\\frac{2}{L}} \\sin\\left( n \\, \\frac{\\pi}{L} x \\right)\\end{equation} \n\nwith n=1,2,3, then we have that\n\n\\begin{equation} \\mathcal{F} v_n = \\lambda_n v_n, \\quad \\text{where the eigenvalues } \\lambda_n = e^{-kT\\left(\\frac{\\pi}{L} n \\right)^2}. \\end{equation}\n\n**Note 1**:\n- Large eigenvalues $\\lambda_n$ corresponds to smooth eigenfunctions $v_n$;\n- Small eigenvalues $\\lambda_n$ corresponds to oscillatory eigenfuctions $v_n$.\n\nThe figure below shows that the eigenvalues $\\lambda_n$ decays extremely fast, that is the matrix $F$ (discretization of the forward model $\\mathcal{F}$) is extremely ill conditioned.\n\n
                                        \n\n\n```python\nT = 1.0\nL = 1.0\nk = 0.005\n\ni = np.arange(1,50)\nlambdas = np.exp(-k*T*np.power(np.pi/L*i,2))\n\nplt.semilogy(i, lambdas, 'ob')\nplt.xlabel('i')\nplt.ylabel('lambda_i')\nplt.title('Eigen Value decay')\nplt.show()\n```\n\n
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                                        \n    \n**Note 2**: The functions $v_n$ ($n=1,2,3, \\ldots$) form an orthonormal basis of $L^2([0,1]$). \n\nThat is, every function $f \\in L^2([0,1])$ can be written as\n\n\\begin{equation} f = \\sum_{n=1}^\\infty \\alpha_n v_n, \\text{ where } \\alpha_n = \\int_0^1 f v_n dx.\\end{equation}\n\nConsider now the noisy problem\n\n\\begin{equation} d = \\mathcal{F}m_{\\rm true} + \\eta, \\end{equation}\n\nwhere\n- $d$ is the data (noisy measurements)\n- $\\eta$ is the noise: $\\eta(x) = \\sum_{n=1}^\\infty \\eta_n v_n(x)$\n- $m_{\\rm true}$ is the true value of the parameter that generated the data\n- $\\mathcal{F}$ is the forward heat equation\n\nThen, the naive solution to the inverse problem $\\mathcal{F}m = d$ is\n\n\\begin{equation} m = \\mathcal{F}^{-1}d = \\mathcal{F}^{-1}\\left( \\mathcal{F}m_{\\rm true} + \\eta \\right) = m_{\\rm true} + \\mathcal{F}^{-1} \\eta = m_{\\rm true} + \\mathcal{F}^{-1} \\sum_{n=1}^{\\infty} \\eta_n v_n = m_{\\rm true} +  \\sum_{n=1}^{\\infty} \\frac{\\eta_n}{\\lambda_n} v_n. \\end{equation}\n\nIf the coefficients $\\eta_n = \\int_0^1 \\eta(x) \\, v_n(x) \\, dx$ do not decay sufficiently fast with respect to the eigenvalues $\\lambda_n$, then the naive solution is unstable.\n\nThis implies that oscillatory components can not reliably be reconstructed from noisy data since they correspond to small eigenvalues.\n\nThis means we must take a different approach even to a relatively simple looking problem which may require filtering or computationally expensive calculations. This is with an even sample of data to use. So we might consider looking to PINNs\n\n
                                        \n\n
                                        \n\n## Next steps\n\nNow we've gone through a Naive manual approach to solving a simple 1D Heat equation we look at the benefits of using neural networks to solve more complex equations starting with the next notebook linked below: \n    \n[1D Heat Equation PINN Example](PINNs_1DHeatEquationExample.ipynb)\n    \n
                                        \n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "6cdf909d84f1871c12359be8010d15428bb52e27", "size": 23578, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Physics_Informed_Neural_Networks/PINNs_1DHeatEquation_nonML.ipynb", "max_stars_repo_name": "wolfiex/LIFD_ENV_ML_NOTEBOOKS", "max_stars_repo_head_hexsha": "324949a841a7a0977b2358d1eaf3eb9b22924ce2", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2021-08-13T21:38:21.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-21T06:16:23.000Z", "max_issues_repo_path": "Physics_Informed_Neural_Networks/PINNs_1DHeatEquation_nonML.ipynb", "max_issues_repo_name": "wolfiex/LIFD_ENV_ML_NOTEBOOKS", "max_issues_repo_head_hexsha": "324949a841a7a0977b2358d1eaf3eb9b22924ce2", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2021-07-26T10:10:12.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-24T12:08:01.000Z", "max_forks_repo_path": "Physics_Informed_Neural_Networks/PINNs_1DHeatEquation_nonML.ipynb", "max_forks_repo_name": "wolfiex/LIFD_ENV_ML_NOTEBOOKS", "max_forks_repo_head_hexsha": "324949a841a7a0977b2358d1eaf3eb9b22924ce2", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-08-13T21:39:34.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-13T21:39:34.000Z", "avg_line_length": 37.3069620253, "max_line_length": 484, "alphanum_fraction": 0.5556875053, "converted": true, "num_tokens": 4876, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813031051514763, "lm_q2_score": 0.7122321842389469, "lm_q1q2_score": 0.4140227802869182}}
{"text": "# [ATM 623: Climate Modeling](../index.ipynb)\n\n[Brian E. J. Rose](http://www.atmos.albany.edu/facstaff/brose/index.html), University at Albany\n\n# Lecture 7: Grey radiation modeling with climlab\n\n### About these notes:\n\nThis document uses the interactive [`Jupyter notebook`](https://jupyter.org) format. The notes can be accessed in several different ways:\n\n- The interactive notebooks are hosted on `github` at https://github.com/brian-rose/ClimateModeling_courseware\n- The latest versions can be viewed as static web pages [rendered on nbviewer](http://nbviewer.ipython.org/github/brian-rose/ClimateModeling_courseware/blob/master/index.ipynb)\n- A complete snapshot of the notes as of May 2017 (end of spring semester) are [available on Brian's website](http://www.atmos.albany.edu/facstaff/brose/classes/ATM623_Spring2017/Notes/index.html).\n\n[Also here is a legacy version from 2015](http://www.atmos.albany.edu/facstaff/brose/classes/ATM623_Spring2015/Notes/index.html).\n\nMany of these notes make use of the `climlab` package, available at https://github.com/brian-rose/climlab\n\n\n```python\n#  Ensure compatibility with Python 2 and 3\nfrom __future__ import print_function, division\n```\n\n## Contents\n\n1. [Introducing `climlab`](#section1)\n2. [Using `climlab` to implement the two-layer leaky greenhouse model](#section2)\n3. [The observed annual, global mean temperature profile](#section3)\n4. [A 30-layer model using the observed temperatures](#section4)\n5. [Radiative forcing in the 30-layer model](#section5)\n6. [Radiative equilibrium in the 30-layer model](#section6)\n7. [Radiative-Convective Equilibrium in the 30-layer model](#section7)\n8. [Putting stratospheric ozone in the grey-gas model](#section8)\n\n____________\n\n\n## 1. Introducing `climlab`\n____________\n\n``climlab`` is a flexible engine for process-oriented climate modeling.\nIt is based on a very general concept of a model as a collection of individual, \ninteracting processes. ``climlab`` defines a base class called ``Process``, which\ncan contain an arbitrarily complex tree of sub-processes (each also some \nsub-class of ``Process``). Every climate process (radiative, dynamical, \nphysical, turbulent, convective, chemical, etc.) can be simulated as a stand-alone\nprocess model given appropriate input, or as a sub-process of a more complex model. \nNew classes of model can easily be defined and run interactively by putting together an\nappropriate collection of sub-processes.\n\n``climlab`` is a work-in-progress, and the code base will evolve substantially over the course of this semester.\nThe latest code can always be found on ``github``:\n\nhttps://github.com/brian-rose/climlab\n\nYou are strongly encouraged to clone the ``climlab`` repository and use ``git`` to keep your local copy up-to-date.\n\nRunning this notebook requires that ``climlab`` is already installed on your system.\n\n\n```python\n%matplotlib inline\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport xarray as xr\nfrom xarray.ufuncs import cos, deg2rad, log\nimport climlab\n```\n\n____________\n\n\n## 2. Using `climlab` to implement the two-layer leaky greenhouse model\n____________\n\nOne of the things that ``climlab`` is set up to do is the grey-radiation modeling we have already been discussing.\n\nSince we already derived a [complete analytical solution to the two-layer leaky greenhouse model](Lecture06 -- Elementary greenhouse models.ipynb), we will use this to validate the `climlab` code.\n\n\n\n### Validation\n\nWe want to verify that the model reproduces the observed OLR given observed temperatures, and the absorptivity that we tuned in the analytical model. The target numbers are:\n\n\\begin{align}\nT_s &= 288 \\text{ K} \\\\\nT_0 &= 275 \\text{ K} \\\\\nT_1 &= 230 \\text{ K} \\\\\n\\end{align}\n\n$$ \\epsilon = 0.586  $$\n\n$$ OLR = 238.5 \\text{ W m}^{-2} $$\n\n\n### Initialize a model in `climlab`\nThe first thing we do is create a new model.\n\nThe following example code is sparsely commented but will hopefully orient you on the basics of defining and working with a `climlab Process` object.\n\n\n```python\n#  Test in a 2-layer atmosphere\ncol = climlab.GreyRadiationModel(num_lev=2)\nprint( col)\n```\n\n    climlab Process of type . \n    State variables and domain shapes: \n      Ts: (1,) \n      Tatm: (2,) \n    The subprocess tree: \n    Untitled: \n       LW: \n       SW: \n       insolation: \n    \n\n\n\n```python\ncol.subprocess\n```\n\n\n\n\n    {'LW': ,\n     'SW': ,\n     'insolation': }\n\n\n\nEvery item in the above dictionary is itself an instance of the `climlab.Process` object:\n\n\n```python\nprint( col.subprocess['LW'])\n```\n\n    climlab Process of type . \n    State variables and domain shapes: \n      Ts: (1,) \n      Tatm: (2,) \n    The subprocess tree: \n    Untitled: \n    \n\n\nThe `state` dictionary holds the state variables of the model. In this case, temperatures:\n\n\n```python\ncol.state\n```\n\n\n\n\n    {'Tatm': Field([200., 278.]), 'Ts': Field([288.])}\n\n\n\nAccess these either through dictionary methods or as attributes of the model object:\n\n\n```python\nprint( col.state['Ts'])\nprint( col.Ts)\ncol.Ts is col.state['Ts']\n```\n\n    [288.]\n    [288.]\n\n\n\n\n\n    True\n\n\n\nNow we are assigning the \"observed\" temperatures to our model state:\n\n\n```python\ncol.Ts[:] = 288.\ncol.Tatm[:] = np.array([230., 275.])\ncol.state\n```\n\n\n\n\n    {'Tatm': Field([230., 275.]), 'Ts': Field([288.])}\n\n\n\n\n```python\nLW = col.subprocess['LW']\nprint( LW)\n```\n\n    climlab Process of type . \n    State variables and domain shapes: \n      Ts: (1,) \n      Tatm: (2,) \n    The subprocess tree: \n    Untitled: \n    \n\n\n\n```python\nLW.absorptivity\n```\n\n\n\n\n    Field([0.47737425, 0.47737425])\n\n\n\n\n```python\n#  copying the tuned value of epsilon from Lecture 6 notes\nLW.absorptivity = 0.586\nLW.absorptivity\n```\n\n\n\n\n    Field([0.586, 0.586])\n\n\n\n\n```python\n#  This does all the calculations that would be performed at each time step, \n#  but doesn't actually update the temperatures\ncol.compute_diagnostics()\n#  Print out the dictionary\ncol.diagnostics\n```\n\n\n\n\n    {'ASR': Field([239.2513]),\n     'LW_absorbed_atm': array([ 20.02990881, -96.98387764]),\n     'LW_absorbed_sfc': Field([-161.57076227]),\n     'LW_down_sfc': array([228.53426769]),\n     'LW_emission': Field([ 92.98664086, 190.03779837]),\n     'LW_up_sfc': Field([390.10502995]),\n     'OLR': array([238.5247311]),\n     'SW_absorbed_atm': array([0., 0.]),\n     'SW_absorbed_sfc': Field([239.2513]),\n     'SW_down_TOA': Field([341.3]),\n     'SW_up_TOA': array([102.0487]),\n     'SW_up_sfc': Field([102.0487]),\n     'absorbed': array([0., 0.]),\n     'absorbed_total': 0.0,\n     'coszen': Field([1.]),\n     'emission': Field([0., 0.]),\n     'emission_sfc': Field([0.]),\n     'flux_from_sfc': Field([102.0487]),\n     'flux_reflected_up': array([  0.    ,   0.    , 102.0487]),\n     'flux_to_sfc': array([341.3]),\n     'flux_to_space': array([102.0487]),\n     'insolation': Field([341.3]),\n     'planetary_albedo': Field([0.299])}\n\n\n\n\n```python\n#  Check OLR against our analytical solution\ncol.OLR\n```\n\n\n\n\n    array([238.5247311])\n\n\n\n\n```python\n# Like the state variables, the diagnostics can also be accessed in two different ways\ncol.diagnostics['OLR']\n```\n\n\n\n\n    array([238.5247311])\n\n\n\n\n```python\ncol.state\n```\n\n\n\n\n    {'Tatm': Field([230., 275.]), 'Ts': Field([288.])}\n\n\n\n\n```python\n# perform a single time step\ncol.step_forward()\n```\n\n\n```python\ncol.state\n```\n\n\n\n\n    {'Tatm': Field([230.33784312, 273.3641795 ]), 'Ts': Field([289.60514636])}\n\n\n\nWe just stepped forward one discreet unit in time. Because we didn't specify a timestep when we created the model, it is set to a default value:\n\n\n```python\ncol.timestep\n```\n\n\n\n\n    86400.0\n\n\n\nwhich is 1 day (expressed in seconds).\n\nNow we will integrate the model out to equilibrium.\n\nWe could easily write a loop to call the `step_forward()` method many times.\n\nOr use a handy shortcut that allows us to specify the integration length in physical time units:\n\n\n```python\n# integrate out to radiative equilibrium\ncol.integrate_years(2.)\n```\n\n    Integrating for 730 steps, 730.4844 days, or 2.0 years.\n    Total elapsed time is 2.0014116660123062 years.\n\n\n\n```python\n# Check for equilibrium\ncol.ASR - col.OLR\n```\n\n\n\n\n    Field([-2.34514516e-07])\n\n\n\n\n```python\n#  The temperatures at radiative equilibrium\ncol.state\n```\n\n\n\n\n    {'Tatm': Field([233.72131685, 262.28540231]), 'Ts': Field([296.38447748])}\n\n\n\nCompare these to the analytical solutions for radiative equilibrium with $\\epsilon = 0.58$:\n\n\\begin{align}\nT_s &= 296.4 \\text{ K} \\\\\nT_0 &= 262.3 \\text{ K} \\\\\nT_1 &= 233.8 \\text{ K} \\\\\n\\end{align}\n\n\nSo it looks like `climlab` agrees with our analytical results to within 0.1 K. That's good.\n\n____________\n\n\n## 3. The observed annual, global mean temperature profile\n____________\n\nWe want to model the OLR in a column whose temperatures match observations. As we've done before, we'll calculate the global, annual mean air temperature from the NCEP Reanalysis data.\n\n\n```python\n## The NOAA ESRL server is shutdown! January 2019\n## This will try to read the data over the internet.\n#ncep_filename = 'air.mon.1981-2010.ltm.nc'\n##  to read over internet\n#ncep_url = \"http://www.esrl.noaa.gov/psd/thredds/dodsC/Datasets/ncep.reanalysis.derived/pressure/\"\n#path = ncep_url\n##  Open handle to data\n#ncep_air = xr.open_dataset( path + ncep_filename, decode_times=False )\nurl = 'http://apdrc.soest.hawaii.edu:80/dods/public_data/Reanalysis_Data/NCEP/NCEP/clima/pressure/air'\nair = xr.open_dataset(url)\n# The name of the vertical axis is different than the NOAA ESRL version..\nncep_air = air.rename({'lev': 'level'})\nprint( ncep_air)\n```\n\n    \n    Dimensions:         (lat: 73, level: 17, lon: 144, time: 12)\n    Coordinates:\n      * time            (time) datetime64[ns] 2001-01-01 2001-02-01 ... 2001-12-01\n      * level           (level) float64 1e+03 925.0 850.0 700.0 ... 30.0 20.0 10.0\n      * lat             (lat) float64 -90.0 -87.5 -85.0 -82.5 ... 85.0 87.5 90.0\n      * lon             (lon) float64 0.0 2.5 5.0 7.5 ... 350.0 352.5 355.0 357.5\n    Data variables:\n        air             (time, level, lat, lon) float32 ...\n        valid_yr_count  (time, level, lat, lon) float32 ...\n    Attributes:\n        title:          NMC reanalysis atlas\n        Conventions:    COARDS\\nGrADS\n        dataType:       Grid\n        documentation:  http://apdrc.soest.hawaii.edu/datadoc/ncep_clima.php\n        history:        Mon Dec 31 09:16:43 HST 2018 : imported by GrADS Data Ser...\n\n\n\n```python\n#  Take global, annual average and convert to Kelvin\nweight = cos(deg2rad(ncep_air.lat)) / cos(deg2rad(ncep_air.lat)).mean(dim='lat')\nTglobal = (ncep_air.air * weight).mean(dim=('lat','lon','time'))\nprint( Tglobal)\n```\n\n    \n    array([ 15.179084,  11.207003,   7.838328,   0.219942,  -6.448343, -14.888846,\n           -25.570469, -39.369689, -46.797913, -53.65224 , -60.563557, -67.006054,\n           -65.532933, -61.486643, -55.853586, -51.59395 , -43.219986])\n    Coordinates:\n      * level    (level) float64 1e+03 925.0 850.0 700.0 ... 50.0 30.0 20.0 10.0\n\n\nWe're going to convert this to degrees Kelvin, using a handy list of pre-defined constants in `climlab.constants`\n\n\n```python\nclimlab.constants.tempCtoK\n```\n\n\n\n\n    273.15\n\n\n\n\n```python\nTglobal += climlab.constants.tempCtoK\nprint( Tglobal)\n```\n\n    \n    array([288.329084, 284.357003, 280.988328, 273.369942, 266.701657, 258.261154,\n           247.579531, 233.780311, 226.352087, 219.49776 , 212.586443, 206.143946,\n           207.617067, 211.663357, 217.296414, 221.55605 , 229.930014])\n    Coordinates:\n      * level    (level) float64 1e+03 925.0 850.0 700.0 ... 50.0 30.0 20.0 10.0\n\n\n\n```python\n#  A handy re-usable routine for making a plot of the temperature profiles\n#  We will plot temperatures with respect to log(pressure) to get a height-like coordinate\n\ndef zstar(lev):\n    return -np.log(lev / climlab.constants.ps)\n\ndef plot_soundings(result_list, name_list, plot_obs=True, fixed_range=True):\n    color_cycle=['r', 'g', 'b', 'y']\n    # col is either a column model object or a list of column model objects\n    #if isinstance(state_list, climlab.Process):\n    #    # make a list with a single item\n    #    collist = [collist]\n    fig, ax = plt.subplots(figsize=(9,9))\n    if plot_obs:\n        ax.plot(Tglobal, zstar(Tglobal.level), color='k', label='Observed')    \n    for i, state in enumerate(result_list):\n        Tatm = state['Tatm']\n        lev = Tatm.domain.axes['lev'].points\n        Ts = state['Ts']\n        ax.plot(Tatm, zstar(lev), color=color_cycle[i], label=name_list[i])\n        ax.plot(Ts, 0, 'o', markersize=12, color=color_cycle[i])\n    #ax.invert_yaxis()\n    yticks = np.array([1000., 750., 500., 250., 100., 50., 20., 10., 5.])\n    ax.set_yticks(-np.log(yticks/1000.))\n    ax.set_yticklabels(yticks)\n    ax.set_xlabel('Temperature (K)', fontsize=14)\n    ax.set_ylabel('Pressure (hPa)', fontsize=14)\n    ax.grid()\n    ax.legend()\n    if fixed_range:\n        ax.set_xlim([200, 300])\n        ax.set_ylim(zstar(np.array([1000., 5.])))\n    #ax2 = ax.twinx()\n    \n    return ax\n```\n\n\n```python\nplot_soundings([],[] );\n```\n\n____________\n\n\n## 4. A 30-layer model using the observed temperatures\n____________\n\n\n\n\n```python\n#  initialize a grey radiation model with 30 levels\ncol = climlab.GreyRadiationModel()\nprint( col)\n```\n\n    climlab Process of type . \n    State variables and domain shapes: \n      Ts: (1,) \n      Tatm: (30,) \n    The subprocess tree: \n    Untitled: \n       LW: \n       SW: \n       insolation: \n    \n\n\n\n```python\ncol.lev\n```\n\n\n\n\n    array([ 16.66666667,  50.        ,  83.33333333, 116.66666667,\n           150.        , 183.33333333, 216.66666667, 250.        ,\n           283.33333333, 316.66666667, 350.        , 383.33333333,\n           416.66666667, 450.        , 483.33333333, 516.66666667,\n           550.        , 583.33333333, 616.66666667, 650.        ,\n           683.33333333, 716.66666667, 750.        , 783.33333333,\n           816.66666667, 850.        , 883.33333333, 916.66666667,\n           950.        , 983.33333333])\n\n\n\n\n```python\n col.lev_bounds\n```\n\n\n\n\n    array([   0.        ,   33.33333333,   66.66666667,  100.        ,\n            133.33333333,  166.66666667,  200.        ,  233.33333333,\n            266.66666667,  300.        ,  333.33333333,  366.66666667,\n            400.        ,  433.33333333,  466.66666667,  500.        ,\n            533.33333333,  566.66666667,  600.        ,  633.33333333,\n            666.66666667,  700.        ,  733.33333333,  766.66666667,\n            800.        ,  833.33333333,  866.66666667,  900.        ,\n            933.33333333,  966.66666667, 1000.        ])\n\n\n\n\n```python\n# interpolate to 30 evenly spaced pressure levels\nlev = col.lev\nTinterp = np.interp(lev, np.flipud(Tglobal.level), np.flipud(Tglobal))\nTinterp\n#  Need to 'flipud' because the interpolation routine \n#  needs the pressure data to be in increasing order\n```\n\n\n\n\n    array([224.34737153, 211.66335696, 206.96234647, 208.29144464,\n           212.5864428 , 217.19398737, 221.78253551, 226.35208723,\n           231.30423641, 236.08018094, 240.6799208 , 245.27966066,\n           249.35980124, 252.92034254, 256.48088384, 259.66790491,\n           262.48140575, 265.29490658, 267.81303779, 270.03579936,\n           272.25856092, 274.21642907, 275.90940379, 277.60237852,\n           279.29535324, 280.98832797, 282.48551703, 283.98270609,\n           285.6810303 , 287.44639955])\n\n\n\n\n```python\n# Initialize model with observed temperatures\ncol.Ts[:] = Tglobal[0]\ncol.Tatm[:] = Tinterp\n```\n\n\n```python\n# This should look just like the observations\nresult_list = [col.state]\nname_list = ['Observed, interpolated']\nplot_soundings(result_list, name_list);\n```\n\n### Tune absorptivity to get observed OLR\n\n\n```python\ncol.compute_diagnostics()\ncol.OLR\n```\n\n\n\n\n    array([263.15004222])\n\n\n\n\n```python\n# Need to tune absorptivity to get OLR = 238.5\nepsarray = np.linspace(0.01, 0.1, 100)\nOLRarray = np.zeros_like(epsarray)\n```\n\n\n```python\nfor i in range(epsarray.size):\n    col.subprocess['LW'].absorptivity = epsarray[i]\n    col.compute_diagnostics()\n    OLRarray[i] = col.OLR\n\nplt.plot(epsarray, OLRarray)\nplt.grid()\nplt.xlabel('epsilon')\nplt.ylabel('OLR')\n```\n\nThe necessary value seems to lie near 0.055 or so.\n\nWe can be more precise with a numerical root-finder.\n\n\n```python\ndef OLRanom(eps):\n    col.subprocess['LW'].absorptivity = eps\n    col.compute_diagnostics()\n    return col.OLR - 238.5\n```\n\n\n```python\n# Use numerical root-finding to get the equilibria\nfrom scipy.optimize import brentq\n# brentq is a root-finding function\n#  Need to give it a function and two end-points\n#  It will look for a zero of the function between those end-points\neps = brentq(OLRanom, 0.01, 0.1)\nprint( eps)\n```\n\n    0.053690752586678686\n\n\n\n```python\ncol.subprocess.LW.absorptivity = eps\ncol.subprocess.LW.absorptivity\n```\n\n\n\n\n    Field([0.05369075, 0.05369075, 0.05369075, 0.05369075, 0.05369075,\n           0.05369075, 0.05369075, 0.05369075, 0.05369075, 0.05369075,\n           0.05369075, 0.05369075, 0.05369075, 0.05369075, 0.05369075,\n           0.05369075, 0.05369075, 0.05369075, 0.05369075, 0.05369075,\n           0.05369075, 0.05369075, 0.05369075, 0.05369075, 0.05369075,\n           0.05369075, 0.05369075, 0.05369075, 0.05369075, 0.05369075])\n\n\n\n\n```python\ncol.compute_diagnostics()\ncol.OLR\n```\n\n\n\n\n    array([238.5])\n\n\n\n____________\n\n\n## 5. Radiative forcing in the 30-layer model\n____________\n\nLet's compute radiative forcing for a **2% increase in absorptivity**.\n\n\n```python\n#  clone our model using a built-in climlab function\ncol2 = climlab.process_like(col)\nprint( col2)\n```\n\n    climlab Process of type . \n    State variables and domain shapes: \n      Ts: (1,) \n      Tatm: (30,) \n    The subprocess tree: \n    Untitled: \n       LW: \n       SW: \n       insolation: \n    \n\n\n\n```python\ncol2.subprocess['LW'].absorptivity *= 1.02\ncol2.subprocess['LW'].absorptivity\n```\n\n\n\n\n    Field([0.05476457, 0.05476457, 0.05476457, 0.05476457, 0.05476457,\n           0.05476457, 0.05476457, 0.05476457, 0.05476457, 0.05476457,\n           0.05476457, 0.05476457, 0.05476457, 0.05476457, 0.05476457,\n           0.05476457, 0.05476457, 0.05476457, 0.05476457, 0.05476457,\n           0.05476457, 0.05476457, 0.05476457, 0.05476457, 0.05476457,\n           0.05476457, 0.05476457, 0.05476457, 0.05476457, 0.05476457])\n\n\n\n\n```python\n#  Radiative forcing by definition is the change in TOA radiative flux,\n# HOLDING THE TEMPERATURES FIXED.\ncol2.Ts - col.Ts\n```\n\n\n\n\n    Field([0.])\n\n\n\n\n```python\ncol2.Tatm - col.Tatm\n```\n\n\n\n\n    Field([0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,\n           0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])\n\n\n\n\n```python\ncol2.compute_diagnostics()\ncol2.OLR\n```\n\n\n\n\n    array([236.65384095])\n\n\n\nThe OLR decreased after we added the extra absorbers, as we expect. Now we can calculate the Radiative Forcing:\n\n\n```python\nRF = -(col2.OLR - col.OLR)\nprint( 'The radiative forcing is %.2f W/m2.' %RF)\n```\n\n    The radiative forcing is 1.85 W/m2.\n\n\n____________\n\n\n## 6. Radiative equilibrium in the 30-layer model\n____________\n\n\n\n```python\nre = climlab.process_like(col)\n```\n\n\n```python\n#  To get to equilibrium, we just time-step the model forward long enough\nre.integrate_years(1.)\n```\n\n    Integrating for 365 steps, 365.2422 days, or 1.0 years.\n    Total elapsed time is 0.9993368783782377 years.\n\n\n\n```python\n#  Check for energy balance\nprint( 'The net downward radiative flux at TOA is %.4f W/m2.' %(re.ASR - re.OLR))\n```\n\n    The net downward radiative flux at TOA is -0.0015 W/m2.\n\n\n\n```python\nresult_list.append(re.state)\nname_list.append('Radiative equilibrium (grey gas)')\nplot_soundings(result_list, name_list)\n```\n\nSome properties of the **radiative equilibrium** temperature profile:\n\n- The surface is warmer than observed.\n- The lower troposphere is colder than observed.\n- Very cold air is sitting immediately above the warm surface.\n- There is no tropopause, no stratosphere.\n\n____________\n\n\n## 7. Radiative-Convective Equilibrium in the 30-layer model\n____________\n\nWe recognize that the large drop in temperature just above the surface is unphysical. Parcels of air in direct contact with the ground will be warmed by mechansisms other than radiative transfer.\n\nThese warm air parcels will then become buoyant, and will convect upward, mixing their heat content with the environment.\n\nWe **parameterize** the statistical effects of this mixing through a **convective adjustment**. \n\nAt each timestep, our model checks for any locations at which the **lapse rate** exceeds some threshold. Unstable layers are removed through an energy-conserving mixing formula.\n\nThis process is assumed to be fast relative to radiative heating. In the model, it is instantaneous.\n\n### Add the convective adjustment as an additional subprocess\n\n\n```python\n#  Here is the existing model\nprint( re)\n```\n\n    climlab Process of type . \n    State variables and domain shapes: \n      Ts: (1,) \n      Tatm: (30,) \n    The subprocess tree: \n    Untitled: \n       LW: \n       SW: \n       insolation: \n    \n\n\n\n```python\n#  First we make a new clone\nrce = climlab.process_like(re)\n#  Then create a new ConvectiveAdjustment process\nconv = climlab.convection.ConvectiveAdjustment(state=rce.state, \n                                               adj_lapse_rate=6.)\n#  And add it to our model\nrce.add_subprocess('Convective Adjustment', conv)\nprint( rce)\n```\n\n    climlab Process of type . \n    State variables and domain shapes: \n      Ts: (1,) \n      Tatm: (30,) \n    The subprocess tree: \n    Untitled: \n       LW: \n       SW: \n       insolation: \n       Convective Adjustment: \n    \n\n\nThis model is exactly like our previous models, except for one additional subprocess called ``Convective Adjustment``. \n\nWe passed a parameter ``adj_lapse_rate`` (in K / km) that sets the neutrally stable lapse rate -- in this case, 6 K / km.\n\nThis number is chosed to very loosely represent the net effect of **moist convection**.\n\n\n```python\n#  Run out to equilibrium\nrce.integrate_years(1.)\n```\n\n    Integrating for 365 steps, 365.2422 days, or 1.0 years.\n    Total elapsed time is 1.9986737567564754 years.\n\n\n\n```python\n#  Check for energy balance\nrce.ASR - rce.OLR\n```\n\n\n\n\n    Field([0.0007796])\n\n\n\n\n```python\nresult_list.append(rce.state)\nname_list.append('Radiatve-Convective equilibrium (grey gas)')\n```\n\n\n```python\nplot_soundings(result_list, name_list)\n```\n\nIntroducing convective adjustment into the model cools the surface quite a bit (compared to Radiative Equilibrium, in green here) -- and warms the lower troposphere. It gives us a MUCH better fit to observations.\n\nBut of course we still have no stratosphere.\n\n____________\n\n\n## 8. Putting stratospheric ozone in the grey-gas model\n____________\n\nOur model has no equivalent of the stratosphere, where temperature increases with height. That's because our model has been completely transparent to shortwave radiation up until now.\n\nWe can load the observed ozone climatology from the input files for the CESM model:\n\n\n```python\ndatapath = \"http://ramadda.atmos.albany.edu:8080/repository/opendap/Top/Users/BrianRose/CESM_runs/\"\nendstr = \"/entry.das\"\n\nozone = xr.open_dataset( datapath + 'som_input/ozone_1.9x2.5_L26_2000clim_c091112.nc' + endstr )\n```\n\n\n```python\nprint( ozone)\n```\n\n    \n    Dimensions:  (ilev: 27, lat: 96, lev: 26, lon: 144, slat: 95, slon: 144, time: 12)\n    Coordinates:\n      * ilev     (ilev) float64 2.194 4.895 9.882 18.05 ... 903.3 956.0 985.1 1e+03\n      * lat      (lat) float64 -90.0 -88.11 -86.21 -84.32 ... 84.32 86.21 88.11 90.0\n      * lev      (lev) float64 3.545 7.389 13.97 23.94 ... 867.2 929.6 970.6 992.6\n      * lon      (lon) float64 0.0 2.5 5.0 7.5 10.0 ... 350.0 352.5 355.0 357.5\n      * slat     (slat) float64 -89.05 -87.16 -85.26 -83.37 ... 85.26 87.16 89.05\n      * slon     (slon) float64 -1.25 1.25 3.75 6.25 ... 348.8 351.2 353.8 356.2\n      * time     (time) object 2000-01-15 00:00:00 ... 2000-12-15 00:00:00\n    Data variables:\n        P0       float64 ...\n        date     (time) int32 ...\n        datesec  (time) int32 ...\n        gw       (lat) float64 ...\n        hyai     (ilev) float64 ...\n        hyam     (lev) float64 ...\n        hybi     (ilev) float64 ...\n        hybm     (lev) float64 ...\n        w_stag   (slat) float64 ...\n        O3       (time, lev, lat, lon) float32 ...\n        PS       (time, lat, lon) float32 ...\n    Attributes:\n        Conventions:                     CF-1.0\n        source:                          CAM\n        case:                            ar5_cam_1850-2000_03\n        title:                           \n        logname:                         lamar\n        host:                            be0207en.ucar.ed\n        Version:                         $Name$\n        revision_Id:                     $Id$\n        initial_file:                    /ptmp/lamar/cam-inputs/ar5_cam_1850_06.c...\n        topography_file:                 /fs/cgd/csm/inputdata/atm/cam/topo/USGS-...\n        history:                         Wed Jan  5 14:56:28 2011: ncks -d time,1...\n        nco_openmp_thread_number:        1\n        NCO:                             4.0.5\n        DODS_EXTRA.Unlimited_Dimension:  time\n\n\nThe pressure levels in this dataset are:\n\n\n```python\nprint( ozone.lev)\n```\n\n    \n    array([  3.544638,   7.388814,  13.967214,  23.944625,  37.23029 ,  53.114605,\n            70.05915 ,  85.439115, 100.514695, 118.250335, 139.115395, 163.66207 ,\n           192.539935, 226.513265, 266.481155, 313.501265, 368.81798 , 433.895225,\n           510.455255, 600.5242  , 696.79629 , 787.70206 , 867.16076 , 929.648875,\n           970.55483 , 992.5561  ])\n    Coordinates:\n      * lev      (lev) float64 3.545 7.389 13.97 23.94 ... 867.2 929.6 970.6 992.6\n    Attributes:\n        long_name:      hybrid level at midpoints (1000*(A+B))\n        units:          level\n        positive:       down\n        standard_name:  atmosphere_hybrid_sigma_pressure_coordinate\n        formula_terms:  a: hyam b: hybm p0: P0 ps: PS\n\n\n### Take the global average of the ozone climatology, and plot it as a function of pressure (or height)\n\n\n```python\n#  Take global, annual average and convert to Kelvin\nweight_ozone = cos(deg2rad(ozone.lat)) / cos(deg2rad(ozone.lat)).mean(dim='lat')\nO3_global = (ozone.O3 * weight_ozone).mean(dim=('lat','lon','time'))\nprint( O3_global)\n```\n\n    \n    array([7.827929e-06, 8.641505e-06, 7.589400e-06, 5.245671e-06, 3.177616e-06,\n           1.823200e-06, 9.807570e-07, 6.228705e-07, 4.476205e-07, 3.344812e-07,\n           2.625703e-07, 2.078981e-07, 1.570746e-07, 1.124255e-07, 8.060050e-08,\n           6.278265e-08, 5.429906e-08, 4.995061e-08, 4.600757e-08, 4.229778e-08,\n           3.805591e-08, 3.387686e-08, 3.121716e-08, 2.978071e-08, 2.879810e-08,\n           2.754299e-08])\n    Coordinates:\n      * lev      (lev) float64 3.545 7.389 13.97 23.94 ... 867.2 929.6 970.6 992.6\n\n\n\n```python\nax = plt.figure(figsize=(10,8)).add_subplot(111)\nax.plot( O3_global * 1.E6, -np.log(ozone.lev/climlab.constants.ps) )\nax.set_xlabel('Ozone (ppm)', fontsize=16)\nax.set_ylabel('Pressure (hPa)', fontsize=16 )\nyticks = np.array([1000., 750., 500., 250., 100., 50., 20., 10., 5.])\nax.set_yticks(-np.log(yticks/1000.))\nax.set_yticklabels(yticks)\nax.grid()\nax.set_title('Global, annual mean ozone concentration', fontsize = 24);\n```\n\nThis shows that most of the ozone is indeed in the stratosphere, and peaks near the top of the stratosphere.\n\nNow create a new column model object **on the same pressure levels as the ozone data**.  We are also going set an adjusted lapse rate of 6 K / km.\n\n\n```python\n# the RadiativeConvectiveModel is pre-defined in climlab\n#  It contains the same components are our previous model\n#   But here we are specifying a different set of vertical levels.\noz_col = climlab.RadiativeConvectiveModel(lev = ozone.lev, adj_lapse_rate=6)\nprint( oz_col)\n```\n\n    climlab Process of type . \n    State variables and domain shapes: \n      Ts: (1,) \n      Tatm: (26,) \n    The subprocess tree: \n    Untitled: \n       LW: \n       SW: \n       insolation: \n       convective adjustment: \n    \n\n\nNow we will do something new: let the column absorb some shortwave radiation. We will assume that the shortwave absorptivity is proportional to the ozone concentration we plotted above. \n\nNow we need to weight the absorptivity by the pressure (mass) of each layer.\n\n\n```python\n# This number is an arbitrary parameter that scales how absorptive we are making the ozone\n# in our grey gas model\nozonefactor = 75\ndp = oz_col.Tatm.domain.lev.delta\nepsSW = O3_global.values * dp * ozonefactor\n```\n\nWe want to use the field `epsSW` as the absorptivity for our SW radiation model.\n\nLet's see what the absorptivity is current set to:\n\n\n```python\nprint( oz_col.subprocess['SW'].absorptivity)\n```\n\n    [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n     0. 0.]\n\n\nIt defaults to zero.\n\nBefore changing this (putting in the ozone), let's take a look at the shortwave absorption in the column:\n\n\n```python\noz_col.compute_diagnostics()\n```\n\n\n```python\noz_col.diagnostics['SW_absorbed_atm']\n```\n\n\n\n\n    array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,\n           0., 0., 0., 0., 0., 0., 0., 0., 0.])\n\n\n\nLet's now put in the ozone:\n\n\n```python\noz_col.subprocess['SW'].absorptivity = epsSW\nprint( oz_col.subprocess['SW'].absorptivity)\n```\n\n    [3.20948549e-03 3.37750296e-03 4.71182551e-03 4.57614201e-03\n     3.47591203e-03 2.24450924e-03 1.18884331e-03 7.11369788e-04\n     5.50761612e-04 4.84170273e-04 4.47141487e-04 4.16507314e-04\n     3.70212131e-04 3.11733073e-04 2.62922861e-04 2.40936639e-04\n     2.45147939e-04 2.65307555e-04 2.87482272e-04 2.95567946e-04\n     2.67120872e-04 2.16427978e-04 1.66169127e-04 1.15468088e-04\n     6.79353134e-05 3.81013280e-05]\n\n\nLet's check how this changes the SW absorption:\n\n\n```python\noz_col.compute_diagnostics()\noz_col.SW_absorbed_atm\n```\n\n\n\n\n    array([1.40571621, 1.47671285, 2.05685574, 1.99236565, 1.50916587,\n           0.97239284, 0.51429089, 0.30750441, 0.23797743, 0.20913838,\n           0.19309134, 0.17981756, 0.1597929 , 0.13452308, 0.11343944,\n           0.10393806, 0.10574109, 0.11442201, 0.12396838, 0.12743553,\n           0.11515129, 0.09328389, 0.07161233, 0.04975714, 0.02927232,\n           0.01641659])\n\n\n\nIt is now non-zero, and largest near the top of the column (also top of the array) where the ozone concentration is highest.\n\nNow it's time to run the model out to radiative-convective equilibrium\n\n\n```python\noz_col.integrate_years(1.)\n```\n\n    Integrating for 365 steps, 365.2422 days, or 1.0 years.\n    Total elapsed time is 0.9993368783782377 years.\n\n\n\n```python\nprint( oz_col.ASR - oz_col.OLR)\n```\n\n    [-0.00396053]\n\n\nAnd let's now see what we got!\n\n\n```python\nresult_list.append(oz_col.state)\nname_list.append('Radiative-Convective equilibrium with O3')\n```\n\n\n```python\n#  Make a plot to compare observations, Radiative Equilibrium, Radiative-Convective Equilibrium, and RCE with ozone!\nplot_soundings(result_list, name_list)\n```\n\nAnd we finally have something that looks looks like the tropopause, with temperature increasing above at approximately the correct rate. \n\nThere are still plenty of discrepancies between this model solution and the observations, including:\n\n- Tropopause temperature is too warm, by about 15 degrees.\n- Surface temperature is too cold\n\nThere are a number of parameters we might adjust if we wanted to improve the fit, including:\n\n- Longwave absorptivity\n- Surface albedo\n\nFeel free to experiment! (That's what models are for, after all).\n\n### The take home message\n\nThe dominant effect of stratospheric ozone is to vastly increase the radiative equilibrium temperature in the ozone layer. The temperature needs to be higher so that the longwave emission can balance the shortwave absorption.\n\nWithout ozone to absorb incoming solar radiation, the **temperature does not increase with height**.\n\nThis simple grey-gas model illustrates this principle very clearly.\n\n
                                        \n[Back to ATM 623 notebook home](../index.ipynb)\n
                                        \n\n____________\n## Version information\n____________\n\n\n\n\n```python\n%load_ext version_information\n%version_information numpy, scipy, matplotlib, xarray, climlab\n```\n\n\n\n\n
                                        SoftwareVersion
                                        Python3.6.2 64bit [GCC 4.2.1 Compatible Clang 4.0.1 (tags/RELEASE_401/final)]
                                        IPython6.1.0
                                        OSDarwin 17.7.0 x86_64 i386 64bit
                                        numpy1.14.2
                                        scipy1.0.0
                                        matplotlib2.0.2
                                        xarray0.11.2
                                        climlab0.7.1.dev5
                                        Tue Jan 15 13:25:17 2019 EST
                                        \n\n\n\n____________\n\n## Credits\n\nThe author of this notebook is [Brian E. J. Rose](http://www.atmos.albany.edu/facstaff/brose/index.html), University at Albany.\n\nIt was developed in support of [ATM 623: Climate Modeling](http://www.atmos.albany.edu/facstaff/brose/classes/ATM623_Spring2015/), a graduate-level course in the [Department of Atmospheric and Envionmental Sciences](http://www.albany.edu/atmos/index.php)\n\nDevelopment of these notes and the [climlab software](https://github.com/brian-rose/climlab) is partially supported by the National Science Foundation under award AGS-1455071 to Brian Rose. Any opinions, findings, conclusions or recommendations expressed here are mine and do not necessarily reflect the views of the National Science Foundation.\n____________\n\n\n```python\n\n```\n", "meta": {"hexsha": "15717e9c9f38a3545031ad356e39cf861291f0b7", "size": 356245, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Lectures/Lecture08 -- Grey radiation modeling with climlab.ipynb", "max_stars_repo_name": "katrinafandrich/ClimateModeling_courseware", "max_stars_repo_head_hexsha": "6f13fd38706cfef91e81f7e7065d9fab6fb8bb2f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Lectures/Lecture08 -- Grey radiation modeling with climlab.ipynb", "max_issues_repo_name": "katrinafandrich/ClimateModeling_courseware", "max_issues_repo_head_hexsha": "6f13fd38706cfef91e81f7e7065d9fab6fb8bb2f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Lectures/Lecture08 -- Grey radiation modeling with climlab.ipynb", "max_forks_repo_name": "katrinafandrich/ClimateModeling_courseware", "max_forks_repo_head_hexsha": "6f13fd38706cfef91e81f7e7065d9fab6fb8bb2f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 154.8218166015, "max_line_length": 71722, "alphanum_fraction": 0.8800741063, "converted": true, "num_tokens": 11270, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813031051514763, "lm_q2_score": 0.7122321781307374, "lm_q1q2_score": 0.41402277673619703}}
{"text": "\n# FFM232, Klassisk fysik och vektorf\u00e4lt - F\u00f6rel\u00e4sningsanteckningar\n\n **[Christian Forss\u00e9n](http://fy.chalmers.se/subatom/tsp/), Institutionen f\u00f6r fysik, Chalmers, G\u00f6teborg, Sverige**\n\nDate: **Oct 3, 2016**\n\n# 9. L\u00f6sningar av Poissons ekvation\n\nVi vet att Poissons ekvation\n$$\n\\Delta \\phi(\\vec{r}) = - \\rho(\\vec{r}),\n$$\nhar entydiga l\u00f6sningar om\n\n$$\n\\begin{equation}\n\\phi|_{\\partial V} = f(\\vec{r}) \\quad  \\mathrm{Dirichlets~randvillkor} \\nonumber \n\\end{equation}\n$$\n\n$$\n\\begin{equation} \n(\\nabla\\phi)|_{\\partial V} \\cdot \\vec{n} = g(\\vec{r}) \\quad  \\mathrm{Neumans~randvillkor} \\nonumber \n\\end{equation}\n$$\n\nd\u00e4r $f$ och $g$ \u00e4r funktioner p\u00e5 randen $\\partial V$.\n\n## L\u00f6sning av Poissons ekvation\n\nVi kommer att betrakta fyra olika l\u00f6sningsmetoder:\n\n### 1. Greensfunktionsmetoden\n\n### 2. Spegling\n\n### 3. Variabelseparation\n\n### 4. Numeriska metoder\n\n* De tre f\u00f6rstn\u00e4mnda \u00e4r analytiska metoder som vi introducerar f\u00f6r att ge en fysikalisk f\u00f6rst\u00e5else av l\u00f6sningarna.\n\n* De numeriska metoderna \u00e4r f\u00f6rst\u00e5s viktigast f\u00f6r praktiska till\u00e4mpningar. Se datoruppgift.\n\n## 1. Greensfunktionsmetoden\n\nVi skall h\u00e4r titta p\u00e5 l\u00f6sningar till Poissons ekvation,\n$$\n\\Delta\\phi(\\vec{r})=-\\rho(\\vec{r}).\n$$ \nVi begr\u00e4nsar oss till homogena randvillkor, dvs. $f,g=0$ p\u00e5 randen.\n\nL\u00f6s f\u00f6rst (med randvillkor!)\n$$\n\\Delta G(\\vec{r},\\vec{r}{\\;}') = -\\delta(\\vec{r} - \\vec{r}{\\;}'),\n$$\ndvs f\u00f6r en punktk\u00e4lla i $\\vec{r} = \\vec{r}{\\;}'$ med $q=1$. \n\n[Comment 1: Notera att Laplaceoperatorn verkar p\u00e5 variabeln $\\vec{r}$ (inte $\\vec{r}{\\;}'$). Dvs., $\\Delta = \\Delta_{\\vec{r}} = \\partial_x^2 + \\partial_y^2 + \\partial_z^2$.]\n\nK\u00e4llf\u00f6rdelningen $\\rho$ kan skrivas som en superposition\n$$\n\\rho(\\vec{r})=\\int_{V'}\\mbox{d}V'\\,\\delta^3(\\vec{r}-\\vec{r}{\\;}')\\rho(\\vec{r}{\\;}').\n$$\nL\u00f6sningen blir en superposition av Greensfunktioner\n$$\n\\phi(\\vec{r})=\\int_{V'}\\mbox{d}V'\\,G(\\vec{r},\\vec{r}{\\;}')\\rho(\\vec{r}{\\;}').\n$$\nDetta visas genom ins\u00e4ttning:\n\n$$\n\\begin{equation}\n\\Delta\\phi(\\vec{r}) = \\Delta\\int_{V'}\\mbox{d}V'\\,G(\\vec{r},\\vec{r}{\\;}')\\rho(\\vec{r}{\\;}')\n=\\int_{V'}\\mbox{d}V'\\,\\Delta G(\\vec{r},\\vec{r}{\\;}')\\rho(\\vec{r}{\\;}') \\nonumber \n\\end{equation}\n$$\n\n$$\n\\begin{equation} \n=-\\int_{V'}\\mbox{d}V'\\,\\delta^3(\\vec{r}-\\vec{r}{\\;}')\\rho(\\vec{r}{\\;}')\n=-\\rho(\\vec{r}).\n\\end{equation}\n$$\n\n* Notera att Greensfunktionen $G$ p\u00e5 ett omr\u00e5de $V$ best\u00e4ms av formen p\u00e5 omr\u00e5det och av randvillkoren p\u00e5 $\\partial V$.\n\n* P\u00e5 $\\mathbf{R}^3$ \u00e4r Greensfunktionen\n\n$$\nG(\\vec{r},\\vec{r}{\\;}')=\\frac{1}{4\\pi|\\vec{r}-\\vec{r}{\\;}'|},\n$$\noch l\u00f6sningen till Poissons ekvation med k\u00e4lla $\\rho$ blir\n$$\n\\phi(\\vec{r})=\\int_{\\mathbf{R}^3}\\mbox{d}V'\\,\\frac{\\rho(\\vec{r}{\\;}')}{4\\pi|\\vec{r}-\\vec{r}{\\;}'|}.\n$$\n[Comment 2: Detta \u00e4r f\u00f6rst\u00e5s samma uttryck som vi h\u00e4rledde genom superposition i kapitel 6.]\n\n---------------------------------------------------------\n### Exempel: linjek\u00e4lla\n\n\nBetrakta en linjek\u00e4lla, $\\rho(\\vec{r})=k\\delta^2(\\vec{\\rho})$, i $\\mathbf{R}^3$.\n\nVi skall integrera \u00f6ver linjek\u00e4llan och introducerar koordinaten\n$$\n\\vec{r}{\\;}' = \\vec{\\rho}{\\;}' + z' \\hat{z}' = \\rho' \\hat{\\rho}' + z' \\hat{z},\n$$\nd\u00e4r vi noterar att det inte beh\u00f6vs n\u00e5got \"prim\" p\u00e5 $z$-riktningen.\n\nVi s\u00e4tter in i \n$$\n\\phi(\\vec{r}) = \\int_{\\mathbf{R}^3}\\rho(\\vec{r}{\\;}')G(\\vec{r},\\vec{r}{\\;}')\\mbox{d}V'\n=\\int_{-\\infty}^{\\infty}dz'\\int_{\\mathbf{R}^2}dS'\n       \\frac{k\\delta^2(\\vec{\\rho}{\\;}')}{4\\pi|\\vec{r}-(\\rho'\\hat\\rho{}'+z'\\hat z)|} .\n$$\n\nIntegralen $\\int dS'$ \u00f6ver $x'$ och $y'$ kan enkelt utf\u00f6ras tack vare deltafunktionen. Resultatet:\n$$\n\\phi(\\vec{r})=\\int_{-\\infty}^{\\infty}dz' \\frac{k}{4\\pi|\\vec{r}-z'\\hat z|}\n$$  \nsom \u00e4r identiskt med den direkta konstruktionen fr\u00e5n kap. 6.\n\n---------------------------------------------------------\n### Exempel: virveltr\u00e5d\n\n\nPoissons ekvation f\u00f6r ett vektorf\u00e4lt,\n$$\n\\Delta\\vec A = -\\vec\\jmath, \n$$\nDivergensfritt f\u00e4lt $\\vec B$ som d\u00e4rmed kan uttryckas som rotationen av en\nvektorpotential, $\\vec B=\\nabla\\times\\vec A$. \n\nL\u00f6sningen blir\n$$\n\\vec A(\\vec{r})=\\int_{\\mathbf{R}^3} \\mbox{d}V'\\, \\frac{\\vec\\jmath(\\vec{r}{\\;}')}{4\\pi|\\vec{r}-\\vec{r}{\\;}'|}.\n$$\nAv speciellt intresse \u00e4r en virveltr\u00e5d med konstant styrka $J$ l\u00e4ngs en o\u00e4ndlig eller sluten kurva $C$. Virvelt\u00e4theten blir $\\vec{jmath}(\\vec{r}{\\;}') = J \\delta^2(\\vec{\\rho}{\\;}') \\mbox{d}\\vec{r}{\\;}' / \\mbox{d}r'$, dvs den \u00e4r riktad l\u00e4ngs kurvan $C$, d\u00e4r vektorn $\\vec{\\rho}{\\;}'$ \u00e4r vinkelr\u00e4t mot kurvan. Vektorpotentialen blir\n$$\n\\vec A(\\vec{r})=\\int_C \\frac{J \\mbox{d}\\vec{r}{\\;}'}{4\\pi|\\vec{r}-\\vec{r}{\\;}'|},\n$$\nd\u00e4r vi kan utf\u00f6ra volymsintegralen \u00f6ver $\\mbox{d}r'\\mbox{d}\\vec{\\rho}{\\;}'$, dvs en riktning l\u00e4ngs kurvan $C$ och en yta vinkelr\u00e4t mot densamma.\n\nResultatet motsvarar t.ex. den EM vektorpotentialen fr\u00e5n en tunn elektrisk ledare med str\u00f6mmen $J$.\n\n---------------------------------------------------------\n\n## 2. Spegling\n\n* Vi s\u00e5g att l\u00f6sningen $\\phi(\\vec{r})$ erh\u00e5lls enkelt om man har tillg\u00e5ng till Greensfunktionen. Men denna \u00e4r ofta sv\u00e5r att finna. \n\n* F\u00f6r vissa geometrier erbjuder *speglingsmetoden* ett v\u00e4ldigt elegant s\u00e4tt att konstruera Greensfunktionen.\n\nBetrakta halvrymden $\\{\\vec{r}:\\,z>0\\}$ med ett homogent randvillkor p\u00e5 planet\n$z=0$: \n* Dirichlets randvillkor: $\\phi=0$, eller \n\n* Neumanns, $\\frac{\\partial \\phi}{\\partial z} = 0$.\n\n[Comment 3: Detta \u00e4r ett bra tillf\u00e4lle att repetera begreppen ekvipotentialytor och f\u00e4ltlinjer (till vektorf\u00e4ltet $-\\nabla\\phi$). Se till att f\u00f6rst\u00e5 att ett randvillkor $\\phi=0$ (Dirichlet) betyder att randen \u00e4r en ekvipotentialyta, och att $\\vec n \\cdot \\nabla\\phi=0$ betyder att f\u00e4ltlinjerna \u00e4r parallella med randen.]\n\n\n\n\n
                                        \n\n\n\n\n\n* Spegelladdningen finns i ett omr\u00e5de som vi inte \u00e4r intresserade av ($z<0$), men hj\u00e4lper till att uppfylla randvillkoren.\n\nMed $\\vec{r}_0 = (x_0,y_0,z_0)$ och $\\vec{r}_1 = (x_0,y_0,-z_0)$ och:\n* $q_1 = q$ uppfylls Neumanns randvillor\n\n* $q_1 = -q$ uppfylls Dirichlets randvillor\n\ndvs potentialen fr\u00e5n den tv\u00e5 punktladdningarna\n$$\n\\phi(\\vec{r}) = \\frac{q}{4 \\pi |\\vec{r} - \\vec{r}_0|} \\pm \\frac{q}{4 \\pi |\\vec{r} - \\vec{r}_1|}\n$$\nI det f\u00f6rra fallet \u00e4r f\u00e4ltlinjerna parallella med $z=0$ planet, i det senare fallet ligger ekvipotentialytan $\\phi=0$ i $z=0$ planet.\n\nGreensfunktionen med Dirichlets homogena randvillkor blir h\u00e4r allts\u00e5\n$$\nG (\\vec{r},\\vec{r}_0)=\\frac{1}{4\\pi|\\vec{r}-\\vec{r}_0|} - \\frac{1} {4\\pi|\\vec{r}-\\vec{r}_1|}.\n$$\n\n[Comment 4: Notera att $G(\\vec{r},\\vec{r}_0)$ uppfyller $\\Delta G(\\vec{r},\\vec{r}_0) = - \\delta(\\vec{r}-\\vec{r}_0)$ i det \u00f6vre halvrummet.]\n\nIntressant nog fungerar speglingsmetoden \u00e4ven f\u00f6r cirklar i tv\u00e5 dimensioner och sf\u00e4rer i tre dimensioner (i det senare fallet dock endast f\u00f6r Dirichlets randvillkor). Se demonstrationsuppgift.\n\n\n\n\n
                                        \n\n\n\n\n\n## 3. Variabelseparation\n\n* Bygger p\u00e5 att man l\u00f6ser ekvationerna stegvis i en variabel i taget. \n\n* Problemet skall *passa bra* ihop med ett visst koordinatsystem.\n\n---------------------------------------------------------\n\n### Exempel: Laplaces ekvation p\u00e5 en cirkelskiva\n\n\n* $\\Delta\\phi=0$, p\u00e5 $\\varrho=\\sqrt{x^2+y^2} 0$, ans\u00e4tt l\u00f6sning $f(\\varrho) = C \\varrho^p$\n\n$$\n\\frac{1}{\\varrho} \\frac{\\partial}{\\partial\\varrho} \\left( \\varrho \\frac{\\partial}{\\partial\\varrho} \\varrho^p \\right) - \\frac{m^2}{\\varrho^2} \\varrho^p = 0 \\quad \\Rightarrow \\quad p^2 \\varrho^{p-2} - m^2 \\varrho^{p-2} = 0 \\quad \\Rightarrow \\quad \np = \\pm m\n$$\nMed l\u00f6sningen $f(\\varrho) = A \\varrho^m + \\frac{B}{\\varrho^m}$, d\u00e4r den andra termen \u00e4r singul\u00e4r i origo (vi skippar denna).\n\nMed randvillkoret fr\u00e5n ovan $h(\\varphi) = \\cos m \\varphi$, $f(a) = \\phi_0$ f\u00e5r vi l\u00f6sningen\n$$\n\\phi(\\vec{r}) = \\phi_0 \\left( \\frac{\\varrho}{a} \\right)^m \\cos m\\varphi,\n$$\nsom ovan.\n\nF\u00f6r ett mer allm\u00e4nt randvillkor kan man (Fourier)-utveckla\n$$\nh(\\varphi) = \\sum_{m=0}^\\infty a_m \\cos(m\\varphi) + b_m \\sin(m\\varphi),\n$$\nvilket ger l\u00f6sningen\n$$\n\\phi(\\vec{r}) = \\sum_{m=0}^\\infty a_m \\left( \\frac{\\varrho}{a} \\right)^m \\cos(m\\varphi) + b_m \\left( \\frac{\\varrho}{a} \\right)^m \\sin(m\\varphi).\n$$\nOBS: ing\u00e5r ej i denna kurs att kunna g\u00f6ra en s\u00e5dan Fourierutveckling.\n\n---------------------------------------------------------\n\n\n[Comment 5: Separationsmetoden kan f\u00f6rst\u00e5s anv\u00e4ndas med fler \u00e4n tv\u00e5 variabler. Vill man t.ex. anv\u00e4nda den i sf\u00e4riska koordinater, hittar man egenfunktioner i tur och ordning i $\\varphi$, $\\theta$ och $r$. Se veckans tal. Eller s\u00e5 hittar man direkt egenfunktioner p\u00e5 $S^2$, s.k. klotytefunktioner.]\n", "meta": {"hexsha": "89b0ccce72369c180103a2bfe5b16e8d3267e068", "size": 16925, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "doc/pub/09-diffekv/ipynb/09-diffekv.ipynb", "max_stars_repo_name": "physics-chalmers/ffm234", "max_stars_repo_head_hexsha": "b37a744e50604ba0956724883714ea3d87929f81", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "doc/pub/09-diffekv/ipynb/09-diffekv.ipynb", "max_issues_repo_name": "physics-chalmers/ffm234", "max_issues_repo_head_hexsha": "b37a744e50604ba0956724883714ea3d87929f81", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/pub/09-diffekv/ipynb/09-diffekv.ipynb", "max_forks_repo_name": "physics-chalmers/ffm234", "max_forks_repo_head_hexsha": "b37a744e50604ba0956724883714ea3d87929f81", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2020-08-06T06:03:59.000Z", "max_forks_repo_forks_event_max_datetime": "2020-11-03T13:36:07.000Z", "avg_line_length": 43.961038961, "max_line_length": 353, "alphanum_fraction": 0.5236041359, "converted": true, "num_tokens": 5036, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802735722128, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.4138900258037189}}
{"text": "\n\n# Tutorial 1: Introduction to Reinforcement Learning\n\n**Week 3, Day 2: Basic Reinforcement Learning (RL)**\n\n**By Neuromatch Academy**\n\n__Content creators:__  Matthew Sargent, Anoop Kulkarni, Sowmya Parthiban, Feryal Behbahani, Jane Wang\n\n__Content reviewers:__ Ezekiel Williams, Mehul Rastogi, Lily Cheng, Roberto Guidotti, Arush Tagade, Kelson Shilling-Scrivo\n\n__Content editors:__ Spiros Chavlis \n\n__Production editors:__ Spiros Chavlis\n\n**Our 2021 Sponsors, including Presenting Sponsor Facebook Reality Labs**\n\n
                                        \n\n---\n# Tutorial Objectives\n\nBy the end of the tutorial, you should be able to:\n\n1. Within the RL framework, be able to identify the different components: environment, agent, states, and actions. \n2. Understand the Bellman equation and components involved. \n3. Implement tabular value-based model-free learning (Q-learning and SARSA).\n4. Discuss real-world applications and ethical issues of RL.\n\nBy completing the Bonus sections, you should be able to:\n1. Run a DQN agent and experiment with different hyperparameters.\n2. Have a high-level understanding of other (nonvalue-based) RL methods.\n\n\n```\n# @title Tutorial slides\n\n# @markdown These are the slides for the videos in this tutorial\n\n# @markdown If you want to locally download the slides, click [here](https://osf.io/m3kqy/download)\nfrom IPython.display import IFrame\nIFrame(src=f\"https://mfr.ca-1.osf.io/render?url=https://osf.io/m3kqy/?direct%26mode=render%26action=download%26mode=render\", width=854, height=480)\n```\n\n\n\n\n\n\n\n\n\n\n---\n# Setup\n\nRun the following *Setup* cells in order to set up needed functions. Don't worry about the code for now!\n\n**Note:** There is an issue with some images not showing up if you're using a Safari browser. Please switch to Chrome if this is the case.\n\n\n```\n# @title Install requirements\nfrom IPython.display import clear_output\n# @markdown we install the acme library, see [here](https://github.com/deepmind/acme) for more info\n\n# @markdown WARNING: There may be errors and warnings reported during the installation.\n# @markdown However, they should be ignored.\n!apt-get install -y xvfb ffmpeg --quiet\n!pip install --upgrade pip --quiet\n!pip install imageio --quiet\n!pip install imageio-ffmpeg\n!pip install gym --quiet\n!pip install enum34 --quiet\n!pip install dm-env --quiet\n!pip install pandas --quiet\n!pip install keras-nightly==2.5.0.dev2021020510 --quiet\n!pip install grpcio==1.34.0 --quiet\n!pip install tensorflow --quiet\n!pip install typing --quiet\n!pip install einops --quiet\n!pip install dm-acme --quiet\n!pip install dm-acme[reverb] --quiet\n!pip install dm-acme[tf] --quiet\n!pip install dm-acme[envs] --quiet\n!pip install dm-env --quiet\n\n!pip install git+https://github.com/NeuromatchAcademy/evaltools --quiet\nfrom evaltools.airtable import AirtableForm\n\n# generate airtable form\natform = AirtableForm('appn7VdPRseSoMXEG','W3D2_T1','https://portal.neuromatchacademy.org/api/redirect/to/3e77471d-4de0-4e43-a026-9cfb603b5197')\nclear_output()\n```\n\n\n```\n# Import modules\nimport gym\nimport enum\nimport copy\nimport time\nimport acme\nimport torch\nimport base64\nimport dm_env\nimport IPython\nimport imageio\nimport warnings\nimport itertools\nimport collections\n\nimport numpy as np\nimport pandas as pd\nimport torch.nn as nn\nimport torch.optim as optim\nimport torch.nn.functional as F\n\nimport matplotlib as mpl\nimport matplotlib.pyplot as plt\n\nimport tensorflow.compat.v2 as tf\n\nfrom acme import specs\nfrom acme import wrappers\nfrom acme.utils import tree_utils\nfrom acme.utils import loggers\nfrom torch.autograd import Variable\nfrom torch.distributions import Categorical\nfrom typing import Callable, Sequence\n\ntf.enable_v2_behavior()\nwarnings.filterwarnings('ignore')\nnp.set_printoptions(precision=3, suppress=1)\n```\n\n\n```\n# @title Figure settings\nimport ipywidgets as widgets       # interactive display\n%matplotlib inline\n%config InlineBackend.figure_format = 'retina'\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/course-content/master/nma.mplstyle\")\nmpl.rc('image', cmap='Blues')\n```\n\n\n```\n# @title Helper Functions\n# @markdown Implement helpers for value visualisation\n\nmap_from_action_to_subplot = lambda a: (2, 6, 8, 4)[a]\nmap_from_action_to_name = lambda a: (\"up\", \"right\", \"down\", \"left\")[a]\n\n\ndef plot_values(values, colormap='pink', vmin=-1, vmax=10):\n  plt.imshow(values, interpolation=\"nearest\",\n             cmap=colormap, vmin=vmin, vmax=vmax)\n  plt.yticks([])\n  plt.xticks([])\n  plt.colorbar(ticks=[vmin, vmax])\n\n\ndef plot_state_value(action_values, epsilon=0.1):\n  q = action_values\n  fig = plt.figure(figsize=(4, 4))\n  vmin = np.min(action_values)\n  vmax = np.max(action_values)\n  v = (1 - epsilon) * np.max(q, axis=-1) + epsilon * np.mean(q, axis=-1)\n  plot_values(v, colormap='summer', vmin=vmin, vmax=vmax)\n  plt.title(\"$v(s)$\")\n\n\ndef plot_action_values(action_values, epsilon=0.1):\n  q = action_values\n  fig = plt.figure(figsize=(8, 8))\n  fig.subplots_adjust(wspace=0.3, hspace=0.3)\n  vmin = np.min(action_values)\n  vmax = np.max(action_values)\n  dif = vmax - vmin\n  for a in [0, 1, 2, 3]:\n    plt.subplot(3, 3, map_from_action_to_subplot(a))\n\n    plot_values(q[..., a], vmin=vmin - 0.05*dif, vmax=vmax + 0.05*dif)\n    action_name = map_from_action_to_name(a)\n    plt.title(r\"$q(s, \\mathrm{\" + action_name + r\"})$\")\n\n  plt.subplot(3, 3, 5)\n  v = (1 - epsilon) * np.max(q, axis=-1) + epsilon * np.mean(q, axis=-1)\n  plot_values(v, colormap='summer', vmin=vmin, vmax=vmax)\n  plt.title(\"$v(s)$\")\n\n\ndef plot_stats(stats, window=10):\n  plt.figure(figsize=(16,4))\n  plt.subplot(121)\n  xline = range(0, len(stats.episode_lengths), window)\n  plt.plot(xline, smooth(stats.episode_lengths, window=window))\n  plt.ylabel('Episode Length')\n  plt.xlabel('Episode Count')\n  plt.subplot(122)\n  plt.plot(xline, smooth(stats.episode_rewards, window=window))\n  plt.ylabel('Episode Return')\n  plt.xlabel('Episode Count')\n```\n\n\n```\n# @title Helper functions\ndef smooth(x, window=10):\n  return x[:window*(len(x)//window)].reshape(len(x)//window, window).mean(axis=1)\n```\n\n\n```\n# @title Set random seed\n\n# @markdown Executing `set_seed(seed=seed)` you are setting the seed\n\n# for DL its critical to set the random seed so that students can have a\n# baseline to compare their results to expected results.\n# Read more here: https://pytorch.org/docs/stable/notes/randomness.html\n\n# Call `set_seed` function in the exercises to ensure reproducibility.\nimport random\nimport torch\n\ndef set_seed(seed=None, seed_torch=True):\n  if seed is None:\n    seed = np.random.choice(2 ** 32)\n  random.seed(seed)\n  np.random.seed(seed)\n  if seed_torch:\n    torch.manual_seed(seed)\n    torch.cuda.manual_seed_all(seed)\n    torch.cuda.manual_seed(seed)\n    torch.backends.cudnn.benchmark = False\n    torch.backends.cudnn.deterministic = True\n\n  print(f'Random seed {seed} has been set.')\n\n\n# In case that `DataLoader` is used\ndef seed_worker(worker_id):\n  worker_seed = torch.initial_seed() % 2**32\n  np.random.seed(worker_seed)\n  random.seed(worker_seed)\n```\n\n\n```\n# @title Set device (GPU or CPU). Execute `set_device()`\n# especially if torch modules used.\n\n# inform the user if the notebook uses GPU or CPU.\n\ndef set_device():\n  device = \"cuda\" if torch.cuda.is_available() else \"cpu\"\n  if device != \"cuda\":\n    print(\"WARNING: For this notebook to perform best, \"\n        \"if possible, in the menu under `Runtime` -> \"\n        \"`Change runtime type.`  select `GPU` \")\n  else:\n    print(\"GPU is enabled in this notebook.\")\n\n  return device\n```\n\n\n```\nSEED = 2021\nset_seed(seed=SEED)\nDEVICE = set_device()\n```\n\n    Random seed 2021 has been set.\n    WARNING: For this notebook to perform best, if possible, in the menu under `Runtime` -> `Change runtime type.`  select `GPU` \n\n\n---\n# Section 1: Introduction to Reinforcement Learning\n\n*Time estimate: ~15mins*\n\n\n```\n# @title Video 1: Introduction to RL\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n  from IPython.display import IFrame\n  class BiliVideo(IFrame):\n    def __init__(self, id, page=1, width=400, height=300, **kwargs):\n      self.id=id\n      src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n      super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n  video = BiliVideo(id=f\"BV18V411p7iK\", width=854, height=480, fs=1)\n  print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n  display(video)\n\nout1 = widgets.Output()\nwith out1:\n  from IPython.display import YouTubeVideo\n  video = YouTubeVideo(id=f\"BWz3scQN50M\", width=854, height=480, fs=1, rel=0)\n  print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n  display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 1: Introduction to RL')\n\ndisplay(out)\n```\n\n\n    Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\n## Acme: a research framework for reinforcement learning\n\n**Acme** is a library of reinforcement learning (RL) agents and agent building blocks by Google DeepMind. Acme strives to expose simple, efficient, and readable agents, that serve both as reference implementations of popular algorithms and as strong baselines, while still providing enough flexibility to do novel research. The design of Acme also attempts to provide multiple points of entry to the RL problem at differing levels of complexity.\n\nFor more information see the github's repository [deepmind/acme](https://github.com/deepmind/acme).\n\n---\n# Section 2: General Formulation of RL Problems and Gridworlds\n\n*Time estimate: ~30mins*\n\n\n```\n# @title Video 2: General Formulation and MDPs\nfrom ipywidgets import widgets\n\nout2 = widgets.Output()\nwith out2:\n  from IPython.display import IFrame\n  class BiliVideo(IFrame):\n    def __init__(self, id, page=1, width=400, height=300, **kwargs):\n      self.id=id\n      src = \"https://player.bilibili.com/player.html?bvid={0}&page={1}\".format(id, page)\n      super(BiliVideo, self).__init__(src, width, height, **kwargs)\n\n  video = BiliVideo(id=f\"BV1k54y1E7Zn\", width=854, height=480, fs=1)\n  print(\"Video available at https://www.bilibili.com/video/{0}\".format(video.id))\n  display(video)\n\nout1 = widgets.Output()\nwith out1:\n  from IPython.display import YouTubeVideo\n  video = YouTubeVideo(id=f\"h6TxAALY5Fc\", width=854, height=480, fs=1, rel=0)\n  print(\"Video available at https://youtube.com/watch?v=\" + video.id)\n  display(video)\n\nout = widgets.Tab([out1, out2])\nout.set_title(0, 'Youtube')\nout.set_title(1, 'Bilibili')\n\n# add event to airtable\natform.add_event('Video 2: General Formulation and MDPs')\n\ndisplay(out)\n```\n\n\n    Tab(children=(Output(), Output()), _titles={'0': 'Youtube', '1': 'Bilibili'})\n\n\nThe agent interacts with the environment in a loop corresponding to the following diagram. The environment defines a set of **actions**  that an agent can take.  The agent takes an action informed by the **observations** it receives, and will get a **reward** from the environment after each action. The goal in RL is to find an agent whose actions maximize the total accumulation of rewards obtained from the environment. \n\n
                                        \n\n\n\n## Section 2.1: The Environment\n\n\n\nFor this practical session we will focus on a **simple grid world** environment,which consists of a 9 x 10 grid of either wall or empty cells, depicted in black and white, respectively. The smiling agent starts from an initial location and needs to navigate to reach the goal square.\n\n
                                        \n\nBelow you will find an implementation of this Gridworld as a `dm_env.Environment`.\n\nThere is no coding in this section, but if you want, you can look over the provided code so that you can familiarize yourself with an example of how to set up a **grid world** environment.\n\n\n\n```\n# @title Implement GridWorld { form-width: \"30%\" }\n# @markdown *Double-click* to inspect the contents of this cell.\n\nclass ObservationType(enum.IntEnum):\n  STATE_INDEX = enum.auto()\n  AGENT_ONEHOT = enum.auto()\n  GRID = enum.auto()\n  AGENT_GOAL_POS = enum.auto()\n\n\nclass GridWorld(dm_env.Environment):\n\n  def __init__(self,\n               layout,\n               start_state,\n               goal_state=None,\n               observation_type=ObservationType.STATE_INDEX,\n               discount=0.9,\n               penalty_for_walls=-5,\n               reward_goal=10,\n               max_episode_length=None,\n               randomize_goals=False):\n    \"\"\"Build a grid environment.\n\n    Simple gridworld defined by a map layout, a start and a goal state.\n\n    Layout should be a NxN grid, containing:\n      * 0: empty\n      * -1: wall\n      * Any other positive value: value indicates reward; episode will terminate\n\n    Args:\n      layout: NxN array of numbers, indicating the layout of the environment.\n      start_state: Tuple (y, x) of starting location.\n      goal_state: Optional tuple (y, x) of goal location. Will be randomly\n        sampled once if None.\n      observation_type: Enum observation type to use. One of:\n        * ObservationType.STATE_INDEX: int32 index of agent occupied tile.\n        * ObservationType.AGENT_ONEHOT: NxN float32 grid, with a 1 where the\n          agent is and 0 elsewhere.\n        * ObservationType.GRID: NxNx3 float32 grid of feature channels.\n          First channel contains walls (1 if wall, 0 otherwise), second the\n          agent position (1 if agent, 0 otherwise) and third goal position\n          (1 if goal, 0 otherwise)\n        * ObservationType.AGENT_GOAL_POS: float32 tuple with\n          (agent_y, agent_x, goal_y, goal_x)\n      discount: Discounting factor included in all Timesteps.\n      penalty_for_walls: Reward added when hitting a wall (should be negative).\n      reward_goal: Reward added when finding the goal (should be positive).\n      max_episode_length: If set, will terminate an episode after this many\n        steps.\n      randomize_goals: If true, randomize goal at every episode.\n    \"\"\"\n    if observation_type not in ObservationType:\n      raise ValueError('observation_type should be a ObservationType instace.')\n    self._layout = np.array(layout)\n    self._start_state = start_state\n    self._state = self._start_state\n    self._number_of_states = np.prod(np.shape(self._layout))\n    self._discount = discount\n    self._penalty_for_walls = penalty_for_walls\n    self._reward_goal = reward_goal\n    self._observation_type = observation_type\n    self._layout_dims = self._layout.shape\n    self._max_episode_length = max_episode_length\n    self._num_episode_steps = 0\n    self._randomize_goals = randomize_goals\n    if goal_state is None:\n      # Randomly sample goal_state if not provided\n      goal_state = self._sample_goal()\n    self.goal_state = goal_state\n\n  def _sample_goal(self):\n    \"\"\"Randomly sample reachable non-starting state.\"\"\"\n    # Sample a new goal\n    n = 0\n    max_tries = 1e5\n    while n < max_tries:\n      goal_state = tuple(np.random.randint(d) for d in self._layout_dims)\n      if goal_state != self._state and self._layout[goal_state] == 0:\n        # Reachable state found!\n        return goal_state\n      n += 1\n    raise ValueError('Failed to sample a goal state.')\n\n  @property\n  def layout(self):\n    return self._layout\n\n  @property\n  def number_of_states(self):\n    return self._number_of_states\n\n  @property\n  def goal_state(self):\n    return self._goal_state\n\n  @property\n  def start_state(self):\n    return self._start_state\n\n  @property\n  def state(self):\n    return self._state\n\n  def set_state(self, x, y):\n    self._state = (y, x)\n\n  @goal_state.setter\n  def goal_state(self, new_goal):\n    if new_goal == self._state or self._layout[new_goal] < 0:\n      raise ValueError('This is not a valid goal!')\n    # Zero out any other goal\n    self._layout[self._layout > 0] = 0\n    # Setup new goal location\n    self._layout[new_goal] = self._reward_goal\n    self._goal_state = new_goal\n\n  def observation_spec(self):\n    if self._observation_type is ObservationType.AGENT_ONEHOT:\n      return specs.Array(\n          shape=self._layout_dims,\n          dtype=np.float32,\n          name='observation_agent_onehot')\n    elif self._observation_type is ObservationType.GRID:\n      return specs.Array(\n          shape=self._layout_dims + (3,),\n          dtype=np.float32,\n          name='observation_grid')\n    elif self._observation_type is ObservationType.AGENT_GOAL_POS:\n      return specs.Array(\n          shape=(4,), dtype=np.float32, name='observation_agent_goal_pos')\n    elif self._observation_type is ObservationType.STATE_INDEX:\n      return specs.DiscreteArray(\n          self._number_of_states, dtype=int, name='observation_state_index')\n\n  def action_spec(self):\n    return specs.DiscreteArray(4, dtype=int, name='action')\n\n  def get_obs(self):\n    if self._observation_type is ObservationType.AGENT_ONEHOT:\n      obs = np.zeros(self._layout.shape, dtype=np.float32)\n      # Place agent\n      obs[self._state] = 1\n      return obs\n    elif self._observation_type is ObservationType.GRID:\n      obs = np.zeros(self._layout.shape + (3,), dtype=np.float32)\n      obs[..., 0] = self._layout < 0\n      obs[self._state[0], self._state[1], 1] = 1\n      obs[self._goal_state[0], self._goal_state[1], 2] = 1\n      return obs\n    elif self._observation_type is ObservationType.AGENT_GOAL_POS:\n      return np.array(self._state + self._goal_state, dtype=np.float32)\n    elif self._observation_type is ObservationType.STATE_INDEX:\n      y, x = self._state\n      return y * self._layout.shape[1] + x\n\n  def reset(self):\n    self._state = self._start_state\n    self._num_episode_steps = 0\n    if self._randomize_goals:\n      self.goal_state = self._sample_goal()\n    return dm_env.TimeStep(\n        step_type=dm_env.StepType.FIRST,\n        reward=None,\n        discount=None,\n        observation=self.get_obs())\n\n  def step(self, action):\n    y, x = self._state\n\n    if action == 0:  # up\n      new_state = (y - 1, x)\n    elif action == 1:  # right\n      new_state = (y, x + 1)\n    elif action == 2:  # down\n      new_state = (y + 1, x)\n    elif action == 3:  # left\n      new_state = (y, x - 1)\n    else:\n      raise ValueError(\n          'Invalid action: {} is not 0, 1, 2, or 3.'.format(action))\n\n    new_y, new_x = new_state\n    step_type = dm_env.StepType.MID\n    if self._layout[new_y, new_x] == -1:  # wall\n      reward = self._penalty_for_walls\n      discount = self._discount\n      new_state = (y, x)\n    elif self._layout[new_y, new_x] == 0:  # empty cell\n      reward = 0.\n      discount = self._discount\n    else:  # a goal\n      reward = self._layout[new_y, new_x]\n      discount = 0.\n      new_state = self._start_state\n      step_type = dm_env.StepType.LAST\n\n    self._state = new_state\n    self._num_episode_steps += 1\n    if (self._max_episode_length is not None and\n        self._num_episode_steps >= self._max_episode_length):\n      step_type = dm_env.StepType.LAST\n    return dm_env.TimeStep(\n        step_type=step_type,\n        reward=np.float32(reward),\n        discount=discount,\n        observation=self.get_obs())\n\n  def plot_grid(self, add_start=True):\n    plt.figure(figsize=(4, 4))\n    plt.imshow(self._layout <= -1, interpolation='nearest')\n    ax = plt.gca()\n    ax.grid(0)\n    plt.xticks([])\n    plt.yticks([])\n    # Add start/goal\n    if add_start:\n      plt.text(\n          self._start_state[1],\n          self._start_state[0],\n          r'$\\mathbf{S}$',\n          fontsize=16,\n          ha='center',\n          va='center')\n    plt.text(\n        self._goal_state[1],\n        self._goal_state[0],\n        r'$\\mathbf{G}$',\n        fontsize=16,\n        ha='center',\n        va='center')\n    h, w = self._layout.shape\n    for y in range(h - 1):\n      plt.plot([-0.5, w - 0.5], [y + 0.5, y + 0.5], '-w', lw=2)\n    for x in range(w - 1):\n      plt.plot([x + 0.5, x + 0.5], [-0.5, h - 0.5], '-w', lw=2)\n\n  def plot_state(self, return_rgb=False):\n    self.plot_grid(add_start=False)\n    # Add the agent location\n    plt.text(\n        self._state[1],\n        self._state[0],\n        u'\ud83d\ude03',\n        # fontname='symbola',\n        fontsize=18,\n        ha='center',\n        va='center',\n    )\n    if return_rgb:\n      fig = plt.gcf()\n      plt.axis('tight')\n      plt.subplots_adjust(0, 0, 1, 1, 0, 0)\n      fig.canvas.draw()\n      data = np.fromstring(fig.canvas.tostring_rgb(), dtype=np.uint8, sep='')\n      w, h = fig.canvas.get_width_height()\n      data = data.reshape((h, w, 3))\n      plt.close(fig)\n      return data\n\n  def plot_policy(self, policy):\n    action_names = [\n        r'$\\uparrow$', r'$\\rightarrow$', r'$\\downarrow$', r'$\\leftarrow$'\n    ]\n    self.plot_grid()\n    plt.title('Policy Visualization')\n    h, w = self._layout.shape\n    for y in range(h):\n      for x in range(w):\n        # if ((y, x) != self._start_state) and ((y, x) != self._goal_state):\n        if (y, x) != self._goal_state:\n          action_name = action_names[policy[y, x]]\n          plt.text(x, y, action_name, ha='center', va='center')\n\n  def plot_greedy_policy(self, q):\n    greedy_actions = np.argmax(q, axis=2)\n    self.plot_policy(greedy_actions)\n\n\ndef build_gridworld_task(task,\n                         discount=0.9,\n                         penalty_for_walls=-5,\n                         observation_type=ObservationType.STATE_INDEX,\n                         max_episode_length=200):\n  \"\"\"Construct a particular Gridworld layout with start/goal states.\n\n  Args:\n      task: string name of the task to use. One of {'simple', 'obstacle',\n        'random_goal'}.\n      discount: Discounting factor included in all Timesteps.\n      penalty_for_walls: Reward added when hitting a wall (should be negative).\n      observation_type: Enum observation type to use. One of:\n        * ObservationType.STATE_INDEX: int32 index of agent occupied tile.\n        * ObservationType.AGENT_ONEHOT: NxN float32 grid, with a 1 where the\n          agent is and 0 elsewhere.\n        * ObservationType.GRID: NxNx3 float32 grid of feature channels.\n          First channel contains walls (1 if wall, 0 otherwise), second the\n          agent position (1 if agent, 0 otherwise) and third goal position\n          (1 if goal, 0 otherwise)\n        * ObservationType.AGENT_GOAL_POS: float32 tuple with\n          (agent_y, agent_x, goal_y, goal_x).\n      max_episode_length: If set, will terminate an episode after this many\n        steps.\n  \"\"\"\n  tasks_specifications = {\n      'simple': {\n          'layout': [\n              [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1],\n              [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n              [-1, 0, 0, 0, -1, -1, 0, 0, 0, -1],\n              [-1, 0, 0, 0, -1, -1, 0, 0, 0, -1],\n              [-1, 0, 0, 0, -1, -1, 0, 0, 0, -1],\n              [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n              [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n              [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n              [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1],\n          ],\n          'start_state': (2, 2),\n          'goal_state': (7, 2)\n      },\n      'obstacle': {\n          'layout': [\n              [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1],\n              [-1, 0, 0, 0, 0, 0, -1, 0, 0, -1],\n              [-1, 0, 0, 0, -1, 0, 0, 0, 0, -1],\n              [-1, 0, 0, 0, -1, -1, 0, 0, 0, -1],\n              [-1, 0, 0, 0, -1, -1, 0, 0, 0, -1],\n              [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n              [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n              [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n              [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1],\n          ],\n          'start_state': (2, 2),\n          'goal_state': (2, 8)\n      },\n      'random_goal': {\n          'layout': [\n              [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1],\n              [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n              [-1, 0, 0, 0, -1, -1, 0, 0, 0, -1],\n              [-1, 0, 0, 0, -1, -1, 0, 0, 0, -1],\n              [-1, 0, 0, 0, -1, -1, 0, 0, 0, -1],\n              [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n              [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n              [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1],\n              [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1],\n          ],\n          'start_state': (2, 2),\n          # 'randomize_goals': True\n      },\n  }\n  return GridWorld(\n      discount=discount,\n      penalty_for_walls=penalty_for_walls,\n      observation_type=observation_type,\n      max_episode_length=max_episode_length,\n      **tasks_specifications[task])\n\n\ndef setup_environment(environment):\n  \"\"\"Returns the environment and its spec.\"\"\"\n\n  # Make sure the environment outputs single-precision floats.\n  environment = wrappers.SinglePrecisionWrapper(environment)\n\n  # Grab the spec of the environment.\n  environment_spec = specs.make_environment_spec(environment)\n\n  return environment, environment_spec\n```\n\n\nWe will use two distinct tabular GridWorlds:\n* `simple` where the goal is at the bottom left of the grid, little navigation required.\n* `obstacle` where the goal is behind an obstacle the agent must avoid.\n\nYou can visualize the grid worlds by running the cell below. \n\nNote that **S** indicates the start state and **G** indicates the goal. \n\n\n\n```\n# Visualise GridWorlds\n\n# Instantiate two tabular environments, a simple task, and one that involves\n# the avoidance of an obstacle.\nsimple_grid = build_gridworld_task(\n    task='simple', observation_type=ObservationType.GRID)\nobstacle_grid = build_gridworld_task(\n    task='obstacle', observation_type=ObservationType.GRID)\n\n# Plot them.\nsimple_grid.plot_grid()\nplt.title('Simple')\n\nobstacle_grid.plot_grid()\nplt.title('Obstacle')\n```\n\nIn this environment, the agent has four possible  **actions**: `up`, `right`, `down`, and `left`.  The **reward** is `-5` for bumping into a wall, `+10` for reaching the goal, and `0` otherwise. The episode ends when the agent reaches the goal, and otherwise continues. The **discount** on continuing steps, is $\\gamma = 0.9$. \n\nBefore we start building an agent to interact with this environment, let's first look at the types of objects the environment either returns (e.g., **observations**) or consumes (e.g., **actions**). The `environment_spec` will show you the form of the **observations**, **rewards** and **discounts** that the environment exposes and the form of the **actions** that can be taken.\n\n\n\n```\n# @title Look at environment_spec { form-width: \"30%\" }\n\n# Note: setup_environment is implemented in the same cell as GridWorld.\nenvironment, environment_spec = setup_environment(simple_grid)\n\nprint('actions:\\n', environment_spec.actions, '\\n')\nprint('observations:\\n', environment_spec.observations, '\\n')\nprint('rewards:\\n', environment_spec.rewards, '\\n')\nprint('discounts:\\n', environment_spec.discounts, '\\n')\n```\n\n    actions:\n     DiscreteArray(shape=(), dtype=int32, name=action, minimum=0, maximum=3, num_values=4) \n    \n    observations:\n     Array(shape=(9, 10, 3), dtype=dtype('float32'), name='observation_grid') \n    \n    rewards:\n     Array(shape=(), dtype=dtype('float32'), name='reward') \n    \n    discounts:\n     BoundedArray(shape=(), dtype=dtype('float32'), name='discount', minimum=0.0, maximum=1.0) \n    \n\n\n\nWe first set the environment to its initial state by calling the `reset()` method which returns the first observation and resets the agent to the starting location.\n\n\n\n```\nenvironment.reset()\nenvironment.plot_state()\n```\n\nNow we want to take an action to interact with the environment. We do this by passing a valid action to the `dm_env.Environment.step()` method which returns a `dm_env.TimeStep` namedtuple with fields `(step_type, reward, discount, observation)`.\n\nLet's take an action and visualise the resulting state of the grid-world. (You'll need to rerun the cell if you pick a new action.)\n\n**Note for kaggle users:** As kaggle does not render the forms automatically the students should be careful to notice the various instructions and manually play around with the values for the variables\n\n\n```\n# @title Pick an action and see the state changing\naction = \"right\" #@param [\"up\", \"right\", \"down\", \"left\"] {type:\"string\"}\n\naction_int = {'up': 0,\n              'right': 1,\n              'down': 2,\n              'left':3 }\naction = int(action_int[action])\ntimestep = environment.step(action)  # pytype: dm_env.TimeStep\nenvironment.plot_state()\n```\n\n\n```\n# @title Run loop  { form-width: \"30%\" }\n# @markdown This function runs an agent in the environment for a number of\n# @markdown episodes, allowing it to learn.\n\n# @markdown *Double-click* to inspect the `run_loop` function.\n\n\ndef run_loop(environment,\n             agent,\n             num_episodes=None,\n             num_steps=None,\n             logger_time_delta=1.,\n             label='training_loop',\n             log_loss=False,\n             ):\n  \"\"\"Perform the run loop.\n\n  We are following the Acme run loop.\n\n  Run the environment loop for `num_episodes` episodes. Each episode is itself\n  a loop which interacts first with the environment to get an observation and\n  then give that observation to the agent in order to retrieve an action. Upon\n  termination of an episode a new episode will be started. If the number of\n  episodes is not given then this will interact with the environment\n  infinitely.\n\n  Args:\n    environment: dm_env used to generate trajectories.\n    agent: acme.Actor for selecting actions in the run loop.\n    num_steps: number of steps to run the loop for. If `None` (default), runs\n      without limit.\n    num_episodes: number of episodes to run the loop for. If `None` (default),\n      runs without limit.\n    logger_time_delta: time interval (in seconds) between consecutive logging\n      steps.\n    label: optional label used at logging steps.\n  \"\"\"\n  logger = loggers.TerminalLogger(label=label, time_delta=logger_time_delta)\n  iterator = range(num_episodes) if num_episodes else itertools.count()\n  all_returns = []\n\n  num_total_steps = 0\n  for episode in iterator:\n    # Reset any counts and start the environment.\n    start_time = time.time()\n    episode_steps = 0\n    episode_return = 0\n    episode_loss = 0\n\n    timestep = environment.reset()\n\n    # Make the first observation.\n    agent.observe_first(timestep)\n\n    # Run an episode.\n    while not timestep.last():\n      # Generate an action from the agent's policy and step the environment.\n      action = agent.select_action(timestep.observation)\n      timestep = environment.step(action)\n\n      # Have the agent observe the timestep and let the agent update itself.\n      agent.observe(action, next_timestep=timestep)\n      agent.update()\n\n      # Book-keeping.\n      episode_steps += 1\n      num_total_steps += 1\n      episode_return += timestep.reward\n\n      if log_loss:\n        episode_loss += agent.last_loss\n\n      if num_steps is not None and num_total_steps >= num_steps:\n        break\n\n    # Collect the results and combine with counts.\n    steps_per_second = episode_steps / (time.time() - start_time)\n    result = {\n        'episode': episode,\n        'episode_length': episode_steps,\n        'episode_return': episode_return,\n    }\n    if log_loss:\n      result['loss_avg'] = episode_loss/episode_steps\n\n    all_returns.append(episode_return)\n\n    # Log the given results.\n    logger.write(result)\n\n    if num_steps is not None and num_total_steps >= num_steps:\n      break\n  return all_returns\n```\n\n\n```\n# @title Implement the evaluation loop { form-width: \"30%\" }\n# @markdown This function runs the agent in the environment for a number of\n# @markdown episodes, without allowing it to learn, in order to evaluate it.\n\n# @markdown *Double-click* to inspect the `evaluate` function.\n\ndef evaluate(environment: dm_env.Environment,\n             agent: acme.Actor,\n             evaluation_episodes: int):\n  frames = []\n\n  for episode in range(evaluation_episodes):\n    timestep = environment.reset()\n    episode_return = 0\n    steps = 0\n    while not timestep.last():\n      frames.append(environment.plot_state(return_rgb=True))\n\n      action = agent.select_action(timestep.observation)\n      timestep = environment.step(action)\n      steps += 1\n      episode_return += timestep.reward\n    print(\n        f'Episode {episode} ended with reward {episode_return} in {steps} steps'\n    )\n  return frames\n\ndef display_video(frames: Sequence[np.ndarray],\n                  filename: str = 'temp.mp4',\n                  frame_rate: int = 12):\n  \"\"\"Save and display video.\"\"\"\n  # Write the frames to a video.\n  with imageio.get_writer(filename, fps=frame_rate) as video:\n    for frame in frames:\n      video.append_data(frame)\n\n  # Read video and display the video.\n  video = open(filename, 'rb').read()\n  b64_video = base64.b64encode(video)\n  video_tag = ('
                                        \n\n\n\n\n```python\n# select single class for inference\nsclass = 1\ndf_cls = data_obj.data.loc[(data_obj.data['label'] == sclass)]\n```\n\n\n```python\n# drop non-data columns\ndf = df_cls.drop(columns=['label','scale','content (% w/w)'])\n```\n\n\n```python\nprint(\"total number of spectra      : {0}\".format(len(data_obj.data)))\nprint(\"total number of x-values     : {0}\".format(len(np.array(df.columns.to_list(), dtype='float32'))))\nprint(\"total number of classes      : {0}\".format(data_obj.classes))\nprint(\"selected class for inference : {0}\".format(sclass))\nprint(\"number of selected spectra   : {0}\".format(len(df)))\nprint(\"\\nclass distribution:\")\nprint(data_obj.data[data_obj.label_column].value_counts())\n```\n\n    total number of spectra      : 310\n    total number of x-values     : 404\n    total number of classes      : 4\n    selected class for inference : 1\n    number of selected spectra   : 70\n    \n    class distribution:\n    4    80\n    3    80\n    2    80\n    1    70\n    Name: label, dtype: int64\n\n\n\n```python\n# number of samples to run inference on\nnsamples = 15\n\n# set peak information for spectrum\npeaks = [7300.0, 7850.0, 8200.0, 8830.0, 9950.0]\n\n# baseline profile\nbase_shape = 'offset'\n\n# plot dataset (nsamples spectra, classes mixed)\nfig.plot_datasets_real(df, peaks, nsamples, savefig='yes', fname=file_basename)\n```\n\n\n```python\n# compare samples of different classes\ndf_lst = []\ndf_lst.append(df)\ndf_lst.append(data_obj.data.loc[(data_obj.data['label'] == 2)].drop(columns=['label','scale','content (% w/w)']))\ndf_lst.append(data_obj.data.loc[(data_obj.data['label'] == 3)].drop(columns=['label','scale','content (% w/w)']))\ndf_lst.append(data_obj.data.loc[(data_obj.data['label'] == 4)].drop(columns=['label','scale','content (% w/w)']))\n\n# plot dataset (nsamples spectra, classes mixed)\nfig.plot_datasets_real_compare(df_lst, nsamples, savefig='yes', fname=file_basename + \"_cmp\")\n```\n\n# Initialize models and run inference\n\n\n```python\nldata  = []\nlpeaks = []\n\n# add dataframes and peakinfo to list (for multiple classes or data per class)\nldata.append(df)\nlpeaks.append(peaks)\n```\n\n\n```python\n# convert pandas data to numpy arrays\nx_val = np.array(df.columns, dtype='float32')\n\n# store dataset y-values in list\ncols = ldata[0].columns\ny_val = [ldata[i][cols].values[:nsamples] for i in range(len(ldata))]\n```\n\n\n```python\n# initialize models and run inference\nmodels = []\ntraces = []\n\nfor i in range(len(ldata)):\n    if conf['peak_info'] == 'yes':\n        plist = np.array(lpeaks[0], dtype='float32').flatten()\n        plist.sort()\n        model_g = mdl.model_pvoigt(xvalues=x_val, observations=y_val[i], npeaks=len(peaks),\n                              mu_peaks=plist, pmodel=conf['prior_model'], baseline=base_shape)\n    else:\n        model_g = mdl.model_pvoigt(xvalues=x_val, observations=y_val[i], npeaks=len(peaks),\n                                              pmodel=conf['prior_model'], baseline=base_shape)\n                \n    models.append(model_g)\n\n    with model_g:\n        if conf['model_mode'] == 'train':\n            print(\"running inference on dataset #{0}/{1}\".format(i+1,len(ldata)))\n            trace_g = pm.sample(conf['nsamples'], init=conf['init_mode'], cores=conf['ncores'])\n            traces.append(trace_g)\n            # save inference results\n            pm.backends.text.dump(out_path + '/traces_%02d' % (i+1), trace_g)\n        else:\n            # load traces from disk\n            print(\"loading dataset #{0}/{1}\".format(i+1,len(ldata)))\n            trace_g = pm.backends.text.load(out_path + '/traces_%02d' % (i+1))\n            traces.append(trace_g)\n```\n\n    loading dataset #1/1\n\n\n# Model visualization\n\n\n```python\n# display first model\npm.model_to_graphviz(models[0])\n```\n\n\n\n\n    \n\n    \n\n\n\n\n```python\n# save model figure as image\nimg = pm.model_to_graphviz(models[0])\nimg.render(filename=file_basename + '_model', format='png');\n```\n\n# Collect results and save\n\n\n```python\n# posterior predictive traces\nppc = [pm.sample_posterior_predictive(traces[i], samples=500, model=models[i]) for i in range(len(traces))]\n```\n\n    /home/johan/VirtualEnv/ppsda/lib/python3.6/site-packages/pymc3/sampling.py:1247: UserWarning: samples parameter is smaller than nchains times ndraws, some draws and/or chains may not be represented in the returned posterior predictive sample\n      \"samples parameter is smaller than nchains times ndraws, some draws \"\n    100%|\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588| 500/500 [00:34<00:00, 14.44it/s]\n\n\n\n```python\n# various plots to inspect the inference results\nvarnames = mdl.get_varnames(traces[0])\n#az.summary(traces[1], varnames)\n```\n\n\n```python\nif conf['model_mode'] == 'train':\n    # collect the results and display\n    df = res.get_results_summary(traces, ppc, y_val, varnames)\nelse:\n    # load results from disk\n    df = pd.read_csv(file_basename + '.csv')\n    df.index += 1\ndf\n```\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n      \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n      \n      \n      \n      \n    \n  \n
                                        r_hatmcseessbfmir2waicepsilon
                                        11.45709715.67241919.9677420.4858960.987058-16487.9374390.062277
                                        \n
                                        \n\n\n\n\n```python\nif conf['model_mode'] == 'train':\n    # save results to .csv\n    df.to_csv(file_basename + '.csv', index=False)\n```\n\n\n```python\ncnf.close(out_path)\n```\n\n# Plot posterior (single dataset)\n\n\n```python\n# dataset to plot\nn_dataset = 0\n\n# trace number in matrix\nn_trace = 0\n\n#fig.plot_posterior_single(x_val, ldata[n_dataset], ppc[n_trace], 10, \n#                            file_basename, hpd_color='yellow', peak_pos=peaks)\nfig.plot_posterior_single(x_val, ldata[n_dataset], ppc[n_trace], 10, \n                             file_basename, hpd_color='orange', peak_pos=peaks, no_data='yes')\n```\n\n\n```python\naz.plot_posterior(traces[n_trace], ['mu', 'amp', 'sigma', 'epsilon'], credible_interval=0.95);\n```\n\n\n```python\naz.plot_posterior(traces[n_trace], ['a0'], credible_interval=0.95);\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "41fbde5f655e451c785a6f9060b73f56d86ab826", "size": 1018319, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "code/scenarios/scenario_f/scenario_data_real_tablets.ipynb", "max_stars_repo_name": "jnispen/PPSDA", "max_stars_repo_head_hexsha": "910261551dd08768a72ab0a3e81bd73c706a143a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-01-07T02:22:25.000Z", "max_stars_repo_stars_event_max_datetime": "2021-01-07T02:22:25.000Z", "max_issues_repo_path": "code/scenarios/scenario_f/scenario_data_real_tablets.ipynb", "max_issues_repo_name": "jnispen/PPSDA", "max_issues_repo_head_hexsha": "910261551dd08768a72ab0a3e81bd73c706a143a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "code/scenarios/scenario_f/scenario_data_real_tablets.ipynb", "max_forks_repo_name": "jnispen/PPSDA", "max_forks_repo_head_hexsha": "910261551dd08768a72ab0a3e81bd73c706a143a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 1023.4361809045, "max_line_length": 281348, "alphanum_fraction": 0.9516674048, "converted": true, "num_tokens": 3977, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.689305616785446, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.4137132071673004}}
{"text": "
                                         UNIVERSIDAD NACIONAL DE INGENIERIA
                                         \n
                                         Inteligencia Artificial
                                         \n
                                         Proyecto Final Parte 01 - Extracci\u00f3n de caracter\u00edsticas 
                                         \n
                                          Cristian Lazo : mecatronico.lazo@gmail.com
                                         \n
                                         Frank Solis : so_ni_mf@hotmail.com   
                                         \n
                                         Diego Quispe  : dquispec@uni.pe   
                                         \n
                                         Docente    : Paul Antonio Cardenas Lizana 
                                         \n\n\n\n--------------------------------\n\n**Resumen  :**  Los usuarios de una l\u00ednea telef\u00f3nica son propensos a portar a otro operador, por ende es necesario saber previamente si esto va a suceder para que la empresa pueda emplear medidas para que esto no suceda. El proyecto presente tiene dos objetivos principales: la clasificaci\u00f3n usuarios que portan o no y la interpretaci\u00f3n del modelo para identificar las razones especificas de la portabilidad de los clientes.\n\n\n\n```\n\n```\n\n--------------------------\n\n \n\n# 1. INDICE\n\n\n* [1. INDICE](#1)\n* [2. DATASET](#2)\n* [3. M\u00c9TRICA](#3)\n* [4. IMPORTANDO LIBRERIAS](#4)\n* [5. CARGAR DATA](#5)\n* [6. DATA EXPLORATION and FEATURES CREATION](#6)\n* [7. PROBANDO PCA PARA REDUCIR LOS FEATURES](#7)\n* [8. USANDO LOS FEATURES HALLADOS EN EL ARCHIVO PROYECTO_FINAL_02](#8)\n* [9. CONTINUAR ARCHIVO PROYECTO_FINAL_02](#9)\n\n\n\n\n \n\n# 2. DATASET\n\nConsta de 6 archivos\n\n- TRAIN_TARGET : 2 variables\n- TRAIN_FUENTE_PLANTAPOST : 10 variables \n- TRAIN_FUENTE_EQUIPOS : 18 variables\n- TRAIN_FUENTE_DATOS : 10 variables\n- TRAIN_FUENTE_VOZ : 29 variables\n- TRAIN_FUENTE_APLICACIONES : 14 variables\n- TRAIN_FUENTE_CONSULTA_PREV_PORTA : 3 variables\n\n\n \n\n# 3. METRICA\n\nLa m\u00e9trica de evaluaci\u00f3n usaremos es el AUC, para entenderlo primero haremos unas definiciones:\n\n\n### CURVA ROC\n\nUna **curva ROC (curva de caracter\u00edstica operativa del recepto)** es un gr\u00e1fico que muestra el rendimiento de un modelo de clasificaci\u00f3n en todos los umbrales de clasificaci\u00f3n. Esta curva representa dos par\u00e1metros:\n- Tasa de verdaderos positivos\n- Tasas de falsos positivos\n\n**Tasa de verdaderos positivos (TPR)** es sin\u00f3nimo de exhaustividad y, por lo tanto, se define de la siguiente manera:\n\n\\begin{equation}\nTPR = \\frac{VP}{VP+FN}\n\\end{equation}\n\n\n**Tasa de falsos positivos (FPR)**  se define de la siguiente manera:\n\n\\begin{equation}\nFPR = \\frac{FP}{FP+VN}\n\\end{equation}\n\n\nUna curva ROC representa TPR frente a FPR en diferentes umbrales de clasificaci\u00f3n. Reducir el umbral de clasificaci\u00f3n clasifica m\u00e1s elementos como positivos, por lo que aumentar\u00e1n tanto los falsos positivos como los verdaderos positivos. En la siguiente figura, se muestra una curva ROC t\u00edpica.\nPara calcular los puntos en una curva ROC, podr\u00edamos evaluar un modelo de regresi\u00f3n log\u00edstica muchas veces con diferentes umbrales de clasificaci\u00f3n, pero esto es ineficiente. Afortunadamente, existe un algoritmo eficiente basado en ordenamiento que puede brindarnos esta informaci\u00f3n, denominado AUC.\n\n\n```\nfrom PIL import Image\nImage.open('/home/liiarpi/Downloads/DATA/MOVISTAR/PAPER/imagenes/roc.png')\n```\n\n### AUC : \u00c1rea bajo la curva ROC \n\n**AUC** significa \"\u00e1rea bajo la curva ROC\". Esto significa que el AUC mide toda el \u00e1rea bidimensional por debajo de la curva ROC completa (piensa en un c\u00e1lculo integral) de (0,0) a (1,1).\n\n\n\n```\nImage.open('/home/liiarpi/Downloads/DATA/MOVISTAR/PAPER/imagenes/auc.png')\n```\n\nEl AUC proporciona una medici\u00f3n agregada del rendimiento en todos los umbrales de clasificaci\u00f3n posibles. Una forma de interpretar el AUC es como la probabilidad de que el modelo clasifique un ejemplo positivo aleatorio m\u00e1s alto que un ejemplo negativo aleatorio. Observa, a modo de ilustraci\u00f3n, los siguientes ejemplos, que est\u00e1n ordenados de izquierda a derecha en orden ascendente con respecto a las predicciones de regresi\u00f3n log\u00edstica:\n\nEl AUC representa la probabilidad de que un ejemplo aleatorio positivo (verde) se posicione a la derecha de un ejemplo aleatorio negativo (rojo).\n\nEl AUC oscila en valor del 0 al 1. Un modelo cuyas predicciones son un 100% incorrectas tiene un AUC de 0.0; otro cuyas predicciones son un 100% correctas tiene un AUC de 1.0.\n\n\n```\nImage.open('/home/liiarpi/Downloads/DATA/MOVISTAR/PAPER/imagenes/cuadri.png')\n\n```\n\n**Razones por la cual usamos AUC como m\u00e9trica :**\n\n- El AUC es **invariable con respecto a la escala.** Mide qu\u00e9 tan bien se clasifican las predicciones, en lugar de sus valores absolutos.\n- El AUC es **invariable con respecto al umbral de clasificaci\u00f3n.** Mide la calidad de las predicciones del modelo, sin tener en cuenta qu\u00e9 umbral de clasificaci\u00f3n se elige.\n\n\n \n\n# 4. IMPORTAR LIBRERIAS\n\n\n```\nfrom librerias3 import *\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\nimport seaborn as sns\nimport pandas as pd\nimport numpy as np\nimport matplotlib.pyplot as pltd\nimport seaborn as sns\nimport gc\nfrom sklearn.model_selection import train_test_split\nimport lightgbm as lgb\nimport warnings\nwarnings.filterwarnings('ignore')\n```\n\n \n\n# 5. CARGAR DATA\n\n\n```\ntrain_path='/home/liiarpi/Downloads/DATA/MOVISTAR/DATA/TRAIN/'\ntarget=pd.read_csv(train_path+'TRAIN_TARGET.txt',sep='|')\ntfa   =pd.read_csv(train_path+'TRAIN_FUENTE_APLICACIONES.txt',sep='|',parse_dates=['PERIODO'])\ntfc   =pd.read_csv(train_path+'TRAIN_FUENTE_CONSULTA_PREV_PORTA.txt',sep='|',parse_dates=['PERIODO'])\ntfd   =pd.read_csv(train_path+'TRAIN_FUENTE_DATOS.txt',sep='|',parse_dates=['PERIODO'])\ntfe   =pd.read_csv(train_path+'TRAIN_FUENTE_EQUIPOS.txt',sep='|',parse_dates=['PERIODO'])\ntfv   =pd.read_csv(train_path+'TRAIN_FUENTE_VOZ.txt',sep='|',parse_dates=['PERIODO'])\ntpp   =pd.read_csv(train_path+'TRAIN_PLANTAPOST.txt',sep='|',parse_dates=['PERIODO'])\n```\n\n\n```\ntarget.head()\n```\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n    \n      \n      \n      \n    \n  \n
                                        ID_CLIENTETARGET
                                        0M4+Yb9VBZlrmolusfk43+6zj9IEiONcbxv8Vo4mDskU=0
                                        17CZN3fc0TkgCt99e2wXhHws0ZdjbidJ13wHbBCWo93s=0
                                        2Gato2h7aHbSHatI1jDDmHEgQsm+L0kVqt59jfZLp6vQ=0
                                        3a7ydviDHC4D3hefkfTJF9UP3sIJ55tM+Y7OW40bIlKc=0
                                        4HY8WUIZd4Bo8uLtyTtFowF+JM3UbqcHEyUaZvOkmp/w=0
                                        \n
                                        \n\n\n\n\n```\ntfc.head()\n```\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n  \n
                                        PERIODOID_CLIENTECANT_CONSULTAS_PREV
                                        02018-12-01Q0Sqxt9/+fI5C96Oj0qJZ92xheja1/7MUmzwG9BEJvU=1
                                        12018-11-01yaomlj6mDsksD3XNE1++B9kjjPmR5IbU/YeZB6/xIt8=1
                                        22018-11-014IMVMkE/j892RjRR74zo+H1/vzj+TzyvDsvDqnMoSd8=1
                                        32018-10-017tzg/k08GC863JggrB3E2X2aJUqV0KbaRIDqZnDezUs=1
                                        42018-10-01Wg7CVsL20twXVpeGqtjkCXsJYJC+fIIJyPvdmphPQeE=1
                                        \n
                                        \n\n\n\n\n```\nnames=tfa.columns.values \nfound_index = np.in1d(names, ['ID_CLIENTE','PERIODO']).nonzero()[0]\nnames = np.delete(names, found_index)+'_mean'\nnames\n```\n\n\n\n\n    array(['N_DIAS_NAVEGACION_mean', 'APP_1_mean', 'APP_2_mean', 'APP_3_mean',\n           'APP_4_mean', 'APP_5_mean', 'APP_6_mean', 'APP_7_mean',\n           'APP_8_mean', 'APP_9_mean', 'APP_10_mean', 'MB_TOTAL_APP_mean'],\n          dtype=object)\n\n\n\n \n\n# 6. Data Exploration and Feature Creation\n\n\n```\ndef procesar_2(names,df):\n    x= df[names]\n    names=x.columns.values\n    for i in range(0,x.shape[1]):\n        namecolumn=names[i]\n        x['alta_'+namecolumn]=(x[namecolumn]>(x[namecolumn].mean()+x[namecolumn].std()))\n        x['media_'+namecolumn]=(x[namecolumn]<=(x[namecolumn].mean()+x[namecolumn].std()))\n        x['media_'+namecolumn]=x['media_'+namecolumn]&(x[namecolumn]>=(x[namecolumn].mean()-x[namecolumn].std()))\n        x['baja_'+namecolumn]=(x[namecolumn]<=(x[namecolumn].mean()-x[namecolumn].std()))\n        x=x.drop([namecolumn],axis=1)\n    x=x.astype('float')\n    return x\n```\n\n\n```\ndef agg_numeric(df, group_var):\n    # Remove id variables other than grouping variable\n    for col in df:\n        if col != group_var and 'ID_CLIENTE' in col:\n            df = df.drop(columns = col)\n            \n    group_ids = df[group_var]\n    numeric_df = df.select_dtypes('number')\n    numeric_df[group_var] = group_ids\n\n    # Group by the specified variable and calculate the statistics\n    agg = numeric_df.groupby(group_var).agg(['count', 'mean', 'max', 'min', 'sum']).reset_index()\n\n    # Need to create new column names\n    columns = [group_var]\n\n    # Iterate through the variables names\n    for var in agg.columns.levels[0]:\n        # Skip the grouping variable\n        if var != group_var:\n            # Iterate through the stat names\n            for stat in agg.columns.levels[1][:-1]:\n                # Make a new column name for the variable and stat\n                columns.append('%s_%s' % (var, stat))\n\n    agg.columns = columns\n    print('NUEVOS FEATURES CREADOS')\n    return agg\ndef agg_numeric2(df, group_var):\n    # Remove id variables other than grouping variable\n    for col in df:\n        if col != group_var and 'ID_CLIENTE' in col:\n            df = df.drop(columns = col)\n            \n    group_ids = df[group_var]\n    numeric_df = df.select_dtypes('number')\n    numeric_df[group_var] = group_ids\n\n    # Group by the specified variable and calculate the statistics\n    agg = numeric_df.groupby(group_var).agg(['mean']).reset_index()\n\n    # Need to create new column names\n    columns = [group_var]\n\n    # Iterate through the variables names\n    for var in agg.columns.levels[0]:\n        # Skip the grouping variable\n        if var != group_var:\n            # Iterate through the stat names\n            for stat in agg.columns.levels[1][:-1]:\n                # Make a new column name for the variable and stat\n                columns.append('%s_%s' % (var, stat))\n    agg.columns = columns\n    print('NUEVOS FEATURES CREADOS')\n    return agg\n```\n\n\n```\ndef procesar_1(path,name):\n    df=pd.read_csv(path+name,sep='|', parse_dates=['PERIODO'])\n    print(df.shape)\n    #names=df.columns.values \n    #found_index = np.in1d(names, ['ID_CLIENTE','PERIODO']).nonzero()[0]\n    #names = np.delete(names, found_index)+'_mean'\n    #names2=df.columns.values\n    #found_index2 = np.in1d(names2, ['PERIODO']).nonzero()[0]\n    #names2 = np.delete(names2, found_index2)\n    \n    #df_date = agg_numeric2(df[['ID_CLIENTE','PERIODO']], group_var = 'ID_CLIENTE')\n    \n    date_attr = ['Year', 'Month', 'Week', 'Day', 'Dayofweek', 'Dayofyear', 'Is_month_end',\n    'Is_month_start', 'Is_quarter_end', 'Is_quarter_start', 'Is_year_end', 'Is_year_start']\n    for n in date_attr:\n        a = getattr(df['PERIODO'].dt, n.lower())\n        #a[a==np.NaN]=0\n        a[(1-np.isnan(a))>0.0001]=0\n        a[a==np.Inf]=0\n        a[a==-np.Inf]=0\n        df['PERIODO_' + n]=a.astype(float)\n        # 2. Agregamos una columan con una representaci\u00f3n num\u00e9rica de la fecha\n        df['PERIODO_elapsed'] = df['PERIODO'].astype(np.int64) // 10 ** 9\n\n    # 3. Eliminamos la variable date\n    df.drop('PERIODO', axis=1, inplace=True)\n    \n    names3=df.columns.values\n    found_index3 = np.in1d(names3, ['ID_CLIENTE']).nonzero()[0]\n    names3 = np.delete(names3, found_index3)+'_mean'\n    \n    #VARIABLES OBJECT FLOAT\n    for n,col in df.items():\n        if not pd.api.types.is_numeric_dtype(col) and n != 'ID_CLIENTE': \n            df[n] = col.astype('category')\n    \n    for n,col in df.items()  : \n        if pd.api.types.is_categorical_dtype(col) and n != 'ID_CLIENTE':\n            df[n] = col.cat.codes+1 \n    \n    names = df.columns.values \n    aa=df.isna().sum() / len(df) # % de data faltante\n    for i in range (0,aa.shape[0]):\n        if(aa[i]>0):\n            print('DATA FALTANTE -',names[i])\n            df[str(names[i])]=df[str(names[i])].fillna(df[str(names[i])].mean())\n   \n    df = agg_numeric(df, group_var = 'ID_CLIENTE')\n    \n    #df = agg_numeric(df[names2], group_var = 'ID_CLIENTE')\n    #df = pd.merge(df,df_date, on='ID_CLIENTE', how='left')\n    #print(df.columns.values)\n    #print(names3)\n    df2 =procesar_2(names3,df)\n  \n    df = pd.concat([df,df2], axis=1)\n    \n    print(df.shape)\n    \n    return df\n```\n\n\n```\na=[2,3,np.NaN,np.Inf]\na[a==np.NaN]=0\na\n\n```\n\n\n\n\n    [0, 3, nan, inf]\n\n\n\n\n```\ntrain_path='/home/liiarpi/Downloads/DATA/MOVISTAR/DATA/TRAIN/'\ndf=procesar_1(train_path,'TRAIN_FUENTE_CONSULTA_PREV_PORTA.txt')\n\n```\n\n    (9249, 3)\n    NUEVOS FEATURES CREADOS\n    (8524, 113)\n\n\n\n```\ntrain_path='/home/liiarpi/Downloads/DATA/MOVISTAR/DATA/TRAIN/'\ndf=procesar_1(train_path,'TRAIN_FUENTE_VOZ.txt')\n```\n\n    (587474, 29)\n    DATA FALTANTE - AIRTIME_IN_TOT\n    DATA FALTANTE - AIRTIME_IN_ON\n    DATA FALTANTE - AIRTIME_IN_OFF\n    DATA FALTANTE - CALLS_IN_TOT\n    DATA FALTANTE - CALLS_IN_ON\n    DATA FALTANTE - CALLS_IN_OFF\n    DATA FALTANTE - DAYS_IN_VOICE_TOT\n    DATA FALTANTE - DAYS_IN_VOICE_ON\n    DATA FALTANTE - DAYS_IN_VOICE_OFF\n    DATA FALTANTE - AIRTIME_OUT_TOT\n    DATA FALTANTE - AIRTIME_OUT_ON\n    DATA FALTANTE - AIRTIME_OUT_OFF\n    DATA FALTANTE - CALLS_OUT_TOT\n    DATA FALTANTE - CALLS_OUT_ON\n    DATA FALTANTE - CALLS_OUT_OFF\n    DATA FALTANTE - DEST_VOICE\n    DATA FALTANTE - DEST_VOICE_ON\n    DATA FALTANTE - DEST_VOICE_OFF\n    DATA FALTANTE - DAYS_OUT_VOICE_TOT\n    DATA FALTANTE - DAYS_OUT_VOICE_ON\n    DATA FALTANTE - DAYS_OUT_VOICE_OFF\n    DATA FALTANTE - CONT_TOT\n    DATA FALTANTE - CONT_ON\n    DATA FALTANTE - CONT_OFF\n    DATA FALTANTE - TOP_CONT_5\n    DATA FALTANTE - TOP_CONT_ON_5\n    DATA FALTANTE - TOP_CONT_OFF_5\n    DATA FALTANTE - PERIODO_Year\n    DATA FALTANTE - PERIODO_Month\n    DATA FALTANTE - PERIODO_Week\n    DATA FALTANTE - PERIODO_Day\n    DATA FALTANTE - PERIODO_Dayofweek\n    DATA FALTANTE - PERIODO_Dayofyear\n    NUEVOS FEATURES CREADOS\n    (210000, 321)\n\n\n\n```\npath='/home/liiarpi/Downloads/DATA/MOVISTAR'\npathtrain=path+'/DATA/TRAIN/'\npathtest =path+'/DATA/TEST/'\npathguardartrain=path+'/NEW_DATA3/NEW_TRAIN/'\npathguardartest =path+'/NEW_DATA3/NEW_TEST/'\n\ndef transformar(pathorigen,path,name):\n    df=procesar_1(pathorigen,name+'.txt')\n    names = df.columns.values\n    df.to_csv(path+name+'.csv',index=None)\n    \n    \n```\n\n\n```\ntransformar(pathtrain,pathguardartrain,'TRAIN_FUENTE_APLICACIONES')\ntransformar(pathtrain,pathguardartrain,'TRAIN_FUENTE_CONSULTA_PREV_PORTA')\ntransformar(pathtrain,pathguardartrain,'TRAIN_FUENTE_DATOS')\ntransformar(pathtrain,pathguardartrain,'TRAIN_FUENTE_EQUIPOS')\ntransformar(pathtrain,pathguardartrain,'TRAIN_FUENTE_VOZ')\ntransformar(pathtrain,pathguardartrain,'TRAIN_PLANTAPOST')\n```\n\n    (513548, 14)\n    NUEVOS FEATURES CREADOS\n    (180063, 201)\n    (9249, 3)\n    NUEVOS FEATURES CREADOS\n    (8524, 113)\n    (629997, 10)\n    DATA FALTANTE - INDICADOR_DATOS_1\n    DATA FALTANTE - INDICADOR_DATOS_2\n    NUEVOS FEATURES CREADOS\n    (209999, 169)\n    (189225, 18)\n    NUEVOS FEATURES CREADOS\n    (189225, 233)\n    (587474, 29)\n    DATA FALTANTE - AIRTIME_IN_TOT\n    DATA FALTANTE - AIRTIME_IN_ON\n    DATA FALTANTE - AIRTIME_IN_OFF\n    DATA FALTANTE - CALLS_IN_TOT\n    DATA FALTANTE - CALLS_IN_ON\n    DATA FALTANTE - CALLS_IN_OFF\n    DATA FALTANTE - DAYS_IN_VOICE_TOT\n    DATA FALTANTE - DAYS_IN_VOICE_ON\n    DATA FALTANTE - DAYS_IN_VOICE_OFF\n    DATA FALTANTE - AIRTIME_OUT_TOT\n    DATA FALTANTE - AIRTIME_OUT_ON\n    DATA FALTANTE - AIRTIME_OUT_OFF\n    DATA FALTANTE - CALLS_OUT_TOT\n    DATA FALTANTE - CALLS_OUT_ON\n    DATA FALTANTE - CALLS_OUT_OFF\n    DATA FALTANTE - DEST_VOICE\n    DATA FALTANTE - DEST_VOICE_ON\n    DATA FALTANTE - DEST_VOICE_OFF\n    DATA FALTANTE - DAYS_OUT_VOICE_TOT\n    DATA FALTANTE - DAYS_OUT_VOICE_ON\n    DATA FALTANTE - DAYS_OUT_VOICE_OFF\n    DATA FALTANTE - CONT_TOT\n    DATA FALTANTE - CONT_ON\n    DATA FALTANTE - CONT_OFF\n    DATA FALTANTE - TOP_CONT_5\n    DATA FALTANTE - TOP_CONT_ON_5\n    DATA FALTANTE - TOP_CONT_OFF_5\n    DATA FALTANTE - PERIODO_Year\n    DATA FALTANTE - PERIODO_Month\n    DATA FALTANTE - PERIODO_Week\n    DATA FALTANTE - PERIODO_Day\n    DATA FALTANTE - PERIODO_Dayofweek\n    DATA FALTANTE - PERIODO_Dayofyear\n    NUEVOS FEATURES CREADOS\n    (210000, 321)\n    (630000, 10)\n    DATA FALTANTE - VALORRENTAMENSUAL\n    DATA FALTANTE - ANTIGUEDAD_POSTPAGO\n    DATA FALTANTE - SUB_TOTAL_FACT\n    DATA FALTANTE - EDAD\n    DATA FALTANTE - DIAS_QUEDANPERMANENCIA\n    NUEVOS FEATURES CREADOS\n    (210000, 169)\n\n\n\n```\ntransformar(pathtest,pathguardartest,'TEST_FUENTE_APLICACIONES')\ntransformar(pathtest,pathguardartest,'TEST_FUENTE_CONSULTA_PREV_PORTA')\ntransformar(pathtest,pathguardartest,'TEST_FUENTE_DATOS')\ntransformar(pathtest,pathguardartest,'TEST_FUENTE_EQUIPOS')\ntransformar(pathtest,pathguardartest,'TEST_FUENTE_VOZ')\ntransformar(pathtest,pathguardartest,'TEST_PLANTAPOST')\n```\n\n    (171983, 14)\n    NUEVOS FEATURES CREADOS\n    (60309, 201)\n    (3583, 3)\n    NUEVOS FEATURES CREADOS\n    (3305, 113)\n    (210000, 10)\n    DATA FALTANTE - INDICADOR_DATOS_1\n    DATA FALTANTE - INDICADOR_DATOS_2\n    NUEVOS FEATURES CREADOS\n    (70000, 169)\n    (62969, 18)\n    NUEVOS FEATURES CREADOS\n    (62969, 233)\n    (194139, 29)\n    DATA FALTANTE - AIRTIME_IN_TOT\n    DATA FALTANTE - AIRTIME_IN_ON\n    DATA FALTANTE - AIRTIME_IN_OFF\n    DATA FALTANTE - CALLS_IN_TOT\n    DATA FALTANTE - CALLS_IN_ON\n    DATA FALTANTE - CALLS_IN_OFF\n    DATA FALTANTE - DAYS_IN_VOICE_TOT\n    DATA FALTANTE - DAYS_IN_VOICE_ON\n    DATA FALTANTE - DAYS_IN_VOICE_OFF\n    DATA FALTANTE - AIRTIME_OUT_TOT\n    DATA FALTANTE - AIRTIME_OUT_ON\n    DATA FALTANTE - AIRTIME_OUT_OFF\n    DATA FALTANTE - CALLS_OUT_TOT\n    DATA FALTANTE - CALLS_OUT_ON\n    DATA FALTANTE - CALLS_OUT_OFF\n    DATA FALTANTE - DEST_VOICE\n    DATA FALTANTE - DEST_VOICE_ON\n    DATA FALTANTE - DEST_VOICE_OFF\n    DATA FALTANTE - DAYS_OUT_VOICE_TOT\n    DATA FALTANTE - DAYS_OUT_VOICE_ON\n    DATA FALTANTE - DAYS_OUT_VOICE_OFF\n    DATA FALTANTE - CONT_TOT\n    DATA FALTANTE - CONT_ON\n    DATA FALTANTE - CONT_OFF\n    DATA FALTANTE - TOP_CONT_5\n    DATA FALTANTE - TOP_CONT_ON_5\n    DATA FALTANTE - TOP_CONT_OFF_5\n    DATA FALTANTE - PERIODO_Year\n    DATA FALTANTE - PERIODO_Month\n    DATA FALTANTE - PERIODO_Week\n    DATA FALTANTE - PERIODO_Day\n    DATA FALTANTE - PERIODO_Dayofweek\n    DATA FALTANTE - PERIODO_Dayofyear\n    NUEVOS FEATURES CREADOS\n    (70000, 321)\n    (210000, 10)\n    DATA FALTANTE - VALORRENTAMENSUAL\n    DATA FALTANTE - ANTIGUEDAD_POSTPAGO\n    DATA FALTANTE - SUB_TOTAL_FACT\n    DATA FALTANTE - EDAD\n    DATA FALTANTE - DIAS_QUEDANPERMANENCIA\n    NUEVOS FEATURES CREADOS\n    (70000, 169)\n\n\n\n```\n\n```\n\n\n```\ntrain_path='/home/liiarpi/Downloads/DATA/MOVISTAR/NEW_DATA3/NEW_TRAIN/'\ntarget=pd.read_csv('/home/liiarpi/Downloads/DATA/MOVISTAR/DATA/TRAIN/TRAIN_TARGET.txt',sep='|')\ntfa=pd.read_csv(train_path+'TRAIN_FUENTE_APLICACIONES.csv')\ntfc=pd.read_csv(train_path+'TRAIN_FUENTE_CONSULTA_PREV_PORTA.csv')\ntfd=pd.read_csv(train_path+'TRAIN_FUENTE_DATOS.csv')\ntfe=pd.read_csv(train_path+'TRAIN_FUENTE_EQUIPOS.csv')\ntfv=pd.read_csv(train_path+'TRAIN_FUENTE_VOZ.csv')\ntpp=pd.read_csv(train_path+'TRAIN_PLANTAPOST.csv')\n```\n\n\n```\nTrainData = pd.merge(target, tfa, how='left')\nTrainData = pd.merge(TrainData, tfc, on='ID_CLIENTE', how='left')\nTrainData = pd.merge(TrainData, tfd, on='ID_CLIENTE', how='left')\nTrainData = pd.merge(TrainData, tfe, on='ID_CLIENTE', how='left')\nTrainData = pd.merge(TrainData, tfv, on='ID_CLIENTE', how='left')\nTrainData = pd.merge(TrainData, tpp, on='ID_CLIENTE', how='left')\n\n```\n\n\n```\ntrain=pd.read_csv('Data_tratada_Train.csv',na_values=['NaN','nan',-1])\n```\n\n\n```\nprint('tama\u00f1o = ', train.shape)\n# Presentamos un ejemplo (de 5 datos) del dataset\ntrain.head()\n```\n\n    tama\u00f1o =  (210000, 578)\n\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n  \n
                                        ID_CLIENTETARGETN_DIAS_NAVEGACION_countN_DIAS_NAVEGACION_meanN_DIAS_NAVEGACION_maxN_DIAS_NAVEGACION_minN_DIAS_NAVEGACION_sumAPP_1_countAPP_1_meanAPP_1_max...baja_SUB_TOTAL_FACT_meanalta_EDAD_meanmedia_EDAD_meanbaja_EDAD_meanalta_DIAS_QUEDANPERMANENCIA_meanmedia_DIAS_QUEDANPERMANENCIA_meanbaja_DIAS_QUEDANPERMANENCIA_meanalta_DEPARTAMENTO_meanmedia_DEPARTAMENTO_meanbaja_DEPARTAMENTO_mean
                                        0M4+Yb9VBZlrmolusfk43+6zj9IEiONcbxv8Vo4mDskU=02.022.50000024.021.045.02.01720.4600001920.41...0.00.01.00.00.01.00.00.01.00.0
                                        17CZN3fc0TkgCt99e2wXhHws0ZdjbidJ13wHbBCWo93s=03.029.66666731.028.089.03.0372.153333517.42...0.00.01.00.00.01.00.00.01.00.0
                                        2Gato2h7aHbSHatI1jDDmHEgQsm+L0kVqt59jfZLp6vQ=03.030.00000031.028.090.03.0257.596667427.18...0.01.00.00.00.00.01.00.01.00.0
                                        3a7ydviDHC4D3hefkfTJF9UP3sIJ55tM+Y7OW40bIlKc=03.028.00000031.026.084.03.0781.6000001042.77...0.00.01.00.00.00.01.01.00.00.0
                                        4HY8WUIZd4Bo8uLtyTtFowF+JM3UbqcHEyUaZvOkmp/w=03.029.00000031.027.087.03.0933.2833331230.82...0.00.01.00.00.01.00.00.00.01.0
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                                        5 rows \u00d7 578 columns
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                                        \n\n\n\n\n```\nX=train.drop(columns=['ID_CLIENTE', 'TARGET'])\ny=train['TARGET']\nX_train,X_test,y_train,y_test = train_test_split(X,y,test_size=0.2,random_state=7)\n```\n\n\n```\n#X_train,X_test,y_train,y_test = train_test_split(X,y,test_size=0.2,random_state=7)\ntrain_data=lgb.Dataset(X_train, y_train)\ntest_data=lgb.Dataset(X_test, y_test, reference=train_data)\n```\n\n\n```\nparams = {\n    'task': 'train',\n    'boosting_type': 'gbdt',\n    'objective': 'binary',\n    'metric': {'auc'},\n    'num_leaves': 20,\n    'num_boost_round': 5000,\n    'min_data_in_leaf': 50,\n    'max_depth': -1,\n    'verbose': -1,\n    'n_jobs': -1,\n    'learning_rate': 0.035\n}\n```\n\n\n```\nmodel = lgb.train(params,train_data,valid_sets=test_data,early_stopping_rounds=80)\n```\n\n    [1]\tvalid_0's auc: 0.618285\n    Training until validation scores don't improve for 80 rounds.\n    [2]\tvalid_0's auc: 0.623808\n    [3]\tvalid_0's auc: 0.626038\n    [4]\tvalid_0's auc: 0.626204\n    [5]\tvalid_0's auc: 0.62651\n    [6]\tvalid_0's auc: 0.627204\n    [7]\tvalid_0's auc: 0.626664\n    [8]\tvalid_0's auc: 0.626893\n    [9]\tvalid_0's auc: 0.627229\n    [10]\tvalid_0's auc: 0.627513\n    [11]\tvalid_0's auc: 0.627644\n    [12]\tvalid_0's auc: 0.627571\n    [13]\tvalid_0's auc: 0.627896\n    [14]\tvalid_0's auc: 0.628177\n    [15]\tvalid_0's auc: 0.628492\n    [16]\tvalid_0's auc: 0.628649\n    [17]\tvalid_0's auc: 0.629023\n    [18]\tvalid_0's auc: 0.62928\n    [19]\tvalid_0's auc: 0.629413\n    [20]\tvalid_0's auc: 0.629706\n    [21]\tvalid_0's auc: 0.629788\n    [22]\tvalid_0's auc: 0.629955\n    [23]\tvalid_0's auc: 0.629956\n    [24]\tvalid_0's auc: 0.630142\n    [25]\tvalid_0's auc: 0.630223\n    [26]\tvalid_0's auc: 0.630383\n    [27]\tvalid_0's auc: 0.630474\n    [28]\tvalid_0's auc: 0.630586\n    [29]\tvalid_0's auc: 0.630575\n    [30]\tvalid_0's auc: 0.630611\n    [31]\tvalid_0's auc: 0.630523\n    [32]\tvalid_0's auc: 0.630594\n    [33]\tvalid_0's auc: 0.630476\n    [34]\tvalid_0's auc: 0.630559\n    [35]\tvalid_0's auc: 0.630627\n    [36]\tvalid_0's auc: 0.630757\n    [37]\tvalid_0's auc: 0.630747\n    [38]\tvalid_0's auc: 0.630775\n    [39]\tvalid_0's auc: 0.630759\n    [40]\tvalid_0's auc: 0.63086\n    [41]\tvalid_0's auc: 0.630907\n    [42]\tvalid_0's auc: 0.630927\n    [43]\tvalid_0's auc: 0.630937\n    [44]\tvalid_0's auc: 0.630976\n    [45]\tvalid_0's auc: 0.631045\n    [46]\tvalid_0's auc: 0.631058\n    [47]\tvalid_0's auc: 0.631028\n    [48]\tvalid_0's auc: 0.630945\n    [49]\tvalid_0's auc: 0.63092\n    [50]\tvalid_0's auc: 0.630967\n    [51]\tvalid_0's auc: 0.63102\n    [52]\tvalid_0's auc: 0.631018\n    [53]\tvalid_0's auc: 0.631077\n    [54]\tvalid_0's auc: 0.631074\n    [55]\tvalid_0's auc: 0.631124\n    [56]\tvalid_0's auc: 0.631105\n    [57]\tvalid_0's auc: 0.631137\n    [58]\tvalid_0's auc: 0.631176\n    [59]\tvalid_0's auc: 0.631205\n    [60]\tvalid_0's auc: 0.631187\n    [61]\tvalid_0's auc: 0.631157\n    [62]\tvalid_0's auc: 0.63114\n    [63]\tvalid_0's auc: 0.631193\n    [64]\tvalid_0's auc: 0.631214\n    [65]\tvalid_0's auc: 0.631231\n    [66]\tvalid_0's auc: 0.63117\n    [67]\tvalid_0's auc: 0.631165\n    [68]\tvalid_0's auc: 0.63113\n    [69]\tvalid_0's auc: 0.631135\n    [70]\tvalid_0's auc: 0.631177\n    [71]\tvalid_0's auc: 0.631123\n    [72]\tvalid_0's auc: 0.631101\n    [73]\tvalid_0's auc: 0.631137\n    [74]\tvalid_0's auc: 0.631149\n    [75]\tvalid_0's auc: 0.631202\n    [76]\tvalid_0's auc: 0.631272\n    [77]\tvalid_0's auc: 0.631296\n    [78]\tvalid_0's auc: 0.631295\n    [79]\tvalid_0's auc: 0.63133\n    [80]\tvalid_0's auc: 0.63136\n    [81]\tvalid_0's auc: 0.631324\n    [82]\tvalid_0's auc: 0.631298\n    [83]\tvalid_0's auc: 0.631338\n    [84]\tvalid_0's auc: 0.631361\n    [85]\tvalid_0's auc: 0.631325\n    [86]\tvalid_0's auc: 0.631308\n    [87]\tvalid_0's auc: 0.631311\n    [88]\tvalid_0's auc: 0.631293\n    [89]\tvalid_0's auc: 0.631289\n    [90]\tvalid_0's auc: 0.631291\n    [91]\tvalid_0's auc: 0.631332\n    [92]\tvalid_0's auc: 0.631387\n    [93]\tvalid_0's auc: 0.631406\n    [94]\tvalid_0's auc: 0.631426\n    [95]\tvalid_0's auc: 0.631444\n    [96]\tvalid_0's auc: 0.631448\n    [97]\tvalid_0's auc: 0.631467\n    [98]\tvalid_0's auc: 0.631443\n    [99]\tvalid_0's auc: 0.63142\n    [100]\tvalid_0's auc: 0.631466\n    [101]\tvalid_0's auc: 0.631534\n    [102]\tvalid_0's auc: 0.631536\n    [103]\tvalid_0's auc: 0.631579\n    [104]\tvalid_0's auc: 0.631558\n    [105]\tvalid_0's auc: 0.631558\n    [106]\tvalid_0's auc: 0.63162\n    [107]\tvalid_0's auc: 0.631678\n    [108]\tvalid_0's auc: 0.631644\n    [109]\tvalid_0's auc: 0.631664\n    [110]\tvalid_0's auc: 0.631685\n    [111]\tvalid_0's auc: 0.631674\n    [112]\tvalid_0's auc: 0.631634\n    [113]\tvalid_0's auc: 0.631619\n    [114]\tvalid_0's auc: 0.631614\n    [115]\tvalid_0's auc: 0.631646\n    [116]\tvalid_0's auc: 0.631596\n    [117]\tvalid_0's auc: 0.631577\n    [118]\tvalid_0's auc: 0.631586\n    [119]\tvalid_0's auc: 0.631562\n    [120]\tvalid_0's auc: 0.631553\n    [121]\tvalid_0's auc: 0.631535\n    [122]\tvalid_0's auc: 0.631559\n    [123]\tvalid_0's auc: 0.631534\n    [124]\tvalid_0's auc: 0.631516\n    [125]\tvalid_0's auc: 0.631528\n    [126]\tvalid_0's auc: 0.631533\n    [127]\tvalid_0's auc: 0.63153\n    [128]\tvalid_0's auc: 0.631457\n    [129]\tvalid_0's auc: 0.631504\n    [130]\tvalid_0's auc: 0.631548\n    [131]\tvalid_0's auc: 0.631563\n    [132]\tvalid_0's auc: 0.631597\n    [133]\tvalid_0's auc: 0.631615\n    [134]\tvalid_0's auc: 0.63161\n    [135]\tvalid_0's auc: 0.63163\n    [136]\tvalid_0's auc: 0.631636\n    [137]\tvalid_0's auc: 0.631645\n    [138]\tvalid_0's auc: 0.631694\n    [139]\tvalid_0's auc: 0.631656\n    [140]\tvalid_0's auc: 0.631689\n    [141]\tvalid_0's auc: 0.631699\n    [142]\tvalid_0's auc: 0.631719\n    [143]\tvalid_0's auc: 0.631699\n    [144]\tvalid_0's auc: 0.631731\n    [145]\tvalid_0's auc: 0.631768\n    [146]\tvalid_0's auc: 0.631793\n    [147]\tvalid_0's auc: 0.631842\n    [148]\tvalid_0's auc: 0.631841\n    [149]\tvalid_0's auc: 0.631811\n    [150]\tvalid_0's auc: 0.631788\n    [151]\tvalid_0's auc: 0.631827\n    [152]\tvalid_0's auc: 0.631851\n    [153]\tvalid_0's auc: 0.631857\n    [154]\tvalid_0's auc: 0.631826\n    [155]\tvalid_0's auc: 0.631838\n    [156]\tvalid_0's auc: 0.631929\n    [157]\tvalid_0's auc: 0.631915\n    [158]\tvalid_0's auc: 0.631917\n    [159]\tvalid_0's auc: 0.63191\n    [160]\tvalid_0's auc: 0.63192\n    [161]\tvalid_0's auc: 0.631869\n    [162]\tvalid_0's auc: 0.63185\n    [163]\tvalid_0's auc: 0.631848\n    [164]\tvalid_0's auc: 0.631891\n    [165]\tvalid_0's auc: 0.631916\n    [166]\tvalid_0's auc: 0.631942\n    [167]\tvalid_0's auc: 0.631955\n    [168]\tvalid_0's auc: 0.631977\n    [169]\tvalid_0's auc: 0.632012\n    [170]\tvalid_0's auc: 0.632086\n    [171]\tvalid_0's auc: 0.631994\n    [172]\tvalid_0's auc: 0.63201\n    [173]\tvalid_0's auc: 0.632015\n    [174]\tvalid_0's auc: 0.632041\n    [175]\tvalid_0's auc: 0.631996\n    [176]\tvalid_0's auc: 0.631965\n    [177]\tvalid_0's auc: 0.631977\n    [178]\tvalid_0's auc: 0.631894\n    [179]\tvalid_0's auc: 0.631881\n    [180]\tvalid_0's auc: 0.631874\n    [181]\tvalid_0's auc: 0.631872\n    [182]\tvalid_0's auc: 0.631861\n    [183]\tvalid_0's auc: 0.631842\n    [184]\tvalid_0's auc: 0.631845\n    [185]\tvalid_0's auc: 0.631858\n    [186]\tvalid_0's auc: 0.631853\n    [187]\tvalid_0's auc: 0.631859\n    [188]\tvalid_0's auc: 0.631776\n    [189]\tvalid_0's auc: 0.631817\n    [190]\tvalid_0's auc: 0.631795\n    [191]\tvalid_0's auc: 0.631777\n    [192]\tvalid_0's auc: 0.631795\n    [193]\tvalid_0's auc: 0.631764\n    [194]\tvalid_0's auc: 0.631737\n    [195]\tvalid_0's auc: 0.63167\n    [196]\tvalid_0's auc: 0.631649\n    [197]\tvalid_0's auc: 0.631653\n    [198]\tvalid_0's auc: 0.631643\n    [199]\tvalid_0's auc: 0.631617\n    [200]\tvalid_0's auc: 0.631655\n    [201]\tvalid_0's auc: 0.63166\n    [202]\tvalid_0's auc: 0.631625\n    [203]\tvalid_0's auc: 0.631694\n    [204]\tvalid_0's auc: 0.631674\n    [205]\tvalid_0's auc: 0.631611\n    [206]\tvalid_0's auc: 0.631589\n    [207]\tvalid_0's auc: 0.631639\n    [208]\tvalid_0's auc: 0.631678\n    [209]\tvalid_0's auc: 0.631716\n    [210]\tvalid_0's auc: 0.631691\n    [211]\tvalid_0's auc: 0.631687\n    [212]\tvalid_0's auc: 0.63165\n    [213]\tvalid_0's auc: 0.631681\n    [214]\tvalid_0's auc: 0.631637\n    [215]\tvalid_0's auc: 0.631623\n    [216]\tvalid_0's auc: 0.631591\n    [217]\tvalid_0's auc: 0.631556\n    [218]\tvalid_0's auc: 0.631645\n    [219]\tvalid_0's auc: 0.631605\n    [220]\tvalid_0's auc: 0.631612\n    [221]\tvalid_0's auc: 0.631553\n    [222]\tvalid_0's auc: 0.631543\n    [223]\tvalid_0's auc: 0.631523\n    [224]\tvalid_0's auc: 0.631519\n    [225]\tvalid_0's auc: 0.631543\n    [226]\tvalid_0's auc: 0.631508\n    [227]\tvalid_0's auc: 0.631524\n    [228]\tvalid_0's auc: 0.63149\n    [229]\tvalid_0's auc: 0.631503\n    [230]\tvalid_0's auc: 0.63148\n    [231]\tvalid_0's auc: 0.63148\n    [232]\tvalid_0's auc: 0.631485\n    [233]\tvalid_0's auc: 0.631475\n    [234]\tvalid_0's auc: 0.631419\n    [235]\tvalid_0's auc: 0.631412\n    [236]\tvalid_0's auc: 0.631386\n    [237]\tvalid_0's auc: 0.631401\n    [238]\tvalid_0's auc: 0.631399\n    [239]\tvalid_0's auc: 0.631349\n    [240]\tvalid_0's auc: 0.631327\n    [241]\tvalid_0's auc: 0.631355\n    [242]\tvalid_0's auc: 0.631305\n    [243]\tvalid_0's auc: 0.631331\n    [244]\tvalid_0's auc: 0.631323\n    [245]\tvalid_0's auc: 0.631348\n    [246]\tvalid_0's auc: 0.631324\n    [247]\tvalid_0's auc: 0.631255\n    [248]\tvalid_0's auc: 0.631278\n    [249]\tvalid_0's auc: 0.631303\n    [250]\tvalid_0's auc: 0.631352\n    Early stopping, best iteration is:\n    [170]\tvalid_0's auc: 0.632086\n\n\n\n```\na=lgb.plot_importance(model,importance_type='split',height=0.8,figsize=(18,80),grid=False,ignore_zero=False)\n```\n\n \n\n# 7.PROBANDO PCA PARA REDUCIR LOS FEATURES\n\n\n```\nfrom sklearn.decomposition import PCA\nfrom sklearn import metrics\n\nfrom sklearn.ensemble import RandomForestClassifier\nfrom sklearn.metrics import roc_auc_score\ndef pcaa(a,xx,yy):\n    print(\"PCA de %d elementos\"%a)\n    pca =PCA(n_components=a)\n    pca.fit(xx)\n    xx=pca.transform(xx)\n    print('pca listo')\n    print('USANDO RANDOMFOREST CLASSIFIER')\n    xx_train, xx_val, yy_train, yy_val = train_test_split(xx, yy, test_size=0.2, random_state=42)\n    mm = RandomForestClassifier(20, n_jobs=-1, oob_score=True,min_samples_split=10)#,max_features=10)\n    mm.fit(xx_train,yy_train)\n    y_train_pred = mm.predict(xx_train)\n    y_val_pred = mm.predict(xx_val)\n    print(f'Scores:')\n    print(f'Train      = {metrics.accuracy_score(yy_train, y_train_pred):.4}')\n    print(f'Validation = {metrics.accuracy_score(yy_val, y_val_pred):.4}')\n    if hasattr(mm, 'oob_score_'): print(f'OOB        = {mm.oob_score_:.4}')\n    print(\"AUC Train \",roc_auc_score(yy_train, y_train_pred))\n    print(\"AUC Test \",roc_auc_score(yy_val, y_val_pred))\n    print('USANDO SVM')\n    '''\n    from sklearn.svm import SVC\n    model_svm = SVC(gamma='auto')\n    model_svm.fit(xx_train,yy_train) \n    y_train_pred = model_svm.predict(xx_train)\n    y_val_pred = model_svm.predict(xx_val)\n    print(\"Accuracy Training Support Vector Machines:\",metrics.accuracy_score(yy_train, y_train_pred))\n    print(\"Accuracy Validation Support Vector Machines:\",metrics.accuracy_score(yy_val, y_val_pred))\n    print(\"AUC Train \",roc_auc_score(yy_train, y_train_pred))\n    print(\"AUC Test \",roc_auc_score(yy_val, y_val_pred))'''\n    \n    print('USANDO LOGISTIC')\n    from sklearn.linear_model import LogisticRegression\n    model_logistic = LogisticRegression(random_state=0, solver='lbfgs')\n    model_logistic.fit(xx_train, yy_train)\n    y_train_pred = model_logistic.predict(xx_train)\n    y_val_pred = model_logistic.predict(xx_val)\n    print(\"Accuracy Training Logistic:\",metrics.accuracy_score(yy_train, y_train_pred))\n    print(\"Accuracy Validation Logistic:\",metrics.accuracy_score(yy_val, y_val_pred))\n    print(\"AUC Train \",roc_auc_score(yy_train, y_train_pred))\n    print(\"AUC Test \",roc_auc_score(yy_val, y_val_pred))\n\n    return mm,pca\n```\n\n\n```\npcaa(10,X,y)\n```\n\n    PCA de 10 elementos\n    pca listo\n    USANDO RANDOMFOREST CLASSIFIER\n    Scores:\n    Train      = 0.9365\n    Validation = 0.9\n    OOB        = 0.8956\n    AUC Train  0.6826190476190477\n    AUC Test  0.5001984126984127\n    USANDO SVM\n    USANDO LOGISTIC\n    Accuracy Training Logistic: 0.8999940476190477\n    Accuracy Validation Logistic: 0.8999761904761905\n    AUC Train  0.49999669312169315\n    AUC Test  0.4999867724867725\n\n\n\n\n\n    (RandomForestClassifier(bootstrap=True, class_weight=None, criterion='gini',\n                 max_depth=None, max_features='auto', max_leaf_nodes=None,\n                 min_impurity_decrease=0.0, min_impurity_split=None,\n                 min_samples_leaf=1, min_samples_split=10,\n                 min_weight_fraction_leaf=0.0, n_estimators=20, n_jobs=-1,\n                 oob_score=True, random_state=None, verbose=0, warm_start=False),\n     PCA(copy=True, iterated_power='auto', n_components=10, random_state=None,\n       svd_solver='auto', tol=0.0, whiten=False))\n\n\n\n\n```\nmodelo,pca_modelo=pcaa(20,X,y)\n```\n\n    PCA de 20 elementos\n    pca listo\n    USANDO RANDOMFOREST CLASSIFIER\n    Scores:\n    Train      = 0.9405\n    Validation = 0.8998\n    OOB        = 0.8949\n    AUC Train  0.7023776455026455\n    AUC Test  0.5000925925925925\n    USANDO SVM\n    USANDO LOGISTIC\n    Accuracy Training Logistic: 0.8999880952380952\n    Accuracy Validation Logistic: 0.8999761904761905\n    AUC Train  0.49999338624338624\n    AUC Test  0.4999867724867725\n\n\n\n```\npcaa(15,X,y)\n```\n\n    PCA de 15 elementos\n    pca listo\n    USANDO RANDOMFOREST CLASSIFIER\n    Scores:\n    Train      = 0.9364\n    Validation = 0.8999\n    OOB        = 0.8962\n    AUC Train  0.6819642857142857\n    AUC Test  0.49992063492063493\n    USANDO SVM\n    USANDO LOGISTIC\n    Accuracy Training Logistic: 0.8793630952380952\n    Accuracy Validation Logistic: 0.8794761904761905\n    AUC Train  0.5106779100529102\n    AUC Test  0.5091269841269841\n\n\n\n\n\n    (RandomForestClassifier(bootstrap=True, class_weight=None, criterion='gini',\n                 max_depth=None, max_features='auto', max_leaf_nodes=None,\n                 min_impurity_decrease=0.0, min_impurity_split=None,\n                 min_samples_leaf=1, min_samples_split=10,\n                 min_weight_fraction_leaf=0.0, n_estimators=20, n_jobs=-1,\n                 oob_score=True, random_state=None, verbose=0, warm_start=False),\n     PCA(copy=True, iterated_power='auto', n_components=15, random_state=None,\n       svd_solver='auto', tol=0.0, whiten=False))\n\n\n\n### No mejoramos el modelo de lightgbm\n\n\n```\ntest_path='/home/liiarpi/Downloads/DATA/MOVISTAR/NEW_DATA3/NEW_TEST/'\nsfa=pd.read_csv(test_path+'TEST_FUENTE_APLICACIONES.csv')\nsfc=pd.read_csv(test_path+'TEST_FUENTE_CONSULTA_PREV_PORTA.csv')\nsfd=pd.read_csv(test_path+'TEST_FUENTE_DATOS.csv')\nsfe=pd.read_csv(test_path+'TEST_FUENTE_EQUIPOS.csv')\nsfv=pd.read_csv(test_path+'TEST_FUENTE_VOZ.csv')\nspp=pd.read_csv(test_path+'TEST_PLANTAPOST.csv')\nenvio=pd.read_csv('/home/liiarpi/Downloads/DATA/MOVISTAR/DATA/TEST/TEST_ENVIO.txt',sep='|')\n```\n\n\n```\nsfa=sfa.reindex(tfa.columns,axis=1)\nsfc=sfc.reindex(tfc.columns,axis=1)\nsfd=sfd.reindex(tfd.columns,axis=1)\nsfe=sfe.reindex(tfe.columns,axis=1)\nsfv=sfv.reindex(tfv.columns,axis=1)\nspp=spp.reindex(tpp.columns,axis=1)\n```\n\n\n```\nenvio=pd.merge(envio,sfa,on='ID_CLIENTE',how='left')\nenvio=pd.merge(envio,sfc,on='ID_CLIENTE',how='left')\nenvio=pd.merge(envio,sfd,on='ID_CLIENTE',how='left')\nenvio=pd.merge(envio,sfe,on='ID_CLIENTE',how='left')\nenvio=pd.merge(envio,sfv,on='ID_CLIENTE',how='left')\nenvio=pd.merge(envio,spp,on='ID_CLIENTE',how='left')\n```\n\n\n```\nenvio['TARGET']=model.predict(envio.drop(columns='ID_CLIENTE'),num_iteration=model.best_iteration)\n```\n\n\n```\nenvio = envio.loc[:,['ID_CLIENTE','TARGET']]\n```\n\n\n```\nenvio.to_csv('/home/liiarpi/Downloads/DATA/MOVISTAR/submissions/sub08_DATA3.csv',index=None, sep=',')\n```\n\n\n```\nenvio['ID_CLIENTE']=envio['ID_CLIENTE'].astype(str)\nenvio['TARGET']=envio['TARGET'].astype('float64')\n```\n\n\n```\npd.read_csv('/home/liiarpi/Downloads/DATA/MOVISTAR/submissions/sub08_DATA3.csv').info()\n\n```\n\n    \n    RangeIndex: 70000 entries, 0 to 69999\n    Data columns (total 2 columns):\n    ID_CLIENTE    70000 non-null object\n    TARGET        70000 non-null float64\n    dtypes: float64(1), object(1)\n    memory usage: 1.1+ MB\n\n\n###  Probando los resultados en la p\u00e1gina oficial obtenemos un 0.67375 en Private Score\n### y obtenemos un 0.66571 en Public Score \n\n\n```\nfrom PIL import Image\nImage.open('1.jpg')\n```\n\n \n\n# 8. USANDO LOS FEATURES HALLADOS EN EL ARCHIVO PROYECTO_FINAL_02\n\n\n```\ntest=pd.read_csv('Data_tratada_test.csv',na_values=['NaN','nan',-1])\n\n```\n\n\n```\nfeatures=['ID_CLIENTE','AIRTIME_IN_TOT_count', 'AIRTIME_OUT_ON_count', 'APP_10_count',\n       'APP_1_count', 'APP_2_count', 'APP_4_count', 'APP_5_count',\n       'APP_6_count', 'APP_7_count', 'APP_8_count', 'CALLS_IN_TOT_count',\n       'CALLS_OUT_ON_count', 'CALLS_OUT_TOT_count', 'CONT_OFF_count',\n       'CONT_ON_count', 'CONT_TOT_count', 'DAYS_IN_VOICE_OFF_sum',\n       'DAYS_IN_VOICE_TOT_count', 'DAYS_OUT_VOICE_OFF_sum',\n       'DEST_VOICE_ON_count', 'DEST_VOICE_count', 'MB_TOTAL_APP_count',\n       'N_DIAS_NAVEGACION_count', 'N_DIAS_NAVEGACION_max',\n       'N_DIAS_NAVEGACION_sum', 'TOP_CONT_5_count', 'TOP_CONT_5_max',\n       'TOP_CONT_5_sum', 'TOP_CONT_OFF_5_count', 'TOP_CONT_OFF_5_max',\n       'TOP_CONT_OFF_5_mean', 'TOP_CONT_OFF_5_sum', 'TOP_CONT_ON_5_count',\n       'TRAF_DATOS_mean', 'TRAF_DATOS_sum', 'alta_CALLS_IN_OFF_mean',\n       'alta_CALLS_OUT_OFF_mean', 'alta_DAYS_IN_VOICE_OFF_mean',\n       'alta_DAYS_OUT_VOICE_OFF_mean', 'alta_DEST_VOICE_OFF_mean',\n       'alta_TOP_CONT_OFF_5_mean', 'baja_DAYS_IN_VOICE_OFF_mean',\n       'baja_DAYS_OUT_VOICE_OFF_mean', 'baja_PANTALLA_mean',\n       'baja_TEC_CHIP_mean', 'media_APP_10_mean', 'media_APP_6_mean',\n       'media_APP_7_mean', 'media_APP_8_mean', 'media_APP_9_mean',\n       'media_N_DIAS_NAVEGACION_mean']\n```\n\n\n```\ntest=test[features]\n```\n\n\n```\nenvio2=pd.read_csv('/home/cristianl/Desktop/MOVISTAR/PAPER2/DATA/TEST/TEST_ENVIO.txt',sep='|')\nenvio2.head(2)\n\n```\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n    \n  \n  \n    \n      \n      \n    \n    \n      \n      \n    \n  \n
                                        ID_CLIENTE
                                        0oaxnYAu2ZFI2ncdPQpJFdFxhoWCa6BBAyIgKRcKELJ4=
                                        10uf2S8ucPs02X3uVr2iIcfnPnCfbfWkGZnlZpNdpR/c=
                                        \n
                                        \n\n\n\n\n```\nenvio2=pd.merge(envio2,test,on='ID_CLIENTE',how='left')\n```\n\n\n```\nenvio2.head(2)\n```\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n  \n
                                        ID_CLIENTEAIRTIME_IN_TOT_countAIRTIME_OUT_ON_countAPP_10_countAPP_1_countAPP_2_countAPP_4_countAPP_5_countAPP_6_countAPP_7_count...baja_DAYS_IN_VOICE_OFF_meanbaja_DAYS_OUT_VOICE_OFF_meanbaja_PANTALLA_meanbaja_TEC_CHIP_meanmedia_APP_10_meanmedia_APP_6_meanmedia_APP_7_meanmedia_APP_8_meanmedia_APP_9_meanmedia_N_DIAS_NAVEGACION_mean
                                        0oaxnYAu2ZFI2ncdPQpJFdFxhoWCa6BBAyIgKRcKELJ4=333.03.03.03.03.03.03.0...0.00.00.00.01.01.01.01.01.01.0
                                        10uf2S8ucPs02X3uVr2iIcfnPnCfbfWkGZnlZpNdpR/c=333.03.03.03.03.03.03.0...0.00.00.00.01.00.01.01.00.01.0
                                        \n
                                        2 rows \u00d7 52 columns
                                        \n
                                        \n\n\n\n\n```\nfor n,col in envio2.items():\n    if  n != 'ID_CLIENTE': \n        print(n)\n        envio2[n][(1-np.isnan(envio2[n]))>0.0001]=0\n        a=np.copy(envio2[n])\n        envio2[n][np.isnan(a)]=0\n        envio2[n][a==np.Inf]=0\n        envio2[n][a==-np.Inf]=0\n```\n\n    AIRTIME_IN_TOT_count\n    AIRTIME_OUT_ON_count\n    APP_10_count\n    APP_1_count\n    APP_2_count\n    APP_4_count\n    APP_5_count\n    APP_6_count\n    APP_7_count\n    APP_8_count\n    CALLS_IN_TOT_count\n    CALLS_OUT_ON_count\n    CALLS_OUT_TOT_count\n    CONT_OFF_count\n    CONT_ON_count\n    CONT_TOT_count\n    DAYS_IN_VOICE_OFF_sum\n    DAYS_IN_VOICE_TOT_count\n    DAYS_OUT_VOICE_OFF_sum\n    DEST_VOICE_ON_count\n    DEST_VOICE_count\n    MB_TOTAL_APP_count\n    N_DIAS_NAVEGACION_count\n    N_DIAS_NAVEGACION_max\n    N_DIAS_NAVEGACION_sum\n    TOP_CONT_5_count\n    TOP_CONT_5_max\n    TOP_CONT_5_sum\n    TOP_CONT_OFF_5_count\n    TOP_CONT_OFF_5_max\n    TOP_CONT_OFF_5_mean\n    TOP_CONT_OFF_5_sum\n    TOP_CONT_ON_5_count\n    TRAF_DATOS_mean\n    TRAF_DATOS_sum\n    alta_CALLS_IN_OFF_mean\n    alta_CALLS_OUT_OFF_mean\n    alta_DAYS_IN_VOICE_OFF_mean\n    alta_DAYS_OUT_VOICE_OFF_mean\n    alta_DEST_VOICE_OFF_mean\n    alta_TOP_CONT_OFF_5_mean\n    baja_DAYS_IN_VOICE_OFF_mean\n    baja_DAYS_OUT_VOICE_OFF_mean\n    baja_PANTALLA_mean\n    baja_TEC_CHIP_mean\n    media_APP_10_mean\n    media_APP_6_mean\n    media_APP_7_mean\n    media_APP_8_mean\n    media_APP_9_mean\n    media_N_DIAS_NAVEGACION_mean\n\n\n\n```\nenvio2.head()\n```\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n  \n
                                        ID_CLIENTEAIRTIME_IN_TOT_countAIRTIME_OUT_ON_countAPP_10_countAPP_1_countAPP_2_countAPP_4_countAPP_5_countAPP_6_countAPP_7_count...baja_DAYS_IN_VOICE_OFF_meanbaja_DAYS_OUT_VOICE_OFF_meanbaja_PANTALLA_meanbaja_TEC_CHIP_meanmedia_APP_10_meanmedia_APP_6_meanmedia_APP_7_meanmedia_APP_8_meanmedia_APP_9_meanmedia_N_DIAS_NAVEGACION_mean
                                        0oaxnYAu2ZFI2ncdPQpJFdFxhoWCa6BBAyIgKRcKELJ4=000.00.00.00.00.00.00.0...0.00.00.00.00.00.00.00.00.00.0
                                        10uf2S8ucPs02X3uVr2iIcfnPnCfbfWkGZnlZpNdpR/c=000.00.00.00.00.00.00.0...0.00.00.00.00.00.00.00.00.00.0
                                        2neSeIS1r7rrJhkWxhJjm8km68p3/uStOqx9fu9MNeIA=000.00.00.00.00.00.00.0...0.00.00.00.00.00.00.00.00.00.0
                                        3WKHM28G6WZiStJaohTxQI6zqlYWpHebYZhpinSLR9Bc=000.00.00.00.00.00.00.0...0.00.00.00.00.00.00.00.00.00.0
                                        43EpIkwtqjUAcHDP3KS9iOFxeoupAt/PhPX6WubaTwcY=000.00.00.00.00.00.00.0...0.00.00.00.00.00.00.00.00.00.0
                                        \n
                                        5 rows \u00d7 52 columns
                                        \n
                                        \n\n\n\n\n```\nenvio3=pca_modelo.transform(envio2.drop(columns='ID_CLIENTE'))\n\n```\n\n\n```\nenvio2['TARGET']=modelo.predict(envio3)\n```\n\n\n```\nenvio2 = envio2.loc[:,['ID_CLIENTE','TARGET']]\n```\n\n\n```\nenvio2.to_csv('/home/cristianl/Desktop/MOVISTAR/PAPER3/subm/subfinal.csv',index=None, sep=',')\n```\n\n###  Probando los resultados en la p\u00e1gina oficial obtenemos un 0.68032 en Private Score\n### y obtenemos un 0.6733 en Public Score a  que nos ubicar\u00eda en el segundo puesto\n\n\n```\nImage.open('2.jpg')\n```\n\n\n```\n\n```\n", "meta": {"hexsha": "8d8d3c7ac111561bfa5b2075df7b4abf924c0fe8", "size": 540735, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Projects-2018-2/Movistar/PROYECTO_IA_01.ipynb", "max_stars_repo_name": "PCL-AI/MT616_2018_2", "max_stars_repo_head_hexsha": "3554b80c33221ffac4c26d30c96106d942b0a15e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2018-12-12T14:13:15.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-23T21:59:27.000Z", "max_issues_repo_path": "Projects-2018-2/Movistar/PROYECTO_IA_01.ipynb", "max_issues_repo_name": "PCL-AI/MT616_2018_2", "max_issues_repo_head_hexsha": "3554b80c33221ffac4c26d30c96106d942b0a15e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Projects-2018-2/Movistar/PROYECTO_IA_01.ipynb", "max_forks_repo_name": "PCL-AI/MT616_2018_2", "max_forks_repo_head_hexsha": "3554b80c33221ffac4c26d30c96106d942b0a15e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 10, "max_forks_repo_forks_event_min_datetime": "2018-09-21T23:45:07.000Z", "max_forks_repo_forks_event_max_datetime": "2021-02-17T05:40:55.000Z", "avg_line_length": 540735.0, "max_line_length": 540735, "alphanum_fraction": 0.9119661202, "converted": true, "num_tokens": 18522, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.7461389986757757, "lm_q1q2_score": 0.413712037520865}}
{"text": "## **pytorch DNN strategy**\n\n\n```python\nimport torch\nfrom torch.autograd import Variable  # \u5c06\u591a\u7ef4\u77e9\u9635\u653e\u5165Variable\uff0c\u53ef\u4ee5\u53cd\u5411\u4f20\u64ad\nimport torch.nn as nn  #\u53c2\u6570\u6269\u5c55\uff0c\u8ba1\u7b97\u6743\u91cd\nimport torch.nn.functional as F\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nfrom sklearn.metrics import accuracy_score  #\u8bc4\u4ef7\u6d4b\u8bd5\u7ed3\u679c\uff0c\u6240\u6709\u5206\u7c7b\u6b63\u786e\u7684\u767e\u5206\u6bd4\nfrom torchvision import datasets\nfrom torchvision import transforms\nfrom tensorflow import keras\nimport sympy\nfrom sympy import Matrix\nimport tushare as ts\n\n%config INlineBackend.figure_format='svg' #  \u4e3a\u4e86\u8ba9\u6570\u5b57\u66f4\u6e05\u6670\u7684\u663e\u793a\u5728notebook\u4e2d\n```\n\n\n```python\ntorch.cuda.is_available()\n```\n\n\n\n\n    False\n\n\n\n\n```python\ndata = ts.get_k_data('hs300',start = '2014-07-01',end = '2021-01-20')\ndata.set_index('date',inplace=True)\ndata\n```\n\n    \u672c\u63a5\u53e3\u5373\u5c06\u505c\u6b62\u66f4\u65b0\uff0c\u8bf7\u5c3d\u5feb\u4f7f\u7528Pro\u7248\u63a5\u53e3\uff1ahttps://waditu.com/document/2\n\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n  \n
                                        openclosehighlowvolumecode
                                        date
                                        2014-07-012169.202164.562171.152157.1353323100.0hs300
                                        2014-07-022164.002170.872171.512155.6159685501.0hs300
                                        2014-07-032169.002180.192184.962164.8463690690.0hs300
                                        2014-07-042180.452178.702183.802174.0752851106.0hs300
                                        2014-07-072178.552176.292186.202171.0850242334.0hs300
                                        .....................
                                        2021-01-145556.215470.465568.025458.68243683380.0hs300
                                        2021-01-155471.395458.085500.635390.27238637435.0hs300
                                        2021-01-185438.165518.525541.465410.78207051869.0hs300
                                        2021-01-195525.975437.525532.485415.72211043425.0hs300
                                        2021-01-205439.915476.435496.055426.54170913260.0hs300
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                                        1601 rows \u00d7 6 columns
                                        \n
                                        \n\n\n\n\n```python\ndata_cleaned = pd.DataFrame()\ndata_cleaned['Close0'] = np.log(data['close'])\ndata_cleaned['H_L'] = data['high']-data['low']\ndata_cleaned['C_O'] = data['close']-data['open']\ndata_cleaned['volume'] = np.log10(data['volume'])\n\ndata_cleaned['Close1'] = data_cleaned['Close0'].shift(1)\ndata_cleaned['Close2'] = data_cleaned['Close0'].shift(2)\ndata_cleaned['Close3'] = data_cleaned['Close0'].shift(3)\ndata_cleaned['Close4'] = data_cleaned['Close0'].shift(4)\ndata_cleaned['Close5'] = data_cleaned['Close0'].shift(5)\n\ndata_cleaned['H_L1'] = data_cleaned['H_L'].shift(1)\ndata_cleaned['H_L2'] = data_cleaned['H_L'].shift(2)\ndata_cleaned['H_L3'] = data_cleaned['H_L'].shift(3)\ndata_cleaned['H_L4'] = data_cleaned['H_L'].shift(4)\ndata_cleaned['H_L5'] = data_cleaned['H_L'].shift(5)\n\ndata_cleaned['C_O1'] = data_cleaned['C_O'].shift(1)\ndata_cleaned['C_O2'] = data_cleaned['C_O'].shift(2)\ndata_cleaned['C_O3'] = data_cleaned['C_O'].shift(3)\ndata_cleaned['C_O4'] = data_cleaned['C_O'].shift(4)\ndata_cleaned['C_O5'] = data_cleaned['C_O'].shift(5)\n\ndata_cleaned['volume1'] = data_cleaned['volume'].shift(1)\ndata_cleaned['volume2'] = data_cleaned['volume'].shift(2)\ndata_cleaned['volume3'] = data_cleaned['volume'].shift(3)\ndata_cleaned['volume4'] = data_cleaned['volume'].shift(4)\ndata_cleaned['volume5'] = data_cleaned['volume'].shift(5)\n\ndata_cleaned['Close0_shift(-1)'] = data_cleaned['Close0'].shift(-1)\ndata_cleaned['label'] =0.5 + 0.5*np.sign(data_cleaned['Close0_shift(-1)']-data_cleaned['Close0'])\ndata_cleaned.head(23)\n```\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n  \n
                                        Close0H_LC_OvolumeClose1Close2Close3Close4Close5H_L1...C_O3C_O4C_O5volume1volume2volume3volume4volume5Close0_shift(-1)label
                                        date
                                        2014-07-017.67997214.02-4.647.726915NaNNaNNaNNaNNaNNaN...NaNNaNNaNNaNNaNNaNNaNNaN7.6828831.0
                                        2014-07-027.68288315.906.877.7758697.679972NaNNaNNaNNaN14.02...NaNNaNNaN7.726915NaNNaNNaNNaN7.6871671.0
                                        2014-07-037.68716720.1211.197.8040767.6828837.679972NaNNaNNaN15.90...NaNNaNNaN7.7758697.726915NaNNaNNaN7.6864840.0
                                        2014-07-047.6864849.73-1.757.7230547.6871677.6828837.679972NaNNaN20.12...-4.64NaNNaN7.8040767.7758697.726915NaNNaN7.6853770.0
                                        2014-07-077.68537715.12-2.267.7010707.6864847.6871677.6828837.679972NaN9.73...6.87-4.64NaN7.7230547.8040767.7758697.726915NaN7.6872961.0
                                        2014-07-087.68729617.425.647.7053817.6853777.6864847.6871677.6828837.67997215.12...11.196.87-4.647.7010707.7230547.8040767.7758697.7269157.6726230.0
                                        2014-07-097.67262330.25-29.437.7925467.6872967.6853777.6864847.6871677.68288317.42...-1.7511.196.877.7053817.7010707.7230547.8040767.7758697.6698920.0
                                        2014-07-107.66989212.25-3.757.7461097.6726237.6872967.6853777.6864847.68716730.25...-2.26-1.7511.197.7925467.7053817.7010707.7230547.8040767.6722971.0
                                        2014-07-117.67229719.9511.697.7879867.6698927.6726237.6872967.6853777.68648412.25...5.64-2.26-1.757.7461097.7925467.7053817.7010707.7230547.6832931.0
                                        2014-07-147.68329326.6222.227.8205887.6722977.6698927.6726237.6872967.68537719.95...-29.435.64-2.267.7879867.7461097.7925467.7053817.7010707.6847751.0
                                        2014-07-157.68477511.064.897.8562197.6832937.6722977.6698927.6726237.68729626.62...-3.75-29.435.647.8205887.7879867.7461097.7925467.7053817.6828830.0
                                        2014-07-167.68288316.48-1.957.9114787.6847757.6832937.6722977.6698927.67262311.06...11.69-3.75-29.437.8562197.8205887.7879867.7461097.7925467.6765060.0
                                        2014-07-177.67650619.85-9.767.7741787.6828837.6847757.6832937.6722977.66989216.48...22.2211.69-3.757.9114787.8562197.8205887.7879867.7461097.6797781.0
                                        2014-07-187.67977830.8716.487.8160647.6765067.6828837.6847757.6832937.67229719.85...4.8922.2211.697.7741787.9114787.8562197.8205887.7879867.6807711.0
                                        2014-07-217.68077112.982.077.7337777.6797787.6765067.6828837.6847757.68329330.87...-1.954.8922.227.8160647.7741787.9114787.8562197.8205887.6928891.0
                                        2014-07-227.69288935.3530.267.8951057.6807717.6797787.6765067.6828837.68477512.98...-9.76-1.954.897.7337777.8160647.7741787.9114787.8562197.6952261.0
                                        2014-07-237.69522619.736.447.9253847.6928897.6807717.6797787.6765067.68288335.35...16.48-9.76-1.957.8951057.7337777.8160647.7741787.9114787.7128951.0
                                        2014-07-247.71289539.9837.188.0771347.6952267.6928897.6807717.6797787.67650619.73...2.0716.48-9.767.9253847.8951057.7337777.8160647.7741787.7233191.0
                                        2014-07-257.72331921.8217.387.9955167.7128957.6952267.6928897.6807717.67977839.98...30.262.0716.488.0771347.9253847.8951057.7337777.8160647.7510021.0
                                        2014-07-287.75100259.0951.528.2549577.7233197.7128957.6952267.6928897.68077121.82...6.4430.262.077.9955168.0771347.9253847.8951057.7337777.7542111.0
                                        2014-07-297.75421126.555.338.1279787.7510027.7233197.7128957.6952267.69288959.09...37.186.4430.268.2549577.9955168.0771347.9253847.8951057.7501880.0
                                        2014-07-307.75018822.98-2.688.0899867.7542117.7510027.7233197.7128957.69522626.55...17.3837.186.448.1279788.2549577.9955168.0771347.9253847.7622771.0
                                        2014-07-317.76227733.9728.288.0452757.7501887.7542117.7510027.7233197.71289522.98...51.5217.3837.188.0899868.1279788.2549577.9955168.0771347.7533660.0
                                        \n
                                        23 rows \u00d7 26 columns
                                        \n
                                        \n\n\n\n\n\n\n```python\n# \u53bb\u6389\u591a\u4f59\u7684\u6570\u636e\u548c\u7a7a\u503c\ndata_cleaned.dropna(inplace=True)\ndata_cleaned.drop(['Close0_shift(-1)'],axis=1,inplace=True)\ndata_cleaned\n```\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n  \n
                                        Close0H_LC_OvolumeClose1Close2Close3Close4Close5H_L1...C_O2C_O3C_O4C_O5volume1volume2volume3volume4volume5label
                                        date
                                        2014-07-087.68729617.425.647.7053817.6853777.6864847.6871677.6828837.67997215.12...-1.7511.196.87-4.647.7010707.7230547.8040767.7758697.7269150.0
                                        2014-07-097.67262330.25-29.437.7925467.6872967.6853777.6864847.6871677.68288317.42...-2.26-1.7511.196.877.7053817.7010707.7230547.8040767.7758690.0
                                        2014-07-107.66989212.25-3.757.7461097.6726237.6872967.6853777.6864847.68716730.25...5.64-2.26-1.7511.197.7925467.7053817.7010707.7230547.8040761.0
                                        2014-07-117.67229719.9511.697.7879867.6698927.6726237.6872967.6853777.68648412.25...-29.435.64-2.26-1.757.7461097.7925467.7053817.7010707.7230541.0
                                        2014-07-147.68329326.6222.227.8205887.6722977.6698927.6726237.6872967.68537719.95...-3.75-29.435.64-2.267.7879867.7461097.7925467.7053817.7010701.0
                                        ..................................................................
                                        2021-01-138.626580109.58-31.298.4175888.6298708.6017488.6116728.6149848.597421178.72...-63.08-30.8685.2731.168.3506758.3651818.3103218.3421708.2872940.0
                                        2021-01-148.607118109.34-85.758.3868268.6265808.6298708.6017488.6116728.614984109.58...176.85-63.08-30.8685.278.4175888.3506758.3651818.3103218.3421700.0
                                        2021-01-158.604852110.36-13.318.3777398.6071188.6265808.6298708.6017488.611672109.34...-31.29176.85-63.08-30.868.3868268.4175888.3506758.3651818.3103211.0
                                        2021-01-188.615865130.6880.368.3160798.6048528.6071188.6265808.6298708.601748110.36...-85.75-31.29176.85-63.088.3777398.3868268.4175888.3506758.3651810.0
                                        2021-01-198.601078116.76-88.458.3243728.6158658.6048528.6071188.6265808.629870130.68...-13.31-85.75-31.29176.858.3160798.3777398.3868268.4175888.3506751.0
                                        \n
                                        1595 rows \u00d7 25 columns
                                        \n
                                        \n\n\n\n\n```python\n# \u5c06\u6570\u636e\u5206\u6210\u8bad\u7ec3\u96c6\u548c\u6d4b\u8bd5\u96c6\ntraning_set = data_cleaned.iloc[:999,:]\ntest_set = data_cleaned.iloc[999:,:]\n```\n\n\n```python\ntraning_set\n```\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n  \n
                                        Close0H_LC_OvolumeClose1Close2Close3Close4Close5H_L1...C_O2C_O3C_O4C_O5volume1volume2volume3volume4volume5label
                                        date
                                        2014-07-087.68729617.425.647.7053817.6853777.6864847.6871677.6828837.67997215.12...-1.7511.196.87-4.647.7010707.7230547.8040767.7758697.7269150.0
                                        2014-07-097.67262330.25-29.437.7925467.6872967.6853777.6864847.6871677.68288317.42...-2.26-1.7511.196.877.7053817.7010707.7230547.8040767.7758690.0
                                        2014-07-107.66989212.25-3.757.7461097.6726237.6872967.6853777.6864847.68716730.25...5.64-2.26-1.7511.197.7925467.7053817.7010707.7230547.8040761.0
                                        2014-07-117.67229719.9511.697.7879867.6698927.6726237.6872967.6853777.68648412.25...-29.435.64-2.26-1.757.7461097.7925467.7053817.7010707.7230541.0
                                        2014-07-147.68329326.6222.227.8205887.6722977.6698927.6726237.6872967.68537719.95...-3.75-29.435.64-2.267.7879867.7461097.7925467.7053817.7010701.0
                                        ..................................................................
                                        2018-08-018.14537396.83-82.647.9559228.1655518.1648188.1665668.1708228.18248935.00...-5.77-12.54-47.07-11.767.8323867.9144337.8462027.9559267.9856890.0
                                        2018-08-028.122953112.25-63.398.0056478.1453738.1655518.1648188.1665668.17082296.83...7.19-5.77-12.54-47.077.9559227.8323867.9144337.8462027.9559260.0
                                        2018-08-038.10629766.14-51.387.8500148.1229538.1453738.1655518.1648188.166566112.25...-82.647.19-5.77-12.548.0056477.9559227.8323867.9144337.8462020.0
                                        2018-08-068.09354586.56-39.557.8919568.1062978.1229538.1453738.1655518.16481866.14...-63.39-82.647.19-5.777.8500148.0056477.9559227.8323867.9144331.0
                                        2018-08-078.122333105.4683.897.9627678.0935458.1062978.1229538.1453738.16555186.56...-51.38-63.39-82.647.197.8919567.8500148.0056477.9559227.8323860.0
                                        \n
                                        999 rows \u00d7 25 columns
                                        \n
                                        \n\n\n\n\n```python\n# \u67e5\u770b\u7ef4\u5ea6\ntraning_set.shape\n```\n\n\n\n\n    (999, 25)\n\n\n\n## **Traning DNN Model**\n\n\n```python\nclass DeepNeuralNetworkMdoel(nn.Module):\n    # constructor of the class\n    def __init__(self):\n        super(DeepNeuralNetworkMdoel,self).__init__()\n        # Fully connected Layer1\n        self.FC_layer1 = nn.Linear(24,48)\n        \n        # Fully connected Layer2\n        self.FC_layer2 = nn.Linear(48,36)\n        \n        # Fully connected Layer3\n        self.FC_layer3 = nn.Linear(36,24)\n        \n        # Fully connected Layer4\n        self.FC_layer4 = nn.Linear(24,12)\n        \n        # Fully connected Layer5\n        self.FC_layer5 = nn.Linear(12,2)\n        \n    # Forward propagation function\n    def forward(self,input_data):  # dim of input_data N * 24\n        z1_= self.FC_layer1(input_data)\n        z1 = torch.sigmoid(z1_)\n        \n        z2_= self.FC_layer2(z1)\n        z2 = torch.sigmoid(z2_)\n        \n        z3_= self.FC_layer3(z2)\n        z3 = torch.sigmoid(z3_)\n        \n        z4_= self.FC_layer4(z3)\n        z4 = torch.sigmoid(z4_)\n        \n        z5_= self.FC_layer5(z4)   # \u6700\u540e\u4e00\u5c42\u4e0d\u52a0\u6fc0\u6d3b\u51fd\u6570,\u540e\u9762\u8c03\u7528entrypyloss\u51fd\u6570\n        \n        return z5_\n```\n\n\n```python\nx = traning_set.iloc[:,0:24]\ny = traning_set.iloc[:,24]\nx = np.array(x)\ny = np.array(y)\nx,y\n```\n\n\n\n\n    (array([[  7.68729573,  17.42      ,   5.64      , ...,   7.80407595,\n               7.77586884,   7.72691539],\n            [  7.67262294,  30.25      , -29.43      , ...,   7.72305408,\n               7.80407595,   7.77586884],\n            [  7.669892  ,  12.25      ,  -3.75      , ...,   7.70106981,\n               7.72305408,   7.80407595],\n            ...,\n            [  8.10629736,  66.14      , -51.38      , ...,   7.83238638,\n               7.91443305,   7.84620163],\n            [  8.09354476,  86.56      , -39.55      , ...,   7.95592151,\n               7.83238638,   7.91443305],\n            [  8.12233266, 105.46      ,  83.89      , ...,   8.00564686,\n               7.95592151,   7.83238638]]),\n     array([0., 0., 1., 1., 1., 0., 0., 1., 1., 1., 1., 1., 1., 1., 1., 0., 1.,\n            0., 1., 0., 0., 0., 1., 1., 0., 1., 0., 1., 1., 1., 0., 0., 1., 0.,\n            0., 1., 0., 1., 1., 1., 1., 1., 1., 0., 0., 0., 1., 0., 0., 1., 1.,\n            1., 0., 1., 1., 0., 1., 1., 1., 1., 1., 0., 0., 0., 1., 0., 0., 1.,\n            0., 0., 0., 0., 0., 1., 1., 1., 1., 1., 1., 0., 1., 0., 1., 0., 1.,\n            0., 1., 0., 0., 0., 0., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,\n            1., 0., 1., 0., 1., 1., 1., 1., 0., 1., 1., 0., 0., 1., 1., 1., 1.,\n            1., 1., 0., 1., 0., 0., 0., 1., 0., 1., 1., 0., 1., 1., 1., 1., 1.,\n            0., 0., 0., 0., 0., 1., 0., 0., 0., 1., 1., 1., 1., 1., 1., 1., 0.,\n            1., 1., 1., 0., 1., 0., 0., 1., 0., 1., 1., 1., 1., 1., 1., 0., 1.,\n            1., 1., 0., 1., 1., 1., 0., 1., 1., 1., 1., 1., 0., 1., 1., 1., 0.,\n            1., 1., 0., 1., 1., 1., 0., 1., 0., 1., 0., 1., 0., 0., 0., 1., 1.,\n            1., 0., 0., 0., 0., 1., 1., 1., 1., 1., 1., 0., 0., 1., 1., 1., 0.,\n            1., 1., 1., 0., 0., 0., 1., 0., 0., 1., 0., 0., 1., 1., 0., 0., 0.,\n            1., 0., 0., 0., 1., 0., 0., 1., 1., 1., 0., 0., 1., 1., 1., 1., 0.,\n            1., 0., 0., 0., 1., 0., 1., 1., 1., 0., 0., 1., 1., 0., 0., 1., 0.,\n            1., 0., 1., 0., 0., 0., 0., 0., 1., 1., 1., 0., 1., 0., 1., 1., 0.,\n            0., 0., 0., 1., 0., 1., 1., 1., 0., 1., 0., 1., 0., 1., 1., 1., 1.,\n            0., 0., 1., 1., 1., 1., 0., 1., 1., 1., 1., 0., 1., 1., 0., 0., 1.,\n            1., 1., 1., 0., 1., 0., 0., 1., 0., 0., 1., 0., 0., 1., 1., 0., 0.,\n            1., 1., 1., 1., 0., 1., 0., 1., 0., 0., 1., 0., 0., 1., 1., 1., 1.,\n            0., 0., 1., 0., 1., 1., 0., 0., 1., 1., 0., 1., 0., 1., 0., 1., 0.,\n            1., 1., 0., 0., 1., 1., 0., 0., 0., 1., 0., 1., 0., 1., 0., 0., 1.,\n            1., 0., 0., 1., 0., 1., 0., 1., 0., 1., 1., 1., 1., 1., 1., 0., 0.,\n            1., 1., 1., 1., 1., 1., 1., 0., 1., 0., 1., 0., 0., 1., 1., 1., 1.,\n            0., 0., 0., 1., 0., 1., 1., 0., 0., 1., 0., 0., 1., 0., 1., 0., 0.,\n            0., 1., 0., 1., 0., 0., 1., 1., 1., 0., 1., 0., 0., 0., 1., 1., 0.,\n            0., 1., 0., 1., 1., 0., 1., 1., 0., 0., 0., 0., 1., 1., 0., 1., 1.,\n            0., 1., 0., 0., 1., 1., 1., 1., 1., 1., 1., 1., 0., 0., 1., 1., 1.,\n            0., 0., 0., 0., 0., 1., 0., 1., 1., 0., 1., 0., 0., 1., 1., 1., 1.,\n            1., 1., 0., 0., 1., 1., 0., 0., 0., 1., 0., 1., 0., 0., 0., 1., 1.,\n            1., 0., 1., 1., 1., 0., 0., 0., 0., 0., 0., 1., 0., 1., 1., 0., 0.,\n            1., 0., 1., 1., 1., 1., 0., 1., 1., 0., 1., 0., 1., 1., 1., 0., 0.,\n            0., 0., 0., 1., 0., 1., 0., 1., 1., 0., 1., 1., 1., 0., 0., 1., 0.,\n            1., 1., 1., 1., 1., 1., 1., 0., 1., 0., 0., 0., 1., 0., 1., 0., 0.,\n            0., 0., 1., 0., 0., 1., 0., 0., 1., 0., 0., 0., 1., 1., 1., 0., 0.,\n            1., 0., 0., 0., 1., 0., 1., 1., 0., 1., 1., 1., 1., 1., 0., 1., 0.,\n            1., 1., 1., 1., 0., 0., 1., 0., 1., 1., 1., 0., 1., 0., 1., 1., 0.,\n            0., 1., 1., 0., 0., 1., 1., 0., 1., 1., 0., 1., 1., 0., 1., 1., 0.,\n            0., 0., 0., 1., 1., 1., 1., 0., 1., 0., 1., 0., 0., 0., 0., 1., 1.,\n            0., 1., 1., 1., 0., 0., 0., 0., 0., 0., 0., 0., 1., 1., 1., 1., 0.,\n            0., 1., 1., 1., 0., 1., 0., 1., 1., 0., 0., 1., 1., 1., 1., 0., 1.,\n            0., 0., 0., 1., 0., 1., 1., 1., 1., 1., 0., 1., 0., 0., 0., 1., 1.,\n            0., 0., 1., 0., 1., 1., 0., 1., 1., 1., 0., 1., 0., 0., 1., 1., 1.,\n            1., 0., 0., 0., 1., 1., 0., 0., 0., 1., 1., 0., 1., 1., 1., 1., 1.,\n            0., 1., 1., 0., 0., 0., 1., 1., 1., 0., 0., 0., 0., 1., 1., 0., 1.,\n            1., 0., 1., 0., 0., 0., 1., 1., 1., 1., 1., 1., 1., 1., 1., 0., 0.,\n            1., 0., 0., 1., 1., 1., 1., 1., 0., 0., 0., 1., 0., 1., 1., 0., 1.,\n            1., 1., 0., 0., 1., 1., 1., 1., 1., 0., 1., 0., 1., 0., 0., 0., 1.,\n            1., 0., 0., 1., 1., 0., 1., 0., 0., 1., 1., 0., 1., 0., 0., 1., 0.,\n            1., 1., 1., 1., 1., 1., 1., 1., 1., 0., 1., 1., 1., 0., 1., 1., 1.,\n            1., 1., 0., 1., 0., 0., 1., 0., 1., 1., 0., 0., 0., 0., 1., 1., 1.,\n            1., 1., 1., 0., 0., 1., 0., 1., 1., 0., 1., 1., 1., 0., 0., 1., 0.,\n            1., 1., 0., 0., 0., 0., 1., 0., 1., 1., 0., 0., 0., 0., 1., 1., 0.,\n            0., 0., 0., 1., 1., 0., 1., 1., 0., 0., 1., 1., 1., 0., 1., 1., 0.,\n            1., 0., 1., 1., 0., 0., 1., 1., 0., 0., 0., 0., 1., 0., 0., 1., 0.,\n            1., 1., 0., 0., 0., 1., 1., 0., 0., 0., 0., 1., 0., 1., 0., 0., 0.,\n            0., 1., 0., 1., 0., 0., 1., 1., 1., 0., 1., 1., 0., 0., 0., 0., 1.,\n            1., 1., 0., 0., 0., 0., 1., 0., 0., 0., 0., 1., 0.]))\n\n\n\n\n```python\nx = Variable(torch.FloatTensor(x))  # x\u4e3a\u6d6e\u70b9\u7c7b\u578b\u7684\u6570\u636e\uff0c\u6240\u4ee5\u7528FloatTensor\ny = Variable(torch.LongTensor(y)) # y\u9700\u8981\u8f6c\u4e3aint\u578b\u6570\u636e\nx.shape,y.shape\n```\n\n\n\n\n    (torch.Size([999, 24]), torch.Size([999]))\n\n\n\n\n```python\nalpha = 1.0\nDNN_Model = DeepNeuralNetworkMdoel()\noptimizer = torch.optim.SGD(DNN_Model.parameters(),lr = alpha)\nloss_function = nn.CrossEntropyLoss()\n\n# \u52a8\u6001\u8c03\u6574\u5b66\u4e60\u7387\ndef adjust_learning_rate(optimizer,epoch):\n    if epoch <=100:\n        lr = alpha\n    elif epoch > 100:\n        lr = alpha / (1 + 0.0008 * (epoch-100))\n    for param_group in optimizer.param_groups:\n        param_group['lr'] = lr\n        \n```\n\n\n```python\nIter_times = 10000000\nloss_list = []\nfor i in range(Iter_times):\n    outputs = DNN_Model.forward(x)  # \u524d\u5411\u4f20\u64ad\n    loss = loss_function(outputs,y) # \u8ba1\u7b97loss\n    loss.backward() # \u53cd\u5411\u4f20\u64ad\n    optimizer.step() # \u66f4\u65b0\u53c2\u6570\n    optimizer.zero_grad() # \u68af\u5ea6\u6e05\u96f6\n    \n    if (i+1) % 500 == 0:\n        print(i+1,'iterations have been completed!')\n        print('     --> Now loss = ',loss)\n        print('===========================')\n    \n    adjust_learning_rate(optimizer,i)\n    loss_list.append(loss)\n    if loss < 0.01:\n        break\n```\n\n    500 iterations have been completed!\n         --> Now loss =  tensor(0.6952, grad_fn=)\n    ===========================\n    1000 iterations have been completed!\n         --> Now loss =  tensor(0.5488, grad_fn=)\n    ===========================\n    1500 iterations have been completed!\n         --> Now loss =  tensor(0.3822, grad_fn=)\n    ===========================\n    2000 iterations have been completed!\n         --> Now loss =  tensor(0.3453, grad_fn=)\n    ===========================\n    2500 iterations have been completed!\n         --> Now loss =  tensor(0.1939, grad_fn=)\n    ===========================\n    3000 iterations have been completed!\n         --> Now loss =  tensor(0.1741, grad_fn=)\n    ===========================\n    3500 iterations have been completed!\n         --> Now loss =  tensor(0.1544, grad_fn=)\n    ===========================\n    4000 iterations have been completed!\n         --> Now loss =  tensor(0.1398, grad_fn=)\n    ===========================\n    4500 iterations have been completed!\n         --> Now loss =  tensor(0.1272, grad_fn=)\n    ===========================\n    5000 iterations have been completed!\n         --> Now loss =  tensor(0.1177, grad_fn=)\n    ===========================\n    5500 iterations have been completed!\n         --> Now loss =  tensor(0.1091, grad_fn=)\n    ===========================\n    6000 iterations have been completed!\n         --> Now loss =  tensor(0.1006, grad_fn=)\n    ===========================\n    6500 iterations have been completed!\n         --> Now loss =  tensor(0.0935, grad_fn=)\n    ===========================\n    7000 iterations have been completed!\n         --> Now loss =  tensor(0.0869, grad_fn=)\n    ===========================\n    7500 iterations have been completed!\n         --> Now loss =  tensor(0.0791, grad_fn=)\n    ===========================\n    8000 iterations have been completed!\n         --> Now loss =  tensor(0.0712, grad_fn=)\n    ===========================\n    8500 iterations have been completed!\n         --> Now loss =  tensor(0.0667, grad_fn=)\n    ===========================\n    9000 iterations have been completed!\n         --> Now loss =  tensor(0.0610, grad_fn=)\n    ===========================\n    9500 iterations have been completed!\n         --> Now loss =  tensor(0.0568, grad_fn=)\n    ===========================\n    10000 iterations have been completed!\n         --> Now loss =  tensor(0.0517, grad_fn=)\n    ===========================\n    10500 iterations have been completed!\n         --> Now loss =  tensor(0.0485, grad_fn=)\n    ===========================\n    11000 iterations have been completed!\n         --> Now loss =  tensor(0.0451, grad_fn=)\n    ===========================\n    11500 iterations have been completed!\n         --> Now loss =  tensor(0.0427, grad_fn=)\n    ===========================\n    12000 iterations have been completed!\n         --> Now loss =  tensor(0.0401, grad_fn=)\n    ===========================\n    12500 iterations have been completed!\n         --> Now loss =  tensor(0.0381, grad_fn=)\n    ===========================\n    13000 iterations have been completed!\n         --> Now loss =  tensor(0.0365, grad_fn=)\n    ===========================\n    13500 iterations have been completed!\n         --> Now loss =  tensor(0.0347, grad_fn=)\n    ===========================\n    14000 iterations have been completed!\n         --> Now loss =  tensor(0.0332, grad_fn=)\n    ===========================\n    14500 iterations have been completed!\n         --> Now loss =  tensor(0.0319, grad_fn=)\n    ===========================\n    15000 iterations have been completed!\n         --> Now loss =  tensor(0.0306, grad_fn=)\n    ===========================\n    15500 iterations have been completed!\n         --> Now loss =  tensor(0.0295, grad_fn=)\n    ===========================\n    16000 iterations have been completed!\n         --> Now loss =  tensor(0.0284, grad_fn=)\n    ===========================\n    16500 iterations have been completed!\n         --> Now loss =  tensor(0.0275, grad_fn=)\n    ===========================\n    17000 iterations have been completed!\n         --> Now loss =  tensor(0.0267, grad_fn=)\n    ===========================\n    17500 iterations have been completed!\n         --> Now loss =  tensor(0.0260, grad_fn=)\n    ===========================\n    18000 iterations have been completed!\n         --> Now loss =  tensor(0.0254, grad_fn=)\n    ===========================\n    18500 iterations have been completed!\n         --> Now loss =  tensor(0.0249, grad_fn=)\n    ===========================\n    19000 iterations have been completed!\n         --> Now loss =  tensor(0.0245, grad_fn=)\n    ===========================\n    19500 iterations have been completed!\n         --> Now loss =  tensor(0.0241, grad_fn=)\n    ===========================\n    20000 iterations have been completed!\n         --> Now loss =  tensor(0.0237, grad_fn=)\n    ===========================\n    20500 iterations have been completed!\n         --> Now loss =  tensor(0.0234, grad_fn=)\n    ===========================\n    21000 iterations have been completed!\n         --> Now loss =  tensor(0.0231, grad_fn=)\n    ===========================\n    21500 iterations have been completed!\n         --> Now loss =  tensor(0.0228, grad_fn=)\n    ===========================\n    22000 iterations have been completed!\n         --> Now loss =  tensor(0.0226, grad_fn=)\n    ===========================\n    22500 iterations have been completed!\n         --> Now loss =  tensor(0.0223, grad_fn=)\n    ===========================\n    23000 iterations have been completed!\n         --> Now loss =  tensor(0.0221, grad_fn=)\n    ===========================\n    23500 iterations have been completed!\n         --> Now loss =  tensor(0.0219, grad_fn=)\n    ===========================\n    24000 iterations have been completed!\n         --> Now loss =  tensor(0.0217, grad_fn=)\n    ===========================\n    24500 iterations have been completed!\n         --> Now loss =  tensor(0.0215, grad_fn=)\n    ===========================\n    25000 iterations have been completed!\n         --> Now loss =  tensor(0.0214, grad_fn=)\n    ===========================\n    25500 iterations have been completed!\n         --> Now loss =  tensor(0.0212, grad_fn=)\n    ===========================\n    26000 iterations have been completed!\n         --> Now loss =  tensor(0.0210, grad_fn=)\n    ===========================\n    26500 iterations have been completed!\n         --> Now loss =  tensor(0.0209, grad_fn=)\n    ===========================\n    27000 iterations have been completed!\n         --> Now loss =  tensor(0.0207, grad_fn=)\n    ===========================\n    27500 iterations have been completed!\n         --> Now loss =  tensor(0.0206, grad_fn=)\n    ===========================\n    28000 iterations have been completed!\n         --> Now loss =  tensor(0.0199, grad_fn=)\n    ===========================\n    28500 iterations have been completed!\n         --> Now loss =  tensor(0.0196, grad_fn=)\n    ===========================\n    29000 iterations have been completed!\n         --> Now loss =  tensor(0.0195, grad_fn=)\n    ===========================\n    29500 iterations have been completed!\n         --> Now loss =  tensor(0.0193, grad_fn=)\n    ===========================\n    30000 iterations have been completed!\n         --> Now loss =  tensor(0.0191, grad_fn=)\n    ===========================\n    30500 iterations have been completed!\n         --> Now loss =  tensor(0.0190, grad_fn=)\n    ===========================\n    31000 iterations have been completed!\n         --> Now loss =  tensor(0.0188, grad_fn=)\n    ===========================\n    31500 iterations have been completed!\n         --> Now loss =  tensor(0.0187, grad_fn=)\n    ===========================\n    32000 iterations have been completed!\n         --> Now loss =  tensor(0.0185, grad_fn=)\n    ===========================\n    32500 iterations have been completed!\n         --> Now loss =  tensor(0.0184, grad_fn=)\n    ===========================\n    33000 iterations have been completed!\n         --> Now loss =  tensor(0.0182, grad_fn=)\n    ===========================\n    33500 iterations have been completed!\n         --> Now loss =  tensor(0.0180, grad_fn=)\n    ===========================\n    34000 iterations have been completed!\n         --> Now loss =  tensor(0.0179, grad_fn=)\n    ===========================\n    34500 iterations have been completed!\n         --> Now loss =  tensor(0.0177, grad_fn=)\n    ===========================\n    35000 iterations have been completed!\n         --> Now loss =  tensor(0.0174, grad_fn=)\n    ===========================\n    35500 iterations have been completed!\n         --> Now loss =  tensor(0.0172, grad_fn=)\n    ===========================\n    36000 iterations have been completed!\n         --> Now loss =  tensor(0.0169, grad_fn=)\n    ===========================\n    36500 iterations have been completed!\n         --> Now loss =  tensor(0.0165, grad_fn=)\n    ===========================\n    37000 iterations have been completed!\n         --> Now loss =  tensor(0.0161, grad_fn=)\n    ===========================\n    37500 iterations have been completed!\n         --> Now loss =  tensor(0.0157, grad_fn=)\n    ===========================\n    38000 iterations have been completed!\n         --> Now loss =  tensor(0.0152, grad_fn=)\n    ===========================\n    38500 iterations have been completed!\n         --> Now loss =  tensor(0.0147, grad_fn=)\n    ===========================\n    39000 iterations have been completed!\n         --> Now loss =  tensor(0.0143, grad_fn=)\n    ===========================\n    39500 iterations have been completed!\n         --> Now loss =  tensor(0.0140, grad_fn=)\n    ===========================\n    40000 iterations have been completed!\n         --> Now loss =  tensor(0.0137, grad_fn=)\n    ===========================\n    40500 iterations have been completed!\n         --> Now loss =  tensor(0.0135, grad_fn=)\n    ===========================\n    41000 iterations have been completed!\n         --> Now loss =  tensor(0.0133, grad_fn=)\n    ===========================\n    41500 iterations have been completed!\n         --> Now loss =  tensor(0.0131, grad_fn=)\n    ===========================\n    42000 iterations have been completed!\n         --> Now loss =  tensor(0.0129, grad_fn=)\n    ===========================\n    42500 iterations have been completed!\n         --> Now loss =  tensor(0.0128, grad_fn=)\n    ===========================\n    43000 iterations have been completed!\n         --> Now loss =  tensor(0.0126, grad_fn=)\n    ===========================\n    43500 iterations have been completed!\n         --> Now loss =  tensor(0.0125, grad_fn=)\n    ===========================\n    44000 iterations have been completed!\n         --> Now loss =  tensor(0.0124, grad_fn=)\n    ===========================\n    44500 iterations have been completed!\n         --> Now loss =  tensor(0.0123, grad_fn=)\n    ===========================\n    45000 iterations have been completed!\n         --> Now loss =  tensor(0.0123, grad_fn=)\n    ===========================\n    45500 iterations have been completed!\n         --> Now loss =  tensor(0.0122, grad_fn=)\n    ===========================\n    46000 iterations have been completed!\n         --> Now loss =  tensor(0.0121, grad_fn=)\n    ===========================\n    46500 iterations have been completed!\n         --> Now loss =  tensor(0.0120, grad_fn=)\n    ===========================\n    47000 iterations have been completed!\n         --> Now loss =  tensor(0.0120, grad_fn=)\n    ===========================\n    47500 iterations have been completed!\n         --> Now loss =  tensor(0.0119, grad_fn=)\n    ===========================\n    48000 iterations have been completed!\n         --> Now loss =  tensor(0.0119, grad_fn=)\n    ===========================\n    48500 iterations have been completed!\n         --> Now loss =  tensor(0.0118, grad_fn=)\n    ===========================\n    49000 iterations have been completed!\n         --> Now loss =  tensor(0.0118, grad_fn=)\n    ===========================\n    49500 iterations have been completed!\n         --> Now loss =  tensor(0.0117, grad_fn=)\n    ===========================\n    50000 iterations have been completed!\n         --> Now loss =  tensor(0.0117, grad_fn=)\n    ===========================\n    50500 iterations have been completed!\n         --> Now loss =  tensor(0.0116, grad_fn=)\n    ===========================\n    51000 iterations have been completed!\n         --> Now loss =  tensor(0.0116, grad_fn=)\n    ===========================\n    51500 iterations have been completed!\n         --> Now loss =  tensor(0.0115, grad_fn=)\n    ===========================\n    52000 iterations have been completed!\n         --> Now loss =  tensor(0.0115, grad_fn=)\n    ===========================\n    52500 iterations have been completed!\n         --> Now loss =  tensor(0.0115, grad_fn=)\n    ===========================\n    53000 iterations have been completed!\n         --> Now loss =  tensor(0.0114, grad_fn=)\n    ===========================\n    53500 iterations have been completed!\n         --> Now loss =  tensor(0.0114, grad_fn=)\n    ===========================\n    54000 iterations have been completed!\n         --> Now loss =  tensor(0.0113, grad_fn=)\n    ===========================\n    54500 iterations have been completed!\n         --> Now loss =  tensor(0.0113, grad_fn=)\n    ===========================\n    55000 iterations have been completed!\n         --> Now loss =  tensor(0.0113, grad_fn=)\n    ===========================\n    55500 iterations have been completed!\n         --> Now loss =  tensor(0.0112, grad_fn=)\n    ===========================\n    56000 iterations have been completed!\n         --> Now loss =  tensor(0.0112, grad_fn=)\n    ===========================\n    56500 iterations have been completed!\n         --> Now loss =  tensor(0.0111, grad_fn=)\n    ===========================\n    57000 iterations have been completed!\n         --> Now loss =  tensor(0.0111, grad_fn=)\n    ===========================\n    57500 iterations have been completed!\n         --> Now loss =  tensor(0.0111, grad_fn=)\n    ===========================\n    58000 iterations have been completed!\n         --> Now loss =  tensor(0.0110, grad_fn=)\n    ===========================\n    58500 iterations have been completed!\n         --> Now loss =  tensor(0.0110, grad_fn=)\n    ===========================\n    59000 iterations have been completed!\n         --> Now loss =  tensor(0.0109, grad_fn=)\n    ===========================\n    59500 iterations have been completed!\n         --> Now loss =  tensor(0.0109, grad_fn=)\n    ===========================\n    60000 iterations have been completed!\n         --> Now loss =  tensor(0.0108, grad_fn=)\n    ===========================\n    60500 iterations have been completed!\n         --> Now loss =  tensor(0.0108, grad_fn=)\n    ===========================\n    61000 iterations have been completed!\n         --> Now loss =  tensor(0.0107, grad_fn=)\n    ===========================\n    61500 iterations have been completed!\n         --> Now loss =  tensor(0.0107, grad_fn=)\n    ===========================\n    62000 iterations have been completed!\n         --> Now loss =  tensor(0.0106, grad_fn=)\n    ===========================\n    62500 iterations have been completed!\n         --> Now loss =  tensor(0.0106, grad_fn=)\n    ===========================\n    63000 iterations have been completed!\n         --> Now loss =  tensor(0.0105, grad_fn=)\n    ===========================\n    63500 iterations have been completed!\n         --> Now loss =  tensor(0.0104, grad_fn=)\n    ===========================\n    64000 iterations have been completed!\n         --> Now loss =  tensor(0.0103, grad_fn=)\n    ===========================\n    64500 iterations have been completed!\n         --> Now loss =  tensor(0.0101, grad_fn=)\n    ===========================\n\n\n\n```python\nimport matplotlib.pyplot as plt\nplt.figure(figsize=(16,8))\nlength = loss_list.__len__()\nprint('The length of loss_list is:',length)\nplt.plot(np.arange(0,4000,1),loss_list[:4000],'black')\nplt.xlabel('epoch')\nplt.ylabel('loss')\nplt.show()\n```\n\nSoftmax\u662f\u4e00\u4e2a\u4ea7\u751f\u6fc0\u6d3b\u4f5c\u7528\u7684\u6982\u7387\u5206\u5e03\u5bc6\u5ea6\u51fd\u6570  \n\u4e3a\u4ec0\u4e48dim\u9700\u8981\u7b49\u4e8e1\uff1f   \n\u9996\u5148dim=1\u610f\u601d\u662f\u4f7f\u5f97\u5728softmax\u64cd\u4f5c\u4e4b\u540e\u5728dim\u8fd9\u4e2a\u7ef4\u5ea6\u76f8\u52a0\u7b49\u4e8e1  \n\n\n```python\nProbability_Calculator = nn.Softmax(dim=1)\npred = []\n# x.detach()\u90fd\u662f\u4ece\u539f\u6709\u8ba1\u7b97\u4e2d\u5206\u79bb\u51fa\u6765\u7684\u4e00\u4e2atensor\u53d8\u91cf,\u4f46\u662f\u6ca1\u6709\u66f4\u6539\u8fd9\u4e2atensor\u65f6\uff0c\u5e76\u4e0d\u4f1a\u5f71\u54cdbackward()\n# numpy()\u5c06tensor\u8f6c\u53d8\u4e3aarray\u6570\u7ec4\nprob = Probability_Calculator(DNN_Model.forward(x)).detach().numpy()   \nprob\n```\n\n\n\n\n    array([[9.9999940e-01, 5.8749430e-07],\n           [9.9939322e-01, 6.0680578e-04],\n           [2.8025184e-04, 9.9971968e-01],\n           ...,\n           [9.9690032e-01, 3.0997563e-03],\n           [5.8808475e-04, 9.9941194e-01],\n           [9.8336065e-01, 1.6639415e-02]], dtype=float32)\n\n\n\n\n```python\nProbability_Calculator = nn.Softmax(dim=1)   # softmax\u7684dim\u8fd9\u4e2a\u53c2\u6570 \u610f\u601d\u5c31\u662f\u4f7f\u5f97\u5728softmax\u64cd\u4f5c\u4e4b\u540e\u5728dim\u8fd9\u4e2a\u7ef4\u5ea6\u76f8\u52a0\u7b49\u4e8e1\npred = []\n# x.detach()\u90fd\u662f\u4ece\u539f\u6709\u8ba1\u7b97\u4e2d\u5206\u79bb\u51fa\u6765\u7684\u4e00\u4e2atensor\u53d8\u91cf,\u4f46\u662f\u6ca1\u6709\u66f4\u6539\u8fd9\u4e2atensor\u65f6\uff0c\u5e76\u4e0d\u4f1a\u5f71\u54cdbackward()\n# numpy()\u5c06tensor\u8f6c\u53d8\u4e3aarray\u6570\u7ec4\nprob = Probability_Calculator(DNN_Model.forward(x)).detach().numpy()\nfor i in range(prob.shape[0]):\n    pred.append(np.argmax(prob[i,:]))\n\npred = np.array(pred)\npred.shape\naccuracy_score(pred,y)\n```\n\n\n\n\n    0.997997997997998\n\n\n\n## **Prediction\uff0cBacktesting\uff0cComputing Gain/loss**\n\n\n```python\nx = test_set.iloc[:,0:24]\ny = test_set.iloc[:,24]\nx = np.array(x)\ny = np.array(y)\nx = Variable(torch.FloatTensor(x))\ny = Variable(torch.LongTensor(y))\n```\n\n\n```python\nProbability_Calculator = nn.Softmax(dim=1)\npred = []\nprob = Probability_Calculator(DNN_Model.forward(x)).detach().numpy()\nfor i in range(prob.shape[0]):\n    if np.max(prob[i,:]) >= 0.95:\n        pred.append(np.argmax(prob[i,:]))\n    else:\n        pred.append(np.nan)\npred = np.array(pred)\npred\n```\n\n\n\n\n    array([ 1.,  1.,  0.,  1.,  1.,  0.,  1.,  0.,  0.,  0.,  1.,  1.,  1.,\n            0.,  1.,  1.,  1.,  0.,  0.,  1.,  1.,  1.,  1.,  0.,  1.,  1.,\n            0.,  0.,  0.,  1.,  1.,  1.,  1.,  1.,  0.,  1.,  1.,  1.,  1.,\n            0.,  1.,  0.,  1.,  1.,  0.,  0.,  1.,  1.,  1.,  1.,  1.,  1.,\n            1.,  1.,  0., nan,  1.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,\n            1.,  1.,  1.,  0.,  0.,  1.,  1.,  0.,  0.,  0.,  1.,  1.,  1.,\n            1.,  1.,  0.,  1.,  0.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,\n            1.,  1.,  0.,  0.,  0.,  1.,  0.,  1., nan, nan,  1.,  1.,  0.,\n            0.,  0.,  0., nan,  0.,  0.,  1.,  1.,  1.,  0.,  0.,  1.,  0.,\n            1.,  0.,  1.,  1.,  1.,  1.,  1.,  1.,  0.,  1.,  0.,  1.,  1.,\n            0.,  0.,  0.,  1.,  1., nan,  1.,  0.,  1., nan,  1.,  0.,  0.,\n           nan,  0.,  1.,  0.,  0.,  1.,  0.,  1.,  0.,  1.,  0., nan,  1.,\n            1.,  1.,  1.,  0.,  1.,  1.,  1.,  0.,  0.,  0.,  1.,  1.,  1.,\n            1.,  1.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  1.,  0., nan,\n            0.,  0.,  0.,  1.,  1.,  1., nan,  0.,  0.,  0.,  1.,  0.,  1.,\n            1.,  0.,  1.,  0.,  0.,  0.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,\n            0.,  1.,  1.,  1.,  0.,  1.,  1.,  1.,  0.,  0.,  1.,  1.,  1.,\n            0.,  0.,  0.,  0.,  1.,  1.,  0.,  0.,  1.,  1., nan,  0.,  0.,\n            1.,  0.,  0.,  1.,  0.,  1.,  0.,  0.,  1.,  1.,  0.,  1.,  1.,\n            1.,  0.,  1.,  1.,  1.,  1.,  1.,  0.,  1.,  0.,  1.,  0.,  1.,\n            0.,  0.,  1.,  1.,  0.,  0.,  0., nan,  1.,  0.,  1.,  1.,  1.,\n            1.,  0.,  1.,  1.,  1.,  0.,  0.,  1.,  0.,  1.,  1.,  1.,  0.,\n            0.,  0.,  0.,  1., nan,  1.,  0., nan,  0.,  1.,  1.,  0.,  1.,\n            1.,  0.,  1.,  1.,  0.,  1.,  1.,  1.,  1.,  0.,  0.,  0.,  0.,\n            1.,  0.,  1.,  0.,  1.,  0.,  0.,  0.,  0.,  1.,  1.,  1.,  0.,\n            1.,  1.,  0.,  0.,  1.,  1.,  1.,  0.,  1.,  1.,  1.,  0.,  1.,\n            0.,  1.,  1.,  1.,  1.,  1.,  0.,  1.,  1.,  1.,  1.,  1.,  0.,\n            1.,  0.,  1.,  0.,  0.,  1.,  1.,  1.,  0.,  1.,  0.,  0.,  0.,\n            1.,  1.,  0.,  0.,  1.,  1.,  1.,  1., nan,  1.,  1., nan,  0.,\n            1.,  1.,  0.,  0.,  0.,  1.,  1.,  0.,  0.,  1.,  1.,  1.,  1.,\n           nan,  1.,  0.,  1.,  1.,  0.,  0.,  1.,  0.,  1.,  1.,  0.,  0.,\n            0.,  0.,  1.,  1.,  1.,  0.,  1.,  0.,  1.,  0.,  1.,  1., nan,\n            1.,  0.,  1.,  0.,  1.,  0.,  1.,  1.,  0.,  0.,  0.,  1.,  1.,\n           nan,  0.,  0.,  1.,  1.,  1.,  1.,  0.,  1.,  1.,  0.,  1.,  1.,\n            0.,  1.,  1.,  0.,  1.,  1.,  0.,  1.,  1.,  0.,  0.,  1.,  1.,\n            1.,  0.,  0.,  1.,  0.,  1.,  0.,  1.,  1.,  1.,  1., nan,  0.,\n            0.,  0., nan,  0.,  1.,  1.,  1.,  1.,  1.,  0.,  0.,  1.,  1.,\n            0.,  1.,  0.,  0., nan,  1.,  0.,  1., nan,  0.,  0., nan,  1.,\n            0.,  0.,  0., nan,  0.,  1.,  0., nan,  1.,  0.,  0.,  1.,  1.,\n            0.,  0.,  1.,  1.,  0.,  1.,  0., nan, nan,  0.,  0.,  1.,  0.,\n            0.,  1.,  0.,  0.,  0.,  1.,  0.,  0.,  1.,  0.,  1.,  1.,  0.,\n            1.,  0.,  1.,  1.,  1.,  1.,  1.,  1., nan,  1.,  1.,  1.,  0.,\n            0.,  1.,  1.,  1.,  1., nan,  1.,  0.,  0.,  1.,  0.,  0.,  1.,\n            1., nan,  0.,  1., nan,  0.,  1.,  1.,  0.,  1.,  0., nan, nan,\n           nan,  1.,  1.,  1.,  1.,  0.,  0.,  0.,  1., nan,  0.,  1.,  1.,\n            1.,  1.,  0.,  1.,  0., nan,  0.,  0.,  1.,  0.,  1.])\n\n\n\n\n```python\npred.shape\n```\n\n\n\n\n    (596,)\n\n\n\n\n```python\nhs300 = np.exp(test_set['Close0'])\npred = 2 * pred-1\nrtn_temp = 1.0\ncum_rtn = []\ncum_rtn.append(rtn_temp)\nfor i in range(pred.shape[0]-1):\n    if pred[i] == 1: #  or pred[i] == -1\n        rtn_temp = rtn_temp * (1.0 + ((hs300[i+1]-hs300[i]) / hs300[i]) * pred[i])\n    else:\n        rtn_temp = rtn_temp\n    cum_rtn.append(rtn_temp)\n```\n\n\n```python\ncum_rtn = np.array(cum_rtn)\ncum_rtn\n```\n\n\n\n\n    array([1.        , 1.02504744, 1.0273072 , 1.0273072 , 1.02202574,\n           0.99750313, 0.99750313, 0.98316189, 0.98316189, 0.98316189,\n           0.98316189, 0.98675221, 0.98832743, 1.01247292, 1.01247292,\n           1.00842323, 0.99785831, 0.99291828, 0.99291828, 0.99291828,\n           0.97350798, 0.96310033, 0.96745702, 0.95341585, 0.95341585,\n           0.94685415, 0.95707076, 0.95707076, 0.95707076, 0.95707076,\n           0.96967293, 0.968985  , 0.99836371, 0.98937973, 1.00033966,\n           1.00033966, 1.01070577, 0.96721925, 0.96656972, 0.96448592,\n           0.96448592, 0.9788786 , 0.9788786 , 0.97090092, 0.97631122,\n           0.97631122, 0.97631122, 1.01845365, 0.99140925, 0.99289476,\n           0.99479758, 0.98836037, 0.95822971, 0.96862206, 0.98218786,\n           0.98218786, 0.98218786, 0.97400558, 0.96812782, 0.96812782,\n           0.96812782, 0.96812782, 0.96812782, 0.96812782, 0.96812782,\n           0.97943444, 0.98405617, 0.99521174, 0.97219675, 0.97219675,\n           0.97219675, 0.9507381 , 0.95006061, 0.95006061, 0.95006061,\n           0.95006061, 0.9606705 , 0.98739507, 0.98944196, 0.98468507,\n           0.96338959, 0.96338959, 0.9522464 , 0.9522464 , 0.95549808,\n           0.97028888, 0.95408169, 0.95266228, 0.94278668, 0.93154591,\n           0.92440064, 0.9129429 , 0.91559488, 0.90929041, 0.90929041,\n           0.90929041, 0.90929041, 0.89687417, 0.89687417, 0.91836099,\n           0.91836099, 0.91836099, 0.9276359 , 0.92589121, 0.92589121,\n           0.92589121, 0.92589121, 0.92589121, 0.92589121, 0.92589121,\n           0.92589121, 0.9135911 , 0.91293133, 0.91808447, 0.91808447,\n           0.91808447, 0.92102289, 0.92102289, 0.93065903, 0.93065903,\n           0.94758766, 0.95442847, 0.97354088, 0.97500533, 0.95682432,\n           0.98750048, 0.98750048, 0.99103751, 0.99103751, 1.01332118,\n           1.0735887 , 1.0735887 , 1.0735887 , 1.0735887 , 1.09709468,\n           1.11008236, 1.11008236, 1.11941447, 1.11941447, 1.07495648,\n           1.07495648, 1.08227666, 1.08227666, 1.08227666, 1.08227666,\n           1.08227666, 1.07727797, 1.07727797, 1.07727797, 1.07641039,\n           1.07641039, 1.06421934, 1.06421934, 1.05995779, 1.05995779,\n           1.05995779, 1.05925363, 1.07282207, 1.08350986, 1.08217622,\n           1.08217622, 1.08494311, 1.06150419, 1.05912498, 1.05912498,\n           1.05912498, 1.05912498, 1.05519657, 1.06777213, 1.04315482,\n           1.04144456, 1.04431572, 1.04431572, 1.04431572, 1.04431572,\n           1.04431572, 0.98331206, 0.98331206, 0.98331206, 0.98331206,\n           1.01902838, 1.01902838, 1.01902838, 1.01902838, 1.01902838,\n           1.01902838, 1.01037906, 1.02406102, 1.01920154, 1.01920154,\n           1.01920154, 1.01920154, 1.01920154, 1.01688407, 1.01688407,\n           1.01370315, 1.01432314, 1.01432314, 1.01393697, 1.01393697,\n           1.01393697, 1.01393697, 1.00625464, 1.004698  , 0.99638047,\n           0.99636411, 0.9998536 , 1.01302643, 1.04371759, 1.04371759,\n           1.04571304, 1.03483469, 1.03293452, 1.03293452, 1.03044836,\n           1.06013685, 1.06050318, 1.06050318, 1.06050318, 1.06600681,\n           1.04125142, 1.03860639, 1.03860639, 1.03860639, 1.03860639,\n           1.03860639, 1.03389433, 1.03329684, 1.03329684, 1.03329684,\n           1.02616571, 1.02839893, 1.02839893, 1.02839893, 1.02839893,\n           1.02725288, 1.02725288, 1.02725288, 1.01871155, 1.01871155,\n           0.99920689, 0.99920689, 0.99920689, 1.01241218, 1.00254546,\n           1.00254546, 0.9935068 , 0.99801935, 1.00116324, 1.00116324,\n           1.02289692, 1.02199034, 1.02037953, 1.02354987, 1.03092934,\n           1.03092934, 1.0449017 , 1.0449017 , 1.04149708, 1.04149708,\n           1.05485435, 1.05485435, 1.05485435, 1.06552776, 1.07182268,\n           1.07182268, 1.07182268, 1.07182268, 1.07182268, 1.06786714,\n           1.06786714, 1.07304289, 1.07696724, 1.08006006, 1.06771346,\n           1.06771346, 1.05947518, 1.05130807, 1.05445832, 1.05445832,\n           1.05445832, 1.05598601, 1.05598601, 1.06609444, 1.07740748,\n           1.07277707, 1.07277707, 1.07277707, 1.07277707, 1.07277707,\n           1.07693456, 1.07693456, 1.0768205 , 1.0768205 , 1.0768205 ,\n           1.0768205 , 1.07158817, 1.07035445, 1.07035445, 1.07732244,\n           1.08400879, 1.08400879, 1.085913  , 1.08077978, 1.08077978,\n           1.08097639, 1.07994905, 1.08157729, 1.07361055, 1.07361055,\n           1.07361055, 1.07361055, 1.07361055, 1.06267736, 1.06267736,\n           1.06636009, 1.06636009, 1.06269515, 1.06269515, 1.06269515,\n           1.06269515, 1.06269515, 1.0708493 , 1.07720645, 1.07529074,\n           1.07529074, 1.07594413, 1.0727103 , 1.0727103 , 1.0727103 ,\n           1.08730436, 1.08487784, 1.08336329, 1.08336329, 1.06983894,\n           1.07681281, 1.07624919, 1.07624919, 1.07519058, 1.07519058,\n           1.07912873, 1.0937908 , 1.09187308, 1.0877479 , 1.09589554,\n           1.09589554, 1.10976638, 1.10945192, 1.12032475, 1.11656723,\n           1.1103953 , 1.1103953 , 1.11195022, 1.11195022, 1.09295119,\n           1.09295119, 1.09295119, 1.00681772, 1.0333724 , 1.04508015,\n           1.04508015, 1.04510427, 1.04510427, 1.04510427, 1.04510427,\n           1.03867537, 1.04596985, 1.04596985, 1.04596985, 1.04437158,\n           1.06843592, 1.06968102, 1.06538888, 1.06538888, 1.05225705,\n           1.05532106, 1.05532106, 1.05532106, 1.06094557, 1.06708871,\n           1.06708871, 1.06708871, 1.06708871, 1.08994081, 1.07544468,\n           1.07544468, 1.07544468, 1.02920838, 1.02419464, 1.00392432,\n           0.9909013 , 0.9909013 , 0.95756312, 0.95756312, 0.98329372,\n           0.97683003, 0.97683003, 0.97683003, 0.98003375, 0.98003375,\n           0.99588728, 0.99020454, 0.99020454, 0.99020454, 0.99020454,\n           0.99020454, 0.98602218, 1.00505296, 0.99760773, 0.99760773,\n           1.00734406, 1.00734406, 0.9954733 , 0.9954733 , 0.99297644,\n           0.98447725, 0.98447725, 0.99127089, 0.99127089, 1.00294713,\n           1.00294713, 1.00005263, 1.00005263, 0.99918469, 0.99919983,\n           0.99919983, 0.99919983, 0.99919983, 1.00177647, 1.01029036,\n           1.01029036, 1.01029036, 1.01029036, 1.01168002, 1.02315921,\n           1.01598372, 1.01889513, 1.01889513, 1.04638171, 1.04960411,\n           1.04960411, 1.04921943, 1.05424132, 1.05424132, 1.06081534,\n           1.05889661, 1.05889661, 1.06080459, 1.04806088, 1.04806088,\n           1.04884929, 1.0558432 , 1.0558432 , 1.0558432 , 1.06092416,\n           1.06535134, 1.05781741, 1.05781741, 1.05781741, 1.07974685,\n           1.07974685, 1.14094374, 1.14094374, 1.15936881, 1.17558395,\n           1.15430052, 1.1785443 , 1.1785443 , 1.1785443 , 1.1785443 ,\n           1.1785443 , 1.1785443 , 1.1785443 , 1.18442566, 1.18392069,\n           1.13195313, 1.13769632, 1.1476979 , 1.1476979 , 1.1476979 ,\n           1.1572864 , 1.17608378, 1.17608378, 1.17640638, 1.17640638,\n           1.17640638, 1.17640638, 1.16567776, 1.16567776, 1.16268558,\n           1.16268558, 1.16268558, 1.16268558, 1.16268558, 1.14759491,\n           1.14759491, 1.14759491, 1.14759491, 1.14759491, 1.14759491,\n           1.1749838 , 1.1749838 , 1.1749838 , 1.17541331, 1.17541331,\n           1.17541331, 1.15056321, 1.1567283 , 1.1567283 , 1.1567283 ,\n           1.16816436, 1.17416514, 1.17416514, 1.16637156, 1.16637156,\n           1.16637156, 1.16637156, 1.16637156, 1.16637156, 1.14399346,\n           1.14399346, 1.14399346, 1.14646275, 1.14646275, 1.14646275,\n           1.14646275, 1.15027545, 1.15027545, 1.15027545, 1.14858315,\n           1.14858315, 1.15775637, 1.15760179, 1.15760179, 1.14318865,\n           1.14318865, 1.14514788, 1.1544055 , 1.1630929 , 1.14418532,\n           1.15039931, 1.16422361, 1.16422361, 1.18150438, 1.18165192,\n           1.20478082, 1.20478082, 1.20478082, 1.20565526, 1.1929784 ,\n           1.20460152, 1.20229752, 1.20229752, 1.21122442, 1.21122442,\n           1.21122442, 1.2037853 , 1.2037853 , 1.2037853 , 1.21875557,\n           1.21373448, 1.21373448, 1.21373448, 1.21132002, 1.21132002,\n           1.21132002, 1.20833889, 1.19213567, 1.19213567, 1.17985604,\n           1.17985604, 1.17985604, 1.17985604, 1.17985604, 1.17573857,\n           1.18676001, 1.16746133, 1.17741989, 1.17741989, 1.17741989,\n           1.17741989, 1.17242835, 1.17242835, 1.17242835, 1.18512389,\n           1.20779722, 1.21885942, 1.24045511, 1.24045511, 1.22820502,\n           1.22820502, 1.22820502, 1.22820502, 1.22820502, 1.24180554,\n           1.24180554])\n\n\n\n\n```python\ntime = test_set.index\ntime\n```\n\n\n\n\n    Index(['2018-08-08', '2018-08-09', '2018-08-10', '2018-08-13', '2018-08-14',\n           '2018-08-15', '2018-08-16', '2018-08-17', '2018-08-20', '2018-08-21',\n           ...\n           '2021-01-06', '2021-01-07', '2021-01-08', '2021-01-11', '2021-01-12',\n           '2021-01-13', '2021-01-14', '2021-01-15', '2021-01-18', '2021-01-19'],\n          dtype='object', name='date', length=596)\n\n\n\n\n```python\nfontTNR = 'Times New Roman'\nplt.figure(figsize=(20,8))\nplt.plot(time,cum_rtn,color='black')\nplt.xticks(test_set.index[::30])\nplt.grid()\n\nplt.title('Backtesting Curve',fontsize = 15 , fontproperties = fontTNR)\nplt.xlabel('Time',fontsize = 20 ,fontproperties = fontTNR)\nplt.ylabel('Cumulative Return',fontsize = 20 ,fontproperties = fontTNR)\n\nplt.xticks(fontsize = 15 , fontproperties = fontTNR,rotation=25)\nplt.yticks(fontsize = 15 , fontproperties = fontTNR)\nplt.show()\n```\n\n\n```python\nDNN_Model.state_dict()\n```\n\n\n\n\n    OrderedDict([('FC_layer1.weight',\n                  tensor([[-0.0861, -0.5635, -0.5108,  ...,  0.2102, -0.1364, -0.0351],\n                          [ 0.3539,  0.0451, -0.6862,  ...,  0.1420,  0.3328,  0.3169],\n                          [-0.0257, -0.1801,  0.1551,  ...,  0.1295, -0.0307,  0.3293],\n                          ...,\n                          [-0.6684, -0.7623, -0.8492,  ..., -0.4728, -0.2793, -0.3192],\n                          [-0.0980,  0.4759,  0.2353,  ..., -0.3693, -0.1182, -0.0647],\n                          [-0.3859,  0.1895, -0.0720,  ..., -0.2732, -0.2560, -0.2556]])),\n                 ('FC_layer1.bias',\n                  tensor([ 0.1029, -0.0978,  0.0527,  0.1733, -0.1503, -0.2079,  0.0588, -0.1384,\n                           0.0311, -0.1918, -0.0311, -0.1319, -0.1119, -0.0931, -0.0207, -0.1940,\n                           0.0717,  0.0722, -0.1249,  0.0102,  0.1022, -0.1759, -0.2364, -0.0517,\n                          -0.0139,  0.0080,  0.0323,  0.1461,  0.0820, -0.1322,  0.1227, -0.0949,\n                           0.0391,  0.1717, -0.2085, -0.1996,  0.2035, -0.0375, -0.1390,  0.0097,\n                           0.0895,  0.1946, -0.1485,  0.1287,  0.0374,  0.0834, -0.0059, -0.0691])),\n                 ('FC_layer2.weight',\n                  tensor([[-0.2359,  0.4644, -0.4710,  ...,  0.1371,  0.0869,  0.1297],\n                          [ 0.1311, -0.2098, -0.0183,  ..., -0.2080,  0.0376, -0.2414],\n                          [-0.0874, -0.2564, -0.1380,  ...,  0.0240,  0.0213, -0.1071],\n                          ...,\n                          [-0.0534, -0.0920, -0.1771,  ..., -0.1108, -0.0409, -0.0428],\n                          [-0.0476, -0.1462, -0.2311,  ..., -0.0941, -0.1929, -0.1422],\n                          [ 0.0466,  0.0469, -0.1201,  ..., -0.1148,  0.0119,  0.0596]])),\n                 ('FC_layer2.bias',\n                  tensor([ 0.0140, -0.0252, -0.1434,  0.1166,  0.2894,  0.0356, -0.0140,  0.0937,\n                          -0.0209,  0.0754, -0.1831, -0.0963, -0.1933, -0.0041, -0.0215,  0.0584,\n                          -0.1314,  0.2693, -0.1370,  0.0254, -0.2048, -0.0788, -0.0190, -0.0352,\n                           0.0127, -0.1445, -0.1950,  0.1544, -0.0668,  0.0474, -0.1389, -0.3142,\n                           0.1560, -0.0246, -0.1696, -0.0040])),\n                 ('FC_layer3.weight',\n                  tensor([[-1.5763e-01,  1.8868e-02, -1.5444e-01, -1.6050e-02, -8.5780e-02,\n                            4.8827e-02,  1.7089e-01, -5.0028e-02, -1.8344e-01, -1.2989e-01,\n                           -1.5731e-02,  1.3388e-01, -1.9808e-02,  1.7498e-01, -1.9790e-01,\n                           -9.7508e-02, -1.9234e-01,  2.0517e-01, -1.9292e-01, -8.1862e-02,\n                           -1.9776e-01, -8.2192e-02, -5.8181e-02,  1.3059e-01,  6.8316e-03,\n                           -3.0491e-02, -1.7550e-01, -7.1686e-02,  4.6621e-02, -1.6485e-01,\n                           -9.9171e-02, -2.2526e-01, -5.9426e-02, -1.3636e-01, -1.3918e-01,\n                            7.2536e-02],\n                          [-4.0250e-01,  1.6652e-01,  1.2620e-01,  3.1609e-01,  3.2940e-01,\n                            1.6882e-01,  1.7309e-01,  8.1675e-02,  1.5435e-02,  3.9453e-02,\n                           -3.2551e-01,  1.4842e-01,  8.7654e-02,  2.6551e-01, -3.4158e-01,\n                           -2.0640e-01, -3.4906e-01,  4.2143e-01, -5.6415e-02, -1.1820e-01,\n                           -3.0409e-01, -4.5581e-02, -7.7152e-02,  1.7091e-01, -1.6446e-02,\n                            2.6684e-02, -3.6083e-01,  3.0810e-01, -3.5481e-01, -3.6783e-02,\n                           -1.4129e-02, -5.4736e-01,  3.6685e-01,  4.5680e-02, -9.9722e-02,\n                           -4.4551e-02],\n                          [-4.0733e-02,  6.4404e-02,  2.2875e-02,  3.3654e-01,  3.8175e-02,\n                           -1.6680e-02,  3.3439e-01, -3.8119e-03, -1.1310e-01, -4.0748e-02,\n                           -2.8271e-01,  2.2232e-04,  6.1828e-02,  2.2192e-01, -2.7651e-01,\n                           -1.3772e-01, -2.7409e-01,  1.7964e-01, -2.5535e-01, -1.0441e-01,\n                           -1.1541e-01,  2.5392e-02,  8.5710e-02, -5.9037e-02, -3.1696e-02,\n                           -1.6340e-01, -2.5830e-01,  9.4642e-02, -3.2512e-01, -1.9128e-02,\n                            7.2878e-02, -3.8466e-01,  4.2642e-01,  1.4116e-02, -6.4068e-02,\n                            1.0759e-01],\n                          [-1.0792e-01, -6.4718e-02,  2.9305e-02,  4.5725e-02, -8.7402e-02,\n                            1.7677e-01, -1.1388e-01, -1.5855e-01, -5.2758e-03, -1.0293e-02,\n                           -1.5989e-01, -4.7772e-02, -3.6837e-03,  1.3837e-01, -1.3800e-01,\n                            9.1490e-02, -1.4130e-01,  1.1627e-01, -1.7613e-01, -1.4890e-01,\n                           -1.1985e-01, -6.1589e-02, -1.5684e-01,  7.8940e-02, -1.9377e-01,\n                            6.4707e-02, -6.6567e-02,  1.6677e-01, -1.7708e-01, -1.3520e-01,\n                           -9.7980e-02, -2.4255e-01,  2.4082e-01, -6.5259e-02, -4.9521e-02,\n                           -1.8615e-02],\n                          [ 2.5957e-01, -7.5980e-02,  9.6908e-03, -2.4372e-01, -4.8096e-01,\n                           -2.4781e-01, -2.2816e-01, -1.3493e-02, -9.3710e-02, -1.4484e-01,\n                            2.4468e-01,  6.8252e-02,  3.5550e-02, -2.3865e-01,  3.4078e-01,\n                            5.4542e-02,  2.6003e-01, -4.2743e-01,  1.9571e-01, -2.1898e-01,\n                            3.6585e-01, -1.3748e-01, -2.3727e-01, -1.4570e-01, -3.0610e-02,\n                            1.5240e-02,  3.6015e-01, -8.9189e-02,  1.4875e-01,  3.0533e-02,\n                            3.0456e-03,  4.5884e-01, -6.1494e-01,  2.7697e-02,  2.0080e-02,\n                            1.6710e-03],\n                          [-2.4429e-01,  1.2640e-01,  1.3220e-01,  2.2486e-01,  1.5426e-01,\n                            1.5394e-01,  2.7562e-01,  1.4497e-02, -2.6055e-02,  1.2952e-03,\n                           -3.0473e-01, -4.2380e-02, -4.5484e-03,  1.4820e-01, -2.7164e-01,\n                            5.3840e-02, -1.9368e-01,  4.2370e-01, -5.8003e-02, -6.1301e-02,\n                           -3.3806e-01,  6.2851e-02,  1.3354e-01,  1.7504e-01,  2.8834e-02,\n                           -1.5985e-01, -3.7975e-01,  3.0278e-02, -3.6712e-01,  5.4724e-02,\n                            3.7224e-02, -3.1202e-01,  4.9635e-01, -1.1616e-01,  7.0275e-02,\n                            6.0680e-03],\n                          [-2.4788e-01,  1.3572e-01,  1.3666e-01,  1.8277e-01,  1.7210e-01,\n                            1.1230e-01,  4.3597e-01, -2.1358e-03, -6.0739e-02, -5.7898e-02,\n                           -3.6411e-01, -5.8389e-03,  7.0215e-02,  2.1818e-01, -2.8247e-01,\n                            1.0756e-01, -1.4074e-01,  5.8754e-01, -2.6964e-01,  2.0395e-01,\n                           -3.2725e-01, -1.7093e-02,  6.9262e-02,  1.0324e-01, -6.2924e-02,\n                           -9.9214e-02, -4.3801e-01,  2.3702e-01, -3.7086e-01,  5.7940e-02,\n                           -1.2366e-01, -3.6793e-01,  6.0831e-01, -5.7057e-02, -1.0810e-01,\n                           -2.0857e-01],\n                          [-2.4936e-01,  1.1129e-01, -8.5846e-02,  1.1629e-01,  2.3204e-01,\n                           -8.3728e-02, -5.0009e-03, -3.6876e-02, -5.6730e-02,  8.9288e-02,\n                           -3.0675e-01, -2.0990e-02,  2.0179e-03,  1.3768e-01, -8.8847e-02,\n                           -1.2821e-01, -2.4594e-01,  2.9177e-02,  1.1291e-02, -7.5023e-03,\n                           -1.9662e-01, -1.1896e-01,  8.1231e-02,  9.6271e-02, -2.1776e-01,\n                            4.3825e-02, -1.1745e-01,  1.5375e-01, -2.7449e-01,  8.7305e-02,\n                           -6.9382e-02, -2.9008e-01,  3.5329e-02,  3.8964e-02,  1.5343e-01,\n                            6.9381e-02],\n                          [ 5.0705e-01,  7.5225e-02, -1.9947e-01, -7.2118e-01, -7.1222e-01,\n                           -4.6810e-01, -6.6338e-01, -2.6533e-01,  9.3366e-03,  1.5560e-01,\n                            6.0585e-01,  2.8834e-02,  8.7921e-02, -1.2617e-01,  3.7753e-01,\n                            1.7388e-01,  3.9981e-01, -8.5104e-01,  2.9579e-01, -3.6954e-02,\n                            3.4440e-01, -1.4266e-01, -2.7917e-01, -2.8903e-01,  2.2271e-01,\n                            2.7432e-01,  7.6578e-01, -3.1120e-01,  5.8330e-01, -2.0584e-01,\n                           -1.1826e-01,  8.7565e-01, -8.4367e-01, -1.4038e-01, -2.1993e-01,\n                            1.6820e-01],\n                          [ 6.1618e-01, -2.0181e-01, -1.1586e-01, -6.5962e-01, -7.5546e-01,\n                           -2.4145e-01, -7.1842e-01, -3.0258e-01,  1.8481e-01, -3.1264e-02,\n                            7.1100e-01, -2.0891e-01, -9.7056e-02, -3.6655e-01,  3.9227e-01,\n                            1.1348e-01,  4.8274e-01, -1.0601e+00,  6.0539e-01, -2.4429e-01,\n                            7.6612e-01, -1.0369e-02, -1.8387e-01, -3.5977e-01,  1.6472e-01,\n                            2.1449e-01,  9.6197e-01, -2.1844e-01,  5.0461e-01, -1.2234e-01,\n                            6.7183e-02,  9.8275e-01, -1.0813e+00,  6.9863e-02, -2.2164e-01,\n                            9.3208e-02],\n                          [-5.7226e-01,  1.8769e-01, -5.3923e-02,  7.7387e-01,  7.7540e-01,\n                            3.4125e-01,  5.3061e-01,  2.2552e-01, -1.8166e-01,  5.1604e-02,\n                           -6.1244e-01,  2.0704e-01,  5.5774e-02,  3.6524e-01, -2.9256e-01,\n                           -1.9852e-01, -5.6431e-01,  9.7928e-01, -5.5009e-01,  2.2640e-01,\n                           -4.6449e-01,  1.4530e-01,  1.9968e-01,  3.0306e-01, -2.6307e-01,\n                           -1.7454e-01, -7.3779e-01,  3.9801e-01, -6.5023e-01,  4.2400e-02,\n                            1.5837e-01, -9.8155e-01,  8.9757e-01,  3.9165e-02,  2.0116e-01,\n                           -2.7800e-01],\n                          [ 6.4124e-01, -2.9317e-01, -2.5212e-01, -1.0591e+00, -8.2375e-01,\n                           -5.8117e-01, -9.2188e-01, -1.4222e-01,  4.4446e-02,  1.3393e-01,\n                            1.0480e+00, -2.2451e-01, -7.7949e-02, -5.0875e-01,  6.1654e-01,\n                            1.5854e-01,  4.8515e-01, -1.3993e+00,  4.7638e-01, -2.7428e-01,\n                            9.0608e-01, -1.0175e-01, -2.6520e-01, -2.0782e-01,  3.1496e-01,\n                            1.1251e-01,  1.1635e+00, -3.7749e-01,  8.8009e-01, -1.9646e-01,\n                            5.0011e-02,  1.3022e+00, -1.2342e+00, -9.8339e-02, -1.1810e-01,\n                            2.1539e-01],\n                          [ 7.1685e-03, -1.2686e-01, -1.1366e-01,  5.2048e-02,  8.9431e-02,\n                            1.3545e-01, -3.1158e-02, -4.1021e-02, -5.7851e-02, -1.8050e-01,\n                           -1.5466e-02, -1.6666e-01,  1.2606e-01, -7.7843e-02, -2.1250e-01,\n                           -1.3819e-01, -8.1280e-02,  1.5420e-01, -2.6689e-01,  5.2791e-02,\n                           -1.0808e-01, -1.3916e-01,  6.3449e-02, -5.7353e-03, -6.5060e-02,\n                            1.8268e-02, -9.4458e-02,  3.9187e-02, -1.6682e-01,  3.6127e-02,\n                           -3.4982e-03, -3.2648e-01,  2.2889e-01,  7.3718e-02,  9.1869e-02,\n                           -2.3548e-02],\n                          [-5.1995e-02,  9.5385e-02, -1.2733e-01,  3.0907e-01,  1.8886e-01,\n                            1.3072e-01,  4.0072e-02, -4.8568e-02, -3.5790e-02, -9.3972e-02,\n                           -1.7419e-01, -9.5049e-02, -8.9847e-02, -7.3447e-02, -3.1358e-01,\n                           -1.7438e-01, -1.5253e-01,  3.3339e-01, -4.8277e-02, -8.9651e-02,\n                           -3.3702e-01,  3.4778e-02,  1.0201e-02, -3.6210e-02, -8.2409e-02,\n                           -7.3123e-02, -3.3187e-01,  4.4632e-02, -7.6358e-02,  4.2277e-02,\n                            2.2176e-02, -2.3865e-01,  3.6393e-01,  8.5710e-03,  1.7082e-01,\n                           -1.0595e-01],\n                          [ 9.8232e-02,  5.8383e-02,  1.2162e-02, -1.7030e-01, -1.9584e-01,\n                           -2.0903e-03, -3.4812e-02, -1.7809e-01, -2.7124e-02, -1.9780e-01,\n                           -8.0016e-03,  1.9516e-02, -3.0012e-02,  8.5232e-02, -1.2433e-01,\n                           -7.3589e-02,  1.0981e-01, -2.0920e-01, -7.4301e-02,  5.8491e-02,\n                            1.2996e-01,  1.0762e-01, -5.5863e-02, -1.8387e-01, -1.9297e-01,\n                            1.0802e-01, -2.3705e-02, -6.5813e-02,  2.9840e-02, -1.8421e-01,\n                           -2.5832e-02,  1.0185e-01, -2.0740e-02,  1.0743e-01,  1.5129e-03,\n                           -8.9038e-02],\n                          [-5.0179e-01,  3.3886e-02, -7.9031e-02,  4.5581e-01,  4.0854e-01,\n                            3.0781e-01,  5.2242e-01,  3.1661e-01, -5.0472e-02,  1.7067e-01,\n                           -5.6787e-01, -8.0417e-02,  1.3836e-01,  1.4746e-01, -1.9306e-01,\n                           -1.2671e-01, -2.6568e-01,  8.4973e-01, -3.4238e-01,  1.6674e-01,\n                           -5.3509e-01,  8.3736e-02,  2.0266e-01,  1.8251e-01,  4.1420e-02,\n                           -2.3186e-02, -6.4609e-01,  3.0461e-01, -5.0640e-01, -8.1449e-02,\n                            1.4625e-02, -7.1546e-01,  8.5928e-01, -6.4718e-02,  2.4350e-01,\n                            7.5128e-03],\n                          [-2.2285e-01,  1.0857e-02, -1.8248e-01, -3.0123e-02,  1.0723e-01,\n                           -1.3167e-01,  3.3189e-02, -8.3115e-02, -1.7852e-01,  7.7904e-02,\n                           -1.9266e-01, -8.3915e-02,  3.2321e-02, -3.3958e-02, -1.8725e-01,\n                           -1.1478e-01,  5.6716e-02, -6.9924e-02, -2.1040e-01,  1.0919e-01,\n                           -2.2569e-01, -1.8240e-01, -1.5931e-02, -1.3928e-01, -7.4190e-02,\n                            4.7199e-02, -3.2381e-02, -1.2039e-01, -2.4302e-01, -9.6816e-02,\n                            3.3323e-02, -1.8928e-01,  1.9644e-01,  1.3776e-01, -5.1122e-02,\n                            1.0238e-02],\n                          [ 3.1542e-01, -1.0071e-01, -3.3809e-02, -3.1980e-01, -1.4546e-01,\n                           -6.3505e-02, -2.0994e-01, -2.1517e-01, -6.4286e-02,  8.2350e-02,\n                            2.7637e-01, -1.6297e-01, -7.3031e-03, -2.0101e-01,  6.0403e-02,\n                           -7.4647e-02,  1.1780e-01, -4.6727e-01, -3.2734e-02, -2.2128e-01,\n                            2.8653e-01, -1.6685e-01, -1.8050e-01, -2.5443e-01, -8.7459e-02,\n                           -6.5968e-03,  4.1845e-01,  3.3995e-02,  2.4856e-01, -1.3881e-02,\n                            1.4678e-01,  3.7109e-01, -3.6217e-01,  5.0945e-02, -1.7698e-01,\n                           -4.1554e-02],\n                          [-8.0044e-01,  6.0991e-02,  2.4030e-01,  1.0383e+00,  1.0808e+00,\n                            6.2563e-01,  9.9345e-01,  4.0953e-01,  4.8197e-02, -9.8052e-02,\n                           -1.0931e+00, -5.2888e-02,  1.0128e-01,  4.1904e-01, -7.3696e-01,\n                           -6.1364e-02, -7.8063e-01,  1.4828e+00, -6.6958e-01,  1.6235e-01,\n                           -7.8602e-01,  1.4799e-03,  1.6296e-01,  3.8643e-01, -1.3420e-01,\n                           -1.1414e-01, -1.2031e+00,  5.6669e-01, -7.6980e-01,  1.6235e-01,\n                            1.2398e-02, -1.4031e+00,  1.3588e+00,  1.0670e-01,  3.1032e-01,\n                            1.8600e-02],\n                          [-7.5362e-01,  5.5423e-02,  1.7893e-01,  7.6729e-01,  8.0874e-01,\n                            5.3784e-01,  5.9990e-01,  2.1773e-01, -9.1679e-02,  1.9330e-02,\n                           -8.4539e-01,  1.5457e-01,  1.3157e-01,  3.5714e-01, -4.8129e-01,\n                           -1.0837e-01, -5.6057e-01,  1.0813e+00, -5.2564e-01,  1.5808e-01,\n                           -7.7431e-01, -3.0707e-02,  8.7690e-02,  4.3069e-01, -3.4133e-02,\n                           -1.1733e-01, -8.0619e-01,  4.1244e-01, -7.6783e-01,  2.2284e-01,\n                            1.7605e-01, -1.1175e+00,  9.8434e-01,  1.1662e-01,  1.3052e-01,\n                            1.7048e-02],\n                          [ 5.6157e-01, -2.0774e-01, -2.7389e-01, -5.5774e-01, -8.0048e-01,\n                           -4.6774e-01, -7.7437e-01, -1.5247e-01,  9.4347e-02, -1.0681e-01,\n                            7.7070e-01, -2.4047e-05,  6.3275e-02, -1.6038e-01,  3.7863e-01,\n                            1.7146e-01,  2.7598e-01, -1.0636e+00,  5.1521e-01, -2.3533e-01,\n                            5.6459e-01,  1.2233e-01, -5.6879e-02, -2.3444e-01,  2.1186e-02,\n                            1.6505e-01,  8.9362e-01, -4.9518e-01,  6.3610e-01, -1.9454e-01,\n                           -1.0285e-01,  8.7712e-01, -1.0878e+00,  8.4451e-02, -7.3794e-03,\n                            2.3018e-01],\n                          [ 5.2103e-01,  9.1095e-02, -1.6315e-01, -5.3979e-01, -5.6779e-01,\n                           -2.4636e-01, -6.2370e-01, -1.0021e-01,  7.9842e-02, -4.2564e-02,\n                            5.2584e-01,  7.6305e-02, -6.6741e-02, -3.5544e-01,  2.4199e-01,\n                           -4.1998e-02,  3.6420e-01, -8.5261e-01,  4.7736e-01, -9.3127e-02,\n                            3.7144e-01,  6.1179e-02, -2.0282e-01, -3.1959e-01, -3.3077e-02,\n                            1.1341e-01,  7.1937e-01, -1.6487e-01,  5.2345e-01, -1.4580e-01,\n                            1.1917e-01,  8.5992e-01, -7.1978e-01, -8.4026e-02, -7.5979e-02,\n                            2.1359e-01],\n                          [-4.5117e-01, -4.0728e-02,  1.6340e-02,  6.0485e-01,  5.7580e-01,\n                            1.3570e-01,  5.0679e-01,  1.4862e-01, -1.0927e-01, -6.9882e-02,\n                           -4.3197e-01,  9.0975e-04,  1.6692e-01,  8.4681e-02, -4.5209e-01,\n                           -1.8310e-01, -3.6066e-01,  7.6379e-01, -2.7570e-01,  2.2056e-01,\n                           -3.9911e-01,  1.4316e-01,  2.3895e-01,  2.4606e-01,  3.6183e-02,\n                           -2.8173e-01, -5.7073e-01,  3.0715e-01, -4.4968e-01,  1.9184e-01,\n                            8.4140e-02, -5.9109e-01,  4.7188e-01,  1.3540e-01,  3.5808e-02,\n                           -5.8094e-02],\n                          [ 2.5161e-02,  7.2590e-02, -1.4838e-01, -3.2675e-01, -3.1266e-01,\n                           -2.1411e-01, -9.1318e-02, -1.1337e-01, -1.0898e-01, -1.4817e-01,\n                            1.9330e-01,  1.0813e-01,  4.8441e-02, -1.0025e-01,  1.0866e-01,\n                            5.5700e-02,  1.0385e-01, -3.6523e-01,  2.7763e-01, -2.0037e-01,\n                            1.1823e-01,  1.5736e-01, -1.8448e-01,  5.2477e-02,  1.2036e-01,\n                            1.3390e-01,  2.3679e-01, -1.6376e-01,  4.2499e-02, -4.8060e-02,\n                           -1.2852e-02,  3.2739e-01, -3.4038e-01,  7.8442e-02, -3.8892e-02,\n                           -8.5504e-02]])),\n                 ('FC_layer3.bias',\n                  tensor([-0.1451,  0.0079, -0.0115,  0.0242,  0.1208, -0.1654, -0.0135, -0.1698,\n                           0.1641, -0.0451, -0.0242, -0.0379,  0.0152, -0.1718, -0.1689, -0.0997,\n                           0.0055,  0.1207, -0.1417, -0.0443,  0.1627, -0.0373, -0.1651, -0.1239])),\n                 ('FC_layer4.weight',\n                  tensor([[-6.2032e-02, -1.4713e-01, -8.9856e-02,  7.9218e-02, -1.2283e-01,\n                           -2.5157e-01, -1.2653e-01, -2.1432e-01, -1.3992e-01, -1.1738e-01,\n                           -1.6877e-01, -4.8216e-02,  8.0311e-02,  1.0261e-01, -5.9118e-02,\n                            7.0779e-02, -1.3477e-02, -2.6327e-01, -1.7546e-01,  6.8196e-03,\n                           -2.3325e-01, -7.9117e-02, -1.0051e-01, -1.0722e-01],\n                          [ 2.1639e-01,  3.2000e-01,  5.0037e-01,  3.1543e-01, -5.6504e-01,\n                            3.7775e-01,  4.9284e-01,  2.9385e-01, -1.0408e+00, -1.3783e+00,\n                            1.1402e+00, -1.6549e+00,  3.1825e-01,  1.7420e-01, -8.8789e-02,\n                            9.6651e-01,  2.5790e-01, -2.7491e-01,  1.5444e+00,  1.2605e+00,\n                           -1.1991e+00, -9.5089e-01,  7.8119e-01, -5.0550e-01],\n                          [ 6.0516e-02,  7.2191e-02,  1.5802e-01,  9.2519e-02, -3.3708e-02,\n                            1.2478e-01, -1.6773e-01, -1.9157e-02, -2.4154e-01, -3.6417e-01,\n                            1.8787e-01, -1.9966e-01, -7.0878e-02, -2.1687e-01,  3.0925e-02,\n                           -3.3352e-02,  2.8121e-02, -3.6361e-01,  4.7108e-02,  2.5743e-01,\n                           -3.8639e-01, -4.9703e-02,  2.0170e-01, -4.7837e-02],\n                          [ 9.3415e-02,  3.9038e-01,  4.5644e-01,  1.5741e-02, -4.7028e-01,\n                            4.5103e-01,  5.4163e-01,  1.3746e-01, -6.6838e-01, -9.9655e-01,\n                            5.8240e-01, -1.1682e+00, -4.7298e-02,  2.9966e-01, -2.1743e-01,\n                            5.7887e-01,  3.6047e-02, -2.6815e-01,  1.2470e+00,  9.5769e-01,\n                           -8.3629e-01, -7.4676e-01,  3.3960e-01, -2.7330e-01],\n                          [ 3.0543e-02, -4.8916e-01, -2.0580e-01, -2.7168e-01,  5.3850e-01,\n                           -4.9789e-01, -6.1361e-01, -1.6785e-01,  9.1953e-01,  1.1141e+00,\n                           -9.3372e-01,  1.4649e+00, -1.7113e-01, -4.0015e-01, -5.8513e-02,\n                           -6.3465e-01, -1.6501e-01,  6.0229e-01, -1.6570e+00, -1.2590e+00,\n                            1.1666e+00,  9.2976e-01, -7.0111e-01,  3.5051e-01],\n                          [ 1.0633e-01, -2.8454e-01, -2.8118e-01, -2.0132e-01, -5.5819e-02,\n                           -7.8870e-02,  1.9714e-02, -4.2597e-02,  1.4777e-01,  3.7150e-01,\n                           -4.2315e-01,  4.3685e-01,  6.3831e-02,  6.1570e-02, -1.6156e-01,\n                           -3.5434e-01, -2.4723e-01,  6.2488e-02, -4.9554e-01, -2.4095e-01,\n                            3.3478e-01,  2.1917e-02, -2.4287e-01,  2.9870e-02],\n                          [ 5.9556e-02, -1.3503e-01, -2.0922e-02, -8.6814e-02,  4.0498e-02,\n                           -9.0152e-02, -2.5226e-01, -2.2193e-01, -2.0231e-01,  2.8868e-02,\n                            1.1160e-01, -1.0689e-01, -2.7715e-01,  1.1711e-01, -1.5687e-01,\n                           -4.7492e-02, -2.4792e-01, -7.1277e-02,  2.0550e-02,  1.3740e-02,\n                           -1.7675e-01, -6.1213e-03,  8.1961e-02, -2.8159e-01],\n                          [-2.3162e-01, -1.4717e-01,  5.5652e-02, -1.9490e-01,  8.4206e-03,\n                           -2.9278e-01, -5.4586e-02,  9.9816e-03,  2.1621e-01,  5.3825e-01,\n                           -2.9789e-01,  7.0983e-01, -2.4044e-02, -5.8823e-02,  4.8796e-02,\n                           -2.0613e-01, -9.5606e-03,  7.5029e-02, -7.2448e-01, -5.2746e-01,\n                            3.4836e-01,  2.1013e-01, -1.9923e-01, -1.2923e-02],\n                          [-2.0249e-02,  3.0672e-01,  2.1209e-01,  8.0248e-02, -4.5953e-01,\n                            1.8632e-01,  4.3298e-01,  2.1034e-01, -1.0567e+00, -8.2499e-01,\n                            9.4889e-01, -1.0286e+00, -3.8023e-03,  4.1586e-01, -1.8947e-01,\n                            7.6165e-01,  3.6232e-02, -3.0432e-01,  1.2662e+00,  9.8726e-01,\n                           -9.8945e-01, -9.2847e-01,  8.2791e-01, -3.2166e-01],\n                          [-1.6355e-01, -7.1227e-01, -2.1366e-01,  9.5916e-04,  6.9108e-01,\n                           -5.4642e-01, -5.5108e-01, -2.3929e-01,  1.2650e+00,  1.2779e+00,\n                           -1.2826e+00,  1.6277e+00, -3.4892e-01, -2.9999e-01,  1.8464e-01,\n                           -1.0074e+00,  9.4502e-02,  6.4880e-01, -1.7828e+00, -1.3357e+00,\n                            1.3621e+00,  8.7730e-01, -8.5667e-01,  3.9720e-01],\n                          [ 1.4073e-01,  3.3857e-01,  1.0371e-01,  3.2180e-02, -8.3751e-02,\n                            1.6141e-01,  1.1849e-01, -1.2933e-01, -4.7865e-01, -6.3727e-01,\n                            3.3267e-01, -7.3078e-01,  2.0280e-02,  2.0269e-01, -1.8950e-01,\n                            4.2790e-01,  1.0871e-01, -2.2768e-01,  6.7758e-01,  3.3401e-01,\n                           -2.6517e-01, -4.0787e-01,  2.2386e-01, -8.4474e-03],\n                          [-1.4583e-01, -4.2622e-01, -4.2170e-01,  4.7281e-02,  5.0980e-01,\n                           -2.7019e-01, -4.9610e-01, -2.4811e-01,  6.1358e-01,  7.2919e-01,\n                           -8.7790e-01,  1.2400e+00, -8.4429e-02, -1.6614e-01,  1.3565e-01,\n                           -4.7334e-01, -1.6465e-01,  2.9387e-01, -1.1166e+00, -8.3574e-01,\n                            7.6809e-01,  7.9646e-01, -5.7243e-01,  3.0396e-01]])),\n                 ('FC_layer4.bias',\n                  tensor([-0.2170, -0.2406, -0.0231,  0.0457,  0.2083,  0.0397, -0.1073, -0.0637,\n                          -0.0361,  0.1476,  0.0649,  0.1438])),\n                 ('FC_layer5.weight',\n                  tensor([[ 0.2527, -2.5937, -0.5486, -1.9344,  2.5169,  0.8707, -0.3714,  0.9929,\n                           -2.0736,  2.8293, -1.1125,  1.9031],\n                          [ 0.3218,  2.6480,  0.2650,  1.7932, -2.3031, -0.3996, -0.1756, -0.7388,\n                            2.0334, -2.7776,  0.8806, -1.6891]])),\n                 ('FC_layer5.bias', tensor([-0.0505, -0.0314]))])\n\n\n\n\n```python\noptimizer.state_dict()\n```\n\n\n\n\n    {'state': {0: {'momentum_buffer': None},\n      1: {'momentum_buffer': None},\n      2: {'momentum_buffer': None},\n      3: {'momentum_buffer': None},\n      4: {'momentum_buffer': None},\n      5: {'momentum_buffer': None},\n      6: {'momentum_buffer': None},\n      7: {'momentum_buffer': None},\n      8: {'momentum_buffer': None},\n      9: {'momentum_buffer': None}},\n     'param_groups': [{'lr': 0.04210455402856372,\n       'momentum': 0,\n       'dampening': 0,\n       'weight_decay': 0,\n       'nesterov': False,\n       'params': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]}]}\n\n\n\n\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "75e4b1f65eeb6f4173dff34964252f13d47247a3", "size": 304690, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "\u673a\u5668\u5b66\u4e60/\u7b97\u6cd5/.ipynb_checkpoints/\u5b66\u4e60\u6a21\u578b--DNN example-checkpoint.ipynb", "max_stars_repo_name": "Yanie1asdfg/Quant-Lectures", "max_stars_repo_head_hexsha": "4e4b84cf2aff290b715a7924277335a23f5e8168", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2020-12-29T07:53:46.000Z", 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{"text": "\n\n_Lambda School Data Science \u2014\u00a0Linear Models_\n\n# Understanding Linear Regression\n\n#### Objectives\n- understand how ordinary least squares regression minimizes the sum of squared errors\n- understand how linear algebra can solve ordinary least squares regression\n- get and interpret coefficients of a linear model\n- visualize a line of best fit in 2D, and hyperplane in 3D\n- use regression metrics: MSE, RMSE, MAE, R^2\n\n#### Extra Links\n- [Statistics 101: Simple Linear Regression](https://www.youtube.com/watch?v=ZkjP5RJLQF4) (20 minute video)\n- [_An Introduction to Statistical Learning_](http://www-bcf.usc.edu/~gareth/ISL/ISLR%20Seventh%20Printing.pdf), Chapter 3.1, Simple Linear Regression, & 3.2, Multiple Linear Regression\n- Priceonomics, [The Discovery of Statistical Regression](https://priceonomics.com/the-discovery-of-statistical-regression/)\n- Priceonomics, [Why the Father of Modern Statistics Didn\u2019t Believe Smoking Caused Cancer](https://priceonomics.com/why-the-father-of-modern-statistics-didnt-believe/)\n- Harvard Business Review, [When to Act on a Correlation, and When Not To](https://hbr.org/2014/03/when-to-act-on-a-correlation-and-when-not-to)\n- [xkcd 552: Correlation](https://www.explainxkcd.com/wiki/index.php/552:_Correlation)\n- [xkcd 1725: Linear Regression](https://www.explainxkcd.com/wiki/index.php/1725:_Linear_Regression)\n\n## What is Linear Regression?\n\n[Linear Regression](https://en.wikipedia.org/wiki/Linear_regression) is a statistical model that seeks to describe the relationship between some y variable and one or more x variables. \n\nIn the simplest case, linear regression seeks to fit a straight line through a cloud of points. This line is referred to as the \"regression line\" or \"line of best fit.\" This line tries to summarize the relationship between our X and Y in a way that enables us to use the equation for that line to make predictions.\n\n\n```python\n%matplotlib inline\nimport seaborn as sns\nx = [10,  8, 13,  9, 11, 14,  6,  4, 12,  7,  5]\ny = [ 8,  6,  7,  8,  8,  9,  7,  4, 10,  4,  5]\nsns.regplot(x, y);\n```\n\n\n\n### Synonyms for \"y variable\" \n- Dependent Variable\n- Response Variable\n- Outcome Variable \n- Predicted Variable\n- Measured Variable\n- Explained Variable\n- Label\n- Target\n\n### Synonyms for \"x variable\"\n- Independent Variable\n- Explanatory Variable\n- Regressor\n- Covariate\n- Feature\n\n## Simple Linear Regresion\n\n#### Making Predictions\n\nSay that we were trying to create a model that captured the relationship between temperature outside and ice cream sales. In Machine Learning our goal is often different that of other flavors of Linear Regression Analysis, because we're trying to fit a model to this data with the intention of making **predictions** on new data (in the future) that we don't have yet.\n\n#### What are we trying to predict?\n\nSo if we had measured ice cream sales and the temprature outside on 11 different days, at the end of our modeling **what would be the thing that we would want to predict? - Ice Cream Sales or Temperature?**\n\nWe would probably want to be measuring temperature with the intention of using that to **forecast** ice cream sales. If we were able to successfully forecast ice cream sales from temperature, this might help us know beforehand how much ice cream to make or how many cones to buy or on which days to open our store, etc. Being able to make predictions accurately has a lot of business implications. This is why making accurate predictions is so valuable (And in large part is why data scientists are paid so well).\n\n#### y variable intuition\n\nWe want the thing that we're trying to predict to serve as our **y** variable. This is why it's sometimes called the \"predicted variable.\" We call it the \"dependent\" variable because our prediction for how much ice cream we're going to sell \"depends\" on the temperature outside. \n\n#### x variable intuition\n\nAll other variables that we use to predict our y variable (we're going to start off just using one) we call our **x** variables. These are called our \"independent\" variables because they don't *depend* on y, they \"explain\" y. Hence they are also referred to as our \"explanatory\" variables.\n\n## Example: Predict presidential election voting\n\n#### Douglas Hibbs, [Background Information on the \u2018Bread and Peace\u2019 Model of Voting in Postwar US Presidential Elections](https://douglas-hibbs.com/background-information-on-bread-and-peace-voting-in-us-presidential-elections/)\n\n> Aggregate two-party vote shares going to candidates of the party holding the presidency during the postwar era are well explained by just two fundamental determinants:\n\n> (1) Positively by weighted-average growth of per capita real disposable personal income over the term.  \n> (2) Negatively by cumulative US military fatalities (scaled to population) owing to unprovoked, hostile deployments of American armed forces in foreign wars.\n\n#### Data sources\n- 1952-2012: Douglas Hibbs, [2014 lecture at Deakin University Melbourne](http://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf), Slide 40\n- 2016, Vote Share: [The American Presidency Project](https://www.presidency.ucsb.edu/statistics/elections)\n- 2016, Recent Growth in Personal Incomes: [The 2016 election economy: the \"Bread and Peace\" model final forecast](https://angrybearblog.com/2016/11/the-2016-election-economy-the-bread-and-peace-model-final-forecast.html)\n- 2016, US Military Fatalities: Assumption that Afghanistan War fatalities in 2012-16 occured at the same rate as 2008-12\n\n\n```python\nimport pandas as pd\n\ncolumns = ['Year','inc_party_candidate','other_candidate','inc_party_vote']\n\ndata = [[1952,\"Stevenson\",\"Eisenhower\",44.6],\n        [1956,\"Eisenhower\",\"Stevenson\",57.76],\n        [1960,\"Nixon\",\"Kennedy\",49.91],\n        [1964,\"Johnson\",\"Goldwater\",61.34],\n        [1968,\"Humphrey\",\"Nixon\",49.60],\n        [1972,\"Nixon\",\"McGovern\",61.79],\n        [1976,\"Ford\",\"Carter\",48.95],\n        [1980,\"Carter\",\"Reagan\",44.70],\n        [1984,\"Reagan\",\"Mondale\",59.17],\n        [1988,\"Bush, Sr.\",\"Dukakis\",53.94],\n        [1992,\"Bush, Sr.\",\"Clinton\",46.55],\n        [1996,\"Clinton\",\"Dole\",54.74],\n        [2000,\"Gore\",\"Bush, Jr.\",50.27],\n        [2004,\"Bush, Jr.\",\"Kerry\",51.24],\n        [2008,\"McCain\",\"Obama\",46.32],\n        [2012,\"Obama\",\"Romney\",52.00], \n        [2016,\"Clinton\",\"Trump\",48.2]]\n        \nvotes = pd.DataFrame(data=data, columns=columns)\n```\n\n\n```python\ncolumns = ['Year','income_growth']\n\ndata = [[1952,2.40],\n        [1956,2.89],\n        [1960, .85],\n        [1964,4.21],\n        [1968,3.02],\n        [1972,3.62],\n        [1976,1.08],\n        [1980,-.39],\n        [1984,3.86],\n        [1988,2.27],\n        [1992, .38],\n        [1996,1.04],\n        [2000,2.36],\n        [2004,1.72],\n        [2008, .10],\n        [2012, .95], \n        [2016, .10]]\n        \ngrowth = pd.DataFrame(data=data, columns=columns)\n```\n\n\n```python\n\"\"\"\nFatalities denotes the cumulative number of American military\nfatalities per millions of US population the in Korea, Vietnam,\nIraq and Afghanistan wars during the presidential terms\npreceding the 1952, 1964, 1968, 1976 and 2004, 2008 and\n2012 elections.\n\nhttp://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf\n\"\"\"\n\ncolumns = ['Year','fatal_per_mil']\n\ndata = [[1952,190],\n        [1956,  0],\n        [1960,  0],\n        [1964,  1],\n        [1968,146],\n        [1972,  0],\n        [1976,  2],\n        [1980,  0],\n        [1984,  0],\n        [1988,  0],\n        [1992,  0],\n        [1996,  0],\n        [2000,  0],\n        [2004,  4],\n        [2008, 14],\n        [2012,  5], \n        [2016,  5]]\n        \ndeaths = pd.DataFrame(data=data, columns=columns)\n```\n\n\n```python\ndf = votes.merge(growth).merge(deaths)  #assumes 'year' is the joining feature\n```\n\n\n```python\ndf.head()\n```\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n  \n
                                        Yearinc_party_candidateother_candidateinc_party_voteincome_growthfatal_per_mil
                                        01952StevensonEisenhower44.602.40190
                                        11956EisenhowerStevenson57.762.890
                                        21960NixonKennedy49.910.850
                                        31964JohnsonGoldwater61.344.211
                                        41968HumphreyNixon49.603.02146
                                        \n
                                        \n\n\n\n### Plot univariate correlations\n[Seaborn tutorial: Visualizing linear relationships](https://seaborn.pydata.org/tutorial/regression.html)\n\n\n```python\n%matplotlib inline\nimport seaborn as sns\n\ntarget = 'inc_party_vote'\nfeatures = ['income_growth', \n            'fatal_per_mil']\n\nfor feature in features:\n    sns.lmplot(x=feature, y=target, data=df)\n```\n\nWe can see from the scatterplot that these data points seem to follow a somewhat linear relationship for the \"Average Recent Growth in Personal Incomes\" feature. This means that we could probably summarize their relationship well by fitting a line of best fit to these points. Lets do it.\n\n\n## The Equation for a Line\n\nA common equation for a line is:\n\n\\begin{align}\ny = mx + b\n\\end{align}\n\nWhere $m$ is the slope of our line and $b$ is the y-intercept. \n\nIf we want to plot a line through our cloud of points we figure out what these two values should be. Linear Regression seeks to **estimate** the slope and intercept values that describe a line that best fits the data points.\n\n## The Anatomy of Linear Regression\n\n- Intercept: The $b$ value in our line equation $y=mx+b$\n- Slope: The $m$ value in our line equation $y=mx+b$. These two values together define our regression line.\n- $\\hat{y}$ : A prediction\n- Line of Best Fit (Regression Line)\n- Predicted (fitted) Values: Points on our regression line\n- Observed Values: Points from our dataset\n- Error: The distance between predicted and observed values.\n\n\nOrdinary Least Squares Regression is a way to solve for $m$ and $b$.\n\nLet's start by seeing what would happen if we just guessed and checked some values for $m$ and $b$. \n\nWhat's the line of \"best\" fit look like? What's the error?\n\n\n\n```python\nx = df['income_growth']\ny = df['inc_party_vote']\n\nm = 0\nb = y.mean()\ny_pred = m*x + b\ny_pred\n```\n\n\n\n\n    0     51.828235\n    1     51.828235\n    2     51.828235\n    3     51.828235\n    4     51.828235\n    5     51.828235\n    6     51.828235\n    7     51.828235\n    8     51.828235\n    9     51.828235\n    10    51.828235\n    11    51.828235\n    12    51.828235\n    13    51.828235\n    14    51.828235\n    15    51.828235\n    16    51.828235\n    Name: income_growth, dtype: float64\n\n\n\n\n```python\nimport matplotlib.pyplot as plt\nfrom sklearn.metrics import mean_absolute_error, r2_score\n\ndef plot_preds(x, y, y_pred):\n  plt.scatter(x, y, label='y_true')\n  plt.plot(x,y_pred, label='y_pred')\n  plt.legend()\n  mae = mean_absolute_error(y,y_pred)\n  r2 = r2_score(y,y_pred)\n  print(f'Mean Absolute Error: {mae}')\n  print(f'R^2 score: {r2}')\n  \n```\n\n\n```python\n'''\n See how R-squared (Coefficient of determination) is calculated\n in https://en.wikipedia.org/wiki/Coefficient_of_determination\n \n R-squared = 1 - SSred / SStot\n'''\n```\n\n\n\n\n    '\\n See how R-squared (Coefficient of determination) is calculated\\n in https://en.wikipedia.org/wiki/Coefficient_of_determination\\n \\n R-squared = 1 - SSred / SStot\\n'\n\n\n\n\n```python\nplot_preds(x, y, y_pred)\n```\n\n\n```python\nm = 4.1\nb = 45\ny_pred = m*x + b\nplot_preds(x, y, y_pred)\nplt.title(\"Guessing LR values\")\n```\n\n## R Squared:  $R^2$\n\nOne final attribute of linear regressions that we're going to talk about today is a measure of goodness of fit known as $R^2$ or R-squared. $R^2$ is a statistical measure of how close the data are fitted to our regression line. A helpful interpretation for the $R^2$ is the percentage of the dependent variable that is explained by the model.\n\nIn other words, the $R^2$ is the percentage of y that is explained by the x variables included in the model. For this reason the $R^2$ is also known as the \"coefficient of determination,\" because it explains how much of y is explained (or determined) by our x varaibles. We won't go into the calculation of $R^2$ today, just know that a higher $R^2$ percentage is nearly always better and indicates a model that fits the data more closely. \n\n## Residual Error \n\nThe residual error is the distance between points in our dataset and our regression line.\n\n\n```python\ndef regression_residuals(df, feature, target, m, b):\n    x = df[feature]\n    y = df[target]\n    y_pred = m*x + b\n    \n    plt.scatter(x, y, label='y_true')\n    plt.plot(x, y_pred, label='y_pred')\n    plt.legend()\n    \n    # Plot residual errors\n    for x, y1, y2 in zip(x, y, y_pred):\n        plt.plot((x, x), (y1, y2), color='grey')\n        \n    mae = mean_absolute_error(y, y_pred) \n    r2 = r2_score(y, y_pred)\n    print('Mean Absolute Error:', mae)\n    print('R^2:', r2)\n        \nregression_residuals(df, feature='income_growth', \n                     target='inc_party_vote', m=3, b=46)\n```\n\n\n```python\nfrom ipywidgets import interact, fixed\n\ninteract(regression_residuals, \n         df=fixed(df), \n         feature=fixed('income_growth'), \n         target=fixed('inc_party_vote'), \n         m=(-10,10,0.5), \n         b=(40,60,0.5));\n```\n\n\n    interactive(children=(FloatSlider(value=0.0, description='m', max=10.0, min=-10.0, step=0.5), FloatSlider(valu\u2026\n\n\n## More Formal Notation\n\n\n\nWe have talked about a line of regression being represented like a regular line $y=mx+b$ but as we get to more complicated versions we're going to need to extend this equation. So lets establish the proper terminology.\n\n**X** - Independent Variable, predictor variable, explanatory variable, regressor, covariate\n\n**Y** - Response variable, predicted variable, measured vairable, explained variable, outcome variable\n\n$\\beta_0$ - \"Beta Naught\" or \"Beta Zero\", the intercept value. This is how much of y would exist if X were zero. This is sometimes represented by the letter \"a\" but I hate that. So it's \"Beta 0\" during my lecture.\n\n$\\beta_1$ - \"Beta One\" The primary coefficient of interest. This values is the slope of the line that is estimated by \"minimizing the sum of the squared errors/residuals\" - We'll get to that. \n\n$\\epsilon$ - \"Epsilon\" The \"error term\", random noise, things outside of our model that affect y.\n\n## Minimizing the Sum of the Squared Error\n\nThe most common method of estimating our $\\beta$ parameters  is what's known as \"Ordinary Least Squares\" (OLS). (There are different methods of arriving at a line of best fit). OLS estimates the parameters that minimize the squared distance between each point in our dataset and our line of best fit. \n\n\\begin{align}\nSSE = \\sum(y_i - \\hat{y})^2\n\\end{align}\n\n\n\n```python\nfrom matplotlib.patches import Rectangle\nimport numpy as np\nfrom sklearn.metrics import mean_squared_error\n\ndef regression_squared_errors(df, feature, target, m, b):\n    x = df[feature]\n    y = df[target]\n    y_pred = m*x + b\n    \n    fig = plt.figure(figsize=(7,7))\n    ax = plt.axes()\n    ax.scatter(x, y, label='y_true')\n    ax.plot(x, y_pred, label='y_pred')\n    ax.legend()\n    \n    # Plot square errors\n    xmin, xmax = ax.get_xlim()\n    ymin, ymax = ax.get_ylim()\n    scale = (xmax-xmin)/(ymax-ymin)\n    for x, y1, y2 in zip(x, y, y_pred):\n        bottom_left = (x, min(y1, y2))\n        height = abs(y1 - y2)\n        width = height * scale\n        ax.add_patch(Rectangle(xy=bottom_left, width=width, height=height, alpha=0.2))\n        \n    mse = mean_squared_error(y, y_pred)\n    rmse = np.sqrt(mse)\n    mae = mean_absolute_error(y, y_pred)\n    r2 = r2_score(y, y_pred)\n    print('Mean Squared Error:', mse)\n    print('Root Mean Squared Error:', rmse)\n    print('Mean Absolute Error:', mae)\n    print('R^2:', r2)\n    \nregression_squared_errors(df, feature='income_growth', \n           target='inc_party_vote', m=3, b=46)\n```\n\n\n```python\ninteract(regression_squared_errors, \n         df=fixed(df), \n         feature=fixed('income_growth'), \n         target=fixed('inc_party_vote'), \n         m=(-10,10,0.5), \n         b=(40,60,0.5));\n```\n\n\n    interactive(children=(FloatSlider(value=0.0, description='m', max=10.0, min=-10.0, step=0.5), FloatSlider(valu\u2026\n\n\n## Hypotheses\n\n\n```python\n# Iterate multiple values of the slope\n\nb = 46\nms = np.arange(-10,10,0.5)\nsses = []\nfeature = 'income_growth'\n\nfor m in ms:\n    predictions = m * df[feature] + b\n    errors = predictions - df[target]\n    square_errors = errors ** 2\n    sse = square_errors.sum()\n    sses.append(sse)\n    \nhypotheses = pd.DataFrame({'Slope': ms})\nhypotheses['Intercept'] = b\nhypotheses['Sum of Square Errors'] = sses\n\nhypotheses\n```\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n    \n  \n
                                        SlopeInterceptSum of Square Errors
                                        0-10.04615215.58740
                                        1-9.54614095.52845
                                        2-9.04613018.88500
                                        3-8.54611985.65705
                                        4-8.04610995.84460
                                        5-7.54610049.44765
                                        6-7.0469146.46620
                                        7-6.5468286.90025
                                        8-6.0467470.74980
                                        9-5.5466698.01485
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                                        \n\n\n\n\n```python\nhypotheses.plot(x='Slope', y='Sum of Square Errors', \n                title=f'Intercept={b}');\n\n#Convex function\n```\n\n## Scikit-learn\n\n#### Jake VanderPlas, [Python Data Science Handbook, Chapter 5.2, Introducing Scikit-Learn](https://jakevdp.github.io/PythonDataScienceHandbook/05.02-introducing-scikit-learn.html), Scikit-Learn's Estimator API\n\n> Most commonly, the steps in using the Scikit-Learn estimator API are as follows (we will step through a handful of detailed examples in the sections that follow).\n\n> 1. Choose a class of model by importing the appropriate estimator class from Scikit-Learn. \n> 2. Choose model hyperparameters by instantiating this class with desired values. \n> 3. Arrange data into a features matrix and target vector following the discussion above.\n> 4. Fit the model to your data by calling the `fit()` method of the model instance.\n> 5. Apply the Model to new data: For supervised learning, often we predict labels for unknown data using the `predict()` method.\n\n\n```python\nfrom sklearn.linear_model import LinearRegression\n\nmodel = LinearRegression()\nfeatures = ['income_growth']\ntarget = 'inc_party_vote'\n\nX = df[features]\ny = df[target]\n\nmodel.fit(X,y)\n```\n\n\n\n\n    LinearRegression(copy_X=True, fit_intercept=True, n_jobs=None, normalize=False)\n\n\n\n\n```python\nmodel.intercept_\n```\n\n\n\n\n    46.499209757741625\n\n\n\n\n```python\nmodel.coef_\n```\n\n\n\n\n    array([2.97417709])\n\n\n\n\n```python\nmodel.predict([[1]])\n```\n\n\n\n\n    array([49.47338685])\n\n\n\n\n```python\nmodel.predict([[4]])\n```\n\n\n\n\n    array([58.39591811])\n\n\n\n\n```python\nmodel.predict([[2]]) - model.predict([[1]])\n```\n\n\n\n\n    array([2.97417709])\n\n\n\n\n```python\nx\n```\n\n\n\n\n    0     2.40\n    1     2.89\n    2     0.85\n    3     4.21\n    4     3.02\n    5     3.62\n    6     1.08\n    7    -0.39\n    8     3.86\n    9     2.27\n    10    0.38\n    11    1.04\n    12    2.36\n    13    1.72\n    14    0.10\n    15    0.95\n    16    0.10\n    Name: income_growth, dtype: float64\n\n\n\n\n```python\nmodel.predict(X)\nm = model.coef_[0]\nb = model.intercept_\n\ny_pred = m*x + b\n#Lines above are the same as y_pred = model.predict(X)\n\nplot_preds(x, y, y_pred)\n\n```\n\n\n```python\n\n```\n\n## Linear Algebra!\n\nThe same result that is found by minimizing the sum of the squared errors can be also found through a linear algebra process known as the \"Least Squares Solution:\"\n\n\\begin{align}\n\\hat{\\beta} = (X^{T}X)^{-1}X^{T}y\n\\end{align}\n\nBefore we can work with this equation in its linear algebra form we have to understand how to set up the matrices that are involved in this equation. \n\n### The $\\beta$ vector\n\nThe $\\beta$ vector represents all the parameters that we are trying to estimate, our $y$ vector and $X$ matrix values are full of data from our dataset. The $\\beta$ vector holds the variables that we are solving for: $\\beta_0$ and $\\beta_1$\n\nNow that we have all of the necessary parts we can set them up in the following equation:\n\n\\begin{align}\ny = X \\beta + \\epsilon\n\\end{align}\n\nSince our $\\epsilon$ value represents **random** error we can assume that it will equal zero on average.\n\n\\begin{align}\ny = X \\beta\n\\end{align}\n\nThe objective now is to isolate the $\\beta$ matrix. We can do this by pre-multiplying both sides by \"X transpose\" $X^{T}$.\n\n\\begin{align}\nX^{T}y =  X^{T}X \\beta\n\\end{align}\n\nSince anything times its transpose will result in a square matrix, if that matrix is then an invertible matrix, then we should be able to multiply both sides by its inverse to remove it from the right hand side. (We'll talk tomorrow about situations that could lead to $X^{T}X$ not being invertible.)\n\n\\begin{align}\n(X^{T}X)^{-1}X^{T}y =  (X^{T}X)^{-1}X^{T}X \\beta\n\\end{align}\n\nSince any matrix multiplied by its inverse results in the identity matrix, and anything multiplied by the identity matrix is itself, we are left with only $\\beta$ on the right hand side:\n\n\\begin{align}\n(X^{T}X)^{-1}X^{T}y = \\hat{\\beta}\n\\end{align}\n\nWe will now call it \"beta hat\" $\\hat{\\beta}$ because it now represents our estimated values for $\\beta_0$ and $\\beta_1$\n\n### Lets calculate our $\\beta$ coefficients with numpy!\n\n\n```python\ndf[feature].values.T\n```\n\n\n\n\n    array([ 2.4 ,  2.89,  0.85,  4.21,  3.02,  3.62,  1.08, -0.39,  3.86,\n            2.27,  0.38,  1.04,  2.36,  1.72,  0.1 ,  0.95,  0.1 ])\n\n\n\n\n```python\nfrom statsmodels.api import add_constant\n```\n\n\n```python\nX = add_constant(df[feature].values)\nprint('X')\nprint(X)\n\ny = df[target].values[:, np.newaxis]\nprint('y')\nprint(y)\n\nX_transpose = X.T\nprint('X Transpose')\nprint(X_transpose)\n\nX_transpose_X = X_transpose @ X\nprint('X Transpose X')\nprint(X_transpose_X)\n\nX_transpose_X_inverse = np.linalg.inv(X_transpose_X)\nprint('X Transpose X Inverse')\nprint(X_transpose_X_inverse)\n\nX_transpose_y = X_transpose @ y\nprint('X Transpose y')\nprint(X_transpose_y)\n\nbeta_hat = X_transpose_X_inverse @ X_transpose_y\nprint('Beta Hat')\nprint(beta_hat)\n```\n\n# Multiple Regression\n\nSimple or bivariate linear regression involves a single $x$ variable and a single $y$ variable. However, we can have many $x$ variables. A linear regression model that involves multiple x variables is known as **Multiple** Regression (NOT MULTIVARIATE).\n\n\\begin{align}\ny = \\beta_0 + \\beta_1 X_1 + \\beta_2 X_2 + ... + \\beta_n X_n + \\epsilon\n\\end{align}\n\n\n```python\nmodel = LinearRegression()\nfeatures = [\n    'income_growth',\n    'fatal_per_mil'\n]\n\ntarget = 'inc_party_vote'\n\nX = df[features]\ny = df[target]\n\nmodel.fit(X,y)\n```\n\n\n```python\npd.Series(model.coef_,features)  #magnitude is not indicative of importance (scaled differently)\n```\n\n## Visualize hyperplane of best fit in 3D\n\n\n```python\n# https://stackoverflow.com/a/47230966\n# Plotly notebook mode with google colaboratory\n# You need to define this function\n# And call it in each offline plotting cell\n\ndef configure_plotly_browser_state():\n    import IPython\n    display(IPython.core.display.HTML('''\n        \n        \n        '''))\n```\n\n\n```python\nimport itertools\nimport plotly.graph_objs as go\nfrom plotly.offline import init_notebook_mode, iplot\ninit_notebook_mode(connected=True)\n\ndef viz3D(fitted_model, X, features, target='', num=100):\n    \"\"\"\n    Visualize model predictions in 3D, for regression model fit on 2 features\n    \n    Parameters\n    ----------\n    fitted_model : scikit-learn model, already fitted\n    X : pandas dataframe, which was used to fit model\n    features : list of strings, column names of the 2 features\n    target : string, name of target\n    num : int, number of grid points for each feature\n    \n    References\n    ----------\n    https://plot.ly/python/3d-charts/\n    \"\"\"\n    feature1, feature2 = features\n    min1, max1 = X[feature1].min(), X[feature1].max()\n    min2, max2 = X[feature2].min(), X[feature2].max()\n    x1 = np.linspace(min1, max1, num)\n    x2 = np.linspace(min2, max2, num)\n    combos = list(itertools.product(x1, x2))\n    Z = fitted_model.predict(combos).reshape(num, num)\n    \n    configure_plotly_browser_state()\n    data = [go.Surface(x=x1, y=x2, z=Z)]\n    layout = go.Layout(\n        scene={'xaxis': {'title': feature1, 'range': [min1,max1], 'showticklabels': True}, \n               'yaxis': {'title': feature2, 'range': [min2,max2], 'showticklabels': True}, \n               'zaxis': {'title': target, 'showticklabels': True}}, \n    )\n    fig = go.Figure(data=data, layout=layout)\n    iplot(fig)\n```\n\n\n```python\nviz3D(model, X, features, target)\n```\n\n## Dimensionality in Linear Regression\n\nMuliple Regression is simply an extension of the bivariate case. The reason why we see the bivariate case demonstrated so often is simply because it's easier to graph and all of the intuition from the bivariate case is the same as we keep on adding explanatory variables.\n\nAs we increase the number of $x$ values in our model we are simply fitting a n-1-dimensional plane to an n-dimensional cloud of points within an n-dimensional hypercube. \n\n## Interpreting Coefficients\n\nOne of Linear Regression's strengths is that the parameters of the model (coefficients) are readily interpretable and useful. Not only do they describe the relationship between x and y but they put a number on just how much x is associated with y. We should be careful to not speak about this relationshiop in terms of causality because these coefficients are in fact correlative measures. We would need a host of additional techniques in order to estimate a causal effect using linear regression (econometrics).\n\n\\begin{align}\n\\hat{\\beta} = \\frac{Cov(x,y)}{Var(y)}\n\\end{align}\n\n## Why is Linear Regression so Important?\n\n### Popularity \n\nLinear Regression is an extremely popular technique that every data scientist **needs** to understand. It's not the most advanced technique and there are supervised learning techniques that will obtain a higher accuracy, but where it lacks in accuracy it makes up for it in interpretability and simplicity.\n\n### Interpretability\n\nFew other models possess coefficients that are so directly linked to their variables with a such a clear interpretation. Tomorrow we're going to learn about ways to make them even easier to interpret.\n\n### Simplicity\n\nA linear regression model can be communicated just by writing out its equation. It's kind of incredible that such high dimensional relationships can be described from just a linear combination of variables and coefficients. \n\n# Assignment\n- Continue to predict New York City apartment rents. This is your last assignment with this dataset.\n- You may select any number of features. You are encouraged to engineer new features.\n- Get and plot your model's coefficients.\n- Report your Root Mean Squared Error, Mean Absolute Error, and R^2 Score, for your Train and Test sets. Share your scores with your cohort on Slack!\n- Fit a model with 2 features, and visualize the plane of best fit in 3D.\n- Commit your notebook to your fork of the repo.\n\n## Stretch Goals\n\nStudy more about Linear Regression. Here are two helpful links. If you find more links, share your favorites with your cohort on Slack.\n\n1. Watch this 20 minute video that just hit 1 million views: Brandon Foltz, Statistics 101: Simple Linear Regression (https://www.youtube.com/watch?v=ZkjP5RJLQF4)\n2. Skim _An Introduction to Statistical Learning_, Chapter 3.1, Simple Linear Regression, & Chapter 3.2, Multiple Linear Regression (http://www-bcf.usc.edu/~gareth/ISL/ISLR%20Seventh%20Printing.pdf)\n\nIn your 3D visualization, can you include the actual datapoints, like in [this notebook](https://nbviewer.jupyter.org/urls/s3.amazonaws.com/datarobotblog/notebooks/multiple_regression_in_python.ipynb)? Can you also include the residual lines from the datapoints to the plane of the best fit, like in _An Introduction to Statistical Learning?_ This would be hard to do, but awesome!\n\n\nCan you get creative with feature engineering? Share with your cohort on Slack. We mentioned some feature ideas at the end of last lesson, but didn't demonstrate how to engineer them. So here are some example solutions:\n\n```python\n# Does apartment have a non-empty description?\ndf['description'] = df['description'].str.strip().fillna('')\ndf['has_description'] = df['description'] != ''\n\n# How long is the description?\ndf['description_length'] = df['description'].str.len()\n\n# How many total perks does each apartment have?\nperk_cols = ['elevator', 'cats_allowed', 'hardwood_floors', 'dogs_allowed',\n             'doorman', 'dishwasher', 'no_fee', 'laundry_in_building',\n             'fitness_center', 'pre-war', 'laundry_in_unit', 'roof_deck',\n             'outdoor_space', 'dining_room', 'high_speed_internet', 'balcony',\n             'swimming_pool', 'new_construction', 'exclusive', 'terrace', \n             'loft', 'garden_patio', 'common_outdoor_space', \n             'wheelchair_access']\ndf['perk_count'] = df[perk_cols].sum(axis=1)\n\n# Are pets allowed?\ndf['pets_allowed'] = (df['cats_allowed']==1) | (df['dogs_allowed']==1)\n```\n\n", "meta": {"hexsha": "1b8423aa1ac58958b4b357872aafa0508dba7770", "size": 542648, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "module3-understanding-linear-regression/understanding_linear_regression.ipynb", "max_stars_repo_name": "martinclehman/DS-Unit-2-Linear-Models", "max_stars_repo_head_hexsha": "4b9e154622535cda5a5c66ee5916fae85f647c6f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "module3-understanding-linear-regression/understanding_linear_regression.ipynb", "max_issues_repo_name": "martinclehman/DS-Unit-2-Linear-Models", "max_issues_repo_head_hexsha": "4b9e154622535cda5a5c66ee5916fae85f647c6f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "module3-understanding-linear-regression/understanding_linear_regression.ipynb", "max_forks_repo_name": "martinclehman/DS-Unit-2-Linear-Models", "max_forks_repo_head_hexsha": "4b9e154622535cda5a5c66ee5916fae85f647c6f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 249.0353373107, "max_line_length": 167150, "alphanum_fraction": 0.9262873907, "converted": true, "num_tokens": 10146, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "# Multi-Cores programming\n# author :\u7530\u5947 ID: SA16225271\n# Asignment2\n\n## 1. parallel prefix-sum\n\u5b9a\u4e49\u5185\u90e8\u6570\u636e\u7ed3\u6784 TreeNode;\n```java\npublic class TreeNode {\n\tint left, right;\n\tint sum;\n\tint fromleft;\n\tTreeNode leftChild, rightChild;\n\t\n\tTreeNode(int left, int right, int sum, int fromleft) {\n\t\tthis.left = left;\n\t\tthis.right = right;\n\t\tthis.sum = sum;\n\t\tthis.fromleft = fromleft;\n\t\tthis.leftChild = null;\n\t\tthis.rightChild = null;\n\t}\n}\n\n```\nBottom-Up \u6784\u5efa\u4e00\u9897Sum Tree;\n```java\nprotected TreeNode compute() {\n\t\tif (high - low <= SEQUENTIAL_CUTOFF) {\n\t\t\tint sum = 0;\n\t\t\tfor (int i = low; i < high; i++) {\n\t\t\t\tsum += input[i];\n\t\t\t}\n\t\t\treturn new TreeNode(low, high, sum, 0);\n\t\t}\n\t\telse {\n\t\t\tint mid = (low+high)/2;\n\t\t\tTreeBuilder leftTask = new TreeBuilder(input, low, mid, SEQUENTIAL_CUTOFF);\n\t\t\tTreeBuilder rightTask = new TreeBuilder(input, mid, high, SEQUENTIAL_CUTOFF);\n\t\t\tleftTask.fork();\n\t\t\tTreeNode rightChild = rightTask.compute();\n\t\t\tTreeNode leftChild = (TreeNode) leftTask.join();\n\t\t\tTreeNode root = new TreeNode(low, high, leftChild.sum+rightChild.sum, 0);\n\t\t\troot.leftChild = leftChild;\n\t\t\troot.rightChild = rightChild;\n\t\t\treturn root;\n\t\t}\n```\nTop-Down \u66f4\u65b0 Sum Tree \u7684 fromLeft \u6570\u636e\u57df;\n```java\n@Override\n\tprotected Object compute() {\n\t\tif (high - low <= SEQUENTIAL_CUTOFF) {\n\t\t\toutput[low] = root.fromleft + input[low];\n\t\t\tfor (int i = low+1; i < high; i++) {\n\t\t\t\toutput[i] = output[i-1] + input[i];\n\t\t\t}\n\t\t}\n\t\telse {\n\t\t\tint mid = (low+high)/2;\n\t\t\tTreePrefix leftTask = new TreePrefix(root.leftChild, root.fromleft, input, output, low, mid, SEQUENTIAL_CUTOFF);\n\t\t\tTreePrefix rightTask = new TreePrefix(root.rightChild, root.fromleft+root.leftChild.sum, input, output, mid, high, SEQUENTIAL_CUTOFF);\n\t\t\tleftTask.fork();\n\t\t\trightTask.compute();\n\t\t\tleftTask.join();\n\t\t}\n\t\treturn null;\n}\n```\n\u6700\u7ec8\u5b9e\u73b0\u5c31\u662f:\n```java\npublic static int[] getPrefixSum(int[] input, int cutoff) {\n\t\tTreeNode root = TreeBuilder.sumTree(input, cutoff);\n\t\treturn TreePrefix.prefixSum(root, input, cutoff);\n\t}\n```\n\n## 2. Parallel Monte PI\n```java\npublic class MontePIParallel extends Thread{\n\tlong left;\n\tlong right;\n\tlong sum = 0;\n\tMontePIParallel(long left, long right) {this.left = left; this.right = right;}\n\t@Override\n\tpublic void run() {\n\t\tdouble x, y;\n\t\t\n\t\tfor (long i = left; i < right; i++) {\n\t\t\tx = Math.random();\n\t\t\ty = Math.random();\n\t\t\tif ((x * x + y * y) <= 1) {\n\t\t\t\tsum ++;\n\t\t\t}\n\t\t}\n\t}\n\t\n\tpublic static double getPI(long n, int threadSize) throws InterruptedException {\n\t\tMontePIParallel[] mpp = new MontePIParallel[threadSize];\n\t\tfor (int i = 0; i < threadSize; i++) {\n\t\t\tmpp[i] = new MontePIParallel(n/threadSize*i, n/threadSize*(i+1));\n\t\t\tmpp[i].start();\n\t\t}\n\t\t\n\t\tint count = 0;\n\t\tfor (int i = 0; i < threadSize; i++) {\n\t\t\tmpp[i].join();\n\t\t\tcount += mpp[i].sum;\n\t\t}\n\t\tdouble PI = 4.0 * count / n;\n\t\treturn PI;\n\t}\n\t\n\tpublic static double getPISerialize(long n) {\n\t\tdouble PI;\n\t\tdouble x,y;\n\t\tlong count = 0;\n\t\tfor (long i = 0; i < n; i++) {\n\t\t\tx = Math.random();\n\t\t\ty = Math.random();\n\t\t\tif ((x * x + y * y) <= 1) {\n\t\t\t\tcount ++;\n\t\t\t}\n\t\t}\n\t\tPI = 4.0 * count / n;\n\t\treturn PI;\n\t}\n\t\n\tpublic static void main(String[] args) throws InterruptedException {\n\t\tdouble PI;\n\t\tlong n = 1L<<40;\n\t\tint threadSize = 1<<3;\n\t\tlong start, end;\n\t\tstart = System.currentTimeMillis();\n\t\tPI = MontePIParallel.getPISerialize(n);\n\t\tend = System.currentTimeMillis();\n\t\tSystem.out.println(\"Serialize PI: \" + PI);\n\t\tSystem.out.println(end - start);\n\t\t\n\t\tstart = System.currentTimeMillis();\n\t\tPI = MontePIParallel.getPI(n, threadSize);\n\t\tend = System.currentTimeMillis();\n\t\tSystem.out.println(\"Parallel PI: \" + PI);\n\t\tSystem.out.println(end - start);\n\t\t\n\t\t\n\t\t\n\t}\n}\n\n```\n\n## 3. Synchronized Block Vs java.util.concurrent.locks\n\u540c\u6b65\u5757\u5b9e\u73b0\u4e00\u4e2a\u8ba1\u6570\u5668;\n```java\npublic class SYNCounter implements Counter{\n\tprivate long count = 0;\n\tSYNCounter(long count) {this.count = count;}\n\tpublic long getAndIncrement() {\n\t\tsynchronized(this) {\n\t\t\treturn this.count++;\n\t\t}\n\t}\n}\n```\n\u91c7\u7528Lock\u5b9e\u73b0\u4e00\u4e2a\u8ba1\u6570\u5668;\n```java\npublic class CASCounter implements Counter{\n\tprivate long count = 0;\n\tprivate Lock lock = new ReentrantLock();\n\tCASCounter(long count) {this.count = count;}\n\tpublic long getAndIncrement() {\n\t\ttry {\n\t\t\tlock.lock();\n\t\t\treturn count ++;\n\t\t\t\n\t\t} finally {\n\t\t\tlock.unlock();\n\t\t}\n\t\t\n\t}\n}\n\n```\n\u5e76\u53d1\u7248\u672c\u7684\u6c42\u7d20\u6570.\u63a5\u53e3\u662f Counter, \u53ef\u4ee5\u4f20\u5165\u4e0d\u540c\u7684 Counter \u5b9e\u73b0\u6765\u83b7\u5f97\u4e0d\u540c\u7684\u6027\u80fd.\n```java\npublic void run() {\n\t\tlong number;\n\t\twhile ((number = counter.getAndIncrement()) <= SIZE) {\n\t\t\tif (isPrime(number)) {\n\t\t\t\tprimes.add(number);\n//\t\t\t\tSystem.out.println(number);\n\t\t\t}\n\t\t}\n\t}\n\t\n\tpublic static boolean isPrime(long number) {\n\t\tif (number <= 1) return false;\n\t\tfor (long i = 2; i <= (long)Math.sqrt(number); i++) {\n\t\t\tif (number % i == 0) return false;\n\t\t}\n\t\treturn true;\n\t}\n\t\n\tpublic static List primeNumbers(Counter counter, int threads, long size) throws InterruptedException {\n\t\tList primes = new Vector();\n\t\tPrimeNumberConcurrency[] pns = new PrimeNumberConcurrency[threads];\n\t\tfor (int i = 0; i < threads; i ++) {\n\t\t\tpns[i] = new PrimeNumberConcurrency(primes, counter, size);\n\t\t\tpns[i].start();\n\t\t}\n\t\t\n\t\tfor (int i = 0; i < threads; i ++) {\n\t\t\tpns[i].join();\n\t\t}\n\t\treturn primes;\n\t}\n```\n\n\n\u7ee7\u7eed\u91c7\u7528\u8ba1\u7b97\u7d20\u6570\u7684\u5e76\u53d1\u7248\u672c, \u4f46\u662f\u4f7f\u7528\u4e0d\u540c\u7684 Counter, \u4e00\u4e2a\u662fJDK \u5e76\u53d1\u5305\u4e2d\u4f7f\u7528 `ReentrantLock` \u5b9e\u73b0\u7684 CASCounter, \u53e6\u4e00\u4e2a\u662f \u4f7f\u7528JVM\u539f\u751f\u5173\u952e\u5b57 `synchronized` \u5b9e\u73b0\u7684 SYNCounter; \u5982\u679c size \u6bd4\u8f83\u5c0f, \u540c\u6b65\u5757\u66f4\u5feb, \u4f46\u662f\u5f53 size \u5f88\u5927\u7684\u65f6\u5019, \u91c7\u7528 ReentrantLock \u7684\u6548\u679c\u8981\u6bd4 synchronized block \u8981\u597d. \u4f46\u662f\u7531\u4e8e \u73b0\u4ee3JVM \u5bf9 synchrinized \u505a\u4e86\u5f88\u591a\u4f18\u5316, \u8fd9\u79cd\u4f18\u52bf\u5e76\u4e0d\u660e\u663e.\n\n## 4. Parallel Merge Sort Time Complexity\n### \u76f8\u5173\u6982\u5ff5\n#### \u5e76\u53d1\u6267\u884c\u73af\u5883\u7684\u62bd\u8c61DAG\n\u6211\u4eec\u53ef\u4ee5\u628a\u591a\u7ebf\u7a0b\u8ba1\u7b97\u62bd\u8c61\u4e3a\u4e00\u4e2aDAG(Directed Acyclic Graph)\u6a21\u578b. $G=(V,E)$. \u5982\u56fe\u662f\u4e00\u4e2a\u9012\u5f52\u6c42\u89e3 Fib \u7a0b\u5e8f\u7684\u62bd\u8c61 DAG.\n\n\u9876\u70b9V\u4e2d\u90fd\u662f\u6307\u4ee4, \u6216\u8005\u8bf4\u975e\u5e76\u884c\u6307\u4ee4\u5e8f\u5217strands.\n\n\u8fb9E\u4ee3\u8868\u7684\u662f\u6307\u4ee4\u4e4b\u95f4\u7684\u4f9d\u8d56\u5173\u7cfb, $(u,v)\\in E.$\u8868\u793a$u$ \u5fc5\u987b\u8981\u5728 $v$ \u4e4b\u524d\u6267\u884c.\n\n - Continuation Edges $(u, v)$ \u662f\u5728\u9876\u70b9\u5185\u90e8\u6c34\u5e73\u753b\u51fa\u7684,\u8868\u793a $v$ \u5728 $u$ \u540e\u9762\u4e32\u884c\u6267\u884c.\n - Call Edges $(u, v)$ \u6307\u5411\u4e0b\u65b9,\u8868\u793a\u8c03\u7528\u5173\u7cfb, $u$\u8c03\u7528\u5b50\u8fc7\u7a0b$v$.\n - Spawn Edges $(u, v)$ \u6307\u5411\u4e0b\u65b9,\u8868\u793a\u7684\u662f $u$ \u751f\u4ea7\u51fa(Spawn)\u4e00\u4e2a $v$, \u5e76\u4e14\u4ed6\u548c $u$ \u662f\u5e76\u884c\u6267\u884c\u7684\u5173\u7cfb.\n - Return Edges \u6307\u5411\u4e0a, \u8868\u793a\u63a5\u4e0b\u6765\u7684strands\u6267\u884c\u9700\u8981\u9996\u5148\u4ece\u4e00\u4e2a\u8fc7\u7a0b\u8c03\u7528\u4e2d\u8fd4\u56de,\u4ea6\u6216\u662f\u5728\u5e76\u884c\u7ed3\u675f\u540e\u540c\u6b65\u6c47\u5408.\n\n#### Work\n$T_1 = \\text{\u7b97\u6cd5\u5728\u53ea\u6709\u4e00\u4e2a\u5904\u7406\u5668\u65f6\u5019\u7684\u6267\u884c\u65f6\u95f4\u5373\u5b8c\u5168\u4e32\u884c\u65f6\u95f4}.$ \u5b8c\u5168\u4e32\u884c\u7684\u6267\u884c\u65f6\u95f4\u5c31\u53eb work.\n\n\u7406\u60f3\u7684P\u4e2a\u5904\u7406\u5668\u7684\u5e76\u884c\u8ba1\u7b97\u673a\u4e00\u6b21\u6700\u591a\u80fd\u591f\u540c\u65f6\u505a P \u4e2a\u5355\u5143\u7684 work. \u56e0\u6b64\u5728 $T_P$ \u65f6\u95f4\u5185, \u6700\u591a\u505a\u7684 work \u91cf\u662f $PT_P$. \u7531\u4e8e\u603b\u7684 work \u662f $T_1$, \u90a3\u4e48\u5fc5\u987b\u8981\u6709 $PT_P \\geq T_1$\u624d\u80fd\u591f\u5b8c\u6210\u539f\u6765\u7684\u4e32\u884c\u4efb\u52a1. \u90a3\u4e48\u6211\u4eec\u5f97\u5230\u4e00\u4e2a\u539f\u5219\u5b9a\u5f8b *work law*:\n$$\nT_P \\geq \\frac{T_1}{P}\n$$\n\n#### Span\n$T_\\infty = \\text{\u7b97\u6cd5\u5728\u6709\u65e0\u9650\u4e2a\u5904\u7406\u5668\u7684\u65f6\u5019\u7684\u6267\u884c\u65f6\u95f4}$. \u5c31\u662f\u5047\u8bbe\u6709\u65e0\u7a77\u4e2a\u5904\u7406\u5668\u7684\u65f6\u5019,\u7b97\u6cd5\u6700\u574f\u6267\u884c\u65f6\u95f4\u53eb span. \n\nspan \u5176\u5b9e\u5c31\u662f\u6709\u5411\u65e0\u73af\u56fe(DAG)\u4e2d\u4ece\u8d77\u70b9\u5230\u7ec8\u70b9\u7684\u6700\u957f\u7684\u8def\u5f84. \u4e5f\u5373\u662f\u5e76\u884c\u7b97\u6cd5\u5fc5\u987b\u4f9d\u8d56\u8fd9\u6837\u4e00\u4e2a\u6700\u957f\u7684\u8def\u5f84\u624d\u80fd\u6267\u884c\u5b8c\u6210.\n\n*span law* \u5c31\u662f\u5b9e\u9645\u751f\u4ea7\u73af\u5883\u53ea\u53ef\u80fd\u662f\u6709\u9650\u4e2a\u5904\u7406\u5668P,\u4ed6\u7684\u6267\u884c\u65f6\u95f4\u80af\u5b9a\u6bd4\u65e0\u7a77\u4e2a\u66f4\u6162, \u90a3\u4e48\u5fc5\u5b9a\u6709\n$$\nT_P \\geq T_\\infty\n$$\n\n#### Speedup\n\u52a0\u901f\u6bd4 $\\frac{T_1}{T_P}$ \u5c31\u5b9a\u4e49\u7684\u662f\u5b9e\u9645\u4f7f\u7528 P \u4e2a\u5904\u7406\u5668\u7684\u65f6\u5019, \u548c\u4f7f\u7528\u4e00\u4e2a\u5904\u7406\u5668\u76f8\u6bd4\u65f6\u95f4\u5c11\u4e86\u591a\u5c11.\n\n\u901a\u8fc7\u4e0a\u8ff0\u7684 *work law* \u53ef\u4ee5\u77e5\u9053, $\\frac{T_1}{T_P} \\leq P$.\n\n\u9700\u8981\u7279\u522b\u6ce8\u610f\u7684\u662f, \u5e76\u884c\u53ea\u662f\u63d0\u4f9b\u5e38\u6570\u4e0a\u7684\u52a0\u901f, \u4e0d\u4f1a\u51fa\u73b0\u6307\u6570\u7ea7\u522b\u6216\u8005\u6570\u91cf\u7ea7\u7684\u52a0\u901f.\n\n>parallelism provides only constant time improvements (the constant being the number of processors) to any algorithm! Parallelism cannot move an algorithm from a higher to lower complexity class (e.g., exponential to polynomial, or quadratic to linear). Parallelism is not a silver bullet: good algorithm design and analysis is still needed.\n\n#### Parallelism\n\u5e76\u884c\u8ba1\u7b97\u590d\u6742\u5ea6\u5206\u6790, \u6211\u4eec\u4e00\u822c\u4f7f\u7528 $\\frac{T_1}{T_\\infty}$\u5373 work \u548c span \u7684\u6bd4\u503c\u6765\u8868\u793a\u5e76\u884c\u8ba1\u7b97\u7684\u590d\u6742\u5ea6.\n\u8be5\u516c\u5f0f\u53ef\u4ee5\u4ece\u4e09\u4e2a\u65b9\u9762\u6765\u7406\u89e3:\n\n - Ratio : The average amount of work that can be performed for each step of parallel execution time.\n - Upper Bound : the maximum possible speedup that can be achieved on any number of processors.\n - Limit: The limit on the possibility of attaining perfect linear speedup. Once the number of processors exceeds the parallelism, the computation cannot possibly achieve perfect linear speedup. The more processors we use beyond parallelism, the less perfect the speedup.\n\n#### MergeSort \u5e76\u884c\u5206\u6790\n\u4e32\u884c\u5f52\u5e76\u6392\u5e8f\u5373\u53ea\u6709\u4e00\u4e2a\u5904\u7406\u5668\u7684\u65f6\u5019, \u5176\u9012\u63a8\u516c\u5f0f\u5982\u4e0b, \u65f6\u95f4\u590d\u6742\u5ea6\u662f $O(nlogn)$;\n$$\nT_1(n) = 2T_1(n/2) + n \\space \\text{(work)}\\tag{1}\n$$\n\u4ee5\u4e0b\u662f\u4e32\u884c\u5f52\u5e76\u6392\u5e8f\u7684\u4f2a\u4ee3\u7801.\n```java\nMerge-Sort(A, l, r)\n    if (l < r) {\n        int mid = (l+r)/2\n        Merge-Sort(A, l, mid)\n        Merge-Sort(A, mid+1, r)\n        Merge(A, l, mid, r)\n    }\n\nMerge(A, l, mid, r)\n    left = A[l...mid] + sentinel\n    right = A[mid+1...r] + sentinel\n    i = 0\n    j = 0\n    k = 0\n    while (i < left.length || j < right.length) {\n        if (left[i] < right[j]) A[k++] = left[i++];\n        else A[k++] = right[j++];\n    }\n```\n\u5f53\u91c7\u7528 Divide-And-Conquer \u5bf9\u5de6\u53f3\u4e24\u90e8\u5206\u7684\u5f52\u5e76\u8fdb\u884c\u5e76\u884c\u65f6(\u5047\u8bbe\u662f\u65e0\u7a77\u4e2a\u5904\u7406\u5668), \u4f46\u662f\u5408\u5e76\u8fc7\u7a0b\u8fd8\u662f $O(n)$; \u6700\u540e\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f$O(n)$\n$$\nT_\\infty(n) = T_\\infty(n/2) + n \\space \\text{(span)}\\tag{2}\n$$\n```java\nMerge-Sort(A, l, r)\n    if (l < r) {\n        int mid = (l+r)/2\n        Spawn Merge-Sort(A, l, mid)\n        Merge-Sort(A, mid+1, r)\n        Sync\n        Merge(A, l, mid, r)\n    }\n    \n```\n\u4ece(1) (2) \u5f97\u5230\u7684\u5e76\u884c\u5ea6\u662f $O(\\frac{nlogn}{n})=O(logn)$.\n\n\u5982\u679c\u6211\u4eec\u8fdb\u4e00\u6b65\u5bf9\u5408\u5e76\u8fc7\u7a0b\u505a\u5e76\u884c\u64cd\u4f5c, \u91c7\u7528\u5de6\u53f3\u5b50\u6570\u7ec4\u4e8c\u5206\u67e5\u627e\u7684\u601d\u60f3,\u53ef\u4ee5\u5c06\u5408\u5e76\u8fc7\u7a0b\u53d8\u4e3a $O(logn)$. \u65b9\u6cd5\u5c31\u662f\u5bf9\u5df2\u7ecf\u6392\u597d\u5e8f\u7684\u5de6\u53f3\u4e24\u90e8\u5206\u5b50\u6570\u7ec4, \u9996\u5148\u627e\u5230\u8f83\u957f\u7684\u5de6\u534a\u90e8\u5206\u5b50\u6570\u7ec4$L$\u7684\u4e2d\u95f4\u503cx, O(1) \u65f6\u95f4\u5373\u53ef\u5b8c\u6210, \u7136\u540e\u518d\u770bx \u5728\u53f3\u534a\u90e8\u5206\u5b50\u6570\u7ec4$R$\u8be5\u63d2\u5165\u7684\u4f4d\u7f6ep, \u91c7\u7528BinarySearch O(logn) \u65f6\u95f4\u5373\u53ef\u5b8c\u6210. \u8fd9\u6837\u5de6\u534a\u90e8\u5206\u5c31\u5206\u6210\u4e86 $L_1$ \u548c $L_2$ \u4e24\u90e8\u5206, \u5206\u522b\u662f\u6bd4x\u5c0f\u7684\u548c\u6bd4x\u5927\u7684. \u540c\u6837, \u53f3\u534a\u90e8\u5206\u5b50\u6570\u7ec4\u5206\u6210 $R_1$ \u548c $R_2$ \u5206\u522b\u662f\u6bd4x\u5c0f\u548c\u6bd4x\u5927\u7684.\u90a3\u4e48\u8fd9\u5c31\u662f\u7c7b\u4f3c\u4e8e\u5feb\u901f\u6392\u5e8f, \u5c06 x \u590d\u5236\u5230\u539f\u6765\u6570\u7ec4\u4e2d\u5e94\u8be5\u5728\u7684\u4f4d\u7f6e(L1+R1+x+L2+R2), \u8fd9\u91cc\u7684\u6392\u5e8f\u90fd\u4e0d\u662f  in-place \u7684, \u5176\u4e2d\u6709\u4e00\u4e2a\u590d\u5236\u6570\u7ec4\u7684\u8fc7\u7a0b, \u5982\u679c\u8fd9\u4e2a\u8fc7\u7a0b\u4f7f\u7528 for \u5faa\u73af, \u90a3\u4e48\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8ba4\u4e3a\u662f O(n) \u4e86, \u56e0\u6b64\u5b50\u6570\u7ec4\u590d\u5236\u8fc7\u7a0b\u6211\u4eec\u76f4\u63a5\u4f7f\u7528\u64cd\u4f5c\u7cfb\u7edf\u57fa\u4e8e\u5185\u5b58\u5757\u7684\u590d\u5236`System.arraycopy`(\u4e5f\u53ef\u4ee5\u4f7f\u7528 `Arrays.copyOf`, \u4ed6\u7684\u5185\u90e8\u5b9e\u9645\u8c03\u7528\u7684\u5c31\u662f `System.arraycopy`), \u4e0d\u4f7f\u7528 for \u5faa\u73af\u8fdb\u884c\u590d\u5236.\n\n```java\n\n// \u6bcf\u4e00\u5c42Merge\u5185\u90e8\u90fd\u7533\u8bf7\u65b0\u7a7a\u95f4\u8bb2A\u590d\u5236\u4e00\u4efdtemp,\u7136\u540e\u6392\u5e8f\u8fc7\u7a0b\u4e2d\u5c06temp\u5408\u5e76\u5199\u4f1aA\n\nParallel-Merge-Sort(A, l, r)\n    if (l < r) {\n        int mid = (l+r)/2\n        Spawn Parallel-Merge-Sort(A, l, mid)\n        Parallel-Merge-Sort(A, mid+1, r)\n        Sync\n        T = Arrays.copyOf(A)\n        Parallel-Merge(T, l, mid, mid+1, r, A, l)\n    }\n    \nParallel-Merge(T, l1, r1, l2, r2, A, pos)\n    n1 = r1 - l1 + 1\n    n2 = r2 - l2 + 1\n    if n1 < n2\n        swap(l1, l2)\n        swap(r1, r2)\n        swap(n1, n2)\n    if n1 == 0\n        return\n    else \n        x = (l1+r1)/2\n        y = BinarySearch(T, l2, r2, T[x])\n        len = x - l1 + y - l2\n        A[pos+len] = T[x]\n        Spawn Parallel-Merge(T, l1, x-1, l2, y-1, A, pos)\n        Parallel-Merge(T, x+1, r1, y, r2, A, pos+len+1)\n        Sync  \n```\n\n\u4ee5\u4e0b\u662f\u5e76\u884c\u8fdb\u884c merge() \u64cd\u4f5c\u7684\u9012\u63a8\u516c\u5f0f;\n$$\nT^{merge}_\\infty(n) = T^{merge}_\\infty(n/2) + logn \\space \\text{(span)}\\tag{3}\n$$\n\u5bf9\u4e8e(3), \u6211\u4eec\u53ef\u4ee5\u505a\u5982\u4e0b\u63a8\u5bfc:\n\n\n\\begin{equation} \\nonumber\n\\begin{split}\nT^{merge}_\\infty(n) &= T^{merge}_\\infty(\\frac{n}{2^1}) + log(\\frac{n}{2^0})\\\\\n     &= T^{merge}_\\infty(\\frac{n}{2^2}) + log(\\frac{n}{2^1}) + log(\\frac{n}{2^0})\\\\\n     &= T^{merge}_\\infty(\\frac{n}{2^3}) + log(\\frac{n}{2^2}) + log(\\frac{n}{2^1}) + log(\\frac{n}{2^0})\\\\\n     &= \\cdots \\\\\n     &= T^{merge}_\\infty(\\frac{n}{2^k}) + log(\\frac{n}{2^{k-1}}) + \\cdots + log(\\frac{n}{2^1}) + log(\\frac{n}{2^0})\\\\\n     &= 1 + log(\\frac{n}{2^{k-1}}) + \\cdots + log(\\frac{n}{2^1}) + log(\\frac{n}{2^0})\n\\end{split}\n\\end{equation}\n\n\u5176\u4e2d\n$$\n\\frac{n}{2^k} = 1; k = logn;\n$$\n\n\u6700\u540e\u53ef\u77e5\n\\begin{equation}\\nonumber\n\\begin{split}\nT^{merge}_\\infty(n) &= log(\\frac{n}{2^{0}}\\frac{n}{2^{1}}\\frac{n}{2^{2}}\\cdots\\frac{n}{2^{k-1}})\\\\\n     &= log(\\frac{n^k}{2^{\\frac{(k)(k-1)}{2}}})\\\\\n     &= klogn + \\frac{k(k-1)}{2}\\\\\n     &= O(log^2n)\n\\end{split}\n\\end{equation}\n\u8fd9\u6837\u4e00\u6765, \u65b0\u7684\u5e76\u884c\u5f52\u5e76\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u9012\u63a8\u516c\u5f0f\u5c31\u662f:\n$$\nT_\\infty(n) = T_\\infty(n/2) + log^2n \\space \\text{(span)}\\tag{4}\n$$\n\u5bf9\u4e8e(4)\u5f0f\u5b50, \u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u9012\u63a8\u51fa\u5176\u65f6\u95f4\u590d\u6742\u5ea6\u662f O(log^3n).\n\u6700\u540e\u5f97\u5230\u5e76\u884c\u8ba1\u7b97\u590d\u6742\u5ea6\u662f$O(\\frac{nlogn}{log^3n})=O(\\frac{n}{log^2n})$\n\n### Java \u5b9e\u73b0\n```java\n@Override\n\tpublic void run() {\n\t\tif (r - l < CUTOFF) {\n\t\t\tArrays.sort(A, l, r+1);\n\t\t\treturn ;\n\t\t}\n\t\telse {\n\t\t\tint mid = (l + r) / 2;\n\t\t\tParallelMergeSort leftMergeSort = new ParallelMergeSort(A, l, mid, CUTOFF);\n\t\t\tParallelMergeSort rightMergeSort = new ParallelMergeSort(A, mid+1, r, CUTOFF);\n\t\t\tleftMergeSort.start();\n\t\t\trightMergeSort.start();\n\t\t\ttry {\n\t\t\t\tleftMergeSort.join();\n\t\t\t\trightMergeSort.join();\n\t\t\t} catch (InterruptedException e) {\n\t\t\t\te.printStackTrace();\n\t\t\t}\n\t\t\tint[] T = Arrays.copyOf(A, A.length);\n\t\t\tParallelMerge pm = new ParallelMerge(T, l, mid, mid+1, r, A, l);\n\t\t\tpm.run();\n\t\t}\n\t}\n\t\n\t\nclass ParallelMerge extends Thread {\n\tint[] A;\n\tint[] T;\n\tint l1,r1,l2,r2,pos;\n\tint CUTOFF;\n\t\n\tpublic ParallelMerge(int[] T, int l1, int r1, int l2, int r2, int[] A, int pos, int cutoff) {\n\t\tthis.A = A;\n\t\tthis.T = T;\n\t\tthis.l1 = l1;\n\t\tthis.r1 = r1;\n\t\tthis.l2 = l2;\n\t\tthis.r2 = r2;\n\t\tthis.pos = pos;\n\t\tthis.CUTOFF = cutoff;\n\t}\n\t\n\t@Override\n\tpublic void run() {\n\t\tint n1 = r1 - l1 + 1;\n\t\tint n2 = r2 - l2 + 1;\n\t\tif (n1 < n2){\n\t\t\tn1 = n1 ^ n2; n2 = n1 ^ n2; n1 = n1 ^ n2;\n\t\t\tr1 = r1 ^ r2; r2 = r1 ^ r2; r1 = r1 ^ r2;\n\t\t\tl1 = l1 ^ l2; l2 = l1 ^ l2; l1 = l1 ^ l2;\n\t\t}\n\t\t\n\t\tif (n1 == 0) {\n\t\t\treturn;\n\t\t}\n\t\tif (n1 + n2 < CUTOFF) {\n\t\t\tint k = 0;\n\t\t\tint i = l1, j = l2;\n\t\t\twhile (i < r1 || j < r2) {\n\t\t\t\tif (i == r1) {\n\t\t\t\t\twhile (j < r2) {\n\t\t\t\t\t\tA[pos+k++] = T[j++];\n\t\t\t\t\t}\n\t\t\t\t\tbreak;\n\t\t\t\t}\n\t\t\t\tif (j == r2) {\n\t\t\t\t\twhile (i < r1) {\n\t\t\t\t\t\tA[pos+k++] = T[i++];\n\t\t\t\t\t}\n\t\t\t\t\tbreak;\n\t\t\t\t}\n\t\t\t\tif (T[i] < T[j]) {\n\t\t\t\t\tA[pos+k++] = T[i++];\n\t\t\t\t}\n\t\t\t\telse {\n\t\t\t\t\tA[pos+k++] = T[j++];\n\t\t\t\t}\n\t\t\t}\n\t\t}\n\t\telse {\n\t\t\tint x = (l1 + r1) / 2;\n\t\t\tint y = Arrays.binarySearch(T, l2, r2+1, T[x]);// return  (-(insertion point) - 1) if no key else >=0\n\t\t\ty = y >= 0 ? y : -(y+1); // \u63d2\u5165\u70b9\n\t\t\tint len = x - l1 + y - l2;\n\t\t\tA[pos+len] = T[x];\n\t\t\tParallelMerge pm1 = new ParallelMerge(T, l1, x-1, l2, y-1, A, pos, CUTOFF);\n\t\t\tParallelMerge pm2 = new ParallelMerge(T, x+1, r1, y, r2, A, pos+len+1, CUTOFF);\n\t\t\tpm1.start();\n\t\t\tpm2.start();\n\t\t\ttry {\n\t\t\t\tpm1.join();\n\t\t\t\tpm2.join();\n\t\t\t} catch (InterruptedException e) {\n\t\t\t\te.printStackTrace();\n\t\t\t}\n\t\t\t\n\t\t}\n\t}\n}\n```\n\n## 5. Mutual and DeadLock Ananysis\n1. \u534f\u8bae\u6ee1\u8db3\u4e92\u65a5.\n\u8bc1\u660e\u5982\u4e0b:\n\u4ece\u4ee3\u7801\u4e2d\u53ef\u4ee5\u770b\u51fa\u6709\u5982\u4e0b\u6b63\u5e38\u8bed\u5e8f:\n$$\nWrite_A[turn=A] -> Read_A[busy=false] -> \n\\\\ Write_A[busy=true] -> Read_A[turn=A] -> CS_A\n$$\n$$\nWrite_B[turn=B] -> Read_B[busy=false] ->\n\\\\ Write_B[busy=true] -> Read_B[turn=B] -> CS_B\n$$\n\u5047\u8bbe\u8be5\u534f\u8bae\u4e0d\u6ee1\u8db3\u4e92\u65a5\u6761\u4ef6, \u4e5f\u5c31\u662f\u8bf4\u5b58\u5728 $CS_A$ \u548c $CS_ B$ \u90fd\u5728\u6267\u884c\u4e34\u754c\u533a\u4ee3\u7801, \u90a3\u4e48\u53ef\u4ee5\u5f97\u51fa\u7684\u662f\n$$\nRead_A[turn=A] -> CS_A\n\\\\ Write_B[turn=B] -> Read_B[busy=false] -> \n\\\\ Write_B[busy=true] -> Read_B[turn=B] -> CS_B\n$$\n\u4f46\u662f\u53ef\u4ee5\u770b\u51fa, $CS_A$ \u5728\u6267\u884c\u7684\u65f6\u5019, busy \u4e0d\u53ef\u80fd\u662f false \u7684. \u4e5f\u5c31\u4e0d\u53ef\u80fd\u53d1\u751f\u540c\u65f6\u8fdb\u5165\u4e34\u754c\u533a\u4e86. \u56e0\u6b64\u8be5\u534f\u8bae\u662f\u6ee1\u8db3\u4e92\u65a5\u6761\u4ef6\u7684.\n\n2. \u4e0d\u6ee1\u8db3\u6b7b\u9501\u548c\u9965\u997f.\n\u56e0\u4e3a\u53ef\u4ee5\u5b58\u5728\u8fd9\u6837\u7684\u7ade\u4e89\u5e8f\u5217, \u5bfc\u81f4\u53cc\u65b9\u6b7b\u5faa\u73af, \u6c38\u8fdc\u4e5f\u65e0\u6cd5\u8fdb\u5165\u4e34\u754c\u533a.\n$$\nRead_A[turn=A] -> Write_B[turn=B] -> Read_B[turn=B] -> Write_A[turn=A]\n$$\n", "meta": {"hexsha": "26a96dfe06a62d42a03941e2e4119d3d01494f4b", "size": 19556, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "MultiCoreReport2.ipynb", "max_stars_repo_name": "kitianFresh/datascience", "max_stars_repo_head_hexsha": "2dee9fe1277ad242d0121ada82dc116d9acb1199", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "MultiCoreReport2.ipynb", "max_issues_repo_name": "kitianFresh/datascience", "max_issues_repo_head_hexsha": "2dee9fe1277ad242d0121ada82dc116d9acb1199", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "MultiCoreReport2.ipynb", "max_forks_repo_name": "kitianFresh/datascience", "max_forks_repo_head_hexsha": "2dee9fe1277ad242d0121ada82dc116d9acb1199", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 35.5563636364, "max_line_length": 503, "alphanum_fraction": 0.5029658417, "converted": true, "num_tokens": 5723, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.7956580903722561, "lm_q1q2_score": 0.4133613429320724}}
{"text": "\n\n# PyTorch tutorial \u306e\u5185\u5bb9\u306b\u3064\u3044\u3066\n\n[https://github.com/pytorch/tutorials/tree/master/beginner_source/nlp](https://github.com/pytorch/tutorials/tree/master/beginner_source/nlp) \u3092\u898b\u308b\u3068\nPyTorch \u3067 \u81ea\u7136\u8a00\u8a9e\u51e6\u7406\u3092\u884c\u3046\u5834\u5408\u306e\u30c1\u30e5\u30fc\u30c8\u30ea\u30a2\u30eb\u306f\u4ee5\u4e0b\u3068\u304a\u308a\u3067\u3042\u308b\n\n```\nDeep Learning for NLP with Pytorch\n----------------------------------\n\n1. pytorch_tutorial.py\n\tIntroduction to PyTorch\n\thttps://pytorch.org/tutorials/beginner/nlp/pytorch_tutorial.html\n\n2. deep_learning_tutorial.py\n\tDeep Learning with PyTorch\n\thttps://pytorch.org/tutorials/beginner/nlp/deep_learning_tutorial.html\n\n3. word_embeddings_tutorial.py\n\tWord Embeddings: Encoding Lexical Semantics\n\thttps://pytorch.org/tutorials/beginner/nlp/word_embeddings_tutorial.html\n\n4. sequence_models_tutorial.py\n\tSequence Models and Long-Short Term Memory Networks\n\thttps://pytorch.org/tutorials/beginner/nlp/sequence_models_tutorial.html\n\n5. advanced_tutorial.py\n\tAdvanced: Making Dynamic Decisions and the Bi-LSTM CRF\n\thttps://pytorch.org/tutorials/beginner/nlp/advanced_tutorial.html\n```\n\n\u4ee5\u4e0b\u3067\u306f\uff0c\u3053\u306e\u3046\u3061\u306e 1 \u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u3066\u3044\u308b\u3002\n\n\n\n```python\n%matplotlib inline\n```\n\n---\n- original: https://colab.research.google.com/github/pytorch/tutorials/blob/gh-pages/_downloads/7e00f82429a7b691e42bb5828dbb036e/pytorch_tutorial.ipynb\n- \u6ce8: \u3053\u306e colab \u30d5\u30a1\u30a4\u30eb\u306f PyTorch \u306e\u516c\u5f0f\u30c1\u30e5\u30fc\u30c8\u30ea\u30a2\u30eb\u304b\u3089\u306e\u7ffb\u8a33\u306b\u624b\u3092\u52a0\u3048\u305f\u3082\u306e\u3067\u3059\u3002\n- date: 2020-0723\n---\n\n# Torch \u306e\u30c6\u30f3\u30bd\u30eb\u30e9\u30a4\u30d6\u30e9\u30ea\u7d39\u4ecb \n\n\u30c6\u30f3\u30bd\u30eb\u3068\u306f 2 \u6b21\u4ee5\u4e0a\u306e\u6b21\u5143\u3092\u6301\u3064\u884c\u5217\u306e\u4e00\u822c\u5316\u3067\u3042\u308a\uff0c\u30c7\u30a3\u30fc\u30d7\u30e9\u30fc\u30cb\u30f3\u30b0\u30e2\u30c7\u30eb\u306f\u3059\u3079\u3066\u30c6\u30f3\u30bd\u30eb\u306b\u95a2\u3059\u308b\u8a08\u7b97\u3067\u3059\u3002\n\u3053\u308c\u304c\u4f55\u3092\u610f\u5473\u3059\u308b\u306e\u304b\u306b\u3064\u3044\u3066\u306f\uff0c\u5f8c\u307b\u3069\u8a73\u3057\u304f\u8aac\u660e\u3057\u307e\u3059\uff0e\u307e\u305a\uff0c\u30c6\u30f3\u30bd\u30eb\u3067\u4f55\u304c\u3067\u304d\u308b\u304b\u3092\u898b\u3066\u307f\u307e\u3057\u3087\u3046\uff0e\n\n\n\n\n```python\n# Author: Robert Guthrie\n\nimport torch\nimport torch.autograd as autograd\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport torch.optim as optim\n\ntorch.manual_seed(1)\n```\n\n\n\n\n    \n\n\n\n### \u30c6\u30f3\u30bd\u30eb\u306e\u4f5c\u6210 \n\n\u30c6\u30f3\u30bd\u30eb\u306f `torch.tesor()` \u95a2\u6570\u3092\u4f7f\u3063\u3066\u4f5c\u6210\u3055\u308c\u308b Python \u306e\u30ea\u30b9\u30c8\u3067\u3059\u3002\n\n\n\n\n\n\n```python\n# torch.tensor(data) \u306b\u3088\u308a\u4efb\u610f\u306e\u30c7\u30fc\u30bf\u3092\u6301\u3064  torch.Tensor \u30aa\u30d6\u30b8\u30a7\u30af\u30c8\u304c\u751f\u6210\u3055\u308c\u307e\u3059\u3002 \nV_data = [1., 2., 3.]\nV = torch.tensor(V_data)\nprint(V)\n\n# \u884c\u5217\u3092\u751f\u6210\u3057\u307e\u3059 \nM_data = [[1., 2., 3.], [4., 5., 6]]\nM = torch.tensor(M_data)\nprint(M)\n\n# 2\uff582\uff582 \u306e 3\u968e\u30c6\u30f3\u30bd\u30eb\u3092\u4f5c\u6210\u3057\u307e\u3059\nT_data = [[[1., 2.], [3., 4.]],\n          [[5., 6.], [7., 8.]]]\nT = torch.tensor(T_data)\nprint(T)\n```\n\n    tensor([1., 2., 3.])\n    tensor([[1., 2., 3.],\n            [4., 5., 6.]])\n    tensor([[[1., 2.],\n             [3., 4.]],\n    \n            [[5., 6.],\n             [7., 8.]]])\n\n\nWhat is a 3D tensor anyway? Think about it like this. If you have a\nvector, indexing into the vector gives you a scalar. If you have a\nmatrix, indexing into the matrix gives you a vector. If you have a 3D\ntensor, then indexing into the tensor gives you a matrix!\n\nA note on terminology:\nwhen I say \"tensor\" in this tutorial, it refers\nto any torch.Tensor object. Matrices and vectors are special cases of\ntorch.Tensors, where their dimension is 1 and 2 respectively. When I am\ntalking about 3D tensors, I will explicitly use the term \"3D tensor\".\n\n\n\n\n\n```python\n# Index into V and get a scalar (0 dimensional tensor)\nprint(V[0])\n# Get a Python number from it\nprint(V[0].item())\n\n# Index into M and get a vector\nprint(M[0])\n\n# Index into T and get a matrix\nprint(T[0])\n```\n\nYou can also create tensors of other data types. To create a tensor of integer types, try\ntorch.tensor([[1, 2], [3, 4]]) (where all elements in the list are integers).\nYou can also specify a data type by passing in ``dtype=torch.data_type``.\nCheck the documentation for more data types, but\nFloat and Long will be the most common.\n\n\n\n\nYou can create a tensor with random data and the supplied dimensionality\nwith torch.randn()\n\n\n\n\n\n```python\nx = torch.randn((3, 4, 5))\nprint(x)\n```\n\nOperations with Tensors\n\n\n\n\n```python\nx = torch.tensor([1., 2., 3.])\ny = torch.tensor([4., 5., 6.])\nz = x + y\nprint(z)\n```\n\nSee [`the documentation`](https://pytorch.org/docs/torch.html) for a complete list of the massive number of operations available to you. \nThey expand beyond just mathematical operations.\n\nOne helpful operation that we will make use of later is concatenation.\n\n\n\n\n\n```python\n# By default, it concatenates along the first axis (concatenates rows)\nx_1 = torch.randn(2, 5)\ny_1 = torch.randn(3, 5)\nz_1 = torch.cat([x_1, y_1])\nprint(z_1)\n\n# Concatenate columns:\nx_2 = torch.randn(2, 3)\ny_2 = torch.randn(2, 5)\n# second arg specifies which axis to concat along\nz_2 = torch.cat([x_2, y_2], 1)\nprint(z_2)\n\n# If your tensors are not compatible, torch will complain.  Uncomment to see the error\n# torch.cat([x_1, x_2])\n```\n\n### Reshaping Tensors\n\nUse the .view() method to reshape a tensor. This method receives heavy\nuse, because many neural network components expect their inputs to have\na certain shape. Often you will need to reshape before passing your data\nto the component.\n\n\n\n\n\n```python\nx = torch.randn(2, 3, 4)\nprint(x)\nprint(x.view(2, 12))  # Reshape to 2 rows, 12 columns\n# Same as above.  If one of the dimensions is -1, its size can be inferred\nprint(x.view(2, -1))\n```\n\nComputation Graphs and Automatic Differentiation\n================================================\n\nThe concept of a computation graph is essential to efficient deep\nlearning programming, because it allows you to not have to write the\nback propagation gradients yourself. A computation graph is simply a\nspecification of how your data is combined to give you the output. Since\nthe graph totally specifies what parameters were involved with which\noperations, it contains enough information to compute derivatives. This\nprobably sounds vague, so let's see what is going on using the\nfundamental flag ``requires_grad``.\n\nFirst, think from a programmers perspective. What is stored in the\ntorch.Tensor objects we were creating above? Obviously the data and the\nshape, and maybe a few other things. But when we added two tensors\ntogether, we got an output tensor. All this output tensor knows is its\ndata and shape. It has no idea that it was the sum of two other tensors\n(it could have been read in from a file, it could be the result of some\nother operation, etc.)\n\nIf ``requires_grad=True``, the Tensor object keeps track of how it was\ncreated. Lets see it in action.\n\n\n\n\n\n```python\n# Tensor factory methods have a ``requires_grad`` flag\nx = torch.tensor([1., 2., 3], requires_grad=True)\n\n# With requires_grad=True, you can still do all the operations you previously\n# could\ny = torch.tensor([4., 5., 6], requires_grad=True)\nz = x + y\nprint(z)\n\n# BUT z knows something extra.\nprint(z.grad_fn)\n```\n\nSo Tensors know what created them. z knows that it wasn't read in from\na file, it wasn't the result of a multiplication or exponential or\nwhatever. And if you keep following z.grad_fn, you will find yourself at\nx and y.\n\nBut how does that help us compute a gradient?\n\n\n\n\n\n```python\n# Lets sum up all the entries in z\ns = z.sum()\nprint(s)\nprint(s.grad_fn)\n```\n\nSo now, what is the derivative of this sum with respect to the first\ncomponent of x? In math, we want\n\n\\begin{align}\\frac{\\partial s}{\\partial x_0}\\end{align}\n\n\n\nWell, s knows that it was created as a sum of the tensor z. z knows\nthat it was the sum x + y. So\n\n\\begin{align}s = \\overbrace{x_0 + y_0}^\\text{$z_0$} + \\overbrace{x_1 + y_1}^\\text{$z_1$} + \\overbrace{x_2 + y_2}^\\text{$z_2$}\\end{align}\n\nAnd so s contains enough information to determine that the derivative\nwe want is 1!\n\nOf course this glosses over the challenge of how to actually compute\nthat derivative. The point here is that s is carrying along enough\ninformation that it is possible to compute it. In reality, the\ndevelopers of Pytorch program the sum() and + operations to know how to\ncompute their gradients, and run the back propagation algorithm. An\nin-depth discussion of that algorithm is beyond the scope of this\ntutorial.\n\n\n\n\nLets have Pytorch compute the gradient, and see that we were right:\n(note if you run this block multiple times, the gradient will increment.\nThat is because Pytorch *accumulates* the gradient into the .grad\nproperty, since for many models this is very convenient.)\n\n\n\n\n\n```python\n# calling .backward() on any variable will run backprop, starting from it.\ns.backward()\nprint(x.grad)\n```\n\nUnderstanding what is going on in the block below is crucial for being a\nsuccessful programmer in deep learning.\n\n\n\n\n\n```python\nx = torch.randn(2, 2)\ny = torch.randn(2, 2)\n# By default, user created Tensors have ``requires_grad=False``\nprint(x.requires_grad, y.requires_grad)\nz = x + y\n# So you can't backprop through z\nprint(z.grad_fn)\n\n# ``.requires_grad_( ... )`` changes an existing Tensor's ``requires_grad``\n# flag in-place. The input flag defaults to ``True`` if not given.\nx = x.requires_grad_()\ny = y.requires_grad_()\n# z contains enough information to compute gradients, as we saw above\nz = x + y\nprint(z.grad_fn)\n# If any input to an operation has ``requires_grad=True``, so will the output\nprint(z.requires_grad)\n\n# Now z has the computation history that relates itself to x and y\n# Can we just take its values, and **detach** it from its history?\nnew_z = z.detach()\n\n# ... does new_z have information to backprop to x and y?\n# NO!\nprint(new_z.grad_fn)\n# And how could it? ``z.detach()`` returns a tensor that shares the same storage\n# as ``z``, but with the computation history forgotten. It doesn't know anything\n# about how it was computed.\n# In essence, we have broken the Tensor away from its past history\n```\n\nYou can also stop autograd from tracking history on Tensors\nwith ``.requires_grad``=True by wrapping the code block in\n``with torch.no_grad():``\n\n\n\n\n```python\nprint(x.requires_grad)\nprint((x ** 2).requires_grad)\n\nwith torch.no_grad():\n\tprint((x ** 2).requires_grad)\n```\n", "meta": {"hexsha": "14b92319bd2ae3e94d658f9366d4ffc4fb030a86", "size": 20364, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/2020_0723pytorch_tutorial.ipynb", "max_stars_repo_name": "project-ccap/project-ccap.github.io", "max_stars_repo_head_hexsha": "867e32af5459ae55d864d9d022d69eac17fbb450", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/2020_0723pytorch_tutorial.ipynb", "max_issues_repo_name": "project-ccap/project-ccap.github.io", "max_issues_repo_head_hexsha": "867e32af5459ae55d864d9d022d69eac17fbb450", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-12-04T11:36:15.000Z", "max_issues_repo_issues_event_max_datetime": "2020-12-04T11:36:15.000Z", "max_forks_repo_path": "notebooks/2020_0723pytorch_tutorial.ipynb", "max_forks_repo_name": "project-ccap/project-ccap.github.io", "max_forks_repo_head_hexsha": "867e32af5459ae55d864d9d022d69eac17fbb450", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2020-07-22T02:58:14.000Z", "max_forks_repo_forks_event_max_datetime": "2020-07-23T07:02:07.000Z", "avg_line_length": 31.6702954899, "max_line_length": 269, "alphanum_fraction": 0.5009330191, "converted": true, "num_tokens": 2892, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "# COVID-19 and Weather Patterns\n\n## Imports\n\n- *os* for interfacing with the operating system\n- *pathlib* for interfacing with the file system\n- *zipfile* for managing archive files\n\n- *numpy* for array processing\n- *pandas* for tabular processing\n- *tensorflow* for tensor processing\n- *keras* for simplified tensor processing\n\n- *matplotlib* for visualization\n- *seaborn* for enhanced visualization\n\n\n```python\n# Custom\nimport data_processing\nimport run\n\n# File System\nimport os\nimport json\nfrom pathlib import Path\nfrom zipfile import ZipFile\nimport pickle\n\n# Processing\nimport gc\nimport numpy as np\nimport pandas as pd\nfrom sympy import *\nfrom sympy.geometry import *\nimport tensorflow\nfrom tensorflow import keras\n\n# Visualization\nimport matplotlib.pyplot as plt\nimport seaborn as sns\n\ntensorflow.__version__\n```\n\n\n\n\n    '2.7.0'\n\n\n\n## Introduction\n\nThe goal of this work is to determine whether or not weather patterns should be considered as a meaningful supporting input when making predictions about new daily COVID-19 cases within a given geospatial extent. Using census, weather, and COVID-19 datasets provided by the Urban Sustain project, the authors attempt to quantify the correlation between particular weather patterns and COVID-19 transmission events.\n\n## Defining Terms\n\n***Adam*** - A modified version of the stochastic gradient descent algorithm, also known as adaptive moment estimation. This algorithm is used to minimize a loss function during the iterative training process.\n\n***ReLU*** - Rectified linear unit. This is a type of activation function intended to introduce non-linearity to our model.\n\n***Urban Sustain Project*** - A joint effort between researchers at Colorado State University, Arizona State University, the University of California-Irvine, and the University of Maryland \u2013 Baltimore County.\n\n## Loading Data\n\n**DEVELOPER NOTE:** Download the five required datasets from Urban Sustain and place them in the cloned repository at ```./data/```. These datasets are also available at a shared OneDrive folder. This logic expects that these files exist at relative path ```../data/``` with respect to this notebook.\n\nWe'll begin by defining a path to our data directory and a list of the datasets that we expect to find there.\n\n\n```python\nDATA_PATH = '../data/' # Point this constant to the location of your data archive files\n\nEXPECTED_DATASETS = [\n    'county_total_population.Colorado.zip',\n    'covid_county.Colorado.zip',\n    'neon_2d_wind.Colorado.zip',\n    'neon_barometric_pressure.Colorado.zip',\n    'neon_single_asp_air_temperature.Colorado.zip'\n]\n```\n\nNext, we will attempt to extract each of these archived datasets into a dedicated subdirectory within the given data directory.\n\n\n```python\nstate = 'Colorado'\n```\n\n\n```python\ndef extract(state):\n    # For each listed dataset string in the EXPECTED_DATASETS constant\n    for datasetName in EXPECTED_DATASETS:\n        try:\n            # Open the given archive file\n            with ZipFile(DATA_PATH + datasetName, 'r') as currentZip:\n                # Build the target directory path for extracted data\n                datasetNameTokens = datasetName.split('.')\n                datasetNameTokens.remove('zip')\n                targetDirectory = DATA_PATH + '.'.join(datasetNameTokens)\n\n                # If the target directory doesn't exist, create it\n                if not os.path.exists(targetDirectory):\n                    Path(targetDirectory).mkdir()\n\n                # Extract all data from the archive file to the target directory\n                currentZip.extractall(targetDirectory)\n        except FileNotFoundError:\n            print(\"Unable to open \" + datasetName + \" at path \" + DATA_PATH + datasetName)\n```\n\n\n```python\nextract(state)\n```\n\n## File Information\n\nFour of the five datasets referenced in this work relate geospatial information to particular events as they occur over time Each is provided by the Urban Sustain project and employs a similar file structure.\n\n- data.json\n- fieldLabels.json\n- linkedGeometry.json\n- README.txt\n\n\n#### New York Times COVID-19 County Dataset\n\nThe included COVID-19 County dataset \n\n\n```python\nstate = 'Colorado'\ncounties = {'Colorado': ['Boulder', 'Boulder County', 'Grand', 'Grand County', 'Larimer', 'Larimer County', 'Logan', 'Logan County', 'Weld', 'Weld County', 'Yuma', 'Yuma County']}\n```\n\n\n```python\nif not os.path.exists(f'../data/control.{state}.pkl'):\n    print(\"Processing COVID dataset\")\n    flattenedCovidDataFrame = pd.json_normalize(json.load(open(Path(f'../data/covid_county.{state}/data.json'))))\n    flattenedCovidGeometryFrame = pd.json_normalize(json.load(open(Path(f'../data/covid_county.{state}/linkedGeometry.json'))))\n\n    print(\"Combining data geometry for COVID dataset\")\n    combinedCovidFrame = flattenedCovidDataFrame.set_index('GISJOIN').join(\n        flattenedCovidGeometryFrame.set_index('GISJOIN'), lsuffix='_data', rsuffix='_geo')\n\n    combinedCovidFrame = combinedCovidFrame[combinedCovidFrame.county.isin(counties[state])]\n    combinedCovidFrame['date'] = pd.to_datetime(combinedCovidFrame['dateString']).dt.date\n    print(\"Finding County Geometries\")\n    county_polygons = data_processing.create_county_polygons(state, combinedCovidFrame)\n    combinedCovidFrame.to_pickle(f'../data/control.{state}.pkl')\nelse:\n    combinedCovidFrame = pd.read_pickle(f'../data/control.{state}.pkl')\n\n```\n\n#### NEON 2D Wind Dataset\n\n\n```python\nif not os.path.exists(f'../data/covidWind.{state}.pkl'):\n    print('Processing Wind data')\n    flattenedWindDataFrame = pd.json_normalize(json.load(open(Path(f'../data/neon_2d_wind.{state}/data.json'))))\n    flattenedWindGeometryFrame = pd.json_normalize(\n        json.load(open(Path(f'../data/neon_2d_wind.{state}/linkedGeometry.json'))))\n    flattenedWindGeometryFrame['county'] = flattenedWindGeometryFrame.apply(\n        lambda row: data_processing.lookup_county_from_geometry(county_polygons, row['geometry.coordinates']), axis=1)\n    combinedWindFrame = flattenedWindDataFrame.set_index('site').join(flattenedWindGeometryFrame.set_index('site'), lsuffix='_data', rsuffix='_geo')\n    combinedWindFrame['date'] = pd.to_datetime(combinedWindFrame['startDateTime']).dt.date\n    combinedWindFrame['datetime'] = pd.to_datetime(combinedWindFrame['startDateTime']).dt.round(\"H\")\n    finalCovidWindFrame = pd.merge(combinedWindFrame, combinedCovidFrame, how='left', left_on=['county', 'date'],\n                                   right_on=['county', 'date'])\n    finalCovidWindFrame = finalCovidWindFrame[finalCovidWindFrame['totalCaseCount'].notna()]\n    finalCovidWindFrame.to_pickle(f'../data/covidWind.{state}.pkl')\n    del finalCovidWindFrame\n    del combinedWindFrame\n    del flattenedWindDataFrame\n    gc.collect()\n```\n\n#### NEON Barometric Pressure Dataset\n\n\n```python\nif not os.path.exists(f'../data/covidPressure.{state}.pkl'):\n    print('Processing Pressure data')\n    flattenedPressureDataFrame = pd.json_normalize(json.load(open(Path(f'../data/neon_barometric_pressure.{state}/data.json'))))\n    flattenedPressureGeometryFrame = pd.json_normalize(json.load(open(Path(f'../data/neon_barometric_pressure.{state}/linkedGeometry.json'))))\n    flattenedPressureGeometryFrame['county'] = flattenedPressureGeometryFrame.apply(\n        lambda row: data_processing.lookup_county_from_geometry(county_polygons, row['geometry.coordinates']), axis=1)\n    combinedPressureFrame = flattenedPressureDataFrame.set_index('site').join(\n        flattenedPressureGeometryFrame.set_index('site'), lsuffix='_data', rsuffix='_geo')\n    combinedPressureFrame['date'] = pd.to_datetime(combinedPressureFrame['startDateTime']).dt.date\n    combinedPressureFrame['datetime'] = pd.to_datetime(combinedPressureFrame['startDateTime']).dt.round(\"H\")\n    finalCovidPressureFrame = pd.merge(combinedPressureFrame, combinedCovidFrame, how='left',\n                                       left_on=['county', 'date'], right_on=['county', 'date'])\n    finalCovidPressureFrame = finalCovidPressureFrame[finalCovidPressureFrame['totalCaseCount'].notna()]\n    finalCovidPressureFrame.to_pickle(f'../data/covidPressure.{state}.pkl')\n    del finalCovidPressureFrame\n    del combinedPressureFrame\n    del flattenedPressureDataFrame\n    gc.collect()\n```\n\n#### NEON Air Temperature\n\n\n```python\nif not os.path.exists(f'../data/covidTemperature.{state}.pkl'):\n    print(\"Processing Temperature data\")\n    flattenedTemperatureDataFrame = pd.json_normalize(json.load(open(Path(f'../data/neon_single_asp_air_temperature.{state}/data.json'))))\n    flattenedTemperatureGeometryFrame = pd.json_normalize(json.load(open(Path(f'../data/neon_single_asp_air_temperature.{state}/linkedGeometry.json'))))\n    flattenedTemperatureGeometryFrame['county'] = flattenedTemperatureGeometryFrame.apply(\n        lambda row: data_processing.lookup_county_from_geometry(county_polygons, row['geometry.coordinates']), axis=1)\n    combinedTemperatureFrame = flattenedTemperatureDataFrame.set_index('site').join(\n        flattenedTemperatureGeometryFrame.set_index('site'), lsuffix='_data', rsuffix='_geo')\n    combinedTemperatureFrame['date'] = pd.to_datetime(combinedTemperatureFrame['startDateTime']).dt.date\n    combinedTemperatureFrame['datetime'] = pd.to_datetime(combinedTemperatureFrame['startDateTime']).dt.round(\"H\")\n    finalCovidTemperatureFrame = pd.merge(combinedTemperatureFrame, combinedCovidFrame, how='left', left_on=['county', 'date'],\n                                          right_on=['county', 'date'])\n    finalCovidTemperatureFrame = finalCovidTemperatureFrame[finalCovidTemperatureFrame['totalCaseCount'].notna()]\n    finalCovidTemperatureFrame.to_pickle(f'../data/covidTemperature.{state}.pkl')\n    del finalCovidTemperatureFrame\n    del combinedTemperatureFrame\n    del flattenedTemperatureDataFrame\n    gc.collect()\n```\n\n#### U.S. Census Total County Population Dataset\n\n\n```python\n# print('Processing Population data')\nflattenedPopulationDataFrame = pd.json_normalize(\n    json.load(open(Path(f'../data/county_total_population.{state}/data.json'))))\nflattenedPopulationGeometryFrame = pd.json_normalize(\n    json.load(open(Path(f'../data/county_total_population.{state}/linkedGeometry.json'))))\n\ncombinedPopulationFrame = flattenedPopulationDataFrame.set_index('GISJOIN').join(flattenedPopulationGeometryFrame.set_index('GISJOIN'), lsuffix='_data', rsuffix='_geo')\ncombinedPopulationFrame = combinedPopulationFrame[combinedPopulationFrame.COUNTY.isin(counties[state])]\ncombinedPopulationFrame['county'] = combinedPopulationFrame['COUNTY'].map({'Boulder County' : 'Boulder', 'Grand County' : 'Grand', 'Larimer County' : 'Larimer', 'Logan County' : 'Logan', 'Weld County' : 'Weld', 'Yuma County' : 'Yuma'})\ncombinedPopulationFrame.to_pickle(f'../data/population.{state}.pkl')\n\ndel flattenedPopulationDataFrame\ndel combinedPopulationFrame\ngc.collect()\n```\n\n\n\n\n    0\n\n\n\n## Data Exploration\n\n\n```python\ncontrol = combinedCovidFrame\nsns.distplot(control.newCaseCount)\n```\n\n\n```python\ncovidTemperature = pd.read_pickle(f'../data/covidTemperature.{state}.pkl')\nsns.distplot(covidTemperature.tempSingleMean)\n```\n\n\n```python\ncombinedCovidFrame['newCaseCount'].mean()\n```\n\n\n\n\n    30.816773504273506\n\n\n\n## Experiments\n\n\n```python\nstate = 'Colorado'\n```\n\n\n```python\nerror_values = {}\n```\n\n\n```python\npop = pd.read_pickle(f'../data/population.{state}.pkl')\n```\n\n#### Control\n\n\n```python\ndf = pd.read_pickle(f'../data/control.{state}.pkl')\nerror_values['control'] = run.run_control(df, pop)\n\ndel df\ngc.collect()\n\nerror_values['control']\n```\n\n#### Experiment One: Wind\n\n\n```python\ndf = pd.read_pickle(f'../data/covidWind.{state}.pkl')\nerror_values['wind'] = run.run_wind(df, pop)\n\ndel df\ngc.collect()\n\nerror_values['wind']\n```\n\n#### Experiment Two: Pressure\n\n\n```python\ndf = pd.read_pickle(f'../data/covidPressure.{state}.pkl')\n\nerror_values['pressure'] =  run.run_pressure(df, pop)\n\ndel df\ngc.collect()\n\nerror_values['pressure']\n```\n\n#### Experiment Three: Temperature\n\n\n```python\ndf = pd.read_pickle(f'../data/covidTemperature.{state}.pkl')\nerror_values['temperature'] = run.run_temperature(df, pop)\n\ndel df\ngc.collect()\n\nerror_values['temperature']\n```\n\n## Results\n\n\n```python\nprint(\"mean_absolute_error\")\nfor dsname, error_value in error_values.items():\n    print(f\"{dsname}\\t\\t{error_value[0]}\")\n    \nprint()\n    \nprint(\"mean_square_error\")\nfor dsname, error_value in error_values.items():\n    print(f\"{dsname}\\t\\t{error_value[2]}\")\n```\n\n    mean_absolute_error\n    control\t\t15.843473434448242\n    wind\t\t13.688925743103027\n    pressure\t\t11.118513107299805\n    temperature\t\t12.56731128692627\n    \n    mean_square_error\n    control\t\t1369.2708740234375\n    wind\t\t630.152099609375\n    pressure\t\t488.0992736816406\n    temperature\t\t575.2431030273438\n\n\n## References\n\n## About this Notebook\n\n**Authors:** Kyle Bassignani, Jeff Borgerson, and Christian Westbrook  \n**Updated On:** 2021-11-20\n", "meta": {"hexsha": "689f7a6d460d57e5c568d8ffed07990b95a4045c", "size": 138074, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "src/COVID-19 and Weather Patterns.ipynb", "max_stars_repo_name": "christian-westbrook/covid-19-and-weather-patterns", "max_stars_repo_head_hexsha": "24d326e3beb6dd86cfc404924813319f28c6a4a6", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-11-08T18:26:46.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-08T18:26:46.000Z", "max_issues_repo_path": "src/COVID-19 and Weather Patterns.ipynb", "max_issues_repo_name": "christian-westbrook/covid-19-and-weather-patterns", "max_issues_repo_head_hexsha": 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{"text": "```python\n# Erasmus+ ICCT project (2018-1-SI01-KA203-047081)\n\n# Toggle cell visibility\n\nfrom IPython.display import HTML\ntag = HTML('''\nPromijeni vidljivost ovdje.''')\ndisplay(tag)\n\n# Hide the code completely\n\n# from IPython.display import HTML\n# tag = HTML('''''')\n# display(tag)\n```\n\n\n\nPromijeni vidljivost ovdje.\n\n\n\n```python\n%matplotlib notebook\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport sympy as sym\nimport scipy.signal as signal\nfrom ipywidgets import widgets, interact\n```\n\n## PID regulator - model zatvorene petlje\n\nProporcionalno-integracijsko-derivacijski (PID) algoritam upravljanja daleko je najpoznatiji i naj\u010de\u0161\u0107e kori\u0161teni algoritam automatskog upravljanja. Njegova prijenosna funkcija je\n\n\\begin{equation}\n    P(s)=K_p \\cdot \\left( 1 + \\frac{1}{T_i s} + T_d s \\right).\n\\end{equation}\n\nFunkcija predstavlja zbroj proporcionalnog, integracijskog i derivacijskog kanala. Ne moraju svi nu\u017eno biti prisutni, pa se koriste i algoritmi upravljanja PI ili PD. U ovom primjeru prikazuje se vremenski odziv P, PI, PD ili PID regulatora za ulazne signale iz skupa: step-funkcija, impuls, rampa i sinus. Regulator je, u ovom slu\u010daju, dio sustava za kontrolu povratne veze. Objekt mo\u017ee biti proporcija nultog, prvog ili drugog reda ili integral nultog ili prvog reda.\n\nGrafovi (dolje) prikazuju:\n1. Izlaz sustava zatvorene petlje za odabrani ulaz, vrstu objekta i odabrani regulator (slika lijevo).\n2. Polo\u017eaj nula i polova prijenosne funkcije rezultiraju\u0107eg sustava zatvorene petlje.\n\n---\n\n### Kako koristiti ovaj interaktivni primjer?\n1. Izaberite izme\u0111u *jedini\u010dna step funkcija*, *jedini\u010dna impulsna funkcija*, *rampa funkcija* i *funkcija sinus* za odabir ulaznog signala.\n2. Kliknite na gumb *P0*, *P1*, *I0* ili *I1* za odabir izme\u0111u sljede\u0107ih objekata: proporcija nultog, prvog ili drugog reda ili integral nultog ili prvog reda. Prijenosna funkcija objekta P0 je $k_p$ (u ovom primjeru $k_p=2$), objekta P1 $\\frac{k_p}{\\tau s+1}$ (u ovom primjeru $k_p=1$ and $\\tau=2$), objekta IO $\\frac{k_i}{s}$ (u ovom primjeru $k_i=\\frac{1}{10}$) i objekta I1 $\\frac{k_i}{s(\\tau s +1}$ (u ovom primjeru $k_i=1$ i $\\tau=10$).\n3. Kliknite na gumb *P*, *PI*, *PD* ili *PID* za odabir izme\u0111u proporcionalnog, proporcionalno-integracijskog, proporcionalno-derivacijskog ili proporcionalno-integracijsko-derivacijskog tipa algoritma upravljanja.\n4. Pomi\u010dite kliza\u010de da biste promijenili vrijednosti proporcionalnog ($K_p$), integracijskog ($T_i$) i derivacijskog ($T_d$) koeficijenta PID regulacije.\n5. Pomi\u010dite kliza\u010d $t_{max}$ za promjenu maksimalne vrijednosti vremena na osi x (na grafu vremenskog odziva).\n\n\n```python\nA = 10\na=0.1\ns, P, I, D = sym.symbols('s, P, I, D')\n\nobj = 1/(A*s)\nPID = P + P/(I*s) + P*D*s#/(a*D*s+1)\nsystem = obj*PID/(1+obj*PID)\nnum = [sym.fraction(system.factor())[0].expand().coeff(s, i) for i in reversed(range(1+sym.degree(sym.fraction(system.factor())[0], gen=s)))]\nden = [sym.fraction(system.factor())[1].expand().coeff(s, i) for i in reversed(range(1+sym.degree(sym.fraction(system.factor())[1], gen=s)))]\n\n# make figure\nfig = plt.figure(figsize=(9.8, 4),num='PID regulator - sustav zatvorene petlje')\nplt.subplots_adjust(wspace=0.3)\n\n# add axes\nax = fig.add_subplot(121)\nax.grid(which='both', axis='both', color='lightgray')\nax.set_title('Vremenski odziv')\nax.set_xlabel('$t$ [s]')\nax.set_ylabel('ulaz, izlaz')\nax.axhline(linewidth=.5, color='k')\nax.axvline(linewidth=.5, color='k')\n\nrlocus = fig.add_subplot(122)\n\n\ninput_type = 'jedini\u010dna impulsna funkcija'\n\n# plot step function and responses (initalisation)\ninput_plot, = ax.plot([],[],'C0', lw=1, label='ulaz')\nresponse_plot, = ax.plot([],[], 'C1', lw=2, label='izlaz')\nax.legend()\n\n\n\n\nrlocus_plot, = rlocus.plot([], [], 'r')\n\nplt.show()\n\ndef update_plot(KP, TI, TD, Time_span):\n    global num, den, input_type\n    \n    num_temp = [float(i.subs(P,KP).subs(I,TI).subs(D,TD)) for i in num]\n    den_temp = [float(i.subs(P,KP).subs(I,TI).subs(D,TD)) for i in den]\n    system = signal.TransferFunction(num_temp, den_temp)\n    zeros = np.roots(num_temp)\n    poles = np.roots(den_temp)\n    \n    rlocus.clear()\n    rlocus.scatter([np.real(i) for i in poles], [np.imag(i) for i in poles], marker='x', color='g', label='pol')\n    rlocus.scatter([np.real(i) for i in zeros], [np.imag(i) for i in zeros], marker='o', color='g', label='nula')\n    rlocus.set_title('Dijagram polova i nula')\n    rlocus.set_xlabel('Re')\n    rlocus.set_ylabel('Im')\n    rlocus.grid(which='both', axis='both', color='lightgray')\n    \n    time = np.linspace(0, Time_span, 300)\n    \n    if input_type == 'jedini\u010dna step funkcija':\n        u = np.ones_like(time)\n        u[0] = 0\n        time, response = signal.step(system, T=time)\n    elif input_type == 'jedini\u010dna impulsna funkcija':\n        u = np.zeros_like(time)\n        u[0] = 10\n        time, response = signal.impulse(system, T=time)\n    elif input_type == 'funkcija sinus':\n        u = np.sin(time*2*np.pi)\n        time, response, _ = signal.lsim(system, U=u, T=time)\n    elif input_type == 'rampa funkcija':\n        u = time\n        time, response, _ = signal.lsim(system, U=u, T=time)\n    else:\n        raise Exception(\"Pogre\u0161ka u programu. Ponovno pokrenite simulaciju.\")\n        \n    response_plot.set_data(time, response)\n    input_plot.set_data(time, u)\n    \n    rlocus.axhline(linewidth=.3, color='k')\n    rlocus.axvline(linewidth=.3, color='k')\n    rlocus.legend()\n    \n    ax.set_ylim([min([np.min(u), min(response),-.1]),min(100,max([max(response)*1.05, 1, 1.05*np.max(u)]))])\n    ax.set_xlim([-0.1,max(time)])\n\n    plt.show()\n\ncontroller_ = PID\nobject_ = obj\n\ndef calc_tf():\n    global num, den, controller_, object_\n    system_func = object_*controller_/(1+object_*controller_)\n    \n    num = [sym.fraction(system_func.factor())[0].expand().coeff(s, i) for i in reversed(range(1+sym.degree(sym.fraction(system_func.factor())[0], gen=s)))]\n    den = [sym.fraction(system_func.factor())[1].expand().coeff(s, i) for i in reversed(range(1+sym.degree(sym.fraction(system_func.factor())[1], gen=s)))]\n    update_plot(Kp_widget.value, Ti_widget.value, Td_widget.value, time_span_widget.value)\n\ndef transfer_func(controller_type):\n    global controller_\n    proportional = P\n    integral = P/(I*s)\n    differential = P*D*s/(a*D*s+1)\n    if controller_type =='P':\n        controller_func = proportional\n        Kp_widget.disabled=False\n        Ti_widget.disabled=True\n        Td_widget.disabled=True\n    elif controller_type =='PI':\n        controller_func = proportional+integral\n        Kp_widget.disabled=False\n        Ti_widget.disabled=False\n        Td_widget.disabled=True\n    elif controller_type == 'PD':\n        controller_func = proportional+differential\n        Kp_widget.disabled=False\n        Ti_widget.disabled=True\n        Td_widget.disabled=False\n    else:\n        controller_func = proportional+integral+differential\n        Kp_widget.disabled=False\n        Ti_widget.disabled=False\n        Td_widget.disabled=False\n    \n    controller_ = controller_func\n    calc_tf()\n    \ndef transfer_func_obj(object_type):\n    global object_\n    if object_type == 'P0':\n        object_ = 2\n    elif object_type == 'P1':\n        object_ = 1/(2*s+1) \n    elif object_type == 'I0':\n        object_ = 1/(10*s)\n    elif object_type == 'I1':\n        object_ = 1/(s*(10*s+1))\n    calc_tf()\n\nstyle = {'description_width': 'initial'}\n\ndef buttons_controller_clicked(event):\n    controller = buttons_controller.options[buttons_controller.index]\n    transfer_func(controller)\nbuttons_controller = widgets.ToggleButtons(\n    options=['P', 'PI', 'PD', 'PID'],\n    description='Odaberite tip algoritma upravljanja:',\n    disabled=False,\n    style=style)\nbuttons_controller.observe(buttons_controller_clicked)\n\ndef buttons_object_clicked(event):\n    object_ = buttons_object.options[buttons_object.index]\n    transfer_func_obj(object_)\nbuttons_object = widgets.ToggleButtons(\n    options=['P0', 'P1', 'I0', 'I1'],\n    description='Odaberite objekt:',\n    disabled=False,\n    style=style)\nbuttons_object.observe(buttons_object_clicked)\n\ndef buttons_input_clicked(event):\n    \n    global input_type\n    input_type = buttons_input.options[buttons_input.index]\n    update_plot(Kp_widget.value, Ti_widget.value, Td_widget.value, time_span_widget.value)\nbuttons_input = widgets.ToggleButtons(\n    options=['jedini\u010dna step funkcija','jedini\u010dna impulsna funkcija', 'rampa funkcija', 'funkcija sinus'],\n    description='Odaberite ulazni signal:',\n    disabled=False,\n    style=style)\nbuttons_input.observe(buttons_input_clicked)\n    \nKp_widget = widgets.IntSlider(value=10,min=1,max=50,step=1,description=r'\\(K_p\\)',\n    disabled=False,continuous_update=True,orientation='horizontal',readout=True,readout_format='.1d')\nTi_widget = widgets.FloatLogSlider(value=1.,min=-3,max=1.1,step=.001,description=r'\\(T_{i} \\)',\n    disabled=False,continuous_update=True,orientation='horizontal',readout=True,readout_format='.3f')\nTd_widget = widgets.FloatLogSlider(value=1.,min=-3,max=1.1,step=.001,description=r'\\(T_{d} \\)',\n    disabled=False,continuous_update=True,orientation='horizontal',readout=True,readout_format='.3f')\n\ntime_span_widget = widgets.FloatSlider(value=10.,min=.5,max=50.,step=0.1,description=r'\\(t_{max} \\)',\n    disabled=False,continuous_update=True,orientation='horizontal',readout=True,readout_format='.1f')\n\ntransfer_func(buttons_controller.options[buttons_controller.index])\ntransfer_func_obj(buttons_object.options[buttons_object.index])\n\ndisplay(buttons_input)\ndisplay(buttons_object)\ndisplay(buttons_controller)\n\ninteract(update_plot, KP=Kp_widget, TI=Ti_widget, TD=Td_widget, Time_span=time_span_widget);\n```\n\n\n    \n\n\n\n\n\n\n\n    ToggleButtons(description='Odaberite ulazni signal:', options=('jedini\u010dna step funkcija', 'jedini\u010dna impulsna \u2026\n\n\n\n    ToggleButtons(description='Odaberite objekt:', options=('P0', 'P1', 'I0', 'I1'), style=ToggleButtonsStyle(desc\u2026\n\n\n\n    ToggleButtons(description='Odaberite tip algoritma upravljanja:', options=('P', 'PI', 'PD', 'PID'), style=Togg\u2026\n\n\n\n    interactive(children=(IntSlider(value=10, description='\\\\(K_p\\\\)', max=50, min=1, readout_format='.1d'), Float\u2026\n\n", "meta": {"hexsha": "660abcf5436091a6b364f8994457e5abf21456f6", "size": 163948, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ICCT_hr/examples/02/TD-16-PID_regulator_model_zatvorene_petlje.ipynb", "max_stars_repo_name": "ICCTerasmus/ICCT", "max_stars_repo_head_hexsha": "fcd56ab6b5fddc00f72521cc87accfdbec6068f6", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2021-05-22T18:42:14.000Z", "max_stars_repo_stars_event_max_datetime": "2021-10-03T14:10:22.000Z", "max_issues_repo_path": "ICCT_hr/examples/02/.ipynb_checkpoints/TD-16-PID_regulator_model_zatvorene_petlje-checkpoint.ipynb", "max_issues_repo_name": "ICCTerasmus/ICCT", "max_issues_repo_head_hexsha": "fcd56ab6b5fddc00f72521cc87accfdbec6068f6", "max_issues_repo_licenses": 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YES\n2. YES", "lm_q1_score": 0.5964331319177487, "lm_q2_score": 0.6926419894793246, "lm_q1q2_score": 0.413114631082894}}
{"text": "# Free Induction Decay - A Real Use Case\n\nThe following will give an example of a complex pulse using many of the features discussed in the previous tutorial examles: We will use two channels, parameters and parameter constraints, parameterized measurements and atomic and non-atomic pulse templates. This is based on real experiments. To see another, a bit more artificial example for a pulse setup use case that offers more verbose explanations, see [Gate Configuration - A Full Use Case](11GateConfigurationExample.ipynb).\n\nWe start by creating some atomic pulse templates using `PointPT` which will be the building blocks for the more complex pulse structure we have in mind.\n\n\n```python\nfrom qupulse.pulses import PointPT, SequencePT, ForLoopPT, RepetitionPT, MappingPT\nimport qupulse.pulses.plotting\nimport numpy as np\nimport sympy as sp\nfrom sympy import sympify as S\n\nchannel_names = ['RFX', 'RFY']\n\nS_init = PointPT([(0,        'S_init'),\n                  ('t_init', 'S_init')],\n                 channel_names=channel_names, identifier='S_init')\n\nmeas_wait = PointPT([(0,             'meas'),\n                     ('t_meas_wait', 'meas')],\n                   channel_names=channel_names)\n\nadprep = PointPT([(0,           'meas'),\n                  ('t_ST_prep', 'ST_plus - ST_jump/2', 'linear'),\n                  ('t_ST_prep', 'ST_plus + ST_jump/2'),\n                  ('t_op',      'op', 'linear')],\n                 parameter_constraints=['Abs(ST_plus - ST_jump/2 - meas) <= Abs(ST_plus - meas)',\n                                        'Abs(ST_plus - ST_jump/2 - meas)/t_ST_prep <= max_ramp_speed',\n                                        'Abs(ST_plus + ST_jump/2 - op)/Abs(t_ST_prep-t_op) <= max_ramp_speed'],\n                 channel_names=channel_names, identifier='adprep')\n\nadread = PointPT([(0,           'op'),\n                  ('t_ST_read', 'ST_plus + ST_jump/2', 'linear'),\n                  ('t_ST_read', 'ST_plus - ST_jump/2'),\n                  ('t_meas_start',      'meas', 'linear'),\n                  ('t_meas_start + t_meas_duration', 'meas')],\n                 parameter_constraints=['Abs(ST_plus - ST_jump/2 - meas) <= Abs(ST_plus - meas)',\n                                        'Abs(ST_plus - ST_jump/2 - meas)/t_ST_read <= max_ramp_speed',\n                                        'Abs(ST_plus + ST_jump/2 - op)/Abs(t_ST_read-t_op) <= max_ramp_speed'],\n                 channel_names=channel_names, identifier='adread',\n                 measurements=[('m', 't_meas_start', 't_meas_duration')])\n\nfree_induction = PointPT([(0, 'op-eps_J'),\n                          ('t_fid', 'op-eps_J')], channel_names=channel_names)\n```\n\nIn the next step, we combine our building blocks into more complex pulses step by step.\nWe first define our core functionality pulse template `stepped_free_induction`.\nThe pulse template `pulse` surrounds our functionality with pulses to reset/initialize our qubit and allow for data acquisition.\nWe will use `pulse` in a `ForLoopPT` `looped_pulse` to perform a parameter sweep. Our final pulse template `experiment` repeats  this whole thing a number of times to allow for statistical aggregating of measurement data and represents the complete pulse template for our experiment.\n\n\n```python\n\n\nstepped_free_induction = MappingPT(free_induction, parameter_mapping={'t_fid': 't_start + i_fid*t_step'}, allow_partial_parameter_mapping=True)\n\npulse = SequencePT(S_init, meas_wait, adprep, stepped_free_induction, adread)\n\nlooped_pulse = ForLoopPT(pulse, loop_index='i_fid', loop_range='N_fid_steps')\n\nexperiment = RepetitionPT(looped_pulse, 'N_repetitions', identifier='free_induction_decay')\n```\n\n\n```python\nprint(experiment.parameter_names)\n```\n\n    {'max_ramp_speed', 't_meas_start', 'ST_jump', 't_ST_read', 'eps_J', 't_init', 'ST_plus', 'N_repetitions', 't_step', 't_start', 'op', 't_meas_duration', 'S_init', 't_meas_wait', 'N_fid_steps', 't_op', 'meas', 't_ST_prep'}\n\n\nLet's use some reasonable (but low) values for our parameters and plot our `experiment` pulse (we set the number of repeititions of `looped_pulse` only to 2 so that the plot does not get too stuffed).\n\nNote that we provide numpy arrays of length 2 for some parameters to assign different values for different channels (see also [The PointPulseTemplate](03PointPulse.ipynb)).\n\n\n```python\n%matplotlib notebook\n\nexample_values = dict(meas=[0, 0],\n                      op=[5, -5],\n                      eps_J=[1, -1],\n                      ST_plus=[2.5, -2.5],\n                      S_init=[-1, -1],\n                      ST_jump=[1, -1],\n                      max_ramp_speed=0.3,\n                      \n                      t_init=5,\n                      \n                      t_meas_wait = 1,\n                      \n                      t_ST_prep = 10,\n                      t_op = 20,\n                      \n                      t_ST_read = 10,\n                      t_meas_start = 20,\n                      t_meas_duration=5,\n                     \n                      t_start=0,\n                      t_step=5,\n                      N_fid_steps=5, N_repetitions=2)\n\n# convert lists to numpy arrays\nexample_values = {k: np.array(v) if isinstance(v, list) else v\n                  for k, v in example_values.items()}\nfrom qupulse.pulses.plotting import plot\n\n_ = plot(experiment, example_values)\n```\n\n\n    \n\n\n\n\n\n\nWe can clearly make out the many repetitions of our basic functionality pulse and also the varying duration between the voltage peaks due to our parameter sweep (as well as the two-fold repetition of the sweep itself).\n\nLet's also quickly plot only a single repetition by setting according parameters for our `experiment` pulse template.\n\n\n```python\nexample_values['N_fid_steps'] = 1\nexample_values['N_repetitions'] = 1\nexample_values['t_start'] = 5\n\n_ = plot(experiment, example_values)\n```\n\n\n    \n\n\n\n\n\n", "meta": {"hexsha": "520fa0e6e9f8a46fffd2b11d08a92c3304624844", "size": 235447, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "doc/source/examples/10FreeInductionDecayExample.ipynb", "max_stars_repo_name": "lankes-fzj/qupulse", "max_stars_repo_head_hexsha": "46f00f70bc998b98ac1ae4721d1a9a1c10b675aa", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "doc/source/examples/10FreeInductionDecayExample.ipynb", "max_issues_repo_name": "lankes-fzj/qupulse", "max_issues_repo_head_hexsha": "46f00f70bc998b98ac1ae4721d1a9a1c10b675aa", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/source/examples/10FreeInductionDecayExample.ipynb", "max_forks_repo_name": "lankes-fzj/qupulse", "max_forks_repo_head_hexsha": "46f00f70bc998b98ac1ae4721d1a9a1c10b675aa", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 132.0510375771, "max_line_length": 113323, "alphanum_fraction": 0.8057949347, "converted": true, "num_tokens": 1363, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702761768249, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.41306924881067597}}
{"text": "```python\n# !/usr/bin/env python3\n# -*- coding: utf-8 -*-\n\"\"\"\nCreated on Tue Sep 19 11:05:23 2017\n\n@author: zhangji\n\"\"\"\n\n%pylab inline\npylab.rcParams['figure.figsize'] = (18.5, 10.5)\nfontsize = 40\n\n%load_ext autoreload\n%autoreload 2\n\nimport os\nfrom datetime import datetime\nfrom time import time\nimport dill\nimport pickle\nimport glob\nimport importlib\nimport numpy as np\nimport scipy as sp\nimport scipy.misc\nimport pandas as pd\nimport re\nimport itertools\nfrom scanf import scanf\nfrom matplotlib import pyplot as plt\nimport matplotlib.ticker as mtick\nfrom matplotlib import colors as mcolors\nfrom matplotlib.colors import ListedColormap, BoundaryNorm, PowerNorm, Normalize\nfrom mpl_toolkits.mplot3d import axes3d, Axes3D\nimport matplotlib\nfrom sympy import symbols, simplify, series, exp\nfrom sympy.matrices import Matrix\nfrom sympy.solvers import solve\nfrom scipy.optimize import leastsq, curve_fit\nfrom scipy import interpolate, integrate, optimize, sparse\nfrom scipy.interpolate import interp1d, interp2d\nfrom IPython.display import display, HTML, Math\nfrom scipy import interpolate\nfrom codeStore import support_fun as spf\nfrom src import slenderBodyTheory as slb\nfrom ecoli_in_pipe import do_slenderbodytheory as do_SLB\nfrom tqdm.notebook import tqdm as tqdm_notebook\nfrom codeStore.support_fun_head_tail import *\n\nPWD = os.getcwd()\nfont = {'size': 20}\nmatplotlib.rc('font', **font)\nnp.set_printoptions(linewidth=110, precision=5)\n\n```\n\n    Populating the interactive namespace from numpy and matplotlib\n\n\n\n```python\ndef load_data(t_dir, t_headle='(.*?).pickle', n_load=None, rand_mode=False):\n    t_path = os.listdir(t_dir)\n    filename_list = [filename for filename in t_path if re.match(t_headle, filename) is not None]\n    n_load = len(filename_list) if n_load is None else n_load\n    assert n_load <= len(filename_list)\n    if rand_mode:\n        tidx = np.random.choice(len(filename_list), n_load, replace=False)\n    else:\n        tidx = np.arange(n_load)\n    use_filename_list = np.array(filename_list)[tidx]\n\n    pickle_path_list = []\n    pickle_path_list = []\n    pickle_path_list = []\n    pickle_path_list = []\n#     intp_X_list = []\n#     intp_t = np.arange(t_start, t_stop, t_step)\n#     for i0, tname in enumerate(tqdm_notebook(use_filename_list)):\n#         tpath = os.path.join(t_dir, tname)\n#         with open(tpath, 'rb') as handle:\n#             tpick = pickle.load(handle)\n#         pickle_path_list.append(tpath)\n#         idx_list.append(i0)\n\n#         Table_t = tpick['Table_t'][1:]\n#         Table_X = tpick['Table_X'][1:]\n#         int_fun_X = interpolate.interp1d(Table_t, Table_X, kind='quadratic', axis=0)\n#         intp_X = int_fun_X(intp_t)\n#         intp_X_list.append(intp_X)\n#     pickle_path_list = np.array(pickle_path_list)\n#     idx_list = np.hstack(idx_list)\n#     intp_X_list = np.dstack(intp_X_list)  # (time, coord, caseid)\n#     return pickle_path_list, idx_list, intp_t, intp_X_list\n```\n\n\n```python\njob_dir = 'fix_case_b'\n\nload_data(job_dir)\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "4b4d18bdb066124894cab1fdb717c6e0f0ce0e49", "size": 4739, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ecoliSphere/database/Untitled.ipynb", "max_stars_repo_name": "pcmagic/stokes_flow", "max_stars_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-11-11T05:00:53.000Z", "max_stars_repo_stars_event_max_datetime": "2018-11-11T05:00:53.000Z", "max_issues_repo_path": "ecoliSphere/database/Untitled.ipynb", "max_issues_repo_name": "pcmagic/stokes_flow", "max_issues_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ecoliSphere/database/Untitled.ipynb", "max_forks_repo_name": "pcmagic/stokes_flow", "max_forks_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.7727272727, "max_line_length": 106, "alphanum_fraction": 0.5726946613, "converted": true, "num_tokens": 805, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.7057850216484837, "lm_q1q2_score": 0.41295563078588027}}
{"text": "# 2 Visualising your data\n\nScientific undertakings are rather pointless if your findings are not communicated to someone else (a colleague, the scientific community, the public, etc.)\n\nThe gold standard of scientific communication is the ***figure*** - a presentation of your findings in pictorial form that maximizes reader comprehension of your ideas. \n\nIn this notebook, we will consider a range of Python visualisations, from simple line plots, to complex dynamic figures. I will attempt (and fail) to be exhaustive, as I hope this notebook can serve as a sort of reference when you come to do your own figures (*\"what was the command for a scatter plot again?\"*)\n\n\"***Why have you tried to cram so much into this notebook?***\"\n\n*Think about learning Python as like learning a new language. Once you have complete mastery of the vocabulary (and the rules for using it) you have total flexibility in how you express yourself... BUT you have to **know** a word exists before you can **use** it. *\n\n*Python is similar - you have total control over all aspects of plotting, but first you need to know the commands that do the different things. Hence, my attempt to be exhaustive here.*\n\n## 2.1D - line plots, histograms, subplots\n\nThe workhorse of Python plotting is the [matplotlib](https://matplotlib.org/index.html) module (a particularly good reference is the [gallery](https://matplotlib.org/gallery.html) - choose your plot type by **thumbnail** sample and click to get the **source code**).\n\nWe can make Python plots **display** in the notebook by calling a special IPython [magic](http://ipython.readthedocs.io/en/stable/interactive/python-ipython-diff.html?highlight=magics#magics) function.\n\n\n```python\n%matplotlib inline\n```\n\n### 2.1.1 The basics\n\nNow, let's **import** the matplotlib module and create an **empty figure and axes**.\n\n\n```python\nfrom matplotlib import pyplot as plt      \nf,ax = plt.subplots(1,1)                  # creates a 'matrix' of subplots, in this case, 1 row and 1 column\n\n# CHANGE THIS PLOT to create two subplots, one on top of the other\n```\n\nThe `subplots` function returns two outputs: a 'figure' object, which we have assigned to the variable `f` and a set of 'axes' objects (in this case, just one), which we have assigned to the variable `ax`.\n\nBy default, both axes plot from 0 to 1, although this will be automatically adjusted as we add lines, points, data, etc.\n\nWe can **resize** the figure and **save a copy** of it to the file system.\n\n\n```python\nf,ax = plt.subplots(1,1)                           # 1 row of subplots, 1 column of subplots\nf.set_size_inches(3,8)                             # 3 inches wide, 8 inches high\nplt.savefig('my_empty_figure.png', dpi = 300)      # dpi is the resolution, 300 is good for many applications\n\n# **to do**\n# CHANGE THIS PLOT so that the figure is short and wide \n```\n\n***Verify that the file `my_empty_figure.png` has been created and corresponds to the awkwardly tall axis above.***\n\n***Modify the code above to produce two SQUARE subplots, side-by-side.*** You will need to modify BOTH the lines `f,ax = plt.subplots(1,1)` and `f.set_size_inches(3,8)`.\n\n### 2.1.2 Line plots\n\nLet's make some simple line plots.\n\n\n```python\nimport numpy as np \nf,ax = plt.subplots(1,1,figsize=(8,4))            # can set figsize here directly\n\n# get some topograpy data\nx,z = np.genfromtxt('../data/topo.csv',delimiter=',',skip_header=1).T\n\nax.plot(x,z)                                      # the 'plot' command is actually a 'method' called on the 'ax' object \n\n```\n\nWant to change the ***color*** of the line?\n\n\n```python\nf,ax = plt.subplots(1,1,figsize=(8,4))\nax.plot(x,z,'r')                                 # r = red, k = black, b = blue, g = green, c = cyan, m = magenta, y = ??  \n\n# **to do**\n# CHANGE THIS PLOT to plot a green line\n```\n\nWant to change the ***style*** of the line? And add a marker?\n\n\n```python\nf,ax = plt.subplots(1,1,figsize=(8,4))\nax.plot(x,z,'r--s')                             # -- = dashed, : = dotted, -. = dash-dot, - = solid\n                                                # s = square, ^ = up triangle, o = circle, . = dot, * = star, p = pentagon\n# **to do**\n# CHANGE THIS PLOT to plot a blue dotted line with circle markers\n```\n\n*\"That plot: 3/10. It doesn't have axes labels or a title.\"*\n\n\n```python\nf,ax = plt.subplots(1,1,figsize=(8,4))\nax.plot(x,z,'r--s')  \n\nax.set_xlabel('x')                                  # add a basic label to the x-axis\nax.set_ylabel('z', size = 12)                       # add a large label to the y-axis\nax.set_title(r'topography along profile $\\alpha$', size = 14)      # add a title using LaTeX math mode\n\n# **to do**\n# CHANGE THIS PLOT so that:\n# - the xlabel reads \"along strike distance / km\"\n# - the ylabel reads \"elevation / m\" \n```\n\n*\"6/10, the plot shouldn't exceed the limits of x axis, it's unseemly.\"*\n\n\n```python\nf,ax = plt.subplots(1,1,figsize=(8,4))\nax.plot(x,z,'r--s')      \nax.set_xlabel('x')                                  \nax.set_ylabel('z', size = 12)                       \nax.set_title(r'topography along profile $\\alpha$', size = 14)       \n\n# set plot limits in x direction to '0' on the left and 'the last element of the array x' on the right\nax.set_xlim([0, x[-1]])      \n\n# **to do**\n# CHANGE THIS PLOT so that the ylimits are set to 300 to 600\n```\n\n*\"8/10 - I'd like a legend and a text annotation for the subplot.\"* \n\nOf course!\n\n\n```python\nf,ax = plt.subplots(1,1,figsize=(8,4))\nax.plot(x,z,'r--s', label = 'profile $\\\\alpha$')                # add a label when you call the plot command      \nax.set_xlabel('x')                                  \nax.set_ylabel('z', size = 12)                       \nax.set_title(r'topography along profile $\\alpha$', size = 14)    \nax.set_xlim([0, x[-1]])     \n\n# add a legend\nax.legend(loc = 1)                                     # 1 = top right, 2 = top left, 0 = default, etc ...\n# add a text annotation in the bottom left (this is great when you're doing subfigures)\nax.text(0.05, 0.05, \"A\", ha = 'left', va = 'bottom', transform=ax.transAxes, size = 15)\n\n# **to do**\n# CHANGE THIS PLOT so that:\n# - the legend label reads \"topography\"\n# - the legend appears in the bottom righthand corner\n# - the text annotation is in the top lefthand corner and reads 'I' \n```\n\n*\"9/10 - Can you put more data on there? And another subplot?\"*\n\n\n```python\n# we'll add a second subplot (1 row, 2 columns) and 'unpack' the axes as 'ax1' and 'ax2'\nf,(ax1,ax2) = plt.subplots(1,2,figsize=(10,4))\nax1.plot(x,z,'r--s', label = 'profile $\\\\alpha$')    \n\nxv = np.linspace(x[0],x[-1],101)\nax1.plot(xv, 110*np.cos(xv/x[-1]*2*np.pi)+420, 'g--', label='not a great model')       # transform and plot the data on the first axis    \n\nax2.plot(x, z/1.e3, 'k-')       # plot a different transformation on the second axis\n\n# let's set the labels for all axes simultaneously using a FOR loop\nfor ax in [ax1,ax2]:\n    ax.set_xlabel('x')                                \n    ax.set_ylabel('z', size = 12)            \n\n# add a legend for the first axis \nax1.legend(loc = 1)       \n# add a text annotation in the two plots\nax1.text(0.05, 0.05, \"A\", ha = 'left', va = 'bottom', transform=ax1.transAxes, size = 15)\nax2.text(0.05, 0.05, \"B\", ha = 'left', va = 'bottom', transform=ax2.transAxes, size = 15)\n\n# **to do**\n# CHANGE THIS PLOT so that:\n# - the 2nd axis plots the line y propto b(1-exp(-a(x-xm)**2))\n# - the 2nd axis has a legend, your choice of labels for the two lines\n```\n\n***10/10***\n\nAs you can see, plotting commands are additive and, depending on the complexity of your figure, will rapidly compound into a quite sizeable Python script.\n\n(*This code also implemented in script: [module 2 - visualisation/line_plot.py](line_plot.py) - try running it at the command line* `ipython line_plot.py`)\n\n#### Replicate the plot below (exercise)\n\n\n\n\n```python\n# **to do**\n# write python commands to produce the same figure as above\n```\n\n### 2.1.3 Histograms\n\nThese are an okay way to present the **'frequency'** of binned values within a dataset. \n\nFirst, let's generate some **random** 'data'.\n\n\n```python\nN = 100                       # number of data points to create\nobs = np.random.randn(N)      # generate normally distributed data\nmean, std = [2.5, 0.3]        # mean and standard deviation to transform the data\nobs = obs*std + mean          # transform the data\nprint(obs)\n```\n\nBefore we plot a histogram, let's look at some useful functions for ***summarising*** the data.\n\n\n```python\n# the basics\nprint('min =', np.min(obs))\nprint('max =', np.max(obs))\nprint('mean =', np.mean(obs))\nprint('std =', np.std(obs))\n\n# more fancy\nprint('LQ =', np.percentile(obs,25))\nprint('median =', np.percentile(obs,50))\nprint('UQ =', np.percentile(obs,75))\nprint('count = ', len(obs))\nprint('sum =', np.sum(obs))\n```\n\nLet's plot a **histogram**: there are two steps and some **decision-making**.\n\n1. Place the data into **bins**. This requires an **expert decision** on how many bins to use (rule-of-thumb, for $N$ data, use $\\sqrt{N}/2$ bins)\n2. Plot the histogram using the `bar` method.\n\n\n```python\n# create the \"bin\" as a vector of evenly spaced points\nNbins = 8                                                         # alternatively, Nbins = int(np.sqrt(N)/2)\nbin_edges = np.linspace(np.min(obs), np.max(obs), Nbins+1)        # create the bins, a vector of bin edges\nh,e = np.histogram(obs, bin_edges)                                # sort the data into their bins\nprint(e)                                                          # bin edges\nprint(h)                                                          # bin heights\n\n# create the histogram\nf,ax = plt.subplots(1,1)\nax.bar(left = e[:-1], height = h, width = e[1]-e[0], color = [0, 0.5, 0], edgecolor = 'k')\n# Above, I have specified the bar 'color' as an [r g b] vector - this gives more flexibility in the type of colors that can\n# be used. Experiment with changing the three values (make sure your values are between 0 and 1 though!)\n\n# **to do**\n# CHANGE THIS PLOT so that:\n# - there are 20 bins\n# - there are 1000 data points\n```\n\n***Is the shape of the plot as you expect? Why is it not perfectly normally distributed?***\n\n***What happens as you increase the number of bins up or down? Describe the trade-offs.***\n\n(*This code also implemented in script: [2_visualisation/histogram.py](histogram.py) - try running it at the command line* `ipython histogram.py`)\n\n#### <homework>\n\nThe task below requires you to selectively copy-paste from the code above and modify for your purposes.\n\n\n```python\n# let's read in the production well data from python101.ipynb\ntime, mf = np.genfromtxt('../data/PW1.dat',delimiter=',',skip_header=True).T\n\n# **to do**\n# - create a series of bin_edges using the min and max of the data 'mf'\n# - plot a histogram of 'mf'\n# - extra: 'detrend' the data first by subtracting the mean of 'mf'\n```\n\n#### </homework>\n\n## 2.2D - scatter plots, contour plots, colour maps, cartopy, KDE, images\n\nMuch of what we covered in the previous section - axis labels, legends, line styles - applies to the plots in this section. For brevity though, I won't include these commands.\n\n### 2.2.1 Scatter plots\n\nThis is a favourite of mine for plotting earthquake locations, because we can modify the **size** and **colour** of the marker to reflect earthquake magnitude and time.\n\nLet's have a look .\n\n***First, ensure you have executed the code below Cell 1.5.2 in the previous notebook - this will download some earthquake data***.\n\n\n```python\n# load the earthquake data downloaded previously\ndata = np.genfromtxt('../1_data/nd_eqs.txt', delimiter = ',', skip_header=1)\n\n# we will pick out four columns: date (col 1), lat (col 4), lon (col 5), magnitude (col 7)\ndate_all = data[:,0]                # extract all the rows, first column\nlat_all = data[:,3]/180.*np.pi      # same, fourth column (coverted from degrees to radians)\nlon_all = data[:,4]/180.*np.pi      # etc\nmag_all = data[:,6]\n\n# convert lat-lon to approximate x-y\nr = 6371                        # radius of earth (km)\nlat_avg = np.mean(lat_all)\nx_all = r*lon_all*np.cos(lat_avg)\ny_all = r*lat_all\n```\n\nLet's just do a really basic x-y scatter plot.\n\n\n```python\nf,ax = plt.subplots(1,1,figsize=(6,6))                # this command should be familiar by now...\nax.scatter(x_all,y_all)\n```\n\nLooks like most of the events are clustered in a single region $410 0.0))      \n\n\n# 'slice' the arrays to keep only the values satisfying the conditions above\ndate = date_all[inds]\nx = x_all[inds]\ny = y_all[inds]\nmag = mag_all[inds]\n\n# plot the clipped data\nf,ax = plt.subplots(1,1,figsize=(6,6))   \nax.scatter(x,y)\nax.set_aspect('equal', adjustable='box')    # this command avoids weird stretching of the axes which distorts the data\n\n# **to do**\n# CHANGE THIS PLOT so that only earthquakes larger than magnitude 1.0 are shown\n```\n\nLet's now modify the **size** of the markers so that they reflect the magnitude of the earthquake.\n\n\n```python\nf,ax = plt.subplots(1,1,figsize=(6,6))   \n\n# convert magnitude to a size between 1 and 100 (largest event is a 100 point circle)\n    # first, rescale magnitudes to range between 0 and 1\ns = (mag - np.min(mag))/(np.max(mag) - np.min(mag))\n    # second, rescale to range between 1 and 100 (this will be the size of the marker)\ns = s*(50-2)+1\nax.scatter(x,y,s)         # passing a third argument now, marker size\nax.set_aspect('equal', adjustable='box')    \n\n# **to do**\n# CHANGE THIS PLOT to implement a DIFFERENT scaling of the sizes\n```\n\nFinally, let's distinguish the events based on the time they occurred. \n\nHowever, before we can do that, we'll need to **convert** the 'datetime' - YYYYMMDD - that we read from the text file from a float to a string, and then **reinterpret** this as a 'decimal year'. For this task, we will use the Python [datetime](https://docs.python.org/3/library/datetime.html) module.\n\n\n```python\n# import the datetime module\nfrom datetime import datetime\n\n# convert the dates from float -> integer -> string\nt0 = datetime.strptime('19910101', '%Y%m%d')           # reference time 1 Jan 1991\ntimes = []                                             # an empty list for storing each date as it is calculated\nfor each_date in date:\n    # first, convert from float -> integer -> string    (converting straight to a string leaves an awkward decimal point)\n    str_date = str(int(each_date))\n    # interpret each datestring \n    t = datetime.strptime(str_date, '%Y%m%d')\n    # take the difference between the datestring and the reference time\n    dt = t - t0\n    # find the total seconds and convert the date to decimal years\n    times.append(dt.total_seconds()/(3600*24*365.25)+1991)\n```\n\nNow let's plot events with colours **corresponding** with earthquake time (and add a colour bar for good measure)\n\n\n```python\nf,ax = plt.subplots(1,1,figsize=(6,6))   \n\nCS = ax.scatter(x,y,s, c = times)            # use the new 'times' array to indicate to scale a color map\nax.set_aspect('equal', adjustable='box')    \nplt.colorbar(CS, ax = ax)                   # create a colorbar and attach it to the 'ax' axis\n```\n\nNot happy with this particular colour scheme? Let's change it (here is the [full selection](https://matplotlib.org/examples/color/colormaps_reference.html)).\n\n***For super-users, consider using a [colour-blind friendly colourmap](http://colorbrewer2.org/#type=sequential&scheme=BuGn&n=3).***\n\n\n```python\nf,ax = plt.subplots(1,1,figsize=(6,6))   \nimport matplotlib\ncoolwarm = matplotlib.cm.get_cmap('coolwarm_r')     # import a new colormap - this one is 'coolwarm', reversed it with '_r'\n\nCS = ax.scatter(x,y,s, c= times, cmap=coolwarm)     # pass the colormap in here\nax.set_aspect('equal', adjustable='box')    \nplt.colorbar(CS, ax = ax)               \n\n# **to do**\n# CHANGE THIS PLOT to use a sequential colormap (your choice)\n```\n\nPhew - you're getting pretty expert at this!\n\n**But we can do more...**\n\nLet's say we have an **image** that outlines the reservoir in which these events are occurring (we do, this image is in the same folder as this notebook, `groningen_reservoir.png`). We can load the image into Python, plot it, and then superimpose the earthquakes over the top.\n\n\n```python\nf,ax = plt.subplots(1,1,figsize=(6,6))   \n\nim = plt.imread('groningen_reservoir.png')                  # use the imread function to read an image file\n                                                            # 'im' is a MxNx3 array of image data (each pixel has three \n                                                            # components, corresponding to an RGB color) \n\nimplot = ax.imshow(im, extent=[430, 475, 5902, 5950])     # plot the image and 'stretch' it to the given x and y limits\n\n# then, let's plot the earthquakes over the top\nCS = ax.scatter(x,y,s, c= times, cmap=coolwarm)     \nax.set_aspect('equal', adjustable='box')    \nplt.colorbar(CS, ax = ax)       \n```\n\nStarting to look pretty good right? You get the idea how a complex figure can be **built up** by just starting with some simple commands and then introducing **incremental improvements**.\n\n(*This code also implemented in script: [2_visualisation/scatter_plot.py](scatter_plot.py) - try running it at the command line* `ipython scatter_plot.py`)\n\nIn the next section, we'll consider a **variation** on this figure, mainly as a vehicle to introduce a few concepts that are important for 2D plotting.\n\n#### <homework>\n\nRead in a set of porosity-permeability-rock type data and use a `scatter` plot to illustrate the poroperm relationship for each rock (color by rock type).\n\n\n```python\n# read in the poroperm data\nphi,k,rt = np.genfromtxt('../data/poroperm.csv',delimiter=',',skip_header=1).T\n\n# **to do**\n# uncomment and complete the commands below to scatter plot permeability vs. porosity\n# color the markers by rock type 'rt'\n\n#f,ax =???\n#ax.set_yscale('log')                       # use a log axes for permeability\n#ax.scatter(???\n\n```\n\n#### </homework>\n\n### 2.2.2 Contour plots\n\nLet's say that instead of plotting the **individual** earthquakes themselves, we'd rather some visualisation of earthquake **density**. \n\nOne way to go about this is the **histogram route** - divide the area (this time 2D instead of 1D) into **rectangular bins**, **count** the earthquakes in each, and then **assign a colour** corresponding to the number of earthquakes.\n\nI have done this in the cell below\n\n\n```python\nf,ax = plt.subplots(1,1)                \nf.set_size_inches(10,10)\nim = plt.imread('groningen_reservoir.png')                  \nimplot = ax.imshow(im, extent=[430, 475, 5902, 5950])     \n\n# use histogram2d to add up EQs in discrete bins\nxbin_edges = np.linspace(430,475,int((475-430)/1)+1)                    # divide x limits into 1 km bins\nybin_edges = np.linspace(5899, 5950, int((5950-5899)/1)+1)              # same for y\ndens, xe, ye = np.histogram2d(x,y, [xbin_edges,ybin_edges])        # create the counting array\ndens = np.fliplr(dens)                                             # flip the matrix columns, left to right\n\n# create an empty RGBA figure (3 color components, 1 'alpha' component = transparency)\n    # create the 'transparency layer'\nalpha_dens = 0.*dens                       # all bins initially transparent\nalpha_dens[np.where(dens>0)] = 1     # any bin with data is not transparent\ndens_scale = dens/np.max(dens)*255         # scale to range [0,255] required for images\nim_dens = np.array([dens_scale, dens_scale, dens_scale, alpha_dens])\nimplot = ax.imshow(im_dens.T, extent=[430, 475, 5899, 5950])    \n\nax.set_aspect('equal', adjustable='box')     \n```\n\nHowever, we can **do better** using something called a [Kernel Density Estimator](http://www.mglerner.com/blog/?p=28). This applies some smoothing to the histogram in an effort to combat arbitrary bin choice.\n\nI will use an implementation of [KDE](http://scikit-learn.org/stable/modules/density.html#kernel-density-estimation) provided in a module called [scikit-learn](http://scikit-learn.org/stable/). This module provides a bunch of machine learning tools.\n\n\n```python\nf,ax = plt.subplots(1,1)                \nf.set_size_inches(10,10)            \nYlOrBr = matplotlib.cm.get_cmap('CMRmap_r')\n\n# import the KDE\nfrom sklearn.neighbors import KernelDensity\ndata = np.vstack([x,y]).T                    # format the data that we will use to train the KDE (the EQ locations)\n\n# Set up a grid for eventual CONTOUR PLOTTING of the KDE\nx0,x1,y0,y1 = [430, 475, 5899, 5950]                   # grid limits\nxg = np.linspace(x0,x1,101)                            # 101 points in the x dir\nyg = np.linspace(y0,y1,101)                            # same in the y\nX, Y = np.meshgrid(xg,yg)                              # create a meshgrid, 2D arrays of x and y coords\nxy = np.vstack([X.ravel(), Y.ravel()]).T               # combine the grid data together\n\n# create the KDE - I will use a Gaussian kernal (that is, normally distributed location errors) with a bandwidth\n# of 1 km (roughly the event location error)\nkde = KernelDensity(bandwidth=1, kernel='gaussian', algorithm='ball_tree')   # set KDE properties\nkde.fit(data)                                                                # train it with the EQ data\n# now we need to evaluate the KDE at our grid locations\nZ = np.exp(kde.score_samples(xy))\nZ = Z.reshape(X.shape)\n# plot contours of the density\nlevels = np.linspace(0, Z.max(), 11)                # the levels at which contours will be drawn\nCS = ax.contourf(X, Y, Z, levels=levels, cmap=YlOrBr, zorder = 1)     # the (filled) contour method\nplt.colorbar(CS,ax=ax)\n\n# plot the reservoir outline over the top\nim = plt.imread('groningen_reservoir.png')  \nim[:,:,-1] = 1.*(im[:,:,0]<1.e-6)\nimplot = ax.imshow(im, extent=[430, 475, 5899, 5950], zorder=2) \n```\n\n#### <neat> Cartopy\n\n\nThe final aspect of 2D plotting to show is a handy little package called [cartopy](http://scitools.org.uk/cartopy/docs/latest/index.html#). You'll have to install this package yourself (a good skill to have).\n\nAssuming you have installed Anaconda, go to the command line and type `conda install -c conda-forge cartopy`. Anaconda will automatically download and install cartopy and any packages it relies on. \n\n**Once you have done this, execute the cell below to plot a map of the world.**\n\n\n```python\n# make cartopy available\nimport cartopy.crs as ccrs\n\n# here, I am positioning the axes myself - this gives you much more flexibility in constructing figures with multiple subplots\nax1 = plt.axes([0.10, 0.15, 0.35, 0.7], projection = ccrs.PlateCarree())    # note the different projections      \nax2 = plt.axes([0.50, 0.15, 0.35, 0.7], projection = ccrs.Mollweide())          \n    \n# adding simple coastlines to each plot\nfor ax in [ax1,ax2]:\n    ax.coastlines()\n\nf = plt.gcf()\nf.set_size_inches(20,20)\n```\n\n\n\nI can zoom the plot in to show everyone's **second favourite** continent after [Zealandia](https://en.wikipedia.org/wiki/Zealandia), adding a little colour for splash.\n\n\n```python\n# code credit: Jeremy Riffault\nimport cartopy.feature as cfeature\nfrom cartopy.mpl.gridliner import LONGITUDE_FORMATTER, LATITUDE_FORMATTER\n\nfigsize = [7.48, 9.06]             # sets the dimensions of the figure to a full page\ntext_size = 10.                    # text size for axis ticks and annotations\n\n# here we create the figure and set some its dimensions\n    # fiddling with margins, extent is not strictly necessary, but does demonstrate the control you have\nfig = plt.figure(figsize=figsize)\nmargin_x = [0.1, 0.02]\nmargin_y = [0.05, 0.02]\nax = plt.axes([margin_x[0], margin_y[0], 1-sum(margin_x), 1-sum(margin_y)], projection=ccrs.PlateCarree())\nx_extent = [112, 154]              # longitude plot limits\ny_center = -25                     # center of lattitude axis (limits computed to be consistent with lon extent) \ny_delta = figsize[1]/figsize[0]*(x_extent[1]-x_extent[0])\ny_extent = [y_center-y_delta/2, y_center+y_delta/2]\nax.set_extent(x_extent+y_extent, ccrs.PlateCarree())\n\n# add lines\nax.coastlines(resolution='50m')\n\n# add coloured land and ocean\nland_50m = cfeature.NaturalEarthFeature('physical', 'land', '50m', edgecolor='k',facecolor=cfeature.COLORS['land'])\nax.add_feature(land_50m, linewidth=.01)\nocean_50m = cfeature.NaturalEarthFeature('physical', 'ocean', '50m', edgecolor='face',facecolor=cfeature.COLORS['water'])\nax.add_feature(ocean_50m);\n```\n\n\n\nWe can do more!\n\n\n```python\n# code credit: Jeremy Riffault\nfig = plt.figure(figsize=figsize)\nax = plt.axes([margin_x[0], margin_y[0], 1-sum(margin_x), 1-sum(margin_y)], projection=ccrs.PlateCarree())\nax.set_extent(x_extent+y_extent, ccrs.PlateCarree())\nax.coastlines(resolution='50m')\nax.add_feature(ocean_50m)\nax.add_feature(land_50m, linewidth=.01)\n\n# add state provinces \nstates_provinces = cfeature.NaturalEarthFeature(category='cultural', name='admin_1_states_provinces_lines', scale='50m', facecolor='none', edgecolor='k')\nax.add_feature(states_provinces, linewidth=.5)\n\n# add locations of two Enhanced Geothermal System projects\nhb = {'lon': 140.45, 'lat':-27.495, 'txt':'Habanero'}\npl = {'lon': 139.7, 'lat':-30.2, 'txt':'Paralana'}\nfor EGS in [hb, pl]:\n    # we'll use 'scatter' to add the points\n    ax.scatter(EGS['lon'], EGS['lat'], marker = 's', c='r', zorder = 5, s = 3, lw=.1, edgecolors='k')\n    # and 'text' to annotate them\n    ax.text(EGS['lon']-0.5, EGS['lat'], EGS['txt'], fontsize = text_size, va='center', ha='right')\n\n# here are some things we can do to pretty up the axes (again, nice to have control of, but not strictly necessary)\nfor axis in ['top','bottom','left','right']:\n    ax.spines[axis].set_linewidth(0.5)\ngl = ax.gridlines(crs=ccrs.PlateCarree(), draw_labels=True, alpha=0., linestyle = '--')\ngl.xlabels_top = False\ngl.ylabels_right = False\ngl.xlines = False\ngl.xformatter = LONGITUDE_FORMATTER\ngl.yformatter = LATITUDE_FORMATTER\ngl.xlabel_style = {'size': text_size}\ngl.ylabel_style = {'size': text_size}\nax.tick_params(direction='out', length=6, width=2)\n```\n\n\n\nFor more resources on the use of Cartopy, refer to the [gallery](http://scitools.org.uk/cartopy/docs/latest/gallery.html) and the [Natural Earth](http://www.naturalearthdata.com/) database.\n\n#### </neat> \n\n## 2.3D - interactive view, meshed surfaces\n\nPlotting in 3D comes up less often than the plot types we've looked at so far. Perhaps that is due to difficulties finding a good **view angle** when looking at 3D data.\n\nHowever, there are times when 3D plots provide a really powerful visual aide. I'll just show some simple examples here. \n\n\n```python\n# import tools for 3D axes\nfrom mpl_toolkits.mplot3d import Axes3D\nfrom matplotlib import cm\nfrom matplotlib import pyplot as plt\nimport numpy as np\n\n# create a grid\nxg = np.linspace(0,1,31)              # evenly spaced grid points\nyg = np.linspace(0,1,31)\nymin,ymax = [0.15,0.85]               # create a smaller subgrid in the y-dir for coloring\ni1 = np.argmin(abs(yg-ymin))\ni2 = np.argmin(abs(yg-ymax))\nyg2 = yg[i1:i2+1]                     # subsample y coords\n[X,Y] = np.meshgrid(xg,yg)            # create the two mesh grids\n[X2,Y2] = np.meshgrid(xg,yg2)\n\n# create a custom surface\n    # parameters\nxm = np.mean(xg)*0.8\nym = np.mean(yg)*1.2\nsx = 0.02*3.\nsy = 0.04*3.\n    # function defining the surface in terms of x, y and parameters\ndef r(X,Y): \n    return (5-np.exp(-((X-xm)**2/sx+(Y-ym)**2/sy)))*(1-(X/4)**2)*(1+(Y/4)**2)\n\n# create a figure with a 3D projection\nfig = plt.figure(figsize=[15,8])\nax = fig.add_subplot(111, projection='3d')\n\n# plot the function as a wireframe over the large grid\nax.plot_wireframe(X, Y, r(X,Y), lw = 0.5, color = 'k')\n    # shade part of the wireframe according to the function value\nCS = ax.plot_surface(X2, Y2, r(X2,Y2), rstride=1, cstride=1,cmap=cm.Oranges, lw = 0.5)\nplt.colorbar(CS, ax=ax)\n```\n\nIt looks fine, but are we happy with that **view angle**? It's the **default** that Python chose - out of all possible angles, how can we expect Python to choose the **best**?\n\nIt would be nicer if we could **modify it in real-time**...\n\n\n```python\n# we'll add this IPython magic command\n%matplotlib notebook\n\nfig = plt.figure(figsize=[10,5])\nax = fig.add_subplot(111, projection='3d')\n\nax.plot_wireframe(X, Y, r(X,Y), lw = 0.5, color = 'k')\nCS = ax.plot_surface(X2, Y2, r(X2,Y2), rstride=1, cstride=1,cmap=cm.Oranges, lw = 0.5)\nplt.colorbar(CS, ax=ax)\n\n# and also a show() command\nplt.show()\n\n# execute this cell and click and drag the subplot below until you find a view you are happy with\n# HINT: depending on your Python version, you'll need to execute the cell twice OR restart the Python kernel (hotkey: 0 0) \n# and then execute the cell above and then this one (I don't know why this is the case...)\n```\n\n(*This code also implemented in script: [2_visualisation/mesh_plot.py](mesh_plot.py) - try running it at the command line* `ipython mesh_plot.py`)\n\n## 2.4D - interactive plots, movies\n\nBecause the fourth dimension is **time**, right?\n\nAnyway, this section is about **moving** plots. There are two flavours we'll cover: interactive Jupyter Notebook plots with slider bars, and movies.\n\n### 2.4.1 Interact\n\nInteractive plots are **excellent** in Jupyter notebooks. The `interact` method essentially **wraps around** any plot written as a **function**, adding slider bars, radio buttons, etc., that can be used to **update** plotting parameters in **real-time**. \n\nLet's look at the example of contamination due to a chemical spill into a groundwater aquifer with local flow velocity, $v$.\n\n\\begin{equation}\nC(x,t) = \\frac{M}{2\\sqrt{\\pi D t}}\\exp\\left(-\\frac{(x-vt)^2}{4Dt}\\right),\n\\end{equation}\n\nwhere $C$ is concentration, $M$ is the mass of spilled contaminant, and $D$ is the diffusion coefficient.\n\n\n```python\n%matplotlib inline\nfrom matplotlib import pyplot as plt\nimport numpy as np\n# plot contamination down stream after two years\nf,ax = plt.subplots(1,1)\nx = np.linspace(0,10,101)                       # a vector of distance values to plot over\nM = 1000.                                       # mass of chemical spilled\nD = 0.1                                         # aquifer diffusivity\nv = 1.                                          # groundwater velocity\nt = 2.                                          # time at which to plot solution\n\n# compute chemical concentration\nC = M/(2*np.sqrt(np.pi*D*t))*np.exp(-(x-v*t)**2/(4*D*t))\nax.plot(x,C, 'r-')\nax.set_xlim([0, 10])\nax.set_ylim([0, 1.e3])\nax.set_xlabel('distance downstream [km]')\nax.set_ylabel('concentration [kg/m^3]')\n```\n\nLet's turn it into a **function**. This is less difficult that you might think.\n\n\n```python\n# a function that plots downstream contamination after two years, with flow velocity v as an input\ndef plot_contamination(v):\n    f,ax = plt.subplots(1,1)\n    x = np.linspace(0,10,101)\n    M = 1000.\n    D = 0.1\n    t = 2.\n    C = M/(2*np.sqrt(np.pi*D*t))*np.exp(-(x-v*t)**2/(4*D*t))\n    ax.plot(x,C, 'r-')\n    ax.set_xlim([0, 10])\n    ax.set_ylim([0, 1.e3])\n    ax.set_xlabel('distance downstream [km]')\n    ax.set_ylabel('concentration [kg/m^3]')\n    \n# call the function with v = 1.5 to produce the same plot as above\nplot_contamination(v=1.5)\n\n# change the value of v and rerun the cell - does the plot update in a manner that you expect?\n```\n\nOkay, now for the **magic**. We'll use `interact` from the ipywidgets module to turn this into an **interactive graph**.\n\n\n```python\n# first, let's get the interact function\nfrom ipywidgets import interact\n\n# then its as simple as one line\ninteract(plot_contamination, v = (0.2, 4.8, 0.2))\n\n# play with the slider bar below and see how the graph is redrawn in real time, each time with an updated value v\n# the first input to 'interact' is the function to be executed\n# the second input says 'create a slider for the parameter v, let it vary between 0.2 and 4.8 in increments of 0.2)\n```\n\n#### <homework>\n\n\n```python\n# CHANGE THE INTERACTIVE PLOT ABOVE to make D the slider parameter, varying between 0.02 and 0.98 in increments of 0.06\n\n# **change the function so D is the input**\ndef plot_contamination(v):\n    f,ax = plt.subplots(1,1)\n    x = np.linspace(0,10,101)\n    M = 1000.\n    D = 0.1\n    t = 2.\n    C = M/(2*np.sqrt(np.pi*D*t))*np.exp(-(x-v*t)**2/(4*D*t))\n    ax.plot(x,C, 'r-')\n    ax.set_xlim([0, 10])\n    ax.set_ylim([0, 1.e3])\n    ax.set_xlabel('distance downstream [km]')\n    ax.set_ylabel('concentration [kg/m^3]')\n    \n# **change the interact call**\ninteract();\n```\n\n\n```python\n# **to do**\n# MAKE ANOTHER CHANGE so that BOTH v and D are slider parameters\n# **your code here**\n```\n\n#### </homework>\n\nWe'll come back to the interact function in the **next notebook** when we look at fitting and calibrating models.\n\n### 2.4.2 Movies\n\nWhen you watch a movie, what does it actually comprise? A **sequence of still images**, flashed in front of you in sequence (usually about 24 each second).\n\nWe shall take a similar approach to creating movies in Python. You can already create still images (that's what this whole notebook is about) so now we just need to extend this idea to creating a **series** of still images, each one different from the previous.\n\nFinally, we will need a way to **turn those images into a movie**.\n\nThe example I'll show is for stress changes on a fault as a rupture propagates along it (an earthquake). Let's create a 5 second movie, in which the position of the rupture front, `a`, goes from 0.25 to 10 km.\n\nFirst, create a function that draws and saves a copy of the image.\n\n\n```python\ndef plot_frame(i,a):\n    ''' Plots a single frame of the fault rupture, at position a.\n    \n        i = frame number\n        a = position of rupture front\n        \n        Stress changes are computed according to linear elastic fracture mechanics.\n        s = s0 for x>>a         (far field stress)\n        s = s0-ds for xa   (decaying stresses with singularity at crack tip)\n    '''\n    f,ax = plt.subplots(1,1)\n    x = np.linspace(0,10.,1001)\n    \n    # calculate the stress field\n    s0 = 25.                      # pre-earthquake stress\n    ds = 3.                       # stress drop\n    K = s0*np.sqrt(np.pi*a)       # stress intensity factor\n    y = s0*np.ones(len(x))        # background stress vector\n    y[np.where(x=a)] = s0+K/np.sqrt(2*np.pi*(x[np.where(x>=a)]-a)*1.e3) \n    \n    # plot the stress changes\n    ax.plot(x,y,'k-')                  # plot stress changes\n    ax.plot([a,a], [s0-ds, 30], 'k:')  # stress singularity\n    ax.set_xlim([x[0], x[-1]])\n    ax.set_ylim([21, 30])\n    ax.set_xlabel('position along fault / km')\n    ax.set_ylabel('stress / MPa')\n    \n    # save the frame in a separate folder and name it using the index i\n    plt.savefig('all_frames/frame{:04d}.png'.format(i), dpi=300)\n    plt.close(f)                   # it will be important to close all these figures when we're looping\n    \n# this part should be familiar!\nimport os\nif not os.path.isdir('all_frames'):\n    os.makedirs('all_frames')\n\nplot_frame(0,1.5)\n```\n\n**Verify that a file called `frame0000.png` has been created in a directory `all_frames`.**  \n\nNow, let's create all the frames we'll need. We'll use a for loop to call `plot_frame` a bunch of times.\n\n\n```python\nFPS = 10           # ten frames per second, let's not be greedy\nsecs = 5           # total seconds\nNframes = secs*FPS  # total frames\n\n# each frame has a different value a, lets create a vector of a values\navals = np.linspace(0.25, 10.0, Nframes)\n\n# then loop over each avalue and create the frame\nfor i,a in enumerate(avals):               # enumerate is a handy function that both loops over the values in an array\n    plot_frame(i,a)                        # AND gives you a corresponding index\n```\n\nOnce this cell has finished running, there should be a large number of figures in the directory `all_frames`. Last step is to 'glue' them all together. We'll use an external command line tool called [ffmpeg](https://www.ffmpeg.org/documentation.html). (Note, if you downloaded this course from GitHub, you'll need to download a copy of ffmpeg.exe and drop it into the `module 2 - visualisation` folder.)\n\nThis isn't a course on use of ffmpeg. I have provided the Python code to call out to ffmpeg to create a video, but if you want to get fancy you'll need to do your own Googling.\n\n**Execute the cell below to create a short movie. (This won't work if you're on Microsoft Azure)**\n\n\n```python\n# we'll use os.system - a handy command that let's us implement command line calls without returning to the command line\nos.system('ffmpeg -framerate {:d} -i all_frames/frame%04d.png earthquake_movie.mp4'.format(FPS))\n```\n\n**Verify that the movie you've created corresponds to the plotted frames.**\n\nLast job - tidy your bedroom!\n\n\n```python\nfrom glob import glob\nfls = glob('all_frames/*.png')\nfor fl in fls: os.remove(fl)\nos.rmdir('all_frames')\n```\n\n(*This code also implemented in script: [2_visualisation/eq_movie.py](eq_movie.py) - try running it at the command line* `ipython eq_movie.py`)\n\n# What now?\n\nYou now have the basic tools to create all sorts of figures. Usually if I write a paper, I'll have a separate Python script to create each figure. Rerunning simulations and getting new output? No problem, just run each figure's Python script and it is automatically updated!\n\nWant some other plotting options? Look into the [seaborn](https://seaborn.pydata.org/), [plotly](https://plot.ly/) and [bqplot](https://bqplot.readthedocs.io/en/stable/) modules.\n\nWe've seen now how Python can (1) manage your data and (2) pretty it up for you. The final task is to look at **Modelling in Python**. \n\n**Open the [Models](../3_models/Models.ipynb) notebook in the `3_models` folder.**\n", "meta": {"hexsha": "9d49bd4f0cde7565cce2ffb91a4564e6bc8150e4", "size": 56908, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "2_visualisation/Visualisation.ipynb", "max_stars_repo_name": "mtoqeerpk/python_for_geoscientists", "max_stars_repo_head_hexsha": "428e2eaeb869f8478a3517d01a5fdff6de30e7d2", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-07-18T15:56:34.000Z", "max_stars_repo_stars_event_max_datetime": "2021-07-18T15:56:34.000Z", "max_issues_repo_path": "2_visualisation/Visualisation.ipynb", "max_issues_repo_name": "mtoqeerpk/python_for_geoscientists", "max_issues_repo_head_hexsha": "428e2eaeb869f8478a3517d01a5fdff6de30e7d2", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "2_visualisation/Visualisation.ipynb", "max_forks_repo_name": "mtoqeerpk/python_for_geoscientists", "max_forks_repo_head_hexsha": "428e2eaeb869f8478a3517d01a5fdff6de30e7d2", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 33.8536585366, "max_line_length": 412, "alphanum_fraction": 0.5577423209, "converted": true, "num_tokens": 10393, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "\n# PHY321: Conservative Forces, Examples and Theory\n\n  \n**[Morten Hjorth-Jensen](http://mhjgit.github.io/info/doc/web/)**, Department of Physics and Astronomy and Facility for Rare Ion Beams (FRIB), Michigan State University, USA and Department of Physics, University of Oslo, Norway\n\nDate: **Mar 3, 2021**\n\nCopyright 1999-2021, [Morten Hjorth-Jensen](http://mhjgit.github.io/info/doc/web/). Released under CC Attribution-NonCommercial 4.0 license\n\n\n\n\n## Aims and Overarching Motivation\n\n### Monday\n\nShort repetition from last week about conservative forces. Discussion\nof conditions for conservative forces and the Earth-Sun gravitional\nforce example.  **Reading suggestion**: Taylor sections 4.3, 4.4 and 4.8.\n\n### Wednesday\n\nPotential curves and discussion of the Earth-Sun example, analytical and numerical considerations.\n**Reading suggestions**: Taylor section 4.6, 4.8 and 4.9.\n\n### Friday\n\nEarth-Sun,  conservative forces and potential energy.\n**Reading suggestion**: Taylor sections 4.8 and 4.9.\n\nIf we get time, we start with harmonic oscillations and Hooke's law. **Reading suggestion**: Taylor section 5.1.\n\n\n## One Figure to Rule All Forces (thx to Julie)\n\n\n\n
                                        Figure 1: 
                                        \n\n\n\n## Repetition from last week: Work, Energy, Momentum and Conservation laws\n\nEnergy conservation is most convenient as a strategy for addressing\nproblems where time does not appear. For example, a particle goes\nfrom position $x_0$ with speed $v_0$, to position $x_f$; what is its\nnew speed? However, it can also be applied to problems where time\ndoes appear, such as in solving for the trajectory $x(t)$, or\nequivalently $t(x)$.\n\n\n\n\n## Energy Conservation\nEnergy is conserved in the case where the potential energy, $V(\\boldsymbol{r})$, depends only on position, and not on time. The force is determined by $V$,\n\n\n
                                        \n\n$$\n\\begin{equation}\n\\boldsymbol{F}(\\boldsymbol{r})=-\\boldsymbol{\\nabla} V(\\boldsymbol{r}).\n\\label{_auto1} \\tag{1}\n\\end{equation}\n$$\n\n## Conservative forces\n\nWe say a force is conservative if it satisfies the following conditions:\n1. The force $\\boldsymbol{F}$ acting on an object only depends on the position $\\boldsymbol{r}$, that is $\\boldsymbol{F}=\\boldsymbol{F}(\\boldsymbol{r})$.\n\n2. For any two points $\\boldsymbol{r}_1$ and $\\boldsymbol{r}_2$, the work done by the force $\\boldsymbol{F}$ on the displacement between these two points is independent of the path taken.\n\n3. Finally, the **curl** of the force is zero $\\boldsymbol{\\nabla}\\times\\boldsymbol{F}=0$.\n\n## Forces and Potentials\n\nThe energy $E$ of a given system is defined as the sum of kinetic and potential energies,\n\n$$\nE=K+V(\\boldsymbol{r}).\n$$\n\nWe define the potential energy at a point $\\boldsymbol{r}$ as the negative work done from a starting point $\\boldsymbol{r}_0$ to a final point $\\boldsymbol{r}$\n\n$$\nV(\\boldsymbol{r})=-W(\\boldsymbol{r}_0\\rightarrow\\boldsymbol{r})= -\\int_{\\boldsymbol{r}_0}^{\\boldsymbol{r}}d\\boldsymbol{r}'\\boldsymbol{F}(\\boldsymbol{r}').\n$$\n\nIf the potential depends on the path taken between these two points there is no unique potential.\n\n\n## Example (relevant for homework 5)\n\nWe study a classical electron which moves in the $x$-direction along a surface. The force from the surface is\n\n$$\n\\boldsymbol{F}(x)=-F_0\\sin{(\\frac{2\\pi x}{b})}\\boldsymbol{e}_1.\n$$\n\nThe constant $b$ represents the distance between atoms at the surface of the material, $F_0$ is a constant and $x$ is the position of the electron.\n\nThis is indeed a conservative force since it depends only on position\nand its **curl** is zero, that is $-\\boldsymbol{\\nabla}\\times \\boldsymbol{F}=0$. This means that energy is conserved and the\nintegral over the work done by the force is independent of the path\ntaken. \n\n## Example Continues\n\n\nUsing the work-energy theorem we can find the work $W$ done when\nmoving an electron from a position $x_0$ to a final position $x$\nthrough the integral\n\n$$\nW=\\int_{x_0}^x \\boldsymbol{F}(x')dx' =  -\\int_{x_0}^x F_0\\sin{(\\frac{2\\pi x'}{b})} dx',\n$$\n\nwhich results in\n\n$$\nW=\\frac{F_0b}{2\\pi}\\left[\\cos{(\\frac{2\\pi x}{b})}-\\cos{(\\frac{2\\pi x_0}{b})}\\right].\n$$\n\nSince this is related to the change in kinetic energy we have, with $v_0$ being the initial velocity at a  time $t_0$,\n\n$$\nv  = \\pm\\sqrt{\\frac{2}{m}\\frac{F_0b}{2\\pi}\\left[\\cos{(\\frac{2\\pi x}{b})}-\\cos{(\\frac{2\\pi x_0}{b})}\\right]+v_0^2}.\n$$\n\n## The potential energy from this example\n\nThe potential energy, due to energy conservation is\n\n$$\nV(x)=V(x_0)+\\frac{1}{2}mv_0^2-\\frac{1}{2}mv^2,\n$$\n\nwith $v$ given by the velocity from above.\n\nWe can now, in order to find a more explicit expression for the\npotential energy at a given value $x$, define a zero level value for\nthe potential. The potential is defined, using the work-energy\ntheorem, as\n\n$$\nV(x)=V(x_0)+\\int_{x_0}^x (-F(x'))dx',\n$$\n\nand if you recall the definition of the indefinite integral, we can rewrite this as\n\n$$\nV(x)=\\int (-F(x'))dx'+C,\n$$\n\nwhere $C$ is an undefined constant. The force is defined as the\ngradient of the potential, and in that case the undefined constant\nvanishes. The constant does not affect the force we derive from the\npotential.\n\nWe have then\n\n$$\nV(x)=V(x_0)-\\int_{x_0}^x \\boldsymbol{F}(x')dx',\n$$\n\nwhich results in\n\n$$\nV(x)=-\\frac{F_0b}{2\\pi}\\left[\\cos{(\\frac{2\\pi x}{b})}-\\cos{(\\frac{2\\pi x_0}{b})}\\right]+V(x_0).\n$$\n\nWe can now define\n\n$$\n-\\frac{F_0b}{2\\pi}\\cos{(\\frac{2\\pi x_0}{b})}=V(x_0),\n$$\n\nwhich gives\n\n$$\nV(x)=-\\frac{F_0b}{2\\pi}\\left[\\cos{(\\frac{2\\pi x}{b})}\\right].\n$$\n\n## Force and Potential\n\nWe have defined work as the energy resulting from a net force acting\non an object (or sseveral objects), that is\n\n$$\nW(\\boldsymbol{r}\\rightarrow \\boldsymbol{r}+d\\boldsymbol{r})= \\boldsymbol{F}(\\boldsymbol{r})d\\boldsymbol{r}.\n$$\n\nIf we write out this for each component we have\n\n$$\nW(\\boldsymbol{r}\\rightarrow \\boldsymbol{r}+d\\boldsymbol{r})=\\boldsymbol{F}(\\boldsymbol{r})d\\boldsymbol{r}=F_xdx+F_ydy+F_zdz.\n$$\n\nThe work done from an initial position to a final one defines also the difference in potential energies\n\n$$\nW(\\boldsymbol{r}\\rightarrow \\boldsymbol{r}+d\\boldsymbol{r})=-\\left[V(\\boldsymbol{r}+d\\boldsymbol{r})-V(\\boldsymbol{r})\\right].\n$$\n\n## Getting to $\\boldsymbol{F}(\\boldsymbol{r})=-\\boldsymbol{\\nabla} V(\\boldsymbol{r})$\n\nWe can write out the differences in potential energies as\n\n$$\nV(\\boldsymbol{r}+d\\boldsymbol{r})-V(\\boldsymbol{r})=V(x+dx,y+dy,z+dz)-V(x,y,z)=dV,\n$$\n\nand using the expression the differential of a multi-variable function $f(x,y,z)$\n\n$$\ndf=\\frac{\\partial f}{\\partial x}dx+\\frac{\\partial f}{\\partial y}dy+\\frac{\\partial f}{\\partial z}dz,\n$$\n\nwe can write the expression for the work done as\n\n$$\nW(\\boldsymbol{r}\\rightarrow \\boldsymbol{r}+d\\boldsymbol{r})=-dV=-\\left[\\frac{\\partial V}{\\partial x}dx+\\frac{\\partial V}{\\partial y}dy+\\frac{\\partial V}{\\partial z}dz \\right].\n$$\n\n## Final expression\n\nComparing the last equation with\n\n$$\nW(\\boldsymbol{r}\\rightarrow \\boldsymbol{r}+d\\boldsymbol{r})=F_xdx+F_ydy+F_zdz,\n$$\n\nwe have\n\n$$\nF_xdx+F_ydy+F_zdz=-\\left[\\frac{\\partial V}{\\partial x}dx+\\frac{\\partial V}{\\partial y}dy+\\frac{\\partial V}{\\partial z}dz \\right],\n$$\n\nleading to\n\n$$\nF_x=-\\frac{\\partial V}{\\partial x},\n$$\n\nand\n\n$$\nF_y=-\\frac{\\partial V}{\\partial y},\n$$\n\nand\n\n$$\nF_z=-\\frac{\\partial V}{\\partial z},\n$$\n\nor just\n\n$$\n\\boldsymbol{F}=-\\frac{\\partial V}{\\partial x}\\boldsymbol{e}_1-\\frac{\\partial V}{\\partial y}\\boldsymbol{e}_2-\\frac{\\partial V}{\\partial z}\\boldsymbol{e}_3=-\\boldsymbol{\\nabla}V(\\boldsymbol{r}).\n$$\n\nAnd this connection is the one we wanted to show.\n\n\n## Net Energy\n\nThe net energy, $E=V+K$ where $K$ is the kinetic energy, is then conserved,\n\n$$\n\\begin{eqnarray}\n\\frac{d}{dt}(K+V)&=&\\frac{d}{dt}\\left(\\frac{m}{2}(v_x^2+v_y^2+v_z^2)+V(\\boldsymbol{r})\\right)\\\\\n\\nonumber\n&=&m\\left(v_x\\frac{dv_x}{dt}+v_y\\frac{dv_y}{dt}+v_z\\frac{dv_z}{dt}\\right)\n+\\partial_xV\\frac{dx}{dt}+\\partial_yV\\frac{dy}{dt}+\\partial_zV\\frac{dz}{dt}\\\\\n\\nonumber\n&=&v_xF_x+v_yF_y+v_zF_z-F_xv_x-F_yv_y-F_zv_z=0.\n\\end{eqnarray}\n$$\n\n## In Vector Notation\n\nThe same proof can be written more compactly with vector notation,\n\n$$\n\\begin{eqnarray}\n\\frac{d}{dt}\\left(\\frac{m}{2}v^2+V(\\boldsymbol{r})\\right)\n&=&m\\boldsymbol{v}\\cdot\\dot{\\boldsymbol{v}}+\\boldsymbol{\\nabla} V(\\boldsymbol{r})\\cdot\\dot{\\boldsymbol{r}}\\\\\n\\nonumber\n&=&\\boldsymbol{v}\\cdot\\boldsymbol{F}-\\boldsymbol{F}\\cdot\\boldsymbol{v}=0.\n\\end{eqnarray}\n$$\n\nInverting the expression for kinetic energy,\n\n\n
                                        \n\n$$\n\\begin{equation}\nv=\\sqrt{2K/m}=\\sqrt{2(E-V)/m},\n\\label{_auto2} \\tag{2}\n\\end{equation}\n$$\n\nallows one to solve for the one-dimensional trajectory $x(t)$, by finding $t(x)$,\n\n\n
                                        \n\n$$\n\\begin{equation}\nt=\\int_{x_0}^x \\frac{dx'}{v(x')}=\\int_{x_0}^x\\frac{dx'}{\\sqrt{2(E-V(x'))/m}}.\n\\label{_auto3} \\tag{3}\n\\end{equation}\n$$\n\nNote this would be much more difficult in higher dimensions, because\nyou would have to determine which points, $x,y,z$, the particles might\nreach in the trajectory, whereas in one dimension you can typically\ntell by simply seeing whether the kinetic energy is positive at every\npoint between the old position and the new position.\n\n\n\n## The Earth-Sun system\n\nWe will now venture into a study of a system which is energy\nconserving. The aim is to see if we (since it is not possible to solve\nthe general equations analytically) we can develop stable numerical\nalgorithms whose results we can trust!\n\nWe solve the equations of motion numerically. We will also compute\nquantities like the energy numerically.\n\nWe start with a simpler case first, the Earth-Sun system  in two dimensions only.  The gravitational force $F_G$ on the earth from the sun is\n\n$$\n\\boldsymbol{F}_G=-\\frac{GM_{\\odot}M_E}{r^3}\\boldsymbol{r},\n$$\n\nwhere $G$ is the gravitational constant,\n\n$$\nM_E=6\\times 10^{24}\\mathrm{Kg},\n$$\n\nthe mass of Earth,\n\n$$\nM_{\\odot}=2\\times 10^{30}\\mathrm{Kg},\n$$\n\nthe mass of the Sun and\n\n$$\nr=1.5\\times 10^{11}\\mathrm{m},\n$$\n\nis the distance between Earth and the Sun. The latter defines what we call an astronomical unit **AU**.\n\n\n## The Earth-Sun system, Newton's Laws\n\nFrom Newton's second law we have then for the $x$ direction\n\n$$\n\\frac{d^2x}{dt^2}=-\\frac{F_{x}}{M_E},\n$$\n\nand\n\n$$\n\\frac{d^2y}{dt^2}=-\\frac{F_{y}}{M_E},\n$$\n\nfor the $y$ direction.\n\nHere we will use  that  $x=r\\cos{(\\theta)}$, $y=r\\sin{(\\theta)}$ and\n\n$$\nr = \\sqrt{x^2+y^2}.\n$$\n\nWe can rewrite\n\n$$\nF_{x}=-\\frac{GM_{\\odot}M_E}{r^2}\\cos{(\\theta)}=-\\frac{GM_{\\odot}M_E}{r^3}x,\n$$\n\nand\n\n$$\nF_{y}=-\\frac{GM_{\\odot}M_E}{r^2}\\sin{(\\theta)}=-\\frac{GM_{\\odot}M_E}{r^3}y,\n$$\n\nfor the $y$ direction.\n\n## The Earth-Sun system, rewriting the Equations\n\nWe can rewrite these two equations\n\n$$\nF_{x}=-\\frac{GM_{\\odot}M_E}{r^2}\\cos{(\\theta)}=-\\frac{GM_{\\odot}M_E}{r^3}x,\n$$\n\nand\n\n$$\nF_{y}=-\\frac{GM_{\\odot}M_E}{r^2}\\sin{(\\theta)}=-\\frac{GM_{\\odot}M_E}{r^3}y,\n$$\n\nas four first-order coupled differential equations\n\n4\n1\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n4\n2\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n4\n3\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\frac{dy}{dt}=v_y.\n$$\n\n## Building a code for the solar system, final coupled equations\n\nThe four coupled differential equations\n\n4\n5\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n4\n6\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n4\n7\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\n\\frac{dy}{dt}=v_y,\n$$\n\ncan be turned into dimensionless equations or we can introduce astronomical units with $1$ AU = $1.5\\times 10^{11}$. \n\nUsing the equations from circular motion (with $r =1\\mathrm{AU}$)\n\n$$\n\\frac{M_E v^2}{r} = F = \\frac{GM_{\\odot}M_E}{r^2},\n$$\n\nwe have\n\n$$\nGM_{\\odot}=v^2r,\n$$\n\nand using that the velocity of Earth (assuming circular motion) is\n$v = 2\\pi r/\\mathrm{yr}=2\\pi\\mathrm{AU}/\\mathrm{yr}$, we have\n\n$$\nGM_{\\odot}= v^2r = 4\\pi^2 \\frac{(\\mathrm{AU})^3}{\\mathrm{yr}^2}.\n$$\n\n## Building a code for the solar system, discretized equations\n\nThe four coupled differential equations can then be discretized using Euler's method as (with step length $h$)\n\n5\n2\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n5\n3\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n5\n4\n \n<\n<\n<\n!\n!\nM\nA\nT\nH\n_\nB\nL\nO\nC\nK\n\n$$\ny_{i+1}=y_i+hv_{y,i},\n$$\n\n## Code Example with Euler's Method\n\nThe code here implements Euler's method for the Earth-Sun system using a more compact way of representing the vectors. Alternatively, you could have spelled out all the variables $v_x$, $v_y$, $x$ and $y$ as one-dimensional arrays.\n\n\n```\n%matplotlib inline\n\n# Common imports\nimport numpy as np\nimport pandas as pd\nfrom math import *\nimport matplotlib.pyplot as plt\nimport os\n\n# Where to save the figures and data files\nPROJECT_ROOT_DIR = \"Results\"\nFIGURE_ID = \"Results/FigureFiles\"\nDATA_ID = \"DataFiles/\"\n\nif not os.path.exists(PROJECT_ROOT_DIR):\n    os.mkdir(PROJECT_ROOT_DIR)\n\nif not os.path.exists(FIGURE_ID):\n    os.makedirs(FIGURE_ID)\n\nif not os.path.exists(DATA_ID):\n    os.makedirs(DATA_ID)\n\ndef image_path(fig_id):\n    return os.path.join(FIGURE_ID, fig_id)\n\ndef data_path(dat_id):\n    return os.path.join(DATA_ID, dat_id)\n\ndef save_fig(fig_id):\n    plt.savefig(image_path(fig_id) + \".png\", format='png')\n\n\nDeltaT = 0.001\n#set up arrays \ntfinal = 10 # in years\nn = ceil(tfinal/DeltaT)\n# set up arrays for t, a, v, and x\nt = np.zeros(n)\nv = np.zeros((n,2))\nr = np.zeros((n,2))\n# Initial conditions as compact 2-dimensional arrays\nr0 = np.array([1.0,0.0])\nv0 = np.array([0.0,2*pi])\nr[0] = r0\nv[0] = v0\nFourpi2 = 4*pi*pi\n# Start integrating using Euler's method\nfor i in range(n-1):\n    # Set up the acceleration\n    # Here you could have defined your own function for this\n    rabs = sqrt(sum(r[i]*r[i]))\n    a =  -Fourpi2*r[i]/(rabs**3)\n    # update velocity, time and position using Euler's forward method\n    v[i+1] = v[i] + DeltaT*a\n    r[i+1] = r[i] + DeltaT*v[i]\n    t[i+1] = t[i] + DeltaT\n# Plot position as function of time    \nfig, ax = plt.subplots()\n#ax.set_xlim(0, tfinal)\nax.set_ylabel('y[AU]')\nax.set_xlabel('x[AU]')\nax.plot(r[:,0], r[:,1])\nfig.tight_layout()\nsave_fig(\"EarthSunEuler\")\nplt.show()\n```\n\n## Problems with Euler's Method\n\nWe notice here that Euler's method doesn't give a stable orbit. It\nmeans that we cannot trust Euler's method. In a deeper way, as we will\nsee in homework 5, Euler's method does not conserve energy. It is an\nexample of an integrator which is not\n[symplectic](https://en.wikipedia.org/wiki/Symplectic_integrator).\n\nHere we present thus two methods, which with simple changes allow us to avoid these pitfalls. The simplest possible extension is the so-called Euler-Cromer method.\nThe changes we need to make to our code are indeed marginal here.\nWe need simply to replace\n\n\n```\n    r[i+1] = r[i] + DeltaT*v[i]\n```\n\nin the above code with the velocity at the new time $t_{i+1}$\n\n\n```\n    r[i+1] = r[i] + DeltaT*v[i+1]\n```\n\nBy this simple caveat we get stable orbits.\nBelow we derive the Euler-Cromer method as well as one of the most utlized algorithms for sovling the above type of problems, the so-called Velocity-Verlet method. \n\n## Deriving the Euler-Cromer Method\n\nLet us repeat Euler's method.\nWe have a differential equation\n\n\n
                                        \n\n$$\n\\begin{equation}\ny'(t_i)=f(t_i,y_i)   \n\\label{_auto4} \\tag{4}\n\\end{equation}\n$$\n\nand if we truncate at the first derivative, we have from the Taylor expansion\n\n\n
                                        \n\n$$\n\\begin{equation}\ny_{i+1}=y(t_i) + (\\Delta t) f(t_i,y_i) + O(\\Delta t^2), \\label{eq:euler} \\tag{5}\n\\end{equation}\n$$\n\nwhich when complemented with $t_{i+1}=t_i+\\Delta t$ forms\nthe algorithm for the well-known Euler method. \nNote that at every step we make an approximation error\nof the order of $O(\\Delta t^2)$, however the total error is the sum over all\nsteps $N=(b-a)/(\\Delta t)$ for $t\\in [a,b]$, yielding thus a global error which goes like\n$NO(\\Delta t^2)\\approx O(\\Delta t)$. \n\nTo make Euler's method more precise we can obviously\ndecrease $\\Delta t$ (increase $N$), but this can lead to loss of numerical precision.\nEuler's method is not recommended for precision calculation,\nalthough it is handy to use in order to get a first\nview on how a solution may look like.\n\nEuler's method is asymmetric in time, since it uses information about the derivative at the beginning\nof the time interval. This means that we evaluate the position at $y_1$ using the velocity\nat $v_0$. A simple variation is to determine $x_{n+1}$ using the velocity at\n$v_{n+1}$, that is (in a slightly more generalized form)\n\n\n
                                        \n\n$$\n\\begin{equation} \ny_{n+1}=y_{n}+ v_{n+1}+O(\\Delta t^2)\n\\label{_auto5} \\tag{6}\n\\end{equation}\n$$\n\nand\n\n\n
                                        \n\n$$\n\\begin{equation}\nv_{n+1}=v_{n}+(\\Delta t) a_{n}+O(\\Delta t^2).\n\\label{_auto6} \\tag{7}\n\\end{equation}\n$$\n\nThe acceleration $a_n$ is a function of $a_n(y_n, v_n, t_n)$ and needs to be evaluated\nas well. This is the Euler-Cromer method.\n\n**Exercise**: go back to the above code with Euler's method and add the Euler-Cromer method. \n\n\n## Deriving the Velocity-Verlet Method\n\nLet us stay with $x$ (position) and $v$ (velocity) as the quantities we are interested in.\n\nWe have the Taylor expansion for the position given by\n\n$$\nx_{i+1} = x_i+(\\Delta t)v_i+\\frac{(\\Delta t)^2}{2}a_i+O((\\Delta t)^3).\n$$\n\nThe corresponding expansion for the velocity is\n\n$$\nv_{i+1} = v_i+(\\Delta t)a_i+\\frac{(\\Delta t)^2}{2}v^{(2)}_i+O((\\Delta t)^3).\n$$\n\nVia Newton's second law we have normally an analytical expression for the derivative of the velocity, namely\n\n$$\na_i= \\frac{d^2 x}{dt^2}\\vert_{i}=\\frac{d v}{dt}\\vert_{i}= \\frac{F(x_i,v_i,t_i)}{m}.\n$$\n\nIf we add to this the corresponding expansion for the derivative of the velocity\n\n$$\nv^{(1)}_{i+1} = a_{i+1}= a_i+(\\Delta t)v^{(2)}_i+O((\\Delta t)^2)=a_i+(\\Delta t)v^{(2)}_i+O((\\Delta t)^2),\n$$\n\nand retain only terms up to the second derivative of the velocity since our error goes as $O(h^3)$, we have\n\n$$\n(\\Delta t)v^{(2)}_i\\approx a_{i+1}-a_i.\n$$\n\nWe can then rewrite the Taylor expansion for the velocity as\n\n$$\nv_{i+1} = v_i+\\frac{(\\Delta t)}{2}\\left( a_{i+1}+a_{i}\\right)+O((\\Delta t)^3).\n$$\n\n## The velocity Verlet method\n\nOur final equations for the position and the velocity become then\n\n$$\nx_{i+1} = x_i+(\\Delta t)v_i+\\frac{(\\Delta t)^2}{2}a_{i}+O((\\Delta t)^3),\n$$\n\nand\n\n$$\nv_{i+1} = v_i+\\frac{(\\Delta t)}{2}\\left(a_{i+1}+a_{i}\\right)+O((\\Delta t)^3).\n$$\n\nNote well that the term $a_{i+1}$ depends on the position at $x_{i+1}$. This means that you need to calculate \nthe position at the updated time $t_{i+1}$ before the computing the next velocity.  Note also that the derivative of the velocity at the time\n$t_i$ used in the updating of the position can be reused in the calculation of the velocity update as well. \n\n\n## Adding the Velocity-Verlet Method\n\nWe can now easily add the Verlet method to our original code as\n\n\n```\nDeltaT = 0.01\n#set up arrays \ntfinal = 10 # in years\nn = ceil(tfinal/DeltaT)\n# set up arrays for t, a, v, and x\nt = np.zeros(n)\nv = np.zeros((n,2))\nr = np.zeros((n,2))\n# Initial conditions as compact 2-dimensional arrays\nr0 = np.array([1.0,0.0])\nv0 = np.array([0.0,2*pi])\nr[0] = r0\nv[0] = v0\nFourpi2 = 4*pi*pi\n# Start integrating using the Velocity-Verlet  method\nfor i in range(n-1):\n    # Set up forces, air resistance FD, note now that we need the norm of the vecto\n    # Here you could have defined your own function for this\n    rabs = sqrt(sum(r[i]*r[i]))\n    a =  -Fourpi2*r[i]/(rabs**3)\n    # update velocity, time and position using the Velocity-Verlet method\n    r[i+1] = r[i] + DeltaT*v[i]+0.5*(DeltaT**2)*a\n    rabs = sqrt(sum(r[i+1]*r[i+1]))\n    anew = -4*(pi**2)*r[i+1]/(rabs**3)\n    v[i+1] = v[i] + 0.5*DeltaT*(a+anew)\n    t[i+1] = t[i] + DeltaT\n# Plot position as function of time    \nfig, ax = plt.subplots()\nax.set_ylabel('y[AU]')\nax.set_xlabel('x[AU]')\nax.plot(r[:,0], r[:,1])\nfig.tight_layout()\nsave_fig(\"EarthSunVV\")\nplt.show()\n```\n\nYou can easily generalize the calculation of the forces by defining a function\nwhich takes in as input the various variables. We leave this as a challenge to you.\n\n\n\n\n## Additional Material: Link between Line Integrals and Conservative forces\n\n\nThe concept of line integrals plays an important role in our discussion of energy conservation,\nour definition of potentials and conservative forces.\n\nLet us remind ourselves of some the basic elements (most of you may\nhave seen this in a calculus course under the general topic of vector\nfields).\n\nWe define a path integration $C$, that is we integrate\nfrom a point $\\boldsymbol{r}_1$ to a point $\\boldsymbol{r}_2$. \nLet us assume that the path $C$ is represented by an arc length $s$. In three dimension we have the following representation of $C$\n\n$$\n\\boldsymbol{r}(s)=x(s)\\boldsymbol{e}_1+y(s)\\boldsymbol{e}_2+z(s)\\boldsymbol{e}_3,\n$$\n\nthen our integral of a function $f(x,y,z)$ along the path $C$ is defined as\n\n$$\n\\int_Cf(x,y,z)ds=\\int_a^bf\\left(x(s),y(s),z(s)\\right)ds,\n$$\n\nwhere the initial and final points are $a$ and $b$, respectively.\n## Exactness and Independence of Path\n\nWith the definition of a line integral, we can in tunrn set up the\ntheorem of independence of integration path.\n\nLet us define\n$f(x,y,z)$, $g(x,y,z)$ and $h(x,y,z)$ to be functions which are\ndefined and continuous in a domain $D$ in space. Then a line integral\nlike the above is said to be independent of path in $D$, if for every\npair of endpoints $a$ and $b$ in $D$ the value of the integral is the\nsame for all paths $C$ in $D$ starting from a point $a$ and ending in\na point $b$. The integral depends thus only on the integration limits\nand not on the path.\n\n## Differential Forms\n\nAn expression of the form\n\n$$\nfdx+gdy+hdz,\n$$\n\nwhere $f$, $g$ and $h$ are functions defined in $D$, is a called a first-order differential form\nin three variables.\nThe form is said to be exact if it is the differential\n\n$$\ndu= \\frac{\\partial u}{\\partial x}dx+\\frac{\\partial u}{\\partial y}dy+\\frac{\\partial u}{\\partial z}dz,\n$$\n\nof a differentiable function $u(x,y,z)$ everywhere in $D$, that is\n\n$$\ndu=fdx+gdy+hdz.\n$$\n\nIt is said to be exact if and only if we can then set\n\n$$\nf=\\frac{\\partial u}{\\partial x},\n$$\n\nand\n\n$$\ng=\\frac{\\partial u}{\\partial y},\n$$\n\nand\n\n$$\nh=\\frac{\\partial u}{\\partial z},\n$$\n\neverywhere in the domain $D$.\n\n## In Vector Language\n\nIn vector language the above means that the differential form\n\n$$\nfdx+gdy+hdz,\n$$\n\nis exact in $D$ if and only if the vector function (it could be a force, or velocity, acceleration or other vectors we encounter in this course)\n\n$$\n\\boldsymbol{F}=f\\boldsymbol{e}_1+g\\boldsymbol{e}_2+h\\boldsymbol{e}_3,\n$$\n\nis the gradient of a function $u(x,y,z)$\n\n$$\n\\boldsymbol{v}=\\boldsymbol{\\nabla}u=\\frac{\\partial u}{\\partial x}\\boldsymbol{e}_1+\\frac{\\partial u}{\\partial y}\\boldsymbol{e}_2+\\frac{\\partial u}{\\partial z}\\boldsymbol{e}_3.\n$$\n\n## Path Independence Theorem\n\nIf this is the case, we can state the path independence theorem which\nstates that with functions $f(x,y,z)$, $g(x,y,z)$ and $h(x,y,z)$ that fulfill the above\nexactness conditions,  the line integral\n\n$$\n\\int_C\\left(fdx+gdy+hdz\\right),\n$$\n\nis independent of path in $D$ if and only if the differential form under the integral sign is exact in $D$.\n\nThis is the path independence theorem. \n\nWe will not give a proof of the theorem. You can find this in any vector analysis chapter in a mathematics textbook.\n\nWe note however that the path integral from a point $p$ to a final point $q$ is given by\n\n$$\n\\int_p^q\\left(fdx+gdy+hdz\\right)=\\int_p^q\\left(\\frac{\\partial u}{\\partial x}dx+\\frac{\\partial u}{\\partial y}dy+\\frac{\\partial u}{\\partial z}dz\\right)=\\int_p^qdu.\n$$\n\nAssume now that we have a dependence on a variable $s$ for $x$, $y$ and $z$. We have then\n\n$$\n\\int_p^qdu=\\int_{s_1}^{s_2}\\frac{du}{ds}ds = u(x(s),y(s),z(s))\\vert_{s=s_1}^{s=s_2}=u(q)-u(p).\n$$\n\nThis last equation\n\n$$\n\\int_p^q\\left(fdx+gdy+hdz\\right)=u(q)-u(p),\n$$\n\nis the analogue of the usual formula\n\n$$\n\\int_a^bf(x)dx=F(x)\\vert_a^b=F(b)-F(a),\n$$\n\nwith $F'(x)=f(x)$.\n\n## Work-Energy Theorem again\n\nWe remember that a the work done by a force\n$\\boldsymbol{F}=f\\boldsymbol{e}_1+g\\boldsymbol{e}_2+h\\boldsymbol{e}_3$ on a displacemnt $d\\boldsymbol{r}$\nis\n\n$$\nW=\\int_C\\boldsymbol{F}d\\boldsymbol{r}=\\int_C(fdx+gdy+hdz).\n$$\n\nFrom the path independence theorem, we know that this has to result in\nthe difference between the two endpoints only. This is exact if and\nonly if the force is the force $\\boldsymbol{F}$ is the gradient of a scalar\nfunction $u$. We call this scalar function, which depends only the\npositions $x,y,z$ for the potential energy $V(x,y,z)=V(\\boldsymbol{r})$.\n\nWe have thus\n\n$$\n\\boldsymbol{F}(\\boldsymbol{r})\\propto \\boldsymbol{\\nabla}V(\\boldsymbol{r}),\n$$\n\nand we define  this as\n\n$$\n\\boldsymbol{F}(\\boldsymbol{r})= -\\boldsymbol{\\nabla}V(\\boldsymbol{r}).\n$$\n\nSuch a force is called **a conservative force**. The above expression can be used to demonstrate\nenergy conservation.\n\n## Additional Theorem\n\nFinally we can define the criterion for exactness and independence of\npath. This theorem states that if $f(x,y,z)$, $g(x,y,z)$ and\n$h(x,y,z)$ are continuous functions with continuous first partial derivatives in the domain $D$,\nthen the line integral\n\n$$\n\\int_C\\left(fdx+gdy+hdz\\right),\n$$\n\nis independent of path in $D$ when\n\n$$\n\\frac{\\partial h}{\\partial y}=\\frac{\\partial g}{\\partial z},\n$$\n\nand\n\n$$\n\\frac{\\partial f}{\\partial z}=\\frac{\\partial h}{\\partial x},\n$$\n\nand\n\n$$\n\\frac{\\partial g}{\\partial x}=\\frac{\\partial f}{\\partial y}.\n$$\n\nThis leads to the **curl** of $\\boldsymbol{F}$ being zero\n\n$$\n\\boldsymbol{\\nabla}\\times\\boldsymbol{F}=\\boldsymbol{\\nabla}\\times\\left(-\\boldsymbol{\\nabla}V(\\boldsymbol{r})\\right)=0!\n$$\n\n## Summarizing\n\nA conservative force $\\boldsymbol{F}$ is a defined as the partial derivative of a scalar potential which depends only on the position,\n\n$$\n\\boldsymbol{F}(\\boldsymbol{r})= -\\boldsymbol{\\nabla}V(\\boldsymbol{r}).\n$$\n\nThis leads to conservation of energy and a path independent line integral as long as the curl of the force is zero, that is\n\n$$\n\\boldsymbol{\\nabla}\\times\\boldsymbol{F}=\\boldsymbol{\\nabla}\\times\\left(-\\boldsymbol{\\nabla}V(\\boldsymbol{r})\\right)=0.\n$$\n", "meta": {"hexsha": "9c5fe428873bb944c726f33ad9d279fad1d640cd", "size": 49480, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "doc/pub/week7/ipynb/week7.ipynb", "max_stars_repo_name": "mhjensen/Physics321", "max_stars_repo_head_hexsha": "f858db36328c9fc127ccb44f62934d8f8749dd9f", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 20, "max_stars_repo_stars_event_min_datetime": "2020-01-09T17:41:16.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-09T00:48:58.000Z", "max_issues_repo_path": "doc/pub/week7/ipynb/week7.ipynb", "max_issues_repo_name": "mhjensen/Physics321", "max_issues_repo_head_hexsha": "f858db36328c9fc127ccb44f62934d8f8749dd9f", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2020-01-08T03:47:53.000Z", "max_issues_repo_issues_event_max_datetime": "2020-12-15T15:02:57.000Z", "max_forks_repo_path": "doc/pub/week7/ipynb/week7.ipynb", "max_forks_repo_name": "mhjensen/Physics321", "max_forks_repo_head_hexsha": "f858db36328c9fc127ccb44f62934d8f8749dd9f", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 33, "max_forks_repo_forks_event_min_datetime": "2020-01-10T20:40:55.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-11T20:28:41.000Z", "avg_line_length": 23.7656099904, "max_line_length": 253, "alphanum_fraction": 0.5100444624, "converted": true, "num_tokens": 8601, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "                                        \n\n# **5.1 Interface behavior governed by damage**\n\n[](https://moodle.rwth-aachen.de/mod/page/view.php?id=551838) part 1\n\n
                                         \n        Starting point 
                                         \n\nHaving described the irreversible behavior of the material using the assumption of frictional stress transfer between the material components which induces permanent **plastic** deformation, we move on to the second possibility, where we idealize the irreversible changes in the material structure as a spring break. This idealization is the basis of **damage** models.\n\n
                                         \n        Where are we heading 
                                         \n\n**Damage functions:** We are going to define several damage functions which are used in damage models available in finite element codes. Their definition is based on an assumed profile of breakage propagation. We can use different types of damage functions that express the state of material as an amount of broken bindings/micro-springs within a unit volume of the material structure. Even though we explain them using a one-dimensional idealization of the bond behavior, it is important to note, that these functions are directly applicable in two- and three-dimensional finite element models. To document this, we demonstrate that the derived damage functions can be used to model a two-dimensional interface with slip and shear stress defined as a vector. \n\n**Damage evolution and energy dissipation:** From a longer perspective, we are preparing the grounds for the discussion of energy dissipation associated with irreversible changes within the material structures. Some of the damage functions presented below already employ the concept of dissipated energy within a damaged unit volume of material. They are used as precursors here to motivate the consideration of energy dissipation in the following tours.\n\nSummarizing, in the present notebook, we are going to\n* describe and visualize examples of damage functions in 1D\n* show how to use an isotropic damage model to represent \n  the material behavior of a 2D interface\n\n# **Motivation and general aspects of damage**\n\n## Fiber bundle behavior described using a damage function\n\nTo provide a motivation for the damage based description of the material behavior let us consider micrographs of a multi-filament yarn cross section embedded in the concrete matrix. Depending on the type of the yarn, the cross section can consist of 800-50000 carbon/glass/basalt filaments  with a diameter ranging between 8-30 $\\mu m$. The micrograph below presents a non-penetrated 800 [tex] yarn. The unit [tex] represents the weight in grams per one km of the yarn\n\\begin{align}\n[\\mathrm{tex}] = [\\mathrm{g}/\\mathrm{km}]\n\\end{align}\nThe yarns can be penetrated by a homogenizing material, most commonly epoxy resin based material.\nThe picture below shows two cross section of a non-penetrated glass yarn. The shapes of the cross section are circular (on the left) and flat (on the right).  \n\n\n\nTo describe the load transfer and the failure process of a yarn, the tree-dimensional representation of the structure has been reproduced using slicing technique in collaboration with IMB, ibac and GIA RWTH Aachen. The pictures show significant geometric irregularity both along the length of the yarn and in the direction of the cross-section.\n\n\n\nThe primary purpose of the reinforcement is to transfer the tensile load. Let us therefore construct an idealization which can be used to describe its tensile response as a result breakage process of the individual elastic filaments. The stress-strain curve measured in a tensile test on a yarn delivers a softening type of response. We use the example of the yarn to describe the relation between tensile stress versus strain as a filament breakage process. However, the same type of idealization applies also for the bond-slip law $\\tau(s)$ that can be regarded as an irregular structure of many springs that fail in a sequence.\n\n\n\n\nNote that the fiber bundle as a system of discrete springs can be applied both \nin tension and in shear. The damage function explained below are applicable in \nboth situations. Moreover, they can be used also in two- and three dimensional\nconfigurations as well.\n\n\n## Mathematical framework for the definition of damage models\n\nThe tensile response of a yarn can be expressed as\n\\begin{align}\n\\sigma &= \\psi E_\\mathrm{b} \\varepsilon = (1 - \\omega) E_\\mathrm{b} \\varepsilon\n\\end{align}\n\nThis expression relates the level of stress directly to the fraction of the broken filaments represented by the dimensionless damage variable \n\\begin{align}\n\\omega \\in (0,1)\n\\end{align}\nThe fraction of unbroken filaments is introduced as a degree of integrity \n\\begin{align}\n\\psi = 1 - \\omega.\n\\end{align}\nThe profile of the integrity and damage functions corresponding to the above stress-strain curve has the following shape.\n\n\n\nIn the elastic regime with no damage, the value of $\\omega$ remains zero $\\omega = 0$. After the breakage of the first filament, it starts to grow up to a complete damage with $\\omega = 1$. Having used the fiber bundles as a suitable picture to  motivate the damage modeling for tensile response, we will return to the bond-slip law $\\tau(s)$ in the sequel with the goal to  mathematically describe the bond and pullout behavior within the damage framework. \n\n\nInstead of explicitly prescribing the nonlinear bond slip law as a nonlinear curve let us prescribe \na nonlinear curve governing the evolution of stiffness.\n\n\n\\begin{equation}\n\\label{EQ:bond_damage_model}\n\\tau = \\left(1 - \\omega(\\kappa)\\right)E_\\mathrm{b} \\; s\n\\end{equation}\n\nwhere $\\omega(\\kappa)$ is the nonlinear damage function and $\\kappa$ is the state variable\nthat is equivalent to maximum slip $s$ (or strain $\\varepsilon$) attained during \nthe loading history.\n\n
                                         \n        Apparent and effective stress 
                                         \n\nBy rearranging the terms in the [above damage equation](#damage_general)  we can introduce the notion of effective stress as \n\n\\begin{equation}\n\\tilde{\\tau} = \\frac{\\tau}{1-\\omega} = E_b \\; s\n\\end{equation}\n\nNote that effective stress $\\tilde{\\tau} \\geq \\tau$. While the apparent stress is related to the original material area, the effective stress is related to the undamaged material area. In other words, it can be interpreted as the stress acting on the remaining fraction of springs. The effective stress is related to the instantaneous, or still effective, cross section and not to the initial cross sectional area of the unit material zone.\n\n## How to identify the shape of the damage function?\n\nThere are several ways how to justify a particular shape of the damage function. \nLet us distinguish three ways in which the damage functions are introduced.\n\n 1. **Definition based on an experiment:** Given a measured $\\sigma(\\epsilon)$ curve, the damage level $\\omega$ can be resolved directly by rearranging the [stress-strain equation](#damage_general), i.e. \n \\begin{align}\n \\omega = \\left(1 - \\dfrac{1}{E_\\mathrm{b}} \\right) \\dfrac{\\tau}{s}\n \\end{align}\n or the integrity function\n \\begin{align}\n \\psi = \\dfrac{\\tau}{E_\\mathrm{b} s}\n \\end{align}\n 2. **Definition based on probabilistic density function of fiber strength**: Let us remark, that a the properties of a damage function as a non-decreasing function within a range (0,1) are equal to a cumulative probability density function. This fact provides the possibility to introduce the damage function as an integrated probability density function of a  filament strength within the fiber bundle model as indicated in the [stress strain response](#sig_eps_damage). \n 3. **Definition based on the amount of dissipated energy**: To narrow down the possible shapes of the damage profile, theoretical arguments based on energetic interpretation of the damage process can be used to scale the damage function to a obtain the desired amount of energy dissipation. This aspect will be addressed more in detail in Tour 6 introducing energy dissipation as an effective means of describing the localization and fracture of material exhibiting damage.\n\n\n```python\n%matplotlib widget\nimport sympy as sp\nimport numpy as np\nimport matplotlib.pylab as plt\nsp.init_printing()\n```\n\n# **Classification of damage evolution functions** \n\n[](https://moodle.rwth-aachen.de/mod/page/view.php?id=551838) part 2\n\n
                                         \n        Let us sort the damage 
                                         \n\n## Linear damage function\n\nWith the image of a material structure as a fiber-bundle with linear elastic fiber response and brittle failure we can consider a case with linear distribution of fiber breaking strain. This corresponds to the case with equidistant breaking strain or slip of all springs constituting the material structure\n\n$$\\Delta s^\\mathrm{break} = s_{i+1}^\\mathrm{break} - s_i^\\mathrm{break} = \\mathrm{constant}$$\n\nIn such a case, the damage function would be a piecewise linear function with a value zero until the first fiber break $s_0$, linear profile until the last spring break $s_\\mathrm{u}$ and plateau at the level 1 afterwards, i.e.\n\n\\begin{align}\n \\omega &= \n \\left\\{\n \\begin{array}{cl}\n 0, & \\kappa \\leq \\kappa_0 \\\\\n \\frac{\\kappa - \\kappa_0}{\\kappa_\\mathrm{u}-\\kappa_0} & \\kappa_0 < \\kappa \\leq \\kappa_\\mathrm{u} \\\\\n 1, & \\mathrm{otherwise}\n \\end{array}\n \\right.\n\\end{align}\n\nTo see this type of damage function in action, let us import the prepared damage material model component `MATS1D5BondSlipD`. This model provides several damage functions in a polymorphic trait `omega_fn`. Let us select the `linear` damage function and construct an  instance of bond-slip model `bs_linear_damage`\n\n\n```python\nfrom bmcs_cross_section.pullout import MATS1D5BondSlipD\nbs_linear_damage = MATS1D5BondSlipD(omega_fn='linear', s_max=0.05)\nbs_linear_damage.omega_fn_\n```\n\n\n\n\n$\\displaystyle \\begin{cases} 0 & \\text{for}\\: \\kappa < \\kappa_{0} \\\\\\frac{\\kappa - \\kappa_{0}}{- \\kappa_{0} + \\kappa_{u}} & \\text{for}\\: \\kappa < \\kappa_{u} \\\\1 & \\text{otherwise} \\end{cases}$\n\n\n\nThe last line in the previous cell renders the symbolic definition of the linear damage function to demonstrate that it is identical with the [above specification](#eq:linear_damage). The attribute of the bond-slip model `omega_fn_` can be instantiated as any other model component to verify the shape of the damage function.\n\n\n```python\nbs_linear_damage.omega_fn_.interact()\n```\n\n\n    VBox(children=(HBox(children=(VBox(children=(Tree(layout=Layout(align_items='stretch', border='solid 1px black\u2026\n\n\nWith the damage function at hand, let us plot the corresponding stress slip function. Let us assume that the loading is monotonically increasing so that we do not need to consider unloading \n$$ \\tau = (1 - \\omega(\\kappa)) E_\\mathrm{b}s$$\nApparently, if $\\omega(\\kappa)$ is linear, then $\\tau$ must be quadratic. Let us verify this by rendering the material model `bs_damage_linear` \n\n\n```python\nbs_linear_damage.interact()\n```\n\n\n    VBox(children=(HBox(children=(VBox(children=(Tree(layout=Layout(align_items='stretch', border='solid 1px black\u2026\n\n\n
                                         \n        Which kind of material structure would exhibit such response? 
                                         \n\nUsing the fiber-bundle analogy, we can design a test which consists of a very large number of fibers \n$i \\in 0 \\ldots N$.\n\n\n\nAll the fibers have the same breaking strain $\\bar{\\varepsilon}$.\nThe length of the fibers follows a linear function\n$$\n  \\ell_i = \\ell_0 + i \\frac{\\ell_\\mathrm{u} - \\ell_0}{N}, \\;\\;\\; i = 0 \\ldots N\n$$\n\nUpon loading, the fibers will break at the control displacement given as\n$$\n u^\\mathrm{break}_i = \\bar{\\varepsilon} \\ell_i\n$$\nwhich exhibit the property of equidistant breakage [postulated above](#equidistant_breaks)\n$$\\Delta u^\\mathrm{break} = u_{i+1}^\\mathrm{break} \n- u_i^\\mathrm{break} = \n\\bar{\\varepsilon} \\frac{\\ell_\\mathrm{u} - \\ell_0}{N} = \\mathrm{constant}$$\n\n
                                         \n        Can the damage function be applied also to a breaking strains within a local material structure? 
                                         \n\nThe fiber length distribution represent an idealization within a unit length \nof the material. Thus, given a nominal length $\\ell$ of the fiber bundle, the above reasoning applies\nalso for strain which is related to the breaking displacement as\n$$\n\\varepsilon^\\mathrm{break}_i = \\frac{u^{\\mathrm{break}}_i}{\\ell} \n$$\nThe linear damage function has been certainly used to provide a mathematical interpretation using the fiber-bundle analogy. It is certainly not applicable for real problems in the simulation of damage evolution as it occurs in a continuum. Let us therefore extend the idea to more general situations.\n\n## Damage function as a cumulative probability density of breaking strain\n\nThe previous example has demonstrated the meaning of the damage function as a distribution of the breaking strains which has been artificially assumed linear. In fact, the linear damage function\n[specified above](#eq:linear_damage) can regarded as a special type of cumulative density function. \nThis brings us to the possibility to capture the breaking strain distribution using an existing \ncumulative probability density function. In particular, we choose the [Weibull distribution](https://en.wikipedia.org/wiki/Weibull_distribution). Before, explaining the reason for this choice \nlet us construct another model component `bs_cdf_damage` with `omega_fn` set to `weibull-CDF`\n\n\n```python\nbs_linear_damage = MATS1D5BondSlipD(omega_fn='weibull-CDF', s_max=0.05)\nbs_linear_damage.omega_fn_\n```\n\n\n\n\n$\\displaystyle \\begin{cases} 0 & \\text{for}\\: \\kappa < 0 \\\\1 - e^{- \\left(\\frac{\\kappa}{\\lambda}\\right)^{m}} & \\text{otherwise} \\end{cases}$\n\n\n\nThe variable $\\kappa$ represents the state variable, i.e. slip or strain.\nThe parameters $\\lambda$ and $m$ denote the so called scale and shape parameters, respectively.\nLet us demonstrate their role by rendering the model component\n\n\n```python\nbs_linear_damage.omega_fn_.interact()\n```\n\n\n    VBox(children=(HBox(children=(VBox(children=(Tree(layout=Layout(align_items='stretch', border='solid 1px black\u2026\n\n\nBy interacting with the model component we can observe that the shape parameter $m$ controls the spread of the breaks along the control slip or strain variable and scale parameter $\\lambda$ shifts the breaking range horizontally. The corresponding bond slip law then has the form\n\n\n```python\nbs_linear_damage.interact()\n```\n\n\n    VBox(children=(HBox(children=(VBox(children=(Tree(layout=Layout(align_items='stretch', border='solid 1px black\u2026\n\n\nThe qualitative shape of the displayed stress-strain curve can be conveniently used to reproduce the experimentally obtained response of a yarn observed in the tensile test.\n\n\n\n
                                         \n        Fiber-bundle and test length - Weibull distribution can capture it 
                                         \n\nThe figure shows an example of the stress-strain curves obtained experimentally in tensile tests on carbon yarns with varied test lengths reported by [Sugimoto et al. (2021)](https://link.springer.com/content/pdf/10.1557/s43578-020-00043-y.pdf). The authors use the Weibull strength distribution to interpret the and to scale the obtained fiber strength characteristics between individual yarn lengths. Since the Weibull probability distribution is based on the concept of a weakest link in a chain, it allows for a scaling of strength given a changed length. Indeed, as the figure above shows, the strength of bundles with the lengths 50, 100 and 150 exhibits a negative trend. This corresponds well with the scaling inherently present in the Weibull probability distribution. Simply speaking, the probability of failure increases with the increasing number of links within the chain. The fact that the length of 10 mm has lower strength is related to the technical difficulties involved in the testing of short yarns discussed e.g. by [Rypl et. al (2015)](../papers/yarn_tensile_test_example.pdf)\n\n
                                         \n        Generalization - exponential damage function provides flexibility 
                                         \n\n[](https://moodle.rwth-aachen.de/mod/page/view.php?id=551838) part 3\n\nThe probabilistic interpretation of damage evolution has shown that an exponential functions provide a flexible tool to adjust the shape of the non-linear softening branch in the stress-strain response. \nThis flexibility has been exploited in many finite-element codes by providing several types of damage functions defined in terms of exponential function.\n\nThis can be documented by the fact that the function\n\\begin{align}\n\\omega(\\kappa) = 1 - \\exp(-\\kappa)\n\\end{align}\napproaches the value 1 for $\\kappa \\rightarrow \\infty$. This function is usually combined with an elastic range $\\kappa < \\kappa_0$ where the damage is implicitly defined as zero. To demonstrate it, let us define it symbolic form using `sympy` \n\n\n```python\nkappa = sp.symbols('kappa')\nomega_kappa_ = 1 - sp.exp(-kappa)\nomega_kappa_\n```\n\n
                                         \n        To see the function 'life', let us use the following code 
                                         \n\n\n```python\nget_omega_kappa = sp.lambdify(kappa, omega_kappa_) # executable expression\nfig, ax = plt.subplots(1,1,figsize=(7,4), tight_layout=True) # plottable area\nfig.canvas.header_visible=False # hide the header\nkappa_arr = np.linspace(0.0,5,100) # generate the slip array from in the range (0, 0.5)\nax.plot(kappa_arr, get_omega_kappa(kappa_arr), linestyle='solid', color='black', label=r'$\\omega$')\nax.fill_between(kappa_arr, get_omega_kappa(kappa_arr), color='gray', alpha=0.2)\nax.set_ylabel(r'$\\omega$'); ax.set_xlabel(r'$\\kappa$');\n```\n\n\n    Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous \u2026\n\n\nThus, the probability density function is only one possible form of the damage function. Let us exemplify three further types of damage function that can be encountered in the finite element codes.\n\n\n## Damage function derived from strength and shape of softening branch\n\nThe first example shows a function which can control the slope damage evolution at the onset of damage $\\kappa = \\kappa_0$ by a parameter controlling its slope of the softening branch in the stress-strain diagram. It has the form\n\\begin{align}\n\\omega(\\kappa) = 1 - \\left[\\frac{\\kappa_0}{\\kappa} \\exp \\left(- \\frac{\\kappa - \\kappa_0}{\\kappa_\\mathrm{f} - \\kappa_0}\\right)\\right]\n\\end{align}\n\n\n```python\nbs_exp_slope = MATS1D5BondSlipD(omega_fn='exp-slope', s_max=0.05)\nbs_exp_slope.omega_fn_.trait_set(kappa_0=0.01, kappa_f=0.04)\nbs_exp_slope.omega_fn_\n```\n\n\n\n\n$\\displaystyle \\begin{cases} 0 & \\text{for}\\: \\kappa \\leq \\kappa_{0} \\\\1 - \\frac{\\kappa_{0} e^{\\frac{- \\kappa + \\kappa_{0}}{- \\kappa_{0} + kappa_\\mathrm{f}}}}{\\kappa} & \\text{otherwise} \\end{cases}$\n\n\n\n\n```python\nbs_exp_slope.interact()\n```\n\n\n    VBox(children=(HBox(children=(VBox(children=(Tree(layout=Layout(align_items='stretch', border='solid 1px black\u2026\n\n\n
                                         \n        When to use this damage format? 
                                         \n\nIn contrast to the cumulative probability function, the above shown function allows for an explicit specification of strength, i.e.\n$$\nf_\\mathrm{t} = E_\\mathrm{b} \\kappa_0\n$$\nand the brittleness controlled by the $\\kappa_f$ which represents the point on the horizontal axes which is intersected by a tangent of the softening branch constructed at the onset of damage, i.e. $s = \\kappa_0$. In the present case, the material strength is equivalent to the onset of inelasticity. Such coupling, however, can be too restrictive. Therefore, an example of a more flexible representation of damage is shown in the next section.\n\n## Damage function used in Abaqus\n\nThe damage function provided in `Abaqus` uses three parameters to control the softening branch.\n\\begin{align}\n\\omega(\\tilde{s}) = \\frac{\\kappa_0}{\\kappa}\\left[ 1 - \\frac{1 - \\exp \\displaystyle\\left(- \\alpha\n\\frac{\\kappa - \\kappa_0}{\\kappa_u - \\kappa_0}\\right)}{1 - \\exp(-\\alpha)}  \\right]\n\\end{align}\n\n\nThis model is available in the catalog of damage function `omega_fn` so that we instantiate it and verify that it is equivalent to the above formula \n\n\n```python\nbs_exp_slope = MATS1D5BondSlipD(omega_fn='abaqus', s_max=0.05)\nbs_exp_slope.omega_fn_.trait_set(kappa_0=0.01, kappa_u=0.04)\nbs_exp_slope.omega_fn_\n```\n\n\n\n\n$\\displaystyle \\begin{cases} 0 & \\text{for}\\: \\kappa \\leq \\kappa_{0} \\\\1 - \\begin{cases} 1 & \\text{for}\\: \\kappa < \\kappa_{0} \\\\\\frac{\\kappa_{0} \\left(1 - \\frac{1 - e^{- \\frac{\\alpha \\left(\\kappa - \\kappa_{0}\\right)}{- \\kappa_{0} + \\kappa_{u}}}}{1 - e^{- \\alpha}}\\right)}{\\kappa} & \\text{for}\\: \\kappa < \\kappa_{u} \\\\0 & \\text{otherwise} \\end{cases} & \\text{otherwise} \\end{cases}$\n\n\n\n\n```python\nbs_exp_slope.interact()\n```\n\n\n    VBox(children=(HBox(children=(VBox(children=(Tree(layout=Layout(align_items='stretch', border='solid 1px black\u2026\n\n\nApparently, the parameter $\\kappa_\\mathrm{u}$ controls the point of ultimate damage at which the stress=strain curve hits the zero level. A small value of the parameter $\\alpha$ renders a linear softening branch. By increasing the value of $\\alpha$ a nonlinear softening with steep decay at the onset of damage is obtained.\n\n\n## Damage function accounting for fracture energy\nThe last example of the damage function controls the softening branch by prescribing the area below the curve denoted as $G_\\mathrm{f}$. This means, a smaller value of $G_\\mathrm{f}$ makes the material more brittle, while a larger value increases the ductility\n\\begin{align}\n\\omega(\\kappa) = 1 - \\displaystyle{\\frac{\\kappa_0}{\\kappa}}\n\\exp\\left(\\frac{2 E_\\mathrm{b} \\kappa_0 ( \\kappa - \\kappa_0) }\n{E_\\mathrm{b} \\kappa^2_0 - 2 G_\\mathrm{f}} \n \\right)\n\\end{align}\n\n\n```python\nbs_fracture_energy = MATS1D5BondSlipD(omega_fn='fracture-energy', s_max=0.05)\nbs_fracture_energy.omega_fn_.trait_set(kappa_0=0.01, G_f=1)\nbs_fracture_energy.omega_fn_\n```\n\n\n\n\n$\\displaystyle 1 - \\begin{cases} 1 & \\text{for}\\: \\kappa < \\kappa_{0} \\\\e^{\\frac{\\left(\\kappa - \\kappa_{0}\\right) \\left(\\sqrt{E_{b}} \\sqrt{- E_{b} \\kappa_{0}^{2} + 4 G_{f}} + E_{b} \\kappa_{0}\\right)}{E_{b} \\kappa_{0}^{2} - 2 G_{f}}} & \\text{otherwise} \\end{cases}$\n\n\n\nEven though the expressions look different, they are equivalent. The implemented version has been rendered using the `sympy` algebra system and can be transformed in to a simpler format given above. Let us test the flexibility of the provided function.\n\n\n```python\nbs_fracture_energy.interact()\n```\n\n\n    VBox(children=(HBox(children=(VBox(children=(Tree(layout=Layout(align_items='stretch', border='solid 1px black\u2026\n\n\nNote that, for large values of $G_\\mathrm{f}$, maximum value of stress can become larger than the stress at the onset of damage, i.e. $\\tau(\\kappa_0)$. Thus, in contrast to the abacus damage function, material strength is not available as a model parameter. However an algebraic rearrangement identifying the maximum stress and resolving $\\kappa_0$ as a function of strength is possible.\n\n
                                         \n        Keep fracture energy in mind, we will revisit it 
                                         \n\nThis shape of the damage function introduces physical interpretation into the control of the softening branch. As we will show in Tour 6, the area below the softening stress-strain curve represents the energy dissipated during the loading process within a unit volume of material.  \n\n## Summary and classification of damage evolution functions\n\n| Name | Symbol | Material parameters |\n| :- | :-: | :-: | \n| linear damage | $\\omega_1(\\kappa)$ | $\\kappa_0, \\kappa_\\mathrm{u}$ | \n| cumulative probability density | $\\omega_2(\\kappa)$ | $\\lambda, m$| \n| softening slope | $\\omega_3(\\kappa)$ | $\\kappa_0, \\kappa_\\mathrm{f}$ | \n| softening slope and curvature | $\\omega_4(\\kappa)$ | $\\kappa_0, \\kappa_\\mathrm{u}, \\alpha$ |\n| fracture energy | $\\omega_5(\\kappa)$ | $\\kappa_0, G_\\mathrm{f}$ |\n \n\n# **Sheet interface - damage in 2D**\n\n[](https://moodle.rwth-aachen.de/mod/page/view.php?id=551838) part 4\n\n\n## Equivalent measure of strain\n\nSo far, the scalar variable $\\kappa$ could be considered equivalent to the maximum slip attained during the loading history. However, it can be defined also for two and three dimensional problems. To simulate the damage in continuum\nproblems, an equivalent strain can be defined in terms of strain tensor invariants. To provide a simple example, \nlet us consider the application of the damage framework \nfor the modeling of two-dimensional interface damage.\n\n\n\nIn this case, equivalent strain can then be defined as the distance from the undeformed configuration with zero slip $s_x, s_y = [0,0]$\n\\begin{align}\n\\kappa =  \\sqrt{ s^2_x(\\theta) + s_y^2(\\theta) }\n\\end{align}\n\n\n```python\ns_x, s_y = sp.symbols('s_x, s_y')\nkappa_ = sp.sqrt( s_x**2 + s_y**2 )\nkappa_\n```\n\nThus, the elastic domain is represented by a circle with the radius $\\kappa_0$ in $s_x$ and $s_y$ plane.\n\n\n```python\nget_kappa = sp.lambdify( (s_x, s_y), kappa_, 'numpy' )\nphi = np.linspace(0, 2*np.pi, 500)\nsx, sy = np.sin(phi), np.cos(phi)\nfig, ax = plt.subplots(1,1, figsize=(4,4), tight_layout=True)\nfig.canvas.header_visible = False\nax.plot(sx,sy,label=r'$\\kappa_0 = 1$')\nax.fill(sx,sy, color='gray', alpha=0.2)\nax.set_xlabel(r'$s_x$'); ax.set_ylabel(r'$s_y$');\nax.set_aspect('equal'); ax.legend();\n```\n\n\n    Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous \u2026\n\n\n
                                         \n        Other examples of equivalent strain definition 
                                         \n\nTo show the flexibility of the the inelastic domain specification using the \nequivalent measure let us provide two more examples which are used\nto define a tensile elastic limit for brittle materials.\nGiven a strain and stress tensors $\\boldsymbol{\\varepsilon}$ and $\\boldsymbol{\\sigma}$, \nthe elastic limit can be introduced using the maximum \nprincipal tensile stress $\\sigma_i$ and divide it by the $E$ modulus as follows.\n\\begin{equation}\\label{eq:rankine_eq}\n\\kappa = \\frac{1}{E} \\mathrm{max}(\\sigma_i), \\qquad i=1,2,3  \n\\end{equation}\nThe principal stresses $\\sigma_i$ are obtained as the eigenvalues of the stress tensor $\\boldsymbol{\\sigma}$.\n\nAnother option is to define a norm of the strain tensor expressed as follows\n\\begin{equation}\\label{eq:mazars_eq}\n\\kappa = ||\\langle\n{\\boldsymbol{\\varepsilon}}\n\\rangle || = \\sqrt{\\langle \\boldsymbol{\\varepsilon} \\rangle  : \\langle \\boldsymbol{\\varepsilon} \\rangle} = \\sqrt{ {\\langle \\varepsilon_1\\rangle}^2   + {\\langle\\varepsilon_2 \\rangle}^2 + {\\langle\\varepsilon_3 \\rangle}^2}\n\\end{equation}\nIn this expression, the McAuley brackets $\\langle . \\rangle$ means the positive part of (.), e.g. $\\langle x \\rangle = (x+|x|)/2$. And $\\varepsilon_1, \\varepsilon_2, \\varepsilon_3$ are the principal strains.\n\n## Reversibility threshold function\n\nWhile plastic models introduce the definition of elastic domain by defining a threshold criterion for the values of stress variables, in damage models, the elastic domain is delimited using an equivalent strain measure $\\kappa$.\n\\begin{align}\nf := \\kappa - \\kappa_0 \\le 0\n\\end{align}\nThis condition is also referred to as **damage initiation criterion** or loading function in the literature.\n\n## How to treat unloading and reloading?\n\nWhile in plastic model, the consistency has been used to derive the evolution equations and to distinguish unloading from loading steps, in damage models the unloading and reloading is distinguished automatically using the following criterion:\n\\begin{align}\n\\kappa(t) = \\max_{\\theta < t} \\kappa(\\theta) = \\max_{\\theta < t} \\sqrt{ s^2_x(\\theta) + s_y^2(\\theta) }\n\\end{align}\n\nThis expression means that the equivalent strain $\\kappa$ can grow only if the new value at time $t$ is larger than any other value of $\\kappa$ attained during the previous time stepping $\\theta < t$. \n\n## Constitutive law between slip and bond stress vectors\n\nTo return to the case of an interface damage, we need to relate the slip vector components to the stress vector components. \n\\begin{align}\n\\tau_x &= (1 - \\omega(\\kappa)) \\, E_b \\, s_x \\\\\n\\tau_y &= (1 - \\omega(\\kappa)) \\, E_b \\, s_y\n\\end{align}\n\nThis kind of definition introduces **isotropic damage**. It uses a single damage variable for both slip directions:\n\n> ** As a consequence, damage increment caused in direction $x$ directly affects the behavior in direction $y$**\n             \n\n
                                         \n        Let see how does the isotropic damage evolves in a single point 
                                         \n\nTo visualize the response of a material point with two control slip  vector components $s_x, s_y$ and two stress vector components $\\tau_x, \\tau_y$ in a 3D diagram, we will plot the loading histories in the space $\\left[s_x, s_y, \\sqrt{\\tau_x^2 + \\tau_y^2}\\right]$. The vertical axis represents the norm of the stress vector. We will make use of an interactive model component `damage_2d_explorer`.\n\n\n```python\nimport damage2d_explorer as de\nbs = MATS1D5BondSlipD(omega_fn='fracture-energy', s_max=0.05)\nbs.omega_fn_.trait_set(G_f=10, kappa_0=0.001)\nexplore = de.Explore(bs=bs, s_min=-0.1, s_max=0.1)\nexplore.interact()\n```\n\n\n    VBox(children=(HBox(children=(VBox(children=(Tree(layout=Layout(align_items='stretch', border='solid 1px black\u2026\n\n\n# **Characterization of damage models**\n\n## Main properties\n * Independent definition of **elastic domain** using equivalent strain measure and **damage evolution** using damage function\n * The definitions of equivalent strain defining the inelastic domain and damage function can be arbitrarily combined\n * **Isotropic damage** - affects all loading directions equally\n * At any state in history, radial unloading returns back to origin with zero stress\n * Suitable to model brittle fracture\n\n## Damage versus plasticity in view of algorithmic treatment\n\nRecall that for models based on plasticity, return mapping was necessary to return back to an admissible state. \nIn case of damage models, the relation between strain and stress can be introduced as an explicit function. \nThis means that  no return-mapping is necessary. Given a strain increment, \nthe corresponding stress can be directly evaluated.\n\n## Damage versus multi-linear models in view of algorithmic treatment\n\nRecall that the multi-linear models of the bond behavior also allowed for an explicit evaluation without return-mapping. The question might arise: What is the added value of a damage model compared to multi-linear elastic models? They both can represent the non-linear response of a monotonically loaded pullout test. The reason is twofold\n\n 1. Damage model can distinguish unloading/reloading by remembering the measure of maximum strain $\\kappa$ attained during history as demonstrated in the example below.\n 2. Damage model can introduce inelasticity in 2D and 3D material models. Multilinear laws are only limited to one-dimensional model. This aspect of damage models is shown in the example of an interface damage model below.\n\n# Open questions\n\n * How is the damage model embedded in a FE simulation\n     * As a material subroutine returning the stress and stiffness for a given slip/strain and $\\kappa_n$\n * What is the instantaneous stiffness at control time $t$? needed in FE simulation\n     * Secant stiffness is available directly as $\\psi E_b$\n     * Consistent algorithmic stiffness can be calculated as $\\left. \\frac{\\partial \\tau}{\\partial s} \\right|_t$\n * Can the damage model represent cracking?\n     * Yes but we need to talk about energy first\n\n\n# Links to manuals of non-linear FE tools\n\nThis brief introduction of damage modeling is meant as an entry point to reading more thorough documentation, e.g. the parts of software manuals describing damage.\n\n - [ABAQUS Manual: available damage functions](https://classes.engineering.wustl.edu/2009/spring/mase5513/abaqus/docs/v6.5/books/usb/default.htm?startat=pt04ch11s06abm39.html)\n - [ATENA Manual: Stress-strain relations for concrete (Pages 18-21)](https://www.cervenka.cz/assets/files/atena-pdf/ATENA_Theory.pdf) \n\n
                                         \n        Exercise X0501: Bond-slip law expressed as damage function \n\n
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                                         \n        Exercise X0502: Bond-slip law expressed as damage function \n
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                                         \n\n\n```python\n\n```\n", "meta": {"hexsha": "0c830542d6028e84a6e5a69107b82f2182aa4fb5", "size": 300178, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tour5_damage_bond/5_1_Introspect_Damage_Evolution_Damage_initiation.ipynb", "max_stars_repo_name": "bmcs-group/bmcs_tutorial", "max_stars_repo_head_hexsha": "4e008e72839fad8820a6b663a20d3f188610525d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tour5_damage_bond/5_1_Introspect_Damage_Evolution_Damage_initiation.ipynb", "max_issues_repo_name": "bmcs-group/bmcs_tutorial", "max_issues_repo_head_hexsha": "4e008e72839fad8820a6b663a20d3f188610525d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tour5_damage_bond/5_1_Introspect_Damage_Evolution_Damage_initiation.ipynb", "max_forks_repo_name": "bmcs-group/bmcs_tutorial", "max_forks_repo_head_hexsha": "4e008e72839fad8820a6b663a20d3f188610525d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 202.4126770061, "max_line_length": 153120, "alphanum_fraction": 0.9009720899, "converted": true, "num_tokens": 8642, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381667555714, "lm_q2_score": 0.7549149868676283, "lm_q1q2_score": 0.41274083597631334}}
{"text": "# Cutting the tests\n\n# Purpose\nCut the tests based on their states.\n\n# Methodology\nLook at the velocities, accelerations and rudder signal to determine the good parts of the tests\n\n# Setup\n\n\n```python\n# %load imports.py\n%matplotlib inline\n%load_ext autoreload\n%autoreload 2\n\n## External packages:\nimport pandas as pd\npd.options.display.max_rows = 999\npd.options.display.max_columns = 999\npd.set_option(\"display.max_columns\", None)\n\nimport numpy as np\nimport os\nimport matplotlib.pyplot as plt\n\nimport plotly.express as px \nimport plotly.graph_objects as go\n\nimport seaborn as sns\nimport sympy as sp\nfrom sympy.physics.mechanics import (dynamicsymbols, ReferenceFrame,\n                                      Particle, Point)\nfrom sympy.physics.vector.printing import vpprint, vlatex\nfrom IPython.display import display, Math, Latex\nfrom src.substitute_dynamic_symbols import run, lambdify\n\nimport pyro\n\nimport sklearn\nimport pykalman\nfrom statsmodels.sandbox.regression.predstd import wls_prediction_std\nimport statsmodels.api as sm\n\nfrom scipy.integrate import solve_ivp\nimport seaborn as sns\n\n## Local packages:\nfrom src.data import mdl\n\n\n\n```\n\n    Duplicate key in file WindowsPath('C:/Users/maa/.matplotlib/stylelib/paper.mplstyle'), line 462 ('figure.figsize   : 5, 3   ## figure size in inches')\n    Duplicate key in file WindowsPath('C:/Users/maa/.matplotlib/stylelib/paper.mplstyle'), line 463 ('figure.dpi       : 100        ## figure dots per inch')\n\n\n\n```python\ndf_runs = mdl.runs()\n```\n\n\n```python\nmask = ((~df_runs['sailing']) &\n        (\n            (df_runs['test_type'] == 'reference speed') |\n            (df_runs['test_type'] == 'rodergrundvinkel') |\n            ((df_runs['series_number'] == 5) & (df_runs['test_number'] == 1)  & (df_runs['run_number'] == 3)) |\n            ((df_runs['series_number'] == 5) & (df_runs['test_number'] == 2)  & (df_runs['run_number'] == 6)) |\n            ((df_runs['series_number'] == 5) & (df_runs['test_number'] == 3)  & (df_runs['run_number'] == 2)) |\n            ((df_runs['series_number'] == 5) & (df_runs['test_number'] == 4)  & (df_runs['run_number'] == 1)) |\n            ((df_runs['series_number'] == 5) & (df_runs['test_number'] == 5)  & (df_runs['run_number'] == 1)) \n            \n       ))\ndf_runs_selected = df_runs.loc[mask].copy()\n```\n\n\n```python\ndef load_run(id):\n    \n    df, units, meta_data = mdl.load(id=id, dir_path='../data/processed/kalman')\n    df.index = df.index.total_seconds()\n    df.index-=df.index[0]\n    df.sort_index(inplace=True)\n    df['-delta'] = -df['delta']\n    df['V'] = np.sqrt(df['u']**2 + df['v']**2)\n    \n    return df, units, meta_data\n\ndf_all = pd.DataFrame()\nfor id,row in df_runs_selected.iterrows():\n    \n    df_, units, meta_data = load_run(id)\n    df_['id'] = id\n    df_['t'] = df_.index\n    df_all = df_all.append(df_, ignore_index=True)\n\n    \ndf_all['thrust'] = df_all['Prop/PS/Thrust'] + df_all['Prop/SB/Thrust']\ndf_all['U'] = df_all['V']\n\ndf_all_clean = df_all.copy()\ndf_all = pd.merge(left=df_all, right = df_runs_selected, how='left', left_on='id', right_index=True)\n```\n\n\n```python\nref_runs = df_all.groupby(by='test_type').get_group('reference speed')\nruns = ref_runs.groupby(by='id')\ndf_cut = pd.DataFrame()\nfor id, group in runs:\n    \n    df_rolling = group.rolling(window=500).std()\n    mask = df_rolling['u1d'] < 0.0004\n    df_ = group.loc[mask].copy()\n    if len(df_) > 300:\n        df_cut = df_cut.append(df_)\n    \n    \n```\n\n\n```python\nruns_cut = df_cut.groupby(by='id')\nfor id, group in runs:\n    \n    fig,ax=plt.subplots()\n    fig.set_size_inches(20,3)\n    \n    meta_data = df_runs_selected.loc[id]\n    title = f'id:{id} ({meta_data[\"test_type\"]})'\n    ax.set_title(title)\n    \n    key='u'\n    group.plot(x='t', y=key,ax=ax)\n    \n    try: \n        group_cut = runs_cut.get_group(id)\n    except:\n        pass\n    else:\n        group_cut.iloc[[0]].plot(x='t', y=key, style='go', label='start', ax=ax)\n        group_cut.iloc[[-1]].plot(x='t', y=key, style='ro', label='stop', ax=ax)\n        \n    ax.set_ylim(df_all[key].min(),df_all[key].max())\n    ax.legend(loc='upper left')\n    \n    ax2 = ax.twinx()\n    key='u1d'\n    group.plot(x='t', y=key,ax=ax2, style='r-', title=title)\n        \n    ax2.set_ylim(df_all[key].min(),df_all[key].max())\n    ax2.get_legend().set_visible(False)\n    ax2.legend(loc='upper right')\n    ax2.grid(True)\n    \n```\n\n\n```python\nref_rud_runs = df_all.groupby(by='test_type').get_group('rodergrundvinkel')\nruns = ref_rud_runs.groupby(by='id')\n\nfor id, group in runs:\n    \n    df_rolling = group.rolling(window=500).std()\n    mask = ((df_rolling['r'] < 0.0004) & (df_rolling['u1d'] < 0.0005))\n    df_ = group.loc[mask].copy()\n    \n    df_ = group.loc[mask].copy()\n    if len(df_) > 300:\n        df_cut = df_cut.append(df_)\n```\n\n\n```python\nruns_cut = df_cut.groupby(by='id')\nfor id, group in runs:\n    \n    fig,ax=plt.subplots()\n    fig.set_size_inches(20,3)\n    \n    meta_data = df_runs_selected.loc[id]\n    title = f'id:{id} ({meta_data[\"test_type\"]})'\n    ax.set_title(title)\n    \n    key='u1d'\n    group.plot(x='t', y=key,ax=ax)\n    \n    try: \n        group_cut = runs_cut.get_group(id)\n    except:\n        pass\n    else:\n        group_cut.iloc[[0]].plot(x='t', y=key, style='go', label='start', ax=ax)\n        group_cut.iloc[[-1]].plot(x='t', y=key, style='ro', label='stop', ax=ax)\n        \n    ax.set_ylim(df_all[key].min(),df_all[key].max())\n    ax.legend(loc='upper left')\n    \n    ax2 = ax.twinx()\n    key='r'\n    group.plot(x='t', y=key,ax=ax2, style='r-', title=title)\n        \n    ax2.set_ylim(df_all[key].min(),df_all[key].max())\n    ax2.get_legend().set_visible(False)\n    ax2.legend(loc='upper right')\n    ax2.grid(True)\n```\n\n\n```python\nmask = ((df_all['test_type']=='reference speed') | \n        (df_all['test_type']=='rodergrundvinkel')\n       )\ndf_man = df_all.loc[~mask]\nfor id,group in df_man.groupby(by='id'):\n    df_ = group.iloc[0:-500].copy()  # \"strange thing may happen in the end\"\n    df_cut = df_cut.append(df_)\n```\n\n\n```python\ndf_cut['test_type'].unique()\n```\n\n\n\n\n    array(['reference speed', 'rodergrundvinkel', 'zigzag', 'turning circle'],\n          dtype=object)\n\n\n\n\n```python\ndf_cut\n```\n\n\n\n\n
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                                        1660050.01.0-0.937778-0.9380110.0651680.37407549.415584-1.243598-1.2441930.0432155.36622850.3759420.0003490.0075030.0500640.013106-0.035665-0.0231818.5135664.837552-0.0990019.5236476.1840800.117744-0.0034900.64577214.89-0.003425-0.0016870.0002319.1560854.73951318.9027250.000741-0.7807940.200363-0.0006982.628581-2.296328-0.002287-4.357649e-03-2.482928e-034.739827-0.765097-0.00793418.9026570.226057-0.0014349.1560660.002323-0.0005100.797656-0.014819-0.0006980.0072680.003489-0.0022870.002323-0.0005100.0185760.8059690.014106-0.000698-3.1438741.516118-0.002287-4.357649e-03-2.482928e-030.0034900.79779422774155.1200019.0055741.0NaN9.005574NaN0.0NaNNaNNaNNaNNaN11.0216310.7977940.722775NaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN0.3313111.253641NaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN0.2063110.206311NaNNaN0.441027NaNNaNNaNNaNNaN0.946602Circle 35SB2020-09-23MDLNaNNaN\\\\sspa.local\\lab\\MeasuredataMDL\\40199079\\005\\0...NaN0.1360920.438908-0.24432167.05.014563M5139-02-ADesign40199079.01.041.25.0M5139-02-A0.96177NaN5.0turning circle0.00.0-0.214False
                                        1660060.01.0-0.937812-0.9379240.0481512.16870649.415584-1.243619-1.2439810.0750990.40687150.375942-0.0222190.0197510.0384480.0189040.0449250.0116198.5180334.728457-0.1010629.5305816.2000990.120292-0.0034970.64577214.89-0.003500-0.0016900.0002289.1559974.73242318.9047280.000741-0.7777980.2019820.000164-0.1165060.144942-0.1906434.337636e-034.336195e-014.732202-0.764666-0.00792118.9047350.225758-0.0014539.1560510.002296-0.0005190.797162-0.0146340.0001640.0072510.003504-0.1906430.002296-0.0005190.0183550.8035090.0118190.0001640.150813-0.108799-0.1906434.337636e-034.336195e-010.0034970.79729622774155.1300049.0045541.0NaN9.004554NaN0.0NaNNaNNaNNaNNaN10.9285560.7972960.722775NaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN0.3313111.253641NaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN0.2063110.206311NaNNaN0.441027NaNNaNNaNNaNNaN0.946602Circle 35SB2020-09-23MDLNaNNaN\\\\sspa.local\\lab\\MeasuredataMDL\\40199079\\005\\0...NaN0.1360920.438908-0.24432167.05.014563M5139-02-ADesign40199079.01.041.25.0M5139-02-A0.96177NaN5.0turning circle0.00.0-0.214False
                                        1660070.01.0-0.937847-0.9380990.0579160.04602449.412958-1.243742-1.2444050.0772733.16635850.3731880.008087-0.1646170.0339310.0034430.028807-0.0025598.5390534.660166-0.1040649.5323436.2605140.123360-0.0034900.64577214.89-0.003425-0.0016930.0002449.1561724.72393818.9067690.000745-0.7831400.203264-0.0045241.1627332.4169880.0102174.310329e-03-4.376807e-014.724264-0.764091-0.00797718.9068210.225697-0.0014699.1561150.002267-0.0005090.796590-0.014779-0.0045240.0073000.0035340.0102170.002267-0.0005090.0185500.8090020.011860-0.004524-0.479600-2.6388950.0102174.310329e-03-4.376807e-010.0034900.79672722774155.1400239.0065951.0NaN9.006595NaN0.0NaNNaNNaNNaNNaN10.9206800.7967270.722775NaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN0.3313111.253641NaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaNNaN0.2063110.206311NaNNaN0.441027NaNNaNNaNNaNNaN0.946602Circle 35SB2020-09-23MDLNaNNaN\\\\sspa.local\\lab\\MeasuredataMDL\\40199079\\005\\0...NaN0.1360920.438908-0.24432167.05.014563M5139-02-ADesign40199079.01.041.25.0M5139-02-A0.96177NaN5.0turning circle0.00.0-0.214False
                                        \n
                                        77520 rows \u00d7 160 columns
                                        \n
                                        \n\n\n\n## Save\n\n\n```python\nsave_dir = '../data/processed/kalman_cut'\nif not os.path.exists(save_dir):\n    os.mkdir(save_dir)\n\nruns = df_all_clean.groupby(by='id')\nfor id, group in df_cut.groupby(by='id'):\n    \n    start_index = group.index[0]\n    stop_index = group.index[-1]\n    df = runs.get_group(id).loc[start_index:stop_index].copy()\n    df.set_index('t', inplace=True)\n    save_name = f'{id}.csv'\n    save_path = os.path.join(save_dir,save_name)\n    df.to_csv(save_path)\n    \n   \n    \n    \n    \n```\n\n\n```python\ndf.head()\n```\n\n\n\n\n
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                                        Arr/Ind/ArrArr/Ind/FriFan/Aft/AngleFan/Aft/AngleOrderFan/Aft/FxFan/Aft/RpmFan/Aft/RpmOrderFan/Fore/AngleFan/Fore/AngleOrderFan/Fore/FxFan/Fore/RpmFan/Fore/RpmOrderHull/Acc/X1Hull/Acc/Y1Hull/Acc/Y2Hull/Acc/Z1Hull/Acc/Z2Hull/Acc/Z3Prop/PS/RpmProp/PS/ThrustProp/PS/TorqueProp/SB/RpmProp/SB/ThrustProp/SB/TorquedeltaRudder/MaxAngleRudder/Ratelab/WaveHeightrollpitchpsix0y0z0x01d_gradienty01d_gradientz01d_gradientx02d_gradienty02d_gradientz02d_gradientpsi1d_gradientpsi2d_gradientx0_filteredx01dx02dy0_filteredy01dy02dpsi_filteredpsi1dpsi2duvwu1dv1dw1drr1dbetau_gradientv_gradientw_gradientu1d_gradientv1d_gradientw1d_gradientr_gradientr1d_gradient-deltaVidFan/Aft/FxOrderFan/ForceFactorFan/Fore/AngleOrder2Fan/Fore/FxOrderFan/Fore/RateWind/GWAWind/CourseWind/CourseTresholdFan/Aft/CalcAngleFan/Fore/CalcAngleWind/EnablethrustU
                                        t
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                                        \n
                                        \n\n\n\n\n```python\nstart_index\n```\n\n\n\n\n    150493\n\n\n\n\n```python\nstop_index\n```\n\n\n\n\n    166007\n\n\n\n\n```python\nindex = list(set(df_runs_selected.index) & set(df_cut['id'].unique()))\ndf_runs = df_runs_selected.loc[index].copy()\nsave_name = 'runs.csv'\nsave_path = os.path.join(save_dir,save_name)\ndf_runs.to_csv(save_path)\n```\n\n\n```python\nindex\n```\n\n\n\n\n    [22605,\n     22607,\n     22608,\n     22610,\n     22611,\n     22612,\n     22613,\n     22614,\n     22615,\n     22631,\n     22632,\n     22633,\n     22634,\n     22635,\n     22764,\n     22636,\n     22638,\n     22637,\n     22639,\n     22770,\n     22772,\n     22773,\n     22774]\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "76fa7dc5f464857777f6530cf780981a7911b678", "size": 879159, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/05.03_cutting.ipynb", "max_stars_repo_name": "martinlarsalbert/wPCC", "max_stars_repo_head_hexsha": "16e0d4cc850d503247916c9f5bd9f0ddb07f8930", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/05.03_cutting.ipynb", "max_issues_repo_name": "martinlarsalbert/wPCC", "max_issues_repo_head_hexsha": "16e0d4cc850d503247916c9f5bd9f0ddb07f8930", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/05.03_cutting.ipynb", "max_forks_repo_name": "martinlarsalbert/wPCC", "max_forks_repo_head_hexsha": "16e0d4cc850d503247916c9f5bd9f0ddb07f8930", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 224.3897396631, "max_line_length": 39504, "alphanum_fraction": 0.8581451137, "converted": true, "num_tokens": 27266, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982315512489, "lm_q2_score": 0.6370307944803832, "lm_q1q2_score": 0.4126674221080794}}
{"text": "```python\n# !/usr/bin/env python3\n# -*- coding: utf-8 -*-\n\"\"\"\nCreated on 20181221\n\n@author: zhangji\n\"\"\"\n%pylab inline\n# pylab.rcParams['figure.figsize'] = (25, 11)\nfontsize = 40\n\nimport numpy as np\nimport math\nimport scipy as sp\nfrom scipy.optimize import leastsq, curve_fit\nfrom scipy import interpolate\nfrom scipy.interpolate import interp1d\nfrom scipy.io import loadmat, savemat\n# import scipy.misc\n\nimport matplotlib\nfrom matplotlib import pyplot as plt\nfrom matplotlib import animation, rc\nimport matplotlib.ticker as mtick\nfrom mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\nfrom mpl_toolkits.mplot3d import Axes3D, axes3d\n\nfrom sympy import symbols, simplify, series, exp\nfrom sympy.matrices import Matrix\nfrom sympy.solvers import solve\n\nfrom IPython.display import display, HTML\nfrom tqdm import tqdm_notebook as tqdm\nimport pandas as pd\nimport re\nfrom scanf import scanf\nimport os\nimport glob\nimport natsort \nfrom shutil import copyfile\n\nfrom codeStore import support_fun as spf\nfrom src.support_class import *\nfrom src import stokes_flow as sf\n\nrc('animation', html='html5')\nPWD = os.getcwd()\nfont = {'size': 20}\nmatplotlib.rc('font', **font)\nnp.set_printoptions(linewidth=90, precision=5)\n\n```\n\n    Populating the interactive namespace from numpy and matplotlib\n\n\n\n```python\ndef _c(ca,i,j,p,q):\n\n    if ca[i,j] > -1:\n        return ca[i,j]\n    elif i == 0 and j == 0:\n        ca[i,j] = np.linalg.norm(p[i]-q[j])\n    elif i > 0 and j == 0:\n        ca[i,j] = max( _c(ca,i-1,0,p,q), np.linalg.norm(p[i]-q[j]) )\n    elif i == 0 and j > 0:\n        ca[i,j] = max( _c(ca,0,j-1,p,q), np.linalg.norm(p[i]-q[j]) )\n    elif i > 0 and j > 0:\n        ca[i,j] = max(                 \\\n            min(                       \\\n                _c(ca,i-1,j,p,q),      \\\n                _c(ca,i-1,j-1,p,q),    \\\n                _c(ca,i,j-1,p,q)       \\\n            ),                         \\\n            np.linalg.norm(p[i]-q[j])  \\\n            )                                                          \n    else:\n        ca[i,j] = float('inf')\n    \n    return ca[i,j]\n\n\ndef frdist(p,q):\n    \"\"\" \n    Computes the discrete Fr\u00e9chet distance between\n    two curves. The Fr\u00e9chet distance between two curves in a\n    metric space is a measure of the similarity between the curves.\n    The discrete Fr\u00e9chet distance may be used for approximately computing\n    the Fr\u00e9chet distance between two arbitrary curves, \n    as an alternative to using the exact Fr\u00e9chet distance between a polygonal\n    approximation of the curves or an approximation of this value.\n    \n    This is a Python 3.* implementation of the algorithm produced\n    in Eiter, T. and Mannila, H., 1994. Computing discrete Fr\u00e9chet distance. Tech. \n    Report CD-TR 94/64, Information Systems Department, Technical University \n    of Vienna.\n    http://www.kr.tuwien.ac.at/staff/eiter/et-archive/cdtr9464.pdf\n\n    Function dF(P, Q): real;\n        input: polygonal curves P = (u1, . . . , up) and Q = (v1, . . . , vq).\n        return: \u03b4dF (P, Q)\n        ca : array [1..p, 1..q] of real;\n        function c(i, j): real;\n            begin\n                if ca(i, j) > \u22121 then return ca(i, j)\n                elsif i = 1 and j = 1 then ca(i, j) := d(u1, v1)\n                elsif i > 1 and j = 1 then ca(i, j) := max{ c(i \u2212 1, 1), d(ui, v1) }\n                elsif i = 1 and j > 1 then ca(i, j) := max{ c(1, j \u2212 1), d(u1, vj ) }\n                elsif i > 1 and j > 1 then ca(i, j) :=\n                max{ min(c(i \u2212 1, j), c(i \u2212 1, j \u2212 1), c(i, j \u2212 1)), d(ui, vj ) }\n                else ca(i, j) = \u221e\n                return ca(i, j);\n            end; /* function c */\n\n        begin\n            for i = 1 to p do for j = 1 to q do ca(i, j) := \u22121.0;\n            return c(p, q);\n        end.\n\n    Parameters\n    ----------\n    P : Input curve - two dimensional array of points\n    Q : Input curve - two dimensional array of points\n\n    Returns\n    -------\n    dist: float64\n        The discrete Fr\u00e9chet distance between curves `P` and `Q`.\n\n    Examples\n    --------\n    >>> from frechetdist import frdist\n    >>> P=[[1,1], [2,1], [2,2]]\n    >>> Q=[[2,2], [0,1], [2,4]]\n    >>> frdist(P,Q)\n    >>> 2.0\n    >>> P=[[1,1], [2,1], [2,2]]\n    >>> Q=[[1,1], [2,1], [2,2]]\n    >>> frdist(P,Q)\n    >>> 0\n    \"\"\"\n    p = np.array(p, np.float64)\n    q = np.array(q, np.float64)\n\n    len_p = len(p)\n    len_q = len(q)\n\n    if len_p == 0 or len_q == 0:\n        raise ValueError('Input curves are empty.')\n\n    if len_p != len_q or len(p[0]) != len(q[0]):\n        raise ValueError('Input curves do not have the same dimensions.')\n\n    ca    = ( np.ones((len_p,len_q), dtype=np.float64) * -1 ) \n\n    dist = _c(ca,len_p-1,len_q-1,p,q)\n    return dist\n\n# Euclidean distance.\ndef euc_dist(pt1,pt2):\n    return math.sqrt((pt2[0]-pt1[0])*(pt2[0]-pt1[0])+(pt2[1]-pt1[1])*(pt2[1]-pt1[1]))\n\ndef _c(ca,i,j,P,Q):\n    if ca[i,j] > -1:\n        return ca[i,j]\n    elif i == 0 and j == 0:\n        ca[i,j] = euc_dist(P[0],Q[0])\n    elif i > 0 and j == 0:\n        ca[i,j] = max(_c(ca,i-1,0,P,Q),euc_dist(P[i],Q[0]))\n    elif i == 0 and j > 0:\n        ca[i,j] = max(_c(ca,0,j-1,P,Q),euc_dist(P[0],Q[j]))\n    elif i > 0 and j > 0:\n        ca[i,j] = max(min(_c(ca,i-1,j,P,Q),_c(ca,i-1,j-1,P,Q),_c(ca,i,j-1,P,Q)),euc_dist(P[i],Q[j]))\n    else:\n        ca[i,j] = float(\"inf\")\n    return ca[i,j]\n\n\"\"\" Computes the discrete frechet distance between two polygonal lines\nAlgorithm: http://www.kr.tuwien.ac.at/staff/eiter/et-archive/cdtr9464.pdf\nP and Q are arrays of 2-element arrays (points)\n\"\"\"\ndef frechetDist(P,Q):\n    ca = np.ones((len(P),len(Q)))\n    ca = np.multiply(ca,-1)\n    return _c(ca,len(P)-1,len(Q)-1,P,Q)\n\ndef read_ecoli_mat(mat_name):\n    mat_contents = loadmat(mat_name)\n    ecoli_U = mat_contents['ecoli_U']\n    ecoli_norm = mat_contents['ecoli_norm']\n    ecoli_center = mat_contents['ecoli_center']\n    return ecoli_center, ecoli_norm, ecoli_U\n\n```\n\n\n```python\nP=[[1,1], [2,1], [2,2]]\nQ=[[1,1], [2,1], [2,2], [2,2]]\nfrechetDist(P,Q)\n```\n\n\n\n\n    0.0\n\n\n\n\n```python\nbase_mat = os.path.join(PWD, 'ecoli_shear1c', 'eq_dt0.010_O5.mat')\ndir_name = 'ecoli_shear1c'\n\nbase_center, base_norm, base_U = read_ecoli_mat(base_mat)\nbase_length = np.linalg.norm((base_center[:-1, :] - base_center[1:, :]), axis=1).sum()\n_, dt0, _ = scanf('%s/eq_dt%f_%s', base_mat)\nt_dir = os.path.join(PWD, dir_name)\nmat_names = glob.glob('%s/*.mat' % t_dir)\nfor mati in natsort.natsorted(mat_names):\n    ecoli_center, ecoli_norm, ecoli_U = read_ecoli_mat(mati)\n    _, dt, _ = scanf('%s/eq_dt%f_%s', mati)\n    scale_cut = int(ecoli_center.shape[0] // (dt / dt0))\n    t_dst = frechetDist(ecoli_center[:scale_cut, :], base_center)\n    print(mati, t_dst, t_dst / base_length)\n    \n```\n\n    /home/zhangji/stokes_flow_master/head_Force/ecoli_shear1c/eq_dt0.010_O1.mat 0.000127075916188 0.00480734833451\n    /home/zhangji/stokes_flow_master/head_Force/ecoli_shear1c/eq_dt0.010_O2.mat 0.000109098699241 0.00412726082036\n    /home/zhangji/stokes_flow_master/head_Force/ecoli_shear1c/eq_dt0.010_O3.mat 7.29694933247e-05 0.00276047407507\n    /home/zhangji/stokes_flow_master/head_Force/ecoli_shear1c/eq_dt0.010_O4.mat 3.65589131018e-05 0.00138304279271\n    /home/zhangji/stokes_flow_master/head_Force/ecoli_shear1c/eq_dt0.010_O5.mat 0.0 0.0\n    /home/zhangji/stokes_flow_master/head_Force/ecoli_shear1c/eq_dt0.050_O1.mat 0.000116317464172 0.00440035047105\n    /home/zhangji/stokes_flow_master/head_Force/ecoli_shear1c/eq_dt0.050_O2.mat 0.000116317464172 0.00440035047105\n    /home/zhangji/stokes_flow_master/head_Force/ecoli_shear1c/eq_dt0.050_O3.mat 0.000225505558207 0.00853099314318\n    /home/zhangji/stokes_flow_master/head_Force/ecoli_shear1c/eq_dt0.050_O4.mat 0.000419755164527 0.0158795572884\n    /home/zhangji/stokes_flow_master/head_Force/ecoli_shear1c/eq_dt0.050_O5.mat 0.000618913347282 0.0234138154459\n    /home/zhangji/stokes_flow_master/head_Force/ecoli_shear1c/eq_dt0.100_O1.mat 0.000261714294386 0.00990078855987\n    /home/zhangji/stokes_flow_master/head_Force/ecoli_shear1c/eq_dt0.100_O2.mat 0.000261714294386 0.00990078855987\n    /home/zhangji/stokes_flow_master/head_Force/ecoli_shear1c/eq_dt0.100_O3.mat 0.000621005620393 0.0234929672314\n    /home/zhangji/stokes_flow_master/head_Force/ecoli_shear1c/eq_dt0.100_O4.mat 0.00103844024888 0.0392847374285\n    /home/zhangji/stokes_flow_master/head_Force/ecoli_shear1c/eq_dt0.100_O5.mat 0.00147326730137 0.0557344721169\n\n\n\n```python\nnp.linalg.norm((base_center[:-1, :] - base_center[1:, :]), axis=1).sum()\n```\n\n\n\n\n    0.026434023281670681\n\n\n", "meta": {"hexsha": "38a6b31c960b090b607f57e6a040948897adbee0", "size": 12356, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "head_Force/FrechetDistance.ipynb", "max_stars_repo_name": "pcmagic/stokes_flow", "max_stars_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-11-11T05:00:53.000Z", "max_stars_repo_stars_event_max_datetime": "2018-11-11T05:00:53.000Z", "max_issues_repo_path": "head_Force/FrechetDistance.ipynb", "max_issues_repo_name": "pcmagic/stokes_flow", "max_issues_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "head_Force/FrechetDistance.ipynb", "max_forks_repo_name": "pcmagic/stokes_flow", "max_forks_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 37.329305136, "max_line_length": 271, "alphanum_fraction": 0.5275169958, "converted": true, "num_tokens": 2832, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.6370307944803831, "lm_q1q2_score": 0.41266741344500923}}
{"text": "Lambda School Data Science\n\n*Unit 2, Sprint 1, Module 2*\n\n---\n\n# Regression 2\n- Do train/test split\n- Use scikit-learn to fit a multiple regression\n- Understand how ordinary least squares regression minimizes the sum of squared errors\n- Define overfitting/underfitting and the bias/variance tradeoff\n\n### Setup\n\nRun the code cell below. You can work locally (follow the [local setup instructions](https://lambdaschool.github.io/ds/unit2/local/)) or on Colab.\n\nLibraries:\n- matplotlib\n- numpy\n- pandas\n- plotly\n- scikit-learn\n\n\n```python\nimport sys\n\n# If you're on Colab:\nif 'google.colab' in sys.modules:\n    DATA_PATH = 'https://raw.githubusercontent.com/LambdaSchool/DS-Unit-2-Applied-Modeling/master/data/'\n\n# If you're working locally:\nelse:\n    DATA_PATH = '../data/'\n    \n# Ignore this Numpy warning when using Plotly Express:\n# FutureWarning: Method .ptp is deprecated and will be removed in a future version. Use numpy.ptp instead.\nimport warnings\nwarnings.filterwarnings(action='ignore', category=FutureWarning, module='numpy')\n```\n\n# Do train/test split\n\n## Overview\n\n### Predict Elections! \ud83c\uddfa\ud83c\uddf8\ud83d\uddf3\ufe0f\n\nHow could we try to predict the 2020 US Presidential election? \n\nAccording to Douglas Hibbs, a political science and economics professor, you can [explain elections with just two features, \"Bread and Peace\":](https://douglas-hibbs.com/background-information-on-bread-and-peace-voting-in-us-presidential-elections/)\n\n> Aggregate two-party vote shares going to candidates of the party holding the presidency during the postwar era are well explained by just two fundamental determinants:\n>\n> (1) Positively by weighted-average growth of per capita real disposable personal income over the term.  \n> (2) Negatively by cumulative US military fatalities (scaled to population) owing to unprovoked, hostile deployments of American armed forces in foreign wars. \n\nLet's look at the data that Hibbs collected and analyzed:\n\n\n```python\nimport pandas as pd\ndf = pd.read_csv(DATA_PATH+'elections/bread_peace_voting.csv')\ndf\n```\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n  \n
                                        YearIncumbent Party CandidateOther CandidateAverage Recent Growth in Personal IncomesUS Military Fatalities per MillionIncumbent Party Vote Share
                                        01952StevensonEisenhower2.4019044.60
                                        11956EisenhowerStevenson2.89057.76
                                        21960NixonKennedy0.85049.91
                                        31964JohnsonGoldwater4.21161.34
                                        41968HumphreyNixon3.0214649.60
                                        51972NixonMcGovern3.62061.79
                                        61976FordCarter1.08248.95
                                        71980CarterReagan-0.39044.70
                                        81984ReaganMondale3.86059.17
                                        91988Bush, Sr.Dukakis2.27053.94
                                        101992Bush, Sr.Clinton0.38046.55
                                        111996ClintonDole1.04054.74
                                        122000GoreBush, Jr.2.36050.27
                                        132004Bush, Jr.Kerry1.72451.24
                                        142008McCainObama0.101446.32
                                        152012ObamaRomney0.95552.00
                                        162016ClintonTrump0.10548.20
                                        \n
                                        \n\n\n\nData Sources & Definitions\n\n- 1952-2012: Douglas Hibbs, [2014 lecture at Deakin University Melbourne](http://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf), Slide 40\n- 2016, Vote Share: [The American Presidency Project](https://www.presidency.ucsb.edu/statistics/elections)\n- 2016, Recent Growth in Personal Incomes: [The 2016 election economy: the \"Bread and Peace\" model final forecast](https://angrybearblog.com/2016/11/the-2016-election-economy-the-bread-and-peace-model-final-forecast.html)\n- 2016, US Military Fatalities: Assumption that Afghanistan War fatalities in 2012-16 occured at the same rate as 2008-12\n\n> Fatalities denotes the cumulative number of American military fatalities per millions of US population the in Korea, Vietnam, Iraq and Afghanistan wars during the presidential terms preceding the 1952, 1964, 1968, 1976 and 2004, 2008 and 2012 elections. \u2014[Hibbs](http://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf), Slide 33\n\nHere we have data from the 1952-2016 elections. We could make a model to predict 1952-2016 election outcomes \u2014 but do we really care about that? \n\nNo, not really. We already know what happened, we don't need to predict it.\n\nThis is explained in [_An Introduction to Statistical Learning_](http://faculty.marshall.usc.edu/gareth-james/ISL/), Chapter 2.2, Assessing Model Accuracy:\n\n> In general, we do not really care how well the method works training on the training data. Rather, _we are interested in the accuracy of the predictions that we obtain when we apply our method to previously unseen test data._ Why is this what we care about? \n>\n> Suppose that we are interested in developing an algorithm to predict a stock\u2019s price based on previous stock returns. We can train the method using stock returns from the past 6 months. But we don\u2019t really care how well our method predicts last week\u2019s stock price. We instead care about how well it will predict tomorrow\u2019s price or next month\u2019s price. \n>\n> On a similar note, suppose that we have clinical measurements (e.g. weight, blood pressure, height, age, family history of disease) for a number of patients, as well as information about whether each patient has diabetes. We can use these patients to train a statistical learning method to predict risk of diabetes based on clinical measurements. In practice, we want this method to accurately predict diabetes risk for _future patients_ based on their clinical measurements. We are not very interested in whether or not the method accurately predicts diabetes risk for patients used to train the model, since we already know which of those patients have diabetes.\n\nSo, we're really interested in the 2020 election \u2014 but we probably don't want to wait until then to evaluate our model.\n\nThere is a way we can estimate now how well our model will generalize in the future. We can't fast-forward time, but we can rewind it...\n\nWe can split our data in **two sets.** For example: \n1. **Train** a model on elections before 2008.\n2. **Test** the model on 2008, 2012, 2016. \n\nThis \"backtesting\" helps us estimate how well the model will predict the next elections going forward, starting in 2020.\n\nThis is explained in [_Forecasting,_ Chapter 3.4,](https://otexts.com/fpp2/accuracy.html) Evaluating forecast accuracy:\n\n> The accuracy of forecasts can only be determined by considering how well a model performs on new data that were not used when fitting the model.\n>\n>When choosing models, it is common practice to separate the available data into two portions, training and test data, where the training data is used to estimate any parameters of a forecasting method and the test data is used to evaluate its accuracy. Because the test data is not used in determining the forecasts, it should provide a reliable indication of how well the model is likely to forecast on new data.\n>\n>\n>\n>The size of the test set is typically about 20% of the total sample, although this value depends on how long the sample is and how far ahead you want to forecast. The following points should be noted.\n>\n>- A model which fits the training data well will not necessarily forecast well.\n>- A perfect fit can always be obtained by using a model with enough parameters.\n>- Over-fitting a model to data is just as bad as failing to identify a systematic pattern in the data.\n>\n>Some references describe the test set as the \u201chold-out set\u201d because these data are \u201cheld out\u201d of the data used for fitting. Other references call the training set the \u201cin-sample data\u201d and the test set the \u201cout-of-sample data\u201d. We prefer to use \u201ctraining data\u201d and \u201ctest data\u201d in this book.\n\n**How should we split: Randomly? Before/after a given date?**\n\nI recommend you all read a great blog post, [How (and why) to create a good validation set](https://www.fast.ai/2017/11/13/validation-sets/), by fast.ai cofounder Rachel Thomas.\n\nShe gives great examples to answer the question \u201cWhen is a random subset not good enough?\u201d I\u2019m not as opposed to random splits as Rachel Thomas seems to be. But it\u2019s worth thinking about the trade-offs!\n\nTime-based and random splits can both be useful, and you\u2019ll get repeated hands-on practice with both during this unit! (She also talks about the distinction between validation & test sets, which we\u2019ll introduce in the last lesson of this Sprint.)\n\n## Follow Along\n\nSplit the data in two sets:\n1. Train on elections before 2008.\n2. Test on 2008 and after.\n\n\n```python\ntrain = df[df['Year'] < 2008]\ntrain.head()\n```\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n  \n
                                        YearIncumbent Party CandidateOther CandidateAverage Recent Growth in Personal IncomesUS Military Fatalities per MillionIncumbent Party Vote Share
                                        01952StevensonEisenhower2.4019044.60
                                        11956EisenhowerStevenson2.89057.76
                                        21960NixonKennedy0.85049.91
                                        31964JohnsonGoldwater4.21161.34
                                        41968HumphreyNixon3.0214649.60
                                        \n
                                        \n\n\n\n\n```python\ntrain.dtypes\n```\n\n\n\n\n    Year                                           int64\n    Incumbent Party Candidate                     object\n    Other Candidate                               object\n    Average Recent Growth in Personal Incomes    float64\n    US Military Fatalities per Million             int64\n    Incumbent Party Vote Share                   float64\n    dtype: object\n\n\n\n\n```python\ntest = df[df['Year'] >= 2008]\ntest.head()\n```\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n    \n  \n
                                        YearIncumbent Party CandidateOther CandidateAverage Recent Growth in Personal IncomesUS Military Fatalities per MillionIncumbent Party Vote Share
                                        142008McCainObama0.101446.32
                                        152012ObamaRomney0.95552.00
                                        162016ClintonTrump0.10548.20
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                                        \n\n\n\nHow many observations (rows) are in the train set? In the test set?\n\n\n```python\nprint(train.shape)\nprint(test.shape)\n```\n\n    (14, 6)\n    (3, 6)\n\n\nNote that this volume of data is at least two orders of magnitude smaller than we usually want to work with for predictive modeling.\n\nThere are other validation techniques that could be used here, such as [time series cross-validation](https://scikit-learn.org/stable/modules/cross_validation.html#time-series-split), or [leave-one-out cross validation](https://scikit-learn.org/stable/modules/cross_validation.html#leave-one-out-loo) for small datasets. However, for this module, let's start simpler, with train/test split. \n\nUsing a tiny dataset is intentional here. It's good for learning because we can see all the data at once.\n\n## Challenge\n\nIn your assignment, you will do train/test split, based on date.\n\n# Use scikit-learn to fit a multiple regression\n\n## Overview\n\nWe've done train/test split, and we're ready to fit a model. \n\nWe'll proceed in 3 steps. The first 2 are review from the previous module. The 3rd is new.\n\n- Begin with baselines (0 features) \n- Simple regression (1 feature)\n- Multiple regression (2 features)\n\n## Follow Along\n\n### Begin with baselines (0 features)\n\nWhat was the average Incumbent Party Vote Share, in the 1952-2004 elections?\n\n\n```python\ntrain['Incumbent Party Vote Share'].mean()\n```\n\n\n\n\n    52.46857142857142\n\n\n\nWhat if we guessed this number for every election? How far off would this be on average?\n\n\n```python\n# Arrange y target vectors\ntarget = 'Incumbent Party Vote Share'\ny_train = train[target]\ny_test = test[target]\n```\n\n\n```python\n# Get mean baseline\nprint('Mean Baseline (using 0 features)')\nguess = y_train.mean()\n```\n\n    Mean Baseline (using 0 features)\n\n\n\n```python\nguess\n```\n\n\n\n\n    52.46857142857142\n\n\n\n\n```python\ntrain['Incumbent Party Vote Share'].mean()\n```\n\n\n\n\n    52.46857142857142\n\n\n\n\n```python\ntrain['Average Recent Growth in Personal Incomes'].mean()\n```\n\n\n\n\n    2.0935714285714284\n\n\n\n\n```python\n# Train Error\nfrom sklearn.metrics import mean_absolute_error\ny_pred = [guess] * len(y_train)\nmae = mean_absolute_error(y_train, y_pred)\nprint(f'Train Error (1952-2004 elections): {mae:.2f} percentage points')\n```\n\n    Train Error (1952-2004 elections): 4.85 percentage points\n\n\n\n```python\nlen(y_train)\n```\n\n\n\n\n    14\n\n\n\n\n```python\n# Test Error\ny_pred = [guess] * len(y_test)\nmae = mean_absolute_error(y_test, y_pred)\nprint(f'Test Error (2008-16 elections): {mae:.2f} percentage points')\n```\n\n    Test Error (2008-16 elections): 3.63 percentage points\n\n\n### Simple regression (1 feature)\n\nMake a scatterplot of the relationship between 1 feature and the target.\n\nWe'll use an economic feature: Average Recent Growth in Personal Incomes. (\"Bread\")\n\n\n```python\nimport pandas as pd\nimport plotly.express as px\n\npx.scatter(\n    train,\n    x='Average Recent Growth in Personal Incomes',\n    y='Incumbent Party Vote Share',\n    text='Year',\n    title='US Presidential Elections, 1952-2004',\n    trendline='ols',  # Ordinary Least Squares\n)\n```\n\n\n
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                                        \n\n\n1952 & 1968 are outliers: The incumbent party got fewer votes than predicted by the regression. What do you think could explain those years? We'll come back to this soon, but first...\n\nUse scikit-learn to fit the simple regression with one feature.\n\nFollow the [5 step process](https://jakevdp.github.io/PythonDataScienceHandbook/05.02-introducing-scikit-learn.html#Basics-of-the-API), and refer to [Scikit-Learn LinearRegression documentation](https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LinearRegression.html).\n\n\n```python\n# 1. Import the appropriate estimator class from Scikit-Learn\nfrom sklearn.linear_model import LinearRegression\n```\n\n\n```python\n# 2. Instantiate this class\nmodel = LinearRegression()\n```\n\n\n```python\n# 3. Arrange X features matrices (already did y target vectors)\nfeatures = ['Average Recent Growth in Personal Incomes']\nX_train = train[features]\nX_test = test[features]\nprint(f'Linear Regression, dependent on: {features}')\n```\n\n    Linear Regression, dependent on: ['Average Recent Growth in Personal Incomes']\n\n\n\n```python\n# 4. Fit the model\nmodel.fit(X_train, y_train)\ny_pred = model.predict(X_train)\nmae = mean_absolute_error(y_train, y_pred)\nprint(f'Train Error: {mae:.2f} percentage points')\n```\n\n    Train Error: 2.65 percentage points\n\n\n\n```python\n# 5. Apply the model to new data\ny_pred = model.predict(X_test)\nmae = mean_absolute_error(y_test, y_pred)\nprint(f'Test Error: {mae:.2f} percentage points')\n```\n\n    Test Error: 1.80 percentage points\n\n\nHow does the error compare to the baseline?\n\n### Multiple regression (2 features)\n\nMake a scatterplot of the relationship between 2 features and the target.\n\nWe'll add another feature: US Military Fatalities per Million. (\"Peace\" or the lack thereof.)\n\nRotate the scatterplot to explore the data. What's different about 1952 & 1968?\n\n\n```python\npx.scatter_3d(\n    train,\n    x='Average Recent Growth in Personal Incomes', \n    y='US Military Fatalities per Million', \n    z='Incumbent Party Vote Share', \n    text='Year', \n    title='US Presidential Elections, 1952-2004'\n)\n```\n\n\n
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                                        \n\n\nUse scikit-learn to fit a multiple regression with two features.\n\n\n```python\n# TODO: Complete this cell\n\n# Re-arrange X features matrices\nfeatures = ['Average Recent Growth in Personal Incomes', \n            'US Military Fatalities per Million']\nprint(f'Linear Regression, dependent on: {features}')\nX_train = train[features]\nX_test = test[features]\n```\n\n    Linear Regression, dependent on: ['Average Recent Growth in Personal Incomes', 'US Military Fatalities per Million']\n\n\n\n```python\n# TODO: Fit the model\nmodel.fit(X_train, y_train)\n\n```\n\n\n\n\n    LinearRegression(copy_X=True, fit_intercept=True, n_jobs=None, normalize=False)\n\n\n\n\n```python\n# TODO: Apply the model to new data\ny_pred = model.predict(X_train)\n\n```\n\n\n```python\nmae = mean_absolute_error(y_train, y_pred)\nprint(f'Train Error: {mae:.2f} percentage points')\n```\n\n    Train Error: 1.33 percentage points\n\n\n\n```python\ny_pred = model.predict(X_test)\nmae = mean_absolute_error(y_test, y_pred)\nprint(f'Test Error: {mae:.2f} percentage points')\n```\n\n    Test Error: 1.63 percentage points\n\n\nHow does the error compare to the prior model?\n\n### Plot the plane of best fit\n\nFor a regression with 1 feature, we plotted the line of best fit in 2D. \n\n(There are many ways to do this. Plotly Express's `scatter` function makes it convenient with its `trendline='ols'` parameter.)\n\nFor a regression with 2 features, we can plot the plane of best fit in 3D!\n\n(Plotly Express has a `scatter_3d` function but it won't plot the plane of best fit for us. But, we can write our own function, with the same \"function signature\" as the Plotly Express API.)\n\n\n```python\nimport itertools\nimport numpy as np\nimport plotly.express as px\nimport plotly.graph_objs as go\nfrom sklearn.linear_model import LinearRegression\n\ndef regression_3d(df, x, y, z, num=100, **kwargs):\n    \"\"\"\n    Visualize linear regression in 3D: 2 features + 1 target\n    \n    df : Pandas DataFrame\n    x : string, feature 1 column in df\n    y : string, feature 2 column in df\n    z : string, target column in df\n    num : integer, number of quantiles for each feature\n    \"\"\"\n    \n    # Plot data\n    fig = px.scatter_3d(df, x, y, z, **kwargs)\n    \n    # Fit Linear Regression\n    features = [x, y]\n    target = z\n    model = LinearRegression()\n    model.fit(df[features], df[target])    \n    \n    # Define grid of coordinates in the feature space\n    xmin, xmax = df[x].min(), df[x].max()\n    ymin, ymax = df[y].min(), df[y].max()\n    xcoords = np.linspace(xmin, xmax, num)\n    ycoords = np.linspace(ymin, ymax, num)\n    coords = list(itertools.product(xcoords, ycoords))\n    \n    # Make predictions for the grid\n    predictions = model.predict(coords)\n    Z = predictions.reshape(num, num).T\n    \n    # Plot predictions as a 3D surface (plane)\n    fig.add_trace(go.Surface(x=xcoords, y=ycoords, z=Z))\n    \n    return fig\n```\n\n\n```python\nregression_3d(\n    train,\n    x='Average Recent Growth in Personal Incomes', \n    y='US Military Fatalities per Million', \n    z='Incumbent Party Vote Share', \n    text='Year', \n    title='US Presidential Elections, 1952-2004'\n)\n```\n\n\n
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                                        \n\n\nWhere are 1952 & 1968 in relation to the plane? Which elections are the biggest outliers now?\n\nRoll over points on the plane to see predicted incumbent party vote share (z axis), dependent on personal income growth (x axis) and military fatatlies per capita (y axis).\n\n### Get and interpret coefficients\n\nDuring the previous module, we got the simple regression's coefficient and intercept. We plugged these numbers into an equation for the line of best fit, in slope-intercept form: $y = mx + b$\n\nLet's review this objective, but now for multiple regression.\n\nWhat's the equation for the plane of best fit?\n\n$y = \\beta_0 + \\beta_1x_1 + \\beta_2x_2$\n\nCan you relate the intercept and coefficients to what you see in the plot above?\n\n\n```python\nmodel.intercept_, model.coef_\n```\n\n\n\n\n    (46.25489966153873, array([ 3.59004735, -0.05315709]))\n\n\n\n\n```python\nbeta0 = model.intercept_\nbeta1, beta2 = model.coef_\nprint(f'y = {beta0} + {beta1}x1 + {beta2}x2')\n```\n\n    y = 46.25489966153873 + 3.5900473494560536x1 + -0.05315709351049327x2\n\n\n\n```python\n# This is easier to read\nprint('Intercept', model.intercept_)\ncoefficients = pd.Series(model.coef_, features)\nprint(coefficients.to_string())\n```\n\n    Intercept 46.25489966153873\n    Average Recent Growth in Personal Incomes    3.590047\n    US Military Fatalities per Million          -0.053157\n\n\nOne of the coefficients is positive, and the other is negative. What does this mean?\n\nLet's look at some scenarios. We'll see that one unit's change in an independent variable results in a coefficient worth of change in the dependent variable.\n\nWhat does the model predict if income growth=0%, fatalities=0\n\n\n```python\nmodel.predict([[0, 0]])\n```\n\n\n\n\n    array([46.25489966])\n\n\n\nIncome growth = 1% (fatalities = 0)\n\n\n```python\nmodel.predict([[1, 0]])\n```\n\n\n\n\n    array([49.84494701])\n\n\n\nThe difference between these predictions = ? \n\n\n```python\nmodel.predict([[1, 0]]) - model.predict([[0, 0]])\n```\n\n\n\n\n    array([3.59004735])\n\n\n\nWhat if... income growth = 2% (fatalities = 0)\n\n\n```python\nmodel.predict([[2, 0]])\n```\n\n\n\n\n    array([53.43499436])\n\n\n\nThe difference between these predictions = ?\n\n\n```python\nmodel.predict([[2, 0]]) - model.predict([[1, 0]])\n```\n\n\n\n\n    array([3.59004735])\n\n\n\nWhat if... (income growth=2%) fatalities = 100\n\n\n```python\nmodel.predict([[2, 100]])\n```\n\n\n\n\n    array([48.11928501])\n\n\n\nThe difference between these predictions = ?\n\n\n```python\nmodel.predict([[2, 100]]) - model.predict([[2, 0]])\n```\n\n\n\n\n    array([-5.31570935])\n\n\n\nWhat if income growth = 3% (fatalities = 100)\n\n\n```python\nmodel.predict([[3, 100]])\n```\n\n\n\n\n    array([51.70933236])\n\n\n\nThe difference between these predictions = ?\n\n\n```python\nmodel.predict([[3, 100]]) - model.predict([[2, 100]])\n```\n\n\n\n\n    array([3.59004735])\n\n\n\nWhat if (income growth = 3%) fatalities = 200\n\n\n```python\nmodel.predict([[3, 200]])\n```\n\n\n\n\n    array([46.39362301])\n\n\n\nThe difference between these predictions = ?\n\n\n```python\nmodel.predict([[3, 200]]) - model.predict([[3, 100]])\n```\n\n\n\n\n    array([-5.31570935])\n\n\n\n## Challenge\n\nIn your assignment, you'll fit a Linear Regression with at least 2 features.\n\n# Understand how ordinary least squares regression minimizes the sum of squared errors\n\n## Overview\n\nSo far, we've evaluated our models by their absolute error. It's an intuitive metric for regression problems.\n\nHowever, ordinary least squares doesn't directly minimize absolute error. Instead, it minimizes squared error.\n\n\n\n\nIn this section, we'll introduce two new regression metrics: \n\n- Squared error\n- $R^2$\n\n\nWe'll demostrate two possible methods to minimize squared error:\n\n- Guess & check\n- Linear Algebra\n\n## Follow Along\n\n### Guess & Check\n\nThis function visualizes squared errors. We'll go back to simple regression with 1 feature, because it's much easier to visualize.\n\nUse the function's m & b parameters to \"fit the model\" manually. Guess & check what values of m & b minimize squared error.\n\n\n```python\nfrom matplotlib.patches import Rectangle\nimport matplotlib.pyplot as plt\nimport numpy as np\nfrom sklearn.metrics import mean_absolute_error, mean_squared_error, r2_score\n\ndef squared_errors(df, feature, target, m, b):\n    \"\"\"\n    Visualize linear regression, with squared errors,\n    in 2D: 1 feature + 1 target.\n    \n    Use the m & b parameters to \"fit the model\" manually.\n    \n    df : Pandas DataFrame\n    feature : string, feature column in df\n    target : string, target column in df\n    m : numeric, slope for linear equation\n    b : numeric, intercept for linear requation\n    \"\"\"\n    \n    # Plot data\n    fig = plt.figure(figsize=(7,7))\n    ax = plt.axes()\n    df.plot.scatter(feature, target, ax=ax)\n    \n    # Make predictions\n    x = df[feature]\n    y = df[target]\n    y_pred = m*x + b\n    \n    # Plot predictions\n    ax.plot(x, y_pred)\n    \n    # Plot squared errors\n    xmin, xmax = ax.get_xlim()\n    ymin, ymax = ax.get_ylim()\n    scale = (xmax-xmin)/(ymax-ymin)\n    for x, y1, y2 in zip(x, y, y_pred):\n        bottom_left = (x, min(y1, y2))\n        height = abs(y1 - y2)\n        width = height * scale\n        ax.add_patch(Rectangle(xy=bottom_left, width=width, height=height, alpha=0.1))\n    \n    # Print regression metrics\n    mse = mean_squared_error(y, y_pred)\n    rmse = np.sqrt(mse)\n    mae = mean_absolute_error(y, y_pred)\n    r2 = r2_score(y, y_pred)\n    print('Mean Squared Error:', mse)\n    print('Root Mean Squared Error:', rmse)\n    print('Mean Absolute Error:', mae)\n    print('R^2:', r2)\n```\n\nHere's what the mean baseline looks like:\n\n\n```python\nfeature = 'Average Recent Growth in Personal Incomes'\nsquared_errors(train, feature, target, m=0, b=y_train.mean())\n```\n\nNotice that $R^2$ is exactly zero. \n\n[$R^2$ represents the proportion of the variance for a dependent variable that is explained by the independent variable(s).](https://en.wikipedia.org/wiki/Coefficient_of_determination)\n\nThe mean baseline uses zero independent variables and explains none of the variance in the dependent variable, so its $R^2$ score is zero.\n\nThe highest possible $R^2$ score is 1. The lowest possible *Train* $R^2$ score with ordinary least squares regression is 0.\n\nIn this demo, it's possible to get a negative Train $R^2$, if you manually set values of m & b that are worse than the mean baseline. But that wouldn't happen in the real world.\n\nHowever, in the real world, it _is_ possible to get a negative *Test/Validation* $R^2$. It means that your *Test/Validation* predictions are worse than if you'd constantly predicted the mean of the *Test/Validation* set.\n\n---\n\nNow that we've visualized the squared errors for the mean baseline, let's guess & check some better values for the m & b parameters:\n\n\n```python\nsquared_errors(train, feature, target, m=3, b=46)\n```\n\nYou can run the function repeatedly, with different values for m & b.\n\nHow do you interpret each metric you see?\n\n- Mean Squared Error\n- Root Mean Squared Error\n- Mean Absolute Error\n- $R^2$\n\nDoes guess & check really get used in machine learning? Sometimes! Some complex functions are hard to minimize, so we use a sophisticated form of guess & check called \"gradient descent\", which you'll learn about in Unit 4.\n\nFortunately, we don't need to use guess & check for ordinary least squares regression. We have a solution, using linear algebra!\n\n\n### Linear Algebra\n\nThe same result that is found by minimizing the sum of the squared errors can be also found through a linear algebra process known as the \"Least Squares Solution:\"\n\n\\begin{align}\n\\hat{\\beta} = (X^{T}X)^{-1}X^{T}y\n\\end{align}\n\nBefore we can work with this equation in its linear algebra form we have to understand how to set up the matrices that are involved in this equation. \n\n#### The $\\beta$ vector\n\nThe $\\beta$ vector represents all the parameters that we are trying to estimate, our $y$ vector and $X$ matrix values are full of data from our dataset. The $\\beta$ vector holds the variables that we are solving for: $\\beta_0$ and $\\beta_1$\n\nNow that we have all of the necessary parts we can set them up in the following equation:\n\n\\begin{align}\ny = X \\beta + \\epsilon\n\\end{align}\n\nSince our $\\epsilon$ value represents **random** error we can assume that it will equal zero on average.\n\n\\begin{align}\ny = X \\beta\n\\end{align}\n\nThe objective now is to isolate the $\\beta$ matrix. We can do this by pre-multiplying both sides by \"X transpose\" $X^{T}$.\n\n\\begin{align}\nX^{T}y =  X^{T}X \\beta\n\\end{align}\n\nSince anything times its transpose will result in a square matrix, if that matrix is then an invertible matrix, then we should be able to multiply both sides by its inverse to remove it from the right hand side. (We'll talk tomorrow about situations that could lead to $X^{T}X$ not being invertible.)\n\n\\begin{align}\n(X^{T}X)^{-1}X^{T}y =  (X^{T}X)^{-1}X^{T}X \\beta\n\\end{align}\n\nSince any matrix multiplied by its inverse results in the identity matrix, and anything multiplied by the identity matrix is itself, we are left with only $\\beta$ on the right hand side:\n\n\\begin{align}\n(X^{T}X)^{-1}X^{T}y = \\hat{\\beta}\n\\end{align}\n\nWe will now call it \"beta hat\" $\\hat{\\beta}$ because it now represents our estimated values for $\\beta_0$ and $\\beta_1$\n\n#### Lets calculate our $\\beta$ parameters with numpy!\n\n\n```python\n# This is NOT something you'll be tested on. It's just a demo.\n\n# X is a matrix. Add column of constants for fitting the intercept.\ndef add_constant(X):\n    constant = np.ones(shape=(len(X),1))\n    return np.hstack((constant, X))\nX = add_constant(train[features].values)\nprint('X')\nprint(X)\n\n# y is a column vector\ny = train[target].values[:, np.newaxis]\nprint('y')\nprint(y)\n\n# Least squares solution in code\nX_transpose = X.T\nX_transpose_X = X_transpose @ X\nX_transpose_X_inverse = np.linalg.inv(X_transpose_X)\nX_transpose_y = X_transpose @ y\nbeta_hat = X_transpose_X_inverse @ X_transpose_y\n\nprint('Beta Hat')\nprint(beta_hat)\n```\n\n    X\n    [[  1.     2.4  190.  ]\n     [  1.     2.89   0.  ]\n     [  1.     0.85   0.  ]\n     [  1.     4.21   1.  ]\n     [  1.     3.02 146.  ]\n     [  1.     3.62   0.  ]\n     [  1.     1.08   2.  ]\n     [  1.    -0.39   0.  ]\n     [  1.     3.86   0.  ]\n     [  1.     2.27   0.  ]\n     [  1.     0.38   0.  ]\n     [  1.     1.04   0.  ]\n     [  1.     2.36   0.  ]\n     [  1.     1.72   4.  ]]\n    y\n    [[44.6 ]\n     [57.76]\n     [49.91]\n     [61.34]\n     [49.6 ]\n     [61.79]\n     [48.95]\n     [44.7 ]\n     [59.17]\n     [53.94]\n     [46.55]\n     [54.74]\n     [50.27]\n     [51.24]]\n    Beta Hat\n    [[46.25489966]\n     [ 3.59004735]\n     [-0.05315709]]\n\n\n\n```python\n# Scikit-learn gave the exact same results!\nmodel.intercept_, model.coef_\n```\n\n\n\n\n    (46.25489966153873, array([ 3.59004735, -0.05315709]))\n\n\n\n# Define overfitting/underfitting and the bias/variance tradeoff\n\n## Overview\n\nRead [_Python Data Science Handbook,_ Chapter 5.3](https://jakevdp.github.io/PythonDataScienceHandbook/05.03-hyperparameters-and-model-validation.html#The-Bias-variance-trade-off). Jake VanderPlas explains overfitting & underfitting:\n\n> Fundamentally, the question of \"the best model\" is about finding a sweet spot in the tradeoff between bias and variance. Consider the following figure, which presents two regression fits to the same dataset:\n> \n>\n>\n> The model on the left attempts to find a straight-line fit through the data. Because the data are intrinsically more complicated than a straight line, the straight-line model will never be able to describe this dataset well. Such a model is said to _underfit_ the data: that is, it does not have enough model flexibility to suitably account for all the features in the data; another way of saying this is that the model has high _bias_.\n>\n> The model on the right attempts to fit a high-order polynomial through the data. Here the model fit has enough flexibility to nearly perfectly account for the fine features in the data, but even though it very accurately describes the training data, its precise form seems to be more reflective of the particular noise properties of the data rather than the intrinsic properties of whatever process generated that data. Such a model is said to _overfit_ the data: that is, it has so much model flexibility that the model ends up accounting for random errors as well as the underlying data distribution; another way of saying this is that the model has high _variance_.\n\nVanderPlas goes on to connect these concepts to the \"bias/variance tradeoff\":\n\n> From the scores associated with these two models, we can make an observation that holds more generally:\n>\n>- For high-bias models, the performance of the model on the validation set is similar to the performance on the training set.\n>\n>- For high-variance models, the performance of the model on the validation set is far worse than the performance on the training set.\n>\n> If we imagine that we have some ability to tune the model complexity, we would expect the training score and validation score to behave as illustrated in the following figure:\n>\n>\n>\n> The diagram shown here is often called a validation curve, and we see the following essential features:\n>\n>- The training score is everywhere higher than the validation score. This is generally the case: the model will be a better fit to data it has seen than to data it has not seen.\n>- For very low model complexity (a high-bias model), the training data is under-fit, which means that the model is a poor predictor both for the training data and for any previously unseen data.\n>- For very high model complexity (a high-variance model), the training data is over-fit, which means that the model predicts the training data very well, but fails for any previously unseen data.\n>- For some intermediate value, the validation curve has a maximum. This level of complexity indicates a suitable trade-off between bias and variance.\n>\n>The means of tuning the model complexity varies from model to model.\n\nSo far, our only \"means of tuning the model complexity\" has been selecting one feature or two features for our linear regression models. But we'll quickly start to select more features, and more complex models, with more \"hyperparameters.\"\n\nThis is just a first introduction to underfitting & overfitting. We'll continue to learn about this topic all throughout this unit.\n\n## Follow Along\n\nLet's make our own Validation Curve, by tuning a new type of model complexity: polynomial degrees in a linear regression.\n\nGo back to the the NYC Tribeca condo sales data\n\n\n```python\n# Read NYC Tribeca condo sales data, from first 4 months of 2019.\n# Dataset has 90 rows, 9 columns.\ndf = pd.read_csv(DATA_PATH+'condos/tribeca.csv')\nassert df.shape == (90, 9)\n\n# Arrange X features matrix & y target vector\nfeatures = ['GROSS_SQUARE_FEET']\ntarget = 'SALE_PRICE'\nX = df[features]\ny = df[target]\n```\n\nDo random [train/test split](https://scikit-learn.org/stable/modules/generated/sklearn.model_selection.train_test_split.html)\n\n\n```python\nfrom sklearn.model_selection import train_test_split\nX_train, X_test, y_train, y_test = train_test_split(X, y, random_state=11)\n```\n\nRepeatedly fit increasingly complex models, and keep track of the scores\n\n\n```python\nfrom IPython.display import display, HTML\nfrom sklearn.linear_model import LinearRegression\nfrom sklearn.metrics import r2_score\nfrom sklearn.pipeline import make_pipeline\nfrom sklearn.preprocessing import PolynomialFeatures\n\n\n# Credit for PolynomialRegression: Jake VanderPlas, Python Data Science Handbook, Chapter 5.3\n# https://jakevdp.github.io/PythonDataScienceHandbook/05.03-hyperparameters-and-model-validation.html#Validation-curves-in-Scikit-Learn\ndef PolynomialRegression(degree=2, **kwargs):\n    return make_pipeline(PolynomialFeatures(degree), \n                         LinearRegression(**kwargs))\n\n\npolynomial_degrees = range(1, 10, 2)\ntrain_r2s = []\ntest_r2s = []\n\nfor degree in polynomial_degrees:\n    model = PolynomialRegression(degree)\n    display(HTML(f'Polynomial degree={degree}'))\n    \n    model.fit(X_train, y_train)\n    train_r2 = model.score(X_train, y_train)\n    test_r2 = model.score(X_test, y_test)\n    display(HTML(f'Train R2 {train_r2:.2f}'))\n    display(HTML(f'Test R2 {test_r2:.2f}'))\n\n    plt.scatter(X_train, y_train, color='blue', alpha=0.5)\n    plt.scatter(X_test, y_test, color='red', alpha=0.5)\n    plt.xlabel(features)\n    plt.ylabel(target)\n    \n    x_domain = np.linspace(X.min(), X.max())\n    curve = model.predict(x_domain)\n    plt.plot(x_domain, curve, color='blue')\n    plt.show()\n    display(HTML('
                                        '))\n    \n    train_r2s.append(train_r2)\n    test_r2s.append(test_r2)\n    \ndisplay(HTML('Validation Curve'))\nplt.plot(polynomial_degrees, train_r2s, color='blue', label='Train')\nplt.plot(polynomial_degrees, test_r2s, color='red', label='Test')\nplt.xlabel('Model Complexity (Polynomial Degree)')\nplt.ylabel('R^2 Score')\nplt.legend()\nplt.show()\n```\n\nAs model complexity increases, what happens to Train $R^2$ and Test $R^2$?\n\n# Review\n\nIn your assignment, you'll continue to **predict how much it costs to rent an apartment in NYC,** using the dataset from renthop.com.\n\n\n- Do train/test split. Use data from April & May 2016 to train. Use data from June 2016 to test.\n- Engineer at least two new features.\n- Fit a linear regression model with at least two features.\n- Get the model's coefficients and intercept.\n- Get regression metrics RMSE, MAE, and $R^2$, for both the train and test sets.\n\nYou've been provided with a separate notebook for your assignment, which has all the instructions and stretch goals. What's the best test MAE you can get? Share your score and features used with your cohort on Slack!\n\n# Sources\n\n#### Train/Test Split\n- James, Witten, Hastie, Tibshirani, [_An Introduction to Statistical Learning_](http://faculty.marshall.usc.edu/gareth-james/ISL/), Chapter 2.2, Assessing Model Accuracy\n- Hyndman, Athanasopoulos, [_Forecasting,_ Chapter 3.4,](https://otexts.com/fpp2/accuracy.html) Evaluating forecast accuracy\n- Rachel Thomas, [How (and why) to create a good validation set](https://www.fast.ai/2017/11/13/validation-sets/)\n\n#### Bias-Variance Tradeoff\n- Jake VanderPlas, [_Python Data Science Handbook,_ Chapter 5.3](https://jakevdp.github.io/PythonDataScienceHandbook/05.03-hyperparameters-and-model-validation.html#The-Bias-variance-trade-off), Hyperparameters and Model Validation\n- StatQuest, [Machine Learning Fundamentals: Bias and Variance](https://youtu.be/EuBBz3bI-aA) (6.5 minutes)\n\n#### \"Bread and Peace\" Background\n- Douglas Hibbs, [Background Information on the \u2018Bread and Peace\u2019 Model of Voting in Postwar US Presidential Elections](https://douglas-hibbs.com/background-information-on-bread-and-peace-voting-in-us-presidential-elections/)\n- Nate Silver, [What Do Economic Models Really Tell Us About Elections?](https://fivethirtyeight.com/features/what-do-economic-models-really-tell-us-about-elections/)\n\n\n#### \"Bread and Peace\" Data Sources & Definitions\n- 1952-2012: Douglas Hibbs, [2014 lecture at Deakin University Melbourne](http://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf), Slide 40\n- 2016, Vote Share: [The American Presidency Project](https://www.presidency.ucsb.edu/statistics/elections)\n- 2016, Recent Growth in Personal Incomes: [The 2016 election economy: the \"Bread and Peace\" model final forecast](https://angrybearblog.com/2016/11/the-2016-election-economy-the-bread-and-peace-model-final-forecast.html)\n- 2016, US Military Fatalities: Assumption that Afghanistan War fatalities in 2012-16 occured at the same rate as 2008-12\n\n> Fatalities denotes the cumulative number of American military fatalities per millions of US population the in Korea, Vietnam, Iraq and Afghanistan wars during the presidential terms preceding the 1952, 1964, 1968, 1976 and 2004, 2008 and 2012 elections. \u2014[Hibbs](http://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf), Slide 33\n\n\n```python\n\n```\n", "meta": {"hexsha": "4131628dce84e31c537e02ecd1f20beb87236b03", "size": 846721, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "module2-regression-2/LS_DS_212.ipynb", "max_stars_repo_name": "nimu77/DS-Unit-2-Linear-Models", "max_stars_repo_head_hexsha": "ebcd0f82a4a3aea7db16433c0d95f248687108ce", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "module2-regression-2/LS_DS_212.ipynb", "max_issues_repo_name": "nimu77/DS-Unit-2-Linear-Models", "max_issues_repo_head_hexsha": "ebcd0f82a4a3aea7db16433c0d95f248687108ce", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "module2-regression-2/LS_DS_212.ipynb", "max_forks_repo_name": "nimu77/DS-Unit-2-Linear-Models", "max_forks_repo_head_hexsha": "ebcd0f82a4a3aea7db16433c0d95f248687108ce", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 51.1397596183, "max_line_length": 197004, "alphanum_fraction": 0.659895054, "converted": true, "num_tokens": 11725, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.6477982043529715, "lm_q1q2_score": 0.41266740478193914}}
{"text": "# Visualization: Calculating with scales\n\n\n```julia\n] activate .\n```\n\n    \u001b[32m\u001b[1mActivating\u001b[22m\u001b[39m environment at `C:\\Users\\carsten\\.julia\\dev\\StableDQMC.jl\\notebooks\\scales_visualizations\\Project.toml`\n\n\n## Using `String` and `Char` (type piracy)\n\n\n```julia\nusing Latexify\n```\n\n\n```julia\n# const small, medium, large = '\u2093', 'x', 'X'\nconst tiny, small, medium, large = 't','s', 'M', 'L'\n# const small, medium, large = '\ud83d\udf84', '\u25cf', '\u2b24'\n# const small, medium, large = '\u25aa', '\u25a0', '\u2b1b'\n```\n\n\n\n\n    ('t', 's', 'M', 'L')\n\n\n\n\n```julia\nconst scale2value = Dict(tiny => 1, small => 2, medium => 3, large => 4)\n```\n\n    WARNING: redefining constant scale2value\n\n\n\n\n\n    Dict{Char,Int64} with 4 entries:\n      'M' => 3\n      'L' => 4\n      's' => 2\n      't' => 1\n\n\n\n\n```julia\nBase.zero(::Type{String}) = \" \"\nBase.zero(::String) = zero(String)\n\nBase.zero(::Type{Char}) = ' '\nBase.zero(::Char) = zero(Char)\n\nfunction Base.:+(x::String, y::String)\n    if occursin(zero(String), x)\n        if occursin(zero(String), y)\n            return zero(String)\n        else\n            return y\n        end\n    elseif occursin(zero(String), y)\n        return x\n    else\n       return sum(scale2value[c] for c in x) < sum(Int.(collect(y))) ? y : x #mean(x) < mean(y) ? y : x \n    end\nend\n\nfunction Base.:+(x::Char, y::Char)\n    if x == zero(Char)\n        if y == zero(Char)\n            return zero(Char)\n        else\n            return y\n        end\n    elseif y == zero(Char)\n        return x\n    else\n       return scale2value[x] > scale2value[y] ? x : y\n    end\nend\n```\n\n\n```julia\nusing LinearAlgebra\nD = diagm(0 => [large, medium, small])\n```\n\n\n\n\n    3\u00d73 Array{Char,2}:\n     'L'  ' '  ' '\n     ' '  'M'  ' '\n     ' '  ' '  's'\n\n\n\n\n```julia\nset_default(convert_unicode = false)\n```\n\n\n```julia\nD |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{ccc}\nL &   &   \\\\\n  & M &   \\\\\n  &   & s \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n\n```julia\nD * [large, medium, small] |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{c}\nLL \\\\\nMM \\\\\nss \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n\n```julia\nD * [small, small, small] |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{c}\nLs \\\\\nMs \\\\\nss \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n\n```julia\nO = zeros(Char, 3,3)\nO |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{ccc}\n  &   &   \\\\\n  &   &   \\\\\n  &   &   \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n\n```julia\nU = fill(small, 3,3)\nU |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{ccc}\ns & s & s \\\\\ns & s & s \\\\\ns & s & s \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n\n```julia\nU * D |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{ccc}\nsL & sM & ss \\\\\nsL & sM & ss \\\\\nsL & sM & ss \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n\n```julia\nD * U * D |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{ccc}\nLsL & LsM & Lss \\\\\nMsL & MsM & Mss \\\\\nssL & ssM & sss \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n\n```julia\nprint(ans)\n```\n\n    \\begin{equation}\n    \\left[\n    \\begin{array}{ccc}\n    LsL & LsM & Lss \\\\\n    MsL & MsM & Mss \\\\\n    ssL & ssM & sss \\\\\n    \\end{array}\n    \\right]\n    \\end{equation}\n\n\n\n```julia\nU*D |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{ccc}\nsL & sM & ss \\\\\nsL & sM & ss \\\\\nsL & sM & ss \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n#### 4x4\n\n\n```julia\nusing LinearAlgebra\nD = diagm(0 => [large, medium, small, tiny])\n```\n\n\n\n\n    4\u00d74 Array{Char,2}:\n     'L'  ' '  ' '  ' '\n     ' '  'M'  ' '  ' '\n     ' '  ' '  's'  ' '\n     ' '  ' '  ' '  't'\n\n\n\n\n```julia\nU = fill(small, 4,4)\nU |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{cccc}\ns & s & s & s \\\\\ns & s & s & s \\\\\ns & s & s & s \\\\\ns & s & s & s \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n\n```julia\nD*U*D |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{cccc}\nLsL & LsM & Lss & Lst \\\\\nMsL & MsM & Mss & Mst \\\\\nssL & ssM & sss & sst \\\\\ntsL & tsM & tss & tst \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n\n```julia\nD\n```\n\n\n\n\n    4\u00d74 Array{Char,2}:\n     'L'  ' '  ' '  ' '\n     ' '  'M'  ' '  ' '\n     ' '  ' '  's'  ' '\n     ' '  ' '  ' '  't'\n\n\n\n\n```julia\nU*D*U |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{cccc}\nsts & sts & sts & sts \\\\\nsts & sts & sts & sts \\\\\nsts & sts & sts & sts \\\\\nsts & sts & sts & sts \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n## Using Sympy\n\n\n```julia\nusing SymPy\nusing LinearAlgebra\n```\n\n\n```julia\nX, x, \u2093 = @vars X x \u2093\n```\n\n\n\n\n    (X, x, \u2093)\n\n\n\n\n```julia\n\n```\n\n\n```julia\nD = diagm([X,x,\u2093])\n```\n\n\n\n\n\\[\\left[ \\begin{array}{rrr}X&0&0\\\\0&x&0\\\\0&0&\u2093\\end{array}\\right]\\]\n\n\n\n\n```julia\nU = fill(x, 3,3)\n```\n\n\n\n\n\\[\\left[ \\begin{array}{rrr}x&x&x\\\\x&x&x\\\\x&x&x\\end{array}\\right]\\]\n\n\n\n\n```julia\nU*D\n```\n\n\n\n\n\\[\\left[ \\begin{array}{rrr}X x&x^{2}&x \u2093\\\\X x&x^{2}&x \u2093\\\\X x&x^{2}&x \u2093\\end{array}\\right]\\]\n\n\n\n\n```julia\n\u26ab, \u2b24, \u25cf, \ud83d\udf84 = @vars \u26ab \u2b24 \u25cf \ud83d\udf84\n```\n\n\n\n\n    (\u26ab, \u2b24, \u25cf, \ud83d\udf84)\n\n\n\n\n```julia\nD = diagm([\u26ab, \u2b24, \u25cf, \ud83d\udf84])\n```\n\n\n\n\n\\[\\left[ \\begin{array}{rrrr}\u26ab&0&0&0\\\\0&\u2b24&0&0\\\\0&0&\u25cf&0\\\\0&0&0&\ud83d\udf84\\end{array}\\right]\\]\n\n\n\n\n```julia\nDiagonal([1000, 100, 10, 1]) |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{cccc}\n1000 & 0 & 0 & 0 \\\\\n0 & 100 & 0 & 0 \\\\\n0 & 0 & 10 & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n\n```julia\n\u2b1b, \u25a0, \u25fe, \u25aa = @vars \u2b1b \u25a0 \u25fe \u25aa\nL, M, s, t = @vars L M s t\n```\n\n\n\n\n    (L, M, s, t)\n\n\n\n\n```julia\nd = [\u2b1b, \u25a0, \u25fe, \u25aa]\nd = [L, M, s, t]\n```\n\n\n\n\n\\[ \\left[ \\begin{array}{r}L\\\\M\\\\s\\\\t\\end{array} \\right] \\]\n\n\n\n\n```julia\nD = diagm(d)\n```\n\n\n\n\n\\[\\left[ \\begin{array}{rrrr}L&0&0&0\\\\0&M&0&0\\\\0&0&s&0\\\\0&0&0&t\\end{array}\\right]\\]\n\n\n\n\n```julia\nU = fill(\u25fe, 4,4)\nU = fill(s, 4,4)\n```\n\n\n\n\n\\[\\left[ \\begin{array}{rrrr}s&s&s&s\\\\s&s&s&s\\\\s&s&s&s\\\\s&s&s&s\\end{array}\\right]\\]\n\n\n\n\n```julia\nR = D*U*D\n```\n\n\n\n\n\\[\\left[ \\begin{array}{rrrr}L^{2} s&L M s&L s^{2}&L s t\\\\L M s&M^{2} s&M s^{2}&M s t\\\\L s^{2}&M s^{2}&s^{3}&s^{2} t\\\\L s t&M s t&s^{2} t&s t^{2}\\end{array}\\right]\\]\n\n\n\n\n```julia\nA = U*U\nA*D\n```\n\n\n\n\n\\[\\left[ \\begin{array}{rrrr}4 L s^{2}&4 M s^{2}&4 s^{3}&4 s^{2} t\\\\4 L s^{2}&4 M s^{2}&4 s^{3}&4 s^{2} t\\\\4 L s^{2}&4 M s^{2}&4 s^{3}&4 s^{2} t\\\\4 L s^{2}&4 M s^{2}&4 s^{3}&4 s^{2} t\\end{array}\\right]\\]\n\n\n\n\n```julia\nA = U*D\nU*A\n```\n\n\n\n\n\\[\\left[ \\begin{array}{rrrr}4 L s^{2}&4 M s^{2}&4 s^{3}&4 s^{2} t\\\\4 L s^{2}&4 M s^{2}&4 s^{3}&4 s^{2} t\\\\4 L s^{2}&4 M s^{2}&4 s^{3}&4 s^{2} t\\\\4 L s^{2}&4 M s^{2}&4 s^{3}&4 s^{2} t\\end{array}\\right]\\]\n\n\n\n\n```julia\nU |> latexify |> print\n```\n\n    \\begin{equation}\n    \\left[\n    \\begin{array}{cccc}\n    s & s & s & s \\\\\n    s & s & s & s \\\\\n    s & s & s & s \\\\\n    s & s & s & s \\\\\n    \\end{array}\n    \\right]\n    \\end{equation}\n\n\n\n```julia\nD |> latexify |> print\n```\n\n    \\begin{equation}\n    \\left[\n    \\begin{array}{cccc}\n    L & 0 & 0 & 0 \\\\\n    0 & M & 0 & 0 \\\\\n    0 & 0 & s & 0 \\\\\n    0 & 0 & 0 & t \\\\\n    \\end{array}\n    \\right]\n    \\end{equation}\n\n\n\n```julia\nD*(U*D) |> latexify |> print\n```\n\n    \\begin{equation}\n    \\left[\n    \\begin{array}{cccc}\n    L^2*s & L*M*s & L*s^2 & L*s*t \\\\\n    L*M*s & M^2*s & M*s^2 & M*s*t \\\\\n    L*s^2 & M*s^2 & s^3 & s^2*t \\\\\n    L*s*t & M*s*t & s^2*t & s*t^2 \\\\\n    \\end{array}\n    \\right]\n    \\end{equation}\n\n\n\n```julia\nU + D |> latexify |> print\n```\n\n    \\begin{equation}\n    \\left[\n    \\begin{array}{cccc}\n    L + s & s & s & s \\\\\n    s & M + s & s & s \\\\\n    s & s & 2*s & s \\\\\n    s & s & s & s + t \\\\\n    \\end{array}\n    \\right]\n    \\end{equation}\n\n\n\n```julia\nX = diagm([1e128, 1e64, 1e0, 1e-64])\n# N = rand(4,4)\n# N[diagind(N)] .= 0\n# X = X+N\n```\n\n\n\n\n    4\u00d74 Array{Float64,2}:\n     1.0e128  0.0     0.0  0.0    \n     0.0      1.0e64  0.0  0.0    \n     0.0      0.0     1.0  0.0    \n     0.0      0.0     0.0  1.0e-64\n\n\n\n\n```julia\nF = svd(X)\n```\n\n\n\n\n    SVD{Float64,Float64,Array{Float64,2}}\n    U factor:\n    4\u00d74 Array{Float64,2}:\n     1.0  0.0  0.0  0.0\n     0.0  1.0  0.0  0.0\n     0.0  0.0  1.0  0.0\n     0.0  0.0  0.0  1.0\n    singular values:\n    4-element Array{Float64,1}:\n     1.0e128\n     1.0e64 \n     1.0    \n     1.0e-64\n    Vt factor:\n    4\u00d74 Array{Float64,2}:\n     1.0  0.0  0.0  0.0\n     0.0  1.0  0.0  0.0\n     0.0  0.0  1.0  0.0\n     0.0  0.0  0.0  1.0\n\n\n\n\n```julia\ns = F.S\n```\n\n\n\n\n    4-element Array{Float64,1}:\n     1.0e128\n     1.0e64 \n     1.0    \n     1.0e-64\n\n\n\n\n```julia\nusing GenericSVD\n```\n\n\n```julia\nXb = big.(X);\n```\n\n\n```julia\nFb = svd(Xb)\n```\n\n    \u250c Warning: keyword `alg` ignored in generic svd!\n    \u2514 @ GenericSVD C:\\Users\\carsten\\.julia\\packages\\GenericSVD\\w8oZE\\src\\GenericSVD.jl:12\n\n\n\n\n\n    SVD{BigFloat,BigFloat,Array{BigFloat,2}}\n    U factor:\n    4\u00d74 Array{BigFloat,2}:\n     -1.0   0.0   0.0   0.0\n      0.0  -1.0   0.0   0.0\n      0.0   0.0  -1.0   0.0\n      0.0   0.0   0.0  -1.0\n    singular values:\n    4-element Array{BigFloat,1}:\n     1.000000000000000075174486916518208627471429064352408213482909102357765925242415e+128\n     1.0000000000000000213204190094543968723012578712679649467743338496e+64               \n     1.0                                                                                  \n     9.999999999999999653057388335469287129029766072807834803722132478776394919962261e-65 \n    Vt factor:\n    4\u00d74 Array{BigFloat,2}:\n     -1.0  -0.0  -0.0  -0.0\n     -0.0  -1.0  -0.0  -0.0\n     -0.0  -0.0  -1.0  -0.0\n     -0.0  -0.0  -0.0  -1.0\n\n\n\n\n```julia\nsb = Fb.S\n```\n\n\n\n\n    4-element Array{BigFloat,1}:\n     1.000000000000000075174486916518208627471429064352408213482909102357765925242415e+128\n     1.0000000000000000213204190094543968723012578712679649467743338496e+64               \n     1.0                                                                                  \n     9.999999999999999653057388335469287129029766072807834803722132478776394919962261e-65 \n\n\n\n\n```julia\ns - sb\n```\n\n\n\n\n    4-element Array{BigFloat,1}:\n     0.0\n     0.0\n     0.0\n     0.0\n\n\n\n\n```julia\nusing StableDQMC\n```\n\n\n```julia\nsvd_inv_one_plus(F).S - svd_inv_one_plus(Fb).S\n```\n\n\n\n\n    4-element Array{BigFloat,1}:\n      1.055257420480776903302833588116958909651870865180872244893734331233485866027102e-146\n     -1.33738421569986748693560322601556473903283452878299156314954071924493402990837e-81  \n      0.0                                                                                  \n      1.000000000000032856963321712779388680454016569740087535451726237621195734616864e-64 \n\n\n\n\n```julia\nsvd_inv_one_plus(F).S\n```\n\n\n\n\n    4-element Array{Float64,1}:\n     9.999999999999999e-129\n     1.0e-64               \n     0.5                   \n     1.0                   \n\n\n\n\n```julia\nFb.U'/Fb.V\n```\n\n\n\n\n    4\u00d74 Array{BigFloat,2}:\n      1.0  -0.0  -0.0  -0.0\n     -0.0   1.0  -0.0  -0.0\n     -0.0  -0.0   1.0  -0.0\n     -0.0  -0.0  -0.0   1.0\n\n\n\n\n```julia\n1 / 1.0000001\n```\n\n\n\n\n    0.9999999000000099\n\n\n\n\n```julia\nDiagonal(Fb.S)\n```\n\n\n\n\n    4\u00d74 Diagonal{BigFloat,Array{BigFloat,1}}:\n     1.0e+128   \u22c5        \u22c5    \u22c5     \n      \u22c5        1.0e+64   \u22c5    \u22c5     \n      \u22c5         \u22c5       1.0   \u22c5     \n      \u22c5         \u22c5        \u22c5   1.0e-64\n\n\n\n\n```julia\nFb.U'/Fb.V + Diagonal(Fb.S)\n```\n\n    search:\n    \n    Couldn't find \u001b[36msetprecision\u001b[39m\n    \u001b[36mFb.U'/Fb.V + Diagonal(Fb.S)\u001b[39m\n    Perhaps you meant setprecision\n\n\n\n\n\nNo documentation found.\n\nBinding `setprecision Fb.U'/Fb.V + Diagonal(Fb.S)` does not exist.\n\n\n\n\n\n```julia\nFtmp = svd(F.U'/F.V + Diagonal(F.S));\n```\n\n\n```julia\n1 ./ Ftmp.S\n```\n\n\n\n\n    4-element Array{Float64,1}:\n     9.999999999999999e-129\n     1.0e-64               \n     0.5                   \n     1.0                   \n\n\n\n\n```julia\nx = big\"1.0000000000000000000000000000000000000000000000000000000000000000000000001\"\n```\n\n\n\n\n    1.000000000000000000000000000000000000000000000000000000000000000000000000100007\n\n\n\n\n```julia\ny = big\"1.0000000000000000000000000000000000000000000000000000000000000000000000000\"\n```\n\n\n\n\n    1.0\n\n\n\n\n```julia\nreldiff(x,y) = abs(x - y)/abs(x)\n```\n\n\n\n\n    reldiff (generic function with 1 method)\n\n\n\n\n```julia\nreldiff(x,y)\n```\n\n\n\n\n    1.00006831867993668761973954571228627033523284220172138438574486184552293070619e-73\n\n\n\n\n```julia\nreldiff(1/x,1/y)\n```\n\n\n\n\n    1.000068318679936687619739545712286270335232842201721384385744861845522930906228e-73\n\n\n\n\n```julia\nDp = diagm([L, M, s, s])\n```\n\n\n\n\n\\[\\left[ \\begin{array}{rrrr}L&0&0&0\\\\0&M&0&0\\\\0&0&s&0\\\\0&0&0&s\\end{array}\\right]\\]\n\n\n\n\n```julia\nDm = diagm([s, s, s, t])\n```\n\n\n\n\n\\[\\left[ \\begin{array}{rrrr}s&0&0&0\\\\0&s&0&0\\\\0&0&s&0\\\\0&0&0&t\\end{array}\\right]\\]\n\n\n\n\n```julia\nA = U * inv(Dp)\n```\n\n\n\n\n\\[\\left[ \\begin{array}{rrrr}\\frac{s}{L}&\\frac{s}{M}&1&1\\\\\\frac{s}{L}&\\frac{s}{M}&1&1\\\\\\frac{s}{L}&\\frac{s}{M}&1&1\\\\\\frac{s}{L}&\\frac{s}{M}&1&1\\end{array}\\right]\\]\n\n\n\n\n```julia\nB = U * Dm\n```\n\n\n\n\n\\[\\left[ \\begin{array}{rrrr}s^{2}&s^{2}&s^{2}&s t\\\\s^{2}&s^{2}&s^{2}&s t\\\\s^{2}&s^{2}&s^{2}&s t\\\\s^{2}&s^{2}&s^{2}&s t\\end{array}\\right]\\]\n\n\n\n\n```julia\nA + B\n```\n\n\n\n\n\\[\\left[ \\begin{array}{rrrr}s^{2} + \\frac{s}{L}&s^{2} + \\frac{s}{M}&s^{2} + 1&s t + 1\\\\s^{2} + \\frac{s}{L}&s^{2} + \\frac{s}{M}&s^{2} + 1&s t + 1\\\\s^{2} + \\frac{s}{L}&s^{2} + \\frac{s}{M}&s^{2} + 1&s t + 1\\\\s^{2} + \\frac{s}{L}&s^{2} + \\frac{s}{M}&s^{2} + 1&s t + 1\\end{array}\\right]\\]\n\n\n\n\n```julia\ninv(Dp) * U |> latexify |> print\n```\n\n    \\begin{equation}\n    \\left[\n    \\begin{array}{cccc}\n    s/L & s/L & s/L & s/L \\\\\n    s/M & s/M & s/M & s/M \\\\\n    1 & 1 & 1 & 1 \\\\\n    1 & 1 & 1 & 1 \\\\\n    \\end{array}\n    \\right]\n    \\end{equation}\n\n\n## Using custom types\n\n\n```julia\n] activate .\n```\n\n    \u001b[32m\u001b[1mActivating\u001b[22m\u001b[39m environment at `C:\\Users\\carsten\\.julia\\dev\\StableDQMC.jl\\notebooks\\scales_visualizations\\Project.toml`\n\n\n\n```julia\n@enum Scale null=1 tiny=2 small=3 medium=4 large=5\nconst unit=small\n# const scale_symbols = [' ', '\u25aa', '\u25fe', '\u25a0', '\u2b1b'];\nconst scale_symbols = [' ', 't', 's', 'M', 'L'];\n# const scale_symbols = [' ', '\u2093', 'u', 'x', 'X'];\n\nsymbol(s::Scale) = scale_symbols[Int(s)]\nBase.isascii(::Scale) = false\nBase.isoverlong(::Scale) = false\nBase.ismalformed(::Scale) = false\nBase.isprint(::Scale) = true\n\n# printing\nBase.show(io::IO, s::Scale) = print(io, symbol(s))\nBase.show(io::IO, ::MIME\"text/plain\", s::Scale) = show(io,s)\nBase.print(io::IO, s::Scale) = show(io, s)\n\n# math\nBase.zero(::Type{Scale}) = null\nBase.iszero(s::Scale) = s === null\n```\n\n    WARNING: redefining constant scale_symbols\n\n\n\n```julia\nscales = null, tiny, small, medium, large\n```\n\n\n\n\n    ( , t, s, M, L)\n\n\n\n\n```julia\nusing Statistics\n\nstruct ScaleString <: AbstractString\n    scales::Vector{Scale}\nend\n\n# constructors\nScaleString(s::Scale) = ScaleString([s])\n\nimport Lazy: @forward\n@forward ScaleString.scales (Base.getindex, Base.length)\n\n# conversion\nBase.convert(::Type{ScaleString}, s::Scale) = ScaleString(s)\nBase.promote_rule(::Type{Scale}, ::Type{ScaleString}) = ScaleString\n\nBase.iterate(s::ScaleString) = iterate(s.scales)\nBase.iterate(s::ScaleString, state::Integer) = iterate(s.scales, state)\n\n# Base.:+(x::ScaleString, y::ScaleString) = ScaleString(vcat(x.scales, null, null, y.scales))\nBase.:*(x::ScaleString, y::ScaleString) = ScaleString(vcat(x.scales, y.scales))\nBase.zero(::Type{ScaleString}) = ScaleString(zero(Scale))\nBase.zero(::ScaleString) = zero(ScaleString)\nfunction Base.:+(x::ScaleString, y::ScaleString)\n    if null in x.scales\n        if null in y.scales\n            # both contain a null\n            return ScaleString([null])\n        else\n            # only x contains a null\n            return y\n        end\n    elseif null in y.scales\n        # only y contains a null\n        return x\n    else\n       return sum(Int.(x.scales)) < sum(Int.(y.scales)) ? y : x #mean(x) < mean(y) ? y : x \n    end\nend\nBase.:*(x::ScaleString, y::Scale) = iszero(y) ? ScaleString([null]) : ScaleString(vcat(x.scales, y))\nBase.:*(x::Scale, y::ScaleString) = iszero(x) ? ScaleString([null]) : ScaleString(vcat(x, y.scales))\n\n# Scale math extension\nBase.:+(x::Scale, y::Scale) = x > y ? x : y\nBase.:/(s::Scale, i::Int64) = /(Int(s), i)\nfunction Base.:*(x::Scale, y::Scale)\n#     if iszero(x)\n#         return ScaleString([y])\n#     elseif iszero(y)\n#         return ScaleString([x])\n#     else\n        return ScaleString([x,y])\n#     end\nend\n```\n\n\n```julia\ntiny * null\n```\n\n\n\n\n    \"t \"\n\n\n\n\n```julia\nA = ScaleString([tiny, small, small])\nB = ScaleString([null, large, small])\n```\n\n\n\n\n    \" Ls\"\n\n\n\n\n```julia\nusing Latexify\n\nLatexify.latexraw(s::Scale) = symbol(s)\nLatexify.latexraw(s::ScaleString) = join(s.scales)\n```\n\n\n```julia\nO = zeros(Scale, 4,4)\n```\n\n\n\n\n    4\u00d74 Array{Scale,2}:\n               \n               \n               \n               \n\n\n\n\n```julia\nU = fill(unit, 4,4)\n```\n\n\n\n\n    4\u00d74 Array{Scale,2}:\n     s  s  s  s\n     s  s  s  s\n     s  s  s  s\n     s  s  s  s\n\n\n\n\n```julia\nU |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{cccc}\ns & s & s & s \\\\\ns & s & s & s \\\\\ns & s & s & s \\\\\ns & s & s & s \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n\n```julia\nU * U |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{cccc}\nss & ss & ss & ss \\\\\nss & ss & ss & ss \\\\\nss & ss & ss & ss \\\\\nss & ss & ss & ss \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n\n```julia\nU * O\n```\n\n\n\n\n    4\u00d74 Array{ScaleString,2}:\n     \" \"  \" \"  \" \"  \" \"\n     \" \"  \" \"  \" \"  \" \"\n     \" \"  \" \"  \" \"  \" \"\n     \" \"  \" \"  \" \"  \" \"\n\n\n\n\n```julia\nusing LinearAlgebra\nD = diagm([large, medium, small, tiny])\n```\n\n\n\n\n    4\u00d74 Array{Scale,2}:\n     L         \n        M      \n           s   \n              t\n\n\n\n\n```julia\nD |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{cccc}\nL &   &   &   \\\\\n  & M &   &   \\\\\n  &   & s &   \\\\\n  &   &   & t \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n\n```julia\nU * D |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{cccc}\nsL & sM & ss & st \\\\\nsL & sM & ss & st \\\\\nsL & sM & ss & st \\\\\nsL & sM & ss & st \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n\n```julia\nD * U * D |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{cccc}\nLsL & LsM & Lss & Lst \\\\\nMsL & MsM & Mss & Mst \\\\\nssL & ssM & sss & sst \\\\\ntsL & tsM & tss & tst \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n\n```julia\nD * [small, small, small, small] |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{c}\nLs \\\\\nMs \\\\\nss \\\\\nts \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n\n```julia\nD * [large, medium, small, tiny] |> latexify\n```\n\n\n\n\n\\begin{equation}\n\\left[\n\\begin{array}{c}\nLL \\\\\nMM \\\\\nss \\\\\ntt \\\\\n\\end{array}\n\\right]\n\\end{equation}\n\n\n\n\n\n```julia\nR = D * U * D\nR[1] == ScaleString([large,unit,large])\n```\n\n\n\n\n    true\n\n\n", "meta": {"hexsha": "6e792cb694d23c9fd8cd309d3cce73e8c3df0e09", "size": 53982, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/scales_visualizations/scales.ipynb", "max_stars_repo_name": "crstnbr/StableDQMC.jl", "max_stars_repo_head_hexsha": "9ebc94358004e5d440060731b6726a6f09474d4d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 21, "max_stars_repo_stars_event_min_datetime": "2019-05-16T00:10:16.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-06T07:08:25.000Z", "max_issues_repo_path": "notebooks/scales_visualizations/scales.ipynb", "max_issues_repo_name": "carstenbauer/StableDQMC.jl", "max_issues_repo_head_hexsha": "9ebc94358004e5d440060731b6726a6f09474d4d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 13, "max_issues_repo_issues_event_min_datetime": "2019-05-23T16:22:14.000Z", "max_issues_repo_issues_event_max_datetime": "2020-08-19T10:46:49.000Z", "max_forks_repo_path": "notebooks/scales_visualizations/scales.ipynb", "max_forks_repo_name": "crstnbr/StableDQMC.jl", "max_forks_repo_head_hexsha": "9ebc94358004e5d440060731b6726a6f09474d4d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2019-11-09T13:12:33.000Z", "max_forks_repo_forks_event_max_datetime": "2020-09-25T00:07:56.000Z", "avg_line_length": 21.6708149338, "max_line_length": 311, "alphanum_fraction": 0.4005223964, "converted": true, "num_tokens": 7376, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660542, "lm_q2_score": 0.6334102705979902, "lm_q1q2_score": 0.41257486108344266}}
{"text": "# Use `ScenarioSeries` to check goodness of fit for a series of simulations\n\nThe `prms_python` module provides the `ScenarioSeries` class to set up and run a number of simulations at once in parallel with a common purpose. One example would be to do some sensitivity analyses of a set of parameters. We'll show this application below. This notebook is compatible with Python 3.\n\n## scale *jh_coef* and *rad_trncf* by multiplicative factors\n\nThe example we'll show modifies *jh_coef* and *rad_trncf* by scaling them. \n\n\n```python\nimport warnings\nimport sys\nsys.path.append('..')\nwarnings.filterwarnings('ignore', module='numpy')\nfrom prms_python import ScenarioSeries\n```\n\n\n```python\nbase_dir = '../test/data/models/lbcd/'\nsimulation_dir = 'jupyternb-scenario-series-example/'\ntitle = 'Jensen-Hays and radiative transfer function sensitivity analysis'\ndescription = '''\nUse title of \\'\"jh_coef\":{jh factor value}|\"rad_trncf\":{rad factor value}\\' so later\nwe can easily generate a dictionary of these factor value combinations.\n'''\n\nsc_series = ScenarioSeries(base_dir, simulation_dir, title, description)\n```\n\nAt this point we have only initialized the series, but we have not created any new scenarios since we have not yet specified what functions will be applied for each of the scenarios. We'll do that next. We need to define a list of dictionaries, each containing the specification for a scenario. First we'll define a function with the signature `_scale_fun(val)` that will itself return a function with the signature `scale(x)` which scales `x` by `val`. This function will then be used to scale all the parameter values by the same amount, `val`.\n\nIn total we will generate sixteen scenarios, with a cartesian product of scaling factors ranging from 0.7 to 1.0. This takes a bit of time since we must copy rather large files, currently in series. \n\n\n```python\nimport numpy as np\n\ndef _scale_fun(val):\n    def scale(x):\n        return x*val\n    \n    return scale\n\nscenario_list = [\n    {\n        'title': '\"jh_coef\":{0:.1f}|\"rad_trncf\":{1:.1f}'.format(jh_val, rad_val),\n        'jh_coef': _scale_fun(jh_val),\n        'rad_trncf': _scale_fun(rad_val)\n    }\n    for jh_val in np.arange(0.7, 1.0, 0.1)\n    for rad_val in np.arange(0.7, 1.0, 0.1)\n]\n\nsc_series.build(scenario_list)\n```\n\nNow with the ScenarioSeries built, that is, the directory structure initialized and scenario data prepared, we can run the scenario series. The runner automatically uses half the available CPUs if the optional kwarg `nproc` is not provided.\n\n\n```python\nsc_series.run()  # alternatively sc_series.run(nproc=8)\n```\n\n## Analyze the data generated by ScenarioSeries\n\n`ScenarioSeries` provides a mapping from UUID string-named directories to titles we used in `series_metadata.json`. Below is a printout of this `series_metadata.json`.\n\n\n```python\nimport json\nmetadata = json.loads(open('jupyternb-scenario-series-example/series_metadata.json').read())\n\nprint(json.dumps(metadata, indent=2))\n```\n\n    {\n      \"title\": \"Jensen-Hays and radiative transfer function sensitivity analysis\",\n      \"description\": \"\\nUse title of '\\\"jh_coef\\\":{jh factor value}|\\\"rad_trncf\\\":{rad factor value}' so later\\nwe can easily generate a dictionary of these factor value combinations.\\n\",\n      \"uuid_title_map\": {\n        \"0a151f07-add4-4e5f-bbfb-50f4f20bc2b8\": \"\\\"jh_coef\\\":0.7|\\\"rad_trncf\\\":0.7\",\n        \"88e3c4b0-6984-47c0-9307-ebd2ae5eff00\": \"\\\"jh_coef\\\":0.7|\\\"rad_trncf\\\":0.8\",\n        \"a5726cd3-7d52-430a-9f93-45217bc862ab\": \"\\\"jh_coef\\\":0.7|\\\"rad_trncf\\\":0.9\",\n        \"99af81ac-b242-44f3-8bf6-9443d0689b11\": \"\\\"jh_coef\\\":0.7|\\\"rad_trncf\\\":1.0\",\n        \"b3b2ef40-a125-4eb2-9f1f-fd159c96a5d5\": \"\\\"jh_coef\\\":0.8|\\\"rad_trncf\\\":0.7\",\n        \"5e8bb47d-503a-40b6-8e2c-a738fdafcc51\": \"\\\"jh_coef\\\":0.8|\\\"rad_trncf\\\":0.8\",\n        \"6610caa1-5c55-4d61-98c8-3ce1bc68ee69\": \"\\\"jh_coef\\\":0.8|\\\"rad_trncf\\\":0.9\",\n        \"3c67d688-ac9b-47c9-83bc-370d9b66796a\": \"\\\"jh_coef\\\":0.8|\\\"rad_trncf\\\":1.0\",\n        \"adca93a4-c6ea-463f-a9d3-e8d0763693e3\": \"\\\"jh_coef\\\":0.9|\\\"rad_trncf\\\":0.7\",\n        \"d0abbd5c-d7a7-471b-bda7-63afea3a4142\": \"\\\"jh_coef\\\":0.9|\\\"rad_trncf\\\":0.8\",\n        \"8793a5ab-f07d-4ef1-877d-f7519fdbdbfd\": \"\\\"jh_coef\\\":0.9|\\\"rad_trncf\\\":0.9\",\n        \"a3394a0f-2525-4c15-a862-f485dcb01a3f\": \"\\\"jh_coef\\\":0.9|\\\"rad_trncf\\\":1.0\",\n        \"1e190ed0-fcb7-466e-a7ab-b357e6f66c4f\": \"\\\"jh_coef\\\":1.0|\\\"rad_trncf\\\":0.7\",\n        \"438b4ebd-2867-4410-81a3-420a906421fd\": \"\\\"jh_coef\\\":1.0|\\\"rad_trncf\\\":0.8\",\n        \"54a18b71-b538-41d3-8cf3-db562182b7d9\": \"\\\"jh_coef\\\":1.0|\\\"rad_trncf\\\":0.9\",\n        \"60941270-3f89-4a48-8a25-2ba0aeb8f844\": \"\\\"jh_coef\\\":1.0|\\\"rad_trncf\\\":1.0\"\n      }\n    }\n\n\nLet's load a single one of these, taking the first element from the `uuid_title_map` and load the `statvar.dat` file into a Pandas DataFrame. We will be comparing the basin_cfs hydrograph in the statvar file to runoff hydrograph in the data input file.\n\nBelow we'll load both an example statvar and an input data file.\n\n\n```python\nimport os\nfrom prms_python import load_statvar\n\nex_uuid = list(metadata['uuid_title_map'].keys()).pop()\nex_statvar = os.path.join('jupyternb-scenario-series-example', ex_uuid, 'outputs', 'statvar.dat')\nstatvar_df = load_statvar(ex_statvar)\n\nstatvar_df.head()\n```\n\n\n\n\n
                                        \n\n\n  \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n  \n  \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n    \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n      \n    \n  \n
                                        statistical_variablesbasin_gwflow_cfs_1basin_sroff_cfs_1basin_ssflow_cfs_1basin_cfs_1sub_cfs_1sub_interflow_1sub_sroff_1sub_gwflow_1sub_cfs_2sub_interflow_2...subinc_snowmelt_1subinc_snowmelt_2subinc_snowmelt_3basin_potet_1swrad_2490basin_ppt_1basin_hortonian_1basin_dunnian_1subinc_snowmelt_1subinc_snowmelt_2
                                        date
                                        1992-10-0112.6851590.01.68878514.3739456.7227780.6351610.06.0876177.1376370.915658...0.00.00.00.112602449.1893010.00.00.00.00.0
                                        1992-10-0212.4045670.01.27903713.6836046.4255540.4789700.05.9465836.7938850.699584...0.00.00.00.109168441.3260800.00.00.00.00.0
                                        1992-10-0312.1335350.00.96963613.1031716.1727680.3618980.05.8108696.5049720.534584...0.00.00.00.069002396.1808470.00.00.00.00.0
                                        1992-10-0411.8718430.00.73570912.6075535.9543160.2739880.05.6803286.2586630.408489...0.00.00.00.055411392.4652400.00.00.00.00.0
                                        1992-10-0511.6192480.00.55864212.1778915.7626540.2078570.05.5547986.0456570.312074...0.00.00.00.056177388.7499390.00.00.00.00.0
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                                        5 rows \u00d7 41 columns
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                                        \n\n\n\nThis also allows us to easily plot the output hydrographs like so. We add some niceness to the plots.\n\n\n```python\n%matplotlib inline\nwarnings.filterwarnings('ignore', module='matplotlib')\nwarnings.filterwarnings('ignore', module='mpltools')\n\nimport matplotlib as mpl\nmpl.style.use('ggplot')\nimport matplotlib.pyplot as plt\nmpl.rcParams['figure.figsize'] = (10.0, 6.0)\nmpl.rcParams['figure.dpi'] = 300.0\nmpl.rcParams['xtick.labelsize'] = 12.0\nmpl.rcParams['ytick.labelsize'] = 12.0\nmpl.rcParams['axes.labelsize'] = 16.0\nmpl.rcParams['axes.titlesize'] = 18.0\nmpl.rcParams['legend.fontsize'] = 16.0\n\nstatvar_df.basin_cfs_1.plot()\nplt.ylabel('Streamflow (cfs)')\n```\n\nNow let's load and plot the data file on top of the simulated streamflow from the statvar file.\n\n\n```python\nfrom prms_python import load_data\n\ndata_path = os.path.join(base_dir, 'data')\ndata_df = load_data(data_path)\n\ndata_df.runoff_1.plot(label='observed')\nstatvar_df.basin_cfs_1.plot(label=metadata['uuid_title_map'][ex_uuid].replace('\"', '').replace('|',', '))\n\nplt.ylabel('Streamflow (cfs)')\nplt.legend()\n```\n\n## Calculate the Nash-Sutcliffe Model Efficiency for one Simulation\n\nIn parameterizing hydrological models, it's necesary to test multiple parameters and calculate the goodness-of-fit for each model run. Goodness-of-fit is a measure of how well the model output matches historical output so that future predictions may be made. A well-known GOF measure in Hydrology is the Nash-Sutcliffe model efficiency, from Nash & Sutcliffe's 1970 paper [River Flow Forecasting Through Conceptual Models Part I - A Discussion of Principles](./nash-sutcliffe-1970.pdf). They elegantly describe the model-fitting problem like so\n\n> To remove subjectivity in fitting the model to the data or in determining the\nparametric values, O'Donnell3) suggested automatic optimisation. This\ninvolves successive changes of parameter values according to some pre-conceived\nrule or pattern of increments which takes into account the results of\nprevious steps and in particular whether or not a change improved the fitting.  \n\nThe equation itself is\n\n\\begin{equation}\nE = 1 - \\frac{\\sum_{t=1}^{T}\\left(Q_o^t - Q_m^t\\right)^2}{\\sum_{t=1}^{T}\\left(Q_o^t - \\overline{Q_o}\\right)^2}\n\\end{equation}\n\nWe have implemented this in the `prms_python/util.py` module. Below we import it and calculate it for the simulated hydrograph shown in the figure above.\n\nNote that the closer to 1 the efficiency is, the more exactly the model accounts for observed variance. A value of 0 for efficiency means that the model does as well as the mean of the hydrograph for predictive purposes. A value below 0 means that the model does worse than the mean for modeling the hydrograph.\n\n\n```python\nfrom prms_python import nash_sutcliffe\n\nnash_sutcliffe(data_df.runoff_1, statvar_df.basin_cfs_1)\n```\n\n\n\n\n    0.5985845573889427\n\n\n\n## Calculate the Nash-Sutcliffe Efficiency for all Simulations\n\nNow we'll iterate over all the statvar outputs and calculate the Nash-Sutcliffe model efficiency for each. To better understand the efficiency landscape across these two parameters, we'll use a matrix plot to see how the efficiency changes across scalings of each of the two parameters, `jh_coef` and `rad_trncf`.\n\n\n```python\nidx_lookup = {'{:.1f}'.format(val): idx for idx, val in enumerate(np.arange(0.7, 1.0, 0.1))}\nprint(idx_lookup)\n```\n\n    {'0.7': 0, '0.8': 1, '0.9': 2, '1.0': 3}\n\n\n\n```python\nnash_sutcliffe_mat = np.zeros((4,4))\n\nmodeled_flows = {\n    title: load_statvar(os.path.join('jupyternb-scenario-series-example', uu, 'outputs', 'statvar.dat')).basin_cfs_1\n    for uu, title in metadata['uuid_title_map'].items()\n}\nprint(list(modeled_flows.keys())[:5])\n```\n\n    ['\"jh_coef\":0.7|\"rad_trncf\":0.7', '\"jh_coef\":0.7|\"rad_trncf\":0.8', '\"jh_coef\":0.7|\"rad_trncf\":0.9', '\"jh_coef\":0.7|\"rad_trncf\":1.0', '\"jh_coef\":0.8|\"rad_trncf\":0.7']\n\n\n\n```python\nobserved = data_df.runoff_1\n\nfor title, hydrograph in modeled_flows.items():\n    param_scalings = eval('{' + title.replace('|', ',') + '}')\n    coord = (idx_lookup[str(param_scalings['jh_coef'])], idx_lookup[str(param_scalings['rad_trncf'])])\n    \n    nash_sutcliffe_mat[coord] = nash_sutcliffe(observed, hydrograph)\n```\n\n\n```python\nimport itertools\nfig, ax = plt.subplots()\n\ncax = ax.matshow(nash_sutcliffe_mat, cmap='viridis')\ntix = [0.7, 0.8, 0.9, 1.0]\nplt.xticks(range(4), tix)\nplt.yticks(range(4), tix)\n\n\nax.xaxis.set_ticks_position('bottom')\nplt.ylabel('jh_coef factor')\nplt.xlabel('rad_trncf factor')\n\nfor i, j in itertools.product(range(4), range(4)):\n    plt.text(j, i, \"%.2f\" % nash_sutcliffe_mat[i, j],\n             horizontalalignment=\"center\", \n             color=\"w\" if nash_sutcliffe_mat[i, j] < .61 else \"k\")\n\nplt.title('Nash-Sutcliffe Matrix')\nplt.grid(b=False)\ncbar = fig.colorbar(cax)\n```\n", "meta": {"hexsha": "a7a93a42a514cabf9036fdaa2ce7029cc642405d", "size": 739557, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/scenario_series.ipynb", "max_stars_repo_name": "JohnVolk/PRMS-Python", "max_stars_repo_head_hexsha": "31ae9c03d16f00186c5d91bf049b6e978d04600a", "max_stars_repo_licenses": ["BSD-3-Clause-Clear"], "max_stars_count": 9, "max_stars_repo_stars_event_min_datetime": "2018-11-22T15:51:38.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-06T01:24:09.000Z", "max_issues_repo_path": "notebooks/scenario_series.ipynb", "max_issues_repo_name": "JohnVolk/PRMS-Python", "max_issues_repo_head_hexsha": "31ae9c03d16f00186c5d91bf049b6e978d04600a", "max_issues_repo_licenses": ["BSD-3-Clause-Clear"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2018-08-21T23:05:51.000Z", "max_issues_repo_issues_event_max_datetime": "2018-09-14T23:11:26.000Z", "max_forks_repo_path": "notebooks/scenario_series.ipynb", "max_forks_repo_name": "JohnVolk/PRMS-Python", "max_forks_repo_head_hexsha": "31ae9c03d16f00186c5d91bf049b6e978d04600a", "max_forks_repo_licenses": ["BSD-3-Clause-Clear"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2019-01-11T14:41:51.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-14T23:51:30.000Z", "avg_line_length": 1038.7036516854, "max_line_length": 311924, "alphanum_fraction": 0.9469804221, "converted": true, "num_tokens": 4744, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "```python\n!pip install tensorflow==2.4.1\n!pip install tensorflow-quantum==0.5.1\n# Cirq linking seems to be broken, so reinstall the core package\n!pip install --no-deps --force-reinstall cirq-core==0.11.0\n```\n\n# **Task III: Part a) implement a quantum variational autoencoder in TF Quantum and show its performance**\n\n### Dataset:\nThe electron-photon dataset (which can be found here) contains 100 samples for training and another 100 for testing, laid out as follows:\n\n\u25cf data[\"x_train\"]: Training dataset of 100 32x32 images containing the particles' energy (100, 32, 32)\n\n\u25cf data[\"y_train\"]:\" Training labels, 0 = \"photon\", 1 = \"electron\" (100,)\n\n\u25cf data[\"x_test\"]: Test dataset of 100 32x32 images containing the particles' energy (100, 32, 32)\n\n\u25cf data[\"y_test\"]:\" Test labels, 0 = \"photon\", 1 = \"electron\" (100,)\n\n## **Quantum Variational Autoencoder**\n\n A limiting factor for nearly\nall of quantum applications, is the amount of quantum\nresources that can be realized in an experiment. Therefore, for\nexperiments now and in the near future, any tool which can reduce the experimental overhead in terms of these resources is\nespecially valuable.This can be done using a technique called autoencoding.\n
                                        \nQuantum Autoencoder is a way of encoding data into a compressed form using suitable training over variational circuits.\n
                                        \n\n
                                        \nThe approach used to solve this task is inspired from this paper[1] where we use a PQC(Parameterised Quantum Circuit) to perform operations on input data.\nAfter this, we trace out a few data qubits and measure its fidelity with a reference state. The PQC circuit is trained to minimize the fidelity loss between the traced out space and the reference state. After training we can say that we have successfully atleast approximately encoded data into a compressed space called latent space.\n\n\n```python\nimport numpy as np\nimport sympy\nimport tensorflow as tf\nimport tensorflow_quantum as tfq\nimport cirq\n%matplotlib inline\nimport matplotlib.pyplot as plt\nfrom cirq.contrib.svg import SVGCircuit\n```\n\n# **Dataset Preprocessing**\n\n\n```python\nfrom google.colab import files\nuploaded = files.upload()\n```\n\n\n\n\n\n Upload widget is only available when the cell has been executed in the\n current browser session. Please rerun this cell to enable.\n \n  \n\n\n    Saving electron-photon.npz to electron-photon.npz\n\n\n\n```python\ndata = np.load('./electron-photon.npz', allow_pickle = True)\n```\n\n\n```python\nfor key in data.keys():\n  print(key)\n  print(data[key])\n  \n```\n\n    x_train\n    [[[0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      ...\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]]\n    \n     [[0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      ...\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]]\n    \n     [[0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      ...\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]]\n    \n     ...\n    \n     [[0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      ...\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]]\n    \n     [[0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      ...\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]]\n    \n     [[0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      ...\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]]]\n    y_train\n    [1. 1. 1. 0. 1. 0. 1. 1. 1. 0. 1. 1. 1. 1. 0. 1. 1. 0. 0. 1. 0. 0. 1. 0.\n     1. 0. 1. 1. 0. 0. 1. 0. 1. 1. 0. 0. 1. 1. 0. 1. 1. 0. 0. 0. 1. 1. 0. 1.\n     1. 0. 1. 1. 0. 0. 1. 1. 1. 1. 1. 1. 1. 1. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0.\n     0. 0. 0. 1. 1. 1. 0. 1. 0. 1. 1. 1. 0. 1. 0. 1. 0. 0. 1. 1. 1. 0. 1. 0.\n     1. 1. 0. 1.]\n    x_test\n    [[[0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      ...\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]]\n    \n     [[0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      ...\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]]\n    \n     [[0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      ...\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]]\n    \n     ...\n    \n     [[0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      ...\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]]\n    \n     [[0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      ...\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]]\n    \n     [[0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      ...\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]\n      [0. 0. 0. ... 0. 0. 0.]]]\n    y_test\n    [0. 0. 0. 0. 0. 1. 0. 1. 0. 0. 0. 0. 1. 0. 0. 1. 0. 0. 1. 1. 1. 1. 0. 1.\n     0. 1. 0. 0. 0. 0. 1. 0. 0. 1. 1. 1. 0. 1. 0. 1. 1. 1. 0. 1. 1. 1. 0. 0.\n     1. 0. 1. 1. 0. 1. 0. 1. 0. 1. 0. 1. 0. 0. 0. 0. 0. 0. 1. 1. 0. 0. 0. 0.\n     0. 0. 1. 1. 1. 0. 1. 0. 1. 0. 1. 1. 0. 0. 0. 1. 1. 1. 0. 0. 0. 0. 1. 1.\n     1. 1. 1. 1.]\n\n\n\n```python\nX_train = data['x_train']\ny_train = data['y_train']\nX_test = data['x_test']\ny_test = data['y_test']\n\n# check\nprint(X_train.shape)\nprint(y_train.shape)\nprint(X_test.shape)\nprint(y_test.shape)\n```\n\n    (100, 32, 32)\n    (100,)\n    (100, 32, 32)\n    (100,)\n\n\n# Feature Reduction\n\nEach image is of size 32x32 and the information in the image is highly sparse(most of pixels are zero). Also after careful preprocessing it is also evident that a very small ammount of information is actually present in the corner pixels and that is what actually distinguishes two or more images to a larger extent. So cropping the images won't be an option. Here, we will apply a classical subroutine called PCA(Principal Component Analysis) to the input data to reduce the feature dimensions from 1024(32x32 in 2D) to some small number of features that best fits our model.\n\n\n```python\nfrom sklearn.decomposition import PCA\n\nn_pca = 10\nX_train_flatten = X_train.reshape(-1,32*32)\nX_test_flatten = X_test.reshape(-1,32*32)\n\npca = PCA(n_components = n_pca)\npca.fit(X_train_flatten)\n\nX_train_transform = pca.transform(X_train_flatten)\nX_test_transform = pca.transform(X_test_flatten)\n\n#check\nprint(X_train_transform.shape)\nprint(X_test_transform.shape)\n\n# Let's check if the pca transform preserves the inherent variance present in this data\nprint(np.cumsum(pca.explained_variance_ratio_)) \n```\n\n    (100, 10)\n    (100, 10)\n    [0.48118138 0.65938324 0.76321054 0.84084606 0.8902277  0.9208717\n     0.94699347 0.9600863  0.96964854 0.9764575 ]\n\n\nWe can see that after transforming the data using PCA, the features reduce from 1024 features to 10 features but retains upto ~97% of its variance. However, we can use more features to get better results  from the model \n\n# **Circuit Architecture**\nIn this task, we are going to use 10 qubits for the data from the dataset.Along with this we use 6 reference qubits and 1 extra qubit for the Swap Test.Total qubits used will be 17 out of which the last 4 qubits form the latent space in which the compressed encoding of data occurs.\n\n\n```python\n# Initializing Qubits\nqubits = cirq.GridQubit.rect(1,17)\n# num_qubits = 9\n```\n\n\n```python\n# Defining standard unitaries\n# encoder unitary\ndef one_qubit_unitary_enc(bit, symbols):\n    \"\"\"Make a Cirq circuit enacting a rotation of the bloch sphere about the X,\n    Y and Z axis, that depends on the values in `symbols`.\n    \"\"\"\n    return cirq.Circuit(\n        cirq.X(bit)**symbols[0],\n        cirq.Y(bit)**symbols[1],\n        cirq.Z(bit)**symbols[2])\n\n# decoder unitary\ndef one_qubit_unitary_dec(bit,symbols):\n   \"\"\"Make a Cirq circuit enacting a rotation of the bloch sphere about the X,\n    Y and Z axis, that depends on the values in `symbols`.\n    \"\"\"\n   return cirq.Circuit(\n        cirq.Z(bit)**symbols[0],\n        cirq.Y(bit)**symbols[1],\n        cirq.X(bit)**symbols[2]\n    )\n\ndef two_qubit_unitary(bits, symbols):\n    \"\"\"Make a Cirq circuit that creates an arbitrary two qubit unitary.\"\"\"\n    circuit = cirq.Circuit()\n    # circuit += one_qubit_unitary_(bits[0], symbols[0:3])\n    # circuit += one_qubit_unitary(bits[1], symbols[3:6])\n    circuit += [cirq.ZZ(*bits)**symbols[0]]\n    circuit += [cirq.YY(*bits)**symbols[1]]\n    circuit += [cirq.XX(*bits)**symbols[2]]\n    # circuit += one_qubit_unitary(bits[0], symbols[9:12])\n    # circuit += one_qubit_unitary(bits[1], symbols[12:])\n    return circuit\n```\n\n\n```python\ndef create_encoder_circuit(qubits,symbols,layers=1):\n  circuit = cirq.Circuit()\n\n  for layer in range(layers):\n    for i in range(len(qubits)):\n      circuit += one_qubit_unitary_enc(qubits[i],symbols[3*i + 3*layer*len(qubits) : 3*(i+1) + 3*layer*len(qubits)])\n    \n    # for i in range(len(qubits)):\n    #   if(i                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  \u250c\u2500\u2500\u2510                                                                                                                                                                                                                  \u250c\u2500\u2500\u2510                                                                                                                                                                                                                  \u250c\u2500\u2500\u2510                                                                                                                                                                                                                  \u250c\u2500\u2500\u2510                                                                                                                                                                                                                  \u250c\u2500\u2500\u2510                                                                                                                                                                                                                  \u250c\u2500\u2500\u2510\n(0, 0): 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        \u2502           \u2502               \u2502            \u2502                 \u2502            \u2502                \u2502                \u2502                              \u2502                    \u2502                \u2502                    \u2502                 \u2502           \u2502               \u2502            \u2502                 \u2502            \u2502                \u2502                \u2502                              \u2502                    \u2502                \u2502                    \u2502                 \u2502           \u2502               \u2502            \u2502                 \u2502            \u2502                \u2502                \u2502                              \u2502                    \u2502                \u2502                    \u2502                 \u2502           \u2502               \u2502            \u2502                 \u2502            \u2502                \u2502                \u2502                        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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500Y^-0.5\u2500\u2500\u2500@\u2500\u2500\u2500Y^0.5\u2500\u2500\u2500\u2500\u2500\u253cT\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500Y^-0.5\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500Y^0.5\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500X^-0.5\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u250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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502            \u2502                \u2502                \u2502                              \u2502                    \u2502                \u2502                    \u2502                 \u2502           \u2502               \u2502            \u2502                 \u2502            \u2502                \u2502                \u2502                              \u2502                    \u2502                \u2502                    \u2502                 \u2502           \u2502               \u2502            \u2502                 \u2502            \u2502                \u2502                \u2502                              \u2502                    \u2502                \u2502                    \u2502                 \u2502           \u2502               \u2502            \u2502                 \u2502            \u2502                \u2502                \u2502                              \u2502                    \u2502                \u2502                    \u2502                 \u2502           \u2502               \u2502            \u2502                 \u2502            \u2502                \u2502                \u2502                              \u2502                    \u2502                \u2502                    \u2502                 \u2502           \u2502               \u2502            \u2502                 \u2502\n(0, 2): \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u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                     \u2502                              \u2502                                              \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502                                              \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502            \u2502                \u2502                \u2502                              \u2502                    \u2502                \u2502                    \u2502                 \u2502           \u2502               \u2502            \u2502                 \u2502            \u2502                \u2502                \u2502                              \u2502                    \u2502                \u2502                    \u2502                 \u2502           \u2502               \u2502            \u2502                 \u2502            \u2502                \u2502                \u2502                              \u2502                    \u2502                \u2502                    \u2502                 \u2502           \u2502               \u2502            \u2502                 \u2502\n(0, 4): \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500\u2500Y^-0.5\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500Y^0.5\u2500\u2500\u2500\u2500\u2500\u253cT\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500Y^-0.5\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500Y^0.5\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500X^-0.5\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u250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                     \u2502                              \u2502                                              \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502                                              \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502                                              \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502            \u2502                \u2502                \u2502                              \u2502                    \u2502                \u2502                    \u2502                 \u2502           \u2502               \u2502            \u2502                 \u2502            \u2502                \u2502                \u2502                              \u2502                    \u2502                \u2502                    \u2502                 \u2502           \u2502               \u2502            \u2502                 \u2502\n(0, 5): \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500\u2500Y^-0.5\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500Y^0.5\u2500\u2500\u2500\u2500\u2500\u253cT\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500Y^-0.5\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500Y^0.5\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500X^-0.5\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502                                              \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502                                              \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502                                              \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502                                              \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502            \u2502                \u2502                \u2502                              \u2502                    \u2502                \u2502                    \u2502                 \u2502           \u2502               \u2502            \u2502                 \u2502\n(0, 6): \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500\u2500Y^-0.5\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500Y^0.5\u2500\u2500\u2500\u2500\u2500\u253cT\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500Y^-0.5\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500Y^0.5\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500X^-0.5\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502                                              \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502                                              \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502                                              \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502                                              \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502                                              \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502\n(0, 7): \u2500\u2500\u2500\u2500X^encoder0\u2500\u2500\u2500Y^encoder1\u2500\u2500\u2500Z^encoder2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500ZZ\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500YY\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500XX\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500@\u2500\u2500\u2500\u2500\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                                                                                                                                                                                                                                                                                                                                                                                                                                     \u2502              \u2502              \u2502                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502                                              \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502                                              \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502                                              \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502                                              \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502\n(0, 8): 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                                       \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502\n(0, 9): 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         \u2502                              \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502\n(0, 10): 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               \u2502                                     \u2502                                      \u2502                           \u2502                              \u2502\n(0, 11): 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                                        \n\n\n\n\n```python\n@tf.function\ndef fidelity_loss(y_true,y_pred):\n # return -1*(tf.math.log(1-y_pred))\n return (1-y_pred)\n```\n\n\n```python\ndef create_model(qubits,symbols,layers=1):\n observables = [cirq.Z(qubits[0])]\n\n circuit = cirq.Circuit()\n circuit += create_full_circuit(qubits,symbols,layers)\n data_input = tf.keras.Input(shape = (),dtype = tf.dtypes.string)\n pqc_circuit = tfq.layers.PQC(circuit,\n observables,\n name='quantum_variational_qutoencoder')(data_input)\n model = tf.keras.Model(inputs = [data_input],outputs = [pqc_circuit])\n\n model.compile(optimizer = tf.keras.optimizers.Adam(lr=0.01),\n loss=[fidelity_loss],\n metrics=['accuracy'])\n return model\n\n```\n\n\n```python\nmodel = create_model(qubits,sympy.symbols('encoder:'+num_symbols),layers)\ntf.keras.utils.plot_model(model,\n show_shapes=True,\n show_layer_names=True,\n dpi=70)\n```\n\n\n```python\n# Hypeyparameters\nbatch = 20\nepochs = 10\n```\n\n# **Training the Model**\n\n\n```python\nH = model.fit(x=train_quantum_data,\n y=y_train,\n batch_size=batch,\n # verbose=verbose,\n epochs = epochs,\n validation_data = (test_quantum_data,y_test))\n```\n\n Epoch 1/10\n 5/5 [==============================] - 32s 7s/step - loss: 0.9807 - accuracy: 0.4571 - val_loss: 0.9217 - val_accuracy: 0.5400\n Epoch 2/10\n 5/5 [==============================] - 30s 6s/step - loss: 0.8879 - accuracy: 0.3987 - val_loss: 0.7717 - val_accuracy: 0.5400\n Epoch 3/10\n 5/5 [==============================] - 30s 6s/step - loss: 0.7132 - accuracy: 0.4458 - val_loss: 0.5508 - val_accuracy: 0.4900\n Epoch 4/10\n 5/5 [==============================] - 30s 6s/step - loss: 0.4949 - accuracy: 0.5157 - val_loss: 0.3550 - val_accuracy: 0.4800\n Epoch 5/10\n 5/5 [==============================] - 30s 6s/step - loss: 0.3042 - accuracy: 0.5572 - val_loss: 0.1949 - val_accuracy: 0.4600\n Epoch 6/10\n 5/5 [==============================] - 31s 6s/step - loss: 0.1643 - accuracy: 0.5158 - val_loss: 0.1057 - val_accuracy: 0.4600\n Epoch 7/10\n 5/5 [==============================] - 30s 6s/step - loss: 0.0825 - accuracy: 0.5478 - val_loss: 0.0576 - val_accuracy: 0.4600\n Epoch 8/10\n 5/5 [==============================] - 32s 7s/step - loss: 0.0479 - accuracy: 0.5033 - val_loss: 0.0360 - val_accuracy: 0.4600\n Epoch 9/10\n 5/5 [==============================] - 30s 6s/step - loss: 0.0328 - accuracy: 0.5658 - val_loss: 0.0372 - val_accuracy: 0.4600\n Epoch 10/10\n 5/5 [==============================] - 30s 6s/step - loss: 0.0352 - accuracy: 0.5394 - val_loss: 0.0333 - val_accuracy: 0.4600\n\n\n\n```python\nplt.plot(H.history['loss'])\nplt.xlabel('Epoch')\nplt.ylabel('Loss')\n```\n\n# **Conclusion**\n1. The image size is 32x32(i.e 1024 features) per sample which we reduced using PCA to 10 features. It is evident reducing features substantially still captures most of the variance(~ 97%) of the dataset. \n2. The circuit depth can become really large if more and more qubits are considered which may introduce errors hence the right tradeoff between number of qubits and encoding performance needs to be found out.\n3. From the training loss it is evident that the circuit learns and encodes the 10 feature data into 4 features(latent space).\n\n# **References**\n[1]-[Quantum autoencoders for efficient compression of quantum data\nJonathan Romero,1\nJonathan P. Olson,1\nand Alan Aspuru-Guzik1,](https://arxiv.org/pdf/1612.02806.pdf)\n\n\n```python\n\n```\n", "meta": {"hexsha": "ec42a5a4a8a92d4d12f24d4ab874cd81a367225d", "size": 440524, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "QMLHEP_testquestion_3a.ipynb", "max_stars_repo_name": "Amey-2002/QMLHEP", "max_stars_repo_head_hexsha": "579c8c8870b4e11dcc027852a693532bfc995fae", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-03-30T09:22:50.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-30T09:22:50.000Z", "max_issues_repo_path": "QMLHEP_testquestion_3a.ipynb", "max_issues_repo_name": "Amey-2002/QMLHEP", "max_issues_repo_head_hexsha": "579c8c8870b4e11dcc027852a693532bfc995fae", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "QMLHEP_testquestion_3a.ipynb", "max_forks_repo_name": "Amey-2002/QMLHEP", "max_forks_repo_head_hexsha": "579c8c8870b4e11dcc027852a693532bfc995fae", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 434.0137931034, "max_line_length": 113232, "alphanum_fraction": 0.5339686374, "converted": true, "num_tokens": 14606, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548511303336, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.41257484800230493}} {"text": "```javascript\n%%javascript\n$('#appmode-leave').hide();\n$('#copy-binder-link').hide();\n$('#visit-repo-link').hide();\n```\n\nCopyright **Paolo Raiteri**, January 2022\n\n# Numerical solution of equilibrium problems\n\nLet's consider a generic chemical reaction \n\n\\begin{equation}\n\\nu_aA + \\nu_bB + \\dots \\leftrightharpoons \\nu_xX + \\nu_yY + \\dots \\tag{1}\n\\end{equation}\n\nif the concentrations of the species are not the equilibrium ones, the reaction will progress (either forward or backwards) to reach the equilibrium conditions, and the free energy of the system will decrease.\n\nAn infinitesimal change of the system's free energy can then be written in terms of the chemical potential, $\\mu$, of all the species involved in the reaction.\n\n\\begin{equation}\n\\mathrm{d}G = \\sum_i \\mu_i \\mathrm{d}n_i \\tag{2}\n\\end{equation}\n\nBecause the concentrations of reactants and products are coupled through the reaction equation, they cannot change independently, _i.e._ if any products are formed, some reactants must be consumed.\nWe can therefore define a unique parameter, $\\xi$, as the _progress of the reaction_ \n\n\\begin{equation}\n\\mathrm{d}\\xi = \\frac{\\mathrm{d}n_i}{ \\nu_i } \\tag{3}\n\\end{equation}\n\nwhere $\\nu_i$ is the stoichiometric coefficient of the species $i$ taken as a positive number for products (formed) and as a negative number for reactants (consumed), and $\\mathrm{d}n_i$ is an infinitesimal change in the concentration of species $i$.\nThus, the free energy change due to the progress of the reaction (in any direction) becomes\n\n\\begin{equation}\n\\mathrm{d}G = \\sum_i \\nu_i \\mu_i \\mathrm{d}\\xi \\tag{4}\n\\end{equation}\n\nAt equilibrium, the concentrations of all species are constant, $\\mathrm{d}n_i=0$, and the reaction is not _progressing_ anymore, $\\mathrm{d}\\xi=0$, hence the free energy is not changing anymore.\nOn the other hand, if the system is out of equilibrium, there is a driving force that pulls the system towards equilibrium. \nA force can always be obtained as the negative of the derivative of the energy with respect to some coordinate.\nIn this case, the driving force that pushes the reaction towards equilibrium can be immediately obtained from equation (4).\n\n\\begin{equation}\n\\frac{\\mathrm{d}G}{\\mathrm{d}\\xi} = \\sum_i \\nu_i \\mu_i = -force \\tag{5}\n\\end{equation}\n\nIf we then substitute the definition of the chemical potential, $\\mu=\\mu^0+RT\\ln x$, where $x$ is the molar fraction of the species, we obtain\n\n\\begin{eqnarray}\n\\frac{\\mathrm{d} G}{\\mathrm{d}\\xi} &=& \\sum_i\\nu_i\\mu_i\\\\ \n&=& \\sum_i\\nu_i \\big[\\mu_i^0 +RT\\ln x_i\\big]\\\\\n&=& \\sum_i\\nu_i\\mu_i^0 +RT\\sum_i\\nu_i\\ln x_i\\\\\n&=& \\Delta_r G^0 +RT\\sum_i\\ln x_i^{\\nu_i}\\\\\n&=& \\Delta_r G^0 +RT\\ln \\prod_i x_i^{\\nu_i}\\\\\n&=& \\Delta_r G^0 +RT\\ln Q\\tag{6}\n\\end{eqnarray}\n\nhere $Q$ is the reaction quotient. By remembering the definition of equilibrium constant, $-RT\\ln K_{eq}=\\Delta_r G^0$ we get\n\n\\begin{equation}\nforce = -\\frac{\\mathrm{d} G}{\\mathrm{d}\\xi} = RT\\ln K_{eq} - RT\\ln Q = RT\\ln\\ \\frac{K_{eq}}{Q} \\tag{7}\n\\end{equation}\n\nThis equation provides us with a conceptual framework to solve *numerically* any equilibrium problem.\nBecause the above equations are differential, they are valid only for infinitesimal changes in the conditions.\nHence each a small change in the concentrations, $\\nu_i\\mathrm{d}c$ the driving force may change significantly.\n\nTherefore we need to set up an iterative procedure where we compute the driving force, change the concentrations, compute the driving force\\dots until the equilibrium is reached.\n\nGiven the concentration of all species, we can compute the driving force using the equation above and alter the concentrations of all species using\n\n\\begin{equation}\n\\mathrm{d}n_i \\approx \\nu_i \\times force \\times \\mathrm{d}c \\tag{8}\n\\end{equation}\n\nwhere $\\nu_i$ are the stoichiometric coefficients of the species and $\\mathrm{d}c$ is an adjustable parameter.\n$\\mathrm{d}c$ must be small enough to maintain the validity of the approximation abover but not too small to allow for a quick convergence of the procedure.\n\n\n## Example #1: Dimerisation reaction\nIn order to elucidate how that procedure works we can take a model dimerisation reaction\n\n\\begin{equation}\n2A \\leftrightharpoons B\n\\end{equation}\n\nwhose equilibrium constant can be written as\n\\begin{equation}\nK_{eq} = \\frac{[B]}{[A]^2} = 0.156\n\\end{equation}\n\nWe now want to calculate the equilibrium concentrations of $[A]_{eq}$ and $[B]_{eq}$ given their initial concentrations $[A]_{0}$ and $[B]_{0}$.\nAlthough this is a simple problem that can be solved analytically, in this workshop we will learn how we can use an iterative method to numerically solve it.\nWe will use a relatively simple minimisation procedure that can be applied to a large number of problems, for which it is not possible or it is too complicated to get an analytic solution.\n\nImagine mixing the reagents and then to be able to monitor the concentration of all the species in the system at very short discrete time intervals (*timesteps*). What you will see is that the concentrations will change and tend to the equilibrium value. As you have learnt in first year, the reaction quotient, $Q$, can be used to decided which way to reaction will proceed, and that at equilibrium the reaction quotient is equal to the equilibrium constant. Hence, as we have discussed in class, the reaction quotient and the equilibrium constant can be use to define a *driving force* that pulls the system towards equilibrium.\n\nThis *driving force* can then be used in conjunction with an *ICE* table to numerically compute the equilibrium concentration of reactant and products.\n\n\nThe working principle of the minimisation procedure that we will employ is\n\n1. compute the reaction quotient\n\\begin{equation}\nQ = \\dfrac{\\mathrm{[B]}_0}{\\mathrm{[A]}^2_0}\n\\end{equation}\n\n2. compute the driving force\n\n\\begin{equation}\nF = RT \\ln\\bigg[\\frac{K_{eq}}{Q}\\bigg]\n\\end{equation}\n\n3. evolve the concentrations using the ICE table\n\n| | [A] | [B]\n| :---: | :--------: |:---------:\n| *I* | [A]$_0$ | [B]$_0$\n| *C* | $-2F\\delta c$ | $+F\\delta c$\n| *E* | [A]$_0$-2$F\\delta c$ | [B]$_0$+$F\\delta c$\n\n4. Repeat until the solution does not change \n\nLet's try to implement this\n\n# [Open empty notebook](Empty.ipynb)\n\n### Critical thinking questions\n1. Verify that the final equilibrium concentrations are independent of the starting conditions, provided that $[A]+[B]$ is constant\n2. What is the effect of changing the values of $\\delta c$ or the number of cycles?\n3. Is the value of RT important?\n\n## Example #2: Dissociation of a mono-protic acid\nAs an exercise let's now try to calculate the final pH of a solution of a 0.15M of acetic acid\n\n\\begin{equation}\n\\mathrm{CH_3COOH \\rightleftharpoons CH_3COO^{-} + H^+} \\qquad pK_{a} = 3.74\n\\end{equation}\n\nRemember that the equilibrium constant for the reation\n\\begin{equation}\nAH \\leftrightharpoons A^- + H^+\n\\end{equation} \nis $K_{eq}=10^{-pK_a}$ and the reaction quotient\n\\begin{equation}\nQ = \\frac{[A^-][H^+]}{[HA]}\n\\end{equation}\n\nWe can then build the ICE table\n\n| | [HA] | [A$^-$] | [H$^+$]\n| :---: |:---------: |:---------: |:---------:\n| *I* | [HA]$_0$ | [A$^-$]$_0$ | [H$^+$]$_0$\n| *C* | $-F_1\\delta c$ | $+F_1\\delta c$ | $+F_1\\delta c$\n| *E* | [HA]$_0-F\\delta c$ | [A$^-$]$_0+F_1\\delta c$ | [H$^+$]$_0+F\\delta c$\n\n\n## Example #3: External titration\nLet's now imagine that the pH of the solution of the previous example is kept constant by external titration.\n\nWhat do you have to change in your program to account for this?\nWhat is the concentration of [AH] if the pH is kept to values of 3, 6, 9 or 12?\n\n## Example #4: Coupled reactions\nLet's now imagine we have two coupled reactions in the system, _e.g._ ascorbic acid in pure water.\n\n\\begin{eqnarray}\nH_2C_6H_6O_6 + H_2O &\\leftrightharpoons& H_3O^+ + HC_6H_6O_6^- &\\qquad\\qquad& pK_{a1} = 4.1\\\\\nHC_6H_6O_6^- + H_2O &\\leftrightharpoons& H_3O^+ + C_6H_6O_6^= &\\qquad\\qquad& pK_{a2} = 11.8 \\\\\n\\end{eqnarray}\n\nFor brevity, let's call ascorbic acid \"H$_2$A\".\n\nWe now have to compute two reaction quotients \n\n\\begin{eqnarray}\nQ_1 = \\frac{[HA^-][H^+]}{[H_2A]} &\\qquad\\qquad& Q_2 = \\frac{[A^=][H^+]}{[HA^-]}\\\\\n\\end{eqnarray}\n\nand use a \"double\" ICE table.\n\n| | [H$_2$A] | [HA$^-$] | [A$^=$] | [H$^+$]\n| :---: | :--------: |:---------: |:---------: |:---------:\n| *I* | [H$_2$A]$_0$ | [HA$^-$]$_0$ | [A$^=$]$_0$ | [H$^+$]$_0$\n| *C1* | $-F_1\\delta c$ | $+F_1\\delta c$ | $-$ | $+F_1\\delta c$\n| *C2* | $-$ | $-F_2\\delta c$ | $+F_2\\delta c$ | $+F_2\\delta c$ \n| *E* | [H$_2$A]$_0-F_1\\delta c$ | [HA$^-$]$_0+(F_1-F_2)\\delta c$ | [A$^=$]$_0+F_2\\delta c$ | [H$^+$]$_0+(F_1+F_2)\\delta c$\n\nLet's now try to implement this in our notebook.\n", "meta": {"hexsha": "8a0cc93dc1d2b60106a0754962ac208f1394ad92", "size": 12035, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "week_09_chemicalEquilibrium/equilibrium.ipynb", "max_stars_repo_name": "blake-armstrong/TeachingNotebook", "max_stars_repo_head_hexsha": "30cdca5bffd552eaecc0368c3e92744c4d6d368c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "week_09_chemicalEquilibrium/equilibrium.ipynb", "max_issues_repo_name": "blake-armstrong/TeachingNotebook", "max_issues_repo_head_hexsha": "30cdca5bffd552eaecc0368c3e92744c4d6d368c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "week_09_chemicalEquilibrium/equilibrium.ipynb", "max_forks_repo_name": "blake-armstrong/TeachingNotebook", "max_forks_repo_head_hexsha": "30cdca5bffd552eaecc0368c3e92744c4d6d368c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 43.9233576642, "max_line_length": 639, "alphanum_fraction": 0.5872870794, "converted": true, "num_tokens": 2733, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073802837477, "lm_q2_score": 0.7490872075132152, "lm_q1q2_score": 0.4124529449329195}} {"text": "# \u57fa\u4e8e\u8109\u51b2\u7684\u53d8\u5206\u91cf\u5b50\u672c\u5f81\u6c42\u89e3\u5668\n\n\n*\u7248\u6743\u6240\u6709 (c) 2021 \u767e\u5ea6\u91cf\u5b50\u8ba1\u7b97\u7814\u7a76\u6240\uff0c\u4fdd\u7559\u6240\u6709\u6743\u5229\u3002*\n\n## \u5185\u5bb9\u6982\u8981\n**\u6ce8\u610f\uff1a\u5b8c\u6574\u8fd0\u884c\u672c\u6559\u7a0b\u7684\u7a0b\u5e8f\u53ef\u80fd\u4f1a\u82b1\u8d39\u8d85\u8fc7 50 \u4e2a Quantum Hub \u70b9\u6570**\n\n\u672c\u6559\u7a0b\u5c06\u4ecb\u7ecd\u5982\u4f55\u5728\u8109\u51b2\u5c42\u9762\u5b9e\u73b0\u53d8\u5206\u91cf\u5b50\u672c\u5f81\u6c42\u89e3\u5668\u7b97\u6cd5\u3002\u672c\u6559\u7a0b\u7684\u5927\u7eb2\u5982\u4e0b\uff1a\n\n- \u53d8\u5206\u91cf\u5b50\u672c\u5f81\u6c42\u89e3\u5668\uff08VQE\uff09\n- \u57fa\u4e8e\u8109\u51b2\u7684\u53d8\u5206\u91cf\u5b50\u672c\u5f81\u6c42\u89e3\u5668\uff08PBVQE\uff09\n- \u51c6\u5907\u5de5\u4f5c\n- \u6784\u9020\u54c8\u5bc6\u987f\u91cf\n- \u4f18\u5316\u53cc\u91cf\u6bd4\u7279\u95e8\u7684\u8109\u51b2\n- \u6784\u9020\u6c22\u5206\u5b50\u7684\u54c8\u5bc6\u987f\u91cf\n- \u6784\u9020\u57fa\u4e8e\u8109\u51b2\u7684\u53c2\u6570\u5316\u7535\u8def\u53ca\u4f18\u5316\n- \u603b\u7ed3\n\n## \u53d8\u5206\u91cf\u5b50\u672c\u5f81\u6c42\u89e3\u5668\uff08VQE\uff09\n\n\u53d8\u5206\u91cf\u5b50\u672c\u5f81\u6c42\u89e3\u5668\uff08Variational Quantum Eigensolver, VQE\uff09\u662f\u5728\u5608\u6742\u4e2d\u578b\u91cf\u5b50\uff08Noisy Intermediate-Scale Quantum, NISQ\uff09\u8ba1\u7b97\u673a\u4e0a\u8fd0\u884c\u7684\u4e00\u79cd\u8fd1\u4f3c\u6c42\u89e3\u5206\u5b50\u57fa\u6001\u80fd\u91cf\u7684\u7b97\u6cd5\u3002\u5b83\u7684\u57fa\u672c\u65b9\u6cd5\u662f\u4f30\u8ba1\u7ed9\u5b9a\u54c8\u5bc6\u987f\u91cf\u7684\u6700\u5c0f\u672c\u5f81\u503c\u5e76\u6c42\u5176\u57fa\u6001\u3002\u5bf9\u4e8e\u8fd1\u671f\u7684\u91cf\u5b50\u8ba1\u7b97\u8bbe\u5907\uff0c\u95e8\u9519\u8bef\u7387\u8f83\u9ad8\u3001\u9000\u76f8\u5e72\u65f6\u95f4\u8f83\u77ed\u4ee5\u53ca\u8fde\u901a\u6027\u8f83\u5dee\u7b49\u95ee\u9898\u9650\u5236\u4e86\u91cf\u5b50\u7535\u8def\u7684\u6df1\u5ea6\u3002\u7136\u800c\uff0cVQE \u7b97\u6cd5\u53ea\u9700\u8981\u6df1\u5ea6\u8f83\u4f4e\u7684\u91cf\u5b50\u7535\u8def\u5373\u53ef\u5b9e\u73b0\uff0c\u56e0\u800c\u5b83\u88ab\u8ba4\u4e3a\u662f\u5229\u7528 NISQ \u8bbe\u5907\u89e3\u51b3\u5b9e\u9645\u95ee\u9898\u7684\u7406\u60f3\u9009\u62e9\u3002\n\nVQE \u7684\u57fa\u672c\u4efb\u52a1\u662f\u5236\u5907\u53c2\u6570\u5316\u7684\u91cf\u5b50\u6001\uff08trail state\uff09$|\\psi(\\vec{\\theta})\\rangle$ \u5e76\u4f30\u8ba1\u51fa\u7ed9\u5b9a\u5206\u5b50\u79bb\u6563\u54c8\u5bc6\u987f\u91cf $\\hat{H}_{\\rm mole}$ \u7684\u57fa\u6001\u80fd\u91cf\u3002\u5176\u4e2d\uff0c\u91cf\u5b50\u6001 $|\\psi(\\vec{\\theta})\\rangle$ \u662f\u7531\u53c2\u6570\u5316\u7684\u91cf\u5b50\u7535\u8def\uff08ansatz\uff09\u751f\u6210\u3002\u5728\u8fd9\u4e2a\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u91c7\u7528\u7ecf\u5178\u7684\u4f18\u5316\u65b9\u6cd5\u6765\u5bfb\u627e\u4e00\u7ec4\u6700\u4f18\u7684\u53c2\u6570 $\\vec{\\theta}^*$\uff0c\u4ee5\u6700\u5c0f\u5316\u671f\u671b\u503c $E = \\langle \\psi(\\vec{\\theta}) | \\hat{H}_{\\rm mole} | \\psi(\\vec{\\theta}) \\rangle$\uff0c\u5373\u5206\u5b50\u54c8\u5bc6\u987f\u91cf $\\hat{H}_{\\rm mole}$ \u7684\u8fd1\u4f3c\u57fa\u6001\u80fd\u91cf $E_0^*$\uff1a\n\n$$\nE_0^* = {\\rm min}_{\\vec{\\theta}} \\langle \\psi(\\vec{\\theta}) | \\hat{H}_{\\rm mole} | \\psi(\\vec{\\theta}) \\rangle.\n$$\n\n\u5728\u672c\u6559\u7a0b\u4e2d\uff0c\u6211\u4eec\u5c06\u4ecb\u7ecd\u5728\u8d85\u5bfc\u5e73\u53f0\u4e0a\u4f7f\u7528 VQE \u8fd1\u4f3c\u6c42\u89e3\u6c22\u5206\u5b50\u57fa\u6001\u80fd\u91cf\u7684\u57fa\u672c\u65b9\u6cd5\u3002\u6211\u4eec\u5c06\u8003\u8651\u591a\u79cd\u975e\u7406\u60f3\u56e0\u7d20\uff0c\u5e76\u4ece\u8109\u51b2\u5c42\u9762\u6a21\u62df VQE \u7b97\u6cd5\u3002\u9996\u5148\uff0c\u6211\u4eec\u6765\u4ecb\u7ecd\u672c\u6559\u7a0b\u4e2d\u6240\u4f7f\u7528\u7684\u53c2\u6570\u5316\u91cf\u5b50\u7535\u8def\u6a21\u677f\uff0c\u5982\u4e0b\u56fe\u6240\u793a\uff1a\n\n\n\n\u5b83\u4e3b\u8981\u7531\u82e5\u5e72\u53c2\u6570\u5316\u7684\u5355\u91cf\u5b50\u6bd4\u7279\u65cb\u8f6c\u95e8\u548c CNOT \u95e8\u7ec4\u6210\u3002\u7531\u4e8e CNOT \u95e8\u5728\u8d85\u5bfc\u5e73\u53f0\u4e0a\u4e0d\u80fd\u76f4\u63a5\u5b9e\u73b0\uff0c\u56e0\u800c\u5728\u672c\u6559\u7a0b\u4e2d\uff0c\u6211\u4eec\u5c06\u4f7f\u7528\u8d85\u5bfc\u5e73\u53f0\u4e2d\u5b9e\u73b0\u6548\u7387\u66f4\u9ad8\u7684\uff08hardware-efficient\uff09\u53cc\u91cf\u5b50\u6bd4\u7279\u95e8\uff0c\u5373 Cross-Resonance \uff08CR\uff09\u95e8\u6765\u4ee3\u66ff CNOT \u95e8\u4f5c\u4e3a\u7ea0\u7f20\u95e8\u3002\u540c\u6837\uff0cCR \u95e8\u4e5f\u53ef\u4ee5\u914d\u5408\u82e5\u5e72\u5355\u91cf\u5b50\u6bd4\u7279\u95e8\u4ea7\u751f\u6700\u5927\u7ea0\u7f20\u6001\u3002\u7406\u60f3 CR \u95e8\u7684\u77e9\u9635\u4e3a\uff1a\n\n$$\n\\begin{equation}\n\\hat{U}_{\\rm CR}(\\alpha) = \\begin{bmatrix}\n\\cos{\\frac{\\alpha}{2}} & -i\\sin{\\frac{\\alpha}{2}} & 0 & 0 \\\\\n-i\\sin{\\frac{\\alpha}{2}} & \\cos{\\frac{\\alpha}{2}} & 0 & 0 \\\\ \n0 & 0 & \\cos{\\frac{\\alpha}{2}} & i\\sin{\\frac{\\alpha}{2}} \\\\\n0 & 0 & i\\sin{\\frac {\\alpha}{2}} & \\cos{\\frac{\\alpha}{2}} \n\\end{bmatrix}.\n\\end{equation}\n$$\n\n\u5728\u8fd9\u91cc\uff0c\u6211\u4eec\u8bbe\u7f6e $\\alpha = -\\pi/2$\u3002\u5173\u4e8e CR \u95e8\u7684\u66f4\u591a\u7ec6\u8282\u8bf7[\u70b9\u51fb\u8fd9\u91cc](https://quanlse.baidu.com/#/doc/tutorial-cr)\u3002\n\n## \u57fa\u4e8e\u8109\u51b2\u7684\u53d8\u5206\u91cf\u5b50\u672c\u5f81\u6c42\u89e3\u5668\uff08PBVQE\uff09\n\n\u5728\u672c\u6559\u7a0b\u4e2d\uff0c\u6211\u4eec\u5c06\u4ece\u8109\u51b2\u5c42\u9762\u7814\u7a76 VQE \u7b97\u6cd5\uff0c\u6211\u4eec\u79f0\u4e4b\u4e3a **pulse-based VQE (PBVQE)**\u3002\u4e0e\u6807\u51c6\u7684 VQE \u7b97\u6cd5\u4e0d\u540c\uff0cPBVQE \u4e0d\u518d\u4f18\u5316\u903b\u8f91\u91cf\u5b50\u7535\u8def\u4e2d\u6bcf\u4e2a\u65cb\u8f6c\u95e8\u7684\u53c2\u6570\uff0c\u800c\u662f\u76f4\u63a5\u5c06\u8109\u51b2\u53c2\u6570\u4f5c\u4e3a\u4f18\u5316\u53c2\u6570\u6765\u6700\u5c0f\u5316\u635f\u5931\u51fd\u6570\uff08\u5373\u57fa\u6001\u80fd\u91cf\uff09\u3002\u4e0b\u56fe\u663e\u793a\u4e86 PBVQE \u548c\u6807\u51c6 VQE \u7b97\u6cd5\u4e4b\u95f4\u7684\u5dee\u5f02\uff1a\n\n\n\n\u4e3a\u4e86\u5b9e\u73b0 PBVQE\uff0c\u6211\u4eec\u9700\u8981\u5c06\u903b\u8f91\u91cf\u5b50\u7535\u8def\u8f6c\u6362\u6210**\u57fa\u4e8e\u8109\u51b2\u7684\u53c2\u6570\u5316\u91cf\u5b50\u7535\u8def\uff08pulse-based ansatz\uff09**\uff0c\u5373\u903b\u8f91\u65cb\u8f6c\u95e8 $R_x(\\theta_m)$ \u548c $R_y(\\theta_m)$ \u5206\u522b\u88ab $X$ \u548c $Y$ \u901a\u9053\u4e0a\u632f\u5e45\u4e0d\u540c\u7684\u63a7\u5236\u8109\u51b2\u6240\u53d6\u4ee3\uff0c\u6211\u4eec\u79f0\u4e4b\u4e3a**\u57fa\u4e8e\u8109\u51b2\u7684\u91cf\u5b50\u95e8\uff08pulse-based gates\uff09**\uff1a\n\n\n\n\u4e0a\u56fe\u4e2d\uff0c$U_{\\rm ENT}$ \u662f\u7528\u4e8e\u4ea7\u751f\u7ea0\u7f20\u7684\u9149\u7b97\u7b26\uff08\u7ec6\u8282\u5c06\u4f1a\u5728\u540e\u9762\u7684\u7ae0\u8282\u4ecb\u7ecd\uff09\u3002\u8fd9\u91cc\uff0c\u6211\u4eec\u4f7f\u7528\u4e00\u79cd\u65b0\u7684\u7b26\u53f7\u6765\u8868\u793a**\u57fa\u4e8e\u8109\u51b2\u7684\u91cf\u5b50\u95e8**\u7684\u53c2\u6570\uff1a\n\n$$\n\\vec{A} = [A_0, \\cdots, A_m, \\cdots, A_{M-1}],\n$$\n\n\u5176\u4e2d\uff0c$M$ \u662f**\u57fa\u4e8e\u8109\u51b2\u7684\u91cf\u5b50\u95e8**\u7684\u4e2a\u6570\uff1b$A_m$ \u8868\u793a\u7b2c $m$ \u4e2a**\u57fa\u4e8e\u8109\u51b2\u7684\u91cf\u5b50\u95e8**\u7684\u9ad8\u65af\u6ce2\u5f62\u7684\u632f\u5e45\uff0c\u56e0\u800c\u8109\u51b2\u5305\u7edc\u7684\u51fd\u6570\u53ef\u4ee5\u5199\u4e3a\uff1a\n\n$$\n\\Omega_m(t) = A_m e^{-(\\frac{t - \\tau_m}{\\sqrt{2} \\sigma_m}) ^2}.\n$$\n\n\u9664\u8109\u51b2\u5f3a\u5ea6\u4ee5\u5916\u7684\u5176\u5b83\u9ad8\u65af\u8109\u51b2\u53c2\u6570\uff0c\u5982\u5bbd\u5ea6 $\\sigma_m$ \u548c\u4e2d\u5fc3\u4f4d\u7f6e $\\tau_m$ \u7b49\u5728\u6574\u4e2a\u8fc7\u7a0b\u4e2d\u90fd\u5c06\u88ab\u56fa\u5b9a\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u6bcf\u4e2a**\u57fa\u4e8e\u8109\u51b2\u7684\u91cf\u5b50\u95e8**\u90fd\u53ea\u6709\u4e00\u4e2a\u9700\u8981\u4f18\u5316\u7684\u53c2\u6570\u3002\u5f15\u5165**\u57fa\u4e8e\u8109\u51b2\u7684\u53c2\u6570\u5316\u91cf\u5b50\u7535\u8def**\u540e\uff0c\u5728\u6bcf\u6b21\u8fed\u4ee3\u4e2d\u6211\u4eec\u4e0d\u518d\u9700\u8981\u4f18\u5316\u4ea7\u751f\u7528\u4e8e\u5b9e\u73b0\u903b\u8f91\u91cf\u5b50\u7535\u8def\u7684\u9a71\u52a8\u8109\u51b2\uff0c\u8fd9\u5927\u5927\u63d0\u9ad8\u4e86 VQE \u7684\u6548\u7387\u548c\u7ed3\u679c\u7684\u51c6\u786e\u6027\u3002\n\n\u5728\u4e0a\u9762\u7684\u7ae0\u8282\u4e2d\uff0c\u6211\u4eec\u7b80\u8981\u4ecb\u7ecd\u4e86\u4f20\u7edf\u7684 VQE \u548c PBVQE\u3002\u5728\u4e0b\u9762\u7684\u90e8\u5206\u4e2d\uff0c\u6211\u4eec\u5c06\u9010\u6b65\u6f14\u793a\u4f7f\u7528\u91cf\u8109\u5b9e\u73b0 PBVQE \u7684\u65b9\u6cd5\u3002\n\n## \u51c6\u5907\u5de5\u4f5c\n\n\u6210\u529f\u5b89\u88c5\u91cf\u8109\u540e\uff0c\u60a8\u53ef\u4ee5\u6309\u7167\u672c\u6559\u7a0b\u8fd0\u884c\u4e0b\u9762\u7684\u91cf\u8109\u7a0b\u5e8f\u3002\u8981\u8fd0\u884c\u6b64\u6559\u7a0b\uff0c\u60a8\u9700\u8981\u4ece\u91cf\u8109\uff08Quanlse\uff09\u548c\u5176\u5b83\u5e38\u7528 Python \u5e93\u5bfc\u5165\u4ee5\u4e0b\u5305\uff1a\n\n\n```python\n# This module creates the Hamiltonian dictionary\nfrom Quanlse.QHamiltonian import QHamiltonian\n\n# These functions help us perform matrix calculations\nfrom Quanlse.Utils.Functions import tensor\nfrom Quanlse.Utils.Infidelity import unitaryInfidelity\n\n# These functions define useful operator matrices\nfrom Quanlse.QOperator import sigmaX, sigmaY, sigmaZ, sigmaI\n\n# This function generates wave data\nfrom Quanlse.QWaveform import gaussian, square\n\n# This function uploads jobs to Quanlse Cloud Service and receives results\nfrom Quanlse.remoteSimulator import remoteSimulatorRunHamiltonian as runHamiltonian\n\n# This module defines matrices of the frequently used quantum gates\nfrom Quanlse.QOperation import FixedGate\n\n# This module saves the PBVQE results\nfrom Quanlse.Define import outputPath\n```\n\n\n```python\n# Import the necessary packages\nimport os\nfrom numpy import linalg, min, random, savez, load, identity, kron\nfrom math import pi\nfrom functools import reduce\nfrom scipy import optimize\n\n# Generate the path of npz file\nlocalFile = os.path.join(outputPath, f'pbvqe.npz')\n```\n\n## \u6784\u9020\u54c8\u5bc6\u987f\u91cf\n\n\u9996\u5148\uff0c\u6211\u4eec\u5b9a\u4e49\u4e00\u4e9b\u5fc5\u8981\u7684\u5e38\u6570\uff0c\u5305\u62ec\u4efb\u610f\u6ce2\u5f62\u53d1\u751f\u5668\uff08arbitrary wave generator, AWG\uff09\u7684\u91c7\u6837\u5468\u671f\u3001\u7cfb\u7edf\u7684\u91cf\u5b50\u6bd4\u7279\u7684\u6570\u91cf\u53ca\u80fd\u7ea7\u3002\n\n\n```python\n# Sampling period (Nano second)\ndt = 2.0\n\n# Number of qubits\nqubits = 4\n\n# System energy level\nlevel = 2\n```\n\n\u7136\u540e\uff0c\u6211\u4eec\u5b9a\u4e49\u8d85\u5bfc\u91cf\u5b50\u6bd4\u7279\u7684\u786c\u4ef6\u53c2\u6570\u3002`freq` \u5217\u8868\u4e2d\u7684\u9879\u5206\u522b\u662f $\\omega_{\\rm q0}, \\omega_{\\rm q1}, \\omega_{\\rm q2}, \\omega_{\\rm q3}$\uff0c\u5373\u6bcf\u4e2a\u91cf\u5b50\u6bd4\u7279\u7684\u8dc3\u8fc1\u9891\u7387\uff1b`coupling` \u5217\u8868\u4e2d\u7684\u9879\u5206\u522b\u4fdd\u5b58\u4e86\u91cf\u5b50\u6bd4\u7279 0-1\u30011-2\u30012-3\u30013-0 \u7684\u8026\u5408\u4fe1\u606f\u3002\u5229\u7528\u65cb\u8f6c\u6ce2\u8fd1\u4f3c\uff08Rotating Wave Approximation, RWA\uff09\uff0c\u6211\u4eec\u5c06\u7cfb\u7edf\u5b9a\u4e49\u5728\u9891\u7387\u4e3a $\\omega_{\\rm RWA} = \\omega_{\\rm q0} = \\omega_{\\rm q2} = 4.914 \\times 2\\pi$ GHz \u7684\u65cb\u8f6c\u5750\u6807\u7cfb\u4e2d\u3002\n\n\n```python\n# Define the hardware parameters of the qubits (GHz)\nfreq = [4.914 * (2 * pi), 5.114 * (2 * pi), 4.914 * (2 * pi), 5.114 * (2 * pi)]\n\n# Define the coupling strength (GHz)\ncoupling = [\n [[0, 1], 0.016 * (2 * pi)],\n [[1, 2], 0.016 * (2 * pi)],\n [[2, 3], 0.016 * (2 * pi)],\n [[3, 0], 0.016 * (2 * pi)]\n]\n\n# Frequency of rotating frame (GHz)\nrwa = 4.914 * (2 * pi)\n```\n\n\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u4e3a\u6240\u6709\u7684\u5355\u91cf\u5b50\u6bd4\u7279\u95e8\u548c\u53cc\u91cf\u5b50\u6bd4\u7279\u95e8\u8bbe\u7f6e\u56fa\u5b9a\u7684\u6267\u884c\u65f6\u95f4\uff1a\n\n\n```python\n# Gate duration time (Nano second)\ntg2q = 200\ntg1q = 64\n```\n\n\u968f\u540e\uff0c\u6211\u4eec\u6839\u636e\u4ee5\u4e0b\u786c\u4ef6\u7ed3\u6784\u4f7f\u7528\u91cf\u8109\u521b\u5efa\u5176\u54c8\u5bc6\u987f\u91cf\uff0c\u6bcf\u4e2a\u91cf\u5b50\u6bd4\u7279\u4e0e\u5176\u76f8\u90bb\u91cf\u5b50\u6bd4\u7279\u8026\u5408\uff0c\u8026\u5408\u5f3a\u5ea6\u4e3a\u4e00\u4e2a\u6052\u5b9a\u503c\uff1a\n\n\n\n\u4e0a\u8ff0\u7cfb\u7edf\u7684\u54c8\u5bc6\u987f\u91cf\u53ef\u4ee5\u5199\u4e3a\uff1a\n$$\n\\hat{H}_{\\rm total} = \\sum_{q=0}^{3} \\delta_{q} \\hat{a}^{\\dagger}_{q}\\hat{a}_{q} + \\frac{1}{2}\\sum_{q=0}^{3}g_{q,(q+1) {\\rm\\ mod}\\ 4}(\\hat{a}_{q}\\hat{a}^{\\dagger}_{(q+1) {\\rm\\ mod}\\ 4}+\\hat{a}^{\\dagger}_{q}\\hat{a}_{(q+1) {\\rm\\ mod}\\ 4}) + \\sum_{q=0}^{3}\\Omega_{q}^x (t) \\hat{\\sigma}_{q}^{x} + \\sum_{q=0}^{3}\\Omega_{q}^y (t) \\hat{\\sigma}_{q}^{y} + \\sum_{q=0}^{3}\\Omega_{q}^z (t) \\hat{\\sigma}_{q}^{z} ,\n$$\n\n\u5176\u4e2d $\\hat{a}_{q}$ \u548c $\\hat{a}^{\\dagger}_{q}$ \u5206\u522b\u662f\u4f5c\u7528\u5728\u7b2c $q$ \u4e2a\u91cf\u5b50\u6bd4\u7279\u7684\u6e6e\u706d\u548c\u4ea7\u751f\u7b97\u7b26\u3002$\\hat{\\sigma}^x_{q}, \\hat{\\sigma}^y_{q}$ \u548c $\\hat{\\sigma}^z_{q}$ \u5206\u522b\u662f\u4f5c\u7528\u5728\u7b2c $q$ \u4e2a\u91cf\u5b50\u6bd4\u7279\u4e0a\u7684\u6ce1\u5229\u7b97\u7b26\u3002$\\delta_{q}=\\omega_{q} - \\omega_{\\rm RWA}$ \u8868\u793a\u7b2c $q$ \u4e2a\u91cf\u5b50\u6bd4\u7279\u7684\u5931\u8c03\u5f3a\u5ea6\uff1b$g_{q,(q+1){\\rm\\ mod}\\ 4}$ \u662f\u7b2c $q$ \u548c\u7b2c $(q+1) {\\rm\\ mod}\\ 4$ \u4e2a\u91cf\u5b50\u6bd4\u7279\u4e4b\u95f4\u7684\u8026\u5408\u5f3a\u5ea6\uff1b $\\Omega_q^{x,y,z}(t)$ \u662f\u4f5c\u7528\u5728\u7b2c $q$ \u4e2a\u91cf\u5b50\u6bd4\u7279\u4e0a\u7684\u78c1\u901a\u8c03\u63a7\u6216\u5fae\u6ce2\u8c03\u63a7\u7684\u5305\u7edc\u51fd\u6570\u3002\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u91cf\u8109\u65b9\u4fbf\u5730\u5b9a\u4e49\u4e0a\u8ff0\u7cfb\u7edf\u7684\u54c8\u5bc6\u987f\u91cf\uff1a\n\n\n```python\n# Create the Hamiltonian\nvqeHam = QHamiltonian(qubits, level, dt)\n\n# Add the coupling terms\nfor item in coupling:\n q0, q1 = item[0][0], item[0][1]\n vqeHam.addCoupling([q0, q1], g=item[1] / 2)\n\nfor qubit in range(qubits):\n # Add the detuning terms\n detuning = freq[qubit] - rwa\n vqeHam.addDrift(sigmaZ, qubit, coef=detuning)\n```\n\n\u5173\u4e8e\u4f7f\u7528\u91cf\u8109\u6784\u5efa\u54c8\u5bc6\u987f\u91cf\u7684\u66f4\u591a\u65b9\u6cd5\uff0c\u53ef\u4ee5\u67e5\u770b\u6559\u7a0b[\u5355\u91cf\u5b50\u6bd4\u7279\u95e8](https://quanlse.baidu.com/#/doc/tutorial-single-qubit)\u3002\n\n## \u4f18\u5316\u53cc\u91cf\u5b50\u6bd4\u7279\u95e8\n\n\u5728\u672c\u6559\u7a0b\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528 CR \u95e8\u4f5c\u4e3a\u7ea0\u7f20\u95e8\uff08\u5173\u4e8e CR \u95e8\u7684\u66f4\u591a\u4fe1\u606f\uff0c\u53ef\u4ee5\u67e5\u770b\u6559\u7a0b\uff1a[Cross-Resonance \u95e8](https://quanlse.baidu.com/#/doc/tutorial-cr)\uff09\u3002\u7531\u4e8e\u5728\u672c\u6559\u7a0b\u4e2d\uff0c\u76f8\u90bb\u91cf\u5b50\u6bd4\u7279\u4e4b\u95f4\u7684\u8026\u5408\u65b9\u5f0f\u4e3a\u76f4\u63a5\u8026\u5408\uff0c\u56e0\u6b64\u5728\u4e00\u4e2a\u91cf\u5b50\u6bd4\u7279\u4e0a\u65bd\u52a0 $X$ \u8109\u51b2\u4f1a\u540c\u65f6\u5f71\u54cd\u5230\u53e6\u5916\u4e24\u4e2a\u76f8\u90bb\u7684\u91cf\u5b50\u6bd4\u7279\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u5728\u8bbe\u8ba1\u8109\u51b2\u65f6\u9700\u8981\u8003\u8651\u8fd9\u4e2a\u56e0\u7d20\uff0c\u4ee5\u6291\u5236\u4e32\u6270\u9020\u6210\u7684\u5f71\u54cd\u3002\n\n\n\n\u5728\u8fd9\u91cc\uff0c\u6211\u4eec\u4f7f\u7528 `vqeHam.subSystem()` \u4ece\u7cfb\u7edf\u54c8\u5bc6\u987f\u91cf `vqeHam` \u4e2d\u63d0\u53d6\u4e24\u4e2a\u7531\u4e09\u4e2a\u91cf\u5b50\u6bd4\u7279\u7ec4\u6210\u7684\u5b50\u7cfb\u7edf\u7528\u4e8e\u4f18\u5316 CR \u95e8\uff0c\u5176\u4e2d\u4e00\u4e2a\u662f\u7531\u91cf\u5b50\u6bd4\u7279 0-1-2 \u7ec4\u6210\u7684\u5b50\u7cfb\u7edf\uff0c\u53e6\u4e00\u4e2a\u662f\u7531\u91cf\u5b50\u6bd4\u7279 1-2-3 \u7ec4\u6210\u7684\u5b50\u7cfb\u7edf\u3002\u5728\u8fd9\u4e9b\u5b50\u7cfb\u7edf\u4e0a\uff0c\u6211\u4eec\u5206\u522b\u8bbe\u7f6e $\\hat{U}_{\\rm goal}=I\\otimes\\hat{U}_{\\rm CR}$ \u4f5c\u4e3a\u76ee\u6807\u9149\u77e9\u9635\u6765\u4f18\u5316\u76f8\u5e94\u8109\u51b2\uff0c\u5373\u5728\u5b50\u7cfb\u7edf\u7684\u7b2c\u4e8c\u548c\u7b2c\u4e09\u4e2a\u91cf\u5b50\u7cfb\u7edf\u4e0a\u751f\u6210\u4e00\u4e2a CR \u95e8\u3002\n\n\u6211\u4eec\u5b9a\u4e49\u51fd\u6570 `makeCrPulse()` \u7528\u4e8e\u751f\u6210 CR \u95e8\u6240\u9700\u7684\u8109\u51b2\u5e8f\u5217\u3002\u6211\u4eec\u5728\u5f53\u524d\u5b50\u7cfb\u7edf\u7684\u7b2c\u4e8c\u4e2a\u91cf\u5b50\u6bd4\u7279\u4e0a\u65bd\u52a0\u9ad8\u65af\u5fae\u6ce2\u9a71\u52a8\u8109\u51b2\uff0c\u540c\u65f6\u56fa\u5b9a\u5176\u5bbd\u5ea6\u548c\u4e2d\u5fc3\u4f4d\u7f6e\uff0c\u5e76\u5c06\u5176\u632f\u5e45\u4f5c\u4e3a\u4f18\u5316\u7684\u7b2c\u4e00\u4e2a\u53c2\u6570\u3002\u7b2c\u4e8c\u4e2a\u9700\u8981\u4f18\u5316\u7684\u53c2\u6570\u662f\u65bd\u52a0\u5728\u7b2c\u4e00\u4e2a\u91cf\u5b50\u6bd4\u7279\u4e0a\u7684\u78c1\u901a\u63a7\u5236\u7684\u632f\u5e45\u3002\u8bf7\u6ce8\u610f\uff0c`tag=\"det\"` \u7684\u9a71\u52a8\u8fd8\u540c\u65f6\u7528\u4e8e\u5c06\u65cb\u8f6c\u5750\u6807\u53c2\u8003\u7cfb\u8f6c\u6362\u4e3a\u7279\u5b9a\u9891\u7387\u3002\n\n\n```python\ndef makeCrPulse(ham, subSys3q, driveFreq, amp, shift, t):\n \"\"\" Assemble the pulses for CR gates \"\"\"\n subHam = ham if subSys3q is None else ham.subSystem(subSys3q)\n subHam.clearWaves()\n subHam.appendWave(sigmaX, 1, gaussian(t, amp, tg2q / 2, tg2q / 8), tag=\"XY\")\n # frame transformation\n subHam.appendWave(sigmaZ, 0, square(t, rwa - driveFreq + shift), tag=\"Z\")\n subHam.appendWave(sigmaZ, 1, square(t, rwa - driveFreq), tag=\"det\")\n subHam.appendWave(sigmaZ, 2, square(t, rwa - driveFreq), tag=\"det\")\n return subHam.job if subSys3q is None else subHam.outputInverseJob(qubits)\n```\n\n\u968f\u540e\uff0c\u6211\u4eec\u5b9a\u4e49\u4e00\u4e2a\u51fd\u6570 `optimize_cr()` \u6765\u8fdb\u884c\u4f18\u5316\u8fc7\u7a0b\uff0c\u5e76\u4fdd\u5b58\u6700\u4f73\u53c2\u6570\u4ee5\u4f9b\u8fdb\u4e00\u6b65\u4f7f\u7528\u3002\n\n\n```python\ndef optimizeCr(subSys3q, driveFreq):\n \"\"\" Realize a CR gate on the second & third qubits \"\"\"\n crHam = vqeHam.subSystem(subSys3q)\n uGoal = tensor([identity(2), FixedGate.CR.getMatrix()])\n\n def crLoss(_x):\n # Clear and add waves\n crHam.clearWaves()\n # Generate and add waves for CR gate implementation\n _crJob = makeCrPulse(crHam, None, driveFreq, _x[0], _x[1], tg2q)\n # Simulate the system's evolution and obtain the infidelity\n unitary = crHam.simulate(job=_crJob)[0][\"unitary\"]\n infidelity = unitaryInfidelity(uGoal, unitary, 3)\n return infidelity\n\n opt = optimize.dual_annealing(crLoss, [(-2, 2), (-0.2, 0.2)], maxiter=60)\n print(\"Min infidelity:\", opt[\"fun\"])\n return opt[\"x\"][0], opt[\"x\"][1]\n\nlhlQ1X, lhlQ0Z = optimizeCr([0, 1, 2], 4.914 * 2 * pi)\nhlhQ1X, hlhQ0Z = optimizeCr([1, 2, 3], 5.114 * 2 * pi)\n```\n\n## \u6784\u9020\u6c22\u5206\u5b50\u7684\u54c8\u5bc6\u987f\u91cf\n\n\u5728\u8fd9\u4e00\u8282\u4e2d\uff0c\u6211\u4eec\u5c06\u4ecb\u7ecd\u5982\u4f55\u5728\u8109\u51b2\u5c42\u9762\u4e0a\u4f30\u8ba1\u6c22\u5206\u5b50\u7684\u57fa\u6001\u80fd\u91cf\u3002\u6211\u4eec\u5c06\u7701\u7565\u8d39\u7c73\u5b50\u2014\u91cf\u5b50\u6bd4\u7279\uff08fermion-to-qubit\uff09\u6620\u5c04\u7684\u5177\u4f53\u7ec6\u8282\uff08\u8bf7\u8bbf\u95ee[\u91cf\u6868](https://github.com/PaddlePaddle/Quantum/blob/master/tutorial/quantum_simulation/VQE_CN.ipynb)\u83b7\u5f97\u66f4\u591a\u76f8\u5173\u4fe1\u606f\uff09\u3002\u9996\u5148\uff0c\u6211\u4eec\u5b9a\u4e49\u4e00\u4e2a\u51fd\u6570 `pauli_str_to_matrix()`\uff0c\u5c06**\u6ce1\u5229\u5b57\u7b26\u4e32**\u8f6c\u6362\u4e3a\u6c22\u5206\u5b50\u7684\u79bb\u6563\u54c8\u5bc6\u987f\u91cf $\\hat{H}_{\\rm mole}$\uff1a\n\n\n```python\ndef pauliStrToMatrix(pauli_str, n):\n \"\"\"\n Convert the Pauli string in Hamiltonian\n \"\"\"\n def NKron(AMatrix, BMatrix, *args):\n return reduce(\n lambda result, index: kron(result, index),\n args,\n kron(AMatrix, BMatrix), )\n pauli_dict = {\n 'i': sigmaI().matrix,\n 'x': sigmaX().matrix,\n 'y': sigmaY().matrix,\n 'z': sigmaZ().matrix\n }\n # Parse pauli_str; 'x0,z1,y4' to 'xziiy'\n new_pauli_str = []\n for coeff, op_str in pauli_str:\n init = list('i' * n)\n op_list = op_str.split(',')\n for op in op_list:\n pos = int(op[1:])\n assert pos < n, 'n is too small'\n init[pos] = op[0]\n new_pauli_str.append([coeff, ''.join(init)])\n\n # Convert new_pauli_str to matrix; 'xziiy' to NKron(x, z, i, i, y)\n matrices = []\n for coeff, op_str in new_pauli_str:\n sub_matrices = []\n for op in op_str:\n sub_matrices.append(pauli_dict[op])\n if len(op_str) == 1:\n matrices.append(coeff * sub_matrices[0])\n else:\n matrices.append(coeff * NKron(sub_matrices[0], sub_matrices[1], *sub_matrices[2:]))\n\n return sum(matrices)\n```\n\n\u5728\u8fd9\u91cc\uff0c\u6211\u4eec\u4f7f\u7528\u539f\u5b50\u95f4\u9694\u4e3a $d=74$ pm \u7684\u6c22\u5206\u5b50\u7a7a\u95f4\u6784\u578b\u6570\u636e\uff0c\u8fd9\u4e9b\u6570\u636e\u6765\u81ea[\u91cf\u6868](https://github.com/PaddlePaddle/Quantum/blob/master/tutorial/quantum_simulation/VQE_CN.ipynb)\u3002\n\n\n```python\ntargetHam = [\n [-0.042078976477822, 'i0'],\n [ 0.177712874651399, 'z0'],\n [ 0.177712874651399, 'z1'],\n [-0.242742805131446, 'z2'],\n [-0.242742805131462, 'z3'],\n [ 0.170597383288005, 'z0,z1'],\n [ 0.044750144015351, 'y0,x1,x2,y3'],\n [-0.044750144015351, 'y0,y1,x2,x3'],\n [-0.044750144015351, 'x0,x1,y2,y3'],\n [ 0.044750144015351, 'x0,y1,y2,x3'],\n [ 0.122933050561837, 'z0,z2'],\n [ 0.167683194577189, 'z0,z3'],\n [ 0.167683194577189, 'z1,z2'],\n [ 0.122933050561837, 'z1,z3'],\n [ 0.176276408043195, 'z2,z3']\n]\nhMatrix = pauliStrToMatrix(targetHam, 4)\n```\n\n\u4e0a\u8ff0\u5206\u5b50\u54c8\u5bc6\u987f\u91cf\u57fa\u6001\u80fd\u91cf\u7684\u7406\u8bba\u503c\u53ef\u4ee5\u901a\u8fc7\u5982\u4e0b\u65b9\u6cd5\u8ba1\u7b97\uff1a\n\n\n```python\n# Calculate the theoretical eigenvalue\neigVal, eigState = linalg.eig(hMatrix)\nminEigH = min(eigVal.real)\nprint(f\"Ground state energy: {minEigH} Ha\")\n```\n\n## \u57fa\u4e8e\u8109\u51b2\u7684\u91cf\u5b50\u7535\u8def\n\n\u9996\u5148\uff0c\u6211\u4eec\u53c2\u8003\u6807\u51c6 VQE \u4e2d\u6700\u5e38\u7528\u7684\u53c2\u6570\u5316\u91cf\u5b50\u7535\u8def\u6a21\u677f\uff0c\u8bbe\u8ba1\u4e86\u4e00\u4e2a\u57fa\u4e8e\u8109\u51b2\u7684\u91cf\u5b50\u7535\u8def\u3002\u4e0b\u56fe\u663e\u793a\u4e86\u8be5\u91cf\u5b50\u7535\u8def\u4e2d\u7684\u4e00\u5c42\uff0c\u5176\u4e2d\uff0c\u6bcf\u4e2a\u91cf\u5b50\u6bd4\u7279\u90fd\u5305\u542b 3 \u4e2a\u5355\u91cf\u5b50\u6bd4\u7279\u95e8\uff0c\u800c\u6bcf\u4e2a\u5355\u91cf\u5b50\u6bd4\u7279\u95e8\u90fd\u6709\u4e00\u4e2a\u53c2\u6570\u4f5c\u4e3a\u9ad8\u65af\u8109\u51b2\u5305\u7edc\u7684\u6700\u5927\u632f\u5e45\uff0c\u8109\u51b2\u5bbd\u5ea6\u548c\u4e2d\u5fc3\u4f4d\u7f6e\u662f\u56fa\u5b9a\u7684\u3002\n\n\n\n\u7531\u4e8e\u8109\u51b2\u7535\u8def\u8f83\u4e3a\u590d\u6742\uff0c\u56e0\u800c\u6211\u4eec\u5b9a\u4e49\u4e86\u4e00\u4e2a\u51fd\u6570 `makeWaveSchedule()` \u4e13\u95e8\u7528\u4e8e\u751f\u6210\u5e76\u6392\u5217\u4e0a\u8ff0\u7535\u8def\u6240\u5bf9\u5e94\u7684\u8109\u51b2\u5e8f\u5217\u3002\u5176\u4e2d\uff0c\u53c2\u6570 `x` \u662f\u4f18\u5316\u53c2\u6570\u5217\u8868\uff08\u5373\u8109\u51b2\u53c2\u6570 $\\vec{A}$\uff09\uff1b`vqeJob` \u662f\u7531 `addWave()` \u751f\u6210\u7684\u6ce2\u5f62\u6570\u636e\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u7528\u6237\u5b9a\u4e49\u6ce2\u5f62\u7684\u8be6\u7ec6\u4fe1\u606f\u3002\n\n\n```python\ndef makeWaveSchedule(x):\n \"\"\" Generate waves for pulse-based circuit \"\"\"\n # Generate pulses for CR gate\n crJob = vqeHam.createJob()\n crJob += makeCrPulse(vqeHam, [3, 0, 1], 5.114 * 2 * pi, hlhQ1X, hlhQ0Z, tg2q)\n crJob += makeCrPulse(vqeHam, [0, 1, 2], 4.914 * 2 * pi, lhlQ1X, lhlQ0Z, tg2q)\n crJob += makeCrPulse(vqeHam, [1, 2, 3], 5.114 * 2 * pi, hlhQ1X, hlhQ0Z, tg2q)\n crJob += makeCrPulse(vqeHam, [2, 3, 0], 4.914 * 2 * pi, lhlQ1X, lhlQ0Z, tg2q)\n # Assemble the pulses\n depth = int(len(x) / 12)\n vqeJob = vqeHam.createJob()\n for d in range(depth):\n gate1QJob = vqeHam.createJob()\n # Add pulses for single-qubit gates\n for q in range(4):\n # X/Y/X controls\n gate1QJob.addWave(sigmaX, q, gaussian(tg1q, x[12 * d + q], tg1q / 2, tg1q / 8), t0=0)\n gate1QJob.addWave(sigmaY, q, gaussian(tg1q, x[12 * d + 4 + q], tg1q / 2, tg1q / 8), t0=tg1q)\n gate1QJob.addWave(sigmaX, q, gaussian(tg1q, x[12 * d + 8 + q], tg1q / 2, tg1q / 8), t0=tg1q * 2)\n # Set detuning\n gate1QJob.addWave(sigmaZ, q, square(tg1q * 3, rwa - freq[q]), t0=0, tag=\"det\")\n vqeJob += gate1QJob\n vqeJob += crJob\n return vqeJob\n```\n\n\u5728\u672c\u6559\u7a0b\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528 `Scipy` \u63d0\u4f9b\u7684\u57fa\u4e8e\u68af\u5ea6\u7684\u4f18\u5316\u65b9\u6cd5\uff08L-BFGS-B\uff09\u6765\u6700\u5c0f\u5316\u76ee\u6807\u51fd\u6570\u3002\u5728\u6bcf\u6b21\u8fed\u4ee3\u4e2d\uff0cL-BFGS-B \u9700\u8981\u7528\u6237\u63d0\u4f9b\u6bcf\u4e2a\u53c2\u6570\u7684\u68af\u5ea6\u4fe1\u606f\uff0c\u5728\u8fd9\u91cc\u6211\u4eec\u4f7f\u7528\u4e24\u70b9\u6709\u9650\u5dee\u5206\u6cd5\u6765\u8fd1\u4f3c\u8ba1\u7b97\u68af\u5ea6\uff1a\n$$\n\\frac{\\partial{\\rm Loss}(\\vec{A})}{\\partial a_m} = \\frac{{\\rm Loss}(a_0, \\cdots, a_m + \\epsilon, \\cdots, a_{M-1}) - {\\rm Loss}(a_0, \\cdots, a_m - \\epsilon, \\cdots, a_{M-1})}{2\\epsilon} ,\n$$\n\n\u5176\u4e2d\uff0c$\\vec{A} = [A_0, \\cdots, A_{M-1}]$ \u662f\u8109\u51b2\u53c2\u6570\u5217\u8868\uff0c$\\epsilon$ \u662f\u4e00\u4e2a\u5f88\u5c0f\u7684\u6b63\u6570\uff0c\u800c\u635f\u5931\u51fd\u6570 ${\\rm Loss}(\\vec{A})$ \u5b9a\u4e49\u4e3a\uff1a\n\n$$\n{\\rm Loss}(\\vec{A}) = \\langle \\psi(\\vec{A}) | \\hat{H}_{\\rm mole} | \\psi(\\vec{A}) \\rangle.\n$$\n\n\u5176\u4e2d\uff0c\u91cf\u5b50\u6001 $\\psi(\\vec{A})$ \u662f\u57fa\u4e8e\u8109\u51b2\u7684\u91cf\u5b50\u7535\u8def\u6240\u4ea7\u751f\u7684\u3002\u6709\u9650\u5dee\u5206\u6cd5\u9700\u8981\u5927\u91cf\u7684\u6837\u672c\uff0c\u4f8b\u5982\uff0c\u5f53\u8109\u51b2\u53c2\u6570\u7684\u53c2\u6570\u4e3a $M$ \u65f6\uff0c\u6211\u4eec\u9700\u8981 $2M$ \u6b21\u91c7\u6837\u6765\u4f30\u8ba1\u8fd1\u4f3c\u68af\u5ea6\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u4f7f\u7528\u91cf\u8109\u4e91\u670d\u52a1\u6765\u52a0\u901f\u8fd9\u4e2a\u8fc7\u7a0b\u3002\n\n\u4e3a\u4e86\u4f7f\u7528\u91cf\u8109\u4e91\u670d\u52a1\uff0c\u6211\u4eec\u9700\u8981\u5bfc\u5165 `Define` \u5e76\u4f20\u5165 token\uff0c\u7528\u6237\u53ef\u4ee5\u5728 [Quantum-hub](http://quantum-hub.baidu.com) \u7533\u8bf7\u83b7\u5f97 token\u3002\n\n\n```python\n# Define the loss function\nimport copy\nfrom Quanlse import Define\nDefine.hubToken = \"\"\n```\n\n\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5b9a\u4e49 VQE \u7684\u635f\u5931\u51fd\u6570\u3002\u5728\u8fd9\u4e2a\u51fd\u6570\u4e2d\uff0c\u6211\u4eec\u6a21\u62df\u4e86\u8109\u51b2\u53c2\u6570\u4e3a $\\vec{x}$ \u65f6\u57fa\u4e8e\u8109\u51b2\u7684\u7535\u8def\u7684\u6f14\u5316\uff0c\u5e76\u7528\u4e0a\u9762\u63d0\u5230\u7684\u6709\u9650\u5dee\u5206\u6cd5\u8ba1\u7b97\u8fd9\u4e00\u70b9\u7684\u68af\u5ea6\u3002\u5728\u6bcf\u6b21\u8fed\u4ee3\u4e2d\uff0c\u6211\u4eec\u5c06\u5f53\u524d\u8109\u51b2\u53c2\u6570\u5217\u8868 $\\vec{x}$ \u8f93\u5165\u5230\u635f\u5931\u51fd\u6570\u4e2d\uff0c\u5e76\u5c06\u6240\u6709\u91c7\u6837\u6240\u9700\u7684\u8109\u51b2\u6570\u636e\u751f\u6210\u5e76\u6253\u5305\u5230 `waveList` \u4e2d\u3002\u6700\u7ec8\uff0c`waveList` \u5305\u542b\u7528\u4e8e\u6c42\u89e3\u68af\u5ea6\u7684 $2M$ \u6b21\u91c7\u6837\u548c\u7528\u4e8e\u83b7\u53d6\u635f\u5931\u503c\u7684 1 \u4e2a\u91c7\u6837\u7684\u8109\u51b2\u6570\u636e\u3002\n\n\u6211\u4eec\u5728\u4e0a\u9762\u7684\u6b65\u9aa4\u4e2d\u5c06\u6240\u6709\u7684\u4efb\u52a1\u96c6\u6210\u5230\u4e00\u4e2a\u5217\u8868\u4e2d\uff0c\u5373 `waveList`\uff0c\u5e76\u901a\u8fc7\u51fd\u6570 `runHamiltonian()` \u5c06\u4efb\u52a1\u5217\u8868\u63d0\u4ea4\u7ed9\u91cf\u8109\u4e91\u670d\u52a1\u3002\u6b63\u5e38\u60c5\u51b5\u4e0b\uff0c\u5927\u7ea6 15 \u5230 20 \u79d2\u540e\uff0c\u6211\u4eec\u5c06\u6536\u5230\u8fd4\u56de\u7ed3\u679c\uff0c\u7ed3\u679c\u5c06\u4f5c\u4e3a JSON \u6587\u4ef6\u4fdd\u5b58\u5230 `Output` \u6587\u4ef6\u5939\u4e2d\u3002\u540c\u65f6\uff0c\u53d8\u91cf `result` \u4f1a\u88ab\u8d4b\u4e88\u4e00\u4e2a\u5217\u8868\uff0c\u5176\u4e2d\u5305\u542b\u4e0e `waveList` \u5bf9\u5e94\u7684\u6240\u6709\u6a21\u62df\u7ed3\u679c\u3002\n\n**\u6ce8\u610f**\uff1a`waveList` \u7684\u6bcf\u4e00\u9879\u90fd\u5305\u542b\u7531 `makeWaveSchedule()` \u51fd\u6570\u751f\u6210\u7684\u57fa\u4e8e\u8109\u51b2\u7684 VQE \u7684\u6240\u6709\u6ce2\u3002\n\n\n```python\ndef loss(x):\n global lossHistory\n # Add wave for current point\n waveList = vqeHam.createJobList()\n waveList.addJob(makeWaveSchedule(x))\n\n # Add wave for calculating gradient\n for xId in range(len(x)):\n xList = copy.deepcopy(x)\n xList[xId] -= 1e-8\n waveList.addJob(makeWaveSchedule(xList))\n xList[xId] += 2 * 1e-8\n waveList.addJob(makeWaveSchedule(xList))\n\n # Simulate the evolution\n result = runHamiltonian(vqeHam, jobList=waveList)\n\n # Calculate the loss function\n lossList = []\n for item in result:\n state = item[\"unitary\"]\n lossVal = (state.conj().T @ hMatrix @ state).real[0][0]\n lossList.append(lossVal)\n\n # Calculate the gradients\n gradient = []\n for index in range(len(x)):\n gradient.append((lossList[2 + 2 * index] - lossList[1 + 2 * index]) / 1e-8 / 2)\n\n print(\"Loss function:\", lossList[0])\n lossHistory.append(lossList[0])\n return lossList[0], gradient\n```\n\n\u7136\u540e\u6211\u4eec\u4f7f\u7528\u7531 `Scipy` \u63d0\u4f9b\u7684 `fmin_l_bfgs_b()` \u51fd\u6570\u6700\u5c0f\u5316\u524d\u9762\u5b9a\u4e49\u7684\u635f\u8017\u51fd\u6570\u3002\n\n**\u6ce8\u610f**\uff1a\u6b64\u4f18\u5316\u53ef\u80fd\u9700\u8981\u8d85\u8fc7 15 \u5206\u949f\u3002\n\n\n```python\ndepth = 3\nlossHistory = []\ninitParas = [random.rand() for _ in range(depth * 12)]\nbounds = [(-1.5, 1.5) for _ in range(depth * 12)]\nx, f, d = optimize.fmin_l_bfgs_b(loss, initParas, fprime=None, bounds=bounds, maxiter=200)\n\n# Save the loss history to a file for further usage\nsavez(localFile, lossHistory)\n```\n\n\n```python\nprint(f\"The estimated ground state energy is: {f} Ha\")\nprint(\"Total iteration:\", d[\"nit\"])\n```\n\n\u53ef\u89c1\uff0c\u6700\u7ec8\u6536\u655b\u7684\u7cbe\u5ea6\u5f88\u9ad8\uff0c\u8fed\u4ee3\u6b21\u6570\u4e3a 72 \u6b21\u3002\u968f\u540e\uff0c\u6211\u4eec\u7ed8\u5236\u5b8c\u6574\u7684\u8fed\u4ee3\u8fc7\u8fc7\u7a0b\uff1a\n\n\n```python\n# Load the loss_history list from the npz file.\nlossHistory = load(localFile)['arr_0']\n\n# Plot the figures\nimport matplotlib.pyplot as plt\nplt.plot(range(len(lossHistory)), lossHistory, label=\"Energy\")\nplt.axhline(minEigH, c=\"gray\", ls=\"--\", lw=1.0)\nplt.xlabel(\"Iteration\")\nplt.ylabel(\"Energy (Ha)\")\nplt.show()\n```\n\n\u6700\u540e\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528 `plot()` \u65b9\u6cd5\u7ed8\u5236\u8109\u51b2\u5e8f\u5217\uff1a\n\n\n```python\n# Print the waveforms.\nmakeWaveSchedule(x).plot(color=['red', 'green', 'blue'])\n```\n\n## \u603b\u7ed3\n\u7528\u6237\u53ef\u4ee5\u901a\u8fc7\u70b9\u51fb\u8fd9\u4e2a\u94fe\u63a5 [tutorial-pbvqe.ipynb](https://github.com/baidu/Quanlse/blob/main/Tutorial/CN/tutorial-pbvqe-cn.ipynb) \u8df3\u8f6c\u5230\u6b64 Jupyter Notebook \u6587\u6863\u76f8\u5e94\u7684 GitHub \u9875\u9762\u5e76\u83b7\u53d6\u76f8\u5173\u4ee3\u7801\u4ee5\u8fd0\u884c\u8be5\u7a0b\u5e8f\u3002\u6211\u4eec\u9f13\u52b1\u7528\u6237\u4f7f\u7528\u91cf\u8109\u5f00\u53d1\u66f4\u591a\u8109\u51b2\u5c42\u7684 NISQ \u7b97\u6cd5\u3002\n\n## \u53c2\u8003\u6587\u732e\n\n\\[1\\] [Peruzzo, Alberto, et al. \"A variational eigenvalue solver on a photonic quantum processor.\" *Nature communications* 5 (2014): 4213.](https://doi.org/10.1038/ncomms5213)\n\n\\[2\\] [Moll, Nikolaj, et al. \"Quantum optimization using variational algorithms on near-term quantum devices.\" *Quantum Science and Technology* 3.3 (2018): 030503.](https://doi.org/10.1088/2058-9565/aab822)\n\n\\[3\\] [Kandala, Abhinav, et al. \"Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets.\" *Nature* 549.7671 (2017): 242-246.](https://doi.org/10.1038/nature23879)\n\n\\[4\\] [Rigetti, Chad, and Michel Devoret. \"Fully microwave-tunable universal gates in superconducting qubits with linear couplings and fixed transition frequencies.\" *Physical Review B* 81.13 (2010): 134507.](https://doi.org/10.1103/PhysRevB.81.134507)\n\n\\[5\\] [Meitei, Oinam Romesh, et al. \"Gate-free state preparation for fast variational quantum eigensolver simulations: ctrl-VQE.\" *arXiv preprint arXiv:2008.04302* (2020).](https://arxiv.org/abs/2008.04302)\n\n\\[6\\] [Wilhelm, Frank K., et al. \"An introduction into optimal control for quantum technologies.\" *arXiv preprint arXiv:2003.10132* (2020).](https://arxiv.org/abs/2003.10132)\n\n\\[7\\] [Krantz, Philip, et al. \"A quantum engineer's guide to superconducting qubits.\" *Applied Physics Reviews* 6.2 (2019): 021318.](https://aip.scitation.org/doi/abs/10.1063/1.5089550)\n", "meta": {"hexsha": "3e6915ba465d07e7dddd6368671e07ef3d5597b2", "size": 25048, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Tutorial/CN/tutorial-pbvqe-cn.ipynb", "max_stars_repo_name": "baidu/Quanlse", "max_stars_repo_head_hexsha": "f84a9afc66a7a404fc1ee2ea0de110f966491279", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 33, "max_stars_repo_stars_event_min_datetime": "2021-01-22T11:15:32.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-18T06:04:42.000Z", "max_issues_repo_path": "Tutorial/CN/tutorial-pbvqe-cn.ipynb", "max_issues_repo_name": "baidu/Quanlse", "max_issues_repo_head_hexsha": "f84a9afc66a7a404fc1ee2ea0de110f966491279", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Tutorial/CN/tutorial-pbvqe-cn.ipynb", "max_forks_repo_name": "baidu/Quanlse", "max_forks_repo_head_hexsha": "f84a9afc66a7a404fc1ee2ea0de110f966491279", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 5, "max_forks_repo_forks_event_min_datetime": "2021-01-25T02:56:06.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-04T13:32:49.000Z", "avg_line_length": 35.8340486409, "max_line_length": 445, "alphanum_fraction": 0.5563717662, "converted": true, "num_tokens": 8304, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872131147275, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.41245293696923524}} {"text": "```octave\n% some housekeeping stuff\nregister_graphics_toolkit (\"gnuplot\");\navailable_graphics_toolkits ();\ngraphics_toolkit (\"gnuplot\")\nclear\n% load packages\npkg load statistics\n% end of housekeeping\n```\n\n error: package statistics is not installed\n error: called from\n load_packages at line 47 column 7\n pkg at line 588 column 7\n\n\n# Chemical speciation modelling\n\n Solving for the equilibrium position of a set of simultaneous reactions subject to the constraints of mass balance and mass action is a common problem in environmental modeling.\nThere exist many available computer programs to solve these types of chemical equilibrium\nproblems, examples include MINEQL (Allison et al., 1991) and PHREEQ (Saini-Eidukat\nand Yahin, 1999). For engineering and scientific practices though these stand-alone pro-\ngrams are not necessarily completely useful. Often, engineers and scientists will use so-called high-level programming languages such as Matlab or Scilab to implement their larger scale problems. Problems such as reactive transport, or simulation of industrial scale processes can be \u201ccoded\u201d into Matlab in a very flexible task-specific manner.\n\nMany processes require the solution of simultaneous equilibria and thus the speciation\ncode presented here was developed for Matlab. In the researcher\u2019s lab this code is used\nto fit experimental data to chemical equilibrium binding constants for metal/organic and\nmetal/phosphate systems but with slight modifications the code could be embedded in Mat-\nlab code for other applications (such as transport modeling).\n\nThe Matlab code was written based on the paper by (Carrayrou et al., 2002). An example calculation for the Fe(III) system in water is presented below.\n\n\n### Iron (III) system\n\nMass balance and mass action for solving chemical equilibrium problems can all be represented in Tableau notation (Smith and Ferris, 2001; Morel and Hering, 1993). For a given defined system it is desirable to determine the equilibrium concentration of all species. Using the iron (III) system as an example a list of species of interest could include H+, OH \u2013 , Fe3+, FeOH2+, Fe(OH) +2 , and Fe(OH) \u20134 . There are other possible species but to keep the discussion simple these are the only ones included here. Also, for dilute solutions water can be assumed as a fixed component. The list includes six species so it is necessary to define six relationships in order to solve this system. The situation can be simplified if it is realized that each of these species are not independent. We can select components from the list of species and use those components to solve the equilibrium problem. For example, if we know H+ (pH) and Fe3+ we can determine the concentration of all other species from their logK values (mass action). In matrix notation we think of pH and Fe3+ as spanning the basis set.\n\nNow we need two equations and two unknowns. First relation is mass balance of iron and\nthe second is proton balance (related to electroneutrality). Now we can write the tableau:\n\n\n| H+ | Fe3+ | logK | species |\n| --- | --- | --- | --- |\n| 1 | 0 | 0 | H$^+$ |\n| 0 | 1 | 0 | Fe$^{3+}$ |\n| -1 | 0 | -14 | OH$^-$ |\n| -1 | 1 | -2.19 | FeOH$^{2+}$ |\n| -2 | 1 | -5.67 | Fe(OH)$_2^+$ |\n| -4 | 1 | -21.6 | Fe(OH)$_4^-$ |\n||||\n| TOTH | FeT | | |\n\nTable 1 completely defines the equilibrium problem. The entries in the columns are the\nstoichiometric coefficients required for formation of each species. For example, there is 1 H+ and 0 Fe3+ in H+. Also, Fe(OH) \u20134 is formed with one iron and removing 4 protons from\nwater. Given the tableau, all that remains is to determine the values for [H+] and [Fe3+] for specified TOTH and FeT. If you multiply across rows it is possible to determine species concentration and if you sum down columns the total values (mass balance) are recovered. This is best understood by writing out some entries:\n\n\\begin{align}\n{[\\mathrm{H}+]}&={[\\mathrm{H}+]}^1\\times{[\\mathrm{Fe}^{3+}]}^0\\times10^0\\\\\n{[\\mathrm{Fe}^{3+}]}&={[\\mathrm{H}+]}^0\\times{[\\mathrm{Fe}^{3+}]}^1\\times10^0\\\\\n{[\\mathrm{OH}^-]}&={[\\mathrm{H}+]}^{-1}\\times{[\\mathrm{Fe}^{3+}]}^0\\times10^{-14} \\label{eqn:Kw}\\\\\n{[\\mathrm{FeOH}^{2+}]}&={[\\mathrm{H}+]}^{-1}\\times{[\\mathrm{Fe}^{3+}]}^1\\times10^{-2.19} \\label{eqn:KH1}\\\\\n&\\text{....}\n\\end{align}\n\nIf equation Kw is rearranged to solve for $\\mathrm{[OH-]}$ then the equation above for [OH-] s obtained. As another example, to understand equation formation of FeOH given above, consider that $K_{H1}$ for the first hydrolysis of iron (III) corresponds to the following reaction:\n\n\\begin{equation}\n\\mathrm{Fe^{3+}} + \\mathrm{H_2O} \\rightleftharpoons \\mathrm{FeOH^{2+}} + \\mathrm{H+} \\quad K_{H1} = \\dfrac{\\mathrm{[FeOH^{2+}]}\\mathrm{[H+]}}{\\mathrm{[Fe^{3+}]}} \n\\end{equation}\n\nif K is rearranged to solve for $\\mathrm{[FeOH^{2+}]}$ then the equation $\\mathrm{[FeOH^{2+}]}={[\\mathrm{H}+]}^{-1}\\times{[\\mathrm{Fe}^{3+}]}^1\\times10^{-2.19}$ is obtained. To explain how the summation down the columns is the mass balance. Consider total iron ....\n\n\\begin{equation}\n\\mathrm{Fe_T}=0\\mathrm{[H+]}+1\\mathrm{[Fe^{3+}]}+0\\mathrm{[OH^-]}+1\\mathrm{[FeOH^{2+}]}+1\\mathrm{[Fe(OH)2^+]}+1\\mathrm{[Fe(OH)4^-]}\n\\end{equation}\n\nnotice that the coefficients are the entries down the iron column in the tableau.\n\nThe problem can easily be expressed in matrix notation. We'll define the $1 \\times 2$ vector of unknown component concentrations as $\\mathbf{X}$ so we can write $\\mathbf{X}=\\bigl( \\begin{smallmatrix}\n\\log\\mathrm{[H+]} & \\log\\mathrm{[Fe^{3+}]} \\end{smallmatrix} \\bigr)$. There is a $6 \\times 1$ vector of species concentrations as well \\begin{equation}\n\\mathbf{C}= \\begin{pmatrix} \n\\mathrm{[H+]} \\\\ \n\\mathrm{[Fe^{3+}]} \\\\\n\\mathrm{[OH-]} \\\\ \n\\mathrm{[FeOH^{2+}]} \\\\ \n\\mathrm{[Fe(OH)2+]} \\\\ \n\\mathrm{[Fe(OH)4-]}\n\\end{pmatrix} \\end{equation}\n\nTotal concentrations are in a $2\\times 1$ vector $\\mathbf{T} = \\bigl( \\begin{smallmatrix}\n\\text{TOTH} \\\\ \\text{Fe}_{\\mathrm{T}} \\end{smallmatrix} \\bigr)$. The logK values are summarized in the $ 6\\times 1$ vector\n\\begin{equation}\n \\mathbf{K}=\\begin{pmatrix}\n0\\\\\n0 \\\\\n\\log{K_w}\\\\\n\\log{K_{H1}} \\\\\n \\log{K_{H2}}\\\\\n \\log{K_{H4}}\\\\\n \\end{pmatrix} \\end{equation}\n\nFinally, we need a $6 \\times 2$ matrix of stoichiometric coefficients\n\\begin{equation}\n\\mathbf{A} = \\begin{pmatrix} 1&0\\\\\n0&1\\\\\n-1 & 0\\\\\n-1 & 1\\\\\n-2 & 1\\\\\n-4 & 1\\\\ \n \\end{pmatrix}\n\\end{equation}\n\nNow the minimization problem is to determine $\\mathbf{X}$ that minimizes the residuals in the mass balance. This calculation is performed as follows:\n\\begin{align}\n \\text{minimize $\\mathbf{R}$ as a function of $\\mathbf{X}$ where}\\quad\n \\mathbf{R}&=\\mathbf{A}'\\times(10^\\mathbf{C})-\\mathbf{T}\\\\\n \\text{and} \\quad \\mathbf{C}&=10^{(\\mathbf{K}+\\mathbf{A}\\times\\mathbf{X}')}\n\\end{align}\n\nMinimization can be performed using all element of $\\mathbf{R}$ using Newton-Raphson method for example or using other nonlinear optimization methods on some summation of $\\mathbf{R}$ such as sum of squares or sum of absolute values of residuals (BrassardandBodurtha:2000). In this case $\\mathbf{R}$ would be a $2 \\times 1$ vector of mass balance residuals.\n\nTo solve this here first we will add some functions. One to return the \"tableau\" and one to adjust for fixed pH (for this example we calculate speciation versus pH, so pH will be fixed). The third function below solves for the minimum in the difference between calculated and actual mass balance. \n\n\n\n```octave\nfunction [KSOLUTION,ASOLUTION,SOLUTIONNAMES] = get_equilib_defn \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nTableau=[...\n%H\t FeIII logK species\n 1\t 0\t 0 {'H'}\n 0\t 1\t 0 {'Fe'}\n-1\t 0\t -14 {'OH'}\n-1 1 -2.19 {'FeOH'}\n-2 1 -5.67 {'FeOH2'}\n-4 1 -21.6 {'FeOH4'}\n];\n\nn=size(Tableau,2);\nASOLUTION=cell2mat(Tableau(:,1:n-2));\nKSOLUTION=cell2mat(Tableau(:,n-1));\nSOLUTIONNAMES=strvcat(Tableau(:,n));\n\nend\n\n```\n\n\n```octave\n% ----------- for fixed pH ----------------\n\nfunction [Ksolution,Asolution]=get_equilib_fixed_pH(KSOLUTION,ASOLUTION,pH)\n\n [N,M]=size(ASOLUTION);\n Ksolution=KSOLUTION-ASOLUTION(:,1)*pH;\n Asolution=[ASOLUTION(:,2:M)];\n\nend\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n```\n\n\n```octave\n%%file nl_massbalancerrnosolid_NR.m\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nfunction [X,F,J,C] = nl_massbalancerrnosolid_NR(X,Asolution,Ksolution,T)\n\n[Nc,Nx]=size(Asolution); %Xsolution=X(1:Nx);\ncriteria=1e-15;\n\nfor i=1:1000\n\nlogC=(Ksolution)+Asolution*log10(X); C=10.^(logC); % calc species\nR=Asolution'*C-T;\n\n% Evaluate the Jacobian \n z=zeros(Nx,Nx); \nfor j=1:Nx \n\tfor k=1:Nx \n\t\tfor i=1:Nc; z(j,k)=z(j,k)+Asolution(i,j)*Asolution(i,k)*C(i)/X(k); end\n \tend\nend\n\nJ = z;\n\ndeltaX=z\\(-1*R);\none_over_del=max([1, -1*deltaX'./(0.5*X')]);\ndel=1/one_over_del; X=X+del*deltaX;\n \ntst=sum(abs(R));\nif tst<=criteria; break; end\n\nend\n\nF=[R]; \n\nend\n\n```\n\n Created file '/run/media/ssmith/T7/ssmith/scottsmith/teaching/CH480_numerical_methods_chemistry/OCTAVE/CH480CH680/nl_massbalancerrnosolid_NR.m'.\n\n\n### make the graph\n\nSo for each pH we will solve the equilibrium problem. First load the tableau, then modify for the pH of that iternation. Then solve. Then store the results and loop.\n\nAt the end make a plot.\n \n\n\n```octave\n%plot -s 600,600 -f 'svg'\n\n%define the equilibrium sytem (The tableau)\n[KSOLUTION,ASOLUTION,SOLUTIONNAMES] = get_equilib_defn;\n% initial guess (just Fe becasue pH is fixed)\nFeguess=[-5.5]; guess=[10.^Feguess];\n\n%set the pH range and total\npH=2:0.1:12; FeT=1e-4; T=[FeT];\n\n%now for each pH solve\nfor i=1:length(pH)\n% adjust for fixed pH\n[Ksolution,Asolution]=get_equilib_fixed_pH(KSOLUTION,ASOLUTION,pH(i));\n% calculate species using NR\n[X,F,J,C] = nl_massbalancerrnosolid_NR(guess,Asolution,Ksolution,T);\nspecies_summary(:,i)=C; err(i)=F;\nend\n\nfor i=1:size(species_summary,1)\ntxt=[SOLUTIONNAMES(i,:),\"=species_summary(i,:);\"]; eval(txt);\nend\n\nh=plot(pH,Fe/T,pH,FeOH/T,pH,FeOH2/T,pH,FeOH4/T);\nset(gca,\"fontsize\",12); set(h,'linewidth',2); set(gca,'linewidth',2);\nh=xlabel('pH'); set(h,'fontsize',12); h=ylabel('fraction of total Fe'); set(h,'fontsize',12);\nlegend('Fe','FeOH','FeOH2','FeOH4','Location','northeast','Orientation','vertical');\naxis([min(pH) max(pH) 0 1])\n\n\n```\n\n\n \n\n \n\n\n\n```octave\nplot(pH,err)\n```\n\n\n \n\n \n\n\n### exercises 7\n\nrecalculate the speciation above but at 4e-5 M added salicylic acid, H2L (There is already 1e-4 M Fe(III).\n\nFe + L = FeL logK=17.55\n\npKa1=2.97 pKa2=13.7\n\nplot species versus pH, and also a plot of the two mass balance errors versus pH.\n\nfor matrix dimensions to match, you will need to put totals in like this T=[FeT; LT];\n\n\n\n\n\n```octave\n\n```\n", "meta": {"hexsha": "e9466668e74877e3d53a0b651033a5bda652a9bb", "size": 49229, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "LECTURE5_chemical_equilibrium.ipynb", "max_stars_repo_name": "skotssd/CH480CH680", "max_stars_repo_head_hexsha": "cdad26d6099a3f254e0da5919f527dbe3012f68c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "LECTURE5_chemical_equilibrium.ipynb", "max_issues_repo_name": "skotssd/CH480CH680", "max_issues_repo_head_hexsha": "cdad26d6099a3f254e0da5919f527dbe3012f68c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "LECTURE5_chemical_equilibrium.ipynb", "max_forks_repo_name": "skotssd/CH480CH680", "max_forks_repo_head_hexsha": "cdad26d6099a3f254e0da5919f527dbe3012f68c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 71.7623906706, "max_line_length": 1702, "alphanum_fraction": 0.5802067887, "converted": true, "num_tokens": 3289, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241632752915, "lm_q2_score": 0.658417487156366, "lm_q1q2_score": 0.4122510982315997}} {"text": "##### Copyright 2020 The TensorFlow Authors.\n\n\n```\n#@title Licensed under the Apache License, Version 2.0 (the \"License\");\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n# https://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\n```\n\n# \u8ba1\u7b97\u68af\u5ea6\n\n\n \n \n \n \n
                                        \u5728 TensorFlow.org \u4e0a\u67e5\u770b \u5728 Google Colab \u8fd0\u884c\n \u5728 Github \u4e0a\u67e5\u770b\u6e90\u4ee3\u7801 \u4e0b\u8f7d\u7b14\u8bb0\u672c
                                        \n\n\u672c\u6559\u7a0b\u63a2\u8ba8\u9002\u7528\u4e8e\u91cf\u5b50\u7535\u8def\u671f\u671b\u503c\u7684\u68af\u5ea6\u8ba1\u7b97\u7b97\u6cd5\u3002\n\n\u8ba1\u7b97\u91cf\u5b50\u7535\u8def\u4e2d\u67d0\u4e2a\u53ef\u89c2\u6d4b\u5bf9\u8c61\u7684\u671f\u671b\u503c\u7684\u68af\u5ea6\u662f\u4e00\u4e2a\u590d\u6742\u7684\u8fc7\u7a0b\u3002\u53ef\u89c2\u6d4b\u5bf9\u8c61\u7684\u671f\u671b\u503c\u5e76\u4e0d\u5177\u5907\u603b\u662f\u6613\u4e8e\u7f16\u5199\u7684\u89e3\u6790\u68af\u5ea6\u516c\u5f0f\u2014\u2014\u8fd9\u4e0d\u540c\u4e8e\u8bf8\u5982\u77e9\u9635\u4e58\u6cd5\u6216\u5411\u91cf\u52a0\u6cd5\u7b49\u5177\u5907\u6613\u4e8e\u7f16\u5199\u7684\u89e3\u6790\u68af\u5ea6\u516c\u5f0f\u7684\u4f20\u7edf\u673a\u5668\u5b66\u4e60\u53d8\u6362\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u8f7b\u677e\u5730\u4e3a\u4e0d\u540c\u7684\u573a\u666f\u91c7\u7528\u4e0d\u540c\u7684\u91cf\u5b50\u68af\u5ea6\u8ba1\u7b97\u65b9\u6cd5\u3002\u672c\u6559\u7a0b\u6bd4\u8f83\u4e86\u4e24\u79cd\u4e0d\u540c\u7684\u5fae\u5206\u65b9\u6848\u3002\n\n## \u8bbe\u7f6e\n\n\n```\n!pip install tensorflow==2.4.1\n```\n\n\u5b89\u88c5 TensorFlow Quantum\uff1a\n\n\n```\n!pip install tensorflow-quantum\n```\n\n\n```\n# Update package resources to account for version changes.\nimport importlib, pkg_resources\nimportlib.reload(pkg_resources)\n```\n\n\u73b0\u5728\uff0c\u5bfc\u5165 TensorFlow \u548c\u6a21\u5757\u4f9d\u8d56\u9879\uff1a\n\n\n```\nimport tensorflow as tf\nimport tensorflow_quantum as tfq\n\nimport cirq\nimport sympy\nimport numpy as np\n\n# visualization tools\n%matplotlib inline\nimport matplotlib.pyplot as plt\nfrom cirq.contrib.svg import SVGCircuit\n```\n\n## 1. \u51c6\u5907\u5de5\u4f5c\n\n\u6211\u4eec\u6765\u66f4\u5177\u4f53\u5730\u8bf4\u660e\u91cf\u5b50\u7535\u8def\u7684\u68af\u5ea6\u8ba1\u7b97\u6982\u5ff5\u3002\u5047\u8bbe\u60a8\u5177\u6709\u5982\u4e0b\u6240\u793a\u7684\u53c2\u6570\u5316\u7535\u8def\uff1a\n\n\n```\nqubit = cirq.GridQubit(0, 0)\nmy_circuit = cirq.Circuit(cirq.Y(qubit)**sympy.Symbol('alpha'))\nSVGCircuit(my_circuit)\n```\n\n\u4ee5\u53ca\u53ef\u89c2\u6d4b\u5bf9\u8c61\uff1a\n\n\n```\npauli_x = cirq.X(qubit)\npauli_x\n```\n\n\u6240\u7528\u7b97\u5b50\u4e3a $\u27e8Y(\\alpha)| X | Y(\\alpha)\u27e9 = \\sin(\\pi \\alpha)$\n\n\n```\ndef my_expectation(op, alpha):\n \"\"\"Compute \u27e8Y(alpha)| `op` | Y(alpha)\u27e9\"\"\"\n params = {'alpha': alpha}\n sim = cirq.Simulator()\n final_state_vector = sim.simulate(my_circuit, params).final_state_vector\n return op.expectation_from_state_vector(final_state_vector, {qubit: 0}).real\n\n\nmy_alpha = 0.3\nprint(\"Expectation=\", my_expectation(pauli_x, my_alpha))\nprint(\"Sin Formula=\", np.sin(np.pi * my_alpha))\n```\n\n\u5982\u679c\u5b9a\u4e49 $f_{1}(\\alpha) = \u27e8Y(\\alpha)| X | Y(\\alpha)\u27e9$\uff0c\u5219 $f_{1}^{'}(\\alpha) = \\pi \\cos(\\pi \\alpha)$\u3002\u8bf7\u53c2\u89c1\u4e0b\u4f8b\uff1a\n\n\n```\ndef my_grad(obs, alpha, eps=0.01):\n grad = 0\n f_x = my_expectation(obs, alpha)\n f_x_prime = my_expectation(obs, alpha + eps)\n return ((f_x_prime - f_x) / eps).real\n\n\nprint('Finite difference:', my_grad(pauli_x, my_alpha))\nprint('Cosine formula: ', np.pi * np.cos(np.pi * my_alpha))\n```\n\n## 2. \u5bf9\u5fae\u5206\u5668\u7684\u9700\u6c42\n\n\u5bf9\u4e8e\u5927\u578b\u7535\u8def\uff0c\u8981\u59cb\u7ec8\u5177\u5907\u53ef\u7cbe\u786e\u8ba1\u7b97\u7ed9\u5b9a\u91cf\u5b50\u7535\u8def\u68af\u5ea6\u7684\u516c\u5f0f\u5e76\u4e0d\u73b0\u5b9e\u3002\u5982\u679c\u7b80\u5355\u7684\u516c\u5f0f\u4e0d\u8db3\u4ee5\u8ba1\u7b97\u68af\u5ea6\uff0c\u5219\u53ef\u4ee5\u4f7f\u7528 `tfq.differentiators.Differentiator` \u7c7b\u6765\u5b9a\u4e49\u7528\u4e8e\u8ba1\u7b97\u7535\u8def\u68af\u5ea6\u7684\u7b97\u6cd5\u3002\u4f8b\u5982\uff0c\u60a8\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u65b9\u6cd5\u5728 TensorFlow Quantum (TFQ) \u4e2d\u91cd\u65b0\u521b\u5efa\u4ee5\u4e0a\u793a\u4f8b\uff1a\n\n\n```\nexpectation_calculation = tfq.layers.Expectation(\n differentiator=tfq.differentiators.ForwardDifference(grid_spacing=0.01))\n\nexpectation_calculation(my_circuit,\n operators=pauli_x,\n symbol_names=['alpha'],\n symbol_values=[[my_alpha]])\n```\n\n\u4f46\u662f\uff0c\u5982\u679c\u60a8\u6539\u4e3a\u57fa\u4e8e\u91c7\u6837\uff08\u5728\u771f\u5b9e\u8bbe\u5907\u4e0a\u8fdb\u884c\uff09\u4f30\u8ba1\u671f\u671b\u503c\uff0c\u5219\u503c\u53ef\u80fd\u4f1a\u6709\u6240\u53d8\u5316\u3002\u8fd9\u610f\u5473\u7740\u60a8\u7684\u4f30\u8ba1\u65b9\u6cd5\u5e76\u4e0d\u5b8c\u5584\uff1a\n\n\n```\nsampled_expectation_calculation = tfq.layers.SampledExpectation(\n differentiator=tfq.differentiators.ForwardDifference(grid_spacing=0.01))\n\nsampled_expectation_calculation(my_circuit,\n operators=pauli_x,\n repetitions=500,\n symbol_names=['alpha'],\n symbol_values=[[my_alpha]])\n```\n\n\u6d89\u53ca\u5230\u68af\u5ea6\u65f6\uff0c\u8fd9\u4f1a\u8fc5\u901f\u52a0\u5267\u9020\u6210\u4e25\u91cd\u7684\u51c6\u786e\u7387\u95ee\u9898\uff1a\n\n\n```\n# Make input_points = [batch_size, 1] array.\ninput_points = np.linspace(0, 5, 200)[:, np.newaxis].astype(np.float32)\nexact_outputs = expectation_calculation(my_circuit,\n operators=pauli_x,\n symbol_names=['alpha'],\n symbol_values=input_points)\nimperfect_outputs = sampled_expectation_calculation(my_circuit,\n operators=pauli_x,\n repetitions=500,\n symbol_names=['alpha'],\n symbol_values=input_points)\nplt.title('Forward Pass Values')\nplt.xlabel('$x$')\nplt.ylabel('$f(x)$')\nplt.plot(input_points, exact_outputs, label='Analytic')\nplt.plot(input_points, imperfect_outputs, label='Sampled')\nplt.legend()\n```\n\n\n```\n# Gradients are a much different story.\nvalues_tensor = tf.convert_to_tensor(input_points)\n\nwith tf.GradientTape() as g:\n g.watch(values_tensor)\n exact_outputs = expectation_calculation(my_circuit,\n operators=pauli_x,\n symbol_names=['alpha'],\n symbol_values=values_tensor)\nanalytic_finite_diff_gradients = g.gradient(exact_outputs, values_tensor)\n\nwith tf.GradientTape() as g:\n g.watch(values_tensor)\n imperfect_outputs = sampled_expectation_calculation(\n my_circuit,\n operators=pauli_x,\n repetitions=500,\n symbol_names=['alpha'],\n symbol_values=values_tensor)\nsampled_finite_diff_gradients = g.gradient(imperfect_outputs, values_tensor)\n\nplt.title('Gradient Values')\nplt.xlabel('$x$')\nplt.ylabel('$f^{\\'}(x)$')\nplt.plot(input_points, analytic_finite_diff_gradients, label='Analytic')\nplt.plot(input_points, sampled_finite_diff_gradients, label='Sampled')\nplt.legend()\n```\n\n\u5728\u8fd9\u91cc\u53ef\u4ee5\u770b\u5230\uff0c\u5c3d\u7ba1\u6709\u9650\u5dee\u5206\u516c\u5f0f\u5728\u89e3\u6790\u793a\u4f8b\u4e2d\u53ef\u4ee5\u5feb\u901f\u8ba1\u7b97\u51fa\u68af\u5ea6\u672c\u8eab\uff0c\u4f46\u5f53\u6d89\u53ca\u5230\u57fa\u4e8e\u91c7\u6837\u7684\u65b9\u6cd5\u65f6\uff0c\u5374\u4ea7\u751f\u4e86\u5927\u91cf\u566a\u58f0\u3002\u5fc5\u987b\u4f7f\u7528\u66f4\u7ec6\u81f4\u7684\u6280\u672f\u6765\u786e\u4fdd\u53ef\u4ee5\u8ba1\u7b97\u51fa\u826f\u597d\u7684\u68af\u5ea6\u3002\u63a5\u4e0b\u6765\uff0c\u60a8\u5c06\u4e86\u89e3\u4e00\u79cd\u901f\u5ea6\u7f13\u6162\u800c\u4e0d\u592a\u9002\u7528\u4e8e\u89e3\u6790\u671f\u671b\u68af\u5ea6\u8ba1\u7b97\u7684\u6280\u672f\uff0c\u4f46\u8be5\u6280\u672f\u5728\u57fa\u4e8e\u5b9e\u9645\u6837\u672c\u7684\u771f\u5b9e\u793a\u4f8b\u4e2d\u5374\u6709\u7740\u51fa\u8272\u7684\u8868\u73b0\uff1a\n\n\n```\n# A smarter differentiation scheme.\ngradient_safe_sampled_expectation = tfq.layers.SampledExpectation(\n differentiator=tfq.differentiators.ParameterShift())\n\nwith tf.GradientTape() as g:\n g.watch(values_tensor)\n imperfect_outputs = gradient_safe_sampled_expectation(\n my_circuit,\n operators=pauli_x,\n repetitions=500,\n symbol_names=['alpha'],\n symbol_values=values_tensor)\n\nsampled_param_shift_gradients = g.gradient(imperfect_outputs, values_tensor)\n\nplt.title('Gradient Values')\nplt.xlabel('$x$')\nplt.ylabel('$f^{\\'}(x)$')\nplt.plot(input_points, analytic_finite_diff_gradients, label='Analytic')\nplt.plot(input_points, sampled_param_shift_gradients, label='Sampled')\nplt.legend()\n```\n\n\u4ece\u4e0a\u9762\u53ef\u4ee5\u770b\u5230\uff0c\u67d0\u4e9b\u5fae\u5206\u5668\u6700\u597d\u7528\u4e8e\u7279\u5b9a\u7684\u7814\u7a76\u573a\u666f\u3002\u901a\u5e38\uff0c\u5728\u66f4\u4e3a\u201c\u771f\u5b9e\u201d\u7684\u73af\u5883\u4e0b\u6d4b\u8bd5\u6216\u5b9e\u73b0\u7b97\u6cd5\u65f6\uff0c\u57fa\u4e8e\u6837\u672c\u7684\u8f83\u6162\u65b9\u6cd5\u5728\u9762\u5bf9\u8bbe\u5907\u566a\u58f0\u7b49\u95ee\u9898\u65f6\u9c81\u68d2\u6027\u66f4\u4f73\uff0c\u56e0\u6b64\u662f\u7406\u60f3\u7684\u5fae\u5206\u5668\u3002\u8bf8\u5982\u6709\u9650\u5dee\u5206\u4e4b\u7c7b\u7684\u8f83\u5feb\u65b9\u6cd5\u975e\u5e38\u9002\u5408\u9762\u5411\u89e3\u6790\u8ba1\u7b97\u4e14\u9700\u8981\u66f4\u9ad8\u541e\u5410\u91cf\u7684\u573a\u666f\uff0c\u4f46\u5c1a\u672a\u8003\u8651\u7b97\u6cd5\u5728\u5b9e\u9645\u8bbe\u5907\u4e0a\u662f\u5426\u53ef\u884c\u3002\n\n## 3. \u591a\u4e2a\u53ef\u89c2\u6d4b\u5bf9\u8c61\n\n\u6211\u4eec\u6765\u5f15\u5165\u4e00\u4e2a\u989d\u5916\u7684\u53ef\u89c2\u6d4b\u5bf9\u8c61\uff0c\u501f\u6b64\u4e86\u89e3 TensorFlow Quantum \u5bf9\u5355\u4e2a\u7535\u8def\u7684\u591a\u4e2a\u53ef\u89c2\u6d4b\u5bf9\u8c61\u7684\u652f\u6301\u60c5\u51b5\u3002\n\n\n```\npauli_z = cirq.Z(qubit)\npauli_z\n```\n\n\u5982\u679c\u6b64\u53ef\u89c2\u6d4b\u5bf9\u8c61\u540c\u6837\u7528\u4e8e\u4e4b\u524d\u7684\u7535\u8def\uff0c\u5219 $f_{2}(\\alpha) = \u27e8Y(\\alpha)| Z | Y(\\alpha)\u27e9 = \\cos(\\pi \\alpha)$ \u4e14 $f_{2}^{'}(\\alpha) = -\\pi \\sin(\\pi \\alpha)$\u3002\u5feb\u901f\u68c0\u67e5\uff1a\n\n\n```\ntest_value = 0.\n\nprint('Finite difference:', my_grad(pauli_z, test_value))\nprint('Sin formula: ', -np.pi * np.sin(np.pi * test_value))\n```\n\n\u7ed3\u679c\u5339\u914d\uff08\u8db3\u591f\u63a5\u8fd1\uff09\u3002\n\n\u73b0\u5728\uff0c\u5982\u679c\u5b9a\u4e49 $g(\\alpha) = f_{1}(\\alpha) + f_{2}(\\alpha)$\uff0c\u5219 $g'(\\alpha) = f_{1}^{'}(\\alpha) + f^{'}_{2}(\\alpha)$\u3002\u5728 TensorFlow Quantum \u4e2d\u4e3a\u7535\u8def\u5b9a\u4e49\u591a\u4e2a\u53ef\u89c2\u6d4b\u5bf9\u8c61\uff0c\u76f8\u5f53\u4e8e\u5411 $g$ \u6dfb\u52a0\u66f4\u591a\u9879\u3002\n\n\u8fd9\u610f\u5473\u7740\uff0c\u7535\u8def\u4e2d\u7279\u5b9a\u7b26\u53f7\u7684\u68af\u5ea6\u7b49\u4e8e\u8be5\u7b26\u53f7\u5e94\u7528\u4e8e\u8be5\u7535\u8def\u7684\u6bcf\u4e2a\u53ef\u89c2\u6d4b\u5bf9\u8c61\u7684\u76f8\u5e94\u68af\u5ea6\u4e4b\u548c\u3002\u8fd9\u4e0e TensorFlow \u68af\u5ea6\u8ba1\u7b97\u548c\u53cd\u5411\u4f20\u64ad\uff08\u5c06\u6240\u6709\u53ef\u89c2\u6d4b\u5bf9\u8c61\u7684\u68af\u5ea6\u603b\u548c\u4f5c\u4e3a\u7279\u5b9a\u7b26\u53f7\u7684\u68af\u5ea6\uff09\u76f8\u517c\u5bb9\u3002\n\n\n```\nsum_of_outputs = tfq.layers.Expectation(\n differentiator=tfq.differentiators.ForwardDifference(grid_spacing=0.01))\n\nsum_of_outputs(my_circuit,\n operators=[pauli_x, pauli_z],\n symbol_names=['alpha'],\n symbol_values=[[test_value]])\n```\n\n\u5728\u8fd9\u91cc\u53ef\u4ee5\u770b\u5230\uff0c\u7b2c\u4e00\u4e2a\u6761\u76ee\u662f\u76f8\u5bf9\u4e8e Pauli X \u7684\u671f\u671b\uff0c\u7b2c\u4e8c\u4e2a\u6761\u76ee\u662f\u76f8\u5bf9\u4e8e Pauli Z \u7684\u671f\u671b\u3002\u73b0\u5728\uff0c\u68af\u5ea6\u8ba1\u7b97\u65b9\u6cd5\u5982\u4e0b\uff1a\n\n\n```\ntest_value_tensor = tf.convert_to_tensor([[test_value]])\n\nwith tf.GradientTape() as g:\n g.watch(test_value_tensor)\n outputs = sum_of_outputs(my_circuit,\n operators=[pauli_x, pauli_z],\n symbol_names=['alpha'],\n symbol_values=test_value_tensor)\n\nsum_of_gradients = g.gradient(outputs, test_value_tensor)\n\nprint(my_grad(pauli_x, test_value) + my_grad(pauli_z, test_value))\nprint(sum_of_gradients.numpy())\n```\n\n\u73b0\u5728\uff0c\u60a8\u5df2\u9a8c\u8bc1\u6bcf\u4e2a\u53ef\u89c2\u6d4b\u5bf9\u8c61\u7684\u68af\u5ea6\u4e4b\u548c\u5373\u4e3a $\\alpha$ \u7684\u68af\u5ea6\u3002\u6240\u6709 TensorFlow Quantum \u5fae\u5206\u5668\u5747\u652f\u6301\u6b64\u884c\u4e3a\uff0c\u4e14\u6b64\u884c\u4e3a\u5728\u4e0e\u5176\u4f59 TensorFlow \u7684\u517c\u5bb9\u6027\u65b9\u9762\u8d77\u7740\u81f3\u5173\u91cd\u8981\u7684\u4f5c\u7528\u3002\n\n## 4. \u9ad8\u7ea7\u7528\u6cd5\n\nTensorFlow Quantum \u5b50\u7c7b `tfq.differentiators.Differentiator` \u4e2d\u5b58\u5728\u7684\u6240\u6709\u5fae\u5206\u5668\u3002\u8981\u5b9e\u73b0\u5fae\u5206\u5668\uff0c\u7528\u6237\u5fc5\u987b\u5b9e\u73b0\u4e24\u4e2a\u63a5\u53e3\u4e4b\u4e00\u3002\u6807\u51c6\u662f\u5b9e\u73b0 `get_gradient_circuits` \uff0c\u5b83\u544a\u8bc9\u57fa\u7c7b\u8981\u6d4b\u91cf\u54ea\u4e9b\u7535\u8def\u4ee5\u83b7\u5f97\u68af\u5ea6\u4f30\u8ba1\u503c\u3002\u6216\u8005\uff0c\u4e5f\u53ef\u4ee5\u91cd\u8f7d `differentiate_analytic` \u548c`differentiate_sampled`\uff1b\u7c7b `tfq.differentiators.Adjoint` \u5c31\u91c7\u7528\u8fd9\u79cd\u65b9\u5f0f\u3002\n\n\u4e0b\u9762\u4f7f\u7528 TensorFlow Quantum \u5b9e\u73b0\u4e00\u4e2a\u7535\u8def\u7684\u68af\u5ea6\u3002\u60a8\u5c06\u4f7f\u7528\u4e00\u4e2a\u53c2\u6570\u8f6c\u79fb\u7684\u5c0f\u793a\u4f8b\u3002\n\n\u56de\u60f3\u4e0a\u6587\u5b9a\u4e49\u7684\u7535\u8def\uff0c$|\\alpha\u27e9 = Y^{\\alpha}|0\u27e9$\u3002\u548c\u4e4b\u524d\u4e00\u6837\uff0c\u53ef\u4ee5\u5b9a\u4e49\u4e00\u4e2a\u51fd\u6570\u4f5c\u4e3a\u8be5\u7535\u8def\u5bf9 $X$ \u53ef\u89c2\u6d4b\u5bf9\u8c61\u7684\u671f\u671b\u503c\uff0c$f(\\alpha) = \u27e8\\alpha|X|\\alpha\u27e9$\u3002\u5bf9\u4e8e\u8be5\u7535\u8def\u4f7f\u7528[\u53c2\u6570\u8f6c\u79fb\u89c4\u5219](https://pennylane.ai/qml/glossary/parameter_shift.html)\uff0c\u60a8\u53ef\u4ee5\u53d1\u73b0\u5bfc\u6570\u662f $$\\frac{\\partial}{\\partial \\alpha} f(\\alpha) = \\frac{\\pi}{2} f\\left(\\alpha + \\frac{1}{2}\\right) - \\frac{ \\pi}{2} f\\left(\\alpha - \\frac{1}{2}\\right)$$\u3002`get_gradient_circuits` \u51fd\u6570\u8fd4\u56de\u8be5\u5bfc\u6570\u7684\u5206\u91cf\u3002\n\n\n```\nclass MyDifferentiator(tfq.differentiators.Differentiator):\n \"\"\"A Toy differentiator for .\"\"\"\n\n def __init__(self):\n pass\n\n def get_gradient_circuits(self, programs, symbol_names, symbol_values):\n \"\"\"Return circuits to compute gradients for given forward pass circuits.\n \n Every gradient on a quantum computer can be computed via measurements\n of transformed quantum circuits. Here, you implement a custom gradient\n for a specific circuit. For a real differentiator, you will need to\n implement this function in a more general way. See the differentiator\n implementations in the TFQ library for examples.\n \"\"\"\n\n # The two terms in the derivative are the same circuit...\n batch_programs = tf.stack([programs, programs], axis=1)\n\n # ... with shifted parameter values.\n shift = tf.constant(1/2)\n forward = symbol_values + shift\n backward = symbol_values - shift\n batch_symbol_values = tf.stack([forward, backward], axis=1)\n \n # Weights are the coefficients of the terms in the derivative.\n num_program_copies = tf.shape(batch_programs)[0]\n batch_weights = tf.tile(tf.constant([[[np.pi/2, -np.pi/2]]]),\n [num_program_copies, 1, 1])\n\n # The index map simply says which weights go with which circuits.\n batch_mapper = tf.tile(\n tf.constant([[[0, 1]]]), [num_program_copies, 1, 1])\n\n return (batch_programs, symbol_names, batch_symbol_values,\n batch_weights, batch_mapper)\n```\n\n`Differentiator` \u57fa\u7c7b\u4f7f\u7528\u4ece `get_gradient_circuits` \u8fd4\u56de\u7684\u5206\u91cf\u6765\u8ba1\u7b97\u5bfc\u6570\uff0c\u5982\u4e0a\u9762\u7684\u53c2\u6570\u8f6c\u79fb\u516c\u5f0f\u6240\u793a\u3002\u73b0\u5728\uff0c\u8fd9\u4e2a\u65b0\u7684\u5fae\u5206\u5668\u53ef\u4ee5\u4e0e\u73b0\u6709 `tfq.layer` \u5bf9\u8c61\u4e00\u8d77\u4f7f\u7528\uff1a\n\n\n```\ncustom_dif = MyDifferentiator()\ncustom_grad_expectation = tfq.layers.Expectation(differentiator=custom_dif)\n\n# Now let's get the gradients with finite diff.\nwith tf.GradientTape() as g:\n g.watch(values_tensor)\n exact_outputs = expectation_calculation(my_circuit,\n operators=[pauli_x],\n symbol_names=['alpha'],\n symbol_values=values_tensor)\n\nanalytic_finite_diff_gradients = g.gradient(exact_outputs, values_tensor)\n\n# Now let's get the gradients with custom diff.\nwith tf.GradientTape() as g:\n g.watch(values_tensor)\n my_outputs = custom_grad_expectation(my_circuit,\n operators=[pauli_x],\n symbol_names=['alpha'],\n symbol_values=values_tensor)\n\nmy_gradients = g.gradient(my_outputs, values_tensor)\n\nplt.subplot(1, 2, 1)\nplt.title('Exact Gradient')\nplt.plot(input_points, analytic_finite_diff_gradients.numpy())\nplt.xlabel('x')\nplt.ylabel('f(x)')\nplt.subplot(1, 2, 2)\nplt.title('My Gradient')\nplt.plot(input_points, my_gradients.numpy())\nplt.xlabel('x')\n```\n\n\u73b0\u5728\uff0c\u53ef\u4ee5\u4f7f\u7528\u8fd9\u4e2a\u65b0\u7684\u5fae\u5206\u5668\u6765\u751f\u6210\u53ef\u5fae\u8fd0\u7b97\u3002\n\n\u8981\u70b9\uff1a\u5982\u679c\u5fae\u5206\u5668\u4e4b\u524d\u5df2\u9644\u52a0\u5230\u4e00\u4e2a\u8fd0\u7b97\uff0c\u90a3\u4e48\u5728\u9644\u52a0\u5230\u65b0\u7684\u8fd0\u7b97\u4e4b\u524d\uff0c\u5fc5\u987b\u5148\u8fdb\u884c\u5237\u65b0\uff0c\u56e0\u4e3a\u4e00\u4e2a\u5fae\u5206\u5668\u4e00\u6b21\u53ea\u80fd\u9644\u52a0\u5230\u4e00\u4e2a\u8fd0\u7b97\u3002\n\n\n```\n# Create a noisy sample based expectation op.\nexpectation_sampled = tfq.get_sampled_expectation_op(\n cirq.DensityMatrixSimulator(noise=cirq.depolarize(0.01)))\n\n# Make it differentiable with your differentiator:\n# Remember to refresh the differentiator before attaching the new op\ncustom_dif.refresh()\ndifferentiable_op = custom_dif.generate_differentiable_op(\n sampled_op=expectation_sampled)\n\n# Prep op inputs.\ncircuit_tensor = tfq.convert_to_tensor([my_circuit])\nop_tensor = tfq.convert_to_tensor([[pauli_x]])\nsingle_value = tf.convert_to_tensor([[my_alpha]])\nnum_samples_tensor = tf.convert_to_tensor([[5000]])\n\nwith tf.GradientTape() as g:\n g.watch(single_value)\n forward_output = differentiable_op(circuit_tensor, ['alpha'], single_value,\n op_tensor, num_samples_tensor)\n\nmy_gradients = g.gradient(forward_output, single_value)\n\nprint('---TFQ---')\nprint('Foward: ', forward_output.numpy())\nprint('Gradient:', my_gradients.numpy())\nprint('---Original---')\nprint('Forward: ', my_expectation(pauli_x, my_alpha))\nprint('Gradient:', my_grad(pauli_x, my_alpha))\n```\n\n\u6210\u529f\uff1a\u73b0\u5728\uff0c\u60a8\u53ef\u4ee5\u4f7f\u7528 TensorFlow Quantum \u63d0\u4f9b\u7684\u6240\u6709\u5fae\u5206\u5668\uff0c\u4ee5\u53ca\u5b9a\u4e49\u81ea\u5df1\u7684\u5fae\u5206\u5668\u4e86\u3002\n", "meta": {"hexsha": "09dcd018554cbccc4d7d5e269bbb76458f6e18ab", "size": 24870, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "site/zh-cn/quantum/tutorials/gradients.ipynb", "max_stars_repo_name": "RedContritio/docs-l10n", "max_stars_repo_head_hexsha": "f69a7c0d2157703a26cef95bac34b39ac0250373", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-03-29T22:32:18.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-29T22:32:18.000Z", "max_issues_repo_path": "site/zh-cn/quantum/tutorials/gradients.ipynb", "max_issues_repo_name": "Juanita-cortez447/docs-l10n", "max_issues_repo_head_hexsha": "edaba1f2b5e329857860db1e937cb1333b6e3f31", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, 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YES\n2. NO", "lm_q1_score": 0.8244619263765706, "lm_q2_score": 0.5, "lm_q1q2_score": 0.4122309631882853}} {"text": "\n\n## This Jupyter notebook is available at https://github.com/dkp-quantum/Tutorials\n\n## Further Information\n\n#### * Qiskit: https://qiskit.org\n\n#### * Qiskit GitHub: https://github.com/Qiskit\n\n## Exercise 2: \n## Design a quantum circuit that creates a 3-qubit entangled state: $\\alpha|000\\rangle+\\beta|111\\rangle$,\n## such that $|\\alpha|^2 = 0.25$, and $|\\beta|^2 = 0.75$. \n## Check the answer with the QASM simulation and by plotting the histogram of the measurement statistics.\n\n### Hint:\n$R_y(\\theta) = \\begin{bmatrix} \\cos(\\theta/2) & -\\sin(\\theta/2)\\\\ \\sin(\\theta/2) & \\cos(\\theta/2)\\end{bmatrix}$ can be implemented by the qiskit code: `qc.ry(theta,q)`\n\n\n```python\n%matplotlib inline\n# Importing standard Qiskit libraries and configuring account\nfrom qiskit import *\nfrom qiskit.visualization import *\nimport numpy as np\n```\n\n\n```python\n# Create a quantum register with 3 qubits\nq3 = QuantumRegister(3,'q')\n# Create a classical register with 3 qubits\nc3 = ClassicalRegister(3,'c')\n\n# Create the quantum circuit with the measurement in one go.\nqc_ex2 = QuantumCircuit(q3,c3,name=\"ex1\")\n\nqc_ex2.ry(2*np.pi/3,0)\nqc_ex2.cx(0,1)\nqc_ex2.cx(1,2)\nqc_ex2.barrier()\nqc_ex2.measure(q3,c3)\nqc_ex2.draw(output='mpl')# Now, add a Toffoli gate\nqc_toff = QuantumCircuit(q3,c3,name=\"ex1\")\n```\n\n\n```python\nbackend_q = Aer.get_backend('qasm_simulator')\n```\n\n\n```python\n# Execute the circuit on the qasm simulator.\njob_ex2 = execute(qc_ex2, backend_q, shots=4096)\n\n# Grab the results from the job.\nresult_ex2 = job_ex2.result()\nplot_histogram(result_ex2.get_counts(qc_ex2))\n```\n\n## 3-qubit gate example: Toffoli gate\n\n\n```python\n# Create a quantum register with 3 qubits\nq3 = QuantumRegister(3,'q')\n# Create a classical register with 3 qubits\nc3 = ClassicalRegister(3,'c')\n\n# Create the quantum circuit without a Toffoli gate\nqc_toff = QuantumCircuit(q3,c3,name=\"ex1\")\n\nqc_toff.ry(2*np.pi/3,0)\nqc_toff.h(1)\nqc_toff.h(2)\nqc_toff.barrier()\nqc_toff.measure(q3,c3)\nqc_toff.draw(output='mpl')\n```\n\n\n```python\n# Execute the circuit on the qasm simulator.\njob_toff = execute(qc_toff, backend_q, shots=4096)\n\n# Grab the results from the job.\nresult_toff = job_toff.result()\nplot_histogram(result_toff.get_counts(qc_toff))\n```\n\n\n```python\n# Now, add a Toffoli gate\nqc_toff = QuantumCircuit(q3,c3,name=\"ex1\")\n\nqc_toff.ry(2*np.pi/3,0)\nqc_toff.h(1)\nqc_toff.h(2)\nqc_toff.ccx(1,2,0)\nqc_toff.barrier()\nqc_toff.measure(q3,c3)\nqc_toff.draw(output='mpl')\n```\n\n\n```python\n# Execute the circuit on the qasm simulator.\njob_toff = execute(qc_toff, backend_q, shots=4096)\n\n# Grab the results from the job.\nresult_toff = job_toff.result()\nplot_histogram(result_toff.get_counts(qc_toff))\n```\n\n## Native single-qubit gates of IBM Q devices\n\nOne way to write a general form of a single qubit unitary:\n
                                        \n
                                        \n$$\nU(\\theta,\\phi,\\lambda)=\\begin{bmatrix} \n \\cos(\\theta/2) & -e^{i\\lambda}\\sin(\\theta/2) \\\\\ne^{i\\phi}\\sin(\\theta/2) & e^{i(\\lambda+\\phi)}\\cos(\\theta/2) \n\\end{bmatrix}\n$$\n***\nNative single qubit gates:\n* `u3`=$U(\\theta,\\phi,\\lambda)$\n* `u2`=$U(\\pi/2,\\phi,\\lambda)$\n* `u1`=$U(0,0,\\lambda)$\n\nNative two qubit gate:\n* controlled-NOT\n\n### But why such form? $\\rightarrow$ This is related to gate implementations on real devices\n\n### Note that $$\nU(\\theta,\\phi,\\lambda)=\\begin{bmatrix} \n \\cos(\\theta/2) & -e^{i\\lambda}\\sin(\\theta/2) \\\\\ne^{i\\phi}\\sin(\\theta/2) & e^{i(\\lambda+\\phi)}\\cos(\\theta/2) \n\\end{bmatrix}\n$$ \n### can be written as:\n### \\begin{align}\nU(\\theta,\\phi,\\lambda)&=R_z(\\phi)R_y(\\theta)R_z(\\lambda)\\\\\n&=R_z(\\phi)R_x(-\\pi/2)R_z(\\theta)R_x(\\pi/2)R_z(\\lambda)\n\\end{align}\n\n### In RF/MW based quantum control, $R_z$ is given for free, and $R_x(\\pm\\pi/2)$ can be calibrated with high precision.\n\n## Running Quantum Circuits on IBM Q\n\n\n### Need IBM Token for running an experiment on a IBM cloud quantum computer: https://quantum-computing.ibm.com\n\n\n```python\n# IBMQ.disable_account()\nprovider = IBMQ.enable_account('TOKEN')\n```\n\n\n```python\nprovider = IBMQ.get_provider(hub='ibm-q-research')\n```\n\n\n```python\nprovider.backends()\n```\n\n\n\n\n [,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ,\n ]\n\n\n\n\n```python\nfrom qiskit.tools.monitor import backend_overview, backend_monitor, job_monitor\nfrom qiskit.tools.visualization import plot_gate_map, plot_error_map\n```\n\n\n```python\n# Retrieve IBM Quantum device information\nbackend_overview()\n```\n\n ibmq_rome ibmq_armonk ibmq_essex\n --------- ----------- ----------\n Num. Qubits: 5 Num. Qubits: 1 Num. Qubits: 5\n Pending Jobs: 4 Pending Jobs: 4 Pending Jobs: 4\n Least busy: False Least busy: False Least busy: False\n Operational: True Operational: True Operational: True\n Avg. T1: 95.4 Avg. T1: 203.0 Avg. T1: 88.1\n Avg. T2: 98.2 Avg. T2: 213.1 Avg. T2: 108.9\n \n \n \n ibmq_burlington ibmq_london ibmq_valencia\n --------------- ----------- -------------\n Num. Qubits: 5 Num. Qubits: 5 Num. Qubits: 5\n Pending Jobs: 8 Pending Jobs: 0 Pending Jobs: 4\n Least busy: False Least busy: True Least busy: False\n Operational: True Operational: True Operational: True\n Avg. T1: 68.4 Avg. T1: 65.3 Avg. T1: 98.7\n Avg. T2: 44.5 Avg. T2: 77.5 Avg. T2: 69.9\n \n \n \n ibmq_ourense ibmq_vigo ibmq_16_melbourne\n ------------ --------- -----------------\n Num. Qubits: 5 Num. Qubits: 5 Num. Qubits: 15\n Pending Jobs: 44 Pending Jobs: 32 Pending Jobs: 17\n Least busy: False Least busy: False Least busy: False\n Operational: True Operational: True Operational: True\n Avg. T1: 111.0 Avg. T1: 80.8 Avg. T1: 51.4\n Avg. T2: 69.4 Avg. T2: 85.3 Avg. T2: 55.2\n \n \n \n ibmqx2\n ------\n Num. Qubits: 5\n Pending Jobs: 12\n Least busy: False\n Operational: True\n Avg. T1: 64.1\n Avg. T2: 42.4\n \n \n \n\n\n\n```python\n# Let's get two quantum devices as an example\nbackend_qx2 = provider.get_backend('ibmqx2')\nbackend_vigo = provider.get_backend('ibmq_vigo')\n```\n\n\n```python\nbackend_monitor(backend_qx2)\n```\n\n ibmqx2\n ======\n Configuration\n -------------\n n_qubits: 5\n operational: True\n status_msg: active\n pending_jobs: 12\n backend_version: 2.1.0\n basis_gates: ['id', 'u1', 'u2', 'u3', 'cx']\n local: False\n simulator: False\n sample_name: sparrow\n open_pulse: False\n backend_name: ibmqx2\n url: None\n max_experiments: 75\n max_shots: 8192\n description: 5 qubit device\n coupling_map: [[0, 1], [0, 2], [1, 0], [1, 2], [2, 0], [2, 1], [2, 3], [2, 4], [3, 2], [3, 4], [4, 2], [4, 3]]\n allow_object_storage: True\n credits_required: True\n n_registers: 1\n allow_q_object: True\n memory: True\n conditional: False\n quantum_volume: 8\n meas_map: [[0, 1, 2, 3, 4]]\n online_date: 2017-01-24T05:00:00+00:00\n \n Qubits [Name / Freq / T1 / T2 / U1 err / U2 err / U3 err / Readout err]\n -----------------------------------------------------------------------\n Q0 / 5.2829 GHz / 76.2028 \u00b5s / 25.66558 \u00b5s / 0.0 / 0.0014 / 0.0028 / 0.0435\n Q1 / 5.24766 GHz / 54.83054 \u00b5s / 24.31219 \u00b5s / 0.0 / 0.00109 / 0.00217 / 0.022\n Q2 / 5.03384 GHz / 54.14951 \u00b5s / 82.47369 \u00b5s / 0.0 / 0.00053 / 0.00107 / 0.012\n Q3 / 5.29225 GHz / 56.5262 \u00b5s / 30.2464 \u00b5s / 0.0 / 0.0006 / 0.00119 / 0.0135\n Q4 / 5.07848 GHz / 79.0084 \u00b5s / 49.4769 \u00b5s / 0.0 / 0.00046 / 0.00092 / 0.0265\n \n Multi-Qubit Gates [Name / Type / Gate Error]\n --------------------------------------------\n cx0_1 / cx / 0.02365\n cx0_2 / cx / 0.01616\n cx1_0 / cx / 0.02365\n cx1_2 / cx / 0.02259\n cx2_0 / cx / 0.01616\n cx2_1 / cx / 0.02259\n cx2_3 / cx / 0.0151\n cx2_4 / cx / 0.01834\n cx3_2 / cx / 0.0151\n cx3_4 / cx / 0.01376\n cx4_2 / cx / 0.01834\n cx4_3 / cx / 0.01376\n\n\n\n```python\nplot_error_map(backend_qx2)\n```\n\n\n```python\nbackend_monitor(backend_vigo)\n```\n\n ibmq_vigo\n =========\n Configuration\n -------------\n n_qubits: 5\n operational: True\n status_msg: active\n pending_jobs: 35\n backend_version: 1.2.0\n basis_gates: ['id', 'u1', 'u2', 'u3', 'cx']\n local: False\n simulator: False\n sample_name: Giraffe\n open_pulse: False\n backend_name: ibmq_vigo\n url: None\n max_experiments: 75\n max_shots: 8192\n description: 5 qubit device Vigo\n coupling_map: [[0, 1], [1, 0], [1, 2], [1, 3], [2, 1], [3, 1], [3, 4], [4, 3]]\n allow_object_storage: True\n credits_required: True\n n_registers: 1\n allow_q_object: True\n memory: True\n conditional: False\n quantum_volume: 16\n meas_map: [[0, 1, 2, 3, 4]]\n online_date: 2019-07-03T04:00:00+00:00\n \n Qubits [Name / Freq / T1 / T2 / U1 err / U2 err / U3 err / Readout err]\n -----------------------------------------------------------------------\n Q0 / 4.79646 GHz / 105.13289 \u00b5s / 13.28137 \u00b5s / 0.0 / 0.00035 / 0.0007 / 0.02\n Q1 / 4.94014 GHz / 89.6182 \u00b5s / 93.21168 \u00b5s / 0.0 / 0.0005 / 0.00101 / 0.025\n Q2 / 4.83351 GHz / 57.54627 \u00b5s / 174.64128 \u00b5s / 0.0 / 0.0004 / 0.00079 / 0.015\n Q3 / 4.80797 GHz / 85.99093 \u00b5s / 98.54726 \u00b5s / 0.0 / 0.00067 / 0.00133 / 0.012\n Q4 / 4.74967 GHz / 65.84481 \u00b5s / 46.96049 \u00b5s / 0.0 / 0.00061 / 0.00123 / 0.026\n \n Multi-Qubit Gates [Name / Type / Gate Error]\n --------------------------------------------\n cx0_1 / cx / 0.00984\n cx1_0 / cx / 0.00984\n cx1_2 / cx / 0.00874\n cx1_3 / cx / 0.01332\n cx2_1 / cx / 0.00874\n cx3_1 / cx / 0.01332\n cx3_4 / cx / 0.00707\n cx4_3 / cx / 0.00707\n\n\n\n```python\nplot_error_map(backend_vigo)\n```\n\n## Let's create a 5-qubit GHZ state, i.e. $ \\frac{|00000\\rangle + |11111\\rangle}{\\sqrt{2}}$.\n\n\n```python\n# Create a 5-qubit GHZ state (i.e. (|00000> + |11111>)/sqrt(2))\nq5 = QuantumRegister(5,'q')\nc5 = ClassicalRegister(5,'c')\nghz5= QuantumCircuit(q5,c5)\n\nghz5.h(0)\nfor i in range(1,5):\n ghz5.cx(0,i)\n\nghz5.barrier()\nghz5.measure(q5,c5)\nghz5.draw(output='mpl')\n```\n\n## Now, let's run it on a real IBMQ device.\n\n\n```python\n# Run the 5-qubit GHZ experiment on a 5-qubit device (try vigo)\njob_exp1 = execute(ghz5, backend=backend_vigo, shots=4096)\njob_monitor(job_exp1)\n```\n\n Job Status: job has successfully run\n\n\n\n```python\n# Grab experimental results\nresult_vigo = job_exp1.result()\ncounts_vigo = result_vigo.get_counts(ghz5)\n```\n\n\n```python\n# Let's also try the same experiment on the 15-qubit device.\njob_exp2 = execute(ghz5, backend=provider.get_backend('ibmq_16_melbourne'), shots=4096)\njob_monitor(job_exp2)\n```\n\n Job Status: job has successfully run\n\n\n\n```python\n# Grab experimental results\nresult_mel = job_exp2.result()\ncounts_mel = result_mel.get_counts(ghz5)\n```\n\n\n```python\n# Now, compare to theory by running it on qasm_simulator\njob_qasm = execute(ghz5,backend=backend_q)\nresult_qasm = job_qasm.result()\ncounts_qasm = result_qasm.get_counts(ghz5)\n\n# Plot both experimental and ideal results\nplot_histogram([counts_qasm,counts_vigo,counts_mel],\n color=['black','green','blue'],\n legend=['QASM','Vigo','Melbourne'],figsize = [20,8])\n```\n\n## Elementary Quantum Protocols\n\n* Superdense coding\n* Quantum teleportation\n\n\n\n\n\n\n\n\n```python\n%matplotlib inline\n# Importing standard Qiskit libraries and configuring account\nfrom qiskit import *\nfrom qiskit.visualization import *\nimport numpy as np\n```\n\n\n```python\n# Define a function that takes a QuantumCircuit (qc)\n# a qubit index (index) and a message string (msg)\ndef encoding(qc, index, msg):\n if msg == 0:\n pass # To send 00 we do nothing\n elif msg == 1:\n qc.x(index) # To send 10 we apply an X-gate\n elif msg == 2:\n qc.z(index) # To send 01 we apply a Z-gate\n elif msg == 3:\n qc.z(index) # To send 11, we apply a Z-gate\n qc.x(index) # followed by an X-gate\n else:\n print(\"Invalid Message. Sending '00'.\")\n```\n\n\n```python\ndef decoding(qc, a, b):\n qc.cx(a,b)\n qc.h(a)\n```\n\n\n```python\ndef qc_sdc(msg):\n\n # Create the quantum circuit with 2 qubits\n qc = QuantumCircuit(2)\n\n # First, an entangled pair is created between Alice and Bob\n # Bob has the first qubit, Alice has the second qubit.\n qc.h(0)\n qc.cx(0,1)\n\n qc.barrier()\n\n # Next, Bob encodes his message onto qubit 0.\n encoding(qc, 0, msg)\n qc.barrier()\n # Bob then sends his qubit to Alice.\n\n # After recieving qubit 0, Alice applies the recovery protocol:\n decoding(qc, 0, 1)\n\n # Finally, Alice measures her qubits to read Bob's message\n qc.measure_all()\n \n return qc\n```\n\n\n```python\nbackend_qasm = Aer.get_backend('qasm_simulator')\n\nrand_msg = np.random.randint(4)\nqc = qc_sdc(rand_msg)\njob_sim = execute(qc, backend_qasm, shots=1024)\nsim_result = job_sim.result()\n\nmeasurement_result = sim_result.get_counts(qc)\nprint(measurement_result)\nplot_histogram(measurement_result)\n```\n\n\n```python\nprint(\"The random message was: %s\" % rand_msg)\nqc.draw(output='mpl')\n```\n\n## Now, let's run it on a real IBMQ device.\n\n\n```python\nfrom qiskit.tools.monitor import backend_overview, backend_monitor, job_monitor\nfrom qiskit.tools.visualization import plot_gate_map, plot_error_map\n```\n\n\n```python\n# Retrieve IBM Quantum device information\nbackend_overview()\n```\n\n ibmq_rome ibmq_armonk ibmq_essex\n --------- ----------- ----------\n Num. Qubits: 5 Num. Qubits: 1 Num. Qubits: 5\n Pending Jobs: 6 Pending Jobs: 2 Pending Jobs: 70\n Least busy: False Least busy: False Least busy: False\n Operational: True Operational: True Operational: True\n Avg. T1: 77.2 Avg. T1: 155.4 Avg. T1: 117.6\n Avg. T2: 109.1 Avg. T2: 152.1 Avg. T2: 152.9\n \n \n \n ibmq_burlington ibmq_london ibmq_valencia\n --------------- ----------- -------------\n Num. Qubits: 5 Num. Qubits: 5 Num. Qubits: 5\n Pending Jobs: 3 Pending Jobs: 11 Pending Jobs: 2\n Least busy: False Least busy: False Least busy: False\n Operational: True Operational: True Operational: True\n Avg. T1: 78.0 Avg. T1: 66.3 Avg. T1: 98.0\n Avg. T2: 76.1 Avg. T2: 81.0 Avg. T2: 65.1\n \n \n \n ibmq_ourense ibmq_vigo ibmq_16_melbourne\n ------------ --------- -----------------\n Num. Qubits: 5 Num. Qubits: 5 Num. Qubits: 15\n Pending Jobs: 1 Pending Jobs: 1 Pending Jobs: 15\n Least busy: True Least busy: False Least busy: False\n Operational: True Operational: True Operational: True\n Avg. T1: 106.2 Avg. T1: 97.4 Avg. T1: 55.2\n Avg. T2: 64.7 Avg. T2: 79.7 Avg. T2: 59.5\n \n \n \n ibmqx2\n ------\n Num. Qubits: 5\n Pending Jobs: 8\n Least busy: False\n Operational: True\n Avg. T1: 61.4\n Avg. T2: 41.4\n \n \n \n\n\n\n```python\n# Run the superdense coding experiment on a 5-qubit device (try london)\njob_sdc = execute(qc, backend=provider.get_backend('ibmq_london'), shots=4096)\njob_monitor(job_sdc)\n```\n\n Job Status: job has successfully run\n\n\n\n```python\nexperiment_result = job_sdc.result().get_counts(qc)\nprint(\"The random message was: %s\" % rand_msg)\nplot_histogram(experiment_result)\n```\n\n\n\n\n\n\n```python\ndef create_bell_pair(qc, a, b):\n # Creates a bell pair in qc using qubits a & b\n qc.h(a) # Put qubit a into state |+>\n qc.cx(a,b) # CNOT with a as control and b as target\n```\n\n\n```python\nqr = QuantumRegister(3) # Protocol uses 3 qubits\ncr1 = ClassicalRegister(1) # and 2 classical bits\ncr2 = ClassicalRegister(1) # in 2 different registers\nteleportation = QuantumCircuit(qr, cr1, cr2)\n```\n\n\n```python\n## STEP 1\n# Entangle qubits q1 and q2\ncreate_bell_pair(teleportation, 1, 2)\nteleportation.barrier()\n\n# And view the circuit so far:\nteleportation.draw(output='mpl')\n```\n\n\n```python\n## STEP 2\n# Bob performs his gates\nteleportation.cx(0,1)\nteleportation.h(0)\nteleportation.barrier()\n\n# And view the circuit so far:\nteleportation.draw(output='mpl')\n```\n\n\n```python\n## STEP 3\n# Bob measures his part\nteleportation.measure(0,0)\nteleportation.measure(1,1)\n\n# And view the circuit so far:\nteleportation.draw(output='mpl')\n```\n\n\n```python\n# This function takes a QuantumCircuit (qc), qubit index\n# and ClassicalRegisters (cr1 & cr2) to decide which gates to apply\ndef alice_recover(qc, index, cr1, cr2):\n # Here we use c_if to control our gates with a classical\n # bit instead of a qubit\n qc.z(index).c_if(cr1, 1) # Apply gates if the registers \n qc.x(index).c_if(cr2, 1) # are in the state '1'\n```\n\n\n```python\n## STEP 4\n# Alice perform recovery\nalice_recover(teleportation, 2, cr1, cr2)\n\n# And view the circuit so far:\nteleportation.draw(output='mpl')\n```\n\n## Let's test Quantum Teleportation with a random state!\n\n\n```python\ndef random_init(qc,r1,r2,index):\n \n ## STEP 0\n # Bob prepares a quantum state to teleport\n # by applying a random rotation around y and z\n qc.ry(r1,index)\n qc.rz(r2,index)\n qc.barrier()\n```\n\n\n```python\ndef quantum_teleportation(qc):\n \n ## STEP 1\n # Entangle qubits q1 and q2\n create_bell_pair(qc, 1, 2)\n qc.barrier()\n\n ## STEP 2\n # Bob performs his gates\n qc.cx(0,1)\n qc.h(0)\n qc.barrier()\n\n ## STEP 3\n # Bob measures his part\n qc.measure(0,0)\n qc.measure(1,1)\n\n ## STEP 4\n # Alice perform recovery\n alice_recover(qc, 2, cr1, cr2)\n```\n\n\n```python\nqr = QuantumRegister(3) # Protocol uses 3 qubits\ncr1 = ClassicalRegister(1) # and 2 classical bits\ncr2 = ClassicalRegister(1) # in 2 different registers\nqc_teleportation = QuantumCircuit(qr, cr1, cr2)\nqc_ref = QuantumCircuit(qr)\n\nr1 = np.random.random()*np.pi\nr2 = np.random.random()*2*np.pi\n\nrandom_init(qc_ref, r1, r2, 0)\n\nrandom_init(qc_teleportation, r1, r2, 0)\n\nquantum_teleportation(qc_teleportation)\n\nqc_teleportation.draw(output='mpl')\n```\n\n\n```python\nbackend_sv = BasicAer.get_backend('statevector_simulator')\n\nin_vector = execute(qc_ref, backend_sv).result().get_statevector()\nout_vector = execute(qc_teleportation, backend_sv).result().get_statevector()\n```\n\n\n```python\nplot_bloch_multivector(in_vector)\n```\n\n\n```python\nplot_bloch_multivector(out_vector)\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "6e70dae5751c0e35f7312f53d49f3941a9cca836", "size": 577881, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "2020 Yonsei-IBM/Yonsei_qiskit_lecture2.ipynb", "max_stars_repo_name": "dkp-quantum/tutorial", "max_stars_repo_head_hexsha": "cfb68dedc2762a39b0760057882f5cfd1f5089a0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-08-05T01:08:42.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-13T15:45:50.000Z", "max_issues_repo_path": "2020 Yonsei-IBM/Yonsei_qiskit_lecture2.ipynb", "max_issues_repo_name": "dkp-quantum/tutorial", "max_issues_repo_head_hexsha": "cfb68dedc2762a39b0760057882f5cfd1f5089a0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "2020 Yonsei-IBM/Yonsei_qiskit_lecture2.ipynb", "max_forks_repo_name": "dkp-quantum/tutorial", "max_forks_repo_head_hexsha": "cfb68dedc2762a39b0760057882f5cfd1f5089a0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2020-08-05T01:05:37.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-21T04:01:28.000Z", "avg_line_length": 352.3664634146, "max_line_length": 121252, "alphanum_fraction": 0.9337631796, "converted": true, "num_tokens": 6546, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.603931819468636, "lm_q2_score": 0.6825737473266735, "lm_q1q2_score": 0.412228005144523}} {"text": "\n# FFM232, Klassisk fysik och vektorf\u00e4lt - F\u00f6rel\u00e4sningsanteckningar\n\n **[Christian Forss\u00e9n](http://fy.chalmers.se/subatom/tsp/), Institutionen f\u00f6r fysik, Chalmers, G\u00f6teborg, Sverige**\n\nDate: **Oct 3, 2016**\n\n# 8. Potentialteori\n\n## Konservativa f\u00e4lt och potentialer\n\nVi har definierat ett konservativt f\u00e4lt som ett f\u00e4lt $\\vec{F}$ s\u00e5dant att\n\n$$\n\\begin{equation}\n \\oint_C \\vec{F}\\cdot \\mbox{d}\\vec{r} = 0\n\\end{equation}\n$$\n\nf\u00f6r varje sluten kurva $C$. Enligt Stokes sats f\u00f6ljer det nu att ett f\u00e4lt som har $\\nabla \\times \\vec{F} = 0$ \u00f6verallt i ett enkelt sammanh\u00e4ngande omr\u00e5de \u00e4r konservativt. Vi har sedan antagit att vi kan skriva\n\n$$\n\\begin{equation}\n\\vec{F} = - \\nabla \\phi,\n\\end{equation}\n$$\n\nd\u00e4r $\\phi$ kallas f\u00f6r det konservativa f\u00e4ltets potential. G\u00e4ller d\u00e5 \n$$\n\\nabla \\times \\vec{F} = 0 \\, \\Longleftrightarrow \\, \\vec{F} = - \\nabla \\phi \\, ??\n$$\nNej, det har vi \u00e4nnu inte visat! Vi har att $\\vec{F} = - \\nabla \\phi \\, \\Rightarrow \\, \\nabla \\times \\vec{F} = 0 $ eftersom $\\nabla \\times ( \\nabla \\phi) = 0$ (se nedan), men det omv\u00e4nda g\u00e4ller inte per automatik. Vi har \u00e4nnu inte visat att konservativa f\u00e4lt alltid kan skrivas som $-\\nabla \\phi$.\n\n### Bevis av $\\nabla \\times ( \\nabla \\phi) = 0$ mha indexnotation\n\n-----------------------\n\nVi kan visa att $\\nabla \\times ( \\nabla \\phi) = 0$ mha indexnotation: Vektorn $\\left[ \\nabla \\times ( \\nabla \\phi) \\right]_i = \\epsilon_{ijk} \\partial_j \\partial_k \\phi = 0$ eftersom den resulterande summan med nio termer $\\sum_{j,k=1}^3$ har tv\u00e5 nollskilda som tar ut varandra. T.ex. f\u00f6r $i=1$ f\u00e5s $\\partial_2\\partial_3 \\phi - \\partial_3\\partial_2 \\phi = 0$.\n\n----------------------------------------------\n\nDefinitionen av ett konservativt f\u00e4lt g\u00f6r dock att vi kan definiera en potential fr\u00e5n\n\n$$\n\\begin{equation}\n \\int_{\\vec{r}_1}^{\\vec{r}_2} \\vec{F} \\cdot \\mbox{d} \\vec{r} = \\phi\\left( \\vec{r}_1 \\right) - \\phi\\left( \\vec{r}_2 \\right).\n\\end{equation}\n$$\n\nNotera tecknet p\u00e5 potentialskillnaden i HL. Det hade g\u00e5tt lika bra med det omv\u00e4nda, men just denna konvention st\u00e4mmer \u00f6verens med energitolkningen i klassisk mekanik.\n\nVi ser nu att f\u00f6r\u00e4ndringen av potentialen mellan punkterna $\\vec{r}$ och $\\vec{r} + \\mbox{d}\\vec{r}$ \u00e4r\n\n$$\n\\begin{equation}\n \\mbox{d}\\phi = -\\vec{F}\\cdot \\mbox{d}\\vec{r},\n\\end{equation}\n$$\n\nmen denna f\u00f6r\u00e4ndring kan ocks\u00e5 skrivas som\n\n$$\n\\begin{equation}\n \\mbox{d}\\phi = \\nabla \\phi \\cdot \\mbox{d}\\vec{r}.\n\\end{equation}\n$$\n\nDet f\u00f6ljer d\u00e4rmed att det finns en potential $\\phi$ s\u00e5 att det konservativa f\u00e4ltet $\\vec{F}$ kan skrivas\n\n$$\n\\begin{equation}\n \\vec{F} = -\\nabla \\phi.\n\\end{equation}\n$$\n\nFaktum \u00e4r att $\\phi$ inte \u00e4r entydigt best\u00e4md. Man kan skapa en ny potential genom att addera en konstant till en potential: $\\phi \\mapsto \\phi + \\phi_0$.\n\n### Exempel: Arbete i klassisk mekanik\n\n-----------------------\n\nF\u00f6r ett konservativt kraftf\u00e4lt $\\vec{F}$ g\u00e4ller att det utr\u00e4ttade arbetet l\u00e4ngs en r\u00f6relsebana motsvarar en *\u00f6kning* av den kinetiska energin \u00e5tf\u00f6ljt av en motsvarade *minskning* av en potentiell energi s\u00e5 att den totala mekaniska energin $E = T + V$ \u00e4r *konserverad*.\n\n[Comment 1: Notera att utr\u00e4ttat arbete kan vara negativt vilket isf inneb\u00e4r en minskning av den kinetiska energin och en \u00f6kning av den potentiella.]\n\nArbetet som utr\u00e4ttas l\u00e4ngs en kurva $C$ motsvaras av kurvintegralen\n\n$$\n\\begin{equation}\n\\int_C \\vec{F} \\cdot \\mbox{d}\\vec{r} = \\left\\{ \\begin{array}{l}\n\\vec{F} = m \\vec{a} = m \\frac{\\mbox{d}\\vec{v}}{\\mbox{d}t} \n\\end{equation}\n$$\n\n$$\n\\begin{equation} \n\\mbox{d}\\vec{r} = \\vec{v} \\mbox{d}t\n\\end{array} \n, \\quad \\frac{\\mbox{d}\\vec{v}}{\\mbox{d}t} \\cdot \\vec{v} = \\frac{1}{2}\\frac{\\mbox{d}}{\\mbox{d}t} (\\vec{v} \\cdot \\vec{v})\n\\right\\} \n\\end{equation}\n$$\n\n$$\n\\begin{equation} \n= \\frac{m}{2} \\int_{t_A}^{t_B} \\frac{\\mbox{d}}{\\mbox{d}t} (\\vec{v} \\cdot \\vec{v}) \\mbox{d}t = \\frac{m v_2^2}{2} - \\frac{m v_1^2}{2} = \\delta T.\n\\end{equation}\n$$\n\nH\u00e4r har vi anv\u00e4nt NII. Tidpunkterna $t_A$ och $t_B$ motsvarar start- respektive sluttiden f\u00f6r r\u00f6relsen.\n\nEtt exempel p\u00e5 ett s\u00e5dant konservativt kraftf\u00e4lt \u00e4r gravitationskraften n\u00e4ra jordytan: $\\vec{F} = - m g \\hat{z}$.\n\nMed v\u00e5r nuvarande kunskap skulle vi uttrycka detta som $\\vec{F} = -\\nabla \\phi$, d\u00e4r vi allts\u00e5 anv\u00e4nder beteckningen $\\phi$ ist\u00e4llet f\u00f6r $V$. F\u00f6r v\u00e5rt exempel finner vi att potentialen $\\phi(\\vec{r}) = m g z$ uppfyller denna likhet.\n\nVidare uppfyller kraftf\u00e4ltet $\\nabla \\times \\vec{F} = 0$. Vi kontrollerar detta f\u00f6r v\u00e5rt exempel: \n$$\n\\begin{vmatrix}\n\\hat{x} & \\hat{y} & \\hat{z} \\\\\n\\frac{\\partial}{\\partial x} & \\frac{\\partial}{\\partial y} & \\frac{\\partial}{\\partial z} \\\\\n\\mbox{0} & \\mbox{0} & -mg \\\\\n\\end{vmatrix}\n= 0\n$$\n\nKurvintegralen\n$$\n\\int_C \\vec{F} \\cdot \\mbox{d}\\vec{r} = \\phi\\left( \\vec{r}_A \\right) - \\phi\\left( \\vec{r}_B \\right),\n$$\nd\u00e4r $A$ och $B$ \u00e4r kurvans start- respektive slutpunkt. H\u00e4r finner vi att potentialskillnaden $-\\delta \\phi = \\phi(z_A) - \\phi(z_B) = mg (z_A-z_B)$. Detta \u00e4r allts\u00e5 den negativa skillnaden i potentiell energi och vi f\u00e5r\n$$\n\\delta T = - \\delta \\phi \\quad \\Rightarrow \\quad \\delta(T+V) = 0.\n$$\n\n--------------------------------------------\n\n## Poissons och Laplaces ekvationer\n\nKonservativa vektorf\u00e4lt \u00e4r allts\u00e5 rotationsfria, men de kan fortfarande ha nollskild divergens. Denna kallas ofta f\u00f6r *k\u00e4llt\u00e4thet*\n\n$$\n\\begin{equation}\n\\rho(\\vec{r}) = \\nabla \\cdot \\vec{F},\n\\end{equation}\n$$\n\noch vi har st\u00f6tt p\u00e5 exempel d\u00e4r denna har tolkats i termer av elektrostatik och massfl\u00f6de. Med $\\vec{F} = -\\nabla \\phi$ f\u00e5r vi *Poissons ekvation*\n\n$$\n\\begin{equation}\n\\nabla \\cdot \\nabla \\phi(\\vec{r}) = \\Delta \\phi(\\vec{r}) = -\\rho(\\vec{r}),\n\\end{equation}\n$$\n\nSpecialfallet av denna ekvation utan k\u00e4lla, dvs divergensfritt, ger *Laplaces ekvation*\n\n$$\n\\begin{equation}\n\\Delta \\phi(\\vec{r}) = 0.\n\\end{equation}\n$$\n\nKom ih\u00e5g att $\\Delta$ \u00e4r Laplacianen vilken kan skrivas\n\n$$\n\\begin{equation}\n\\Delta \\phi = \\left( \\frac{\\partial^2}{\\partial x^2} + \\frac{\\partial^2}{\\partial y^2} + \\frac{\\partial^2}{\\partial z^2} \\right) \\phi,\n\\end{equation}\n$$\n\ni cartesiska koordinater. Den ser annorlunda ut i kroklinjiga koordinatsystem.\n\n[Comment 2: Notera att $\\vec{F} = -m g \\hat{z}$ \u00e4r divergensfritt och att potentialen $\\phi = m g z$ uppfyller Laplaces ekvation $\\Delta \\phi = 0$.]\n\n[Rita 3: Skissa g\u00e4rna hur ett f\u00e4lt som uppfyller Laplaces ekvation beter sig. Titta p\u00e5 ett f\u00e4lt $\\phi(x,y)$ och anv\u00e4nd Taylorexpansion f\u00f6r att teckna beteendet n\u00e4ra en punkt $\\vec{r}_0$. Villkoret $\\frac{\\partial^2\\phi}{\\partial x^2} + \\frac{\\partial^2\\phi}{\\partial y^2} = 0$ ger en andragradskurva \n$$\n\\phi(\\vec{r}) = c_0 + c_x (x-x_0) + c_y (y-y_0) + c_{xy} (x-x_0)(y-y_0)+ \\frac{c_{ii}}{2}(x-x_0)^2 - \\frac{c_{ii}}{2}(y-y_0)^2,\n$$\nd\u00e4r t.ex. $c_0 = \\phi(\\vec{r}_0)$, $c_x = d\\phi/dx(\\vec{r}_0)$, etc. Det viktiga \u00e4r allts\u00e5 att expansionen slutar med den kvadratiska termen och att dessa \u00e4r lika, men motriktade, i $x$- och $y$-led.\n]\n\nNotera att Poissons och Laplaces ekvationer \u00e4r exempel p\u00e5 *differentialekvationer*. F\u00f6r att l\u00f6sa dessa i ett omr\u00e5de beh\u00f6ver man ocks\u00e5 veta randvillkor f\u00f6r f\u00e4ltet $\\phi$. \n\n### Exempel: L\u00f6sningar till Laplaces ekvation\n\n------------------------\n\n\n\n
                                        \n\n

                                        L\u00f6sningen ($\\phi = xy$) till Laplaces ekvation i tv\u00e5 dimensioner p\u00e5 ett kvadratiskt omr\u00e5de med randvillkor enligt figuren.

                                        \n\n\n\n\n\n\n\n
                                        \n\n

                                        L\u00f6sningen ($\\phi = x^2 - y^2$) till Laplaces ekvation i tv\u00e5 dimensioner p\u00e5 ett cirkul\u00e4rt omr\u00e5de med vinkelberoende randvillkor enligt figuren.

                                        \n\n\n\n\n\nTv\u00e5 exempel p\u00e5 l\u00f6sningar till Laplaces ekvation i ett omr\u00e5de med givna randvillkor visas i figurerna ref{fig:xy} och ref{fig:x2y2}.\n\n------------------------------------------------\n\n\nVi kommer t.ex. se att (ett station\u00e4rt) temperaturf\u00e4lt uppfyller n\u00e5gon av dessa ekvationer (Laplaces ekvation om det inte finns en v\u00e4rmek\u00e4lla). F\u00f6r att t.ex. r\u00e4kna ut temperaturf\u00e4ltet inuti ett v\u00e4rmeisolerande f\u00f6nster beh\u00f6ver vi veta randvillkor f\u00f6r f\u00e4ltet p\u00e5 glasets in- och utsida.\n\nOlika tekniker f\u00f6r l\u00f6sning av dessa ekvationer presenteras i kapitel 9.\n\n## Divergensfria f\u00e4lt\n\nKan man erh\u00e5lla *divergensfria* f\u00e4lt fr\u00e5n n\u00e5gon potential p\u00e5 liknande s\u00e4tt som vi just gjorde f\u00f6r rotationsfria f\u00e4lt?\n\nVi betraktar ett divergensfritt f\u00e4lt $\\vec{G}$\n$$\n\\nabla \\cdot \\vec{G} = 0,\n$$\noch n\u00f6jer oss med att konstatera att divergensfriheten uppenbarligen uppfylls om f\u00e4ltet kan skrivas\n\n$$\n\\begin{equation}\n\\vec{G} = \\nabla \\times \\vec{A}.\n\\end{equation}\n$$\n\n[Kommentar 4: Visa detta om ni \u00e4r os\u00e4kra. G\u00e4rna med indexnotation.\n$$\n\\nabla \\cdot (\\nabla \\times \\vec{A}) = \\partial_i \\epsilon_{ijk} \\partial_j A_k,\n$$\nVilket \u00e4r en summa med 27 termer, vara sex \u00e4r nollskilda och dessa tar ut varandra parvis.\n]\n\nVektorf\u00e4ltet $\\vec{A}$ kallas d\u00e5 f\u00f6r en *vektorpotential*.\n\n### Standardexempel: Statiskt magnetf\u00e4lt\n\n------------------------\n\nEtt statiskt magnetf\u00e4lt $\\vec{B}(\\vec{r})$ \u00e4r divergensfritt och uppfyller $\\vec{B}(\\vec{r}) = \\nabla \\times \\vec{A}(\\vec{r})$ d\u00e4r $\\vec{A}$ kallas f\u00f6r den elektromagnetiska vektorpotentialen.\n\n---------------------------------------------\n\nVi fann tidigare att skal\u00e4ra potentialer hade en invarians genom att vi kunde addera en konstant term $\\phi \\mapsto \\phi + \\phi_0$ utan att \u00e4ndra f\u00e4ltstyrkan.\n\nP\u00e5 samma s\u00e4tt har vektorpotentialen en invarians\n\n$$\n\\begin{equation}\n\\vec{A}(\\vec{r}) \\mapsto \\vec{A}(\\vec{r}) + \\nabla \\Lambda(\\vec{r}),\n\\end{equation}\n$$\n\nd\u00e4r $\\Lambda$ kallas f\u00f6r en Gaugeparameter och invariansen kallas f\u00f6r *Gaugeinvarians*. \n\n[Kommentar 5: Detta m\u00e5 verka som en kuriositet, men just Gaugeinvarians \u00e4r av fundamental betydelse f\u00f6r v\u00e5r teoretiska f\u00f6rst\u00e5else av elektromagnetiska, svaga och starka krafter.]\n\n[Kommentar 6: Ofta anv\u00e4nder man Gaugeinvariansen till att skapa en vektorpotential som \u00e4r divergensfri. Dvs man v\u00e4ljer $\\Lambda$ s\u00e5 att $\\Delta \\Lambda = -\\nabla \\cdot \\vec{A}$.]\n\nRotationen kallas ofta f\u00f6r virvelt\u00e4thet\n$$\n\\nabla \\times \\vec{G} = \\vec{j} \\quad \\Rightarrow \\nabla \\times (\\nabla \\times \\vec{A}) = \\vec{j}.\n$$\nVi anv\u00e4nder sambandet $\\nabla \\times \\left( \\nabla \\times \\vec{A} \\right) = \\nabla \\left( \\nabla \\cdot \\vec{A} \\right) - \\Delta \\vec{A}$ vilket allts\u00e5 ger\n$$\n\\Delta \\vec{A} - \\nabla \\left( \\nabla \\cdot \\vec{A} \\right) = -\\vec{j}.\n$$\nGenom att v\u00e4lja Gaugeparametern s\u00e5 att $\\nabla \\cdot \\vec{A} = 0$ f\u00e5r vi Poissons ekvation f\u00f6r vektorpotentialen\n\n$$\n\\begin{equation}\n\\Delta \\vec{A} = -\\vec{j}.\n\\end{equation}\n$$\n\n## Potentialer f\u00f6r godtyckliga vektorf\u00e4lt\n\n* Betrakta ett f\u00e4lt med b\u00e5de k\u00e4llor och virvlar, dvs $\\nabla \\cdot \\vec{H} = \\rho \\neq 0$ och $\\nabla \\times \\vec{H} = \\vec{j} \\neq 0$.\n\n* Detta allm\u00e4nna vektorf\u00e4lt kan vi dela upp i tv\u00e5 delar $\\vec{H} = \\vec{F} + \\vec{G}$ d\u00e4r\n\n$$\n\\begin{equation}\n\\nabla \\cdot \\vec{F} = \\rho \\quad \\nabla \\cdot \\vec{G} = 0 \n\\end{equation}\n$$\n\n$$\n\\begin{equation} \n\\nabla \\times \\vec{F} = 0 \\quad \\nabla \\times \\vec{G} = \\vec{j}\n\\end{equation}\n$$\n\n* Dvs, f\u00e4ltet $\\vec{H}$ kan skrivas som summan av ett rotationsfritt f\u00e4lt $\\vec{F}$ och ett divergensfritt f\u00e4lt $\\vec{G}$ som representeras av potentialerna $\\phi$ och $\\vec{A}$.\n\n$$\n\\vec{H} = \\vec{F} + \\vec{G} = -\\nabla \\phi + \\nabla \\times \\vec{A}.\n$$\n\n### Exempel: Stagnationsstr\u00f6m, konservativt hastighetsf\u00e4lt\n\n---------------\n\nBetrakta (det rotationsfria) hastighetsf\u00e4ltet som ges av potentialen\n$$\n\\phi(x,y) = \\frac{A}{2}(x^2 - y^2),\n$$\nd\u00e4r $A$ \u00e4r en positiv konstant. Hastighetsf\u00e4ltet blir\n$$\n\\vec{v} = -\\nabla \\phi \\quad \\Rightarrow \n\\left\\{\n\\begin{array}{l}\nv_x = -A x \\\\\nv_y = A y\n\\end{array}\n\\right.\n$$\nVi noterar att detta f\u00e4lt ocks\u00e5 \u00e4r divergensfritt. \n\nEn f\u00e4ltbild (potential och f\u00e4ltlinjer) visas nedan. Notera att det finns omr\u00e5den d\u00e4r hastigheten \u00e4r noll. Dessa kallas stagnationspunkter. Vi noterar ocks\u00e5 att f\u00e4ltlinjerna blir parallella med $x$- och $y$-axlarna n\u00e4r vi kommer tillr\u00e4ckligt n\u00e4ra. Vi kan d\u00e4rf\u00f6r t\u00e4nka oss dessa som fasta v\u00e4ggar och att v\u00e5rt hastighetsf\u00e4lt beskriver str\u00f6mmen vid ett h\u00f6rn.\n\n\n\n\n

                                        \n\n\n\n\n\nI verkligheten kommer dock friktionen n\u00e4ra v\u00e4ggen att skapa virvlar, och v\u00e5r rotationsfria approximation ger en s\u00e4mre beskrivning.\n\n------------------------------------\n\n## Standardexempel p\u00e5 k\u00e4ll- och virvelf\u00f6rdelningar\n\n### Punktk\u00e4lla med styrkan $q$ i origo\n\n* vektorf\u00e4lt \n\n$$\n\\vec F=\\frac{q}{4\\pi r^2}\\hat r.\n$$\n* rotationsfritt, men med en k\u00e4lla $\\rho=q\\delta^3(\\vec{r})$. F\u00e4ltet har en potential\n\n$$\n\\phi=\\frac{q}{4\\pi r},\n$$\nsom uppfyller Poissons ekvation med k\u00e4llan $\\rho=q\\delta^3(\\vec{r})$, \n$$\n\\Delta\\phi=-q\\delta^3(\\vec{r}).\n$$\n\n### Linjek\u00e4lla p\u00e5 $z$-axeln med konstant styrka $k$\n\n* vektorf\u00e4lt\n\n$$\n\\vec F=\\frac{k}{2\\pi\\varrho}\\hat\\varrho\n$$\n* Motsvarande potential \u00e4r \n\n$$\n\\phi=-\\frac{k}{2\\pi}\\log\\frac{\\varrho}{\\varrho_0},\n$$\n* Potentialen uppfyller Poissons ekvation:\n\n$$\n\\Delta\\phi=-k\\delta^2(\\vec{\\varrho}).\n$$\n\n### Virveltr\u00e5d p\u00e5 $z$-axeln med styrka $J$\n\n* f\u00e4ltet \n\n$$\n\\vec G=\\frac{J}{2\\pi\\varrho}\\hat\\varphi.\n$$\n* Vektorpotentialen \u00e4r (t.ex., med tanke p\u00e5 gaugeinvarians)\n\n$$\n\\vec A=-\\frac{J\\hat z}{2\\pi}\\log\\frac{\\varrho}{\\varrho_0},\n$$\n(kontrollera; $\\nabla\\times\\vec A$ i cylindriska koordinater)\n$$\n\\nabla\\times\\vec A=\\frac{1}{\\varrho}\n\t\\begin{vmatrix}\n \\hat\\varrho & \\varrho\\hat\\varphi & \\hat z \\\\ \\frac{\\partial}{\\partial \\varrho} & \n\\frac{\\partial}{\\partial \\varphi} &\n \\frac{\\partial}{\\partial z} \\\\ 0\\mbox{} & 0\\mbox{} & -\\frac{J}{2\\pi}\\log\\varrho \\\\\n \\end{vmatrix}\n =\\frac{J}{2\\pi\\varrho}\\hat\\varphi.\n$$\n* Notera att denna vektorpotential uppfyller $\\nabla\\cdot\\vec A=0$, och d\u00e4rf\u00f6r ges virvelf\u00f6rdelningen av \n\n$$\n\\nabla \\times \\vec{G} = \\vec{\\jmath} = -\\Delta\\vec A.\n$$ \n* Poissons ekvation, \n\n$$\n\\Delta\\vec A=-J\\hat z\\delta^2(\\vec{\\varrho}).\n$$\n\n\n## Randv\u00e4rdesproblem\n\n[Comment 7: Att ta reda p\u00e5 precis \"hur mycket\" villkor, och av vilket slag, man b\u00f6r l\u00e4gga p\u00e5 f\u00e4ltet p\u00e5 randen $\\partial V$ \u00e4r ju ett matematiskt problem, men det matematiska svaret p\u00e5 fr\u00e5gan b\u00f6r ocks\u00e5 vara ett svar inom fysik, s\u00e5 att en given fysikalisk f\u00f6ruts\u00e4ttning ger en unik l\u00f6sning (f\u00e4ltkonfiguration).]\n\n* V\u00e5rt randv\u00e4rdesproblem best\u00e5r av den partiella differentialekvationen $\\Delta\\phi=-\\rho$ samt n\u00e5gra randvillkor \n\n* Vilka randvillkor ger en *unik* l\u00f6sning? Eller hur skall \"bra\" randvillkor formuleras? \n\n* Antag att $\\phi_1$ och $\\phi_2$ b\u00e5da \u00e4r l\u00f6sningar. \n\n* $\\psi=\\phi_1-\\phi_2$, uppfyller Laplaces ekvation, $\\Delta\\psi=0$. \n\n* En trivial l\u00f6sning, $\\psi=\\mathrm{konstant}$, inneb\u00e4r att l\u00f6sningen till Poissons ekvation \u00e4r unik \n\n[Comment 8: Eftersom $\\phi_1$ och $\\phi_2$ ger samma vektorf\u00e4lt om de enbart skiljer p\u00e5 en konstant.]\n\nBetrakta nu identiteten\n$$\n\\nabla\\cdot(\\psi\\nabla\\psi)=\\nabla\\psi\\cdot\\nabla\\psi+\\psi\\Delta\\psi\n=\\nabla\\psi\\cdot\\nabla\\psi=|\\nabla\\psi|^2,\n$$\nsom g\u00e4ller n\u00e4r $\\Delta\\psi=0$. Till\u00e4mpa nu Gauss sats p\u00e5 vektorf\u00e4ltet $\\psi\\nabla\\psi$. \n$$\n\\int_V|\\nabla\\psi|^2dV=\\int_{\\partial V}\\psi\\nabla\\psi\\cdot d\\vec S.\n$$\n* VL \u00e4r positivt semidefinit, och noll endast om $\\psi = \\mathrm{konstant}$. \n\n* Randvillkor som g\u00f6r HL till noll inneb\u00e4r d\u00e4rf\u00f6r det vi vill.\n\nYtintegralen i h\u00f6gerledet \u00e4r \n$$\n\\int_{\\partial V}\\psi(\\nabla\\psi\\cdot\\vec n)dS.\n$$ \nTv\u00e5 faktorer i integranden: $\\psi$ och $\\nabla\\psi\\cdot\\vec n$. \n\n[Comment 9: Den andra faktorn \u00e4r \"normalderivatan\" vid randen, allts\u00e5 riktningsderivatan i normalens riktning.]\n\n[Comment 10: L\u00f6sningen till Laplaces ekvation \u00e4r trivial (konstant) om den ena eller den andra \u00e4r noll p\u00e5 randen.]\n\n### Dirichlets randvillkor:\n\n$$\n\\psi = 0 \\mathrm{~p\u00e5~} \\partial V \\quad \\Rightarrow \\quad \\psi = 0 \\mathrm{~i~} V\n$$\nDetta ger att\n$$\n\\phi|_{\\partial V}=f,\n$$\nd\u00e4r $f$ \u00e4r en funktion p\u00e5 randen\n\n### Neumanns randvillkor:\n\n$$\n(\\nabla\\psi) \\cdot \\vec n=0 \\mathrm{~p\u00e5~} \\partial V \\quad \\Rightarrow \\quad \\psi = \\mathrm{konstant} \\mathrm{~i~} V\n$$\nDetta ger att\n$$\n(\\nabla\\phi)|_{\\partial V}\\cdot\\vec n=g,\n$$\nd\u00e4r $g$ \u00e4r en funktion p\u00e5 randen. \n\n### Sammanfattning:\n\nPoissons ekvation i volymen $V$ med n\u00e5gon k\u00e4llf\u00f6rdelning $\\rho$ har en unik l\u00f6sning (s\u00e5n\u00e4r\nsom p\u00e5 en ointressant konstant) f\u00f6r dessa tv\u00e5 typer av randvillkor.\n", "meta": {"hexsha": "437827f7bf4cb713a1182eb2e2f62244da8ce0b3", "size": 23978, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "doc/pub/08-potentialteori/ipynb/08-potentialteori.ipynb", "max_stars_repo_name": "physics-chalmers/ffm234", "max_stars_repo_head_hexsha": "b37a744e50604ba0956724883714ea3d87929f81", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "doc/pub/08-potentialteori/ipynb/08-potentialteori.ipynb", "max_issues_repo_name": "physics-chalmers/ffm234", "max_issues_repo_head_hexsha": "b37a744e50604ba0956724883714ea3d87929f81", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/pub/08-potentialteori/ipynb/08-potentialteori.ipynb", "max_forks_repo_name": "physics-chalmers/ffm234", "max_forks_repo_head_hexsha": "b37a744e50604ba0956724883714ea3d87929f81", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2020-08-06T06:03:59.000Z", "max_forks_repo_forks_event_max_datetime": "2020-11-03T13:36:07.000Z", "avg_line_length": 36.6076335878, "max_line_length": 389, "alphanum_fraction": 0.5399532905, "converted": true, "num_tokens": 6340, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.7431680086124812, "lm_q1q2_score": 0.4120647112790773}} {"text": "### Recurrent Neural Network\n\nLet us now move on to Recurrent Neural Network (RNN). Recurrent neural network is good in handling sequential data because they have a memory component which enables this network to remember past (few) information making it better for a model requiring varying length inputs and outputs.\n\n>For example, consider the two sentences \u201cI went to Nepal in 2009\u201d and \u201cIn 2009,I went to Nepal.\u201d If we ask a machine learning model to read each sentence and extract the year in which the narrator went to Nepal, we would like it to recognize the year 2009 as the relevant piece of information, whether it appears in the sixth word or the second word of the sentence. Suppose that we trained a feedforward network that processes sentences of \ufb01xed length. A traditional fully connected feedforward network would have separate parameters for each input feature, so itwould need to learn all of the rules of the language separately at each position in the sentence.\n

                                        - Ian Goodfellow, Yoshua Bengio, and Aaron Courville

                                        \n\nIn this post, we'll implement a simple RNN, applying it to the problem of [part-of-speech (POS) tagging](https://en.wikipedia.org/wiki/Part-of-speech_tagging) problem. \n\n\n```julia\nusing PyPlot\n```\n\n### Understanding the Data\n\nThe data used in solving the POS tagging problem was taken from Graham Neubig's [NLP Programming Tutorial](http://www.phontron.com/teaching.php). Training dataset is a collection of sentences where each word already has a POS tag attached to it : \n\n$$word1\\_tag1\\;word2\\_tag2\\;\\;...\\;\\; .\\_.$$\n\nThe function readWordTagData, reads such a file and outputs an array of :\n\n- unique words\n- unique tags\n- whole dataset split into tuples containing word and its associated tag\n\nArrays of unique words and tags will help in [one-hot vector representation](http://lakshgupta.github.io/2015/07/12/VectorRepresentationofWords1/) which we'll feed to our neural network. In order to handle unknown words and unknown tags, I have also added $UNK\\_W$ and $UNK\\_T$ to these arrays.\n\n\n```julia\n# https://github.com/JuliaLang/julia/issues/14099\nconst spaces = filter(isspace, Char(0):Char(0x10FFFF));\n```\n\n\n```julia\nfunction readWordTagData(filePath)\n file = open(filePath);\n vocabSet = Set{AbstractString}();\n tagSet = Set{AbstractString}();\n # read line\n for ln in eachline(file)\n word_tag = split(ln, spaces);\n # remove \"\"\n word_tag = word_tag[word_tag .!= \"\"]\n # separate word from tag\n for token in word_tag\n tokenSplit = split(token, \"_\");\n push!(vocabSet, tokenSplit[1]);\n push!(tagSet, tokenSplit[2]);\n end\n end\n close(file);\n # to handle unknown words\n push!(vocabSet, \"UNK_W\");\n # to handle unknown tags\n push!(tagSet, \"UNK_T\");\n #println(vocabSet)\n #println(tagSet)\n vocab = collect(vocabSet);\n tags = collect(tagSet);\n # prepare data array\n data = Tuple{AbstractString , AbstractString }[];\n file = open(filePath);\n # read line\n for ln in eachline(file)\n word_tag = split(ln, spaces);\n # remove \"\"\n word_tag = word_tag[word_tag .!= \"\"]\n # separate word from tag\n for token in word_tag\n tokenSplit = split(token, \"_\");\n push!(data, (tokenSplit[1], tokenSplit[2]));\n end\n end\n close(file);\n #println(length(data))\n return vocab, tags, data;\nend\n```\n\n\n\n\n readWordTagData (generic function with 1 method)\n\n\n\n### Setting up RNN\n\nLooking at a time step $t$, an RNN receives an input $x_t$ along with the hidden state($h_{t-1}$) computed in the previous time step ($t-1$). If you unroll the RNN over few time steps then it becomes easier to understand and train the network, similar to a feedforward neural network. Any network with connections making a cycle can be unrolled into a series of feedforward network representing each time step. All these time steps share the respective weights and biases over the network and our task would be to learn these parameters using the backpropagation algorithm. It'll become clear once we get to the code.\n \n\n\n```julia\n# read data\nvocabListTrain, tagListTrain, dataTrain = readWordTagData(\"data/pos/wiki-en-train.norm_pos\");\n# define the network\ninputLayerSize = length(vocabListTrain);\nhiddenLayerSize = 100;\noutputLayerSize = length(tagListTrain);\nlearningRate = 1e-3;\ndecayRate = 0.9;\nepsVal = 1e-5;\n# initialize weights and biases\nWxh = randn(inputLayerSize, hiddenLayerSize)*0.01; # input to hidden\nWhh = randn(hiddenLayerSize, hiddenLayerSize)*0.01; # hidden to hidden\nBh = zeros(1, hiddenLayerSize); # hidden bias\nWhy = randn(hiddenLayerSize, outputLayerSize)*0.01; # hidden to output\nBy = zeros(1, outputLayerSize); # output bias\n```\n\n\n\n\n 1x43 Array{Float64,2}:\n 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 \u2026 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n\n\n\n### Backpropagation Through Time (BPTT)\n\nIn order to learn the weights for our recurrent network, we have to generalize our backpropagation algorithm and apply it to our recurrent neural networks. This variant we are going to use is known as [backpropagation through time (BPTT)](http://mail.werbos.com/Neural/BTT.pdf) since it works on sequences in time. \n\nIn the forward process, we'll take one-hot vector representation of a word and will try to predict the normalized tag probability. For that we are using the tanh activation function at the hidden layer and the softmax at the output layer. Since our network is an RNN, at time step $1$ for the first sequence, the hidden state can be a zero matrix and in the next time step we can use whatever we have calculated in the previous time step i.e., the value of the last hidden state of the first sequence can be used at time step $1$ for the next sequence and so on. \n\n$$a_t = W_{xh}*x_{t} + W_{hh}*h_{t-1} + B_h$$\n$$h_t = tanh(a_t)$$\n$$z_t = W_{hy}*h_t + B_y$$\n$$\\hat y_t = softmax(z_t)$$\n\nWe'll save the hidden state values and the normalized probabilities at the each time step to use during the backward process. \n\nFor each time step we define out cross entropy cost function as:\n\n$$J_t(W_xh, W_hh, W_hy) = J_t(W) = - y_t* \\log (\\hat y_t)$$\n\nSince our input is a sequence we define the cost for the whole sequence as:\n\n$$J(W) = \\sum_{t}J_t(W) = - \\sum_{t} y_t* \\log (\\hat y_t)$$\n\n\n\n```julia\nfunction forwardRNN(x::Array{Vector{Float64},1}, y::Array{Vector{Float64},1}, \n h::Array{Array{Float64,2},1}, p::Array{Vector{Float64},1}, hPrev::Array{Float64,2})\n \n global Wxh, Whh, Why, Bh, By;\n cost = 0;\n # for each time t in x\n # unrolling RNN -> Feedforward NN step\n for time in 1:length(x)\n if time == 1\n h[time] = tanh(x[time]'*Wxh + hPrev*Whh .+ Bh);\n else\n h[time] = tanh(x[time]'*Wxh + h[time-1]*Whh .+ Bh);\n end\n # output layer\n score = h[time]*Why .+ By;\n p_softmax = exp(score) / sum(exp(score));\n p[time] = vec(p_softmax); # output probability distribution (at time t)\n cost += -sum(log(y[time]'*p[time])) # assuming y is a one-hot vector\n end\n return cost;\nend\n```\n\n\n\n\n forwardRNN (generic function with 1 method)\n\n\n\nIn the backward process, we apply the chain rule as usual to backpropagate the errors and calculate the gradients. We already defined our cost function as :\n\n$$J(W) = - \\sum_{t} y_t* \\log (\\hat y_t)$$\n\nSince all output units contribute to the error of each hidden unit we sum up all the gradients calculated at each time step in the sequence and use it to update the parameters. So our parameter gradients becomes :\n\n$$\\frac{\\partial J_t}{\\partial W_{hy}} = \\sum_t \\frac{\\partial J}{\\partial z_t}*\\underbrace{\\frac{\\partial z_t}{\\partial W_{hy}}}_{h_t}$$\n\n\n$$\\frac{\\partial J_t}{\\partial W_{hh}} = \\sum_t \\frac{\\partial J}{\\partial h_t}*\\underbrace{\\frac{\\partial h_t}{\\partial W_{hh}}}_{(1-h_t^2)*h_{t-1}}$$\n\n\n$$\\frac{\\partial J_t}{\\partial W_{xh}} = \\sum_t \\frac{\\partial J}{\\partial h_t}*\\underbrace{\\frac{\\partial h_t}{\\partial W_{xh}}}_{(1-h_t^2)*x_{t}}$$\n\n\nwhere, \n\n$$\\frac{\\partial J}{\\partial z_t} = \\underbrace{\\frac{\\partial J}{\\partial J_t}}_{= 1} * \\frac{\\partial J_t}{\\partial z_t}$$\n\n\nFrom the [previous post](http://lakshgupta.github.io/2016/01/16/NeuralNetwork2/) we know that:\n\n$$\\frac{\\partial J_t}{\\partial z_t} = \\hat y_t - y_t $$\n\nPoint to note here is that change in $z_t$ does not affect the cost at other time steps. But that is not the case with $h_t$. A change in $h$ at time step $t$ is going to change the cost at $t+1$, $t+2$ ...$T$. In other words, $h_t$ is going to affect $\\hat y_t$ and $h_{t+1}$, which in turn will propagate forward and affect $h$ at other time steps. \n\nSo let us look at the end of the sequence, i.e., at time step $T$:\n$$\\begin{aligned} \n\\frac{\\partial J}{\\partial h_T} &= \\frac{\\partial J}{\\partial z_T}*\\frac{\\partial z_T}{\\partial h_T} \\\\\n&= (\\hat y_T - y_T)*W_{hy}\n\\end{aligned}\n$$\n\nand at other time step $t$ in the sequence,\n\n$$\\begin{aligned} \n\\underbrace{\\frac{\\partial J}{\\partial h_t}}_{dh} &= \\underbrace{\\left (\\frac{\\partial J}{\\partial z_t}*\\frac{\\partial z_t}{\\partial h_t} \\right )}_{a} + \\underbrace{\\left (\\frac{\\partial J}{\\partial h_{t+1}}*\\frac{\\partial h_{t+1}}{\\partial h_t}\\right )}_{dhnext}\\end{aligned}\n$$\n\nWe can work out $a$ from the other equations above. In $dhnext$, the value of $\\frac{\\partial J}{\\partial h_{t+1}}$ will backpropagate since we would have already calculated this at time step $(t+1)$ before coming on to the time step $t$. For the rest :\n$$\\frac{\\partial h_{t+1}}{\\partial h_t} = W_{hh}*(1-h_t^2)$$\n\n\nA similar but a different way of working out the equations can be seen in Richard Sochers's [Recurrent Neural Network](http://cs224d.stanford.edu/lectures/CS224d-Lecture7.pdf) lecture slide.\n >\n\n\nIn the code we are also using gradient clipping to handle the gradient explosion problem. I first observed this being used in Andrej Karpathy's [min-char-rnn.py](https://gist.github.com/karpathy/d4dee566867f8291f086), and then in [this](http://arxiv.org/abs/1211.5063) paper.\n>We propose a gradient norm\nclipping strategy to deal with exploding gradients\nand a soft constraint for the vanishing\ngradients problem. \n

                                        - Razvan Pascanu, Tomas Mikolov, Yoshua Bengio

                                        \n\n\n```julia\nfunction backwardRNN(x::Array{Vector{Float64},1}, y::Array{Vector{Float64},1}, \n h::Array{Array{Float64,2},1}, p::Array{Vector{Float64},1}, hPrev::Array{Float64,2})\n \n global Wxh, Whh, Why, Bh, By;\n \n dWxh = zeros(size(Wxh));\n dWhh = zeros(size(Whh));\n dBh = zeros(size(Bh));\n dWhy = zeros(size(Why));\n dBy = zeros(size(By));\n \n dh = zeros(size(Bh)); # error from the following time step\n dhnext = zeros(size(h[1]));\n for time in length(x):-1:1\n # output layer error\n dy = p[time] - y[time]; #assuming y is a one hot vector\n # output gradient\n dWhy = dWhy + (dy * h[time])';\n dBy = dBy + dy';\n # backpropagate\n dh = (Why*dy)' + dhnext;\n dhRaw = (1 - (h[time].*h[time])) .* dh;\n # hidden layer gradient\n dWxh = dWxh + (x[time] * dhRaw);\n dBh = dBh + dhRaw;\n if time == 1\n dWhh = dWhh + (hPrev' * dhRaw);\n else\n dWhh = dWhh + (h[time-1]' * dhRaw);\n end\n dhnext = dhRaw*Whh;\n end\n # clip to mitigate exploding gradients\n dWxh = clamp(dWxh, -5, 5);\n dWhh = clamp(dWhh, -5, 5);\n dBh = clamp(dBh, -5, 5);\n dWhy = clamp(dWhy, -5, 5);\n dBy = clamp(dBy, -5, 5);\n \n return dWxh, dWhh, dBh, dWhy, dBy;\nend\n```\n\n\n\n\n backwardRNN (generic function with 1 method)\n\n\n\n### Gradient Check\n\nThe implementation of backpropagation algorithm hasn't changed much over these series of posts but it's always good to keep a tool in hand to check whether our code is correct or not. We first go through the backpropagation step to calculate the analytical gradients. Then we perturb an element in the weight matrix and go through the forward step to calculate the cost calculating the numerical gradient as :\n\n$$\\begin{align}\n\\frac{d}{d\\theta}J(\\theta) = \\lim_{\\epsilon \\rightarrow 0}\n\\frac{J(\\theta+ \\epsilon) - J(\\theta-\\epsilon)}{2 \\epsilon}.\n\\end{align}$$\n\nBy comparing the relative error between the analytical gradient and the numerical gradient we can be sure that there is no bug in calculating the gradients. It should be very very small, less than $1e^{-5}$.\n\n>Use relative error for the comparison. What are the details of comparing the numerical gradient $f\u2032n$ and analytic gradient $f\u2032a$? That is, how do we know if the two are not compatible? You might be tempted to keep track of the difference $\u2223fa\u2032\u2212fn\u2032\u2223$ or its square and define the gradient check as failed if that difference is above a threshold. However, this is problematic. For example, consider the case where their difference is 1e-4. This seems like a very appropriate difference if the two gradients are about 1.0, so we'd consider the two gradients to match. But if the gradients were both on order of 1e-5 or lower, then we'd consider 1e-4 to be a huge difference and likely a failure. Hence, it is always more appropriate to consider the relative error :\n>\n$$\\frac{\\mid f'_a - f'_n \\mid}{\\max(\\mid f'_a \\mid, \\mid f'_n \\mid)}$$\n>\nwhich considers their ratio of the differences to the ratio of the absolute values of both gradients. Notice that normally the relative error formula only includes one of the two terms (either one), but I prefer to max (or add) both to make it symmetric and to prevent dividing by zero in the case where one of the two is zero (which can often happen, especially with ReLUs). However, one must explicitly keep track of the case where both are zero and pass the gradient check in that edge case. In practice :\n>\n- relative error > 1e-2 usually means the gradient is probably wrong\n- 1e-2 > relative error > 1e-4 should make you feel uncomfortable\n- 1e-4 > relative error is usually okay for objectives with kinks. But if there are no kinks (e.g. use of tanh nonlinearities and softmax), then 1e-4 is too high.\n- 1e-7 and less you should be happy.\n>\nAlso keep in mind that the deeper the network, the higher the relative errors will be. So if you are gradient checking the input data for a 10-layer network, a relative error of 1e-2 might be okay because the errors build up on the way. Conversely, an error of 1e-2 for a single differentiable function likely indicates incorrect gradient.\n

                                        - Andrej Karpathy

                                        \n\nIts been also suggested to:\n- not perform the gradient check at the first iteration\n- turn off regularization\n- check some of the dimensions of the gradient and assume that the others are correct\n\n\n```julia\n# gradient checking\nfunction gradCheck(inputs::Array{Vector{Float64},1}, targets::Array{Vector{Float64},1},\n h::Array{Array{Float64,2},1}, p::Array{Vector{Float64},1}, hPrev::Array{Float64,2})\n \n paramNameList = [\"Wxh\", \"Whh\", \"Bh\", \"Why\", \"By\"];\n # collect paramters\n global Wxh, Whh, Why, Bh, By;\n paramList = [x for x=(Wxh, Whh, Bh, Why, By)];\n num_checks = 2;\n delta = 1e-5;\n # collect parameter gradients\n cost = forwardRNN(inputs, targets, h, p, hPrev);\n dWxh, dWhh, dBh, dWhy, dBy = backwardRNN(inputs, targets, h, p, hPrev);\n dParamList = [x for x=(dWxh, dWhh, dBh, dWhy, dBy)];\n for (param,dparam,name) in zip(paramList, dParamList, paramNameList)\n # validate the size of the parameter and its gradient\n s0 = size(dparam);\n s1 = size(param);\n if s0 != s1\n println(\"Error dims dont match: \", s0,\" and \",s1);\n end\n println(name)\n for i in 1:num_checks\n ri = rand(1:length(param));\n # evaluate cost at [x + delta] and [x - delta]\n old_val = param[ri];\n param[ri] = old_val + delta;\n cg0 = forwardRNN(inputs, targets, h, p, hPrev);\n param[ri] = old_val - delta;\n cg1 = forwardRNN(inputs, targets, h, p, hPrev);\n param[ri] = old_val # reset old value for this parameter\n # fetch both numerical and analytic gradient\n grad_analytic = dparam[ri];\n grad_numerical = (cg0 - cg1) / ( 2 * delta );\n \n rel_error = abs(grad_analytic - grad_numerical) / abs(grad_numerical + grad_analytic);\n println(grad_numerical,\", \", grad_analytic, \" => \",rel_error);\n # rel_error should be on order of 1e-7 or less\n if rel_error > 1e-5\n error(\"Gradient check failed.\");\n end\n end\n println(\"Gradient check passed.\")\n end\nend\n```\n\n\n\n\n gradCheck (generic function with 1 method)\n\n\n\nI ran the gradient check for a sample dataset and used the sequence length of 1. This operation is commented out in the code below because it is very computation expensive. The train method loops over each sequence, and updates the weights/biases matrices using the backpropagation algorithm to calculate the gradients for that sequence.\n\n\n```julia\n\nfunction train(data::Array{Tuple{AbstractString,AbstractString},1}, vocabList::Array{AbstractString,1} \n , tagList::Array{AbstractString,1}, numItr::Int64, seqLength::Int64, sampleCostAtItr::Int64)\n \n global Wxh, Whh, Why, Bh, By;\n numIterations = numItr * length(data);\n costList = []; # store cost per sampled iteration\n ptr = 1;\n p = [zeros(length(tagList)) for i in 1:seqLength];\n h = Array{Float64,2}[zeros(1,hiddenLayerSize) for i in 1:seqLength]; # hidden layers (at time t)\n hPrev = zeros(1, hiddenLayerSize);\n for itr in 1:numIterations\n # take care of the sequence length\n if ptr+seqLength-1 > length(data) # whenever we are looking at the data from the start\n # reset RNN memory\n hPrev = zeros(1, hiddenLayerSize);\n # go from start of data\n ptr = 1 \n end\n # generate sequence\n seqData = data[ptr:ptr+seqLength-1];\n x = Vector{Float64}[];\n y = Vector{Float64}[];\n for word_tag in seqData\n word = word_tag[1];\n tag = word_tag[2];\n # convert to one-hot vectors\n # words\n wordVec = zeros(length(vocabList));\n findWord = collect(vocabList.==word)\n if length(findWord[findWord.==true]) == 0\n # unknown word: UNK_W\n findWord[length(findWord)] = true;\n end\n wordVec[findWord] = 1;\n # tags\n tagVec = zeros(length(tagList));\n findTag = collect(tagList.==tag)\n if length(findTag[findTag.==true]) == 0\n # unknown tag: UNK_T\n findTag[length(findTag)] = true;\n end\n tagVec[findTag] = 1;\n # push to the sequence\n push!(x , wordVec);\n push!(y , tagVec);\n end\n # gradient check\n #gradCheck(x, y, h, p, hPrev);\n \n # feedforward\n cost = forwardRNN(x, y, h, p, hPrev);\n # sample cost\n if itr%sampleCostAtItr == 1\n push!(costList, cost);\n end\n # backpropagate\n dWxh, dWhh, dBh, dWhy, dBy = backwardRNN(x, y, h, p, hPrev);\n \n # update weights\n Wxh += -learningRate * dWxh;\n Whh += -learningRate * dWhh;\n Bh += -learningRate * dBh;\n Why += -learningRate * dWhy;\n By += -learningRate * dBy;\n \n # previous output as hidden input for the next sequence\n hPrev = h[length(h)]; \n \n # move data pointer\n ptr += seqLength;\n end\n return costList;\nend\n```\n\n\n\n\n train (generic function with 1 method)\n\n\n\nWe'll run the training step for the whole data 20 times, using an input sequence of size 2. Since there will be lot of iteration we'll sample the cost at every 1000 interation.\n\n\n```julia\n# MAIN\n# number of steps to unroll the RNN for\nseqLen = 2 \n# run through the data n times\nnumIterOverData = 20;\n# sample cost after each n iteration\nsampleCostAtEveryItr = 1000;\nJ = train(dataTrain, vocabListTrain, tagListTrain, numIterOverData, seqLen, sampleCostAtEveryItr);\n```\n\n\n```julia\n# plot the cost per iteration\nsampleIdxJ = [1+sampleCostAtEveryItr*i for i in 0:length(J)-1]\nplot(sampleIdxJ, J)\nxlabel(\"Sampled Iterations\")\nylabel(\"Cost\")\ngrid(\"on\")\n```\n\nThe cost in the graph seems to be decreasing with respect to the iteration. The cost is not smoothly decreasing because we are kind of running a mini-batch gradient descent hence it is just an estimate of the cost over the whole dataset. It's in my TODO list to investigate it more thoroughly some day since the graph looks a bit ugly. For now, let's check how it is performing. To find the accuracy, we'll use the learned model and go through the forward step to make the prediction.\n\n\n```julia\nfunction findAccuracy(data::Array{Tuple{AbstractString,AbstractString},1}, vocabList::Array{AbstractString,1} \n , tagList::Array{AbstractString,1}, seqLength::Int64)\n \n correct = 0;\n ptr=1;\n p = [zeros(length(tagList)) for i in 1:seqLength];\n h = Array{Float64,2}[zeros(1,hiddenLayerSize) for i in 1: seqLength]; # hidden layers (at time t)\n hPrev = zeros(1,hiddenLayerSize);\n for i in 1:length(data)/seqLength\n \n # prepare inputs (we're sweeping from left to right in steps seq_length long)\n if ptr+seqLength-1 > length(data) # whenever we are looking at the data from the start\n break # return if data is alread read\n end\n # generate sequence\n seqData = data[ptr:ptr+seqLength-1];\n x = Vector{Float64}[];\n y = Vector{Float64}[];\n for word_tag in seqData\n word = word_tag[1];\n tag = word_tag[2];\n # convert to one-hot vectors\n # words\n wordVec = zeros(length(vocabList));\n findWord = collect(vocabList.==word)\n if length(findWord[findWord.==true]) == 0\n # unknown word: UNK_W\n findWord[length(findWord)] = true;\n end\n wordVec[findWord] = 1;\n # tags\n tagVec = zeros(length(tagList));\n findTag = collect(tagList.==tag)\n if length(findTag[findTag.==true]) == 0\n # unknown tag: UNK_T\n findTag[length(findTag)] = true;\n end\n tagVec[findTag] = 1;\n # push to the sequence\n push!(x , wordVec);\n push!(y , tagVec);\n end\n # feedforward\n cost = forwardRNN(x, y, h, p, hPrev);\n hPrev = h[size(h,1)];\n \n prediction = [indmax(p[j]) for j in 1:length(seqData)];\n truth = [indmax(y[j]) for j in 1:length(seqData)];\n # accuracy\n for j in 1:length(seqData)\n if truth[j] == prediction[j]\n correct = correct + 1;\n end\n end\n ptr += seqLength; # move data pointer\n end\n accuracy = correct/length(data)*100;\n return accuracy;\nend\n```\n\n\n\n\n findAccuracy (generic function with 1 method)\n\n\n\n\n```julia\naccuracy = findAccuracy(dataTrain, vocabListTrain, tagListTrain, seqLen);\nprintln(\"accuracy: \", accuracy);\n```\n\n accuracy: 90.99910251585072\n\n\n### Test data\n\nThe test data consists of two files, one with regular sentences in English and other with their corresponding tags. I am just mapping the words and tags to produce an array of tuples similar to what we used in the training step.\n\n\n```julia\nfunction readData(filePath)\n file = open(filePath);\n # read line\n wordList = [];\n for ln in eachline(file)\n words = split(ln, spaces);\n # remove \"\"\n words = words[words .!= \"\"]\n # append to the list\n wordList = [words; wordList];\n end\n close(file);\n return wordList;\nend\n```\n\n\n\n\n readData (generic function with 1 method)\n\n\n\n\n```julia\n# read test data\nwordListTest = readData(\"data/pos/wiki-en-test.norm\");\ntagListTest = readData(\"data/pos/wiki-en-test.pos\"); \ndataTest = Tuple{AbstractString , AbstractString }[];\n\nfor i in 1:length(wordListTest)\n push!(dataTest, (wordListTest[i], tagListTest[i]));\nend\n```\n\n\n```julia\n# will use vocab list and tag list created at the training time\naccuracy = findAccuracy(dataTest, vocabListTrain, tagListTrain, seqLen);\nprintln(\"accuracy: \", accuracy);\n```\n\n accuracy: 85.1194389655928\n\n\nAs you can see, RNN performed quite well on both the train and the test data set. But RNN has it own problems. It suffers from vanishing/exploding gradient problem.\n\n>As introduced in Bengio et al. (1994), the exploding\ngradients problem refers to the large increase in the\nnorm of the gradient during training. Such events are\ncaused by the explosion of the long term components,\nwhich can grow exponentially more then short term\nones. The vanishing gradients problem refers to the\nopposite behaviour, when long term components go\nexponentially fast to norm 0, making it impossible for\nthe model to learn correlation between temporally distant\nevents.\n

                                        - Razvan Pascanu, Tomas Mikolov, Yoshua Bengio

                                        \n\nIn order to tackle these problems, people have come up with few other variants of RNN such as [LSTM (Long-Short Term Memory)](https://en.wikipedia.org/wiki/Long_short-term_memory) and GRU (Gated Recurrent Unit). Maybe I'll cover some of them in the coming posts. Till then, keep it constructive and enjoy!\n\n

                                        References:

                                        \n\n- [The Unreasonable Effectiveness of Recurrent Neural Networks](http://karpathy.github.io/2015/05/21/rnn-effectiveness/)\n- [Recurrent\tNeural\tNetworks - cs224d](http://cs224d.stanford.edu/lectures/CS224d-Lecture7.pdf)\n- [Sequence Modeling: Recurrent and Recursive Nets](http://www.deeplearningbook.org/contents/rnn.html)\n- [A Critical Review of Recurrent Neural Networks for Sequence Learning](http://arxiv.org/abs/1506.00019)\n- [Backpropagation Through Time: What it does and how to do it](http://mail.werbos.com/Neural/BTT.pdf)\n- [A guide to recurrent neural networks and backpropagation](http://faculty.ksu.edu.sa/ghulam/Documents/Downloads/RecurrentNeuralNetwork.pdf)\n- [Lec [5.1]: Deep Learning, Recurrent neural network](https://youtu.be/AvyhbrQptHk)\n- [Training RNNs](http://www.cs.toronto.edu/~rgrosse/csc321/lec10.pdf)\n- [On the difficulty of training Recurrent Neural Networks\n](http://arxiv.org/abs/1211.5063)\n- [Deep Learning Lecture 12: Recurrent Neural Nets and LSTMs](https://youtu.be/56TYLaQN4N8)\n- [NLP Programming Tutorial](http://www.phontron.com/slides/nlp-programming-en-08-rnn.pdf)\n- [Statistical Language Models based on Neural Networks](http://www.fit.vutbr.cz/~imikolov/rnnlm/thesis.pdf)\n- [A Simple Way to Initialize Recurrent Networks of Rectified Linear Units](http://arxiv.org/abs/1504.00941)\n\n", "meta": {"hexsha": "2a196a0da9e9340fa965f2d27022250664561a03", "size": 137159, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/Recurrent Neural Network.ipynb", "max_stars_repo_name": "lakshgupta/lakshgupta.github.io", "max_stars_repo_head_hexsha": "5527262278e6695ae053db03be848c90b7056a55", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/Recurrent Neural Network.ipynb", "max_issues_repo_name": "lakshgupta/lakshgupta.github.io", "max_issues_repo_head_hexsha": "5527262278e6695ae053db03be848c90b7056a55", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/Recurrent Neural Network.ipynb", "max_forks_repo_name": "lakshgupta/lakshgupta.github.io", "max_forks_repo_head_hexsha": "5527262278e6695ae053db03be848c90b7056a55", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 157.835443038, "max_line_length": 100654, "alphanum_fraction": 0.8597248449, "converted": true, "num_tokens": 7436, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593452091673, "lm_q2_score": 0.6619228758499942, "lm_q1q2_score": 0.4120200798805564}} {"text": "\t#========================================================================\n\t# Copyright 2019 Science Technology Facilities Council\n\t# Copyright 2019 University of Manchester\n\t#\n\t# This work is part of the Core Imaging Library developed by Science Technology\n\t# Facilities Council and University of Manchester\n\t#\n\t# Licensed under the Apache License, Version 2.0 (the \"License\");\n\t# you may not use this file except in compliance with the License.\n\t# You may obtain a copy of the License at\n\t#\n\t# http://www.apache.org/licenses/LICENSE-2.0.txt\n\n\t#\n\t# Unless required by applicable law or agreed to in writing, software\n\t# distributed under the License is distributed on an \"AS IS\" BASIS,\n\t# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n\t# See the License for the specific language governing permissions and\n\t# limitations under the License.\n\t#\n\t#=========================================================================\n\n

                                        Multi-Channel Reconstruction

                                        \n\n

                                        Learning Objectives

                                        \n\nBy the end of this notebook, you will be able to:\n\n- Identify the key differences in building Image/Acquisition Geometries and Operators for multi-channel datasets \n- Build your own reconstructions using FDK, CGLS and PDHG\n- Determine optimum regularisation parameters based on reconstruction method\n- Evaluate the effectiveness of each reconstruction routine using energy profiles and elemental maps.\n\n### Prerequisites:\n\n- Acquisition/Image Geometry, Acquisition Data\n- AstraProjectorMC/3DMC\n- FDK, CGLS, PDHG, TV\n- BlockFramework\n\n### Background:\n\nConventional X-ray detectors only measure the variable density of objects they pass through, giving no insight into what materials are actually inside. This is because standard detectors measure only the number of photons that arrive at each point on the detector.\n\nFor multi-channel imaging, one can use an energy-sensitive X-ray detector, which measures the energy of every X-ray photon that arrives at each individual pixel. This provides an additional layer of information which can provide important insight on a sample's composition or structure. However, adapted reconstruction routines are required to account for the extra energy-based dimension.\n\nThe additional energy dimension is stored as a histogram of energy 'channels', indicating the number of X-ray photons detected by a pixel within a fine energy range. Typically 200+ channels are acquired, however in order to speed up computation time, we will restrict our dataset to just 40 channels, where the dominant energy signals are known to appear. \n \n\n\n```python\nfrom __future__ import absolute_import\nfrom __future__ import division\nfrom __future__ import print_function\n\nfrom utilities import islicer, link_islicer\nfrom utilities.show_utilities import channel_to_energy, show\n\nfrom ccpi.framework import ImageGeometry, AcquisitionGeometry, AcquisitionData, ImageData, BlockDataContainer\n\nimport numpy as np \nimport matplotlib.pyplot as plt\nimport h5py\nimport os\nimport sys\n\nfrom ccpi.io import NEXUSDataReader\nfrom ccpi.optimisation.algorithms import PDHG, CGLS\n\nfrom ccpi.optimisation.operators import BlockOperator, Gradient\nfrom ccpi.optimisation.functions import L2NormSquared, KullbackLeibler,\\\n MixedL21Norm, BlockFunction, IndicatorBox\n\nfrom ccpi.astra.utils import *\nfrom ccpi.astra.processors import FBP\nfrom ccpi.astra.operators import AstraProjectorMC, AstraProjector3DMC\n```\n\n\n```python\nfrom ccpi.framework.TestData import data_dir\npathname = data_dir\nfilename = 'sinogram_centered_channels100_140.h5'\n\npath = os.path.join(pathname , filename)\narrays = {}\n\nwith h5py.File(path, 'r+') as f: \n for k, v in f.items():\n arrays[k] = np.array(v)\n X = arrays['SC'] \n```\n\nThe sample we will look at in this notebook is an iodine-stained lizard head. The use of elemental staining is common in biology and medicine, by acting as a contrast agent to provide improved visibility of internal structures for X-ray imaging. Iodine is a popular choice in the clinical and research fields, due to its energy-based properties falling in the typical range for diagnostic X-ray imaging.\n\nThe sample was scanned in a lab-based X-ray CT cone-beam system at 50keV, 0.8W, using an energy-sensitive detector to acquire a full 4D dataset. The detector consisted of an 80 x 80 pixel array, with pixel sizes of 250 $\\mu$m x 250 $\\mu$m. A source-sample distance of 233.0 mm and a sample-detector distance of 245.0 mm gave a geometric magnification of 2.05x. The sample was scanned for 60 projections over a full 360$^{\\circ}$ rotation, with 60s exposure time per projection.\n\nA diagram of the style of setup used for spectral imaging is shown below from a paper by [C.K.Egan *et al*, 2015](https://www.nature.com/articles/srep15979#):\n\n \n\n\nAs shown by the diagram, each pixel stores its own energy channel histogram, with characteristic signals such as 'Absorption edges' (caused by photoelectric absorption of X-ray photons by the iodine in our sample) producing sharp rises in signal. These are the key features we look for when analysing spectral data.\n\n\n

                                        Setting up Acquisition/Image Geometry

                                        \n\nFirst we need to setup the geometries based on the acquisition system.\nThese are currently ordered based on the 4D raw dataset.\n\nRun the code to see the current dimensions.\n\n\n```python\nprint(X.shape)\n```\n\nWe can allocate these separately\n\n\n```python\nnum_channels = X.shape[0]\nnum_pixels_h = X.shape[3]\nnum_pixels_v = X.shape[1]\nnum_angles = X.shape[2]\n```\n\nWe set the angles used based on the parameters chosen for data acquisition. An offset is also applied (this just gives us the freedom to adjust the rotation of our reconstructed images).\n\n\n```python\nangles = np.linspace(-180-55,180-55,num_angles,endpoint=False)*np.pi/180\n```\n\n\nWe encode all this information into `AcquisitionGeometry` and `ImageGeometry`.\n\nRecall that the Geometry for scanning with a cone-beam X-ray source is defined by the following parameters:\n(See **Notebook00_Building_Blocks** for more information)\n\n\n\n\n```python\n# Define acquisition geometry from the detector system acquisition settings\n# with reordering of the dimension labels to match the raw data\n\ndistance_source_center = 233.0 # [mm]\ndistance_center_detector = 245.0 # [mm]\ndetector_pixel_size = 0.25 # [mm]\n\nag = AcquisitionGeometry('cone',\n '3D',\n angles,\n pixel_num_h=num_pixels_h,\n pixel_size_h=detector_pixel_size,\n pixel_num_v=num_pixels_v,\n pixel_size_v=detector_pixel_size, \n dist_source_center=distance_source_center, \n dist_center_detector=distance_center_detector,\n channels=num_channels,\n dimension_labels = ['channel', 'vertical', 'angle', 'horizontal'])\n\n# Create the 4D acquisition data\ndata = ag.allocate()\ndata.fill(X)\n\n# Calculate the geometric magnification to scale the voxel size relative\n# to the detector pixel size.\nmag = (ag.dist_source_center + ag.dist_center_detector)/ag.dist_source_center\n\n# Define Image Geoemtry\nig = ImageGeometry(voxel_num_x=ag.pixel_num_h, \n voxel_num_y=ag.pixel_num_h,\n voxel_num_z=ag.pixel_num_h,\n voxel_size_x=ag.pixel_size_h/mag, \n voxel_size_y=ag.pixel_size_h/mag, \n voxel_size_z=ag.pixel_size_h/mag, \n channels=num_channels)\n```\n\n\n```python\nprint(data)\n```\n\nWe can use an interactive image slicer `islicer` to provide a visualisation of the `AcquisitionData` in any dimension below. Simply by taking a data subset in a particular dimension, we can then visualise the data in any other given dimension.\n\nRun the code below to see three such examples:\n\n1) Projection radiographs for each of the 60 rotation angles acquired in a single channel\n\n2) The sinogram for each energy channel for the central slice subset\n\n3) The spectral signals acquired in each energy channel for a single projection angle \n\n\n**Note: You can adjust the look of your reconstructions by varying certain parameters** \n - by removing `cmap`, you return to the default colour map of 'gray'\n - the default scaling runs from 0.0 through the full data range, vary this using e.g. `minmax = (0.0 , 2.0)`\n\n\n\n```python\nislicer(data.subset(channel=20), direction='angle', title = 'Projection Angle', cmap='inferno')\nislicer(data.subset(vertical=40), direction='channel', title = 'Sinogram Channel', cmap='inferno')\nislicer(data.subset(angle=40), direction='channel', title = 'Channel', cmap='inferno')\n```\n\nWe setup the tomography operator for 3D multi-channel data using the `AcquisitionGeometry` and `ImageGeometry`\n\n\n\n```python\nA3DMC = AstraProjector3DMC(ig, ag)\n```\n\n# FDK Reconstruction\n\nOne of the simplest, and most common, means of image reconstruction for X-ray CT is the use of Filtered BackProjection (FBP). In the case of many lab-based X-ray sources, which utilise a cone-beam rather than parallel- or fan-beams, we use a specific case of FBP: The [Feldkamp-Davis-Kress (FDK)](https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-1-6-612) algorithm.\n\nThe function `FBP` is capable of handling reconstructions for both parallel-beam and cone-beam geometries in 2D and 3D. When supplied with both Acquisition and Image Geometry (`ag`, `ig`), the function recognises and performs the appropriate form of FBP, in this case FDK.\nRun the code below to see a reconstruction using the FDK algorithm. Here we use a `ram-lak` filter as part of the reconstruction.\n\n\n```python\n# Setup geometry for FDK\nfdk = FBP(ig, ag, 'ram-lak') \n# Provide dataset\nfdk.set_input(data)\n# Store results\nrecon = fdk.get_output()\n# Check dimensions\nprint('Shape: {}'.format(recon.shape))\n```\n\n\n```python\nrecon.dimension_labels\n```\n\nWe can see we now have three spatial dimensions to our reconstructed data (each of size 80), along with our 40 energy channels. By selecting a single slice of a spatial dimension, we can view the resulting 80x80 reconstructed image for each energy channel using `islicer`.\n\n\n```python\n# Show results\nislicer(recon.subset(vertical=46),direction='channel',\n title='FDK: Channel', cmap='inferno', minmax=(0.0,0.4))\n```\n\nWhile some features of the lizard head can be seen in the reconstructed images, much of the signal is shrouded by noise. \nIn the next section, we will explore the first iterative algorithm - CGLS.\n\n## Running the CGLS algorithm on a 4D dataset\n\nAs the next step, we will begin with a standard CGLS algorithm, applied to our 4D dataset\n\nWe initialise the operator based on the dimensions of the 4D dataset\n\nLet's test the CGLS algorithm for just 10 iterations\n\n\n```python\n# We initialise \nx_init = A3DMC.volume_geometry.allocate()\n\n# Run the CGLS for 10 iterations\ncgls = CGLS(x_init = x_init, operator = A3DMC, data = data, max_iteration = 100)\ncgls.run(10)\n```\n\nWe can use custom-made functions to visualise the quality of our reconstructions. Here we are looking at three commonly used views of the reconstructed dataset (axial, coronal and sagittal), and how these vary for different energy channels. Here we use the `show` function for multi-channel datasets, providing parameters to vary:\n - Title, Figure/font size\n - Channel number you wish to view (Currently takes one value for 4D data)\n - Colour map\n - Min/max colour map scaling\n\nRun the code below to see reconstructions for the 10th, 20th and 30th channel in our truncated dataset, with the X-ray energies these correspond to.\n\n\n```python\n# Plot axial, coronal and sagittal views\n# Plotter automatically converts channel number to energy\nshow(cgls.get_output(), title='CGLS 4D Reconstruction', show_channels=10, \n cmap='inferno', figure_size = [15,6], font_size=[25,20], minmax=(0.0,0.3)) \nshow(cgls.get_output(), title='CGLS 4D Reconstruction', show_channels=20, \n cmap='inferno', figure_size = [15,6], font_size=[25,20], minmax=(0.0,0.3)) \nshow(cgls.get_output(), title='CGLS 4D Reconstruction', show_channels=30, \n cmap='inferno', figure_size = [15,6], font_size=[25,20], minmax=(0.0,0.3)) \n```\n\nAs a result of performing our reconstruction, we change the shape and dimensions of our 4D dataset, as we now have 3 spatial dimensions along with the energy channels.\nRun the following code to see the new shape and corresponding dimension labels.\n\n\n```python\nprint('Shape = ', cgls.get_output().shape)\nprint('Labels = ', cgls.get_output().dimension_labels)\n```\n\nYou can use these labels to once more explore visualisations with the islicer function. Try varying the subset dimension and the direction using the label names above.\ne.g. try: \n `islicer(cgls.get_output().subset(vertical=46),direction='channel',title='Axial: Channel', cmap='inferno', minmax = (0.0,0.4))`\n\n\n```python\n# Examples slicer\nislicer(cgls.get_output().subset(vertical=46),direction='channel',\n title='Axial View: Channel', cmap='inferno', minmax = (0.0,0.4))\n```\n\nNow we can see how the visualisation changes if we increase the number of iterations. Using the `show` function, we can visualise the reconstructed image slices at chosen iteration intervals. We'll run it for 30 iterations.\n\n\n```python\n# Initialise \nx_init = A3DMC.volume_geometry.allocate()\n\n# Set up max iterations and step size\nmax_iter = 30\nstep = 10\n\n# Set up algorithm\ncgls2 = CGLS(x_init = x_init, operator = A3DMC, data = data, \n max_iteration = max_iter, update_objective_interval = 10)\n\nfor i in range(0, max_iter // step):\n cgls2.run(step)\n \n # get and visusualise the results\n cgls_out = cgls2.get_output()\n show(cgls_out, title='Iteration {},'.format((i+1) * step) + ' Objective Function: {}'.format(cgls2.loss[-1]) + '\\n',\\\n show_channels=20,cmap='inferno', figure_size=[15,6], font_size=[25,20], minmax=(0.0,0.4))\n```\n\nThese images highlight the instability of a basic CGLS algorithm, where the **number of iterations** effectively acts as the algorithm's **regularisation parameter**. As a result, too many iterations leads to a divergence from an optimum solution, and while the objective function continues to reduce, only additional noise is being contributed. \n\nIn the next section, we will look at a different algorithm: Primal-Dual Hybrid Gradient (PDHG), combined with a Total Variation (TV) regularising factor, to see if this improves our reconstructed data quality.\n\n## Total Variation Regularisation in 4D Volume using PDHG\n\nThe PDHG algorithm and Total Variation regularisation were covered extensively in **Notebook_04_PDHG_Tomography**, but below gives a brief recap of the basics behind each.\n\n### Recap on PDHG\n\nPDHG aims to solve problems of the form:\n\n$$ \\begin{equation} \\min_{u} \\mathcal{F}(K u) + \\mathcal{G}(u) \\label{min_problem} \\end{equation} $$\n\nIn order to setup and run PDHG, we need to define the following:\n\n - The operator $K$.\n - The function $\\mathcal{F}$ and $\\mathcal{G}$.\n - Step-sizes $\\sigma$ and $\\tau$ such that $\\sigma\\tau\\|K\\|^{2}<1$.\n \nThen we can setup PDHG:\n\n`pdhg = PDHG(f = F, g = G, operator = K, tau = tau, sigma = sigma, max_iteration = maxiter)`\n\n\n\n\n### Applying Total Variation Regularisation\n\n\n\nFor this notebook, we will be applying Total Variation (TV) regularisation to our PDHG algorithm. TV is a non-smooth regulariser, \n\n$$ \\underset{u}{\\operatorname{argmin}} \\alpha\\,\\mathrm{TV}(u) + \\frac{1}{2} \\| \\mathcal{A} u - g\\|^{2}_{2} + \\mathbb{I}_{\\{u>0\\}}(u) $$\n\nwhere,\n\n- The Total Variation is taken as a `MixedL21Norm()`, such that $$\\mathrm{TV}(u) = \\|\\nabla u \\|_{2,1} = \\sum \\sqrt{ (\\partial_{y}u)^{2} + (\\partial_{x}u)^{2} }$$ \n- $g$ is the Acquisition data obtained from the detector \n- $\\mathcal{A}$ is the projection operator ( Radon transform ) that maps from an image-space to an acquisition space, i.e., \n$\\mathcal{A} : X \\rightarrow Y$. \n- $\\alpha$ is the regularising parameter that measures a trade-off between the fidelity and the regulariser terms \n- $\\mathbb{I}_{\\{u>0\\}}(u) : = \n\\begin{cases}\n0, \\mbox u>0\\\\\n\\infty, \\mbox otherwise\n\\quad\n\\end{cases}\n$ is a positivity constraint for the minimiser, $u$.\n\n## Setting up PDHG for TV\n\nWe define K as a `BlockOperator`, containing the Gradient and Projection operator:\n\n$$ K = \n\\begin{bmatrix}\n\\nabla\\\\\nA\n\\end{bmatrix}\n$$\nK = `BlockOperator(Gradient, A)`\n\nThe function $\\mathcal{F}$, is a `BlockFunction` with\n\na function $\\alpha\\|\\cdot\\|_{2,1}\\quad\\Longleftrightarrow\\quad$ `MixedL21Norm()` term that represents the Total variation regularisation ,\n\na function $\\|\\cdot -g \\|_{2}^{2}\\quad\\Longleftrightarrow\\quad$ `L2NormSquared(data)` term that represents the data fitting.\n\nHence, $\\mathcal{F} = [f_{1}, f_{2}] \\quad\\Longleftrightarrow\\quad $ `F = BlockFunction(MixedL21Norm(), L2NormSquared(data))`\n\nFinally, we have the function $\\mathcal{G} = \\mathbb{I}_{\\{u>0\\}}(u) \\quad\\Longleftrightarrow\\quad$ `G = IndicatorBox(lower=0)`\n\nAgain, we can verify that with the above setting we can express our problem into this form, for $x=u$\n\n$$ \\begin{align} \\underset{u}{\\operatorname{argmin}}\\alpha\\|\\nabla u\\|_{2,1} + \\frac{1}{2}\\|\\mathcal{A} u - g\\|^{2}_{2} + \\mathbb{I}_{\\{u>0\\}}(u) \\\\= \\underset{u}{\\operatorname{argmin}} f_{1}(\\nabla u) + f_{2}(\\mathcal{A}u) + \\mathbb{I}_{\\{u>0\\}}(u) \\\\ = \\underset{u}{\\operatorname{argmin}} \\mathcal{F}( \\begin{bmatrix} \\nabla \\\\ \\mathcal{A} \\end{bmatrix}u) + \\mathbb{I}_{\\{u>0\\}}(u)\\\\ = \\underset{u}{\\operatorname{argmin}} \\mathcal{F}(Ku) + \\mathcal{G}(u)\\\\ = \\underset{x}{\\operatorname{argmin}} \\mathcal{F}(Kx) + \\mathcal{G}(x) \\end{align} $$\n\n\n```python\n# The operator K is a Block Operator that contains the gradient and the tomography operator \nop1 = Gradient(ig)\nop2 = A3DMC\n\n# Set up a BlockOperator K\nK = BlockOperator(op1, op2, shape=(2,1)) \n\n```\n\n$\\mathcal{F}$ is a `BlockFunction` of a fidelity term and our TV regularising term:\n\n- $f_{1}$ = $\\alpha\\|\\cdot\\|_{2,1}\\quad\\Longleftrightarrow\\quad$ `MixedL21Norm()` term that represents the Total variation regularisation ,\n\n- $f_{2}$ = $\\|\\cdot -g \\|_{2}^{2}\\quad\\Longleftrightarrow\\quad$ `L2NormSquared(data)` term that represents the data fitting.\n\nTherefore as $f_{1}$ and $f_{2}$ act on each element of $K$, we end up with \n$$ \\mathcal{F}(Ku) = \\mathcal{F}(\n\\begin{bmatrix}\n\\nabla u \\\\\nA u\n\\end{bmatrix}) = ( f_{1}(\\nabla u), f_{2}(Au) ) $$\n\n\n```python\nalpha = 0.05\nf1 = alpha * MixedL21Norm() \nf2 = 0.5 * L2NormSquared(b = data) \n\nF = BlockFunction(f1, f2) \n```\n\nNext, we have our positivity constraint function $\\mathcal{G} = \\mathbb{I}_{\\{u>0\\}}(u) \\quad\\Longleftrightarrow\\quad$ `G = IndicatorBox(lower=0)`.\n\n\n```python\nG = IndicatorBox(lower = 0) \n```\n\nFinally, we compute the operator norm of $K$ (`normK = operator.norm()`), and set our step sizes, $\\sigma$ and $\\tau$.\n\n\n```python\n# Compute the operator norm for K\nnormK = K.norm()\n\n# Define the step sizes sigma and tau\nsigma = 1\ntau = 1/(sigma*normK**2)\n```\n\nNow we can setup and run the PDHG algorithm. Here it will run for 100 iterations.\n\nDue to the increased complexity of the algorithm, combined with the reconstruction of a 4D dataset, the runtime for 100 iterations alone will be around **4 minutes**. However, we may require many more than 100 iterations to reach an optimum solution. Further, we may wish to reconstruct hundreds of energy channels, rather than just the 40 we use here.\n\nTherefore, consideration must be taken on the number of iterations you perform based on your choice of algorithm and size of your dataset.\n\n**Scroll past the cell below** if you wish to save yourself 4 minutes! \nWe have provided a final reconstruction after 1000 iterations of PDHG, which required around 40 minutes of runtime.\n\n\n```python\npdhg_S_100 = PDHG(f = F, g = G, operator = K, tau = tau, sigma = sigma,\n max_iteration = 100, update_objective_interval = 25)\npdhg_S_100.run()\n\nshow(pdhg_S_100.get_output(), title='PDHG TV 100 Iterations', \n show_channels=20, cmap='inferno', figure_size = [15,6], font_size=[25,20], minmax=(0.0,0.4)) \n```\n\n### 1000 Iterations\n\nWe load in the 1000 iteration result using the `NEXUSDataReader()` we saw in **Notebook 03_3D_Diamond_dataset**.\n\n\n```python\n# Load in 1000 iteration dataset\nreader = NEXUSDataReader()\nreader.set_up(nexus_file = 'pdhg_S_1000.nxs')\npdhg_S_1000 = reader.load_data()\n\n# Show the result\nshow(pdhg_S_1000, title='PDHG TV 1000 Iterations', \n show_channels=20, cmap='inferno', figure_size = [15,6], font_size=[25,20], minmax=(0.0,0.4)) \n```\n\n## Exercise 1: Change correlation to Space-Channel\n\nWe can make some adjustments to our framework to see how these affect the reconstruction.\nOur gradient component is capable of smoothing our reconstruction both in the spatial domain, but also in the energy domain, by comparing reconstructions to its neighbouring channel.\n\nTry changing the gradient component of our operator, $\\nabla$, so that it is correlated in both the spatial and energy channel domain. We can do this by simply setting: \n \n `op1 = Gradient(ig, correlation='SpaceChannels')` \n\nKeep the tomography operator, $A$ the same. \n\nThere are three numerical parameters that have an impact on the resulting reconstruction: alpha ($\\alpha$), sigma ($\\sigma$) and tau ($\\tau$). For our original PDHG reconstruction, these optimum parameters were: \n$\\alpha$ = 0.05, \n$\\sigma$ = 1, \n$\\tau$ = `1/(sigma*normK**2)`.\n\nFor this test, we'll adjust our parameters slightly to: \n$\\alpha$ = 0.07, \n$\\sigma$ = 1, \n$\\tau$ = `1/(sigma*normK**2)`. \n\nSee if you notice any difference when you visualise the new reconstruction.\n\n\n```python\n# Setup BlockOperator, K\nop1 = Gradient(ig, correlation = \"SpaceChannels\")\nop2 = A3DMC\n\nK = BlockOperator(op1, op2, shape=(2,1) ) \n\n# Define regularising parameter\nalpha = 0.07\n\n# Setup BlockFunction, F\nf1 = alpha * MixedL21Norm()\nf2 = 0.5 * L2NormSquared(b = data) \nF = BlockFunction(f1, f2)\n\n# Positivity Constraint Function\nG = IndicatorBox(lower = 0)\n\n# Compute the operator norm for K\nnormK = K.norm()\n\n# Define the step sizes sigma and tau\nsigma = 1\ntau = 1/(sigma*normK**2)\n```\n\nOnce you've made these changes, have a go at running the reconstruction and see what influence your changes\nhad.\n\n\n```python\npdhgSC = PDHG(f = F, g = G, operator = K, tau = tau, sigma = sigma,\n max_iteration = 100, update_objective_interval = 25)\npdhgSC.run()\n```\n\n\n```python\nshow(pdhgSC.get_output(), title='PDHG TV Reconstruction', show_channels=20, \n cmap='inferno', figure_size = [15,6], font_size=[25,20], minmax=(0.0,0.4)) \n```\n\nWe can directly compare the two types of PDHG reconstruction using `islicer`. As we move across energy channels, see if there are any major differences between the constructions: \n 1) PDHG TV using a Spatial correlation for 1000 iterations \n 2) PDHG TV using a Space-Channel correlation for 100 iterations\n \n**Note:** If you also ran the earlier code for 100 iterations of PDHG TV using a Spatial correlation, feel free to add in an extra line to compare this too: \n`i3 = islicer(pdhg_S_100.get_output().subset(vertical=40),direction='channel',title='100 Iteration Space Corr.: Channel', cmap='inferno', minmax = (0.0,0.4))`\n\n\n\n```python\ni1 = islicer(pdhg_S_1000.subset(vertical=46),direction='channel',\n title='1000 Iteration Space Corr.: Channel', cmap='inferno', minmax = (0.0,0.4))\ni2 = islicer(pdhgSC.get_output().subset(vertical=46),direction='channel',\n title='100 Iteration Space-Channel Corr.: Channel',cmap='inferno', minmax = (0.0,0.4))\n#i3 = islicer(pdhg_S_100.get_output().subset(vertical=46),direction='channel',\n# title='100 Iteration Space Corr.: Channel', cmap='inferno', minmax = (0.0,0.4))\nlink_islicer(i1,i2)\n```\n\n# Spatial/Energy Profiles\n\nOne way we can directly observe the effect of applying Spatial/Energy Correlation to our algorithms is through the use of pixel profiles across our reconstructed images in each domain respectively.\n\nBy plotting signal values across spatial pixels, or across energy channels, the smoothing effect of this correlation feature is seen. We will demonstrate each of these below.\n\n## Extracting Spatial Profiles\n\nBy identifying a region of our reconstructed slices were the Iodine signals are strong, we can plot along the pixels in this region and compare the line profiles for each algorithm covered here.\n\nGiven the strong signals given by the eyes of the lizard head sample, we will look across this pixel row for a single channel, given by subset values of:\n - `vertical = 46`\n - `horizontal_y = 33`\n - `channel = 20`\n \n**Feel free to adjust these values in the cell below if you wish to explore different regions of the reconstructed images.**\n\n\n```python\nplt.figure(figsize=[12,8])\n\n# Extract line profiles for each algorithm\nplt.plot(recon.subset(vertical = 46, horizontal_y = 33, channel = 20).as_array())\nplt.plot(cgls.get_output().subset(vertical = 46, horizontal_y = 33, channel = 20).as_array())\nplt.plot(pdhg_S_1000.subset(vertical = 46, horizontal_y = 33, channel = 20).as_array())\nplt.plot(pdhgSC.get_output().subset(vertical = 46, horizontal_y = 33, channel = 20).as_array())\n\n# Label and add key\nplt.xlabel('Pixels X', fontsize=20)\nplt.ylabel('Optical Density (no units)', fontsize=20)\nplt.legend(labels=['FDK', 'Simple CGLS', 'Space Corr. PDHG TV', 'Space-Channel Corr. PDHG TV'])\n```\n\nWe can see the reduced noise fluctuations for the Space-correlated PDHG algorithms, compared to the significant noise seen in both FDK and CGLS reconstructions.\n\n## Extracting Energy Profiles\n\nOnce we have chosen and optimised our reconstruction routine, we can exploit this additional, energy dimension further by identifying the spectral signals that lie within.\n\nFirst we can plot out energy profiles for any region-of-interest in our reconstructed slices. This allows us to find, for instance, an absorption edge.\nIodine has a known absorption 'K-edge' of 33.169 keV, so we can test the accuracy of reconstruction algorithms by seeing where this edge falls in each case. A plot of an idealised, theoretical Iodine K-edge is shown below, falling precisely at 33.169 keV.\n\nWe start by extracting a 3x3 set of pixels from each of the 3D multi-channel datasets. We can choose the pixels with the highest signal by looking at our previous reconstructed images.\n\n\n```python\n# Collect 3x3 regions of interest for each dataset\n\n# FDK Recon\n# Sum along z-direction (vertical)\nfdk_ROI1 = np.mean(fdk.get_output().as_array()[:,44:47,:,:],1)\n# Sum along y-direction (horizontal_y)\nfdk_ROI2 = np.mean(fdk_ROI1[:,31:34,:],1)\n# Sum along x-direction (horizontal_x)\nfdk_ROI3 = np.mean(fdk_ROI2[:,14:17],1)\n\n# 3D multi-channel 10 iteration CGLS\ncgls_ROI1 = np.mean(cgls.get_output().as_array()[:,44:47,:,:],1)\ncgls_ROI2 = np.mean(cgls_ROI1[:,31:34,:],1)\ncgls_ROI3 = np.mean(cgls_ROI2[:,14:17],1)\n\n# 3D multi-channel space correlated PDHG\npdhg_S_1000_ROI1 = np.mean(pdhg_S_1000.as_array()[:,44:47,:],1)\npdhg_S_1000_ROI2 = np.mean(pdhg_S_1000_ROI1[:,31:34],1)\npdhg_S_1000_ROI3 = np.mean(pdhg_S_1000_ROI2[:,14:17],1)\n\n# 3D multi-channel space-channel correlated PDHG TV\npdhgSC_ROI1 = np.mean(pdhgSC.get_output().as_array()[:,44:47,:],1)\npdhgSC_ROI2 = np.mean(pdhgSC_ROI1[:,31:34],1)\npdhgSC_ROI3 = np.mean(pdhgSC_ROI2[:,14:17],1)\n```\n\n\n```python\n# Set channel numbers for reduced dataset (here 100 to 140)\nchannel_no = np.linspace(100,140,num_channels)\n# Apply linear calibration equation to convert channels to energy\ne_keV = 0.2786*channel_no + 0.8575\n\nplt.figure(figsize=(10,8))\n\nplt.plot(e_keV,fdk_ROI3)\nplt.plot(e_keV,cgls_ROI3) \nplt.plot(e_keV,pdhg_S_1000_ROI3)\nplt.plot(e_keV,pdhgSC_ROI3)\n\nplt.plot((33.169,33.169),plt.ylim(0.0,0.4))\n\nplt.legend(labels=['FDK', 'Simple CGLS', 'Space Corr. 1000 Iter. PDHG', \n 'Space-Channel Corr. 100 Iter. PDHG', 'I K-edge 33.169 keV'])\nplt.ylim(0.1,0.3)\nplt.xlim([29,39])\n\nplt.ylabel('Optical Density (no units)',fontsize=20)\nplt.xlabel('Energy (keV)',fontsize=20)\n```\n\nFrom our plots, we can see all algorithms experience a sharp rise in signal value due to the iodine absorption K-edge. Compared to the theoretical value line, each of the cases match well with the expected position of the edge.\nAn important factor to note is the increased smoothness of the signal for PDHG TV, when a 'Space-Channel' correlation is applied to our gradient operator for these methods. This correlation enforces our operator to use the previous energy channel as a reference point for reconstructing data in the next, resulting in a smoother transition across channels.\n\n## Exercise 2: Move to 2D Spatial Geometry with Channels\n### Defining 2D `Acquisition/Image Geometry` and `Projector`\n\nFrom our 4D dataset, we can reduce the dimensionality by taking a single slice along a particular spatial dimension. \nWe can go from a shape of \n\n (Channel, Vertical, Angle, Horizontal) = (40, 80, 60, 80) \nto a shape of \n\n (Channel, Angle, Horizontal) = (40, 60, 80) \nby taking a vertical slice subset.\n\nLet's start by reducing our original `AcquisitionData`. For simplicity, choose a central slice subset by setting: \n\n`data2DMC = data.subset(vertical=40)` \n`ag2d = data2DMC.geometry` \n\nThis will set our new `AcquisitionGeometry`.\n\nWe must then alter the dimensionality of `ag2d`. Currently this is stored as `'3D'`, but as we now use only 2 spatial dimensions, we must change it to `'2D'`. So we can do: \n \n`ag2d.dimension = '2D'`\n\n\n```python\ndata2DMC = data.subset(vertical=40)\nag2d = data2DMC.geometry\nag2d.dimension = '2D'\n```\n\nWe still need to manually adjust the `ImageGeometry` parameters we used for the [4D dataset](#4D_Acquisition_Image_Geometry). \nBelow we have the parameters used for the original dataset. \nWhich parts of this geometry do we no longer need? \n**Delete** what's not needed and then run the code.\n\n\n```python\nig2d = ImageGeometry(voxel_num_x=ig.voxel_num_x,\n voxel_num_y=ig.voxel_num_y,\n voxel_size_x=ag.pixel_size_h/mag,\n voxel_size_y=ag.pixel_size_h/mag,\n channels = ig.channels)\n```\n\nNow we can setup the tomography operator for 2D multi-channel data using the redefined `AcquisitionGeometry` and `ImageGeometry` for 2D. Let's call it `A2DMC`.\n\nRemember: Last time we used [AstraProjector3DMC](#A3DMC) as we had a 3D multi-channel dataset. Here we instead use `AstraProjectorMC()`.\n\nNote: While `AstraProjector3DMC` required the use of `gpu`, we can setup our 2D multi-channel tomography operator using either `gpu` or `cpu`. The default is the use of `gpu`.\n\n\n```python\nA2DMC = AstraProjectorMC(ig2d, ag2d ,'cpu')\n```\n\n\n```python\ndata2DMC.dimension_labels\n\n```\n\n## Exercise 3: Running 2D plus channel CGLS\n\nIn this exercise, use what you've learned about setting up reconstruction routines, and perform a simple CGLS using our 2D multi-channel data. Then run it for 10 iterations.\n\nRemember, the key parameters you need are:\n - initialisation x_init (`A2DMC.volume_geometry.allocate()`)\n - operator (`A2DMC`)\n - data (`data2DMC`)\n \n \n\n\n```python\nx_init = A2DMC.volume_geometry.allocate()\n\ncgls2d = CGLS(x_init = x_init, operator = A2DMC,\n data = data2DMC, max_iteration = 100)\ncgls2d.run(10)\n\n```\n\nYou can check the results of your reconstruction by running the function `show` below, giving you axial views for two energy channels. Or run an interactive slicer across energy channels.\n\n\n```python\n# Show reconstruction \nshow(cgls2d.get_output(), title='2D Multi-Channel CGLS', show_channels=[10,20,30], \n cmap='inferno', figure_size = [15,6], font_size=[25,20],minmax=(0.0,0.4))\n```\n\n\n```python\nislicer(cgls2d.get_output(),direction='channel',title='Channel',cmap='inferno',minmax=(0.0,0.4))\n```\n\n## Exercise 4: Running 2D plus channel regularised CGLS\n\nHere we will expand upon what you learned about BlockFrameworks. In particular, we will cover both `BlockOperators` and `BlockDataContainers`, which were covered in **Notebook 02_Tikhonov_Block_Framework**.\n\nBelow gives a brief definition of each type:\n\n`BlockDataContainer` holds datacontainers as a column vector.\n\n`BlockOperator` is a matrix of operators.\n\n### Setting up Regularised CGLS\n\nFor our regularisation, we wish to solve:\n\n$$\\underset{u}{\\mathrm{argmin}}\\begin{Vmatrix}\\binom{\\alpha \\nabla}{A} u - \\binom{0}{g}\\end{Vmatrix}^2_2$$\nWith the definitions:\n\n$\\tilde{A} = \\binom{\\alpha \\nabla}{A} \\quad\\Longleftrightarrow\\quad$ `BlockOperator(alpha*Gradient(ig2d),A2DMC)` \n(where $\\alpha$ is the regularisation parameter)\n\n$\\tilde{g} = \\binom{0}{g} \\quad\\Longleftrightarrow\\quad$ `BlockDataContainer(op1.range_geometry().allocate(),data2DMC)`\n\nthis can now be recognised as a least squares problem:\n\n$$\\underset{u}{\\mathrm{argmin}}\\begin{Vmatrix}\\tilde{A} u - \\tilde{g}\\end{Vmatrix}^2_2$$\nand being a least squares problem, it can be solved using CGLS with $\\tilde{A}$ as operator and $\\tilde{g}$ as data.\n\n### Build `BlockOperator`, $\\tilde{A} = \\binom{\\alpha \\nabla}{A}$\n\nUsing the 2D multi-channel data and geometry constructed above, build a `BlockOperator`, $\\tilde{A}$, applying a Space-Channel correlation. \nChoose a regularisation value around $\\alpha$ = 1.5 as a starting point.\n\n\n\n```python\n# Setup the BlockOperator\nop1 = Gradient(ig2d, correlation='SpaceChannels')\nop2 = A2DMC\nalpha = 1.5\nA_block = BlockOperator(alpha*op1,op2)\n```\n\n### Build `BlockDataContainer`\n\nNext build a `BlockDataContainer`, $\\tilde{g}$, containing an array with the range of the regularising operator, $\\alpha \\nabla$, and our reduced `AcquisitionData` (`data2DMC`).\n\n\n```python\n# Setup the BlockDataContainer\ng_block = BlockDataContainer(op1.range_geometry().allocate(),data2DMC)\n```\n\n### Initialise and Run\n\nNow we can initialise the `BlockOperator` and run the algorithm for 50 iterations.\n\n\n```python\n# Initialise the BlockOperator\n\nx_init = A_block.domain_geometry().allocate()\n\n# Setup and run the regularised CGLS algorithm\n\ncgls_reg = CGLS(x_init = x_init, operator = A_block, data = g_block,\n max_iteration = 100,update_objective_interval = 10)\ncgls_reg.run(50)\n```\n\nCheck the reconstruction below. Does the value of $\\alpha$ need changing? Click [here](#Choosing_alpha) if you want to go back and vary your initial value of $\\alpha$ to see if you can improve the reconstruction.\n\n\n```python\n# Show reconstruction \nshow(cgls_reg.get_output(), show_channels=[10,20,30], title='2D Multi-Channel Regularised CGLS', \n cmap='inferno', figure_size = [15,6], font_size=[25,20], minmax=(0.0,0.4))\n```\n\nWhile still noisy, we can see a significant improvement in contrast to the CGLS algorithm when no regularisation is applied. \nNext we will try the PDHG algorithm using TV regularisation to see if this fares any better.\n\n## Exercise 5: 2D Multi-channel Total Variation with PDHG\n\nNow let's test our reduced dataset with the PDHG algorithm using Total Variation regularisation.\n\nBased on what our construction looked like for the 4D version, try to recreate this for our 2D multi-channel data. You'll need to adjust the `BlockOperator`, $K$ and `BlockFunction`, $\\mathcal{F}$, accordingly.\n\nPlay around with the numeric parameters ($\\alpha$, $\\sigma$ and $\\tau$) as well, and see how they affect the reconstruction here. Is there a big change when moving from 3D multi-channel to 2D multi-channel?\n\n\n```python\n# Setup the BlockOperator, K\nop1 = Gradient(ig2d, correlation='SpaceChannels')\nop2 = A2DMC\nK = BlockOperator(op1,op2)\n\n# Define the regularising parameter, alpha\nalpha = 0.03\n\n# Setup the BlockFunction, F\nf1 = alpha * MixedL21Norm()\nf2 = 0.5 * L2NormSquared(b=data2DMC)\nF = BlockFunction(f1,f2)\n\n# Define indicator function that provides a positivity constraint\nG = IndicatorBox(lower=0)\n\n# Compute the operator norm for K\nnormK = K.norm()\n\n# Define the step sizes sigma and tau\nsigma = 1\ntau = 1/(sigma*normK**2) \n```\n\n\n```python\npdhg_2d = PDHG(f=F, g=G, operator=K, tau=tau, sigma=sigma, \n max_iteration = 1000, update_objective_interval = 25)\npdhg_2d.run(500)\n```\n\n\n```python\n# Show reconstruction \nshow(pdhg_2d.get_output(), show_channels=[10,20,30], title='2D Multi-Channel TV PDHG',\n cmap='inferno', figure_size = [15,6], font_size=[25,20], minmax=(0.0,0.4))\n```\n\n### Comparing 2D Multi-channel Methods\n\nWe can bring together the 2D multi-channel reconstructions we have performed for:\n - Simple CGLS\n - Regularised CGLS\n - PDHG with TV \n \nUsing the `islicer` tool, we can directly compare these reconstructions side-by-side and evaluate the level of noise removal and spatial/energy smoothing offered by each method.\n\n\n```python\ni1 = islicer(cgls2d.get_output(),direction='channel',title='Simple CGLS: Channel',\n cmap='inferno',minmax=(0.0,0.4))\ni2 = islicer(cgls_reg.get_output(),direction='channel',title='Regularised CGLS: Channel',\n cmap='inferno',minmax=(0.0,0.4))\ni3 = islicer(pdhg_2d.get_output(),direction='channel',title='PDHG TV: Channel',\n cmap='inferno',minmax = (0.0,0.4))\nlink_islicer(i1,i2,i3)\n```\n\n## Energy profiles\n\nOnce more we can extract energy profiles, to determine the K-edge position in our 2D multi-channel datasets.\nIn this case, we extract a 3x3 set of pixels by only summing along two spatial domains, y (`horizontal_y`) and x (`horizontal_x`).\n\n\n```python\n# Collect 3x3 regions of interest for each dataset\n\n# 2d multi-channel simple CGLS\n# Sum along y-direction\ncgls2d_ROI1 = np.mean(cgls2d.get_output().as_array()[:,33:36,:],1)\n# Sum along x-direction\ncgls2d_ROI2 = np.mean(cgls2d_ROI1[:,60:63],1)\n\n# 2d multi-channel regularised CGLS\ncgls_reg_ROI1 = np.mean(cgls_reg.get_output().as_array()[:,33:36,:],1)\ncgls_reg_ROI2 = np.mean(cgls_reg_ROI1[:,60:63],1)\n\n# 2d multi-channel PDHG TV\npdhg_2d_ROI1 = np.mean(pdhg_2d.get_output().as_array()[:,33:36,:],1)\npdhg_2d_ROI2 = np.mean(pdhg_2d_ROI1[:,60:63],1)\n\n```\n\n\n```python\n# Set channel numbers for reduced dataset\nchannel_no = np.linspace(100,140,num_channels)\n# Apply linear calibration equation to convert channels to energy\ne_keV = 0.2786*channel_no + 0.8575\n\nplt.figure(figsize=(10,8))\n\nplt.plot(e_keV,cgls2d_ROI2) \nplt.plot(e_keV,cgls_reg_ROI2)\nplt.plot(e_keV,pdhg_2d_ROI2)\n\nplt.plot((33.169,33.169),plt.ylim(0.0,0.4))\n\nplt.legend(labels=['Simple CGLS', 'Regularised (Space-Channel) CGLS',\n '(Space-Channel) PDHG TV', 'I K-edge 33.169 keV'])\nplt.ylim(0.1,0.4)\nplt.xlim([29,39])\n\nplt.ylabel('Optical Density (no units)',fontsize=20)\nplt.xlabel('Energy (keV)',fontsize=20)\n```\n\n## Constructing Elemental Maps\n\nOnce we know have identified the position of this edge in our energy profile, we can narrow our dataset and produce an 'Iodine map'. That is, we can select only the energy channels occupied by the absorption edge, so all reconstructed signal is now due only to the iodine contrast agent. This method is known as **'K-edge subtraction'**, which you can read about in more detail in papers such as that by [C.K.Egan *et al*, 2015](https://www.nature.com/articles/srep15979#). \nA basic concept is shown below for an energy profile plot. The hashed area highlights the energy range we are interested in, corresponding to the absorption edge.\n\n\n\nBased on our plots, we will estimate the start and end of the edge to occur at approximately 32.5 keV and 34 keV respectively.\n\n\n```python\n# Calculate energy channels corresponding to start and end of the K-edge\ne_keV = np.array([32.5,34])\nchannel_no = ((e_keV-0.8575)/0.2786)-100\n\n# Display the channels corresponding to the start and end of the K-edge\nprint(\"Start of edge = channel\",int(channel_no[0]))\nprint(\"End of edge = channel\",int(channel_no[1]))\n\n```\n\n\n```python\n# 2d multi-channel simple CGLS\n# Sum over all pixels for channels of interest\ncgls2d_COI = np.mean(cgls2d.get_output().as_array()[int(channel_no[0]):int(channel_no[1]),:,:],0)\n\n# 2d multi-channel regularised CGLS\ncgls_reg_COI = np.mean(cgls_reg.get_output().as_array()[int(channel_no[0]):int(channel_no[1]),:,:],0)\n\n# 2d multi-channel PDHG TV\npdhg_2d_COI = np.mean(pdhg_2d.get_output().as_array()[int(channel_no[0]):int(channel_no[1]),:,:],0)\n\n```\n\nBy stacking our iodine maps as an array, we can use `islicer` to move between each, directly comparing the quality of feature definition and degree of background noise for each algorithm.\n\n\n```python\n# Collect together iodine maps\nstack_maps = (np.array([cgls2d_COI, cgls_reg_COI, pdhg_2d_COI]))\ntitles = ['Simple CGLS', 'Regularised CGLS', 'Space-Channel PDHG TV']\nislicer(stack_maps, title=titles, direction=0, cmap='inferno',minmax = (0.0,0.4))\n```\n\n

                                        Conclusions

                                        \n\nThis notebook focused on bringing together the core elements of the CIL framework you have learned in the previous courses, and seeing how these may be applied to a multi-channel dataset. \n\nWe looked at the key differences in building Image/Acquisition Geometries and Operators for multi-channel datasets, and how these vary for both 2D and 3D multi-channel.\n\nWe have covered three key algorithms (FDK, CGLS and PDHG) and shown the differences between these for both 2D and 3D multi-channel. This also allowed us to explore some of the ways in which we can vary these reconstruction routines, and how making these adjustments can have significant effects on the quality of your reconstructions.\n\nFinally we have shown the ability to extract spectral properties from the reconstruction, providing an insight to the elemental composition of the sample of interest using energy profiles and elemental maps.\n\n**Click the link below to see the potential for full volume, white-beam reconstructions using both CGLS (Left) and PDHG TV (Right)** \n[4D Lizard Full Volume Reconstruction](images/Lizard_gif.gif \"segment\")\n", "meta": {"hexsha": "573c3982b64e0f9f90c1cc08090d18b6c88f2524", "size": 58678, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "training/2019_SynergisticSymposium/05_Multi_Channel_Reconstruction_solutions.ipynb", "max_stars_repo_name": "Sophilyplum/CIL-Demos", "max_stars_repo_head_hexsha": "f1a2d4b7675540f5e56accb255a2ea66a1931079", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2021-05-21T11:34:58.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-05T14:06:17.000Z", "max_issues_repo_path": "training/2019_SynergisticSymposium/05_Multi_Channel_Reconstruction_solutions.ipynb", "max_issues_repo_name": "Sophilyplum/CIL-Demos", "max_issues_repo_head_hexsha": "f1a2d4b7675540f5e56accb255a2ea66a1931079", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 33, "max_issues_repo_issues_event_min_datetime": "2019-06-25T20:44:51.000Z", "max_issues_repo_issues_event_max_datetime": "2021-05-07T12:25:32.000Z", "max_forks_repo_path": "training/2019_SynergisticSymposium/05_Multi_Channel_Reconstruction_solutions.ipynb", "max_forks_repo_name": "Sophilyplum/CIL-Demos", "max_forks_repo_head_hexsha": "f1a2d4b7675540f5e56accb255a2ea66a1931079", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2019-12-09T16:55:20.000Z", "max_forks_repo_forks_event_max_datetime": "2020-06-30T10:31:17.000Z", "avg_line_length": 38.5785667324, "max_line_length": 600, "alphanum_fraction": 0.6092061761, "converted": true, "num_tokens": 11347, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.6619228758499942, "lm_q1q2_score": 0.41202007060879564}} {"text": "

                                        BANK CUSTOMER CHURN PREDICTION


                                        \nCustomer churn can be defined as the customer terminates any relationship with a company that provides services either online or offline. Churn prediction can be referred to as the prediction of customers who are likely to cancel a subscription, product or service.
                                        \n
                                        \n

                                        :::::Importing following libraries:::::

                                        \n
                                          \n
                                        1. Pandas:: For the data manipulation and analysis.
                                          2. \n
                                        2. Matplotlib:: For the data visualization.
                                          3. \n
                                        3. Keras:: For building the neural network built on the Tensorflow backend.
                                          4. \n
                                        4. Warning:: For dealing with the warnings coming while execution of the lines of code.
                                          5. \n
                                        \n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\nimport warnings\nwarnings.filterwarnings('ignore')\nfrom sklearn.preprocessing import LabelEncoder, OneHotEncoder\nimport keras\nfrom keras.models import Sequential\nfrom keras.layers import Dense\nfrom keras.layers import Dropout\n```\n\n Using TensorFlow backend.\n\n\n

                                        ::::: Importing dataset:::::


                                        \nThe data is imported using the pandas (alias name pd) pre-defined function read_csv() as our data file format is csv (comma-seprated values) in the dataset variable.\n\n\n```python\ndataset = pd.read_csv('Churn_Modelling.csv')\n```\n\n

                                        :::::The snapshot of the imported data:::::


                                        \nAs the dataset variable of DATA_FRAME type so we can use the head() function to show the top 5 rows/tuples of the whole dataset.
                                        \n

                                        :::::The featuers and Class_Label of the data:::::


                                        \n
                                          \n
                                        1. RowNumber: The index number of the row.
                                          2. \n
                                        2. CusomerId: The customer ID.
                                          3. \n
                                        3. Surname: The last name of the customer.
                                          4. \n
                                        4. CreditScore: The credit score given by the bank.
                                          5. \n
                                        5. Geography: Country that customer belongs.\\begin{equation}\nGeography \\: \\: \\epsilon \\: \\: R^{\\{France,Germany,Spain\\}}\n\\end{equation}
                                          6. \n
                                        6. Gender: The gender of the customer. \\begin{equation}\nGender \\: \\: \\epsilon \\: \\: R^{\\{Male\\:,\\:Female\\}}\n\\end{equation}
                                          7. \n
                                        7. Age:The age of the customer.
                                          8. \n
                                        8. Tenure:Number of years customer is with the bank.
                                          9. \n
                                        9. Balance:The current balance of the account.
                                          10. \n
                                        10. NumOfProducts:The number of the products taken by the customer.\\begin{equation}\nNumOfProducts \\: \\: \\epsilon \\: \\: R^{\\{1\\:,\\:2\\:,\\:3\\:,\\:4\\:\\}}\n\\end{equation}
                                          11. \n
                                        11. HasCrCard: Is customer owing a credit card or not. \\begin{equation}\nHasCrCard \\: \\: \\epsilon \\: \\: R^{\\{\\:0\\: = \\:No\\:,\\: 1\\: =\\: Yes\\:\\}}\n\\end{equation}
                                          12. \n
                                        12. IsActiveMember:Is customer is active or not.\\begin{equation}\nIsActiveMember \\: \\: \\epsilon \\: \\: R^{\\{\\:0\\: = \\:No\\:,\\: 1\\: =\\: Yes\\:\\}}\n\\end{equation}
                                          13. \n
                                        13. EstimatedSalary:The annual salary of the customers.
                                          14. \n
                                        14. Exited:The CLASS LABEL whether customer still with bank or not.\\begin{equation}\nExited \\: \\: \\epsilon \\: \\: R^{\\{\\:0\\: = \\:No\\:,\\: 1\\: =\\: Yes\\:\\}}\n\\end{equation}
                                          15. \n
                                        \n\n\n```python\ndataset.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        RowNumberCustomerIdSurnameCreditScoreGeographyGenderAgeTenureBalanceNumOfProductsHasCrCardIsActiveMemberEstimatedSalaryExited
                                        0115634602Hargrave619FranceFemale4220.00111101348.881
                                        1215647311Hill608SpainFemale41183807.86101112542.580
                                        2315619304Onio502FranceFemale428159660.80310113931.571
                                        3415701354Boni699FranceFemale3910.0020093826.630
                                        4515737888Mitchell850SpainFemale432125510.8211179084.100
                                        \n
                                        \n\n\n\n

                                        :::::Dropping the unsignificant featuers:::::


                                        \nThe function dataset_name.drop([\"LIST_OF_FEATUERS\"], axis = 0/1) will drop the columns (when axis=1) and rows (when axis=0).

                                        \nThe data divison into the dataset featuers and class label.\n\n\n```python\nX = dataset.iloc[:, 3:13].values\n```\n\n\n```python\ny = dataset.iloc[:, 13].values\n```\n\n\n```python\nprint(X[1])\nprint(y)\n```\n\n [608 'Spain' 'Female' 41 1 83807.86 1 0 1 112542.58]\n [1 0 1 ... 1 1 0]\n\n\n

                                        :::::The shape of the feature dataset:::::

                                        \n\n\n```python\nX.shape\n```\n\n\n\n\n (10000, 10)\n\n\n\n

                                        :::::The shape of the class label dataset:::::

                                        \n\n\n```python\ny.shape\n```\n\n\n\n\n (10000,)\n\n\n\n

                                        :::::The conversion of the categorical features into numerical using the One-Hot coding Technique.:::::


                                        \nThe conversion of the categorical featuers into numerical featuers using the one-hot encoding technique in which each unique value in the feature will be converted into a seperate column.\n\n\n```python\nlabelencoder_X_1 = LabelEncoder()\n```\n\n\n```python\nX[:, 1] = labelencoder_X_1.fit_transform(X[:, 1])\n```\n\n\n```python\nlabelencoder_X_2 = LabelEncoder()\n```\n\n\n```python\nX[:, 2] = labelencoder_X_2.fit_transform(X[:, 2])\n```\n\n\n```python\nonehotencoder = OneHotEncoder(categorical_features = [1])\nX = onehotencoder.fit_transform(X).toarray()\nX = X[:, 1:]\n```\n\n C:\\Users\\ARPIT\\Anaconda3\\lib\\site-packages\\sklearn\\preprocessing\\_encoders.py:451: DeprecationWarning: The 'categorical_features' keyword is deprecated in version 0.20 and will be removed in 0.22. You can use the ColumnTransformer instead.\n \"use the ColumnTransformer instead.\", DeprecationWarning)\n\n\n\n```python\nX[1]\n```\n\n\n\n\n array([0.0000000e+00, 1.0000000e+00, 6.0800000e+02, 0.0000000e+00,\n 4.1000000e+01, 1.0000000e+00, 8.3807860e+04, 1.0000000e+00,\n 0.0000000e+00, 1.0000000e+00, 1.1254258e+05])\n\n\n\n:::::The data division into training and testing:::::
                                        \nThe data is divided into training and testing into 80-20 ratio.\n\n\n```python\nfrom sklearn.model_selection import train_test_split\nX_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.2, random_state = 0)\n```\n\n\n```python\nX_train[1:5]\n```\n\n\n\n\n array([[1.0000000e+00, 0.0000000e+00, 4.2700000e+02, 1.0000000e+00,\n 4.2000000e+01, 1.0000000e+00, 7.5681520e+04, 1.0000000e+00,\n 1.0000000e+00, 1.0000000e+00, 5.7098000e+04],\n [0.0000000e+00, 0.0000000e+00, 5.3500000e+02, 0.0000000e+00,\n 2.9000000e+01, 2.0000000e+00, 1.1236734e+05, 1.0000000e+00,\n 1.0000000e+00, 0.0000000e+00, 1.8563076e+05],\n [0.0000000e+00, 1.0000000e+00, 6.5400000e+02, 1.0000000e+00,\n 4.0000000e+01, 5.0000000e+00, 1.0568363e+05, 1.0000000e+00,\n 1.0000000e+00, 0.0000000e+00, 1.7361709e+05],\n [0.0000000e+00, 1.0000000e+00, 8.5000000e+02, 0.0000000e+00,\n 5.7000000e+01, 8.0000000e+00, 1.2677630e+05, 2.0000000e+00,\n 1.0000000e+00, 1.0000000e+00, 1.3229849e+05]])\n\n\n\n\n```python\ny_train[1:5]\n```\n\n\n\n\n array([0, 0, 0, 0], dtype=int64)\n\n\n\n\n```python\nfrom sklearn.preprocessing import StandardScaler\nsc = StandardScaler()\nX_train = sc.fit_transform(X_train)\nX_test = sc.transform(X_test)\n```\n\n

                                        ::::: The Keras Sequential Model:::::


                                        \nThe Sequential model is a linear stack of layers.\n\n\n```python\nclassifier = Sequential()\n```\n\n

                                        ::::: Initializing the weights and bias for the neural network model:::::


                                        \nWe have taken the initial vector values from normal/gaussian distribution (mean = 0 and standard deviation = 0.05) for the neural network weights and bias.\n\n\n```python\ninitializers = keras.initializers.RandomNormal(mean=0.0, stddev=1.0, seed=None)\n```\n\n

                                        :::::The Input Layer:::::

                                        \n
                                          \n
                                        1. The Neurons: 11
                                          2. \n
                                        2. The weight matrix initial value: Normal Distribution
                                          3. \n
                                        3. The Bias initial values: Normal Distribution
                                          4. \n
                                        4. Activation Function: Rectified Linear Unit (ReLU)
                                          5. \n
                                        \n\n\n```python\nclassifier.add(Dense(units = 11, kernel_initializer=initializers ,bias_initializer=initializers, activation = 'relu'))\n```\n\n\n```python\nclassifier.add(Dropout(0.5))\n```\n\n

                                        :::::The Hidden Layer (L1):::::

                                        \n
                                          \n
                                        1. Neurons: 8
                                          2. \n
                                        2. The weight matrix initial value: Normal Distribution
                                          3. \n
                                        3. The Bias initial values: Normal Distribution
                                          4. \n
                                        4. Activation Function: Rectified Linear Unit (ReLU)
                                          5. \n
                                        \n\n\n```python\nclassifier.add(Dense( units = 8, kernel_initializer=initializers ,bias_initializer=initializers, activation = 'relu'))\n```\n\n

                                        :::::The Hidden Layer (L2):::::

                                        \n
                                          \n
                                        1. Neurons: 6
                                          2. \n
                                        2. The weight matrix initial value: Normal Distribution
                                          3. \n
                                        3. The Bias initial values: Normal Distribution
                                          4. \n
                                        4. Activation Function: Rectified Linear Unit (ReLU)
                                          5. \n
                                        \n\n\n```python\nclassifier.add(Dense(units = 6, kernel_initializer=initializers ,bias_initializer=initializers, activation = 'relu'))\n```\n\n

                                        :::::The Hidden Layer (L3):::::

                                        \n
                                          \n
                                        1. Neurons: 4
                                          2. \n
                                        2. The weight matrix initial value: Normal Distribution
                                          3. \n
                                        3. The Bias initial values: Normal Distribution
                                          4. \n
                                        4. Activation Function: Rectified Linear Unit (ReLU)
                                          5. \n
                                        \n\n\n```python\nclassifier.add(Dense(units = 4, kernel_initializer=initializers ,bias_initializer=initializers, activation = 'relu'))\n```\n\n

                                        :::::The Output Layer:::::

                                        \n
                                          \n
                                        1. Neurons: 1
                                          2. \n
                                        2. The weight matrix initial value: Normal Distribution
                                          3. \n
                                        3. The Bias initial values: Normal Distribution
                                          4. \n
                                        4. Activation Function: Sigmoid
                                          5. \n
                                        \n\n\n```python\nclassifier.add(Dense(units = 1, kernel_initializer=initializers ,bias_initializer=initializers, activation = 'sigmoid'))\n```\n\n

                                        :::::The Weight's Optimizer:::::


                                        \n Adam is an optimization algorithm that can used instead of the classical stochastic gradient descent procedure to update network weights iterative based in training data.
                                        \n The method computes individual adaptive learning rates for different parameters from estimates of first and second moments of the gradients.
                                        \n
                                          \n
                                        1. Learning Rate (0.001): The proportion that weights are updated (e.g. 0.001). Larger values (e.g. 0.3) results in faster initial learning before the rate is updated. Smaller values (e.g. 1.0E-5) slow learning right down during training
                                          2. \n
                                        2. beta_1(0.9): The exponential decay rate for the first moment estimates (e.g. 0.9).
                                          3. \n
                                        3. beta_2(0.999): The exponential decay rate for the second-moment estimates (e.g. 0.999). This value should be set close to 1.0 on problems with a sparse gradient (e.g. NLP and computer vision problems).
                                          4. \n
                                        \n\n\n```python\nopti = keras.optimizers.Adam(lr = 0.0001, beta_1 = 0.9, beta_2 = 0.999, amsgrad=False)\n```\n\n

                                        :::::The Fiting Of The Neural Network:::::


                                        \nThis method of the keras library will fit the all the layers of the neural network.\n
                                          \n
                                        1. Optimizer: Adam Optimizer
                                          2. \n
                                        2. Loss Function: Binary Cross Entropy
                                          3. \n
                                        3. Metrics: Accuracy
                                          4. \n
                                        \n\n\n```python\nclassifier.compile(optimizer = opti, loss = 'binary_crossentropy', metrics = ['accuracy'])\n```\n\n

                                        :::::Training of the Neural Network

                                        \n
                                          \n
                                        1. The Batch Size: 128
                                          2. \n
                                        2. Number of Epochs:10000
                                          3. \n
                                        \n\n\n```python\nhistory = classifier.fit(X_train, y_train, validation_split=0.10, batch_size = 32, epochs = 10000)\n```\n\n Train on 7200 samples, validate on 800 samples\n Epoch 1/10000\n 7200/7200 [==============================] - 3s 393us/step - loss: 0.3738 - acc: 0.8458 - val_loss: 0.3689 - val_acc: 0.8200\n Epoch 2/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3704 - acc: 0.8457 - val_loss: 0.3696 - val_acc: 0.8213\n Epoch 3/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3732 - acc: 0.8431 - val_loss: 0.3709 - val_acc: 0.8187\n Epoch 4/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3651 - acc: 0.8451 - val_loss: 0.3702 - val_acc: 0.8187\n Epoch 5/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3729 - acc: 0.8433 - val_loss: 0.3708 - val_acc: 0.8187\n Epoch 6/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3743 - acc: 0.8431 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 7/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3741 - acc: 0.8488 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 8/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3724 - acc: 0.8457 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 9/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3674 - acc: 0.8475 - val_loss: 0.3699 - val_acc: 0.8187\n Epoch 10/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3711 - acc: 0.8467 - val_loss: 0.3700 - val_acc: 0.8187\n Epoch 11/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3707 - acc: 0.8446 - val_loss: 0.3706 - val_acc: 0.8187\n Epoch 12/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3688 - acc: 0.8465 - val_loss: 0.3694 - val_acc: 0.8200\n Epoch 13/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3692 - acc: 0.8478 - val_loss: 0.3707 - val_acc: 0.8200\n Epoch 14/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3690 - acc: 0.8483 - val_loss: 0.3707 - val_acc: 0.8200\n Epoch 15/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3728 - acc: 0.8482 - val_loss: 0.3709 - val_acc: 0.8187\n Epoch 16/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3678 - acc: 0.8442 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 17/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3669 - acc: 0.8490 - val_loss: 0.3703 - val_acc: 0.8200\n Epoch 18/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3688 - acc: 0.8482 - val_loss: 0.3707 - val_acc: 0.8200\n Epoch 19/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3662 - acc: 0.8497 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 20/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3728 - acc: 0.8472 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 21/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3674 - acc: 0.8478 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 22/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3721 - acc: 0.8456 - val_loss: 0.3697 - val_acc: 0.8200\n Epoch 23/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3696 - acc: 0.8462 - val_loss: 0.3703 - val_acc: 0.8200\n Epoch 24/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3739 - acc: 0.8432 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 25/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3698 - acc: 0.8483 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 26/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3729 - acc: 0.8460 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 27/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3680 - acc: 0.8511 - val_loss: 0.3697 - val_acc: 0.8213\n Epoch 28/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3696 - acc: 0.8474 - val_loss: 0.3705 - val_acc: 0.8187\n Epoch 29/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3673 - acc: 0.8476 - val_loss: 0.3703 - val_acc: 0.8187\n Epoch 30/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3741 - acc: 0.8451 - val_loss: 0.3712 - val_acc: 0.8187\n Epoch 31/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3676 - acc: 0.8460 - val_loss: 0.3707 - val_acc: 0.8187\n Epoch 32/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3724 - acc: 0.8454 - val_loss: 0.3704 - val_acc: 0.8187\n Epoch 33/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3679 - acc: 0.8500 - val_loss: 0.3701 - val_acc: 0.8187\n Epoch 34/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3666 - acc: 0.8493 - val_loss: 0.3702 - val_acc: 0.8200\n Epoch 35/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3712 - acc: 0.8458 - val_loss: 0.3710 - val_acc: 0.8187\n Epoch 36/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3688 - acc: 0.8504 - val_loss: 0.3697 - val_acc: 0.8237\n Epoch 37/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3732 - acc: 0.8446 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 38/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3694 - acc: 0.8485 - val_loss: 0.3688 - val_acc: 0.8250\n Epoch 39/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3681 - acc: 0.8471 - val_loss: 0.3694 - val_acc: 0.8237\n Epoch 40/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3661 - acc: 0.8521 - val_loss: 0.3692 - val_acc: 0.8237\n Epoch 41/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3698 - acc: 0.8464 - val_loss: 0.3695 - val_acc: 0.8225\n Epoch 42/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3650 - acc: 0.8522 - val_loss: 0.3688 - val_acc: 0.8237\n Epoch 43/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3713 - acc: 0.8478 - val_loss: 0.3698 - val_acc: 0.8237\n Epoch 44/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3683 - acc: 0.8468 - val_loss: 0.3696 - val_acc: 0.8213\n Epoch 45/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3674 - acc: 0.8499 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 46/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3702 - acc: 0.8444 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 47/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3720 - acc: 0.8478 - val_loss: 0.3704 - val_acc: 0.8213\n Epoch 48/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3723 - acc: 0.8469 - val_loss: 0.3696 - val_acc: 0.8237\n Epoch 49/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3746 - acc: 0.8451 - val_loss: 0.3700 - val_acc: 0.8250\n Epoch 50/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3673 - acc: 0.8494 - val_loss: 0.3692 - val_acc: 0.8250\n Epoch 51/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3632 - acc: 0.8486 - val_loss: 0.3692 - val_acc: 0.8237\n Epoch 52/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3676 - acc: 0.8482 - val_loss: 0.3687 - val_acc: 0.8263\n Epoch 53/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3694 - acc: 0.8475 - val_loss: 0.3691 - val_acc: 0.8250\n Epoch 54/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3734 - acc: 0.8432 - val_loss: 0.3700 - val_acc: 0.8250\n Epoch 55/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3660 - acc: 0.8500 - val_loss: 0.3697 - val_acc: 0.8225\n Epoch 56/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3699 - acc: 0.8474 - val_loss: 0.3695 - val_acc: 0.8225\n Epoch 57/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3675 - acc: 0.8476 - val_loss: 0.3701 - val_acc: 0.8200\n Epoch 58/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3694 - acc: 0.8440 - val_loss: 0.3687 - val_acc: 0.8237\n Epoch 59/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3724 - acc: 0.8461 - val_loss: 0.3701 - val_acc: 0.8200\n Epoch 60/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3678 - acc: 0.8472 - val_loss: 0.3698 - val_acc: 0.8200\n Epoch 61/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3640 - acc: 0.8504 - val_loss: 0.3699 - val_acc: 0.8200\n Epoch 62/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3696 - acc: 0.8444 - val_loss: 0.3701 - val_acc: 0.8200\n Epoch 63/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3687 - acc: 0.8501 - val_loss: 0.3702 - val_acc: 0.8200\n Epoch 64/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8468 - val_loss: 0.3698 - val_acc: 0.8200\n Epoch 65/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3660 - acc: 0.8483 - val_loss: 0.3700 - val_acc: 0.8200\n Epoch 66/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3696 - acc: 0.8478 - val_loss: 0.3707 - val_acc: 0.8200\n Epoch 67/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3680 - acc: 0.8468 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 68/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3720 - acc: 0.8435 - val_loss: 0.3697 - val_acc: 0.8187\n Epoch 69/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3719 - acc: 0.8485 - val_loss: 0.3687 - val_acc: 0.8237\n Epoch 70/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3702 - acc: 0.8457 - val_loss: 0.3696 - val_acc: 0.8200\n Epoch 71/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3692 - acc: 0.8486 - val_loss: 0.3697 - val_acc: 0.8200\n Epoch 72/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3720 - acc: 0.8482 - val_loss: 0.3697 - val_acc: 0.8187\n Epoch 73/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3740 - acc: 0.8469 - val_loss: 0.3696 - val_acc: 0.8237\n Epoch 74/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3725 - acc: 0.8458 - val_loss: 0.3710 - val_acc: 0.8187\n Epoch 75/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3701 - acc: 0.8450 - val_loss: 0.3710 - val_acc: 0.8187\n Epoch 76/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3653 - acc: 0.8483 - val_loss: 0.3700 - val_acc: 0.8187\n Epoch 77/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3710 - acc: 0.8475 - val_loss: 0.3690 - val_acc: 0.8237\n Epoch 78/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3692 - acc: 0.8451 - val_loss: 0.3690 - val_acc: 0.8250\n Epoch 79/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3707 - acc: 0.8444 - val_loss: 0.3688 - val_acc: 0.8237\n Epoch 80/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3679 - acc: 0.8468 - val_loss: 0.3701 - val_acc: 0.8200\n Epoch 81/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3665 - acc: 0.8515 - val_loss: 0.3688 - val_acc: 0.8225\n Epoch 82/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3663 - acc: 0.8496 - val_loss: 0.3692 - val_acc: 0.8213\n Epoch 83/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3692 - acc: 0.8482 - val_loss: 0.3691 - val_acc: 0.8237\n Epoch 84/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3651 - acc: 0.8514 - val_loss: 0.3695 - val_acc: 0.8237\n Epoch 85/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3696 - acc: 0.8488 - val_loss: 0.3699 - val_acc: 0.8213\n Epoch 86/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3701 - acc: 0.8467 - val_loss: 0.3698 - val_acc: 0.8213\n Epoch 87/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3695 - acc: 0.8456 - val_loss: 0.3696 - val_acc: 0.8213\n Epoch 88/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3713 - acc: 0.8482 - val_loss: 0.3704 - val_acc: 0.8187\n Epoch 89/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3706 - acc: 0.8451 - val_loss: 0.3696 - val_acc: 0.8187\n Epoch 90/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3701 - acc: 0.8468 - val_loss: 0.3699 - val_acc: 0.8187\n Epoch 91/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3702 - acc: 0.8469 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 92/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3683 - acc: 0.8506 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 93/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3687 - acc: 0.8476 - val_loss: 0.3713 - val_acc: 0.8187\n Epoch 94/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3668 - acc: 0.8482 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 95/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3689 - acc: 0.8472 - val_loss: 0.3693 - val_acc: 0.8200\n Epoch 96/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3749 - acc: 0.8422 - val_loss: 0.3699 - val_acc: 0.8200\n Epoch 97/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3725 - acc: 0.8456 - val_loss: 0.3699 - val_acc: 0.8200\n Epoch 98/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3641 - acc: 0.8515 - val_loss: 0.3694 - val_acc: 0.8200\n Epoch 99/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3690 - acc: 0.8464 - val_loss: 0.3692 - val_acc: 0.8200\n Epoch 100/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3671 - acc: 0.8497 - val_loss: 0.3699 - val_acc: 0.8200\n Epoch 101/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3702 - acc: 0.8439 - val_loss: 0.3708 - val_acc: 0.8187\n Epoch 102/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3702 - acc: 0.8506 - val_loss: 0.3695 - val_acc: 0.8200\n Epoch 103/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3679 - acc: 0.8465 - val_loss: 0.3699 - val_acc: 0.8187\n Epoch 104/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3685 - acc: 0.8465 - val_loss: 0.3698 - val_acc: 0.8200\n Epoch 105/10000\n 7200/7200 [==============================] - 1s 111us/step - loss: 0.3666 - acc: 0.8494 - val_loss: 0.3697 - val_acc: 0.8187\n Epoch 106/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3673 - acc: 0.8503 - val_loss: 0.3700 - val_acc: 0.8187\n Epoch 107/10000\n 7200/7200 [==============================] - 1s 97us/step - loss: 0.3724 - acc: 0.8472 - val_loss: 0.3706 - val_acc: 0.8187\n Epoch 108/10000\n 7200/7200 [==============================] - 1s 121us/step - loss: 0.3681 - acc: 0.8476 - val_loss: 0.3698 - val_acc: 0.8187\n Epoch 109/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3701 - acc: 0.8444 - val_loss: 0.3692 - val_acc: 0.8200\n Epoch 110/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3669 - acc: 0.8494 - val_loss: 0.3693 - val_acc: 0.8200\n Epoch 111/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3703 - acc: 0.8461 - val_loss: 0.3689 - val_acc: 0.8200\n Epoch 112/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3706 - acc: 0.8454 - val_loss: 0.3688 - val_acc: 0.8200\n Epoch 113/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3684 - acc: 0.8461 - val_loss: 0.3690 - val_acc: 0.8250\n Epoch 114/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3712 - acc: 0.8486 - val_loss: 0.3703 - val_acc: 0.8200\n Epoch 115/10000\n 7200/7200 [==============================] - 1s 86us/step - loss: 0.3715 - acc: 0.8460 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 116/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3674 - acc: 0.8501 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 117/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3720 - acc: 0.8460 - val_loss: 0.3702 - val_acc: 0.8200\n Epoch 118/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3689 - acc: 0.8475 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 119/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3661 - acc: 0.8486 - val_loss: 0.3707 - val_acc: 0.8200\n Epoch 120/10000\n 7200/7200 [==============================] - 1s 91us/step - loss: 0.3651 - acc: 0.8492 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 121/10000\n 7200/7200 [==============================] - 1s 91us/step - loss: 0.3664 - acc: 0.8483 - val_loss: 0.3685 - val_acc: 0.8250\n Epoch 122/10000\n 7200/7200 [==============================] - 1s 102us/step - loss: 0.3664 - acc: 0.8503 - val_loss: 0.3698 - val_acc: 0.8200\n Epoch 123/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3676 - acc: 0.8474 - val_loss: 0.3695 - val_acc: 0.8200\n Epoch 124/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3701 - acc: 0.8488 - val_loss: 0.3698 - val_acc: 0.8200\n Epoch 125/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3660 - acc: 0.8494 - val_loss: 0.3696 - val_acc: 0.8200\n Epoch 126/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3708 - acc: 0.8446 - val_loss: 0.3692 - val_acc: 0.8200\n Epoch 127/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3729 - acc: 0.8465 - val_loss: 0.3689 - val_acc: 0.8237\n Epoch 128/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3698 - acc: 0.8428 - val_loss: 0.3689 - val_acc: 0.8200\n Epoch 129/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3705 - acc: 0.8460 - val_loss: 0.3694 - val_acc: 0.8200\n Epoch 130/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3675 - acc: 0.8500 - val_loss: 0.3697 - val_acc: 0.8200\n Epoch 131/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3700 - acc: 0.8465 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 132/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3679 - acc: 0.8511 - val_loss: 0.3702 - val_acc: 0.8200\n Epoch 133/10000\n 7200/7200 [==============================] - 1s 97us/step - loss: 0.3708 - acc: 0.8464 - val_loss: 0.3702 - val_acc: 0.8200\n Epoch 134/10000\n 7200/7200 [==============================] - 1s 89us/step - loss: 0.3715 - acc: 0.8476 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 135/10000\n 7200/7200 [==============================] - 1s 75us/step - loss: 0.3706 - acc: 0.8465 - val_loss: 0.3703 - val_acc: 0.8200\n Epoch 136/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3723 - acc: 0.8458 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 137/10000\n 7200/7200 [==============================] - 1s 93us/step - loss: 0.3677 - acc: 0.8471 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 138/10000\n 7200/7200 [==============================] - 1s 90us/step - loss: 0.3705 - acc: 0.8474 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 139/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3720 - acc: 0.8456 - val_loss: 0.3699 - val_acc: 0.8200\n Epoch 140/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3701 - acc: 0.8467 - val_loss: 0.3691 - val_acc: 0.8200\n Epoch 141/10000\n 7200/7200 [==============================] - 1s 93us/step - loss: 0.3692 - acc: 0.8489 - val_loss: 0.3690 - val_acc: 0.8200\n Epoch 142/10000\n 7200/7200 [==============================] - 1s 83us/step - loss: 0.3680 - acc: 0.8483 - val_loss: 0.3699 - val_acc: 0.8200\n Epoch 143/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3722 - acc: 0.8499 - val_loss: 0.3696 - val_acc: 0.8200\n Epoch 144/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3703 - acc: 0.8454 - val_loss: 0.3691 - val_acc: 0.8200\n Epoch 145/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3696 - acc: 0.8476 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 146/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3679 - acc: 0.8460 - val_loss: 0.3703 - val_acc: 0.8200\n Epoch 147/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3698 - acc: 0.8446 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 148/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3694 - acc: 0.8457 - val_loss: 0.3697 - val_acc: 0.8200\n Epoch 149/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3704 - acc: 0.8472 - val_loss: 0.3696 - val_acc: 0.8200\n Epoch 150/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3720 - acc: 0.8460 - val_loss: 0.3696 - val_acc: 0.8200\n Epoch 151/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3692 - acc: 0.8496 - val_loss: 0.3703 - val_acc: 0.8200\n Epoch 152/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3675 - acc: 0.8469 - val_loss: 0.3696 - val_acc: 0.8225\n Epoch 153/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3717 - acc: 0.8481 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 154/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3649 - acc: 0.8486 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 155/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3735 - acc: 0.8438 - val_loss: 0.3704 - val_acc: 0.8213\n Epoch 156/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3702 - acc: 0.8464 - val_loss: 0.3710 - val_acc: 0.8213\n Epoch 157/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3688 - acc: 0.8447 - val_loss: 0.3705 - val_acc: 0.8213\n Epoch 158/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3687 - acc: 0.8468 - val_loss: 0.3703 - val_acc: 0.8213\n Epoch 159/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3656 - acc: 0.8468 - val_loss: 0.3703 - val_acc: 0.8200\n Epoch 160/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3711 - acc: 0.8447 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 161/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3698 - acc: 0.8482 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 162/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3706 - acc: 0.8444 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 163/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3675 - acc: 0.8489 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 164/10000\n 7200/7200 [==============================] - 1s 106us/step - loss: 0.3640 - acc: 0.8481 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 165/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3719 - acc: 0.8454 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 166/10000\n 7200/7200 [==============================] - 1s 80us/step - loss: 0.3708 - acc: 0.8440 - val_loss: 0.3707 - val_acc: 0.8200\n Epoch 167/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3709 - acc: 0.8485 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 168/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3647 - acc: 0.8492 - val_loss: 0.3703 - val_acc: 0.8200\n Epoch 169/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3648 - acc: 0.8511 - val_loss: 0.3696 - val_acc: 0.8200\n Epoch 170/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3715 - acc: 0.8512 - val_loss: 0.3700 - val_acc: 0.8187\n Epoch 171/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3717 - acc: 0.8421 - val_loss: 0.3710 - val_acc: 0.8187\n Epoch 172/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3676 - acc: 0.8501 - val_loss: 0.3702 - val_acc: 0.8187\n Epoch 173/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3723 - acc: 0.8433 - val_loss: 0.3697 - val_acc: 0.8187\n Epoch 174/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3674 - acc: 0.8490 - val_loss: 0.3698 - val_acc: 0.8200\n Epoch 175/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3740 - acc: 0.8464 - val_loss: 0.3692 - val_acc: 0.8200\n Epoch 176/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3645 - acc: 0.8500 - val_loss: 0.3686 - val_acc: 0.8200\n Epoch 177/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3674 - acc: 0.8485 - val_loss: 0.3691 - val_acc: 0.8200\n Epoch 178/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3714 - acc: 0.8461 - val_loss: 0.3688 - val_acc: 0.8200\n Epoch 179/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3700 - acc: 0.8439 - val_loss: 0.3690 - val_acc: 0.8200\n Epoch 180/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3706 - acc: 0.8467 - val_loss: 0.3689 - val_acc: 0.8187\n Epoch 181/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3715 - acc: 0.8483 - val_loss: 0.3702 - val_acc: 0.8187\n Epoch 182/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3704 - acc: 0.8453 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 183/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3713 - acc: 0.8489 - val_loss: 0.3701 - val_acc: 0.8213\n Epoch 184/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3679 - acc: 0.8490 - val_loss: 0.3713 - val_acc: 0.8187\n Epoch 185/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3709 - acc: 0.8453 - val_loss: 0.3699 - val_acc: 0.8225\n Epoch 186/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3679 - acc: 0.8468 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 187/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3668 - acc: 0.8497 - val_loss: 0.3702 - val_acc: 0.8200\n Epoch 188/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3693 - acc: 0.8468 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 189/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3713 - acc: 0.8467 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 190/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3644 - acc: 0.8515 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 191/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3653 - acc: 0.8483 - val_loss: 0.3702 - val_acc: 0.8200\n Epoch 192/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3755 - acc: 0.8428 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 193/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3667 - acc: 0.8485 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 194/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3703 - acc: 0.8483 - val_loss: 0.3700 - val_acc: 0.8200\n Epoch 195/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3664 - acc: 0.8503 - val_loss: 0.3701 - val_acc: 0.8225\n Epoch 196/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3664 - acc: 0.8474 - val_loss: 0.3704 - val_acc: 0.8225\n Epoch 197/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3736 - acc: 0.8446 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 198/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3711 - acc: 0.8447 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 199/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3685 - acc: 0.8469 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 200/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3700 - acc: 0.8488 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 201/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3689 - acc: 0.8490 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 202/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3722 - acc: 0.8451 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 203/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3698 - acc: 0.8486 - val_loss: 0.3703 - val_acc: 0.8200\n Epoch 204/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3748 - acc: 0.8453 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 205/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3686 - acc: 0.8462 - val_loss: 0.3698 - val_acc: 0.8200\n Epoch 206/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3672 - acc: 0.8469 - val_loss: 0.3692 - val_acc: 0.8200\n Epoch 207/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3674 - acc: 0.8471 - val_loss: 0.3700 - val_acc: 0.8200\n Epoch 208/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3685 - acc: 0.8489 - val_loss: 0.3680 - val_acc: 0.8237\n Epoch 209/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3652 - acc: 0.8471 - val_loss: 0.3682 - val_acc: 0.8250\n Epoch 210/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3729 - acc: 0.8469 - val_loss: 0.3686 - val_acc: 0.8275\n Epoch 211/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3741 - acc: 0.8449 - val_loss: 0.3700 - val_acc: 0.8200\n Epoch 212/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3740 - acc: 0.8472 - val_loss: 0.3696 - val_acc: 0.8200\n Epoch 213/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3735 - acc: 0.8460 - val_loss: 0.3692 - val_acc: 0.8200\n Epoch 214/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3749 - acc: 0.8406 - val_loss: 0.3699 - val_acc: 0.8200\n Epoch 215/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3670 - acc: 0.8479 - val_loss: 0.3696 - val_acc: 0.8200\n Epoch 216/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3708 - acc: 0.8462 - val_loss: 0.3696 - val_acc: 0.8200\n Epoch 217/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3688 - acc: 0.8454 - val_loss: 0.3685 - val_acc: 0.8213\n Epoch 218/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3695 - acc: 0.8453 - val_loss: 0.3691 - val_acc: 0.8200\n Epoch 219/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3705 - acc: 0.8468 - val_loss: 0.3695 - val_acc: 0.8200\n Epoch 220/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3712 - acc: 0.8479 - val_loss: 0.3694 - val_acc: 0.8200\n Epoch 221/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3689 - acc: 0.8476 - val_loss: 0.3677 - val_acc: 0.8263\n Epoch 222/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3674 - acc: 0.8478 - val_loss: 0.3690 - val_acc: 0.8213\n Epoch 223/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3708 - acc: 0.8476 - val_loss: 0.3692 - val_acc: 0.8250\n Epoch 224/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3675 - acc: 0.8489 - val_loss: 0.3692 - val_acc: 0.8237\n Epoch 225/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3659 - acc: 0.8486 - val_loss: 0.3702 - val_acc: 0.8213\n Epoch 226/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3682 - acc: 0.8464 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 227/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3734 - acc: 0.8438 - val_loss: 0.3703 - val_acc: 0.8200\n Epoch 228/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3702 - acc: 0.8464 - val_loss: 0.3691 - val_acc: 0.8213\n Epoch 229/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3739 - acc: 0.8425 - val_loss: 0.3695 - val_acc: 0.8213\n Epoch 230/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3709 - acc: 0.8450 - val_loss: 0.3702 - val_acc: 0.8200\n Epoch 231/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3698 - acc: 0.8468 - val_loss: 0.3702 - val_acc: 0.8213\n Epoch 232/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3681 - acc: 0.8471 - val_loss: 0.3702 - val_acc: 0.8213\n Epoch 233/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3664 - acc: 0.8494 - val_loss: 0.3707 - val_acc: 0.8213\n Epoch 234/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3661 - acc: 0.8496 - val_loss: 0.3694 - val_acc: 0.8250\n Epoch 235/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3632 - acc: 0.8496 - val_loss: 0.3702 - val_acc: 0.8237\n Epoch 236/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3656 - acc: 0.8499 - val_loss: 0.3698 - val_acc: 0.8250\n Epoch 237/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3656 - acc: 0.8510 - val_loss: 0.3701 - val_acc: 0.8263\n Epoch 238/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3694 - acc: 0.8488 - val_loss: 0.3696 - val_acc: 0.8263\n Epoch 239/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3667 - acc: 0.8489 - val_loss: 0.3689 - val_acc: 0.8263\n Epoch 240/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3718 - acc: 0.8496 - val_loss: 0.3701 - val_acc: 0.8250\n Epoch 241/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3691 - acc: 0.8462 - val_loss: 0.3701 - val_acc: 0.8250\n Epoch 242/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3679 - acc: 0.8474 - val_loss: 0.3699 - val_acc: 0.8250\n Epoch 243/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3738 - acc: 0.8458 - val_loss: 0.3706 - val_acc: 0.8250\n Epoch 244/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3722 - acc: 0.8443 - val_loss: 0.3703 - val_acc: 0.8250\n Epoch 245/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3755 - acc: 0.8462 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 246/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3684 - acc: 0.8454 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 247/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3710 - acc: 0.8446 - val_loss: 0.3703 - val_acc: 0.8213\n Epoch 248/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3724 - acc: 0.8483 - val_loss: 0.3706 - val_acc: 0.8213\n Epoch 249/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3749 - acc: 0.8472 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 250/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3690 - acc: 0.8442 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 251/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3716 - acc: 0.8479 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 252/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3684 - acc: 0.8500 - val_loss: 0.3704 - val_acc: 0.8225\n Epoch 253/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3726 - acc: 0.8486 - val_loss: 0.3703 - val_acc: 0.8213\n Epoch 254/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3681 - acc: 0.8478 - val_loss: 0.3714 - val_acc: 0.8213\n Epoch 255/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3706 - acc: 0.8488 - val_loss: 0.3709 - val_acc: 0.8225\n Epoch 256/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3678 - acc: 0.8486 - val_loss: 0.3715 - val_acc: 0.8237\n Epoch 257/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3697 - acc: 0.8476 - val_loss: 0.3706 - val_acc: 0.8263\n Epoch 258/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3734 - acc: 0.8461 - val_loss: 0.3709 - val_acc: 0.8263\n Epoch 259/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3694 - acc: 0.8461 - val_loss: 0.3710 - val_acc: 0.8263\n Epoch 260/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3698 - acc: 0.8444 - val_loss: 0.3715 - val_acc: 0.8237\n Epoch 261/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3682 - acc: 0.8481 - val_loss: 0.3709 - val_acc: 0.8263\n Epoch 262/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3714 - acc: 0.8465 - val_loss: 0.3717 - val_acc: 0.8237\n Epoch 263/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3713 - acc: 0.8457 - val_loss: 0.3720 - val_acc: 0.8237\n Epoch 264/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3696 - acc: 0.8475 - val_loss: 0.3716 - val_acc: 0.8237\n Epoch 265/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3690 - acc: 0.8476 - val_loss: 0.3711 - val_acc: 0.8225\n Epoch 266/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3692 - acc: 0.8494 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 267/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3704 - acc: 0.8451 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 268/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3648 - acc: 0.8503 - val_loss: 0.3703 - val_acc: 0.8225\n Epoch 269/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3647 - acc: 0.8529 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 270/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3666 - acc: 0.8464 - val_loss: 0.3698 - val_acc: 0.8250\n Epoch 271/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3718 - acc: 0.8471 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 272/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3679 - acc: 0.8467 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 273/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3619 - acc: 0.8497 - val_loss: 0.3707 - val_acc: 0.8200\n Epoch 274/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3704 - acc: 0.8472 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 275/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3685 - acc: 0.8456 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 276/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3704 - acc: 0.8467 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 277/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3675 - acc: 0.8493 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 278/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3657 - acc: 0.8469 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 279/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3666 - acc: 0.8475 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 280/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3721 - acc: 0.8458 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 281/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3674 - acc: 0.8475 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 282/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3690 - acc: 0.8429 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 283/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3708 - acc: 0.8436 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 284/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3734 - acc: 0.8453 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 285/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3672 - acc: 0.8486 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 286/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3722 - acc: 0.8447 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 287/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3682 - acc: 0.8481 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 288/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3685 - acc: 0.8500 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 289/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3705 - acc: 0.8449 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 290/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3755 - acc: 0.8447 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 291/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3746 - acc: 0.8438 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 292/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3722 - acc: 0.8469 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 293/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3701 - acc: 0.8485 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 294/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3690 - acc: 0.8458 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 295/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3678 - acc: 0.8496 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 296/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3659 - acc: 0.8469 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 297/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3674 - acc: 0.8462 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 298/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3668 - acc: 0.8483 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 299/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3764 - acc: 0.8418 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 300/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3651 - acc: 0.8474 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 301/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3691 - acc: 0.8486 - val_loss: 0.3697 - val_acc: 0.8200\n Epoch 302/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3706 - acc: 0.8454 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 303/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3700 - acc: 0.8443 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 304/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3701 - acc: 0.8451 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 305/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3641 - acc: 0.8507 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 306/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3678 - acc: 0.8453 - val_loss: 0.3710 - val_acc: 0.8237\n Epoch 307/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3701 - acc: 0.8446 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 308/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3694 - acc: 0.8474 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 309/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3718 - acc: 0.8483 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 310/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3761 - acc: 0.8461 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 311/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3648 - acc: 0.8486 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 312/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3713 - acc: 0.8435 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 313/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3729 - acc: 0.8457 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 314/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3740 - acc: 0.8454 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 315/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3699 - acc: 0.8462 - val_loss: 0.3713 - val_acc: 0.8175\n Epoch 316/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3707 - acc: 0.8450 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 317/10000\n 7200/7200 [==============================] - 1s 87us/step - loss: 0.3720 - acc: 0.8456 - val_loss: 0.3707 - val_acc: 0.8213\n Epoch 318/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3670 - acc: 0.8464 - val_loss: 0.3713 - val_acc: 0.8213\n Epoch 319/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3686 - acc: 0.8496 - val_loss: 0.3715 - val_acc: 0.8250\n Epoch 320/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3675 - acc: 0.8472 - val_loss: 0.3711 - val_acc: 0.8250\n Epoch 321/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3717 - acc: 0.8464 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 322/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3718 - acc: 0.8458 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 323/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3649 - acc: 0.8489 - val_loss: 0.3720 - val_acc: 0.8175\n Epoch 324/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3716 - acc: 0.8494 - val_loss: 0.3718 - val_acc: 0.8175\n Epoch 325/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3671 - acc: 0.8482 - val_loss: 0.3705 - val_acc: 0.8263\n Epoch 326/10000\n 7200/7200 [==============================] - 1s 102us/step - loss: 0.3721 - acc: 0.8462 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 327/10000\n 7200/7200 [==============================] - 1s 144us/step - loss: 0.3702 - acc: 0.8486 - val_loss: 0.3715 - val_acc: 0.8225\n Epoch 328/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3705 - acc: 0.8464 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 329/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3689 - acc: 0.8464 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 330/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3678 - acc: 0.8511 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 331/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3724 - acc: 0.8461 - val_loss: 0.3705 - val_acc: 0.8263\n Epoch 332/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3690 - acc: 0.8486 - val_loss: 0.3709 - val_acc: 0.8213\n Epoch 333/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3655 - acc: 0.8479 - val_loss: 0.3707 - val_acc: 0.8225\n Epoch 334/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3676 - acc: 0.8485 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 335/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3703 - acc: 0.8456 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 336/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3718 - acc: 0.8451 - val_loss: 0.3696 - val_acc: 0.8237\n Epoch 337/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3715 - acc: 0.8447 - val_loss: 0.3703 - val_acc: 0.8200\n Epoch 338/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3735 - acc: 0.8451 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 339/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3699 - acc: 0.8464 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 340/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3681 - acc: 0.8475 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 341/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3675 - acc: 0.8486 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 342/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3745 - acc: 0.8429 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 343/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3690 - acc: 0.8490 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 344/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3733 - acc: 0.8462 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 345/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3655 - acc: 0.8469 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 346/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3679 - acc: 0.8489 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 347/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3721 - acc: 0.8464 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 348/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3715 - acc: 0.8474 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 349/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3691 - acc: 0.8468 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 350/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3687 - acc: 0.8512 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 351/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3691 - acc: 0.8471 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 352/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3686 - acc: 0.8467 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 353/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3691 - acc: 0.8481 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 354/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3689 - acc: 0.8488 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 355/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3688 - acc: 0.8489 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 356/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3716 - acc: 0.8464 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 357/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3666 - acc: 0.8461 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 358/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3723 - acc: 0.8438 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 359/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3674 - acc: 0.8476 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 360/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3736 - acc: 0.8429 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 361/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3707 - acc: 0.8457 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 362/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3708 - acc: 0.8454 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 363/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3646 - acc: 0.8501 - val_loss: 0.3728 - val_acc: 0.8175\n Epoch 364/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3689 - acc: 0.8456 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 365/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3687 - acc: 0.8475 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 366/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3682 - acc: 0.8474 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 367/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3676 - acc: 0.8476 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 368/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3696 - acc: 0.8474 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 369/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3641 - acc: 0.8499 - val_loss: 0.3720 - val_acc: 0.8175\n Epoch 370/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3713 - acc: 0.8465 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 371/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3672 - acc: 0.8492 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 372/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3738 - acc: 0.8435 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 373/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3682 - acc: 0.8482 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 374/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3675 - acc: 0.8500 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 375/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3697 - acc: 0.8465 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 376/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3734 - acc: 0.8454 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 377/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3746 - acc: 0.8421 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 378/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3688 - acc: 0.8454 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 379/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3690 - acc: 0.8483 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 380/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3638 - acc: 0.8518 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 381/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3694 - acc: 0.8456 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 382/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3685 - acc: 0.8464 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 383/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3738 - acc: 0.8438 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 384/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3725 - acc: 0.8471 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 385/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3701 - acc: 0.8479 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 386/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3674 - acc: 0.8504 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 387/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3693 - acc: 0.8433 - val_loss: 0.3700 - val_acc: 0.8200\n Epoch 388/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3697 - acc: 0.8481 - val_loss: 0.3702 - val_acc: 0.8200\n Epoch 389/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3707 - acc: 0.8492 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 390/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3729 - acc: 0.8483 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 391/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3684 - acc: 0.8447 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 392/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3711 - acc: 0.8439 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 393/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3655 - acc: 0.8489 - val_loss: 0.3698 - val_acc: 0.8200\n Epoch 394/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3673 - acc: 0.8481 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 395/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3710 - acc: 0.8454 - val_loss: 0.3700 - val_acc: 0.8200\n Epoch 396/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3684 - acc: 0.8478 - val_loss: 0.3699 - val_acc: 0.8200\n Epoch 397/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3708 - acc: 0.8465 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 398/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3696 - acc: 0.8471 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 399/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3776 - acc: 0.8419 - val_loss: 0.3707 - val_acc: 0.8200\n Epoch 400/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3701 - acc: 0.8472 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 401/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3763 - acc: 0.8428 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 402/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3691 - acc: 0.8458 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 403/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3694 - acc: 0.8454 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 404/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3697 - acc: 0.8460 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 405/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3689 - acc: 0.8467 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 406/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3666 - acc: 0.8447 - val_loss: 0.3707 - val_acc: 0.8200\n Epoch 407/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3707 - acc: 0.8472 - val_loss: 0.3698 - val_acc: 0.8200\n Epoch 408/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3711 - acc: 0.8458 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 409/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3667 - acc: 0.8483 - val_loss: 0.3700 - val_acc: 0.8200\n Epoch 410/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3722 - acc: 0.8488 - val_loss: 0.3695 - val_acc: 0.8200\n Epoch 411/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3664 - acc: 0.8472 - val_loss: 0.3692 - val_acc: 0.8213\n Epoch 412/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3706 - acc: 0.8458 - val_loss: 0.3703 - val_acc: 0.8200\n Epoch 413/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3701 - acc: 0.8453 - val_loss: 0.3700 - val_acc: 0.8200\n Epoch 414/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3664 - acc: 0.8464 - val_loss: 0.3698 - val_acc: 0.8200\n Epoch 415/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3764 - acc: 0.8429 - val_loss: 0.3701 - val_acc: 0.8200\n Epoch 416/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3714 - acc: 0.8433 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 417/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3727 - acc: 0.8426 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 418/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3690 - acc: 0.8458 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 419/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3745 - acc: 0.8475 - val_loss: 0.3701 - val_acc: 0.8200\n Epoch 420/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3658 - acc: 0.8465 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 421/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3689 - acc: 0.8488 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 422/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3636 - acc: 0.8500 - val_loss: 0.3707 - val_acc: 0.8200\n Epoch 423/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3688 - acc: 0.8458 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 424/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3698 - acc: 0.8447 - val_loss: 0.3704 - val_acc: 0.8213\n Epoch 425/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3690 - acc: 0.8481 - val_loss: 0.3695 - val_acc: 0.8213\n Epoch 426/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3671 - acc: 0.8497 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 427/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3662 - acc: 0.8482 - val_loss: 0.3701 - val_acc: 0.8200\n Epoch 428/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3702 - acc: 0.8443 - val_loss: 0.3707 - val_acc: 0.8200\n Epoch 429/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3663 - acc: 0.8454 - val_loss: 0.3703 - val_acc: 0.8200\n Epoch 430/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3762 - acc: 0.8415 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 431/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3671 - acc: 0.8469 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 432/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3668 - acc: 0.8478 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 433/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3705 - acc: 0.8467 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 434/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3696 - acc: 0.8451 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 435/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3683 - acc: 0.8436 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 436/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3700 - acc: 0.8468 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 437/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3696 - acc: 0.8474 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 438/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3700 - acc: 0.8450 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 439/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3719 - acc: 0.8425 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 440/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3705 - acc: 0.8428 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 441/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3682 - acc: 0.8488 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 442/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3700 - acc: 0.8454 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 443/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3692 - acc: 0.8510 - val_loss: 0.3719 - val_acc: 0.8175\n Epoch 444/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3718 - acc: 0.8476 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 445/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3668 - acc: 0.8456 - val_loss: 0.3701 - val_acc: 0.8187\n Epoch 446/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3720 - acc: 0.8474 - val_loss: 0.3700 - val_acc: 0.8187\n Epoch 447/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3639 - acc: 0.8507 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 448/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3739 - acc: 0.8460 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 449/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3696 - acc: 0.8465 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 450/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3686 - acc: 0.8471 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 451/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3750 - acc: 0.8454 - val_loss: 0.3711 - val_acc: 0.8187\n Epoch 452/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3757 - acc: 0.8446 - val_loss: 0.3716 - val_acc: 0.8187\n Epoch 453/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3682 - acc: 0.8478 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 454/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3701 - acc: 0.8504 - val_loss: 0.3707 - val_acc: 0.8213\n Epoch 455/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3677 - acc: 0.8501 - val_loss: 0.3710 - val_acc: 0.8175\n Epoch 456/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3665 - acc: 0.8499 - val_loss: 0.3715 - val_acc: 0.8187\n Epoch 457/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3677 - acc: 0.8485 - val_loss: 0.3711 - val_acc: 0.8175\n Epoch 458/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3677 - acc: 0.8464 - val_loss: 0.3714 - val_acc: 0.8187\n Epoch 459/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3677 - acc: 0.8490 - val_loss: 0.3709 - val_acc: 0.8187\n Epoch 460/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3714 - acc: 0.8468 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 461/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3682 - acc: 0.8478 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 462/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3732 - acc: 0.8443 - val_loss: 0.3709 - val_acc: 0.8187\n Epoch 463/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3717 - acc: 0.8454 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 464/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3664 - acc: 0.8488 - val_loss: 0.3711 - val_acc: 0.8187\n Epoch 465/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3684 - acc: 0.8474 - val_loss: 0.3712 - val_acc: 0.8187\n Epoch 466/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3714 - acc: 0.8475 - val_loss: 0.3712 - val_acc: 0.8187\n Epoch 467/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3704 - acc: 0.8512 - val_loss: 0.3713 - val_acc: 0.8187\n Epoch 468/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3713 - acc: 0.8460 - val_loss: 0.3712 - val_acc: 0.8187\n Epoch 469/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3708 - acc: 0.8457 - val_loss: 0.3697 - val_acc: 0.8225\n Epoch 470/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3689 - acc: 0.8435 - val_loss: 0.3704 - val_acc: 0.8187\n Epoch 471/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3652 - acc: 0.8490 - val_loss: 0.3700 - val_acc: 0.8200\n Epoch 472/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3689 - acc: 0.8433 - val_loss: 0.3697 - val_acc: 0.8200\n Epoch 473/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3728 - acc: 0.8460 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 474/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3706 - acc: 0.8494 - val_loss: 0.3697 - val_acc: 0.8200\n Epoch 475/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3700 - acc: 0.8489 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 476/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3698 - acc: 0.8465 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 477/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3707 - acc: 0.8457 - val_loss: 0.3707 - val_acc: 0.8200\n Epoch 478/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3681 - acc: 0.8468 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 479/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3737 - acc: 0.8457 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 480/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3699 - acc: 0.8467 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 481/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3732 - acc: 0.8425 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 482/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3677 - acc: 0.8475 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 483/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3695 - acc: 0.8460 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 484/10000\n 7200/7200 [==============================] - 1s 92us/step - loss: 0.3638 - acc: 0.8529 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 485/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3653 - acc: 0.8504 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 486/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3681 - acc: 0.8485 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 487/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3692 - acc: 0.8472 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 488/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3706 - acc: 0.8469 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 489/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3708 - acc: 0.8464 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 490/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3709 - acc: 0.8478 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 491/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3693 - acc: 0.8507 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 492/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3667 - acc: 0.8488 - val_loss: 0.3714 - val_acc: 0.8187\n Epoch 493/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3704 - acc: 0.8449 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 494/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3643 - acc: 0.8507 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 495/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3697 - acc: 0.8456 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 496/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3683 - acc: 0.8481 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 497/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3661 - acc: 0.8501 - val_loss: 0.3715 - val_acc: 0.8187\n Epoch 498/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3730 - acc: 0.8442 - val_loss: 0.3718 - val_acc: 0.8187\n Epoch 499/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3714 - acc: 0.8453 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 500/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3697 - acc: 0.8483 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 501/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3715 - acc: 0.8474 - val_loss: 0.3710 - val_acc: 0.8187\n Epoch 502/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3625 - acc: 0.8494 - val_loss: 0.3711 - val_acc: 0.8187\n Epoch 503/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3719 - acc: 0.8474 - val_loss: 0.3716 - val_acc: 0.8187\n Epoch 504/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3665 - acc: 0.8507 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 505/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3651 - acc: 0.8475 - val_loss: 0.3707 - val_acc: 0.8200\n Epoch 506/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3718 - acc: 0.8481 - val_loss: 0.3707 - val_acc: 0.8200\n Epoch 507/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3740 - acc: 0.8456 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 508/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3676 - acc: 0.8475 - val_loss: 0.3716 - val_acc: 0.8187\n Epoch 509/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3709 - acc: 0.8467 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 510/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3701 - acc: 0.8474 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 511/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3719 - acc: 0.8460 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 512/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3729 - acc: 0.8435 - val_loss: 0.3732 - val_acc: 0.8175\n Epoch 513/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3662 - acc: 0.8503 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 514/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3661 - acc: 0.8483 - val_loss: 0.3715 - val_acc: 0.8187\n Epoch 515/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3689 - acc: 0.8494 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 516/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3676 - acc: 0.8499 - val_loss: 0.3709 - val_acc: 0.8225\n Epoch 517/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3755 - acc: 0.8444 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 518/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3688 - acc: 0.8486 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 519/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3697 - acc: 0.8522 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 520/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3653 - acc: 0.8482 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 521/10000\n 7200/7200 [==============================] - 1s 80us/step - loss: 0.3696 - acc: 0.8464 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 522/10000\n 7200/7200 [==============================] - 1s 82us/step - loss: 0.3718 - acc: 0.8451 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 523/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3728 - acc: 0.8458 - val_loss: 0.3703 - val_acc: 0.8225\n Epoch 524/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3717 - acc: 0.8464 - val_loss: 0.3706 - val_acc: 0.8213\n Epoch 525/10000\n 7200/7200 [==============================] - 1s 82us/step - loss: 0.3706 - acc: 0.8450 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 526/10000\n 7200/7200 [==============================] - 1s 87us/step - loss: 0.3722 - acc: 0.8476 - val_loss: 0.3714 - val_acc: 0.8225\n Epoch 527/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3718 - acc: 0.8468 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 528/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3700 - acc: 0.8457 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 529/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3698 - acc: 0.8461 - val_loss: 0.3717 - val_acc: 0.8225\n Epoch 530/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3656 - acc: 0.8506 - val_loss: 0.3714 - val_acc: 0.8225\n Epoch 531/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3640 - acc: 0.8471 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 532/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3646 - acc: 0.8510 - val_loss: 0.3710 - val_acc: 0.8237\n Epoch 533/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3671 - acc: 0.8494 - val_loss: 0.3711 - val_acc: 0.8250\n Epoch 534/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3705 - acc: 0.8488 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 535/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3710 - acc: 0.8450 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 536/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3671 - acc: 0.8476 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 537/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3664 - acc: 0.8481 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 538/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3710 - acc: 0.8454 - val_loss: 0.3740 - val_acc: 0.8200\n Epoch 539/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3709 - acc: 0.8500 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 540/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3710 - acc: 0.8450 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 541/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3724 - acc: 0.8450 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 542/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3692 - acc: 0.8472 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 543/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3679 - acc: 0.8483 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 544/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3743 - acc: 0.8462 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 545/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3684 - acc: 0.8478 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 546/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3684 - acc: 0.8468 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 547/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3724 - acc: 0.8413 - val_loss: 0.3750 - val_acc: 0.8200\n Epoch 548/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3711 - acc: 0.8458 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 549/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3659 - acc: 0.8494 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 550/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3717 - acc: 0.8438 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 551/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3691 - acc: 0.8476 - val_loss: 0.3730 - val_acc: 0.8200\n Epoch 552/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3661 - acc: 0.8494 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 553/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3754 - acc: 0.8417 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 554/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3708 - acc: 0.8478 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 555/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3725 - acc: 0.8440 - val_loss: 0.3712 - val_acc: 0.8225\n Epoch 556/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3691 - acc: 0.8461 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 557/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3731 - acc: 0.8471 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 558/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3645 - acc: 0.8483 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 559/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3743 - acc: 0.8462 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 560/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3637 - acc: 0.8507 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 561/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3726 - acc: 0.8451 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 562/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3726 - acc: 0.8476 - val_loss: 0.3730 - val_acc: 0.8200\n Epoch 563/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3674 - acc: 0.8469 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 564/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3680 - acc: 0.8485 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 565/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3686 - acc: 0.8481 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 566/10000\n 7200/7200 [==============================] - 1s 88us/step - loss: 0.3725 - acc: 0.8469 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 567/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3673 - acc: 0.8474 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 568/10000\n 7200/7200 [==============================] - 1s 83us/step - loss: 0.3688 - acc: 0.8497 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 569/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3723 - acc: 0.8450 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 570/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3680 - acc: 0.8471 - val_loss: 0.3735 - val_acc: 0.8200\n Epoch 571/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3706 - acc: 0.8446 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 572/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3661 - acc: 0.8488 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 573/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3680 - acc: 0.8451 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 574/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3696 - acc: 0.8462 - val_loss: 0.3707 - val_acc: 0.8213\n Epoch 575/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3654 - acc: 0.8500 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 576/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3667 - acc: 0.8460 - val_loss: 0.3705 - val_acc: 0.8225\n Epoch 577/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3681 - acc: 0.8465 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 578/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3698 - acc: 0.8450 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 579/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3610 - acc: 0.8526 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 580/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3717 - acc: 0.8469 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 581/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3708 - acc: 0.8489 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 582/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3669 - acc: 0.8468 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 583/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3689 - acc: 0.8471 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 584/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3666 - acc: 0.8482 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 585/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3677 - acc: 0.8503 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 586/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3678 - acc: 0.8481 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 587/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3699 - acc: 0.8471 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 588/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3716 - acc: 0.8451 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 589/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3658 - acc: 0.8476 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 590/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3736 - acc: 0.8426 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 591/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3685 - acc: 0.8485 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 592/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3714 - acc: 0.8421 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 593/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3706 - acc: 0.8486 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 594/10000\n 7200/7200 [==============================] - 1s 82us/step - loss: 0.3693 - acc: 0.8457 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 595/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3665 - acc: 0.8485 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 596/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3694 - acc: 0.8497 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 597/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3711 - acc: 0.8475 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 598/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3730 - acc: 0.8478 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 599/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3719 - acc: 0.8469 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 600/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3637 - acc: 0.8496 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 601/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3707 - acc: 0.8454 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 602/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3708 - acc: 0.8474 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 603/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3724 - acc: 0.8464 - val_loss: 0.3708 - val_acc: 0.8213\n Epoch 604/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3732 - acc: 0.8460 - val_loss: 0.3714 - val_acc: 0.8225\n Epoch 605/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3695 - acc: 0.8471 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 606/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3676 - acc: 0.8496 - val_loss: 0.3726 - val_acc: 0.8175\n Epoch 607/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3718 - acc: 0.8465 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 608/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3737 - acc: 0.8447 - val_loss: 0.3734 - val_acc: 0.8175\n Epoch 609/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3664 - acc: 0.8472 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 610/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3719 - acc: 0.8447 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 611/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3707 - acc: 0.8462 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 612/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3708 - acc: 0.8446 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 613/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3645 - acc: 0.8474 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 614/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3698 - acc: 0.8447 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 615/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3672 - acc: 0.8481 - val_loss: 0.3740 - val_acc: 0.8200\n Epoch 616/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3696 - acc: 0.8474 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 617/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3652 - acc: 0.8492 - val_loss: 0.3730 - val_acc: 0.8200\n Epoch 618/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3659 - acc: 0.8511 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 619/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3669 - acc: 0.8479 - val_loss: 0.3719 - val_acc: 0.8225\n Epoch 620/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3651 - acc: 0.8503 - val_loss: 0.3716 - val_acc: 0.8225\n Epoch 621/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3666 - acc: 0.8472 - val_loss: 0.3711 - val_acc: 0.8237\n Epoch 622/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3718 - acc: 0.8486 - val_loss: 0.3720 - val_acc: 0.8213\n Epoch 623/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3696 - acc: 0.8471 - val_loss: 0.3721 - val_acc: 0.8225\n Epoch 624/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3677 - acc: 0.8497 - val_loss: 0.3710 - val_acc: 0.8225\n Epoch 625/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3692 - acc: 0.8450 - val_loss: 0.3702 - val_acc: 0.8225\n Epoch 626/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3706 - acc: 0.8483 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 627/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3656 - acc: 0.8511 - val_loss: 0.3709 - val_acc: 0.8237\n Epoch 628/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3628 - acc: 0.8512 - val_loss: 0.3698 - val_acc: 0.8250\n Epoch 629/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3722 - acc: 0.8436 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 630/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3637 - acc: 0.8517 - val_loss: 0.3704 - val_acc: 0.8225\n Epoch 631/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3683 - acc: 0.8503 - val_loss: 0.3707 - val_acc: 0.8200\n Epoch 632/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3737 - acc: 0.8429 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 633/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3691 - acc: 0.8442 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 634/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3723 - acc: 0.8447 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 635/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3691 - acc: 0.8462 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 636/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3675 - acc: 0.8512 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 637/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3709 - acc: 0.8469 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 638/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3684 - acc: 0.8482 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 639/10000\n 7200/7200 [==============================] - ETA: 0s - loss: 0.3679 - acc: 0.847 - 0s 61us/step - loss: 0.3689 - acc: 0.8467 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 640/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3690 - acc: 0.8458 - val_loss: 0.3700 - val_acc: 0.8200\n Epoch 641/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3687 - acc: 0.8486 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 642/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3723 - acc: 0.8464 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 643/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3729 - acc: 0.8439 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 644/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3644 - acc: 0.8481 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 645/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3680 - acc: 0.8471 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 646/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3727 - acc: 0.8460 - val_loss: 0.3707 - val_acc: 0.8213\n Epoch 647/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3698 - acc: 0.8474 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 648/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3637 - acc: 0.8514 - val_loss: 0.3709 - val_acc: 0.8213\n Epoch 649/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3732 - acc: 0.8450 - val_loss: 0.3707 - val_acc: 0.8237\n Epoch 650/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3675 - acc: 0.8501 - val_loss: 0.3699 - val_acc: 0.8263\n Epoch 651/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3715 - acc: 0.8461 - val_loss: 0.3712 - val_acc: 0.8175\n Epoch 652/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3710 - acc: 0.8474 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 653/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3645 - acc: 0.8514 - val_loss: 0.3709 - val_acc: 0.8225\n Epoch 654/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3660 - acc: 0.8483 - val_loss: 0.3703 - val_acc: 0.8237\n Epoch 655/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3678 - acc: 0.8482 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 656/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3679 - acc: 0.8476 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 657/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3662 - acc: 0.8488 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 658/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3734 - acc: 0.8444 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 659/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3693 - acc: 0.8511 - val_loss: 0.3710 - val_acc: 0.8250\n Epoch 660/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3661 - acc: 0.8472 - val_loss: 0.3717 - val_acc: 0.8213\n Epoch 661/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3694 - acc: 0.8511 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 662/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3764 - acc: 0.8414 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 663/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3654 - acc: 0.8508 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 664/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3736 - acc: 0.8469 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 665/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3695 - acc: 0.8476 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 666/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3655 - acc: 0.8489 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 667/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3701 - acc: 0.8450 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 668/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3684 - acc: 0.8474 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 669/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3714 - acc: 0.8460 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 670/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3720 - acc: 0.8450 - val_loss: 0.3712 - val_acc: 0.8213\n Epoch 671/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3700 - acc: 0.8457 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 672/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3702 - acc: 0.8438 - val_loss: 0.3723 - val_acc: 0.8175\n Epoch 673/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3671 - acc: 0.8483 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 674/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3658 - acc: 0.8504 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 675/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3711 - acc: 0.8507 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 676/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3703 - acc: 0.8454 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 677/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3664 - acc: 0.8490 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 678/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3655 - acc: 0.8482 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 679/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3715 - acc: 0.8468 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 680/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3692 - acc: 0.8474 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 681/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3748 - acc: 0.8461 - val_loss: 0.3709 - val_acc: 0.8213\n Epoch 682/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3636 - acc: 0.8483 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 683/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3683 - acc: 0.8481 - val_loss: 0.3707 - val_acc: 0.8213\n Epoch 684/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3653 - acc: 0.8501 - val_loss: 0.3702 - val_acc: 0.8200\n Epoch 685/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3728 - acc: 0.8443 - val_loss: 0.3703 - val_acc: 0.8200\n Epoch 686/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3710 - acc: 0.8468 - val_loss: 0.3703 - val_acc: 0.8213\n Epoch 687/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3680 - acc: 0.8471 - val_loss: 0.3707 - val_acc: 0.8200\n Epoch 688/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3727 - acc: 0.8429 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 689/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3653 - acc: 0.8486 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 690/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3635 - acc: 0.8517 - val_loss: 0.3697 - val_acc: 0.8200\n Epoch 691/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3652 - acc: 0.8476 - val_loss: 0.3701 - val_acc: 0.8200\n Epoch 692/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3661 - acc: 0.8489 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 693/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3698 - acc: 0.8481 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 694/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3729 - acc: 0.8447 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 695/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3688 - acc: 0.8456 - val_loss: 0.3707 - val_acc: 0.8200\n Epoch 696/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3722 - acc: 0.8475 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 697/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3654 - acc: 0.8481 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 698/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3658 - acc: 0.8446 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 699/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3726 - acc: 0.8444 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 700/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3681 - acc: 0.8485 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 701/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3691 - acc: 0.8474 - val_loss: 0.3699 - val_acc: 0.8187\n Epoch 702/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3667 - acc: 0.8489 - val_loss: 0.3702 - val_acc: 0.8200\n Epoch 703/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3694 - acc: 0.8475 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 704/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3756 - acc: 0.8435 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 705/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3675 - acc: 0.8472 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 706/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3683 - acc: 0.8478 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 707/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3691 - acc: 0.8467 - val_loss: 0.3703 - val_acc: 0.8200\n Epoch 708/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3693 - acc: 0.8475 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 709/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3730 - acc: 0.8444 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 710/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3677 - acc: 0.8486 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 711/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3646 - acc: 0.8492 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 712/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3653 - acc: 0.8476 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 713/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3718 - acc: 0.8424 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 714/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3688 - acc: 0.8467 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 715/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3710 - acc: 0.8461 - val_loss: 0.3707 - val_acc: 0.8225\n Epoch 716/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3689 - acc: 0.8465 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 717/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3739 - acc: 0.8444 - val_loss: 0.3722 - val_acc: 0.8175\n Epoch 718/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3730 - acc: 0.8467 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 719/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3676 - acc: 0.8476 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 720/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3716 - acc: 0.8458 - val_loss: 0.3710 - val_acc: 0.8213\n Epoch 721/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3687 - acc: 0.8465 - val_loss: 0.3702 - val_acc: 0.8200\n Epoch 722/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3681 - acc: 0.8440 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 723/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3661 - acc: 0.8475 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 724/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3716 - acc: 0.8442 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 725/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3688 - acc: 0.8482 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 726/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3740 - acc: 0.8460 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 727/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3706 - acc: 0.8469 - val_loss: 0.3720 - val_acc: 0.8175\n Epoch 728/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3663 - acc: 0.8493 - val_loss: 0.3724 - val_acc: 0.8175\n Epoch 729/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3709 - acc: 0.8474 - val_loss: 0.3732 - val_acc: 0.8163\n Epoch 730/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3651 - acc: 0.8483 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 731/10000\n 7200/7200 [==============================] - 1s 82us/step - loss: 0.3679 - acc: 0.8471 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 732/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3705 - acc: 0.8447 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 733/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3709 - acc: 0.8450 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 734/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3710 - acc: 0.8460 - val_loss: 0.3736 - val_acc: 0.8175\n Epoch 735/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3706 - acc: 0.8472 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 736/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3636 - acc: 0.8507 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 737/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3667 - acc: 0.8504 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 738/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3709 - acc: 0.8512 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 739/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3720 - acc: 0.8446 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 740/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3671 - acc: 0.8486 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 741/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3726 - acc: 0.8465 - val_loss: 0.3710 - val_acc: 0.8237\n Epoch 742/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3718 - acc: 0.8475 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 743/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3718 - acc: 0.8478 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 744/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3730 - acc: 0.8422 - val_loss: 0.3718 - val_acc: 0.8187\n Epoch 745/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3682 - acc: 0.8474 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 746/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3734 - acc: 0.8449 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 747/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3677 - acc: 0.8483 - val_loss: 0.3710 - val_acc: 0.8225\n Epoch 748/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3653 - acc: 0.8501 - val_loss: 0.3709 - val_acc: 0.8225\n Epoch 749/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3701 - acc: 0.8474 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 750/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3686 - acc: 0.8469 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 751/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3666 - acc: 0.8507 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 752/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3729 - acc: 0.8411 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 753/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3735 - acc: 0.8443 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 754/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3701 - acc: 0.8476 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 755/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3702 - acc: 0.8475 - val_loss: 0.3731 - val_acc: 0.8175\n Epoch 756/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3704 - acc: 0.8481 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 757/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3693 - acc: 0.8475 - val_loss: 0.3711 - val_acc: 0.8237\n Epoch 758/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3690 - acc: 0.8450 - val_loss: 0.3713 - val_acc: 0.8213\n Epoch 759/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3694 - acc: 0.8475 - val_loss: 0.3713 - val_acc: 0.8225\n Epoch 760/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3667 - acc: 0.8494 - val_loss: 0.3725 - val_acc: 0.8175\n Epoch 761/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3680 - acc: 0.8475 - val_loss: 0.3724 - val_acc: 0.8225\n Epoch 762/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3759 - acc: 0.8458 - val_loss: 0.3720 - val_acc: 0.8213\n Epoch 763/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3719 - acc: 0.8454 - val_loss: 0.3730 - val_acc: 0.8200\n Epoch 764/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3712 - acc: 0.8436 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 765/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3703 - acc: 0.8468 - val_loss: 0.3719 - val_acc: 0.8175\n Epoch 766/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3745 - acc: 0.8442 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 767/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3708 - acc: 0.8451 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 768/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3694 - acc: 0.8450 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 769/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3723 - acc: 0.8442 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 770/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3684 - acc: 0.8468 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 771/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3598 - acc: 0.8522 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 772/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3698 - acc: 0.8482 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 773/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3708 - acc: 0.8465 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 774/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3752 - acc: 0.8453 - val_loss: 0.3714 - val_acc: 0.8187\n Epoch 775/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3710 - acc: 0.8456 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 776/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3736 - acc: 0.8478 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 777/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3686 - acc: 0.8451 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 778/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3689 - acc: 0.8521 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 779/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3650 - acc: 0.8496 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 780/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3667 - acc: 0.8469 - val_loss: 0.3703 - val_acc: 0.8237\n Epoch 781/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3653 - acc: 0.8482 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 782/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3727 - acc: 0.8432 - val_loss: 0.3710 - val_acc: 0.8213\n Epoch 783/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3709 - acc: 0.8476 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 784/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3711 - acc: 0.8471 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 785/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3726 - acc: 0.8429 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 786/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3673 - acc: 0.8472 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 787/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3672 - acc: 0.8490 - val_loss: 0.3708 - val_acc: 0.8213\n Epoch 788/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3684 - acc: 0.8472 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 789/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3694 - acc: 0.8474 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 790/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3657 - acc: 0.8489 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 791/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3690 - acc: 0.8475 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 792/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3752 - acc: 0.8469 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 793/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3737 - acc: 0.8428 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 794/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3693 - acc: 0.8479 - val_loss: 0.3717 - val_acc: 0.8175\n Epoch 795/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3683 - acc: 0.8469 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 796/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3700 - acc: 0.8456 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 797/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3673 - acc: 0.8478 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 798/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3689 - acc: 0.8481 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 799/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3698 - acc: 0.8486 - val_loss: 0.3711 - val_acc: 0.8213\n Epoch 800/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3676 - acc: 0.8478 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 801/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3704 - acc: 0.8496 - val_loss: 0.3710 - val_acc: 0.8237\n Epoch 802/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3684 - acc: 0.8490 - val_loss: 0.3712 - val_acc: 0.8213\n Epoch 803/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3651 - acc: 0.8517 - val_loss: 0.3709 - val_acc: 0.8250\n Epoch 804/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3710 - acc: 0.8483 - val_loss: 0.3715 - val_acc: 0.8187\n Epoch 805/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3716 - acc: 0.8450 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 806/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3680 - acc: 0.8460 - val_loss: 0.3716 - val_acc: 0.8175\n Epoch 807/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3692 - acc: 0.8444 - val_loss: 0.3720 - val_acc: 0.8175\n Epoch 808/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3689 - acc: 0.8501 - val_loss: 0.3721 - val_acc: 0.8175\n Epoch 809/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3644 - acc: 0.8501 - val_loss: 0.3721 - val_acc: 0.8175\n Epoch 810/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3665 - acc: 0.8501 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 811/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3645 - acc: 0.8521 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 812/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3717 - acc: 0.8472 - val_loss: 0.3723 - val_acc: 0.8175\n Epoch 813/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3722 - acc: 0.8474 - val_loss: 0.3731 - val_acc: 0.8175\n Epoch 814/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3712 - acc: 0.8444 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 815/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3707 - acc: 0.8439 - val_loss: 0.3707 - val_acc: 0.8200\n Epoch 816/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3648 - acc: 0.8504 - val_loss: 0.3707 - val_acc: 0.8187\n Epoch 817/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3691 - acc: 0.8461 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 818/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3703 - acc: 0.8456 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 819/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3719 - acc: 0.8458 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 820/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3704 - acc: 0.8492 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 821/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3704 - acc: 0.8472 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 822/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3729 - acc: 0.8451 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 823/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3726 - acc: 0.8436 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 824/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3705 - acc: 0.8481 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 825/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3709 - acc: 0.8467 - val_loss: 0.3704 - val_acc: 0.8225\n Epoch 826/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3661 - acc: 0.8468 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 827/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3679 - acc: 0.8472 - val_loss: 0.3702 - val_acc: 0.8200\n Epoch 828/10000\n 7200/7200 [==============================] - ETA: 0s - loss: 0.3724 - acc: 0.846 - 0s 61us/step - loss: 0.3730 - acc: 0.8461 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 829/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3689 - acc: 0.8467 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 830/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3667 - acc: 0.8456 - val_loss: 0.3701 - val_acc: 0.8200\n Epoch 831/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3671 - acc: 0.8512 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 832/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3711 - acc: 0.8454 - val_loss: 0.3701 - val_acc: 0.8200\n Epoch 833/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3646 - acc: 0.8504 - val_loss: 0.3688 - val_acc: 0.8263\n Epoch 834/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3682 - acc: 0.8444 - val_loss: 0.3701 - val_acc: 0.8200\n Epoch 835/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3693 - acc: 0.8457 - val_loss: 0.3698 - val_acc: 0.8200\n Epoch 836/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3700 - acc: 0.8449 - val_loss: 0.3702 - val_acc: 0.8200\n Epoch 837/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3696 - acc: 0.8476 - val_loss: 0.3701 - val_acc: 0.8200\n Epoch 838/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3693 - acc: 0.8454 - val_loss: 0.3696 - val_acc: 0.8213\n Epoch 839/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3698 - acc: 0.8460 - val_loss: 0.3692 - val_acc: 0.8200\n Epoch 840/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3678 - acc: 0.8469 - val_loss: 0.3698 - val_acc: 0.8200\n Epoch 841/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3691 - acc: 0.8457 - val_loss: 0.3697 - val_acc: 0.8200\n Epoch 842/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3699 - acc: 0.8467 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 843/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3719 - acc: 0.8485 - val_loss: 0.3713 - val_acc: 0.8187\n Epoch 844/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3676 - acc: 0.8476 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 845/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3675 - acc: 0.8464 - val_loss: 0.3711 - val_acc: 0.8187\n Epoch 846/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3666 - acc: 0.8497 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 847/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3721 - acc: 0.8429 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 848/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3688 - acc: 0.8476 - val_loss: 0.3704 - val_acc: 0.8200\n Epoch 849/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3683 - acc: 0.8501 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 850/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3739 - acc: 0.8467 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 851/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3699 - acc: 0.8450 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 852/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3671 - acc: 0.8486 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 853/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3699 - acc: 0.8468 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 854/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3707 - acc: 0.8432 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 855/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3725 - acc: 0.8464 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 856/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3700 - acc: 0.8508 - val_loss: 0.3714 - val_acc: 0.8237\n Epoch 857/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3668 - acc: 0.8506 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 858/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3708 - acc: 0.8438 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 859/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3730 - acc: 0.8456 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 860/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3704 - acc: 0.8485 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 861/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3734 - acc: 0.8451 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 862/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3722 - acc: 0.8467 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 863/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3677 - acc: 0.8488 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 864/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3658 - acc: 0.8500 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 865/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3651 - acc: 0.8519 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 866/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3656 - acc: 0.8512 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 867/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3686 - acc: 0.8460 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 868/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3705 - acc: 0.8481 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 869/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3733 - acc: 0.8493 - val_loss: 0.3726 - val_acc: 0.8225\n Epoch 870/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3726 - acc: 0.8460 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 871/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3678 - acc: 0.8492 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 872/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3663 - acc: 0.8490 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 873/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3708 - acc: 0.8453 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 874/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3687 - acc: 0.8469 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 875/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3679 - acc: 0.8475 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 876/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3669 - acc: 0.8474 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 877/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3664 - acc: 0.8439 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 878/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3697 - acc: 0.8444 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 879/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3699 - acc: 0.8429 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 880/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3693 - acc: 0.8479 - val_loss: 0.3720 - val_acc: 0.8213\n Epoch 881/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3733 - acc: 0.8446 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 882/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3646 - acc: 0.8508 - val_loss: 0.3710 - val_acc: 0.8225\n Epoch 883/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3666 - acc: 0.8474 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 884/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3675 - acc: 0.8483 - val_loss: 0.3706 - val_acc: 0.8237\n Epoch 885/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3683 - acc: 0.8462 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 886/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3712 - acc: 0.8442 - val_loss: 0.3710 - val_acc: 0.8225\n Epoch 887/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3666 - acc: 0.8492 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 888/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3664 - acc: 0.8488 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 889/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3711 - acc: 0.8431 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 890/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3677 - acc: 0.8472 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 891/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3623 - acc: 0.8507 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 892/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3644 - acc: 0.8514 - val_loss: 0.3707 - val_acc: 0.8225\n Epoch 893/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3676 - acc: 0.8504 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 894/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3682 - acc: 0.8451 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 895/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3685 - acc: 0.8475 - val_loss: 0.3693 - val_acc: 0.8225\n Epoch 896/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3739 - acc: 0.8449 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 897/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3667 - acc: 0.8476 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 898/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3719 - acc: 0.8454 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 899/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3669 - acc: 0.8472 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 900/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3695 - acc: 0.8476 - val_loss: 0.3703 - val_acc: 0.8225\n Epoch 901/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3687 - acc: 0.8469 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 902/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3695 - acc: 0.8476 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 903/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3639 - acc: 0.8499 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 904/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3712 - acc: 0.8454 - val_loss: 0.3714 - val_acc: 0.8225\n Epoch 905/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3736 - acc: 0.8479 - val_loss: 0.3720 - val_acc: 0.8213\n Epoch 906/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3686 - acc: 0.8475 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 907/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3666 - acc: 0.8454 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 908/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3671 - acc: 0.8472 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 909/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3664 - acc: 0.8486 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 910/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3690 - acc: 0.8454 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 911/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3711 - acc: 0.8442 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 912/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3710 - acc: 0.8457 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 913/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3694 - acc: 0.8460 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 914/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3733 - acc: 0.8462 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 915/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3670 - acc: 0.8488 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 916/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3694 - acc: 0.8464 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 917/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3710 - acc: 0.8458 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 918/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3699 - acc: 0.8479 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 919/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3688 - acc: 0.8475 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 920/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3695 - acc: 0.8462 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 921/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3672 - acc: 0.8472 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 922/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3694 - acc: 0.8461 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 923/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3763 - acc: 0.8439 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 924/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3706 - acc: 0.8464 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 925/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3661 - acc: 0.8519 - val_loss: 0.3714 - val_acc: 0.8213\n Epoch 926/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3685 - acc: 0.8458 - val_loss: 0.3707 - val_acc: 0.8225\n Epoch 927/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3752 - acc: 0.8438 - val_loss: 0.3709 - val_acc: 0.8225\n Epoch 928/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3739 - acc: 0.8435 - val_loss: 0.3711 - val_acc: 0.8187\n Epoch 929/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3730 - acc: 0.8460 - val_loss: 0.3704 - val_acc: 0.8225\n Epoch 930/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3656 - acc: 0.8476 - val_loss: 0.3696 - val_acc: 0.8213\n Epoch 931/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3663 - acc: 0.8496 - val_loss: 0.3702 - val_acc: 0.8225\n Epoch 932/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3678 - acc: 0.8479 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 933/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3675 - acc: 0.8496 - val_loss: 0.3698 - val_acc: 0.8225\n Epoch 934/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3708 - acc: 0.8467 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 935/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3712 - acc: 0.8440 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 936/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3660 - acc: 0.8483 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 937/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3723 - acc: 0.8465 - val_loss: 0.3722 - val_acc: 0.8225\n Epoch 938/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3692 - acc: 0.8490 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 939/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3670 - acc: 0.8488 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 940/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3686 - acc: 0.8488 - val_loss: 0.3714 - val_acc: 0.8213\n Epoch 941/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3660 - acc: 0.8488 - val_loss: 0.3705 - val_acc: 0.8225\n Epoch 942/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3720 - acc: 0.8446 - val_loss: 0.3711 - val_acc: 0.8225\n Epoch 943/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3647 - acc: 0.8521 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 944/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3712 - acc: 0.8462 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 945/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3684 - acc: 0.8476 - val_loss: 0.3717 - val_acc: 0.8225\n Epoch 946/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3696 - acc: 0.8488 - val_loss: 0.3718 - val_acc: 0.8187\n Epoch 947/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3763 - acc: 0.8447 - val_loss: 0.3713 - val_acc: 0.8187\n Epoch 948/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3674 - acc: 0.8482 - val_loss: 0.3707 - val_acc: 0.8225\n Epoch 949/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3643 - acc: 0.8515 - val_loss: 0.3711 - val_acc: 0.8213\n Epoch 950/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3757 - acc: 0.8433 - val_loss: 0.3716 - val_acc: 0.8213\n Epoch 951/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3707 - acc: 0.8486 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 952/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3693 - acc: 0.8493 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 953/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3761 - acc: 0.8433 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 954/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3689 - acc: 0.8486 - val_loss: 0.3731 - val_acc: 0.8213\n Epoch 955/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3725 - acc: 0.8449 - val_loss: 0.3727 - val_acc: 0.8225\n Epoch 956/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3667 - acc: 0.8499 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 957/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3713 - acc: 0.8481 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 958/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3725 - acc: 0.8456 - val_loss: 0.3721 - val_acc: 0.8213\n Epoch 959/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3689 - acc: 0.8478 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 960/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3733 - acc: 0.8444 - val_loss: 0.3725 - val_acc: 0.8213\n Epoch 961/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3660 - acc: 0.8468 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 962/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3712 - acc: 0.8428 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 963/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3750 - acc: 0.8419 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 964/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3645 - acc: 0.8504 - val_loss: 0.3721 - val_acc: 0.8213\n Epoch 965/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3700 - acc: 0.8446 - val_loss: 0.3728 - val_acc: 0.8213\n Epoch 966/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3738 - acc: 0.8447 - val_loss: 0.3726 - val_acc: 0.8213\n Epoch 967/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3652 - acc: 0.8464 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 968/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3681 - acc: 0.8485 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 969/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3698 - acc: 0.8471 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 970/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3740 - acc: 0.8454 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 971/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3641 - acc: 0.8485 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 972/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3707 - acc: 0.8472 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 973/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3732 - acc: 0.8447 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 974/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3704 - acc: 0.8462 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 975/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3645 - acc: 0.8503 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 976/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3686 - acc: 0.8488 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 977/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3652 - acc: 0.8492 - val_loss: 0.3718 - val_acc: 0.8213\n Epoch 978/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3689 - acc: 0.8469 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 979/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3650 - acc: 0.8499 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 980/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3689 - acc: 0.8447 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 981/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3710 - acc: 0.8485 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 982/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3687 - acc: 0.8493 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 983/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3675 - acc: 0.8476 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 984/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3645 - acc: 0.8511 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 985/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3656 - acc: 0.8483 - val_loss: 0.3708 - val_acc: 0.8225\n Epoch 986/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3704 - acc: 0.8475 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 987/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3689 - acc: 0.8481 - val_loss: 0.3703 - val_acc: 0.8225\n Epoch 988/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3710 - acc: 0.8482 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 989/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3696 - acc: 0.8465 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 990/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3669 - acc: 0.8478 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 991/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3710 - acc: 0.8447 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 992/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3714 - acc: 0.8482 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 993/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3684 - acc: 0.8467 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 994/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3665 - acc: 0.8507 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 995/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3678 - acc: 0.8465 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 996/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3674 - acc: 0.8449 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 997/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3741 - acc: 0.8453 - val_loss: 0.3703 - val_acc: 0.8200\n Epoch 998/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3669 - acc: 0.8479 - val_loss: 0.3699 - val_acc: 0.8225\n Epoch 999/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3707 - acc: 0.8431 - val_loss: 0.3700 - val_acc: 0.8225\n Epoch 1000/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3634 - acc: 0.8529 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 1001/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3659 - acc: 0.8485 - val_loss: 0.3708 - val_acc: 0.8225\n Epoch 1002/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3691 - acc: 0.8469 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 1003/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3695 - acc: 0.8478 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 1004/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3699 - acc: 0.8457 - val_loss: 0.3711 - val_acc: 0.8225\n Epoch 1005/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3653 - acc: 0.8488 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 1006/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3672 - acc: 0.8488 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 1007/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3697 - acc: 0.8456 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 1008/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3681 - acc: 0.8486 - val_loss: 0.3711 - val_acc: 0.8225\n Epoch 1009/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3689 - acc: 0.8465 - val_loss: 0.3703 - val_acc: 0.8225\n Epoch 1010/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3673 - acc: 0.8458 - val_loss: 0.3707 - val_acc: 0.8213\n Epoch 1011/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3701 - acc: 0.8467 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 1012/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3675 - acc: 0.8490 - val_loss: 0.3711 - val_acc: 0.8225\n Epoch 1013/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3718 - acc: 0.8458 - val_loss: 0.3711 - val_acc: 0.8225\n Epoch 1014/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3664 - acc: 0.8483 - val_loss: 0.3706 - val_acc: 0.8213\n Epoch 1015/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3674 - acc: 0.8485 - val_loss: 0.3706 - val_acc: 0.8200\n Epoch 1016/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3697 - acc: 0.8476 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 1017/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3698 - acc: 0.8475 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 1018/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3772 - acc: 0.8408 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 1019/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3667 - acc: 0.8479 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 1020/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3687 - acc: 0.8454 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 1021/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8464 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 1022/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3643 - acc: 0.8503 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 1023/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3688 - acc: 0.8493 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 1024/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3673 - acc: 0.8461 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 1025/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3661 - acc: 0.8496 - val_loss: 0.3703 - val_acc: 0.8225\n Epoch 1026/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3689 - acc: 0.8488 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 1027/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3696 - acc: 0.8461 - val_loss: 0.3715 - val_acc: 0.8225\n Epoch 1028/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3684 - acc: 0.8482 - val_loss: 0.3715 - val_acc: 0.8225\n Epoch 1029/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3640 - acc: 0.8499 - val_loss: 0.3709 - val_acc: 0.8225\n Epoch 1030/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3697 - acc: 0.8488 - val_loss: 0.3713 - val_acc: 0.8225\n Epoch 1031/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3714 - acc: 0.8467 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 1032/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3681 - acc: 0.8489 - val_loss: 0.3719 - val_acc: 0.8225\n Epoch 1033/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3719 - acc: 0.8472 - val_loss: 0.3726 - val_acc: 0.8225\n Epoch 1034/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3664 - acc: 0.8486 - val_loss: 0.3723 - val_acc: 0.8213\n Epoch 1035/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3751 - acc: 0.8439 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 1036/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3695 - acc: 0.8469 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 1037/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3705 - acc: 0.8468 - val_loss: 0.3713 - val_acc: 0.8225\n Epoch 1038/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8489 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 1039/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3694 - acc: 0.8475 - val_loss: 0.3715 - val_acc: 0.8225\n Epoch 1040/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3703 - acc: 0.8454 - val_loss: 0.3712 - val_acc: 0.8225\n Epoch 1041/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3714 - acc: 0.8472 - val_loss: 0.3708 - val_acc: 0.8225\n Epoch 1042/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3681 - acc: 0.8485 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 1043/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3653 - acc: 0.8472 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 1044/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3721 - acc: 0.8435 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 1045/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3676 - acc: 0.8454 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 1046/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3686 - acc: 0.8489 - val_loss: 0.3702 - val_acc: 0.8225\n Epoch 1047/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3678 - acc: 0.8471 - val_loss: 0.3701 - val_acc: 0.8237\n Epoch 1048/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3702 - acc: 0.8446 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 1049/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3655 - acc: 0.8481 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 1050/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3692 - acc: 0.8450 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 1051/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3737 - acc: 0.8432 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 1052/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3702 - acc: 0.8486 - val_loss: 0.3737 - val_acc: 0.8200\n Epoch 1053/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3684 - acc: 0.8496 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 1054/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3690 - acc: 0.8475 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 1055/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3642 - acc: 0.8478 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 1056/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3663 - acc: 0.8489 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 1057/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3676 - acc: 0.8461 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 1058/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3671 - acc: 0.8467 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 1059/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3656 - acc: 0.8485 - val_loss: 0.3709 - val_acc: 0.8225\n Epoch 1060/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3609 - acc: 0.8486 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 1061/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3687 - acc: 0.8457 - val_loss: 0.3711 - val_acc: 0.8200\n Epoch 1062/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3708 - acc: 0.8450 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 1063/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3699 - acc: 0.8450 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 1064/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3667 - acc: 0.8511 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 1065/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3694 - acc: 0.8507 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 1066/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3682 - acc: 0.8483 - val_loss: 0.3715 - val_acc: 0.8225\n Epoch 1067/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3688 - acc: 0.8481 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 1068/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3669 - acc: 0.8492 - val_loss: 0.3703 - val_acc: 0.8213\n Epoch 1069/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3682 - acc: 0.8483 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 1070/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3679 - acc: 0.8504 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 1071/10000\n 7200/7200 [==============================] - ETA: 0s - loss: 0.3764 - acc: 0.839 - 0s 62us/step - loss: 0.3762 - acc: 0.8404 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 1072/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3691 - acc: 0.8457 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 1073/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3682 - acc: 0.8486 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 1074/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3715 - acc: 0.8471 - val_loss: 0.3713 - val_acc: 0.8225\n Epoch 1075/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3639 - acc: 0.8511 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 1076/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3670 - acc: 0.8517 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 1077/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3718 - acc: 0.8471 - val_loss: 0.3703 - val_acc: 0.8225\n Epoch 1078/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3689 - acc: 0.8483 - val_loss: 0.3715 - val_acc: 0.8225\n Epoch 1079/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3723 - acc: 0.8472 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 1080/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3685 - acc: 0.8494 - val_loss: 0.3714 - val_acc: 0.8225\n Epoch 1081/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3709 - acc: 0.8478 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 1082/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3667 - acc: 0.8496 - val_loss: 0.3708 - val_acc: 0.8200\n Epoch 1083/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3697 - acc: 0.8462 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 1084/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3721 - acc: 0.8467 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 1085/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3719 - acc: 0.8471 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 1086/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3696 - acc: 0.8472 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 1087/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3760 - acc: 0.8438 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 1088/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3620 - acc: 0.8485 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 1089/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3683 - acc: 0.8478 - val_loss: 0.3735 - val_acc: 0.8200\n Epoch 1090/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3698 - acc: 0.8450 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 1091/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3755 - acc: 0.8425 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 1092/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3672 - acc: 0.8471 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 1093/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3684 - acc: 0.8465 - val_loss: 0.3730 - val_acc: 0.8200\n Epoch 1094/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3717 - acc: 0.8443 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 1095/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3703 - acc: 0.8471 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 1096/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3753 - acc: 0.8450 - val_loss: 0.3735 - val_acc: 0.8200\n Epoch 1097/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3699 - acc: 0.8467 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 1098/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3678 - acc: 0.8474 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 1099/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3679 - acc: 0.8474 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 1100/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3670 - acc: 0.8507 - val_loss: 0.3735 - val_acc: 0.8200\n Epoch 1101/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3681 - acc: 0.8529 - val_loss: 0.3720 - val_acc: 0.8225\n Epoch 1102/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3717 - acc: 0.8474 - val_loss: 0.3719 - val_acc: 0.8213\n Epoch 1103/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3664 - acc: 0.8492 - val_loss: 0.3724 - val_acc: 0.8225\n Epoch 1104/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3716 - acc: 0.8482 - val_loss: 0.3725 - val_acc: 0.8237\n Epoch 1105/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3677 - acc: 0.8468 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 1106/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3723 - acc: 0.8471 - val_loss: 0.3730 - val_acc: 0.8200\n Epoch 1107/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3705 - acc: 0.8488 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 1108/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3715 - acc: 0.8489 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 1109/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3685 - acc: 0.8481 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1110/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3707 - acc: 0.8450 - val_loss: 0.3718 - val_acc: 0.8187\n Epoch 1111/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3669 - acc: 0.8482 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 1112/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3704 - acc: 0.8474 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 1113/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3682 - acc: 0.8468 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 1114/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3653 - acc: 0.8504 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 1115/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3711 - acc: 0.8465 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 1116/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8492 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 1117/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3714 - acc: 0.8461 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 1118/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3723 - acc: 0.8447 - val_loss: 0.3709 - val_acc: 0.8200\n Epoch 1119/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3688 - acc: 0.8468 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 1120/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3750 - acc: 0.8439 - val_loss: 0.3712 - val_acc: 0.8225\n Epoch 1121/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3741 - acc: 0.8439 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 1122/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3667 - acc: 0.8485 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 1123/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3642 - acc: 0.8490 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 1124/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3702 - acc: 0.8482 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 1125/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3698 - acc: 0.8457 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 1126/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3689 - acc: 0.8449 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 1127/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3710 - acc: 0.8460 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 1128/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3746 - acc: 0.8447 - val_loss: 0.3712 - val_acc: 0.8225\n Epoch 1129/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3644 - acc: 0.8519 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 1130/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3707 - acc: 0.8451 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 1131/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3657 - acc: 0.8508 - val_loss: 0.3717 - val_acc: 0.8225\n Epoch 1132/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3717 - acc: 0.8450 - val_loss: 0.3716 - val_acc: 0.8225\n Epoch 1133/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3705 - acc: 0.8469 - val_loss: 0.3714 - val_acc: 0.8225\n Epoch 1134/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3613 - acc: 0.8532 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 1135/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3660 - acc: 0.8489 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 1136/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3672 - acc: 0.8468 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 1137/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3683 - acc: 0.8490 - val_loss: 0.3709 - val_acc: 0.8237\n Epoch 1138/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3719 - acc: 0.8453 - val_loss: 0.3714 - val_acc: 0.8225\n Epoch 1139/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3662 - acc: 0.8483 - val_loss: 0.3708 - val_acc: 0.8237\n Epoch 1140/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3718 - acc: 0.8465 - val_loss: 0.3705 - val_acc: 0.8237\n Epoch 1141/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3736 - acc: 0.8429 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 1142/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3739 - acc: 0.8465 - val_loss: 0.3725 - val_acc: 0.8175\n Epoch 1143/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3694 - acc: 0.8469 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 1144/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8485 - val_loss: 0.3739 - val_acc: 0.8175\n Epoch 1145/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3703 - acc: 0.8465 - val_loss: 0.3732 - val_acc: 0.8175\n Epoch 1146/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3693 - acc: 0.8439 - val_loss: 0.3743 - val_acc: 0.8175\n Epoch 1147/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3741 - acc: 0.8453 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 1148/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3679 - acc: 0.8483 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 1149/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3674 - acc: 0.8469 - val_loss: 0.3729 - val_acc: 0.8225\n Epoch 1150/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3712 - acc: 0.8440 - val_loss: 0.3721 - val_acc: 0.8225\n Epoch 1151/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3682 - acc: 0.8486 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 1152/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3682 - acc: 0.8489 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 1153/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3687 - acc: 0.8462 - val_loss: 0.3740 - val_acc: 0.8200\n Epoch 1154/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3692 - acc: 0.8479 - val_loss: 0.3745 - val_acc: 0.8200\n Epoch 1155/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3736 - acc: 0.8440 - val_loss: 0.3742 - val_acc: 0.8200\n Epoch 1156/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3695 - acc: 0.8436 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 1157/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3726 - acc: 0.8453 - val_loss: 0.3742 - val_acc: 0.8200\n Epoch 1158/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3708 - acc: 0.8472 - val_loss: 0.3724 - val_acc: 0.8225\n Epoch 1159/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3700 - acc: 0.8456 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 1160/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3681 - acc: 0.8475 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 1161/10000\n 7200/7200 [==============================] - ETA: 0s - loss: 0.3680 - acc: 0.850 - 0s 62us/step - loss: 0.3679 - acc: 0.8508 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 1162/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3682 - acc: 0.8482 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 1163/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3676 - acc: 0.8451 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 1164/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3695 - acc: 0.8482 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 1165/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3737 - acc: 0.8454 - val_loss: 0.3705 - val_acc: 0.8263\n Epoch 1166/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3736 - acc: 0.8431 - val_loss: 0.3713 - val_acc: 0.8225\n Epoch 1167/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3725 - acc: 0.8429 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 1168/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3722 - acc: 0.8486 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 1169/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3663 - acc: 0.8479 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 1170/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3694 - acc: 0.8435 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 1171/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3690 - acc: 0.8460 - val_loss: 0.3743 - val_acc: 0.8200\n Epoch 1172/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3697 - acc: 0.8450 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 1173/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3774 - acc: 0.8432 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 1174/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3664 - acc: 0.8500 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 1175/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3663 - acc: 0.8482 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 1176/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3652 - acc: 0.8489 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 1177/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3669 - acc: 0.8493 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1178/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3653 - acc: 0.8494 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 1179/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3750 - acc: 0.8442 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1180/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3696 - acc: 0.8493 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1181/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3728 - acc: 0.8428 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1182/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3682 - acc: 0.8507 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1183/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3725 - acc: 0.8457 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 1184/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3675 - acc: 0.8506 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 1185/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3684 - acc: 0.8506 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 1186/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3715 - acc: 0.8425 - val_loss: 0.3730 - val_acc: 0.8200\n Epoch 1187/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3719 - acc: 0.8471 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 1188/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3727 - acc: 0.8482 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 1189/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3689 - acc: 0.8468 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 1190/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3694 - acc: 0.8483 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 1191/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3679 - acc: 0.8488 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 1192/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3660 - acc: 0.8460 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 1193/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3741 - acc: 0.8478 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 1194/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3722 - acc: 0.8483 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 1195/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3733 - acc: 0.8471 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 1196/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3723 - acc: 0.8446 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 1197/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3671 - acc: 0.8497 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 1198/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3721 - acc: 0.8429 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 1199/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3714 - acc: 0.8460 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 1200/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3687 - acc: 0.8474 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1201/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3681 - acc: 0.8465 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 1202/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3623 - acc: 0.8507 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 1203/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3672 - acc: 0.8478 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 1204/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3684 - acc: 0.8503 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 1205/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3663 - acc: 0.8481 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 1206/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3639 - acc: 0.8511 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 1207/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3705 - acc: 0.8465 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 1208/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3708 - acc: 0.8458 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 1209/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3725 - acc: 0.8444 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 1210/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3694 - acc: 0.8497 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1211/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3691 - acc: 0.8450 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 1212/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3700 - acc: 0.8469 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 1213/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3709 - acc: 0.8460 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1214/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3662 - acc: 0.8497 - val_loss: 0.3720 - val_acc: 0.8225\n Epoch 1215/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3665 - acc: 0.8496 - val_loss: 0.3720 - val_acc: 0.8225\n Epoch 1216/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3662 - acc: 0.8504 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 1217/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3704 - acc: 0.8465 - val_loss: 0.3730 - val_acc: 0.8200\n Epoch 1218/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3702 - acc: 0.8462 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 1219/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3690 - acc: 0.8457 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 1220/10000\n 7200/7200 [==============================] - ETA: 0s - loss: 0.3710 - acc: 0.847 - 0s 62us/step - loss: 0.3712 - acc: 0.8469 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 1221/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3731 - acc: 0.8433 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 1222/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3740 - acc: 0.8453 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1223/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3675 - acc: 0.8467 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 1224/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3682 - acc: 0.8447 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 1225/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3683 - acc: 0.8489 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 1226/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3666 - acc: 0.8500 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 1227/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3632 - acc: 0.8510 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 1228/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3722 - acc: 0.8444 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 1229/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3728 - acc: 0.8444 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 1230/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3691 - acc: 0.8465 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 1231/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3682 - acc: 0.8490 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 1232/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3674 - acc: 0.8474 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 1233/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3699 - acc: 0.8486 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 1234/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3738 - acc: 0.8433 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 1235/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3658 - acc: 0.8500 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 1236/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3734 - acc: 0.8476 - val_loss: 0.3730 - val_acc: 0.8200\n Epoch 1237/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3690 - acc: 0.8493 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 1238/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3703 - acc: 0.8478 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 1239/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3738 - acc: 0.8440 - val_loss: 0.3730 - val_acc: 0.8200\n Epoch 1240/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3667 - acc: 0.8471 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 1241/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3669 - acc: 0.8493 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 1242/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3729 - acc: 0.8426 - val_loss: 0.3740 - val_acc: 0.8200\n Epoch 1243/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3700 - acc: 0.8476 - val_loss: 0.3743 - val_acc: 0.8200\n Epoch 1244/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3698 - acc: 0.8472 - val_loss: 0.3747 - val_acc: 0.8200\n Epoch 1245/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3680 - acc: 0.8489 - val_loss: 0.3744 - val_acc: 0.8200\n Epoch 1246/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3648 - acc: 0.8511 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 1247/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3658 - acc: 0.8462 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 1248/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3696 - acc: 0.8461 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 1249/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3700 - acc: 0.8489 - val_loss: 0.3735 - val_acc: 0.8200\n Epoch 1250/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3676 - acc: 0.8481 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1251/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3712 - acc: 0.8450 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 1252/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3710 - acc: 0.8471 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 1253/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3675 - acc: 0.8507 - val_loss: 0.3720 - val_acc: 0.8213\n Epoch 1254/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3667 - acc: 0.8481 - val_loss: 0.3725 - val_acc: 0.8213\n Epoch 1255/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3698 - acc: 0.8438 - val_loss: 0.3728 - val_acc: 0.8213\n Epoch 1256/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3678 - acc: 0.8476 - val_loss: 0.3721 - val_acc: 0.8225\n Epoch 1257/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3679 - acc: 0.8469 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 1258/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3657 - acc: 0.8469 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 1259/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3633 - acc: 0.8479 - val_loss: 0.3716 - val_acc: 0.8213\n Epoch 1260/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3709 - acc: 0.8447 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 1261/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3718 - acc: 0.8458 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1262/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3692 - acc: 0.8464 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 1263/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3651 - acc: 0.8478 - val_loss: 0.3715 - val_acc: 0.8187\n Epoch 1264/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3749 - acc: 0.8410 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1265/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3691 - acc: 0.8485 - val_loss: 0.3715 - val_acc: 0.8187\n Epoch 1266/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3656 - acc: 0.8512 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1267/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3678 - acc: 0.8475 - val_loss: 0.3703 - val_acc: 0.8237\n Epoch 1268/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3667 - acc: 0.8485 - val_loss: 0.3708 - val_acc: 0.8213\n Epoch 1269/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3666 - acc: 0.8485 - val_loss: 0.3705 - val_acc: 0.8213\n Epoch 1270/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8488 - val_loss: 0.3716 - val_acc: 0.8187\n Epoch 1271/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3678 - acc: 0.8469 - val_loss: 0.3712 - val_acc: 0.8213\n Epoch 1272/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3655 - acc: 0.8476 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 1273/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3702 - acc: 0.8443 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1274/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3701 - acc: 0.8462 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1275/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3716 - acc: 0.8471 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1276/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3715 - acc: 0.8446 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1277/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8479 - val_loss: 0.3726 - val_acc: 0.8225\n Epoch 1278/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3707 - acc: 0.8468 - val_loss: 0.3729 - val_acc: 0.8225\n Epoch 1279/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3722 - acc: 0.8439 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 1280/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3691 - acc: 0.8483 - val_loss: 0.3726 - val_acc: 0.8213\n Epoch 1281/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3725 - acc: 0.8447 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 1282/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3728 - acc: 0.8461 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1283/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3672 - acc: 0.8512 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1284/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3648 - acc: 0.8493 - val_loss: 0.3709 - val_acc: 0.8225\n Epoch 1285/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3654 - acc: 0.8529 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1286/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3731 - acc: 0.8449 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1287/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3677 - acc: 0.8486 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1288/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3717 - acc: 0.8467 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1289/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3666 - acc: 0.8474 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1290/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3699 - acc: 0.8471 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 1291/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3712 - acc: 0.8478 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1292/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3697 - acc: 0.8467 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 1293/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3699 - acc: 0.8469 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1294/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3674 - acc: 0.8488 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1295/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3668 - acc: 0.8497 - val_loss: 0.3711 - val_acc: 0.8187\n Epoch 1296/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3685 - acc: 0.8475 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1297/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3744 - acc: 0.8451 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1298/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3655 - acc: 0.8474 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1299/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3706 - acc: 0.8450 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1300/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3645 - acc: 0.8497 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1301/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3716 - acc: 0.8450 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1302/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3714 - acc: 0.8469 - val_loss: 0.3726 - val_acc: 0.8213\n Epoch 1303/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3653 - acc: 0.8494 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1304/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3717 - acc: 0.8465 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 1305/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3694 - acc: 0.8501 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 1306/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3742 - acc: 0.8454 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1307/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3691 - acc: 0.8460 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1308/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3657 - acc: 0.8494 - val_loss: 0.3716 - val_acc: 0.8187\n Epoch 1309/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3661 - acc: 0.8497 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1310/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3722 - acc: 0.8436 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1311/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3684 - acc: 0.8465 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1312/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3704 - acc: 0.8476 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 1313/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3632 - acc: 0.8508 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 1314/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3686 - acc: 0.8469 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1315/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3712 - acc: 0.8460 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 1316/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3678 - acc: 0.8462 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 1317/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3663 - acc: 0.8497 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1318/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3728 - acc: 0.8429 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1319/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3712 - acc: 0.8458 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 1320/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3671 - acc: 0.8500 - val_loss: 0.3717 - val_acc: 0.8225\n Epoch 1321/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3681 - acc: 0.8461 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1322/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3675 - acc: 0.8462 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1323/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3735 - acc: 0.8436 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 1324/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3699 - acc: 0.8464 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1325/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3690 - acc: 0.8474 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1326/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3696 - acc: 0.8449 - val_loss: 0.3709 - val_acc: 0.8187\n Epoch 1327/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3711 - acc: 0.8438 - val_loss: 0.3697 - val_acc: 0.8187\n Epoch 1328/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3742 - acc: 0.8454 - val_loss: 0.3713 - val_acc: 0.8187\n Epoch 1329/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3702 - acc: 0.8464 - val_loss: 0.3714 - val_acc: 0.8187\n Epoch 1330/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3682 - acc: 0.8475 - val_loss: 0.3711 - val_acc: 0.8187\n Epoch 1331/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3694 - acc: 0.8449 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1332/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3657 - acc: 0.8489 - val_loss: 0.3718 - val_acc: 0.8187\n Epoch 1333/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3664 - acc: 0.8504 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1334/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3722 - acc: 0.8415 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1335/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3677 - acc: 0.8465 - val_loss: 0.3712 - val_acc: 0.8187\n Epoch 1336/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3755 - acc: 0.8454 - val_loss: 0.3716 - val_acc: 0.8187\n Epoch 1337/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3708 - acc: 0.8461 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 1338/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3682 - acc: 0.8472 - val_loss: 0.3718 - val_acc: 0.8187\n Epoch 1339/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3723 - acc: 0.8497 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1340/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3657 - acc: 0.8497 - val_loss: 0.3709 - val_acc: 0.8213\n Epoch 1341/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3705 - acc: 0.8458 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 1342/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3700 - acc: 0.8464 - val_loss: 0.3716 - val_acc: 0.8213\n Epoch 1343/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3682 - acc: 0.8472 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1344/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3740 - acc: 0.8461 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1345/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3708 - acc: 0.8462 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1346/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3722 - acc: 0.8425 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 1347/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3676 - acc: 0.8454 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 1348/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3698 - acc: 0.8460 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 1349/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3720 - acc: 0.8426 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1350/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3704 - acc: 0.8450 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1351/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3760 - acc: 0.8440 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1352/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3682 - acc: 0.8490 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1353/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8506 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1354/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3713 - acc: 0.8460 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 1355/10000\n 7200/7200 [==============================] - 1s 75us/step - loss: 0.3675 - acc: 0.8462 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1356/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3663 - acc: 0.8483 - val_loss: 0.3716 - val_acc: 0.8187\n Epoch 1357/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3684 - acc: 0.8489 - val_loss: 0.3715 - val_acc: 0.8187\n Epoch 1358/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3695 - acc: 0.8476 - val_loss: 0.3714 - val_acc: 0.8187\n Epoch 1359/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3677 - acc: 0.8478 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 1360/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3640 - acc: 0.8476 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1361/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3666 - acc: 0.8489 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 1362/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3692 - acc: 0.8478 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 1363/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3657 - acc: 0.8518 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 1364/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3637 - acc: 0.8519 - val_loss: 0.3715 - val_acc: 0.8187\n Epoch 1365/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3669 - acc: 0.8478 - val_loss: 0.3712 - val_acc: 0.8187\n Epoch 1366/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3658 - acc: 0.8476 - val_loss: 0.3706 - val_acc: 0.8187\n Epoch 1367/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3667 - acc: 0.8478 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1368/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3697 - acc: 0.8450 - val_loss: 0.3714 - val_acc: 0.8187\n Epoch 1369/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3735 - acc: 0.8465 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1370/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3720 - acc: 0.8458 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1371/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3688 - acc: 0.8462 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 1372/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3657 - acc: 0.8506 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1373/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3732 - acc: 0.8451 - val_loss: 0.3717 - val_acc: 0.8213\n Epoch 1374/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3691 - acc: 0.8492 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1375/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3672 - acc: 0.8504 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1376/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3640 - acc: 0.8489 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1377/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3685 - acc: 0.8528 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1378/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3665 - acc: 0.8492 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1379/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3675 - acc: 0.8474 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1380/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3698 - acc: 0.8481 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1381/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3704 - acc: 0.8449 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 1382/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3710 - acc: 0.8431 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 1383/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3677 - acc: 0.8479 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1384/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3660 - acc: 0.8457 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1385/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3740 - acc: 0.8464 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1386/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3692 - acc: 0.8451 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1387/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3722 - acc: 0.8461 - val_loss: 0.3718 - val_acc: 0.8187\n Epoch 1388/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3692 - acc: 0.8465 - val_loss: 0.3711 - val_acc: 0.8187\n Epoch 1389/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3643 - acc: 0.8472 - val_loss: 0.3714 - val_acc: 0.8187\n Epoch 1390/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3696 - acc: 0.8501 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 1391/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3672 - acc: 0.8468 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 1392/10000\n 7200/7200 [==============================] - 1s 80us/step - loss: 0.3740 - acc: 0.8451 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 1393/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3690 - acc: 0.8464 - val_loss: 0.3711 - val_acc: 0.8213\n Epoch 1394/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3679 - acc: 0.8494 - val_loss: 0.3714 - val_acc: 0.8187\n Epoch 1395/10000\n 7200/7200 [==============================] - ETA: 0s - loss: 0.3701 - acc: 0.848 - 0s 63us/step - loss: 0.3681 - acc: 0.8496 - val_loss: 0.3717 - val_acc: 0.8213\n Epoch 1396/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3645 - acc: 0.8525 - val_loss: 0.3704 - val_acc: 0.8225\n Epoch 1397/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3705 - acc: 0.8460 - val_loss: 0.3702 - val_acc: 0.8237\n Epoch 1398/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3676 - acc: 0.8469 - val_loss: 0.3707 - val_acc: 0.8225\n Epoch 1399/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3697 - acc: 0.8485 - val_loss: 0.3702 - val_acc: 0.8225\n Epoch 1400/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3683 - acc: 0.8478 - val_loss: 0.3709 - val_acc: 0.8187\n Epoch 1401/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3716 - acc: 0.8438 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1402/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3669 - acc: 0.8460 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 1403/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3691 - acc: 0.8486 - val_loss: 0.3716 - val_acc: 0.8187\n Epoch 1404/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3689 - acc: 0.8468 - val_loss: 0.3713 - val_acc: 0.8187\n Epoch 1405/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3688 - acc: 0.8467 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1406/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3710 - acc: 0.8467 - val_loss: 0.3714 - val_acc: 0.8187\n Epoch 1407/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3769 - acc: 0.8458 - val_loss: 0.3710 - val_acc: 0.8187\n Epoch 1408/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3670 - acc: 0.8483 - val_loss: 0.3714 - val_acc: 0.8187\n Epoch 1409/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3705 - acc: 0.8460 - val_loss: 0.3716 - val_acc: 0.8187\n Epoch 1410/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3705 - acc: 0.8481 - val_loss: 0.3710 - val_acc: 0.8213\n Epoch 1411/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3690 - acc: 0.8485 - val_loss: 0.3699 - val_acc: 0.8213\n Epoch 1412/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3706 - acc: 0.8454 - val_loss: 0.3714 - val_acc: 0.8187\n Epoch 1413/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3672 - acc: 0.8433 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1414/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3669 - acc: 0.8488 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1415/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3620 - acc: 0.8504 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1416/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3686 - acc: 0.8486 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1417/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3714 - acc: 0.8453 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 1418/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3650 - acc: 0.8492 - val_loss: 0.3713 - val_acc: 0.8187\n Epoch 1419/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3676 - acc: 0.8476 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1420/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3714 - acc: 0.8481 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1421/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3724 - acc: 0.8444 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1422/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3691 - acc: 0.8468 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 1423/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3663 - acc: 0.8506 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1424/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3713 - acc: 0.8465 - val_loss: 0.3716 - val_acc: 0.8187\n Epoch 1425/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3718 - acc: 0.8456 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 1426/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3707 - acc: 0.8472 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1427/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3706 - acc: 0.8504 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1428/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3727 - acc: 0.8449 - val_loss: 0.3715 - val_acc: 0.8187\n Epoch 1429/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3722 - acc: 0.8481 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1430/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3698 - acc: 0.8483 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1431/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3698 - acc: 0.8474 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1432/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3652 - acc: 0.8501 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1433/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3661 - acc: 0.8489 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1434/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3741 - acc: 0.8407 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1435/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3695 - acc: 0.8468 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1436/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3701 - acc: 0.8458 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1437/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3720 - acc: 0.8456 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1438/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3672 - acc: 0.8497 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1439/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3707 - acc: 0.8492 - val_loss: 0.3713 - val_acc: 0.8225\n Epoch 1440/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3715 - acc: 0.8453 - val_loss: 0.3719 - val_acc: 0.8225\n Epoch 1441/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3719 - acc: 0.8467 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1442/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3714 - acc: 0.8440 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1443/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3741 - acc: 0.8456 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1444/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3672 - acc: 0.8485 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1445/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3743 - acc: 0.8474 - val_loss: 0.3730 - val_acc: 0.8213\n Epoch 1446/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3683 - acc: 0.8456 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 1447/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3659 - acc: 0.8490 - val_loss: 0.3733 - val_acc: 0.8237\n Epoch 1448/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3634 - acc: 0.8514 - val_loss: 0.3728 - val_acc: 0.8213\n Epoch 1449/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3679 - acc: 0.8462 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 1450/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3673 - acc: 0.8461 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1451/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3740 - acc: 0.8444 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1452/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3723 - acc: 0.8467 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 1453/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8468 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1454/10000\n 7200/7200 [==============================] - 1s 84us/step - loss: 0.3686 - acc: 0.8489 - val_loss: 0.3716 - val_acc: 0.8213\n Epoch 1455/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3753 - acc: 0.8440 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 1456/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3753 - acc: 0.8432 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 1457/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3666 - acc: 0.8443 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1458/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3663 - acc: 0.8481 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1459/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3695 - acc: 0.8481 - val_loss: 0.3718 - val_acc: 0.8187\n Epoch 1460/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3709 - acc: 0.8460 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1461/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3710 - acc: 0.8489 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1462/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3680 - acc: 0.8492 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 1463/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3732 - acc: 0.8442 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 1464/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3731 - acc: 0.8439 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1465/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3712 - acc: 0.8492 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1466/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3616 - acc: 0.8478 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1467/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3740 - acc: 0.8467 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1468/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3664 - acc: 0.8488 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1469/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3645 - acc: 0.8493 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 1470/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3661 - acc: 0.8492 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1471/10000\n 7200/7200 [==============================] - 1s 84us/step - loss: 0.3758 - acc: 0.8440 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1472/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3745 - acc: 0.8474 - val_loss: 0.3713 - val_acc: 0.8187\n Epoch 1473/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3693 - acc: 0.8504 - val_loss: 0.3716 - val_acc: 0.8187\n Epoch 1474/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3694 - acc: 0.8456 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1475/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3679 - acc: 0.8462 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1476/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3715 - acc: 0.8443 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 1477/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3681 - acc: 0.8446 - val_loss: 0.3716 - val_acc: 0.8187\n Epoch 1478/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3710 - acc: 0.8478 - val_loss: 0.3716 - val_acc: 0.8187\n Epoch 1479/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3686 - acc: 0.8488 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1480/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3713 - acc: 0.8438 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 1481/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3712 - acc: 0.8503 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1482/10000\n 7200/7200 [==============================] - ETA: 0s - loss: 0.3675 - acc: 0.848 - 0s 69us/step - loss: 0.3675 - acc: 0.8475 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 1483/10000\n 7200/7200 [==============================] - 1s 83us/step - loss: 0.3692 - acc: 0.8492 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1484/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3673 - acc: 0.8461 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1485/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3706 - acc: 0.8458 - val_loss: 0.3714 - val_acc: 0.8187\n Epoch 1486/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3693 - acc: 0.8474 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1487/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3678 - acc: 0.8447 - val_loss: 0.3715 - val_acc: 0.8187\n Epoch 1488/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3679 - acc: 0.8460 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1489/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3667 - acc: 0.8457 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1490/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3713 - acc: 0.8479 - val_loss: 0.3718 - val_acc: 0.8187\n Epoch 1491/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3742 - acc: 0.8449 - val_loss: 0.3709 - val_acc: 0.8213\n Epoch 1492/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3650 - acc: 0.8483 - val_loss: 0.3705 - val_acc: 0.8200\n Epoch 1493/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3660 - acc: 0.8454 - val_loss: 0.3700 - val_acc: 0.8225\n Epoch 1494/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3680 - acc: 0.8472 - val_loss: 0.3707 - val_acc: 0.8213\n Epoch 1495/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3742 - acc: 0.8432 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1496/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3766 - acc: 0.8447 - val_loss: 0.3716 - val_acc: 0.8187\n Epoch 1497/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3699 - acc: 0.8469 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1498/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3662 - acc: 0.8496 - val_loss: 0.3718 - val_acc: 0.8187\n Epoch 1499/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3664 - acc: 0.8492 - val_loss: 0.3716 - val_acc: 0.8187\n Epoch 1500/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3676 - acc: 0.8488 - val_loss: 0.3718 - val_acc: 0.8187\n Epoch 1501/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3687 - acc: 0.8458 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 1502/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3686 - acc: 0.8504 - val_loss: 0.3713 - val_acc: 0.8213\n Epoch 1503/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3714 - acc: 0.8454 - val_loss: 0.3719 - val_acc: 0.8213\n Epoch 1504/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3667 - acc: 0.8506 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 1505/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3696 - acc: 0.8458 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1506/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3697 - acc: 0.8515 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 1507/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3632 - acc: 0.8496 - val_loss: 0.3713 - val_acc: 0.8225\n Epoch 1508/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3707 - acc: 0.8499 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1509/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3736 - acc: 0.8451 - val_loss: 0.3713 - val_acc: 0.8225\n Epoch 1510/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3646 - acc: 0.8496 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1511/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3719 - acc: 0.8501 - val_loss: 0.3712 - val_acc: 0.8237\n Epoch 1512/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3692 - acc: 0.8488 - val_loss: 0.3722 - val_acc: 0.8213\n Epoch 1513/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3677 - acc: 0.8493 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 1514/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3714 - acc: 0.8444 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1515/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3688 - acc: 0.8483 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 1516/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3647 - acc: 0.8486 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1517/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3669 - acc: 0.8483 - val_loss: 0.3717 - val_acc: 0.8225\n Epoch 1518/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3716 - acc: 0.8467 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 1519/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3717 - acc: 0.8444 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1520/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3691 - acc: 0.8494 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1521/10000\n 7200/7200 [==============================] - 1s 85us/step - loss: 0.3671 - acc: 0.8464 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1522/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3716 - acc: 0.8435 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1523/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3682 - acc: 0.8472 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1524/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3697 - acc: 0.8497 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 1525/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3669 - acc: 0.8482 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 1526/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3686 - acc: 0.8462 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 1527/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3679 - acc: 0.8476 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 1528/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3715 - acc: 0.8449 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1529/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3716 - acc: 0.8438 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 1530/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3603 - acc: 0.8529 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 1531/10000\n 7200/7200 [==============================] - 1s 88us/step - loss: 0.3713 - acc: 0.8469 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1532/10000\n 7200/7200 [==============================] - 1s 90us/step - loss: 0.3706 - acc: 0.8457 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 1533/10000\n 7200/7200 [==============================] - 1s 91us/step - loss: 0.3737 - acc: 0.8435 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 1534/10000\n 7200/7200 [==============================] - 1s 75us/step - loss: 0.3655 - acc: 0.8481 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1535/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3666 - acc: 0.8494 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 1536/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3673 - acc: 0.8490 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1537/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3704 - acc: 0.8492 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1538/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3714 - acc: 0.8456 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 1539/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3660 - acc: 0.8462 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1540/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3669 - acc: 0.8489 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1541/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3718 - acc: 0.8447 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 1542/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3701 - acc: 0.8476 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 1543/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3678 - acc: 0.8485 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1544/10000\n 7200/7200 [==============================] - 0s 55us/step - loss: 0.3700 - acc: 0.8442 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 1545/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3679 - acc: 0.8476 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1546/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3656 - acc: 0.8483 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 1547/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3720 - acc: 0.8461 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 1548/10000\n 7200/7200 [==============================] - 1s 84us/step - loss: 0.3672 - acc: 0.8483 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 1549/10000\n 7200/7200 [==============================] - 1s 83us/step - loss: 0.3665 - acc: 0.8485 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1550/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3678 - acc: 0.8446 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1551/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3735 - acc: 0.8444 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 1552/10000\n 7200/7200 [==============================] - 1s 90us/step - loss: 0.3695 - acc: 0.8482 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 1553/10000\n 7200/7200 [==============================] - 1s 82us/step - loss: 0.3665 - acc: 0.8501 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 1554/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3696 - acc: 0.8462 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1555/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3688 - acc: 0.8489 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1556/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3652 - acc: 0.8496 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 1557/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3731 - acc: 0.8446 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1558/10000\n 7200/7200 [==============================] - 0s 56us/step - loss: 0.3693 - acc: 0.8471 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 1559/10000\n 7200/7200 [==============================] - 0s 56us/step - loss: 0.3666 - acc: 0.8499 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 1560/10000\n 7200/7200 [==============================] - 0s 55us/step - loss: 0.3707 - acc: 0.8465 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 1561/10000\n 7200/7200 [==============================] - 0s 56us/step - loss: 0.3675 - acc: 0.8486 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 1562/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3661 - acc: 0.8503 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 1563/10000\n 7200/7200 [==============================] - 0s 55us/step - loss: 0.3697 - acc: 0.8451 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1564/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3714 - acc: 0.8443 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1565/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3749 - acc: 0.8447 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 1566/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3673 - acc: 0.8461 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 1567/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3665 - acc: 0.8476 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 1568/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3693 - acc: 0.8474 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1569/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3733 - acc: 0.8443 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1570/10000\n 7200/7200 [==============================] - 0s 56us/step - loss: 0.3671 - acc: 0.8478 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 1571/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3653 - acc: 0.8496 - val_loss: 0.3714 - val_acc: 0.8225\n Epoch 1572/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3691 - acc: 0.8444 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 1573/10000\n 7200/7200 [==============================] - 1s 100us/step - loss: 0.3686 - acc: 0.8481 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1574/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3711 - acc: 0.8438 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1575/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3679 - acc: 0.8464 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1576/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3721 - acc: 0.8446 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 1577/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3651 - acc: 0.8482 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 1578/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3694 - acc: 0.8464 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 1579/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3727 - acc: 0.8461 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 1580/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3660 - acc: 0.8479 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 1581/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3659 - acc: 0.8482 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 1582/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3692 - acc: 0.8454 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1583/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3663 - acc: 0.8440 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 1584/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3691 - acc: 0.8460 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1585/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3730 - acc: 0.8443 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1586/10000\n 7200/7200 [==============================] - 0s 56us/step - loss: 0.3708 - acc: 0.8474 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 1587/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3724 - acc: 0.8479 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 1588/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3699 - acc: 0.8457 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1589/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3687 - acc: 0.8479 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 1590/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3703 - acc: 0.8471 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1591/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3743 - acc: 0.8424 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 1592/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3690 - acc: 0.8446 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1593/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3686 - acc: 0.8482 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1594/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3664 - acc: 0.8475 - val_loss: 0.3716 - val_acc: 0.8187\n Epoch 1595/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3706 - acc: 0.8471 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 1596/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3701 - acc: 0.8488 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1597/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3740 - acc: 0.8428 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1598/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3724 - acc: 0.8464 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1599/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3692 - acc: 0.8469 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1600/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3671 - acc: 0.8494 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1601/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3678 - acc: 0.8506 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1602/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3666 - acc: 0.8471 - val_loss: 0.3708 - val_acc: 0.8187\n Epoch 1603/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3677 - acc: 0.8479 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1604/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3708 - acc: 0.8457 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1605/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3689 - acc: 0.8476 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1606/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3681 - acc: 0.8497 - val_loss: 0.3715 - val_acc: 0.8213\n Epoch 1607/10000\n 7200/7200 [==============================] - 1s 94us/step - loss: 0.3657 - acc: 0.8475 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1608/10000\n 7200/7200 [==============================] - 1s 99us/step - loss: 0.3718 - acc: 0.8438 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1609/10000\n 7200/7200 [==============================] - 1s 84us/step - loss: 0.3680 - acc: 0.8488 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1610/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3697 - acc: 0.8482 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1611/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3702 - acc: 0.8464 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1612/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3680 - acc: 0.8483 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 1613/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3680 - acc: 0.8489 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1614/10000\n 7200/7200 [==============================] - 1s 87us/step - loss: 0.3720 - acc: 0.8469 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1615/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3698 - acc: 0.8472 - val_loss: 0.3718 - val_acc: 0.8187\n Epoch 1616/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3720 - acc: 0.8465 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1617/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3647 - acc: 0.8514 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1618/10000\n 7200/7200 [==============================] - 1s 86us/step - loss: 0.3672 - acc: 0.8474 - val_loss: 0.3709 - val_acc: 0.8237\n Epoch 1619/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3676 - acc: 0.8467 - val_loss: 0.3705 - val_acc: 0.8213\n Epoch 1620/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3675 - acc: 0.8450 - val_loss: 0.3706 - val_acc: 0.8213\n Epoch 1621/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3727 - acc: 0.8433 - val_loss: 0.3707 - val_acc: 0.8213\n Epoch 1622/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3712 - acc: 0.8465 - val_loss: 0.3714 - val_acc: 0.8187\n Epoch 1623/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3670 - acc: 0.8488 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 1624/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3685 - acc: 0.8488 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1625/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3665 - acc: 0.8492 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1626/10000\n 7200/7200 [==============================] - 1s 138us/step - loss: 0.3701 - acc: 0.8469 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 1627/10000\n 7200/7200 [==============================] - 1s 142us/step - loss: 0.3693 - acc: 0.8446 - val_loss: 0.3720 - val_acc: 0.8213\n Epoch 1628/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3722 - acc: 0.8458 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 1629/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3649 - acc: 0.8489 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1630/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3709 - acc: 0.8490 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 1631/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3686 - acc: 0.8464 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1632/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3721 - acc: 0.8479 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1633/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3708 - acc: 0.8439 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 1634/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3654 - acc: 0.8507 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 1635/10000\n 7200/7200 [==============================] - 0s 56us/step - loss: 0.3683 - acc: 0.8468 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1636/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3661 - acc: 0.8478 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1637/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3653 - acc: 0.8494 - val_loss: 0.3715 - val_acc: 0.8187\n Epoch 1638/10000\n 7200/7200 [==============================] - 0s 54us/step - loss: 0.3710 - acc: 0.8453 - val_loss: 0.3714 - val_acc: 0.8187\n Epoch 1639/10000\n 7200/7200 [==============================] - 0s 55us/step - loss: 0.3692 - acc: 0.8476 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1640/10000\n 7200/7200 [==============================] - 0s 54us/step - loss: 0.3732 - acc: 0.8462 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 1641/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3683 - acc: 0.8479 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1642/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3736 - acc: 0.8481 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 1643/10000\n 7200/7200 [==============================] - 0s 55us/step - loss: 0.3711 - acc: 0.8475 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 1644/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3688 - acc: 0.8464 - val_loss: 0.3728 - val_acc: 0.8213\n Epoch 1645/10000\n 7200/7200 [==============================] - 0s 56us/step - loss: 0.3681 - acc: 0.8472 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 1646/10000\n 7200/7200 [==============================] - 0s 56us/step - loss: 0.3674 - acc: 0.8479 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 1647/10000\n 7200/7200 [==============================] - 0s 56us/step - loss: 0.3705 - acc: 0.8461 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1648/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3728 - acc: 0.8462 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 1649/10000\n 7200/7200 [==============================] - 0s 55us/step - loss: 0.3732 - acc: 0.8443 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1650/10000\n 7200/7200 [==============================] - 0s 56us/step - loss: 0.3741 - acc: 0.8446 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1651/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3758 - acc: 0.8433 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 1652/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3698 - acc: 0.8462 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1653/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3654 - acc: 0.8481 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1654/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3684 - acc: 0.8499 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1655/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3691 - acc: 0.8462 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1656/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3640 - acc: 0.8514 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 1657/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3675 - acc: 0.8492 - val_loss: 0.3715 - val_acc: 0.8187\n Epoch 1658/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3643 - acc: 0.8482 - val_loss: 0.3712 - val_acc: 0.8200\n Epoch 1659/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3721 - acc: 0.8454 - val_loss: 0.3715 - val_acc: 0.8187\n Epoch 1660/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3747 - acc: 0.8426 - val_loss: 0.3714 - val_acc: 0.8187\n Epoch 1661/10000\n 7200/7200 [==============================] - 0s 56us/step - loss: 0.3684 - acc: 0.8468 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1662/10000\n 7200/7200 [==============================] - 0s 54us/step - loss: 0.3760 - acc: 0.8440 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1663/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3638 - acc: 0.8512 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 1664/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3711 - acc: 0.8464 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1665/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3700 - acc: 0.8475 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1666/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3694 - acc: 0.8458 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 1667/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3674 - acc: 0.8533 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 1668/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3723 - acc: 0.8456 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 1669/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3645 - acc: 0.8503 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 1670/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3697 - acc: 0.8465 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1671/10000\n 7200/7200 [==============================] - 0s 56us/step - loss: 0.3646 - acc: 0.8474 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 1672/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3631 - acc: 0.8486 - val_loss: 0.3718 - val_acc: 0.8187\n Epoch 1673/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3735 - acc: 0.8453 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 1674/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3665 - acc: 0.8493 - val_loss: 0.3714 - val_acc: 0.8187\n Epoch 1675/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3648 - acc: 0.8508 - val_loss: 0.3704 - val_acc: 0.8237\n Epoch 1676/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3696 - acc: 0.8481 - val_loss: 0.3707 - val_acc: 0.8225\n Epoch 1677/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3704 - acc: 0.8453 - val_loss: 0.3707 - val_acc: 0.8187\n Epoch 1678/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3717 - acc: 0.8444 - val_loss: 0.3704 - val_acc: 0.8187\n Epoch 1679/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3673 - acc: 0.8456 - val_loss: 0.3715 - val_acc: 0.8187\n Epoch 1680/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3752 - acc: 0.8458 - val_loss: 0.3718 - val_acc: 0.8187\n Epoch 1681/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3728 - acc: 0.8456 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1682/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3720 - acc: 0.8432 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1683/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3636 - acc: 0.8511 - val_loss: 0.3713 - val_acc: 0.8225\n Epoch 1684/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3707 - acc: 0.8499 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1685/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3758 - acc: 0.8428 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 1686/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3698 - acc: 0.8457 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1687/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3655 - acc: 0.8490 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1688/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3682 - acc: 0.8471 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1689/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3667 - acc: 0.8461 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1690/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3672 - acc: 0.8490 - val_loss: 0.3718 - val_acc: 0.8237\n Epoch 1691/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3668 - acc: 0.8457 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 1692/10000\n 7200/7200 [==============================] - 1s 93us/step - loss: 0.3691 - acc: 0.8456 - val_loss: 0.3719 - val_acc: 0.8225\n Epoch 1693/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3701 - acc: 0.8458 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 1694/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3679 - acc: 0.8476 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 1695/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3726 - acc: 0.8457 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 1696/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3742 - acc: 0.8457 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1697/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3687 - acc: 0.8456 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1698/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3668 - acc: 0.8479 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1699/10000\n 7200/7200 [==============================] - 0s 56us/step - loss: 0.3729 - acc: 0.8436 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1700/10000\n 7200/7200 [==============================] - 1s 84us/step - loss: 0.3703 - acc: 0.8471 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 1701/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3650 - acc: 0.8479 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 1702/10000\n 7200/7200 [==============================] - 0s 55us/step - loss: 0.3711 - acc: 0.8471 - val_loss: 0.3710 - val_acc: 0.8200\n Epoch 1703/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3698 - acc: 0.8458 - val_loss: 0.3716 - val_acc: 0.8200\n Epoch 1704/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3701 - acc: 0.8447 - val_loss: 0.3711 - val_acc: 0.8213\n Epoch 1705/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3736 - acc: 0.8443 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1706/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3703 - acc: 0.8461 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 1707/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3691 - acc: 0.8485 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 1708/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3676 - acc: 0.8489 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1709/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3699 - acc: 0.8474 - val_loss: 0.3706 - val_acc: 0.8225\n Epoch 1710/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3635 - acc: 0.8503 - val_loss: 0.3703 - val_acc: 0.8250\n Epoch 1711/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3753 - acc: 0.8457 - val_loss: 0.3704 - val_acc: 0.8237\n Epoch 1712/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3630 - acc: 0.8499 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 1713/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3688 - acc: 0.8469 - val_loss: 0.3714 - val_acc: 0.8250\n Epoch 1714/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3673 - acc: 0.8483 - val_loss: 0.3726 - val_acc: 0.8213\n Epoch 1715/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3686 - acc: 0.8490 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1716/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3715 - acc: 0.8472 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1717/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3701 - acc: 0.8481 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1718/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3682 - acc: 0.8488 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1719/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3724 - acc: 0.8436 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1720/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3701 - acc: 0.8468 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 1721/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3676 - acc: 0.8493 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1722/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3696 - acc: 0.8454 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 1723/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3620 - acc: 0.8494 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1724/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3654 - acc: 0.8488 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 1725/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3715 - acc: 0.8474 - val_loss: 0.3727 - val_acc: 0.8213\n Epoch 1726/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3651 - acc: 0.8446 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 1727/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3710 - acc: 0.8464 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1728/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3636 - acc: 0.8482 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 1729/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3697 - acc: 0.8472 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 1730/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3676 - acc: 0.8465 - val_loss: 0.3716 - val_acc: 0.8187\n Epoch 1731/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3681 - acc: 0.8471 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 1732/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3707 - acc: 0.8476 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 1733/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3682 - acc: 0.8494 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 1734/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3721 - acc: 0.8471 - val_loss: 0.3725 - val_acc: 0.8225\n Epoch 1735/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3709 - acc: 0.8451 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 1736/10000\n 7200/7200 [==============================] - 0s 55us/step - loss: 0.3667 - acc: 0.8465 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1737/10000\n 7200/7200 [==============================] - 0s 56us/step - loss: 0.3678 - acc: 0.8486 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1738/10000\n 7200/7200 [==============================] - 1s 82us/step - loss: 0.3661 - acc: 0.8472 - val_loss: 0.3723 - val_acc: 0.8213\n Epoch 1739/10000\n 7200/7200 [==============================] - 1s 97us/step - loss: 0.3693 - acc: 0.8492 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 1740/10000\n 7200/7200 [==============================] - 1s 94us/step - loss: 0.3654 - acc: 0.8450 - val_loss: 0.3713 - val_acc: 0.8200\n Epoch 1741/10000\n 7200/7200 [==============================] - 1s 91us/step - loss: 0.3708 - acc: 0.8472 - val_loss: 0.3714 - val_acc: 0.8200\n Epoch 1742/10000\n 7200/7200 [==============================] - 1s 93us/step - loss: 0.3654 - acc: 0.8515 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 1743/10000\n 7200/7200 [==============================] - 1s 84us/step - loss: 0.3689 - acc: 0.8438 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1744/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3661 - acc: 0.8512 - val_loss: 0.3715 - val_acc: 0.8187\n Epoch 1745/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3699 - acc: 0.8456 - val_loss: 0.3711 - val_acc: 0.8187\n Epoch 1746/10000\n 7200/7200 [==============================] - 1s 85us/step - loss: 0.3694 - acc: 0.8485 - val_loss: 0.3709 - val_acc: 0.8187\n Epoch 1747/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3695 - acc: 0.8483 - val_loss: 0.3715 - val_acc: 0.8187\n Epoch 1748/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3715 - acc: 0.8469 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1749/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3713 - acc: 0.8458 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1750/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3664 - acc: 0.8472 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1751/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3710 - acc: 0.8461 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1752/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3697 - acc: 0.8482 - val_loss: 0.3723 - val_acc: 0.8237\n Epoch 1753/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3690 - acc: 0.8511 - val_loss: 0.3727 - val_acc: 0.8237\n Epoch 1754/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3695 - acc: 0.8461 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 1755/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3674 - acc: 0.8501 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 1756/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3686 - acc: 0.8462 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1757/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3686 - acc: 0.8464 - val_loss: 0.3730 - val_acc: 0.8200\n Epoch 1758/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3728 - acc: 0.8456 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 1759/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3659 - acc: 0.8474 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 1760/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3695 - acc: 0.8476 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 1761/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3732 - acc: 0.8462 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 1762/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3757 - acc: 0.8447 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 1763/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3632 - acc: 0.8508 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 1764/10000\n 7200/7200 [==============================] - 1s 90us/step - loss: 0.3729 - acc: 0.8458 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 1765/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3639 - acc: 0.8525 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1766/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3673 - acc: 0.8493 - val_loss: 0.3723 - val_acc: 0.8225\n Epoch 1767/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3737 - acc: 0.8446 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 1768/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3711 - acc: 0.8447 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1769/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3696 - acc: 0.8465 - val_loss: 0.3715 - val_acc: 0.8213\n Epoch 1770/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3714 - acc: 0.8468 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1771/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3677 - acc: 0.8478 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1772/10000\n 7200/7200 [==============================] - 1s 82us/step - loss: 0.3684 - acc: 0.8469 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1773/10000\n 7200/7200 [==============================] - 1s 99us/step - loss: 0.3674 - acc: 0.8503 - val_loss: 0.3714 - val_acc: 0.8187\n Epoch 1774/10000\n 7200/7200 [==============================] - 1s 80us/step - loss: 0.3667 - acc: 0.8511 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 1775/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3664 - acc: 0.8472 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1776/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3726 - acc: 0.8444 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1777/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3646 - acc: 0.8481 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1778/10000\n 7200/7200 [==============================] - 1s 84us/step - loss: 0.3690 - acc: 0.8492 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1779/10000\n 7200/7200 [==============================] - 1s 82us/step - loss: 0.3717 - acc: 0.8453 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1780/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3687 - acc: 0.8468 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1781/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3668 - acc: 0.8488 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1782/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3693 - acc: 0.8494 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 1783/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3658 - acc: 0.8488 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1784/10000\n 7200/7200 [==============================] - 1s 94us/step - loss: 0.3698 - acc: 0.8433 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1785/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3658 - acc: 0.8501 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 1786/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3674 - acc: 0.8440 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 1787/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3701 - acc: 0.8469 - val_loss: 0.3706 - val_acc: 0.8250\n Epoch 1788/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3719 - acc: 0.8419 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1789/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3719 - acc: 0.8497 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1790/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3686 - acc: 0.8469 - val_loss: 0.3718 - val_acc: 0.8187\n Epoch 1791/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3707 - acc: 0.8471 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1792/10000\n 7200/7200 [==============================] - 1s 99us/step - loss: 0.3697 - acc: 0.8464 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1793/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3703 - acc: 0.8468 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 1794/10000\n 7200/7200 [==============================] - 1s 75us/step - loss: 0.3686 - acc: 0.8453 - val_loss: 0.3716 - val_acc: 0.8213\n Epoch 1795/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3662 - acc: 0.8475 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 1796/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3675 - acc: 0.8483 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 1797/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3672 - acc: 0.8468 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1798/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3689 - acc: 0.8468 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 1799/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3692 - acc: 0.8483 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 1800/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3715 - acc: 0.8476 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1801/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3693 - acc: 0.8457 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 1802/10000\n 7200/7200 [==============================] - 1s 104us/step - loss: 0.3694 - acc: 0.8472 - val_loss: 0.3732 - val_acc: 0.8213\n Epoch 1803/10000\n 7200/7200 [==============================] - 1s 80us/step - loss: 0.3697 - acc: 0.8469 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1804/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3656 - acc: 0.8489 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1805/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3705 - acc: 0.8474 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 1806/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3686 - acc: 0.8478 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 1807/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3728 - acc: 0.8475 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1808/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3676 - acc: 0.8497 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 1809/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3702 - acc: 0.8471 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 1810/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3727 - acc: 0.8449 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 1811/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3671 - acc: 0.8486 - val_loss: 0.3721 - val_acc: 0.8225\n Epoch 1812/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3711 - acc: 0.8457 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 1813/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3683 - acc: 0.8468 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 1814/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3683 - acc: 0.8469 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 1815/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3653 - acc: 0.8475 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 1816/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3651 - acc: 0.8506 - val_loss: 0.3731 - val_acc: 0.8213\n Epoch 1817/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3724 - acc: 0.8447 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 1818/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3690 - acc: 0.8482 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 1819/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3679 - acc: 0.8479 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1820/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3721 - acc: 0.8471 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 1821/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3727 - acc: 0.8467 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1822/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3708 - acc: 0.8429 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1823/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3704 - acc: 0.8497 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 1824/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3651 - acc: 0.8468 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 1825/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3695 - acc: 0.8488 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1826/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3673 - acc: 0.8465 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1827/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3661 - acc: 0.8486 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 1828/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3703 - acc: 0.8475 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 1829/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3665 - acc: 0.8489 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 1830/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3682 - acc: 0.8474 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 1831/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3693 - acc: 0.8468 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1832/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3759 - acc: 0.8433 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1833/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3672 - acc: 0.8482 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1834/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3707 - acc: 0.8446 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1835/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3745 - acc: 0.8440 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 1836/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3695 - acc: 0.8501 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 1837/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3667 - acc: 0.8478 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 1838/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3697 - acc: 0.8464 - val_loss: 0.3741 - val_acc: 0.8200\n Epoch 1839/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3663 - acc: 0.8490 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 1840/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3684 - acc: 0.8488 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 1841/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3670 - acc: 0.8494 - val_loss: 0.3737 - val_acc: 0.8200\n Epoch 1842/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3702 - acc: 0.8472 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 1843/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3733 - acc: 0.8447 - val_loss: 0.3744 - val_acc: 0.8200\n Epoch 1844/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3669 - acc: 0.8503 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1845/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3658 - acc: 0.8489 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1846/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3685 - acc: 0.8476 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 1847/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3683 - acc: 0.8476 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 1848/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3715 - acc: 0.8462 - val_loss: 0.3728 - val_acc: 0.8225\n Epoch 1849/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3684 - acc: 0.8471 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 1850/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3656 - acc: 0.8511 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1851/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3702 - acc: 0.8479 - val_loss: 0.3721 - val_acc: 0.8237\n Epoch 1852/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3672 - acc: 0.8475 - val_loss: 0.3719 - val_acc: 0.8225\n Epoch 1853/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3706 - acc: 0.8468 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1854/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3743 - acc: 0.8450 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1855/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3674 - acc: 0.8471 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 1856/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3659 - acc: 0.8511 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1857/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3700 - acc: 0.8492 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 1858/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3701 - acc: 0.8471 - val_loss: 0.3736 - val_acc: 0.8213\n Epoch 1859/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3661 - acc: 0.8482 - val_loss: 0.3742 - val_acc: 0.8200\n Epoch 1860/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3719 - acc: 0.8458 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 1861/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3676 - acc: 0.8453 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 1862/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3664 - acc: 0.8508 - val_loss: 0.3735 - val_acc: 0.8225\n Epoch 1863/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3655 - acc: 0.8483 - val_loss: 0.3724 - val_acc: 0.8225\n Epoch 1864/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3708 - acc: 0.8457 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 1865/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3726 - acc: 0.8464 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 1866/10000\n 7200/7200 [==============================] - 1s 75us/step - loss: 0.3645 - acc: 0.8492 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 1867/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3699 - acc: 0.8472 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 1868/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3738 - acc: 0.8435 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 1869/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3656 - acc: 0.8496 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 1870/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3674 - acc: 0.8475 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 1871/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3742 - acc: 0.8446 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 1872/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3725 - acc: 0.8451 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 1873/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3683 - acc: 0.8497 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 1874/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3701 - acc: 0.8478 - val_loss: 0.3732 - val_acc: 0.8213\n Epoch 1875/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3677 - acc: 0.8497 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1876/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3658 - acc: 0.8499 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 1877/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3652 - acc: 0.8468 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1878/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3653 - acc: 0.8496 - val_loss: 0.3720 - val_acc: 0.8213\n Epoch 1879/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3659 - acc: 0.8469 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 1880/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3645 - acc: 0.8489 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1881/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3714 - acc: 0.8493 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 1882/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3681 - acc: 0.8442 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1883/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3682 - acc: 0.8485 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 1884/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3678 - acc: 0.8468 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 1885/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3675 - acc: 0.8476 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 1886/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3661 - acc: 0.8464 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 1887/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3691 - acc: 0.8478 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 1888/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3709 - acc: 0.8456 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1889/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3689 - acc: 0.8462 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 1890/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3687 - acc: 0.8504 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 1891/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3718 - acc: 0.8440 - val_loss: 0.3727 - val_acc: 0.8213\n Epoch 1892/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3657 - acc: 0.8486 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 1893/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3686 - acc: 0.8457 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1894/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3694 - acc: 0.8456 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 1895/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3684 - acc: 0.8474 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 1896/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3640 - acc: 0.8482 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 1897/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3653 - acc: 0.8479 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 1898/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3663 - acc: 0.8476 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1899/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3647 - acc: 0.8485 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1900/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3761 - acc: 0.8425 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 1901/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3692 - acc: 0.8462 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 1902/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3705 - acc: 0.8476 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 1903/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3678 - acc: 0.8454 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 1904/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3648 - acc: 0.8508 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 1905/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3700 - acc: 0.8457 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 1906/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3676 - acc: 0.8488 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1907/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3692 - acc: 0.8456 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 1908/10000\n 7200/7200 [==============================] - 1s 82us/step - loss: 0.3666 - acc: 0.8489 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1909/10000\n 7200/7200 [==============================] - 1s 87us/step - loss: 0.3711 - acc: 0.8464 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 1910/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3726 - acc: 0.8462 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 1911/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3687 - acc: 0.8486 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 1912/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3693 - acc: 0.8453 - val_loss: 0.3725 - val_acc: 0.8213\n Epoch 1913/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3667 - acc: 0.8478 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1914/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3650 - acc: 0.8489 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 1915/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3737 - acc: 0.8444 - val_loss: 0.3732 - val_acc: 0.8213\n Epoch 1916/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3683 - acc: 0.8489 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 1917/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3681 - acc: 0.8467 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 1918/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3683 - acc: 0.8471 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1919/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3688 - acc: 0.8493 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 1920/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3688 - acc: 0.8454 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 1921/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3715 - acc: 0.8433 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 1922/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3646 - acc: 0.8494 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 1923/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3697 - acc: 0.8453 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1924/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3702 - acc: 0.8444 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 1925/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3755 - acc: 0.8467 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 1926/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3714 - acc: 0.8471 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 1927/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3680 - acc: 0.8476 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 1928/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3695 - acc: 0.8449 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 1929/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3667 - acc: 0.8472 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 1930/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3706 - acc: 0.8439 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 1931/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3680 - acc: 0.8485 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 1932/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3698 - acc: 0.8461 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 1933/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3630 - acc: 0.8506 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1934/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3729 - acc: 0.8449 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 1935/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3677 - acc: 0.8481 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1936/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3692 - acc: 0.8500 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1937/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3678 - acc: 0.8493 - val_loss: 0.3724 - val_acc: 0.8213\n Epoch 1938/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3688 - acc: 0.8483 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1939/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3636 - acc: 0.8465 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1940/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3672 - acc: 0.8483 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1941/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3685 - acc: 0.8468 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 1942/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3687 - acc: 0.8442 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 1943/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3681 - acc: 0.8474 - val_loss: 0.3723 - val_acc: 0.8213\n Epoch 1944/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3661 - acc: 0.8464 - val_loss: 0.3730 - val_acc: 0.8200\n Epoch 1945/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3663 - acc: 0.8469 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 1946/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3699 - acc: 0.8453 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 1947/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3727 - acc: 0.8471 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1948/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3622 - acc: 0.8515 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1949/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3663 - acc: 0.8483 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1950/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3704 - acc: 0.8486 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 1951/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3709 - acc: 0.8468 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 1952/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3672 - acc: 0.8476 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 1953/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3667 - acc: 0.8522 - val_loss: 0.3712 - val_acc: 0.8213\n Epoch 1954/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3680 - acc: 0.8462 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 1955/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3681 - acc: 0.8490 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1956/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3684 - acc: 0.8490 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 1957/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3637 - acc: 0.8506 - val_loss: 0.3724 - val_acc: 0.8213\n Epoch 1958/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3699 - acc: 0.8444 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 1959/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3678 - acc: 0.8492 - val_loss: 0.3722 - val_acc: 0.8213\n Epoch 1960/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3716 - acc: 0.8428 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1961/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3696 - acc: 0.8449 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1962/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3715 - acc: 0.8425 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 1963/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3655 - acc: 0.8497 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 1964/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3606 - acc: 0.8510 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 1965/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3681 - acc: 0.8468 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1966/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3696 - acc: 0.8454 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 1967/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3752 - acc: 0.8419 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 1968/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3696 - acc: 0.8453 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 1969/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3718 - acc: 0.8457 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 1970/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3679 - acc: 0.8483 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 1971/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3634 - acc: 0.8499 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 1972/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3702 - acc: 0.8472 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 1973/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3700 - acc: 0.8471 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1974/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3707 - acc: 0.8475 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 1975/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3710 - acc: 0.8464 - val_loss: 0.3728 - val_acc: 0.8213\n Epoch 1976/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3701 - acc: 0.8444 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 1977/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3702 - acc: 0.8449 - val_loss: 0.3727 - val_acc: 0.8225\n Epoch 1978/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3709 - acc: 0.8478 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 1979/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3705 - acc: 0.8446 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 1980/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3666 - acc: 0.8503 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 1981/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3661 - acc: 0.8483 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 1982/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3683 - acc: 0.8482 - val_loss: 0.3724 - val_acc: 0.8213\n Epoch 1983/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3698 - acc: 0.8469 - val_loss: 0.3735 - val_acc: 0.8225\n Epoch 1984/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3654 - acc: 0.8514 - val_loss: 0.3725 - val_acc: 0.8225\n Epoch 1985/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3658 - acc: 0.8486 - val_loss: 0.3721 - val_acc: 0.8225\n Epoch 1986/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3741 - acc: 0.8457 - val_loss: 0.3735 - val_acc: 0.8200\n Epoch 1987/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3722 - acc: 0.8447 - val_loss: 0.3742 - val_acc: 0.8200\n Epoch 1988/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3740 - acc: 0.8475 - val_loss: 0.3740 - val_acc: 0.8200\n Epoch 1989/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3679 - acc: 0.8490 - val_loss: 0.3715 - val_acc: 0.8275\n Epoch 1990/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3730 - acc: 0.8458 - val_loss: 0.3724 - val_acc: 0.8237\n Epoch 1991/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3662 - acc: 0.8511 - val_loss: 0.3721 - val_acc: 0.8225\n Epoch 1992/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3712 - acc: 0.8474 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 1993/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3674 - acc: 0.8468 - val_loss: 0.3732 - val_acc: 0.8175\n Epoch 1994/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3696 - acc: 0.8481 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 1995/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3662 - acc: 0.8490 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 1996/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3683 - acc: 0.8486 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 1997/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3722 - acc: 0.8449 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 1998/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3669 - acc: 0.8450 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 1999/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3676 - acc: 0.8465 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 2000/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3626 - acc: 0.8507 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2001/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3676 - acc: 0.8478 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 2002/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3701 - acc: 0.8481 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 2003/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3776 - acc: 0.8436 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 2004/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3671 - acc: 0.8479 - val_loss: 0.3718 - val_acc: 0.8213\n Epoch 2005/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3681 - acc: 0.8451 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 2006/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3681 - acc: 0.8478 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 2007/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3688 - acc: 0.8449 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 2008/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3665 - acc: 0.8482 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2009/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3679 - acc: 0.8500 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2010/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3677 - acc: 0.8500 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2011/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3718 - acc: 0.8433 - val_loss: 0.3735 - val_acc: 0.8200\n Epoch 2012/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3671 - acc: 0.8492 - val_loss: 0.3738 - val_acc: 0.8175\n Epoch 2013/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3689 - acc: 0.8474 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 2014/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3742 - acc: 0.8454 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2015/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3721 - acc: 0.8464 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2016/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3783 - acc: 0.8408 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 2017/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3705 - acc: 0.8453 - val_loss: 0.3751 - val_acc: 0.8175\n Epoch 2018/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3681 - acc: 0.8465 - val_loss: 0.3747 - val_acc: 0.8175\n Epoch 2019/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3716 - acc: 0.8451 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2020/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3709 - acc: 0.8469 - val_loss: 0.3741 - val_acc: 0.8175\n Epoch 2021/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3659 - acc: 0.8485 - val_loss: 0.3747 - val_acc: 0.8175\n Epoch 2022/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3718 - acc: 0.8474 - val_loss: 0.3753 - val_acc: 0.8175\n Epoch 2023/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3681 - acc: 0.8479 - val_loss: 0.3735 - val_acc: 0.8175\n Epoch 2024/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3661 - acc: 0.8501 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2025/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3681 - acc: 0.8458 - val_loss: 0.3741 - val_acc: 0.8175\n Epoch 2026/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3663 - acc: 0.8490 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2027/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3690 - acc: 0.8469 - val_loss: 0.3719 - val_acc: 0.8225\n Epoch 2028/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3693 - acc: 0.8508 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 2029/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3654 - acc: 0.8471 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 2030/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3659 - acc: 0.8479 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2031/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3699 - acc: 0.8472 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2032/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3683 - acc: 0.8468 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 2033/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3644 - acc: 0.8506 - val_loss: 0.3717 - val_acc: 0.8213\n Epoch 2034/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3675 - acc: 0.8460 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 2035/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3746 - acc: 0.8440 - val_loss: 0.3740 - val_acc: 0.8175\n Epoch 2036/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3711 - acc: 0.8456 - val_loss: 0.3744 - val_acc: 0.8175\n Epoch 2037/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3722 - acc: 0.8444 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 2038/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3734 - acc: 0.8461 - val_loss: 0.3735 - val_acc: 0.8175\n Epoch 2039/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3708 - acc: 0.8460 - val_loss: 0.3732 - val_acc: 0.8175\n Epoch 2040/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3646 - acc: 0.8511 - val_loss: 0.3726 - val_acc: 0.8175\n Epoch 2041/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3677 - acc: 0.8453 - val_loss: 0.3732 - val_acc: 0.8175\n Epoch 2042/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3681 - acc: 0.8483 - val_loss: 0.3729 - val_acc: 0.8175\n Epoch 2043/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3702 - acc: 0.8439 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2044/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3716 - acc: 0.8442 - val_loss: 0.3745 - val_acc: 0.8175\n Epoch 2045/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3739 - acc: 0.8433 - val_loss: 0.3756 - val_acc: 0.8175\n Epoch 2046/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3632 - acc: 0.8503 - val_loss: 0.3755 - val_acc: 0.8175\n Epoch 2047/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3705 - acc: 0.8462 - val_loss: 0.3748 - val_acc: 0.8175\n Epoch 2048/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3668 - acc: 0.8483 - val_loss: 0.3750 - val_acc: 0.8175\n Epoch 2049/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3720 - acc: 0.8442 - val_loss: 0.3745 - val_acc: 0.8175\n Epoch 2050/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3683 - acc: 0.8464 - val_loss: 0.3750 - val_acc: 0.8175\n Epoch 2051/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3712 - acc: 0.8483 - val_loss: 0.3743 - val_acc: 0.8163\n Epoch 2052/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3664 - acc: 0.8482 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 2053/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3620 - acc: 0.8499 - val_loss: 0.3742 - val_acc: 0.8175\n Epoch 2054/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3668 - acc: 0.8486 - val_loss: 0.3741 - val_acc: 0.8175\n Epoch 2055/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3667 - acc: 0.8492 - val_loss: 0.3737 - val_acc: 0.8175\n Epoch 2056/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3656 - acc: 0.8468 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 2057/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3700 - acc: 0.8485 - val_loss: 0.3729 - val_acc: 0.8225\n Epoch 2058/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3692 - acc: 0.8488 - val_loss: 0.3727 - val_acc: 0.8250\n Epoch 2059/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3743 - acc: 0.8462 - val_loss: 0.3742 - val_acc: 0.8175\n Epoch 2060/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3701 - acc: 0.8454 - val_loss: 0.3737 - val_acc: 0.8225\n Epoch 2061/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3650 - acc: 0.8499 - val_loss: 0.3746 - val_acc: 0.8175\n Epoch 2062/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3745 - acc: 0.8435 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2063/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3675 - acc: 0.8489 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 2064/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3687 - acc: 0.8474 - val_loss: 0.3750 - val_acc: 0.8200\n Epoch 2065/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3649 - acc: 0.8481 - val_loss: 0.3753 - val_acc: 0.8200\n Epoch 2066/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3745 - acc: 0.8461 - val_loss: 0.3757 - val_acc: 0.8200\n Epoch 2067/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3669 - acc: 0.8504 - val_loss: 0.3752 - val_acc: 0.8200\n Epoch 2068/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3679 - acc: 0.8490 - val_loss: 0.3751 - val_acc: 0.8200\n Epoch 2069/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3719 - acc: 0.8464 - val_loss: 0.3752 - val_acc: 0.8200\n Epoch 2070/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3718 - acc: 0.8439 - val_loss: 0.3751 - val_acc: 0.8200\n Epoch 2071/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3735 - acc: 0.8467 - val_loss: 0.3740 - val_acc: 0.8200\n Epoch 2072/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3651 - acc: 0.8490 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 2073/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3701 - acc: 0.8481 - val_loss: 0.3743 - val_acc: 0.8200\n Epoch 2074/10000\n 7200/7200 [==============================] - ETA: 0s - loss: 0.3671 - acc: 0.848 - 0s 62us/step - loss: 0.3662 - acc: 0.8485 - val_loss: 0.3741 - val_acc: 0.8200\n Epoch 2075/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3650 - acc: 0.8506 - val_loss: 0.3737 - val_acc: 0.8200\n Epoch 2076/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8496 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 2077/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3692 - acc: 0.8507 - val_loss: 0.3730 - val_acc: 0.8225\n Epoch 2078/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8483 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 2079/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3711 - acc: 0.8467 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2080/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3719 - acc: 0.8469 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2081/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3709 - acc: 0.8460 - val_loss: 0.3726 - val_acc: 0.8213\n Epoch 2082/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3705 - acc: 0.8451 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 2083/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3699 - acc: 0.8464 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2084/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3730 - acc: 0.8446 - val_loss: 0.3745 - val_acc: 0.8200\n Epoch 2085/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3686 - acc: 0.8454 - val_loss: 0.3743 - val_acc: 0.8200\n Epoch 2086/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3689 - acc: 0.8488 - val_loss: 0.3748 - val_acc: 0.8200\n Epoch 2087/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3699 - acc: 0.8447 - val_loss: 0.3748 - val_acc: 0.8200\n Epoch 2088/10000\n 7200/7200 [==============================] - 1s 75us/step - loss: 0.3677 - acc: 0.8496 - val_loss: 0.3735 - val_acc: 0.8225\n Epoch 2089/10000\n 7200/7200 [==============================] - 1s 87us/step - loss: 0.3698 - acc: 0.8483 - val_loss: 0.3730 - val_acc: 0.8225\n Epoch 2090/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3722 - acc: 0.8462 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 2091/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3679 - acc: 0.8485 - val_loss: 0.3744 - val_acc: 0.8200\n Epoch 2092/10000\n 7200/7200 [==============================] - 1s 75us/step - loss: 0.3643 - acc: 0.8483 - val_loss: 0.3730 - val_acc: 0.8213\n Epoch 2093/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3684 - acc: 0.8469 - val_loss: 0.3753 - val_acc: 0.8175\n Epoch 2094/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3717 - acc: 0.8450 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 2095/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3695 - acc: 0.8429 - val_loss: 0.3760 - val_acc: 0.8175\n Epoch 2096/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3676 - acc: 0.8467 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 2097/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3639 - acc: 0.8492 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 2098/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3665 - acc: 0.8488 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 2099/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3716 - acc: 0.8440 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2100/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3716 - acc: 0.8465 - val_loss: 0.3764 - val_acc: 0.8175\n Epoch 2101/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3693 - acc: 0.8493 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 2102/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3668 - acc: 0.8475 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 2103/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3724 - acc: 0.8450 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2104/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3702 - acc: 0.8476 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2105/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3676 - acc: 0.8499 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 2106/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3712 - acc: 0.8475 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 2107/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3681 - acc: 0.8456 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2108/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3695 - acc: 0.8464 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2109/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3692 - acc: 0.8481 - val_loss: 0.3736 - val_acc: 0.8213\n Epoch 2110/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3744 - acc: 0.8435 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 2111/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3724 - acc: 0.8417 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2112/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3675 - acc: 0.8488 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2113/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3673 - acc: 0.8490 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 2114/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3665 - acc: 0.8479 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2115/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3726 - acc: 0.8451 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 2116/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3689 - acc: 0.8478 - val_loss: 0.3745 - val_acc: 0.8200\n Epoch 2117/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3700 - acc: 0.8469 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 2118/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3671 - acc: 0.8492 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2119/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3676 - acc: 0.8471 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2120/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3730 - acc: 0.8462 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2121/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3695 - acc: 0.8501 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2122/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3716 - acc: 0.8444 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 2123/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3715 - acc: 0.8482 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2124/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3672 - acc: 0.8493 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2125/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3722 - acc: 0.8456 - val_loss: 0.3730 - val_acc: 0.8213\n Epoch 2126/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3690 - acc: 0.8457 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2127/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3713 - acc: 0.8435 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2128/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3688 - acc: 0.8456 - val_loss: 0.3736 - val_acc: 0.8175\n Epoch 2129/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3721 - acc: 0.8456 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 2130/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3665 - acc: 0.8475 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 2131/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3692 - acc: 0.8444 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 2132/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3700 - acc: 0.8488 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2133/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3680 - acc: 0.8497 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2134/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3682 - acc: 0.8454 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2135/10000\n 7200/7200 [==============================] - 1s 148us/step - loss: 0.3679 - acc: 0.8454 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 2136/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3672 - acc: 0.8515 - val_loss: 0.3729 - val_acc: 0.8213\n Epoch 2137/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3681 - acc: 0.8468 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2138/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3684 - acc: 0.8442 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 2139/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3657 - acc: 0.8456 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 2140/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3671 - acc: 0.8468 - val_loss: 0.3757 - val_acc: 0.8175\n Epoch 2141/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3649 - acc: 0.8490 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2142/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3712 - acc: 0.8481 - val_loss: 0.3738 - val_acc: 0.8213\n Epoch 2143/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3691 - acc: 0.8457 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2144/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3657 - acc: 0.8501 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2145/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3659 - acc: 0.8467 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 2146/10000\n 7200/7200 [==============================] - 1s 92us/step - loss: 0.3714 - acc: 0.8460 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 2147/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3715 - acc: 0.8489 - val_loss: 0.3725 - val_acc: 0.8213\n Epoch 2148/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3681 - acc: 0.8483 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2149/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3666 - acc: 0.8510 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2150/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3693 - acc: 0.8479 - val_loss: 0.3741 - val_acc: 0.8200\n Epoch 2151/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3723 - acc: 0.8449 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2152/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3720 - acc: 0.8457 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 2153/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3702 - acc: 0.8483 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 2154/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3724 - acc: 0.8421 - val_loss: 0.3722 - val_acc: 0.8213\n Epoch 2155/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3729 - acc: 0.8467 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 2156/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3701 - acc: 0.8490 - val_loss: 0.3714 - val_acc: 0.8213\n Epoch 2157/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3658 - acc: 0.8442 - val_loss: 0.3723 - val_acc: 0.8213\n Epoch 2158/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3744 - acc: 0.8458 - val_loss: 0.3720 - val_acc: 0.8213\n Epoch 2159/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3712 - acc: 0.8457 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 2160/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3718 - acc: 0.8449 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 2161/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3660 - acc: 0.8467 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 2162/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3695 - acc: 0.8474 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2163/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3667 - acc: 0.8478 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 2164/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3658 - acc: 0.8501 - val_loss: 0.3724 - val_acc: 0.8213\n Epoch 2165/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3707 - acc: 0.8449 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 2166/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3674 - acc: 0.8482 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 2167/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3710 - acc: 0.8496 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 2168/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3740 - acc: 0.8436 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 2169/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3703 - acc: 0.8458 - val_loss: 0.3747 - val_acc: 0.8175\n Epoch 2170/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3690 - acc: 0.8479 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2171/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3712 - acc: 0.8457 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 2172/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3684 - acc: 0.8475 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2173/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3708 - acc: 0.8457 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2174/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3704 - acc: 0.8460 - val_loss: 0.3739 - val_acc: 0.8175\n Epoch 2175/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3686 - acc: 0.8462 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 2176/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3713 - acc: 0.8460 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 2177/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3693 - acc: 0.8467 - val_loss: 0.3725 - val_acc: 0.8250\n Epoch 2178/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3706 - acc: 0.8465 - val_loss: 0.3735 - val_acc: 0.8225\n Epoch 2179/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3689 - acc: 0.8450 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2180/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3642 - acc: 0.8469 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2181/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3750 - acc: 0.8439 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2182/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3700 - acc: 0.8442 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 2183/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3687 - acc: 0.8474 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2184/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3735 - acc: 0.8422 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2185/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3676 - acc: 0.8474 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 2186/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3722 - acc: 0.8488 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 2187/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3676 - acc: 0.8453 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 2188/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3660 - acc: 0.8492 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 2189/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3719 - acc: 0.8447 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2190/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3666 - acc: 0.8489 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 2191/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3672 - acc: 0.8468 - val_loss: 0.3728 - val_acc: 0.8225\n Epoch 2192/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3702 - acc: 0.8493 - val_loss: 0.3724 - val_acc: 0.8250\n Epoch 2193/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3757 - acc: 0.8454 - val_loss: 0.3723 - val_acc: 0.8263\n Epoch 2194/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3686 - acc: 0.8478 - val_loss: 0.3730 - val_acc: 0.8250\n Epoch 2195/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3715 - acc: 0.8471 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 2196/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3746 - acc: 0.8446 - val_loss: 0.3745 - val_acc: 0.8175\n Epoch 2197/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3695 - acc: 0.8478 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 2198/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3685 - acc: 0.8472 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2199/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3691 - acc: 0.8460 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 2200/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3708 - acc: 0.8444 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 2201/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3691 - acc: 0.8461 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2202/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3640 - acc: 0.8511 - val_loss: 0.3725 - val_acc: 0.8213\n Epoch 2203/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3718 - acc: 0.8431 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 2204/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3694 - acc: 0.8454 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2205/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3702 - acc: 0.8460 - val_loss: 0.3725 - val_acc: 0.8213\n Epoch 2206/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3681 - acc: 0.8460 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 2207/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3714 - acc: 0.8450 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2208/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3623 - acc: 0.8486 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 2209/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3696 - acc: 0.8440 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2210/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3674 - acc: 0.8494 - val_loss: 0.3721 - val_acc: 0.8213\n Epoch 2211/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3702 - acc: 0.8457 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 2212/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3699 - acc: 0.8458 - val_loss: 0.3745 - val_acc: 0.8175\n Epoch 2213/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3732 - acc: 0.8472 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 2214/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3645 - acc: 0.8500 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 2215/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3715 - acc: 0.8458 - val_loss: 0.3737 - val_acc: 0.8200\n Epoch 2216/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3654 - acc: 0.8500 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 2217/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8461 - val_loss: 0.3727 - val_acc: 0.8213\n Epoch 2218/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3709 - acc: 0.8468 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 2219/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3720 - acc: 0.8462 - val_loss: 0.3730 - val_acc: 0.8213\n Epoch 2220/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3700 - acc: 0.8462 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2221/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3635 - acc: 0.8504 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 2222/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3690 - acc: 0.8481 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2223/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3698 - acc: 0.8451 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2224/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3686 - acc: 0.8478 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2225/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3714 - acc: 0.8457 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 2226/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3741 - acc: 0.8417 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2227/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3707 - acc: 0.8428 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 2228/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3696 - acc: 0.8469 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 2229/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3659 - acc: 0.8471 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2230/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3663 - acc: 0.8512 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2231/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3673 - acc: 0.8494 - val_loss: 0.3748 - val_acc: 0.8200\n Epoch 2232/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3728 - acc: 0.8456 - val_loss: 0.3740 - val_acc: 0.8200\n Epoch 2233/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3696 - acc: 0.8456 - val_loss: 0.3743 - val_acc: 0.8200\n Epoch 2234/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3659 - acc: 0.8492 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 2235/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3680 - acc: 0.8468 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 2236/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3666 - acc: 0.8460 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 2237/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3696 - acc: 0.8485 - val_loss: 0.3737 - val_acc: 0.8200\n Epoch 2238/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3703 - acc: 0.8488 - val_loss: 0.3726 - val_acc: 0.8250\n Epoch 2239/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3688 - acc: 0.8474 - val_loss: 0.3737 - val_acc: 0.8225\n Epoch 2240/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3685 - acc: 0.8447 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 2241/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3697 - acc: 0.8475 - val_loss: 0.3744 - val_acc: 0.8200\n Epoch 2242/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3600 - acc: 0.8504 - val_loss: 0.3758 - val_acc: 0.8200\n Epoch 2243/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3726 - acc: 0.8460 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 2244/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3687 - acc: 0.8457 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 2245/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3654 - acc: 0.8476 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2246/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3673 - acc: 0.8472 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2247/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3708 - acc: 0.8440 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2248/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3712 - acc: 0.8456 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 2249/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3716 - acc: 0.8465 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 2250/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3682 - acc: 0.8446 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 2251/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3714 - acc: 0.8488 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 2252/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3691 - acc: 0.8446 - val_loss: 0.3735 - val_acc: 0.8200\n Epoch 2253/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3727 - acc: 0.8456 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 2254/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3740 - acc: 0.8425 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 2255/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8461 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 2256/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3682 - acc: 0.8468 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2257/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3686 - acc: 0.8467 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2258/10000\n 7200/7200 [==============================] - 1s 110us/step - loss: 0.3688 - acc: 0.8469 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 2259/10000\n 7200/7200 [==============================] - 1s 91us/step - loss: 0.3714 - acc: 0.8449 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 2260/10000\n 7200/7200 [==============================] - 1s 82us/step - loss: 0.3700 - acc: 0.8481 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 2261/10000\n 7200/7200 [==============================] - 1s 84us/step - loss: 0.3692 - acc: 0.8485 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 2262/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3689 - acc: 0.8464 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 2263/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3655 - acc: 0.8493 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 2264/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3677 - acc: 0.8478 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 2265/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3675 - acc: 0.8456 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2266/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3687 - acc: 0.8499 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 2267/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3675 - acc: 0.8476 - val_loss: 0.3715 - val_acc: 0.8225\n Epoch 2268/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3688 - acc: 0.8460 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 2269/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3700 - acc: 0.8462 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 2270/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3634 - acc: 0.8497 - val_loss: 0.3720 - val_acc: 0.8187\n Epoch 2271/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3735 - acc: 0.8411 - val_loss: 0.3715 - val_acc: 0.8213\n Epoch 2272/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3695 - acc: 0.8461 - val_loss: 0.3718 - val_acc: 0.8237\n Epoch 2273/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3732 - acc: 0.8475 - val_loss: 0.3724 - val_acc: 0.8225\n Epoch 2274/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3677 - acc: 0.8489 - val_loss: 0.3717 - val_acc: 0.8200\n Epoch 2275/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3690 - acc: 0.8465 - val_loss: 0.3721 - val_acc: 0.8187\n Epoch 2276/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3667 - acc: 0.8462 - val_loss: 0.3715 - val_acc: 0.8213\n Epoch 2277/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3690 - acc: 0.8489 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 2278/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3674 - acc: 0.8439 - val_loss: 0.3735 - val_acc: 0.8200\n Epoch 2279/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3696 - acc: 0.8444 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 2280/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3681 - acc: 0.8475 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 2281/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3660 - acc: 0.8464 - val_loss: 0.3714 - val_acc: 0.8213\n Epoch 2282/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3664 - acc: 0.8482 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 2283/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3679 - acc: 0.8474 - val_loss: 0.3726 - val_acc: 0.8213\n Epoch 2284/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8510 - val_loss: 0.3728 - val_acc: 0.8213\n Epoch 2285/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3708 - acc: 0.8472 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 2286/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3724 - acc: 0.8433 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2287/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3643 - acc: 0.8474 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 2288/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3675 - acc: 0.8456 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2289/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3736 - acc: 0.8460 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 2290/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3647 - acc: 0.8490 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 2291/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3669 - acc: 0.8494 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 2292/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3682 - acc: 0.8479 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2293/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3704 - acc: 0.8461 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 2294/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3689 - acc: 0.8461 - val_loss: 0.3731 - val_acc: 0.8213\n Epoch 2295/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3705 - acc: 0.8449 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 2296/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3667 - acc: 0.8476 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2297/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3681 - acc: 0.8497 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2298/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3680 - acc: 0.8489 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2299/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3685 - acc: 0.8465 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 2300/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3690 - acc: 0.8512 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 2301/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3661 - acc: 0.8489 - val_loss: 0.3728 - val_acc: 0.8263\n Epoch 2302/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3715 - acc: 0.8456 - val_loss: 0.3724 - val_acc: 0.8250\n Epoch 2303/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3724 - acc: 0.8436 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2304/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3654 - acc: 0.8504 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 2305/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3745 - acc: 0.8467 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 2306/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3698 - acc: 0.8496 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 2307/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3681 - acc: 0.8494 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 2308/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3663 - acc: 0.8476 - val_loss: 0.3727 - val_acc: 0.8213\n Epoch 2309/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3699 - acc: 0.8458 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2310/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3678 - acc: 0.8490 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 2311/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3704 - acc: 0.8460 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2312/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3703 - acc: 0.8462 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2313/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3680 - acc: 0.8507 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 2314/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3691 - acc: 0.8471 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 2315/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3720 - acc: 0.8474 - val_loss: 0.3723 - val_acc: 0.8213\n Epoch 2316/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3662 - acc: 0.8503 - val_loss: 0.3728 - val_acc: 0.8213\n Epoch 2317/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3702 - acc: 0.8450 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2318/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3687 - acc: 0.8481 - val_loss: 0.3722 - val_acc: 0.8225\n Epoch 2319/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3692 - acc: 0.8458 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 2320/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3712 - acc: 0.8456 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 2321/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3729 - acc: 0.8449 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2322/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3711 - acc: 0.8461 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2323/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3684 - acc: 0.8453 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 2324/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3708 - acc: 0.8442 - val_loss: 0.3738 - val_acc: 0.8175\n Epoch 2325/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3645 - acc: 0.8518 - val_loss: 0.3729 - val_acc: 0.8213\n Epoch 2326/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3674 - acc: 0.8503 - val_loss: 0.3728 - val_acc: 0.8213\n Epoch 2327/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3690 - acc: 0.8472 - val_loss: 0.3725 - val_acc: 0.8213\n Epoch 2328/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3749 - acc: 0.8449 - val_loss: 0.3725 - val_acc: 0.8213\n Epoch 2329/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3697 - acc: 0.8485 - val_loss: 0.3729 - val_acc: 0.8213\n Epoch 2330/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3723 - acc: 0.8469 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2331/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3677 - acc: 0.8478 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2332/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3708 - acc: 0.8475 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 2333/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3735 - acc: 0.8453 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2334/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3697 - acc: 0.8464 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2335/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3684 - acc: 0.8475 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2336/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3710 - acc: 0.8440 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2337/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3648 - acc: 0.8489 - val_loss: 0.3724 - val_acc: 0.8213\n Epoch 2338/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3701 - acc: 0.8435 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 2339/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3697 - acc: 0.8474 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 2340/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3677 - acc: 0.8469 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2341/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3685 - acc: 0.8454 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 2342/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3673 - acc: 0.8485 - val_loss: 0.3724 - val_acc: 0.8213\n Epoch 2343/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3689 - acc: 0.8493 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 2344/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3683 - acc: 0.8462 - val_loss: 0.3726 - val_acc: 0.8187\n Epoch 2345/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3694 - acc: 0.8449 - val_loss: 0.3725 - val_acc: 0.8213\n Epoch 2346/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3655 - acc: 0.8474 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2347/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3704 - acc: 0.8481 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2348/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3673 - acc: 0.8512 - val_loss: 0.3718 - val_acc: 0.8187\n Epoch 2349/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3693 - acc: 0.8492 - val_loss: 0.3718 - val_acc: 0.8200\n Epoch 2350/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3684 - acc: 0.8446 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2351/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3691 - acc: 0.8442 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 2352/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3709 - acc: 0.8479 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 2353/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3669 - acc: 0.8460 - val_loss: 0.3715 - val_acc: 0.8187\n Epoch 2354/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3691 - acc: 0.8453 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 2355/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3726 - acc: 0.8444 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 2356/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3664 - acc: 0.8449 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 2357/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3658 - acc: 0.8481 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 2358/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3697 - acc: 0.8442 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 2359/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3678 - acc: 0.8490 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 2360/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3628 - acc: 0.8504 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 2361/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3671 - acc: 0.8460 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 2362/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3653 - acc: 0.8476 - val_loss: 0.3721 - val_acc: 0.8213\n Epoch 2363/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3712 - acc: 0.8429 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2364/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3698 - acc: 0.8499 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 2365/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3695 - acc: 0.8449 - val_loss: 0.3742 - val_acc: 0.8200\n Epoch 2366/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3683 - acc: 0.8488 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 2367/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3702 - acc: 0.8486 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 2368/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3707 - acc: 0.8443 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2369/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3672 - acc: 0.8481 - val_loss: 0.3729 - val_acc: 0.8213\n Epoch 2370/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3690 - acc: 0.8462 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2371/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3631 - acc: 0.8485 - val_loss: 0.3744 - val_acc: 0.8200\n Epoch 2372/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3666 - acc: 0.8472 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 2373/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3671 - acc: 0.8483 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2374/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3704 - acc: 0.8486 - val_loss: 0.3733 - val_acc: 0.8225\n Epoch 2375/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3672 - acc: 0.8461 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2376/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3674 - acc: 0.8501 - val_loss: 0.3726 - val_acc: 0.8225\n Epoch 2377/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3667 - acc: 0.8499 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2378/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3666 - acc: 0.8442 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 2379/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3679 - acc: 0.8482 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2380/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3718 - acc: 0.8482 - val_loss: 0.3729 - val_acc: 0.8213\n Epoch 2381/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3662 - acc: 0.8499 - val_loss: 0.3727 - val_acc: 0.8213\n Epoch 2382/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3689 - acc: 0.8457 - val_loss: 0.3727 - val_acc: 0.8225\n Epoch 2383/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3759 - acc: 0.8407 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 2384/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3698 - acc: 0.8483 - val_loss: 0.3715 - val_acc: 0.8225\n Epoch 2385/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3682 - acc: 0.8464 - val_loss: 0.3720 - val_acc: 0.8225\n Epoch 2386/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3660 - acc: 0.8492 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 2387/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3726 - acc: 0.8474 - val_loss: 0.3715 - val_acc: 0.8213\n Epoch 2388/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3728 - acc: 0.8453 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 2389/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3711 - acc: 0.8462 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2390/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3714 - acc: 0.8440 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 2391/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3646 - acc: 0.8486 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 2392/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3675 - acc: 0.8447 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 2393/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3656 - acc: 0.8481 - val_loss: 0.3720 - val_acc: 0.8225\n Epoch 2394/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3734 - acc: 0.8451 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 2395/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3634 - acc: 0.8492 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 2396/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3656 - acc: 0.8467 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2397/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3717 - acc: 0.8433 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 2398/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3627 - acc: 0.8497 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 2399/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3736 - acc: 0.8419 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 2400/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3692 - acc: 0.8451 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 2401/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3723 - acc: 0.8433 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 2402/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3641 - acc: 0.8481 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 2403/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3669 - acc: 0.8494 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 2404/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3691 - acc: 0.8457 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 2405/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3680 - acc: 0.8497 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 2406/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3662 - acc: 0.8479 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 2407/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3646 - acc: 0.8458 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2408/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3746 - acc: 0.8451 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 2409/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3722 - acc: 0.8460 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2410/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3688 - acc: 0.8482 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 2411/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3732 - acc: 0.8464 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 2412/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3687 - acc: 0.8485 - val_loss: 0.3724 - val_acc: 0.8200\n Epoch 2413/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3684 - acc: 0.8462 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 2414/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3705 - acc: 0.8472 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2415/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3739 - acc: 0.8449 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2416/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3695 - acc: 0.8458 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 2417/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3651 - acc: 0.8471 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2418/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3697 - acc: 0.8450 - val_loss: 0.3730 - val_acc: 0.8213\n Epoch 2419/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3754 - acc: 0.8461 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 2420/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3710 - acc: 0.8451 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2421/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3698 - acc: 0.8490 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2422/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3684 - acc: 0.8475 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 2423/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3700 - acc: 0.8474 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2424/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3665 - acc: 0.8471 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 2425/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3741 - acc: 0.8411 - val_loss: 0.3746 - val_acc: 0.8175\n Epoch 2426/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3654 - acc: 0.8493 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 2427/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3683 - acc: 0.8482 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2428/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3713 - acc: 0.8447 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 2429/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3676 - acc: 0.8483 - val_loss: 0.3724 - val_acc: 0.8187\n Epoch 2430/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3674 - acc: 0.8454 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2431/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3687 - acc: 0.8469 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 2432/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3647 - acc: 0.8456 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 2433/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3672 - acc: 0.8489 - val_loss: 0.3721 - val_acc: 0.8213\n Epoch 2434/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3713 - acc: 0.8456 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 2435/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3647 - acc: 0.8500 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 2436/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3714 - acc: 0.8462 - val_loss: 0.3718 - val_acc: 0.8263\n Epoch 2437/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3720 - acc: 0.8444 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 2438/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3700 - acc: 0.8453 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 2439/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3678 - acc: 0.8488 - val_loss: 0.3721 - val_acc: 0.8250\n Epoch 2440/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3673 - acc: 0.8465 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 2441/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3728 - acc: 0.8469 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2442/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3694 - acc: 0.8483 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 2443/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3685 - acc: 0.8475 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 2444/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3690 - acc: 0.8474 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 2445/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3692 - acc: 0.8469 - val_loss: 0.3720 - val_acc: 0.8213\n Epoch 2446/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3649 - acc: 0.8511 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 2447/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3722 - acc: 0.8454 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 2448/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3661 - acc: 0.8468 - val_loss: 0.3717 - val_acc: 0.8213\n Epoch 2449/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3681 - acc: 0.8501 - val_loss: 0.3717 - val_acc: 0.8213\n Epoch 2450/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3658 - acc: 0.8486 - val_loss: 0.3713 - val_acc: 0.8213\n Epoch 2451/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3695 - acc: 0.8457 - val_loss: 0.3708 - val_acc: 0.8213\n Epoch 2452/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3665 - acc: 0.8494 - val_loss: 0.3725 - val_acc: 0.8213\n Epoch 2453/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3725 - acc: 0.8481 - val_loss: 0.3722 - val_acc: 0.8187\n Epoch 2454/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3682 - acc: 0.8488 - val_loss: 0.3714 - val_acc: 0.8225\n Epoch 2455/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3684 - acc: 0.8481 - val_loss: 0.3722 - val_acc: 0.8200\n Epoch 2456/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3699 - acc: 0.8497 - val_loss: 0.3713 - val_acc: 0.8213\n Epoch 2457/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3654 - acc: 0.8485 - val_loss: 0.3713 - val_acc: 0.8225\n Epoch 2458/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3689 - acc: 0.8471 - val_loss: 0.3715 - val_acc: 0.8200\n Epoch 2459/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3739 - acc: 0.8482 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 2460/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3692 - acc: 0.8501 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 2461/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3754 - acc: 0.8442 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2462/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3657 - acc: 0.8481 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 2463/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3679 - acc: 0.8476 - val_loss: 0.3725 - val_acc: 0.8187\n Epoch 2464/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3720 - acc: 0.8444 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2465/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3688 - acc: 0.8458 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 2466/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3731 - acc: 0.8443 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 2467/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3679 - acc: 0.8490 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 2468/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3704 - acc: 0.8471 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 2469/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3677 - acc: 0.8457 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2470/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3750 - acc: 0.8443 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 2471/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3742 - acc: 0.8461 - val_loss: 0.3729 - val_acc: 0.8225\n Epoch 2472/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3713 - acc: 0.8469 - val_loss: 0.3726 - val_acc: 0.8225\n Epoch 2473/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3732 - acc: 0.8442 - val_loss: 0.3724 - val_acc: 0.8213\n Epoch 2474/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3745 - acc: 0.8438 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2475/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3693 - acc: 0.8474 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2476/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8457 - val_loss: 0.3740 - val_acc: 0.8175\n Epoch 2477/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3715 - acc: 0.8439 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 2478/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3685 - acc: 0.8475 - val_loss: 0.3723 - val_acc: 0.8225\n Epoch 2479/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3703 - acc: 0.8468 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 2480/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3717 - acc: 0.8464 - val_loss: 0.3725 - val_acc: 0.8213\n Epoch 2481/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3723 - acc: 0.8467 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2482/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3672 - acc: 0.8467 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 2483/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3712 - acc: 0.8457 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 2484/10000\n 7200/7200 [==============================] - ETA: 0s - loss: 0.3677 - acc: 0.847 - 0s 62us/step - loss: 0.3668 - acc: 0.8481 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 2485/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3716 - acc: 0.8471 - val_loss: 0.3737 - val_acc: 0.8200\n Epoch 2486/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3713 - acc: 0.8468 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 2487/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3740 - acc: 0.8432 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 2488/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3696 - acc: 0.8446 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 2489/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3708 - acc: 0.8432 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 2490/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3720 - acc: 0.8442 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 2491/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3643 - acc: 0.8510 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2492/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3677 - acc: 0.8488 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 2493/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3684 - acc: 0.8512 - val_loss: 0.3723 - val_acc: 0.8213\n Epoch 2494/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3682 - acc: 0.8464 - val_loss: 0.3730 - val_acc: 0.8213\n Epoch 2495/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3660 - acc: 0.8510 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 2496/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3698 - acc: 0.8492 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2497/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3707 - acc: 0.8450 - val_loss: 0.3741 - val_acc: 0.8200\n Epoch 2498/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3737 - acc: 0.8447 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2499/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3666 - acc: 0.8464 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2500/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3675 - acc: 0.8476 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 2501/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3655 - acc: 0.8507 - val_loss: 0.3731 - val_acc: 0.8213\n Epoch 2502/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3675 - acc: 0.8458 - val_loss: 0.3735 - val_acc: 0.8200\n Epoch 2503/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3634 - acc: 0.8490 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 2504/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3695 - acc: 0.8450 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 2505/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3724 - acc: 0.8426 - val_loss: 0.3748 - val_acc: 0.8200\n Epoch 2506/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3700 - acc: 0.8432 - val_loss: 0.3743 - val_acc: 0.8200\n Epoch 2507/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3725 - acc: 0.8456 - val_loss: 0.3743 - val_acc: 0.8200\n Epoch 2508/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3643 - acc: 0.8479 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2509/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3708 - acc: 0.8461 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2510/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3711 - acc: 0.8478 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 2511/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3756 - acc: 0.8432 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2512/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3717 - acc: 0.8436 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2513/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3715 - acc: 0.8478 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2514/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3696 - acc: 0.8451 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2515/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3672 - acc: 0.8475 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 2516/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3680 - acc: 0.8478 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 2517/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3719 - acc: 0.8449 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 2518/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3734 - acc: 0.8449 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2519/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3721 - acc: 0.8438 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 2520/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3702 - acc: 0.8447 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 2521/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3696 - acc: 0.8438 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 2522/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3676 - acc: 0.8471 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2523/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3672 - acc: 0.8482 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 2524/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3688 - acc: 0.8482 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 2525/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3692 - acc: 0.8475 - val_loss: 0.3742 - val_acc: 0.8200\n Epoch 2526/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3698 - acc: 0.8461 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 2527/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3636 - acc: 0.8517 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2528/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3676 - acc: 0.8497 - val_loss: 0.3743 - val_acc: 0.8225\n Epoch 2529/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3675 - acc: 0.8467 - val_loss: 0.3743 - val_acc: 0.8200\n Epoch 2530/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3633 - acc: 0.8497 - val_loss: 0.3729 - val_acc: 0.8225\n Epoch 2531/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3674 - acc: 0.8485 - val_loss: 0.3738 - val_acc: 0.8213\n Epoch 2532/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3699 - acc: 0.8494 - val_loss: 0.3724 - val_acc: 0.8250\n Epoch 2533/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3654 - acc: 0.8485 - val_loss: 0.3728 - val_acc: 0.8250\n Epoch 2534/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3706 - acc: 0.8443 - val_loss: 0.3738 - val_acc: 0.8213\n Epoch 2535/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3751 - acc: 0.8428 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 2536/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3655 - acc: 0.8485 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 2537/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3650 - acc: 0.8479 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 2538/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3726 - acc: 0.8442 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2539/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3728 - acc: 0.8438 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 2540/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3666 - acc: 0.8468 - val_loss: 0.3744 - val_acc: 0.8200\n Epoch 2541/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3679 - acc: 0.8453 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 2542/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3697 - acc: 0.8464 - val_loss: 0.3735 - val_acc: 0.8200\n Epoch 2543/10000\n 7200/7200 [==============================] - ETA: 0s - loss: 0.3708 - acc: 0.846 - 0s 69us/step - loss: 0.3713 - acc: 0.8465 - val_loss: 0.3725 - val_acc: 0.8213\n Epoch 2544/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3728 - acc: 0.8475 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 2545/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3660 - acc: 0.8493 - val_loss: 0.3737 - val_acc: 0.8200\n Epoch 2546/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3681 - acc: 0.8450 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2547/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3750 - acc: 0.8450 - val_loss: 0.3742 - val_acc: 0.8175\n Epoch 2548/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3726 - acc: 0.8460 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 2549/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3687 - acc: 0.8440 - val_loss: 0.3744 - val_acc: 0.8175\n Epoch 2550/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3712 - acc: 0.8447 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 2551/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3663 - acc: 0.8465 - val_loss: 0.3732 - val_acc: 0.8213\n Epoch 2552/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3664 - acc: 0.8506 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 2553/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3687 - acc: 0.8503 - val_loss: 0.3732 - val_acc: 0.8213\n Epoch 2554/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3742 - acc: 0.8436 - val_loss: 0.3746 - val_acc: 0.8200\n Epoch 2555/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3702 - acc: 0.8439 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 2556/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3693 - acc: 0.8482 - val_loss: 0.3720 - val_acc: 0.8213\n Epoch 2557/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3699 - acc: 0.8467 - val_loss: 0.3722 - val_acc: 0.8213\n Epoch 2558/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3700 - acc: 0.8486 - val_loss: 0.3730 - val_acc: 0.8213\n Epoch 2559/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3698 - acc: 0.8476 - val_loss: 0.3716 - val_acc: 0.8225\n Epoch 2560/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3709 - acc: 0.8454 - val_loss: 0.3713 - val_acc: 0.8225\n Epoch 2561/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3678 - acc: 0.8486 - val_loss: 0.3717 - val_acc: 0.8225\n Epoch 2562/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3653 - acc: 0.8501 - val_loss: 0.3709 - val_acc: 0.8237\n Epoch 2563/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3645 - acc: 0.8496 - val_loss: 0.3713 - val_acc: 0.8213\n Epoch 2564/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3724 - acc: 0.8465 - val_loss: 0.3713 - val_acc: 0.8237\n Epoch 2565/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3664 - acc: 0.8478 - val_loss: 0.3715 - val_acc: 0.8213\n Epoch 2566/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3673 - acc: 0.8490 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 2567/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3670 - acc: 0.8507 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 2568/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3606 - acc: 0.8503 - val_loss: 0.3727 - val_acc: 0.8263\n Epoch 2569/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3660 - acc: 0.8488 - val_loss: 0.3737 - val_acc: 0.8200\n Epoch 2570/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3717 - acc: 0.8456 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 2571/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3661 - acc: 0.8485 - val_loss: 0.3727 - val_acc: 0.8237\n Epoch 2572/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3696 - acc: 0.8467 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 2573/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3734 - acc: 0.8432 - val_loss: 0.3745 - val_acc: 0.8200\n Epoch 2574/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3680 - acc: 0.8492 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2575/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3714 - acc: 0.8464 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 2576/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3659 - acc: 0.8483 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2577/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3683 - acc: 0.8507 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 2578/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3670 - acc: 0.8462 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2579/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3682 - acc: 0.8499 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2580/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3707 - acc: 0.8456 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 2581/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3657 - acc: 0.8478 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2582/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3697 - acc: 0.8456 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2583/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3690 - acc: 0.8479 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 2584/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3754 - acc: 0.8479 - val_loss: 0.3737 - val_acc: 0.8200\n Epoch 2585/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3664 - acc: 0.8460 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2586/10000\n 7200/7200 [==============================] - 1s 82us/step - loss: 0.3678 - acc: 0.8465 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 2587/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3702 - acc: 0.8482 - val_loss: 0.3738 - val_acc: 0.8213\n Epoch 2588/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3668 - acc: 0.8486 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2589/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3727 - acc: 0.8444 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 2590/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3727 - acc: 0.8482 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 2591/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3657 - acc: 0.8475 - val_loss: 0.3735 - val_acc: 0.8200\n Epoch 2592/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3697 - acc: 0.8457 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 2593/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3695 - acc: 0.8471 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2594/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3656 - acc: 0.8504 - val_loss: 0.3722 - val_acc: 0.8237\n Epoch 2595/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3741 - acc: 0.8457 - val_loss: 0.3737 - val_acc: 0.8175\n Epoch 2596/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3738 - acc: 0.8397 - val_loss: 0.3735 - val_acc: 0.8175\n Epoch 2597/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3696 - acc: 0.8438 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 2598/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3662 - acc: 0.8493 - val_loss: 0.3732 - val_acc: 0.8175\n Epoch 2599/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3683 - acc: 0.8488 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2600/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3647 - acc: 0.8467 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2601/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3681 - acc: 0.8485 - val_loss: 0.3752 - val_acc: 0.8175\n Epoch 2602/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3692 - acc: 0.8483 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 2603/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3705 - acc: 0.8465 - val_loss: 0.3736 - val_acc: 0.8213\n Epoch 2604/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3658 - acc: 0.8494 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 2605/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3679 - acc: 0.8472 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 2606/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3696 - acc: 0.8481 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 2607/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3644 - acc: 0.8500 - val_loss: 0.3735 - val_acc: 0.8200\n Epoch 2608/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3726 - acc: 0.8451 - val_loss: 0.3737 - val_acc: 0.8200\n Epoch 2609/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3695 - acc: 0.8464 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 2610/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3714 - acc: 0.8464 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 2611/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3696 - acc: 0.8464 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 2612/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3686 - acc: 0.8475 - val_loss: 0.3734 - val_acc: 0.8237\n Epoch 2613/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3638 - acc: 0.8492 - val_loss: 0.3731 - val_acc: 0.8237\n Epoch 2614/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3624 - acc: 0.8501 - val_loss: 0.3744 - val_acc: 0.8175\n Epoch 2615/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3685 - acc: 0.8478 - val_loss: 0.3735 - val_acc: 0.8200\n Epoch 2616/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3690 - acc: 0.8461 - val_loss: 0.3740 - val_acc: 0.8175\n Epoch 2617/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3700 - acc: 0.8453 - val_loss: 0.3744 - val_acc: 0.8175\n Epoch 2618/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3697 - acc: 0.8446 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 2619/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3690 - acc: 0.8478 - val_loss: 0.3747 - val_acc: 0.8175\n Epoch 2620/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3704 - acc: 0.8446 - val_loss: 0.3739 - val_acc: 0.8175\n Epoch 2621/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3660 - acc: 0.8479 - val_loss: 0.3732 - val_acc: 0.8175\n Epoch 2622/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3715 - acc: 0.8465 - val_loss: 0.3731 - val_acc: 0.8175\n Epoch 2623/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3683 - acc: 0.8447 - val_loss: 0.3745 - val_acc: 0.8175\n Epoch 2624/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3663 - acc: 0.8494 - val_loss: 0.3737 - val_acc: 0.8175\n Epoch 2625/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3713 - acc: 0.8433 - val_loss: 0.3746 - val_acc: 0.8175\n Epoch 2626/10000\n 7200/7200 [==============================] - 1s 126us/step - loss: 0.3724 - acc: 0.8486 - val_loss: 0.3747 - val_acc: 0.8175\n Epoch 2627/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3696 - acc: 0.8478 - val_loss: 0.3742 - val_acc: 0.8175\n Epoch 2628/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3687 - acc: 0.8458 - val_loss: 0.3745 - val_acc: 0.8175\n Epoch 2629/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3695 - acc: 0.8443 - val_loss: 0.3745 - val_acc: 0.8175\n Epoch 2630/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3713 - acc: 0.8446 - val_loss: 0.3747 - val_acc: 0.8175\n Epoch 2631/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3720 - acc: 0.8457 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 2632/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3725 - acc: 0.8433 - val_loss: 0.3726 - val_acc: 0.8175\n Epoch 2633/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3659 - acc: 0.8532 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 2634/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3672 - acc: 0.8474 - val_loss: 0.3721 - val_acc: 0.8175\n Epoch 2635/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3701 - acc: 0.8440 - val_loss: 0.3719 - val_acc: 0.8200\n Epoch 2636/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3682 - acc: 0.8464 - val_loss: 0.3723 - val_acc: 0.8175\n Epoch 2637/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3747 - acc: 0.8438 - val_loss: 0.3710 - val_acc: 0.8213\n Epoch 2638/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3694 - acc: 0.8483 - val_loss: 0.3717 - val_acc: 0.8187\n Epoch 2639/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3673 - acc: 0.8475 - val_loss: 0.3739 - val_acc: 0.8175\n Epoch 2640/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3706 - acc: 0.8460 - val_loss: 0.3736 - val_acc: 0.8175\n Epoch 2641/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3695 - acc: 0.8489 - val_loss: 0.3732 - val_acc: 0.8175\n Epoch 2642/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3713 - acc: 0.8439 - val_loss: 0.3729 - val_acc: 0.8175\n Epoch 2643/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3695 - acc: 0.8453 - val_loss: 0.3739 - val_acc: 0.8175\n Epoch 2644/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3719 - acc: 0.8432 - val_loss: 0.3734 - val_acc: 0.8175\n Epoch 2645/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3692 - acc: 0.8486 - val_loss: 0.3720 - val_acc: 0.8175\n Epoch 2646/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3691 - acc: 0.8493 - val_loss: 0.3734 - val_acc: 0.8175\n Epoch 2647/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3707 - acc: 0.8453 - val_loss: 0.3732 - val_acc: 0.8175\n Epoch 2648/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3710 - acc: 0.8457 - val_loss: 0.3736 - val_acc: 0.8175\n Epoch 2649/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3674 - acc: 0.8485 - val_loss: 0.3721 - val_acc: 0.8225\n Epoch 2650/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3671 - acc: 0.8493 - val_loss: 0.3734 - val_acc: 0.8175\n Epoch 2651/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3725 - acc: 0.8439 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 2652/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3667 - acc: 0.8479 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 2653/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3705 - acc: 0.8440 - val_loss: 0.3741 - val_acc: 0.8175\n Epoch 2654/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3733 - acc: 0.8424 - val_loss: 0.3743 - val_acc: 0.8163\n Epoch 2655/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3696 - acc: 0.8476 - val_loss: 0.3742 - val_acc: 0.8163\n Epoch 2656/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3710 - acc: 0.8454 - val_loss: 0.3748 - val_acc: 0.8163\n Epoch 2657/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3646 - acc: 0.8489 - val_loss: 0.3736 - val_acc: 0.8175\n Epoch 2658/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3706 - acc: 0.8475 - val_loss: 0.3746 - val_acc: 0.8175\n Epoch 2659/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3731 - acc: 0.8482 - val_loss: 0.3743 - val_acc: 0.8175\n Epoch 2660/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3712 - acc: 0.8450 - val_loss: 0.3743 - val_acc: 0.8175\n Epoch 2661/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3711 - acc: 0.8460 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2662/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3708 - acc: 0.8464 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2663/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3664 - acc: 0.8465 - val_loss: 0.3746 - val_acc: 0.8175\n Epoch 2664/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3668 - acc: 0.8486 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 2665/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3711 - acc: 0.8476 - val_loss: 0.3729 - val_acc: 0.8213\n Epoch 2666/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3663 - acc: 0.8494 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 2667/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3699 - acc: 0.8451 - val_loss: 0.3723 - val_acc: 0.8187\n Epoch 2668/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3655 - acc: 0.8493 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 2669/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3653 - acc: 0.8510 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2670/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3782 - acc: 0.8439 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2671/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3703 - acc: 0.8467 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 2672/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3705 - acc: 0.8464 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 2673/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3712 - acc: 0.8464 - val_loss: 0.3723 - val_acc: 0.8200\n Epoch 2674/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3637 - acc: 0.8536 - val_loss: 0.3731 - val_acc: 0.8175\n Epoch 2675/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3679 - acc: 0.8458 - val_loss: 0.3728 - val_acc: 0.8175\n Epoch 2676/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3696 - acc: 0.8497 - val_loss: 0.3719 - val_acc: 0.8213\n Epoch 2677/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3702 - acc: 0.8485 - val_loss: 0.3717 - val_acc: 0.8213\n Epoch 2678/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3703 - acc: 0.8447 - val_loss: 0.3729 - val_acc: 0.8175\n Epoch 2679/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3686 - acc: 0.8482 - val_loss: 0.3723 - val_acc: 0.8225\n Epoch 2680/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3681 - acc: 0.8458 - val_loss: 0.3724 - val_acc: 0.8225\n Epoch 2681/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3711 - acc: 0.8467 - val_loss: 0.3726 - val_acc: 0.8225\n Epoch 2682/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3685 - acc: 0.8490 - val_loss: 0.3713 - val_acc: 0.8213\n Epoch 2683/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3732 - acc: 0.8444 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 2684/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3717 - acc: 0.8444 - val_loss: 0.3726 - val_acc: 0.8200\n Epoch 2685/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3717 - acc: 0.8439 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 2686/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3660 - acc: 0.8488 - val_loss: 0.3725 - val_acc: 0.8225\n Epoch 2687/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3661 - acc: 0.8510 - val_loss: 0.3716 - val_acc: 0.8237\n Epoch 2688/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3684 - acc: 0.8471 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 2689/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3726 - acc: 0.8464 - val_loss: 0.3737 - val_acc: 0.8200\n Epoch 2690/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3655 - acc: 0.8501 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2691/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3745 - acc: 0.8435 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 2692/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3702 - acc: 0.8472 - val_loss: 0.3743 - val_acc: 0.8200\n Epoch 2693/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3709 - acc: 0.8475 - val_loss: 0.3729 - val_acc: 0.8225\n Epoch 2694/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3656 - acc: 0.8492 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 2695/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3637 - acc: 0.8510 - val_loss: 0.3743 - val_acc: 0.8175\n Epoch 2696/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3708 - acc: 0.8485 - val_loss: 0.3744 - val_acc: 0.8200\n Epoch 2697/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3691 - acc: 0.8493 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 2698/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3690 - acc: 0.8468 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 2699/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3666 - acc: 0.8461 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2700/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3657 - acc: 0.8518 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 2701/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3705 - acc: 0.8460 - val_loss: 0.3737 - val_acc: 0.8175\n Epoch 2702/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3743 - acc: 0.8438 - val_loss: 0.3758 - val_acc: 0.8175\n Epoch 2703/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3703 - acc: 0.8465 - val_loss: 0.3742 - val_acc: 0.8175\n Epoch 2704/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3672 - acc: 0.8458 - val_loss: 0.3753 - val_acc: 0.8175\n Epoch 2705/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3665 - acc: 0.8472 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2706/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3700 - acc: 0.8440 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 2707/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3682 - acc: 0.8458 - val_loss: 0.3739 - val_acc: 0.8175\n Epoch 2708/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3730 - acc: 0.8413 - val_loss: 0.3728 - val_acc: 0.8175\n Epoch 2709/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3694 - acc: 0.8475 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 2710/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3706 - acc: 0.8456 - val_loss: 0.3735 - val_acc: 0.8175\n Epoch 2711/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3695 - acc: 0.8490 - val_loss: 0.3738 - val_acc: 0.8175\n Epoch 2712/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3682 - acc: 0.8504 - val_loss: 0.3732 - val_acc: 0.8175\n Epoch 2713/10000\n 7200/7200 [==============================] - 1s 88us/step - loss: 0.3679 - acc: 0.8474 - val_loss: 0.3738 - val_acc: 0.8175\n Epoch 2714/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3662 - acc: 0.8482 - val_loss: 0.3732 - val_acc: 0.8175\n Epoch 2715/10000\n 7200/7200 [==============================] - 1s 91us/step - loss: 0.3674 - acc: 0.8479 - val_loss: 0.3721 - val_acc: 0.8225\n Epoch 2716/10000\n 7200/7200 [==============================] - 1s 110us/step - loss: 0.3715 - acc: 0.8474 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 2717/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3657 - acc: 0.8492 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2718/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3664 - acc: 0.8496 - val_loss: 0.3748 - val_acc: 0.8175\n Epoch 2719/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3663 - acc: 0.8462 - val_loss: 0.3741 - val_acc: 0.8175\n Epoch 2720/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3652 - acc: 0.8442 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2721/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3707 - acc: 0.8486 - val_loss: 0.3743 - val_acc: 0.8175\n Epoch 2722/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3680 - acc: 0.8486 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 2723/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3659 - acc: 0.8517 - val_loss: 0.3737 - val_acc: 0.8200\n Epoch 2724/10000\n 7200/7200 [==============================] - 1s 80us/step - loss: 0.3689 - acc: 0.8481 - val_loss: 0.3733 - val_acc: 0.8225\n Epoch 2725/10000\n 7200/7200 [==============================] - 1s 147us/step - loss: 0.3742 - acc: 0.8451 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 2726/10000\n 7200/7200 [==============================] - 1s 97us/step - loss: 0.3693 - acc: 0.8482 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2727/10000\n 7200/7200 [==============================] - 1s 106us/step - loss: 0.3671 - acc: 0.8494 - val_loss: 0.3729 - val_acc: 0.8225\n Epoch 2728/10000\n 7200/7200 [==============================] - 1s 104us/step - loss: 0.3702 - acc: 0.8471 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2729/10000\n 7200/7200 [==============================] - 1s 84us/step - loss: 0.3688 - acc: 0.8507 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 2730/10000\n 7200/7200 [==============================] - 1s 156us/step - loss: 0.3672 - acc: 0.8469 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 2731/10000\n 7200/7200 [==============================] - 1s 107us/step - loss: 0.3679 - acc: 0.8460 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 2732/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3677 - acc: 0.8486 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2733/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3705 - acc: 0.8449 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2734/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3629 - acc: 0.8525 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2735/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3722 - acc: 0.8443 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2736/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3677 - acc: 0.8479 - val_loss: 0.3729 - val_acc: 0.8213\n Epoch 2737/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3730 - acc: 0.8444 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2738/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3673 - acc: 0.8469 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2739/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3685 - acc: 0.8497 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 2740/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3690 - acc: 0.8464 - val_loss: 0.3734 - val_acc: 0.8225\n Epoch 2741/10000\n 7200/7200 [==============================] - ETA: 0s - loss: 0.3665 - acc: 0.848 - 0s 68us/step - loss: 0.3672 - acc: 0.8481 - val_loss: 0.3726 - val_acc: 0.8225\n Epoch 2742/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3683 - acc: 0.8482 - val_loss: 0.3728 - val_acc: 0.8225\n Epoch 2743/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3680 - acc: 0.8474 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2744/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3678 - acc: 0.8450 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2745/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3684 - acc: 0.8471 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2746/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3684 - acc: 0.8450 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 2747/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3657 - acc: 0.8506 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 2748/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3683 - acc: 0.8482 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2749/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3686 - acc: 0.8467 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2750/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3694 - acc: 0.8468 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 2751/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3716 - acc: 0.8492 - val_loss: 0.3736 - val_acc: 0.8213\n Epoch 2752/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3643 - acc: 0.8521 - val_loss: 0.3736 - val_acc: 0.8213\n Epoch 2753/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3654 - acc: 0.8501 - val_loss: 0.3736 - val_acc: 0.8225\n Epoch 2754/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3637 - acc: 0.8512 - val_loss: 0.3735 - val_acc: 0.8225\n Epoch 2755/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3685 - acc: 0.8490 - val_loss: 0.3734 - val_acc: 0.8225\n Epoch 2756/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3624 - acc: 0.8511 - val_loss: 0.3733 - val_acc: 0.8225\n Epoch 2757/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3708 - acc: 0.8460 - val_loss: 0.3727 - val_acc: 0.8225\n Epoch 2758/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3709 - acc: 0.8444 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 2759/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3704 - acc: 0.8465 - val_loss: 0.3724 - val_acc: 0.8225\n Epoch 2760/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3704 - acc: 0.8482 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 2761/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3680 - acc: 0.8474 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 2762/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3734 - acc: 0.8433 - val_loss: 0.3729 - val_acc: 0.8225\n Epoch 2763/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3641 - acc: 0.8499 - val_loss: 0.3727 - val_acc: 0.8213\n Epoch 2764/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3694 - acc: 0.8492 - val_loss: 0.3725 - val_acc: 0.8237\n Epoch 2765/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3675 - acc: 0.8481 - val_loss: 0.3727 - val_acc: 0.8263\n Epoch 2766/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3637 - acc: 0.8511 - val_loss: 0.3728 - val_acc: 0.8213\n Epoch 2767/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3694 - acc: 0.8486 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 2768/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3692 - acc: 0.8467 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 2769/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3663 - acc: 0.8506 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 2770/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3704 - acc: 0.8456 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 2771/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3692 - acc: 0.8464 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2772/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3650 - acc: 0.8478 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2773/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3672 - acc: 0.8499 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2774/10000\n 7200/7200 [==============================] - 1s 86us/step - loss: 0.3716 - acc: 0.8450 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2775/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3641 - acc: 0.8493 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2776/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3725 - acc: 0.8458 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2777/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3735 - acc: 0.8464 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 2778/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3628 - acc: 0.8508 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2779/10000\n 7200/7200 [==============================] - 1s 83us/step - loss: 0.3720 - acc: 0.8439 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2780/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3671 - acc: 0.8494 - val_loss: 0.3744 - val_acc: 0.8200\n Epoch 2781/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3661 - acc: 0.8483 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2782/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3754 - acc: 0.8456 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2783/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3689 - acc: 0.8476 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 2784/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3713 - acc: 0.8483 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 2785/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3680 - acc: 0.8486 - val_loss: 0.3740 - val_acc: 0.8200\n Epoch 2786/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3710 - acc: 0.8478 - val_loss: 0.3739 - val_acc: 0.8213\n Epoch 2787/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3692 - acc: 0.8468 - val_loss: 0.3731 - val_acc: 0.8213\n Epoch 2788/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3682 - acc: 0.8489 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2789/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3677 - acc: 0.8499 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 2790/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3725 - acc: 0.8468 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 2791/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3717 - acc: 0.8479 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 2792/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3695 - acc: 0.8476 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2793/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3706 - acc: 0.8458 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2794/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3722 - acc: 0.8451 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2795/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3719 - acc: 0.8462 - val_loss: 0.3755 - val_acc: 0.8175\n Epoch 2796/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3678 - acc: 0.8492 - val_loss: 0.3732 - val_acc: 0.8213\n Epoch 2797/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3699 - acc: 0.8467 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2798/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3684 - acc: 0.8456 - val_loss: 0.3727 - val_acc: 0.8213\n Epoch 2799/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3708 - acc: 0.8475 - val_loss: 0.3736 - val_acc: 0.8213\n Epoch 2800/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3691 - acc: 0.8472 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 2801/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3710 - acc: 0.8467 - val_loss: 0.3728 - val_acc: 0.8213\n Epoch 2802/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3689 - acc: 0.8464 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 2803/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3683 - acc: 0.8478 - val_loss: 0.3722 - val_acc: 0.8213\n Epoch 2804/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3671 - acc: 0.8510 - val_loss: 0.3721 - val_acc: 0.8237\n Epoch 2805/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3700 - acc: 0.8457 - val_loss: 0.3730 - val_acc: 0.8213\n Epoch 2806/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3684 - acc: 0.8439 - val_loss: 0.3727 - val_acc: 0.8213\n Epoch 2807/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3664 - acc: 0.8474 - val_loss: 0.3726 - val_acc: 0.8213\n Epoch 2808/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3685 - acc: 0.8478 - val_loss: 0.3728 - val_acc: 0.8213\n Epoch 2809/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3686 - acc: 0.8493 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 2810/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3700 - acc: 0.8458 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 2811/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3679 - acc: 0.8496 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2812/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3702 - acc: 0.8486 - val_loss: 0.3729 - val_acc: 0.8225\n Epoch 2813/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3687 - acc: 0.8488 - val_loss: 0.3725 - val_acc: 0.8225\n Epoch 2814/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3658 - acc: 0.8476 - val_loss: 0.3731 - val_acc: 0.8213\n Epoch 2815/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3690 - acc: 0.8488 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 2816/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3696 - acc: 0.8469 - val_loss: 0.3738 - val_acc: 0.8213\n Epoch 2817/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3726 - acc: 0.8447 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 2818/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3622 - acc: 0.8511 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2819/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3666 - acc: 0.8478 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2820/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3744 - acc: 0.8454 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 2821/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3684 - acc: 0.8472 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2822/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3694 - acc: 0.8443 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 2823/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3655 - acc: 0.8512 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 2824/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3689 - acc: 0.8474 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2825/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3713 - acc: 0.8447 - val_loss: 0.3737 - val_acc: 0.8200\n Epoch 2826/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3731 - acc: 0.8460 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2827/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3663 - acc: 0.8485 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2828/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3678 - acc: 0.8478 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2829/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3705 - acc: 0.8453 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2830/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3701 - acc: 0.8474 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 2831/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3719 - acc: 0.8454 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2832/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3665 - acc: 0.8486 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2833/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3711 - acc: 0.8453 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2834/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3689 - acc: 0.8500 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2835/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3679 - acc: 0.8465 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 2836/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3697 - acc: 0.8497 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2837/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3685 - acc: 0.8497 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 2838/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3675 - acc: 0.8497 - val_loss: 0.3728 - val_acc: 0.8213\n Epoch 2839/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3714 - acc: 0.8476 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 2840/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3698 - acc: 0.8458 - val_loss: 0.3721 - val_acc: 0.8213\n Epoch 2841/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3711 - acc: 0.8438 - val_loss: 0.3729 - val_acc: 0.8213\n Epoch 2842/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3723 - acc: 0.8433 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 2843/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3739 - acc: 0.8450 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2844/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3672 - acc: 0.8488 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 2845/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3678 - acc: 0.8485 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 2846/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3663 - acc: 0.8507 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2847/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3691 - acc: 0.8478 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 2848/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3692 - acc: 0.8471 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 2849/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3697 - acc: 0.8469 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2850/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3685 - acc: 0.8467 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 2851/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3700 - acc: 0.8479 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 2852/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3689 - acc: 0.8485 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2853/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3668 - acc: 0.8492 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 2854/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3648 - acc: 0.8497 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 2855/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3711 - acc: 0.8457 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 2856/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3715 - acc: 0.8464 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 2857/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3662 - acc: 0.8489 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2858/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3724 - acc: 0.8486 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 2859/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3736 - acc: 0.8450 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 2860/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3681 - acc: 0.8469 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2861/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3735 - acc: 0.8456 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2862/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3725 - acc: 0.8444 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 2863/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3679 - acc: 0.8456 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 2864/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3663 - acc: 0.8504 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 2865/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3725 - acc: 0.8453 - val_loss: 0.3738 - val_acc: 0.8213\n Epoch 2866/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3714 - acc: 0.8439 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 2867/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3659 - acc: 0.8475 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 2868/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3721 - acc: 0.8440 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 2869/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3693 - acc: 0.8478 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 2870/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3695 - acc: 0.8496 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 2871/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3710 - acc: 0.8446 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 2872/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3696 - acc: 0.8479 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 2873/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3656 - acc: 0.8474 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 2874/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3680 - acc: 0.8460 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 2875/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3697 - acc: 0.8449 - val_loss: 0.3719 - val_acc: 0.8213\n Epoch 2876/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3716 - acc: 0.8438 - val_loss: 0.3725 - val_acc: 0.8213\n Epoch 2877/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3634 - acc: 0.8483 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 2878/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3699 - acc: 0.8486 - val_loss: 0.3731 - val_acc: 0.8213\n Epoch 2879/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3704 - acc: 0.8482 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 2880/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3663 - acc: 0.8478 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2881/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3706 - acc: 0.8488 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2882/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3683 - acc: 0.8479 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 2883/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3653 - acc: 0.8512 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 2884/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3654 - acc: 0.8479 - val_loss: 0.3732 - val_acc: 0.8213\n Epoch 2885/10000\n 7200/7200 [==============================] - 1s 147us/step - loss: 0.3716 - acc: 0.8440 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2886/10000\n 7200/7200 [==============================] - 1s 80us/step - loss: 0.3723 - acc: 0.8456 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 2887/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3695 - acc: 0.8494 - val_loss: 0.3749 - val_acc: 0.8213\n Epoch 2888/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3668 - acc: 0.8468 - val_loss: 0.3748 - val_acc: 0.8213\n Epoch 2889/10000\n 7200/7200 [==============================] - 1s 75us/step - loss: 0.3692 - acc: 0.8486 - val_loss: 0.3749 - val_acc: 0.8200\n Epoch 2890/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3679 - acc: 0.8454 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 2891/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3674 - acc: 0.8486 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 2892/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3675 - acc: 0.8458 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 2893/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3725 - acc: 0.8443 - val_loss: 0.3736 - val_acc: 0.8213\n Epoch 2894/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3686 - acc: 0.8440 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2895/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3685 - acc: 0.8506 - val_loss: 0.3739 - val_acc: 0.8213\n Epoch 2896/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3704 - acc: 0.8450 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2897/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3682 - acc: 0.8458 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 2898/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3670 - acc: 0.8464 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2899/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3699 - acc: 0.8468 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2900/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3723 - acc: 0.8450 - val_loss: 0.3757 - val_acc: 0.8175\n Epoch 2901/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3722 - acc: 0.8469 - val_loss: 0.3748 - val_acc: 0.8175\n Epoch 2902/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3713 - acc: 0.8449 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2903/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3738 - acc: 0.8442 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2904/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3709 - acc: 0.8453 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2905/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3696 - acc: 0.8471 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 2906/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3686 - acc: 0.8471 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 2907/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3725 - acc: 0.8425 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 2908/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3717 - acc: 0.8446 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2909/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3675 - acc: 0.8500 - val_loss: 0.3719 - val_acc: 0.8213\n Epoch 2910/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3761 - acc: 0.8422 - val_loss: 0.3731 - val_acc: 0.8213\n Epoch 2911/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3688 - acc: 0.8486 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 2912/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3681 - acc: 0.8460 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 2913/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3723 - acc: 0.8443 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 2914/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3679 - acc: 0.8461 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 2915/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3675 - acc: 0.8453 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 2916/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3638 - acc: 0.8521 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 2917/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3669 - acc: 0.8478 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2918/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3679 - acc: 0.8471 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2919/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3680 - acc: 0.8471 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2920/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3650 - acc: 0.8467 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 2921/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3675 - acc: 0.8461 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 2922/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3713 - acc: 0.8453 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 2923/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3713 - acc: 0.8476 - val_loss: 0.3732 - val_acc: 0.8200\n Epoch 2924/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3708 - acc: 0.8439 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2925/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3610 - acc: 0.8507 - val_loss: 0.3728 - val_acc: 0.8225\n Epoch 2926/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3691 - acc: 0.8468 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2927/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3688 - acc: 0.8486 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 2928/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3718 - acc: 0.8450 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2929/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3683 - acc: 0.8461 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 2930/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3661 - acc: 0.8474 - val_loss: 0.3730 - val_acc: 0.8213\n Epoch 2931/10000\n 7200/7200 [==============================] - 1s 85us/step - loss: 0.3684 - acc: 0.8462 - val_loss: 0.3730 - val_acc: 0.8213\n Epoch 2932/10000\n 7200/7200 [==============================] - 1s 84us/step - loss: 0.3678 - acc: 0.8471 - val_loss: 0.3723 - val_acc: 0.8237\n Epoch 2933/10000\n 7200/7200 [==============================] - 1s 90us/step - loss: 0.3654 - acc: 0.8524 - val_loss: 0.3721 - val_acc: 0.8200\n Epoch 2934/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3689 - acc: 0.8503 - val_loss: 0.3718 - val_acc: 0.8225\n Epoch 2935/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3706 - acc: 0.8474 - val_loss: 0.3728 - val_acc: 0.8200\n Epoch 2936/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3716 - acc: 0.8476 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2937/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3703 - acc: 0.8492 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2938/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3690 - acc: 0.8458 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2939/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3678 - acc: 0.8486 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 2940/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3691 - acc: 0.8436 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2941/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3692 - acc: 0.8472 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2942/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3693 - acc: 0.8499 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 2943/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3711 - acc: 0.8449 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 2944/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3674 - acc: 0.8464 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 2945/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3658 - acc: 0.8488 - val_loss: 0.3714 - val_acc: 0.8225\n Epoch 2946/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3687 - acc: 0.8474 - val_loss: 0.3716 - val_acc: 0.8213\n Epoch 2947/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3713 - acc: 0.8462 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 2948/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3715 - acc: 0.8460 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 2949/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3715 - acc: 0.8490 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2950/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3705 - acc: 0.8460 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 2951/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3695 - acc: 0.8486 - val_loss: 0.3732 - val_acc: 0.8175\n Epoch 2952/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3697 - acc: 0.8457 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 2953/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3707 - acc: 0.8462 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 2954/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3683 - acc: 0.8492 - val_loss: 0.3719 - val_acc: 0.8187\n Epoch 2955/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3689 - acc: 0.8471 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 2956/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3699 - acc: 0.8475 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 2957/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3721 - acc: 0.8461 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 2958/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3658 - acc: 0.8521 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 2959/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3642 - acc: 0.8489 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 2960/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3707 - acc: 0.8467 - val_loss: 0.3725 - val_acc: 0.8225\n Epoch 2961/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3708 - acc: 0.8438 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2962/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3694 - acc: 0.8451 - val_loss: 0.3752 - val_acc: 0.8175\n Epoch 2963/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3678 - acc: 0.8450 - val_loss: 0.3750 - val_acc: 0.8175\n Epoch 2964/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3687 - acc: 0.8442 - val_loss: 0.3758 - val_acc: 0.8175\n Epoch 2965/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3690 - acc: 0.8493 - val_loss: 0.3752 - val_acc: 0.8175\n Epoch 2966/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3700 - acc: 0.8461 - val_loss: 0.3763 - val_acc: 0.8175\n Epoch 2967/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3696 - acc: 0.8468 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 2968/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3658 - acc: 0.8506 - val_loss: 0.3750 - val_acc: 0.8200\n Epoch 2969/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3662 - acc: 0.8496 - val_loss: 0.3753 - val_acc: 0.8175\n Epoch 2970/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3713 - acc: 0.8469 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2971/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3666 - acc: 0.8486 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2972/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3703 - acc: 0.8453 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 2973/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3650 - acc: 0.8510 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 2974/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3687 - acc: 0.8446 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 2975/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3703 - acc: 0.8449 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 2976/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3686 - acc: 0.8492 - val_loss: 0.3765 - val_acc: 0.8187\n Epoch 2977/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3666 - acc: 0.8490 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 2978/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3692 - acc: 0.8489 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 2979/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3640 - acc: 0.8479 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 2980/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3684 - acc: 0.8467 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 2981/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3684 - acc: 0.8467 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 2982/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3696 - acc: 0.8500 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 2983/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3648 - acc: 0.8499 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2984/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3655 - acc: 0.8496 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 2985/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3652 - acc: 0.8503 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 2986/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3653 - acc: 0.8511 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 2987/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3695 - acc: 0.8461 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2988/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3670 - acc: 0.8471 - val_loss: 0.3743 - val_acc: 0.8213\n Epoch 2989/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3689 - acc: 0.8464 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 2990/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3661 - acc: 0.8464 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 2991/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3690 - acc: 0.8474 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 2992/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3716 - acc: 0.8467 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 2993/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3672 - acc: 0.8468 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 2994/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3696 - acc: 0.8465 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 2995/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3684 - acc: 0.8479 - val_loss: 0.3738 - val_acc: 0.8213\n Epoch 2996/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3702 - acc: 0.8471 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 2997/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3674 - acc: 0.8499 - val_loss: 0.3754 - val_acc: 0.8213\n Epoch 2998/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3659 - acc: 0.8501 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 2999/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3669 - acc: 0.8508 - val_loss: 0.3746 - val_acc: 0.8213\n Epoch 3000/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3630 - acc: 0.8514 - val_loss: 0.3736 - val_acc: 0.8213\n Epoch 3001/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3692 - acc: 0.8474 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 3002/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3686 - acc: 0.8482 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3003/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3665 - acc: 0.8506 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3004/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3672 - acc: 0.8512 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 3005/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3770 - acc: 0.8414 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 3006/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3705 - acc: 0.8456 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3007/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3658 - acc: 0.8511 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 3008/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3719 - acc: 0.8468 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 3009/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3694 - acc: 0.8462 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3010/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3665 - acc: 0.8500 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3011/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3703 - acc: 0.8444 - val_loss: 0.3739 - val_acc: 0.8213\n Epoch 3012/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3686 - acc: 0.8461 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3013/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3698 - acc: 0.8481 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 3014/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3700 - acc: 0.8449 - val_loss: 0.3742 - val_acc: 0.8213\n Epoch 3015/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3675 - acc: 0.8489 - val_loss: 0.3741 - val_acc: 0.8200\n Epoch 3016/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3668 - acc: 0.8471 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 3017/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3678 - acc: 0.8475 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3018/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3670 - acc: 0.8476 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3019/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3705 - acc: 0.8468 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3020/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3724 - acc: 0.8461 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3021/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3658 - acc: 0.8496 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3022/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3639 - acc: 0.8499 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3023/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3686 - acc: 0.8465 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3024/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3660 - acc: 0.8451 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3025/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3715 - acc: 0.8468 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3026/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3702 - acc: 0.8479 - val_loss: 0.3746 - val_acc: 0.8213\n Epoch 3027/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3703 - acc: 0.8501 - val_loss: 0.3744 - val_acc: 0.8213\n Epoch 3028/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3661 - acc: 0.8500 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3029/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3744 - acc: 0.8456 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3030/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3672 - acc: 0.8514 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3031/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3692 - acc: 0.8464 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 3032/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3706 - acc: 0.8478 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 3033/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3717 - acc: 0.8417 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3034/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3663 - acc: 0.8496 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3035/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3705 - acc: 0.8482 - val_loss: 0.3776 - val_acc: 0.8187\n Epoch 3036/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3703 - acc: 0.8476 - val_loss: 0.3765 - val_acc: 0.8187\n Epoch 3037/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3689 - acc: 0.8468 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 3038/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3639 - acc: 0.8501 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 3039/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3679 - acc: 0.8504 - val_loss: 0.3772 - val_acc: 0.8187\n Epoch 3040/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3671 - acc: 0.8478 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3041/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3684 - acc: 0.8499 - val_loss: 0.3752 - val_acc: 0.8213\n Epoch 3042/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3706 - acc: 0.8476 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 3043/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3688 - acc: 0.8486 - val_loss: 0.3765 - val_acc: 0.8187\n Epoch 3044/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3701 - acc: 0.8476 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 3045/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3687 - acc: 0.8464 - val_loss: 0.3772 - val_acc: 0.8187\n Epoch 3046/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3707 - acc: 0.8456 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 3047/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3755 - acc: 0.8460 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 3048/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3683 - acc: 0.8490 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3049/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3755 - acc: 0.8424 - val_loss: 0.3742 - val_acc: 0.8200\n Epoch 3050/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3689 - acc: 0.8478 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3051/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3674 - acc: 0.8482 - val_loss: 0.3732 - val_acc: 0.8213\n Epoch 3052/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3680 - acc: 0.8511 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 3053/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3645 - acc: 0.8511 - val_loss: 0.3727 - val_acc: 0.8213\n Epoch 3054/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3673 - acc: 0.8464 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 3055/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3671 - acc: 0.8475 - val_loss: 0.3729 - val_acc: 0.8213\n Epoch 3056/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3673 - acc: 0.8457 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3057/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3666 - acc: 0.8507 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3058/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3699 - acc: 0.8457 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3059/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3664 - acc: 0.8522 - val_loss: 0.3742 - val_acc: 0.8200\n Epoch 3060/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3739 - acc: 0.8429 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3061/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3712 - acc: 0.8453 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3062/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3728 - acc: 0.8450 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3063/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3701 - acc: 0.8471 - val_loss: 0.3730 - val_acc: 0.8200\n Epoch 3064/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3646 - acc: 0.8504 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 3065/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3689 - acc: 0.8482 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 3066/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3739 - acc: 0.8444 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 3067/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3660 - acc: 0.8496 - val_loss: 0.3731 - val_acc: 0.8213\n Epoch 3068/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3700 - acc: 0.8476 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3069/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3700 - acc: 0.8438 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3070/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3712 - acc: 0.8471 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 3071/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3668 - acc: 0.8518 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3072/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3701 - acc: 0.8456 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3073/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3689 - acc: 0.8493 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3074/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3651 - acc: 0.8507 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3075/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3711 - acc: 0.8468 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 3076/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3699 - acc: 0.8512 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 3077/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3658 - acc: 0.8460 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3078/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3738 - acc: 0.8506 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 3079/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3712 - acc: 0.8494 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3080/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3688 - acc: 0.8494 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3081/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3671 - acc: 0.8482 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3082/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3721 - acc: 0.8485 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 3083/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3669 - acc: 0.8494 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3084/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3654 - acc: 0.8469 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 3085/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3696 - acc: 0.8451 - val_loss: 0.3736 - val_acc: 0.8213\n Epoch 3086/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3733 - acc: 0.8440 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3087/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3671 - acc: 0.8486 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3088/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3682 - acc: 0.8468 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3089/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3722 - acc: 0.8475 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3090/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3708 - acc: 0.8447 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3091/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3684 - acc: 0.8506 - val_loss: 0.3724 - val_acc: 0.8213\n Epoch 3092/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3710 - acc: 0.8447 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 3093/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3647 - acc: 0.8492 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 3094/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3693 - acc: 0.8467 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 3095/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3683 - acc: 0.8478 - val_loss: 0.3740 - val_acc: 0.8175\n Epoch 3096/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3709 - acc: 0.8446 - val_loss: 0.3744 - val_acc: 0.8175\n Epoch 3097/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3715 - acc: 0.8474 - val_loss: 0.3729 - val_acc: 0.8200\n Epoch 3098/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3652 - acc: 0.8486 - val_loss: 0.3736 - val_acc: 0.8175\n Epoch 3099/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3729 - acc: 0.8460 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3100/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3683 - acc: 0.8485 - val_loss: 0.3742 - val_acc: 0.8175\n Epoch 3101/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3673 - acc: 0.8457 - val_loss: 0.3741 - val_acc: 0.8175\n Epoch 3102/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3685 - acc: 0.8511 - val_loss: 0.3730 - val_acc: 0.8200\n Epoch 3103/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3710 - acc: 0.8450 - val_loss: 0.3747 - val_acc: 0.8175\n Epoch 3104/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3688 - acc: 0.8464 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 3105/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3634 - acc: 0.8506 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3106/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3755 - acc: 0.8393 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3107/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3661 - acc: 0.8488 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 3108/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3680 - acc: 0.8499 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3109/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3646 - acc: 0.8510 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 3110/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3722 - acc: 0.8444 - val_loss: 0.3739 - val_acc: 0.8175\n Epoch 3111/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3723 - acc: 0.8449 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3112/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3669 - acc: 0.8471 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3113/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3667 - acc: 0.8482 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3114/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3699 - acc: 0.8485 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3115/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3726 - acc: 0.8460 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3116/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3655 - acc: 0.8500 - val_loss: 0.3744 - val_acc: 0.8175\n Epoch 3117/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3677 - acc: 0.8489 - val_loss: 0.3744 - val_acc: 0.8175\n Epoch 3118/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3738 - acc: 0.8451 - val_loss: 0.3756 - val_acc: 0.8175\n Epoch 3119/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3687 - acc: 0.8476 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 3120/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3621 - acc: 0.8493 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3121/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3701 - acc: 0.8450 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3122/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3678 - acc: 0.8468 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 3123/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3728 - acc: 0.8472 - val_loss: 0.3741 - val_acc: 0.8175\n Epoch 3124/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3659 - acc: 0.8490 - val_loss: 0.3723 - val_acc: 0.8237\n Epoch 3125/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3686 - acc: 0.8472 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 3126/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3649 - acc: 0.8499 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3127/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3746 - acc: 0.8454 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3128/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3674 - acc: 0.8490 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3129/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3651 - acc: 0.8503 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3130/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3711 - acc: 0.8454 - val_loss: 0.3750 - val_acc: 0.8175\n Epoch 3131/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3648 - acc: 0.8478 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 3132/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3704 - acc: 0.8457 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 3133/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3697 - acc: 0.8435 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 3134/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3666 - acc: 0.8469 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3135/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3680 - acc: 0.8465 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3136/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3679 - acc: 0.8485 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3137/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3733 - acc: 0.8446 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3138/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3668 - acc: 0.8471 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3139/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3740 - acc: 0.8457 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 3140/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3734 - acc: 0.8422 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3141/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3751 - acc: 0.8451 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3142/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3663 - acc: 0.8461 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3143/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3646 - acc: 0.8510 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3144/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3666 - acc: 0.8460 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3145/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3731 - acc: 0.8467 - val_loss: 0.3745 - val_acc: 0.8200\n Epoch 3146/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3712 - acc: 0.8457 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 3147/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3689 - acc: 0.8493 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3148/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3668 - acc: 0.8490 - val_loss: 0.3741 - val_acc: 0.8200\n Epoch 3149/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3714 - acc: 0.8482 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3150/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3625 - acc: 0.8496 - val_loss: 0.3737 - val_acc: 0.8200\n Epoch 3151/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3688 - acc: 0.8462 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3152/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3636 - acc: 0.8488 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 3153/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3655 - acc: 0.8504 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3154/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3702 - acc: 0.8469 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 3155/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3651 - acc: 0.8485 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3156/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3666 - acc: 0.8461 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3157/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3707 - acc: 0.8453 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 3158/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3697 - acc: 0.8481 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 3159/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3722 - acc: 0.8451 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3160/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3720 - acc: 0.8467 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3161/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3694 - acc: 0.8460 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 3162/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3648 - acc: 0.8481 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 3163/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3686 - acc: 0.8488 - val_loss: 0.3739 - val_acc: 0.8213\n Epoch 3164/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3694 - acc: 0.8464 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 3165/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3719 - acc: 0.8472 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 3166/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3670 - acc: 0.8490 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 3167/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3636 - acc: 0.8489 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 3168/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3664 - acc: 0.8483 - val_loss: 0.3736 - val_acc: 0.8200\n Epoch 3169/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3700 - acc: 0.8447 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3170/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3736 - acc: 0.8439 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3171/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3669 - acc: 0.8486 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3172/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3722 - acc: 0.8474 - val_loss: 0.3731 - val_acc: 0.8213\n Epoch 3173/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3663 - acc: 0.8496 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3174/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3671 - acc: 0.8471 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3175/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3713 - acc: 0.8476 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3176/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3679 - acc: 0.8460 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3177/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3689 - acc: 0.8479 - val_loss: 0.3751 - val_acc: 0.8175\n Epoch 3178/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3662 - acc: 0.8479 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3179/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3653 - acc: 0.8503 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3180/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3700 - acc: 0.8444 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3181/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3717 - acc: 0.8481 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3182/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3680 - acc: 0.8458 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 3183/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3682 - acc: 0.8467 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 3184/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3651 - acc: 0.8485 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3185/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3695 - acc: 0.8458 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3186/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3683 - acc: 0.8478 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 3187/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3703 - acc: 0.8471 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3188/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3665 - acc: 0.8494 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 3189/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3671 - acc: 0.8494 - val_loss: 0.3744 - val_acc: 0.8213\n Epoch 3190/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3682 - acc: 0.8461 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3191/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3689 - acc: 0.8469 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 3192/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3665 - acc: 0.8504 - val_loss: 0.3735 - val_acc: 0.8200\n Epoch 3193/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3699 - acc: 0.8465 - val_loss: 0.3727 - val_acc: 0.8213\n Epoch 3194/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3659 - acc: 0.8476 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 3195/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3672 - acc: 0.8465 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 3196/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3680 - acc: 0.8469 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 3197/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3650 - acc: 0.8486 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 3198/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3705 - acc: 0.8469 - val_loss: 0.3732 - val_acc: 0.8213\n Epoch 3199/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3681 - acc: 0.8482 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3200/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3634 - acc: 0.8511 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3201/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3713 - acc: 0.8449 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3202/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3719 - acc: 0.8476 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 3203/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3651 - acc: 0.8475 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 3204/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3681 - acc: 0.8478 - val_loss: 0.3738 - val_acc: 0.8213\n Epoch 3205/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3683 - acc: 0.8475 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3206/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3703 - acc: 0.8457 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 3207/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3679 - acc: 0.8488 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 3208/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3708 - acc: 0.8476 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3209/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3641 - acc: 0.8493 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 3210/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3691 - acc: 0.8478 - val_loss: 0.3732 - val_acc: 0.8213\n Epoch 3211/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3678 - acc: 0.8465 - val_loss: 0.3729 - val_acc: 0.8213\n Epoch 3212/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3703 - acc: 0.8451 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3213/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3669 - acc: 0.8511 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3214/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3681 - acc: 0.8499 - val_loss: 0.3744 - val_acc: 0.8213\n Epoch 3215/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3701 - acc: 0.8454 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3216/10000\n 7200/7200 [==============================] - 1s 83us/step - loss: 0.3731 - acc: 0.8465 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3217/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3703 - acc: 0.8456 - val_loss: 0.3743 - val_acc: 0.8200\n Epoch 3218/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3724 - acc: 0.8436 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3219/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3674 - acc: 0.8508 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3220/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3668 - acc: 0.8488 - val_loss: 0.3747 - val_acc: 0.8200\n Epoch 3221/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3665 - acc: 0.8506 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 3222/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3686 - acc: 0.8469 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3223/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3706 - acc: 0.8462 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 3224/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3722 - acc: 0.8462 - val_loss: 0.3739 - val_acc: 0.8213\n Epoch 3225/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3671 - acc: 0.8522 - val_loss: 0.3747 - val_acc: 0.8213\n Epoch 3226/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3665 - acc: 0.8481 - val_loss: 0.3743 - val_acc: 0.8213\n Epoch 3227/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3678 - acc: 0.8496 - val_loss: 0.3739 - val_acc: 0.8213\n Epoch 3228/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3665 - acc: 0.8494 - val_loss: 0.3731 - val_acc: 0.8213\n Epoch 3229/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3713 - acc: 0.8418 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 3230/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3713 - acc: 0.8458 - val_loss: 0.3731 - val_acc: 0.8213\n Epoch 3231/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3698 - acc: 0.8472 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3232/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3669 - acc: 0.8469 - val_loss: 0.3745 - val_acc: 0.8213\n Epoch 3233/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3679 - acc: 0.8504 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3234/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3702 - acc: 0.8490 - val_loss: 0.3753 - val_acc: 0.8200\n Epoch 3235/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3746 - acc: 0.8454 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 3236/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3684 - acc: 0.8457 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 3237/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3741 - acc: 0.8424 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3238/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3707 - acc: 0.8458 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 3239/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3694 - acc: 0.8482 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 3240/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3677 - acc: 0.8453 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 3241/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3669 - acc: 0.8450 - val_loss: 0.3726 - val_acc: 0.8213\n Epoch 3242/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3663 - acc: 0.8476 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 3243/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3661 - acc: 0.8492 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3244/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3699 - acc: 0.8485 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 3245/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3688 - acc: 0.8471 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 3246/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3670 - acc: 0.8468 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 3247/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3662 - acc: 0.8494 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3248/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3663 - acc: 0.8467 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3249/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3704 - acc: 0.8449 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 3250/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3676 - acc: 0.8456 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 3251/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3723 - acc: 0.8454 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3252/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3714 - acc: 0.8438 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3253/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3679 - acc: 0.8456 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3254/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3742 - acc: 0.8457 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3255/10000\n 7200/7200 [==============================] - 0s 57us/step - loss: 0.3722 - acc: 0.8451 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3256/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3688 - acc: 0.8479 - val_loss: 0.3742 - val_acc: 0.8213\n Epoch 3257/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3668 - acc: 0.8519 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3258/10000\n 7200/7200 [==============================] - 1s 118us/step - loss: 0.3699 - acc: 0.8460 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3259/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3675 - acc: 0.8461 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3260/10000\n 7200/7200 [==============================] - 1s 82us/step - loss: 0.3718 - acc: 0.8475 - val_loss: 0.3746 - val_acc: 0.8200\n Epoch 3261/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3702 - acc: 0.8488 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 3262/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3624 - acc: 0.8510 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3263/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3726 - acc: 0.8449 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 3264/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3702 - acc: 0.8503 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 3265/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3693 - acc: 0.8479 - val_loss: 0.3742 - val_acc: 0.8213\n Epoch 3266/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3665 - acc: 0.8497 - val_loss: 0.3731 - val_acc: 0.8213\n Epoch 3267/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3637 - acc: 0.8514 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 3268/10000\n 7200/7200 [==============================] - 1s 91us/step - loss: 0.3669 - acc: 0.8453 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3269/10000\n 7200/7200 [==============================] - 1s 90us/step - loss: 0.3700 - acc: 0.8465 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3270/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3666 - acc: 0.8474 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3271/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3688 - acc: 0.8456 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3272/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3694 - acc: 0.8532 - val_loss: 0.3722 - val_acc: 0.8225\n Epoch 3273/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3691 - acc: 0.8469 - val_loss: 0.3728 - val_acc: 0.8213\n Epoch 3274/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3732 - acc: 0.8465 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 3275/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3699 - acc: 0.8474 - val_loss: 0.3736 - val_acc: 0.8213\n Epoch 3276/10000\n 7200/7200 [==============================] - 1s 99us/step - loss: 0.3698 - acc: 0.8479 - val_loss: 0.3738 - val_acc: 0.8213\n Epoch 3277/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3680 - acc: 0.8457 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 3278/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3691 - acc: 0.8444 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 3279/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3721 - acc: 0.8449 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3280/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3693 - acc: 0.8521 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3281/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3669 - acc: 0.8472 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3282/10000\n 7200/7200 [==============================] - 1s 89us/step - loss: 0.3703 - acc: 0.8435 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 3283/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3672 - acc: 0.8485 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 3284/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3663 - acc: 0.8490 - val_loss: 0.3745 - val_acc: 0.8213\n Epoch 3285/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3696 - acc: 0.8489 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3286/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3665 - acc: 0.8507 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3287/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3675 - acc: 0.8506 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3288/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3692 - acc: 0.8489 - val_loss: 0.3742 - val_acc: 0.8213\n Epoch 3289/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3667 - acc: 0.8501 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 3290/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3713 - acc: 0.8457 - val_loss: 0.3742 - val_acc: 0.8213\n Epoch 3291/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3699 - acc: 0.8464 - val_loss: 0.3764 - val_acc: 0.8175\n Epoch 3292/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3685 - acc: 0.8456 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 3293/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3690 - acc: 0.8515 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 3294/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3676 - acc: 0.8485 - val_loss: 0.3763 - val_acc: 0.8175\n Epoch 3295/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3709 - acc: 0.8456 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3296/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3687 - acc: 0.8486 - val_loss: 0.3748 - val_acc: 0.8200\n Epoch 3297/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3697 - acc: 0.8472 - val_loss: 0.3756 - val_acc: 0.8175\n Epoch 3298/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3703 - acc: 0.8467 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3299/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3681 - acc: 0.8457 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 3300/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3688 - acc: 0.8486 - val_loss: 0.3754 - val_acc: 0.8213\n Epoch 3301/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3720 - acc: 0.8471 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3302/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3723 - acc: 0.8468 - val_loss: 0.3751 - val_acc: 0.8213\n Epoch 3303/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3669 - acc: 0.8468 - val_loss: 0.3744 - val_acc: 0.8213\n Epoch 3304/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3632 - acc: 0.8514 - val_loss: 0.3753 - val_acc: 0.8175\n Epoch 3305/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3679 - acc: 0.8451 - val_loss: 0.3749 - val_acc: 0.8213\n Epoch 3306/10000\n 7200/7200 [==============================] - 1s 91us/step - loss: 0.3685 - acc: 0.8489 - val_loss: 0.3764 - val_acc: 0.8175\n Epoch 3307/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3648 - acc: 0.8500 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 3308/10000\n 7200/7200 [==============================] - 1s 80us/step - loss: 0.3743 - acc: 0.8436 - val_loss: 0.3755 - val_acc: 0.8213\n Epoch 3309/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3704 - acc: 0.8494 - val_loss: 0.3758 - val_acc: 0.8213\n Epoch 3310/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3701 - acc: 0.8475 - val_loss: 0.3743 - val_acc: 0.8213\n Epoch 3311/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3687 - acc: 0.8465 - val_loss: 0.3752 - val_acc: 0.8213\n Epoch 3312/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3668 - acc: 0.8503 - val_loss: 0.3746 - val_acc: 0.8213\n Epoch 3313/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3658 - acc: 0.8465 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 3314/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3632 - acc: 0.8501 - val_loss: 0.3754 - val_acc: 0.8200\n Epoch 3315/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3678 - acc: 0.8467 - val_loss: 0.3750 - val_acc: 0.8200\n Epoch 3316/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3729 - acc: 0.8444 - val_loss: 0.3755 - val_acc: 0.8175\n Epoch 3317/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3681 - acc: 0.8462 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3318/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3700 - acc: 0.8442 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3319/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3734 - acc: 0.8439 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3320/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3661 - acc: 0.8475 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3321/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3748 - acc: 0.8450 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 3322/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3689 - acc: 0.8460 - val_loss: 0.3773 - val_acc: 0.8187\n Epoch 3323/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3688 - acc: 0.8456 - val_loss: 0.3767 - val_acc: 0.8187\n Epoch 3324/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3690 - acc: 0.8449 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 3325/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3682 - acc: 0.8483 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3326/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3737 - acc: 0.8440 - val_loss: 0.3756 - val_acc: 0.8200\n Epoch 3327/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3655 - acc: 0.8507 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 3328/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8497 - val_loss: 0.3755 - val_acc: 0.8213\n Epoch 3329/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3688 - acc: 0.8464 - val_loss: 0.3770 - val_acc: 0.8175\n Epoch 3330/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3682 - acc: 0.8469 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3331/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3689 - acc: 0.8483 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 3332/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3694 - acc: 0.8482 - val_loss: 0.3767 - val_acc: 0.8187\n Epoch 3333/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3677 - acc: 0.8485 - val_loss: 0.3770 - val_acc: 0.8187\n Epoch 3334/10000\n 7200/7200 [==============================] - 1s 143us/step - loss: 0.3677 - acc: 0.8464 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 3335/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3696 - acc: 0.8464 - val_loss: 0.3770 - val_acc: 0.8187\n Epoch 3336/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3711 - acc: 0.8433 - val_loss: 0.3781 - val_acc: 0.8187\n Epoch 3337/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3693 - acc: 0.8443 - val_loss: 0.3771 - val_acc: 0.8187\n Epoch 3338/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3751 - acc: 0.8462 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3339/10000\n 7200/7200 [==============================] - 1s 96us/step - loss: 0.3677 - acc: 0.8471 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 3340/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3699 - acc: 0.8450 - val_loss: 0.3769 - val_acc: 0.8187\n Epoch 3341/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3710 - acc: 0.8472 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 3342/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3703 - acc: 0.8447 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 3343/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3672 - acc: 0.8499 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 3344/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3687 - acc: 0.8497 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 3345/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3653 - acc: 0.8501 - val_loss: 0.3749 - val_acc: 0.8213\n Epoch 3346/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3665 - acc: 0.8506 - val_loss: 0.3772 - val_acc: 0.8187\n Epoch 3347/10000\n 7200/7200 [==============================] - 1s 87us/step - loss: 0.3636 - acc: 0.8496 - val_loss: 0.3772 - val_acc: 0.8187\n Epoch 3348/10000\n 7200/7200 [==============================] - 1s 87us/step - loss: 0.3739 - acc: 0.8464 - val_loss: 0.3764 - val_acc: 0.8187\n Epoch 3349/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3723 - acc: 0.8442 - val_loss: 0.3764 - val_acc: 0.8187\n Epoch 3350/10000\n 7200/7200 [==============================] - 1s 85us/step - loss: 0.3664 - acc: 0.8479 - val_loss: 0.3777 - val_acc: 0.8187\n Epoch 3351/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3737 - acc: 0.8417 - val_loss: 0.3767 - val_acc: 0.8187\n Epoch 3352/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3673 - acc: 0.8500 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 3353/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3650 - acc: 0.8489 - val_loss: 0.3765 - val_acc: 0.8187\n Epoch 3354/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3691 - acc: 0.8447 - val_loss: 0.3765 - val_acc: 0.8187\n Epoch 3355/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3699 - acc: 0.8479 - val_loss: 0.3779 - val_acc: 0.8187\n Epoch 3356/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3656 - acc: 0.8492 - val_loss: 0.3774 - val_acc: 0.8213\n Epoch 3357/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3670 - acc: 0.8469 - val_loss: 0.3762 - val_acc: 0.8213\n Epoch 3358/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3657 - acc: 0.8481 - val_loss: 0.3758 - val_acc: 0.8213\n Epoch 3359/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3661 - acc: 0.8485 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 3360/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3700 - acc: 0.8468 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3361/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3692 - acc: 0.8479 - val_loss: 0.3768 - val_acc: 0.8175\n Epoch 3362/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3677 - acc: 0.8496 - val_loss: 0.3763 - val_acc: 0.8175\n Epoch 3363/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3626 - acc: 0.8500 - val_loss: 0.3748 - val_acc: 0.8213\n Epoch 3364/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3690 - acc: 0.8472 - val_loss: 0.3756 - val_acc: 0.8200\n Epoch 3365/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3726 - acc: 0.8494 - val_loss: 0.3754 - val_acc: 0.8213\n Epoch 3366/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3697 - acc: 0.8443 - val_loss: 0.3764 - val_acc: 0.8187\n Epoch 3367/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3669 - acc: 0.8511 - val_loss: 0.3768 - val_acc: 0.8175\n Epoch 3368/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3602 - acc: 0.8501 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 3369/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3732 - acc: 0.8497 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 3370/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3683 - acc: 0.8464 - val_loss: 0.3767 - val_acc: 0.8187\n Epoch 3371/10000\n 7200/7200 [==============================] - 1s 83us/step - loss: 0.3714 - acc: 0.8462 - val_loss: 0.3767 - val_acc: 0.8187\n Epoch 3372/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3672 - acc: 0.8471 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3373/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3684 - acc: 0.8485 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3374/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3699 - acc: 0.8474 - val_loss: 0.3742 - val_acc: 0.8213\n Epoch 3375/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3685 - acc: 0.8472 - val_loss: 0.3738 - val_acc: 0.8213\n Epoch 3376/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3665 - acc: 0.8446 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3377/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3669 - acc: 0.8478 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3378/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3612 - acc: 0.8522 - val_loss: 0.3738 - val_acc: 0.8213\n Epoch 3379/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3667 - acc: 0.8500 - val_loss: 0.3736 - val_acc: 0.8225\n Epoch 3380/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3694 - acc: 0.8475 - val_loss: 0.3745 - val_acc: 0.8213\n Epoch 3381/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3742 - acc: 0.8444 - val_loss: 0.3744 - val_acc: 0.8213\n Epoch 3382/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3708 - acc: 0.8511 - val_loss: 0.3748 - val_acc: 0.8213\n Epoch 3383/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3644 - acc: 0.8503 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 3384/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3682 - acc: 0.8467 - val_loss: 0.3767 - val_acc: 0.8187\n Epoch 3385/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3711 - acc: 0.8457 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3386/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3697 - acc: 0.8482 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3387/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3668 - acc: 0.8493 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 3388/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3611 - acc: 0.8517 - val_loss: 0.3741 - val_acc: 0.8200\n Epoch 3389/10000\n 7200/7200 [==============================] - 1s 86us/step - loss: 0.3697 - acc: 0.8489 - val_loss: 0.3746 - val_acc: 0.8200\n Epoch 3390/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3677 - acc: 0.8479 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 3391/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3730 - acc: 0.8457 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3392/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3700 - acc: 0.8476 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3393/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3648 - acc: 0.8476 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3394/10000\n 7200/7200 [==============================] - 1s 88us/step - loss: 0.3702 - acc: 0.8457 - val_loss: 0.3735 - val_acc: 0.8225\n Epoch 3395/10000\n 7200/7200 [==============================] - 1s 117us/step - loss: 0.3662 - acc: 0.8468 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3396/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3724 - acc: 0.8462 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3397/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3754 - acc: 0.8461 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3398/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3655 - acc: 0.8472 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3399/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3725 - acc: 0.8461 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 3400/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3670 - acc: 0.8511 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 3401/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3698 - acc: 0.8474 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3402/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3652 - acc: 0.8507 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 3403/10000\n 7200/7200 [==============================] - 1s 132us/step - loss: 0.3702 - acc: 0.8454 - val_loss: 0.3769 - val_acc: 0.8187\n Epoch 3404/10000\n 7200/7200 [==============================] - 1s 98us/step - loss: 0.3659 - acc: 0.8483 - val_loss: 0.3769 - val_acc: 0.8187\n Epoch 3405/10000\n 7200/7200 [==============================] - 1s 93us/step - loss: 0.3724 - acc: 0.8467 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 3406/10000\n 7200/7200 [==============================] - 1s 80us/step - loss: 0.3680 - acc: 0.8475 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 3407/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3689 - acc: 0.8478 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 3408/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3642 - acc: 0.8529 - val_loss: 0.3745 - val_acc: 0.8225\n Epoch 3409/10000\n 7200/7200 [==============================] - 1s 116us/step - loss: 0.3726 - acc: 0.8457 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3410/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3703 - acc: 0.8467 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3411/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3704 - acc: 0.8444 - val_loss: 0.3768 - val_acc: 0.8187\n Epoch 3412/10000\n 7200/7200 [==============================] - 1s 87us/step - loss: 0.3715 - acc: 0.8457 - val_loss: 0.3770 - val_acc: 0.8187\n Epoch 3413/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3685 - acc: 0.8472 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 3414/10000\n 7200/7200 [==============================] - 1s 91us/step - loss: 0.3714 - acc: 0.8435 - val_loss: 0.3745 - val_acc: 0.8225\n Epoch 3415/10000\n 7200/7200 [==============================] - 1s 91us/step - loss: 0.3625 - acc: 0.8519 - val_loss: 0.3750 - val_acc: 0.8213\n Epoch 3416/10000\n 7200/7200 [==============================] - 1s 135us/step - loss: 0.3657 - acc: 0.8488 - val_loss: 0.3768 - val_acc: 0.8187\n Epoch 3417/10000\n 7200/7200 [==============================] - 1s 82us/step - loss: 0.3672 - acc: 0.8483 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 3418/10000\n 7200/7200 [==============================] - 1s 101us/step - loss: 0.3657 - acc: 0.8494 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 3419/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3691 - acc: 0.8444 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 3420/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3664 - acc: 0.8500 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3421/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3631 - acc: 0.8508 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3422/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3685 - acc: 0.8454 - val_loss: 0.3764 - val_acc: 0.8187\n Epoch 3423/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3679 - acc: 0.8464 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3424/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3745 - acc: 0.8451 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 3425/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3683 - acc: 0.8488 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 3426/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3730 - acc: 0.8433 - val_loss: 0.3767 - val_acc: 0.8187\n Epoch 3427/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3725 - acc: 0.8468 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 3428/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3619 - acc: 0.8522 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3429/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3705 - acc: 0.8467 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 3430/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3663 - acc: 0.8481 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 3431/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3691 - acc: 0.8469 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3432/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3620 - acc: 0.8504 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 3433/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3663 - acc: 0.8512 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3434/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3646 - acc: 0.8501 - val_loss: 0.3743 - val_acc: 0.8200\n Epoch 3435/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3718 - acc: 0.8447 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3436/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3673 - acc: 0.8465 - val_loss: 0.3740 - val_acc: 0.8200\n Epoch 3437/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3712 - acc: 0.8456 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3438/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3722 - acc: 0.8446 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 3439/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3692 - acc: 0.8469 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 3440/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3714 - acc: 0.8442 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3441/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3700 - acc: 0.8450 - val_loss: 0.3771 - val_acc: 0.8187\n Epoch 3442/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3653 - acc: 0.8503 - val_loss: 0.3765 - val_acc: 0.8175\n Epoch 3443/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3669 - acc: 0.8497 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3444/10000\n 7200/7200 [==============================] - 1s 130us/step - loss: 0.3697 - acc: 0.8488 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3445/10000\n 7200/7200 [==============================] - 1s 102us/step - loss: 0.3686 - acc: 0.8450 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 3446/10000\n 7200/7200 [==============================] - 1s 80us/step - loss: 0.3647 - acc: 0.8476 - val_loss: 0.3744 - val_acc: 0.8213\n Epoch 3447/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3688 - acc: 0.8461 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 3448/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3718 - acc: 0.8414 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 3449/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3685 - acc: 0.8464 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 3450/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3720 - acc: 0.8458 - val_loss: 0.3770 - val_acc: 0.8187\n Epoch 3451/10000\n 7200/7200 [==============================] - 1s 105us/step - loss: 0.3694 - acc: 0.8472 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 3452/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3654 - acc: 0.8476 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3453/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3681 - acc: 0.8483 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 3454/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3603 - acc: 0.8492 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3455/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3669 - acc: 0.8486 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3456/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3638 - acc: 0.8497 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 3457/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3678 - acc: 0.8488 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3458/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3630 - acc: 0.8506 - val_loss: 0.3769 - val_acc: 0.8187\n Epoch 3459/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3725 - acc: 0.8443 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3460/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3696 - acc: 0.8468 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3461/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3714 - acc: 0.8453 - val_loss: 0.3764 - val_acc: 0.8187\n Epoch 3462/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3616 - acc: 0.8531 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3463/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3669 - acc: 0.8497 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 3464/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3678 - acc: 0.8462 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 3465/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3635 - acc: 0.8528 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3466/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3717 - acc: 0.8451 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3467/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3660 - acc: 0.8462 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3468/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3696 - acc: 0.8460 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 3469/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3694 - acc: 0.8492 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3470/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3651 - acc: 0.8499 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 3471/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3661 - acc: 0.8494 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3472/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3703 - acc: 0.8485 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3473/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3693 - acc: 0.8483 - val_loss: 0.3743 - val_acc: 0.8213\n Epoch 3474/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3661 - acc: 0.8489 - val_loss: 0.3751 - val_acc: 0.8213\n Epoch 3475/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3662 - acc: 0.8501 - val_loss: 0.3749 - val_acc: 0.8213\n Epoch 3476/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3694 - acc: 0.8469 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 3477/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3683 - acc: 0.8482 - val_loss: 0.3759 - val_acc: 0.8175\n Epoch 3478/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3672 - acc: 0.8494 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3479/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3684 - acc: 0.8471 - val_loss: 0.3757 - val_acc: 0.8175\n Epoch 3480/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3693 - acc: 0.8476 - val_loss: 0.3755 - val_acc: 0.8175\n Epoch 3481/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3690 - acc: 0.8456 - val_loss: 0.3758 - val_acc: 0.8175\n Epoch 3482/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3707 - acc: 0.8460 - val_loss: 0.3747 - val_acc: 0.8175\n Epoch 3483/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3719 - acc: 0.8422 - val_loss: 0.3749 - val_acc: 0.8175\n Epoch 3484/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3720 - acc: 0.8467 - val_loss: 0.3752 - val_acc: 0.8200\n Epoch 3485/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3639 - acc: 0.8475 - val_loss: 0.3744 - val_acc: 0.8213\n Epoch 3486/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3692 - acc: 0.8456 - val_loss: 0.3744 - val_acc: 0.8200\n Epoch 3487/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3604 - acc: 0.8508 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 3488/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3689 - acc: 0.8446 - val_loss: 0.3745 - val_acc: 0.8213\n Epoch 3489/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3672 - acc: 0.8469 - val_loss: 0.3744 - val_acc: 0.8213\n Epoch 3490/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3722 - acc: 0.8486 - val_loss: 0.3749 - val_acc: 0.8213\n Epoch 3491/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3678 - acc: 0.8468 - val_loss: 0.3756 - val_acc: 0.8175\n Epoch 3492/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3666 - acc: 0.8494 - val_loss: 0.3760 - val_acc: 0.8175\n Epoch 3493/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3715 - acc: 0.8451 - val_loss: 0.3757 - val_acc: 0.8200\n Epoch 3494/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3685 - acc: 0.8447 - val_loss: 0.3751 - val_acc: 0.8175\n Epoch 3495/10000\n 7200/7200 [==============================] - 1s 123us/step - loss: 0.3683 - acc: 0.8464 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 3496/10000\n 7200/7200 [==============================] - 1s 94us/step - loss: 0.3656 - acc: 0.8471 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3497/10000\n 7200/7200 [==============================] - 1s 84us/step - loss: 0.3680 - acc: 0.8493 - val_loss: 0.3746 - val_acc: 0.8200\n Epoch 3498/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3619 - acc: 0.8524 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3499/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3693 - acc: 0.8462 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 3500/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3658 - acc: 0.8497 - val_loss: 0.3732 - val_acc: 0.8213\n Epoch 3501/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3697 - acc: 0.8451 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3502/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3699 - acc: 0.8468 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3503/10000\n 7200/7200 [==============================] - 1s 80us/step - loss: 0.3689 - acc: 0.8476 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3504/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3684 - acc: 0.8415 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 3505/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3703 - acc: 0.8451 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3506/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3693 - acc: 0.8471 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 3507/10000\n 7200/7200 [==============================] - 1s 80us/step - loss: 0.3684 - acc: 0.8506 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3508/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3721 - acc: 0.8446 - val_loss: 0.3745 - val_acc: 0.8213\n Epoch 3509/10000\n 7200/7200 [==============================] - 1s 82us/step - loss: 0.3647 - acc: 0.8528 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 3510/10000\n 7200/7200 [==============================] - 1s 98us/step - loss: 0.3672 - acc: 0.8476 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 3511/10000\n 7200/7200 [==============================] - 1s 83us/step - loss: 0.3713 - acc: 0.8442 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3512/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3701 - acc: 0.8435 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3513/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3660 - acc: 0.8468 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3514/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3647 - acc: 0.8512 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3515/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3635 - acc: 0.8533 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3516/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3660 - acc: 0.8508 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 3517/10000\n 7200/7200 [==============================] - 1s 83us/step - loss: 0.3661 - acc: 0.8490 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 3518/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3661 - acc: 0.8499 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 3519/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3646 - acc: 0.8499 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 3520/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3657 - acc: 0.8474 - val_loss: 0.3725 - val_acc: 0.8200\n Epoch 3521/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3739 - acc: 0.8462 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3522/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3698 - acc: 0.8472 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3523/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3703 - acc: 0.8483 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3524/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3698 - acc: 0.8449 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 3525/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3652 - acc: 0.8486 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3526/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3685 - acc: 0.8469 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3527/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3699 - acc: 0.8469 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 3528/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3684 - acc: 0.8471 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3529/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3705 - acc: 0.8497 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 3530/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3663 - acc: 0.8493 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3531/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3687 - acc: 0.8447 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3532/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3696 - acc: 0.8478 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 3533/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3679 - acc: 0.8464 - val_loss: 0.3764 - val_acc: 0.8187\n Epoch 3534/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3718 - acc: 0.8443 - val_loss: 0.3771 - val_acc: 0.8187\n Epoch 3535/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3677 - acc: 0.8481 - val_loss: 0.3772 - val_acc: 0.8187\n Epoch 3536/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3704 - acc: 0.8469 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3537/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3667 - acc: 0.8475 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3538/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3656 - acc: 0.8485 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3539/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3624 - acc: 0.8521 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 3540/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3703 - acc: 0.8447 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 3541/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3675 - acc: 0.8472 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3542/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3657 - acc: 0.8454 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3543/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3728 - acc: 0.8449 - val_loss: 0.3767 - val_acc: 0.8187\n Epoch 3544/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3689 - acc: 0.8478 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3545/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3693 - acc: 0.8453 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 3546/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3643 - acc: 0.8494 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3547/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3733 - acc: 0.8486 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3548/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3667 - acc: 0.8500 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3549/10000\n 7200/7200 [==============================] - 1s 92us/step - loss: 0.3671 - acc: 0.8524 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3550/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3676 - acc: 0.8482 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3551/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3677 - acc: 0.8478 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3552/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3708 - acc: 0.8458 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3553/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3672 - acc: 0.8485 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3554/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3656 - acc: 0.8501 - val_loss: 0.3730 - val_acc: 0.8225\n Epoch 3555/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3674 - acc: 0.8483 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3556/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3692 - acc: 0.8476 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3557/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3671 - acc: 0.8482 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3558/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3677 - acc: 0.8475 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3559/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3678 - acc: 0.8488 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3560/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3708 - acc: 0.8446 - val_loss: 0.3752 - val_acc: 0.8175\n Epoch 3561/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3679 - acc: 0.8489 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3562/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3693 - acc: 0.8465 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 3563/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3701 - acc: 0.8464 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3564/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3720 - acc: 0.8485 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3565/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3665 - acc: 0.8493 - val_loss: 0.3745 - val_acc: 0.8175\n Epoch 3566/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3674 - acc: 0.8499 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 3567/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3714 - acc: 0.8469 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3568/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3674 - acc: 0.8462 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 3569/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3741 - acc: 0.8422 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3570/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3638 - acc: 0.8493 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3571/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3642 - acc: 0.8490 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3572/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3680 - acc: 0.8482 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3573/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3720 - acc: 0.8436 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3574/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3710 - acc: 0.8457 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 3575/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3687 - acc: 0.8438 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3576/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3696 - acc: 0.8488 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3577/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3680 - acc: 0.8479 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 3578/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3653 - acc: 0.8493 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3579/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3648 - acc: 0.8494 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3580/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3694 - acc: 0.8472 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3581/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3628 - acc: 0.8517 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3582/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3681 - acc: 0.8515 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3583/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3687 - acc: 0.8472 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 3584/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3680 - acc: 0.8436 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3585/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3669 - acc: 0.8460 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 3586/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3665 - acc: 0.8481 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3587/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3701 - acc: 0.8461 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3588/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3673 - acc: 0.8483 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3589/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3717 - acc: 0.8462 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3590/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3631 - acc: 0.8501 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3591/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3647 - acc: 0.8483 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 3592/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3632 - acc: 0.8483 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 3593/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3678 - acc: 0.8485 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 3594/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3730 - acc: 0.8424 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3595/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3706 - acc: 0.8431 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3596/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3717 - acc: 0.8465 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 3597/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3670 - acc: 0.8496 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 3598/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3685 - acc: 0.8476 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 3599/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3681 - acc: 0.8458 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3600/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3692 - acc: 0.8482 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 3601/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3659 - acc: 0.8481 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 3602/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8482 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 3603/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3667 - acc: 0.8488 - val_loss: 0.3755 - val_acc: 0.8175\n Epoch 3604/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3672 - acc: 0.8488 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3605/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3698 - acc: 0.8457 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3606/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3664 - acc: 0.8492 - val_loss: 0.3735 - val_acc: 0.8187\n Epoch 3607/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3712 - acc: 0.8454 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 3608/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3684 - acc: 0.8500 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3609/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3700 - acc: 0.8469 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3610/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3686 - acc: 0.8481 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3611/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3675 - acc: 0.8472 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3612/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3635 - acc: 0.8489 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3613/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3673 - acc: 0.8465 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 3614/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3655 - acc: 0.8504 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 3615/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3667 - acc: 0.8488 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 3616/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3753 - acc: 0.8414 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3617/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3644 - acc: 0.8518 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3618/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3689 - acc: 0.8490 - val_loss: 0.3728 - val_acc: 0.8225\n Epoch 3619/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3702 - acc: 0.8457 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 3620/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3681 - acc: 0.8462 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 3621/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3701 - acc: 0.8460 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3622/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3692 - acc: 0.8501 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3623/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3690 - acc: 0.8456 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3624/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3699 - acc: 0.8482 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3625/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3644 - acc: 0.8500 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 3626/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3747 - acc: 0.8406 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3627/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3655 - acc: 0.8490 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3628/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3647 - acc: 0.8492 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3629/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3742 - acc: 0.8436 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3630/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3694 - acc: 0.8449 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3631/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3713 - acc: 0.8440 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3632/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3742 - acc: 0.8474 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3633/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3731 - acc: 0.8454 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3634/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3692 - acc: 0.8501 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 3635/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3652 - acc: 0.8492 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 3636/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3697 - acc: 0.8460 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 3637/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3678 - acc: 0.8457 - val_loss: 0.3729 - val_acc: 0.8187\n Epoch 3638/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3665 - acc: 0.8475 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 3639/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3633 - acc: 0.8494 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 3640/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3669 - acc: 0.8485 - val_loss: 0.3728 - val_acc: 0.8187\n Epoch 3641/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3661 - acc: 0.8494 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3642/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3710 - acc: 0.8453 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3643/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3721 - acc: 0.8462 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3644/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3671 - acc: 0.8486 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3645/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3677 - acc: 0.8486 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3646/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3662 - acc: 0.8508 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 3647/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3658 - acc: 0.8488 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3648/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3688 - acc: 0.8451 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3649/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3633 - acc: 0.8504 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3650/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3725 - acc: 0.8462 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 3651/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3658 - acc: 0.8510 - val_loss: 0.3721 - val_acc: 0.8225\n Epoch 3652/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3704 - acc: 0.8460 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 3653/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3723 - acc: 0.8481 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3654/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3682 - acc: 0.8492 - val_loss: 0.3727 - val_acc: 0.8187\n Epoch 3655/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3692 - acc: 0.8439 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3656/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3699 - acc: 0.8475 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3657/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3637 - acc: 0.8461 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3658/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3618 - acc: 0.8468 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3659/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3638 - acc: 0.8492 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 3660/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3707 - acc: 0.8460 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3661/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3655 - acc: 0.8485 - val_loss: 0.3764 - val_acc: 0.8187\n Epoch 3662/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3718 - acc: 0.8468 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3663/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3734 - acc: 0.8439 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3664/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3614 - acc: 0.8486 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3665/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3698 - acc: 0.8447 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3666/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3695 - acc: 0.8468 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3667/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3678 - acc: 0.8482 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 3668/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3679 - acc: 0.8478 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 3669/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3693 - acc: 0.8435 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3670/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3675 - acc: 0.8485 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3671/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3680 - acc: 0.8478 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3672/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3692 - acc: 0.8472 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3673/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3684 - acc: 0.8492 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3674/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3671 - acc: 0.8514 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3675/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3718 - acc: 0.8482 - val_loss: 0.3740 - val_acc: 0.8237\n Epoch 3676/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3642 - acc: 0.8476 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 3677/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3680 - acc: 0.8481 - val_loss: 0.3732 - val_acc: 0.8225\n Epoch 3678/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3751 - acc: 0.8426 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3679/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3650 - acc: 0.8532 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3680/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3721 - acc: 0.8467 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3681/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3669 - acc: 0.8483 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3682/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3709 - acc: 0.8469 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3683/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3642 - acc: 0.8501 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3684/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3686 - acc: 0.8446 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3685/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3735 - acc: 0.8425 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3686/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3679 - acc: 0.8488 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3687/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3725 - acc: 0.8433 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 3688/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3700 - acc: 0.8451 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3689/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3700 - acc: 0.8444 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3690/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3670 - acc: 0.8469 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3691/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3653 - acc: 0.8504 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3692/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3689 - acc: 0.8438 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3693/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3699 - acc: 0.8478 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3694/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3709 - acc: 0.8483 - val_loss: 0.3744 - val_acc: 0.8213\n Epoch 3695/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3704 - acc: 0.8479 - val_loss: 0.3743 - val_acc: 0.8213\n Epoch 3696/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3629 - acc: 0.8496 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 3697/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3689 - acc: 0.8461 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3698/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3688 - acc: 0.8497 - val_loss: 0.3739 - val_acc: 0.8213\n Epoch 3699/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3669 - acc: 0.8510 - val_loss: 0.3731 - val_acc: 0.8213\n Epoch 3700/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3630 - acc: 0.8521 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 3701/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3682 - acc: 0.8489 - val_loss: 0.3736 - val_acc: 0.8213\n Epoch 3702/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3648 - acc: 0.8525 - val_loss: 0.3722 - val_acc: 0.8263\n Epoch 3703/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3663 - acc: 0.8449 - val_loss: 0.3723 - val_acc: 0.8250\n Epoch 3704/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3723 - acc: 0.8460 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 3705/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3727 - acc: 0.8482 - val_loss: 0.3730 - val_acc: 0.8213\n Epoch 3706/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3688 - acc: 0.8464 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3707/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3659 - acc: 0.8492 - val_loss: 0.3731 - val_acc: 0.8213\n Epoch 3708/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3675 - acc: 0.8493 - val_loss: 0.3730 - val_acc: 0.8213\n Epoch 3709/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3708 - acc: 0.8472 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3710/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3701 - acc: 0.8467 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 3711/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3722 - acc: 0.8453 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 3712/10000\n 7200/7200 [==============================] - 2s 233us/step - loss: 0.3682 - acc: 0.8479 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 3713/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3620 - acc: 0.8504 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 3714/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3677 - acc: 0.8475 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 3715/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3691 - acc: 0.8464 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 3716/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3701 - acc: 0.8436 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 3717/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3685 - acc: 0.8471 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3718/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3736 - acc: 0.8457 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3719/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3673 - acc: 0.8468 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3720/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3690 - acc: 0.8449 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3721/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3691 - acc: 0.8468 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3722/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3670 - acc: 0.8503 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3723/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3641 - acc: 0.8476 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3724/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3701 - acc: 0.8440 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3725/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3683 - acc: 0.8496 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3726/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3702 - acc: 0.8468 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3727/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3668 - acc: 0.8489 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3728/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3676 - acc: 0.8478 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3729/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3709 - acc: 0.8458 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3730/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3699 - acc: 0.8429 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3731/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3687 - acc: 0.8446 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3732/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3660 - acc: 0.8493 - val_loss: 0.3720 - val_acc: 0.8213\n Epoch 3733/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3627 - acc: 0.8475 - val_loss: 0.3720 - val_acc: 0.8200\n Epoch 3734/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3715 - acc: 0.8475 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 3735/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3745 - acc: 0.8426 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 3736/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3636 - acc: 0.8490 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3737/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3696 - acc: 0.8464 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 3738/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3677 - acc: 0.8501 - val_loss: 0.3729 - val_acc: 0.8225\n Epoch 3739/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3728 - acc: 0.8446 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 3740/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3682 - acc: 0.8453 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3741/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3682 - acc: 0.8482 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 3742/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3732 - acc: 0.8440 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 3743/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3675 - acc: 0.8476 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3744/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3708 - acc: 0.8436 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 3745/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3678 - acc: 0.8493 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3746/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3623 - acc: 0.8518 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3747/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3699 - acc: 0.8464 - val_loss: 0.3740 - val_acc: 0.8200\n Epoch 3748/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3674 - acc: 0.8479 - val_loss: 0.3743 - val_acc: 0.8200\n Epoch 3749/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3690 - acc: 0.8438 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 3750/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3710 - acc: 0.8488 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 3751/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3665 - acc: 0.8453 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 3752/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3678 - acc: 0.8479 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 3753/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3716 - acc: 0.8433 - val_loss: 0.3733 - val_acc: 0.8250\n Epoch 3754/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3724 - acc: 0.8424 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3755/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3724 - acc: 0.8464 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3756/10000\n 7200/7200 [==============================] - 1s 83us/step - loss: 0.3660 - acc: 0.8508 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3757/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3642 - acc: 0.8490 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3758/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3687 - acc: 0.8479 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3759/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3723 - acc: 0.8500 - val_loss: 0.3735 - val_acc: 0.8225\n Epoch 3760/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3686 - acc: 0.8486 - val_loss: 0.3728 - val_acc: 0.8250\n Epoch 3761/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3643 - acc: 0.8488 - val_loss: 0.3729 - val_acc: 0.8225\n Epoch 3762/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3686 - acc: 0.8489 - val_loss: 0.3731 - val_acc: 0.8225\n Epoch 3763/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3692 - acc: 0.8489 - val_loss: 0.3733 - val_acc: 0.8200\n Epoch 3764/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3670 - acc: 0.8476 - val_loss: 0.3730 - val_acc: 0.8200\n Epoch 3765/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3741 - acc: 0.8415 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 3766/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3699 - acc: 0.8460 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 3767/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3652 - acc: 0.8481 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3768/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3707 - acc: 0.8504 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3769/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3651 - acc: 0.8511 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 3770/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3694 - acc: 0.8435 - val_loss: 0.3768 - val_acc: 0.8187\n Epoch 3771/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3710 - acc: 0.8461 - val_loss: 0.3772 - val_acc: 0.8187\n Epoch 3772/10000\n 7200/7200 [==============================] - 1s 80us/step - loss: 0.3666 - acc: 0.8494 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 3773/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3708 - acc: 0.8443 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 3774/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3715 - acc: 0.8453 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3775/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3673 - acc: 0.8478 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 3776/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3704 - acc: 0.8457 - val_loss: 0.3769 - val_acc: 0.8187\n Epoch 3777/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3680 - acc: 0.8450 - val_loss: 0.3773 - val_acc: 0.8187\n Epoch 3778/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3688 - acc: 0.8451 - val_loss: 0.3765 - val_acc: 0.8187\n Epoch 3779/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3657 - acc: 0.8494 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 3780/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3719 - acc: 0.8438 - val_loss: 0.3765 - val_acc: 0.8187\n Epoch 3781/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3637 - acc: 0.8514 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 3782/10000\n 7200/7200 [==============================] - 1s 101us/step - loss: 0.3632 - acc: 0.8493 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3783/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3694 - acc: 0.8486 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3784/10000\n 7200/7200 [==============================] - 1s 75us/step - loss: 0.3707 - acc: 0.8443 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3785/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3706 - acc: 0.8472 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3786/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3690 - acc: 0.8438 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3787/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3696 - acc: 0.8481 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3788/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3709 - acc: 0.8465 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 3789/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3656 - acc: 0.8472 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3790/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3702 - acc: 0.8457 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 3791/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3734 - acc: 0.8436 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3792/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3685 - acc: 0.8471 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3793/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3692 - acc: 0.8469 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 3794/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3712 - acc: 0.8460 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3795/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3666 - acc: 0.8497 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 3796/10000\n 7200/7200 [==============================] - 1s 87us/step - loss: 0.3688 - acc: 0.8499 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 3797/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3689 - acc: 0.8476 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 3798/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3745 - acc: 0.8432 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3799/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3669 - acc: 0.8492 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3800/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3686 - acc: 0.8451 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3801/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3707 - acc: 0.8429 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3802/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3690 - acc: 0.8471 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 3803/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3687 - acc: 0.8500 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3804/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3703 - acc: 0.8456 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3805/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3657 - acc: 0.8488 - val_loss: 0.3727 - val_acc: 0.8200\n Epoch 3806/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3684 - acc: 0.8436 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3807/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3718 - acc: 0.8458 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3808/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3707 - acc: 0.8457 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 3809/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3654 - acc: 0.8515 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 3810/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3683 - acc: 0.8472 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 3811/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3732 - acc: 0.8426 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 3812/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3678 - acc: 0.8506 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3813/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3692 - acc: 0.8474 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3814/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3709 - acc: 0.8483 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3815/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3705 - acc: 0.8467 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3816/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3748 - acc: 0.8476 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 3817/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3723 - acc: 0.8461 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 3818/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3650 - acc: 0.8525 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 3819/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3667 - acc: 0.8490 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3820/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3696 - acc: 0.8462 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 3821/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3686 - acc: 0.8486 - val_loss: 0.3734 - val_acc: 0.8225\n Epoch 3822/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3689 - acc: 0.8476 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3823/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3689 - acc: 0.8508 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3824/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3702 - acc: 0.8453 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3825/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3760 - acc: 0.8425 - val_loss: 0.3772 - val_acc: 0.8187\n Epoch 3826/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3668 - acc: 0.8483 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 3827/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3706 - acc: 0.8489 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3828/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3698 - acc: 0.8468 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3829/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3680 - acc: 0.8468 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 3830/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3657 - acc: 0.8507 - val_loss: 0.3744 - val_acc: 0.8213\n Epoch 3831/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3676 - acc: 0.8508 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3832/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3749 - acc: 0.8425 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3833/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3672 - acc: 0.8483 - val_loss: 0.3751 - val_acc: 0.8200\n Epoch 3834/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3720 - acc: 0.8476 - val_loss: 0.3742 - val_acc: 0.8213\n Epoch 3835/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3681 - acc: 0.8469 - val_loss: 0.3752 - val_acc: 0.8213\n Epoch 3836/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3728 - acc: 0.8446 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3837/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3723 - acc: 0.8456 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3838/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3696 - acc: 0.8479 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3839/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3708 - acc: 0.8461 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 3840/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3736 - acc: 0.8451 - val_loss: 0.3727 - val_acc: 0.8213\n Epoch 3841/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3674 - acc: 0.8490 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 3842/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3652 - acc: 0.8488 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3843/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3674 - acc: 0.8472 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 3844/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3648 - acc: 0.8471 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 3845/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3670 - acc: 0.8510 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3846/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3660 - acc: 0.8499 - val_loss: 0.3741 - val_acc: 0.8200\n Epoch 3847/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3743 - acc: 0.8447 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3848/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3686 - acc: 0.8488 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3849/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3702 - acc: 0.8446 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3850/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3696 - acc: 0.8488 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3851/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3667 - acc: 0.8478 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3852/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3690 - acc: 0.8468 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3853/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3678 - acc: 0.8444 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 3854/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3711 - acc: 0.8469 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 3855/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3707 - acc: 0.8461 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3856/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3703 - acc: 0.8481 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 3857/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3649 - acc: 0.8469 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3858/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3659 - acc: 0.8468 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3859/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3691 - acc: 0.8456 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3860/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3709 - acc: 0.8469 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3861/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3682 - acc: 0.8481 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 3862/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3723 - acc: 0.8468 - val_loss: 0.3739 - val_acc: 0.8213\n Epoch 3863/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3700 - acc: 0.8478 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3864/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3663 - acc: 0.8478 - val_loss: 0.3767 - val_acc: 0.8187\n Epoch 3865/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3651 - acc: 0.8482 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 3866/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3622 - acc: 0.8504 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 3867/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3653 - acc: 0.8483 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3868/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3726 - acc: 0.8467 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3869/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3718 - acc: 0.8465 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3870/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3701 - acc: 0.8497 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 3871/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3687 - acc: 0.8468 - val_loss: 0.3773 - val_acc: 0.8187\n Epoch 3872/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3671 - acc: 0.8458 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 3873/10000\n 7200/7200 [==============================] - 1s 89us/step - loss: 0.3707 - acc: 0.8458 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 3874/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3669 - acc: 0.8493 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3875/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3656 - acc: 0.8490 - val_loss: 0.3739 - val_acc: 0.8213\n Epoch 3876/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3698 - acc: 0.8483 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3877/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3712 - acc: 0.8444 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 3878/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3674 - acc: 0.8492 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 3879/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3649 - acc: 0.8471 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3880/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3638 - acc: 0.8507 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 3881/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3697 - acc: 0.8444 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 3882/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3739 - acc: 0.8435 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3883/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3695 - acc: 0.8436 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3884/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3667 - acc: 0.8515 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 3885/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3638 - acc: 0.8490 - val_loss: 0.3728 - val_acc: 0.8225\n Epoch 3886/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3683 - acc: 0.8460 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3887/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3716 - acc: 0.8472 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3888/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3706 - acc: 0.8475 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3889/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3725 - acc: 0.8436 - val_loss: 0.3767 - val_acc: 0.8187\n Epoch 3890/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3699 - acc: 0.8481 - val_loss: 0.3752 - val_acc: 0.8213\n Epoch 3891/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3676 - acc: 0.8508 - val_loss: 0.3756 - val_acc: 0.8213\n Epoch 3892/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3667 - acc: 0.8467 - val_loss: 0.3752 - val_acc: 0.8213\n Epoch 3893/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3679 - acc: 0.8451 - val_loss: 0.3746 - val_acc: 0.8213\n Epoch 3894/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3639 - acc: 0.8499 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 3895/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3696 - acc: 0.8439 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 3896/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3671 - acc: 0.8453 - val_loss: 0.3746 - val_acc: 0.8200\n Epoch 3897/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3693 - acc: 0.8468 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 3898/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3684 - acc: 0.8460 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 3899/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3700 - acc: 0.8454 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3900/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3734 - acc: 0.8465 - val_loss: 0.3737 - val_acc: 0.8200\n Epoch 3901/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3651 - acc: 0.8514 - val_loss: 0.3740 - val_acc: 0.8225\n Epoch 3902/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3694 - acc: 0.8471 - val_loss: 0.3737 - val_acc: 0.8200\n Epoch 3903/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3699 - acc: 0.8482 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 3904/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3658 - acc: 0.8478 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3905/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3677 - acc: 0.8475 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3906/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3711 - acc: 0.8450 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3907/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3711 - acc: 0.8440 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3908/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3697 - acc: 0.8453 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3909/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3721 - acc: 0.8462 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 3910/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3664 - acc: 0.8475 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 3911/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3706 - acc: 0.8481 - val_loss: 0.3732 - val_acc: 0.8225\n Epoch 3912/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3691 - acc: 0.8478 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 3913/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3719 - acc: 0.8440 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3914/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3725 - acc: 0.8438 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 3915/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3668 - acc: 0.8485 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3916/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3723 - acc: 0.8462 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 3917/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3686 - acc: 0.8503 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 3918/10000\n 7200/7200 [==============================] - ETA: 0s - loss: 0.3666 - acc: 0.848 - 1s 69us/step - loss: 0.3650 - acc: 0.8489 - val_loss: 0.3770 - val_acc: 0.8175\n Epoch 3919/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3660 - acc: 0.8490 - val_loss: 0.3767 - val_acc: 0.8187\n Epoch 3920/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3720 - acc: 0.8474 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 3921/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3675 - acc: 0.8472 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 3922/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3685 - acc: 0.8468 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 3923/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3701 - acc: 0.8458 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 3924/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3651 - acc: 0.8482 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 3925/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3699 - acc: 0.8485 - val_loss: 0.3736 - val_acc: 0.8213\n Epoch 3926/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3714 - acc: 0.8433 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 3927/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3628 - acc: 0.8494 - val_loss: 0.3748 - val_acc: 0.8200\n Epoch 3928/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3664 - acc: 0.8467 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3929/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3670 - acc: 0.8481 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3930/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3680 - acc: 0.8490 - val_loss: 0.3744 - val_acc: 0.8200\n Epoch 3931/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3693 - acc: 0.8467 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 3932/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3664 - acc: 0.8488 - val_loss: 0.3737 - val_acc: 0.8200\n Epoch 3933/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3709 - acc: 0.8503 - val_loss: 0.3733 - val_acc: 0.8225\n Epoch 3934/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3705 - acc: 0.8475 - val_loss: 0.3730 - val_acc: 0.8187\n Epoch 3935/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3728 - acc: 0.8446 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 3936/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3690 - acc: 0.8482 - val_loss: 0.3736 - val_acc: 0.8213\n Epoch 3937/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3696 - acc: 0.8451 - val_loss: 0.3738 - val_acc: 0.8213\n Epoch 3938/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3708 - acc: 0.8475 - val_loss: 0.3742 - val_acc: 0.8213\n Epoch 3939/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3665 - acc: 0.8486 - val_loss: 0.3746 - val_acc: 0.8213\n Epoch 3940/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3654 - acc: 0.8488 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 3941/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3641 - acc: 0.8467 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3942/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3705 - acc: 0.8465 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3943/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3656 - acc: 0.8497 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3944/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3723 - acc: 0.8431 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 3945/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3689 - acc: 0.8479 - val_loss: 0.3748 - val_acc: 0.8200\n Epoch 3946/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3716 - acc: 0.8472 - val_loss: 0.3749 - val_acc: 0.8213\n Epoch 3947/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3665 - acc: 0.8503 - val_loss: 0.3754 - val_acc: 0.8213\n Epoch 3948/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3597 - acc: 0.8546 - val_loss: 0.3750 - val_acc: 0.8213\n Epoch 3949/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3657 - acc: 0.8460 - val_loss: 0.3761 - val_acc: 0.8200\n Epoch 3950/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3672 - acc: 0.8488 - val_loss: 0.3757 - val_acc: 0.8213\n Epoch 3951/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3673 - acc: 0.8460 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3952/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3687 - acc: 0.8493 - val_loss: 0.3762 - val_acc: 0.8213\n Epoch 3953/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3723 - acc: 0.8444 - val_loss: 0.3769 - val_acc: 0.8187\n Epoch 3954/10000\n 7200/7200 [==============================] - 1s 80us/step - loss: 0.3683 - acc: 0.8461 - val_loss: 0.3769 - val_acc: 0.8187\n Epoch 3955/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3644 - acc: 0.8494 - val_loss: 0.3778 - val_acc: 0.8187\n Epoch 3956/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3685 - acc: 0.8481 - val_loss: 0.3764 - val_acc: 0.8187\n Epoch 3957/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3667 - acc: 0.8500 - val_loss: 0.3754 - val_acc: 0.8213\n Epoch 3958/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3702 - acc: 0.8456 - val_loss: 0.3769 - val_acc: 0.8187\n Epoch 3959/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3689 - acc: 0.8462 - val_loss: 0.3767 - val_acc: 0.8200\n Epoch 3960/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3707 - acc: 0.8462 - val_loss: 0.3779 - val_acc: 0.8187\n Epoch 3961/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3733 - acc: 0.8443 - val_loss: 0.3779 - val_acc: 0.8187\n Epoch 3962/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3699 - acc: 0.8439 - val_loss: 0.3773 - val_acc: 0.8187\n Epoch 3963/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3632 - acc: 0.8506 - val_loss: 0.3773 - val_acc: 0.8200\n Epoch 3964/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3622 - acc: 0.8531 - val_loss: 0.3764 - val_acc: 0.8213\n Epoch 3965/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3718 - acc: 0.8469 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3966/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3611 - acc: 0.8479 - val_loss: 0.3768 - val_acc: 0.8187\n Epoch 3967/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3688 - acc: 0.8481 - val_loss: 0.3764 - val_acc: 0.8187\n Epoch 3968/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3718 - acc: 0.8449 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3969/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3710 - acc: 0.8474 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 3970/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3683 - acc: 0.8490 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 3971/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3680 - acc: 0.8508 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3972/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3734 - acc: 0.8465 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 3973/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3683 - acc: 0.8485 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3974/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3691 - acc: 0.8450 - val_loss: 0.3758 - val_acc: 0.8200\n Epoch 3975/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3660 - acc: 0.8515 - val_loss: 0.3746 - val_acc: 0.8200\n Epoch 3976/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3684 - acc: 0.8460 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3977/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3729 - acc: 0.8453 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 3978/10000\n 7200/7200 [==============================] - 1s 144us/step - loss: 0.3678 - acc: 0.8485 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3979/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3681 - acc: 0.8471 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 3980/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3658 - acc: 0.8490 - val_loss: 0.3768 - val_acc: 0.8187\n Epoch 3981/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3765 - acc: 0.8426 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 3982/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3642 - acc: 0.8479 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3983/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3677 - acc: 0.8482 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3984/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3722 - acc: 0.8471 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 3985/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3683 - acc: 0.8458 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 3986/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3741 - acc: 0.8432 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 3987/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3658 - acc: 0.8488 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 3988/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3706 - acc: 0.8456 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 3989/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3675 - acc: 0.8469 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 3990/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3724 - acc: 0.8440 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 3991/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3700 - acc: 0.8442 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 3992/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3697 - acc: 0.8479 - val_loss: 0.3735 - val_acc: 0.8237\n Epoch 3993/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3683 - acc: 0.8474 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 3994/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3660 - acc: 0.8478 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 3995/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3639 - acc: 0.8519 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3996/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3688 - acc: 0.8471 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 3997/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3668 - acc: 0.8501 - val_loss: 0.3732 - val_acc: 0.8237\n Epoch 3998/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3680 - acc: 0.8489 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 3999/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3703 - acc: 0.8447 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 4000/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3630 - acc: 0.8525 - val_loss: 0.3729 - val_acc: 0.8213\n Epoch 4001/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3674 - acc: 0.8483 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 4002/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3724 - acc: 0.8456 - val_loss: 0.3734 - val_acc: 0.8225\n Epoch 4003/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3608 - acc: 0.8515 - val_loss: 0.3718 - val_acc: 0.8250\n Epoch 4004/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3653 - acc: 0.8496 - val_loss: 0.3726 - val_acc: 0.8225\n Epoch 4005/10000\n 7200/7200 [==============================] - 1s 80us/step - loss: 0.3664 - acc: 0.8472 - val_loss: 0.3739 - val_acc: 0.8213\n Epoch 4006/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3658 - acc: 0.8478 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 4007/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3675 - acc: 0.8489 - val_loss: 0.3740 - val_acc: 0.8200\n Epoch 4008/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3655 - acc: 0.8469 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 4009/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3721 - acc: 0.8461 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 4010/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3680 - acc: 0.8471 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 4011/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3713 - acc: 0.8469 - val_loss: 0.3728 - val_acc: 0.8225\n Epoch 4012/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3678 - acc: 0.8501 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 4013/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3623 - acc: 0.8511 - val_loss: 0.3722 - val_acc: 0.8225\n Epoch 4014/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3657 - acc: 0.8507 - val_loss: 0.3717 - val_acc: 0.8225\n Epoch 4015/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3690 - acc: 0.8451 - val_loss: 0.3731 - val_acc: 0.8200\n Epoch 4016/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3650 - acc: 0.8496 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 4017/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3669 - acc: 0.8504 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 4018/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3708 - acc: 0.8460 - val_loss: 0.3724 - val_acc: 0.8225\n Epoch 4019/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3691 - acc: 0.8449 - val_loss: 0.3732 - val_acc: 0.8213\n Epoch 4020/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3676 - acc: 0.8453 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4021/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3662 - acc: 0.8499 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 4022/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3725 - acc: 0.8457 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 4023/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3715 - acc: 0.8504 - val_loss: 0.3750 - val_acc: 0.8213\n Epoch 4024/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3678 - acc: 0.8479 - val_loss: 0.3737 - val_acc: 0.8250\n Epoch 4025/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3658 - acc: 0.8490 - val_loss: 0.3740 - val_acc: 0.8200\n Epoch 4026/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3690 - acc: 0.8475 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 4027/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3626 - acc: 0.8508 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 4028/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3652 - acc: 0.8490 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4029/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3711 - acc: 0.8453 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 4030/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3668 - acc: 0.8508 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 4031/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3678 - acc: 0.8485 - val_loss: 0.3741 - val_acc: 0.8225\n Epoch 4032/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3681 - acc: 0.8494 - val_loss: 0.3740 - val_acc: 0.8237\n Epoch 4033/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3661 - acc: 0.8497 - val_loss: 0.3746 - val_acc: 0.8200\n Epoch 4034/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3696 - acc: 0.8451 - val_loss: 0.3744 - val_acc: 0.8225\n Epoch 4035/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3654 - acc: 0.8489 - val_loss: 0.3752 - val_acc: 0.8200\n Epoch 4036/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3645 - acc: 0.8442 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 4037/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3674 - acc: 0.8493 - val_loss: 0.3752 - val_acc: 0.8200\n Epoch 4038/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3724 - acc: 0.8483 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 4039/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3637 - acc: 0.8492 - val_loss: 0.3751 - val_acc: 0.8200\n Epoch 4040/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3685 - acc: 0.8503 - val_loss: 0.3752 - val_acc: 0.8213\n Epoch 4041/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3686 - acc: 0.8504 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4042/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3721 - acc: 0.8465 - val_loss: 0.3748 - val_acc: 0.8250\n Epoch 4043/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3715 - acc: 0.8476 - val_loss: 0.3763 - val_acc: 0.8225\n Epoch 4044/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3635 - acc: 0.8489 - val_loss: 0.3761 - val_acc: 0.8225\n Epoch 4045/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3683 - acc: 0.8490 - val_loss: 0.3769 - val_acc: 0.8187\n Epoch 4046/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3688 - acc: 0.8483 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 4047/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3670 - acc: 0.8511 - val_loss: 0.3755 - val_acc: 0.8237\n Epoch 4048/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3666 - acc: 0.8500 - val_loss: 0.3736 - val_acc: 0.8263\n Epoch 4049/10000\n 7200/7200 [==============================] - 1s 80us/step - loss: 0.3687 - acc: 0.8469 - val_loss: 0.3746 - val_acc: 0.8213\n Epoch 4050/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3661 - acc: 0.8494 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 4051/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3684 - acc: 0.8478 - val_loss: 0.3734 - val_acc: 0.8225\n Epoch 4052/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3676 - acc: 0.8486 - val_loss: 0.3746 - val_acc: 0.8213\n Epoch 4053/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3685 - acc: 0.8486 - val_loss: 0.3751 - val_acc: 0.8213\n Epoch 4054/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3708 - acc: 0.8474 - val_loss: 0.3753 - val_acc: 0.8200\n Epoch 4055/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3655 - acc: 0.8500 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 4056/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3689 - acc: 0.8462 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4057/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3707 - acc: 0.8469 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 4058/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3673 - acc: 0.8489 - val_loss: 0.3748 - val_acc: 0.8213\n Epoch 4059/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3635 - acc: 0.8499 - val_loss: 0.3751 - val_acc: 0.8200\n Epoch 4060/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3694 - acc: 0.8453 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4061/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3703 - acc: 0.8469 - val_loss: 0.3763 - val_acc: 0.8200\n Epoch 4062/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3666 - acc: 0.8462 - val_loss: 0.3765 - val_acc: 0.8187\n Epoch 4063/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3660 - acc: 0.8497 - val_loss: 0.3747 - val_acc: 0.8200\n Epoch 4064/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3711 - acc: 0.8462 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 4065/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3727 - acc: 0.8438 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 4066/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3729 - acc: 0.8468 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 4067/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3684 - acc: 0.8486 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4068/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3661 - acc: 0.8472 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 4069/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3673 - acc: 0.8449 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4070/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3699 - acc: 0.8481 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4071/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3685 - acc: 0.8497 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 4072/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3675 - acc: 0.8461 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4073/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3700 - acc: 0.8481 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4074/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3658 - acc: 0.8483 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 4075/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3672 - acc: 0.8471 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4076/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3668 - acc: 0.8472 - val_loss: 0.3774 - val_acc: 0.8187\n Epoch 4077/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3739 - acc: 0.8424 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 4078/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3692 - acc: 0.8486 - val_loss: 0.3771 - val_acc: 0.8187\n Epoch 4079/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3687 - acc: 0.8450 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 4080/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3671 - acc: 0.8467 - val_loss: 0.3767 - val_acc: 0.8187\n Epoch 4081/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3721 - acc: 0.8439 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 4082/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3710 - acc: 0.8449 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 4083/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3686 - acc: 0.8472 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4084/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3692 - acc: 0.8472 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 4085/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3692 - acc: 0.8472 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4086/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3672 - acc: 0.8483 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 4087/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3697 - acc: 0.8436 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4088/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3682 - acc: 0.8493 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4089/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3702 - acc: 0.8468 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4090/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3680 - acc: 0.8490 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 4091/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3721 - acc: 0.8462 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4092/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3707 - acc: 0.8453 - val_loss: 0.3749 - val_acc: 0.8213\n Epoch 4093/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3658 - acc: 0.8500 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 4094/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3655 - acc: 0.8493 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 4095/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3710 - acc: 0.8479 - val_loss: 0.3741 - val_acc: 0.8225\n Epoch 4096/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3716 - acc: 0.8456 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4097/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3686 - acc: 0.8453 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 4098/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3646 - acc: 0.8504 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 4099/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3698 - acc: 0.8451 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4100/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3697 - acc: 0.8461 - val_loss: 0.3742 - val_acc: 0.8213\n Epoch 4101/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3707 - acc: 0.8464 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4102/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3687 - acc: 0.8488 - val_loss: 0.3747 - val_acc: 0.8213\n Epoch 4103/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3692 - acc: 0.8462 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 4104/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3658 - acc: 0.8492 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 4105/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3699 - acc: 0.8433 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 4106/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3628 - acc: 0.8500 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4107/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3699 - acc: 0.8471 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4108/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3706 - acc: 0.8469 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4109/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3654 - acc: 0.8478 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 4110/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3695 - acc: 0.8465 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 4111/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3685 - acc: 0.8464 - val_loss: 0.3738 - val_acc: 0.8213\n Epoch 4112/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3640 - acc: 0.8492 - val_loss: 0.3731 - val_acc: 0.8213\n Epoch 4113/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3670 - acc: 0.8500 - val_loss: 0.3726 - val_acc: 0.8213\n Epoch 4114/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3729 - acc: 0.8432 - val_loss: 0.3742 - val_acc: 0.8200\n Epoch 4115/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3719 - acc: 0.8439 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4116/10000\n 7200/7200 [==============================] - 1s 94us/step - loss: 0.3746 - acc: 0.8428 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 4117/10000\n 7200/7200 [==============================] - 1s 85us/step - loss: 0.3667 - acc: 0.8496 - val_loss: 0.3752 - val_acc: 0.8200\n Epoch 4118/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3653 - acc: 0.8485 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 4119/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3685 - acc: 0.8454 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 4120/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3677 - acc: 0.8497 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 4121/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3696 - acc: 0.8471 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 4122/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3692 - acc: 0.8468 - val_loss: 0.3744 - val_acc: 0.8213\n Epoch 4123/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3696 - acc: 0.8471 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 4124/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3660 - acc: 0.8506 - val_loss: 0.3742 - val_acc: 0.8213\n Epoch 4125/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3705 - acc: 0.8483 - val_loss: 0.3736 - val_acc: 0.8213\n Epoch 4126/10000\n 7200/7200 [==============================] - ETA: 0s - loss: 0.3709 - acc: 0.847 - 0s 62us/step - loss: 0.3707 - acc: 0.8478 - val_loss: 0.3739 - val_acc: 0.8213\n Epoch 4127/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3648 - acc: 0.8501 - val_loss: 0.3746 - val_acc: 0.8213\n Epoch 4128/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3715 - acc: 0.8494 - val_loss: 0.3753 - val_acc: 0.8213\n Epoch 4129/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3619 - acc: 0.8515 - val_loss: 0.3747 - val_acc: 0.8213\n Epoch 4130/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3656 - acc: 0.8479 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 4131/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3668 - acc: 0.8471 - val_loss: 0.3756 - val_acc: 0.8200\n Epoch 4132/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3685 - acc: 0.8493 - val_loss: 0.3752 - val_acc: 0.8200\n Epoch 4133/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3670 - acc: 0.8465 - val_loss: 0.3752 - val_acc: 0.8200\n Epoch 4134/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3682 - acc: 0.8460 - val_loss: 0.3767 - val_acc: 0.8187\n Epoch 4135/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3712 - acc: 0.8460 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 4136/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3714 - acc: 0.8443 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 4137/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3690 - acc: 0.8460 - val_loss: 0.3732 - val_acc: 0.8213\n Epoch 4138/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3702 - acc: 0.8439 - val_loss: 0.3733 - val_acc: 0.8225\n Epoch 4139/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3716 - acc: 0.8469 - val_loss: 0.3750 - val_acc: 0.8200\n Epoch 4140/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3628 - acc: 0.8514 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4141/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3749 - acc: 0.8443 - val_loss: 0.3756 - val_acc: 0.8200\n Epoch 4142/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3697 - acc: 0.8506 - val_loss: 0.3752 - val_acc: 0.8213\n Epoch 4143/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3651 - acc: 0.8493 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 4144/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3666 - acc: 0.8475 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4145/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3640 - acc: 0.8512 - val_loss: 0.3741 - val_acc: 0.8200\n Epoch 4146/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3722 - acc: 0.8433 - val_loss: 0.3745 - val_acc: 0.8213\n Epoch 4147/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3719 - acc: 0.8442 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 4148/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3675 - acc: 0.8493 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 4149/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3713 - acc: 0.8446 - val_loss: 0.3771 - val_acc: 0.8187\n Epoch 4150/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3668 - acc: 0.8479 - val_loss: 0.3771 - val_acc: 0.8187\n Epoch 4151/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3694 - acc: 0.8469 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4152/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3680 - acc: 0.8483 - val_loss: 0.3750 - val_acc: 0.8213\n Epoch 4153/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3733 - acc: 0.8439 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4154/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3704 - acc: 0.8449 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 4155/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3685 - acc: 0.8488 - val_loss: 0.3734 - val_acc: 0.8225\n Epoch 4156/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3653 - acc: 0.8476 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 4157/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3671 - acc: 0.8483 - val_loss: 0.3729 - val_acc: 0.8250\n Epoch 4158/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3682 - acc: 0.8476 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4159/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3728 - acc: 0.8439 - val_loss: 0.3753 - val_acc: 0.8200\n Epoch 4160/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3685 - acc: 0.8490 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 4161/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3661 - acc: 0.8464 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 4162/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3757 - acc: 0.8431 - val_loss: 0.3736 - val_acc: 0.8213\n Epoch 4163/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3718 - acc: 0.8467 - val_loss: 0.3738 - val_acc: 0.8213\n Epoch 4164/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3661 - acc: 0.8483 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 4165/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3736 - acc: 0.8447 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4166/10000\n 7200/7200 [==============================] - 1s 89us/step - loss: 0.3664 - acc: 0.8471 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4167/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3674 - acc: 0.8469 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4168/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3702 - acc: 0.8458 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 4169/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3668 - acc: 0.8479 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4170/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3715 - acc: 0.8449 - val_loss: 0.3746 - val_acc: 0.8200\n Epoch 4171/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3673 - acc: 0.8476 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4172/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3680 - acc: 0.8496 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 4173/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3747 - acc: 0.8418 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4174/10000\n 7200/7200 [==============================] - 1s 91us/step - loss: 0.3660 - acc: 0.8496 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 4175/10000\n 7200/7200 [==============================] - 1s 82us/step - loss: 0.3688 - acc: 0.8464 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 4176/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3684 - acc: 0.8468 - val_loss: 0.3735 - val_acc: 0.8200\n Epoch 4177/10000\n 7200/7200 [==============================] - 1s 75us/step - loss: 0.3667 - acc: 0.8493 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 4178/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3703 - acc: 0.8461 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4179/10000\n 7200/7200 [==============================] - 1s 86us/step - loss: 0.3639 - acc: 0.8501 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4180/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3682 - acc: 0.8451 - val_loss: 0.3739 - val_acc: 0.8200\n Epoch 4181/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3695 - acc: 0.8489 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 4182/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3704 - acc: 0.8457 - val_loss: 0.3735 - val_acc: 0.8200\n Epoch 4183/10000\n 7200/7200 [==============================] - 1s 84us/step - loss: 0.3738 - acc: 0.8428 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4184/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3733 - acc: 0.8438 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4185/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3658 - acc: 0.8494 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 4186/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3661 - acc: 0.8479 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4187/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3688 - acc: 0.8497 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 4188/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3631 - acc: 0.8521 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 4189/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3667 - acc: 0.8504 - val_loss: 0.3738 - val_acc: 0.8213\n Epoch 4190/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3632 - acc: 0.8496 - val_loss: 0.3736 - val_acc: 0.8213\n Epoch 4191/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3631 - acc: 0.8486 - val_loss: 0.3739 - val_acc: 0.8213\n Epoch 4192/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3737 - acc: 0.8458 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4193/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3667 - acc: 0.8488 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 4194/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3664 - acc: 0.8496 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4195/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3730 - acc: 0.8464 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4196/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3663 - acc: 0.8500 - val_loss: 0.3730 - val_acc: 0.8213\n Epoch 4197/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3786 - acc: 0.8457 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 4198/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3694 - acc: 0.8460 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 4199/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3671 - acc: 0.8478 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 4200/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3646 - acc: 0.8489 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 4201/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3720 - acc: 0.8468 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 4202/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3647 - acc: 0.8506 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 4203/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3666 - acc: 0.8504 - val_loss: 0.3729 - val_acc: 0.8225\n Epoch 4204/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3711 - acc: 0.8436 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4205/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3697 - acc: 0.8429 - val_loss: 0.3769 - val_acc: 0.8175\n Epoch 4206/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3662 - acc: 0.8475 - val_loss: 0.3767 - val_acc: 0.8187\n Epoch 4207/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3703 - acc: 0.8497 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 4208/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3684 - acc: 0.8492 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 4209/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3685 - acc: 0.8489 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 4210/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3731 - acc: 0.8456 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4211/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3723 - acc: 0.8461 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 4212/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3683 - acc: 0.8474 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4213/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3723 - acc: 0.8443 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4214/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3666 - acc: 0.8469 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4215/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3708 - acc: 0.8475 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 4216/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3731 - acc: 0.8443 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 4217/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3714 - acc: 0.8468 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4218/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3668 - acc: 0.8479 - val_loss: 0.3763 - val_acc: 0.8175\n Epoch 4219/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3747 - acc: 0.8456 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 4220/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3649 - acc: 0.8510 - val_loss: 0.3750 - val_acc: 0.8213\n Epoch 4221/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3700 - acc: 0.8467 - val_loss: 0.3748 - val_acc: 0.8213\n Epoch 4222/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3650 - acc: 0.8492 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 4223/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3654 - acc: 0.8488 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 4224/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3715 - acc: 0.8462 - val_loss: 0.3731 - val_acc: 0.8237\n Epoch 4225/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3682 - acc: 0.8489 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 4226/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3659 - acc: 0.8483 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 4227/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3719 - acc: 0.8460 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4228/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3708 - acc: 0.8438 - val_loss: 0.3749 - val_acc: 0.8200\n Epoch 4229/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3658 - acc: 0.8485 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4230/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3661 - acc: 0.8493 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4231/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3668 - acc: 0.8478 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 4232/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3661 - acc: 0.8464 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 4233/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3631 - acc: 0.8507 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4234/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3718 - acc: 0.8458 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 4235/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3680 - acc: 0.8492 - val_loss: 0.3749 - val_acc: 0.8200\n Epoch 4236/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3721 - acc: 0.8458 - val_loss: 0.3748 - val_acc: 0.8213\n Epoch 4237/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3662 - acc: 0.8483 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 4238/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3681 - acc: 0.8492 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4239/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3692 - acc: 0.8465 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 4240/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3664 - acc: 0.8432 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 4241/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3722 - acc: 0.8483 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 4242/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3704 - acc: 0.8438 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 4243/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3673 - acc: 0.8497 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 4244/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3705 - acc: 0.8454 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 4245/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3654 - acc: 0.8488 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 4246/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3693 - acc: 0.8447 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 4247/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3701 - acc: 0.8456 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 4248/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3670 - acc: 0.8465 - val_loss: 0.3732 - val_acc: 0.8187\n Epoch 4249/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3662 - acc: 0.8517 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 4250/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3705 - acc: 0.8503 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4251/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3704 - acc: 0.8474 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 4252/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3654 - acc: 0.8522 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 4253/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3711 - acc: 0.8457 - val_loss: 0.3748 - val_acc: 0.8213\n Epoch 4254/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3675 - acc: 0.8460 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 4255/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3713 - acc: 0.8453 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4256/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3683 - acc: 0.8478 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 4257/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3635 - acc: 0.8503 - val_loss: 0.3748 - val_acc: 0.8200\n Epoch 4258/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3710 - acc: 0.8467 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4259/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3707 - acc: 0.8472 - val_loss: 0.3739 - val_acc: 0.8213\n Epoch 4260/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3669 - acc: 0.8462 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 4261/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3714 - acc: 0.8450 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 4262/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3712 - acc: 0.8461 - val_loss: 0.3738 - val_acc: 0.8200\n Epoch 4263/10000\n 7200/7200 [==============================] - ETA: 0s - loss: 0.3645 - acc: 0.847 - 0s 68us/step - loss: 0.3633 - acc: 0.8476 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 4264/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3731 - acc: 0.8464 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4265/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3682 - acc: 0.8482 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 4266/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3628 - acc: 0.8510 - val_loss: 0.3726 - val_acc: 0.8213\n Epoch 4267/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3682 - acc: 0.8492 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 4268/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3734 - acc: 0.8457 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 4269/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3663 - acc: 0.8500 - val_loss: 0.3728 - val_acc: 0.8213\n Epoch 4270/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3688 - acc: 0.8488 - val_loss: 0.3731 - val_acc: 0.8213\n Epoch 4271/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3685 - acc: 0.8472 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4272/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3679 - acc: 0.8464 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4273/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3690 - acc: 0.8489 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4274/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3704 - acc: 0.8458 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 4275/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3718 - acc: 0.8439 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 4276/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3702 - acc: 0.8462 - val_loss: 0.3734 - val_acc: 0.8200\n Epoch 4277/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3669 - acc: 0.8501 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 4278/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3685 - acc: 0.8485 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 4279/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3674 - acc: 0.8500 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4280/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3636 - acc: 0.8492 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 4281/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8479 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 4282/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3729 - acc: 0.8465 - val_loss: 0.3767 - val_acc: 0.8187\n Epoch 4283/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3692 - acc: 0.8444 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 4284/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3696 - acc: 0.8461 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 4285/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3672 - acc: 0.8462 - val_loss: 0.3745 - val_acc: 0.8213\n Epoch 4286/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3689 - acc: 0.8471 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 4287/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3703 - acc: 0.8461 - val_loss: 0.3765 - val_acc: 0.8187\n Epoch 4288/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3713 - acc: 0.8476 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 4289/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3705 - acc: 0.8474 - val_loss: 0.3752 - val_acc: 0.8200\n Epoch 4290/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3724 - acc: 0.8426 - val_loss: 0.3751 - val_acc: 0.8200\n Epoch 4291/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3705 - acc: 0.8479 - val_loss: 0.3746 - val_acc: 0.8200\n Epoch 4292/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3742 - acc: 0.8469 - val_loss: 0.3742 - val_acc: 0.8213\n Epoch 4293/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3737 - acc: 0.8457 - val_loss: 0.3747 - val_acc: 0.8250\n Epoch 4294/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3685 - acc: 0.8472 - val_loss: 0.3752 - val_acc: 0.8200\n Epoch 4295/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3661 - acc: 0.8483 - val_loss: 0.3741 - val_acc: 0.8225\n Epoch 4296/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3705 - acc: 0.8446 - val_loss: 0.3746 - val_acc: 0.8225\n Epoch 4297/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3676 - acc: 0.8453 - val_loss: 0.3752 - val_acc: 0.8200\n Epoch 4298/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3711 - acc: 0.8456 - val_loss: 0.3744 - val_acc: 0.8225\n Epoch 4299/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3635 - acc: 0.8508 - val_loss: 0.3754 - val_acc: 0.8213\n Epoch 4300/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3689 - acc: 0.8515 - val_loss: 0.3752 - val_acc: 0.8200\n Epoch 4301/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3644 - acc: 0.8472 - val_loss: 0.3748 - val_acc: 0.8225\n Epoch 4302/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3722 - acc: 0.8456 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4303/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3682 - acc: 0.8456 - val_loss: 0.3765 - val_acc: 0.8187\n Epoch 4304/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3671 - acc: 0.8512 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 4305/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3714 - acc: 0.8469 - val_loss: 0.3771 - val_acc: 0.8187\n Epoch 4306/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3670 - acc: 0.8462 - val_loss: 0.3772 - val_acc: 0.8187\n Epoch 4307/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3682 - acc: 0.8482 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 4308/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3680 - acc: 0.8483 - val_loss: 0.3756 - val_acc: 0.8200\n Epoch 4309/10000\n 7200/7200 [==============================] - ETA: 1s - loss: 0.3654 - acc: 0.847 - 20s 3ms/step - loss: 0.3659 - acc: 0.8464 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 4310/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3676 - acc: 0.8496 - val_loss: 0.3773 - val_acc: 0.8187\n Epoch 4311/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3598 - acc: 0.8494 - val_loss: 0.3767 - val_acc: 0.8187\n Epoch 4312/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3678 - acc: 0.8481 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 4313/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3697 - acc: 0.8479 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 4314/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3695 - acc: 0.8507 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4315/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3720 - acc: 0.8460 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 4316/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3640 - acc: 0.8493 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 4317/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3699 - acc: 0.8465 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4318/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3642 - acc: 0.8489 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4319/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3708 - acc: 0.8482 - val_loss: 0.3746 - val_acc: 0.8200\n Epoch 4320/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3705 - acc: 0.8462 - val_loss: 0.3746 - val_acc: 0.8213\n Epoch 4321/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3664 - acc: 0.8464 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 4322/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3700 - acc: 0.8438 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 4323/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3699 - acc: 0.8478 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 4324/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3716 - acc: 0.8468 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 4325/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3713 - acc: 0.8481 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 4326/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3715 - acc: 0.8443 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 4327/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3691 - acc: 0.8442 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 4328/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3738 - acc: 0.8462 - val_loss: 0.3779 - val_acc: 0.8187\n Epoch 4329/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3705 - acc: 0.8444 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 4330/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3701 - acc: 0.8492 - val_loss: 0.3749 - val_acc: 0.8200\n Epoch 4331/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3662 - acc: 0.8518 - val_loss: 0.3742 - val_acc: 0.8213\n Epoch 4332/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3719 - acc: 0.8428 - val_loss: 0.3753 - val_acc: 0.8213\n Epoch 4333/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3683 - acc: 0.8471 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 4334/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3681 - acc: 0.8471 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4335/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3699 - acc: 0.8442 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4336/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3698 - acc: 0.8469 - val_loss: 0.3733 - val_acc: 0.8213\n Epoch 4337/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3684 - acc: 0.8490 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 4338/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3665 - acc: 0.8500 - val_loss: 0.3742 - val_acc: 0.8213\n Epoch 4339/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3709 - acc: 0.8451 - val_loss: 0.3748 - val_acc: 0.8200\n Epoch 4340/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3740 - acc: 0.8438 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 4341/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3690 - acc: 0.8442 - val_loss: 0.3753 - val_acc: 0.8200\n Epoch 4342/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3647 - acc: 0.8497 - val_loss: 0.3749 - val_acc: 0.8200\n Epoch 4343/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3692 - acc: 0.8456 - val_loss: 0.3747 - val_acc: 0.8213\n Epoch 4344/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3737 - acc: 0.8454 - val_loss: 0.3750 - val_acc: 0.8213\n Epoch 4345/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3701 - acc: 0.8494 - val_loss: 0.3749 - val_acc: 0.8213\n Epoch 4346/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3690 - acc: 0.8494 - val_loss: 0.3744 - val_acc: 0.8213\n Epoch 4347/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3711 - acc: 0.8472 - val_loss: 0.3747 - val_acc: 0.8213\n Epoch 4348/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3732 - acc: 0.8421 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4349/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3675 - acc: 0.8478 - val_loss: 0.3742 - val_acc: 0.8213\n Epoch 4350/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3680 - acc: 0.8488 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4351/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3763 - acc: 0.8432 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 4352/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3655 - acc: 0.8478 - val_loss: 0.3746 - val_acc: 0.8213\n Epoch 4353/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3645 - acc: 0.8508 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4354/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3683 - acc: 0.8472 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 4355/10000\n 7200/7200 [==============================] - 1s 181us/step - loss: 0.3686 - acc: 0.8446 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 4356/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3674 - acc: 0.8467 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 4357/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3718 - acc: 0.8424 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 4358/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3673 - acc: 0.8458 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 4359/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3680 - acc: 0.8485 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 4360/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3685 - acc: 0.8464 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 4361/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3648 - acc: 0.8517 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 4362/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3684 - acc: 0.8485 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4363/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3717 - acc: 0.8458 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 4364/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3665 - acc: 0.8475 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 4365/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3622 - acc: 0.8494 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 4366/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3758 - acc: 0.8433 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 4367/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3691 - acc: 0.8476 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4368/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3680 - acc: 0.8517 - val_loss: 0.3749 - val_acc: 0.8200\n Epoch 4369/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3686 - acc: 0.8485 - val_loss: 0.3743 - val_acc: 0.8200\n Epoch 4370/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3737 - acc: 0.8453 - val_loss: 0.3742 - val_acc: 0.8225\n Epoch 4371/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3674 - acc: 0.8482 - val_loss: 0.3735 - val_acc: 0.8225\n Epoch 4372/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3729 - acc: 0.8442 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 4373/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3672 - acc: 0.8489 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4374/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3679 - acc: 0.8482 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 4375/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3717 - acc: 0.8469 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 4376/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3662 - acc: 0.8510 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4377/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3690 - acc: 0.8467 - val_loss: 0.3764 - val_acc: 0.8187\n Epoch 4378/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3694 - acc: 0.8460 - val_loss: 0.3744 - val_acc: 0.8200\n Epoch 4379/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3689 - acc: 0.8469 - val_loss: 0.3736 - val_acc: 0.8237\n Epoch 4380/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3676 - acc: 0.8485 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 4381/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3698 - acc: 0.8500 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4382/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3686 - acc: 0.8475 - val_loss: 0.3736 - val_acc: 0.8213\n Epoch 4383/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3713 - acc: 0.8446 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 4384/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3637 - acc: 0.8512 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 4385/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3707 - acc: 0.8467 - val_loss: 0.3735 - val_acc: 0.8200\n Epoch 4386/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3686 - acc: 0.8471 - val_loss: 0.3731 - val_acc: 0.8237\n Epoch 4387/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3685 - acc: 0.8436 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 4388/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3678 - acc: 0.8485 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4389/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3651 - acc: 0.8485 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 4390/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3661 - acc: 0.8460 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 4391/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3675 - acc: 0.8488 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 4392/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3721 - acc: 0.8440 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4393/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3687 - acc: 0.8482 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 4394/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3712 - acc: 0.8475 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4395/10000\n 7200/7200 [==============================] - 1s 82us/step - loss: 0.3697 - acc: 0.8440 - val_loss: 0.3736 - val_acc: 0.8187\n Epoch 4396/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3698 - acc: 0.8474 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 4397/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3706 - acc: 0.8468 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 4398/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3734 - acc: 0.8458 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 4399/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3651 - acc: 0.8489 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 4400/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3687 - acc: 0.8493 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 4401/10000\n 7200/7200 [==============================] - 1s 75us/step - loss: 0.3701 - acc: 0.8464 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 4402/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3683 - acc: 0.8458 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 4403/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3662 - acc: 0.8488 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 4404/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3705 - acc: 0.8494 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 4405/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3692 - acc: 0.8478 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4406/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3724 - acc: 0.8432 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 4407/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3701 - acc: 0.8475 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 4408/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3686 - acc: 0.8468 - val_loss: 0.3745 - val_acc: 0.8213\n Epoch 4409/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3676 - acc: 0.8444 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 4410/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3712 - acc: 0.8494 - val_loss: 0.3732 - val_acc: 0.8213\n Epoch 4411/10000\n 7200/7200 [==============================] - 1s 100us/step - loss: 0.3724 - acc: 0.8456 - val_loss: 0.3754 - val_acc: 0.8200\n Epoch 4412/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3739 - acc: 0.8444 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 4413/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3670 - acc: 0.8482 - val_loss: 0.3752 - val_acc: 0.8213\n Epoch 4414/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3661 - acc: 0.8481 - val_loss: 0.3749 - val_acc: 0.8200\n Epoch 4415/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3646 - acc: 0.8521 - val_loss: 0.3733 - val_acc: 0.8225\n Epoch 4416/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3673 - acc: 0.8467 - val_loss: 0.3739 - val_acc: 0.8213\n Epoch 4417/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3681 - acc: 0.8476 - val_loss: 0.3735 - val_acc: 0.8225\n Epoch 4418/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3747 - acc: 0.8442 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 4419/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3675 - acc: 0.8501 - val_loss: 0.3746 - val_acc: 0.8213\n Epoch 4420/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3710 - acc: 0.8449 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4421/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3683 - acc: 0.8479 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4422/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3687 - acc: 0.8465 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 4423/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3682 - acc: 0.8472 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4424/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3720 - acc: 0.8442 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 4425/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3672 - acc: 0.8481 - val_loss: 0.3744 - val_acc: 0.8213\n Epoch 4426/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3716 - acc: 0.8442 - val_loss: 0.3741 - val_acc: 0.8200\n Epoch 4427/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3755 - acc: 0.8429 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4428/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3694 - acc: 0.8475 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4429/10000\n 7200/7200 [==============================] - 1s 85us/step - loss: 0.3689 - acc: 0.8467 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 4430/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3668 - acc: 0.8504 - val_loss: 0.3744 - val_acc: 0.8200\n Epoch 4431/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3692 - acc: 0.8468 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 4432/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3684 - acc: 0.8476 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 4433/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3622 - acc: 0.8511 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 4434/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3744 - acc: 0.8451 - val_loss: 0.3742 - val_acc: 0.8200\n Epoch 4435/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3679 - acc: 0.8479 - val_loss: 0.3729 - val_acc: 0.8225\n Epoch 4436/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3719 - acc: 0.8457 - val_loss: 0.3730 - val_acc: 0.8225\n Epoch 4437/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3673 - acc: 0.8506 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4438/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3675 - acc: 0.8494 - val_loss: 0.3746 - val_acc: 0.8213\n Epoch 4439/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3657 - acc: 0.8494 - val_loss: 0.3764 - val_acc: 0.8187\n Epoch 4440/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3714 - acc: 0.8474 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4441/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3665 - acc: 0.8483 - val_loss: 0.3737 - val_acc: 0.8225\n Epoch 4442/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3661 - acc: 0.8478 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 4443/10000\n 7200/7200 [==============================] - 4s 608us/step - loss: 0.3698 - acc: 0.8443 - val_loss: 0.3771 - val_acc: 0.8187\n Epoch 4444/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3690 - acc: 0.8475 - val_loss: 0.3771 - val_acc: 0.8187\n Epoch 4445/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3636 - acc: 0.8518 - val_loss: 0.3753 - val_acc: 0.8200\n Epoch 4446/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3700 - acc: 0.8467 - val_loss: 0.3747 - val_acc: 0.8200\n Epoch 4447/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3667 - acc: 0.8471 - val_loss: 0.3745 - val_acc: 0.8225\n Epoch 4448/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3686 - acc: 0.8482 - val_loss: 0.3738 - val_acc: 0.8237\n Epoch 4449/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3682 - acc: 0.8449 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 4450/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3689 - acc: 0.8471 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4451/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3653 - acc: 0.8483 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4452/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3655 - acc: 0.8488 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4453/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3712 - acc: 0.8454 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 4454/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3678 - acc: 0.8485 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 4455/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3676 - acc: 0.8478 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 4456/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3715 - acc: 0.8465 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4457/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3697 - acc: 0.8493 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 4458/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3638 - acc: 0.8511 - val_loss: 0.3766 - val_acc: 0.8187\n Epoch 4459/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3678 - acc: 0.8453 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4460/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3701 - acc: 0.8469 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 4461/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3685 - acc: 0.8462 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 4462/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3653 - acc: 0.8450 - val_loss: 0.3741 - val_acc: 0.8225\n Epoch 4463/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3715 - acc: 0.8468 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 4464/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3669 - acc: 0.8453 - val_loss: 0.3742 - val_acc: 0.8200\n Epoch 4465/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3645 - acc: 0.8493 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 4466/10000\n 7200/7200 [==============================] - 1s 82us/step - loss: 0.3694 - acc: 0.8494 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 4467/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3697 - acc: 0.8457 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4468/10000\n 7200/7200 [==============================] - 1s 84us/step - loss: 0.3684 - acc: 0.8490 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 4469/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3676 - acc: 0.8482 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 4470/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3688 - acc: 0.8478 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 4471/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3706 - acc: 0.8496 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4472/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3645 - acc: 0.8485 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4473/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3666 - acc: 0.8489 - val_loss: 0.3739 - val_acc: 0.8213\n Epoch 4474/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3710 - acc: 0.8428 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 4475/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3721 - acc: 0.8456 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4476/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3722 - acc: 0.8474 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4477/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3689 - acc: 0.8476 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4478/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3659 - acc: 0.8497 - val_loss: 0.3744 - val_acc: 0.8213\n Epoch 4479/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3681 - acc: 0.8489 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4480/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3706 - acc: 0.8467 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4481/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3701 - acc: 0.8457 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4482/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3781 - acc: 0.8431 - val_loss: 0.3748 - val_acc: 0.8213\n Epoch 4483/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3653 - acc: 0.8535 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 4484/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3730 - acc: 0.8457 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 4485/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3700 - acc: 0.8458 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 4486/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3682 - acc: 0.8481 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4487/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3711 - acc: 0.8461 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 4488/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3730 - acc: 0.8458 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 4489/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3663 - acc: 0.8483 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 4490/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3686 - acc: 0.8458 - val_loss: 0.3733 - val_acc: 0.8237\n Epoch 4491/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3700 - acc: 0.8456 - val_loss: 0.3729 - val_acc: 0.8237\n Epoch 4492/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3683 - acc: 0.8469 - val_loss: 0.3730 - val_acc: 0.8225\n Epoch 4493/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3662 - acc: 0.8485 - val_loss: 0.3726 - val_acc: 0.8225\n Epoch 4494/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3692 - acc: 0.8439 - val_loss: 0.3738 - val_acc: 0.8225\n Epoch 4495/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3658 - acc: 0.8494 - val_loss: 0.3732 - val_acc: 0.8225\n Epoch 4496/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3686 - acc: 0.8517 - val_loss: 0.3736 - val_acc: 0.8225\n Epoch 4497/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3677 - acc: 0.8482 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 4498/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3676 - acc: 0.8451 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 4499/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3725 - acc: 0.8438 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 4500/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3702 - acc: 0.8479 - val_loss: 0.3764 - val_acc: 0.8187\n Epoch 4501/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3643 - acc: 0.8483 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 4502/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3688 - acc: 0.8474 - val_loss: 0.3739 - val_acc: 0.8213\n Epoch 4503/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3735 - acc: 0.8436 - val_loss: 0.3729 - val_acc: 0.8213\n Epoch 4504/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3655 - acc: 0.8465 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 4505/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3714 - acc: 0.8469 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 4506/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3687 - acc: 0.8479 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4507/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3670 - acc: 0.8429 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4508/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3679 - acc: 0.8454 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4509/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3651 - acc: 0.8479 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 4510/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3657 - acc: 0.8508 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 4511/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3712 - acc: 0.8453 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 4512/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3752 - acc: 0.8450 - val_loss: 0.3743 - val_acc: 0.8200\n Epoch 4513/10000\n 7200/7200 [==============================] - 1s 162us/step - loss: 0.3721 - acc: 0.8451 - val_loss: 0.3742 - val_acc: 0.8213\n Epoch 4514/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3718 - acc: 0.8483 - val_loss: 0.3747 - val_acc: 0.8213\n Epoch 4515/10000\n 7200/7200 [==============================] - 3s 375us/step - loss: 0.3666 - acc: 0.8476 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 4516/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3744 - acc: 0.8424 - val_loss: 0.3745 - val_acc: 0.8213\n Epoch 4517/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3654 - acc: 0.8443 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 4518/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3677 - acc: 0.8469 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4519/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3684 - acc: 0.8492 - val_loss: 0.3734 - val_acc: 0.8187\n Epoch 4520/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3665 - acc: 0.8501 - val_loss: 0.3736 - val_acc: 0.8237\n Epoch 4521/10000\n 7200/7200 [==============================] - 0s 58us/step - loss: 0.3669 - acc: 0.8488 - val_loss: 0.3735 - val_acc: 0.8237\n Epoch 4522/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3710 - acc: 0.8469 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4523/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3686 - acc: 0.8458 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 4524/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3712 - acc: 0.8471 - val_loss: 0.3731 - val_acc: 0.8237\n Epoch 4525/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3639 - acc: 0.8500 - val_loss: 0.3732 - val_acc: 0.8237\n Epoch 4526/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3697 - acc: 0.8465 - val_loss: 0.3743 - val_acc: 0.8213\n Epoch 4527/10000\n 7200/7200 [==============================] - 1s 166us/step - loss: 0.3704 - acc: 0.8467 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 4528/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3675 - acc: 0.8508 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 4529/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3658 - acc: 0.8472 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4530/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3714 - acc: 0.8444 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4531/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3688 - acc: 0.8467 - val_loss: 0.3757 - val_acc: 0.8187\n Epoch 4532/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3710 - acc: 0.8458 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4533/10000\n 7200/7200 [==============================] - 1s 74us/step - loss: 0.3697 - acc: 0.8462 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 4534/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3638 - acc: 0.8476 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 4535/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3697 - acc: 0.8436 - val_loss: 0.3740 - val_acc: 0.8187\n Epoch 4536/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3646 - acc: 0.8503 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 4537/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3677 - acc: 0.8499 - val_loss: 0.3731 - val_acc: 0.8187\n Epoch 4538/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3672 - acc: 0.8478 - val_loss: 0.3730 - val_acc: 0.8200\n Epoch 4539/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3696 - acc: 0.8500 - val_loss: 0.3732 - val_acc: 0.8213\n Epoch 4540/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3693 - acc: 0.8471 - val_loss: 0.3733 - val_acc: 0.8237\n Epoch 4541/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3675 - acc: 0.8486 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 4542/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3709 - acc: 0.8493 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 4543/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3662 - acc: 0.8476 - val_loss: 0.3738 - val_acc: 0.8225\n Epoch 4544/10000\n 7200/7200 [==============================] - 1s 76us/step - loss: 0.3714 - acc: 0.8444 - val_loss: 0.3758 - val_acc: 0.8187\n Epoch 4545/10000\n 7200/7200 [==============================] - ETA: 0s - loss: 0.3677 - acc: 0.846 - 19s 3ms/step - loss: 0.3692 - acc: 0.8457 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4546/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3671 - acc: 0.8478 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 4547/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3657 - acc: 0.8492 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 4548/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3678 - acc: 0.8519 - val_loss: 0.3737 - val_acc: 0.8225\n Epoch 4549/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3665 - acc: 0.8482 - val_loss: 0.3743 - val_acc: 0.8200\n Epoch 4550/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3696 - acc: 0.8460 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 4551/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3675 - acc: 0.8488 - val_loss: 0.3727 - val_acc: 0.8263\n Epoch 4552/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3709 - acc: 0.8489 - val_loss: 0.3740 - val_acc: 0.8200\n Epoch 4553/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3656 - acc: 0.8507 - val_loss: 0.3726 - val_acc: 0.8250\n Epoch 4554/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3654 - acc: 0.8486 - val_loss: 0.3722 - val_acc: 0.8237\n Epoch 4555/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3720 - acc: 0.8443 - val_loss: 0.3734 - val_acc: 0.8213\n Epoch 4556/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3671 - acc: 0.8471 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4557/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3714 - acc: 0.8474 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4558/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3694 - acc: 0.8478 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4559/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3684 - acc: 0.8481 - val_loss: 0.3738 - val_acc: 0.8187\n Epoch 4560/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3667 - acc: 0.8475 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 4561/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3727 - acc: 0.8457 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 4562/10000\n 7200/7200 [==============================] - 0s 60us/step - loss: 0.3706 - acc: 0.8469 - val_loss: 0.3741 - val_acc: 0.8200\n Epoch 4563/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3709 - acc: 0.8462 - val_loss: 0.3756 - val_acc: 0.8187\n Epoch 4564/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3711 - acc: 0.8461 - val_loss: 0.3739 - val_acc: 0.8187\n Epoch 4565/10000\n 7200/7200 [==============================] - 0s 62us/step - loss: 0.3703 - acc: 0.8468 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 4566/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3735 - acc: 0.8428 - val_loss: 0.3735 - val_acc: 0.8213\n Epoch 4567/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3670 - acc: 0.8497 - val_loss: 0.3724 - val_acc: 0.8250\n Epoch 4568/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3719 - acc: 0.8467 - val_loss: 0.3733 - val_acc: 0.8187\n Epoch 4569/10000\n 7200/7200 [==============================] - 1s 98us/step - loss: 0.3665 - acc: 0.8483 - val_loss: 0.3744 - val_acc: 0.8187\n Epoch 4570/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3663 - acc: 0.8510 - val_loss: 0.3741 - val_acc: 0.8200\n Epoch 4571/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3647 - acc: 0.8451 - val_loss: 0.3741 - val_acc: 0.8187\n Epoch 4572/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3678 - acc: 0.8476 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4573/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3705 - acc: 0.8479 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 4574/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3646 - acc: 0.8486 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4575/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3655 - acc: 0.8485 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 4576/10000\n 7200/7200 [==============================] - 0s 59us/step - loss: 0.3696 - acc: 0.8472 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4577/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3671 - acc: 0.8476 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 4578/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3740 - acc: 0.8471 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 4579/10000\n 7200/7200 [==============================] - 0s 61us/step - loss: 0.3689 - acc: 0.8454 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4580/10000\n 7200/7200 [==============================] - 0s 63us/step - loss: 0.3695 - acc: 0.8457 - val_loss: 0.3751 - val_acc: 0.8200\n Epoch 4581/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3669 - acc: 0.8494 - val_loss: 0.3749 - val_acc: 0.8200\n Epoch 4582/10000\n 7200/7200 [==============================] - 1s 83us/step - loss: 0.3683 - acc: 0.8453 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 4583/10000\n 7200/7200 [==============================] - 1s 89us/step - loss: 0.3636 - acc: 0.8485 - val_loss: 0.3754 - val_acc: 0.8187\n Epoch 4584/10000\n 7200/7200 [==============================] - 1s 86us/step - loss: 0.3674 - acc: 0.8492 - val_loss: 0.3743 - val_acc: 0.8213\n Epoch 4585/10000\n 7200/7200 [==============================] - 1s 81us/step - loss: 0.3720 - acc: 0.8479 - val_loss: 0.3746 - val_acc: 0.8213\n Epoch 4586/10000\n 7200/7200 [==============================] - 1s 85us/step - loss: 0.3676 - acc: 0.8518 - val_loss: 0.3729 - val_acc: 0.8225\n Epoch 4587/10000\n 7200/7200 [==============================] - 1s 78us/step - loss: 0.3698 - acc: 0.8471 - val_loss: 0.3747 - val_acc: 0.8213\n Epoch 4588/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3698 - acc: 0.8462 - val_loss: 0.3762 - val_acc: 0.8187\n Epoch 4589/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3651 - acc: 0.8499 - val_loss: 0.3743 - val_acc: 0.8213\n Epoch 4590/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3706 - acc: 0.8446 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4591/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3696 - acc: 0.8489 - val_loss: 0.3750 - val_acc: 0.8187\n Epoch 4592/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3688 - acc: 0.8486 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 4593/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3711 - acc: 0.8476 - val_loss: 0.3760 - val_acc: 0.8187\n Epoch 4594/10000\n 7200/7200 [==============================] - 1s 79us/step - loss: 0.3651 - acc: 0.8503 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4595/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3726 - acc: 0.8475 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 4596/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3665 - acc: 0.8506 - val_loss: 0.3741 - val_acc: 0.8213\n Epoch 4597/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3705 - acc: 0.8483 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 4598/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3703 - acc: 0.8461 - val_loss: 0.3744 - val_acc: 0.8213\n Epoch 4599/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3690 - acc: 0.8478 - val_loss: 0.3763 - val_acc: 0.8187\n Epoch 4600/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3719 - acc: 0.8489 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4601/10000\n 7200/7200 [==============================] - 1s 72us/step - loss: 0.3685 - acc: 0.8429 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4602/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3710 - acc: 0.8467 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 4603/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3711 - acc: 0.8438 - val_loss: 0.3751 - val_acc: 0.8187\n Epoch 4604/10000\n 7200/7200 [==============================] - 1s 77us/step - loss: 0.3740 - acc: 0.8458 - val_loss: 0.3756 - val_acc: 0.8200\n Epoch 4605/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3676 - acc: 0.8475 - val_loss: 0.3767 - val_acc: 0.8187\n Epoch 4606/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3675 - acc: 0.8492 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4607/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3717 - acc: 0.8451 - val_loss: 0.3755 - val_acc: 0.8187\n Epoch 4608/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3708 - acc: 0.8428 - val_loss: 0.3761 - val_acc: 0.8187\n Epoch 4609/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3677 - acc: 0.8510 - val_loss: 0.3752 - val_acc: 0.8187\n Epoch 4610/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3659 - acc: 0.8496 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 4611/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3678 - acc: 0.8492 - val_loss: 0.3746 - val_acc: 0.8187\n Epoch 4612/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3686 - acc: 0.8453 - val_loss: 0.3743 - val_acc: 0.8187\n Epoch 4613/10000\n 7200/7200 [==============================] - 0s 69us/step - loss: 0.3656 - acc: 0.8482 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 4614/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3660 - acc: 0.8469 - val_loss: 0.3737 - val_acc: 0.8187\n Epoch 4615/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3656 - acc: 0.8478 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 4616/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3676 - acc: 0.8438 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 4617/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3718 - acc: 0.8458 - val_loss: 0.3743 - val_acc: 0.8200\n Epoch 4618/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3629 - acc: 0.8489 - val_loss: 0.3742 - val_acc: 0.8187\n Epoch 4619/10000\n 7200/7200 [==============================] - 0s 65us/step - loss: 0.3678 - acc: 0.8486 - val_loss: 0.3737 - val_acc: 0.8213\n Epoch 4620/10000\n 7200/7200 [==============================] - 1s 70us/step - loss: 0.3681 - acc: 0.8492 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 4621/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3745 - acc: 0.8444 - val_loss: 0.3744 - val_acc: 0.8213\n Epoch 4622/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3653 - acc: 0.8490 - val_loss: 0.3745 - val_acc: 0.8187\n Epoch 4623/10000\n 7200/7200 [==============================] - 0s 64us/step - loss: 0.3670 - acc: 0.8488 - val_loss: 0.3748 - val_acc: 0.8187\n Epoch 4624/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3697 - acc: 0.8475 - val_loss: 0.3749 - val_acc: 0.8187\n Epoch 4625/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3648 - acc: 0.8488 - val_loss: 0.3742 - val_acc: 0.8213\n Epoch 4626/10000\n 7200/7200 [==============================] - 1s 69us/step - loss: 0.3683 - acc: 0.8467 - val_loss: 0.3759 - val_acc: 0.8187\n Epoch 4627/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3664 - acc: 0.8490 - val_loss: 0.3742 - val_acc: 0.8200\n Epoch 4628/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3707 - acc: 0.8475 - val_loss: 0.3740 - val_acc: 0.8213\n Epoch 4629/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3671 - acc: 0.8472 - val_loss: 0.3747 - val_acc: 0.8187\n Epoch 4630/10000\n 7200/7200 [==============================] - 0s 68us/step - loss: 0.3694 - acc: 0.8492 - val_loss: 0.3747 - val_acc: 0.8200\n Epoch 4631/10000\n 7200/7200 [==============================] - 1s 71us/step - loss: 0.3653 - acc: 0.8503 - val_loss: 0.3746 - val_acc: 0.8200\n Epoch 4632/10000\n 7200/7200 [==============================] - 1s 75us/step - loss: 0.3691 - acc: 0.8472 - val_loss: 0.3752 - val_acc: 0.8175\n Epoch 4633/10000\n 7200/7200 [==============================] - 1s 120us/step - loss: 0.3723 - acc: 0.8458 - val_loss: 0.3736 - val_acc: 0.8225\n Epoch 4634/10000\n 7200/7200 [==============================] - 1s 114us/step - loss: 0.3698 - acc: 0.8432 - val_loss: 0.3741 - val_acc: 0.8200\n Epoch 4635/10000\n 7200/7200 [==============================] - 1s 73us/step - loss: 0.3696 - acc: 0.8457 - val_loss: 0.3753 - val_acc: 0.8187\n Epoch 4636/10000\n 7200/7200 [==============================] - 0s 66us/step - loss: 0.3655 - acc: 0.8458 - val_loss: 0.3756 - val_acc: 0.8175\n Epoch 4637/10000\n 7200/7200 [==============================] - 0s 67us/step - loss: 0.3701 - acc: 0.8481 - val_loss: 0.3750 - val_acc: 0.8175\n Epoch 4638/10000\n 4896/7200 [===================>..........] - ETA: 0s - loss: 0.3688 - acc: 0.8474\n\n\n```python\n# Plot training & validation accuracy values\nplt.plot(history.history['acc'])\nplt.plot(history.history['val_acc'])\nplt.title('Model accuracy')\nplt.ylabel('Accuracy')\nplt.xlabel('Epoch')\nplt.legend(['Train', 'Test'], loc='upper left')\nplt.show()\n\n# Plot training & validation loss values\nplt.plot(history.history['loss'])\nplt.plot(history.history['val_loss'])\nplt.title('Model loss')\nplt.ylabel('Loss')\nplt.xlabel('Epoch')\nplt.legend(['Train', 'Test'], loc='upper left')\nplt.show()\n```\n\n

                                        :::::The Testing of the Neural Network:::::


                                        \nWe are testing the model with 2000 testing sample extracted randomly from the dataset during the spliting of dataset.\n\n\n```python\ny_pred = classifier.predict(X_test)\ny_pred = (y_pred > 0.5)\nprint(y_pred)\n```\n\n

                                        :::::The Evaluation of the Model:::::


                                        \nAs, we are doing the binary classification so we have used the confusion matrix for the evaluation.
                                        \nA confusion matrix, in predictive analytics, is a two-by-two table that tells us the rate of false positives, false negatives, true positives and true negatives for a test or predictor. \n\n\n```python\nfrom sklearn.metrics import confusion_matrix\ncm = confusion_matrix(y_test, y_pred)\nprint(cm)\n```\n\n

                                        ::::Various Measures Calculated using the Confusion Matrix:::::

                                        \n
                                          \n
                                        1. Accuracy
                                          2. \n
                                        2. Recall: Recall can be defined as the ratio of the total number of correctly classified positive examples divide to the total number of positive examples. High Recall indicates the class is correctly recognized (small number of FN).
                                          3. \n
                                        3. Precision: Precision is calculated by the division of the total number of correctly classified positive examples by the total number of predicted positive examples. High Precision indicates an example labeled as positive is indeed positive (small number of FP).
                                          4. \n
                                        4. F-Measure: F-measure which uses Harmonic Mean in place of Arithmetic Mean as it punishes the extreme values more. The F-Measure will always be nearer to the smaller value of Precision or Recall.
                                          5. \n
                                        \n\n\n```python\ntotal_test_sample = 2000\nAccuracy = ((cm[0][0]+cm[1][1])/total_test_sample)*100\nRecall = (cm[0][0]/(cm[0][0]+cm[1][0]))*100\nPrecision = (cm[0][0]/(cm[0][0]+cm[0][1]))*100\n```\n\n\n```python\nRecall_1 = (cm[0][0]/(cm[0][0]+cm[1][0]))\nPrecision_1 = (cm[0][0]/(cm[0][0]+cm[0][1]))\n```\n\n\n```python\nF = (Recall_1+Precision_1)/(2*Recall_1*Precision_1)\n```\n\n\n```python\nprint(\"********** CONFUSION MATRIX MEASURES**********\")\nprint(\"The accuracy is:::::\",Accuracy,\"%\")\nprint(\"\\n\")\nprint(\"**********************************************\")\nprint(\"The Recall is:::::\", Recall,\"%\")\nprint(\"\\n\")\nprint(\"**********************************************\")\nprint(\"The Precision is:::::\",Precision,\"%\")\nprint(\"\\n\")\nprint(\"**********************************************\")\nprint(\"The F-Measure is:::::\",F)\n```\n", "meta": {"hexsha": "eb1fbff59e5865c0d1e5320417b4e619ec44ff8d", "size": 790152, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Churn NN Model.ipynb", "max_stars_repo_name": "sharmarahulraj03/Customer-Churn-Prediction-", "max_stars_repo_head_hexsha": "7e24ff90bd1186d52a842bf592dcc1d603837156", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Churn NN Model.ipynb", "max_issues_repo_name": "sharmarahulraj03/Customer-Churn-Prediction-", "max_issues_repo_head_hexsha": "7e24ff90bd1186d52a842bf592dcc1d603837156", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Churn NN Model.ipynb", "max_forks_repo_name": "sharmarahulraj03/Customer-Churn-Prediction-", "max_forks_repo_head_hexsha": "7e24ff90bd1186d52a842bf592dcc1d603837156", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 74.1509009009, "max_line_length": 290, "alphanum_fraction": 0.4658040478, "converted": true, "num_tokens": 301326, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.41178333558441516}} {"text": "**TODO**:\n- fix pyspice to do not only R sweeps but also other parameter sweeps. The issue appears to be in the `plot` module that read the return from ngspice server results.\n - need to experiment with ngspice server, since I have already confirmed R sweep just using ngspice alone\n - run simulation from pyspice step by step and grab results without `plot`\n - fix `plot`\n \n- use what we have done\n- find max power and efficincy with the SPICE\n- add usage of standard resistor values from sweep \n\n\n```python\n#Library import statements\n\nfrom skidl.pyspice import *\n#can you say cheeky \nimport PySpice as pspice\n#becouse it's written by a kiwi you know\nimport lcapy as kiwi\n\nimport sympy as sym\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nimport warnings\n\n\nfrom IPython.display import YouTubeVideo, display\n\nimport traceback\n```\n\n WARNING: KICAD_SYMBOL_DIR environment variable is missing, so the default KiCad symbol libraries won't be searched.\n\n\n#notebook specific loading control statements \n%matplotlib inline\n#tool to log notebook internals\n#https://github.com/jrjohansson/version_information\n%load_ext version_information\n%version_information skidl, PySpice,lcapy, sympy, numpy, matplotlib, pandas, scipy\n\n# Maximum Power Transfer Theorem\n\nThe Maximum Power transfer theorem for DC states that when the Load is equal to the Thevenin resistance of the circuit($R_{L}=R_{th}$) the power delivered to the load is maximized which is proven via taking the derivative of $$P=i^2 R_L=(\\dfrac{V_{th}}{R_{th}+R_L})^2 R_L$$ and setting it to zero and solving. The details of which are given by ALL ABOUT ELECTRONICS in his YT video [Maximum Power Transfer Theorem for DC Circuits (with Examples)](https://www.youtube.com/watch?v=RbII8o49Hvs). But suffice to say we have already done the work to find this in the last two sections of this chapter by finding the Thevenin and Norton values of the circuit or when permitted using the .tf to find the DC transfer function which also gives us the input resistance. But here we want to show how to just sweep the load resistance itself (or any resistor in the circuit) and then find the peak in the resulting data that will give us the maximum power while also measuring the efficiency of the load where then we can find the ideal optimal load by finding the intersection of the two curves. We can do this by again utilizing the machinery we have developed to easily find the Thevenin and Norton equivalent circuits for circuits with multiple sources. Further, we want to then translate that semi-arbitrary value to a value we can use in the real world where the values you can get for a resistor are finite. So, let\u2019s start with a theoretical model and then work with Examples 2 and 3 from ALL ABOUT ELECTRONICS video and develop a tool to automatically do all this for us.\n\n\n## Max Power Delivered vs Most Efficient Power Delivered\n\n\n```python\nefficiency, powerload, powersource, current, voltage, Rth, RL =sym.symbols(r\"\\eta, P_L, P_S, i, v, R_{th}, R_L\")\nefficiency, powerload, powersource, current, voltage, Rth, RL\n```\n\nFor a DC Thevenin reduced circuit (one that contains only a Thevenin voltage source, Thevenin equivalent resistor, and equivalent load resistor) we know that the current will be\n\n\n```python\nithev_eq=sym.Eq(current, voltage/(Rth+RL)); ithev_eq\n```\n\nand the power in the load will be \n\n\n```python\npowerload_eq=sym.Eq(powerload, current**2 * RL); powerload_eq\n```\n\nand thus the power the load receives reduces to\n\n\n```python\npowerload_eq=powerload_eq.subs({ithev_eq.lhs: ithev_eq.rhs}); powerload_eq\n```\n\nUsing the first example from [Maximum Power Transfer Theorem for DC Circuits (with Examples)](https://www.youtube.com/watch?v=RbII8o49Hvs) we know and can find the Thevenin equivalent easily, and thus the max power that the source can supply. And so if we then plot the values of the power of the load vs the resistance of the load we get\n\n\n```python\nsubs={Rth:8, voltage:32}\n```\n\n\n```python\npowerload_lam=sym.lambdify(RL, powerload_eq.rhs.subs(subs))\n\nRL_sweep=np.linspace(0, 100)\nplt.plot(RL_sweep, powerload_lam(RL_sweep), label=f'${powerload}$ vs ${Rth}$')\nplt.plot(8, powerload_lam(subs[Rth]), 'ro', label=f'Max Load Power {powerload_lam(subs[Rth])} [watts] @ {subs[Rth]} [Ohms]')\nplt.xlabel(f\"${RL}$[Ohms]\"); plt.ylabel(f\"${powerload}$[Watts]\")\nplt.legend()\nplt.grid()\nplt.title('Load Power vs Load Resistance');\n```\n\nBut this is not our efficient load since our Thevenin equivalent source of the voltage and current source given by this example can deliver a total power of 128 [watts] when the load is short-circuited. So, then what is the most efficient load. While efficiency is defined as. Where we are looking at the global sources and global loads to our circuit which we have made easy in this example by looking only at a Thevenin reduced DC circuit\n\n\n```python\nefficiency_eq=sym.Eq(efficiency, powerload/powersource); efficiency_eq\n```\n\nthe power from the source with a completed Thevenin circuit with load is the sum of the power in the Thevenin and load resistances\n\n\n```python\npowersource_eq=sym.Eq(powersource, current**2 *Rth +current**2 *RL); powersource_eq\n```\n\nSubstituting the expression for the current in the Thevenin circuit we have\n\n\n```python\npowersource_eq=sym.simplify(powersource_eq.subs({ithev_eq.lhs: ithev_eq.rhs})); powersource_eq\n```\n\nplotting this as a function of the load we can see how the load affects the power supplied on the Thevenin equivalent source\n\n\n```python\npowersource_lam=sym.lambdify(RL, powersource_eq.rhs.subs(subs))\n\nplt.plot(RL_sweep, powersource_lam(RL_sweep), label=f'${powerload}$ vs ${Rth}$')\nplt.plot(8, powersource_lam(subs[Rth]), 'ro', label=f'Source Power {powersource_lam(subs[Rth])} [watts] @ ${Rth}$ ({subs[Rth]} [Ohms])')\nplt.xlabel(f\"${RL}$[Ohms]\"); plt.ylabel(f\"${powerload}$[Watts]\")\nplt.legend()\nplt.grid()\nplt.title('Source Power vs Load Resistance');\n```\n\nso then we find for the case of a DC Thevenin circuit our efficiency is\n\n\n```python\nefficiency_eq=efficiency_eq.subs({powersource_eq.lhs:powersource_eq.rhs, powerload_eq.lhs:powerload_eq.rhs }); efficiency_eq\n```\n\nSo then we can compare the efficiency (which is only defined between 0 and 1) to the power transferred to the load normalized to the maximum power at the Thevenin resistance as follows\n\n\n```python\npowerload_norm_lam=sym.lambdify(RL, powerload_eq.rhs.subs(subs)/powerload_eq.rhs.subs(subs).subs({RL:subs[Rth]}))\n\nefficiency_lam=sym.lambdify(RL, efficiency_eq.rhs.subs(subs))\n\n\nplt.plot(RL_sweep, powerload_norm_lam(RL_sweep), label=f'Normalized ${powerload}$ vs ${RL}$')\nplt.plot(subs[Rth], powerload_norm_lam(subs[Rth]), 'ro', label=f'Max Load Power {powerload_lam(subs[Rth])} [watts] @ {subs[Rth]} [Ohms]')\n\nplt.plot(RL_sweep, efficiency_lam(RL_sweep), label=f'${efficiency}$ vs ${RL}$')\n\nplt.plot(subs[Rth], efficiency_lam(subs[Rth]), 'r*', \n label=f'${efficiency}={efficiency_lam(subs[Rth])}$ at $\\max[{powerload}]$')\n\n\nplt.xlabel(f\"${RL}$[Ohms]\"); \nplt.ylabel(r'$\\dfrac{'+f\"{powerload}\"+'}{\\max['+f\"{powerload}\"+\"]}$\")\n\nplt.legend()\nplt.grid()\nplt.title(f'Maxium Load Power Transfer & Efficiency vs ${RL}$');\n```\n\nSo then if we need to optimize the DC maximum power to the load, we know it to be the load resistance equal the Thevenin resistance; but if we could sacrifice the maximum power to the load in order max out the efficiency of our total circuit while minimizing the loss of power delivered to load. We then can then find said load via the intersection of the two curves above thusly\n\n\n```python\n#%%writefile -a DC_1_Codes.py\n#chapter 1 section 6 findIntersection function\n#Assist function to find the intersection of two functions\n\n#from https://glowingpython.blogspot.com/2011/05/hot-to-find-intersection-of-two.html \n#load fslove from scipy's optimize module\nfrom scipy.optimize import fsolve\n\n#helper function to find the intersection of two functions with an initial guess\ndef findIntersection(fun1,fun2,x0):\n \"\"\"\n Aid function to find the intersection point of two curves\n from: https://glowingpython.blogspot.com/2011/05/hot-to-find-intersection-of-two.html \n \n Args:\n func1(function or class): the first function whose curve is \n used to find the intersection of the two curves\n \n func2(function or class): the second function whose curve is \n used to find the intersection of the two curves\n \n x0 (float); initial guess of the intersection of the two functions\n \n Returns:\n Returns array of float that are the intersections of the two functions, \n this is not very robust and thus one should read `fsolve`'s documentation \n for caveats of usage\n \"\"\"\n return fsolve(lambda x : fun1(x) - fun2(x),x0)\n```\n\n\n```python\n#find the find the intersection and round value to three digits and; get just the first intersection\noptimal_point=np.around(findIntersection(powerload_norm_lam, efficiency_lam, subs[Rth]), 3)[0]\noptimal_point\n```\n\n\n```python\nplt.plot(RL_sweep, powerload_norm_lam(RL_sweep), label=f'${powerload}$ vs ${RL}$')\nplt.plot(subs[Rth], powerload_norm_lam(subs[Rth]), 'ro', label=f'Max ${powerload}$ {powerload_lam(subs[Rth])} [watts] @ {subs[Rth]} [Ohms]')\n\nplt.plot(RL_sweep, efficiency_lam(RL_sweep), label=f'${efficiency}$ vs ${RL}$')\nplt.plot(optimal_point, efficiency_lam(optimal_point), 'go', \n label=f'Mutualy Optimal Point @${RL}={optimal_point}$[Ohm]; ${efficiency}={efficiency_lam(optimal_point)}$; ${powerload}={powerload_lam(optimal_point)}$ [watts]')\n\nplt.xlabel(f\"${RL}$[Ohms]\")\nplt.ylabel(r'$\\dfrac{'+f\"{powerload}\"+'}{\\max['+f\"{powerload}\"+\"]}$\")\n\nplt.legend()\nplt.grid()\nplt.title(f'Maximum Load Power Transfer & Efficiency vs ${RL}$ with cross optimization');\n```\n\nThus, by moving to the intersection of the Power to the load and the efficiency of the circuit at the load from the Maximum power to the load point. We have a reduction of the amount of power delivered to the load of\n\n\n```python\npowerload_at_mpow=powerload_lam(subs[Rth])\npowerload_at_mp=powerload_lam(optimal_point)\nf\"{powerload_at_mpow}[watts] to {powerload_at_mp}[watts]; a {powerload_at_mp-powerload_at_mpow}[watt] change\"\n```\n\n\n\n\n '32.0[watts] to 24.0[watts]; a -8.0[watt] change'\n\n\n\nAnd have gained an efficiency boost of\n\n\n```python\nefficiency_at_mpow=efficiency_lam(subs[Rth])\nefficiency_at_mp=efficiency_lam(optimal_point)\nf\"{efficiency_at_mpow*100}% to {efficiency_at_mp*100}%; a {(efficiency_at_mp-efficiency_at_mpow)*100}% gain\"\n```\n\n\n\n\n '50.0% to 75.0%; a 25.0% gain'\n\n\n\nSo now the task at hand is to move beyond this ideal theoretical example to having SPICE find the values we need and then building a testbench class in python that will do the data analysis along with all the SPICE work we just showed automatically for nearly any DC circuit. Where AC circuits and circuits that rely on Transient circuit effects (think switch-mode power supplies) will have to be analyzed separately when we cross those bridges.\n\n## Cannot move on due to the issue in pyspice that prevents sweeping anything but Current and Voltage Sources; see TODO above\n\n## Example 2 from \"Maximum Power Transfer Theorem for DC Circuits (with Examples)\" @ ~4:47 min \n\n\n```python\nYouTubeVideo('RbII8o49Hvs', width=500, height=400, start=287)\n```\n\n\n\n\n\n\n\n\n\n\n\n```python\nreset()\nnet_1=Net('N1'); net_2=Net('N2'); net_3=Net('N3'); net_4=Net('N4')\n\n#voltage source bottom left\nvs=V(dc_value=100@u_V); vs['p', 'n']+=net_1, gnd\n#restors on the center leg\nrleft=R(ref='left', value=4@u_Ohm); rleft[1, 2]+=net_1, net_2\nrright=R(ref='right', value=4@u_Ohm); rright[1, 2]+=net_2, net_3\n\n#vcvs and resistor on top leg\nvcvs=E(voltage_gain=1)\n#vcvs inputs; outputs\nvcvs['ip', 'in']+=net_2, net_1; vcvs['op', 'on']+=net_1, net_4\nrtop=R(ref='top', value=4@u_Ohm); rtop[1, 2]+=net_4, net_3\n\n#load with dummy resistance\nrload=R(ref='load', value=1@u_Ohm); rload[1, 2]+=net_3, gnd\n\n\ncirc=generate_netlist()\nprint(circ)\n```\n\n .title \n V1 N1 0 100V\n Rleft N1 N2 4Ohm\n Rright N2 N3 4Ohm\n E1 N1 N4 N2 N1 1\n Rtop N4 N3 4Ohm\n Rload N3 0 1Ohm\n \n\n\n \n No errors or warnings found during netlist generation.\n \n\n\n## Example 3 from \"Maximum Power Transfer Theorem for DC Circuits (with Examples)\" @ ~8:46 min \n\n\n```python\nYouTubeVideo('RbII8o49Hvs', width=500, height=400, start=526)\n```\n\n\n\n\n\n\n\n\n\n\n\n```python\nreset()\nnet_1=Net('N1'); net_2=Net('N2'); net_3=Net('N3')\n\nvs=V(dc_value=10@u_V); vs['p', 'n']+=net_1, gnd\n\n#center T network\nrleft=R(ref='left', value=30@u_Ohm); rleft[1, 2]+=net_1, net_2\nrcenter=R(ref='center', value=30@u_Ohm); rcenter[1, 2]+=net_2, gnd\nrright=R(ref='right', value=30@u_Ohm); rright[1, 2]+=net_2, net_3\n\n#rvar with dummy resistance\nrtest=R(ref='test', value=1@u_Ohm); rtest[1, 2]+=net_1, net_3\n\nrload=R(ref='load', value=10@u_Ohm); rload[1, 2]+=net_3, gnd\n\ncirc=generate_netlist()\nprint(circ)\n```\n\n .title \n V1 N1 0 10V\n Rleft N1 N2 30Ohm\n Rcenter N2 0 30Ohm\n Rright N2 N3 30Ohm\n Rtest N1 N3 1Ohm\n Rload N3 0 10Ohm\n \n\n\n \n No errors or warnings found during netlist generation.\n \n\n\n## Citations:\n[1] ALL ABOUT ELECTRONICS. \"Maximum Power Transfer Theorem for DC Circuits (with Examples),\" YouTube, May 20, 2017. [Video file]. Available: https://youtu.be/RbII8o49Hvs. [Accessed: Nov 30, 2020].\n\n[2] @JustGlowing, \u201cHow to find the intersection of two functions,\u201d The Glowing Python, 10-May-2011. [Online]. Available: https://glowingpython.blogspot.com/2011/05/hot-to-find-intersection-of-two.html. [Accessed: 20-Nov-2020].\n\n\n```python\n\n```\n", "meta": {"hexsha": "9a5a31c1cbeadb1288a1bf96ae4afa15aa31332e", "size": 195980, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "_build/html/_sources/DC_1/DC_6_DC_MaxPower_MaxEfficiency.ipynb", "max_stars_repo_name": "PyLCARS/Python-and-SPICE-Book", "max_stars_repo_head_hexsha": "0bf02aa16d97115cea955d33a7aab7e02f8d3453", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2021-01-04T23:56:51.000Z", "max_stars_repo_stars_event_max_datetime": "2021-02-22T13:22:30.000Z", "max_issues_repo_path": "_build/html/_sources/DC_1/DC_6_DC_MaxPower_MaxEfficiency.ipynb", "max_issues_repo_name": "PyLCARS/Python-and-SPICE-Book", "max_issues_repo_head_hexsha": "0bf02aa16d97115cea955d33a7aab7e02f8d3453", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "_build/html/_sources/DC_1/DC_6_DC_MaxPower_MaxEfficiency.ipynb", "max_forks_repo_name": "PyLCARS/Python-and-SPICE-Book", "max_forks_repo_head_hexsha": "0bf02aa16d97115cea955d33a7aab7e02f8d3453", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 217.9977753059, "max_line_length": 34400, "alphanum_fraction": 0.9124706603, "converted": true, "num_tokens": 3901, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.6992544335934765, "lm_q1q2_score": 0.4117833355844151}} {"text": "# Experiment 2: Airtrack I---Kinematics\n### Objectives\n- To learn good lab practice\n- To learn how to estimate and propagate uncertainties\n- To learn how position, velocity, and acceleration are related\n- To learn how to use the Airtrack\n- To learn how to use PASCO Capstone Data Acquisition and Analysis software\n\n### Equipment\n- One Airtrack, Blower, Cart, and Acoustic Sensor\n- One wooden reflector\n- One 2x4 \u201cshim\u201d\n- One 30 cm plastic ruler\n- One two-meter stick\n\n### Safety\n- **Do not attempt to push the carts faster than 1 m/s.**\n- **Be careful placing the carts on the track.**\n- **Do not damage the track.** Please place paper underneath the carts when they are resting on the track without the air turned on.\n- **Do not slide the carts on the track without the air turned on.**\n\n# Introduction\n\nBefore one can study the action of forces on objects, one must be able to describe motion. This description of motion is called $kinematics$. We will be learning how to use two tools to investigate the meanings of the three kinematic equations for motion in one dimension: \nThe definition of average velocity: \n\n\\begin{equation}\nv_{avg} \\equiv \\frac{\\Delta x}{\\Delta t}\n\\label{eq:v_avg}\n\\end{equation}\n\nThe definition of average acceleration: \n\n\\begin{equation}\na_{avg} \\equiv \\frac{\\Delta v}{\\Delta t}\n\\label{eq:a_avg}\n\\end{equation}\n\nAnd the position of an object undergoing constant acceleration: \n\n\\begin{equation}\nx(t) = \\frac{1}{2} a t^2 + v_o t + x_o\n\\label{eq:x_of_t}\n\\end{equation}\n\nIn this lab, your job will be to compare different measurements of velocity and acceleration for an air-cart experiencing both no acceleration and constant acceleration.\n\n# Theory\n\n### Measurement of Velocity and Acceleration\nAn acoustic sensor will measure the position of a cart on the airtrack at specific intervals of time. By use of the definition of velocity (Equation $\\eqref{eq:v_avg}$), we can compute the average velocity between two measurements. Notice that if we plot the position versus time, the fraction is the same as the rise divided by the run-- or the slope of the graph. By plotting *x* vs. *t*, we then find the slope of the graph and determine the velocity.\n\nThis same technique works for determining acceleration. However, notice that the variables to plot are now *velocity* versus time.\n\n### Analysis of Uncertainties\nIt will be important to know what the uncertainties are in our computed velocities and accelerations.\n\nFirst, recall that a difference in position or time is a subtraction of two quantities: \n\\begin{equation} \n\\Delta x = x_2 - x_1 \\nonumber\n\\end{equation}\n\nTherefore, we can use (R1) to compute the uncertainty in the difference: \n\\begin{equation}\n(\\delta(\\Delta x))^2 = (\\delta x_2)^2 - (\\delta x_1)^2 \\nonumber\n\\end{equation}\n\nHowever, in this case, the uncertainties $\\delta x_1$ and $\\delta x_2$ should be the same, so we can say that the uncertainty in our computed difference is: \n\\begin{equation} \n\\delta(\\Delta x) = \\sqrt{2} (\\delta x) \\nonumber\n\\end{equation} \n\nwhere $\\delta x$ is the uncertainty in a single position measurement. We can say the same for time differences: \n\\begin{equation} \n\\delta(\\Delta t) = \\sqrt{2} (\\delta t) \\nonumber\n\\end{equation}\n\nwhere $\\delta t$ is the uncertainty in a single time measurement. Notice that the uncertainty in the difference is larger than the uncertainty in each individual measurement.\n\nComputing the uncertainty in velocity, $\\delta v$, is a straightforward application of (R2). Looking at Equation $\\eqref{eq:v_avg}$ we have: \n\\begin{equation} \n\\left(\\frac{\\delta v}{v}\\right)^2 = \\left(\\frac{\\delta (\\Delta x)}{\\Delta x}\\right)^2 + \\left(\\frac{\\delta (\\Delta t)}{\\Delta t}\\right)^2 \\nonumber\n\\end{equation}\n\nThis equation needs some work to become useful. Capstone will take data points at constant $\\Delta t$. But because $\\Delta x$ may be different for each measurement, we should try to remove it from our equation. We do this by using the definition of velocity to replace $\\Delta x$ with $v \\Delta t$: \n\\begin{equation} \n\\left(\\frac{\\delta v}{v}\\right)^2 = \\left(\\frac{\\delta (\\Delta x)}{v \\Delta t}\\right)^2 + \\left(\\frac{\\delta (\\Delta t)}{\\Delta t}\\right)^2 \\nonumber\n\\end{equation}\n\nFactoring out $\\Delta t$ so that we need only use it once: \n\\begin{equation} \n\\left(\\frac{\\delta v}{v}\\right)^2 = \\frac{1}{(\\Delta t)^2} \\left(\\left(\\frac{\\delta (\\Delta x)}{v}\\right)^2 + \\left(\\delta (\\Delta t)\\right)^2 \\right) \\nonumber\n\\end{equation}\n\nWe may also multiply through by $v^2$: \n\\begin{equation} \n\\left(\\delta v\\right)^2 = \\frac{1}{(\\Delta t)^2} \\left(\\left(\\delta (\\Delta x)\\right)^2 + v^2 \\left(\\delta (\\Delta t)\\right)^2 \\right)\n\\label{eq:v_unc}\n\\end{equation}\n\nNow we have an equation to find our uncertainty in velocity based on things we know. Notice that the uncertainty in the velocity *increases* with the velocity. Also notice that the uncertainty is *not zero* when the velocity is zero.\nDoes this make sense?\n\n# Experimental Procedure\n\n### Leveling the Airtrack\nIn order to study unaccelerated motion, we must ensure that gravity does not interfere. To do this, we must level the airtrack. The best method for leveling the airtrack is to use the cart itself to verify that it is not pulled one way or the other by gravity. \n\n1. Turn on the blower (you should be able to feel the air coming out of the drilled holes) and place a cart at the 100 cm mark.\n2. Place your fingers at the front and rear of the cart so that it remains at rest.\n3. Release the cart without pushing it.\n\nIf the cart tends to move to one side, you will need to raise that side (or lower the other side). This can be accomplished by rotating the feet on the two-foot support. If this is still not sufficient, you may need to place a folded sheet of paper under one of the two supports until the cart favors neither side. It\u2019s OK if the cart slowly oscillates.\n\n### Configuring the Acoustic Sensor\nPlace a wooden reflector into the center top hole in the cart. You will now need to place the sensor so that it properly reads the position of the reflector.\n\n1. Place the sensor at the opposite end from where the air hose is placed.\n2. Place a cart at that end of the airtrack and set the sensor at the same height as the center of the wooden reflector.\n3. Ensure that the sensor is centered over the track and pointed straight down the track.\n4. Ensure that the top switch is in \u201ccart\u201d mode (*not* \u201cstick figure\u201d mode).\n\nThere are two plugs which must be inserted into the PASCO 850 interface. The yellow plug goes into Digital Input 1 and the black plug goes into Digital Input 2.\n\n### Using PASCO Capstone DAA\nCapstone will be used throughout the physics laboratories to collect data, so you should spend some time now learning how to use it. To set it up for the first time, the PASCO 850 interface must be turned on (press and hold the power button until it beeps and the green light is lit) and connected to the computer.\n\n1. Start Capstone.\n2. Click \u201cHardware Setup\u201d on the left-side \u201cTools\u201d palette. A picture of the PASCO 850 interface will appear.\n3. Click on the Digital Input 1 port and select \u201cMotion Sensor II.\u201d\n4. Click on \u201cProperties\u201d in the lower left. You will need to enter the proper speed of sound. The speed of sound is (331+0.6T) m/s where T is the air temperature in Celcius. Click \u201cOK.\u201d\n5. Click on \u201cHardware Setup\u201d to close the window.\n6. In the right-side palette, drag \u201cGraph\u201d to the main window. A large graph will appear.\n\nAt the very bottom of the window, notice the sample rate (in Hz). This number tells you how many data points are taken every second. You will need this number later.\n\nWhen you click \u201cRecord,\u201d Capstone will start taking data on all the sensors attached to the 850 interface. When you click \u201cStop,\u201d Capstone will stop taking data. Every Record/Stop sequence is considered a \u201crun.\u201d All of the data runs you gather are listed in the \u201cData Summary\u201d tab on the left-side palette. Clicking \u201cData Summary\u201d a second time will close that window.\n\n### Testing the Acoustic Sensor using Capstone\nA very quick test of the sensor would be to click \u201cRecord\u201d and move your hand towards and away from the sensor a few times and then click \u201cStop.\u201d By clicking `` button and select a data source, then click Record/Stop.\n6. Click each button on the graph to see what it does.\n7. Learn how to pan and zoom your graph.\n8. Drag the legend around the graph.\n9. Delete all your graphs/tables (select the display by clicking on it, then click on the \u201cDelete Selected Display\u201d icon in the upper Capstone toolbar).\n10. Delete every data run and start over by clicking the arrow next to \u201cDelete Last Run\u201d in the bottom tool palette and selecting \u201cDelete All Runs.\u201d\n\n\n```python\n# Sensor test (Sawtooth plot)\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\nfrom sklearn import linear_model\n\nreg = linear_model.LinearRegression()\n\n# Reads the name of the csv file and gets the data\ndf = pd.read_csv(\"./Demo files/Sawtooth test.csv\")\n\n# Prints information about the file\n#df.info()\n\n# Sets the size of the graph\nfig = plt.figure(figsize=(7,5), dpi= 100)\n\n# Defines the x and y values\ntime = df.filter(['Time (s)'], axis=1).dropna()\nposition = df.filter(['Position (m)'], axis=1).dropna()\n\n# Creates the base plot with titles\ndf.plot(x='Time (s)', y='Position (m)', kind ='scatter')\nplt.ylabel('Position (m)')\nplt.title('Sawtooth Graph')\n\n# Gets the data values for x and y\nx_data = time.values.reshape(-1, 1)\ny_data = position.values.reshape(-1, 1)\n\n# Takes the x and y values to make a line\nreg.fit(x_data,y_data)\ny_fit = x_data*reg.coef_ + reg.intercept_\n\n# Add a dotted line to simulate the sawtooth shape \nplt.plot(x_data,y_data,'b--',label='Linear Fit')\n\n# Adds the legend to the plot\nplt.legend()\n\n# Displays the plot\nplt.show()\n\nprint(\"Slope = (slope) m/s\")\nprint(\"Y-int = (y-int) m\")\n```\n\n### Acoustic Sensor Uncertainty\nThe acoustic sensor can only measure distance so well. How well? You\u2019ll need to determine that. The best way to do this is to take many distance measurements at a fixed location and find the standard deviation of those measurements. Here is one method of doing this:\n\nTurn the air off so that the cart doesn\u2019t move. At three locations on your airtrack (near the sensor, at the 100 cm mark, and at the far end from the sensor):\n1. $Carefully$ place the cart at its location on a sheet of paper and press \u201cRecord.\u201d\n2. After 30 seconds, press \u201cStop.\u201d\n3. Drag a \u201cTable\u201d from the right tool palette onto the main window.\n4. In one column, click the `` button and select \u201cPosition (m).\u201d\n6. The cell above should show your last data run. If not, click it and select your last data run.\n7. On the table, click the arrow next to the summation sign ($\\Sigma$) and make sure \u201cMean\u201d and \u201cStandard Deviation\u201d are selected. Then click the summation sign ($\\Sigma$) itself.\n8. Read the standard deviation off the table at the bottom. This is your $\\delta x$.\n\nIf your standard deviation is greater than 0.001 m (1.0 mm) then you should try to fine-tune the alignment of your sensor. It is very possible to get less than one millimeter uncertainty at the far end of the track. By looking at the\ndisplayed significant digits, your table also tells you what the uncertainty is for a single time measurement. Once you are done aligning your set-up, record your new worst (highest) position uncertainty (this will probably be at\nyour far position) as well as your uncertainty in time, and the uncertainties for any differences the computer would calculate between two adjacent points.\n\n\n```python\n# Raw Data 1\n\nimport pandas as pd\nimport numpy as np\nimport ipywidgets as widgets\n\n#### Enter Raw Data Here!!!!!!!!!!!!!! ####\n\nuncertainty_x = 0.001\nuncertainty_t = 0.002\n\nuncertainty_Delta_x = 0.001\nuncertainty_Delta_t = 0.003\n\n###########################################\n\nprint(\"\ud835\udeffx =\", uncertainty_x, \" \ud835\udefft =\", uncertainty_t)\nprint(\"\")\nprint(\"\ud835\udeff(\u0394x) =\", uncertainty_Delta_x, \" \ud835\udeff(\u0394t) =\", uncertainty_Delta_t)\nprint(\"\")\n```\n\n \ud835\udeffx = 0.001 \ud835\udefft = 0.002\n \n \ud835\udeff(\u0394x) = 0.001 \ud835\udeff(\u0394t) = 0.003\n \n\n\nWhat is the difference in time between two $successive$ data points? (You can either look at your table or recall the Sample Rate from above).\n\n\n```python\n# Raw Data 2\n\n#### Enter Raw Data Here!!!!!!!!!!!!!! ####\n\nDelta_t = 0.005\n\n###########################################\n\nprint(\"\u0394t =\", Delta_t)\n```\n\n \u0394t = 0.005\n\n\n### Unaccelerated Motion\nThis first part of the experiment is to compare the position of the cart as a function of time with its velocity and verify equation $\\eqref{eq:v_avg}$:\n\n1. Start fresh: delete ALL data sets and close all data windows.\n2. Drag a \u201cGraph\u201d from the right tool palette to your data area. Select \u201cPosition (m)\u201d for your y-axis.\n3. Turn on the air and remove the paper from under the cart.\n4. With your cart at the far end of the track, give it a good push towards the sensor.\n5. Immediately after the cart bounces off the near bumper, press \u201cRecord.\u201d\n6. Immediately BEFORE the cart strikes the far bumper, press \u201cStop.\u201d\n7. Turn off the air.\n8. Rename this run \u201cLevel Airtrack\u201d.\n9. Right-click on one of the data points and, in the window that pops up, un-check \u201cShow Connecting Lines.\u201d\n10. Move the legend so that it does not cover any data points.\n11. Type an appropriate title for this graph and print it in Landscape mode. You will need to make the grid lines show up clearly in your graph. To do this, right click on the plot and select \u201cPlot Area Properties.\u201d\n12. Click on the \u201cPosition (m)\u201d axis label and select \u201cVelocity (m/s)\u201d instead.\n13. Repeat steps 9\u201411.\n\nIt is now up to you to draw the best-fit line for the Position vs. Time graph by hand and find its slope as well as draw the best-fit ***horizontal*** line for the Velocity vs. Time graph and find its height. Use the methods outlined in your Lab Manual for this.\n\n\n```python\n# Position vs. Time Graph\n\n# Reads the name of the csv file and gets the data\ndf = pd.read_csv(\"./Demo files/Position vs Time test.csv\")\n\n# Prints information about the file\n#df.info()\n\nprint(\"\")\n\n# Defines the x and y values\ntime = df.filter(['Time (s)'], axis=1).dropna()\nposition = df.filter(['Position (m)'], axis=1).dropna()\n\n# Creates the base plot with titles\ndf.plot(x='Time (s)', y='Position (m)', kind ='scatter')\nplt.ylabel('Position (m)')\nplt.title('Position vs. Time')\n\n# Gets the data values for x and y\nx_data = time.values.reshape(-1, 1)\ny_data = position.values.reshape(-1, 1)\n\n# Takes the x and y values to make a trendline\nreg.fit(x_data,y_data)\ny_fit = x_data*reg.coef_ + reg.intercept_\n\n# Adds the trendline to the plot\nplt.plot(x_data,y_fit,'r--',label='Linear Fit')\n\n# Adds the legend to the plot\nplt.legend()\n\n# Displays the plot\nplt.show()\n\nprint(\"Slope = (slope) m/s\")\nprint(\"Y-int = (y-int) m\")\n```\n\n\n```python\n# Velocity vs. Time Graph (unaccelerated)\n\n# Reads the name of the csv file and gets the data\ndf = pd.read_csv(\"./Demo files/Unacc_Velocity vs Time test.csv\")\n\n# Prints information about the file\n#df.info()\n\n# Sets the size of the graph\nfig = plt.figure(figsize=(7,5), dpi= 100, facecolor='w', edgecolor='k')\n\n# Defines the x and y values\ntime = df.filter(['Time (s)'], axis=1).dropna()\nposition = df.filter(['Velocity (m/s)'], axis=1).dropna()\n\n# Creates the base plot with titles\ndf.plot(x='Time (s)', y='Velocity (m/s)', kind ='scatter')\nplt.ylabel('Velocity (m/s)')\nplt.title('Velocity vs. Time')\n\n# Gets the data values for x and y\nx_data = time.values.reshape(-1, 1)\ny_data = position.values.reshape(-1, 1)\n\n# Takes the x and y values to make a trendline\nreg.fit(x_data,y_data)\ny_fit = x_data*reg.coef_ + reg.intercept_\n\n# Adds the trendline to the plot\nplt.plot(x_data,y_fit,'r--',label='Linear Fit')\n\n# Adds the legend to the plot\nplt.legend()\n\n# Displays the plot\nplt.show()\n\nprint(\"Slope = (slope) m/s\")\nprint(\"Y-int = (y-int) m\")\nprint(\"\")\n```\n\nFor the slope of your Position vs. Time graph (these values represent your uncertainty in **reading** the graph---i.e. how well can you locate a point on the graph \u201cby eye?\u201d):\n\n\n```python\n# Raw Data 3\n\n#### Enter Raw Data Here!!!!!!!!!!!!!! ####\n\nuncertainty_x = 0.001\nuncertainty_t = 0.002\n\nuncertainty_Delta_x = 0.001\nuncertainty_Delta_t = 0.003\n\n###########################################\n\nprint(\"\ud835\udeffx =\", uncertainty_x, \" \ud835\udefft =\", uncertainty_t)\nprint(\"\")\nprint(\"\ud835\udeff(\u0394x) =\", uncertainty_Delta_x, \" \ud835\udeff(\u0394t) =\", uncertainty_Delta_t)\n```\n\n \ud835\udeffx = 0.001 \ud835\udefft = 0.002\n \n \ud835\udeff(\u0394x) = 0.001 \ud835\udeff(\u0394t) = 0.003\n\n\n### Accelerated Motion\nNow that you\u2019ve gotten some data to investigate unaccelerated motion, you need to collect data on accelerated motion. To do this, you will need to allow gravity to provide the acceleration by tilting the track. \n\n1. Measure the distance between the supports on the airtrack. This will be your base length.\n2. Measure the thickness of a 2x4 \u201cshim.\u201d This will be your height change.\n3. Place the 2x4 \u201cshim\u201d under the foot nearest the acoustic sensor.\n4. You will need to re-align the acoustic sensor using the methods described earlier.\n5. Re-measure the uncertainty in position at the end of the track furthest from the sensor and try to reduce that uncertainty below 1 mm. Record this uncertainty in your Data Sheet.\n\nWhen you are ready to take data:\n\n1. Remove everything except one graph from the data area and make sure the y-axis is set to \u201cAcceleration ($m / s^2$).\u201d\n2. Turn on the air, remove the paper, and allow the cart to slide down the airtrack and bounce off the far bumper.\n3. **START** taking data ***immediately after*** the cart strikes the bumper.\n4. **STOP** taking data ***just before*** the cart strikes the bumper again.\n5. Turn off the air.\n6. Rename this data set as \u201cTilted Airtrack.\u201d\n\nThis time, let\u2019s allow the Capstone software to analyze our data:\n\n1. Click the arrow beside the summation symbol ($\\Sigma$) and make sure that \u201cMean\u201d and \u201cStandard Deviation\u201d are selected. Then click the summation symbol ($\\Sigma$).\n2. Move the legend so that it does not obscure the data points.\n3. Type in an appropriate title and print this graph in Landscape Mode.\n4. Click the summation symbol ($\\Sigma$) again to remove the fit.\n5. Click the axis button labeled \u201cAcceleration ($m / s^2$)\u201d and select \u201cVelocity (m/s)\u201d\n6. Click the arrow next to to the \u201cFit\u201d button and select \u201cLinear.\u201d Then click the \u201cFit\u201d button.\n7. Move the legend and linear fit windows so that they do not obscure the data points.\n8. Type in an appropriate title and print this graph in Landscape Mode.\n9. Click the \u201cFit\u201d button to remove the fit.\n10. Click the axis button labeled \u201cVelocity (m/s)\u201d and select \u201cPosition (m).\u201d\n11. Click the arrow next to to the \u201cFit\u201d button, unselect \u201cLinear\u201d and select \u201cQuadratic.\u201d Then click the \u201cFit\u201d button.\n12. Move the legend and quadratic fit windows so that they do not obscure the data points.\n13. Type in an appropriate title and print this graph in Landscape Mode. \n\n\n\n```python\n# Acceleration vs Time Graph (Tilted Airtrack)\n\n# Reads the name of the csv file and gets the data\ndf = pd.read_csv(\"./Demo files/Acceleration vs Time test.csv\")\n\n# Prints information about the file\n#df.info()\n\nprint(\"\")\n\n# Defines the x and y values\ntime = df.filter(['Time (s)'], axis=1).dropna()\nposition = df.filter(['Acceleration (m/s^2)'], axis=1).dropna()\n\n# Creates the base plot with titles\ndf.plot(x='Time (s)', y='Acceleration (m/s^2)', kind ='scatter')\nplt.ylabel('Acceleration (m/s^2)')\nplt.title('Acceleration vs Time')\n\n# Gets the data values for x and y\nx_data = time.values.reshape(-1, 1)\ny_data = position.values.reshape(-1, 1)\n\n# Takes the x and y values to make a line\nreg.fit(x_data,y_data)\ny_fit = x_data*reg.coef_ + reg.intercept_\n\n# Add a dotted line to simulate the sawtooth shape \nplt.plot(x_data,y_data,'b--',label='Linear Fit')\n\n# Adds the legend to the plot\nplt.legend()\n\n# Displays the plot\nplt.show()\n\nprint(\"Slope = (slope) m/s\")\nprint(\"Y-int = (y-int) m\")\nprint(\"\")\n```\n\n\n```python\n# Position vs. Time Graph (accelerated)\n\n# Reads the name of the csv file and gets the data\ndf = pd.read_csv(\"./Demo files/Position vs Time test.csv\")\n\n# Prints information about the file\n#df.info()\n\nprint(\"\")\n\n# Defines the x and y values\ntime = df.filter(['Time (s)'], axis=1).dropna()\nposition = df.filter(['Position (m)'], axis=1).dropna()\n\n# Creates the base plot with titles\ndf.plot(x='Time (s)', y='Position (m)', kind ='scatter')\nplt.ylabel('Position (m)')\nplt.title('Position vs. Time')\n\n# Gets the data values for x and y\nx_data = time.values.reshape(-1, 1)\ny_data = position.values.reshape(-1, 1)\n\n# Takes the x and y values to make a trendline\nreg.fit(x_data,y_data)\ny_fit = x_data*reg.coef_ + reg.intercept_\n\n# Adds the trendline to the plot\nplt.plot(x_data,y_fit,'r--',label='Linear Fit')\n\n# Adds the legend to the plot\nplt.legend()\n\n# Displays the plot\nplt.show()\n\nprint(\"Slope = (slope) m/s\")\nprint(\"Y-int = (y-int) m\")\nprint(\"\")\n```\n\n\n```python\n# Velocity vs. Time Graph (Accelerated Motion)\n\n# Reads the name of the csv file and gets the data\ndf = pd.read_csv(\"./Demo files/Velocity vs Time test.csv\")\n\n# Prints information about the file\n#df.info()\n\nprint(\"\")\n\n# Defines the x and y values\ntime = df.filter(['Time (s)'], axis=1).dropna()\nvelocity = df.filter(['Velocity (m/s)'], axis=1).dropna()\n\n# Creates the base plot with titles\ndf.plot(x='Time (s)', y='Velocity (m/s)', kind ='scatter')\nplt.ylabel('Velocity (m/s)')\nplt.title('Velocity vs. Time')\n\n# Gets the data values for x and y\n#x_data = time.values.reshape(-1, 1)\n#y_data = position.values.reshape(-1, 1)\n\n# Takes the x and y values to make a trendline\n#reg.fit(x_data,y_data)\n#y_fit = x_data*reg.coef_ + reg.intercept_\n\n# Adds the trendline to the plot\n#plt.plot(x_data,y_fit,'r--',label='Linear Fit')\n\n# Adds the legend to the plot\n#plt.legend()\n\n# Displays the plot\nplt.show()\n\nprint(\"Slope = (slope) m/s\")\nprint(\"Y-int = (y-int) m\")\nprint(\"\")\n```\n\n\n```python\n# Raw Data 4\n\n#### Enter Raw Data Here!!!!!!!!!!!!!! ####\n\n# The distance between the two supports and the uncertainty in that distance\nd = 25.0\nuncertainty_d = 0.002\n\n# The thickness of the 2x4 and the uncertainty in that thickness\nh = 1.0\nuncertainty_h = 0.003\n\n# Your new uncertainty in cart position\nuncertainty_x = 0.005\n\n# The angle your track was tilted\ntheta = 25.0\n\n###########################################\n\nprint(\"d =\", d, \" \ud835\udeffd =\", uncertainty_d)\nprint(\"\")\nprint(\"h =\", h, \" \ud835\udeffh =\", uncertainty_h)\nprint(\"\")\nprint(\"\ud835\udeffx =\", uncertainty_x)\nprint(\"\")\nprint(\"\u03b8 =\", theta)\n```\n\n d = 25.0 \ud835\udeffd = 0.002\n \n h = 1.0 \ud835\udeffh = 0.003\n \n \ud835\udeffx = 0.005\n \n \u03b8 = 25.0\n\n\n***Print out all six graphs.***\n\n# Your Lab Report\n\n## EXPERIMENTAL PROCEDURE Section\nYou will need to write a few paragraphs to carefully describe how you set up your experiment. Include the following specific items:\n- What did you do to ensure that your zero-acceleration and constant-acceleration experiments were free from unwanted sources of acceleration?\n- How did you configure your measurement equipment so that you got the most accuracy?\n- How did you compute the best-line fits and slopes for your ***x*** **vs.** ***t*** and ***v*** **vs.** ***t*** graphs?\n- How did you set the track and determine the angle for the constant-acceleration experiment?\n- How did you re-calibrate your measurement equipment after changing experiments?\n\n**Remember:** This section is a description of what you did---not a set of instructions. Include a description of any problems you encountered and how you solved them.\n\n## Data\nReport the following data (with units and uncertainties):\n\n- Velocity of cart without acceleration\n- Angle that you tilted the airtrack\n- Average Acceleration from the Acceleration vs. Time graph\n- Average Acceleration calculated form the slope of the Velocity vs. Time graph\n- Average Acceleration computed from the parabolic fit to your Position vs. Time graph\n\nInclude all six of your graphs in order: Sawtooth, two unaccelerated motion graphs, and three accelerated motion graphs.\n\n## Results and Conclusions\n**Unaccelerated Motion Experiment** \n- Why does the time between peaks in the saw-tooth pattern increase? Will this affect your results?\n- Compare the slope of the Position vs. Time graph to the average value of the velocities in the velocity vs. time graph. Use the % difference formula and cite your uncertainties. Do the two numbers overlap if you include your uncertainties?\n- Using equation $\\eqref{eq:v_unc}$, compute the uncertainty in the speed of the cart at the speed you measured.\n\n**Accelerated Motion Experiment** \nCreate the following table in your report and fill in the entries:\n\n\n```python\n# Table 1: Percent Differences Between Different Methods of Computing Acceleration\n\n# Create an empty numpy array to hold the raw data\nraw_data_1 = np.empty((4,5), dtype = object)\n\n# Set the column identifiers\nraw_data_1[0][0]= 1\nraw_data_1[1][0]= 2\nraw_data_1[2][0]= 3\nraw_data_1[3][0]= 4\n\n# Create a Pandas dataframe, and convert the cylinder number column to integer format\ndf1 = pd.DataFrame(raw_data_1, columns=[\" \", \n \"Average Acceleration from Acceleration vs. Time graph\",\n \"Acceleration Calculated from Slope of Velocity vs. Time Graph\",\n \"Acceleration Computed from Parabolic Fit of Position vs. Time Graph\",\n \"The value of (g sin \u03b8 )\"])\n\n\ndf1[' '] = ['Av. Acceleration from a vs. t graph', \n 'Acceleration from Slope of v vs. t Graph',\n 'Accelerationfrom Parabolic Fit of m vs. t Graph.', \n 'The value of (g sin \u03b8 )']\n\n\n#### Enter Raw Data Here!!!!!!!!!!!!!! ####\n\n# Replace the 0's with your percent differences for the table \ndf1['Average Acceleration from Acceleration vs. Time graph'] = ['N/A',0,0, 0]\n\ndf1['Acceleration Calculated from Slope of Velocity vs. Time Graph'] = [0,'N/A',0, 0]\n\ndf1['Acceleration Computed from Parabolic Fit of Position vs. Time Graph'] = [0,0,'N/A', 0]\n\ndf1['The value of (g sin \u03b8 )'] = [0,0,0, 'N/A']\n\n###########################################\n\ndisplay(df1)\nprint (\"Table 1: Percent Differences Between Different Methods of Computing Acceleration\")\nprint(\"\")\n```\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Average Acceleration from Acceleration vs. Time graphAcceleration Calculated from Slope of Velocity vs. Time GraphAcceleration Computed from Parabolic Fit of Position vs. Time GraphThe value of (g sin \u03b8 )
                                        0Av. Acceleration from a vs. t graphN/A000
                                        1Acceleration from Slope of v vs. t Graph0N/A00
                                        2Accelerationfrom Parabolic Fit of m vs. t Graph.00N/A0
                                        3The value of (g sin \u03b8 )000N/A
                                        \n
                                        \n\n\n Table 1: Percent Differences Between Different Methods of Computing Acceleration\n \n\n\nTo compare the \u201cA\u201d value to the acceleration, you will need to know that the A, B, and C values are the constants for the following fit to the data: \n\n\\begin{equation}\nx(t) = At^2 + Bt + C \\nonumber\n\\end{equation}\n\nCompare this equation to equation $\\eqref{eq:x_of_t}$ to determine the acceleration, *a*.\n\n**Errors/Uncertainties** \nDid you have any systematic errors (i.e. were there sources of acceleration in either experiment that you could not remove)? \n\nHow could you reduce your errors/uncertainties?\n\n## Appendix\n- Include your signed raw data sheet\n- Include sample calculations for your best-line fit and slope for your unaccelerated motion graphs as well as the angle your track was set to for the accelerated motion.\n", "meta": {"hexsha": "e036cc143fccc2b0e135924bd48d8da584f64409", "size": 136833, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "PHYS201/PHYS201L2-1.ipynb", "max_stars_repo_name": "JNichols-19/PhysicsLabs", "max_stars_repo_head_hexsha": "289cb0d07408afde252fe2cabad17fc0b4d987c8", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "PHYS201/PHYS201L2-1.ipynb", "max_issues_repo_name": "JNichols-19/PhysicsLabs", "max_issues_repo_head_hexsha": "289cb0d07408afde252fe2cabad17fc0b4d987c8", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "PHYS201/PHYS201L2-1.ipynb", "max_forks_repo_name": "JNichols-19/PhysicsLabs", "max_forks_repo_head_hexsha": "289cb0d07408afde252fe2cabad17fc0b4d987c8", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 111.7916666667, "max_line_length": 19672, "alphanum_fraction": 0.830289477, "converted": true, "num_tokens": 8020, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804478040616, "lm_q2_score": 0.6654105521116443, "lm_q1q2_score": 0.41174303940919116}} {"text": "## Imports\n\nWe use the SymPy library to work with mathematical aspects of symbolic provenance polynomials, such as multiplication, addition, substitution, etc.\n\n\n```python\n# !pip install sympy\n```\n\n\n```python\n# PW-explorer\nfrom PW_explorer.load_worlds import load_worlds\nfrom PW_explorer.run_clingo import run_clingo\nfrom PW_explorer.export import PWEExport\nfrom PW_explorer.visualize import PWEVisualization\nfrom PWE_NB_Extension.helper import ASPRules\n# Other utilities\nfrom copy import deepcopy\nimport pandas as pd\nimport sympy\nfrom functools import reduce\n```\n\n\n```python\n# Graph-Visualization\nimport networkx as nx\nfrom nxpd import draw\nimport nxpd\nfrom nxpd import nxpdParams\nnxpdParams['show'] = 'ipynb'\n```\n\n\n```python\n%load_ext PWE_NB_Extension\n```\n\n\n\n## Helpers\n\n\n```python\ndef simple_3hop_viz(pw_rel_dfs, gv_md, prov=False):\n g = nx.MultiDiGraph()\n \n g.graph['rankdir'] = 'LR'\n hop_edges_color = 'black'\n thop_edges_color = 'blue'\n thop_df_name = 'thop_3' if prov else 'thop_2'\n \n if thop_df_name in pw_rel_dfs:\n for i, row in pw_rel_dfs[thop_df_name].iterrows():\n g.add_edge(row['x1'], row['x2'], color=thop_edges_color, constraint='false')\n for i, row in pw_rel_dfs['hop_3'].iterrows():\n g.add_edge(row['head'], row['tail'], color=hop_edges_color, style='dotted')\n else:\n for i, row in pw_rel_dfs['hop_3'].iterrows():\n g.add_edge(row['head'], row['tail'], color=hop_edges_color, label=row['hop_name'])\n return g\n```\n\n## Provenance Polynomials\n\n#### Consider a database D of named hops:\n\n\n```python\n%%clingo --donot-display_input -lci hop_db_a --donot-run\n\n% schema hop(head, tail, hop_name)\nhop(a,a, p).\nhop(a,b, q).\nhop(b,a, r).\nhop(b,c, s).\n```\n\nWe can visualize it easily as follows:\n\n\n```python\nhop_db_a_asp_out, hop_db_a_md = run_clingo(hop_db_a)\nhop_db_a_pw_rel_dfs, _, _ = load_worlds(hop_db_a_asp_out, hop_db_a_md, silent=True)\ndraw(simple_3hop_viz(hop_db_a_pw_rel_dfs, hop_db_a_md['graphviz']))\n```\n\n#### Now consider the traditional 3hop query:\n\n\n```python\n%%clingo --donot-display_input -lci thop_query --donot-run\n\nthop(X,Y) :- hop(X,Z1, P1), hop(Z1,Z2, P2), hop(Z2,Y, P3).\n```\n\nOn our hop database D, this yields the following output:\n\n\n```python\n%clingo -l hop_db_a thop_query --donot-display_input -exp thop_D\n```\n\n Output:\n\n\n\n\n \n\n\n\n\n
                                        1 Answer: 1\n2 hop(a,a,p) hop(a,b,q) hop(b,a,r) hop(b,c,s) thop(a,a) thop(b,a) thop(a,b) thop(b,b) thop(a,c) thop(b,c)\n3 SATISFIABLE\n4 \n5 Models       : 1\n6 Calls        : 1\n7 Time         : 0.001s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s)\n8 CPU Time     : 0.001s\n
                                        \n\n\n\n\n```python\nthop_D['pw_rel_dfs'], _, thop_D['pw_objs'] = load_worlds(thop_D['asp_soln'], thop_D['meta_data'], silent=True)\ndraw(simple_3hop_viz(thop_D['pw_rel_dfs'], thop_D['meta_data']['graphviz']))\n```\n\nAs we can see, we get 6 3hop outputs: (a,a), (a,b), (a,c), (b,a), (b,b), (b,c) \n\n#### Now consider the provenance-enhanced version of the same query:\n\ni.e. we 'record' the hops that build each of the 3hop\n\n\n```python\n%%clingo --donot-display_input --donot-run -lci thop_prov_query\n\n%schema thop(start, dest, prov)\nthop(X,Y, f(P1,P2,P3)) :- hop(X,Z1, P1), hop(Z1,Z2, P2), hop(Z2,Y, P3).\n```\n\nNote that the 3hop/2 query is a projection of this query, i.e., we can get the outputs using:\n\nthop(X,Y) :- thop(X,Y,\\_).\n\nOn running it with our database D, we get the following result:\n\n\n```python\n%clingo -l hop_db_a thop_prov_query --donot-display_input -exp thop_prov_D\n```\n\n Output:\n\n\n\n\n \n\n\n\n\n
                                        1 Answer: 1\n2 hop(a,a,p) hop(a,b,q) hop(b,a,r) hop(b,c,s) thop(a,a,f(p,p,p)) thop(b,a,f(r,p,p)) thop(a,a,f(q,r,p)) thop(a,b,f(p,p,q)) thop(b,b,f(r,p,q)) thop(a,b,f(q,r,q)) thop(a,a,f(p,q,r)) thop(b,a,f(r,q,r)) thop(a,c,f(p,q,s)) thop(b,c,f(r,q,s))\n3 SATISFIABLE\n4 \n5 Models       : 1\n6 Calls        : 1\n7 Time         : 0.001s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s)\n8 CPU Time     : 0.001s\n
                                        \n\n\n\n\n```python\nrel = 'thop_3'\n\nthop_prov_D['pw_rel_dfs'], thop_prov_D['rel_schemas'], thop_prov_D['pw_objs'] \\\n= load_worlds(thop_prov_D['asp_soln'], thop_prov_D['meta_data'], internal_facts_as_string=False)\n```\n\n Number of Models: 1\n\n\n\n```python\nthop_prov_D['pw_rel_dfs'][rel]\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        pwstartdestprov
                                        01aa[f, p, p, p]
                                        11ba[f, r, p, p]
                                        21aa[f, q, r, p]
                                        31ab[f, p, p, q]
                                        41bb[f, r, p, q]
                                        51ab[f, q, r, q]
                                        61aa[f, p, q, r]
                                        71ba[f, r, q, r]
                                        81ac[f, p, q, s]
                                        91bc[f, r, q, s]
                                        \n
                                        \n\n\n\n\n```python\nthop_prov_D['pw_rel_dfs']['hop_3']\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        pwheadtailhop_name
                                        01aap
                                        11abq
                                        21bar
                                        31bcs
                                        \n
                                        \n\n\n\nThe provenance information captured here (in the 'prov' column) can help us create the __\"provenance polynomials\"__\n\n\n```python\n# Helper Functions\n\ndef prov_mono(row):\n provs = sympy.symbols(row['prov'][1:])\n expr = reduce(lambda x, y: x*y, provs)\n return expr\n\ndef list_to_freq_list(l):\n if len(l) <= 0: \n return []\n l_ = [[l[0],1]]\n for i in range(1, len(l)):\n if l[i] == l_[-1][0]:\n l_[-1][1] += 1\n else:\n l_.append([l[i],1])\n return l_\n\ndef prov_mono_ord(row):\n f_list = list_to_freq_list(row['prov'][1:])\n f_list_exprs = list(map(lambda p: '{}^{}'.format(p[0],p[1]) if p[1]>1 else '{}'.format(p[0]), f_list))\n expr = \"${}$\".format(\" \".join(f_list_exprs))\n return expr\n```\n\nFirst, we create the provenance monomials:\n\n\n```python\nthop_prov_D['pw_rel_dfs'][rel]['prov_mono_ord'] = thop_prov_D['pw_rel_dfs'][rel].apply(prov_mono_ord, axis=1)\nthop_prov_D['pw_rel_dfs'][rel]\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        pwstartdestprovprov_mono_ord
                                        01aa[f, p, p, p]$p^3$
                                        11ba[f, r, p, p]$r p^2$
                                        21aa[f, q, r, p]$q r p$
                                        31ab[f, p, p, q]$p^2 q$
                                        41bb[f, r, p, q]$r p q$
                                        51ab[f, q, r, q]$q r q$
                                        61aa[f, p, q, r]$p q r$
                                        71ba[f, r, q, r]$r q r$
                                        81ac[f, p, q, s]$p q s$
                                        91bc[f, r, q, s]$r q s$
                                        \n
                                        \n\n\n\nNote that the above provenance monomials (prov_mono_ord) are 'ordered' i.e. we preserve the order of the hops that make up the 3hop. In traditional provenance polynomials, order is not preserved, as shown below:\n\n\n```python\nthop_prov_D['pw_rel_dfs'][rel]['prov_mono'] = thop_prov_D['pw_rel_dfs'][rel].apply(prov_mono, axis=1)\nthop_prov_D['pw_rel_dfs'][rel]['prov_mono_'] = thop_prov_D['pw_rel_dfs'][rel].apply(lambda row: '${}$'.format(sympy.latex(row['prov_mono'])), axis=1)\nthop_prov_D['pw_rel_dfs'][rel][['pw', 'start', 'dest', 'prov', 'prov_mono_ord', 'prov_mono_']]\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        pwstartdestprovprov_mono_ordprov_mono_
                                        01aa[f, p, p, p]$p^3$$p^{3}$
                                        11ba[f, r, p, p]$r p^2$$p^{2} r$
                                        21aa[f, q, r, p]$q r p$$p q r$
                                        31ab[f, p, p, q]$p^2 q$$p^{2} q$
                                        41bb[f, r, p, q]$r p q$$p q r$
                                        51ab[f, q, r, q]$q r q$$q^{2} r$
                                        61aa[f, p, q, r]$p q r$$p q r$
                                        71ba[f, r, q, r]$r q r$$q r^{2}$
                                        81ac[f, p, q, s]$p q s$$p q s$
                                        91bc[f, r, q, s]$r q s$$q r s$
                                        \n
                                        \n\n\n\nNow, we can form the _provenance polynomials_ for each of the 3hop solutions by using a group-by (start-dest pairs) on the monomials and then using a simple summation aggregation function:\n\n\n```python\nthop_prov_D_prov_poly = thop_prov_D['pw_rel_dfs'][rel].groupby(['start', 'dest']).agg(\n prov_poly=('prov_mono', lambda x: '$'+sympy.latex(sum(x))+'$'))\nthop_prov_D_prov_poly.style.set_properties(subset=['prov_poly'], **{'width': '100px'})\n```\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        prov_poly
                                        start dest
                                        aa$p^{3} + 2 p q r$
                                        b$p^{2} q + q^{2} r$
                                        c$p q s$
                                        ba$p^{2} r + q r^{2}$
                                        b$p q r$
                                        c$q r s$
                                        \n\n\n\n#### The provenance polynomials can be used for various use-cases.\n\nFor instance, say we want to know what happens when we slightly modify the database D, say delete the hop 'p' (i.e. hop(a,a)).\n\nOne way to do it would be to modify the database D:\n\n(let's call this modified database Db)\n\n\n```python\n%%clingo --donot-display_input -lci hop_db_b --donot-run\n\n% schema hop(head, tail, hop_name)\n\n% hop(a,a, p). --> DELETE\nhop(a,b, q).\nhop(b,a, r).\nhop(b,c, s).\n```\n\nAnd re-run the query:\n\n\n```python\n%clingo -l hop_db_b thop_query --donot-display_input -exp thop_Db\n```\n\n Output:\n\n\n\n\n \n\n\n\n\n
                                        1 Answer: 1\n2 hop(a,b,q) hop(b,a,r) hop(b,c,s) thop(a,b) thop(b,a) thop(b,c)\n3 SATISFIABLE\n4 \n5 Models       : 1\n6 Calls        : 1\n7 Time         : 0.001s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s)\n8 CPU Time     : 0.001s\n
                                        \n\n\n\nAs we can see, there are 3 outputs: 3hop(a,b), 3hop(b,a), 3hop(b,c).\n\nOr, we could delete some of the results in Datalog using the previous result, using a procedure like shown below:\n\n\n```python\n%%clingo -lci thop_remove_p --donot-run --donot-display_input\n\ndel_thop(X,Y, f(P1,P2,P3)) :- thop(X,Y, f(P1,P2,P3)), P1=p.\ndel_thop(X,Y, f(P1,P2,P3)) :- thop(X,Y, f(P1,P2,P3)), P2=p. \ndel_thop(X,Y, f(P1,P2,P3)) :- thop(X,Y, f(P1,P2,P3)), P3=p.\n\nnew_thop(X,Y, F) :- thop(X,Y, F), not del_thop(X,Y, F).\n#show new_thop/3.\n```\n\nHere, the thop/3 logical atoms come from materializing our earlier results from trop_prov_D experiments into a database of logical atoms, as shown below:\n\n\n```python\nthop_prov_D_facts = PWEExport.export_as_asp_facts(thop_prov_D['pw_objs'], include_pw_ids=False)\nASPRules(thop_prov_D_facts)\n```\n\n\n\n \n\n\n\n\n\n\n
                                         1 hop(a,a,p).\n 2 hop(a,b,q).\n 3 hop(b,a,r).\n 4 hop(b,c,s).\n 5 thop(a,a,f(p,p,p)).\n 6 thop(b,a,f(r,p,p)).\n 7 thop(a,a,f(q,r,p)).\n 8 thop(a,b,f(p,p,q)).\n 9 thop(b,b,f(r,p,q)).\n10 thop(a,b,f(q,r,q)).\n11 thop(a,a,f(p,q,r)).\n12 thop(b,a,f(r,q,r)).\n13 thop(a,c,f(p,q,s)).\n14 thop(b,c,f(r,q,s)).\n
                                        \n\n\n\n\nIn this way, we don't need to re-run the 3hop query itself, just do further computations on the already available results. \n\nNow we run the 'thop_remove_p' rules along with the set of logical facts above, to get the desired results:\n\n\n```python\n%clingo -l thop_remove_p thop_prov_D_facts --donot-display_input\n```\n\n Output:\n\n\n\n\n \n\n\n\n\n
                                        1 Answer: 1\n2 new_thop(a,b,f(q,r,q)) new_thop(b,a,f(r,q,r)) new_thop(b,c,f(r,q,s))\n3 SATISFIABLE\n4 \n5 Models       : 1\n6 Calls        : 1\n7 Time         : 0.002s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s)\n8 CPU Time     : 0.002s\n
                                        \n\n\n\nAs we can see, this also gives us the same 3 results. However, even this requires re-running some sort of querying the provenance information in Datalog.\n\n#### Using Provenance Polynomials\nWe can get the desired result without ever re-running the query, by simply substituting p=0 in the provenance monomials and polynomials:\n\nRecall that the provenance monomials on the original database D looked as follows:\n\n\n```python\nthop_prov_D['pw_rel_dfs'][rel][['pw', 'start', 'dest', 'prov_mono_']]\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        pwstartdestprov_mono_
                                        01aa$p^{3}$
                                        11ba$p^{2} r$
                                        21aa$p q r$
                                        31ab$p^{2} q$
                                        41bb$p q r$
                                        51ab$q^{2} r$
                                        61aa$p q r$
                                        71ba$q r^{2}$
                                        81ac$p q s$
                                        91bc$q r s$
                                        \n
                                        \n\n\n\nNow, we simply substitute p=0 in these provenance monomials:\n\n\n```python\nthop_prov_Db = deepcopy(thop_prov_D)\nsubstitutions = {'p':0}\n\nthop_prov_Db['pw_rel_dfs'][rel]['prov_mono'] \\\n= thop_prov_Db['pw_rel_dfs'][rel].apply(lambda row: row['prov_mono'].subs(substitutions).evalf(), axis=1)\n\nthop_prov_Db['pw_rel_dfs'][rel]['prov_mono_'] \\\n= thop_prov_Db['pw_rel_dfs'][rel].apply(lambda row: '${}$'.format(sympy.latex(row['prov_mono'])), axis=1)\n\nthop_prov_Db['pw_rel_dfs'][rel][['pw', 'start', 'dest', 'prov_mono_']]\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        pwstartdestprov_mono_
                                        01aa$0$
                                        11ba$0$
                                        21aa$0$
                                        31ab$0$
                                        41bb$0$
                                        51ab$q^{2} r$
                                        61aa$0$
                                        71ba$q r^{2}$
                                        81ac$0$
                                        91bc$q r s$
                                        \n
                                        \n\n\n\nAnd removing the 0 monomials:\n\n\n```python\ndrop_cond = thop_prov_Db['pw_rel_dfs'][rel].prov_mono == 0\nthop_prov_Db['pw_rel_dfs'][rel].drop(thop_prov_Db['pw_rel_dfs'][rel][drop_cond].index, axis=0, inplace=True)\nthop_prov_Db['pw_rel_dfs'][rel][['pw', 'start', 'dest', 'prov_mono_']]\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        pwstartdestprov_mono_
                                        51ab$q^{2} r$
                                        71ba$q r^{2}$
                                        91bc$q r s$
                                        \n
                                        \n\n\n\nAs we can see, we get the results, along with the correct provenance information, without re-running the query or any other Datalog operation.\n\nLike before, we can compute the provenance polynomials by grouping by the variables in the head (start, dest) and aggregating the provenance monomials by adding them:\n\n\n```python\nthop_prov_Db['pw_rel_dfs'][rel].groupby(['start', 'dest']).agg(prov_poly=('prov_mono', \n lambda x: '$'+sympy.latex(sum(x))+'$'))\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        prov_poly
                                        startdest
                                        ab$q^{2} r$
                                        ba$q r^{2}$
                                        c$q r s$
                                        \n
                                        \n\n\n\n#### Provenance Polynomials have other interestings uses as well\n\nFor instance, say each hop had a corresponding 'work' value, i.e. amount of work required to make the hop.\nThen, one might be interested in finding the most/minimum amount of work required for each 3hop/2 (start-dest pair). This can be done using __tropical semirings__ i.e. we treat multiplication as addition and addition as the max/min operator.\n\n\n```python\n%%clingo --donot-display_input -lci hop_db_a_work --donot-run\n\n% schema hop_work(hop_name, work)\nhop_work(p, 0).\nhop_work(q, 5).\nhop_work(r, 10).\nhop_work(s, 15).\n```\n\nOne way to compute the desired 3hops would be to re-run the entire operation with a Datalog program like the one below:\n\n(say for the min operation)\n\n\n```python\n%%clingo --donot-display_input -l hop_db_a hop_db_a_work\n\nthop(X,Y, WORK) :- hop(X,Z1,P1), hop(Z1,Z2,P2), hop(Z2,Y,P3), \n hop_work(P1,WP1), hop_work(P2,WP2), hop_work(P3,WP3), WORK=WP1+WP2+WP3.\n\nthop_opt(X,Y, WORK) :- #min{W: thop(X,Y, W)} = WORK, thop(X,Y,_).\n\n#show thop_opt/3.\n```\n\n Output:\n\n\n\n\n \n\n\n\n\n
                                        1 Answer: 1\n2 thop_opt(a,a,0) thop_opt(a,b,5) thop_opt(a,c,20) thop_opt(b,a,10) thop_opt(b,b,15) thop_opt(b,c,30)\n3 SATISFIABLE\n4 \n5 Models       : 1\n6 Calls        : 1\n7 Time         : 0.002s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s)\n8 CPU Time     : 0.002s\n
                                        \n\n\n\nNote that since we do not have any provenance information, we, for example, know that there exists a sequence of 3 hops that takes 0 work to get from a to a, but we don't know which of the 3 (we know from earlier that there are 3 ways to go from a to a in 3 hops) 3hop(a,a) sequences achieves this.\n\nOr, using the provenance information from earlier, we can do the following:\n\n\n```python\n%%clingo --donot-display_input -l thop_prov_D_facts hop_db_a_work\n\nthop(X,Y, f(P1,P2,P3), WORK) :- thop(X,Y, f(P1,P2,P3)), \n hop_work(P1,WP1), hop_work(P2,WP2), hop_work(P3,WP3), WORK=WP1+WP2+WP3.\n\nthop_opt(X,Y, WORK) :- #min{W: thop(X,Y, _, W)} = WORK, thop(X,Y,_,_).\nthop_opt(X,Y, PROV, WORK) :- thop_opt(X,Y, WORK), thop(X,Y,PROV,WORK).\n\n#show thop_opt/4.\n```\n\n Output:\n\n\n\n\n \n\n\n\n\n
                                        1 Answer: 1\n2 thop_opt(a,a,f(p,p,p),0) thop_opt(b,a,f(r,p,p),10) thop_opt(a,b,f(p,p,q),5) thop_opt(b,b,f(r,p,q),15) thop_opt(a,c,f(p,q,s),20) thop_opt(b,c,f(r,q,s),30)\n3 SATISFIABLE\n4 \n5 Models       : 1\n6 Calls        : 1\n7 Time         : 0.002s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s)\n8 CPU Time     : 0.002s\n
                                        \n\n\n\nAs we can see, the answers match exactly, and we never had to re-evaluate the original 3hop query on the database. In addition, we know the hops that make up this optimized 3hop.\n\nAnother way to do this is directly in Python using the provenance monomials as follows:\n\nFirst, we import the work values for the hops into Python and make a Python dictionary to enable substitution into the provenance monomials.\n\n\n```python\nhop_db_a_work_asp_out, hop_db_a_work_md = run_clingo(hop_db_a_work)\nhop_db_a_work_pw_rel_dfs, _, _ = load_worlds(hop_db_a_work_asp_out, hop_db_a_work_md, silent=True)\nhop_work_map = {row['hop_name']: int(row['work']) for _, row in hop_db_a_work_pw_rel_dfs['hop_work_2'].iterrows()}\nhop_work_map\n```\n\n\n\n\n {'p': 0, 'q': 5, 'r': 10, 's': 15}\n\n\n\n\n```python\ndef tropical_semiring_mono(row):\n provs = sympy.symbols(row['prov'][1:])\n expr = reduce(lambda x, y: x+y, provs)\n return expr\n```\n\nNow, we can build our tropical semiring expressions:\n\n\n```python\nthop_opt_prov_D = deepcopy(thop_prov_D)\n\nthop_opt_prov_D['pw_rel_dfs'][rel]['prov_trop_mono'] = thop_opt_prov_D['pw_rel_dfs'][rel].apply(tropical_semiring_mono, axis=1)\nthop_opt_prov_D['pw_rel_dfs'][rel]['prov_trop_mono_'] = thop_opt_prov_D['pw_rel_dfs'][rel].apply(lambda row: '${}$'.format(sympy.latex(row['prov_trop_mono'])), axis=1)\nthop_opt_prov_D['pw_rel_dfs'][rel][['pw', 'start', 'dest', 'prov', 'prov_trop_mono_']]\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        pwstartdestprovprov_trop_mono_
                                        01aa[f, p, p, p]$3 p$
                                        11ba[f, r, p, p]$2 p + r$
                                        21aa[f, q, r, p]$p + q + r$
                                        31ab[f, p, p, q]$2 p + q$
                                        41bb[f, r, p, q]$p + q + r$
                                        51ab[f, q, r, q]$2 q + r$
                                        61aa[f, p, q, r]$p + q + r$
                                        71ba[f, r, q, r]$q + 2 r$
                                        81ac[f, p, q, s]$p + q + s$
                                        91bc[f, r, q, s]$q + r + s$
                                        \n
                                        \n\n\n\nTo get the optimal solutions, we can now aggregate based on the min/max value of the prov_expr_ after grouping by the start-dest pairs:\n\n\n```python\nthop_opt_prov_D_trop_poly = thop_opt_prov_D['pw_rel_dfs'][rel].groupby(['start', 'dest']).agg(\n prov_trop_poly_min=('prov_trop_mono', lambda x: '$'+sympy.latex(sympy.Min(*x))+'$'),\n prov_trop_poly_max=('prov_trop_mono', lambda x: '$'+sympy.latex(sympy.Max(*x))+'$'))\n\nthop_opt_prov_D_trop_poly.style.set_properties(subset=['prov_trop_poly_min', 'prov_trop_poly_max'], **{'width': '150px'})\n```\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        prov_trop_poly_min prov_trop_poly_max
                                        start dest
                                        aa$\\min\\left(3 p, p + q + r\\right)$$\\max\\left(3 p, p + q + r\\right)$
                                        b$\\min\\left(2 p + q, 2 q + r\\right)$$\\max\\left(2 p + q, 2 q + r\\right)$
                                        c$p + q + s$$p + q + s$
                                        ba$\\min\\left(2 p + r, q + 2 r\\right)$$\\max\\left(2 p + r, q + 2 r\\right)$
                                        b$p + q + r$$p + q + r$
                                        c$q + r + s$$q + r + s$
                                        \n\n\n\nNow, we can substitute the work values to get the numerical solution:\n\n\n```python\nthop_opt_prov_D['pw_rel_dfs'][rel]['prov_trop_mono'] \\\n= thop_opt_prov_D['pw_rel_dfs'][rel].apply(lambda row: row['prov_trop_mono'].subs(hop_work_map).evalf(), axis=1)\n\nthop_opt_prov_D['pw_rel_dfs'][rel]['prov_trop_mono_'] \\\n= thop_opt_prov_D['pw_rel_dfs'][rel].apply(lambda row: '${}$'.format(sympy.latex(row['prov_trop_mono'])), axis=1)\n\nthop_opt_prov_D['pw_rel_dfs'][rel][['pw', 'start', 'dest', 'prov_trop_mono_']]\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        pwstartdestprov_trop_mono_
                                        01aa$0$
                                        11ba$10.0$
                                        21aa$15.0$
                                        31ab$5.0$
                                        41bb$15.0$
                                        51ab$20.0$
                                        61aa$15.0$
                                        71ba$25.0$
                                        81ac$20.0$
                                        91bc$30.0$
                                        \n
                                        \n\n\n\n\n```python\nthop_opt_prov_D['pw_rel_dfs'][rel].groupby(['start', 'dest']).agg(prov_trop_poly_min=('prov_trop_mono', \n lambda x: '$'+sympy.latex(min(x))+'$'),\n prov_trop_poly_max=('prov_trop_mono', \n lambda x: '$'+sympy.latex(max(x))+'$'))\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        prov_trop_poly_minprov_trop_poly_max
                                        startdest
                                        aa$0$$15.0$
                                        b$5.0$$20.0$
                                        c$20.0$$20.0$
                                        ba$10.0$$25.0$
                                        b$15.0$$15.0$
                                        c$30.0$$30.0$
                                        \n
                                        \n\n\n\nAs we can see, this matches the earlier solutions exactly, and required no further Datalog computation, just pure mathematical substitution and manipulation.\n\nOne can repeat the demonstrations above with minor modifications for other variants of the 3hop query such as:\n\n\n```python\n%%clingo --donot-display_input --donot-run\n\n%schema thop(start, prov)\nthop(X, f(P1,P2,P3)) :- e(X,Z1, P1), e(Z1,Z2, P2), e(Z2,Y, P3).\n\n#show thop/2.\n```\n\n\n```python\n%%clingo --donot-display_input --donot-run\n\n%schema thop(dest, prov)\nthop(Y, f(P1,P2,P3)) :- e(X,Z1, P1), e(Z1,Z2, P2), e(Z2,Y, P3).\n\n#show thop/2.\n```\n\n\n```python\n%%clingo --donot-display_input --donot-run\n\n%schema thop(dest, prov)\nthop(Z1, Z2, f(P1,P2,P3)) :- e(X,Z1, P1), e(Z1,Z2, P2), e(Z2,Y, P3).\n\n#show thop/3.\n```\n\n\n```python\n%%clingo --donot-display_input --donot-run\n\n%schema thop(dest, prov)\nthop(X, Z1, Z2, Y, f(P1,P2,P3)) :- e(X,Z1, P1), e(Z1,Z2, P2), e(Z2,Y, P3).\n\n#show thop/5.\n```\n\nAnd so on......\n\n#### However, provenance polynomials do have some limitations.\n\nFor instance, since they don't preserve order, one cannot tell, say (p,q,r) from (q,r,p), since they're both reduced to $pqr$.\n\n\n```python\n\n```\n", "meta": {"hexsha": "a8c971f6a57b605b88bea5a83a98d2468f38e281", "size": 163583, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Query Analysis/Provenance Polynomial.ipynb", "max_stars_repo_name": "sahil1105/ASP-Public", "max_stars_repo_head_hexsha": "4466d91b1ee684e8a23e08413804d370bb644f43", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Query Analysis/Provenance Polynomial.ipynb", "max_issues_repo_name": "sahil1105/ASP-Public", "max_issues_repo_head_hexsha": "4466d91b1ee684e8a23e08413804d370bb644f43", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Query Analysis/Provenance Polynomial.ipynb", "max_forks_repo_name": "sahil1105/ASP-Public", "max_forks_repo_head_hexsha": "4466d91b1ee684e8a23e08413804d370bb644f43", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 54.0412950116, "max_line_length": 13744, "alphanum_fraction": 0.5463831816, "converted": true, "num_tokens": 31463, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.665410558746814, "lm_q1q2_score": 0.4117430341590912}} {"text": "# Extended Kalman Smoother\n\n\n```python\n# %load imports.py\n%load_ext autoreload\n%autoreload 2\n%reload_kedro\n%config Completer.use_jedi = False ## (To fix autocomplete)\n\nimport pandas as pd\nfrom src.models.vmm import ModelSimulator\nimport matplotlib.pyplot as plt\nimport matplotlib\n#matplotlib.rcParams[\"figure.figsize\"] = (5,5)\nplt.style.use('presentation')\nfrom src.visualization.plot import track_plots, plot, captive_plot\nimport kedro\nimport numpy as np\nimport os.path\nimport anyconfig\n\n\nfrom myst_nb import glue\nfrom src.symbols import *\nimport src.symbols as symbols\nfrom src.system_equations import *\n\nfrom IPython.display import display, Math, Latex, Markdown\nfrom sympy.physics.vector.printing import vpprint, vlatex\n\nfrom src.models.regression import MotionRegression\n\nfrom src.parameters import df_parameters\np = df_parameters[\"symbol\"]\n\n# Read configs:\nconf_path = os.path.join(\"../../conf/base/\")\nruns_globals_path = os.path.join(\n conf_path,\n \"runs_globals.yml\",\n)\n\nruns_globals = anyconfig.load(runs_globals_path)\nmodel_test_ids = runs_globals[\"model_test_ids\"]\n\njoin_globals_path = os.path.join(\n conf_path,\n \"join_globals.yml\",\n)\n\njoins = runs_globals[\"joins\"]\njoin_runs_dict = anyconfig.load(join_globals_path)\n\nglobals_path = os.path.join(\n conf_path,\n \"globals.yml\",\n)\nglobal_variables = anyconfig.load(globals_path)\n\n\n\nvmm_names = global_variables[\"vmms\"]\nonly_joined = global_variables[\n \"only_joined\"\n] # (regress/predict with only models from joined runs)S\n\nvmms = {}\nfor vmm_name in vmm_names:\n vmms[vmm_name] = catalog.load(vmm_name)\n\n```\n\n UsageError: Line magic function `%reload_kedro` not found.\n\n\n\n```python\nid = 22774\ndata = catalog.load(f\"{ id }.data\")\ndata_ek_smooth = catalog.load(f\"{ id }.data_ek_smooth\")\ndata_ek_filter = catalog.load(f\"{ id }.data_ek_filter\")\ndata['r1d'] = np.gradient(data['r'], data.index)\n```\n\n 2022-03-28 14:43:55,585 - kedro.io.data_catalog - INFO - Loading data from `22774.data` (CSVDataSet)...\n 2022-03-28 14:43:55,819 - kedro.io.data_catalog - INFO - Loading data from `22774.data_ek_smooth` (CSVDataSet)...\n 2022-03-28 14:43:55,869 - kedro.io.data_catalog - INFO - Loading data from `22774.data_ek_filter` (CSVDataSet)...\n\n\n\n```python\ndataframes = {\n 'raw' : data,\n #'EKF' : data_ek_filter,\n 'EKS' : data_ek_smooth,\n \n}\nfig = plot(dataframes, keys=['psi','r','r1d'], ncols=3, fig_size=matplotlib.rcParams[\"figure.figsize\"]);\nfig.axes[0].set_ylabel(r'$\\psi$')\nfig.axes[2].set_ylabel(r'$\\dot{r}$')\nfig.axes[2].set_xlabel('time');\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "8ee7c94dcfa12b8e92eb040bc442bc731c333eea", "size": 173519, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "_build/jupyter_execute/06.01_EKS.ipynb", "max_stars_repo_name": "martinlarsalbert/NMUW2022", "max_stars_repo_head_hexsha": "13b951e0a3965780c38bcd42124c28a92ebadf53", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "_build/jupyter_execute/06.01_EKS.ipynb", "max_issues_repo_name": "martinlarsalbert/NMUW2022", "max_issues_repo_head_hexsha": "13b951e0a3965780c38bcd42124c28a92ebadf53", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "_build/jupyter_execute/06.01_EKS.ipynb", "max_forks_repo_name": "martinlarsalbert/NMUW2022", "max_forks_repo_head_hexsha": "13b951e0a3965780c38bcd42124c28a92ebadf53", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 854.7733990148, "max_line_length": 168200, "alphanum_fraction": 0.9520686495, "converted": true, "num_tokens": 718, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.6859494421679929, "lm_q1q2_score": 0.4116988702302605}} {"text": "```python\nfrom IPython.core.display import HTML\nHTML(\"\"\"\n\n\"\"\")\n```\n\n\n\n\n\n\n\n\n\n\n# *Circuitos El\u00e9tricos I - Semana 12.1*\n\n\n```python\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport sympy as sp\nfrom utils import round_expr, symdisp, symplot\n\nfrom sympy.polys.partfrac import apart\n\n# temp workaround\nimport warnings\nfrom matplotlib import MatplotlibDeprecationWarning\nwarnings.filterwarnings('ignore', category=MatplotlibDeprecationWarning)\n\nplt.rcParams['figure.figsize'] = 6, 4\nplt.rcParams['legend.fontsize'] = 13\nplt.rcParams['lines.linewidth'] = 2\nplt.rcParams['axes.grid'] = False\n```\n\n\n```python\n# transformada de Laplace\ndef L(f,t,s):\n return sp.laplace_transform(f, t, s, noconds=True)\n\n# transformada inversa de Laplace\ndef invL(F,s,t):\n return sp.re(sp.inverse_laplace_transform(F, s, t, noconds=True))\n\n# fun\u00e7\u00f5es para aux\u00edlio na expans\u00e3o em fra\u00e7\u00f5es parciais\ndef adjustCoeff(expr): \n coeff = expr.as_numer_denom()\n c0 = sp.poly(coeff[1].cancel()).coeffs()[0]\n \n return (coeff[0].cancel()/c0)/(coeff[1].cancel()/c0)\n\ndef partFrac(expr, Ndigits):\n expr = expr.cancel()\n expr = apart(adjustCoeff(expr), s, full=True).doit()\n \n return sp.N(expr, Ndigits)\n\nsp.init_printing()\n```\n\n#### Definindo algumas vari\u00e1veis simb\u00f3licas de interesse\n\n\n```python\ns = sp.symbols('s')\na = sp.symbols('a', real=True, positive=True)\nomega, t = sp.symbols('omega, t', real=True)\ninfty = sp.oo\n```\n\n### Problema 1\n\nO circuito da figura a seguir est\u00e1 em regime estacion\u00e1rio no momento em que chave \u00e9 aberta. Sabe-se que $v(t)=\\mathrm{12\\;V}$.\n\n\n\na. Determine $I_0(s)$ e $I_1(s)$.\\\nb. Verifique a consist\u00eancia das respostas do item a. com os teoremas do valor inicial e do valor final.\\\nc. Determine a fun\u00e7\u00e3o de transfer\u00eancia $H_0(s)$ entre $V(s)$ e $I_0(s)$.\\\nd. Determine a fun\u00e7\u00e3o de transfer\u00eancia $H_1(s)$ entre $V(s)$ e $I_1(s)$.\\\ne. Determine a fun\u00e7\u00e3o de transfer\u00eancia $H_c(s)$ entre $V(s)$ e $V_c(s)$.\\\nf. Determine a $v_c(t)$ para o caso em que a tens\u00e3o aplicada $v(t)$ em $\\mathrm{V}$ corresponde ao gr\u00e1fico a seguir:\n\n\n\na. Determinando $I_0(s)$ e $I_1(s)$:\n\n\n```python\nI0, I1, s = sp.symbols('I0, I1, s')\n\n# define os sistema de equa\u00e7\u00f5es\neq1 = sp.Eq((-2*s**2)*I0 + (2*s**2 + 10*s + 250)*I1, 2.4*s + 12) \neq2 = sp.Eq((4*s+50)*I0 -2*s*I1, -2.4) \n\n# resolve o sistema\nsoluc = sp.solve([eq1, eq2],[I0, I1], dict=True)\nsoluc\n\nI0 = [sol[I0] for sol in soluc]\nI1 = [sol[I1] for sol in soluc]\n\nI0 = I0[0]\nI1 = I1[0]\n\nprint('Correntes no dom\u00ednio de Laplace: \\n')\nsymdisp('I_0(s) =', I0, 'As')\nsymdisp('I_1(s) =', I1, 'As')\n```\n\n Correntes no dom\u00ednio de Laplace: \n \n\n\n\n$\\displaystyle I_0(s) =- \\frac{150.0}{s^{3} + 35.0 s^{2} + 375.0 s + 3125.0}\\;As$\n\n\n\n$\\displaystyle I_1(s) =\\frac{6.0 s^{2} + 210.0 s + 750.0}{5.0 s^{3} + 175.0 s^{2} + 1875.0 s + 15625.0}\\;As$\n\n\nb. Checando a consist\u00eancia das solu\u00e7\u00f5es\n\n**Teorema do valor inicial (TVI)**\n\n$$\nf(0^+) = \\lim_{t \\to 0^+}f(t) = \\lim_{s \\to \\infty}sF(s)\n$$\n\n\n\n```python\ni0_0_tvi = sp.limit(s*I0, s, infty)\ni1_0_tvi = sp.limit(s*I1, s, infty)\n\n\nsymdisp('i_0(0^+) = ', i0_0_tvi, ' A' )\nsymdisp('i_1(0^+) = ', i1_0_tvi, ' A' )\n```\n\n\n$\\displaystyle i_0(0^+) = 0\\; A$\n\n\n\n$\\displaystyle i_1(0^+) = 1.2\\; A$\n\n\n**Teorema do valor final (TVF)**\n\n$$\nf(\\infty) = \\lim_{t \\to \\infty}f(t) = \\lim_{s \\to 0}sF(s)\n$$\n\n\n\n\n```python\ni0_inf_tvf = sp.limit(s*I0, s, 0)\ni1_inf_tvf = sp.limit(s*I1, s, 0)\n\n\nsymdisp('i_0(\\infty) = ', i0_inf_tvf, ' A' )\nsymdisp('i_1(\\infty) = ', i1_inf_tvf, ' A' )\n```\n\n\n$\\displaystyle i_0(\\infty) = 0\\; A$\n\n\n\n$\\displaystyle i_1(\\infty) = 0\\; A$\n\n\n\n```python\nC = 4e-3 # F\n\n# Calculando Vc\nVc = (1/(s*C))*I1\n\nVc = Vc.simplify()\n\nsymdisp('V_c(s) =', adjustCoeff(Vc).simplify(), 'Vs')\n```\n\n\n$\\displaystyle V_c(s) =\\frac{300.0 s^{2} + 10500.0 s + 37500.0}{s \\left(1.0 s^{3} + 35.0 s^{2} + 375.0 s + 3125.0\\right)}\\;Vs$\n\n\n\n```python\n sp.limit(s*Vc, s, 0)\n```\n\n\n```python\nnp.roots([1, 35, 375, 3125, 0])\n```\n\n\n\n\n array([-25. +0.j, -5.+10.j, -5.-10.j, 0. +0.j])\n\n\n\n\n```python\npartFrac(Vc, 4)\n```\n\n\n```python\nvc = invL(partFrac(Vc, 4), s, t)\n\nsymdisp('v_c(t) = ', vc, ' V')\n```\n\n\n$\\displaystyle v_c(t) = 12.0 \\theta\\left(t\\right) + 3.0 e^{- 25.0 t} \\theta\\left(t\\right) + 30.0 e^{- 5.0 t} \\sin{\\left(10.0 t \\right)} \\theta\\left(t\\right) - 15.0 e^{- 5.0 t} \\cos{\\left(10.0 t \\right)} \\theta\\left(t\\right)\\; V$\n\n\n\n```python\n# plota fun\u00e7\u00f5es no dom\u00ednio do tempo\nintervalo = np.arange(-1, 4, 0.01)\nsymplot(t, vc, intervalo, 'vc(t)')\n```\n\nc-e) Determinando $H_0(s)$, $H_0(s)$ e $H_c(s)$.\n\n\n```python\nV = 12/s\n\nH0 = I0/V\n\nsymdisp('H_0(s) =', H0,)\n```\n\n\n$\\displaystyle H_0(s) =- \\frac{12.5 s}{s^{3} + 35.0 s^{2} + 375.0 s + 3125.0}\\; $\n\n\n\n```python\nH1 = I1/V\nH1 = adjustCoeff(H1)\n\nsymdisp('H_1(s) =', H1,)\n```\n\n\n$\\displaystyle H_1(s) =\\frac{0.1 s^{3} + 3.5 s^{2} + 12.5 s}{1.0 s^{3} + 35.0 s^{2} + 375.0 s + 3125.0}\\; $\n\n\n\n```python\nHc = Vc/V\nHc = adjustCoeff(Hc)\n\nsymdisp('H_c(s) =', Hc,)\n```\n\n\n$\\displaystyle H_c(s) =\\frac{25.0 s^{2} + 875.0 s + 3125.0}{1.0 s^{3} + 35.0 s^{2} + 375.0 s + 3125.0}\\; $\n\n\nf. $v_c(t)=?$\n\n\n```python\nv = 2*(sp.Heaviside(t)-sp.Heaviside(t-1)) - 1*(sp.Heaviside(t-1)-sp.Heaviside(t-2))\n\nsymdisp('v(t) =', v, ' V')\n\n# plota fun\u00e7\u00f5es no dom\u00ednio do tempo\nintervalo = np.arange(-1, 3, 0.01)\nsymplot(t, v, intervalo, 'v(t)')\n```\n\n\n```python\n# determina Vc(s) via fun\u00e7\u00e3o de transfer\u00eancia\nV = L(v, t, s)\nVc = Hc*V\n\nsymdisp('V_c(s) =', Vc.simplify(), ' Vs')\n```\n\n\n$\\displaystyle V_c(s) =\\frac{\\left(25.0 s^{2} + 875.0 s + 3125.0\\right) \\left(2 e^{2 s} - 3 e^{s} + 1\\right) e^{- 2 s}}{s \\left(1.0 s^{3} + 35.0 s^{2} + 375.0 s + 3125.0\\right)}\\; Vs$\n\n\n\n```python\n# fun\u00e7\u00e3o auxiliar Va(s)\nP = (25*s**2 + 875*s + 3125)/(s**4 + 35*s**3 + 375*s**2 + 3125*s)\nP = partFrac(P, 10)\n\n# encontra va(t)\np = invL(P, s, t)\np = p.expand()\n\nsymdisp('P(s) =', P , ' Vs')\nsymdisp('p(t) =', p, ' V')\n```\n\n\n$\\displaystyle P(s) =\\frac{-0.625 + 1.25 i}{s + 5.0 + 10.0 i} + \\frac{-0.625 - 1.25 i}{s + 5.0 - 10.0 i} + \\frac{0.25}{s + 25.0} + \\frac{1}{s}\\; Vs$\n\n\n\n$\\displaystyle p(t) =1.0 \\theta\\left(t\\right) + 0.25 e^{- 25.0 t} \\theta\\left(t\\right) + 2.5 e^{- 5.0 t} \\sin{\\left(10.0 t \\right)} \\theta\\left(t\\right) - 1.25 e^{- 5.0 t} \\cos{\\left(10.0 t \\right)} \\theta\\left(t\\right)\\; V$\n\n\n\n```python\nvc = 2*p - 3*p.subs({t:sp.UnevaluatedExpr(t-1)}) + p.subs({t:sp.UnevaluatedExpr(t-2)})\n\nsymdisp('v_c(t) =', vc, ' V')\n```\n\n\n$\\displaystyle v_c(t) =2.0 \\theta\\left(t\\right) + 1.0 \\theta\\left(t - 2\\right) + 0.25 \\theta\\left(t - 2\\right) e^{- 25.0 \\left(t - 2\\right)} + 2.5 \\theta\\left(t - 2\\right) e^{- 5.0 \\left(t - 2\\right)} \\sin{\\left(10.0 \\left(t - 2\\right) \\right)} - 1.25 \\theta\\left(t - 2\\right) \\cos{\\left(10.0 \\left(t - 2\\right) \\right)} e^{- 5.0 \\left(t - 2\\right)} - 3 \\left(1.0 \\theta\\left(t - 1\\right) + 0.25 \\theta\\left(t - 1\\right) e^{- 25.0 \\left(t - 1\\right)} + 2.5 \\theta\\left(t - 1\\right) e^{- 5.0 \\left(t - 1\\right)} \\sin{\\left(10.0 \\left(t - 1\\right) \\right)} - 1.25 \\theta\\left(t - 1\\right) \\cos{\\left(10.0 \\left(t - 1\\right) \\right)} e^{- 5.0 \\left(t - 1\\right)}\\right) + 0.5 e^{- 25.0 t} \\theta\\left(t\\right) + 5.0 e^{- 5.0 t} \\sin{\\left(10.0 t \\right)} \\theta\\left(t\\right) - 2.5 e^{- 5.0 t} \\cos{\\left(10.0 t \\right)} \\theta\\left(t\\right)\\; V$\n\n\n\n```python\n# plota fun\u00e7\u00f5es no dom\u00ednio do tempo\nintervalo = np.arange(-1, 3, 0.01)\nsymplot(t, [v, vc], intervalo, ['v(t)','vc(t)'])\n```\n", "meta": {"hexsha": "143871fa61c128dde476e8982355de882884f9cc", "size": 65942, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Jupyter notebooks/Circuitos Eletricos I - Semana 12.1.ipynb", "max_stars_repo_name": "Jefferson-Lopes/ElectricCircuits", "max_stars_repo_head_hexsha": "bf2075dc0731cacece75f7b0b378c180630bdf85", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 9, "max_stars_repo_stars_event_min_datetime": "2021-05-19T18:36:53.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-18T16:30:17.000Z", "max_issues_repo_path": "Jupyter notebooks/Circuitos Eletricos I - Semana 12.1.ipynb", "max_issues_repo_name": "Jefferson-Lopes/ElectricCircuits", "max_issues_repo_head_hexsha": "bf2075dc0731cacece75f7b0b378c180630bdf85", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Jupyter notebooks/Circuitos Eletricos I - Semana 12.1.ipynb", "max_forks_repo_name": "Jefferson-Lopes/ElectricCircuits", "max_forks_repo_head_hexsha": "bf2075dc0731cacece75f7b0b378c180630bdf85", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 10, "max_forks_repo_forks_event_min_datetime": "2021-06-25T12:52:40.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-11T14:25:48.000Z", "avg_line_length": 74.5107344633, "max_line_length": 16460, "alphanum_fraction": 0.8044796943, "converted": true, "num_tokens": 3202, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5156199306096344, "lm_q2_score": 0.7981867801399695, "lm_q1q2_score": 0.4115610121892986}} {"text": "# Problema de selecci\u00f3n de portafolio con preferencias media-varianza\n\n\n\nEn la clase pasada hablamos acerca de:\n- preferencias,\n- funciones de utilidad,\n- la actitud de los inversionistas de cara al riesgo,\n- la aversi\u00f3n al riesgo, entre otros.\n\nTodas ellas son piezas que necesitamos para responder la pregunta de \u00bfc\u00f3mo un inversionista toma la decisi\u00f3n \u00f3ptima de selecci\u00f3n de portafolio?\n\nEn esta clase al fin estamos listos para ensamblar estos conceptos y escribir el problema de selecci\u00f3n de portafolios. \n\nEn el camino aprenderemos acerca del concepto de **utilidad esperada**, que nos permite trabajar con incertidumbre en el modelado econ\u00f3mico (una de las ideas m\u00e1s importantes en econom\u00eda). Esta idea tiene m\u00e1s de 60 a\u00f1os, y b\u00e1sicamente dice que los individuos, cuando est\u00e1n de cara a incertidumbre, maximizan el valor esperado de su utilidad (solo cierto si somos homo economicus). \n\nAdem\u00e1s del concepto de utilidad esperada, aprenderemos acerca de **preferencias media-varianza**. Es decir, supondremos que los inversionistas toman decisiones basados en un tipo particular de preferencias.\n\nCon lo anterior, estableceremos el problema de selecci\u00f3n de portafolios.\n\n**Objetivos:**\n\n- \u00bfQu\u00e9 es utilidad esperada?\n- \u00bfQu\u00e9 son las preferencias media-varianza?\n- Funciones de utilidad media-varianza.\n- Enunciado y soluci\u00f3n del problema b\u00e1sico de selecci\u00f3n de portafolio.\n\n*Referencia:*\n- Notas del curso \"Portfolio Selection and Risk Management\", Rice University, disponible en Coursera.\n___\n\n## 1. Utilidad esperada\n- B\u00e1sicamente combina las probabilidades de los resultados con c\u00f3mo los inversionistas se sienten con dichos resultados.\n- En otras palabras, la utilidad esperada multiplica la probabilidad de suceso de un evento con la utilidad que genera dicho evento.\n\nRecordemos que las *funciones de utilidad* permiten a los inversionistas expresar c\u00f3mo se sienten con los resultados, especialmente en los malos ratos. \n\nEntonces la *utilidad esperada* es una herramienta que nos permite cuantificar c\u00f3mo nos sentimos en nuestros malos momentos econ\u00f3micos, capturando el riesgo con la probabilidad de ocurrencia de dichos malos momentos. \n\nDado este marco de trabajo, cualquier decisi\u00f3n se puede escribir como la maximizaci\u00f3n de la utilidad esperada:\n\\begin{align}\n\\max_{\\theta} & \\quad E[U(W)], \\\\\n\\end{align}\nmediante la escogencia de cierta variable $\\theta$ (gastos, planes de ahorro, compra de activos, planes de producci\u00f3n, etc.).\n\nPara nuestros prop\u00f3sitos, la variable de decisi\u00f3n ser\u00e1n los pesos o ponderaciones del portafolio.\n\nAdicionalmente, en el contexto de la decisi\u00f3n de distribuci\u00f3n de la riqueza entre activos, el problema de maximizaci\u00f3n tendr\u00e1 com\u00fanmente las siguientes restricciones:\n- universo de inversi\u00f3n,\n- posici\u00f3n en los activos dados.\n\n**Ejemplo.** \n\nSupongamos que un inversionista debe determinar la composici\u00f3n \u00f3ptima de su portafolio, que contiene activos y bonos. Supongamos que son los \u00fanicos instrumentos disponibles.\n\nSean:\n- $w_s$: peso o ponderaci\u00f3n de activos en el portafolio,\n- $w_b$: peso o ponderaci\u00f3n de bonos en el portafolio,\n- $r_s$: rendimiento de los activos, y\n- $r_b$: rendimiento de los bonos.\n\nDe manera que podemos escribir el problema de selecci\u00f3n de portafolios como la maximizaci\u00f3n de la utilidad esperade de nuestra riqueza futura, la cual depender\u00e1 de nuestros rendimientos:\n \n\\begin{align}\n\\max_{w_s,w_b} &\\quad E[U(W)]\\\\\n\\text{s. a.} &\\quad W=W_0(1+w_sr_s+w_br_b)\\\\\n &\\quad w_s+w_b=1\n\\end{align}\n\n\nPreguntas:\n- \u00bfQu\u00e9 significan las restricciones?\n- Ya que tenemos planteado este problema b\u00e1sico, \u00bfqu\u00e9 har\u00eda falta para empezar a resolverlo?\n___\n\n## 2. Preferencias media-varianza\n\n### 2.1. Utilidad media-varianza\n\nEntonces, \u00bfqu\u00e9 funciones de utilidad deber\u00edamos de usar en este problema de selecci\u00f3n de portafolios?\n\n- La respuesta es: **preferencias media-varianza**.\n- \u00c9stas ser\u00e1n representadas en t\u00e9rminos de funciones de utilidad como: **utilidad media-varianza**.\n\nUsamos la *utilidad media-varianza* en el problema de selecci\u00f3n de portafolios dado que \u00e9sta decribe el \"trade-off\" entre riesgo y rendimiento que enfrentan los inversionistas. La *utilidad media-varianza* est\u00e1 dada por la siguiente expresi\u00f3n:\n\n$$U=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2,$$\n\ndonde\n- $E[r_p]$ es el rendimiento esperado del portafolio,\n- $\\sigma_p^2$ es la varianza del portafolio, y\n- $\\gamma$ es el coeficiente de aversi\u00f3n al riesgo. \n\n#### Intuici\u00f3n acerca de la funci\u00f3n de utilidad media-varianza:\n- S\u00f3lo se preocupa por medias :) y varianzas :(.\n- Incrementa con: rendimiento esperado del portafolio.\n- Decrece con: varianza del portafolio (riesgo).\n- Malos tiempos: rendimientos son bajos y las volatilidades son altas.\n- Conecta bastante bien con la teor\u00eda moderna de portafolios, la cual caracteriza los rendimientos con medias y varianzas \u00fanicamente.\n- Criticada por su limitaci\u00f3n: supone que los inversionistas s\u00f3lo se preocupan por medias y varianzas.\n\n### 2.2. Curvas de indiferencia\n\n*\u00bfRecuerdan las **curvas de nivel** que se ven en c\u00e1lculo de varias variables?*\n- Bien, ac\u00e1 nos servir\u00e1n para representar la utilidad media-varianza gr\u00e1ficamente.\n- En el contexto de utilidad media-varianza, las curvas de nivel se llaman **curvas de indiferencia**.\n\nDados ciertos niveles de utilidad $U_1>U_2>U_3$, las curvas de indiferencia relativas a estos niveles de utilidad, son los lugares geom\u00e9tricos en el espacio de rendimiento esperado vs. volatilidad representados por las siguientes expresiones\n\n$$U_1=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2\\Rightarrow E[r_p]=\\frac{1}{2}\\gamma\\sigma_p^2+U_1,$$\n\n$$U_2=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2\\Rightarrow E[r_p]=\\frac{1}{2}\\gamma\\sigma_p^2+U_2,$$\n\n$$U_3=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2\\Rightarrow E[r_p]=\\frac{1}{2}\\gamma\\sigma_p^2+U_3.$$\n\n**Gr\u00e1ficamente**\n\n\n```python\n# Importar numpy y pyplot\nimport numpy as np\nimport matplotlib.pyplot as plt\n%matplotlib inline\n```\n\n\n```python\n# Coeficiente de aversi\u00f3n al riesgo (entre 1 y 10 com\u00fanmente)\ng= 9 #elegido en clase\n# Niveles de utilidad 3, 2, 1\nU1, U2, U3 = 0.15, 0.1, 0.05\n# Vector de volatilidades (sugerido 1%-60%)\nsp = np.linspace(0.01, 0.6, 100)\n# Curvas de indiferencia\nE1= 0.5*g*sp**2+U1\nE2= 0.5*g*sp**2+U2\nE3= 0.5*g*sp**2+U3\n```\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(8,6))\nplt.plot(sp, E1, label='$U_1=0.15$')\nplt.plot(sp, E2,label='$U_2=0.1$')\nplt.plot(sp, E3,label='$U_3=0.05$')\nplt.legend(loc=4)\nplt.xlabel('Volatilidad $\\sigma$')\nplt.ylabel('Rendimiento esperado($E[r_p]$)')\nplt.grid()\n```\n\nVa en aumento hacia ariba y hacia la izquierda.\n\n\nBueno, \u00bfy porqu\u00e9 se llaman curvas de indiferencia?, \u00bfqu\u00e9 representa una cuva de indiferencia?\n\n- Porque sobre una misma curva el nivel de utilidad es el mismo (es indiferente).\n- Son todas las combinaciones de riesgo y rendimiento que producen un mismo nivel de utilidad.\n\nVolviendo al problema de selecci\u00f3n de portafolios, queremos la utilidad m\u00e1s alta.\n- \u00bfCu\u00e1l de las anteriores curvas de indiferencia corresponde a la utilidad m\u00e1s alta?\n- Intuitivamente, \u00bfporqu\u00e9?\n- Curvas de indiferencia para niveles de utilidad m\u00e1s altos, estar\u00e1n...\n\nNotamos adem\u00e1s que las anteriores curvas de indiferencia son *paralelas* una con otra. Claro, las dibujamos con el mismo coeficiente de aversi\u00f3n al riesgo.\n\n\u00bfC\u00f3mo cambian estas curvas para coeficientes de aversi\u00f3n al riesgo m\u00e1s altos?\n\n\n```python\n# Coeficientes de aversi\u00f3n al riesgo (entre 1 y 10 com\u00fanmente)\ng1, g2, g3= 3,5,7\n# Nivel de utilidad\nU= 0.15\n# Vector de volatilidades (sugerido 1%-60%)\nsp_new= np.linspace(0.01,0.6,100)\n# Curvas de indiferencia\nE1= 0.5*g1*sp**2+U\nE2= 0.5*g2*sp**2+U\nE3= 0.5*g3*sp**2+U\n```\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(8,6))\nplt.plot(sp, E1, label='$\\gamma_1=3$')\nplt.plot(sp, E2,label='$\\gamma_2=5$')\nplt.plot(sp, E3,label='$\\gamma_3=7$')\nplt.legend(loc=4)\nplt.xlabel('Volatilidad $\\sigma$')\nplt.ylabel('Rendimiento esperado($E[r_p]$)')\nplt.grid()\n```\n\n\u00bfC\u00f3mo interpretamos las anteriores gr\u00e1ficas?, \u00bfqu\u00e9 pasa con las personas m\u00e1s aversas al riesgo?\n- Se puede ver de dos maneras: para un mismo nivel de rendimiento esperado, una persona m\u00e1s aversa al riesgo soporta un nivel menor de riesgo; equivalentemente, para un mismo nivel de riesgo, una persona m\u00e1s aversa al riesgo requerir\u00e1 un nivel de rendimiento esperado m\u00e1s alto.\n\nCon todo lo anterior, el problema de selecci\u00f3n de portafolios se puede plantear como *\"encontrar la curva de indeferencia m\u00e1s alta dado el conjunto de oportunidades de inversi\u00f3n y restricciones\"*.\n\n## 3. Problema de selecci\u00f3n de portafolios: una ilustraci\u00f3n\n\nAhora ilustraremos el problema de selecci\u00f3n de portafolios con algunos datos. \n- Por ahora solo queremos ilustrar gr\u00e1ficamente c\u00f3mo se resuelve este problema. Trabajar en la intuici\u00f3n.\n- En las siguientes dos clases nos enfocaremos en c\u00f3mo resolverlo anal\u00edticamente.\n\nAc\u00e1 tenemos el rendimiento medio anual y la volatilidad para dos instrumentos usando datos de EU: instrumentos de deuda (bonos) y acciones. Supondremos que el inversionista solo puede invertir en estas dos clases de instrumentos.\n\n\n```python\n# Importamos pandas\nimport pandas as pd\n```\n\n\n```python\n# Datos\ndata=pd.DataFrame(index=['Stocks', 'Bonds', 'CorrSB']\n , columns=['Mean', 'Std'])\ndata['Mean']=[0.119, 0.59, 0.113]\ndata['Std']=[0.1915, 0.0833, None]\ndata\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        MeanStd
                                        Stocks0.1190.1915
                                        Bonds0.5900.0833
                                        CorrSB0.113NaN
                                        \n
                                        \n\n\n\nEntonces, \u00bfcu\u00e1l es la distribuci\u00f3n de riqueza \u00f3ptima?, o m\u00e1s bien, \u00bfcu\u00e1l es la composici\u00f3n \u00f3ptima del portafolio para un inversionista dado su nivel de aversi\u00f3n al riesgo?\n\n**Primero.** Recordamos que, para dos activos, podemos trazar la frontera de m\u00ednima varianza tomando todas las posibles combinaciones de los dos activos.\n\nDe nuevo, sean:\n- $w_s=w$: peso o ponderaci\u00f3n de activos en el portafolio,\n- $w_b=1-w$: peso o ponderaci\u00f3n de bonos en el portafolio,\n- $r_s$: rendimiento de los activos, y\n- $r_b$: rendimiento de los bonos.\n\nEntonces\n\n$$E[r_p]=wE[r_{s}]+(1-w)E[r_b]$$\n\n$$\\sigma_p^2=w^2\\sigma_{s}^2+(1-w)^2\\sigma_b^2+2w(1-w)\\rho_{s,b}\\sigma_s\\sigma_b$$\n\n\n```python\n# Vector de w variando entre 0 y 1 con n pasos\nw= np.linspace(0, 1, 101)\n# Rendimientos esperados individuales\nEs = data.loc['Stocks', 'Mean']\nEb = data.loc['Bonds', 'Mean']\n# Volatilidades individuales\nss= data.loc['Stocks', 'Std']\nsb= data.loc['Bonds', 'Std']\n# Correlacion\nrsb= data.loc['CorrSB', 'Mean']\nEs, Eb, ss, sb, rsb\n```\n\n\n\n\n (0.119, 0.59, 0.1915, 0.0833, 0.113)\n\n\n\n\n```python\n# Crear un DataFrame cuyas columnas sean rendimiento\n# y volatilidad del portafolio para cada una de las w\n# generadas\nportafolios= pd.DataFrame(data={'Media': w*Es+(1-w)*Eb,\n 'Vol':(w**2*ss**2+(1-w)**2*sb**2+2*w*(1-w)*rsb*ss*sb)**0.5})\nportafolios.head(),portafolios.tail()\n```\n\n\n\n\n ( Media Vol\n 0 0.59000 0.083300\n 1 0.58529 0.082705\n 2 0.58058 0.082155\n 3 0.57587 0.081650\n 4 0.57116 0.081191, Media Vol\n 96 0.13784 0.184246\n 97 0.13313 0.186054\n 98 0.12842 0.187866\n 99 0.12371 0.189681\n 100 0.11900 0.191500)\n\n\n\ncuando $w_s$ es igual entonces $w_b$ es igual a $1$ y estamos invirtiendo todo en bonos\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(6,4))\nplt.plot(portafolios['Vol'], portafolios['Media'], label='Port')\nplt.legend(loc=3)\nplt.xlabel('Volatilidad $\\sigma$')\nplt.ylabel('Rendimiento esperado($E[r_p]$)')\nplt.grid()\n```\n\n**Segundo.** Graficamos en la misma ventana, curvas de indiferencia.\n\n\n```python\n# Niveles de utilidad 0.06, 0.07, 0.09\nU4, U5, U6= 0.06, 0.07, 0.09\n# Coeficiente de aversi\u00f3n al riesgo\ng=3\n# Curvas de indiferencia\nE1= 0.5*g*sp**2+U4\nE2= 0.5*g*sp**2+U5\nE3= 0.5*g*sp**2+U6\n```\n\n\n```python\n# Gr\u00e1fica\n# Gr\u00e1fica\nplt.figure(figsize=(8,8))\nplt.plot(portafolios['Vol'], portafolios['Media'], label='Port')\n\nplt.plot(portafolios['Vol'], E1, label='$U_4=0.15$')\nplt.plot(portafolios['Vol'], E2, label='$U_5=0.1$')\nplt.plot(portafolios['Vol'], E3, label='$U_6=0.05$')\n\nplt.legend(loc=4)\nplt.xlabel('Volatilidad $\\sigma$')\nplt.ylabel('Rendimiento esperado($E[r_p]$)')\nplt.grid()\nplt.show()\n```\n\n**Tercero.** La elecci\u00f3n \u00f3ptima est\u00e1 dada por la curva de indiferencia para el nivel de utilidad m\u00e1s alto que es tangente a la frontera media-varianza.\n- Claramente, esta selecci\u00f3n depende del coeficiente de aversi\u00f3n al riesgo.\n\n\n```python\n# Gr\u00e1fica con zoom\n# Gr\u00e1fica\nplt.figure(figsize=(8,8))\nplt.plot(portafolios['Vol'], portafolios['Media'], label='Port')\n\nplt.plot(portafolios['Vol'], E1, label='$U_4=0.15$')\nplt.plot(portafolios['Vol'], E2, label='$U_5=0.1$')\nplt.plot(portafolios['Vol'], E3, label='$U_6=0.05$')\n\nplt.legend(loc=4)\nplt.xlabel('Volatilidad $\\sigma$')\nplt.ylabel('Rendimiento esperado($E[r_p]$)')\nplt.grid()\nplt.axis([0.11, 0.15, 0.08, 0.11])\nplt.show()\n```\n\nEl portafolio que maximiza mi vol\u00e1tilidad es aquel que sea tangente a nuestra curva de m\u00ednima varianza \n\n# Anuncios parroquiales\n\n## 1. Quiz la siguiente clase.\n## 2. Un par de art\u00edculos del WSJ y el NYT que discuten herramientas disponibles para la medici\u00f3n de su propia tolerancia al riesgo:\n- [Art\u00edculo 1](https://www.nytimes.com/2016/02/13/your-money/as-stocks-fall-its-time-to-measure-your-risk-tolerance.html)\n- [Art\u00edculo 2](https://www.wsj.com/articles/check-your-tolerance-for-investment-risk-now-before-markets-sag-1405619939)\n\n\n\n
                                        \nCreated with Jupyter by Esteban Jim\u00e9nez Rodr\u00edguez.\n
                                        \n", "meta": {"hexsha": "34e816ed9495f847a65fb8fbceb9dfa944fdb082", "size": 169828, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Modulo3/.ipynb_checkpoints/Clase12_ProblemaSeleccionPortafolio-checkpoint.ipynb", "max_stars_repo_name": "ariadnagalindom/portafolios", "max_stars_repo_head_hexsha": "5dd192773b22cbb53b291030afc248e482de97b1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Modulo3/.ipynb_checkpoints/Clase12_ProblemaSeleccionPortafolio-checkpoint.ipynb", "max_issues_repo_name": "ariadnagalindom/portafolios", "max_issues_repo_head_hexsha": "5dd192773b22cbb53b291030afc248e482de97b1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Modulo3/.ipynb_checkpoints/Clase12_ProblemaSeleccionPortafolio-checkpoint.ipynb", "max_forks_repo_name": "ariadnagalindom/portafolios", "max_forks_repo_head_hexsha": "5dd192773b22cbb53b291030afc248e482de97b1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 234.5690607735, "max_line_length": 39380, "alphanum_fraction": 0.8859375368, "converted": true, "num_tokens": 4514, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5312093733737562, "lm_q2_score": 0.7745833789613196, "lm_q1q2_score": 0.4114659513637693}} {"text": "# !!! D . R . A . F . T !!!\n\n# Chromatic Adaptation\n\nAdaptation is the process of change by which an organism becomes better suited to its environment. \n\nIn the context of colour science, it is defined as the process by which the state of the visual system is modified by previous and present exposure to stimuli that may have various luminance values, spectral distributions and angular subtenses. [1]\n\nThe three essential types of adaptation for human vision are:\n\n* Dark adaptation\n* Light adaptation\n* Chromatic adaptation\n\n### Dark Adaptation\n\nDark adaptation happens when the luminance level decreases and as the result of the lack of illumination, the observer visual system adapts to the stimulus and the visual sensitivity increases.\n\nWhen entering a dark area, initially, the [cone cells](http://en.wikipedia.org/wiki/Cone_cell) sensitivity increases to reach full adaptation after around 10 minutes, then the [rod cells](http://en.wikipedia.org/wiki/Rod_cell) sensitivity outperforms the cones and reach complete adaptation after 30 minutes. [2] [3] [4]\n\nA notable effect of rod cells driving the visual system at low luminance level is that humans become colour-blind in those low lighting conditions.\n\n### Light Adaptation\n\nLight adaptation is similar to dark adaption but instead the visual sensitivity decreases with luminance level increase.\n\nThe adaptation process happening when entering a bright area is faster than dark adaptation and occurs in around 5 minutes. The rod cells first saturate as [rhodopsin](http://en.wikipedia.org/wiki/Rhodopsin) *photobleaches*, while the cone cells continue to adapt reaching peak sensitivity after 5-10 minutes. [4] [5] [6] \n\n### Chromatic Adaptation\n\nChromatic adaptation is defined as the visual process whereby approximate compensation is made for changes in the colours of stimuli, especially in the case of changes in illuminants. [7] \n\nChromatic adaptation controls the independent sensitivity of the three cones cells type and is the most inportant adaptation mechanism in colour appearance. A white object viewed under different lighting conditions (daylight, tungsten or incandescent lighting) will retain its white appearance because the sensitivity of the cone cells is independently adjusted to compensate for the changes in energy level at the wavelengths ranges they are sensitive to.\n\nChromatic adaptation can be thought of as analogous to automatic white balancing feature of a digital still camera.[8]\n\n## Johannes Von Kries Hypothesis\n\nChromatic adaptation models are based on physiological plausible representations of cone responses signals. All modern chromatic adaptation models build upon Von Kries (1902) hypothesis: [9]\n\n> This can be conceived in the sense that the individual components present in the organ of vision are completely independent of one another and each is fatigued or adapted exclusively according to its own function (Translation by MacAdam 1970: 118).\n\nThe modern interpretation of Von Kries (1902) hypothesis, the *Von Kries Coefficient Law*, defines the following relationship: [10]\n\n$$\n\\begin{equation}\n\\begin{vmatrix}\nL_a\\\\\nM_a\\\\\nS_a\n\\end{vmatrix}=\n\\begin{vmatrix}\nk_L&0.0&0.0\\\\\n0.0&k_M&0.0\\\\\n0.0&0.0&k_S\n\\end{vmatrix}\n\\begin{vmatrix}\nL\\\\\nM\\\\\nS\n\\end{vmatrix}\n\\end{equation}\n$$\n\nor\n\n$$\n\\begin{equation}\n\\begin{aligned}\nL_a&\\ =k_L\\cdot L\\\\\nM_a&\\ =k_M\\cdot M\\\\\nS_a&\\ =k_S\\cdot S\n\\end{aligned}\n\\end{equation}\n$$\n\nwith\n\n$$\n\\begin{equation}\nk_L=\\cfrac{1}{L_{max}},\\qquad k_M=\\cfrac{1}{M_{max}},\\qquad k_S=\\cfrac{1}{S_{max}}\n\\end{equation}\n$$\n\nor\n\n$$\n\\begin{equation}\nk_L=\\cfrac{1}{L_{white}},\\qquad k_M=\\cfrac{1}{M_{white}},\\qquad k_S=\\cfrac{1}{S_{white}}\n\\end{equation}\n$$\n\nwhere $L_a$, $M_a$ and $S_a$ are the post-adaptation cones signals and $k_L$, $k_M$ and $k_S$ are the coefficients used to scale the initial cone signals $L$, $M$ and $S$.\n\nGiven the above equations, the *Von Kries Coefficient Law* can be used to compute *corresponding colour stimuli* between two different viewing conditions.\n\nThe *CIE* defines *corresponding colour stimuli* as pairs of colour stimuli that have same colour appearance when one is seen in one set of adaptation conditions and the other is seen in a different set. [11]\n\nComputing the *corresponding colour stimuli* is done as follows:\n\nPost-adaptation signals are computed for the two viewing conditions, then, the first viewing condition post-adaptation signals are set equal to the second viewing condition post-adaptation signals, finally the model is reversed for the second viewing condition:\n\n$$\n\\begin{equation}\n\\begin{vmatrix}\nX_2\\\\\nY_2\\\\\nZ_2\n\\end{vmatrix}=\n\\textbf{M}^{-1}\n\\begin{vmatrix}\nL_{max2}&0.0&0.0\\\\\n0.0&M_{max2}&0.0\\\\\n0.0&0.0&S_{max2}\n\\end{vmatrix}\n\\begin{vmatrix}\nk_L&0.0&0.0\\\\\n0.0&k_M&0.0\\\\\n0.0&0.0&k_S\n\\end{vmatrix}\n\\textbf{M}\n\\begin{vmatrix}\nX\\\\\nY\\\\\nZ\n\\end{vmatrix}\n\\end{equation}\n$$\n\nwith\n\n$$\n\\begin{equation}\nL_2=\\biggl(\\cfrac{L_1}{L_{max1}}\\biggr)L_{max2},\\qquad M_2=\\biggl(\\cfrac{M_1}{M_{max1}}\\biggr)M_{max2},\\qquad S_2=\\biggl(\\cfrac{S_1}{S_{max1}}\\biggr)S_{max2}\n\\end{equation}\n$$\n\nwhere $\\textbf{M}$ is the *chromatic adaptation transform* matrix converting from *CIE XYZ* tristimulus values to relative cone responses $LMS$.\n\n[Colour](https://github.com/colour-science/colour/) defines the following *chromatic adaptation transforms*:\n\n\n```python\nimport colour\n\nprint(sorted(colour.CHROMATIC_ADAPTATION_TRANSFORMS))\n```\n\n ['Bianco', 'Bianco PC', 'Bradford', 'CAT02', 'CAT02_BRILL_CAT', 'CMCCAT2000', 'CMCCAT97', 'Fairchild', 'Sharp', 'Von Kries', 'XYZ Scaling']\n\n\n## Johannes Von Kries Chromatic Adaptation Transform\n\n## Bibliography\n\n1. ^ CIE. (n.d.). 17-18 adaptation. Retrieved September 07, 2014, from http://eilv.cie.co.at/term/18\n2. ^ Wikipedia. (n.d.). Dark adaptation. Retrieved September 07, 2014, from http://en.wikipedia.org/wiki/Adaptation_(eye)#Dark_Adaptation\n3. ^ Fairchild, M. D. (2013). Dark Adaptation. In *Color Appearance Models* (3rd ed., p. 985,3749). Wiley. ASIN:B00DAYO8E2\n4. ^ Kalloniatis, M., & Luu, C. (n.d.). Light and Dark Adaptation. Retrieved September 07, 2014, from http://webvision.med.utah.edu/book/part-viii-gabac-receptors/light-and-dark-adaptation/\n5. ^ Wikipedia. (n.d.). Light adaptation. Retrieved September 07, 2014, from http://en.wikipedia.org/wiki/Adaptation_(eye)#Light_Adaptation\n6. ^ Fairchild, M. D. (2013). Light Adaptation. In *Color Appearance Models* (3rd ed., p. 1006,3728). Wiley. ASIN:B00DAYO8E2\n7. ^ CIE. (n.d.). 17-140 chromatic adaptation. Retrieved September 07, 2014, from http://eilv.cie.co.at/term/140\n8. ^ Fairchild, M. D. (2013). Chromatic Adaptation. In *Color Appearance Models* (3rd ed., p. 1029,3749). Wiley. ASIN:B00DAYO8E2\n9. ^ Fairchild, M. D. (2013). VON KRIES MODEL. In *Color Appearance Models* (3rd ed., p. 4204). Wiley. ASIN:B00DAYO8E2\n10. ^ Fairchild, M. D. (2013). Equations 9.1-9.11. In *Color Appearance Models* (3rd ed., pp. 4225\u20134262). Wiley. ASIN:B00DAYO8E2\n11. ^ CIE. (n.d.). 17-260 corresponding colour stimuli. Retrieved September 07, 2014, from http://eilv.cie.co.at/term/260\n", "meta": {"hexsha": "384227df5586ac84bb430d7665e2117691c452ed", "size": 11536, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/adaptation/vonkries.ipynb", "max_stars_repo_name": "colour-science/colour-notebooks", "max_stars_repo_head_hexsha": "f227bb1ebc041812de4048ae20e2b702ffb3150d", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 13, "max_stars_repo_stars_event_min_datetime": "2016-11-23T22:13:24.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-28T14:52:13.000Z", "max_issues_repo_path": "notebooks/adaptation/vonkries.ipynb", "max_issues_repo_name": "colour-science/colour-ipython", "max_issues_repo_head_hexsha": "f227bb1ebc041812de4048ae20e2b702ffb3150d", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2015-07-13T19:38:16.000Z", "max_issues_repo_issues_event_max_datetime": "2015-12-14T06:30:04.000Z", "max_forks_repo_path": "notebooks/adaptation/vonkries.ipynb", "max_forks_repo_name": "colour-science/colour-ipython", "max_forks_repo_head_hexsha": "f227bb1ebc041812de4048ae20e2b702ffb3150d", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2016-10-06T16:18:40.000Z", "max_forks_repo_forks_event_max_datetime": "2020-08-01T10:04:27.000Z", "avg_line_length": 42.2564102564, "max_line_length": 515, "alphanum_fraction": 0.6153779473, "converted": true, "num_tokens": 2550, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.6688802537704064, "lm_q1q2_score": 0.41142012088868163}} {"text": "```python\n# !/usr/bin/env python3\n# -*- coding: utf-8 -*-\n\"\"\"\nCreated on 20191125\n\n@author: zhangji\n\ntest the linear relationship\nU_t =?= U_sh + U_wm\nU_t is the total velocity\nU_sh is the velocity induced by shear flow\nU_wm is the active velocity. \n\"\"\"\n\n# %pylab inline\n# pylab.rcParams['figure.figsize'] = (25, 11)\n# fontsize = 40\n\n# import numpy as np\n# import scipy as sp\n# from scipy.optimize import leastsq, curve_fit\n# from scipy import interpolate\n# from scipy.interpolate import interp1d\n# from scipy.io import loadmat, savemat\n# # import scipy.misc\n\n# import matplotlib\n# from matplotlib import pyplot as plt\n# from matplotlib import animation, rc\n# import matplotlib.ticker as mtick\n# from mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\n# from mpl_toolkits.mplot3d import Axes3D, axes3d\n\n# from sympy import symbols, simplify, series, exp\n# from sympy.matrices import Matrix\n# from sympy.solvers import solve\n\n# from IPython.display import display, HTML\n# from tqdm import tqdm_notebook as tqdm\n# import pandas as pd\n# import re\n# from scanf import scanf\n# import os\n# import glob\n\n# from codeStore import support_fun as spf\n# from src.support_class import *\n# from src import stokes_flow as sf\n\n# rc('animation', html='html5')\n# PWD = os.getcwd()\n# font = {'size': 20}\n# matplotlib.rc('font', **font)\n# np.set_printoptions(linewidth=90, precision=5)\n\n%load_ext autoreload\n%autoreload 2\n\nimport os\nimport glob\nimport re\nimport pandas as pd\nfrom scanf import scanf\nimport natsort \nimport numpy as np\nimport scipy as sp\nfrom scipy.optimize import leastsq, curve_fit\nfrom scipy import interpolate, spatial, sparse, optimize\n# from scipy.interpolate import interp1d\nfrom scipy.io import loadmat, savemat\n# import scipy.misc\nimport importlib\nfrom IPython.display import display, HTML\nimport pickle\n\nimport matplotlib\nfrom matplotlib import pyplot as plt\nfrom matplotlib import colors as mcolors\nfrom matplotlib import animation, rc\nimport matplotlib.ticker as mtick\nfrom mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\nfrom mpl_toolkits.mplot3d import Axes3D, axes3d\nfrom matplotlib import ticker, cm\nfrom mpl_toolkits.axes_grid1 import make_axes_locatable\n\nfrom time import time\nfrom src import support_class as spc\nfrom src import jeffery_model as jm\nfrom codeStore import support_fun as spf\nfrom codeStore import support_fun_table as spf_tb\n\n# %matplotlib notebook\n\n%matplotlib inline\nrc('animation', html='html5')\nrc('text', usetex=True)\nparams = {'text.latex.preamble': [r'\\usepackage{bm}', r'\\usepackage{amsmath}']}\nplt.rcParams.update(params)\nfontsize = 40\nfigsize = (30, 16)\nPWD = os.getcwd()\n```\n\n\n```python\nfig = plt.figure(figsize=(2, 2))\nfig.patch.set_facecolor('white')\nax0 = fig.add_subplot(1, 1, 1)\n```\n\n\n```python\nfrom shutil import copyfile\n\npickle_name = 'curve0.5_baseFlow'\nmdf_pickle_name = 'curve0.5_baseFlow_mdf'\n\nwith open('%s.pickle' % pickle_name, 'rb') as handle:\n pickle_dict = pickle.load(handle)\ndisplay(np.vstack(pickle_dict['uw_Base_list'])[1:6, :3])\ndisplay(np.vstack(pickle_dict['uw_Base_list'])[1:6, 3:])\ndisplay(np.vstack(pickle_dict['uw_Base_list'])[9, :])\n\nt1 = pickle_dict['uw_Base_list'].copy()\nfor i0 in (1, 2, 3, 4, 5):\n t1[i0] = np.zeros_like(t1[i0])\nt1[1][0] = pickle_dict['uw_Base_list'][1][0]\nt1[2][0] = pickle_dict['uw_Base_list'][2][0]\nt1[3][1] = pickle_dict['uw_Base_list'][3][1]\nt1[4][2] = pickle_dict['uw_Base_list'][4][2]\nt1[3][5] = pickle_dict['uw_Base_list'][3][5]\nt1[4][4] = pickle_dict['uw_Base_list'][4][4]\nt1[5][3] = pickle_dict['uw_Base_list'][5][3]\ndisplay(np.vstack(t1)[1:6, :3])\ndisplay(np.vstack(t1)[1:6, 3:])\ndisplay(np.vstack(t1)[9, :])\n\npickle_dict['uw_Base_list'] = t1\ntname = '%s.pickle' % mdf_pickle_name\nwith open(tname, 'wb') as handle:\n pickle.dump(pickle_dict, handle, protocol=pickle.HIGHEST_PROTOCOL)\ncopyfile(tname, os.path.join(os.getcwd(), os.pardir, os.pardir, 'src', tname))\nprint('save table_data to %s' % tname)\n\n```\n\n\n array([[ 4.54421792e-01, -1.70841659e-05, 2.15117240e-07],\n [-1.32320423e-01, -2.18376942e-05, -3.18737211e-06],\n [-3.17170214e-06, 8.07184556e-01, -4.39438280e-06],\n [-1.20039731e-07, 4.35344191e-07, 6.05959068e-02],\n [ 1.44139640e-07, 4.88388708e-07, 2.97835889e-06]])\n\n\n\n array([[-1.41011613e-07, 2.90499422e-07, 1.08864381e-05],\n [-1.44997826e-06, -4.02168914e-06, 1.34990679e-05],\n [-1.80105930e-06, -5.43934238e-06, -5.48793178e-01],\n [ 1.20565217e-06, -8.85711172e-01, -2.20900596e-08],\n [ 9.67713920e-01, 3.64893044e-06, -5.05933269e-07]])\n\n\n\n array([0., 0., 0., 0., 0., 0.])\n\n\n\n array([[ 0.45442179, 0. , 0. ],\n [-0.13232042, 0. , 0. ],\n [ 0. , 0.80718456, 0. ],\n [ 0. , 0. , 0.06059591],\n [ 0. , 0. , 0. ]])\n\n\n\n array([[ 0. , 0. , 0. ],\n [ 0. , 0. , 0. ],\n [ 0. , 0. , -0.54879318],\n [ 0. , -0.88571117, 0. ],\n [ 0.96771392, 0. , 0. ]])\n\n\n\n array([0., 0., 0., 0., 0., 0.])\n\n\n save table_data to curve0.5_baseFlow_mdf.pickle\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "65900665fc34f59ec7c01792475abd6d674505b2", "size": 10544, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "head_Force/loop_table/baseFlow_curve.ipynb", "max_stars_repo_name": "pcmagic/stokes_flow", "max_stars_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-11-11T05:00:53.000Z", "max_stars_repo_stars_event_max_datetime": "2018-11-11T05:00:53.000Z", "max_issues_repo_path": "head_Force/loop_table/baseFlow_curve.ipynb", "max_issues_repo_name": "pcmagic/stokes_flow", "max_issues_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "head_Force/loop_table/baseFlow_curve.ipynb", "max_forks_repo_name": "pcmagic/stokes_flow", "max_forks_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 37.5231316726, "max_line_length": 2064, "alphanum_fraction": 0.6091616085, "converted": true, "num_tokens": 1762, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.668880247169804, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.41142011682873125}} {"text": "```python\nfrom IPython.display import HTML\ntag = HTML('''\nToggle cell visibility here.''')\ndisplay(tag)\n```\n\n\n\nToggle cell visibility here.\n\n\n\n```python\n# Erasmus+ ICCT project (2018-1-SI01-KA203-047081)\n\n%matplotlib notebook\nimport numpy as np\nimport math\nimport matplotlib.pyplot as plt\nfrom scipy import signal\nimport ipywidgets as widgets\nimport control as c\nimport sympy as sym\nfrom IPython.display import Latex, display, Markdown # For displaying Markdown and LaTeX code\nfrom fractions import Fraction\nimport matplotlib.patches as patches\n```\n\n## Dominant pole approximation\n\nWhen studying the behaviour of systems, we often approximate systems by a dominant pole or a pair of dominant complex poles. This example demonstrates this property.\n\nA second-order system is defined by the following transfer function:\n\n\\begin{equation}\n G(s)=\\frac{\\alpha\\beta}{(s+\\alpha)(s+\\beta)}=\\frac{1}{(\\frac{1}{\\alpha}s+1)(\\frac{1}{\\beta}s+1)},\n\\end{equation}\n\nwhere $\\beta=1$, and $\\alpha$ is iterable.\n\nA third-order system is defined by the following transfer function:\n\n\\begin{equation}\n G(s)=\\frac{\\alpha{\\omega_0}^2}{\\big(s+\\alpha\\big)\\big(s^2+2\\zeta\\omega_0s+\\omega_0^2\\big)}=\\frac{1}{(\\frac{1}{\\alpha}s+1)(\\frac{1}{\\omega_0^2}s^2+\\frac{2\\zeta\\alpha}{\\omega_0}s+1)},\n\\end{equation}\n\nwhere $\\beta=1$, $\\omega_0=4.1$ and $\\zeta=0.24$, and $\\alpha$ is iterable.\n\n---\n\n### How to use this notebook?\n\nToggle between the second- and third-order system and move the slider to change the location of the moveable pole $\\alpha$.\n\nThis notebook is based on the following [tutorial](https://lpsa.swarthmore.edu/PZXferStepBode/DomPole.html \"The Dominant Pole Approximation\") by Prof. Erik Cheever.\n\n\n```python\n# System selector buttons\nstyle = {'description_width': 'initial'}\ntypeSelect = widgets.ToggleButtons(\n options=[('second-order system', 0), ('third-order system', 1),],\n description='Select: ',style=style)\n```\n\n\n```python\ndisplay(typeSelect)\ncontinuous_update=False\n\n# set up plot \n\nfig, ax = plt.subplots(2,1,figsize=[9.8,7],num='Dominant pole approximation')\nplt.subplots_adjust(hspace=0.35)\nax[0].grid(True)\nax[1].grid(True)\n# ax[2].grid(which='both', axis='both', color='lightgray')\nax[0].axhline(y=0,color='k',lw=.8)\nax[1].axhline(y=0,color='k',lw=.8)\nax[0].axvline(x=0,color='k',lw=.8)\nax[1].axvline(x=0,color='k',lw=.8)\nax[0].set_xlabel('Re')\nax[0].set_ylabel('Im')\nax[0].set_xlim([-10,0.5])\nax[1].set_xlim([-0.5,20])\nax[1].set_xlabel('$t$ [s]')\nax[1].set_ylabel('input, output')\nax[0].set_title('Pole-zero plot')\nax[1].set_title('Time response')\n\nplotzero, = ax[0].plot([], [])\nresponse, = ax[1].plot([], [])\nresponseAdom, = ax[1].plot([], [])\nresponseBdom, = ax[1].plot([], [])\n\nax[1].step([0,50],[0,1],color='C0',label='input')\n\n# generate x values\n \ndef response_func(a,index):\n \n global plotzero, response, responseAdom, responseBdom \n# global bodePlot, bodePlotAdom, bodePlotBdom\n\n t = np.linspace(0, 50, 1000)\n \n if index==0:\n b=1\n num=a*b\n den=([1,a+b,a*b])\n tf_sys=c.TransferFunction(num,den)\n poles_sys,zeros_sys=c.pzmap(tf_sys, Plot=False)\n tout, yout = c.step_response(tf_sys,t)\n den1=([1,a])\n tf_sys1=c.TransferFunction(a,den1)\n toutA, youtA = c.step_response(tf_sys1,t)\n den2=([1,b])\n tf_sys2=c.TransferFunction(b,den2)\n toutB, youtB = c.step_response(tf_sys2,t)\n mag, phase, omega = c.bode_plot(tf_sys, Plot=False) # Bode-plot\n magA, phase, omegaA = c.bode_plot(tf_sys1, Plot=False) # Bode-plot\n magB, phase, omegaB = c.bode_plot(tf_sys2, Plot=False) # Bode-plot\n s=sym.Symbol('s')\n eq=(a*b/((s+a)*(s+b)))\n eq1=1/(((1/a)*s+1)*((1/b)*s+1))\n display(Markdown('Moveable pole (purple curve) $\\\\alpha$ is equal to %.1f, fixed pole (red curve) $b$ is equal to %i; The transfer function is equal to:'%(a,1)))\n display(eq),display(Markdown('or')),display(eq1)\n\n elif index==1:\n omega0=4.1\n zeta=0.24\n num=a*omega0**2\n den=([1,2*zeta*omega0+a,omega0**2+2*zeta*omega0*a,a*omega0**2])\n tf_sys=c.TransferFunction(num,den)\n poles_sys,zeros_sys=c.pzmap(tf_sys, Plot=False)\n tout, yout = c.step_response(tf_sys,t)\n den1=([1,a])\n tf_sys1=c.TransferFunction(a,den1)\n toutA, youtA = c.step_response(tf_sys1,t)\n den2=([1,2*zeta*omega0,omega0**2])\n tf_sys2=c.TransferFunction(omega0**2,den2)\n toutB, youtB = c.step_response(tf_sys2,t)\n mag, phase, omega = c.bode_plot(tf_sys, Plot=False) # Bode-plot\n magA, phase, omegaA = c.bode_plot(tf_sys1, Plot=False) # Bode-plot\n magB, phase, omegaB = c.bode_plot(tf_sys2, Plot=False) # Bode-plot\n s=sym.Symbol('s')\n eq=(a*omega0**2/((s+a)*(s**2+2*zeta*omega0*s+omega0*omega0)))\n eq1=1/(((1/a)*s+1)*((1/(omega0*omega0))*s*s+(2*zeta*a/omega0)*s+1))\n \n display(Markdown('Moveable pole (purple curve) $\\\\alpha$ is equal to %.1f, fixed poles (red curve) $\\\\beta$ are equal to $1\\pm4j$ ($\\omega_0$ is set to 4.1, $\\zeta$ is set to 0.24). The transfer function is equal to:'%(a)))\n display(eq),display(Markdown('or')),display(eq1)\n \n ax[0].lines.remove(plotzero)\n ax[1].lines.remove(response)\n ax[1].lines.remove(responseAdom)\n ax[1].lines.remove(responseBdom)\n \n plotzero, = ax[0].plot(np.real(poles_sys), np.imag(poles_sys), 'xg', markersize=10, label = 'pole')\n response, = ax[1].plot(tout,yout,color='C1',label='system response',lw=3)\n responseAdom, = ax[1].plot(toutA,youtA,color='C4',label='response due to the moveable pole (pair) only')\n responseBdom, = ax[1].plot(toutB,youtB,color='C3',label='response due to the fixed pole only')\n\n ax[0].legend()\n ax[1].legend()\n \na_slider=widgets.FloatSlider(value=0.1, min=0.1, max=10, step=.1,\n description='$\\\\alpha$:',disabled=False,continuous_update=False,\n orientation='horizontal',readout=True,readout_format='.2f',)\n\ninput_data=widgets.interactive_output(response_func,{'a':a_slider,'index':typeSelect})\n\ndef update_slider(index):\n global a_slider\n \n aval=[0.1,0.1]\n a_slider.value=aval[index] \n\ninput_data2=widgets.interactive_output(update_slider,{'index':typeSelect})\n\ndisplay(a_slider,input_data)\n\n```\n\n\n ToggleButtons(description='Select: ', options=(('second-order system', 0), ('third-order system', 1)), style=T\u2026\n\n\n\n \n\n\n\n\n\n\n\n FloatSlider(value=0.1, continuous_update=False, description='$\\\\alpha$:', max=10.0, min=0.1)\n\n\n\n Output()\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "831135ec256dae62ed1b3de72fb030fa780354a1", "size": 155993, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ICCT_en/examples/02/.ipynb_checkpoints/TD-12-Dominant-pole-approximation-checkpoint.ipynb", "max_stars_repo_name": "ICCTerasmus/ICCT", "max_stars_repo_head_hexsha": "fcd56ab6b5fddc00f72521cc87accfdbec6068f6", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2021-05-22T18:42:14.000Z", "max_stars_repo_stars_event_max_datetime": "2021-10-03T14:10:22.000Z", "max_issues_repo_path": "ICCT/ENG/examples/02/TD-12-Dominant-pole-approximation.ipynb", "max_issues_repo_name": "tuxsaurus/ICCT", "max_issues_repo_head_hexsha": "30d1aea4fb056c9736c9b4c5a0f50fff14fa6382", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ICCT/ENG/examples/02/TD-12-Dominant-pole-approximation.ipynb", "max_forks_repo_name": "tuxsaurus/ICCT", "max_forks_repo_head_hexsha": "30d1aea4fb056c9736c9b4c5a0f50fff14fa6382", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2021-05-24T11:40:09.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-29T16:36:18.000Z", "avg_line_length": 141.0424954792, "max_line_length": 110053, "alphanum_fraction": 0.8265947831, "converted": true, "num_tokens": 2104, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.668880247169804, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.41142011682873125}} {"text": "```python\nimport bh\nimport sympy as sy\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport matplotlib.cm as cm\nimport cmocean.cm as ccm\nimport matplotlib.transforms as mt\nimport scipy.interpolate as si\nimport out\nplt.style.use('dark_background')\n```\n\n\n```python\nth0 = 80\nalpha = np.linspace(0, 2*np.pi, 1000)\nr_vals = np.arange(6, 60, 0.5)\nn_vals = [0, 1]\nm = 1\n\nfix, ax = plt.subplots(figsize=(40, 40))\nfig = out.generate_image(ax, alpha, r_vals, th0, n_vals, m, None);\n```\n\n\n```python\nfig.savefig('blackhole.png', dpi=300)\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "4a1ed2a0b68ca47b7b867a89e88295afada50ec9", "size": 73870, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Example.ipynb", "max_stars_repo_name": "peytondmurray/bhsim", "max_stars_repo_head_hexsha": "772b9312d9c91700053416f8bee81607ed4731fd", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Example.ipynb", "max_issues_repo_name": "peytondmurray/bhsim", "max_issues_repo_head_hexsha": "772b9312d9c91700053416f8bee81607ed4731fd", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Example.ipynb", "max_forks_repo_name": "peytondmurray/bhsim", "max_forks_repo_head_hexsha": "772b9312d9c91700053416f8bee81607ed4731fd", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 794.3010752688, "max_line_length": 71920, "alphanum_fraction": 0.9570867741, "converted": true, "num_tokens": 175, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943822145997, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.41137874629869786}} {"text": "#
                                        Applied Stochastic Processes HW_01
                                        \n\n**
                                        11510691 \u7a0b\u8fdc\u661f
                                        **\n\n## Question 1\n\n$\\DeclareMathOperator*{\\argmin}{argmin}\n\\newcommand{\\using}[1]{\\stackrel{\\mathrm{#1}}{=}}\n\\newcommand{\\ffrac}{\\displaystyle \\frac}\n\\newcommand{\\space}{\\text{ }}\n\\newcommand{\\bspace}{\\;\\;\\;\\;}\n\\newcommand{\\QQQ}{\\boxed{?\\:}}\n\\newcommand{\\CB}[1]{\\left\\{ #1 \\right\\}}\n\\newcommand{\\SB}[1]{\\left[ #1 \\right]}\n\\newcommand{\\P}[1]{\\left( #1 \\right)}\n\\newcommand{\\dd}{\\mathrm{d}}\n\\newcommand{\\Tran}[1]{{#1}^{\\mathrm{T}}}\n\\newcommand{\\d}[1]{\\displaystyle{#1}}\n\\newcommand{\\EE}[2][\\,\\!]{\\mathbb{E}_{#1}\\left[#2\\right]}\n\\newcommand{\\Var}[2][\\,\\!]{\\mathrm{Var}_{#1}\\left[#2\\right]}\n\\newcommand{\\Cov}[2][\\,\\!]{\\mathrm{Cov}_{#1}\\left(#2\\right)}\n\\newcommand{\\Corr}[2][\\,\\!]{\\mathrm{Corr}_{#1}\\left(#2\\right)}\n\\newcommand{\\I}[1]{\\mathrm{I}\\left( #1 \\right)}\n\\newcommand{\\N}[1]{\\mathrm{N} \\left( #1 \\right)}\n\\newcommand{\\ow}{\\text{otherwise}}\n\\P{\\text{a}}$\n\n$\\bspace \\EE{Y} = \\d{\\int_{-\\infty}^{\\infty} \\sqrt{x} \\cdot 2x \\;\\dd{x} = \\int_{0}^{1} 2x^{1.5} \\;\\dd{x} = \\Big. \\ffrac{4} {5}x^{2.5}}\\Big|_{0}^{1} = 0.8$\n\n$\\P{\\text{b}}$\n\n$\\bspace \\begin{align}\nF_Y\\P{y} &= P\\CB{Y \\leq y} = P\\CB{\\sqrt{X} \\leq y} = P\\CB{0 \\leq x \\leq y^2} = \\int_{0}^{y^2} 2x \\;\\dd{x}\\\\\n&= \\begin{cases}\n1, &\\text{if } 1 \\leq y \\\\[0.5em]\ny^4, &\\text{if } 0 < y < 1 \\\\[0.5em]\n0, &\\ow\n\\end{cases}\n\\end{align}$\n\n$\\bspace$and thus the **pdf** of $Y$ is the derivative of the **cdf** which is $4y^3$ if $0 < y <1$ and $0$ otherwise.\n\n## Question 2\n\n$\\P{\\text{a}}$ \n\n$\\bspace$Given the conditions we can directly write that $P\\CB{X = m} = e^{-\\lambda_1}\\ffrac{\\lambda_1^m} {m!}$ and $P\\CB{Y = n} = e^{-\\lambda_2}\\ffrac{\\lambda_2^n} {n!}$. Then:\n\n$\\bspace \\begin{align}\nP\\CB{X+Y = f} &= \\sum_{j=0}^{f} P\\CB{X+Y=f \\mid Y = f-j}\\cdot P\\CB{Y=f-j} \\\\\n&= \\sum_{j=0}^{f} P\\CB{X=j }\\cdot P\\CB{Y=f-j} \\\\\n&= \\sum_{j=0}^{f}e^{-\\lambda_1}\\ffrac{\\lambda_1^j} {j!} \\cdot e^{-\\lambda_2}\\ffrac{\\lambda_2^{\\P{f-j}}} {\\P{f-j}!}\\\\\n&= e^{-\\lambda_1-\\lambda_2}\\ffrac{\\lambda_1^f} {f!} \\sum_{j=0}^{f} \\ffrac{f!} {j!\\P{f-j}!}\\lambda_1^{\\P{j-f}}\\lambda_2^{\\P{f-j}} \\\\\n&= e^{-\\lambda_1-\\lambda_2}\\ffrac{\\lambda_1^f} {f!} \\sum_{j=0}^{f} \\binom{f} {j} \\P{\\ffrac{\\lambda_2} {\\lambda_1}}^{f-j} \\\\\n&= e^{-\\lambda_1-\\lambda_2}\\ffrac{\\lambda_1^f} {f!} \\P{1+\\ffrac{\\lambda_2} {\\lambda_1}}^{f} = e^{-\\P{\\lambda_1 + \\lambda_2}} \\ffrac{\\P{\\lambda_1 + \\lambda_2}^f} {f!}\n\\end{align}$\n\n$\\P{\\text{b}}$\n\n$\\bspace$ Similarly, we have $P\\CB{X = m} = \\d{n_1 \\choose m}p^{m}(1-p)^{\\d{n_1-m}}$ and $P\\CB{Y = n} = \\d{n_2 \\choose n}p^{n}(1-p)^{\\d{n_2-n}}$. Then\n\n$\\bspace \\begin{align}\nP\\CB{X+Y = f} &= \\sum_{j=0}^{f} P\\CB{X+Y=f \\mid Y = f-j}\\cdot P\\CB{Y=f-j} \\\\\n&= \\sum_{j=0}^{f} P\\CB{X=j }\\cdot P\\CB{Y=f-j} \\\\\n&= \\sum_{j=0}^{f} \\d{n_1 \\choose j}p^{j}(1-p)^{\\d{n_1-j}} \\cdot \\d{n_2 \\choose f-j}p^{f-j}(1-p)^{\\d{n_2-f+j}}\\\\\n&= p^{\\d{f}}(1-p)^{\\d{n_1+n_2-f}} \\sum_{j=0}^{f} \\d{n_1 \\choose j}\\d{n_2 \\choose f-j}\\\\\n&= p^{\\d{f}}(1-p)^{\\d{n_1+n_2-f}} \\binom{n_1 + n_2} {f}\n\\end{align}$\n\n## Question 3\n\n$\\P{\\text{a}}$\n\n$\\bspace\nf_X\\P{x} = \\d{\\int_{-\\infty}^{\\infty} f\\P{x,y}\\;\\dd{y} = \\int_{0}^{1} x+y \\;\\dd{y} = \\frac{1} {2} + x}$\n\n$\\P{\\text{b}}$\n\n$\\bspace$We first need to get the marginal distribution of $Y$: $f_Y\\P{y} = \\d{\\int_{-\\infty}^{\\infty} f\\P{x,y}\\;\\dd{x} = \\int_{0}^{1} x+y \\;\\dd{x} = \\frac{1} {2} + y}$. Then $\\EE{X} = \\d{\\int_{-\\infty}^{\\infty}x\\cdot f_X\\P{x}\\;\\dd{x} = \\int_{0}^{1} \\frac{x} {2} + x^2 \\;\\dd{x}=\\frac{7} {12}}$ and $\\EE{X^2} = \\d{\\int_{-\\infty}^{\\infty}x^2\\cdot f_X\\P{x}\\;\\dd{x} = \\int_{0}^{1} \\frac{x^2} {2} + x^3 \\;\\dd{x}=\\frac{5} {12}}$ thus $\\Var{X} = \\ffrac{11} {144}$. And same results for $Y$.\n\n$\\bspace$Next, $\\EE{XY} = \\d{\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}xy\\cdot \\P{x+y}\\;\\dd{x}\\dd{y} = \\int_{0}^{1}\\int_{0}^{1}x^2y+xy^2\\;\\dd{x}\\dd{y} = \\frac{1} {3}}$, thus $\\Cov{X,Y} = \\EE{XY} - \\EE{X}\\EE{Y} = -\\ffrac{1} {144}$.\n\n$\\bspace$Finally, $\\rho = \\ffrac{\\Cov{X,Y}} {\\sqrt{\\Var{X} \\Var{Y}}} = \\ffrac{-\\ffrac{1} {144}} {\\sqrt{\\ffrac{11} {144} \\cdot \\ffrac{11} {144}}} = -\\ffrac{1} {11}$\n\n## Question 4\n\n$\\bspace$Since $X$ and $Y$ are independent thus their joint density function is:\n\n$$f_{X,Y}\\P{x,y} = f_X\\P{x} \\cdot f_Y\\P{y} = \\ffrac{1} {2\\pi \\sigma^2}\\exp\\CB{-\\ffrac{\\P{x-\\mu}^2 + \\P{y-\\mu}^2} {2\\sigma^2}}$$\n\n$\\bspace$Then we let $\\begin{cases}\nS \\,= X+Y\\\\\nT = X-Y\n\\end{cases}$ thus we can obtain the Jacobian determinant\n\n$$J = \\begin{vmatrix}\n\\;\\ffrac{\\partial s}{\\partial x} & \\ffrac{\\partial s}{\\partial y}\\;\\\\ \n\\;\\ffrac{\\partial t}{\\partial x} & \\ffrac{\\partial t}{\\partial y}\\;\n\\end{vmatrix} = \\begin{vmatrix}\n1 & 1\\\\ \n1 & -1\n\\end{vmatrix} = -2$$\n\n$\\bspace$And with this we can derive the joint density function for $S$ and $T$:\n\n$$f_{S,T}\\P{s,t} = f_{X,Y}\\P{x,y} \\cdot \\left|J\\right|^{-1} = \\ffrac{1} {4\\pi \\sigma^2}\\exp\\CB{-\\ffrac{\\P{s + t-2\\mu}^2 + \\P{s-t-2\\mu}^2} {8\\sigma^2}}$$\n\n$\\bspace$We can simplify that and obtain:\n\n$$f_{S,T}\\P{s,t} = \\CB{\\ffrac{1} {2\\sigma\\sqrt{\\pi}}\\exp\\CB{-\\ffrac{\\P{s-2\\mu}^2} {4\\sigma^2}}}\\cdot\\CB{\\ffrac{1} {2\\sigma\\sqrt{\\pi}}\\exp\\CB{-\\ffrac{t^2 + 4\\mu^2} {4\\sigma^2}}}$$\n\n$\\bspace$Two seperate terms with relying only on $s$ and $t$, respectively. Thus we can draw the conclusion that $S$ and $T$ are independent...\n\n## Question 5\n\n$\\bspace \\begin{align}\n\\EE{X\\mid Y=y} &= \\int_{-\\infty}^{\\infty} x \\cdot P\\CB{X=x \\mid Y = y} \\;\\dd{x} \\\\\n&= \\int_{-\\infty}^{\\infty} x \\cdot \\ffrac{P\\CB{X=x ,Y = y}}{P\\CB{Y=y}} \\;\\dd{x} \\\\\n&= \\int_{-\\infty}^{\\infty} x \\cdot \\ffrac{P\\CB{X=x}\\cdot P\\CB{Y = y}}{P\\CB{Y=y}} \\;\\dd{x} \\\\\n&\\using{{P\\CB{Y=y}}>0} \\int_{-\\infty}^{\\infty} x \\cdot P\\CB{X=x} \\;\\dd{x} =\\EE{X}\n\\end{align}$\n\n## Question 6\n\n$\\bspace \\begin{align}\n\\EE{X \\mid Y=2} &= \\sum_{x=1,2}\\sum_{z = 1,2} x\\cdot p\\P{x,2,z} \\\\\n&= 1 \\times\\P{p\\P{1,2,1} + p\\P{1,2,2}} + 2 \\times \\P{p\\P{2,2,1} + p\\P{2,2,2}} \\\\\n&= \\ffrac{1} {16} + 0 + 2\\P{0 + \\ffrac{1} {4}} = \\ffrac{9} {16}\n\\end{align}$\n\n$\\bspace \\begin{align}\n\\EE{X \\mid Y=2,Z=1} &= \\sum_{x=1,2} x\\cdot p\\P{x,2,1} \\\\\n&= 1 \\times p\\P{1,2,1} + 2 \\times p\\P{2,2,1} \\\\\n&= \\ffrac{1} {16} + 2\\times 0 = \\ffrac{1} {16}\n\\end{align}$\n", "meta": {"hexsha": "549607e0997a4f3994c29cc323530f7e46f2d6bd", "size": 9286, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Probability and Statistics/Applied Random Process/HW/HW_01.ipynb", "max_stars_repo_name": "XavierOwen/Notes", "max_stars_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2018-11-27T10:31:08.000Z", "max_stars_repo_stars_event_max_datetime": "2019-01-20T03:11:58.000Z", "max_issues_repo_path": "Probability and Statistics/Applied Random Process/HW/HW_01.ipynb", "max_issues_repo_name": "XavierOwen/Notes", "max_issues_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Probability and Statistics/Applied Random Process/HW/HW_01.ipynb", "max_forks_repo_name": "XavierOwen/Notes", "max_forks_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-07-14T19:57:23.000Z", "max_forks_repo_forks_event_max_datetime": "2020-07-14T19:57:23.000Z", "avg_line_length": 42.0180995475, "max_line_length": 535, "alphanum_fraction": 0.4554167564, "converted": true, "num_tokens": 2967, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.7577943658046609, "lm_q1q2_score": 0.4113787373903446}} {"text": "```python\n import json\n !pip install pycrypto\n from base64 import b64encode\n !pip install pycryptodomex\n from Crypto.Random import get_random_bytes\n from Cryptodome.Cipher import AES\n import itertools\nimport random\nimport math \nimport time\nimport sympy\n#algoritman\u0131n genel mant\u0131\u011f\u0131 kullan\u0131labilecek t\u00fcm string de\u011ferleri aras\u0131ndan o \u00e7al\u0131\u015fma seans\u0131 i\u00e7in belirlenmi\u015f key ile enkripte edip enkripte edilmi\u015f halde \n#patternlar aramakt\u0131r uygun bulunan stringler set data typ\u0131nda return edilir ve kontrol etmek i\u00e7in de key tekrar uygulan\u0131r AES_S\u0130V algoritmas\u0131 Ayn\u0131 key ve string de\u011feri ile bire bir ve tersi olan bi fonksiyon olarak \u00e7al\u0131\u015f\u0131r \n#Aessiv sayesinde kullan\u0131lan pattern tersine m\u00fchendislik ile k\u0131rlmas\u0131n\u0131 \u00e7ok zorla\u015ft\u0131r\u0131r brute force korumas\u0131 i\u00e7inde valid pattern iyi se\u00e7ilmelidir \n#a\u015fa\u011f\u0131da belirtilen de\u011ferler i\u00e7in brut forcun \u00e7al\u0131\u015fma olas\u0131l\u0131\u011f\u0131 0.00026087612114774017\n# key = b'7\\xb2\\xb17\\xe3\\xe7\\xd8\\xfay\\xe51\\xee\\x18\\x0bn\\xc5\\x8b^\\x88 \\xcc,\\xe3\\x8b\\xf9(\\xe8\\x01d\\x9a\\x9c\\xa8'\n# nonce = b'p\\xfb\\xda#\\x86\\x88\\xaf\\xce;\\t\\x95\\xa0\"\\xd4PY'\n#Kodlar\u0131n \u00fcretilme s\u00fcresi uzun ancak \u00fcretilen kodlara bak\u0131p valid kod \u00e7\u0131kar\u0131lmas\u0131 \u00e7ok d\u00fc\u015f\u00fck olas\u0131l\u0131k\n```\n\n\n```python\nkey = get_random_bytes(16 * 2)\nnonce = get_random_bytes(16)\nheader = b\"header\"\nchars = \"ACDEFGHKLMNPRTXYZ234579234579\"\nchars = list(chars)\nrandom.shuffle(chars)\nchars = \"\".join(chars)\n\n```\n\n\n```python\ndef encrypt(ip):\n data = str.encode(ip)\n cipher = AES.new(key, AES.MODE_SIV, nonce=nonce) \n cipher.update(header)\n ciphertext, tag = cipher.encrypt_and_digest(data)\n return str(ciphertext)\n\ndef ChekCode(const):\n if(len(const) != 8):\n return False\n sum=0\n av = 0\n t = const\n m = encrypt(t)\n m = str(m)\n l = len(m)\n for i in m:\n av += ord(i)\n if(i in [\"1\",\"2\",\"3\",\"4\",\"5\",\"6\",\"7\",\"8\",\"9\"]):\n sum+=int(i)\n \n if((av+sum!=2)and(sympy.isprime(av) and sympy.isprime(sum) and m.count(\"a\") > 3) ):\n return True\n else:\n return False\n\n\ndef GenerateCodes():\n all = 0\n list_of_codes = []\n list_of_codes =set(list_of_codes)\n \n for i in itertools.combinations(chars, 8):\n counter = len(list_of_codes)\n all +=1\n txt = str(\"\".join(i))\n #if(all%100000 == 0):\n #print(counter / all)\n if(counter==1000):\n print(counter / all)\n return list_of_codes\n if(ChekCode(txt)):\n counter += 1\n list_of_codes.add(\"\".join(i))\n return list_of_codes\n\n\n\n\n\n```\n\n\n```python\na=GenerateCodes()\n```\n\n\n```python\nzz=0\nfor i in a:\n if(ChekCode(i)):\n zz+=1\nif(zz==len(a)):\n print(\"worksss\")\n```\n\n\n```python\nprint(a)\n```\n\n\n```python\nf = open(\"genarated.txt\", \"w\")\nfor i in a:\n f.write(i+\"\\n\")\nf.close()\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "e41909bf8ba2cfca480b62b7c63819f1b4ec8bcc", "size": 5564, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "caase-study/Soru1/Copy_of_testcasecaizensoru1F.ipynb", "max_stars_repo_name": "okanskaya23/Kaizen-test-case", "max_stars_repo_head_hexsha": "cd142227b15e1711adabc1f784daf7f1fdc7c590", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "caase-study/Soru1/Copy_of_testcasecaizensoru1F.ipynb", "max_issues_repo_name": "okanskaya23/Kaizen-test-case", "max_issues_repo_head_hexsha": "cd142227b15e1711adabc1f784daf7f1fdc7c590", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "caase-study/Soru1/Copy_of_testcasecaizensoru1F.ipynb", "max_forks_repo_name": "okanskaya23/Kaizen-test-case", "max_forks_repo_head_hexsha": "cd142227b15e1711adabc1f784daf7f1fdc7c590", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 29.2842105263, "max_line_length": 237, "alphanum_fraction": 0.4653127247, "converted": true, "num_tokens": 881, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799928900257127, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.4113031390291399}} {"text": "# LSTM for Part-of-Speech Tagging\n\nIn this section, we will use an LSTM to predict part-of-speech tags for words. What exactly is part-of-speech tagging?\n\nPart of speech tagging is the process of determining the *category* of a word from the words in its surrounding context. You can think of part of speech tagging as a way to go from words to their [Mad Libs](https://en.wikipedia.org/wiki/Mad_Libs#Format) categories. Mad Libs are incomplete short stories that have many words replaced by blanks. Each blank has a specified word-category, such as `\"noun\"`, `\"verb\"`, `\"adjective\"`, and so on. One player asks another to fill in these blanks (prompted only by the word-category) until they have created a complete, silly story of their own. Here is an example of such categories:\n\n```text\nToday, you'll be learning how to [verb]. It may be a [adjective] process, but I think it will be rewarding! \nIf you want to take a break you should [verb] and treat yourself to some [plural noun].\n```\n... and a set of possible words that fall into those categories:\n```text\nToday, you'll be learning how to code. It may be a challenging process, but I think it will be rewarding! \nIf you want to take a break you should stretch and treat yourself to some puppies.\n```\n\n\n### Why Tag Speech?\n\nTagging parts of speech is often used to help disambiguate natural language phrases because it can be done quickly and with high accuracy. It can help answer: what subject is someone talking about? Tagging can be used for many NLP tasks like creating new sentences using a sequence of tags that make sense together, filling in a Mad Libs style game, and determining correct pronunciation during speech synthesis. It is also used in information retrieval, and for word disambiguation (ex. determining when someone says *right* like the direction versus *right* like \"that's right!\").\n\n---\n\n\n### Preparing the Data\n\nNow, we know that neural networks do not do well with words as input and so our first step will be to prepare our training data and map each word to a numerical value. \n\nWe start by creating a small set of training data, you can see that this is a few simple sentences broken down into a list of words and their corresponding word-tags. Note that the sentences are turned into lowercase words using `lower()` and then split into separate words using `split()`, which splits the sentence by whitespace characters.\n\n#### Words to indices\n\nThen, from this training data, we create a dictionary that maps each unique word in our vocabulary to a numerical value; a unique index `idx`. We do the same for each word-tag, for example: a noun will be represented by the number `1`.\n\n\n```python\n# import resources\nimport torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport torch.optim as optim\nimport matplotlib.pyplot as plt\n\n%matplotlib inline\n```\n\n\n```python\n# training sentences and their corresponding word-tags\ntraining_data = [\n (\"The cat ate the cheese\".lower().split(), [\"DET\", \"NN\", \"V\", \"DET\", \"NN\"]),\n (\"She read that book\".lower().split(), [\"NN\", \"V\", \"DET\", \"NN\"]),\n (\"The dog loves art\".lower().split(), [\"DET\", \"NN\", \"V\", \"NN\"]),\n (\"The elephant answers the phone\".lower().split(), [\"DET\", \"NN\", \"V\", \"DET\", \"NN\"])\n]\n\n# create a dictionary that maps words to indices\nword2idx = {}\nfor sent, tags in training_data:\n for word in sent:\n if word not in word2idx:\n word2idx[word] = len(word2idx)\n\n# create a dictionary that maps tags to indices\ntag2idx = {\"DET\": 0, \"NN\": 1, \"V\": 2}\n```\n\nNext, print out the created dictionary to see the words and their numerical values! \n\nYou should see every word in our training set and its index value. Note that the word \"the\" only appears once because our vocabulary only includes *unique* words.\n\n\n```python\n# print out the created dictionary\nprint(word2idx)\n```\n\n {'the': 0, 'cat': 1, 'ate': 2, 'cheese': 3, 'she': 4, 'read': 5, 'that': 6, 'book': 7, 'dog': 8, 'loves': 9, 'art': 10, 'elephant': 11, 'answers': 12, 'phone': 13}\n\n\n\n```python\nimport numpy as np\n\n# a helper function for converting a sequence of words to a Tensor of numerical values\n# will be used later in training\ndef prepare_sequence(seq, to_idx):\n '''This function takes in a sequence of words and returns a \n corresponding Tensor of numerical values (indices for each word).'''\n idxs = [to_idx[w] for w in seq]\n idxs = np.array(idxs)\n return torch.from_numpy(idxs)\n\n```\n\n\n```python\n# check out what prepare_sequence does for one of our training sentences:\nexample_input = prepare_sequence(\"The dog answers the phone\".lower().split(), word2idx)\nprint(example_input)\n```\n\n tensor([ 0, 8, 12, 0, 13])\n\n\n---\n## Creating the Model\n\nOur model will assume a few things:\n1. Our input is broken down into a sequence of words, so a sentence will be [w1, w2, ...]\n2. These words come from a larger list of words that we already know (a vocabulary)\n3. We have a limited set of tags, `[NN, V, DET]`, which mean: a noun, a verb, and a determinant (words like \"the\" or \"that\"), respectively\n4. We want to predict\\* a tag for each input word\n\n\\* To do the prediction, we will pass an LSTM over a test sentence and apply a softmax function to the hidden state of the LSTM; the result is a vector of tag scores from which we can get the predicted tag for a word based on the *maximum* value in this distribution of tag scores. \n\nMathematically, we can represent any tag prediction $\\hat{y}_i$ as: \n\n\\begin{align}\\hat{y}_i = \\text{argmax}_j \\ (\\log \\text{Softmax}(Ah_i + b))_j\\end{align}\n\nWhere $A$ is a learned weight and $b$, a learned bias term, and the hidden state at timestep $i$ is $h_i$. \n\n\n### Word embeddings\n\nWe know that an LSTM takes in an expected input size and hidden_dim, but sentences are rarely of a consistent size, so how can we define the input of our LSTM?\n\nWell, at the very start of this net, we'll create an `Embedding` layer that takes in the size of our vocabulary and returns a vector of a specified size, `embedding_dim`, for each word in an input sequence of words. It's important that this be the first layer in this net. You can read more about this embedding layer in [the PyTorch documentation](https://pytorch.org/tutorials/beginner/nlp/word_embeddings_tutorial.html#word-embeddings-in-pytorch).\n\nPictured below is the expected architecture for this tagger model.\n\n\n\n\n\n```python\nclass LSTMTagger(nn.Module):\n\n def __init__(self, embedding_dim, hidden_dim, vocab_size, tagset_size):\n ''' Initialize the layers of this model.'''\n super(LSTMTagger, self).__init__()\n \n self.hidden_dim = hidden_dim\n\n # embedding layer that turns words into a vector of a specified size\n self.word_embeddings = nn.Embedding(vocab_size, embedding_dim)\n \n # the LSTM takes embedded word vectors (of a specified size) as inputs \n # and outputs hidden states of size hidden_dim\n self.lstm = nn.LSTM(embedding_dim, hidden_dim)\n\n # the linear layer that maps the hidden state output dimension \n # to the number of tags we want as output, tagset_size (in this case this is 3 tags)\n self.hidden2tag = nn.Linear(hidden_dim, tagset_size)\n \n # initialize the hidden state (see code below)\n self.hidden = self.init_hidden()\n\n \n def init_hidden(self):\n ''' At the start of training, we need to initialize a hidden state;\n there will be none because the hidden state is formed based on perviously seen data.\n So, this function defines a hidden state with all zeroes and of a specified size.'''\n # The axes dimensions are (n_layers, batch_size, hidden_dim)\n return (torch.zeros(1, 1, self.hidden_dim),\n torch.zeros(1, 1, self.hidden_dim))\n\n def forward(self, sentence):\n ''' Define the feedforward behavior of the model.'''\n # create embedded word vectors for each word in a sentence\n embeds = self.word_embeddings(sentence)\n \n # get the output and hidden state by passing the lstm over our word embeddings\n # the lstm takes in our embeddings and hiddent state\n lstm_out, self.hidden = self.lstm(\n embeds.view(len(sentence), 1, -1), self.hidden)\n \n # get the scores for the most likely tag for a word\n tag_outputs = self.hidden2tag(lstm_out.view(len(sentence), -1))\n tag_scores = F.log_softmax(tag_outputs, dim=1)\n \n return tag_scores\n\n```\n\n\n```python\n?nn.Embedding\n```\n\n\n```python\n\n```\n\n## Define how the model trains\n\nTo train the model, we have to instantiate it and define the loss and optimizers that we want to use.\n\nFirst, we define the size of our word embeddings. The `EMBEDDING_DIM` defines the size of our word vectors for our simple vocabulary and training set; we will keep them small so we can see how the weights change as we train.\n\n**Note: the embedding dimension for a complex dataset will usually be much larger, around 64, 128, or 256 dimensional.**\n\n\n#### Loss and Optimization\n\nSince our LSTM outputs a series of tag scores with a softmax layer, we will use `NLLLoss`. In tandem with a softmax layer, NLL Loss creates the kind of cross entropy loss that we typically use for analyzing a distribution of class scores. We'll use standard gradient descent optimization, but you are encouraged to play around with other optimizers!\n\n\n```python\n# the embedding dimension defines the size of our word vectors\n# for our simple vocabulary and training set, we will keep these small\nEMBEDDING_DIM = 8\nHIDDEN_DIM = 8\n\n# instantiate our model\nmodel = LSTMTagger(EMBEDDING_DIM, HIDDEN_DIM, len(word2idx), len(tag2idx))\n\n# define our loss and optimizer\nloss_function = nn.NLLLoss()\noptimizer = optim.SGD(model.parameters(), lr=0.1)\n\n```\n\nJust to check that our model has learned something, let's first look at the scores for a sample test sentence *before* our model is trained. Note that the test sentence *must* be made of words from our vocabulary otherwise its words cannot be turned into indices.\n\nThe scores should be Tensors of length 3 (for each of our tags) and there should be scores for each word in the input sentence.\n\nFor the test sentence, \"The cheese loves the elephant\", we know that this has the tags (DET, NN, V, DET, NN) or `[0, 1, 2, 0, 1]`, but our network does not yet know this. In fact, in this case, our model starts out with a hidden state of all zeroes and so all the scores and the predicted tags should be low, random, and about what you'd expect for a network that is not yet trained!\n\n\n```python\ntest_sentence = \"The cheese loves the elephant\".lower().split()\n\n# see what the scores are before training\n# element [i,j] of the output is the *score* for tag j for word i.\n# to check the initial accuracy of our model, we don't need to train, so we use model.eval()\ninputs = prepare_sequence(test_sentence, word2idx)\ninputs = inputs\ntag_scores = model(inputs)\nprint(tag_scores)\n\n# tag_scores outputs a vector of tag scores for each word in an inpit sentence\n# to get the most likely tag index, we grab the index with the maximum score!\n# recall that these numbers correspond to tag2idx = {\"DET\": 0, \"NN\": 1, \"V\": 2}\n_, predicted_tags = torch.max(tag_scores, 1)\nprint('\\n')\nprint('Predicted tags: \\n',predicted_tags)\n```\n\n tensor([[-1.0379, -1.2264, -1.0429],\n [-1.1214, -1.2944, -0.9160],\n [-1.1577, -1.1224, -1.0209],\n [-1.0650, -1.2073, -1.0321],\n [-1.0938, -1.1371, -1.0662]], grad_fn=)\n \n \n Predicted tags: \n tensor([0, 2, 2, 2, 2])\n\n\n---\n## Train the Model\n\nLoop through all our training data for multiple epochs (again we are using a small epoch value for this simple training data). This loop:\n\n1. Prepares our model for training by zero-ing the gradients\n2. Initializes the hidden state of our LSTM\n3. Prepares our data for training\n4. Runs a forward pass on our inputs to get tag_scores\n5. Calculates the loss between tag_scores and the true tag\n6. Updates the weights of our model using backpropagation\n\nIn this example, we are printing out the average epoch loss, every 20 epochs; you should see it decrease over time.\n\n\n```python\n# normally these epochs take a lot longer \n# but with our toy data (only 3 sentences), we can do many epochs in a short time\nn_epochs = 300\n\nfor epoch in range(n_epochs):\n \n epoch_loss = 0.0\n \n # get all sentences and corresponding tags in the training data\n for sentence, tags in training_data:\n \n # zero the gradients\n model.zero_grad()\n\n # zero the hidden state of the LSTM, this detaches it from its history\n model.hidden = model.init_hidden()\n\n # prepare the inputs for processing by out network, \n # turn all sentences and targets into Tensors of numerical indices\n sentence_in = prepare_sequence(sentence, word2idx)\n targets = prepare_sequence(tags, tag2idx)\n\n # forward pass to get tag scores\n tag_scores = model(sentence_in)\n\n # compute the loss, and gradients \n loss = loss_function(tag_scores, targets)\n epoch_loss += loss.item()\n loss.backward()\n \n # update the model parameters with optimizer.step()\n optimizer.step()\n \n # print out avg loss per 20 epochs\n if(epoch%20 == 19):\n print(\"Epoch: %d, loss: %1.5f\" % (epoch+1, epoch_loss/len(training_data)))\n\n```\n\n Epoch: 20, loss: 0.92172\n Epoch: 40, loss: 0.60580\n Epoch: 60, loss: 0.30329\n Epoch: 80, loss: 0.15031\n Epoch: 100, loss: 0.08358\n Epoch: 120, loss: 0.05338\n Epoch: 140, loss: 0.03819\n Epoch: 160, loss: 0.02937\n Epoch: 180, loss: 0.02370\n Epoch: 200, loss: 0.01977\n Epoch: 220, loss: 0.01691\n Epoch: 240, loss: 0.01474\n Epoch: 260, loss: 0.01304\n Epoch: 280, loss: 0.01167\n Epoch: 300, loss: 0.01055\n\n\n## Testing\n\nSee how your model performs *after* training. Compare this output with the scores from before training, above.\n\nAgain, for the test sentence, \"The cheese loves the elephant\", we know that this has the tags (DET, NN, V, DET, NN) or `[0, 1, 2, 0, 1]`. Let's see if our model has learned to find these tags!\n\n\n```python\ntest_sentence = \"The cheese loves the elephant\".lower().split()\n\n# see what the scores are after training\ninputs = prepare_sequence(test_sentence, word2idx)\ninputs = inputs\ntag_scores = model(inputs)\nprint(tag_scores)\n\n# print the most likely tag index, by grabbing the index with the maximum score!\n# recall that these numbers correspond to tag2idx = {\"DET\": 0, \"NN\": 1, \"V\": 2}\n_, predicted_tags = torch.max(tag_scores, 1)\nprint('\\n')\nprint('Predicted tags: \\n',predicted_tags)\n```\n\n tensor([[-1.0144e-02, -4.7545e+00, -6.5156e+00],\n [-6.5128e+00, -3.6229e-03, -6.1507e+00],\n [-5.7125e+00, -3.8703e+00, -2.4453e-02],\n [-1.3425e-02, -4.7353e+00, -5.3914e+00],\n [-6.5335e+00, -8.8856e-03, -4.9073e+00]], grad_fn=)\n \n \n Predicted tags: \n tensor([0, 1, 2, 0, 1])\n\n\n## Great job!\n\nTo improve this model, see if you can add sentences to this model and create a more robust speech tagger. Try to initialize the hidden state in a different way or play around with the optimizers and see if you can decrease model loss even faster.\n\n\n```python\n\n```\n", "meta": {"hexsha": "216c32e278be127f770fbf2a82ece9e6602adda0", "size": 20801, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "CVND_Exercises/2_4_LSTMs/2. LSTM Training, Part of Speech Tagging.ipynb", "max_stars_repo_name": "nz-is/Vision_ND", "max_stars_repo_head_hexsha": "cb79b2cdf28962c9410e7eb2e7fdc8e55691a20b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2019-11-28T10:38:10.000Z", "max_stars_repo_stars_event_max_datetime": "2019-12-15T03:43:30.000Z", "max_issues_repo_path": "CVND_Exercises/2_4_LSTMs/2. 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YES", "lm_q1_score": 0.7217432182679956, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.41128728657020397}} {"text": "# Introduction\n\nWelcome to the Scientific Python tutorial! The goal is to present to you the set of simple tools that might be of use in research or education. This tutorial is not meant to be a comprehensive introduction to the language or the plethora of scientific libraries. The only goal is to show that Python is useful and easy to learn. In place of usual passage from basics to advanced features of the language, we are going to use a problem-based approach. By that, I mean that first, we formulate a (scientific) problem and then try to solve it using Python. Appropriate syntax, tricks, and libraries appear along the way.\n\n___\n___\n___\n\n# Shotgun sequencing\n\n*Links:*\n + [wikipedia](https://en.wikipedia.org/wiki/Shotgun_sequencing)\n + [towardsdatascience](https://towardsdatascience.com/dna-sequence-data-analysis-starting-off-in-bioinformatics-3dba4cea04f)\n\nThe first problem is a toy version of DNA sequence reconstruction from the set of overlapping chunks. When one tries to extract the sequence, what she gets is the collection of fragments like that\n\n```\nAGCTCTTTGA\n TTGACCGTAATGA\n ATGATTACA\n ACAAAAT\n```\n\nThe resulting string is\n\n```\nAGCTCTTTGACCGTAATGATTACAAAAT\n```\n\nTo make the problem manageable, we formulate it as follows.\n\n---\n\n\n**Shotgun sequencing**\n\nGiven a set of strings and an estimation of minimum overlap,\none needs to restore a unique sequence that produced the set.\n\n---\n\nThe problem is complicated, so we decompose it on the smaller ones.\n\n## Subproblem 1\n\n*Given two strings `s1` and `s2`, find the first position starting from which the end of the string `s1` coincides with the beginning of the string `s2`.*\n\nFor example if\n\n```\ns1 = 'GCTATATAC'\ns2 = 'TATACGTCTG'\n```\n\nit is easy to see that the answer is $4$:\n\n```\n012345678\nGCTATATAC\n TATACGTCTG\n 4\n```\n\nTo solve this problem, we need a basic python syntax, which we now introduce.\n\n---\n\n## Python: variables (int, double, bool, string), loops (for, while), boolean operations, conditions, list and methods (pop, append, insert, extend)\n\n\n```\n# # represents a comment\n# Shift+Enter executes a cell\n\n\n# some of build-in types below:\n# integer\n# double\n# complex (1j means imaginary unit)\n# string\n# string\n# bool\n# bool\n\na = 4\nb = 6.6\nc = 12 + 1j*6\ns1 = 'GCTATATAC'\ns2 = 'TATACGTCTG'\nT = True\nF = False\n\n# print \n\nH = 'Hello!'\nprint(H)\n```\n\n Hello!\n\n\n\n```\n# access a particular element of a string\n\nprint(s1[0])\nprint(s1[1])\nprint(s1[5])\nprint(s1[-1]) # counting from the other side \nprint(s1[-2])\nprint(s1[-5])\n\n# access an interval starting from a (included) to b (excluded) s[a:b]\nprint(s1[0:5])\nprint(s1[:5])\nprint(s2[2], s2[2:6], s2[6])\n\n# find a length of the string\n\nprint(len(s1), len(s2))\n```\n\n G\n C\n A\n C\n A\n T\n GCTAT\n GCTAT\n T TACG T\n 9 10\n\n\n\n```\n# for loop\n# it is possible to iterate over special variables\n# among them is range(start, stop, step)\n\nstart = 10\nstop = 20\nstep = 2\nfor i in range(start, stop, step): # the interval is [start, stop) (right end is excluded)\n print(i)\n\nprint('\\n') # new line\n\nfor i in range(1, 11):\n print(i)\n```\n\n 10\n 12\n 14\n 16\n 18\n \n \n 1\n 2\n 3\n 4\n 5\n 6\n 7\n 8\n 9\n 10\n\n\n\n```\n# for loop\n# runs over the string too\n\nfor s in s1:\n print(s)\n\nprint('\\n')\n\nfor i in range(len(s1)):\n print(s1[i])\n```\n\n G\n C\n T\n A\n T\n A\n T\n A\n C\n \n \n G\n C\n T\n A\n T\n A\n T\n A\n C\n\n\nBoolean opeartions in Python\n\n|Operation|Python syntax|\n|:-------:|:-----------:|\n|$\\leq$|`<=`|\n|$\\geq$|`>=`|\n|$=$|`==`|\n|$\\neq$|`!=`|\n|conjunction|`and`/`*`|\n|disjunction|`or`|\n\n\n```\n# Boolean opeartions\n\na = 14\nb = -13\nc = 22.7\nd = 'op'\ne = 'oqop'\nf = 'oqopop'\n\nprint('a < 22?', a < 22)\nprint('a < 10?', a < 10)\nprint('a > b?', a > b)\nprint('a*b + a^2 >= a - b?', a*b + a**2 >= a - b) # a**b means a^b\nprint('12 < a <= 14?', 12 < a <= 14)\nprint(d == e)\nprint(d == e[:2])\nprint(d == e[2:])\nprint(f == e + d) # you can add stings\n```\n\n a < 22? True\n a < 10? False\n a > b? True\n a*b + a^2 >= a - b? False\n 12 < a <= 14? True\n False\n False\n True\n True\n\n\n\n```\n# conjunction and disjunction\n\nprint(1 == 1 and 3>2)\nprint(1 == 1 and 3<2)\nprint(1 == 1 or 3<2)\n```\n\n True\n False\n True\n\n\n\n```\n# while loop\n\na = 14\nprint(a)\n\nwhile a<22:\n a += 1\n print(a)\n\nprint('\\n')\n\ns = 'wo'\ntarget = 'wooooooooooooooooooo'\np = 0\n\nwhile s != target:\n s += 'o'\n p += 1\n\nprint('We are out of the while loop. Total number of iterations is', p)\nprint('\\nVariable s is', s)\nprint('Is s equals target?', s==target)\n```\n\n 14\n 15\n 16\n 17\n 18\n 19\n 20\n 21\n 22\n \n \n We are out of the while loop. Total number of iterations is 18\n \n Variable s is wooooooooooooooooooo\n Is s equals target? True\n\n\n\n```\n# while loop supports multiple conditions\n\ns = 14\nb = -12\n\nprint('s', 'b')\nwhile bb:\n print(\"a>b\")\nelse:\n print(\"a<=b\")\n\nif a>b:\n print(\"a>b\")\nelif a==b:\n print(\"a=b\")\nelse:\n print(\"amin_overlap`, it returns `True` and concatenated string.\n+ In case there is an overlap, and one string is a substring of another, it returns `True` and the largest string regardless of the size of the overlap compare to `min_overlap`.\n+ In case there is no overlap, it returns `False` and `'no overlap'` string.\n\nRemember that previously we cared about the order, i.e., the functions worked like that\n\n```\ns1 = 'GTCA'\ns2 = 'CA'\n\ndetect_overlap(s1, s2) -> 2\ndetect_overlap(s2, s1) -> 3/4\n```\n\nNow we do not care about the order. The function should concatenate strings anyway.\n\n\n```\n# hint\n\ndef divide(a, b):\n return a/b, b/a\n\nprint(divide(4, 5))\n```\n\n (0.8, 1.25)\n\n\n\n```\n### your solution is here\n\ndef merge_me(s1, s2, min_overlap=1):\n N = detect_overlap(s1, s2)\n if N == 3/4:\n return False, s1\n else:\n if len(s2) <= len(s1) - N:\n return True, s1\n elif len(s1) - N < min_overlap:\n return False, s1\n else:\n return True, s1[:N]+s2 \n\ndef merge_me_if_you_can(s1, s2, min_overlap=1):\n merged1, target = merge_me(s1, s2, min_overlap=min_overlap)\n merged2 = False\n if not merged1:\n merged2, target = merge_me(s2, s1, min_overlap=min_overlap)\n if merged1 or merged2:\n return merged1 or merged2, target\n else:\n return merged1, 'no overlap'\n\n###\n```\n\n\n```\n# tests\n\ns1 = 'GTTGACCA'\ns2 = 'TGA'\n\n### your solution is here\nprint(merge_me_if_you_can(s1, s2, min_overlap=100)) # min_overlap = 100\n###\n\nprint('\\n')\n\ns1 = 'GTTGACCA'\ns2 = 'ACCAGTTGACC'\n\n### your solution is here\nprint(merge_me_if_you_can(s1, s2, min_overlap=4)) # min_overlap = 4\nprint(merge_me_if_you_can(s1, s2, min_overlap=5)) # min_overlap = 5\nprint(merge_me_if_you_can(s1, s2, min_overlap=8)) # min_overlap = 8\n###\n\nprint('\\n')\n\ns1 = 'GTTGACCA'\ns2 = 'CCATTTTTTT'\n\n### your solution is here\nprint(merge_me_if_you_can(s1, s2, min_overlap=3)) # min_overlap = 3\nprint(merge_me_if_you_can(s1, s2, min_overlap=4)) # min_overlap = 4\n###\n```\n\n (True, 'GTTGACCA')\n \n \n (True, 'GTTGACCAGTTGACC')\n (True, 'ACCAGTTGACCA')\n (False, 'no overlap')\n \n \n (True, 'GTTGACCATTTTTTT')\n (False, 'no overlap')\n\n\n---\n\nNow we are ready to implement the function that glue short chunks of DNA into the larger strand.\n\n## Shotgun sequencing: solution\n\nSuppose the variable `chopped_DNA` contains a set of substrings.\n\n```\nchopped_DNA = ['AGCTTCC', 'CCTGA', ...]\n```\n\nThe strategy is to define a `target` variable and a `mask`. The former contains a concatenated string. The later represents yet unused elements. We move along the `chopped_DNA,` using only elements with the nonzero `mask`, and trying to concatenate the target with each component. In case we succeed, the value of the `mask` becomes `0`, and the `target` becomes the concatenated string. We stop either when the mask consists of zeros or when the number of iteration is too large, in case it is impossible to obtain a single string from available pieces.\n\n\n```\ndef shotgun_DNA(chopped_DNA, rounds=100, min_overlap=1):\n target = chopped_DNA[0]\n mask = [1 for i in range(len(chopped_DNA))]\n mask[0] = 0\n N = 1\n while N0:\n N += 1\n for i in range(len(chopped_DNA)):\n if mask[i] == 1:\n merged, s1 = merge_me_if_you_can(target, chopped_DNA[i], min_overlap=min_overlap)\n if merged:\n target = s1\n mask[i] = 0\n return target\n```\n\n\n```\nsequence = 'GTCTGACACCCTAGATCGATGTACCAGTGACCACGTATAGACACAGATAGACACACGTATTGGTGTCCCGTGCGT'\nchopped_DNA = [sequence[i:j] for i, j in zip([0, 4, 13, 30, 50], [10, 21, 39, 61, 75])]\n\nfor i, j in zip([2, 3, 0, 2], [1, 2, 3, 1]):\n dummy = chopped_DNA[i]\n chopped_DNA[i] = chopped_DNA[j]\n chopped_DNA[j] = dummy\n\nfor chunk in chopped_DNA:\n print(chunk)\n```\n\n GACACCCTAGATCGATG\n CCACGTATAGACACAGATAGACACACGTATT\n GATCGATGTACCAGTGACCACGTATA\n GTCTGACACC\n ACACACGTATTGGTGTCCCGTGCGT\n\n\n\n```\nfor overlap in range(1, 10):\n reconstructed_straid = shotgun_DNA(chopped_DNA, rounds=100, min_overlap=overlap)\n print('Minimal overlap =', overlap, 'Reconstructed = Real?', sequence == reconstructed_straid)\n```\n\n Minimal overlap = 1 Reconstructed = Real? False\n Minimal overlap = 2 Reconstructed = Real? False\n Minimal overlap = 3 Reconstructed = Real? True\n Minimal overlap = 4 Reconstructed = Real? True\n Minimal overlap = 5 Reconstructed = Real? True\n Minimal overlap = 6 Reconstructed = Real? True\n Minimal overlap = 7 Reconstructed = Real? False\n Minimal overlap = 8 Reconstructed = Real? False\n Minimal overlap = 9 Reconstructed = Real? False\n\n\n---\n---\n---\n\n# [Ballooning spiders](https://arxiv.org/abs/1309.4731?context=physics)\n\nPlease find a link above on the arXiv paper about Grossamer spiders. [It turns out that spiders can fly using electric field of the Earth atmosphere](https://www.theatlantic.com/science/archive/2018/07/the-electric-flight-of-spiders/564437/). What you need howerver is to reproduce the figure (FIG. 2) from [the article](https://arxiv.org/abs/1309.4731?context=physics).\n\nAuthor estimates the overall charge from the balance of electrastatic and gravitation forces ($H_{eq}$ is an equilibrium height):\n\n\\begin{equation}\n Q = \\frac{mg}{E_0}e^{\\alpha H_{eq}},\n\\end{equation}\n\nwith\n\n\\begin{equation}\n \\begin{split}\n E_0 &= 120 \\frac{V}{m},\\\\\n \\alpha &= 3\\times 10^{-4}\\frac{1}{m},\\\\\n g &= 9.8 \\frac{m}{s^2}.\n \\end{split}\n\\end{equation}\n\nYou goal is to produce comparable figure, i.e. the dependence between equilibrium altitude ($x$) and total charge in nano Coulombs ($y$) for masses $0.1$, $0.3$, $1$, $3$, $10$, $30$, $100$ mg on the same plot. It would be nice to reproduce logarithmic grid too.\n\n---\n\nHopefully, the goal is clear, but we need to store data, compute exponentials, and produce plots. To achieve that, we now introduce a `numpy`, `matplotlib`, and new Python syntax.\n\n## Python: import, namespaces, IPython magic, numpy and matplotlib, f-strings\n\nPython is popular in part because it comes with an extensive set of libraries for any imaginable problem. To use the library, you need to download it, install and show Python where it stored. Usually, everything works in a more or less automatic fashion. Libraries that we are going to use now are already available, so the only thing we need is to import them.\n\nBelow is a simple import statement.\n\n\n```\n# import library using namespace\n\nimport numpy as np\n```\n\nAfter that, one can call functions from the library using a namespace, as we show in the next line.\n\n\n```\n# this function create an array (resembles list [a, b, c, d]) of a given size which contains 1\n\nN = 10\n\na = np.ones(N)\nprint(a)\nprint('\\n')\n\n# arrays can be of more peculiar shape\nM = 3\nK = 2\n\nb = np.ones((N, M))\nprint(b)\nprint('\\n')\n\nc = np.ones((N, M, K))\nprint(c)\n```\n\n [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]\n \n \n [[1. 1. 1.]\n [1. 1. 1.]\n [1. 1. 1.]\n [1. 1. 1.]\n [1. 1. 1.]\n [1. 1. 1.]\n [1. 1. 1.]\n [1. 1. 1.]\n [1. 1. 1.]\n [1. 1. 1.]]\n \n \n [[[1. 1.]\n [1. 1.]\n [1. 1.]]\n \n [[1. 1.]\n [1. 1.]\n [1. 1.]]\n \n [[1. 1.]\n [1. 1.]\n [1. 1.]]\n \n [[1. 1.]\n [1. 1.]\n [1. 1.]]\n \n [[1. 1.]\n [1. 1.]\n [1. 1.]]\n \n [[1. 1.]\n [1. 1.]\n [1. 1.]]\n \n [[1. 1.]\n [1. 1.]\n [1. 1.]]\n \n [[1. 1.]\n [1. 1.]\n [1. 1.]]\n \n [[1. 1.]\n [1. 1.]\n [1. 1.]]\n \n [[1. 1.]\n [1. 1.]\n [1. 1.]]]\n\n\n\n```\n# shape function allows you to know dimensionality\n\nprint(a.shape)\n\nprint(b.shape)\n\nprint(c.shape)\n```\n\n (10,)\n (10, 3)\n (10, 3, 2)\n\n\nThere are a lot of functions available for numpy array creation and manipulation. Arguably the best way to learn them is to solve your problems using help from [Google](https://www.google.com/?client=safari), [StackExchange](https://stackexchange.com), and [Documentation](https://numpy.org/doc/1.17/).\n\n\n```\n# for example there is a function that perform summation along a given axis\n\na = np.array([\n [[1, 1], [2, 2], [3, 3], [4, 4]],\n [[4, 4], [5, 5], [6, 6], [7, 7]],\n [[7, 7], [8, 8], [9, 9], [10, 10]]\n ])\n\nprint(np.sum(a, axis=0), '\\n')\nprint(np.sum(a, axis=1), '\\n')\nprint(np.sum(a, axis=2), '\\n')\n```\n\n [[12 12]\n [15 15]\n [18 18]\n [21 21]] \n \n [[10 10]\n [22 22]\n [34 34]] \n \n [[ 2 4 6 8]\n [ 8 10 12 14]\n [14 16 18 20]] \n \n\n\nYou can add arrays and apply functions from the numpy library to them (in case the sizes are comparable).\n\n\n```\n# below we create a set of points with uniform distances between them\n\nx = np.linspace(0, 1, 10)\nprint(x, '\\n')\n\n# compute sin and cos\nf1 = np.cos(np.pi*x)\nprint(f1, '\\n')\nf2 = np.sin(np.pi*x)\nprint(f2, '\\n')\n\n# and check the trigonometric identity\ng = f1**2 + f2**2\nprint(g)\n\n# np.pi is a variables stored in numpy library that contains a value of pi\nprint(np.pi)\n```\n\n [0. 0.11111111 0.22222222 0.33333333 0.44444444 0.55555556\n 0.66666667 0.77777778 0.88888889 1. ] \n \n [ 1. 0.93969262 0.76604444 0.5 0.17364818 -0.17364818\n -0.5 -0.76604444 -0.93969262 -1. ] \n \n [0.00000000e+00 3.42020143e-01 6.42787610e-01 8.66025404e-01\n 9.84807753e-01 9.84807753e-01 8.66025404e-01 6.42787610e-01\n 3.42020143e-01 1.22464680e-16] \n \n [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]\n 3.141592653589793\n\n\n`Numpy` operates much faster then build-in python function in case an array is large enough. To see that, we create an array and a list and then measure the time of a sum function.\n\n\n```\nM = int(1e8) # scientific notation for 100000000 produce a float 100000000.0, we convert it to int\na = np.ones(M)\nb = [1 for i in range(M)]\n```\n\n\n```\n%timeit sum(b)\n```\n\n 1 loop, best of 3: 692 ms per loop\n\n\n\n```\n%timeit np.sum(a)\n```\n\n 10 loops, best of 3: 128 ms per loop\n\n\nLets also measure the time for the loop.\n\n\n```\nK = int(1e3)\na = np.ones((K, K))\nb = [[1]*K for i in range(K)]\n```\n\n\n```\n%%timeit\n\nres = 0\nfor i in range(K):\n for j in range(K):\n res += a[i][j]\n```\n\n 1 loop, best of 3: 301 ms per loop\n\n\n\n```\n%%timeit\n\nnp.sum(a)\n```\n\n The slowest run took 4.66 times longer than the fastest. This could mean that an intermediate result is being cached.\n 1000 loops, best of 3: 358 \u00b5s per loop\n\n\nAs you see, with the growth of dimensionality of the data, the difference between `Numpy` and pure Python becames more and more relevant.\n\nThere are other libraries with fast containers:\n\n+ [xarray](http://xarray.pydata.org/en/stable/)\n+ [TensorFlow](https://www.tensorflow.org)\n+ [Pytorch](https://pytorch.org)\n+ [Jax](https://jax.readthedocs.io/en/latest/)\n+ [Pandas](https://pandas.pydata.org)\n\nNow we turn to the visualization with matplotlib. [The library](https://matplotlib.org) is vast and complicated, but routine tasks are easy to perform.\n\n\n```\n# the dot below represents a directory\n# so the structure of the package is matplotlib/pyplot.py\n\nimport matplotlib.pyplot as plt\n\n# magic command that shows us all plots right here\n%matplotlib inline\n```\n\nNow, we are ready to produce the first plot.\n\n\n```\n# create relevant objects and fix the size of the figure\nfig, ax = plt.subplots(1, 1, figsize = (8, 6))\n\n# create data for the plot\nx = np.linspace(0, 1, 100)\nf1 = np.cos(2*np.pi*x)\nf2 = np.sin(4*np.pi*x)\n\n# draw\nax.plot(x, f1, color='black', dashes=[8, 4, 4, 1], label=r'$\\cos(2\\pi x)$') # r is for raw\nax.plot(x, f2, color='blue', label= '$\\\\sin(4\\\\pi x)$') # without r you should use \\\\ to represent \\\nax.set_title('Basic figure')\nax.legend(loc=3);\n```\n\nExample with more figures stacked together.\n\n\n```\n# create four figures\nfig, ax = plt.subplots(2, 2, figsize = (16, 12))\n\nfig.suptitle('Subplots')\n\nx = np.linspace(0, 1, 100)\nfor i in range(2):\n for j in range(2):\n f1 = np.cos((i+3)*np.pi*x)\n f2 = np.sin((j+2)*np.pi*x)\n\n # f if for format\n ax[i, j].plot(x, f1, color=[i*0.3 + 0.1, j*0.2 + 0.4, i], dashes=[8, 4, 4, 1], label=fr'$\\cos({i+3}\\pi x)$')\n ax[i, j].plot(x, f2, color=[j*0.5 + 0.4, i*0.3 + 0.1, i], label=fr'$\\sin({j+2}\\pi x)$')\n ax[i, j].set_title(f'Position [{i}, {j}]')\n ax[i, j].legend()\n```\n\nSome comments about `f-strings`.\n\n\n```\n# that is the basic syntax\n\nN, M = 3, 4\n\ns1 = f'Mark has {N} apples.\\n'\ns2 = f'Kate has {M} apples.\\n'\ns3 = f'Together they have {N+M} apples.'\n\nprint(s1 + s2 + s3)\n```\n\n Mark has 3 apples.\n Kate has 4 apples.\n Together they have 7 apples.\n\n\n\n```\n# you can put string in this curly brackets\n\nName1 = 'Ann'\nName2 = 'John'\n\ns1 = f'{Name1} wakes up early.\\n'\ns2 = f'{Name2} sleeps all day long.'\n\nprint(s1+s2)\n```\n\n Ann wakes up early.\n John sleeps all day long.\n\n\n\n```\n# or a double\n\npi = 3.1416926\n\ns = f'Pi up to three significant digits {pi:.3}, and up to six {pi:.6}.'\n\nprint(s)\n```\n\n Pi up to three significant digits 3.14, and up to six 3.14169.\n\n\nThere are other libraries for visualization:\n\n+ [seaborn](https://seaborn.pydata.org)\n+ [bokeh](https://bokeh.pydata.org/en/latest/index.html)\n+ [plotly](https://plot.ly/python/)\n+ also you can read this post in [towardsdatascience](https://towardsdatascience.com/interactive-visualizations-in-jupyter-notebook-3be02ab2b8cd) to producce **interactive plots**\n\n\n\n---\n\n## Ballooning spiders: solution\n\nThe text of the problem for your convenience is below.\n\nPlease find a link above on the arXiv paper about Grossamer spiders. [It turns out that spiders can fly using electric field of the Earth atmosphere](https://www.theatlantic.com/science/archive/2018/07/the-electric-flight-of-spiders/564437/). What you need howerver is to reproduce the figure (FIG. 2) from [the article](https://arxiv.org/abs/1309.4731?context=physics).\n\nAuthor estimates the overall charge from the balance of electrastatic and gravitation forces ($H_{eq}$ is an equilibrium height):\n\n\\begin{equation}\n Q = \\frac{mg}{E_0}e^{\\alpha H_{eq}},\n\\end{equation}\n\nwith\n\n\\begin{equation}\n \\begin{split}\n E_0 &= 120 \\frac{V}{m},\\\\\n \\alpha &= 3\\times 10^{-4}\\frac{1}{m},\\\\\n g &= 9.8 \\frac{m}{s^2}.\n \\end{split}\n\\end{equation}\n\nYou goal is to produce comparable figure, i.e. the dependence between equilibrium altitude ($x$) and total charge in nano Coulombs ($y$) for masses $0.1$, $0.3$, $1$, $3$, $10$, $30$, $100$ mg on the same plot. It would be nice to reproduce logarithmic grid too.\n\nRelevant links\n\n+ [logspace](https://docs.scipy.org/doc/numpy/reference/generated/numpy.logspace.html)\n\n+ [logscale](https://matplotlib.org/3.1.1/api/_as_gen/matplotlib.pyplot.yscale.html)\n\n+ [grid](https://matplotlib.org/3.1.1/api/_as_gen/matplotlib.pyplot.grid.html)\n\n+ [text](https://matplotlib.org/3.1.1/api/_as_gen/matplotlib.pyplot.text.html)\n\n+ [axes properties](https://matplotlib.org/3.1.1/api/axes_api.html)\n\nFor a start, you may want to produce a single curve for the mass $0.1\\text{ mg}$ on the log-log plot,\n\n\n```\n### your solution is here\n\nH = np.logspace(0, 3.5, 100)\nm = 0.1\ng = 9.8\nE = 120\nalpha = 3*1e-4\n\nq = 1e3*0.1*g*np.exp(alpha*H)/E\n\nplt.figure(figsize=(10, 7))\n\nplt.plot(H, q, label=f'm = {m}mg')\nplt.xscale('log')\nplt.yscale('log')\nplt.grid(which='both', linestyle='--', linewidth=0.5)\nplt.xlabel('equilibrium altitude, m')\nplt.ylabel('total charge, nC')\nplt.text(10**2*0.3, q[50]*1.1, f'{m} mg');\n\n###\n```\n\nand then implement a general solution.\n\n\n```\n### your solution is here\n\nH = np.logspace(0, 3.5, 100)\nM = [0.1, 0.3, 1, 3, 10, 30, 100]\ng = 9.8\nE = 120\nalpha = 3*1e-4\nQ = []\nfor m in M:\n x = 1e3*m*g*np.exp(alpha*H)/E\n Q.append(x)\n\nplt.figure(figsize=(10, 7))\n\nfor q, m in zip(Q, M):\n plt.plot(H, q, label=f'm = {m}mg')\n plt.xscale('log')\n plt.yscale('log')\n plt.grid(which='both', linestyle='--', linewidth=0.5)\n plt.xlabel('equilibrium altitude, m')\n plt.ylabel('total charge, nC')\n plt.ylim(bottom=10**0.5, top=10**(4.5))\n plt.text(10**2*0.3, q[50]*0.7, f'{m} mg')\n\n###\n```\n\n---\n---\n---\n\n# Chemical bonds\n\nThis problem is a simple illustration of the concept of the chemical bond. Schr\u00f6dinger equation does not contain free parameters, so it defines all chemistry in closed form. Too bad, we can not solve this equation for most physical systems, neither analytically nor numerically.\n\nHere we consider a one-dimensional case of stationary Schr\u00f6dinger for the toy potential with two minima.\n\n\\begin{equation}\n \\begin{split}\n &\\left(-\\frac{d^2}{d\\widetilde{x}^2} + \\widetilde{V}(\\widetilde{x})\\right) \\psi(\\widetilde{x}) = \\widetilde{E} \\psi(\\widetilde{x}),~\\left\\|\\psi\\right\\|_{2}<\\infty;\\\\\n &\\widetilde{V}(\\widetilde{x}) = \\frac{\\alpha l_0^4 \\left(\\widetilde{x}-a\\big/ l_0\\right)^2\\left(\\widetilde{x}+a\\big/ l_0\\right)^2 - V_0}{E_0},~\\widetilde{x} = \\frac{x}{l_0},~\\widetilde{E}=\\frac{E}{E_{0}},~E_0 = \\frac{\\hbar^2}{2m l_{0}^2}.\n \\end{split}\n\\end{equation}\n\nAlternatively, in the usual fashion, we proclaim that energy is measured in the units of $E_0$, and the length is measured in the units of $l_0$. With that we rewrite\n\n\\begin{equation}\n \\begin{split}\n &\\left(-\\frac{d^2}{dx^2} + V(x)\\right) \\psi(x) = E \\psi(x),~\\left\\|\\psi\\right\\|_{2}<\\infty;\\\\\n &V(x) = \\alpha\\left(x^2-a^2\\right)^2 - V_0.\n \\end{split}\n\\end{equation}\n\n**We want to investigate the dependence of the ground state energy from a distance between potential wells.**\n\nIn our setting, potential wells represent the attractive potential of the protons in nuclei. The particle under consideration is an electron that forms (or does not from) a bond.\n\nTo solve this problem, we use finite differences approximation for the otherwise intractable differential operator\n\n\\begin{equation}\n -\\frac{d^2}{dx^2}\\psi(x) \\sim \\frac{2\\psi(x_i) - \\psi(x_{i+1}) - \\psi(x_{i-1})}{\\Delta x^2},\n\\end{equation}\n\nand for potential energy we have\n\n\\begin{equation}\n V(x)\\psi(x) \\sim V(x_i)\\psi(x_i).\n\\end{equation}\n\nWith that, we need to solve the usual eigenvalue problem\n\n\\begin{equation}\n Ax_i = \\lambda_i x_i.\n\\end{equation}\n\n---\n\n## Subproblem 1\n\nFor a start, I ask you to plot the potential for different values of parameters:\n\n+ $a \\in \\left[0, 6\\right]$ four points in linear scale, $\\alpha = 0.001$, $V_0 = 40$ \n+ $a = 5$, $\\alpha = \\left[-6, -4\\right]$ four points in $\\log$-scale, $V_0 = 40$ \n+ $a = 5$, $\\alpha = 0.001$, $V_0 \\in \\left[35, 45\\right]$ four points in linear scale \n\nUse [this function](https://matplotlib.org/3.1.1/api/_as_gen/matplotlib.pyplot.subplots_adjust.html) if you need to adjust subplots.\n\nTo remove unnecessary subplots, consider using `fig.delaxes()`.\n\n\n```\nx = np.linspace(-8, 8, 100)\n\n### your solution is here\n\nV = lambda x, a, alpha, V_0: alpha*(x**2 - a**2)**2 - V_0\n\nfig, ax = plt.subplots(2, 2, figsize=(12, 8))\nfig.delaxes(ax[1, 1])\nplt.subplots_adjust(wspace=0.4, hspace=0.4)\n\nfor a in np.linspace(0, 6, 4):\n ax[0, 0].plot(x, V(x, a, 0.001, 40), label=f'a = {a}')\n ax[0, 0].set_title(r'$\\alpha = 0.001$, $V_0 = 40$')\n ax[0, 0].set_xlabel('$x$')\n ax[0, 0].set_ylabel('$V(x)$')\n ax[0, 0].legend()\n\nfor alpha in np.logspace(-6, -4, 4):\n ax[0, 1].plot(x, V(x, 5, alpha, 40), label=fr'$\\alpha = {alpha:.2}$')\n ax[0, 1].set_title(r'$a = 5$, $V_0 = 40$')\n ax[0, 1].set_xlabel('$x$')\n ax[0, 1].set_ylabel('$V(x)$')\n ax[0, 1].legend()\n\nfor V_0 in np.linspace(35, 45, 4):\n ax[1, 0].plot(x, V(x, 5, 0.001, V_0), label=fr'$V_0 = {V_0:.2}$')\n ax[1, 0].set_title(r'$a = 5$, $\\alpha = 0.001$')\n ax[1, 0].set_xlabel('$x$')\n ax[1, 0].set_ylabel('$V(x)$')\n ax[1, 0].legend()\n\n###\n```\n\n---\n\n## Subproblem 2\n\nNow we want to build a matrix of the linear operator. Write the function that takes the number of points, parameters of the potential, and return the matrix of the linear operator with the vector of coordinates.\n\nDefault values of parameters\n\n\\begin{equation}\n a = 5,~\\alpha = 0.001,~V_0 = 40.\n\\end{equation}\n\nMinimal and maximal value of $x$ are $\\pm8$.\n\nThis function can be of interest:\n\n+ [eye](https://docs.scipy.org/doc/numpy/reference/generated/numpy.eye.html)\n\n\n```\ndef hamiltonian(N, a=5, alpha=1e-3, V_0=40):\n ### your solution is here\n x = np.linspace(-8, 8, N)\n delta_x = x[1] - x[0]\n\n V = alpha*(x**2-a**2)**2 - V_0\n A = (2*np.eye(N) - np.eye(N, k=-1) - np.eye(N, k=1))/delta_x**2 + np.eye(N)*V\n ###\n return A, x\n```\n\n---\n\nLet's try to find eigenvalues and eigenvectors using standard linear algebra library.\n\n\n```\nA, x = hamiltonian(1000)\nl, vec = np.linalg.eigh(A)\n```\n\nEigenvalues are in ascending order and aligned with eigenvectors. Below I plot the first two probability densities and all eigenvalues.\n\n\n```\nfig, ax = plt.subplots(3, 2, figsize=(12, 12))\nfig.delaxes(ax[1, 1])\nfig.delaxes(ax[2, 1])\nplt.subplots_adjust(wspace=0.4, hspace=0.4)\n\nax[0, 0].plot(x, vec[:, 0])\nax[0, 0].set_title(f'Ground state with energy {l[0]:.5}')\nax[0, 0].set_xlabel('$x$')\nax[0, 0].set_ylabel('$|\\Psi_1(x)|^2$')\n\nax[0, 1].plot(l)\nax[0, 1].set_title(r'Energy')\nax[0, 1].set_xlabel('$n$')\nax[0, 1].set_ylabel('$E_n$')\n\nax[1, 0].plot(x, vec[:, 1])\nax[1, 0].set_title(f'First exited state with energy {l[1]:.5}')\nax[1, 0].set_xlabel('$x$')\nax[1, 0].set_ylabel('$|\\Psi_2(x)|^2$')\n\nV_0 = 40\na = 5\nalpha = 1e-3\nV = alpha*(x**2-a**2)**2 - V_0\nax[2, 0].plot(x, V, label='potential')\nax[2, 0].plot(x, V*0 + l[0], label='ground energy')\nax[2, 0].set_title(f'Potential')\nax[2, 0].set_xlabel('$x$')\nax[2, 0].set_ylabel('$V(x)$')\nax[2, 0].legend();\n```\n\n## Chemical bonds: solution\n\nThat is the problem:\n\nWe want to investigate the dependence of the ground state energy from a distance between potential wells.\n\n+ Scan distances between $0$ and $4$.\n+ Use at least $25$ points.\n+ Produce a neat figure.\n\n\n```\n### your solution is here\n\nL = []\ndistances = np.linspace(0, 4, 25)\nfor a in distances:\n A, x = hamiltonian(1000, a=a)\n l, vectors = np.linalg.eigh(A)\n L.append(l[0])\n\nplt.plot(distances, L)\nplt.xlabel('the distance between wells')\nplt.ylabel('ground state energy');\n\n###\n```\n\n---\n---\n---\n# Snowball Earth\n\nIn this problem, we investigate the properties of a simple zero-dimensional energy balance model. Sun heats the Earth, and the portion of the energy is reflected due to nonzero albedo $\\alpha$. The tricky point is that the albedo itself is a function of temperature (soon, you will see the explicit dependence). Being in thermodynamic equilibrium, Earth radiates according to the Stefan-Boltzmann law.\n\n\\begin{equation}\n C\\frac{dT}{dt} = (1 - \\alpha(T))Q - \\epsilon \\sigma T^4,\n\\end{equation}\n\n\\begin{equation}\n \\epsilon = 0.6,~\\sigma = 5.67\\times 10^{-8}\\frac{W}{m^2 K^4},~Q = 342\\frac{W}{m^2},~T\\in\\left[200K, 350K\\right],~C=1 \\frac{W}{m^2 K};\n\\end{equation}\n\nThe temperature $T$ that you see above is an average value over the globe and altitude. $C$ just fixes a timescale.\n\nAlbedo of the planet is the function of average temperature\n\n\\begin{equation}\n \\alpha(T) = 0.5 - 0.2\\tanh\\left(\\frac{T - 265}{10}\\right).\n\\end{equation}\n\n\n\n```\nT = np.linspace(200, 350, 100)\nalpha = 0.5 - 0.2*np.tanh((T - 265)/10)\nplt.plot(T, alpha)\nplt.xlabel('T, K')\nplt.ylabel('albedo')\nplt.text(200, 0.65, 'snowball Earth')\nplt.text(310, 0.33, 'open water')\nplt.title('Albedo with simple feedback.');\n```\n\nThis dependence represents a feedback mechanism: once the temperature falls below zero, the snow and ice reflect more radiation than the open water when everything melts due to the rise of temperature.\n\nOur goasl in this problem are\n\n+ to investigate the dynamics and equilibrium state as well as their dependence on the input parameters\n+ to consider the forcing scenario, when the Sun activity changes with time\n\nHowever, first, we need to discuss some additional syntax.\n\n---\n\n## Python: dictionaries, more advanced functions (unknown number of arguments, keyword arguments, lambda), scipy\n\nThere are more build-in types, as you probably suspect. The one that we consider now is a dictionary.\n\n\n```\n# dictionary has keys and values\n\napple_pie = {\n 'brown sugar': '1/2 cup',\n 'flour': {'tablespoon': 3},\n 'cinnamon': {'teaspoon': 1},\n 'ginger': {'teaspoon': 1/4},\n 375: 'bake at for 25 minutes',\n (6, 7): 'cups thinly sliced peeled tart apples'\n}\n\nprint(apple_pie.keys()) # this is how you can see keys\nprint(apple_pie.values()) # and this gives you values\n```\n\n dict_keys(['brown sugar', 'flour', 'cinnamon', 'ginger', 375, (6, 7)])\n dict_values(['1/2 cup', {'tablespoon': 3}, {'teaspoon': 1}, {'teaspoon': 0.25}, 'bake at for 25 minutes', 'cups thinly sliced peeled tart apples'])\n\n\n\n```\n# if you want to add another key and a value\n\napple_pie['Beat egg white'] = 'until foamy'\n\nprint(apple_pie.keys())\n```\n\n dict_keys(['brown sugar', 'flour', 'cinnamon', 'ginger', 375, (6, 7), 'Beat egg white'])\n\n\n\n```\n# you can iterate over keys\n\nfor key in apple_pie.keys():\n print(key, ':', apple_pie[key])\n```\n\n brown sugar : 1/2 cup\n flour : {'tablespoon': 3}\n cinnamon : {'teaspoon': 1}\n ginger : {'teaspoon': 0.25}\n 375 : bake at for 25 minutes\n (6, 7) : cups thinly sliced peeled tart apples\n Beat egg white : until foamy\n\n\n\n```\n# there are other methods for dictionaries\n\nitem = apple_pie.popitem()\nprint(item)\n\nfor key in apple_pie.keys():\n print(key, ':', apple_pie[key])\n\n# but we are not willing to cover them here\n# we will use dictionaries to organize our code better as you will see soon\n```\n\n ('Beat egg white', 'until foamy')\n brown sugar : 1/2 cup\n flour : {'tablespoon': 3}\n cinnamon : {'teaspoon': 1}\n ginger : {'teaspoon': 0.25}\n 375 : bake at for 25 minutes\n (6, 7) : cups thinly sliced peeled tart apples\n\n\nWe know how to define a function, but what if the number of arguments is unavailable in advanced?\n\n\n```\n# that will do the job\n\ndef multiply(*args):\n output = 1\n for argument in args:\n output *= argument\n return output\n\n\nprint(multiply(1, 2, 3), '\\n')\n\nprint(multiply(1, 2, 3, 4, 5), '\\n')\n\nprint(multiply(5), '\\n')\n```\n\n 6 \n \n 120 \n \n 5 \n \n\n\nThe same trick works for keword arguments.\n\n\n```\n# the main difference is that now we are working with dictionaries\n\ndef copycat(*args, **kwargs):\n if len(args) != 0:\n print('Here are values of my arguments')\n for argument in args:\n print(argument)\n if len(kwargs) != 0:\n print('Here you see keword argument')\n for key in kwargs.keys():\n print(key, kwargs[key])\n\ncopycat(a=4, b=12, c=32, cat='bark', dog='meow')\n\nprint('\\n')\n\ncopycat(12, 13, 14, [1, 2, 3], 'boom', z22 = 13, pr44 = copycat)\n```\n\n Here you see keword argument\n a 4\n b 12\n c 32\n cat bark\n dog meow\n \n \n Here are values of my arguments\n 12\n 13\n 14\n [1, 2, 3]\n boom\n Here you see keword argument\n z22 13\n pr44 \n\n\n\n```\n# you can pass arguments the same way you used them in the definition of the function\n\ncopycat(*[4, 12, 32], **{'cat': 'bark', 'dog': 'meow'})\n```\n\n Here are values of my arguments\n 4\n 12\n 32\n Here you see keword argument\n cat bark\n dog meow\n\n\nLambda!\n\n\n```\n# suppose you are so smart that you can write a function in a single line\n# the usual def statement looks too clumsy, lambda is more concise\n\ng = lambda a, b: a+b\n\nprint(g(2, 3))\nprint(g(12, 15))\n\nG = lambda **args: [print(k, args[k]) for k in args.keys()][0]\n\nG(copy=14, cat=13)\n\nprint('\\n')\n\nG(**{'cat': 'bark', 'dog': 'meow'})\n```\n\n 5\n 27\n copy 14\n cat 13\n \n \n cat bark\n dog meow\n\n\n\n```\n# another application is to produce a new function from a previously defined\n\ndef greetings(Name, form):\n if form is 'polite':\n print(f'It is a pleasure to have you with us, dear {Name}.\\n')\n elif form is 'rude':\n print(\"Hey, who's let this asshole in?\\n\")\n else:\n print(f'Hi, {Name}.\\n')\n\ngreetings('Mike', 'polite')\ngreetings('Mike', 'rude')\ngreetings('Mike', 'wahtever')\n\n# now we define a new function that is always polite\n\npolite_greetings = lambda N, f='polite': greetings(N, f)\n\npolite_greetings('Mike')\npolite_greetings('Zebra')\n```\n\n It is a pleasure to have you with us, dear Mike.\n \n Hey, who's let this asshole in?\n \n Hi, Mike.\n \n It is a pleasure to have you with us, dear Mike.\n \n It is a pleasure to have you with us, dear Zebra.\n \n\n\n[Scipy](https://docs.scipy.org/doc/scipy-1.3.0/reference/) is another extensive library with a lot of useful functionality. We are going to use their implementation of Newton's method.\n\n\n```\nfrom scipy.optimize import newton\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n%matplotlib inline\n\nx0 = np.linspace(0, 10, 20)\n\nf = lambda x, a=0.5: a*x - np.cos(2*x)\ndf = lambda x, a=0.5: a + 2*np.sin(2*x)\nd2f = lambda x, a=0.5: 4*np.cos(2*x)\n\nnewton?\n```\n\n\n```\nparams = {\n 'fprime': df,\n 'tol': 1e-5,\n 'maxiter': 100,\n 'fprime2': d2f,\n 'rtol': 1e-8\n}\n\nresult = newton(f, x0, **params)\n\nfig, ax = plt.subplots(1, 2, figsize=(10, 4))\nax[0].plot(result[(result<=2)*(result>=-2)], 'x')\nax[0].set_title('Redundant results within a reasonable range')\n\nind = np.unique(np.round(result[(result<=2)*(result>=-2)], 4), return_index=True)\nax[1].plot(result[ind[1]], 'x', color='red')\nax[1].set_title('Unique results within a reasonable range');\n```\n\nAnother function from Scipy that we are going to use is [odeint](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.odeint.html).\n\nAs an example we solve Van der Pol equation\n\n\\begin{equation}\n \\begin{split}\n &\\dot{x} = y,\\\\\n &\\dot{y} = \\mu(1 - x^2)y - x.\\\\\n \\end{split}\n\\end{equation}\n\n\n```\nfrom scipy.integrate import odeint\n\nVan_der_Pol = lambda xy, t, mu: (xy[1], mu*(1 - xy[0]**2)*xy[1] - xy[0])\n\nt = np.linspace(0, 100, 1000)\nsol = odeint(Van_der_Pol, [0., 0.1], t, args=(21.43, ))\nplt.plot(t, sol[:, 0])\nplt.xlabel('time')\nplt.ylabel('amplitude');\n```\n\n---\n\n## Subproblem 1\n\nHere are all equations again.\n\n\\begin{equation}\n C\\frac{dT}{dt} = (1 - \\alpha(T))Q - \\epsilon \\sigma T^4,\n\\end{equation}\n\n\\begin{equation}\n \\epsilon = 0.6,~\\sigma = 5.67\\times 10^{-8}\\frac{W}{m^2 K^4},~Q = 342\\frac{W}{m^2},~T\\in\\left[200K, 350K\\right],~C=1 \\frac{W}{m^2 K};\n\\end{equation}\n\nThe temperature $T$ that you see above is an average value over the globe and altitude. $C$ just fixes a timescale.\n\nAlbedo of the planet is the function of average temperature\n\n\\begin{equation}\n \\alpha(T) = 0.5 - 0.2\\tanh\\left(\\frac{T - 265}{10}\\right).\n\\end{equation}\n\nFor a start lets try to solve this equation from different starting points for the standard set of parameters. \n\nTake the set of initial temperatures to be $T_0 \\in \\left[200, 350\\right]$ with six uniformly distributed points, and find solution $T(t)$ on the time interval $\\left[0, 2\\right]$.\n\n\n```\n# Define right hand side with optional parameters (Q, epsilon)\n\n### your solution is here\n\ndef g(T, t, Q=342, epsilon=0.6, C=1):\n alpha = 0.5 - 0.2*np.tanh((T - 265)/10)\n sigma = 5.67*1e-8\n return ((1 - alpha)*Q - epsilon*sigma*T**4)/C\n\n###\n```\n\n\n```\n# Solve equation for the starting points specified above and produce a plot with trajectories\n\n### your solution is here\n\nfig, ax = plt.subplots(1, 1)\nt = np.linspace(0, 2, 100)\nT0 = np.linspace(200, 350, 6)\n\nfor t0 in T0:\n sol = odeint(g, t0, t)\n ax.plot(t, sol, color='black')\n\n###\n```\n\n## Subproblem 2\n\nIn this problem, I ask you to produce four plots with shared axes. Each plot should contain six trajectories alike ones in the previous problem. Below you can see the values of parameters for each simulation.\n\n||$~~~0~~~$|$~~~1~~~$|\n|:---:|:---:|:---:|\n|0|$Q = 342\\times 1.5$|$\\epsilon= 0.7$|\n|1|$Q = 342$|$\\epsilon = 0.3$|\n\nParameters that are not specified should be set to default values.\n\n\n```\n### your solution is here\n\nparameters = {\n 'Q': 342,\n 'epsilon': 0.6,\n 'C': 1\n}\n\nfig, ax = plt.subplots(2, 2, figsize=(16, 10), sharex=True, sharey=True)\nt = np.linspace(0, 2, 100)\nT0 = np.linspace(200, 350, 6)\n\nfor i, q in zip(range(2), [342*1.5, 342.0]):\n parameters['Q'] = q\n g_ = lambda T, t: g(T, t, **parameters)\n ax[i, 0].set_title(f'Q = {q:.2}')\n if i==1: ax[i, 0].set_xlabel('time')\n ax[i, 0].set_ylabel('temperature, K')\n for T in T0:\n sol = odeint(g_, T, t)\n ax[i, 0].plot(t, sol)\n\nparameters['Q'] = 342\nfor i, e in zip(range(2), [0.7, 0.3]):\n parameters['epsilon'] = e\n g_ = lambda T, t: g(T, t, **parameters)\n ax[i, 1].set_title(fr'$\\epsilon = {e:.2}$')\n if i==1: ax[i, 1].set_xlabel('time')\n for T in T0:\n sol = odeint(g_, T, t)\n ax[i, 1].plot(t, sol)\n\n###\n```\n\nYou can see that the number of steady states depends on the value of parameters. Starting from now, we take $\\epsilon$ to be fixed to the default value and vary $Q$.\n\n## Subproblem 3\n\nNow we are ready to produce the first bifurcation diagram. On the $x$-axis, we put the values of $Q$ from $0.5$ to $1.5$ of the default value. On the $y$-axis, we plot all steady states that can be achieved, starting from different initial conditions. To do that, I propose you to use the same starting temperatures as before, integrate till $t=4$, and put all endpoints on the $y$-axis.\n\n\n```\n### your solution is here\n\nt = np.linspace(0, 4, 400)\nT0 = np.linspace(200, 350, 6)\nparameters['epsilon'] = 0.6\n\nfor q in np.linspace(342*0.5, 342*1.5, 20):\n for t0 in T0:\n parameters['Q'] = q\n g_ = lambda T, t: g(T, t, **parameters)\n sol = odeint(g_, t0, t)\n plt.plot(q, sol[-1], '.', color='black')\nplt.xlabel('$Q$')\nplt.ylabel('$T$ steady state');\n###\n```\n\nThe meaning of this graph will be explained later to the full extent. For now, it is enough to notice that there are values of Q correspond to two steady states and also a single equilibrium.\n\n## Snowball Earth: solution part 1\n\nHere are all equations again.\n\n\\begin{equation}\n C\\frac{dT}{dt} = (1 - \\alpha(T))Q - \\epsilon \\sigma T^4,\n\\end{equation}\n\n\\begin{equation}\n \\epsilon = 0.6,~\\sigma = 5.67\\times 10^{-8}\\frac{W}{m^2 K^4},~Q = 342\\frac{W}{m^2},~T\\in\\left[200K, 350K\\right],~C=1 \\frac{W}{m^2 K};\n\\end{equation}\n\n\n\\begin{equation}\n \\alpha(T) = 0.5 - 0.2\\tanh\\left(\\frac{T - 265}{10}\\right).\n\\end{equation}\n\nHowever, the picture that we have is incomplete. That is because we are missing unstable equilibria. To find them, we should compute all roots of the right-hand side for each value of Q. And that is what we are going to do next! More specifically, it would help if you fill the gaps in my code below.\n\n\n\n```\nQ = np.linspace(0.6*342, 1.4*342, 20)\nT0 = np.linspace(200, 350, 10)\n\n### your solution is here\n\ndef RHS(T, Q=342):\n alpha = 0.5 - 0.2*np.tanh((T - 265)/10)\n epsilon = 0.6\n sigma = 5.67*1e-8\n return (1 - alpha)*Q - epsilon*sigma*T**4\n\ndef dRHS(T, Q=342):\n epsilon = 0.6\n sigma = 5.67*1e-8\n return 0.02*Q/np.cosh((T - 265)/10)**2 - 4*epsilon*sigma*T**3\n\ndef d2RHS(T, Q=342):\n epsilon = 0.6\n sigma = 5.67*1e-8\n return -0.004*Q*np.sinh((T - 265)/10)/np.cosh((T - 265)/10)**3 - 12*epsilon*sigma*T**2\n\n### \n\nS = []\nfor q in Q:\n f = lambda T, Q=q: RHS(T, Q)\n df = lambda T, Q=q: dRHS(T, Q)\n d2f = lambda T, Q=q: d2RHS(T, Q)\n solution = newton(f, T0, fprime=df, fprime2=d2f, maxiter=100)\n s = np.unique(np.round(solution, 2))\n s = s[s>0]\n S.append(s)\n```\n\n\n```\n### plot bifurcation diagram\nfor s, q in zip(S, Q):\n for root in s:\n plt.plot(q, root, 'o', color='black')\nplt.xlabel('$Q$')\nplt.ylabel('Equilibrium T');\n\n###\n```\n\n## Snowball Earth: solution part 2 (forcing)\n\nSuppose the energy flux from the Sun starts to change gradually. We model this situation by the time-dependent term\n\\begin{equation}\n Q(t) = Q_0\\left(\\left[1 - \\frac{\\gamma}{2}\\right] + \\frac{\\gamma}{1 + e^{-\\beta t}}\\right).\n\\end{equation}\n\nThere are two parameters $\\beta>0$ and $\\gamma\\in \\mathbb{R}$. For large $t$, $Q(t)$ reaches $\\sim Q_0\\left(1 + \\frac{\\gamma}{2}\\right)$.\n\nThere are two parameters $\\beta>0$ and $\\gamma\\in \\mathbb{R}$. For large $t$, $Q(t)$ reaches $\\sim Q_0\\left(1 + \\frac{\\gamma}{2}\\right)$. The first parameter $\\beta$ defines the speed of change.\n\nI ask you to obtain a trajectory for the following set of parameters and initial conditions:\n\n+ $T_0 = 200~\\text{K}$\n+ $\\gamma = 1$\n+ $\\beta = 1\\big/10$\n+ $Q_0, \\epsilon$ default values\n+ $t_0 = 0$, $t_1 = 20$\n\nWhen you get the trajectory, produce three graphs\n\n+ The trajectory itself\n+ Below the previous one $Q(t)$\n+ To the right from the first one put the bifurcation diagram\n\n\n```\n# define new right-hand side\n### your solution is here\n\ndef g(T, t, beta, gamma):\n alpha = 0.5 - 0.2*np.tanh((T - 265)/10)\n Q = 342*((1 - gamma/2) + gamma/(1 + np.exp(-beta*t)))\n epsilon = 0.6\n sigma = 5.67*1e-8\n return (1 - alpha)*Q - epsilon*sigma*T**4\n\n###\n```\n\n\n```\n### your solution is here\n\nt = np.linspace(0, 20, 1000)\nT0 = 200\n\nfig, ax = plt.subplots(2, 2, figsize=(13, 8))\nT_t = odeint(g, T0, t, args=(1/10, 1)).reshape(-1,)\nfig.delaxes(ax[1, 1])\nax[0, 0].plot(t, T_t)\nax[0, 0].set_ylabel('T, K')\n\nQ_ = lambda t, beta=1/10, gamma=1: 342*((1 - gamma/2) + gamma/(1 + np.exp(-beta*t)))\nax[1, 0].plot(t, Q_(t))\nax[1, 0].set_ylabel('Q')\nax[1, 0].set_xlabel('t')\n\nfor s, q in zip(S, Q):\n for root in s:\n ax[0, 1].plot(q, root, 'o', color='black')\nax[0, 1].set_xlabel('$Q$')\nax[0, 1].set_ylabel('Equilibrium T')\n\nax[1, 0].set_xlabel('time');\n\n###\n```\n\nThis illustrates how forcing can change result in abrupt changes in the behavior when the previous equilibrium point ceases to exist. \n\nDid we have this scenario in our past? [Yep](https://en.wikipedia.org/wiki/Snowball_Earth).\n\n---\n---\n---\n# Newton's fractals\n\nNewton algorithm and ones alike seek the solutions of the nonlinear equation\n\n\\begin{equation}\n g(x) = 0.\n\\end{equation}\n\nThat how Newton's algorithm works: from the initial approximation $x_0$, we construct a better approximation via the solution of a linear model\n\n\\begin{equation}\n 0 = g(x_0) + \\left(x_{\\text{new}} - x_0\\right)\\left.\\frac{d g (x)}{d x}\\right|_{x=x_0} \\Longrightarrow x_{\\text{new}} = x_{0} - g(x_0)\\big/\\left.\\frac{d g (x)}{d x}\\right|_{x=x_0}.\n\\end{equation}\n\nNewton's method is from the class of iterative algorithms, so once you run it from some starting point $x_0$, it produces a series of points:\n\n\\begin{equation}\n x_0 \\longrightarrow x_1 \\longrightarrow \\dots\\longrightarrow x_n\\longrightarrow \\dots\\longrightarrow x_{\\infty}.\n\\end{equation}\n\nThe limit of this sequence (if any) depends on the initial point.\n\nFor a given root $x^{\\star}$, we define a **basin of attraction** as a set of all starting points, which lead to this root as a limit of Newton's algorithm.\n\n---\n\nOur goal is to study the basins of attraction for different functions. To do that, we are going to run Newton's algorithm on multiply starting points and color them according to their basins of attraction.\n\n---\n\n## Subproblem 1\n\nWrite a function that **takes**\n\n1. a function\n2. it's derivative (which is another function)\n3. $x_0$ - initial guess\n4. $N_{\\max}$ - maximal number of iterations\n5. `tolerance` (optional)\n6. `verbose` (optional)\n\n`tolerance` should be used as a stopping criterion and if `verbose == True` printout should contain progress.\n\nand **returns**: approximate solution for $f(x)=0$.\n\n\n```\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n%matplotlib inline\n\ndef Newton(f, df, x0, tolerance=1e-10, verbose=False, N_max=100):\n ### your solution is here\n error = 100\n i = 0\n while error>tolerance and i L1:\n\t\twhile y<0.5 and (PHI(x,y,z) < PHI_RL):#(PHI(x,y,z) < PHI_RL) and (x**2+y**2+z**2)**0.5 < max(-L2,L3):\n\t\t\twhile z<0.5 and (PHI(x,y,z) < PHI_RL):# (PHI(x,y,z) < PHI_RL) and (x**2+y**2+z**2)**0.5 < max(-L2,L3):\n\t\t\t\tV = V + dx*dy*dz\n\t\t\t\tz = z + dz\n\t\t\tz = 0.\n\t\t\ty = y + dy\n\t\tz = 0.\n\t\ty = 0.\n\t\tx = x - dx\n\n\tprint (q, (3.*V/np.pi)**(1./3.))\n\t\n```\n\n\n```\nsns.set()\n```\n\n\n```\nn = 1\nfor z in [-0.35,-0.3,-0.25,-0.2,-0.15,-0.1,-0.05,0,0.05,0.1,0.15,0.2,0.25,0.3,0.35]:\n\t\tx = L2\n\t\ty = 0.7*L2\n\t\tdx = 0.005\n\t\tdy = 0.005\n\n\t\ttabx = []\n\t\ttaby = []\n\t\ttabc = []\n\n\t\twhile x < L3:\n\t\t\twhile y < 0.7*L3:\n\t\t\t\tif PHI(x,y,z) > -7 and PHI(x,y,z)< PHI_RL:\n\t\t\t\t\ttabx.append(x)\n\t\t\t\t\ttaby.append(y)\n\t\t\t\t\ttabc.append(PHI(x,y,z))\n\t\t\t\ty = y + dy\n\t\t\ty = 0.7*L2\n\t\t\tx = x + dx\n\n\t\tplt.clf()\n\t\tplt.scatter(tabx, taby, c=tabc,lw = 0)\n\t\tplt.title(\"q = \"+str(q)+\" z = \"+str(z))\n\t\tplt.xlabel(\"x\")\n\t\tplt.ylabel(\"y\")\n\t\tplt.xlim(L2,L3)\n\t\tplt.clim(-7,PHI_RL)\n\t\tplt.ylim(0.7*L2,0.7*L3)\n\t\tplt.colorbar()\n\t\tnStr=str(n)\n\t\tnStr=nStr.rjust(5,'0')\n\t\tplt.savefig('img'+nStr+'.png')\n\t\tn = n + 1\n\t\tplt.show()\n\t\n\n```\n\n\n```\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport matplotlib.animation as animation\nfrom matplotlib.animation import PillowWriter\n\nfig = plt.figure()\n\ndef f(x, y):\n return np.sin(x) + np.cos(y)\n\nx = np.linspace(0, 2 * np.pi, 120)\ny = np.linspace(0, 2 * np.pi, 100).reshape(-1, 1)\n# ims is a list of lists, each row is a list of artists to draw in the\n# current frame; here we are just animating one artist, the image, in\n# each frame\nims = []\nfor i in range(20):\n x += np.pi / 15.\n y += np.pi / 20.\n im = plt.imshow(f(x, y))\n ims.append([im])\n\nani = animation.ArtistAnimation(fig, ims, interval=50, blit=True,\n repeat_delay=500)\n\nwriter = PillowWriter(fps=20)\nani.save(\"demo2.gif\", writer=writer)\n\nplt.show()\n\n```\n\n\n \n\n\n\n
                                        \n\n\n\n```\n\n```\n", "meta": {"hexsha": "427e2667bdeef8f3423c9bf357ed2f282d6680ce", "size": 765756, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Week 4/7-11-2020/Roche_lobe.ipynb", "max_stars_repo_name": "the-halfbloodprince/Cosmic-Quarks-Updates", "max_stars_repo_head_hexsha": "5cf8ab4c9fe35e4443461abaa78d8745503d2096", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2020-11-02T11:35:31.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-30T12:20:51.000Z", "max_issues_repo_path": "Week 4/7-11-2020/Roche_lobe.ipynb", "max_issues_repo_name": "the-halfbloodprince/Cosmic-Quarks-Updates", "max_issues_repo_head_hexsha": "5cf8ab4c9fe35e4443461abaa78d8745503d2096", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-12-06T06:07:45.000Z", "max_issues_repo_issues_event_max_datetime": "2020-12-06T06:07:45.000Z", "max_forks_repo_path": "Week 4/7-11-2020/Roche_lobe.ipynb", "max_forks_repo_name": "the-halfbloodprince/Cosmic-Quarks-Updates", "max_forks_repo_head_hexsha": "5cf8ab4c9fe35e4443461abaa78d8745503d2096", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-11-22T09:57:31.000Z", "max_forks_repo_forks_event_max_datetime": "2020-11-22T09:57:31.000Z", "avg_line_length": 580.1181818182, "max_line_length": 148066, "alphanum_fraction": 0.922429599, "converted": true, "num_tokens": 1658, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056040203136, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.41112471014367374}} {"text": "

                                        \n\n

                                        Massimo Nocentini
                                        \n\n
                                        September 27, 2016: refactoring and sync\n                                        \n
                                        \n

                                        \n\n
                                        \n
                                        Abstract
                                        \nIn this notebook we study the *Pascal triangle*, locking at it recursively: using it's $A$-sequence, we perform a series of unfolding using the main recurrence relation, where subscripts dependends on *two* dimensions, as rewriting rule. This is a natural enhancement to the case of recurrences where subscripts have *one* dimension only, as the *Fibonacci* sequence; on the other hand, Pascal array is a deeply studied and well know triangle, yet simple, toy we can play with.\n
                                        \n
                                        \n\n\n```python\n%run \"../src/start_session.py\"\n%run \"../src/recurrences.py\"\n```\n\n\n```python\nimport oeis\n```\n\n# Stirling array $\\mathcal{S}$, of the second kind\n\nThis notebook studies the Riordan array $\\mathcal{P}$, aka the *Pascal triangle*, defined according to the following definition:\n\n$$\\mathcal{P}=\\left(\\frac{1}{1-t}, \\frac{t}{1-t}\\right)$$\n\nwith $A$-sequence $A(t)=1+t$ and $Z$-sequence $Z(t)=1$.\n\n\n```python\ns = oeis.oeis_search(id=48993)\n```\n\n *\n\n\n```python\ns()\n```\n\n\n\n\n_Results for query: https://oeis.org/search?q=id%3AA048993&start=0&fmt=json_
                                        A048993: Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0<=k<=n.
                                        \n\nby _N. J. A. Sloane_, Dec 11 1999\n\n_Keywords_: `nonn,tabl,nice`\n\n_Data_:\n\n$$\n\\begin{array}{c|ccccccccccc}\nn, k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\\\n\\hline\n0 & 1 & & & & & & & & & \\\\1 & 0 & 1 & & & & & & & & \\\\2 & 0 & 1 & 1 & & & & & & & \\\\3 & 0 & 1 & 3 & 1 & & & & & & \\\\4 & 0 & 1 & 7 & 6 & 1 & & & & & \\\\5 & 0 & 1 & 15 & 25 & 10 & 1 & & & & \\\\6 & 0 & 1 & 31 & 90 & 65 & 15 & 1 & & & \\\\7 & 0 & 1 & 63 & 301 & 350 & 140 & 21 & 1 & & \\\\8 & 0 & 1 & 127 & 966 & 1701 & 1050 & 266 & 28 & 1 & \\\\9 & 0 & 1 & 255 & 3025 & 7770 & 6951 & 2646 & 462 & 36 & 1\n\\end{array}\n$$\n\n\n_Comments_:\n - Also known as Stirling set numbers.\n - S(n,k) enumerates partitions of an n-set into k nonempty subsets.\n - The o.g.f. for the sequence of diagonal k (k=0 for the main diagonal) is G(k,x)= ((x^k)/(1-x)^(2*k+1))*sum(A008517(k,m+1)*x^m,m=0..k-1). A008517 is the second-order Eulerian triangle. - _Wolfdieter Lang_, Oct 14 2005.\n - From _Philippe Del\u00e9ham_, Nov 14 2007: \n - Sum_{k, 0<=k<=n}S(n,k)*x^k = B_n(x), where B_n(x) = Bell polynomials. The first few Bell polynomials are:\n - B_0(x) = 1;\n - B_1(x) = 0 + x;\n - B_2(x) = 0 + x + x^2;\n - B_3(x) = 0 + x + 3x^2 + x^3;\n - B_4(x) = 0 + x + 7x^2 + 6x^3 + x^4;\n - B_5(x) = 0 + x + 15x^2 + 25x^3 + 10x^4 + x^5;\n - B_6(x) = 0 + x + 31x^2 + 90x^3 + 65x^4 + 15x^5 + x^6;\n - This is the Sheffer triangle (1, exp(x) - 1), an exponential (binomial) convolution triangle. The a-sequence is given by A006232/A006233 (Cauchy sequence). The z-sequence is the zero sequence. See the link under A006232 for the definition and use of these sequences. The row sums give A000110 (Bell), and the alternating row sums give A000587 (see the Philippe Del\u00e9ham formulas and crossreferences below). - _Wolfdieter Lang_, Oct 16 2014\n - Also the inverse Bell transform of the factorial numbers (A000142). For the definition of the Bell transform see A264428 and for cross-references A265604. - _Peter Luschny_, Dec 31 2015\n\n_Formulae_:\n - S(n, k) = k*S(n-1, k) + S(n-1, k-1), n>0; S(0, k) = 0, k>0; S(0, 0)=1.\n - Equals [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, ..] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is Del\u00e9ham's operator defined in A084938.\n - Sum_{k = 0..n} x^k*S(n, k) = A213170(n), A000587(n), A000007(n), A000110(n), A001861(n), A027710(n), A078944(n), A144180(n), A144223(n), A144263(n) respectively for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7. - _Philippe Del\u00e9ham_, May 09 2004, Feb 16 2013\n - S(n, k) = sum{i=0..k, (-1)^(k+i)binomial(k, i)i^n/k!}. - _Paul Barry_, Aug 05 2004\n - Sum(k*S(n,k), k=0..n)=B(n+1)-B(n), where B(q) are the Bell numbers (A000110). - _Emeric Deutsch_, Nov 01 2006\n - Equals the inverse binomial transform of A008277. - _Gary W. Adamson_, Jan 29 2008\n - G.f.: 1/(1-xy/(1-x/(1-xy/(1-2x/(1-xy/1-3x/(1-xy/(1-4x/(1-xy/(1-5x/(1-... (continued fraction equivalent to Deleham DELTA construction). - _Paul Barry_, Dec 06 2009\n - G.f.: 1/Q(0), where Q(k) = 1 -(y+k)*x - (k+1)*y*x^2/Q(k+1) ; (continued fraction). - _Sergei N. Gladkovskii_, Nov 09 2013\n - Inverse of padded A008275 (padded just as A048993 = padded A008277). - _Tom Copeland_, Apr 25 2014\n - E.g.f. for the row polynomials s(n,x) = sum(S(n,k)*x^k, k=0..n) is exp(x*(exp(z)-1)) (Sheffer property). E.g.f. for the k-th column sequence with k leading zeros is ((exp(x)-1)^k)/k! (Sheffer property). - _Wolfdieter Lang_, Oct 16 2014\n\n_Cross references_:\n - See especially A008277 which is the main entry for this triangle.\n - Cf. A008275, A039810-A039813, A048994.\n - A000110(n) = sum(S(n, k)) k=0..n, n >= 0. Cf. A085693.\n - Cf. A084938, A106800 (mirror image), A213061.\n\n_Links_:\n - David W. Wilson, A048993/b048993.txt\">Table of n, a(n) for n = 0..10010\n - M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].\n - V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.\n - P. Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Toda Chain Equations, Journal of Integer Sequences, 17 (2014), #14.2.3.\n - P. Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.\n - R. M. Dickau, Stirling numbers of the second kind\n - G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela and P. Blasiak, One-parameter groups and combinatorial physics, arXiv:quant-ph/0401126, 2004.\n - FindStat - Combinatorial Statistic Finder, The number of blocks in the set partition.\n - W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.\n - C. M. Ringel, The Catalan combinatorics of the hereditary artin algebras, arXiv preprint arXiv:1502.06553, 2015\n - X.-T. Su, D.-Y. Yang, W.-W. Zhang, A note on the generalized factorial, Australasian Journal of Combinatorics, Volume 56 (2013), Pages 133-137.\n\n_References_:\n - M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.\n - L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 310.\n - J. H. Conway and R. K. Guy, The Book of Numbers, Springer, p. 92.\n - F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.\n - R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 244.\n - J. Riordan, An Introduction to Combinatorial Analysis, p. 48.\n\n\n\n\n```python\nfrom sympy.functions.combinatorial.numbers import stirling\n```\n\n\n```python\ns = IndexedBase('s')\nn, k = symbols('n k')\n\nstirling_recurrence_spec = recurrence_spec(recurrence_eq=Eq(s[n+1, k+1], s[n, k] + (k+1)*s[n, k+1]), \n recurrence_symbol=s, \n variables=[n,k])\n```\n\n\n```python\nstirling_recurrence_spec\n```\n\n\n\n\n$\\left(\\Theta, \\Gamma\\right)_{n,k}^{s}$ where:
                                        • $\\Theta = \\left\\{ s_{n + 1,k + 1} = \\left(k + 1\\right) s_{n,k + 1} + s_{n,k} \\right\\}$
                                        • $\\Gamma = \\left\\{\\begin{array}{c}\\end{array}\\right\\}$
                                        \n\n\n\n\n```python\nunfolded = stirling_recurrence_spec.unfold(depth=1)\n```\n\n\n```python\nunfolded\n```\n\n\n\n\n$\\left(\\Theta, \\Gamma\\right)_{n,k}^{s}$ where:
                                        • $\\Theta = \\left\\{ s_{n + 1,k + 1} = k \\left(\\left(k + 1\\right) s_{n - 1,k + 1} + s_{n - 1,k}\\right) + k s_{n - 1,k} + \\left(k + 1\\right) s_{n - 1,k + 1} + s_{n - 1,k} + s_{n - 1,k - 1} \\right\\}$
                                        • $\\Gamma = \\left\\{\\begin{array}{c}s_{n,k} = k s_{n - 1,k} + s_{n - 1,k - 1}\\\\s_{n,k + 1} = \\left(k + 1\\right) s_{n - 1,k + 1} + s_{n - 1,k}\\end{array}\\right\\}$
                                        \n\n\n\n\n```python\ninstantiated = unfolded.instantiate(strategy=raw(substitutions={n:9,k:4}))\ninstantiated\n```\n\n\n\n\n$\\left(\\Theta, \\Gamma\\right)_{n,k}^{s}$ where:
                                        • $\\Theta = \\left\\{ s_{10,5} = s_{8,3} + 9 s_{8,4} + 25 s_{8,5} \\right\\}$
                                        • $\\Gamma = \\left\\{\\begin{array}{c}s_{9,4} = s_{8,3} + 4 s_{8,4}\\\\s_{9,5} = s_{8,4} + 5 s_{8,5}\\end{array}\\right\\}$
                                        \n\n\n\n\n```python\nknown_binomials = {s[n,k]:stirling(n,k) for n in [10,9,8] for k in range(2,6)}\n\nchecked = instantiated.instantiate(strategy=raw(substitutions=known_binomials))\nchecked\n```\n\n\n\n\n$\\left(\\Theta, \\Gamma\\right)_{n,k}^{s}$ where:
                                        • $\\Theta = \\left\\{ \\mathrm{True} \\right\\}$
                                        • $\\Gamma = \\left\\{\\begin{array}{c}\\end{array}\\right\\}$
                                        \n\n\n\n\n```python\nchecked.subsume()\n```\n\n\n\n\n$\\left(\\Theta, \\Gamma\\right)_{n,k}^{s}$ where:
                                        • $\\Theta = \\left\\{ \\mathrm{True} \\right\\}$
                                        • $\\Gamma = \\left\\{\\begin{array}{c}\\end{array}\\right\\}$
                                        \n\n\n\n\n```python\nbased_recurrence_spec = unfolded.instantiate(strategy=based(arity=doubly_indexed()))\nbased_recurrence_spec\n```\n\n\n\n\n$\\left(\\Theta, \\Gamma\\right)_{n,k}^{s}$ where:
                                        • $\\Theta = \\left\\{ s_{4,2} = s_{2,0} + 3 s_{2,1} + 4 s_{2,2} \\right\\}$
                                        • $\\Gamma = \\left\\{\\begin{array}{c}s_{3,1} = s_{2,0} + s_{2,1}\\\\s_{3,2} = s_{2,1} + 2 s_{2,2}\\end{array}\\right\\}$
                                        \n\n\n\n\n```python\nbased_recurrence_spec.subsume()\n```\n\n\n\n\n$\\left(\\Theta, \\Gamma\\right)_{n,k}^{s}$ where:
                                        • $\\Theta = \\left\\{ s_{4,2} = s_{2,0} + 3 s_{2,1} + 4 s_{2,2} \\right\\}$
                                        • $\\Gamma = \\left\\{\\begin{array}{c}s_{3,1} = s_{2,0} + s_{2,1}\\\\s_{3,2} = s_{2,1} + 2 s_{2,2}\\end{array}\\right\\}$
                                        \n\n\n\n\n```python\nipython_latex_description(rec_spec=stirling_recurrence_spec, depths=range(9), based_instantiation=False,\n arity=doubly_indexed())\n```\n\n\n\n\n\\begin{array}{c}s_{n + 1,k + 1} = k s_{n,k + 1} + s_{n,k} + s_{n,k + 1}\\\\\ns_{n + 1,k + 1} = k^{2} s_{n - 1,k + 1} + 2 k s_{n - 1,k} + 2 k s_{n - 1,k + 1} + s_{n - 1,k} + s_{n - 1,k - 1} + s_{n - 1,k + 1}\\\\\ns_{n + 1,k + 1} = k^{3} s_{n - 2,k + 1} + 3 k^{2} s_{n - 2,k} + 3 k^{2} s_{n - 2,k + 1} + 3 k s_{n - 2,k} + 3 k s_{n - 2,k - 1} + 3 k s_{n - 2,k + 1} + s_{n - 2,k} + s_{n - 2,k - 2} + s_{n - 2,k + 1}\\\\\ns_{n + 1,k + 1} = k^{4} s_{n - 3,k + 1} + 4 k^{3} s_{n - 3,k} + 4 k^{3} s_{n - 3,k + 1} + 6 k^{2} s_{n - 3,k} + 6 k^{2} s_{n - 3,k - 1} + 6 k^{2} s_{n - 3,k + 1} + 4 k s_{n - 3,k} + 4 k s_{n - 3,k - 2} + 4 k s_{n - 3,k + 1} + s_{n - 3,k} + s_{n - 3,k - 3} - 2 s_{n - 3,k - 2} + s_{n - 3,k - 1} + s_{n - 3,k + 1}\\\\\ns_{n + 1,k + 1} = k^{5} s_{n - 4,k + 1} + 5 k^{4} s_{n - 4,k} + 5 k^{4} s_{n - 4,k + 1} + 10 k^{3} s_{n - 4,k} + 10 k^{3} s_{n - 4,k - 1} + 10 k^{3} s_{n - 4,k + 1} + 10 k^{2} s_{n - 4,k} + 10 k^{2} s_{n - 4,k - 2} + 10 k^{2} s_{n - 4,k + 1} + 5 k s_{n - 4,k} + 5 k s_{n - 4,k - 3} - 10 k s_{n - 4,k - 2} + 5 k s_{n - 4,k - 1} + 5 k s_{n - 4,k + 1} + s_{n - 4,k} + s_{n - 4,k - 4} - 5 s_{n - 4,k - 3} + 5 s_{n - 4,k - 2} + s_{n - 4,k + 1}\\\\\ns_{n + 1,k + 1} = k^{6} s_{n - 5,k + 1} + 6 k^{5} s_{n - 5,k} + 6 k^{5} s_{n - 5,k + 1} + 15 k^{4} s_{n - 5,k} + 15 k^{4} s_{n - 5,k - 1} + 15 k^{4} s_{n - 5,k + 1} + 20 k^{3} s_{n - 5,k} + 20 k^{3} s_{n - 5,k - 2} + 20 k^{3} s_{n - 5,k + 1} + 15 k^{2} s_{n - 5,k} + 15 k^{2} s_{n - 5,k - 3} - 30 k^{2} s_{n - 5,k - 2} + 15 k^{2} s_{n - 5,k - 1} + 15 k^{2} s_{n - 5,k + 1} + 6 k s_{n - 5,k} + 6 k s_{n - 5,k - 4} - 30 k s_{n - 5,k - 3} + 30 k s_{n - 5,k - 2} + 6 k s_{n - 5,k + 1} + s_{n - 5,k} + s_{n - 5,k - 5} - 9 s_{n - 5,k - 4} + 20 s_{n - 5,k - 3} - 10 s_{n - 5,k - 2} + s_{n - 5,k - 1} + s_{n - 5,k + 1}\\\\\ns_{n + 1,k + 1} = k^{7} s_{n - 6,k + 1} + 7 k^{6} s_{n - 6,k} + 7 k^{6} s_{n - 6,k + 1} + 21 k^{5} s_{n - 6,k} + 21 k^{5} s_{n - 6,k - 1} + 21 k^{5} s_{n - 6,k + 1} + 35 k^{4} s_{n - 6,k} + 35 k^{4} s_{n - 6,k - 2} + 35 k^{4} s_{n - 6,k + 1} + 35 k^{3} s_{n - 6,k} + 35 k^{3} s_{n - 6,k - 3} - 70 k^{3} s_{n - 6,k - 2} + 35 k^{3} s_{n - 6,k - 1} + 35 k^{3} s_{n - 6,k + 1} + 21 k^{2} s_{n - 6,k} + 21 k^{2} s_{n - 6,k - 4} - 105 k^{2} s_{n - 6,k - 3} + 105 k^{2} s_{n - 6,k - 2} + 21 k^{2} s_{n - 6,k + 1} + 7 k s_{n - 6,k} + 7 k s_{n - 6,k - 5} - 63 k s_{n - 6,k - 4} + 140 k s_{n - 6,k - 3} - 70 k s_{n - 6,k - 2} + 7 k s_{n - 6,k - 1} + 7 k s_{n - 6,k + 1} + s_{n - 6,k} + s_{n - 6,k - 6} - 14 s_{n - 6,k - 5} + 56 s_{n - 6,k - 4} - 70 s_{n - 6,k - 3} + 21 s_{n - 6,k - 2} + s_{n - 6,k + 1}\\\\\ns_{n + 1,k + 1} = k^{8} s_{n - 7,k + 1} + 8 k^{7} s_{n - 7,k} + 8 k^{7} s_{n - 7,k + 1} + 28 k^{6} s_{n - 7,k} + 28 k^{6} s_{n - 7,k - 1} + 28 k^{6} s_{n - 7,k + 1} + 56 k^{5} s_{n - 7,k} + 56 k^{5} s_{n - 7,k - 2} + 56 k^{5} s_{n - 7,k + 1} + 70 k^{4} s_{n - 7,k} + 70 k^{4} s_{n - 7,k - 3} - 140 k^{4} s_{n - 7,k - 2} + 70 k^{4} s_{n - 7,k - 1} + 70 k^{4} s_{n - 7,k + 1} + 56 k^{3} s_{n - 7,k} + 56 k^{3} s_{n - 7,k - 4} - 280 k^{3} s_{n - 7,k - 3} + 280 k^{3} s_{n - 7,k - 2} + 56 k^{3} s_{n - 7,k + 1} + 28 k^{2} s_{n - 7,k} + 28 k^{2} s_{n - 7,k - 5} - 252 k^{2} s_{n - 7,k - 4} + 560 k^{2} s_{n - 7,k - 3} - 280 k^{2} s_{n - 7,k - 2} + 28 k^{2} s_{n - 7,k - 1} + 28 k^{2} s_{n - 7,k + 1} + 8 k s_{n - 7,k} + 8 k s_{n - 7,k - 6} - 112 k s_{n - 7,k - 5} + 448 k s_{n - 7,k - 4} - 560 k s_{n - 7,k - 3} + 168 k s_{n - 7,k - 2} + 8 k s_{n - 7,k + 1} + s_{n - 7,k} + s_{n - 7,k - 7} - 20 s_{n - 7,k - 6} + 126 s_{n - 7,k - 5} - 294 s_{n - 7,k - 4} + 231 s_{n - 7,k - 3} - 42 s_{n - 7,k - 2} + s_{n - 7,k - 1} + s_{n - 7,k + 1}\\\\\ns_{n + 1,k + 1} = k^{9} s_{n - 8,k + 1} + 9 k^{8} s_{n - 8,k} + 9 k^{8} s_{n - 8,k + 1} + 36 k^{7} s_{n - 8,k} + 36 k^{7} s_{n - 8,k - 1} + 36 k^{7} s_{n - 8,k + 1} + 84 k^{6} s_{n - 8,k} + 84 k^{6} s_{n - 8,k - 2} + 84 k^{6} s_{n - 8,k + 1} + 126 k^{5} s_{n - 8,k} + 126 k^{5} s_{n - 8,k - 3} - 252 k^{5} s_{n - 8,k - 2} + 126 k^{5} s_{n - 8,k - 1} + 126 k^{5} s_{n - 8,k + 1} + 126 k^{4} s_{n - 8,k} + 126 k^{4} s_{n - 8,k - 4} - 630 k^{4} s_{n - 8,k - 3} + 630 k^{4} s_{n - 8,k - 2} + 126 k^{4} s_{n - 8,k + 1} + 84 k^{3} s_{n - 8,k} + 84 k^{3} s_{n - 8,k - 5} - 756 k^{3} s_{n - 8,k - 4} + 1680 k^{3} s_{n - 8,k - 3} - 840 k^{3} s_{n - 8,k - 2} + 84 k^{3} s_{n - 8,k - 1} + 84 k^{3} s_{n - 8,k + 1} + 36 k^{2} s_{n - 8,k} + 36 k^{2} s_{n - 8,k - 6} - 504 k^{2} s_{n - 8,k - 5} + 2016 k^{2} s_{n - 8,k - 4} - 2520 k^{2} s_{n - 8,k - 3} + 756 k^{2} s_{n - 8,k - 2} + 36 k^{2} s_{n - 8,k + 1} + 9 k s_{n - 8,k} + 9 k s_{n - 8,k - 7} - 180 k s_{n - 8,k - 6} + 1134 k s_{n - 8,k - 5} - 2646 k s_{n - 8,k - 4} + 2079 k s_{n - 8,k - 3} - 378 k s_{n - 8,k - 2} + 9 k s_{n - 8,k - 1} + 9 k s_{n - 8,k + 1} + s_{n - 8,k} + s_{n - 8,k - 8} - 27 s_{n - 8,k - 7} + 246 s_{n - 8,k - 6} - 924 s_{n - 8,k - 5} + 1407 s_{n - 8,k - 4} - 735 s_{n - 8,k - 3} + 85 s_{n - 8,k - 2} + s_{n - 8,k + 1}\\\\\\end{array}\n\n\n\n\n```python\ns = oeis.oeis_search(seq=[1,1,4,3,1,27,19,6,1,256,175,55,10,1])\n```\n\n *\n\n\n```python\ns()\n```\n\n\n\n\n_Results for query: https://oeis.org/search?q=1%2C+1%2C+4%2C+3%2C+1%2C+27%2C+19%2C+6%2C+1%2C+256%2C+175%2C+55%2C+10%2C+1&start=0&fmt=json_
                                        A039621: Triangle of Lehmer-Comtet numbers of 2nd kind.
                                        \n\nby _Len Smiley_\n\n_Keywords_: `tabl,sign`\n\n_Data_:\n\n$$\n\\begin{array}{c|ccccccccccc}\nn, k & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\\\n\\hline\n0 & 1 & & & & & & & & & \\\\1 & -1 & 1 & & & & & & & & \\\\2 & 4 & -3 & 1 & & & & & & & \\\\3 & -27 & 19 & -6 & 1 & & & & & & \\\\4 & 256 & -175 & 55 & -10 & 1 & & & & & \\\\5 & -3125 & 2101 & -660 & 125 & -15 & 1 & & & & \\\\6 & 46656 & -31031 & 9751 & -1890 & 245 & -21 & 1 & & & \\\\7 & -823543 & 543607 & -170898 & 33621 & -4550 & 434 & -28 & 1 & & \\\\8 & 16777216 & -11012415 & 3463615 & -688506 & 95781 & -9702 & 714 & -36 & 1 & \\\\9 & -387420489\n\\end{array}\n$$\n\n\n_Comments_:\n - Also the Bell transform of (-n)^n adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 16 2016\n\n_Formulae_:\n - (k-1)!*a(n, k) = Sum_{i=0..k-1}((-1)^(n-k-i)*binomial(k-1, i)*(n-i-1)^(n-1)).\n\n_Cross references_:\n - Cf. A008296 (matrix inverse). Also A045531 (for column |a(n, 2)|). A185164.\n\n_Links_:\n - D. H. Lehmer, Numbers Associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.\n\n\n\n\n\n```python\ns = oeis.oeis_search(id=264428)\n```\n\n *\n\n\n```python\ns()\n```\n\n\n\n\n_Results for query: https://oeis.org/search?q=id%3AA264428&start=0&fmt=json_
                                        A264428: Triangle read by rows, Bell transform of Bell numbers.
                                        \n\nby _Peter Luschny_, Nov 13 2015\n\n_Keywords_: `nonn,tabl`\n\n_Data_:\n\n$$\n\\begin{array}{c|ccccccccccc}\nn, k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\\\n\\hline\n0 & 1 & & & & & & & & & \\\\1 & 0 & 1 & & & & & & & & \\\\2 & 0 & 1 & 1 & & & & & & & \\\\3 & 0 & 2 & 3 & 1 & & & & & & \\\\4 & 0 & 5 & 11 & 6 & 1 & & & & & \\\\5 & 0 & 15 & 45 & 35 & 10 & 1 & & & & \\\\6 & 0 & 52 & 205 & 210 & 85 & 15 & 1 & & & \\\\7 & 0 & 203 & 1029 & 1330 & 700 & 175 & 21 & 1 & & \\\\8 & 0 & 877 & 5635 & 8946 & 5845 & 1890 & 322 & 28 & 1 & \\\\9 & 0 & 4140 & 33387 & 63917 & 50358 & 20055 & 4410 & 546 & 36 & 1\n\\end{array}\n$$\n\n\n_Comments_:\n - Consider the sequence S0 -> T0 -> S1 -> T1 -> S2 -> T2 -> ... Here Sn -> Tn indicates the Bell transform mapping a sequence Sn to a triangle Tn as defined in the link and Tn -> S{n+1} the operator associating a triangle with the sequence of its row sums. If\n - S0 = A000012 = <1,1,1,...> then\n - T0 = A048993 # Stirling subset numbers,\n - S1 = A000110 # Bell numbers,\n - T1 = A264428 # Bell transform of Bell numbers,\n - S2 = A187761 # second-order Bell numbers,\n - T2 = A264430 # Bell transform of second-order Bell numbers,\n - S3 = A264432 # third-order Bell numbers.\n - This construction is closely related to permutations trees and A179455. Sn is A179455_col(n+1) prepended by A179455_diag(k) = k! for k <= n. In other words, Sn 'converges' to n! for n -> inf.\n - Given a sequence (s(n))n>=0 with s(0) = 0 and with e.g.f. B(x) = Sum_{n >= 1} s(n)*x^n/n!, then the Bell matrix associated with s(n) equals the exponential Riordan array [1, B(x)] belonging to the Lagrange subgroup of the exponential Riordan group. Omitting the first row and column from the Bell matrix produces the exponential Riordan array [d/dx(B(x)), B(x)] belonging to the Derivative subgroup of the exponential Riordan group. - _Peter Bala_, Jun 07 2016\n\n_Formulae_:\n - From _Peter Bala_, Jun 07 2016: \n - E.g.f. exp(t*B(x)), where B(x) = Integral_{u = 0..x} exp(exp(u) - 1) du = x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 15*x^5/5! + 52*x^6/6! + ....\n - Row polynomial recurrence: R(n+1,t) = t*Sum_{k = 0 ..n} binomial(n,k)*Bell(k)* R(n-k,t) with R(0,t) = 1. \n\n_Cross references_:\n - Cf. A000012, A000110, A000217, A000914, A027801, A048993, A051836, A179455, A187761 (row sums), A264430, A264432, A265312.\n\n_Links_:\n - G. C. Greubel, A264428/b264428.txt\">Table of n, a(n) for n = 0..1325\n - Peter Luschny, The Bell transform\n - Peter Luschny, Permutation Trees\n\n\n\n\n---\n
                                        This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.\n", "meta": {"hexsha": "7c2b523111b2c8f543fc709433f70b48f079cff0", "size": 32101, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/stirling-II-array-doubly-indexed-unfolding.ipynb", "max_stars_repo_name": "massimo-nocentini/Ph.D", "max_stars_repo_head_hexsha": "7b5174c669d2c1acfe4538e69338064d8acfbe92", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/stirling-II-array-doubly-indexed-unfolding.ipynb", "max_issues_repo_name": "massimo-nocentini/Ph.D", "max_issues_repo_head_hexsha": "7b5174c669d2c1acfe4538e69338064d8acfbe92", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/stirling-II-array-doubly-indexed-unfolding.ipynb", "max_forks_repo_name": "massimo-nocentini/Ph.D", "max_forks_repo_head_hexsha": "7b5174c669d2c1acfe4538e69338064d8acfbe92", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 52.1967479675, "max_line_length": 1306, "alphanum_fraction": 0.5064328214, "converted": true, "num_tokens": 10507, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819874558604, "lm_q2_score": 0.6723316860482763, "lm_q1q2_score": 0.41105148244574474}} {"text": "```python\n# import numpy as np\n\n# # !/usr/bin/env python3\n# # -*- coding: utf-8 -*-\n# \"\"\"\n# Created on 20181219\n\n# @author: zhangji\n\n# Trajection of a ellipse, Jeffery equation. \n# \"\"\"\n\n# %pylab inline\n# pylab.rcParams['figure.figsize'] = (25, 11)\n# fontsize = 40\n\n# import numpy as np\n# import scipy as sp\n# from scipy.optimize import leastsq, curve_fit\n# from scipy import interpolate\n# from scipy.interpolate import interp1d\n# from scipy.io import loadmat, savemat\n# # import scipy.misc\n\n# import matplotlib\n# from matplotlib import pyplot as plt\n# from matplotlib import animation, rc\n# import matplotlib.ticker as mtick\n# from mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\n# from mpl_toolkits.mplot3d import Axes3D, axes3d\n\n# from sympy import symbols, simplify, series, exp\n# from sympy.matrices import Matrix\n# from sympy.solvers import solve\n\n# from IPython.display import display, HTML\n# from tqdm import tqdm_notebook as tqdm\n# import pandas as pd\n# import re\n# from scanf import scanf\n# import os\n# import glob\n\n# from codeStore import support_fun as spf\n# from src.support_class import *\n# from src import stokes_flow as sf\n\n# rc('animation', html='html5')\n# PWD = os.getcwd()\n# font = {'size': 20}\n# matplotlib.rc('font', **font)\n# np.set_printoptions(linewidth=90, precision=5)\n\nimport os\nimport glob\nimport natsort \nimport numpy as np\nimport scipy as sp\nfrom scipy.optimize import leastsq, curve_fit\nfrom scipy import interpolate\nfrom scipy import spatial\n# from scipy.interpolate import interp1d\nfrom scipy.io import loadmat, savemat\n# import scipy.misc\nimport importlib\nfrom IPython.display import display, HTML\n\nimport matplotlib\nfrom matplotlib import pyplot as plt\nfrom matplotlib import colors as mcolors\nfrom matplotlib import animation, rc\nimport matplotlib.ticker as mtick\nfrom mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\nfrom mpl_toolkits.mplot3d import Axes3D, axes3d\n\nfrom time import time\nfrom src.support_class import *\nfrom src import jeffery_model as jm\nfrom codeStore import support_fun as spf\n\n# %matplotlib notebook\n\nrc('animation', html='html5')\nfontsize = 40\nPWD = os.getcwd()\n```\n\n\n```python\nrc1 = np.array((0, 1, 0))\nrc2 = np.array((-np.sqrt(3) / 2, - 1 / 2, 0))\nrc3 = np.array((np.sqrt(3) / 2, - 1 / 2, 0))\nnc1 = np.array((-1, np.sqrt(3), np.sqrt(5))) / 3\nnc2 = np.array((-1, -np.sqrt(3), np.sqrt(5))) / 3\nnc3 = np.array((2, 0, np.sqrt(5))) / 3\nrc_all = np.vstack((rc1, rc2, rc3))\nnc_all = np.vstack((nc1, nc2, nc3))\n\n# %matplotlib notebook\n%matplotlib inline \nfig = plt.figure(figsize=np.array((7, 4))*2)\nfig.patch.set_facecolor('white')\nax0 = axes3d.Axes3D(fig)\nax0.plot(rc_all[:, 0], rc_all[:, 1], rc_all[:, 2], 'ob', ms=fontsize*0.5)\nax0.plot(np.hstack((rc1[0], rc2[0], rc3[0], rc1[0])), \n np.hstack((rc1[1], rc2[1], rc3[1], rc1[1])), \n np.hstack((rc1[2], rc2[2], rc3[2], rc1[2])), '-b')\nax0.quiver(rc_all[:, 0], rc_all[:, 1], rc_all[:, 2], nc_all[:, 0], nc_all[:, 1], nc_all[:, 2], \n length=0.01*fontsize, arrow_length_ratio=0.2, )\nax0.plot(np.zeros(1), np.zeros(1), np.zeros(1), '.r', ms=fontsize*0.5)\nax0.quiver(np.zeros(1), np.zeros(1), np.zeros(1), np.zeros(1), np.zeros(1), np.ones(1), \n length=0.01*fontsize, arrow_length_ratio=0.2, colors='r')\ntxticks = np.array([-1, -0.5, 0, 0.5, 1])\nax0.set_xticks(txticks)\nax0.set_xticklabels(txticks)\nax0.set_xlim(txticks.min(), txticks.max())\ntyticks = np.array([-1, -0.5, 0, 0.5, 1])\nax0.set_yticks(tyticks)\nax0.set_yticklabels(tyticks)\nax0.set_ylim(tyticks.min(), tyticks.max())\ntzticks = np.array([-1, -0.5, 0, 0.5, 1])\nax0.set_zticks(tzticks)\nax0.set_zticklabels(tzticks)\nax0.set_zlim(tzticks.min(), tzticks.max())\nax0.set_xlabel('x')\nax0.set_ylabel('y')\nax0.set_zlabel('z')\nax0.view_init(50, -30)\n# plt.tight_layout()\n\n```\n\n\n```python\nnp.linspace(np.pi / 2, 0, 10, endpoint=False)\n```\n\n\n\n\n array([ 1.5708 , 1.41372, 1.25664, 1.09956, 0.94248, 0.7854 , 0.62832, 0.47124,\n 0.31416, 0.15708])\n\n\n\n\n```python\nprint('%s Current process uses: %08.3fs %s' % ('#' * 30, 10, '#' * 30))\n\n```\n\n ############################## Current process uses: 0010.000s ##############################\n\n", "meta": {"hexsha": "e9a36065a2b6d06823740cfc32f32ac9ff652c3b", "size": 191971, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "HelicodsParticles/Sketch.ipynb", "max_stars_repo_name": "pcmagic/stokes_flow", "max_stars_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-11-11T05:00:53.000Z", "max_stars_repo_stars_event_max_datetime": "2018-11-11T05:00:53.000Z", "max_issues_repo_path": "HelicodsParticles/Sketch.ipynb", "max_issues_repo_name": "pcmagic/stokes_flow", "max_issues_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "HelicodsParticles/Sketch.ipynb", "max_forks_repo_name": "pcmagic/stokes_flow", "max_forks_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 888.7546296296, "max_line_length": 185352, "alphanum_fraction": 0.9484453381, "converted": true, "num_tokens": 1383, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.611381973294151, "lm_q2_score": 0.6723316991792861, "lm_q1q2_score": 0.4110514809524415}} {"text": "# 1\uc870 \uc778\uacf5\uc9c0\ub2a5\uc744 \ud65c\uc6a9\ud55c \ub9c8\uc2a4\ud06c \ucc29\uc6a9 \uac10\uc9c0 \ubaa8\ub378 (\uace0\uc724\ud64d, \ubc15\uc2e0, \uc548\ud61c\ube48)\n\n## \ud504\ub85c\uc81d\ud2b8\ub97c \ud558\uac8c \ub41c \uacc4\uae30\n\n - \uc804\uad6d\uc801\uc73c\ub85c \ud655\uc9c4\uc774 \ub2e4\uc2dc \ub298\uc5b4\ub098\uace0 \uc788\ub294 \ucd94\uc138\uc774\uace0 \ub9c8\uc2a4\ud06c \ubbf8\ucc29\uc6a9\uc790\uc5d0 \ub300\ud55c \ucc98\ubc8c\uc774 \uac15\ud654\ub418\uace0 \uc788\uc2b5\ub2c8\ub2e4. \ud558\uc9c0\ub9cc \ub9c8\uc2a4\ud06c\ub97c \uc62c\ubc14\ub978 \ubc29\ubc95\uc73c\ub85c \ucc29\uc6a9\ud55c \uc0ac\ub78c\ub4e4 \uc0ac\uc774\uc5d0 \uadf8\ub807\uc9c0 \uc54a\uc740 \uc0ac\ub78c\ub4e4\uc774 \uc874\uc7ac\ud569\ub2c8\ub2e4. \uadf8\ub7f0 \uc774\ub4e4\uc744 \ud1b5\ud574 COVID19\uc758 \uac10\uc5fc\uc774 \uc804\ud30c\ub418\ub294 \uac83\uc744 \uc608\ubc29\ud558\uace0 \ub9c8\uc2a4\ud06c \ucc29\uc6a9\uc5d0 \ub300\ud55c \uac10\uc2dc\uc5ed\uc774 \uc5c6\ub354\ub77c\ub3c4 \ub9c8\uc2a4\ud06c \ucc29\uc6a9\uc744 \uad8c\uc7a5\ud558\ub294 \uc778\uacf5\uc9c0\ub2a5\uc744 \ub9cc\ub4e4\uc5b4\ubcf4\uace0 \uc2f6\uc5b4\uc11c \uc774 \ud504\ub85c\uc81d\ud2b8\ub97c \ud558\uac8c \ub418\uc5c8\uc2b5\ub2c8\ub2e4.\n\n## \ud504\ub85c\uc81d\ud2b8 \uac1c\uc694\n\n - \uae30\uc874\uc758 \ub9c8\uc2a4\ud06c \ucc29\uc6a9 \uac10\uc9c0 \uc778\uacf5\uc9c0\ub2a5\uc744 \uc774\uc6a9\ud558\uc5ec \uc2e4\uc2dc\uac04 \uc601\uc0c1 \uc18d\uc5d0\uc11c \uc778\uc6d0 \uc218\ub97c \ud654\uba74\uc5d0 \ud45c\uc2dc\ud558\uace0 \uac01\uac01\uc758 \ub9c8\uc2a4\ud06c \ucc29\uc6a9 \ube44\uc728\uc744 \ud45c\uc2dc\ud558\uba70 \ub9c8\uc2a4\ud06c\ub97c \ucc29\uc6a9\ud558\uc9c0 \uc54a\uc740 \uc0ac\ub78c\uc774 \uc11e\uc5ec\uc788\ub294 \uacbd\uc6b0\ub97c \ud0d0\uc9c0\ud560 \uc218 \uc788\ub294 \uc778\uacf5\uc9c0\ub2a5\uc744 \ubaa9\ud45c\ub85c \ud558\uace0 \uc788\uc2b5\ub2c8\ub2e4. \n\n## \uc778\uacf5\uc9c0\ub2a5 \ud559\uc2b5 \ub2e8\uacc4\n\n#### \uccab\ubc88\uc9f8 \ub2e8\uacc4: \ubaa8\ub4c8 \ubd88\ub7ec\uc624\uae30\n\n **\uc0ac\uc6a9\ub41c \ubaa8\ub4c8**\n```pyrthon\n - keras # tensorflow \uc704\uc5d0\uc11c \uc791\ub3d9\ud558\uba70 \ub354 \uc0c1\uc704 \uc218\uc900\uc758 \uae30\ub2a5\uc744 \uc81c\uacf5\n - sklearn # machine learning\uc758 \ubd84\uc11d\uc5d0\uc11c \uc911\uc694\n - imutils # Open CV\uac00 \ucc98\ub9ac\ud558\uae30 \ud798\ub4e0 \uc774\ubbf8\uc9c0\ub098 \ube44\ub514\uc624 \uc2a4\ud2b8\ub9bc \ud30c\uc77c\uc758 \ucc98\ub9ac\ub97c \ubcf4\uc644\n - matplotlib # \ud30c\uc774\uc36c\uc5d0\uc11c \uadf8\ub798\ud504 \ud45c\uc2dc\n - numpy # \uc800 \uc218\uc900\uc758 \uc5b8\uc5b4\ub85c \ud30c\uc774\uc36c\uc758 \ub9ac\uc2a4\ud2b8\ubcf4\ub2e4 \ud6a8\uc728\uc801\n - argparse # \ub2e4\uc591\ud55c \uc778\uc790 \uad00\ub9ac\n - os # \ucef4\ud4e8\ud130\uc758 \ub514\ub809\ud1a0\ub9ac, \ud30c\uc77c\uc744 \uc774\uc6a9\uac00\ub2a5\ud558\uac8c \ub9cc\ub4dc\ub294 \uae30\ub2a5\n ```\n\n```python\nfrom tensorflow.keras.preprocessing.image import ImageDataGenerator\nfrom tensorflow.keras.applications import MobileNetV2\nfrom tensorflow.keras.layers import AveragePooling2D\nfrom tensorflow.keras.layers import Dropout\nfrom tensorflow.keras.layers import Flatten\nfrom tensorflow.keras.layers import Dense\nfrom tensorflow.keras.layers import Input\nfrom tensorflow.keras.models import Model \nfrom tensorflow.keras.optimizers import Adam\nfrom tensorflow.keras.applications.mobilenet_v2 import preprocess_input\nfrom tensorflow.keras.preprocessing.image import img_to_array\nfrom tensorflow.keras.preprocessing.image import load_img\nfrom tensorflow.keras.utils import to_categorical\nfrom sklearn.preprocessing import LabelBinarizer \nfrom sklearn.model_selection import train_test_split\nfrom sklearn.metrics import classification_report\nfrom imutils import paths\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport argparse\nimport os \n```\n\n#### \ub450\ubc88\uc9f8 \ub2e8\uacc4: \uc678\ubd80 \ub370\uc774\ud130 \ubd88\ub7ec\uc624\uae30\n\n```python\nap = argparse.ArgumentParser()\nap.add_argument(\"-d\", \"--dataset\", required=True, \n help=\"path to input dataset\") # --dataset \ub370\uc774\ud130 \ubd88\ub7ec\uc624\uae30\nap.add_argument(\"-p\", \"--plot\", type=str, default=\"plot.png\", \n help=\"path to output loss/accuracy plot\") # --plot\uc758 \ub370\uc774\ud130\ub97c \ubd88\ub7ec\uc640 plot.png\uc5d0 \uc800\uc7a5\nap.add_argument(\"-m\", \"--model\", type=str, default=\"mask_detector.model\", \n help=\"path to output face mask detector model\") # --model\uc758 \ub370\uc774\ud130\ub97c \ubd88\ub7ec\uc640 mask_detector.model\uc5d0 \uc800\uc7a5\nargs = vars(ap.parse_args()) # args\uc5d0 \uac01 \ub370\uc774\ud130\ub4e4\uc744 \uc5c5\ub370\uc774\ud2b8\n```\n\n#### \uc138\ubc88\uc9f8 \ub2e8\uacc4: \uc778\uacf5\uc9c0\ub2a5 \ud559\uc2b5\uc5d0 \ud544\uc694\ud55c \uc124\uc815\n\n```python\nINIT_LR = 1e-4 # \ud559\uc2b5\ub960: 0.0001\nEPOCHS = 20 # \uc5d0\ud3ed\nBS = 32 # \ubc30\uce58 \ud06c\uae30\n\nprint(\"[INFO] loading images...\") \nimagePaths = list(paths.list_images(args[\"dataset\"])) # dataset\uc758 \uc774\ubbf8\uc9c0 \ub370\uc774\ud130\ub4e4\uc744 \ubc30\uc5f4\ub85c \ub9cc\ub4e4\uae30\ndata = [] # \ubb38\uc81c\uc758 \ubcf4\uae30\uc640 \uac19\uc740 \uc5ed\ud560\uc744 \ud560 \ubc30\uc5f4\nlabels = [] # \ubb38\uc81c\uc758 \uc815\ub2f5\uacfc \uac19\uc740 \uc5ed\ud560\uc744 \ud560 \ubc30\uc5f4\n\nfor imagePath in imagePaths:\n label = imagePath.split(os.path.sep)[-2]\n image = load_img(imagePath, target_size=(224, 224))\n image = img_to_array(image)\n image = preprocess_input(image)\n data.append(image)\n labels.append(label)\n\n# \uac01\uac01\uc758 \ubc30\uc5f4\uc744 \ub118\ud30c\uc774 \ubc30\uc5f4\ub85c \ub9cc\ub4e4\uae30\ndata = np.array(data, dtype=\"float32\")\nlabels = np.array(labels) \n\nlb = LabelBinarizer()\nlabels = lb.fit_transform(labels)\nlabels = to_categorical(labels) \n\n# data\uc640 lables\ubc30\uc5f4\uc744 \ub098\ub204\uae30\n(trainX, testX, trainY, testY) = train_test_split(data, labels, \n test_size=0.20, stratify=labels, random_state=42) \n\n# \uc774\ubbf8\uc9c0 \ub370\uc774\ud130 \uc804\ucc98\ub9ac \uacfc\uc815\naug = ImageDataGenerator(\n rotation_range=20,\n zoom_range=0.15,\n width_shift_range=0.2,\n height_shift_range=0.2,\n shear_range=0.15,\n horizontal_flip=True,\n fill_mode=\"nearest\")\n```\n\n#### \ub124\ubc88\uc9f8 \ub2e8\uacc4: \ubaa8\ub378 \uc124\uc815\n\n```python\n#\ubaa8\ubc14\uc77c \ud658\uacbd\uc5d0\uc11c \uc791\ub3d9\uac00\ub2a5\ud558\ub3c4\ub85d \ucd5c\uc801\ud654\nbaseModel = MobileNetV2(weights=\"imagenet\", include_top=False, input_tensor=Input(shape=(224, 224, 3)))\n\n#\ud65c\uc131\ud654 \ud568\uc218\ub85c relu\uc640 softmax\uac00 \uc0ac\uc6a9\ub428\nheadModel = baseModel.output\nheadModel = AveragePooling2D(pool_size=(7, 7))(headModel)\nheadModel = Flatten(name=\"flatten\")(headModel)\nheadModel = Dense(128, activation=\"relu\")(headModel)\nheadModel = Dropout(0.5)(headModel)\nheadModel = Dense(2, activation=\"softmax\")(headModel)\n\n#\ubaa8\ub378 \uc0dd\uc131\nmodel = Model(inputs=baseModel.input, outputs=headModel)\n\nfor layer in baseModel.layers:\n layer.trainable = False\n```\n\n#### \ub2e4\uc12f\ubc88\uc9f8 \ub2e8\uacc4: \ud559\uc2b5 \uc2dc\uc791\n\n```python\n# \ubaa8\ub378 \uac00\uc838\uc624\uae30\nprint(\"[INFO] compiling model...\")\nopt = Adam(lr=INIT_LR, decay=INIT_LR / EPOCHS)\nmodel.compile(loss=\"binary_crossentropy\", optimizer=opt, metrics=[\"accuracy\"])\n\n# \ud559\uc2b5\nprint(\"[INFO] training head...\")\nH = model.fit(\n aug.flow(trainX, trainY, batch_size=BS),\n steps_per_epoch=len(trainX) // BS,\n validation_data=(testX, testY),\n\tvalidation_steps=len(testX) // BS,\n epochs=EPOCHS)\n\n# \uc608\uce21\uac12 \uad6c\ud558\uae30 \ubc0f \ucd9c\ub825\nprint(\"[INFO] evaluating network...\")\npredIdxs = model.predict(testX, batch_size=BS)\npredIdxs = np.argmax(predIdxs, axis=1)\nprint(classification_report(testY.argmax(axis=1), predIdxs, target_names=lb.classes_))\n\n# mask_detector.model\uc5d0 \uc800\uc7a5\nprint(\"[INFO] saving mask detector model...\")\nmodel.save(args[\"model\"], save_format=\"h5\")\n```\n\n#### \uc5ec\uc12f\ubc88\uc9f8 \ub2e8\uacc4: \uc2dc\uac01\ud654\n\n```python\n# plot.png\uc5d0 \uc800\uc7a5\nN = EPOCHS\nplt.style.use(\"ggplot\")\nplt.figure()\nplt.plot(np.arange(0, N), H.history[\"loss\"], label=\"train_loss\")\nplt.plot(np.arange(0, N), H.history[\"val_loss\"], label=\"val_loss\")\nplt.plot(np.arange(0, N), H.history[\"accuracy\"], label=\"train_acc\")\nplt.plot(np.arange(0, N), H.history[\"val_accuracy\"], label=\"val_acc\")\nplt.title(\"Training Loss and Accuracy\")\nplt.xlabel(\"Epoch #\")\nplt.ylabel(\"Loss/Accuracy\")\nplt.legend(loc=\"lower left\")\nplt.savefig(args[\"plot\"])\n```\n\n## \uc601\uc0c1\uc744 \ud1b5\ud55c \ub370\uc774\ud130 \uac80\uc99d\n\n#### 1) Import packages\n\n```python\nfrom tensorflow.keras.applications.mobilenet_v2 import preprocess_input\nfrom tensorflow.keras.preprocessing.image import img_to_array\nfrom tensorflow.keras.models import load_model\nfrom imutils.video import VideoStream\nimport numpy as np\nimport argparse\nimport imutils\nimport time\nimport cv2\nimport os\n```\n\n#### 2) Argument setting\n\n```python\nap = argparse.ArgumentParser() # --\ub97c \uc778\uc2dd\ud568. \uc678\ubd80 \ub370\uc774\ud130 \ubd88\ub7ec\ub4e4\uc774\uae30\n\nap.add_argument(\"-f\", -face\"-\", type=str, #\uc678\ubd80\uc5d0\uc11c --face, --model, \uadf8\ub9ac\uace0 --confidence\uc758 \ub370\uc774\ud130\ub97c \ubc1b\uc544\uc634.\n default=\"face_detector\",\n help=\"path to face detector model directory\") #--face\uc758 \ub370\uc774\ud130\ub294 face_detector\uc5d0 \ucd94\uac00\ud569\ub2c8\ub2e4.\n \nap.add_argument(\"-m\", \"--model\", type=str,\n default=\"mask_detector.model\",\n help=\"path to trained face mask detector model\") #--model\uc758 \ub370\uc774\ud130\ub294 mask_detector.model\uc5d0 \ucd94\uac00\ud569\ub2c8\ub2e4.\n \nap.add_argument(\"-c\", \"--confidence\", type=float, default=0.5,\n help=\"minimum probability to filter weak detections\") \n \nargs = vars(ap.parse_args()) #\ub9c8\uc9c0\ub9c9\uc73c\ub85c args\uc5d0 \uac01\uac01\uc758 \ub370\uc774\ud130\ub4e4\uc744 \uc5c5\ub370\uc774\ud2b8\ud574\uc90d\ub2c8\ub2e4\n```\n\n#### 3) detect_and_predict_mask \ud568\uc218 \uc815\uc758\n\n```python\ndef detect_and_predict_mask(frame, faceNet, maskNet):\n (h, w) = frame.shape[:2] #frame\ubc30\uc5f4\uc758 1,2\ubc88\uc9f8 index\uac12\uc744 h(\uc138\ub85c),w(\uac00\ub85c)\uc5d0 \uc800\uc7a5 #frame\uc740 \uc2e4\uc2dc\uac04 \ube44\ub514\uc624\uc758 \ud654\uba74\n blob = cv2.dnn.blobFromImage(frame, 1.0, (300, 300),(104.0, 177.0, 123.0)) #\uc774\ubbf8\uc9c0 \uac00\uacf5\ud574\uc11c blob\uc5d0 \uc800\uc7a5\n\n faceNet.setInput(blob) #blob\uc5d0 \uc800\uc7a5\ub41c \uac00\uacf5\ud55c \uc774\ubbf8\uc9c0\ub97c faceNet\uc5d0 \uc800\uc7a5\n detections = faceNet.forward() #\uc21c\ubc29\ud5a5\uc73c\ub85c \ub525\ub7ec\ub2dd \ub124\ud2b8\uc6cc\ud06c\ub97c \uc2e4\ud589\ud574 detection\uc5d0 \uc800\uc7a5\n\n faces = [] #\uc5bc\uad74\uc744 \ubaa8\uc73c\ub294 \ub9ac\uc2a4\ud2b8\n locs = [] #box\uc758 \uc88c\ud45c\ub97c \ubaa8\uc73c\ub294 \ub9ac\uc2a4\ud2b8\n preds = [] #\ub9c8\uc2a4\ud06c \uc608\uce21\ud55c \uac83\uc744 \ubaa8\uc73c\ub294 \ub9ac\uc2a4\ud2b8\n \n for i in range(0, detections.shape[2]): #200\ubc88 \ubc18\ubcf5\n confidence = detections[0, 0, i, 2] #\uc5bc\uad74\uc744 \uc778\uc2dd\ud558\uace0 \ub9c8\uc2a4\ud06c\ub97c \ucc29\uc6a9\ud588\ub294\uc9c0\uc5d0 \ub300\ud55c \ucd94\uce21 \uc815\ud655\ub3c4\ub97c confidence\uc5d0 \uc800\uc7a5\n #detections\ub294 \uc2e4\uc2dc\uac04\uc73c\ub85c \ucc0d\ud788\ub294 \ud654\uba74\uc774 \uc800\uc7a5\ub418\ub294 \uac83\n\n if confidence > args[\"confidence\"]: #\uae30\ubcf8\uac12\uc774 0.5\ub85c \uc124\uc815. confidence\uac00 \ud06c\uba74 \ub9c8\uc2a4\ud06c\ub97c \uc798 \uc4f4 \uac83.\n box = detections[0, 0, i, 3:7] * np.array([w, h, w, h] #detection\uc5d0\uc11c \uc5bc\uad74\ud06c\uae30\ub97c \uc774\uc6a9\ud574 box\uc758 \ub124 \ubaa8\ud241\uc774\ub97c box\uc5d0 \uc800\uc7a5.\n (startX, startY, endX, endY) = box.astype(\"int\") #box\uc758 \ub450 \ubaa8\ud241\uc774 \uac12\uc744 \uc815\uc218\ub85c \ubc14\uafd4\uc11c \uc88c\ud45c\ub85c \uc9c0\uc815\n (startX, startY) = (max(0, startX), max(0, startY)) #\uc88c\ud45c\uac00 \uc74c\uc218\uac00 \uc548\ub098\uc624\uac8c \ud558\uae30 \uc704\ud568.\n (endX, endY) = (min(w - 1, endX), min(h - 1, endY)) #box\uac00 \ud654\uba74 \ubc16\uc73c\ub85c \uc548\ub098\uac00\uac8c \ud558\uae30 \uc704\ud568.\n\n face = frame[startY:endY, startX:endX] #frame(\ubc30\uacbd+\uc0ac\ub78c)\uc5d0\uc11c box\ud06c\uae30 \ub9cc\ud074\uc744 slicing \ud55c \uac83\uc744 face\uc5d0 \uc800\uc7a5.\n face = cv2.cvtColor(face, cv2.COLOR_BGR2RGB)\n face = cv2.resize(face, (224, 224)) \n face = img_to_array(face)\n face = preprocess_input(face)\n\n faces.append(face) #\uac00\uacf5\ub41c face\uac00 faces\uc5d0 \ucd94\uac00\ub428.\n locs.append((startX, startY, endX, endY)) #locs\uc5d0\ub294 box\uc758 \uc88c\ud45c\uac00 \ucd94\uac00\ub428.\n\n if len(faces) > 0: #\uc800\uc7a5\ub41c \uc5bc\uad74\uc758 \uac1c\uc218\uac00 1\uac1c \uc774\uc0c1\uc774\uba74 \uc5bc\uad74 \uc0ac\uc9c4\ub4e4\uc744 \ub118\ud30c\uc774 \uc5b4\ub808\uc774\ub85c \ub9cc\ub4e6.\n faces = np.array(faces, dtype=\"float32\")\n preds = maskNet.predict(faces, batch_size=32) #\uc608\uce21\ud55c \uac83(\ub9c8\uc2a4\ud06c\ub97c \ucc29\uc6a9\ud588\uc744 \ud655\ub960\uacfc \ucc29\uc6a9\ud558\uc9c0 \uc54a\uc558\uc744 \ud655\ub960) \n\n return (locs,preds) #box\uc758 \uc88c\ud45c\ub4e4\uacfc \uc608\uce21\ud55c \uac12\uc744 return\n```\n\n#### 4) face_detector\ud3f4\ub354 \ub0b4\uc758 \ud30c\uc77c\ub4e4\uc744 \ubd88\ub7ec\uc640 faceNet\uc5d0 \uc800\uc7a5\n\n```python\nprint(\"[INFO] loading face detector model...\")\n\nprototxtPath = os.path.sep.join([args[\"face\"], \"deploy.prototxt\"]) #\"deploy.prototxt\"\ud30c\uc77c\uc744 \ubd88\ub7ec\uc640\uc11c prototxtPath\uc5d0 \uc800\uc7a5 \n\nweightsPath = os.path.sep.join([args[\"face\"],\"res10_300x300_ssd_iter_140000.caffemodel\"])\n\nfaceNet = cv2.dnn.readNet(prototxtPath, weightsPath) #dnn(\ub525\ub7ec\ub2dd \ub124\ud2b8\uc6cc\ud06c)\ub97c \uc0ac\uc6a9\ud574 prototxtPath\uc640 weightPath\ub97c faceNet\uc5d0 \uc800\uc7a5\n```\n\n#### 5) mask_detector.model\uc744 \ubd88\ub7ec\uc640 maskNet\uc5d0 \uc800\uc7a5, \ube44\ub514\uc624 \uc2dc\uc791\n\n```python\nprint(\"[INFO] loading face mask detector model...\")\nmaskNet = load_model(args[\"model\"]) #mask_detector.model\uc744 \ubd88\ub7ec\uc640\uc11c model\uc5d0 \uc800\uc7a5\ud558\uace0, \uadf8 model\uc744 maskNet\uc774\ub77c\ub294 \ubcc0\uc218\uc5d0 \uc800\uc7a5.\n\nprint(\"[INFO] starting video stream...\")\nvs = VideoStream(src=0).start() #video stream\uc744 \ucd08\uae30\ud654\ud558\uace0, \uce74\uba54\ub77c \uc13c\uc11c\ub97c warm up\uc2dc\ud0b4.\ntime.sleep(2.0)\n```\n\n#### 6) \uc2e4\uc2dc\uac04 video\ub97c \uc77d\uc5b4\uc624\uace0, q\ud0a4\ub97c \ub204\ub974\uba74 video stop\n\n```python\nwhile True:\n\n frame = vs.read() #\uc2e4\uc2dc\uac04 \ud654\uba74\uc774 frame\n frame = imutils.resize(frame, width=1100) #frame\ud06c\uae30 \uc870\uc815\n\n (locs, preds) = detect_and_predict_mask(frame, faceNet, maskNet) #frame(\ud654\uba74)\uacfc 4,5\ubc88\uc5d0\uc11c \uc5bb\uc740 faceNet,maskNet\uc744 \ud568\uc218\uc5d0 \ud1b5\uacfc\uc2dc\ud0b4.\n\n maskCount = 0 #\ucd94\uac00\ud55c \ubd80\ubd84\n nonMaskCount = 0 #\ucd94\uac00\ud55c \ubd80\ubd84 #\uc2e4\uc2dc\uac04\uc73c\ub85c \uc0ac\ub78c \uc218\ub97c \uc138\uae30 \uc704\ud574 \ub9e4\ubc88 \ucd08\uae30\ud654\ud568. \n for (box, pred) in zip(locs, preds): #box\uc758 \uc88c\ud45c\ub4e4\uc774 \uc788\ub294 locs\ubcc0\uc218\uc640 \uc608\uce21\ud55c \uac83\uc774 \ubaa8\uc5ec\uc788\ub294 pred\ubcc0\uc218\uc5d0\uc11c \uac01\uac01 \ud558\ub098\uc529 \uc138\ud2b8\ub85c \ubcf4\ub0c4. \n (startX, startY, endX, endY) = box #box\uc758 2\uc810\uc758 \uc88c\ud45c\n (mask, withoutMask) = pred #\ub9c8\uc2a4\ud06c\ub97c \ucc29\uc6a9\ud588\uc744 \ud655\ub960, \ucc29\uc6a9\ud558\uc9c0 \uc54a\uc558\uc744 \ud655\ub960\uc744 pred\uc5d0 \uc800\uc7a5\n\n if mask > withoutMask: #\ucd94\uac00\ud55c \ubd80\ubd84 #\ubc18\ubcf5\ubb38\uc5d0 \ub4e4\uc5b4\uc628 \uc0ac\ub78c\uc774 \ub9c8\uc2a4\ud06c\ub97c \ucc29\uc6a9\ud588\uc744 \ud655\ub960\uc774 \ub354 \ub192\ub2e4\uba74 \n maskCount += 1 #\ucd94\uac00\ud55c \ubd80\ubd84\n else: #\ucd94\uac00\ud55c \ubd80\ubd84\n nonMaskCount += 1 #\ucd94\uac00\ud55c \ubd80\ubd84 #\uc774\ub807\uac8c \ub9e4 \uc21c\uac04 \ub9c8\uc2a4\ud06c\ub97c \ucc29\uc6a9\ud55c \uc0ac\ub78c\uacfc \ucc29\uc6a9\ud558\uc9c0 \uc54a\uc744 \uc0ac\ub78c\uc758 \uc218\ub97c \ud5e4\uc544\ub9b0\ub2e4. \n\n label = \"Mask\" if mask > withoutMask else \"No Mask\" #\ub9c8\uc2a4\ud06c\ub97c \ucc29\uc6a9\ud588\uc744 \ud655\ub960\uc774 \ub192\ub2e4\uba74 label\uc5d0 Mask, \uc544\ub2c8\uba74 No Mask\n color = (0, 255, 0) if label == \"Mask\" else (0, 0, 255) #\uc804\uc790\uc758 \uacbd\uc6b0\ub77c\uba74 \ucd08\ub85d\uc0c9\uc73c\ub85c, \ud6c4\uc790\ub77c\uba74 \ube68\uac04\uc0c9\uc73c\ub85c \uae00\uc790\ub97c \ud45c\uc2dc\ud55c\ub2e4.\n label = \"{}: {:.2f}%\".format(label, max(mask, withoutMask) * 100) \n label2 = \"Good Job\" if mask > withoutMask else \"Warning[Wear Mask]\" #\ucd94\uac00\ud55c \ubd80\ubd84 #\ucc29\uc6a9 \uc5ec\ubd80\uc5d0 \ub300\ud55c \ubc18\uc751\uc744 \ud574\uc90c.\n \n cv2.putText(frame, label, (startX, startY - 10) #frame\uc5d0 \uc704\uc758 label\uafb8\uba70\uc11c \ubcf4\uc774\uac8c \ud568., \n cv2.FONT_HERSHEY_SIMPLEX, 0.45, color, 2) \n cv2.putText(frame, label2, (startX, endY + 20), #\ucd94\uac00\ud55c \ubd80\ubd84 #frame\uc5d0 \uc704\uc758 label2\ub97c \uafb8\uba70\uc11c \ucd9c\ub825\n cv2.FONT_HERSHEY_SIMPLEX, 0.8, color, 2) \n cv2.rectangle(frame, (startX, startY), (endX, endY),color, 2) #frame\uc5d0 \uafb8\ubbfc box\ub97c \uc62c\ub9bc.\n\n color2 = (0, 255, 0) if nonMaskCount == 0 else (0, 0, 255) #\ucd94\uac00\ud55c \ubd80\ubd84 #\ud55c \uba85\uc774\ub77c\ub3c4 \ub9c8\uc2a4\ud06c\ub97c \ucc29\uc6a9\ud558\uc9c0 \uc54a\uc558\uc73c\uba74 \ube68\uac04\uc0c9\uc73c\ub85c color\uc9c0\uc815\n label3 = \"{}/{}\".format(maskCount, nonMaskCount + maskCount) #\ucd94\uac00\ud55c \ubd80\ubd84 #\uba87 \uba85 \uc911, \uba87 \uba85\uc774 \ub9c8\uc2a4\ud06c\ub97c \ucc29\uc6a9\ud588\ub294\uc9c0\uc5d0 \ub300\ud55c \ud655\ub960 \uacc4\uc0b0\n cv2.putText(frame, label3, (100, 130),cv2.FONT_HERSHEY_SIMPLEX, 0.45, color2, 2) #\ucd94\uac00\ud55c \ubd80\ubd84 #frame\uc704\uc5d0 color2\uc0c9\uc744 \uc774\uc6a9\ud574 label3\uc744 \ubcf4\uc5ec\uc90c.\n\n label4 = \"# of Wearing Masks/# of Detected\" #\ucd94\uac00\ud55c \ubd80\ubd84\n cv2.putText(frame, label4, (100, 100), #\ucd94\uac00\ud55c \ubd80\ubd84\n cv2.FONT_HERSHEY_SIMPLEX, 0.45, (255, 0, 0), 2) #frame\uc704\uc5d0 label4\ub97c \ubcf4\uc5ec\uc90c.\n \n cv2.imshow(\"Frame\", frame)\n key = cv2.waitKey(1) & 0xFF #\uc0ac\uc6a9\uc790\uac00 \uc885\ub8cc\ud560 \ub54c \uae4c\uc9c0 \uae30\ub2e4\ub9bc.\n\n if key == ord(\"q\"):\n break #\uc0ac\uc6a9\uc790\uac00 'q'\ud0a4\ub97c \ub204\ub974\uba74 \uc2e4\uc2dc\uac04 \ube44\ub514\uc624 \uc885\ub8cc\ub41c. \n \ncv2.destroyAllWindows()\n\nvs.stop() \n```\n\n## \ub370\ubaa8 \uac00\uc774\ub4dc \ubb38\uc11c\n\n#### 1) \ud504\ub85c\uc81d\ud2b8 \ub2e4\uc6b4\ub85c\ub4dc\n\n\uc544\ub798 \uba85\ub839\uc5b4\ub97c \uc2e4\ud589\ud558\uc5ec \ud504\ub85c\uc81d\ud2b8\ub97c \ub2e4\uc6b4\ub85c\ub4dc\ud55c\ub2e4.\n```\n$ git clone https://github.com/younhong/COVID19-Mask-Detection-Model\n```\n\n\n> \ub9cc\uc57d git\uc774 \uc124\uce58\ub418\uc5b4 \uc788\uc9c0 \uc54a\ub2e4\uba74 [\ub9c1\ud06c](https://github.com/younhong/COVID19-Mask-Detection-Model)\ub85c \ub4e4\uc5b4\uac00\uc11c \uc218\ub3d9\uc73c\ub85c zip \ud30c\uc77c\uc744 \ub2e4\uc6b4\ub85c\ub4dc\n\n#### 2) \ud658\uacbd \uc124\uc815\n\n\ud658\uacbd\uc124\uc815: \uac00\uc0c1\ud658\uacbd\uc744 \uc124\uc815 \nWhy?\n> \uac00\uc0c1\ud658\uacbd\uc744 \uc124\uce58\ud558\uc9c0 \uc54a\uc73c\uba74 \uae30\uc874\uc758 \ud30c\uc774\uc36c \uc124\uce58\ub41c \ud658\uacbd\uacfc \ucda9\ub3cc\ud558\uc5ec \uc758\ub3c4\uce58 \uc54a\uc740 \uc5d0\ub7ec\uac00 \ub0a0 \uc218 \uc788\ub2e4.\n\n##### Step1) \uac00\uc0c1\ud658\uacbd \uc0dd\uc131\uc5d0 \ud544\uc694\ud55c \ud328\ud0a4\uc9c0 \uc124\uce58\nMac\uc758 \uacbd\uc6b0\n```\n$ pip install virtualenv virtualenvwrapper\n```\n\nWindow\uc758 \uacbd\uc6b0\n```\n$ pip install virtualenv virtualenvwrapper-win\n```\n\n##### Step2) \ud658\uacbd\ubcc0\uc218 \uc124\uc815\nMac\uc758 \uacbd\uc6b0\n- \uc544\ub798 \uba85\ub839\uc5b4\ub85c virtualenvwrapper.sh\uac00 \uc124\uce58\ub41c \uc704\uce58 \ud655\uc778(\uacb0\uacfc\uac12\uc744 \uae30\uc5b5\ud574\ub458 \uac83)\n```\n$ find / -name virtualenvwrapper.sh 2>/dev/null\n```\n> \n> (\uacbd\ub85c\uac00 \ub450 \uac00\uc9c0 \uc774\uc0c1 \ub098\uc624\uba74 \ub458 \uc911\uc5d0 \ud558\ub098 \uc120\ud0dd\ud574\uc11c \ubcf5\uc0ac)\n\n- \uc544\ub798 \uba85\ub839\uc5b4\ub85c .zshrc \ud30c\uc77c\uc744 \uc5f4\uace0 \ud658\uacbd \ubcc0\uc218\ub97c \ucd94\uac00(\uc544\ub798 \uc774\ubbf8\uc9c0 \ub9c8\uc9c0\ub9c9 3\uc904 \ucc38\uace0) \n```\n$ vi ~/.zshrc\n``` \n> \ub9c8\uc9c0\ub9c9 \uc904\uc758 source \ub4a4\uc758 \uacbd\ub85c\ub97c \uc704\uc5d0\uc11c \uccb4\ud06c\ud55c \uacbd\ub85c\ub85c \ubcc0\uacbd\n> \n\n- \uc218\uc815\ud55c \ub0b4\uc6a9 \uc801\uc6a9\n```\n$ source ~/.zshrc\n```\n\n- \uc2e4\ud589 \ud6c4 \uc7ac\ubd80\ud305\n\nWindow\uc758 \uacbd\uc6b0\n- \uc544\ub798 \uba85\ub839\uc5b4\ub85c \ud658\uacbd\ubcc0\uc218 \uacbd\ub85c\ub97c \ud504\ub85c\uc81d\ud2b8 \uacbd\ub85c\ub85c \uc124\uc815\n```\n$ setx WORKON_HOME=\"\ud504\ub85c\uc81d\ud2b8\uac00 \uc124\uce58\ub41c \uacbd\ub85c\" \n```\n- \uc7ac\ubd80\ud305\n\n##### Step3) \uac00\uc0c1\ud658\uacbd \uc0dd\uc131\n```\n$ mkvirtualenv test(test \ub300\uc2e0 \ub2e4\ub978 \uc774\ub984 \uc368\ub3c4 \uad1c\ucc2e\uc74c)\n```\n> \n\n> (\ub9e5) \uc774\ubbf8\uc9c0 \ub9e8 \uc544\ub798\uc5d0 \ud45c\uc2dc\ub41c \ubd80\ubd84\uacfc \uac19\uc774 \uc124\uc815\ud55c \ud658\uacbd \uc774\ub984\uc774 \ud45c\uc2dc\ub418\uba74 \uc131\uacf5\n\n> (\uc708\ub3c4\uc6b0) ls\ub97c \ud130\ubbf8\ub110\uc5d0\uc11c \uc785\ub825 \ud6c4 \uc124\uc815\ud55c \ud658\uacbd\uba85\uc774 \ud3f4\ub354\ub85c \uc0dd\uc131\ub418\uc5c8\ub2e4\uba74 \uc131\uacf5\n\n##### Step4) \uc124\uce58\ub41c \uac00\uc0c1\ud658\uacbd\uc5d0 \ud504\ub85c\uc81d\ud2b8 \uc2e4\ud589\uc5d0 \ud544\uc694\ud55c \ud30c\uc774\uc36c \ubaa8\ub4c8 \uc124\uce58\n```\n$ pip install -r requirements.txt\n```\n\n#### 3) \ub370\uc774\ud130 \ud559\uc2b5 \ub2e8\uacc4\n\n```\n$ python train_mask_detector.py --dataset dataset\n```\n\n> dataset \ud3f4\ub354 \uc548\uc5d0 \uc788\ub294 3800\uac1c\uc758 \uc774\ubbf8\uc9c0\ub97c \uac00\uc838\uc57c 20\ubc88\uc758 epoch \ub3d9\uc548 \uac01\uac01 96\ubc88\uc758 step\uc744 \uac70\uce58\uba70 \ud559\uc2b5\uc744 \uc9c4\ud589\n\n#### 4) \ub370\uc774\ud130 \ud559\uc2b5 \uacb0\uacfc\uc5d0 \ub300\ud55c \ucd94\uac00 \uc124\uba85\n\n\n\n\n\n \n\n- (\ud3b8\uc758\uc0c1 TP, TN, FP, FN\uc73c\ub85c \ud45c\uc2dc\ud569\ub2c8\ub2e4)\n\n##### 4-1) Precision\n> \ubaa8\ub378\uc774 True\ub77c\uace0 \uc608\uce21\ud55c \uac12 \uc911 \uc2e4\uc81c \uacb0\uacfc\uac00 True\uc758 \uac12\uc744 \uac00\uc9c0\ub294 \ub2f5\uc758 \ube44\uc728\n\n\\begin{align}\n\\frac{TP}{(TP + FP)}\n\\end{align}\n\n##### 4-2) Recall\n> \uc2e4\uc81c True\uc778 \uac12 \uc911 \ubaa8\ub378\uc774 True\ub77c\uace0 \uc608\uce21\ud55c \ub2f5\uc758 \ube44\uc728\n\n\\begin{align}\n\\frac{TP}{(TP + FN)}\n\\end{align}\n\n##### 4-3) F1\n> Precision\uacfc Recall\uc758 \uc870\ud654\ud3c9\uade0\n\n\\begin{align}\n\\frac{2 * precision * recall}{(precision + recall)}\n\\end{align}\n\n##### 4-4) Accuracy\n> True\ub77c\uace0 \ub9de\uac8c \uc608\uce21\ud55c \uac83\uacfc False\ub77c\uace0 \ub9de\uac8c \uc608\uce21\ud55c \uac83\uc774 \uc804\uccb4\uc5d0\uc11c \ucc28\uc9c0\ud558\ub294 \ube44\uc728\n\n\\begin{align}\n\\frac{TP + TN}{(TP + FP + TN + FN)}\n\\end{align}\n\n##### 4-5) \uc790\ub8cc \ucd9c\ucc98\n> [\ucd9c\ucc98](https://eunsukimme.github.io/ml/2019/10/21/Accuracy-Recall-Precision-F1-score/)\n\n#### 5) \ub370\uc774\ud130 \uac80\uc99d \ub2e8\uacc4\n\n\uc2e4\uc2dc\uac04 \uc6f9\ucea0 \uc601\uc0c1\uc744 \ud1b5\ud55c \uac80\uc99d\n\n```\n$ python detect_mask_video.py\n```\n\n## \ubcf4\uc644\ud560 \uc810\n\n#### 1) \ud55c\uc815\ub41c \ub370\uc774\ud130\n- \ud604\uc7ac \ud559\uc2b5\uc5d0 \uc0ac\uc6a9\ud55c \uac74 3800\uac1c\uc758 \uc774\ubbf8\uc9c0. \ubaa8\ub4e0 \uc0c1\ud669\uc744 \ucee4\ubc84\ud558\uae30\ub294 \ubd80\uc871\ud558\ub2e4. \ub354 \uc815\ud655\ud55c \uac80\uc99d\uc744 \uc704\ud574\uc11c\ub294 \ub354 \ub9ce\uc740 \ud559\uc2b5 \ub370\uc774\ud130\uac00 \ud544\uc694\ud558\ub2e4.\n\n#### 2) \ub9c8\uc2a4\ud06c \ubd88\ub7c9 \ucc29\uc6a9\uc790 \uac10\uc9c0\n- \uc774 \ud504\ub85c\uc81d\ud2b8\ub294 \ub9c8\uc2a4\ud06c \ucc29\uc6a9 \uc5ec\ubd80\ub9cc \uac10\uc9c0\ud55c\ub2e4. \ub9c8\uc2a4\ud06c\ub97c \ucf54 \ub05d\uae4c\uc9c0 \uc62c\ub9ac\uc9c0 \uc54a\uc740 \uc0ac\ub78c\ub4e4\uc744 \ubaa8\ub450 \uac10\uc9c0\ud558\uace0 \uc2f6\ub2e4\uba74 \uadf8 \ubd80\ubd84\uc5d0 \ud574\ub2f9\ud558\ub294 \ub370\uc774\ud130\uc640 \ub808\uc774\ube14\uc774 \ud544\uc694\ud558\ub2e4. \uae30\uc874\uc5d0 \ud559\uc2b5\uc5d0 \uc0ac\uc6a9\ud55c \ub370\uc774\ud130\ub9cc\ud07c \ub9ce\uc740 \uc218\uc758 \ub370\uc774\ud130\uac00 \uc8fc\uc5b4\uc9c4\ub2e4\uba74 \uac00\ub2a5\ud560 \uac83\uc73c\ub85c \ud310\ub2e8\ub41c\ub2e4.\n\n#### 3) \uad00\ub9ac\uc790\uc5d0\uac8c \uc54c\ub9bc\n- \ud604\uc7ac\ub294 \ub9c8\uc2a4\ud06c \ubd88\ub7c9\uc790\ub97c \uac10\uc9c0\ud558\uace0 \uacbd\uace0\ub97c \uc8fc\ub294 \uac83\uc5d0 \ub05d\ub09c\ub2e4. \uc790\ub3d9\uc73c\ub85c \uad00\ub9ac\uc790\uc5d0\uac8c \uba54\uc2dc\uc9c0\ub97c \ubcf4\ub0bc \uc218 \uc788\ub3c4\ub85d \ubcf4\uc644\ud558\uba74 \uc88b\uc744 \uac83 \uac19\ub2e4.\n\n## \uae30\ub300\ud6a8\uacfc\n\n- \uc885\uad50\uc2dc\uc124 \ub4f1 \uc9d1\ub2e8 \uac10\uc5fc\uc758 \uc704\ud5d8\uc774 \uc788\ub294 \uc7a5\uc18c\uc5d0\uc11c \ub9c8\uc2a4\ud06c \ubbf8\ucc29\uc6a9\uc790\ub97c \uac10\uc9c0\ud558\uc5ec \uac74\ubb3c\uc758 \ucd9c\uc785\uc774\ub098 \uc774\uc6a9\uc744 \uc81c\ud55c\ud558\uac70\ub098 \uaca9\ub9ac \uc870\ucde8\ub97c \ucde8\ud560 \uc218 \uc788\ub2e4. \n\n- \ub2e4\uc218\uc758 \uc778\uc6d0\uc774 \uac10\uc9c0\ub418\ub354\ub77c\ub3c4 \uc2e4\uc2dc\uac04\uc73c\ub85c \ub9c8\uc2a4\ud06c \ubbf8\ucc29\uc6a9\uc790\uc758 \uc218\ub97c \uad6c\ubd84\ud574\ub0bc \uc218 \uc788\ub2e4.\n", "meta": {"hexsha": "d433c667f792317660838e1a49ffa50a4932066f", "size": 22116, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Mask-Detector-AI-Model.ipynb", "max_stars_repo_name": "Younhong/COVID19-Mask-Detection-Model", "max_stars_repo_head_hexsha": "5f7478141588073e3984ed8cbd588d0d07d29abd", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Mask-Detector-AI-Model.ipynb", "max_issues_repo_name": "Younhong/COVID19-Mask-Detection-Model", "max_issues_repo_head_hexsha": "5f7478141588073e3984ed8cbd588d0d07d29abd", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Mask-Detector-AI-Model.ipynb", "max_forks_repo_name": "Younhong/COVID19-Mask-Detection-Model", "max_forks_repo_head_hexsha": "5f7478141588073e3984ed8cbd588d0d07d29abd", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.6740638003, "max_line_length": 215, "alphanum_fraction": 0.5196238018, "converted": true, "num_tokens": 6060, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.4110511474443969}} {"text": "\n
                                        \n \n \n \n
                                        \n\n# Python for Scientific Computing\n\n## Contents\n\n- [Python for Scientific Computing](#Python-for-Scientific-Computing) \n - [Overview](#Overview) \n - [Scientific Libraries](#Scientific-Libraries) \n - [The Need for Speed](#The-Need-for-Speed) \n - [Vectorization](#Vectorization) \n - [Beyond Vectorization](#Beyond-Vectorization) \n\nIn addition to what\u2019s in Anaconda, this lecture will need the following libraries:\n\n\n```python\n!pip install --upgrade quantecon\n```\n\n Requirement already up-to-date: quantecon in /usr/local/lib/python3.7/site-packages (0.4.6)\n Requirement already satisfied, skipping upgrade: scipy>=1.0.0 in /usr/local/lib/python3.7/site-packages (from quantecon) (1.2.1)\n Requirement already satisfied, skipping upgrade: numba>=0.38 in /usr/local/lib/python3.7/site-packages (from quantecon) (0.46.0)\n Requirement already satisfied, skipping upgrade: sympy in /usr/local/lib/python3.7/site-packages (from quantecon) (1.5)\n Requirement already satisfied, skipping upgrade: numpy in /usr/local/lib/python3.7/site-packages (from quantecon) (1.16.2)\n Requirement already satisfied, skipping upgrade: requests in /usr/local/lib/python3.7/site-packages (from quantecon) (2.21.0)\n Requirement already satisfied, skipping upgrade: llvmlite>=0.30.0dev0 in /usr/local/lib/python3.7/site-packages (from numba>=0.38->quantecon) (0.30.0)\n Requirement already satisfied, skipping upgrade: mpmath>=0.19 in /usr/local/lib/python3.7/site-packages (from sympy->quantecon) (1.1.0)\n Requirement already satisfied, skipping upgrade: idna<2.9,>=2.5 in /usr/local/lib/python3.7/site-packages (from requests->quantecon) (2.8)\n Requirement already satisfied, skipping upgrade: chardet<3.1.0,>=3.0.2 in /usr/local/lib/python3.7/site-packages (from requests->quantecon) (3.0.4)\n Requirement already satisfied, skipping upgrade: urllib3<1.25,>=1.21.1 in /usr/local/lib/python3.7/site-packages (from requests->quantecon) (1.24.1)\n Requirement already satisfied, skipping upgrade: certifi>=2017.4.17 in /usr/local/lib/python3.7/site-packages (from requests->quantecon) (2018.11.29)\n \u001b[33mYou are using pip version 19.0.3, however version 19.3.1 is available.\n You should consider upgrading via the 'pip install --upgrade pip' command.\u001b[0m\n Note: you may need to restart the kernel to use updated packages.\n\n\n## Scientific Libraries\n\nLet\u2019s briefly review Python\u2019s scientific libraries, starting with why we need\nthem.\n\n### The Role of Scientific Libraries\n\nOne obvious reason we use scientific libraries is because they implement\nroutines we want to use.\n\nFor example, it\u2019s almost always better to use an existing routine for root\nfinding than to write a new one from scratch.\n\n(For standard algorithms, efficiency is maximized if the community can coordinate on a\ncommon set of implementations, written by experts and tuned by users to be as fast and robust as possible.)\n\nBut this is not the only reason that we use Python\u2019s scientific libraries.\n\nAnother is that pure Python, while flexible and elegant, is not fast.\n\nSo we need libraries that are designed to accelerate execution of Python code.\n\nAs we\u2019ll see below, there are now Python libraries that can do this extremely well.\n\n### Python\u2019s Scientific Ecosystem\n\nIn terms of popularity, the big four in the world of scientific Python\nlibraries are\n\n- NumPy \n- SciPy \n- Matplotlib \n- Pandas \n\n\nFor us, there\u2019s another (relatively new) library that will also be essential for\nnumerical computing:\n\n- Numba \n\n\nOver the next few lectures we\u2019ll see how to use these libraries.\n\nBut first, let\u2019s quickly review how they fit together.\n\n- NumPy forms the foundations by providing a basic array data type (think of\n vectors and matrices) and functions for acting on these arrays (e.g., matrix\n multiplication). \n- SciPy builds on NumPy by adding the kinds of numerical methods that are\n routinely used in science (interpolation, optimization, root finding, etc.). \n- Matplotlib is used to generate figures, with a focus on plotting data stored in NumPy arrays. \n- Pandas provides types and functions for empirical work (e.g., manipulating data). \n- Numba accellerates execution via JIT compilation \u2014 we\u2019ll learn about this\n soon. \n\n## The Need for Speed\n\nNow let\u2019s discuss execution speed.\n\nHigher-level languages like Python are optimized for humans.\n\nThis means that the programmer can leave many details to the runtime environment\n\n- specifying variable types \n- memory allocation/deallocation, etc. \n\n\nThe upside is that, compared to low-level languages, Python is typically faster to write, less error-prone and easier to debug.\n\nThe downside is that Python is harder to optimize \u2014 that is, turn into fast machine code \u2014 than languages like C or Fortran.\n\nIndeed, the standard implementation of Python (called CPython) cannot match the speed of compiled languages such as C or Fortran.\n\nDoes that mean that we should just switch to C or Fortran for everything?\n\nThe answer is: No, no and one hundred times no!\n\n(This is what you should say to the senior professor insisting that the model\nneeds to be rewritten in Fortran or C++.)\n\nThere are two reasons why:\n\nFirst, for any given program, relatively few lines are ever going to\nbe time-critical.\n\nHence it is far more efficient to write most of our code in a high productivity language like Python.\n\nSecond, even for those lines of code that *are* time-critical, we can now achieve the same speed as C or Fortran using Python\u2019s scientific libraries.\n\n### Where are the Bottlenecks?\n\nBefore we learn how to do this, let\u2019s try to understand why plain vanila\nPython is slower than C or Fortran.\n\nThis will, in turn, help us figure out how to speed things up.\n\n#### Dynamic Typing\n\n\n\nConsider this Python operation\n\n\n```python\na, b = 10, 10\na + b\n```\n\nEven for this simple operation, the Python interpreter has a fair bit of work to do.\n\nFor example, in the statement `a + b`, the interpreter has to know which\noperation to invoke.\n\nIf `a` and `b` are strings, then `a + b` requires string concatenation\n\n\n```python\na, b = 'foo', 'bar'\na + b\n```\n\nIf `a` and `b` are lists, then `a + b` requires list concatenation\n\n\n```python\na, b = ['foo'], ['bar']\na + b\n```\n\n(We say that the operator `+` is *overloaded* \u2014 its action depends on the\ntype of the objects on which it acts)\n\nAs a result, Python must check the type of the objects and then call the correct operation.\n\nThis involves substantial overheads.\n\n#### Static Types\n\n\n\nCompiled languages avoid these overheads with explicit, static types.\n\nFor example, consider the following C code, which sums the integers from 1 to 10\n\n```c\n#include \n\nint main(void) {\n int i;\n int sum = 0;\n for (i = 1; i <= 10; i++) {\n sum = sum + i;\n }\n printf(\"sum = %d\\n\", sum);\n return 0;\n}\n```\n\n\nThe variables `i` and `sum` are explicitly declared to be integers.\n\nHence, the meaning of addition here is completely unambiguous.\n\n### Data Access\n\nAnother drag on speed for high-level languages is data access.\n\nTo illustrate, let\u2019s consider the problem of summing some data \u2014 say, a collection of integers.\n\n#### Summing with Compiled Code\n\nIn C or Fortran, these integers would typically be stored in an array, which\nis a simple data structure for storing homogeneous data.\n\nSuch an array is stored in a single contiguous block of memory\n\n- In modern computers, memory addresses are allocated to each byte (one byte = 8 bits). \n- For example, a 64 bit integer is stored in 8 bytes of memory. \n- An array of $ n $ such integers occupies $ 8n $ **consecutive** memory slots. \n\n\nMoreover, the compiler is made aware of the data type by the programmer.\n\n- In this case 64 bit integers \n\n\nHence, each successive data point can be accessed by shifting forward in memory\nspace by a known and fixed amount.\n\n- In this case 8 bytes \n\n#### Summing in Pure Python\n\nPython tries to replicate these ideas to some degree.\n\nFor example, in the standard Python implementation (CPython), list elements are placed in memory locations that are in a sense contiguous.\n\nHowever, these list elements are more like pointers to data rather than actual data.\n\nHence, there is still overhead involved in accessing the data values themselves.\n\nThis is a considerable drag on speed.\n\nIn fact, it\u2019s generally true that memory traffic is a major culprit when it comes to slow execution.\n\nLet\u2019s look at some ways around these problems.\n\n## Vectorization\n\n\n\nThere is a clever method called **vectorization** that can be\nused to speed up high level languages in numerical applications.\n\nThe key idea is to send array processing operations in batch to pre-compiled\nand efficient native machine code.\n\nThe machine code itself is typically compiled from carefully optimized C or Fortran.\n\nFor example, when working in a high level language, the operation of inverting a large matrix can be subcontracted to efficient machine code that is pre-compiled for this purpose and supplied to users as part of a package.\n\nThis clever idea dates back to MATLAB, which uses vectorization extensively.\n\nVectorization can greatly accelerate many numerical computations (but not all,\nas we shall see).\n\nLet\u2019s see how vectorization works in Python, using NumPy.\n\n### Operations on Arrays\n\n\n\nFirst, let\u2019s run some imports\n\n\n```python\nimport random\nimport numpy as np\nimport quantecon as qe\n```\n\nNext let\u2019s try some non-vectorized code, which uses a native Python loop to generate,\nsquare and then sum a large number of random variables:\n\n\n```python\nn = 1_000_000\n```\n\n\n```python\n%%time\n\ny = 0 # Will accumulate and store sum\nfor i in range(n):\n x = random.uniform(0, 1)\n y += x**2\n```\n\nThe following vectorized code achieves the same thing.\n\n\n```python\n%%time\n\nx = np.random.uniform(0, 1, n)\ny = np.sum(x**2)\n```\n\nAs you can see, the second code block runs much faster. Why?\n\nThe second code block breaks the loop down into three basic operations\n\n1. draw `n` uniforms \n1. square them \n1. sum them \n\n\nThese are sent as batch operators to optimized machine code.\n\nApart from minor overheads associated with sending data back and forth, the result is C or Fortran-like speed.\n\nWhen we run batch operations on arrays like this, we say that the code is *vectorized*.\n\nVectorized code is typically fast and efficient.\n\nIt is also surprisingly flexible, in the sense that many operations can be vectorized.\n\nThe next section illustrates this point.\n\n\n\n\n### Universal Functions\n\n\n\nMany functions provided by NumPy are so-called *universal functions* \u2014 also called [ufuncs](https://docs.scipy.org/doc/numpy/reference/ufuncs.html).\n\nThis means that they\n\n- map scalars into scalars, as expected \n- map arrays into arrays, acting element-wise \n\n\nFor example, `np.cos` is a ufunc:\n\n\n```python\nnp.cos(1.0)\n```\n\n\n```python\nnp.cos([0, 1, 2, 7.5])\n```\n\nBy exploiting ufuncs, many operations can be vectorized.\n\nFor example, consider the problem of maximizing a function $ f $ of two\nvariables $ (x,y) $ over the square $ [-a, a] \\times [-a, a] $.\n\nFor $ f $ and $ a $ let\u2019s choose\n\n$$\nf(x,y) = \\frac{\\cos(x^2 + y^2)}{1 + x^2 + y^2}\n\\quad \\text{and} \\quad\na = 3\n$$\n\nHere\u2019s a plot of $ f $\n\n\n```python\nimport matplotlib.pyplot as plt\n%matplotlib inline\nfrom mpl_toolkits.mplot3d.axes3d import Axes3D\nfrom matplotlib import cm\n\n%matplotlib inline\n\ndef f(x, y):\n return np.cos(x**2 + y**2) / (1 + x**2 + y**2)\n\nxgrid = np.linspace(-3, 3, 50)\nygrid = xgrid\nx, y = np.meshgrid(xgrid, ygrid)\n\nfig = plt.figure(figsize=(8, 6))\nax = fig.add_subplot(111, projection='3d')\nax.plot_surface(x,\n y,\n f(x, y),\n rstride=2, cstride=2,\n cmap=cm.jet,\n alpha=0.7,\n linewidth=0.25)\nax.set_zlim(-0.5, 1.0)\nplt.show()\n```\n\nTo maximize it, we\u2019re going to use a naive grid search:\n\n1. Evaluate $ f $ for all $ (x,y) $ in a grid on the square. \n1. Return the maximum of observed values. \n\n\nThe grid will be\n\n\n```python\ngrid = np.linspace(-3, 3, 1000)\n```\n\nHere\u2019s a non-vectorized version that uses Python loops.\n\n\n```python\n%%time\n\nm = -np.inf\n\nfor x in grid:\n for y in grid:\n z = f(x, y)\n if z > m:\n m = z\n```\n\nAnd here\u2019s a vectorized version\n\n\n```python\n%%time\n\nx, y = np.meshgrid(grid, grid)\nnp.max(f(x, y))\n```\n\nIn the vectorized version, all the looping takes place in compiled code.\n\nAs you can see, the second version is **much** faster.\n\n(We\u2019ll make it even faster again later on, using more scientific programming tricks.)\n\n\n\n\n## Beyond Vectorization\n\nAt its best, vectorization yields fast, simple code.\n\nHowever, it\u2019s not without disadvantages.\n\nOne issue is that it can be highly memory-intensive.\n\nFor example, the vectorized maximization routine above is far more memory\nintensive than the non-vectorized version that preceded it.\n\nThis is because vectorization tends to create many intermediate arrays before\nproducing the final calculation.\n\nAnother issue is that not all algorithms can be vectorized.\n\nIn these kinds of settings, we need to go back to loops.\n\nFortunately, there are alternative ways to speed up Python loops that work in\nalmost any setting.\n\nFor example, in the last few years, a new Python library called [Numba](http://numba.pydata.org/) has appeared that solves the main problems\nwith vectorization listed above.\n\nIt does so through something called **just in time (JIT) compilation**,\nwhich can generate extremely fast and efficient code.\n\nWe\u2019ll learn how to use Numba [soon](numba.ipynb).\n", "meta": {"hexsha": "7fe4ae0a07e086b5742b29afa48d830ab588f5c6", "size": 128744, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "day1/need_for_speed.ipynb", "max_stars_repo_name": "varunsatish/summer_course_2019", "max_stars_repo_head_hexsha": "1fc14ad2e7bddb7671bb638d0c6e0c7412d8051f", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "day1/need_for_speed.ipynb", "max_issues_repo_name": "varunsatish/summer_course_2019", "max_issues_repo_head_hexsha": "1fc14ad2e7bddb7671bb638d0c6e0c7412d8051f", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "day1/need_for_speed.ipynb", "max_forks_repo_name": "varunsatish/summer_course_2019", "max_forks_repo_head_hexsha": "1fc14ad2e7bddb7671bb638d0c6e0c7412d8051f", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 167.8539765319, "max_line_length": 106348, "alphanum_fraction": 0.8999098987, "converted": true, "num_tokens": 3452, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166047041654, "lm_q2_score": 0.8031738010682209, "lm_q1q2_score": 0.41099737046996876}} {"text": "\n\n# Linear diffusion exercise with Landlab\n\nThis notebook is adapted from *Landscape Evolution Modeling with CHILD* by Gregory Tucker and Stephen Lancaster. This notebook was created by Nicole Gasparini at Tulane University.\n\n
                                        \nFor tutorials on learning Landlab, click here: https://github.com/landlab/landlab/wiki/Tutorials\n
                                        \n\n\n**What is this notebook?**\n\nThis notebook illustrates the evolution of landforms dominated by processes that result in linear diffusion of sediment. In other words, the downhill flow of soil is proportional to the (downhill) gradient of the land surface multiplied by a transport coefficient.\n\nThe notebook first illustrates a simple example of a diffusing hillslope. We then provide a number of exercises for students to do on their own. This set of exercises is recomended for students in a quantitative geomorphology class, who have been introduced to the linear diffusion equation in class. \n\n**Application of linear diffusion transport law:**\n\nFor relatively gentle, soil-mantled slopes, there is reasonably strong support for a transport law of the form:\n\\begin{equation}\nq_s = -D \\nabla z\n\\end{equation}\nwhere ${q}_s$ is the transport rate with dimensions of L$^2$T$^{-1}$; $D$ is a transport coefficient with dimensions of L$^2$T$^{-1}$; and $z$ is elevation. $\\nabla z$ is the gradient in elevation. If distance is increasing downslope, $\\nabla z$ is negative downslope, hence the negative in front of $D$. \n \nChanges in elevation, or erosion, are calculated from conservation of mass:\n\\begin{equation}\n\\frac{dz}{dt} = U-\\nabla q_s\n\\end{equation}\nwhere $U$ is the rock uplift rate, with dimensions LT$^{-1}$.\n\n**How will we explore this with Landlab?**\n\nWe will use the Landlab component *LinearDiffuser*, which implements the equations above, to explore how hillslopes evolve when linear diffusion describes hillslope sediment transport. We will explore both steady state, here defined as erosion rate equal to rock uplift rate, and also how a landscape gets to steady state.\n\nThe first example illustrates how to set-up the model and evolve a hillslope to steady state, along with how to plot some variables of interest. We assume that you have knowledge of how to derive the steady-state form of a uniformly uplifting, steady-state, diffusive hillslope. For more information on hillslope sediment transport laws, this paper is a great overview:\n\nRoering, Joshua J. (2008) \"How well can hillslope evolution models \u201cexplain\u201d topography? Simulating soil transport and production with high-resolution topographic data.\" Geological Society of America Bulletin.\n\nBased on the first example, you are asked to first think about what will happen as you change a parameter, and then you explore this numerically by changing the code.\n\nStart at the top by reading each block of text and sequentially running each code block (shift - enter OR got to the _Cell_ pulldown menu at the top and choose _Run Cells_). \n\nRemember that you can always go to the _Kernel_ pulldown menu at the top and choose _Restart & Clear Output_ or _Restart & Run All_ if you change things and want to start afresh. If you just change one code block and rerun only that code block, only the parts of the code in that code block will be updated. (E.g. if you change parameters but don't reset the code blocks that initialize run time or topography, then these values will not be reset.) \n\n**Now on to the code example**\n\nImport statements. You should not need to edit this.\n\n\n```python\n# Code Block 1\n\nfrom landlab import RasterModelGrid\nfrom landlab.components import FlowAccumulator, LinearDiffuser\nfrom landlab.plot.imshow import imshow_grid\nfrom matplotlib.pyplot import (\n figure, show, plot, xlabel, ylabel, title, legend, ylim\n)\nimport numpy as np\n```\n\nWe will create a grid with 41 rows and 5 columns, and dx is 5 m (a long, narrow, hillslope). The initial elevation is 0 at all nodes.\n\nWe set-up boundary conditions so that material can leave the hillslope at the two short ends.\n\n\n```python\n# Code Block 2\n\n# setup grid\nmg = RasterModelGrid((41, 5), 5.)\nz_vals = mg.add_zeros('topographic__elevation', at='node')\n\n# initialize some values for plotting\nycoord_rast = mg.node_vector_to_raster(mg.node_y)\nys_grid = ycoord_rast[:, 2]\n\n# set boundary condition.\nmg.set_closed_boundaries_at_grid_edges(True, False, True, False)\n```\n\nNow we initialize the *LinearDiffuser* component. \n\n\n```python\n# Code Block 3\n\nD = 0.01 # initial value of 0.01 m^2/yr\nlin_diffuse = LinearDiffuser(mg, linear_diffusivity=D)\n```\n\nWe now initialize a few more parameters.\n\n\n```python\n# Code Block 4\n\n# Uniform rate of rock uplift\nuplift_rate = 0.0001 # meters/year, originally set to 0.0001\n\n# Total time in years that the model will run for.\nruntime = 1000000 # years, originally set to 1,000,000\n\n# Stability criteria for timestep dt. Coefficient can be changed\n# depending on our tolerance for stability vs tolerance for run time.\ndt = 0.5 * mg.dx * mg.dx / D\n\n# nt is number of time steps\nnt = int(runtime // dt)\n\n# Below is to keep track of time for labeling plots\ntime_counter = 0\n\n# length of uplift over a single time step, meters\nuplift_per_step = uplift_rate * dt\n```\n\nNow we figure out the analytical solution for the elevation of the steady-state profile.\n\n\n```python\n# Code Block 5\n\nys = np.arange(mg.number_of_node_rows*mg.dx-mg.dx)\n\n# location of divide or ridge crest -> middle of grid \n# based on boundary conds.\ndivide_loc = (mg.number_of_node_rows*mg.dx-mg.dx)/2\n\n# half-width of the ridge\nhalf_width = (mg.number_of_node_rows*mg.dx-mg.dx)/2\n\n# analytical solution for elevation under linear diffusion at steady state\nzs = (uplift_rate/(2*D)) * \\\n (np.power(half_width, 2) - np.power(ys - divide_loc, 2))\n```\n\nBefore we evolve the landscape, let's look at the initial topography. (This is just verifying that it is flat with zero elevation.)\n\n\n```python\n# Code Block 6\n\nfigure(1)\nimshow_grid(mg, 'topographic__elevation')\ntitle('initial topography')\nfigure(2)\nelev_rast = mg.node_vector_to_raster(\n mg.at_node['topographic__elevation'])\nplot(ys_grid, elev_rast[:, 2], 'r-', label='model')\nplot(ys, zs, 'k--', label='analytical solution')\nylim((-5,50)) #may want to change upper limit if D changes\nxlabel('horizontal distance (m)')\nylabel('vertical distance (m)')\nlegend(loc='lower center')\n_ = title('initial topographic cross section')\n```\n\nNow we are ready to evolve the landscape and compare it to the steady state solution.\n\nBelow is the time loop that does all the calculations. \n\n\n```python\n# Code Block 7\n\nfor i in range(nt):\n mg['node']['topographic__elevation'][mg.core_nodes] += uplift_per_step\n lin_diffuse.run_one_step(dt)\n time_counter += dt\n\n # All landscape evolution is the first two lines of loop.\n # Below is simply for plotting the topography halfway through the run\n if i == int(nt // 2):\n figure(1)\n imshow_grid(mg, 'topographic__elevation')\n title('topography at time %s, with D = %s'%(time_counter,D))\n \n figure(2)\n elev_rast = mg.node_vector_to_raster(\n mg.at_node['topographic__elevation']\n )\n plot(ys_grid, elev_rast[:, 2], 'k-', label='model')\n plot(ys, zs, 'g--', label='analytical solution - SS')\n plot(ys, zs*0.75, 'b--', label='75% of analytical solution')\n plot(ys, zs*0.5, 'r--', label='50% of analytical solution')\n xlabel('horizontal distance (m)')\n ylabel('vertical distance (m)')\n legend(loc='lower center')\n title(\n 'topographic__elevation at time %s, with D = %s'\n %(time_counter,D)\n )\n```\n\nNow we plot the final cross-section.\n\n\n```python\n# Code Block 8\n\nelev_rast = mg.node_vector_to_raster(mg.at_node['topographic__elevation'])\n\nplot(ys_grid, elev_rast[:, 2], 'k-', label='model')\nplot(ys, zs, 'g--', label='analytical solution - SS')\nplot(ys, zs * 0.75, 'b--', label='75% of analytical solution')\nplot(ys, zs * 0.5, 'r--', label='50% of analytical solution')\nxlabel('horizontal distance (m)')\nylabel('vertical distance (m)')\nlegend(loc='lower center')\n_ = title('topographic cross section at time %s, with D = %s'%(time_counter,D))\n```\n\nNow we plot the steepest slope in the downward direction across the landscape.\n\n(To calculate the steepest slope at a location, we need to route flow across the landscape.)\n\n\n```python\n# Code Block 9\n\nfr = FlowAccumulator(mg, flow_director='FlowDirectorD8') # intializing flow routing\nfr.run_one_step()\n\nplot(\n mg.node_y[mg.core_nodes],\n mg.at_node['topographic__steepest_slope'][mg.core_nodes],\n 'k-'\n)\nxlabel('horizontal distance (m)')\nylabel('topographic slope (m/m)')\n_ = title('slope of the hillslope at time %s, with D = %s'%(time_counter,D))\n```\n\nHas the landscape reached steady state yet? How do you know?\n\n\n\n\n\n\n\nAnswer: Not quite, but it is getting close. Go back and rerun Code Blocks 7, 8 and 9 (time loop and plotting). (Remember you can rerun a cell with shift-return, or from the cell pull-down menu.) Has it reached steady state yet? \n\n**What to do and hand in:**\n1. In the example illustrated here ($D$ = 0.01 m$^2$yr$^{-1}$ and $U$ = 0.0001 m yr$^{-1}$). Restart everything, and use the model to determine how long it takes for the landscape to go from a flat to reach 50%, 75% and 100% of its steady-state morphology. Does the landscape approach steady state linearly in time? (You can run the time loop (Code Block 7) multiple times without running other code blocks again to continually evolve the landscape. You will initially want to rerun all the code blocks and change the value of **run_time** (Code Block 4). Determining the correct value of **run_time** to use will take some iteration.)\n2. What do you think will happen when you increase $D$ (Code Block 3) by a factor of 10? Will the time to steady state differ? If yes, how? Will the topography be different? If yes, how and why? What does it mean physically, about processes, if $D$ increases? Answer these questions before running any code. \n3. Now set $D$ = 0.1 m$^2$yr$^{-1}$ and rerun landscape evolution from an initial flat. Illustrate the final steady state topography and record the time to steady state. Discuss how the landscape differs from the results in question 1. Discuss how the results are similar to or different from your intuition. It is OK if your intuition was wrong! \n4. What do you think will happen when you increase **uplift_rate** (Code Block 4) by a factor of 10? Will the time to steady state differ? If yes, how? Will the topography be different? If yes, how and why? Answer these questions first, and then rerun the code with **uplift_rate** = 0.001 m yr$^{-1}$. (Make sure you change $D$ - Code Block 3 - back to the original value of 0.01 m$^2$yr$^{-1}$ and restart from a flat surface.) Illustrate the final steady state topography. Discuss how these results differ from the results in question 1 and how the results match (or do not) your intuition. It is OK if your intuition was wrong.\n\nYou should hand in a typed document that answers the above questions with supporting plots. Plots should be embedded in the text, or, if they all fall at the end, they need to be clearly labeled, e.g. each plot has a figure number and plots are referred to by figure number in the text.\n\nOther questions you can explore.\n\n1. What happens to time to steady state as you increase the length of your hillslope? \n2. Does grid resolution affect your answers? If so, how?\n\n", "meta": {"hexsha": "3f2a94ab9d044c948a062b81615b4c62d327bc2c", "size": 165427, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/hillslope_diffusion_class_notebook.ipynb", "max_stars_repo_name": "nathanlyons/UT_Landlab_Clinic", "max_stars_repo_head_hexsha": "68f1b0b3363d4c29d35b8278caf744b3082d5ade", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-01-22T15:03:39.000Z", "max_stars_repo_stars_event_max_datetime": "2020-01-22T15:03:39.000Z", "max_issues_repo_path": "notebooks/hillslope_diffusion_class_notebook.ipynb", "max_issues_repo_name": "landlab/psu-clinic-2020", "max_issues_repo_head_hexsha": "f683e382adcfc4e4c16a5d2fa6fa2120d75bc4c7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2020-09-09T21:14:43.000Z", "max_issues_repo_issues_event_max_datetime": "2020-09-10T01:20:41.000Z", "max_forks_repo_path": "notebooks/hillslope_diffusion_class_notebook.ipynb", "max_forks_repo_name": "landlab/psu-clinic-2020", "max_forks_repo_head_hexsha": "f683e382adcfc4e4c16a5d2fa6fa2120d75bc4c7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2020-09-10T00:44:07.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-25T02:34:16.000Z", "avg_line_length": 351.2250530786, "max_line_length": 41848, "alphanum_fraction": 0.9280165874, "converted": true, "num_tokens": 2927, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585786300048, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.41098395367676116}} {"text": "# Parameter Estimation in Bayesian Networks\n\n\n\n> Throughout this point we have assumed that we already have a graphical model (both, the structure and parameters) as an input. But, as you may suspect by now, this is not the case in real-life problems.\n>\n> One way to obtain a graphical model is to elicit both the network and the parameters by hand with the help of an expert. This is a nontrivial task (recall the knowledge engineering section you read), and may require a lot of work (weeks or even months, depending on the size of the network). Moreover, the access to an expert can be a non plausible assumption.\n>\n> On the other hand, it is common to have access to a set of examples generated from the distribution (process) we want to model. Hence, in this class and in the following, we will focus on constructing a model from a set of data instances: **model learning**.\n\n> **Objetives:**\n> - To describe the problem of model learning.\n> - To derive the maximum likelihood parameter estimators for Bayesian Networks.\n> - To derive the maximum aposteriori parameter estimators for Bayesian Networks.\n\n> **References:**\n> - Probabilistic Graphical Models: Principles and Techniques, By Daphne Koller and Nir Friedman. Ch. 17.\n> - Mastering Probabilistic Graphical Models Using Python, By Ankur Ankan and Abinash Panda. Ch. 5.\n> - Probabilistic Graphical Models Specialization, offered through Coursera. Prof. Daphne Koller.\n\n\n

                                        Imagen recuperada de: https://upload.wikimedia.org/wikipedia/commons/d/d5/Hey_Machine_Learning_Logo.png.

                                        \n\n___\n\n# 1. Learning overview\n\nThe complete process of learning can be summarized in the following diagram:\n\n\n```python\nfrom IPython.display import Image\n```\n\n\n```python\nImage(\"figures/pgm_construction.png\")\n```\n\n## 1.1. Why would we want to learn a probabilistic graphical model?\n\n### 1. To answer general probabilistic queries about new instances.\n\nWhat will we optimize? **The training set likelihood**\n\n$$P(\\mathcal{D} : \\mathcal{M}).$$\n\n- If a model makes the data more likely, then it was more likely to have generated this data set.\n \nHowever, what we really care about is *new data*, not the data that we have seen before.\n\n- So, we evaluate the model on a separate test set: **Test set likelihood**\n \n $$P(\\mathcal{D}' : \\mathcal{M}).$$\n\n### 2. To perform a specific prediction task on new instances.\n\nIn this setting, we want to predict target variables $\\bar{y}$ from some observed variables $\\bar{x}$.\n\nWhat will we optimize? An specialized objective **recall, balanced accuracy, ...**\n\n- However, from a mathematical and algorithmical point of view, it is convenient to optimize the training set likelihood\n \n $$P(\\mathcal{D} : \\mathcal{M}) = \\prod_{m} P(\\bar{d}[m] : \\mathcal{M}),$$\n \n or the conditional likelihood\n \n $$\\prod_{m} P(\\bar{y}[m] | \\bar{x}[m] : \\mathcal{M}).$$\n \nFinally, the model should be evaluated over the **test set** and the **true specialized objective**.\n\n### 3. To discover some underlying knowledge about $\\mathcal{M}^*$.\n\nIn this case, we would like to:\n\n- Distinguish between direct and indirect dependencies (edges).\n- Learn the directionality of the edges.\n \nWhat will we optimize? **The training set likelihood**\n\n- Very convenient from a mathematical and algorithmical point of view.\n- Poor representative for structural accuracy.\n\nThe model should be evaluated by comparing to prior knowledge.\n\n## 1.2. How do we deal with overfitting?\n\n1. If we select $\\mathcal{M}$ optimizing according training set likelihood, then it will overfit to statistical noise.\n\n2. Parameter overfitting:\n - The parameters fit to random noise in training data.\n - Solution: use regularization, or equivalently, parameter priors.\n\n3. Structural overfitting:\n - Training likelihood always benefits more complex structures.\n - Solution: penalize model complexity.\n\n## 1.3. Why learning a PGM?\n\n1. Can incorporate prior knowledge into the model.\n\n2. Learn one model, use it for what you want.\n\n3. Framework for knowledge discovery and validation.\n\n___\n# 2. Maximum likelihood parameter estimation in BNs\n\nWe begin with a short introduction to maximum likelihood estimation (MLE), since it is the **key building block for parameter estimation**.\n\n## 2.1. Maximum likelihood estimation\n\nConsider the a simple biased coin, which follows a Bernoulli distribution:\n\n- Random variable $X$ with $\\mathrm{Val}(X)=\\{x^0,x^1\\}$.\n\n- $P(X=x^1)=\\theta$ and $P(X=x^0)=1-\\theta$.\n\nMoreover, we have a set of independent and identically distributed (IID) data instances sampled from $P$:\n\n$$\\mathcal{D} = \\{x[1], x[2], \\dots x[M]\\}$$\n\n**IID**:\n- Tosses are independent of each other.\n- Tosses are sampled from the same distribution.\n\nHow can we represent IID as a PGM?\n\n\n```python\nImage(\"figures/iid.png\")\n```\n\n$$P(X[d] | \\theta) = \\left\\{\\begin{array}{ccc} \\theta & \\text{if} & X[d]=x^1\\\\\n 1-\\theta & \\text{if} & X[d]=x^0\n\\end{array}\\right.$$\n\nOur **goal** is to estimate the parameter $\\theta \\in [0, 1]$ so that the data instances $\\mathcal{D} = \\{x[1], x[2], \\dots x[M]\\}$ were likely to be predicted.\n\nHow do we evaluate the **likelihood**?\n\nWe use the likelihood of $\\mathcal{D}$ given $\\theta$:\n\n$$\\mathcal{L}(\\theta : \\mathcal{D}) = P(\\mathcal{D} | \\theta) = \\prod_{d=1}^{M} P(x[m] | \\theta),$$\n\nbecause the samples are IID.\n\nIf, for example, the data instances are \n\n$$\\mathcal{D} = \\{x^1, x^0, x^1, x^1, x^0\\},$$\n\nwhat would the likelihood function be?\n\n$$\\mathcal{L}(\\theta : \\mathcal{D}) %= \\theta (1 - \\theta) \\theta \\theta (1 - \\theta) = \\theta^3 (1 - \\theta)^2.$$\n\n\n```python\n# Import numpy\n\n# Import matplotlib.pyplot\n\n```\n\n\n```python\n# Compute the likelihood function of the above data instances\n\n```\n\n\n```python\n# Draw the likelihood function\n\n```\n\n\n```python\n\n```\n\n\n\n\n 0.6\n\n\n\nWhat is the **optimal** value for theta?\n\n- As intuition dictates it is ... $\\frac{3}{5}=0.6$\n\n### More generally ...\n\n- We will have $M_1$ appearances of $x^1$ and $M_0$ appearances of $x^0$, with $M_1+M_0=M$.\n\n- And we will find $\\theta$ that maximizes the likelihood:\n \n $$\\mathcal{L}(\\theta:\\mathcal{D})=\\theta^{M_1}(1-\\theta)^{M_0}.$$\n \n#### Maximum likelihood estimation (MLE) principle: choose $\\theta$ to maximize $\\mathcal{L}(\\theta:\\mathcal{D})$\n\n- Since the $\\log$ is an increasing function, the above is equivalent to maximize the log-likelihood\n\n $$l(\\theta:\\mathcal{D}) = \\log \\mathcal{L}(\\theta:\\mathcal{D}) = M_1 \\log(\\theta) + M_0 \\log(1 - \\theta).$$\n \n- Whose maximizer is obtained by (see in the whiteboard):\n\n $$\n \\begin{align}\n \\frac{\\partial l}{\\partial \\theta} = 0 %\\Leftrightarrow \n %& M_1 \\frac{1}{\\hat{\\theta}} - M_0 \\frac{1}{1-\\hat{\\theta}} = 0 \\\\\n %& M_1(1-\\hat{\\theta}) - M_0 \\hat{\\theta} = 0 \\\\\n %& \\hat{\\theta} = \\frac{M_1}{M_0 + M_1}\n \\end{align}\n $$\n\nWhich fits very nicely with our frequentist interpretation of probability.\n\n\n```python\n# Generate N samples from a Bernoulli distribution\n\n```\n\n\n```python\n\n```\n\n\n\n\n array([0, 1, 1, 1, 1, 1, 1, 1, 1, 0])\n\n\n\n\n```python\n# Estimate the parameter theta\n\n```\n\n\n\n\n (0.8, 0.7)\n\n\n\n### What happens with a multinomial?\n\nIn this case we have:\n\n- Random variable $X$ with $\\mathrm{Val}(X)=\\{x^1, \\dots, x^{k}\\}$.\n\n- $P(X=x^i)=\\theta_i$ for $i\\in\\{1,\\dots,k\\}$.\n\n- $\\sum_{i=1}^{k}\\theta_i=1$.\n\nWe assume that the data instances \n\n$$\\mathcal{D} = \\{x[1], \\dots, x[M]\\}$$\n\nare such that:\n\n- $x^1$ appears $M_1$ times,\n- ...,\n- $x^i$ appears $M_i$ times,\n- ...,\n- $x^k$ appears $M_k$ times.\n\nwith $M=\\sum_{j=1}^{k}M_j$.\n\nThus, (**homework**) one can show that the maximum likelihood estimator (MLE) of $\\theta_i$ is:\n\n$$\\hat{\\theta_i} = \\frac{M_i}{\\sum_{j=1}^{k} M_j} = \\frac{M_i}{M},$$\n\nthis is, the fraction of times $x^i$ appears in the data.\n\n\n```python\n# Generate N samples from a multinomial distribution\n\n```\n\n\n```python\n# Estimate the parameters theta_i\n\n```\n\n\n\n\n array([0.2, 0. , 0.6, 0.2])\n\n\n\n### What happens with a Gaussian?\n\nIn this case we have:\n\n- Random variable $X \\sim \\mathcal{N}(\\mu, \\sigma^2)$.\n\n- $p(x;\\mu,\\sigma^2)=\\frac{1}{\\sqrt{2 \\pi}} \\exp\\left(-\\frac{(x-\\mu)^2}{2\\sigma^2}\\right)$.\n\nWe assume that the data instances \n\n$$\\mathcal{D} = \\{x[1], \\dots, x[M]\\}$$\n\nare drawn from this distribution.\n\nOne can show then (**homework**) that the MLE estimators of $\\mu$ and $\\sigma$ are:\n\n$$\\hat{\\mu} = \\frac{1}{M} \\sum_{j=1}^{M}x[j] \\qquad \\text{and} \\qquad \\hat{\\sigma}=\\sqrt{\\frac{1}{M}\\sum_{j=1}^{M}(x[j]-\\hat{\\mu})^2}.$$\n\n\n```python\n# Generate N samples from a Gaussian distribution\n\n```\n\n\n```python\n# Estimate the parameters mu and sigma\n\n```\n\n\n\n\n (4.674731576916187, 0.7437894283375306)\n\n\n\n## 2.2. Maximum likelihood estimation for BNs\n\nNow, let's apply the MLE principle for estimating the parameters in a Bayesian network.\n\nWe assume that we have a set of independent and identically distributed (IID) data instances sampled from the BN $X \\to Y$:\n\n$$\\mathcal{D} = \\{(x[1], y[1]), (x[2], y[2]), \\dots, (x[M], y[M])\\}$$\n\nAs above, we can give a PGM interpretation to IID data:\n\n\n```python\nImage(\"figures/iid_bn.png\")\n```\n\nNow, we have the parameters $\\theta_X$ and $\\theta_{Y|X}$, which we summarize in a single parameter set $\\Theta$.\n\nThe likelihood function is (see in the whiteboard):\n\n$$\n\\begin{align}\n\\mathcal{L}(\\Theta : \\mathcal{D}) %& = \\prod_{d=1}^{M} P(x[d], y[d] | \\Theta) \\\\\n %& = \\prod_{d=1}^{M} P(x[d] | \\Theta) P(y[d] | x[d], \\Theta) \\\\\n %& = \\left(\\prod_{d=1}^{M} P(x[d] | \\Theta)\\right) \\left(\\prod_{d=1}^{M} P(y[d] | x[d], \\Theta)\\right) \\\\\n %& = \\underbrace{\\left(\\prod_{d=1}^{M} P(x[d] | \\theta_X)\\right)}_{\\text{local likelihood}} \\underbrace{\\left(\\prod_{d=1}^{M} P(y[d] | x[d], \\theta_{Y|X})\\right)}_{\\text{local likelihood}}\n\\end{align}\n$$\n\nThis can be generalized:\n\n$$\n\\begin{align}\n\\mathcal{L}(\\Theta : \\mathcal{D}) & = \\prod_{d=1}^{M} P(\\bar{x}[d] | \\Theta) \\\\\n & = \\prod_{d=1}^{M} \\prod_{i=1}^{n} P(x_i[d] | \\mathrm{Pa}x_i[d], \\Theta) \\\\\n & = \\prod_{i=1}^{n} \\underbrace{\\prod_{d=1}^{M} P(x_i[d] | \\mathrm{Pa}x_i[d], \\theta_{X_i|\\mathrm{Pa}X_i})}_{\\text{local likelihood } \\mathcal{L}_i(\\theta_{X_i|\\mathrm{Pa}X_i} : \\mathcal{D})} \\\\\n & = \\prod_{i=1}^{n} \\mathcal{L}_i(\\theta_{X_i|\\mathrm{Pa}X_i} : \\mathcal{D})\n\\end{align}\n$$\n\nThus, if the parameters $\\theta_{X_i|\\mathrm{Pa}X_i}$ are disjoint, then the MLE can be computed by maximizing each local likelihood separately :).\n\n### MLE for table CPDs\n\nUsing the above, the local likelihood ($\\bar{U}=\\mathrm{Pa}X$, see in the whiteboard)\n\n$$\n\\begin{align}\n\\prod_{d=1}^{M} P(x[d] | \\bar{u}[d], \\theta_{X|\\bar{U}}) & = \\prod_{x \\in \\mathrm{Val}(X),\\bar{u} \\in \\mathrm{Val}(U)} \\prod_{d: x[d]=x, \\bar{u}[d]=\\bar{u}} \\underbrace{P(x[d] | \\bar{u}[d], \\theta_{X|\\bar{U}})}_{\\theta_{x|\\bar{u}}} \\\\\n& = \\prod_{x \\in \\mathrm{Val}(X),\\bar{u} \\in \\mathrm{Val}(U)} \\theta_{x | \\bar{u}}^{M(x,\\bar{u})},\n\\end{align}\n$$\n\nwhere $M(x,\\bar{u})$ is the number of times that the sample is consistent with $x, \\bar{u}$.\n\nApplying the same principle as in mutinomial distributions:\n\n$$\\hat{\\theta}_{x|\\bar{u}} = \\frac{M(x,\\bar{u})}{\\sum_{x'} M(x',\\bar{u})} = \\frac{M(x,\\bar{u})}{M(\\bar{u})},$$\n\nthe MLE estimate corresponds to the count of data instances consistent with $x,\\bar{u}$ over the count of data instances consistent with $\\bar{u}$.\n\n### How is this done in `pgmpy`?\n\nConsider the simple model $$X \\to Y:$$\n\n\n```python\n# Import pgmpy.models.BayesianModel\n\n# Import pgmpy.estimators.MaximumLikelihoodEstimator\n\n```\n\n\n```python\n# Import pandas\n\n```\n\n\n```python\n\n```\n\n Help on built-in function randint:\n \n randint(...) method of numpy.random.mtrand.RandomState instance\n randint(low, high=None, size=None, dtype='l')\n \n Return random integers from `low` (inclusive) to `high` (exclusive).\n \n Return random integers from the \"discrete uniform\" distribution of\n the specified dtype in the \"half-open\" interval [`low`, `high`). If\n `high` is None (the default), then results are from [0, `low`).\n \n .. note::\n New code should use the ``integers`` method of a ``default_rng()``\n instance instead; see `random-quick-start`.\n \n Parameters\n ----------\n low : int or array-like of ints\n Lowest (signed) integers to be drawn from the distribution (unless\n ``high=None``, in which case this parameter is one above the\n *highest* such integer).\n high : int or array-like of ints, optional\n If provided, one above the largest (signed) integer to be drawn\n from the distribution (see above for behavior if ``high=None``).\n If array-like, must contain integer values\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. Default is None, in which case a\n single value is returned.\n dtype : dtype, optional\n Desired dtype of the result. All dtypes are determined by their\n name, i.e., 'int64', 'int', etc, so byteorder is not available\n and a specific precision may have different C types depending\n on the platform. The default value is `np.int_`.\n \n .. versionadded:: 1.11.0\n \n Returns\n -------\n out : int or ndarray of ints\n `size`-shaped array of random integers from the appropriate\n distribution, or a single such random int if `size` not provided.\n \n See Also\n --------\n random_integers : similar to `randint`, only for the closed\n interval [`low`, `high`], and 1 is the lowest value if `high` is\n omitted.\n Generator.integers: which should be used for new code.\n \n Examples\n --------\n >>> np.random.randint(2, size=10)\n array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) # random\n >>> np.random.randint(1, size=10)\n array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])\n \n Generate a 2 x 4 array of ints between 0 and 4, inclusive:\n \n >>> np.random.randint(5, size=(2, 4))\n array([[4, 0, 2, 1], # random\n [3, 2, 2, 0]])\n \n Generate a 1 x 3 array with 3 different upper bounds\n \n >>> np.random.randint(1, [3, 5, 10])\n array([2, 2, 9]) # random\n \n Generate a 1 by 3 array with 3 different lower bounds\n \n >>> np.random.randint([1, 5, 7], 10)\n array([9, 8, 7]) # random\n \n Generate a 2 by 4 array using broadcasting with dtype of uint8\n \n >>> np.random.randint([1, 3, 5, 7], [[10], [20]], dtype=np.uint8)\n array([[ 8, 6, 9, 7], # random\n [ 1, 16, 9, 12]], dtype=uint8)\n \n\n\n\n```python\n# Generate some random data for X -> Y\n\n```\n\n\n```python\n# Wrap data in a dataframe\n\n```\n\n\n\n\n
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                                        \n\n\n\n\n```python\n# Instantiate model\n\n```\n\n\n```python\n# Fit using MLE\n\n```\n\n\n```python\n# Show CPDs\n\n```\n\n +------+-----+\n | X(0) | 0.5 |\n +------+-----+\n | X(1) | 0.5 |\n +------+-----+\n\n\n\n```python\n\n```\n\n +------+------+------+\n | X | X(0) | X(1) |\n +------+------+------+\n | Y(0) | 0.8 | 0.6 |\n +------+------+------+\n | Y(1) | 0.2 | 0.4 |\n +------+------+------+\n\n\n### Fragmentation and overfitting\n\nWe have that\n\n$$\\hat{\\theta}_{x|\\bar{u}} = \\frac{M(x,\\bar{u})}{\\sum_{x'} M(x',\\bar{u})} = \\frac{M(x,\\bar{u})}{M(\\bar{u})}.$$\n\n- Note that the number of possible joint assignments $x, \\bar{u}$ grows exponentially with the size of $|\\bar{U}|$.\n\n- This is, if $X$ has a lot of parents there will be few instances of $x, \\bar{u}$.\n\n- This will lead to very poor parameter estimates $\\hat{\\theta}_{x|\\bar{u}}$\n\nThis phenomenon is called **fragmentation**.\n\nHence, when the amount of data is limited, we often obtan better generalization performances with simpler structures, **even when wrong**.\n\n___\n\n# 3. Bayesian parameter estimation for BNs\n\nWe motivate the study of **Bayesian estimation** with a remarkable limitation of MLE:\n\n- Assume a coin is tossed 10 times, and it comes out $x^1$ 7 out of the the 10 tosses.\n\n $$\\hat{\\theta}_{MLE} = 0.7$$\n\n- Assume a coin is tossed 10000 times, and it comes out $x^1$ 7000 out of the the 10000 tosses.\n\n $$\\hat{\\theta}_{MLE} = 0.7$$\n\nThe MLE in the second scenario is much more plausible than the MLE in the first one.\n\nHowever, the MLE has not the ability to distinguish between these scenarios.\n\n## 3.1. Bayesian estimation\n\nLet's revisit the simple parameter estimation example:\n\n\n```python\nImage(\"figures/iid.png\")\n```\n\n**Bayesian paradigm:** If we are uncertain about something, then we should view it as a Random Variable over which we have a distribution that is updated over time as more data is acquired.\n\nUnder this paradigm, the parameter $\\theta$ will be a Random Variable itself.\n\n- Given a fixed $\\theta$, the tosses are independent.\n\n- If $\\theta$ is not known, then tosses are not marginally independent:\n - Each toss tells us something about $\\theta$.\n\nJoint probability model:\n\n$$\n\\begin{align}\nP(x[1], \\dots, x[M], \\theta) & = P(x[1], \\dots, x[M] | \\theta) P(\\theta) \\\\\n & = P(\\theta) \\underbrace{\\prod_{d=1}^{M} P(x[d] | \\theta)}_{\\text{likelihood function}} \\\\\n & = P(\\theta) \\theta^{M_1} (1 - \\theta)^{M_0}.\n\\end{align}\n$$\n\nHence, the posterior:\n\n$$\n\\begin{align}\nP(\\theta | x[1], \\dots, x[M]) & = \\frac{P(x[1], \\dots, x[M] | \\theta) P(\\theta)}{P(x[1], \\dots, x[M])} \\\\\n & \\propto P(x[1], \\dots, x[M] | \\theta) P(\\theta) \\\\\n & = \\theta^{M_1} (1 - \\theta)^{M_0} P(\\theta)\n\\end{align}\n$$\n\n- $P(x[1], \\dots, x[M] | \\theta)$ is the likelihood function, which we already know.\n\n- $P(\\theta)$ is the **prior distribution**.\n\n- $P(x[1], \\dots, x[M])$ is a normalizing constant with respect to $\\theta$.\n\n### Prior: Dirichlet distribution\n\nIf $X$ is a multinomial distribution over $k$ values, then a good assumption for the distribution of the parameters $\\bar{\\theta} = [\\theta_1, \\dots, \\theta_k]$ is the [**Dirichlet distribution**](https://en.wikipedia.org/wiki/Dirichlet_distribution).\n\n$\\bar{\\theta} \\sim \\mathrm{Dir}(\\bar{\\alpha})$, where $\\bar{\\alpha}=[\\alpha_1, \\dots, \\alpha_k]$, if\n\n$$P(\\bar{\\theta}; \\bar{\\alpha}) = \\frac{1}{Z} \\prod_{i=1}^{k} \\theta_i^{\\alpha_i - 1},$$\n\nwhere $\\sum_{i=1}^{k}\\theta_i=1$ and,\n\n$$Z = \\frac{\\prod_{i=1}^{k} \\Gamma(\\alpha_i)}{\\Gamma(\\sum_{i=1}^{k}\\alpha_i)},$$\n\nwhere $\\Gamma(x)= \\int_{0}^{\\infty}t^{x-1}e^{-t} \\mathrm{d}t$ is the continuous extension of the factorial.\n\n- Intuitively, the hyperparameters $\\alpha_i$ correspond to the number of samples we have previously seen.\n\n- The Dirichlet distribution is a multivariate generalization of the Beta distribution.\n\n\n```python\n# Special function gamma\n\n```\n\n\n```python\n# Beta distribution\n\n```\n\n\n```python\n# Plot the beta distribution for different values of alpha1 and alpha0\n\n```\n\n /home/esteban/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:4: RuntimeWarning: divide by zero encountered in power\n after removing the cwd from sys.path.\n\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n#### Now, having this Dirichlet prior, how is the posterior?\n\nAssume that $X$ is multinomial.\n\nWe have that:\n\n$$\nP(\\bar{\\theta} | \\mathcal{D}) \\propto P(\\mathcal{D} | \\bar{\\theta}) P(\\bar{\\theta})\n$$\n\n- The likelihood function is: $P(\\mathcal{D} | \\bar{\\theta}) = \\prod_{i=1}^{k} \\theta_i^{M_i}$.\n\n- The prior is: $P(\\bar{\\theta}) \\propto \\prod_{i=1}^{k} \\theta_i^{\\alpha_i - 1}$\n\nThen,\n\n$$\nP(\\bar{\\theta} | \\mathcal{D}) \\propto \\prod_{i=1}^{k} \\theta_i^{\\alpha_i + M_i - 1}.\n$$\n\nThis is, the posterior also Dirichlet: $(\\bar{\\theta} | \\mathcal{D}) \\sim \\mathrm{Dir}(\\alpha_1 + M_1, \\dots, \\alpha_k + M_k)$.\n\n### In this sense ...\n\nBefore seeing data instances:\n\n$$P(X=x^i | \\theta) = \\frac{\\alpha_i}{\\sum_{j=1}^{k}\\alpha_j} = \\frac{\\alpha_i}{\\alpha}.$$\n\nAfter seeing data instances:\n\n$$P(X[M+1]=x^i | \\theta, x[1], \\dots, x[M]) = \\frac{\\alpha_i + M_i}{\\alpha + M}.$$\n\n\n- The term $\\alpha=\\sum_{j=1}^{k}\\alpha_j$ is known as **equivalent sample size**.\n - Larger $\\alpha$ means that we are more confident in our prior.\n\n### What is the effect?\n\n\n```python\n# Generate random numbers form a Bernoulli distribution\n\n```\n\n\n```python\n# Obtain MLE estimation at each step\n\n```\n\n\n```python\n# Obtain Bayesian estimation for different values of alpha\n# alpha1 = alpha2 = alpha_i\n# Equivalent sample size alpha = alpha1 + alpha2 = 2 * alpha_i\n\n```\n\n\n```python\n# Plot\n\n```\n\n- Bayesian estimates are less sensitive to noise in the data.\n\n- Bayesian estimates $\\to$ MLE as number of samples $\\to \\infty$.\n\nThe ideas in BNs are very similar. Please, take a look after Chapter 17.\n\n### How is this done in `pgmpy`?\n\nConsider the simple model $$X \\to Y:$$\n\n\n```python\n# Import pgmpy.estimators.BayesianEstimator\n\n```\n\n\n```python\n\n```\n\n Help on class BayesianEstimator in module pgmpy.estimators.BayesianEstimator:\n \n class BayesianEstimator(pgmpy.estimators.base.ParameterEstimator)\n | BayesianEstimator(model, data, **kwargs)\n | \n | Method resolution order:\n | BayesianEstimator\n | pgmpy.estimators.base.ParameterEstimator\n | pgmpy.estimators.base.BaseEstimator\n | builtins.object\n | \n | Methods defined here:\n | \n | __init__(self, model, data, **kwargs)\n | Class used to compute parameters for a model using Bayesian Parameter Estimation.\n | See `MaximumLikelihoodEstimator` for constructor parameters.\n | \n | estimate_cpd(self, node, prior_type='BDeu', pseudo_counts=[], equivalent_sample_size=5)\n | Method to estimate the CPD for a given variable.\n | \n | Parameters\n | ----------\n | node: int, string (any hashable python object)\n | The name of the variable for which the CPD is to be estimated.\n | \n | prior_type: 'dirichlet', 'BDeu', 'K2',\n | string indicting which type of prior to use for the model parameters.\n | - If 'prior_type' is 'dirichlet', the following must be provided:\n | 'pseudo_counts' = dirichlet hyperparameters; 2-D array of shape\n | (node_card, product of parents_card) with a \"virtual\" count for\n | each variable state in the CPD.\n | The virtual counts are added to the actual state counts found in the data.\n | (if a list is provided, a lexicographic ordering of states is assumed)\n | - If 'prior_type' is 'BDeu', then an 'equivalent_sample_size'\n | must be specified instead of 'pseudo_counts'. This is equivalent to\n | 'prior_type=dirichlet' and using uniform 'pseudo_counts' of\n | `equivalent_sample_size/(node_cardinality*np.prod(parents_cardinalities))`.\n | - A prior_type of 'K2' is a shorthand for 'dirichlet' + setting every pseudo_count to 1,\n | regardless of the cardinality of the variable.\n | \n | Returns\n | -------\n | CPD: TabularCPD\n | \n | Examples\n | --------\n | >>> import pandas as pd\n | >>> from pgmpy.models import BayesianModel\n | >>> from pgmpy.estimators import BayesianEstimator\n | >>> data = pd.DataFrame(data={'A': [0, 0, 1], 'B': [0, 1, 0], 'C': [1, 1, 0]})\n | >>> model = BayesianModel([('A', 'C'), ('B', 'C')])\n | >>> estimator = BayesianEstimator(model, data)\n | >>> cpd_C = estimator.estimate_cpd('C', prior_type=\"dirichlet\", pseudo_counts=[1, 2])\n | >>> print(cpd_C)\n | \u2552\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2555\n | \u2502 A \u2502 A(0) \u2502 A(0) \u2502 A(1) \u2502 A(1) \u2502\n | \u251c\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2524\n | \u2502 B \u2502 B(0) \u2502 B(1) \u2502 B(0) \u2502 B(1) \u2502\n | \u251c\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2524\n | \u2502 C(0) \u2502 0.25 \u2502 0.25 \u2502 0.5 \u2502 0.3333333333333333 \u2502\n | \u251c\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2524\n | \u2502 C(1) \u2502 0.75 \u2502 0.75 \u2502 0.5 \u2502 0.6666666666666666 \u2502\n | \u2558\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255b\n | \n | get_parameters(self, prior_type='BDeu', equivalent_sample_size=5, pseudo_counts=None)\n | Method to estimate the model parameters (CPDs).\n | \n | Parameters\n | ----------\n | prior_type: 'dirichlet', 'BDeu', or 'K2'\n | string indicting which type of prior to use for the model parameters.\n | - If 'prior_type' is 'dirichlet', the following must be provided:\n | 'pseudo_counts' = dirichlet hyperparameters; a dict containing, for each variable, a 2-D\n | array of the shape (node_card, product of parents_card) with a \"virtual\" count for each\n | variable state in the CPD, that is added to the state counts.\n | (lexicographic ordering of states assumed)\n | - If 'prior_type' is 'BDeu', then an 'equivalent_sample_size'\n | must be specified instead of 'pseudo_counts'. This is equivalent to\n | 'prior_type=dirichlet' and using uniform 'pseudo_counts' of\n | `equivalent_sample_size/(node_cardinality*np.prod(parents_cardinalities))` for each node.\n | 'equivalent_sample_size' can either be a numerical value or a dict that specifies\n | the size for each variable seperately.\n | - A prior_type of 'K2' is a shorthand for 'dirichlet' + setting every pseudo_count to 1,\n | regardless of the cardinality of the variable.\n | \n | Returns\n | -------\n | parameters: list\n | List of TabularCPDs, one for each variable of the model\n | \n | Examples\n | --------\n | >>> import numpy as np\n | >>> import pandas as pd\n | >>> from pgmpy.models import BayesianModel\n | >>> from pgmpy.estimators import BayesianEstimator\n | >>> values = pd.DataFrame(np.random.randint(low=0, high=2, size=(1000, 4)),\n | ... columns=['A', 'B', 'C', 'D'])\n | >>> model = BayesianModel([('A', 'B'), ('C', 'B'), ('C', 'D')])\n | >>> estimator = BayesianEstimator(model, values)\n | >>> estimator.get_parameters(prior_type='BDeu', equivalent_sample_size=5)\n | [,\n | ,\n | ,\n | ]\n | \n | ----------------------------------------------------------------------\n | Methods inherited from pgmpy.estimators.base.ParameterEstimator:\n | \n | state_counts(self, variable, **kwargs)\n | Return counts how often each state of 'variable' occured in the data.\n | If the variable has parents, counting is done conditionally\n | for each state configuration of the parents.\n | \n | Parameters\n | ----------\n | variable: string\n | Name of the variable for which the state count is to be done.\n | \n | complete_samples_only: bool\n | Specifies how to deal with missing data, if present. If set to `True` all rows\n | that contain `np.NaN` somewhere are ignored. If `False` then\n | every row where neither the variable nor its parents are `np.NaN` is used.\n | Desired default behavior can be passed to the class constructor.\n | \n | Returns\n | -------\n | state_counts: pandas.DataFrame\n | Table with state counts for 'variable'\n | \n | Examples\n | --------\n | >>> import pandas as pd\n | >>> from pgmpy.models import BayesianModel\n | >>> from pgmpy.estimators import ParameterEstimator\n | >>> model = BayesianModel([('A', 'C'), ('B', 'C')])\n | >>> data = pd.DataFrame(data={'A': ['a1', 'a1', 'a2'],\n | 'B': ['b1', 'b2', 'b1'],\n | 'C': ['c1', 'c1', 'c2']})\n | >>> estimator = ParameterEstimator(model, data)\n | >>> estimator.state_counts('A')\n | A\n | a1 2\n | a2 1\n | >>> estimator.state_counts('C')\n | A a1 a2\n | B b1 b2 b1 b2\n | C\n | c1 1 1 0 0\n | c2 0 0 1 0\n | \n | ----------------------------------------------------------------------\n | Methods inherited from pgmpy.estimators.base.BaseEstimator:\n | \n | test_conditional_independence(self, X, Y, Zs=[], method='chi_square', tol=0.01, **kwargs)\n | \n | ----------------------------------------------------------------------\n | Data descriptors inherited from pgmpy.estimators.base.BaseEstimator:\n | \n | __dict__\n | dictionary for instance variables (if defined)\n | \n | __weakref__\n | list of weak references to the object (if defined)\n \n\n\n\n```python\n# Instantiate estimator\n\n```\n\n\n```python\n# Fit CPD of X\n\n```\n\n +------+-----+\n | X(0) | 0.5 |\n +------+-----+\n | X(1) | 0.5 |\n +------+-----+\n\n\n\n```python\n# Fit CPD of Y\n\n```\n\n +------+--------------------+---------------------+\n | X | X(0) | X(1) |\n +------+--------------------+---------------------+\n | Y(0) | 0.7142857142857143 | 0.5714285714285714 |\n +------+--------------------+---------------------+\n | Y(1) | 0.2857142857142857 | 0.42857142857142855 |\n +------+--------------------+---------------------+\n\n\n# Announcements\n\n## 1. Quiz.\n\n\n\n
                                        \nCreated with Jupyter by Esteban Jim\u00e9nez Rodr\u00edguez.\n
                                        \n", "meta": {"hexsha": "4985d6ecb7da07136201437f8eaaf0dfbd9ddeff", "size": 351240, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Modulo3/Clase10/ParameterEstimationBN.ipynb", "max_stars_repo_name": "esjimenezro/mgp_online_public", "max_stars_repo_head_hexsha": "b2d2a49c1c8730d1e507144ac4f65ec6842a5d94", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Modulo3/Clase10/ParameterEstimationBN.ipynb", "max_issues_repo_name": "esjimenezro/mgp_online_public", "max_issues_repo_head_hexsha": "b2d2a49c1c8730d1e507144ac4f65ec6842a5d94", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Modulo3/Clase10/ParameterEstimationBN.ipynb", "max_forks_repo_name": "esjimenezro/mgp_online_public", "max_forks_repo_head_hexsha": "b2d2a49c1c8730d1e507144ac4f65ec6842a5d94", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 6, "max_forks_repo_forks_event_min_datetime": "2021-08-18T01:07:56.000Z", "max_forks_repo_forks_event_max_datetime": "2021-09-07T04:06:28.000Z", "avg_line_length": 206.1267605634, "max_line_length": 172008, "alphanum_fraction": 0.8965920738, "converted": true, "num_tokens": 9373, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.7461389986757757, "lm_q1q2_score": 0.4108296283839711}} {"text": "# Problema de selecci\u00f3n de portafolio con preferencias media-varianza\n\n\n\nEn la clase pasada hablamos acerca de:\n- preferencias,\n- funciones de utilidad,\n- la actitud de los inversionistas de cara al riesgo,\n- la aversi\u00f3n al riesgo, entre otros.\n\nTodas ellas son piezas que necesitamos para responder la pregunta de \u00bfc\u00f3mo un inversionista toma la decisi\u00f3n \u00f3ptima de selecci\u00f3n de portafolio?\n\nEn esta clase al fin estamos listos para ensamblar estos conceptos y escribir el problema de selecci\u00f3n de portafolios. \n\nEn el camino aprenderemos acerca del concepto de **utilidad esperada**, que nos permite trabajar con incertidumbre en el modelado econ\u00f3mico (una de las ideas m\u00e1s importantes en econom\u00eda). Esta idea tiene m\u00e1s de 60 a\u00f1os, y b\u00e1sicamente dice que los individuos, cuando est\u00e1n de cara a incertidumbre, maximizan el valor esperado de su utilidad (solo cierto si somos homo economicus). \n\nAdem\u00e1s del concepto de utilidad esperada, aprenderemos acerca de **preferencias media-varianza**. Es decir, supondremos que los inversionistas toman decisiones basados en un tipo particular de preferencias.\n\nCon lo anterior, estableceremos el problema de selecci\u00f3n de portafolios.\n\n**Objetivos:**\n\n- \u00bfQu\u00e9 es utilidad esperada?\n- \u00bfQu\u00e9 son las preferencias media-varianza?\n- Funciones de utilidad media-varianza.\n- Enunciado y soluci\u00f3n del problema b\u00e1sico de selecci\u00f3n de portafolio.\n\n*Referencia:*\n- Notas del curso \"Portfolio Selection and Risk Management\", Rice University, disponible en Coursera.\n___\n\n## 1. Utilidad esperada\n- B\u00e1sicamente combina las probabilidades de los resultados con c\u00f3mo los inversionistas se sienten con dichos resultados.\n- En otras palabras, la utilidad esperada multiplica la probabilidad de suceso de un evento con la utilidad que genera dicho evento.\n\nRecordemos que las *funciones de utilidad* permiten a los inversionistas expresar c\u00f3mo se sienten con los resultados, especialmente en los malos ratos. \n\nEntonces la *utilidad esperada* es una herramienta que nos permite cuantificar c\u00f3mo nos sentimos en nuestros malos momentos econ\u00f3micos, capturando el riesgo con la probabilidad de ocurrencia de dichos malos momentos. \n\nDado este marco de trabajo, cualquier decisi\u00f3n se puede escribir como la maximizaci\u00f3n de la utilidad esperada:\n\\begin{align}\n\\max_{\\theta} & \\quad E[U(W)], \\\\\n\\end{align}\nmediante la escogencia de cierta variable $\\theta$ (gastos, planes de ahorro, compra de activos, planes de producci\u00f3n, etc.).\n\nPara nuestros prop\u00f3sitos, la variable de decisi\u00f3n ser\u00e1n los pesos o ponderaciones del portafolio.\n\nAdicionalmente, en el contexto de la decisi\u00f3n de distribuci\u00f3n de la riqueza entre activos, el problema de maximizaci\u00f3n tendr\u00e1 com\u00fanmente las siguientes restricciones:\n- universo de inversi\u00f3n,\n- posici\u00f3n en los activos dados.\n\n**Ejemplo.** \n\nSupongamos que un inversionista debe determinar la composici\u00f3n \u00f3ptima de su portafolio, que contiene activos y bonos. Supongamos que son los \u00fanicos instrumentos disponibles.\n\nSean:\n- $w_s$: peso o ponderaci\u00f3n de activos en el portafolio,\n- $w_b$: peso o ponderaci\u00f3n de bonos en el portafolio,\n- $r_s$: rendimiento de los activos, y\n- $r_b$: rendimiento de los bonos.\n\nDe manera que podemos escribir el problema de selecci\u00f3n de portafolios como la maximizaci\u00f3n de la utilidad esperade de nuestra riqueza futura, la cual depender\u00e1 de nuestros rendimientos:\n \n\\begin{align}\n\\max_{w_s,w_b} &\\quad E[U(W)]\\\\\n\\text{s. a.} &\\quad W=W_0(1+w_sr_s+w_br_b)\\\\\n &\\quad w_s+w_b=1\n\\end{align}\n\n\nPreguntas:\n- \u00bfQu\u00e9 significan las restricciones?\n- Ya que tenemos planteado este problema b\u00e1sico, \u00bfqu\u00e9 har\u00eda falta para empezar a resolverlo?\n___\n\n## 2. Preferencias media-varianza\n\n### 2.1. Utilidad media-varianza\n\nEntonces, \u00bfqu\u00e9 funciones de utilidad deber\u00edamos de usar en este problema de selecci\u00f3n de portafolios?\n\n- La respuesta es: **preferencias media-varianza**.\n- \u00c9stas ser\u00e1n representadas en t\u00e9rminos de funciones de utilidad como: **utilidad media-varianza**.\n\nUsamos la *utilidad media-varianza* en el problema de selecci\u00f3n de portafolios dado que \u00e9sta decribe el \"trade-off\" entre riesgo y rendimiento que enfrentan los inversionistas. La *utilidad media-varianza* est\u00e1 dada por la siguiente expresi\u00f3n:\n\n$$U=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2,$$\n\ndonde\n- $E[r_p]$ es el rendimiento esperado del portafolio,\n- $\\sigma_p^2$ es la varianza del portafolio, y\n- $\\gamma$ es el coeficiente de aversi\u00f3n al riesgo. \n\n#### Intuici\u00f3n acerca de la funci\u00f3n de utilidad media-varianza:\n- S\u00f3lo se preocupa por medias :) y varianzas :(.\n- Incrementa con: rendimiento esperado del portafolio.\n- Decrece con: varianza del portafolio.\n- Malos tiempos: rendimientos son bajos y las volatilidades son altas.\n- Conecta bastante bien con la teor\u00eda moderna de portafolios, la cual caracteriza los rendimientos con medias y varianzas \u00fanicamente.\n- Criticada por su limitaci\u00f3n: supone que los inversionistas s\u00f3lo se preocupan por medias y varianzas.\n\n### 2.2. Curvas de indiferencia\n\n*\u00bfRecuerdan las curvas de nivel que se ven en c\u00e1lculo de varias variables?*\n- Bien, ac\u00e1 nos servir\u00e1n para representar la utilidad media-varianza gr\u00e1ficamente.\n- En el contexto de utilidad media-varianza, las curvas de nivel se llaman **curvas de indiferencia**.\n\nDados ciertos niveles de utilidad $U_1>U_2>U_3$, las curvas de indiferencia relativas a estos niveles de utilidad, son los lugares geom\u00e9tricos en el espacio de rendimiento esperado vs. volatilidad representados por las siguientes expresiones\n\n$$U_1=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2\\Rightarrow E[r_p]=\\frac{1}{2}\\gamma\\sigma_p^2+U_1,$$\n\n$$U_2=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2\\Rightarrow E[r_p]=\\frac{1}{2}\\gamma\\sigma_p^2+U_2,$$\n\n$$U_3=E[r_p]-\\frac{1}{2}\\gamma\\sigma_p^2\\Rightarrow E[r_p]=\\frac{1}{2}\\gamma\\sigma_p^2+U_3.$$\n\n**Gr\u00e1ficamente**\n\n\n```python\n# Importar numpy y pyplot\nimport numpy as np\nimport matplotlib.pyplot as plt\n%matplotlib inline\n```\n\n\n```python\n# Coeficiente de aversi\u00f3n al riesgo (entre 1 y 10 com\u00fanmente)\ng = 7\n# Niveles de utilidad \nU1, U2, U3 = 3, 2, 1\n# Vector de volatilidades (sugerido 1%-60%)\nsp = np.linspace(0.01, 0.60, 100)\n# Curvas de indiferencia\nErp1, Erp2, Erp3 = 0.5*g*sp**2+U1, 0.5*g*sp**2+U2, 0.5*g*sp**2+U3\n```\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(10,8))\nplt.plot(sp, Erp1, lw=5, label='U1')\nplt.plot(sp, Erp2, lw=5, label='U2')\nplt.plot(sp, Erp3, lw=5, label='U3')\nplt.grid()\nplt.xlabel('Volatilidad $\\sigma_p$')\nplt.ylabel('Rendimiento esperado $E[r_p]$')\nplt.legend(loc='best')\n```\n\nBueno, \u00bfy porqu\u00e9 se llaman curvas de indiferencia?, \u00bfqu\u00e9 representa una cuva de indiferencia?\n\n- Porque sobre una misma curva el nivel de utilidad es el mismo (es indiferente).\n- Son todas las combinaciones de riesgo y rendimiento que producen un mismo nivel de utilidad.\n\nVolviendo al problema de selecci\u00f3n de portafolios, queremos la utilidad m\u00e1s alta.\n- \u00bfCu\u00e1l de las anteriores curvas de indiferencia corresponde a la utilidad m\u00e1s alta?\n- Intuitivamente, \u00bfporqu\u00e9?\n- Curvas de indiferencia para niveles de utilidad m\u00e1s altos, estar\u00e1n...\n\nNotamos adem\u00e1s que las anteriores curvas de indiferencia son *paralelas* una con otra. Claro, las dibujamos con el mismo coeficiente de aversi\u00f3n al riesgo.\n\n\u00bfC\u00f3mo cambian estas curvas para coeficientes de aversi\u00f3n al riesgo m\u00e1s altos?\n\n\n```python\n# Coeficientes de aversi\u00f3n al riesgo (entre 1 y 10 com\u00fanmente)\ng1, g2, g3 = 3, 5, 7\n# Nivel de utilidad\nU = 1\n# Vector de volatilidades (sugerido 1%-60%)\nsp = np.linspace(0.01, 0.60, 100)\n# Curvas de indiferencia\nErp1, Erp2, Erp3 = 0.5*g1*sp**2+U, 0.5*g2*sp**2+U, 0.5*g3*sp**2+U\n```\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(10,8))\nplt.plot(sp, Erp1, lw=5, label='$\\gamma_1$')\nplt.plot(sp, Erp2, lw=5, label='$\\gamma_2$')\nplt.plot(sp, Erp3, lw=5, label='$\\gamma_3$')\nplt.grid()\nplt.xlabel('Volatilidad $\\sigma_p$')\nplt.ylabel('Rendimiento esperado $E[r_p]$')\nplt.legend(loc='best')\n```\n\n\u00bfC\u00f3mo interpretamos las anteriores gr\u00e1ficas?, \u00bfqu\u00e9 pasa con las personas m\u00e1s aversas al riesgo?\n- Se puede ver de dos maneras: para un mismo nivel de rendimiento esperado, una persona m\u00e1s aversa al riesgo soporta un nivel menor de riesgo; equivalentemente, para un mismo nivel de riesgo, una persona m\u00e1s aversa al riesgo requerir\u00e1 un nivel de rendimiento esperado m\u00e1s alto.\n\nCon todo lo anterior, el problema de selecci\u00f3n de portafolios se puede plantear como *\"encontrar la curva de indeferencia m\u00e1s alta dado el conjunto de oportunidades de inversi\u00f3n y restricciones\"*.\n\n## 3. Problema de selecci\u00f3n de portafolios: una ilustraci\u00f3n\n\nAhora ilustraremos el problema de selecci\u00f3n de portafolios con algunos datos. \n- Por ahora solo queremos ilustrar gr\u00e1ficamente c\u00f3mo se resuelve este problema. Trabajar en la intuici\u00f3n.\n- En las siguientes dos clases nos enfocaremos en c\u00f3mo resolverlo anal\u00edticamente.\n\nAc\u00e1 tenemos el rendimiento medio anual y la volatilidad para dos instrumentos usando datos de EU: instrumentos de deuda (bonos) y acciones. Supondremos que el inversionista solo puede invertir en estas dos clases de instrumentos.\n\n\n```python\n# Importamos pandas\nimport pandas as pd\n```\n\n\n```python\n# Datos\ndata=pd.DataFrame(index=['Stocks','Bonds', 'CorrSB'], columns=['Mean', 'Std'])\ndata['Mean'] = np.array([0.119,0.0591,0.113])\ndata['Std'] = np.array([0.1915,0.0833,None])\ndata.round(4)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        MeanStd
                                        Stocks0.11900.1915
                                        Bonds0.05910.0833
                                        CorrSB0.1130None
                                        \n
                                        \n\n\n\nEntonces, \u00bfcu\u00e1l es la distribuci\u00f3n de riqueza \u00f3ptima?, o m\u00e1s bien, \u00bfcu\u00e1l es la composici\u00f3n \u00f3ptima del portafolio para un inversionista dado su nivel de aversi\u00f3n al riesgo?\n\n**Primero.** Recordamos que, para dos activos, podemos trazar la frontera de m\u00ednima varianza tomando todas las posibles combinaciones de los dos activos.\n\nDe nuevo, sean:\n- $w_s=w$: peso o ponderaci\u00f3n de activos en el portafolio,\n- $w_b=1-w$: peso o ponderaci\u00f3n de bonos en el portafolio,\n- $r_s$: rendimiento de los activos, y\n- $r_b$: rendimiento de los bonos.\n\nEntonces\n\n$$E[r_p]=wE[r_{s}]+(1-w)E[r_b]$$\n\n$$\\sigma_p^2=w^2\\sigma_{s}^2+(1-w)^2\\sigma_b^2+2w(1-w)\\rho_{s,b}\\sigma_s\\sigma_b$$\n\n\n```python\n# Vector de w variando entre 0 y 1 con n pasos\nn = 100\nw = np.linspace(0, 1, n)\n# Rendimientos esperados individuales\nErs = data.loc['Stocks', 'Mean']\nErb = data.loc['Bonds', 'Mean']\n# Volatilidades individuales\nss = data.loc['Stocks', 'Std']\nsb = data.loc['Bonds', 'Std']\n# Correlacion\nrsb = data.loc['CorrSB', 'Mean']\n```\n\n\n```python\n# Crear un DataFrame cuyas columnas sean rendimiento\n# y volatilidad del portafolio para cada una de las w\n# generadas\nportafolios = pd.DataFrame(index = w, columns=['Erp', 'sp'])\nportafolios.loc[:,'Erp'] = w*Ers+(1-w)*Erb\nportafolios.loc[:,'sp'] = ((w*ss)**2+((1-w)*sb)**2+2*w*(1-w)*ss*sb*rsb)**0.5\nportafolios\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Erpsp
                                        0.0000000.05910.0833
                                        0.0101010.05970510.0826995
                                        0.0202020.06031010.0821443
                                        0.0303030.06091520.0816354
                                        0.0404040.06152020.0811735
                                        0.0505050.06212530.0807596
                                        0.0606060.06273030.0803944
                                        0.0707070.06333540.0800784
                                        0.0808080.06394040.0798124
                                        0.0909090.06454550.0795967
                                        0.1010100.06515050.0794319
                                        0.1111110.06575560.0793183
                                        0.1212120.06636060.0792559
                                        0.1313130.06696570.0792451
                                        0.1414140.06757070.0792858
                                        0.1515150.06817580.0793778
                                        0.1616160.06878080.0795212
                                        0.1717170.06938590.0797154
                                        0.1818180.06999090.0799603
                                        0.1919190.0705960.0802553
                                        0.2020200.0712010.0805999
                                        0.2121210.07180610.0809934
                                        0.2222220.07241110.0814352
                                        0.2323230.07301620.0819244
                                        0.2424240.07362120.0824602
                                        0.2525250.07422630.0830417
                                        0.2626260.07483130.083668
                                        0.2727270.07543640.0843381
                                        0.2828280.07604140.0850509
                                        0.2929290.07664650.0858053
                                        .........
                                        0.7070710.1014540.140272
                                        0.7171720.1020590.141944
                                        0.7272730.1026640.143625
                                        0.7373740.1032690.145314
                                        0.7474750.1038740.147012
                                        0.7575760.1044790.148717
                                        0.7676770.1050840.150431
                                        0.7777780.1056890.152152
                                        0.7878790.1062940.15388
                                        0.7979800.1068990.155616
                                        0.8080810.1075040.157358
                                        0.8181820.1081090.159107
                                        0.8282830.1087140.160862
                                        0.8383840.1093190.162623
                                        0.8484850.1099240.16439
                                        0.8585860.1105290.166163
                                        0.8686870.1111340.167942
                                        0.8787880.1117390.169726
                                        0.8888890.1123440.171515
                                        0.8989900.1129490.173309
                                        0.9090910.1135550.175108
                                        0.9191920.114160.176912
                                        0.9292930.1147650.178721
                                        0.9393940.115370.180534
                                        0.9494950.1159750.182352
                                        0.9595960.116580.184173
                                        0.9696970.1171850.185999
                                        0.9797980.117790.187829
                                        0.9898990.1183950.189663
                                        1.0000000.1190.1915
                                        \n

                                        100 rows \u00d7 2 columns

                                        \n
                                        \n\n\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(8,6))\nplt.plot(portafolios.loc[:,'sp'], portafolios.loc[:,'Erp'], label='Portafolios')\nplt.grid()\nplt.xlabel('Volatilidad $\\sigma_p$')\nplt.ylabel('Rendimiento esperado $E[r_p]$')\nplt.legend(loc='best')\n```\n\n**Segundo.** Graficamos en la misma ventana, curvas de indiferencia.\n\n\n```python\n# Niveles de utilidad 0.07, 0.08, 0.09\nU1, U2, U3 = 0.06, 0.07, 0.0725\n# Coeficiente de aversi\u00f3n al riesgo\ng = 3\n# Curvas de indiferencia\nsp = portafolios.loc[:,'sp']\nErp1, Erp2, Erp3 = 0.5*g*sp**2+U1, 0.5*g*sp**2+U2, 0.5*g*sp**2+U3\n```\n\n\n```python\n# Gr\u00e1fica\nplt.figure(figsize=(8,6))\nplt.plot(portafolios.loc[:,'sp'], portafolios.loc[:,'Erp'], label='Portafolios')\nplt.plot(sp, Erp1, lw=2, label='U1')\nplt.plot(sp, Erp2, lw=2, label='U2')\nplt.plot(sp, Erp3, lw=2, label='U3')\nplt.grid()\nplt.xlabel('Volatilidad $\\sigma_p$')\nplt.ylabel('Rendimiento esperado $E[r_p]$')\nplt.legend(loc='best')\n```\n\n**Tercero.** La elecci\u00f3n \u00f3ptima est\u00e1 dada por la curva de indiferencia para el nivel de utilidad m\u00e1s alto que es tangente a la frontera media-varianza.\n- Claramente, esta selecci\u00f3n depende del coeficiente de aversi\u00f3n al riesgo.\n\n\n```python\n# Gr\u00e1fica con zoom\n# Gr\u00e1fica\nplt.figure(figsize=(8,6))\nplt.plot(portafolios.loc[:,'sp'], portafolios.loc[:,'Erp'], label='Portafolios')\nplt.plot(sp, Erp1, lw=2, label='U1')\nplt.plot(sp, Erp2, lw=2, label='U2')\nplt.plot(sp, Erp3, lw=2, label='U3')\nplt.grid()\nplt.axis([0.12, 0.14, 0.09, 0.11])\nplt.xlabel('Volatilidad $\\sigma_p$')\nplt.ylabel('Rendimiento esperado $E[r_p]$')\nplt.legend(loc='best')\n```\n\n# Anuncios parroquiales\n\n## 1. Quiz la siguiente clase.\n## 2. Segunda entrega de la tarea 5 para el mi\u00e9rcoles 27 de Marzo.\n## 3. Un par de art\u00edculos del WSJ y el NYT que discuten herramientas disponibles para la medici\u00f3n de su propia tolerancia al riesgo:\n- [Art\u00edculo 1](https://www.nytimes.com/2016/02/13/your-money/as-stocks-fall-its-time-to-measure-your-risk-tolerance.html)\n- [Art\u00edculo 2](https://www.wsj.com/articles/check-your-tolerance-for-investment-risk-now-before-markets-sag-1405619939)\n\n\n\n
                                        \nCreated with Jupyter by Esteban Jim\u00e9nez Rodr\u00edguez.\n
                                        \n", "meta": {"hexsha": "6bbef290a01e015d6b39e39a779b3c81ccba0dd9", "size": 215097, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Modulo3/Clase12_ProblemaSeleccionPortafolio.ipynb", "max_stars_repo_name": "apaolavr/PorInv2019-1", "max_stars_repo_head_hexsha": "a3280af05d6c3cf489f3748b01f3d34bab321c9d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Modulo3/Clase12_ProblemaSeleccionPortafolio.ipynb", "max_issues_repo_name": "apaolavr/PorInv2019-1", "max_issues_repo_head_hexsha": "a3280af05d6c3cf489f3748b01f3d34bab321c9d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Modulo3/Clase12_ProblemaSeleccionPortafolio.ipynb", "max_forks_repo_name": "apaolavr/PorInv2019-1", "max_forks_repo_head_hexsha": "a3280af05d6c3cf489f3748b01f3d34bab321c9d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 199.5333951763, "max_line_length": 43332, "alphanum_fraction": 0.8866790332, "converted": true, "num_tokens": 6957, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.746138993030751, "lm_q1q2_score": 0.4108296252757789}} {"text": "```python\nfrom IPython.display import Image\nImage('../../Python_probability_statistics_machine_learning_2E.png',width=200)\n```\n\nThis chapter takes a geometric view of probability theory and relates it to\nfamiliar concepts in linear algebra and geometry. This approach connects your\nnatural geometric intuition to the key abstractions in probability that can\nhelp\nguide your reasoning. This is particularly important in probability\nbecause it\nis easy to be misled. We need a bit of rigor and some\nintuition to guide us.\n\nIn\ngrade school, you were introduced to the natural numbers (i.e., `1,2,3,..`)\nand\nyou learned how to manipulate them by operations like addition,\nsubtraction, and\nmultiplication. Later, you were introduced to positive and\nnegative numbers and\nwere again taught how to manipulate them. Ultimately, you\nwere introduced to the\ncalculus of the real line, and learned how to\ndifferentiate, take limits, and so\non. This progression provided more\nabstractions, but also widened the field of\nproblems you could successfully\ntackle. The same is true of probability. One way\nto think about probability is\nas a new number concept that allows you to tackle\nproblems that have a special\nkind of *uncertainty* built into them. Thus, the\nkey idea is that there is some\nnumber, say $x$, with a traveling companion, say,\n$f(x)$, and this companion\nrepresents the uncertainties about the value of $x$\nas if looking at the number\n$x$ through a frosted window. The degree of opacity\nof the window is\nrepresented by $f(x)$. If we want to manipulate $x$, then we\nhave to figure\nout what to do with $f(x)$. For example if we want $y= 2 x $,\nthen we have to\nunderstand how $f(x)$ generates $f(y)$. \n\nWhere is the *random*\npart? To conceptualize this, we need still another\nanalogy: think about a\nbeehive with the swarm around it representing $f(x)$,\nand the hive itself, which\nyou can barely see through the swarm, as $x$. The\nrandom piece is you don't\nknow *which* bee in particular is going to sting you!\nOnce this happens the\nuncertainty evaporates.\nUp until that happens, all we have is a concept of a\nswarm (i.e., density of\nbees) which represents a *potentiality* of which bee\nwill ultimately sting.\nIn summary, one way to think about probability is as a\nway of carrying through\nmathematical reasoning (e.g., adding, subtracting,\ntaking\nlimits) with a notion of potentiality that is so-transformed by these\noperations.\n\n## Understanding Probability Density\n\nIn order to understand the\nheart of modern probability, which is built\non the Lesbesgue theory of\nintegration, we need to extend the concept\nof integration from basic calculus.\nTo begin, let us consider the\nfollowing piecewise function\n\n$$\nf(x) = \\begin{cases}\n 1 & \\mbox{if } 0 < x \\leq 1 \\\\\\\n2 & \\mbox{if } 1 < x \\leq 2 \\\\\\\n 0 & \\mbox{otherwise }\n\\end{cases}\n$$\n\n as shown in [Figure](#fig:intro_001). In calculus, you learned\nRiemann\nintegration, which you can apply here as\n\n\n\n\n\n

                                        Simple piecewise-constant function.

                                        \n\n\n\n\n$$\n\\int_0^2 f(x) dx = 1 + 2 = 3\n$$\n\n which has the usual interpretation as the area of the two rectangles\nthat make\nup $f(x)$. So far, so good.\n\nWith Lesbesgue integration, the idea is very\nsimilar except that we\nfocus on the y-axis instead of moving along the x-axis.\nThe question\nis given $f(x) = 1$, what is the set of $x$ values for which this\nis\ntrue? For our example, this is true whenever $x\\in (0,1]$. So now we\nhave a\ncorrespondence between the values of the function (namely, `1`\nand `2`) and the\nsets of $x$ values for which this is true, namely,\n$\\lbrace (0,1] \\rbrace$ and\n$\\lbrace (1,2] \\rbrace$, respectively. To\ncompute the integral, we simply take\nthe function values (i.e., `1,2`)\nand some way of measuring the size of the\ncorresponding interval\n(i.e., $\\mu$) as in the following:\n\n$$\n\\int_0^2 f d\\mu = 1 \\mu(\\lbrace (0,1] \\rbrace) + 2 \\mu(\\lbrace (1,2] \\rbrace)\n$$\n\nWe have suppressed some of the notation above to emphasize generality. Note\nthat\nwe obtain the same value of the integral as in the Riemann case when\n$\\mu((0,1])\n= \\mu((1,2]) = 1$. By introducing the $\\mu$ function as a way of\nmeasuring the\nintervals above, we have introduced another degree of freedom in\nour\nintegration. This accommodates many weird functions that are not tractable\nusing\nthe usual Riemann theory, but we refer you to a proper introduction to\nLesbesgue\nintegration for further study [[jones2001lebesgue]](#jones2001lebesgue).\nNonetheless,\nthe key step in the above discussion is the introduction of the\n$\\mu$ function,\nwhich we will encounter again as the so-called probability\ndensity function.\n\n## Random Variables\n\nMost introductions to probability jump\nstraight into *random variables* and\nthen explain how to compute complicated\nintegrals. The problem with this\napproach is that it skips over some of the\nimportant subtleties that we will now\nconsider. Unfortunately, the term *random\nvariable* is not very descriptive. A\nbetter term is *measurable function*. To\nunderstand why this is a better term,\nwe have to dive into the formal\nconstructions of probability by way of a simple\nexample.\n\nConsider tossing a\nfair six-sided die. There are only six outcomes possible,\n\n$$\n\\Omega=\\lbrace 1,2,3,4,5,6 \\rbrace\n$$\n\nAs we know, if the die is fair, then the probability of each outcome is $1/6$.\nTo say this formally, the measure of each set (i.e., $\\lbrace 1 \\rbrace,\\lbrace\n2 \\rbrace,\\ldots,\\lbrace 6 \\rbrace$) is $\\mu(\\lbrace 1 \\rbrace ) =\\mu(\\lbrace 2\n\\rbrace ) \\ldots = \\mu(\\lbrace 6 \\rbrace ) = 1/6$. In this case, the $\\mu$\nfunction we discussed earlier is the usual *probability* mass function, denoted\nby\n$\\mathbb{P}$. The measurable function maps a set into a\nnumber on the real\nline. For example, $ \\lbrace 1 \\rbrace \\mapsto 1 $ is\none such function.\n\nNow,\nhere's where things get interesting. Suppose you were asked to construct a\nfair\ncoin from the fair die. In other words, we want to throw the die and then\nrecord\nthe outcomes as if we had just tossed a fair coin. How could we do this?\nOne way\nwould be to define a measurable function that says if the die comes up\n`3` or\nless, then we declare *heads* and otherwise declare *tails*. This has\nsome\nstrong intuition behind it, but let's articulate it in terms of formal\ntheory.\nThis strategy creates two different non-overlapping sets $\\lbrace\n1,2,3 \\rbrace$\nand $\\lbrace 4,5,6 \\rbrace$. Each set has the same probability\n*measure*,\n\n$$\n\\begin{eqnarray*}\n\\mathbb{P}(\\lbrace 1,2,3 \\rbrace) & = & 1/2 \\\\\\\n\\mathbb{P}(\\lbrace 4,5,6 \\rbrace) & = & 1/2\n\\end{eqnarray*}\n$$\n\n And the problem is solved. Everytime the die comes up\n$\\lbrace 1,2,3 \\rbrace$,\nwe record heads and record tails otherwise.\n\nIs this the only way to construct a\nfair coin experiment from a\nfair die? Alternatively, we can define the sets as\n$\\lbrace 1 \\rbrace$,\n$\\lbrace 2 \\rbrace$, $\\lbrace 3,4,5,6 \\rbrace$. If we\ndefine the corresponding\nmeasure for each set as the following\n\n$$\n\\begin{eqnarray*}\n\\mathbb{P}(\\lbrace 1 \\rbrace) & = & 1/2 \\\\\\\n\\mathbb{P}(\\lbrace 2 \\rbrace) & = & 1/2 \\\\\\\n\\mathbb{P}(\\lbrace 3,4,5,6 \\rbrace)\n& = & 0\n\\end{eqnarray*}\n$$\n\n then, we have another solution to the fair coin problem. To\nimplement this,\nall we do is ignore every time the die shows `3,4,5,6` and\nthrow again. This is\nwasteful, but it solves the problem. Nonetheless,\nwe hope you can see how the\ninterlocking pieces of the theory provide a\nframework for carrying the notion of\nuncertainty/potentiality from one problem\nto the next (e.g., from the fair die\nto the fair coin). \n\nLet's consider a slightly more interesting problem where we\ntoss two dice. We\nassume that each throw is *independent*, meaning that the\noutcome of one does\nnot influence the other. What are the sets in this case?\nThey are all pairs\nof possible outcomes from two throws as shown below,\n\n$$\n\\Omega = \\lbrace (1,1),(1,2),\\ldots,(5,6),(6,6) \\rbrace\n$$\n\n What are the measures of each of these sets? By virtue of the\nindependence\nclaim, the measure of each is the product of the respective measures\nof each\nelement. For instance,\n\n$$\n\\mathbb{P}((1,2)) = \\mathbb{P}(\\lbrace 1 \\rbrace) \\mathbb{P}(\\lbrace 2\n\\rbrace) = \\frac{1}{6^2}\n$$\n\n With all that established, we can ask the following\nquestion: what is the\nprobability that the sum of the dice equals\nseven? As before, the first thing to\ndo is characterize the\nmeasurable function for this as $X:(a,b) \\mapsto (a+b)$.\nNext, we\nassociate all of the $(a,b)$ pairs with their sum. We can create a\nPython dictionary for this as shown,\n\n\n```python\nd={(i,j):i+j for i in range(1,7) for j in range(1,7)}\n```\n\nThe next step is to collect all of the $(a,b)$ pairs that sum to\neach of the\npossible values from two to twelve.\n\n\n```python\nfrom collections import defaultdict\ndinv = defaultdict(list)\nfor i,j in d.items():\n dinv[j].append(i)\n```\n\n**Programming Tip.**\n\nThe `defaultdict` object from the built-in collections\nmodule creates dictionaries with\ndefault values when it encounters a new key.\nOtherwise, we would have had to\ncreate default values manually for a regular\ndictionary.\n\n\n\n For example, `dinv[7]` contains the following list of pairs that\nsum to seven,\n\n\n```python\ndinv[7]\n```\n\n\n\n\n [(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)]\n\n\n\nThe next step is to compute the probability measured for each of these items.\nUsing the independence assumption, this means we have to compute the sum of the\nproducts of the individual item probabilities in `dinv`. Because we know that\neach outcome is equally likely, the probability of every term in the sum equals\n$1/36$. Thus, all\nwe have to do is count the number of items in the\ncorresponding list for each\nkey in `dinv` and divide by `36`. For example,\n`dinv[11]` contains `[(5, 6),\n(6, 5)]`. The probability of `5+6=6+5=11` is the\nprobability of this set which\nis composed of the sum of the probabilities of the\nindividual elements\n`{(5,6),(6,5)}`. In this case, we have $\\mathbb{P}(11) =\n\\mathbb{P}(\\lbrace\n(5,6) \\rbrace)+ \\mathbb{P}(\\lbrace (6,5) \\rbrace) = 1/36 +\n1/36 = 2/36$.\nRepeating this procedure for all the elements, we derive the\nprobability mass\nfunction as shown below,\n\n\n```python\nX={i:len(j)/36. for i,j in dinv.items()}\nprint(X)\n```\n\n {2: 0.027777777777777776, 3: 0.05555555555555555, 4: 0.08333333333333333, 5: 0.1111111111111111, 6: 0.1388888888888889, 7: 0.16666666666666666, 8: 0.1388888888888889, 9: 0.1111111111111111, 10: 0.08333333333333333, 11: 0.05555555555555555, 12: 0.027777777777777776}\n\n\n**Programming Tip.**\n\nIn the preceding code note that `36.` is written with\nthe\ntrailing decimal mark. This is a good habit to get into \nbecause the default\ndivision operation changed between Python 2.x and \nPython 3.x. In Python 2.x\ndivision is integer division by default,\nand it is floating-point division in\nPython 3.x.\n\n\n\nThe above example exposes the elements of probability theory that\nare in play for this simple problem while deliberately suppressing some of the\ngory technical details. With this framework, we can ask other questions like\nwhat is the probability that half the product of three dice will exceed the\ntheir sum? We can solve this using the same method as in the following. First,\nlet's create the first mapping,\n\n\n```python\nd={(i,j,k):((i*j*k)/2>i+j+k) for i in range(1,7) \n for j in range(1,7) \n for k in range(1,7)}\n```\n\nThe keys of this dictionary are the triples and the values are the\nlogical\nvalues of whether or not half the product of three dice exceeds their sum.\nNow,\nwe do the inverse mapping to collect the corresponding lists,\n\n\n```python\ndinv = defaultdict(list)\nfor i,j in d.items(): \n dinv[j].append(i)\n```\n\nNote that `dinv` contains only two keys, `True` and `False`. Again,\nbecause the\ndice are independent, the probability of any triple is $1/6^3$.\nFinally, we\ncollect this for each outcome as in the following,\n\n\n```python\nX={i:len(j)/6.0**3 for i,j in dinv.items()}\nprint(X)\n```\n\n {False: 0.37037037037037035, True: 0.6296296296296297}\n\n\nThus, the probability of half the product of three dice exceeding their sum is\n`136/(6.0**3) = 0.63`. The set that is induced by the random variable has only\ntwo elements in it, `True` and `False`, with $\\mathbb{P}(\\mbox{True})=136/216$\nand $\\mathbb{P}(\\mbox{False})=1-136/216$.\n\nAs a final example to exercise\nanother layer of generality, let is consider the\nfirst problem with the two dice\nwhere we want the probability of a\nseven, but this time one of the dice is no\nlonger fair. The distribution for\nthe unfair die is the following:\n\n$$\n\\begin{eqnarray*}\n\\mathbb{P}(\\lbrace 1\\rbrace)=\\mathbb{P}(\\lbrace 2\n\\rbrace)=\\mathbb{P}(\\lbrace 3 \\rbrace) = \\frac{1}{9} \\\\\\\n\\mathbb{P}(\\lbrace\n4\\rbrace)=\\mathbb{P}(\\lbrace 5 \\rbrace)=\\mathbb{P}(\\lbrace 6 \\rbrace) =\n\\frac{2}{9} \n\\end{eqnarray*}\n$$\n\nFrom our earlier work, we know the elements corresponding to the sum of seven\nare the following:\n\n$$\n\\lbrace (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) \\rbrace\n$$\n\n Because we still have the independence assumption, all we need to\nchange is\nthe probability computation of each of elements. For example, given\nthat the\nfirst die is the unfair one, we have\n\n$$\n\\mathbb{P}((1,6)) = \\mathbb{P}(1)\\mathbb{P}(6) = \\frac{1}{9} \\times\n\\frac{1}{6}\n$$\n\n and likewise for $(2,5)$ we have the following:\n\n$$\n\\mathbb{P}((2,5)) = \\mathbb{P}(2)\\mathbb{P}(5) = \\frac{1}{9} \\times\n\\frac{1}{6}\n$$\n\n and so forth. Summing all of these gives the following:\n\n$$\n\\mathbb{P}_X(7) = \\frac{1}{9} \\times \\frac{1}{6}\n+\\frac{1}{9} \\times \\frac{1}{6} \n +\\frac{1}{9} \\times\n\\frac{1}{6} \n +\\frac{2}{9} \\times \\frac{1}{6}\n+\\frac{2}{9} \\times \\frac{1}{6} \n +\\frac{2}{9} \\times\n\\frac{1}{6} = \\frac{1}{6}\n$$\n\n Let's try computing this using Pandas instead\nof Python dictionaries. First, we\nconstruct\na `DataFrame` object with an index of tuples\nconsisting of all pairs\nof possible dice outcomes.\n\n\n```python\nfrom pandas import DataFrame\nd=DataFrame(index=[(i,j) for i in range(1,7) for j in range(1,7)],\n columns=['sm','d1','d2','pd1','pd2','p'])\n```\n\nNow, we can populate the columns that we set up above\nwhere the outcome of the\nfirst die is the `d1` column and\nthe outcome of the second die is `d2`,\n\n\n```python\nd.d1=[i[0] for i in d.index]\nd.d2=[i[1] for i in d.index]\n```\n\nNext, we compute the sum of the dices in the `sm`\ncolumn,\n\n\n```python\nd.sm=list(map(sum,d.index))\n```\n\nWith that established, the DataFrame now looks like\nthe following:\n\n\n```python\nd.head(5) # show first five lines\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        smd1d2pd1pd2p
                                        (1, 1)211NaNNaNNaN
                                        (1, 2)312NaNNaNNaN
                                        (1, 3)413NaNNaNNaN
                                        (1, 4)514NaNNaNNaN
                                        (1, 5)615NaNNaNNaN
                                        \n
                                        \n\n\n\nNext, we fill out the probabilities for each face of the\nunfair die (`d1`) and\nthe fair die (`d2`),\n\n\n```python\nd.loc[d.d1<=3,'pd1']=1/9.\nd.loc[d.d1 > 3,'pd1']=2/9.\nd.pd2=1/6.\nd.head(10)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        smd1d2pd1pd2p
                                        (1, 1)2110.1111110.166667NaN
                                        (1, 2)3120.1111110.166667NaN
                                        (1, 3)4130.1111110.166667NaN
                                        (1, 4)5140.1111110.166667NaN
                                        (1, 5)6150.1111110.166667NaN
                                        (1, 6)7160.1111110.166667NaN
                                        (2, 1)3210.1111110.166667NaN
                                        (2, 2)4220.1111110.166667NaN
                                        (2, 3)5230.1111110.166667NaN
                                        (2, 4)6240.1111110.166667NaN
                                        \n
                                        \n\n\n\nFinally, we can compute the joint probabilities\nfor the sum of the shown faces\nas the following:\n\n\n```python\nd.p = d.pd1 * d.pd2\nd.head(5)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        smd1d2pd1pd2p
                                        (1, 1)2110.1111110.1666670.0185185
                                        (1, 2)3120.1111110.1666670.0185185
                                        (1, 3)4130.1111110.1666670.0185185
                                        (1, 4)5140.1111110.1666670.0185185
                                        (1, 5)6150.1111110.1666670.0185185
                                        \n
                                        \n\n\n\nWith all that established, we can compute the\ndensity of all the dice outcomes\nby using `groupby` as in the\nfollowing,\n\n\n```python\nd.groupby('sm')['p'].sum()\n```\n\n\n\n\n sm\n 2 0.018519\n 3 0.037037\n 4 0.055556\n 5 0.092593\n 6 0.129630\n 7 0.166667\n 8 0.148148\n 9 0.129630\n 10 0.111111\n 11 0.074074\n 12 0.037037\n Name: p, dtype: float64\n\n\n\nThese examples have shown how the theory of probability\nbreaks down sets and\nmeasurements of those sets and how these can be\ncombined to develop the\nprobability mass functions for new random\nvariables. \n\n## Continuous Random\nVariables\n\nThe same ideas work with continuous variables but managing the sets\nbecomes trickier because the real line, unlike discrete sets, has many\nlimiting\nproperties already built into it that have to be handled\ncarefully.\nNonetheless, let's start with an example that should\nillustrate the analogous\nideas. Suppose a random variable $X$ is\nuniformly distributed on the unit\ninterval. What is the probability\nthat the variable takes on values less than\n1/2? \n\nIn order to build intuition onto the discrete case, let's go back to our\ndice-throwing experiment with the fair dice. The sum of the values of the dice\nis a measurable function,\n\n$$\nY \\colon \\lbrace 1,2,\\dots,6 \\rbrace^2 \\mapsto \\lbrace 2,3,\\ldots, 12 \\rbrace\n$$\n\n That is, $Y$ is a mapping of the cartesian product of sets to a\ndiscrete set of\noutcomes. In order to compute probabilities of the set of\noutcomes, we need to\nderive the probability measure for $Y$, $\\mathbb{P}_Y$,\nfrom the corresponding\nprobability measures for each die. Our previous discussion\nwent through the\nmechanics of that. This means that\n\n$$\n\\mathbb{P}_Y \\colon \\lbrace 2,3,\\ldots,12 \\rbrace \\mapsto [0,1]\n$$\n\n Note there is a separation between the function definition and where the\ntarget\nitems of the function are measured in probability. More bluntly,\n\n$$\nY \\colon A \\mapsto B\n$$\n\n with,\n\n$$\n\\mathbb{P}_Y \\colon B \\mapsto [0,1]\n$$\n\n Thus, to compute $\\mathbb{P}_Y$, which is derived \nfrom other random variables,\nwe have to express the equivalence classes\nin $B$ in terms of their progenitor\n$A$ sets. \n\nThe situation for continuous variables follows the same pattern, but\nwith many more deep technicalities that we are going to skip. For the continuous\ncase, the random variable is now,\n\n$$\nX \\colon \\mathbb{R} \\mapsto \\mathbb{R}\n$$\n\n with corresponding probability measure,\n\n$$\n\\mathbb{P}_X \\colon \\mathbb{R} \\mapsto [0,1]\n$$\n\n But where are the corresponding sets here? Technically, these are the\n*Borel*\nsets, but we can just think of them as intervals. Returning to our\nquestion,\nwhat is the probability that a uniformly distributed random variable\non the unit\ninterval takes values less than $1/2$? Rephrasing this question\naccording to the\nframework, we have the following:\n\n$$\nX \\colon [0,1] \\mapsto [0,1]\n$$\n\n with corresponding,\n\n$$\n\\mathbb{P}_X \\colon [0,1] \\mapsto [0,1]\n$$\n\n To answer the question, by the definition of the uniform random\nvariable on\nthe unit interval, we compute the following integral,\n\n$$\n\\mathbb{P}_X([0,1/2]) = \\mathbb{P}_X(0 < X < 1/2) = \\int_0^{1/2} dx = 1/2\n$$\n\n where the above integral's $dx$ sweeps through intervals of the\n$B$-type. The\nmeasure of any $dx$ interval (i.e., $A$-type set) is equal to\n$dx$, by\ndefinition of the uniform random variable. To get all the moving parts\ninto one\nnotationally rich integral, we can also write this as,\n\n$$\n\\mathbb{P}_X(0 < X < 1/2) = \\int_0^{ 1/2 } d\\mathbb{P}_X(dx) = 1/2\n$$\n\nNow, let's consider a slightly more complicated and interesting example. As\nbefore, suppose we have a uniform random variable, $X$ and let us introduce\nanother random variable defined,\n\n$$\nY = 2 X\n$$\n\n Now, what is the probability that $0 < Y < \\frac{1}{2}$? \nTo express this in\nour framework, we write,\n\n$$\nY \\colon [0,1] \\mapsto [0,2]\n$$\n\n with corresponding,\n\n$$\n\\mathbb{P}_Y \\colon [0,2] \\mapsto [0,1]\n$$\n\n To answer the question, we need to measure the set $[0,1/2]$, with\nthe\nprobability measure for $Y$, $\\mathbb{P}_Y([0,1/2])$. How can we do this?\nBecause $Y$ is derived from the $X$ random variable, as with the fair-dice\nthrowing experiment, we have to create a set of equivalences in the target\nspace\n(i.e., $B$-type sets) that reflect back on the input space (i.e.,\n$A$-type\nsets). That is, what is the interval $[0,1/2]$ equivalent to in terms\nof the $X$\nrandom variable? Because, functionally, $Y=2 X$, then the $B$-type\ninterval\n$[0,1/2]$ corresponds to the $A$-type interval $[0,1/4]$. From the\nprobability\nmeasure of $X$, we compute this with the integral,\n\n$$\n\\mathbb{P}_Y([0,1/2]) =\\mathbb{P}_X([0,1/4])= \\int_0^{1/4} dx = 1/4\n$$\n\nNow, let's up the ante and consider the following random variable,\n\n$$\nY = X^2\n$$\n\n where now $X$ is still uniformly distributed, but now over the\ninterval\n$[-1/2,1/2]$. We can express this in our framework as,\n\n$$\nY \\colon [-1/2,1/2] \\mapsto [0,1/4]\n$$\n\n with corresponding,\n\n$$\n\\mathbb{P}_Y \\colon [0,1/4] \\mapsto [0,1]\n$$\n\n What is the $\\mathbb{P}_Y(Y < 1/8)$? In other words, what is the\nmeasure of\nthe set $B_Y= [0,1/8]$? As before, because $X$ is derived from our\nuniformly\ndistributed random variable, we have to reflect the $B_Y$ set onto\nsets of the\n$A$-type. The thing to recognize is that because $X^2$\nis symmetric about zero,\nall $B_Y$ sets reflect back into two sets.\nThis means that for any set $B_Y$, we\nhave the correspondence $B_Y = A_X^+ \\cup\nA_X^{-}$. So, we have,\n\n$$\nB_Y=\\Big\\lbrace 00$. In this case, $Y>X$ because $Z$ cannot be positive\notherwise. For the density function, we are interested in the set \n$\\lbrace 0\n< Z < z \\rbrace $. We want to compute\n\n$$\n\\mathbb{P}(Z X$. For $Z < z $,\nwe have $Y > X(1/z+1)$.\nPutting this together gives\n\n$$\nA_1 = \\lbrace \\max (X,X(1/z+1)) < Y < 1 \\rbrace\n$$\n\n Integrating this over $Y$ as follows,\n\n$$\n\\int_0^1\\lbrace\\max(X,X(1/z+1)) \\frac{X}{1-X}\n$$\n\n and integrating this one more time over $X$ gives\n\n$$\n\\int_0^{\\frac{z}{1+z}} \\frac{-X+z-Xz}{z} dX = \\frac{z}{2(z+1)} \\mbox{ where }\nz > 0\n$$\n\n Note that this is the computation for the *probability*\nitself, not the\nprobability density function. To get that, all we have\nto do is differentiate\nthe last expression to obtain\n\n$$\nf_Z(z) = \\frac{1}{(z+1)^2} \\mbox{ where } z > 0\n$$\n\n Now we need to compute this density using the same process\nfor when $z < -1$.\nWe want the interval $ Z < z $ for when $z < -1$.\nFor a fixed $z$, this is\nequivalent to $ X(1+1/z) < Y$. Because $z$\nis negative, this also means that $Y\n< X$. Under these terms, we\nhave the following integral,\n\n$$\n\\int_0^1 \\lbrace X(1/z+1) 0 \\\\\\\n\\frac{1}{2 z^2} & \\mbox{if } z < -1 \\\\\\\n 0 & \\mbox{otherwise }\n\\end{cases}\n$$\n\n We will leave it as an exercise to show that this\nintegrates out to one.\n\n\n## Independent Random Variables\n\nIndependence is a standard assumption.\nMathematically, the\nnecessary and sufficient condition for independence between\ntwo\nrandom variables $X$ and $Y$ is the following:\n\n$$\n\\mathbb{P}(X,Y) = \\mathbb{P}(X)\\mathbb{P}(Y)\n$$\n\n Two random variables $X$ and $Y$ \nare *uncorrelated* if,\n\n$$\n\\mathbb{E}\\left( (X-\\overline{X})(Y-\\overline{Y}) \\right)=0 \n$$\n\n where $\\overline{X}=\\mathbb{E}(X)$ Note that uncorrelated random\nvariables are\nsometimes called *orthogonal* random variables. Uncorrelatedness\nis a weaker\nproperty than independence, however. For example, consider the\ndiscrete random\nvariables $X$ and $Y$ uniformly distributed over the set\n$\\lbrace 1,2,3 \\rbrace$\nwhere\n\n$$\nX = \n\\begin{cases} \n1 & \\mbox{if } \\omega =1 \\\\\\\n0 & \\mbox{if } \\omega =2\n\\\\\\\n-1 & \\mbox{if } \\omega =3\n\\end{cases}\n$$\n\n and also,\n\n$$\nY = \n\\begin{cases} \n0 & \\mbox{if } \\omega =1 \\\\\\\n1 & \\mbox{if } \\omega =2\n\\\\\\\n0 & \\mbox{if } \\omega =3\n\\end{cases}\n$$\n\n Thus, $\\mathbb{E}(X)=0$ and $\\mathbb{E}(X Y)=0$, so\n$X$ and $Y$ are\nuncorrelated. However, we have\n\n$$\n\\mathbb{P}(X=1,Y=1)=0\\neq \\mathbb{P}(X=1)\\mathbb{P}(Y=1)=\\frac{1}{9}\n$$\n\n So, these two random variables are *not* independent.\nThus, uncorrelatedness\ndoes not imply independence, generally, but\nthere is the important case of\nGaussian random variables for which\nit does. To see this, consider the\nprobability density function\nfor two zero-mean, unit-variance Gaussian random\nvariables $X$ and\n$Y$,\n\n$$\nf_{X,Y}(x,y) = \\frac{e^{\\frac{x^2-2 \\rho x\n y+y^2}{2\n\\left(\\rho^2-1\\right)}}}{2 \\pi \n \\sqrt{1-\\rho^2}}\n$$\n\n where $\\rho:=\\mathbb{E}(X Y)$ is the correlation coefficient. In\nthe\nuncorrelated case where $\\rho=0$, the probability density function factors\ninto\nthe following,\n\n$$\nf_{X,Y}(x,y)=\\frac{e^{-\\frac{1}{2}\\left(x^2+y^2\\right)}}{2\\pi}=\\frac{e^{-\\frac{x^2}{2}}}{\\sqrt{2\\pi}}\\frac{e^{-\\frac{y^2}{2}}}{\\sqrt{2\\pi}}\n=f_X(x)f_Y(y)\n$$\n\n which means that $X$ and $Y$ are independent.\n\nIndependence and conditional\nindependence are closely related, as in the following:\n\n$$\n\\mathbb{P}(X,Y\\vert Z) =\\mathbb{P}(X\\vert Z) \\mathbb{P}(Y\\vert Z)\n$$\n\n which says that $X$ and $Y$ and independent conditioned\non $Z$. Conditioning\nindependent random variables can break\ntheir independence. For example, consider\ntwo independent\nBernoulli-distributed random variables, $X_1, X_2\\in\\lbrace 0,1\n\\rbrace$. We define $Z=X_1+X_2$. Note that $Z\\in \\lbrace\n0,1,2 \\rbrace$. In the\ncase where $Z=1$, we have,\n\n$$\n\\begin{align*}\n\\mathbb{P}(X_1\\vert Z=1) &>0 \\\\\\\n\\mathbb{P}(X_2\\vert Z=1) &>0\n\\end{align*}\n$$\n\n Even though $X_1,X_2$ are independent,\nafter conditioning on $Z$, we have the\nfollowing,\n\n$$\n\\mathbb{P}(X_1=1,X_2=1\\vert Z=1)=0\\neq \\mathbb{P}(X_1=1\\vert\nZ=1)\\mathbb{P}(X_2=1\\vert Z=1)\n$$\n\n Thus, conditioning on $Z$ breaks the independence of\n$X_1,X_2$. This also works\nin the opposite direction ---\nconditioning can make dependent random variables\nindependent.\nDefine $Z_n=\\sum_i^n X_i$ with $X_i$ independent, integer-valued\nrandom variables. The $Z_n$ variables are \ndependent because they stack the\nsame telescoping set of\n$X_i$ variables. Consider the following,\n\n\n
                                        \n\n$$\n\\begin{equation}\n\\mathbb{P}(Z_1=i,Z_3=j\\vert Z_2=k) =\n\\frac{\\mathbb{P}(Z_1=i,Z_2=k,Z_3=j)}{\\mathbb{P}(Z_2 =k)}\n\\label{_auto1} \\tag{1}\n\\end{equation}\n$$\n\n\n
                                        \n\n$$\n\\begin{equation} \\\n=\\frac{\\mathbb{P}(X_1 =i)\\mathbb{P}(X_2 =k-i)\\mathbb{P}(X_3\n=j-k) }{\\mathbb{P}(Z_2 =k)}\n\\end{equation} \n\\label{eq:condIndep} \\tag{2}\n$$\n\n where the factorization comes from the independence of\nthe $X_i$ variables.\nUsing the definition of conditional\nprobability,\n\n$$\n\\mathbb{P}(Z_1=i\\vert Z_2)=\\frac{\\mathbb{P}(Z_1=i,Z_2=k)}{\\mathbb{P}(Z_2=k)}\n$$\n\n We can continue to expand Equation [2](#eq:condIndep),\n\n$$\n\\begin{align*}\n\\mathbb{P}(Z_1=i,Z_3=j\\vert Z_2=k) &=\\mathbb{P}(Z_1 =i\\vert\nZ_2) \\frac{\\mathbb{P}( X_3 =j-k)\\mathbb{P}( Z_2 =k)}{\\mathbb{P}( Z_2 =k)}\\\\\\\n&=\\mathbb{P}(Z_1 =i\\vert Z_2)\\mathbb{P}(Z_3 =j\\vert Z_2)\n\\end{align*}\n$$\n\n where $\\mathbb{P}(X_3=j-k)\\mathbb{P}(Z_2=k)=\n\\mathbb{P}(Z_3=j,Z_2)$. Thus, we\nsee that dependence between\nrandom variables can be broken by conditioning to\ncreate\nconditionally independent random variables. As we have just\nwitnessed,\nunderstanding how conditioning influences independence\nis important and is the\nmain topic of\nstudy in Probabilistic Graphical Models, a field\nwith many\nalgorithms and concepts to extract these\nnotions of conditional independence\nfrom graph-based\nrepresentations of random variables.\n\n\n## Classic Broken Rod\nExample\n\nLet's do one last example to exercise fluency in our methods by\nconsidering the following classic problem: given a rod of unit-length,\nbroken\nindependently and randomly at two places, what is the\nprobability that you can\nassemble the three remaining pieces into a\ntriangle? The first task is to find a\nrepresentation of a triangle as\nan easy-to-apply constraint. What we want is\nsomething like the\nfollowing:\n\n$$\n\\mathbb{P}(\\mbox{ triangle exists }) = \\int_0^1 \\int_0^1 \\lbrace \\mbox{\ntriangle exists } \\rbrace dX dY\n$$\n\n where $X$ and $Y$ are independent and uniformly distributed\nin the unit-\ninterval. Heron's formula for the area of the triangle,\n\n$$\n\\mbox{ area } = \\sqrt{(s-a)(s-b)(s-c)s}\n$$\n\n where $s = (a+b+c)/2$ is what we need. The idea is that this\nyields a valid\narea only when each of the terms under the square root is\ngreater than or equal\nto zero. Thus, suppose that we have\n\n$$\n\\begin{eqnarray*}\na & = & X \\\\\\\nb & = & Y-X \\\\\\\nc & = & 1-Y\n\\end{eqnarray*}\n$$\n\n assuming that $Y>X$. Thus, the criterion for a valid triangle boils down\nto\n\n$$\n\\lbrace (s > a) \\wedge (s > b) \\wedge (s > c) \\wedge (XX$. By symmetry, we get the same result for $X>Y$. Thus, the\nfinal\nresult is the following:\n\n$$\n\\mathbb{P}(\\mbox{ triangle exists }) = \\frac{1}{8}+\\frac{1}{8} = \\frac{1}{4}\n$$\n\nWe can quickly check using this result using Python for the case $Y>X$ using\nthe\nfollowing code:\n\n\n```python\nimport numpy as np\nx,y = np.random.rand(2,1000) # uniform rv\na,b,c = x,(y-x),1-y # 3 sides\ns = (a+b+c)/2\nnp.mean((s>a) & (s>b) & (s>c) & (y>x)) # approx 1/8=0.125\n```\n\n\n\n\n 0.129\n\n\n\n**Programming Tip.**\n\nThe chained logical `&` symbols above tell Numpy that the\nlogical operation\nshould be considered element-wise.\n", "meta": {"hexsha": "f9ba434148e003faba5220b2d22f4eb53228de5f", "size": 232463, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "chapter/probability/intro.ipynb", "max_stars_repo_name": "derakding/Python-for-Probability-Statistics-and-Machine-Learning-2E", "max_stars_repo_head_hexsha": "9d12a298d43ae285d9549a79bb5544cf0a9b7516", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 224, "max_stars_repo_stars_event_min_datetime": "2019-05-07T08:56:01.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-25T15:50:41.000Z", "max_issues_repo_path": "chapter/probability/intro.ipynb", "max_issues_repo_name": "derakding/Python-for-Probability-Statistics-and-Machine-Learning-2E", "max_issues_repo_head_hexsha": "9d12a298d43ae285d9549a79bb5544cf0a9b7516", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 9, "max_issues_repo_issues_event_min_datetime": "2019-08-27T12:57:17.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-21T15:45:13.000Z", "max_forks_repo_path": "chapter/probability/intro.ipynb", "max_forks_repo_name": "derakding/Python-for-Probability-Statistics-and-Machine-Learning-2E", "max_forks_repo_head_hexsha": "9d12a298d43ae285d9549a79bb5544cf0a9b7516", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 73, "max_forks_repo_forks_event_min_datetime": "2019-05-25T07:15:47.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-07T00:22:37.000Z", "avg_line_length": 131.558007923, "max_line_length": 176652, "alphanum_fraction": 0.8531766346, "converted": true, "num_tokens": 11927, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5039061705290805, "lm_q2_score": 0.8152324915965392, "lm_q1q2_score": 0.41080068293129285}} {"text": "# A short & practical introduction to Tensor Flow!\n\nPart 4\n\nThe goal of this notebook is to train a LSTM character prediction model over [Text8](http://mattmahoney.net/dc/textdata) data.\n\nThis is a personal wrap-up of all the material provided by [Google's Deep Learning course on Udacity](https://www.udacity.com/course/deep-learning--ud730), so all credit goes to them. \n\nAuthor: Pablo M. Olmos (olmos@tsc.uc3m.es)\n\nDate: March 2017\n\n\n```python\n# These are all the modules we'll be using later. Make sure you can import them\n# before proceeding further.\nfrom __future__ import print_function\nimport os\nimport numpy as np\nimport random\nimport string\nimport tensorflow as tf\nimport zipfile\nfrom six.moves import range\nfrom six.moves.urllib.request import urlretrieve\n```\n\n\n```python\n# Lets check what version of tensorflow we have installed. The provided scripts should run with tf 1.0 and above\n\nprint(tf.__version__)\n```\n\n\n```python\nurl = 'http://mattmahoney.net/dc/'\n\ndef maybe_download(filename, expected_bytes):\n \"\"\"Download a file if not present, and make sure it's the right size.\"\"\"\n if not os.path.exists(filename):\n filename, _ = urlretrieve(url + filename, filename)\n statinfo = os.stat(filename)\n if statinfo.st_size == expected_bytes:\n print('Found and verified %s' % filename)\n else:\n print(statinfo.st_size)\n raise Exception(\n 'Failed to verify ' + filename + '. Can you get to it with a browser?')\n return filename\n\n\nfilename = maybe_download('XXX/textWordEmbeddings/text8.zip', 31344016) ## Change according to the folder where you saved the dataset provided\n```\n\n\n```python\ndef read_data(filename):\n with zipfile.ZipFile(filename) as f:\n name = f.namelist()[0]\n data = tf.compat.as_str(f.read(name))\n return data\n \ntext = read_data(filename)\nprint('Data size %d' % len(text))\n```\n\n\n```python\ntext[0:20]\n```\n\nCreate a small validation set\n\n\n```python\nvalid_size = 1000\nvalid_text = text[:valid_size]\ntrain_text = text[valid_size:]\ntrain_size = len(train_text)\nprint(train_size, train_text[:64])\nprint(valid_size, valid_text[:64])\n```\n\nUtility functions to map characters to vocabulary IDs and back\n\n\n```python\nvocabulary_size = len(string.ascii_lowercase) + 1 # [a-z] + ' '\nfirst_letter = ord(string.ascii_lowercase[0])\n\ndef char2id(char):\n if char in string.ascii_lowercase:\n return ord(char) - first_letter + 1\n elif char == ' ':\n return 0\n else:\n print('Unexpected character: %s' % char)\n return 0\n```\n\n\n```python\ndef id2char(dictid):\n if dictid > 0:\n return chr(dictid + first_letter - 1)\n else:\n return ' '\n \nprint(char2id('a'), char2id('z'), char2id(' '), char2id('\u00ef'))\nprint(id2char(1), id2char(26), id2char(0))\n```\n\nFunction to generate a training batch for the LSTM model.\n\n\n```python\nbatch_size=64 ## Number of batches, but also number of segments in which we divide the text. We read batch_size \n ## batches in parallel, each read from a different segment. The implementation is not obvious, the\n ## key seems to be the zip function inside the for loop below\n \nnum_unrollings=10 ## Each sequence is num_unrolling character long\n\n### NOW I GET IT!! Every batch is a batch_size times 27 (num letters) matrix. Every row correspond to a letter. Each letter \n### comes from a different sequence of (num_unrollings) so that the 64 letters cannot be read together.\n## In the next batch, we have the following letter for each of the 64 training sequences!!\n\nclass BatchGenerator(object):\n \n def __init__(self, text, batch_size, num_unrollings):\n self._text = text\n self._text_size = len(text)\n self._batch_size = batch_size\n self._num_unrollings = num_unrollings\n segment = self._text_size // batch_size #We split the text into batch_size pieces\n self._cursor = [ offset * segment for offset in range(batch_size)] #Cursor pointing every piece\n self._last_batch = self._next_batch()\n \n #\n def _next_batch(self):\n \"\"\"Generate a single batch from the current cursor position in the data.\"\"\"\n batch = np.zeros(shape=(self._batch_size, vocabulary_size), dtype=np.float)\n for b in range(self._batch_size):\n batch[b, char2id(self._text[self._cursor[b]])] = 1.0 #One hot encoding\n #print(self._text[self._cursor[b]])\n self._cursor[b] = (self._cursor[b] + 1) % self._text_size\n return batch\n \n def next(self):\n \"\"\"Generate the next array of batches from the data. The array consists of\n the last batch of the previous array, followed by num_unrollings new ones.\n \"\"\"\n batches = [self._last_batch]\n for step in range(self._num_unrollings):\n batches.append(self._next_batch())\n self._last_batch = batches[-1]\n return batches\n \n \ndef characters(probabilities):\n \"\"\"Turn a 1-hot encoding or a probability distribution over the possible\n characters back into its (mostl likely) character representation.\"\"\"\n return [id2char(c) for c in np.argmax(probabilities, 1)]\n\ndef batches2string(batches):\n \"\"\"Convert a sequence of batches back into their (most likely) string\n representation.\"\"\"\n s = [''] * batches[0].shape[0]\n for b in batches:\n s = [''.join(x) for x in zip(s, characters(b))] #Clever! The ZIP is the key function here!\n return s \n\ntrain_batches = BatchGenerator(train_text, batch_size, 10)\nvalid_batches = BatchGenerator(valid_text, 1, 1)\n\n\n```\n\n\n```python\nprint(batches2string(train_batches.next()))\nprint(batches2string(train_batches.next()))\n```\n\n\n```python\n#OK with this one\ndef logprob(predictions, labels):\n \"\"\"Log-probability of the true labels in a predicted batch.\"\"\"\n predictions[predictions < 1e-10] = 1e-10\n return np.sum(np.multiply(labels, -np.log(predictions))) / labels.shape[0]\n\n#OK with this one\ndef sample_distribution(distribution):\n \"\"\"Sample one element from a distribution assumed to be an array of normalized\n probabilities.\n \"\"\"\n \n r = random.uniform(0,1)\n s = 0\n for i in range(len(distribution)):\n s += distribution[i]\n if s >= r:\n return i\n return len(distribution) - 1\n\n#OK with this one\ndef sample(prediction):\n \"\"\"Turn a (column) prediction into 1-hot encoded samples.\"\"\"\n p = np.zeros(shape=[1, vocabulary_size], dtype=np.float)\n p[0, sample_distribution(prediction[0])] = 1.0\n return p\n\n\ndef random_distribution():\n \"\"\"Generate a random column of probabilities.\"\"\"\n b = np.random.uniform(0.0, 1.0, size=[1, vocabulary_size])\n return b / np.sum(b, 1)[:, None]\n\n```\n\n\n```python\ntrain_batches.next()[0].shape\n```\n\n# Simple LSTM Model\n\nRecall the fundamental model\n\n\n\n\nAlso, the un-regularized cost function is\n\n\\begin{align}\nJ(\\boldsymbol{\\theta})=\\frac{1}{N}\\sum_{n=1}^N\\sum_{t=1}^{T_n}d(\\boldsymbol{y}_t^{(n)},\\sigma(\\boldsymbol{h}_t^{(n)}))\n\\end{align}\nwhere $d(\\cdot,\\cdot)$ is the cross-entropy loss function.\n\nAbout the TF implementation below, see the following excellent [post](http://www.thushv.com/sequential_modelling/long-short-term-memory-lstm-networks-implementing-with-tensorflow-part-2/)\n\n> \nNow calculating logits for softmax is a little bit tricky. This a temporal (time-based) network. So after each processing each num_unrolling batches through the LSTM cell, we update h_{t-1}=h_t and c_{t-1}=c_t before calculating logits and the loss. This is done by using tf.control_dependencies. What this does is that, logits will not be calculated until saved_output and saved_states are updated. Finally, as you can see, num_unrolling acts as the amount of history we are remembering.\n\nIn other words, in the computation graph everytime something is updated, all the dependent op nodes are updated and this is propagated through the graph. If we want to wait until the very end to compute the loss, we wait using the command tf.control_dependencies.\n\nAbout the zip() and zip(*) operators, see this [post](https://docs.python.org/2/library/functions.html#zip)\n\n\n```python\nnum_nodes = 64\n\ngraph = tf.Graph()\nwith graph.as_default():\n \n # Parameters:\n \n #i(t) parameters\n # Input gate: input, previous output, and bias.\n ix = tf.Variable(tf.truncated_normal([vocabulary_size, num_nodes], -0.1, 0.1)) ##W^ix\n im = tf.Variable(tf.truncated_normal([num_nodes, num_nodes], -0.1, 0.1)) ## W^ih\n ib = tf.Variable(tf.zeros([1, num_nodes])) ##b_i\n \n #f(t) parameters\n # Forget gate: input, previous output, and bias.\n fx = tf.Variable(tf.truncated_normal([vocabulary_size, num_nodes], -0.1, 0.1)) ##W^fx\n fm = tf.Variable(tf.truncated_normal([num_nodes, num_nodes], -0.1, 0.1)) ##W^fh\n fb = tf.Variable(tf.zeros([1, num_nodes])) ##b_f\n \n #g(t) parameters\n # Memory cell: input, state and bias. \n cx = tf.Variable(tf.truncated_normal([vocabulary_size, num_nodes], -0.1, 0.1)) ##W^gx\n cm = tf.Variable(tf.truncated_normal([num_nodes, num_nodes], -0.1, 0.1)) ##W^gh\n cb = tf.Variable(tf.zeros([1, num_nodes])) ##b_g\n \n #o(t) parameters\n # Output gate: input, previous output, and bias.\n ox = tf.Variable(tf.truncated_normal([vocabulary_size, num_nodes], -0.1, 0.1)) ##W^ox\n om = tf.Variable(tf.truncated_normal([num_nodes, num_nodes], -0.1, 0.1)) ##W^oh\n ob = tf.Variable(tf.zeros([1, num_nodes])) ##b_o\n \n # Variables saving state across unrollings.\n saved_output = tf.Variable(tf.zeros([batch_size, num_nodes]), trainable=False) #h(t)\n saved_state = tf.Variable(tf.zeros([batch_size, num_nodes]), trainable=False) #s(t)\n \n \n # Classifier weights and biases (over h(t) to labels)\n w = tf.Variable(tf.truncated_normal([num_nodes, vocabulary_size], -0.1, 0.1))\n b = tf.Variable(tf.zeros([vocabulary_size]))\n \n # Definition of the cell computation.\n def lstm_cell(i, o, state):\n \"\"\"Create a LSTM cell. See e.g.: http://arxiv.org/pdf/1402.1128v1.pdf\n Note that in this formulation, we omit the various connections between the\n previous state and the gates.\"\"\"\n input_gate = tf.sigmoid(tf.matmul(i, ix) + tf.matmul(o, im) + ib)\n forget_gate = tf.sigmoid(tf.matmul(i, fx) + tf.matmul(o, fm) + fb)\n update = tf.matmul(i, cx) + tf.matmul(o, cm) + cb \n state = forget_gate * state + input_gate * tf.tanh(update) #tf.tanh(update) is g(t)\n output_gate = tf.sigmoid(tf.matmul(i, ox) + tf.matmul(o, om) + ob)\n return output_gate * tf.tanh(state), state #h(t) is output_gate * tf.tanh(state)\n\n # Input data. Now it makes sense!!!\n \n train_data = list()\n for _ in range(num_unrollings + 1):\n train_data.append(tf.placeholder(tf.float32, shape=[batch_size,vocabulary_size]))\n train_inputs = train_data[:num_unrollings]\n train_labels = train_data[1:] # labels are inputs shifted by one time step.\n\n # Unrolled LSTM loop.\n \n outputs = list()\n output = saved_output\n aux = output\n state = saved_state\n for i in train_inputs:\n output, state = lstm_cell(i, output, state)\n outputs.append(output)\n\n # State saving across unrollings.\n with tf.control_dependencies([saved_output.assign(output),saved_state.assign(state)]):\n #Classifier.\n logits = tf.nn.xw_plus_b(tf.concat(axis=0,values=outputs), w, b)\n loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(labels=tf.concat(axis=0, values=train_labels),logits=logits))\n\n # Optimizer.\n \n \"\"\"Next, we are implementing the optimizer. Remember! we should use \u201cgradient clipping\u201d (tf.clip_by_global_norm) \n to avoid \u201cExploding gradient\u201d phenomenon. Also, we decay the learning_rate over time.\"\"\"\n global_step = tf.Variable(0)\n \n learning_rate = tf.train.exponential_decay(10.0, global_step, 5000, 0.1, staircase=True)\n optimizer = tf.train.GradientDescentOptimizer(learning_rate)\n \n \"\"\" optimizer.compute_gradients(loss) yields (gradient, value) tuples. gradients, v = zip(*optimizer.compute_gradients(loss))\n performs a transposition, creating a list of gradients and a list of values.\n gradients, _ = tf.clip_by_global_norm(gradients, 1.25)\n then clips the gradients, and optimizer = optimizer.apply_gradients(zip(gradients, v), global_step=global_step)\n re-zips the gradient and value lists back into an iterable of (gradient, value) \n tuples which is then passed to the optimizer.apply_gradients method.\"\"\"\n \n gradients, v = zip(*optimizer.compute_gradients(loss))\n gradients, _ = tf.clip_by_global_norm(gradients, 1.25)\n optimizer = optimizer.apply_gradients(zip(gradients, v), global_step=global_step)\n\n # Predictions.\n train_prediction = tf.nn.softmax(logits)\n \n # Sampling and validation eval: batch 1, no unrolling.\n sample_input = tf.placeholder(tf.float32, shape=[1, vocabulary_size])\n saved_sample_output = tf.Variable(tf.zeros([1, num_nodes]))\n saved_sample_state = tf.Variable(tf.zeros([1, num_nodes]))\n # Create an op that groups multiple operations.\n reset_sample_state = tf.group(saved_sample_output.assign(tf.zeros([1, num_nodes])),\n saved_sample_state.assign(tf.zeros([1, num_nodes])))\n \n sample_output, sample_state = lstm_cell(sample_input, saved_sample_output, saved_sample_state)\n \n with tf.control_dependencies([saved_sample_output.assign(sample_output),saved_sample_state.assign(sample_state)]):\n sample_prediction = tf.nn.softmax(tf.nn.xw_plus_b(sample_output, w, b))\n\n```\n\n\n```python\nnum_steps = 1001\nsummary_frequency = 100\n\nwith tf.Session(graph=graph) as session:\n tf.global_variables_initializer().run()\n print('Initialized')\n mean_loss = 0\n for step in range(num_steps):\n batches = train_batches.next()\n feed_dict = dict()\n for i in range(num_unrollings + 1):\n feed_dict[train_data[i]] = batches[i]\n _, l, predictions, lr = session.run(\n [optimizer, loss, train_prediction, learning_rate], feed_dict=feed_dict)\n mean_loss += l\n if step % summary_frequency == 0:\n if step > 0:\n mean_loss /= summary_frequency\n # The mean loss is an estimate of the loss over the last few batches.\n print(\n 'Average loss at step %d: %f learning rate: %f' % (step, mean_loss, lr))\n mean_loss = 0\n labels = np.concatenate(list(batches)[1:])\n print('Minibatch perplexity: %.2f' % float(\n np.exp(logprob(predictions, labels))))\n if step % (summary_frequency * 10) == 0:\n # Generate some samples.\n print('=' * 80)\n for _ in range(5):\n feed = sample(random_distribution())\n sentence = characters(feed)[0]\n reset_sample_state.run()\n for _ in range(79):\n prediction = sample_prediction.eval({sample_input: feed})\n feed = sample(prediction)\n sentence += characters(feed)[0]\n print(sentence)\n print('=' * 80)\n # Measure validation set perplexity.\n reset_sample_state.run()\n valid_logprob = 0\n for _ in range(valid_size):\n b = valid_batches.next()\n predictions = sample_prediction.eval({sample_input: b[0]})\n valid_logprob = valid_logprob + logprob(predictions, b[1])\n print('Validation set perplexity: %.2f' % float(np.exp(\n valid_logprob / valid_size)))\n```\n\n\n```python\nbatches = train_batches.next()\n```\n\n\n```python\nbatches[0]\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "da5d555c95e11342314081565c1c6900762a9171", "size": 22156, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Notebooks/Part_4/.ipynb_checkpoints/LSTMs-checkpoint.ipynb", "max_stars_repo_name": "olmosUC3M/Introduction-to-Tensor-Flow-and-Deep-Learning", "max_stars_repo_head_hexsha": "3d173606f273f6b3e2bf3cbdccea1c4fe59af71f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2018-03-05T14:19:15.000Z", "max_stars_repo_stars_event_max_datetime": "2020-09-13T23:53:08.000Z", "max_issues_repo_path": "Notebooks/Part_4/.ipynb_checkpoints/LSTMs-checkpoint.ipynb", "max_issues_repo_name": "olmosUC3M/Introduction-to-Tensor-Flow-and-Deep-Learning", "max_issues_repo_head_hexsha": "3d173606f273f6b3e2bf3cbdccea1c4fe59af71f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Notebooks/Part_4/.ipynb_checkpoints/LSTMs-checkpoint.ipynb", "max_forks_repo_name": "olmosUC3M/Introduction-to-Tensor-Flow-and-Deep-Learning", "max_forks_repo_head_hexsha": "3d173606f273f6b3e2bf3cbdccea1c4fe59af71f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-03-31T20:26:47.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-31T20:26:47.000Z", "avg_line_length": 37.8735042735, "max_line_length": 497, "alphanum_fraction": 0.5566437985, "converted": true, "num_tokens": 3880, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.6757646010190476, "lm_q1q2_score": 0.4106372595160937}} {"text": "# Tutorial: Unit Conversions\nMicromagnetic simulations provide a method of simulating the magnetic structure in real materials. However, in order to do this we need to be able to convert the information we have about the system into the parameters that micromagnetics use.\n\n# CGS to SI\n`Ubermag` uses SI units for all of the system parameters. Experimentally, material parameters are can often be measured in CGS units SI units, or a mixture of the two. Firstly, we show a convinent way of converting from CGS to SI units and other useful quantities.\n\nTo do this, we make use of `astropy.units`, a package aimed at the astrophysics community for its unit converting functionality. `astropy` is not installed by default. We can install it (e.g. with `pip`) directly from the notebook. Generally, you will not install `astropy` from inside the notebook (using either `pip` or `conda`) but for the sake of demonstration we run the command in here.\n\n\n```python\n# NBVAL_IGNORE_OUTPUT\n!pip install astropy\n```\n\n Requirement already satisfied: astropy in /opt/miniconda3/envs/ubermagdev/lib/python3.8/site-packages (4.3.1)\n Requirement already satisfied: pyerfa>=1.7.3 in /opt/miniconda3/envs/ubermagdev/lib/python3.8/site-packages (from astropy) (2.0.0)\n Requirement already satisfied: numpy>=1.17 in /opt/miniconda3/envs/ubermagdev/lib/python3.8/site-packages (from astropy) (1.21.2)\n\n\nNow we can import `astropy`.\n\n\n```python\nfrom astropy import units as u\nimport numpy as np\n```\n\nTo initalise a value with units, one has to simply multiply the value by the units provided.\n\n\n```python\nfield = 100 * u.Gauss\nfield\n```\n\n\n\n\n$100 \\; \\mathrm{G}$\n\n\n\nThese units remain with the variable throughout operations. \n\n\n```python\nfield2 = field * 10\nfield2\n```\n\n\n\n\n$1000 \\; \\mathrm{G}$\n\n\n\nWe can easily convert to SI units by using\n\n\n```python\nfield2.si\n```\n\n\n\n\n$0.1 \\; \\mathrm{T}$\n\n\n\nMagnetic moment can be expressed using the following CGS units\n\n\n```python\nmagnetic_moment = 1 * u.erg/u.Gauss\nmagnetic_moment\n```\n\n\n\n\n$1 \\; \\mathrm{\\frac{erg}{G}}$\n\n\n\nand simplified using\n\n\n```python\nmagnetic_moment.si\n```\n\n\n\n\n$0.001 \\; \\mathrm{m^{2}\\,A}$\n\n\n\nIf we wish to detach the numervical value for its units, we can do this using `.value`\n\n\n```python\nmagnetic_moment.si.value\n```\n\n\n\n\n 0.001\n\n\n\n## Equivalencies\n\nThe `astropy.units` also enables conversions of units with different equivalencies i.e. temperature and energy. To use this we can create a variable with the relevant units of temperature and use the `to` function to convert to the relevant units with the relevant equivalency.\n\nFor example, if an exchange interaction has a temperature of 4.15 K we can calculate the equivalent energy in J.\n\n\n```python\nt_k = 4.15 * u.K\nt_k.to(u.J, equivalencies=u.temperature_energy()) \n```\n\n\n\n\n$5.7296934 \\times 10^{-23} \\; \\mathrm{J}$\n\n\n\nWhile `astropy` handles a variety of units and conversions, it does not currently have an equivalency for magnetic induction and magnetic field strength i.e. B to H . As this is a very useful for the magnetism community, we have provided the conversion here.\n\n\n```python\nfrom astropy import constants as const\ninduction_field = [(u.T, u.A/u.m,\n lambda x: x / const.mu0,\n lambda x: const.mu0 / x)]\n```\n\n\n```python\nfield = 100 * u.Gauss\nfield.to(u.A/u.m, equivalencies=induction_field)\n```\n\n\n\n\n$7957.7472 \\; \\mathrm{\\frac{A}{m}}$\n\n\n\n# Parameters\nHere we will describe some select methods for relating atomistic parameters, micromagnetic parameters, and experimental results.\n\nNOTE: Different definitions of the exchange Hamiltonian will lead to different conversion factors. Here, we use the atomistic exchange Hamiltonian\n\\begin{equation}\n{\\cal H}_{ex} = -\\frac{1}{2}\\sum_{i\\neq j} J_{ij} {\\bf S}_i \\cdot {\\bf S}_j,\n\\end{equation}\nwhere $\\lvert{\\bf S}_i \\rvert = \\lvert {\\bf S}_j \\rvert = 1$ are the normaised spin vectors, and $J_{ij}$ is the exchange between the.\n\n## Exchange\n### Atomistic\nThe atomistic exchange $J$ can be obtained from the Curie temperature $T_\\text{C}$ of a material by using\n\\begin{equation}\nJ = \\frac{3k_\\text{B}T_\\text{C}}{\\epsilon z},\n\\end{equation}\nwhere $k_\\text{B}$ is the Boltzmann constant, $z$ is the number of nearest neighbours, and $\\epsilon$ is\na structural depended correction factor. The values of this correction factor has been calculated in Table I of [Garanin 1996](https://doi.org/10.1103/PhysRevB.53.11593).\n\n\n```python\nfrom scipy import constants\ndef Tc_to_J(Tc, e, z):\n return Tc*3*constants.k/(e*z)\n```\n\nFor example, a system with a $T_c$ of 100 K and a fcc cubic structure.\n\n\n```python\nTc = 100\ne = 0.808\nz = 12\nJ = Tc_to_J(Tc, e, z)\nJ\n```\n\n\n\n\n 4.271810024752475e-22\n\n\n\n### Micromagnetic\nThe micromagnetic exchange correlation constant $A$ can be related to atomistic exchange using\n\\begin{equation}\nA = \\frac{zJl^2}{12V},\n\\end{equation}\nwhere $J$ is the Heisenberg exchange, $z$ is the number of nearest neighbour atoms, and $l$ is the distance between neighbor atoms and $V$ is the crystal volume per magnetic atom.\n\n\n```python\ndef J_to_A(J, z, l, V):\n return J*l*l*z/(12*V)\n```\n\n\n```python\nl = 6.84e-10\nV = 2.24e-28\nA = J_to_A(J, z, l, V)\nA\n```\n\n\n\n\n 8.922285495270508e-13\n\n\n\nThe micromagnetic exchange correlation constant can be obtained directly from $T_c$ using \n\\begin{equation}\nA = \\frac{k_\\text{B}T_\\text{C}l^2}{4\\epsilon V}.\n\\end{equation}\n\n\n```python\ndef Tc_to_A(Tc, e, l, V):\n return Tc * constants.k * l * l / (4 * e* V)\n```\n\n\n```python\nA = Tc_to_A(Tc, e, l, V)\nA\n```\n\n\n\n\n 8.922285495270509e-13\n\n\n\n## DMI\n### Atomistic\n\nThe Dzyaloshinskii\u2013Moriya interaction is given by\n\\begin{equation}\n{\\cal H}_{ex} = -\\frac{1}{2}\\sum_{i\\neq j} {\\bf d}_{ij} \\cdot \\left( {\\bf S}_i \\times {\\bf S}_j \\right),\n\\end{equation}\nwhere ${\\bf d}_{ij}$ is the atomistic DMI vector. \n\n### Micromagentic\nThe micromagnetic DMI constant $D$ can be related to atomistic DMI using\n\\begin{equation}\nD = \\frac{zdl}{12V},\n\\end{equation}\nwhere $d$ is the atomistic DMI, $z$ is the number of nearest neighbour atoms, and $l$ is the distance between neighbor atoms and $V$ is the crystal volume per magnetic atom.\n\n\n```python\ndef d_to_D(d, z, l, V):\n return d*l*z/(12*V)\n```\n\n\n```python\nd = 2.01e-23\nD = d_to_D(d, z, l, V)\nD\n```\n\n\n\n\n 6.137678571428571e-05\n\n\n\nThe DMI constants are not easy to measure experimentally, however they can be be calculated from the helical period P using\n\\begin{equation}\nP = \\frac{4\\pi A}{|D|},\n\\end{equation}\nwhere $A$ and $D$ are the micromagnetic exchange and DMI respectively.\n\n\n```python\ndef P_to_D(P, A):\n return 4*np.pi*A/P\n\ndef D_to_P(D, A):\n return 4*np.pi*A/abs(D)\n```\n\nFor a system with a micromagentic exchange of $6\\times 10^{-14}$ Jm$^{-1}$ and a helical period of 20 nm\n\n\n```python\nA = 6e-14\nD = P_to_D(20e-9, A)\nD\n```\n\n\n\n\n 3.7699111843077517e-05\n\n\n\nFor atomistic simulations, this can be converted into\n\\begin{equation}\nP = \\frac{4\\pi J l}{|d|},\n\\end{equation}\nwhere $J$ is the Heisenberg exchange, $d$ is the atomistic DMI, and $l$ is the distance between neighbor atoms.\n\n## Saturation Magnetisation\n### Micromagnetics\nThe saturation magnetisation is often measured in $\\mu_\\text{B}/f.u.$ but is needed in A/m in micromagnetics. A simple converstion can be used\n\\begin{equation}\nM_s [ \\text{A}/ \\text{m}]= \\frac{\\mu_\\text{B} M_s[\\mu_\\text{B}/f.u.]}{V},\n\\end{equation}\nwhere $M_s[\\mu_\\text{B}/f.u.]$ is the saturation magnetisation in $\\mu_B$ per formula unit, $\\mu_B$ is the Bohr magneton in J/m, and $V$ is the volume of the formula unit in m$^3$.\n\n\n```python\ndef Ms_muB_to_Am(Ms, V):\n return constants.value('Bohr magneton') * Ms/V\n```\n\n\n```python\nMs = Ms_muB_to_Am(0.8, 2.24375e-28)\nprint(Ms)\n```\n\n 33066.10835716991\n\n\n### Atomistic\nIn atomistic simulations the saturation magnetisation $M_s$ in micromagnetic simulations can be related to the magnetic moment $\\mu$ simply by\n\\begin{equation}\n\\mu = M_s V,\n\\end{equation}\nwhere $V$ is the crystal volume per magnetic atom.\n\n\n```python\ndef Ms_to_mu(Ms, V):\n return Ms*V\n```\n\n\n```python\nMs = 6e5\nmu = Ms_to_mu(Ms, V)\nmu\n```\n\n\n\n\n 1.3440000000000002e-22\n\n\n\n## Anisotropy\n### Micromagnetic\nAnisotropy can can be measured experimentally in a variety of different ways. The results torque magnetometry, for example, can give correct value for the anisotropy in units of Jm$^{-3}$.\n\n### Atomistic\nSimilarly to the saturation magnetisation the conversion between micromagnetic $K$ and atomistic anisotropy $k$ is simply volume weighted\n\\begin{equation}\nk = K V,\n\\end{equation}\nwhere $V$ is the crystal volume per magnetic atom.\nThis atomistic anisotropy $k$ can also be calculated from the difference in energy of $J$ in different directions. i.e. $J_{\\perp} = 6\\times 10^{-23}$ J and $J_{\\parallel} = 5 \\times 10^{-23}$ J gives an atomistic anisotropy $k=1\\times 10^{-23}$ J.\n\n\n```python\ndef K_to_k(K, V):\n return Ms*V\n```\n\n\n```python\nK = 1.2e6\nk = K_to_k(K, V)\nk\n```\n\n\n\n\n 1.3440000000000002e-22\n\n\n\n# Worked Example\n\nHere FeGe will be used as example for how to obtain micromagnetic parameters. FeGe has a cubic crystal structure with four Ge and four Fe atoms per unit cell with a lattice constant of $a=\t\n4.6995$ \u00c5 and the distance between Fe atoms is 2.881 \u00c5 \\[[Wilhelm 2007](http://doi.org/10.1016/j.stam.2007.04.004)\\]. The saturation magnetisation is $1.07 \\mu_\\text{B}/f.u.$ \\[[Yamada 2003](https://doi.org/10.1016%2FS0921-4526%2802%2902471-7)\\] and magnetic ordering temperature is 278 K \\[[Lebech 1989](https://iopscience.iop.org/article/10.1088/0953-8984/1/35/010/meta)\\]. The helical period of FeGe is $\\sim 70$ nm \\[[Yu 2011](https://doi.org/10.1038/nmat2916)\\].\n\n\n```python\na = 4.6995e-10\nl = 2.881e-10\nMs_orig = 1.07\nTc = 278\nP = 70e-9\n```\n\nVolume per magnetic atom\n\n\n```python\nV = (a**3)/4\nV\n```\n\n\n\n\n 2.594746713121875e-29\n\n\n\nSaturation magnetisation\n\n\n```python\nMs = Ms_muB_to_Am(Ms_orig, V)\nMs\n```\n\n\n\n\n 382433.88973569183\n\n\n\nExchange\n\n\n```python\ne = 0.644\nA = Tc_to_A(Tc, e, l, V)\nA\n```\n\n\n\n\n 4.76621650209023e-12\n\n\n\nDMI\n\n\n```python\nD = P_to_D(P, A)\nD\n```\n\n\n\n\n 0.000855629185622006\n\n\n", "meta": {"hexsha": "5552bb9748a5e7964b69b07004acea1c5faf4497", "size": 22664, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/Unit_converter.ipynb", "max_stars_repo_name": "ubermag/exsim", "max_stars_repo_head_hexsha": "35e7a88716a9ed2c9a34f4c93c628560a597b57f", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/Unit_converter.ipynb", "max_issues_repo_name": "ubermag/exsim", "max_issues_repo_head_hexsha": "35e7a88716a9ed2c9a34f4c93c628560a597b57f", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2021-06-10T13:42:08.000Z", "max_issues_repo_issues_event_max_datetime": "2021-07-21T08:57:50.000Z", "max_forks_repo_path": "docs/Unit_converter.ipynb", "max_forks_repo_name": "ubermag/exsim", "max_forks_repo_head_hexsha": "35e7a88716a9ed2c9a34f4c93c628560a597b57f", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 23.4374353671, "max_line_length": 484, "alphanum_fraction": 0.5279738793, "converted": true, "num_tokens": 3185, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.6757645944891559, "lm_q1q2_score": 0.4106372555481191}} {"text": "```python\n# \uadf8\ub798\ud504, \uc218\ud559 \uae30\ub2a5 \ucd94\uac00\n# Add graph and math features\nimport pylab as py\nimport numpy as np\nimport numpy.linalg as nl\n# \uae30\ud638 \uc5f0\uc0b0 \uae30\ub2a5 \ucd94\uac00\n# Add symbolic operation capability\nimport sympy as sy\n\n\n```\n\n\n```python\nsy.init_printing()\n\n\n```\n\n# \ub8fd\uac8c-\ucfe0\ud0c0\ubc95 (RK4) : \uace0\ucc28 \uc0c1\ubbf8\ubc29
                                        Runge-Kutta Method (RK4) : Higher Order ODE\n\n\n\n## \ub2e8\uc9c4\uc790
                                        Simple Pendulum\n\n\n\n\ub2e4\uc74c \ubbf8\ubd84 \ubc29\uc815\uc2dd\uc740 \ub2e8\uc9c4\uc790\uc758 \uc6b4\ub3d9\uc744 \ubb18\uc0ac\ud55c\ub2e4.
                                        \nFollowing differential equation describes the motion of a simple pendulum.
                                        \nRef : Wikipedia contributors, 'Pendulum (mathematics)', Wikipedia, The Free Encyclopedia, 2 June 2018, 13:28 UTC, [accessed 5 August 2018]\n\n\n\n$$\n\\frac{d^2\\theta}{dt^2} + \\frac{g}{l}sin\\theta = 0\n$$\n\n\n\n\uc0c1\ud0dc\ubcc0\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4\uace0 \uac00\uc815\ud558\uc790.
                                        \nLet's assume that the state variables are as follows.\n\n\n\n$$\n\\mathbf{x}\n=\n\\begin{pmatrix}\nx_0\\\\\nx_1\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n\\theta\\\\\n\\frac{d}{dt}\\theta\n\\end{pmatrix}\n$$\n\n\n\n\uc0c1\ud0dc\ubcc0\uc218\uc758 \ubbf8\ubd84\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.
                                        Differentiation of the state variables are as follows.\n\n\n\n$$\n\\frac{d}{dt}\n\\begin{pmatrix}\n x_0\\\\\n x_1\n\\end{pmatrix} \n=\n\\begin{pmatrix}\n x_1\\\\\n -\\frac{g}{l}sinx_0\n\\end{pmatrix} \n$$\n\n\n\npython \ud568\uc218\ub85c \uad6c\ud604\ud574 \ubcf4\uba74 \ub2e4\uc74c\uacfc \uac19\uc744 \uac83\uc774\ub2e4.
                                        \nOne possible python implementation would be as follows.\n\n\n\n\n```python\ng_mpsps = 9.8\nl_m = 0.3\n\nlegends = ('$\\\\theta(deg)$', '$\\\\frac{d}{dt}\\\\theta(deg/s)$')\nylabel = ''\n\n# Initial state\nx_0 = np.array([np.deg2rad(90), 0])\n\ndef pendulum_NL(x, t):\n \"\"\"\n Parameters\n ==========\n x: array of theta and d(theta)/dt\n t: time value\n \n Return Value\n ============\n One dimensional array of dx/dt\n \"\"\"\n \n return np.array([x[1], (-g_mpsps/l_m)*np.sin(x[0])])\n\n\n```\n\n\ub8fd\uac8c-\ucfe0\ud0c0\ubc95\uc744 \uc624\uc77c\ub7ec\ubc95, \ud6c8\ubc95\uacfc \ube44\uad50\ud574\ubcf4\uc790.
                                        Let's compare the Runge-Kutta method with Euler method, and Heun's method.\n\n\n\n\n```python\ndef rk4_step(f, x0, t0, t1):\n \"\"\"\n One time step of Runge-Kutta method\n\n f: dx_dt function\n x0 : initial condition\n t0 : this step time\n t1 : next step time\n \"\"\"\n delta_t = (t1 - t0)\n delta_t_half = delta_t * 0.5\n t_half = t0 + delta_t_half\n \n # Step 1\n s1 = f(x0, t0)\n\n # Step 2\n s2 = f(x0 + s1 * delta_t_half, t_half)\n\n # Step 3\n s3 = f(x0 + s2 * delta_t_half, t_half)\n\n # Step 4\n s4 = f(x0 + s3 * delta_t, t1)\n\n # Step 5\n s = (1.0 / 6.0) * (s1 + (s2 + s3) * 2 + s4)\n\n # Step 6\n x1 = x0 + s * delta_t\n\n return x1\n\n\n```\n\n\n```python\ndef rk4(f, t_array, x_0):\n time_list = [t_array[0]]\n result_list = [x_0]\n\n x_i = x_0\n\n for k, t_i in enumerate(t_array[:-1]):\n # time step\n x_i_plus_1 = rk4_step(f, x_i, t_i, t_array[k+1])\n\n time_list.append(t_array[k+1])\n result_list.append(x_i_plus_1)\n \n x_i = x_i_plus_1\n\n return time_list, result_list\n\n\n```\n\n\n```python\ndef euler(f, t_array, x_0):\n time_list = [t_array[0]]\n result_list = [x_0]\n\n x_i = x_0\n\n for k, t_i in enumerate(t_array[:-1]):\n # time step\n delta_t = t_array[k+1] - t_array[k]\n\n # slope\n s_i = f(x_i, t_i)\n\n # x[i + 1]\n x_i_plus_1 = x_i + s_i * delta_t\n\n time_list.append(t_array[k+1])\n result_list.append(x_i_plus_1)\n \n x_i = x_i_plus_1\n\n return time_list, result_list\n\n\n```\n\n\n```python\ndef heun(f, t_array, x_0):\n time_list = [t_array[0]]\n result_list = [x_0]\n\n x_i = x_0\n\n for k, t_i in enumerate(t_array[:-1]):\n # time step\n delta_t = t_array[k+1] - t_array[k]\n\n # slope at i\n s_i = f(x_i, t_i)\n\n # x[i + 1] by Forward Euler\n x_i_plus_1 = x_i + s_i * delta_t\n \n # slope at i + 1\n s_i_plus_1 = f(x_i_plus_1, t_array[k+1])\n \n # average of slope\n s_average = (s_i + s_i_plus_1) * 0.5\n \n # x[i + 1] by Modified Euler\n x_i_plus_1_m = x_i + s_average * delta_t\n\n time_list.append(t_array[k+1])\n result_list.append(x_i_plus_1_m)\n \n x_i = x_i_plus_1_m\n\n return time_list, result_list\n\n\n```\n\n\n```python\n\npy.figure(figsize=(12, 12))\n\n\n# Time array\ndelta_t = 0.001\n\nt_sec_array = np.arange(0, 6 + delta_t*0.5, delta_t)\n\n# *** Euler ***\nt_euler, x_euler = euler(pendulum_NL, t_sec_array, x_0)\npy.plot(t_euler, x_euler, 'o', label='Euler')\n\n# *** Heun ***\nt_heun, x_heun = heun(pendulum_NL, t_sec_array, x_0)\npy.plot(t_heun, x_heun, '.', label='Heun')\n\n# *** RK4 ***\nt_rk4, x_rk4 = rk4(pendulum_NL, t_sec_array, x_0)\npy.plot(t_rk4, x_rk4, '-', label='RK4')\n\npy.xlabel('t(sec)')\npy.ylabel('x(m)')\n\npy.legend(loc=0)\npy.grid(True)\n\n\n```\n\n## 4\ucc28 \uc120\ud615 \uc0c1\ubbf8\ubc29
                                        Fourth Order Linear ODE\n\n\n\n$$\n \\frac{d^4x}{dt^4} \n + 12 \\frac{d^3x}{dt^3} \n + 54 \\frac{d^2x}{dt^2} \n + 108 \\frac{dx}{dt} \n + 80 x = 0\n$$\n\n\n\n$$\n\\mathbf{q} = \\begin{pmatrix}q_0 & q_1 & q_2 & q_3 \\end{pmatrix}^T = \\begin{pmatrix}x & \\frac{dx}{dt} & \\frac{d^2x}{dt^2} & \\frac{d^3x}{dt^3} \\end{pmatrix}^T\n$$\n\n\n\n$$\n\\frac{d\\mathbf{q}}{dt}\n=\n\\begin{bmatrix}\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n- 80 & - 108 & - 54 & -12\n\\end{bmatrix}\n\\begin{pmatrix}\nq_0 \\\\ q_1 \\\\ q_2 \\\\ q_3\n\\end{pmatrix}\n=\n\\mathbf{Aq}\n$$\n\n\n\n\n```python\nmatrix_A = np.matrix([\n [0, 1, 0, 0],\n [0, 0, 1, 0],\n [0, 0, 0, 1],\n [-80, -108, -54, -12],]\n)\n\nlegends = (f'$q_{k}$' for k in range(matrix_A.shape[0]))\n\nylabel = '$\\mathbf{q}$'\n\ndef fourth_order(q, t):\n \"\"\"\n Parameters\n ==========\n q: array of q_0, q_1, q_2, and q_3\n t: time value\n \n Return Value\n ============\n One dimensional array of dq/dt\n \"\"\"\n\n q_column = np.matrix(q).T\n qdot_column = matrix_A * q_column\n \n qdot_array = np.array(qdot_column.T).flatten()\n \n return qdot_array\n\n# Initial state\nx_0 = np.array([1, 0, 0, 0])\n\n\n```\n\n\n```python\n# Time array\ndelta_t = 0.08\nt_sec_array = np.arange(0, 6 + delta_t*0.5, delta_t)\n\n# *** Euler ***\nt_euler, x_euler_ode4 = euler(fourth_order, t_sec_array, x_0)\n\n# *** Heun ***\nt_heun, x_heun_ode4 = heun(fourth_order, t_sec_array, x_0)\n\n# *** RK4 ***\nt_rk4, x_rk4_ode4 = rk4(fourth_order, t_sec_array, x_0)\n\n# Convert to NumPy arrays\nx_euler_ode4_array, x_heun_ode4_array, x_rk4_ode4_array = (\n np.array(x_euler_ode4), \n np.array(x_heun_ode4), \n np.array(x_rk4_ode4))\n\npy.figure(figsize=(10, 10))\n\nfor i_state in range(x_euler_ode4_array.shape[1]):\n py.subplot(x_euler_ode4_array.shape[1], 1, i_state+1)\n py.plot(t_euler, x_euler_ode4_array[:, i_state], '-', label='Euler')\n py.plot(t_heun, x_heun_ode4_array[:, i_state], '-', label='Heun')\n py.plot(t_rk4, x_rk4_ode4_array[:, i_state], '-', label='RK4')\n\n py.xlabel('t(sec)')\n py.ylabel(f'$q_{i_state}$')\n\n py.legend(loc=1)\n py.grid(True)\n\n\n```\n\n## \ubd80\ub85d: \ub2e8\uc9c4\uc790\uc758 \ubc29\ud5a5\uc7a5
                                        Appendix: The direction field of a simple pendulum\n\n\n\n\uac01 \uc0c1\ud0dc\uc5d0\uc11c \uc0c1\ud0dc\ubcc0\uc218\uc758 \ubcc0\ud654\uc758 \ubc29\ud5a5 $\\left(\\frac{d}{dt}\\theta, \\frac{d^2}{dt^2}\\theta \\right)$ \uc744 \ud45c\uc2dc\ud574 \ubcf4\uc790.
                                        At each state, let's present the direction of state variable change $\\left(\\frac{d}{dt}\\theta, \\frac{d^2}{dt^2}\\theta \\right)$.\n\n\n\n\n```python\ndef ode_slopes_2states_cartesian(func, theta_rad_list, theta_dot_rad_list, time_list):\n \"\"\"\n Plot field of arrows indicating derivatives of the state\n :param func:\n :param theta_rad_list:\n :param theta_dot_rad_list:\n :param time_list:\n :return:\n \"\"\"\n\n # cartesian coordinate\n y_rad = np.meshgrid(theta_rad_list, theta_dot_rad_list)\n\n # derivatives of state at each point\n y_rad_dot = func(y_rad, time_list)\n\n # color\n color_mesh = np.sqrt(y_rad_dot[0] * y_rad_dot[0] + y_rad_dot[1] * y_rad_dot[1])\n\n py.figure(figsize=(18, 18))\n py.axis('equal')\n py.quiver(py.rad2deg(y_rad[0]), py.rad2deg(y_rad[1]), py.rad2deg(y_rad_dot[0]), py.rad2deg(y_rad_dot[1]), color_mesh, angles='xy')\n l, r, b, t = py.axis()\n x_span, y2_mesh = r - l, t - b\n py.axis([l - 0.05 * x_span, r + 0.05 * x_span, b - 0.05 * y2_mesh, t + 0.05 * y2_mesh])\n py.grid()\n\n\n```\n\n\n```python\ntime_list = []\n\n# list of theta\ntheta_deg_array = np.arange(-540, 540+1, 30)\ntheta_rad_list = np.deg2rad(theta_deg_array)\n\n# list of theta_dot\ntheta_dot_deg_array = np.arange(-540, 540+1, 45)\ntheta_dot_rad_list = np.deg2rad(theta_dot_deg_array)\n\n# plot the direction filed\node_slopes_2states_cartesian(pendulum_NL, theta_rad_list, theta_dot_rad_list, time_list)\n\n# Convert pendulum solution curves to NumPy arrays\nx_euler_array, x_heun_array, x_rk4_array = (\n np.array(x_euler), np.array(x_heun), np.array(x_rk4))\n\n# Plot the solution curves\npy.plot(py.rad2deg(x_euler_array[:, 0]), py.rad2deg(x_euler_array[:, 1]), '-', label='Euler', alpha=0.7)\npy.plot(py.rad2deg(x_heun_array[:, 0]), py.rad2deg(x_heun_array[:, 1]), '.-', label='Heun', alpha=0.7)\npy.plot(py.rad2deg(x_rk4_array[:, 0]), py.rad2deg(x_rk4_array[:, 1]), '-', label='RK4')\n\nax = py.gca()\n\nxlims = py.xlim(left=theta_deg_array[0], right=theta_deg_array[-1])\n# http://matplotlib.1069221.n5.nabble.com/How-do-I-set-grid-spacing-td9968.html\nax.set_xticks(np.hstack([np.arange(0, xlims[1]+1, 90), np.arange(-90, xlims[0]-1, -90)]))\n\nylims = py.ylim(bottom=theta_dot_deg_array[0], top=theta_dot_deg_array[-1])\n# http://matplotlib.1069221.n5.nabble.com/How-do-I-set-grid-spacing-td9968.html\nax.set_yticks(np.hstack([np.arange(0, ylims[1]+1, 90), np.arange(-90, ylims[0]-1, -90)]))\n\npy.legend(loc=0)\npy.xlabel('$\\\\theta(deg)$')\npy.ylabel('$\\\\frac{d}{dt}\\\\theta(deg/sec)$')\npy.title('Simple pendulum')\n\npy.savefig('pendulum_direction_field.svg')\n\n\n```\n\n## \ub3c4\uc804 \uacfc\uc81c
                                        Try This\n\n\n\n\ub2e4\uc74c 2\uacc4 \uc120\ud615 \uc0c1\ubbf8\ubd84 \ubc29\uc815\uc2dd\uc758 \uc218\uce58\ud574\ub97c RK4\ubc95\uc73c\ub85c \uad6c\ud558\uc2dc\uc624:
                                        \nFind the numerical solutions of the following second order linear ordinary equation using RK4 Method:\n\n$$\n\\begin{align}\n\\frac{d^2}{dt^2}x(t) + 2\\frac{d}{dt}x(t) + x(t) &= 0 \\\\\nx(0) &= 0 \\\\\n\\frac{d}{dt}x(0) &= 1\n\\end{align}\n$$\n\n\n\n\ub2e4\uc74c 2\uacc4 \uc120\ud615 \uc0c1\ubbf8\ubd84 \ubc29\uc815\uc2dd\uc758 \uc218\uce58\ud574\ub97c RK4\ubc95\uc73c\ub85c \uad6c\ud558\uc2dc\uc624:
                                        \nFind the numerical solutions of the following second order linear ordinary equation using RK4 Method:\n\n$$\n\\begin{align}\n\\frac{d^2}{dt^2}x(t) + \\frac{d}{dt}x(t) + 4x(t) &= sin(t[rad]) \\\\\nx(0) &= 0 \\\\\n\\frac{d}{dt}x(0) &= 0\n\\end{align}\n$$\n\n\n\n## Final Bell
                                        \ub9c8\uc9c0\ub9c9 \uc885\n\n\n\n\n```python\n# stackoverfow.com/a/24634221\nimport os\nos.system(\"printf '\\a'\");\n\n\n```\n\n\n```python\n\n\n\n```\n", "meta": {"hexsha": "b15d42c546a18372dfe07a32162dfef0a9aba81a", "size": 17254, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "50_ode/35_Runge_Kutta_Higher_Order.ipynb", "max_stars_repo_name": "cv2316eca19a/nmisp", "max_stars_repo_head_hexsha": "731a01bc687f3380ddf370d45c3a286f754b45cb", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "50_ode/35_Runge_Kutta_Higher_Order.ipynb", "max_issues_repo_name": "cv2316eca19a/nmisp", "max_issues_repo_head_hexsha": "731a01bc687f3380ddf370d45c3a286f754b45cb", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "50_ode/35_Runge_Kutta_Higher_Order.ipynb", "max_forks_repo_name": "cv2316eca19a/nmisp", "max_forks_repo_head_hexsha": "731a01bc687f3380ddf370d45c3a286f754b45cb", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 25.7138599106, "max_line_length": 240, "alphanum_fraction": 0.470673467, "converted": true, "num_tokens": 3758, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757645879592642, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.41063725158014436}} {"text": "# Ikeda $B_e$ assumtion.\n\n\n```python\nfrom rolldecayestimators import equations\n```\n\n Duplicate key in file WindowsPath('C:/Users/maa/.matplotlib/stylelib/paper.mplstyle'), line 461 ('figure.figsize : 5, 3 ## figure size in inches')\n Duplicate key in file WindowsPath('C:/Users/maa/.matplotlib/stylelib/paper.mplstyle'), line 462 ('figure.dpi : 100 ## figure dots per inch')\n\n\n# Purpose\nThe quadratic or cubic model can be expressed using the linearized equivalent damping ($B_e$) according to .:\n\n\n```python\nequations.B_e_equation\n```\n\n\n\n\n$\\displaystyle B_{e} = B_{1} + \\frac{8 B_{2} \\omega_{0} \\phi_{a}}{3 \\pi}$\n\n\n\n\n```python\nequations.B_e_equation_cubic\n```\n\n\n\n\n$\\displaystyle B_{e} = B_{1} + \\frac{8 B_{2} \\omega_{0} \\phi_{a}}{3 \\pi} + 0.75 B_{3} \\omega_{0}^{2} \\phi_{a}^{2}$\n\n\n\nBut I have some doubt about the validity of this, which will be investigated in this notebook.\n\n# Methodology\nA quadratic and cubic model from Simplified Ikeda will be used to calculate $B_e$. $B_e$ will also be obtained from Roll-decay simulations with these models, will the value be the same?\n\n# WIP - improvements\n(WORK IN PROGRESS)\nUse this section only if the notebook is not final.\n\nNotable TODOs:\n* todo 1\n* todo 2\n* todo 3\n\n## Results\nDescribe and comment the most important results.\n\n# Suggested next steps\nState suggested next steps, based on results obtained in this notebook.\n\n# Setup\n\n\n```python\n# %load imports.py\n\"\"\"\nThese is the standard setup for the notebooks.\n\"\"\"\n\n%matplotlib inline\n%load_ext autoreload\n%autoreload 2\n\nfrom jupyterthemes import jtplot\njtplot.style(theme='onedork', context='notebook', ticks=True, grid=False)\n\nimport pandas as pd\npd.options.display.max_rows = 999\npd.options.display.max_columns = 999\npd.set_option(\"display.max_columns\", None)\nimport numpy as np\nimport os\nimport matplotlib.pyplot as plt\nfrom collections import OrderedDict\n#plt.style.use('paper')\n\n#import data\nimport copy\nfrom mdldb.run import Run\n\nfrom sklearn.pipeline import Pipeline\nfrom rolldecayestimators.transformers import CutTransformer, LowpassFilterDerivatorTransformer, ScaleFactorTransformer, OffsetTransformer\nfrom rolldecayestimators.direct_estimator_cubic import EstimatorQuadraticB, EstimatorCubic\nfrom rolldecayestimators.ikeda_estimator import IkedaQuadraticEstimator\nimport rolldecayestimators.equations as equations\nimport rolldecayestimators.lambdas as lambdas\nfrom rolldecayestimators.substitute_dynamic_symbols import lambdify\nimport rolldecayestimators.symbols as symbols\nimport sympy as sp\n\nfrom sklearn.metrics import r2_score\nfrom src.data import database\nfrom mdldb import tables\n\n```\n\n\n```python\nfrom rolldecayestimators.simplified_ikeda_class import SimplifiedIkeda\nimport rolldecayestimators\n```\n\n\n```python\ndf_ikeda = database.load(rolldecay_table_name='rolldecay_simplified_ikeda_unlimited', limit_score=0.95, \n exclude_table_name='rolldecay_exclude')\n```\n\n\n```python\ndf_ikeda.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        model_numberloading_condition_idship_speedB_44_1B_F_1B_W_1B_E_1B_BK_1B_L_1B_44_2B_F_2B_W_2B_E_2B_BK_2B_L_2B_3omega0omega0_fftzetadscorephi_startphi_stopmean_dampingB_1B_2B_eidproject_numberseries_numberrun_numbertest_numbership_nameascii_namecommentfile_path_asciifile_path_ascii_tempfile_path_logfile_path_hdf5datetest_typefacilityangle1angle2K\u00f6rfallstypnameproject_pathlcgkggmCWTFTABWLKXXKZZBTT1CPVolumeA0RHscale_factorlppbeamABULBBKXTWINDCLRVDESRHBLASKEGPDARHCFPAIXPDTDESRTYPESFPBKLBKBPROTDLSKEGRRXSKEGNDESARBRBRAIRUDPTYPEXRUDAIHSKEGRSKEGLOAship_type_id
                                        run_id
                                        148483541-A10713.015.7186151.1014384.2187190.4687863.5928816.33679221.2862811.1014384.2187190.9375728.6917606.336792None2.4881412.4881410.0266150.4969910.968976-0.1384160.006405None10.15094938.09097621.286281148482013652213213541-A20136522-ser001-k032-100hz.ascKursstyrning Roll decayN:\\Gamla_Projekt\\ascii_files\\20136522-ser001-k...None\\\\sspa.local\\gbg\\MDLdata\\Gamla_Projekt\\Gamla_P...None2014-12-29roll decayMDLNoneNoneKursstyrning20136522_Hudong-Zhonghua 170k class LNGCS:\\2013\\20136522-Hudong-Zhonghua-170k-class-LNG-4.19018.383.80.84711.5011.5046.9715.565.323.360.7669116472.00.98998.555.000284.0046.975.1-7.6651.08.419.89.2130.00.9910.0NoneNone20000.01.0None92.130.433.08.328.02.320-123.067.849.400.00.02.01.0-142.000None7.758.0290.04.0
                                        55093338-B440.05.0622261.0812531.6383192.3426540.0000000.0000007.4048801.0812531.6383194.6853070.0000000.000000None2.4504422.4504420.0044660.1181530.9859860.153420-0.016432None2.71957214.6822707.40488055092009542111813338-B20095421-ser001-k018-100hz.ascKursstyrning Rolldecay 0.0 knN:\\Gamla_Projekt\\ascii_files\\20095421-ser001-k...None\\\\sspa.local\\gbg\\MDLdata\\Gamla_Projekt\\gamla_p...None2010-09-02roll decayMDLNoneNoneKursstyrning20095421_STX_Polar_Supply\\\\sspa.local\\gbg\\ProjektArkiv\\2009\\20095421_ST...-2.3279.002.20.8306.746.7422.008.330.35NaN0.632511260.00.98905.123.889121.4022.00NaN0.0000.0NaN14.0NaNNaN0.990NaNNoneNoneNaN1.0None0.000.00NaN4.3NaNNaNNaNNaN12.45NaNNaN2.01.0NaNNoneNaNNaN133.48.0
                                        55103338-B443.08.2412471.0812531.7207271.6458100.0000003.7934579.8870571.0812531.7207273.2916200.0000003.793457None2.4504422.4504420.0108300.0842830.9843640.151097-0.002482None6.59543710.4734469.88705755102009542111913338-B20095421-ser001-k019-100hz.ascKursstyrning Rolldecay 3.0 knN:\\Gamla_Projekt\\ascii_files\\20095421-ser001-k...None\\\\sspa.local\\gbg\\MDLdata\\Gamla_Projekt\\gamla_p...None2010-09-02roll decayMDLNoneNoneKursstyrning20095421_STX_Polar_Supply\\\\sspa.local\\gbg\\ProjektArkiv\\2009\\20095421_ST...-2.3279.002.20.8306.746.7422.008.330.35NaN0.632511260.00.98905.123.889121.4022.00NaN0.0000.0NaN14.0NaNNaN0.990NaNNoneNoneNaN1.0None0.000.00NaN4.3NaNNaNNaNNaN12.45NaNNaN2.01.0NaNNoneNaNNaN133.48.0
                                        34623250220.09.6876311.0707265.4667920.6499402.5001730.00000014.0046941.0707265.4667921.2998806.1672960.000000None2.6892032.6892030.0090630.2202520.996212-0.1558700.003981None5.37056824.26686814.0046943462200746041151325020074604-ser001-k015-100hz.ascKursstyrning RolldecayN:\\Gamla_Projekt\\ascii_files\\20074604-ser001-k...None\\\\sspa.local\\gbg\\MDLdata\\Gamla_Projekt\\gamla_p...None2008-01-17roll decayMDLNoneNoneKursstyrning20074604-1\\\\sspa.local\\gbg\\ProjektArkiv\\2007\\200746046.40115.105.1NaN15.2515.2548.0017.863.301.95NaN161867.00.998512.056.463263.7548.0067.019.7800.00.0NaN12.1130.00.87016.9NoneNone16730.01.0None65.900.451.08.326.02.708-114.0NaN88.400.00.01.01.0-131.875None8.42.0285.01.0
                                        55113338-B4414.027.8183881.1004898.7532600.2618400.00000017.70279828.0802271.1004898.7532600.5236800.00000017.702798None2.5384072.5384070.0468720.0138480.964556-0.1515570.002947None27.5565481.60365428.08022755112009542112013338-B20095421-ser001-k020-100hz.ascKursstyrning Rolldecay 14.0 knN:\\Gamla_Projekt\\ascii_files\\20095421-ser001-k...None\\\\sspa.local\\gbg\\MDLdata\\Gamla_Projekt\\gamla_p...None2010-09-02roll decayMDLNoneNoneKursstyrning20095421_STX_Polar_Supply\\\\sspa.local\\gbg\\ProjektArkiv\\2009\\20095421_ST...-2.3279.002.20.8306.746.7422.008.330.35NaN0.632511260.00.98905.123.889121.4022.00NaN0.0000.0NaN14.0NaNNaN0.990NaNNoneNoneNaN1.0None0.000.00NaN4.3NaNNaNNaNNaN12.45NaNNaN2.01.0NaNNoneNaNNaN133.48.0
                                        \n
                                        \n\n\n\n\n```python\nrow = df_ikeda.iloc[0]\n\ndb = database.get_db()\nrun = db.session.query(Run).get(int(row.id))\nrun = database.load_run(run, save_as_example=False, prefer_hdf5=True)\n```\n\n c:\\dev\\evaluation\\signal_lab\\mdl_to_evaluation.py:106: UserWarning: Pandas doesn't allow columns to be created via a new attribute name - see https://pandas.pydata.org/pandas-docs/stable/indexing.html#attribute-access\n df_.units = units\n\n\n\n```python\nlowpass_filter = LowpassFilterDerivatorTransformer(cutoff=2, minimum_score=0.99)\nscaler = ScaleFactorTransformer(scale_factor=None) # dummy value None for now\ncutter = CutTransformer(phi_max=np.deg2rad(9), phi_min=np.deg2rad(0.25), phi1d_start_tolerance=0.015)\noffset_transformer = OffsetTransformer()\n```\n\n\n```python\nsteps = [\n ('filter',lowpass_filter),\n ('cutter', cutter), \n# ('offset_transformer',offset_transformer),\n]\n\ndf = run.df.copy()\n\npreprosessor = Pipeline(steps=steps)\npreprosessor.fit(X=df[['phi']])\nX = preprosessor.transform(df[['phi']])\n```\n\n\n```python\ndata = row.copy()\nscale_factor=run.model.scale_factor\nrho=1000\ng=9.81\ndata['rho']=rho\ndata['g']=g\n\ndata['lpp']/=scale_factor\ndata['TA']/=scale_factor\ndata['TF']/=scale_factor\ndata['beam']/=scale_factor\ndata['BKL']/=scale_factor\ndata['BKB']/=scale_factor\ndata['kg']/=scale_factor\ndata['Volume']/=scale_factor**3\ndata['gm']/=scale_factor\ndata['V']=data['ship_speed']*1.852/3.6/np.sqrt(scale_factor) #[m/s]\ndata['KXX']/=scale_factor\n\nestimator_ikeda_quadratic_db = IkedaQuadraticEstimator.load(data=data, X=X)\n```\n\n\n\n\n```python\nestimator_ikeda_quadratic = IkedaQuadraticEstimator(**data,\n verify_input=False, limit_inputs=False)\nestimator_ikeda_quadratic.fit(X=X)\n```\n\n\n```python\nfig,ax=plt.subplots()\nestimator_ikeda_quadratic_db.plot_fit(ax=ax)\nestimator_ikeda_quadratic.plot_fit(ax=ax, model_test=False)\n\nfig,ax=plt.subplots()\nestimator_ikeda_quadratic_db.plot_damping(ax=ax)\nestimator_ikeda_quadratic.plot_damping(ax=ax, include_model_test=False)\n```\n\n## Calculate for each $\\phi_a$\n\n\n```python\nestimator_ikeda_quadratic_db.plot_omega0()\n```\n\n\n```python\nestimator_ikeda_quadratic.calculate_amplitudes_and_damping()\nX_pred = estimator_ikeda_quadratic.predict(X=X)\nX_amplitudes = rolldecayestimators.measure.calculate_amplitudes_and_damping(X=X_pred)\n```\n\n\n```python\nX_amplitudes.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        phiphi1dphi2dphi_azeta_nB_nomega0
                                        10.2929120.2054375.045877e-161.2718210.1027180.0568110.1136222.483525
                                        11.5578850.1708877.274997e-161.0579280.0854430.0535220.1070442.483964
                                        12.8224120.1437661.916327e-160.8900360.0718830.0509150.1018292.484656
                                        14.0866800.1220842.853078e-160.7558100.0610420.0489090.0978182.485092
                                        15.3507630.1044052.222614e-160.6463570.0522030.0472060.0944122.485439
                                        \n
                                        \n\n\n\n\n```python\ndata\n```\n\n\n\n\n model_number 3541-A\n loading_condition_id 107\n ship_speed 13\n B_44_1 15.7186\n B_F_1 1.10144\n B_W_1 4.21872\n B_E_1 0.468786\n B_BK_1 3.59288\n B_L_1 6.33679\n B_44_2 21.2863\n B_F_2 1.10144\n B_W_2 4.21872\n B_E_2 0.937572\n B_BK_2 8.69176\n B_L_2 6.33679\n B_3 None\n omega0 2.48814\n omega0_fft 2.48814\n zeta 0.0266151\n d 0.496991\n score 0.968976\n phi_start -0.138416\n phi_stop 0.00640469\n mean_damping None\n B_1 10.1509\n B_2 38.091\n B_e 21.2863\n id 14848\n project_number 20136522\n series_number 1\n run_number 32\n test_number 1\n ship_name 3541-A\n ascii_name 20136522-ser001-k032-100hz.asc\n comment Kursstyrning Roll decay\n file_path_ascii N:\\Gamla_Projekt\\ascii_files\\20136522-ser001-k...\n file_path_ascii_temp None\n file_path_log \\\\sspa.local\\gbg\\MDLdata\\Gamla_Projekt\\Gamla_P...\n file_path_hdf5 None\n date 2014-12-29\n test_type roll decay\n facility MDL\n angle1 None\n angle2 None\n K\u00f6rfallstyp Kursstyrning\n name 20136522_Hudong-Zhonghua 170k class LNGC\n project_path S:\\2013\\20136522-Hudong-Zhonghua-170k-class-LNG\n lcg -4.19\n kg 0.334182\n gm 0.0690909\n CW 0.847\n TF 0.209091\n TA 0.209091\n BWL 46.97\n KXX 0.281818\n KZZ 65.32\n BTT1 3.36\n CP 0.7669\n Volume 0.700057\n A0 0.9899\n RH 8.5\n scale_factor 55\n lpp 5.16364\n beam 0.854\n ABULB 5.1\n BKX -7.665\n TWIN 1\n DCLR 8.4\n VDES 19.8\n RHBL 9.2\n ASKEG 130\n PD 0.991\n ARH 0\n CFP None\n AIX None\n PDTDES 20000\n RTYPE 1\n SFP None\n BKL 1.67509\n BKB 0.00781818\n PROT 3\n D 8.3\n LSKEG 28\n RR 2.32\n XSKEG -123\n NDES 67.8\n AR 49.4\n BR 0\n BRA 0\n IRUD 2\n PTYPE 1\n XRUD -142\n AI None\n HSKEG 7.7\n RSKEG 58\n LOA 290\n ship_type_id 4\n rho 1000\n g 9.81\n V 0.90178\n Name: 14848, dtype: object\n\n\n\n\n```python\ndata['draught']=(data['TA']+data['TF'])/2\ndata['volume']=data['Volume']\n```\n\n\n```python\ndata2 = data.copy()\n```\n\n\n```python\nN = len(X_amplitudes)\ndata_ = np.tile(data2.values,(N,1))\ninputs_raw = pd.DataFrame(data=data_, columns=data2.index)\n```\n\n\n```python\nfor key,value in inputs_raw.items():\n try: \n inputs_raw[key] =inputs_raw[key].astype(float)\n except:\n continue\n \n \n```\n\n\n```python\ninputs_raw.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        model_numberloading_condition_idship_speedB_44_1B_F_1B_W_1B_E_1B_BK_1B_L_1B_44_2B_F_2B_W_2B_E_2B_BK_2B_L_2B_3omega0omega0_fftzetadscorephi_startphi_stopmean_dampingB_1B_2B_eidproject_numberseries_numberrun_numbertest_numbership_nameascii_namecommentfile_path_asciifile_path_ascii_tempfile_path_logfile_path_hdf5datetest_typefacilityangle1angle2K\u00f6rfallstypnameproject_pathlcgkggmCWTFTABWLKXXKZZBTT1CPVolumeA0RHscale_factorlppbeamABULBBKXTWINDCLRVDESRHBLASKEGPDARHCFPAIXPDTDESRTYPESFPBKLBKBPROTDLSKEGRRXSKEGNDESARBRBRAIRUDPTYPEXRUDAIHSKEGRSKEGLOAship_type_idrhogVdraughtvolume
                                        03541-A107.013.015.7186151.1014384.2187190.4687863.5928816.33679221.2862811.1014384.2187190.9375728.691766.336792NaN2.4881412.4881410.0266150.4969910.968976-0.1384160.006405NaN10.15094938.09097621.28628114848.020136522.01.032.01.03541-A20136522-ser001-k032-100hz.ascKursstyrning Roll decayN:\\Gamla_Projekt\\ascii_files\\20136522-ser001-k...NaN\\\\sspa.local\\gbg\\MDLdata\\Gamla_Projekt\\Gamla_P...NaN2014-12-29roll decayMDLNaNNaNKursstyrning20136522_Hudong-Zhonghua 170k class LNGCS:\\2013\\20136522-Hudong-Zhonghua-170k-class-LNG-4.190.3341820.0690910.8470.2090910.20909146.970.28181865.323.360.76690.7000570.98998.555.05.1636360.8545.1-7.6651.08.419.89.2130.00.9910.0NaNNaN20000.01.0NaN1.6750910.0078183.08.328.02.32-123.067.849.40.00.02.01.0-142.0NaN7.758.0290.04.01000.09.810.901780.2090910.700057
                                        13541-A107.013.015.7186151.1014384.2187190.4687863.5928816.33679221.2862811.1014384.2187190.9375728.691766.336792NaN2.4881412.4881410.0266150.4969910.968976-0.1384160.006405NaN10.15094938.09097621.28628114848.020136522.01.032.01.03541-A20136522-ser001-k032-100hz.ascKursstyrning Roll decayN:\\Gamla_Projekt\\ascii_files\\20136522-ser001-k...NaN\\\\sspa.local\\gbg\\MDLdata\\Gamla_Projekt\\Gamla_P...NaN2014-12-29roll decayMDLNaNNaNKursstyrning20136522_Hudong-Zhonghua 170k class LNGCS:\\2013\\20136522-Hudong-Zhonghua-170k-class-LNG-4.190.3341820.0690910.8470.2090910.20909146.970.28181865.323.360.76690.7000570.98998.555.05.1636360.8545.1-7.6651.08.419.89.2130.00.9910.0NaNNaN20000.01.0NaN1.6750910.0078183.08.328.02.32-123.067.849.40.00.02.01.0-142.0NaN7.758.0290.04.01000.09.810.901780.2090910.700057
                                        23541-A107.013.015.7186151.1014384.2187190.4687863.5928816.33679221.2862811.1014384.2187190.9375728.691766.336792NaN2.4881412.4881410.0266150.4969910.968976-0.1384160.006405NaN10.15094938.09097621.28628114848.020136522.01.032.01.03541-A20136522-ser001-k032-100hz.ascKursstyrning Roll decayN:\\Gamla_Projekt\\ascii_files\\20136522-ser001-k...NaN\\\\sspa.local\\gbg\\MDLdata\\Gamla_Projekt\\Gamla_P...NaN2014-12-29roll decayMDLNaNNaNKursstyrning20136522_Hudong-Zhonghua 170k class LNGCS:\\2013\\20136522-Hudong-Zhonghua-170k-class-LNG-4.190.3341820.0690910.8470.2090910.20909146.970.28181865.323.360.76690.7000570.98998.555.05.1636360.8545.1-7.6651.08.419.89.2130.00.9910.0NaNNaN20000.01.0NaN1.6750910.0078183.08.328.02.32-123.067.849.40.00.02.01.0-142.0NaN7.758.0290.04.01000.09.810.901780.2090910.700057
                                        33541-A107.013.015.7186151.1014384.2187190.4687863.5928816.33679221.2862811.1014384.2187190.9375728.691766.336792NaN2.4881412.4881410.0266150.4969910.968976-0.1384160.006405NaN10.15094938.09097621.28628114848.020136522.01.032.01.03541-A20136522-ser001-k032-100hz.ascKursstyrning Roll decayN:\\Gamla_Projekt\\ascii_files\\20136522-ser001-k...NaN\\\\sspa.local\\gbg\\MDLdata\\Gamla_Projekt\\Gamla_P...NaN2014-12-29roll decayMDLNaNNaNKursstyrning20136522_Hudong-Zhonghua 170k class LNGCS:\\2013\\20136522-Hudong-Zhonghua-170k-class-LNG-4.190.3341820.0690910.8470.2090910.20909146.970.28181865.323.360.76690.7000570.98998.555.05.1636360.8545.1-7.6651.08.419.89.2130.00.9910.0NaNNaN20000.01.0NaN1.6750910.0078183.08.328.02.32-123.067.849.40.00.02.01.0-142.0NaN7.758.0290.04.01000.09.810.901780.2090910.700057
                                        43541-A107.013.015.7186151.1014384.2187190.4687863.5928816.33679221.2862811.1014384.2187190.9375728.691766.336792NaN2.4881412.4881410.0266150.4969910.968976-0.1384160.006405NaN10.15094938.09097621.28628114848.020136522.01.032.01.03541-A20136522-ser001-k032-100hz.ascKursstyrning Roll decayN:\\Gamla_Projekt\\ascii_files\\20136522-ser001-k...NaN\\\\sspa.local\\gbg\\MDLdata\\Gamla_Projekt\\Gamla_P...NaN2014-12-29roll decayMDLNaNNaNKursstyrning20136522_Hudong-Zhonghua 170k class LNGCS:\\2013\\20136522-Hudong-Zhonghua-170k-class-LNG-4.190.3341820.0690910.8470.2090910.20909146.970.28181865.323.360.76690.7000570.98998.555.05.1636360.8545.1-7.6651.08.419.89.2130.00.9910.0NaNNaN20000.01.0NaN1.6750910.0078183.08.328.02.32-123.067.849.40.00.02.01.0-142.0NaN7.758.0290.04.01000.09.810.901780.2090910.700057
                                        \n
                                        \n\n\n\n\n```python\ninputs = inputs_raw.copy()\ninputs['w']=inputs['omega0'].astype(float)\ninputs['fi_a']=np.array(X_amplitudes['phi_a'])\ninputs['g']=9.81\n\n```\n\n\n```python\ninputs.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        model_numberloading_condition_idship_speedB_44_1B_F_1B_W_1B_E_1B_BK_1B_L_1B_44_2B_F_2B_W_2B_E_2B_BK_2B_L_2B_3omega0omega0_fftzetadscorephi_startphi_stopmean_dampingB_1B_2B_eidproject_numberseries_numberrun_numbertest_numbership_nameascii_namecommentfile_path_asciifile_path_ascii_tempfile_path_logfile_path_hdf5datetest_typefacilityangle1angle2K\u00f6rfallstypnameproject_pathlcgkggmCWTFTABWLKXXKZZBTT1CPVolumeA0RHscale_factorlppbeamABULBBKXTWINDCLRVDESRHBLASKEGPDARHCFPAIXPDTDESRTYPESFPBKLBKBPROTDLSKEGRRXSKEGNDESARBRBRAIRUDPTYPEXRUDAIHSKEGRSKEGLOAship_type_idrhogVdraughtvolumewfi_a
                                        03541-A107.013.015.7186151.1014384.2187190.4687863.5928816.33679221.2862811.1014384.2187190.9375728.691766.336792NaN2.4881412.4881410.0266150.4969910.968976-0.1384160.006405NaN10.15094938.09097621.28628114848.020136522.01.032.01.03541-A20136522-ser001-k032-100hz.ascKursstyrning Roll decayN:\\Gamla_Projekt\\ascii_files\\20136522-ser001-k...NaN\\\\sspa.local\\gbg\\MDLdata\\Gamla_Projekt\\Gamla_P...NaN2014-12-29roll decayMDLNaNNaNKursstyrning20136522_Hudong-Zhonghua 170k class LNGCS:\\2013\\20136522-Hudong-Zhonghua-170k-class-LNG-4.190.3341820.0690910.8470.2090910.20909146.970.28181865.323.360.76690.7000570.98998.555.05.1636360.8545.1-7.6651.08.419.89.2130.00.9910.0NaNNaN20000.01.0NaN1.6750910.0078183.08.328.02.32-123.067.849.40.00.02.01.0-142.0NaN7.758.0290.04.01000.09.810.901780.2090910.7000572.4881410.102718
                                        13541-A107.013.015.7186151.1014384.2187190.4687863.5928816.33679221.2862811.1014384.2187190.9375728.691766.336792NaN2.4881412.4881410.0266150.4969910.968976-0.1384160.006405NaN10.15094938.09097621.28628114848.020136522.01.032.01.03541-A20136522-ser001-k032-100hz.ascKursstyrning Roll decayN:\\Gamla_Projekt\\ascii_files\\20136522-ser001-k...NaN\\\\sspa.local\\gbg\\MDLdata\\Gamla_Projekt\\Gamla_P...NaN2014-12-29roll decayMDLNaNNaNKursstyrning20136522_Hudong-Zhonghua 170k class LNGCS:\\2013\\20136522-Hudong-Zhonghua-170k-class-LNG-4.190.3341820.0690910.8470.2090910.20909146.970.28181865.323.360.76690.7000570.98998.555.05.1636360.8545.1-7.6651.08.419.89.2130.00.9910.0NaNNaN20000.01.0NaN1.6750910.0078183.08.328.02.32-123.067.849.40.00.02.01.0-142.0NaN7.758.0290.04.01000.09.810.901780.2090910.7000572.4881410.085443
                                        23541-A107.013.015.7186151.1014384.2187190.4687863.5928816.33679221.2862811.1014384.2187190.9375728.691766.336792NaN2.4881412.4881410.0266150.4969910.968976-0.1384160.006405NaN10.15094938.09097621.28628114848.020136522.01.032.01.03541-A20136522-ser001-k032-100hz.ascKursstyrning Roll decayN:\\Gamla_Projekt\\ascii_files\\20136522-ser001-k...NaN\\\\sspa.local\\gbg\\MDLdata\\Gamla_Projekt\\Gamla_P...NaN2014-12-29roll decayMDLNaNNaNKursstyrning20136522_Hudong-Zhonghua 170k class LNGCS:\\2013\\20136522-Hudong-Zhonghua-170k-class-LNG-4.190.3341820.0690910.8470.2090910.20909146.970.28181865.323.360.76690.7000570.98998.555.05.1636360.8545.1-7.6651.08.419.89.2130.00.9910.0NaNNaN20000.01.0NaN1.6750910.0078183.08.328.02.32-123.067.849.40.00.02.01.0-142.0NaN7.758.0290.04.01000.09.810.901780.2090910.7000572.4881410.071883
                                        33541-A107.013.015.7186151.1014384.2187190.4687863.5928816.33679221.2862811.1014384.2187190.9375728.691766.336792NaN2.4881412.4881410.0266150.4969910.968976-0.1384160.006405NaN10.15094938.09097621.28628114848.020136522.01.032.01.03541-A20136522-ser001-k032-100hz.ascKursstyrning Roll decayN:\\Gamla_Projekt\\ascii_files\\20136522-ser001-k...NaN\\\\sspa.local\\gbg\\MDLdata\\Gamla_Projekt\\Gamla_P...NaN2014-12-29roll decayMDLNaNNaNKursstyrning20136522_Hudong-Zhonghua 170k class LNGCS:\\2013\\20136522-Hudong-Zhonghua-170k-class-LNG-4.190.3341820.0690910.8470.2090910.20909146.970.28181865.323.360.76690.7000570.98998.555.05.1636360.8545.1-7.6651.08.419.89.2130.00.9910.0NaNNaN20000.01.0NaN1.6750910.0078183.08.328.02.32-123.067.849.40.00.02.01.0-142.0NaN7.758.0290.04.01000.09.810.901780.2090910.7000572.4881410.061042
                                        43541-A107.013.015.7186151.1014384.2187190.4687863.5928816.33679221.2862811.1014384.2187190.9375728.691766.336792NaN2.4881412.4881410.0266150.4969910.968976-0.1384160.006405NaN10.15094938.09097621.28628114848.020136522.01.032.01.03541-A20136522-ser001-k032-100hz.ascKursstyrning Roll decayN:\\Gamla_Projekt\\ascii_files\\20136522-ser001-k...NaN\\\\sspa.local\\gbg\\MDLdata\\Gamla_Projekt\\Gamla_P...NaN2014-12-29roll decayMDLNaNNaNKursstyrning20136522_Hudong-Zhonghua 170k class LNGCS:\\2013\\20136522-Hudong-Zhonghua-170k-class-LNG-4.190.3341820.0690910.8470.2090910.20909146.970.28181865.323.360.76690.7000570.98998.555.05.1636360.8545.1-7.6651.08.419.89.2130.00.9910.0NaNNaN20000.01.0NaN1.6750910.0078183.08.328.02.32-123.067.849.40.00.02.01.0-142.0NaN7.758.0290.04.01000.09.810.901780.2090910.7000572.4881410.052203
                                        \n
                                        \n\n\n\n\n```python\ndef calculate(inputs, IkedaClass=SimplifiedIkeda):\n si = IkedaClass(**inputs) \n output = pd.DataFrame(index=inputs.index)\n output['B_44_hat'] = si.calculate_B44()\n output['B_W0'] =si.calculate_B_W0()\n output['B_W'] =si.calculate_B_W()\n output['B_F'] =si.calculate_B_F()\n output['B_E'] =si.calculate_B_E()\n output['B_BK'] =si.calculate_B_BK()\n output['B_L'] =si.calculate_B_L()\n output['Bw_div_Bw0'] =si.calculate_Bw_div_Bw0()\n \n return output\n```\n\n\n```python\noutput = calculate(inputs=inputs, IkedaClass=SimplifiedIkeda)\n```\n\n\n```python\noutput.head()\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        B_44_hatB_W0B_WB_FB_EB_BKB_LBw_div_Bw0
                                        00.0102220.0002480.0017240.0004520.0011600.0042970.0025896.959384
                                        10.0094670.0002480.0017240.0004520.0009650.0037370.0025896.959384
                                        20.0088960.0002480.0017240.0004520.0008120.0033190.0025896.959384
                                        30.0084520.0002480.0017240.0004520.0006890.0029970.0025896.959384
                                        40.0080990.0002480.0017240.0004520.0005890.0027440.0025896.959384
                                        \n
                                        \n\n\n\n\n```python\noutput['B_44'] = lambdas.B_from_hat_lambda(B_44_hat=output['B_44_hat'], Disp=inputs['volume'], \n beam=inputs['beam'],\n g=inputs['g'], rho=inputs['rho'])\n```\n\n\n```python\nfig,ax=plt.subplots()\nx = inputs['fi_a']\ny = output['B_44']\nax.plot(x,y)\nax.set_xlabel('$\\phi_a$ [rad]')\nax.set_ylabel('$B_{44}$')\n\n```\n\n\n```python\nnp.array(output['B_44'])/np.array(X_amplitudes['B_n'])\n```\n\n\n\n\n array([220.15619203, 216.43961076, 213.78553395, 211.45020088,\n 209.92964181, 209.14145408, 208.29162558, 207.59734788,\n 207.36961231, 206.97347589, 206.39749995, 206.38231801,\n 206.68514401, 206.46781746, 205.99165004, 206.03015756,\n 206.28829518, 206.13001126, 205.87092656, 206.10920137,\n 206.4537088 , 206.27099344, 205.86343667, 205.67919139,\n 205.86164755, 206.26391714, 206.52329792, nan])\n\n\n\n\n```python\nequations.C_equation_linear\n```\n\n\n\n\n$\\displaystyle C = GM g m$\n\n\n\n\n```python\nequations.A44\n```\n\n\n\n\n$\\displaystyle \\frac{GM g m}{\\omega_{0}^{2}}$\n\n\n\n\n```python\nmass = inputs.Volume*inputs.rho\nA44_tot = lambdas.A44_lambda(GM=inputs.gm, g=inputs.g, m=mass, omega0=inputs.w)\nA44_mass = mass*data.KXX**2\nKXX_tot = np.sqrt(A44_tot/mass)\n(KXX_tot - data.KXX)/data.KXX\n```\n\n\n\n\n 0 0.174089\n 1 0.174089\n 2 0.174089\n 3 0.174089\n 4 0.174089\n 5 0.174089\n 6 0.174089\n 7 0.174089\n 8 0.174089\n 9 0.174089\n 10 0.174089\n 11 0.174089\n 12 0.174089\n 13 0.174089\n 14 0.174089\n 15 0.174089\n 16 0.174089\n 17 0.174089\n 18 0.174089\n 19 0.174089\n 20 0.174089\n 21 0.174089\n 22 0.174089\n 23 0.174089\n 24 0.174089\n 25 0.174089\n 26 0.174089\n 27 0.174089\n dtype: float64\n\n\n\n\n```python\nresults = estimator_ikeda_quadratic.result_for_database()\nresults\n```\n\n\n\n\n {'zeta': 0.036588024189946715,\n 'd': 0.5442643525426031,\n 'omega0': 2.48814138164312,\n 'score': 0.9042684962009301,\n 'phi_start': -0.1381428102953512,\n 'phi_stop': 0.0079412480965742,\n 'omega0_fft': 2.48814138164312,\n 'B_44_1': 20.039803465353693,\n 'B_44_2': 26.125010601697632,\n 'B_F_1': 1.1014377904648913,\n 'B_F_2': 1.1014377904648913,\n 'B_W_1': 4.218718560323839,\n 'B_W_2': 4.218718560323839,\n 'B_E_1': 0.4678599368133092,\n 'B_E_2': 0.9357198736266183,\n 'B_BK_1': 7.914995241710201,\n 'B_BK_2': 13.532342441240832,\n 'B_L_1': 6.336791936041452,\n 'B_L_2': 6.336791936041452,\n 'B_1': 13.954596329009753,\n 'B_2': 41.71412718172688,\n 'B_e': 26.125010601697632}\n\n\n\n\n```python\nequations.extinction_equation\n```\n\n\n\n\n$\\displaystyle \\phi_{a} = \\phi_{0}{\\left(t \\right)} e^{- \\omega_{0} t \\zeta}$\n\n\n\n\n```python\nfig,ax=plt.subplots()\nX_amplitudes.plot(y='phi_a', ax=ax)\n```\n\n\n```python\nt = X_amplitudes.index\ny = np.log(X_amplitudes['phi_a'])\n\nfig,ax=plt.subplots()\nax.plot(t,y)\n```\n\n\n```python\nsp.Eq(symbols.zeta,sp.solve(equations.extinction_equation,symbols.zeta)[0])\n```\n\n\n\n\n$\\displaystyle \\zeta = \\frac{\\log{\\left(\\frac{\\phi_{0}{\\left(t \\right)}}{\\phi_{a}} \\right)}}{\\omega_{0} t}$\n\n\n\n\n```python\nzeta_lambda = lambdify(sp.solve(equations.extinction_equation,symbols.zeta)[0])\nphi_0 = X_amplitudes['phi_a'].iloc[0]\nt = X_amplitudes.index - X_amplitudes.index[0]\nX_amplitudes['zeta2'] = zeta_lambda(omega0=X_amplitudes['omega0'],phi_0=phi_0, phi_a=X_amplitudes['phi_a'],\n t=t)\n```\n\n\n```python\nomega0 = inputs.iloc[0]['omega0']\n\nfor i in range(len(X_amplitudes)-1):\n \n row1 = X_amplitudes.iloc[i]\n row2 = X_amplitudes.iloc[i+1]\n t_ = row2.name - row1.name\n B_n = zeta_lambda(omega0=omega0,phi_0=row1['phi_a'], phi_a=row2['phi_a'],\n t=t_)\n \n X_amplitudes.loc[row2.name,'B_n2'] = B_n\n```\n\n\n```python\nX_amplitudes['B_n3'] = np.array(output['B_44']/(A44_tot))\n```\n\n\n```python\nfig,ax=plt.subplots()\nX_amplitudes.plot(x='phi_a',y='zeta2',ax=ax)\nX_amplitudes.plot(x='phi_a',y='B_n', ax=ax)\n\nX_amplitudes['B_n/2'] = X_amplitudes['B_n']/2\nX_amplitudes.plot(x='phi_a',y='B_n/2', ax=ax)\n\nX_amplitudes.plot(x='phi_a',y='B_n2', ax=ax)\n#X_amplitudes.plot(x='phi_a',y='B_n3', ax=ax)\n```\n", "meta": {"hexsha": "1e0d9d07bcec6c4ba9057cd0a665e93f71df4ef3", "size": 375120, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/02.1_ikeda_Be_assumption.ipynb", "max_stars_repo_name": "rddaz2013/Prediction-of-roll-motion-using-fully-nonlinear-potential-flow-and-Ikedas-method", "max_stars_repo_head_hexsha": "ac0a27e31d64edc8ae8912b6ed10005029868c90", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/02.1_ikeda_Be_assumption.ipynb", "max_issues_repo_name": "rddaz2013/Prediction-of-roll-motion-using-fully-nonlinear-potential-flow-and-Ikedas-method", "max_issues_repo_head_hexsha": "ac0a27e31d64edc8ae8912b6ed10005029868c90", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/02.1_ikeda_Be_assumption.ipynb", "max_forks_repo_name": "rddaz2013/Prediction-of-roll-motion-using-fully-nonlinear-potential-flow-and-Ikedas-method", "max_forks_repo_head_hexsha": "ac0a27e31d64edc8ae8912b6ed10005029868c90", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-06-05T15:38:54.000Z", "max_forks_repo_forks_event_max_datetime": "2021-06-05T15:38:54.000Z", "avg_line_length": 100.3531300161, "max_line_length": 67776, "alphanum_fraction": 0.7430155684, "converted": true, "num_tokens": 26058, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.6926419894793248, "lm_q1q2_score": 0.41050577289013396}} {"text": "```python\n%matplotlib inline\n```\n\n\nWrite your own GNN module\n=========================\n\nSometimes, your model goes beyond simply stacking existing GNN modules.\nFor example, you would like to invent a new way of aggregating neighbor\ninformation by considering node importance or edge weights.\n\nBy the end of this tutorial you will be able to\n\n- Understand DGL\u2019s message passing APIs.\n- Implement GraphSAGE convolution module by your own.\n\nThis tutorial assumes that you already know :doc:`the basics of training a\nGNN for node classification <1_introduction>`.\n\n(Time estimate: 10 minutes)\n\n\n\n```python\nimport dgl\nimport torch\nimport torch.nn as nn\nimport torch.nn.functional as F\n```\n\n Using backend: pytorch\n\n\nMessage passing and GNNs\n------------------------\n\nDGL follows the *message passing paradigm* inspired by the Message\nPassing Neural Network proposed by `Gilmer et\nal. `__ Essentially, they found many\nGNN models can fit into the following framework:\n\n\\begin{align}m_{u\\to v}^{(l)} = M^{(l)}\\left(h_v^{(l-1)}, h_u^{(l-1)}, e_{u\\to v}^{(l-1)}\\right)\\end{align}\n\n\\begin{align}m_{v}^{(l)} = \\sum_{u\\in\\mathcal{N}(v)}m_{u\\to v}^{(l)}\\end{align}\n\n\\begin{align}h_v^{(l)} = U^{(l)}\\left(h_v^{(l-1)}, m_v^{(l)}\\right)\\end{align}\n\nwhere DGL calls $M^{(l)}$ the *message function*, $\\sum$ the\n*reduce function* and $U^{(l)}$ the *update function*. Note that\n$\\sum$ here can represent any function and is not necessarily a\nsummation.\n\n\n\n\nFor example, the `GraphSAGE convolution (Hamilton et al.,\n2017) `__\ntakes the following mathematical form:\n\n\\begin{align}h_{\\mathcal{N}(v)}^k\\leftarrow \\text{Average}\\{h_u^{k-1},\\forall u\\in\\mathcal{N}(v)\\}\\end{align}\n\n\\begin{align}h_v^k\\leftarrow \\text{ReLU}\\left(W^k\\cdot \\text{CONCAT}(h_v^{k-1}, h_{\\mathcal{N}(v)}^k) \\right)\\end{align}\n\nYou can see that message passing is directional: the message sent from\none node $u$ to other node $v$ is not necessarily the same\nas the other message sent from node $v$ to node $u$ in the\nopposite direction.\n\nAlthough DGL has builtin support of GraphSAGE via\n:class:`dgl.nn.SAGEConv `,\nhere is how you can implement GraphSAGE convolution in DGL by your own.\n\n\n\n\n\n```python\nimport dgl.function as fn\n\nclass SAGEConv(nn.Module):\n \"\"\"Graph convolution module used by the GraphSAGE model.\n \n Parameters\n ----------\n in_feat : int\n Input feature size.\n out_feat : int\n Output feature size.\n \"\"\"\n def __init__(self, in_feat, out_feat):\n super(SAGEConv, self).__init__()\n # A linear submodule for projecting the input and neighbor feature to the output.\n self.linear = nn.Linear(in_feat * 2, out_feat)\n \n def forward(self, g, h):\n \"\"\"Forward computation\n \n Parameters\n ----------\n g : Graph\n The input graph.\n h : Tensor\n The input node feature.\n \"\"\"\n with g.local_scope():\n g.ndata['h'] = h\n # update_all is a message passing API.\n g.update_all(message_func=fn.copy_u('h', 'm'), reduce_func=fn.mean('m', 'h_N'))\n h_N = g.ndata['h_N']\n h_total = torch.cat([h, h_N], dim=1)\n return self.linear(h_total)\n```\n\nThe central piece in this code is the\n:func:`g.update_all `\nfunction, which gathers and averages the neighbor features. There are\nthree concepts here:\n\n* Message function ``fn.copy_u('h', 'm')`` that\n copies the node feature under name ``'h'`` as *messages* sent to\n neighbors.\n\n* Reduce function ``fn.mean('m', 'h_N')`` that averages\n all the received messages under name ``'m'`` and saves the result as a\n new node feature ``'h_N'``.\n\n* ``update_all`` tells DGL to trigger the\n message and reduce functions for all the nodes and edges.\n\n\n\n\nAfterwards, you can stack your own GraphSAGE convolution layers to form\na multi-layer GraphSAGE network.\n\n\n\n\n\n```python\nclass Model(nn.Module):\n def __init__(self, in_feats, h_feats, num_classes):\n super(Model, self).__init__()\n self.conv1 = SAGEConv(in_feats, h_feats)\n self.conv2 = SAGEConv(h_feats, num_classes)\n \n def forward(self, g, in_feat):\n h = self.conv1(g, in_feat)\n h = F.relu(h)\n h = self.conv2(g, h)\n return h\n```\n\nTraining loop\n~~~~~~~~~~~~~\nThe following code for data loading and training loop is directly copied\nfrom the introduction tutorial.\n\n\n\n\n\n```python\nimport dgl.data\n\ndataset = dgl.data.CoraGraphDataset()\ng = dataset[0]\n\ndef train(g, model):\n optimizer = torch.optim.Adam(model.parameters(), lr=0.01)\n all_logits = []\n best_val_acc = 0\n best_test_acc = 0\n\n features = g.ndata['feat']\n labels = g.ndata['label']\n train_mask = g.ndata['train_mask']\n val_mask = g.ndata['val_mask']\n test_mask = g.ndata['test_mask']\n for e in range(200):\n # Forward\n logits = model(g, features)\n\n # Compute prediction\n pred = logits.argmax(1)\n\n # Compute loss\n # Note that we should only compute the losses of the nodes in the training set,\n # i.e. with train_mask 1.\n loss = F.cross_entropy(logits[train_mask], labels[train_mask])\n\n # Compute accuracy on training/validation/test\n train_acc = (pred[train_mask] == labels[train_mask]).float().mean()\n val_acc = (pred[val_mask] == labels[val_mask]).float().mean()\n test_acc = (pred[test_mask] == labels[test_mask]).float().mean()\n\n # Save the best validation accuracy and the corresponding test accuracy.\n if best_val_acc < val_acc:\n best_val_acc = val_acc\n best_test_acc = test_acc\n\n # Backward\n optimizer.zero_grad()\n loss.backward()\n optimizer.step()\n all_logits.append(logits.detach())\n\n if e % 5 == 0:\n print('In epoch {}, loss: {:.3f}, val acc: {:.3f} (best {:.3f}), test acc: {:.3f} (best {:.3f})'.format(\n e, loss, val_acc, best_val_acc, test_acc, best_test_acc))\n\nmodel = Model(g.ndata['feat'].shape[1], 16, dataset.num_classes)\ntrain(g, model)\n```\n\n NumNodes: 2708\n NumEdges: 10556\n NumFeats: 1433\n NumClasses: 7\n NumTrainingSamples: 140\n NumValidationSamples: 500\n NumTestSamples: 1000\n Done loading data from cached files.\n In epoch 0, loss: 1.953, val acc: 0.156 (best 0.156), test acc: 0.144 (best 0.144)\n In epoch 5, loss: 1.868, val acc: 0.304 (best 0.304), test acc: 0.298 (best 0.298)\n In epoch 10, loss: 1.706, val acc: 0.576 (best 0.576), test acc: 0.594 (best 0.594)\n In epoch 15, loss: 1.463, val acc: 0.640 (best 0.640), test acc: 0.644 (best 0.644)\n In epoch 20, loss: 1.151, val acc: 0.664 (best 0.664), test acc: 0.682 (best 0.682)\n In epoch 25, loss: 0.813, val acc: 0.700 (best 0.700), test acc: 0.718 (best 0.718)\n In epoch 30, loss: 0.514, val acc: 0.730 (best 0.730), test acc: 0.738 (best 0.738)\n In epoch 35, loss: 0.297, val acc: 0.754 (best 0.754), test acc: 0.759 (best 0.759)\n In epoch 40, loss: 0.164, val acc: 0.760 (best 0.762), test acc: 0.758 (best 0.755)\n In epoch 45, loss: 0.092, val acc: 0.758 (best 0.762), test acc: 0.754 (best 0.755)\n In epoch 50, loss: 0.055, val acc: 0.762 (best 0.762), test acc: 0.753 (best 0.755)\n In epoch 55, loss: 0.035, val acc: 0.760 (best 0.762), test acc: 0.756 (best 0.755)\n In epoch 60, loss: 0.024, val acc: 0.760 (best 0.762), test acc: 0.755 (best 0.755)\n In epoch 65, loss: 0.018, val acc: 0.758 (best 0.762), test acc: 0.754 (best 0.755)\n In epoch 70, loss: 0.014, val acc: 0.758 (best 0.762), test acc: 0.752 (best 0.755)\n In epoch 75, loss: 0.012, val acc: 0.756 (best 0.762), test acc: 0.750 (best 0.755)\n In epoch 80, loss: 0.010, val acc: 0.760 (best 0.762), test acc: 0.754 (best 0.755)\n In epoch 85, loss: 0.009, val acc: 0.760 (best 0.762), test acc: 0.753 (best 0.755)\n In epoch 90, loss: 0.008, val acc: 0.760 (best 0.762), test acc: 0.754 (best 0.755)\n In epoch 95, loss: 0.007, val acc: 0.760 (best 0.762), test acc: 0.754 (best 0.755)\n In epoch 100, loss: 0.007, val acc: 0.764 (best 0.764), test acc: 0.755 (best 0.755)\n In epoch 105, loss: 0.006, val acc: 0.764 (best 0.764), test acc: 0.754 (best 0.755)\n In epoch 110, loss: 0.006, val acc: 0.764 (best 0.764), test acc: 0.755 (best 0.755)\n In epoch 115, loss: 0.005, val acc: 0.764 (best 0.764), test acc: 0.755 (best 0.755)\n In epoch 120, loss: 0.005, val acc: 0.764 (best 0.766), test acc: 0.755 (best 0.755)\n In epoch 125, loss: 0.005, val acc: 0.764 (best 0.766), test acc: 0.757 (best 0.755)\n In epoch 130, loss: 0.004, val acc: 0.764 (best 0.766), test acc: 0.757 (best 0.755)\n In epoch 135, loss: 0.004, val acc: 0.764 (best 0.766), test acc: 0.759 (best 0.755)\n In epoch 140, loss: 0.004, val acc: 0.764 (best 0.766), test acc: 0.759 (best 0.755)\n In epoch 145, loss: 0.004, val acc: 0.764 (best 0.766), test acc: 0.759 (best 0.755)\n In epoch 150, loss: 0.004, val acc: 0.762 (best 0.766), test acc: 0.759 (best 0.755)\n In epoch 155, loss: 0.003, val acc: 0.762 (best 0.766), test acc: 0.759 (best 0.755)\n In epoch 160, loss: 0.003, val acc: 0.764 (best 0.766), test acc: 0.759 (best 0.755)\n In epoch 165, loss: 0.003, val acc: 0.764 (best 0.766), test acc: 0.758 (best 0.755)\n In epoch 170, loss: 0.003, val acc: 0.766 (best 0.766), test acc: 0.759 (best 0.755)\n In epoch 175, loss: 0.003, val acc: 0.766 (best 0.766), test acc: 0.759 (best 0.755)\n In epoch 180, loss: 0.003, val acc: 0.766 (best 0.766), test acc: 0.759 (best 0.755)\n In epoch 185, loss: 0.003, val acc: 0.766 (best 0.766), test acc: 0.758 (best 0.755)\n In epoch 190, loss: 0.002, val acc: 0.766 (best 0.766), test acc: 0.758 (best 0.755)\n In epoch 195, loss: 0.002, val acc: 0.766 (best 0.766), test acc: 0.758 (best 0.755)\n\n\nMore customization\n------------------\n\nIn DGL, we provide many built-in message and reduce functions under the\n``dgl.function`` package. You can find more details in `the API\ndoc `.\n\n\n\n\nThese APIs allow one to quickly implement new graph convolution modules.\nFor example, the following implements a new ``SAGEConv`` that aggregates\nneighbor representations using a weighted average. Note that ``edata``\nmember can hold edge features which can also take part in message\npassing.\n\n\n\n\n\n```python\nclass WeightedSAGEConv(nn.Module):\n \"\"\"Graph convolution module used by the GraphSAGE model with edge weights.\n \n Parameters\n ----------\n in_feat : int\n Input feature size.\n out_feat : int\n Output feature size.\n \"\"\"\n def __init__(self, in_feat, out_feat):\n super(WeightedSAGEConv, self).__init__()\n # A linear submodule for projecting the input and neighbor feature to the output.\n self.linear = nn.Linear(in_feat * 2, out_feat)\n \n def forward(self, g, h, w):\n \"\"\"Forward computation\n \n Parameters\n ----------\n g : Graph\n The input graph.\n h : Tensor\n The input node feature.\n w : Tensor\n The edge weight.\n \"\"\"\n with g.local_scope():\n g.ndata['h'] = h\n g.edata['w'] = w\n g.update_all(message_func=fn.u_mul_e('h', 'w', 'm'), reduce_func=fn.mean('m', 'h_N'))\n h_N = g.ndata['h_N']\n h_total = torch.cat([h, h_N], dim=1)\n return self.linear(h_total)\n```\n\nBecause the graph in this dataset does not have edge weights, we\nmanually assign all edge weights to one in the ``forward()`` function of\nthe model. You can replace it with your own edge weights.\n\n\n\n\n\n```python\nclass Model(nn.Module):\n def __init__(self, in_feats, h_feats, num_classes):\n super(Model, self).__init__()\n self.conv1 = WeightedSAGEConv(in_feats, h_feats)\n self.conv2 = WeightedSAGEConv(h_feats, num_classes)\n \n def forward(self, g, in_feat):\n h = self.conv1(g, in_feat, torch.ones(g.num_edges()).to(g.device))\n h = F.relu(h)\n h = self.conv2(g, h, torch.ones(g.num_edges()).to(g.device))\n return h\n \nmodel = Model(g.ndata['feat'].shape[1], 16, dataset.num_classes)\ntrain(g, model)\n```\n\nEven more customization by user-defined function\n------------------------------------------------\n\nDGL allows user-defined message and reduce function for the maximal\nexpressiveness. Here is a user-defined message function that is\nequivalent to ``fn.u_mul_e('h', 'w', 'm')``.\n\n\n\n\n\n```python\ndef u_mul_e_udf(edges):\n return {'m' : edges.src['h'] * edges.data['w']}\n```\n\n``edges`` has three members: ``src``, ``data`` and ``dst``, representing\nthe source node feature, edge feature, and destination node feature for\nall edges.\n\n\n\n\nYou can also write your own reduce function. For example, the following\nis equivalent to the builtin ``fn.sum('m', 'h')`` function that sums up\nthe incoming messages:\n\n\n\n\n\n```python\ndef sum_udf(nodes):\n return {'h': nodes.mailbox['m'].sum(1)}\n```\n\nIn short, DGL will group the nodes by their in-degrees, and for each\ngroup DGL stacks the incoming messages along the second dimension. You \ncan then perform a reduction along the second dimension to aggregate\nmessages.\n\nFor more details on customizing message and reduce function with\nuser-defined function, please refer to the `API\nreference `.\n\n\n\n\nBest practice of writing custom GNN modules\n-------------------------------------------\n\nDGL recommends the following practice ranked by preference:\n\n- Use ``dgl.nn`` modules.\n- Use ``dgl.nn.functional`` functions which contain lower-level complex\n operations such as computing a softmax for each node over incoming\n edges.\n- Use ``update_all`` with builtin message and reduce functions.\n- Use user-defined message or reduce functions.\n\n\n\n\nWhat\u2019s next?\n------------\n\n- `Writing Efficient Message Passing\n Code `.\n\n\n\n\n\n```python\n# Thumbnail Courtesy: Representation Learning on Networks, Jure Leskovec, WWW 2018\n# sphinx_gallery_thumbnail_path = '_static/blitz_3_message_passing.png'\n```\n", "meta": {"hexsha": "68588f0d75f0847d0bc4d8f664458565ea6673cf", "size": 20009, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "chenchencode/GCN/3_message_passing.ipynb", "max_stars_repo_name": "ChenChenGith/argoverse-api-ccuse", "max_stars_repo_head_hexsha": "2e2aed64d4a286821aece806134054c6d6e1d3cb", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "chenchencode/GCN/3_message_passing.ipynb", "max_issues_repo_name": "ChenChenGith/argoverse-api-ccuse", "max_issues_repo_head_hexsha": "2e2aed64d4a286821aece806134054c6d6e1d3cb", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "chenchencode/GCN/3_message_passing.ipynb", "max_forks_repo_name": "ChenChenGith/argoverse-api-ccuse", "max_forks_repo_head_hexsha": "2e2aed64d4a286821aece806134054c6d6e1d3cb", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 35.539964476, "max_line_length": 138, "alphanum_fraction": 0.5331101005, "converted": true, "num_tokens": 4337, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665855647395, "lm_q2_score": 0.6926419894793246, "lm_q1q2_score": 0.41050576292347957}} {"text": "# training an unsupervised VAE with APOGEE DR14 spectra\nthis notebooke takes you through the building and training of a fairly deep VAE. I have not actually done too much work with DR14, so it may pose some potential difficulties, but this should be a good start. The training is suited to be put into a python script and run from the command line.\n\nIf you're inclined, you may want to experiment with the model architecture, but I'm pretty sure this will work.\n\n\n```python\nimport numpy as np\nimport time\nimport h5py\nimport keras\nimport matplotlib.pyplot as plt\n\nimport sys\n\nfrom keras.layers import (Input, Dense, Lambda, Flatten, Reshape, BatchNormalization, Activation, \n Dropout, Conv1D, UpSampling1D, MaxPooling1D, ZeroPadding1D, LeakyReLU)\nfrom keras.engine.topology import Layer\nfrom keras.optimizers import Adam\nfrom keras.models import Model\nfrom keras import backend as K\n\nplt.switch_backend('agg')\n```\n\n Using TensorFlow backend.\n\n\n### load data\nthis is file dependent\n\nthis particular function expects the aspcap dr14 h5 file that can be downloaded from vos:starnet/public\n\n\n```python\n# Define edges of detectors (for APOGEE)\nblue_chip_begin = 322\nblue_chip_end = 3242\ngreen_chip_begin = 3648\ngreen_chip_end = 6048 \nred_chip_begin = 6412\nred_chip_end = 8306 \n\n# function for loading data\ndef load_train_data_weighted(data_file,indices=None):\n \n # grab all\n if indices is None:\n with h5py.File(data_file,\"r\") as F:\n ap_spectra = F['spectrum'][:]\n ap_err_spectra = F['error_spectrum'][:]\n # grab a batch\n else: \n with h5py.File(data_file, \"r\") as F:\n indices_bool = np.ones((len(F['spectrum']),),dtype=bool)\n indices_bool[:] = False\n indices_bool[indices] = True \n\n ap_spectra = F['spectrum'][indices_bool,:]\n ap_err_spectra = F['error_spectrum'][indices_bool,:]\n \n # combine chips \n ap_spectra = np.hstack((ap_spectra[:,blue_chip_begin:blue_chip_end],\n ap_spectra[:,green_chip_begin:green_chip_end],\n ap_spectra[:,red_chip_begin:red_chip_end]))\n # set nan values to zero\n ap_spectra[np.isnan(ap_spectra)]=0.\n\n\n ap_err_spectra = np.hstack((ap_err_spectra[:,blue_chip_begin:blue_chip_end],\n ap_err_spectra[:,green_chip_begin:green_chip_end],\n ap_err_spectra[:,red_chip_begin:red_chip_end]))\n \n \n\n return ap_spectra,ap_err_spectra\n```\n\n\n```python\n# function for reshaping spectra into appropriate format for CNN\ndef cnn_reshape(spectra):\n return spectra.reshape(spectra.shape[0],spectra.shape[1],1)\n```\n\nset some model hyper-parameters\n\n\n```python\nimg_cols, img_chns = 7214, 1 \nnum_fluxes=7214\ninput_shape=(num_fluxes,1)\n\n# z_dims is the dimension of the latent space\nz_dims = 64\nbatch_size = 64\nepsilon_std = 1.0\nlearning_rate = 0.001\ndecay = 0.0\npadding=u'same'\nkernel_init = keras.initializers.RandomNormal(mean=0.0, stddev=0.01)\nbias_init = keras.initializers.Zeros()\n```\n\n## Zero-Augmentation\n\n\nIn an effort to evenly distribute the weighting of the VAE, throughout training, a *zero-augmentation* technique was applied to the training spectra samples - both synthetic and observed. The zero-augmentation is implemented as the first layer in the encoder where a zero-augmentation mask is sent as an input along with the input spectrum and the two are multiplied together. The zero-augmentation mask is the same size as the input spectrum vector and is composed of ones and zeros. For the APOGEE wave-grid, the spectral region is divided into seven *chunks* and for each input spectrum a random 0-3 of these chunks are assigned to be zeros while the remainder of the zero-augmentation mask is made up of ones. This means for a given spectrum, the input for training may be 4/7ths, 5/7ths, 6/7ths, or the entire spectrum. This augmentation is done randomly throughout training, meaning that each spectrum will be randomly assigned a different zero-augmentation mask at every iteration.\n\n\n```python\n# zero-augmentation layer (a trick I use to input chunks of zeros into the input spectra)\nclass ZeroAugmentLayer(Layer):\n def __init__(self, **kwargs):\n self.is_placeholder = True\n super(ZeroAugmentLayer, self).__init__(**kwargs)\n\n def zero_agument(self, x_real, zero_mask):\n return x_real*zero_mask\n\n def call(self, inputs):\n x_real = inputs[0]\n zero_mask = inputs[1]\n x_augmented = self.zero_agument(x_real, zero_mask)\n\n return x_augmented\n\n# a function for creating the zero-masks used during training\ndef create_zero_mask(spectra,min_chunks,max_chunks,chunk_size,dataset=None,ones_padded=False):\n if dataset is None:\n zero_mask = np.ones_like(spectra)\n elif dataset=='apogee':\n zero_mask = np.ones((spectra.shape[0],7214))\n elif dataset=='segue':\n zero_mask = np.ones((spectra.shape[0],3688))\n \n num_spec = zero_mask.shape[0]\n len_spec = zero_mask.shape[1]\n num_bins = len_spec/chunk_size\n remainder = len_spec%chunk_size\n spec_sizes = np.array([chunk_size for i in range(num_bins)])\n spec_sizes[-1]=spec_sizes[-1]+remainder\n \n\n num_bins_removed = np.random.randint(min_chunks,max_chunks+1,size=(num_spec,))\n for i, mask in enumerate(zero_mask):\n bin_indx_removed = np.random.choice(num_bins, num_bins_removed[i], replace=False)\n for indx in bin_indx_removed:\n if indx==0:\n mask[indx*spec_sizes[indx]:(indx+1)*spec_sizes[indx]]=0.\n else:\n mask[indx*spec_sizes[indx-1]:indx*spec_sizes[indx-1]+spec_sizes[indx]]=0.\n\n return zero_mask\n```\n\n# build encoder\ntakes spectra (x) and zero-augmentation mask as inputs and outputs latent distribution (z_mean and z_log_var)\n\n\n```python\ndef build_encoder(input_1,input_2):\n \n # zero-augment input spectrum\n x = ZeroAugmentLayer()([input_1,input_2])\n \n # first conv block\n x = Conv1D(filters=16, kernel_size=8, strides=1, kernel_initializer=kernel_init,\n bias_initializer=bias_init, padding=padding)(x)\n x = LeakyReLU(0.1)(x)\n x = BatchNormalization()(x)\n x = Dropout(0.2)(x)\n \n # second conv bloack\n x = Conv1D(filters=16, kernel_size=8, strides=1, kernel_initializer=kernel_init,\n bias_initializer=bias_init, padding=padding)(x)\n x = LeakyReLU(0.1)(x)\n x = BatchNormalization()(x)\n \n # maxpooling layer and flatten\n x = MaxPooling1D(pool_size=4, strides=4, padding='valid')(x) \n x = Flatten()(x)\n x = Dropout(0.2)(x)\n \n # intermediate dense block\n x = Dense(256)(x)\n x = LeakyReLU(0.3)(x)\n x = BatchNormalization()(x)\n x = Dropout(0.3)(x)\n \n # latent distribution output\n z_mean = Dense(z_dims)(x)\n z_log_var = Dense(z_dims)(x)\n \n \n return Model([input_1,input_2],[z_mean,z_log_var])\n\n\n# function for obtaining a latent sample given a distribution\ndef sampling(args, latent_dim=z_dims, epsilon_std=epsilon_std):\n z_mean, z_log_var = args\n \n epsilon = K.random_normal(shape=(z_dims,),\n mean=0., stddev=epsilon_std)\n \n return z_mean + K.exp(z_log_var) * epsilon\n```\n\n# build decoder\ntakes z (latent variables) as an input and outputs a stellar spectrum\n\n\n```python\ndef build_decoder(inputs):\n \n # input fully-connected block\n x = Dense(256)(inputs)\n x = LeakyReLU(0.1)(x)\n x = BatchNormalization()(x)\n x = Dropout(0.2)(x)\n \n # intermediate fully-connected block\n w = input_shape[0] // (2 ** 3)\n x = Dense(w * 16)(x)\n x = LeakyReLU(0.1)(x)\n x = BatchNormalization()(x)\n x = Dropout(0.2)(x)\n \n # reshape for convolutional blocks\n x = Reshape((w, 16))(x)\n\n # first deconv block\n x = UpSampling1D(size=4)(x)\n x = Conv1D(kernel_initializer=kernel_init,bias_initializer=bias_init,padding=\"same\", \n filters=16,kernel_size=8)(x)\n x = LeakyReLU(0.1)(x)\n x = BatchNormalization()(x)\n x = Dropout(0.1)(x)\n \n # zero-padding to get x in the right dimension to create the spectra\n x = ZeroPadding1D(padding=(2,1))(x)\n \n # second deconv block\n x = UpSampling1D(size=2)(x)\n x = Conv1D(kernel_initializer=kernel_init,bias_initializer=bias_init,padding=\"same\", \n filters=16,kernel_size=8)(x)\n x = LeakyReLU(0.1)(x)\n x = BatchNormalization()(x)\n \n # output conv layer\n x = Conv1D(kernel_initializer=kernel_init,bias_initializer=bias_init,padding=\"same\", \n filters=1,kernel_size=8,activation='linear')(x)\n\n return Model(inputs,x)\n```\n\n## build models\n\n\n```python\n# encoder and predictor input placeholders\ninput_spec = Input(shape=input_shape)\ninput_mask = Input(shape=input_shape)\n\n# error spectra placeholder\ninput_err_spec = Input(shape=input_shape)\n\n# decoder input placeholder\ninput_z = Input(shape=(z_dims,))\n\nmodel_name='vae_test'\nstart_e = 0\n\n# if you want to continue training from a certain epoch, you can uncomment the load models lines \n# and comment out the build_encoder, build_decoder lines\n'''\nencoder = keras.models.load_model('models/encoder_'+model_name+'_epoch_'+str(start_e)+'.h5',\n custom_objects={'ZeroAugmentLayer':ZeroAugmentLayer})\ndecoder = keras.models.load_model('models/decoder_'+model_name+'_epoch_'+str(start_e)+'.h5',\n custom_objects={'ZeroAugmentLayer':ZeroAugmentLayer})\n\n'''\n# encoder model\nencoder = build_encoder(input_spec, input_mask)\n\n# decoder layers\ndecoder = build_decoder(input_z)\n#'''\n\nencoder.summary()\ndecoder.summary()\n\n\n# outputs for encoder\nz_mean, z_log_var = encoder([input_spec, input_mask])\n\n# sample from latent distribution given z_mean and z_log_var\nz = Lambda(sampling, output_shape=(z_dims,))([z_mean, z_log_var])\n\n# outputs for decoder\noutput_spec = decoder(z)\n\n```\n\n __________________________________________________________________________________________________\n Layer (type) Output Shape Param # Connected to \n ==================================================================================================\n input_1 (InputLayer) (None, 7214, 1) 0 \n __________________________________________________________________________________________________\n input_2 (InputLayer) (None, 7214, 1) 0 \n __________________________________________________________________________________________________\n zero_augment_layer_1 (ZeroAugme [(None, 7214, 1), (N 0 input_1[0][0] \n input_2[0][0] \n __________________________________________________________________________________________________\n conv1d_1 (Conv1D) (None, 7214, 16) 144 zero_augment_layer_1[0][0] \n __________________________________________________________________________________________________\n leaky_re_lu_1 (LeakyReLU) (None, 7214, 16) 0 conv1d_1[0][0] \n __________________________________________________________________________________________________\n batch_normalization_1 (BatchNor (None, 7214, 16) 64 leaky_re_lu_1[0][0] \n __________________________________________________________________________________________________\n dropout_1 (Dropout) (None, 7214, 16) 0 batch_normalization_1[0][0] \n __________________________________________________________________________________________________\n conv1d_2 (Conv1D) (None, 7214, 16) 2064 dropout_1[0][0] \n __________________________________________________________________________________________________\n leaky_re_lu_2 (LeakyReLU) (None, 7214, 16) 0 conv1d_2[0][0] \n __________________________________________________________________________________________________\n batch_normalization_2 (BatchNor (None, 7214, 16) 64 leaky_re_lu_2[0][0] \n __________________________________________________________________________________________________\n max_pooling1d_1 (MaxPooling1D) (None, 1803, 16) 0 batch_normalization_2[0][0] \n __________________________________________________________________________________________________\n flatten_1 (Flatten) (None, 28848) 0 max_pooling1d_1[0][0] \n __________________________________________________________________________________________________\n dropout_2 (Dropout) (None, 28848) 0 flatten_1[0][0] \n __________________________________________________________________________________________________\n dense_1 (Dense) (None, 256) 7385344 dropout_2[0][0] \n __________________________________________________________________________________________________\n leaky_re_lu_3 (LeakyReLU) (None, 256) 0 dense_1[0][0] \n __________________________________________________________________________________________________\n batch_normalization_3 (BatchNor (None, 256) 1024 leaky_re_lu_3[0][0] \n __________________________________________________________________________________________________\n dropout_3 (Dropout) (None, 256) 0 batch_normalization_3[0][0] \n __________________________________________________________________________________________________\n dense_2 (Dense) (None, 64) 16448 dropout_3[0][0] \n __________________________________________________________________________________________________\n dense_3 (Dense) (None, 64) 16448 dropout_3[0][0] \n ==================================================================================================\n Total params: 7,421,600\n Trainable params: 7,421,024\n Non-trainable params: 576\n __________________________________________________________________________________________________\n _________________________________________________________________\n Layer (type) Output Shape Param # \n =================================================================\n input_4 (InputLayer) (None, 64) 0 \n _________________________________________________________________\n dense_4 (Dense) (None, 256) 16640 \n _________________________________________________________________\n leaky_re_lu_4 (LeakyReLU) (None, 256) 0 \n _________________________________________________________________\n batch_normalization_4 (Batch (None, 256) 1024 \n _________________________________________________________________\n dropout_4 (Dropout) (None, 256) 0 \n _________________________________________________________________\n dense_5 (Dense) (None, 14416) 3704912 \n _________________________________________________________________\n leaky_re_lu_5 (LeakyReLU) (None, 14416) 0 \n _________________________________________________________________\n batch_normalization_5 (Batch (None, 14416) 57664 \n _________________________________________________________________\n dropout_5 (Dropout) (None, 14416) 0 \n _________________________________________________________________\n reshape_1 (Reshape) (None, 901, 16) 0 \n _________________________________________________________________\n up_sampling1d_1 (UpSampling1 (None, 3604, 16) 0 \n _________________________________________________________________\n conv1d_3 (Conv1D) (None, 3604, 16) 2064 \n _________________________________________________________________\n leaky_re_lu_6 (LeakyReLU) (None, 3604, 16) 0 \n _________________________________________________________________\n batch_normalization_6 (Batch (None, 3604, 16) 64 \n _________________________________________________________________\n dropout_6 (Dropout) (None, 3604, 16) 0 \n _________________________________________________________________\n zero_padding1d_1 (ZeroPaddin (None, 3607, 16) 0 \n _________________________________________________________________\n up_sampling1d_2 (UpSampling1 (None, 7214, 16) 0 \n _________________________________________________________________\n conv1d_4 (Conv1D) (None, 7214, 16) 2064 \n _________________________________________________________________\n leaky_re_lu_7 (LeakyReLU) (None, 7214, 16) 0 \n _________________________________________________________________\n batch_normalization_7 (Batch (None, 7214, 16) 64 \n _________________________________________________________________\n conv1d_5 (Conv1D) (None, 7214, 1) 129 \n =================================================================\n Total params: 3,784,625\n Trainable params: 3,755,217\n Non-trainable params: 29,408\n _________________________________________________________________\n\n\n## create loss function\n\nThis VAE has two loss functions that are minimized simultaneously:\n\n1. a weighted mean-squared-error to analyze the predicted spectra:\n\\begin{equation}\nmse = \\frac{1}{N}\\sum{\\frac{(x_{true}-x_{pred})^2}{(x_{err})^2}}\n\\end{equation}\n\n\n\n\n2. a relative entropy, KL (Kullback\u2013Leibler divergence) loss to keep the latent variables within a similar distribuition:\n\\begin{equation}\nKL = \\frac{1}{N}\\sum{-\\frac{1}{2}(1.0+z_{log\\_var} - z_{avg}^2 - e^{z_{log\\_var}})}\n\\end{equation}\n\n\n\n```python\n# loss for evaluating the regenerated spectra and the latent distribution\nclass VAE_LossLayer_weighted(Layer):\n __name__ = u'vae_labeled_loss_layer'\n\n def __init__(self, **kwargs):\n self.is_placeholder = True\n super(VAE_LossLayer_weighted, self).__init__(**kwargs)\n\n def lossfun(self, x_true, x_pred, z_avg, z_log_var, x_err):\n mse = K.mean(K.square((x_true - x_pred)/x_err))\n kl_loss_x = K.mean(-0.5 * K.sum(1.0 + z_log_var - K.square(z_avg) - K.exp(z_log_var), axis=-1))\n\n return mse + kl_loss_x\n\n def call(self, inputs):\n # inputs for the layer:\n x_true = inputs[0]\n x_pred = inputs[1]\n z_avg = inputs[2]\n z_log_var = inputs[3]\n x_err = inputs[4]\n \n # calculate loss\n loss = self.lossfun(x_true, x_pred, z_avg, z_log_var, x_err)\n \n # add loss to model\n self.add_loss(loss, inputs=inputs)\n \n # returned value not really used for anything\n return x_true\n \n# dummy loss to give zeros, hence no gradients to train\n# the real loss is computed as the layer shown above and therefore this dummy loss is just \n# used to satisfy keras notation when compiling the model\ndef zero_loss(y_true, y_pred):\n return K.zeros_like(y_true)\n```\n\n## build and compile model trainer\n\n\n```python\n# create loss layer\nvae_loss = VAE_LossLayer_weighted()([input_spec, output_spec, z_mean, z_log_var, input_err_spec])\n\n# build trainer with spectra, zero-masks, and error spectra as inputs. output is the final loss layer\nvae = Model(inputs=[input_spec, input_mask, input_err_spec], outputs=[vae_loss])\n\n# compile trainer\nvae.compile(loss=[zero_loss],\n optimizer=Adam(lr=1.0e-4, beta_1=0.5))\n\nvae.summary()\n```\n\n __________________________________________________________________________________________________\n Layer (type) Output Shape Param # Connected to \n ==================================================================================================\n input_1 (InputLayer) (None, 7214, 1) 0 \n __________________________________________________________________________________________________\n input_2 (InputLayer) (None, 7214, 1) 0 \n __________________________________________________________________________________________________\n model_1 (Model) [(None, 64), (None, 7421600 input_1[0][0] \n input_2[0][0] \n __________________________________________________________________________________________________\n lambda_1 (Lambda) (None, 64) 0 model_1[1][0] \n model_1[1][1] \n __________________________________________________________________________________________________\n model_2 (Model) (None, 7214, 1) 3784625 lambda_1[0][0] \n __________________________________________________________________________________________________\n input_3 (InputLayer) (None, 7214, 1) 0 \n __________________________________________________________________________________________________\n vae__loss_layer_weighted_1 (VAE [(None, 7214, 1), (N 0 input_1[0][0] \n model_2[1][0] \n model_1[1][0] \n model_1[1][1] \n input_3[0][0] \n ==================================================================================================\n Total params: 11,206,225\n Trainable params: 11,176,241\n Non-trainable params: 29,984\n __________________________________________________________________________________________________\n\n\n\n```python\n# a model that encodes and then decodes a spectrum (this is used to plot the intermediate results during training)\ngen_x_to_x = Model([input_spec,input_mask], output_spec)\ngen_x_to_x.compile(loss='mse',\n optimizer=Adam(lr=1.0e-4, beta_1=0.5))\n```\n\n\n```python\n# a function to display the time remaining or elapsed\ndef time_format(t):\n m, s = divmod(t, 60)\n m = int(m)\n s = int(s)\n if m == 0:\n return u'%d sec' % s\n else:\n return u'%d min' % (m)\n \n \n \n# function for training on a batch\ndef train_on_batch(x_batch,x_err_batch): \n \n # create zero-augmentation mask for batch\n zero_mask = create_zero_mask(x_batch,0,3,1030,dataset=None,ones_padded=False)\n \n # train on batch\n loss = [vae.train_on_batch([cnn_reshape(x_batch),\n cnn_reshape(zero_mask),cnn_reshape(x_err_batch)], \n cnn_reshape(x_batch))]\n\n losses = {'vae_loss': loss[0]}\n return losses\n```\n\n\n```python\ndef fit_model(model_name, data_file, epochs, reporter):\n \n # get the number of spectra in the data_file\n with h5py.File(data_file, \"r\") as F:\n num_data_ap = len(F['spectrum'])\n \n # lets use 90% of the samples for training\n num_data_train_ap = int(num_data_ap*0.9)\n \n # the remainder will be grabbed for testing the model throughout training\n test_indices_range_ap = [num_data_train_ap,num_data_ap]\n \n \n # loop through the number of epochs\n for e in xrange(start_e,epochs):\n \n # create a randomized array of indices to grab batches of the spectra\n perm_ap = np.random.permutation(num_data_train_ap) \n \n start_time = time.time()\n \n # loop through the batches \n losses_=[]\n for b in xrange(0, num_data_train_ap, batchsize): \n \n # determine current batch size\n bsize = min(batchsize, num_data_train_ap - b)\n \n # grab a batch of indices\n indx_batch = perm_ap[b:b+bsize]\n \n # load a batch of data\n x_batch, x_err_batch= load_train_data_weighted(data_file,indices=indx_batch)\n \n # train on batch\n losses = train_on_batch(x_batch,x_err_batch)\n \n losses_.append(losses)\n \n # Print current status\n ratio = 100.0 * (b + bsize) / num_data_train_ap\n print unichr(27) + u\"[2K\",; sys.stdout.write(u'')\n print u'\\rEpoch #%d | %d / %d (%6.2f %%) ' % \\\n (e + 1, b + bsize, num_data_train_ap, ratio),; sys.stdout.write(u'')\n\n for k in reporter:\n if k in losses:\n print u'| %s = %5.3f ' % (k, losses[k]),; sys.stdout.write(u'')\n\n # Compute ETA\n elapsed_time = time.time() - start_time\n eta = elapsed_time / (b + bsize) * (num_data_train_ap - (b + bsize))\n print u'| ETA: %s ' % time_format(eta),; sys.stdout.write(u'')\n\n sys.stdout.flush()\n\n \n \n print u''\n \n # Print epoch status\n ratio = 100.0\n print unichr(27) + u\"[2K\",; sys.stdout.write(u'')\n print u'\\rEpoch #%d | %d / %d (%6.2f %%) ' % \\\n (e + 1, num_data_train_ap, num_data_train_ap, ratio),; sys.stdout.write(u'')\n\n losses_all = {}\n for k in losses_[0].iterkeys():\n losses_all[k] = tuple(d[k] for d in losses_)\n for k in reporter:\n if k in losses_all:\n losses_all[k]=np.sum(losses_all[k])/len(losses_)\n for k in reporter:\n if k in losses_all:\n print u'| %s = %5.3f ' % (k, losses_all[k]),; sys.stdout.write(u'')\n \n\n # save loss to evaluate progress \n myfile = open(model_name+'.txt', 'a')\n for k in reporter:\n if k in losses:\n myfile.write(\"%s,\" % losses[k])\n myfile.write(\"\\n\")\n myfile.close() \n \n # Compute Time Elapsed\n elapsed_time = time.time() - start_time\n eta = elapsed_time\n print u'| TE: %s ' % time_format(eta),; sys.stdout.write(u'')\n\n #sys.stdout.flush()\n print('\\n')\n \n \n # save models\n encoder.save('models/encoder_'+model_name+'_epoch_'+str(e)+'.h5')\n decoder.save('models/decoder_'+model_name+'_epoch_'+str(e)+'.h5')\n \n \n # plot results for a test set to evaluate how the vae is able to reproduce a spectrum\n test_sample_indices = np.random.choice(range(test_indices_range_ap[0],test_indices_range_ap[1]), 5, replace=False)\n sample_orig,_, = load_train_data_weighted(data_file,indices=test_sample_indices)\n zero_mask_test = create_zero_mask(sample_orig,0,3,1030)\n test_x = gen_x_to_x.predict([cnn_reshape(sample_orig),cnn_reshape(zero_mask_test)])\n sample_orig_aug = sample_orig*zero_mask_test\n \n sample_diff = sample_orig-test_x.reshape(test_x.shape[0],test_x.shape[1])\n\n \n # save test results\n fig, axes = plt.subplots(20,1,figsize=(70, 20))\n for i in range(len(test_sample_indices)):\n # original spectrum\n axes[i*4].plot(sample_orig[i],c='r')\n axes[i*4].set_ylim((0.4,1.2))\n # input zero-augmented spectrum\n axes[1+4*i].plot(sample_orig_aug[i],c='g')\n axes[1+4*i].set_ylim((0.4,1.2))\n # regenerated spectrum\n axes[2+4*i].plot(test_x[i],c='b')\n axes[2+4*i].set_ylim((0.4,1.2))\n # residual between original and regenerated spectra\n axes[3+4*i].plot(sample_diff[i],c='m')\n axes[3+4*i].set_ylim((-0.3,0.3))\n # save results\n plt.savefig('results/test_sample_ap_'+model_name+'_epoch_'+str(e)+'.jpg')\n plt.close('all')\n```\n\n# train model\n\nyou can experiment with the number of epochs. I suggest starting with fewer and seeing if the results are adequate. if not, continue training. The models and results are saved in models/ and results/ after each epoch, so you can run analyses throughout training.\n\n\n```python\nreporter=['vae_loss']\nepochs=30\nbatchsize=64\nif start_e>0:\n start_e=start_e+1\n\ndata_file = '/data/stars/aspcapStar_combined_main_dr14.h5'\n\nfit_model(model_name,data_file, epochs,reporter)\n```\n\n Epoch #1 | 24896 / 233481 ( 10.66 %) | vae_loss = 5541.463 | ETA: 201 min [2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \u001b[2K \n\n## analyze results\n\nNote, this is a dummy result. I haven't trained the models for a proper epoch yet\n\n\n```python\nimport numpy as np\nimport h5py\nimport keras\nimport matplotlib.pyplot as plt\n\nimport sys\n\nfrom keras.layers import (Input, Lambda)\nfrom keras.engine.topology import Layer\nfrom keras import backend as K\n\n\n%matplotlib inline\n```\n\n Using TensorFlow backend.\n\n\n\n```python\n# Define edges of detectors (for APOGEE)\nblue_chip_begin = 322\nblue_chip_end = 3242\ngreen_chip_begin = 3648\ngreen_chip_end = 6048 \nred_chip_begin = 6412\nred_chip_end = 8306 \n\n# function for loading data\ndef load_train_data_weighted(data_file,indices=None):\n \n # grab all\n if indices is None:\n with h5py.File(data_file,\"r\") as F:\n ap_spectra = F['spectrum'][:]\n ap_err_spectra = F['error_spectrum'][:]\n # grab a batch\n else: \n with h5py.File(data_file, \"r\") as F:\n indices_bool = np.ones((len(F['spectrum']),),dtype=bool)\n indices_bool[:] = False\n indices_bool[indices] = True \n\n ap_spectra = F['spectrum'][indices_bool,:]\n ap_err_spectra = F['error_spectrum'][indices_bool,:]\n \n # combine chips \n ap_spectra = np.hstack((ap_spectra[:,blue_chip_begin:blue_chip_end],\n ap_spectra[:,green_chip_begin:green_chip_end],\n ap_spectra[:,red_chip_begin:red_chip_end]))\n # set nan values to zero\n ap_spectra[np.isnan(ap_spectra)]=0.\n\n\n ap_err_spectra = np.hstack((ap_err_spectra[:,blue_chip_begin:blue_chip_end],\n ap_err_spectra[:,green_chip_begin:green_chip_end],\n ap_err_spectra[:,red_chip_begin:red_chip_end]))\n \n \n\n return ap_spectra,ap_err_spectra\n\n\n# zero-augmentation layer (a trick I use to input chunks of zeros into the input spectra)\nclass ZeroAugmentLayer(Layer):\n def __init__(self, **kwargs):\n self.is_placeholder = True\n super(ZeroAugmentLayer, self).__init__(**kwargs)\n\n def zero_agument(self, x_real, zero_mask):\n return x_real*zero_mask\n\n def call(self, inputs):\n x_real = inputs[0]\n zero_mask = inputs[1]\n x_augmented = self.zero_agument(x_real, zero_mask)\n\n return x_augmented\n\n# a function for creating the zero-masks used during training\ndef create_zero_mask(spectra,min_chunks,max_chunks,chunk_size,dataset=None,ones_padded=False):\n if dataset is None:\n zero_mask = np.ones_like(spectra)\n elif dataset=='apogee':\n zero_mask = np.ones((spectra.shape[0],7214))\n elif dataset=='segue':\n zero_mask = np.ones((spectra.shape[0],3688))\n \n num_spec = zero_mask.shape[0]\n len_spec = zero_mask.shape[1]\n num_bins = len_spec/chunk_size\n remainder = len_spec%chunk_size\n spec_sizes = np.array([chunk_size for i in range(num_bins)])\n spec_sizes[-1]=spec_sizes[-1]+remainder\n \n\n num_bins_removed = np.random.randint(min_chunks,max_chunks+1,size=(num_spec,))\n for i, mask in enumerate(zero_mask):\n bin_indx_removed = np.random.choice(num_bins, num_bins_removed[i], replace=False)\n for indx in bin_indx_removed:\n if indx==0:\n mask[indx*spec_sizes[indx]:(indx+1)*spec_sizes[indx]]=0.\n else:\n mask[indx*spec_sizes[indx-1]:indx*spec_sizes[indx-1]+spec_sizes[indx]]=0.\n\n return zero_mask\n\n# function for reshaping spectra into appropriate format for CNN\ndef cnn_reshape(spectra):\n return spectra.reshape(spectra.shape[0],spectra.shape[1],1)\n```\n\n\n```python\nlosses = np.zeros((1,))\nwith open(\"vae_test.txt\", \"r\") as f:\n for i,line in enumerate(f):\n currentline = np.array(line.split(\",\")[0],dtype=float)\n if i ==0:\n losses[0]=currentline.reshape((1,))\n else:\n losses = np.hstack((losses,currentline.reshape((1,))))\n```\n\n\n```python\nplt.plot(losses[0:16],label='vae_loss')\nplt.legend()\nplt.show()\n```\n\n\n```python\n# function for encoding a spectrum into the latent space\ndef encode_spectrum(model_name,epoch,spectra):\n\n encoder = keras.models.load_model('models/encoder_'+model_name+'_epoch_'+str(epoch)+'.h5',\n custom_objects={'ZeroAugmentLayer':ZeroAugmentLayer})\n z_avg,z_log_var = encoder.predict([cnn_reshape(spectra),cnn_reshape(np.ones_like(spectra))])\n \n return z_avg, z_log_var\n```\n\n\n```python\ndata_file = '/data/stars/aspcapStar_combined_main_dr14.h5'\n\ntest_range = [0,30000]\n\n\ntest_sample_indices = np.random.choice(range(0,30000), 5000, replace=False)\nsample_x,_, = load_train_data_weighted(data_file,indices=test_sample_indices) \n```\n\n\n```python\nmodel_name = 'vae_test'\nepoch=16\n\nz_avg, z_log_var = encode_spectrum(model_name,epoch,sample_x)\n```\n\n /opt/conda/lib/python2.7/site-packages/keras/models.py:252: UserWarning: No training configuration found in save file: the model was *not* compiled. Compile it manually.\n warnings.warn('No training configuration found in save file: '\n\n\n## t-sne\n\nan example of an unsupervised clustering method on the latent space.\n\n\n```python\nfrom tsne import bh_sne\n\n\nperplex=80\n\nt_data = z_avg\n\n# convert data to float64 matrix. float64 is need for bh_sne\nt_data = np.asarray(t_data).astype('float64')\nt_data = t_data.reshape((t_data.shape[0], -1))\n```\n\n\n```python\n# perform t-SNE embedding\nvis_data = bh_sne(t_data, perplexity=perplex)\n```\n\n\n```python\n# separate 2D into x and y axes information\nvis_x = vis_data[:, 0]\nvis_y = vis_data[:, 1]\n```\n\n\n```python\nfig = plt.figure(figsize=(10, 10))\nsynth_ap = plt.scatter(vis_x, vis_y, marker='o', c='r',label='APOGEE', alpha=0.4)\n\n\nplt.tick_params(\n axis='x', \n which='both', \n bottom='off', \n top='off', \n labelbottom='off')\n\nplt.tick_params(\n axis='y', \n which='both', \n right='off', \n left='off', \n labelleft='off')\n\nplt.legend(fontsize=30)\nplt.tight_layout()\nplt.show()\n```\n\nyou could also use a colour gradient based on, say, the star's Teff to see how hot and cool stars clump together\n\n\n```python\n\n```\n", "meta": {"hexsha": "3d3d190a35b9b83e9eceea5f99075ee263c6a359", "size": 288830, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "VAE/StarNet_VAE.ipynb", "max_stars_repo_name": "sandeepmbm/starnet", "max_stars_repo_head_hexsha": "60ca225f73ea582c661869774887f1cc848d50f5", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": 19, "max_stars_repo_stars_event_min_datetime": "2017-08-11T19:19:47.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-07T05:45:35.000Z", "max_issues_repo_path": "VAE/StarNet_VAE.ipynb", "max_issues_repo_name": "sandeepmbm/starnet", "max_issues_repo_head_hexsha": "60ca225f73ea582c661869774887f1cc848d50f5", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2018-06-07T22:53:51.000Z", "max_issues_repo_issues_event_max_datetime": "2018-06-07T22:53:51.000Z", "max_forks_repo_path": "VAE/StarNet_VAE.ipynb", "max_forks_repo_name": "sandeepmbm/starnet", "max_forks_repo_head_hexsha": "60ca225f73ea582c661869774887f1cc848d50f5", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": 7, "max_forks_repo_forks_event_min_datetime": "2018-01-23T00:18:24.000Z", "max_forks_repo_forks_event_max_datetime": "2021-06-21T14:09:15.000Z", "avg_line_length": 238.899917287, "max_line_length": 226912, "alphanum_fraction": 0.8904788284, "converted": true, "num_tokens": 10202, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419704455589, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.4105057616094567}} {"text": "# Surfinpy\n\n#### Tutorial 3 - Pressure\n\nChemical potential can be converted to pressure values using\n\n\\begin{align}\nP & = \\frac{\\mu_O}{k_B T} ,\n\\end{align}\n\nwhere P is the pressure, $\\mu$ is the chemical potential of oxygen, $k_B$ is the Boltzmnann constant and T is the temperature. \n\n\n```python\nimport matplotlib.pyplot as plt\nfrom surfinpy import mu_vs_mu\nfrom surfinpy import utils as ut\nfrom surfinpy import data\n```\n\n\n```python\nOxygen_exp = ut.fit_nist(\"O2.txt\")[298]\nWater_exp = ut.fit_nist(\"H2O.txt\")[298]\nOxygen_corrected = (-9.08 + -0.86 + Oxygen_exp) / 2 \nWater_corrected = -14.84 + 0.55 + Water_exp\n```\n\n\n```python\nbulk = data.ReferenceDataSet(cation = 1, anion = 2, energy = -780.0, funits = 4)\n\npure = data.DataSet(cation = 24, x = 48, y = 0, area = 60.0, energy = -575.0, label = \"0.00 $TiO_2$\", nspecies = 1)\nH2O = data.DataSet(cation = 24, x = 48, y = 2, area = 60.0, energy = -612.0, label = \"0.16 $TiO_2$\", nspecies = 1)\nH2O_2 = data.DataSet(cation = 24, x = 48, y = 4, area = 60.0, energy = -640.0, label = \"0.32 $TiO_2$\", nspecies = 1)\nH2O_3 = data.DataSet(cation = 24, x = 48, y = 8, area = 60.0, energy = -676.0, label = \"0.64 $TiO_2$\", nspecies = 1)\nVo = data.DataSet(cation = 24, x = 46, y = 0, area = 60.0, energy = -558.0, label = \"0.00 $TiO_1.9$\", nspecies = 1)\nH2O_Vo_1 = data.DataSet(cation = 24, x = 46, y = 2, area = 60.0, energy = -594.0, label = \"0.00 $TiO_1.9$\", nspecies = 1)\nH2O_Vo_2 = data.DataSet(cation = 24, x = 46, y = 4, area = 60.0, energy = -624.0, label = \"0.16 $TiO_1.9$\", nspecies = 1)\nH2O_Vo_3 = data.DataSet(cation = 24, x = 46, y = 6, area = 60.0, energy = -640.0, label = \"0.32 $TiO_1.9$\", nspecies = 1)\nH2O_Vo_4 = data.DataSet(cation = 24, x = 46, y = 8, area = 60.0, energy = -670.0, label = \"0.64 $TiO_1.9$\", nspecies = 1)\n\ndata = [pure, Vo, H2O, H2O_Vo_1, H2O_2, H2O_Vo_2, H2O_3, H2O_Vo_3, H2O_Vo_4]\n\n```\n\n\n```python\ndeltaX = {'Range': [ -12, -6], 'Label': 'O'}\ndeltaY = {'Range': [ -19, -12], 'Label': 'H_2O'}\n```\n\n\n```python\nsystem, SE = mu_vs_mu.calculate(data, bulk, deltaX, deltaY, x_energy=Oxygen_corrected, y_energy=Water_corrected)\n```\n\nAs before we can generate a basic plot of oxygen chemical potential vs water chemical potential at 298 K\n\n\n```python\nax = system.plot_phase(temperature=298, cbar_title=\"$H_2O$ $/$ $nm^2$\", figsize=(6, 5))\nplt.savefig(\"../../../docs/source/Figures/Surfaces_4.png\", dpi=600)\nplt.show()\n```\n\nWe can also generate the same plot but with the $\\mu$ values converted to pressure.\n\n\n```python\nsystem.plot_pressure(temperature=298, cbar_title=\"$H_2O$ $/$ $nm^2$\", figsize=(6, 5))\nplt.savefig(\"../../../docs/source/Figures/Surfaces_5.png\", dpi=600)\nplt.show()\n```\n\nFinally, we can also combine these two plots into one\n\n\n```python\nsystem.plot_mu_p(temperature=298, cbar_title=\"$H_2O$ $/$ $nm^2$\", figsize=(6, 5))\nplt.savefig(\"../../../docs/source/Figures/Surfaces_6.png\", dpi=600)\nplt.show()\n```\n", "meta": {"hexsha": "3b3d45c2d6af23b3bd67f6788c5b3e9c5822e10e", "size": 72869, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/Notebooks/Surfaces/Tutorial_3.ipynb", "max_stars_repo_name": "awvwgk/SurfinPy", "max_stars_repo_head_hexsha": "b094d8af592b79b73cb31a42f4be6b5cd0ac38f3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "examples/Notebooks/Surfaces/Tutorial_3.ipynb", "max_issues_repo_name": "awvwgk/SurfinPy", "max_issues_repo_head_hexsha": "b094d8af592b79b73cb31a42f4be6b5cd0ac38f3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "examples/Notebooks/Surfaces/Tutorial_3.ipynb", "max_forks_repo_name": "awvwgk/SurfinPy", "max_forks_repo_head_hexsha": "b094d8af592b79b73cb31a42f4be6b5cd0ac38f3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 307.4641350211, "max_line_length": 31822, "alphanum_fraction": 0.9316856276, "converted": true, "num_tokens": 1118, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239836484143, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.4104232915927526}} {"text": "[The 1st IAA-CSIC Severo Ochoa School on Statistics, Data Mining and Machine Learning](https://www.granadacongresos.com/sostat)\n\n[Zeljko Ivezic, University of Washington](http://faculty.washington.edu/ivezic/) \n\n[This notebook](https://github.com/carmensg/IAA_School2019/tree/master/lectures/Day3-ZeljkoIvezic/notebooks/classification.ipynb)\n### November 6, 2019 \n\n# Lecture 3: Classification\n\n### o Introduction to classification\n### o Classification loss (risk)\n### o Receiver Operating Characteristic curves\n### o Generative Classification and examples\n### o Discriminative Classification and examples\n### o Decision Trees\n\n\n\n```python\n# first things first...\nimport numpy as np\nfrom matplotlib import pyplot as plt\nfrom scipy import stats\nfrom scipy import integrate\nfrom scipy.stats import norm\nfrom scipy.stats import cauchy\n# astroML tools \nfrom astroML.plotting import hist\nfrom astroML.plotting.mcmc import convert_to_stdev\nfrom astroML.plotting import setup_text_plots\nfrom astroML.plotting.mcmc import convert_to_stdev\nsetup_text_plots(fontsize=8, usetex=True)\nimport warnings; warnings.simplefilter('ignore')\n```\n\n## Classification and density estimation\n\nIn density estimation we estimate joint probability distributions from multivariate data sets to identify the inherent clustering. This is essentially unsupervised classification \n\nIf we have labels for some of these data points (e.g., an object is tall, short, red, or blue) we can develop a relationship between the label and the properties of a source. This is supervised classification \n\nHere \u201csupervised\u201d means that there is prior information about the number and properties of clusters: for a training sample, we know the so-called \u201cclass labels\u201d (for each data point in the training sample, we know to which cluster it belongs; characterizing these known clusters is typically easier than finding and characterizing unknown clusters)\n\n\nClassification, regression, and density estimation are all related. For example, the regression function \n$\u0177 =f(y|x\u20d7 )$ is the best estimated value of $y$ given a value of $x\u20d7$. \n\nIn classification $y$ is categorical and $f(y|x\u20d7)$ is the called the discriminant function\n\nUsing density estimation for classification is referred to as **generative classification** (we have a full model of the density for each class or we have a model which describes how data could be generated from each class).\n\nClassification that finds the decision boundary that separates classes is called **discriminative classification** \n\nBoth have their place and role in astrophysical classification. \n\n\n ## Classification loss: how well are we doing\n\nThe first question we need to address is how we score (defined the success of our classification)\n\nWe can define a _loss function_. A zero-one loss function assigns a value of one for a misclassification and zero for a correct classification (i.e. we will want to minimize the loss).\n\nIf $\\hat{y}$ is the best guess value of $y$, the classification loss, $L(y,\\widehat{y})$, is\n\n> $L(y,\\widehat{y}) = \\delta(y \\neq \\widehat{y})$\n\nwhich means\n\n>$\\begin{eqnarray} L(y,\\hat{y}) & = & \\left\\{ \\begin{array}{cl} 1 & \\mbox{if $y\\neq\\hat{y}$}, \\\\ 0 & \\mbox{otherwise.} \t \\end{array} \\right. \\end{eqnarray}$\n\n\nThe expectation (mean) value of the loss $\\mathbb{E} \\left[ L(y,\\hat{y}) \\right] = p(y\\neq \\hat{y})$ is called the classification risk \n\nThis is related to regression loss functions: $L(y, \\hat{y}) = (y - \\hat{y})^2$ and risk $\\mathbb{E}[(y - \\hat{y})^2]$.\n\nWe can then define:\n\n> $ {\\rm completeness} = \\frac{\\rm true\\ positives}\n {\\rm true\\ positives + false\\ negatives}\n$\n\n> $ {\\rm contamination} = \\frac{\\rm false\\ positives}\n {\\rm true\\ positives + false\\ positives}\n$\n\nor\n\n> $ {\\rm true\\ positive\\ rate} = \\frac{\\rm true\\ positives}\n {\\rm true\\ positives + false\\ negatives}\n$\n\n> $ {\\rm false\\ positive\\ rate} = \\frac{\\rm false\\ positives}\n {\\rm true\\ negatives + false\\ positives}\n$\n\n## Comparing the performance of classifiers\n\nBest performance is a bit of a subjective topic (e.g. star-galaxy separation for correlation function studies or Galactic streams studies). We trade contamination as a function of completeness and this is science dependent.\n\n**ROC curves: Receiver Operating Characteristic curves**\n\n- Plot the true-positive vs the false-positive rate\n\n- Initially used to analyze radar results in WWII (a very productive era for statistics...).\n\n- One concern about ROC curves is that they are sensitive to the relative sample sizes (if there are many more background events than source events small false positive results can dominate a signal). For these cases we we can plot efficiency (1 - contamination) vs completeness\n\n\n\n\n\n\n```python\n### Modeled after astroML book figure 9.17: \n### https://www.astroml.org/book_figures/chapter9/fig_ROC_curve.html \nfrom __future__ import print_function\nfrom sklearn.naive_bayes import GaussianNB\nfrom sklearn.discriminant_analysis import (LinearDiscriminantAnalysis,\n QuadraticDiscriminantAnalysis)\nfrom sklearn.linear_model import LogisticRegression\nfrom sklearn.neighbors import KNeighborsClassifier\nfrom sklearn.tree import DecisionTreeClassifier\nfrom sklearn.metrics import roc_curve\nfrom astroML.classification import GMMBayes\nfrom astroML.utils import split_samples, completeness_contamination\nfrom astroML.datasets import fetch_rrlyrae_combined\nsetup_text_plots(fontsize=16, usetex=True)\n\n#----------------------------------------------------------------------\n# get data and split into training & testing sets\nX, y = fetch_rrlyrae_combined()\ny = y.astype(int)\n(X_train, X_test), (y_train, y_test) = split_samples(X, y, [0.75, 0.25],\n random_state=0)\n#------------------------------------------------------------\n# Fit all the models to the training data\ndef compute_models(*args):\n names = []\n probs = []\n for classifier, kwargs in args:\n print(classifier.__name__)\n clf = classifier(**kwargs)\n clf.fit(X_train, y_train)\n y_probs = clf.predict_proba(X_test)[:, 1]\n\n names.append(classifier.__name__)\n probs.append(y_probs)\n\n return names, probs\n\n\nnames, probs = compute_models((GaussianNB, {}),\n (LinearDiscriminantAnalysis, {}),\n (QuadraticDiscriminantAnalysis, {}),\n (LogisticRegression,\n dict(class_weight='balanced')),\n (KNeighborsClassifier,\n dict(n_neighbors=10)),\n (DecisionTreeClassifier,\n dict(random_state=0, max_depth=12,\n criterion='entropy')),\n (GMMBayes, dict(n_components=3, tol=1E-5,\n covariance_type='full')))\n\n#------------------------------------------------------------\n# Plot ROC curves and completeness/efficiency\nfig = plt.figure(figsize=(14, 7))\nfig.subplots_adjust(left=0.1, right=0.95, bottom=0.15, top=0.9, wspace=0.25)\n\n\n# ax2 will show roc curves\nax1 = plt.subplot(121)\n\n# ax1 will show completeness/efficiency\nax2 = plt.subplot(122)\n\nlabels = dict(GaussianNB='GNB',\n LinearDiscriminantAnalysis='LDA',\n QuadraticDiscriminantAnalysis='QDA',\n KNeighborsClassifier='KNN',\n DecisionTreeClassifier='DT',\n GMMBayes='GMMB',\n LogisticRegression='LR')\n\nthresholds = np.linspace(0, 1, 1001)[:-1]\n\n# iterate through and show results\nfor name, y_prob in zip(names, probs):\n fpr, tpr, thresh = roc_curve(y_test, y_prob)\n\n # add (0, 0) as first point\n fpr = np.concatenate([[0], fpr])\n tpr = np.concatenate([[0], tpr])\n\n ax1.plot(fpr, tpr, label=labels[name])\n\n comp = np.zeros_like(thresholds)\n cont = np.zeros_like(thresholds)\n for i, t in enumerate(thresholds):\n y_pred = (y_prob >= t)\n comp[i], cont[i] = completeness_contamination(y_pred, y_test)\n ax2.plot(1 - cont, comp, label=labels[name])\n\nax1.set_xlim(0, 0.04)\nax1.set_ylim(0, 1.02)\nax1.xaxis.set_major_locator(plt.MaxNLocator(5))\nax1.set_xlabel('false positive rate')\nax1.set_ylabel('true positive rate')\nax1.legend(loc=4)\n\nax2.set_xlabel('efficiency')\nax2.set_ylabel('completeness')\nax2.set_xlim(0, 1.0)\nax2.set_ylim(0.2, 1.02)\n\nplt.show()\n```\n\n## Generative Classification\n\nIn generative classifiers we model class-conditional densities $p_k(\\vec{x})$ given $p(\\vec{x}|y=y_k)$ \n\n$p(y=y_k)$, or $\\pi_k$ for short, is the probability of any point having class $k$ (equivalent to the prior probability of the class $k$). \n\nOur goal is to estimate the $p_k$'s \n\n The discriminative function \n\n$\\hat{y} = f(y|\\vec{x})$ represents the best guess of $y$ given a value of $\\vec{x}$.\n\n$f(y|\\vec{x})$ is the _discriminant function_\n\nFor a simple 2-class example\n\n> $\\begin{eqnarray}\ng(\\vec{x}) & = & \\int y \\, p(y|\\vec{x}) \\, dy \\\\\n% & = & \\int y p(y|\\vec{x}) \\, dy \\\\\n & = & 1 \\cdot p(y=1 | \\vec{x}) + 0 \\cdot p(y=0 | \\vec{x}) = p(y=1 | \\vec{x}).\n% & = & p(y=1 | \\vec{x})\n\\end{eqnarray}\n$\n\nFrom Bayes rule\n\n> $\\begin{eqnarray} g(\\vec{x}) & = & \\frac{p(\\vec{x}|y=1) \\, p(y=1)}{p(\\vec{x}|y=1) \\, p(y=1) + p(\\vec{x}|y=0) \\, p(y=0)} \\\\ & = & \\frac{\\pi_1 p_1(\\vec{x})}{\\pi_1 p_1(\\vec{x}) + \\pi_0 p_0(\\vec{x})} \\end{eqnarray}$\n\n Bayes Classifier \n\nThe discriminant function gives a binary predictor called a Bayes classifier\n\n>$\\begin{eqnarray} \\widehat{y} & = & \\left\\{ \\begin{array}{cl} \t 1 & \\mbox{if $g(\\vec{x}) > 1/2$}, \\\\ \t 0 & \\mbox{otherwise,} \t \\end{array} \t \\right. \\\\ & = & \\left\\{ \\begin{array}{cl} 1 & \\mbox{if $p(y=1|\\vec{x}) > p(y=0|\\vec{x})$}, \\\\ 0 & \\mbox{otherwise,} \\end{array} \\right. \\\\ & = & \\left\\{ \\begin{array}{cl} 1 & \\mbox{if $\\pi_1 p_1(\\vec{x}) > \\pi_0 p_0(\\vec{x})$}, \\\\ 0 & \\mbox{otherwise.} \\end{array} \\right.\\end{eqnarray}$\n\n Decision Boundary \n\nA set of $x$ values at which each class is equally likely;\n\n>$\n\\pi_1 p_1(\\vec{x}) = \\pi_2 p_2(\\vec{x});\n$\n\n> $g_1(\\vec{x}) = g_2(\\vec{x})$; $g_1(\\vec{x}) - g_2(\\vec{x}) = 0$; $g(\\vec{x}) = 1/2$; in a two-class problem\n \n\n\n## Simplest classifier: Naive Bayes\n\nWe want $p(x_1,x_2,x_3...x_n|y)$ but if we assume that all attributes are conditionally independent this simplifies to\n\n> $ p(x^i,x^j|y_k) = p(x^i|y)p(x^j|y_k)$\n \nwhich can be written as\n\n> $ p({x^0,x^1,x^2,\\ldots,x^N}|y_k) = \\prod_i p(x^i|y_k)$\n\nFrom Bayes' rule and conditional independence we get\n\n> $\n p(y_k | {x^0,x^1,\\ldots,x^N}) =\n \\frac{\\prod_i p(x^i|y_k) p(y_k)}\n {\\sum_j \\prod_i p(x^i|y_j) p(y_j)}.\n$\n\nWe calculate the most likely value of $y$ by maximizing over $y_k$,\n\n## $ \\widehat{y} = \\arg \\max_{y_k} \\frac{\\prod_i p(x^i|y_k) p(y_k)}\n {\\sum_j \\prod_i p(x^i|y_j) p(y_j)},\n$\n\nor\n\n## $\\widehat{y} = \\arg \\max_{y_k} \\frac{\\prod_i p_k(x^i) \\pi_k}\n {\\sum_j \\prod_i p_j(x^i) \\pi_j}.\n$\n\n## The rest of the class is now just estimating densities\n\n> $p(\\vec{x}|y=y_k)$ and $\\pi_k$ are learned from a set of training data.\n- $\\pi_k$ is just the frequency of the class $k$ in the training set\n- $p(\\vec{x}|y=y_k)$ is just the density (probability) of a object with class $k$ having the attributes $x$\n\n\n\nIf the training set does not cover the full parameter space $p_k(x^i)$ can be $0$ for some value of $y_k$ and $x^i$. The posterior probability is then $p(y_k|\\{x^i\\}) = 0/0$ which is unfortunate! The trick is _Laplace smoothing_ : an offset $\\alpha$ is added to the probability of each bin $p(\\vec{x}|y=y_k)$ for all $i, k$ (equivalent to the addition of a Bayesian prior to the naive Bayes classifier).\n\n## Gaussian Naive Bayes\n\nIn Gaussian naive Bayes $p_k(x^i)$ are modeled as one-dimensional normal distributions, with means $\\mu^i_k$ and widths $\\sigma^i_k$. The naive Bayes estimator is then\n\n# $\\hat{y} = \\arg\\max_{y_k}\\left[\\ln \\pi_k - \\frac{1}{2}\\sum_{i=1}^N\\left(2\\pi(\\sigma^i_k)^2 + \\frac{(x^i - \\mu^i_k)^2}{(\\sigma^i_k)^2} \\right) \\right]$\n\n Note: this is the log of the Bayes criterion with no normalization constant \n\n\n```python\nfrom matplotlib import colors\nfrom astroML.datasets import fetch_imaging_sample\nfrom astroML.plotting.tools import draw_ellipse\nfrom sklearn.naive_bayes import GaussianNB\n\ndef get_stars_and_galaxies(Nstars=10000, Ngals=10000):\n \"\"\"Get the subset of star/galaxy data to plot\"\"\"\n data = fetch_imaging_sample()\n\n objtype = data['type']\n\n stars = data[objtype == 6][:Nstars]\n \n galaxies = data[objtype == 3][:Ngals]\n\n return np.concatenate([stars,galaxies]), np.concatenate([np.zeros(len(stars)),np.ones(len(galaxies))])\n\ndata, y = get_stars_and_galaxies(Nstars=10000, Ngals=10000)\n\n# select r model mag and psf - model mag as columns\nX = np.column_stack((data['rRaw'], data['rRawPSF'] - data['rRaw']))\n\n#------------------------------------------------------------\n# Fit the Naive Bayes classifier\nclf = GaussianNB()\nclf.fit(X, y)\n\n# predict the classification probabilities on a grid\nxlim = (14, 23)\nylim = (-1, 5)\n\nxx, yy = np.meshgrid(np.linspace(xlim[0], xlim[1], 71),\n np.linspace(ylim[0], ylim[1], 81))\nZ = clf.predict_proba(np.c_[xx.ravel(), yy.ravel()])\nZ = Z[:, 1].reshape(xx.shape)\n\n#------------------------------------------------------------\n# Plot the results\nfig = plt.figure(figsize=(10,10))\nax = fig.add_subplot(111)\nax.scatter(X[:, 0], X[:, 1], c=y, cmap=plt.cm.Oranges, zorder=2)\n\nax.contour(xx, yy, Z, [0.5], linewidths=2., colors='blue')\n\nax.set_xlim(xlim)\nax.set_ylim(ylim)\n\nax.set_xlabel('$x$')\nax.set_ylabel('$y$')\n\nplt.show()\n```\n\n## GMM and Bayes classification\n\nThe natural extension to the Gaussian assumptions is to use GMM's to learn the density distribution. \n\nThe number of Gaussian components $K$ must be chosen for each class independently\n\n\n\n```python\n### Modeled after astroML book figure 4.2: \n### https://www.astroml.org/book_figures/chapter4/fig_GMM_1D.html\n\nfrom sklearn.mixture import GaussianMixture\nsetup_text_plots(fontsize=16, usetex=True)\n#------------------------------------------------------------\n# Set up the dataset.\n# We'll create our dataset by drawing samples from Gaussians.\nrandom_state = np.random.RandomState(seed=1)\nX = np.concatenate([random_state.normal(-1, 1.5, 350),\n random_state.normal(0, 1, 500),\n random_state.normal(3, 0.5, 150)]).reshape(-1, 1)\n\n#------------------------------------------------------------\n# Learn the best-fit GaussianMixture models\n# Here we'll use scikit-learn's GaussianMixture model. The fit() method\n# uses an Expectation-Maximization approach to find the best\n# mixture of Gaussians for the data\n\n# fit models with 1-10 components\nN = np.arange(1, 11)\nmodels = [None for i in range(len(N))]\n\nfor i in range(len(N)):\n models[i] = GaussianMixture(N[i]).fit(X)\n\n# compute the AIC and the BIC\nAIC = [m.aic(X) for m in models]\nBIC = [m.bic(X) for m in models]\n\n#------------------------------------------------------------\n# Plot the results\n# We'll use three panels:\n# 1) data + best-fit mixture\n# 2) AIC and BIC vs number of components\n# 3) probability that a point came from each component\n\nfig = plt.figure(figsize=(15, 5.2))\nfig.subplots_adjust(left=0.12, right=0.97,\n bottom=0.21, top=0.9, wspace=0.5)\n\n\n# plot 1: data + best-fit mixture\nax = fig.add_subplot(131)\nM_best = models[np.argmin(AIC)]\n\nx = np.linspace(-6, 6, 1000)\nlogprob = M_best.score_samples(x.reshape(-1, 1))\nresponsibilities = M_best.predict_proba(x.reshape(-1, 1))\npdf = np.exp(logprob)\npdf_individual = responsibilities * pdf[:, np.newaxis]\n \nax.hist(X, 30, histtype='stepfilled', alpha=0.4, normed=True)\nax.plot(x, pdf, '-k')\nax.plot(x, pdf_individual, '--k')\nax.text(0.04, 0.96, \"Best-fit Mixture\",\n ha='left', va='top', transform=ax.transAxes)\nax.set_xlabel('$x$')\nax.set_ylabel('$p(x)$')\n\n\n# plot 2: AIC and BIC\nax = fig.add_subplot(132)\nax.plot(N, AIC, '-k', label='AIC')\nax.plot(N, BIC, '--k', label='BIC')\nax.set_xlabel('n. components')\nax.set_ylabel('information criterion')\nax.legend(loc=2)\n\n\n# plot 3: posterior probabilities for each component\nax = fig.add_subplot(133)\n\np = responsibilities\np = p[:, (1, 0, 2)] # rearrange order so the plot looks better\np = p.cumsum(1).T\n\nax.fill_between(x, 0, p[0], color='gray', alpha=0.3)\nax.fill_between(x, p[0], p[1], color='gray', alpha=0.5)\nax.fill_between(x, p[1], 1, color='gray', alpha=0.7)\nax.set_xlim(-6, 6)\nax.set_ylim(0, 1)\nax.set_xlabel('$x$')\nax.set_ylabel(r'$p({\\rm class}|x)$')\n\nax.text(-5, 0.3, 'class 1', rotation='vertical')\nax.text(0, 0.5, 'class 2', rotation='vertical')\nax.text(3, 0.3, 'class 3', rotation='vertical')\n\nplt.show()\n```\n\n\n```python\n### Modeled after astroML book figure 9.3: \n### https://www.astroml.org/book_figures/chapter9/fig_rrlyrae_naivebayes.html \nfrom __future__ import print_function\n\nimport numpy as np\nfrom matplotlib import pyplot as plt\n\nfrom sklearn.naive_bayes import GaussianNB\nfrom astroML.datasets import fetch_rrlyrae_combined\nfrom astroML.utils import split_samples\nfrom astroML.utils import completeness_contamination\nsetup_text_plots(fontsize=16, usetex=True)\n\n#----------------------------------------------------------------------\n# get data and split into training & testing sets\nX, y = fetch_rrlyrae_combined()\nX = X[:, [1, 0, 2, 3]] # rearrange columns for better 1-color results\n(X_train, X_test), (y_train, y_test) = split_samples(X, y, [0.75, 0.25],\n random_state=0)\n\nN_tot = len(y)\nN_st = np.sum(y == 0)\nN_rr = N_tot - N_st\nN_train = len(y_train)\nN_test = len(y_test)\nN_plot = 5000 + N_rr\n\n#----------------------------------------------------------------------\n# perform Naive Bayes\nclassifiers = []\npredictions = []\nNcolors = np.arange(1, X.shape[1] + 1)\n\norder = np.array([1, 0, 2, 3])\n\nfor nc in Ncolors:\n clf = GaussianNB()\n clf.fit(X_train[:, :nc], y_train)\n y_pred = clf.predict(X_test[:, :nc])\n\n classifiers.append(clf)\n predictions.append(y_pred)\n\ncompleteness, contamination = completeness_contamination(predictions, y_test)\n\nprint(\"completeness\", completeness)\nprint(\"contamination\", contamination)\n\n#------------------------------------------------------------\n# Compute the decision boundary\nclf = classifiers[1]\nxlim = (0.7, 1.35)\nylim = (-0.15, 0.4)\n\nxx, yy = np.meshgrid(np.linspace(xlim[0], xlim[1], 81),\n np.linspace(ylim[0], ylim[1], 71))\n\nZ = clf.predict_proba(np.c_[yy.ravel(), xx.ravel()])\nZ = Z[:, 1].reshape(xx.shape)\n\n#----------------------------------------------------------------------\n# plot the results\nfig = plt.figure(figsize=(15, 7.5))\nfig.subplots_adjust(bottom=0.15, top=0.95, hspace=0.0,\n left=0.1, right=0.95, wspace=0.2)\n\n# left plot: data and decision boundary\nax = fig.add_subplot(121)\nim = ax.scatter(X[-N_plot:, 1], X[-N_plot:, 0], c=y[-N_plot:],\n s=4, lw=0, cmap=plt.cm.binary, zorder=2)\nim.set_clim(-0.5, 1)\n\nim = ax.imshow(Z, origin='lower', aspect='auto',\n cmap=plt.cm.binary, zorder=1,\n extent=xlim + ylim)\nim.set_clim(0, 1.5)\nax.contour(xx, yy, Z, [0.5], colors='k')\n\nax.set_xlim(xlim)\nax.set_ylim(ylim)\n\nax.set_xlabel('$u-g$')\nax.set_ylabel('$g-r$')\n\n# Plot completeness vs Ncolors\nax = plt.subplot(222)\nax.plot(Ncolors, completeness, 'o-k', ms=6)\n\nax.xaxis.set_major_locator(plt.MultipleLocator(1))\nax.yaxis.set_major_locator(plt.MultipleLocator(0.2))\nax.xaxis.set_major_formatter(plt.NullFormatter())\n\nax.set_ylabel('completeness')\nax.set_xlim(0.5, 4.5)\nax.set_ylim(-0.1, 1.1)\nax.grid(True)\n\n# Plot contamination vs Ncolors\nax = plt.subplot(224)\nax.plot(Ncolors, contamination, 'o-k', ms=6)\n\nax.xaxis.set_major_locator(plt.MultipleLocator(1))\nax.yaxis.set_major_locator(plt.MultipleLocator(0.2))\nax.xaxis.set_major_formatter(plt.FormatStrFormatter('%i'))\n\nax.set_xlabel('N colors')\nax.set_ylabel('contamination')\nax.set_xlim(0.5, 4.5)\nax.set_ylim(-0.1, 1.1)\nax.grid(True)\n\nplt.show()\n\n```\n\n## K-nearest neighbours\n\nAs with density estimation (and kernel density estimation) the intuitive justification is that $p(y|x) \\approx p(y|x')$ if $x'$ is very close to $x$. \n\nThe number of neighbors, $K$, regulates the complexity of the classification. In simplest form, a majority rule classification is adopted, where each of the $K$ points votes on the classification. Increasing $K$ decreases the variance in the classification but at the expense of an increase in the bias. \n\nWeights can be assigned to individual votes by weighting the vote by the distance to the nearest point.\n\n\n```python\n### Modeled after astroML book figure 9.7: \n### https://www.astroml.org/book_figures/chapter9/fig_rrlyrae_knn.html\nfrom __future__ import print_function\nfrom sklearn.neighbors import KNeighborsClassifier\nfrom astroML.datasets import fetch_rrlyrae_combined\nfrom astroML.utils import split_samples\nfrom astroML.utils import completeness_contamination\nsetup_text_plots(fontsize=16, usetex=True)\n\n#----------------------------------------------------------------------\n# get data and split into training & testing sets\nX, y = fetch_rrlyrae_combined()\nX = X[:, [1, 0, 2, 3]] # rearrange columns for better 1-color results\n(X_train, X_test), (y_train, y_test) = split_samples(X, y, [0.75, 0.25],\n random_state=0)\n\nN_tot = len(y)\nN_st = np.sum(y == 0)\nN_rr = N_tot - N_st\nN_train = len(y_train)\nN_test = len(y_test)\nN_plot = 5000 + N_rr\n\n#----------------------------------------------------------------------\n# perform Classification\n\nclassifiers = []\npredictions = []\nNcolors = np.arange(1, X.shape[1] + 1)\nkvals = [1, 10]\n\nfor k in kvals:\n classifiers.append([])\n predictions.append([])\n for nc in Ncolors:\n clf = KNeighborsClassifier(n_neighbors=k)\n clf.fit(X_train[:, :nc], y_train)\n y_pred = clf.predict(X_test[:, :nc])\n\n classifiers[-1].append(clf)\n predictions[-1].append(y_pred)\n\ncompleteness, contamination = completeness_contamination(predictions, y_test)\n\nprint(\"completeness\", completeness)\nprint(\"contamination\", contamination)\n\n#------------------------------------------------------------\n# Compute the decision boundary\nclf = classifiers[1][1]\nxlim = (0.7, 1.35)\nylim = (-0.15, 0.4)\n\nxx, yy = np.meshgrid(np.linspace(xlim[0], xlim[1], 71),\n np.linspace(ylim[0], ylim[1], 81))\n\nZ = clf.predict(np.c_[yy.ravel(), xx.ravel()])\nZ = Z.reshape(xx.shape)\n\n#----------------------------------------------------------------------\n# plot the results\nfig = plt.figure(figsize=(15, 7.5))\nfig.subplots_adjust(bottom=0.15, top=0.95, hspace=0.0,\n left=0.1, right=0.95, wspace=0.2)\n\n# left plot: data and decision boundary\nax = fig.add_subplot(121)\nim = ax.scatter(X[-N_plot:, 1], X[-N_plot:, 0], c=y[-N_plot:],\n s=4, lw=0, cmap=plt.cm.binary, zorder=2)\nim.set_clim(-0.5, 1)\n\nim = ax.imshow(Z, origin='lower', aspect='auto',\n cmap=plt.cm.binary, zorder=1,\n extent=xlim + ylim)\nim.set_clim(0, 2)\n\nax.contour(xx, yy, Z, [0.5], colors='k')\n\nax.set_xlim(xlim)\nax.set_ylim(ylim)\n\nax.set_xlabel('$u-g$')\nax.set_ylabel('$g-r$')\n\nax.text(0.02, 0.02, \"k = %i\" % kvals[1],\n transform=ax.transAxes)\n\n# plot completeness vs Ncolors\nax = fig.add_subplot(222)\n\nax.plot(Ncolors, completeness[0], 'o-k', ms=6, label='k=%i' % kvals[0])\nax.plot(Ncolors, completeness[1], '^--k', ms=6, label='k=%i' % kvals[1])\n\nax.xaxis.set_major_locator(plt.MultipleLocator(1))\nax.yaxis.set_major_locator(plt.MultipleLocator(0.2))\nax.xaxis.set_major_formatter(plt.NullFormatter())\n\nax.set_ylabel('completeness')\nax.set_xlim(0.5, 4.5)\nax.set_ylim(-0.1, 1.1)\nax.grid(True)\n\n# plot contamination vs Ncolors\nax = fig.add_subplot(224)\nax.plot(Ncolors, contamination[0], 'o-k', ms=6, label='k=%i' % kvals[0])\nax.plot(Ncolors, contamination[1], '^--k', ms=6, label='k=%i' % kvals[1])\nax.legend(loc='lower right',\n bbox_to_anchor=(1.0, 0.79))\n\nax.xaxis.set_major_locator(plt.MultipleLocator(1))\nax.yaxis.set_major_locator(plt.MultipleLocator(0.2))\nax.xaxis.set_major_formatter(plt.FormatStrFormatter('%i'))\nax.set_xlabel('N colors')\nax.set_ylabel('contamination')\nax.set_xlim(0.5, 4.5)\nax.set_ylim(-0.1, 1.1)\nax.grid(True)\n\nplt.show()\n```\n\n# Discriminative classification\n\nRather than a probabilistc description we directly model the decision boundary between two or more classes\n\nFor a two class example $\\{0,1\\}$, the discriminant function is given by $g(x) = p(y=1 | x)$. Once we have it we can use the rule\n\n## $ \\widehat{y} = \\left\\{ \\begin{array}{cl} \t 1 & \\mbox{if $g(x) > 1/2$}, \\\\ \t 0 & \\mbox{otherwise,} \t \\end{array} \t \\right.$\n\nto perform classification.\n\n# Logistic regression\n\nAssuming a binomial logistic regression and consider the linear model\n\n## $\n\\begin{eqnarray}\n p(y=1|x) &=& \\frac{\\exp\\left[\\sum_j \\theta_j x^j\\right]}\n {1 + \\exp\\left[\\sum_j \\theta_j x^j\\right]}\\nonumber\\\\\n &=& p({\\theta}),\n\\end{eqnarray}\n$\n\nwhere we define\n\n## $\n\\mbox{logit}(p_i) = \\log\\left(\\frac{p_i}{1-p_i}\\right)\n= \\sum_j \\theta_j x_i^j.\n$\n\nThe name logistic regression comes from the fact that the function $e^x/(1+e^x)$ is called the logistic function. \n- useful for categorical regression as values cannot go above or below 1 and 0.\n\nBecause $y$ is binary, it can be modeled as a Bernoulli distribution with (conditional) likelihood function\n\n## $\nL(\\beta) = \\prod_{i=1}^N p_i(\\beta)^{y_i} (1-p_i(\\beta))^{1-y_i}.\n$\n\nlogistic regression is related to linear discriminant analysis (LDA). In LDA\n\n## \\begin{eqnarray}\n\\log\\left( \\frac{p(y=1|x)}{p(y=0|x)} \\right) & = &\n - \\frac{1}{2}(\\mu_0+\\mu_1)^T \\Sigma^{-1} (\\mu_1-\\mu_0) \\\\ \\nonumber\n & + & \\log\\left( \\frac{\\pi_0}{\\pi_1} \\right) + x^T \\Sigma^{-1} (\\mu_1-\\mu_0) \\\\ \\nonumber\n & = & \\alpha_0 + \\alpha^T x.\n\\end{eqnarray}\n\nIn logistic regression the model is by assumption\n\n## $\n\\log\\left( \\frac{p(y=1|x)}{p(y=0|x)} \\right) = \\beta_0 + \\beta^T x.\n$\n\nLogistic regression minimizes classification error rather than density estimation error.\n\n\n\n\n## Support Vector Machines\n \n\n\nFind the hyperplane that maximizes the distance of the closest point from either class. This distance is the margin (width of the line before it hits a point). We want the line that maximizes the margin (m).\n\nThe points on the margin are called _support vectors_\n\nIf we assume $y \\in \\{-1,1\\}$, (+1 is maximum margin, -1 is minimum, 0 is the decision boundary)\n\nThe maximum is then just when $\\beta_0 + \\beta^T x_i = 1$ etc\n\nThe hyperplane which maximizes the margin is given by finding\n\n> \\begin{equation}\n\\max_{\\beta_0,\\beta}(m) \\;\\;\\;\n \\mbox{subject to} \\;\\;\\; \\frac{1}{||\\beta||} y_i ( \\beta_0 + \\beta^T x_i )\n \\geq m \\,\\,\\, \\forall \\, i.\n\\end{equation}\n\nThe constraints can be written as $y_i ( \\beta_0 + \\beta^T x_i ) \\geq m ||\\beta|| $. \n\nThus the optimization problem is equivalent to minimizing\n> \\begin{equation}\n\\frac{1}{2} ||\\beta|| \\;\\;\\; \\mbox{subject to} \\;\\;\\; y_i\n ( \\beta_0 + \\beta^T x_i ) \\geq 1 \\,\\,\\, \\forall \\, i.\n\\end{equation}\n\nThis optimization is a _quadratic programming_ problem (quadratic objective function with linear constraints)\n\nNote that because SVM uses a metric which maximizes the margin rather than a measure over all points in the data sets, it is similar in spirit to the rank-based estimators \n\n- The median of a distribution is unaffected by even large perturbations of outlying points, as long as those perturbations do not cross the boundary.\n- In the same way, once the support vectors are determined, changes to the positions or numbers of points beyond the margin will not change the decision boundary. For this reason, SVM can be a very powerful tool for discriminative classification.\n\n- This is why there is a high completeness compared to the other methods: it does not matter that the background sources outnumber the RR Lyrae stars by a factor of $\\sim$200 to 1. It simply determines the best boundary between the small RR Lyrae clump and the large background clump.\n- This completeness, however, comes at the cost of a relatively large contamination level.\n\n- SVM is not scale invariant so it often worth rescaling the data to [0,1] or to whiten it to have a mean of 0 and variance 1 (remember to do this to the test data as well!)\n- The data dont need to be separable (we can put a constraint in minimizing the number of \"failures\")\n\n\n```python\n### Modeled after astroML book figure 9.10: \n### https://www.astroml.org/book_figures/chapter9/fig_rrlyrae_svm.html\nfrom __future__ import print_function\nfrom sklearn.svm import SVC\nfrom astroML.utils.decorators import pickle_results\nfrom astroML.datasets import fetch_rrlyrae_combined\nfrom astroML.utils import split_samples\nfrom astroML.utils import completeness_contamination\nsetup_text_plots(fontsize=16, usetex=True)\n\n#----------------------------------------------------------------------\n# get data and split into training & testing sets\nX, y = fetch_rrlyrae_combined()\nX = X[:, [1, 0, 2, 3]] # rearrange columns for better 1-color results\n\n# SVM takes several minutes to run, and is order[N^2]\n# truncating the dataset can be useful for experimentation.\n#X = X[::5]\n#y = y[::5]\n\n(X_train, X_test), (y_train, y_test) = split_samples(X, y, [0.75, 0.25],\n random_state=0)\n\nN_tot = len(y)\nN_st = np.sum(y == 0)\nN_rr = N_tot - N_st\nN_train = len(y_train)\nN_test = len(y_test)\nN_plot = 5000 + N_rr\n\n#----------------------------------------------------------------------\n# Fit SVM\nNcolors = np.arange(1, X.shape[1] + 1)\n\n\n@pickle_results('SVM_rrlyrae.pkl')\ndef compute_SVM(Ncolors):\n classifiers = []\n predictions = []\n\n for nc in Ncolors:\n # perform support vector classification\n clf = SVC(kernel='linear', class_weight='balanced')\n clf.fit(X_train[:, :nc], y_train)\n y_pred = clf.predict(X_test[:, :nc])\n\n classifiers.append(clf)\n predictions.append(y_pred)\n\n return classifiers, predictions\n\nclassifiers, predictions = compute_SVM(Ncolors)\n\ncompleteness, contamination = completeness_contamination(predictions, y_test)\n\nprint(\"completeness\", completeness)\nprint(\"contamination\", contamination)\n\n#------------------------------------------------------------\n# compute the decision boundary\nclf = classifiers[1]\nw = clf.coef_[0]\na = -w[0] / w[1]\nyy = np.linspace(-0.1, 0.4)\nxx = a * yy - clf.intercept_[0] / w[1]\n\n#----------------------------------------------------------------------\n# plot the results\nfig = plt.figure(figsize=(15, 7.5))\nfig.subplots_adjust(bottom=0.15, top=0.95, hspace=0.0,\n left=0.1, right=0.95, wspace=0.2)\n\n# left plot: data and decision boundary\nax = fig.add_subplot(121)\nax.plot(xx, yy, '-k')\nim = ax.scatter(X[-N_plot:, 1], X[-N_plot:, 0], c=y[-N_plot:],\n s=4, lw=0, cmap=plt.cm.binary, zorder=2)\nim.set_clim(-0.5, 1)\n\nax.set_xlim(0.7, 1.35)\nax.set_ylim(-0.15, 0.4)\n\nax.set_xlabel('$u-g$')\nax.set_ylabel('$g-r$')\n\n# plot completeness vs Ncolors\nax = fig.add_subplot(222)\nax.plot(Ncolors, completeness, 'o-k', ms=6)\n\nax.xaxis.set_major_locator(plt.MultipleLocator(1))\nax.yaxis.set_major_locator(plt.MultipleLocator(0.2))\nax.xaxis.set_major_formatter(plt.NullFormatter())\n\nax.set_ylabel('completeness')\nax.set_xlim(0.5, 4.5)\nax.set_ylim(-0.1, 1.1)\nax.grid(True)\n\n# plot contamination vs Ncolors\nax = fig.add_subplot(224)\nax.plot(Ncolors, contamination, 'o-k', ms=6)\n\nax.xaxis.set_major_locator(plt.MultipleLocator(1))\nax.yaxis.set_major_locator(plt.MultipleLocator(0.2))\nax.xaxis.set_major_formatter(plt.FormatStrFormatter('%i'))\n\nax.set_xlabel('N colors')\nax.set_ylabel('contamination')\nax.set_xlim(0.5, 4.5)\nax.set_ylim(-0.1, 1.1)\nax.grid(True)\n\nplt.show()\n```\n\n\n```python\n### Modeled after astroML book figure 10.23: \n### https://www.astroml.org/book_figures/chapter10/fig_LINEAR_SVM.html\nfrom __future__ import print_function\n\nimport numpy as np\nfrom matplotlib import pyplot as plt\nfrom sklearn.svm import SVC\nfrom sklearn.model_selection import train_test_split\nfrom astroML.datasets import fetch_LINEAR_geneva\n\n#----------------------------------------------------------------------\n# This function adjusts matplotlib settings for a uniform feel in the textbook.\n# Note that with usetex=True, fonts are rendered with LaTeX. This may\n# result in an error if LaTeX is not installed on your system. In that case,\n# you can set usetex to False.\nif \"setup_text_plots\" not in globals():\n from astroML.plotting import setup_text_plots\nsetup_text_plots(fontsize=8, usetex=True)\n\ndata = fetch_LINEAR_geneva()\n\nattributes = [('gi', 'logP'),\n ('gi', 'logP', 'ug', 'iK', 'JK', 'amp', 'skew')]\nlabels = ['$u-g$', '$g-i$', '$i-K$', '$J-K$',\n r'$\\log(P)$', 'amplitude', 'skew']\ncls = 'LCtype'\nNtrain = 3000\n\n#------------------------------------------------------------\n# Create attribute arrays\nX = []\ny = []\n\nfor attr in attributes:\n X.append(np.vstack([data[a] for a in attr]).T)\n LCtype = data[cls].copy()\n\n # there is no #3. For a better color scheme in plots,\n # we'll set 6->3\n LCtype[LCtype == 6] = 3\n y.append(LCtype)\n\n\n#@pickle_results('LINEAR_SVM.pkl')\ndef compute_SVM_results(i_train, i_test):\n classifiers = []\n predictions = []\n Xtests = []\n ytests = []\n Xtrains = []\n ytrains = []\n\n for i in range(len(attributes)):\n Xtrain = X[i][i_train]\n Xtest = X[i][i_test]\n ytrain = y[i][i_train]\n ytest = y[i][i_test]\n\n clf = SVC(kernel='linear', class_weight=None)\n clf.fit(Xtrain, ytrain)\n y_pred = clf.predict(Xtest)\n\n classifiers.append(clf)\n predictions.append(y_pred)\n\n return classifiers, predictions\n\n\ni = np.arange(len(data))\ni_train, i_test = train_test_split(i, random_state=0, train_size=2000)\nclfs, ypred = compute_SVM_results(i_train, i_test)\n\n\n#------------------------------------------------------------\n# Plot the results\nfig = plt.figure(figsize=(5, 5))\nfig.subplots_adjust(hspace=0.1, wspace=0.1)\n\nclass_labels = []\n\nfor i in range(2):\n Xtest = X[i][i_test]\n ytest = y[i][i_test]\n amp = data['amp'][i_test]\n\n # Plot the resulting classifications\n ax1 = fig.add_subplot(221 + 2 * i)\n ax1.scatter(Xtest[:, 0], Xtest[:, 1],\n c=ypred[i], edgecolors='none', s=4, linewidths=0)\n\n ax1.set_ylabel(r'$\\log(P)$')\n\n ax2 = plt.subplot(222 + 2 * i)\n ax2.scatter(amp, Xtest[:, 1],\n c=ypred[i], edgecolors='none', s=4, lw=0)\n\n #------------------------------\n # set axis limits\n ax1.set_xlim(-0.6, 2.1)\n ax2.set_xlim(0.1, 1.5)\n ax1.set_ylim(-1.5, 0.5)\n ax2.set_ylim(-1.5, 0.5)\n\n ax2.yaxis.set_major_formatter(plt.NullFormatter())\n if i == 0:\n ax1.xaxis.set_major_formatter(plt.NullFormatter())\n ax2.xaxis.set_major_formatter(plt.NullFormatter())\n else:\n ax1.set_xlabel(r'$g-i$')\n ax2.set_xlabel(r'$A$')\n\n#------------------------------------------------------------\n# Second figure\nfig = plt.figure(figsize=(5, 5))\nfig.subplots_adjust(left=0.11, right=0.95, wspace=0.3)\n\nattrs = ['skew', 'ug', 'iK', 'JK']\nlabels = ['skew', '$u-g$', '$i-K$', '$J-K$']\nylims = [(-1.8, 2.2), (0.6, 2.9), (0.1, 2.6), (-0.2, 1.2)]\n\nfor i in range(4):\n ax = fig.add_subplot(221 + i)\n ax.scatter(data['gi'][i_test], data[attrs[i]][i_test],\n c=ypred[1], edgecolors='none', s=4, lw=0)\n ax.set_xlabel('$g-i$')\n ax.set_ylabel(labels[i])\n\n ax.set_xlim(-0.6, 2.1)\n ax.set_ylim(ylims[i])\n\n#------------------------------------------------------------\n# Save the results\n#\n# run the script as\n#\n# >$ python fig_LINEAR_clustering.py --save\n#\n# to output the data file showing the cluster labels of each point\nimport sys\nif len(sys.argv) > 1 and sys.argv[1] == '--save':\n filename = 'cluster_labels_svm.dat'\n\n print(\"Saving cluster labels to\", filename)\n\n from astroML.datasets.LINEAR_sample import ARCHIVE_DTYPE\n new_data = np.zeros(len(data),\n dtype=(ARCHIVE_DTYPE + [('2D_cluster_ID', 'i4'),\n ('7D_cluster_ID', 'i4')]))\n\n # switch the labels back 3->6\n for i in range(2):\n ypred[i][ypred[i] == 3] = 6\n\n # need to put labels back in order\n class_labels = [-999 * np.ones(len(data)) for i in range(2)]\n for i in range(2):\n class_labels[i][i_test] = ypred[i]\n\n for name in data.dtype.names:\n new_data[name] = data[name]\n new_data['2D_cluster_ID'] = class_labels[0]\n new_data['7D_cluster_ID'] = class_labels[1]\n\n fmt = ('%.6f %.6f %.3f %.3f %.3f %.3f %.7f %.3f %.3f '\n '%.3f %.2f %i %i %s %i %i\\n')\n\n\n F = open(filename, 'w')\n F.write('# ra dec ug gi iK JK '\n 'logP Ampl skew kurt magMed nObs LCtype '\n 'LINEARobjectID 2D_cluster_ID 7D_cluster_ID\\n')\n for line in new_data:\n F.write(fmt % tuple(line[col] for col in line.dtype.names))\n F.close()\n\nplt.show()\n```\n\n## Decision Trees\n\nThe hierarchical application of decision boundaries lead to _decision trees_\n\nTree structure:\n- top node contains the entire data set\n- at each branch the data are subdivided into two child nodes \n- split is based on a predefined decision boundary (usually axis aligned)\n- splitting repeats, recursively, until we reach a predefined stopping criteria \n\n\n\nThe \"leaf nodes\" record the fraction of points that have one classification or the other\n\nApplication of the tree to classification is simple (a series of binary decisions). The fraction of points from the training set classified as one class or the other (in the leaf node) defines the class associated with that leaf node.\n\nDecision trees are simple to interpret (a set of questions)\n\n## Splitting Criteria\n\nIn order to build a decision tree we must choose the feature and\nvalue on which we wish to split the data.\n\n\n## Information content or entropy of the data\n\n> $\n E(x) = -\\sum_i p_i(x) \\ln (p_i(x)),\n$\n\nwhere $i$ is the class and $p_i(x)$ is the probability of that class\ngiven the training data. \n\nInformation gain is the reduction in entropy due to the partitioning of the data \n\nFor a binary split $IG(x)$ is\n\n> $ IG(x|x_i) = E(x) - \\sum_{i=0}^{1} \\frac{N_i}{N} E(x_i), $\n\nwhere $N_i$ is the number of points, $x_i$, in the $i$th class,\nand $E(x_i)$ is the entropy associated with that class\n\nSearch for the split is undertaken in a greedy fashion (one attribute at a time) where we sort the data on feature $i$ and maximize the information gain for a given split point, $s$,\n\n>$\nIG(x|s) = E(x) - \\arg\\max_s \\left( \\frac{N(x|xnaive Bayes                                        , \n- linear discriminant analysis (LDA),\n- logistic regression, \n- linear support vector machines, \n- quadratic discriminant analysis (QDA),\n- linear ensembles of linear models. \n\nFor non-parametric models accuracy increases as:\n- decision trees\n- $K$-nearest-neighbor, \n- neural networks\n- kernel discriminant analysis,\n- kernelized support vector machines\n- random forests\n- boosting\n\nNaive Bayes and its variants are by far the easiest to compute. Linear support vector machines are more expensive, though several fast algorithms exist. Random forests can be easily parallelized. \n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "3326507eb8f8b6ed74d7ed53838a8e08315ff3ff", "size": 60593, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "lectures/Day3-ZeljkoIvezic/notebooks/classification.ipynb", "max_stars_repo_name": "lgalbany/IAA_School2019", "max_stars_repo_head_hexsha": "d0ae7401804bc45be55f3f6cf90fc3137a6fd833", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "lectures/Day3-ZeljkoIvezic/notebooks/classification.ipynb", "max_issues_repo_name": "lgalbany/IAA_School2019", "max_issues_repo_head_hexsha": "d0ae7401804bc45be55f3f6cf90fc3137a6fd833", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "lectures/Day3-ZeljkoIvezic/notebooks/classification.ipynb", "max_forks_repo_name": "lgalbany/IAA_School2019", "max_forks_repo_head_hexsha": "d0ae7401804bc45be55f3f6cf90fc3137a6fd833", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2019-10-18T05:11:00.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-23T13:42:04.000Z", "avg_line_length": 37.3339494763, "max_line_length": 641, "alphanum_fraction": 0.5228656776, "converted": true, "num_tokens": 12315, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.6406358411176238, "lm_q1q2_score": 0.41041428971546606}} {"text": "#
                                        Econometrics HW_10
                                        \n\n**
                                        11510691 \u7a0b\u8fdc\u661f$\\DeclareMathOperator*{\\argmin}{argmin}\n\\DeclareMathOperator*{\\argmax}{argmax}\n\\DeclareMathOperator*{\\plim}{plim}\n\\newcommand{\\ffrac}{\\displaystyle \\frac}\n\\newcommand{\\d}[1]{\\displaystyle{#1}}\n\\newcommand{\\space}{\\text{ }}\n\\newcommand{\\bspace}{\\;\\;\\;\\;}\n\\newcommand{\\bbspace}{\\;\\;\\;\\;\\;\\;\\;\\;}\n\\newcommand{\\QQQ}{\\boxed{?\\:}}\n\\newcommand{\\void}{\\left.\\right.}\n\\newcommand{\\Tran}[1]{{#1}^{\\mathrm{T}}}\n\\newcommand{\\CB}[1]{\\left\\{ #1 \\right\\}}\n\\newcommand{\\SB}[1]{\\left[ #1 \\right]}\n\\newcommand{\\P}[1]{\\left( #1 \\right)}\n\\newcommand{\\abs}[1]{\\left| #1 \\right|}\n\\newcommand{\\norm}[1]{\\left\\| #1 \\right\\|}\n\\newcommand{\\given}[1]{\\left. #1 \\right|}\n\\newcommand{\\using}[1]{\\stackrel{\\mathrm{#1}}{=}}\n\\newcommand{\\asim}{\\overset{\\text{a}}{\\sim}}\n\\newcommand{\\RR}{\\mathbb{R}}\n\\newcommand{\\EE}{\\mathbb{E}}\n\\newcommand{\\II}{\\mathbb{I}}\n\\newcommand{\\NN}{\\mathbb{N}}\n\\newcommand{\\ZZ}{\\mathbb{Z}}\n\\newcommand{\\QQ}{\\mathbb{Q}}\n\\newcommand{\\PP}{\\mathbb{P}}\n\\newcommand{\\AcA}{\\mathcal{A}}\n\\newcommand{\\FcF}{\\mathcal{F}}\n\\newcommand{\\AsA}{\\mathscr{A}}\n\\newcommand{\\FsF}{\\mathscr{F}}\n\\newcommand{\\dd}{\\mathrm{d}}\n\\newcommand{\\I}[1]{\\mathrm{I}\\left( #1 \\right)}\n\\newcommand{\\N}[1]{\\mathcal{N}\\left( #1 \\right)}\n\\newcommand{\\Exp}[1]{\\mathrm{E}\\left[ #1 \\right]}\n\\newcommand{\\Var}[1]{\\mathrm{Var}\\left[ #1 \\right]}\n\\newcommand{\\Avar}[1]{\\mathrm{Avar}\\left[ #1 \\right]}\n\\newcommand{\\Cov}[1]{\\mathrm{Cov}\\left( #1 \\right)}\n\\newcommand{\\Corr}[1]{\\mathrm{Corr}\\left( #1 \\right)}\n\\newcommand{\\ExpH}{\\mathrm{E}}\n\\newcommand{\\VarH}{\\mathrm{Var}}\n\\newcommand{\\AVarH}{\\mathrm{Avar}}\n\\newcommand{\\CovH}{\\mathrm{Cov}}\n\\newcommand{\\CorrH}{\\mathrm{Corr}}\n\\newcommand{\\ow}{\\text{otherwise}}\n\\newcommand{\\FSD}{\\text{FSD}}\n\\void^\\dagger$
                                        **\n\n## Question 10.e1\n\n$\\bspace$Let the three middle estimated equations be\n\n$$\n\\begin{align}\ny_t &= \\hat\\alpha_0 + \\hat\\alpha_1 t + \\ddot y_t \\\\\nx_{t1} &= \\hat\\eta_0 + \\hat\\eta_1 t +\\ddot x_{t1}\\\\\nx_{t2} &= \\hat\\xi_0 + \\hat\\xi_1 t + \\ddot x_{t2}\n\\end{align}$$\n\n$\\bspace$Plug these into the overall estimated model: $y_t = \\hat\\beta_0 + \\hat\\beta_1 x_{t1} + \\hat\\beta_2 x_{t2} + \\hat\\beta_3 t + \\hat u_t$ and we obtain\n\n$$\\begin{align}\n\\hat\\alpha_0 + \\hat\\alpha_1 t + \\ddot y_t &= \\hat\\beta_0 + \\hat\\beta_1 \\P{\\hat\\eta_0 + \\hat\\eta_1 t +\\ddot x_{t1}} + \\hat\\beta_2 \\P{\\hat\\xi_0 + \\hat\\xi_1 t + \\ddot x_{t2}} + \\hat\\beta_3 t + \\hat u_t\\\\\n&= \\P{\\hat\\beta_0 + \\hat\\beta_1\\hat\\eta_0 + \\hat\\beta_2 \\hat\\xi_0} + \\hat\\beta_1 \\ddot x_{t1} + \\hat\\beta_2 \\ddot x_{t2} + \\P{\\hat\\beta_1\\hat\\eta_1+\\hat\\beta_2\\hat\\xi_1 + \\hat\\beta_3} t + \\hat u_t\n\\end{align}$$\n\n$\\bspace$So now we tend to prove $\\left\\{\\begin{align}\n\\hat\\alpha_1 &= \\hat\\beta_1\\hat\\eta_1+\\hat\\beta_2\\hat\\xi_1 + \\hat\\beta_3\\\\\n\\hat\\alpha_0 &= \\hat\\beta_0 + \\hat\\beta_1\\hat\\eta_0 + \\hat\\beta_2 \\hat\\xi_0\n\\end{align}\\right.$\n\n$$\\begin{align}\n\\hat\\alpha_1 &= \\ffrac{\\d{\\sum_{t=1}^n \\P{t - \\bar t}y_t}}{\\d{\\sum_{t=1}^n \\P{t - \\bar t}^2}}\\\\\n&= \\ffrac{\\d{\\sum_{t=1}^n \\P{t - \\bar t}\\P{\\hat\\beta_0 + \\hat\\beta_1 x_{t1} + \\hat\\beta_2 x_{t2} + \\hat\\beta_3 t + \\hat u_t}}}{\\d{\\sum_{t=1}^n \\P{t - \\bar t}^2}}\\\\\n&= \\ffrac{\\d{\\sum_{t=1}^n \\P{t - \\bar t}\\hat\\beta_0}}{\\d{\\sum_{t=1}^n \\P{t - \\bar t}^2}} + \\ffrac{\\d{\\sum_{t=1}^n \\P{t - \\bar t}\\hat\\beta_1 x_{t1}}}{\\d{\\sum_{t=1}^n \\P{t - \\bar t}^2}} + \\ffrac{\\d{\\sum_{t=1}^n \\P{t - \\bar t}\\hat\\beta_2 x_{t2}}}{\\d{\\sum_{t=1}^n \\P{t - \\bar t}^2}} + \\ffrac{\\d{\\sum_{t=1}^n \\P{t - \\bar t}\\hat\\beta_3 t}}{\\d{\\sum_{t=1}^n \\P{t - \\bar t}^2}} + \\ffrac{\\d{\\sum_{t=1}^n \\P{t - \\bar t}\\hat u_t}}{\\d{\\sum_{t=1}^n \\P{t - \\bar t}^2}}\\\\\n&= \\hat\\beta_0 \\cdot 0 + \\hat\\beta_1 \\hat\\eta_1 + \\hat\\beta_2 \\hat\\xi_1 +\\hat\\beta_3 \\cdot 1 + 0 = \\hat\\beta_1\\hat\\eta_1+\\hat\\beta_2\\hat\\xi_1 + \\hat\\beta_3\n\\end{align}$$\n\n$\\bspace$For the other one, we use the average of dependent variable to derive the equation.\n\n$$\\begin{align}\n\\hat\\alpha_0 &= \\bar y - \\P{\\hat\\beta_1 \\hat\\eta_1 +\\hat\\beta_2 \\hat\\xi_1 +\\hat\\beta_3}\\bar t\\\\\n&= \\hat\\beta_0 + \\hat\\beta_1\\P{\\bar x_1 - \\hat\\eta_1 \\bar t} + \\hat\\beta_2\\P{\\bar x_2 - \\xi_1 \\bar t} + \\bar{ \\hat u}\\\\\n&= \\hat\\beta_0 + \\hat\\beta_1\\P{\\bar x_1 - \\P{\\bar x_1 - \\hat\\eta_0}} + \\hat\\beta_2\\P{\\bar x_2 -\\P{\\bar x_2-\\xi_0}} + 0 = \\hat\\beta_0 + \\hat\\beta_1\\hat\\eta_0 + \\hat\\beta_2 \\hat\\xi_0\n\\end{align}$$\n\n## Question 10.2\n\n$\\bspace$$\\text{TS}.2$ is violated, the reason accounting for this is the following. From the model for $\\text{gGDP}_t$ we have \n\n$$\\begin{align}\n\\text{int}_t &= \\gamma_0 + \\gamma_1 \\P{\\text{gGDP}_{t-1} - 3} + v_t\\\\\n&= \\gamma_0 + \\gamma_1 \\P{\\alpha_0 + \\delta_0 \\text{int}_{t-1} + \\delta_1 \\text{int}_{t-2}+ u_{t-1} - 3} + v_t\\\\\n&= \\P{\\gamma_0 + \\gamma_1 \\alpha_0 - 3\\gamma_1}+ \\gamma_1\\delta_0 \\text{int}_{t-1} + \\gamma_1\\delta_1 \\text{int}_{t-2} + \\gamma_1 u_{t-1} + v_t\n\\end{align}$$\n\n$\\bspace$Given that $u_t$ is uncorrelated with $\\text{int}_t$ and $\\text{int}_{t-1}$, we claim that $u_{t-1}$ is uncorrelated with $\\text{int}_{t-1}$ and $\\text{int}_{t-2}$\n$$\\begin{align}\n\\Cov{\\text{int}_t,u_{t-1}} &= \\Cov{\\P{\\gamma_0 + \\gamma_1 \\alpha_0 - 3\\gamma_1}+ \\gamma_1\\delta_0 \\text{int}_{t-1} + \\gamma_1\\delta_1 \\text{int}_{t-2} + \\gamma_1 u_{t-1} + v_t , u_{t-1}}\\\\\n&= \\gamma_1 \\Cov{u_{t-1},u_{t-1}}>0\n\\end{align}$$\n\n$\\bspace$So the two are correlated.\n\n## Question 10.3\n\n$\\bspace$The conclusion is obvious. Since **LRP** is equalto $\\sum_i \\delta_i = \\delta_0 + \\delta_1 + \\delta_2$, w have\n\n$$y^* = \\alpha_0 +\\P{\\delta_0 + \\delta_1 + \\delta_2}z^* = \\alpha_0 + \\text{LRP}\\cdot z^*\\Rightarrow \\Delta y^* = \\text{LRP}\\cdot \\Delta z^*$$\n\n$\\bspace$Following this we interprete **LRP** as *equilibrium propensity*.\n\n## Question 11.6\n\n$\\P{1}$\n\n$\\bspace$We first calculate $t = \\ffrac{1.104 - 1}{0.039} \\approx 2.667$. From the statistic table with $df = 120$, the critical value for two-sided hypothesis is $2.62$, so we assert that we rejuct $H_0$ against $H_1:\\beta_1\\neq 1$ at %$7%$ significant level.\n\n$\\bspace$Pratically, however, the difference is more than $10\\%$, so the estimator is different.\n\n$\\P{2}$\n\n$\\bspace$Still, we calculate $t = \\ffrac{1.053 - 1}{0.039} \\approx 1.359$. Under this we can't reject $H_0$. As for the lagged term, the $t$ statistic is calculated as $t = \\ffrac{0.480}{0.109} = 4.404$, so we conclude that this term is significant. And judging from the coefficient of the estimated model, we'd better invest in six-month T-bills, since\n\n$$\\widehat{\\text{hy6}_t} - \\text{hy3}_{t-1} = -0.123 + 0.053\\text{hy3}_{t-1} + 0.480\\P{\\text{r6}_{t-1} - \\text{r3}_{t-1}}$$\n\n$\\P{3}$\n\n$\\bspace$It violates $\\text{TS}.2$ and consequently $t$ test fails. There's unit root behavior.\n\n$\\P{4}$\n\n$\\bspace$We put $3$ seasonal dummy variables out of $4$ in the model. Then do $F$ test.\n\n## Question 11.7\n\n$\\P{1}$\n\n$$\\begin{align}\ny_t &= y_{t-1} + \\lambda\\P{\\gamma_0 + \\gamma_1 x_t + e_{t} - y_{t-1}} + a_t\\\\\n&= \\lambda \\gamma_0 + \\P{1-\\lambda}y_{t-1} + \\lambda \\gamma_1 x_t + a_t + \\lambda e_t \\\\\n&:= \\beta_0 + \\beta_1 y_{t-1} + \\beta_2 x_t + u_t \n\\end{align}$$\n\n$\\bspace$where $\\beta_0 := \\lambda \\gamma_0$, $\\beta_1 := 1-\\lambda$, $\\beta_2 := \\lambda \\gamma_1$ and $u_t := a_t + \\lambda e_t$.\n\n$\\P{2}$\n\n$\\bspace$We could use OLS to do regression of $y_t$ on $x_t$ and $y_{t-1}$. And Given that $\\Exp{e_t\\mid \\P{\\mathbf x,\\mathbf y}} = \\Exp{a_t\\mid \\P{\\mathbf x,\\mathbf y}} = 0$, we java\n\n$$\\Exp{u_t\\mid \\P{\\mathbf x,\\mathbf y}} = 0$$\n\n$\\bspace$This shows that $\\text{TS}.5$ is satisfied. Thus we have Gauss-Markov Assumptions satisfied so that after estimation we can find the $t$ statistic and test certain hypothesis. From the weak dependency and stationaryness we also claim the consistency of thie estimation.\n\n$\\P{3}$\n\n$$\\hat\\beta_1 = 0.7\\Rightarrow \\hat\\lambda = 1-\\hat\\beta_1 = 0.3 \\Rightarrow \\hat\\gamma_1 = \\ffrac{\\hat\\beta_2}{\\hat\\lambda} = 0.6667$$\n\n***\n", "meta": {"hexsha": "44acb5a6b8aeca9c613b1b86f20b6680fde46400", "size": 11322, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "FinMath/Econometrics/HW/HW_10.ipynb", "max_stars_repo_name": "XavierOwen/Notes", "max_stars_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2018-11-27T10:31:08.000Z", "max_stars_repo_stars_event_max_datetime": "2019-01-20T03:11:58.000Z", "max_issues_repo_path": "FinMath/Econometrics/HW/HW_10.ipynb", "max_issues_repo_name": "XavierOwen/Notes", "max_issues_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "FinMath/Econometrics/HW/HW_10.ipynb", "max_forks_repo_name": "XavierOwen/Notes", "max_forks_repo_head_hexsha": "d262a9103b29ee043aa198b475654aabd7a2818d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-07-14T19:57:23.000Z", "max_forks_repo_forks_event_max_datetime": "2020-07-14T19:57:23.000Z", "avg_line_length": 46.0243902439, "max_line_length": 521, "alphanum_fraction": 0.5166931638, "converted": true, "num_tokens": 3397, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.6442251064863695, "lm_q1q2_score": 0.4103912314092214}} {"text": "# Reproducing Vitaly Feldman MNIST Results\n\nThe goal of this notebook is to reproduce the results in https://arxiv.org/abs/2008.11193 which are based on the MNIST model from https://github.com/pytorch/opacus/blob/master/examples/mnist.py\n\n\n```python\nimport numpy as np\nimport syft as sy\nfrom syft.core.tensor.tensor import Tensor\nfrom syft.core.adp.entity import Entity\nfrom syft.core.adp.adversarial_accountant import AdversarialAccountant\nfrom collections import defaultdict\n\ndata_batch = np.random.rand(4,2)\nlabel_batch = np.random.rand(4,10) \n\nclass DataSubjectGuardianOfTheGalaxy():\n \"\"\"One of these per domain\"\"\"\n \n def __init__(self, enforce_unique_attrs=['name', 'email']):\n \"\"\"\"\"\"\n self.enforce_unique_attrs = enforce_unique_attrs\n self.entity_id2obj = {}\n \n self.unique_attr2entity_id = defaultdict(dict)\n \n # verfiy_key-to-accountant mapping\n self.accountants = {}\n \n #\u00a0the idea with the symbol factory is that a session with a data scientist\n # will have a unique SymbolFactory for the length of that session (this is\n # different from an accountant which a data scientist is stuck with for life).\n # A Symbol Factory is an object which produces symbols (mapped to prime numbers)\n # whenever a PhiTensor is turned into a GammaTensor. Thus, the Symbol Factory\n # can be re-initialized every time a data scientist creates a new session on a worker. However,\n #\u00a0if ever the data scientist were to save some intermediate variables across sessions or,\n # to transfer a variable from one of their workers to another,\n # they would also need to bring across the symbol factory associated with those variables.\n # The main issue is that a symbol factory is what allows the privacy logic in gammatensor to\n # know when it's dealing with the same symbol twice or two different symbols. \n \n def register_data_subject(self, **attributes):\n \"\"\"\"\"\"\n \n for attr in self.enforce_unique_attrs:\n if attr not in attributes.keys():\n raise Exception(\"The DataSubjectGuardian for your node requires you to \"+\\\n \"specify a unique \" + str(attr) + \" for every data subject. \"+\\\n \"Please specify email= in register_entity() method.\")\n \n ent = Entity(**attributes)\n\n for attr in self.enforce_unique_attrs:\n try:\n self.unique_attr2entity_id[attr][ent.attributes[attr]] = ent.id\n except KeyError as e:\n print(e)\n raise KeyError(\"'\" + str(attr) + \"' attribute must be unique '\" + str(attributes[attr]) + \\\n \"' has already been claimed by another entity.\")\n \n return ent\n \n @property\n def symbol_factory(self):\n return self._symbol_factory\n\nimport sympy as sp \nclass PrimeFactory:\n\n \"\"\"IMPORTANT: it's very important that two tensors be able to tell that\n they are indeed referencing the EXACT same PrimeFactory. At present this is done\n by ensuring that it is literally the same python object. In the future, we will probaby\n need to formalize this. However, the main way this could go wrong is if we created some\n alternate way for checking to see if two prime factories 'sortof looked the same' but which\n in fact weren't the EXACT same object. This could lead to security leaks wherein two tensors\n think two different symbols in fact are the same symbol.\"\"\"\n\n def __init__(self):\n self.prev_prime = 1\n\n def next(self):\n self.prev_prime = sp.nextprime(self.prev_prime)\n return self.prev_prime\n \nfrom syft.core.adp.scalar import GammaScalar\n \nclass VirtualMachinePrivateScalarManager:\n\n def __init__(self):\n self.prime_factory = PrimeFactory()\n self.prime2symbol = {}\n \n def get_symbol(self,min_val, value, max_val, entity):\n gs = GammaScalar(min_val=min_val,\n value=value,\n max_val=max_val,\n entity=entity)\n gs.prime = self.prime_factory.next()\n self.prime2symbol[gs.prime] = gs\n return gs.prime\n\n \ndsg = DataSubjectGuardianOfTheGalaxy()\nbob = dsg.register_data_subject(name=\"Bob\", email=\"bob@openmined.org\")\nkritika = dsg.register_data_subject(name=\"Kritika\", email=\"kritika@openmined.org\")\nmadhava = dsg.register_data_subject(name=\"Madhava\", email=\"madhava@openmined.org\")\ntudor = dsg.register_data_subject(name=\"Tudor\", email=\"tudor@openmined.org\")\n\nentities = [bob,kritika,madhava,tudor]\n\n# needs to come from the VirtualWorker/Session/etc.\nscalar_manager = VirtualMachinePrivateScalarManager()\n\ndata = Tensor(data_batch).private(0.01,1,entities=entities, scalar_manager=scalar_manager).autograd(requires_grad=True)\ndata2 = Tensor(data_batch).private(0.02,2,entities=[bob,bob,bob,bob], scalar_manager=scalar_manager).autograd(requires_grad=True)\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\ny = data.sum(0)\ny2 = data2.sum(0)\n```\n\n\n```python\nacc = AdversarialAccountant(20)\n```\n\n\n```python\ny.child.child.flat_scalars\n```\n\n\n\n\n [,\n ]\n\n\n\n\n```python\n_y = y.publish(acc=acc, sigma=1.5)\n# _y2 = y2.publish(acc=acc, sigma=1.5)\nacc.print_ledger()\n```\n\n \t16.754948874946987\n \t16.754948874946987\n \t16.754948874946987\n \t16.754948874946987\n\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\nfrom syft.core.tensor.passthrough import HANDLED_FUNCTIONS\n```\n\n\n```python\n\ndata_batch = np.random.rand(4,28*28)\nlabel_batch = np.random.rand(4,10) \n\nbob = Entity(unique_name=\"Bob\")\n\ndata = Tensor(data_batch).autograd(requires_grad=True)\ntarget = Tensor(label_batch).autograd(requires_grad=True)\n\nweights = Tensor(np.random.rand(28*28,10)).autograd(requires_grad=True)\n\nfor i in range(10):\n pred = data.dot(weights)\n diff = target - pred\n pre_loss = np.square(diff)\n loss = np.mean(pre_loss)\n extraneous_thing = -diff\n loss.backward()\n\n weights = weights - (weights.grad * 0.01)\n print(loss)\n```\n\n Tensor(child=AutogradTensor(child=[39593.85306277]))\n Tensor(child=AutogradTensor(child=[12606.85187116]))\n Tensor(child=AutogradTensor(child=[4015.70540841]))\n Tensor(child=AutogradTensor(child=[1280.66337406]))\n Tensor(child=AutogradTensor(child=[409.85003185]))\n Tensor(child=AutogradTensor(child=[132.50026923]))\n Tensor(child=AutogradTensor(child=[44.08106506]))\n Tensor(child=AutogradTensor(child=[15.81375355]))\n Tensor(child=AutogradTensor(child=[6.70270769]))\n Tensor(child=AutogradTensor(child=[3.69698355]))\n\n\n\n```python\nimport pymbolic as pmbl\nimport numpy as np\n\nvs = list()\nfor i in range(2*28*28):\n x = pmbl.var(\"s\"+str(i))\n vs.append(x)\n \nws = list()\nfor i in range(10*28*28):\n w = pmbl.var(\"w\"+str(i))\n ws.append(x) \n \ndata = np.array(vs).reshape(2,28*28)\nweights = np.array(ws).reshape(28*28,10)\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n%%timeit\npred = data.dot(weights)\n```\n\n 58.3 ms \u00b1 270 \u00b5s per loop (mean \u00b1 std. dev. of 7 runs, 10 loops each)\n\n\n\n```python\nfrom syft.lib.adp.tensor import Tensor\nfrom syft.lib.adp.tensor import make_entities\n\n\ndata = Tensor(np.random.rand(2,28*28))\ntarget = Tensor(np.random.rand(28*28,10))\nentities = make_entities(n=len(data))\ndata = data.private(min_val=-0.42421296, max_val=2.8214867, entities=entities)\ndata = data.reshape(2,28*28)\nweights = np.random.rand(28 * 28, 10)\n```\n\n making entities\n\n\n\n```python\n%%timeit -n1 -r1\nout = data.dot(weights)\n```\n\n 467 ms \u00b1 0 ns per loop (mean \u00b1 std. dev. of 1 run, 1 loop each)\n\n\n\n```python\nimport cocos.device as cd\nfrom cocos.symbolic import (\n LambdifiedMatrixExpression, \\\n find_length_of_state_vectors\n)\ncd.info()\nfrom cocos import numerics as cn\n```\n\n Cocos running on ArrayFire v3.5.1 (OpenCL 64bit)\n [0] Apple: AMD Radeon Pro Vega 56 Compute Engine | OpenCL | compute version 1.2\n\n\n\n```python\ndata = cn.random.rand(500000,28*28)\nweights = cn.random.rand(28*28,10)\n```\n\n\n```python\n%%timeit -n1 -r1\nout = data.dot(weights)\n```\n\n 297 \u00b5s \u00b1 0 ns per loop (mean \u00b1 std. dev. of 1 run, 1 loop each)\n\n\n##### out\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\nout2 = np.dot(data,weights)\n```\n\n\n```python\nsp = out2[0][0].value.poly\n```\n\n\n```python\nclass ScalarTensor(np.ndarray):\n def __new__(\n cls,\n data\n ):\n obj = np.asarray(data).view(cls)\n return obj\n \n def __array_finalize__(self, obj):\n print(\"finalizing\")\n if obj is None:\n return\n\n def __array_wrap__(self, out_arr, context=None):\n \n print(context)\n \n # out_arr.view calls __array_finalize__\n output = out_arr.view(self.__class__)\n \n return output\n```\n\n\n```python\nimport cocos.device as cd\nfrom cocos.symbolic import (\n LambdifiedMatrixExpression, \\\n find_length_of_state_vectors\n)\ncd.info()\n```\n\n Cocos running on ArrayFire v3.5.1 (OpenCL 64bit)\n [0] Apple: AMD Radeon Pro Vega 56 Compute Engine | OpenCL | compute version 1.2\n\n\n\n```python\nimport sympy as sym\nimport numpy as np\n```\n\n\n```python\n# x1, x2, x3, t = sym.symbols('x1, x2, x3, t')\n# argument_symbols = (x1, x2, x3)\n# g = sym.Function('g')\n# f = sym.Matrix([[x1 + x2], [(g(t) * x1 + x3) ** 2], [sym.exp(x1 + x2 + g(t))]])\n# jacobian_f = f.jacobian([x1, x2, x3])\n\n# def numeric_time_function(t: float):\n# return np.log(t)\n\n# jacobian_f_lambdified \\\n# = LambdifiedMatrixExpression(\n# argument_symbols=argument_symbols,\n# time_symbol=t,\n# symbolic_matrix_expression=jacobian_f,\n# symbolic_time_function_name_to_numeric_time_function_map={'g': numeric_time_function})\n\n# n = 10000000\n# X_gpu = cn.random.rand(n, 3)\n# X_cpu = np.array(X_gpu)\n```\n\n\n```python\n\n```\n\n\n```python\n# %%timeit\n# result = X_gpu.sum()\n```\n\n\n```python\n# %%timeit\n# result2 = X_cpu.sum()\n```\n\n\n```python\nsingle_poly = np.array([1,0,1])\n\nsingle_poly.shape\n```\n\n\n\n\n (3,)\n\n\n\n\n```python\nn_polys = (np.random.rand(10,3) > 0.5).astype(int)\nn_polys\n```\n\n\n\n\n array([[1, 0, 0],\n [0, 1, 1],\n [0, 1, 1],\n [0, 1, 1],\n [0, 1, 1],\n [0, 1, 0],\n [0, 0, 1],\n [0, 1, 1],\n [0, 0, 0],\n [1, 1, 0]])\n\n\n\n\n```python\nx = data.flatten()[0:5]\n```\n\n\n```python\nimport autograd.numpy as np # Thinly-wrapped version of Numpy\nfrom autograd import grad\n\ndef taylor_sine(x): # Taylor approximation to sine function\n ans = currterm = x\n i = 0\n while np.abs(currterm) > 0.001:\n currterm = -currterm * x**2 / ((2 * i + 3) * (2 * i + 2))\n ans = ans + currterm\n i += 1\n return ans\n\ngrad_sine = grad(taylor_sine)\nprint(\"Gradient of sin(pi) is\", grad_sine(x))\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n# %%timeit\n# jacobian_f_numeric_gpu = \\\n# (jacobian_f_lambdified\n# .evaluate_with_kwargs(x1=X_gpu[:, 0],\n# x2=X_gpu[:, 1],\n# x3=X_gpu[:, 2],\n# t=1.0))\n```\n\n\n```python\n# %%timeit\n# jacobian_f_numeric_cpu = \\\n# (jacobian_f_lambdified\n# .evaluate_with_kwargs(x1=X_cpu[:, 0],\n# x2=X_cpu[:, 1],\n# x3=X_cpu[:, 2],\n# t=1.0))\n```\n\n\n```python\nprint(f'numerical results from cpu and gpu match: '\n f'{np.allclose(jacobian_f_numeric_gpu, jacobian_f_numeric_cpu)}')\n```\n\n numerical results from cpu and gpu match: True\n\n\n\n```python\n\n```\n\n\n```python\nimport symengine\nfrom symengine import var\n```\n\n\n```python\nimport uuid\n```\n\n\n```python\nsyms = \"\"\nfor i in range(784):\n id = str(uuid.uuid4()).replace(\"-\",\"\")\n syms += 's'+id+str(i)+' '\n```\n\n\n```python\nv = var(syms)\n```\n\n\n```python\n%%timeit -n1 -r1\nresult = np.sum(v)\n```\n\n 63.8 ms \u00b1 0 ns per loop (mean \u00b1 std. dev. of 1 run, 1 loop each)\n\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n 58 ms \u00b1 291 \u00b5s per loop (mean \u00b1 std. dev. of 7 runs, 10 loops each)\n\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n%%timeit -n1 -r1\nout = np.sum(vs)\n```\n\n 2.12 ms \u00b1 0 ns per loop (mean \u00b1 std. dev. of 1 run, 1 loop each)\n\n\n\n```python\n\n```\n\n\n```python\nsyms = \"\"\nfor i in range(784):\n id = str(uuid.uuid4()).replace(\"-\",\"\")\n syms += 's'+id+str(i)+','\n```\n\n\n```python\nfrom sympy import symbols\n```\n\n\n```python\nsymbols = symbols(syms[:-1])\n```\n\n\n```python\n%%timeit -n1 -r1\nout = np.sum(symbols)\n```\n\n 2.01 s \u00b1 0 ns per loop (mean \u00b1 std. dev. of 1 run, 1 loop each)\n\n\n\n```python\n%%timeit -n1 -r1\nout = np.sum([[x.value.poly for x in row] for row in data[0].tolist()])\n```\n\n 2.05 s \u00b1 0 ns per loop (mean \u00b1 std. dev. of 1 run, 1 loop each)\n\n\n\n```python\n# %%timeit -n1 -r1\n# result = data[0].sum()\n```\n\n\n```python\n%%timeit -n1 -r1\nresult = data[1].sum()\n```\n\n 2.06 s \u00b1 0 ns per loop (mean \u00b1 std. dev. of 1 run, 1 loop each)\n\n\n\n```python\n%%timeit -n1 -r1\nresult = data[0].sum()\n```\n\n 2.07 s \u00b1 0 ns per loop (mean \u00b1 std. dev. of 1 run, 1 loop each)\n\n\n\n```python\n%%timeit -n1 -r1\nresult = data[1].sum()\n```\n\n 2.06 s \u00b1 0 ns per loop (mean \u00b1 std. dev. of 1 run, 1 loop each)\n\n\n\n```python\n\n```\n\n\n```python\nacc = AdversarialAccountant(max_budget=7000)\n```\n\n\n```python\nout = result.publish(acc=acc,sigma=1.5)\nacc.print_ledger()\n```\n\n\n```python\nacc.print_ledger()\n```\n\n \t2.365443354618569\n\n\n\n```python\n=\n\nx = Tensor(np.array([[1,1],[1,0],[0,1],[0,0]])).private(min_val=0, max_val=1, entities=entities, is_discrete=True)\ny = Tensor(np.array([[1],[1],[0],[0]])).private(min_val=0, max_val=1, entities=entities, is_discrete=False)\n\n_weights = Tensor(np.random.uniform(size=(2,1)))\n```\n\n\n```python\nweights = _weights + 0\n# acc = AdversarialAccountant(max_budget=7) # causes error at end of budget\n\n\nfor i in range(3):\n batch_loss = 0\n\n pred = x.dot(weights)\n pre_loss = np.square(y-pred)\n loss = np.mean(pre_loss, axis=1)\n# loss = np.mean(pre_loss) \n ledger = loss.backward(accumulate_grads=True)\n weight_grad = (weights.grad*0.3)\n weight_grad_noise = weight_grad.publish(acc=acc, sigma=0.005)\n weights = weights - weight_grad_noise\n ledger.zero_grads()\n batch_loss += loss.value\n print(np.sum([loss.value]))\n \nacc.print_ledger()\n\n```\n\n 0.17169736596498625\n 0.07441205306476974\n 0.04001946718047936\n \t602.4247883171074\n \t602.4247883171074\n \t602.4247883171074\n \t602.4247883171074\n\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\nx = PhiScalar(0,0.01,1)\n```\n\n\n```python\nout = x * x\n```\n\n\n```python\nout.publish(acc=acc)\n```\n\n \n found one!\n\n\n\n\n\n [0.703353940208477]\n\n\n\n\n```python\nacc.print_ledger()\n```\n\n \t0.0\n \t0.0002737137892581691\n\n\n\n```python\nx.entity\n```\n\n\n```python\nimport sympy as sym\n```\n\n\n```python\na,b = sym.symbols('a,b')\n```\n\n\n```python\ny = a + b\n```\n\n\n```python\nfrom sympy.core.numbers import Number\nclass Float2(Number):\n \n def __init__(self, val):\n self.val = val\n \n def __add__(self, other):\n print(\"adding\")\n return Float(self.val - other.val)\n```\n\n\n```python\nout = y.subs({a:Float2(3)})\nout\n```\n\n\n\n\n$\\displaystyle b + 3$\n\n\n\n\n```python\nout.subs({b:Float2(2)})\n```\n\n\n\n\n$\\displaystyle 5$\n\n\n\n\n```python\n# loss[0].value.poly\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n\n\n$\\displaystyle 24ffce1fe8ab44a583abecf0b2e1ca69_{c06e7e291fdb4cb6b5ec0b9abbabf1bf} + 3 \\cdot 3175e5ebc4944304bb4a1a010fa5003d_{162afbde9a87417a9a49c14568ea1c11} + 4 \\cdot 5ec1e3788c314340a6ea45342375281e_{162afbde9a87417a9a49c14568ea1c11} + 6 \\cdot 68ed3172c1f6417faa4f5ca24611e12d_{d59083b02ef74bdf8122e0c51454aa71} + 6 \\cdot 98d310c205b743be9af907ff4646238b_{b63effdec6ba47bf93a126d0b5cc7086} + d082ac7660e84d0c806997c2dbe4f600_{b63effdec6ba47bf93a126d0b5cc7086} + 5 eeffa282abe44326ae5fc1b8a6f8c6a0_{d59083b02ef74bdf8122e0c51454aa71} + 2 fdefb3e070394d5cbf22820ac0ab7b00_{c06e7e291fdb4cb6b5ec0b9abbabf1bf}$\n\n\n\n\n```python\n\n```\n\n\n```python\n## first-pass methods to try to get into AutogradTensor\n\nops = ['T',\n '__abs__',\n# '__add__',\n# '__and__',\n# '__array__',\n# '__array_finalize__',\n# '__array_function__',\n# '__array_interface__',\n# '__array_prepare__',\n# '__array_priority__',\n# '__array_struct__',\n# '__array_ufunc__',\n# '__array_wrap__',\n# '__bool__',\n# '__class__',\n# '__complex__',\n# '__contains__',\n# '__copy__',\n# '__deepcopy__',\n# '__delattr__',\n# '__delitem__',\n# '__dir__',\n '__divmod__',\n# '__doc__',\n '__eq__',\n# '__float__',\n '__floordiv__',\n# '__format__',\n '__ge__',\n# '__getattribute__',\n '__getitem__',\n '__gt__',\n# '__hash__',\n# '__iadd__',\n# '__iand__', # trigger an error for inline \n# '__ifloordiv__',\n# '__ilshift__',\n# '__imatmul__',\n# '__imod__',\n# '__imul__',\n '__index__',\n# '__init__',\n# '__init_subclass__',\n# '__int__',\n '__invert__',\n# '__ior__',\n# '__ipow__',\n# '__irshift__',\n# '__isub__',\n '__iter__',\n# '__itruediv__',\n# '__ixor__',\n '__le__',\n '__len__',\n '__lshift__',\n '__lt__',\n '__matmul__',\n# '__mod__',\n '__mul__',\n '__ne__',\n '__neg__',\n# '__new__',\n# '__or__',\n# '__pos__',\n '__pow__',\n '__radd__',\n# '__rand__',\n# '__rdivmod__',\n# '__reduce__',\n# '__reduce_ex__',\n '__repr__',\n '__rfloordiv__',\n '__rlshift__',\n '__rmatmul__',\n# '__rmod__',\n '__rmul__',\n# '__ror__',\n '__rpow__',\n '__rrshift__',\n '__rshift__',\n '__rsub__',\n '__rtruediv__',\n# '__rxor__',\n# '__setattr__',\n# '__setitem__',\n# '__setstate__',\n '__sizeof__',\n '__str__',\n '__sub__',\n# '__subclasshook__',\n '__truediv__',\n# '__xor__',\n# 'all',\n# 'any',\n 'argmax',\n 'argmin',\n# 'argpartition',\n 'argsort',\n# 'astype',\n# 'base',\n# 'byteswap',\n 'choose',\n 'clip',\n# 'compress',\n# 'conj',\n# 'conjugate',\n 'copy',\n# 'ctypes',\n 'cumprod',\n 'cumsum',\n# 'data',\n 'diagonal',\n 'dot',\n# 'dtype',\n# 'dump',\n# 'dumps',\n# 'fill',\n# 'flags',\n 'flat',\n 'flatten',\n# 'getfield',\n# 'imag',\n 'item',\n 'itemset',\n 'itemsize',\n 'max',\n 'mean',\n 'min',\n# 'nbytes',\n 'ndim',\n# 'newbyteorder',\n# 'nonzero',\n# 'partition',\n 'prod',\n# 'ptp',\n# 'put',\n# 'ravel',\n# 'real',\n 'repeat',\n 'reshape',\n 'resize',\n# 'round',\n# 'searchsorted',\n# 'setfield',\n# 'setflags',\n# 'shape',\n# 'size',\n 'sort',\n 'squeeze',\n 'std',\n# 'strides',\n 'sum',\n 'swapaxes',\n 'take',\n# 'tobytes',\n# 'tofile',\n# 'tolist',\n# 'tostring',\n# 'trace',\n 'transpose',\n# 'var',\n# 'view'\n ]\n```\n", "meta": {"hexsha": "cd7bd6f1b87972fa3e3c7427ef0b732e2f5f4b58", "size": 89865, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "packages/syft/examples/experimental/adversarial_accountant/paper_experiments/MNIST Experiment v1.ipynb", "max_stars_repo_name": "callezenwaka/PySyft", "max_stars_repo_head_hexsha": "2545c302441cfe727ec095c4f9aa136bff02be32", "max_stars_repo_licenses": ["Apache-1.1"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2022-02-18T03:48:27.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-05T06:13:57.000Z", "max_issues_repo_path": "packages/syft/examples/experimental/adversarial_accountant/paper_experiments/MNIST Experiment v1.ipynb", "max_issues_repo_name": "callezenwaka/PySyft", "max_issues_repo_head_hexsha": "2545c302441cfe727ec095c4f9aa136bff02be32", "max_issues_repo_licenses": ["Apache-1.1"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2021-11-17T15:34:03.000Z", "max_issues_repo_issues_event_max_datetime": "2021-12-08T14:39:10.000Z", "max_forks_repo_path": "packages/syft/examples/experimental/adversarial_accountant/paper_experiments/MNIST Experiment v1.ipynb", "max_forks_repo_name": "callezenwaka/PySyft", "max_forks_repo_head_hexsha": "2545c302441cfe727ec095c4f9aa136bff02be32", "max_forks_repo_licenses": ["Apache-1.1"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-08-20T10:22:43.000Z", "max_forks_repo_forks_event_max_datetime": "2020-08-20T10:22:43.000Z", "avg_line_length": 54.3647912886, "max_line_length": 1663, "alphanum_fraction": 0.6267957492, "converted": true, "num_tokens": 5822, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307806984444, "lm_q2_score": 0.6442251133170357, "lm_q1q2_score": 0.41039122688189505}} {"text": " Trusted Notebook\" width=\"500 px\" align=\"left\">\n\n## _*Relaxation and Decoherence*_ \n\nThe latest version of this notebook is available on https://github.com/qiskit/qiskit-tutorial.\n\n***\n### Contributors\nMartin Sandberg, Hanhee Paik, Antonio C\u00f3rcoles, Doug McClure, and Jay Gambetta\n\n## Introduction\n\nIn an ideal world, quantum systems would be well-isolated from their environment, which prevents unwanted dynamics of the quantum information we encode in them. For example, suppose we prepared a qubit in the $|1\\rangle$ state, but through interaction with the environment, the state is flipped to $|0\\rangle$. That flip could affect the outcome of a quantum algorithm that's being run using that qubit, meaning the answers we get out of the quantum device would change. For this reason, we seek to isolate quantum computers from the surrounding environment.\n\nHowever, perfect isolation is not possible: after all, we have to be able to control the quantum computer, which means coupling it to external systems to manipulate quantum information. This tradeoff is sometimes referred to as the \"Tao of quantum computing\". Because our controls introduce coupling between qubits and the environment, we expect some unwanted interactions can occur.\n\nThese unwanted interactions introduce _noise_ into the qubits, which affects their behavior. The rate of these interactions sets characteristic timescales over which information encoded in qubits can be reliably stored and manipulated. (If the interaction has a rate $\\Gamma$, the characteristic timescale is $\\sim 1/\\Gamma$.) In this tutorial, we discuss two timescales that arise from energy relaxation and decoherence -- usually referred to as $T_{1}$ and $T_{2}$, respectively -- and show how they can be measured.\n\n**Contents**\n\n[Measuring $T_{1}$ time](#t1)\n\nMeasuring $T_{2}^{\\star}$ time\n\nCode imports\n==============\n\n\n```python\nimport qiskit as qk\nimport numpy as np\nfrom scipy.optimize import curve_fit\nfrom qiskit.tools.qcvv.fitters import exp_fit_fun, osc_fit_fun, plot_coherence\nfrom qiskit.tools.monitor import job_monitor\n```\n\n\n```python\n# Load saved IBMQ accounts\nqk.IBMQ.load_accounts()\n```\n\n\n```python\n# backend and token settings\nbackend = qk.IBMQ.get_backend('ibmq_16_melbourne') # the device to run on\nshots = 1024 # the number of shots in the experiment \n```\n\n\n```python\n# This function is called in code below.\ndef pad_QId(circuit,N,qr):\n r'''A function for padding a circuit with identity gates.\n \n Args:\n circuit: The quantum circuit that the gates should be appended to\n N: The number of identity gates to add\n qr: The qubit register that the gates should be added to\n \n Returns:\n circuit: The original circuit object, but with N identity\n gates added to the qubit register qr\n '''\n for j in range(N):\n circuit.barrier(qr)\n circuit.iden(qr)\n return circuit \n```\n\n\n\n# Measuring $T_1$ time\n\n**Theory**\n\nThe $T_{1}$ time is the characteristic timescale over which the state of a qubit damps toward the $|0\\rangle$ state. Given an arbitrary initial single-qubit state $\\rho(0)$, represented by a $2\\times 2$ matrix as\n$$\\rho(0) = \\begin{pmatrix}\\rho_{00} & \\rho_{01} \\\\ \\rho_{01}^{\\star} & \\rho_{11}\\end{pmatrix},$$\nunder amplitude damping noise, the state of the changes as\n$$\\rho(t) = \\begin{pmatrix}\\rho_{00} + (1-e^{-\\Gamma_{1}t})\\rho_{11} & e^{-\\Gamma_{1}t/2}\\rho_{01} \\\\ e^{-\\Gamma_{1}t/2}\\rho_{01}^{\\star} & e^{-\\Gamma_{1} t}\\rho_{11}\\end{pmatrix} \\underset{t\\rightarrow \\infty}{\\longrightarrow} |0\\rangle\\langle 0|.$$\n\nNotice that amplitude damping noise also removes any coherences between $|0\\rangle$ and $|1\\rangle$ of the state (the off-diagonal elements.) The rate at which this _decoherence_ occurs is half that of $\\Gamma_{1}$.\n\nThe time evolution of the state under amplitude damping noise can be derived as the continuous-time limit of an amplitude damping channel\n$$\\mathcal{E}[\\rho] = M_{0} \\rho M_{0}^{\\dagger} + M_{1}\\rho M_{1}^{\\dagger},$$\nwhere\n$$M_{0} = \\begin{pmatrix} 1 & 0 \\\\0& \\sqrt{1-p}\\end{pmatrix}~,~M_{1} = \\begin{pmatrix} 0 & \\sqrt{p} \\\\ 0 & 0 \\end{pmatrix},$$\nand the probability of decay $p$ is $\\Gamma_{1}\\Delta t$.\n\nThe decay rate $\\Gamma_{1}$ sets a natural time scale for the decay process; namely, $\\Gamma_{1}^{-1}$. This number is often called the $T_{1}$ time.\n\nNotice that the probability of the qubit remaining in the $|1\\rangle$ state is given by\n\n$$P_{1}(t) = \\mathrm{Tr}\\left[|1\\rangle\\langle 1| \\rho(t)\\right] = e^{-\\Gamma_{1} t}\\rho_{11}.$$\n\nIf the qubit was prepared in the $|1\\rangle$ state, then $P_{1}(t) =e^{-\\Gamma_{1} t}$.\n\nA simple way of estimating the $T_{1}$ time is to collect statistics about the decay curve for $P_{1}(t)$ when the qubit is initialized to $|1\\rangle$. This can be done by choosing a variety of times $t_{1}, t_{2}, \\cdots t_{N}$, and then running the following experiment many times:\n* Prepare the qubit in $|1\\rangle$.\n* Wait a delay time $t_{j}$.\n* Measure the qubit in the $|0\\rangle, |1\\rangle$ basis.\n\nAn estimate of $P_{1}(t_{j})$ is the number of times the qubit was observed to be in $|1\\rangle$, divided by the total number of times the experiment was repeated. Given several estimated values of $P_{1}$ for a variety of $(t_{j})$, we can fit the resulting decay curve is fit to an exponential and extract an estimate of $\\Gamma_{1}$, and hence, the $T_{1}$ time.\n\nThe IBM Q Experience does not currently support delays of arbitrary length, so for now, we just append a series of identity operations after the initial excitation pulse. Each identity operation has the same duration of a single-qubit gate and is followed by a -shorter- buffer time. These parameters are backend-dependent.\n\n**Code**\n\nThe code blocks below walk through constructing the requisite experiments to estimate the $T_{1}$ time of a qubit, sending those experiments to an IBMQ backend, and then fitting the data the backend sends back.\n\n\n```python\n# Select qubit whose T1 is to be measured\nqubit = 1\n\n# Creating registers\nqr = qk.QuantumRegister(5)\ncr = qk.ClassicalRegister(5)\n\n# the delay times are all set in terms of single-qubit gates\n# so we need to calculate the time from these parameters\npulse_length=100 # single-qubit gate time \nbuffer_length=10 # spacing between pulses\nunit = 'ns'\n\nsteps = 10\ngates_per_step = 120\nmax_gates = (steps-1)*gates_per_step+1\ntot_length = buffer_length+pulse_length\ntime_per_step = gates_per_step*tot_length\nqc_dict = {}\nfor ii in range(steps):\n step_num = 'step_%s'%(str(ii))\n qc_dict.update({step_num:qk.QuantumCircuit(qr, cr)})\n qc_dict[step_num].x(qr[qubit])\n qc_dict[step_num]=pad_QId(qc_dict[step_num],gates_per_step*ii,qr[qubit])\n qc_dict[step_num].barrier(qr[qubit])\n qc_dict[step_num].measure(qr[qubit], cr[qubit])\n\ncircuits=list(qc_dict.values()) \n```\n\n\n```python\n# run the program\nt1_job = qk.execute(circuits, backend, shots=shots)\njob_monitor(t1_job)\n```\n\n\n HTML(value=\"

                                        Job Status: job is being initialized

                                        \")\n\n\n\n```python\n# arrange the data from the run\n\nresult_t1 = t1_job.result()\nkeys_0_1=list(result_t1.get_counts(qc_dict['step_0']).keys())# get the key of the excited state '00010' \ndata=np.zeros(len(qc_dict.keys())) # numpy array for data\nsigma_data = np.zeros(len(qc_dict.keys()))\n\n# change unit from ns to microseconds\nplot_factor=1\nif unit.find('ns')>-1:\n plot_factor=1000\n punit='$\\\\mu$s'\nxvals=time_per_step*np.linspace(0,len(qc_dict.keys()),len(qc_dict.keys()))/plot_factor # calculate the time steps in microseconds \n\nfor ii,key in enumerate(qc_dict.keys()):\n # get the data in terms of counts for the excited state normalized to the total number of counts\n data[ii]=float(result_t1.get_counts(qc_dict[key])[keys_0_1[0]])/shots\n sigma_data[ii] = np.sqrt(data[ii]*(1-data[ii]))/np.sqrt(shots)\n\n# fit the data to an exponential \nfitT1, fcov = curve_fit(exp_fit_fun, xvals, data, bounds=([-1,2,0], [1., 500, 1])) \nferr = np.sqrt(np.diag(fcov))\n\nplot_coherence(xvals, data, sigma_data, fitT1, exp_fit_fun, punit, 'T$_1$ ', qubit)\n\nprint(\"a: \" + str(round(fitT1[0],2)) + u\" \\u00B1 \" + str(round(ferr[0],2)))\nprint(\"T1: \" + str(round(fitT1[1],2))+ \" \u00b5s\" + u\" \\u00B1 \" + str(round(ferr[1],2)) + ' \u00b5s')\nprint(\"c: \" + str(round(fitT1[2],2)) + u\" \\u00B1 \" + str(round(ferr[2],2)))\n```\n\nThe last calibration of $T_1$ was measured to be\n\n\n```python\nprint(backend.properties().qubits[qubit][0].value, backend.properties().qubits[qubit][0].unit)\n```\n\n 72.38826037862532 \u00b5s\n\n\n\n\n# Measuring $T_2^*$ time\n\n**Theory**\n\nAmplitude damping noise affects the off-diagonal elements of the density matrix in addition to the on-diagonal elements. However, there are other noise processes that only affect the off-diagonal elements, while keeping the on-diagonal elements the same. These kinds of noise processes cause _decoherence_.\n\nAs a simple example of decoherence, consider the pure superposition state\n$$|\\psi(\\theta)\\rangle = \\frac{1}{\\sqrt{2}}\\left(|0\\rangle + e^{i\\theta}|1\\rangle\\right).$$\nExpressed as a density matrix, this state is\n$$\\rho(\\theta) = |\\psi(\\theta)\\rangle \\langle\\psi(\\theta)| = \\frac{1}{2}\\begin{pmatrix}1 & e^{-i\\theta} \\\\ e^{i\\theta} & 1\\end{pmatrix}.$$\n\nThis state has _coherence_ between $|0\\rangle$ and $|1\\rangle$, which manifests itself in the non-zero off-diagonal terms. If the state had _decohered_, those off-diagonal terms would be zero:\n$$\\rho_{\\mathrm{decohered}} = \\frac{1}{2}\\begin{pmatrix}1 & 0 \\\\ 0 & 1\\end{pmatrix}.$$\nWhen the state has decohered, it can be written as a classical _mixture_:\n$$\\rho_{\\mathrm{decohered}} = \\frac{1}{2}\\left(|0\\rangle \\langle 0| + |1\\rangle \\langle 1|\\right).$$\n\nOne mechanism by which decoherence happens is _dephasing_. Under dephasing noise, the state of the qubit evolves as\n$$\\rho(t) = \\begin{pmatrix}\\rho_{00} & e^{-\\Gamma_{2}t}\\rho_{01} \\\\ e^{-\\Gamma_{2}t}\\rho_{01}^{\\star} & \\rho_{11}\\end{pmatrix} \\underset{t\\rightarrow \\infty}{\\longrightarrow} \\begin{pmatrix}\\rho_{00} & 0\\\\ 0& \\rho_{11}\\end{pmatrix}.$$\n\nThe time evolution of $\\rho$ under dephasing noise can be derived as the continuous-time limit of the following noise channel:\n$$\\mathcal{E}[\\rho] = M_{0}\\rho M_{0}^{\\dagger} + M_{1} \\rho M_{1}^{\\dagger} + M_{2}\\rho M_{2}^{\\dagger},$$\nwhere\n$$M_{0} =\\sqrt{1-p}I~,~M_{1} = \\sqrt{p}\\begin{pmatrix}1 &0 \\\\ 0 & 0 \\end{pmatrix}~,~M_{2} = \\sqrt{p}\\begin{pmatrix}0 & 0 \\\\ 0 & 1\\end{pmatrix}.$$\n\n\nThe rate of decay in the coherences can be measured by the following experiment:\n\n* Prepare the qubit in the $|+\\rangle$ state, which can be done by initializing the qubit to $|0\\rangle$ and applying a Hadamard gate, $H$.\n* Wait a delay time $t_{j}$.\n* Measure the qubit in the $|\\pm\\rangle$ basis, which can be done by applying a Hadamard and then measuring in the computational basis.\n\nIf decoherence processes are present, then after a delay time $t_{j}$, the state of the qubit is\n\n$$\\rho(t_{j}) = \\frac{1}{2}\\begin{pmatrix}1 & e^{-\\Gamma_{2}t_{j}} \\\\ e^{-\\Gamma_{2}t_{j}} & 1\\end{pmatrix}.$$\n\nMeasuring in the $|\\pm\\rangle$ basis, the probability of observing the outcome $|+\\rangle$ is given by\n\n$$P_{+}(t_{j}) = \\mathrm{Tr}\\left(|+\\rangle \\langle + | \\rho(t_{j})\\right) = \\frac{1}{2}\\left(1 + e^{-\\Gamma_{2}t_{j}}\\right).$$\n\nAgain, by estimating $P_{+}(t_{j})$ for a variety of $t_{j}$, we can then fit a decay curve to extract an estimate of $\\Gamma_{2}$.\n\nIn the actual experiment, we change the phase of the pulse before the measurement in order to create oscillations in the observed dynamics of $P_{+}(t_{j})$. If we just did two Hadamard gates separated by a delay, we would observe a decay of characteristic time $T^*_2$, but with a strong dependence on any deviation of the calibrated qubit frequency from the actual one. By implementing the qubit pulses with different phases, we shift the frequency dependence into the oscillating feature of the dynamics, and can fit the decaying envelope for a more faithful measure of the coherence time.\n\n
                                        \n\nThere is one subtle point of note. In the discussion of $T_{1}$ time above, we saw that amplitude damping noise also causes the off-diagonal elements to decay (at a rate $\\Gamma_{1}/2$). Suppose a qubit was affected by only amplitude damping noise, and dephasing noise was absent. In this scenaro, _the rate of decay of coherences can be non-zero, even in the absense of dephasing noise_.\n\nFor this reason, it's important to recognize there are many noise processes contributing to the rate at which coherences decay. In this tutorial, we assume that the total rate of decoherence, $\\Gamma$, can be decomposed into a sum of independent rates:\n\n$$\\Gamma = \\Gamma_{T_{1}} + \\Gamma_{2} + \\Gamma_{\\mathrm{other}}. $$\n\nPhenomenologically, the rate $\\Gamma_{\\mathrm{other}}$ quantifies the rate of decoherence due to other noise processes in addition to pure amplitude damping and pure dephasing about the $Z$-axis. Note that because general noise can cause dephasing about the $Z$-axis -- in addition to doing other things to the qubit -- echo sequences are typically used to help mitigate the effects of those kind(s) of noise on $T_{2}^{\\star}$. (Echo sequences are discussed below, in the sections $T_{2}$ echo and CPMG measurement.)\n\nIf decoherence at a rate $\\Gamma$ is taking place, then the state of the qubit changes as\n$$\\begin{equation}\n\\rho(t) = \\begin{pmatrix}\\rho_{00} & e^{-\\Gamma t}\\rho_{01} \\\\ e^{-\\Gamma t}\\rho_{01}^{\\star} & 1-\\rho_{00}\\end{pmatrix}.\n\\end{equation}$$\n\nThe timescale associated with this decay rate is called $T_{2}$, and is given by $T_{2} = 1/\\Gamma$.\n\n\n$T_{2}$ relates to the other timescales introduced previously as\n$$T_{2} = \\left(\\frac{2}{T_{1}} + \\frac{1}{T_{2}} + \\frac{1}{T_{\\mathrm{other}}}\\right)^{-1} = T_{2}^{\\star}\\left( 1 + \\frac{2T_{2}^{\\star}}{T_{1}} + \\frac{T^{\\star}_{2}}{T_{\\mathrm{other}}}\\right)^{-1} \\leq T_{2}^{\\star},$$\nwhere we've defined $T_{\\mathrm{other}} = 1 /\\Gamma_{\\mathrm{other}}$.\n\n
                                        \n\n\n```python\n# Select qubit on which to measure T2*\nqubit = 1\n\n# Creating registers\nqr = qk.QuantumRegister(5)\ncr = qk.ClassicalRegister(5)\n\nsteps = 35\ngates_per_step = 20\nmax_gates = (steps-1)*gates_per_step+2\n\nnum_osc = 5\ntot_length = buffer_length+pulse_length\ntime_per_step = gates_per_step*tot_length\nqc_dict = {}\nfor ii in range(steps):\n step_num = 'step_%s'%(str(ii))\n qc_dict.update({step_num:qk.QuantumCircuit(qr, cr)})\n qc_dict[step_num].h(qr[qubit])\n qc_dict[step_num]=pad_QId(qc_dict[step_num],gates_per_step*ii,qr[qubit])\n qc_dict[step_num].u1(2*np.pi*num_osc*ii/(steps-1),qr[qubit])\n qc_dict[step_num].h(qr[qubit])\n qc_dict[step_num].barrier(qr[qubit])\n qc_dict[step_num].measure(qr[qubit], cr[qubit])\ncircuits = list(qc_dict.values()) \n\n```\n\n\n```python\n# run the program\nt2star_job = qk.execute(circuits, backend, shots=shots)\njob_monitor(t2star_job)\n```\n\n\n HTML(value=\"

                                        Job Status: job is being initialized

                                        \")\n\n\n\n```python\n# arrange the data from the run\n\nresult_t2star = t2star_job.result()\nkeys_0_1 = list(result_t2star.get_counts(qc_dict['step_0']).keys())# get the key of the excited state '00010' \n\n# change unit from ns to microseconds\nplot_factor = 1\nif unit.find('ns') > -1:\n plot_factor = 1000\n punit = '$\\\\mu$s'\nxvals = time_per_step*np.linspace(0,len(qc_dict.keys()),len(qc_dict.keys()))/plot_factor # calculate the time steps \n\n\ndata = np.zeros(len(qc_dict.keys())) # numpy array for data\nsigma_data = np.zeros(len(qc_dict.keys()))\n\nfor ii,key in enumerate(qc_dict.keys()):\n # get the data in terms of counts for the excited state normalized to the total number of counts\n data[ii] = float(result_t2star.get_counts(qc_dict[key])[keys_0_1[0]])/shots\n sigma_data[ii] = np.sqrt(data[ii]*(1-data[ii]))/np.sqrt(shots)\n \nfitT2s, fcov = curve_fit(osc_fit_fun, xvals, data, p0=[0.5, 100, 1/10, np.pi, 0],\n bounds=([0.3,0,0,0,0], [0.5, 200, 1/2,2*np.pi,1]))\nferr = np.sqrt(np.diag(fcov))\n\nplot_coherence(xvals, data, sigma_data, fitT2s, osc_fit_fun, punit, '$T_2^*$ ', qubit)\n\nprint(\"a: \" + str(round(fitT2s[0],2)) + u\" \\u00B1 \" + str(round(ferr[0],2)))\nprint(\"T2*: \" + str(round(fitT2s[1],2))+ \" \u00b5s\"+ u\" \\u00B1 \" + str(round(ferr[1],2)) + ' \u00b5s')\nprint(\"f: \" + str(round(10**3*fitT2s[2],3)) + 'kHz' + u\" \\u00B1 \" + str(round(10**6*ferr[2],3)) + 'kHz')\nprint(\"phi: \" + str(round(fitT2s[3],2)) + u\" \\u00B1 \" + str(round(ferr[3],2)))\nprint(\"c: \" + str(round(fitT2s[4],2)) + u\" \\u00B1 \" + str(round(ferr[4],2)))\n```\n\n\n\n# Measurement of $T_2$ Echo\n\nWe have referred to the previous experiment's characteristic time as $T^*_2$ and not $T_2$ by analogy to nuclear magnetic resonance (NMR). Indeed, one can isolate different frequency components to the decoherence process by devising increasingly elaborated pulse sequences. To illustrate the analogy with NMR, one can think about an ensemble of nuclear spins precessing in an external DC magnetic field. Due to field inhomogeneities, each spin might precess with a slightly different Larmor frequency. This certainly will affect the observed coherence time of the ensemble. However, it is possible to echo away this low-frequency decoherence process by applying a pi-pulse to the system halfway through the delay. The effect of this pi-pulse is to reverse the direction of the precession of each individual spin due to field inhomogeneities. Thus, the spins that had precessed more now start precessing in the opposite direction faster than the spins that had precessed less, and after an equal delay, all the spins in the system recover the initial coherence, except for other, higher-frequency, decoherence mechanisms.\n\nHere, we are measuring only a single qubit rather than an ensemble of spins. Consequently coherence measurements require averaging an ensemble of measurements in order to eliminate projection noise, and run-to-run fluctuations in the qubit's frequency which will similarly manifest themselves as decoherence if they are not canceled out. By running this $T_2$ echo sequence, we can therefore remove low-frequency components of the decoherence.\n\n\n```python\n# Select qubit to measure T2 echo on\nqubit = 1\n\n# Creating registers\nqr = qk.QuantumRegister(5)\ncr = qk.ClassicalRegister(5)\n\nsteps = 18\ngates_per_step = 28\ntot_length = buffer_length+pulse_length\nmax_gates = (steps-1)*2*gates_per_step+3\ntime_per_step = (2*gates_per_step)*tot_length\nqc_dict = {}\nfor ii in range(steps):\n step_num = 'step_%s'%(str(ii))\n qc_dict.update({step_num:qk.QuantumCircuit(qr, cr)})\n qc_dict[step_num].h(qr[qubit])\n qc_dict[step_num]=pad_QId(qc_dict[step_num],gates_per_step*ii,qr[qubit])\n qc_dict[step_num].x(qr[qubit])\n qc_dict[step_num]=pad_QId(qc_dict[step_num],gates_per_step*ii,qr[qubit])\n qc_dict[step_num].h(qr[qubit])\n qc_dict[step_num].barrier(qr[qubit])\n qc_dict[step_num].measure(qr[qubit], cr[qubit])\ncircuits = list(qc_dict.values()) \n\n```\n\n\n```python\n# run the program\nt2echo_job = qk.execute(circuits, backend, shots=shots)\njob_monitor(t2echo_job)\n```\n\n\n HTML(value=\"

                                        Job Status: job is being initialized

                                        \")\n\n\n\n```python\n# arrange the data from the run\n\nresult_t2echo = t2echo_job.result()\nkeys_0_1 = list(result_t2echo.get_counts(qc_dict['step_0']).keys())# get the key of the excited state '00010' \n\n\n# change unit from ns to microseconds\nplot_factor=1\nif unit.find('ns')>-1:\n plot_factor=1000\n punit='$\\\\mu$s'\nxvals=time_per_step*np.linspace(0,len(qc_dict.keys()),len(qc_dict.keys()))/plot_factor # calculate the time steps \n\n\ndata=np.zeros(len(qc_dict.keys())) # numpy array for data\nsigma_data = np.zeros(len(qc_dict.keys()))\n\n\nfor ii,key in enumerate(qc_dict.keys()):\n # get the data in terms of counts for the excited state normalized to the total number of counts\n data[ii]=float(result_t2echo.get_counts(qc_dict[key])[keys_0_1[0]])/shots\n sigma_data[ii] = np.sqrt(data[ii]*(1-data[ii]))/np.sqrt(shots)\n \nfitT2e, fcov = curve_fit(exp_fit_fun, xvals, data, bounds=([-1,10,0], [1, 150, 1])) \nferr = np.sqrt(np.diag(fcov))\n\nplot_coherence(xvals, data, sigma_data, fitT2e, exp_fit_fun, punit, '$T_{2echo}$ ', qubit)\n\nprint(\"a: \" + str(round(fitT2e[0],2)) + u\" \\u00B1 \" + str(round(ferr[0],2)))\nprint(\"T2: \" + str(round(fitT2e[1],2))+ ' \u00b5s' + u\" \\u00B1 \" + str(round(ferr[1],2)) + ' \u00b5s')\nprint(\"c: \" + str(round(fitT2e[2],2)) + u\" \\u00B1 \" + str(round(ferr[2],2)))\n```\n\nThe last calibration of $T_2$ was measured to be\n\n\n```python\nprint(backend.properties().qubits[qubit][1].value, backend.properties().qubits[qubit][1].unit)\n```\n\n 111.62868245151734 \u00b5s\n\n\n\n\n## CPMG measurement\n \nAs explained above, the echo sequence removes low-frequency decoherence mechanisms. This noise-filtering procedure can be extended with increased number of pi-pulses within the delay. In the following experiment, we implement an echo experiment with seven pi-pulses during the delay between the initial and final pulses. This kind of echo with several pi-pulses is referred to as a CPMG experiment, after Carr, Purcell, Meiboom, and Gill. \n\n\n```python\n# Select qubit for CPMG measurement of T2\nqubit = 1\n\n# Creating registers\nqr = qk.QuantumRegister(5)\ncr = qk.ClassicalRegister(5)\n\nsteps = 10\ngates_per_step = 18\nnum_echo = 5 # has to be odd number to end up in excited state at the end\ntot_length = buffer_length+pulse_length\ntime_per_step = ((num_echo+1)*gates_per_step+num_echo)*tot_length\nmax_gates = num_echo*(steps-1)*gates_per_step+num_echo+2\nqc_dict = {}\nfor ii in range(steps):\n step_num='step_%s'%(str(ii))\n qc_dict.update({step_num:qk.QuantumCircuit(qr, cr)})\n qc_dict[step_num].h(qr[qubit])\n for iii in range(num_echo):\n qc_dict[step_num]=pad_QId(qc_dict[step_num], gates_per_step*ii, qr[qubit])\n qc_dict[step_num].x(qr[qubit])\n qc_dict[step_num]=pad_QId(qc_dict[step_num], gates_per_step*ii, qr[qubit])\n qc_dict[step_num].h(qr[qubit])\n qc_dict[step_num].barrier(qr[qubit])\n qc_dict[step_num].measure(qr[qubit], cr[qubit])\ncircuits=list(qc_dict.values())\n\n```\n\n\n```python\n# run the program\nt2cpmg_job = qk.execute(circuits, backend, shots=shots)\njob_monitor(t2cpmg_job)\n```\n\n\n HTML(value=\"

                                        Job Status: job is being initialized

                                        \")\n\n\n\n```python\n# arrange the data from the run\n\nresult_t2cpmg = t2cpmg_job.result()\nkeys_0_1 = list(result_t2cpmg.get_counts(qc_dict['step_0']).keys())# get the key of the excited state '00001' \n\n\n# change unit from ns to microseconds\nplot_factor = 1\nif unit.find('ns') > -1:\n plot_factor = 1000\n punit = '$\\\\mu$s'\nxvals = time_per_step*np.linspace(0,len(qc_dict.keys()),len(qc_dict.keys()))/plot_factor # calculate the time steps \n\n\ndata = np.zeros(len(qc_dict.keys())) # numpy array for data\nsigma_data = np.zeros(len(qc_dict.keys()))\n\nfor ii,key in enumerate(qc_dict.keys()):\n # get the data in terms of counts for the excited state normalized to the total number of counts\n data[ii] = float(result_t2cpmg.get_counts(qc_dict[key])[keys_0_1[0]])/shots\n sigma_data[ii] = np.sqrt(data[ii]*(1-data[ii]))/np.sqrt(shots)\n \nfitT2cpmg, fcov = curve_fit(exp_fit_fun, xvals, data, bounds=([-1,10,0], [1, 150, 1])) \nferr = np.sqrt(np.diag(fcov))\n\nplot_coherence(xvals, data, sigma_data, fitT2cpmg, exp_fit_fun, punit, '$T_{2cpmg}$ ', qubit)\n\nprint(\"a: \" + str(round(fitT2cpmg[0],2)) + u\" \\u00B1 \" + str(round(ferr[0],2)))\nprint(\"T2: \" + str(round(fitT2cpmg[1],2))+ ' \u00b5s' + u\" \\u00B1 \" + str(round(ferr[1],2)) + ' \u00b5s')\nprint(\"c: \" + str(round(fitT2cpmg[2],2)) + u\" \\u00B1 \" + str(round(ferr[2],2)))\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "1dcae18832c48ec9c55f3f89e8ca83aace76c0bb", "size": 135501, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "qiskit/ignis/relaxation_and_decoherence.ipynb", "max_stars_repo_name": "juanfran/qiskit-tutorials", "max_stars_repo_head_hexsha": "319bdebf31b3970996d96ed60ea55bd4a9691f50", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2017-11-09T16:33:14.000Z", "max_stars_repo_stars_event_max_datetime": "2018-02-26T00:42:17.000Z", "max_issues_repo_path": "qiskit/ignis/relaxation_and_decoherence.ipynb", "max_issues_repo_name": "juanfran/qiskit-tutorials", "max_issues_repo_head_hexsha": "319bdebf31b3970996d96ed60ea55bd4a9691f50", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2019-04-12T07:43:25.000Z", "max_issues_repo_issues_event_max_datetime": "2020-02-07T13:32:18.000Z", "max_forks_repo_path": "qiskit/ignis/relaxation_and_decoherence.ipynb", "max_forks_repo_name": "juanfran/qiskit-tutorials", "max_forks_repo_head_hexsha": "319bdebf31b3970996d96ed60ea55bd4a9691f50", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2017-12-28T17:07:19.000Z", "max_forks_repo_forks_event_max_datetime": "2017-12-28T17:07:19.000Z", "avg_line_length": 146.6461038961, "max_line_length": 29848, "alphanum_fraction": 0.8658386285, "converted": true, "num_tokens": 7124, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442250928250375, "lm_q2_score": 0.6370307944803832, "lm_q1q2_score": 0.4103912227065323}} {"text": "# Model Training-Evaluation-Selection\n\n## Import Libraries\n\n\n```python\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nimport seaborn as sns\nimport time\nfrom matplotlib.dates import date2num\n%matplotlib inline\nsns.set_style('darkgrid')\n```\n\n\n```python\nfrom sklearn.preprocessing import MinMaxScaler\nfrom tensorflow.keras.preprocessing import sequence\n\nfrom tensorflow.keras.layers import Dense\nfrom tensorflow.keras.layers import Input, LSTM\nfrom tensorflow.keras.losses import mean_squared_error\n\nfrom tensorflow.keras.models import Model\nfrom tensorflow.keras.models import load_model\nimport h5py\n```\n\n## Load Data\n\n\n```python\nconsumption= pd.read_pickle('../Data_Cleaned/consumption.pkl')\ngeneration= pd.read_pickle('../Data_Cleaned/generation.pkl')\ninstalled= pd.read_pickle('../Data_Cleaned/installed.pkl')\nprice= pd.read_pickle('../Data_Cleaned/price.pkl')\n```\n\n\n```python\ndfs=[consumption,generation,installed,price]\nfor df in dfs:\n df.set_index('timestamp', inplace=True)\n```\n\n## Resample Data\n\n\n```python\n# resampling data by hour\nconsumption= consumption.resample('h').sum()\ngeneration = generation.resample('h').sum()\n# For simplicity, we only predict total electricity generation\ngeneration= generation.sum(axis=1).to_frame(name='Electricity Generation [MWh]')\n```\n\n\n```python\n# save data\nconsumption.to_pickle('../Data_Cleaned/consumption_ready_for_forcast.pkl')\ngeneration.to_pickle('../Data_Cleaned/generation_ready_for_forcast.pkl')\n```\n\n## Stateful vs. Stateless LSTM\n* **Stateless**: LSTM updates parameters on batch 1 and then initiates cell states (meaning - memory, usually with zeros) for batch 2\n* **Stateful**: it uses batch 1 last output cell sates as initial states for batch 2.\n\n**When to use which?**\n* When sequences in batches are related to each other (e.g. prices of one commodity), we should better use stateful mode\n* Else, when one sequence represents a complete sentence, we should go with stateless mode\n\n## Batch Size\n\nBatch-size: which batch-size to choose?\nVery important decision!\n\nImagine, you must learn to recognize a bird... You are presented images of different birds.\n\nWhat would you prefer:\n\n1. To see the one image at a time, make your notes about special bird quilities (set your weights) and then see another bird and so on\n2. OR may be you would better learn if you see - let's say 5 - bird images at ones. May be then you can faster notice the bird's intrinsic properties?\nI'd say - the second method is more efficient for humans. We need more examples of an entitiy, that we have to distinguish.\n\nSo the machines! Therefore we select a batch size of 64. Later in programming assigment we will see how the batch size impacts the prediction accuracy.\n\n\n```python\n# defining the batch size\nbatch_size = 64\n```\n\n## Test and Training Set\n\nWith **stateful** LSTMs the trainings-set size must be divisible without remainder by the batch-size (modulo = 0).\nWe split our datasets with 80-20 ratio. That means we use four years recorded data (2015-2018) to train our model and then use 2019 data to evaluate our model. Once we find the best model, we use all data to create a model wich can predict values of year 2020.\n\n\n```python\nprint('\\n==== Electricity Consumption Dataset ====')\n# dataframe size\nlength = len(consumption)\nprint('Dataframe size: ',length)\n# training set size (80%)\nlength *= 1 - 0.2\nprint('Training set size (80%)', length)\n\nprint('\\n==== Electricity Generation Dataset ====')\n# dataframe size\nlength = len(generation)\nprint('Dataframe size: ',length)\n# training set size (80%)\nlength *= 1 - 0.2\nprint('Training set size (80%)', length,'\\n')\n```\n\n \n ==== Electricity Consumption Dataset ====\n Dataframe size: 43824\n Training set size (80%) 35059.200000000004\n \n ==== Electricity Generation Dataset ====\n Dataframe size: 43824\n Training set size (80%) 35059.200000000004 \n \n\n\n\n```python\nsns.set_style('darkgrid')\nplt.figure(figsize=(15,4))\nplt.plot(consumption[:35059], label='Training Set')\nplt.plot(consumption[35059:], label='Test Set')\nplt.title('\\nElectricity Consumption: Training and Test Sets\\n', fontsize=20 ,fontweight='bold')\nplt.ylabel('Electricity Unit [MWh]', fontsize=15)\nplt.xlabel('')\nplt.legend(loc='upper right')\nplt.show()\n```\n\n\n```python\nsns.set_style('darkgrid')\nplt.figure(figsize=(15,4))\nplt.plot(generation[:35059], label='Training Set')\nplt.plot(generation[35059:], label='Test Set')\nplt.title('\\nElectricity Generation: Training and Test Sets\\n', fontsize=20 ,fontweight='bold')\nplt.ylabel('Electricity Unit [MWh]', fontsize=15)\nplt.xlabel('')\nplt.legend(loc='upper right')\nplt.show()\n```\n\nBut as we can see, 80% of data is not divisable by our batch size. So we write a function to find the nearest divisable index.\n\n\n```python\n# Function to calculate training size regarding batch_size\ndef get_train_length(dataset, batch_size, test_percent):\n # substract test_percent to be excluded from training, reserved for testset\n length = len(dataset)\n length *= 1 - test_percent\n train_length_values = []\n for x in range(int(length) - 200,int(length)): \n modulo=x%batch_size\n if (modulo == 0):\n train_length_values.append(x)\n return (max(train_length_values))\n```\n\n\n```python\nlength = get_train_length(consumption, batch_size, 0.2)\nprint('\\n==== Electricity Consumption Dataset ====')\nprint('Divisible training set size (almost 80%)', length)\nlength = get_train_length(generation, batch_size, 0.2)\nprint('\\n==== Electricity Generation Dataset ====')\nprint('Divisible training set size (almost 80%)', length,'\\n')\n```\n\n \n ==== Electricity Consumption Dataset ====\n Divisible training set size (almost 80%) 35008\n \n ==== Electricity Generation Dataset ====\n Divisible training set size (almost 80%) 35008 \n \n\n\nAs the size of both dataframes are the same, I don't assign different parameters for length.\n\n## Adding Timesteps\n\n\n```python\n# Set time steps to 7 days\ntimesteps=24*7\n# Adding timesteps * 2\nupper_train = length + timesteps*2\n\n# Set y_train variable for consumption df\nconsumption_train_df = consumption[0:upper_train]\nconsumption_y_train = consumption_train_df.iloc[:,].values\nprint('\\nTraining Sets Shapes after Adding Timesteps:\\n')\nprint('==== Electricity Consumption Dataset ====')\nprint('Training Set Shape:',consumption_y_train.shape)\n\n# Set y_train variable for generation df\ngeneration_train_df = generation[0:upper_train]\ngeneration_y_train = generation_train_df.iloc[:,].values\nprint('\\n==== Electricity Generation Dataset ====')\nprint('Training Set Shape:',generation_y_train.shape,'\\n')\n```\n\n \n Training Sets Shapes after Adding Timesteps:\n \n ==== Electricity Consumption Dataset ====\n Training Set Shape: (35344, 1)\n \n ==== Electricity Generation Dataset ====\n Training Set Shape: (35344, 1) \n \n\n\n## Feature Scaling\n\n\n```python\n#scale between 0 and 1. the weights are esier to find.\nsc = MinMaxScaler(feature_range = (0, 1))\nconsumption_y_train_scaled = sc.fit_transform(np.float64(consumption_y_train))\ngeneration_y_train_scaled = sc.fit_transform(np.float64(generation_y_train))\n```\n\n\n```python\nprint('\\nElectricity Generation Dataframe:\\n')\nprint('Before Scaling:\\n\\n', generation_y_train[:3],'\\n')\nprint('After Scaling:\\n\\n', generation_y_train_scaled[:3],'\\n')\n```\n\n \n Electricity Generation Dataframe:\n \n Before Scaling:\n \n [[50432.]\n [48756.]\n [48040.]] \n \n After Scaling:\n \n [[0.28926323]\n [0.2644761 ]\n [0.25388685]] \n \n\n\n## Creating a data structure with n timesteps\n\n\n```python\n# Empty Lists to store X_train and y_train\nconsumption_X_train_matrix = []\nconsumption_y_train_matrix = []\n# Creating a data structure with n timesteps\nfor i in range(timesteps, length + timesteps):\n #create X_train matrix\n #24*7 items per array (timestep) \n consumption_X_train_matrix.append(consumption_y_train_scaled[i-timesteps:i,0])\n #create Y_train matrix\n #24*7 items per array (timestep)\n consumption_y_train_matrix.append(consumption_y_train_scaled[i:i+timesteps,0])\n \n# reapeat all of these steps fot generation dataframe\ngeneration_X_train_matrix = []\ngeneration_y_train_matrix = []\nfor i in range(timesteps, length + timesteps):\n generation_X_train_matrix.append(generation_y_train_scaled[i-timesteps:i,0])\n generation_y_train_matrix.append(generation_y_train_scaled[i:i+timesteps,0])\n```\n\n\n```python\n# Check shape\nprint('X_train sets shape:', np.array(consumption_X_train_matrix).shape)\nprint('y_train sets shape:', np.array(consumption_y_train_matrix).shape)\n```\n\n X_train sets shape: (35008, 168)\n y_train sets shape: (35008, 168)\n\n\n## Reshape\n\n\n```python\n# Turn list into numpy array\nconsumption_X_train_matrix = np.array(consumption_X_train_matrix)\nconsumption_y_train_matrix = np.array(consumption_y_train_matrix)\n# reshape arrays\nconsumption_X_train_reshaped = np.reshape(consumption_X_train_matrix, \n (consumption_X_train_matrix.shape[0], \n consumption_X_train_matrix.shape[1], 1))\nconsumption_y_train_reshaped = np.reshape(consumption_y_train_matrix, \n (consumption_y_train_matrix.shape[0], \n consumption_y_train_matrix.shape[1], 1))\n\n# Repeat the same stes for generatin dataframe\ngeneration_X_train_matrix = np.array(generation_X_train_matrix)\ngeneration_y_train_matrix = np.array(generation_y_train_matrix)\ngeneration_X_train_reshaped = np.reshape(generation_X_train_matrix, \n (generation_X_train_matrix.shape[0], \n generation_X_train_matrix.shape[1], 1))\ngeneration_y_train_reshaped = np.reshape(generation_y_train_matrix, \n (generation_y_train_matrix.shape[0], \n generation_y_train_matrix.shape[1], 1))\n```\n\n\n```python\n# Check shapes\nprint('X_train sets shape:', generation_X_train_reshaped.shape)\nprint('y_train sets shape:', generation_y_train_reshaped.shape)\n```\n\n X_train sets shape: (35008, 168, 1)\n y_train sets shape: (35008, 168, 1)\n\n\n## Building the LSTM\n\n\n```python\n# Initialising the LSTM Model with MSE Loss-Function\n# Using Functional API, each layer output is the input of next layer\n\n# Input\ninputs = Input(batch_shape=(batch_size,timesteps,1))\n\n# Layer 1: LSTM \nlstm_1 = LSTM(12, \n activation='tanh', \n recurrent_activation='sigmoid', \n stateful=True, \n return_sequences=True)(inputs)\n# Layer 2: LSTM \nlstm_2 = LSTM(12, \n activation='tanh', \n recurrent_activation='sigmoid', \n stateful=True, \n return_sequences=True)(lstm_1)\n# Output\noutput = Dense(units = 1)(lstm_2)\n\n# Sticking all layers into a Model\nregressor = Model(inputs=inputs, outputs = output)\n\n#adam is fast starting off and then gets slower and more precise\nregressor.compile(optimizer='adam', loss = mean_squared_error)\n```\n\n\n```python\n# Check the model summary\nregressor.summary()\n```\n\n Model: \"model\"\n _________________________________________________________________\n Layer (type) Output Shape Param # \n =================================================================\n input_1 (InputLayer) [(64, 168, 1)] 0 \n _________________________________________________________________\n lstm (LSTM) (64, 168, 12) 672 \n _________________________________________________________________\n lstm_1 (LSTM) (64, 168, 12) 1200 \n _________________________________________________________________\n dense (Dense) (64, 168, 1) 13 \n =================================================================\n Total params: 1,885\n Trainable params: 1,885\n Non-trainable params: 0\n _________________________________________________________________\n\n\nThis is how the param numbers are calculated:\n\n\\begin{equation}\n \\textbf{PARAMETERS} = \\textbf4 \\times \\textbf{ LSTM outputs size} \\times (\\textbf{weights LSTM inputs size} + \\textbf{weights LSTM outputs size} + 1 \\textbf{ bias variable})\n \\end{equation}\n\n\n```python\n# For example for lstm layers\nprint('1st LSTM layer Params: ',4*12*(1+12+1))\nprint('2st LSTM layer Params: ',4*12*(12+12+1))\n```\n\n 1st LSTM layer Params: 672\n 2st LSTM layer Params: 1200\n\n\n## Run LSTM Model for Electricity Consumption\n\nThis model is so heavy that I cannot run it on my local computer. Therefore I ran it on IBM Watson Studio, then saved it to IBM Cloud Open Storage and then downloaded it. Now, we can import the model and evaluate it.\n\n```python\nepochs = 5\n\n# start time\nstart=time.time()\n\n#Statefull\nfor i in range(epochs):\n print(\"\\nEpoch: \" + str(i))\n #run through all data but the cell, hidden state are used for the next batch.\n regressor.fit(consumption_X_train_reshaped, consumption_y_train_reshaped, \n shuffle=False, epochs = 1, batch_size = batch_size)\n #resets only the states but the weights, cell and hidden are kept.\n regressor.reset_states()\n\n# duration of training the model\nduration=time.time()-start\n```\n\n\n```python\nregressor_loaded = load_model(filepath=\"../Models/LSTM_initial_Model_Consumption.h5\")\n```\n\n## Test Set\n\n\n```python\n# Function that extarct a divisable index regarding batch\ndef get_test_length(dataset, batch_size):\n test_length_values = []\n for x in range(len(dataset) - 500, len(dataset) - timesteps*2): \n modulo=(x-upper_train)%batch_size\n if (modulo == 0):\n test_length_values.append(x)\n #print(x)\n return (max(test_length_values))\n```\n\n\n```python\ntest_length = get_test_length(consumption, batch_size)\nupper_test = test_length + timesteps*2\ntestset_length = test_length - upper_train\nprint('Test Set Size:' ,testset_length)\n```\n\n Test Set Size: 8128\n\n\n\n```python\nprint('Train Set Endpoint: ', upper_train)\nprint('Test Set Endpoint: ', upper_test)\nprint('Size of Dataframe: ',len(consumption))\n```\n\n Train Set Endpoint: 35344\n Test Set Endpoint: 43808\n Size of Dataframe: 43824\n\n\n\n```python\n# construct test set\n\n#subsetting\ncon_test = consumption[upper_train:upper_test] \ncon_test_set = con_test.iloc[:,].values\n\n#scaling\nconsumption_X_test_scaled = sc.fit_transform(np.float64(con_test_set))\n\n#creating input data\nconsumption_X_test_matrix = []\nfor i in range(timesteps, testset_length + timesteps):\n consumption_X_test_matrix.append(consumption_X_test_scaled[i-timesteps:i, 0])\nconsumption_X_test_matrix = np.array(consumption_X_test_matrix)\n\n\n#reshaping\nconsumption_X_test_reshaped = np.reshape(consumption_X_test_matrix, \n (consumption_X_test_matrix.shape[0], \n consumption_X_test_matrix.shape[1], 1))\n\n```\n\n\n```python\nprint('Test Set Shape:', consumption_X_test_reshaped.shape)\n```\n\n Test Set Shape: (8128, 168, 1)\n\n\n## Prediction\n\n\n```python\nconsumption_y_hat = regressor_loaded.predict(consumption_X_test_reshaped, batch_size=batch_size)\nregressor_loaded.reset_states()\n\nprint('Predicted Set Shape:', consumption_y_hat.shape)\n```\n\n Predicted Set Shape: (8128, 168, 1)\n\n\n\n```python\n#reshaping\nconsumption_y_hat = np.reshape(consumption_y_hat, \n (consumption_y_hat.shape[0], \n consumption_y_hat.shape[1]))\n\nprint('Predicted Set Shape After reshaping:', consumption_y_hat.shape)\n```\n\n Predicted Set Shape After reshaping: (8128, 168)\n\n\n\n```python\n#inverse transform\nconsumption_y_hat = sc.inverse_transform(consumption_y_hat)\n```\n\n\n```python\n#creating y_pred data\nconsumption_y_pred = []\n\nfor j in range(0, testset_length - timesteps):\n consumption_y_pred = np.append(consumption_y_pred, consumption_y_hat[j, timesteps-1])\n\n# reshaping\nconsumption_y_pred = np.reshape(consumption_y_pred, (consumption_y_pred.shape[0], 1))\n\nprint('Predicted Values: ', consumption_y_pred.shape)\n```\n\n Predicted Values: (7960, 1)\n\n\n## Evaluating\n\n\n```python\nfrom sklearn.metrics import mean_squared_error\n```\n\n\n```python\nRMSE= np.sqrt(mean_squared_error(consumption_y_pred[0:len(consumption_y_pred) - timesteps].flatten(),\n con_test_set[timesteps:len(consumption_y_pred)].flatten()))\nprint(RMSE)\n```\n\n 3651.061804982601\n\n\n\n```python\nx_index = consumption[upper_train+timesteps:upper_test].index\n```\n\n\n```python\n# Visualising the results\nfig= plt.figure(figsize=(15,5))\nax = fig.add_subplot(111)\n\nplt.plot(x_index,con_test_set[timesteps:len(consumption_y_pred)].astype(float), \n color = 'red', label = 'Real Values')\nplt.plot(x_index,consumption_y_pred[0:len(consumption_y_pred) - timesteps].astype(float), \n color = 'blue', label = 'Predicted Values')\n\nplt.title('\\nActual vs Predicted Values for Electricity\\n Consumption in Germany in 2019 using LSTM\\n', \n fontsize=20 ,fontweight='bold')\nplt.xlabel('')\nplt.ylabel('Electricity Unit [MWh]', fontsize=15)\nplt.legend()\nplt.show()\n```\n\n\n```python\n# Visualising the results\nfig= plt.figure(figsize=(15,5))\nax = fig.add_subplot(111)\n\nplt.plot(x_index,con_test_set[timesteps:len(consumption_y_pred)].astype(float), \n color = 'red', label = 'Real Values')\nplt.plot(x_index,consumption_y_pred[0:len(consumption_y_pred) - timesteps].astype(float), \n color = 'blue', label = 'Predicted Values')\n\nplt.title('\\nActual vs Predicted Values for Electricity\\n Consumption in Germany in February 2019 using LSTM\\n', \n fontsize=20 ,fontweight='bold')\nplt.xlabel('')\nplt.xlim(date2num(pd.to_datetime(['2019-02-01','2019-03-01'])))\nplt.ylabel('Electricity Unit [MWh]', fontsize=15)\nplt.legend()\nplt.show()\n```\n\n## Run LSTM Model for Electricity Generation\n\nThe model is ran on IBM Watson Studio. It is then stored to IBM Object Storage and now we can download it and load it.\n\n\n```python\nregressor_loaded = load_model(filepath=\"../Models/LSTM_initial_Model_Generation.h5\")\n```\n\n## Test Set\n\n\n```python\ntest_length = get_test_length(generation, batch_size)\nupper_test = test_length + timesteps*2\ntestset_length = test_length - upper_train\nprint('Test Set Size:' ,testset_length)\n```\n\n Test Set Size: 8128\n\n\n\n```python\nprint('Train Set Endpoint: ', upper_train)\nprint('Test Set Endpoint: ', upper_test)\nprint('Size of Dataframe: ',len(generation))\n```\n\n Train Set Endpoint: 35344\n Test Set Endpoint: 43808\n Size of Dataframe: 43824\n\n\n\n```python\n# construct test set\n\n#subsetting\ngen_test = generation[upper_train:upper_test] \ngen_test_set = gen_test.iloc[:,].values\n\n#scaling\ngeneration_X_test_scaled = sc.fit_transform(np.float64(gen_test_set))\n\n#creating input data\ngeneration_X_test_matrix = []\nfor i in range(timesteps, testset_length + timesteps):\n generation_X_test_matrix.append(generation_X_test_scaled[i-timesteps:i, 0])\ngeneration_X_test_matrix = np.array(generation_X_test_matrix)\n\n\n#reshaping\ngeneration_X_test_reshaped = np.reshape(generation_X_test_matrix, \n (generation_X_test_matrix.shape[0], \n generation_X_test_matrix.shape[1], 1))\n\n```\n\n\n```python\nprint('Test Set Shape:', generation_X_test_reshaped.shape)\n```\n\n Test Set Shape: (8128, 168, 1)\n\n\n## Prediction\n\n\n```python\ngeneration_y_hat = regressor_loaded.predict(generation_X_test_reshaped, batch_size=batch_size)\nregressor_loaded.reset_states()\n\nprint('Predicted Set Shape:', generation_y_hat.shape)\n```\n\n Predicted Set Shape: (8128, 168, 1)\n\n\n\n```python\n#reshaping\ngeneration_y_hat = np.reshape(generation_y_hat, \n (generation_y_hat.shape[0], \n generation_y_hat.shape[1]))\n\nprint('Predicted Set Shape After reshaping:', generation_y_hat.shape)\n```\n\n Predicted Set Shape After reshaping: (8128, 168)\n\n\n\n```python\n#inverse transform\ngeneration_y_hat = sc.inverse_transform(generation_y_hat)\n```\n\n\n```python\n#creating y_pred data\ngeneration_y_pred = []\n\nfor j in range(0, testset_length - timesteps):\n generation_y_pred = np.append(generation_y_pred, generation_y_hat[j, timesteps-1])\n\n# reshaping\ngeneration_y_pred = np.reshape(generation_y_pred, (generation_y_pred.shape[0], 1))\n\nprint('Predicted Values: ', generation_y_pred.shape)\n```\n\n Predicted Values: (7960, 1)\n\n\n## Evaluating\n\n\n```python\nfrom sklearn.metrics import mean_squared_error\n```\n\n\n```python\ny_true= gen_test_set[timesteps:len(generation_y_pred)].flatten()\ny_pred= generation_y_pred[0:len(generation_y_pred) - timesteps].flatten()\n```\n\n\n```python\nRMSE= np.sqrt(mean_squared_error(y_true,y_pred))\nprint('RMSE:',RMSE)\n```\n\n RMSE: 5803.153205714402\n\n\n\n```python\nmean= np.mean(y_true)\nprint('Loss: ' ,round((RMSE/mean)*100,2),'%')\n```\n\n Loss: 10.05 %\n\n\n\n```python\n# Visualising the results\nfig= plt.figure(figsize=(15,5))\nax = fig.add_subplot(111)\n\nplt.plot(x_index,gen_test_set[timesteps:len(generation_y_pred)].astype(float), \n color = 'red', label = 'Real Values')\nplt.plot(x_index,generation_y_pred[0:len(generation_y_pred) - timesteps].astype(float), \n color = 'blue', label = 'Predicted Values')\n\nplt.title('\\nActual vs Predicted Values for Electricity\\n Generation in Germany in 2019 using LSTM\\n', \n fontsize=20 ,fontweight='bold')\nplt.xlabel('')\nplt.ylabel('Electricity Unit [MWh]', fontsize=15)\nplt.legend()\nplt.show()\n```\n\n\n```python\n# Visualising the results\nfig= plt.figure(figsize=(15,5))\nax = fig.add_subplot(111)\n\nplt.plot(x_index,gen_test_set[timesteps:len(generation_y_pred)].astype(float), \n color = 'red', label = 'Real Values')\nplt.plot(x_index,generation_y_pred[0:len(generation_y_pred) - timesteps].astype(float), \n color = 'blue', label = 'Predicted Values')\n\nplt.title('\\nActual vs Predicted Values for Electricity\\n Generation in Germany in August 2019 using LSTM\\n', \n fontsize=20 ,fontweight='bold')\nplt.xlabel('')\nplt.xlim(date2num(pd.to_datetime(['2019-08-01','2019-09-01'])))\nplt.ylabel('Electricity Unit [MWh]', fontsize=15)\nplt.legend()\nplt.show()\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "67c7d5676d60545e12bd09e69cd3aabc9233a65c", "size": 494234, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Notebooks/.ipynb_checkpoints/03. 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YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.6370307806984444, "lm_q1q2_score": 0.41039122253055055}} {"text": "\n\n# Neuromatch Academy: Week 3, Day 1, Tutorial 3\n# Real Neurons: Synaptic transmission - Models of static and dynamic synapses\n__Content creators:__ Qinglong Gu, Songtin Li, John Murray, Richard Naud, Arvind Kumar\n\n__Content reviewers:__ Maryam Vaziri-Pashkam, Ella Batty, Lorenzo Fontolan, Richard Gao, Matthew Krause, Spiros Chavlis, Michael Waskom\n\n---\n# Tutorial Objectives\nSynapses connect neurons into neural networks or circuits. Specialized electrical synapses make direct, physical connections between neurons. In this tutorial, however, we will focus on **chemical synapses**, which are more common in the brain. These synapses do not physically join neurons. Instead, a spike in the presynaptic cell causes a chemical, or neurotransmitter, to be released into a small space between the neurons called the synaptic cleft. Once the chemical diffuses across that space, it changes the pearmeability of the postsynaptic membrane, which may result in a positive or negative change in the membrane voltage.\n\nIn this tutorial, we will model chemical synaptic transmission and study some interesting effects produced by **static synapses** and **dynamic synapses**.\n\nFirst, we will start by writing code to simulate static synapses -- whose weight is always fixed. \nNext, we will extend the model and model **dynamic synapses** -- whose synaptic strength is dependent on the recent spike history: synapses can either progressively increase or decrease the size of their effects on the post-synaptic neuron, based on the recent firing rate of its presynaptic partners. This feature of synapses in the brain is called **Short-Term Plasticity** and causes synapses to undergo *Facilitation* or *Depression*. \n\nOur goals for this tutorial are to:\n\n- simulate static synapses and study how excitation and inhibition affect the patterns in the neurons' spiking output\n- define mean- or fluctuation-driven regimes\n- simulate short-term dynamics of synapses (facilitation and depression)\n- study how a change in pre-synaptic firing history affects the synaptic weights (i.e., PSP amplitude)\n\n\n```python\n#@title Video 1: Static and dynamic synapses\nfrom IPython.display import YouTubeVideo\nvideo = YouTubeVideo(id='Hbz2lj2AO_0', width=854, height=480, fs=1)\nprint(\"Video available at https://youtube.com/watch?v=\" + video.id)\nvideo\n```\n\n Video available at https://youtube.com/watch?v=Hbz2lj2AO_0\n\n\n\n\n\n\n\n\n\n\n\n---\n# Setup\n\n\n\n```python\n# Import libraries\nimport matplotlib.pyplot as plt\nimport numpy as np\n```\n\n\n```python\n# @title Figure Settings\nimport ipywidgets as widgets # interactive display\n%config InlineBackend.figure_format='retina'\n# use NMA plot style\nplt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/course-content/master/nma.mplstyle\")\nmy_layout = widgets.Layout()\n```\n\n\n```python\n# @title Helper functions\n\n\ndef my_GWN(pars, mu, sig, myseed=False):\n \"\"\"\n Args:\n pars : parameter dictionary\n mu : noise baseline (mean)\n sig : noise amplitute (standard deviation)\n myseed : random seed. int or boolean\n the same seed will give the same random number sequence\n\n Returns:\n I : Gaussian White Noise (GWN) input\n \"\"\"\n\n # Retrieve simulation parameters\n dt, range_t = pars['dt'], pars['range_t']\n Lt = range_t.size\n\n # set random seed\n # you can fix the seed of the random number generator so that the results\n # are reliable. However, when you want to generate multiple realizations\n # make sure that you change the seed for each new realization.\n if myseed:\n np.random.seed(seed=myseed)\n else:\n np.random.seed()\n\n # generate GWN\n # we divide here by 1000 to convert units to seconds.\n I_GWN = mu + sig * np.random.randn(Lt) / np.sqrt(dt / 1000.)\n\n return I_GWN\n\n\ndef Poisson_generator(pars, rate, n, myseed=False):\n \"\"\"\n Generates poisson trains\n\n Args:\n pars : parameter dictionary\n rate : noise amplitute [Hz]\n n : number of Poisson trains\n myseed : random seed. int or boolean\n\n Returns:\n pre_spike_train : spike train matrix, ith row represents whether\n there is a spike in ith spike train over time\n (1 if spike, 0 otherwise)\n \"\"\"\n\n # Retrieve simulation parameters\n dt, range_t = pars['dt'], pars['range_t']\n Lt = range_t.size\n\n # set random seed\n if myseed:\n np.random.seed(seed=myseed)\n else:\n np.random.seed()\n\n # generate uniformly distributed random variables\n u_rand = np.random.rand(n, Lt)\n\n # generate Poisson train\n poisson_train = 1. * (u_rand < rate * (dt / 1000.))\n\n return poisson_train \n\n\ndef default_pars(**kwargs):\n pars = {}\n\n ### typical neuron parameters###\n pars['V_th'] = -55. # spike threshold [mV]\n pars['V_reset'] = -75. # reset potential [mV]\n pars['tau_m'] = 10. # membrane time constant [ms]\n pars['g_L'] = 10. # leak conductance [nS]\n pars['V_init'] = -65. # initial potential [mV]\n pars['E_L'] = -75. # leak reversal potential [mV]\n pars['tref'] = 2. # refractory time (ms)\n\n ### simulation parameters ###\n pars['T'] = 400. # Total duration of simulation [ms]\n pars['dt'] = .1 # Simulation time step [ms]\n\n ### external parameters if any ###\n for k in kwargs:\n pars[k] = kwargs[k]\n\n pars['range_t'] = np.arange(0, pars['T'], pars['dt']) # Vector of discretized time points [ms]\n \n return pars\n\n\ndef my_illus_LIFSYN(pars, v_fmp, v):\n \"\"\"\n Illustartion of FMP and membrane voltage\n\n Args:\n pars : parameters dictionary\n v_fmp : free membrane potential, mV\n v : membrane voltage, mV\n\n Returns:\n plot of membrane voltage and FMP, alongside with the spiking threshold\n and the mean FMP (dashed lines)\n \"\"\"\n\n plt.figure(figsize=(14, 5))\n plt.plot(pars['range_t'], v_fmp, 'r', lw=1.,\n label='Free mem. pot.', zorder=2)\n plt.plot(pars['range_t'], v, 'b', lw=1.,\n label='True mem. pot', zorder=1, alpha=0.7)\n plt.axhline(-55, 0, 1, color='k', lw=2., ls='--',\n label='Spike Threshold', zorder=1)\n plt.axhline(np.mean(v_fmp), 0, 1, color='r', lw=2., ls='--',\n label='Mean Free Mem. Pot.', zorder=1)\n plt.xlabel('Time (ms)')\n plt.ylabel('V (mV)')\n plt.legend(loc=[1.02, 0.68])\n plt.show()\n\n\ndef my_illus_STD(Poi_or_reg=False, rate=20., U0=0.5,\n tau_d=100., tau_f=50., plot_out=True):\n \"\"\"\n Only for one presynaptic train\n\n Args:\n Poi_or_reg : Poisson or regular input spiking trains\n rate : Rate of input spikes, Hz\n U0 : synaptic release probability at rest\n tau_d : synaptic depression time constant of x [ms]\n tau_f : synaptic facilitation time constantr of u [ms]\n plot_out : whether ot not to plot, True or False\n\n Returns:\n Nothing.\n \"\"\"\n\n T_simu = 10.0 * 1000 / (1.0 * rate) # 10 spikes in the time window\n pars = default_pars(T=T_simu)\n dt = pars['dt']\n\n if Poi_or_reg:\n # Poisson type spike train\n pre_spike_train = Poisson_generator(pars, rate, n=1)\n pre_spike_train = pre_spike_train.sum(axis=0)\n else:\n # Regular firing rate\n isi_num = int((1e3/rate)/dt) # number of dt\n pre_spike_train = np.zeros(len(pars['range_t']))\n pre_spike_train[::isi_num] = 1.\n\n u, R, g = dynamic_syn(g_bar=1.2, tau_syn=5., U0=U0,\n tau_d=tau_d, tau_f=tau_f,\n pre_spike_train=pre_spike_train,\n dt=pars['dt'])\n\n if plot_out:\n plt.figure(figsize=(12, 6))\n\n plt.subplot(221)\n plt.plot(pars['range_t'], R, 'b', label='R')\n plt.plot(pars['range_t'], u, 'r', label='u')\n plt.legend(loc='best')\n plt.xlim((0, pars['T']))\n plt.ylabel(r'$R$ or $u$ (a.u)')\n plt.subplot(223)\n spT = pre_spike_train > 0\n t_sp = pars['range_t'][spT] #spike times \n plt.plot(t_sp, 0. * np.ones(len(t_sp)), 'k|', ms=18, markeredgewidth=2)\n plt.xlabel('Time (ms)');\n plt.xlim((0, pars['T']))\n plt.yticks([])\n plt.title('Presynaptic spikes')\n\n plt.subplot(122)\n plt.plot(pars['range_t'], g, 'r', label='STP synapse')\n plt.xlabel('Time (ms)')\n plt.ylabel('g (nS)')\n plt.xlim((0, pars['T']))\n\n plt.tight_layout()\n\n if not Poi_or_reg:\n return g[isi_num], g[9*isi_num]\n\n\ndef plot_volt_trace(pars, v, sp):\n \"\"\"\n Plot trajetory of membrane potential for a single neuron\n\n Args:\n pars : parameter dictionary\n v : volt trajetory\n sp : spike train\n\n Returns:\n figure of the membrane potential trajetory for a single neuron\n \"\"\"\n\n V_th = pars['V_th']\n dt = pars['dt']\n if sp.size:\n sp_num = (sp/dt).astype(int) - 1\n v[sp_num] += 10\n\n plt.plot(pars['range_t'], v, 'b')\n plt.axhline(V_th, 0, 1, color='k', ls='--', lw=1.)\n plt.xlabel('Time (ms)')\n plt.ylabel('V (mV)')\n```\n\nIn the `Helper Function`:\n\n- Gaussian white noise generator: `my_GWN(pars, mu, sig, myseed=False)`\n- Poissonian spike train generator: `Poisson_generator(pars, rate, n, myseed=False)`\n- default parameter function (as before) and other plotting utilities\n\n---\n# Section 1: Static synapses\n\n## Section 1.1: Simulate synaptic conductance dynamics\n\nSynaptic input _in vivo_ consists of a mixture of **excitatory** neurotransmitters, which depolarizes the cell and drives it towards spike threshold, and **inhibitory** neurotransmitters that hyperpolarize it, driving it away from spike threshold. These chemicals cause specific ion channels on the postsynaptic neuron to open, resulting in a change in that neuron's conductance and, therefore, the flow of current in or out of the cell.\n\nThis process can be modelled by assuming that the presynaptic neuron's spiking activity produces transient changes in the postsynaptic neuron's conductance ($g_{\\rm syn}(t)$). Typically, the conductance transient is modeled as an exponential function. \n\nSuch conductance transients can be generated using a simple ordinary differential equation (ODE):\n\n\\\\\n\n\\begin{eqnarray}\n\\frac{dg_{\\rm syn}(t)}{dt} &=& \\bar{g}_{\\rm syn} \\sum_k \\delta(t-t_k) -g_{\\rm syn}(t)/\\tau_{\\rm syn}\n\\end{eqnarray}\n\n\\\\\n\nwhere $\\bar{g}_{\\rm syn}$ (often referred to as synaptic weight) is the maximum conductance elicited by each incoming spike, and $\\tau_{\\rm syn}$ is the synaptic time constant. Note that the summation runs over all spikes received by the neuron at time $t_k$.\n\nOhm's law allows us to convert conductance changes to the current as:\n\n\\\\\n\n\\begin{align}\nI_{\\rm syn}(t) = g_{\\rm syn}(t)(V(t)-E_{\\rm syn}) \\\\\n\\end{align}\n\n\\\\\n\nThe reversal potential $E_{\\rm syn}$ determines the direction of current flow and the excitatory or inhibitory nature of the synapse. \n\n**Thus, incoming spikes are filtered by an exponential-shaped kernel, effectively low-pass filtering the input. In other words, synaptic input is not white noise, but it is, in fact, colored noise, where the color (spectrum) of the noise is determined by the synaptic time constants of both excitatory and inhibitory synapses.**\n\nIn a neuronal network, the total synaptic input current $I_{\\rm syn}$ is the sum of both excitatory and inhibitory inputs. Assuming the total excitatory and inhibitory conductances received at time $t$ are $g_E(t)$ and $g_I(t)$, and their corresponding reversal potentials are $E_E$ and $E_I$, respectively, then the total synaptic current can be described as: \n\n\\\\\n\n\\begin{align}\nI_{\\rm syn}(V(t),t) = -g_E(t) (V-E_E) - g_I(t) (V-E_I)\n\\end{align}\n\n\\\\\n\nAccordingly, the membrane potential dynamics of the LIF neuron under synaptic current drive become:\n\n\\\\\n\n\\begin{eqnarray}\n\\tau_m\\frac{dV(t)}{dt} = -(V(t)-E_L) - \\frac{g_E(t)}{g_L} (V(t)-E_E) - \\frac{g_I(t)}{g_L} (V(t)-E_I) + \\frac{I_{\\rm inj}}{g_L}\\quad (2)\n\\end{eqnarray}\n\n\\\\\n\n$I_{\\rm inj}$ is an external current injected in the neuron, which is under experimental control; it can be GWN, DC, or anything else.\n\nWe will use Eq. (2) to simulate the conductance-based LIF neuron model below.\n\nIn the previous tutorials, we saw how the output of a single neuron (spike count/rate and spike time irregularity) changes when we stimulate the neuron with DC and GWN, respectively. Now, we are in a position to study how the neuron behaves when it is bombarded with both excitatory and inhibitory spikes trains -- as happens *in vivo*.\n\nWhat kind of input is a neuron receiving? When we do not know, we chose the simplest option. The simplest model of input spikes is given when every input spike arrives independently of other spikes, i.e., we assume that the input is Poissonian.\n\n## Section 1.2: Simulate LIF neuron with conductance-based synapses\n\nWe are now ready to simulate a LIF neuron with conductance-based synaptic inputs! The following code defines the LIF neuron with synaptic input modeled as conductance transients.\n\n\n```python\n# @markdown Execute this cell to get a function for conductance-based LIF neuron (run_LIF_cond)\n\ndef run_LIF_cond(pars, I_inj, pre_spike_train_ex, pre_spike_train_in):\n \"\"\"\n Conductance-based LIF dynamics\n\n Args:\n pars : parameter dictionary\n I_inj : injected current [pA]. The injected current here\n can be a value or an array\n pre_spike_train_ex : spike train input from presynaptic excitatory neuron\n pre_spike_train_in : spike train input from presynaptic inhibitory neuron\n\n Returns:\n rec_spikes : spike times\n rec_v : mebrane potential\n gE : postsynaptic excitatory conductance\n gI : postsynaptic inhibitory conductance\n\n \"\"\"\n\n # Retrieve parameters\n V_th, V_reset = pars['V_th'], pars['V_reset']\n tau_m, g_L = pars['tau_m'], pars['g_L']\n V_init, E_L = pars['V_init'], pars['E_L']\n gE_bar, gI_bar = pars['gE_bar'], pars['gI_bar']\n VE, VI = pars['VE'], pars['VI']\n tau_syn_E, tau_syn_I = pars['tau_syn_E'], pars['tau_syn_I']\n tref = pars['tref']\n dt, range_t = pars['dt'], pars['range_t']\n Lt = range_t.size\n\n # Initialize\n tr = 0.\n v = np.zeros(Lt)\n v[0] = V_init\n gE = np.zeros(Lt)\n gI = np.zeros(Lt)\n Iinj = I_inj * np.ones(Lt) # ensure Iinj has length Lt\n\n if pre_spike_train_ex.max() == 0:\n pre_spike_train_ex_total = np.zeros(Lt)\n else:\n pre_spike_train_ex_total = pre_spike_train_ex.sum(axis=0) * np.ones(Lt)\n\n if pre_spike_train_in.max() == 0:\n pre_spike_train_in_total = np.zeros(Lt)\n else:\n pre_spike_train_in_total = pre_spike_train_in.sum(axis=0) * np.ones(Lt)\n\n # simulation\n rec_spikes = [] # recording spike times\n for it in range(Lt - 1):\n if tr > 0:\n v[it] = V_reset\n tr = tr - 1\n elif v[it] >= V_th: # reset voltage and record spike event\n rec_spikes.append(it)\n v[it] = V_reset\n tr = tref / dt\n\n # update the synaptic conductance\n gE[it + 1] = gE[it] - (dt / tau_syn_E) * gE[it] + gE_bar * pre_spike_train_ex_total[it + 1]\n gI[it + 1] = gI[it] - (dt / tau_syn_I) * gI[it] + gI_bar * pre_spike_train_in_total[it + 1]\n\n # calculate the increment of the membrane potential\n dv = (dt / tau_m) * (-(v[it] - E_L) \\\n - (gE[it + 1] / g_L) * (v[it] - VE) \\\n - (gI[it + 1] / g_L) * (v[it] - VI) + Iinj[it] / g_L)\n\n # update membrane potential\n v[it+1] = v[it] + dv\n\n rec_spikes = np.array(rec_spikes) * dt\n\n return v, rec_spikes, gE, gI\n\n\nprint(help(run_LIF_cond))\n```\n\n Help on function run_LIF_cond in module __main__:\n \n run_LIF_cond(pars, I_inj, pre_spike_train_ex, pre_spike_train_in)\n Conductance-based LIF dynamics\n \n Args:\n pars : parameter dictionary\n I_inj : injected current [pA]. The injected current here\n can be a value or an array\n pre_spike_train_ex : spike train input from presynaptic excitatory neuron\n pre_spike_train_in : spike train input from presynaptic inhibitory neuron\n \n Returns:\n rec_spikes : spike times\n rec_v : mebrane potential\n gE : postsynaptic excitatory conductance\n gI : postsynaptic inhibitory conductance\n \n None\n\n\n### Exercise 1: Measure the mean free membrane potential\n\nLet's simulate the conductance-based LIF neuron with presynaptic spike trains generated by a `Poisson_generator` with rate 10 Hz for both excitatory and inhibitory inputs. Here, we choose 80 excitatory presynaptic spike trains and 20 inhibitory ones.\n\nPreviously, we've already learned that $CV_{\\rm ISI}$ can describe the irregularity of the output spike pattern. Now, we will introduce a new descriptor of the neuron membrane, i.e., the **Free Membrane Potential (FMP)** -- the membrane potential of the neuron when its spike threshold is removed. \n\nAlthough this is completely artificial, calculating this quantity allows us to get an idea of how strong the input is. We are mostly interested in knowing the mean and standard deviation (std.) of the FMP. In the exercise, you can visualize the FMP and membrane voltage with spike threshold.\n\n\n```python\n# To complete the exercise, uncomment the code and fill the missing parts (...)\npars = default_pars(T=1000.)\n# Add parameters\npars['gE_bar'] = 2.4 # [nS]\npars['VE'] = 0. # [mV] excitatory reversal potential\npars['tau_syn_E'] = 2. # [ms]\npars['gI_bar'] = 2.4 # [nS]\npars['VI'] = -80. # [mV] inhibitory reversal potential\npars['tau_syn_I'] = 5. # [ms]\n\n# generate presynaptic spike trains\npre_spike_train_ex = Poisson_generator(pars, rate=10, n=80)\npre_spike_train_in = Poisson_generator(pars, rate=10, n=20)\n\n# simulate conductance-based LIF model\nv, rec_spikes, gE, gI = run_LIF_cond(pars, 0, pre_spike_train_ex,\n pre_spike_train_in)\ndt, range_t = pars['dt'], pars['range_t']\nif rec_spikes.size:\n sp_num = (rec_spikes / dt).astype(int) - 1\n v[sp_num] = 10 # draw nicer spikes\n\n####################################################################\n## TODO for students: measure the free membrane potential\n# In order to measure the free membrane potential, first,\n# you should prevent the firing of the LIF neuron\n# How to prevent a LIF neuron from firing? Increse the threshold pars['V_th'].\n####################################################################\n# Change the threshold\npars['V_th'] = 1e5\n# Calculate FMP\nv_fmp, _, _, _ = run_LIF_cond(pars, 0, pre_spike_train_ex, pre_spike_train_in)\n\n# uncomment when you have filled the exercise\nmy_illus_LIFSYN(pars, v_fmp, v)\n```\n\n\n```python\n# to_remove solution\npars = default_pars(T=1000.)\n# Add parameters\npars['gE_bar'] = 2.4 # [nS]\npars['VE'] = 0. # [mV] excitatory reversal potential\npars['tau_syn_E'] = 2. # [ms]\npars['gI_bar'] = 2.4 # [nS]\npars['VI'] = -80. # [mV] inhibitory reversal potential\npars['tau_syn_I'] = 5. # [ms]\n\n# generate presynaptic spike trains\npre_spike_train_ex = Poisson_generator(pars, rate=10, n=80)\npre_spike_train_in = Poisson_generator(pars, rate=10, n=20)\n\n# simulate conductance-based LIF model\nv, rec_spikes, gE, gI = run_LIF_cond(pars, 0, pre_spike_train_ex,\n pre_spike_train_in)\ndt, range_t = pars['dt'], pars['range_t']\nif rec_spikes.size:\n sp_num = (rec_spikes / dt).astype(int) - 1\n v[sp_num] = 10 # draw nicer spikes\n\n# Change the threshold\npars['V_th'] = 1e3\n# Calculate FMP\nv_fmp, _, _, _ = run_LIF_cond(pars, 0, pre_spike_train_ex, pre_spike_train_in)\n\nwith plt.xkcd():\n my_illus_LIFSYN(pars, v_fmp, v)\n```\n\n### Interactive Demo: Conductance-based LIF Explorer with different E/I input\n\nIn the following, we can investigate how varying the ratio of excitatory to inhibitory inputs changes the firing rate and the spike time regularity (see the output text). \n\nTo change both the excitatory and inhibitory inputs, we will vary their firing rates. *However, if you wish, you can vary the strength and/or the number of these connections as well.* \n\nPay close attention to the mean free membrane potential (red dotted line) and its location with respect to the spike threshold (black dotted line). Try to develop a heuristic about the mean of the FMP and spike time irregularity ($CV_{\\rm ISI}$)\n\n\n```python\n# @title\n\n# @markdown Make sure you execute this cell to enable the widget!\n\nmy_layout.width = '450px'\n@widgets.interact(\n inh_rate=widgets.FloatSlider(20., min=10., max=60., step=5.,\n layout=my_layout),\n exc_rate=widgets.FloatSlider(10., min=2., max=20., step=2.,\n layout=my_layout)\n)\n\n\ndef EI_isi_regularity(exc_rate, inh_rate):\n\n pars = default_pars(T=1000.)\n # Add parameters\n pars['gE_bar'] = 3. # [nS]\n pars['VE'] = 0. # [mV] excitatory reversal potential\n pars['tau_syn_E'] = 2. # [ms]\n pars['gI_bar'] = 3. # [nS]\n pars['VI'] = -80. # [mV] inhibitory reversal potential\n pars['tau_syn_I'] = 5. # [ms]\n\n pre_spike_train_ex = Poisson_generator(pars, rate=exc_rate, n=80)\n pre_spike_train_in = Poisson_generator(pars, rate=inh_rate, n=20) # 4:1\n\n # Lets first simulate a neuron with identical input but with no spike \n # threshold by setting the threshold to a very high value\n # so that we can look at the free membrane potential\n pars['V_th'] = 1e3\n v_fmp, rec_spikes, gE, gI = run_LIF_cond(pars, 0, pre_spike_train_ex,\n pre_spike_train_in)\n\n # Now simulate a LIP with a regular spike threshold\n pars['V_th'] = -55. \n v, rec_spikes, gE, gI = run_LIF_cond(pars, 0, pre_spike_train_ex,\n pre_spike_train_in)\n dt, range_t = pars['dt'], pars['range_t']\n if rec_spikes.size:\n sp_num = (rec_spikes / dt).astype(int) - 1\n v[sp_num] = 10 # draw nicer spikes\n\n spike_rate = 1e3 * len(rec_spikes) / pars['T']\n\n cv_isi = 0.\n if len(rec_spikes) > 3:\n isi = np.diff(rec_spikes)\n cv_isi = np.std(isi) / np.mean(isi)\n\n print('\\n')\n plt.figure(figsize=(15, 10))\n plt.subplot(211)\n plt.text(500, -35, f'Spike rate = {spike_rate:.3f} (sp/s), Mean of Free Mem Pot = {np.mean(v_fmp):.3f}',\n fontsize=16, fontweight='bold', horizontalalignment='center',\n verticalalignment='bottom')\n plt.text(500, -38.5, f'CV ISI = {cv_isi:.3f}, STD of Free Mem Pot = {np.std(v_fmp):.3f}',\n fontsize=16, fontweight='bold', horizontalalignment='center',\n verticalalignment='bottom')\n\n plt.plot(pars['range_t'], v_fmp, 'r', lw=1.,\n label='Free mem. pot.', zorder=2)\n plt.plot(pars['range_t'], v, 'b', lw=1.,\n label='mem. pot with spk thr', zorder=1, alpha=0.7)\n plt.axhline(pars['V_th'], 0, 1, color='k', lw=1., ls='--',\n label='Spike Threshold', zorder=1)\n plt.axhline(np.mean(v_fmp),0, 1, color='r', lw=1., ls='--',\n label='Mean Free Mem. Pot.', zorder=1)\n plt.ylim(-76, -39) \n plt.xlabel('Time (ms)')\n plt.ylabel('V (mV)')\n plt.legend(loc=[1.02, 0.68])\n\n plt.subplot(223)\n plt.plot(pars['range_t'][::3], gE[::3], 'r', lw=1)\n plt.xlabel('Time (ms)')\n plt.ylabel(r'$g_E$ (nS)')\n\n plt.subplot(224)\n plt.plot(pars['range_t'][::3], gI[::3], 'b', lw=1)\n plt.xlabel('Time (ms)')\n plt.ylabel(r'$g_I$ (nS)')\n\n plt.tight_layout()\n```\n\n\n interactive(children=(FloatSlider(value=10.0, description='exc_rate', layout=Layout(width='450px'), max=20.0, \u2026\n\n\n**Mean-driven and Fluctuation-driven regimes**\n\nIf we look at the figure above, we note that when the mean FMP is above spike threshold, the fluctuations in the FMP are rather small, and the neuron spikes in a fairly regular fashion. This regime, where the mean FMP is above the spike threshold, is called **mean-driven regime**. \n\n\nWhen the mean FMP is below the spike threshold, the fluctuations in the FMP are large, and the neuron's spikes are driven by these fluctuations. As a consequence, the neuron spikes in more Poisson-like fashion. This regime, where the mean FMP is below the spike threshold, and spikes are driven by the fluctuations, is called **fluctuation-driven regime**. \n\n## Think!\n\n- How much can you increase the spike pattern variability? Under what condition(s) the neuron may also respond with Poisson-type spikes? Note that we injected Poisson-type spikes. (Think of the answer in terms of the ratio of the exc. and inh. input spike rates.)\n\n- Link to the balance of excitation and inhibition: one of the definitions of excitation and inhibition balance is that mean free membrane potential remains constant as excitatory and inhibitory input rates are increased. What do you think happens to the neuron firing rate as we change excitatory and inhibitory rates while keeping the neuron in balance? See [Kuhn, Aertsen, and Rotter (2004)](https://www.jneurosci.org/content/jneuro/24/10/2345.full.pdf) for much more on this.\n\n\n```python\n# to_remove explanation\n\n\"\"\"\nDiscussion: \n\n1. We can push the neuron to spike almost like a Poisson neuron. Of course given \nthat there is a refractoriness it will never spike completely like a Poisson process. \nPoisson type spike irregularity will be achieved when mean is small (far from the \nspike threshold) and fluctuations are large. This will achieved when excitatory \nand inhibitory rates are balanced -- i.e. ratio of exc and inh. spike rate is \nconstant as you vary the inout rate.\n\n2. Firing rate will increase because fluctuations will increase as we increase \nexc. and inh. rates. But if synapses are modelled as conductance as opposed to \ncurrents, fluctuations may start decrease at high input rates because neuron time \nconstant will drop.\n\n\"\"\";\n```\n\n---\n# Section 2: Short-term synaptic plasticity\nAbove, we modeled synapses with fixed weights. Now we will explore synapses whose weight change in some input conditions. \n\nShort-term plasticity (STP) is a phenomenon in which synaptic efficacy changes over time in a way that reflects the history of presynaptic activity. Two types of STP, with opposite effects on synaptic efficacy, have been experimentally observed. They are known as Short-Term Depression (STD) and Short-Term Facilitation (STF).\n\nThe mathematical model (_for more information see [here](http://www.scholarpedia.org/article/Short-term_synaptic_plasticity)_) of STP is based on the concept of a limited pool of synaptic resources available for transmission ($R$), such as, for example, the overall amount of synaptic vesicles at the presynaptic terminals. The amount of presynaptic resource changes in a dynamic fashion depending on the recent history of spikes. \n\nFollowing a presynaptic spike, (i) the fraction $u$ (release probability) of the available pool to be utilized increases due to spike-induced calcium influx to the presynaptic terminal, after which (ii) $u$ is consumed to increase the post-synaptic conductance. Between spikes, $u$ decays back to zero with time constant $\\tau_f$ and $R$ recovers to 1 with time constant $\\tau_d$. In summary, the dynamics of excitatory (subscript $E$) STP are given by:\n\n\\\\\n\n\\begin{eqnarray}\n&& \\frac{du_E}{dt} &=& -\\frac{u_E}{\\tau_f} + U_0(1-u_E^-)\\delta(t-t_{\\rm sp}) \\\\[.5mm]\n&& \\frac{dR_E}{dt} &=& \\frac{1-R_E}{\\tau_d} - u_E^+ R_E^- \\delta(t-t_{\\rm sp}) \\qquad (6) \\\\[.5mm] \n&& \\frac{dg_E(t)}{dt} &=& -\\frac{g_E}{\\tau_E} + \\bar{g}_E u_E^+ R_E^- \\sum_k \\delta(t-t_{\\rm k})\n\\end{eqnarray}\n\n\\\\\n\nwhere $U_0$ is a constant determining the increment of $u$ produced by a spike. $u_E^-$ and $R_E^-$ denote the corresponding values just before the spike arrives, whereas $u_E^+$ refers to the moment right after the spike. $\\bar{g}_E$ denotes the maximum excitatory conductane, and $g_E(t)$ is calculated for all spiketimes $k$, and decays over time with a time constant $\\tau_{E}$. Similarly, one can obtain the dynamics of inhibitory STP (i.e., by replacing the subscript $E$ with $I$).\n\n\nThe interplay between the dynamics of $u$ and $R$ determines whether the joint effect of $uR$ is dominated by *depression* or *facilitation*. In the parameter regime of $\\tau_d \\gg \\tau_f$ and for large $U_0$, an initial spike incurs a large drop in $R$ that takes a long time to recover; therefore, the synapse is STD-dominated. In the regime of $\\tau_d \\ll \\tau_f$ and for small $U_0$, the synaptic efficacy is increased gradually by spikes, and consequently, the synapse is STF-dominated. This phenomenological model successfully reproduces the kinetic dynamics of depressed and facilitated synapses observed in many cortical areas.\n\n## Exercise 2: Compute $du$, $dR$ and $dg$\n\nAs we learned in several previous tutorials, the Euler numerical integration method involves the calculation of each derivative at step $n$:\n\n\\\\\n\n\\begin{eqnarray}\ndu_E &=& -\\frac{u_E[t]}{\\tau_f} dt + U_0(1-u_E[t])\\cdot \\text{sp_or_not[t+dt]} \\\\\ndR_E &=& \\frac{1-R_E[t]}{\\tau_d} dt - u_E[t+dt]R_E[t]\\cdot \\text{sp_or_not[t+dt]} \\\\\ndg_E &=& -\\frac{g_E[t]}{\\tau_{E}} dt + \\bar{g}_Eu_E[t+dt]R_E[t]\\cdot \\text{sp_or_not[t+dt]} \\\\\n\\end{eqnarray}\n\n\\\\\n\nwhere $\\text{sp_or_not}=1$ if there's a spike in the time window $dt$, and $\\text{sp_or_not}=0$ otherwise. In addition, note that any spike train generated by our `Poisson_generator` is binary. Then, the values are updated:\n\n\\\\\n\n\\begin{eqnarray}\n u_E[t+dt] &=& u_E[t] + du_E \\\\\n R_E[t+dt] &=& R_E[t] + dR_E \\\\\n g_E[t+dt] &=& g_E[t] + dg_E \\\\\n\\end{eqnarray}\n\n\\\\\n\nSimilarly, one can obtain the dynamics of inhibitory conductance.\n\n\n\n```python\ndef dynamic_syn(g_bar, tau_syn, U0, tau_d, tau_f, pre_spike_train, dt):\n \"\"\"\n Short-term synaptic plasticity\n\n Args:\n g_bar : synaptic conductance strength\n tau_syn : synaptic time constant [ms]\n U0 : synaptic release probability at rest\n tau_d : synaptic depression time constant of x [ms]\n tau_f : synaptic facilitation time constantr of u [ms]\n pre_spike_train : total spike train (number) input\n from presynaptic neuron\n dt : time step [ms]\n\n Returns:\n u : usage of releasable neurotransmitter\n R : fraction of synaptic neurotransmitter resources available\n g : postsynaptic conductance\n\n \"\"\"\n\n Lt = len(pre_spike_train)\n # Initialize\n u = np.zeros(Lt)\n R = np.zeros(Lt)\n R[0] = 1.\n g = np.zeros(Lt)\n\n # simulation\n for it in range(Lt - 1):\n\n #########################################################################\n ## TODO for students: compute du, dx and dg, remove NotImplementedError #\n # Note pre_spike_train[i] is binary, i.e., sp_or_not in the i-th timebin\n # Fill out function and remove\n # raise NotImplementedError(\"Student excercise: compute the STP dynamics\")\n #########################################################################\n # Compute du\n du = - (u[it] * dt) / tau_f + U0 * (1-u[it]) * pre_spike_train[it+1]\n u[it + 1] = u[it] + du\n # Compute dR\n dR = (1 - R[it]) * dt / tau_d - u[it+1] * R[it] * pre_spike_train[it+1] \n R[it + 1] = R[it] + dR\n # Compute dg\n dg = - (g[it] * dt) / tau_syn + g_bar * u[it+1] * R[it] * pre_spike_train[it+1]\n g[it + 1] = g[it] + dg\n\n return u, R, g\n\n\n# Uncomment this line after completing the dynamic_syn function\n_ = my_illus_STD(Poi_or_reg=False, rate=20., U0=0.5, tau_d=100., tau_f=50.)\n```\n\n\n```python\n# to_remove solution\ndef dynamic_syn(g_bar, tau_syn, U0, tau_d, tau_f, pre_spike_train, dt):\n \"\"\"\n Short-term synaptic plasticity\n\n Args:\n g_bar : synaptic conductance strength\n tau_syn : synaptic time constant [ms]\n U0 : synaptic release probability at rest\n tau_d : synaptic depression time constant of x [ms]\n tau_f : synaptic facilitation time constantr of u [ms]\n pre_spike_train : total spike train (number) input\n from presynaptic neuron\n dt : time step [ms]\n\n Returns:\n u : usage of releasable neurotransmitter\n R : fraction of synaptic neurotransmitter resources available\n g : postsynaptic conductance\n\n \"\"\"\n\n Lt = len(pre_spike_train)\n # Initialize\n u = np.zeros(Lt)\n R = np.zeros(Lt)\n R[0] = 1.\n g = np.zeros(Lt)\n\n # simulation\n for it in range(Lt - 1):\n # Compute du\n du = -(dt / tau_f) * u[it] + U0 * (1.0 - u[it]) * pre_spike_train[it + 1]\n u[it + 1] = u[it] + du\n # Compute dR\n dR = (dt / tau_d) * (1.0 - R[it]) - u[it + 1] * R[it] * pre_spike_train[it + 1]\n R[it + 1] = R[it] + dR\n # Compute dg\n dg = -(dt / tau_syn) * g[it] + g_bar * R[it] * u[it + 1] * pre_spike_train[it + 1]\n g[it + 1] = g[it] + dg\n\n return u, R, g\n\n\nwith plt.xkcd():\n _ = my_illus_STD(Poi_or_reg=False, rate=20., U0=0.5, tau_d=100., tau_f=50.)\n```\n\n## Section 2.1: Short-term syaptic depression (STD)\n\n\n### Interactive Demo: STD Explorer with input rate\nBelow, an interactive demo that shows how Short-term synaptic depression (STD) changes for different firing rates of the presynaptic spike train and how the amplitude synaptic conductance $g$ changes with every incoming spike until it reaches its stationary state.\n\nDoes it matter if the neuron fires in a Poisson manner, rather than regularly?\n\n**Note:** `Poi_or_Reg=1`: for *Posisson type* and `Poi_or_Reg=0`: for *regular* presynaptic spikes.\n\n\n```python\n#@title\n\n#@markdown Make sure you execute this cell to enable the widget!\n\n\ndef my_STD_diff_rate(rate, Poi_or_Reg):\n _ = my_illus_STD(Poi_or_reg=Poi_or_Reg, rate=rate)\n\n\n_ = widgets.interact(my_STD_diff_rate, rate=(10., 100.1, 5.),\n Poi_or_Reg=(0, 1, 1))\n```\n\n\n interactive(children=(FloatSlider(value=55.0, description='rate', max=100.1, min=10.0, step=5.0), IntSlider(va\u2026\n\n\n\n```python\n# to_remove explanation\n\n\"\"\"\nDiscussion: \n\nIncreasing the input rate, we decrease the synaptic efficacy, i.e., the synaptic \nconductance decreases. This is the case for both Poisson or a regular spiking input.\nIn case of regular spiking, the synaptic conductance reaches a steady state. This \nwill not happen in the case of Poisson type spikes.\n\"\"\";\n```\n\n### Synaptic depression and presynaptic firing rate\nOnce, I asked an experimentalist about the experimental values of the PSP amplitude produced by a connection between two neocortical excitatory neurons. She asked: \"At what frequency?\" I was confused, but you will understand her question, now that you know that PSP amplitude depends on the spike history, and therefore on the spike rate of the presynaptic neuron. \n\nHere, we will study how the ratio of the synaptic conductance corresponding to the first and 10th spikes change as a function of the presynaptic firing rate (experimentalists often take the ratio of first and second PSPs). \n\nFor computational efficiency, we assume that the presynaptic spikes are regular. This assumption means that we do not have to run multiple trials. \n\n\n```python\n# @markdown STD conductance ratio with different input rate\n# Regular firing rate\ninput_rate = np.arange(5., 40.1, 5.)\ng_1 = np.zeros(len(input_rate)) # record the the PSP at 1st spike\ng_2 = np.zeros(len(input_rate)) # record the the PSP at 10th spike\n\nfor ii in range(len(input_rate)):\n g_1[ii], g_2[ii] = my_illus_STD(Poi_or_reg=False, rate=input_rate[ii],\n plot_out=False, U0=0.5, tau_d=100., tau_f=50)\n\nplt.figure(figsize=(11, 4.5))\n\nplt.subplot(121)\nplt.plot(input_rate, g_1, 'm-o', label='1st Spike')\nplt.plot(input_rate, g_2, 'c-o', label='10th Spike')\nplt.xlabel('Rate [Hz]')\nplt.ylabel('Conductance [nS]')\nplt.legend()\n\nplt.subplot(122)\nplt.plot(input_rate, g_2 / g_1, 'b-o')\nplt.xlabel('Rate [Hz]')\nplt.ylabel(r'Conductance ratio $g_{10}/g_{1}$')\nplt.tight_layout()\n```\n\nAs we increase the input rate the ratio of the first to tenth spike is increased, because the tenth spike conductance becomes smaller. This is a clear evidence of synaptic depression, as using the same amount of current has a smaller effect on the neuron.\n\n## Section 2.2: Short-term synaptic facilitation (STF)\n\n### Interactive Demo: STF explorer with input rate\nBelow, we see an illustration of a short-term facilitation example. Take note of the change in the synaptic variables: `U_0`, `tau_d`, and `tau_f`.\n\n- for STD, `U0=0.5, tau_d=100., tau_f=50.`\n\n- for STP, `U0=0.2, tau_d=100., tau_f=750.`\n \nHow does the synaptic conductance change as we change the input rate? What do you observe in the case of a regular input and a Poisson type one? \n\n\n```python\n# @title\n\n# @markdown Make sure you execute this cell to enable the widget!\n\n\ndef my_STD_diff_rate(rate, Poi_or_Reg):\n _ = my_illus_STD(Poi_or_reg=Poi_or_Reg, rate=rate, U0=0.2, tau_d=100., tau_f=750.)\n\n\n_ = widgets.interact(my_STD_diff_rate, rate=(4., 40.1, 2.),\n Poi_or_Reg=(0, 1, 1))\n```\n\n\n interactive(children=(FloatSlider(value=22.0, description='rate', max=40.1, min=4.0, step=2.0), IntSlider(valu\u2026\n\n\n### Synaptic facilitation and presynaptic firing rate\n\nHere, we will study how the ratio of the synaptic conductance corresponding to the $1^{st}$ and $10^{th}$ spike changes as a function of the presynaptic rate. \n\n\n```python\n# @title STF conductance ratio with different input rates\n# Regular firing rate\ninput_rate = np.arange(2., 40.1, 2.)\ng_1 = np.zeros(len(input_rate)) # record the the PSP at 1st spike\ng_2 = np.zeros(len(input_rate)) # record the the PSP at 10th spike\n\nfor ii in range(len(input_rate)):\n g_1[ii], g_2[ii] = my_illus_STD(Poi_or_reg=False, rate=input_rate[ii],\n plot_out=False,\n U0=0.2, tau_d=100., tau_f=750.)\n\nplt.figure(figsize=(11, 4.5))\n\nplt.subplot(121)\nplt.plot(input_rate, g_1, 'm-o', label='1st Spike')\nplt.plot(input_rate, g_2, 'c-o', label='10th Spike')\nplt.xlabel('Rate [Hz]')\nplt.ylabel('Conductance [nS]')\nplt.legend()\n\nplt.subplot(122)\nplt.plot(input_rate, g_2 / g_1, 'b-o',)\nplt.xlabel('Rate [Hz]')\nplt.ylabel(r'Conductance ratio $g_{10}/g_{1}$')\nplt.tight_layout()\n```\n\n## Think!\n\nWhy does the ratio of the first and tenth spike conductance changes in a non-monotonic fashion for synapses with STF, even though it decreases monotonically for synapses with STD?\n\n\n```python\n# to_remove explanation\n\n\"\"\"\nDiscussion: \n\nBecause we have a facilitatory synapses, as the input rate increases synaptic \nresources released per spike also increase. Therefore, we expect that the synaptic \nconductance will increase with input rate. However, total synaptic resources are \nfinite. And they recover in a finite time. Therefore, at high frequency inputs \nsynaptic resources are rapidly deleted at a higher rate than their recovery, so \nafter first few spikes, only a small number of synaptic resources are left. This \nresults in decrease in the steady-state synaptic conductance at high frequency inputs. \n\"\"\";\n```\n\n---\n# Summary\n\nCongratulations! You have just finished the last tutorial of this day. Here, we saw how to model conductance-based synapses and also how to incorporate short-term dynamics in synaptic weights. \n\nWe covered the:\n\n- static synapses and how excitation and inhibition affect the neuronal output\n- mean- or fluctuation-driven regimes\n- short-term dynamics of synapses (both facilitation and depression)\n\nFinally, we incorporated all the aforementioned tools to study how a change in presynaptic firing history affects the synaptic weights!\n\nThere are many interesting things that you can try on your own to develop a deeper understanding of biological synapses. A couple of those are mentioned below in the optional boxes -- if you have time.\n\nBut now it is time to explore another important feature of biological synapses, i.e., spike timing dependent synaptic plasticity (go to the next tutorial).\n\n---\n# Bonus 1: Conductance-based LIF with STP\n\n\nPreviously, we looked only at how the presynaptic firing rate affects the presynaptic resource availability and thereby the synaptic conductance. It is straightforward to imagine that, while the synaptic conductances are changing, the output of the postsynaptic neuron will change as well. \n\nSo, let's put the STP on synapses impinging on an LIF neuron and see what happens. \n\n\n```python\n# @title Function for conductance-based LIF neuron with STP-synapses\n\ndef run_LIF_cond_STP(pars, I_inj, pre_spike_train_ex, pre_spike_train_in):\n \"\"\"\n conductance-based LIF dynamics\n\n Args:\n pars : parameter dictionary\n I_inj : injected current [pA]\n The injected current here can be a value or an array\n pre_spike_train_ex : spike train input from presynaptic excitatory neuron (binary)\n pre_spike_train_in : spike train input from presynaptic inhibitory neuron (binary)\n\n Returns:\n rec_spikes : spike times\n rec_v : mebrane potential\n gE : postsynaptic excitatory conductance\n gI : postsynaptic inhibitory conductance\n\n \"\"\"\n\n # Retrieve parameters\n V_th, V_reset = pars['V_th'], pars['V_reset']\n tau_m, g_L = pars['tau_m'], pars['g_L']\n V_init, V_L = pars['V_init'], pars['E_L']\n gE_bar, gI_bar = pars['gE_bar'], pars['gI_bar']\n U0E, tau_dE, tau_fE = pars['U0_E'], pars['tau_d_E'], pars['tau_f_E']\n U0I, tau_dI, tau_fI = pars['U0_I'], pars['tau_d_I'], pars['tau_f_I']\n VE, VI = pars['VE'], pars['VI']\n tau_syn_E, tau_syn_I = pars['tau_syn_E'], pars['tau_syn_I']\n tref = pars['tref']\n\n dt, range_t = pars['dt'], pars['range_t']\n Lt = range_t.size\n\n nE = pre_spike_train_ex.shape[0]\n nI = pre_spike_train_in.shape[0]\n\n # compute conductance Excitatory synapses\n uE = np.zeros((nE, Lt))\n RE = np.zeros((nE, Lt))\n gE = np.zeros((nE, Lt))\n for ie in range(nE):\n u, R, g = dynamic_syn(gE_bar, tau_syn_E, U0E, tau_dE, tau_fE,\n pre_spike_train_ex[ie, :], dt)\n\n uE[ie, :], RE[ie, :], gE[ie, :] = u, R, g\n\n gE_total = gE.sum(axis=0)\n\n # compute conductance Inhibitory synapses\n uI = np.zeros((nI, Lt))\n RI = np.zeros((nI, Lt))\n gI = np.zeros((nI, Lt))\n for ii in range(nI):\n u, R, g = dynamic_syn(gI_bar, tau_syn_I, U0I, tau_dI, tau_fI,\n pre_spike_train_in[ii, :], dt)\n\n uI[ii, :], RI[ii, :], gI[ii, :] = u, R, g\n\n gI_total = gI.sum(axis=0)\n\n # Initialize\n v = np.zeros(Lt)\n v[0] = V_init\n Iinj = I_inj * np.ones(Lt) # ensure I has length Lt\n\n # simulation\n rec_spikes = [] # recording spike times\n tr = 0.\n for it in range(Lt - 1):\n if tr > 0:\n v[it] = V_reset\n tr = tr - 1\n elif v[it] >= V_th: # reset voltage and record spike event\n rec_spikes.append(it)\n v[it] = V_reset\n tr = tref / dt\n\n # calculate the increment of the membrane potential\n dv = (dt / tau_m) * (-(v[it] - V_L) \\\n - (gE_total[it + 1] / g_L) * (v[it] - VE) \\\n - (gI_total[it + 1] / g_L) * (v[it] - VI) + Iinj[it] / g_L)\n\n # update membrane potential\n v[it+1] = v[it] + dv\n\n rec_spikes = np.array(rec_spikes) * dt\n\n return v, rec_spikes, uE, RE, gE, RI, RI, gI\n\n\nprint(help(run_LIF_cond_STP))\n```\n\n Help on function run_LIF_cond_STP in module __main__:\n \n run_LIF_cond_STP(pars, I_inj, pre_spike_train_ex, pre_spike_train_in)\n conductance-based LIF dynamics\n \n Args:\n pars : parameter dictionary\n I_inj : injected current [pA]\n The injected current here can be a value or an array\n pre_spike_train_ex : spike train input from presynaptic excitatory neuron (binary)\n pre_spike_train_in : spike train input from presynaptic inhibitory neuron (binary)\n \n Returns:\n rec_spikes : spike times\n rec_v : mebrane potential\n gE : postsynaptic excitatory conductance\n gI : postsynaptic inhibitory conductance\n \n None\n\n\n## Simulation of a postsynaptic neuron with STP synapses driven by Poisson type spike trains\n\nHere we have assumed that both excitatory and inhibitory synapses show short-term depression. Change the nature of synapses and study how spike pattern variability changes.\nIn the interactive demo, `tau_d = 500*tau_ratio (ms)` and `tau_f = 300*tau_ratio (ms)`.\n\nYou should compare the output of this neuron with what you observed in the previous tutorial when synapses were assumed to be static. \n\n_Note: it will take slightly longer time to run each case_\n\n### Interactive Demo: LIF with STP Explorer\n\n\n```python\n# @title\n\n# @markdown Make sure you execute this cell to enable the widget!\n\n\ndef LIF_STP(tau_ratio):\n pars = default_pars(T=1000)\n pars['gE_bar'] = 1.2 * 4 # [nS]\n pars['VE'] = 0. # [mV]\n pars['tau_syn_E'] = 5. # [ms]\n pars['gI_bar'] = 1.6 * 4 # [nS]\n pars['VI'] = -80. # [ms]\n pars['tau_syn_I'] = 10. # [ms]\n\n # here we assume that both Exc and Inh synapses have synaptic depression\n pars['U0_E'] = 0.45\n pars['tau_d_E'] = 500. * tau_ratio # [ms]\n pars['tau_f_E'] = 300. * tau_ratio # [ms]\n\n pars['U0_I'] = 0.45\n pars['tau_d_I'] = 500. * tau_ratio # [ms]\n pars['tau_f_I'] = 300. * tau_ratio # [ms]\n\n pre_spike_train_ex = Poisson_generator(pars, rate=15, n=80)\n pre_spike_train_in = Poisson_generator(pars, rate=15, n=20) # 4:1\n\n v, rec_spikes, uE, RE, gE, uI, RI, gI = run_LIF_cond_STP(pars, 0,\n pre_spike_train_ex,\n pre_spike_train_in)\n\n t_plot_range = pars['range_t'] > 200\n\n plt.figure(figsize=(11, 7))\n plt.subplot(211)\n plot_volt_trace(pars, v, rec_spikes)\n\n plt.subplot(223)\n plt.plot(pars['range_t'][t_plot_range], gE.sum(axis=0)[t_plot_range], 'r')\n plt.xlabel('Time (ms)')\n plt.ylabel(r'$g_E$ (nS)')\n\n plt.subplot(224)\n plt.plot(pars['range_t'][t_plot_range], gI.sum(axis=0)[t_plot_range], 'b')\n plt.xlabel('Time (ms)')\n plt.ylabel(r'$g_I$ (nS)')\n\n plt.tight_layout()\n\n\n_ = widgets.interact(LIF_STP, tau_ratio=(0.2, 1.1, 0.2))\n```\n\n\n interactive(children=(FloatSlider(value=0.6000000000000001, description='tau_ratio', max=1.1, min=0.2, step=0.\u2026\n\n\nWhen we vary the tau_ratio we are increasing `tau_f` and `tau_d` i.e. by increasing `tau_ratio` we are increasing the synaptic depression. The effect is same on both Exc and Inh conductances.\nThis is visible as a clear decrease in the firing rate of the neuron from 300-400ms onwards.\n\nNot much happens in the beginning because synaptic depression takes some time to become visible.\n\nIt is curious that while both excitatory and inhibitory conductances have depressed but output firing rate has still decreased.\n\nThere are two explanations of this:\n1. excitation has depressed more than the inhibition from their starting values.\n2. because synaptic conductances have depressed, membrane fluctuation size has decreased.\n\nWhich is more likely reason? Think.\n\n---\n# Bonus 2: STP Synapse Parameter Exploration\n\nVary the parameters of the above simulation and observe the spiking pattern of the postsynaptic neuron. \nWill the neuron show higher irregularity if the synapses have STP? If yes, what should be the nature of STP on static and dynamic synapses, respectively? \n\nCalculate the $CV_{\\rm ISI}$ for different `tau_ratio` after simulating the LIF neuron with STP (Hint:`run_LIF_cond_STP` help you understand the irregularity).\n\n\n## Functional implications of short-term dynamics of synapses\nAs you have seen above, if the firing rate is stationary, the synaptic conductance quickly reaches a fixed point. On the other hand, if the firing rate transiently changes, synaptic conductance will vary -- even if the change is as short as a single inter-spike-interval. Such small changes can be observed in a single neuron when input spikes are regular and periodic. If the input spikes are Poissonian, then one may have to perform an average over several neurons.\n\n_Come up with other functions that short-term dynamics of synapses can be used to implement and implement them._\n", "meta": {"hexsha": "3dcde0c4bffc536c85ed4a59d2888a766e4805fa", "size": 1039188, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "tutorials/W3D1_RealNeurons/W3D1_Tutorial3.ipynb", "max_stars_repo_name": "RajanDasgupta/course-content", "max_stars_repo_head_hexsha": "fd82ea612802786a5934d38f198ce02585ffdc19", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tutorials/W3D1_RealNeurons/W3D1_Tutorial3.ipynb", "max_issues_repo_name": "RajanDasgupta/course-content", "max_issues_repo_head_hexsha": "fd82ea612802786a5934d38f198ce02585ffdc19", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tutorials/W3D1_RealNeurons/W3D1_Tutorial3.ipynb", "max_forks_repo_name": "RajanDasgupta/course-content", "max_forks_repo_head_hexsha": "fd82ea612802786a5934d38f198ce02585ffdc19", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 337.6179337232, "max_line_length": 232436, "alphanum_fraction": 0.92244233, "converted": true, "num_tokens": 13339, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702761768249, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.41029001192879994}} {"text": "Lambda School Data Science\n\n*Unit 2, Sprint 1, Module 2*\n\n---\n\n# Regression 2\n- Do train/test split\n- Use scikit-learn to fit a multiple regression\n- Understand how ordinary least squares regression minimizes the sum of squared errors\n- Define overfitting/underfitting and the bias/variance tradeoff\n\n### Setup\n\nRun the code cell below. You can work locally (follow the [local setup instructions](https://lambdaschool.github.io/ds/unit2/local/)) or on Colab.\n\nLibraries:\n- matplotlib\n- numpy\n- pandas\n- plotly\n- scikit-learn\n\n\n```python\nimport sys\n\n# If you're on Colab:\nif 'google.colab' in sys.modules:\n DATA_PATH = 'https://raw.githubusercontent.com/LambdaSchool/DS-Unit-2-Applied-Modeling/master/data/'\n\n# If you're working locally:\nelse:\n DATA_PATH = '../data/'\n \n# Ignore this Numpy warning when using Plotly Express:\n# FutureWarning: Method .ptp is deprecated and will be removed in a future version. Use numpy.ptp instead.\nimport warnings\nwarnings.filterwarnings(action='ignore', category=FutureWarning, module='numpy')\n```\n\n# Do train/test split\n\n## Overview\n\n### Predict Elections! \ud83c\uddfa\ud83c\uddf8\ud83d\uddf3\ufe0f\n\nHow could we try to predict the 2020 US Presidential election? \n\nAccording to Douglas Hibbs, a political science and economics professor, you can [explain elections with just two features, \"Bread and Peace\":](https://douglas-hibbs.com/background-information-on-bread-and-peace-voting-in-us-presidential-elections/)\n\n> Aggregate two-party vote shares going to candidates of the party holding the presidency during the postwar era are well explained by just two fundamental determinants:\n>\n> (1) Positively by weighted-average growth of per capita real disposable personal income over the term. \n> (2) Negatively by cumulative US military fatalities (scaled to population) owing to unprovoked, hostile deployments of American armed forces in foreign wars. \n\nLet's look at the data that Hibbs collected and analyzed:\n\n\n```python\nimport pandas as pd\ndf = pd.read_csv(DATA_PATH+'elections/bread_peace_voting.csv')\ndf\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        YearIncumbent Party CandidateOther CandidateAverage Recent Growth in Personal IncomesUS Military Fatalities per MillionIncumbent Party Vote Share
                                        01952StevensonEisenhower2.4019044.60
                                        11956EisenhowerStevenson2.89057.76
                                        21960NixonKennedy0.85049.91
                                        31964JohnsonGoldwater4.21161.34
                                        41968HumphreyNixon3.0214649.60
                                        51972NixonMcGovern3.62061.79
                                        61976FordCarter1.08248.95
                                        71980CarterReagan-0.39044.70
                                        81984ReaganMondale3.86059.17
                                        91988Bush, Sr.Dukakis2.27053.94
                                        101992Bush, Sr.Clinton0.38046.55
                                        111996ClintonDole1.04054.74
                                        122000GoreBush, Jr.2.36050.27
                                        132004Bush, Jr.Kerry1.72451.24
                                        142008McCainObama0.101446.32
                                        152012ObamaRomney0.95552.00
                                        162016ClintonTrump0.10548.20
                                        \n
                                        \n\n\n\nData Sources & Definitions\n\n- 1952-2012: Douglas Hibbs, [2014 lecture at Deakin University Melbourne](http://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf), Slide 40\n- 2016, Vote Share: [The American Presidency Project](https://www.presidency.ucsb.edu/statistics/elections)\n- 2016, Recent Growth in Personal Incomes: [The 2016 election economy: the \"Bread and Peace\" model final forecast](https://angrybearblog.com/2016/11/the-2016-election-economy-the-bread-and-peace-model-final-forecast.html)\n- 2016, US Military Fatalities: Assumption that Afghanistan War fatalities in 2012-16 occured at the same rate as 2008-12\n\n> Fatalities denotes the cumulative number of American military fatalities per millions of US population the in Korea, Vietnam, Iraq and Afghanistan wars during the presidential terms preceding the 1952, 1964, 1968, 1976 and 2004, 2008 and 2012 elections. \u2014[Hibbs](http://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf), Slide 33\n\nHere we have data from the 1952-2016 elections. We could make a model to predict 1952-2016 election outcomes \u2014 but do we really care about that? \n\nNo, not really. We already know what happened, we don't need to predict it.\n\nThis is explained in [_An Introduction to Statistical Learning_](http://faculty.marshall.usc.edu/gareth-james/ISL/), Chapter 2.2, Assessing Model Accuracy:\n\n> In general, we do not really care how well the method works training on the training data. Rather, _we are interested in the accuracy of the predictions that we obtain when we apply our method to previously unseen test data._ Why is this what we care about? \n>\n> Suppose that we are interested in developing an algorithm to predict a stock\u2019s price based on previous stock returns. We can train the method using stock returns from the past 6 months. But we don\u2019t really care how well our method predicts last week\u2019s stock price. We instead care about how well it will predict tomorrow\u2019s price or next month\u2019s price. \n>\n> On a similar note, suppose that we have clinical measurements (e.g. weight, blood pressure, height, age, family history of disease) for a number of patients, as well as information about whether each patient has diabetes. We can use these patients to train a statistical learning method to predict risk of diabetes based on clinical measurements. In practice, we want this method to accurately predict diabetes risk for _future patients_ based on their clinical measurements. We are not very interested in whether or not the method accurately predicts diabetes risk for patients used to train the model, since we already know which of those patients have diabetes.\n\nSo, we're really interested in the 2020 election \u2014 but we probably don't want to wait until then to evaluate our model.\n\nThere is a way we can estimate now how well our model will generalize in the future. We can't fast-forward time, but we can rewind it...\n\nWe can split our data in **two sets.** For example: \n1. **Train** a model on elections before 2008.\n2. **Test** the model on 2008, 2012, 2016. \n\nThis \"backtesting\" helps us estimate how well the model will predict the next elections going forward, starting in 2020.\n\nThis is explained in [_Forecasting,_ Chapter 3.4,](https://otexts.com/fpp2/accuracy.html) Evaluating forecast accuracy:\n\n> The accuracy of forecasts can only be determined by considering how well a model performs on new data that were not used when fitting the model.\n>\n>When choosing models, it is common practice to separate the available data into two portions, training and test data, where the training data is used to estimate any parameters of a forecasting method and the test data is used to evaluate its accuracy. Because the test data is not used in determining the forecasts, it should provide a reliable indication of how well the model is likely to forecast on new data.\n>\n>\n>\n>The size of the test set is typically about 20% of the total sample, although this value depends on how long the sample is and how far ahead you want to forecast. The following points should be noted.\n>\n>- A model which fits the training data well will not necessarily forecast well.\n>- A perfect fit can always be obtained by using a model with enough parameters.\n>- Over-fitting a model to data is just as bad as failing to identify a systematic pattern in the data.\n>\n>Some references describe the test set as the \u201chold-out set\u201d because these data are \u201cheld out\u201d of the data used for fitting. Other references call the training set the \u201cin-sample data\u201d and the test set the \u201cout-of-sample data\u201d. We prefer to use \u201ctraining data\u201d and \u201ctest data\u201d in this book.\n\n**How should we split: Randomly? Before/after a given date?**\n\nI recommend you all read a great blog post, [How (and why) to create a good validation set](https://www.fast.ai/2017/11/13/validation-sets/), by fast.ai cofounder Rachel Thomas.\n\nShe gives great examples to answer the question \u201cWhen is a random subset not good enough?\u201d I\u2019m not as opposed to random splits as Rachel Thomas seems to be. But it\u2019s worth thinking about the trade-offs!\n\nTime-based and random splits can both be useful, and you\u2019ll get repeated hands-on practice with both during this unit! (She also talks about the distinction between validation & test sets, which we\u2019ll introduce in the last lesson of this Sprint.)\n\n## Follow Along\n\nSplit the data in two sets:\n1. Train on elections before 2008.\n2. Test on 2008 and after.\n\n\n```python\n# Splitting with slicing syntax\ntrain = df[:14]\ntest = df[14:]\n```\n\n\n```python\n# Splitting with dataframe filtering\ntrain = df[df['Year'] < 2008]\ntest = df[df['Year'] >= 2008]\n```\n\nHow many observations (rows) are in the train set? In the test set?\n\n\n```python\n\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        YearIncumbent Party CandidateOther CandidateAverage Recent Growth in Personal IncomesUS Military Fatalities per MillionIncumbent Party Vote Share
                                        142008McCainObama0.101446.32
                                        152012ObamaRomney0.95552.00
                                        162016ClintonTrump0.10548.20
                                        \n
                                        \n\n\n\nNote that this volume of data is at least two orders of magnitude smaller than we usually want to work with for predictive modeling.\n\nThere are other validation techniques that could be used here, such as [time series cross-validation](https://scikit-learn.org/stable/modules/cross_validation.html#time-series-split), or [leave-one-out cross validation](https://scikit-learn.org/stable/modules/cross_validation.html#leave-one-out-loo) for small datasets. However, for this module, let's start simpler, with train/test split. \n\nUsing a tiny dataset is intentional here. It's good for learning because we can see all the data at once.\n\n## Challenge\n\nIn your assignment, you will do train/test split, based on date.\n\n# Use scikit-learn to fit a multiple regression\n\n## Overview\n\nWe've done train/test split, and we're ready to fit a model. \n\nWe'll proceed in 3 steps. The first 2 are review from the previous module. The 3rd is new.\n\n- Begin with baselines (0 features) \n- Simple regression (1 feature)\n- Multiple regression (2 features)\n\n## Follow Along\n\n### Begin with baselines (0 features)\n\nWhat was the average Incumbent Party Vote Share, in the 1952-2004 elections?\n\n\n```python\ntrain['Incumbent Party Vote Share'].mean()\n```\n\n\n\n\n 52.46857142857142\n\n\n\nWhat if we guessed this number for every election? How far off would this be on average?\n\n\n```python\n# Arrange y target vectors\ntarget = 'Incumbent Party Vote Share'\ny_train = train[target]\ny_test = test[target]\n```\n\n\n```python\n# Get mean baseline\nprint('Mean Baseline (using 0 features)')\nguess = y_train.mean()\nprint(guess)\n```\n\n Mean Baseline (using 0 features)\n 52.46857142857142\n\n\n\n```python\n# Train Error\nfrom sklearn.metrics import mean_absolute_error\ny_pred = [guess] * len(y_train)\nmae = mean_absolute_error(y_train, y_pred)\nprint(f'Train Error (1952-2004 elections): {mae:.2f} percentage points')\n```\n\n Train Error (1952-2004 elections): 4.85 percentage points\n\n\n\n```python\n# Test Error\ny_pred = [guess] * len(y_test)\nmae = mean_absolute_error(y_test, y_pred)\nprint(f'Test Error (2008-16 elections): {mae:.2f} percentage points')\n```\n\n Test Error (2008-16 elections): 3.63 percentage points\n\n\n### Simple regression (1 feature)\n\nMake a scatterplot of the relationship between 1 feature and the target.\n\nWe'll use an economic feature: Average Recent Growth in Personal Incomes. (\"Bread\")\n\n\n```python\nimport pandas as pd\nimport plotly.express as px\n\npx.scatter(\n train,\n x='Average Recent Growth in Personal Incomes',\n y='Incumbent Party Vote Share',\n text='Year',\n title='US Presidential Elections, 1952-2004',\n trendline='ols', # Ordinary Least Squares\n)\n```\n\n\n\n\n\n
                                        \n \n \n \n
                                        \n \n
                                        \n\n\n\n\n1952 & 1968 are outliers: The incumbent party got fewer votes than predicted by the regression. What do you think could explain those years? We'll come back to this soon, but first...\n\nUse scikit-learn to fit the simple regression with one feature.\n\nFollow the [5 step process](https://jakevdp.github.io/PythonDataScienceHandbook/05.02-introducing-scikit-learn.html#Basics-of-the-API), and refer to [Scikit-Learn LinearRegression documentation](https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LinearRegression.html).\n\n\n```python\n# 1. Import the appropriate estimator class from Scikit-Learn\nfrom sklearn.linear_model import LinearRegression\n```\n\n\n```python\n# 2. Instantiate this class\nmodel = LinearRegression()\n```\n\n\n```python\n# 3. Arrange X features matrices (already did y target vectors)\nfeatures = ['Average Recent Growth in Personal Incomes']\nX_train = train[features]\nX_test = test[features]\nprint(f'Linear Regression, dependent on: {features}')\n```\n\n Linear Regression, dependent on: ['Average Recent Growth in Personal Incomes']\n\n\n\n```python\n# 4. Fit the model\nmodel.fit(X_train, y_train)\ny_pred = model.predict(X_train)\nmae = mean_absolute_error(y_train, y_pred)\nprint(f'Train Error: {mae:.2f} percentage points')\n```\n\n Train Error: 2.65 percentage points\n\n\n\n```python\n# 5. Apply the model to new data\ny_pred = model.predict(X_test)\nmae = mean_absolute_error(y_test, y_pred)\nprint(f'Test Error: {mae:.2f} percentage points')\n```\n\n Test Error: 1.80 percentage points\n\n\nHow does the error compare to the baseline?\n\n### Multiple regression (2 features)\n\nMake a scatterplot of the relationship between 2 features and the target.\n\nWe'll add another feature: US Military Fatalities per Million. (\"Peace\" or the lack thereof.)\n\nRotate the scatterplot to explore the data. What's different about 1952 & 1968?\n\n\n```python\npx.scatter_3d(\n train,\n x='Average Recent Growth in Personal Incomes', \n y='US Military Fatalities per Million', \n z='Incumbent Party Vote Share', \n text='Year', \n title='US Presidential Elections, 1952-2004',\n \n)\n```\n\n\n\n\n\n
                                        \n \n \n \n
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                                        \n\n\n\n\nUse scikit-learn to fit a multiple regression with two features.\n\n\n```python\n# Re-arrange X features matrices\nfeatures = ['Average Recent Growth in Personal Incomes', \n 'US Military Fatalities per Million']\nprint(f'Linear Regression, dependent on: {features}')\n\nX_train = train[features]\nX_test = test[features]\n```\n\n Linear Regression, dependent on: ['Average Recent Growth in Personal Incomes', 'US Military Fatalities per Million']\n\n\n\n```python\n# Fit the model\nmodel.fit(X_train, y_train)\n```\n\n\n\n\n LinearRegression(copy_X=True, fit_intercept=True, n_jobs=None, normalize=False)\n\n\n\n\n```python\n# Check train error\ny_pred = model.predict(X_train)\nmae = mean_absolute_error(y_train, y_pred)\nprint(f'Train Error: {mae:.2f} percentage points')\n```\n\n Train Error: 1.33 percentage points\n\n\n\n```python\n# Apply the model to new data\ny_pred = model.predict(X_test)\nmae = mean_absolute_error(y_test, y_pred)\nprint(f'Test Error: {mae:.2f} percentage points')\n```\n\n Test Error: 1.63 percentage points\n\n\nHow does the error compare to the prior model?\n\n### Plot the plane of best fit\n\nFor a regression with 1 feature, we plotted the line of best fit in 2D. \n\n(There are many ways to do this. Plotly Express's `scatter` function makes it convenient with its `trendline='ols'` parameter.)\n\nFor a regression with 2 features, we can plot the plane of best fit in 3D!\n\n(Plotly Express has a `scatter_3d` function but it won't plot the plane of best fit for us. But, we can write our own function, with the same \"function signature\" as the Plotly Express API.)\n\n\n```python\nimport itertools\nimport numpy as np\nimport plotly.express as px\nimport plotly.graph_objs as go\nfrom sklearn.linear_model import LinearRegression\n\ndef regression_3d(df, x, y, z, num=100, **kwargs):\n \"\"\"\n Visualize linear regression in 3D: 2 features + 1 target\n \n df : Pandas DataFrame\n x : string, feature 1 column in df\n y : string, feature 2 column in df\n z : string, target column in df\n num : integer, number of quantiles for each feature\n \"\"\"\n \n # Plot data\n fig = px.scatter_3d(df, x, y, z, **kwargs)\n \n # Fit Linear Regression\n features = [x, y]\n target = z\n model = LinearRegression()\n model.fit(df[features], df[target]) \n \n # Define grid of coordinates in the feature space\n xmin, xmax = df[x].min(), df[x].max()\n ymin, ymax = df[y].min(), df[y].max()\n xcoords = np.linspace(xmin, xmax, num)\n ycoords = np.linspace(ymin, ymax, num)\n coords = list(itertools.product(xcoords, ycoords))\n \n # Make predictions for the grid\n predictions = model.predict(coords)\n Z = predictions.reshape(num, num).T\n \n # Plot predictions as a 3D surface (plane)\n fig.add_trace(go.Surface(x=xcoords, y=ycoords, z=Z))\n \n return fig\n```\n\n\n```python\nregression_3d(\n train,\n x='Average Recent Growth in Personal Incomes', \n y='US Military Fatalities per Million', \n z='Incumbent Party Vote Share', \n text='Year', \n title='US Presidential Elections, 1952-2004'\n)\n```\n\n\n\n\n\n
                                        \n \n \n \n
                                        \n \n
                                        \n\n\n\n\nWhere are 1952 & 1968 in relation to the plane? Which elections are the biggest outliers now?\n\nRoll over points on the plane to see predicted incumbent party vote share (z axis), dependent on personal income growth (x axis) and military fatatlies per capita (y axis).\n\n### Get and interpret coefficients\n\nDuring the previous module, we got the simple regression's coefficient and intercept. We plugged these numbers into an equation for the line of best fit, in slope-intercept form: $y = mx + b$\n\nLet's review this objective, but now for multiple regression.\n\nWhat's the equation for the plane of best fit?\n\n$y = \\beta_0 + \\beta_1x_1 + \\beta_2x_2$\n\nCan you relate the intercept and coefficients to what you see in the plot above?\n\n\n```python\nmodel.intercept_, model.coef_\n```\n\n\n\n\n (46.25489966153873, array([ 3.59004735, -0.05315709]))\n\n\n\n\n```python\nbeta0 = model.intercept_\nbeta1, beta2 = model.coef_\nprint(f'y = {beta0} + {beta1}x1 + {beta2}x2')\n```\n\n y = 46.25489966153873 + 3.5900473494560536x1 + -0.05315709351049324x2\n\n\n\n```python\n# This is easier to read\nprint('Intercept', model.intercept_)\ncoefficients = pd.Series(model.coef_, features)\nprint(coefficients.to_string())\n```\n\n Intercept 46.25489966153873\n Average Recent Growth in Personal Incomes 3.590047\n US Military Fatalities per Million -0.053157\n\n\n\n```python\n# This does not exactly match correlation\ndf.corr()['Incumbent Party Vote Share']\n```\n\n\n\n\n Year -0.219182\n Average Recent Growth in Personal Incomes 0.766281\n US Military Fatalities per Million -0.364377\n Incumbent Party Vote Share 1.000000\n Name: Incumbent Party Vote Share, dtype: float64\n\n\n\nOne of the coefficients is positive, and the other is negative. What does this mean?\n\nLet's look at some scenarios. We'll see that one unit's change in an independent variable results in a coefficient worth of change in the dependent variable.\n\nWhat does the model predict if income growth=0%, fatalities=0\n\n\n```python\nmodel.predict([[0, 0]])\n```\n\n\n\n\n array([46.25489966])\n\n\n\nIncome growth = 1% (fatalities = 0)\n\n\n```python\nmodel.predict([[1, 0]])\n```\n\n\n\n\n array([49.84494701])\n\n\n\nThe difference between these predictions = ? \n\n\n```python\nmodel.predict([[1, 0]]) - model.predict([[0, 0]])\n```\n\n\n\n\n array([3.59004735])\n\n\n\nWhat if... income growth = 2% (fatalities = 0)\n\n\n```python\nmodel.predict([[2, 0]])\n```\n\n\n\n\n array([53.43499436])\n\n\n\nThe difference between these predictions = ?\n\n\n```python\nmodel.predict([[2, 0]]) - model.predict([[1, 0]])\n```\n\n\n\n\n array([3.59004735])\n\n\n\nWhat if... (income growth=2%) fatalities = 100\n\n\n```python\nmodel.predict([[2, 100]])\n```\n\n\n\n\n array([48.11928501])\n\n\n\nThe difference between these predictions = ?\n\n\n```python\nmodel.predict([[2, 100]]) - model.predict([[2, 0]])\n```\n\n\n\n\n array([-5.31570935])\n\n\n\nWhat if income growth = 3% (fatalities = 100)\n\n\n```python\nmodel.predict([[3, 100]])\n```\n\n\n\n\n array([51.70933236])\n\n\n\nThe difference between these predictions = ?\n\n\n```python\nmodel.predict([[3, 100]]) - model.predict([[2, 100]])\n```\n\n\n\n\n array([3.59004735])\n\n\n\nWhat if (income growth = 3%) fatalities = 200\n\n\n```python\nmodel.predict([[3, 200]])\n```\n\n\n\n\n array([46.39362301])\n\n\n\nThe difference between these predictions = ?\n\n\n```python\nmodel.predict([[3, 200]]) - model.predict([[3, 100]])\n```\n\n\n\n\n array([-5.31570935])\n\n\n\n## Challenge\n\nIn your assignment, you'll fit a Linear Regression with at least 2 features.\n\n# Understand how ordinary least squares regression minimizes the sum of squared errors\n\n## Overview\n\nSo far, we've evaluated our models by their absolute error. It's an intuitive metric for regression problems.\n\nHowever, ordinary least squares doesn't directly minimize absolute error. Instead, it minimizes squared error.\n\n\n\n\nIn this section, we'll introduce two new regression metrics: \n\n- Squared error\n- $R^2$\n\n\nWe'll demostrate two possible methods to minimize squared error:\n\n- Guess & check\n- Linear Algebra\n\n## Follow Along\n\n### Guess & Check\n\nThis function visualizes squared errors. We'll go back to simple regression with 1 feature, because it's much easier to visualize.\n\nUse the function's m & b parameters to \"fit the model\" manually. Guess & check what values of m & b minimize squared error.\n\n\n```python\nfrom matplotlib.patches import Rectangle\nimport matplotlib.pyplot as plt\nimport numpy as np\nfrom sklearn.metrics import mean_absolute_error, mean_squared_error, r2_score\n\ndef squared_errors(df, feature, target, m, b):\n \"\"\"\n Visualize linear regression, with squared errors,\n in 2D: 1 feature + 1 target.\n \n Use the m & b parameters to \"fit the model\" manually.\n \n df : Pandas DataFrame\n feature : string, feature column in df\n target : string, target column in df\n m : numeric, slope for linear equation\n b : numeric, intercept for linear requation\n \"\"\"\n \n # Plot data\n fig = plt.figure(figsize=(7,7))\n ax = plt.axes()\n df.plot.scatter(feature, target, ax=ax)\n \n # Make predictions\n x = df[feature]\n y = df[target]\n y_pred = m*x + b\n \n # Plot predictions\n ax.plot(x, y_pred)\n \n # Plot squared errors\n xmin, xmax = ax.get_xlim()\n ymin, ymax = ax.get_ylim()\n scale = (xmax-xmin)/(ymax-ymin)\n for x, y1, y2 in zip(x, y, y_pred):\n bottom_left = (x, min(y1, y2))\n height = abs(y1 - y2)\n width = height * scale\n ax.add_patch(Rectangle(xy=bottom_left, width=width, height=height, alpha=0.1))\n \n # Print regression metrics\n mse = mean_squared_error(y, y_pred)\n rmse = np.sqrt(mse)\n mae = mean_absolute_error(y, y_pred)\n r2 = r2_score(y, y_pred)\n print('Mean Squared Error:', mse)\n print('Root Mean Squared Error:', rmse)\n print('Mean Absolute Error:', mae)\n print('R^2:', r2)\n```\n\nHere's what the mean baseline looks like:\n\n\n```python\nfeature = 'Average Recent Growth in Personal Incomes'\nsquared_errors(train, feature, target, m=0, b=y_train.mean())\n```\n\nNotice that $R^2$ is exactly zero. \n\n[$R^2$ represents the proportion of the variance for a dependent variable that is explained by the independent variable(s).](https://en.wikipedia.org/wiki/Coefficient_of_determination)\n\nThe mean baseline uses zero independent variables and explains none of the variance in the dependent variable, so its $R^2$ score is zero.\n\nThe highest possible $R^2$ score is 1. The lowest possible *Train* $R^2$ score with ordinary least squares regression is 0.\n\nIn this demo, it's possible to get a negative Train $R^2$, if you manually set values of m & b that are worse than the mean baseline. But that wouldn't happen in the real world.\n\nHowever, in the real world, it _is_ possible to get a negative *Test/Validation* $R^2$. It means that your *Test/Validation* predictions are worse than if you'd constantly predicted the mean of the *Test/Validation* set.\n\n---\n\nNow that we've visualized the squared errors for the mean baseline, let's guess & check some better values for the m & b parameters:\n\n\n```python\nsquared_errors(train, feature, target, m=3, b=46)\n```\n\nYou can run the function repeatedly, with different values for m & b.\n\nHow do you interpret each metric you see?\n\n- Mean Squared Error\n- Root Mean Squared Error\n- Mean Absolute Error\n- $R^2$\n\nDoes guess & check really get used in machine learning? Sometimes! Some complex functions are hard to minimize, so we use a sophisticated form of guess & check called \"gradient descent\", which you'll learn about in Unit 4.\n\nFortunately, we don't need to use guess & check for ordinary least squares regression. We have a solution, using linear algebra!\n\n\n### Linear Algebra\n\nThe same result that is found by minimizing the sum of the squared errors can be also found through a linear algebra process known as the \"Least Squares Solution:\"\n\n\\begin{align}\n\\hat{\\beta} = (X^{T}X)^{-1}X^{T}y\n\\end{align}\n\nBefore we can work with this equation in its linear algebra form we have to understand how to set up the matrices that are involved in this equation. \n\n#### The $\\beta$ vector\n\nThe $\\beta$ vector represents all the parameters that we are trying to estimate, our $y$ vector and $X$ matrix values are full of data from our dataset. The $\\beta$ vector holds the variables that we are solving for: $\\beta_0$ and $\\beta_1$\n\nNow that we have all of the necessary parts we can set them up in the following equation:\n\n\\begin{align}\ny = X \\beta + \\epsilon\n\\end{align}\n\nSince our $\\epsilon$ value represents **random** error we can assume that it will equal zero on average.\n\n\\begin{align}\ny = X \\beta\n\\end{align}\n\nThe objective now is to isolate the $\\beta$ matrix. We can do this by pre-multiplying both sides by \"X transpose\" $X^{T}$.\n\n\\begin{align}\nX^{T}y = X^{T}X \\beta\n\\end{align}\n\nSince anything times its transpose will result in a square matrix, if that matrix is then an invertible matrix, then we should be able to multiply both sides by its inverse to remove it from the right hand side. (We'll talk tomorrow about situations that could lead to $X^{T}X$ not being invertible.)\n\n\\begin{align}\n(X^{T}X)^{-1}X^{T}y = (X^{T}X)^{-1}X^{T}X \\beta\n\\end{align}\n\nSince any matrix multiplied by its inverse results in the identity matrix, and anything multiplied by the identity matrix is itself, we are left with only $\\beta$ on the right hand side:\n\n\\begin{align}\n(X^{T}X)^{-1}X^{T}y = \\hat{\\beta}\n\\end{align}\n\nWe will now call it \"beta hat\" $\\hat{\\beta}$ because it now represents our estimated values for $\\beta_0$ and $\\beta_1$\n\n#### Lets calculate our $\\beta$ parameters with numpy!\n\n\n```python\n# This is NOT something you'll be tested on. It's just a demo.\n\n# X is a matrix. Add column of constants for fitting the intercept.\ndef add_constant(X):\n constant = np.ones(shape=(len(X),1))\n return np.hstack((constant, X))\nX = add_constant(train[features].values)\nprint('X')\nprint(X)\n\n# y is a column vector\ny = train[target].values[:, np.newaxis]\nprint('y')\nprint(y)\n\n# Least squares solution in code\nX_transpose = X.T\nX_transpose_X = X_transpose @ X\nX_transpose_X_inverse = np.linalg.inv(X_transpose_X)\nX_transpose_y = X_transpose @ y\nbeta_hat = X_transpose_X_inverse @ X_transpose_y\n\nprint('Beta Hat')\nprint(beta_hat)\n```\n\n X\n [[ 1. 2.4 190. ]\n [ 1. 2.89 0. ]\n [ 1. 0.85 0. ]\n [ 1. 4.21 1. ]\n [ 1. 3.02 146. ]\n [ 1. 3.62 0. ]\n [ 1. 1.08 2. ]\n [ 1. -0.39 0. ]\n [ 1. 3.86 0. ]\n [ 1. 2.27 0. ]\n [ 1. 0.38 0. ]\n [ 1. 1.04 0. ]\n [ 1. 2.36 0. ]\n [ 1. 1.72 4. ]]\n y\n [[44.6 ]\n [57.76]\n [49.91]\n [61.34]\n [49.6 ]\n [61.79]\n [48.95]\n [44.7 ]\n [59.17]\n [53.94]\n [46.55]\n [54.74]\n [50.27]\n [51.24]]\n Beta Hat\n [[46.25489966]\n [ 3.59004735]\n [-0.05315709]]\n\n\n\n```python\n# Scikit-learn gave the exact same results!\nmodel.intercept_, model.coef_\n```\n\n\n\n\n (46.25489966153873, array([ 3.59004735, -0.05315709]))\n\n\n\n# Define overfitting/underfitting and the bias/variance tradeoff\n\n## Overview\n\nRead [_Python Data Science Handbook,_ Chapter 5.3](https://jakevdp.github.io/PythonDataScienceHandbook/05.03-hyperparameters-and-model-validation.html#The-Bias-variance-trade-off). Jake VanderPlas explains overfitting & underfitting:\n\n> Fundamentally, the question of \"the best model\" is about finding a sweet spot in the tradeoff between bias and variance. Consider the following figure, which presents two regression fits to the same dataset:\n> \n>\n>\n> The model on the left attempts to find a straight-line fit through the data. Because the data are intrinsically more complicated than a straight line, the straight-line model will never be able to describe this dataset well. Such a model is said to _underfit_ the data: that is, it does not have enough model flexibility to suitably account for all the features in the data; another way of saying this is that the model has high _bias_.\n>\n> The model on the right attempts to fit a high-order polynomial through the data. Here the model fit has enough flexibility to nearly perfectly account for the fine features in the data, but even though it very accurately describes the training data, its precise form seems to be more reflective of the particular noise properties of the data rather than the intrinsic properties of whatever process generated that data. Such a model is said to _overfit_ the data: that is, it has so much model flexibility that the model ends up accounting for random errors as well as the underlying data distribution; another way of saying this is that the model has high _variance_.\n\nVanderPlas goes on to connect these concepts to the \"bias/variance tradeoff\":\n\n> From the scores associated with these two models, we can make an observation that holds more generally:\n>\n>- For high-bias models, the performance of the model on the validation set is similar to the performance on the training set.\n>\n>- For high-variance models, the performance of the model on the validation set is far worse than the performance on the training set.\n>\n> If we imagine that we have some ability to tune the model complexity, we would expect the training score and validation score to behave as illustrated in the following figure:\n>\n>\n>\n> The diagram shown here is often called a validation curve, and we see the following essential features:\n>\n>- The training score is everywhere higher than the validation score. This is generally the case: the model will be a better fit to data it has seen than to data it has not seen.\n>- For very low model complexity (a high-bias model), the training data is under-fit, which means that the model is a poor predictor both for the training data and for any previously unseen data.\n>- For very high model complexity (a high-variance model), the training data is over-fit, which means that the model predicts the training data very well, but fails for any previously unseen data.\n>- For some intermediate value, the validation curve has a maximum. This level of complexity indicates a suitable trade-off between bias and variance.\n>\n>The means of tuning the model complexity varies from model to model.\n\nSo far, our only \"means of tuning the model complexity\" has been selecting one feature or two features for our linear regression models. But we'll quickly start to select more features, and more complex models, with more \"hyperparameters.\"\n\nThis is just a first introduction to underfitting & overfitting. We'll continue to learn about this topic all throughout this unit.\n\n## Follow Along\n\nLet's make our own Validation Curve, by tuning a new type of model complexity: polynomial degrees in a linear regression.\n\nGo back to the the NYC Tribeca condo sales data\n\n\n```python\n# Read NYC Tribeca condo sales data, from first 4 months of 2019.\n# Dataset has 90 rows, 9 columns.\ndf = pd.read_csv(DATA_PATH+'condos/tribeca.csv')\nassert df.shape == (90, 9)\n\n# Arrange X features matrix & y target vector\nfeatures = ['GROSS_SQUARE_FEET']\ntarget = 'SALE_PRICE'\nX = df[features]\ny = df[target]\n```\n\nDo random [train/test split](https://scikit-learn.org/stable/modules/generated/sklearn.model_selection.train_test_split.html)\n\n\n```python\nfrom sklearn.model_selection import train_test_split\nX_train, X_test, y_train, y_test = train_test_split(X, y, random_state=11)\n```\n\nRepeatedly fit increasingly complex models, and keep track of the scores\n\n\n```python\nfrom IPython.display import display, HTML\nfrom sklearn.linear_model import LinearRegression\nfrom sklearn.metrics import r2_score\nfrom sklearn.pipeline import make_pipeline\nfrom sklearn.preprocessing import PolynomialFeatures\n\n\n# Credit for PolynomialRegression: Jake VanderPlas, Python Data Science Handbook, Chapter 5.3\n# https://jakevdp.github.io/PythonDataScienceHandbook/05.03-hyperparameters-and-model-validation.html#Validation-curves-in-Scikit-Learn\ndef PolynomialRegression(degree=2, **kwargs):\n return make_pipeline(PolynomialFeatures(degree), \n LinearRegression(**kwargs))\n\n\npolynomial_degrees = range(1, 10, 2)\ntrain_r2s = []\ntest_r2s = []\n\nfor degree in polynomial_degrees:\n model = PolynomialRegression(degree)\n display(HTML(f'Polynomial degree={degree}'))\n \n model.fit(X_train, y_train)\n train_r2 = model.score(X_train, y_train)\n test_r2 = model.score(X_test, y_test)\n display(HTML(f'Train R2 {train_r2:.2f}'))\n display(HTML(f'Test R2 {test_r2:.2f}'))\n\n plt.scatter(X_train, y_train, color='blue', alpha=0.5)\n plt.scatter(X_test, y_test, color='red', alpha=0.5)\n plt.xlabel(features)\n plt.ylabel(target)\n \n x_domain = np.linspace(X.min(), X.max())\n curve = model.predict(x_domain)\n plt.plot(x_domain, curve, color='blue')\n plt.show()\n display(HTML('
                                        '))\n \n train_r2s.append(train_r2)\n test_r2s.append(test_r2)\n \ndisplay(HTML('Validation Curve'))\nplt.plot(polynomial_degrees, train_r2s, color='blue', label='Train')\nplt.plot(polynomial_degrees, test_r2s, color='red', label='Test')\nplt.xlabel('Model Complexity (Polynomial Degree)')\nplt.ylabel('R^2 Score')\nplt.legend()\nplt.show()\n```\n\nAs model complexity increases, what happens to Train $R^2$ and Test $R^2$?\n\n# Review\n\nIn your assignment, you'll continue to **predict how much it costs to rent an apartment in NYC,** using the dataset from renthop.com.\n\n\n- Do train/test split. Use data from April & May 2016 to train. Use data from June 2016 to test.\n- Engineer at least two new features.\n- Fit a linear regression model with at least two features.\n- Get the model's coefficients and intercept.\n- Get regression metrics RMSE, MAE, and $R^2$, for both the train and test sets.\n\nYou've been provided with a separate notebook for your assignment, which has all the instructions and stretch goals. What's the best test MAE you can get? Share your score and features used with your cohort on Slack!\n\n# Sources\n\n#### Train/Test Split\n- James, Witten, Hastie, Tibshirani, [_An Introduction to Statistical Learning_](http://faculty.marshall.usc.edu/gareth-james/ISL/), Chapter 2.2, Assessing Model Accuracy\n- Hyndman, Athanasopoulos, [_Forecasting,_ Chapter 3.4,](https://otexts.com/fpp2/accuracy.html) Evaluating forecast accuracy\n- Rachel Thomas, [How (and why) to create a good validation set](https://www.fast.ai/2017/11/13/validation-sets/)\n\n#### Bias-Variance Tradeoff\n- Jake VanderPlas, [_Python Data Science Handbook,_ Chapter 5.3](https://jakevdp.github.io/PythonDataScienceHandbook/05.03-hyperparameters-and-model-validation.html#The-Bias-variance-trade-off), Hyperparameters and Model Validation\n- StatQuest, [Machine Learning Fundamentals: Bias and Variance](https://youtu.be/EuBBz3bI-aA) (6.5 minutes)\n\n#### \"Bread and Peace\" Background\n- Douglas Hibbs, [Background Information on the \u2018Bread and Peace\u2019 Model of Voting in Postwar US Presidential Elections](https://douglas-hibbs.com/background-information-on-bread-and-peace-voting-in-us-presidential-elections/)\n- Nate Silver, [What Do Economic Models Really Tell Us About Elections?](https://fivethirtyeight.com/features/what-do-economic-models-really-tell-us-about-elections/)\n\n\n#### \"Bread and Peace\" Data Sources & Definitions\n- 1952-2012: Douglas Hibbs, [2014 lecture at Deakin University Melbourne](http://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf), Slide 40\n- 2016, Vote Share: [The American Presidency Project](https://www.presidency.ucsb.edu/statistics/elections)\n- 2016, Recent Growth in Personal Incomes: [The 2016 election economy: the \"Bread and Peace\" model final forecast](https://angrybearblog.com/2016/11/the-2016-election-economy-the-bread-and-peace-model-final-forecast.html)\n- 2016, US Military Fatalities: Assumption that Afghanistan War fatalities in 2012-16 occured at the same rate as 2008-12\n\n> Fatalities denotes the cumulative number of American military fatalities per millions of US population the in Korea, Vietnam, Iraq and Afghanistan wars during the presidential terms preceding the 1952, 1964, 1968, 1976 and 2004, 2008 and 2012 elections. \u2014[Hibbs](http://www.douglas-hibbs.com/HibbsArticles/HIBBS-PRESVOTE-SLIDES-MELBOURNE-Part1-2014-02-26.pdf), Slide 33\n", "meta": {"hexsha": "b886d86d382c824c8651f11b051643df8867ff17", "size": 488828, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "module2-regression-2/LS_DS_212.ipynb", "max_stars_repo_name": "jonDuke/DS-Unit-2-Linear-Models", "max_stars_repo_head_hexsha": "5486caca2e5539e9aa2b679242341f9c5a9bdeaa", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "module2-regression-2/LS_DS_212.ipynb", "max_issues_repo_name": "jonDuke/DS-Unit-2-Linear-Models", "max_issues_repo_head_hexsha": "5486caca2e5539e9aa2b679242341f9c5a9bdeaa", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "module2-regression-2/LS_DS_212.ipynb", "max_forks_repo_name": "jonDuke/DS-Unit-2-Linear-Models", "max_forks_repo_head_hexsha": "5486caca2e5539e9aa2b679242341f9c5a9bdeaa", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 148.5347918566, "max_line_length": 197108, "alphanum_fraction": 0.774597609, "converted": true, "num_tokens": 11207, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.6513548646660542, "lm_q1q2_score": 0.4102067451239238}} {"text": "\n\n\n```python\n# -*- coding: utf-8 -*-\n\n%matplotlib inline\n!wget https://project-ccap.github.io/data/tiny_celeba.tgz -O tiny_celeba.tgz\n!tar -zxf tiny_celeba.tgz\n!pip install japanize_matplotlib\n```\n\n# DCGAN Tutorial\n\n**Author**: `Nathan Inkawhich` \n\n\n# \u306f\u3058\u3081\u306b\n\n\n\u672c\u7269\u306e\u6709\u540d\u4eba\u306e\u5199\u771f\u3092\u96c6\u3081\u305f\u30c7\u30fc\u30bf\u30bb\u30c3\u30c8 celeba \u3092\u7528\u3044\u305f\uff0c\u6709\u540d\u4eba\u3092\u751f\u6210\u3059\u308b\u751f\u6210\u6575\u5bfe\u30cd\u30c3\u30c8\u30ef\u30fc\u30af (GAN)\u3002\n\n\u3053\u3053\u3067\u306e\u30b3\u30fc\u30c9\u306e\u307b\u3068\u3093\u3069\u306f `pytorch/examples` \u306b\u3042\u308b dcgan \u306e\u5b9f\u88c5\u304b\u3089\u306e\u3082\u306e\u3067\u3042\u308b\u3002\n\n# 1. \u751f\u6210\u6575\u5bfe\u30cd\u30c3\u30c8\u30ef\u30fc\u30af Generative Adversarial Networks\n\n## 1.1 GAN \u3068\u306f\uff1f\n\n\nGAN \u306f\uff0c \u5b66\u7fd2\u30c7\u30fc\u30bf\u306e\u5206\u5e03\u3092\u628a\u63e1\u3059\u308b\u3053\u3068\u3067\uff0c \u305d\u306e\u540c\u3058\u5206\u5e03\u304b\u3089\u65b0\u3057\u3044\u30c7\u30fc\u30bf\u3092\u751f\u6210\u3067\u304d\u308b\u3088\u3046\u306b\uff0c \u6df1\u5c64\u5b66\u7fd2\u30e2\u30c7\u30eb\u306b\u6559\u3048\u308b\u305f\u3081\u306e\u67a0\u7d44\u307f\u3067\u3042\u308b\u3002\nGAN \u306f 2014 \u5e74 \u306b Ian Goodfellow \u306b\u3088\u3063\u3066\u767a\u660e\u3055\u308c\uff0c \u8ad6\u6587 [Generative Adversarial Nets](https://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf) \u3067\u521d\u3081\u3066\u8aac\u660e\u3055\u308c\u305f\u3002\nGAN \u306f *\u751f\u6210\u5668 generator* \u3068 *\u8b58\u5225\u5668 discriminator* \u3068\u3044\u3046 2 \u3064\u306e\u7570\u306a\u308b\u30e2\u30c7\u30eb\u3067\u69cb\u6210\u3055\u308c\u3066\u3044\u308b\u3002\n\u751f\u6210\u5668\u306e\u4ed5\u4e8b\u306f\uff0c \u5b66\u7fd2\u753b\u50cf\u306b\u4f3c\u305f \u300c\u507d\u300d\u306e\u753b\u50cf\u3092\u751f\u6210\u3059\u308b\u3053\u3068\u3067\u3042\u308b\u3002\n\u8b58\u5225\u5668\u306e\u5f79\u5272\u306f\uff0c \u3042\u308b\u753b\u50cf\u3092\u898b\u3066\uff0c \u305d\u308c\u304c\u672c\u7269\u306e\u8a13\u7df4\u753b\u50cf\u306a\u306e\u304b\uff0c \u751f\u6210\u5668\u304b\u3089\u306e\u507d\u7269\u306e\u753b\u50cf\u306a\u306e\u304b\u3092\u51fa\u529b\u3059\u308b\u3053\u3068\u3067\u3042\u308b\u3002\n\u8b58\u5225\u5668\u306e\u4ed5\u4e8b\u306f\uff0c\u3042\u308b\u753b\u50cf\u3092\u898b\u3066\u3001\u305d\u308c\u304c\u672c\u7269\u306e\u8a13\u7df4\u753b\u50cf\u304b\uff0c \u751f\u6210\u5668\u306e\u507d\u7269\u306e\u753b\u50cf\u304b\u3092\u51fa\u529b\u3059\u308b\u3053\u3068\u3067\u3042\u308b\u3002\n\u8a13\u7df4\u4e2d\uff0c \u751f\u6210\u5668\u306f\u5e38\u306b\u8b58\u5225\u5668\u3092\u51fa\u3057\u629c\u3053\u3046\u3068\uff0c \u3088\u308a\u826f\u3044\u507d\u7269\u3092\u751f\u6210\u3057\uff0c \u8b58\u5225\u5668\u306f\u3088\u308a\u826f\u3044\u63a2\u5075\u306b\u306a\u3063\u3066\u672c\u7269\u3068\u507d\u7269\u306e\u753b\u50cf\u3092\u6b63\u3057\u304f\u5206\u985e\u3059\u308b\u3088\u3046\u306b\u52aa\u529b\u3059\u308b\u3002\n\u3053\u306e\u30b2\u30fc\u30e0\u306e\u5747\u8861\u306f\uff0c \u751f\u6210\u5668\u304c\u3042\u305f\u304b\u3082\u8a13\u7df4\u30c7\u30fc\u30bf\u304b\u3089\u76f4\u63a5\u6765\u305f\u304b\u306e\u3088\u3046\u306a\u5b8c\u74a7\u306a\u507d\u7269\u3092\u751f\u6210\u3057\u3066\u3044\u308b\u3068\u304d\u3067\uff0c \u8b58\u5225\u5668\u306f\u751f\u6210\u5668\u306e\u51fa\u529b\u304c\u672c\u7269\u304b\u507d\u7269\u304b\u3092\u5e38\u306b 50 \uff05\u306e\u4fe1\u983c\u5ea6\u3067\u63a8\u6e2c\u3059\u308b\u3053\u3068\u306b\u306a\u308b\u3002\n\n\u3053\u3053\u3067\u306e\u8868\u8a18\u3092\uff0c \u8b58\u5225\u5668\u304b\u3089\u5b9a\u7fa9\u3059\u308b\u3002\n$x$ \u3092\u753b\u50cf\u306e\u30c7\u30fc\u30bf\u3068\u3059\u308b\u3002\n$D(x)$ \u306f\u8b58\u5225\u5668\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3067 $x$ \u304c\u751f\u6210\u5668\u3067\u306f\u306a\u304f\u5b66\u7fd2\u30c7\u30fc\u30bf\u304b\u3089\u6765\u305f\u3082\u306e\u3067\u3042\u308b\u78ba\u7387\u3092 (\u30b9\u30ab\u30e9\u30fc) \u3067\u51fa\u529b\u3059\u308b\u3002\n\u3053\u3053\u3067\u306f\uff0c \u753b\u50cf\u3092\u6271\u3063\u3066\u3044\u308b\u306e\u3067 $D(x)$ \u306e\u5165\u529b\u306f CHW \u30b5\u30a4\u30ba 3x64x64 \u306e\u753b\u50cf\u3067\u3042\u308b\u3002\n\u76f4\u611f\u7684\u306b\u306f $D(x)$ \u306f $x$ \u304c\u5b66\u7fd2\u30c7\u30fc\u30bf\u304b\u3089\u6765\u305f\u5834\u5408\u306f HIGH $x$ \u304c\u751f\u6210\u5668\u304b\u3089\u6765\u305f\u5834\u5408\u306f LOW \u306b\u306a\u308b\u306f\u305a\u3067\u3042\u308b\u3002\n$D(x)$ \u306f\uff0c \u5f93\u6765\u306e 2 \u5024\u5206\u985e\u5668\u3068\u8003\u3048\u308b\u3053\u3068\u3082\u3067\u304d\u308b\u3002\n\n\n\n\u751f\u6210\u5668\u306e\u8868\u8a18\u6cd5\u3068\u3057\u3066 $z$ \u3092\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u304b\u3089\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3055\u308c\u305f\u6f5c\u5728\u7a7a\u9593\u30d9\u30af\u30c8\u30eb\u3068\u3057\u307e\u3059\u3002\n$G(z)$ \u306f\uff0c \u6f5c\u5728\u30d9\u30af\u30c8\u30eb $z$ \u3092\u30c7\u30fc\u30bf\u7a7a\u9593\u306b\u30de\u30c3\u30d4\u30f3\u30b0\u3059\u308b\u751f\u6210\u5668\u95a2\u6570\u3092\u8868\u3059\u3002\n$G$ \u306e\u76ee\u7684\u306f\uff0c \u5b66\u7fd2\u30c7\u30fc\u30bf\u306e\u5206\u5e03\uff08$p_{data}$\uff09\u3092\u63a8\u5b9a\u3057\uff0c \u305d\u306e\u63a8\u5b9a\u5206\u5e03\uff08$p_g$\uff09\u304b\u3089\u507d\u306e\u30b5\u30f3\u30d7\u30eb\u3092\u751f\u6210\u3059\u308b\u3053\u3068\u3067\u3042\u308b\u3002\n\n\n\u3064\u307e\u308a $D(G(z))$ \u306f\uff0c \u751f\u6210\u5668 $G$ \u306e\u51fa\u529b\u304c\u5b9f\u50cf\u3067\u3042\u308b\u78ba\u7387(\u30b9\u30ab\u30e9\u30fc) \u3067\u3042\u308b\u3002\n[Goodfellow \u306e \u8ad6\u6587](https://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf) \u306b\u8a18\u8f09\u3055\u308c\u3066\u3044\u308b\u3088\u3046\u306b\uff0c $D$ \u3068 $G$ \u306f\u30df\u30cb\u30de\u30c3\u30af\u30b9\u30b2\u30fc\u30e0\u3092\u884c\u3044\uff0c $D$ \u306f\u672c\u7269\u3068\u507d\u7269\u3092\u6b63\u3057\u304f\u5206\u985e\u3059\u308b\u78ba\u7387 ($logD(x)$) \u3092\u6700\u5927\u5316\u3057\u3088\u3046\u3068\u3057\uff0c $G$ \u306f $D$ \u306e\u51fa\u529b\u304c\u507d\u7269\u3067\u3042\u308b\u3068\u4e88\u6e2c\u3059\u308b\u78ba\u7387 ($log(1-D(G(x)))$) \u3092\u6700\u5c0f\u5316\u3057\u3088\u3046\u3068\u3057\u3059\u308b\u3002\n\u8ad6\u6587\u3088\u308a GAN \u306e\u640d\u5931\u95a2\u6570\u306f\n\n\n\\begin{align}\n\\underset{G}{\\text{min}} \\underset{D}{\\text{max}}V(D,G) = \\mathbb{E}_{x\\sim p_{data}(x)}\\big[\\log D(x)\\big] + \\mathbb{E}_{z\\sim p_{z}(z)}\\big[\\log(1-D(G(z)))\\big]\\tag{1}\n\\end{align}\n\n\u7406\u8ad6\u7684\u306b\u306f\uff0c \u3053\u306e\u30df\u30cb\u30de\u30c3\u30af\u30b9\u30b2\u30fc\u30e0\u306e\u89e3\u306f $p_g = p_{data}$ \u3068\u306a\u308a\uff0c \u8b58\u5225\u5668\u306f\u5165\u529b\u304c\u672c\u7269\u304b\u507d\u7269\u304b\u3092\u30e9\u30f3\u30c0\u30e0\u306b\u63a8\u6e2c\u3059\u308b\u3053\u3068\u306b\u306a\u308b\u3002\n\u3057\u304b\u3057 GAN \u306e\u53ce\u675f\u7406\u8ad6\u306f\u307e\u3060\u76db\u3093\u306b\u7814\u7a76\u3055\u308c\u3066\u304a\u308a\uff0c \u73fe\u5b9f\u306b\u306f\u30e2\u30c7\u30eb\u304c\u3053\u306e\u70b9\u307e\u3067\u5b66\u7fd2\u3059\u308b\u3068\u306f\u9650\u3089\u306a\u3044\u3002\n\n\n# 2. DCGAN\u3068\u306f\uff1f\n\nDCGAN \u306f\u4e0a\u8ff0\u306e GAN \u3092\u76f4\u63a5\u62e1\u5f35\u3057\u305f\u3082\u3060\u304c\uff0c \u8b58\u5225\u5668\u3068\u751f\u6210\u5668\u306b\u305d\u308c\u305e\u308c\u7573\u307f\u8fbc\u307f\u5c64\u3068\u7573\u307f\u8fbc\u307f\u5909\u63db\u5c64\u3092\u660e\u793a\u7684\u306b\u4f7f\u7528\u3057\u3066\u3044\u308b\u3002\nDCGAN \u306f Radford\u3089\u304c\u8ad6\u6587 [Unsupervised Representation Learning With Deep Convolutional Generative Adversarial Networks](https://arxiv.org/pdf/1511.06434.pdf) \u3067\u521d\u3081\u3066\u767a\u8868\u3057\u305f\u3002\n\u8b58\u5225\u5668\u306f strided [`convolution`](https://pytorch.org/docs/stable/nn.html#torch.nn.Conv2d) \u5c64\uff0c [`batch norm`](https://pytorch.org/docs/stable/nn.html#torch.nn.BatchNorm2d) \u5c64\uff0c\n[`LeakyReLU`](https://pytorch.org/docs/stable/nn.html#torch.nn.LeakyReLU) \u6d3b\u6027\u5316\u95a2\u6570\u3067\u69cb\u6210\u3055\u308c\u3066\u3044\u308b\u3002\n\u5165\u529b\u306f 3x64x64 \u306e\u5165\u529b\u753b\u50cf\u3067\u3042\u308a\uff0c\u51fa\u529b\u306f\u5165\u529b\u304c\u5b9f\u30c7\u30fc\u30bf\u306e\u5206\u5e03\u304b\u3089\u306e\u3082\u306e\u3067\u3042\u308b\u3068\u3044\u3046\u30b9\u30ab\u30e9\u30fc\u78ba\u7387\u3067\u3042\u308b\u3002\n\u751f\u6210\u5668\u306f\uff0c[`convolutional-transpose`](https://pytorch.org/docs/stable/nn.html#torch.nn.ConvTranspose2d) \u5c64\uff0c\u30d0\u30c3\u30c1\u30ce\u30eb\u30e0\u5c64\uff0c[`ReLU`](https://pytorch.org/docs/stable/nn.html#relu) \u6d3b\u6027\u5316\u95a2\u6570\u3067\u69cb\u6210\u3055\u308c\u3066\u3044\u308b\u3002\n\u5165\u529b\u306f\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u304b\u3089\u5f15\u304d\u51fa\u3055\u308c\u305f\u6f5c\u5728\u7684\u306a\u30d9\u30af\u30c8\u30eb $z$ \u3067\uff0c\u51fa\u529b\u306f 3x64x64 \u306e RGB \u753b\u50cf\u3067\u3042\u308b\u3002\nstrided conv-transpose \u5c64\u306b\u3088\u308a\uff0c\u6f5c\u5728\u30d9\u30af\u30c8\u30eb\u306f\u753b\u50cf\u3068\u540c\u3058\u5f62\u72b6\u306e\u30dc\u30ea\u30e5\u30fc\u30e0\u306b\u5909\u63db\u3055\u308c\u308b\u3002\n\u3053\u306e\u8ad6\u6587\u3067\u306f\uff0c \u6700\u9069\u5316\u624b\u6cd5\u306e\u8a2d\u5b9a\u65b9\u6cd5\uff0c \u640d\u5931\u95a2\u6570\u306e\u8a08\u7b97\u65b9\u6cd5\uff0c \u30e2\u30c7\u30eb\u306e\u91cd\u307f\u306e\u521d\u671f\u5316\u65b9\u6cd5\u306a\u3069\u306e\u30d2\u30f3\u30c8\u3082\u7d39\u4ecb\u3055\u308c\u3066\u3044\u308b\u304c\uff0c \u3053\u308c\u3089\u306b\u3064\u3044\u3066\u306f\u5f8c\u306e\u7bc0\u3067\u8aac\u660e\u3059\u308b\u3002\n\n\n\n```python\nfrom __future__ import print_function\n#%matplotlib inline\nimport argparse\nimport os\nimport random\nimport torch\nimport torch.nn as nn\nimport torch.nn.parallel\nimport torch.backends.cudnn as cudnn\nimport torch.optim as optim\nimport torch.utils.data\nimport torchvision.datasets as dset\nimport torchvision.transforms as transforms\nimport torchvision.utils as vutils\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport matplotlib.animation as animation\n\nngpu = 1\ndevice = torch.device(\"cuda:0\" if (torch.cuda.is_available() and ngpu > 0) else \"cpu\")\n\nimport japanize_matplotlib\nfrom IPython.display import HTML\n\n# Set random seed for reproducibility\nmanualSeed = 999\n#manualSeed = random.randint(1, 10000) # use if you want new results\nprint(\"Random Seed: \", manualSeed)\nrandom.seed(manualSeed)\ntorch.manual_seed(manualSeed)\n\n# from https://discuss.pytorch.org/t/how-to-cast-a-tensor-to-another-type/2713/28\ntorch.set_default_dtype(torch.float32)\n```\n\n Random Seed: 999\n\n\n# 3. \u5165\u529b\n\n\n\u5b9f\u884c\u3059\u308b\u305f\u3081\u306e\u5165\u529b\u3092\u5b9a\u7fa9\u3059\u308b\u3002\n\n\n- **dataroot** - \u30c7\u30fc\u30bf\u30bb\u30c3\u30c8\u306e\u30d5\u30a9\u30eb\u30c0\u306e\u30eb\u30fc\u30c8\u3078\u306e\u30d1\u30b9\u3002\u30c7\u30fc\u30bf\u30bb\u30c3\u30c8\u306b\u3064\u3044\u3066\u306f\uff0c \u6b21\u7bc0\u3067\u8a73\u3057\u304f\u8aac\u660e\u3059\u308b\u3002\n- **workers** - DataLoader \u3067\u30c7\u30fc\u30bf\u3092\u8aad\u307f\u8fbc\u3080\u305f\u3081\u306e\u30ef\u30fc\u30ab\u30fc\u30b9\u30ec\u30c3\u30c9\u6570\u3002\n- **batch_size** - \u5b66\u7fd2\u306b\u4f7f\u7528\u3059\u308b\u30d0\u30c3\u30c1\u30b5\u30a4\u30ba\u3002DCGAN \u8ad6\u6587\u3067\u306f\u30d0\u30c3\u30c1\u30b5\u30a4\u30ba 128 \u3092\u4f7f\u7528\u3057\u3066\u308b\u3002\n- **image_size** - \u5b66\u7fd2\u306b\u4f7f\u7528\u3059\u308b\u753b\u50cf\u306e\u7a7a\u9593\u30b5\u30a4\u30ba\u3002\n \u3053\u306e\u5b9f\u88c5\u3067\u306f\uff0c\u30c7\u30d5\u30a9\u30eb\u30c8\u3067 64x64 \u306b\u8a2d\u5b9a\u3055\u308c\u3066\u3044\u308b\u3002\n\u5225\u306e\u30b5\u30a4\u30ba\u306b\u3057\u305f\u3044\u5834\u5408\u306f\uff0cD \u3068 G \u306e\u69cb\u9020\u3092\u5909\u66f4\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002\n \u8a73\u7d30\u306b\u3064\u3044\u3066\u306f\uff0c \u3092\u53c2\u7167\u3002\n- **nc** - \u5165\u529b\u753b\u50cf\u306e\u8272\u30c1\u30e3\u30f3\u30cd\u30eb\u6570\uff0c \u30ab\u30e9\u30fc\u753b\u50cf\u306e\u5834\u5408\u306f 3 \n- **nz** - \u6f5c\u5728\u7684\u306a\u30d9\u30af\u30c8\u30eb\u306e\u9577\u3055\u3002\n- **ngf** - \u751f\u6210\u5668\u3067\u51e6\u7406\u3055\u308c\u308b\u7279\u5fb4\u5730\u56f3\u306e\u6df1\u3055\u306b\u95a2\u9023\n- **ndf** - \u8b58\u5225\u5668\u3092\u901a\u3057\u3066\u4f1d\u642c\u3055\u308c\u308b\u7279\u5fb4\u5730\u56f3\u306e\u6df1\u3055\u3092\u8a2d\u5b9a\n- **num_epochs** - \u5b9f\u884c\u3059\u308b\u5b66\u7fd2\u30a8\u30dd\u30c3\u30af\u6570\u3002\u9577\u304f\u5b66\u7fd2\u3059\u308c\u3070\u826f\u3044\u7d50\u679c\u304c\u5f97\u3089\u308c\u308b\u3060\u308d\u3046\u304c\uff0c\u305d\u308c\u3060\u3051\u6642\u9593\u3082\u304b\u304b\u308b\u3002\n- **lr** - \u5b66\u7fd2\u6642\u306e\u5b66\u7fd2\u7387\u3002DCGAN \u8ad6\u6587\u306b\u8a18\u8f09\u3055\u308c\u3066\u3044\u308b\u3088\u3046\u306b\uff0c\u3053\u306e\u6570\u5024\u306f 0.0002 \u3068\u3059\u308b\u3002\n- **\u03b21** - Adam \u6700\u9069\u5316 \u306e \u03b21 \u30cf\u30a4\u30d1\u30fc\u30d1\u30e9\u30e1\u30fc\u30bf\u3002\u8ad6\u6587\u306b\u3042\u308b\u3088\u3046\u306b\u3001\u3053\u306e\u6570\u5024\u306f 0.5 \u3067\u3042\u308b\u3079\u304d\u3002\n- **ngpu** - \u5229\u7528\u53ef\u80fd\u306a GPU \u6570\u3002\u3053\u306e\u5024\u304c 0 \u306e\u5834\u5408\uff0c\u30b3\u30fc\u30c9\u306f CPU \u30e2\u30fc\u30c9\u3067\u5b9f\u884c\u3055\u308c\u308b\u3002\u3053\u306e\u6570\u5b57\u304c 0 \u3088\u308a\u5927\u304d\u3044\u5834\u5408\uff0c\u305d\u306e\u6570\u306e GPU \u3067\u5b9f\u884c\u3055\u308c\u308b\u3002\n\n\n\n\n\n```python\n#!ls tiny_celeba\n```\n\n\n```python\n# Root directory for dataset\ndataroot = \"data/celeba\"\ndataroot = 'tiny_celeba' #img_align_celeba'\n\n# Number of workers for dataloader\nworkers = 2\n#workers = 4\n\n# Batch size during training\nbatch_size = 128\n\n# Spatial size of training images. All images will be resized to this\n# size using a transformer.\nimage_size = 64\n\n# Number of channels in the training images. For color images this is 3\nnc = 3\n\n# Size of z latent vector (i.e. size of generator input)\nnz = 100\n\n# Size of feature maps in generator\nngf = 64\n\n# Size of feature maps in discriminator\nndf = 64\n\n# Number of training epochs\nnum_epochs = 20\n\n# Learning rate for optimizers\nlr = 0.0002\n\n# Beta1 hyperparam for Adam optimizers\nbeta1 = 0.5\n\n# Number of GPUs available. Use 0 for CPU mode.\nngpu = 1\n#ngpu = 0\n```\n\n# 4. \u30c7\u30fc\u30bf\n\n\n\u3053\u306e\u30c1\u30e5\u30fc\u30c8\u30ea\u30a2\u30eb\u3067\u306f [Celeb-A Faces dataset](http://mmlab.ie.cuhk.edu.hk/projects/CelebA.html) \u3092\u4f7f\u3046\u3002\n\u30ea\u30f3\u30af\u5148\u306e\u30b5\u30a4\u30c8\uff0c\u307e\u305f\u306f [Google Drive](https://drive.google.com/drive/folders/0B7EVK8r0v71pTUZsaXdaSnZBZzg) \u3067\u30c0\u30a6\u30f3\u30ed\u30fc\u30c9\u3067\u304d\u308b\u3002\n\n\u30c7\u30fc\u30bf\u30bb\u30c3\u30c8\u306f *img_align_celeba.zip*\u3068\u3044\u3046\u30d5\u30a1\u30a4\u30eb\u540d\u3067\u30c0\u30a6\u30f3\u30ed\u30fc\u30c9\u3055\u308c\u308b\u3002\n\u30c0\u30a6\u30f3\u30ed\u30fc\u30c9\u304c\u5b8c\u4e86\u3057\u305f\u3089 *celeba* \u3068\u3044\u3046\u540d\u524d\u306e\u30c7\u30a3\u30ec\u30af\u30c8\u30ea\u3092\u4f5c\u6210\u3057\uff0c \u305d\u306e\u30c7\u30a3\u30ec\u30af\u30c8\u30ea\u306b zip \u30d5\u30a1\u30a4\u30eb\u3092\u89e3\u51cd\u3059\u308b\u3002\n\u305d\u3057\u3066\u3001\u3053\u306e\u30ce\u30fc\u30c8\u30d6\u30c3\u30af\u306e *dataroot* \u5165\u529b\u3092\uff0c \u5148\u307b\u3069\u4f5c\u6210\u3057\u305f *celeba* \u30c7\u30a3\u30ec\u30af\u30c8\u30ea\u306b\u8a2d\u5b9a\u3057\u3066\u304f\u3060\u3055\u3044\u3002\n\u7d50\u679c\u3068\u3057\u3066\uff0c \u30c7\u30a3\u30ec\u30af\u30c8\u30ea\u69cb\u9020\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b\u3002\n\n\n```\n/path/to/celeba\n -> img_align_celeba \n -> 188242.jpg\n -> 173822.jpg\n -> 284702.jpg\n -> 537394.jpg\n ...\n```\n\n\u3053\u308c\u306f\u91cd\u8981\u306a\u30b9\u30c6\u30c3\u30d7\u3067\u3042\u308b\u3002\n\u306a\u305c\u306a\u3089 ImageFolder \u30c7\u30fc\u30bf\u30bb\u30c3\u30c8\u30af\u30e9\u30b9\u3092\u4f7f\u7528\u3059\u308b\u305f\u3081\uff0c \u30c7\u30fc\u30bf\u30bb\u30c3\u30c8\u306e\u30eb\u30fc\u30c8\u30d5\u30a9\u30eb\u30c0\u306b\u30b5\u30d6\u30c7\u30a3\u30ec\u30af\u30c8\u30ea\u304c\u3042\u308b\u3053\u3068\u304c\u5fc5\u8981\u3060\u304b\u3089\u3067\u3042\u308b\u3002\n\u3053\u308c\u3067\uff0c \u30c7\u30fc\u30bf\u30bb\u30c3\u30c8\u306e\u4f5c\u6210\uff0c \u30c7\u30fc\u30bf\u30ed\u30fc\u30c0\u306e\u4f5c\u6210\uff0c \u5b9f\u884c\u3059\u308b\u30c7\u30d0\u30a4\u30b9\u306e\u8a2d\u5b9a\uff0c \u305d\u3057\u3066\u6700\u5f8c\u306b\u8a13\u7df4\u30c7\u30fc\u30bf\u306e\u4e00\u90e8\u3092\u53ef\u8996\u5316\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002 \n\n\n\n\n```python\n!mkdir tiny_celeba/img_align_celeba\n!mv tiny_celeba/*.jpg tiny_celeba/img_align_celeba/\n```\n\n\n```python\n# We can use an image folder dataset the way we have it setup.\n# Create the dataset\ndataset = dset.ImageFolder(root=dataroot,\n transform=transforms.Compose([\n transforms.Resize(image_size),\n transforms.CenterCrop(image_size),\n transforms.ToTensor(),\n transforms.Normalize((0.5, 0.5, 0.5), (0.5, 0.5, 0.5)),\n ]))\n# Create the dataloader\ndataloader = torch.utils.data.DataLoader(dataset, batch_size=batch_size,\n shuffle=True, num_workers=workers)\n\n# Decide which device we want to run on\ndevice = torch.device(\"cuda:0\" if (torch.cuda.is_available() and ngpu > 0) else \"cpu\")\n\n# Plot some training images\nreal_batch = next(iter(dataloader))\nplt.figure(figsize=(14,14))\nplt.axis(\"off\")\nplt.title(\"Training Images\")\nplt.imshow(np.transpose(vutils.make_grid(real_batch[0].to(device)[:64], padding=2, normalize=True).cpu(),(1,2,0)))\n```\n\n\n```python\nreal_batch = next(iter(dataloader))\nplt.figure(figsize=(24,12))\nplt.axis(\"off\")\nplt.title(\"Training Images\")\nplt.imshow(np.transpose(vutils.make_grid(real_batch[0].to(device)[:24], padding=2, normalize=True).cpu(),(1,2,0)))\n```\n\n\n```python\nplt.imshow(real_batch[0][0].numpy().transpose(1,2,0).clip(0,1))\n```\n\n# 5. \u5b9f\u88c5\n\n\n\u5165\u529b\u30d1\u30e9\u30e1\u30fc\u30bf\u306e\u8a2d\u5b9a\u3068\u30c7\u30fc\u30bf\u30bb\u30c3\u30c8\u306e\u6e96\u5099\u304c\u3067\u304d\u305f\u3089\uff0c\u3044\u3088\u3044\u3088\u5b9f\u88c5\u306b\u5165\u308b\u3002\n\u307e\u305a\uff0c \u521d\u671f\u5316\u306e\u65b9\u6cd5\u304b\u3089\u59cb\u3081\u3066\uff0c \u751f\u6210\u5668\uff0c \u8b58\u5225\u5668\uff0c \u640d\u5931\u95a2\u6570\uff0c \u5b66\u7fd2\u30eb\u30fc\u30d7\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u8aac\u660e\u3059\u308b\u3002\n\n\n# 6. \u91cd\u307f\u306e\u521d\u671f\u5316\n\n\nDCGAN \u306e\u8ad6\u6587\u304b\u3089\uff0c \u8457\u8005\u306f\u3059\u3079\u3066\u306e\u30e2\u30c7\u30eb\u306e\u91cd\u307f\u3092 `mean=0`, `stdev=0.02` \u306e\u6b63\u898f\u5206\u5e03\u304b\u3089\u30e9\u30f3\u30c0\u30e0\u306b\u521d\u671f\u5316\u3059\u308b\u3053\u3068\u3092\u6307\u5b9a\u3057\u3066\u3044\u308b\u3002\n`weights_init` \u95a2\u6570\u306f\uff0c \u521d\u671f\u5316\u3055\u308c\u305f\u30e2\u30c7\u30eb\u3092\u5165\u529b\u3068\u3057\uff0c \u3053\u306e\u57fa\u6e96\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\uff0c \u3059\u3079\u3066\u306e\u7573\u307f\u8fbc\u307f\u5c64\uff0c \u7573\u307f\u8fbc\u307f\u5909\u63db\u5c64\uff0c \u30d0\u30c3\u30c1\u6b63\u898f\u5316\u5c64\u3092\u518d\u521d\u671f\u5316\u3059\u308b\u3002\n\u3053\u306e\u95a2\u6570\u306f\uff0c \u521d\u671f\u5316\u5f8c\u3059\u3050\u306b\u30e2\u30c7\u30eb\u306b\u9069\u7528\u3055\u308c\u308b\u3002\n\n\n\n```python\n# custom weights initialization called on netG and netD\ndef weights_init(m):\n classname = m.__class__.__name__\n if classname.find('Conv') != -1:\n nn.init.normal_(m.weight.data, 0.0, 0.02)\n elif classname.find('BatchNorm') != -1:\n nn.init.normal_(m.weight.data, 1.0, 0.02)\n nn.init.constant_(m.bias.data, 0)\n```\n\n# 7. \u751f\u6210\u5668\n\n\n\u751f\u6210\u5668\u3067\u3042\u308b $G$ \u306f\uff0c \u6f5c\u5728\u7a7a\u9593\u30d9\u30af\u30c8\u30eb ($z$) \u3092\u30c7\u30fc\u30bf\u7a7a\u9593\u306b\u30de\u30c3\u30d4\u30f3\u30b0\u3059\u308b\u3088\u3046\u306b\u8a2d\u8a08\u3055\u308c\u3066\u3044\u308b\u3002\n\u30c7\u30fc\u30bf\u306f\u753b\u50cf\u306a\u306e\u3067 $z$ \u3092\u30c7\u30fc\u30bf\u7a7a\u9593\u306b\u5909\u63db\u3059\u308b\u3068\u3044\u3046\u3053\u3068\u306f\uff0c \u6700\u7d42\u7684\u306b\u5b66\u7fd2\u753b\u50cf\u3068\u540c\u3058\u30b5\u30a4\u30ba\uff083x64x64\uff09\u306e RGB \u753b\u50cf\u3092\u4f5c\u6210\u3059\u308b\u3053\u3068\u306b\u306a\u308b\u3002\n$z$ \u3092\u30c7\u30fc\u30bf\u7a7a\u9593\u306b\u5909\u63db\u3059\u308b\u3068\u3044\u3046\u3053\u3068\u306f\uff0c \u6700\u7d42\u7684\u306b\u5b66\u7fd2\u753b\u50cf\u3068\u540c\u3058\u30b5\u30a4\u30ba\uff083x64x64\uff09\u306e RGB \u753b\u50cf\u3092\u4f5c\u6210\u3059\u308b\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002\n\u5b9f\u969b\u306b\u306f 2 \u6b21\u5143\u306e\u7573\u307f\u8fbc\u307f\u5909\u63db\u5c64\u3068 2 \u6b21\u5143\u306e\u30d0\u30c3\u30c1\u30fb\u30ce\u30eb\u30e0\u5c64\uff0c\u304a\u3088\u3073 ReLU \u6d3b\u6027\u5316\u3092\u7d44\u307f\u5408\u308f\u305b\u3066\uff0c\u3053\u306e\u5909\u63db\u3092\u884c\u3063\u3066\u3044\u308b\u3002\n\u751f\u6210\u5668\u306e\u51fa\u529b\u306f\uff0ctanh \u95a2\u6570\u3092\u4ecb\u3057\u3066\u5165\u529b\u30c7\u30fc\u30bf\u306e\u7bc4\u56f2 $[-1,1]$ \u306b\u623b\u3055\u308c\u308b\u3002\n\u3053\u308c\u306f DCGAN \u8ad6\u6587\u306e\u91cd\u8981\u306a\u8ca2\u732e\u3067\u3042\u308b\u305f\u3081\uff0cconv-transpose \u5c64\u306e\u5f8c\u306e\u30d0\u30c3\u30c1\u30ce\u30eb\u30e0\u95a2\u6570\u306e\u5b58\u5728\u306f\u6ce8\u76ee\u306b\u5024\u3059\u308b\u3002\n\u3053\u308c\u306f DCGAN \u8ad6\u6587\u306e\u91cd\u8981\u306a\u8ca2\u732e\u3067\u3042\u308b\u3002\n\u3053\u308c\u3089\u306e\u5c64\u306f\uff0c \u5b66\u7fd2\u4e2d\u306e\u52fe\u914d\u306e\u6d41\u308c\u3092\u52a9\u3051\u308b\u3002\nDCGAN \u8ad6\u6587\u306b\u63b2\u8f09\u3055\u308c\u3066\u3044\u308b\u751f\u6210\u5668\u306e\u30a4\u30e1\u30fc\u30b8\u3092\u4ee5\u4e0b\u306b\u793a\u3059\u3002\n\n \n\n
                                        \n
                                        \nRadford \u3089\u306e\u8ad6\u6587\u3088\u308a \n
                                        \n\n\u5165\u529b \u7bc0\u3067\u8a2d\u5b9a\u3057\u305f\u5165\u529b (*nz*, *ngf*, *nc*) \u304c\uff0c \u30b3\u30fc\u30c9\u4e0a\u306e\u751f\u6210\u5668\u30a2\u30fc\u30ad\u30c6\u30af\u30c1\u30e3\u306b\u3069\u306e\u3088\u3046\u306b\u5f71\u97ff\u3059\u308b\u304b\u306b\u6ce8\u76ee\u3002\n*nz*\u306f $z$ \u5165\u529b\u30d9\u30af\u30c8\u30eb\u306e\u9577\u3055\uff0c*ngf*\u306f\uff0c\u751f\u6210\u5668\u3092\u4ecb\u3057\u3066\u4f1d\u642c\u3055\u308c\u308b\u7279\u5fb4\u5730\u56f3\u306e\u30b5\u30a4\u30ba\uff0c*nc* \u306f\uff0c\u51fa\u529b\u753b\u50cf\u306e\u30c1\u30e3\u30f3\u30cd\u30eb\u6570 (RGB \u753b\u50cf\u306e\u5834\u5408\u306f 3 \u306b\u8a2d\u5b9a) \u3067\u3042\u308b\u3002\n\u4ee5\u4e0b\u306f\uff0c\u3053\u306e\u751f\u6210\u5668\u306e\u30b3\u30fc\u30c9\u3067\u3042\u308b\u3002\n\n\n\n\n\n```python\n# Generator Code\n\nclass Generator(nn.Module):\n def __init__(self, ngpu):\n super(Generator, self).__init__()\n self.ngpu = ngpu\n self.main = nn.Sequential(\n # input is Z, going into a convolution\n nn.ConvTranspose2d( nz, ngf * 8, 4, 1, 0, bias=False),\n nn.BatchNorm2d(ngf * 8),\n nn.ReLU(True),\n # state size. (ngf*8) x 4 x 4\n nn.ConvTranspose2d(ngf * 8, ngf * 4, 4, 2, 1, bias=False),\n nn.BatchNorm2d(ngf * 4),\n nn.ReLU(True),\n # state size. (ngf*4) x 8 x 8\n nn.ConvTranspose2d( ngf * 4, ngf * 2, 4, 2, 1, bias=False),\n nn.BatchNorm2d(ngf * 2),\n nn.ReLU(True),\n # state size. (ngf*2) x 16 x 16\n nn.ConvTranspose2d( ngf * 2, ngf, 4, 2, 1, bias=False),\n nn.BatchNorm2d(ngf),\n nn.ReLU(True),\n # state size. (ngf) x 32 x 32\n nn.ConvTranspose2d( ngf, nc, 4, 2, 1, bias=False),\n nn.Tanh()\n # state size. (nc) x 64 x 64\n )\n\n def forward(self, input):\n return self.main(input)\n```\n\n\u3053\u308c\u3067\uff0c \u751f\u6210\u5668\u3092\u5b9f\u4f53\u5316\u3057\u3066 `weights_init` \u95a2\u6570\u3092\u9069\u7528\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\n\u5370\u5237\u3055\u308c\u305f\u30e2\u30c7\u30eb\u3092\u898b\u3066\uff0c \u751f\u6210\u5668\u30aa\u30d6\u30b8\u30a7\u30af\u30c8\u304c\u3069\u306e\u3088\u3046\u306b\u69cb\u6210\u3055\u308c\u3066\u3044\u308b\u304b\u3092\u78ba\u8a8d\u3057\u3066\u307f\u3088\u3046\u3002\n\n\n\n\n\n```python\n# Create the generator\nnetG = Generator(ngpu).to(device)\n\n# Handle multi-gpu if desired\nif (device.type == 'cuda') and (ngpu > 1):\n netG = nn.DataParallel(netG, list(range(ngpu)))\n\n# Apply the weights_init function to randomly initialize all weights\n# to mean=0, stdev=0.2.\nnetG.apply(weights_init)\n\n# Print the model\nprint(netG)\n```\n\n# 8. \u8b58\u5225\u5668\n\n\n\u524d\u8ff0\u306e\u3088\u3046\u306b\uff0c\u8b58\u5225\u5668\u3067\u3042\u308b $D$ \u306f\uff0c\u753b\u50cf\u3092\u5165\u529b\u3068\u3057\uff0c \u305d\u306e\u5165\u529b\u753b\u50cf\u304c (\u507d\u7269\u3067\u306f\u306a\u304f) \u672c\u7269\u3067\u3042\u308b\u304b\u3069\u3046\u304b\u306e\u78ba\u7387\u3092\u30b9\u30ab\u30e9\u30fc\u3067\u51fa\u529b\u3059\u308b 2 \u5024\u5206\u985e\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3067\u3042\u308b\u3002\n\u3053\u3053\u3067\u306f $D$ \u306f 3x64x64 \u306e\u5165\u529b\u753b\u50cf\u3092\u53d7\u3051\u53d6\u308a\uff0cConv2d\uff0cBatchNorm2d\uff0cLeakyReLU \u306e\u5404\u5c64\u3067\u51e6\u7406\u3057\uff0c\u30b7\u30b0\u30e2\u30a4\u30c9\u6d3b\u6027\u5316\u95a2\u6570\u306b\u3088\u3063\u3066\u6700\u7d42\u7684\u306a\u78ba\u7387\u3092\u51fa\u529b\u3057\u3066\u3044\u308b\u3002\n\u3053\u306e\u30a2\u30fc\u30ad\u30c6\u30af\u30c1\u30e3\u306f\uff0c \u5fc5\u8981\u306b\u5fdc\u3058\u3066\u5c64\u3092\u5897\u3084\u3059\u3053\u3068\u304c\u3067\u304d\u308b\uff0cstrided convolution, BatchNorm, LeakyReLU\u3092\u4f7f\u7528\u3059\u308b\u3053\u3068\u306b\u610f\u5473\u304c\u3042\u308b\u3002\nDCGAN \u306e\u8ad6\u6587\u3067\u306f\uff0c \u30c0\u30a6\u30f3\u30b5\u30f3\u30d7\u30eb\u306b\u30d7\u30fc\u30ea\u30f3\u30b0\u3067\u306f\u306a\u304f strided convolution \u3092\u4f7f\u7528\u3059\u308b\u3053\u3068\u306f\uff0c \u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306b\u72ec\u81ea\u306e\u30d7\u30fc\u30ea\u30f3\u30b0\u95a2\u6570\u3092\u5b66\u7fd2\u3055\u305b\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u305f\u3081\uff0c \u826f\u3044\u65b9\u6cd5\u3067\u3042\u308b\u3068\u8ff0\u3079\u3089\u308c\u3066\u3044\u308b\u3002\n\u307e\u305f\uff0c\u30d0\u30c3\u30c1\u6b63\u5247\u5316\u3068\u30ea\u30fc\u30ad\u30fc ReLU \u306f $G$ \u3068 $D$ \u306e\u4e21\u65b9\u306e\u5b66\u7fd2\u904e\u7a0b\u306b\u3068\u3063\u3066\u91cd\u8981\u306a\uff0c \u5065\u5168\u306a\u52fe\u914d\u306e\u6d41\u308c\u3092\u4fc3\u9032\u3059\u308b\u3002\n\n\n\n\n\u8b58\u5225\u6a5f\u306e\u30b3\u30fc\u30c9 \n\n\n```python\nclass Discriminator(nn.Module):\n def __init__(self, ngpu):\n super(Discriminator, self).__init__()\n self.ngpu = ngpu\n self.main = nn.Sequential(\n # input is (nc) x 64 x 64\n nn.Conv2d(nc, ndf, 4, 2, 1, bias=False),\n nn.LeakyReLU(0.2, inplace=True),\n # state size. (ndf) x 32 x 32\n nn.Conv2d(ndf, ndf * 2, 4, 2, 1, bias=False),\n nn.BatchNorm2d(ndf * 2),\n nn.LeakyReLU(0.2, inplace=True),\n # state size. (ndf*2) x 16 x 16\n nn.Conv2d(ndf * 2, ndf * 4, 4, 2, 1, bias=False),\n nn.BatchNorm2d(ndf * 4),\n nn.LeakyReLU(0.2, inplace=True),\n # state size. (ndf*4) x 8 x 8\n nn.Conv2d(ndf * 4, ndf * 8, 4, 2, 1, bias=False),\n nn.BatchNorm2d(ndf * 8),\n nn.LeakyReLU(0.2, inplace=True),\n # state size. (ndf*8) x 4 x 4\n nn.Conv2d(ndf * 8, 1, 4, 1, 0, bias=False),\n nn.Sigmoid()\n )\n\n def forward(self, input):\n return self.main(input)\n```\n\n\u3053\u308c\u3067\uff0c \u751f\u6210\u5668\u3068\u540c\u69d8\u306b\uff0c\u5224\u5225\u5668\u3092\u4f5c\u6210\u3057 `weights_init` \u95a2\u6570\u3092\u9069\u7528\u3057\u3066\uff0c\u30e2\u30c7\u30eb\u306e\u69cb\u9020\u3092\u8868\u793a\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\n\n\n\n\n\n\n```python\n# Create the Discriminator\nnetD = Discriminator(ngpu).to(device)\n\n# Handle multi-gpu if desired\nif (device.type == 'cuda') and (ngpu > 1):\n netD = nn.DataParallel(netD, list(range(ngpu)))\n \n# Apply the weights_init function to randomly initialize all weights\n# to mean=0, stdev=0.2.\nnetD.apply(weights_init)\n\n# Print the model\nprint(netD)\n```\n\n# 8. \u640d\u5931\u95a2\u6570\u3068\u6700\u9069\u5316\u95a2\u6570\n\n\n$D$ \u3068 $G$ \u306e\u8a2d\u5b9a\u304c\u3067\u304d\u305f\u3089\uff0c \u640d\u5931\u95a2\u6570\u3068\u6700\u9069\u5316\u95a2\u6570\u3092\u4f7f\u3063\u3066\u5b66\u7fd2\u65b9\u6cd5\u3092\u6307\u5b9a\u3059\u308b\u3002\n\u3053\u3053\u3067\u306f\uff0c2 \u5024\u5316\u4ea4\u5dee\u30a8\u30f3\u30c8\u30ed\u30d4\u30fc\u640d\u5931 ([`BCELoss`](https://pytorch.org/docs/stable/nn.html#torch.nn.BCELoss) \u95a2\u6570\u3092\u7528\u3044\u308b\u3002\nPyTorch \u3067\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u308b:\n\n\n\\begin{align}\n\\ell(x, y) = L = \\{l_1,\\dots,l_N\\}^\\top, \\quad\\\\\nl_n = - \\left[y_n \\cdot \\log x_n + (1 - y_n) \\cdot \\log (1 - x_n) \\right]\n\\end{align}\n\n\u3053\u306e\u95a2\u6570\u306f\uff0c \u76ee\u7684\u95a2\u6570\u306e\u4e21\u5bfe\u6570\u6210\u5206 (\u3059\u306a\u308f\u3061 $\\log(D(x))$ \u3068 $\\log(1-D(G(z)))$) \u306e\u8a08\u7b97\u3092\u63d0\u4f9b\u3059\u308b\u3053\u3068\u306b\u6ce8\u76ee\u3002\nBCE \u65b9\u7a0b\u5f0f\u306e\u3069\u306e\u90e8\u5206\u3092 $y$ \u306e\u5165\u529b\u306b\u4f7f\u7528\u3059\u308b\u304b\u3092\u6307\u5b9a\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\n\u3053\u308c\u306f\u4e0b\u306e\u8a13\u7df4\u30eb\u30fc\u30d7\u3067\u5b9f\u73fe\u3055\u308c\u308b\u304c\uff0c \u3069\u306e\u3088\u3046\u306b\u3057\u3066\u8a08\u7b97\u3057\u305f\u3044\u6210\u5206\u3092\u9078\u629e\u3067\u304d\u308b\u304b\u3092\u7406\u89e3\u3059\u308b\u3053\u3068\u306f\u91cd\u8981\u3067\u3042\u308b\u3002\n\u3053\u3053\u3067\u91cd\u8981\u306a\u306e\u306f $y$ \u3092\u5909\u66f4\u3059\u308b\u3060\u3051\u3067\uff0c\u3069\u306e\u6210\u5206\u3092\u8a08\u7b97\u3059\u308b\u304b\u3092\u9078\u629e\u3067\u304d\u308b\u3053\u3068\u3067\u3042\u308b\u3002\n\n\n\u6b21\u306b\uff0c \u672c\u7269\u306e\u30e9\u30d9\u30eb\u3092 1\uff0c \u507d\u7269\u306e\u30e9\u30d9\u30eb\u3092 0 \u3068\u5b9a\u7fa9\u3059\u308b\u3002\n\u3053\u308c\u3089\u306e\u30e9\u30d9\u30eb\u306f $D$ \u3068 $G$ \u306e\u640d\u5931\u3092\u8a08\u7b97\u3059\u308b\u969b\u306b\u4f7f\u7528\u3055\u308c\uff0c \u3053\u308c\u306f\u5143\u306e GAN \u8ad6\u6587\u3067\u3082\u4f7f\u7528\u3055\u308c\u3066\u3044\u308b\u6163\u7fd2\u3067\u3042\u308b\u3002\n\u6700\u5f8c\u306b $D$ \u7528\u3068 $G$ \u7528\u306e 2 \u3064\u306e\u5225\u3005\u306e\u6700\u9069\u5316\u95a2\u6570\u3092\u8a2d\u5b9a\u3059\u308b\u3002\nDCGAN \u8ad6\u6587\u3067\u6307\u5b9a\u3055\u308c\u3066\u3044\u308b\u3088\u3046\u306b\uff0c \u3069\u3061\u3089\u3082\u5b66\u7fd2\u7387 0.0002, Beta1 =0.5 \u306e Adam \u6700\u9069\u5316\u3067\u3042\u308b\u3002\n\u751f\u6210\u5668\u306e\u5b66\u7fd2\u306e\u9032\u884c\u72b6\u6cc1\u3092\u628a\u63e1\u3059\u308b\u305f\u3081\u306b\uff0c \u30ac\u30a6\u30b9\u5206\u5e03 (\u3064\u307e\u308a \u56fa\u5b9a\u30ce\u30a4\u30ba) \u304b\u3089\u5f15\u304d\u51fa\u3055\u308c\u305f\u6f5c\u5728\u7684\u306a\u30d9\u30af\u30c8\u30eb\u306e\u56fa\u5b9a\u30d0\u30c3\u30c1\u3092\u751f\u6210\u3059\u308b\u3002\n\u5b66\u7fd2\u30eb\u30fc\u30d7\u3067\u306f\uff0c\u3053\u306e \u56fa\u5b9a\u30ce\u30a4\u30ba \u3092\u5b9a\u671f\u7684\u306b $G$ \u306b\u5165\u529b\u3057\uff0c\u53cd\u5fa9\u3057\u3066\u3044\u304f\u3046\u3061\u306b\n\u30ce\u30a4\u30ba\u306e\u4e2d\u304b\u3089\u753b\u50cf\u304c\u5f62\u6210\u3055\u308c\u3066\u3044\u304f\u69d8\u5b50\u3092\u898b\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\n\n\n\n\n```python\n# Initialize BCELoss function\ncriterion = nn.BCELoss()\n\n# Create batch of latent vectors that we will use to visualize\n# the progression of the generator\nfixed_noise = torch.randn(64, nz, 1, 1, device=device)\n\n# Establish convention for real and fake labels during training\nreal_label = 1\nfake_label = 0\n\n# Setup Adam optimizers for both G and D\noptimizerD = optim.Adam(netD.parameters(), lr=lr, betas=(beta1, 0.999))\noptimizerG = optim.Adam(netG.parameters(), lr=lr, betas=(beta1, 0.999))\n```\n\n# 9. \u8a13\u7df4\n\n\n\u6700\u5f8c\u306b GAN \u306e\u67a0\u7d44\u307f\u306e\u3059\u3079\u3066\u306e\u90e8\u5206\u304c\u5b9a\u7fa9\u3055\u308c\u305f\u306e\u3067\uff0c\u305d\u308c\u3092\u8a13\u7df4\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\n\u30cf\u30a4\u30d1\u30fc\u30d1\u30e9\u30e1\u30fc\u30bf\u306e\u8a2d\u5b9a\u3092\u9593\u9055\u3048\u308b\u3068\uff0c\u4f55\u304c\u60aa\u304b\u3063\u305f\u306e\u304b\u307b\u3068\u3093\u3069\u8aac\u660e\u3067\u304d\u306a\u3044\u307e\u307e\u30e2\u30fc\u30c9\u5d29\u58ca\u3057\u3066\u3057\u307e\u3046\u306e\u3067 GAN \u306e\u8a13\u7df4\u306f\u3084\u3084\u82b8\u8853\u7684\u306a\u5f62\u5f0f\u3067\u3042\u308b\u3053\u3068\u306b\u7559\u610f\u3057\u3066\u6b32\u3057\u3044\u3002\n\u3053\u3053\u3067\u306f Goodfellow \u306e\u8ad6\u6587\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0 1 \u306b\u5fe0\u5b9f\u306b\u5f93\u3044\u3064\u3064 [ganhacks](https://github.com/soumith/ganhacks) \u3067\u793a\u3055\u308c\u3066\u3044\u308b\u3044\u304f\u3064\u304b\u306e\u30d9\u30b9\u30c8\u30d7\u30e9\u30af\u30c6\u30a3\u30b9\u3092\u9075\u5b88\u3059\u308b\u3002\n\u5177\u4f53\u7684\u306b\u306f \u300c\u672c\u7269\u3068\u507d\u7269\u306e\u753b\u50cf\u306b\u5bfe\u3057\u3066\u7570\u306a\u308b\u30df\u30cb\u30d0\u30c3\u30c1\u3092\u4f5c\u6210\u3059\u308b\u300d\u3053\u3068\u3068\uff0cG \u306e\u76ee\u7684\u95a2\u6570\u304c $\\log D(G(z))$ \u3092\u6700\u5927\u5316\u3059\u308b\u3088\u3046\u306b\u8abf\u6574\u3059\u308b\u3053\u3068\u3067\u3042\u308b\u3002\n\u5b66\u7fd2\u306f\u5927\u304d\u304f 2 \u3064\u306e\u30d1\u30fc\u30c8\u306b\u5206\u304b\u308c\u3066\u3044\u308b\u3002\n\u30d1\u30fc\u30c81 \u3067\u306f \u8b58\u5225\u5668\u306e\u66f4\u65b0\uff0c \u30d1\u30fc\u30c82 \u3067\u306f\u751f\u6210\u5668\u306e\u66f4\u65b0\u3092\u884c\u3046\u3002\n\n\n**\u30d1\u30fc\u30c81 - \u8b58\u5225\u5668\u306e\u8a13\u7df4**\n\n\n\u8b58\u5225\u5668\u3092\u5b66\u7fd2\u3059\u308b\u76ee\u7684\u306f\uff0c \u4e0e\u3048\u3089\u308c\u305f\u5165\u529b\u3092\u6b63\u3057\u304f\u672c\u7269\u304b\u507d\u7269\u304b\u306b\u5206\u985e\u3059\u308b\u78ba\u7387\u3092\u6700\u5927\u5316\u3059\u308b\u3053\u3068\u3067\u3042\u308b\u3002\nGoodfellow \u306e\u8a00\u8449\u3092\u501f\u308a\u308c\u3070\u300c\u78ba\u7387\u7684\u52fe\u914d\u3092\u4e0a\u6607\u3055\u305b\u3066\u8b58\u5225\u5668\u3092\u66f4\u65b0\u3059\u308b\u300d\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002\n\u5b9f\u969b\u306b\u306f $\\log(D(x)) + \\log(1-D(G(x))$ \u3092\u6700\u5927\u5316\u3057\u305f\u3044\u3002\nganhacks \u304b\u3089\u306e\u5225\u306e\u30df\u30cb\u30d0\u30c3\u30c1\u306e\u63d0\u6848\u306b\u3088\u308a\uff0c \u3053\u308c\u3092 2 \u3064\u306e\u30b9\u30c6\u30c3\u30d7\u3067\u8a08\u7b97\u3059\u308b\u3002\n\u307e\u305a\uff0c\u8a13\u7df4\u30bb\u30c3\u30c8\u304b\u3089\u5b9f\u30b5\u30f3\u30d7\u30eb\u306e\u30d0\u30c3\u30c1\u3092\u69cb\u7bc9\u3057\uff0c$D$ \u3092\u524d\u5411\u304d\u7d4c\u8def\u3067\u51e6\u7406\u3057\uff0c\u640d\u5931 ($\\log(D(x))$) \u3092\u8a08\u7b97\u3057\uff0c\u9006\u5411\u7d4c\u8def\u3067\u52fe\u914d\u3092\u8a08\u7b97\u3059\u308b\u3002\n\u6b21\u306b\uff0c\u73fe\u5728\u306e\u751f\u6210\u5668\u3067\u507d\u306e\u30b5\u30f3\u30d7\u30eb\u306e\u30d0\u30c3\u30c1\u3092\u4f5c\u6210\u3057\uff0c\u3053\u306e\u30d0\u30c3\u30c1\u3092 $D$ \u306b\u524d\u5411\u304d\u7d4c\u8def\u306b\u6e21\u3057\uff0c\u640d\u5931 ($\\log(1-D(G(z)))$) \u3092\u8a08\u7b97\u3059\u308b\u3002\n\u305d\u3057\u3066\uff0c\u9006\u5411\u7d4c\u8def\u51e6\u7406\u3067\u52fe\u914d\u3092\u84c4\u7a4d\u3059\u308b\u3002\n\u3053\u3053\u3067\uff0c \u3059\u3079\u3066\u306e\u672c\u7269\u306e\u30d0\u30c3\u30c1\u3068\u3059\u3079\u3066\u306e\u507d\u7269\u306e\u30d0\u30c3\u30c1\u306e\u4e21\u65b9\u304b\u3089\u84c4\u7a4d\u3055\u308c\u305f\u52fe\u914d\u3092\u4f7f\u3063\u3066\uff0c \u8b58\u5225\u6a5f\u306e\u6700\u9069\u5316\u95a2\u6570\u306e\u30b9\u30c6\u30c3\u30d7\u3092\u547c\u3073\u51fa\u3059\u3002\n\n\n**\u30d1\u30fc\u30c82 - \u751f\u6210\u5668\u306e\u8a13\u7df4**\n\n\n\u30aa\u30ea\u30b8\u30ca\u30eb\u8ad6\u6587\u3067\u8ff0\u3079\u3089\u308c\u3066\u3044\u308b\u3088\u3046\u306b\uff0c \u6211\u3005\u306f\uff0c\u3088\u308a\u826f\u3044\u507d\u7269\u3092\u751f\u6210\u3059\u308b\u305f\u3081\u306b $\\log(1-D(G(z)))$ \u3092\u6700\u5c0f\u5316\u3059\u308b\u3053\u3068\u3067\u751f\u6210\u5668\u3092\u8a13\u7df4\u3057\u305f\u3044\u3068\u8003\u3048\u3066\u3044\u308b\u3002\n\u524d\u8ff0\u3057\u305f\u3088\u3046\u306b Goodfellow \u306f\u3053\u306e\u65b9\u6cd5\u3067\u306f\uff0c \u7279\u306b\u5b66\u7fd2\u306e\u521d\u671f\u6bb5\u968e\u3067\u306f\u5341\u5206\u306a\u52fe\u914d\u304c\u5f97\u3089\u308c\u306a\u3044\u3053\u3068\u3092\u793a\u3057\u305f\u3002\n\u305d\u3053\u3067\uff0c\u4ee3\u308f\u308a\u306b $\\log(D(G(z)))$ \u3092\u6700\u5927\u5316\u3059\u308b\u3053\u3068\u306b\u3057\u305f\u3002\n\u3053\u306e\u30b3\u30fc\u30c9\u3067\u306f\uff0c\u30d1\u30fc\u30c81 \u306e\u751f\u6210\u5668\u306e\u51fa\u529b\u3092\u8b58\u5225\u5668\u3067\u5206\u985e\u3057\uff0cG \u306e\u640d\u5931\u3092\u8a08\u7b97\u3057\uff0c \u5b9f\u30e9\u30d9\u30eb\u3092 GT* \u3068\u3057\u3066\u4f7f\u7528\u3057\uff0c \u9006\u5411\u7d4c\u8def\u51e6\u7406\u3067 G \u306e\u52fe\u914d\u3092\u8a08\u7b97\u3057\uff0c \u6700\u5f8c\u306b\u6700\u9069\u5316\u95a2\u6570\u306e\u30b9\u30c6\u30c3\u30d7\u3067 G \u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u3092\u66f4\u65b0\u3059\u308b\u3053\u3068\u3067\uff0c\u3053\u308c\u3092\u5b9f\u73fe\u3057\u3066\u3044\u308b\u3002\n\u5b9f\u30e9\u30d9\u30eb\u3092\u640d\u5931\u95a2\u6570\u306e GT \u30e9\u30d9\u30eb\u3068\u3057\u3066\u4f7f\u7528\u3059\u308b\u306e\u306f\u76f4\u611f\u7684\u3067\u306f\u306a\u3044\u304b\u3082\u3057\u308c\u306a\u3044\u304c\uff0c \u3053\u308c\u306b\u3088\u308a BCELoss \u306e$\\log(x)$ \u90e8\u5206\uff08$\\log(1-x)$ \u90e8\u5206\u3067\u306f\u306a\u304f) \u3092\u4f7f\u7528\u3059\u308b\u3053\u3068\u304c\u3067\u304d\uff0c\u3053\u308c\u306f\u307e\u3055\u306b\u6211\u3005\u304c\u671b\u3080\u3082\u306e\u3067\u3042\u308b\u3002\n\n\n\u6700\u5f8c\u306b\uff0c\u7d71\u8a08\u60c5\u5831\u306e\u5831\u544a\u3092\u884c\u3046\u3002\n\u5404\u30a8\u30dd\u30c3\u30af\u306e\u7d42\u308f\u308a\u306b\uff0c\u56fa\u5b9a\u30ce\u30a4\u30ba\u306e\u30d0\u30c3\u30c1\u3092\u30b8\u30a7\u30cd\u30ec\u30fc\u30bf\u30fc\u306b\u901a\u3057\u3066 G \u306e\u8a13\u7df4\u306e\u9032\u6357\u72b6\u6cc1\u3092\u8996\u899a\u7684\u306b\u78ba\u8a8d\u3059\u308b\u3002\n\u8a13\u7df4\u306e\u7d71\u8a08\u60c5\u5831\u3092\u5831\u544a\u306f:\n\n\n- **Loss_D** - \u3059\u3079\u3066\u306e\u672c\u7269\u306e\u30d0\u30c3\u30c1\u3068\u3059\u3079\u3066\u306e\u507d\u7269\u306e\u30d0\u30c3\u30c1\u306e\u640d\u5931\u306e\u5408\u8a08\u3068\u3057\u3066\u8a08\u7b97\u3055\u308c\u305f\u8b58\u5225\u5668\u306e\u640d\u5931 ($\\log(D(x)) + \\log(D(G(z)))$).\n- **Loss_G** - \u751f\u6210\u5668\u306e\u640d\u5931\u306f $\\log(D(G(z)))$ \u3068\u3057\u3066\u8a08\u7b97\u3055\u308c\u308b\u3002\n- **D(x)** - \u3059\u3079\u3066\u306e\u73fe\u5b9f\u306e\u30d0\u30c3\u30c1\u306b\u5bfe\u3059\u308b\u8b58\u5225\u5668\u306e (\u30d0\u30c3\u30c1\u5168\u4f53\u306e) \u5e73\u5747\u51fa\u529b\u3002\u3053\u308c\u306f 1 \u306b\u8fd1\u3044\u5024\u304b\u3089\u59cb\u307e\u308a G \u304c\u826f\u304f\u306a\u308b\u3068\u7406\u8ad6\u7684\u306b\u306f 0.5 \u306b\u53ce\u675f\u3059\u308b\u306f\u305a\u3067 \u3042\u308b\u3002\u3053\u306e\u7406\u7531\u3092\u8003\u3048\u3066\u307f\u3088\u3046\u3002\n- **D(G(z))** - \u3059\u3079\u3066\u306e\u507d\u306e\u30d0\u30c3\u30c1\u306b\u5bfe\u3059\u308b\u8b58\u5225\u5668\u306e\u5e73\u5747\u51fa\u529b\u3002\u6700\u521d\u306e\u6570\u5b57\u306f D \u304c\u66f4\u65b0\u3055\u308c\u308b\u524d\u3067 2 \u756a\u76ee\u306e\u6570\u5b57\u306f D \u304c\u66f4\u65b0\u3055\u308c\u305f\u5f8c\u3002\u3053\u308c\u3089\u306e\u6570\u5024\u306f 0 \u306b\u8fd1\u3044\u3068\u3053\u308d\u304b\u3089\u59cb\u307e\u308a G \u304c\u826f\u304f\u306a\u308b\u306b\u3064\u308c\u3066 0.5 \u306b\u53ce\u675f\u3059\u308b\u306f\u305a\u3067\u3042\u308b\u3002\u3053\u306e\u7406\u7531\u3092\u8003\u3048\u3066\u307f\u3088\u3046\u3002\n\n\n \n**\u6ce8** \u3053\u306e\u30b9\u30c6\u30c3\u30d7\u306f\uff0c \u4f55\u56de\u30a8\u30dd\u30c3\u30af\u3092\u5b9f\u884c\u3057\u305f\u304b\uff0c \u30c7\u30fc\u30bf\u30bb\u30c3\u30c8\u304b\u3089\u3044\u304f\u3064\u304b\u306e\u30c7\u30fc\u30bf\u3092\u524a\u9664\u3057\u305f\u304b\u306b\u3088\u3063\u3066\uff0c\u6642\u9593\u304c\u304b\u304b\u308b\u3053\u3068\u304c\u3042\u308b\u3002\n\n\n\n\n\n```python\n# \u8a13\u7df4\u30eb\u30fc\u30d7\n\n# Lists to keep track of progress\nimg_list = []\nG_losses = []\nD_losses = []\niters = 0\n\nprint(\"Starting Training Loop...\")\n# For each epoch\nfor epoch in range(num_epochs):\n # For each batch in the dataloader\n for i, data in enumerate(dataloader, 0):\n \n ############################\n # (1) Update D network: maximize log(D(x)) + log(1 - D(G(z)))\n ###########################\n ## Train with all-real batch\n netD.zero_grad()\n # Format batch\n real_cpu = data[0].to(device)\n b_size = real_cpu.size(0)\n #label = torch.full((b_size,), real_label, device=device)\n label = torch.full((b_size,), real_label, device=device).float()\n\n # Forward pass real batch through D\n #output = netD(real_cpu).view(-1)\n output = netD(real_cpu).view(-1).float()\n\n # Calculate loss on all-real batch\n errD_real = criterion(output, label)\n # Calculate gradients for D in backward pass\n errD_real.backward()\n D_x = output.mean().item()\n\n ## Train with all-fake batch\n # Generate batch of latent vectors\n noise = torch.randn(b_size, nz, 1, 1, device=device)\n # Generate fake image batch with G\n fake = netG(noise)\n label.fill_(fake_label)\n # Classify all fake batch with D\n output = netD(fake.detach()).view(-1)\n # Calculate D's loss on the all-fake batch\n errD_fake = criterion(output, label)\n # Calculate the gradients for this batch\n errD_fake.backward()\n D_G_z1 = output.mean().item()\n # Add the gradients from the all-real and all-fake batches\n errD = errD_real + errD_fake\n # Update D\n optimizerD.step()\n\n ############################\n # (2) Update G network: maximize log(D(G(z)))\n ###########################\n netG.zero_grad()\n label.fill_(real_label) # fake labels are real for generator cost\n # Since we just updated D, perform another forward pass of all-fake batch through D\n output = netD(fake).view(-1)\n # Calculate G's loss based on this output\n errG = criterion(output, label)\n # Calculate gradients for G\n errG.backward()\n D_G_z2 = output.mean().item()\n # Update G\n optimizerG.step()\n \n # Output training stats\n if i % 50 == 0:\n print(f'[{epoch}/{num_epochs}][{i}/{len(dataloader)}]\\t',\n f'\u8b58\u5225\u5668\u640d\u5931: {errD.item():.3f}\\t\u751f\u6210\u5668\u640d\u5931: {errG.item():.3f}\\t',\n f'D(x):{D_x:.3f}\\tD(G(z)): {D_G_z1:.3f} / {D_G_z2:.3f}')\n \n # Save Losses for plotting later\n G_losses.append(errG.item())\n D_losses.append(errD.item())\n \n # Check how the generator is doing by saving G's output on fixed_noise\n if (iters % 500 == 0) or ((epoch == num_epochs-1) and (i == len(dataloader)-1)):\n with torch.no_grad():\n fake = netG(fixed_noise).detach().cpu()\n img_list.append(vutils.make_grid(fake, padding=2, normalize=True))\n \n iters += 1\n```\n\n# 10. \u7d50\u679c\n\n\n\u6700\u5f8c\u306b\uff0c\u6211\u3005\u306e\u7d50\u679c\u3092\u78ba\u8a8d\u3057\u3088\u3046\u3002\n\u3053\u3053\u3067\u306f 3 \u3064\u306e\u7570\u306a\u308b\u7d50\u679c\u3092\u898b\u3066\u307f\u3088\u3046\u3002\n\u307e\u305a D \u3068 G \u306e\u640d\u5931\u304c\u8a13\u7df4\u4e2d\u306b\u3069\u306e\u3088\u3046\u306b\u5909\u5316\u3057\u305f\u304b\u3092\u898b\u308b\u3002\n\u6b21\u306b \u56fa\u5b9a\u30ce\u30a4\u30ba\u30d0\u30c3\u30c1\u3067\u306e G \u306e\u51fa\u529b\u3092\u30a8\u30dd\u30c3\u30af\u3054\u3068\u306b\u53ef\u8996\u5316\u3059\u308b\u3002\n\u305d\u3057\u3066 3 \u3064\u76ee\u306f\uff0c\u5b9f\u30c7\u30fc\u30bf\u306e\u30d0\u30c3\u30c1\u3068 G \u306e\u507d\u30c7\u30fc\u30bf\u306e\u30d0\u30c3\u30c1\u3092\u6bd4\u8f03\u3059\u308b\u3002\n\n\n## 10.1 \u640d\u5931\u3068\u30c8\u30ec\u30fc\u30cb\u30f3\u30b0\u56de\u6570\u306e\u95a2\u4fc2\n\n\n\u4ee5\u4e0b\u306f D \u3068 G \u306e\u640d\u5931\u3068\u8a13\u7df4\u306e\u53cd\u5fa9\u56de\u6570\u3092\u30d7\u30ed\u30c3\u30c8\u3057\u305f\u3082\u306e\u3067\u3042\u308b\u3002\n\n\n\n\n```python\nplt.figure(figsize=(14,8))\nplt.title(\"\u751f\u6210\u5668\u3068\u8b58\u5225\u6a5f\u306e\u640d\u5931\u5024\u306e\u5909\u5316\")\nplt.plot(G_losses,label=\"\u751f\u6210\u5668\", color='green')\nplt.plot(D_losses,label=\"\u8b58\u5225\u5668\", color='blue')\nplt.xlabel(\"\u53cd\u5fa9\u56de\u6570\")\nplt.ylabel(\"\u640d\u5931\")\nplt.legend()\nplt.show()\n```\n\n## 10.2 \u8b58\u5225\u6a5f\u306e\u5b66\u7fd2\u306e\u8996\u899a\u5316\n\n\n\u5b66\u7fd2\u306e\u5404\u30a8\u30dd\u30c3\u30af\u306e\u5f8c\uff0c \u56fa\u5b9a\u30ce\u30a4\u30ba\u30d0\u30c3\u30c1\u306b\u751f\u6210\u5668\u306e\u51fa\u529b\u3092\u4fdd\u5b58\u3057\u305f\u3053\u3068\u3092\u601d\u3044\u51fa\u305b\u3002\n\u3053\u3053\u3067\u306f G \u306e\u5b66\u7fd2\u7d4c\u904e\u3092\u30a2\u30cb\u30e1\u30fc\u30b7\u30e7\u30f3\u3067\u53ef\u8996\u5316\u3057\u3066\u307f\u3088\u3046\u3002\n\u518d\u751f\u30dc\u30bf\u30f3\u3092\u62bc\u3059\u3068\u30a2\u30cb\u30e1\u30fc\u30b7\u30e7\u30f3\u304c\u59cb\u307e\u308b\u3002\n\n\n\n```python\n#%%capture\nfig = plt.figure(figsize=(8,8))\nplt.axis(\"off\")\nims = [[plt.imshow(np.transpose(i,(1,2,0)), animated=True)] for i in img_list]\nani = animation.ArtistAnimation(fig, ims, interval=1000, repeat_delay=1000, blit=True)\n\nHTML(ani.to_jshtml())\n```\n\n## 10.3 \u672c\u7269\u306e\u753b\u50cf\u3068\u507d\u7269\u306e\u753b\u50cf\u306e\u6bd4\u8f03\n\n\u6700\u5f8c\u306b\uff0c\u672c\u7269\u306e\u753b\u50cf\u3068\u507d\u7269\u306e\u753b\u50cf\u3092\u4e26\u3079\u3066\u898b\u3066\u307f\u307e\u3088\u3046\u3002\n\n\n\n```python\n# Grab a batch of real images from the dataloader\nreal_batch = next(iter(dataloader))\n\n# Plot the real images\nplt.figure(figsize=(15,15))\nplt.subplot(1,2,1)\nplt.axis(\"off\")\nplt.title(\"Real Images\")\nplt.imshow(np.transpose(vutils.make_grid(real_batch[0].to(device)[:64], padding=5, normalize=True).cpu(),(1,2,0)))\n\n# Plot the fake images from the last epoch\nplt.subplot(1,2,2)\nplt.axis(\"off\")\nplt.title(\"Fake Images\")\nplt.imshow(np.transpose(img_list[-1],(1,2,0)))\nplt.show()\n```\n\n\n\n\n\n```python\n#help(netD)\n```\n\n\n```python\nprint(netD.state_dict(), netG.state_dict())\n#print(optimizerD.state_dict(), optimizerG.state_dict())\n#!pwd\n```\n\n\n```python\nPATH = '2020-0217_011dcgan_faces2.pth'\ntorch.save({\n 'netD_state_dict': netD.state_dict(),\n 'netG_state_dict': netG.state_dict(),\n 'optimizerD_state_dict': optimizerD.state_dict(),\n 'optimizerG_state_dict': optimizerG.state_dict(),\n }, PATH)\n```\n\n\n```python\n#!ls -t | head\n!file 2020-0217_011dcgan_faces.pth\n```\n\n\n```python\nprint(list(netD.parameters())[5].size())\n```\n\n\n```python\nnp.load()\n```\n", "meta": {"hexsha": "d32c37ba05fc238b9756feac1b1f85cc1ee337a6", "size": 53603, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "2021notebooks/2021_1011dcgan_tiny_celeba.ipynb", "max_stars_repo_name": "project-ccap/project-ccap.github.io", "max_stars_repo_head_hexsha": "867e32af5459ae55d864d9d022d69eac17fbb450", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "2021notebooks/2021_1011dcgan_tiny_celeba.ipynb", "max_issues_repo_name": "project-ccap/project-ccap.github.io", "max_issues_repo_head_hexsha": "867e32af5459ae55d864d9d022d69eac17fbb450", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-12-04T11:36:15.000Z", 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YES\n2. YES", "lm_q1_score": 0.679178692681616, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.4101776332988075}} {"text": "# **Quantum K-Means**\n\n\n```python\npip install qiskit-ibmq-provider\n```\n\n Requirement already satisfied: qiskit-ibmq-provider in /opt/conda/lib/python3.8/site-packages (0.18.3)\n Requirement already satisfied: requests>=2.19 in /opt/conda/lib/python3.8/site-packages (from qiskit-ibmq-provider) (2.27.1)\n Requirement already satisfied: websocket-client>=1.0.1 in /opt/conda/lib/python3.8/site-packages (from qiskit-ibmq-provider) (1.3.1)\n Requirement already satisfied: numpy>=1.13 in /opt/conda/lib/python3.8/site-packages (from qiskit-ibmq-provider) (1.22.3)\n Requirement already satisfied: python-dateutil>=2.8.0 in /opt/conda/lib/python3.8/site-packages (from qiskit-ibmq-provider) (2.8.2)\n Requirement already satisfied: qiskit-terra>=0.18.0 in /opt/conda/lib/python3.8/site-packages (from qiskit-ibmq-provider) (0.19.2)\n Requirement already satisfied: urllib3>=1.21.1 in /opt/conda/lib/python3.8/site-packages (from qiskit-ibmq-provider) (1.26.8)\n Requirement already satisfied: requests-ntlm>=1.1.0 in /opt/conda/lib/python3.8/site-packages (from qiskit-ibmq-provider) (1.1.0)\n Requirement already satisfied: six>=1.5 in /opt/conda/lib/python3.8/site-packages (from python-dateutil>=2.8.0->qiskit-ibmq-provider) (1.16.0)\n Requirement already satisfied: scipy>=1.5 in /opt/conda/lib/python3.8/site-packages (from qiskit-terra>=0.18.0->qiskit-ibmq-provider) (1.8.0)\n Requirement already satisfied: tweedledum<2.0,>=1.1 in /opt/conda/lib/python3.8/site-packages (from qiskit-terra>=0.18.0->qiskit-ibmq-provider) (1.1.1)\n Requirement already satisfied: ply>=3.10 in /opt/conda/lib/python3.8/site-packages (from qiskit-terra>=0.18.0->qiskit-ibmq-provider) (3.11)\n Requirement already satisfied: dill>=0.3 in /opt/conda/lib/python3.8/site-packages (from qiskit-terra>=0.18.0->qiskit-ibmq-provider) (0.3.4)\n Requirement already satisfied: stevedore>=3.0.0 in /opt/conda/lib/python3.8/site-packages (from qiskit-terra>=0.18.0->qiskit-ibmq-provider) (3.5.0)\n Requirement already satisfied: symengine>=0.8 in /opt/conda/lib/python3.8/site-packages (from qiskit-terra>=0.18.0->qiskit-ibmq-provider) (0.9.2)\n Requirement already satisfied: python-constraint>=1.4 in /opt/conda/lib/python3.8/site-packages (from qiskit-terra>=0.18.0->qiskit-ibmq-provider) (1.4.0)\n Requirement already satisfied: psutil>=5 in /opt/conda/lib/python3.8/site-packages (from qiskit-terra>=0.18.0->qiskit-ibmq-provider) (5.9.0)\n Requirement already satisfied: sympy>=1.3 in /opt/conda/lib/python3.8/site-packages (from qiskit-terra>=0.18.0->qiskit-ibmq-provider) (1.10)\n Requirement already satisfied: retworkx>=0.10.1 in /opt/conda/lib/python3.8/site-packages (from qiskit-terra>=0.18.0->qiskit-ibmq-provider) (0.11.0)\n Requirement already satisfied: certifi>=2017.4.17 in /opt/conda/lib/python3.8/site-packages (from requests>=2.19->qiskit-ibmq-provider) (2021.10.8)\n Requirement already satisfied: charset-normalizer~=2.0.0 in /opt/conda/lib/python3.8/site-packages (from requests>=2.19->qiskit-ibmq-provider) (2.0.12)\n Requirement already satisfied: idna<4,>=2.5 in /opt/conda/lib/python3.8/site-packages (from requests>=2.19->qiskit-ibmq-provider) (3.3)\n Requirement already satisfied: ntlm-auth>=1.0.2 in /opt/conda/lib/python3.8/site-packages (from requests-ntlm>=1.1.0->qiskit-ibmq-provider) (1.5.0)\n Requirement already satisfied: cryptography>=1.3 in /opt/conda/lib/python3.8/site-packages (from requests-ntlm>=1.1.0->qiskit-ibmq-provider) (36.0.1)\n Requirement already satisfied: cffi>=1.12 in /opt/conda/lib/python3.8/site-packages (from cryptography>=1.3->requests-ntlm>=1.1.0->qiskit-ibmq-provider) (1.15.0)\n Requirement already satisfied: pycparser in /opt/conda/lib/python3.8/site-packages (from cffi>=1.12->cryptography>=1.3->requests-ntlm>=1.1.0->qiskit-ibmq-provider) (2.21)\n Requirement already satisfied: pbr!=2.1.0,>=2.0.0 in /opt/conda/lib/python3.8/site-packages (from stevedore>=3.0.0->qiskit-terra>=0.18.0->qiskit-ibmq-provider) (5.8.1)\n Requirement already satisfied: mpmath>=0.19 in /opt/conda/lib/python3.8/site-packages (from sympy>=1.3->qiskit-terra>=0.18.0->qiskit-ibmq-provider) (1.2.1)\n Note: you may need to restart the kernel to use updated packages.\n\n\n\n```python\npip install matlab\n```\n\n Collecting matlab\n Downloading matlab-0.1.tar.gz (538 bytes)\n Building wheels for collected packages: matlab\n Building wheel for matlab (setup.py) ... \u001b[?25ldone\n \u001b[?25h Created wheel for matlab: filename=matlab-0.1-py3-none-any.whl size=1180 sha256=6f3feb961dd6f5a1288d05aa85864287831ad165f88923f7cd72e85e39990ea9\n Stored in directory: /home/jovyan/.cache/pip/wheels/9d/e3/ca/f9444a09793775674f6d6c3389b6192c72eff2503085389009\n Successfully built matlab\n Installing collected packages: matlab\n Successfully installed matlab-0.1\n Note: you may need to restart the kernel to use updated packages.\n\n\n\n```python\nimport matplotlib.pyplot as plot\nimport pandas as pand\nfrom qiskit import QuantumRegister, ClassicalRegister\nfrom qiskit import QuantumCircuit\nfrom qiskit import Aer, execute\nfrom numpy import pi\n```\n\n :219: RuntimeWarning: scipy._lib.messagestream.MessageStream size changed, may indicate binary incompatibility. Expected 56 from C header, got 64 from PyObject\n\n\n\n```python\nfigure, axis = plot.subplots()\naxis.set(xlabel='Feature 1', ylabel='Feature 2')\n```\n\n\n```python\ndata_input = pand.read_csv('kmeans_input.csv',\n usecols=['Feature 1', 'Feature 2', 'Class'])\n```\n\n\n```python\nisRed = data_input['Class'] == 'Red'\nisGreen = data_input['Class'] == 'Green'\nisBlack = data_input['Class'] == 'Black'\n```\n\n\n```python\n# Filter data\nredData = data_input[isRed].drop(['Class'], axis=1)\ngreenData = data_input[isGreen].drop(['Class'], axis=1)\nblackData = data_input[isBlack].drop(['Class'], axis=1)\n```\n\n\n```python\ny_p = 0.141\nx_p = -0.161\n```\n\n\n```python\nxgc = sum(redData['Feature 1']) / len(redData['Feature 1'])\nxbc = sum(greenData['Feature 1']) / len(greenData['Feature 1'])\nxkc = sum(blackData['Feature 1']) / len(blackData['Feature 1'])\n```\n\n\n```python\n# Finding the y-coords of the centroids\nygc = sum(redData['Feature 2']) / len(redData['Feature 2'])\nybc = sum(greenData['Feature 2']) / len(greenData['Feature 2'])\nykc = sum(blackData['Feature 2']) / len(blackData['Feature 2'])\n```\n\n\n```python\n# Plotting the centroids\nplot.plot(xgc, ygc, 'rx')\nplot.plot(xbc, ybc, 'gx')\nplot.plot(xkc, ykc, 'kx')\n```\n\n\n```python\nplot.plot(x_p, y_p, 'bo')\n```\n\n\n```python\n# Setting the axis ranges\nplot.axis([-1, 1, -1, 1])\n```\n\n\n```python\nplot.show()\n```\n\n\n```python\n# Calculating theta and phi values\nphi_list = [((x + 1) * pi / 2) for x in [x_p, xgc, xbc, xkc]]\ntheta_list = [((x + 1) * pi / 2) for x in [y_p, ygc, ybc, ykc]]\n```\n\n\n```python\nquantumregister = QuantumRegister(3, 'quantumregister')\n```\n\n\n```python\nclassicregister = ClassicalRegister(1, 'classicregister')\nquantum_circuit = QuantumCircuit(quantumregister, classicregister, name='qc')\nbackend = Aer.get_backend('qasm_simulator')\nquantum_results_list = []\n```\n\n\n```python\nfor i in range(1, 4):\n quantum_circuit.h(quantumregister[2])\n\n \n quantum_circuit.u3(theta_list[0], phi_list[0], 0, quantumregister[0]) \n quantum_circuit.u3(theta_list[i], phi_list[i], 0, quantumregister[1]) \n\n quantum_circuit.cswap(quantumregister[2], quantumregister[0], quantumregister[1])\n quantum_circuit.h(quantumregister[2])\n\n quantum_circuit.measure(quantumregister[2], classicregister[0])\n\n quantum_circuit.reset(quantumregister)\n\n job = execute(quantum_circuit, backend=backend, shots=1024)\n result = job.result().get_counts(quantum_circuit)\n quantum_results_list.append(result['1'])\n\nprint(quantum_results_list)\n```\n\n /tmp/ipykernel_110/1224122024.py:5: DeprecationWarning: The QuantumCircuit.u3 method is deprecated as of 0.16.0. It will be removed no earlier than 3 months after the release date. You should use QuantumCircuit.u instead, which acts identically. Alternatively, you can decompose u3 in terms of QuantumCircuit.p and QuantumCircuit.sx: u3(\u03f4,\u03c6,\u03bb) = p(\u03c6+\u03c0) sx p(\u03f4+\u03c0) sx p(\u03bb) (2 pulses on hardware).\n quantum_circuit.u3(theta_list[0], phi_list[0], 0, quantumregister[0])\n /tmp/ipykernel_110/1224122024.py:6: DeprecationWarning: The QuantumCircuit.u3 method is deprecated as of 0.16.0. It will be removed no earlier than 3 months after the release date. You should use QuantumCircuit.u instead, which acts identically. Alternatively, you can decompose u3 in terms of QuantumCircuit.p and QuantumCircuit.sx: u3(\u03f4,\u03c6,\u03bb) = p(\u03c6+\u03c0) sx p(\u03f4+\u03c0) sx p(\u03bb) (2 pulses on hardware).\n quantum_circuit.u3(theta_list[i], phi_list[i], 0, quantumregister[1])\n\n\n [79, 64, 115]\n\n\n /tmp/ipykernel_110/1224122024.py:5: DeprecationWarning: The QuantumCircuit.u3 method is deprecated as of 0.16.0. It will be removed no earlier than 3 months after the release date. You should use QuantumCircuit.u instead, which acts identically. Alternatively, you can decompose u3 in terms of QuantumCircuit.p and QuantumCircuit.sx: u3(\u03f4,\u03c6,\u03bb) = p(\u03c6+\u03c0) sx p(\u03f4+\u03c0) sx p(\u03bb) (2 pulses on hardware).\n quantum_circuit.u3(theta_list[0], phi_list[0], 0, quantumregister[0])\n /tmp/ipykernel_110/1224122024.py:6: DeprecationWarning: The QuantumCircuit.u3 method is deprecated as of 0.16.0. It will be removed no earlier than 3 months after the release date. You should use QuantumCircuit.u instead, which acts identically. Alternatively, you can decompose u3 in terms of QuantumCircuit.p and QuantumCircuit.sx: u3(\u03f4,\u03c6,\u03bb) = p(\u03c6+\u03c0) sx p(\u03f4+\u03c0) sx p(\u03bb) (2 pulses on hardware).\n quantum_circuit.u3(theta_list[i], phi_list[i], 0, quantumregister[1])\n\n\n\n```python\nclass_list = ['Red', 'Green', 'Black']\nquantum_p_class = class_list[quantum_results_list.index(min(quantum_results_list))]\ndistances_list = [((x_p - i[0])**2 + (y_p - i[1])**2)**0.5 for i in [(xgc, ygc), (xbc, ybc), (xkc, ykc)]]\nclassical_p_class = class_list[distances_list.index(min(distances_list))]\n```\n\n\n```python\nprint(\"\"\"using quantumdistance algorithm,\n the new data point is related to the\"\"\", quantum_p_class, \n 'class.\\n')\nprint('Euclidean distances are listed: ', distances_list, '\\n')\nprint(\"\"\"based on euclidean distance calculations,\n the new data point is related to the\"\"\", classical_p_class, \n 'class.')\n```\n\n using quantumdistance algorithm,\n the new data point is related to the Green class.\n \n Euclidean distances are listed: [0.520285324797846, 0.4905204028376394, 0.7014755294377704] \n \n based on euclidean distance calculations,\n the new data point is related to the Green class.\n\n\n\n```python\nfigure, axis = plot.subplots()\naxis.set(xlabel='Feature 1', ylabel='Feature 2')\n\n\ndata_input = pand.read_csv('kmeans_input.csv',\n usecols=['Feature 1', 'Feature 2', 'Class'])\n\n\nisRed = data_input['Class'] == 'Red'\nisGreen = data_input['Class'] == 'Green'\nisBlack = data_input['Class'] == 'Black'\n\n# Filter data\nredData = data_input[isRed].drop(['Class'], axis=1)\ngreenData = data_input[isGreen].drop(['Class'], axis=1)\nblackData = data_input[isBlack].drop(['Class'], axis=1)\n\n\ny_p = 0.141\nx_p = -0.161\n\nxgc = sum(redData['Feature 1']) / len(redData['Feature 1'])\nxbc = sum(greenData['Feature 1']) / len(greenData['Feature 1'])\nxkc = sum(blackData['Feature 1']) / len(blackData['Feature 1'])\n\n# Finding the y-coords of the centroids\nygc = sum(redData['Feature 2']) / len(redData['Feature 2'])\nybc = sum(greenData['Feature 2']) / len(greenData['Feature 2'])\nykc = sum(blackData['Feature 2']) / len(blackData['Feature 2'])\n\n# Plotting the centroids\nplot.plot(xgc, ygc, 'rx')\nplot.plot(xbc, ybc, 'gx')\nplot.plot(xkc, ykc, 'kx')\n\n\nplot.plot(x_p, y_p, 'bo')\n\n# Setting the axis ranges\nplot.axis([-1, 1, -1, 1])\n\nplot.show()\n\n# Calculating theta and phi values\nphi_list = [((x + 1) * pi / 2) for x in [x_p, xgc, xbc, xkc]]\ntheta_list = [((x + 1) * pi / 2) for x in [y_p, ygc, ybc, ykc]]\n\nquantumregister = QuantumRegister(3, 'quantumregister')\n\n\nclassicregister = ClassicalRegister(1, 'classicregister')\n\nquantum_circuit = QuantumCircuit(quantumregister, classicregister, name='qc')\n\n\nbackend = Aer.get_backend('qasm_simulator')\n\n\nquantum_results_list = []\n\n\nfor i in range(1, 4):\n quantum_circuit.h(quantumregister[2])\n\n \n quantum_circuit.u3(theta_list[0], phi_list[0], 0, quantumregister[0]) \n quantum_circuit.u3(theta_list[i], phi_list[i], 0, quantumregister[1]) \n\n quantum_circuit.cswap(quantumregister[2], quantumregister[0], quantumregister[1])\n quantum_circuit.h(quantumregister[2])\n\n quantum_circuit.measure(quantumregister[2], classicregister[0])\n\n quantum_circuit.reset(quantumregister)\n\n job = execute(quantum_circuit, backend=backend, shots=1024)\n result = job.result().get_counts(quantum_circuit)\n quantum_results_list.append(result['1'])\n\nprint(quantum_results_list)\n\n\n\nclass_list = ['Red', 'Green', 'Black']\n\n\nquantum_p_class = class_list[quantum_results_list.index(min(quantum_results_list))]\n\n\ndistances_list = [((x_p - i[0])**2 + (y_p - i[1])**2)**0.5 for i in [(xgc, ygc), (xbc, ybc), (xkc, ykc)]]\nclassical_p_class = class_list[distances_list.index(min(distances_list))]\n\nprint(\"\"\"using quantumdistance algorithm,\n the new data point is related to the\"\"\", quantum_p_class, \n 'class.\\n')\nprint('Euclidean distances are listed: ', distances_list, '\\n')\nprint(\"\"\"based on euclidean distance calculations,\n the new data point is related to the\"\"\", classical_p_class, \n 'class.')\n```\n", "meta": {"hexsha": "d5fcf26583768d9097330812e8cf812e9d1411ed", "size": 59253, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Solving QC/Quantum K-Means.ipynb", "max_stars_repo_name": "thirasit/Quantum-Computing", "max_stars_repo_head_hexsha": "32be3646e3af14e2868d9325660996927efe30d2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Solving QC/Quantum K-Means.ipynb", "max_issues_repo_name": "thirasit/Quantum-Computing", "max_issues_repo_head_hexsha": "32be3646e3af14e2868d9325660996927efe30d2", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Solving QC/Quantum K-Means.ipynb", "max_forks_repo_name": "thirasit/Quantum-Computing", "max_forks_repo_head_hexsha": "32be3646e3af14e2868d9325660996927efe30d2", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 93.3118110236, "max_line_length": 9692, "alphanum_fraction": 0.8256122053, "converted": true, "num_tokens": 4232, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353744, "lm_q2_score": 0.7341195269001831, "lm_q1q2_score": 0.40987874893708626}} {"text": "#
                                        Block 5: One-dimensional optimal transport
                                        \n###
                                        Alfred Galichon (NYU)
                                        \n##
                                        `math+econ+code' masterclass on matching models, optimal transport and applications
                                        \n
                                        \u00a9 2018-2019 by Alfred Galichon. Support from NSF grant DMS-1716489 is acknowledged. Julie Lenoir and James Nesbit contributed.
                                        \n\n## Optimal transport I: One-dimensional matching\n\n### Learning Objectives\n\n* Copulas and comonotonicity\n\n* Positive Assortative Matching\n\n* The Wage Equation\n\n### References\n\n* [OTME], Ch. 4 and App. C\n\n* Complements:\n\n * Becker (1973). A Theory of Marriage: Part I. *JPE*.\n\n * Sattinger (1993). Assignment Models of the Distribution of Earnings. *JEL*.\n\n * Gabaix and Landier (2008). Why has CEO Pay Increased So Much? *QJE*.\n\n * Tervi0 (2008). The Difference That CEOs Make: An Assignment Model Approach. *AER*.\n\n\n## Motivation\n\nToday we shall study one-dimensional models of matching. The application we shall envision is to the literature on CEO compensation: talented CEOs arguably generate more output if they run large firms than if they run a small business. Hence in these models, the CEO's talent is leveraged by the firm size.\n\nThe output of a match is CEO's talent multiplied by a term capturing the firm's size. This surplus function belongs to a class which leads to positive assortative matching: the optimal matching will consists of assigning the most talented CEO with the biggest firm, etc.\n\nThe model we shall follows a literature in family economics, see Becker (1973), and labor economics: Sattinger (1993), Tervio (2008), Gabaix and Landier (2008). Our application will be based on Gabaix and Landier's CEO compensation data. The difference with the empirical literature on marriage is that one observes typically less relevant characteristics on CEOs, but we observe the transfers, which are typically not observed whithin the couple.\n\n## Data\n\nFor each firm, we observe the size of the firm (market cap) and the compensation.\n\nDistribution of CEO talent (to be defined later) is calibrated in their paper and we will take it as given; so will be the production function.\n\nOur exercise will consist of predicting the CEO compensation. (Actually GL do the opposite: they use the distribution of wages to calibrate the production function).\n\n\n```R\nthePath = getwd()\ndata = as.matrix(read.csv(paste0(thePath, \"/../data_mec_optim/wages_gabaix-landier/data_Gabaix_Landier.csv\"), header = TRUE))\nhead(data)\n```\n\n\n\n\n\n\t\n\n\n\t\n\t\n\t\n\t\n\t\n\t\n\n
                                        A matrix: 6 \u00d7 5 of type dbl
                                        yearamktcapnFIRMcompnCEO
                                        200017522.9633322024.619 74
                                        199119737.49129 NA NA
                                        199223442.23107 NA NA
                                        199323647.43118 NA NA
                                        199422099.51126 5147.516175
                                        199522952.51154 6280.350208
                                        \n\n\n\n## Positive assortative matching\n\nAssume $\\mathcal{X}=\\mathcal{Y}=\\mathbb{R}$. This implies that both workers and firms are characterized by scalar characteristics $x$ and $y$. Restrictive, but most of the applied literature in Economics to this day has \nfocused on this case, which already generates interesting economic insights.\n\nAssume that there is the same number of workers and firms, and this number is normalized to one. Let $P$ be the probability distribution of the workers types $x$, and $Q$ the probability distribution of the firms types $y$.\n\nE.g. the literature on CEO compensation studies the matching problem of firms and managers: each manager has\\ a measure of talent $x\\in\\mathbb{R}_{+}$ (extra return generated), and each firm has market capitalization $y\\in\\mathbb{R}_{+}$. Then, the economic value generated by a manager of talent $x$ running a firm of size $y$ is $\\Phi\\left( x,y\\right) =xy$. Assume manager of talent $x$ is assigned to firm of size $y=T\\left(x\\right) $. The constraint on the assignment map $T$ is that $T\\#P=Q$, which means that each firm is run by a manager and each manager runs a firm. Then the total value created is\n\n\\begin{align*}\n\\mathbb{E}_{P}\\left[ \\Phi\\left( X,T\\left( X\\right) \\right) \\right]\n=\\mathbb{E}\\left[ XT\\left( X\\right) \\right] .\n\\end{align*}\n\nUnder a natural assuption on $\\Phi$, called \\emph{supermodularity}, we will see that the optimal coupling is such that $Y=T\\left( X\\right) $, where $T$ is nondecreasing. $T$ will be given an explicit characterization.\n\n## Copulas\n\nRecall that given a probability distribution $P$ on the real line, one can define the \\emph{quantile} of that distribution, denoted $F_{P}^{-1}$, which is a map from $\\left[ 0,1\\right] $ to $R$, nondecreasing and continuous from the right, and such that if $U\\sim\\mathcal{U}:=\\mathcal{U} \\left(\\left[0,1\\right]\\right)$, then\n\n\n\\begin{align*}\nF_{P}^{-1}\\left( U\\right) \\sim P.\n\\end{align*}\n\n$F_{P}^{-1}$ is the generalized inverse of the c.d.f. of $P$, $F_{P}$, the proper inverse when $P$ has a positive density, in which case $U=F_{P}\\left(X\\right) $. Note also that if $X\\sim P$, then $F_{P}\\left( X\\right) $ has a uniform distribution if and only if $P$ has no mass point.\n\n[Quantile representation](#quantileRep) extends to the case of bivariate distributions: for $\\pi\\in\\mathcal{M}\\left( P,Q\\right) $, there exists a pair $\\left( U,V\\right) $ of uniform random variables such that\n\n\n\\begin{align*}\n\\left( F_{P}^{-1}\\left( U\\right) ,F_{Q}^{-1}\\left( V\\right) \\right)\n\\sim\\pi,\n\\end{align*}\n\nand the c.d.f. associated with the distribution of $\\left( U,V\\right) $ is called the *copula* associated with distribution $\\pi$.\n\n## Comonotonicity\n\nA pair of random variables $\\left( X,Y\\right)$ is *comonotone* if there is $U\\sim\\mathcal{U}$ such that $X=F_{P}^{-1}\\left( U\\right)$ and $Y=F_{Q}^{-1}\\left( U\\right) $. Equivalently, $X$ and $Y$ are said to exhibit *Positive Assortative Matching (PAM)*.\n\nThe copula associated with a pair of comonotone random variables is the c.d.f. associated with $\\left( U,U\\right) $, which is $F\\left( u,v\\right) =\\min\\left( u,v\\right) $. This copula is called the *upper Fr\u00e9chet-Hoeffding copula*.\n\nNote that when the cdf of $X$ is continuous, there is a much simpler equivalent statement of comonotonicity:\n\n---\n**Lemma**\nIf the distribution of $X$ has no mass points, then $X$ and $Y$ are comonotone if and only if there exists a nondecreasing map $T$ such that $Y=T\\left( X\\right) $. Moreover, one can choose $T\\left(x\\right) =F_{Q}^{-1}\\left( F_{P}\\left( x\\right) \\right).$\n\n---\n\n---\n**Proof**\n\nConsider $U\\sim\\mathcal{U}$ such that $X=F_{P}^{-1}\\left( U\\right) $ and $Y=F_{Q}^{-1}\\left( U\\right) $. If the distribution of $X$ has no mass point, then $U=F_{P}\\left( X\\right) $. Hence, $Y=F_{Q}^{-1}\\left(F_{P}\\left( X\\right) \\right) $.\n\n---\n\n### Supermodular surplus\n\nAssume $\\Phi$ is \\emph{supermodular}, that is, for every scalars $x$, $x^{\\prime}$, $y$ and $y^{\\prime}$,\n\n\n\\begin{align*}\n\\Phi\\left( x\\vee x^{\\prime},y\\vee y^{\\prime}\\right) +\\Phi\\left( x\\wedge\nx^{\\prime},y\\wedge y^{\\prime}\\right) \\geq\\Phi\\left( x,y\\right) +\\Phi\\left(\nx^{\\prime},y^{\\prime}\\right) , \\label{supermodularPhi}%\n\\end{align*}\n\nwhere $x\\vee x^{\\prime}$ and $x\\wedge x^{\\prime}$ denote respectively the maximum and the minimum between scalars $x$ and $x^{\\prime}$. When $\\Phi$ is twice continuously differentiable (which we will assume from now on), this is equivalent to\n\n\\begin{align*}\n\\frac{\\partial^{2}\\Phi\\left( x,y\\right) }{\\partial x\\partial y}\\geq0.\n\\label{SupermodDiff}%\n\\end{align*}\n\nAssume that there are two types of workers $\\mathcal{X=}\\left\\{\\underline{x},\\overline{x}\\right\\} $ and firms $\\mathcal{Y=}\\left\\{ \\underline{y},\\overline{y}\\right\\} $. An equivalent restatement of the [supermodular condition](#supermodularPhi) is then\n\n\n\\begin{align*}\n\\overline{x}\\geq \\underline{x} \\text{ and }\\overline{y}\\geq \\underline{y} \\text{ implies }\\Phi\\left(\n\\overline{x},\\overline{y}\\right) +\\Phi\\left( \\underline{x},\\underline{y}\\right)\n\\geq\\Phi\\left( \\overline{x},\\underline{y}\\right) +\\Phi\\left( \\underline{x},\\overline\n{y}\\right) \\label{PAMequiv}%\n\\end{align*}\n\nwhich asserts that the total output created is higher if the high types match together and the low types match together (assortative matching) rather than if mixed high/low pairs are formed.\n\nThe following examples of surplus functions are supermodular:\n\n1. Cobb-Douglas function: $\\Phi\\left( x,y\\right) =x^{a}y^{b}$ ($x,y\\geq 0$), with $a,b\\geq 0$,\n\n2. General multiplicative form: $\\Phi\\left( x,y\\right) =\\zeta\\left(x\\right) \\xi\\left( y\\right) $ with $\\zeta$ and $\\xi$ nondecreasing,\n\n3. Leontieff: $\\Phi\\left( x,y\\right) =\\min\\left( x,y\\right)$,\n\n4. C.E.S. function: $\\Phi\\left( x,y\\right) =\\left( x^{-\\rho}+y^{-\\rho}\\right) ^{-1/\\rho}$, $\\rho\\geq 0$,\n\n5. $\\Phi\\left( x,y\\right) =\\phi\\left( x-y\\right) $ where $\\phi$ is concave; in particular, $\\Phi\\left( x,y\\right) =-\\left\\vert x-y\\right\\vert ^{p}$, $p\\geq1$ or $\\Phi\\left( x,y\\right) =-\\left( x-y-k\\right) ^{+}$,\n\n6. $\\Phi\\left( x,y\\right) =\\phi\\left( x+y\\right) $, where $\\phi$ convex.\n\n## Rearrangement theorem\n\n---\n**Theorem**\n\n1. Assume that $\\Phi$ is supermodular. Then the primal of the Monge-Kantorovich problem\n\n \n \\begin{align*}\n \\sup_{\\pi\\in\\mathcal{M}\\left( P,Q\\right) }\\mathbb{E}_{\\pi}\\left[\n \\Phi\\left( X,Y\\right) \\right]\n \\end{align*}\n\n has a comonotone solution.\n\n2. Conversely, if [MK primal](#mkpb1d) has a comonotone solution for any choice of probability distributions $P$ and $Q$ on the real line, then $\\Phi$ is supermodular.\n\n3. If, in addition, $P$ has no mass points, then there is an optimal assignment which is is pure and satisfies $Y=T\\left( X\\right)$ where\n\n \n \\begin{align*}\n T\\left( x\\right) =F_{Q}^{-1}\\circ F_{P}\\left( x\\right).\n \\end{align*}\n\n---\nThe proof of part 1. is based on the following lemma.\n\n---\n\n**Lemma**\n\nLet $Z_{1}$ and $Z_{2}$ be two Bernoulli random variables of respective success probability $p_{1}$ and $p_{2}$. Then $\\mathbb{E}\\left[ Z_{1}Z_{2}\\right] \\leq\\min\\left( p_{1},p_{2}\\right)$.\n\n---\n---\n**Proof**\n\nAs $Z_{2}\\leq1$, $\\mathbb{E}\\left[ Z_{1}Z_{2}\\right] \\leq\\mathbb{E}\\left[Z_{1}\\right] =p_{1}$. Similarly $\\mathbb{E}\\left[ Z_{1}Z_{2}\\right]\\leq\\mathbb{E}\\left[ Z_{2}\\right] =p_{2}$. Thus, $\\mathbb{E}\\left[Z_{1}Z_{2}\\right] \\leq\\min\\left( p_{1},p_{2}\\right)$.\n\n---\n\nWe are now ready to sketch the proof of the [Rearrangement Theorem](#thm:MKsupermod).\n\n---\n**Proof**\n\n1. Take $U\\sim\\mathcal{U}$,and $X=F_{P}^{-1}\\left( U\\right) $ and $Y=F_{Q}^{-1}\\left( U\\right)$. By the [quantile representation](#quantileRep), $X\\sim P$ and $Y\\sim Q$ and $\\left( X,Y\\right)$ is comonotone by definition. The proof is in three steps.\n\n **Step 1.** For $a,b\\in\\mathbb{R}$, consider surplus function $\\phi_{ab}\\left(x,y\\right) :=1\\left\\{ x\\geq a\\right\\} 1\\left\\{ y\\geq b\\right\\}$, and let $Z_{1}=1\\left\\{ X\\geq a\\right\\} $ and $Z_{2}=1\\left\\{ Y\\geq b\\right\\}$. $Z_{1}$ and $Z_{2}$ are two Bernoulli random variables of respective success probability $p_{1}=1-F_{P}\\left( a\\right) $ and $p_{2}=1-F_{Q}\\left(b\\right) $, thus $\\mathbb{E}\\left[ Z_{1}Z_{2}\\right] \\leq\\min\\left(p_{1},p_{2}\\right) $, but a straightforward calculation shows that the inequality actually holds as an equality. Hence $\\left( X,Y\\right)$, which is comonotone, is optimal for each surplus function $\\phi_{ab}$.\n\n **Step 2.** Assume $\\mathcal{X}=\\left[\\underline{x},\\overline{x}\\right]$ and $\\mathcal{Y} = \\left[ \\underline{y}, \\overline{y}\\right]$ are compact intervals. Then\n\n \\begin{align*}\nF\\left( x,y\\right) =\\frac{\\Phi\\left( x,y\\right) -\\Phi\\left( \\underline{x},y\\right) -\\Phi\\left( x,\\underline{y}\\right) +\\Phi\\left( \\underline{x},\\underline{y}\\right) }{\\Phi\\left( \\overline{x},\\overline{y}\\right) -\\Phi\\left(\\underline{x},\\overline{y}\\right) -\\Phi\\left( \\overline{x},\\underline{y}\\right)+\\Phi\\left( \\underline{x},\\underline{y}\\right) }\n \\end{align*}\n\n is a c.d.f. associated to a probability measure $\\zeta$, and hence \n\n \\begin{align*}\nF\\left(x,y\\right) = \\int\\phi_{ab}\\left( x,y\\right) d\\zeta\\left( a,b\\right)$.\n \\end{align*}\n \n As a result, if $\\pi\\in\\mathcal{M}\\left( p,q\\right)$ is the distribution of $\\left( X,Y\\right) $ where $X$ and $Y$ are comonotone, then\n \n \\begin{align*}\n\\int F\\left( x,y\\right) d\\pi\\left( x,y\\right) \\geq\\int F\\left(\nx,y\\right) d\\tilde{\\pi}\\left( x,y\\right)\n \\end{align*}\n \n for every $\\tilde{\\pi}\\in\\mathcal{M}\\left( p,q\\right) $. But as $F$ is of the form $F\\left( x,y\\right) =K\\Phi\\left( x,y\\right) +f\\left( x\\right) +g\\left( y\\right) +c$ with $K>0$, and because $\\int\\left\\{ f\\left(x\\right) +g\\left( y\\right) +c\\right\\} d\\pi\\left( x,y\\right)=\\int\\left\\{ f\\left( x\\right) +g\\left( y\\right) +c\\right\\} d\\tilde{\\pi}\\left( x,y\\right) $ for every $\\tilde{\\pi}\\in\\mathcal{M}\\left( p,q\\right) $, it results that\n \n \\begin{align*}\n\\int\\Phi\\left( x,y\\right) d\\pi\\left( x,y\\right) \\geq\\int\\Phi\\left(x,y\\right) d\\tilde{\\pi}\\left( x,y\\right) ~\\forall\\tilde{\\pi}\\in \\mathcal{M}\\left( p,q\\right)\n \\end{align*}\n which completes step 2.\n \n **Step 3.** When $\\mathcal{X}$ and $\\mathcal{Y}$ are the real line, the result still holds by approximation.\n\n2. The converse follows by taking for $P$ the discrete probability with two mass points \\b{x} and $\\overline{x}$ with probability 1/2 each, and $Q$ the discrete probability with two mass points \\b{y} and $\\overline{y}$ also each with probability 1/2. Then if [MK primal](#mkpb1d) has a solution such that $F_{P}^{-1}\\left( U\\right) $ and $Y=F_{Q}^{-1}\\left( U\\right) $, for $U\\sim\\mathcal{U}\\left( \\left[ 0,1\\right] \\right) $, it follows that [condition](#PAMequiv) holds.\n\n3. follows from (i) and [Lemma](#lem:comonotone).\n\n---\n\n\nNote that the assumptions made in the [Rearrangement Theorem](#thm:MKsupermod) do not guarantee that all the optimal assignments are comonotone. Indeed, the trivial example where $\\Phi\\left( x,y\\right) =0$ for every $x$ and $y$ provides an example of supermodular surplus function, for which any assignment is optimal. For this reason, we provide a strengthening of the previous result, which ensures uniqueness. We will assume $\\Phi$ is strictly supermodular, that is if both $\\overline{x}>\\underline{x}$ and $\\overline{y}>\\underline{y}$ hold, then $\\Phi\\left(\\overline{x},\\overline{y}\\right) +\\Phi\\left(\\underline{x},\\underline{y}\\right) >\\Phi\\left(\\overline{x},\\underline{y}\\right) +\\Phi\\left( \\underline{x},\\overline{y}\\right)$.\n\n---\n**Theorem**\n\nAssume that $\\Phi$ is strictly supermodular, and $P$ has no mass point. Then the primal [Monge-Kantorovich problem](#mkpb1d) has a unique optimal assignment, and this assignment is characterized by $Y=T\\left( X\\right) $ where $T$ is given by [this](#mongeMap1d).\n\n---\n\n\n### The wage equation\n\nAssume $\\left(u,v\\right)$ is a solution to the dual of the Monge-Kantorovich problem\n\n\n\\begin{align}\n\\inf \\,& \\mathbb{E}_{P}\\left[ u\\left( X\\right) \\right] +\\mathbb{E}_{Q}\\left[ v\\left( Y\\right) \\right] \\\\\ns.t.~ & u\\left( x\\right) +v\\left( y\\right) \\geq\\Phi\\left( x,y\\right)\n\\end{align}\n\nThen $v\\left(y\\right)$ is interpreted as the value of the problem of a firm of type $y$, choosing the optimal worker $x$. Then the firm's program is\n\n\\begin{align*}\nv\\left( y\\right) =\\max_{x}\\left\\{ \\Phi\\left( x,y\\right) -u\\left(\nx\\right) \\right\\}\n\\end{align*}\n\nthus by first order conditions, one is led to the *wage equation*\n\n\n\\begin{align*}\nu^{\\prime}\\left( x\\right) =\\frac{\\partial\\Phi}{\\partial x}\\left( x,T\\left(\nx\\right) \\right) , \\label{WageEq}%\n\\end{align*}\nwhere $T$ is given by [this](#mongeMap1d).\n\n---\n**Theorem**\n\n1. Assume $\\Phi$ is supermodular and continuously differentiable with respect to its first variable. Assume $P$ has no mass point. Then the [dual Monge-Kantorovich problem](#mkdual1d) has a solution $\\left(u,v\\right)$. Further, $u$ solves the [wage equation](#WageEq). Hence, $u$ is determined up to a constant $c$ by\n\n \\begin{align*}\n u\\left( x\\right) =c+\\int_{x_{0}}^{x}\\frac{\\partial\\Phi}{\\partial x}\\left(t,T\\left( t\\right) \\right) dt. \n \\end{align*}\n\n2. Assume further that $Q$ has no mass point, and that $\\Phi$ is also continuously differentiable with respect to its second variable. Then $v$ is given by\n\n \\begin{align*}\n v\\left( y\\right) =c^{\\prime}+\\int_{T\\left( x_{0}\\right) }^{y}\\frac{\\partial\\Phi}{\\partial y}\\left( T^{-1}\\left( z\\right) ,z\\right) dz,\n \\end{align*}\n\n where $c$ and $c^{\\prime}$ are related by $c+c^{\\prime}=\\Phi\\left(x_{0},T\\left( x_{0}\\right) \\right).$\n\n---\n\n### Interpretation: imperfect competition and rents\n\nIt is important to distinguish between $\\partial\\Phi\\left( x,T\\left(x\\right) \\right) /\\partial x$, which is the partial derivative of $\\Phi\\left( x,y\\right) $ applied at $\\left( x,y\\right) =\\left( x,T\\left(x\\right) \\right) $, and $d\\Phi\\left( x,T\\left( x\\right) \\right) /dx$,which is the total derivative of $\\Phi\\left( x,T\\left( x\\right) \\right)$ with respect to $x$. One has\n\n\\begin{align*}\n\\frac{d\\Phi\\left( x,T\\left( x\\right) \\right) }{dx}=\\frac{\\partial \\Phi\\left( x,T\\left( x\\right) \\right) }{\\partial x}+\\frac{\\partial \\Phi\\left( x,T\\left( x\\right) \\right) }{\\partial y}T^{\\prime}\\left(x\\right)\n\\end{align*}\n\nThis decomposition has an interesting interpretation in terms of differential rent. The total derivative $d\\Phi\\left( x,T\\left( x\\right) \\right) /dx$ is the marginal increase in value between a firm run by a manager of talent $x$ and a firm run by a manager of talent $x+dx$. This differential value is split between the manager's differential rent ($\\partial\\Phi\\left( x,T\\left( x\\right) \\right) /\\partial x=u^{\\prime}\\left( x\\right) $) and the firm's differential rent ( $\\left( \\partial \\Phi\\left( x,T\\left( x\\right) \\right) /\\partial y\\right) T^{\\prime }\\left( x\\right) =dv\\left( T\\left( x\\right) \\right) /dx$).\n\nThis discussion highlights the nature of the assignment model, which is a model of imperfect competition. In this model, managers compete against each other, and likewise on the other side of the market. Managers are imperfect substitutes for each other. This imperfect competition is the source of the rent, as made apparent by [wage equation formula](#WageEq).\n\n## Application\n\nGabaix and Landier (2008) assume that\n\n\\begin{align*}\n\\Phi\\left( x,y\\right) =Cxy^{\\gamma}\n\\end{align*}\n\nwhere $C$ and $\\gamma$ are parameters.\n\nIf $X$ is the talent of the CEO, then the distribution of $X$ can be parameterized through its quantile by\n\n\\begin{align*}\nQ_{X}\\left( t\\right) =X^{\\max}-\\frac{B}{\\beta N}\\left( N\\left( 1-t\\right) \\right) ^{\\beta}\n\\end{align*}\n\n(such a distribution is consistent with extreme value theory). $N$ is the number of firms and $\\beta$ is a parameter.\n\nIf $Y$ is the size of the firm, then its distribution is assumed to be Pareto with distribution $1/\\alpha$, i.e.\n\n\\begin{align*}\nQ_{Y}\\left( t\\right) =A\\left( N\\left( 1-t\\right) \\right) ^{-\\alpha}%\n\\end{align*}\n\n(they find that $\\alpha=1$, i.e. Zipf's law, fits the data quite well).\n\nThe wage of CEO indexed by $t\\in\\left[ 0,1\\right] $ is $w\\left(t\\right) $; firm $y=Q_{Y}\\left( t\\right) $'s problem is\n\n\\begin{align*}\n\\max_{t\\in\\left[ 0,1\\right] }\\left\\{ CQ_{X}\\left( t\\right) y^{\\gamma}-w\\left( t\\right) \\right\\} =\\max_{t\\in\\left[ 0,1\\right] }\\left\\{-C\\frac{B}{\\beta N}\\left( N\\left( 1-t\\right) \\right) ^{\\beta}y^{\\gamma}-w\\left( t\\right) \\right\\}\n\\end{align*}\n\nand by FOC:\n\n\\begin{align*}\nw^{\\prime}\\left( t\\right) & =BCy^{\\gamma}\\left( N\\left( 1-t\\right)\n\\right) ^{\\beta-1}\\\\\n& =A^{\\gamma}BC\\left( N\\left( 1-t\\right) \\right) ^{-\\alpha\\gamma+\\beta-1}%\n\\end{align*}\n\nwhere it is assumed $\\alpha\\gamma>\\beta$. When $N$ is large, this yields\n\n\\begin{align*}\nw\\left( t\\right) \\approx\\frac{A^{\\gamma}BC}{\\alpha\\gamma-\\beta}\\left(N\\left( 1-t\\right) \\right) ^{-\\alpha\\gamma+\\beta}\n\\end{align*}\n\n\"Superstar\" effect (Rosen, 1981): if $\\beta>0$, then the wages are unbounded, even though talent is bounded -- talent is very valuable at the top end of the distribution.\n\nWe shall use Gabaix-Landier's calibrated values\n\\begin{align*}\n\\beta\\simeq2/3,\\gamma\\simeq1,BC=2.8\\ast10^{-6}\n\\end{align*}\n\n\n```R\nbeta = 2/3\nB = 1\nC = 2.8e-06\nalpha = gamma = 1\nX_max = 0\n```\n\nWe will focus on data from year 2003.\n\n\n```R\nyear = 2003\ndata2003 = data[data[, 1] == year, , drop = FALSE]\n```\n\nWe will normalize compensation to insure compensation and mktcap are in the same order of magnitude. We need to get rid of the `NA`'s in the data (separately for compensation and market cap) and sort in decreasing order.\n\n\n```R\ncomp = data2003[, 4]/1000 # insures same order of magnitude\ncomp = comp[which(!is.na(comp))]\ncomp = comp[order(comp, decreasing = T)]\n\nmktcap = data2003[, 2]\nmktcap = mktcap[which(!is.na(mktcap))]\nmktcap = mktcap[order(mktcap, decreasing = T)]\n```\n\n\n```R\nN = length(mktcap)\nn = 1:N\n```\n\nSo in order compute the wage function, it remains to find the value of $A$. This will simply be a regression of $\\log(S)$ on $n$. Regress firm rank against mktcap:\n\n\n```R\nols = lm(log(mktcap) ~ log(n))\nsummary(ols)\n```\n\n\n \n Call:\n lm(formula = log(mktcap) ~ log(n))\n \n Residuals:\n Min 1Q Median 3Q Max \n -1.41075 -0.05161 0.03054 0.06535 0.14352 \n \n Coefficients:\n Estimate Std. Error t value Pr(>|t|) \n (Intercept) 15.514000 0.027602 562.1 <2e-16 ***\n log(n) -0.982167 0.005195 -189.0 <2e-16 ***\n ---\n Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n \n Residual standard error: 0.1133 on 498 degrees of freedom\n Multiple R-squared: 0.9863,\tAdjusted R-squared: 0.9862 \n F-statistic: 3.574e+04 on 1 and 498 DF, p-value: < 2.2e-16\n\n\n\n\n```R\nA = unname(exp(ols$coefficients[1]))\n```\n\n\n```R\nwage = function(x) {\n return((A^gamma * B * C)/(alpha * gamma - beta) * x^(-alpha * gamma + beta))\n}\n```\n\nWe can compute the theoretical wage function and plot it against the realized wage function.\n\n\n```R\nols$coefficients[1]\n```\n\n\n(Intercept): 15.5140000823903\n\n\n\n```R\n# Wage estimation\n\nW = mapply(wage, 1:N)\n\nplot(W, type = \"l\", xlab = \"Rank\")\npoints(1:N, comp, pch = 3, col = \"blue\")\n```\n\n\n```R\n\n```\n", "meta": {"hexsha": "27dacfd858f5692f2c2bd0dccc97f79ab97f22ff", "size": 41755, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ipynb_r_mec_optim/B05_onedimensionaltransport.ipynb", "max_stars_repo_name": "math-econ-code/mec_optim_2020-01", "max_stars_repo_head_hexsha": "2deb505aa92f3e3eff32284c05ba8bb590f0140a", "max_stars_repo_licenses": ["AFL-1.1"], "max_stars_count": 8, "max_stars_repo_stars_event_min_datetime": "2020-01-20T13:39:04.000Z", "max_stars_repo_stars_event_max_datetime": "2020-12-19T04:11:51.000Z", "max_issues_repo_path": "ipynb_r_mec_optim/B05_onedimensionaltransport.ipynb", "max_issues_repo_name": "math-econ-code/mec_optim_2020-01", "max_issues_repo_head_hexsha": "2deb505aa92f3e3eff32284c05ba8bb590f0140a", "max_issues_repo_licenses": ["AFL-1.1"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ipynb_r_mec_optim/B05_onedimensionaltransport.ipynb", "max_forks_repo_name": "math-econ-code/mec_optim_2020-01", "max_forks_repo_head_hexsha": "2deb505aa92f3e3eff32284c05ba8bb590f0140a", "max_forks_repo_licenses": ["AFL-1.1"], "max_forks_count": 12, "max_forks_repo_forks_event_min_datetime": "2020-03-17T21:09:24.000Z", "max_forks_repo_forks_event_max_datetime": "2021-06-03T17:06:54.000Z", "avg_line_length": 57.5137741047, "max_line_length": 9036, "alphanum_fraction": 0.6351574662, "converted": true, "num_tokens": 7980, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.734119521083126, "lm_q1q2_score": 0.4098787456892663}} {"text": "Text provided under a Creative Commons Attribution license, CC-BY. All code is made available under the FSF-approved MIT license. (c) Yves Dubief, 2016. NSF for support via NSF-CBET award #1258697.\n\n```python\n%matplotlib inline \n\nfrom IPython.display import clear_output\n\nimport schemdraw as schem\nimport schemdraw.elements as e\n\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport math\nimport scipy.constants as sc\n\nimport sympy as sym\n\n\n \nfont = {'family' : 'serif',\n 'color' : 'black',\n 'weight' : 'normal',\n 'size' : 18,\n }\n\n\n```\n\n# Concept\n\n\nOne dimensional materials are materials with two dimensions much larger than the third so that properties within the materials have negligible variations in the first two dimensions. In this chapter, we focus on the heat transfer inside 1-D solids or fluids at rest (i.e. no convective motion).\n\nConsider a 1D material of dimension $L$, ($0\\leq x\\leq L$). Imagine that this the wall thickness of a house made of one material. The outside is at constant temperature $T_{w,o}$ and the inside at $T_{w,i}>T_{w,o}$. Typically the owner heats up the house to maintain $T_{w,i}$ and the weather controls $T_{w,o}$. You can view the problem as:\n
                                          \n
                                        1. the outside air extracts heat from the house,
                                          2. \n
                                        2. the house warms the outside air.
                                          3. \n
                                        \n\nIn any case, there is heat exchange between the outside and inside through the outside and inside surfaces of our wall. This is called heat transfer. Obviously, if inside and outside are at the same temperature, there is no heat transfer. This observation suggests that temperature gradients are necessary for heat transfer to occur. An analog property is the viscous stress in fluid mechanics which can only be non-zero when at least one velocity gradient is non-zero. Another observation that we have all made is that certain material transfer heat more than other: Licking a metal pole on a cold winter day is really bad idea; if the pole was made of wood, the scene below (from the movie A Christmas Story) would not be funny. Of course there is a little more to the story than just temperature of the air.\n\n\n\n\n\nEach material has a specific thermal conductivity, $k$, or ability to conduct heat. Going back to the wall of our house, with heat flowing or conducting from the inside to the outside, the question is how to quantify heat?\n\n# Heat Flux and Heat Transfer\n\n\nThe guiding principle of heat is basic law of energy: energy must be conserved. Here we are referring to thermal energy or enthalpy:\n

                                        \n$$\nh=\\rho C_p T\n$$\n

                                        \n\nwhere $\\rho$ and $C_p$ are the density and the specific heat of the wall material. We now construct a discretized model of the local thermal energy variation in time and space. We divide our wall in $N$ elements with the objective to compute the evolution of temperature (enthalpy) throughout the wall by applying conservation of energy to each element (or control volume).\n\n\n\nIn one element, we define the temperature at the center, as the average temperature over the entire cell. Since we limit our present discussion to 1D, the total enthalpy over the cell is $h\\Delta x\\Delta y\\Delta z$. The conservation of energy dictates that\n

                                        \n$$\n\\left[\\begin{array}{1}\n\\text{growth or decay of}\\\\\n\\text{enthalpy over time}\n\\end{array}\n\\right] = \n\\left[\\text{energy coming in}\\right] - \\left[\\text{energy going out}\\right]+\\left[\\text{internal energy sources}\\right]\n$$\n

                                        \nwhere **internal energy sources** may be heat production or destruction from chemical reactions, mechanical systems (friction) or electrical resistance. Here after the total internal energy source per unit volume is defined as $\\dot{q}$\n\nThe left hand side can simply be interpreted as\n$$\n\\frac{h(x_i,t+\\Delta t)-h(x_i,t)}{\\Delta t}\\Delta x\\Delta y\\Delta z=\\rho C_p\\frac{T(x_i,t+\\Delta t)-T(x_i,t)}{\\Delta t}\\Delta x\\Delta y\\Delta z\n$$\n\n\n\nIn the light of the previous discussion, we shall now derive an expression of the energy going in and out (RHS of the conservation energy). We speculated before that heat transfer should be related to temperature gradient, for instance at the interface between cell $i$ and $i+1$, this gradient is\n$$\n\\frac{T(x_{i+1},t)-T(x_i,t)}{x_{i+1}-x_{i}}=\\frac{T(x_{i+1},t)-T(x_i,t)}{\\Delta x}\n$$\nThis is an approximation of the temperature gradient that becomes exact when $\\Delta x\\rightarrow 0$.\n\nHow does the heat flow? It is sort of a philosiphical question. Does the cold air cool the house or does the inside of the house warms the air? Let's be practical: We need energy to keep our houses warm in the water, so the heat appears to escape the house. The transport heat is therefore against the gradient of temperature. We therefore define the heat flux\n$$\nq''(x_{i+1/2},t)=-k\\frac{T(x_{i+1},t)-T(x_i,t)}{\\Delta x}\n$$\nas our energy in or out. The thermal conductivity $k$ is material dependent and necessary to achieve the dimensions of an energy flux (W/m2). The heat rate across the interface at $x_{i+1/2}$ is the integral over the surface area. Assuming (thanks to 1D) that $q''$ is constant over that surface, the heat rate is\n$$\nq(x_{i+1/2},t)=-k\\frac{T(x_{i+1},t)-T(x_i,t)}{\\Delta x}\\Delta y\\Delta z\n$$\nThe units of $k$ are therefore W/(m.K). Temperature can be expressed here in Kelvin or degree Centigrade (since this is a difference) but not in Farhenheit. From now on we will stick to Kelvin and Centigrade as much as possible.\n\nOur balance of energy now becomes:\n$$\n\\begin{split}\n&\\rho C_p\\frac{T(x_i,t+\\Delta t)-T(x_i,t)}{\\Delta t}\\Delta x\\Delta y\\Delta z\\\\\n&-\\left(-k\\frac{T(x_{i},t)-T(x_{i-1},t)}{\\Delta x}\\right)\\Delta y\\Delta z+\\left(-k\\frac{T(x_{i+1},t)-T(x_i,t)}{\\Delta x}\\right)\\Delta y\\Delta z=\\dot{q}\\Delta x\\Delta y\\Delta z\n\\end{split}\n$$\nwhich can be recast as:\n$$\n\\rho C_p \\frac{T(x_i,t+\\Delta t)-T(x_i,t)}{\\Delta t}\\Delta x\\Delta y\\Delta z=k\\frac{T(x_{i+1},t)-2T(x_i,t)+T(x_{i-1},t)}{\\Delta x}\\Delta y\\Delta z+\\dot{q}\n$$\nand simplified to \n$$\n\\rho C_p \\frac{T(x_i,t+\\Delta t)-T(x_i,t)}{\\Delta t}=k\\frac{T(x_{i+1},t)-2T(x_i,t)+T(x_{i-1},t)}{\\Delta x^2}+\\dot{q}\n$$\nIn the absence of heat generation, or internal energy source ($\\dot{q}=0$). The steady state solution is independent of time, hence the governing equation of each cell in our wall:\n$$\n-T(x_{i+1},t)+2T(x_i,t)-T(x_{i-1},t)=0\n$$\nHow do we solve this problem? The solution at $x_i$ depends on the solution at $x_{i+1}$ and $x_{i-1}$, which imposes that we treat the problem as a compact system of equations written in a matrix form:\n$$\n\\left(\\begin{array}{cccccccccc}\n1 & 0 & 0 & \\ldots & \\ldots& \\ldots & 0\\\\\n-1 & 2 & -1 & \\ddots & & & \\vdots\\\\\n0& \\ddots & \\ddots & \\ddots& \\ddots& & \\vdots\\\\\n\\vdots &\\ddots & -1 & 2 & -1 & \\ddots & \\vdots\\\\\n\\vdots & &\\ddots & \\ddots &\\ddots &\\ddots & 0\\\\\n\\vdots & & & \\ddots &-1 & 2 & -1 \\\\\n0 & \\ldots & \\ldots & \\ldots & 0 &0 & 1\\\\\n\\end{array}\n\\right)\\cdot\n\\left(\\begin{array}{c}\nT(x_{0})\\\\\nT(x_{1})\\\\\n\\vdots \\\\\nT(x_{i})\\\\\n\\vdots \\\\\nT(x_{N-2})\\\\\nT(x_{N-1})\n\\end{array}\n\\right)=\n\\left(\\begin{array}{c}\nT_{w,i}\\\\\n0\\\\\n\\vdots \\\\\n\\vdots \\\\\n\\vdots \\\\\n0\\\\\nT_{w,o}\n\\end{array}\n\\right)\n$$\n\n\n```python\n\n\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nL = 0.025 #wall thickness is 2.5 cm\nN = 5 # Number of cells in the wall\nTi = 20. #\nTo = -20. #\ndx = float(L)/float(N-1) #cell size\nx = np.array([i*dx for i in range(N)])\nA = np.zeros((N, N)) # pre-allocate [A] array\nb = np.zeros((N, 1)) # pre-allocate {b} column vector\n\nA[0, 0] = 1.\nb[0, 0] = Ti\nA[N-1,N-1] = 1.\nb[N-1,0] = To\n\nfor i in range(1, N-1):\n A[i, i-1] = -1 # node-1\n A[i, i] = 2 # node\n A[i, i+1] = -1 # node+1\n \n#print('A \\n', A)\n#print('B \\n', b)\n\n#---- Solve using numpy.linalg.solve\n\nT = np.linalg.solve(A, b) # solve A*x = B for x\n\n#print('T \\n', T)\n\nplt.figure(figsize=(3,2.5), dpi=150)\nplt.plot(x,T, lw=2, label='Wall temperature')\nplt.xlim([0,L])\nplt.ylim([To,Ti])\nplt.xlabel('$x$ (m)')\nplt.ylabel('$T$ ($^\\circ$C)')\nplt.legend()\nplt.show\n\n```\n\nHow is the distribution of temperature inside our wall? Use the code to change the inside and/or outside temperature, is the distribution other than linear? Increase the number of cells inside the wall, do you see any change?\n\nUsing our intuitive approach and a model of the heat flux, our simulation suggests that the temperature distribution is linear inside our wall resulting in the following heat flux\n

                                        \n$$\nq''=-k\\frac{T_{w,i}-T_{w,o}}{L}\n$$\n

                                        \nand the heat transfer rate is $q=q''A$ where $A$ is the surface area of the wall. Heat flux is defined by a phenomologic law, called Fourier's law which was derived from observations and experiments. Fourier's law in 3D is\n

                                        \n$$\n\\vec{q}''=-k\\vec{\\nabla}T=-k\\left(\\begin{array}{c}\\frac{\\partial T}{\\partial x}\\\\\\frac{\\partial T}{\\partial y}\\\\\\frac{\\partial T}{\\partial z}\\end{array}\\right)\n$$\n

                                        \nand for one dimension problem:\n

                                        \n$$\nq''=-k\\frac{dT}{dx}\n$$\n

                                        \n\nConsider now a practical application. Pinewood has a thermal conductivity of 0.26 $\\text{W}/\\text{m}.\\text{K}$. The heat transfer rate through a one squared meter wood panel is 260 W. For an insulating material (fiberglass-type) with a thermal conductivity of 0.035 $\\text{W}/\\text{m}.\\text{K}$, the heat transfer rate drops to 35 W. \n\nThe problem is that one cannot build houses out of insulation material. Now consider a wall is 2.5cm of wood and 1cm of insulation. The wall is no longer homogenous: 2.5cm of the domain has a thermal conductivity of 0.26, the rest has a thermal conductivity of 0.035. Let us assume that for 1D steady-state conduction the heat flux within the same material of thermal conductivity $k$ between point $a$ and $b$ separated by a length $L$ is\n$$\nq''=-k\\frac{T_a-T_b}{L}\n$$\nWe are here assuming that the distribution of temperature is linear between $a$ and $b$.\n\nConservation of energy imposes that the same energy (heat) flowing through the wood has to flow through the insulation. Our system is 1D so the energy is nowhere to go but through the walls and it needs to be conserved! (Yes I am repeating myself, I am just making sure you remember that energy is always conserved). The equality of heat flux writes:\n$$\nq_\\text{wood}''=q_\\text{insulation}'' \\Rightarrow -k_\\text{wood}\\frac{T_{w,i}-T_1}{L_\\text{wood}}=-k_\\text{insulation}\\frac{T_1-T_{w,o}}{L_\\text{insulation}}\n$$\nwhich leads to the determination of the interface temperature $T_1$\n$$\nT_1=\\frac{1}{\\frac{k_\\text{wood}}{L_\\text{wood}}+\\frac{k_\\text{insulation}}{L_\\text{insulation}}}\\left(\\frac{k_\\text{wood}}{L_\\text{wood}}T_{w,i}+\\frac{k_\\text{insulation}}{L_\\text{insulation}}T_{w,o}\\right)\n$$\nThe temperature distribution is plotted below and the heat flux is calculated.\n\n\n\n```python\nimport math\nk_wood=0.26 #W/(m.K)\nk_insulation=0.035 #W/(m.K)\nL_wood=0.025 #m\nL_insulation=0.01 #m\nL_wall=L_wood+L_insulation\nT_wi=20. #C\nT_wo=-5. #C\nArea=1. #m^2\nx=np.array([0., L_wood, L_wall])\nT_1=(k_wood/L_wood*T_wi+k_insulation/L_insulation*T_wo)/(k_wood/L_wood+k_insulation/L_insulation)\nT=np.array([T_wi, T_1, T_wo])\nplt.figure(figsize=(3,2.5), dpi=150)\nplt.plot(x,T, lw=2, label='Wall temperature')\nplt.xlim([0,L_wall])\nplt.ylim([T_wo,T_wi])\nplt.xlabel('$x$ (m)')\nplt.ylabel('$T$ ($^\\circ$C)')\nplt.legend()\nplt.show\nq_wall=k_wood/L_wood*(T_wi-T_1)*Area\nprint ('Heat rate through wall: {0:.2f} W'.format(q_wall))\n```\n\nThis is still a lousy house. What would the heat rate be if we add sheetrock on the inside and bricks on the outside?\n\nTo summarize, we have 1cm of sheetrock (sr) with $k_{sr}=0.10$W/(m.K), 2.5cm of pinewood (pw), 1cm of insulation (in) and 5 cm of brick (br) with $k_{br}=0.6$W/(m.K). The system of equations for heat rate is\n\\begin{eqnarray}\n-\\frac{k_\\text{sr}A}{L_\\text{sr}}\\left(T_{w,i}-T_1\\right)&=&\n-\\frac{k_\\text{pw}A}{L_\\text{pw}}\\left(T_1-T_2\\right)\\\\\n-\\frac{k_\\text{pw}A}{L_\\text{pw}}\\left(T_1-T_2\\right)&=&\n-\\frac{k_\\text{in}A}{L_\\text{in}}\\left(T_2-T_3\\right)\\\\\n-\\frac{k_\\text{in}A}{L_\\text{in}}\\left(T_2-T_3\\right)&=&\n-\\frac{k_\\text{br}A}{L_\\text{br}}\\left(T_3-T_{w,o}\\right)\n\\end{eqnarray}\nWe now define the following\n\\begin{eqnarray}\nR_\\text{sr}&=&\\frac{L_\\text{sr}}{k_\\text{sr}A}\\\\\nR_\\text{pw}&=&\\frac{L_\\text{pw}}{k_\\text{pw}A}\\\\\nR_\\text{in}&=&\\frac{L_\\text{in}}{k_\\text{in}A}\\\\\nR_\\text{br}&=&\\frac{L_\\text{br}}{k_\\text{br}A}\n\\end{eqnarray}\nand explain later why. We can recast the system into a matrix form\n$$\n\\left(\\begin{array}{ccccc}\n1 & 0 & 0 & 0 & 0\\\\\n-\\frac{1}{R_\\text{sr}} & \\frac{1}{R_\\text{sr}}+\\frac{1}{R_\\text{pw}} & -\\frac{1}{R_\\text{pw}} & 0 & 0 \\\\\n0 & -\\frac{1}{R_\\text{pw}} & \\frac{1}{R_\\text{pw}}+\\frac{1}{R_\\text{in}} & -\\frac{1}{R_\\text{in}} &0 \\\\\n0 & 0 & -\\frac{1}{R_\\text{in}} & \\frac{1}{R_\\text{in}}+\\frac{1}{R_\\text{br}} & \\frac{1}{R_\\text{br}} \\\\\n0 & 0 & 0 & 0& 1\n\\end{array}\\right)\n\\left(\\begin{array}{c}\nT_{w,i}\\\\\nT_1\\\\\nT_2\\\\\nT_3\\\\\nT_{w,o}\n\\end{array}\\right)=\n\\left(\\begin{array}{c}\n20\\\\\n0\\\\\n0\\\\\n0\\\\\n-5\n\\end{array}\\right)\n$$\n\n\n```python\nk_sr=0.1\nL_sr=0.01\nk_pw=0.26\nL_pw=0.025\nk_in=0.035\nL_in=0.01\nk_br=0.6\nL_br=0.05\nArea=1.\nT_wi=25.\nT_wo=-5.\nR_sr=L_sr/(k_sr*Area)\nR_pw=L_pw/(k_pw*Area)\nR_in=L_in/(k_in*Area)\nR_br=L_br/(k_br*Area)\nA = np.zeros((5, 5))\nb = np.zeros((5, 1))\nA[0, 0] = 1.\nb[0, 0] = T_wi\nA[1,0] = -1./R_sr\nA[1,1] = 1./R_sr+1./R_pw\nA[1,2] = -1./R_pw\nA[2,1] = -1./R_pw\nA[2,2] = 1./R_pw+1./R_in\nA[2,3] = -1./R_in\nA[3,2] = -1./R_in\nA[3,3] = 1./R_in+1./R_br\nA[3,4] = -1./R_br\nA[4,4] = 1.\nb[4,0] = T_wo\nprint(A)\n\nT = np.linalg.solve(A, b)\nprint(T)\nx = np.zeros((5,1))\nx[1,0] = L_sr\nx[2,0] = x[1,0]+L_pw\nx[3,0] = x[2,0]+L_in\nx[4,0] = x[3,0]+L_br\n\nL=x[4,0]\n\nplt.figure(figsize=(3,2.5), dpi=150)\nplt.plot(x,T, lw=2, label='Wall temperature')\nplt.xlim([0,L])\nplt.ylim([T_wo,T_wi])\nplt.xlabel('$x$ (m)')\nplt.ylabel('$T$ ($^\\circ$C)')\nplt.legend()\nplt.show\n\nq_wall=(T_wi-T[1,0])/R_sr\nprint ('Heat rate through wall in sr: {0:.2f} W'.format(q_wall))\nq_wall=(T[1,0]-T[2,0])/R_pw\nprint ('Heat rate through wall in pw: {0:.2f} W'.format(q_wall))\nq_wall=(T[2,0]-T[3,0])/R_in\nprint ('Heat rate through wall in in: {0:.2f} W'.format(q_wall))\nq_wall=(T[3,0]-T[4,0])/R_br\nprint ('Heat rate through wall in br: {0:.2f} W'.format(q_wall))\n```\n\nThe significance of \n

                                        \n$$\nR=\\frac{L}{kA}\n$$\n

                                        \ncomes from the analogy between heat rate, temperature, thermal conductivity and current $I$, potential difference $\\Delta V$, and conductivity $\\sigma$. Ohm's law in a wire of area $A$ and length $L$ between points $1$ and $2$ states that\n$$\nV_1-V_2=R I\\text{ with }R=\\frac{L}{\\sigma A}\\text{ or } I=\\frac{\\Delta V}{R}\n$$\n\nIn heat transfer $R$ is the thermal resistance\n

                                        \n$$\nR=\\frac{\\Delta T}{q}\n$$\n

                                        \nIn the case of composite materials, such as the one we have just computed, the total resistance is the sum of all the resistance\n

                                        \n$$\nR_\\text{tot}=\\sum_{i=0}^{N-1}R_i\n$$\n

                                        \nand the heat rate through the wall is \n

                                        \n$$\nq=\\frac{T_0-T_{N-1}}{R_\\text{tot}}\n$$\n

                                        \nIf you are not concerned by the temperature distribution through the wall, this method is faster than the matrix one.\n\nFor heat transfer per unit surface area, or heat flux, the thermal resistance for a plane wall is defined as\n$$\nR''=\\frac{L}{k}\n$$\n\n\n```python\nR_tot=R_sr+R_pw+R_in+R_br\nq_wall=(T_wi-T_wo)/R_tot\nprint ('Heat rate through wall using the total thermal resistance: {0:.2f} W'.format(q_wall))\n```\n\n Heat rate through wall using the total thermal resistance: 53.08 W\n\n\n## What you need to remember\n\n* Units of thermal conductivity\n* **Fourier's law** (not only because Fourier is French and lived in Grenoble)\n* The following definitions and units\n * **Heat flux**: heat transfer per unit surface area $q''\\, [W/m^2]$\n * **Heat rate per unit length**: $q'\\, [W/m]$\n * **Heat rate**: $q\\, [W]$\n* Conservation of energy applied to a solid or a volume of fluid at rest\n* There is a analogy between heat transfer and electricity. \n\n\n```python\n\n```\n", "meta": {"hexsha": "e9e85bda4def30ea36279ab803331ab1f115d1ff", "size": 100583, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Lecture 3/Lecture 3.ipynb", "max_stars_repo_name": "CarlGriffinsteed/UVM-ME144-Heat-Transfer", "max_stars_repo_head_hexsha": "9c477449d6ba5d6a9ee7c57f1c0ed4aab0ce4cca", "max_stars_repo_licenses": ["CC-BY-3.0"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2017-06-02T20:31:22.000Z", "max_stars_repo_stars_event_max_datetime": "2021-04-05T13:52:33.000Z", "max_issues_repo_path": "Lecture 3/Lecture 3.ipynb", "max_issues_repo_name": "CarlGriffinsteed/UVM-ME144-Heat-Transfer", "max_issues_repo_head_hexsha": "9c477449d6ba5d6a9ee7c57f1c0ed4aab0ce4cca", "max_issues_repo_licenses": ["CC-BY-3.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Lecture 3/Lecture 3.ipynb", "max_forks_repo_name": "CarlGriffinsteed/UVM-ME144-Heat-Transfer", "max_forks_repo_head_hexsha": "9c477449d6ba5d6a9ee7c57f1c0ed4aab0ce4cca", "max_forks_repo_licenses": ["CC-BY-3.0"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2019-01-24T17:43:41.000Z", "max_forks_repo_forks_event_max_datetime": "2021-07-25T18:08:34.000Z", "avg_line_length": 164.3513071895, "max_line_length": 27240, "alphanum_fraction": 0.8659216766, "converted": true, "num_tokens": 5416, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.6959583376458153, "lm_q1q2_score": 0.4098423005068695}} {"text": "

                                        \n \n\n

                                        \n\n## Interactive Simple Kriging Demonstration\n\n\n### Michael Pyrcz, Associate Professor, University of Texas at Austin \n\n##### [Twitter](https://twitter.com/geostatsguy) | [GitHub](https://github.com/GeostatsGuy) | [Website](http://michaelpyrcz.com) | [GoogleScholar](https://scholar.google.com/citations?user=QVZ20eQAAAAJ&hl=en&oi=ao) | [Book](https://www.amazon.com/Geostatistical-Reservoir-Modeling-Michael-Pyrcz/dp/0199731446) | [YouTube](https://www.youtube.com/channel/UCLqEr-xV-ceHdXXXrTId5ig) | [LinkedIn](https://www.linkedin.com/in/michael-pyrcz-61a648a1)\n\n\n### The Interactive Workflow\n\nHere's a simple workflow for calculating the simple kriging estimate and the estimation variance for a local uncertainty model\n\n* we use a 'toy problem' with only 3 data for speed and interpretability of the results\n\n#### Spatial Estimation\n\nConsider the case of making an estimate at some unsampled location, $\ud835\udc67(\\bf{u}_0)$, where $z$ is the property of interest (e.g. porosity etc.) and $\ud835\udc2e_0$ is a location vector describing the unsampled location.\n\nHow would you do this given data, $\ud835\udc67($**\ud835\udc2e**$_1)$, $\ud835\udc67($**\ud835\udc2e**$_2)$, and $\ud835\udc67($**\ud835\udc2e**$_3)$?\n\nIt would be natural to use a set of linear weights to formulate the estimator given the available data.\n\n\\begin{equation}\nz^{*}(\\bf{u}) = \\sum^{n}_{\\alpha = 1} \\lambda_{\\alpha} z(\\bf{u}_{\\alpha})\n\\end{equation}\n\nWe could add an unbiasedness constraint to impose the sum of the weights equal to one. What we will do is assign the remainder of the weight (one minus the sum of weights) to the global average; therefore, if we have no informative data we will estimate with the global average of the property of interest.\n\n\\begin{equation}\nz^{*}(\\bf{u}) = \\sum^{n}_{\\alpha = 1} \\lambda_{\\alpha} z(\\bf{u}_{\\alpha}) + \\left(1-\\sum^{n}_{\\alpha = 1} \\lambda_{\\alpha} \\right) \\overline{z}\n\\end{equation}\n\nWe will make a stationarity assumption, so let's assume that we are working with residuals, $y$. \n\n\\begin{equation}\ny^{*}(\\bf{u}) = z^{*}(\\bf{u}) - \\overline{z}(\\bf{u})\n\\end{equation}\n\nIf we substitute this form into our estimator the estimator simplifies, since the mean of the residual is zero.\n\n\\begin{equation}\ny^{*}(\\bf{u}) = \\sum^{n}_{\\alpha = 1} \\lambda_{\\alpha} y(\\bf{u}_{\\alpha})\n\\end{equation}\n\nwhile satisfying the unbaisedness constraint. \n\n#### Kriging\n\nNow the next question is what weights should we use? \n\nWe could use equal weighting, $\\lambda = \\frac{1}{n}$, and the estimator would be the average of the local data applied for the spatial estimate. This would not be very informative.\n\nWe could assign weights considering the spatial context of the data and the estimate:\n\n* **spatial continuity** as quantified by the variogram (and covariance function)\n* **redundancy** the degree of spatial continuity between all of the available data with themselves \n* **closeness** the degree of spatial continuity between the avaiable data and the estimation location\n\nThe kriging approach accomplishes this, calculating the best linear unbiased weights for the local data to estimate at the unknown location. The derivation of the kriging system and the resulting linear set of equations is available in the lecture notes. Furthermore kriging provides a measure of the accuracy of the estimate! This is the kriging estimation variance (sometimes just called the kriging variance).\n\n\\begin{equation}\n\\sigma^{2}_{E}(\\bf{u}) = C(0) - \\sum^{n}_{\\alpha = 1} \\lambda_{\\alpha} C(\\bf{u}_0 - \\bf{u}_{\\alpha})\n\\end{equation}\n\nWhat is 'best' about this estimate? Kriging estimates are best in that they minimize the above estimation variance. \n\n#### Properties of Kriging\n\nHere are some important properties of kriging:\n\n* **Exact interpolator** - kriging estimates with the data values at the data locations\n* **Kriging variance** can be calculated before getting the sample information, as the kriging estimation variance is not dependent on the values of the data nor the kriging estimate, i.e. the kriging estimator is homoscedastic. \n* **Spatial context** - kriging takes into account, furthermore to the statements on spatial continuity, closeness and redundancy we can state that kriging accounts for the configuration of the data and structural continuity of the variable being estimated.\n* **Scale** - kriging may be generalized to account for the support volume of the data and estimate. We will cover this later.\n* **Multivariate** - kriging may be generalized to account for multiple secondary data in the spatial estimate with the cokriging system. We will cover this later.\n* **Smoothing effect** of kriging can be forecast. We will use this to build stochastic simulations later.\n\n#### Spatial Continuity \n\n**Spatial Continuity** is the correlation between values over distance.\n\n* No spatial continuity \u2013 no correlation between values over distance, random values at each location in space regardless of separation distance.\n\n* Homogenous phenomenon have perfect spatial continuity, since all values as the same (or very similar) they are correlated. \n\nWe need a statistic to quantify spatial continuity! A convenient method is the Semivariogram.\n\n#### The Semivariogram\n\nFunction of difference over distance.\n\n* The expected (average) squared difference between values separated by a lag distance vector (distance and direction), $h$:\n\n\\begin{equation}\n\\gamma(\\bf{h}) = \\frac{1}{2 N(\\bf{h})} \\sum^{N(\\bf{h})}_{\\alpha=1} (z(\\bf{u}_\\alpha) - z(\\bf{u}_\\alpha + \\bf{h}))^2 \n\\end{equation}\n\nwhere $z(\\bf{u}_\\alpha)$ and $z(\\bf{u}_\\alpha + \\bf{h})$ are the spatial sample values at tail and head locations of the lag vector respectively.\n\n* Calculated over a suite of lag distances to obtain a continuous function.\n\n* the $\\frac{1}{2}$ term converts a variogram into a semivariogram, but in practice the term variogram is used instead of semivariogram.\n* We prefer the semivariogram because it relates directly to the covariance function, $C_x(\\bf{h})$ and univariate variance, $\\sigma^2_x$:\n\n\\begin{equation}\nC_x(\\bf{h}) = \\sigma^2_x - \\gamma(\\bf{h})\n\\end{equation}\n\nNote the correlogram is related to the covariance function as:\n\n\\begin{equation}\n\\rho_x(\\bf{h}) = \\frac{C_x(\\bf{h})}{\\sigma^2_x}\n\\end{equation}\n\nThe correlogram provides of function of the $\\bf{h}-\\bf{h}$ scatter plot correlation vs. lag offset $\\bf{h}$. \n\n\\begin{equation}\n-1.0 \\le \\rho_x(\\bf{h}) \\le 1.0\n\\end{equation}\n\n#### Objective \n\nIn the PGE 383: Stochastic Subsurface Modeling class I want to provide hands-on experience with building subsurface modeling workflows. Python provides an excellent vehicle to accomplish this. I have coded a package called GeostatsPy with GSLIB: Geostatistical Library (Deutsch and Journel, 1998) functionality that provides basic building blocks for building subsurface modeling workflows. \n\nThe objective is to remove the hurdles of subsurface modeling workflow construction by providing building blocks and sufficient examples. This is not a coding class per se, but we need the ability to 'script' workflows working with numerical methods. \n\n#### Getting Started\n\nHere's the steps to get setup in Python with the GeostatsPy package:\n\n1. Install Anaconda 3 on your machine (https://www.anaconda.com/download/). \n2. From Anaconda Navigator (within Anaconda3 group), go to the environment tab, click on base (root) green arrow and open a terminal. \n3. In the terminal type: pip install geostatspy. \n4. Open Jupyter and in the top block get started by copy and pasting the code block below from this Jupyter Notebook to start using the geostatspy functionality. \n\nYou will need to copy the data file to your working directory. They are available here:\n\n* Tabular data - sample_data.csv at https://git.io/fh4gm.\n\nThere are exampled below with these functions. You can go here to see a list of the available functions, https://git.io/fh4eX, other example workflows and source code. \n\n#### Load the required libraries\n\nThe following code loads the required libraries.\n\n\n```python\nimport geostatspy.GSLIB as GSLIB # GSLIB utilies, visualization and wrapper\nimport geostatspy.geostats as geostats # GSLIB methods convert to Python \n```\n\nWe will also need some standard packages. These should have been installed with Anaconda 3.\n\n\n```python\n%matplotlib inline\n#import os # to set current working directory \nimport sys # supress output to screen for interactive variogram modeling\nimport io\nimport numpy as np # arrays and matrix math\nimport pandas as pd # DataFrames\nimport matplotlib.pyplot as plt # plotting\nfrom matplotlib.pyplot import cm # color maps\nfrom matplotlib.patches import Ellipse # plot an ellipse\nimport math # sqrt operator\nfrom scipy.stats import norm\nfrom ipywidgets import interactive # widgets and interactivity\nfrom ipywidgets import widgets \nfrom ipywidgets import Layout\nfrom ipywidgets import Label\nfrom ipywidgets import VBox, HBox\n```\n\nIf you get a package import error, you may have to first install some of these packages. This can usually be accomplished by opening up a command window on Windows and then typing 'python -m pip install [package-name]'. More assistance is available with the respective package docs. \n\n#### Simple, Simple Kriging Function\n\nLet's write a fast Python function to take data points and unknown location and provide the:\n\n* **simple kriging estimate**\n\n* **simple kriging variance / estimation variance**\n\n* **simple kriging weights**\n\nThis provides a fast method for small datasets, with less parameters (no search parameters) and the ability to see the simple kriging weights \n\n\n```python\ndef simple_simple_krige(df,xcol,ycol,vcol,dfl,xlcol,ylcol,vario,skmean):\n# load the variogram\n nst = vario['nst']; pmx = 9999.9\n cc = np.zeros(nst); aa = np.zeros(nst); it = np.zeros(nst)\n ang = np.zeros(nst); anis = np.zeros(nst)\n nug = vario['nug']; sill = nug \n cc[0] = vario['cc1']; sill = sill + cc[0]\n it[0] = vario['it1']; ang[0] = vario['azi1']; \n aa[0] = vario['hmaj1']; anis[0] = vario['hmin1']/vario['hmaj1'];\n if nst == 2:\n cc[1] = vario['cc2']; sill = sill + cc[1]\n it[1] = vario['it2']; ang[1] = vario['azi2']; \n aa[1] = vario['hmaj2']; anis[1] = vario['hmin2']/vario['hmaj2']; \n\n# set up the required matrices\n rotmat, maxcov = geostats.setup_rotmat(nug,nst,it,cc,ang,pmx) \n ndata = len(df); a = np.zeros([ndata,ndata]); r = np.zeros(ndata); s = np.zeros(ndata); rr = np.zeros(ndata)\n nest = len(dfl)\n\n est = np.zeros(nest); var = np.full(nest,sill); weights = np.zeros([nest,ndata])\n\n# Make and solve the kriging matrix, calculate the kriging estimate and variance \n for iest in range(0,nest):\n for idata in range(0,ndata):\n for jdata in range(0,ndata):\n a[idata,jdata] = geostats.cova2(df[xcol].values[idata],df[ycol].values[idata],df[xcol].values[jdata],df[ycol].values[jdata],\n nst,nug,pmx,cc,aa,it,ang,anis,rotmat,maxcov)\n r[idata] = geostats.cova2(df[xcol].values[idata],df[ycol].values[idata],dfl[xlcol].values[iest],dfl[ylcol].values[iest],\n nst,nug,pmx,cc,aa,it,ang,anis,rotmat,maxcov)\n rr[idata] = r[idata]\n \n s = geostats.ksol_numpy(ndata,a,r) \n sumw = 0.0\n for idata in range(0,ndata): \n sumw = sumw + s[idata]\n weights[iest,idata] = s[idata]\n est[iest] = est[iest] + s[idata]*df[vcol].values[idata]\n var[iest] = var[iest] - s[idata]*rr[idata]\n est[iest] = est[iest] + (1.0-sumw)*skmean\n return est,var,weights \n```\n\n#### Setup the Spatial Problem\n\nHere you can specify the values of the 3 data samples (v1, v2, v3) and the global mean (vmean).\n\n\n```python\nv1 = 25.0 # value of sample data #1\nv2 = 43.0 # value of sample data #1 \nv3 = 56.0 # value of sample data #1 \nvalue = [v1,v2,v3] # make a data value list\nvmean = 38.0 # value of the global mean for simple kriging\nsill = 25.0 # sill, variance of the dataset \nvmin = np.min(value) - 3 * np.sqrt(sill) # minimum value for plotting\nvmax = np.max(value) + 3 * np.sqrt(sill) # maximum value for plotting\n```\n\n#### Interactive Simple Kriging Method\n\nThe following code includes:\n\n* dashboard with variogram model data locations \n\n* plots of variogram model, data locations with point scaled by weights and uncertainty distribution at the unknown location\n\n\n```python\nimport warnings; warnings.simplefilter('ignore')\n\n# interactive calculation of the sample set (control of source parametric distribution and number of samples)\nstyle = {'description_width': 'initial'}\nl = widgets.Text(value=' Simple Kriging, Michael Pyrcz, Associate Professor, The University of Texas at Austin',layout=Layout(width='950px', height='30px'))\nnug = widgets.FloatSlider(min = 0, max = 1.0, value = 0.0, step = 0.1, description = 'nug',orientation='vertical',\n layout=Layout(width='25px', height='200px'))\nnug.style.handle_color = 'gray'\nit1 = widgets.Dropdown(options=['Spherical', 'Exponential', 'Gaussian'],value='Spherical',\n description='Type1:',disabled=False,layout=Layout(width='180px', height='30px'), style=style,continuous_update=False)\n\nazi = widgets.FloatSlider(min=0, max = 360, value = 0, step = 22.5, description = 'azi',\n orientation='vertical',layout=Layout(width='40px', height='200px'),continuous_update=False)\nazi.style.handle_color = 'gray'\nhmaj1 = widgets.FloatSlider(min=0.01, max = 10000.0, value = 100.0, step = 25.0, description = 'hmaj1',\n orientation='vertical',layout=Layout(width='40px', height='200px'),continuous_update=False)\nhmaj1.style.handle_color = 'gray'\nhmin1 = widgets.FloatSlider(min = 0.01, max = 10000.0, value = 100.0, step = 25.0, description = 'hmin1',\n orientation='vertical',layout=Layout(width='40px', height='200px'),continuous_update=False)\nhmin1.style.handle_color = 'gray'\nuikvar = widgets.HBox([nug,it1,azi,hmaj1,hmin1],) # basic widget formatting \n\nx1 = widgets.FloatSlider(min=0.0, max = 1000.0, value = 100.0, step = 1.0, description = 'x1',orientation='horizontal',\n layout=Layout(width='150px', height='30px'),readout_format = '.0f',style=style,continuous_update=False)\nx1.style.handle_color = 'blue'\ny1 = widgets.FloatSlider(min=0.0, max = 1000.0, value = 200.0, step = 1.0, description = 'y1',orientation='vertical',\n layout=Layout(width='70px', height='180px'),readout_format = '.0f',style=style,continuous_update=False)\ny1.style.handle_color = 'blue'\nuik1 = widgets.VBox([x1,y1],)\n\nx2 = widgets.FloatSlider(min=0.0, max = 1000.0, value = 500.0, step = 1.0, description = 'x2',orientation='horizontal',\n layout=Layout(width='150px', height='30px'),readout_format = '.0f',style=style,continuous_update=False)\nx2.style.handle_color = 'red'\ny2 = widgets.FloatSlider(min=0.0, max = 1000.0, value = 800.0, step = 1.0, description = 'y2',orientation='vertical',\n layout=Layout(width='70px', height='180px'),readout_format = '.0f',style=style,continuous_update=False)\ny2.style.handle_color = 'red'\nuik2 = widgets.VBox([x2,y2],)\n\nx3 = widgets.FloatSlider(min=0.0, max = 1000.0, value = 700.0, step = 1.0, description = 'x3',orientation='horizontal',\n layout=Layout(width='150px', height='30px'),readout_format = '.0f',style=style,continuous_update=False)\nx3.style.handle_color = 'green'\ny3 = widgets.FloatSlider(min=0.0, max = 1000.0, value = 200.0, step = 1.0, description = 'y3',orientation='vertical',\n layout=Layout(width='70px', height='180px'),readout_format = '.0f',style=style,continuous_update=False)\ny3.style.handle_color = 'green'\nuik3 = widgets.VBox([x3,y3],)\n\nxu = widgets.FloatSlider(min=0.0, max = 1000.0, value = 500.0, step = 1.0, description = '?',orientation='horizontal',\n layout=Layout(width='150px', height='30px'),readout_format = '.0f',style=style,continuous_update=False)\nxu.style.handle_color = 'gray'\nyu = widgets.FloatSlider(min=0.0, max = 1000.0, value = 500.0, step = 1.0, description = '?',orientation='vertical',\n layout=Layout(width='70px', height='180px'),readout_format = '.0f',style=style,continuous_update=False)\nyu.style.handle_color = 'gray'\nuiku = widgets.VBox([xu,yu],)\n\nuipars = widgets.HBox([uikvar,uik1,uik2,uik3,uiku],) \nuik = widgets.VBox([l,uipars],)\n\ndef convert_type(it):\n if it == 'Spherical': \n return 1\n elif it == 'Exponential':\n return 2\n else: \n return 3\n\ndef f_make_krige(nug,it1,azi,hmaj1,hmin1,x1,y1,x2,y2,x3,y3,xu,yu,): # function to take parameters, make sample and plot\n text_trap = io.StringIO()\n sys.stdout = text_trap\n it1 = convert_type(it1)\n nst = 1; xlag = 10; nlag = int(hmaj1/xlag); c1 = 1.0-nug\n vario = GSLIB.make_variogram(nug,nst,it1,c1,azi,hmaj1,hmin1) # make model object\n index_maj,h_maj,gam_maj,cov_maj,ro_maj = geostats.vmodel(nlag,xlag,azi,vario) # project the model in the major azimuth # project the model in the 135 azimuth\n index_min,h_min,gam_min,cov_min,ro_min = geostats.vmodel(nlag,xlag,azi+90.0,vario) # project the model in the minor azimuth\n \n clist = ['blue','red','green']\n x = [x1,x2,x3]; y = [y1,y2,y3]; \n df = pd.DataFrame({'X':x,'Y':y,'Value':value})\n\n xl = [xu,0,1]; yl = [yu,0,1]; value1 = [0,0,0]\n dfl = pd.DataFrame({'X':xl,'Y':yl, 'Value':value1})\n \n sk_est, sk_var, sk_weights = simple_simple_krige(df,'X','Y','Value',dfl,'X','Y',vario,skmean=vmean)\n if sk_var[0] == 0: # sk_std is used for plotting only, leave with sill = 1.0\n sk_std = 0.0\n else:\n sk_std = math.sqrt(sk_var[0])\n sk_var[0] = sk_var[0]*sill \n xlag = 10.0; nlag = int(hmaj1/xlag)\n \n plt.subplot(1,3,1)\n plt.plot([0,hmaj1*1.5],[1.0,1.0],color = 'black')\n plt.plot(h_maj,gam_maj,color = 'black',label = 'Major ' + str(azi)) \n plt.plot(h_min,gam_min,color = 'black',label = 'Minor ' + str(azi+90.0))\n deltas = [22.5, 45, 67.5]; \n ndelta = len(deltas); hd = np.zeros(ndelta); gamd = np.zeros(ndelta);\n color=iter(cm.plasma(np.linspace(0,1,ndelta)))\n for delta in deltas:\n index,hd,gamd,cov,ro = geostats.vmodel(nlag,xlag,azi+delta,vario);\n c=next(color)\n plt.plot(hd,gamd,color = c,label = 'Azimuth ' + str(azi+delta))\n plt.xlabel(r'Lag Distance $\\bf(h)$, (m)')\n plt.ylabel(r'$\\gamma \\bf(h)$')\n plt.title('Interpolated NSCORE Porosity Variogram Models')\n plt.xlim([0,hmaj1*1.5])\n plt.ylim([0,1.4])\n plt.legend(loc='upper left')\n \n plt.subplot(1,3,2)\n if sk_weights[0,0] > 0.01:\n plt.scatter(x1,y1,color = 'blue', edgecolors = 'black', s = sk_weights[0,0]*1000, alpha = 0.3)\n else:\n plt.scatter(x1,y1,color = 'blue', edgecolors = 'black', marker = 'x', alpha = 0.3)\n if sk_weights[0,1] > 0.01: \n plt.scatter(x2,y2,color = 'red', edgecolors = 'black', s = sk_weights[0,1]*1000,alpha = 0.3)\n else:\n plt.scatter(x2,y2,color = 'red', edgecolors = 'black', marker = 'x',alpha = 0.3)\n if sk_weights[0,0] > 0.01:\n plt.scatter(x3,y3,color = 'green', edgecolors = 'black', s = sk_weights[0,2]*1000, alpha = 0.3)\n else:\n plt.scatter(x3,y3,color = 'green', edgecolors = 'black', marker = 'x', alpha = 0.3)\n \n if (1-sk_std) > 0.01:\n scatter = plt.scatter(xu,yu,color = 'gray', edgecolors = 'black', s = (1-sk_std)*1000,alpha = 0.3)\n else: \n scatter = plt.scatter(xu,yu,color = 'blue', edgecolors = 'black', marker = 'x', alpha = 0.3)\n \n ax = plt.gca()\n plt.xlabel('X(m)'); plt.ylabel('Y(m)')\n plt.title('Simple Kriging - Data and Unknown Locations')\n plt.xlim([0,1000])\n plt.ylim([0,1000])\n for i, txt in enumerate(np.round(sk_weights[0],2)):\n plt.annotate('$\\lambda$' + ' ' + '$=$' + str(txt), (x[i]+20, y[i]+20),color = clist[i])\n plt.annotate(i+1, (x[i]+46, y[i]+15),fontsize=6,color = clist[i])\n for i, txt in enumerate(value):\n plt.annotate('$z($u' + ' ' + '$ )$ = ' + str(txt), (x[i]+20, y[i]-40),color = clist[i])\n plt.annotate(i+1, (x[i]+80, y[i]-45),fontsize=6,color = clist[i])\n plt.annotate('$\\sum \\lambda_{\\\\alpha} = $'+ str(np.round(np.sum(sk_weights[0]),2)), (xu+20, yu+20))\n plt.annotate('$z^*($u$)$ = '+ str(np.round(sk_est[0],2)), (xu+20, yu-40))\n plt.annotate('Mean Weight = ' + str(np.round(1.0 - np.sum(sk_weights[0]),2)), (20, 20))\n plt.annotate('?', (xu-20, yu + 30))\n\n ax = plt.gca()\n ellipse = Ellipse((xu, yu),width=hmin1*2.0,height=hmaj1*2.0,angle = 360-azi,facecolor='gray',alpha = 0.03)\n ax.add_patch(ellipse)\n ellipse = Ellipse((x1, y1),width=hmin1*2.0,height=hmaj1*2.0,angle = 360-azi,facecolor='blue',alpha = 0.03)\n ax.add_patch(ellipse)\n ellipse = Ellipse((x2, y2),width=hmin1*2.0,height=hmaj1*2.0,angle = 360-azi,facecolor='red',alpha = 0.03)\n ax.add_patch(ellipse)\n ellipse = Ellipse((x3, y3),width=hmin1*2.0,height=hmaj1*2.0,angle = 360-azi,facecolor='green',alpha = 0.03)\n ax.add_patch(ellipse)\n \n samples = norm.rvs(sk_est[0],math.sqrt(sk_var[0]),1000,random_state=73073) \n \n plt.subplot(1,3,3)\n plt.hist(samples,histtype = 'stepfilled',cumulative = True, bins = np.linspace(vmin,vmax,200),alpha=0.2,color=\"red\",edgecolor='black',density=True)\n plt.xlim([vmin,vmax]); plt.ylim([0,1.0])\n plt.title('Kriging Uncertainty Model at Unknown Location')\n plt.xlabel('Value'); plt.ylabel('Frequency')\n \n ax = plt.gca()\n ax.annotate('Simple Kriging Estimate = ' + str(np.round(sk_est[0],2)), (0.05*(vmax-vmin)+vmin, 0.9))\n ax.annotate('Simple Kriging Variance = ' + str(np.round(sk_var[0],2)), (0.05*(vmax-vmin)+vmin, 0.83))\n ax.annotate('Local P10 = ' + str(np.round(np.percentile(samples,10),2)), (0.05*(vmax-vmin)+vmin, 0.76))\n ax.annotate('Local P90 = ' + str(np.round(np.percentile(samples,90),2)), (0.05*(vmax-vmin)+vmin, 0.69))\n plt.subplots_adjust(left=0.0, bottom=0.0, right=2.2, top=0.9, wspace=0.3, hspace=0.3)\n plt.show()\n \n# connect the function to make the samples and plot to the widgets \ninteractive_plot = widgets.interactive_output(f_make_krige, {'nug':nug, 'it1':it1, 'azi':azi, 'hmaj1':hmaj1, 'hmin1':hmin1, \n 'x1':x1, 'y1':y1, 'x2':x2, 'y2':y2, 'x3':x3, 'y3':y3, 'xu':xu, 'yu':yu})\ninteractive_plot.clear_output(wait = True) # reduce flickering by delaying plot updating\n```\n\n### Interactive Simple Kriging Demostration\n\n* select the variogram model and the data locations and observe the outputs from simple kriging \n\n#### Michael Pyrcz, Associate Professor, University of Texas at Austin \n\n##### [Twitter](https://twitter.com/geostatsguy) | [GitHub](https://github.com/GeostatsGuy) | [Website](http://michaelpyrcz.com) | [GoogleScholar](https://scholar.google.com/citations?user=QVZ20eQAAAAJ&hl=en&oi=ao) | [Book](https://www.amazon.com/Geostatistical-Reservoir-Modeling-Michael-Pyrcz/dp/0199731446) | [YouTube](https://www.youtube.com/channel/UCLqEr-xV-ceHdXXXrTId5ig) | [LinkedIn](https://www.linkedin.com/in/michael-pyrcz-61a648a1) | [GeostatsPy](https://github.com/GeostatsGuy/GeostatsPy)\n\n### The Inputs\n\nSelect the variogram model and the data locations:\n\n* **nug**: nugget effect\n\n* **c1 **: contributions of the sill\n\n* **hmaj1 / hmin1 **: range in the major and minor direction\n\n* **(x1, y1),...(x3,y3) **: spatial data locations \n\n\n```python\ndisplay(uik, interactive_plot) # display the interactive plot\n```\n\n\n VBox(children=(Text(value=' Simple Kriging, Michael Pyrcz, Associ\u2026\n\n\n\n Output(outputs=({'output_type': 'display_data', 'data': {'text/plain': '
                                        ', 'i\u2026\n\n\n#### Comments\n\nThis was an interactive demonstration of simple kriging for spatial data analytics. Much more could be done, I have other demonstrations on the basics of working with DataFrames, ndarrays, univariate statistics, plotting data, declustering, data transformations and many other workflows available at https://github.com/GeostatsGuy/PythonNumericalDemos and https://github.com/GeostatsGuy/GeostatsPy. \n \n#### The Author:\n\n### Michael Pyrcz, Associate Professor, University of Texas at Austin \n*Novel Data Analytics, Geostatistics and Machine Learning Subsurface Solutions*\n\nWith over 17 years of experience in subsurface consulting, research and development, Michael has returned to academia driven by his passion for teaching and enthusiasm for enhancing engineers' and geoscientists' impact in subsurface resource development. \n\nFor more about Michael check out these links:\n\n#### [Twitter](https://twitter.com/geostatsguy) | [GitHub](https://github.com/GeostatsGuy) | [Website](http://michaelpyrcz.com) | [GoogleScholar](https://scholar.google.com/citations?user=QVZ20eQAAAAJ&hl=en&oi=ao) | [Book](https://www.amazon.com/Geostatistical-Reservoir-Modeling-Michael-Pyrcz/dp/0199731446) | [YouTube](https://www.youtube.com/channel/UCLqEr-xV-ceHdXXXrTId5ig) | [LinkedIn](https://www.linkedin.com/in/michael-pyrcz-61a648a1)\n\n#### Want to Work Together?\n\nI hope this content is helpful to those that want to learn more about subsurface modeling, data analytics and machine learning. Students and working professionals are welcome to participate.\n\n* Want to invite me to visit your company for training, mentoring, project review, workflow design and / or consulting? I'd be happy to drop by and work with you! \n\n* Interested in partnering, supporting my graduate student research or my Subsurface Data Analytics and Machine Learning consortium (co-PIs including Profs. Foster, Torres-Verdin and van Oort)? My research combines data analytics, stochastic modeling and machine learning theory with practice to develop novel methods and workflows to add value. We are solving challenging subsurface problems!\n\n* I can be reached at mpyrcz@austin.utexas.edu.\n\nI'm always happy to discuss,\n\n*Michael*\n\nMichael Pyrcz, Ph.D., P.Eng. Associate Professor The Hildebrand Department of Petroleum and Geosystems Engineering, Bureau of Economic Geology, The Jackson School of Geosciences, The University of Texas at Austin\n\n#### More Resources Available at: [Twitter](https://twitter.com/geostatsguy) | [GitHub](https://github.com/GeostatsGuy) | [Website](http://michaelpyrcz.com) | [GoogleScholar](https://scholar.google.com/citations?user=QVZ20eQAAAAJ&hl=en&oi=ao) | [Book](https://www.amazon.com/Geostatistical-Reservoir-Modeling-Michael-Pyrcz/dp/0199731446) | [YouTube](https://www.youtube.com/channel/UCLqEr-xV-ceHdXXXrTId5ig) | [LinkedIn](https://www.linkedin.com/in/michael-pyrcz-61a648a1) \n \n\n\n```python\n\n```\n", "meta": {"hexsha": "9dea096fead3ddd674a0bb33d6b7d04625839622", "size": 34322, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Workflows/Interactive_Simple_Kriging.ipynb", "max_stars_repo_name": "GeostatsGuy/GeostatsPy_Course_2", "max_stars_repo_head_hexsha": "843f2817e563e82402a7a41bfbbf1268ef93a17a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2021-03-24T15:50:00.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-08T18:48:49.000Z", "max_issues_repo_path": "Workflows/Interactive_Simple_Kriging.ipynb", "max_issues_repo_name": "GeostatsGuy/GeostatsPy_Course_2", "max_issues_repo_head_hexsha": "843f2817e563e82402a7a41bfbbf1268ef93a17a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Workflows/Interactive_Simple_Kriging.ipynb", "max_forks_repo_name": "GeostatsGuy/GeostatsPy_Course_2", "max_forks_repo_head_hexsha": "843f2817e563e82402a7a41bfbbf1268ef93a17a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2021-04-19T19:34:17.000Z", "max_forks_repo_forks_event_max_datetime": "2021-07-06T22:42:02.000Z", "avg_line_length": 55.1800643087, "max_line_length": 512, "alphanum_fraction": 0.5868247771, "converted": true, "num_tokens": 7848, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593171945417, "lm_q2_score": 0.6584174938590245, "lm_q1q2_score": 0.40983810365642975}} {"text": "```python\nimport os\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nimport torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nfrom torch.autograd import Variable\n```\n\n\n```python\ndef list_all_files(rootdir, key):\n import os\n _files = []\n list = os.listdir(rootdir) # \u5217\u51fa\u6587\u4ef6\u5939\u4e0b\u6240\u6709\u7684\u76ee\u5f55\u4e0e\u6587\u4ef6\n for i in range(0, len(list)):\n path = os.path.join(rootdir, list[i])\n if os.path.isdir(path):\n _files.extend(list_all_files(path, key))\n if os.path.isfile(path) and key in path:\n _files.append(path)\n return _files\n```\n\n\n```python\ndef load_data():\n global df, df_ind\n root = '.'\n key = '10_'\n files = list_all_files(root, key)\n for f in files:\n yield extract_signal(f)\n```\n\n\n```python\ndef extract_signal(f):\n data = pd.read_table(f, header=None, skiprows=1)\n rawdata = np.array(data.iloc[:, 19:20])\n force_flag = np.array(data.iloc[:, 2])\n tds = np.where(np.diff(force_flag) == 1)[0]\n # print(len(tds))\n x_data = np.array([np.diff(rawdata[i - 3:i + 13, :], axis=0).T.flatten() for i in tds])\n x_data = x_data[np.max(x_data, axis=1) > 20]\n x_data = x_data.T\n x_data = np.apply_along_axis(\n lambda x: (x - np.min(x)) / (np.max(x) - np.min(x)), 0, x_data\n )\n x_data = (x_data - 0.5) * 2\n # print(x_data.max(), x_data.min())\n return x_data\n```\n\n\n```python\ndef make_data():\n sample = np.random.choice(datas.shape[0], batch_size, False)\n return datas[sample]\n```\n\n\n```python\ndef make_noise():\n return np.random.uniform(-1, 1, (batch_size, generator_len))\n\n```\n\n\\begin{equation}\n\\operatorname*{\\Sigma}\\limits_{t=r+1}^s (x_{t-1}-x_t)^2\n\\end{equation}\n\n\\begin{equation}\nL=\\operatorname*{\\mathbb{E}}\\limits_{\\tilde{x}\\sim\\mathbb{P}_g} [D(\\tilde{x})]-\\operatorname*{\\mathbb{E}}\\limits_{x\\sim\\mathbb{P}_r}[D(x)]+\\lambda \\operatorname*{\\mathbb{E}}_{\\hat{x}\\sim\\mathbb{P}_{\\hat{x}}}[(\\left\\|\\nabla_{\\tilde{x}}D(\\hat{x})\\right\\|_2 -1)^2]\n\\end{equation}\n\n\n```python\ndef train():\n d_optim = torch.optim.Adam(D.parameters(), d_lr, betas=(0.5, 0.9))\n g_optim = torch.optim.Adam(G.parameters(), g_lr, betas=(0.5, 0.9))\n\n plt.ion()\n wd = []\n\n for epoch in range(epochs):\n D.train(), G.train()\n for ci in range(critic_iters):\n data_batch = make_data()\n gen_batch = make_noise()\n data_batch, gen_batch = Variable(torch.FloatTensor(data_batch)), \\\n Variable(torch.FloatTensor(gen_batch))\n d_loss = -torch.mean(D(data_batch)) + torch.mean(D(G(gen_batch))) + calc_gradient_penalty(data_batch,\n G(gen_batch))\n wasserstein_distance = -torch.mean(D(G(gen_batch))) + torch.mean(D(data_batch))\n# print(wasserstein_distance.item())\n\n # d_loss = -torch.mean(torch.log(D(data_batch)) + torch.log(1 - D(G(gen_batch))))\n # g_loss = torch.mean(torch.log(1 - D(G(gen_batch))))\n\n d_optim.zero_grad()\n d_loss.backward(retain_graph=True)\n d_optim.step()\n\n data_batch = make_data()\n gen_batch = make_noise()\n data_batch, gen_batch = Variable(torch.FloatTensor(data_batch)), \\\n Variable(torch.FloatTensor(gen_batch))\n g_loss = -torch.mean(D(G(gen_batch)))\n g_optim.zero_grad()\n g_loss.backward()\n g_optim.step()\n\n if epoch % 50 == 0:\n D.eval(), G.eval()\n plt.clf()\n plt.suptitle('epoch=%d, w-dist=%.6f' % (epoch, wasserstein_distance.item()))\n wd.append(wasserstein_distance.item())\n for i in range(16):\n plt.subplot(4, 4, i + 1)\n gen_diff = G(gen_batch).detach().numpy()\n gen_raw = np.hstack((np.cumsum(gen_diff[:, :int(data_len / 2)], axis=1),\n np.cumsum(gen_diff[:, int(data_len / 2):], axis=1)))\n plt.plot(gen_raw[i])\n plt.xlim((0, data_len))\n plt.ylim((-2, 2))\n plt.pause(0.01)\n plt.ioff()\n plt.figure()\n plt.plot(wd)\n plt.show()\n```\n\nfor each sample of real x and generated x, we make\n$$ \\tilde{x}=\\alpha\\cdot x\\_real+(1-\\alpha)\\cdot x\\_gen $$\nwhere $ \\alpha $ comes from a uniform distribution.\n\\begin{equation}\nGradient Penalty = \\lambda\\cdot(\\left\\|\\nabla_{\\tilde{x}}D(\\tilde{X})\\right\\|_2^2-1)^2\n\\end{equation}\n\n\n```python\ndef calc_gradient_penalty(x_real, x_gen):\n alpha = torch.rand(batch_size, 1)\n alpha = alpha.expand(x_real.size())\n x_hat = alpha * x_real + (1 - alpha) * x_gen\n D_x = D(x_hat)\n gradients = torch.autograd.grad(\n outputs=D_x,\n inputs=x_hat,\n grad_outputs=torch.ones(D_x.size()),\n create_graph=True,\n retain_graph=True,\n only_inputs=True\n )[0]\n # print(gradients)\n gradient_penalty = gp_lambda * ((gradients.norm(2, dim=1) - 1) ** 2).mean()\n # print(gradient_penalty)\n return gradient_penalty\n```\n\n\n```python\ndatas = load_data()\ndatas = np.hstack([d for d in datas]).T\n\nbatch_size = 32\ngenerator_len = 20\ndata_len = 15\nepochs = 200000\nd_lr = 0.000001\ng_lr = 0.000001\ngp_lambda = 0.1\ncritic_iters = 5\n\nD = nn.Sequential(\n nn.Linear(data_len, 32),\n # nn.Dropout(0.5),\n nn.ReLU(),\n nn.Linear(32, 16),\n # nn.Dropout(0.5),\n nn.ReLU(),\n nn.Linear(16, 4),\n # nn.Dropout(0.5),\n nn.ReLU(),\n nn.Linear(4, 1)\n)\n\nG = nn.Sequential(\n nn.Linear(generator_len, 30),\n nn.ReLU(),\n nn.Linear(30, 30),\n nn.ReLU(),\n nn.Linear(30, data_len),\n nn.Tanh()\n)\n\nx = np.tile(np.linspace(-1, 1, data_len), [batch_size, 1])\n# make_data()\ntrain()\ntorch.save(D, 'D.model')\ntorch.save(G, 'G.model')\n\n# D_ = torch.load('D.model')\n# G_ = torch.load('G.model')\n# print(D_, G_)\n# batch_size = 1000\n# gen_data = make_noise()\n# print(gen_data.shape)\n# gen_data = G_(Variable(torch.FloatTensor(gen_data))).detach().numpy()\n# plt.ion()\n# for i in range(gen_data.shape[0]):\n# plt.cla()\n# plt.plot(np.cumsum(gen_data[i, :int(data_len / 2)]))\n# plt.plot(np.cumsum(gen_data[i, int(data_len / 2):]))\n# # gen_raw = np.hstack((np.cumsum(gen_data[:, :int(data_len / 2)], axis=1),\n# # np.cumsum(gen_data[:, int(data_len / 2):], axis=1)))\n# # plt.plot(gen_raw[i])\n# plt.pause(0.2)\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "fc272865021f924f140ec50d6a5994433920b202", "size": 12451, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "pytorch/exercises/wgan/improved wgan.ipynb", "max_stars_repo_name": "wangyendt/deeplearning_models", "max_stars_repo_head_hexsha": "47883b6c65b8d05a0d1c5737f1552df6476ded34", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-06-04T11:10:27.000Z", "max_stars_repo_stars_event_max_datetime": "2020-06-04T11:10:27.000Z", "max_issues_repo_path": "pytorch/exercises/wgan/improved wgan.ipynb", "max_issues_repo_name": "wangyendt/deeplearning_models", "max_issues_repo_head_hexsha": "47883b6c65b8d05a0d1c5737f1552df6476ded34", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "pytorch/exercises/wgan/improved wgan.ipynb", "max_forks_repo_name": "wangyendt/deeplearning_models", "max_forks_repo_head_hexsha": "47883b6c65b8d05a0d1c5737f1552df6476ded34", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 29.5047393365, "max_line_length": 296, "alphanum_fraction": 0.4989157497, "converted": true, "num_tokens": 1854, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.40967280973104375}} {"text": "# [Introductory applied machine learning (INFR10069)](https://www.learn.ed.ac.uk/webapps/blackboard/execute/content/blankPage?cmd=view&content_id=_2651677_1&course_id=_53633_1)\n\n# Lab 5: Neural Networks\n\n*by [James Owers](https://jamesowers.github.io/), University of Edinburgh 2017*\n\n1. [Introduction](#Introduction)\n * [Lab Outline](#Lab-Outline)\n * [The Data](#The-Data)\n1. [Part 1 - Introducing the Neural Network Model](#Part-1---Introducing-the-Neural-Network-Model)\n * [Resources to Watch and Read pt. 1](##Resources-to-Watch-and-Read-pt.-1)\n * [Model Design](#Model-Design)\n * [The Cost Space](#The-Cost-Space)\n1. [Part 2 - Fitting the Model & Optimisation](#Part-2---Fitting-the-Model-&-Optimisation)\n * [Resources to Watch and Read pt. 2](#Resources-to-Watch-and-Read-pt.-2)\n * [Finding the Best Parameters](#Finding-the-Best-Parameters)\n * [Gradient Descent](#Gradient-Descent)\n * [Backpropagation](#Backpropagation)\n1. [Part 3 - Implementation From Scratch](#Part-3---Implementation-From-Scratch!)\n1. [Part 4 - Implementation With Sklearn](#Part-4---Implementation-with-Sklearn)\n1. [Moar?!](#Please-sir...I-want-some-more)\n\n## Import packages\n\n\n```python\n# https://docs.python.org/2/library/__future__.html\n# make printing and division act like python 3\nfrom __future__ import division, print_function\n\n# General\nimport sys\nimport os\nimport copy\nfrom IPython.display import Image, HTML\n\n# Data structures\nimport numpy as np \nimport pandas as pd\n\n# Modelling\nfrom sklearn.neural_network import MLPClassifier\nfrom sklearn.model_selection import train_test_split\nfrom scipy.optimize import check_grad\n\n# Plotting\nimport matplotlib.pyplot as plt\nimport seaborn as sns\n\n# Local module adjacent to this notebook\nimport iaml\nfrom iaml.data import load_letters\n\n# http://ipython.readthedocs.io/en/stable/interactive/magics.html\n%matplotlib inline\n```\n\n## Introduction\n\nThis lab:\n1. introduces a simple neural network model in a supervised learning setting\n1. provides impetus to understand the fitting procedure of that, and other networks\n1. encourages you to implement a model from scratch\n1. models the same problem with the sklearn package\n1. makes you think about what you've done!\n\nIt does not discuss in detail:\n1. any of the plethora of different activation functions you can use e.g. RELUs, SELUs, Tanh, ...\n1. how to initialise the parameters and why that matters\n1. issues with the fitting process e.g. local optima, and how to avoid them e.g. learning rate schedulers, momentum, RMSProp, Adam, cyclic learning rates\n1. issues with model complexity e.g. overfitting, and solutions such as dropout, regularisation, or using [shedloads of data](https://what-if.xkcd.com/63/)\n1. other tricks for speeding up and stablising fitting such as batch sizes, weight norm, layer norm\n1. deep networks and their tricks like skip connections, pooling, convolutions\n1. nor other more complex architectures like CNNs, RNNs, LSTMs, GANs, etc. etc.\n1. many, many, MANY other things (that probably were published, like, [yesterday](https://arxiv.org/abs/1711.04340v1))\n\nHowever, if you understand what is in this notebook well, **you will have the ability to understand [all of these things](https://i.imgflip.com/1zn8p9.jpg)**.\n\n### Lab outline\n\nI provide you with a function that creates data then link you to some excellent resources to learn the basics. These resources are superb, short, and free. I highly, highly recommend setting aside a couple of hours to give them a good watch/read and, at the very least, use them for reference. \n\nAfter you have had a crack at the problems, I'll release the solutions. The solutions, particularly to part 3, walk you through the process of coding a simple neural neural network in detail.\n\nParts 3 & 4 are practical, parts 1 & 2 are links to external resources to read. Whilst I recommend you soak up some context first with 1 & 2, feel free to jump in at the deep end and get your hands dirty with part 3 or 4.\n\n### The Data\n\nThroughout this lab we are going to be using a simple classification example: the TC classification problem (not to be confused with the real [TC](https://www.youtube.com/watch?v=NToYkBYezZA)). This is a small toy problem where we, initially, try to distinguish between 3x3 grids that look like Ts and Cs. Let's create the dataset and have a look...\n\nI have written a function `load_letters()` to generate synthetic data. For now, you will use the data generated below, but later you have opportunity to play with generating different data if you like. The function is located in the `iaml` module adjacent to this notebook - feel free to check out the code but I advise you **do not edit it**. Run (and don't edit) the next few cells to create and observe the data.\n\n\n```python\nbounds = [-1, 1]\nX, y, y_labels = load_letters(categories=['T', 'C'], \n num_obs=[50, 50],\n bounds=bounds,\n beta_params=[[1, 8], [8, 1]],\n shuffle=True, \n random_state=42)\n```\n\nLet's print the data (I'm just creating a Pandas DataFrame for display, I probably wont use this object again)\n\n\n```python\npd.set_option(\"max_rows\", 10)\ndf = pd.DataFrame(\n np.hstack(\n [np.around(X,2), \n y[:, np.newaxis], \n np.array([y_labels[ii] for ii in y])[:, np.newaxis]\n ]\n ),\n columns = ['x{}'.format(ii) for ii in range(9)] + ['Class (numeric)', 'Class Label']\n)\ndf\n```\n\n\n```python\npd.reset_option(\"max_rows\")\n```\n\nThe data are arranged as vectors for your convenience, but they're really `3 x 3` images. Here's a function to plot them.\n\n\n```python\ndef plot_grid(x, shape=None, **heatmap_params):\n \"\"\"Function for reshaping and plotting vector data.\n If shape not given, assumed square.\n \"\"\"\n if shape is None:\n width = int(np.sqrt(len(x)))\n if width == np.sqrt(len(x)):\n shape = (width, width)\n else:\n print('Data not square, supply shape argument')\n sns.heatmap(x.reshape(shape), annot=True, **heatmap_params)\n```\n\n\n```python\nfor ii in range(3):\n plt.figure()\n plot_grid(X[ii], vmin=bounds[0], vmax=bounds[1], cmap='Greys')\n plt.title('Observation {}: Class = {} (numeric label {})'.format(ii, y_labels[y[ii]], y[ii]))\n plt.show()\n```\n\nFinally, let's make the train and test split. Again, don't alter this code.\n\n\n```python\nX_train, X_test, y_train, y_test = train_test_split(\n X, y, test_size=0.5, random_state=42)\n```\n\n\n```python\nX_train, X_valid, y_train, y_valid = train_test_split(\n X_train, y_train, test_size=0.33, random_state=42)\n```\n\n\n```python\n[dd.shape for dd in [X_train, X_valid, X_test, y_train, y_valid, y_test]]\n```\n\n## Part 1 - Introducing the Neural Network Model\n\n### Resources to Watch and Read pt. 1\n\n**Reading/watching time:** 30 minutes\n\nFirst, watch this video from 3 Blue 1 Brown: [But what *is* a Neural Network? | Deep learning, chapter 1](https://www.youtube.com/watch?v=aircAruvnKk)\n\nIf you prefer reading, try 2 sections of Nielsen's Book Chapter 1:\n* [Sigmoid Neurons](http://neuralnetworksanddeeplearning.com/chap1.html#sigmoid_neurons)\n* and [The Architecture of Neural Networks](http://neuralnetworksanddeeplearning.com/chap1.html#the_architecture_of_neural_networks)\n\n### Model Design\n\nJust so as there's something in this notebook to quickly reference - here's a nice illustration of what's going on in a neural net. Within the calculation of the $z$'s you'll see the learned **parameters**: $w$'s and $b$'s - these are the weights and biases respectively. *N.B. I omit the bias $b$ parameters in the Part 3 implementation.* The functions $g$ are the activation functions.\n\n\n\n### The Cost Space\n\nWhen we talk about the cost space, loss$^*$ space, or cost surface, we are talking about a function that changes with respect to the parameters. This function determines how well the network is performing - a low cost is good, a high cost is bad. A simple example for two parameters is shown below. **Our goal is to update the parameters such that we find the global minimum of the cost function.**\n\n$^*$ 'loss' and 'cost' are interchangeable terms - you'll see them both around but I try to stick to 'cost'!\n\n\n\nN.B. The cost function is often referred to with different letters e.g. $J(w)$, $C(\\theta)$, $\\mathcal{L}(x)$, and $E(w)$\n\n## Part 2 - Fitting the Model & Optimisation\n\n### Resources to Watch and Read pt. 2\n\n**Watching/reading time:** ~1 hour\n\nFirst, watch these two videos from 3 Blue 1 Brown:\n1. [Gradient descent, how neural networks learn | Deep learning, chapter 2](https://www.youtube.com/watch?v=IHZwWFHWa-w)\n2. [What is backpropagation and what is it actually doing? | Deep learning, chapter 3](https://www.youtube.com/watch?v=Ilg3gGewQ5U)\n\nThis will take you just over half an hour (if you watch at 1x speed). They are really excellent and well worth the time investment.\n\nAgain, if you prefer reading try Nielsen's section [Learning with Gradient Descent](http://neuralnetworksanddeeplearning.com/chap1.html#learning_with_gradient_descent)\n\n### Finding the Best Parameters\n\nSo, we've got a function, let's call it $C(\\theta)$ that puts a number on how well the neural network is doing. We provide the function with the parameters $\\theta$ and it spits out the cost$^*$. We could just randomly chose values for $\\theta$ and select the ones that result in the best cost...but that might take a long time. We'd also need to define a way to randomly select parameters as well. What if the best parameter setting is very unlikely to be selected?\n\n**Calculus to the rescue!** The cost $C(\\theta)$ is a function and, whilst we can't see the surface without evaluating it everywhere (expensive!), we can calculate the derivative with respect to the parameters $\\frac{\\partial C(\\theta)}{\\partial \\theta}$. The derivative **tells you how the function value changes if you change $\\theta$**. \n\nFor example, imagine $\\theta$ is 1D and I tell you that $\\frac{\\partial C(\\theta)}{\\partial \\theta} = 10\\theta$. This means that if I increase $theta$ by 2, the cost function will go up by 20. Which way will you update $\\theta$? You want to *decrease* the cost, so you would want to *decrease* $\\theta$ by some amount.\n\nThe only thing we need to do is choose a cost function $C(\\theta)$ that has a derivative function $\\frac{\\partial C(\\theta)}{\\partial \\theta}$...and that is easy!\n\n$^*$It's much easier if you imagine $\\theta$ as just one number to start with, but the maths is basically the same as $\\theta$ becomes a vector (or matrix) of numbers.\n\n### Gradient Descent\n\nSo how do we actually update the parameters?! All update the parameters in the opposite direction to the gradient; you always try to take a step 'downhill'. Here's the formula:\n\n$$\n\\theta \\leftarrow \\theta - \\eta \\frac{\\partial C(\\theta)}{\\partial \\theta}\n$$\n\nwhere \"$\\leftarrow$\" means \"update from\", and $\\eta$ is the \"learning rate\" - a hyperparameter you can choose. If you increase $\\eta$ you make bigger updates to $\\theta$, and vice versa. \n\nThere are many more complicated ways to update the parameters using the gradient of the cost function, but they all have this same starting point.\n\nBelow is an example cost surface. A few things to note:\n\n* The axes should be labelled $\\theta_0$ (1, -1.5) and $\\theta_1$ (-1, 1) on the 'flat' axes, and $C(\\theta)$ (-4, 4) on the vertical axis\n* The surface is shown - we don't have direct access to this in reality. To show it, the creator has queried the cost function *at every [$\\theta_0$, $\\theta_1$] location* and plotted it\n* The animated balls rolling along the surface are different gradient descent algorithms - each frame of the GIF shows one update. The equation shown above is SGD - the GIF highlights a potential issue with the algorithm!\n\n\n\n\nVisualisation by [Alec Radford](https://blog.openai.com/tag/alec-radford/), summarised excellently in [this blog post](http://ruder.io/optimizing-gradient-descent/).\n\n### Backpropagation\n\n**Reading/watching time:** 1 hour\n\nRight...it's time for some derivatives. If you've been liking the videos - go ahead and watch the next in the series:\n\n1. [Backpropagation calculus | Appendix to deep learning chapter 3](https://www.youtube.com/watch?v=tIeHLnjs5U8)\n\nIf you have time, I recommend now having a crack at reading half of [Nielsen Chapter 2](http://neuralnetworksanddeeplearning.com/chap2.html), up to and including the section entitled [The Backpropagation Algorithm](http://neuralnetworksanddeeplearning.com/chap2.html#the_backpropagation_algorithm).\n\nI'm just going to write out some derivatives you're going to find useful for Part 3 below:\n\n$$\n\\begin{align}\nz^{(L)} &= W^{(L)}a^{(L-1)} \\\\\n\\frac{\\partial z^{(L)}}{\\partial W} &= a^{(L-1)}\n\\end{align}\n$$\n\n$$\n\\begin{align}\n\\text{linear}[z] &= z \\\\\n\\frac{\\partial \\text{linear}[z]}{\\partial z} &= 1 \\\\\n\\end{align}\n$$\n\n$$\n\\begin{align}\n\\text{sigmoid}[z] = \\sigma[z] &= \\frac{1}{1 + e^{-z}} = \\frac{e^{z}}{e^{z} + 1}\\\\\n\\frac{\\partial \\sigma[z]}{\\partial z} &= \\frac{e^{z}}{e^{z} + 1} - (\\frac{e^{z}}{e^{z} + 1})^2 \\\\\n &= \\frac{e^{z}}{e^{z} + 1} ( 1 - \\frac{e^{z}}{e^{z} + 1} ) \\\\\n &= \\sigma[z] (1 - \\sigma[z])\n\\end{align}\n$$\n\n$$\n\\begin{align}\n\\text{crossentropy}[y, a] = C[y, a] &= - \\frac{1}{N} \\sum_{i=1}^N y_i \\log a_i + (1-y_i)\\log(1-a_i) \\\\\n\\frac{\\partial C[y_i, a_i]}{\\partial a_i} &= \\frac{1 - y_i}{1 - a_i} + \\frac{y_i}{a_i}\n\\end{align}\n$$\n\nAnd finally, this is all backpropagation really is...\n$$\n\\begin{align}\n\\frac{\\partial C[y_i, a_i]}{\\partial w_j} &= \\frac{\\partial a_i}{\\partial w_j}\\frac{\\partial C[y_i, a_i]}{\\partial a_i}\\\\\n &= \\frac{\\partial z_k}{\\partial w_j}\\frac{\\partial a_i}{\\partial z_k}\\frac{\\partial C[y_i, a_i]}{\\partial a_i}\\\\\n\\end{align}\n$$\n\n\nChallenge: derive these yourself.\n\n#### Reading extension\n\nFor more on gradient based optimisers [check out this blog post](http://ruder.io/optimizing-gradient-descent/)\n\nFor another look at backpropagation - try [Christopher Olah's blog](http://colah.github.io/posts/2015-08-Backprop/)\n\n## Part 3 - Implementation From Scratch!\n\n### ========== Question 3.1 ==========\n\nFirst thing is first: **don't get stuck on this**. I recommend you attempt this question for an hour and, if you don't get anywhere, move on to Question 3.2. You can even move straight on to Part 4. It's exactly the same problem addressed here in 3.1, but using sklearn instead of coding it yourself.\n\n#### Model Specification\n\n\n\nWe are going to fit a very small neural network to classify the TC data. Here is the specification of the model:\n\n1. Input of size 9\n1. Hidden layer of size 3\n * Linear activation function\n1. Output layer of size 1\n * Logistic activation function\n\nAs for the **cost function**: use Cross-Entropy. However, if you're getting bogged down with derivatives, feel free to try squared error to start with (this is what Nielsen and 3 Blue 1 Brown start with in their tutorials). Squared error is [not necessarily the right cost function to use](https://jamesmccaffrey.wordpress.com/2013/11/05/why-you-should-use-cross-entropy-error-instead-of-classification-error-or-mean-squared-error-for-neural-network-classifier-training/) but it will still work!\n\nFor a given input vector $x$, we can predict an output probability $a^{(2)}$ (were the $^{(2)}$ indicates the layer number, *not a power* - I'm following 3 Blue 1 Brown notation as best I can) using the following formula:\n\n$$\n\\begin{align}\na^{(2)} &= f^{(2)}[z^{(2)}] \\\\\n &= f^{(2)}[W^{(2)}a^{(1)}] \\\\\n &= f^{(2)}[W^{(2)}f^{(1)}[z^{(1)}]] \\\\\n &= f^{(2)}[W^{(2)}f^{(1)}[W^{(1)}a^{(0)}]] \\\\\n &= f^{(2)}[W^{(2)}f^{(1)}[W^{(1)}x]] \\\\\n &= \\sigma[W^{(2)}(W^{(1)}x)]\n\\end{align}\n$$ \n\nwhere:\n\n* $f^{(2)}$ is the activation function of the output layer (a sigmoid function $\\sigma[]$)\n* $f^{(1)}$ is the activation function of the hidden layer (the identity - 'linear activation')\n* $W^{(2)}$ and $W^{(1)}$ are the parameters to learn\n* $a^{(L)} = f^{(L)}[z^{(L)}]$ are the activations **exiting** layer $^{(L)}$\n* $z^{(L)} = W^{(L)}a^{(L-1)}$ is the pre-activation weighted sum calculated **within** layer $^{(L)}$\n\nThe formula for the Cross-Entropy cost function is:\n\n$$\nC(a) = - \\frac{1}{N} \\sum_{i=1}^N y_i \\log a_i + (1-y_i)\\log(1-a_i)\n$$\n\nNotice how only one term in the sum is ever non-zero because $y_i$ is only ever 0 or 1. In our case, $N$ is the number of data observations in the dataset.\n\n##### Parameters\n\nThe parameters of the model are two matrices:\n\n1. $W^{(2)}$ - $3 \\times 9$ matrix\n * used within the hidden layer (the $1^{st}$ layer) to get $z^{(1)} = W^{(1)}x$ for some $9 \\times 1$ input vector $x$. $z^{(1)}$ is thus $3 \\times 1$.\n1. $W^{(1)}$ - $1 \\times 3$ matrix\n * used within the output layer (the $2^{nd}$ layer) to get $z^{(2)} = W^{(2)}a^{(1)}$ for some $3 \\times 1$ input vector $a^{(1)}$. $z^{(2)}$ is thus $1 \\times 1$.\n\n**Note that I'm not asking you to fit *bias parameters*.**\n\nYou'll often see parameters referred to as $\\theta$, it's a catch all term. In our case it's just a list of all the weights, $\\theta = [W^{(1)}, W^{(2)}]$. **We have 3 x 9 + 3 x 1 = 30 parameters to learn in total.**\n\n##### Advice\n\nYou can use any of the equations and code I've given you or linked you to in this lab but **you do not have to!** You're free to code as you please. Personally, since this is a simple example, I did not do anything fancy (I didn't create any objects with methods and attributes). I simply:\n* created a list containing the two parameter matrices `theta = [W1, W2]`\n* created a function to do prediction (the forward pass)\n* created a function to do the backward pass (updating the weights)\n * This is the tricky bit - I coded functions that are the [relevant derivatives](#http://localhost:8888/notebooks/10_Lab_5_Neural_Networks.ipynb#Backpropagation), and wrote code to iteratively pass back the 'deltas' - (I think Nielsen's equations [here](http://neuralnetworksanddeeplearning.com/chap2.html#the_backpropagation_algorithm) are very useful) \n* wrote a training loop which called these two main functions\n * each epoch calls the forward pass to predict, then the backward pass to update the parameters.\n\nWhen the training was finished, my \"model\" was simply the parameters I had fitted, along with the 'forward pass' function - a function which uses those weights to predict a probability for any input data.\n\n**You do not have to code it up like me**, you can do it however you like! The point of this part is for you to explore, code up all the equations, understand how to calculate the loss, and how to use that loss to update the parameters of the model by backpropagation.\n\n**Debugging**: You're probably going to have issues particularly in the backprop section. You are welcome to make use of the `scipy.optimize.check_grad()` function. This takes as input a function f, g: a function that is (supposed to be) the function's derivative. \n\nIf you didn't watch it already, now is a great time to take 10 minutes and watch [Backpropagation calculus | Appendix to deep learning chapter 3](https://www.youtube.com/watch?v=tIeHLnjs5U8)\n\n#### ===== What you actually need to do for this question! =====\n\nWrite a training loop which uses gradient descent to learn the parameters. Each iteration of the loop is called an **epoch**. Run your code for *no more than 100 epochs*. You should be able to achieve 100% accuracy on this problem. \n\nIn this case, for simplicity, you may initialise the weights to be samples from a normal distribution mean 0 variance 1, but please note that this [is not necessarily good practice](https://intoli.com/blog/neural-network-initialization/)!\n\n**Do not code up a grid search for the learning rate hyperparameter**. You may instead play with the learning rate manually until you are happy. Try small values first like 0.0001 (if your backprop code is correct you **should** see your cost decreasing every epoch). Since this problem is so simple, a range of values should work. Again, with real data, you *must* do a search over hyperparameters, but here we are focussed on *coding* a working model.\n\nTo test whether or not what you have written has worked, please output the following:\n1. After the training loop:\n 1. plot a graph of training and validation loss against epoch number\n 1. print or plot the final parameters you have learned using a Hinton diagram - feel free to use [code you can find online](http://bfy.tw/F74s)\n 1. pick one weight parameter and produce a plot of its value against epoch number\n * Extension: do that for all the weights **leaving one specific input node** (i.e. the weights for one pixel of the input data)\n 1. use your model to:\n 1. print a few of the validation data examples and their predicted probabilities\n 1. print the output for a T and C with no noise (you can make that input data yourself)\n 1. print the output of a few random binary vectors i.e. 9x1 vectors of only 0s and 1s (again, you can make that input data yourself)\n\n1. Within the training loop:\n 1. print the training and validation crossentropy loss **and** percentage accuracy every epoch\n 1. save the value of the training and validation losses for every epoch [for the plot after the loop]\n 1. save the value of a weight parameter of your choice [for the plot after the loop]\n\n#### ===== Example outputs =====\n\nBelow I give you some examples of what I'd like you to produce. **I produced these using a learning rate of 0.003, 100 epochs, and weights initialised with N(0,1) with a random seed of 42**. I found that you could learn faster i.e. you can use a larger learning rate, but I wanted to make smooth plots for you. \n\nYou don't need to produce plots exactly like this, you can do them how you like, but try and display the same information. You can also use my plots for checking (if you use the same settings as me).\n\n##### 1A\n\n\n\n##### 1B\n\n\n\n\n\n##### 1C\n\n\n\n##### 1D\n\n\n\n\n\n\n\n\n\n\n\n##### 2A\n\n\n\n\n```python\n# Your code goes here\n```\n\n#### Forward pass\n\n\n```python\ndef sigmoid(z):\n \"\"\"Sigmoid activation function\n \"\"\"\n return 1 / (1 + np.exp(-z))\n\n\ndef linear(z):\n \"\"\"Linear activation function i.e. identity\n \"\"\"\n return z\n\n\ndef predict(X, params, activation_funs, return_all=False):\n \"\"\"Performs the 'forward pass' of the network which\n calculates the output $a^{(L)}$, the predicted probability for y i.e.\n the activation of the final layer\n \n Arguments\n ---------\n X : numpy array, The data, assumed N x D where N is the number of \n observations and D the dimensionality\n params : list, each element is the weight parameters for successive layers\n activation_funs : list, each element is the activation function\n corresponding to each layer\n return_all : bool, whether to return a list of all the activations\n from each layer (set to True), or just to give the final predictions\n (Default: False)\n Returns\n -------\n if return_all is True:\n [activations, zs] : list, activations and zs are lists of numpy arrays\n if return_all is False:\n a : numpy array, the predicted probabilities i.e. the activations\n that are output from the final layer\n \"\"\"\n activations = [X.T]\n zs = []\n for ii, (W, f) in enumerate(zip(params, activation_funs)):\n a_incoming = activations[ii]\n z = W.dot(a_incoming)\n a_outgoing = f(z)\n zs.append(copy.deepcopy(z))\n activations.append(copy.deepcopy(a_outgoing))\n if return_all:\n return [activations, zs]\n else:\n return activations[-1]\n\n\ndef crossentropy_cost(y, a):\n \"\"\"Calculates the crossentropy cost (aka loss) for predictions a (the \n activation output from the final layer) against true labels y. Assumes y is \n binary data in [0, 1].\n \n See https://goo.gl/Aw7Q3i FMI.\n \"\"\"\n nr_obs = len(y)\n # np.nan_to_num v useful for avoiding log(0) issues\n costs = (-y * np.log(a)) - ((1 - y) * np.log(1 - a)) \n # if the true label y = 1:\n # the first bracket gets very large as a goes to 1\n # and the second bracket is 0\n # if the true label y = 0:\n # the second bracket gets very large as a goes to 0\n # and the first bracket is 0\n return np.nan_to_num(np.sum(costs)) / nr_obs\n\n\ndef squared_cost(y, a):\n \"\"\"Calculates the sum of squared errors cost (aka loss) for predictions\n a (the activation output from the final layer) against true labels y.\n \"\"\"\n nr_obs = len(y)\n costs = (a - y)**2 \n return np.sum(costs) / nr_obs\n```\n\n#### Backward pass (calculating the gradients and updating the parameters)\n\n\n```python\ndef crossentropy_cost_derivative(y, a):\n \"\"\"Derivative of the crossentropy cost function w.r.t activation a.\n \"\"\"\n # np.nan_to_num will help avoid division by 0 issues (generally as a -> y)\n cost_derivs = np.nan_to_num((1 - y) / (1 - a)) - np.nan_to_num(y / a)\n return np.nan_to_num(cost_derivs)\n\n\ndef squared_cost_derivative(y, a):\n \"\"\"Derivative of the MSE cost function w.r.t activation a.\n \"\"\"\n nr_obs = len(y)\n return 2*(a - y) / nr_obs\n\n\ndef sigmoid_derivative(z):\n \"\"\"Derivative of sigmoid activation function w.r.t. z\n \"\"\"\n return sigmoid(z) * (1 - sigmoid(z))\n\n\ndef linear_derivative(z):\n \"\"\"Derivative of linear activation function i.e. identity w.r.t. z\n \"\"\"\n return 1\n\n\ndef backward_pass(y, theta, fwd_pass_data, cost_fun_derivative, \n activation_funs_derivatives, learning_rate=0.001):\n \"\"\"Performs the backward pass through the network.\n Returns updated parameters\n \n Arguments\n ---------\n y : array, the outcome data\n theta : list, each element is the weight parameters for successive layers\n fwd_pass_data : list, the output data from the forward pass [activations, zs]\n activations and zs are lists of numpy arrays corresponding to the activations\n and intermediatry zs for each neuron in each layer. These are used for \n *evaluating* the derivatives. **Create these with the predict function**\n cost_fun_derivative : function, the function which returns the value of\n the cost function's derivative when supplied with the true labels y, and\n final layer activations a\n activation_funs_derivatives : list, list of functions. Each function returns the\n value of the derivative of the activation function of the layer (the index in the\n list indicates the layer number excluding the input layer)\n learning_rate : float, hyperparameter which determines the size of the step in the \n direction of the negative gradient you should take when updating the parameters \n \n Returns\n -------\n theta: list, same as input but updated\n \n \"\"\"\n nr_layers = len(theta)\n activations, zs = fwd_pass_data\n \n ## Get the initial delta from the final layer (needs cost function derivative)\n delta = cost_fun_derivative(y, activations[-1]) * activation_funs_derivatives[-1](zs[-1])\n nablas = [None, None] # https://en.wikipedia.org/wiki/Nabla_symbol\n # These are the gradient values used to update the parameters\n nablas[-1] = np.dot(delta, activations[-2].transpose())\n \n ## Iterate *backwards* through the layers, passing deltas back and evaluating\n ## the derivatives w.r.t. the parameters (uses the forward pass activations and z's)\n ## N.B. for our initial model with 2 layers, this loop only has 1 iteration!\n for ii in range(2, nr_layers+1):\n W = theta[-ii + 1] # weights applied to the activations leaving this layer\n a_incoming = activations[-ii - 1] # activations coming into this layer\n z = zs[-ii] # the z's calculated within this layer\n act_fun_deriv = activation_funs_derivatives[-ii]\n \n delta = np.dot(W.transpose(), delta) * act_fun_deriv(z)\n nablas[-2] = np.dot(delta, a_incoming.transpose())\n \n ## Update the parameters for each layer by gradient descent\n for ii in range(len(theta)):\n nabla = nablas[ii]\n W = theta[ii]\n theta[ii] = W - (learning_rate * nabla)\n return theta\n```\n\n\n```python\ndef get_accuracy(y, a, decision_boundary=.5):\n \"\"\"Convenience function for getting accuracy\"\"\"\n y_hat = (a > decision_boundary).astype(int)\n correct_predictions = np.sum(y == y_hat)\n accuracy = correct_predictions / len(y)\n return accuracy\n```\n\n#### Training Loop\n\n\n```python\n# Settings\nrandom_seed = 42\nnr_epochs = 101 # The way I've written code means 11 epochs amounts to 10 parameter updates\nactivation_funs = [linear, sigmoid]\nactivation_funs_derivatives = [linear_derivative, sigmoid_derivative]\ncost_fun = crossentropy_cost\ncost_fun_derivative = crossentropy_cost_derivative\ndecision_boundary = .5\nlearning_rate = 0.003 # You can actually go a lot faster than this\n # I chose to go slower for clearer graphs!\n\n# Weight initialisation - N(0, 1): quick and dirty\n## a better approach: http://neuralnetworksanddeeplearning.com/chap3.html#weight_initialization\nnp.random.seed(random_seed)\n# W1 = 1/np.sqrt(9) * np.random.randn(3, 9)\n# W2 = 1/np.sqrt(3) * np.random.randn(1, 3)\nW1 = np.random.randn(3, 9)\nW2 = np.random.randn(1, 3)\nparams = [W1, W2]\n\n# Things to store\ncost_per_epoch = dict(train=[], valid=[])\naccuracy_per_epoch = dict(train=[], valid=[])\nparams_per_epoch = []\n\n\n# Training loop\nprint('Beginning Training!\\n{}\\n'.format('-'*len('Beginning Training!')))\nprint(' {:^10} | {:^10} | {:^10} | {:^10} | {:^10} '.\\\n format('Epoch', 'Train Cost', 'Valid Cost', 'Train Acc', 'Valid Acc'))\nprint(' {0} {0} {0} {0} {0} '.format(10*'-'))\n\nfor ii in range(nr_epochs):\n # WARNING: \n # If your network is large and/or you are doing many epochs, it's not \n # sensible to save the parameters every epoch - for this toy problem it's ok\n params_per_epoch.append(copy.deepcopy(params)) # without deepcopy, python will\n # use a pointer only, resulting\n # in a final parameter list with\n # nr_epochs identical sets of params!\n # Forward pass\n fwd_pass_data = predict(X_train, params, activation_funs, return_all=True)\n activations = fwd_pass_data[0]\n\n a_out = dict(train=activations[-1], # this is just to avoid calculating train output activations twice\n valid=predict(X_valid, params, activation_funs, return_all=False))\n for dataset_name, (XX, yy) in dict(train=(X_train, y_train), \n valid=(X_valid, y_valid)).iteritems():\n aa = a_out[dataset_name]\n accuracy_per_epoch[dataset_name].append(get_accuracy(yy, aa, decision_boundary))\n cost_per_epoch[dataset_name].append(cost_fun(yy, aa))\n \n print(' {:>10d} | {:<10.7f} | {:<10.7f} | {:<10.7f} | {:<10.7f} '.\\\n format(ii, cost_per_epoch['train'][ii], cost_per_epoch['valid'][ii], \n accuracy_per_epoch['train'][ii], accuracy_per_epoch['valid'][ii]))\n \n # Don't do backward pass on the final epoch (just to avoid repeating code) \n if ii == nr_epochs - 1:\n break\n \n # Backward pass\n params = backward_pass(y_train, params, fwd_pass_data, cost_fun_derivative, \n activation_funs_derivatives, learning_rate=learning_rate)\n```\n\n#### Results\n\n##### 1A ---------------------------------\n\n\n```python\nplt.figure()\nfor dataset_name in ('train', 'valid'):\n plt.plot(cost_per_epoch[dataset_name], label=dataset_name)\nplt.xlabel('Epoch')\nplt.ylabel('Training cost')\nplt.title('Training and Validation Cost')\nplt.legend() \nplt.savefig('./img/cost_per_epoch.png', dpi=100, bbox_inches='tight')\nplt.show()\n```\n\n##### 1B ---------------------------------\n\n\n```python\ndef hinton(matrix, max_weight=None, ax=None):\n \"\"\"Draw Hinton diagram for visualizing a weight matrix.\n Source: https://matplotlib.org/examples/specialty_plots/hinton_demo.html\n \"\"\"\n ax = ax if ax is not None else plt.gca()\n\n if not max_weight:\n max_weight = 2 ** np.ceil(np.log(np.abs(matrix).max()) / np.log(2))\n\n ax.patch.set_facecolor('gray')\n ax.set_aspect('equal', 'box')\n ax.xaxis.set_major_locator(plt.NullLocator())\n ax.yaxis.set_major_locator(plt.NullLocator())\n\n for (x, y), w in np.ndenumerate(matrix):\n color = 'white' if w > 0 else 'black'\n size = np.sqrt(np.abs(w) / max_weight)\n rect = plt.Rectangle([x - size / 2, y - size / 2], size, size,\n facecolor=color, edgecolor=color)\n ax.add_patch(rect)\n\n ax.autoscale_view()\n ax.invert_yaxis()\n\nfinal_params = params_per_epoch[-1]\n\nfor ii, W in enumerate(final_params):\n plt.figure()\n hinton(W.T)\n plt.title('$W^{{({})}}$'.format(ii + 1))\n plt.tight_layout()\n plt.savefig('img/hinton_W{}.png'.format(ii + 1), dpi=100, bbox_inches='tight')\n plt.show()\n\nW1 = final_params[0]\nfor ii in range(3):\n plt.figure()\n hinton(W1[ii, :].reshape(3, 3).T)\n plt.title('$W^{{({})}}_{}$'.format(1, ii))\n plt.savefig('img/hinton_W{}_{}.png'.format(1, ii), dpi=100, bbox_inches='tight')\n plt.tight_layout()\n plt.show()\n```\n\n##### 1C ---------------------------------\n\n\n```python\n# I disregard my instructions, and show the plots for weights leaving every input data pixel (seeing as there aren't many)\n# You need only plot a single weight over time, I'm just showing off.\nW1_per_epoch = [W1 for W1, W2 in params_per_epoch]\nfor ii in range(3):\n for jj in range(3):\n plt.figure()\n plt.title('Weights multiplying pixel at ({}, {})'.format(ii, jj))\n idx = ii + 3*jj\n weights_over_time = [W1[:, idx] for W1 in W1_per_epoch]\n lines = plt.plot(weights_over_time)\n plt.legend(handles=lines, labels=['Weight for hidden unit {}'.format(kk) for kk in range(3)],\n bbox_to_anchor=[1.5, .6])\n plt.savefig('img/W1_x{}__per_epoch.png'.format(idx), dpi=100, bbox_inches='tight')\n plt.show()\n```\n\n##### 1D ---------------------------------\n\n\n```python\ndata = [X_valid, y_valid]\n# nr_examples = y_valid.shape[0]\nnr_examples = 3\nfor ii in range(nr_examples):\n xx, yy = data[0][ii, :], data[1][ii]\n yy = np.array([yy])\n plt.figure()\n plot_grid(xx, vmin=bounds[0], vmax=bounds[1], cmap='Greys')\n aa = predict(xx, params, activation_funs)[0]\n y_hat = (aa > decision_boundary).astype(int)\n cost = cost_fun(yy, aa)\n plt.title('Validation Example {}\\n'\n 'y={}, a2={:.5f}, $\\hat{{y}}$={}, cost={:.5f}'.format(ii, yy[0], aa, y_hat, cost))\n plt.savefig('img/predict_valid_{}.png'.format(ii), dpi=100, bbox_inches='tight')\n plt.show()\n```\n\n\n```python\nXX = np.array([2*iaml.data.LETTERMATS['T']-1,\n 2*iaml.data.LETTERMATS['C']-1,\n np.random.randn(9),\n np.random.randn(9),\n np.random.randn(9),\n 10*np.random.randn(9)])\nyy = np.array([0, 1, 0, 1, 0, 1])\nnames = ['No noise T', 'No noise C', 'N(0, 1) sample 1', 'N(0, 1) sample 2', 'N(0, 1) sample 3', 'N(0, 10) sample']\ndata = [XX, yy]\nnr_examples = len(yy)\nfor ii in range(nr_examples):\n xx, yy = data[0][ii, :], data[1][ii]\n yy = np.array([yy])\n plt.figure()\n plot_grid(xx, vmin=bounds[0], vmax=bounds[1], cmap='Greys')\n aa = predict(xx, params, activation_funs)[0]\n y_hat = (aa > decision_boundary).astype(int)\n cost = cost_fun(yy, aa)\n plt.title('{}\\n'\n 'y={}, a2={:.5f}, $\\hat{{y}}$={}, cost={:.5f}'.format(names[ii], yy[0], aa, y_hat, cost))# \n plt.savefig('img/predict_valid_{}.png'.format(names[ii]), dpi=100, bbox_inches='tight')\n plt.show()\n```\n\n##### ------------ Below here is some extra stuff, I didn't ask you for it ------------\n\n#### A quick forward pass test\n\nHere I test a few random parameter initialisations to see if the forward pass is working. **I'm not doing any learning here**, I'm just randomly picking values from a normal distribution for all the weights. The reason I'm doing this is to see if the cost function values look sensible, and to check if my code works!\n\nUltimately, as you'll see, I end up concluding that learning is going to be pretty easy for any initialisation we start on!\n\n\n```python\nactivation_funs = [linear, sigmoid]\ncost_fun = crossentropy_cost\ncosts = []\nparam_list = []\nfor ii in range(50):\n np.random.seed(ii)\n W1 = np.random.randn(3, 9)\n W2 = np.random.randn(1, 3)\n params = [W1, W2]\n a2 = predict(X_train, params, activation_funs)\n cost = cost_fun(y_train, a2)\n costs.append(cost)\n param_list.append(params)\nplt.figure()\nplt.plot(costs, '.')\nplt.xlabel('Random initialisation nr')\nplt.ylabel('Crossentropy cost')\nplt.show()\n```\n\nHmm...that's interesting. We seem to have some random weight initialisations with really low costs! Let's check out the predictions we get using the best performing parameters from our random search of weights...\n\n\n```python\nidx_min = np.argmin(costs)\nparams_min = param_list[idx_min]\n```\n\n\n```python\nfor ii in range(3):\n plt.figure()\n plot_grid(X_train[ii], vmin=bounds[0], vmax=bounds[1], cmap='Greys')\n aa2 = predict(X_train[ii], params_min, activation_funs)\n plt.title('y = {}, a2 = {:.5f}'.format(y_train[ii], aa2[0]))\n plt.show()\n```\n\nHey that looks pretty good! I wonder if it's that good for all the letters. Let's check out the worst cases.\n\n\n```python\na2 = predict(X_train, params_min, activation_funs)\nlosses = (-y_train * np.log(a2)) - ((1-y_train) * np.log(1-a2)) # crossentropy for each data item\nlosses = losses.flatten()\nplt.figure()\nplt.plot(losses.flatten(), '.')\nplt.xlabel('Data item')\nplt.ylabel('Cost for data item')\nplt.title('Cost for each data item using best params')\nplt.show()\n```\n\n\n```python\nloss_order = np.argsort(losses.flatten())[::-1]\nprint('Worst predictions:')\nfor ii in range(3):\n idx = loss_order[ii]\n xx, yy = X_train[idx, :], y_train[idx]\n plt.figure()\n plot_grid(xx, vmin=bounds[0], vmax=bounds[1], cmap='Greys')\n aa2 = predict(xx, params_min, activation_funs)\n plt.title('y={}, a2={:.5f} loss={:.5f}'.format(yy, aa2[0], losses[idx]))\n plt.show()\n```\n\nNot bad! I reckon the accuracy will be 100% since a decision boundary at .5 (as is normal) would result in the correct classification for the very worst cases...let's check just in case!\n\n\n```python\ndecision_boundary = .5\ny_hat = (a2 > decision_boundary).astype(int)\ncorrect_predictions = np.sum(y_train == y_hat)\naccuracy = correct_predictions / len(y_train)\nprint('Training Accuracy = {}'.format(accuracy))\n```\n\nThe predictions look good - losses appear to be small and we have 100% accuracy! Apparently we have found a good parameter setting from a random search...so this problem *should* be easy to optimise! I don't expect you to have done this quick check, but it's a good example for how checking your code can lead to some informative results in itself. \n
                                        \n
                                        \nThe above check turned out to be invaluable for me: my solution contained bugs in the backprop function (my cost function derivative was wrong) and, because I had a good idea about reasonable cost values from this analysis, I was able to better diagnose my issues.                                        \n\n#### Backward pass tests\n\n\n```python\ntest_vals = np.linspace(-10, 10, 100)[:, np.newaxis]\n```\n\n\n```python\nnp.sum([check_grad(linear, linear_derivative, xx) for xx in test_vals])\n```\n\n\n```python\nnp.sum([check_grad(sigmoid, sigmoid_derivative, xx) for xx in test_vals])\n```\n\n\n```python\ntest_vals = np.linspace(0.01, .99, 100)[:, np.newaxis]\nc_1 = lambda x: crossentropy_cost(np.array([1]), x)\nc_0 = lambda x: crossentropy_cost(np.array([0]), x)\nc_deriv1 = lambda x: crossentropy_cost_derivative(np.array([1]), x)\nc_deriv0 = lambda x: crossentropy_cost_derivative(np.array([0]), x)\n```\n\n\n```python\nnp.sum([check_grad(c_1, c_deriv1, xx) for xx in test_vals])\n```\n\n\n```python\nnp.sum([check_grad(c_0, c_deriv0, xx) for xx in test_vals])\n```\n\n### ========== Question 3.2 ==========\n\nDid you need a network this large to do this classification task? Give the values for the parameters of a network with no hidden layers, one output node, and an output activation function of a sigmoid that would get 100% accuracy. This network only has 9 parameters.\n\n*Your answer goes here*\n\nThe 'C' character data should be close to -1 in the centre square, whereas the 'T' data is close to 1. This means that all we need to do is return that value (passed through a sigmoid) to get 100% accuracy! Any weight vector with a positive value in index 4 (the $5^{th}$ entry) and zeros everywhere else will produce the desired result. \n\nIn fact, there don't even need to be zeros everywhere else, so long as the sum is always >.5 for T and <.5 for C.\n\n### ========== Question 3.3 ==========\n\nYou should recognise the model described in question 3.2. What is it?\n\n*Your answer goes here*\n\nIt's a logistic regression!\n\n### ========== Question 3.4 ==========\n\nWhy did I create input data, `X`, that was between [-1, 1] i.e. why wasn't it between [0, 1] like normal?! Would the model specified in question 3.1 above have worked if `X` was in [0, 1]? Explain why or why not.\n\n*Hint: if you're stuck, you can try it out by generating some new data and trying to fit it.*\n\n*Your answer goes here*\n\nNo, it wouldn't have worked. The reason is that, regardless of how negative the learned parameters are, if you input a 0, multiplying it will never make it negative. Therefore, as the calculations are passed forward to the final sigmoid activation, the output activations will always be >= 0.5. The cost would still probably push us in the correct direction but your network would get very frustrated trying to predict outputs of `0` (`T` data).\n\nThis could be solved by including a bias parameter at each layer. Now you can make your hidden layer activation negative despite the input being 0. To simplify things, I chose to change the data instead!!! Below is some code that shows empirically that it doesn't work:\n\n\n```python\nbounds = [0, 1]\nX_, y_, y_labels_ = load_letters(categories=['T', 'C'], \n num_obs=[25, 25],\n bounds=bounds,\n beta_params=[[1, 8], [8, 1]],\n shuffle=True, \n random_state=42)\n\nX_tr, X_val, y_tr, y_val = train_test_split(\n X_, y_, test_size=0.33, random_state=42)\n```\n\n\n```python\nnp.random.seed(random_seed)\nW1 = np.random.randn(3, 9)\nW2 = np.random.randn(1, 3)\nparams2 = [W1, W2]\n\n# Things to store\ncost_per_epoch = dict(train=[], valid=[])\naccuracy_per_epoch = dict(train=[], valid=[])\nparams2_per_epoch = []\n\n# Training loop\nprint('Beginning Training!\\n{}\\n'.format('-'*len('Beginning Training!')))\nprint(' {:^10} | {:^10} | {:^10} | {:^10} | {:^10} '.\\\n format('Epoch', 'Train Cost', 'Valid Cost', 'Train Acc', 'Valid Acc'))\nprint(' {0} {0} {0} {0} {0} '.format(10*'-'))\n\nfor ii in range(nr_epochs):\n # WARNING: \n # If your network is large and/or you are doing many epochs, it's not \n # sensible to save the parameters every epoch - for this toy problem it's ok\n params2_per_epoch.append(copy.deepcopy(params2)) # without deepcopy, python will\n # use a pointer only, resulting\n # in a final parameter list with\n # nr_epochs identical sets of params!\n # Forward pass\n fwd_pass_data = predict(X_tr, params2, activation_funs, return_all=True)\n activations = fwd_pass_data[0]\n\n a_out = dict(train=activations[-1], # this is just to avoid calculating train output activations twice\n valid=predict(X_valid, params2, activation_funs, return_all=False))\n for dataset_name, (XX, yy) in dict(train=(X_tr, y_tr), \n valid=(X_val, y_val)).iteritems():\n aa = a_out[dataset_name]\n accuracy_per_epoch[dataset_name].append(get_accuracy(yy, aa, decision_boundary))\n cost_per_epoch[dataset_name].append(cost_fun(yy, aa))\n \n print(' {:>10d} | {:<10.7f} | {:<10.7f} | {:<10.7f} | {:<10.7f} '.\\\n format(ii, cost_per_epoch['train'][ii], cost_per_epoch['valid'][ii], \n accuracy_per_epoch['train'][ii], accuracy_per_epoch['valid'][ii]))\n \n # Don't do backward pass on the final epoch (just to avoid repeating code) \n if ii == nr_epochs - 1:\n break\n \n # Backward pass\n params2 = backward_pass(y_train, params2, fwd_pass_data, cost_fun_derivative, \n activation_funs_derivatives, learning_rate=learning_rate)\n```\n\n### ========== Question 3.5 [EXTENSION] ==========\n\nCreate a dataset which makes the problem harder. Have a look at the dataset generation code. You can use the arguments to create data with:\n* more letters (make the problem a multiclass classification)\n * You'll need to implement the multiclass version of the sigmoid for the output activation function - [the softmax](https://en.wikipedia.org/wiki/Softmax_function) (and of course it's derivative) \n* increase the noise on the data\n\nSome other things you could implement:\n* include rotated letters in the data\n* make larger data (bigger than 3x3)\n* make the letters non-centred e.g. 5x5 data with 3x3 letters in 1 of 9 different places\n\nYou'll probably need to adapt the code you wrote in 3.1, but you can probably copy and paste most of it. For an additional challenge: introduce [bias parameters](http://neuralnetworksanddeeplearning.com/chap1.html) and create your `X` data in range [0, 1] (i.e. set the bounds argument to [0, 1])...\n\nSome other things to try if you get code happy:\n* Implement stochastic gradient descent updates (updating parameters every training example, as opposed to every epoch) - tip: randomise data order each epoch\n* Implement batch gradient descent updates - tip: randomise data order each epoch\n\n**Requirements**:\n1. Describe the modelling problem and your input data. Plot some examples of the data\n1. Write down the model specification (I should be able to reproduce your model with this description):\n * number of nodes in each layer\n * a description of the parameters to learn (and a total number of parameters)\n * the activation functions used for each layer\n * cost function used\n1. All the outputs asked for in Question 3.1: loss per epoch plot, final parameters, a weight against epoch plot, and example predictions\n\n*Your answer goes here*\n\n\n```python\n# Your code goes here\n```\n\n## Part 4 - Implementation with Sklearn\n\n### ========== Question 4.1 ==========\n\nIf you did Question 3.1, this should be a breeze! Use the same data and perform the same modelling task. This time you can use Sklearn's Neural Network object [MLPClassifier](http://scikit-learn.org/stable/modules/generated/sklearn.neural_network.MLPClassifier.html#sklearn.neural_network.MLPClassifier).\n\n\nBefore you begin, read the [introduction](http://scikit-learn.org/stable/modules/neural_networks_supervised.html) (sections 1.17.1 and 1.17.2 at a minimum, 1.17.5, 1.17.6, 1.17.7 are recommended).\n\n\n```python\n# Your code goes here\nmlp = MLPClassifier(hidden_layer_sizes=(3, ), \n activation='identity',\n solver='sgd',\n alpha=0, \n batch_size=X_train.shape[0],\n learning_rate='constant',\n learning_rate_init=0.3, \n max_iter=100,\n shuffle=False, \n random_state=None,\n tol=0.0001, \n verbose=True, \n warm_start=False, \n momentum=0, \n nesterovs_momentum=False, \n early_stopping=False, \n validation_fraction=0)\nmlp.fit(X_train, y_train)\n```\n\n### ========== Question 4.2 ==========\n\nThe learned parameters are stored in the fitted sklearn [MLPClassifier](http://scikit-learn.org/stable/modules/generated/sklearn.neural_network.MLPClassifier.html#sklearn.neural_network.MLPClassifier) object **as two separate attributes**.\n\n1. Print the parameters learned by your fitted model\n1. Print the total number of parameters learned\n\nLook at the number of parameters described in question 3.1 (you do not need to have done this question 3.1 - just read its description). Below the code:\n\n1. Explain why the number of parameters learned by sklearn is different from the number specified in 3.1?\n\n\n```python\n# Your code goes here\nfinal_Ws = [mat.T for mat in mlp.coefs_] # Just transposing so we can use same code as above\nfinal_bees = mlp.intercepts_\n\nfor ii, W in enumerate(final_Ws):\n plt.figure()\n hinton(W.T)\n plt.title('$W^{{({})}}$'.format(ii + 1))\n plt.tight_layout()\n plt.show()\n\nfor ii, b in enumerate(final_bees):\n plt.figure()\n hinton(b[np.newaxis, :])\n plt.title('$b^{{({})}}$'.format(ii + 1))\n plt.tight_layout()\n plt.show()\n \nW1 = final_Ws[0]\nfor ii in range(3):\n plt.figure()\n hinton(W1[ii, :].reshape(3, 3).T)\n plt.title('$W^{{({})}}_{}$'.format(1, ii))\n plt.savefig('img/hinton_W{}_{}.png'.format(1, ii), dpi=100, bbox_inches='tight')\n plt.tight_layout()\n plt.show()\n```\n\n*Your answer goes here*\n\nThe sklearn object will fit bias parameters by default (it calls them intercepts). We did not fit them in our code in 3.1. You can read more about the bias parameter and how to fit it [here](http://neuralnetworksanddeeplearning.com/chap1.html).\n\n# [Please sir...I want some more](https://www.youtube.com/watch?v=Ex2r86G0sdc)\n\nWell done, you successfully covered the basics of Neural Networks!\n\nIf you enjoyed this lab, you'll love another course @ Edinburgh: [Machine Learning Practical](https://github.com/CSTR-Edinburgh/mlpractical). Check it out.\n\n### Next steps\n\nThe first thing to do, if you haven't already, is do the extension question 3.5. **In particular, you should implement bias parameters in your model code**.\n\nNext, go back to the very top of the notebook where I detail things I will not cover. Pick some words you don't understand (perhaps along with the keyword 'example' or 'introduction') and have fun reading/watching some tutorials about them online. Code up what you have learned; if you can code it up without peeking, you know you have understood it very well indeed. Another good \"starter for 10\" google is \"a review of neural networks for [images|text|music|bat detection|captioning images|generation|...]\".\n\nHere are some things that you might find fun to read:\n* [Visualising networks learning](http://playground.tensorflow.org/#activation=tanh&batchSize=10&dataset=circle®Dataset=reg-plane&learningRate=0.03®ularizationRate=0&noise=5&networkShape=3&seed=0.42978&showTestData=false&discretize=false&percTrainData=50&x=true&y=true&xTimesY=false&xSquared=false&ySquared=false&cosX=false&sinX=false&cosY=false&sinY=false&collectStats=false&problem=classification&initZero=false&hideText=false)\n* [Trying to understand what features are learned by Deep Nets](https://distill.pub/2017/feature-visualization/)\n* [Modelling sound waves](https://deepmind.com/blog/wavenet-generative-model-raw-audio/)\n * ...and using that to [encode instruments](https://magenta.tensorflow.org/nsynth)\n* An [Introduction to LSTMs](http://colah.github.io/posts/2015-08-Understanding-LSTMs/) and their [unreasonable effectiveness](http://karpathy.github.io/2015/05/21/rnn-effectiveness/)\n* How to encode the entire meaning of a word [in a few numbers](http://colah.github.io/posts/2014-07-NLP-RNNs-Representations/)\n* [Convolutions for text data?!](http://www.wildml.com/2015/11/understanding-convolutional-neural-networks-for-nlp/)\n\n### Learning resources\n\nAlso:\n* [there](http://ufldl.stanford.edu/tutorial/supervised/MultiLayerNeuralNetworks/)\n* [are](http://neuralnetworksanddeeplearning.com/chap1.html)\n* [literally](https://www.coursera.org/learn/machine-learning)\n* [so](https://www.coursera.org/learn/neural-networks)\n* [many](http://deeplearning.net/)\n* [learning](http://datasciencemasters.org/)\n* [resources](https://metacademy.org/graphs/concepts/backpropagation)\n* [online!](http://www.deeplearningbook.org/)\n\n(about neural nets etc.)\n\nIn all seriousness, make sure you check out [metacademy](https://metacademy.org/). You can search for a topic and it gives you a list of free resources, an estimated time you need to understand it, and prerequisite topics.\n\n# Attributions\n\nParts of this lab were inspired by D. E. Rumelhart, G. E. Hinton, and R. J. Williams, Parallel distributed processing: Explorations\nin the microstructure of cognition, vol. 1, MIT Press, Cambridge, MA, USA, 1986,\npp. 318\u2013362.\n\n\nThanks also to:\n* [3 Blue 1 Brown](https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw)\n* [Michael Nielsen](http://neuralnetworksanddeeplearning.com)\n* [Christopher Olah](http://colah.github.io/)\n\nfor producing some excellent visualisations and learning resources and providing them free of charge.\n\nAdditionally, many thanks to the developers of open source software, in particular:\n* [Numpy](http://www.numpy.org/)\n* [Scipy](https://www.scipy.org/)\n* [Sklearn](http://scikit-learn.org/stable/)\n* [Matplotlib](https://matplotlib.org/)\n* [Jupyter](http://jupyter.org/)\n* and of course [Python](https://www.python.org/) itself!\n\nyour work is invaluable and appreciated.\n\n# Credits\n\nThis lab was created by [James Owers](https://jamesowers.github.io/) in November 2017 and reviewed by [Patric Fulop](https://www.inf.ed.ac.uk/people/students/Patric_Fulop.html).\n", "meta": {"hexsha": "f3dcbac73b429dd303d13ad8211599145ee764cf", "size": 76525, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "10_Lab_5_Neural_Networks_solution.ipynb", "max_stars_repo_name": "CaesarZhang070497/Iaml", "max_stars_repo_head_hexsha": "cb13d2aa50c37563d50eaf380542578994effd91", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 59, "max_stars_repo_stars_event_min_datetime": "2017-09-18T13:14:48.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-23T13:52:25.000Z", "max_issues_repo_path": "10_Lab_5_Neural_Networks_solution.ipynb", "max_issues_repo_name": "pyian/iaml2017", "max_issues_repo_head_hexsha": "cb13d2aa50c37563d50eaf380542578994effd91", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 5, "max_issues_repo_issues_event_min_datetime": "2017-09-22T14:27:50.000Z", "max_issues_repo_issues_event_max_datetime": "2018-08-23T09:57:59.000Z", "max_forks_repo_path": "10_Lab_5_Neural_Networks_solution.ipynb", "max_forks_repo_name": "pyian/iaml2017", "max_forks_repo_head_hexsha": "cb13d2aa50c37563d50eaf380542578994effd91", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 91, "max_forks_repo_forks_event_min_datetime": "2017-09-18T15:42:06.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-21T16:19:21.000Z", "avg_line_length": 37.8461918892, "max_line_length": 522, "alphanum_fraction": 0.5726102581, "converted": true, "num_tokens": 14060, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573377, "lm_q2_score": 0.6825737279551494, "lm_q1q2_score": 0.4096727960927612}} {"text": "\n\n\n# `GiRaFFE_NRPy`: Main Driver, staggered prescription\n\n## Author: Patrick Nelson\n\n\n\n**Notebook Status:** Validation in progress \n\n**Validation Notes:** This code assembles the various parts needed for GRFFE evolution in order.\n\n### NRPy+ Source Code for this module:\n* [GiRaFFE_NRPy/GiRaFFE_NRPy_Main_Driver_staggered.py](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_Main_Driver_staggered.py)\n\n### Other critical files (in alphabetical order): \n* [GiRaFFE_NRPy/GiRaFFE_NRPy_staggered_Afield_flux.py](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_staggered_Afield_flux.py) [\\[**tutorial**\\]](Tutorial-GiRaFFE_NRPy_staggered-Afield_flux.ipynb) Generates the expressions to find the flux term of the induction equation.\n* [GiRaFFE_NRPy/GiRaFFE_NRPy_staggered_A2B.py](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_staggered_A2B.py) [\\[**tutorial**\\]](Tutorial-GiRaFFE_NRPy_staggered-A2B.ipynb) Generates the driver to compute the magnetic field from the vector potential/\n* [GiRaFFE_NRPy/GiRaFFE_NRPy_BCs.py](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_BCs.py) [\\[**tutorial**\\]](Tutorial-GiRaFFE_NRPy-BCs.ipynb) Generates the code to apply boundary conditions to the vector potential, scalar potential, and three-velocity.\n* [GiRaFFE_NRPy/GiRaFFE_NRPy_C2P_P2C.py](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_C2P_P2C.py) [\\[**tutorial**\\]](Tutorial-GiRaFFE_NRPy-C2P_P2C.ipynb) Generates the conservative-to-primitive and primitive-to-conservative solvers.\n* [GiRaFFE_NRPy/GiRaFFE_NRPy_Metric_Face_Values.py](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_Metric_Face_Values.py) [\\[**tutorial**\\]](Tutorial-GiRaFFE_NRPy-Metric_Face_Values.ipynb) Generates code to interpolate metric gridfunctions to cell faces.\n* [GiRaFFE_NRPy/GiRaFFE_NRPy_PPM.py](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_PPM.py) [\\[**tutorial**\\]](Tutorial-GiRaFFE_NRPy-PPM.ipynb) Genearates code to reconstruct primitive variables on cell faces.\n* [GiRaFFE_NRPy/GiRaFFE_NRPy_Source_Terms.py](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_Source_Terms.py) [\\[**tutorial**\\]](Tutorial-GiRaFFE_NRPy-Source_Terms.ipynb) Genearates code to compute the $\\tilde{S}_i$ source term.\n* [GiRaFFE_NRPy/GiRaFFE_NRPy_staggered_Source_Terms.py](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_staggered_Source_Terms.py) [\\[**tutorial**\\]](Tutorial-GiRaFFE_NRPy_staggered-Source_Terms.ipynb) Genearates code to compute the $A_i$ gauge term and $\\psi^6 \\Phi$ right-hand sides.\n* [GiRaFFE_NRPy/Stilde_flux.py](../../edit/in_progress/GiRaFFE_NRPy/Stilde_flux.py) [\\[**tutorial**\\]](Tutorial-GiRaFFE_NRPy-Stilde_flux.ipynb) Generates the expressions to find the flux term of the Poynting flux evolution equation.\n* [../GRFFE/equations.py](../../edit/GRFFE/equations.py) [\\[**tutorial**\\]](../Tutorial-GRFFE_Equations-Cartesian.ipynb) Generates code necessary to compute the source terms.\n* [../GRHD/equations.py](../../edit/GRHD/equations.py) [\\[**tutorial**\\]](../Tutorial-GRHD_Equations-Cartesian.ipynb) Generates code necessary to compute the source terms.\n\n## Introduction: \nHaving written all the various algorithms that will go into evolving the GRFFE equations forward through time, we are ready to write a start-to-finish module to do so. However, to help keep things more organized, we will first create a dedicated module to assemble the various functions we need to run, in order, to perform the evolution. This will reduce the length of the standalone C code, improving that notebook's readability.\n\nThis notebook does this for the *staggered prescription*.\n\n\n# Table of Contents\n$$\\label{prelim}$$\n\nDuring a given RK substep, we will perform the following steps in this order, based on the order used in the original `GiRaFFE`:\n0. [Step 0](#prelim): Preliminaries\n1. [Step 1](#rhs): Calculate the right-hand sides\n 1. [Step 1.a](#source): Calculate the source terms of $\\partial_t A_i$, $\\partial_t \\tilde{S}_i$, and $\\partial_t [\\sqrt{\\gamma} \\Phi]$ right-hand sides\n 1. [**GiRaFFE_NRPy/GiRaFFE_NRPy_staggered_Source_Terms.py**](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_staggered_Source_Terms.py), [**GiRaFFE_NRPy/GiRaFFE_NRPy_Source_Terms**](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_Source_Terms.py)\n 1. [Step 1.b](#flux): Calculate the Flux terms\n 1. In each direction: \n 1. Interpolate the metric gridfunctions to cell faces\n 1. [**GiRaFFE_NRPy/GiRaFFE_NRPy_Metric_Face_Values.py**](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_Metric_Face_Values.py)\n 1. Reconstruct primitives $\\bar{v}^i$ and $B^i$ on cell faces with the piecewise-parabolic method\n 1. [**GiRaFFE_NRPy/GiRaFFE_NRPy_PPM.py**](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_PPM.py)\n 1. Compute the fluxes of $\\tilde{S}_i$ and $A_i$ and add the appropriate combinations to the evolution equation right-hand sides\n 1. [**GiRaFFE_NRPy/Stilde_flux.py**](../../edit/in_progress/GiRaFFE_NRPy/Stilde_flux.py), [**GiRaFFE_NRPy/GiRaFFE_NRPy_staggered_Afield_flux.py**](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_staggered_Afield_flux.py)\n1. [Step 2](#poststep): Recover the primitive variables and apply boundary conditions (post-step)\n 1. [Step 2.a](#potential_bc): Apply boundary conditions to $A_i$ and $\\sqrt{\\gamma} \\Phi$\n 1. [**GiRaFFE_NRPy/GiRaFFE_NRPy_BCs.py**](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_BCs.py)\n 1. [Step 2.b](#a2b): Compute $B^i$ from $A_i$\n 1. [**GiRaFFE_NRPy/GiRaFFE_NRPy_staggered_A2B.py**](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_staggered_A2B.py)\n 1. [Step 2.c](#c2p): Run the Conservative-to-Primitive solver\n 1. This applies fixes to $\\tilde{S}_i$, then computes $\\bar{v}^i$. A current sheet prescription is then applied to $\\bar{v}^i$, and $\\tilde{S}_i$ is recomputed to be consistent.\n 1. [**GiRaFFE_NRPy/GiRaFFE_NRPy_C2P_P2C.py**](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_C2P_P2C.py)\n 1. [Step 2.d](#velocity_bc): Apply outflow boundary conditions to $\\bar{v}^i$\n 1. [**GiRaFFE_NRPy/GiRaFFE_NRPy_BCs.py**](../../edit/in_progress/GiRaFFE_NRPy/GiRaFFE_NRPy_BCs.py)\n 1. [Step 2.e](#bssn_interp): Workaround to interpolate BSSN instead of ADM\n bssn_interp\n1. [Step 3](#write_out): Write out the C code function\n1. [Step 3](#code_validation): Self-Validation against `GiRaFFE_NRPy_Main_Drive.py`\n1. [Step 5](#latex_pdf_output): Output this notebook to $\\LaTeX$-formatted PDF file\n\n\n\n\n# Step 0: Preliminaries \\[Back to [top](#toc)\\]\n$$\\label{prelim}$$\n\nWe begin by importing the NRPy+ core functionality. We also import the GRHD module.\n\n\n```python\n# Step 0: Add NRPy's directory to the path\n# https://stackoverflow.com/questions/16780014/import-file-from-parent-directory\nimport os,sys\nnrpy_dir_path = os.path.join(\"..\")\nif nrpy_dir_path not in sys.path:\n sys.path.append(nrpy_dir_path)\n\nfrom outputC import outCfunction, lhrh # NRPy+: Core C code output module\nimport finite_difference as fin # NRPy+: Finite difference C code generation module\nimport NRPy_param_funcs as par # NRPy+: Parameter interface\nimport grid as gri # NRPy+: Functions having to do with numerical grids\nimport loop as lp # NRPy+: Generate C code loops\nimport indexedexp as ixp # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support\nimport reference_metric as rfm # NRPy+: Reference metric support\nimport cmdline_helper as cmd # NRPy+: Multi-platform Python command-line interface\n\nthismodule = \"GiRaFFE_NRPy_Main_Driver\"\n\npar.set_parval_from_str(\"finite_difference::FD_CENTDERIVS_ORDER\",2)\n\nout_dir = os.path.join(\"GiRaFFE_standalone_Ccodes\")\ncmd.mkdir(out_dir)\n\nCoordSystem = \"Cartesian\"\n\npar.set_parval_from_str(\"reference_metric::CoordSystem\",CoordSystem)\nrfm.reference_metric() # Create ReU, ReDD needed for rescaling B-L initial data, generating BSSN RHSs, etc.\n\noutCparams = \"outCverbose=False,CSE_sorting=none\"\n```\n\n\n\n# Step 1: Calculate the right-hand sides \\[Back to [top](#toc)\\]\n$$\\label{rhs}$$\n\n\nIn the method of lines using Runge-Kutta methods, each timestep involves several \"RK substeps\" during which we will run the same set of function calls. These can be divided into two groups: one in which the RHSs themselves are calculated, and a second in which boundary conditions are applied and auxiliary variables updated (the post-step). Here, we focus on that first group.\n\nThe gauge terms of our evolution equations consist of two derivative terms: the Lorentz gauge term of $\\partial_t A_k$, which is $\\partial_k (\\alpha \\Phi - \\beta^j A_j)$ and the non-damping, flux-like term of $\\partial_t [\\psi^6 \\Phi]$, which is $\\partial_j (\\alpha\\sqrt{\\gamma}A^j - \\beta^j [\\sqrt{\\gamma} \\Phi])$. We compute these terms first, after we register all the gridfunctions we will need for GRFFE in the staggered prescription.\n\n\n```python\nimport GRHD.equations as GRHD # NRPy+: Generate general relativistic hydrodynamics equations\n\ngammaDD = ixp.register_gridfunctions_for_single_rank2(\"AUXEVOL\",\"gammaDD\",\"sym01\",DIM=3)\nbetaU = ixp.register_gridfunctions_for_single_rank1(\"AUXEVOL\",\"betaU\",DIM=3)\nalpha = gri.register_gridfunctions(\"AUXEVOL\",\"alpha\")\nixp.register_gridfunctions_for_single_rank1(\"EVOL\",\"AD\")\nBU = ixp.register_gridfunctions_for_single_rank1(\"AUXEVOL\",\"BU\")\nixp.register_gridfunctions_for_single_rank1(\"AUXEVOL\",\"BstaggerU\")\nValenciavU = ixp.register_gridfunctions_for_single_rank1(\"AUXEVOL\",\"ValenciavU\")\ngri.register_gridfunctions(\"EVOL\",\"psi6Phi\")\nStildeD = ixp.register_gridfunctions_for_single_rank1(\"EVOL\",\"StildeD\")\n\ngri.register_gridfunctions(\"AUXEVOL\",\"psi6_temp\")\ngri.register_gridfunctions(\"AUXEVOL\",\"psi6center\")\ngri.register_gridfunctions(\"AUXEVOL\",\"cmax_x\")\ngri.register_gridfunctions(\"AUXEVOL\",\"cmin_x\")\ngri.register_gridfunctions(\"AUXEVOL\",\"cmax_y\")\ngri.register_gridfunctions(\"AUXEVOL\",\"cmin_y\")\ngri.register_gridfunctions(\"AUXEVOL\",\"cmax_z\")\ngri.register_gridfunctions(\"AUXEVOL\",\"cmin_z\")\n\nphi = gri.register_gridfunctions(\"AUXEVOL\",\"phi\") # Needed only for ADM-BSSN-ADM workaround\ngammaUUxx,gammaUUyy,gammaUUzz = gri.register_gridfunctions(\"AUXEVOL\",[\"gammaUUxx\",\"gammaUUyy\",\"gammaUUzz\"])\ngamma_faceUUxx,gamma_faceUUyy,gamma_faceUUzz = gri.register_gridfunctions(\"AUXEVOL\",[\"gamma_faceUUxx\",\"gamma_faceUUyy\",\"gamma_faceUUzz\"])\n```\n\n\n\n## Step 1.a: Calculate the source terms of $\\partial_t A_i$, $\\partial_t \\tilde{S}_i$, and $\\partial_t [\\sqrt{\\gamma} \\Phi]$ right-hand sides \\[Back to [top](#toc)\\]\n$$\\label{source}$$\n\nWe will now calculate the terms on the RHS of $A_i$ and $[\\sqrt{\\gamma} \\Phi]$ that involve the divergence and gradient operators. We also compute the other term in the RHS of $[\\sqrt{\\gamma} \\Phi]$, which is a straightforward damping term. Documentation for this can be found in [Tutorial-GiRaFFE_NRPy_staggered-Source_Terms](Tutorial-GiRaFFE_NRPy_staggered-Source_Terms.ipynb).\n\n\n```python\nimport GiRaFFE_NRPy.GiRaFFE_NRPy_staggered_Source_Terms as stgsrc\n\nsubdir = \"RHSs\"\nstgsrc.GiRaFFE_NRPy_Source_Terms(os.path.join(out_dir,subdir))\n```\n\nWe also need to compute the source term of the $\\tilde{S}_i$ evolution equation. This term involves derivatives of the four metric, so we can save some effort here by taking advantage of the interpolations done of the metric gridfunctions to the cell faces, which will allow us to take a finite-difference derivative with the accuracy of a higher order and the computational cost of a lower order. However, it will require some more complicated coding, detailed in [Tutorial-GiRaFFE_NRPy-Source_Terms](Tutorial-GiRaFFE_NRPy-Source_Terms.ipynb) \n\n\n```python\nimport GiRaFFE_NRPy.GiRaFFE_NRPy_Source_Terms as source\n\n# Declare this symbol:\nsqrt4pi = par.Cparameters(\"REAL\",thismodule,\"sqrt4pi\",\"sqrt(4.0*M_PI)\")\n\nsource.write_out_functions_for_StildeD_source_term(os.path.join(out_dir,subdir),outCparams,gammaDD,betaU,alpha,\n ValenciavU,BU,sqrt4pi)\n```\n\n Output C function calculate_StildeD0_source_term() to file GiRaFFE_standalone_Ccodes/RHSs/calculate_StildeD0_source_term.h\n Output C function calculate_StildeD1_source_term() to file GiRaFFE_standalone_Ccodes/RHSs/calculate_StildeD1_source_term.h\n Output C function calculate_StildeD2_source_term() to file GiRaFFE_standalone_Ccodes/RHSs/calculate_StildeD2_source_term.h\n\n\n\n\n## Step 1.b: Calculate the Flux terms \\[Back to [top](#toc)\\]\n$$\\label{flux}$$\n\nNow, we will compute the flux terms of $\\partial_t A_i$ and $\\partial_t \\tilde{S}_i$. To do so, we will first need to interpolate the metric gridfunctions to cell faces and to reconstruct the primitives on the cell faces using the code detailed in [Tutorial-GiRaFFE_NRPy-Metric_Face_Values](Tutorial-GiRaFFE_NRPy-Metric_Face_Values.ipynb) and in [Tutorial-GiRaFFE_NRPy-PPM](Tutorial-GiRaFFE_NRPy-PPM.ipynb).\n\n\n```python\nsubdir = \"FCVAL\"\ncmd.mkdir(os.path.join(out_dir, subdir))\nimport GiRaFFE_NRPy.GiRaFFE_NRPy_Metric_Face_Values as FCVAL\nFCVAL.GiRaFFE_NRPy_FCVAL(os.path.join(out_dir,subdir))\n```\n\n\n```python\nsubdir = \"PPM\"\ncmd.mkdir(os.path.join(out_dir, subdir))\nimport GiRaFFE_NRPy.GiRaFFE_NRPy_PPM as PPM\nPPM.GiRaFFE_NRPy_PPM(os.path.join(out_dir,subdir))\n```\n\nHere, we will write the function to compute the electric field contribution to the induction equation RHS. This is coded with documentation in [Tutorial-GiRaFFE_NRPy_staggered-Afield_flux](Tutorial-GiRaFFE_NRPy_staggered-Afield_flux.ipynb). For the $i^{\\rm th}$ component of the electric field, we will need reconstrutions in the $j^{\\rm th}$ and $k^{\\rm th}$ direction. These will be computed in the driver function, [below](#write_out).\n\n\n```python\nimport GiRaFFE_NRPy.GiRaFFE_NRPy_staggered_Afield_flux as Af\n\n# We will pass values of the gridfunction on the cell faces into the function. This requires us\n# to declare them as C parameters in NRPy+. We will denote this with the _face infix/suffix.\nalpha_face = gri.register_gridfunctions(\"AUXEVOL\",\"alpha_face\")\ngamma_faceDD = ixp.register_gridfunctions_for_single_rank2(\"AUXEVOL\",\"gamma_faceDD\",\"sym01\")\nbeta_faceU = ixp.register_gridfunctions_for_single_rank1(\"AUXEVOL\",\"beta_faceU\")\n\n# We'll need some more gridfunctions, now, to represent the reconstructions of BU and ValenciavU\n# on the right and left faces\nValenciav_rU = ixp.register_gridfunctions_for_single_rank1(\"AUXEVOL\",\"Valenciav_rU\",DIM=3)\nB_rU = ixp.register_gridfunctions_for_single_rank1(\"AUXEVOL\",\"B_rU\",DIM=3)\nValenciav_lU = ixp.register_gridfunctions_for_single_rank1(\"AUXEVOL\",\"Valenciav_lU\",DIM=3)\nB_lU = ixp.register_gridfunctions_for_single_rank1(\"AUXEVOL\",\"B_lU\",DIM=3)\n\nsubdir = \"RHSs\"\nAf.GiRaFFE_NRPy_Afield_flux(os.path.join(out_dir, subdir))\n```\n\nWe must do something similar here, albeit much simpler. For instance, the $x$ component of $\\partial_t \\tilde{S}_i$ will be a finite difference of the flux throught the faces in the $\\pm x$ direction; for further detail, see [Tutorial-GiRaFFE_NRPy-Stilde_flux](Tutorial-GiRaFFE_NRPy-Stilde_flux.ipynb).\n\n\n```python\nixp.register_gridfunctions_for_single_rank1(\"AUXEVOL\",\"Stilde_flux_HLLED\")\n\nimport GiRaFFE_NRPy.Stilde_flux as Sf\nSf.generate_C_code_for_Stilde_flux(os.path.join(out_dir,subdir), True, alpha_face,gamma_faceDD,beta_faceU,\n Valenciav_rU,B_rU,Valenciav_lU,B_lU, sqrt4pi, write_cmax_cmin=True)\n```\n\n Output C function calculate_Stilde_rhsD() to file GiRaFFE_standalone_Ccodes/RHSs/calculate_Stilde_rhsD.h\n\n\n\n\n# Step 2: Recover the primitive variables and apply boundary conditions \\[Back to [top](#toc)\\]\n$$\\label{poststep}$$\n\nWith the RHSs computed, we can now recover the primitive variables, which are the Valencia three-velocity $\\bar{v}^i$ and the magnetic field $B^i$. We can also apply boundary conditions to the vector potential and velocity. By doing this at each RK substep, we can help ensure the accuracy of the following substeps. \n\n\n\n## Step 2.a: Apply boundary conditions to $A_i$ and $\\sqrt{\\gamma} \\Phi$ \\[Back to [top](#toc)\\]\n$$\\label{potential_bc}$$\n\nFirst, we will apply boundary conditions to the vector potential, $A_i$, and the scalar potential $\\sqrt{\\gamma} \\Phi$. The file we generate here contains both functions we need for BCs, as documented in [Tutorial-GiRaFFE_NRPy-BCs](Tutorial-GiRaFFE_NRPy-BCs.ipynb).\n\n\n```python\nsubdir = \"boundary_conditions\"\ncmd.mkdir(os.path.join(out_dir,subdir))\nimport GiRaFFE_NRPy.GiRaFFE_NRPy_BCs as BC\nBC.GiRaFFE_NRPy_BCs(os.path.join(out_dir,subdir))\n```\n\n\n\n## Step 2.b: Compute $B^i$ from $A_i$ \\[Back to [top](#toc)\\]\n$$\\label{a2b}$$\n\nNow, we will calculate the magnetic field as the curl of the vector potential at all points in our domain; we will need these at both cell centers and faces, as detailed in [Tutorial-GiRaFFE_NRPy_staggered-A2B](Tutorial-GiRaFFE_NRPy-A2B.ipynb).\n\n\n```python\nsubdir = \"A2B\"\ncmd.mkdir(os.path.join(out_dir,subdir))\nimport GiRaFFE_NRPy.GiRaFFE_NRPy_staggered_A2B as A2B\nA2B.GiRaFFE_NRPy_A2B(os.path.join(out_dir,subdir))\n```\n\n\n\n## Step 2.c: Run the Conservative-to-Primitive solver \\[Back to [top](#toc)\\]\n$$\\label{c2p}$$\n\nWith these functions, we apply fixes to the Poynting flux, and use that to update the three-velocity. Then, we apply our current sheet prescription to the velocity, and recompute the Poynting flux to agree with the now-fixed velocity. More detail can be found in [Tutorial-GiRaFFE_NRPy-C2P_P2C](Tutorial-GiRaFFE_NRPy-C2P_P2C.ipynb).\n\n\n```python\nimport GiRaFFE_NRPy.GiRaFFE_NRPy_C2P_P2C as C2P_P2C\nC2P_P2C.GiRaFFE_NRPy_C2P(StildeD,BU,gammaDD,betaU,alpha)\n\nvalues_to_print = [\n lhrh(lhs=gri.gfaccess(\"in_gfs\",\"StildeD0\"),rhs=C2P_P2C.outStildeD[0]),\n lhrh(lhs=gri.gfaccess(\"in_gfs\",\"StildeD1\"),rhs=C2P_P2C.outStildeD[1]),\n lhrh(lhs=gri.gfaccess(\"in_gfs\",\"StildeD2\"),rhs=C2P_P2C.outStildeD[2]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"ValenciavU0\"),rhs=C2P_P2C.ValenciavU[0]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"ValenciavU1\"),rhs=C2P_P2C.ValenciavU[1]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"ValenciavU2\"),rhs=C2P_P2C.ValenciavU[2])\n ]\n\nsubdir = \"C2P\"\ncmd.mkdir(os.path.join(out_dir,subdir))\ndesc = \"Apply fixes to \\tilde{S}_i and recompute the velocity to match with current sheet prescription.\"\nname = \"GiRaFFE_NRPy_cons_to_prims\"\noutCfunction(\n outfile = os.path.join(out_dir,subdir,name+\".h\"), desc=desc, name=name,\n params =\"const paramstruct *params,REAL *xx[3],REAL *auxevol_gfs,REAL *in_gfs\",\n body = fin.FD_outputC(\"returnstring\",values_to_print,params=outCparams),\n loopopts =\"AllPoints,Read_xxs\",\n rel_path_to_Cparams=os.path.join(\"../\"))\n```\n\n Output C function GiRaFFE_NRPy_cons_to_prims() to file GiRaFFE_standalone_Ccodes/C2P/GiRaFFE_NRPy_cons_to_prims.h\n\n\n\n```python\n# TINYDOUBLE = par.Cparameters(\"REAL\",thismodule,\"TINYDOUBLE\",1e-100)\n\nC2P_P2C.GiRaFFE_NRPy_P2C(gammaDD,betaU,alpha, ValenciavU,BU, sqrt4pi)\n\nvalues_to_print = [\n lhrh(lhs=gri.gfaccess(\"in_gfs\",\"StildeD0\"),rhs=C2P_P2C.StildeD[0]),\n lhrh(lhs=gri.gfaccess(\"in_gfs\",\"StildeD1\"),rhs=C2P_P2C.StildeD[1]),\n lhrh(lhs=gri.gfaccess(\"in_gfs\",\"StildeD2\"),rhs=C2P_P2C.StildeD[2]),\n ]\n\ndesc = \"Recompute StildeD after current sheet fix to Valencia 3-velocity to ensure consistency between conservative & primitive variables.\"\nname = \"GiRaFFE_NRPy_prims_to_cons\"\noutCfunction(\n outfile = os.path.join(out_dir,subdir,name+\".h\"), desc=desc, name=name,\n params =\"const paramstruct *params,REAL *auxevol_gfs,REAL *in_gfs\",\n body = fin.FD_outputC(\"returnstring\",values_to_print,params=outCparams),\n loopopts =\"AllPoints\",\n rel_path_to_Cparams=os.path.join(\"../\"))\n```\n\n Output C function GiRaFFE_NRPy_prims_to_cons() to file GiRaFFE_standalone_Ccodes/C2P/GiRaFFE_NRPy_prims_to_cons.h\n\n\n\n\n## Step 2.d: Apply outflow boundary conditions to $\\bar{v}^i$ \\[Back to [top](#toc)\\]\n$$\\label{velocity_bc}$$\n\nNow, we can apply outflow boundary conditions to the Valencia three-velocity. This specific type of boundary condition helps avoid numerical error \"flowing\" into our grid. \n\nThis function has already been generated [above](#potential_bc).\n\n\n\n## Step 2.e: Workaround to interpolate BSSN instead of ADM $\\bar{v}^i$ \\[Back to [top](#toc)\\]\n$$\\label{bssn_interp}$$\n\nThe original `GiRaFFE` converted its metric to BSSN, interpolated that to metric faces, and then converted back to ADM; we'll have to do the same in order to verify round-off level agreement.\n\n\n```python\nimport sympy as sp # SymPy: The Python computer algebra package upon which NRPy+ depends\n# First calculate the conformal factor psi^4 = detgamma^(1/3)\n_gammaUU, gammaDET = ixp.symm_matrix_inverter3x3(gammaDD) # _gammaUU unused.\npsi4 = sp.cbrt(gammaDET)\nphi_expression = sp.Rational(1,4)*sp.log(psi4)\n# Rescale gammaDD: gammabarDD = gammaDD/psi4\ngammabarDD = ixp.zerorank2(DIM=3)\nfor i in range(3):\n for j in range(3):\n gammabarDD[i][j] = gammaDD[i][j]/psi4\ngammabarUUxx = gammaUUxx*psi4\ngammabarUUyy = gammaUUyy*psi4\ngammabarUUzz = gammaUUzz*psi4\n# Generate a kernel to convert to BSSN:\n# We'll convert the metric in place to ensure compatibility with our metric face interpolator\nvalues_to_print = [\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"gammaDD00\"),rhs=gammabarDD[0][0]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"gammaDD01\"),rhs=gammabarDD[0][1]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"gammaDD02\"),rhs=gammabarDD[0][2]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"gammaDD11\"),rhs=gammabarDD[1][1]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"gammaDD12\"),rhs=gammabarDD[1][2]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"gammaDD22\"),rhs=gammabarDD[2][2]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"phi\"),rhs=phi_expression),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"gammaUUxx\"),rhs=gammabarUUxx),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"gammaUUyy\"),rhs=gammabarUUyy),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"gammaUUzz\"),rhs=gammabarUUzz),\n ]\n\ndesc = \"Convert ADM metric to BSSN\"\nname = \"Workaround_ADM_to_BSSN\"\noutCfunction(\n outfile = os.path.join(out_dir,name+\".h\"), desc=desc, name=name,\n params =\"const paramstruct *params,REAL *auxevol_gfs\",\n body = fin.FD_outputC(\"returnstring\",values_to_print,params=outCparams),\n loopopts =\"AllPoints\",\n rel_path_to_Cparams=os.path.join(\"./\"))\n\nrescaled_gammaDD = ixp.zerorank2(DIM=3)\nfor i in range(3):\n for j in range(3):\n # Here, gammaDD actually represents gammabarDD, but recall that we converted in place.\n rescaled_gammaDD[i][j] = gammaDD[i][j]*sp.exp(4*phi)\nrescaled_gammaUUxx = gammaUUxx/sp.exp(4*phi)\nrescaled_gammaUUyy = gammaUUyy/sp.exp(4*phi)\nrescaled_gammaUUzz = gammaUUzz/sp.exp(4*phi)\n# We'll convert the metric in place to ensure compatibility with our metric face interpolator\nvalues_to_print = [\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"gammaDD00\"),rhs=rescaled_gammaDD[0][0]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"gammaDD01\"),rhs=rescaled_gammaDD[0][1]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"gammaDD02\"),rhs=rescaled_gammaDD[0][2]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"gammaDD11\"),rhs=rescaled_gammaDD[1][1]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"gammaDD12\"),rhs=rescaled_gammaDD[1][2]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"gammaDD22\"),rhs=rescaled_gammaDD[2][2]),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"gammaUUxx\"),rhs=rescaled_gammaUUxx),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"gammaUUyy\"),rhs=rescaled_gammaUUyy),\n lhrh(lhs=gri.gfaccess(\"auxevol_gfs\",\"gammaUUzz\"),rhs=rescaled_gammaUUzz)\n ]\n\nC_code_kernel = fin.FD_outputC(\"returnstring\",values_to_print,params=outCparams)\\\n\nC_face_kernel = C_code_kernel.replace(\"GAMMA\",\"GAMMA_FACE\").replace(\"PHIGF\",\"PSI6_TEMPGF\")\n\ndesc = \"Convert BSSN metric to ADM\"\nname = \"Workaround_BSSN_to_ADM\"\nCcode_function = outCfunction(\n outfile = \"returnstring\", desc=desc, name=name,\n params =\"const paramstruct *params,REAL *auxevol_gfs\",\n body = C_code_kernel+\"\\n\"+C_face_kernel,\nloopopts =\"InteriorPoints\",\nrel_path_to_Cparams=os.path.join(\"./\")).replace(\"NGHOSTS+Nxx0\",\"NGHOSTS+Nxx0+1\").replace(\"NGHOSTS+Nxx1\",\"NGHOSTS+Nxx1+1\").replace(\"NGHOSTS+Nxx2\",\"NGHOSTS+Nxx2+1\")\nwith open(os.path.join(out_dir,name+\".h\"),\"w\") as file:\n file.write(Ccode_function)\n```\n\n Output C function Workaround_ADM_to_BSSN() to file GiRaFFE_standalone_Ccodes/Workaround_ADM_to_BSSN.h\n\n\n\n\n# Step 3: Write out the C code function \\[Back to [top](#toc)\\]\n$$\\label{write_out}$$\n\nNow, we have generated all the functions we will need for the `GiRaFFE` evolution. So, we will now assemble our evolution driver. This file will first `#include` all of the files we just generated for easy access. Then, we will write a function that calls these functions in the correct order, iterating over the flux directions as necessary. \n\n\n```python\n%%writefile $out_dir/GiRaFFE_NRPy_Main_Driver.h\n// Structure to track ghostzones for PPM:\ntypedef struct __gf_and_gz_struct__ {\n REAL *gf;\n int gz_lo[4],gz_hi[4];\n} gf_and_gz_struct;\n#define WORKAROUND_ENABLED\n\n// Include ALL functions needed for evolution\n#include \"PPM/reconstruct_set_of_prims_PPM_GRFFE_NRPy.c\"\n#include \"FCVAL/interpolate_metric_gfs_to_cell_faces.h\"\n#include \"RHSs/calculate_StildeD0_source_term.h\"\n#include \"RHSs/calculate_StildeD1_source_term.h\"\n#include \"RHSs/calculate_StildeD2_source_term.h\"\n#include \"RHSs/Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.h\"\n#include \"RHSs/A_i_rhs_no_gauge_terms.h\"\n#include \"A2B/compute_B_and_Bstagger_from_A.h\"\n#include \"RHSs/calculate_Stilde_flux_D0.h\"\n#include \"RHSs/calculate_Stilde_flux_D1.h\"\n#include \"RHSs/calculate_Stilde_flux_D2.h\"\n#include \"RHSs/calculate_Stilde_rhsD.h\"\n#include \"boundary_conditions/GiRaFFE_boundary_conditions.h\"\n#include \"C2P/GiRaFFE_NRPy_cons_to_prims.h\"\n#include \"C2P/GiRaFFE_NRPy_prims_to_cons.h\"\n\nvoid workaround_Valencia_to_Drift_velocity(const paramstruct *params, REAL *vU0, const REAL *alpha, const REAL *betaU0, const REAL flux_dirn) {\n#include \"set_Cparameters.h\"\n // Converts Valencia 3-velocities to Drift 3-velocities for testing. The variable argument\n // vu0 is any Valencia 3-velocity component or reconstruction thereof.\n#pragma omp parallel for\n for (int i2 = 2*(flux_dirn==3);i2 < Nxx_plus_2NGHOSTS2-1*(flux_dirn==3);i2++) for (int i1 = 2*(flux_dirn==2);i1 < Nxx_plus_2NGHOSTS1-1*(flux_dirn==2);i1++) for (int i0 = 2*(flux_dirn==1);i0 < Nxx_plus_2NGHOSTS0-1*(flux_dirn==1);i0++) {\n int ii = IDX3S(i0,i1,i2);\n // Here, we modify the velocity in place.\n vU0[ii] = alpha[ii]*vU0[ii]-betaU0[ii];\n }\n}\n\nvoid workaround_Drift_to_Valencia_velocity(const paramstruct *params, REAL *vU0, const REAL *alpha, const REAL *betaU0, const REAL flux_dirn) {\n#include \"set_Cparameters.h\"\n // Converts Drift 3-velocities to Valencia 3-velocities for testing. The variable argument\n // vu0 is any drift (i.e. IllinoisGRMHD's definition) 3-velocity component or reconstruction thereof.\n#pragma omp parallel for\n for (int i2 = 2*(flux_dirn==3);i2 < Nxx_plus_2NGHOSTS2-1*(flux_dirn==3);i2++) for (int i1 = 2*(flux_dirn==2);i1 < Nxx_plus_2NGHOSTS1-1*(flux_dirn==2);i1++) for (int i0 = 2*(flux_dirn==1);i0 < Nxx_plus_2NGHOSTS0-1*(flux_dirn==1);i0++) {\n int ii = IDX3S(i0,i1,i2);\n // Here, we modify the velocity in place.\n vU0[ii] = (vU0[ii]+betaU0[ii])/alpha[ii];\n }\n}\n\nvoid GiRaFFE_NRPy_RHSs(const paramstruct *restrict params,REAL *restrict auxevol_gfs,REAL *restrict in_gfs,REAL *restrict rhs_gfs) {\n#include \"set_Cparameters.h\"\n // First thing's first: initialize the RHSs to zero!\n#pragma omp parallel for\n for(int ii=0;iiMAXNUMINTERP) {CCTK_VError(VERR_DEF_PARAMS,\"Error: Didn't allocate enough space for interp_vars[].\"); }\n // We are FINISHED with v{x,y,z}{r,l} and P{r,l} so we use these 8 gridfunctions' worth of space as temp storage.\n Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs(params,interp_vars,\n in_gfs+Nxxp2NG012*PSI6PHIGF,\n auxevol_gfs+Nxxp2NG012*VALENCIAV_RU0GF, // WARNING:\n auxevol_gfs+Nxxp2NG012*VALENCIAV_RU1GF, // ALL VARIABLES\n auxevol_gfs+Nxxp2NG012*VALENCIAV_RU2GF, // ON THESE LINES\n auxevol_gfs+Nxxp2NG012*VALENCIAV_LU0GF, // ARE OVERWRITTEN\n auxevol_gfs+Nxxp2NG012*VALENCIAV_LU1GF, // FOR TEMP STORAGE\n auxevol_gfs+Nxxp2NG012*VALENCIAV_LU2GF, // .\n auxevol_gfs+Nxxp2NG012*VALENCIAV_RRU0GF, // .\n auxevol_gfs+Nxxp2NG012*VALENCIAV_RLU0GF, // .\n rhs_gfs+Nxxp2NG012*PSI6PHIGF,\n rhs_gfs+Nxxp2NG012*AD0GF,\n rhs_gfs+Nxxp2NG012*AD1GF,\n rhs_gfs+Nxxp2NG012*AD2GF);\n/*#pragma omp parallel for\n for(int k=0;k\n\n# Step 4: Self-Validation against `GiRaFFE_NRPy_Main_Drive.py` \\[Back to [top](#toc)\\]\n$$\\label{code_validation}$$\n\nTo validate the code in this tutorial we check for agreement between the files\n\n1. that were generated in this tutorial and\n1. those that are generated in the module `GiRaFFE_NRPy_Main_Driver.py`\n\n\n\n```python\ngri.glb_gridfcs_list = []\n# Define the directory that we wish to validate against:\nvaldir = os.path.join(\"GiRaFFE_validation_Ccodes\")\ncmd.mkdir(valdir)\n\nimport GiRaFFE_NRPy.GiRaFFE_NRPy_Main_Driver_staggered as md\nmd.GiRaFFE_NRPy_Main_Driver_generate_all(valdir)\n\n\n```\n\n Output C function calculate_StildeD0_source_term() to file GiRaFFE_validation_Ccodes/RHSs/calculate_StildeD0_source_term.h\n Output C function calculate_StildeD1_source_term() to file GiRaFFE_validation_Ccodes/RHSs/calculate_StildeD1_source_term.h\n Output C function calculate_StildeD2_source_term() to file GiRaFFE_validation_Ccodes/RHSs/calculate_StildeD2_source_term.h\n Output C function calculate_Stilde_rhsD() to file GiRaFFE_validation_Ccodes/RHSs/calculate_Stilde_rhsD.h\n Output C function GiRaFFE_NRPy_cons_to_prims() to file GiRaFFE_validation_Ccodes/C2P/GiRaFFE_NRPy_cons_to_prims.h\n Output C function GiRaFFE_NRPy_prims_to_cons() to file GiRaFFE_validation_Ccodes/C2P/GiRaFFE_NRPy_prims_to_cons.h\n\n\nWith both sets of codes generated, we can now compare them against each other.\n\n\n```python\nimport difflib\nimport sys\n\nprint(\"Printing difference between original C code and this code...\")\n# Open the files to compare\nfiles = [\"GiRaFFE_NRPy_Main_Driver.h\",\n \"RHSs/Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.h\",\n \"PPM/reconstruct_set_of_prims_PPM_GRFFE_NRPy.c\",\n \"PPM/loop_defines_reconstruction_NRPy.h\",\n \"FCVAL/interpolate_metric_gfs_to_cell_faces.h\",\n \"RHSs/calculate_StildeD0_source_term.h\",\n \"RHSs/calculate_StildeD1_source_term.h\",\n \"RHSs/calculate_StildeD2_source_term.h\",\n \"RHSs/calculate_Stilde_flux_D0.h\",\n \"RHSs/calculate_Stilde_flux_D1.h\",\n \"RHSs/calculate_Stilde_flux_D2.h\",\n \"A2B/compute_B_and_Bstagger_from_A.h\",\n \"boundary_conditions/GiRaFFE_boundary_conditions.h\",\n \"A2B/compute_B_and_Bstagger_from_A.h\",\n \"C2P/GiRaFFE_NRPy_cons_to_prims.h\",\n \"C2P/GiRaFFE_NRPy_prims_to_cons.h\"]\n\nfor file in files:\n print(\"Checking file \" + file)\n with open(os.path.join(valdir,file)) as file1, open(os.path.join(out_dir,file)) as file2:\n # Read the lines of each file\n file1_lines = file1.readlines()\n file2_lines = file2.readlines()\n num_diffs = 0\n for line in difflib.unified_diff(file1_lines, file2_lines, fromfile=os.path.join(valdir,file), tofile=os.path.join(out_dir,file)):\n sys.stdout.writelines(line)\n num_diffs = num_diffs + 1\n if num_diffs == 0:\n print(\"No difference. TEST PASSED!\")\n else:\n print(\"ERROR: Disagreement found with .py file. See differences above.\")\n sys.exit(1)\n```\n\n Printing difference between original C code and this code...\n Checking file GiRaFFE_NRPy_Main_Driver.h\n No difference. TEST PASSED!\n Checking file RHSs/Lorenz_psi6phi_rhs__add_gauge_terms_to_A_i_rhs.h\n No difference. TEST PASSED!\n Checking file PPM/reconstruct_set_of_prims_PPM_GRFFE_NRPy.c\n No difference. TEST PASSED!\n Checking file PPM/loop_defines_reconstruction_NRPy.h\n No difference. TEST PASSED!\n Checking file FCVAL/interpolate_metric_gfs_to_cell_faces.h\n No difference. TEST PASSED!\n Checking file RHSs/calculate_StildeD0_source_term.h\n No difference. TEST PASSED!\n Checking file RHSs/calculate_StildeD1_source_term.h\n No difference. TEST PASSED!\n Checking file RHSs/calculate_StildeD2_source_term.h\n No difference. TEST PASSED!\n Checking file RHSs/calculate_Stilde_flux_D0.h\n No difference. TEST PASSED!\n Checking file RHSs/calculate_Stilde_flux_D1.h\n No difference. TEST PASSED!\n Checking file RHSs/calculate_Stilde_flux_D2.h\n No difference. TEST PASSED!\n Checking file A2B/compute_B_and_Bstagger_from_A.h\n No difference. TEST PASSED!\n Checking file boundary_conditions/GiRaFFE_boundary_conditions.h\n No difference. TEST PASSED!\n Checking file A2B/compute_B_and_Bstagger_from_A.h\n No difference. TEST PASSED!\n Checking file C2P/GiRaFFE_NRPy_cons_to_prims.h\n No difference. TEST PASSED!\n Checking file C2P/GiRaFFE_NRPy_prims_to_cons.h\n No difference. TEST PASSED!\n\n\n\n\n# Step 4: Output this notebook to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n$$\\label{latex_pdf_output}$$\n\nThe following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n[Tutorial-GiRaFFE_NRPy_Main_Driver](TTutorial-GiRaFFE_NRPy_Main_Driver.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means.)\n\n\n```python\nimport cmdline_helper as cmd # NRPy+: Multi-platform Python command-line interface\ncmd.output_Jupyter_notebook_to_LaTeXed_PDF(\"Tutorial-GiRaFFE_NRPy_Main_Driver\")\n```\n\n Created Tutorial-GiRaFFE_NRPy_Main_Driver.tex, and compiled LaTeX file to\n PDF file Tutorial-GiRaFFE_NRPy_Main_Driver.pdf\n\n", "meta": {"hexsha": "ca2e652b6b0a36014e0d002cb71d7448aa3a7560", "size": 81451, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "in_progress/Tutorial-GiRaFFE_NRPy_Main_Driver_staggered.ipynb", "max_stars_repo_name": "stevenrbrandt/nrpytutorial", "max_stars_repo_head_hexsha": "219af363f810cc46ea8955a9d28cf075f2252582", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": 66, "max_stars_repo_stars_event_min_datetime": "2018-06-26T22:18:09.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-09T21:12:33.000Z", "max_issues_repo_path": "in_progress/Tutorial-GiRaFFE_NRPy_Main_Driver_staggered.ipynb", "max_issues_repo_name": "stevenrbrandt/nrpytutorial", "max_issues_repo_head_hexsha": "219af363f810cc46ea8955a9d28cf075f2252582", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": 14, "max_issues_repo_issues_event_min_datetime": "2020-02-13T16:09:29.000Z", "max_issues_repo_issues_event_max_datetime": "2021-11-12T14:59:59.000Z", "max_forks_repo_path": "in_progress/Tutorial-GiRaFFE_NRPy_Main_Driver_staggered.ipynb", "max_forks_repo_name": "stevenrbrandt/nrpytutorial", "max_forks_repo_head_hexsha": "219af363f810cc46ea8955a9d28cf075f2252582", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": 30, "max_forks_repo_forks_event_min_datetime": "2019-01-09T09:57:51.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-08T18:45:08.000Z", "avg_line_length": 59.4099197666, "max_line_length": 551, "alphanum_fraction": 0.6236387521, "converted": true, "num_tokens": 22241, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8128673133042217, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.4096088549953926}} {"text": "```python\n# Erasmus+ ICCT project (2018-1-SI01-KA203-047081)\n\n# Toggle cell visibility\n\nfrom IPython.display import HTML\ntag = HTML('''\nPromijeni vidljivost ovdje.''')\ndisplay(tag)\n\n# Hide the code completely\n\n# from IPython.display import HTML\n# tag = HTML('''''')\n# display(tag)\n```\n\n\n\nPromijeni vidljivost ovdje.\n\n\n\n```python\n%matplotlib notebook\nimport scipy.signal as signal\nimport matplotlib.pyplot as plt\nfrom ipywidgets import widgets\nfrom ipywidgets import interact\nimport numpy as np\nimport sympy as sym\n```\n\n## PID regulator - vremenski odziv\n\nProporcionalno-integracijsko-derivacijski (PID) algoritam upravljanja daleko je najpoznatiji i naj\u010de\u0161\u0107e kori\u0161teni algoritam automatskog upravljanja. Njegova prijenosna funkcija je\n\n\\begin{equation}\n P(s)=K_p \\cdot \\left( 1 + \\frac{1}{T_i s} + T_d s \\right).\n\\end{equation}\n\nFunkcija predstavlja zbroj proporcionalnog, integracijskog i derivacijskog kanala. Ne moraju svi nu\u017eno biti prisutni, pa se koriste i algoritmi upravljanja PI ili PD. U ovom primjeru prikazuje se vremenski odziv P, PI, PD ili PID regulatora za ulazne signale iz skupa: step-funkcija, impuls, rampa i sinus.\n\n---\n\n### Kako koristiti ovaj interaktivni primjer?\n1. Izaberite izme\u0111u *jedini\u010dna step funkcija*, *jedini\u010dna impulsna funkcija*, *rampa funkcija* i *funkcija sinus* za odabir ulaznog signala.\n2. Kliknite na gumb *P*, *PI*, *PD* ili *PID* za odabir izme\u0111u proporcionalnog, proporcionalno-integracijskog, proporcionalno-derivacijskog ili proporcionalno-integracijsko-derivacijskog tipa algoritma upravljanja.\n3. Pomi\u010dite kliza\u010de da biste promijenili vrijednosti proporcionalnog ($K_p$), integracijskog ($T_i$) i derivacijskog ($T_d$) koeficijenta PID regulacije.\n4. Pomi\u010dite kliza\u010d $t_{max}$ za promjenu maksimalne vrijednosti vremena na osi x.\n\n\n```python\na = 0.1\n\n# make figure\nfig = plt.figure(figsize=(9.8, 5),num='PID regulator')\n# add axes\nax = fig.add_subplot(111)\nax.grid(which='both', axis='both', color='lightgray')\nax.set_title('Vremenski odziv')\n# plot step function and responses (initalisation)\ninput_plot, = ax.plot([],[],'C0', linewidth=1,label='ulaz')\nresponse_plot, = ax.plot([],[], 'C1', linewidth=2,label='izlaz')\nax.axhline(linewidth=.5, color='k')\nax.axvline(linewidth=.5, color='k')\nax.legend()\n\nax.set_xlabel('$t$ [s]')\nax.set_ylabel('ulaz, izlaz')\nplt.show()\n\nP, I, D, s = sym.symbols('P, I, D, s')\n\ninput_type = 'jedini\u010dna step funkcija' #input function\nTime_span = 10 # max time on x-axis plot\n\n#initialize global variables\nKP = 1.\nTI = 1.\nTD = 1.\nnum = []\nden = []\n\ndef update_plot():\n global num, den, input_type, Time_span\n num_temp = [float(i.subs(P,KP).subs(I,TI).subs(D,TD)) for i in num]\n den_temp = [float(i.subs(P,KP).subs(I,TI).subs(D,TD)) for i in den]\n \n system = signal.TransferFunction(num_temp, den_temp)\n \n #time, response = signal.step(system) #only for setting time borders (for nicer plot. could also calculate dominant frequency)\n #time = np.linspace(0,time[-1],1000)\n time = np.linspace(0, Time_span, 600)\n \n if input_type == 'jedini\u010dna step funkcija':\n u = np.ones_like(time)\n u = np.concatenate((np.array([0]),u))\n time, response = signal.step(system, T=time)\n time = np.concatenate((np.array([0]), time))\n response = np.concatenate((np.array([0]), response))\n elif input_type == 'jedini\u010dna impulsna funkcija':\n u = np.zeros_like(time)\n u = np.concatenate((np.array([10]), u))\n time, response = signal.impulse(system, T=time)\n time = np.concatenate((np.array([0]), time))\n response = np.concatenate((np.array([0]), response))\n elif input_type == 'funkcija sinus':\n u = np.sin(time*2*np.pi)\n time, response, _ = signal.lsim(system, U=u, T=time)\n elif input_type == 'rampa funkcija':\n u = time\n time, response, _ = signal.lsim(system, U=u, T=time)\n else:\n raise Exception(\"Gre\u0161ka u programu. Ponovno pokrenite simulaciju.\")\n \n response_plot.set_data(time, response)\n input_plot.set_data(time, u)\n ax.set_ylim([min([np.min(u), min(response),-.1]),min(100,max([max(response)*1.05, 1, 1.05*np.max(u[1:])]))])\n ax.set_xlim([-0.1,max(time)])\n plt.show()\n \n\ndef transfer_func(controller_type):\n global num, den\n proportional = P\n integral = P/(I*s)\n differential = P*D*s/(a*D*s+1)\n if controller_type =='P':\n controller_func = proportional\n Kp_widget.disabled=False\n Ti_widget.disabled=True\n Td_widget.disabled=True\n elif controller_type =='PI':\n controller_func = proportional+integral\n Kp_widget.disabled=False\n Ti_widget.disabled=False\n Td_widget.disabled=True\n elif controller_type == 'PD':\n controller_func = proportional+differential\n Kp_widget.disabled=False\n Ti_widget.disabled=True\n Td_widget.disabled=False\n else:\n controller_func = proportional+integral+differential\n Kp_widget.disabled=False\n Ti_widget.disabled=False\n Td_widget.disabled=False\n system_func = controller_func\n \n num = [sym.fraction(system_func.factor())[0].expand().coeff(s, i) for i in reversed(range(1+sym.degree(sym.fraction(system_func.factor())[0], gen=s)))]\n den = [sym.fraction(system_func.factor())[1].expand().coeff(s, i) for i in reversed(range(1+sym.degree(sym.fraction(system_func.factor())[1], gen=s)))]\n update_plot()\n \ndef func(Kp, Ti, Td, time_span):\n global KP, TI, TD, Time_span\n KP = Kp\n TI = Ti\n TD = Td\n Time_span = time_span\n update_plot()\n \nstyle = {'description_width': 'initial'}\n\ndef buttons_controller_clicked(event):\n controller = buttons_controller.options[buttons_controller.index]\n transfer_func(controller)\nbuttons_controller = widgets.ToggleButtons(\n options=['P', 'PI', 'PD', 'PID'],\n description='Odaberite tip algoritma upravljanja:',\n disabled=False,\n style=style)\nbuttons_controller.observe(buttons_controller_clicked)\n\ndef buttons_input_clicked(event):\n global input_type\n input_type = buttons_input.options[buttons_input.index]\n update_plot()\nbuttons_input = widgets.ToggleButtons(\n options=['jedini\u010dna step funkcija','jedini\u010dna impulsna funkcija', 'rampa funkcija', 'funkcija sinus'],\n description='Odaberite ulazni signal:',\n disabled=False,\n style=style)\nbuttons_input.observe(buttons_input_clicked)\n\n\nKp_widget = widgets.IntSlider(value=20,min=1,max=100,step=1,description=r'\\(K_p \\)',\n disabled=False,continuous_update=True,orientation='horizontal',readout=True,readout_format='.1d')\nTi_widget = widgets.FloatSlider(value=.1,min=0.001,max=3.,step=0.001,description=r'\\(T_{i} \\)',\n disabled=False,continuous_update=True,orientation='horizontal',readout=True,readout_format='.3f')\nTd_widget = widgets.FloatSlider(value=.1,min=0.001,max=3.,step=0.001,description=r'\\(T_{d} \\)',\n disabled=False,continuous_update=True,orientation='horizontal',readout=True,readout_format='.3f')\n\ntime_span_widget = widgets.FloatSlider(value=10.,min=.5,max=50.,step=0.1,description=r'\\(t_{max} \\)',\n disabled=False,continuous_update=True,orientation='horizontal',readout=True,readout_format='.1f')\n\ntransfer_func('P')\n\ndisplay(buttons_input)\ndisplay(buttons_controller)\n\ninteract(func, Kp=Kp_widget, Ti=Ti_widget, Td=Td_widget, time_span=time_span_widget);\n```\n\n\n \n\n\n\n\n\n\n\n ToggleButtons(description='Odaberite ulazni signal:', options=('jedini\u010dna step funkcija', 'jedini\u010dna impulsna \u2026\n\n\n\n ToggleButtons(description='Odaberite tip algoritma upravljanja:', options=('P', 'PI', 'PD', 'PID'), style=Togg\u2026\n\n\n\n interactive(children=(IntSlider(value=20, description='\\\\(K_p \\\\)', min=1, readout_format='.1d'), FloatSlider(\u2026\n\n", "meta": {"hexsha": "fbe1be2ee2a349078c128ba8f22dbd9faf63913e", "size": 106369, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "ICCT_hr/examples/02/TD-15-PID_regulator_vremenski_odziv.ipynb", "max_stars_repo_name": 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"max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2021-05-24T11:40:09.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-29T16:36:18.000Z", "avg_line_length": 94.9723214286, "max_line_length": 59267, "alphanum_fraction": 0.7581720238, "converted": true, "num_tokens": 2276, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804478040617, "lm_q2_score": 0.6619228825191872, "lm_q1q2_score": 0.40958493765697795}} {"text": "```python\n%matplotlib inline\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom sympy import symbols, lambdify\nfrom sympy.plotting import plot\n\nn = np.arange(0, 100, 1)\nSPLIT = False\nSHOW_TEXT = False\nNAMES = \"PROV\", \"PROV-Dictionary\", \"Versioned-PROV\"\n```\n\n# Comparison\n\n## List Definition\n\n\n```python\ndef list_definition(ax, show_text=True, split=True):\n N = symbols(\"n\")\n prov_nodes = N + 1\n prov_edges = 2*N + 1\n dic_nodes = N + 1 + 1\n dic_edges = N + 2\n ver_nodes = 0*N\n ver_edges = N\n \n \n colors = \"r\", \"b\", \"g\"\n nodes = prov_nodes, dic_nodes, ver_nodes\n edges = prov_edges, dic_edges, ver_edges\n delta_edges = (-25, -5), (-35, -6), (-30, -40)\n delta_nodes = (-17, -30), (-25, 5), (-10, 7)\n \n if split:\n main_loops = [\n (edges, delta_edges, \"{} edges\", \"{}\"),\n (nodes, delta_nodes, \"{} nodes\", \"{}--\"),\n ]\n else:\n addeq = (x + y for x, y in zip(nodes, edges))\n deltas = [(-17, -3), (-13, 8), (-3, 5)]\n main_loops = [\n (addeq, deltas, \"{}\", \"{}\"),\n ]\n\n for elements, deltas, text, colormod in main_loops:\n for name, eq, color, delta in zip(NAMES, elements, colors, deltas):\n fn = lambdify((N,), eq, 'numpy')\n ax.plot(n, n*0 + fn(n), colormod.format(color), label=text.format(name))\n if show_text:\n x = n[-1]\n y = fn(x)\n dx, dy = delta\n ax.text(x + dx, y + dy, text.format(eq), fontsize=10, color=color)\n \n ax.set_xlabel(\"List Size\")\n ax.set_ylabel(\"Statements\")\n\n\nfig1 = plt.figure()\nax = fig1.add_subplot(111)\nlist_definition(ax, SHOW_TEXT, SPLIT)\nplt.legend()\nplt.savefig(\"../generated/comparison/list.svg\")\nplt.savefig(\"../generated/comparison/list.png\")\nplt.savefig(\"../generated/comparison/list.pdf\")\n\nplt.show()\n```\n\n## Reference assignment\n\n\n\n```python\ndef reference_assignment(ax, show_text=True, split=True):\n N = symbols(\"n\")\n prov_edges = N\n dic_edges = 0*N + 1\n ver_edges = 0*N\n \n colors = \"r\", \"b\", \"g\"\n edges = prov_edges, dic_edges, ver_edges\n \n if split:\n delta_edges = (-17, -3), (-15, 4), (-15, -5)\n main_loops = [\n (edges, delta_edges, \"{} edges\", \"{}\"),\n ]\n else:\n delta_edges = (-3, 1), (-3, 4), (-3, -5)\n main_loops = [\n (edges, delta_edges, \"{}\", \"{}\"),\n ]\n\n for elements, deltas, text, colormod in main_loops:\n for name, eq, color, delta in zip(NAMES, elements, colors, deltas):\n fn = lambdify((N,), eq, 'numpy')\n ax.plot(n, n*0 + fn(n), colormod.format(color), label=text.format(name))\n if show_text:\n x = n[-1]\n y = fn(x)\n dx, dy = delta\n ax.text(x + dx, y + dy, text.format(eq), fontsize=10, color=color)\n \n ax.set_xlabel(\"List Size\")\n ax.set_ylabel(\"PROV-N Statements\")\n\nfig1 = plt.figure()\nax = fig1.add_subplot(111)\nreference_assignment(ax, SHOW_TEXT, SPLIT)\nplt.legend()\nplt.savefig(\"../generated/comparison/assign.svg\")\nplt.savefig(\"../generated/comparison/assign.png\")\nplt.savefig(\"../generated/comparison/assign.pdf\")\nplt.show()\n```\n\n## Both\n\n\n```python\nf, (ax0, ax1) = plt.subplots(1, 2, sharey=False)\nlist_definition(ax0, SHOW_TEXT, SPLIT)\nreference_assignment(ax1, SHOW_TEXT, SPLIT)\nlgd = ax0.legend(\n loc='center', ncol=6 if SPLIT else 3,\n bbox_to_anchor=(1.08, 1.15), fontsize=16)\n\nax0.set_ylabel(\"Statements\", fontsize=16)\nax0.set_xlabel(\"List Size\\n(A)\", fontsize=16)\nax1.set_ylabel(\"\")\nax1.set_xlabel(\"List Size\\n(B)\", fontsize=16)\nf.set_size_inches(12, 3)\n#plt.tight_layout()\nplt.savefig(\"../generated/paper/comp_list_and_assign.svg\", bbox_extra_artists=(lgd,), bbox_inches='tight')\nplt.savefig(\"../generated/paper/comp_list_and_assign.png\", bbox_extra_artists=(lgd,), bbox_inches='tight')\nplt.savefig(\"../generated/paper/comp_list_and_assign.pdf\", bbox_extra_artists=(lgd,), bbox_inches='tight')\nplt.show()\n```\n\n## Assignment to part of structures\n\n\n\n```python\nfrom mpl_toolkits.mplot3d import Axes3D\nR, N = symbols(\"r n\")\nprov_nodes = R + N*0\ndic_nodes = R + N*0\nver_nodes = N*0\nprov_edges = R * N + 2 * R\ndic_edges = 3 * R + N*0\nver_edges = 2 + R*0 + N*0\ncolors = \"r\", \"b\", \"g\"\nnodes = prov_nodes, dic_nodes, ver_nodes\nedges = prov_edges, dic_edges, ver_edges\n\n\ndef use_xy(fn, params, k, delta):\n x, z = params(n[-1])\n y = fn(x, z)\n dx, dy = delta\n return x + dx, y + dy\n\ndef use_zy(fn, params, k, delta):\n x, z = params(n[-1])\n y = fn(x, z)\n dx, dy = delta\n return z + dx, y + dy\n\ndef part_assign(ax, params, use=use_xy, show_text=True, split=True, xlabel=\"List Size\", ylabel=\"Statements\", delta_nodes=[], delta_edges=[], delta_sum=[]):\n delta_edges = delta_edges or ((-45, -5), (-35, 34), (-20, 13))\n delta_nodes = delta_nodes or ((-19, 30), (-19, 23), (-20, 7))\n \n if split:\n main_loops = [\n (edges, delta_edges, \"{} edges\", \"{}\"),\n (nodes, delta_nodes, \"{} nodes\", \"{}--\"),\n ]\n else:\n addeq = (x + y for x, y in zip(nodes, edges))\n delta_sum = delta_sum or [(-17, -3), (-13, 8), (-3, 5)]\n main_loops = [\n (addeq, delta_sum, \"{}\", \"{}\"),\n ]\n\n for elements, deltas, text, colormod in main_loops:\n for name, eq, color, delta in zip(NAMES, elements, colors, deltas):\n fn = lambdify((N, R), eq, 'numpy')\n ax.plot(n, n*0 + fn(*params(n)), colormod.format(color), label=text.format(name))\n if show_text:\n ax.text(*use(fn, params, n[-1], delta), text.format(eq), fontsize=10, color=color)\n \n ax.set_xlabel(xlabel, fontsize=16)\n ax.set_ylabel(ylabel, fontsize=16)\n\n\ndef prov(n, r, count, color, axis=[]):\n return axis + [count * (4 + (r * n + 3 * r)), color]\n\ndef dic(n, r, count, color, axis=[]):\n return axis + [count * (4 + (3 * r + 1 + n * 0)), color]\n\ndef ver(n, r, count, color, axis=[]):\n return axis + [count * (4 + (2 + n * 0 + r * 0)), color]\n\ndef to_matrix(fn):\n def new_fn(x, y):\n value = fn(x, y)\n if not isinstance(value, (int, float)):\n return value\n result = np.zeros(shape=(len(x), len(y)))\n result.fill(value)\n return result\n return new_fn\n \nplt.tight_layout()\n\nf, (\n (ax00, ax01, ax02, ax03),\n (ax10, ax11, ax12, ax13),\n) = plt.subplots(2, 4, sharey=False, figsize=(15, 6))\n\n\npart_assign(\n ax00, (lambda x: (x, 1)),\n xlabel=\"List Size\\nFixed: 1 reference\",\n delta_nodes=[(-19, 30), (-19, 23), (-20, 7)],\n delta_edges=[(-50, -5), (-35, 34), (-20, 13)],\n delta_sum=[(-30, -3), (-18, 15), (-3, 7)],\n split=SPLIT, show_text=SHOW_TEXT\n)\n\npart_assign(\n ax01, (lambda x: (x, 20)),\n xlabel=\"List Size\\nFixed: 20 references\", ylabel=\"\",\n delta_nodes=[(-19, 30 * 20), (-19, 23 * 20), (-20, 7 * 20)],\n delta_edges=[(-50, -5 * 20), (-35, 35 * 20), (-20, 15 * 20)],\n delta_sum=[(-30, -3 * 20), (-18, 15 * 20), (-3, 7 * 20)],\n split=SPLIT, show_text=SHOW_TEXT\n)\n\npart_assign(\n ax10, (lambda x: (1, x)), use=use_zy,\n xlabel=\"List References\\nFixed: 1 element\",\n delta_nodes=[(-19, 30), (-19, 12), (-20, 8)],\n delta_edges=[(-58, -40), (-45, -15), (-20, 30)],\n delta_sum=[(-37, -40), (-27, -15), (-3, 10)],\n split=SPLIT, show_text=SHOW_TEXT\n)\n\npart_assign(\n ax11, (lambda x: (20, x)), use=use_zy,\n xlabel=\"List References\\nFixed: 20 elements\", ylabel=\"\",\n delta_nodes=[(-19, 400), (-19, 200), (-20, -100)],\n delta_edges=[(-47, -100), (-70, -50), (-20, 100)],\n delta_sum=[(-30, -100), (-18, 150), (-3, 100)],\n split=SPLIT, show_text=SHOW_TEXT\n)\nplt.tight_layout()\nax02.axis('off')\nax03.axis('off')\nax12.axis('off')\nax13.axis('off')\n\n\n\ndic = to_matrix(lambdify((N, R), dic_nodes + dic_edges, 'numpy'))\nprov = to_matrix(lambdify((N, R), prov_nodes + prov_edges, 'numpy'))\nver = to_matrix(lambdify((N, R), ver_nodes + ver_edges, 'numpy'))\n\n\n_r, _n = np.meshgrid(np.arange(0, 100, 1), n)\nax3 = f.add_subplot(122, projection='3d')\nax3.plot_surface(_r, _n, dic(_n, _r), alpha=0.4)\nax3.plot_surface(_r, _n, prov(_n, _r), alpha=0.4)\nax3.plot_surface(_r, _n, ver(_n, _r), alpha=0.4)\nax3.set_xlabel(\"List Size\", fontsize=16)\nax3.set_ylabel(\"List References\", fontsize=16)\nax3.set_zlabel(\"\\nStatements\", fontsize=16)\n\n\nlgd = ax01.legend(\n loc='center', ncol=6 if SPLIT else 3,\n bbox_to_anchor=(1.08, 1.2), fontsize=16)\n\n\nplt.savefig(\"../generated/comparison/part.svg\", bbox_extra_artists=(lgd,), bbox_inches='tight')\nplt.savefig(\"../generated/comparison/part.png\", bbox_extra_artists=(lgd,), bbox_inches='tight')\nplt.savefig(\"../generated/comparison/part.pdf\", bbox_extra_artists=(lgd,), bbox_inches='tight')\nplt.show();\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "bf1aeba344e2d5820a1d06e35d1d7be35d459b06", "size": 300637, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "notebooks/Comparison.ipynb", "max_stars_repo_name": "dew-uff/versioned-prov", "max_stars_repo_head_hexsha": "cf86ce5cc4d5a072ccbaeee90e201b84690989ce", "max_stars_repo_licenses": ["Unlicense", "MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-03-26T15:19:37.000Z", "max_stars_repo_stars_event_max_datetime": "2018-03-26T15:19:37.000Z", "max_issues_repo_path": "notebooks/Comparison.ipynb", "max_issues_repo_name": "dew-uff/versioned-prov", "max_issues_repo_head_hexsha": "cf86ce5cc4d5a072ccbaeee90e201b84690989ce", "max_issues_repo_licenses": ["Unlicense", "MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/Comparison.ipynb", "max_forks_repo_name": "dew-uff/versioned-prov", "max_forks_repo_head_hexsha": "cf86ce5cc4d5a072ccbaeee90e201b84690989ce", "max_forks_repo_licenses": ["Unlicense", "MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 680.1742081448, "max_line_length": 211292, "alphanum_fraction": 0.9339203092, "converted": true, "num_tokens": 2749, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.66192288918838, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.4095849324769682}} {"text": "# Constructing Models\n\nIn this notebook example, a step-by-step approach of building a simple model$^1$ of trafficking of high-energy phosphate bonds is demonstrated. Illustrated below is a graphical view of the full system along with the reaction rate equations and numerical values:\n\n\n\nThe example starts by creating a model of the \"use\", \"distr\", and \"form\" reactions. Then the model is expanded to include the remaining metabolites, reactions, and any additional information that should be defined in the model. \n\n## Creating a Model\n\n\n```python\nimport numpy as np\nimport pandas as pd\n\nfrom mass import (\n MassConfiguration, MassMetabolite, MassModel, MassReaction)\nfrom mass.util.matrix import left_nullspace, nullspace\n\nmass_config = MassConfiguration()\n```\n\nThe first step to creating the model is to define the `MassModel` object. A `MassModel` only requires an identifier to be initialized. For best practice, it is recommended to utilize SBML compliant identifiers for all objects.\n\n\n```python\nmodel = MassModel(\"Phosphate_Trafficking\")\n```\n\nThe model is initially empty.\n\n\n```python\nprint(\"Number of metabolites: {0}\".format(len(model.metabolites)))\nprint(\"Number of initial conditions: {0}\".format(len(model.initial_conditions)))\nprint(\"Number of reactions: {0}\".format(len(model.reactions)))\n```\n\n Number of metabolites: 0\n Number of initial conditions: 0\n Number of reactions: 0\n\n\nThe next step is to create `MassMetabolite` and `MassReaction` objects to represent the metabolites and reactions that should exist in the model.\n\n### Defining metabolites\n\nTo create a `MassMetabolite`, a unique identifier is required. The `formula` and `charge` attributes are set to ensure mass and charge balancing of reactions in which the metabolite is a participant. The `compartment` attribute indicates where the metabolite is located. In this model, all metabolites exist in a single compartment, abbreviated as \"c\". \n\n\n```python\natp_c = MassMetabolite(\n \"atp_c\",\n name=\"ATP\",\n formula=\"C10H12N5O13P3\",\n charge=-4,\n compartment=\"c\")\n\nadp_c = MassMetabolite(\n \"adp_c\",\n name=\"ADP\",\n formula=\"C10H12N5O10P2\",\n charge=-3,\n compartment=\"c\")\n\namp_c = MassMetabolite(\n \"amp_c\",\n name=\"AMP\",\n formula=\"C10H12N5O7P\",\n charge=-2,\n compartment=\"c\")\n```\n\nThe metabolite concentrations can be defined as the initial conditions for the metabolites using the `initial_condition` attribute. As previously stated, the concentrations are $\\text{[ATP]}=1.6$, $\\text{[ADP]}=0.4$, and $\\text{[AMP]}=0.1$.\n\n\n```python\natp_c.initial_condition = 1.6\nadp_c.ic = 0.4 # Alias for initial_condition\namp_c.ic = 0.1\n\nfor metabolite in [atp_c, adp_c, amp_c]:\n print(\"{0}: {1}\".format(metabolite.id, metabolite.initial_condition))\n```\n\n atp_c: 1.6\n adp_c: 0.4\n amp_c: 0.1\n\n\nThe metabolites are currently not a part of any reaction. Consequently, the `ordinary_differential_equation` attribute is `None`. \n\n\n```python\nprint(atp_c.ordinary_differential_equation)\nprint(adp_c.ode) # Alias for ordinary_differential_equation\nprint(amp_c.ode) \n```\n\n None\n None\n None\n\n\nThe next step is to create the reactions in which the metabolites participate.\n\n### Defining reactions\n\nJust like `MassMetabolite` objects, a unique identifier is also required to create a `MassReaction`. The `reversible` attribute determines whether the reaction can proceed in both the forward and reverse directions, or only in the forward direction. \n\n\n```python\ndistr = MassReaction(\"distr\", name=\"Distribution\", reversible=True)\nuse = MassReaction(\"use\", name=\"ATP Utilization\", reversible=False)\nform = MassReaction(\"form\", name=\"ATP Formation\", reversible=False)\n```\n\nMetabolites are added to reactions using a dictionary of metabolite objects and their stoichiometric coefficients. A group of metabolites can be added either all at once or one at a time. A negative coefficient indicates the metabolite is a reactant, while a positive coefficient indicates the metabolite is a product.\n\n\n```python\ndistr.add_metabolites({\n adp_c: -2,\n atp_c: 1,\n amp_c: 1})\n\nuse.add_metabolites({\n atp_c: -1,\n adp_c: 1})\n\nform.add_metabolites({\n adp_c: -1,\n atp_c: 1})\n\nfor reaction in [distr, use, form]:\n print(reaction)\n```\n\n distr: 2 adp_c <=> amp_c + atp_c\n use: atp_c --> adp_c\n form: adp_c --> atp_c\n\n\nOnce the reactions are created, their parameters can be defined. As stated earlier, the distribution reaction is considerably faster when compared to other reactions in the model. The forward rate constant $k^{\\rightarrow}$, represented as `kf`, can be set as $k^{\\rightarrow}_{distr}=1000\\ \\text{min}^{-1}$. The equilibrium constant $K_{eq}$, represented as `Keq`, is approximately $K_{distr}=1$.\n\n\n```python\ndistr.forward_rate_constant = 1000\ndistr.equilibrium_constant = 1\ndistr.parameters # Return defined mass action kinetic parameters\n```\n\n\n\n\n {'kf_distr': 1000, 'Keq_distr': 1}\n\n\n\nAs shown earlier, the forward rate constants are set as $k^{\\rightarrow}_{use}=6.25\\ \\text{min}^{-1}$ and $k^{\\rightarrow}_{form}=25\\ \\text{min}^{-1}$.\nThe `kf_str` attribute can be used to get the identifier of the forward rate constant as a string.\n\n\n```python\nuse.forward_rate_constant = 6.25\nform.kf = 25 # Alias for forward_rate_constant\n\nprint(\"{0}: {1}\".format(use.kf_str, use.kf))\nprint(\"{0}: {1}\".format(form.kf_str, form.kf))\n```\n\n kf_use: 6.25\n kf_form: 25\n\n\nReactions can be added to the model using the `add_reactions()` method. Adding the reactions to the model also adds the associated metabolites and genes.\n\n\n```python\nmodel.add_reactions([distr, use, form])\n\nprint(\"Number of metabolites: {0}\".format(len(model.metabolites)))\nprint(\"Number of initial conditions: {0}\".format(len(model.initial_conditions)))\nprint(\"Number of reactions: {0}\".format(len(model.reactions)))\n```\n\n Number of metabolites: 3\n Number of initial conditions: 3\n Number of reactions: 3\n\n\nThe stoichiometric matrix of the model is automatically constructed with the addition of the reactions and metabolites to the model. It can be accessed through the `stoichiometric_matrix` property (alias `S`).\n\n\n```python\nprint(model.S)\n```\n\n [[-2. 1. -1.]\n [ 1. -1. 1.]\n [ 1. 0. 0.]]\n\n\nThe stoichiometric matrix attribute can be updated and stored in various formats using the `update_S()` method. For example, the stoichiometric matrix can be converted and stored as a `pandas.DataFrame`.\n\n\n```python\nmodel.update_S(array_type=\"DataFrame\", dtype=np.int_, update_model=True)\nmodel.S\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        distruseform
                                        adp_c-21-1
                                        atp_c1-11
                                        amp_c100
                                        \n
                                        \n\n\n\nAssociating the metabolites with reactions allows for the mass action reaction rate expressions to be generated based on the stoichiometry.\n\n\n```python\nprint(distr.rate)\n```\n\n kf_distr*(adp_c(t)**2 - amp_c(t)*atp_c(t)/Keq_distr)\n\n\nGeneration of the reaction rates also allows for the metabolite ODEs to be generated.\n\n\n```python\nprint(atp_c.ode)\n```\n\n kf_distr*(adp_c(t)**2 - amp_c(t)*atp_c(t)/Keq_distr) + kf_form*adp_c(t) - kf_use*atp_c(t)\n\n\nThe `nullspace()` method can be used to obtain the null space of the stoichiometric matrix. The nullspace reflects the pathways through the system.\n\n\n```python\nns = nullspace(model.S) # Get the null space\n# Divide by the minimum and round to nearest integer\nns = np.rint(ns / np.min(ns[np.nonzero(ns)]))\npd.DataFrame(\n ns, index=model.reactions.list_attr(\"id\"), # Rows represent reactions\n columns=[\"Pathway 1\"], dtype=np.int_)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Pathway 1
                                        distr0
                                        use1
                                        form1
                                        \n
                                        \n\n\n\nIn a similar fashion the left nullspace can be obtained using the `left_nullspace` function. The left nullspace represents the conserved moieties in the model.\n\n\n```python\nlns = left_nullspace(model.S)\n# Divide by the minimum and round to nearest integer\nlns = np.rint(lns / np.min(lns[np.nonzero(lns)]))\npd.DataFrame(\n lns, index=[\"Total AxP\"],\n columns=model.metabolites.list_attr(\"id\"), # Columns represent metabolites\n dtype=np.int_)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        adp_catp_camp_c
                                        Total AxP111
                                        \n
                                        \n\n\n\n## Expanding an Existing Model\n\nNow, the existing model is expanded to include a buffer reaction, where a phosphagen is utilized to store a high-energy phosphate in order to buffer the ATP/ADP ratio as needed. Because the buffer molecule represents a generic phosphagen, there is no chemical formula for the molecule. Therefore, the buffer molecule can be represented as a moiety in the `formula` attribute using square brackets.\n\n\n```python\nb = MassMetabolite(\n \"B\",\n name=\"Phosphagen buffer (Free)\",\n formula=\"[B]\",\n charge=0,\n compartment=\"c\")\n\nbp = MassMetabolite(\n \"BP\",\n name=\"Phosphagen buffer (Loaded)\",\n formula=\"[B]-PO3\",\n charge=-1,\n compartment=\"c\")\n\nbuffer = MassReaction(\"buffer\", name=\"ATP Buffering\")\n```\n\nWhen adding metabolites to the reaction, the `get_by_id()` method is used to add already existing metabolites in the model to the reaction.\n\n\n```python\nbuffer.add_metabolites({\n b: -1,\n model.metabolites.get_by_id(\"atp_c\"): -1,\n model.metabolites.get_by_id(\"adp_c\"): 1,\n bp: 1})\n\n# Add reaction to model\nmodel.add_reactions([buffer])\n```\n\nFor this reaction, $k^{\\rightarrow}_{buffer}=1000\\ \\text{min}^{-1}$ and $K_{buffer}=1$. Because the reaction has already been added to the model, the `MassModel.update_parameters()` method can be used to update the reaction parameters using a dictionary:\n\n\n```python\nmodel.update_parameters({\n buffer.kf_str: 1000,\n buffer.Keq_str: 1})\n\nbuffer.parameters\n```\n\n\n\n\n {'kf_buffer': 1000, 'Keq_buffer': 1}\n\n\n\nBy adding the reaction to the model, the left nullspace expanded to include a conservation pool for the total buffer in the system.\n\n\n```python\nlns = left_nullspace(model.S)\nfor i, row in enumerate(lns):\n # Divide by the minimum and round to nearest integer\n lns[i] = np.rint(row / np.min(row[np.nonzero(row)]))\npd.DataFrame(lns, index=[\"Total AxP\", \"Total Buffer\"],\n columns=model.metabolites.list_attr(\"id\"),\n dtype=np.int_)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        adp_catp_camp_cBBP
                                        Total AxP11100
                                        Total Buffer00011
                                        \n
                                        \n\n\n\n### Performing symbolic calculations\n\nAlthough the concentrations for the free and loaded buffer molecules are currently unknown, the total amount of buffer is known and set as $B_{total} = 10$. Because the buffer reaction is assumed to be at equilibrium, it becomes possible to solve for the concentrations of the free and loaded buffer molecules.\n\nBelow, the symbolic capabilities of **SymPy** are used to solve for the steady state concentrations of the buffer molecules.\n\n\n```python\nfrom sympy import Eq, Symbol, pprint, solve\n\nfrom mass.util import strip_time\n```\n\nThe first step is to define the equation for the total buffer pool symbolically:\n\n\n```python\nbuffer_total_equation = Eq(Symbol(\"B\") + Symbol(\"BP\"), 10)\npprint(buffer_total_equation)\n```\n\n B + BP = 10\n\n\nThe equation for the reaction rate at equilibrium is also defined. The `strip_time()` function is used to strip time dependency from the equation.\n\n\n```python\nbuffer_rate_equation = Eq(0, strip_time(buffer.rate))\n# Substitute defined concentration values into equation\nbuffer_rate_equation = buffer_rate_equation.subs({\n \"atp_c\": atp_c.initial_condition,\n \"adp_c\": adp_c.initial_condition,\n \"kf_buffer\": buffer.kf,\n \"Keq_buffer\": buffer.Keq})\npprint(buffer_rate_equation)\n```\n\n 0 = 1600.0\u22c5B - 400.0\u22c5BP\n\n\nThese two equations can be solved to get the buffer concentrations:\n\n\n```python\nbuffer_sol = solve([buffer_total_equation, buffer_rate_equation],\n [Symbol(\"B\"), Symbol(\"BP\")])\nbuffer_sol\n```\n\n\n\n\n {B: 2.00000000000000, BP: 8.00000000000000}\n\n\n\nBecause the metabolites already exist in the model, their initial conditions can be updated to the calculated concentrations using the `MassModel.update_initial_conditions()` method.\n\n\n```python\n# Replace the symbols in the dict\nfor met_symbol, concentration in buffer_sol.items():\n metabolite = model.metabolites.get_by_id(str(met_symbol))\n # Make value as a float\n buffer_sol[metabolite] = float(buffer_sol.pop(met_symbol))\n\nmodel.update_initial_conditions(buffer_sol)\nmodel.initial_conditions\n```\n\n\n\n\n {: 0.4,\n : 1.6,\n : 0.1,\n : 2.0,\n : 8.0}\n\n\n\n### Adding boundary reactions\n\nAfter adding the buffer reactions, the next step is to define the AMP source and demand reactions. The `add_boundary()` method is employed to create and add a boundary reaction to a model.\n\n\n```python\namp_drain = model.add_boundary(\n model.metabolites.amp_c,\n boundary_type=\"demand\",\n reaction_id=\"amp_drain\")\n\namp_in = model.add_boundary(\n model.metabolites.amp_c,\n boundary_type=\"demand\",\n reaction_id=\"amp_in\")\n\nprint(amp_drain)\nprint(amp_in)\n```\n\n amp_drain: amp_c --> \n amp_in: amp_c --> \n\n\nWhen a boundary reaction is created, a 'boundary metabolite' is also created as a proxy metabolite. The proxy metabolite is the external metabolite concentration (i.e., boundary condition) without instantiating a new `MassMetabolite` object to represent the external metabolite.\n\n\n```python\namp_in.boundary_metabolite\n```\n\n\n\n\n 'amp_b'\n\n\n\nThe value of the 'boundary metabolite' can be set using the `MassModel.add_boundary_conditions()` method. Boundary conditions are accessed through the `MassModel.boundary_conditions` attribute.\n\n\n```python\nmodel.add_boundary_conditions({amp_in.boundary_metabolite: 1})\nmodel.boundary_conditions\n```\n\n\n\n\n {'amp_b': 1.0}\n\n\n\nThe automatic generation of the boundary reaction can be useful. However, sometimes the reaction stoichiometry needs to be switched in order to be intuitive. In this case, the stoichiometry of the AMP source reaction should be reversed to show that AMP enters the system, which is accomplished by using the `MassReaction.reverse_stoichiometry()` method.\n\n\n```python\namp_in.reverse_stoichiometry(inplace=True)\nprint(amp_in)\n```\n\n amp_in: --> amp_c\n\n\nNote that the addition of these two reactions adds an another pathway to the null space:\n\n\n```python\nns = nullspace(model.S).T\nfor i, row in enumerate(ns):\n # Divide by the minimum to get all integers\n ns[i] = np.rint(row / np.min(row[np.nonzero(row)]))\nns = ns.T\npd.DataFrame(\n ns, index=model.reactions.list_attr(\"id\"), # Rows represent reactions\n columns=[\"Path 1\", \"Path 2\"], dtype=np.int_)\n```\n\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        Path 1Path 2
                                        distr00
                                        use10
                                        form10
                                        buffer00
                                        amp_drain01
                                        amp_in01
                                        \n
                                        \n\n\n\n### Defining custom rates\n\nIn this model, the rate for the AMP source reaction should remain at a fixed input value. However, the current rate expression for the AMP source reaction is dependent on an external AMP metabolite that exists as a boundary condition:\n\n\n```python\nprint(amp_in.rate)\n```\n\n amp_b*kf_amp_in\n\n\nTherefore, the rate can be set as a fixed input by using a custom rate expression. Custom rate expressions can be set for reactions in a model using the `MassModel.add_custom_rate()` method as follows: by passing the reaction object, a string representation of the custom rate expression, and a dictionary containing any custom parameter associated with the rate.\n\n\n```python\nmodel.add_custom_rate(amp_in, custom_rate=\"b1\",\n custom_parameters={\"b1\": 0.03})\nprint(model.rates[amp_in])\n```\n\n b1\n\n\n## Ensuring Model Completeness\n### Inspecting rates and ODEs\n\nAccording to the [network schema](#Constructing-Models) at the start of the notebook, the network has been fully reconstructed. The reaction rates and metabolite ODEs can be inspected to ensure that the model was built without any issues.\n\nThe `MassModel.rates` property is used to return a dictionary containing reactions and symbolic expressions of their rates. The model always prioritizes custom rate expressions over automatically generated mass action rates.\n\n\n```python\nfor reaction, rate in model.rates.items():\n print(\"{0}: {1}\".format(reaction.id, rate))\n```\n\n distr: kf_distr*(adp_c(t)**2 - amp_c(t)*atp_c(t)/Keq_distr)\n use: kf_use*atp_c(t)\n form: kf_form*adp_c(t)\n buffer: kf_buffer*(B(t)*atp_c(t) - BP(t)*adp_c(t)/Keq_buffer)\n amp_drain: kf_amp_drain*amp_c(t)\n amp_in: b1\n\n\nSimilarly, the model can access the ODEs for metabolites using the `ordinary_differential_equations` property (alias `odes`) to return a dictionary of metabolites and symbolic expressions of their ODEs.\n\n\n```python\nfor metabolite, ode in model.odes.items():\n print(\"{0}: {1}\".format(metabolite.id, ode))\n```\n\n adp_c: kf_buffer*(B(t)*atp_c(t) - BP(t)*adp_c(t)/Keq_buffer) - 2*kf_distr*(adp_c(t)**2 - amp_c(t)*atp_c(t)/Keq_distr) - kf_form*adp_c(t) + kf_use*atp_c(t)\n atp_c: -kf_buffer*(B(t)*atp_c(t) - BP(t)*adp_c(t)/Keq_buffer) + kf_distr*(adp_c(t)**2 - amp_c(t)*atp_c(t)/Keq_distr) + kf_form*adp_c(t) - kf_use*atp_c(t)\n amp_c: b1 - kf_amp_drain*amp_c(t) + kf_distr*(adp_c(t)**2 - amp_c(t)*atp_c(t)/Keq_distr)\n B: -kf_buffer*(B(t)*atp_c(t) - BP(t)*adp_c(t)/Keq_buffer)\n BP: kf_buffer*(B(t)*atp_c(t) - BP(t)*adp_c(t)/Keq_buffer)\n\n\n### Compartments\n\nCompartments, defined in metabolites, are recognized by the model and can be viewed in the `compartments` attribute.\n\n\n```python\nmodel.compartments\n```\n\n\n\n\n {'c': ''}\n\n\n\nFor this model, \"c\" is an abbreviation for \"compartment\". The `compartments` attribute can be updated to reflect this mapping using a `dict`:\n\n\n```python\nmodel.compartments = {\"c\": \"compartment\"}\nmodel.compartments\n```\n\n\n\n\n {'c': 'compartment'}\n\n\n\n### Units\n`Unit` and `UnitDefinition` objects are implemented as per the [SBML Unit](http://sbml.org/Software/libSBML/5.18.0/docs/python-api/classlibsbml_1_1_unit.html) and [SBML UnitDefinition](http://sbml.org/Software/libSBML/5.18.0/docs/python-api/classlibsbml_1_1_unit_definition.html) specifications.\nIt can be useful for comparative reasons to create `Unit` and `UnitDefinition` objects for the model (e.g., amount, volume, time) to provide additional context. However, the model does not maintain unit consistency automatically. It is the responsibility of the users to ensure consistency among units and associated numerical values in a model.\n\n\n```python\nfrom mass import Unit, UnitDefinition\nfrom mass.core.units import print_defined_unit_values\n```\n\nSBML defines units using a compositional approach. The `Unit` objects represent references to base units. A `Unit` has four attributes: `kind`, `exponent`, `scale`, and `multiplier`. The `kind` attribute indicates the base unit. Valid base units are viewed using the `print_defined_unit_values()` function.\n\n\n```python\nprint_defined_unit_values(\"BaseUnitKinds\")\n```\n\n \u2552\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2555\n \u2502 SBML Base Unit Kinds \u2502\n \u255e\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2561\n \u2502 Base Unit SBML Value \u2502\n \u2502 ------------- ------------ \u2502\n \u2502 ampere 0 \u2502\n \u2502 avogadro 1 \u2502\n \u2502 becquerel 2 \u2502\n \u2502 candela 3 \u2502\n \u2502 coulomb 5 \u2502\n \u2502 dimensionless 6 \u2502\n \u2502 farad 7 \u2502\n \u2502 gram 8 \u2502\n \u2502 gray 9 \u2502\n \u2502 henry 10 \u2502\n \u2502 hertz 11 \u2502\n \u2502 item 12 \u2502\n \u2502 joule 13 \u2502\n \u2502 katal 14 \u2502\n \u2502 kelvin 15 \u2502\n \u2502 kilogram 16 \u2502\n \u2502 liter 17 \u2502\n \u2502 litre 18 \u2502\n \u2502 lumen 19 \u2502\n \u2502 lux 20 \u2502\n \u2502 meter 21 \u2502\n \u2502 metre 22 \u2502\n \u2502 mole 23 \u2502\n \u2502 newton 24 \u2502\n \u2502 ohm 25 \u2502\n \u2502 pascal 26 \u2502\n \u2502 radian 27 \u2502\n \u2502 second 28 \u2502\n \u2502 siemens 29 \u2502\n \u2502 sievert 30 \u2502\n \u2502 steradian 31 \u2502\n \u2502 tesla 32 \u2502\n \u2502 volt 33 \u2502\n \u2502 watt 34 \u2502\n \u2502 weber 35 \u2502\n \u2502 invalid 36 \u2502\n \u2558\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255b\n\n\nThe `exponent`, `scale` and `multiplier` attributes indicate how the base unit should be transformed. For this model, the unit for concentration, \"Millimolar\", which is represented as millimole per liter and composed of the following base units:\n\n\n```python\nmillimole = Unit(kind=\"mole\", exponent=1, scale=-3, multiplier=1)\nper_liter = Unit(kind=\"liter\", exponent=-1, scale=1, multiplier=1)\n```\n\nCombinations of `Unit` objects are contained inside a `UnitDefintion` object. `UnitDefinition` objects have three attributes: an `id`, an optional `name` to represent the combination, and a `list_of_units` attribute that contain references to the `Unit` objects. The concentration unit \u201cMillimolar\u201d is abbreviated as \u201cmM\u201d and defined as follows:\n\n\n```python\nconcentration_unit = UnitDefinition(id=\"mM\", name=\"Millimolar\",\n list_of_units=[millimole, per_liter])\nprint(\"{0}:\\n{1!r}\\n{2!r}\".format(\n concentration_unit.name, *concentration_unit.list_of_units))\n```\n\n Millimolar:\n \n \n\n\n`UnitDefinition` objects also have the `UnitDefinition.create_unit()` method to directly create `Unit` objects within the `UnitDefintion`.\n\n\n```python\ntime_unit = UnitDefinition(id=\"min\", name=\"Minute\")\ntime_unit.create_unit(kind=\"second\", exponent=1, scale=1, multiplier=60)\nprint(time_unit)\nprint(time_unit.list_of_units)\n```\n\n min\n []\n\n\nOnce created, `UnitDefintion` objects are added to the model:\n\n\n```python\nmodel.add_units([concentration_unit, time_unit])\nmodel.units\n```\n\n\n\n\n [,\n ]\n\n\n\n### Checking model completeness\n\nOnce constructed, the model should be checked for completeness.\n\n\n```python\nfrom mass import qcqa_model\n```\n\nThe `qcqa_model()` function can be used to print a report about the model's completeness based on the set kwargs. The `qcqa_model()` function is used to ensure that all numerical values necessary for simulating the model are defined by setting the `parameters` and `concentrations` kwargs as `True`.\n\n\n```python\nqcqa_model(model, parameters=True, concentrations=True)\n```\n\n \u2552\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2555\n \u2502 MODEL ID: Phosphate_Trafficking \u2502\n \u2502 SIMULATABLE: False \u2502\n \u2502 PARAMETERS NUMERICALY CONSISTENT: True \u2502\n \u255e\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2561\n \u2502 ============================================ \u2502\n \u2502 MISSING PARAMETERS \u2502\n \u2502 ============================================ \u2502\n \u2502 Reaction Parameters \u2502\n \u2502 --------------------- \u2502\n \u2502 amp_drain: kf \u2502\n \u2502 ============================================ \u2502\n \u2558\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255b\n\n\nAs shown in the report above, the forward rate constant for the AMP drain reaction was never defined. Therefore, the forward rate constant is defined, and the model is checked again.\n\n\n```python\namp_drain.kf = 0.03\n\nqcqa_model(model, parameters=True, concentrations=True)\n```\n\n \u2552\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2555\n \u2502 MODEL ID: Phosphate_Trafficking \u2502\n \u2502 SIMULATABLE: True \u2502\n \u2502 PARAMETERS NUMERICALY CONSISTENT: True \u2502\n \u255e\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2561\n \u2558\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255b\n\n\nNow, the report shows that the model is not missing any values necessary for simulation. See [Checking Model Quality](./quality_assurance.ipynb) for more information on quality assurance functions and the `qcqa` submodule.\n\n## Additional Examples\nFor additional examples on constructing models, see the following: \n\n* [Constructing Glycolysis](../gallery/workflows/constructing_glycolysis.ipynb)\n\n$^1$ Trafficking model is created from Chapter 8 of Systems Biology: Simulation of Dynamic Network States\n", "meta": {"hexsha": "bdc9ef94558fbc070c1d65c2debf517ae5fdb595", "size": 59587, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "docs/tutorials/constructing_models.ipynb", "max_stars_repo_name": "SBRG/MASSpy", "max_stars_repo_head_hexsha": "1315c1d40be8feb8731c8143dbc9ba43bf8c78ff", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 13, "max_stars_repo_stars_event_min_datetime": "2020-07-13T00:48:13.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-27T15:42:15.000Z", "max_issues_repo_path": "docs/tutorials/constructing_models.ipynb", "max_issues_repo_name": "SBRG/MASSpy", "max_issues_repo_head_hexsha": "1315c1d40be8feb8731c8143dbc9ba43bf8c78ff", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 10, "max_issues_repo_issues_event_min_datetime": "2021-02-17T18:07:43.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-23T16:22:14.000Z", "max_forks_repo_path": "docs/tutorials/constructing_models.ipynb", "max_forks_repo_name": "SBRG/MASSpy", "max_forks_repo_head_hexsha": "1315c1d40be8feb8731c8143dbc9ba43bf8c78ff", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2020-01-15T00:48:49.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-04T07:01:17.000Z", "avg_line_length": 29.9582704877, "max_line_length": 407, "alphanum_fraction": 0.5335056304, "converted": true, "num_tokens": 7558, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.6619228758499942, "lm_q1q2_score": 0.409584924223436}} {"text": "# LASSO Regresion on covid-19 cases in NYS\n\n## Load modulues\n\n\n```python\n%matplotlib widget\nimport os\nimport requests\nimport urllib.parse\nimport json\nimport io\nfrom zipfile import ZipFile\nfrom datetime import datetime, timedelta\n\nimport pandas as pd\nimport numpy as np\nimport statsmodels.api as sm\nimport scipy\n\nimport seaborn as sns\nimport matplotlib.pyplot as plt\nimport matplotlib.dates as mdates\nfrom statsmodels.sandbox.regression.predstd import wls_prediction_std\nfrom sklearn import linear_model\nfrom scipy import stats as sps\nfrom sklearn.feature_selection import SelectFromModel\nfrom IPython.display import display\n\nlocator = mdates.AutoDateLocator()\nformatter = mdates.ConciseDateFormatter(locator)\n\nsns.set_style(\"whitegrid\")\nGAMMA = 1/7.5\n\nevents = {}\ndf_dict = {}\ndataset_info = {}\n#may 15 last day\n```\n\n\n```python\ndsname = 'New York'\ndfs = []\noffset = 0\nwhile offset >=0:\n url = \"https://health.data.ny.gov/resource/xdss-u53e.csv/?$limit=5000&$offset={}\".format(offset)\n df = pd.read_csv(url)\n dfs.append(df)\n if len(df)==5000:\n offset += 5000 \n else:\n offset = -1\ndfraw = pd.concat(dfs)\n#'test_date=2020-03-15T00:00:00.000'\n\n#dfraw = dfraw.rename(columns={'new_positives': 'Positives', 'total_number_of_tests': 'Tests', 'test_date': 'date'})\nprint(len(dfraw))\n#dfraw['date'] = pd.to_datetime(dfraw['date'])\n\n#counties = (df.groupby('date')['Tests']>0).count()\ndf = dfraw.groupby('test_date').sum()\ndfraw.to_csv('NYS_all_data.csv')\n\n\n\n```\n\n 8184\n\n\n## Important dates\n\n\n```python\n# intervention dates\n# 03/10/2020 school close in New Rochelle\n# https://www.governor.ny.gov/news/during-novel-coronavirus-briefing-governor-cuomo-announces-new-mass-gatherings-regulations\n# 03/12/2020 mass gathering reduced to 500 max. 50% occupancy\n\n\n# 03/17/2020 close of gyms, restaurants and bars, movie theaters, mass gathering up to 50. https://www.governor.ny.gov/news/amid-lack-federal-direction-governor-cuomo-governor-murphy-and-governor-lamont-announce\nbars = pd.to_datetime('03-16-2020 20:00', dayfirst=False)\n# 03/18/2020 school clousure http://www.nysed.gov/news/2020/state-education-department-issues-updated-guidance-schools-regarding-novel-coronavirus\nschools = pd.to_datetime('03-18-2020 20:00', dayfirst=False)\n\n# https://www.governor.ny.gov/news/amid-ongoing-covid-19-pandemic-governor-cuomo-announces-deployment-1000-bed-hospital-ship-usns\n# 03/20/2020 00:00 50% of the workforce\nworkforce_50 = pd.to_datetime('03-20-2020 20:00', dayfirst=False)\n# 03/22/2020 20:00 ny_pause \nny_pause = pd.to_datetime('22-03-2020 00:00', dayfirst=True)\n# CDC masks https://www.npr.org/sections/goatsandsoda/2020/04/10/829890635/why-there-so-many-different-guidelines-for-face-masks-for-the-public\nmasks_cdc = pd.to_datetime('03-04-2020 00:00', dayfirst=True)\nmask_employers = pd.to_datetime('12-04-2020 00:00', dayfirst=True)\nmask_public = pd.to_datetime('17-04-2020 00:00', dayfirst=True)\n\nevents_list = [bars, schools, workforce_50, ny_pause, masks_cdc, mask_employers, mask_public]\nevents['New York'] = events_list\n```\n\n## Get data\n\n\n```python\ndsname = 'Alaska'\nurl = 'https://opendata.arcgis.com/datasets/72c6d13ea1e9420ea398724bdd10372f_0.geojson'\nif os.path.isfile('data/alaska.csv'):\n r = requests.get(url)\n data = json.loads(r.content)\n df = pd.DataFrame([x['properties'] for x in data['features']])\n df['date'] = pd.to_datetime(df['Date_'].str[:-3], format='%Y/%m/%d 12:00:00')\n df = df.rename(columns={'daily_positive': 'Positives', 'daily_negative': 'Negatives', 'daily_tests': 'Tests'})\n df = df.set_index('date')\n #df.index = df.index.round('D')\n df['Date'] = df.index\n df.to_csv('data/alaska.csv')\n\nelse:\n df = pd.read_csv('data/alaska.csv', index_col='date', parse_dates=['date', 'Date'])\ndf.Positives = df.Positives.astype(float)\ndf.Tests = df.Tests.astype(float)\ndf.Negatives = df.Negatives.astype(float)\ndf = df[['Date', 'Positives', 'Negatives', 'Tests']]\ndf['Odds'] = df.Positives /df.Negatives\ndf = df[df['Date']>'2020-03-23']\ndf_dict[dsname] = df\ndataset_info[dsname] = {'Link to source': url, \n 'Link to information': 'https://coronavirus-response-alaska-dhss.hub.arcgis.com/datasets/daily-test-positivity/data',\n 'Type': 'API'}\n```\n\n\n```python\ndsname = 'Colorado'\nurl = 'https://opendata.arcgis.com/datasets/566216cf203e400f8cbf2e6b4354bc57_0.geojson'\nif not os.path.isfile('data/colorado.csv'):\n r = requests.get(url)\n data = json.loads(r.content)\n df = pd.DataFrame([x['properties'] for x in data['features']])\n #df = df.rename(columns={'Cases': 'Positives', 'Tested': 'Tests'})\n df['Date'] = pd.to_datetime(df['Date'])\n df['date'] = pd.to_datetime(df['Date'])\n df = df.set_index('date')\n df = df.sort_index()\n df.to_csv('data/colorado.csv')\nelse:\n df = pd.read_csv('data/colorado.csv', index_col='date', parse_dates=['date', 'Date'])\ndf['Tests'] = df['Tested'].diff()\ndf['Positives'] = df['Cases'].diff()\ndf['Negatives'] = df['Tests'] - df['Positives']\ndf = df[['Date', 'Positives', 'Negatives', 'Tests']]\ndf['Odds'] = df['Positives'] /df['Negatives']\ndf = df[df.Odds.notna()]\ndf_dict[dsname] = df\ndataset_info[dsname] = {'Link to source': url, 'Link to information': 'https://data-cdphe.opendata.arcgis.com/datasets/cdphe-covid19-daily-state-statistics/data', 'Type': 'API'}\n```\n\n\n```python\ndsname = 'Connecticut'\n\n# reopening phase 1 may 20\n# https://portal.ct.gov/Coronavirus/Covid-19-Knowledge-Base/Reopen-plan\n\nif not os.path.isfile('data/connecticut.csv'):\n dfs = []\n offset = 0\n while offset >=0:\n url = 'https://data.ct.gov/resource/qfkt-uahj.csv?$limit=5000&$offset={}'.format(offset)\n df = pd.read_csv(url)\n dfs.append(df)\n if len(df)==5000:\n offset += 5000 \n else:\n offset = -1\n dfcounty = pd.concat(dfs)\n dfcounty = dfcounty.rename(columns={'number_of_positives': 'Positives', 'number_of_tests': 'Tests', 'number_of_negatives': 'Negatives'})\n dfcounty['Tests'] = dfcounty['Tests'] - dfcounty['number_of_indeterminates']\n dfcounty['date'] = pd.to_datetime(dfcounty['date'])\n df = dfcounty.groupby('date').sum()\n df['Odds'] = df.Positives / df.Negatives\n df['Date'] = pd.to_datetime(df.index)\n df.to_csv('data/connecticut.csv')\n #df = df[df['Date'] <= '2020-05-19']\n df = df[df['Date'] > '2020-03-19']\nelse:\n df = pd.read_csv('data/connecticut.csv', index_col='date', parse_dates=['date', 'Date'])\n #df = df[df['Date'] <= '2020-05-19']\n df = df[df['Date'] > '2020-03-19']\n #df = df.set_index('date')\ndf['Date'] = pd.to_datetime(df['Date'])\ndf.index = pd.to_datetime(df.index)\n\n# last days are not accurate. See S.I.\ndf = df[df['Date']'2020-03-22']\ndf_dict[dsname] = df\ndataset_info[dsname] = {'Link to source': url,\n 'Link to information': 'https://myhealthycommunity.dhss.delaware.gov/locations/state',\n 'Type': 'File'}\n```\n\n\n```python\ndsname = 'Indiana'\nurl = 'https://hub.mph.in.gov/dataset/ab9d97ab-84e3-4c19-97f8-af045ee51882/resource/182b6742-edac-442d-8eeb-62f96b17773e/download/covid_report_date.xlsx'\nif not os.path.isfile('data/indiana.csv'):\n r = requests.get(url)\n df =pd.read_excel(io.BytesIO(r.content))\n df['date'] = pd.to_datetime(df['DATE'])\n df = df.set_index('date')\n df = df.sort_index()\n df['Date'] = df.index\n df.to_csv('data/indiana.csv')\nelse:\n df = pd.read_csv('data/indiana.csv', index_col='date', parse_dates=['date', 'Date'])\ndf['Tests'] = df['COVID_TEST']\ndf['Positives'] = df['DAILY_BASE_CASES']\ndf = df[['Date', 'Tests', 'Positives']]\ndf['Negatives'] = df['Tests'] - df['Positives']\ndf['Odds'] = df.Positives / df.Negatives\ndf = df[df['Date'] > '2020-03-18']\ndf = df[df['Date']'2020-03-21']\ndf_dict[dsname] = df\ndataset_info[dsname] = {'Link to source': ' Downloadable from dashboard',\n 'Link to information': 'https://coronavirus.iowa.gov/',\n 'Type': 'File'}\n```\n\n\n```python\ndsname = 'Massachusetts'\n# massachusets reopening on May 18\n# https://www.mass.gov/doc/reopening-massachusetts-may-18-2020/download\nyesterday = datetime.today() - timedelta(1)\nif not os.path.isfile('data/massachusetts.csv'):\n #yesterday = datetime.today() - timedelta(10)\n yesterday_str = datetime.strftime(yesterday, '%B-%d-%Y').lower()\n fn = 'data/{}.zip'.format(dsname)\n \n url2 = 'https://www.mass.gov/doc/covid-19-raw-data-june-10-2020/download'\n url = 'https://www.mass.gov/doc/covid-19-raw-data-{}/download'.format(yesterday_str)\n print(url)\n print(url2)\n myfile = requests.get(url, allow_redirects=True)\n #open(fn, 'wb').write(myfile.content)\n zf = ZipFile(io.BytesIO(myfile.content))\n csvf = zf.open('TestingByDate.csv')\n df = pd.read_csv(csvf)\n\n # https://www.mass.gov/doc/covid-19-raw-data-may-27-2020/download\n #df = pd.read_csv('data/massachusetts/COVID-19-Dashboard-Files-05-24-2020/TestingByDate.csv',\n # usecols=['Date', 'New', 'Positive'])\n\n df = df.rename(columns={'Molecular Positive New': 'Positives', 'Molecular New': 'Tests'})\n df['Negatives'] = df.Tests - df.Positives\n df = df.query('Tests>100')\n df['date'] = pd.to_datetime(df['Date'])\n df= df.set_index('date')\n df['Date'] = pd.to_datetime(df['Date'])\n df['Odds'] = df.Positives / df.Negatives\n #df = df[df['Date']<'2020-05-15']\n df.to_csv('data/massachusetts.csv')\n #df = df[df['Date']<'2020-05-18']\n df = df[df['Date']>'2020-03-15']\nelse:\n df = pd.read_csv('data/massachusetts.csv')\n df['Date'] = pd.to_datetime(df['Date'])\n df['date'] = pd.to_datetime(df['date'])\n df = df.set_index('date')\n #df = df[df['Date']<'2020-05-18']\n df = df[df['Date']>'2020-03-15']\n\ndf = df[df['Date']'2020-03-18']\ndf = df[df['Date']=0:\n url = \"https://health.data.ny.gov/resource/xdss-u53e.csv/?$limit=5000&$offset={}\".format(offset)\n df = pd.read_csv(url, usecols=['test_date', 'total_number_of_tests', 'new_positives'])\n dfs.append(df)\n if len(df)==5000:\n offset += 5000 \n else:\n offset = -1\n dfraw = pd.concat(dfs)\n #'test_date=2020-03-15T00:00:00.000'\n\n dfraw = dfraw.rename(columns={'new_positives': 'Positives', 'total_number_of_tests': 'Tests', 'test_date': 'date'})\n print(len(dfraw))\n dfraw['date'] = pd.to_datetime(dfraw['date'])\n #counties = (df.groupby('date')['Tests']>0).count()\n df = dfraw.groupby('date').sum()\n df['Odds'] = df.Positives / (df.Tests - df.Positives)\n df['Date'] = pd.to_datetime(df.index)\n df.to_csv('data/ny.csv')\n df = df[df['Date'] >= '2020-03-15']\n # last date of full NYS PAUSE\n #df = df[df['Date'] <= '2020-05-15']\n\nelse:\n df = pd.read_csv('data/ny.csv')\n df['date'] =pd.to_datetime(df['date'])\n df['Date'] = pd.to_datetime(df['Date'])\n df = df.set_index('date')\n df = df[df['Date'] >= '2020-03-15']\n # last date of full NYS PAUSE\n #df = df[df['Date'] <= '2020-05-15']\ndf['Negatives'] = df.Tests - df.Positives\ndf_dict[dsname] = df\ndataset_info[dsname] = {'Link to source': url,\n 'Link to information': 'https://health.data.ny.gov/Health/New-York-State-Statewide-COVID-19-Testing/xdss-u53e',\n 'Type': 'API'}\n```\n\n 8184\n\n\n\n```python\ndsname = 'Rhode Island'\n#https://docs.google.com/spreadsheets/d/1n-zMS9Al94CPj_Tc3K7Adin-tN9x1RSjjx2UzJ4SV7Q/edit#gid=590763272\nurl = 'https://docs.google.com/spreadsheets/d/1n-zMS9Al94CPj_Tc3K7Adin-tN9x1RSjjx2UzJ4SV7Q/export?format=csv&id=1n-zMS9Al94CPj_Tc3K7Adin-tN9x1RSjjx2UzJ4SV7Q&gid=590763272'\ndf = pd.read_csv(url, \n usecols=['New positive labs', 'New negative labs', 'Date'])\n#df = pd.read_csv('data/COVID-19 Rhode Island Data - COVID Trends.csv', \n# usecols=['New positive labs', 'New negative labs', 'Date'])\n\ndf = df.rename(columns={'New positive labs': 'Positives', 'New negative labs': 'Negatives'})\ndf['Tests'] = df.Positives + df.Negatives\ndf = df.query('Tests>100')\ndf['date'] = pd.to_datetime(df['Date'])\ndf= df.set_index('date')\ndf['Date'] = pd.to_datetime(df['Date'])\ndf['Odds'] = df.Positives / df.Negatives\ndf = df[df['Date']>='2020-04-01']\ndf_dict[dsname] = df\ndataset_info[dsname] = {'Link to source': url,\n 'Link to information': 'https://docs.google.com/spreadsheets/d/1n-zMS9Al94CPj_Tc3K7Adin-tN9x1RSjjx2UzJ4SV7Q/edit#gid=590763272',\n 'Type': 'File'}\n```\n\n\n```python\ndsname = 'Tennessee'\nurl = 'https://www.tn.gov/content/dam/tn/health/documents/cedep/novel-coronavirus/datasets/Public-Dataset-Daily-Case-Info.XLSX'\nif not os.path.isfile('data/tennessee.csv'):\n df = pd.read_excel(url,\n usecols=['DATE', 'NEW_CASES', 'POS_TESTS', 'NEG_TESTS', 'TOTAL_TESTS', 'NEW_TESTS'])\n df['date'] = pd.to_datetime(df['DATE'])\n df = df.set_index('date')\n df['Date'] = df.index\n df.to_csv('data/tennessee.csv')\nelse:\n df = pd.read_csv('data/tennessee.csv', index_col='date', parse_dates=['date', 'Date'])\ndf['Positives'] = df['POS_TESTS'].diff()\ndf['Negatives'] = df['NEG_TESTS'].diff()\ndf['Tests'] = df.Positives + df.Negatives\ndf['Odds'] = df.Positives / df.Negatives\ndf = df[['Date', 'Positives', 'Negatives', 'Tests','Odds']]\ndf = df.dropna()\n\n# data on 2020 06 12 unrelaible. see notes\ndf[df['Date']=='2020-06-12'] = np.nan\ndf_dict[dsname] = df\ndataset_info[dsname] = {'Link to source': url,\n 'Link to information': 'https://www.tn.gov/health/cedep/ncov/data/downloadable-datasets.html',\n 'Type': 'File'}\n```\n\n\n```python\ndsname = 'Texas'\nurl = 'https://www.dshs.texas.gov/coronavirus/TexasCOVID19CaseCountData.xlsx'\nif not os.path.isfile('data/texas.csv'):\n cases = pd.read_excel(url, sheet_name='Trends', skiprows=[0], skipfooter=2)\n tests = pd.read_excel(url, sheet_name='Tests by day', skiprows=[0], skipfooter=2)\n cases = cases[cases['Date']!='06/16/20-TDCJ*']\n cases = cases[cases['Date']!='06/16/2020-TDCJ*']\n cases['date'] = pd.to_datetime(cases['Date'])\n cases['Date'] = pd.to_datetime(cases['Date'])\n #tests['date'] = tests['Date']\n tests = tests.set_index('Date')\n tests = tests.loc[tests['Viral Tests']!='.']\n tests['Tests'] = tests['Viral Tests'].diff()\n cases['Positives'] = cases['Daily\\nNew\\nCases']\n df = cases.join(tests, on='date', how='inner')\n df = df.dropna()\n df = df.set_index('date')\n df.to_csv('data/texas.csv')\nelse:\n df = pd.read_csv('data/texas.csv', index_col='date', parse_dates=['date', 'Date'])\ndf['Negatives'] = df.Tests + df.Positives\ndf['Odds'] = df.Positives / df.Negatives\ndf = df[['Date', 'Positives', 'Negatives', 'Tests', 'Odds']]\ndf = df.dropna()\ndf_dict[dsname] = df\ndataset_info[dsname] = {'Link to source': url,\n 'Link to information': 'https://www.dshs.texas.gov/coronavirus/',\n 'Type': 'File'}\n```\n\n\n```python\ndsname = 'Virginia'\nif not os.path.isfile('data/virginia.csv'):\n #df = pd.read_csv('https://www.vdh.virginia.gov/content/uploads/sites/182/2020/05/VDH-COVID-19-PublicUseDataset-Tests_by-LabReportDate.csv')\n df = pd.read_csv('data/VDH-COVID-19-PublicUseDataset-Tests_by-LabReportDate.csv')\n df = df.rename(columns={'Number of Positive PCR Tests': 'Positives', 'Number of PCR Testing Encounters': 'Tests'})\n df = df[df['Lab Report Date']!='Not Reported']\n df['date'] = pd.to_datetime(df['Lab Report Date'])\n df = df.groupby('date').sum()\n df['Date'] = df.index\n df.to_csv('data/virginia.csv')\nelse:\n df = pd.read_csv('data/virginia.csv', index_col='date', parse_dates=['date', 'Date'])\ndf = df[['Date', 'Tests', 'Positives']]\ndf['Negatives'] = df.Tests + df.Positives\ndf['Odds'] = df.Positives / df.Negatives\ndf = df[df['Date']>'2020-03-22']\n\n# lab report day might be lagging \ndf = df[df['Date']0]\n# on the 30th they changed the definition of negative case\ndf = df[df['Date']>'2020-03-30']\ndf_dict[dsname] = df\ndataset_info[dsname] = {'Link to source': url,\n 'Link to information': 'https://data.dhsgis.wi.gov/datasets/covid-19-historical-data-table/data?where=GEO%20%3D%20%27State%27',\n 'Type': 'API'}\n```\n\n\n```python\ndsname = 'Wyoming'\ndf = pd.read_csv('data/wyoming.csv', encoding='utf-16', sep='\\t', thousands=',')\ndf = df.rename(columns={'All': 'Tests'})\ndf['date'] = pd.to_datetime(df['Spec Coll Dt'])\ndf = df.set_index('date')\npos = df.loc[df.Lab.isna(),'Positivity']\nnwphl = df.loc[df.Lab=='Non-WPHL', 'Tests']\nwphl = df.loc[df.Lab=='WPHL', 'Tests']\ndf['Tests'] = nwphl + wphl\ndf['Date'] = df.index\n\ndf['Positivity'] = df.Positivity.str[:-1].astype(float)/100\n#df['Tests'] = df['Tests'].str.replace(',', '').astype(float)\ndf['Positives'] = (df['Tests'] * df['Positivity']).round(0)\ndf['Negatives'] = df.Tests - df.Positives\ndf['Odds'] = df.Positives / df.Negatives\ndf = df[['Date', 'Positives', 'Negatives', 'Tests', 'Odds']]\ndf = df.dropna()\ndf = df[df['Odds']>0]\ndf = df[df['Date']>'2020-03-16']\ndf_dict[dsname] = df\ndataset_info[dsname] = {'Link to source': 'Downloadable from dashboard',\n 'Link to information': 'https://health.wyo.gov/publichealth/infectious-disease-epidemiology-unit/disease/novel-coronavirus/covid-19-testing-data/',\n 'Type': 'API'}\n```\n\n\n```python\n# df = pd.concat(df_dict).reset_index()\n\n# df = df.rename(columns={'level_0': 'State'})\n# df['Odds'] = df['Odds'].astype(float)\n# df['Tests'] = df['Tests'].astype(float)\n# df['Positives'] = df['Positives'].astype(float)\n# df['Tests (right)'] = df['Tests']\n\n```\n\n\n```python\n# dataset_info_df = pd.DataFrame(dataset_info, ).T.reset_index().rename(columns={'index': 'Dataset'})\n# pd.options.display.max_colwidth = 150\n\n# #df.loc[df['Link to source']=='Downloadable from dashboard', 'Link to source'] = df['Link to information']\n# with open('dataset_table.txt', 'w') as fp:\n# dataset_info_df[['Dataset', 'Link to information', 'Type']].to_latex(fp, index=False, label='tab:datasets', caption='Information about the source of the datasets used in this work')\n```\n\n\n```python\n# pd.options.display.max_colwidth\n```\n\n\n```python\n# dsname = 'Argentina'\n# df1 = pd.read_csv('data/argentina/argentina_tests.csv', parse_dates=[0],\n# index_col=0)\n# df2 = pd.read_csv('data/argentina/argentina_tests2.csv',\n# parse_dates=[0],\n# index_col=0)\n# df1['Date'] = df1.index\n# df2['Date'] = df2.index\n# df2['Positives'] = df2.confirmed.diff()\n# df = df1.merge(df2, on='Date', how='outer').fillna(0)\n# df['Positives'] = df[['new_confirmed', 'Positives']].max(axis=1)\n# df['Tests'] = df[['new_tests_x', 'new_tests_y']].max(axis=1)\n# df = df[['Date', 'Positives', 'Tests']].set_index('Date')\n# df['Date'] = df.index\n# df = df.sort_index()\n\n# df = df[df['Date'] > '2020-04-17']\n# df['Odds'] = df.Positives / (df.Tests-df.Positives)\n\n# ax = df.plot.scatter(x='Date', y='Odds')\n# ax.set_yscale('log')\n# df.tail()\n```\n\n## Plots\n\nI we plot the number of positive tests we can see that the data is noisy.\nBut, if we take into account the number of people tested each day, the data looks way more clean.\n\n\n```python\nplt.close(1)\n\ncurrent_palette = sns.color_palette()\n\ndf = pd.concat(df_dict).reset_index()\n\ndf = df.rename(columns={'level_0': 'State'})\ndf['Odds'] = df['Odds'].astype(float)\ndf['Tests'] = df['Tests'].astype(float)\ndf['Positives'] = df['Positives'].astype(float)\ndf['Tests (right)'] = df['Tests']\n#df = df[df['Odds']>0]\n\ng = sns.FacetGrid(df, col='State', col_wrap=3, xlim=[df.Date.min(), df.Date.max()], sharey=False, aspect=1.3, )\nlegend_and_hand = []\ndef myplot(data, y=None, secondary_y=None, legend=None, color=None):\n ax = data.plot(x='Date', y=y[0], ax=plt.gca(), label=y[0])\n legend_and_hand.append(ax.get_legend_handles_labels())\n ax = data.plot(x='Date', y=y[1], secondary_y=secondary_y, ax=ax, label=y[1])\n legend_and_hand.append(ax.get_legend_handles_labels())\n return\ng.map_dataframe(myplot, y=['Positives', 'Tests (right)'], secondary_y=['Tests (right)'])\ng.axes[0].xaxis.set_major_locator(locator)\ng.axes[0].xaxis.set_major_formatter(formatter)\n\ng.add_legend({l[0]: h[0] for h,l in legend_and_hand[:2]}, loc='lower center', bbox_to_anchor=(0.7, 0), ncol=2)\ng.set_titles('{col_name}')\nplt.tight_layout()\nplt.savefig('figs/all_state_testes.jpg', dpi=300)\n```\n\n\n Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous \u2026\n\n\n## Relationship between the total number of infected individuals and positive tests\n\nAs has been shown previously [[1]](https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0002185), the number of new infected individuals in a given day $k_t$ is given by:\n$$\nk_t = k_{t-1} e^{(R_{t-1}-1)\\gamma}\n$$\n\nwhere $R_t$ is the effective reproductuve number and $\\gamma^{-1}$ is the infectious period estimated as 7 days accoring to [2].\n\nThe following derivation was suggested to my by Will Meierjurgen Farr on this GitHub [Issue](https://github.com/k-sys/covid-19/issues/45#issuecomment-623782130). \nSince we do not have access to the total number of infected indiviudals, but only to the population being tested, we have to use some statisticals assumtions on this populations.\nIf we asume that the people being tested, in a given day, is a sample of the population with COVID-19-like sympoms we can state that:\n\n$$\nn_{t} = [P_t(CV|sympoms) P_t(sympoms) +P_t(not CV|sympoms)P_t(sympoms)]Nf_t \n$$\n\nwhere $n_{t}$ is the number of people tested, $P_t(CV|sympoms)$ is the probablity of a pacient being positive for coronavirus given that the she is sympomatic, $P_t(sympoms)$ is the probablity of having COVID-like sympoms, $P_t(not CV|sympoms)$ is the probability of a pacient being coronavirus negative given he has COVID-19-like sympoms, $N$ is the total population, and $f_t$ is the fraction of people with sympoms that is selected to be tested (this number can be different each day, for example if the number of tests availabes changes).\nAlso, note that the probability of a test being positive in a given day is $Positive_t=P_t(CV|sympoms) P_t(sympoms) N f_t$\n\n\nNow, if we assume that $P_t(sympoms|CV)=cte$ we can use Bayes theorem to show that:\n\n$$\nP_t(CV|sympoms) P_t(sympoms) \\propto P_t(CV) = \\frac{k_t}{N}\n$$\n\nThen:\n$$\nP_t(CV|sympoms) P_t(sympoms) \\propto k_t\n$$\n\nFinally, if we assume that $P_t(not CV|sympoms)P_t(sympoms)=cte$:\n$$\nOdds_t = \\frac{P(CV|sympoms) P(sympoms)Nf_t}{P_t(not CV|sympoms)P_t(sympoms)Nf_t}\n$$\n$$\nOdds_t = \\frac{P(CV|sympoms) P(sympoms)}{P_t(not CV|sympoms)P_t(sympoms)}\n$$\n$$\nOdds_t \\propto k_t\n$$\n\n\\begin{equation}\nOdds_t = Odds_{t-1} e^{(R_{t-1}-1)\\gamma} (1)\n\\end{equation}\n\nWe used three hypothesis. First, constant population $N$ (for $P_t \\propto k_t$ and for the evolution of $k_t$). Second, that the tested population is a random sample from the population with COVID19-like sympoms ($n_t = [P_t(CV|sympoms) P_t(sympoms) +P_t(not CV|sympoms)P_t(sympoms)]Nf_t$) this is not the case when people is being tested based on contacts for example. And third, that $P_t(not CV|sympoms)P_t(sympoms)=cte$, this is equivalent to say that the number of people with covid-19-like sympoms but without the coronavirus (for example people with the flu) is constant, or at least it changes are negligible compared with the changes in the amount of sympomatic people with coronavirus.\n\n## Linearization\n\nDefining\n\n$$\nb_i = e^{(R_{i-1}-1)\\gamma} (2)\n$$\n\nWe can write [1] as:\n\n\\begin{equation}\nodd_i = b_{i-1} * odd_{i-1} (3)\n\\end{equation}\n\nNow, instead of using $b_i$ as the parameters to estimate we decompose each $b$ as follows:\n\n$$\nb_i = \\prod_{j=0}^{i} a_j (4)\n$$\n\nNow, the $a_j$ represent the rate of change of the variable $b_i$. Next, we replace the [4] in [3]\n$$\nodd_i = \\prod_{j=0}^{i-1} a_j * odd_{i-1}\n$$\n$$\nodd_i = \\prod_{j=0}^{i-1} a_j * \\prod_{j=0}^{i-2} a_j * odd_{i-2}\n$$\n$$\nodd_i = \\prod_{k=0}^{i-1}\\prod_{j=0}^{k} a_j * odd_{0}\n$$\n$$\nodd_i = \\prod_{j=0}^{i-1} a_j^{i-j} * odd_{0}\n$$\n\nNow, we liniarize this result and we generalize it to the case where $i=0$ using the $max$ function:\n\n$$\nlog(odd_i) = \\sum_{j=0}^{max(i-1, 0)} (i-j)log(a_j) + log(odd_{0}) (5)\n$$\n\nWe can write [5] as a linear problem with the following definitions:\n\n$$\ny = X \\beta + \\beta_0\n$$\n\n$$\ny_i = log(odd_i)\n$$\n\n$$\nX_{i,j} = max(i-j, 0)\n$$\n\n$$\n\\beta_i = log(a_i) (6)\n$$\n\nNow if we apply a LASSO regression we will find the solution that minimize the following cost function\n\n$$\nErr = \\sum (y-\\hat{y})^2 + \\alpha |\\beta|_1\n$$\n\nHopefully, this solution will be sparse which means that most of the $\\beta_i$ will be $0$, and hence $a_i=1$.\nThis is equivalent to say that the $b_i$ are almost constant except at the values whete $a_i \\neq 1$.\n\n\n\n[1] Bettencourt, L. M. A., & Ribeiro, R. M. (2008). Real time bayesian estimation of the epidemic potential of emerging infectious diseases. PLoS ONE, 3(5). https://doi.org/10.1371/journal.pone.0002185\n\n[2] Sanche, S., Lin, Y. T., Xu, C., Romero-Severson, E., Hengartner, N., & Ke, R. (2020). High Contagiousness and Rapid Spread of Severe Acute Respiratory Syndrome Coronavirus 2. Emerging Infectious Diseases, 26(7). https://doi.org/10.3201/eid2607.200282\n\n\n\n## Classes\n\nThis cell contains the main class: LassoICSelector. Its main method is fit_best_alpha. It works as follows:\n```\nFor each alpha value:\n 1. Fits a lasso regression to the data\n 2. Selectes the first non zero variable from each chunck\n 3. Fits a linear regression with the selected variables\n 4. Excludes all non sifgificative (p-value>0.05) variables and fits a linear model again\n\nThe linear model with less AIC from step 4 is selected.\n```\n\n\n\n\n```python\nclass FirstInChunkSelector(object):\n '''Selects first element from each non zero chunk.'''\n\n def __init__(self, clf):\n self.clf = clf\n self.coef = None\n self.mask = None\n\n def select_coef(self):\n n_features = len(self.clf.coef_)\n no_zero = np.zeros(n_features+1)\n no_zero[1:] = self.clf.coef_ != 0\n #v = np.hstack([np.zeros(np.int(1/GAMMA-2)), np.ones(np.int(1/GAMMA-1))])\n #no_zero[1:] = np.convolve(self.clf.coef_ != 0, v, mode='same') > 0\n self.mask = np.diff(no_zero)>0\n self.mask[0] = True\n self.coef = self.clf.coef_[self.mask]\n return self.coef\n\n def transform(self, X):\n self.select_coef()\n return X[:, self.mask]\n\n def get_support(self):\n self.select_coef()\n return self.mask\n\n def get_number_of_features(self):\n self.select_coef()\n return sum(self.mask)\n\n\nclass LassoICSelector(object):\n \"\"\"LASSO regression with FirstInChunk selector.\"\"\"\n\n def __init__(self, X, y, criterion, alpha=0.05):\n self.lasso = linear_model.LassoLars(alpha=0, max_iter=100000)\n self.criterion = criterion\n self.selector = FirstInChunkSelector(self.lasso)\n self.OLS = sm.OLS\n #self.OLS = sm.RLM\n self.ols = self.OLS(y, X)\n\n self.ols_results = None\n self.X = X\n self.y = y\n self.final_ols = False\n self.alpha = alpha\n\n def transform_to_ols(self, X):\n '''Selects only the features of X are used by OLS.\n Also, adds a coloumn with ones for the intercept.\n '''\n\n X_new = self.selector.transform(X)\n if self.final_ols:\n X_new = X[:, self.support]\n X_new_with_cte = np.hstack([X_new, np.ones((X_new.shape[0], 1))])\n return X_new_with_cte\n\n def fit(self, X, y):\n '''Selects features and fits the OLS.'''\n\n # select features\n X_new = self.transform_to_ols(X)\n\n # fit ols\n self.ols = self.OLS(y, X_new)\n self.ols_results = self.ols.fit()\n\n # iteratively remove non signicative variables and fit again\n mask = self.ols_results.pvalues < self.alpha / len(self.ols_results.pvalues)\n mask[0] = True\n Xnew = self.transform_to_ols(X)\n Xnew = Xnew[:, mask]\n self.support = self.selector.get_support()\n self.ols = self.OLS(y, Xnew)\n self.ols_results = self.ols.fit()\n while any(self.ols_results.pvalues[1:] >= self.alpha / len(self.ols_results.pvalues)):\n mask.values[mask.values] = (self.ols_results.pvalues < self.alpha / len(self.ols_results.pvalues)).values\n mask[0] = True\n Xnew = self.transform_to_ols(X)\n Xnew = Xnew[:, mask]\n self.support = self.selector.get_support()\n self.ols = self.OLS(y, Xnew)\n self.ols_results = self.ols.fit()\n\n self.support[self.support] = mask[:-1]\n\n def fit_best_alpha(self, X, y):\n '''returns the model with the lowst cirterion.'''\n\n self.lasso.fit(X, y)\n alphas = self.lasso.alphas_\n self.criterions_ = np.zeros(len(alphas))\n self.log_liklehods = np.zeros(len(alphas))\n \n \n for i, alpha in enumerate(alphas):\n self.lasso.coef_ = self.lasso.coef_path_[:, i]\n self.fit(X, y)\n self.criterions_[i], self.log_liklehods[i] = self.get_criterion(self.ols.exog, y)\n \n # we use a list of tuples to find the minimum cirterion value.\n # If there are ties, we use the maximum alpha value.\n criterions_idx = list(zip(self.criterions_, alphas, range(len(alphas))))\n criterion, alpha, idx = min(criterions_idx, key=lambda x: (x[0], -x[1]))\n \n self.lasso.coef_ = self.lasso.coef_path_[:, idx]\n self.lasso.alpha = alpha\n self.fit(X, y)\n self.final_ols = True\n\n def predict(self, X):\n '''Predicts y useing the OLS fit.'''\n\n return self.ols.predict(self.ols_results.params, X)\n\n def log_liklihood(self, X, y):\n '''Computes the log liklihood assuming normally distributed errors.'''\n\n eps64 = np.finfo('float64').eps\n\n # residuals\n R = y - self.predict(X)\n sigma2 = np.var(R)\n\n loglike = -0.5 * len(R) * np.log(sigma2)\n loglike -= 0.5 * len(R) * np.log(2*np.pi) - 0.5*len(R) + 0.5\n return loglike\n\n def get_criterion(self, X, y):\n '''Computes AIC or BIC criterion.'''\n\n n_samples = X.shape[0]\n if self.criterion == 'aic':\n K = 2 # AIC\n elif self.criterion == 'bic':\n K = np.log(n_samples)\n else:\n raise ValueError('criterion should be either bic or aic')\n\n log_like = self.log_liklihood(X, y)\n df = X.shape[1]\n\n aic = K * df - 2*log_like\n self.criterion_ = aic\n\n return self.criterion_, log_like\n```\n\n## Fit\nNow, we create the linear system and fit the model\n\n\n```python\nplt.close('all')\nlics_dict = {}\n# fig, axes = plt.subplots(nrows=17, ncols=3, figsize=(12, 8))\ngof_list = []\nfor i, state in enumerate(df.State.unique()):\n dfstate = df[df['State']==state]\n #dfstate = dfstate[(dfstate.Odds.notna()) & (dfstate.Odds!=0)]\n # create the independent and the dependent variables\n y = np.log(dfstate['Odds'])\n X = np.tri(len(y))\n X = np.cumsum(X, axis=0)[:, 1:]\n X = X[(dfstate.Odds.notna()) & (dfstate.Odds!=0), :]\n y = y[(dfstate.Odds.notna()) & (dfstate.Odds!=0)]\n\n # create lasso instance\n lics = LassoICSelector(X, y.values, 'bic', alpha=0.01)\n\n # fit\n lics.fit_best_alpha(X, y)\n lics_dict[state] = lics\n\n # ax = sns.lineplot(lics.lasso.alphas_, lics.criterions_, ax=axes[i, 2])\n # ax.vlines(lics.lasso.alpha, min(lics.criterions_), max(lics.criterions_))\n # ax.set_ylabel('BIC')\n # ax.set_xlabel('Alpha')\n # ax.set_xscale('log')\n # axes[i, 0].plot(lics.lasso.alphas_, lics.lasso.coef_path_.T)\n # axes[i, 0].set_xscale('log')\n # axes[i, 0].set_title(state)\n # axes[i, 0].set_xlabel('Alpha')\n # axes[i, 0].set_ylabel('Coefficient Value')\n # axes[i, 0].vlines(lics.lasso.alpha, lics.lasso.coef_path_.min(), lics.lasso.coef_path_.max())\n # axes[i, 1].plot(lics.lasso.coef_)\n # axes[i, 1].scatter(np.arange(len(lics.lasso.coef_))[lics.support], lics.lasso.coef_[lics.support])\n # axes[i, 1].set_xlabel('Coefficient #')\n # axes[i, 1].set_ylabel('Coefficient Value')\n \n gof_list.append({\"State\":state, '$R^2$':lics.ols_results.rsquared, \"N\": int(lics.ols_results.nobs), \"df_model\": int(lics.ols_results.df_model), \"fvalue\":lics.ols_results.fvalue, \"f_pvalue\":lics.ols_results.f_pvalue})\n #print('{:<13} & {:.2f} & {:n} & {:n} & {:.2f} & {:.2e}'.format(state, lics.ols_results.rsquared, lics.ols_results.nobs, lics.ols_results.df_model, lics.ols_results.fvalue, lics.ols_results.f_pvalue))\n #print(state)\n #display(lics.ols_results.summary())\ngof = pd.DataFrame(gof_list).sort_values('$R^2$', ascending=False).rename(columns={'df_model': 'D.F.', 'fvalue': 'F-statistic', 'f_pvalue': 'p-value'})\nout = gof.to_latex(index=False,formatters={\"$R^2$\": \"{:0.3f}\".format, 'fvalue':\"{:0.2f}\".format, 'f_pvalue': \"{:.2e}\".format},\n bold_rows=True, escape=False, label='tab:gof', caption='Goodness of fit for each dataset.')\nprint(out)\n```\n\n /home/andres/anaconda3/lib/python3.7/site-packages/pandas/core/series.py:679: RuntimeWarning: divide by zero encountered in log\n result = getattr(ufunc, method)(*inputs, **kwargs)\n /home/andres/anaconda3/lib/python3.7/site-packages/pandas/core/series.py:679: RuntimeWarning: divide by zero encountered in log\n result = getattr(ufunc, method)(*inputs, **kwargs)\n /home/andres/anaconda3/lib/python3.7/site-packages/pandas/core/series.py:679: RuntimeWarning: divide by zero encountered in log\n result = getattr(ufunc, method)(*inputs, **kwargs)\n\n\n \\begin{table}\n \\centering\n \\caption{Goodness of fit for each dataset.}\n \\label{tab:gof}\n \\begin{tabular}{lrrrrr}\n \\toprule\n State & $R^2$ & N & D.F. & F-statistic & p-value \\\\\n \\midrule\n New York & 0.997 & 119 & 10 & 3992.658934 & 7.877846e-134 \\\\\n Massachusetts & 0.987 & 92 & 4 & 1711.946879 & 8.455634e-82 \\\\\n Michigan & 0.983 & 89 & 5 & 987.866820 & 2.324393e-72 \\\\\n Connecticut & 0.981 & 88 & 3 & 1438.831575 & 4.534263e-72 \\\\\n Rhode Island & 0.948 & 98 & 2 & 864.776556 & 1.088772e-61 \\\\\n Virginia & 0.918 & 86 & 3 & 305.468584 & 2.184829e-44 \\\\\n Delaware & 0.886 & 92 & 3 & 228.607283 & 2.038413e-41 \\\\\n Iowa & 0.872 & 86 & 3 & 185.719315 & 1.862787e-36 \\\\\n Wisconsin & 0.811 & 83 & 2 & 171.824297 & 1.105016e-29 \\\\\n Colorado & 0.804 & 96 & 2 & 191.334575 & 1.095911e-33 \\\\\n Minnesota & 0.640 & 84 & 2 & 71.849515 & 1.131865e-18 \\\\\n Alaska & 0.586 & 107 & 2 & 73.626389 & 1.202176e-20 \\\\\n Indiana & 0.472 & 93 & 3 & 26.540667 & 2.359064e-12 \\\\\n Tennessee & 0.398 & 88 & 2 & 28.101998 & 4.283584e-10 \\\\\n Wyoming & 0.203 & 84 & 2 & 10.312295 & 1.023745e-04 \\\\\n Texas & 0.079 & 34 & 1 & 2.752898 & 1.068493e-01 \\\\\n \\bottomrule\n \\end{tabular}\n \\end{table}\n \n\n\nLets copy the fitted values to a dataframe, and calculate the parameters and erros of the model.\n\n\n```python\ndata_list = []\nfor state in df.State.unique():\n print(state)\n lics = lics_dict[state]\n data = df[df['State']==state].copy()\n #data = data[(data.Odds.notna()) & (data.Odds!=0)]\n # yhat = lics.ols_results.fittedvalues\n y = np.log(data['Odds'])\n X = np.tri(len(y))\n X = np.cumsum(X, axis=0)[:, 1:]\n X = X[(data.Odds.notna()) & (data.Odds!=0), :]\n y = y[(data.Odds.notna()) & (data.Odds!=0)]\n data = data[(data.Odds.notna()) & (data.Odds!=0)]\n Xols = lics.transform_to_ols(X)\n yhat = lics.ols.predict(lics.ols_results.params, Xols)\n # from equation 5\n odds_hat = np.exp(yhat)\n\n # the error in yhat is\n # Xols = lics.transform_to_ols(X)\n (yhat_std, yhat_l, yhat_u) = wls_prediction_std(lics.ols_results, Xols)\n\n # propagation of errors\n #oddshat_std = np.array([exp_x_sigma(mu, s)[0] for mu, s in zip(yhat, yhat_std)])#odds_hat*yhat_std\n #oddshat_std = exp_x_sigma(yhat, yhat_std)\n oddshat_l = np.exp(yhat-2*yhat_std)\n oddshat_u = np.exp(yhat+2*yhat_std)\n data.loc[:, 'odds_hat'] = odds_hat\n #data.loc[:, 'oddshat_std'] = oddshat_std\n #data.loc[:, 'oddshat_l'] = odds_hat - 2*oddshat_std\n #data.loc[:, 'oddshat_u'] = odds_hat + 2*oddshat_std\n data.loc[:, 'oddshat_l'] = oddshat_l\n data.loc[:, 'oddshat_u'] = oddshat_u\n\n # use coefficients to calculate Rt\n coef = np.zeros(len(data))\n coef_std = np.zeros_like(coef) * np.nan\n ind = np.squeeze(np.argwhere(lics.support))\n\n # we do not use the last coefficient since it's the intercept (=log(odds_0))\n coef[ind] = lics.ols_results.params[:-1]\n\n # using equation 2, 4 and 6\n data.loc[:, 'R'] = np.cumsum(coef)/GAMMA+1\n\n # get covarinace matrix of coefficients\n cov = lics.ols_results.cov_params().values\n\n # since the values of Rts are a sum of variables, we use the formula\n # of the sum of gaussian variables with a known covariance matrix\n stds = [np.sqrt(cov[:n, :n].sum()) for n in range(1, cov.shape[0])]\n if len(stds)==1:\n stds = stds[0]\n coef_std[ind] = stds\n\n # error propagation formula\n data.loc[:, 'Rstd'] = coef_std / GAMMA\n\n data['Rstd'] = data['Rstd'].fillna(method='ffill')\n data['R_l'] = data['R'] - 2*data['Rstd']\n data['R_u'] = data['R'] + 2*data['Rstd']\n\n r_index = data.R.diff() != 0\n Rts = data.loc[r_index, ['Date', 'R', 'R_l', 'R_u']]\n Rts['delta'] = Rts['R_u'] - Rts['R_l']\n display(Rts)\n data_list.append(data)\ndata = pd.concat(data_list)\n```\n\n Alaska\n\n\n /home/andres/anaconda3/lib/python3.7/site-packages/pandas/core/series.py:679: RuntimeWarning: divide by zero encountered in log\n result = getattr(ufunc, method)(*inputs, **kwargs)\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        DateRR_lR_udelta
                                        02020-03-240.6146590.5511200.6781980.127078
                                        582020-05-211.2616451.1936541.3296370.135983
                                        \n
                                        \n\n\n Colorado\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        DateRR_lR_udelta
                                        1102020-03-181.2405051.1520751.3289360.176861
                                        1382020-04-150.7110520.6799420.7421620.062220
                                        \n
                                        \n\n\n Connecticut\n\n\n /home/andres/anaconda3/lib/python3.7/site-packages/pandas/core/series.py:679: RuntimeWarning: divide by zero encountered in log\n result = getattr(ufunc, method)(*inputs, **kwargs)\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        DateRR_lR_udelta
                                        2062020-03-201.9754271.7827462.1681080.385362
                                        2152020-03-290.9510310.8784391.0236230.145184
                                        2302020-04-130.6061560.5916900.6206220.028932
                                        \n
                                        \n\n\n Delaware\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        DateRR_lR_udelta
                                        2952020-03-231.4975091.4217111.5733070.151596
                                        3242020-04-210.5810460.5448540.6172380.072385
                                        3762020-06-121.0914790.7899411.3930160.603075
                                        \n
                                        \n\n\n Indiana\n\n\n /home/andres/anaconda3/lib/python3.7/site-packages/pandas/core/series.py:679: RuntimeWarning: divide by zero encountered in log\n result = getattr(ufunc, method)(*inputs, **kwargs)\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        DateRR_lR_udelta
                                        3872020-03-191.9711421.6798362.2624470.582611
                                        4032020-04-040.8142500.7683770.8601240.091748
                                        4762020-06-166.5104753.9387249.0822265.143501
                                        \n
                                        \n\n\n Iowa\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        DateRR_lR_udelta
                                        4812020-03-221.3888671.3392671.4384670.099200
                                        5132020-04-230.6021670.5435510.6607820.117231
                                        5362020-05-160.8217630.7677320.8757950.108063
                                        \n
                                        \n\n\n Massachusetts\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        DateRR_lR_udelta
                                        5672020-03-161.8930451.8079771.9781130.170136
                                        5792020-03-281.1032721.0609471.1455980.084650
                                        5952020-04-130.6811100.6703400.6918800.021540
                                        6502020-06-070.3152840.1937820.4367860.243003
                                        \n
                                        \n\n\n Michigan\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        DateRR_lR_udelta
                                        6592020-03-191.3728421.2940771.4516070.157529
                                        6762020-04-050.4368840.3989570.4748110.075854
                                        7032020-05-020.7667590.7257730.8077440.081971
                                        7292020-05-280.4451900.3220200.5683600.246340
                                        7402020-06-081.2911521.0113471.5709570.559609
                                        \n
                                        \n\n\n Minnesota\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        DateRR_lR_udelta
                                        7482020-03-281.4624611.3464541.5784690.232015
                                        7782020-04-270.6486310.5896970.7075660.117869
                                        \n
                                        \n\n\n New York\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        DateRR_lR_udelta
                                        8322020-03-151.8760191.8109561.9410820.130126
                                        8462020-03-290.7295730.6862230.7729230.086701
                                        8622020-04-140.4295670.4041400.4549940.050855
                                        8862020-05-080.5025550.4810100.5241000.043090
                                        9152020-06-060.7564870.6860650.8269090.140844
                                        9262020-06-171.2099781.0771521.3428040.265653
                                        9342020-06-250.163532-0.2624690.5895340.852003
                                        9372020-06-281.9442961.5622752.3263160.764041
                                        9412020-07-02-0.583596-1.2667690.0995771.366346
                                        9432020-07-041.1828590.9778241.3878950.410071
                                        \n
                                        \n\n\n Rhode Island\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        DateRR_lR_udelta
                                        9512020-04-011.0633580.9782061.1485090.170303
                                        9692020-04-190.7410790.7269300.7552270.028296
                                        \n
                                        \n\n\n Tennessee\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        DateRR_lR_udelta
                                        10492020-03-250.8237080.7752390.8721760.096937
                                        10992020-05-141.1961401.1281251.2641550.136030
                                        \n
                                        \n\n\n Texas\n\n\n\n
                                        \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                        DateRR_lR_udelta
                                        11382020-05-161.1779630.9634451.3924810.429037
                                        \n
                                        \n\n\n Virginia\n\n\n\n
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                                        DateRR_lR_udelta
                                        11722020-03-231.1276131.1030381.1521890.049151
                                        11992020-04-190.9054420.8877800.9231040.035324
                                        12302020-05-200.8066080.7820330.8311840.049151
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                                        \n\n\n Wisconsin\n\n\n\n
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                                        DateRR_lR_udelta
                                        12582020-03-311.0091250.9432011.0750500.131849
                                        12842020-04-260.7974180.7710250.8238110.052787
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                                        \n\n\n Wyoming\n\n\n\n
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                                        DateRR_lR_udelta
                                        13412020-03-172.1743021.5005212.8480831.347562
                                        13502020-03-260.8955470.8448870.9462060.101319
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                                        \n\n\n\n```python\nres = lics_dict['Delaware'].ols_results\ndir(res)\n#res.params\npd.Series(2*np.diag(res.cov_params())**0.5, index=res.cov_params().index)\n```\n\n\n\n\n x1 0.010106\n x2 0.013502\n x3 0.042731\n const 0.215323\n dtype: float64\n\n\n\n\n```python\nplt.close(1)\nstacked = data.set_index(['Date', 'State'])[['Odds', 'odds_hat', 'oddshat_u', 'oddshat_l']]\nstacked = stacked.stack().reset_index().rename(columns={'level_2': 'variable', 0: 'val'})\nstacked['Branch'] = 'Fit'\nstacked.loc[stacked['variable'] == 'Odds', 'Branch'] = 'Data'\nstacked = stacked.set_index(['Date', 'State', 'Branch', 'variable']).unstack().reset_index()\nstacked.columns = ['Date', 'State', 'Branch', 'Odds', 'odds_hat', 'oddshat_u', 'oddshat_l']\nstacked = stacked.sort_values(['State', 'Date'])\n#stacked = stacked[stacked.State.isin(['Alaska', 'Colorado', 'Connecticut'])]\n\n\n\ng1 = sns.FacetGrid(stacked, col='State', col_wrap=3, hue='Branch', aspect=1.5, dropna=False, hue_order=['Data', 'Fit'], sharey=False, col_order=gof.State)\nlegend_and_hand = []\ndef myplot(data, x=None, y=None, color=None, label=None):\n ax = plt.gca()\n ax = data.plot(x=x, y=y, ax=ax, label=label)\n legend_and_hand.append(ax.get_legend_handles_labels())\n return ax\n#g1.map_dataframe(myplot, x='Date', y='odds_hat')\ng1.map_dataframe(sns.lineplot, x='Date', y='odds_hat')\ng1.map_dataframe(sns.scatterplot, x='Date', y='Odds')\n\ng1.map(plt.fill_between, 'Date', 'oddshat_l', 'oddshat_u', alpha=0.1)\nhl = g1.axes[0].get_legend_handles_labels()\n\nhandlers = [hl[0][1], hl[0][0]]\nlabels = [hl[1][1], hl[1][0]]\n\nlh = {l:h for h,l in zip(handlers, labels)}\ng1.add_legend(lh, loc='lower center', bbox_to_anchor=(0.7, 0), title='', ncol=3)#\n\n# plt.yscale('log')\n# plt.xlabel('')\ng1.axes[0].xaxis.set_major_locator(locator)\ng1.axes[0].xaxis.set_major_formatter(formatter)\ng1.set(xlim=[data['Date'].min(), data.loc[data.odds_hat.notna(), 'Date'].max()], ylabel='Odds', yscale='log')\ng1.set_titles(\"{col_name}\")\nplt.tight_layout()\nplt.savefig('figs/all_states_OddsL1.jpg', dpi=300)\nplt.show()\n```\n\n\n Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous \u2026\n\n\n\n```python\nplt.close('all')\n\ng1 = sns.FacetGrid(data.set_index('date'), col='State', col_wrap=3, aspect=1.5, dropna=False, ylim=[0, 2.5])\ndef myplot(data, y=None, color=None):\n data.plot(x='Date', y=y, ax=plt.gca())\ng1.map_dataframe(sns.lineplot, x='Date', y='R')\n#g1.map_dataframe(myplot, y='R')\ng1.map(plt.fill_between, 'Date', 'R_u', 'R_l', alpha=0.2)\n#ax = data.plot(x='Date', y='R', legend=False)\n# lines = []\n# for ax, dsname in zip(g1.axes[:, 0], data.State.unique()):\n# for line in events[dsname]:\n# if line < pd.to_datetime('03-04-2020 00:00', dayfirst=True):\n# color = current_palette[1]\n# label = 'Mobility restrictions'\n# elif line == pd.to_datetime('03-04-2020 00:00', dayfirst=True):\n# color = 'k'\n# label = 'Masks (CDC)'\n# else:\n# label = 'Masks (State)'\n# color = current_palette[2]\n# lines.append(ax.axvline(line, 0,1, linestyle='--', color=color, label=label))\n\n\ng1.set(ylabel='$R_t$')\ng1.set(yscale='linear')\ng#1.axes[0, 0].legend([lines[0], lines[4], lines[5]], ['Mobility restrictions', 'Masks (CDC)', 'Masks (Local)'])\n#print(g1.axes[0, 0].legend().get_legend_handler_map())\ng1.axes[0].xaxis.set_major_locator(locator)\ng1.axes[0].xaxis.set_major_formatter(formatter)\nplt.xlabel('')\ng1.set_titles('{col_name}')\nplt.tight_layout()\nplt.savefig('figs/all_states_Rt.jpg', dpi=300)\n```\n\n\n Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous \u2026\n\n", "meta": {"hexsha": "26008ffaf9a6c534dd415749dc7ca33971ae46c2", "size": 101014, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "Binomial_L1_LR_AIC_states_all.ipynb", "max_stars_repo_name": "ababino/corona", "max_stars_repo_head_hexsha": "dbccecbb7213777b48cc6e9273a3b62bf2190bfd", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-08-05T05:02:00.000Z", "max_stars_repo_stars_event_max_datetime": "2020-08-05T05:02:00.000Z", "max_issues_repo_path": "Binomial_L1_LR_AIC_states_all.ipynb", "max_issues_repo_name": "ababino/corona", "max_issues_repo_head_hexsha": "dbccecbb7213777b48cc6e9273a3b62bf2190bfd", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Binomial_L1_LR_AIC_states_all.ipynb", "max_forks_repo_name": "ababino/corona", "max_forks_repo_head_hexsha": "dbccecbb7213777b48cc6e9273a3b62bf2190bfd", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-03-28T23:03:08.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-28T23:03:08.000Z", "avg_line_length": 36.1927624507, "max_line_length": 706, "alphanum_fraction": 0.4760429247, "converted": true, "num_tokens": 21606, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878696277513, "lm_q2_score": 0.665410558746814, "lm_q1q2_score": 0.40928596300738945}} {"text": "# TTcrystal - Define the crystal parameters\n\nIn this file the usage of TTcrystal class is demonstrated. TTcrystal holds all the necessary information about the crystal, it's reflection and deformation that is passed as an input to the TT-solver. Let's start by doing some imports:\n\n\n```python\nimport sys\nimport os.path\n\nsys.path.insert(1, '..')\n\nfrom pyTTE import TTcrystal, Quantity\n```\n\nTTcrystal object can be initialized either by passing the parameters of the crystal as keyword arguments, or by reading them from a file. Let's examine the former case first.\n\nThe initialization of the class requires at least the following three parameters: _crystal_, _hkl_, _thickness_. This initializes a symmetric Bragg case of reflection $(hkl)$ of a perfect, non-deformed crystal. For example:\n\n\n```python\nxtal = TTcrystal(crystal = 'Si', hkl=[6,6,0], thickness = Quantity(1,'mm'))\n\nprint(xtal)\n```\n\n Crystal: Si\n Crystallographic parameters:\n a = 0.543069 nm, b = 0.543069 nm, c = 0.543069 nm\n alpha = 90.0 deg, beta = 90.0 nm, gamma = 90.0 deg\n Direct primitive vectors (before rotations, in nm):\n a1 = [0.5431 0. 0. ]\n a2 = [0. 0.5431 0. ]\n a3 = [0. 0. 0.5431]\n Reciprocal primitive vectors (before rotations, in 1/nm):\n b1 = [11.5698 -0. -0. ]\n b2 = [ 0. 11.5698 -0. ]\n b3 = [ 0. 0. 11.5698]\n \n Reflection: [6, 6, 0]\n Asymmetry angle: 0 deg\n In-plane rotation angle: 0 deg\n Crystal directions parallel to the Cartesian axes (after rotations):\n x || [ 0.7071 -0.7071 -1. ]\n y || [-0.7071 0.7071 -1. ]\n z || [ 1. 1. -0.]\n \n Crystal thickness: 1 mm\n Debye-Waller factor: 1.0\n \n Deformation model: anisotropic toroidal, fixed shape (built-in)\n Meridional bending radius: inf m\n Sagittal bending radius: inf m\n Material elastic isotropy: anisotropic\n Compliance matrix S (with rotations applied):\n [[ 0.0055 -0.0008 -0.0013 0. 0. 0.0009]\n [-0.0008 0.0055 -0.0013 -0. 0. 0.0009]\n [-0.0013 -0.0013 0.0059 -0. -0. -0.0018]\n [ 0. -0. -0. 0.0161 -0.0035 0. ]\n [-0. 0. -0. -0.0035 0.0161 0. ]\n [ 0.0009 0.0009 -0.0018 0. 0. 0.0179]] GPa^-1\n\n\nThe crystallographic parameters are read from _xraylib_. The elastic tensor data is saved in _pyTTE.elastic_tensors.py_.\n\nTTcrystal has also many optional parameters to define e.g. asymmetry angle, in plane rotation, deformation andd so on. For extensive list use `help(TTcrystal)`. As an example, a Ge(555) reflection in the Laue case with 5 degree asymmetry and the Debye-Waller factor of 0.8 is defined as follows:\n\n\n```python\nxtal = TTcrystal(crystal = 'Ge', hkl=[5,5,5], thickness = Quantity(1,'mm'), asymmetry = Quantity(95,'deg'), debye_waller = 0.8)\n\nprint(xtal)\n```\n\n Crystal: Ge\n Crystallographic parameters:\n a = 0.565735 nm, b = 0.565735 nm, c = 0.565735 nm\n alpha = 90.0 deg, beta = 90.0 nm, gamma = 90.0 deg\n Direct primitive vectors (before rotations, in nm):\n a1 = [0.5657 0. 0. ]\n a2 = [0. 0.5657 0. ]\n a3 = [0. 0. 0.5657]\n Reciprocal primitive vectors (before rotations, in 1/nm):\n b1 = [11.1062 -0. -0. ]\n b2 = [ 0. 11.1062 -0. ]\n b3 = [ 0. 0. 11.1062]\n \n Reflection: [5, 5, 5]\n Asymmetry angle: 95 deg\n In-plane rotation angle: 0 deg\n Crystal directions parallel to the Cartesian axes (after rotations):\n x || [0.8097 0.949 1. ]\n y || [-0.2679 1. -0.7321]\n z || [-1. 0.1916 0.6278]\n \n Crystal thickness: 1 mm\n Debye-Waller factor: 0.8\n \n Deformation model: anisotropic toroidal, fixed shape (built-in)\n Meridional bending radius: inf m\n Sagittal bending radius: inf m\n Material elastic isotropy: anisotropic\n Compliance matrix S (with rotations applied):\n [[ 0.0065 -0.0009 -0.0012 -0.0003 0.0005 -0. ]\n [-0.0009 0.0073 -0.002 -0.0001 0.0015 0.0017]\n [-0.0012 -0.002 0.0076 0.0004 -0.002 -0.0016]\n [-0.0003 -0.0001 0.0004 0.0178 -0.0032 0.003 ]\n [ 0.0005 0.0015 -0.002 -0.0032 0.0209 -0.0006]\n [-0. 0.0017 -0.0016 0.003 -0.0006 0.0222]] GPa^-1\n\n\n(Note that the symmetric Laue case would be defined by `asymmetry = Quantity(90,'deg')`).\n\nIt is also possible to adjust the crystal parameters after initialization:\n\n\n```python\nxtal = TTcrystal(crystal = 'Si', hkl=[6,6,0], thickness = Quantity(1,'mm'))\n\nprint('BEFORE ADJUSTMENT:')\nprint(xtal)\n\nxtal.set_thickness(Quantity(500,'um'))\nxtal.set_in_plane_rotation(Quantity(-45,'deg'))\n\nprint('\\nAFTER ADJUSTMENT:')\nprint(xtal)\n```\n\n BEFORE ADJUSTMENT:\n Crystal: Si\n Crystallographic parameters:\n a = 0.543069 nm, b = 0.543069 nm, c = 0.543069 nm\n alpha = 90.0 deg, beta = 90.0 nm, gamma = 90.0 deg\n Direct primitive vectors (before rotations, in nm):\n a1 = [0.5431 0. 0. ]\n a2 = [0. 0.5431 0. ]\n a3 = [0. 0. 0.5431]\n Reciprocal primitive vectors (before rotations, in 1/nm):\n b1 = [11.5698 -0. -0. ]\n b2 = [ 0. 11.5698 -0. ]\n b3 = [ 0. 0. 11.5698]\n \n Reflection: [6, 6, 0]\n Asymmetry angle: 0 deg\n In-plane rotation angle: 0 deg\n Crystal directions parallel to the Cartesian axes (after rotations):\n x || [ 0.7071 -0.7071 -1. ]\n y || [-0.7071 0.7071 -1. ]\n z || [ 1. 1. -0.]\n \n Crystal thickness: 1 mm\n Debye-Waller factor: 1.0\n \n Deformation model: anisotropic toroidal, fixed shape (built-in)\n Meridional bending radius: inf m\n Sagittal bending radius: inf m\n Material elastic isotropy: anisotropic\n Compliance matrix S (with rotations applied):\n [[ 0.0055 -0.0008 -0.0013 0. 0. 0.0009]\n [-0.0008 0.0055 -0.0013 -0. 0. 0.0009]\n [-0.0013 -0.0013 0.0059 -0. -0. -0.0018]\n [ 0. -0. -0. 0.0161 -0.0035 0. ]\n [-0. 0. -0. -0.0035 0.0161 0. ]\n [ 0.0009 0.0009 -0.0018 0. 0. 0.0179]] GPa^-1\n \n AFTER ADJUSTMENT:\n Crystal: Si\n Crystallographic parameters:\n a = 0.543069 nm, b = 0.543069 nm, c = 0.543069 nm\n alpha = 90.0 deg, beta = 90.0 nm, gamma = 90.0 deg\n Direct primitive vectors (before rotations, in nm):\n a1 = [0.5431 0. 0. ]\n a2 = [0. 0.5431 0. ]\n a3 = [0. 0. 0.5431]\n Reciprocal primitive vectors (before rotations, in 1/nm):\n b1 = [11.5698 -0. -0. ]\n b2 = [ 0. 11.5698 -0. ]\n b3 = [ 0. 0. 11.5698]\n \n Reflection: [6, 6, 0]\n Asymmetry angle: 0 deg\n In-plane rotation angle: -45 deg\n Crystal directions parallel to the Cartesian axes (after rotations):\n x || [-0. -0. -1.]\n y || [-1. 1. -0.]\n z || [ 1. 1. -0.]\n \n Crystal thickness: 500 um\n Debye-Waller factor: 1.0\n \n Deformation model: anisotropic toroidal, fixed shape (built-in)\n Meridional bending radius: inf m\n Sagittal bending radius: inf m\n Material elastic isotropy: anisotropic\n Compliance matrix S (with rotations applied):\n [[ 0.0077 -0.0021 -0.0021 -0. 0. 0. ]\n [-0.0021 0.0059 -0.0004 0. -0. 0. ]\n [-0.0021 -0.0004 0.0059 -0. -0. 0. ]\n [-0. 0. 0. 0.0196 0. -0. ]\n [ 0. -0. -0. -0. 0.0126 0. ]\n [ 0. 0. 0. -0. 0. 0.0126]] GPa^-1\n\n\n### Kinematical diffraction conditions\n\n`TTcrystal` implements two functions `bragg_energy` and `bragg_angle` that can be used to calculate the photon energy corresponding to the given Bragg angle and vice versa.\n\n\n```python\nincidence_angle = Quantity(85,'deg')\nphoton_energy = Quantity(9.723,'keV')\n\nprint('Energy of photons corresponding to the Bragg angle '+ str(incidence_angle) +': ' + str(xtal.bragg_energy(incidence_angle)))\nprint('Bragg angle corresponding to the photon energy '+ str(photon_energy) +': ' + str(xtal.bragg_angle(photon_energy)))\n```\n\n Energy of photons corresponding to the Bragg angle 85 deg: 9.723049764800313 keV\n Bragg angle corresponding to the photon energy 9.723 keV: 85.0033530351427 deg\n\n\n### Elastic constants\n\nCurrently (v. 1.0) _pyTTE_ contains elastic tensors only for a handful of crystals that are used most often. In other cases a KeyError will be raised.\n\n\n```python\ntry:\n TTcrystal(crystal = 'NaCl', hkl=[6,6,0], thickness = Quantity(1,'mm'))\nexcept KeyError as ke:\n print(ke)\n```\n\n \"Elastic parameters for 'NaCl' not found!\"\n\n\nIn such cases the elastic parameters can be given as input. For example, in the isotropic case Young's modulus and Poisson's ratio are given as follows:\n\n\n```python\nxtal = TTcrystal(crystal = 'NaCl', hkl=[1,0,0], thickness = Quantity(1,'mm'), E = Quantity(39.98,'GPa'), nu = 0.26)\n\nprint(xtal)\n```\n\n Crystal: NaCl\n Crystallographic parameters:\n a = 0.563978 nm, b = 0.563978 nm, c = 0.563978 nm\n alpha = 90.0 deg, beta = 90.0 nm, gamma = 90.0 deg\n Direct primitive vectors (before rotations, in nm):\n a1 = [0.564 0. 0. ]\n a2 = [0. 0.564 0. ]\n a3 = [0. 0. 0.564]\n Reciprocal primitive vectors (before rotations, in 1/nm):\n b1 = [11.1408 -0. -0. ]\n b2 = [ 0. 11.1408 -0. ]\n b3 = [ 0. 0. 11.1408]\n \n Reflection: [1, 0, 0]\n Asymmetry angle: 0 deg\n In-plane rotation angle: 0 deg\n Crystal directions parallel to the Cartesian axes (after rotations):\n x || [-0. 0. -1.]\n y || [0. 1. 0.]\n z || [ 1. -0. -0.]\n \n Crystal thickness: 1 mm\n Debye-Waller factor: 1.0\n \n Deformation model: isotropic toroidal (built-in)\n Meridional bending radius: inf m\n Sagittal bending radius: inf m\n Material elastic isotropy: isotropic\n Young's modulus E: 39.98 GPa\n Poisson's ratio nu: 0.26\n\n\n### Deformation\n\nPyTTE has three different built-in deformation models for toroidal bending, one for isotropic materials and two for anisotropic. The main parameters defining the deformation are elastic constants (either $E$ and $\\mu$, or $S$) and the bending radii. The meridional and sagittal bending radii are given with the keywords `Rx` and `Ry`, respectively. In the case of spherical bending, a single keyword `R` can be used. \n\n\n```python\nxtal = TTcrystal(crystal = 'Si', hkl=[6,6,0], thickness = Quantity(1,'mm'), Rx = Quantity(0.5,'m'), Ry = 'inf')\nprint(xtal)\n```\n\n Crystal: Si\n Crystallographic parameters:\n a = 0.543069 nm, b = 0.543069 nm, c = 0.543069 nm\n alpha = 90.0 deg, beta = 90.0 nm, gamma = 90.0 deg\n Direct primitive vectors (before rotations, in nm):\n a1 = [0.5431 0. 0. ]\n a2 = [0. 0.5431 0. ]\n a3 = [0. 0. 0.5431]\n Reciprocal primitive vectors (before rotations, in 1/nm):\n b1 = [11.5698 -0. -0. ]\n b2 = [ 0. 11.5698 -0. ]\n b3 = [ 0. 0. 11.5698]\n \n Reflection: [6, 6, 0]\n Asymmetry angle: 0 deg\n In-plane rotation angle: 0 deg\n Crystal directions parallel to the Cartesian axes (after rotations):\n x || [ 0.7071 -0.7071 -1. ]\n y || [-0.7071 0.7071 -1. ]\n z || [ 1. 1. -0.]\n \n Crystal thickness: 1 mm\n Debye-Waller factor: 1.0\n \n Deformation model: anisotropic toroidal, fixed shape (built-in)\n Meridional bending radius: 0.5 m\n Sagittal bending radius: inf m\n Material elastic isotropy: anisotropic\n Compliance matrix S (with rotations applied):\n [[ 0.0055 -0.0008 -0.0013 0. 0. 0.0009]\n [-0.0008 0.0055 -0.0013 -0. 0. 0.0009]\n [-0.0013 -0.0013 0.0059 -0. -0. -0.0018]\n [ 0. -0. -0. 0.0161 -0.0035 0. ]\n [-0. 0. -0. -0.0035 0.0161 0. ]\n [ 0.0009 0.0009 -0.0018 0. 0. 0.0179]] GPa^-1\n\n\nFor anisotropic crystals, there is also an additional keyword `fix_to_axes`. `fix_to_axes = 'torques'` is used when the wafer is bent by two orthogonal torques which act about $x$- and $y$-axes, respectively. The deformation field is still determined by the curvature radii `Rx` and `Ry` in the $x$- and $y$-directions but due to the non-diagonal elements of $S$ these may not be the main axis of curvature. This situation is encountered _e.g._ when a free-standing crystal slab is bend by its ends. If `Rx` or `Ry` is `None`, then the corresponding torque is set to zero and the radius of curvature is determined via anticlastic bending.\n\nThe other option is `fix_to_axes = 'shape'` which fixes `Rx` and `Ry` as the main radii of curvatures and finds the torques needed for such deformation by letting them rotate in the $xy$-plane. This is the case when the wafer is forced to adopt a specific case _e.g._ that of a substrate. In this case `None` values of `Rx` or `Ry` are interpreted as `inf`:s.\n\n\n```python\nxtal = TTcrystal(crystal = 'Si', hkl=[6,6,0], thickness = Quantity(1,'mm'), Rx = Quantity(0.5,'m'), Ry = None, fix_to_axes = 'torques')\nprint(xtal)\n```\n\n Crystal: Si\n Crystallographic parameters:\n a = 0.543069 nm, b = 0.543069 nm, c = 0.543069 nm\n alpha = 90.0 deg, beta = 90.0 nm, gamma = 90.0 deg\n Direct primitive vectors (before rotations, in nm):\n a1 = [0.5431 0. 0. ]\n a2 = [0. 0.5431 0. ]\n a3 = [0. 0. 0.5431]\n Reciprocal primitive vectors (before rotations, in 1/nm):\n b1 = [11.5698 -0. -0. ]\n b2 = [ 0. 11.5698 -0. ]\n b3 = [ 0. 0. 11.5698]\n \n Reflection: [6, 6, 0]\n Asymmetry angle: 0 deg\n In-plane rotation angle: 0 deg\n Crystal directions parallel to the Cartesian axes (after rotations):\n x || [ 0.7071 -0.7071 -1. ]\n y || [-0.7071 0.7071 -1. ]\n z || [ 1. 1. -0.]\n \n Crystal thickness: 1 mm\n Debye-Waller factor: 1.0\n \n Deformation model: anisotropic toroidal, fixed torques (built-in)\n Meridional bending radius: 0.5 m\n Sagittal bending radius: None\n Material elastic isotropy: anisotropic\n Compliance matrix S (with rotations applied):\n [[ 0.0055 -0.0008 -0.0013 0. 0. 0.0009]\n [-0.0008 0.0055 -0.0013 -0. 0. 0.0009]\n [-0.0013 -0.0013 0.0059 -0. -0. -0.0018]\n [ 0. -0. -0. 0.0161 -0.0035 0. ]\n [-0. 0. -0. -0.0035 0.0161 0. ]\n [ 0.0009 0.0009 -0.0018 0. 0. 0.0179]] GPa^-1\n\n\nFor isotropic material, the main radii of curvature always follow the bending torques, so there is no difference between `fix_to_axes = 'torques'` and `fix_to_axes = 'shape'`. \n\n\n```python\nxtal = TTcrystal(crystal = 'Si', hkl=[6,6,0], thickness = Quantity(1,'mm'), Rx = Quantity(0.5,'m'), Ry = None, fix_to_axes = 'torques', E = Quantity(160, 'GPa'),nu = 0.27)\nprint(xtal)\n```\n\n Crystal: Si\n Crystallographic parameters:\n a = 0.543069 nm, b = 0.543069 nm, c = 0.543069 nm\n alpha = 90.0 deg, beta = 90.0 nm, gamma = 90.0 deg\n Direct primitive vectors (before rotations, in nm):\n a1 = [0.5431 0. 0. ]\n a2 = [0. 0.5431 0. ]\n a3 = [0. 0. 0.5431]\n Reciprocal primitive vectors (before rotations, in 1/nm):\n b1 = [11.5698 -0. -0. ]\n b2 = [ 0. 11.5698 -0. ]\n b3 = [ 0. 0. 11.5698]\n \n Reflection: [6, 6, 0]\n Asymmetry angle: 0 deg\n In-plane rotation angle: 0 deg\n Crystal directions parallel to the Cartesian axes (after rotations):\n x || [ 0.7071 -0.7071 -1. ]\n y || [-0.7071 0.7071 -1. ]\n z || [ 1. 1. -0.]\n \n Crystal thickness: 1 mm\n Debye-Waller factor: 1.0\n \n Deformation model: isotropic toroidal (built-in)\n Meridional bending radius: 0.5 m\n Sagittal bending radius: None\n Material elastic isotropy: isotropic\n Young's modulus E: 160 GPa\n Poisson's ratio nu: 0.27\n\n\nArbitrary deformation fields can be used by defining a custom function that takes in the $x$ and $z$ coordinates in micrometer and returns 2x2 array $J$ so that\n\n$\\begin{equation}\nJ = \\left[\\begin{matrix}\n\\frac{\\partial u_x}{\\partial x} & \\frac{\\partial u_x}{\\partial z} \\\\\n\\frac{\\partial u_z}{\\partial x} & \\frac{\\partial u_z}{\\partial z} \n\\end{matrix}\\right]\n\\end{equation}$\n\nThe custom Jacobian is added after initialization using `set_deformation`.\n\n\n```python\nfrom pyTTE.deformation import isotropic_plate\n\nujac = isotropic_plate(1,1,0.27,1e-4)[0]\n\nxtal = TTcrystal(crystal = 'Si', hkl=[6,6,0], thickness = Quantity(1,'mm'))\nxtal.set_deformation(ujac)\n\nprint(xtal)\n```\n\n Crystal: Si\n Crystallographic parameters:\n a = 0.543069 nm, b = 0.543069 nm, c = 0.543069 nm\n alpha = 90.0 deg, beta = 90.0 nm, gamma = 90.0 deg\n Direct primitive vectors (before rotations, in nm):\n a1 = [0.5431 0. 0. ]\n a2 = [0. 0.5431 0. ]\n a3 = [0. 0. 0.5431]\n Reciprocal primitive vectors (before rotations, in 1/nm):\n b1 = [11.5698 -0. -0. ]\n b2 = [ 0. 11.5698 -0. ]\n b3 = [ 0. 0. 11.5698]\n \n Reflection: [6, 6, 0]\n Asymmetry angle: 0 deg\n In-plane rotation angle: 0 deg\n Crystal directions parallel to the Cartesian axes (after rotations):\n x || [ 0.7071 -0.7071 -1. ]\n y || [-0.7071 0.7071 -1. ]\n z || [ 1. 1. -0.]\n \n Crystal thickness: 1 mm\n Debye-Waller factor: 1.0\n \n Deformation model: custom Jacobian (bending radii and elastic parameters neglected)\n Meridional bending radius: inf m\n Sagittal bending radius: inf m\n Material elastic isotropy: anisotropic\n Compliance matrix S (with rotations applied):\n [[ 0.0055 -0.0008 -0.0013 0. 0. 0.0009]\n [-0.0008 0.0055 -0.0013 -0. 0. 0.0009]\n [-0.0013 -0.0013 0.0059 -0. -0. -0.0018]\n [ 0. -0. -0. 0.0161 -0.0035 0. ]\n [-0. 0. -0. -0.0035 0.0161 0. ]\n [ 0.0009 0.0009 -0.0018 0. 0. 0.0179]] GPa^-1\n\n\n## Initialization from a file\n\nTTcrystal parameters can be written in a file, the path of which is passed as an argument to the constructor.\n\n\n```python\n'''\nContents of TTcrystal_init.inp:\n\ncrystal LiF\nhkl 2 0 0\nthickness 200 um\nasymmetry 2.5 deg\nRx 1 m\n'''\n\nprint(TTcrystal(filepath = 'TTcrystal_init.inp'))\n```\n\n Crystal: LiF\n Crystallographic parameters:\n a = 0.402629 nm, b = 0.402629 nm, c = 0.402629 nm\n alpha = 90.0 deg, beta = 90.0 nm, gamma = 90.0 deg\n Direct primitive vectors (before rotations, in nm):\n a1 = [0.4026 0. 0. ]\n a2 = [0. 0.4026 0. ]\n a3 = [0. 0. 0.4026]\n Reciprocal primitive vectors (before rotations, in 1/nm):\n b1 = [15.6054 -0. -0. ]\n b2 = [ 0. 15.6054 -0. ]\n b3 = [ 0. 0. 15.6054]\n \n Reflection: [2, 0, 0]\n Asymmetry angle: 2.5 deg\n In-plane rotation angle: 0 deg\n Crystal directions parallel to the Cartesian axes (after rotations):\n x || [ 0.0437 0. -1. ]\n y || [0. 1. 0.]\n z || [ 1. -0. 0.0437]\n \n Crystal thickness: 200.0 um\n Debye-Waller factor: 1.0\n \n Deformation model: anisotropic toroidal, fixed shape (built-in)\n Meridional bending radius: 1.0 m\n Sagittal bending radius: inf m\n Material elastic isotropy: anisotropic\n Compliance matrix S (with rotations applied):\n [[ 0.0116 -0.0034 -0.0034 0. -0.0006 -0. ]\n [-0.0034 0.0116 -0.0034 -0. 0. 0. ]\n [-0.0034 -0.0034 0.0116 0. 0.0006 0. ]\n [ 0. -0. 0. 0.0157 0. 0. ]\n [-0.0006 -0. 0.0006 0. 0.0158 0. ]\n [-0. 0. 0. -0. 0. 0.0157]] GPa^-1\n\n", "meta": {"hexsha": "95ccc623265ea6f2ee9bf3c60ee1dfbfd11fe477", "size": 26741, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "examples/TTcrystal.ipynb", "max_stars_repo_name": "aripekka/pyTTE", "max_stars_repo_head_hexsha": "198d0232913e38f3d0a8ff034497640a730b4e8b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2019-05-08T03:01:53.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-12T15:00:16.000Z", "max_issues_repo_path": "examples/TTcrystal.ipynb", "max_issues_repo_name": "aripekka/pyTTE", "max_issues_repo_head_hexsha": "198d0232913e38f3d0a8ff034497640a730b4e8b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 9, "max_issues_repo_issues_event_min_datetime": "2017-12-21T12:47:04.000Z", "max_issues_repo_issues_event_max_datetime": "2021-06-01T11:01:02.000Z", 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YES\n2. YES", "lm_q1_score": 0.7279754489059774, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.40925069059673047}} {"text": "```python\nA=[]\nwith open(\"latex_custom_command.txt\") as file: # Use file to refer to the file object\n cmd=file.readline()\n #print(cmd)\n while(cmd):\n A.append(cmd)\n cmd=file.readline()\n \nA\n \n```\n\n\n\n\n ['\\\\newcommand{\\\\pd}{\\\\partial}\\n',\n '\\\\newcommand{\\\\bi}{\\\\mathbb i}\\n',\n '\\\\newcommand{\\\\bC}{\\\\mathbb C}\\n',\n '\\\\newcommand{\\\\bE}{\\\\mathbb E}\\n',\n '\\\\newcommand{\\\\bF}{\\\\mathbb F}\\n',\n '\\\\newcommand{\\\\bN}{\\\\mathbb N}\\n',\n '\\\\newcommand{\\\\bP}{\\\\mathbb P}\\n',\n '\\\\newcommand{\\\\bQ}{\\\\mathbb Q}\\n',\n '\\\\newcommand{\\\\bR}{\\\\mathbb R}\\n',\n '\\\\newcommand{\\\\bZ}{\\\\mathbb Z}\\n',\n '\\\\newcommand{\\\\cA}{\\\\mathcal A}\\n',\n '\\\\newcommand{\\\\cB}{\\\\mathcal B}\\n',\n '\\\\newcommand{\\\\cC}{\\\\mathcal C}\\n',\n '\\\\newcommand{\\\\cD}{\\\\mathcal D}\\n',\n '\\\\newcommand{\\\\cE}{\\\\mathcal E}\\n',\n '\\\\newcommand{\\\\cF}{\\\\mathcal F}\\n',\n '\\\\newcommand{\\\\cM}{\\\\mathcal M}\\n',\n '\\\\newcommand{\\\\cO}{\\\\mathcal O}\\n',\n '\\\\newcommand{\\\\cP}{\\\\mathcal P}\\n',\n '\\\\newcommand{\\\\cT}{\\\\mathcal T}\\n',\n '\\\\newcommand{\\\\half}{\\\\frac{1}{2}}\\n',\n '\\\\newcommand{\\\\cV}{\\\\mathcal V}\\n',\n '\\\\newcommand{\\\\cW}{\\\\mathcal W}\\n',\n '\\\\newcommand{\\\\bx}{\\\\boldsymbol}\\n',\n '\\\\newcommand{\\\\ud}{\\\\mathrm{d}}\\n',\n '\\\\newcommand{\\\\uD}{\\\\mathrm{D}}\\n',\n '\\\\newcommand{\\\\fh}{\\\\mathfrak{h}}\\n',\n '\\\\newcommand{\\\\fg}{\\\\mathfrak{g}}\\n',\n '\\\\newcommand{\\\\ft}{\\\\mathfrak{t}}\\n',\n '\\\\newcommand{\\\\fU}{\\\\mathfrak{U}}\\n',\n '\\\\newcommand{\\\\sG}{\\\\mathscr{G}}\\n',\n '\\\\newcommand{\\\\sA}{\\\\mathscr{A}}\\n',\n '\\\\newcommand{\\\\sB}{\\\\mathscr{B}}\\n',\n '\\\\newcommand{\\\\sC}{\\\\mathscr{C}}\\n',\n '\\\\newcommand{\\\\sE}{\\\\mathscr{E}}\\n',\n '\\\\newcommand{\\\\sH}{\\\\mathscr{H}}\\n',\n '\\\\newcommand{\\\\sO}{\\\\mathscr{O}}\\n',\n '\\\\newcommand{\\\\sM}{\\\\mathscr{M}}\\n',\n '\\\\newcommand{\\\\sS}{\\\\mathscr{S}}\\n',\n '\\\\newcommand{\\\\sU}{\\\\mathscr{U}}\\n',\n '\\\\newcommand{\\\\rH}{\\\\mathrm{H}}\\n',\n '\\\\newcommand{\\\\sR}{\\\\mathscr{R}}\\n',\n '\\\\newcommand{\\\\sT}{\\\\mathscr{T}}\\n',\n '\\\\newcommand{\\\\sD}{\\\\mathscr{D}}\\n',\n '\\\\newcommand{\\\\loto}{\\\\longrightarrow}\\n',\n '\\\\newcommand{\\\\lmto}{\\\\longmapsto}\\n',\n '\\\\newcommand{\\\\gal}{\\\\alpha}\\n',\n '\\\\newcommand{\\\\gbe}{\\\\beta}\\n',\n '\\\\newcommand{\\\\gga}{\\\\gamma}\\n',\n '\\\\newcommand{\\\\gde}{\\\\delta}\\n',\n '\\\\newcommand{\\\\gep}{\\\\epsilon}\\n',\n '\\\\newcommand{\\\\vp}{\\\\varphi}\\n',\n '\\\\newcommand{\\\\pbs}{\\\\bar\\\\pd^*}\\n',\n '\\\\newcommand{\\\\pb}{\\\\bar\\\\pd}\\n',\n '\\\\DeclareMathOperator{\\\\ad}{ad}\\n',\n '\\\\DeclareMathOperator{\\\\Ad}{Ad}\\n',\n '\\\\DeclareMathOperator{\\\\Aut}{Aut}\\n',\n '\\\\DeclareMathOperator{\\\\End}{End}\\n',\n '\\\\DeclareMathOperator{\\\\Ker}{Ker}\\n',\n '\\\\DeclareMathOperator{\\\\Rank}{Rank}\\n',\n '\\\\DeclareMathOperator{\\\\Coker}{Coker}\\n',\n '\\\\DeclareMathOperator{\\\\im}{im}\\n',\n '\\\\DeclareMathOperator{\\\\Tr}{Tr}\\n',\n '\\\\DeclareMathOperator{\\\\D}{D}\\n',\n '\\\\newcommand{\\\\be}{\\\\begin{equation}}\\n',\n '\\\\newcommand{\\\\ee}{\\\\end{equation}}\\n',\n '\\\\newcommand{\\\\bes}{\\\\begin{equation*}}\\n',\n '\\\\newcommand{\\\\ees}{\\\\end{equation*}}\\n',\n '\\\\newcommand{\\\\bea}{\\\\begin{eqnarray}}\\n',\n '\\\\newcommand{\\\\eea}{\\\\end{eqnarray}}\\n',\n '\\\\newcommand{\\\\beas}{\\\\begin{eqnarray*}}\\n',\n '\\\\newcommand{\\\\eeas}{\\\\end{eqnarray*}}\\n',\n '\\\\newcommand{\\\\ben}{\\\\begin{eqnarray*}}\\n',\n '\\\\newcommand{\\\\een}{\\\\end{eqnarray*}}\\n',\n '\\\\newcommand{\\\\bst}{\\\\begin{split}}\\n',\n '\\\\newcommand{\\\\est}{\\\\end{split}}\\n',\n '\\\\newcommand{\\\\bp}{\\\\begin{proof}}\\n',\n '\\\\newcommand{\\\\ep}{\\\\end{proof}}\\n',\n '\\\\newcommand{\\\\bex}{\\\\begin{exercise}}\\n',\n '\\\\newcommand{\\\\eex}{\\\\end{exercise}}\\n',\n '\\\\newcommand{\\\\bz}{\\\\bar z}']\n\n\n\n\n```python\n# pd: \"{\\\\partial}\",\n#\\newcommand{\\sech}{\\mathop{\\rm sech}\\nolimits}\n#'\\\\DeclareMathOperator{\\\\Rank}{Rank}\\n',\nimport re\n\ndef op_replacer(m):\n return '\\\\newcommand{\\\\'+m.group(1)+'}{\\\\mathop{\\\\\\\\rm '+m.group(2)+'}\\\\\\\\nolimits}'\n\n\ndef cmd_replacer(m):\n return m.group(1)+':\\\"{\\\\\\\\'+m.group(2)+'}\\\",\\n'\n\nC=[]\nfor x in A:\n C.append(re.sub(r'\\\\DeclareMathOperator\\{\\\\(\\w+)\\}\\{(\\w+)\\}\\n?',op_replacer,x))\n \nC \nB=[]\nfor x in C:\n B.append(re.sub(r'\\\\newcommand\\{\\\\(\\w+)\\}\\{\\\\(.+)\\}\\n?',cmd_replacer,x))\n \nB\n```\n\n\n\n\n ['pd:\"{\\\\\\\\partial}\",\\n',\n 'bi:\"{\\\\\\\\mathbb i}\",\\n',\n 'bC:\"{\\\\\\\\mathbb C}\",\\n',\n 'bE:\"{\\\\\\\\mathbb E}\",\\n',\n 'bF:\"{\\\\\\\\mathbb F}\",\\n',\n 'bN:\"{\\\\\\\\mathbb N}\",\\n',\n 'bP:\"{\\\\\\\\mathbb P}\",\\n',\n 'bQ:\"{\\\\\\\\mathbb Q}\",\\n',\n 'bR:\"{\\\\\\\\mathbb R}\",\\n',\n 'bZ:\"{\\\\\\\\mathbb Z}\",\\n',\n 'cA:\"{\\\\\\\\mathcal A}\",\\n',\n 'cB:\"{\\\\\\\\mathcal B}\",\\n',\n 'cC:\"{\\\\\\\\mathcal C}\",\\n',\n 'cD:\"{\\\\\\\\mathcal D}\",\\n',\n 'cE:\"{\\\\\\\\mathcal E}\",\\n',\n 'cF:\"{\\\\\\\\mathcal F}\",\\n',\n 'cM:\"{\\\\\\\\mathcal M}\",\\n',\n 'cO:\"{\\\\\\\\mathcal O}\",\\n',\n 'cP:\"{\\\\\\\\mathcal P}\",\\n',\n 'cT:\"{\\\\\\\\mathcal T}\",\\n',\n 'half:\"{\\\\\\\\frac{1}{2}}\",\\n',\n 'cV:\"{\\\\\\\\mathcal V}\",\\n',\n 'cW:\"{\\\\\\\\mathcal W}\",\\n',\n 'bx:\"{\\\\\\\\boldsymbol}\",\\n',\n 'ud:\"{\\\\\\\\mathrm{d}}\",\\n',\n 'uD:\"{\\\\\\\\mathrm{D}}\",\\n',\n 'fh:\"{\\\\\\\\mathfrak{h}}\",\\n',\n 'fg:\"{\\\\\\\\mathfrak{g}}\",\\n',\n 'ft:\"{\\\\\\\\mathfrak{t}}\",\\n',\n 'fU:\"{\\\\\\\\mathfrak{U}}\",\\n',\n 'sG:\"{\\\\\\\\mathscr{G}}\",\\n',\n 'sA:\"{\\\\\\\\mathscr{A}}\",\\n',\n 'sB:\"{\\\\\\\\mathscr{B}}\",\\n',\n 'sC:\"{\\\\\\\\mathscr{C}}\",\\n',\n 'sE:\"{\\\\\\\\mathscr{E}}\",\\n',\n 'sH:\"{\\\\\\\\mathscr{H}}\",\\n',\n 'sO:\"{\\\\\\\\mathscr{O}}\",\\n',\n 'sM:\"{\\\\\\\\mathscr{M}}\",\\n',\n 'sS:\"{\\\\\\\\mathscr{S}}\",\\n',\n 'sU:\"{\\\\\\\\mathscr{U}}\",\\n',\n 'rH:\"{\\\\\\\\mathrm{H}}\",\\n',\n 'sR:\"{\\\\\\\\mathscr{R}}\",\\n',\n 'sT:\"{\\\\\\\\mathscr{T}}\",\\n',\n 'sD:\"{\\\\\\\\mathscr{D}}\",\\n',\n 'loto:\"{\\\\\\\\longrightarrow}\",\\n',\n 'lmto:\"{\\\\\\\\longmapsto}\",\\n',\n 'gal:\"{\\\\\\\\alpha}\",\\n',\n 'gbe:\"{\\\\\\\\beta}\",\\n',\n 'gga:\"{\\\\\\\\gamma}\",\\n',\n 'gde:\"{\\\\\\\\delta}\",\\n',\n 'gep:\"{\\\\\\\\epsilon}\",\\n',\n 'vp:\"{\\\\\\\\varphi}\",\\n',\n 'pbs:\"{\\\\\\\\bar\\\\pd^*}\",\\n',\n 'pb:\"{\\\\\\\\bar\\\\pd}\",\\n',\n 'ad:\"{\\\\\\\\mathop{\\\\\\\\rm ad}\\\\\\\\nolimits}\",\\n',\n 'Ad:\"{\\\\\\\\mathop{\\\\\\\\rm Ad}\\\\\\\\nolimits}\",\\n',\n 'Aut:\"{\\\\\\\\mathop{\\\\\\\\rm Aut}\\\\\\\\nolimits}\",\\n',\n 'End:\"{\\\\\\\\mathop{\\\\\\\\rm End}\\\\\\\\nolimits}\",\\n',\n 'Ker:\"{\\\\\\\\mathop{\\\\\\\\rm Ker}\\\\\\\\nolimits}\",\\n',\n 'Rank:\"{\\\\\\\\mathop{\\\\\\\\rm Rank}\\\\\\\\nolimits}\",\\n',\n 'Coker:\"{\\\\\\\\mathop{\\\\\\\\rm Coker}\\\\\\\\nolimits}\",\\n',\n 'im:\"{\\\\\\\\mathop{\\\\\\\\rm im}\\\\\\\\nolimits}\",\\n',\n 'Tr:\"{\\\\\\\\mathop{\\\\\\\\rm Tr}\\\\\\\\nolimits}\",\\n',\n 'D:\"{\\\\\\\\mathop{\\\\\\\\rm D}\\\\\\\\nolimits}\",\\n',\n 'be:\"{\\\\\\\\begin{equation}}\",\\n',\n 'ee:\"{\\\\\\\\end{equation}}\",\\n',\n 'bes:\"{\\\\\\\\begin{equation*}}\",\\n',\n 'ees:\"{\\\\\\\\end{equation*}}\",\\n',\n 'bea:\"{\\\\\\\\begin{eqnarray}}\",\\n',\n 'eea:\"{\\\\\\\\end{eqnarray}}\",\\n',\n 'beas:\"{\\\\\\\\begin{eqnarray*}}\",\\n',\n 'eeas:\"{\\\\\\\\end{eqnarray*}}\",\\n',\n 'ben:\"{\\\\\\\\begin{eqnarray*}}\",\\n',\n 'een:\"{\\\\\\\\end{eqnarray*}}\",\\n',\n 'bst:\"{\\\\\\\\begin{split}}\",\\n',\n 'est:\"{\\\\\\\\end{split}}\",\\n',\n 'bp:\"{\\\\\\\\begin{proof}}\",\\n',\n 'ep:\"{\\\\\\\\end{proof}}\",\\n',\n 'bex:\"{\\\\\\\\begin{exercise}}\",\\n',\n 'eex:\"{\\\\\\\\end{exercise}}\",\\n',\n 'bz:\"{\\\\\\\\bar z}\",\\n']\n\n\n\n\n```python\ntext=\"\".join(B)\nwith open(\"math_jax_macro.txt\", \"w\") as file:\n file.write(text)\n```\n\n\n```python\n\n```\n", "meta": {"hexsha": "e040ab683870dcba634dbb088872061b61fb0a90", "size": 11302, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "_posts/Latex_cmd2tex_macro.ipynb", "max_stars_repo_name": "jyguo1729/blog", "max_stars_repo_head_hexsha": "21c25f5e97338c74a0929d90f2edba3bd5d8e43c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "_posts/Latex_cmd2tex_macro.ipynb", "max_issues_repo_name": "jyguo1729/blog", "max_issues_repo_head_hexsha": "21c25f5e97338c74a0929d90f2edba3bd5d8e43c", "max_issues_repo_licenses": ["MIT"], 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YES\n2. YES", "lm_q1_score": 0.5851011686727231, "lm_q2_score": 0.6992544085240402, "lm_q1q2_score": 0.40913457162696965}} {"text": "```python\n# !/usr/bin/env python3\n# -*- coding: utf-8 -*-\n\"\"\"\nCreated on 20191125\n\n@author: zhangji\n\ntest the linear relationship\nU_t =?= U_sh + U_wm\nU_t is the total velocity\nU_sh is the velocity induced by shear flow\nU_wm is the active velocity. \n\"\"\"\n\n# %pylab inline\n# pylab.rcParams['figure.figsize'] = (25, 11)\n# fontsize = 40\n\n# import numpy as np\n# import scipy as sp\n# from scipy.optimize import leastsq, curve_fit\n# from scipy import interpolate\n# from scipy.interpolate import interp1d\n# from scipy.io import loadmat, savemat\n# # import scipy.misc\n\n# import matplotlib\n# from matplotlib import pyplot as plt\n# from matplotlib import animation, rc\n# import matplotlib.ticker as mtick\n# from mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\n# from mpl_toolkits.mplot3d import Axes3D, axes3d\n\n# from sympy import symbols, simplify, series, exp\n# from sympy.matrices import Matrix\n# from sympy.solvers import solve\n\n# from IPython.display import display, HTML\n# from tqdm import tqdm_notebook as tqdm\n# import pandas as pd\n# import re\n# from scanf import scanf\n# import os\n# import glob\n\n# from codeStore import support_fun as spf\n# from src.support_class import *\n# from src import stokes_flow as sf\n\n# rc('animation', html='html5')\n# PWD = os.getcwd()\n# font = {'size': 20}\n# matplotlib.rc('font', **font)\n# np.set_printoptions(linewidth=90, precision=5)\n\n%load_ext autoreload\n%autoreload 2\n\nimport os\nimport glob\nimport re\nimport pandas as pd\nfrom scanf import scanf\nimport natsort \nimport numpy as np\nimport scipy as sp\nfrom scipy.optimize import leastsq, curve_fit\nfrom scipy import interpolate\nfrom scipy import spatial\n# from scipy.interpolate import interp1d\nfrom scipy.io import loadmat, savemat\n# import scipy.misc\nimport importlib\nfrom IPython.display import display, HTML\nimport pickle\n\nimport matplotlib\nfrom matplotlib import pyplot as plt\nfrom matplotlib import colors as mcolors\nfrom matplotlib import animation, rc\nimport matplotlib.ticker as mtick\nfrom mpl_toolkits.axes_grid1.inset_locator import inset_axes, zoomed_inset_axes\nfrom mpl_toolkits.mplot3d import Axes3D, axes3d\nfrom matplotlib import ticker, cm\nfrom mpl_toolkits.axes_grid1 import make_axes_locatable\n\nfrom time import time\nfrom src import support_class as spc\nfrom src import jeffery_model as jm\nfrom codeStore import support_fun as spf\nfrom codeStore import support_fun_table as spf_tb\n\n# %matplotlib notebook\n\n%matplotlib inline\nrc('animation', html='html5')\nrc('text', usetex=True)\nparams = {'text.latex.preamble': [r'\\usepackage{bm}', r'\\usepackage{amsmath}']}\nplt.rcParams.update(params)\nfontsize = 40\nfigsize = (30, 16)\nPWD = os.getcwd()\n```\n\n\n```python\nfrom src.objComposite import *\nfrom src import stokes_flow as sf\nfrom src.geo import *\n\nfig = plt.figure(figsize=(2, 2))\nfig.patch.set_facecolor('white')\nax0 = fig.add_subplot(1, 1, 1)\n```\n\n\n```python\n%matplotlib notebook\nproblem_kwargs = {'nth': 10, \n 'hfct': 1, \n 'eh': -1, \n 'ch': 5, \n 'rh11': 1, \n 'rh12': 1, \n 'rh2': 0.1, \n 'ph': 6, \n 'n_tail': 1, \n 'with_cover': 2, \n 'with_T_geo': 0, \n 'left_hand': 0, \n 'rT2': 0.1, \n 'center': np.zeros(3), \n 'matrix_method': 'pf', \n 'zoom_factor': 1} \n\ntail_obj_list = create_ecoli_tail(moveh=np.zeros(3), **problem_kwargs)\ntobj = sf.StokesFlowObj()\ntobj.combine(tail_obj_list)\ntobj.node_rotation(norm=np.array([0, 1, 0]), theta=np.pi / 2)\ntobj2 = tobj.copy()\ntobj.node_rotation(norm=np.array([0, 0, 1]), theta=np.pi)\ntobj3 = tobj.copy()\ntobj3.node_rotation(norm=np.array([0, 1, 0]), theta=np.pi)\n\ntobj.show_u_nodes()\n# tobj2.show_u_nodes()\n# tobj3.show_u_nodes()\n\n```\n\n\n \n\n\n\n\n\n\n\n\n\n True\n\n\n\n\n```python\nnp.pi / 4\n```\n\n\n\n\n 0.7853981633974483\n\n\n\n\n```python\nfrom HelicodsParticles.obj_helicoid_disk import create_helicoid_dsk_comp\n%matplotlib notebook\n\nproblem_kwargs = {'helicoid_r': 1, \n 'helicoid_r1': 1, \n 'helicoid_r2': 0.2, \n 'helicoid_ds': 0.01, \n 'helicoid_th_loc': np.pi / 4, \n 'helicoid_ndsk_each': 4, \n 'matrix_method': 'rs'}\nhelicoid_comp = create_helicoid_dsk_comp(**problem_kwargs)\nhelicoid_comp = create_helicoid_comp(**problem_kwargs)\nhelicoid_comp.show_u_nodes(linestyle='')\n```\n\n\n \n\n\n\n\n\n\n\n\n\n True\n\n\n\n\n```python\n# helicoid_comp = create_helicoid_comp(**problem_kwargs)\nhelicoid_comp = create_helicoid_dsk_comp(**problem_kwargs)\nhelicoid_comp.show_u_nodes(linestyle='')\ntobj = sf.StokesFlowObj()\ntobj.combine([helicoid_comp.get_obj_list()[i0] for i0 in [0, 3, 6, 9]])\ntobj.combine([helicoid_comp.get_obj_list()[i0] for i0 in [0, 1, 2]])\ntobj.show_u_nodes(linestyle='')\n# for i0, tobj in enumerate(helicoid_comp.get_obj_list()):\n# t1 = tobj.get_u_geo().get_geo_norm()\n# print('%3d, [%6.2f, %6.2f, %6.2f]' % (i0, t1[2], t1[0], t1[1]))\n# print(tobj.get_u_geo().get_center())\n# print()\n\n```\n\n\n \n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n True\n\n\n\n\n```python\n[helicoid_comp.get_obj_list()[i0] for i0 in [0, 3, 6, 9]]\n```\n\n\n\n\n [general obj (index 1),\n general obj (index 4),\n general obj (index 7),\n general obj (index 10)]\n\n\n\n\n```python\nfor tobj in helicoid_comp.get_obj_list():\n tgeo = tobj.get_u_geo()\n print(tgeo.get_center())\n```\n\n [0.70710678 0.70710678 0. ]\n [7.07106781e-01 4.32978028e-17 7.07106781e-01]\n [ 4.32978028e-17 7.07106781e-01 -7.07106781e-01]\n [-0.70710678 0.70710678 0. ]\n [-7.07106781e-01 4.32978028e-17 7.07106781e-01]\n [-4.32978028e-17 7.07106781e-01 7.07106781e-01]\n [-0.70710678 -0.70710678 0. ]\n [-7.07106781e-01 -4.32978028e-17 -7.07106781e-01]\n [-4.32978028e-17 -7.07106781e-01 7.07106781e-01]\n [ 0.70710678 -0.70710678 0. ]\n [ 7.07106781e-01 -4.32978028e-17 -7.07106781e-01]\n [ 4.32978028e-17 -7.07106781e-01 -7.07106781e-01]\n\n\n\n```python\n%matplotlib notebook\n\nproblem_kwargs = {'helicoid_r': 1, \n 'helicoid_r2': 0.2, \n 'helicoid_ds': 0.01, \n 'helicoid_th_loc': np.pi / 4, \n 'helicoid_ndsk_each': 4, \n 'matrix_method': 'rs'}\nnamehandle='helicoid'\n\nr1 = problem_kwargs['helicoid_r']\nr2 = problem_kwargs['helicoid_r2']\nds = problem_kwargs['helicoid_ds']\nth_loc = problem_kwargs['helicoid_th_loc']\nmatrix_method = problem_kwargs['matrix_method']\n\ntgeo = regularizeDisk()\ntgeo.create_ds(ds, r2)\ntgeo.node_rotation(norm=np.array([1, 0, 0]), theta=th_loc)\ntobj = sf.StokesFlowObj()\ntobj.set_matrix_method(matrix_method) # the geo is regularizeDisk\ntobj.set_data(f_geo=tgeo, u_geo=tgeo, name=namehandle)\n# tobj.show_u_nodes(linestyle='')\n\nhelicoid_comp = obj2helicoid_comp(tobj, **problem_kwargs)\nhelicoid_comp.show_u_nodes(linestyle='')\n\n```\n\n\n \n\n\n\n\n\n\n\n\n\n True\n\n\n\n\n```python\n%matplotlib notebook\nproblem_kwargs = {'nth': 20, \n 'hfct': 1, \n 'eh': -1, \n 'ch': 0.7, \n 'rh11': 0.1, \n 'rh12': 0.1, \n 'rh2': 0.01, \n 'ph': 0, \n 'n_tail': 1, \n 'with_cover': 2, \n 'with_T_geo': 0, \n 'left_hand': 0, \n 'rT2': 0.1, \n 'center': np.zeros(3), \n 'matrix_method': 'pf', \n 'zoom_factor': 1, \n 'helicoid_r': 100, \n 'helicoid_ndsk_each': 4, } \n\ntail_obj_list = create_ecoli_tail(moveh=np.zeros(3), **problem_kwargs)\ntobj = sf.StokesFlowObj()\ntobj.combine(tail_obj_list)\n# tobj.node_rotation(norm=np.array([1, 0, 0]), theta=th_loc)\ntobj.show_u_nodes()\n\n# helicoid_comp = obj2helicoid_comp(tobj, **problem_kwargs)\n# helicoid_comp.show_u_nodes(linestyle='')\n\n```\n\n\n \n\n\n\n\n\n\n\n\n\n True\n\n\n\n\n```python\nfrom scipy.spatial.transform import Rotation as spR\nfrom src.support_class import *\n\ne0 = np.eye(3)\ntq1 = spR.from_matrix(e0).as_quat()\nquat1 = Quaternion()\nquat1.set_wxyz(tq1[3], tq1[0], tq1[1], tq1[2])\ntq2 = spR.from_rotvec(np.array((0, 0, 0))).as_quat()\nquat2 = Quaternion()\nquat2.set_wxyz(tq2[3], tq2[0], tq2[1], tq2[2])\nquat3 = quat2.mul(quat1)\n```\n\n\n```python\n%matplotlib inline\n\nph, ch, rh1, rh2, theta0 = 4, 4, 1, 0.1, 0\nn_segment, check_nth = 50 * ch, 0\n\nhlx1_geo = slb_helix(ph, ch, rh1, rh2, theta0=theta0)\nhlx1_geo.create_nSegment(n_segment, check_nth=check_nth)\n\nfor _ in range(10):\n rot_th = 2 * np.pi * np.random.sample(1)\n rot_norm = np.random.sample(3)\n rotation_origin = 100 * np.random.sample(3)\n rot_norm = rot_norm / np.linalg.norm(rot_norm)\n displacement = 100 * np.random.sample(3)\n hlx1_geo.move(displacement)\n hlx1_geo.node_rotation(rot_norm, theta=rot_th, rotation_origin=rotation_origin)\n# print(hlx1_geo.get_geo_norm())\nhlx1_geo.show_nodes()\ntnodes = hlx1_geo.xc_fun(hlx1_geo.s_list)\ntgeo = base_geo()\ntgeo.set_nodes(tnodes, 0)\ntgeo.show_nodes()\n```\n\n\n```python\n5 / 2 / np.sqrt(2)\n```\n\n\n\n\n 1.7677669529663687\n\n\n\n\n```python\n%matplotlib notebook\n\nproblem_kwargs = {'helicoid_r': 20, \n 'helicoid_ndsk_each': 4, } \nrs, ds = 1, 0.1\n\nsphere_geo0 = sphere_geo()\nsphere_geo0.create_delta(ds, rs)\nsphere_geo1 = sphere_geo0.copy()\ntmove = 5 \nsphere_geo0.move(np.array((0, 0, tmove / 2)))\nsphere_geo1.move(np.array((0, 0, -tmove / 2)))\ndumb_geo = base_geo()\ndumb_geo.combine([sphere_geo0, sphere_geo1])\ndumb_geo.node_rotation(norm=np.array((1, 0, 0)), theta=np.pi / 3)\n# dumb_geo = sphere_geo0\ntobj = sf.StokesFlowObj()\ntobj.set_data(dumb_geo, dumb_geo, 'helicoid_dumb')\n\nhelicoid_comp = obj2helicoid_comp(tobj, **problem_kwargs)\nhelicoid_obj_list = np.array(helicoid_comp.get_obj_list())[[0, 3, 6, 9]]\nhelicoid_center = helicoid_comp.get_center()\nhelicoid_comp.show_u_nodes(linestyle='')\n\ntobj = sf.StokesFlowObj()\ntobj.combine([i0 for i0 in helicoid_obj_list])\ntobj.show_u_nodes(linestyle='')\n\n```\n\n\n \n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n True\n\n\n\n\n```python\nhelicoid_obj_list = [i0 for i0 in helicoid_obj_list]\nlist(tube_flatten((helicoid_obj_list,)))\n```\n\n\n\n\n [general obj (index 1),\n general obj (index 4),\n general obj (index 7),\n general obj (index 10)]\n\n\n\n\n```python\n\n```\n", "meta": {"hexsha": "d8d7865d999f0a50ca4c0fc5dae4b397c0b06201", "size": 767425, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "HelicodsParticles/helicoid.ipynb", "max_stars_repo_name": "pcmagic/stokes_flow", "max_stars_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-11-11T05:00:53.000Z", "max_stars_repo_stars_event_max_datetime": "2018-11-11T05:00:53.000Z", "max_issues_repo_path": "HelicodsParticles/helicoid.ipynb", "max_issues_repo_name": "pcmagic/stokes_flow", "max_issues_repo_head_hexsha": "464d512d3739eee77b33d1ebf2f27dae6cfa0423", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": 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YES\n2. YES", "lm_q1_score": 0.685949442167993, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.4091229839707209}} {"text": "# TensorFlow 2.0 Practical\n\nThis notebook is based on the Tensorflow without a PhD codelab by Martin Gorner and was modified to run on TensorFlow 2.0.\n\n### Get Started\n\n \n \n
                                        \n Run in Google Colab\n \n View source on GitHub\n
                                        \n\n\n## 1. Introduction\nIn this practical, you will learn how to build and train a neural network that recognises handwritten digits. Along the way, as you enhance your neural network to achieve 99% accuracy, you will also discover the tools of the trade that deep learning professionals use to train their models efficiently.\n\nThis codelab uses the MNIST dataset, a collection of 60,000 labeled digits that has kept generations of PhDs busy for almost two decades. You will solve the problem with less than 100 lines of Python / TensorFlow code.\n\nWhat you'll learn:\n* What is a neural network and how to train it\n* How to build a basic 1-layer neural network using TensorFlow\n* How to add more layers\n* Training tips and tricks: overfitting, dropout, learning rate decay...\n* How to troubleshoot deep neural networks\n* How to build convolutional networks\n\nWhat you'll need:\n* Python 2 or 3 (Python 3 recommended)\n* TensorFlow\n\n### Running on GPU\nFor this practical, you will need to use a GPU to speed up training. To do this, go to the \"Runtime\" menu in Colab, select \"Change runtime type\" and then in the popup menu, choose \"GPU\" in the \"Hardware accelelator\" box. This is all you need to do, Colab and Tensorflow will take care of the rest!\n\n### Requirements installation\nFirst, we need o install TensorFlow 2.0 and download ngrock (ngrock is only used in order to run TensorBoard on Google Colab).\n\n\n```\n#@title Dependencies and Imports (RUN ME!) { display-mode: \"form\" }\n!pip install -q tensorflow-gpu==2.0.0-alpha0\n\n! wget https://bin.equinox.io/c/4VmDzA7iaHb/ngrok-stable-linux-amd64.zip\n! unzip ngrok-stable-linux-amd64.zip\n\nfrom __future__ import absolute_import, division, print_function\n\nimport os\nimport datetime\nimport numpy as np\n\nimport tensorflow as tf\n```\n\n### TensorBoard\nTensorBoard is a tool used to monitor the training and investigate the model and the results.\n\nRun the next cell in order to launch TensorBoard on background. Then, click on the link to access TensorBoard.\n\n\n```\n#@title Start TensorBoard (RUN ME!) { display-mode: \"form\" }\nLOG_DIR = \"/tmp/log\"\nget_ipython().system_raw(\n 'tensorboard --logdir {} --host 0.0.0.0 --port 6006 &'\n .format(LOG_DIR)\n)\n```\n\n## 2. The Data\nIn this practical, we use the MNIST dataset consisting of 70,000 greyscale images and their labels.\nThe idea is to train a classifier to identify the class value (what handwritten digit it is) given the image. We train and tune a model on the 60,000 training images and then evaluate how well it classifies the 10,000 test images that the model did not see during training. This task is an example of a supervised learning problem, where we are given both input and labels (targets) to learn from. This is in contrast to unsupervised learning where we only have inputs from which to learn patterns or reinforcement learning where an agent learns how to maximise a reward signal through interaction with its environment.\n\n### Aside: Train/Validation/Test Split\nWhen we build machine learning models, the goal is to build a model that will perform well on future data that we have not seen yet. We say that we want our models to be able to generalise well from whatever training data we can collect and do have available, to whatever data we will be applying them to in future. To do this, we split whatever data we have available into a training set, a validation set and a test set. The idea is that we train our model and use the performance on the validation set to make any adjustments to the model and its hyperparameters, but then we report the final accuracy on the test set. The test set (which we never train on), therefore acts as a proxy for our future data.\n\n### Load the dataset\nExecute the next cell in order to download the mnist dataset and prepare the training and test sets.\n\n\n```\n# Load the mnist dataset\nmnist = tf.keras.datasets.mnist\n\n# Split the dataset in train/test sets\n(x_train, y_train),(x_test, y_test) = mnist.load_data()\n\n# Normalize the inputs\nx_train, x_test = x_train / 255.0, x_test / 255.0\n```\n\n## 3. Theory: a 1-layer neural network\n\n\n\nHandwritten digits in the MNIST dataset are 28x28 pixel greyscale images. The simplest approach for classifying them is to use the 28x28=784 pixels as inputs for a 1-layer neural network.\n\n\n\nEach \"neuron\" in a neural network does a weighted sum of all of its inputs, adds a constant called the \"bias\" and then feeds the result through some non-linear activation function.\n\nHere we design a 1-layer neural network with 10 output neurons since we want to classify digits into 10 classes (0 to 9).\n\nFor a classification problem, an activation function that works well is softmax. Applying softmax on a vector is done by taking the exponential of each element and then normalising the vector (using any norm, for example the ordinary euclidean length of the vector).\n\n\n\n***Why is \"softmax\" called softmax ? The exponential is a steeply increasing function. It will increase differences between the elements of the vector. It also quickly produces large values. Then, as you normalise the vector, the largest element, which dominates the norm, will be normalised to a value close to 1 while all the other elements will end up divided by a large value and normalised to something close to 0. The resulting vector clearly shows which was its largest element, the \"max\", but retains the original relative order of its values, hence the \"soft\".***\n\nWe will now summarise the behaviour of this single layer of neurons into a simple formula using a matrix multiply. Let us do so directly for a \"mini-batch\" of 100 images as the input, producing 100 predictions (10-element vectors) as the output.\n\n\n\nUsing the first column of weights in the weights matrix W, we compute the weighted sum of all the pixels of the first image. This sum corresponds to the first neuron. Using the second column of weights, we do the same for the second neuron and so on until the 10th neuron. We can then repeat the operation for the remaining 99 images. If we call X the matrix containing our 100 images, all the weighted sums for our 10 neurons, computed on 100 images are simply X.W (matrix multiply).\n\nEach neuron must now add its bias (a constant). Since we have 10 neurons, we have 10 bias constants. We will call this vector of 10 values b. It must be added to each line of the previously computed matrix. Using a bit of magic called \"broadcasting\" we will write this with a simple plus sign.\n\n***\"Broadcasting\" is a standard trick used in Python and numpy, its scientific computation library. It extends how normal operations work on matrices with incompatible dimensions. \"Broadcasting add\" means \"if you are adding two matrices but you cannot because their dimensions are not compatible, try to replicate the small one as much as needed to make it work.\"***\n\nWe finally apply the softmax activation function and obtain the formula describing a 1-layer neural network, applied to 100 images:\n\n\n\n***By the way, what is a \"tensor\"?\nA \"tensor\" is like a matrix but with an arbitrary number of dimensions. A 1-dimensional tensor is a vector. A 2-dimensions tensor is a matrix. And then you can have tensors with 3, 4, 5 or more dimensions.***\n\n## 4. Theory: gradient descent\n\nNow that our neural network produces predictions from input images, we need to measure how good they are, i.e. the distance between what the network tells us and what we know to be the truth. Remember that we have true labels for all the images in this dataset.\n\nAny distance would work, the ordinary euclidian distance is fine but for classification problems one distance, called the \"cross-entropy\" is more efficient.\n\n\n\n***\"One-hot\" encoding means that you represent the label \"6\" by using a vector of 10 values, all zeros but the 6th value which is 1. It is handy here because the format is very similar to how our neural network outputs ts predictions, also as a vector of 10 values.***\n\n\"Training\" the neural network actually means using training images and labels to adjust weights and biases so as to minimise the cross-entropy loss function. Here is how it works.\n\nThe cross-entropy is a function of weights, biases, pixels of the training image and its known label.\n\nIf we compute the partial derivatives of the cross-entropy relatively to all the weights and all the biases we obtain a \"gradient\", computed for a given image, label and present value of weights and biases. Remember that we have 7850 weights and biases so computing the gradient sounds like a lot of work. Fortunately, TensorFlow will do it for us.\n\nThe mathematical property of a gradient is that it points \"up\". Since we want to go where the cross-entropy is low, we go in the opposite direction. We update weights and biases by a fraction of the gradient and do the same thing again using the next batch of training images. Hopefully, this gets us to the bottom of the pit where the cross-entropy is minimal.\n\n\n\nIn this picture, cross-entropy is represented as a function of 2 weights. In reality, there are many more. The gradient descent algorithm follows the path of steepest descent into a local minimum. The training images are changed at each iteration too so that we converge towards a local minimum that works for all images.\n\n***\"Learning rate\": you cannot update your weights and biases by the whole length of the gradient at each iteration. It would be like trying to get to the bottom of a valley while wearing seven-league boots. You would be jumping from one side of the valley to the other. To get to the bottom, you need to do smaller steps, i.e. use only a fraction of the gradient, typically in the 1/1000th region. We call this fraction the \"learning rate\".***\n\nTo sum it up, here is how the training loop looks like:\n```\nTraining digits and labels => loss function => gradient (partial derivatives) => steepest descent => update weights and biases => repeat with next mini-batch of training images and labels\n```\n***Why work with \"mini-batches\" of 100 images and labels ?***\n\n***You can definitely compute your gradient on just one example image and update the weights and biases immediately (it's called \"stochastic gradient descent\" in scientific literature). Doing so on 100 examples gives a gradient that better represents the constraints imposed by different example images and is therefore likely to converge towards the solution faster. The size of the mini-batch is an adjustable parameter though. There is another, more technical reason: working with batches also means working with larger matrices and these are usually easier to optimise on GPUs.***\n\n## 5. Lab: let's jump into the code\n\n### Define the model\nIn this section we'll build a classifier. A **classifier** is a function that takes an object's characteristics (or \"features\") as inputs and outputs a prediction of the class (or group) that the object belongs to. It may make a single prediction for each input or it may output some score (for example a probability) for each of the possible classes. Specifically, we will build a classifier that takes in (a batch of) 28 x 28 Fashion MNIST images as we've seen above, and outputs predictions about which class the image belongs to. \n\nFor each (batch of) input images, we will use a **feed-forward neural network** to compute un-normalised scores (also known as **logits**) for each of the 10 possible classes that the image could belong to. We can then **classify** the image as belonging to the class which receives the highest score, or we can quantify the model's \"confidence\" about the classifications by converting the scores into a probability distribution. \n\nA feed-forward neural network consisting of $N$ layers, applied to an input vector $\\mathbf{x}$ can be defined as:\n\n\\begin{equation}\n\\mathbf{f_0} = \\mathbf{x} \\\\\n\\mathbf{f_i} = \\sigma_i(\\mathbf{W_if_{i-1}} + \\mathbf{b_i}) \\ \\ \\ i \\in [1, N]\n\\end{equation}\n\nEach layer has a particular number, $m_i$, of neurons. The parameters of a layer consist of a matrix $\\mathbf{W_i} \\in \\mathbb{R}^{m_i \\times m_{i-1}}$ and bias vector $\\mathbf{b_i} \\in \\mathbb{R}^{m_i}$. Each layer also has a non-linear activation function $\\sigma_i$. \n\n**QUESTION**: Why do you think the activation functions need to be *non-linear*? What would happen if they were *linear*? **HINT**: If you're stuck, consider the very simplest case of an identity activation (which essentially does nothing) and ignore the biases. \n\n### Aside: Activation functions\n\nActivation functions are a core ingredient in deep neural networks. In fact they are what allows us to make use of multiple layers in a neural network. There are a number of different activation functions, each of which are more or less useful under different circumstances. The four activation functions that you are most likely to encounter are, arguably, [ReLU](https://www.tensorflow.org/api_docs/python/tf/keras/layers/ReLU), [Tanh](https://www.tensorflow.org/api_docs/python/tf/keras/activations/tanh), [Sigmoid](https://www.tensorflow.org/api_docs/python/tf/keras/activations/sigmoid), and [Softmax](https://www.tensorflow.org/api_docs/python/tf/keras/layers/Softmax). \n\nReLU, has in recent years, become the default choice for use in MLPs and Convolutional Neural Networks (CNNs). ReLU has two advantages over Tanh and Sigmoid: it is computationally much more efficient, and, it allows us to use deeper networks because it does not suffer from [vanishing gradients](https://en.wikipedia.org/wiki/Vanishing_gradient_problem). As a result of their success, a number of ReLU variants, such as [LeakyRelu](https://www.tensorflow.org/api_docs/python/tf/keras/layers/LeakyReLU) and [PReLU](https://www.tensorflow.org/api_docs/python/tf/keras/layers/PReLU), have been developed.\n\nSigmoid and Softmax activations are often found after the last layer in binary and multi-class classification networks, respectively, as they transform the outputs of the network in such a way that we can interpret them as class probabilities.\n\nBoth Tanh and Sigmoid are found in LSTM and GRU recurrent neural networks, which we will find out more about in the coming days. They are also useful in MLPs and CNNs where we want the output to be bounded between -1 and 1 (Tanh) or 0 and 1 (Sigmoid).\n\nRead more about activation functions [here](https://towardsdatascience.com/activation-functions-neural-networks-1cbd9f8d91d6). \n\n### Create a 1-layer neural network\nWe configure the feed-forward neural network part of our classifier using the [Keras Layers API](https://www.tensorflow.org/api_docs/python/tf/keras/layers). This API consists of various reusable building-blocks that allow us to define many different neural network architectures (similar to how we defined a data pipeline earlier!). \n\nHere we use the [Sequential](https://www.tensorflow.org/api_docs/python/tf/keras/Sequential) component which allows us to wrap together a sequence of layers. An important point to note here is that we are **configuring** our neural network architecture as a pipeline. We can think of the resulting ```model`` variable as a *function* that takes a batch of images as inputs and outputs a batch of logits. \n\n\n```\n# Define the model\nmodel = tf.keras.models.Sequential([\n# flatten the images into a single line of pixels\n tf.keras.layers.Flatten(input_shape=(28, 28), name='flatten_input'),\n# apply softmax as an activation function\n tf.keras.layers.Dense(10, activation=tf.nn.softmax, name='logits')\n])\n```\n\nThe following summary shows how many parameters each layer is made up of (the number of entries in the weight matrics and bias vectors). Note that a value of ```None``` in a particular dimension of a shape means that the shape will dynamically adapt based on the shape of the inputs. In particular, the output shape of the ```flatten_input``` layer will be $[N, 784]$ when the batch of inputs passed to the model has shape $[N, 28, 28]$\n\n\n```\nmodel.summary()\n```\n\n### Define the loss\nAs we did in the previous practical, we need to specify a loss function for our classifier. This tells us how good our model's predictions are compared to the actual labels (the targets), with a lower loss meaning better predictions. The standard loss function to use with a **multi-class classifier** is the **cross-entropy loss** also known as the \"negative log likelihood\". Suppose we have a classification problem with $C$ classes. A classifier is trained to predict a probability distribution $p(y | X_i)$ for each input $X_i$ from a batch of $N$ examples. The vector $p(y|X_i)$ is $C$ dimensional, sums to one, and we use $p(y|X_i)_c$ to denote the $c$th component of $p(y|X_i)$. The true class for example $i$ in the batch is $y_i$ and we define the indicator function $\\mathbb{1}[y_i=c]$ to be 1 whenever $y_i = c$ and $0$ otherwise. This classifier has a cross-entropy loss of\n\n$- \\frac{1}{N}\\sum_{i=1}^N \\sum_{c=1}^C log( p(y|X_i)_c) \\mathbb{1}[y_i=c]$\n\n**NOTE**: The indicator is one for the true class label, and zero everywhere else. So in that sum, the indicator just \"lifts out\" the $log(p(y|X_i))$ values for all true classes. So the above expression is minimised (note the negative at the front) when the model places all its probability mass on the true labels for all the examples. Remember that log(1)=0 , thus the closer all probabilities of $y_i = c$ are to one, the lower the loss will be and the better the model will be performing.\n\n**QUESTION**: \n* Why do you think this is a *good* loss function?\n* Can you think of any potential issues with this loss function?\n\nFortunately we don't need to write this function ourselves as Tensorflow provides a version called \n\n```tf.nn.sparse_softmax_cross_entropy_with_logits```. \n\n**NOTE**: This function actually computes the cross entropy loss directly from the un-normalised logits, rather than from the probability distribution for numerical stability.\n\nBy the way, for training data in which the labels are themselves distributions rather than exact values, this definition of cross-entropy still works, where the indicator function is replaced with the corresponding probability of each class for that example. This might be important when we are not sure whether the training data has been labelled correctly, or when the data was labelled by a human who gave their answer along with a degree of confidence that the answer was correct.\n\n### Train the model\nNow that we have our data, data processing pipeline, our neural network architecture and the corresponding loss that we want to minimise, we need to **train** the model using batched stochastic gradient descent. We do this in multiple **epochs**, which is a single iteration through the entire training dataset. Briefly, in each epoch we loop over all the batches of images and labels, and for each batch we perform the following steps:\n* Get the **predictions** of the model on the current batch of images\n* Compute the **average loss** values across the batch, telling us how good these predictions are / how close they are to the true targets.\n* Compute the **gradient of the average loss** (or the average gradient of the losses in the batch) with respect to each of the model's parameters: This tells us the direction to move in \"parameter space\" to decrease the loss value\n* **Adjust the parameters** by taking a small step in the direction of each component of the gradient (where the learning rate controls the size of the step)\n\nDuring training we also track some metrics, such as the loss and accuracy to see how well the classifier is doing. Note that the cell below may take a few minutes to run!\n\n\n```\ndef train_model():\n model.compile(optimizer=tf.keras.optimizers.SGD(learning_rate=0.005),\n loss='sparse_categorical_crossentropy',\n metrics=['accuracy'])\n\n log_dir = os.path.join(\"/tmp/log\", \"1-layer-fully-connected-\"+datetime.datetime.now().strftime(\"%Y%m%d-%H%M%S\"))\n tensorboard_callback = tf.keras.callbacks.TensorBoard(log_dir, histogram_freq=1)\n\n model.fit(x=x_train, \n y=y_train,\n batch_size=100,\n epochs=300, \n validation_data=(x_test, y_test), \n callbacks=[tensorboard_callback])\n\ntrain_model()\n```\n\n\n```\n#@title Visualize in TensorBoard (RUN ME!) { display-mode: \"form\" }\n\nget_ipython().system_raw('./ngrok http 6006 &')\n! curl -s http://localhost:4040/api/tunnels | python3 -c \"import sys, json; print(json.load(sys.stdin)['tunnels'][0]['public_url'])\"\n```\n\n## 6. Add more Layers\n\n\n\nTo improve the recognition accuracy we will add more layers to the neural network. The neurons in the second layer, instead of computing weighted sums of pixels will compute weighted sums of neuron outputs from the previous layer. Here is for example a 5-layer fully connected neural network:\n\n\n\nWe keep softmax as the activation function on the last layer because that is what works best for classification. On intermediate layers however we will use the the most classical activation function: the sigmoid:\n\n\n\n**Your task in this section is to add four intermediate layers to your model to increase its performance.**\n\n\n```\n# Define the model\nmodel = tf.keras.models.Sequential([\n# flatten the images into a single line of pixels\n tf.keras.layers.Flatten(input_shape=(28, 28), name='flatten_input'),\n\n # YOUR CODE GOES HERE\n \n# add output layer and apply softmax as an activation function\n tf.keras.layers.Dense(10, activation=tf.nn.softmax, name='logits')\n])\n\nmodel.summary()\n```\n\n\n```\n#@title Solution\n\n# Define the model\nmodel = tf.keras.models.Sequential([\n# flatten the images into a single line of pixels\n tf.keras.layers.Flatten(input_shape=(28, 28), name='flatten_input'),\n# add the first hidden layer\n tf.keras.layers.Dense(200, activation=tf.nn.sigmoid, name='hidden_layer_1'),\n# add the second hidden layer\n tf.keras.layers.Dense(100, activation=tf.nn.sigmoid, name='hidden_layer_2'),\n# add the third hidden layer\n tf.keras.layers.Dense(60, activation=tf.nn.sigmoid, name='hidden_layer_3'),\n# add the fourth hidden layer\n tf.keras.layers.Dense(30, activation=tf.nn.sigmoid, name='hidden_layer_4'),\n# add output layer and apply softmax as an activation function\n tf.keras.layers.Dense(10, activation=tf.nn.softmax, name='logits')\n])\n\nmodel.summary()\n```\n\n### Train the model\n\n\n```\ndef train_model():\n model.compile(optimizer=tf.keras.optimizers.SGD(learning_rate=0.005),\n loss='sparse_categorical_crossentropy',\n metrics=['accuracy'])\n\n log_dir = os.path.join(\"/tmp/log\", \"five-layers-\" + datetime.datetime.now().strftime(\"%Y%m%d-%H%M%S\"))\n tensorboard_callback = tf.keras.callbacks.TensorBoard(log_dir, histogram_freq=1)\n\n model.fit(x=x_train, \n y=y_train,\n batch_size=100,\n epochs=300, \n validation_data=(x_test, y_test), \n callbacks=[tensorboard_callback])\n\ntrain_model()\n```\n\n\n```\n#@title Visualize in TensorBoard (RUN ME!) { display-mode: \"form\" }\n\nget_ipython().system_raw('./ngrok http 6006 &')\n! curl -s http://localhost:4040/api/tunnels | python3 -c \"import sys, json; print(json.load(sys.stdin)['tunnels'][0]['public_url'])\"\n```\n\n## 7. Lab: special care for deep networks\n\n\n\nAs layers were added, neural networks tended to converge with more difficulties. But we know today how to make them behave. Here are a couple of 1-line updates that will help if you see an accuracy curve like this:\n\n\n\n**Relu activation function**\nThe sigmoid activation function is actually quite problematic in deep networks. It squashes all values between 0 and 1 and when you do so repeatedly, neuron outputs and their gradients can vanish entirely. It was mentioned for historical reasons but modern networks use the RELU (Rectified Linear Unit) which looks like this:\n\n\n\n**TO DO:**\n\n**Now, replace all your sigmoids with RELUs now and you will get faster initial convergence and avoid problems later as we add layers. Simply swap tf.nn.sigmoid with tf.nn.relu in your code.**\n\n\n```\n# Define the model\nmodel = tf.keras.models.Sequential([\n# flatten the images into a single line of pixels\n tf.keras.layers.Flatten(input_shape=(28, 28), name='flatten_input'),\n# add the first hidden layer\n tf.keras.layers.Dense(200, activation=tf.nn.sigmoid, name='hidden_layer_1'),\n# add the second hidden layer\n tf.keras.layers.Dense(100, activation=tf.nn.sigmoid, name='hidden_layer_2'),\n# add the third hidden layer\n tf.keras.layers.Dense(60, activation=tf.nn.sigmoid, name='hidden_layer_3'),\n# add the fourth hidden layer\n tf.keras.layers.Dense(30, activation=tf.nn.sigmoid, name='hidden_layer_4'),\n# add output layer and apply softmax as an activation function\n tf.keras.layers.Dense(10, activation=tf.nn.softmax, name='logits')\n])\n\nmodel.summary()\n```\n\n\n```\n#@title Solution\n\n# Define the model\nmodel = tf.keras.models.Sequential([\n# flatten the images into a single line of pixels\n tf.keras.layers.Flatten(input_shape=(28, 28), name='flatten_input'),\n# add the first hidden layer\n tf.keras.layers.Dense(200, activation=tf.nn.relu, name='hidden_layer_1'),\n# add the second hidden layer\n tf.keras.layers.Dense(100, activation=tf.nn.relu, name='hidden_layer_2'),\n# add the third hidden layer\n tf.keras.layers.Dense(60, activation=tf.nn.relu, name='hidden_layer_3'),\n# add the fourth hidden layer\n tf.keras.layers.Dense(30, activation=tf.nn.relu, name='hidden_layer_4'),\n# add output layer and apply softmax as an activation function\n tf.keras.layers.Dense(10, activation=tf.nn.softmax, name='logits')\n])\n\nmodel.summary()\n```\n\n**A better optimizer**\n\nIn very high dimensional spaces like here - we have in the order of 10K weights and biases - \"saddle points\" are frequent. These are points that are not local minima but where the gradient is nevertheless zero and the gradient descent optimizer stays stuck there. TensorFlow has a full array of available optimizers, including some that work with an amount of inertia and will safely sail past saddle points.\n\n**TO DO:**\n\n**Replace your tf.train.GradientDescentOptimiser with a tf.train.AdamOptimizer now.**\n\n\n```\ndef train_model():\n model.compile(optimizer=tf.keras.optimizers.SGD(learning_rate=0.005), ## change the optimizer here\n loss='sparse_categorical_crossentropy',\n metrics=['accuracy'])\n\n log_dir = os.path.join(\"/tmp/log\", \"relu-and-adam-\" + datetime.datetime.now().strftime(\"%Y%m%d-%H%M%S\"))\n tensorboard_callback = tf.keras.callbacks.TensorBoard(log_dir, histogram_freq=1)\n\n model.fit(x=x_train, \n y=y_train,\n batch_size=100,\n epochs=300, \n validation_data=(x_test, y_test), \n callbacks=[tensorboard_callback])\n\ntrain_model()\n```\n\n\n```\n#@title Solution\n\ndef train_model():\n model.compile(optimizer=tf.keras.optimizers.Adam(learning_rate=0.003),\n loss='sparse_categorical_crossentropy',\n metrics=['accuracy'])\n\n log_dir = os.path.join(\"/tmp/log\", \"relu-and-adam-\" + datetime.datetime.now().strftime(\"%Y%m%d-%H%M%S\"))\n tensorboard_callback = tf.keras.callbacks.TensorBoard(log_dir, histogram_freq=1)\n\n model.fit(x=x_train, \n y=y_train,\n batch_size=100,\n epochs=300, \n validation_data=(x_test, y_test), \n callbacks=[tensorboard_callback])\n\ntrain_model()\n```\n\n\n```\n#@title Visualize in TensorBoard (RUN ME!) { display-mode: \"form\" }\n\nget_ipython().system_raw('./ngrok http 6006 &')\n! curl -s http://localhost:4040/api/tunnels | python3 -c \"import sys, json; print(json.load(sys.stdin)['tunnels'][0]['public_url'])\"\n```\n\n## 8. Lab: dropout, overfitting\n\n\n\nYou will have noticed that cross-entropy curves for test and training data start disconnecting after a couple thousand iterations. The learning algorithm works on training data only and optimises the training cross-entropy accordingly. It never sees test data so it is not surprising that after a while its work no longer has an effect on the test cross-entropy which stops dropping and sometimes even bounces back up.\n\n\n\nThis does not immediately affect the real-world recognition capabilities of your model but it will prevent you from running many iterations and is generally a sign that the training is no longer having a positive effect. This disconnect is usually labeled \"overfitting\" and when you see it, you can try to apply a regularisation technique called \"dropout\".\n\n\n\nIn dropout, at each training iteration, you drop random neurons from the network. You choose a probability pkeep for a neuron to be kept, usually between 50% and 75%, and then at each iteration of the training loop, you randomly remove neurons with all their weights and biases. Different neurons will be dropped at each iteration (and you also need to boost the output of the remaining neurons in proportion to make sure activations on the next layer do not shift).\n\n**TO DO:**\n\n**Now, add dropout to each intermediate layer of your network.**\n\n\n```\n# Define the model\nmodel = tf.keras.models.Sequential([\n# flatten the images into a single line of pixels\n tf.keras.layers.Flatten(input_shape=(28, 28), name='flatten_input'),\n# add the first hidden layer\n tf.keras.layers.Dense(200, activation=tf.nn.relu, name='hidden_layer_1'),\n# add the second hidden layer\n tf.keras.layers.Dense(100, activation=tf.nn.relu, name='hidden_layer_2'),\n# add the third hidden layer\n tf.keras.layers.Dense(60, activation=tf.nn.relu, name='hidden_layer_3'),\n# add the fourth hidden layer\n tf.keras.layers.Dense(30, activation=tf.nn.relu, name='hidden_layer_4'),\n# add output layer and apply softmax as an activation function\n tf.keras.layers.Dense(10, activation=tf.nn.softmax, name='logits')\n])\n\nmodel.summary()\n```\n\n\n```\n#@title Solution\n\n# Define the model\nmodel = tf.keras.models.Sequential([\n# flatten the images into a single line of pixels\n tf.keras.layers.Flatten(input_shape=(28, 28), name='flatten_input'),\n# add dropout\n tf.keras.layers.Dropout(0.25),\n# add the first hidden layer\n tf.keras.layers.Dense(200, activation=tf.nn.relu, name='hidden_layer_1'),\n# add dropout\n tf.keras.layers.Dropout(0.25),\n# add the second hidden layer\n tf.keras.layers.Dense(100, activation=tf.nn.relu, name='hidden_layer_2'),\n# add dropout\n tf.keras.layers.Dropout(0.25),\n# add the third hidden layer\n tf.keras.layers.Dense(60, activation=tf.nn.relu, name='hidden_layer_3'),\n# add dropout\n tf.keras.layers.Dropout(0.25),\n# add the fourth hidden layer\n tf.keras.layers.Dense(30, activation=tf.nn.relu, name='hidden_layer_4'),\n# add dropout\n tf.keras.layers.Dropout(0.25),\n# add output layer and apply softmax as an activation function\n tf.keras.layers.Dense(10, activation=tf.nn.softmax, name='logits')\n])\n\nmodel.summary()\n```\n\n### Train the model with dropout\n\n\n```\ndef train_model():\n model.compile(optimizer=tf.keras.optimizers.Adam(learning_rate=0.005),\n loss='sparse_categorical_crossentropy',\n metrics=['accuracy'])\n\n log_dir = os.path.join(\"/tmp/log\", \"dropout-\" + datetime.datetime.now().strftime(\"%Y%m%d-%H%M%S\"))\n tensorboard_callback = tf.keras.callbacks.TensorBoard(log_dir, histogram_freq=1)\n\n model.fit(x=x_train, \n y=y_train,\n batch_size=100,\n epochs=300, \n validation_data=(x_test, y_test), \n callbacks=[tensorboard_callback])\n\ntrain_model()\n```\n\n\n```\n#@title Visualize in TensorBoard (RUN ME!) { display-mode: \"form\" }\n\nget_ipython().system_raw('./ngrok http 6006 &')\n! curl -s http://localhost:4040/api/tunnels | python3 -c \"import sys, json; print(json.load(sys.stdin)['tunnels'][0]['public_url'])\"\n```\n\n\n\nYou should see that the test loss is largely brought back under control, noise reappears (unsurprisingly given how dropout works) but in this case at least, the test accuracy remains unchanged which is a little disappointing. There must be another reason for the \"overfitting\".\n\nBefore we continue, a recap of all the tools we have tried so far:\n\n\n\nWhatever we do, we do not seem to be able to break the 98% barrier in a significant way and our loss curves still exhibit the \"overfitting\" disconnect. What is really \"overfitting\" ? Overfitting happens when a neural network learns \"badly\", in a way that works for the training examples but not so well on real-world data. There are regularisation techniques like dropout that can force it to learn in a better way but overfitting also has deeper roots.\n\n\n\nBasic overfitting happens when a neural network has too many degrees of freedom for the problem at hand. Imagine we have so many neurons that the network can store all of our training images in them and then recognise them by pattern matching. It would fail on real-world data completely. A neural network must be somewhat constrained so that it is forced to generalise what it learns during training.\n\nIf you have very little training data, even a small network can learn it by heart. Generally speaking, you always need lots of data to train neural networks.\n\nFinally, if you have done everything well, experimented with different sizes of network to make sure its degrees of freedom are constrained, applied dropout, and trained on lots of data you might still be stuck at a performance level that nothing seems to be able to improve. This means that your neural network, in its present shape, is not capable of extracting more information from your data, as in our case here.\n\nRemember how we are using our images, all pixels flattened into a single vector ? That was a really bad idea. Handwritten digits are made of shapes and we discarded the shape information when we flattened the pixels. However, there is a type of neural network that can take advantage of shape information: convolutional networks. Let us try them.\n\n## 9. Theory: convolutional networks\n\n\n\nIn a layer of a convolutional network, one \"neuron\" does a weighted sum of the pixels just above it, across a small region of the image only. It then acts normally by adding a bias and feeding the result through its activation function. The big difference is that each neuron reuses the same weights whereas in the fully-connected networks seen previously, each neuron had its own set of weights.\n\nIn the animation above, you can see that by sliding the patch of weights across the image in both directions (a convolution) you obtain as many output values as there were pixels in the image (some padding is necessary at the edges though).\n\nTo generate one plane of output values using a patch size of 4x4 and a color image as the input, as in the animation, we need 4x4x3=48 weights. That is not enough. To add more degrees of freedom, we repeat the same thing with a different set of weights.\n\n\n\nThe two (or more) sets of weights can be rewritten as one by adding a dimension to the tensor and this gives us the generic shape of the weights tensor for a convolutional layer. Since the number of input and output channels are parameters, we can start stacking and chaining convolutional layers.\n\n\n\nOne last issue remains. We still need to boil the information down. In the last layer, we still want only 10 neurons for our 10 classes of digits. Traditionally, this was done by a \"max-pooling\" layer. Even if there are simpler ways today, \"max-pooling\" helps understand intuitively how convolutional networks operate: if you assume that during training, our little patches of weights evolve into filters that recognise basic shapes (horizontal and vertical lines, curves, ...) then one way of boiling useful information down is to keep through the layers the outputs where a shape was recognised with the maximum intensity. In practice, in a max-pool layer neuron outputs are processed in groups of 2x2 and only the one max one retained.\n\nThere is a simpler way though: if you slide the patches across the image with a stride of 2 pixels instead of 1, you also obtain fewer output values. This approach has proven just as effective and today's convolutional networks use convolutional layers only.\n\n\n\nLet us build a convolutional network for handwritten digit recognition. We will use three convolutional layers at the top, our traditional softmax readout layer at the bottom and connect them with one fully-connected layer:\n\n\n\nNotice that the second and third convolutional layers have a stride of two which explains why they bring the number of output values down from 28x28 to 14x14 and then 7x7. The sizing of the layers is done so that the number of neurons goes down roughly by a factor of two at each layer: 28x28x4\u22483000 \u2192 14x14x8\u22481500 \u2192 7x7x12\u2248500 \u2192 200.\n\n## 10. Lab: the 99% challenge\nA good approach to sizing your neural networks is to implement a network that is a little too constrained, then give it a bit more degrees of freedom and add dropout to make sure it is not overfitting. This ends up with a fairly optimal network for your problem.\n\nHere for example, we used only 4 patches in the first convolutional layer. If you accept that those patches of weights evolve during training into shape recognisers, you can intuitively see that this might not be enough for our problem. Handwritten digits are mode from more than 4 elemental shapes.\n\nSo let us bump up the patch sizes a little, increase the number of patches in our convolutional layers from 4, 8, 12 to 6, 12, 24 and then add dropout on the fully-connected layer. Why not on the convolutional layers? Their neurons reuse the same weights, so dropout, which effectively works by freezing some weights during one training iteration, would not work on them.\n\n\n\n**Go for it and break the 99% limit. Increase the patch sizes and channel numbers as on the picture above and add dropout on the convolutional layer.**\n\n\n```\n# Define the model\nmodel = tf.keras.models.Sequential([\n# First convolutional layer\n tf.keras.layers.Conv2D(filters=6, kernel_size=6, padding='same', strides=1, activation='relu', input_shape=(28,28,1), name='conv_layer_1'),\n\n #YOUR CODE GOES HERE! ADD TWO CONVOLUTIONAL LAYERS\n \n#flatten layer\n tf.keras.layers.Flatten(name='flatten_layer'),\n# add a fully connected layer\n tf.keras.layers.Dense(200, activation=tf.nn.relu, name='fc_layer_4'),\n\n #YOR CODE GOES HERE! ADD DROPOUT\n \n# add output layer and apply softmax as an activation function\n tf.keras.layers.Dense(10, activation=tf.nn.softmax, name='logits')\n])\n\nmodel.summary()\n```\n\n\n```\n#@title Solution\n\n# Define the model\nmodel = tf.keras.models.Sequential([\n# First convolutional layer\n tf.keras.layers.Conv2D(filters=6, kernel_size=6, padding='same', strides=1, activation='relu', input_shape=(28,28,1), name='conv_layer_1'),\n# Second convolutional layer\n tf.keras.layers.Conv2D(filters=12, kernel_size=5, padding='same', strides=2, activation='relu', name='conv_layer_2'),\n# Third convolutional layer\n tf.keras.layers.Conv2D(filters=24, kernel_size=4, padding='same', strides=2, activation='relu', name='conv_layer_3'),\n#flatten layer\n tf.keras.layers.Flatten(name='flatten_layer'),\n# add a fully connected layer\n tf.keras.layers.Dense(200, activation=tf.nn.relu, name='fc_layer_4'),\n# add dropout\n tf.keras.layers.Dropout(0.35, name='dropout_layer'),\n# add output layer and apply softmax as an activation function\n tf.keras.layers.Dense(10, activation=tf.nn.softmax, name='logits')\n])\n\nmodel.summary()\n```\n\n### Train the model\n\n\n```\n(x_train, y_train),(x_test, y_test) = mnist.load_data()\nx_train, x_test = x_train / 255.0, x_test / 255.0\nx_train, x_test = np.expand_dims(x_train,-1), np.expand_dims(x_test,-1)\n\ndef train_model():\n model.compile(optimizer=tf.keras.optimizers.Adam(learning_rate=0.003, decay=0.001),\n loss='sparse_categorical_crossentropy',\n metrics=['accuracy'])\n\n log_dir = os.path.join(\"/tmp/log\", \"convolutional-and-lr-decay-\" + datetime.datetime.now().strftime(\"%Y%m%d-%H%M%S\"))\n tensorboard_callback = tf.keras.callbacks.TensorBoard(log_dir, histogram_freq=1)\n\n model.fit(x=x_train, \n y=y_train,\n batch_size=32,\n epochs=300, \n validation_data=(x_test, y_test), \n callbacks=[tensorboard_callback])\n\ntrain_model()\n```\n\n\n```\n#@title Visualize in TensorBoard (RUN ME!) { display-mode: \"form\" }\n\nget_ipython().system_raw('./ngrok http 6006 &')\n! curl -s http://localhost:4040/api/tunnels | python3 -c \"import sys, json; print(json.load(sys.stdin)['tunnels'][0]['public_url'])\"\n```\n", "meta": {"hexsha": "916e41d83d06ca0acf27871372493c7c2cdeefa0", "size": 60133, "ext": "ipynb", "lang": "Jupyter Notebook", "max_stars_repo_path": "colabs/Tensorflow_2_0_practical.ipynb", "max_stars_repo_name": "KalifiaBillal/tensorflow-without-a-phd", "max_stars_repo_head_hexsha": "a614e885e4fbc186e9b05566f003833472aae3a5", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "colabs/Tensorflow_2_0_practical.ipynb", "max_issues_repo_name": "KalifiaBillal/tensorflow-without-a-phd", "max_issues_repo_head_hexsha": "a614e885e4fbc186e9b05566f003833472aae3a5", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "colabs/Tensorflow_2_0_practical.ipynb", "max_forks_repo_name": "KalifiaBillal/tensorflow-without-a-phd", "max_forks_repo_head_hexsha": "a614e885e4fbc186e9b05566f003833472aae3a5", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 56.0942164179, "max_line_length": 902, "alphanum_fraction": 0.6170821346, "converted": true, "num_tokens": 9452, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802735722128, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.4089413415541111}} {"text": "
                                        \n

                                        ILI285 - Computaci\u00f3n Cient\u00edfica I / INF285 - Computaci\u00f3n Cient\u00edfica

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                                        Interpolation: Splines

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                                        [[S]cientific [C]omputing [T]eam](#acknowledgements)

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                                        Version: 1.21

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                                        \n\n## Table of Contents\n* [Introduction](#intro)\n* [Splines](#sp)\n* [Properties](#pr)\n* [Solving](#so)\n* [The additional Property](#ad)\n* [Exercises](#ex)\n* [Acknowledgements](#acknowledgements)\n\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\n%matplotlib inline\nimport scipy as sp\nfrom scipy import interpolate\nimport ipywidgets as widgets\nimport matplotlib as mpl\nmpl.rcParams['font.size'] = 14\nmpl.rcParams['axes.labelsize'] = 20\nmpl.rcParams['xtick.labelsize'] = 14\nmpl.rcParams['ytick.labelsize'] = 14\nfrom scipy.interpolate import CubicSpline\nM=8\n```\n\n
                                        \n## Introduction\n\nPreviously in our jupyter notebooks, we learn about interpolation. Methods like Newton's Divided Difference, Lagrange, among others. Other alternative for interpolate a set of data points is using ** Cubic Splines**.\nThis technique, avoids the Runge's Phenomenon and creates a 3-degree polynomial easily.\n\n
                                        \n## Splines\n\nThe most common spline is the linear spline. Given a set of points $(x_{1},y_{1}), (x_{2},y_{2}),...,(x_{n},y_{n}) $, this spline connects each point creating a non-smooth curve. However, this polynomial haves a problem. It's no smooth curve! For to avoid this problem, **the cubic splines creates a set of 3-degree polynomial (specifically n-1 polynomials)**... a much better curve.\n\n \n\n\n```python\n# Code based on Example from: https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.CubicSpline.html#scipy.interpolate.CubicSpline\n\n# The data\nx = np.linspace(0,2*np.pi,12)\ny = np.sin(x)\n\n# Building interpolation object\ncs = CubicSpline(x, y)\n\n# Defining a finer mesh to plot the function\nxx = np.linspace(0,2*np.pi,100)\n\n#Interpolating the date with the spline\nyy = cs(xx)\nyy1 = cs(xx, 1)\nyy2 = cs(xx, 2)\nyy3 = cs(xx, 3)\nyy4 = cs(xx, 4)\n\n# Plotting the splines and its derivatives\nplt.figure(figsize=(M,M))\nplt.plot(x,y,'.',markersize=20,label=r'Data Points')\nplt.plot(xx,yy, linewidth=2, label=r'S$(x)$')\nplt.plot(xx,yy1, linewidth=2, label=r'$\\frac{d}{dx}$S$(x)$')\nplt.plot(xx,yy2, linewidth=2, label=r'$\\frac{d^2}{dx^2}$S$(x)$')\nplt.plot(xx,yy3, linewidth=2, label=r'$\\frac{d^3}{dx^3}$S$(x)$')\nplt.plot(xx,yy4, linewidth=2, label=r'$\\frac{d^4}{dx^4}$S$(x)$')\n\nplt.title(r'Cubic Spline is defined as S$(x)$')\nplt.axis('tight')\nplt.xlabel(r'$x$')\nplt.ylabel(r'$y$')\nplt.grid(True)\nplt.legend(loc='center left', bbox_to_anchor=(1, 0.5))\nplt.show()\n\n```\n\nThe orange curve is generated with cubic splines (using the scipy implementation). The other colors are the **derivatives** of the Cubic Spline as indicated in the legend.\nHowever, if we thinks about this curve, we can say that exists an **infinitely many** quantity of polynomials such that meets all the points. Our goal is to create an unique polynomial. Given this condition, there is 4 properties that defines the cubic spline we are looking for. \n\n
                                        \n## Properties of Splines\n\nWhen we want creates a spline of **n** data points, we obtains a set of **n-1** 3-degree polynomials. For example:\nGiven a set of points $(x_{1},y_{1}), (x_{2},y_{2}),...,(x_{n},y_{n})$, the splines is:\n\n\\begin{equation} S_{1}(x) = y_{1} + b_{1}(x-x_{1}) + c_{1}(x-x_{1})^{2} + d_{1}(x-x_{1})^{3} \\\\ \n S_{2}(x) = y_{2} + b_{2}(x-x_{2}) + c_{2}(x-x_{2})^{2} + d_{2}(x-x_{2})^{3} \\\\\n ... \\\\\n ... \\\\\n ... \\\\\n S_{n-1}(x) = y_{n-1} + b_{n-1}(x-x_{n-1}) + c_{n-1}(x-x_{n-1})^{2} + d_{n-1}(x-x_{n-1})^{3}\n\\end{equation}\n\nThus, our goal is obtains the $y, b, c$ and $d$ coefficients. With this values, we are creating the spline $S(x)$ that meets all the data points. This spline have the next properties:\n\n## Property 1 (Are the points connected?)\n\nThe first property checks if each x-coordinate reachs the correspondant y-coordinate in the spline $S(x)$. Thus, checks if the spline meets the data points.\n\n$$S_{i}(x_{i}) = y_{i}$$ $$ S_{i}(x_{i+1}) = y_{i+1}$$ \n\n$$i \\in [1,n-1]$$\n\n\n## Property 2 (Slope Continuity)\n\nThe second property ensures that the slopes of the splines at the points' neighborhood to be equals. This guarantees the smoothly of $S(x)$\n\n$$S'_{i-1}(x_{i}) = S'_{i}(x_{i})$$\n\n$$i \\in [2,n-1]$$\n\n## Property 3 (Curvature Continuity)\n\nThis property ensure that the curvature between differents polynomials will be equal at the neighborhood of points, avoiding abrupt changes of the curve at the data points. \n\n$$S''_{i-1}(x_{i}) = S''_{i}(x_{i})$$\n\n$$i \\in [2,n-1]$$\n\n
                                        \n## Solving the system \n\nIf we have **n points**, we know that our splines will be composed of **n-1 curves** $S_{i}(x)$. We have too, **(3n-3) unknowns variables** ($b_{i}, c_{i}, d_{i}$ for each spline). However, we can build a system of equations for find this variables. How can i do this? Easy.. Using the previous properties!\n\nUsing the previously defined splines for n points:\n\\begin{equation} S_{1}(x) = y_{1} + b_{1}(x-x_{1}) + c_{1}(x-x_{1})^{2} + d_{1}(x-x_{1})^{3} \\\\ \n S_{2}(x) = y_{2} + b_{2}(x-x_{2}) + c_{2}(x-x_{2})^{2} + d_{2}(x-x_{2})^{3} \\\\\n \\vdots \\\\\n S_{n-1}(x) = y_{n-1} + b_{n-1}(x-x_{n-1}) + c_{n-1}(x-x_{n-1})^{2} + d_{n-1}(x-x_{n-1})^{3}\n\\end{equation}\n\nWe need too, the first derivatives of this curves:\n\\begin{equation} S'_{1}(x) = b_{1} + 2c_{1}(x-x_{1}) + 3d_{1}(x-x_{1})^{2} \\\\ \n S'_{1}(x) = b_{2} + 2c_{2}(x-x_{2}) + 3d_{2}(x-x_{2})^{2} \\\\\n \\vdots \\\\\n S'_{n-1}(x) = b_{n-1} + 2c_{n-1}(x-x_{n-1}) + 3d_{n-1}(x-x_{n-1})^{2} \\\\\n\\end{equation}\n\nAnd its second derivatives:\n\\begin{equation} S''_{1}(x) = 2c_{1} + 6d_{1}(x-x_{1}) \\\\ \n S''_{2}(x) = 2c_{2} + 6d_{2}(x-x_{2}) \\\\ \n \\vdots \\\\\n S''_{n-1}(x) = 2c_{n-1} + 6d_{n-1}(x-x_{n-1}) \\\\ \n\\end{equation}\n\nUsing the first property, we get **(n-1) equations}**:\n\n\\begin{equation} b_{1}(x_{2}-x_{1}) + c_{1}(x_{2}-x_{1})^2 + d_{1}(x_{2}-x_{1})^3 = y_{2} - y_{1} \\hspace{1cm}(1)\\\\ \n b_{2}(x_{3}-x_{2}) + c_{2}(x_{3}-x_{2})^2 + d_{2}(x_{3}-x_{2})^3 = y_{3} - y_{2} \\hspace{1cm}(2)\\\\ \n \\vdots\\\\\n b_{n-1}(x_{n}-x_{n-1}) + c_{n-1}(x_{n}-x_{n-1})^2 + d_{n-1}(x_{n}-x_{n-1})^3 = y_{n} - y_{n-1} \\hspace{1cm}(n-1) \n\\end{equation}\n\nUsing the second property, we get **(n-2) equations**:\n\n\\begin{equation} b_{1}+2c_{1}(x_{2}-x_{1}) + 3d_{1}(x_{2}-x_{1})^2 - b_{2}= 0 \\hspace{1cm}(1)\\\\ \n b_{2}+2c_{2}(x_{3}-x_{2}) + 3d_{2}(x_{3}-x_{2})^2 - b_{3}= 0 \\hspace{1cm}(2)\\\\ \n \\vdots\\\\\n b_{n-2}+2c_{n-2}(x_{n-1}-x_{n-2}) + 3d_{n-2}(x_{n-1}-x_{n-2})^2 -b_{n-1}=0 \\hspace{1cm}(n-2)\\\\ \n\\end{equation}\n\nUsing the third property, we get **(n-2) equations**:\n\n\\begin{equation} 2c_{1}+6d_{1}(x_{2}-x_{1}) - 2c_{2} = 0 \\hspace{1cm}(1)\\\\ \n 2c_{2}+6d_{2}(x_{3}-x_{2}) - 2c_{3}=0 \\hspace{1cm}(2)\\\\ \n \\vdots\\\\\n 2c_{n-2}+6d_{n-2}(x_{n-1}-x_{n-2}) - 2c_{n-1} = 0 \\hspace{1cm}(n-2)\\\\ \n\\end{equation}\n\nIf we adds all our equations, we obtains **(3n-5) equations**. Clearly, the matrix in that system is not square (we need 2 equations more). For this, **we have another property**, that defines the edges conditions of the splines\n\n\n\n